# Properties

 Label 1134.2.f.h Level $1134$ Weight $2$ Character orbit 1134.f Analytic conductor $9.055$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + ( -1 + \zeta_{6} ) q^{7} + q^{8} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + ( -1 + \zeta_{6} ) q^{7} + q^{8} -3 q^{10} + ( 6 - 6 \zeta_{6} ) q^{11} + \zeta_{6} q^{13} -\zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} + 3 q^{17} + 2 q^{19} + ( 3 - 3 \zeta_{6} ) q^{20} + 6 \zeta_{6} q^{22} + 6 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} - q^{26} + q^{28} + ( 9 - 9 \zeta_{6} ) q^{29} + 10 \zeta_{6} q^{31} -\zeta_{6} q^{32} + ( -3 + 3 \zeta_{6} ) q^{34} -3 q^{35} -7 q^{37} + ( -2 + 2 \zeta_{6} ) q^{38} + 3 \zeta_{6} q^{40} -6 \zeta_{6} q^{41} + ( 4 - 4 \zeta_{6} ) q^{43} -6 q^{44} -6 q^{46} + ( -6 + 6 \zeta_{6} ) q^{47} -\zeta_{6} q^{49} -4 \zeta_{6} q^{50} + ( 1 - \zeta_{6} ) q^{52} -6 q^{53} + 18 q^{55} + ( -1 + \zeta_{6} ) q^{56} + 9 \zeta_{6} q^{58} + 6 \zeta_{6} q^{59} + ( -11 + 11 \zeta_{6} ) q^{61} -10 q^{62} + q^{64} + ( -3 + 3 \zeta_{6} ) q^{65} -2 \zeta_{6} q^{67} -3 \zeta_{6} q^{68} + ( 3 - 3 \zeta_{6} ) q^{70} + 12 q^{71} -7 q^{73} + ( 7 - 7 \zeta_{6} ) q^{74} -2 \zeta_{6} q^{76} + 6 \zeta_{6} q^{77} + ( -2 + 2 \zeta_{6} ) q^{79} -3 q^{80} + 6 q^{82} + ( -6 + 6 \zeta_{6} ) q^{83} + 9 \zeta_{6} q^{85} + 4 \zeta_{6} q^{86} + ( 6 - 6 \zeta_{6} ) q^{88} + 3 q^{89} - q^{91} + ( 6 - 6 \zeta_{6} ) q^{92} -6 \zeta_{6} q^{94} + 6 \zeta_{6} q^{95} + ( -14 + 14 \zeta_{6} ) q^{97} + q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} + 3q^{5} - q^{7} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{4} + 3q^{5} - q^{7} + 2q^{8} - 6q^{10} + 6q^{11} + q^{13} - q^{14} - q^{16} + 6q^{17} + 4q^{19} + 3q^{20} + 6q^{22} + 6q^{23} - 4q^{25} - 2q^{26} + 2q^{28} + 9q^{29} + 10q^{31} - q^{32} - 3q^{34} - 6q^{35} - 14q^{37} - 2q^{38} + 3q^{40} - 6q^{41} + 4q^{43} - 12q^{44} - 12q^{46} - 6q^{47} - q^{49} - 4q^{50} + q^{52} - 12q^{53} + 36q^{55} - q^{56} + 9q^{58} + 6q^{59} - 11q^{61} - 20q^{62} + 2q^{64} - 3q^{65} - 2q^{67} - 3q^{68} + 3q^{70} + 24q^{71} - 14q^{73} + 7q^{74} - 2q^{76} + 6q^{77} - 2q^{79} - 6q^{80} + 12q^{82} - 6q^{83} + 9q^{85} + 4q^{86} + 6q^{88} + 6q^{89} - 2q^{91} + 6q^{92} - 6q^{94} + 6q^{95} - 14q^{97} + 2q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$325$$ $$407$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 1.50000 + 2.59808i 0 −0.500000 + 0.866025i 1.00000 0 −3.00000
757.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.50000 2.59808i 0 −0.500000 0.866025i 1.00000 0 −3.00000
 $$n$$: 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.h 2
3.b odd 2 1 1134.2.f.i 2
9.c even 3 1 1134.2.a.e yes 1
9.c even 3 1 inner 1134.2.f.h 2
9.d odd 6 1 1134.2.a.d 1
9.d odd 6 1 1134.2.f.i 2
36.f odd 6 1 9072.2.a.b 1
36.h even 6 1 9072.2.a.v 1
63.l odd 6 1 7938.2.a.bc 1
63.o even 6 1 7938.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.a.d 1 9.d odd 6 1
1134.2.a.e yes 1 9.c even 3 1
1134.2.f.h 2 1.a even 1 1 trivial
1134.2.f.h 2 9.c even 3 1 inner
1134.2.f.i 2 3.b odd 2 1
1134.2.f.i 2 9.d odd 6 1
7938.2.a.d 1 63.o even 6 1
7938.2.a.bc 1 63.l odd 6 1
9072.2.a.b 1 36.f odd 6 1
9072.2.a.v 1 36.h even 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 3 T_{5} + 9$$ $$T_{11}^{2} - 6 T_{11} + 36$$ $$T_{13}^{2} - T_{13} + 1$$ $$T_{17} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$9 - 3 T + T^{2}$$
$7$ $$1 + T + T^{2}$$
$11$ $$36 - 6 T + T^{2}$$
$13$ $$1 - T + T^{2}$$
$17$ $$( -3 + T )^{2}$$
$19$ $$( -2 + T )^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$81 - 9 T + T^{2}$$
$31$ $$100 - 10 T + T^{2}$$
$37$ $$( 7 + T )^{2}$$
$41$ $$36 + 6 T + T^{2}$$
$43$ $$16 - 4 T + T^{2}$$
$47$ $$36 + 6 T + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$36 - 6 T + T^{2}$$
$61$ $$121 + 11 T + T^{2}$$
$67$ $$4 + 2 T + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$( 7 + T )^{2}$$
$79$ $$4 + 2 T + T^{2}$$
$83$ $$36 + 6 T + T^{2}$$
$89$ $$( -3 + T )^{2}$$
$97$ $$196 + 14 T + T^{2}$$