Newspace parameters
| Level: | \( N \) | \(=\) | \( 256 = 2^{8} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 256.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(79.9705665239\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.50765497344.1 |
|
|
|
| Defining polynomial: |
\( x^{6} + 101x^{4} + 2512x^{2} + 36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{22} \) |
| Twist minimal: | no (minimal twist has level 128) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 101x^{4} + 2512x^{2} + 36 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} - 95\nu^{3} - 1854\nu ) / 44 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} - 95\nu^{3} - 2206\nu ) / 33 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4\nu^{4} - 204\nu^{2} + 31 ) / 11 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 73\nu^{5} + 7639\nu^{3} + 197118\nu ) / 66 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -20\nu^{4} - 1724\nu^{2} - 23517 ) / 33 \)
|
| \(\nu\) | \(=\) |
\( ( -3\beta_{2} + 4\beta_1 ) / 32 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{5} + 5\beta_{3} - 2152 ) / 64 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 12\beta_{4} + 1053\beta_{2} - 820\beta_1 ) / 128 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 153\beta_{5} - 431\beta_{3} + 110248 ) / 64 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -1140\beta_{4} - 77787\beta_{2} + 42604\beta_1 ) / 128 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(255\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 129.1 |
|
0 | − | 62.2585i | 0 | − | 203.009i | 0 | 534.535 | 0 | −1689.12 | 0 | ||||||||||||||||||||||||||||||||||
| 129.2 | 0 | − | 51.2165i | 0 | 145.477i | 0 | 192.521 | 0 | −436.129 | 0 | ||||||||||||||||||||||||||||||||||||
| 129.3 | 0 | − | 2.95798i | 0 | 362.486i | 0 | −715.055 | 0 | 2178.25 | 0 | ||||||||||||||||||||||||||||||||||||
| 129.4 | 0 | 2.95798i | 0 | − | 362.486i | 0 | −715.055 | 0 | 2178.25 | 0 | ||||||||||||||||||||||||||||||||||||
| 129.5 | 0 | 51.2165i | 0 | − | 145.477i | 0 | 192.521 | 0 | −436.129 | 0 | ||||||||||||||||||||||||||||||||||||
| 129.6 | 0 | 62.2585i | 0 | 203.009i | 0 | 534.535 | 0 | −1689.12 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 256.8.b.l | 6 | |
| 4.b | odd | 2 | 1 | 256.8.b.k | 6 | ||
| 8.b | even | 2 | 1 | inner | 256.8.b.l | 6 | |
| 8.d | odd | 2 | 1 | 256.8.b.k | 6 | ||
| 16.e | even | 4 | 1 | 128.8.a.a | ✓ | 3 | |
| 16.e | even | 4 | 1 | 128.8.a.d | yes | 3 | |
| 16.f | odd | 4 | 1 | 128.8.a.b | yes | 3 | |
| 16.f | odd | 4 | 1 | 128.8.a.c | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 128.8.a.a | ✓ | 3 | 16.e | even | 4 | 1 | |
| 128.8.a.b | yes | 3 | 16.f | odd | 4 | 1 | |
| 128.8.a.c | yes | 3 | 16.f | odd | 4 | 1 | |
| 128.8.a.d | yes | 3 | 16.e | even | 4 | 1 | |
| 256.8.b.k | 6 | 4.b | odd | 2 | 1 | ||
| 256.8.b.k | 6 | 8.d | odd | 2 | 1 | ||
| 256.8.b.l | 6 | 1.a | even | 1 | 1 | trivial | |
| 256.8.b.l | 6 | 8.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(256, [\chi])\):
|
\( T_{3}^{6} + 6508T_{3}^{4} + 10224432T_{3}^{2} + 88962624 \)
|
|
\( T_{7}^{3} - 12T_{7}^{2} - 416976T_{7} + 73585600 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 6508 T^{4} + \cdots + 88962624 \)
|
| $5$ |
\( T^{6} + \cdots + 114603876302400 \)
|
| $7$ |
\( (T^{3} - 12 T^{2} + \cdots + 73585600)^{2} \)
|
| $11$ |
\( T^{6} + \cdots + 34\!\cdots\!84 \)
|
| $13$ |
\( T^{6} + \cdots + 29\!\cdots\!24 \)
|
| $17$ |
\( (T^{3} + \cdots - 5603612104824)^{2} \)
|
| $19$ |
\( T^{6} + \cdots + 32\!\cdots\!36 \)
|
| $23$ |
\( (T^{3} + \cdots - 36954823141312)^{2} \)
|
| $29$ |
\( T^{6} + \cdots + 40\!\cdots\!44 \)
|
| $31$ |
\( (T^{3} + \cdots + 17\!\cdots\!52)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 38\!\cdots\!36 \)
|
| $41$ |
\( (T^{3} + \cdots + 31\!\cdots\!36)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 25\!\cdots\!00 \)
|
| $47$ |
\( (T^{3} + \cdots - 41\!\cdots\!92)^{2} \)
|
| $53$ |
\( T^{6} + \cdots + 18\!\cdots\!00 \)
|
| $59$ |
\( T^{6} + \cdots + 45\!\cdots\!96 \)
|
| $61$ |
\( T^{6} + \cdots + 44\!\cdots\!84 \)
|
| $67$ |
\( T^{6} + \cdots + 17\!\cdots\!64 \)
|
| $71$ |
\( (T^{3} + \cdots - 35\!\cdots\!40)^{2} \)
|
| $73$ |
\( (T^{3} + \cdots + 48\!\cdots\!92)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots - 13\!\cdots\!32)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 98\!\cdots\!64 \)
|
| $89$ |
\( (T^{3} + \cdots + 95\!\cdots\!24)^{2} \)
|
| $97$ |
\( (T^{3} + \cdots - 83\!\cdots\!64)^{2} \)
|
| show more | |
| show less |