# Properties

 Label 1134.2.h.g Level $1134$ Weight $2$ Character orbit 1134.h Analytic conductor $9.055$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + 3 q^{5} + ( 2 + \zeta_{6} ) q^{7} + q^{8} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + 3 q^{5} + ( 2 + \zeta_{6} ) q^{7} + q^{8} + ( -3 + 3 \zeta_{6} ) q^{10} + ( 4 - 4 \zeta_{6} ) q^{13} + ( -3 + 2 \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} + 4 \zeta_{6} q^{19} -3 \zeta_{6} q^{20} + 6 q^{23} + 4 q^{25} + 4 \zeta_{6} q^{26} + ( 1 - 3 \zeta_{6} ) q^{28} + 3 \zeta_{6} q^{29} -8 \zeta_{6} q^{31} -\zeta_{6} q^{32} -6 \zeta_{6} q^{34} + ( 6 + 3 \zeta_{6} ) q^{35} -8 \zeta_{6} q^{37} -4 q^{38} + 3 q^{40} + ( 6 - 6 \zeta_{6} ) q^{41} -8 \zeta_{6} q^{43} + ( -6 + 6 \zeta_{6} ) q^{46} + ( -6 + 6 \zeta_{6} ) q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} + ( -4 + 4 \zeta_{6} ) q^{50} -4 q^{52} + ( -9 + 9 \zeta_{6} ) q^{53} + ( 2 + \zeta_{6} ) q^{56} -3 q^{58} + 3 \zeta_{6} q^{59} + ( 10 - 10 \zeta_{6} ) q^{61} + 8 q^{62} + q^{64} + ( 12 - 12 \zeta_{6} ) q^{65} + 10 \zeta_{6} q^{67} + 6 q^{68} + ( -9 + 6 \zeta_{6} ) q^{70} + 6 q^{71} + ( 7 - 7 \zeta_{6} ) q^{73} + 8 q^{74} + ( 4 - 4 \zeta_{6} ) q^{76} + ( -17 + 17 \zeta_{6} ) q^{79} + ( -3 + 3 \zeta_{6} ) q^{80} + 6 \zeta_{6} q^{82} + 12 \zeta_{6} q^{83} + ( -18 + 18 \zeta_{6} ) q^{85} + 8 q^{86} -6 \zeta_{6} q^{89} + ( 12 - 8 \zeta_{6} ) q^{91} -6 \zeta_{6} q^{92} -6 \zeta_{6} q^{94} + 12 \zeta_{6} q^{95} + 10 \zeta_{6} q^{97} + ( -8 + 3 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} + 6q^{5} + 5q^{7} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{4} + 6q^{5} + 5q^{7} + 2q^{8} - 3q^{10} + 4q^{13} - 4q^{14} - q^{16} - 6q^{17} + 4q^{19} - 3q^{20} + 12q^{23} + 8q^{25} + 4q^{26} - q^{28} + 3q^{29} - 8q^{31} - q^{32} - 6q^{34} + 15q^{35} - 8q^{37} - 8q^{38} + 6q^{40} + 6q^{41} - 8q^{43} - 6q^{46} - 6q^{47} + 11q^{49} - 4q^{50} - 8q^{52} - 9q^{53} + 5q^{56} - 6q^{58} + 3q^{59} + 10q^{61} + 16q^{62} + 2q^{64} + 12q^{65} + 10q^{67} + 12q^{68} - 12q^{70} + 12q^{71} + 7q^{73} + 16q^{74} + 4q^{76} - 17q^{79} - 3q^{80} + 6q^{82} + 12q^{83} - 18q^{85} + 16q^{86} - 6q^{89} + 16q^{91} - 6q^{92} - 6q^{94} + 12q^{95} + 10q^{97} - 13q^{98} + O(q^{100})$$

## Character values

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

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

## 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 3.00000 0 2.50000 + 0.866025i 1.00000 0 −1.50000 + 2.59808i
541.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 3.00000 0 2.50000 0.866025i 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.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.g 2
3.b odd 2 1 1134.2.h.k 2
7.c even 3 1 1134.2.e.j 2
9.c even 3 1 378.2.g.a 2
9.c even 3 1 1134.2.e.j 2
9.d odd 6 1 378.2.g.f yes 2
9.d odd 6 1 1134.2.e.f 2
21.h odd 6 1 1134.2.e.f 2
63.g even 3 1 inner 1134.2.h.g 2
63.g even 3 1 2646.2.a.bc 1
63.h even 3 1 378.2.g.a 2
63.j odd 6 1 378.2.g.f yes 2
63.k odd 6 1 2646.2.a.r 1
63.n odd 6 1 1134.2.h.k 2
63.n odd 6 1 2646.2.a.b 1
63.s even 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.c even 3 1
378.2.g.a 2 63.h even 3 1
378.2.g.f yes 2 9.d odd 6 1
378.2.g.f yes 2 63.j odd 6 1
1134.2.e.f 2 9.d odd 6 1
1134.2.e.f 2 21.h odd 6 1
1134.2.e.j 2 7.c even 3 1
1134.2.e.j 2 9.c even 3 1
1134.2.h.g 2 1.a even 1 1 trivial
1134.2.h.g 2 63.g even 3 1 inner
1134.2.h.k 2 3.b odd 2 1
1134.2.h.k 2 63.n odd 6 1
2646.2.a.b 1 63.n odd 6 1
2646.2.a.m 1 63.s even 6 1
2646.2.a.r 1 63.k odd 6 1
2646.2.a.bc 1 63.g even 3 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -3 + T )^{2}$$
$7$ $$7 - 5 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$16 - 4 T + T^{2}$$
$17$ $$36 + 6 T + T^{2}$$
$19$ $$16 - 4 T + T^{2}$$
$23$ $$( -6 + T )^{2}$$
$29$ $$9 - 3 T + T^{2}$$
$31$ $$64 + 8 T + T^{2}$$
$37$ $$64 + 8 T + T^{2}$$
$41$ $$36 - 6 T + T^{2}$$
$43$ $$64 + 8 T + T^{2}$$
$47$ $$36 + 6 T + T^{2}$$
$53$ $$81 + 9 T + T^{2}$$
$59$ $$9 - 3 T + T^{2}$$
$61$ $$100 - 10 T + T^{2}$$
$67$ $$100 - 10 T + T^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$49 - 7 T + T^{2}$$
$79$ $$289 + 17 T + T^{2}$$
$83$ $$144 - 12 T + T^{2}$$
$89$ $$36 + 6 T + T^{2}$$
$97$ $$100 - 10 T + T^{2}$$
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