Newspace parameters
| Level: | \( N \) | \(=\) | \( 832 = 2^{6} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 832.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(49.0895891248\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{6} - 110x^{4} + 2925x^{2} - 10400 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{7} \) |
| Twist minimal: | no (minimal twist has level 416) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 110x^{4} + 2925x^{2} - 10400 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 55\nu ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 35\nu^{2} - 800 ) / 55 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 90\nu^{3} + 1455\nu ) / 110 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{4} - 90\nu^{2} + 1235 ) / 55 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{3} + 37 \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{2} + 55\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -35\beta_{5} + 90\beta_{3} + 2095 \)
|
| \(\nu^{5}\) | \(=\) |
\( 110\beta_{4} + 450\beta_{2} + 3495\beta_1 \)
|
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 | −8.40941 | 0 | −5.33769 | 0 | 23.8083 | 0 | 43.7182 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −5.92375 | 0 | −15.5783 | 0 | −13.5605 | 0 | 8.09087 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.04717 | 0 | 12.9160 | 0 | 24.4801 | 0 | −22.8091 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.04717 | 0 | 12.9160 | 0 | −24.4801 | 0 | −22.8091 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 5.92375 | 0 | −15.5783 | 0 | 13.5605 | 0 | 8.09087 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 8.40941 | 0 | −5.33769 | 0 | −23.8083 | 0 | 43.7182 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(13\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 832.4.a.bh | 6 | |
| 4.b | odd | 2 | 1 | inner | 832.4.a.bh | 6 | |
| 8.b | even | 2 | 1 | 416.4.a.k | ✓ | 6 | |
| 8.d | odd | 2 | 1 | 416.4.a.k | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 416.4.a.k | ✓ | 6 | 8.b | even | 2 | 1 | |
| 416.4.a.k | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 832.4.a.bh | 6 | 1.a | even | 1 | 1 | trivial | |
| 832.4.a.bh | 6 | 4.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(832))\):
|
\( T_{3}^{6} - 110T_{3}^{4} + 2925T_{3}^{2} - 10400 \)
|
|
\( T_{5}^{3} + 8T_{5}^{2} - 187T_{5} - 1074 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 110 T^{4} + \cdots - 10400 \)
|
| $5$ |
\( (T^{3} + 8 T^{2} + \cdots - 1074)^{2} \)
|
| $7$ |
\( T^{6} - 1350 T^{4} + \cdots - 62465000 \)
|
| $11$ |
\( T^{6} - 4448 T^{4} + \cdots - 109905536 \)
|
| $13$ |
\( (T + 13)^{6} \)
|
| $17$ |
\( (T^{3} - 4 T^{2} + \cdots - 123642)^{2} \)
|
| $19$ |
\( T^{6} + \cdots - 6623929143936 \)
|
| $23$ |
\( T^{6} + \cdots - 11447832674304 \)
|
| $29$ |
\( (T^{3} + 126 T^{2} + \cdots - 6999032)^{2} \)
|
| $31$ |
\( T^{6} + \cdots - 145782374400 \)
|
| $37$ |
\( (T^{3} + 444 T^{2} + \cdots + 729542)^{2} \)
|
| $41$ |
\( (T^{3} - 326 T^{2} + \cdots + 1436032)^{2} \)
|
| $43$ |
\( T^{6} + \cdots - 102336053354496 \)
|
| $47$ |
\( T^{6} + \cdots - 40725552143016 \)
|
| $53$ |
\( (T^{3} + 890 T^{2} + \cdots - 17311680)^{2} \)
|
| $59$ |
\( T^{6} + \cdots - 10\!\cdots\!36 \)
|
| $61$ |
\( (T^{3} + 1250 T^{2} + \cdots + 42323840)^{2} \)
|
| $67$ |
\( T^{6} + \cdots - 115567712614016 \)
|
| $71$ |
\( T^{6} + \cdots - 28\!\cdots\!84 \)
|
| $73$ |
\( (T^{3} - 458 T^{2} + \cdots + 184823784)^{2} \)
|
| $79$ |
\( T^{6} + \cdots - 23\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots - 17\!\cdots\!96 \)
|
| $89$ |
\( (T^{3} + 410 T^{2} + \cdots - 170348040)^{2} \)
|
| $97$ |
\( (T^{3} + 1590 T^{2} + \cdots - 236023960)^{2} \)
|
| show more | |
| show less |