Properties

Label 1260.1.cw.d
Level $1260$
Weight $1$
Character orbit 1260.cw
Analytic conductor $0.629$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -20
Inner twists $4$

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

Newform invariants

Self dual: no
Analytic conductor: \(0.628821915918\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.79380.2
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.31752000.4

$q$-expansion

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

\(f(q)\) \(=\) \( q + q^{2} -\zeta_{6} q^{3} + q^{4} + \zeta_{6}^{2} q^{5} -\zeta_{6} q^{6} + q^{7} + q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10})\) \( q + q^{2} -\zeta_{6} q^{3} + q^{4} + \zeta_{6}^{2} q^{5} -\zeta_{6} q^{6} + q^{7} + q^{8} + \zeta_{6}^{2} q^{9} + \zeta_{6}^{2} q^{10} -\zeta_{6} q^{12} + q^{14} + q^{15} + q^{16} + \zeta_{6}^{2} q^{18} + \zeta_{6}^{2} q^{20} -\zeta_{6} q^{21} -\zeta_{6}^{2} q^{23} -\zeta_{6} q^{24} -\zeta_{6} q^{25} + q^{27} + q^{28} + 2 \zeta_{6}^{2} q^{29} + q^{30} + q^{32} + \zeta_{6}^{2} q^{35} + \zeta_{6}^{2} q^{36} + \zeta_{6}^{2} q^{40} -2 \zeta_{6} q^{41} -\zeta_{6} q^{42} -\zeta_{6}^{2} q^{43} -\zeta_{6} q^{45} -\zeta_{6}^{2} q^{46} - q^{47} -\zeta_{6} q^{48} + q^{49} -\zeta_{6} q^{50} + q^{54} + q^{56} + 2 \zeta_{6}^{2} q^{58} + q^{60} - q^{61} + \zeta_{6}^{2} q^{63} + q^{64} - q^{67} - q^{69} + \zeta_{6}^{2} q^{70} + \zeta_{6}^{2} q^{72} + \zeta_{6}^{2} q^{75} + \zeta_{6}^{2} q^{80} -\zeta_{6} q^{81} -2 \zeta_{6} q^{82} + 2 \zeta_{6}^{2} q^{83} -\zeta_{6} q^{84} -\zeta_{6}^{2} q^{86} + 2 q^{87} + \zeta_{6} q^{89} -\zeta_{6} q^{90} -\zeta_{6}^{2} q^{92} - q^{94} -\zeta_{6} q^{96} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - q^{3} + 2q^{4} - q^{5} - q^{6} + 2q^{7} + 2q^{8} - q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - q^{3} + 2q^{4} - q^{5} - q^{6} + 2q^{7} + 2q^{8} - q^{9} - q^{10} - q^{12} + 2q^{14} + 2q^{15} + 2q^{16} - q^{18} - q^{20} - q^{21} + q^{23} - q^{24} - q^{25} + 2q^{27} + 2q^{28} - 2q^{29} + 2q^{30} + 2q^{32} - q^{35} - q^{36} - q^{40} - 2q^{41} - q^{42} + q^{43} - q^{45} + q^{46} - 2q^{47} - q^{48} + 2q^{49} - q^{50} + 2q^{54} + 2q^{56} - 2q^{58} + 2q^{60} - 2q^{61} - q^{63} + 2q^{64} - 2q^{67} - 2q^{69} - q^{70} - q^{72} - q^{75} - q^{80} - q^{81} - 2q^{82} - 2q^{83} - q^{84} + q^{86} + 4q^{87} + q^{89} - q^{90} + q^{92} - 2q^{94} - q^{96} + 2q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(-\zeta_{6}\) \(-1\) \(-1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
