# Properties

 Label 1134.2.e.c Level $1134$ Weight $2$ Character orbit 1134.e Analytic conductor $9.055$ Analytic rank $1$ 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: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + ( 1 - 3 \zeta_{6} ) q^{7} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + ( 1 - 3 \zeta_{6} ) q^{7} - q^{8} -6 \zeta_{6} q^{11} -5 \zeta_{6} q^{13} + ( -1 + 3 \zeta_{6} ) q^{14} + q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} + 4 \zeta_{6} q^{19} + 6 \zeta_{6} q^{22} + ( -6 + 6 \zeta_{6} ) q^{23} + 5 \zeta_{6} q^{25} + 5 \zeta_{6} q^{26} + ( 1 - 3 \zeta_{6} ) q^{28} + ( -6 + 6 \zeta_{6} ) q^{29} - q^{31} - q^{32} + ( 6 - 6 \zeta_{6} ) q^{34} + \zeta_{6} q^{37} -4 \zeta_{6} q^{38} + 6 \zeta_{6} q^{41} + ( 1 - \zeta_{6} ) q^{43} -6 \zeta_{6} q^{44} + ( 6 - 6 \zeta_{6} ) q^{46} -6 q^{47} + ( -8 + 3 \zeta_{6} ) q^{49} -5 \zeta_{6} q^{50} -5 \zeta_{6} q^{52} + ( 6 - 6 \zeta_{6} ) q^{53} + ( -1 + 3 \zeta_{6} ) q^{56} + ( 6 - 6 \zeta_{6} ) q^{58} -6 q^{59} - q^{61} + q^{62} + q^{64} - q^{67} + ( -6 + 6 \zeta_{6} ) q^{68} + 12 q^{71} + ( -2 + 2 \zeta_{6} ) q^{73} -\zeta_{6} q^{74} + 4 \zeta_{6} q^{76} + ( -18 + 12 \zeta_{6} ) q^{77} - q^{79} -6 \zeta_{6} q^{82} + ( -6 + 6 \zeta_{6} ) q^{83} + ( -1 + \zeta_{6} ) q^{86} + 6 \zeta_{6} q^{88} + ( -15 + 10 \zeta_{6} ) q^{91} + ( -6 + 6 \zeta_{6} ) q^{92} + 6 q^{94} + ( -17 + 17 \zeta_{6} ) q^{97} + ( 8 - 3 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} - q^{7} - 2q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} - q^{7} - 2q^{8} - 6q^{11} - 5q^{13} + q^{14} + 2q^{16} - 6q^{17} + 4q^{19} + 6q^{22} - 6q^{23} + 5q^{25} + 5q^{26} - q^{28} - 6q^{29} - 2q^{31} - 2q^{32} + 6q^{34} + q^{37} - 4q^{38} + 6q^{41} + q^{43} - 6q^{44} + 6q^{46} - 12q^{47} - 13q^{49} - 5q^{50} - 5q^{52} + 6q^{53} + q^{56} + 6q^{58} - 12q^{59} - 2q^{61} + 2q^{62} + 2q^{64} - 2q^{67} - 6q^{68} + 24q^{71} - 2q^{73} - q^{74} + 4q^{76} - 24q^{77} - 2q^{79} - 6q^{82} - 6q^{83} - q^{86} + 6q^{88} - 20q^{91} - 6q^{92} + 12q^{94} - 17q^{97} + 13q^{98} + O(q^{100})$$

## 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 0 0 −0.500000 2.59808i −1.00000 0 0
919.1 −1.00000 0 1.00000 0 0 −0.500000 + 2.59808i −1.00000 0 0
 $$n$$: 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.c 2
3.b odd 2 1 1134.2.e.m 2
7.c even 3 1 1134.2.h.n 2
9.c even 3 1 378.2.g.e yes 2
9.c even 3 1 1134.2.h.n 2
9.d odd 6 1 378.2.g.b 2
9.d odd 6 1 1134.2.h.d 2
21.h odd 6 1 1134.2.h.d 2
63.g even 3 1 378.2.g.e yes 2
63.h even 3 1 inner 1134.2.e.c 2
63.h even 3 1 2646.2.a.h 1
63.i even 6 1 2646.2.a.w 1
63.j odd 6 1 1134.2.e.m 2
63.j odd 6 1 2646.2.a.x 1
63.n odd 6 1 378.2.g.b 2
63.t odd 6 1 2646.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.g.b 2 9.d odd 6 1
378.2.g.b 2 63.n odd 6 1
378.2.g.e yes 2 9.c even 3 1
378.2.g.e yes 2 63.g even 3 1
1134.2.e.c 2 1.a even 1 1 trivial
1134.2.e.c 2 63.h even 3 1 inner
1134.2.e.m 2 3.b odd 2 1
1134.2.e.m 2 63.j odd 6 1
1134.2.h.d 2 9.d odd 6 1
1134.2.h.d 2 21.h odd 6 1
1134.2.h.n 2 7.c even 3 1
1134.2.h.n 2 9.c even 3 1
2646.2.a.g 1 63.t odd 6 1
2646.2.a.h 1 63.h even 3 1
2646.2.a.w 1 63.i even 6 1
2646.2.a.x 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 $$S_{2}^{\mathrm{new}}(1134, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 + T + T^{2}$$
$11$ $$36 + 6 T + T^{2}$$
$13$ $$25 + 5 T + T^{2}$$
$17$ $$36 + 6 T + T^{2}$$
$19$ $$16 - 4 T + T^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$36 + 6 T + T^{2}$$
$31$ $$( 1 + T )^{2}$$
$37$ $$1 - T + T^{2}$$
$41$ $$36 - 6 T + T^{2}$$
$43$ $$1 - T + T^{2}$$
$47$ $$( 6 + T )^{2}$$
$53$ $$36 - 6 T + T^{2}$$
$59$ $$( 6 + T )^{2}$$
$61$ $$( 1 + T )^{2}$$
$67$ $$( 1 + T )^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$4 + 2 T + T^{2}$$
$79$ $$( 1 + T )^{2}$$
$83$ $$36 + 6 T + T^{2}$$
$89$ $$T^{2}$$
$97$ $$289 + 17 T + T^{2}$$