Newspace parameters
| Level: | \( N \) | \(=\) | \( 1352 = 2^{3} \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1352.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(79.7705823278\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{5} - 2x^{4} - 83x^{3} + 112x^{2} + 804x + 144 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 104) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 83x^{3} + 112x^{2} + 804x + 144 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 3\nu^{2} - 66\nu + 96 ) / 8 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 77\nu^{2} + 118\nu + 432 ) / 24 \)
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| \(\beta_{4}\) | \(=\) |
\( ( \nu^{3} + 5\nu^{2} - 66\nu - 176 ) / 8 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{2} + 34 \)
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| \(\nu^{3}\) | \(=\) |
\( 3\beta_{4} + 5\beta_{2} + 66\beta _1 + 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( 83\beta_{4} + 24\beta_{3} - 67\beta_{2} + 14\beta _1 + 2198 \)
|
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 | −7.78626 | 0 | 7.31325 | 0 | 0.135778 | 0 | 33.6258 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −3.21876 | 0 | −9.41803 | 0 | −19.8746 | 0 | −16.6396 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.18449 | 0 | 18.9843 | 0 | 9.32401 | 0 | −25.5970 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 3.54428 | 0 | −10.1595 | 0 | 21.9598 | 0 | −14.4381 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 9.27625 | 0 | 2.27999 | 0 | −28.5449 | 0 | 59.0489 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(13\) | \( -1 \) |
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 | 1352.4.a.l | 5 | |
| 13.b | even | 2 | 1 | 1352.4.a.k | 5 | ||
| 13.d | odd | 4 | 2 | 104.4.f.a | ✓ | 10 | |
| 39.f | even | 4 | 2 | 936.4.c.a | 10 | ||
| 52.f | even | 4 | 2 | 208.4.f.e | 10 | ||
| 104.j | odd | 4 | 2 | 832.4.f.k | 10 | ||
| 104.m | even | 4 | 2 | 832.4.f.l | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 104.4.f.a | ✓ | 10 | 13.d | odd | 4 | 2 | |
| 208.4.f.e | 10 | 52.f | even | 4 | 2 | ||
| 832.4.f.k | 10 | 104.j | odd | 4 | 2 | ||
| 832.4.f.l | 10 | 104.m | even | 4 | 2 | ||
| 936.4.c.a | 10 | 39.f | even | 4 | 2 | ||
| 1352.4.a.k | 5 | 13.b | even | 2 | 1 | ||
| 1352.4.a.l | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1352))\):
|
\( T_{3}^{5} - 3T_{3}^{4} - 81T_{3}^{3} + 139T_{3}^{2} + 776T_{3} - 976 \)
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\( T_{5}^{5} - 9T_{5}^{4} - 265T_{5}^{3} + 841T_{5}^{2} + 12824T_{5} - 30288 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
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| $3$ |
\( T^{5} - 3 T^{4} + \cdots - 976 \)
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| $5$ |
\( T^{5} - 9 T^{4} + \cdots - 30288 \)
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| $7$ |
\( T^{5} + 17 T^{4} + \cdots - 15772 \)
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| $11$ |
\( T^{5} + 4 T^{4} + \cdots + 131072 \)
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| $13$ |
\( T^{5} \)
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| $17$ |
\( T^{5} + \cdots + 1264419564 \)
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| $19$ |
\( T^{5} + 46 T^{4} + \cdots + 148852896 \)
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| $23$ |
\( T^{5} + 90 T^{4} + \cdots + 786703872 \)
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| $29$ |
\( T^{5} + \cdots + 77894516736 \)
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| $31$ |
\( T^{5} + \cdots + 106655862784 \)
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| $37$ |
\( T^{5} + \cdots + 157797094832 \)
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| $41$ |
\( T^{5} + \cdots - 67714940928 \)
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| $43$ |
\( T^{5} + \cdots - 156039593008 \)
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| $47$ |
\( T^{5} + \cdots + 1378246395812 \)
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| $53$ |
\( T^{5} + \cdots - 3566476576 \)
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| $59$ |
\( T^{5} + \cdots + 6732388323936 \)
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| $61$ |
\( T^{5} + \cdots - 159706376192 \)
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| $67$ |
\( T^{5} + \cdots - 5761248471808 \)
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| $71$ |
\( T^{5} + \cdots - 40179792804948 \)
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| $73$ |
\( T^{5} + \cdots - 41114038915584 \)
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| $79$ |
\( T^{5} + \cdots + 26292758095872 \)
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| $83$ |
\( T^{5} + \cdots + 17826052367712 \)
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| $89$ |
\( T^{5} + \cdots - 195373952257536 \)
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| $97$ |
\( T^{5} + \cdots - 8511025938432 \)
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