Newspace parameters
| Level: | \( N \) | \(=\) | \( 6005 = 5 \cdot 1201 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6005.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(47.9501664138\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.2777.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 4x^{2} + x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 4x^{2} + x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−1.26308 | 2.12152 | −0.404635 | −1.00000 | −2.67964 | −2.67964 | 3.03724 | 1.50084 | 1.26308 | ||||||||||||||||||||||||||||||
| 1.2 | −0.515722 | −0.702588 | −1.73403 | −1.00000 | 0.362340 | 0.362340 | 1.92572 | −2.50637 | 0.515722 | |||||||||||||||||||||||||||||||
| 1.3 | 1.18264 | −2.96664 | −0.601352 | −1.00000 | −3.50848 | −3.50848 | −3.07647 | 5.80096 | −1.18264 | |||||||||||||||||||||||||||||||
| 1.4 | 2.59615 | −0.452290 | 4.74002 | −1.00000 | −1.17422 | −1.17422 | 7.11351 | −2.79543 | −2.59615 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(1201\) | \( +1 \) |
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 | 6005.2.a.c | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6005.2.a.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 2T_{2}^{3} - 3T_{2}^{2} + 3T_{2} + 2 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6005))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 2 T^{3} + \cdots + 2 \)
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| $3$ |
\( T^{4} + 2 T^{3} + \cdots - 2 \)
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| $5$ |
\( (T + 1)^{4} \)
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| $7$ |
\( T^{4} + 7 T^{3} + \cdots - 4 \)
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| $11$ |
\( T^{4} + 4 T^{3} + \cdots - 32 \)
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| $13$ |
\( T^{4} - 2 T^{3} + \cdots + 1 \)
|
| $17$ |
\( T^{4} + T^{3} + \cdots + 67 \)
|
| $19$ |
\( T^{4} - 11 T^{3} + \cdots + 8 \)
|
| $23$ |
\( T^{4} + 9 T^{3} + \cdots + 8 \)
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| $29$ |
\( T^{4} + 8 T^{3} + \cdots - 536 \)
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| $31$ |
\( T^{4} + 5 T^{3} + \cdots + 92 \)
|
| $37$ |
\( T^{4} - 4 T^{3} + \cdots + 943 \)
|
| $41$ |
\( T^{4} - 3 T^{3} + \cdots + 376 \)
|
| $43$ |
\( T^{4} + 19 T^{3} + \cdots - 44 \)
|
| $47$ |
\( T^{4} - 4 T^{3} + \cdots - 23 \)
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| $53$ |
\( T^{4} - 28 T^{3} + \cdots + 1486 \)
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| $59$ |
\( T^{4} + 9 T^{3} + \cdots + 1013 \)
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| $61$ |
\( T^{4} + 23 T^{3} + \cdots - 41 \)
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| $67$ |
\( T^{4} + 11 T^{3} + \cdots + 2 \)
|
| $71$ |
\( T^{4} - 27 T^{3} + \cdots + 472 \)
|
| $73$ |
\( T^{4} - 18 T^{3} + \cdots - 6016 \)
|
| $79$ |
\( T^{4} - 11 T^{3} + \cdots - 8 \)
|
| $83$ |
\( T^{4} + 2 T^{3} + \cdots - 128 \)
|
| $89$ |
\( T^{4} - T^{3} + \cdots + 14648 \)
|
| $97$ |
\( T^{4} - 8 T^{3} + \cdots + 1 \)
|
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