Newspace parameters
| Level: | \( N \) | \(=\) | \( 208 = 2^{4} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 208.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(159.815381556\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 679393x^{2} - 65654928x + 60501589200 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 26) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 679393x^{2} - 65654928x + 60501589200 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -41\nu^{3} + 13644\nu^{2} + 18665879\nu - 2615930010 ) / 457074 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13\nu^{3} + 6822\nu^{2} - 7713301\nu - 2957773608 ) / 457074 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 41\beta_{3} + 13\beta_{2} + 161\beta _1 + 339717 \)
|
| \(\nu^{3}\) | \(=\) |
\( 13644\beta_{3} - 6822\beta_{2} + 508843\beta _1 + 49248018 \)
|
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 | −667.100 | 0 | −3756.22 | 0 | −11669.8 | 0 | 267876. | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −416.896 | 0 | 13888.3 | 0 | −25219.0 | 0 | −3344.54 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 265.922 | 0 | 1587.51 | 0 | −17685.6 | 0 | −106432. | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 818.074 | 0 | −11025.6 | 0 | 60418.5 | 0 | 492099. | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(13\) | \( -1 \) |
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 | 208.12.a.d | 4 | |
| 4.b | odd | 2 | 1 | 26.12.a.d | ✓ | 4 | |
| 12.b | even | 2 | 1 | 234.12.a.l | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 26.12.a.d | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 208.12.a.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 234.12.a.l | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 679393T_{3}^{2} - 65654928T_{3} + 60501589200 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(208))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + \cdots + 60501589200 \)
|
| $5$ |
\( T^{4} + \cdots + 913101596252100 \)
|
| $7$ |
\( T^{4} + \cdots - 31\!\cdots\!00 \)
|
| $11$ |
\( T^{4} + \cdots + 92\!\cdots\!00 \)
|
| $13$ |
\( (T - 371293)^{4} \)
|
| $17$ |
\( T^{4} + \cdots + 23\!\cdots\!84 \)
|
| $19$ |
\( T^{4} + \cdots + 40\!\cdots\!80 \)
|
| $23$ |
\( T^{4} + \cdots - 28\!\cdots\!56 \)
|
| $29$ |
\( T^{4} + \cdots + 10\!\cdots\!80 \)
|
| $31$ |
\( T^{4} + \cdots + 24\!\cdots\!20 \)
|
| $37$ |
\( T^{4} + \cdots - 33\!\cdots\!88 \)
|
| $41$ |
\( T^{4} + \cdots - 48\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots + 81\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots + 10\!\cdots\!24 \)
|
| $53$ |
\( T^{4} + \cdots + 18\!\cdots\!04 \)
|
| $59$ |
\( T^{4} + \cdots + 15\!\cdots\!20 \)
|
| $61$ |
\( T^{4} + \cdots + 53\!\cdots\!76 \)
|
| $67$ |
\( T^{4} + \cdots - 22\!\cdots\!80 \)
|
| $71$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots + 26\!\cdots\!60 \)
|
| $79$ |
\( T^{4} + \cdots + 35\!\cdots\!40 \)
|
| $83$ |
\( T^{4} + \cdots - 16\!\cdots\!80 \)
|
| $89$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 24\!\cdots\!52 \)
|
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