Newspace parameters
| Level: | \( N \) | \(=\) | \( 64 = 2^{6} \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 64.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(14.7234613517\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{6})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{17} \) |
| Twist minimal: | no (minimal twist has level 32) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 16\nu^{3} - 12\nu^{2} + 48\nu ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -64\nu^{2} ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -64\nu^{3} + 192\nu ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( 4\beta_{3} - 3\beta_{2} + 16\beta_1 ) / 512 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{2} ) / 64 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -12\beta_{3} - 9\beta_{2} + 48\beta_1 ) / 512 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(63\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 63.1 |
|
0 | − | 51.1918i | 0 | −142.767 | 0 | 217.616i | 0 | −1891.60 | 0 | |||||||||||||||||||||||||||||
| 63.2 | 0 | − | 27.1918i | 0 | 170.767 | 0 | − | 374.384i | 0 | −10.3959 | 0 | |||||||||||||||||||||||||||||
| 63.3 | 0 | 27.1918i | 0 | 170.767 | 0 | 374.384i | 0 | −10.3959 | 0 | |||||||||||||||||||||||||||||||
| 63.4 | 0 | 51.1918i | 0 | −142.767 | 0 | − | 217.616i | 0 | −1891.60 | 0 | ||||||||||||||||||||||||||||||
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 | 64.7.c.e | 4 | |
| 3.b | odd | 2 | 1 | 576.7.g.l | 4 | ||
| 4.b | odd | 2 | 1 | inner | 64.7.c.e | 4 | |
| 8.b | even | 2 | 1 | 32.7.c.b | ✓ | 4 | |
| 8.d | odd | 2 | 1 | 32.7.c.b | ✓ | 4 | |
| 12.b | even | 2 | 1 | 576.7.g.l | 4 | ||
| 16.e | even | 4 | 1 | 256.7.d.d | 4 | ||
| 16.e | even | 4 | 1 | 256.7.d.g | 4 | ||
| 16.f | odd | 4 | 1 | 256.7.d.d | 4 | ||
| 16.f | odd | 4 | 1 | 256.7.d.g | 4 | ||
| 24.f | even | 2 | 1 | 288.7.g.b | 4 | ||
| 24.h | odd | 2 | 1 | 288.7.g.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 32.7.c.b | ✓ | 4 | 8.b | even | 2 | 1 | |
| 32.7.c.b | ✓ | 4 | 8.d | odd | 2 | 1 | |
| 64.7.c.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 64.7.c.e | 4 | 4.b | odd | 2 | 1 | inner | |
| 256.7.d.d | 4 | 16.e | even | 4 | 1 | ||
| 256.7.d.d | 4 | 16.f | odd | 4 | 1 | ||
| 256.7.d.g | 4 | 16.e | even | 4 | 1 | ||
| 256.7.d.g | 4 | 16.f | odd | 4 | 1 | ||
| 288.7.g.b | 4 | 24.f | even | 2 | 1 | ||
| 288.7.g.b | 4 | 24.h | odd | 2 | 1 | ||
| 576.7.g.l | 4 | 3.b | odd | 2 | 1 | ||
| 576.7.g.l | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 3360T_{3}^{2} + 1937664 \)
acting on \(S_{7}^{\mathrm{new}}(64, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 3360 T^{2} + 1937664 \)
|
| $5$ |
\( (T^{2} - 28 T - 24380)^{2} \)
|
| $7$ |
\( T^{4} + \cdots + 6637686784 \)
|
| $11$ |
\( T^{4} + \cdots + 1222201600 \)
|
| $13$ |
\( (T^{2} - 4348 T + 4111876)^{2} \)
|
| $17$ |
\( (T^{2} + 2652 T - 26651580)^{2} \)
|
| $19$ |
\( T^{4} + \cdots + 1640407654656 \)
|
| $23$ |
\( T^{4} + \cdots + 33\!\cdots\!00 \)
|
| $29$ |
\( (T^{2} + 38788 T - 85139708)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 17\!\cdots\!04 \)
|
| $37$ |
\( (T^{2} + 62628 T - 219749820)^{2} \)
|
| $41$ |
\( (T^{2} + 104924 T + 2638622020)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 70\!\cdots\!24 \)
|
| $47$ |
\( T^{4} + \cdots + 74\!\cdots\!16 \)
|
| $53$ |
\( (T^{2} - 34332 T - 659098428)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 19\!\cdots\!16 \)
|
| $61$ |
\( (T^{2} + 119108 T - 30268152188)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $71$ |
\( T^{4} + \cdots + 31\!\cdots\!24 \)
|
| $73$ |
\( (T^{2} - 109764 T - 174293877372)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 15\!\cdots\!24 \)
|
| $83$ |
\( T^{4} + \cdots + 62\!\cdots\!00 \)
|
| $89$ |
\( (T^{2} + 1861500 T + 848539402500)^{2} \)
|
| $97$ |
\( (T^{2} - 1459172 T - 12900710204)^{2} \)
|
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