Properties

 Label 1134.2.f.g Level $1134$ Weight $2$ Character orbit 1134.f 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.f (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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + 2 \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} + q^{8} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + 2 \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} + q^{8} -2 q^{10} + ( 4 - 4 \zeta_{6} ) q^{11} -6 \zeta_{6} q^{13} + \zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} + 2 q^{17} -4 q^{19} + ( 2 - 2 \zeta_{6} ) q^{20} + 4 \zeta_{6} q^{22} -8 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} + 6 q^{26} - q^{28} + ( 2 - 2 \zeta_{6} ) q^{29} -\zeta_{6} q^{32} + ( -2 + 2 \zeta_{6} ) q^{34} + 2 q^{35} -10 q^{37} + ( 4 - 4 \zeta_{6} ) q^{38} + 2 \zeta_{6} q^{40} + 6 \zeta_{6} q^{41} + ( 4 - 4 \zeta_{6} ) q^{43} -4 q^{44} + 8 q^{46} -\zeta_{6} q^{49} + \zeta_{6} q^{50} + ( -6 + 6 \zeta_{6} ) q^{52} + 6 q^{53} + 8 q^{55} + ( 1 - \zeta_{6} ) q^{56} + 2 \zeta_{6} q^{58} -4 \zeta_{6} q^{59} + ( -6 + 6 \zeta_{6} ) q^{61} + q^{64} + ( 12 - 12 \zeta_{6} ) q^{65} -4 \zeta_{6} q^{67} -2 \zeta_{6} q^{68} + ( -2 + 2 \zeta_{6} ) q^{70} + 8 q^{71} + 10 q^{73} + ( 10 - 10 \zeta_{6} ) q^{74} + 4 \zeta_{6} q^{76} -4 \zeta_{6} q^{77} -2 q^{80} -6 q^{82} + ( 4 - 4 \zeta_{6} ) q^{83} + 4 \zeta_{6} q^{85} + 4 \zeta_{6} q^{86} + ( 4 - 4 \zeta_{6} ) q^{88} -6 q^{89} -6 q^{91} + ( -8 + 8 \zeta_{6} ) q^{92} -8 \zeta_{6} q^{95} + ( 14 - 14 \zeta_{6} ) q^{97} + q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} + 2 q^{5} + q^{7} + 2 q^{8} + O(q^{10})$$ $$2 q - q^{2} - q^{4} + 2 q^{5} + q^{7} + 2 q^{8} - 4 q^{10} + 4 q^{11} - 6 q^{13} + q^{14} - q^{16} + 4 q^{17} - 8 q^{19} + 2 q^{20} + 4 q^{22} - 8 q^{23} + q^{25} + 12 q^{26} - 2 q^{28} + 2 q^{29} - q^{32} - 2 q^{34} + 4 q^{35} - 20 q^{37} + 4 q^{38} + 2 q^{40} + 6 q^{41} + 4 q^{43} - 8 q^{44} + 16 q^{46} - q^{49} + q^{50} - 6 q^{52} + 12 q^{53} + 16 q^{55} + q^{56} + 2 q^{58} - 4 q^{59} - 6 q^{61} + 2 q^{64} + 12 q^{65} - 4 q^{67} - 2 q^{68} - 2 q^{70} + 16 q^{71} + 20 q^{73} + 10 q^{74} + 4 q^{76} - 4 q^{77} - 4 q^{80} - 12 q^{82} + 4 q^{83} + 4 q^{85} + 4 q^{86} + 4 q^{88} - 12 q^{89} - 12 q^{91} - 8 q^{92} - 8 q^{95} + 14 q^{97} + 2 q^{98} + O(q^{100})$$

