Properties

Label 2520.1.ef.d
Level $2520$
Weight $1$
Character orbit 2520.ef
Analytic conductor $1.258$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
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.ef (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.25764383184\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.829785600.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

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

Character values

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

\(n\) \(281\) \(631\) \(1081\) \(1261\) \(2017\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{12}^{2}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
739.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 1.00000i −1.00000 0 0.500000 0.866025i
739.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 1.00000i −1.00000 0 0.500000 0.866025i
2179.1 0.500000 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 1.00000i −1.00000 0 0.500000 + 0.866025i
2179.2 0.500000 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 1.00000i −1.00000 0 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
7.c even 3 1 inner
15.d odd 2 1 inner
24.f even 2 1 inner
105.o odd 6 1 inner
168.v even 6 1 inner
280.bi 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.ef.d yes 4
3.b odd 2 1 2520.1.ef.c 4
5.b even 2 1 2520.1.ef.c 4
7.c even 3 1 inner 2520.1.ef.d yes 4
8.d odd 2 1 2520.1.ef.c 4
15.d odd 2 1 inner 2520.1.ef.d yes 4
21.h odd 6 1 2520.1.ef.c 4
24.f even 2 1 inner 2520.1.ef.d yes 4
35.j even 6 1 2520.1.ef.c 4
40.e odd 2 1 CM 2520.1.ef.d yes 4
56.k odd 6 1 2520.1.ef.c 4
105.o odd 6 1 inner 2520.1.ef.d yes 4
120.m even 2 1 2520.1.ef.c 4
168.v even 6 1 inner 2520.1.ef.d yes 4
280.bi odd 6 1 inner 2520.1.ef.d yes 4
840.cv even 6 1 2520.1.ef.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2520.1.ef.c 4 3.b odd 2 1
2520.1.ef.c 4 5.b even 2 1
2520.1.ef.c 4 8.d odd 2 1
2520.1.ef.c 4 21.h odd 6 1
2520.1.ef.c 4 35.j even 6 1
2520.1.ef.c 4 56.k odd 6 1
2520.1.ef.c 4 120.m even 2 1
2520.1.ef.c 4 840.cv even 6 1
2520.1.ef.d yes 4 1.a even 1 1 trivial
2520.1.ef.d yes 4 7.c even 3 1 inner
2520.1.ef.d yes 4 15.d odd 2 1 inner
2520.1.ef.d yes 4 24.f even 2 1 inner
2520.1.ef.d yes 4 40.e odd 2 1 CM
2520.1.ef.d yes 4 105.o odd 6 1 inner
2520.1.ef.d yes 4 168.v even 6 1 inner
2520.1.ef.d yes 4 280.bi odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2520, [\chi])\):

\( T_{11}^{4} + 3 T_{11}^{2} + 9 \)
\( T_{13}^{2} - 3 \)
\( T_{23}^{2} - T_{23} + 1 \)

Hecke characteristic polynomials

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