Newspace parameters
Level: | \( N \) | \(=\) | \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2520.hf (of order \(12\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.25764383184\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{12})\) |
Coefficient field: | \(\Q(\zeta_{48})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} - x^{8} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
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 \(\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times\).
\(n\) | \(281\) | \(631\) | \(1081\) | \(1261\) | \(2017\) |
\(\chi(n)\) | \(-\zeta_{48}^{16}\) | \(1\) | \(-1\) | \(-1\) | \(-\zeta_{48}^{12}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
293.1 |
|
−0.258819 | + | 0.965926i | −0.382683 | + | 0.923880i | −0.866025 | − | 0.500000i | −0.991445 | + | 0.130526i | −0.793353 | − | 0.608761i | −0.965926 | − | 0.258819i | 0.707107 | − | 0.707107i | −0.707107 | − | 0.707107i | 0.130526 | − | 0.991445i | ||||||||||||||||||||||||||||||||||||||||||||||||
293.2 | −0.258819 | + | 0.965926i | 0.382683 | − | 0.923880i | −0.866025 | − | 0.500000i | 0.991445 | − | 0.130526i | 0.793353 | + | 0.608761i | −0.965926 | − | 0.258819i | 0.707107 | − | 0.707107i | −0.707107 | − | 0.707107i | −0.130526 | + | 0.991445i | |||||||||||||||||||||||||||||||||||||||||||||||||
293.3 | 0.258819 | − | 0.965926i | −0.923880 | − | 0.382683i | −0.866025 | − | 0.500000i | −0.130526 | − | 0.991445i | −0.608761 | + | 0.793353i | 0.965926 | + | 0.258819i | −0.707107 | + | 0.707107i | 0.707107 | + | 0.707107i | −0.991445 | − | 0.130526i | |||||||||||||||||||||||||||||||||||||||||||||||||
293.4 | 0.258819 | − | 0.965926i | 0.923880 | + | 0.382683i | −0.866025 | − | 0.500000i | 0.130526 | + | 0.991445i | 0.608761 | − | 0.793353i | 0.965926 | + | 0.258819i | −0.707107 | + | 0.707107i | 0.707107 | + | 0.707107i | 0.991445 | + | 0.130526i | |||||||||||||||||||||||||||||||||||||||||||||||||
797.1 | −0.965926 | − | 0.258819i | −0.923880 | + | 0.382683i | 0.866025 | + | 0.500000i | −0.793353 | − | 0.608761i | 0.991445 | − | 0.130526i | −0.258819 | + | 0.965926i | −0.707107 | − | 0.707107i | 0.707107 | − | 0.707107i | 0.608761 | + | 0.793353i | |||||||||||||||||||||||||||||||||||||||||||||||||
797.2 | −0.965926 | − | 0.258819i | 0.923880 | − | 0.382683i | 0.866025 | + | 0.500000i | 0.793353 | + | 0.608761i | −0.991445 | + | 0.130526i | −0.258819 | + | 0.965926i | −0.707107 | − | 0.707107i | 0.707107 | − | 0.707107i | −0.608761 | − | 0.793353i | |||||||||||||||||||||||||||||||||||||||||||||||||
797.3 | 0.965926 | + | 0.258819i | −0.382683 | − | 0.923880i | 0.866025 | + | 0.500000i | 0.608761 | − | 0.793353i | −0.130526 | − | 0.991445i | 0.258819 | − | 0.965926i | 0.707107 | + | 0.707107i | −0.707107 | + | 0.707107i | 0.793353 | − | 0.608761i | |||||||||||||||||||||||||||||||||||||||||||||||||
