Newspace parameters
Level: | \( N \) | \(=\) | \( 30 = 2 \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 30.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(15.4510750849\) |
Analytic rank: | \(1\) |
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 |
|
16.0000 | −81.0000 | 256.000 | 625.000 | −1296.00 | −10336.0 | 4096.00 | 6561.00 | 10000.0 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(3\) | \(1\) |
\(5\) | \(-1\) |
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 | 30.10.a.e | ✓ | 1 |
3.b | odd | 2 | 1 | 90.10.a.a | 1 | ||
4.b | odd | 2 | 1 | 240.10.a.j | 1 | ||
5.b | even | 2 | 1 | 150.10.a.e | 1 | ||
5.c | odd | 4 | 2 | 150.10.c.c | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
30.10.a.e | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
90.10.a.a | 1 | 3.b | odd | 2 | 1 | ||
150.10.a.e | 1 | 5.b | even | 2 | 1 | ||
150.10.c.c | 2 | 5.c | odd | 4 | 2 | ||
240.10.a.j | 1 | 4.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7} + 10336 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(30))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T - 16 \)
$3$
\( T + 81 \)
$5$
\( T - 625 \)
$7$
\( T + 10336 \)
$11$
\( T - 27420 \)
$13$
\( T + 169762 \)
$17$
\( T + 385086 \)
$19$
\( T + 637084 \)
$23$
\( T + 1298400 \)
$29$
\( T - 7162974 \)
$31$
\( T + 7031872 \)
$37$
\( T - 1926038 \)
$41$
\( T - 8896074 \)
$43$
\( T - 32429444 \)
$47$
\( T - 17206440 \)
$53$
\( T + 20642154 \)
$59$
\( T + 63193380 \)
$61$
\( T + 63758050 \)
$67$
\( T - 145261964 \)
$71$
\( T + 367656840 \)
$73$
\( T - 252486218 \)
$79$
\( T + 185523712 \)
$83$
\( T + 467897652 \)
$89$
\( T - 579096378 \)
$97$
\( T + 1314516862 \)
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