Newspace parameters
| Level: | \( N \) | \(=\) | \( 7406 = 2 \cdot 7 \cdot 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7406.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(59.1372077370\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.596217024.1 |
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| Defining polynomial: |
\( x^{6} - 15x^{4} - 6x^{3} + 57x^{2} + 24x - 48 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{6} - 15x^{4} - 6x^{3} + 57x^{2} + 24x - 48 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 5 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 4\nu^{4} + 7\nu^{3} - 30\nu^{2} - 9\nu + 36 ) / 8 \)
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| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 9\nu^{3} + 12\nu^{2} + 15\nu - 14 ) / 2 \)
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| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 11\nu^{3} + 14\nu^{2} + 27\nu - 18 ) / 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 5 \)
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| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + \beta_{2} + 7\beta _1 + 3 \)
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| \(\nu^{4}\) | \(=\) |
\( -\beta_{5} + 2\beta_{4} + 4\beta_{3} + 10\beta_{2} + 13\beta _1 + 37 \)
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| \(\nu^{5}\) | \(=\) |
\( -11\beta_{5} + 15\beta_{4} + 8\beta_{3} + 17\beta_{2} + 62\beta _1 + 55 \)
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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 | −2.49020 | 1.00000 | 4.22225 | 2.49020 | 1.00000 | −1.00000 | 3.20109 | −4.22225 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.26639 | 1.00000 | 0.534336 | 2.26639 | 1.00000 | −1.00000 | 2.13651 | −0.534336 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.50154 | 1.00000 | −0.230514 | 1.50154 | 1.00000 | −1.00000 | −0.745387 | 0.230514 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.817039 | 1.00000 | 0.915012 | −0.817039 | 1.00000 | −1.00000 | −2.33245 | −0.915012 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 2.03587 | 1.00000 | −3.76792 | −2.03587 | 1.00000 | −1.00000 | 1.14478 | 3.76792 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 3.40521 | 1.00000 | −1.67316 | −3.40521 | 1.00000 | −1.00000 | 8.59546 | 1.67316 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(7\) | \( -1 \) |
| \(23\) | \( -1 \) |
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 | 7406.2.a.bj | yes | 6 |
| 23.b | odd | 2 | 1 | 7406.2.a.bi | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7406.2.a.bi | ✓ | 6 | 23.b | odd | 2 | 1 | |
| 7406.2.a.bj | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7406))\):
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\( T_{3}^{6} - 15T_{3}^{4} - 6T_{3}^{3} + 57T_{3}^{2} + 24T_{3} - 48 \)
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\( T_{5}^{6} - 18T_{5}^{4} - 6T_{5}^{3} + 30T_{5}^{2} - 6T_{5} - 3 \)
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\( T_{11}^{6} - 33T_{11}^{4} - 64T_{11}^{3} + 9T_{11}^{2} + 48T_{11} + 16 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
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| $3$ |
\( T^{6} - 15 T^{4} + \cdots - 48 \)
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| $5$ |
\( T^{6} - 18 T^{4} + \cdots - 3 \)
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| $7$ |
\( (T - 1)^{6} \)
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| $11$ |
\( T^{6} - 33 T^{4} + \cdots + 16 \)
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| $13$ |
\( T^{6} + 6 T^{5} + \cdots - 23 \)
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| $17$ |
\( T^{6} + 6 T^{5} + \cdots - 128 \)
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| $19$ |
\( T^{6} + 6 T^{5} + \cdots - 128 \)
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| $23$ |
\( T^{6} \)
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| $29$ |
\( T^{6} - 6 T^{5} + \cdots + 208 \)
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| $31$ |
\( T^{6} + 6 T^{5} + \cdots + 64 \)
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| $37$ |
\( T^{6} + 12 T^{5} + \cdots + 8464 \)
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| $41$ |
\( T^{6} + 6 T^{5} + \cdots + 9 \)
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| $43$ |
\( T^{6} + 24 T^{5} + \cdots - 119600 \)
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| $47$ |
\( T^{6} - 12 T^{5} + \cdots - 9584 \)
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| $53$ |
\( T^{6} - 24 T^{5} + \cdots + 28176 \)
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| $59$ |
\( T^{6} - 132 T^{4} + \cdots + 1024 \)
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| $61$ |
\( T^{6} + 36 T^{5} + \cdots - 30576 \)
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| $67$ |
\( T^{6} - 300 T^{4} + \cdots - 871424 \)
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| $71$ |
\( T^{6} + 6 T^{5} + \cdots - 87152 \)
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| $73$ |
\( T^{6} - 141 T^{4} + \cdots - 1187 \)
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| $79$ |
\( T^{6} + 24 T^{5} + \cdots + 108496 \)
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| $83$ |
\( T^{6} + 12 T^{5} + \cdots + 435264 \)
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| $89$ |
\( T^{6} + 42 T^{5} + \cdots - 414179 \)
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| $97$ |
\( T^{6} - 6 T^{5} + \cdots - 20763 \)
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