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Properties

Label 244.1.u.a
Level $244$
Weight $1$
Character orbit 244.u
Analytic conductor $0.122$
Analytic rank $0$
Dimension $8$
Projective image $D_{15}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [244,1,Mod(15,244)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(244, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([15, 14]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("244.15");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 244 = 2^{2} \cdot 61 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 244.u (of order \(30\), degree \(8\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.121771863082\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{15}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{15} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{30}^{14} q^{2} - \zeta_{30}^{13} q^{4} + (\zeta_{30}^{10} + \zeta_{30}^{6}) q^{5} + \zeta_{30}^{12} q^{8} - \zeta_{30}^{3} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{30}^{14} q^{2} - \zeta_{30}^{13} q^{4} + (\zeta_{30}^{10} + \zeta_{30}^{6}) q^{5} + \zeta_{30}^{12} q^{8} - \zeta_{30}^{3} q^{9} + ( - \zeta_{30}^{9} - \zeta_{30}^{5}) q^{10} + (\zeta_{30}^{8} + \zeta_{30}^{2}) q^{13} - \zeta_{30}^{11} q^{16} + ( - \zeta_{30}^{7} + \zeta_{30}^{4}) q^{17} + \zeta_{30}^{2} q^{18} + (\zeta_{30}^{8} + \zeta_{30}^{4}) q^{20} + (\zeta_{30}^{12} - \zeta_{30}^{5} - \zeta_{30}) q^{25} + ( - \zeta_{30}^{7} - \zeta_{30}) q^{26} + ( - \zeta_{30}^{11} - \zeta_{30}^{9}) q^{29} + \zeta_{30}^{10} q^{32} + (\zeta_{30}^{6} - \zeta_{30}^{3}) q^{34} - \zeta_{30} q^{36} - \zeta_{30}^{6} q^{37} + ( - \zeta_{30}^{7} - \zeta_{30}^{3}) q^{40} + (\zeta_{30}^{14} - \zeta_{30}^{13}) q^{41} + ( - \zeta_{30}^{13} - \zeta_{30}^{9}) q^{45} - \zeta_{30}^{7} q^{49} + ( - \zeta_{30}^{11} + \zeta_{30}^{4} + 1) q^{50} + (\zeta_{30}^{6} + 1) q^{52} + ( - \zeta_{30}^{5} + \zeta_{30}^{4}) q^{53} + (\zeta_{30}^{10} + \zeta_{30}^{8}) q^{58} + \zeta_{30}^{14} q^{61} - \zeta_{30}^{9} q^{64} + (\zeta_{30}^{14} + \cdots - \zeta_{30}^{3}) q^{65} + \cdots + \zeta_{30}^{6} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + q^{4} - 6 q^{5} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} + q^{4} - 6 q^{5} - 2 q^{8} - 2 q^{9} - 6 q^{10} + 2 q^{13} + q^{16} + 2 q^{17} + q^{18} + 2 q^{20} - 5 q^{25} + 2 q^{26} - q^{29} - 4 q^{32} - 4 q^{34} + q^{36} + 2 q^{37} - q^{40} + 2 q^{41} - q^{45} + q^{49} + 10 q^{50} + 6 q^{52} - 3 q^{53} - 3 q^{58} + q^{61} - 2 q^{64} - 2 q^{65} - 3 q^{68} + 8 q^{72} - q^{73} + 4 q^{74} - q^{80} - 2 q^{81} - q^{82} - q^{85} - 3 q^{89} - q^{90} - q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/244\mathbb{Z}\right)^\times\).

\(n\) \(123\) \(185\)
\(\chi(n)\) \(-1\) \(-\zeta_{30}^{13}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
−0.104528 + 0.994522i
0.669131 − 0.743145i
−0.978148 − 0.207912i
0.669131 + 0.743145i
−0.104528 − 0.994522i
0.913545 − 0.406737i
−0.978148 + 0.207912i
0.913545 + 0.406737i
−0.104528 − 0.994522i 0 −0.978148 + 0.207912i −1.30902 − 1.45381i 0 0 0.309017 + 0.951057i 0.309017 − 0.951057i −1.30902 + 1.45381i
83.1 0.669131 + 0.743145i 0 −0.104528 + 0.994522i −0.190983 + 0.0850311i 0 0 −0.809017 + 0.587785i −0.809017 − 0.587785i −0.190983 − 0.0850311i
103.1 −0.978148 + 0.207912i 0 0.913545 − 0.406737i −0.190983 + 1.81708i 0 0 −0.809017 + 0.587785i −0.809017 − 0.587785i −0.190983 − 1.81708i
147.1 0.669131 − 0.743145i 0 −0.104528 − 0.994522i −0.190983 − 0.0850311i 0 0 −0.809017 − 0.587785i −0.809017 + 0.587785i −0.190983 + 0.0850311i
179.1 −0.104528 + 0.994522i 0 −0.978148 − 0.207912i −1.30902 + 1.45381i 0 0 0.309017 − 0.951057i 0.309017 + 0.951057i −1.30902 − 1.45381i
195.1 0.913545 + 0.406737i 0 0.669131 + 0.743145i −1.30902 + 0.278240i 0 0 0.309017 + 0.951057i 0.309017 − 0.951057i −1.30902 − 0.278240i
199.1 −0.978148 − 0.207912i 0 0.913545 + 0.406737i −0.190983 − 1.81708i 0 0 −0.809017 − 0.587785i −0.809017 + 0.587785i −0.190983 + 1.81708i
239.1 0.913545 − 0.406737i 0 0.669131 − 0.743145i −1.30902 − 0.278240i 0 0 0.309017 − 0.951057i 0.309017 + 0.951057i −1.30902 + 0.278240i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
61.i even 15 1 inner
244.u odd 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 244.1.u.a ✓ 8
3.b odd 2 1 2196.1.dp.a 8
4.b odd 2 1 CM 244.1.u.a ✓ 8
8.b even 2 1 3904.1.dm.a 8
8.d odd 2 1 3904.1.dm.a 8
12.b even 2 1 2196.1.dp.a 8
61.i even 15 1 inner 244.1.u.a ✓ 8
183.t odd 30 1 2196.1.dp.a 8
244.u odd 30 1 inner 244.1.u.a ✓ 8
488.bl odd 30 1 3904.1.dm.a 8
488.bp even 30 1 3904.1.dm.a 8
732.bl even 30 1 2196.1.dp.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
244.1.u.a ✓ 8 1.a even 1 1 trivial
244.1.u.a ✓ 8 4.b odd 2 1 CM
244.1.u.a ✓ 8 61.i even 15 1 inner
244.1.u.a ✓ 8 244.u odd 30 1 inner
2196.1.dp.a 8 3.b odd 2 1
2196.1.dp.a 8 12.b even 2 1
2196.1.dp.a 8 183.t odd 30 1
2196.1.dp.a 8 732.bl even 30 1
3904.1.dm.a 8 8.b even 2 1
3904.1.dm.a 8 8.d odd 2 1
3904.1.dm.a 8 488.bl odd 30 1
3904.1.dm.a 8 488.bp even 30 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(244, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 6 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} - T^{3} + 2 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + T^{7} + 5 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} - T^{7} + T^{5} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + T^{7} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{8} + T^{7} + 4 T^{5} + \cdots + 1 \) Copy content Toggle raw display
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