Properties

Label 1134.3.q.c
Level $1134$
Weight $3$
Character orbit 1134.q
Analytic conductor $30.899$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1134.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.8992619785\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12745506816.5
Defining polynomial: \(x^{8} - 8 x^{6} + 55 x^{4} - 72 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + ( 2 - 2 \beta_{2} ) q^{4} + ( \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{5} + \beta_{5} q^{7} + 2 \beta_{6} q^{8} +O(q^{10})\) \( q + \beta_{4} q^{2} + ( 2 - 2 \beta_{2} ) q^{4} + ( \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{5} + \beta_{5} q^{7} + 2 \beta_{6} q^{8} + ( 2 + 4 \beta_{1} ) q^{10} + ( -4 \beta_{3} - 2 \beta_{4} ) q^{11} + ( 8 - 4 \beta_{1} - 8 \beta_{2} - 4 \beta_{5} ) q^{13} + \beta_{7} q^{14} -4 \beta_{2} q^{16} + ( -2 \beta_{3} - 13 \beta_{6} - 2 \beta_{7} ) q^{17} + 20 q^{19} + ( 4 \beta_{3} + 2 \beta_{4} ) q^{20} + ( -4 - 8 \beta_{1} + 4 \beta_{2} - 8 \beta_{5} ) q^{22} + ( 2 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} ) q^{23} + ( 33 \beta_{2} - 8 \beta_{5} ) q^{25} + ( -4 \beta_{3} + 8 \beta_{6} - 4 \beta_{7} ) q^{26} -2 \beta_{1} q^{28} + ( 4 \beta_{3} - 19 \beta_{4} ) q^{29} + ( -4 + 8 \beta_{1} + 4 \beta_{2} + 8 \beta_{5} ) q^{31} + ( -4 \beta_{4} + 4 \beta_{6} ) q^{32} + ( 26 \beta_{2} - 4 \beta_{5} ) q^{34} + ( \beta_{3} + 14 \beta_{6} + \beta_{7} ) q^{35} + 38 q^{37} + 20 \beta_{4} q^{38} + ( 4 + 8 \beta_{1} - 4 \beta_{2} + 8 \beta_{5} ) q^{40} + ( -27 \beta_{4} + 27 \beta_{6} - 6 \beta_{7} ) q^{41} + ( -20 \beta_{2} - 24 \beta_{5} ) q^{43} + ( -8 \beta_{3} - 4 \beta_{6} - 8 \beta_{7} ) q^{44} + ( 4 + 8 \beta_{1} ) q^{46} + 12 \beta_{4} q^{47} + ( -7 + 7 \beta_{2} ) q^{49} + ( 33 \beta_{4} - 33 \beta_{6} - 8 \beta_{7} ) q^{50} + ( -16 \beta_{2} - 8 \beta_{5} ) q^{52} + ( -24 \beta_{3} + 3 \beta_{6} - 24 \beta_{7} ) q^{53} + ( -116 - 16 \beta_{1} ) q^{55} -2 \beta_{3} q^{56} + ( -38 + 8 \beta_{1} + 38 \beta_{2} + 8 \beta_{5} ) q^{58} + ( -20 \beta_{4} + 20 \beta_{6} - 8 \beta_{7} ) q^{59} + ( -58 \beta_{2} + 16 \beta_{5} ) q^{61} + ( 8 \beta_{3} - 4 \beta_{6} + 8 \beta_{7} ) q^{62} -8 q^{64} + ( 12 \beta_{3} - 48 \beta_{4} ) q^{65} + ( 48 - 32 \beta_{1} - 48 \beta_{2} - 32 \beta_{5} ) q^{67} + ( 26 \beta_{4} - 26 \beta_{6} - 4 \beta_{7} ) q^{68} + ( -28 \beta_{2} + 2 \beta_{5} ) q^{70} + ( 4 \beta_{3} + 2 \beta_{6} + 4 \beta_{7} ) q^{71} + ( -24 + 20 \beta_{1} ) q^{73} + 38 \beta_{4} q^{74} + ( 40 - 40 \beta_{2} ) q^{76} + ( 28 \beta_{4} - 28 \beta_{6} - 2 \beta_{7} ) q^{77} + ( -76 \beta_{2} + 16 \beta_{5} ) q^{79} + ( 8 \beta_{3} + 4 \beta_{6} + 8 \beta_{7} ) q^{80} + ( -54 + 12 \beta_{1} ) q^{82} + ( 8 \beta_{3} + 64 \beta_{4} ) q^{83} + ( -82 - 56 \beta_{1} + 82 \beta_{2} - 56 \beta_{5} ) q^{85} + ( -20 \beta_{4} + 20 \beta_{6} - 24 \beta_{7} ) q^{86} + ( 8 \beta_{2} - 16 \beta_{5} ) q^{88} + ( -18 \beta_{3} - 51 \beta_{6} - 18 \beta_{7} ) q^{89} + ( 28 - 8 \beta_{1} ) q^{91} + ( 8 \beta_{3} + 4 \beta_{4} ) q^{92} + ( 24 - 24 \beta_{2} ) q^{94} + ( 20 \beta_{4} - 20 \beta_{6} - 40 \beta_{7} ) q^{95} + ( 72 \beta_{2} + 44 \beta_{5} ) q^{97} -7 \beta_{6} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{4} + O(q^{10}) \) \( 8q + 8q^{4} + 16q^{10} + 32q^{13} - 16q^{16} + 160q^{19} - 16q^{22} + 132q^{25} - 16q^{31} + 104q^{34} + 304q^{37} + 16q^{40} - 80q^{43} + 32q^{46} - 28q^{49} - 64q^{52} - 928q^{55} - 152q^{58} - 232q^{61} - 64q^{64} + 192q^{67} - 112q^{70} - 192q^{73} + 160q^{76} - 304q^{79} - 432q^{82} - 328q^{85} + 32q^{88} + 224q^{91} + 96q^{94} + 288q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 8 x^{6} + 55 x^{4} - 72 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{6} - 148 \)\()/55\)
\(\beta_{2}\)\(=\)\((\)\( -8 \nu^{6} + 55 \nu^{4} - 440 \nu^{2} + 576 \)\()/495\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 533 \nu \)\()/165\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - 203 \nu \)\()/165\)
\(\beta_{5}\)\(=\)\((\)\( -23 \nu^{6} + 220 \nu^{4} - 1265 \nu^{2} + 1656 \)\()/495\)
\(\beta_{6}\)\(=\)\((\)\( -8 \nu^{7} + 55 \nu^{5} - 341 \nu^{3} + 81 \nu \)\()/297\)
\(\beta_{7}\)\(=\)\((\)\( 79 \nu^{7} - 605 \nu^{5} + 4345 \nu^{3} - 5688 \nu \)\()/1485\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{5} - 4 \beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{7} + 11 \beta_{6} + 5 \beta_{3}\)\()/2\)
\(\nu^{4}\)\(=\)\(8 \beta_{5} - 23 \beta_{2}\)
\(\nu^{5}\)\(=\)\((\)\(31 \beta_{7} + 79 \beta_{6} - 79 \beta_{4}\)\()/2\)
\(\nu^{6}\)\(=\)\(-55 \beta_{1} - 148\)
\(\nu^{7}\)\(=\)\((\)\(-533 \beta_{4} - 203 \beta_{3}\)\()/2\)

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1\) \(1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1
−2.23256 + 1.28897i
1.00781 0.581861i
−1.00781 + 0.581861i
2.23256 1.28897i
−2.23256 1.28897i
1.00781 + 0.581861i
−1.00781 0.581861i
2.23256 + 1.28897i
−1.22474 + 0.707107i 0 1.00000 1.73205i −7.70549 4.44876i 0 −1.32288 2.29129i 2.82843i 0 12.5830
701.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 5.25600 + 3.03455i 0 1.32288 + 2.29129i 2.82843i 0 −8.58301
701.3 1.22474 0.707107i 0 1.00000 1.73205i −5.25600 3.03455i 0 1.32288 + 2.29129i 2.82843i 0 −8.58301
701.4 1.22474 0.707107i 0 1.00000 1.73205i 7.70549 + 4.44876i 0 −1.32288 2.29129i 2.82843i 0 12.5830
1079.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −7.70549 + 4.44876i 0 −1.32288 + 2.29129i 2.82843i 0 12.5830
1079.2 −1.22474 0.707107i 0 1.00000 + 1.73205i 5.25600 3.03455i 0 1.32288 2.29129i 2.82843i 0 −8.58301
1079.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −5.25600 + 3.03455i 0 1.32288 2.29129i 2.82843i 0 −8.58301
1079.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 7.70549 4.44876i 0 −1.32288 + 2.29129i 2.82843i 0 12.5830
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1079.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.3.q.c 8
3.b odd 2 1 inner 1134.3.q.c 8
9.c even 3 1 126.3.b.a 4
9.c even 3 1 inner 1134.3.q.c 8
9.d odd 6 1 126.3.b.a 4
9.d odd 6 1 inner 1134.3.q.c 8
