# Properties

 Label 1134.2.l.f Level 1134 Weight 2 Character orbit 1134.l Analytic conductor 9.055 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$1134 = 2 \cdot 3^{4} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1134.l (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: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ 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 primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{24}^{6} q^{2} - q^{4} + ( -2 \zeta_{24} + \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{5} + ( -\zeta_{24} - 2 \zeta_{24}^{3} + \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{7} -\zeta_{24}^{6} q^{8} +O(q^{10})$$ $$q + \zeta_{24}^{6} q^{2} - q^{4} + ( -2 \zeta_{24} + \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{6} + \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} + ( 1 - 2 \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{10} + ( 3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{11} + ( \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{13} + ( \zeta_{24} - \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{14} + q^{16} + ( -2 \zeta_{24} - 2 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + 4 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{17} + ( -4 - \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{19} + ( 2 \zeta_{24} - \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{20} + 3 \zeta_{24}^{4} q^{22} + ( -3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{23} + ( 6 \zeta_{24} - 4 \zeta_{24}^{4} + 6 \zeta_{24}^{7} ) q^{25} + ( -\zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{26} + ( \zeta_{24} + 2 \zeta_{24}^{3} - \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{28} + ( -3 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{29} + ( -1 + 3 \zeta_{24} - 3 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{31} + \zeta_{24}^{6} q^{32} + ( -2 - 2 \zeta_{24} - \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{34} + ( -4 \zeta_{24} + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 7 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{35} + ( 3 \zeta_{24} - 4 \zeta_{24}^{4} + 3 \zeta_{24}^{7} ) q^{37} + ( \zeta_{24} - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{38} + ( -1 + 2 \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{4} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{40} + ( 2 \zeta_{24} - 4 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{41} + ( -4 + 3 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{43} + ( -3 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{44} + ( 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{46} + ( \zeta_{24} + 4 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 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} + ( -6 \zeta_{24} + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{50} + ( -\zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{52} + ( -3 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{53} + ( 3 + 3 \zeta_{24} - 3 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{55} + ( -\zeta_{24} + \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{56} + ( 3 + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{4} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{58} + ( -4 \zeta_{24} + 2 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{59} + ( -2 - \zeta_{24} + \zeta_{24}^{3} + 4 \zeta_{24}^{4} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{61} + ( -3 \zeta_{24} - 2 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{62} - q^{64} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{65} -10 q^{67} + ( 2 \zeta_{24} + 2 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - 4 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{68} + ( 5 - 4 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{70} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{71} + ( 