# Properties

 Label 1134.2.f.p 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: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 378) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + 4 \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} - q^{8} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + 4 \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} - q^{8} + 4 q^{10} + ( -4 + 4 \zeta_{6} ) q^{11} -3 \zeta_{6} q^{13} -\zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} + 7 q^{17} + 2 q^{19} + ( 4 - 4 \zeta_{6} ) q^{20} + 4 \zeta_{6} q^{22} -\zeta_{6} q^{23} + ( -11 + 11 \zeta_{6} ) q^{25} -3 q^{26} - q^{28} + ( 1 - \zeta_{6} ) q^{29} + 9 \zeta_{6} q^{31} + \zeta_{6} q^{32} + ( 7 - 7 \zeta_{6} ) q^{34} + 4 q^{35} + 2 q^{37} + ( 2 - 2 \zeta_{6} ) q^{38} -4 \zeta_{6} q^{40} + 6 \zeta_{6} q^{41} + ( -11 + 11 \zeta_{6} ) q^{43} + 4 q^{44} - q^{46} + ( -6 + 6 \zeta_{6} ) q^{47} -\zeta_{6} q^{49} + 11 \zeta_{6} q^{50} + ( -3 + 3 \zeta_{6} ) q^{52} + 9 q^{53} -16 q^{55} + ( -1 + \zeta_{6} ) q^{56} -\zeta_{6} q^{58} -5 \zeta_{6} q^{59} + ( 6 - 6 \zeta_{6} ) q^{61} + 9 q^{62} + q^{64} + ( 12 - 12 \zeta_{6} ) q^{65} -7 \zeta_{6} q^{67} -7 \zeta_{6} q^{68} + ( 4 - 4 \zeta_{6} ) q^{70} + 7 q^{71} -14 q^{73} + ( 2 - 2 \zeta_{6} ) q^{74} -2 \zeta_{6} q^{76} + 4 \zeta_{6} q^{77} + ( 6 - 6 \zeta_{6} ) q^{79} -4 q^{80} + 6 q^{82} + ( -4 + 4 \zeta_{6} ) q^{83} + 28 \zeta_{6} q^{85} + 11 \zeta_{6} q^{86} + ( 4 - 4 \zeta_{6} ) q^{88} + 3 q^{89} -3 q^{91} + ( -1 + \zeta_{6} ) q^{92} + 6 \zeta_{6} q^{94} + 8 \zeta_{6} q^{95} + ( 8 - 8 \zeta_{6} ) q^{97} - q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{4} + 4q^{5} + q^{7} - 2q^{8} + O(q^{10})$$ $$2q + q^{2} - q^{4} + 4q^{5} + q^{7} - 2q^{8} + 8q^{10} - 4q^{11} - 3q^{13} - q^{14} - q^{16} + 14q^{17} + 4q^{19} + 4q^{20} + 4q^{22} - q^{23} - 11q^{25} - 6q^{26} - 2q^{28} + q^{29} + 9q^{31} + q^{32} + 7q^{34} + 8q^{35} + 4q^{37} + 2q^{38} - 4q^{40} + 6q^{41} - 11q^{43} + 8q^{44} - 2q^{46} - 6q^{47} - q^{49} + 11q^{50} - 3q^{52} + 18q^{53} - 32q^{55} - q^{56} - q^{58} - 5q^{59} + 6q^{61} + 18q^{62} + 2q^{64} + 12q^{65} - 7q^{67} - 7q^{68} + 4q^{70} + 14q^{71} - 28q^{73} + 2q^{74} - 2q^{76} + 4q^{77} + 6q^{79} - 8q^{80} + 12q^{82} - 4q^{83} + 28q^{85} + 11q^{86} + 4q^{88} + 6q^{89} - 6q^{91} - q^{92} + 6q^{94} + 8q^{95} + 8q^{97} - 2q^{98} + O(q^{100})$$

## 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 2.00000 + 3.46410i 0 0.500000 0.866025i −1.00000 0 4.00000
757.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 2.00000 3.46410i 0 0.500000 + 0.866025i −1.00000 0 4.00000
 $$n$$: 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.p 2
3.b odd 2 1 1134.2.f.a 2
9.c even 3 1 378.2.a.a 1
9.c even 3 1 inner 1134.2.f.p 2
9.d odd 6 1 378.2.a.h yes 1
9.d odd 6 1 1134.2.f.a 2
36.f odd 6 1 3024.2.a.a 1
36.h even 6 1 3024.2.a.bd 1
45.h odd 6 1 9450.2.a.bc 1
45.j even 6 1 9450.2.a.dv 1
63.l odd 6 1 2646.2.a.o 1
63.o even 6 1 2646.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.a 1 9.c even 3 1
378.2.a.h yes 1 9.d odd 6 1
1134.2.f.a 2 3.b odd 2 1
1134.2.f.a 2 9.d odd 6 1
1134.2.f.p 2 1.a even 1 1 trivial
1134.2.f.p 2 9.c even 3 1 inner
2646.2.a.o 1 63.l odd 6 1
2646.2.a.p 1 63.o even 6 1
3024.2.a.a 1 36.f odd 6 1
3024.2.a.bd 1 36.h even 6 1
9450.2.a.bc 1 45.h odd 6 1
9450.2.a.dv 1 45.j even 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 4 T_{5} + 16$$ $$T_{11}^{2} + 4 T_{11} + 16$$ $$T_{13}^{2} + 3 T_{13} + 9$$ $$T_{17} - 7$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ 1
$5$ $$1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4}$$
$7$ $$1 - T + T^{2}$$
$11$ $$1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4}$$
$13$ $$1 + 3 T - 4 T^{2} + 39 T^{3} + 169 T^{4}$$
$17$ $$( 1 - 7 T + 17 T^{2} )^{2}$$
$19$ $$( 1 - 2 T + 19 T^{2} )^{2}$$
$23$ $$1 + T - 22 T^{2} + 23 T^{3} + 529 T^{4}$$
$29$ $$1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4}$$
$31$ $$1 - 9 T + 50 T^{2} - 279 T^{3} + 961 T^{4}$$
$37$ $$( 1 - 2 T + 37 T^{2} )^{2}$$
$41$ $$1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4}$$
$43$ $$1 + 11 T + 78 T^{2} + 473 T^{3} + 1849 T^{4}$$
$47$ $$1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4}$$
$53$ $$( 1 - 9 T + 53 T^{2} )^{2}$$
$59$ $$1 + 5 T - 34 T^{2} + 295 T^{3} + 3481 T^{4}$$
$61$ $$1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4}$$
$67$ $$1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 7 T + 71 T^{2} )^{2}$$
$73$ $$( 1 + 14 T + 73 T^{2} )^{2}$$
$79$ $$1 - 6 T - 43 T^{2} - 474 T^{3} + 6241 T^{4}$$
$83$ $$1 + 4 T - 67 T^{2} + 332 T^{3} + 6889 T^{4}$$
$89$ $$( 1 - 3 T + 89 T^{2} )^{2}$$
$97$ $$1 - 8 T - 33 T^{2} - 776 T^{3} + 9409 T^{4}$$