797.4 | 0.965926 | + | 0.258819i | 0.382683 | + | 0.923880i | 0.866025 | + | 0.500000i | −0.608761 | + | 0.793353i | 0.130526 | + | 0.991445i | 0.258819 | − | 0.965926i | 0.707107 | + | 0.707107i | −0.707107 | + | 0.707107i | −0.793353 | + | 0.608761i | |||||||||||||||||||||||||||||||||||||||||||||||||
1973.1 | −0.965926 | + | 0.258819i | −0.923880 | − | 0.382683i | 0.866025 | − | 0.500000i | −0.793353 | + | 0.608761i | 0.991445 | + | 0.130526i | −0.258819 | − | 0.965926i | −0.707107 | + | 0.707107i | 0.707107 | + | 0.707107i | 0.608761 | − | 0.793353i | |||||||||||||||||||||||||||||||||||||||||||||||||
1973.2 | −0.965926 | + | 0.258819i | 0.923880 | + | 0.382683i | 0.866025 | − | 0.500000i | 0.793353 | − | 0.608761i | −0.991445 | − | 0.130526i | −0.258819 | − | 0.965926i | −0.707107 | + | 0.707107i | 0.707107 | + | 0.707107i | −0.608761 | + | 0.793353i | |||||||||||||||||||||||||||||||||||||||||||||||||
1973.3 | 0.965926 | − | 0.258819i | −0.382683 | + | 0.923880i | 0.866025 | − | 0.500000i | 0.608761 | + | 0.793353i | −0.130526 | + | 0.991445i | 0.258819 | + | 0.965926i | 0.707107 | − | 0.707107i | −0.707107 | − | 0.707107i | 0.793353 | + | 0.608761i | |||||||||||||||||||||||||||||||||||||||||||||||||
1973.4 | 0.965926 | − | 0.258819i | 0.382683 | − | 0.923880i | 0.866025 | − | 0.500000i | −0.608761 | − | 0.793353i | 0.130526 | − | 0.991445i | 0.258819 | + | 0.965926i | 0.707107 | − | 0.707107i | −0.707107 | − | 0.707107i | −0.793353 | − | 0.608761i | |||||||||||||||||||||||||||||||||||||||||||||||||
2477.1 | −0.258819 | − | 0.965926i | −0.382683 | − | 0.923880i | −0.866025 | + | 0.500000i | −0.991445 | − | 0.130526i | −0.793353 | + | 0.608761i | −0.965926 | + | 0.258819i | 0.707107 | + | 0.707107i | −0.707107 | + | 0.707107i | 0.130526 | + | 0.991445i | |||||||||||||||||||||||||||||||||||||||||||||||||
2477.2 | −0.258819 | − | 0.965926i | 0.382683 | + | 0.923880i | −0.866025 | + | 0.500000i | 0.991445 | + | 0.130526i | 0.793353 | − | 0.608761i | −0.965926 | + | 0.258819i | 0.707107 | + | 0.707107i | −0.707107 | + | 0.707107i | −0.130526 | − | 0.991445i | |||||||||||||||||||||||||||||||||||||||||||||||||
2477.3 | 0.258819 | + | 0.965926i | −0.923880 | + | 0.382683i | −0.866025 | + | 0.500000i | −0.130526 | + | 0.991445i | −0.608761 | − | 0.793353i | 0.965926 | − | 0.258819i | −0.707107 | − | 0.707107i | 0.707107 | − | 0.707107i | −0.991445 | + | 0.130526i | |||||||||||||||||||||||||||||||||||||||||||||||||
2477.4 | 0.258819 | + | 0.965926i | 0.923880 | − | 0.382683i | −0.866025 | + | 0.500000i | 0.130526 | − | 0.991445i | 0.608761 | + | 0.793353i | 0.965926 | − | 0.258819i | −0.707107 | − | 0.707107i | 0.707107 | − | 0.707107i | 0.991445 | − | 0.130526i | |||||||||||||||||||||||||||||||||||||||||||||||||
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.l | even | 12 | 1 | inner |
