Newspace parameters
| Level: | \( N \) | \(=\) | \( 7110 = 2 \cdot 3^{2} \cdot 5 \cdot 79 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7110.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(56.7736358371\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{17 +8 \sqrt{3}})\) |
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|
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| Defining polynomial: |
\( x^{4} - 2x^{3} - 7x^{2} + 8x + 4 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2370) |
| 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
\( x^{4} - 2x^{3} - 7x^{2} + 8x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 4 ) / 2 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 6\nu + 2 ) / 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + \beta _1 + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 2\beta_{2} + 7\beta _1 + 2 \)
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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 |
|
−1.00000 | 0 | 1.00000 | 1.00000 | 0 | −2.44247 | −1.00000 | 0 | −1.00000 | ||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | 0 | 1.00000 | 1.00000 | 0 | −1.61023 | −1.00000 | 0 | −1.00000 | |||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0 | 1.00000 | 1.00000 | 0 | −0.121816 | −1.00000 | 0 | −1.00000 | |||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0 | 1.00000 | 1.00000 | 0 | 4.17452 | −1.00000 | 0 | −1.00000 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(79\) | \( -1 \) |
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 | 7110.2.a.bn | 4 | |
| 3.b | odd | 2 | 1 | 2370.2.a.z | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2370.2.a.z | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 7110.2.a.bn | 4 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7110))\):
|
\( T_{7}^{4} - 13T_{7}^{2} - 18T_{7} - 2 \)
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\( T_{11}^{4} - 6T_{11}^{3} + 5T_{11}^{2} + 12T_{11} - 8 \)
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\( T_{13}^{4} + 2T_{13}^{3} - 39T_{13}^{2} - 148T_{13} - 104 \)
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\( T_{17}^{4} - 6T_{17}^{3} - 49T_{17}^{2} + 246T_{17} + 262 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{4} \)
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| $3$ |
\( T^{4} \)
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| $5$ |
\( (T - 1)^{4} \)
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| $7$ |
\( T^{4} - 13 T^{2} + \cdots - 2 \)
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| $11$ |
\( T^{4} - 6 T^{3} + \cdots - 8 \)
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| $13$ |
\( T^{4} + 2 T^{3} + \cdots - 104 \)
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| $17$ |
\( T^{4} - 6 T^{3} + \cdots + 262 \)
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| $19$ |
\( T^{4} + 6 T^{3} + \cdots - 32 \)
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| $23$ |
\( T^{4} - 2 T^{3} + \cdots + 472 \)
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| $29$ |
\( T^{4} - 18 T^{3} + \cdots + 214 \)
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| $31$ |
\( T^{4} + 2 T^{3} + \cdots - 488 \)
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| $37$ |
\( T^{4} + 6 T^{3} + \cdots + 1656 \)
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| $41$ |
\( T^{4} - 16 T^{3} + \cdots - 3488 \)
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| $43$ |
\( T^{4} + 8 T^{3} + \cdots - 2696 \)
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| $47$ |
\( T^{4} + 4 T^{3} + \cdots + 628 \)
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| $53$ |
\( T^{4} - 20 T^{3} + \cdots - 3776 \)
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| $59$ |
\( T^{4} - 22 T^{3} + \cdots + 568 \)
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| $61$ |
\( T^{4} + 22 T^{3} + \cdots - 416 \)
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| $67$ |
\( (T - 2)^{4} \)
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| $71$ |
\( T^{4} - 14 T^{3} + \cdots - 512 \)
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| $73$ |
\( T^{4} + 12 T^{3} + \cdots + 52 \)
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| $79$ |
\( (T - 1)^{4} \)
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| $83$ |
\( T^{4} - 8 T^{3} + \cdots + 436 \)
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| $89$ |
\( T^{4} - 26 T^{3} + \cdots - 8 \)
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| $97$ |
\( T^{4} - 295 T^{2} + \cdots + 20884 \)
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