Newspace parameters
Level: | \( N \) | \(=\) | \( 2312 = 2^{3} \cdot 17^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 2312.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(136.412415933\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 8) |
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 | 4.00000 | 0 | 2.00000 | 0 | −24.0000 | 0 | −11.0000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(17\) | \(1\) |
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 | 2312.4.a.a | 1 | |
17.b | even | 2 | 1 | 8.4.a.a | ✓ | 1 | |
51.c | odd | 2 | 1 | 72.4.a.c | 1 | ||
68.d | odd | 2 | 1 | 16.4.a.a | 1 | ||
85.c | even | 2 | 1 | 200.4.a.g | 1 | ||
85.g | odd | 4 | 2 | 200.4.c.e | 2 | ||
119.d | odd | 2 | 1 | 392.4.a.e | 1 | ||
119.h | odd | 6 | 2 | 392.4.i.b | 2 | ||
119.j | even | 6 | 2 | 392.4.i.g | 2 | ||
136.e | odd | 2 | 1 | 64.4.a.b | 1 | ||
136.h | even | 2 | 1 | 64.4.a.d | 1 | ||
153.h | even | 6 | 2 | 648.4.i.h | 2 | ||
153.i | odd | 6 | 2 | 648.4.i.e | 2 | ||
187.b | odd | 2 | 1 | 968.4.a.a | 1 | ||
204.h | even | 2 | 1 | 144.4.a.e | 1 | ||
221.b | even | 2 | 1 | 1352.4.a.a | 1 | ||
255.h | odd | 2 | 1 | 1800.4.a.d | 1 | ||
255.o | even | 4 | 2 | 1800.4.f.u | 2 | ||
272.k | odd | 4 | 2 | 256.4.b.g | 2 | ||
272.r | even | 4 | 2 | 256.4.b.a | 2 | ||
340.d | odd | 2 | 1 | 400.4.a.g | 1 | ||
340.r | even | 4 | 2 | 400.4.c.i | 2 | ||
408.b | odd | 2 | 1 | 576.4.a.k | 1 | ||
408.h | even | 2 | 1 | 576.4.a.j | 1 | ||
476.e | even | 2 | 1 | 784.4.a.e | 1 | ||
680.h | even | 2 | 1 | 1600.4.a.o | 1 | ||
680.k | odd | 2 | 1 | 1600.4.a.bm | 1 | ||
748.f | even | 2 | 1 | 1936.4.a.l | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8.4.a.a | ✓ | 1 | 17.b | even | 2 | 1 | |
16.4.a.a | 1 | 68.d | odd | 2 | 1 | ||
64.4.a.b | 1 | 136.e | odd | 2 | 1 | ||
64.4.a.d | 1 | 136.h | even | 2 | 1 | ||
72.4.a.c | 1 | 51.c | odd | 2 | 1 | ||
144.4.a.e | 1 | 204.h | even | 2 | 1 | ||
200.4.a.g | 1 | 85.c | even | 2 | 1 | ||
200.4.c.e | 2 | 85.g | odd | 4 | 2 | ||
256.4.b.a | 2 | 272.r | even | 4 | 2 | ||
256.4.b.g | 2 | 272.k | odd | 4 | 2 | ||
392.4.a.e | 1 | 119.d | odd | 2 | 1 | ||
392.4.i.b | 2 | 119.h | odd | 6 | 2 | ||
392.4.i.g | 2 | 119.j | even | 6 | 2 | ||
400.4.a.g | 1 | 340.d | odd | 2 | 1 | ||
400.4.c.i | 2 | 340.r | even | 4 | 2 | ||
576.4.a.j | 1 | 408.h | even | 2 | 1 | ||
576.4.a.k | 1 | 408.b | odd | 2 | 1 | ||
648.4.i.e | 2 | 153.i | odd | 6 | 2 | ||
648.4.i.h | 2 | 153.h | even | 6 | 2 | ||
784.4.a.e | 1 | 476.e | even | 2 | 1 | ||
968.4.a.a | 1 | 187.b | odd | 2 | 1 | ||
1352.4.a.a | 1 | 221.b | even | 2 | 1 | ||
1600.4.a.o | 1 | 680.h | even | 2 | 1 | ||
1600.4.a.bm | 1 | 680.k | odd | 2 | 1 | ||
1800.4.a.d | 1 | 255.h | odd | 2 | 1 | ||
1800.4.f.u | 2 | 255.o | even | 4 | 2 | ||
1936.4.a.l | 1 | 748.f | even | 2 | 1 | ||
2312.4.a.a | 1 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} - 4 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2312))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T - 4 \)
$5$
\( T - 2 \)
$7$
\( T + 24 \)
$11$
\( T - 44 \)
$13$
\( T - 22 \)
$17$
\( T \)
$19$
\( T - 44 \)
$23$
\( T - 56 \)
$29$
\( T + 198 \)
$31$
\( T - 160 \)
$37$
\( T - 162 \)
$41$
\( T - 198 \)
$43$
\( T - 52 \)
$47$
\( T - 528 \)
$53$
\( T + 242 \)
$59$
\( T + 668 \)
$61$
\( T + 550 \)
$67$
\( T - 188 \)
$71$
\( T + 728 \)
$73$
\( T + 154 \)
$79$
\( T - 656 \)
$83$
\( T - 236 \)
$89$
\( T - 714 \)
$97$
\( T - 478 \)
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