# Properties

 Label 1134.2.m.g Level $1134$ Weight $2$ Character orbit 1134.m Analytic conductor $9.055$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$9.05503558921$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 42) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{2} + ( 1 - \zeta_{24}^{4} ) q^{4} + ( 2 \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{5} + ( \zeta_{24} + 2 \zeta_{24}^{3} + \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{7} -\zeta_{24}^{6} q^{8} +O(q^{10})$$ $$q + ( \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{2} + ( 1 - \zeta_{24}^{4} ) q^{4} + ( 2 \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{5} + ( \zeta_{24} + 2 \zeta_{24}^{3} + \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{7} -\zeta_{24}^{6} q^{8} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{10} + ( 2 \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{13} + ( 2 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{14} -\zeta_{24}^{4} q^{16} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{17} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{19} + ( \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{20} -6 \zeta_{24}^{2} q^{23} -\zeta_{24}^{4} q^{25} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{26} + ( 1 - \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{28} + ( -6 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{29} -\zeta_{24}^{2} q^{32} + ( 2 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{34} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 6 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{35} -2 q^{37} + ( -\zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{38} + ( -2 \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{40} + ( 4 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{41} -4 \zeta_{24}^{4} q^{43} -6 q^{46} + ( 2 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{47} + ( 5 + 4 \zeta_{24} + 2 \zeta_{24}^{3} - 5 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{49} -\zeta_{24}^{2} q^{50} + ( \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{52} + 6 \zeta_{24}^{6} q^{53} + ( \zeta_{24} + \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{56} + ( -6 + 6 \zeta_{24}^{4} ) q^{58} + ( 10 \zeta_{24} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} - 5 \zeta_{24}^{7} ) q^{59} + ( -5 \zeta_{24} - 10 \zeta_{24}^{3} + 10 \zeta_{24}^{5} + 5 \zeta_{24}^{7} ) q^{61} - q^{64} + ( 6 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{65} + ( -8 + 8 \zeta_{24}^{4} ) q^{67} + ( 4 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{68} + ( \zeta_{24} + 2 \zeta_{24}^{3} - 6 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{70} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{73} + ( -2 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{74} + ( 2 \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{76} + 10 \zeta_{24}^{4} q^{79} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{80} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{82} + ( \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{83} + ( 12 - 12 \zeta_{24}^{4} ) q^{85} -4 \zeta_{24}^{2} q^{86} + ( 6 + \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{91} + ( -6 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{92} + ( -4 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{94} + 6 \zeta_{24}^{2} q^{95} + ( 2 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{97} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 5 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{4} + 4q^{7} + O(q^{10})$$ $$8q + 4q^{4} + 4q^{7} - 4q^{16} - 4q^{25} + 8q^{28} - 16q^{37} - 16q^{43} - 48q^{46} + 20q^{49} - 24q^{58} - 8q^{64} - 32q^{67} - 24q^{70} + 40q^{79} + 48q^{85} + 48q^{91} + O(q^{100})$$

