Properties

Label 1260.2.c.a
Level $1260$
Weight $2$
Character orbit 1260.c
Analytic conductor $10.061$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(10.0611506547\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \zeta_{12} - \zeta_{12}^{2} ) q^{2} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{4} + \zeta_{12}^{3} q^{5} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + ( 2 - 2 \zeta_{12}^{3} ) q^{8} +O(q^{10})\) \( q + ( 1 + \zeta_{12} - \zeta_{12}^{2} ) q^{2} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{4} + \zeta_{12}^{3} q^{5} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + ( 2 - 2 \zeta_{12}^{3} ) q^{8} + ( -1 + \zeta_{12} + \zeta_{12}^{2} ) q^{10} + ( -1 + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{11} + ( -2 + 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{13} + ( 3 - \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{14} + ( 4 - 4 \zeta_{12}^{2} ) q^{16} + ( 2 - 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{17} + 6 q^{19} + 2 \zeta_{12}^{2} q^{20} + ( -1 + \zeta_{12} + 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{22} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{23} - q^{25} + ( -1 + \zeta_{12} + 5 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{26} + ( 4 - 6 \zeta_{12}^{2} ) q^{28} + ( -1 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{29} -6 q^{31} + ( 4 \zeta_{12} - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{32} + ( -5 + 5 \zeta_{12} + \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{34} + ( 1 + 2 \zeta_{12}^{2} ) q^{35} + ( -6 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{37} + ( 6 + 6 \zeta_{12} - 6 \zeta_{12}^{2} ) q^{38} + ( 2 + 2 \zeta_{12}^{3} ) q^{40} + ( 2 - 4 \zeta_{12}^{2} ) q^{41} -2 \zeta_{12}^{3} q^{43} + ( 2 \zeta_{12} + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{44} + ( -4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{46} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{47} + ( 3 - 8 \zeta_{12}^{2} ) q^{49} + ( -1 - \zeta_{12} + \zeta_{12}^{2} ) q^{50} + ( 4 \zeta_{12} + 6 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{52} -2 q^{53} + ( -2 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{55} + ( -2 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{56} + ( 3 + 3 \zeta_{12} + 5 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{58} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{59} + ( 2 - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{61} + ( -6 - 6 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{62} -8 \zeta_{12}^{3} q^{64} + ( -3 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{65} + ( 2 - 4 \zeta_{12}^{2} ) q^{67} + ( -4 \zeta_{12} + 6 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{68} + ( 3 + \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{70} + ( 2 - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{71} + ( 4 - 8 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{73} + ( -4 - 4 \zeta_{12} + 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{74} + ( 12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{76} + ( 2 + 4 \zeta_{12} + 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{77} + ( -5 + 10 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{79} + 4 \zeta_{12} q^{80} + ( -2 + 2 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{82} + ( -12 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{83} + ( -3 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{85} + ( 2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{86} + ( 2 + 4 \zeta_{12} + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{88} + ( 2 - 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{89} + ( 3 + 8 \zeta_{12} + 6 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{91} + ( -4 \zeta_{12} - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{92} + ( 1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{94} + 