Newspace parameters
| Level: | \( N \) | \(=\) | \( 7865 = 5 \cdot 11^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7865.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(62.8023411897\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.6224.1 |
|
|
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| Defining polynomial: |
\( x^{4} - 6x^{2} - 2x + 5 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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
\( x^{4} - 6x^{2} - 2x + 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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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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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \)
|
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 |
|
−2.03032 | 1.66000 | 2.12221 | 1.00000 | −3.37033 | 2.00000 | −0.248119 | −0.244414 | −2.03032 | ||||||||||||||||||||||||||||||
| 1.2 | 0.134632 | 3.31116 | −1.98187 | 1.00000 | 0.445787 | 2.00000 | −0.536087 | 7.96375 | 0.134632 | |||||||||||||||||||||||||||||||
| 1.3 | 1.57942 | −2.45170 | 0.494582 | 1.00000 | −3.87228 | 2.00000 | −2.37769 | 3.01084 | 1.57942 | |||||||||||||||||||||||||||||||
| 1.4 | 2.31627 | −0.519450 | 3.36509 | 1.00000 | −1.20318 | 2.00000 | 3.16190 | −2.73017 | 2.31627 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(11\) | \( -1 \) |
| \(13\) | \( -1 \) |
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 | 7865.2.a.n | yes | 4 |
| 11.b | odd | 2 | 1 | 7865.2.a.m | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7865.2.a.m | ✓ | 4 | 11.b | odd | 2 | 1 | |
| 7865.2.a.n | yes | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7865))\):
|
\( T_{2}^{4} - 2T_{2}^{3} - 4T_{2}^{2} + 8T_{2} - 1 \)
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\( T_{3}^{4} - 2T_{3}^{3} - 8T_{3}^{2} + 10T_{3} + 7 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 2 T^{3} + \cdots - 1 \)
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| $3$ |
\( T^{4} - 2 T^{3} + \cdots + 7 \)
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| $5$ |
\( (T - 1)^{4} \)
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| $7$ |
\( (T - 2)^{4} \)
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| $11$ |
\( T^{4} \)
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| $13$ |
\( (T - 1)^{4} \)
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| $17$ |
\( T^{4} - 2 T^{3} + \cdots - 5 \)
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| $19$ |
\( T^{4} - 4 T^{3} + \cdots + 32 \)
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| $23$ |
\( T^{4} + 10 T^{3} + \cdots - 533 \)
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| $29$ |
\( T^{4} - 16 T^{3} + \cdots - 691 \)
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| $31$ |
\( T^{4} - 14 T^{3} + \cdots + 4 \)
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| $37$ |
\( T^{4} + 18 T^{3} + \cdots - 7444 \)
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| $41$ |
\( T^{4} + 6 T^{3} + \cdots - 100 \)
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| $43$ |
\( T^{4} - 18 T^{3} + \cdots - 925 \)
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| $47$ |
\( T^{4} - 18 T^{3} + \cdots - 12244 \)
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| $53$ |
\( T^{4} + 6 T^{3} + \cdots + 631 \)
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| $59$ |
\( T^{4} - 2 T^{3} + \cdots + 76 \)
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| $61$ |
\( T^{4} - 4 T^{3} + \cdots + 1561 \)
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| $67$ |
\( T^{4} - 18 T^{3} + \cdots - 292 \)
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| $71$ |
\( T^{4} + 30 T^{3} + \cdots - 4228 \)
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| $73$ |
\( T^{4} + 8 T^{3} + \cdots + 14000 \)
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| $79$ |
\( T^{4} - 24 T^{3} + \cdots + 605 \)
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| $83$ |
\( T^{4} + 30 T^{3} + \cdots - 2708 \)
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| $89$ |
\( T^{4} - 24 T^{3} + \cdots + 416 \)
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| $97$ |
\( T^{4} - 12 T^{3} + \cdots - 1184 \)
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