Properties

Label 2520.1.eq.a
Level $2520$
Weight $1$
Character orbit 2520.eq
Analytic conductor $1.258$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -40
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2520.eq (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: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q -\zeta_{24}^{10} q^{2} -\zeta_{24}^{8} q^{4} -\zeta_{24}^{4} q^{5} + \zeta_{24}^{9} q^{7} -\zeta_{24}^{6} q^{8} +O(q^{10})\) \( q -\zeta_{24}^{10} q^{2} -\zeta_{24}^{8} q^{4} -\zeta_{24}^{4} q^{5} + \zeta_{24}^{9} q^{7} -\zeta_{24}^{6} q^{8} -\zeta_{24}^{2} q^{10} + ( -\zeta_{24} - \zeta_{24}^{3} ) q^{11} + ( -\zeta_{24} - \zeta_{24}^{11} ) q^{13} + \zeta_{24}^{7} q^{14} -\zeta_{24}^{4} q^{16} -\zeta_{24}^{10} q^{19} - q^{20} + ( -\zeta_{24} + \zeta_{24}^{11} ) q^{22} + ( -1 + \zeta_{24}^{8} ) q^{23} + \zeta_{24}^{8} q^{25} + ( -\zeta_{24}^{9} + \zeta_{24}^{11} ) q^{26} + \zeta_{24}^{5} q^{28} -\zeta_{24}^{2} q^{32} + \zeta_{24} q^{35} + ( -\zeta_{24}^{9} + \zeta_{24}^{11} ) q^{37} -\zeta_{24}^{8} q^{38} + \zeta_{24}^{10} q^{40} + ( \zeta_{24} - \zeta_{24}^{11} ) q^{41} + ( \zeta_{24}^{9} + \zeta_{24}^{11} ) q^{44} + ( \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{46} + ( -\zeta_{24}^{2} - \zeta_{24}^{6} ) q^{47} -\zeta_{24}^{6} q^{49} + \zeta_{24}^{6} q^{50} + ( -\zeta_{24}^{7} + \zeta_{24}^{9} ) q^{52} + \zeta_{24}^{2} q^{53} + ( \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{55} + \zeta_{24}^{3} q^{56} + ( -\zeta_{24}^{5} - \zeta_{24}^{11} ) q^{59} - q^{64} + ( -\zeta_{24}^{3} + \zeta_{24}^{5} ) q^{65} -\zeta_{24}^{11} q^{70} + ( -\zeta_{24}^{7} + \zeta_{24}^{9} ) q^{74} -\zeta_{24}^{6} q^{76} + ( 1 - \zeta_{24}^{10} ) q^{77} + \zeta_{24}^{8} q^{80} + ( -\zeta_{24}^{9} - \zeta_{24}^{11} ) q^{82} + ( \zeta_{24}^{7} + \zeta_{24}^{9} ) q^{88} + ( -\zeta_{24} - \zeta_{24}^{7} ) q^{89} + ( \zeta_{24}^{8} - \zeta_{24}^{10} ) q^{91} + ( \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{92} + ( -1 - \zeta_{24}^{4} ) q^{94} -\zeta_{24}^{2} q^{95} -\zeta_{24}^{4} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{4} - 4q^{5} + O(q^{10}) \) \( 8q + 4q^{4} - 4q^{5} - 4q^{16} - 8q^{20} - 12q^{23} - 4q^{25} + 4q^{38} - 8q^{64} + 8q^{77} - 4q^{80} - 4q^{91} - 12q^{94} - 4q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(1081\) \(1261\) \(2017\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{24}^{4}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
899.1
−0.258819 + 0.965926i
0.258819 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.866025 + 0.500000i 0 0.500000 0.866025i −0.500000 0.866025i 0 −0.707107 0.707107i 1.00000i 0 0.866025 + 0.500000i
899.2 −0.866025 + 0.500000i 0 0.500000 0.866025i −0.500000 0.866025i 0 0.707107 + 0.707107i 1.00000i 0 0.866025 + 0.500000i
899.3 0.866025 0.500000i 0 0.500000 0.866025i −0.500000 0.866025i 0 −0.707107 + 0.707107i 1.00000i 0 −0.866025 0.500000i
