# Properties

 Label 1134.2.e.g 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: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + ( 3 - 3 \zeta_{6} ) q^{5} + ( 2 + \zeta_{6} ) q^{7} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + ( 3 - 3 \zeta_{6} ) q^{5} + ( 2 + \zeta_{6} ) q^{7} - q^{8} + ( -3 + 3 \zeta_{6} ) q^{10} -3 \zeta_{6} q^{11} -2 \zeta_{6} q^{13} + ( -2 - \zeta_{6} ) q^{14} + q^{16} + ( 6 - 6 \zeta_{6} ) q^{17} -2 \zeta_{6} q^{19} + ( 3 - 3 \zeta_{6} ) q^{20} + 3 \zeta_{6} q^{22} + ( -6 + 6 \zeta_{6} ) q^{23} -4 \zeta_{6} q^{25} + 2 \zeta_{6} q^{26} + ( 2 + \zeta_{6} ) q^{28} + ( 9 - 9 \zeta_{6} ) q^{29} -7 q^{31} - q^{32} + ( -6 + 6 \zeta_{6} ) q^{34} + ( 9 - 6 \zeta_{6} ) q^{35} + 10 \zeta_{6} q^{37} + 2 \zeta_{6} q^{38} + ( -3 + 3 \zeta_{6} ) q^{40} + ( 4 - 4 \zeta_{6} ) q^{43} -3 \zeta_{6} q^{44} + ( 6 - 6 \zeta_{6} ) q^{46} -12 q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} + 4 \zeta_{6} q^{50} -2 \zeta_{6} q^{52} + ( -3 + 3 \zeta_{6} ) q^{53} -9 q^{55} + ( -2 - \zeta_{6} ) q^{56} + ( -9 + 9 \zeta_{6} ) q^{58} + 3 q^{59} -4 q^{61} + 7 q^{62} + q^{64} -6 q^{65} + 2 q^{67} + ( 6 - 6 \zeta_{6} ) q^{68} + ( -9 + 6 \zeta_{6} ) q^{70} + ( -2 + 2 \zeta_{6} ) q^{73} -10 \zeta_{6} q^{74} -2 \zeta_{6} q^{76} + ( 3 - 9 \zeta_{6} ) q^{77} + 5 q^{79} + ( 3 - 3 \zeta_{6} ) q^{80} + ( 9 - 9 \zeta_{6} ) q^{83} -18 \zeta_{6} q^{85} + ( -4 + 4 \zeta_{6} ) q^{86} + 3 \zeta_{6} q^{88} -6 \zeta_{6} q^{89} + ( 2 - 6 \zeta_{6} ) q^{91} + ( -6 + 6 \zeta_{6} ) q^{92} + 12 q^{94} -6 q^{95} + ( 13 - 13 \zeta_{6} ) q^{97} + ( -3 - 5 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} + 3q^{5} + 5q^{7} - 2q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} + 3q^{5} + 5q^{7} - 2q^{8} - 3q^{10} - 3q^{11} - 2q^{13} - 5q^{14} + 2q^{16} + 6q^{17} - 2q^{19} + 3q^{20} + 3q^{22} - 6q^{23} - 4q^{25} + 2q^{26} + 5q^{28} + 9q^{29} - 14q^{31} - 2q^{32} - 6q^{34} + 12q^{35} + 10q^{37} + 2q^{38} - 3q^{40} + 4q^{43} - 3q^{44} + 6q^{46} - 24q^{47} + 11q^{49} + 4q^{50} - 2q^{52} - 3q^{53} - 18q^{55} - 5q^{56} - 9q^{58} + 6q^{59} - 8q^{61} + 14q^{62} + 2q^{64} - 12q^{65} + 4q^{67} + 6q^{68} - 12q^{70} - 2q^{73} - 10q^{74} - 2q^{76} - 3q^{77} + 10q^{79} + 3q^{80} + 9q^{83} - 18q^{85} - 4q^{86} + 3q^{88} - 6q^{89} - 2q^{91} - 6q^{92} + 24q^{94} - 12q^{95} + 13q^{97} - 11q^{98} + O(q^{100})$$

## 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 1.50000 2.59808i 0 2.50000 + 0.866025i −1.00000 0 −1.50000 + 2.59808i
919.1 −1.00000 0 1.00000 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
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.g 2
3.b odd 2 1 1134.2.e.k 2
7.c even 3 1 1134.2.h.j 2
9.c even 3 1 126.2.g.d yes 2
9.c even 3 1 1134.2.h.j 2
9.d odd 6 1 126.2.g.a 2
