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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 9\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.
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 | 726.000 | 0 | − | 3055.34i | 0 | −2187.00 | 0 | ||||||||||||||||||||||
| 31.2 | 0 | 46.7654i | 0 | 726.000 | 0 | 3055.34i | 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.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 144.9.g.c | 2 | ||
| 4.b | odd | 2 | 1 | inner | 48.9.g.b | ✓ | 2 |
| 8.b | even | 2 | 1 | 192.9.g.a | 2 | ||
| 8.d | odd | 2 | 1 | 192.9.g.a | 2 | ||
| 12.b | even | 2 | 1 | 144.9.g.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 48.9.g.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 48.9.g.b | ✓ | 2 | 4.b | odd | 2 | 1 | inner |
| 144.9.g.c | 2 | 3.b | odd | 2 | 1 | ||
| 144.9.g.c | 2 | 12.b | even | 2 | 1 | ||
| 192.9.g.a | 2 | 8.b | even | 2 | 1 | ||
| 192.9.g.a | 2 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} - 726 \)
acting on \(S_{9}^{\mathrm{new}}(48, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} + 2187 \)
$5$
\( (T - 726)^{2} \)
$7$
\( T^{2} + 9335088 \)
$11$
\( T^{2} + 176394672 \)
$13$
\( (T - 39034)^{2} \)
$17$
\( (T + 65814)^{2} \)
$19$
\( T^{2} + 16966924848 \)
$23$
\( T^{2} + 252077329152 \)
$29$
\( (T - 202062)^{2} \)
$31$
\( T^{2} + 1429542270000 \)
$37$
\( (T + 1876030)^{2} \)
$41$
\( (T - 3091050)^{2} \)
$43$
\( T^{2} + 5125154792112 \)
$47$
\( T^{2} + 40407933129408 \)
$53$
\( (T + 1066482)^{2} \)
$59$
\( T^{2} + 33218525693232 \)
$61$
\( (T - 17154194)^{2} \)
$67$
\( T^{2} + \cdots + 752268658081968 \)
$71$
\( T^{2} + 15\!\cdots\!12 \)
$73$
\( (T + 53286014)^{2} \)
$79$
\( T^{2} + \cdots + 333778633635888 \)
$83$
\( T^{2} + 60669350784432 \)
$89$
\( (T - 86667234)^{2} \)
$97$
\( (T + 73901822)^{2} \)
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