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: |
\( x^{3} - 1841764x - 103489260 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{21}\cdot 3\cdot 5 \) |
| 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
\( x^{3} - 1841764x - 103489260 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -128\nu^{2} + 20096\nu + 157163741 ) / 361 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5248\nu^{2} + 43535744\nu - 6443718435 ) / 361 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 41\beta _1 + 14 ) / 122880 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 157\beta_{2} - 340123\beta _1 + 150877193558 ) / 122880 \)
|
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 | −253079. | 0 | 1.36864e8 | 0 | 4.69307e9 | 0 | −3.00944e10 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −156216. | 0 | −1.90634e8 | 0 | −4.80640e9 | 0 | −6.97396e10 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 442003. | 0 | 2.22890e7 | 0 | −8.79699e8 | 0 | 1.01223e11 | 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.j | 3 | |
| 4.b | odd | 2 | 1 | 64.24.a.i | 3 | ||
| 8.b | even | 2 | 1 | 16.24.a.d | 3 | ||
| 8.d | odd | 2 | 1 | 8.24.a.b | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.24.a.b | ✓ | 3 | 8.d | odd | 2 | 1 | |
| 16.24.a.d | 3 | 8.b | even | 2 | 1 | ||
| 64.24.a.i | 3 | 4.b | odd | 2 | 1 | ||
| 64.24.a.j | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 32708T_{3}^{2} - 141374520144T_{3} - 17474586120020160 \)
acting on \(S_{24}^{\mathrm{new}}(\Gamma_0(64))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + \cdots - 17\!\cdots\!60 \)
|
| $5$ |
\( T^{3} + \cdots + 58\!\cdots\!00 \)
|
| $7$ |
\( T^{3} + \cdots - 19\!\cdots\!72 \)
|
| $11$ |
\( T^{3} + \cdots - 50\!\cdots\!04 \)
|
| $13$ |
\( T^{3} + \cdots - 16\!\cdots\!56 \)
|
| $17$ |
\( T^{3} + \cdots + 14\!\cdots\!00 \)
|
| $19$ |
\( T^{3} + \cdots + 17\!\cdots\!64 \)
|
| $23$ |
\( T^{3} + \cdots - 27\!\cdots\!52 \)
|
| $29$ |
\( T^{3} + \cdots + 36\!\cdots\!28 \)
|
| $31$ |
\( T^{3} + \cdots + 27\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots + 23\!\cdots\!32 \)
|
| $41$ |
\( T^{3} + \cdots - 90\!\cdots\!16 \)
|
| $43$ |
\( T^{3} + \cdots + 50\!\cdots\!04 \)
|
| $47$ |
\( T^{3} + \cdots - 39\!\cdots\!12 \)
|
| $53$ |
\( T^{3} + \cdots - 26\!\cdots\!92 \)
|
| $59$ |
\( T^{3} + \cdots - 14\!\cdots\!96 \)
|
| $61$ |
\( T^{3} + \cdots + 52\!\cdots\!00 \)
|
| $67$ |
\( T^{3} + \cdots + 19\!\cdots\!24 \)
|
| $71$ |
\( T^{3} + \cdots + 47\!\cdots\!40 \)
|
| $73$ |
\( T^{3} + \cdots + 37\!\cdots\!32 \)
|
| $79$ |
\( T^{3} + \cdots - 57\!\cdots\!40 \)
|
| $83$ |
\( T^{3} + \cdots - 44\!\cdots\!48 \)
|
| $89$ |
\( T^{3} + \cdots - 40\!\cdots\!76 \)
|
| $97$ |
\( T^{3} + \cdots - 35\!\cdots\!52 \)
|
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