Properties

Label 1260.2.s.f
Level $1260$
Weight $2$
Character orbit 1260.s
Analytic conductor $10.061$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{5} + ( -\beta_{1} - \beta_{3} ) q^{7} +O(q^{10})\) \( q -\beta_{2} q^{5} + ( -\beta_{1} - \beta_{3} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} ) q^{11} -\beta_{3} q^{13} + ( 1 + \beta_{1} + \beta_{2} ) q^{17} + ( -2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{19} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{23} + ( -1 - \beta_{2} ) q^{25} + ( -5 - \beta_{3} ) q^{29} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{31} -\beta_{1} q^{35} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{37} + ( 3 - 3 \beta_{3} ) q^{41} + ( 2 - 3 \beta_{3} ) q^{43} -6 \beta_{2} q^{47} + ( -7 - 7 \beta_{2} ) q^{49} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{53} + ( 1 - \beta_{3} ) q^{55} + ( 3 - 3 \beta_{1} + 3 \beta_{2} ) q^{59} + 8 \beta_{2} q^{61} + ( -\beta_{1} - \beta_{3} ) q^{65} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{67} + ( -11 - \beta_{3} ) q^{71} + 5 \beta_{1} q^{73} + ( 7 - \beta_{3} ) q^{77} + ( -2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{79} + ( 3 + 3 \beta_{3} ) q^{83} + ( 1 - \beta_{3} ) q^{85} + ( -5 \beta_{1} + \beta_{2} - 5 \beta_{3} ) q^{89} -7 \beta_{2} q^{91} + ( 3 - 2 \beta_{1} + 3 \beta_{2} ) q^{95} + 8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} + O(q^{10}) \) \( 4 q + 2 q^{5} + 2 q^{11} + 2 q^{17} - 6 q^{19} - 2 q^{23} - 2 q^{25} - 20 q^{29} - 2 q^{31} + 4 q^{37} + 12 q^{41} + 8 q^{43} + 12 q^{47} - 14 q^{49} - 4 q^{53} + 4 q^{55} + 6 q^{59} - 16 q^{61} + 4 q^{67} - 44 q^{71} + 28 q^{77} + 6 q^{79} + 12 q^{83} + 4 q^{85} - 2 q^{89} + 14 q^{91} + 6 q^{95} + 32 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/7\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(7 \beta_{2}\)
\(\nu^{3}\)\(=\)\(7 \beta_{3}\)

Character values

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

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−1.32288 + 2.29129i
1.32288 2.29129i
−1.32288 2.29129i
1.32288 + 2.29129i
0 0 0 0.500000 + 0.866025i 0 −1.32288 2.29129i 0 0 0
361.2 0 0 0 0.500000 + 0.866025i 0 1.32288 + 2.29129i 0 0 0
541.1 0 0 0 0.500000 0.866025i 0 −1.32288 + 2.29129i 0 0 0
541.2 0 0 0 0.500000 0.866025i 0 1.32288 2.29129i 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.f 4
3.b odd 2 1 420.2.q.c 4
7.c even 3 1 inner 1260.2.s.f 4
7.c even 3 1 8820.2.a.be 2
7.d odd 6 1 8820.2.a.bj 2
12.b even 2 1 1680.2.bg.q 4
15.d odd 2 1 2100.2.q.h 4
15.e even 4 2 2100.2.bc.e 8
21.c even 2 1 2940.2.q.t 4
21.g even 6 1 2940.2.a.m 2
21.g even 6 1 2940.2.q.t 4
21.h odd 6 1 420.2.q.c 4
21.h odd 6 1 2940.2.a.s 2
84.n even 6 1 1680.2.bg.q 4
105.o odd 6 1 2100.2.q.h 4
105.x even 12 2 2100.2.bc.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.c 4 3.b odd 2 1
420.2.q.c 4 21.h odd 6 1
1260.2.s.f 4 1.a even 1 1 trivial
1260.2.s.f 4 7.c even 3 1 inner
1680.2.bg.q 4 12.b even 2 1
1680.2.bg.q 4 84.n even 6 1
2100.2.q.h 4 15.d odd 2 1
2100.2.q.h 4 105.o odd 6 1
2100.2.bc.e 8 15.e even 4 2
2100.2.bc.e 8 105.x even 12 2
2940.2.a.m 2 21.g even 6 1
2940.2.a.s 2 21.h odd 6 1
2940.2.q.t 4 21.c even 2 1
2940.2.q.t 4 21.g even 6 1
8820.2.a.be 2 7.c even 3 1
8820.2.a.bj 2 7.d odd 6 1

Hecke kernels

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

\( T_{11}^{4} - 2 T_{11}^{3} + 10 T_{11}^{2} + 12 T_{11} + 36 \)
\( T_{13}^{2} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( 49 + 7 T^{2} + T^{4} \)
$11$ \( 36 + 12 T + 10 T^{2} - 2 T^{3} + T^{4} \)
$13$ \( ( -7 + T^{2} )^{2} \)
$17$ \( 36 + 12 T + 10 T^{2} - 2 T^{3} + T^{4} \)
$19$ \( 361 - 114 T + 55 T^{2} + 6 T^{3} + T^{4} \)
$23$ \( 36 - 12 T + 10 T^{2} + 2 T^{3} + T^{4} \)
$29$ \( ( 18 + 10 T + T^{2} )^{2} \)
$31$ \( 729 - 54 T + 31 T^{2} + 2 T^{3} + T^{4} \)
$37$ \( 9 + 12 T + 19 T^{2} - 4 T^{3} + T^{4} \)
$41$ \( ( -54 - 6 T + T^{2} )^{2} \)
$43$ \( ( -59 - 4 T + T^{2} )^{2} \)
$47$ \( ( 36 - 6 T + T^{2} )^{2} \)
$53$ \( 576 - 96 T + 40 T^{2} + 4 T^{3} + T^{4} \)
$59$ \( 2916 + 324 T + 90 T^{2} - 6 T^{3} + T^{4} \)
$61$ \( ( 64 + 8 T + T^{2} )^{2} \)
$67$ \( 9 + 12 T + 19 T^{2} - 4 T^{3} + T^{4} \)
$71$ \( ( 114 + 22 T + T^{2} )^{2} \)
$73$ \( 30625 + 175 T^{2} + T^{4} \)
$79$ \( 361 + 114 T + 55 T^{2} - 6 T^{3} + T^{4} \)
$83$ \( ( -54 - 6 T + T^{2} )^{2} \)
$89$ \( 30276 - 348 T + 178 T^{2} + 2 T^{3} + T^{4} \)
$97$ \( ( -8 + T )^{4} \)
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