Newspace parameters
Level: | \( N \) | \(=\) | \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 3800.cq (of order \(18\), degree \(6\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.89644704801\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{18})\) |
Coefficient field: | \(\Q(\zeta_{36})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} - x^{6} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{9}\) |
Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).
\(n\) | \(401\) | \(951\) | \(1901\) | \(1977\) |
\(\chi(n)\) | \(-\zeta_{36}^{2}\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
99.1 |
|
−0.342020 | + | 0.939693i | −1.50881 | + | 0.266044i | −0.766044 | − | 0.642788i | 0 | 0.266044 | − | 1.50881i | 0 | 0.866025 | − | 0.500000i | 1.26604 | − | 0.460802i | 0 | ||||||||||||||||||||||||||||||||||||||||||
99.2 | 0.342020 | − | 0.939693i | 1.50881 | − | 0.266044i | −0.766044 | − | 0.642788i | 0 | 0.266044 | − | 1.50881i | 0 | −0.866025 | + | 0.500000i | 1.26604 | − | 0.460802i | 0 | |||||||||||||||||||||||||||||||||||||||||||
499.1 | −0.342020 | − | 0.939693i | −1.50881 | − | 0.266044i | −0.766044 | + | 0.642788i | 0 | 0.266044 | + | 1.50881i | 0 | 0.866025 | + | 0.500000i | 1.26604 | + | 0.460802i | 0 | |||||||||||||||||||||||||||||||||||||||||||
499.2 | 0.342020 | + | 0.939693i | 1.50881 | + | 0.266044i | −0.766044 | + | 0.642788i | 0 | 0.266044 | + | 1.50881i | 0 | −0.866025 | − | 0.500000i | 1.26604 | + | 0.460802i | 0 | |||||||||||||||||||||||||||||||||||||||||||
899.1 | −0.642788 | − | 0.766044i | 0.118782 | − | 0.326352i | −0.173648 | + | 0.984808i | 0 | −0.326352 | + | 0.118782i | 0 | 0.866025 | − | 0.500000i | 0.673648 | + | 0.565258i | 0 | |||||||||||||||||||||||||||||||||||||||||||
899.2 | 0.642788 | + | 0.766044i | −0.118782 | + | 0.326352i | −0.173648 | + | 0.984808i | 0 | −0.326352 | + | 0.118782i | 0 | −0.866025 | + | 0.500000i | 0.673648 | + | 0.565258i | 0 | |||||||||||||||||||||||||||||||||||||||||||
1099.1 | −0.642788 | + | 0.766044i | 0.118782 | + | 0.326352i | −0.173648 | − | 0.984808i | 0 | −0.326352 | − | 0.118782i | 0 | 0.866025 | + | 0.500000i | 0.673648 | − | 0.565258i | 0 | |||||||||||||||||||||||||||||||||||||||||||
1099.2 | 0.642788 | − | 0.766044i | −0.118782 | − | 0.326352i | −0.173648 | − | 0.984808i | 0 | −0.326352 | − | 0.118782i | 0 | −0.866025 | − | 0.500000i | 0.673648 | − | 0.565258i | 0 | |||||||||||||||||||||||||||||||||||||||||||
1499.1 | −0.984808 | − | 0.173648i | 1.20805 | − | 1.43969i | 0.939693 | + | 0.342020i | 0 | −1.43969 | + | 1.20805i | 0 | −0.866025 | − | 0.500000i | −0.439693 | − | 2.49362i | 0 | |||||||||||||||||||||||||||||||||||||||||||
1499.2 | 0.984808 | + | 0.173648i | −1.20805 | + | 1.43969i | 0.939693 | + | 0.342020i | 0 | −1.43969 | + | 1.20805i | 0 | 0.866025 | + | 0.500000i | −0.439693 | − | 2.49362i | 0 | |||||||||||||||||||||||||||||||||||||||||||
2099.1 | −0.984808 | + | 0.173648i | 1.20805 | + | 1.43969i | 0.939693 | − | 0.342020i | 0 | −1.43969 | − | 1.20805i | 0 | −0.866025 | + | 0.500000i | −0.439693 | + | 2.49362i | 0 | |||||||||||||||||||||||||||||||||||||||||||
