Newspace parameters
| Level: | \( N \) | \(=\) | \( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7350.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(58.6900454856\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.10304.1 |
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| Defining polynomial: |
\( x^{4} - 2x^{3} - 7x^{2} + 8x + 8 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 1470) |
| 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
\( x^{4} - 2x^{3} - 7x^{2} + 8x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 4\nu ) / 2 \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 4 ) / 2 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} + 8\nu - 2 ) / 2 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta _1 + 1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 4\beta_{2} + \beta _1 + 9 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{3} + 4\beta_{2} + 9\beta _1 + 13 ) / 2 \)
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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 |
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1.00000 | −1.00000 | 1.00000 | 0 | −1.00000 | 0 | 1.00000 | 1.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.00000 | 1.00000 | 0 | −1.00000 | 0 | 1.00000 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.00000 | 1.00000 | 0 | −1.00000 | 0 | 1.00000 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −1.00000 | 1.00000 | 0 | −1.00000 | 0 | 1.00000 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(7\) | \( +1 \) |
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 | 7350.2.a.dt | 4 | |
| 5.b | even | 2 | 1 | 7350.2.a.ds | 4 | ||
| 5.c | odd | 4 | 2 | 1470.2.g.j | ✓ | 8 | |
| 7.b | odd | 2 | 1 | 7350.2.a.du | 4 | ||
| 35.c | odd | 2 | 1 | 7350.2.a.dr | 4 | ||
| 35.f | even | 4 | 2 | 1470.2.g.k | yes | 8 | |
| 35.k | even | 12 | 4 | 1470.2.n.k | 16 | ||
| 35.l | odd | 12 | 4 | 1470.2.n.l | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1470.2.g.j | ✓ | 8 | 5.c | odd | 4 | 2 | |
| 1470.2.g.k | yes | 8 | 35.f | even | 4 | 2 | |
| 1470.2.n.k | 16 | 35.k | even | 12 | 4 | ||
| 1470.2.n.l | 16 | 35.l | odd | 12 | 4 | ||
| 7350.2.a.dr | 4 | 35.c | odd | 2 | 1 | ||
| 7350.2.a.ds | 4 | 5.b | even | 2 | 1 | ||
| 7350.2.a.dt | 4 | 1.a | even | 1 | 1 | trivial | |
| 7350.2.a.du | 4 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7350))\):
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\( T_{11}^{4} - 18T_{11}^{2} - 16T_{11} + 32 \)
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\( T_{13}^{4} - 26T_{13}^{2} - 48T_{13} - 16 \)
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\( T_{17}^{4} + 4T_{17}^{3} - 32T_{17}^{2} - 8T_{17} + 124 \)
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\( T_{19}^{4} + 12T_{19}^{3} + 12T_{19}^{2} - 224T_{19} - 448 \)
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\( T_{23}^{4} - 4T_{23}^{3} - 36T_{23}^{2} + 128 \)
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\( T_{31}^{4} + 16T_{31}^{3} + 78T_{31}^{2} + 128T_{31} + 64 \)
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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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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} \)
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| $11$ |
\( T^{4} - 18 T^{2} + \cdots + 32 \)
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| $13$ |
\( T^{4} - 26 T^{2} + \cdots - 16 \)
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| $17$ |
\( T^{4} + 4 T^{3} + \cdots + 124 \)
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| $19$ |
\( T^{4} + 12 T^{3} + \cdots - 448 \)
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| $23$ |
\( T^{4} - 4 T^{3} + \cdots + 128 \)
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| $29$ |
\( T^{4} + 8 T^{3} + \cdots + 1568 \)
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| $31$ |
\( T^{4} + 16 T^{3} + \cdots + 64 \)
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| $37$ |
\( T^{4} + 8 T^{3} + \cdots + 200 \)
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| $41$ |
\( T^{4} + 12 T^{3} + \cdots - 376 \)
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| $43$ |
\( T^{4} - 4 T^{3} + \cdots + 736 \)
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| $47$ |
\( T^{4} + 12 T^{3} + \cdots - 1024 \)
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| $53$ |
\( T^{4} - 20 T^{3} + \cdots + 64 \)
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| $59$ |
\( T^{4} + 20 T^{3} + \cdots - 3008 \)
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| $61$ |
\( T^{4} + 12 T^{3} + \cdots - 648 \)
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| $67$ |
\( T^{4} + 4 T^{3} + \cdots + 32 \)
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| $71$ |
\( T^{4} + 20 T^{3} + \cdots + 2944 \)
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| $73$ |
\( T^{4} - 12 T^{3} + \cdots - 248 \)
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| $79$ |
\( T^{4} + 8 T^{3} + \cdots - 3584 \)
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| $83$ |
\( T^{4} + 8 T^{3} + \cdots + 128 \)
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| $89$ |
\( T^{4} + 44 T^{3} + \cdots + 5048 \)
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| $97$ |
\( T^{4} + 20 T^{3} + \cdots - 10376 \)
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