## Character values

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

 $$n$$ $$325$$ $$407$$ $$\chi(n)$$ $$-1$$ $$1 - \zeta_{24}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
377.1
 −0.258819 + 0.965926i 0.258819 − 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i
−0.866025 + 0.500000i 0 0.500000 0.866025i −1.22474 + 2.12132i 0 2.62132 0.358719i 1.00000i 0 2.44949i
377.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.22474 2.12132i 0 −1.62132 + 2.09077i 1.00000i 0 2.44949i
377.3 0.866025 0.500000i 0 0.500000 0.866025i −1.22474 + 2.12132i 0 −1.62132 + 2.09077i 1.00000i 0 2.44949i
377.4 0.866025 0.500000i 0 0.500000 0.866025i 1.22474 2.12132i 0 2.62132 0.358719i 1.00000i 0 2.44949i
755.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −1.22474 2.12132i 0 2.62132 + 0.358719i 1.00000i 0 2.44949i
755.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.22474 + 2.12132i 0 −1.62132 2.09077i 1.00000i 0 2.44949i
755.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.22474 2.12132i 0 −1.62132 2.09077i 1.00000i 0 2.44949i
755.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.22474 + 2.12132i 0 2.62132 + 0.358719i 1.00000i 0 2.44949i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 755.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
21.c even 2 1 inner
63.l odd 6 1 inner
63.o even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.2.m.g 8
3.b odd 2 1 inner 1134.2.m.g 8
7.b odd 2 1 inner 1134.2.m.g 8
9.c even 3 1 42.2.d.a 4
9.c even 3 1 inner 1134.2.m.g 8
9.d odd 6 1 42.2.d.a 4
9.d odd 6 1 inner 1134.2.m.g 8
21.c even 2 1 inner 1134.2.m.g 8
36.f odd 6 1 336.2.k.b 4
36.h even 6 1 336.2.k.b 4
45.h odd 6 1 1050.2.b.b 4
45.j even 6 1 1050.2.b.b 4
45.k odd 12 1 1050.2.d.b 4
45.k odd 12 1 1050.2.d.e 4
45.l even 12 1 1050.2.d.b 4
45.l even 12 1 1050.2.d.e 4
63.g even 3 1 294.2.f.b 8
63.h even 3 1 294.2.f.b 8
63.i even 6 1 294.2.f.b 8
63.j odd 6 1 294.2.f.b 8
63.k odd 6 1 294.2.f.b 8
63.l odd 6 1 42.2.d.a 4
63.l odd 6 1 inner 1134.2.m.g 8
63.n odd 6 1 294.2.f.b 8
63.o even 6 1 42.2.d.a 4
63.o even 6 1 inner 1134.2.m.g 8
63.s even 6 1 294.2.f.b 8
63.t odd 6 1 294.2.f.b 8
72.j odd 6 1 1344.2.k.c 4
72.l even 6 1 1344.2.k.d 4
72.n even 6 1 1344.2.k.c 4
72.p odd 6 1 1344.2.k.d 4
252.s odd 6 1 336.2.k.b 4
252.bi even 6 1 336.2.k.b 4
315.z even 6 1 1050.2.b.b 4
315.bg odd 6 1 1050.2.b.b 4
315.cb even 12 1 1050.2.d.b 4
315.cb even 12 1 1050.2.d.e 4
315.cf odd 12 1 1050.2.d.b 4
315.cf odd 12 1 1050.2.d.e 4
504.be even 6 1 1344.2.k.d 4
504.bn odd 6 1 1344.2.k.c 4
504.cc even 6 1 1344.2.k.c 4
504.co odd 6 1 1344.2.k.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.d.a 4 9.c even 3 1
42.2.d.a 4 9.d odd 6 1
42.2.d.a 4 63.l odd 6 1
42.2.d.a 4 63.o even 6 1
294.2.f.b 8 63.g even 3 1
294.2.f.b 8 63.h even 3 1
294.2.f.b 8 63.i even 6 1
294.2.f.b 8 63.j odd 6 1
294.2.f.b 8 63.k odd 6 1
294.2.f.b 8 63.n odd 6 1
294.2.f.b 8 63.s even 6 1
294.2.f.b 8 63.t odd 6 1
336.2.k.b 4 36.f odd 6 1
336.2.k.b 4 36.h even 6 1
336.2.k.b 4 252.s odd 6 1
336.2.k.b 4 252.bi even 6 1
1050.2.b.b 4 45.h odd 6 1
1050.2.b.b 4 45.j even 6 1
1050.2.b.b 4 315.z even 6 1
1050.2.b.b 4 315.bg odd 6 1
1050.2.d.b 4 45.k odd 12 1
1050.2.d.b 4 45.l even 12 1
1050.2.d.b 4 315.cb even 12 1
1050.2.d.b 4 315.cf odd 12 1
1050.2.d.e 4 45.k odd 12 1
1050.2.d.e 4 45.l even 12 1
1050.2.d.e 4 315.cb even 12 1
1050.2.d.e 4 315.cf odd 12 1
1134.2.m.g 8 1.a even 1 1 trivial
1134.2.m.g 8 3.b odd 2 1 inner
1134.2.m.g 8 7.b odd 2 1 inner
1134.2.m.g 8 9.c even 3 1 inner
1134.2.m.g 8 9.d odd 6 1 inner
1134.2.m.g 8 21.c even 2 1 inner
1134.2.m.g 8 63.l odd 6 1 inner
1134.2.m.g 8 63.o even 6 1 inner
1344.2.k.c 4 72.j odd 6 1
1344.2.k.c 4 72.n even 6 1
1344.2.k.c 4 504.bn odd 6 1
1344.2.k.c 4 504.cc even 6 1
1344.2.k.d 4 72.l even 6 1
1344.2.k.d 4 72.p odd 6 1
1344.2.k.d 4 504.be even 6 1
1344.2.k.d 4 504.co odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 6 T_{5}^{2} + 36$$ $$T_{11}$$ $$T_{13}^{4} - 6 T_{13}^{2} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ 1
$5$ $$( 1 - 4 T^{2} - 9 T^{4} - 100 T^{6} + 625 T^{8} )^{2}$$
$7$ $$( 1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4} )^{2}$$
$11$ $$( 1 + 11 T^{2} + 121 T^{4} )^{4}$$
$13$ $$( 1 + 20 T^{2} + 231 T^{4} + 3380 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$( 1 + 10 T^{2} + 289 T^{4} )^{4}$$
$19$ $$( 1 - 32 T^{2} + 361 T^{4} )^{4}$$
$23$ $$( 1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 + 22 T^{2} - 357 T^{4} + 18502 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 + 31 T^{2} + 961 T^{4} )^{4}$$
$37$ $$( 1 + 2 T + 37 T^{2} )^{8}$$
$41$ $$( 1 - 58 T^{2} + 1683 T^{4} - 97498 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 + 4 T - 27 T^{2} + 172 T^{3} + 1849 T^{4} )^{4}$$
$47$ $$( 1 - 70 T^{2} + 2691 T^{4} - 154630 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 - 70 T^{2} + 2809 T^{4} )^{4}$$
$59$ $$( 1 + 32 T^{2} - 2457 T^{4} + 111392 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 28 T^{2} - 2937 T^{4} - 104188 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} )^{4}$$
$71$ $$( 1 - 71 T^{2} )^{8}$$
$73$ $$( 1 - 14 T + 73 T^{2} )^{4}( 1 + 14 T + 73 T^{2} )^{4}$$
$79$ $$( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} )^{4}$$
$83$ $$( 1 - 160 T^{2} + 18711 T^{4} - 1102240 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 89 T^{2} )^{8}$$
$97$ $$( 1 + 170 T^{2} + 19491 T^{4} + 1599530 T^{6} + 88529281 T^{8} )^{2}$$