Properties

Label 1134.2.g.e
Level $1134$
Weight $2$
Character orbit 1134.g
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.g (of order \(3\), degree \(2\), 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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} -3 \zeta_{6} q^{5} + ( -2 + 3 \zeta_{6} ) q^{7} - q^{8} +O(q^{10})\) \( q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} -3 \zeta_{6} q^{5} + ( -2 + 3 \zeta_{6} ) q^{7} - q^{8} + ( 3 - 3 \zeta_{6} ) q^{10} + ( 3 - 3 \zeta_{6} ) q^{11} - q^{13} + ( -3 + \zeta_{6} ) q^{14} -\zeta_{6} q^{16} + ( -3 + 3 \zeta_{6} ) q^{17} + 7 \zeta_{6} q^{19} + 3 q^{20} + 3 q^{22} + 9 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} -\zeta_{6} q^{26} + ( -1 - 2 \zeta_{6} ) q^{28} + 3 q^{29} + ( -8 + 8 \zeta_{6} ) q^{31} + ( 1 - \zeta_{6} ) q^{32} -3 q^{34} + ( 9 - 3 \zeta_{6} ) q^{35} + \zeta_{6} q^{37} + ( -7 + 7 \zeta_{6} ) q^{38} + 3 \zeta_{6} q^{40} + 3 q^{41} - q^{43} + 3 \zeta_{6} q^{44} + ( -9 + 9 \zeta_{6} ) q^{46} + ( -5 - 3 \zeta_{6} ) q^{49} -4 q^{50} + ( 1 - \zeta_{6} ) q^{52} + ( -3 + 3 \zeta_{6} ) q^{53} -9 q^{55} + ( 2 - 3 \zeta_{6} ) q^{56} + 3 \zeta_{6} q^{58} -2 \zeta_{6} q^{61} -8 q^{62} + q^{64} + 3 \zeta_{6} q^{65} + ( 4 - 4 \zeta_{6} ) q^{67} -3 \zeta_{6} q^{68} + ( 3 + 6 \zeta_{6} ) q^{70} + 12 q^{71} + ( -11 + 11 \zeta_{6} ) q^{73} + ( -1 + \zeta_{6} ) q^{74} -7 q^{76} + ( 3 + 6 \zeta_{6} ) q^{77} + 16 \zeta_{6} q^{79} + ( -3 + 3 \zeta_{6} ) q^{80} + 3 \zeta_{6} q^{82} -9 q^{83} + 9 q^{85} -\zeta_{6} q^{86} + ( -3 + 3 \zeta_{6} ) q^{88} -3 \zeta_{6} q^{89} + ( 2 - 3 \zeta_{6} ) q^{91} -9 q^{92} + ( 21 - 21 \zeta_{6} ) q^{95} - q^{97} + ( 3 - 8 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} - 3q^{5} - q^{7} - 2q^{8} + O(q^{10}) \) \( 2q + q^{2} - q^{4} - 3q^{5} - q^{7} - 2q^{8} + 3q^{10} + 3q^{11} - 2q^{13} - 5q^{14} - q^{16} - 3q^{17} + 7q^{19} + 6q^{20} + 6q^{22} + 9q^{23} - 4q^{25} - q^{26} - 4q^{28} + 6q^{29} - 8q^{31} + q^{32} - 6q^{34} + 15q^{35} + q^{37} - 7q^{38} + 3q^{40} + 6q^{41} - 2q^{43} + 3q^{44} - 9q^{46} - 13q^{49} - 8q^{50} + q^{52} - 3q^{53} - 18q^{55} + q^{56} + 3q^{58} - 2q^{61} - 16q^{62} + 2q^{64} + 3q^{65} + 4q^{67} - 3q^{68} + 12q^{70} + 24q^{71} - 11q^{73} - q^{74} - 14q^{76} + 12q^{77} + 16q^{79} - 3q^{80} + 3q^{82} - 18q^{83} + 18q^{85} - q^{86} - 3q^{88} - 3q^{89} + q^{91} - 18q^{92} + 21q^{95} - 2q^{97} - 2q^{98} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i −1.50000 + 2.59808i 0 −0.500000 2.59808i −1.00000 0 1.50000 + 2.59808i
487.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.50000 2.59808i 0 −0.500000 + 2.59808i −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
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 1134.2.g.e 2
3.b odd 2 1 1134.2.g.c 2
7.c even 3 1 inner 1134.2.g.e 2
7.c even 3 1 7938.2.a.m 1
7.d odd 6 1 7938.2.a.b 1
9.c even 3 1 126.2.e.a 2
9.c even 3 1 126.2.h.b yes 2
