Newspace parameters
Level: | \( N \) | \(=\) | \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 980.p (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.489083712380\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
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: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{4}\) |
Projective field: | Galois closure of 4.2.137200.1 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(197\) | \(491\) |
\(\chi(n)\) | \(-\zeta_{24}^{4}\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
79.1 |
|
−0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | −0.965926 | + | 0.258819i | 0 | 0 | − | 1.00000i | 0.500000 | − | 0.866025i | 0.965926 | + | 0.258819i | |||||||||||||||||||||||||||||||
79.2 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0.965926 | − | 0.258819i | 0 | 0 | − | 1.00000i | 0.500000 | − | 0.866025i | −0.965926 | − | 0.258819i | ||||||||||||||||||||||||||||||||
79.3 | 0.866025 | + | 0.500000i | 0 | 0.500000 | + | 0.866025i | −0.258819 | − | 0.965926i | 0 | 0 | 1.00000i | 0.500000 | − | 0.866025i | 0.258819 | − | 0.965926i | |||||||||||||||||||||||||||||||||
79.4 | 0.866025 | + | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0.258819 | + | 0.965926i | 0 | 0 | 1.00000i | 0.500000 | − | 0.866025i | −0.258819 | + | 0.965926i | |||||||||||||||||||||||||||||||||
459.1 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | −0.965926 | − | 0.258819i | 0 | 0 | 1.00000i | 0.500000 | + | 0.866025i | 0.965926 | − | 0.258819i | |||||||||||||||||||||||||||||||||
459.2 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0.965926 | + | 0.258819i | 0 | 0 | 1.00000i | 0.500000 | + | 0.866025i | −0.965926 | + | 0.258819i | |||||||||||||||||||||||||||||||||
459.3 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | −0.258819 | + | 0.965926i | 0 | 0 | − | 1.00000i | 0.500000 | + | 0.866025i | 0.258819 | + | 0.965926i | ||||||||||||||||||||||||||||||||
459.4 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0.258819 | − | 0.965926i | 0 | 0 | − | 1.00000i | 0.500000 | + | 0.866025i | −0.258819 | − | 0.965926i | ||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-1}) \) |
5.b | even | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
7.d | odd | 6 | 1 | inner |
20.d | odd | 2 | 1 | inner |
28.d | even | 2 | 1 | inner |
28.f | even | 6 | 1 | inner |
28.g | odd | 6 | 1 | inner |
35.c | odd | 2 | 1 | inner |
35.i | odd | 6 | 1 | inner |
35.j | even | 6 | 1 | inner |
140.c | even | 2 | 1 | inner |
140.p | odd | 6 | 1 | inner |
140.s | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 980.1.p.c | 8 | |
4.b | odd | 2 | 1 | CM | 980.1.p.c | 8 | |
5.b | even | 2 | 1 | inner | 980.1.p.c | 8 | |
7.b | odd | 2 | 1 | inner | 980.1.p.c | 8 | |
7.c | even | 3 | 1 | 980.1.f.e | ✓ | 4 | |
