# Properties

 Label 1134.2.e.e 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + ( 1 - \zeta_{6} ) q^{5} + ( -2 - \zeta_{6} ) q^{7} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + ( 1 - \zeta_{6} ) q^{5} + ( -2 - \zeta_{6} ) q^{7} - q^{8} + ( -1 + \zeta_{6} ) q^{10} + 5 \zeta_{6} q^{11} + ( 2 + \zeta_{6} ) q^{14} + q^{16} + ( -4 + 4 \zeta_{6} ) q^{17} -8 \zeta_{6} q^{19} + ( 1 - \zeta_{6} ) q^{20} -5 \zeta_{6} q^{22} + ( -4 + 4 \zeta_{6} ) q^{23} + 4 \zeta_{6} q^{25} + ( -2 - \zeta_{6} ) q^{28} + ( -5 + 5 \zeta_{6} ) q^{29} + 3 q^{31} - q^{32} + ( 4 - 4 \zeta_{6} ) q^{34} + ( -3 + 2 \zeta_{6} ) q^{35} + 4 \zeta_{6} q^{37} + 8 \zeta_{6} q^{38} + ( -1 + \zeta_{6} ) q^{40} + ( -2 + 2 \zeta_{6} ) q^{43} + 5 \zeta_{6} q^{44} + ( 4 - 4 \zeta_{6} ) q^{46} + 6 q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} -4 \zeta_{6} q^{50} + ( -9 + 9 \zeta_{6} ) q^{53} + 5 q^{55} + ( 2 + \zeta_{6} ) q^{56} + ( 5 - 5 \zeta_{6} ) q^{58} + 11 q^{59} -6 q^{61} -3 q^{62} + q^{64} -2 q^{67} + ( -4 + 4 \zeta_{6} ) q^{68} + ( 3 - 2 \zeta_{6} ) q^{70} -2 q^{71} + ( -10 + 10 \zeta_{6} ) q^{73} -4 \zeta_{6} q^{74} -8 \zeta_{6} q^{76} + ( 5 - 15 \zeta_{6} ) q^{77} + 3 q^{79} + ( 1 - \zeta_{6} ) q^{80} + ( -7 + 7 \zeta_{6} ) q^{83} + 4 \zeta_{6} q^{85} + ( 2 - 2 \zeta_{6} ) q^{86} -5 \zeta_{6} q^{88} -6 \zeta_{6} q^{89} + ( -4 + 4 \zeta_{6} ) q^{92} -6 q^{94} -8 q^{95} + ( -7 + 7 \zeta_{6} ) q^{97} + ( -3 - 5 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} + q^{5} - 5q^{7} - 2q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} + q^{5} - 5q^{7} - 2q^{8} - q^{10} + 5q^{11} + 5q^{14} + 2q^{16} - 4q^{17} - 8q^{19} + q^{20} - 5q^{22} - 4q^{23} + 4q^{25} - 5q^{28} - 5q^{29} + 6q^{31} - 2q^{32} + 4q^{34} - 4q^{35} + 4q^{37} + 8q^{38} - q^{40} - 2q^{43} + 5q^{44} + 4q^{46} + 12q^{47} + 11q^{49} - 4q^{50} - 9q^{53} + 10q^{55} + 5q^{56} + 5q^{58} + 22q^{59} - 12q^{61} - 6q^{62} + 2q^{64} - 4q^{67} - 4q^{68} + 4q^{70} - 4q^{71} - 10q^{73} - 4q^{74} - 8q^{76} - 5q^{77} + 6q^{79} + q^{80} - 7q^{83} + 4q^{85} + 2q^{86} - 5q^{88} - 6q^{89} - 4q^{92} - 12q^{94} - 16q^{95} - 7q^{97} - 11q^{98} + O(q^{100})$$

