Newspace parameters
| Level: | \( N \) | \(=\) | \( 7410 = 2 \cdot 3 \cdot 5 \cdot 13 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7410.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(59.1691478978\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.85688.1 |
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|
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| Defining polynomial: |
\( x^{4} - x^{3} - 10x^{2} - 2x + 8 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{4} - x^{3} - 10x^{2} - 2x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 8\nu - 2 ) / 2 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 3\nu^{2} + 6\nu - 8 ) / 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 5 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_{2} + 9\beta _1 + 7 \)
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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 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | −3.69457 | −1.00000 | 1.00000 | −1.00000 | ||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | −0.794255 | −1.00000 | 1.00000 | −1.00000 | |||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | 1.18164 | −1.00000 | 1.00000 | −1.00000 | |||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | 2.30718 | −1.00000 | 1.00000 | −1.00000 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(13\) | \( -1 \) |
| \(19\) | \( -1 \) |
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 | 7410.2.a.bo | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7410.2.a.bo | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7410))\):
|
\( T_{7}^{4} + T_{7}^{3} - 10T_{7}^{2} + 2T_{7} + 8 \)
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\( T_{11}^{4} - 4T_{11}^{3} - 13T_{11}^{2} + 52T_{11} - 4 \)
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\( T_{17}^{4} + T_{17}^{3} - 30T_{17}^{2} + 32T_{17} + 64 \)
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\( T_{23}^{4} - 5T_{23}^{3} - 36T_{23}^{2} + 128T_{23} + 256 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{4} \)
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| $3$ |
\( (T + 1)^{4} \)
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| $5$ |
\( (T - 1)^{4} \)
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| $7$ |
\( T^{4} + T^{3} - 10 T^{2} + \cdots + 8 \)
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| $11$ |
\( T^{4} - 4 T^{3} + \cdots - 4 \)
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| $13$ |
\( (T - 1)^{4} \)
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| $17$ |
\( T^{4} + T^{3} + \cdots + 64 \)
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| $19$ |
\( (T - 1)^{4} \)
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| $23$ |
\( T^{4} - 5 T^{3} + \cdots + 256 \)
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| $29$ |
\( T^{4} - 12 T^{3} + \cdots - 448 \)
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| $31$ |
\( T^{4} - 12 T^{3} + \cdots - 448 \)
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| $37$ |
\( T^{4} + 9 T^{3} + \cdots + 688 \)
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| $41$ |
\( T^{4} - 12 T^{3} + \cdots + 8 \)
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| $43$ |
\( T^{4} - T^{3} + \cdots + 1916 \)
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| $47$ |
\( T^{4} - T^{3} + \cdots + 1172 \)
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| $53$ |
\( T^{4} + T^{3} + \cdots + 1688 \)
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| $59$ |
\( T^{4} - 24 T^{3} + \cdots + 128 \)
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| $61$ |
\( T^{4} + 6 T^{3} + \cdots + 64 \)
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| $67$ |
\( T^{4} - 5 T^{3} + \cdots - 392 \)
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| $71$ |
\( T^{4} - 4 T^{3} + \cdots + 5242 \)
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| $73$ |
\( T^{4} + 17 T^{3} + \cdots - 14188 \)
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| $79$ |
\( T^{4} - 12 T^{3} + \cdots + 436 \)
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| $83$ |
\( T^{4} - 2 T^{3} + \cdots - 2656 \)
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| $89$ |
\( T^{4} - 10 T^{3} + \cdots + 368 \)
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| $97$ |
\( T^{4} - 4 T^{3} + \cdots + 1136 \)
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