# Properties

 Label 1134.2.e.f Level $1134$ Weight $2$ Character orbit 1134.e 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.e (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: $$\Z[a_1, \ldots, a_{13}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + ( 3 - 3 \zeta_{6} ) q^{5} + ( 1 - 3 \zeta_{6} ) q^{7} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + ( 3 - 3 \zeta_{6} ) q^{5} + ( 1 - 3 \zeta_{6} ) q^{7} - q^{8} + ( -3 + 3 \zeta_{6} ) q^{10} + 4 \zeta_{6} q^{13} + ( -1 + 3 \zeta_{6} ) q^{14} + q^{16} + ( 6 - 6 \zeta_{6} ) q^{17} + 4 \zeta_{6} q^{19} + ( 3 - 3 \zeta_{6} ) q^{20} + ( 6 - 6 \zeta_{6} ) q^{23} -4 \zeta_{6} q^{25} -4 \zeta_{6} q^{26} + ( 1 - 3 \zeta_{6} ) q^{28} + ( -3 + 3 \zeta_{6} ) q^{29} + 8 q^{31} - q^{32} + ( -6 + 6 \zeta_{6} ) q^{34} + ( -6 - 3 \zeta_{6} ) q^{35} -8 \zeta_{6} q^{37} -4 \zeta_{6} q^{38} + ( -3 + 3 \zeta_{6} ) q^{40} -6 \zeta_{6} q^{41} + ( -8 + 8 \zeta_{6} ) q^{43} + ( -6 + 6 \zeta_{6} ) q^{46} -6 q^{47} + ( -8 + 3 \zeta_{6} ) q^{49} + 4 \zeta_{6} q^{50} + 4 \zeta_{6} q^{52} + ( 9 - 9 \zeta_{6} ) q^{53} + ( -1 + 3 \zeta_{6} ) q^{56} + ( 3 - 3 \zeta_{6} ) q^{58} + 3 q^{59} -10 q^{61} -8 q^{62} + q^{64} + 12 q^{65} -10 q^{67} + ( 6 - 6 \zeta_{6} ) q^{68} + ( 6 + 3 \zeta_{6} ) q^{70} -6 q^{71} + ( 7 - 7 \zeta_{6} ) q^{73} + 8 \zeta_{6} q^{74} + 4 \zeta_{6} q^{76} + 17 q^{79} + ( 3 - 3 \zeta_{6} ) q^{80} + 6 \zeta_{6} q^{82} + ( -12 + 12 \zeta_{6} ) q^{83} -18 \zeta_{6} q^{85} + ( 8 - 8 \zeta_{6} ) q^{86} + 6 \zeta_{6} q^{89} + ( 12 - 8 \zeta_{6} ) q^{91} + ( 6 - 6 \zeta_{6} ) q^{92} + 6 q^{94} + 12 q^{95} + ( 10 - 10 \zeta_{6} ) q^{97} + ( 8 - 3 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} + 3q^{5} - q^{7} - 2q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} + 3q^{5} - q^{7} - 2q^{8} - 3q^{10} + 4q^{13} + q^{14} + 2q^{16} + 6q^{17} + 4q^{19} + 3q^{20} + 6q^{23} - 4q^{25} - 4q^{26} - q^{28} - 3q^{29} + 16q^{31} - 2q^{32} - 6q^{34} - 15q^{35} - 8q^{37} - 4q^{38} - 3q^{40} - 6q^{41} - 8q^{43} - 6q^{46} - 12q^{47} - 13q^{49} + 4q^{50} + 4q^{52} + 9q^{53} + q^{56} + 3q^{58} + 6q^{59} - 20q^{61} - 16q^{62} + 2q^{64} + 24q^{65} - 20q^{67} + 6q^{68} + 15q^{70} - 12q^{71} + 7q^{73} + 8q^{74} + 4q^{76} + 34q^{79} + 3q^{80} + 6q^{82} - 12q^{83} - 18q^{85} + 8q^{86} + 6q^{89} + 16q^{91} + 6q^{92} + 12q^{94} + 24q^{95} + 10q^{97} + 13q^{98} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
865.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 0 1.00000 1.50000 2.59808i 0 −0.500000 2.59808i −1.00000 0 −1.50000 + 2.59808i
919.1 −1.00000 0 1.00000 1.50000 + 2.59808i 0 −0.500000 + 2.59808i −1.00000 0 −1.50000 2.59808i
 $$n$$: 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
63.h 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.e.f 2
3.b odd 2 1 1134.2.e.j 2
7.c even 3 1 1134.2.h.k 2
9.c even 3 1 378.2.g.f yes 2
9.c even 3 1 1134.2.h.k 2
9.d odd 6 1 378.2.g.a 2
9.d odd 6 1 1134.2.h.g 2
21.h odd 6 1 1134.2.h.g 2
63.g even 3 1 378.2.g.f yes 2
63.h even 3 1 inner 1134.2.e.f 2
63.h even 3 1 2646.2.a.b 1
63.i even 6 1 2646.2.a.r 1
63.j odd 6 1 1134.2.e.j 2
63.j odd 6 1 2646.2.a.bc 1
63.n odd 6 1 378.2.g.a 2
63.t odd 6 1 2646.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.g.a 2 9.d odd 6 1
378.2.g.a 2 63.n odd 6 1
378.2.g.f yes 2 9.c even 3 1
378.2.g.f yes 2 63.g even 3 1
1134.2.e.f 2 1.a even 1 1 trivial
1134.2.e.f 2 63.h even 3 1 inner
1134.2.e.j 2 3.b odd 2 1
1134.2.e.j 2 63.j odd 6 1
1134.2.h.g 2 9.d odd 6 1
1134.2.h.g 2 21.h odd 6 1
1134.2.h.k 2 7.c even 3 1
1134.2.h.k 2 9.c even 3 1
2646.2.a.b 1 63.h even 3 1
2646.2.a.m 1 63.t odd 6 1
2646.2.a.r 1 63.i even 6 1
2646.2.a.bc 1 63.j odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 3 T_{5} + 9$$ $$T_{11}$$ $$T_{17}^{2} - 6 T_{17} + 36$$ $$T_{23}^{2} - 6 T_{23} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$9 - 3 T + T^{2}$$
$7$ $$7 + T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$16 - 4 T + T^{2}$$
$17$ $$36 - 6 T + T^{2}$$
$19$ $$16 - 4 T + T^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$9 + 3 T + T^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$64 + 8 T + T^{2}$$
$41$ $$36 + 6 T + T^{2}$$
$43$ $$64 + 8 T + T^{2}$$
$47$ $$( 6 + T )^{2}$$
$53$ $$81 - 9 T + T^{2}$$
$59$ $$( -3 + T )^{2}$$
$61$ $$( 10 + T )^{2}$$
$67$ $$( 10 + T )^{2}$$
$71$ $$( 6 + T )^{2}$$
$73$ $$49 - 7 T + T^{2}$$
$79$ $$( -17 + T )^{2}$$
$83$ $$144 + 12 T + T^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$100 - 10 T + T^{2}$$