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: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 53560x - 70812 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{21}\cdot 3^{4}\cdot 5\cdot 7 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - 53560x - 70812 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 768\nu^{2} + 948864\nu - 27422720 ) / 11 \)
|
| \(\beta_{2}\) | \(=\) |
\( -3072\nu^{2} + 75264\nu + 109690880 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 44\beta_1 ) / 3870720 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -353\beta_{2} + 308\beta _1 + 39488716800 ) / 1105920 \)
|
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 | −2.28989e7 | 0 | 1.04414e9 | 0 | 3.48678e9 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −59049.0 | 0 | 995860. | 0 | −1.18014e9 | 0 | 3.48678e9 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | −59049.0 | 0 | 1.70699e7 | 0 | 4.07437e8 | 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.b | ✓ | 3 |
| 3.b | odd | 2 | 1 | 72.22.a.e | 3 | ||
| 4.b | odd | 2 | 1 | 48.22.a.k | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 24.22.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 48.22.a.k | 3 | 4.b | odd | 2 | 1 | ||
| 72.22.a.e | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 4833126T_{5}^{2} - 396686171190900T_{5} + 389262902874527045000 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(24))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} \)
$3$
\( (T + 59049)^{3} \)
$5$
\( T^{3} + 4833126 T^{2} + \cdots + 38\!\cdots\!00 \)
$7$
\( T^{3} - 271431024 T^{2} + \cdots + 50\!\cdots\!00 \)
$11$
\( T^{3} - 61194658188 T^{2} + \cdots - 90\!\cdots\!00 \)
$13$
\( T^{3} + 594486202422 T^{2} + \cdots - 47\!\cdots\!16 \)
$17$
\( T^{3} - 1424819519334 T^{2} + \cdots - 16\!\cdots\!52 \)
$19$
\( T^{3} + 35094825681804 T^{2} + \cdots - 23\!\cdots\!64 \)
$23$
\( T^{3} + 94911043073592 T^{2} + \cdots - 20\!\cdots\!88 \)
$29$
\( T^{3} + \cdots - 34\!\cdots\!24 \)
$31$
\( T^{3} + \cdots - 39\!\cdots\!00 \)
$37$
\( T^{3} + \cdots + 74\!\cdots\!76 \)
$41$
\( T^{3} + \cdots + 15\!\cdots\!40 \)
$43$
\( T^{3} + \cdots - 51\!\cdots\!56 \)
$47$
\( T^{3} + \cdots - 12\!\cdots\!00 \)
$53$
\( T^{3} + \cdots - 61\!\cdots\!40 \)
$59$
\( T^{3} + \cdots - 34\!\cdots\!00 \)
$61$
\( T^{3} + \cdots - 15\!\cdots\!24 \)
$67$
\( T^{3} + \cdots + 17\!\cdots\!24 \)
$71$
\( T^{3} + \cdots - 72\!\cdots\!00 \)
$73$
\( T^{3} + \cdots + 15\!\cdots\!00 \)
$79$
\( T^{3} + \cdots + 31\!\cdots\!36 \)
$83$
\( T^{3} + \cdots - 57\!\cdots\!32 \)
$89$
\( T^{3} + \cdots + 32\!\cdots\!88 \)
$97$
\( T^{3} + \cdots + 19\!\cdots\!32 \)
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