9.d odd 6 1 378.2.e.b 2
9.d odd 6 1 378.2.h.a 2
21.g even 6 1 7938.2.a.be 1
21.h odd 6 1 1134.2.g.c 2
21.h odd 6 1 7938.2.a.t 1
36.f odd 6 1 1008.2.q.a 2
36.f odd 6 1 1008.2.t.f 2
36.h even 6 1 3024.2.q.f 2
36.h even 6 1 3024.2.t.a 2
63.g even 3 1 126.2.e.a 2
63.g even 3 1 882.2.f.i 2
63.h even 3 1 126.2.h.b yes 2
63.h even 3 1 882.2.f.i 2
63.i even 6 1 2646.2.f.a 2
63.i even 6 1 2646.2.h.d 2
63.j odd 6 1 378.2.h.a 2
63.j odd 6 1 2646.2.f.d 2
63.k odd 6 1 882.2.e.c 2
63.k odd 6 1 882.2.f.g 2
63.l odd 6 1 882.2.e.c 2
63.l odd 6 1 882.2.h.i 2
63.n odd 6 1 378.2.e.b 2
63.n odd 6 1 2646.2.f.d 2
63.o even 6 1 2646.2.e.g 2
63.o even 6 1 2646.2.h.d 2
63.s even 6 1 2646.2.e.g 2
63.s even 6 1 2646.2.f.a 2
63.t odd 6 1 882.2.f.g 2
63.t odd 6 1 882.2.h.i 2
252.o even 6 1 3024.2.q.f 2
252.u odd 6 1 1008.2.t.f 2
252.bb even 6 1 3024.2.t.a 2
252.bl odd 6 1 1008.2.q.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.a 2 9.c even 3 1
126.2.e.a 2 63.g even 3 1
126.2.h.b yes 2 9.c even 3 1
126.2.h.b yes 2 63.h even 3 1
378.2.e.b 2 9.d odd 6 1
378.2.e.b 2 63.n odd 6 1
378.2.h.a 2 9.d odd 6 1
378.2.h.a 2 63.j odd 6 1
882.2.e.c 2 63.k odd 6 1
882.2.e.c 2 63.l odd 6 1
882.2.f.g 2 63.k odd 6 1
882.2.f.g 2 63.t odd 6 1
882.2.f.i 2 63.g even 3 1
882.2.f.i 2 63.h even 3 1
882.2.h.i 2 63.l odd 6 1
882.2.h.i 2 63.t odd 6 1
1008.2.q.a 2 36.f odd 6 1
1008.2.q.a 2 252.bl odd 6 1
1008.2.t.f 2 36.f odd 6 1
1008.2.t.f 2 252.u odd 6 1
1134.2.g.c 2 3.b odd 2 1
1134.2.g.c 2 21.h odd 6 1
1134.2.g.e 2 1.a even 1 1 trivial
1134.2.g.e 2 7.c even 3 1 inner
2646.2.e.g 2 63.o even 6 1
2646.2.e.g 2 63.s even 6 1
2646.2.f.a 2 63.i even 6 1
2646.2.f.a 2 63.s even 6 1
2646.2.f.d 2 63.j odd 6 1
2646.2.f.d 2 63.n odd 6 1
2646.2.h.d 2 63.i even 6 1
2646.2.h.d 2 63.o even 6 1
3024.2.q.f 2 36.h even 6 1
3024.2.q.f 2 252.o even 6 1
3024.2.t.a 2 36.h even 6 1
3024.2.t.a 2 252.bb even 6 1
7938.2.a.b 1 7.d odd 6 1
7938.2.a.m 1 7.c even 3 1
7938.2.a.t 1 21.h odd 6 1
7938.2.a.be 1 21.g even 6 1

Hecke kernels

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

\( T_{5}^{2} + 3 T_{5} + 9 \)
\( T_{11}^{2} - 3 T_{11} + 9 \)
\( T_{17}^{2} + 3 T_{17} + 9 \)
\( T_{23}^{2} - 9 T_{23} + 81 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ 1
$5$ \( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \)
$7$ \( 1 + T + 7 T^{2} \)
$11$ \( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \)
$13$ \( ( 1 + T + 13 T^{2} )^{2} \)
$17$ \( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \)
$23$ \( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} \)
$29$ \( ( 1 - 3 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \)
$41$ \( ( 1 - 3 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + T + 43 T^{2} )^{2} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 9 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + T + 97 T^{2} )^{2} \)
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