# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$9.05503558921$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: yes 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{12}^{2} ) q^{2} -\zeta_{12}^{2} q^{4} + ( \zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + ( 1 - \zeta_{12}^{2} ) q^{7} - q^{8} +O(q^{10})$$ $$q + ( 1 - \zeta_{12}^{2} ) q^{2} -\zeta_{12}^{2} q^{4} + ( \zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + ( 1 - \zeta_{12}^{2} ) q^{7} - q^{8} + ( -2 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{10} + ( -1 - 3 \zeta_{12} + \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{11} + ( 2 \zeta_{12} + 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{13} -\zeta_{12}^{2} q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + 7 q^{17} + ( -1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{19} + ( -2 + \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{20} + ( 3 \zeta_{12} + \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{22} + ( 3 \zeta_{12} - \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{23} + ( -2 + 4 \zeta_{12} + 2 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{25} + ( 3 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{26} - q^{28} + ( -5 + 2 \zeta_{12} + 5 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{29} + ( -3 \zeta_{12} - 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{31} + \zeta_{12}^{2} q^{32} + ( 7 - 7 \zeta_{12}^{2} ) q^{34} + ( -2 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{35} + ( 2 + 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{37} + ( -1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{38} + ( -\zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{40} + ( 2 \zeta_{12} - 6 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{41} + ( -2 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{43} + ( 1 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{44} + ( -1 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{46} + ( -3 - \zeta_{12} + 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{47} -\zeta_{12}^{2} q^{49} + ( -4 \zeta_{12} + 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{50} + ( 3 + 2 \zeta_{12} - 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{52} + ( -6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{53} + ( -7 + 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{55} + ( -1 + \zeta_{12}^{2} ) q^{56} + ( -2 \zeta_{12} + 5 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{58} + ( -3 \zeta_{12} + \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{59} + ( 3 - 4 \zeta_{12} - 3 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{61} + ( -3 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{62} + q^{64} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{65} + ( \zeta_{12} + 5 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{67} -7 \zeta_{12}^{2} q^{68} + ( -2 + \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{70} + ( 10 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{71} + ( 10 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{73} + ( 2 + 5 \zeta_{12} - 2 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{74} + ( -\zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{76} + ( 3 \zeta_{12} + \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{77} + ( -3 + 7 \zeta_{12} + 3 \zeta_{12}^{2} - 14 \zeta_{12}^{3} ) q^{79} + ( 2 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{80} + ( -6 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{82} + ( -1 - 9 \zeta_{12} + \zeta_{12}^{2} + 18 \zeta_{12}^{3} ) q^{83} + ( 7 \zeta_{12} - 14 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{85} + ( -2 \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{86} + ( 1 + 3 \zeta_{12} - \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{88} + ( 3 