Properties

Label 1260.1.bj.d
Level $1260$
Weight $1$
Character orbit 1260.bj
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.bj (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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.79380.1
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.

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

Character values

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

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(-\zeta_{6}\) \(-1\) \(-1\) \(\zeta_{6}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i 0.500000 0.866025i −1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
319.1 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i 1.00000 −0.500000 0.866025i 0.500000 + 0.866025i −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.g even 3 1 inner
1260.bj 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.bj.d yes 2
3.b odd 2 1 3780.1.bj.a 2
4.b odd 2 1 1260.1.bj.a 2
5.b even 2 1 1260.1.bj.a 2
7.c even 3 1 1260.1.cw.b yes 2
9.c even 3 1 1260.1.cw.b yes 2
9.d odd 6 1 3780.1.cw.d 2
12.b even 2 1 3780.1.bj.c 2
15.d odd 2 1 3780.1.bj.c 2
20.d odd 2 1 CM 1260.1.bj.d yes 2
21.h odd 6 1 3780.1.cw.d 2
28.g odd 6 1 1260.1.cw.c yes 2
35.j even 6 1 1260.1.cw.c yes 2
36.f odd 6 1 1260.1.cw.c yes 2
36.h even 6 1 3780.1.cw.a 2
45.h odd 6 1 3780.1.cw.a 2
45.j even 6 1 1260.1.cw.c yes 2
60.h even 2 1 3780.1.bj.a 2
63.g even 3 1 inner 1260.1.bj.d yes 2
63.n odd 6 1 3780.1.bj.a 2
84.n even 6 1 3780.1.cw.a 2
105.o odd 6 1 3780.1.cw.a 2
140.p odd 6 1 1260.1.cw.b yes 2
180.n even 6 1 3780.1.cw.d 2
180.p odd 6 1 1260.1.cw.b yes 2
252.o even 6 1 3780.1.bj.c 2
252.bl odd 6 1 1260.1.bj.a 2
315.v odd 6 1 3780.1.bj.c 2
315.bo even 6 1 1260.1.bj.a 2
420.ba even 6 1 3780.1.cw.d 2
1260.bj odd 6 1 inner 1260.1.bj.d yes 2
1260.dh even 6 1 3780.1.bj.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.1.bj.a 2 4.b odd 2 1
1260.1.bj.a 2 5.b even 2 1
1260.1.bj.a 2 252.bl odd 6 1
1260.1.bj.a 2 315.bo even 6 1
1260.1.bj.d yes 2 1.a even 1 1 trivial
1260.1.bj.d yes 2 20.d odd 2 1 CM
1260.1.bj.d yes 2 63.g even 3 1 inner
1260.1.bj.d yes 2 1260.bj odd 6 1 inner
1260.1.cw.b yes 2 7.c even 3 1
1260.1.cw.b yes 2 9.c even 3 1
1260.1.cw.b yes 2 140.p odd 6 1
1260.1.cw.b yes 2 180.p odd 6 1
1260.1.cw.c yes 2 28.g odd 6 1
1260.1.cw.c yes 2 35.j even 6 1
1260.1.cw.c yes 2 36.f odd 6 1
1260.1.cw.c yes 2 45.j even 6 1
3780.1.bj.a 2 3.b odd 2 1
3780.1.bj.a 2 60.h even 2 1
3780.1.bj.a 2 63.n odd 6 1
3780.1.bj.a 2 1260.dh even 6 1
3780.1.bj.c 2 12.b even 2 1
3780.1.bj.c 2 15.d odd 2 1
3780.1.bj.c 2 252.o even 6 1
3780.1.bj.c 2 315.v odd 6 1
3780.1.cw.a 2 36.h even 6 1
3780.1.cw.a 2 45.h odd 6 1
3780.1.cw.a 2 84.n even 6 1
3780.1.cw.a 2 105.o odd 6 1
3780.1.cw.d 2 9.d odd 6 1
3780.1.cw.d 2 21.h odd 6 1
3780.1.cw.d 2 180.n even 6 1
3780.1.cw.d 2 420.ba even 6 1

Hecke kernels

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

Hecke characteristic polynomials

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