7.c | even | 3 | 1 | inner | 980.1.p.c | 8 | |
7.d | odd | 6 | 1 | 980.1.f.e | ✓ | 4 | |
7.d | odd | 6 | 1 | inner | 980.1.p.c | 8 | |
20.d | odd | 2 | 1 | inner | 980.1.p.c | 8 | |
28.d | even | 2 | 1 | inner | 980.1.p.c | 8 | |
28.f | even | 6 | 1 | 980.1.f.e | ✓ | 4 | |
28.f | even | 6 | 1 | inner | 980.1.p.c | 8 | |
28.g | odd | 6 | 1 | 980.1.f.e | ✓ | 4 | |
28.g | odd | 6 | 1 | inner | 980.1.p.c | 8 | |
35.c | odd | 2 | 1 | inner | 980.1.p.c | 8 | |
35.i | odd | 6 | 1 | 980.1.f.e | ✓ | 4 | |
35.i | odd | 6 | 1 | inner | 980.1.p.c | 8 | |
35.j | even | 6 | 1 | 980.1.f.e | ✓ | 4 | |
35.j | even | 6 | 1 | inner | 980.1.p.c | 8 | |
140.c | even | 2 | 1 | inner | 980.1.p.c | 8 | |
140.p | odd | 6 | 1 | 980.1.f.e | ✓ | 4 | |
140.p | odd | 6 | 1 | inner | 980.1.p.c | 8 | |
140.s | even | 6 | 1 | 980.1.f.e | ✓ | 4 | |
140.s | even | 6 | 1 | inner | 980.1.p.c | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
980.1.f.e | ✓ | 4 | 7.c | even | 3 | 1 | |
980.1.f.e | ✓ | 4 | 7.d | odd | 6 | 1 | |
980.1.f.e | ✓ | 4 | 28.f | even | 6 | 1 | |
980.1.f.e | ✓ | 4 | 28.g | odd | 6 | 1 | |
980.1.f.e | ✓ | 4 | 35.i | odd | 6 | 1 | |
980.1.f.e | ✓ | 4 | 35.j | even | 6 | 1 | |
980.1.f.e | ✓ | 4 | 140.p | odd | 6 | 1 | |
980.1.f.e | ✓ | 4 | 140.s | even | 6 | 1 | |
980.1.p.c | 8 | 1.a | even | 1 | 1 | trivial | |
980.1.p.c | 8 | 4.b | odd | 2 | 1 | CM | |
980.1.p.c | 8 | 5.b | even | 2 | 1 | inner | |
980.1.p.c | 8 | 7.b | odd | 2 | 1 | inner | |
980.1.p.c | 8 | 7.c | even | 3 | 1 | inner | |
980.1.p.c | 8 | 7.d | odd | 6 | 1 | inner | |
980.1.p.c | 8 | 20.d | odd | 2 | 1 | inner | |
980.1.p.c | 8 | 28.d | even | 2 | 1 | inner | |
980.1.p.c | 8 | 28.f | even | 6 | 1 | inner | |
980.1.p.c | 8 | 28.g | odd | 6 | 1 | inner | |
980.1.p.c | 8 | 35.c | odd | 2 | 1 | inner | |
980.1.p.c | 8 | 35.i | odd | 6 | 1 | inner | |
980.1.p.c | 8 | 35.j | even | 6 | 1 | inner | |
980.1.p.c | 8 | 140.c | even | 2 | 1 | inner | |
980.1.p.c | 8 | 140.p | odd | 6 | 1 | inner | |
980.1.p.c | 8 | 140.s | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} \)
acting on \(S_{1}^{\mathrm{new}}(980, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - T^{2} + 1)^{2} \)
$3$
\( T^{8} \)
$5$
\( T^{8} - T^{4} + 1 \)
$7$
\( T^{8} \)
$11$
\( T^{8} \)
$13$
\( (T^{2} + 2)^{4} \)
$17$
\( (T^{4} - 2 T^{2} + 4)^{2} \)
$19$
\( T^{8} \)
$23$
\( T^{8} \)
$29$
\( T^{8} \)
$31$
\( T^{8} \)
$37$
\( (T^{4} - 4 T^{2} + 16)^{2} \)
$41$
\( (T^{2} - 2)^{4} \)
$43$
\( T^{8} \)
$47$
\( T^{8} \)
$53$
\( T^{8} \)
$59$
\( T^{8} \)
$61$
\( (T^{4} + 2 T^{2} + 4)^{2} \)
$67$
\( T^{8} \)
$71$
\( T^{8} \)
$73$
\( (T^{4} - 2 T^{2} + 4)^{2} \)
$79$
\( T^{8} \)
$83$
\( T^{8} \)
$89$
\( (T^{4} + 2 T^{2} + 4)^{2} \)
$97$
\( (T^{2} + 2)^{4} \)
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