# Properties

 Label 1134.2.g.d Level $1134$ Weight $2$ Character orbit 1134.g Analytic conductor $9.055$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1134 = 2 \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1134.g (of order $$3$$, degree $$2$$, 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 126) 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + 3 \zeta_{6} q^{5} + ( 2 + \zeta_{6} ) q^{7} + q^{8} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + 3 \zeta_{6} q^{5} + ( 2 + \zeta_{6} ) q^{7} + q^{8} + ( 3 - 3 \zeta_{6} ) q^{10} + ( 3 - 3 \zeta_{6} ) q^{11} + 5 q^{13} + ( 1 - 3 \zeta_{6} ) q^{14} -\zeta_{6} q^{16} + ( -3 + 3 \zeta_{6} ) q^{17} -5 \zeta_{6} q^{19} -3 q^{20} -3 q^{22} + 3 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} -5 \zeta_{6} q^{26} + ( -3 + 2 \zeta_{6} ) q^{28} -3 q^{29} + ( 4 - 4 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} + 3 q^{34} + ( -3 + 9 \zeta_{6} ) q^{35} + 7 \zeta_{6} q^{37} + ( -5 + 5 \zeta_{6} ) q^{38} + 3 \zeta_{6} q^{40} -9 q^{41} + 11 q^{43} + 3 \zeta_{6} q^{44} + ( 3 - 3 \zeta_{6} ) q^{46} + ( 3 + 5 \zeta_{6} ) q^{49} + 4 q^{50} + ( -5 + 5 \zeta_{6} ) q^{52} + ( 3 - 3 \zeta_{6} ) q^{53} + 9 q^{55} + ( 2 + \zeta_{6} ) q^{56} + 3 \zeta_{6} q^{58} + ( -12 + 12 \zeta_{6} ) q^{59} -2 \zeta_{6} q^{61} -4 q^{62} + q^{64} + 15 \zeta_{6} q^{65} + ( 4 - 4 \zeta_{6} ) q^{67} -3 \zeta_{6} q^{68} + ( 9 - 6 \zeta_{6} ) q^{70} + ( -11 + 11 \zeta_{6} ) q^{73} + ( 7 - 7 \zeta_{6} ) q^{74} + 5 q^{76} + ( 9 - 6 \zeta_{6} ) q^{77} -8 \zeta_{6} q^{79} + ( 3 - 3 \zeta_{6} ) q^{80} + 9 \zeta_{6} q^{82} + 3 q^{83} -9 q^{85} -11 \zeta_{6} q^{86} + ( 3 - 3 \zeta_{6} ) q^{88} -15 \zeta_{6} q^{89} + ( 10 + 5 \zeta_{6} ) q^{91} -3 q^{92} + ( 15 - 15 \zeta_{6} ) q^{95} - q^{97} + ( 5 - 8 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} + 3q^{5} + 5q^{7} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{4} + 3q^{5} + 5q^{7} + 2q^{8} + 3q^{10} + 3q^{11} + 10q^{13} - q^{14} - q^{16} - 3q^{17} - 5q^{19} - 6q^{20} - 6q^{22} + 3q^{23} - 4q^{25} - 5q^{26} - 4q^{28} - 6q^{29} + 4q^{31} - q^{32} + 6q^{34} + 3q^{35} + 7q^{37} - 5q^{38} + 3q^{40} - 18q^{41} + 22q^{43} + 3q^{44} + 3q^{46} + 11q^{49} + 8q^{50} - 5q^{52} + 3q^{53} + 18q^{55} + 5q^{56} + 3q^{58} - 12q^{59} - 2q^{61} - 8q^{62} + 2q^{64} + 15q^{65} + 4q^{67} - 3q^{68} + 12q^{70} - 11q^{73} + 7q^{74} + 10q^{76} + 12q^{77} - 8q^{79} + 3q^{80} + 9q^{82} + 6q^{83} - 18q^{85} - 11q^{86} + 3q^{88} - 15q^{89} + 25q^{91} - 6q^{92} + 15q^{95} - 2q^{97} + 2q^{98} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
163.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 1.50000 2.59808i 0 2.50000 0.866025i 1.00000 0 1.50000 + 2.59808i
487.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.50000 + 2.59808i 0 2.50000 + 0.866025i 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
7.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.g.d 2
3.b odd 2 1 1134.2.g.f 2
7.c even 3 1 inner 1134.2.g.d 2
7.c even 3 1 7938.2.a.r 1
7.d odd 6 1 7938.2.a.bd 1
9.c even 3 1 126.2.e.b 2
9.c even 3 1 126.2.h.a yes 2
9.d odd 6 1 378.2.e.a 2