Character values

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

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
379.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 1.00000 + 1.73205i 0 0.500000 0.866025i 1.00000 0 −2.00000
757.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.00000 1.73205i 0 0.500000 + 0.866025i 1.00000 0 −2.00000
 $$n$$: 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
9.c 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.f.g 2
3.b odd 2 1 1134.2.f.j 2
9.c even 3 1 42.2.a.a 1
9.c even 3 1 inner 1134.2.f.g 2
9.d odd 6 1 126.2.a.a 1
9.d odd 6 1 1134.2.f.j 2
36.f odd 6 1 336.2.a.d 1
36.h even 6 1 1008.2.a.j 1
45.h odd 6 1 3150.2.a.bo 1
45.j even 6 1 1050.2.a.i 1
45.k odd 12 2 1050.2.g.a 2
45.l even 12 2 3150.2.g.r 2
63.g even 3 1 294.2.e.c 2
63.h even 3 1 294.2.e.c 2
63.i even 6 1 882.2.g.j 2
63.j odd 6 1 882.2.g.h 2
63.k odd 6 1 294.2.e.a 2
63.l odd 6 1 294.2.a.g 1
63.n odd 6 1 882.2.g.h 2
63.o even 6 1 882.2.a.b 1
63.s even 6 1 882.2.g.j 2
63.t odd 6 1 294.2.e.a 2
72.j odd 6 1 4032.2.a.e 1
72.l even 6 1 4032.2.a.m 1
72.n even 6 1 1344.2.a.q 1
72.p odd 6 1 1344.2.a.i 1
99.h odd 6 1 5082.2.a.d 1
117.t even 6 1 7098.2.a.f 1
144.v odd 12 2 5376.2.c.e 2
144.x even 12 2 5376.2.c.bc 2
180.p odd 6 1 8400.2.a.k 1
252.n even 6 1 2352.2.q.n 2
252.s odd 6 1 7056.2.a.k 1
252.u odd 6 1 2352.2.q.i 2
252.bi even 6 1 2352.2.a.l 1
252.bj even 6 1 2352.2.q.n 2
252.bl odd 6 1 2352.2.q.i 2
315.bg odd 6 1 7350.2.a.f 1
504.be even 6 1 9408.2.a.bw 1
504.bn odd 6 1 9408.2.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.a.a 1 9.c even 3 1
126.2.a.a 1 9.d odd 6 1
294.2.a.g 1 63.l odd 6 1
294.2.e.a 2 63.k odd 6 1
294.2.e.a 2 63.t odd 6 1
294.2.e.c 2 63.g even 3 1
294.2.e.c 2 63.h even 3 1
336.2.a.d 1 36.f odd 6 1
882.2.a.b 1 63.o even 6 1
882.2.g.h 2 63.j odd 6 1
882.2.g.h 2 63.n odd 6 1
882.2.g.j 2 63.i even 6 1
882.2.g.j 2 63.s even 6 1
1008.2.a.j 1 36.h even 6 1
1050.2.a.i 1 45.j even 6 1
1050.2.g.a 2 45.k odd 12 2
1134.2.f.g 2 1.a even 1 1 trivial
1134.2.f.g 2 9.c even 3 1 inner
1134.2.f.j 2 3.b odd 2 1
1134.2.f.j 2 9.d odd 6 1
1344.2.a.i 1 72.p odd 6 1
1344.2.a.q 1 72.n even 6 1
2352.2.a.l 1 252.bi even 6 1
2352.2.q.i 2 252.u odd 6 1
2352.2.q.i 2 252.bl odd 6 1
2352.2.q.n 2 252.n even 6 1
2352.2.q.n 2 252.bj even 6 1
3150.2.a.bo 1 45.h odd 6 1
3150.2.g.r 2 45.l even 12 2
4032.2.a.e 1 72.j odd 6 1
4032.2.a.m 1 72.l even 6 1
5082.2.a.d 1 99.h odd 6 1
5376.2.c.e 2 144.v odd 12 2
5376.2.c.bc 2 144.x even 12 2
7056.2.a.k 1 252.s odd 6 1
7098.2.a.f 1 117.t even 6 1
7350.2.a.f 1 315.bg odd 6 1
8400.2.a.k 1 180.p odd 6 1
9408.2.a.n 1 504.bn odd 6 1
9408.2.a.bw 1 504.be even 6 1

Hecke kernels

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

 $$T_{5}^{2} - 2 T_{5} + 4$$ $$T_{11}^{2} - 4 T_{11} + 16$$ $$T_{13}^{2} + 6 T_{13} + 36$$ $$T_{17} - 2$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$4 - 2 T + T^{2}$$
$7$ $$1 - T + T^{2}$$
$11$ $$16 - 4 T + T^{2}$$
$13$ $$36 + 6 T + T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$64 + 8 T + T^{2}$$
$29$ $$4 - 2 T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$( 10 + T )^{2}$$
$41$ $$36 - 6 T + T^{2}$$
$43$ $$16 - 4 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$16 + 4 T + T^{2}$$
$61$ $$36 + 6 T + T^{2}$$
$67$ $$16 + 4 T + T^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$( -10 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$16 - 4 T + T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$196 - 14 T + T^{2}$$