Newspace parameters
| Level: | \( N \) | \(=\) | \( 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 37.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(0.295446487479\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 | 1.00000 | −2.00000 | 0 | 0 | −1.00000 | 0 | −2.00000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(37\) | \(-1\) |
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 | 37.2.a.b | ✓ | 1 |
| 3.b | odd | 2 | 1 | 333.2.a.b | 1 | ||
| 4.b | odd | 2 | 1 | 592.2.a.a | 1 | ||
| 5.b | even | 2 | 1 | 925.2.a.b | 1 | ||
| 5.c | odd | 4 | 2 | 925.2.b.e | 2 | ||
| 7.b | odd | 2 | 1 | 1813.2.a.b | 1 | ||
| 8.b | even | 2 | 1 | 2368.2.a.d | 1 | ||
| 8.d | odd | 2 | 1 | 2368.2.a.m | 1 | ||
| 11.b | odd | 2 | 1 | 4477.2.a.a | 1 | ||
| 12.b | even | 2 | 1 | 5328.2.a.k | 1 | ||
| 13.b | even | 2 | 1 | 6253.2.a.b | 1 | ||
| 15.d | odd | 2 | 1 | 8325.2.a.p | 1 | ||
| 37.b | even | 2 | 1 | 1369.2.a.c | 1 | ||
| 37.d | odd | 4 | 2 | 1369.2.b.a | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 37.2.a.b | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
| 333.2.a.b | 1 | 3.b | odd | 2 | 1 | ||
| 592.2.a.a | 1 | 4.b | odd | 2 | 1 | ||
| 925.2.a.b | 1 | 5.b | even | 2 | 1 | ||
| 925.2.b.e | 2 | 5.c | odd | 4 | 2 | ||
| 1369.2.a.c | 1 | 37.b | even | 2 | 1 | ||
| 1369.2.b.a | 2 | 37.d | odd | 4 | 2 | ||
| 1813.2.a.b | 1 | 7.b | odd | 2 | 1 | ||
| 2368.2.a.d | 1 | 8.b | even | 2 | 1 | ||
| 2368.2.a.m | 1 | 8.d | odd | 2 | 1 | ||
| 4477.2.a.a | 1 | 11.b | odd | 2 | 1 | ||
| 5328.2.a.k | 1 | 12.b | even | 2 | 1 | ||
| 6253.2.a.b | 1 | 13.b | even | 2 | 1 | ||
| 8325.2.a.p | 1 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(37))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T - 1 \)
$5$
\( T \)
$7$
\( T + 1 \)
$11$
\( T - 3 \)
$13$
\( T + 4 \)
$17$
\( T - 6 \)
$19$
\( T - 2 \)
$23$
\( T - 6 \)
$29$
\( T + 6 \)
$31$
\( T + 4 \)
$37$
\( T - 1 \)
$41$
\( T + 9 \)
$43$
\( T - 8 \)
$47$
\( T - 3 \)
$53$
\( T + 3 \)
$59$
\( T - 12 \)
$61$
\( T - 8 \)
$67$
\( T + 4 \)
$71$
\( T + 15 \)
$73$
\( T - 11 \)
$79$
\( T + 10 \)
$83$
\( T - 9 \)
$89$
\( T - 6 \)
$97$
\( T - 8 \)
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