# Properties

 Label 1134.2.e.a Level $1134$ Weight $2$ Character orbit 1134.e 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.e (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: $$\Z[a_1, \ldots, a_{13}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + ( -3 + 3 \zeta_{6} ) q^{5} + ( -2 + 3 \zeta_{6} ) q^{7} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + ( -3 + 3 \zeta_{6} ) q^{5} + ( -2 + 3 \zeta_{6} ) q^{7} - q^{8} + ( 3 - 3 \zeta_{6} ) q^{10} -3 \zeta_{6} q^{11} + 4 \zeta_{6} q^{13} + ( 2 - 3 \zeta_{6} ) q^{14} + q^{16} + 4 \zeta_{6} q^{19} + ( -3 + 3 \zeta_{6} ) q^{20} + 3 \zeta_{6} q^{22} -4 \zeta_{6} q^{25} -4 \zeta_{6} q^{26} + ( -2 + 3 \zeta_{6} ) q^{28} + ( -9 + 9 \zeta_{6} ) q^{29} - q^{31} - q^{32} + ( -3 - 6 \zeta_{6} ) q^{35} -8 \zeta_{6} q^{37} -4 \zeta_{6} q^{38} + ( 3 - 3 \zeta_{6} ) q^{40} + ( 10 - 10 \zeta_{6} ) q^{43} -3 \zeta_{6} q^{44} -6 q^{47} + ( -5 - 3 \zeta_{6} ) q^{49} + 4 \zeta_{6} q^{50} + 4 \zeta_{6} q^{52} + ( 3 - 3 \zeta_{6} ) q^{53} + 9 q^{55} + ( 2 - 3 \zeta_{6} ) q^{56} + ( 9 - 9 \zeta_{6} ) q^{58} + 3 q^{59} -10 q^{61} + q^{62} + q^{64} -12 q^{65} -10 q^{67} + ( 3 + 6 \zeta_{6} ) q^{70} -6 q^{71} + ( -2 + 2 \zeta_{6} ) q^{73} + 8 \zeta_{6} q^{74} + 4 \zeta_{6} q^{76} + ( 9 - 3 \zeta_{6} ) q^{77} - q^{79} + ( -3 + 3 \zeta_{6} ) q^{80} + ( 9 - 9 \zeta_{6} ) q^{83} + ( -10 + 10 \zeta_{6} ) q^{86} + 3 \zeta_{6} q^{88} -6 \zeta_{6} q^{89} + ( -12 + 4 \zeta_{6} ) q^{91} + 6 q^{94} -12 q^{95} + ( 1 - \zeta_{6} ) q^{97} + ( 5 + 3 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} - 3q^{5} - q^{7} - 2q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} - 3q^{5} - q^{7} - 2q^{8} + 3q^{10} - 3q^{11} + 4q^{13} + q^{14} + 2q^{16} + 4q^{19} - 3q^{20} + 3q^{22} - 4q^{25} - 4q^{26} - q^{28} - 9q^{29} - 2q^{31} - 2q^{32} - 12q^{35} - 8q^{37} - 4q^{38} + 3q^{40} + 10q^{43} - 3q^{44} - 12q^{47} - 13q^{49} + 4q^{50} + 4q^{52} + 3q^{53} + 18q^{55} + q^{56} + 9q^{58} + 6q^{59} - 20q^{61} + 2q^{62} + 2q^{64} - 24q^{65} - 20q^{67} + 12q^{70} - 12q^{71} - 2q^{73} + 8q^{74} + 4q^{76} + 15q^{77} - 2q^{79} - 3q^{80} + 9q^{83} - 10q^{86} + 3q^{88} - 6q^{89} - 20q^{91} + 12q^{94} - 24q^{95} + q^{97} + 13q^{98} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
865.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 0 1.00000 −1.50000 + 2.59808i 0 −0.500000 + 2.59808i −1.00000 0 1.50000 2.59808i
919.1 −1.00000 0 1.00000 −1.50000 2.59808i 0 −0.500000 2.59808i −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.h 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.e.a 2
3.b odd 2 1 1134.2.e.p 2
7.c even 3 1 1134.2.h.p 2
9.c even 3 1 42.2.e.b 2
9.c even 3 1 1134.2.h.p 2
9.d odd 6 1 126.2.g.b 2
9.d odd 6 1 1134.2.h.a 2
21.h odd 6 1 1134.2.h.a 2
36.f odd 6 1 336.2.q.d 2
36.h even 6 1 1008.2.s.n 2
45.j even 6 1 1050.2.i.e 2
45.k odd 12 2 1050.2.o.b 4
63.g even 3 1 42.2.e.b 2
63.h even 3 1 294.2.a.d 1
63.h even 3 1 inner 1134.2.e.a 2
63.i even 6 1 882.2.a.k 1
63.j odd 6 1 882.2.a.g 1
63.j odd 6 1 1134.2.e.p 2
63.k odd 6 1 294.2.e.f 2
63.l odd 6 1 294.2.e.f 2
63.n odd 6 1 126.2.g.b 2
63.o even 6 1 882.2.g.b 2
63.s even 6 1 882.2.g.b 2
63.t odd 6 1 294.2.a.a 1
72.n even 6 1 1344.2.q.v 2
72.p odd 6 1 1344.2.q.j 2
252.n even 6 1 2352.2.q.m 2
252.o even 6 1 1008.2.s.n 2
252.r odd 6 1 7056.2.a.bz 1
252.u odd 6 1 2352.2.a.m 1
252.bb even 6 1 7056.2.a.g 1
252.bi even 6 1 2352.2.q.m 2
252.bj even 6 1 2352.2.a.n 1
252.bl odd 6 1 336.2.q.d 2
315.q odd 6 1 7350.2.a.cw 1
315.r even 6 1 7350.2.a.ce 1
315.bo even 6 1 1050.2.i.e 2
315.ch odd 12 2 1050.2.o.b 4
504.w even 6 1 1344.2.q.v 2
504.ba odd 6 1 1344.2.q.j 2
504.bf even 6 1 9408.2.a.bm 1
504.bp odd 6 1 9408.2.a.db 1
504.ce odd 6 1 9408.2.a.bu 1
504.cq even 6 1 9408.2.a.d 1

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

## Hecke kernels

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

 $$T_{5}^{2} + 3 T_{5} + 9$$ $$T_{11}^{2} + 3 T_{11} + 9$$ $$T_{17}$$ $$T_{23}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ 1
$5$ $$1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}$$
$7$ $$1 + T + 7 T^{2}$$
$11$ $$1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4}$$
$13$ $$1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4}$$
$17$ $$1 - 17 T^{2} + 289 T^{4}$$
$19$ $$1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4}$$
$23$ $$1 - 23 T^{2} + 529 T^{4}$$
$29$ $$1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4}$$
$31$ $$( 1 + T + 31 T^{2} )^{2}$$
$37$ $$1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4}$$
$41$ $$1 - 41 T^{2} + 1681 T^{4}$$
$43$ $$1 - 10 T + 57 T^{2} - 430 T^{3} + 1849 T^{4}$$
$47$ $$( 1 + 6 T + 47 T^{2} )^{2}$$
$53$ $$1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4}$$
$59$ $$( 1 - 3 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 10 T + 61 T^{2} )^{2}$$
$67$ $$( 1 + 10 T + 67 T^{2} )^{2}$$
$71$ $$( 1 + 6 T + 71 T^{2} )^{2}$$
$73$ $$1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4}$$
$79$ $$( 1 + T + 79 T^{2} )^{2}$$
$83$ $$1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4}$$
$89$ $$1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4}$$
$97$ $$1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4}$$