Properties

Label 1134.2.g.l
Level $1134$
Weight $2$
Character orbit 1134.g
Analytic conductor $9.055$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Defining polynomial: \(x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{4} ) q^{2} -\beta_{4} q^{4} -\beta_{2} q^{5} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{7} + q^{8} +O(q^{10})\) \( q + ( -1 + \beta_{4} ) q^{2} -\beta_{4} q^{4} -\beta_{2} q^{5} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{7} + q^{8} + ( \beta_{2} + \beta_{3} ) q^{10} + ( -\beta_{2} - \beta_{3} ) q^{11} + ( -3 - 2 \beta_{1} + \beta_{3} ) q^{13} + ( -1 - \beta_{2} - \beta_{3} - \beta_{5} ) q^{14} + ( -1 + \beta_{4} ) q^{16} + ( 2 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} ) q^{17} + ( -2 + 2 \beta_{4} - 3 \beta_{5} ) q^{19} -\beta_{3} q^{20} + \beta_{3} q^{22} + ( 2 - 2 \beta_{4} - \beta_{5} ) q^{23} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{25} + ( 3 + \beta_{2} - 3 \beta_{4} + 2 \beta_{5} ) q^{26} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{28} + ( 3 \beta_{1} + 2 \beta_{3} ) q^{29} + ( -\beta_{1} - \beta_{2} - \beta_{3} + 6 \beta_{4} + \beta_{5} ) q^{31} -\beta_{4} q^{32} + ( -2 - 2 \beta_{1} ) q^{34} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{35} + ( 1 - \beta_{4} ) q^{37} + ( -3 \beta_{1} - 2 \beta_{4} + 3 \beta_{5} ) q^{38} -\beta_{2} q^{40} + ( 1 + \beta_{1} + 2 \beta_{3} ) q^{41} + ( 2 - 3 \beta_{1} + 3 \beta_{3} ) q^{43} + \beta_{2} q^{44} + ( -\beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{46} + ( 3 - 3 \beta_{2} - 3 \beta_{4} - 3 \beta_{5} ) q^{47} + ( 2 \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{49} + ( -1 + \beta_{1} - 2 \beta_{3} ) q^{50} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{52} + ( \beta_{1} - \beta_{2} - \beta_{3} - 5 \beta_{4} - \beta_{5} ) q^{53} + ( -4 - \beta_{1} + 2 \beta_{3} ) q^{55} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{56} + ( 2 \beta_{2} - 3 \beta_{5} ) q^{58} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} + \beta_{5} ) q^{59} + ( 3 - 2 \beta_{2} - 3 \beta_{4} - \beta_{5} ) q^{61} + ( -6 + \beta_{1} + \beta_{3} ) q^{62} + q^{64} + ( 2 + 5 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{65} + ( \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{67} + ( 2 - 2 \beta_{4} + 2 \beta_{5} ) q^{68} + ( -4 + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{70} + ( -4 - 7 \beta_{1} + 2 \beta_{3} ) q^{71} + ( \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 8 \beta_{4} - \beta_{5} ) q^{73} + \beta_{4} q^{74} + ( 2 + 3 \beta_{1} ) q^{76} + ( 4 - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{77} + ( \beta_{2} - 4 \beta_{5} ) q^{79} + ( \beta_{2} + \beta_{3} ) q^{80} + ( -1 + 2 \beta_{2} + \beta_{4} - \beta_{5} ) q^{82} + ( 1 - 2 \beta_{1} + 3 \beta_{3} ) q^{83} + ( 2 + 2 \beta_{1} + 2 \beta_{3} ) q^{85} + ( -2 + 3 \beta_{2} + 2 \beta_{4} + 3 \beta_{5} ) q^{86} + ( -\beta_{2} - \beta_{3} ) q^{88} + ( 2 - \beta_{2} - 2 \beta_{4} - 4 \beta_{5} ) q^{89} + ( -7 - 2 \beta_{1} - 4 \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{91} + ( -2 + \beta_{1} ) q^{92} + ( -3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{94} + ( 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{95} + ( -8 + 2 \beta_{1} + 2 \beta_{3} ) q^{97} + ( 1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 3q^{2} - 3q^{4} - q^{5} + 2q^{7} + 6q^{8} + O(q^{10}) \) \( 6q - 3q^{2} - 3q^{4} - q^{5} + 2q^{7} + 6q^{8} - q^{10} + q^{11} - 16q^{13} - 4q^{14} - 3q^{16} + 4q^{17} - 3q^{19} + 2q^{20} - 2q^{22} + 7q^{23} + 2q^{25} + 8q^{26} + 2q^{28} - 10q^{29} + 20q^{31} - 3q^{32} - 8q^{34} + 13q^{35} + 3q^{37} - 3q^{38} - q^{40} + 12q^{43} + q^{44} + 7q^{46} + 9q^{47} - 4q^{50} + 8q^{52} - 15q^{53} - 26q^{55} + 2q^{56} + 5q^{58} + 14q^{59} + 8q^{61} - 40q^{62} + 6q^{64} + 12q^{65} + q^{67} + 4q^{68} - 23q^{70} - 14q^{71} + 19q^{73} + 3q^{74} + 6q^{76} + 23q^{77} + 5q^{79} - q^{80} + 4q^{83} + 4q^{85} - 6q^{86} + q^{88} + 9q^{89} - 46q^{91} - 14q^{92} + 9q^{94} + 4q^{95} - 56q^{97} + 12q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - \nu + 2 \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 8 \nu^{3} + 5 \nu^{2} - 18 \nu + 6 \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{4} - 2 \nu^{3} + 6 \nu^{2} - 5 \nu + 3 \)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 9 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 22 \nu^{2} + 30 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_{1} + 2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 4 \beta_{1} - 4\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(7 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + \beta_{1} - 10\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(16 \beta_{5} + 11 \beta_{4} + 8 \beta_{3} + 10 \beta_{2} - 17 \beta_{1} + 5\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-14 \beta_{5} - 16 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} - 23 \beta_{1} + 47\)\()/3\)

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(1\)

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 + 1.41036i
0.500000 2.05195i
0.500000 0.224437i
0.500000 1.41036i
0.500000 + 2.05195i
0.500000 + 0.224437i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −1.59097 + 2.75564i 0 1.85185 1.88962i 1.00000 0 −1.59097 2.75564i
163.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.296790 0.514055i 0 −2.25729 + 1.38008i 1.00000 0 0.296790 + 0.514055i
163.3 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.794182 1.37556i 0 1.40545 + 2.24159i 1.00000 0 0.794182 + 1.37556i
487.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −1.59097 2.75564i 0 1.85185 + 1.88962i 1.00000 0 −1.59097 + 2.75564i
487.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0.296790 + 0.514055i 0 −2.25729 1.38008i 1.00000 0 0.296790 0.514055i
487.3 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0.794182 + 1.37556i 0 1.40545 2.24159i 1.00000 0 0.794182 1.37556i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 487.3
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.l 6
3.b odd 2 1 1134.2.g.m 6
