Newspace parameters
| Level: | \( N \) | \(=\) | \( 961 = 31^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 961.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(56.7008355155\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.27702880.2 |
|
|
|
| Defining polynomial: |
\( x^{4} - 134x^{2} + 4120 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{4} - 134x^{2} + 4120 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 70 ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 70\nu ) / 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 6\beta_{2} + 70 \)
|
| \(\nu^{3}\) | \(=\) |
\( 6\beta_{3} + 70\beta_1 \)
|
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 |
|
−2.70156 | −6.91308 | −0.701562 | 3.10469 | 18.6761 | −7.70156 | 23.5078 | 20.7906 | −8.38750 | ||||||||||||||||||||||||||||||
| 1.2 | −2.70156 | 6.91308 | −0.701562 | 3.10469 | −18.6761 | −7.70156 | 23.5078 | 20.7906 | −8.38750 | |||||||||||||||||||||||||||||||
| 1.3 | 3.70156 | −9.28490 | 5.70156 | −16.1047 | −34.3686 | −1.29844 | −8.50781 | 59.2094 | −59.6125 | |||||||||||||||||||||||||||||||
| 1.4 | 3.70156 | 9.28490 | 5.70156 | −16.1047 | 34.3686 | −1.29844 | −8.50781 | 59.2094 | −59.6125 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(31\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 961.4.a.d | ✓ | 4 |
| 31.b | odd | 2 | 1 | inner | 961.4.a.d | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 961.4.a.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 961.4.a.d | ✓ | 4 | 31.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(961))\):
|
\( T_{2}^{2} - T_{2} - 10 \)
|
|
\( T_{3}^{4} - 134T_{3}^{2} + 4120 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T - 10)^{2} \)
|
| $3$ |
\( T^{4} - 134T^{2} + 4120 \)
|
| $5$ |
\( (T^{2} + 13 T - 50)^{2} \)
|
| $7$ |
\( (T^{2} + 9 T + 10)^{2} \)
|
| $11$ |
\( T^{4} - 1530 T^{2} + 412000 \)
|
| $13$ |
\( T^{4} - 7014 T^{2} + 5640280 \)
|
| $17$ |
\( T^{4} - 12356 T^{2} + 15837280 \)
|
| $19$ |
\( (T^{2} + 119 T + 2300)^{2} \)
|
| $23$ |
\( T^{4} - 33564 T^{2} + 139486720 \)
|
| $29$ |
\( T^{4} - 70230 T^{2} + 548887000 \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( T^{4} - 51786 T^{2} + 454122880 \)
|
| $41$ |
\( (T^{2} - 539 T + 41624)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 12231163480 \)
|
| $47$ |
\( (T^{2} - 454 T + 49520)^{2} \)
|
| $53$ |
\( T^{4} + \cdots + 12103719520 \)
|
| $59$ |
\( (T^{2} + 521 T - 32600)^{2} \)
|
| $61$ |
\( T^{4} + \cdots + 237658183000 \)
|
| $67$ |
\( (T^{2} - 464 T - 158720)^{2} \)
|
| $71$ |
\( (T^{2} - 315 T - 162000)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 188274112000 \)
|
| $79$ |
\( T^{4} + \cdots + 229899708000 \)
|
| $83$ |
\( T^{4} + \cdots + 168438932320 \)
|
| $89$ |
\( T^{4} + \cdots + 70615152000 \)
|
| $97$ |
\( (T^{2} + 631 T + 61400)^{2} \)
|
| show more | |
| show less |