## Character values

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

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

## 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 0.500000 0.866025i 0 −2.50000 0.866025i −1.00000 0 −0.500000 + 0.866025i
919.1 −1.00000 0 1.00000 0.500000 + 0.866025i 0 −2.50000 + 0.866025i −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.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.e 2
3.b odd 2 1 1134.2.e.l 2
7.c even 3 1 1134.2.h.l 2
9.c even 3 1 126.2.g.c 2
9.c even 3 1 1134.2.h.l 2
9.d odd 6 1 42.2.e.a 2
9.d odd 6 1 1134.2.h.e 2
21.h odd 6 1 1134.2.h.e 2
36.f odd 6 1 1008.2.s.k 2
36.h even 6 1 336.2.q.b 2
45.h odd 6 1 1050.2.i.l 2
45.l even 12 2 1050.2.o.a 4
63.g even 3 1 126.2.g.c 2
63.h even 3 1 882.2.a.c 1
63.h even 3 1 inner 1134.2.e.e 2
63.i even 6 1 294.2.a.f 1
63.j odd 6 1 294.2.a.e 1
63.j odd 6 1 1134.2.e.l 2
63.k odd 6 1 882.2.g.i 2
63.l odd 6 1 882.2.g.i 2
63.n odd 6 1 42.2.e.a 2
63.o even 6 1 294.2.e.b 2
63.s even 6 1 294.2.e.b 2
63.t odd 6 1 882.2.a.d 1
72.j odd 6 1 1344.2.q.g 2
72.l even 6 1 1344.2.q.s 2
252.o even 6 1 336.2.q.b 2
252.r odd 6 1 2352.2.a.f 1
252.s odd 6 1 2352.2.q.u 2
252.u odd 6 1 7056.2.a.w 1
252.bb even 6 1 2352.2.a.t 1
252.bj even 6 1 7056.2.a.bl 1
252.bl odd 6 1 1008.2.s.k 2
252.bn odd 6 1 2352.2.q.u 2
315.v odd 6 1 1050.2.i.l 2
315.bq even 6 1 7350.2.a.q 1
315.br odd 6 1 7350.2.a.bl 1
315.bx even 12 2 1050.2.o.a 4
504.bi odd 6 1 9408.2.a.ce 1
504.bt even 6 1 9408.2.a.q 1
504.ca even 6 1 9408.2.a.z 1
504.cm odd 6 1 9408.2.a.cr 1
504.cy even 6 1 1344.2.q.s 2
504.db odd 6 1 1344.2.q.g 2

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

## Hecke kernels

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

 $$T_{5}^{2} - T_{5} + 1$$ $$T_{11}^{2} - 5 T_{11} + 25$$ $$T_{17}^{2} + 4 T_{17} + 16$$ $$T_{23}^{2} + 4 T_{23} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ 1
$5$ $$1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4}$$
$7$ $$1 + 5 T + 7 T^{2}$$
$11$ $$1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4}$$
$13$ $$1 - 13 T^{2} + 169 T^{4}$$
$17$ $$1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4}$$
$19$ $$( 1 + T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$$
$23$ $$1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4}$$
$29$ $$1 + 5 T - 4 T^{2} + 145 T^{3} + 841 T^{4}$$
$31$ $$( 1 - 3 T + 31 T^{2} )^{2}$$
$37$ $$1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4}$$
$41$ $$1 - 41 T^{2} + 1681 T^{4}$$
$43$ $$1 + 2 T - 39 T^{2} + 86 T^{3} + 1849 T^{4}$$
$47$ $$( 1 - 6 T + 47 T^{2} )^{2}$$
$53$ $$1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4}$$
$59$ $$( 1 - 11 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 6 T + 61 T^{2} )^{2}$$
$67$ $$( 1 + 2 T + 67 T^{2} )^{2}$$
$71$ $$( 1 + 2 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} )$$
$79$ $$( 1 - 3 T + 79 T^{2} )^{2}$$
$83$ $$1 + 7 T - 34 T^{2} + 581 T^{3} + 6889 T^{4}$$
$89$ $$1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4}$$
$97$ $$1 + 7 T - 48 T^{2} + 679 T^{3} + 9409 T^{4}$$