Newspace parameters
Level: | \( N \) | \(=\) | \( 240 = 2^{4} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 240.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(74.9724061162\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 30) |
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 | 27.0000 | 0 | −125.000 | 0 | 988.000 | 0 | 729.000 | 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 | 240.8.a.j | 1 | |
4.b | odd | 2 | 1 | 30.8.a.d | ✓ | 1 | |
12.b | even | 2 | 1 | 90.8.a.c | 1 | ||
20.d | odd | 2 | 1 | 150.8.a.i | 1 | ||
20.e | even | 4 | 2 | 150.8.c.a | 2 | ||
60.h | even | 2 | 1 | 450.8.a.x | 1 | ||
60.l | odd | 4 | 2 | 450.8.c.r | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
30.8.a.d | ✓ | 1 | 4.b | odd | 2 | 1 | |
90.8.a.c | 1 | 12.b | even | 2 | 1 | ||
150.8.a.i | 1 | 20.d | odd | 2 | 1 | ||
150.8.c.a | 2 | 20.e | even | 4 | 2 | ||
240.8.a.j | 1 | 1.a | even | 1 | 1 | trivial | |
450.8.a.x | 1 | 60.h | even | 2 | 1 | ||
450.8.c.r | 2 | 60.l | odd | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7} - 988 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(240))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T - 27 \)
$5$
\( T + 125 \)
$7$
\( T - 988 \)
$11$
\( T - 8040 \)
$13$
\( T + 3334 \)
$17$
\( T - 6582 \)
$19$
\( T - 27436 \)
$23$
\( T + 48600 \)
$29$
\( T + 132414 \)
$31$
\( T + 254408 \)
$37$
\( T - 519434 \)
$41$
\( T - 92394 \)
$43$
\( T - 234532 \)
$47$
\( T - 1277640 \)
$53$
\( T + 835278 \)
$59$
\( T - 3068760 \)
$61$
\( T + 1009330 \)
$67$
\( T + 3082172 \)
$71$
\( T - 3666720 \)
$73$
\( T - 1122866 \)
$79$
\( T - 4128808 \)
$83$
\( T + 4586556 \)
$89$
\( T + 5763678 \)
$97$
\( T - 6747554 \)
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