Newspace parameters
Level: | \( N \) | \(=\) | \( 475 = 5^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 475.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(3.79289409601\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 19) |
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 | 2.00000 | −2.00000 | 0 | 0 | 1.00000 | 0 | 1.00000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \(1\) |
\(19\) | \(-1\) |
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 | 475.2.a.b | 1 | |
3.b | odd | 2 | 1 | 4275.2.a.i | 1 | ||
4.b | odd | 2 | 1 | 7600.2.a.c | 1 | ||
5.b | even | 2 | 1 | 19.2.a.a | ✓ | 1 | |
5.c | odd | 4 | 2 | 475.2.b.a | 2 | ||
15.d | odd | 2 | 1 | 171.2.a.b | 1 | ||
19.b | odd | 2 | 1 | 9025.2.a.d | 1 | ||
20.d | odd | 2 | 1 | 304.2.a.f | 1 | ||
35.c | odd | 2 | 1 | 931.2.a.a | 1 | ||
35.i | odd | 6 | 2 | 931.2.f.b | 2 | ||
35.j | even | 6 | 2 | 931.2.f.c | 2 | ||
40.e | odd | 2 | 1 | 1216.2.a.b | 1 | ||
40.f | even | 2 | 1 | 1216.2.a.o | 1 | ||
55.d | odd | 2 | 1 | 2299.2.a.b | 1 | ||
60.h | even | 2 | 1 | 2736.2.a.c | 1 | ||
65.d | even | 2 | 1 | 3211.2.a.a | 1 | ||
85.c | even | 2 | 1 | 5491.2.a.b | 1 | ||
95.d | odd | 2 | 1 | 361.2.a.b | 1 | ||
95.h | odd | 6 | 2 | 361.2.c.a | 2 | ||
95.i | even | 6 | 2 | 361.2.c.c | 2 | ||
95.o | odd | 18 | 6 | 361.2.e.e | 6 | ||
95.p | even | 18 | 6 | 361.2.e.d | 6 | ||
105.g | even | 2 | 1 | 8379.2.a.j | 1 | ||
285.b | even | 2 | 1 | 3249.2.a.d | 1 | ||
380.d | even | 2 | 1 | 5776.2.a.c | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
19.2.a.a | ✓ | 1 | 5.b | even | 2 | 1 | |
171.2.a.b | 1 | 15.d | odd | 2 | 1 | ||
304.2.a.f | 1 | 20.d | odd | 2 | 1 | ||
361.2.a.b | 1 | 95.d | odd | 2 | 1 | ||
361.2.c.a | 2 | 95.h | odd | 6 | 2 | ||
361.2.c.c | 2 | 95.i | even | 6 | 2 | ||
361.2.e.d | 6 | 95.p | even | 18 | 6 | ||
361.2.e.e | 6 | 95.o | odd | 18 | 6 | ||
475.2.a.b | 1 | 1.a | even | 1 | 1 | trivial | |
475.2.b.a | 2 | 5.c | odd | 4 | 2 | ||
931.2.a.a | 1 | 35.c | odd | 2 | 1 | ||
931.2.f.b | 2 | 35.i | odd | 6 | 2 | ||
931.2.f.c | 2 | 35.j | even | 6 | 2 | ||
1216.2.a.b | 1 | 40.e | odd | 2 | 1 | ||
1216.2.a.o | 1 | 40.f | even | 2 | 1 | ||
2299.2.a.b | 1 | 55.d | odd | 2 | 1 | ||
2736.2.a.c | 1 | 60.h | even | 2 | 1 | ||
3211.2.a.a | 1 | 65.d | even | 2 | 1 | ||
3249.2.a.d | 1 | 285.b | even | 2 | 1 | ||
4275.2.a.i | 1 | 3.b | odd | 2 | 1 | ||
5491.2.a.b | 1 | 85.c | even | 2 | 1 | ||
5776.2.a.c | 1 | 380.d | even | 2 | 1 | ||
7600.2.a.c | 1 | 4.b | odd | 2 | 1 | ||
8379.2.a.j | 1 | 105.g | even | 2 | 1 | ||
9025.2.a.d | 1 | 19.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(475))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T - 2 \)
$5$
\( T \)
$7$
\( T - 1 \)
$11$
\( T - 3 \)
$13$
\( T - 4 \)
$17$
\( T - 3 \)
$19$
\( T - 1 \)
$23$
\( T \)
$29$
\( T - 6 \)
$31$
\( T + 4 \)
$37$
\( T + 2 \)
$41$
\( T + 6 \)
$43$
\( T - 1 \)
$47$
\( T - 3 \)
$53$
\( T + 12 \)
$59$
\( T + 6 \)
$61$
\( T + 1 \)
$67$
\( T - 4 \)
$71$
\( T - 6 \)
$73$
\( T - 7 \)
$79$
\( T - 8 \)
$83$
\( T + 12 \)
$89$
\( T - 12 \)
$97$
\( T + 8 \)
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