Newspace parameters
Level: | \( N \) | \(=\) | \( 3672 = 2^{3} \cdot 3^{3} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 3672.df (of order \(24\), degree \(8\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.83256672639\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | \(\Q(\zeta_{24})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - x^{4} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 1224) |
Projective image: | \(D_{24}\) |
Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\) |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3672\mathbb{Z}\right)^\times\).
\(n\) | \(137\) | \(649\) | \(919\) | \(1837\) |
\(\chi(n)\) | \(-\zeta_{24}^{4}\) | \(-\zeta_{24}^{3}\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 |
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−0.965926 | + | 0.258819i | 0 | 0.866025 | − | 0.500000i | 0 | 0 | 0 | −0.707107 | + | 0.707107i | 0 | 0 | ||||||||||||||||||||||||||||||||||||
451.1 | 0.258819 | + | 0.965926i | 0 | −0.866025 | + | 0.500000i | 0 | 0 | 0 | −0.707107 | − | 0.707107i | 0 | 0 | |||||||||||||||||||||||||||||||||||||
739.1 | 0.965926 | + | 0.258819i | 0 | 0.866025 | + | 0.500000i | 0 | 0 | 0 | 0.707107 | + | 0.707107i | 0 | 0 | |||||||||||||||||||||||||||||||||||||
1171.1 | −0.258819 | + | 0.965926i | 0 | −0.866025 | − | 0.500000i | 0 | 0 | 0 | 0.707107 | − | 0.707107i | 0 | 0 | |||||||||||||||||||||||||||||||||||||
1963.1 | −0.258819 | − | 0.965926i | 0 | −0.866025 | + | 0.500000i | 0 | 0 | 0 | 0.707107 | + | 0.707107i | 0 | 0 | |||||||||||||||||||||||||||||||||||||
2395.1 | 0.965926 | − | 0.258819i | 0 | 0.866025 | − | 0.500000i | 0 | 0 | 0 | 0.707107 | − | 0.707107i | 0 | 0 | |||||||||||||||||||||||||||||||||||||
2467.1 | 0.258819 | − | 0.965926i | 0 | −0.866025 | − | 0.500000i | 0 | 0 | 0 | −0.707107 | + | 0.707107i | 0 | 0 | |||||||||||||||||||||||||||||||||||||
2899.1 | −0.965926 | − | 0.258819i | 0 | 0.866025 | + | 0.500000i | 0 | 0 | 0 | −0.707107 | − | 0.707107i | 0 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-2}) \) |
153.r | even | 24 | 1 | inner |
1224.cp | odd | 24 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3672.1.df.b | 8 | |
3.b | odd | 2 | 1 | 1224.1.cp.b | yes | 8 | |
8.d | odd | 2 | 1 | CM | 3672.1.df.b | 8 | |
9.c | even | 3 | 1 | 3672.1.df.a | 8 | ||
9.d | odd | 6 | 1 | 1224.1.cp.a | ✓ | 8 | |
17.d | even | 8 | 1 | 3672.1.df.a | 8 | ||
24.f | even | 2 | 1 | 1224.1.cp.b | yes | 8 | |
51.g | odd | 8 | 1 | 1224.1.cp.a | ✓ | 8 | |
72.l | even | 6 | 1 | 1224.1.cp.a | ✓ | 8 | |
72.p | odd | 6 | 1 | 3672.1.df.a | 8 | ||
136.p | odd | 8 | 1 | 3672.1.df.a | 8 | ||
153.q | odd | 24 | 1 | 1224.1.cp.b | yes | 8 | |
153.r | even | 24 | 1 | inner | 3672.1.df.b | 8 | |
408.bd | even | 8 | 1 | 1224.1.cp.a | ✓ | 8 | |
1224.cn | even | 24 | 1 | 1224.1.cp.b | yes | 8 | |
1224.cp | odd | 24 | 1 | inner | 3672.1.df.b | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1224.1.cp.a | ✓ | 8 | 9.d | odd | 6 | 1 | |
1224.1.cp.a | ✓ | 8 | 51.g | odd | 8 | 1 | |
1224.1.cp.a | ✓ | 8 | 72.l | even | 6 | 1 | |
1224.1.cp.a | ✓ | 8 | 408.bd | even | 8 | 1 | |
1224.1.cp.b | yes | 8 | 3.b | odd | 2 | 1 | |
1224.1.cp.b | yes | 8 | 24.f | even | 2 | 1 | |
1224.1.cp.b | yes | 8 | 153.q | odd | 24 | 1 | |
1224.1.cp.b | yes | 8 | 1224.cn | even | 24 | 1 | |
3672.1.df.a | 8 | 9.c | even | 3 | 1 | ||
3672.1.df.a | 8 | 17.d | even | 8 | 1 | ||
3672.1.df.a | 8 | 72.p | odd | 6 | 1 | ||
3672.1.df.a | 8 | 136.p | odd | 8 | 1 | ||
3672.1.df.b | 8 | 1.a | even | 1 | 1 | trivial | |
3672.1.df.b | 8 | 8.d | odd | 2 | 1 | CM | |
3672.1.df.b | 8 | 153.r | even | 24 | 1 | inner | |
3672.1.df.b | 8 | 1224.cp | odd | 24 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{8} - 8T_{11}^{7} + 28T_{11}^{6} - 56T_{11}^{5} + 69T_{11}^{4} - 52T_{11}^{3} + 22T_{11}^{2} - 4T_{11} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(3672, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} - T^{4} + 1 \)
$3$
\( T^{8} \)
$5$
\( T^{8} \)
$7$
\( T^{8} \)
$11$
\( T^{8} - 8 T^{7} + 28 T^{6} - 56 T^{5} + \cdots + 1 \)
$13$
\( T^{8} \)
$17$
\( T^{8} - T^{4} + 1 \)
$19$
\( (T^{4} + 9)^{2} \)
$23$
\( T^{8} \)
$29$
\( T^{8} \)
$31$
\( T^{8} \)
$37$
\( T^{8} \)
$41$
\( T^{8} - 2 T^{6} + 4 T^{5} + 5 T^{4} + \cdots + 1 \)
$43$
\( (T^{4} + 2 T^{3} + 5 T^{2} + 4 T + 1)^{2} \)
$47$
\( T^{8} \)
$53$
\( T^{8} \)
$59$
\( (T^{4} + 2 T^{3} + 5 T^{2} + 4 T + 1)^{2} \)
$61$
\( T^{8} \)
$67$
\( T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1 \)
$71$
\( T^{8} \)
$73$
\( T^{8} - 2 T^{6} + 4 T^{5} + 2 T^{4} + \cdots + 1 \)
$79$
\( T^{8} \)
$83$
\( (T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4)^{2} \)
$89$
\( (T^{2} + 2)^{4} \)
$97$
\( T^{8} - 4 T^{7} + 10 T^{6} - 16 T^{5} + \cdots + 1 \)
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