Newspace parameters
Level: | \( N \) | \(=\) | \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 3640.gq (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.81659664598\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{6}\) |
Projective field: | Galois closure of 6.2.7131795944000.1 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3640\mathbb{Z}\right)^\times\).
\(n\) | \(521\) | \(561\) | \(911\) | \(1457\) | \(1821\) |
\(\chi(n)\) | \(-\zeta_{12}^{2}\) | \(\zeta_{12}^{2}\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
459.1 |
|
− | 1.00000i | 0 | −1.00000 | 0.866025 | − | 0.500000i | 0 | − | 1.00000i | 1.00000i | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | ||||||||||||||||||||||
459.2 | 1.00000i | 0 | −1.00000 | −0.866025 | + | 0.500000i | 0 | 1.00000i | − | 1.00000i | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | ||||||||||||||||||||||||
3299.1 | − | 1.00000i | 0 | −1.00000 | −0.866025 | − | 0.500000i | 0 | − | 1.00000i | 1.00000i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | |||||||||||||||||||||||
3299.2 | 1.00000i | 0 | −1.00000 | 0.866025 | + | 0.500000i | 0 | 1.00000i | − | 1.00000i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | ||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
40.e | odd | 2 | 1 | CM by \(\Q(\sqrt{-10}) \) |
5.b | even | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
91.k | even | 6 | 1 | inner |
455.bz | even | 6 | 1 | inner |
728.br | odd | 6 | 1 | inner |
3640.gq | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3640.1.gq.b | yes | 4 |
5.b | even | 2 | 1 | inner | 3640.1.gq.b | yes | 4 |
7.c | even | 3 | 1 | 3640.1.dk.a | ✓ | 4 | |
8.d | odd | 2 | 1 | inner | 3640.1.gq.b | yes | 4 |
13.e | even | 6 | 1 | 3640.1.dk.a | ✓ | 4 | |
35.j | even | 6 | 1 | 3640.1.dk.a | ✓ | 4 | |
40.e | odd | 2 | 1 | CM | 3640.1.gq.b | yes | 4 |
56.k | odd | 6 | 1 | 3640.1.dk.a | ✓ | 4 | |
65.l | even | 6 | 1 | 3640.1.dk.a | ✓ | 4 | |
91.k | even | 6 | 1 | inner | 3640.1.gq.b | yes | 4 |
104.p | odd | 6 | 1 | 3640.1.dk.a | ✓ | 4 | |
280.bi | odd | 6 | 1 | 3640.1.dk.a | ✓ | 4 | |
455.bz | even | 6 | 1 | inner | 3640.1.gq.b | yes | 4 |
520.cd | odd | 6 | 1 | 3640.1.dk.a | ✓ | 4 | |
728.br | odd | 6 | 1 | inner | 3640.1.gq.b | yes | 4 |
3640.gq | odd | 6 | 1 | inner | 3640.1.gq.b | yes | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3640.1.dk.a | ✓ | 4 | 7.c | even | 3 | 1 | |
3640.1.dk.a | ✓ | 4 | 13.e | even | 6 | 1 | |
3640.1.dk.a | ✓ | 4 | 35.j | even | 6 | 1 | |
3640.1.dk.a | ✓ | 4 | 56.k | odd | 6 | 1 | |
3640.1.dk.a | ✓ | 4 | 65.l | even | 6 | 1 | |
3640.1.dk.a | ✓ | 4 | 104.p | odd | 6 | 1 | |
3640.1.dk.a | ✓ | 4 | 280.bi | odd | 6 | 1 | |
3640.1.dk.a | ✓ | 4 | 520.cd | odd | 6 | 1 | |
3640.1.gq.b | yes | 4 | 1.a | even | 1 | 1 | trivial |
3640.1.gq.b | yes | 4 | 5.b | even | 2 | 1 | inner |
3640.1.gq.b | yes | 4 | 8.d | odd | 2 | 1 | inner |
3640.1.gq.b | yes | 4 | 40.e | odd | 2 | 1 | CM |
3640.1.gq.b | yes | 4 | 91.k | even | 6 | 1 | inner |
3640.1.gq.b | yes | 4 | 455.bz | even | 6 | 1 | inner |
3640.1.gq.b | yes | 4 | 728.br | odd | 6 | 1 | inner |
3640.1.gq.b | yes | 4 | 3640.gq | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11} \)
acting on \(S_{1}^{\mathrm{new}}(3640, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 1)^{2} \)
$3$
\( T^{4} \)
$5$
\( T^{4} - T^{2} + 1 \)
$7$
\( (T^{2} + 1)^{2} \)
$11$
\( T^{4} \)
$13$
\( (T^{2} + 1)^{2} \)
$17$
\( T^{4} \)
$19$
\( (T^{2} - 3 T + 3)^{2} \)
$23$
\( (T^{2} - 3)^{2} \)
$29$
\( T^{4} \)
$31$
\( T^{4} \)
$37$
\( (T^{2} + 1)^{2} \)
$41$
\( (T^{2} + 3 T + 3)^{2} \)
$43$
\( T^{4} \)
$47$
\( T^{4} - T^{2} + 1 \)
$53$
\( T^{4} \)
$59$
\( T^{4} \)
$61$
\( T^{4} \)
$67$
\( T^{4} \)
$71$
\( T^{4} \)
$73$
\( T^{4} \)
$79$
\( T^{4} \)
$83$
\( T^{4} \)
$89$
\( (T^{2} + 3)^{2} \)
$97$
\( T^{4} \)
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