# Properties

 Label 1134.2.h.b Level $1134$ Weight $2$ Character orbit 1134.h 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.h (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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -2 q^{5} + ( -3 + \zeta_{6} ) q^{7} + q^{8} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -2 q^{5} + ( -3 + \zeta_{6} ) q^{7} + q^{8} + ( 2 - 2 \zeta_{6} ) q^{10} + 5 q^{11} + ( -6 + 6 \zeta_{6} ) q^{13} + ( 2 - 3 \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( 4 - 4 \zeta_{6} ) q^{17} + 4 \zeta_{6} q^{19} + 2 \zeta_{6} q^{20} + ( -5 + 5 \zeta_{6} ) q^{22} -4 q^{23} - q^{25} -6 \zeta_{6} q^{26} + ( 1 + 2 \zeta_{6} ) q^{28} -7 \zeta_{6} q^{29} -3 \zeta_{6} q^{31} -\zeta_{6} q^{32} + 4 \zeta_{6} q^{34} + ( 6 - 2 \zeta_{6} ) q^{35} -8 \zeta_{6} q^{37} -4 q^{38} -2 q^{40} + ( 6 - 6 \zeta_{6} ) q^{41} -8 \zeta_{6} q^{43} -5 \zeta_{6} q^{44} + ( 4 - 4 \zeta_{6} ) q^{46} + ( -6 + 6 \zeta_{6} ) q^{47} + ( 8 - 5 \zeta_{6} ) q^{49} + ( 1 - \zeta_{6} ) q^{50} + 6 q^{52} + ( 6 - 6 \zeta_{6} ) q^{53} -10 q^{55} + ( -3 + \zeta_{6} ) q^{56} + 7 q^{58} -7 \zeta_{6} q^{59} + 3 q^{62} + q^{64} + ( 12 - 12 \zeta_{6} ) q^{65} -10 \zeta_{6} q^{67} -4 q^{68} + ( -4 + 6 \zeta_{6} ) q^{70} -4 q^{71} + ( -13 + 13 \zeta_{6} ) q^{73} + 8 q^{74} + ( 4 - 4 \zeta_{6} ) q^{76} + ( -15 + 5 \zeta_{6} ) q^{77} + ( 3 - 3 \zeta_{6} ) q^{79} + ( 2 - 2 \zeta_{6} ) q^{80} + 6 \zeta_{6} q^{82} + 7 \zeta_{6} q^{83} + ( -8 + 8 \zeta_{6} ) q^{85} + 8 q^{86} + 5 q^{88} -6 \zeta_{6} q^{89} + ( 12 - 18 \zeta_{6} ) q^{91} + 4 \zeta_{6} q^{92} -6 \zeta_{6} q^{94} -8 \zeta_{6} q^{95} + 5 \zeta_{6} q^{97} + ( -3 + 8 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} - 4q^{5} - 5q^{7} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{4} - 4q^{5} - 5q^{7} + 2q^{8} + 2q^{10} + 10q^{11} - 6q^{13} + q^{14} - q^{16} + 4q^{17} + 4q^{19} + 2q^{20} - 5q^{22} - 8q^{23} - 2q^{25} - 6q^{26} + 4q^{28} - 7q^{29} - 3q^{31} - q^{32} + 4q^{34} + 10q^{35} - 8q^{37} - 8q^{38} - 4q^{40} + 6q^{41} - 8q^{43} - 5q^{44} + 4q^{46} - 6q^{47} + 11q^{49} + q^{50} + 12q^{52} + 6q^{53} - 20q^{55} - 5q^{56} + 14q^{58} - 7q^{59} + 6q^{62} + 2q^{64} + 12q^{65} - 10q^{67} - 8q^{68} - 2q^{70} - 8q^{71} - 13q^{73} + 16q^{74} + 4q^{76} - 25q^{77} + 3q^{79} + 2q^{80} + 6q^{82} + 7q^{83} - 8q^{85} + 16q^{86} + 10q^{88} - 6q^{89} + 6q^{91} + 4q^{92} - 6q^{94} - 8q^{95} + 5q^{97} + 2q^{98} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −2.00000 0 −2.50000 + 0.866025i 1.00000 0 1.00000 1.73205i
541.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −2.00000 0 −2.50000 0.866025i 1.00000 0 1.00000 + 1.73205i
 $$n$$: 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.g 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.h.b 2
3.b odd 2 1 1134.2.h.o 2
7.c even 3 1 1134.2.e.o 2
9.c even 3 1 378.2.g.c 2
9.c even 3 1 1134.2.e.o 2
9.d odd 6 1 378.2.g.d yes 2
9.d odd 6 1 1134.2.e.b 2
21.h odd 6 1 1134.2.e.b 2
63.g even 3 1 inner 1134.2.h.b 2
63.g even 3 1 2646.2.a.t 1
63.h even 3 1 378.2.g.c 2
63.j odd 6 1 378.2.g.d yes 2
63.k odd 6 1 2646.2.a.bb 1
63.n odd 6 1 1134.2.h.o 2
63.n odd 6 1 2646.2.a.k 1
63.s even 6 1 2646.2.a.c 1

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

## Hecke kernels

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

 $$T_{5} + 2$$ $$T_{11} - 5$$ $$T_{17}^{2} - 4 T_{17} + 16$$ $$T_{23} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( 2 + T )^{2}$$
$7$ $$7 + 5 T + T^{2}$$
$11$ $$( -5 + T )^{2}$$
$13$ $$36 + 6 T + T^{2}$$
$17$ $$16 - 4 T + T^{2}$$
$19$ $$16 - 4 T + T^{2}$$
$23$ $$( 4 + T )^{2}$$
$29$ $$49 + 7 T + T^{2}$$
$31$ $$9 + 3 T + T^{2}$$
$37$ $$64 + 8 T + T^{2}$$
$41$ $$36 - 6 T + T^{2}$$
$43$ $$64 + 8 T + T^{2}$$
$47$ $$36 + 6 T + T^{2}$$
$53$ $$36 - 6 T + T^{2}$$
$59$ $$49 + 7 T + T^{2}$$
$61$ $$T^{2}$$
$67$ $$100 + 10 T + T^{2}$$
$71$ $$( 4 + T )^{2}$$
$73$ $$169 + 13 T + T^{2}$$
$79$ $$9 - 3 T + T^{2}$$
$83$ $$49 - 7 T + T^{2}$$
$89$ $$36 + 6 T + T^{2}$$
$97$ $$25 - 5 T + T^{2}$$