499.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 −0.500000 + 0.866025i 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i 1.00000 1.00000 −0.500000 0.866025i −0.500000 0.866025i
1159.1 1.00000 −0.500000 0.866025i 1.00000 −0.500000 + 0.866025i −0.500000 0.866025i 1.00000 1.00000 −0.500000 + 0.866025i −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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
63.h even 3 1 inner
1260.cw 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.cw.d yes 2
3.b odd 2 1 3780.1.cw.b 2
4.b odd 2 1 1260.1.cw.a yes 2
5.b even 2 1 1260.1.cw.a yes 2
7.c even 3 1 1260.1.bj.b 2
9.c even 3 1 1260.1.bj.b 2
9.d odd 6 1 3780.1.bj.d 2
12.b even 2 1 3780.1.cw.c 2
15.d odd 2 1 3780.1.cw.c 2
20.d odd 2 1 CM 1260.1.cw.d yes 2
21.h odd 6 1 3780.1.bj.d 2
28.g odd 6 1 1260.1.bj.c yes 2
35.j even 6 1 1260.1.bj.c yes 2
36.f odd 6 1 1260.1.bj.c yes 2
36.h even 6 1 3780.1.bj.b 2
45.h odd 6 1 3780.1.bj.b 2
45.j even 6 1 1260.1.bj.c yes 2
60.h even 2 1 3780.1.cw.b 2
63.h even 3 1 inner 1260.1.cw.d yes 2
63.j odd 6 1 3780.1.cw.b 2
84.n even 6 1 3780.1.bj.b 2
105.o odd 6 1 3780.1.bj.b 2
140.p odd 6 1 1260.1.bj.b 2
180.n even 6 1 3780.1.bj.d 2
180.p odd 6 1 1260.1.bj.b 2
252.u odd 6 1 1260.1.cw.a yes 2
252.bb even 6 1 3780.1.cw.c 2
315.r even 6 1 1260.1.cw.a yes 2
315.br odd 6 1 3780.1.cw.c 2
420.ba even 6 1 3780.1.bj.d 2
1260.bx even 6 1 3780.1.cw.b 2
1260.cw odd 6 1 inner 1260.1.cw.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.1.bj.b 2 7.c even 3 1
1260.1.bj.b 2 9.c even 3 1
1260.1.bj.b 2 140.p odd 6 1
1260.1.bj.b 2 180.p odd 6 1
1260.1.bj.c yes 2 28.g odd 6 1
1260.1.bj.c yes 2 35.j even 6 1
1260.1.bj.c yes 2 36.f odd 6 1
1260.1.bj.c yes 2 45.j even 6 1
1260.1.cw.a yes 2 4.b odd 2 1
1260.1.cw.a yes 2 5.b even 2 1
1260.1.cw.a yes 2 252.u odd 6 1
1260.1.cw.a yes 2 315.r even 6 1
1260.1.cw.d yes 2 1.a even 1 1 trivial
1260.1.cw.d yes 2 20.d odd 2 1 CM
1260.1.cw.d yes 2 63.h even 3 1 inner
1260.1.cw.d yes 2 1260.cw odd 6 1 inner
3780.1.bj.b 2 36.h even 6 1
3780.1.bj.b 2 45.h odd 6 1
3780.1.bj.b 2 84.n even 6 1
3780.1.bj.b 2 105.o odd 6 1
3780.1.bj.d 2 9.d odd 6 1
3780.1.bj.d 2 21.h odd 6 1
3780.1.bj.d 2 180.n even 6 1
3780.1.bj.d 2 420.ba even 6 1
3780.1.cw.b 2 3.b odd 2 1
3780.1.cw.b 2 60.h even 2 1
3780.1.cw.b 2 63.j odd 6 1
3780.1.cw.b 2 1260.bx even 6 1
3780.1.cw.c 2 12.b even 2 1
3780.1.cw.c 2 15.d odd 2 1
3780.1.cw.c 2 252.bb even 6 1
3780.1.cw.c 2 315.br odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{2} - T_{23} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1260, [\chi])\).

Hecke characteristic polynomials

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