315.cf | odd | 12 | 1 | inner |
360.br | even | 12 | 1 | inner |
2520.hf | odd | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2520.1.hf.b | yes | 16 |
5.c | odd | 4 | 1 | 2520.1.hf.a | ✓ | 16 | |
7.b | odd | 2 | 1 | inner | 2520.1.hf.b | yes | 16 |
8.b | even | 2 | 1 | inner | 2520.1.hf.b | yes | 16 |
9.d | odd | 6 | 1 | 2520.1.hf.a | ✓ | 16 | |
35.f | even | 4 | 1 | 2520.1.hf.a | ✓ | 16 | |
40.i | odd | 4 | 1 | 2520.1.hf.a | ✓ | 16 | |
45.l | even | 12 | 1 | inner | 2520.1.hf.b | yes | 16 |
56.h | odd | 2 | 1 | CM | 2520.1.hf.b | yes | 16 |
63.o | even | 6 | 1 | 2520.1.hf.a | ✓ | 16 | |
72.j | odd | 6 | 1 | 2520.1.hf.a | ✓ | 16 | |
280.s | even | 4 | 1 | 2520.1.hf.a | ✓ | 16 | |
315.cf | odd | 12 | 1 | inner | 2520.1.hf.b | yes | 16 |
360.br | even | 12 | 1 | inner | 2520.1.hf.b | yes | 16 |
504.cc | even | 6 | 1 | 2520.1.hf.a | ✓ | 16 | |
2520.hf | odd | 12 | 1 | inner | 2520.1.hf.b | yes | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2520.1.hf.a | ✓ | 16 | 5.c | odd | 4 | 1 | |
2520.1.hf.a | ✓ | 16 | 9.d | odd | 6 | 1 | |
2520.1.hf.a | ✓ | 16 | 35.f | even | 4 | 1 | |
2520.1.hf.a | ✓ | 16 | 40.i | odd | 4 | 1 | |
2520.1.hf.a | ✓ | 16 | 63.o | even | 6 | 1 | |
2520.1.hf.a | ✓ | 16 | 72.j | odd | 6 | 1 | |
2520.1.hf.a | ✓ | 16 | 280.s | even | 4 | 1 | |
2520.1.hf.a | ✓ | 16 | 504.cc | even | 6 | 1 | |
2520.1.hf.b | yes | 16 | 1.a | even | 1 | 1 | trivial |
2520.1.hf.b | yes | 16 | 7.b | odd | 2 | 1 | inner |
2520.1.hf.b | yes | 16 | 8.b | even | 2 | 1 | inner |
2520.1.hf.b | yes | 16 | 45.l | even | 12 | 1 | inner |
2520.1.hf.b | yes | 16 | 56.h | odd | 2 | 1 | CM |
2520.1.hf.b | yes | 16 | 315.cf | odd | 12 | 1 | inner |
2520.1.hf.b | yes | 16 | 360.br | even | 12 | 1 | inner |
2520.1.hf.b | yes | 16 | 2520.hf | odd | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{23}^{4} + 2T_{23}^{3} + 5T_{23}^{2} + 4T_{23} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(2520, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{8} - T^{4} + 1)^{2} \)
$3$
\( (T^{8} + 1)^{2} \)
$5$
\( T^{16} - T^{8} + 1 \)
$7$
\( (T^{8} - T^{4} + 1)^{2} \)
$11$
\( T^{16} \)
$13$
\( T^{16} - 12 T^{12} + 140 T^{8} + \cdots + 16 \)
$17$
\( T^{16} \)
$19$
\( (T^{8} + 8 T^{6} + 20 T^{4} + 16 T^{2} + \cdots + 1)^{2} \)
$23$
\( (T^{4} + 2 T^{3} + 5 T^{2} + 4 T + 1)^{4} \)
$29$
\( T^{16} \)
$31$
\( T^{16} \)
$37$
\( T^{16} \)
$41$
\( T^{16} \)
$43$
\( T^{16} \)
$47$
\( T^{16} \)
$53$
\( T^{16} \)
$59$
\( (T^{8} + 4 T^{6} + 14 T^{4} + 8 T^{2} + 4)^{2} \)
$61$
\( T^{16} + 8 T^{14} + 44 T^{12} + 128 T^{10} + \cdots + 1 \)
$67$
\( T^{16} \)
$71$
\( (T^{4} + 4 T^{2} + 1)^{4} \)
$73$
\( T^{16} \)
$79$
\( (T^{8} - 4 T^{6} + 15 T^{4} - 4 T^{2} + 1)^{2} \)
$83$
\( T^{16} - 12 T^{12} + 140 T^{8} + \cdots + 16 \)
$89$
\( T^{16} \)
$97$
\( T^{16} \)
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