Newspace parameters
Level: | \( N \) | \(=\) | \( 304 = 2^{4} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 304.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(156.570894194\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 38) |
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 | 119.000 | 0 | −684.000 | 0 | −9149.00 | 0 | −5522.00 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(19\) | \(1\) |
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 | 304.10.a.b | 1 | |
4.b | odd | 2 | 1 | 38.10.a.a | ✓ | 1 | |
12.b | even | 2 | 1 | 342.10.a.a | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
38.10.a.a | ✓ | 1 | 4.b | odd | 2 | 1 | |
304.10.a.b | 1 | 1.a | even | 1 | 1 | trivial | |
342.10.a.a | 1 | 12.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} - 119 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(304))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T - 119 \)
$5$
\( T + 684 \)
$7$
\( T + 9149 \)
$11$
\( T + 5790 \)
$13$
\( T + 179881 \)
$17$
\( T + 594093 \)
$19$
\( T + 130321 \)
$23$
\( T - 1744767 \)
$29$
\( T - 4314387 \)
$31$
\( T + 160232 \)
$37$
\( T + 21943090 \)
$41$
\( T - 294816 \)
$43$
\( T - 39393148 \)
$47$
\( T + 46596360 \)
$53$
\( T - 22121703 \)
$59$
\( T + 33070233 \)
$61$
\( T - 188535938 \)
$67$
\( T - 20769067 \)
$71$
\( T - 232299978 \)
$73$
\( T + 3022183 \)
$79$
\( T - 446379406 \)
$83$
\( T + 794022846 \)
$89$
\( T - 90999336 \)
$97$
\( T + 123974170 \)
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