Properties

Label 1134.2.h.j
Level $1134$
Weight $2$
Character orbit 1134.h
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.h (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 126)
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -3 q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} - q^{8} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -3 q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} - q^{8} + ( -3 + 3 \zeta_{6} ) q^{10} + 3 q^{11} + ( -2 + 2 \zeta_{6} ) q^{13} + ( -1 + 3 \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( 6 - 6 \zeta_{6} ) q^{17} -2 \zeta_{6} q^{19} + 3 \zeta_{6} q^{20} + ( 3 - 3 \zeta_{6} ) q^{22} + 6 q^{23} + 4 q^{25} + 2 \zeta_{6} q^{26} + ( 2 + \zeta_{6} ) q^{28} + 9 \zeta_{6} q^{29} + 7 \zeta_{6} q^{31} + \zeta_{6} q^{32} -6 \zeta_{6} q^{34} + ( 9 - 6 \zeta_{6} ) q^{35} + 10 \zeta_{6} q^{37} -2 q^{38} + 3 q^{40} + 4 \zeta_{6} q^{43} -3 \zeta_{6} q^{44} + ( 6 - 6 \zeta_{6} ) q^{46} + ( 12 - 12 \zeta_{6} ) q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + ( 4 - 4 \zeta_{6} ) q^{50} + 2 q^{52} + ( -3 + 3 \zeta_{6} ) q^{53} -9 q^{55} + ( 3 - 2 \zeta_{6} ) q^{56} + 9 q^{58} -3 \zeta_{6} q^{59} + ( 4 - 4 \zeta_{6} ) q^{61} + 7 q^{62} + q^{64} + ( 6 - 6 \zeta_{6} ) q^{65} -2 \zeta_{6} q^{67} -6 q^{68} + ( 3 - 9 \zeta_{6} ) q^{70} + ( -2 + 2 \zeta_{6} ) q^{73} + 10 q^{74} + ( -2 + 2 \zeta_{6} ) q^{76} + ( -9 + 6 \zeta_{6} ) q^{77} + ( -5 + 5 \zeta_{6} ) q^{79} + ( 3 - 3 \zeta_{6} ) q^{80} + 9 \zeta_{6} q^{83} + ( -18 + 18 \zeta_{6} ) q^{85} + 4 q^{86} -3 q^{88} -6 \zeta_{6} q^{89} + ( 2 - 6 \zeta_{6} ) q^{91} -6 \zeta_{6} q^{92} -12 \zeta_{6} q^{94} + 6 \zeta_{6} q^{95} + 13 \zeta_{6} q^{97} + ( -3 - 5 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} - 6q^{5} - 4q^{7} - 2q^{8} + O(q^{10}) \) \( 2q + q^{2} - q^{4} - 6q^{5} - 4q^{7} - 2q^{8} - 3q^{10} + 6q^{11} - 2q^{13} + q^{14} - q^{16} + 6q^{17} - 2q^{19} + 3q^{20} + 3q^{22} + 12q^{23} + 8q^{25} + 2q^{26} + 5q^{28} + 9q^{29} + 7q^{31} + q^{32} - 6q^{34} + 12q^{35} + 10q^{37} - 4q^{38} + 6q^{40} + 4q^{43} - 3q^{44} + 6q^{46} + 12q^{47} + 2q^{49} + 4q^{50} + 4q^{52} - 3q^{53} - 18q^{55} + 4q^{56} + 18q^{58} - 3q^{59} + 4q^{61} + 14q^{62} + 2q^{64} + 6q^{65} - 2q^{67} - 12q^{68} - 3q^{70} - 2q^{73} + 20q^{74} - 2q^{76} - 12q^{77} - 5q^{79} + 3q^{80} + 9q^{83} - 18q^{85} + 8q^{86} - 6q^{88} - 6q^{89} - 2q^{91} - 6q^{92} - 12q^{94} + 6q^{95} + 13q^{97} - 11q^{98} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i −3.00000 0 −2.00000 + 1.73205i −1.00000 0 −1.50000 + 2.59808i
541.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −3.00000 0 −2.00000 1.73205i −1.00000 0 −1.50000 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g 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.h.j 2
3.b odd 2 1 1134.2.h.f 2
7.c even 3 1 1134.2.e.g 2
9.c even 3 1 126.2.g.d yes 2
9.c even 3 1 1134.2.e.g 2
9.d odd 6 1 126.2.g.a 2
9.d odd 6 1 1134.2.e.k 2
21.h odd 6 1 1134.2.e.k 2
36.f odd 6 1 1008.2.s.o 2
36.h even 6 1 1008.2.s.b 2
63.g even 3 1 882.2.a.a 1
63.g even 3 1 inner 1134.2.h.j 2
63.h even 3 1 126.2.g.d yes 2
63.i even 6 1 882.2.g.e 2
63.j odd 6 1 126.2.g.a 2
63.k odd 6 1 882.2.a.e 1
63.l odd 6 1 882.2.g.g 2
63.n odd 6 1 882.2.a.j 1
63.n odd 6 1 1134.2.h.f 2
63.o even 6 1 882.2.g.e 2
63.s even 6 1 882.2.a.h 1
63.t odd 6 1 882.2.g.g 2
252.n even 6 1 7056.2.a.bx 1
252.o even 6 1 7056.2.a.by 1
252.u odd 6 1 1008.2.s.o 2
252.bb even 6 1 1008.2.s.b 2
252.bl odd 6 1 7056.2.a.e 1
252.bn odd 6 1 7056.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.g.a 2 9.d odd 6 1
126.2.g.a 2 63.j odd 6 1
126.2.g.d yes 2 9.c even 3 1
126.2.g.d yes 2 63.h even 3 1
882.2.a.a 1 63.g even 3 1
882.2.a.e 1 63.k odd 6 1
882.2.a.h 1 63.s even 6 1
882.2.a.j 1 63.n odd 6 1
882.2.g.e 2 63.i even 6 1
882.2.g.e 2 63.o even 6 1
882.2.g.g 2 63.l odd 6 1
882.2.g.g 2 63.t odd 6 1
1008.2.s.b 2 36.h even 6 1
1008.2.s.b 2 252.bb even 6 1
1008.2.s.o 2 36.f odd 6 1
1008.2.s.o 2 252.u odd 6 1
1134.2.e.g 2 7.c even 3 1
1134.2.e.g 2 9.c even 3 1
1134.2.e.k 2 9.d odd 6 1
1134.2.e.k 2 21.h odd 6 1
1134.2.h.f 2 3.b odd 2 1
1134.2.h.f 2 63.n odd 6 1
1134.2.h.j 2 1.a even 1 1 trivial
1134.2.h.j 2 63.g even 3 1 inner
7056.2.a.e 1 252.bl odd 6 1
7056.2.a.h 1 252.bn odd 6 1
7056.2.a.bx 1 252.n even 6 1
7056.2.a.by 1 252.o even 6 1

Hecke kernels

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

\( T_{5} + 3 \)
\( T_{11} - 3 \)
\( T_{17}^{2} - 6 T_{17} + 36 \)
\( T_{23} - 6 \)