2 - 4 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{73} + ( -3 \zeta_{24} + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{74} + ( 4 + \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{76} + ( -6 \zeta_{24} + 3 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{77} + ( -7 + 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{79} + ( -2 \zeta_{24} + \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{80} + ( 8 + 2 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{82} + ( -4 \zeta_{24} - \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{83} + ( -3 \zeta_{24} - 3 \zeta_{24}^{7} ) q^{85} + ( -4 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{86} -3 \zeta_{24}^{4} q^{88} + ( 6 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{89} + ( -6 + 2 \zeta_{24} + \zeta_{24}^{3} + 6 \zeta_{24}^{4} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{91} + ( 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{92} + ( -2 + \zeta_{24} - \zeta_{24}^{3} + 4 \zeta_{24}^{4} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{94} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{95} + ( 5 + 4 \zeta_{24} + 2 \zeta_{24}^{3} + 5 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{97} + ( 4 \zeta_{24} + 5 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{4} + 4q^{7} + O(q^{10})$$ $$8q - 8q^{4} + 4q^{7} + 12q^{10} + 8q^{16} - 24q^{19} + 12q^{22} - 16q^{25} - 4q^{28} - 24q^{34} - 16q^{37} - 12q^{40} - 16q^{43} + 20q^{49} + 12q^{58} - 8q^{64} - 80q^{67} + 48q^{70} + 24q^{73} + 24q^{76} - 56q^{79} + 48q^{82} - 12q^{88} - 24q^{91} + 60q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
215.1
 0.965926 − 0.258819i −0.965926 + 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i 0.258819 − 0.965926i −0.258819 + 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i
1.00000i 0 −1.00000 −0.358719 0.621320i 0 −1.62132 2.09077i 1.00000i 0 −0.621320 + 0.358719i
215.2 1.00000i 0 −1.00000 2.09077 + 3.62132i 0 2.62132 + 0.358719i 1.00000i 0 3.62132 2.09077i
215.3 1.00000i 0 −1.00000 −2.09077 3.62132i 0 2.62132 + 0.358719i 1.00000i 0 3.62132 2.09077i
215.4 1.00000i 0 −1.00000 0.358719 + 0.621320i 0 −1.62132 2.09077i 1.00000i 0 −0.621320 + 0.358719i
269.1 1.00000i 0 −1.00000 −2.09077 + 3.62132i 0 2.62132 0.358719i 1.00000i 0 3.62132 + 2.09077i
269.2 1.00000i 0 −1.00000 0.358719 0.621320i 0 −1.62132 + 2.09077i 1.00000i 0 −0.621320 0.358719i
269.3 1.00000i 0 −1.00000 −0.358719 + 0.621320i 0 −1.62132 + 2.09077i 1.00000i 0 −0.621320 0.358719i
269.4 1.00000i 0 −1.00000 2.09077 3.62132i 0 2.62132 0.358719i 1.00000i 0 3.62132 + 2.09077i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 269.4 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
63.i even 6 1 inner
63.t odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.2.l.f 8
3.b odd 2 1 inner 1134.2.l.f 8
7.d odd 6 1 1134.2.t.e 8
9.c even 3 1 126.2.k.a 8
9.c even 3 1 1134.2.t.e 8
9.d odd 6 1 126.2.k.a 8
9.d odd 6 1 1134.2.t.e 8
21.g even 6 1 1134.2.t.e 8
36.f odd 6 1 1008.2.bt.c 8
36.h even 6 1 1008.2.bt.c 8
45.h odd 6 1 3150.2.bf.a 8
45.j even 6 1 3150.2.bf.a 8
45.k odd 12 1 3150.2.bp.b 8
45.k odd 12 1 3150.2.bp.e 8
45.l even 12 1 3150.2.bp.b 8
45.l even 12 1 3150.2.bp.e 8
63.g even 3 1 882.2.k.a 8
63.h even 3 1 882.2.d.a 8
63.i even 6 1 882.2.d.a 8
63.i even 6 1 inner 1134.2.l.f 8
63.j odd 6 1 882.2.d.a 8
63.k odd 6 1 126.2.k.a 8
63.l odd 6 1 882.2.k.a 8
63.n odd 6 1 882.2.k.a 8
63.o even 6 1 882.2.k.a 8
63.s even 6 1 126.2.k.a 8
63.t odd 6 1 882.2.d.a 8
63.t odd 6 1 inner 1134.2.l.f 8
252.n even 6 1 1008.2.bt.c 8
252.r odd 6 1 7056.2.k.f 8
252.u odd 6 1 7056.2.k.f 8
252.bb even 6 1 7056.2.k.f 8
252.bj even 6 1 7056.2.k.f 8
252.bn odd 6 1 1008.2.bt.c 8
315.u even 6 1 3150.2.bf.a 8
315.bn odd 6 1 3150.2.bf.a 8
315.bw odd 12 1 3150.2.bp.b 8
315.bw odd 12 1 3150.2.bp.e 8
