Newspace parameters
| Level: | \( N \) | \(=\) | \( 54 = 2 \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 13 \) |
| Character orbit: | \([\chi]\) | \(=\) | 54.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(49.3556661329\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 9235x^{2} + 9236x + 21362886 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{6} \) |
| 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
\( x^{4} - 2x^{3} - 9235x^{2} + 9236x + 21362886 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 64\nu^{3} - 96\nu^{2} - 295264\nu + 147648 ) / 18489 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -288\nu^{3} + 432\nu^{2} + 3991104\nu - 1995624 ) / 6163 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8\nu^{3} + 1996800\nu^{2} - 2033720\nu - 9221259360 ) / 18489 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{2} + 27\beta _1 + 432 ) / 864 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{3} + \beta_{2} + 13\beta _1 + 1995192 ) / 432 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 12\beta_{3} + 9230\beta_{2} + 374205\beta _1 + 5985360 ) / 864 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 53.1 |
|
− | 45.2548i | 0 | −2048.00 | − | 14467.5i | 0 | 97235.1 | 92681.9i | 0 | −654723. | ||||||||||||||||||||||||||||
| 53.2 | − | 45.2548i | 0 | −2048.00 | 27059.6i | 0 | 38506.9 | 92681.9i | 0 | 1.22458e6 | ||||||||||||||||||||||||||||||
| 53.3 | 45.2548i | 0 | −2048.00 | − | 27059.6i | 0 | 38506.9 | − | 92681.9i | 0 | 1.22458e6 | |||||||||||||||||||||||||||||
| 53.4 | 45.2548i | 0 | −2048.00 | 14467.5i | 0 | 97235.1 | − | 92681.9i | 0 | −654723. | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 54.13.b.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 54.13.b.b | ✓ | 4 |
| 9.c | even | 3 | 2 | 162.13.d.c | 8 | ||
| 9.d | odd | 6 | 2 | 162.13.d.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 54.13.b.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 54.13.b.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 162.13.d.c | 8 | 9.c | even | 3 | 2 | ||
| 162.13.d.c | 8 | 9.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 941530752T_{5}^{2} + 153259847530905600 \)
acting on \(S_{13}^{\mathrm{new}}(54, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 2048)^{2} \)
$3$
\( T^{4} \)
$5$
\( T^{4} + 941530752 T^{2} + \cdots + 15\!\cdots\!00 \)
$7$
\( (T^{2} - 135742 T + 3744223105)^{2} \)
$11$
\( T^{4} + 3058161293952 T^{2} + \cdots + 21\!\cdots\!00 \)
$13$
\( (T^{2} - 348142 T - 38357910879215)^{2} \)
$17$
\( T^{4} + 886383248261760 T^{2} + \cdots + 57\!\cdots\!36 \)
$19$
\( (T^{2} + 43938482 T - 71954800437263)^{2} \)
$23$
\( T^{4} + \cdots + 15\!\cdots\!76 \)
$29$
\( T^{4} + \cdots + 17\!\cdots\!96 \)
$31$
\( (T^{2} + 1191267068 T + 33\!\cdots\!80)^{2} \)
$37$
\( (T^{2} + 286643570 T - 18\!\cdots\!75)^{2} \)
$41$
\( T^{4} + \cdots + 93\!\cdots\!00 \)
$43$
\( (T^{2} + 7558366172 T - 51\!\cdots\!48)^{2} \)
$47$
\( T^{4} + \cdots + 11\!\cdots\!00 \)
$53$
\( T^{4} + \cdots + 14\!\cdots\!24 \)
$59$
\( T^{4} + \cdots + 15\!\cdots\!24 \)
$61$
\( (T^{2} - 29181433198 T - 19\!\cdots\!15)^{2} \)
$67$
\( (T^{2} - 154487577550 T + 55\!\cdots\!61)^{2} \)
$71$
\( T^{4} + \cdots + 58\!\cdots\!84 \)
$73$
\( (T^{2} + 89435351714 T - 41\!\cdots\!27)^{2} \)
$79$
\( (T^{2} + 452549584418 T + 49\!\cdots\!17)^{2} \)
$83$
\( T^{4} + \cdots + 20\!\cdots\!96 \)
$89$
\( T^{4} + \cdots + 60\!\cdots\!44 \)
$97$
\( (T^{2} + 2835640618178 T + 20\!\cdots\!45)^{2} \)
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