# Properties

 Label 1134.2.h.f 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 126) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + 3 q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} + q^{8} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + 3 q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} + q^{8} + ( -3 + 3 \zeta_{6} ) q^{10} -3 q^{11} + ( -2 + 2 \zeta_{6} ) q^{13} + ( 1 - 3 \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} -2 \zeta_{6} q^{19} -3 \zeta_{6} q^{20} + ( 3 - 3 \zeta_{6} ) q^{22} -6 q^{23} + 4 q^{25} -2 \zeta_{6} q^{26} + ( 2 + \zeta_{6} ) q^{28} -9 \zeta_{6} q^{29} + 7 \zeta_{6} q^{31} -\zeta_{6} q^{32} -6 \zeta_{6} q^{34} + ( -9 + 6 \zeta_{6} ) q^{35} + 10 \zeta_{6} q^{37} + 2 q^{38} + 3 q^{40} + 4 \zeta_{6} q^{43} + 3 \zeta_{6} q^{44} + ( 6 - 6 \zeta_{6} ) q^{46} + ( -12 + 12 \zeta_{6} ) q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + ( -4 + 4 \zeta_{6} ) q^{50} + 2 q^{52} + ( 3 - 3 \zeta_{6} ) q^{53} -9 q^{55} + ( -3 + 2 \zeta_{6} ) q^{56} + 9 q^{58} + 3 \zeta_{6} q^{59} + ( 4 - 4 \zeta_{6} ) q^{61} -7 q^{62} + q^{64} + ( -6 + 6 \zeta_{6} ) q^{65} -2 \zeta_{6} q^{67} + 6 q^{68} + ( 3 - 9 \zeta_{6} ) q^{70} + ( -2 + 2 \zeta_{6} ) q^{73} -10 q^{74} + ( -2 + 2 \zeta_{6} ) q^{76} + ( 9 - 6 \zeta_{6} ) q^{77} + ( -5 + 5 \zeta_{6} ) q^{79} + ( -3 + 3 \zeta_{6} ) q^{80} -9 \zeta_{6} q^{83} + ( -18 + 18 \zeta_{6} ) q^{85} -4 q^{86} -3 q^{88} + 6 \zeta_{6} q^{89} + ( 2 - 6 \zeta_{6} ) q^{91} + 6 \zeta_{6} q^{92} -12 \zeta_{6} q^{94} -6 \zeta_{6} q^{95} + 13 \zeta_{6} q^{97} + ( 3 + 5 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} + 6q^{5} - 4q^{7} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{4} + 6q^{5} - 4q^{7} + 2q^{8} - 3q^{10} - 6q^{11} - 2q^{13} - q^{14} - q^{16} - 6q^{17} - 2q^{19} - 3q^{20} + 3q^{22} - 12q^{23} + 8q^{25} - 2q^{26} + 5q^{28} - 9q^{29} + 7q^{31} - q^{32} - 6q^{34} - 12q^{35} + 10q^{37} + 4q^{38} + 6q^{40} + 4q^{43} + 3q^{44} + 6q^{46} - 12q^{47} + 2q^{49} - 4q^{50} + 4q^{52} + 3q^{53} - 18q^{55} - 4q^{56} + 18q^{58} + 3q^{59} + 4q^{61} - 14q^{62} + 2q^{64} - 6q^{65} - 2q^{67} + 12q^{68} - 3q^{70} - 2q^{73} - 20q^{74} - 2q^{76} + 12q^{77} - 5q^{79} - 3q^{80} - 9q^{83} - 18q^{85} - 8q^{86} - 6q^{88} + 6q^{89} - 2q^{91} + 6q^{92} - 12q^{94} - 6q^{95} + 13q^{97} + 11q^{98} + O(q^{100})$$

## 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 −2.00000 + 1.73205i 1.00000 0 −1.50000 + 2.59808i
541.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 3.00000 0 −2.00000 1.73205i 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.f 2
3.b odd 2 1 1134.2.h.j 2
7.c even 3 1 1134.2.e.k 2
9.c even 3 1 126.2.g.a 2
9.c even 3 1 1134.2.e.k 2
9.d odd 6 1 126.2.g.d yes 2
9.d odd 6 1 1134.2.e.g 2
21.h odd 6 1 1134.2.e.g 2
36.f odd 6 1 1008.2.s.b 2
36.h even 6 1 1008.2.s.o 2
63.g even 3 1 882.2.a.j 1
63.g even 3 1 inner 1134.2.h.f 2
63.h even 3 1 126.2.g.a 2
63.i even 6 1 882.2.g.g 2
63.j odd 6 1 126.2.g.d yes 2
63.k odd 6 1 882.2.a.h 1
63.l odd 6 1 882.2.g.e 2
63.n odd 6 1 882.2.a.a 1
63.n odd 6 1 1134.2.h.j 2
63.o even 6 1 882.2.g.g 2
63.s even 6 1 882.2.a.e 1
63.t odd 6 1 882.2.g.e 2
252.n even 6 1 7056.2.a.h 1
252.o even 6 1 7056.2.a.e 1
252.u odd 6 1 1008.2.s.b 2
252.bb even 6 1 1008.2.s.o 2
252.bl odd 6 1 7056.2.a.by 1
252.bn odd 6 1 7056.2.a.bx 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.g.a 2 9.c even 3 1
126.2.g.a 2 63.h even 3 1
126.2.g.d yes 2 9.d odd 6 1
126.2.g.d yes 2 63.j odd 6 1
882.2.a.a 1 63.n odd 6 1
882.2.a.e 1 63.s even 6 1
882.2.a.h 1 63.k odd 6 1
882.2.a.j 1 63.g even 3 1
882.2.g.e 2 63.l odd 6 1
882.2.g.e 2 63.t odd 6 1
882.2.g.g 2 63.i even 6 1
882.2.g.g 2 63.o even 6 1
1008.2.s.b 2 36.f odd 6 1
1008.2.s.b 2 252.u odd 6 1
1008.2.s.o 2 36.h even 6 1
1008.2.s.o 2 252.bb even 6 1
1134.2.e.g 2 9.d odd 6 1
1134.2.e.g 2 21.h odd 6 1
1134.2.e.k 2 7.c even 3 1
1134.2.e.k 2 9.c even 3 1
1134.2.h.f 2 1.a even 1 1 trivial
1134.2.h.f 2 63.g even 3 1 inner
1134.2.h.j 2 3.b odd 2 1
1134.2.h.j 2 63.n odd 6 1
7056.2.a.e 1 252.o even 6 1
7056.2.a.h 1 252.n even 6 1
7056.2.a.bx 1 252.bn odd 6 1
7056.2.a.by 1 252.bl odd 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -3 + T )^{2}$$
$7$ $$7 + 4 T + T^{2}$$
$11$ $$( 3 + T )^{2}$$
$13$ $$4 + 2 T + T^{2}$$
$17$ $$36 + 6 T + T^{2}$$
$19$ $$4 + 2 T + T^{2}$$
$23$ $$( 6 + T )^{2}$$
$29$ $$81 + 9 T + T^{2}$$
$31$ $$49 - 7 T + T^{2}$$
$37$ $$100 - 10 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$16 - 4 T + T^{2}$$
$47$ $$144 + 12 T + T^{2}$$
$53$ $$9 - 3 T + T^{2}$$
$59$ $$9 - 3 T + T^{2}$$
$61$ $$16 - 4 T + T^{2}$$
$67$ $$4 + 2 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$4 + 2 T + T^{2}$$
$79$ $$25 + 5 T + T^{2}$$
$83$ $$81 + 9 T + T^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$169 - 13 T + T^{2}$$