7.c even 3 1 inner 1134.2.g.l 6
7.c even 3 1 7938.2.a.ca 3
7.d odd 6 1 7938.2.a.bz 3
9.c even 3 1 378.2.e.d 6
9.c even 3 1 378.2.h.c 6
9.d odd 6 1 126.2.e.c 6
9.d odd 6 1 126.2.h.d yes 6
21.g even 6 1 7938.2.a.bw 3
21.h odd 6 1 1134.2.g.m 6
21.h odd 6 1 7938.2.a.bv 3
36.f odd 6 1 3024.2.q.g 6
36.f odd 6 1 3024.2.t.h 6
36.h even 6 1 1008.2.q.g 6
36.h even 6 1 1008.2.t.h 6
63.g even 3 1 378.2.e.d 6
63.g even 3 1 2646.2.f.l 6
63.h even 3 1 378.2.h.c 6
63.h even 3 1 2646.2.f.l 6
63.i even 6 1 882.2.f.o 6
63.i even 6 1 882.2.h.p 6
63.j odd 6 1 126.2.h.d yes 6
63.j odd 6 1 882.2.f.n 6
63.k odd 6 1 2646.2.e.p 6
63.k odd 6 1 2646.2.f.m 6
63.l odd 6 1 2646.2.e.p 6
63.l odd 6 1 2646.2.h.o 6
63.n odd 6 1 126.2.e.c 6
63.n odd 6 1 882.2.f.n 6
63.o even 6 1 882.2.e.o 6
63.o even 6 1 882.2.h.p 6
63.s even 6 1 882.2.e.o 6
63.s even 6 1 882.2.f.o 6
63.t odd 6 1 2646.2.f.m 6
63.t odd 6 1 2646.2.h.o 6
252.o even 6 1 1008.2.q.g 6
252.u odd 6 1 3024.2.t.h 6
252.bb even 6 1 1008.2.t.h 6
252.bl odd 6 1 3024.2.q.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.c 6 9.d odd 6 1
126.2.e.c 6 63.n odd 6 1
126.2.h.d yes 6 9.d odd 6 1
126.2.h.d yes 6 63.j odd 6 1
378.2.e.d 6 9.c even 3 1
378.2.e.d 6 63.g even 3 1
378.2.h.c 6 9.c even 3 1
378.2.h.c 6 63.h even 3 1
882.2.e.o 6 63.o even 6 1
882.2.e.o 6 63.s even 6 1
882.2.f.n 6 63.j odd 6 1
882.2.f.n 6 63.n odd 6 1
882.2.f.o 6 63.i even 6 1
882.2.f.o 6 63.s even 6 1
882.2.h.p 6 63.i even 6 1
882.2.h.p 6 63.o even 6 1
1008.2.q.g 6 36.h even 6 1
1008.2.q.g 6 252.o even 6 1
1008.2.t.h 6 36.h even 6 1
1008.2.t.h 6 252.bb even 6 1
1134.2.g.l 6 1.a even 1 1 trivial
1134.2.g.l 6 7.c even 3 1 inner
1134.2.g.m 6 3.b odd 2 1
1134.2.g.m 6 21.h odd 6 1
2646.2.e.p 6 63.k odd 6 1
2646.2.e.p 6 63.l odd 6 1
2646.2.f.l 6 63.g even 3 1
2646.2.f.l 6 63.h even 3 1
2646.2.f.m 6 63.k odd 6 1
2646.2.f.m 6 63.t odd 6 1
2646.2.h.o 6 63.l odd 6 1
2646.2.h.o 6 63.t odd 6 1
3024.2.q.g 6 36.f odd 6 1
3024.2.q.g 6 252.bl odd 6 1
3024.2.t.h 6 36.f odd 6 1
3024.2.t.h 6 252.u odd 6 1
7938.2.a.bv 3 21.h odd 6 1
7938.2.a.bw 3 21.g even 6 1
7938.2.a.bz 3 7.d odd 6 1
7938.2.a.ca 3 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 \(S_{2}^{\mathrm{new}}(1134, [\chi])\):

\( T_{5}^{6} + T_{5}^{5} + 7 T_{5}^{4} - 12 T_{5}^{3} + 33 T_{5}^{2} - 18 T_{5} + 9 \)
\( T_{11}^{6} - T_{11}^{5} + 7 T_{11}^{4} + 12 T_{11}^{3} + 33 T_{11}^{2} + 18 T_{11} + 9 \)
\( T_{17}^{6} - 4 T_{17}^{5} + 28 T_{17}^{4} + 240 T_{17}^{2} - 288 T_{17} + 576 \)
\( T_{23}^{6} - 7 T_{23}^{5} + 37 T_{23}^{4} - 78 T_{23}^{3} + 123 T_{23}^{2} - 36 T_{23} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{3} \)
$3$ 1
$5$ \( 1 + T - 8 T^{2} - 17 T^{3} + 23 T^{4} + 52 T^{5} - 11 T^{6} + 260 T^{7} + 575 T^{8} - 2125 T^{9} - 5000 T^{10} + 3125 T^{11} + 15625 T^{12} \)
$7$ \( 1 - 2 T + 2 T^{2} + 19 T^{3} + 14 T^{4} - 98 T^{5} + 343 T^{6} \)
$11$ \( 1 - T - 26 T^{2} + 23 T^{3} + 407 T^{4} - 202 T^{5} - 4853 T^{6} - 2222 T^{7} + 49247 T^{8} + 30613 T^{9} - 380666 T^{10} - 161051 T^{11} + 1771561 T^{12} \)
