Newspace parameters
| Level: | \( N \) | \(=\) | \( 64 = 2^{6} \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 64.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(178.865500344\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
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|
|
| Defining polynomial: |
\( x^{6} - 318697x^{4} + 1583195235x^{2} - 409146984675 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{71}\cdot 3^{8}\cdot 5^{2}\cdot 7^{3} \) |
| Twist minimal: | no (minimal twist has level 32) |
| 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
\( x^{6} - 318697x^{4} + 1583195235x^{2} - 409146984675 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -47156\nu^{5} + 15021220952\nu^{3} - 74887530551460\nu ) / 7837401845 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -4096\nu^{4} + 1315939840\nu^{2} - 5444665549312 ) / 131489 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -754936592\nu^{5} + 241280458537184\nu^{3} - 1371447701859821520\nu ) / 23512205535 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2392064\nu^{4} - 749120024576\nu^{2} + 1119962756206592 ) / 131489 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 8350798908\nu^{5} - 2647669458615240\nu^{3} + 8809678491675595020\nu ) / 1567480369 \)
|
| \(\nu\) | \(=\) |
\( ( -7\beta_{5} + 1629\beta_{3} - 14891181\beta_1 ) / 7927234560 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + 584\beta_{2} + 15664594944 ) / 147456 \)
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| \(\nu^{3}\) | \(=\) |
\( ( -301741\beta_{5} + 116782623\beta_{3} - 890379107727\beta_1 ) / 1585446912 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2570195\beta_{4} + 1463125048\beta_{2} + 38692999923892224 ) / 1179648 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -93894216823\beta_{5} + 36682908346413\beta_{3} - 279157993433458557\beta_1 ) / 1585446912 \)
|
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 | −183460. | 0 | 3.17375e7 | 0 | −1.21539e9 | 0 | 2.31974e10 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −149353. | 0 | −3.99877e7 | 0 | 6.55061e8 | 0 | 1.18461e10 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −37979.1 | 0 | 4.30662e6 | 0 | 6.31379e8 | 0 | −9.01794e9 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 37979.1 | 0 | 4.30662e6 | 0 | −6.31379e8 | 0 | −9.01794e9 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 149353. | 0 | −3.99877e7 | 0 | −6.55061e8 | 0 | 1.18461e10 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 183460. | 0 | 3.17375e7 | 0 | 1.21539e9 | 0 | 2.31974e10 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 64.22.a.q | 6 | |
| 4.b | odd | 2 | 1 | inner | 64.22.a.q | 6 | |
| 8.b | even | 2 | 1 | 32.22.a.e | ✓ | 6 | |
| 8.d | odd | 2 | 1 | 32.22.a.e | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 32.22.a.e | ✓ | 6 | 8.b | even | 2 | 1 | |
| 32.22.a.e | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 64.22.a.q | 6 | 1.a | even | 1 | 1 | trivial | |
| 64.22.a.q | 6 | 4.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 57406595328T_{3}^{4} + 831507715552040386560T_{3}^{2} - 1082940467249991685988627251200 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(64))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} + \cdots - 10\!\cdots\!00 \)
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| $5$ |
\( (T^{3} + \cdots + 54\!\cdots\!00)^{2} \)
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| $7$ |
\( T^{6} + \cdots - 25\!\cdots\!00 \)
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| $11$ |
\( T^{6} + \cdots - 60\!\cdots\!00 \)
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| $13$ |
\( (T^{3} + \cdots - 82\!\cdots\!12)^{2} \)
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| $17$ |
\( (T^{3} + \cdots + 10\!\cdots\!72)^{2} \)
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| $19$ |
\( T^{6} + \cdots - 15\!\cdots\!00 \)
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| $23$ |
\( T^{6} + \cdots - 23\!\cdots\!00 \)
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| $29$ |
\( (T^{3} + \cdots + 28\!\cdots\!00)^{2} \)
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| $31$ |
\( T^{6} + \cdots - 18\!\cdots\!00 \)
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| $37$ |
\( (T^{3} + \cdots - 28\!\cdots\!40)^{2} \)
|
| $41$ |
\( (T^{3} + \cdots - 12\!\cdots\!00)^{2} \)
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| $43$ |
\( T^{6} + \cdots - 11\!\cdots\!00 \)
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| $47$ |
\( T^{6} + \cdots - 32\!\cdots\!00 \)
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| $53$ |
\( (T^{3} + \cdots + 31\!\cdots\!96)^{2} \)
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| $59$ |
\( T^{6} + \cdots - 13\!\cdots\!00 \)
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| $61$ |
\( (T^{3} + \cdots - 17\!\cdots\!00)^{2} \)
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| $67$ |
\( T^{6} + \cdots - 14\!\cdots\!00 \)
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| $71$ |
\( T^{6} + \cdots - 31\!\cdots\!00 \)
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| $73$ |
\( (T^{3} + \cdots + 15\!\cdots\!96)^{2} \)
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| $79$ |
\( T^{6} + \cdots - 70\!\cdots\!00 \)
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| $83$ |
\( T^{6} + \cdots - 15\!\cdots\!00 \)
|
| $89$ |
\( (T^{3} + \cdots - 81\!\cdots\!00)^{2} \)
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| $97$ |
\( (T^{3} + \cdots + 11\!\cdots\!72)^{2} \)
|
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