2099.2 | 0.984808 | − | 0.173648i | −1.20805 | − | 1.43969i | 0.939693 | − | 0.342020i | 0 | −1.43969 | − | 1.20805i | 0 | 0.866025 | − | 0.500000i | −0.439693 | + | 2.49362i | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-2}) \) |
5.b | even | 2 | 1 | inner |
19.e | even | 9 | 1 | inner |
40.e | odd | 2 | 1 | inner |
95.p | even | 18 | 1 | inner |
152.u | odd | 18 | 1 | inner |
760.bz | odd | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3800.1.cq.a | 12 | |
5.b | even | 2 | 1 | inner | 3800.1.cq.a | 12 | |
5.c | odd | 4 | 1 | 3800.1.cv.b | ✓ | 6 | |
5.c | odd | 4 | 1 | 3800.1.cv.d | yes | 6 | |
8.d | odd | 2 | 1 | CM | 3800.1.cq.a | 12 | |
19.e | even | 9 | 1 | inner | 3800.1.cq.a | 12 | |
40.e | odd | 2 | 1 | inner | 3800.1.cq.a | 12 | |
40.k | even | 4 | 1 | 3800.1.cv.b | ✓ | 6 | |
40.k | even | 4 | 1 | 3800.1.cv.d | yes | 6 | |
95.p | even | 18 | 1 | inner | 3800.1.cq.a | 12 | |
95.q | odd | 36 | 1 | 3800.1.cv.b | ✓ | 6 | |
95.q | odd | 36 | 1 | 3800.1.cv.d | yes | 6 | |
152.u | odd | 18 | 1 | inner | 3800.1.cq.a | 12 | |
760.bz | odd | 18 | 1 | inner | 3800.1.cq.a | 12 | |
760.cp | even | 36 | 1 | 3800.1.cv.b | ✓ | 6 | |
760.cp | even | 36 | 1 | 3800.1.cv.d | yes | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3800.1.cq.a | 12 | 1.a | even | 1 | 1 | trivial | |
3800.1.cq.a | 12 | 5.b | even | 2 | 1 | inner | |
3800.1.cq.a | 12 | 8.d | odd | 2 | 1 | CM | |
3800.1.cq.a | 12 | 19.e | even | 9 | 1 | inner | |
3800.1.cq.a | 12 | 40.e | odd | 2 | 1 | inner | |
3800.1.cq.a | 12 | 95.p | even | 18 | 1 | inner | |
3800.1.cq.a | 12 | 152.u | odd | 18 | 1 | inner | |
3800.1.cq.a | 12 | 760.bz | odd | 18 | 1 | inner | |
3800.1.cv.b | ✓ | 6 | 5.c | odd | 4 | 1 | |
3800.1.cv.b | ✓ | 6 | 40.k | even | 4 | 1 | |
3800.1.cv.b | ✓ | 6 | 95.q | odd | 36 | 1 | |
3800.1.cv.b | ✓ | 6 | 760.cp | even | 36 | 1 | |
3800.1.cv.d | yes | 6 | 5.c | odd | 4 | 1 | |
3800.1.cv.d | yes | 6 | 40.k | even | 4 | 1 | |
3800.1.cv.d | yes | 6 | 95.q | odd | 36 | 1 | |
3800.1.cv.d | yes | 6 | 760.cp | even | 36 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} - 3T_{3}^{10} + 12T_{3}^{8} - 46T_{3}^{6} + 60T_{3}^{4} + 12T_{3}^{2} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(3800, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} - T^{6} + 1 \)
$3$
\( T^{12} - 3 T^{10} + 12 T^{8} - 46 T^{6} + \cdots + 1 \)
$5$
\( T^{12} \)
$7$
\( T^{12} \)
$11$
\( (T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + 3 T + 1)^{2} \)
$13$
\( T^{12} \)
$17$
\( T^{12} - 64T^{6} + 4096 \)
$19$
\( (T^{6} - T^{3} + 1)^{2} \)
$23$
\( T^{12} \)
$29$
\( T^{12} \)
$31$
\( T^{12} \)
$37$
\( T^{12} \)
$41$
\( (T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + 3 T^{2} + \cdots + 1)^{2} \)
$43$
\( T^{12} - T^{6} + 1 \)
$47$
\( T^{12} \)
$53$
\( T^{12} \)
$59$
\( (T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + 12 T^{2} + \cdots + 1)^{2} \)
$61$
\( T^{12} \)
$67$
\( T^{12} + 6 T^{10} + 21 T^{8} + 35 T^{6} + \cdots + 1 \)
$71$
\( T^{12} \)
$73$
\( T^{12} - 3 T^{10} - 6 T^{8} + 8 T^{6} + \cdots + 1 \)
$79$
\( T^{12} \)
$83$
\( T^{12} - 6 T^{10} + 27 T^{8} - 52 T^{6} + \cdots + 1 \)
$89$
\( (T^{6} + T^{3} + 1)^{2} \)
$97$
\( T^{12} - 3 T^{10} - 6 T^{8} + 8 T^{6} + \cdots + 1 \)
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