Newspace parameters
| Level: | \( N \) | \(=\) | \( 256 = 2^{8} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 256.d (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(104.288924176\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{39})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 19x^{2} + 100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{12} \) |
| 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
\( x^{4} - 19x^{2} + 100 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 9\nu ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -8\nu^{3} + 232\nu ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( 64\nu^{2} - 608 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 8\beta_1 ) / 32 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 608 ) / 64 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 9\beta_{2} + 232\beta_1 ) / 32 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 127.1 |
|
0 | −63.9200 | 0 | − | 217.680i | 0 | − | 1726.24i | 0 | −2475.24 | 0 | ||||||||||||||||||||||||||||
| 127.2 | 0 | −63.9200 | 0 | 217.680i | 0 | 1726.24i | 0 | −2475.24 | 0 | |||||||||||||||||||||||||||||||
| 127.3 | 0 | 135.920 | 0 | − | 581.680i | 0 | − | 2670.24i | 0 | 11913.2 | 0 | |||||||||||||||||||||||||||||
| 127.4 | 0 | 135.920 | 0 | 581.680i | 0 | 2670.24i | 0 | 11913.2 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 256.9.d.h | 4 | |
| 4.b | odd | 2 | 1 | 256.9.d.b | 4 | ||
| 8.b | even | 2 | 1 | 256.9.d.b | 4 | ||
| 8.d | odd | 2 | 1 | inner | 256.9.d.h | 4 | |
| 16.e | even | 4 | 1 | 32.9.c.a | ✓ | 4 | |
| 16.e | even | 4 | 1 | 64.9.c.f | 4 | ||
| 16.f | odd | 4 | 1 | 32.9.c.a | ✓ | 4 | |
| 16.f | odd | 4 | 1 | 64.9.c.f | 4 | ||
| 48.i | odd | 4 | 1 | 288.9.g.b | 4 | ||
| 48.k | even | 4 | 1 | 288.9.g.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 32.9.c.a | ✓ | 4 | 16.e | even | 4 | 1 | |
| 32.9.c.a | ✓ | 4 | 16.f | odd | 4 | 1 | |
| 64.9.c.f | 4 | 16.e | even | 4 | 1 | ||
| 64.9.c.f | 4 | 16.f | odd | 4 | 1 | ||
| 256.9.d.b | 4 | 4.b | odd | 2 | 1 | ||
| 256.9.d.b | 4 | 8.b | even | 2 | 1 | ||
| 256.9.d.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 256.9.d.h | 4 | 8.d | odd | 2 | 1 | inner | |
| 288.9.g.b | 4 | 48.i | odd | 4 | 1 | ||
| 288.9.g.b | 4 | 48.k | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 72T_{3} - 8688 \)
acting on \(S_{9}^{\mathrm{new}}(256, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} - 72 T - 8688)^{2} \)
|
| $5$ |
\( T^{4} + \cdots + 16032624400 \)
|
| $7$ |
\( T^{4} + \cdots + 21247232118784 \)
|
| $11$ |
\( (T^{2} - 20104 T + 84259600)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 44\!\cdots\!96 \)
|
| $17$ |
\( (T^{2} + 195996 T + 7808724420)^{2} \)
|
| $19$ |
\( (T^{2} + 332376 T + 26288332944)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 17\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 12\!\cdots\!24 \)
|
| $31$ |
\( T^{4} + \cdots + 20\!\cdots\!24 \)
|
| $37$ |
\( T^{4} + \cdots + 14\!\cdots\!00 \)
|
| $41$ |
\( (T^{2} + \cdots - 13944688274300)^{2} \)
|
| $43$ |
\( (T^{2} + \cdots - 17476345855472)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 55\!\cdots\!16 \)
|
| $53$ |
\( T^{4} + \cdots + 34\!\cdots\!04 \)
|
| $59$ |
\( (T^{2} + \cdots - 115904724604016)^{2} \)
|
| $61$ |
\( T^{4} + \cdots + 10\!\cdots\!84 \)
|
| $67$ |
\( (T^{2} + \cdots - 41584895095280)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 58\!\cdots\!64 \)
|
| $73$ |
\( (T^{2} + \cdots - 107748446132028)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 58\!\cdots\!04 \)
|
| $83$ |
\( (T^{2} + \cdots + 11\!\cdots\!80)^{2} \)
|
| $89$ |
\( (T^{2} + \cdots - 933201241117500)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots - 19\!\cdots\!76)^{2} \)
|
| show more | |
| show less |