Newspace parameters
Level: | \( N \) | \(=\) | \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2520.fw (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.25764383184\) |
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: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{12}\) |
Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times\).
\(n\) | \(281\) | \(631\) | \(1081\) | \(1261\) | \(2017\) |
\(\chi(n)\) | \(-\zeta_{24}^{4}\) | \(1\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
349.1 |
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−0.866025 | − | 0.500000i | −0.707107 | + | 0.707107i | 0.500000 | + | 0.866025i | 0.707107 | + | 0.707107i | 0.965926 | − | 0.258819i | 0.866025 | + | 0.500000i | − | 1.00000i | − | 1.00000i | −0.258819 | − | 0.965926i | ||||||||||||||||||||||||||
349.2 | −0.866025 | − | 0.500000i | 0.707107 | − | 0.707107i | 0.500000 | + | 0.866025i | −0.707107 | − | 0.707107i | −0.965926 | + | 0.258819i | 0.866025 | + | 0.500000i | − | 1.00000i | − | 1.00000i | 0.258819 | + | 0.965926i | |||||||||||||||||||||||||||
349.3 | 0.866025 | + | 0.500000i | −0.707107 | − | 0.707107i | 0.500000 | + | 0.866025i | 0.707107 | − | 0.707107i | −0.258819 | − | 0.965926i | −0.866025 | − | 0.500000i | 1.00000i | 1.00000i | 0.965926 | − | 0.258819i | |||||||||||||||||||||||||||||
349.4 | 0.866025 | + | 0.500000i | 0.707107 | + | 0.707107i | 0.500000 | + | 0.866025i | −0.707107 | + | 0.707107i | 0.258819 | + | 0.965926i | −0.866025 | − | 0.500000i | 1.00000i | 1.00000i | −0.965926 | + | 0.258819i | |||||||||||||||||||||||||||||
2029.1 | −0.866025 | + | 0.500000i | −0.707107 | − | 0.707107i | 0.500000 | − | 0.866025i | 0.707107 | − | 0.707107i | 0.965926 | + | 0.258819i | 0.866025 | − | 0.500000i | 1.00000i | 1.00000i | −0.258819 | + | 0.965926i | |||||||||||||||||||||||||||||
2029.2 | −0.866025 | + | 0.500000i | 0.707107 | + | 0.707107i | 0.500000 | − | 0.866025i | −0.707107 | + | 0.707107i | −0.965926 | − | 0.258819i | 0.866025 | − | 0.500000i | 1.00000i | 1.00000i | 0.258819 | − | 0.965926i | |||||||||||||||||||||||||||||
2029.3 | 0.866025 | − | 0.500000i | −0.707107 | + | 0.707107i | 0.500000 | − | 0.866025i | 0.707107 | + | 0.707107i | −0.258819 | + | 0.965926i | −0.866025 | + | 0.500000i | − | 1.00000i | − | 1.00000i | 0.965926 | + | 0.258819i | |||||||||||||||||||||||||||
2029.4 | 0.866025 | − | 0.500000i | 0.707107 | − | 0.707107i | 0.500000 | − | 0.866025i | −0.707107 | − | 0.707107i | 0.258819 | − | 0.965926i | −0.866025 | + | 0.500000i | − | 1.00000i | − | 1.00000i | −0.965926 | − | 0.258819i | |||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
56.h | odd | 2 | 1 | CM by \(\Q(\sqrt{-14}) \) |
7.b | odd | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
45.j | even | 6 | 1 | inner |
315.bg | odd | 6 | 1 | inner |
360.bk | even | 6 | 1 | inner |
2520.fw | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2520.1.fw.a | ✓ | 8 |
5.b | even | 2 | 1 | 2520.1.fw.b | yes | 8 | |
7.b | odd | 2 | 1 | inner | 2520.1.fw.a | ✓ | 8 |
8.b | even | 2 | 1 | inner | 2520.1.fw.a | ✓ | 8 |
9.c | even | 3 | 1 | 2520.1.fw.b | yes | 8 | |
35.c | odd | 2 | 1 | 2520.1.fw.b | yes | 8 | |
40.f | even | 2 | 1 | 2520.1.fw.b | yes | 8 | |
45.j | even | 6 | 1 | inner | 2520.1.fw.a | ✓ | 8 |
56.h | odd | 2 | 1 | CM | 2520.1.fw.a | ✓ | 8 |
63.l | odd | 6 | 1 | 2520.1.fw.b | yes | 8 | |
72.n | even | 6 | 1 | 2520.1.fw.b | yes | 8 | |
280.c | odd | 2 | 1 | 2520.1.fw.b | yes | 8 | |
315.bg | odd | 6 | 1 | inner | 2520.1.fw.a | ✓ | 8 |
360.bk | even | 6 | 1 | inner | 2520.1.fw.a | ✓ | 8 |
504.bn | odd | 6 | 1 | 2520.1.fw.b | yes | 8 | |
2520.fw | odd | 6 | 1 | inner | 2520.1.fw.a | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2520.1.fw.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
2520.1.fw.a | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
2520.1.fw.a | ✓ | 8 | 8.b | even | 2 | 1 | inner |
2520.1.fw.a | ✓ | 8 | 45.j | even | 6 | 1 | inner |
2520.1.fw.a | ✓ | 8 | 56.h | odd | 2 | 1 | CM |
2520.1.fw.a | ✓ | 8 | 315.bg | odd | 6 | 1 | inner |
2520.1.fw.a | ✓ | 8 | 360.bk | even | 6 | 1 | inner |
2520.1.fw.a | ✓ | 8 | 2520.fw | odd | 6 | 1 | inner |
2520.1.fw.b | yes | 8 | 5.b | even | 2 | 1 | |
2520.1.fw.b | yes | 8 | 9.c | even | 3 | 1 | |
2520.1.fw.b | yes | 8 | 35.c | odd | 2 | 1 | |
2520.1.fw.b | yes | 8 | 40.f | even | 2 | 1 | |
2520.1.fw.b | yes | 8 | 63.l | odd | 6 | 1 | |
2520.1.fw.b | yes | 8 | 72.n | even | 6 | 1 | |
2520.1.fw.b | yes | 8 | 280.c | odd | 2 | 1 | |
2520.1.fw.b | yes | 8 | 504.bn | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{23}^{2} - 3T_{23} + 3 \)
acting on \(S_{1}^{\mathrm{new}}(2520, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - T^{2} + 1)^{2} \)
$3$
\( (T^{4} + 1)^{2} \)
$5$
\( (T^{4} + 1)^{2} \)
$7$
\( (T^{4} - T^{2} + 1)^{2} \)
$11$
\( T^{8} \)
$13$
\( (T^{4} - 2 T^{2} + 4)^{2} \)
$17$
\( T^{8} \)
$19$
\( (T^{4} - 4 T^{2} + 1)^{2} \)
$23$
\( (T^{2} - 3 T + 3)^{4} \)
$29$
\( T^{8} \)
$31$
\( T^{8} \)
$37$
\( T^{8} \)
$41$
\( T^{8} \)
$43$
\( T^{8} \)
$47$
\( T^{8} \)
$53$
\( T^{8} \)
$59$
\( (T^{4} + 2 T^{2} + 4)^{2} \)
$61$
\( T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1 \)
$67$
\( T^{8} \)
$71$
\( (T^{2} - 3)^{4} \)
$73$
\( T^{8} \)
$79$
\( (T^{4} + 3 T^{2} + 9)^{2} \)
$83$
\( (T^{4} - 2 T^{2} + 4)^{2} \)
$89$
\( T^{8} \)
$97$
\( T^{8} \)
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