9.d odd 6 1 378.2.h.b 2
21.g even 6 1 7938.2.a.c 1
21.h odd 6 1 1134.2.g.f 2
21.h odd 6 1 7938.2.a.o 1
36.f odd 6 1 1008.2.q.e 2
36.f odd 6 1 1008.2.t.c 2
36.h even 6 1 3024.2.q.a 2
36.h even 6 1 3024.2.t.f 2
63.g even 3 1 126.2.e.b 2
63.g even 3 1 882.2.f.e 2
63.h even 3 1 126.2.h.a yes 2
63.h even 3 1 882.2.f.e 2
63.i even 6 1 2646.2.f.i 2
63.i even 6 1 2646.2.h.f 2
63.j odd 6 1 378.2.h.b 2
63.j odd 6 1 2646.2.f.e 2
63.k odd 6 1 882.2.e.h 2
63.k odd 6 1 882.2.f.a 2
63.l odd 6 1 882.2.e.h 2
63.l odd 6 1 882.2.h.e 2
63.n odd 6 1 378.2.e.a 2
63.n odd 6 1 2646.2.f.e 2
63.o even 6 1 2646.2.e.e 2
63.o even 6 1 2646.2.h.f 2
63.s even 6 1 2646.2.e.e 2
63.s even 6 1 2646.2.f.i 2
63.t odd 6 1 882.2.f.a 2
63.t odd 6 1 882.2.h.e 2
252.o even 6 1 3024.2.q.a 2
252.u odd 6 1 1008.2.t.c 2
252.bb even 6 1 3024.2.t.f 2
252.bl odd 6 1 1008.2.q.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.b 2 9.c even 3 1
126.2.e.b 2 63.g even 3 1
126.2.h.a yes 2 9.c even 3 1
126.2.h.a yes 2 63.h even 3 1
378.2.e.a 2 9.d odd 6 1
378.2.e.a 2 63.n odd 6 1
378.2.h.b 2 9.d odd 6 1
378.2.h.b 2 63.j odd 6 1
882.2.e.h 2 63.k odd 6 1
882.2.e.h 2 63.l odd 6 1
882.2.f.a 2 63.k odd 6 1
882.2.f.a 2 63.t odd 6 1
882.2.f.e 2 63.g even 3 1
882.2.f.e 2 63.h even 3 1
882.2.h.e 2 63.l odd 6 1
882.2.h.e 2 63.t odd 6 1
1008.2.q.e 2 36.f odd 6 1
1008.2.q.e 2 252.bl odd 6 1
1008.2.t.c 2 36.f odd 6 1
1008.2.t.c 2 252.u odd 6 1
1134.2.g.d 2 1.a even 1 1 trivial
1134.2.g.d 2 7.c even 3 1 inner
1134.2.g.f 2 3.b odd 2 1
1134.2.g.f 2 21.h odd 6 1
2646.2.e.e 2 63.o even 6 1
2646.2.e.e 2 63.s even 6 1
2646.2.f.e 2 63.j odd 6 1
2646.2.f.e 2 63.n odd 6 1
2646.2.f.i 2 63.i even 6 1
2646.2.f.i 2 63.s even 6 1
2646.2.h.f 2 63.i even 6 1
2646.2.h.f 2 63.o even 6 1
3024.2.q.a 2 36.h even 6 1
3024.2.q.a 2 252.o even 6 1
3024.2.t.f 2 36.h even 6 1
3024.2.t.f 2 252.bb even 6 1
7938.2.a.c 1 21.g even 6 1
7938.2.a.o 1 21.h odd 6 1
7938.2.a.r 1 7.c even 3 1
7938.2.a.bd 1 7.d odd 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ 1
$5$ $$1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4}$$
$7$ $$1 - 5 T + 7 T^{2}$$
$11$ $$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 5 T + 13 T^{2} )^{2}$$
$17$ $$1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}$$
$19$ $$1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4}$$
$23$ $$1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 3 T + 29 T^{2} )^{2}$$
$31$ $$( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$
$37$ $$1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 9 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - 11 T + 43 T^{2} )^{2}$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4}$$
$59$ $$1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4}$$
$61$ $$1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4}$$
$67$ $$1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4}$$
$79$ $$1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4}$$
$83$ $$( 1 - 3 T + 83 T^{2} )^{2}$$
$89$ $$1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + T + 97 T^{2} )^{2}$$
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