Properties

Label 1260.2.s.a
Level $1260$
Weight $2$
Character orbit 1260.s
Analytic conductor $10.061$
Analytic rank $0$
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: \(0\)
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 420)
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.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{5} + ( -2 - \zeta_{6} ) q^{7} +O(q^{10})\) \( q -\zeta_{6} q^{5} + ( -2 - \zeta_{6} ) q^{7} + ( -4 + 4 \zeta_{6} ) q^{11} + 7 q^{13} + ( -6 + 6 \zeta_{6} ) q^{17} -3 \zeta_{6} q^{19} -2 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + 2 q^{29} + ( -7 + 7 \zeta_{6} ) q^{31} + ( -1 + 3 \zeta_{6} ) q^{35} + 7 \zeta_{6} q^{37} + 8 q^{41} + 5 q^{43} + 10 \zeta_{6} q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} + ( -8 + 8 \zeta_{6} ) q^{53} + 4 q^{55} + ( 10 - 10 \zeta_{6} ) q^{59} + 6 \zeta_{6} q^{61} -7 \zeta_{6} q^{65} + ( -3 + 3 \zeta_{6} ) q^{67} + ( -15 + 15 \zeta_{6} ) q^{73} + ( 12 - 8 \zeta_{6} ) q^{77} -\zeta_{6} q^{79} -8 q^{83} + 6 q^{85} + 2 \zeta_{6} q^{89} + ( -14 - 7 \zeta_{6} ) q^{91} + ( -3 + 3 \zeta_{6} ) q^{95} -10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} - 5 q^{7} + O(q^{10}) \) \( 2 q - q^{5} - 5 q^{7} - 4 q^{11} + 14 q^{13} - 6 q^{17} - 3 q^{19} - 2 q^{23} - q^{25} + 4 q^{29} - 7 q^{31} + q^{35} + 7 q^{37} + 16 q^{41} + 10 q^{43} + 10 q^{47} + 11 q^{49} - 8 q^{53} + 8 q^{55} + 10 q^{59} + 6 q^{61} - 7 q^{65} - 3 q^{67} - 15 q^{73} + 16 q^{77} - q^{79} - 16 q^{83} + 12 q^{85} + 2 q^{89} - 35 q^{91} - 3 q^{95} - 20 q^{97} + O(q^{100}) \)

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 −2.50000 0.866025i 0 0 0
541.1 0 0 0 −0.500000 + 0.866025i 0 −2.50000 + 0.866025i 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.a 2
3.b odd 2 1 420.2.q.b 2
7.c even 3 1 inner 1260.2.s.a 2
7.c even 3 1 8820.2.a.bb 1
7.d odd 6 1 8820.2.a.l 1
12.b even 2 1 1680.2.bg.j 2
15.d odd 2 1 2100.2.q.d 2
15.e even 4 2 2100.2.bc.d 4
21.c even 2 1 2940.2.q.c 2
21.g even 6 1 2940.2.a.k 1
21.g even 6 1 2940.2.q.c 2
21.h odd 6 1 420.2.q.b 2
21.h odd 6 1 2940.2.a.b 1
84.n even 6 1 1680.2.bg.j 2
105.o odd 6 1 2100.2.q.d 2
105.x even 12 2 2100.2.bc.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.b 2 3.b odd 2 1
420.2.q.b 2 21.h odd 6 1
1260.2.s.a 2 1.a even 1 1 trivial
1260.2.s.a 2 7.c even 3 1 inner
1680.2.bg.j 2 12.b even 2 1
1680.2.bg.j 2 84.n even 6 1
2100.2.q.d 2 15.d odd 2 1
2100.2.q.d 2 105.o odd 6 1
2100.2.bc.d 4 15.e even 4 2
2100.2.bc.d 4 105.x even 12 2
2940.2.a.b 1 21.h odd 6 1
2940.2.a.k 1 21.g even 6 1
2940.2.q.c 2 21.c even 2 1
2940.2.q.c 2 21.g even 6 1
8820.2.a.l 1 7.d odd 6 1
8820.2.a.bb 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 \(S_{2}^{\mathrm{new}}(1260, [\chi])\):

\( T_{11}^{2} + 4 T_{11} + 16 \)
\( T_{13} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( 7 + 5 T + T^{2} \)
$11$ \( 16 + 4 T + T^{2} \)
$13$ \( ( -7 + T )^{2} \)
$17$ \( 36 + 6 T + T^{2} \)
$19$ \( 9 + 3 T + T^{2} \)
$23$ \( 4 + 2 T + T^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( 49 + 7 T + T^{2} \)
$37$ \( 49 - 7 T + T^{2} \)
$41$ \( ( -8 + T )^{2} \)
$43$ \( ( -5 + T )^{2} \)
$47$ \( 100 - 10 T + T^{2} \)
$53$ \( 64 + 8 T + T^{2} \)
$59$ \( 100 - 10 T + T^{2} \)
$61$ \( 36 - 6 T + T^{2} \)
$67$ \( 9 + 3 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 225 + 15 T + T^{2} \)
$79$ \( 1 + T + T^{2} \)
$83$ \( ( 8 + T )^{2} \)
$89$ \( 4 - 2 T + T^{2} \)
$97$ \( ( 10 + T )^{2} \)
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