Newspace parameters
Level: | \( N \) | \(=\) | \( 117 = 3^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 117.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(0.934249703649\) |
Analytic rank: | \(1\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 39) |
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 |
|
−1.00000 | 0 | −1.00000 | −2.00000 | 0 | −4.00000 | 3.00000 | 0 | 2.00000 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(13\) | \(-1\) |
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 | 117.2.a.a | 1 | |
3.b | odd | 2 | 1 | 39.2.a.a | ✓ | 1 | |
4.b | odd | 2 | 1 | 1872.2.a.h | 1 | ||
5.b | even | 2 | 1 | 2925.2.a.p | 1 | ||
5.c | odd | 4 | 2 | 2925.2.c.e | 2 | ||
7.b | odd | 2 | 1 | 5733.2.a.e | 1 | ||
8.b | even | 2 | 1 | 7488.2.a.bl | 1 | ||
8.d | odd | 2 | 1 | 7488.2.a.by | 1 | ||
9.c | even | 3 | 2 | 1053.2.e.d | 2 | ||
9.d | odd | 6 | 2 | 1053.2.e.b | 2 | ||
12.b | even | 2 | 1 | 624.2.a.i | 1 | ||
13.b | even | 2 | 1 | 1521.2.a.e | 1 | ||
13.d | odd | 4 | 2 | 1521.2.b.b | 2 | ||
15.d | odd | 2 | 1 | 975.2.a.f | 1 | ||
15.e | even | 4 | 2 | 975.2.c.f | 2 | ||
21.c | even | 2 | 1 | 1911.2.a.f | 1 | ||
24.f | even | 2 | 1 | 2496.2.a.e | 1 | ||
24.h | odd | 2 | 1 | 2496.2.a.q | 1 | ||
33.d | even | 2 | 1 | 4719.2.a.c | 1 | ||
39.d | odd | 2 | 1 | 507.2.a.a | 1 | ||
39.f | even | 4 | 2 | 507.2.b.a | 2 | ||
39.h | odd | 6 | 2 | 507.2.e.b | 2 | ||
39.i | odd | 6 | 2 | 507.2.e.a | 2 | ||
39.k | even | 12 | 4 | 507.2.j.e | 4 | ||
156.h | even | 2 | 1 | 8112.2.a.s | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
39.2.a.a | ✓ | 1 | 3.b | odd | 2 | 1 | |
117.2.a.a | 1 | 1.a | even | 1 | 1 | trivial | |
507.2.a.a | 1 | 39.d | odd | 2 | 1 | ||
507.2.b.a | 2 | 39.f | even | 4 | 2 | ||
507.2.e.a | 2 | 39.i | odd | 6 | 2 | ||
507.2.e.b | 2 | 39.h | odd | 6 | 2 | ||
507.2.j.e | 4 | 39.k | even | 12 | 4 | ||
624.2.a.i | 1 | 12.b | even | 2 | 1 | ||
975.2.a.f | 1 | 15.d | odd | 2 | 1 | ||
975.2.c.f | 2 | 15.e | even | 4 | 2 | ||
1053.2.e.b | 2 | 9.d | odd | 6 | 2 | ||
1053.2.e.d | 2 | 9.c | even | 3 | 2 | ||
1521.2.a.e | 1 | 13.b | even | 2 | 1 | ||
1521.2.b.b | 2 | 13.d | odd | 4 | 2 | ||
1872.2.a.h | 1 | 4.b | odd | 2 | 1 | ||
1911.2.a.f | 1 | 21.c | even | 2 | 1 | ||
2496.2.a.e | 1 | 24.f | even | 2 | 1 | ||
2496.2.a.q | 1 | 24.h | odd | 2 | 1 | ||
2925.2.a.p | 1 | 5.b | even | 2 | 1 | ||
2925.2.c.e | 2 | 5.c | odd | 4 | 2 | ||
4719.2.a.c | 1 | 33.d | even | 2 | 1 | ||
5733.2.a.e | 1 | 7.b | odd | 2 | 1 | ||
7488.2.a.bl | 1 | 8.b | even | 2 | 1 | ||
7488.2.a.by | 1 | 8.d | odd | 2 | 1 | ||
8112.2.a.s | 1 | 156.h | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(117))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T + 1 \)
$3$
\( T \)
$5$
\( T + 2 \)
$7$
\( T + 4 \)
$11$
\( T + 4 \)
$13$
\( T - 1 \)
$17$
\( T + 2 \)
$19$
\( T \)
$23$
\( T \)
$29$
\( T - 10 \)
$31$
\( T - 4 \)
$37$
\( T + 2 \)
$41$
\( T + 6 \)
$43$
\( T + 12 \)
$47$
\( T \)
$53$
\( T + 6 \)
$59$
\( T + 12 \)
$61$
\( T + 2 \)
$67$
\( T + 8 \)
$71$
\( T \)
$73$
\( T - 2 \)
$79$
\( T - 8 \)
$83$
\( T + 4 \)
$89$
\( T - 2 \)
$97$
\( T - 10 \)
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