# Properties

 Label 1134.2.f.c 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: no (minimal twist has level 378) 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.

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

## 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 −0.500000 0.866025i 0 0.500000 0.866025i 1.00000 0 1.00000
757.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.500000 + 0.866025i 1.00000 0 1.00000
 $$n$$: 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.c 2
3.b odd 2 1 1134.2.f.n 2
9.c even 3 1 378.2.a.f yes 1
9.c even 3 1 inner 1134.2.f.c 2
9.d odd 6 1 378.2.a.c 1
9.d odd 6 1 1134.2.f.n 2
36.f odd 6 1 3024.2.a.t 1
36.h even 6 1 3024.2.a.m 1
45.h odd 6 1 9450.2.a.dc 1
45.j even 6 1 9450.2.a.bx 1
63.l odd 6 1 2646.2.a.v 1
63.o even 6 1 2646.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.c 1 9.d odd 6 1
378.2.a.f yes 1 9.c even 3 1
1134.2.f.c 2 1.a even 1 1 trivial
1134.2.f.c 2 9.c even 3 1 inner
1134.2.f.n 2 3.b odd 2 1
1134.2.f.n 2 9.d odd 6 1
2646.2.a.i 1 63.o even 6 1
2646.2.a.v 1 63.l odd 6 1
3024.2.a.m 1 36.h even 6 1
3024.2.a.t 1 36.f odd 6 1
9450.2.a.bx 1 45.j even 6 1
9450.2.a.dc 1 45.h odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} + T_{5} + 1$$ $$T_{11}^{2} + 5 T_{11} + 25$$ $$T_{13}$$ $$T_{17} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$1 - T + T^{2}$$
$11$ $$25 + 5 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$1 - T + T^{2}$$
$29$ $$16 + 4 T + T^{2}$$
$31$ $$81 - 9 T + T^{2}$$
$37$ $$( -5 + T )^{2}$$
$41$ $$81 - 9 T + T^{2}$$
$43$ $$100 - 10 T + T^{2}$$
$47$ $$36 + 6 T + T^{2}$$
$53$ $$( -12 + T )^{2}$$
$59$ $$196 - 14 T + T^{2}$$
$61$ $$T^{2}$$
$67$ $$64 - 8 T + T^{2}$$
$71$ $$( 13 + T )^{2}$$
$73$ $$( 2 + T )^{2}$$
$79$ $$36 + 6 T + T^{2}$$
$83$ $$16 - 4 T + T^{2}$$
$89$ $$( 9 + T )^{2}$$
$97$ $$256 + 16 T + T^{2}$$