Newspace parameters
| Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 400.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(64.1535279252\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 20) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
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 | 22.0000 | 0 | 0 | 0 | 218.000 | 0 | 241.000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
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 | 400.6.a.m | 1 | |
| 4.b | odd | 2 | 1 | 100.6.a.a | 1 | ||
| 5.b | even | 2 | 1 | 80.6.a.b | 1 | ||
| 5.c | odd | 4 | 2 | 400.6.c.c | 2 | ||
| 12.b | even | 2 | 1 | 900.6.a.b | 1 | ||
| 15.d | odd | 2 | 1 | 720.6.a.l | 1 | ||
| 20.d | odd | 2 | 1 | 20.6.a.a | ✓ | 1 | |
| 20.e | even | 4 | 2 | 100.6.c.a | 2 | ||
| 40.e | odd | 2 | 1 | 320.6.a.c | 1 | ||
| 40.f | even | 2 | 1 | 320.6.a.n | 1 | ||
| 60.h | even | 2 | 1 | 180.6.a.e | 1 | ||
| 60.l | odd | 4 | 2 | 900.6.d.h | 2 | ||
| 140.c | even | 2 | 1 | 980.6.a.b | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 20.6.a.a | ✓ | 1 | 20.d | odd | 2 | 1 | |
| 80.6.a.b | 1 | 5.b | even | 2 | 1 | ||
| 100.6.a.a | 1 | 4.b | odd | 2 | 1 | ||
| 100.6.c.a | 2 | 20.e | even | 4 | 2 | ||
| 180.6.a.e | 1 | 60.h | even | 2 | 1 | ||
| 320.6.a.c | 1 | 40.e | odd | 2 | 1 | ||
| 320.6.a.n | 1 | 40.f | even | 2 | 1 | ||
| 400.6.a.m | 1 | 1.a | even | 1 | 1 | trivial | |
| 400.6.c.c | 2 | 5.c | odd | 4 | 2 | ||
| 720.6.a.l | 1 | 15.d | odd | 2 | 1 | ||
| 900.6.a.b | 1 | 12.b | even | 2 | 1 | ||
| 900.6.d.h | 2 | 60.l | odd | 4 | 2 | ||
| 980.6.a.b | 1 | 140.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} - 22 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T - 22 \)
$5$
\( T \)
$7$
\( T - 218 \)
$11$
\( T - 480 \)
$13$
\( T - 622 \)
$17$
\( T + 186 \)
$19$
\( T - 1204 \)
$23$
\( T + 3186 \)
$29$
\( T - 5526 \)
$31$
\( T + 9356 \)
$37$
\( T + 5618 \)
$41$
\( T + 14394 \)
$43$
\( T + 370 \)
$47$
\( T - 16146 \)
$53$
\( T - 4374 \)
$59$
\( T - 11748 \)
$61$
\( T - 13202 \)
$67$
\( T + 11542 \)
$71$
\( T - 29532 \)
$73$
\( T + 33698 \)
$79$
\( T + 31208 \)
$83$
\( T + 38466 \)
$89$
\( T - 119514 \)
$97$
\( T + 94658 \)
show more
show less