6 \zeta_{12}^{3} q^{95} + ( -6 + 12 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{97} + ( -5 + 3 \zeta_{12} - 3 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} + 8q^{8} + O(q^{10}) \) \( 4q + 2q^{2} + 8q^{8} - 2q^{10} + 10q^{14} + 8q^{16} + 24q^{19} + 4q^{20} + 2q^{22} - 4q^{25} + 6q^{26} + 4q^{28} - 4q^{29} - 24q^{31} - 8q^{32} - 18q^{34} + 8q^{35} - 24q^{37} + 12q^{38} + 8q^{40} + 8q^{44} - 8q^{46} - 4q^{49} - 2q^{50} + 12q^{52} - 8q^{53} - 8q^{55} - 16q^{56} + 22q^{58} - 12q^{62} - 12q^{65} + 12q^{68} + 10q^{70} + 16q^{77} - 12q^{82} - 48q^{83} - 12q^{85} + 4q^{86} + 16q^{88} + 24q^{91} - 8q^{92} + 6q^{94} - 26q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
811.1
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.366025 1.36603i 0 −1.73205 + 1.00000i 1.00000i 0 −1.73205 + 2.00000i 2.00000 + 2.00000i 0 −1.36603 + 0.366025i
811.2 −0.366025 + 1.36603i 0 −1.73205 1.00000i 1.00000i 0 −1.73205 2.00000i 2.00000 2.00000i 0 −1.36603 0.366025i
811.3 1.36603 0.366025i 0 1.73205 1.00000i 1.00000i 0 1.73205 2.00000i 2.00000 2.00000i 0 0.366025 + 1.36603i
811.4 1.36603 + 0.366025i 0 1.73205 + 1.00000i 1.00000i 0 1.73205 + 2.00000i 2.00000 + 2.00000i 0 0.366025 1.36603i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.c.a 4
3.b odd 2 1 140.2.g.a 4
4.b odd 2 1 1260.2.c.b 4
7.b odd 2 1 1260.2.c.b 4
12.b even 2 1 140.2.g.b yes 4
15.d odd 2 1 700.2.g.f 4
15.e even 4 1 700.2.c.b 4
15.e even 4 1 700.2.c.e 4
21.c even 2 1 140.2.g.b yes 4
21.g even 6 1 980.2.o.b 4
21.g even 6 1 980.2.o.d 4
21.h odd 6 1 980.2.o.a 4
21.h odd 6 1 980.2.o.c 4
24.f even 2 1 2240.2.k.b 4
24.h odd 2 1 2240.2.k.a 4
28.d even 2 1 inner 1260.2.c.a 4
60.h even 2 1 700.2.g.g 4
60.l odd 4 1 700.2.c.c 4
60.l odd 4 1 700.2.c.f 4
84.h odd 2 1 140.2.g.a 4
84.j odd 6 1 980.2.o.a 4
84.j odd 6 1 980.2.o.c 4
84.n even 6 1 980.2.o.b 4
84.n even 6 1 980.2.o.d 4
105.g even 2 1 700.2.g.g 4
105.k odd 4 1 700.2.c.c 4
105.k odd 4 1 700.2.c.f 4
168.e odd 2 1 2240.2.k.a 4
168.i even 2 1 2240.2.k.b 4
420.o odd 2 1 700.2.g.f 4
420.w even 4 1 700.2.c.b 4
420.w even 4 1 700.2.c.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.a 4 3.b odd 2 1
140.2.g.a 4 84.h odd 2 1
140.2.g.b yes 4 12.b even 2 1
140.2.g.b yes 4 21.c even 2 1
700.2.c.b 4 15.e even 4 1
700.2.c.b 4 420.w even 4 1
700.2.c.c 4 60.l odd 4 1
700.2.c.c 4 105.k odd 4 1
700.2.c.e 4 15.e even 4 1
700.2.c.e 4 420.w even 4 1
700.2.c.f 4 60.l odd 4 1
700.2.c.f 4 105.k odd 4 1
700.2.g.f 4 15.d odd 2 1
700.2.g.f 4 420.o odd 2 1
700.2.g.g 4 60.h even 2 1
700.2.g.g 4 105.g even 2 1
980.2.o.a 4 21.h odd 6 1
980.2.o.a 4 84.j odd 6 1
980.2.o.b 4 21.g even 6 1
980.2.o.b 4 84.n even 6 1
980.2.o.c 4 21.h odd 6 1
980.2.o.c 4 84.j odd 6 1
980.2.o.d 4 21.g even 6 1
980.2.o.d 4 84.n even 6 1
1260.2.c.a 4 1.a even 1 1 trivial
1260.2.c.a 4 28.d even 2 1 inner
1260.2.c.b 4 4.b odd 2 1
1260.2.c.b 4 7.b odd 2 1
2240.2.k.a 4 24.h odd 2 1
2240.2.k.a 4 168.e odd 2 1
2240.2.k.b 4 24.f even 2 1
2240.2.k.b 4 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1260, [\chi])\):

\( T_{11}^{4} + 14 T_{11}^{2} + 1 \)
\( T_{19} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( 49 + 2 T^{2} + T^{4} \)
$11$ \( 1 + 14 T^{2} + T^{4} \)
$13$ \( 9 + 42 T^{2} + T^{4} \)
$17$ \( 9 + 42 T^{2} + T^{4} \)
$19$ \( ( -6 + T )^{4} \)
$23$ \( 64 + 32 T^{2} + T^{4} \)
$29$ \( ( -47 + 2 T + T^{2} )^{2} \)
$31$ \( ( 6 + T )^{4} \)
$37$ \( ( 24 + 12 T + T^{2} )^{2} \)
$41$ \( ( 12 + T^{2} )^{2} \)
$43$ \( ( 4 + T^{2} )^{2} \)
$47$ \( ( -3 + T^{2} )^{2} \)
$53$ \( ( 2 + T )^{4} \)
$59$ \( ( -12 + T^{2} )^{2} \)
$61$ \( 576 + 96 T^{2} + T^{4} \)
$67$ \( ( 12 + T^{2} )^{2} \)
$71$ \( 16 + 56 T^{2} + T^{4} \)
$73$ \( 144 + 168 T^{2} + T^{4} \)
$79$ \( 1521 + 222 T^{2} + T^{4} \)
$83$ \( ( 132 + 24 T + T^{2} )^{2} \)
$89$ \( 576 + 96 T^{2} + T^{4} \)
$97$ \( 9801 + 234 T^{2} + T^{4} \)
show more
show less