Newspace parameters
| Level: | \( N \) | \(=\) | \( 24 = 2^{3} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 24 \) |
| Character orbit: | \([\chi]\) | \(=\) | 24.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(80.4489689628\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{3} - 43285816x - 66694276650 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{21}\cdot 3^{4}\cdot 5 \) |
| 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
\( x^{3} - 43285816x - 66694276650 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 9600\nu^{2} + 4055040\nu - 277029222400 ) / 1873 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -31872\nu^{2} + 152248320\nu + 919737018368 ) / 1873 \)
|
| \(\nu\) | \(=\) |
\( ( 25\beta_{2} + 83\beta_1 ) / 2211840 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -11\beta_{2} + 413\beta _1 + 66487013376 ) / 2304 \)
|
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 | 177147. | 0 | −1.51825e8 | 0 | −3.35467e9 | 0 | 3.13811e10 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 177147. | 0 | −1.58062e7 | 0 | 8.56279e9 | 0 | 3.13811e10 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 177147. | 0 | 1.22619e8 | 0 | −4.92683e9 | 0 | 3.13811e10 | 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 | 24.24.a.c | ✓ | 3 |
| 4.b | odd | 2 | 1 | 48.24.a.j | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 24.24.a.c | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 48.24.a.j | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 45012150T_{5}^{2} - 18155059753998900T_{5} - 294258907498613722675000 \)
acting on \(S_{24}^{\mathrm{new}}(\Gamma_0(24))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T - 177147)^{3} \)
|
| $5$ |
\( T^{3} + \cdots - 29\!\cdots\!00 \)
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| $7$ |
\( T^{3} + \cdots - 14\!\cdots\!00 \)
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| $11$ |
\( T^{3} + \cdots - 34\!\cdots\!80 \)
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| $13$ |
\( T^{3} + \cdots - 60\!\cdots\!68 \)
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| $17$ |
\( T^{3} + \cdots - 56\!\cdots\!64 \)
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| $19$ |
\( T^{3} + \cdots - 13\!\cdots\!76 \)
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| $23$ |
\( T^{3} + \cdots - 22\!\cdots\!44 \)
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| $29$ |
\( T^{3} + \cdots - 10\!\cdots\!76 \)
|
| $31$ |
\( T^{3} + \cdots - 31\!\cdots\!60 \)
|
| $37$ |
\( T^{3} + \cdots - 13\!\cdots\!48 \)
|
| $41$ |
\( T^{3} + \cdots - 54\!\cdots\!60 \)
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| $43$ |
\( T^{3} + \cdots - 37\!\cdots\!08 \)
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| $47$ |
\( T^{3} + \cdots + 70\!\cdots\!00 \)
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| $53$ |
\( T^{3} + \cdots + 53\!\cdots\!20 \)
|
| $59$ |
\( T^{3} + \cdots + 38\!\cdots\!20 \)
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| $61$ |
\( T^{3} + \cdots - 67\!\cdots\!84 \)
|
| $67$ |
\( T^{3} + \cdots - 10\!\cdots\!72 \)
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| $71$ |
\( T^{3} + \cdots + 48\!\cdots\!40 \)
|
| $73$ |
\( T^{3} + \cdots + 18\!\cdots\!00 \)
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| $79$ |
\( T^{3} + \cdots + 41\!\cdots\!64 \)
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| $83$ |
\( T^{3} + \cdots - 39\!\cdots\!16 \)
|
| $89$ |
\( T^{3} + \cdots - 14\!\cdots\!68 \)
|
| $97$ |
\( T^{3} + \cdots + 53\!\cdots\!44 \)
|
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