Newspace parameters
| Level: | \( N \) | \(=\) | \( 7935 = 3 \cdot 5 \cdot 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7935.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(63.3612940039\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.4507648.1 |
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| Defining polynomial: |
\( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 1 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 4\nu^{2} + 2\nu + 2 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 4\nu^{3} + 6\nu^{2} + 4\nu - 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 5\beta_{2} + 6\beta _1 + 7 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + 6\beta_{3} + 8\beta_{2} + 18\beta _1 + 8 \)
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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 |
|
−2.44785 | 1.00000 | 3.99195 | 1.00000 | −2.44785 | 1.75717 | −4.87599 | 1.00000 | −2.44785 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.17619 | 1.00000 | −0.616586 | 1.00000 | −1.17619 | −4.49962 | 3.07759 | 1.00000 | −1.17619 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.656184 | 1.00000 | −1.56942 | 1.00000 | −0.656184 | 3.56472 | 2.34220 | 1.00000 | −0.656184 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.264627 | 1.00000 | −1.92997 | 1.00000 | 0.264627 | 0.526121 | −1.03998 | 1.00000 | 0.264627 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.83237 | 1.00000 | 1.35758 | 1.00000 | 1.83237 | −1.47931 | −1.17715 | 1.00000 | 1.83237 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.18322 | 1.00000 | 2.76645 | 1.00000 | 2.18322 | −1.86907 | 1.67333 | 1.00000 | 2.18322 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(23\) | \( +1 \) |
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 | 7935.2.a.bg | yes | 6 |
| 23.b | odd | 2 | 1 | 7935.2.a.bf | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7935.2.a.bf | ✓ | 6 | 23.b | odd | 2 | 1 | |
| 7935.2.a.bg | yes | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7935))\):
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\( T_{2}^{6} - 8T_{2}^{4} + 14T_{2}^{2} + 4T_{2} - 2 \)
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\( T_{7}^{6} + 2T_{7}^{5} - 19T_{7}^{4} - 24T_{7}^{3} + 63T_{7}^{2} + 54T_{7} - 41 \)
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\( T_{11}^{6} + 12T_{11}^{5} + 34T_{11}^{4} - 24T_{11}^{3} - 78T_{11}^{2} - 28T_{11} + 2 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 8 T^{4} + \cdots - 2 \)
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| $3$ |
\( (T - 1)^{6} \)
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| $5$ |
\( (T - 1)^{6} \)
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| $7$ |
\( T^{6} + 2 T^{5} + \cdots - 41 \)
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| $11$ |
\( T^{6} + 12 T^{5} + \cdots + 2 \)
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| $13$ |
\( T^{6} + 4 T^{5} + \cdots - 562 \)
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| $17$ |
\( T^{6} + 14 T^{5} + \cdots + 647 \)
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| $19$ |
\( T^{6} + 8 T^{5} + \cdots + 82 \)
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| $23$ |
\( T^{6} \)
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| $29$ |
\( T^{6} + 14 T^{5} + \cdots + 9457 \)
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| $31$ |
\( T^{6} + 6 T^{5} + \cdots + 193 \)
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| $37$ |
\( T^{6} - 2 T^{5} + \cdots + 1823 \)
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| $41$ |
\( T^{6} + 2 T^{5} + \cdots - 33287 \)
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| $43$ |
\( T^{6} + 12 T^{5} + \cdots - 932 \)
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| $47$ |
\( T^{6} - 8 T^{5} + \cdots + 10718 \)
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| $53$ |
\( T^{6} + 14 T^{5} + \cdots - 5233 \)
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| $59$ |
\( T^{6} + 14 T^{5} + \cdots + 961 \)
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| $61$ |
\( T^{6} - 8 T^{5} + \cdots - 5294 \)
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| $67$ |
\( T^{6} - 2 T^{5} + \cdots - 336689 \)
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| $71$ |
\( T^{6} + 10 T^{5} + \cdots - 113911 \)
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| $73$ |
\( T^{6} + 4 T^{5} + \cdots + 5198 \)
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| $79$ |
\( T^{6} - 16 T^{5} + \cdots + 83168 \)
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| $83$ |
\( T^{6} - 10 T^{5} + \cdots - 593 \)
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| $89$ |
\( T^{6} + 16 T^{5} + \cdots + 8264 \)
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| $97$ |
\( T^{6} + 8 T^{5} + \cdots - 385988 \)
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