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: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.998068.1 |
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| Defining polynomial: |
\( x^{5} - 2x^{4} - 6x^{3} + 10x^{2} + 3x - 2 \)
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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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 6x^{3} + 10x^{2} + 3x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - 5\nu - 1 \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 7\nu^{2} + 7\nu + 4 ) / 2 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 7\nu^{2} + 3\nu + 6 ) / 2 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 8\nu \)
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| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + 1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} - 2\beta_{3} + \beta _1 + 7 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{3} + 5\beta_{2} + 2\beta _1 + 7 ) / 2 \)
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| \(\nu^{4}\) | \(=\) |
\( ( 7\beta_{4} - 12\beta_{3} + 2\beta_{2} + 9\beta _1 + 41 ) / 2 \)
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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 | 1.00000 | 1.00000 | −1.00000 | −1.00000 | −4.85184 | −1.00000 | 1.00000 | 1.00000 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | 1.00000 | 1.00000 | −1.00000 | −1.00000 | −3.14266 | −1.00000 | 1.00000 | 1.00000 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 1.00000 | 1.00000 | −1.00000 | −1.00000 | −1.38739 | −1.00000 | 1.00000 | 1.00000 | ||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 1.00000 | 1.00000 | −1.00000 | −1.00000 | −0.699450 | −1.00000 | 1.00000 | 1.00000 | ||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 1.00000 | 1.00000 | −1.00000 | −1.00000 | 1.08134 | −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.bv | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7410.2.a.bv | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7410))\):
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\( T_{7}^{5} + 9T_{7}^{4} + 22T_{7}^{3} + 4T_{7}^{2} - 28T_{7} - 16 \)
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\( T_{11}^{5} - T_{11}^{4} - 36T_{11}^{3} - 16T_{11}^{2} + 336T_{11} + 512 \)
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\( T_{17}^{5} - 4T_{17}^{4} - 52T_{17}^{3} + 144T_{17}^{2} + 576T_{17} - 256 \)
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\( T_{23}^{5} - T_{23}^{4} - 84T_{23}^{3} + 48T_{23}^{2} + 1344T_{23} - 512 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{5} \)
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| $3$ |
\( (T - 1)^{5} \)
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| $5$ |
\( (T + 1)^{5} \)
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| $7$ |
\( T^{5} + 9 T^{4} + \cdots - 16 \)
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| $11$ |
\( T^{5} - T^{4} + \cdots + 512 \)
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| $13$ |
\( (T + 1)^{5} \)
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| $17$ |
\( T^{5} - 4 T^{4} + \cdots - 256 \)
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| $19$ |
\( (T - 1)^{5} \)
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| $23$ |
\( T^{5} - T^{4} + \cdots - 512 \)
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| $29$ |
\( T^{5} - 4 T^{4} + \cdots + 32 \)
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| $31$ |
\( T^{5} + 2 T^{4} + \cdots - 3200 \)
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| $37$ |
\( T^{5} + 22 T^{4} + \cdots - 1184 \)
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| $41$ |
\( T^{5} - 8 T^{4} + \cdots - 6736 \)
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| $43$ |
\( T^{5} - 3 T^{4} + \cdots - 4688 \)
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| $47$ |
\( T^{5} + 6 T^{4} + \cdots + 37376 \)
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| $53$ |
\( T^{5} + 2 T^{4} + \cdots + 106016 \)
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| $59$ |
\( T^{5} + 6 T^{4} + \cdots - 2048 \)
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| $61$ |
\( T^{5} - 18 T^{4} + \cdots - 1312 \)
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| $67$ |
\( T^{5} + 17 T^{4} + \cdots - 15424 \)
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| $71$ |
\( T^{5} - 7 T^{4} + \cdots + 3776 \)
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| $73$ |
\( T^{5} + 14 T^{4} + \cdots + 5552 \)
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| $79$ |
\( T^{5} + 13 T^{4} + \cdots + 64 \)
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| $83$ |
\( T^{5} + 2 T^{4} + \cdots + 128 \)
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| $89$ |
\( T^{5} + 6 T^{4} + \cdots + 17792 \)
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| $97$ |
\( T^{5} + 34 T^{4} + \cdots - 247136 \)
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