Newspace parameters
| Level: | \( N \) | \(=\) | \( 403 = 13 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 403.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(3.21797120146\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{8} - 3x^{7} - 7x^{6} + 19x^{5} + 21x^{4} - 31x^{3} - 29x^{2} - x + 1 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{7} - 7x^{6} + 19x^{5} + 21x^{4} - 31x^{3} - 29x^{2} - x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - 3\nu^{2} + 9\nu + 3 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 3\nu^{3} + 9\nu^{2} + 3\nu \)
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| \(\beta_{6}\) | \(=\) |
\( \nu^{7} - 4\nu^{6} - 3\nu^{5} + 21\nu^{4} + 3\nu^{3} - 30\nu^{2} - 9\nu + 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{7} - 3\nu^{6} - 7\nu^{5} + 20\nu^{4} + 18\nu^{3} - 34\nu^{2} - 20\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 + 3 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 3\beta_{3} + 6\beta_{2} + 9\beta _1 + 15 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 3\beta_{4} + 12\beta_{3} + 12\beta_{2} + 30\beta _1 + 27 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{7} - \beta_{6} + 4\beta_{5} + 13\beta_{4} + 36\beta_{3} + 43\beta_{2} + 69\beta _1 + 90 \)
|
| \(\nu^{7}\) | \(=\) |
\( 4\beta_{7} - 3\beta_{6} + 19\beta_{5} + 40\beta_{4} + 114\beta_{3} + 109\beta_{2} + 201\beta _1 + 205 \)
|
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 |
|
−2.66697 | 1.08287 | 5.11275 | −1.31345 | −2.88800 | 1.98184 | −8.30164 | −1.82738 | 3.50294 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.57233 | −3.30456 | 4.61688 | −4.29347 | 8.50042 | 0.180190 | −6.73148 | 7.92012 | 11.0442 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.75921 | 2.94965 | 1.09483 | −3.30948 | −5.18907 | −3.55859 | 1.59239 | 5.70046 | 5.82207 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.24762 | −2.42435 | −0.443454 | 0.848757 | 3.02465 | −1.14028 | 3.04849 | 2.87746 | −1.05892 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.841125 | 0.162061 | −1.29251 | 0.443680 | −0.136313 | 0.552392 | 2.76941 | −2.97374 | −0.373191 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.933096 | 1.42306 | −1.12933 | −3.66529 | 1.32785 | −3.96149 | −2.91997 | −0.974910 | −3.42006 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.39479 | −1.21460 | −0.0545724 | −0.0316546 | −1.69410 | −2.20383 | −2.86569 | −1.52475 | −0.0441514 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.75938 | −1.67414 | 1.09540 | −3.67909 | −2.94544 | 4.14977 | −1.59152 | −0.197253 | −6.47291 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(13\) | \( +1 \) |
| \(31\) | \( +1 \) |
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 | 403.2.a.d | ✓ | 8 |
| 3.b | odd | 2 | 1 | 3627.2.a.q | 8 | ||
| 4.b | odd | 2 | 1 | 6448.2.a.bf | 8 | ||
| 13.b | even | 2 | 1 | 5239.2.a.j | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 403.2.a.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 3627.2.a.q | 8 | 3.b | odd | 2 | 1 | ||
| 5239.2.a.j | 8 | 13.b | even | 2 | 1 | ||
| 6448.2.a.bf | 8 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 5T_{2}^{7} - 30T_{2}^{5} - 24T_{2}^{4} + 54T_{2}^{3} + 54T_{2}^{2} - 28T_{2} - 29 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(403))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 5 T^{7} + \cdots - 29 \)
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| $3$ |
\( T^{8} + 3 T^{7} + \cdots + 12 \)
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| $5$ |
\( T^{8} + 15 T^{7} + \cdots + 3 \)
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| $7$ |
\( T^{8} + 4 T^{7} + \cdots + 29 \)
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| $11$ |
\( T^{8} + 5 T^{7} + \cdots - 14580 \)
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| $13$ |
\( (T + 1)^{8} \)
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| $17$ |
\( T^{8} + 11 T^{7} + \cdots - 1940 \)
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| $19$ |
\( T^{8} + 9 T^{7} + \cdots + 3557 \)
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| $23$ |
\( T^{8} - 86 T^{6} + \cdots + 3452 \)
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| $29$ |
\( T^{8} + 12 T^{7} + \cdots - 315444 \)
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| $31$ |
\( (T + 1)^{8} \)
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| $37$ |
\( T^{8} + 9 T^{7} + \cdots - 852812 \)
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| $41$ |
\( T^{8} + 25 T^{7} + \cdots + 18859 \)
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| $43$ |
\( T^{8} - 7 T^{7} + \cdots - 149092 \)
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| $47$ |
\( T^{8} + 17 T^{7} + \cdots - 120144 \)
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| $53$ |
\( T^{8} + 15 T^{7} + \cdots - 828 \)
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| $59$ |
\( T^{8} + 15 T^{7} + \cdots - 1314177 \)
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| $61$ |
\( T^{8} - 11 T^{7} + \cdots + 493348 \)
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| $67$ |
\( T^{8} - 18 T^{7} + \cdots + 52672 \)
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| $71$ |
\( T^{8} + 7 T^{7} + \cdots + 79141 \)
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| $73$ |
\( T^{8} - 24 T^{7} + \cdots - 173100 \)
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| $79$ |
\( T^{8} - 33 T^{7} + \cdots - 10460864 \)
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| $83$ |
\( T^{8} + 13 T^{7} + \cdots + 3798748 \)
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| $89$ |
\( T^{8} + 23 T^{7} + \cdots - 462276 \)
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| $97$ |
\( T^{8} + 17 T^{7} + \cdots + 2246337 \)
|
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