Newspace parameters
| Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 48.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(134.149125258\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{2} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 797544 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{2}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 12) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2880\sqrt{3190177}\). We also show the integral \(q\)-expansion of the trace form.
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 | 59049.0 | 0 | −1.13060e7 | 0 | 8.09970e8 | 0 | 3.48678e9 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 59049.0 | 0 | 4.01339e7 | 0 | −1.31964e9 | 0 | 3.48678e9 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 48.22.a.i | 2 | |
| 4.b | odd | 2 | 1 | 12.22.a.b | ✓ | 2 | |
| 12.b | even | 2 | 1 | 36.22.a.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 12.22.a.b | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 36.22.a.b | 2 | 12.b | even | 2 | 1 | ||
| 48.22.a.i | 2 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 28827900T_{5} - 453753148117500 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(48))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( (T - 59049)^{2} \)
$5$
\( T^{2} + \cdots - 453753148117500 \)
$7$
\( T^{2} + 509669728 T - 10\!\cdots\!04 \)
$11$
\( T^{2} - 76050855288 T + 11\!\cdots\!36 \)
$13$
\( T^{2} + 809326043300 T + 13\!\cdots\!00 \)
$17$
\( T^{2} - 831385897668 T - 98\!\cdots\!44 \)
$19$
\( T^{2} + 29817568652920 T + 22\!\cdots\!00 \)
$23$
\( T^{2} + 62680641406128 T - 11\!\cdots\!04 \)
$29$
\( T^{2} + \cdots - 68\!\cdots\!24 \)
$31$
\( T^{2} + \cdots + 85\!\cdots\!76 \)
$37$
\( T^{2} + \cdots - 10\!\cdots\!36 \)
$41$
\( T^{2} + \cdots + 30\!\cdots\!00 \)
$43$
\( T^{2} + \cdots + 15\!\cdots\!76 \)
$47$
\( T^{2} + \cdots + 98\!\cdots\!16 \)
$53$
\( T^{2} + \cdots + 81\!\cdots\!44 \)
$59$
\( T^{2} + \cdots - 88\!\cdots\!64 \)
$61$
\( T^{2} + \cdots + 75\!\cdots\!84 \)
$67$
\( T^{2} + \cdots - 37\!\cdots\!24 \)
$71$
\( T^{2} + \cdots - 77\!\cdots\!00 \)
$73$
\( T^{2} + \cdots - 77\!\cdots\!16 \)
$79$
\( T^{2} + \cdots - 86\!\cdots\!96 \)
$83$
\( T^{2} + \cdots + 25\!\cdots\!96 \)
$89$
\( T^{2} + \cdots + 96\!\cdots\!00 \)
$97$
\( T^{2} + \cdots + 34\!\cdots\!64 \)
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