# Properties

 Label 1134.2.h.c Level $1134$ Weight $2$ Character orbit 1134.h Analytic conductor $9.055$ Analytic rank $1$ 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.h (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.05503558921$$ Analytic rank: $$1$$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -3 + 2 \zeta_{6} ) q^{7} + q^{8} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -3 + 2 \zeta_{6} ) q^{7} + q^{8} + ( 4 - 4 \zeta_{6} ) q^{13} + ( 1 - 3 \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} -2 \zeta_{6} q^{19} -3 q^{23} -5 q^{25} + 4 \zeta_{6} q^{26} + ( 2 + \zeta_{6} ) q^{28} + 6 \zeta_{6} q^{29} -5 \zeta_{6} q^{31} -\zeta_{6} q^{32} -6 \zeta_{6} q^{34} -8 \zeta_{6} q^{37} + 2 q^{38} + ( -3 + 3 \zeta_{6} ) q^{41} -2 \zeta_{6} q^{43} + ( 3 - 3 \zeta_{6} ) q^{46} + ( 3 - 3 \zeta_{6} ) q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + ( 5 - 5 \zeta_{6} ) q^{50} -4 q^{52} + ( -6 + 6 \zeta_{6} ) q^{53} + ( -3 + 2 \zeta_{6} ) q^{56} -6 q^{58} -12 \zeta_{6} q^{59} + ( -8 + 8 \zeta_{6} ) q^{61} + 5 q^{62} + q^{64} -8 \zeta_{6} q^{67} + 6 q^{68} -15 q^{71} + ( -11 + 11 \zeta_{6} ) q^{73} + 8 q^{74} + ( -2 + 2 \zeta_{6} ) q^{76} + ( 1 - \zeta_{6} ) q^{79} -3 \zeta_{6} q^{82} + 2 q^{86} -9 \zeta_{6} q^{89} + ( -4 + 12 \zeta_{6} ) q^{91} + 3 \zeta_{6} q^{92} + 3 \zeta_{6} q^{94} -2 \zeta_{6} q^{97} + ( 3 + 5 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} - 4q^{7} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{4} - 4q^{7} + 2q^{8} + 4q^{13} - q^{14} - q^{16} - 6q^{17} - 2q^{19} - 6q^{23} - 10q^{25} + 4q^{26} + 5q^{28} + 6q^{29} - 5q^{31} - q^{32} - 6q^{34} - 8q^{37} + 4q^{38} - 3q^{41} - 2q^{43} + 3q^{46} + 3q^{47} + 2q^{49} + 5q^{50} - 8q^{52} - 6q^{53} - 4q^{56} - 12q^{58} - 12q^{59} - 8q^{61} + 10q^{62} + 2q^{64} - 8q^{67} + 12q^{68} - 30q^{71} - 11q^{73} + 16q^{74} - 2q^{76} + q^{79} - 3q^{82} + 4q^{86} - 9q^{89} + 4q^{91} + 3q^{92} + 3q^{94} - 2q^{97} + 11q^{98} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 −2.00000 + 1.73205i 1.00000 0 0
541.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −2.00000 1.73205i 1.00000 0 0
 $$n$$: 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.g 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.h.c 2
3.b odd 2 1 1134.2.h.m 2
7.c even 3 1 1134.2.e.n 2
9.c even 3 1 1134.2.e.n 2
9.c even 3 1 1134.2.g.b 2
9.d odd 6 1 1134.2.e.d 2
9.d odd 6 1 1134.2.g.g yes 2
21.h odd 6 1 1134.2.e.d 2
63.g even 3 1 inner 1134.2.h.c 2
63.g even 3 1 7938.2.a.y 1
63.h even 3 1 1134.2.g.b 2
63.j odd 6 1 1134.2.g.g yes 2
63.k odd 6 1 7938.2.a.z 1
63.n odd 6 1 1134.2.h.m 2
63.n odd 6 1 7938.2.a.g 1
63.s even 6 1 7938.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.e.d 2 9.d odd 6 1
1134.2.e.d 2 21.h odd 6 1
1134.2.e.n 2 7.c even 3 1
1134.2.e.n 2 9.c even 3 1
1134.2.g.b 2 9.c even 3 1
1134.2.g.b 2 63.h even 3 1
1134.2.g.g yes 2 9.d odd 6 1
1134.2.g.g yes 2 63.j odd 6 1
1134.2.h.c 2 1.a even 1 1 trivial
1134.2.h.c 2 63.g even 3 1 inner
1134.2.h.m 2 3.b odd 2 1
1134.2.h.m 2 63.n odd 6 1
7938.2.a.g 1 63.n odd 6 1
7938.2.a.h 1 63.s even 6 1
7938.2.a.y 1 63.g even 3 1
7938.2.a.z 1 63.k odd 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 + 4 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$16 - 4 T + T^{2}$$
$17$ $$36 + 6 T + T^{2}$$
$19$ $$4 + 2 T + T^{2}$$
$23$ $$( 3 + T )^{2}$$
$29$ $$36 - 6 T + T^{2}$$
$31$ $$25 + 5 T + T^{2}$$
$37$ $$64 + 8 T + T^{2}$$
$41$ $$9 + 3 T + T^{2}$$
$43$ $$4 + 2 T + T^{2}$$
$47$ $$9 - 3 T + T^{2}$$
$53$ $$36 + 6 T + T^{2}$$
$59$ $$144 + 12 T + T^{2}$$
$61$ $$64 + 8 T + T^{2}$$
$67$ $$64 + 8 T + T^{2}$$
$71$ $$( 15 + T )^{2}$$
$73$ $$121 + 11 T + T^{2}$$
$79$ $$1 - T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$81 + 9 T + T^{2}$$
$97$ $$4 + 2 T + T^{2}$$