Newspace parameters
| Level: | \( N \) | \(=\) | \( 1296 = 2^{4} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1296.g (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(133.967472157\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-19}, \sqrt{-39})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 29x^{2} + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{4} \) |
| Twist minimal: | yes |
| 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
\( x^{4} + 29x^{2} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( 18\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 36\nu^{3} + 954\nu ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} + 29 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 18 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 29 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{2} - 53\beta_1 ) / 36 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).
| \(n\) | \(325\) | \(1135\) | \(1217\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1135.1 |
|
0 | 0 | 0 | −15.2213 | 0 | − | 16.9749i | 0 | 0 | 0 | |||||||||||||||||||||||||||||
| 1135.2 | 0 | 0 | 0 | −15.2213 | 0 | 16.9749i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1135.3 | 0 | 0 | 0 | 39.2213 | 0 | − | 95.4351i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1135.4 | 0 | 0 | 0 | 39.2213 | 0 | 95.4351i | 0 | 0 | 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 | 1296.5.g.e | yes | 4 |
| 3.b | odd | 2 | 1 | 1296.5.g.c | ✓ | 4 | |
| 4.b | odd | 2 | 1 | inner | 1296.5.g.e | yes | 4 |
| 12.b | even | 2 | 1 | 1296.5.g.c | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1296.5.g.c | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 1296.5.g.c | ✓ | 4 | 12.b | even | 2 | 1 | |
| 1296.5.g.e | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 1296.5.g.e | yes | 4 | 4.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 24T_{5} - 597 \)
acting on \(S_{5}^{\mathrm{new}}(1296, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} - 24 T - 597)^{2} \)
|
| $7$ |
\( T^{4} + 9396 T^{2} + 2624400 \)
|
| $11$ |
\( T^{4} + 34020 T^{2} + 114318864 \)
|
| $13$ |
\( (T^{2} + 76 T - 5225)^{2} \)
|
| $17$ |
\( (T^{2} + 78 T - 10335)^{2} \)
|
| $19$ |
\( T^{4} + \cdots + 66014052624 \)
|
| $23$ |
\( T^{4} + \cdots + 135709718544 \)
|
| $29$ |
\( (T^{2} + 780 T - 311025)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 1535061528576 \)
|
| $37$ |
\( (T^{2} - 268 T - 308825)^{2} \)
|
| $41$ |
\( (T^{2} + 1152 T - 1828980)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 2938769632656 \)
|
| $47$ |
\( T^{4} + \cdots + 586657956096 \)
|
| $53$ |
\( (T^{2} + 1116 T - 8331660)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 7176140884224 \)
|
| $61$ |
\( (T^{2} - 1612 T - 11681345)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 264469947051024 \)
|
| $71$ |
\( T^{4} + \cdots + 647993069781264 \)
|
| $73$ |
\( (T^{2} + 3742 T + 2833741)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 3447869640336 \)
|
| $83$ |
\( T^{4} + \cdots + 37403303198976 \)
|
| $89$ |
\( (T^{2} - 10122 T + 21333705)^{2} \)
|
| $97$ |
\( (T^{2} + 4264 T - 14901380)^{2} \)
|
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