Newspace parameters
| Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 24 \) |
| Character orbit: | \([\chi]\) | \(=\) | 48.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(160.897937926\) |
| 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: |
\( x^{3} - x^{2} - 313478447x - 3858843765 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{21}\cdot 3^{5} \) |
| Twist minimal: | no (minimal twist has level 24) |
| 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
\( x^{3} - x^{2} - 313478447x - 3858843765 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -192\nu^{2} + 1771392\nu + 40124650816 ) / 391 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8448\nu^{2} + 95024640\nu - 1765542291200 ) / 391 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 44\beta _1 + 147456 ) / 442368 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4613\beta_{2} - 247460\beta _1 + 46224277954560 ) / 221184 \)
|
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 | −1.57121e8 | 0 | −8.81163e9 | 0 | 3.13811e10 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 177147. | 0 | 3.07262e6 | 0 | 6.69271e9 | 0 | 3.13811e10 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 177147. | 0 | 7.68230e7 | 0 | −1.22170e9 | 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 | 48.24.a.l | 3 | |
| 4.b | odd | 2 | 1 | 24.24.a.b | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 24.24.a.b | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 48.24.a.l | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 77225166T_{5}^{2} - 12317220461555220T_{5} + 37088028673642780985000 \)
acting on \(S_{24}^{\mathrm{new}}(\Gamma_0(48))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T - 177147)^{3} \)
|
| $5$ |
\( T^{3} + \cdots + 37\!\cdots\!00 \)
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| $7$ |
\( T^{3} + \cdots - 72\!\cdots\!80 \)
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| $11$ |
\( T^{3} + \cdots + 47\!\cdots\!44 \)
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| $13$ |
\( T^{3} + \cdots + 18\!\cdots\!92 \)
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| $17$ |
\( T^{3} + \cdots - 11\!\cdots\!12 \)
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| $19$ |
\( T^{3} + \cdots + 34\!\cdots\!56 \)
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| $23$ |
\( T^{3} + \cdots - 25\!\cdots\!20 \)
|
| $29$ |
\( T^{3} + \cdots - 21\!\cdots\!08 \)
|
| $31$ |
\( T^{3} + \cdots + 54\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots + 74\!\cdots\!00 \)
|
| $41$ |
\( T^{3} + \cdots - 70\!\cdots\!96 \)
|
| $43$ |
\( T^{3} + \cdots - 35\!\cdots\!84 \)
|
| $47$ |
\( T^{3} + \cdots - 23\!\cdots\!00 \)
|
| $53$ |
\( T^{3} + \cdots - 35\!\cdots\!00 \)
|
| $59$ |
\( T^{3} + \cdots + 15\!\cdots\!28 \)
|
| $61$ |
\( T^{3} + \cdots - 12\!\cdots\!76 \)
|
| $67$ |
\( T^{3} + \cdots - 38\!\cdots\!12 \)
|
| $71$ |
\( T^{3} + \cdots - 58\!\cdots\!52 \)
|
| $73$ |
\( T^{3} + \cdots + 47\!\cdots\!32 \)
|
| $79$ |
\( T^{3} + \cdots - 29\!\cdots\!08 \)
|
| $83$ |
\( T^{3} + \cdots + 93\!\cdots\!96 \)
|
| $89$ |
\( T^{3} + \cdots + 16\!\cdots\!60 \)
|
| $97$ |
\( T^{3} + \cdots + 12\!\cdots\!16 \)
|
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