Newspace parameters
| Level: | \( N \) | \(=\) | \( 24 = 2^{3} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 24.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(67.0745626289\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{537541}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 134385 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{2}\cdot 7 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4032\sqrt{537541}\). 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 | −3.80644e6 | 0 | −1.01851e9 | 0 | 3.48678e9 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 59049.0 | 0 | 2.57551e7 | 0 | 3.59057e8 | 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 | 24.22.a.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 72.22.a.a | 2 | ||
| 4.b | odd | 2 | 1 | 48.22.a.h | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 24.22.a.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 48.22.a.h | 2 | 4.b | odd | 2 | 1 | ||
| 72.22.a.a | 2 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 21948620T_{5} - 98034943473500 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(24))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( (T - 59049)^{2} \)
$5$
\( T^{2} - 21948620 T - 98034943473500 \)
$7$
\( T^{2} + 659451408 T - 36\!\cdots\!60 \)
$11$
\( T^{2} + 29910896872 T - 14\!\cdots\!48 \)
$13$
\( T^{2} - 4468866812 T - 15\!\cdots\!28 \)
$17$
\( T^{2} + 17665404721820 T + 51\!\cdots\!64 \)
$19$
\( T^{2} + 22467979297496 T + 11\!\cdots\!88 \)
$23$
\( T^{2} - 120995273049584 T - 18\!\cdots\!80 \)
$29$
\( T^{2} + \cdots + 24\!\cdots\!64 \)
$31$
\( T^{2} + \cdots - 70\!\cdots\!60 \)
$37$
\( T^{2} + \cdots - 40\!\cdots\!80 \)
$41$
\( T^{2} + \cdots + 13\!\cdots\!88 \)
$43$
\( T^{2} + \cdots - 29\!\cdots\!44 \)
$47$
\( T^{2} + \cdots - 91\!\cdots\!60 \)
$53$
\( T^{2} + \cdots - 26\!\cdots\!20 \)
$59$
\( T^{2} + \cdots - 12\!\cdots\!72 \)
$61$
\( T^{2} + \cdots - 43\!\cdots\!92 \)
$67$
\( T^{2} + \cdots - 32\!\cdots\!52 \)
$71$
\( T^{2} + \cdots - 16\!\cdots\!52 \)
$73$
\( T^{2} + \cdots + 86\!\cdots\!16 \)
$79$
\( T^{2} + \cdots - 28\!\cdots\!76 \)
$83$
\( T^{2} + \cdots + 21\!\cdots\!32 \)
$89$
\( T^{2} + \cdots + 66\!\cdots\!60 \)
$97$
\( T^{2} + \cdots + 26\!\cdots\!16 \)
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