# Properties

 Label 1134.2.h.e 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 42) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} + q^{8} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} + q^{8} + ( -1 + \zeta_{6} ) q^{10} + 5 q^{11} + ( -1 + 3 \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( 4 - 4 \zeta_{6} ) q^{17} -8 \zeta_{6} q^{19} -\zeta_{6} q^{20} + ( -5 + 5 \zeta_{6} ) q^{22} -4 q^{23} -4 q^{25} + ( -2 - \zeta_{6} ) q^{28} + 5 \zeta_{6} q^{29} -3 \zeta_{6} q^{31} -\zeta_{6} q^{32} + 4 \zeta_{6} q^{34} + ( 3 - 2 \zeta_{6} ) q^{35} + 4 \zeta_{6} q^{37} + 8 q^{38} + q^{40} -2 \zeta_{6} q^{43} -5 \zeta_{6} q^{44} + ( 4 - 4 \zeta_{6} ) q^{46} + ( 6 - 6 \zeta_{6} ) q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + ( 4 - 4 \zeta_{6} ) q^{50} + ( 9 - 9 \zeta_{6} ) q^{53} + 5 q^{55} + ( 3 - 2 \zeta_{6} ) q^{56} -5 q^{58} + 11 \zeta_{6} q^{59} + ( 6 - 6 \zeta_{6} ) q^{61} + 3 q^{62} + q^{64} + 2 \zeta_{6} q^{67} -4 q^{68} + ( -1 + 3 \zeta_{6} ) q^{70} + 2 q^{71} + ( -10 + 10 \zeta_{6} ) q^{73} -4 q^{74} + ( -8 + 8 \zeta_{6} ) q^{76} + ( 15 - 10 \zeta_{6} ) q^{77} + ( -3 + 3 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{80} + 7 \zeta_{6} q^{83} + ( 4 - 4 \zeta_{6} ) q^{85} + 2 q^{86} + 5 q^{88} + 6 \zeta_{6} q^{89} + 4 \zeta_{6} q^{92} + 6 \zeta_{6} q^{94} -8 \zeta_{6} q^{95} -7 \zeta_{6} q^{97} + ( 3 + 5 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} + 2q^{5} + 4q^{7} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{4} + 2q^{5} + 4q^{7} + 2q^{8} - q^{10} + 10q^{11} + q^{14} - q^{16} + 4q^{17} - 8q^{19} - q^{20} - 5q^{22} - 8q^{23} - 8q^{25} - 5q^{28} + 5q^{29} - 3q^{31} - q^{32} + 4q^{34} + 4q^{35} + 4q^{37} + 16q^{38} + 2q^{40} - 2q^{43} - 5q^{44} + 4q^{46} + 6q^{47} + 2q^{49} + 4q^{50} + 9q^{53} + 10q^{55} + 4q^{56} - 10q^{58} + 11q^{59} + 6q^{61} + 6q^{62} + 2q^{64} + 2q^{67} - 8q^{68} + q^{70} + 4q^{71} - 10q^{73} - 8q^{74} - 8q^{76} + 20q^{77} - 3q^{79} - q^{80} + 7q^{83} + 4q^{85} + 4q^{86} + 10q^{88} + 6q^{89} + 4q^{92} + 6q^{94} - 8q^{95} - 7q^{97} + 11q^{98} + O(q^{100})$$

