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: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 9138x^{2} + 12641832 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{17} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 9138x^{2} + 12641832 \)
:
| \(\beta_{1}\) | \(=\) |
\( 8\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 128\nu^{3} - 814776\nu ) / 297 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 16\nu^{2} - 73104 ) / 33 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 33\beta_{3} + 73104 ) / 16 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 297\beta_{2} + 101847\beta_1 ) / 128 \)
|
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 | −689.973 | 0 | 1696.32 | 0 | 30915.7 | 0 | 298916. | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −329.802 | 0 | −12216.3 | 0 | −87187.3 | 0 | −68377.9 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 329.802 | 0 | −12216.3 | 0 | 87187.3 | 0 | −68377.9 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 689.973 | 0 | 1696.32 | 0 | −30915.7 | 0 | 298916. | 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.h | 4 | |
| 4.b | odd | 2 | 1 | inner | 256.12.a.h | 4 | |
| 8.b | even | 2 | 1 | 256.12.a.i | 4 | ||
| 8.d | odd | 2 | 1 | 256.12.a.i | 4 | ||
| 16.e | even | 4 | 2 | 128.12.b.c | ✓ | 8 | |
| 16.f | odd | 4 | 2 | 128.12.b.c | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 128.12.b.c | ✓ | 8 | 16.e | even | 4 | 2 | |
| 128.12.b.c | ✓ | 8 | 16.f | odd | 4 | 2 | |
| 256.12.a.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 256.12.a.h | 4 | 4.b | odd | 2 | 1 | inner | |
| 256.12.a.i | 4 | 8.b | even | 2 | 1 | ||
| 256.12.a.i | 4 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(256))\):
|
\( T_{3}^{4} - 584832T_{3}^{2} + 51780943872 \)
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\( T_{5}^{2} + 10520T_{5} - 20722800 \)
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\( T_{7}^{4} - 8557398016T_{7}^{2} + 7265483746147565568 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} + \cdots + 51780943872 \)
|
| $5$ |
\( (T^{2} + 10520 T - 20722800)^{2} \)
|
| $7$ |
\( T^{4} + \cdots + 72\!\cdots\!68 \)
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| $11$ |
\( T^{4} + \cdots + 21\!\cdots\!28 \)
|
| $13$ |
\( (T^{2} - 490536 T - 979828793200)^{2} \)
|
| $17$ |
\( (T^{2} + \cdots + 13386395366276)^{2} \)
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| $19$ |
\( T^{4} + \cdots + 90\!\cdots\!92 \)
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| $23$ |
\( T^{4} + \cdots + 22\!\cdots\!00 \)
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| $29$ |
\( (T^{2} + \cdots + 21\!\cdots\!08)^{2} \)
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| $31$ |
\( T^{4} + \cdots + 51\!\cdots\!88 \)
|
| $37$ |
\( (T^{2} + \cdots - 17\!\cdots\!28)^{2} \)
|
| $41$ |
\( (T^{2} + \cdots + 21\!\cdots\!36)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 76\!\cdots\!68 \)
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| $47$ |
\( T^{4} + \cdots + 33\!\cdots\!00 \)
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| $53$ |
\( (T^{2} + \cdots - 19\!\cdots\!04)^{2} \)
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| $59$ |
\( T^{4} + \cdots + 19\!\cdots\!12 \)
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| $61$ |
\( (T^{2} + \cdots - 47\!\cdots\!04)^{2} \)
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| $67$ |
\( T^{4} + \cdots + 14\!\cdots\!48 \)
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| $71$ |
\( T^{4} + \cdots + 29\!\cdots\!08 \)
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| $73$ |
\( (T^{2} + \cdots + 11\!\cdots\!36)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 30\!\cdots\!08 \)
|
| $83$ |
\( T^{4} + \cdots + 17\!\cdots\!32 \)
|
| $89$ |
\( (T^{2} + \cdots + 35\!\cdots\!72)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots - 42\!\cdots\!80)^{2} \)
|
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