Properties

Label 2520.1.ek.a
Level $2520$
Weight $1$
Character orbit 2520.ek
Analytic conductor $1.258$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -24
Inner twists $4$

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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.ek (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.25764383184\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.3630312000.4

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{2} -\zeta_{6} q^{4} + \zeta_{6} q^{5} -\zeta_{6} q^{7} + q^{8} +O(q^{10})\) \( q + \zeta_{6}^{2} q^{2} -\zeta_{6} q^{4} + \zeta_{6} q^{5} -\zeta_{6} q^{7} + q^{8} - q^{10} + ( -1 + \zeta_{6}^{2} ) q^{11} + q^{14} + \zeta_{6}^{2} q^{16} -\zeta_{6}^{2} q^{20} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{22} + \zeta_{6}^{2} q^{25} + \zeta_{6}^{2} q^{28} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{29} + ( -1 + \zeta_{6}^{2} ) q^{31} -\zeta_{6} q^{32} -\zeta_{6}^{2} q^{35} + \zeta_{6} q^{40} + ( 1 + \zeta_{6} ) q^{44} + \zeta_{6}^{2} q^{49} -\zeta_{6} q^{50} -\zeta_{6} q^{53} + ( -1 - \zeta_{6} ) q^{55} -\zeta_{6} q^{56} + ( -1 - \zeta_{6} ) q^{58} + \zeta_{6} q^{59} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{62} + q^{64} + \zeta_{6} q^{70} + 2 \zeta_{6} q^{73} + ( 1 + \zeta_{6} ) q^{77} + \zeta_{6}^{2} q^{79} - q^{80} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{83} + ( -1 + \zeta_{6}^{2} ) q^{88} - q^{97} -\zeta_{6} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} + q^{5} - q^{7} + 2q^{8} + O(q^{10}) \) \( 2q - q^{2} - q^{4} + q^{5} - q^{7} + 2q^{8} - 2q^{10} - 3q^{11} + 2q^{14} - q^{16} + q^{20} - q^{25} - q^{28} - 3q^{31} - q^{32} + q^{35} + q^{40} + 3q^{44} - q^{49} - q^{50} - q^{53} - 3q^{55} - q^{56} - 3q^{58} + q^{59} + 2q^{64} + q^{70} + 2q^{73} + 3q^{77} - q^{79} - 2q^{80} - 3q^{88} - 2q^{97} - q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(281\) \(631\) \(1081\) \(1261\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}^{2}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
829.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i 0.500000 0.866025i 0 −0.500000 + 0.866025i 1.00000 0 −1.00000
1909.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.500000 + 0.866025i 0 −0.500000 0.866025i 1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{11}^{2} + 3 T_{11} + 3 \)
\( T_{53}^{2} + T_{53} + 1 \)

Hecke characteristic polynomials

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