Newspace parameters
Level: | \( N \) | \(=\) | \( 10 = 2 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 12 \) |
Character orbit: | \([\chi]\) | \(=\) | 10.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(7.68343180560\) |
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 |
|
−32.0000 | −12.0000 | 1024.00 | 3125.00 | 384.000 | −14176.0 | −32768.0 | −177003. | −100000. | |||||||||||||||||||||
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 | 10.12.a.a | ✓ | 1 |
3.b | odd | 2 | 1 | 90.12.a.g | 1 | ||
4.b | odd | 2 | 1 | 80.12.a.d | 1 | ||
5.b | even | 2 | 1 | 50.12.a.d | 1 | ||
5.c | odd | 4 | 2 | 50.12.b.c | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
10.12.a.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
50.12.a.d | 1 | 5.b | even | 2 | 1 | ||
50.12.b.c | 2 | 5.c | odd | 4 | 2 | ||
80.12.a.d | 1 | 4.b | odd | 2 | 1 | ||
90.12.a.g | 1 | 3.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} + 12 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(10))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T + 32 \)
$3$
\( T + 12 \)
$5$
\( T - 3125 \)
$7$
\( T + 14176 \)
$11$
\( T + 756348 \)
$13$
\( T + 905482 \)
$17$
\( T - 2803794 \)
$19$
\( T + 5428660 \)
$23$
\( T + 10236672 \)
$29$
\( T + 197498010 \)
$31$
\( T + 44362288 \)
$37$
\( T - 576737054 \)
$41$
\( T - 930058362 \)
$43$
\( T - 1605598988 \)
$47$
\( T + 1803684456 \)
$53$
\( T - 1558674798 \)
$59$
\( T + 9501997020 \)
$61$
\( T - 6736320422 \)
$67$
\( T - 8402906564 \)
$71$
\( T + 4806306168 \)
$73$
\( T - 7462713338 \)
$79$
\( T + 20644540720 \)
$83$
\( T + 68013349212 \)
$89$
\( T - 69871323210 \)
$97$
\( T - 39960952514 \)
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