Newspace parameters
| Level: | \( N \) | \(=\) | \( 500 = 2^{2} \cdot 5^{3} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 500.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(29.5009550029\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
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| Defining polynomial: |
\( x^{6} - 66x^{4} - 34x^{3} + 1084x^{2} + 842x - 3631 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 5^{2} \) |
| 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
\( x^{6} - 66x^{4} - 34x^{3} + 1084x^{2} + 842x - 3631 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 28\nu^{4} - 718\nu^{3} - 907\nu^{2} + 24312\nu + 8698 ) / 8418 \)
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| \(\beta_{2}\) | \(=\) |
\( ( 818\nu^{5} - 1859\nu^{4} - 44026\nu^{3} + 68486\nu^{2} + 421209\nu - 372146 ) / 42090 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 901\nu^{5} - 4183\nu^{4} - 47567\nu^{3} + 185857\nu^{2} + 465723\nu - 1275067 ) / 42090 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -1687\nu^{5} + 5146\nu^{4} + 93524\nu^{3} - 267409\nu^{2} - 949386\nu + 2517934 ) / 42090 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 512\nu^{5} - 1709\nu^{4} - 28030\nu^{3} + 77156\nu^{2} + 326571\nu - 572678 ) / 8418 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{5} + \beta_{4} - 2\beta_{3} - 2\beta_{2} - \beta _1 - 1 ) / 10 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 7\beta_{4} - 6\beta_{3} - 11\beta_{2} - 7\beta _1 + 215 ) / 10 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 33\beta_{5} + 14\beta_{4} - 38\beta_{3} - 33\beta_{2} - 74\beta _1 + 41 ) / 5 \)
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| \(\nu^{4}\) | \(=\) |
\( ( 82\beta_{5} - 349\beta_{4} - 542\beta_{3} - 382\beta_{2} - 427\beta _1 + 7101 ) / 10 \)
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| \(\nu^{5}\) | \(=\) |
\( ( 525\beta_{5} + 157\beta_{4} - 758\beta_{3} - 391\beta_{2} - 1567\beta _1 + 1523 ) / 2 \)
|
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 |
|
0 | −7.96155 | 0 | 0 | 0 | 0.0689265 | 0 | 36.3863 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −7.11328 | 0 | 0 | 0 | −15.3866 | 0 | 23.5988 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.08545 | 0 | 0 | 0 | 16.3532 | 0 | −25.8218 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.26710 | 0 | 0 | 0 | −24.0678 | 0 | −21.8603 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 4.45839 | 0 | 0 | 0 | 14.7973 | 0 | −7.12278 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 9.26390 | 0 | 0 | 0 | −34.7650 | 0 | 58.8198 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( +1 \) |
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 | 500.4.a.c | yes | 6 |
| 4.b | odd | 2 | 1 | 2000.4.a.l | 6 | ||
| 5.b | even | 2 | 1 | 500.4.a.b | ✓ | 6 | |
| 5.c | odd | 4 | 2 | 500.4.c.c | 12 | ||
| 20.d | odd | 2 | 1 | 2000.4.a.m | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 500.4.a.b | ✓ | 6 | 5.b | even | 2 | 1 | |
| 500.4.a.c | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 500.4.c.c | 12 | 5.c | odd | 4 | 2 | ||
| 2000.4.a.l | 6 | 4.b | odd | 2 | 1 | ||
| 2000.4.a.m | 6 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 2T_{3}^{5} - 111T_{3}^{4} + 214T_{3}^{3} + 2589T_{3}^{2} - 8222T_{3} + 5756 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(500))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} - 2 T^{5} + \cdots + 5756 \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( T^{6} + 43 T^{5} + \cdots - 214729 \)
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| $11$ |
\( T^{6} + 15 T^{5} + \cdots + 730959875 \)
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| $13$ |
\( T^{6} + 87 T^{5} + \cdots - 95925879 \)
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| $17$ |
\( T^{6} + \cdots - 5244962896 \)
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| $19$ |
\( T^{6} + \cdots - 1220233819 \)
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| $23$ |
\( T^{6} + \cdots - 24927796656 \)
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| $29$ |
\( T^{6} + \cdots - 11561139063884 \)
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| $31$ |
\( T^{6} + \cdots + 10595036078116 \)
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| $37$ |
\( T^{6} + \cdots + 12360260449936 \)
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| $41$ |
\( T^{6} + \cdots - 2501624923189 \)
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| $43$ |
\( T^{6} + \cdots - 10915805558000 \)
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| $47$ |
\( T^{6} + \cdots - 244373573098211 \)
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| $53$ |
\( T^{6} + \cdots - 39\!\cdots\!31 \)
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| $59$ |
\( T^{6} + \cdots + 623930671754649 \)
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| $61$ |
\( T^{6} + \cdots - 34\!\cdots\!76 \)
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| $67$ |
\( T^{6} + \cdots + 15\!\cdots\!76 \)
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| $71$ |
\( T^{6} + \cdots + 870909015303524 \)
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| $73$ |
\( T^{6} + \cdots + 40\!\cdots\!16 \)
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| $79$ |
\( T^{6} + \cdots + 12472751663244 \)
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| $83$ |
\( T^{6} + \cdots - 22\!\cdots\!16 \)
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| $89$ |
\( T^{6} + \cdots - 30\!\cdots\!24 \)
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| $97$ |
\( T^{6} + \cdots + 39\!\cdots\!76 \)
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