315.cg even 12 1 3150.2.bp.b 8
315.cg even 12 1 3150.2.bp.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.k.a 8 9.c even 3 1
126.2.k.a 8 9.d odd 6 1
126.2.k.a 8 63.k odd 6 1
126.2.k.a 8 63.s even 6 1
882.2.d.a 8 63.h even 3 1
882.2.d.a 8 63.i even 6 1
882.2.d.a 8 63.j odd 6 1
882.2.d.a 8 63.t odd 6 1
882.2.k.a 8 63.g even 3 1
882.2.k.a 8 63.l odd 6 1
882.2.k.a 8 63.n odd 6 1
882.2.k.a 8 63.o even 6 1
1008.2.bt.c 8 36.f odd 6 1
1008.2.bt.c 8 36.h even 6 1
1008.2.bt.c 8 252.n even 6 1
1008.2.bt.c 8 252.bn odd 6 1
1134.2.l.f 8 1.a even 1 1 trivial
1134.2.l.f 8 3.b odd 2 1 inner
1134.2.l.f 8 63.i even 6 1 inner
1134.2.l.f 8 63.t odd 6 1 inner
1134.2.t.e 8 7.d odd 6 1
1134.2.t.e 8 9.c even 3 1
1134.2.t.e 8 9.d odd 6 1
1134.2.t.e 8 21.g even 6 1
3150.2.bf.a 8 45.h odd 6 1
3150.2.bf.a 8 45.j even 6 1
3150.2.bf.a 8 315.u even 6 1
3150.2.bf.a 8 315.bn odd 6 1
3150.2.bp.b 8 45.k odd 12 1
3150.2.bp.b 8 45.l even 12 1
3150.2.bp.b 8 315.bw odd 12 1
3150.2.bp.b 8 315.cg even 12 1
3150.2.bp.e 8 45.k odd 12 1
3150.2.bp.e 8 45.l even 12 1
3150.2.bp.e 8 315.bw odd 12 1
3150.2.bp.e 8 315.cg even 12 1
7056.2.k.f 8 252.r odd 6 1
7056.2.k.f 8 252.u odd 6 1
7056.2.k.f 8 252.bb even 6 1
7056.2.k.f 8 252.bj even 6 1

## Hecke kernels

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

 $$T_{5}^{8} + 18 T_{5}^{6} + 315 T_{5}^{4} + 162 T_{5}^{2} + 81$$ $$T_{11}^{4} - 9 T_{11}^{2} + 81$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{4}$$
$3$ 
$5$ $$( 1 - 2 T^{2} + 25 T^{4} )^{2}( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} )$$
$7$ $$( 1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4} )^{2}$$
$11$ $$( 1 + 13 T^{2} + 48 T^{4} + 1573 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 + 20 T^{2} + 231 T^{4} + 3380 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$1 - 32 T^{2} + 478 T^{4} + 1024 T^{6} - 81341 T^{8} + 295936 T^{10} + 39923038 T^{12} - 772402208 T^{14} + 6975757441 T^{16}$$
$19$ $$( 1 + 12 T + 92 T^{2} + 528 T^{3} + 2487 T^{4} + 10032 T^{5} + 33212 T^{6} + 82308 T^{7} + 130321 T^{8} )^{2}$$
$23$ $$( 1 + 28 T^{2} + 255 T^{4} + 14812 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$1 + 62 T^{2} + 1849 T^{4} + 19406 T^{6} + 39940 T^{8} + 16320446 T^{10} + 1307762569 T^{12} + 36879045902 T^{14} + 500246412961 T^{16}$$
$31$ $$( 1 - 10 T^{2} + 1299 T^{4} - 9610 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 + 8 T - 8 T^{2} - 16 T^{3} + 1447 T^{4} - 592 T^{5} - 10952 T^{6} + 405224 T^{7} + 1874161 T^{8} )^{2}$$
$41$ $$1 - 20 T^{2} + 1546 T^{4} + 90160 T^{6} - 2184845 T^{8} + 151558960 T^{10} + 4368626506 T^{12} - 95002084820 T^{14} + 7984925229121 T^{16}$$
$43$ $$( 1 + 8 T - 20 T^{2} - 16 T^{3} + 2455 T^{4} - 688 T^{5} - 36980 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 + 152 T^{2} + 9906 T^{4} + 335768 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$1 + 158 T^{2} + 13753 T^{4} + 883694 T^{6} + 47672164 T^{8} + 2482296446 T^{10} + 108517785193 T^{12} + 3501969058382 T^{14} + 62259690411361 T^{16}$$
$59$ $$( 1 + 38 T^{2} + 6171 T^{4} + 132278 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 208 T^{2} + 17970 T^{4} - 773968 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 + 10 T + 67 T^{2} )^{8}$$
$71$ $$( 1 - 176 T^{2} + 15234 T^{4} - 887216 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 - 12 T + 182 T^{2} - 1608 T^{3} + 16131 T^{4} - 117384 T^{5} + 969878 T^{6} - 4668204 T^{7} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 + 14 T + 189 T^{2} + 1106 T^{3} + 6241 T^{4} )^{4}$$
$83$ $$1 - 278 T^{2} + 44473 T^{4} - 5291174 T^{6} + 496693924 T^{8} - 36450897686 T^{10} + 2110613909833 T^{12} - 90889423796582 T^{14} + 2252292232139041 T^{16}$$
$89$ $$( 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 30 T + 545 T^{2} - 7350 T^{3} + 79716 T^{4} - 712950 T^{5} + 5127905 T^{6} - 27380190 T^{7} + 88529281 T^{8} )^{2}$$