Properties

Label 1134.2.f.c
Level $1134$
Weight $2$
Character orbit 1134.f
Analytic conductor $9.055$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.05503558921\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 378)
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.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -\zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} + q^{8} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -\zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} + q^{8} + q^{10} + ( -5 + 5 \zeta_{6} ) q^{11} + \zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} + 2 q^{17} - q^{19} + ( -1 + \zeta_{6} ) q^{20} -5 \zeta_{6} q^{22} + \zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} - q^{28} + ( -4 + 4 \zeta_{6} ) q^{29} + 9 \zeta_{6} q^{31} -\zeta_{6} q^{32} + ( -2 + 2 \zeta_{6} ) q^{34} - q^{35} + 5 q^{37} + ( 1 - \zeta_{6} ) q^{38} -\zeta_{6} q^{40} + 9 \zeta_{6} q^{41} + ( 10 - 10 \zeta_{6} ) q^{43} + 5 q^{44} - q^{46} + ( -6 + 6 \zeta_{6} ) q^{47} -\zeta_{6} q^{49} + 4 \zeta_{6} q^{50} + 12 q^{53} + 5 q^{55} + ( 1 - \zeta_{6} ) q^{56} -4 \zeta_{6} q^{58} + 14 \zeta_{6} q^{59} -9 q^{62} + q^{64} + 8 \zeta_{6} q^{67} -2 \zeta_{6} q^{68} + ( 1 - \zeta_{6} ) q^{70} -13 q^{71} -2 q^{73} + ( -5 + 5 \zeta_{6} ) q^{74} + \zeta_{6} q^{76} + 5 \zeta_{6} q^{77} + ( -6 + 6 \zeta_{6} ) q^{79} + q^{80} -9 q^{82} + ( 4 - 4 \zeta_{6} ) q^{83} -2 \zeta_{6} q^{85} + 10 \zeta_{6} q^{86} + ( -5 + 5 \zeta_{6} ) q^{88} -9 q^{89} + ( 1 - \zeta_{6} ) q^{92} -6 \zeta_{6} q^{94} + \zeta_{6} q^{95} + ( -16 + 16 \zeta_{6} ) q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - q^{5} + q^{7} + 2 q^{8} + O(q^{10}) \) \( 2 q - q^{2} - q^{4} - q^{5} + q^{7} + 2 q^{8} + 2 q^{10} - 5 q^{11} + q^{14} - q^{16} + 4 q^{17} - 2 q^{19} - q^{20} - 5 q^{22} + q^{23} + 4 q^{25} - 2 q^{28} - 4 q^{29} + 9 q^{31} - q^{32} - 2 q^{34} - 2 q^{35} + 10 q^{37} + q^{38} - q^{40} + 9 q^{41} + 10 q^{43} + 10 q^{44} - 2 q^{46} - 6 q^{47} - q^{49} + 4 q^{50} + 24 q^{53} + 10 q^{55} + q^{56} - 4 q^{58} + 14 q^{59} - 18 q^{62} + 2 q^{64} + 8 q^{67} - 2 q^{68} + q^{70} - 26 q^{71} - 4 q^{73} - 5 q^{74} + q^{76} + 5 q^{77} - 6 q^{79} + 2 q^{80} - 18 q^{82} + 4 q^{83} - 2 q^{85} + 10 q^{86} - 5 q^{88} - 18 q^{89} + q^{92} - 6 q^{94} + q^{95} - 16 q^{97} + 2 q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −0.500000 0.866025i 0 0.500000 0.866025i 1.00000 0 1.00000
757.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.500000 + 0.866025i 1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.2.f.c 2
3.b odd 2 1 1134.2.f.n 2
9.c even 3 1 378.2.a.f yes 1
9.c even 3 1 inner 1134.2.f.c 2
9.d odd 6 1 378.2.a.c 1
9.d odd 6 1 1134.2.f.n 2
36.f odd 6 1 3024.2.a.t 1
36.h even 6 1 3024.2.a.m 1
45.h odd 6 1 9450.2.a.dc 1
45.j even 6 1 9450.2.a.bx 1
63.l odd 6 1 2646.2.a.v 1
63.o even 6 1 2646.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.c 1 9.d odd 6 1
378.2.a.f yes 1 9.c even 3 1
1134.2.f.c 2 1.a even 1 1 trivial
1134.2.f.c 2 9.c even 3 1 inner
1134.2.f.n 2 3.b odd 2 1
1134.2.f.n 2 9.d odd 6 1
2646.2.a.i 1 63.o even 6 1
2646.2.a.v 1 63.l odd 6 1
3024.2.a.m 1 36.h even 6 1
3024.2.a.t 1 36.f odd 6 1
9450.2.a.bx 1 45.j even 6 1
9450.2.a.dc 1 45.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1134, [\chi])\):

\( T_{5}^{2} + T_{5} + 1 \)
\( T_{11}^{2} + 5 T_{11} + 25 \)
\( T_{13} \)
\( T_{17} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( 1 - T + T^{2} \)
$11$ \( 25 + 5 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( 1 - T + T^{2} \)
$29$ \( 16 + 4 T + T^{2} \)
$31$ \( 81 - 9 T + T^{2} \)
$37$ \( ( -5 + T )^{2} \)
$41$ \( 81 - 9 T + T^{2} \)
$43$ \( 100 - 10 T + T^{2} \)
$47$ \( 36 + 6 T + T^{2} \)
$53$ \( ( -12 + T )^{2} \)
$59$ \( 196 - 14 T + T^{2} \)
$61$ \( T^{2} \)
$67$ \( 64 - 8 T + T^{2} \)
$71$ \( ( 13 + T )^{2} \)
$73$ \( ( 2 + T )^{2} \)
$79$ \( 36 + 6 T + T^{2} \)
$83$ \( 16 - 4 T + T^{2} \)
$89$ \( ( 9 + T )^{2} \)
$97$ \( 256 + 16 T + T^{2} \)
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