Newspace parameters
| Level: | \( N \) | \(=\) | \( 6664 = 2^{3} \cdot 7^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6664.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(53.2123079070\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{7} - 11x^{5} - 3x^{4} + 27x^{3} + 6x^{2} - 12x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 952) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 11x^{5} - 3x^{4} + 27x^{3} + 6x^{2} - 12x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 11\nu^{4} - \nu^{3} + 25\nu^{2} - 6\nu - 6 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - 11\nu^{4} - 3\nu^{3} + 27\nu^{2} + 4\nu - 10 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( -2\nu^{6} + \nu^{5} + 21\nu^{4} - 4\nu^{3} - 48\nu^{2} + 11\nu + 14 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{6} + 2\nu^{5} + 33\nu^{4} - 13\nu^{3} - 85\nu^{2} + 32\nu + 34 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} - \beta_{5} - 2\beta_{4} + \beta_{3} + 9\beta_{2} + 2\beta _1 + 17 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{6} - 8\beta_{4} + 11\beta_{3} + 13\beta_{2} + 33\beta _1 + 15 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11\beta_{6} - 11\beta_{5} - 23\beta_{4} + 14\beta_{3} + 75\beta_{2} + 33\beta _1 + 119 \)
|
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 |
|
0 | −2.43200 | 0 | −2.60963 | 0 | 0 | 0 | 2.91461 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.91766 | 0 | −1.87473 | 0 | 0 | 0 | 0.677431 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.594018 | 0 | 1.77288 | 0 | 0 | 0 | −2.64714 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.386094 | 0 | 3.79399 | 0 | 0 | 0 | −2.85093 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.824520 | 0 | −2.60113 | 0 | 0 | 0 | −2.32017 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.51862 | 0 | −0.798372 | 0 | 0 | 0 | −0.693805 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.98664 | 0 | 1.31699 | 0 | 0 | 0 | 5.92000 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( +1 \) |
| \(17\) | \( +1 \) |
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 | 6664.2.a.w | 7 | |
| 7.b | odd | 2 | 1 | 6664.2.a.x | 7 | ||
| 7.c | even | 3 | 2 | 952.2.q.d | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 952.2.q.d | ✓ | 14 | 7.c | even | 3 | 2 | |
| 6664.2.a.w | 7 | 1.a | even | 1 | 1 | trivial | |
| 6664.2.a.x | 7 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6664))\):
|
\( T_{3}^{7} - 11T_{3}^{5} - 3T_{3}^{4} + 27T_{3}^{3} + 6T_{3}^{2} - 12T_{3} - 4 \)
|
|
\( T_{5}^{7} + T_{5}^{6} - 18T_{5}^{5} - 25T_{5}^{4} + 74T_{5}^{3} + 98T_{5}^{2} - 87T_{5} - 90 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} - 11 T^{5} + \cdots - 4 \)
|
| $5$ |
\( T^{7} + T^{6} + \cdots - 90 \)
|
| $7$ |
\( T^{7} \)
|
| $11$ |
\( T^{7} - 5 T^{6} + \cdots + 892 \)
|
| $13$ |
\( T^{7} - 11 T^{6} + \cdots + 36 \)
|
| $17$ |
\( (T + 1)^{7} \)
|
| $19$ |
\( T^{7} + 2 T^{6} + \cdots + 8506 \)
|
| $23$ |
\( T^{7} - 9 T^{6} + \cdots - 12032 \)
|
| $29$ |
\( T^{7} - 9 T^{6} + \cdots - 8616 \)
|
| $31$ |
\( T^{7} + 2 T^{6} + \cdots - 592 \)
|
| $37$ |
\( T^{7} - 8 T^{6} + \cdots + 158776 \)
|
| $41$ |
\( T^{7} - 6 T^{6} + \cdots - 36744 \)
|
| $43$ |
\( T^{7} - 10 T^{6} + \cdots - 6184 \)
|
| $47$ |
\( T^{7} + T^{6} + \cdots - 254264 \)
|
| $53$ |
\( T^{7} - 3 T^{6} + \cdots + 378232 \)
|
| $59$ |
\( T^{7} + 4 T^{6} + \cdots - 128448 \)
|
| $61$ |
\( T^{7} + 32 T^{6} + \cdots - 40080 \)
|
| $67$ |
\( T^{7} - 31 T^{6} + \cdots + 6858 \)
|
| $71$ |
\( T^{7} - 3 T^{6} + \cdots - 1803 \)
|
| $73$ |
\( T^{7} + 7 T^{6} + \cdots - 70504 \)
|
| $79$ |
\( T^{7} - 8 T^{6} + \cdots + 173956 \)
|
| $83$ |
\( T^{7} - 24 T^{6} + \cdots - 7744 \)
|
| $89$ |
\( T^{7} - 4 T^{6} + \cdots + 21952 \)
|
| $97$ |
\( T^{7} - 15 T^{6} + \cdots + 18413568 \)
|
| show more | |
| show less |