9.d odd 6 1 1134.2.h.f 2
21.h odd 6 1 1134.2.h.f 2
36.f odd 6 1 1008.2.s.o 2
36.h even 6 1 1008.2.s.b 2
63.g even 3 1 126.2.g.d yes 2
63.h even 3 1 882.2.a.a 1
63.h even 3 1 inner 1134.2.e.g 2
63.i even 6 1 882.2.a.h 1
63.j odd 6 1 882.2.a.j 1
63.j odd 6 1 1134.2.e.k 2
63.k odd 6 1 882.2.g.g 2
63.l odd 6 1 882.2.g.g 2
63.n odd 6 1 126.2.g.a 2
63.o even 6 1 882.2.g.e 2
63.s even 6 1 882.2.g.e 2
63.t odd 6 1 882.2.a.e 1
252.o even 6 1 1008.2.s.b 2
252.r odd 6 1 7056.2.a.h 1
252.u odd 6 1 7056.2.a.e 1
252.bb even 6 1 7056.2.a.by 1
252.bj even 6 1 7056.2.a.bx 1
252.bl odd 6 1 1008.2.s.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.g.a 2 9.d odd 6 1
126.2.g.a 2 63.n odd 6 1
126.2.g.d yes 2 9.c even 3 1
126.2.g.d yes 2 63.g even 3 1
882.2.a.a 1 63.h even 3 1
882.2.a.e 1 63.t odd 6 1
882.2.a.h 1 63.i even 6 1
882.2.a.j 1 63.j odd 6 1
882.2.g.e 2 63.o even 6 1
882.2.g.e 2 63.s even 6 1
882.2.g.g 2 63.k odd 6 1
882.2.g.g 2 63.l odd 6 1
1008.2.s.b 2 36.h even 6 1
1008.2.s.b 2 252.o even 6 1
1008.2.s.o 2 36.f odd 6 1
1008.2.s.o 2 252.bl odd 6 1
1134.2.e.g 2 1.a even 1 1 trivial
1134.2.e.g 2 63.h even 3 1 inner
1134.2.e.k 2 3.b odd 2 1
1134.2.e.k 2 63.j odd 6 1
1134.2.h.f 2 9.d odd 6 1
1134.2.h.f 2 21.h odd 6 1
1134.2.h.j 2 7.c even 3 1
1134.2.h.j 2 9.c even 3 1
7056.2.a.e 1 252.u odd 6 1
7056.2.a.h 1 252.r odd 6 1
7056.2.a.bx 1 252.bj even 6 1
7056.2.a.by 1 252.bb even 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 3 T_{5} + 9$$ $$T_{11}^{2} + 3 T_{11} + 9$$ $$T_{17}^{2} - 6 T_{17} + 36$$ $$T_{23}^{2} + 6 T_{23} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ 1
$5$ $$1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4}$$
$7$ $$1 - 5 T + 7 T^{2}$$
$11$ $$1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$
$17$ $$1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4}$$
$19$ $$1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4}$$
$23$ $$1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4}$$
$29$ $$1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4}$$
$31$ $$( 1 + 7 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 11 T + 37 T^{2} )( 1 + T + 37 T^{2} )$$
$41$ $$1 - 41 T^{2} + 1681 T^{4}$$
$43$ $$1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4}$$
$47$ $$( 1 + 12 T + 47 T^{2} )^{2}$$
$53$ $$1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4}$$
$59$ $$( 1 - 3 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 4 T + 61 T^{2} )^{2}$$
$67$ $$( 1 - 2 T + 67 T^{2} )^{2}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4}$$
$79$ $$( 1 - 5 T + 79 T^{2} )^{2}$$
$83$ $$1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4}$$
$89$ $$1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4}$$
$97$ $$1 - 13 T + 72 T^{2} - 1261 T^{3} + 9409 T^{4}$$