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{89} + ( 3 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{91} + ( -1 + 3 \zeta_{12} + \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{92} + ( \zeta_{12} + 3 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{94} + ( -3 \zeta_{12} + 5 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{95} + ( -4 - 4 \zeta_{12} + 4 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{97} - q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} - 2q^{4} - 4q^{5} + 2q^{7} - 4q^{8} + O(q^{10})$$ $$4q + 2q^{2} - 2q^{4} - 4q^{5} + 2q^{7} - 4q^{8} - 8q^{10} - 2q^{11} + 6q^{13} - 2q^{14} - 2q^{16} + 28q^{17} - 4q^{19} - 4q^{20} + 2q^{22} - 2q^{23} - 4q^{25} + 12q^{26} - 4q^{28} - 10q^{29} - 6q^{31} + 2q^{32} + 14q^{34} - 8q^{35} + 8q^{37} - 2q^{38} + 4q^{40} - 12q^{41} - 4q^{43} + 4q^{44} - 4q^{46} - 6q^{47} - 2q^{49} + 4q^{50} + 6q^{52} - 24q^{53} - 28q^{55} - 2q^{56} + 10q^{58} + 2q^{59} + 6q^{61} - 12q^{62} + 4q^{64} + 10q^{67} - 14q^{68} - 4q^{70} + 40q^{71} + 40q^{73} + 4q^{74} + 2q^{76} + 2q^{77} - 6q^{79} + 8q^{80} - 24q^{82} - 2q^{83} - 28q^{85} + 4q^{86} + 2q^{88} + 12q^{89} + 12q^{91} - 2q^{92} + 6q^{94} + 10q^{95} - 8q^{97} - 4q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$325$$ $$407$$ $$\chi(n)$$ $$1$$ $$-\zeta_{12}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
379.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0.500000 0.866025i 0 −0.500000 0.866025i −1.86603 3.23205i 0 0.500000 0.866025i −1.00000 0 −3.73205
379.2 0.500000 0.866025i 0 −0.500000 0.866025i −0.133975 0.232051i 0 0.500000 0.866025i −1.00000 0 −0.267949
757.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.86603 + 3.23205i 0 0.500000 + 0.866025i −1.00000 0 −3.73205
757.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.133975 + 0.232051i 0 0.500000 + 0.866025i −1.00000 0 −0.267949
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.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.f.s 4
3.b odd 2 1 1134.2.f.r 4
9.c even 3 1 1134.2.a.l 2
9.c even 3 1 inner 1134.2.f.s 4
9.d odd 6 1 1134.2.a.m yes 2
9.d odd 6 1 1134.2.f.r 4
36.f odd 6 1 9072.2.a.bp 2
36.h even 6 1 9072.2.a.y 2
63.l odd 6 1 7938.2.a.bg 2
63.o even 6 1 7938.2.a.bt 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.a.l 2 9.c even 3 1
1134.2.a.m yes 2 9.d odd 6 1
1134.2.f.r 4 3.b odd 2 1
1134.2.f.r 4 9.d odd 6 1
1134.2.f.s 4 1.a even 1 1 trivial
1134.2.f.s 4 9.c even 3 1 inner
7938.2.a.bg 2 63.l odd 6 1
7938.2.a.bt 2 63.o even 6 1
9072.2.a.y 2 36.h even 6 1
9072.2.a.bp 2 36.f odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 4 T_{5}^{3} + 15 T_{5}^{2} + 4 T_{5} + 1$$ $$T_{11}^{4} + 2 T_{11}^{3} + 30 T_{11}^{2} - 52 T_{11} + 676$$ $$T_{13}^{4} - 6 T_{13}^{3} + 39 T_{13}^{2} + 18 T_{13} + 9$$ $$T_{17} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$1 + 4 T + 15 T^{2} + 4 T^{3} + T^{4}$$
$7$ $$( 1 - T + T^{2} )^{2}$$
$11$ $$676 - 52 T + 30 T^{2} + 2 T^{3} + T^{4}$$
$13$ $$9 + 18 T + 39 T^{2} - 6 T^{3} + T^{4}$$
$17$ $$( -7 + T )^{4}$$
$19$ $$( -2 + 2 T + T^{2} )^{2}$$
$23$ $$676 - 52 T + 30 T^{2} + 2 T^{3} + T^{4}$$
$29$ $$169 + 130 T + 87 T^{2} + 10 T^{3} + T^{4}$$
$31$ $$324 - 108 T + 54 T^{2} + 6 T^{3} + T^{4}$$
$37$ $$( -71 - 4 T + T^{2} )^{2}$$
$41$ $$576 + 288 T + 120 T^{2} + 12 T^{3} + T^{4}$$
$43$ $$64 - 32 T + 24 T^{2} + 4 T^{3} + T^{4}$$
$47$ $$36 + 36 T + 30 T^{2} + 6 T^{3} + T^{4}$$
$53$ $$( 24 + 12 T + T^{2} )^{2}$$
$59$ $$676 + 52 T + 30 T^{2} - 2 T^{3} + T^{4}$$
$61$ $$1521 + 234 T + 75 T^{2} - 6 T^{3} + T^{4}$$
$67$ $$484 - 220 T + 78 T^{2} - 10 T^{3} + T^{4}$$
$71$ $$( 88 - 20 T + T^{2} )^{2}$$
$73$ $$( 97 - 20 T + T^{2} )^{2}$$
$79$ $$19044 - 828 T + 174 T^{2} + 6 T^{3} + T^{4}$$
$83$ $$58564 - 484 T + 246 T^{2} + 2 T^{3} + T^{4}$$
$89$ $$( -39 - 6 T + T^{2} )^{2}$$
$97$ $$1024 - 256 T + 96 T^{2} + 8 T^{3} + T^{4}$$