Newspace parameters
| Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 48.g (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(19.5541732829\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{1801})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 451x^{2} + 450x + 202500 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3^{2} \) |
| 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
\( x^{4} - x^{3} + 451x^{2} + 450x + 202500 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 451\nu^{2} - 451\nu + 101025 ) / 101475 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 48\nu^{3} + 32424 ) / 451 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8\nu^{3} - 8\nu^{2} + 7208\nu + 1800 ) / 75 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + 24\beta _1 + 24 ) / 96 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 21624\beta _1 - 21624 ) / 96 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 451\beta_{2} - 32424 ) / 48 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).
| \(n\) | \(17\) | \(31\) | \(37\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
0 | − | 46.7654i | 0 | −1084.52 | 0 | − | 426.979i | 0 | −2187.00 | 0 | ||||||||||||||||||||||||||||
| 31.2 | 0 | − | 46.7654i | 0 | 952.517 | 0 | 3101.27i | 0 | −2187.00 | 0 | ||||||||||||||||||||||||||||||
| 31.3 | 0 | 46.7654i | 0 | −1084.52 | 0 | 426.979i | 0 | −2187.00 | 0 | |||||||||||||||||||||||||||||||
| 31.4 | 0 | 46.7654i | 0 | 952.517 | 0 | − | 3101.27i | 0 | −2187.00 | 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 | 48.9.g.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 144.9.g.i | 4 | ||
| 4.b | odd | 2 | 1 | inner | 48.9.g.c | ✓ | 4 |
| 8.b | even | 2 | 1 | 192.9.g.c | 4 | ||
| 8.d | odd | 2 | 1 | 192.9.g.c | 4 | ||
| 12.b | even | 2 | 1 | 144.9.g.i | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 48.9.g.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 48.9.g.c | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 144.9.g.i | 4 | 3.b | odd | 2 | 1 | ||
| 144.9.g.i | 4 | 12.b | even | 2 | 1 | ||
| 192.9.g.c | 4 | 8.b | even | 2 | 1 | ||
| 192.9.g.c | 4 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 132T_{5} - 1033020 \)
acting on \(S_{9}^{\mathrm{new}}(48, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( (T^{2} + 2187)^{2} \)
$5$
\( (T^{2} + 132 T - 1033020)^{2} \)
$7$
\( T^{4} + 9800160 T^{2} + \cdots + 1753442078976 \)
$11$
\( T^{4} + 254051424 T^{2} + \cdots + 10\!\cdots\!44 \)
$13$
\( (T^{2} - 7316 T - 920257436)^{2} \)
$17$
\( (T^{2} - 166452 T + 2938893732)^{2} \)
$19$
\( T^{4} + 86956856160 T^{2} + \cdots + 18\!\cdots\!96 \)
$23$
\( T^{4} + 130419942912 T^{2} + \cdots + 11\!\cdots\!00 \)
$29$
\( (T^{2} - 1171788 T + 169738484580)^{2} \)
$31$
\( T^{4} + 2873741432544 T^{2} + \cdots + 49\!\cdots\!84 \)
$37$
\( (T^{2} - 2157892 T - 2332762137980)^{2} \)
$41$
\( (T^{2} - 4517748 T + 4295512060452)^{2} \)
$43$
\( T^{4} + 26254406590560 T^{2} + \cdots + 54\!\cdots\!16 \)
$47$
\( T^{4} + 2880843373440 T^{2} + \cdots + 14\!\cdots\!16 \)
$53$
\( (T^{2} + 21093300 T + 108678054443364)^{2} \)
$59$
\( T^{4} + 78425893959264 T^{2} + \cdots + 44\!\cdots\!24 \)
$61$
\( (T^{2} + 24074204 T + 111045415899460)^{2} \)
$67$
\( T^{4} + \cdots + 28\!\cdots\!96 \)
$71$
\( T^{4} + \cdots + 54\!\cdots\!36 \)
$73$
\( (T^{2} + 10607740 T - 670117553322236)^{2} \)
$79$
\( T^{4} + \cdots + 43\!\cdots\!00 \)
$83$
\( T^{4} + \cdots + 62\!\cdots\!44 \)
$89$
\( (T^{2} + 6337788 T - 11\!\cdots\!64)^{2} \)
$97$
\( (T^{2} - 131576900 T + 651598379321476)^{2} \)
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