$13$ \( ( 1 + 8 T + 40 T^{2} + 139 T^{3} + 520 T^{4} + 1352 T^{5} + 2197 T^{6} )^{2} \)
$17$ \( 1 - 4 T - 23 T^{2} + 68 T^{3} + 410 T^{4} - 220 T^{5} - 8111 T^{6} - 3740 T^{7} + 118490 T^{8} + 334084 T^{9} - 1920983 T^{10} - 5679428 T^{11} + 24137569 T^{12} \)
$19$ \( 1 + 3 T - 12 T^{2} - 67 T^{3} - 153 T^{4} + 54 T^{5} + 6315 T^{6} + 1026 T^{7} - 55233 T^{8} - 459553 T^{9} - 1563852 T^{10} + 7428297 T^{11} + 47045881 T^{12} \)
$23$ \( 1 - 7 T - 32 T^{2} + 83 T^{3} + 2423 T^{4} - 3946 T^{5} - 46865 T^{6} - 90758 T^{7} + 1281767 T^{8} + 1009861 T^{9} - 8954912 T^{10} - 45054401 T^{11} + 148035889 T^{12} \)
$29$ \( ( 1 + 5 T + 21 T^{2} - 73 T^{3} + 609 T^{4} + 4205 T^{5} + 24389 T^{6} )^{2} \)
$31$ \( 1 - 20 T + 186 T^{2} - 1398 T^{3} + 10342 T^{4} - 62234 T^{5} + 331987 T^{6} - 1929254 T^{7} + 9938662 T^{8} - 41647818 T^{9} + 171774906 T^{10} - 572583020 T^{11} + 887503681 T^{12} \)
$37$ \( ( 1 - 11 T + 37 T^{2} )^{3}( 1 + 10 T + 37 T^{2} )^{3} \)
$41$ \( ( 1 + 90 T^{2} - 9 T^{3} + 3690 T^{4} + 68921 T^{6} )^{2} \)
$43$ \( ( 1 - 6 T + 60 T^{2} - 389 T^{3} + 2580 T^{4} - 11094 T^{5} + 79507 T^{6} )^{2} \)
$47$ \( 1 - 9 T - 6 T^{2} + 531 T^{3} - 2433 T^{4} - 3438 T^{5} + 104623 T^{6} - 161586 T^{7} - 5374497 T^{8} + 55130013 T^{9} - 29278086 T^{10} - 2064105063 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 + 15 T + 33 T^{3} + 13635 T^{4} + 60360 T^{5} - 225155 T^{6} + 3199080 T^{7} + 38300715 T^{8} + 4912941 T^{9} + 6272932395 T^{11} + 22164361129 T^{12} \)
$59$ \( 1 - 14 T - 20 T^{2} + 154 T^{3} + 11666 T^{4} - 35126 T^{5} - 499301 T^{6} - 2072434 T^{7} + 40609346 T^{8} + 31628366 T^{9} - 242347220 T^{10} - 10008940186 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 - 8 T - 114 T^{2} + 342 T^{3} + 13762 T^{4} - 13214 T^{5} - 937217 T^{6} - 806054 T^{7} + 51208402 T^{8} + 77627502 T^{9} - 1578425874 T^{10} - 6756770408 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 - T - 88 T^{2} - 243 T^{3} + 2035 T^{4} + 14290 T^{5} + 72259 T^{6} + 957430 T^{7} + 9135115 T^{8} - 73085409 T^{9} - 1773298648 T^{10} - 1350125107 T^{11} + 90458382169 T^{12} \)
$71$ \( ( 1 + 7 T + 15 T^{2} - 599 T^{3} + 1065 T^{4} + 35287 T^{5} + 357911 T^{6} )^{2} \)
$73$ \( 1 - 19 T + 134 T^{2} - 27 T^{3} - 5759 T^{4} + 41986 T^{5} - 314903 T^{6} + 3064978 T^{7} - 30689711 T^{8} - 10503459 T^{9} + 3805364294 T^{10} - 39388360267 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 - 5 T - 138 T^{2} + 123 T^{3} + 11347 T^{4} + 21118 T^{5} - 1048937 T^{6} + 1668322 T^{7} + 70816627 T^{8} + 60643797 T^{9} - 5375111178 T^{10} - 15385281995 T^{11} + 243087455521 T^{12} \)
$83$ \( ( 1 - 2 T + 186 T^{2} - 185 T^{3} + 15438 T^{4} - 13778 T^{5} + 571787 T^{6} )^{2} \)
$89$ \( 1 - 9 T - 144 T^{2} + 1197 T^{3} + 16101 T^{4} - 73314 T^{5} - 1141967 T^{6} - 6524946 T^{7} + 127536021 T^{8} + 843847893 T^{9} - 9034882704 T^{10} - 50256535041 T^{11} + 496981290961 T^{12} \)
$97$ \( ( 1 + 28 T + 503 T^{2} + 5680 T^{3} + 48791 T^{4} + 263452 T^{5} + 912673 T^{6} )^{2} \)
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