Newspace parameters
Level: | \( N \) | \(=\) | \( 32 = 2^{5} \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 32.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(7.36173067584\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-1}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4i\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).
\(n\) | \(5\) | \(31\) |
\(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 |
|
0 | − | 4.00000i | 0 | 50.0000 | 0 | − | 184.000i | 0 | 713.000 | 0 | ||||||||||||||||||||||
31.2 | 0 | 4.00000i | 0 | 50.0000 | 0 | 184.000i | 0 | 713.000 | 0 | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 32.7.c.a | ✓ | 2 |
3.b | odd | 2 | 1 | 288.7.g.a | 2 | ||
4.b | odd | 2 | 1 | inner | 32.7.c.a | ✓ | 2 |
8.b | even | 2 | 1 | 64.7.c.b | 2 | ||
8.d | odd | 2 | 1 | 64.7.c.b | 2 | ||
12.b | even | 2 | 1 | 288.7.g.a | 2 | ||
16.e | even | 4 | 1 | 256.7.d.a | 2 | ||
16.e | even | 4 | 1 | 256.7.d.c | 2 | ||
16.f | odd | 4 | 1 | 256.7.d.a | 2 | ||
16.f | odd | 4 | 1 | 256.7.d.c | 2 | ||
24.f | even | 2 | 1 | 576.7.g.i | 2 | ||
24.h | odd | 2 | 1 | 576.7.g.i | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
32.7.c.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
32.7.c.a | ✓ | 2 | 4.b | odd | 2 | 1 | inner |
64.7.c.b | 2 | 8.b | even | 2 | 1 | ||
64.7.c.b | 2 | 8.d | odd | 2 | 1 | ||
256.7.d.a | 2 | 16.e | even | 4 | 1 | ||
256.7.d.a | 2 | 16.f | odd | 4 | 1 | ||
256.7.d.c | 2 | 16.e | even | 4 | 1 | ||
256.7.d.c | 2 | 16.f | odd | 4 | 1 | ||
288.7.g.a | 2 | 3.b | odd | 2 | 1 | ||
288.7.g.a | 2 | 12.b | even | 2 | 1 | ||
576.7.g.i | 2 | 24.f | even | 2 | 1 | ||
576.7.g.i | 2 | 24.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 16 \)
acting on \(S_{7}^{\mathrm{new}}(32, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} + 16 \)
$5$
\( (T - 50)^{2} \)
$7$
\( T^{2} + 33856 \)
$11$
\( T^{2} + 4443664 \)
$13$
\( (T - 2242)^{2} \)
$17$
\( (T - 2898)^{2} \)
$19$
\( T^{2} + 64834704 \)
$23$
\( T^{2} + 345365056 \)
$29$
\( (T + 20990)^{2} \)
$31$
\( T^{2} + 2197734400 \)
$37$
\( (T - 402)^{2} \)
$41$
\( (T - 13330)^{2} \)
$43$
\( T^{2} + 35760400 \)
$47$
\( T^{2} + 20962406656 \)
$53$
\( (T - 171570)^{2} \)
$59$
\( T^{2} + 21702003856 \)
$61$
\( (T + 270878)^{2} \)
$67$
\( T^{2} + 186288918544 \)
$71$
\( T^{2} + 1493667904 \)
$73$
\( (T + 696606)^{2} \)
$79$
\( T^{2} + 449028649216 \)
$83$
\( T^{2} + 90781690000 \)
$89$
\( (T + 161598)^{2} \)
$97$
\( (T - 520306)^{2} \)
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