Newspace parameters
| Level: | \( N \) | \(=\) | \( 64 = 2^{6} \) |
| Weight: | \( k \) | \(=\) | \( 24 \) |
| Character orbit: | \([\chi]\) | \(=\) | 64.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(214.530583901\) |
| 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: |
\( x^{3} - 166408x - 10560732 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{21}\cdot 3^{3} \) |
| Twist minimal: | no (minimal twist has level 8) |
| 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
\( x^{3} - 166408x - 10560732 \)
:
| \(\beta_{1}\) | \(=\) |
\( 1152\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 24576\nu^{2} - 2326656\nu - 2726428672 ) / 25 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 1152 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 75\beta_{2} + 6059\beta _1 + 8179286016 ) / 73728 \)
|
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 | −574251. | 0 | −8.41392e7 | 0 | 5.95738e9 | 0 | 2.35621e11 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 3705.18 | 0 | 6.91268e7 | 0 | −6.89672e9 | 0 | −9.41295e10 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 356598. | 0 | −8.06163e7 | 0 | 9.58726e9 | 0 | 3.30186e10 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
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 | 64.24.a.h | 3 | |
| 4.b | odd | 2 | 1 | 64.24.a.k | 3 | ||
| 8.b | even | 2 | 1 | 16.24.a.e | 3 | ||
| 8.d | odd | 2 | 1 | 8.24.a.a | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.24.a.a | ✓ | 3 | 8.d | odd | 2 | 1 | |
| 16.24.a.e | 3 | 8.b | even | 2 | 1 | ||
| 64.24.a.h | 3 | 1.a | even | 1 | 1 | trivial | |
| 64.24.a.k | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 213948T_{3}^{2} - 205582806864T_{3} + 758732719429440 \)
acting on \(S_{24}^{\mathrm{new}}(\Gamma_0(64))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + \cdots + 758732719429440 \)
|
| $5$ |
\( T^{3} + \cdots - 46\!\cdots\!00 \)
|
| $7$ |
\( T^{3} + \cdots + 39\!\cdots\!32 \)
|
| $11$ |
\( T^{3} + \cdots + 90\!\cdots\!76 \)
|
| $13$ |
\( T^{3} + \cdots + 17\!\cdots\!96 \)
|
| $17$ |
\( T^{3} + \cdots - 74\!\cdots\!00 \)
|
| $19$ |
\( T^{3} + \cdots - 27\!\cdots\!36 \)
|
| $23$ |
\( T^{3} + \cdots + 27\!\cdots\!12 \)
|
| $29$ |
\( T^{3} + \cdots - 40\!\cdots\!32 \)
|
| $31$ |
\( T^{3} + \cdots + 11\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots + 17\!\cdots\!88 \)
|
| $41$ |
\( T^{3} + \cdots + 30\!\cdots\!84 \)
|
| $43$ |
\( T^{3} + \cdots + 24\!\cdots\!36 \)
|
| $47$ |
\( T^{3} + \cdots + 12\!\cdots\!92 \)
|
| $53$ |
\( T^{3} + \cdots - 14\!\cdots\!68 \)
|
| $59$ |
\( T^{3} + \cdots + 13\!\cdots\!64 \)
|
| $61$ |
\( T^{3} + \cdots - 98\!\cdots\!00 \)
|
| $67$ |
\( T^{3} + \cdots + 14\!\cdots\!76 \)
|
| $71$ |
\( T^{3} + \cdots - 46\!\cdots\!40 \)
|
| $73$ |
\( T^{3} + \cdots - 12\!\cdots\!12 \)
|
| $79$ |
\( T^{3} + \cdots - 62\!\cdots\!60 \)
|
| $83$ |
\( T^{3} + \cdots - 57\!\cdots\!32 \)
|
| $89$ |
\( T^{3} + \cdots - 12\!\cdots\!16 \)
|
| $97$ |
\( T^{3} + \cdots + 45\!\cdots\!52 \)
|
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