Properties

Label 1260.2.s.b
Level $1260$
Weight $2$
Character orbit 1260.s
Analytic conductor $10.061$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(10.0611506547\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} +O(q^{10})\) \( q -\zeta_{6} q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} + ( -2 + 2 \zeta_{6} ) q^{11} -6 q^{13} + ( 2 - 2 \zeta_{6} ) q^{17} -9 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} -3 q^{29} + ( -2 + 2 \zeta_{6} ) q^{31} + ( 3 - 2 \zeta_{6} ) q^{35} -8 \zeta_{6} q^{37} -5 q^{41} + q^{43} + 8 \zeta_{6} q^{47} + ( -8 + 3 \zeta_{6} ) q^{49} + ( 4 - 4 \zeta_{6} ) q^{53} + 2 q^{55} + ( -8 + 8 \zeta_{6} ) q^{59} -7 \zeta_{6} q^{61} + 6 \zeta_{6} q^{65} + ( 3 - 3 \zeta_{6} ) q^{67} -8 q^{71} + ( -14 + 14 \zeta_{6} ) q^{73} + ( -4 - 2 \zeta_{6} ) q^{77} -4 \zeta_{6} q^{79} + q^{83} -2 q^{85} + 13 \zeta_{6} q^{89} + ( 6 - 18 \zeta_{6} ) q^{91} -10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{5} + q^{7} + O(q^{10}) \) \( 2q - q^{5} + q^{7} - 2q^{11} - 12q^{13} + 2q^{17} - 9q^{23} - q^{25} - 6q^{29} - 2q^{31} + 4q^{35} - 8q^{37} - 10q^{41} + 2q^{43} + 8q^{47} - 13q^{49} + 4q^{53} + 4q^{55} - 8q^{59} - 7q^{61} + 6q^{65} + 3q^{67} - 16q^{71} - 14q^{73} - 10q^{77} - 4q^{79} + 2q^{83} - 4q^{85} + 13q^{89} - 6q^{91} - 20q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −0.500000 0.866025i 0 0.500000 + 2.59808i 0 0 0
541.1 0 0 0 −0.500000 + 0.866025i 0 0.500000 2.59808i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.s.b 2
3.b odd 2 1 140.2.i.b 2
7.c even 3 1 inner 1260.2.s.b 2
7.c even 3 1 8820.2.a.w 1
7.d odd 6 1 8820.2.a.k 1
12.b even 2 1 560.2.q.a 2
15.d odd 2 1 700.2.i.a 2
15.e even 4 2 700.2.r.c 4
21.c even 2 1 980.2.i.a 2
21.g even 6 1 980.2.a.i 1
21.g even 6 1 980.2.i.a 2
21.h odd 6 1 140.2.i.b 2
21.h odd 6 1 980.2.a.a 1
84.j odd 6 1 3920.2.a.d 1
84.n even 6 1 560.2.q.a 2
84.n even 6 1 3920.2.a.bi 1
105.o odd 6 1 700.2.i.a 2
105.o odd 6 1 4900.2.a.v 1
105.p even 6 1 4900.2.a.a 1
105.w odd 12 2 4900.2.e.b 2
105.x even 12 2 700.2.r.c 4
105.x even 12 2 4900.2.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.i.b 2 3.b odd 2 1
140.2.i.b 2 21.h odd 6 1
560.2.q.a 2 12.b even 2 1
560.2.q.a 2 84.n even 6 1
700.2.i.a 2 15.d odd 2 1
700.2.i.a 2 105.o odd 6 1
700.2.r.c 4 15.e even 4 2
700.2.r.c 4 105.x even 12 2
980.2.a.a 1 21.h odd 6 1
980.2.a.i 1 21.g even 6 1
980.2.i.a 2 21.c even 2 1
980.2.i.a 2 21.g even 6 1
1260.2.s.b 2 1.a even 1 1 trivial
1260.2.s.b 2 7.c even 3 1 inner
3920.2.a.d 1 84.j odd 6 1
3920.2.a.bi 1 84.n even 6 1
4900.2.a.a 1 105.p even 6 1
4900.2.a.v 1 105.o odd 6 1
4900.2.e.b 2 105.w odd 12 2
4900.2.e.c 2 105.x even 12 2
8820.2.a.k 1 7.d odd 6 1
8820.2.a.w 1 7.c even 3 1

Hecke kernels

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

\( T_{11}^{2} + 2 T_{11} + 4 \)
\( T_{13} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( 7 - T + T^{2} \)
$11$ \( 4 + 2 T + T^{2} \)
$13$ \( ( 6 + T )^{2} \)
$17$ \( 4 - 2 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( 81 + 9 T + T^{2} \)
$29$ \( ( 3 + T )^{2} \)
$31$ \( 4 + 2 T + T^{2} \)
$37$ \( 64 + 8 T + T^{2} \)
$41$ \( ( 5 + T )^{2} \)
$43$ \( ( -1 + T )^{2} \)
$47$ \( 64 - 8 T + T^{2} \)
$53$ \( 16 - 4 T + T^{2} \)
$59$ \( 64 + 8 T + T^{2} \)
$61$ \( 49 + 7 T + T^{2} \)
$67$ \( 9 - 3 T + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( 196 + 14 T + T^{2} \)
$79$ \( 16 + 4 T + T^{2} \)
$83$ \( ( -1 + T )^{2} \)
$89$ \( 169 - 13 T + T^{2} \)
$97$ \( ( 10 + T )^{2} \)
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