## 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 1.00000 0 2.00000 1.73205i 1.00000 0 −0.500000 + 0.866025i
541.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.00000 0 2.00000 + 1.73205i 1.00000 0 −0.500000 0.866025i
 $$n$$: 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.e 2
3.b odd 2 1 1134.2.h.l 2
7.c even 3 1 1134.2.e.l 2
9.c even 3 1 42.2.e.a 2
9.c even 3 1 1134.2.e.l 2
9.d odd 6 1 126.2.g.c 2
9.d odd 6 1 1134.2.e.e 2
21.h odd 6 1 1134.2.e.e 2
36.f odd 6 1 336.2.q.b 2
36.h even 6 1 1008.2.s.k 2
45.j even 6 1 1050.2.i.l 2
45.k odd 12 2 1050.2.o.a 4
63.g even 3 1 294.2.a.e 1
63.g even 3 1 inner 1134.2.h.e 2
63.h even 3 1 42.2.e.a 2
63.i even 6 1 882.2.g.i 2
63.j odd 6 1 126.2.g.c 2
63.k odd 6 1 294.2.a.f 1
63.l odd 6 1 294.2.e.b 2
63.n odd 6 1 882.2.a.c 1
63.n odd 6 1 1134.2.h.l 2
63.o even 6 1 882.2.g.i 2
63.s even 6 1 882.2.a.d 1
63.t odd 6 1 294.2.e.b 2
72.n even 6 1 1344.2.q.g 2
72.p odd 6 1 1344.2.q.s 2
252.n even 6 1 2352.2.a.f 1
252.o even 6 1 7056.2.a.w 1
252.u odd 6 1 336.2.q.b 2
252.bb even 6 1 1008.2.s.k 2
252.bi even 6 1 2352.2.q.u 2
252.bj even 6 1 2352.2.q.u 2
252.bl odd 6 1 2352.2.a.t 1
252.bn odd 6 1 7056.2.a.bl 1
315.r even 6 1 1050.2.i.l 2
315.bn odd 6 1 7350.2.a.q 1
315.bo even 6 1 7350.2.a.bl 1
315.bt odd 12 2 1050.2.o.a 4
504.w even 6 1 9408.2.a.ce 1
504.ba odd 6 1 9408.2.a.q 1
504.ce odd 6 1 1344.2.q.s 2
504.cq even 6 1 1344.2.q.g 2
504.cw odd 6 1 9408.2.a.z 1
504.cz even 6 1 9408.2.a.cr 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.a 2 9.c even 3 1
42.2.e.a 2 63.h even 3 1
126.2.g.c 2 9.d odd 6 1
126.2.g.c 2 63.j odd 6 1
294.2.a.e 1 63.g even 3 1
294.2.a.f 1 63.k odd 6 1
294.2.e.b 2 63.l odd 6 1
294.2.e.b 2 63.t odd 6 1
336.2.q.b 2 36.f odd 6 1
336.2.q.b 2 252.u odd 6 1
882.2.a.c 1 63.n odd 6 1
882.2.a.d 1 63.s even 6 1
882.2.g.i 2 63.i even 6 1
882.2.g.i 2 63.o even 6 1
1008.2.s.k 2 36.h even 6 1
1008.2.s.k 2 252.bb even 6 1
1050.2.i.l 2 45.j even 6 1
1050.2.i.l 2 315.r even 6 1
1050.2.o.a 4 45.k odd 12 2
1050.2.o.a 4 315.bt odd 12 2
1134.2.e.e 2 9.d odd 6 1
1134.2.e.e 2 21.h odd 6 1
1134.2.e.l 2 7.c even 3 1
1134.2.e.l 2 9.c even 3 1
1134.2.h.e 2 1.a even 1 1 trivial
1134.2.h.e 2 63.g even 3 1 inner
1134.2.h.l 2 3.b odd 2 1
1134.2.h.l 2 63.n odd 6 1
1344.2.q.g 2 72.n even 6 1
1344.2.q.g 2 504.cq even 6 1
1344.2.q.s 2 72.p odd 6 1
1344.2.q.s 2 504.ce odd 6 1
2352.2.a.f 1 252.n even 6 1
2352.2.a.t 1 252.bl odd 6 1
2352.2.q.u 2 252.bi even 6 1
2352.2.q.u 2 252.bj even 6 1
7056.2.a.w 1 252.o even 6 1
7056.2.a.bl 1 252.bn odd 6 1
7350.2.a.q 1 315.bn odd 6 1
7350.2.a.bl 1 315.bo even 6 1
9408.2.a.q 1 504.ba odd 6 1
9408.2.a.z 1 504.cw odd 6 1
9408.2.a.ce 1 504.w even 6 1
9408.2.a.cr 1 504.cz even 6 1

## Hecke kernels

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

 $$T_{5} - 1$$ $$T_{11} - 5$$ $$T_{17}^{2} - 4 T_{17} + 16$$ $$T_{23} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$7 - 4 T + T^{2}$$
$11$ $$( -5 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$16 - 4 T + T^{2}$$
$19$ $$64 + 8 T + T^{2}$$
$23$ $$( 4 + T )^{2}$$
$29$ $$25 - 5 T + T^{2}$$
$31$ $$9 + 3 T + T^{2}$$
$37$ $$16 - 4 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$4 + 2 T + T^{2}$$
$47$ $$36 - 6 T + T^{2}$$
$53$ $$81 - 9 T + T^{2}$$
$59$ $$121 - 11 T + T^{2}$$
$61$ $$36 - 6 T + T^{2}$$
$67$ $$4 - 2 T + T^{2}$$
$71$ $$( -2 + T )^{2}$$
$73$ $$100 + 10 T + T^{2}$$
$79$ $$9 + 3 T + T^{2}$$
$83$ $$49 - 7 T + T^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$49 + 7 T + T^{2}$$