Newspace parameters
Level: | \( N \) | \(=\) | \( 112 = 2^{4} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 112.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(17.9629878191\) |
Analytic rank: | \(1\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 7) |
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 | 14.0000 | 0 | −56.0000 | 0 | 49.0000 | 0 | −47.0000 | 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 | 112.6.a.g | 1 | |
3.b | odd | 2 | 1 | 1008.6.a.y | 1 | ||
4.b | odd | 2 | 1 | 7.6.a.a | ✓ | 1 | |
7.b | odd | 2 | 1 | 784.6.a.c | 1 | ||
8.b | even | 2 | 1 | 448.6.a.c | 1 | ||
8.d | odd | 2 | 1 | 448.6.a.m | 1 | ||
12.b | even | 2 | 1 | 63.6.a.e | 1 | ||
20.d | odd | 2 | 1 | 175.6.a.b | 1 | ||
20.e | even | 4 | 2 | 175.6.b.a | 2 | ||
28.d | even | 2 | 1 | 49.6.a.a | 1 | ||
28.f | even | 6 | 2 | 49.6.c.b | 2 | ||
28.g | odd | 6 | 2 | 49.6.c.c | 2 | ||
44.c | even | 2 | 1 | 847.6.a.b | 1 | ||
84.h | odd | 2 | 1 | 441.6.a.k | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
7.6.a.a | ✓ | 1 | 4.b | odd | 2 | 1 | |
49.6.a.a | 1 | 28.d | even | 2 | 1 | ||
49.6.c.b | 2 | 28.f | even | 6 | 2 | ||
49.6.c.c | 2 | 28.g | odd | 6 | 2 | ||
63.6.a.e | 1 | 12.b | even | 2 | 1 | ||
112.6.a.g | 1 | 1.a | even | 1 | 1 | trivial | |
175.6.a.b | 1 | 20.d | odd | 2 | 1 | ||
175.6.b.a | 2 | 20.e | even | 4 | 2 | ||
441.6.a.k | 1 | 84.h | odd | 2 | 1 | ||
448.6.a.c | 1 | 8.b | even | 2 | 1 | ||
448.6.a.m | 1 | 8.d | odd | 2 | 1 | ||
784.6.a.c | 1 | 7.b | odd | 2 | 1 | ||
847.6.a.b | 1 | 44.c | even | 2 | 1 | ||
1008.6.a.y | 1 | 3.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} - 14 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(112))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T - 14 \)
$5$
\( T + 56 \)
$7$
\( T - 49 \)
$11$
\( T + 232 \)
$13$
\( T + 140 \)
$17$
\( T + 1722 \)
$19$
\( T - 98 \)
$23$
\( T + 1824 \)
$29$
\( T - 3418 \)
$31$
\( T - 7644 \)
$37$
\( T + 10398 \)
$41$
\( T + 17962 \)
$43$
\( T + 10880 \)
$47$
\( T + 9324 \)
$53$
\( T - 2262 \)
$59$
\( T - 2730 \)
$61$
\( T - 25648 \)
$67$
\( T - 48404 \)
$71$
\( T - 58560 \)
$73$
\( T - 68082 \)
$79$
\( T + 31784 \)
$83$
\( T - 20538 \)
$89$
\( T + 50582 \)
$97$
\( T + 58506 \)
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