899.4 0.866025 0.500000i 0 0.500000 0.866025i −0.500000 0.866025i 0 0.707107 0.707107i 1.00000i 0 −0.866025 0.500000i
1979.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.500000 + 0.866025i 0 −0.707107 + 0.707107i 1.00000i 0 0.866025 0.500000i
1979.2 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.707107 0.707107i 1.00000i 0 0.866025 0.500000i
1979.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.500000 + 0.866025i 0 −0.707107 0.707107i 1.00000i 0 −0.866025 + 0.500000i
1979.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.707107 + 0.707107i 1.00000i 0 −0.866025 + 0.500000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1979.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
15.d odd 2 1 inner
21.g even 6 1 inner
24.f even 2 1 inner
35.i odd 6 1 inner
56.m even 6 1 inner
840.ct 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.eq.a 8
3.b odd 2 1 2520.1.eq.b yes 8
5.b even 2 1 2520.1.eq.b yes 8
7.d odd 6 1 2520.1.eq.b yes 8
8.d odd 2 1 2520.1.eq.b yes 8
15.d odd 2 1 inner 2520.1.eq.a 8
21.g even 6 1 inner 2520.1.eq.a 8
24.f even 2 1 inner 2520.1.eq.a 8
35.i odd 6 1 inner 2520.1.eq.a 8
40.e odd 2 1 CM 2520.1.eq.a 8
56.m even 6 1 inner 2520.1.eq.a 8
105.p even 6 1 2520.1.eq.b yes 8
120.m even 2 1 2520.1.eq.b yes 8
168.be odd 6 1 2520.1.eq.b yes 8
280.ba even 6 1 2520.1.eq.b yes 8
840.ct odd 6 1 inner 2520.1.eq.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2520.1.eq.a 8 1.a even 1 1 trivial
2520.1.eq.a 8 15.d odd 2 1 inner
2520.1.eq.a 8 21.g even 6 1 inner
2520.1.eq.a 8 24.f even 2 1 inner
2520.1.eq.a 8 35.i odd 6 1 inner
2520.1.eq.a 8 40.e odd 2 1 CM
2520.1.eq.a 8 56.m even 6 1 inner
2520.1.eq.a 8 840.ct odd 6 1 inner
2520.1.eq.b yes 8 3.b odd 2 1
2520.1.eq.b yes 8 5.b even 2 1
2520.1.eq.b yes 8 7.d odd 6 1
2520.1.eq.b yes 8 8.d odd 2 1
2520.1.eq.b yes 8 105.p even 6 1
2520.1.eq.b yes 8 120.m even 2 1
2520.1.eq.b yes 8 168.be odd 6 1
2520.1.eq.b yes 8 280.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{2} + 3 T_{23} + 3 \) acting on \(S_{1}^{\mathrm{new}}(2520, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( ( 1 + T + T^{2} )^{4} \)
$7$ \( ( 1 + T^{4} )^{2} \)
$11$ \( 1 - 4 T^{2} + 15 T^{4} - 4 T^{6} + T^{8} \)
$13$ \( ( 1 + 4 T^{2} + T^{4} )^{2} \)
$17$ \( T^{8} \)
$19$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$23$ \( ( 3 + 3 T + T^{2} )^{4} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( 1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8} \)
$41$ \( ( 1 - 4 T^{2} + T^{4} )^{2} \)
$43$ \( T^{8} \)
$47$ \( ( 9 + 3 T^{2} + T^{4} )^{2} \)
$53$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$59$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$61$ \( T^{8} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( T^{8} \)
$79$ \( T^{8} \)
$83$ \( T^{8} \)
$89$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$97$ \( T^{8} \)
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