Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [52,26,Mod(1,52)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(52, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 26, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("52.1");
S:= CuspForms(chi, 26);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 52 = 2^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 26 \) |
| Character orbit: | \([\chi]\) | \(=\) | 52.a (trivial) |
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: | yes |
| Analytic conductor: | \(205.918325575\) |
| Analytic rank: | \(1\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 5815509829300 x^{10} + \cdots - 40\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{68}\cdot 3^{10}\cdot 5^{5}\cdot 13^{5} \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{12} - 5815509829300 x^{10} + \cdots - 40\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 20\!\cdots\!25 \nu^{11} + \cdots - 49\!\cdots\!28 ) / 92\!\cdots\!92 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 41\!\cdots\!39 \nu^{11} + \cdots - 18\!\cdots\!04 ) / 92\!\cdots\!92 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 49\!\cdots\!31 \nu^{11} + \cdots + 33\!\cdots\!52 ) / 15\!\cdots\!32 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 55\!\cdots\!75 \nu^{11} + \cdots - 57\!\cdots\!84 ) / 92\!\cdots\!92 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 11\!\cdots\!13 \nu^{11} + \cdots + 13\!\cdots\!76 ) / 92\!\cdots\!92 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 84\!\cdots\!83 \nu^{11} + \cdots + 74\!\cdots\!60 ) / 46\!\cdots\!96 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 16\!\cdots\!13 \nu^{11} + \cdots + 35\!\cdots\!80 ) / 51\!\cdots\!44 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 58\!\cdots\!99 \nu^{11} + \cdots - 18\!\cdots\!28 ) / 92\!\cdots\!92 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 56\!\cdots\!27 \nu^{11} + \cdots + 10\!\cdots\!36 ) / 46\!\cdots\!96 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 59\!\cdots\!47 \nu^{11} + \cdots - 84\!\cdots\!84 ) / 46\!\cdots\!96 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 149\beta_{2} + 99148\beta _1 + 969251638217 \)
|
| \(\nu^{3}\) | \(=\) |
\( 19 \beta_{11} + 482 \beta_{10} + 1410 \beta_{9} + 2211 \beta_{8} + 802 \beta_{7} - 6350 \beta_{6} + \cdots + 96\!\cdots\!15 \)
|
| \(\nu^{4}\) | \(=\) |
\( 333172350 \beta_{11} + 1783577184 \beta_{10} - 715449633 \beta_{9} + 1698424848 \beta_{8} + \cdots + 17\!\cdots\!72 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 642173448063383 \beta_{11} + \cdots + 76\!\cdots\!33 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 27\!\cdots\!77 \beta_{11} + \cdots + 38\!\cdots\!17 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 30\!\cdots\!86 \beta_{11} + \cdots + 32\!\cdots\!75 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 28\!\cdots\!69 \beta_{11} + \cdots + 97\!\cdots\!16 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 10\!\cdots\!18 \beta_{11} + \cdots + 11\!\cdots\!65 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 13\!\cdots\!62 \beta_{11} + \cdots + 26\!\cdots\!97 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 34\!\cdots\!75 \beta_{11} + \cdots + 37\!\cdots\!03 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −1.67969e6 | 0 | 3.14060e8 | 0 | 2.87128e10 | 0 | 1.97407e12 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.15621e6 | 0 | −6.41693e8 | 0 | 1.32175e10 | 0 | 4.89537e11 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −706925. | 0 | 4.63801e8 | 0 | −2.43719e10 | 0 | −3.47546e11 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −637841. | 0 | 3.16365e8 | 0 | −3.12648e10 | 0 | −4.40448e11 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −180323. | 0 | −7.37835e8 | 0 | 6.72971e10 | 0 | −8.14772e11 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −148407. | 0 | 1.58863e8 | 0 | 5.51751e10 | 0 | −8.25264e11 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | −19767.5 | 0 | −7.56537e8 | 0 | −4.33277e10 | 0 | −8.46898e11 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 408924. | 0 | 1.06316e9 | 0 | 4.88528e8 | 0 | −6.80070e11 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 966284. | 0 | −6.11952e8 | 0 | −1.95730e10 | 0 | 8.64156e10 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 1.33104e6 | 0 | −3.48639e8 | 0 | 4.12704e10 | 0 | 9.24385e11 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 1.35768e6 | 0 | 3.36645e8 | 0 | 4.54626e9 | 0 | 9.96018e11 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 1.36487e6 | 0 | 1.56721e8 | 0 | −5.30518e10 | 0 | 1.01558e12 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(13\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 52.26.a.a | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 52.26.a.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} - 899640 T_{3}^{11} - 5444556769900 T_{3}^{10} + \cdots - 45\!\cdots\!00 \)
acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(52))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + \cdots - 45\!\cdots\!00 \)
|
| $5$ |
\( T^{12} + \cdots - 31\!\cdots\!00 \)
|
| $7$ |
\( T^{12} + \cdots - 44\!\cdots\!12 \)
|
| $11$ |
\( T^{12} + \cdots + 35\!\cdots\!36 \)
|
| $13$ |
\( (T - 23298085122481)^{12} \)
|
| $17$ |
\( T^{12} + \cdots - 37\!\cdots\!72 \)
|
| $19$ |
\( T^{12} + \cdots - 18\!\cdots\!00 \)
|
| $23$ |
\( T^{12} + \cdots - 65\!\cdots\!52 \)
|
| $29$ |
\( T^{12} + \cdots - 18\!\cdots\!00 \)
|
| $31$ |
\( T^{12} + \cdots - 13\!\cdots\!00 \)
|
| $37$ |
\( T^{12} + \cdots + 43\!\cdots\!00 \)
|
| $41$ |
\( T^{12} + \cdots + 22\!\cdots\!52 \)
|
| $43$ |
\( T^{12} + \cdots - 25\!\cdots\!64 \)
|
| $47$ |
\( T^{12} + \cdots + 79\!\cdots\!00 \)
|
| $53$ |
\( T^{12} + \cdots + 18\!\cdots\!56 \)
|
| $59$ |
\( T^{12} + \cdots - 67\!\cdots\!00 \)
|
| $61$ |
\( T^{12} + \cdots - 51\!\cdots\!56 \)
|
| $67$ |
\( T^{12} + \cdots + 16\!\cdots\!00 \)
|
| $71$ |
\( T^{12} + \cdots - 29\!\cdots\!00 \)
|
| $73$ |
\( T^{12} + \cdots - 16\!\cdots\!80 \)
|
| $79$ |
\( T^{12} + \cdots + 46\!\cdots\!00 \)
|
| $83$ |
\( T^{12} + \cdots - 11\!\cdots\!00 \)
|
| $89$ |
\( T^{12} + \cdots - 61\!\cdots\!00 \)
|
| $97$ |
\( T^{12} + \cdots + 41\!\cdots\!76 \)
|
| show more | |
| show less |