Newspace parameters
| Level: | \( N \) | \(=\) | \( 256 = 2^{8} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 256.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(196.695854223\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{6} - 1640x^{4} + 512875x^{2} - 42187500 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{41}\cdot 3\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 128) |
| 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
\( x^{6} - 1640x^{4} + 512875x^{2} - 42187500 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 4\nu^{5} - 5560\nu^{3} + 811500\nu ) / 9375 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -172\nu^{5} + 255080\nu^{3} - 44734500\nu ) / 3125 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 584\nu^{5} - 987760\nu^{3} + 354719000\nu ) / 3125 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -16\nu^{4} + 21840\nu^{2} - 3065375 ) / 125 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -704\nu^{4} + 1088960\nu^{2} - 204847875 ) / 375 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 11\beta_{2} + 981\beta_1 ) / 40960 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{5} - 44\beta_{4} + 559771 ) / 1024 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 123\beta_{3} + 2953\beta_{2} + 327063\beta_1 ) / 8192 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 4095\beta_{5} - 68060\beta_{4} + 567903415 ) / 1024 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 130395\beta_{3} + 3658345\beta_{2} + 434013495\beta_1 ) / 8192 \)
|
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 | −580.453 | 0 | −6948.91 | 0 | 45098.0 | 0 | 159779. | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −501.129 | 0 | 10865.5 | 0 | −32440.9 | 0 | 73983.7 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −150.224 | 0 | −2288.63 | 0 | −47323.2 | 0 | −154580. | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 150.224 | 0 | −2288.63 | 0 | 47323.2 | 0 | −154580. | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 501.129 | 0 | 10865.5 | 0 | 32440.9 | 0 | 73983.7 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 580.453 | 0 | −6948.91 | 0 | −45098.0 | 0 | 159779. | 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 | 256.12.a.k | 6 | |
| 4.b | odd | 2 | 1 | inner | 256.12.a.k | 6 | |
| 8.b | even | 2 | 1 | 256.12.a.j | 6 | ||
| 8.d | odd | 2 | 1 | 256.12.a.j | 6 | ||
| 16.e | even | 4 | 2 | 128.12.b.f | ✓ | 12 | |
| 16.f | odd | 4 | 2 | 128.12.b.f | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 128.12.b.f | ✓ | 12 | 16.e | even | 4 | 2 | |
| 128.12.b.f | ✓ | 12 | 16.f | odd | 4 | 2 | |
| 256.12.a.j | 6 | 8.b | even | 2 | 1 | ||
| 256.12.a.j | 6 | 8.d | odd | 2 | 1 | ||
| 256.12.a.k | 6 | 1.a | even | 1 | 1 | trivial | |
| 256.12.a.k | 6 | 4.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(256))\):
|
\( T_{3}^{6} - 610624T_{3}^{4} + 97883320320T_{3}^{2} - 1909477048320000 \)
|
|
\( T_{5}^{3} - 1628T_{5}^{2} - 84467280T_{5} - 172799611200 \)
|
|
\( T_{7}^{6} - 5325722624T_{7}^{4} + 9052003049663365120T_{7}^{2} - 4793438289348518026936320000 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + \cdots - 19\!\cdots\!00 \)
|
| $5$ |
\( (T^{3} - 1628 T^{2} + \cdots - 172799611200)^{2} \)
|
| $7$ |
\( T^{6} + \cdots - 47\!\cdots\!00 \)
|
| $11$ |
\( T^{6} + \cdots - 40\!\cdots\!00 \)
|
| $13$ |
\( (T^{3} + \cdots - 38\!\cdots\!00)^{2} \)
|
| $17$ |
\( (T^{3} + \cdots - 94\!\cdots\!92)^{2} \)
|
| $19$ |
\( T^{6} + \cdots - 49\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots - 38\!\cdots\!00 \)
|
| $29$ |
\( (T^{3} + \cdots + 65\!\cdots\!72)^{2} \)
|
| $31$ |
\( T^{6} + \cdots - 75\!\cdots\!00 \)
|
| $37$ |
\( (T^{3} + \cdots + 37\!\cdots\!92)^{2} \)
|
| $41$ |
\( (T^{3} + \cdots + 37\!\cdots\!00)^{2} \)
|
| $43$ |
\( T^{6} + \cdots - 33\!\cdots\!00 \)
|
| $47$ |
\( T^{6} + \cdots - 44\!\cdots\!00 \)
|
| $53$ |
\( (T^{3} + \cdots - 66\!\cdots\!96)^{2} \)
|
| $59$ |
\( T^{6} + \cdots - 46\!\cdots\!00 \)
|
| $61$ |
\( (T^{3} + \cdots + 30\!\cdots\!24)^{2} \)
|
| $67$ |
\( T^{6} + \cdots - 28\!\cdots\!00 \)
|
| $71$ |
\( T^{6} + \cdots - 16\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} + \cdots + 13\!\cdots\!48)^{2} \)
|
| $79$ |
\( T^{6} + \cdots - 26\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots - 78\!\cdots\!00 \)
|
| $89$ |
\( (T^{3} + \cdots - 82\!\cdots\!00)^{2} \)
|
| $97$ |
\( (T^{3} + \cdots - 11\!\cdots\!68)^{2} \)
|
| show more | |
| show less |