Properties

Label 2520.1.ef.a
Level $2520$
Weight $1$
Character orbit 2520.ef
Analytic conductor $1.258$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -40
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.ef (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: no (minimal twist has level 280)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.1960.1
Artin image $C_6\times S_3$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
739.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 −1.00000 1.00000 0 −0.500000 + 0.866025i
2179.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 −1.00000 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
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.a 2
3.b odd 2 1 280.1.bi.b yes 2
5.b even 2 1 2520.1.ef.b 2
7.c even 3 1 inner 2520.1.ef.a 2
8.d odd 2 1 2520.1.ef.b 2
12.b even 2 1 1120.1.by.b 2
15.d odd 2 1 280.1.bi.a 2
15.e even 4 2 1400.1.ba.a 4
21.c even 2 1 1960.1.bi.b 2
21.g even 6 1 1960.1.i.b 1
21.g even 6 1 1960.1.bi.b 2
21.h odd 6 1 280.1.bi.b yes 2
21.h odd 6 1 1960.1.i.a 1
24.f even 2 1 280.1.bi.a 2
24.h odd 2 1 1120.1.by.a 2
35.j even 6 1 2520.1.ef.b 2
40.e odd 2 1 CM 2520.1.ef.a 2
56.k odd 6 1 2520.1.ef.b 2
60.h even 2 1 1120.1.by.a 2
84.n even 6 1 1120.1.by.b 2
105.g even 2 1 1960.1.bi.a 2
105.o odd 6 1 280.1.bi.a 2
105.o odd 6 1 1960.1.i.d 1
105.p even 6 1 1960.1.i.c 1
105.p even 6 1 1960.1.bi.a 2
105.x even 12 2 1400.1.ba.a 4
120.i odd 2 1 1120.1.by.b 2
120.m even 2 1 280.1.bi.b yes 2
120.q odd 4 2 1400.1.ba.a 4
168.e odd 2 1 1960.1.bi.a 2
168.s odd 6 1 1120.1.by.a 2
168.v even 6 1 280.1.bi.a 2
168.v even 6 1 1960.1.i.d 1
168.be odd 6 1 1960.1.i.c 1
168.be odd 6 1 1960.1.bi.a 2
280.bi odd 6 1 inner 2520.1.ef.a 2
420.ba even 6 1 1120.1.by.a 2
840.b odd 2 1 1960.1.bi.b 2
840.cg odd 6 1 1120.1.by.b 2
840.ct odd 6 1 1960.1.i.b 1
840.ct odd 6 1 1960.1.bi.b 2
840.cv even 6 1 280.1.bi.b yes 2
840.cv even 6 1 1960.1.i.a 1
840.dp odd 12 2 1400.1.ba.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.1.bi.a 2 15.d odd 2 1
280.1.bi.a 2 24.f even 2 1
280.1.bi.a 2 105.o odd 6 1
280.1.bi.a 2 168.v even 6 1
280.1.bi.b yes 2 3.b odd 2 1
280.1.bi.b yes 2 21.h odd 6 1
280.1.bi.b yes 2 120.m even 2 1
280.1.bi.b yes 2 840.cv even 6 1
1120.1.by.a 2 24.h odd 2 1
1120.1.by.a 2 60.h even 2 1
1120.1.by.a 2 168.s odd 6 1
1120.1.by.a 2 420.ba even 6 1
1120.1.by.b 2 12.b even 2 1
1120.1.by.b 2 84.n even 6 1
1120.1.by.b 2 120.i odd 2 1
1120.1.by.b 2 840.cg odd 6 1
1400.1.ba.a 4 15.e even 4 2
1400.1.ba.a 4 105.x even 12 2
1400.1.ba.a 4 120.q odd 4 2
1400.1.ba.a 4 840.dp odd 12 2
1960.1.i.a 1 21.h odd 6 1
1960.1.i.a 1 840.cv even 6 1
1960.1.i.b 1 21.g even 6 1
1960.1.i.b 1 840.ct odd 6 1
1960.1.i.c 1 105.p even 6 1
1960.1.i.c 1 168.be odd 6 1
1960.1.i.d 1 105.o odd 6 1
1960.1.i.d 1 168.v even 6 1
1960.1.bi.a 2 105.g even 2 1
1960.1.bi.a 2 105.p even 6 1
1960.1.bi.a 2 168.e odd 2 1
1960.1.bi.a 2 168.be odd 6 1
1960.1.bi.b 2 21.c even 2 1
1960.1.bi.b 2 21.g even 6 1
1960.1.bi.b 2 840.b odd 2 1
1960.1.bi.b 2 840.ct odd 6 1
2520.1.ef.a 2 1.a even 1 1 trivial
2520.1.ef.a 2 7.c even 3 1 inner
2520.1.ef.a 2 40.e odd 2 1 CM
2520.1.ef.a 2 280.bi odd 6 1 inner
2520.1.ef.b 2 5.b even 2 1
2520.1.ef.b 2 8.d odd 2 1
2520.1.ef.b 2 35.j even 6 1
2520.1.ef.b 2 56.k odd 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

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