Newspace parameters
| Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 315.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(162.236288392\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 1462x^{2} + 568x + 469504 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 105) |
| 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
\( x^{4} - x^{3} - 1462x^{2} + 568x + 469504 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 818\nu + 264 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 731 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 731 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_{2} + 819\beta _1 + 467 \)
|
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 |
|
−26.9927 | 0 | 216.604 | −625.000 | 0 | −2401.00 | 7973.53 | 0 | 16870.4 | ||||||||||||||||||||||||||||||
| 1.2 | −17.7320 | 0 | −197.578 | −625.000 | 0 | −2401.00 | 12582.2 | 0 | 11082.5 | |||||||||||||||||||||||||||||||
| 1.3 | 25.9121 | 0 | 159.434 | −625.000 | 0 | −2401.00 | −9135.70 | 0 | −16195.0 | |||||||||||||||||||||||||||||||
| 1.4 | 35.8126 | 0 | 770.540 | −625.000 | 0 | −2401.00 | 9258.97 | 0 | −22382.9 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(7\) | \( +1 \) |
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 | 315.10.a.f | 4 | |
| 3.b | odd | 2 | 1 | 105.10.a.c | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.10.a.c | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 315.10.a.f | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 17T_{2}^{3} - 1354T_{2}^{2} + 11960T_{2} + 444160 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(315))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 17 T^{3} + \cdots + 444160 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T + 625)^{4} \)
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| $7$ |
\( (T + 2401)^{4} \)
|
| $11$ |
\( T^{4} + \cdots - 63\!\cdots\!36 \)
|
| $13$ |
\( T^{4} + \cdots + 19\!\cdots\!60 \)
|
| $17$ |
\( T^{4} + \cdots + 34\!\cdots\!04 \)
|
| $19$ |
\( T^{4} + \cdots - 44\!\cdots\!16 \)
|
| $23$ |
\( T^{4} + \cdots + 95\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 63\!\cdots\!56 \)
|
| $31$ |
\( T^{4} + \cdots + 52\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots + 35\!\cdots\!44 \)
|
| $41$ |
\( T^{4} + \cdots + 27\!\cdots\!56 \)
|
| $43$ |
\( T^{4} + \cdots - 23\!\cdots\!40 \)
|
| $47$ |
\( T^{4} + \cdots - 50\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 94\!\cdots\!44 \)
|
| $59$ |
\( T^{4} + \cdots - 15\!\cdots\!60 \)
|
| $61$ |
\( T^{4} + \cdots + 23\!\cdots\!16 \)
|
| $67$ |
\( T^{4} + \cdots - 64\!\cdots\!64 \)
|
| $71$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots - 10\!\cdots\!76 \)
|
| $79$ |
\( T^{4} + \cdots - 22\!\cdots\!04 \)
|
| $83$ |
\( T^{4} + \cdots + 21\!\cdots\!76 \)
|
| $89$ |
\( T^{4} + \cdots - 36\!\cdots\!36 \)
|
| $97$ |
\( T^{4} + \cdots + 22\!\cdots\!64 \)
|
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