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: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 1253x^{2} - 1039x + 42996 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 5^{2} \) |
| 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
\( x^{4} - x^{3} - 1253x^{2} - 1039x + 42996 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 626 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 1185\nu - 1424 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 626 \)
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| \(\nu^{3}\) | \(=\) |
\( 4\beta_{3} + 1185\beta _1 + 1424 \)
|
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 |
|
−30.9278 | 0 | 444.529 | 625.000 | 0 | 2401.00 | 2086.72 | 0 | −19329.9 | ||||||||||||||||||||||||||||||
| 1.2 | −3.41896 | 0 | −500.311 | 625.000 | 0 | 2401.00 | 3461.05 | 0 | −2136.85 | |||||||||||||||||||||||||||||||
| 1.3 | 8.50890 | 0 | −439.599 | 625.000 | 0 | 2401.00 | −8097.06 | 0 | 5318.06 | |||||||||||||||||||||||||||||||
| 1.4 | 38.8379 | 0 | 996.380 | 625.000 | 0 | 2401.00 | 18812.3 | 0 | 24273.7 | |||||||||||||||||||||||||||||||
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.e | 4 | |
| 3.b | odd | 2 | 1 | 105.10.a.d | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.10.a.d | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 315.10.a.e | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 13T_{2}^{3} - 1190T_{2}^{2} + 6344T_{2} + 34944 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(315))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 13 T^{3} + \cdots + 34944 \)
|
| $3$ |
\( T^{4} \)
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| $5$ |
\( (T - 625)^{4} \)
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| $7$ |
\( (T - 2401)^{4} \)
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| $11$ |
\( T^{4} + \cdots + 12\!\cdots\!56 \)
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| $13$ |
\( T^{4} + \cdots + 73\!\cdots\!64 \)
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| $17$ |
\( T^{4} + \cdots - 34\!\cdots\!36 \)
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| $19$ |
\( T^{4} + \cdots + 59\!\cdots\!36 \)
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| $23$ |
\( T^{4} + \cdots + 21\!\cdots\!36 \)
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| $29$ |
\( T^{4} + \cdots - 11\!\cdots\!96 \)
|
| $31$ |
\( T^{4} + \cdots + 34\!\cdots\!44 \)
|
| $37$ |
\( T^{4} + \cdots - 24\!\cdots\!16 \)
|
| $41$ |
\( T^{4} + \cdots + 57\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots - 11\!\cdots\!76 \)
|
| $47$ |
\( T^{4} + \cdots + 90\!\cdots\!76 \)
|
| $53$ |
\( T^{4} + \cdots + 16\!\cdots\!56 \)
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| $59$ |
\( T^{4} + \cdots - 14\!\cdots\!64 \)
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| $61$ |
\( T^{4} + \cdots - 79\!\cdots\!04 \)
|
| $67$ |
\( T^{4} + \cdots - 13\!\cdots\!24 \)
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| $71$ |
\( T^{4} + \cdots - 14\!\cdots\!24 \)
|
| $73$ |
\( T^{4} + \cdots + 18\!\cdots\!96 \)
|
| $79$ |
\( T^{4} + \cdots + 12\!\cdots\!56 \)
|
| $83$ |
\( T^{4} + \cdots - 43\!\cdots\!84 \)
|
| $89$ |
\( T^{4} + \cdots - 10\!\cdots\!36 \)
|
| $97$ |
\( T^{4} + \cdots - 69\!\cdots\!36 \)
|
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