Properties

Label 1260.1.cf.a
Level $1260$
Weight $1$
Character orbit 1260.cf
Analytic conductor $0.629$
Analytic rank $0$
Dimension $8$
Projective image $S_{4}$
CM/RM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1260.cf (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.628821915918\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.132300.3

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{24}^{7} q^{5} -\zeta_{24}^{10} q^{7} +O(q^{10})\) \( q -\zeta_{24}^{7} q^{5} -\zeta_{24}^{10} q^{7} -\zeta_{24}^{6} q^{13} + ( \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{17} + \zeta_{24}^{4} q^{19} + ( -\zeta_{24} - \zeta_{24}^{7} ) q^{23} -\zeta_{24}^{2} q^{25} + ( -\zeta_{24}^{3} - \zeta_{24}^{9} ) q^{29} + \zeta_{24}^{8} q^{31} -\zeta_{24}^{5} q^{35} -\zeta_{24}^{10} q^{37} + \zeta_{24}^{6} q^{43} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{47} -\zeta_{24}^{8} q^{49} + ( \zeta_{24}^{5} - \zeta_{24}^{11} ) q^{59} -\zeta_{24} q^{65} + \zeta_{24}^{2} q^{67} -\zeta_{24}^{2} q^{73} -\zeta_{24}^{4} q^{79} + ( 1 + \zeta_{24}^{6} ) q^{85} + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{89} -\zeta_{24}^{4} q^{91} -\zeta_{24}^{11} q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q + 4 q^{19} - 4 q^{31} + 4 q^{49} - 4 q^{79} + 8 q^{85} - 4 q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-\zeta_{24}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
809.1
−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
0.258819 + 0.965926i
0 0 0 −0.965926 0.258819i 0 −0.866025 + 0.500000i 0 0 0
809.2 0 0 0 −0.258819 + 0.965926i 0 0.866025 0.500000i 0 0 0
809.3 0 0 0 0.258819 0.965926i 0 0.866025 0.500000i 0 0 0
809.4 0 0 0 0.965926 + 0.258819i 0 −0.866025 + 0.500000i 0 0 0
989.1 0 0 0 −0.965926 + 0.258819i 0 −0.866025 0.500000i 0 0 0
989.2 0 0 0 −0.258819 0.965926i 0 0.866025 + 0.500000i 0 0 0
989.3 0 0 0 0.258819 + 0.965926i 0 0.866025 + 0.500000i 0 0 0
989.4 0 0 0 0.965926 0.258819i 0 −0.866025 0.500000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 989.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.c even 3 1 inner
15.d odd 2 1 inner
21.h odd 6 1 inner
35.j even 6 1 inner
105.o odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.1.cf.a 8
3.b odd 2 1 inner 1260.1.cf.a 8
5.b even 2 1 inner 1260.1.cf.a 8
7.c even 3 1 inner 1260.1.cf.a 8
15.d odd 2 1 inner 1260.1.cf.a 8
21.h odd 6 1 inner 1260.1.cf.a 8
35.j even 6 1 inner 1260.1.cf.a 8
105.o odd 6 1 inner 1260.1.cf.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.1.cf.a 8 1.a even 1 1 trivial
1260.1.cf.a 8 3.b odd 2 1 inner
1260.1.cf.a 8 5.b even 2 1 inner
1260.1.cf.a 8 7.c even 3 1 inner
1260.1.cf.a 8 15.d odd 2 1 inner
1260.1.cf.a 8 21.h odd 6 1 inner
1260.1.cf.a 8 35.j even 6 1 inner
1260.1.cf.a 8 105.o odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1260, [\chi])\).

Hecke characteristic polynomials

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