Newspace parameters
| Level: | \( N \) | \(=\) | \( 784 = 2^{4} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 784.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(125.740914733\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{37}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 7) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{37}\). We also show the integral \(q\)-expansion of the trace form.
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 | −10.0828 | 0 | 79.8276 | 0 | 0 | 0 | −141.338 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 2.08276 | 0 | −41.8276 | 0 | 0 | 0 | −238.662 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 784.6.a.t | 2 | |
| 4.b | odd | 2 | 1 | 49.6.a.e | 2 | ||
| 7.b | odd | 2 | 1 | 784.6.a.ba | 2 | ||
| 7.d | odd | 6 | 2 | 112.6.i.c | 4 | ||
| 12.b | even | 2 | 1 | 441.6.a.m | 2 | ||
| 28.d | even | 2 | 1 | 49.6.a.d | 2 | ||
| 28.f | even | 6 | 2 | 7.6.c.a | ✓ | 4 | |
| 28.g | odd | 6 | 2 | 49.6.c.f | 4 | ||
| 84.h | odd | 2 | 1 | 441.6.a.n | 2 | ||
| 84.j | odd | 6 | 2 | 63.6.e.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.6.c.a | ✓ | 4 | 28.f | even | 6 | 2 | |
| 49.6.a.d | 2 | 28.d | even | 2 | 1 | ||
| 49.6.a.e | 2 | 4.b | odd | 2 | 1 | ||
| 49.6.c.f | 4 | 28.g | odd | 6 | 2 | ||
| 63.6.e.d | 4 | 84.j | odd | 6 | 2 | ||
| 112.6.i.c | 4 | 7.d | odd | 6 | 2 | ||
| 441.6.a.m | 2 | 12.b | even | 2 | 1 | ||
| 441.6.a.n | 2 | 84.h | odd | 2 | 1 | ||
| 784.6.a.t | 2 | 1.a | even | 1 | 1 | trivial | |
| 784.6.a.ba | 2 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 8T_{3} - 21 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(784))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} + 8T - 21 \)
$5$
\( T^{2} - 38T - 3339 \)
$7$
\( T^{2} \)
$11$
\( T^{2} + 424T + 25371 \)
$13$
\( T^{2} - 924T + 184436 \)
$17$
\( T^{2} - 2346 T + 731241 \)
$19$
\( T^{2} + 360 T - 2806573 \)
$23$
\( T^{2} - 12T - 176121 \)
$29$
\( T^{2} + 7052 T - 5697324 \)
$31$
\( T^{2} - 3548 T - 15768249 \)
$37$
\( T^{2} - 11090 T + 7655325 \)
$41$
\( T^{2} + 3500 T - 24814188 \)
$43$
\( T^{2} - 12680 T - 26638832 \)
$47$
\( T^{2} + 22956 T + 32835159 \)
$53$
\( T^{2} - 3042 T - 119976147 \)
$59$
\( T^{2} + 65808 T + 1072240659 \)
$61$
\( T^{2} - 42486 T - 187196443 \)
$67$
\( T^{2} + 42312 T + 416327579 \)
$71$
\( T^{2} - 2208 T - 175265856 \)
$73$
\( T^{2} - 50506 T - 1373102199 \)
$79$
\( T^{2} + 9004 T - 977520209 \)
$83$
\( T^{2} + 104328 T + 1959796944 \)
$89$
\( T^{2} - 26666 T - 3061016343 \)
$97$
\( T^{2} + 209132 T + 10932626964 \)
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