Newspace parameters
| Level: | \( N \) | \(=\) | \( 112 = 2^{4} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 112.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(159.816725712\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 7385x^{2} - 167250x - 586980 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{14}\cdot 3\cdot 5\cdot 7 \) |
| Twist minimal: | no (minimal twist has level 7) |
| 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
\( x^{4} - 2x^{3} - 7385x^{2} - 167250x - 586980 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 25\nu^{3} + 127\nu^{2} - 93050\nu - 3836328 ) / 1707 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -145\nu^{3} + 2905\nu^{2} + 994890\nu + 8567618 ) / 569 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -89\nu^{3} - 7007\nu^{2} + 1041370\nu + 37510536 ) / 1707 \)
|
| \(\nu\) | \(=\) |
\( ( 5\beta_{3} + 3\beta_{2} + 70\beta _1 + 2274 ) / 4480 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -125\beta_{3} + 65\beta_{2} + 686\beta _1 + 3309814 ) / 896 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4357\beta_{3} + 1903\beta_{2} + 109802\beta _1 + 122372906 ) / 896 \)
|
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 | −5331.04 | 0 | 305120. | 0 | 823543. | 0 | 1.40711e7 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −4148.79 | 0 | −142812. | 0 | 823543. | 0 | 2.86353e6 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −3391.24 | 0 | −75579.8 | 0 | 823543. | 0 | −2.84842e6 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 4317.07 | 0 | −78958.0 | 0 | 823543. | 0 | 4.28816e6 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( -1 \) |
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 | 112.16.a.f | 4 | |
| 4.b | odd | 2 | 1 | 7.16.a.b | ✓ | 4 | |
| 12.b | even | 2 | 1 | 63.16.a.e | 4 | ||
| 28.d | even | 2 | 1 | 49.16.a.d | 4 | ||
| 28.f | even | 6 | 2 | 49.16.c.g | 8 | ||
| 28.g | odd | 6 | 2 | 49.16.c.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.16.a.b | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 49.16.a.d | 4 | 28.d | even | 2 | 1 | ||
| 49.16.c.f | 8 | 28.g | odd | 6 | 2 | ||
| 49.16.c.g | 8 | 28.f | even | 6 | 2 | ||
| 63.16.a.e | 4 | 12.b | even | 2 | 1 | ||
| 112.16.a.f | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 8554T_{3}^{3} - 1299540T_{3}^{2} - 159263455176T_{3} - 323802472565376 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(112))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + \cdots - 323802472565376 \)
|
| $5$ |
\( T^{4} + \cdots - 26\!\cdots\!00 \)
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| $7$ |
\( (T - 823543)^{4} \)
|
| $11$ |
\( T^{4} + \cdots - 15\!\cdots\!44 \)
|
| $13$ |
\( T^{4} + \cdots + 30\!\cdots\!20 \)
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| $17$ |
\( T^{4} + \cdots + 13\!\cdots\!56 \)
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| $19$ |
\( T^{4} + \cdots - 18\!\cdots\!00 \)
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| $23$ |
\( T^{4} + \cdots + 46\!\cdots\!56 \)
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| $29$ |
\( T^{4} + \cdots + 39\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots - 40\!\cdots\!92 \)
|
| $37$ |
\( T^{4} + \cdots - 86\!\cdots\!76 \)
|
| $41$ |
\( T^{4} + \cdots - 75\!\cdots\!36 \)
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| $43$ |
\( T^{4} + \cdots + 59\!\cdots\!60 \)
|
| $47$ |
\( T^{4} + \cdots + 32\!\cdots\!08 \)
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| $53$ |
\( T^{4} + \cdots - 29\!\cdots\!32 \)
|
| $59$ |
\( T^{4} + \cdots - 68\!\cdots\!00 \)
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| $61$ |
\( T^{4} + \cdots - 53\!\cdots\!36 \)
|
| $67$ |
\( T^{4} + \cdots - 21\!\cdots\!68 \)
|
| $71$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots - 16\!\cdots\!52 \)
|
| $79$ |
\( T^{4} + \cdots + 92\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 15\!\cdots\!88 \)
|
| $89$ |
\( T^{4} + \cdots - 24\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 10\!\cdots\!68 \)
|
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