Newspace parameters
| Level: | \( N \) | \(=\) | \( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7440.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(59.4086991038\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.8468.1 |
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| Defining polynomial: |
\( x^{4} - x^{3} - 5x^{2} + 3x + 4 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 465) |
| 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
\( x^{4} - x^{3} - 5x^{2} + 3x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - 3\nu - 1 \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
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| \(\nu\) | \(=\) |
\( ( -\beta_{3} - \beta_{2} + \beta_1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + \beta _1 + 6 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{3} - 3\beta_{2} + 5\beta _1 + 2 ) / 2 \)
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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 |
|
0 | 1.00000 | 0 | 1.00000 | 0 | −4.20813 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 1.00000 | 0 | 1.00000 | 0 | −2.31451 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.00000 | 0 | 1.00000 | 0 | −0.946946 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.00000 | 0 | 1.00000 | 0 | 3.46958 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(31\) | \( -1 \) |
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 | 7440.2.a.bz | 4 | |
| 4.b | odd | 2 | 1 | 465.2.a.h | ✓ | 4 | |
| 12.b | even | 2 | 1 | 1395.2.a.k | 4 | ||
| 20.d | odd | 2 | 1 | 2325.2.a.v | 4 | ||
| 20.e | even | 4 | 2 | 2325.2.c.p | 8 | ||
| 60.h | even | 2 | 1 | 6975.2.a.bo | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 465.2.a.h | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 1395.2.a.k | 4 | 12.b | even | 2 | 1 | ||
| 2325.2.a.v | 4 | 20.d | odd | 2 | 1 | ||
| 2325.2.c.p | 8 | 20.e | even | 4 | 2 | ||
| 6975.2.a.bo | 4 | 60.h | even | 2 | 1 | ||
| 7440.2.a.bz | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7440))\):
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\( T_{7}^{4} + 4T_{7}^{3} - 10T_{7}^{2} - 46T_{7} - 32 \)
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\( T_{11}^{4} + 6T_{11}^{3} - 12T_{11}^{2} - 56T_{11} + 32 \)
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\( T_{13}^{4} - 2T_{13}^{3} - 50T_{13}^{2} + 94T_{13} + 4 \)
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\( T_{17}^{4} + 10T_{17}^{3} + 20T_{17}^{2} - 52T_{17} - 136 \)
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\( T_{19} + 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( (T - 1)^{4} \)
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| $5$ |
\( (T - 1)^{4} \)
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| $7$ |
\( T^{4} + 4 T^{3} + \cdots - 32 \)
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| $11$ |
\( T^{4} + 6 T^{3} + \cdots + 32 \)
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| $13$ |
\( T^{4} - 2 T^{3} + \cdots + 4 \)
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| $17$ |
\( T^{4} + 10 T^{3} + \cdots - 136 \)
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| $19$ |
\( (T + 4)^{4} \)
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| $23$ |
\( T^{4} + 6 T^{3} + \cdots + 2048 \)
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| $29$ |
\( T^{4} - 12 T^{2} + \cdots - 4 \)
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| $31$ |
\( (T - 1)^{4} \)
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| $37$ |
\( T^{4} - 8 T^{3} + \cdots + 388 \)
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| $41$ |
\( T^{4} + 6 T^{3} + \cdots - 704 \)
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| $43$ |
\( T^{4} + 2 T^{3} + \cdots + 928 \)
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| $47$ |
\( T^{4} - 2 T^{3} + \cdots - 736 \)
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| $53$ |
\( T^{4} + 6 T^{3} + \cdots - 8 \)
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| $59$ |
\( T^{4} + 4 T^{3} + \cdots - 808 \)
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| $61$ |
\( T^{4} - 168 T^{2} + \cdots + 848 \)
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| $67$ |
\( T^{4} - 2 T^{3} + \cdots + 184 \)
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| $71$ |
\( T^{4} + 22 T^{3} + \cdots - 1808 \)
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| $73$ |
\( T^{4} + 32 T^{3} + \cdots + 788 \)
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| $79$ |
\( T^{4} + 24 T^{3} + \cdots - 24064 \)
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| $83$ |
\( T^{4} - 6 T^{3} + \cdots - 464 \)
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| $89$ |
\( T^{4} + 14 T^{3} + \cdots - 92 \)
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| $97$ |
\( T^{4} + 14 T^{3} + \cdots - 64 \)
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