Newspace parameters
Level: | \( N \) | \(=\) | \( 392 = 2^{3} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 392.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(3.13013575923\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 56) |
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 | 1.00000 | 0 | 1.00000 | 0 | 0 | 0 | −2.00000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(7\) | \(-1\) |
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 | 392.2.a.e | 1 | |
3.b | odd | 2 | 1 | 3528.2.a.j | 1 | ||
4.b | odd | 2 | 1 | 784.2.a.c | 1 | ||
5.b | even | 2 | 1 | 9800.2.a.s | 1 | ||
7.b | odd | 2 | 1 | 392.2.a.c | 1 | ||
7.c | even | 3 | 2 | 392.2.i.b | 2 | ||
7.d | odd | 6 | 2 | 56.2.i.b | ✓ | 2 | |
8.b | even | 2 | 1 | 3136.2.a.i | 1 | ||
8.d | odd | 2 | 1 | 3136.2.a.t | 1 | ||
12.b | even | 2 | 1 | 7056.2.a.u | 1 | ||
21.c | even | 2 | 1 | 3528.2.a.p | 1 | ||
21.g | even | 6 | 2 | 504.2.s.c | 2 | ||
21.h | odd | 6 | 2 | 3528.2.s.q | 2 | ||
28.d | even | 2 | 1 | 784.2.a.h | 1 | ||
28.f | even | 6 | 2 | 112.2.i.a | 2 | ||
28.g | odd | 6 | 2 | 784.2.i.h | 2 | ||
35.c | odd | 2 | 1 | 9800.2.a.be | 1 | ||
35.i | odd | 6 | 2 | 1400.2.q.d | 2 | ||
35.k | even | 12 | 4 | 1400.2.bh.a | 4 | ||
56.e | even | 2 | 1 | 3136.2.a.j | 1 | ||
56.h | odd | 2 | 1 | 3136.2.a.u | 1 | ||
56.j | odd | 6 | 2 | 448.2.i.b | 2 | ||
56.m | even | 6 | 2 | 448.2.i.d | 2 | ||
84.h | odd | 2 | 1 | 7056.2.a.bj | 1 | ||
84.j | odd | 6 | 2 | 1008.2.s.g | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
56.2.i.b | ✓ | 2 | 7.d | odd | 6 | 2 | |
112.2.i.a | 2 | 28.f | even | 6 | 2 | ||
392.2.a.c | 1 | 7.b | odd | 2 | 1 | ||
392.2.a.e | 1 | 1.a | even | 1 | 1 | trivial | |
392.2.i.b | 2 | 7.c | even | 3 | 2 | ||
448.2.i.b | 2 | 56.j | odd | 6 | 2 | ||
448.2.i.d | 2 | 56.m | even | 6 | 2 | ||
504.2.s.c | 2 | 21.g | even | 6 | 2 | ||
784.2.a.c | 1 | 4.b | odd | 2 | 1 | ||
784.2.a.h | 1 | 28.d | even | 2 | 1 | ||
784.2.i.h | 2 | 28.g | odd | 6 | 2 | ||
1008.2.s.g | 2 | 84.j | odd | 6 | 2 | ||
1400.2.q.d | 2 | 35.i | odd | 6 | 2 | ||
1400.2.bh.a | 4 | 35.k | even | 12 | 4 | ||
3136.2.a.i | 1 | 8.b | even | 2 | 1 | ||
3136.2.a.j | 1 | 56.e | even | 2 | 1 | ||
3136.2.a.t | 1 | 8.d | odd | 2 | 1 | ||
3136.2.a.u | 1 | 56.h | odd | 2 | 1 | ||
3528.2.a.j | 1 | 3.b | odd | 2 | 1 | ||
3528.2.a.p | 1 | 21.c | even | 2 | 1 | ||
3528.2.s.q | 2 | 21.h | odd | 6 | 2 | ||
7056.2.a.u | 1 | 12.b | even | 2 | 1 | ||
7056.2.a.bj | 1 | 84.h | odd | 2 | 1 | ||
9800.2.a.s | 1 | 5.b | even | 2 | 1 | ||
9800.2.a.be | 1 | 35.c | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(392))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T - 1 \)
$5$
\( T - 1 \)
$7$
\( T \)
$11$
\( T - 3 \)
$13$
\( T - 6 \)
$17$
\( T - 5 \)
$19$
\( T + 1 \)
$23$
\( T + 7 \)
$29$
\( T - 2 \)
$31$
\( T - 5 \)
$37$
\( T - 3 \)
$41$
\( T - 2 \)
$43$
\( T + 4 \)
$47$
\( T + 5 \)
$53$
\( T + 1 \)
$59$
\( T + 15 \)
$61$
\( T - 5 \)
$67$
\( T + 9 \)
$71$
\( T \)
$73$
\( T + 7 \)
$79$
\( T - 1 \)
$83$
\( T + 12 \)
$89$
\( T + 7 \)
$97$
\( T - 2 \)
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