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: | \(6\) |
| Coefficient field: | 6.6.997608784.1 |
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| Defining polynomial: |
\( x^{6} - 16x^{4} + 72x^{2} - 4x - 80 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 16x^{4} + 72x^{2} - 4x - 80 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 5 \)
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| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 12\nu^{3} - 24\nu + 4 ) / 4 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 12\nu^{3} + 32\nu - 4 ) / 4 \)
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| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 10\nu^{2} + 16 ) / 2 \)
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| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{4} + 2\nu^{3} + 10\nu^{2} - 12\nu - 16 ) / 2 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta _1 + 5 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 3\beta_{3} + 3\beta_{2} \)
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| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 10\beta _1 + 34 \)
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| \(\nu^{5}\) | \(=\) |
\( 12\beta_{5} + 12\beta_{4} + 24\beta_{3} + 20\beta_{2} + 4 \)
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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 | −5.01051 | 1.00000 | 1.00000 | −1.00000 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | 1.00000 | 1.00000 | −1.00000 | 1.00000 | −1.65456 | 1.00000 | 1.00000 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 1.00000 | 1.00000 | −1.00000 | 1.00000 | −1.24994 | 1.00000 | 1.00000 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.00000 | 1.00000 | −1.00000 | 1.00000 | −0.362700 | 1.00000 | 1.00000 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 1.00000 | 1.00000 | −1.00000 | 1.00000 | 3.82019 | 1.00000 | 1.00000 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 1.00000 | 1.00000 | −1.00000 | 1.00000 | 4.45752 | 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.ca | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7410.2.a.ca | ✓ | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7410))\):
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\( T_{7}^{6} - 32T_{7}^{4} - 4T_{7}^{3} + 200T_{7}^{2} + 248T_{7} + 64 \)
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\( T_{11}^{6} - T_{11}^{5} - 40T_{11}^{4} + 8T_{11}^{3} + 352T_{11}^{2} - 96T_{11} - 704 \)
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\( T_{17}^{6} - 9T_{17}^{5} - 24T_{17}^{4} + 252T_{17}^{3} + 224T_{17}^{2} - 976T_{17} + 528 \)
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\( T_{23}^{6} - 8T_{23}^{5} - 80T_{23}^{4} + 512T_{23}^{3} + 1792T_{23}^{2} - 5632T_{23} - 2048 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
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| $3$ |
\( (T - 1)^{6} \)
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| $5$ |
\( (T + 1)^{6} \)
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| $7$ |
\( T^{6} - 32 T^{4} + \cdots + 64 \)
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| $11$ |
\( T^{6} - T^{5} + \cdots - 704 \)
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| $13$ |
\( (T - 1)^{6} \)
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| $17$ |
\( T^{6} - 9 T^{5} + \cdots + 528 \)
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| $19$ |
\( (T + 1)^{6} \)
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| $23$ |
\( T^{6} - 8 T^{5} + \cdots - 2048 \)
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| $29$ |
\( T^{6} - 8 T^{5} + \cdots + 51744 \)
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| $31$ |
\( T^{6} - 116 T^{4} + \cdots + 128 \)
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| $37$ |
\( T^{6} - 11 T^{5} + \cdots - 32 \)
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| $41$ |
\( T^{6} - 12 T^{5} + \cdots + 400 \)
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| $43$ |
\( T^{6} - 20 T^{5} + \cdots + 15280 \)
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| $47$ |
\( T^{6} - 3 T^{5} + \cdots - 88 \)
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| $53$ |
\( T^{6} + 13 T^{5} + \cdots - 158304 \)
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| $59$ |
\( T^{6} - 2 T^{5} + \cdots + 7744 \)
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| $61$ |
\( T^{6} - 18 T^{5} + \cdots - 2176 \)
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| $67$ |
\( T^{6} - 6 T^{5} + \cdots + 150400 \)
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| $71$ |
\( T^{6} - 11 T^{5} + \cdots + 4840 \)
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| $73$ |
\( T^{6} - 9 T^{5} + \cdots - 2920 \)
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| $79$ |
\( T^{6} + 5 T^{5} + \cdots - 15536 \)
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| $83$ |
\( T^{6} - 20 T^{5} + \cdots + 148384 \)
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| $89$ |
\( T^{6} - 12 T^{5} + \cdots + 2640 \)
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| $97$ |
\( T^{6} - 12 T^{5} + \cdots - 123456 \)
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