Newspace parameters
Level: | \( N \) | \(=\) | \( 112 = 2^{4} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 112.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(25.7660573654\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-510}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} + 510 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | no (minimal twist has level 7) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-510}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).
\(n\) | \(15\) | \(17\) | \(85\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
97.1 |
|
0 | − | 45.1664i | 0 | − | 45.1664i | 0 | −133.000 | − | 316.165i | 0 | −1311.00 | 0 | ||||||||||||||||||||
97.2 | 0 | 45.1664i | 0 | 45.1664i | 0 | −133.000 | + | 316.165i | 0 | −1311.00 | 0 | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 112.7.c.b | 2 | |
4.b | odd | 2 | 1 | 7.7.b.b | ✓ | 2 | |
7.b | odd | 2 | 1 | inner | 112.7.c.b | 2 | |
8.b | even | 2 | 1 | 448.7.c.c | 2 | ||
8.d | odd | 2 | 1 | 448.7.c.d | 2 | ||
12.b | even | 2 | 1 | 63.7.d.d | 2 | ||
20.d | odd | 2 | 1 | 175.7.d.e | 2 | ||
20.e | even | 4 | 2 | 175.7.c.c | 4 | ||
28.d | even | 2 | 1 | 7.7.b.b | ✓ | 2 | |
28.f | even | 6 | 2 | 49.7.d.d | 4 | ||
28.g | odd | 6 | 2 | 49.7.d.d | 4 | ||
56.e | even | 2 | 1 | 448.7.c.d | 2 | ||
56.h | odd | 2 | 1 | 448.7.c.c | 2 | ||
84.h | odd | 2 | 1 | 63.7.d.d | 2 | ||
140.c | even | 2 | 1 | 175.7.d.e | 2 | ||
140.j | odd | 4 | 2 | 175.7.c.c | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
7.7.b.b | ✓ | 2 | 4.b | odd | 2 | 1 | |
7.7.b.b | ✓ | 2 | 28.d | even | 2 | 1 | |
49.7.d.d | 4 | 28.f | even | 6 | 2 | ||
49.7.d.d | 4 | 28.g | odd | 6 | 2 | ||
63.7.d.d | 2 | 12.b | even | 2 | 1 | ||
63.7.d.d | 2 | 84.h | odd | 2 | 1 | ||
112.7.c.b | 2 | 1.a | even | 1 | 1 | trivial | |
112.7.c.b | 2 | 7.b | odd | 2 | 1 | inner | |
175.7.c.c | 4 | 20.e | even | 4 | 2 | ||
175.7.c.c | 4 | 140.j | odd | 4 | 2 | ||
175.7.d.e | 2 | 20.d | odd | 2 | 1 | ||
175.7.d.e | 2 | 140.c | even | 2 | 1 | ||
448.7.c.c | 2 | 8.b | even | 2 | 1 | ||
448.7.c.c | 2 | 56.h | odd | 2 | 1 | ||
448.7.c.d | 2 | 8.d | odd | 2 | 1 | ||
448.7.c.d | 2 | 56.e | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 2040 \)
acting on \(S_{7}^{\mathrm{new}}(112, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} + 2040 \)
$5$
\( T^{2} + 2040 \)
$7$
\( T^{2} + 266T + 117649 \)
$11$
\( (T + 874)^{2} \)
$13$
\( T^{2} + 4898040 \)
$17$
\( T^{2} + 35544960 \)
$19$
\( T^{2} + 9712440 \)
$23$
\( (T + 4738)^{2} \)
$29$
\( (T - 11146)^{2} \)
$31$
\( T^{2} + 754114560 \)
$37$
\( (T - 3002)^{2} \)
$41$
\( T^{2} + 3311075040 \)
$43$
\( (T + 31418)^{2} \)
$47$
\( T^{2} + 5248544640 \)
$53$
\( (T + 76406)^{2} \)
$59$
\( T^{2} + 12821499960 \)
$61$
\( T^{2} + 75684573240 \)
$67$
\( (T + 495242)^{2} \)
$71$
\( (T - 184406)^{2} \)
$73$
\( T^{2} + 3717900000 \)
$79$
\( (T - 534934)^{2} \)
$83$
\( T^{2} + 511007615160 \)
$89$
\( T^{2} + 396306401760 \)
$97$
\( T^{2} + 663312168960 \)
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