36.f odd 6 1 1008.3.d.a 4
36.h even 6 1 1008.3.d.a 4
45.h odd 6 1 3150.3.e.e 4
45.j even 6 1 3150.3.e.e 4
45.k odd 12 2 3150.3.c.b 8
45.l even 12 2 3150.3.c.b 8
63.g even 3 1 882.3.s.e 8
63.h even 3 1 882.3.s.e 8
63.i even 6 1 882.3.s.i 8
63.j odd 6 1 882.3.s.e 8
63.k odd 6 1 882.3.s.i 8
63.l odd 6 1 882.3.b.f 4
63.n odd 6 1 882.3.s.e 8
63.o even 6 1 882.3.b.f 4
63.s even 6 1 882.3.s.i 8
63.t odd 6 1 882.3.s.i 8
72.j odd 6 1 4032.3.d.i 4
72.l even 6 1 4032.3.d.j 4
72.n even 6 1 4032.3.d.i 4
72.p odd 6 1 4032.3.d.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.b.a 4 9.c even 3 1
126.3.b.a 4 9.d odd 6 1
882.3.b.f 4 63.l odd 6 1
882.3.b.f 4 63.o even 6 1
882.3.s.e 8 63.g even 3 1
882.3.s.e 8 63.h even 3 1
882.3.s.e 8 63.j odd 6 1
882.3.s.e 8 63.n odd 6 1
882.3.s.i 8 63.i even 6 1
882.3.s.i 8 63.k odd 6 1
882.3.s.i 8 63.s even 6 1
882.3.s.i 8 63.t odd 6 1
1008.3.d.a 4 36.f odd 6 1
1008.3.d.a 4 36.h even 6 1
1134.3.q.c 8 1.a even 1 1 trivial
1134.3.q.c 8 3.b odd 2 1 inner
1134.3.q.c 8 9.c even 3 1 inner
1134.3.q.c 8 9.d odd 6 1 inner
3150.3.c.b 8 45.k odd 12 2
3150.3.c.b 8 45.l even 12 2
3150.3.e.e 4 45.h odd 6 1
3150.3.e.e 4 45.j even 6 1
4032.3.d.i 4 72.j odd 6 1
4032.3.d.i 4 72.n even 6 1
4032.3.d.j 4 72.l even 6 1
4032.3.d.j 4 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 116 T_{5}^{6} + 10540 T_{5}^{4} - 338256 T_{5}^{2} + 8503056 \) acting on \(S_{3}^{\mathrm{new}}(1134, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 - 2 T^{2} + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( 8503056 - 338256 T^{2} + 10540 T^{4} - 116 T^{6} + T^{8} \)
$7$ \( ( 49 + 7 T^{2} + T^{4} )^{2} \)
$11$ \( 2176782336 - 21648384 T^{2} + 168640 T^{4} - 464 T^{6} + T^{8} \)
$13$ \( ( 2304 + 768 T + 304 T^{2} - 16 T^{3} + T^{4} )^{2} \)
$17$ \( ( 79524 + 788 T^{2} + T^{4} )^{2} \)
$19$ \( ( -20 + T )^{8} \)
$23$ \( 2176782336 - 21648384 T^{2} + 168640 T^{4} - 464 T^{6} + T^{8} \)
$29$ \( 61505984016 - 469223568 T^{2} + 3331660 T^{4} - 1892 T^{6} + T^{8} \)
$31$ \( ( 186624 - 3456 T + 496 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$37$ \( ( -38 + T )^{8} \)
$41$ \( 828311133456 - 3571295184 T^{2} + 14487660 T^{4} - 3924 T^{6} + T^{8} \)
$43$ \( ( 13191424 - 145280 T + 5232 T^{2} + 40 T^{3} + T^{4} )^{2} \)
$47$ \( ( 82944 - 288 T^{2} + T^{4} )^{2} \)
$53$ \( ( 64738116 + 16164 T^{2} + T^{4} )^{2} \)
$59$ \( 84934656 - 31260672 T^{2} + 11496448 T^{4} - 3392 T^{6} + T^{8} \)
$61$ \( ( 2471184 + 182352 T + 11884 T^{2} + 116 T^{3} + T^{4} )^{2} \)
$67$ \( ( 23658496 + 466944 T + 14080 T^{2} - 96 T^{3} + T^{4} )^{2} \)
$71$ \( ( 46656 + 464 T^{2} + T^{4} )^{2} \)
$73$ \( ( -2224 + 48 T + T^{2} )^{4} \)
$79$ \( ( 15872256 + 605568 T + 19120 T^{2} + 152 T^{3} + T^{4} )^{2} \)
$83$ \( 2833604941971456 - 967537852416 T^{2} + 277135360 T^{4} - 18176 T^{6} + T^{8} \)
$89$ \( ( 443556 + 19476 T^{2} + T^{4} )^{2} \)
$97$ \( ( 70023424 + 1204992 T + 29104 T^{2} - 144 T^{3} + T^{4} )^{2} \)
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