# Properties

 Label 1134.2.l.d Level $1134$ Weight $2$ Character orbit 1134.l Analytic conductor $9.055$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1134 = 2 \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1134.l (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.05503558921$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 378) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{3} q^{2} - q^{4} + ( 3 - 2 \zeta_{12}^{2} ) q^{7} -\zeta_{12}^{3} q^{8} +O(q^{10})$$ $$q + \zeta_{12}^{3} q^{2} - q^{4} + ( 3 - 2 \zeta_{12}^{2} ) q^{7} -\zeta_{12}^{3} q^{8} + 3 \zeta_{12} q^{11} + ( -2 - 2 \zeta_{12}^{2} ) q^{13} + ( 2 \zeta_{12} + \zeta_{12}^{3} ) q^{14} + q^{16} + ( -3 + 3 \zeta_{12}^{2} ) q^{22} + ( 5 - 5 \zeta_{12}^{2} ) q^{25} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{26} + ( -3 + 2 \zeta_{12}^{2} ) q^{28} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{29} + ( 1 - 2 \zeta_{12}^{2} ) q^{31} + \zeta_{12}^{3} q^{32} + ( 8 - 8 \zeta_{12}^{2} ) q^{37} + ( -6 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{41} -4 \zeta_{12}^{2} q^{43} -3 \zeta_{12} q^{44} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{47} + ( 5 - 8 \zeta_{12}^{2} ) q^{49} + 5 \zeta_{12} q^{50} + ( 2 + 2 \zeta_{12}^{2} ) q^{52} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{53} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{56} + 9 \zeta_{12}^{2} q^{58} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{59} + ( -8 + 16 \zeta_{12}^{2} ) q^{61} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{62} - q^{64} + 2 q^{67} + 12 \zeta_{12}^{3} q^{71} + ( 6 - 3 \zeta_{12}^{2} ) q^{73} + 8 \zeta_{12} q^{74} + ( 9 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{77} -13 q^{79} + ( -6 - 6 \zeta_{12}^{2} ) q^{82} + ( 3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{83} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{86} + ( 3 - 3 \zeta_{12}^{2} ) q^{88} + ( 6 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{89} + ( -10 + 2 \zeta_{12}^{2} ) q^{91} + ( -6 + 12 \zeta_{12}^{2} ) q^{94} + ( -10 + 5 \zeta_{12}^{2} ) q^{97} + ( 8 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 8 q^{7} + O(q^{10})$$ $$4 q - 4 q^{4} + 8 q^{7} - 12 q^{13} + 4 q^{16} - 6 q^{22} + 10 q^{25} - 8 q^{28} + 16 q^{37} - 8 q^{43} + 4 q^{49} + 12 q^{52} + 18 q^{58} - 4 q^{64} + 8 q^{67} + 18 q^{73} - 52 q^{79} - 36 q^{82} + 6 q^{88} - 36 q^{91} - 30 q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$325$$ $$407$$ $$\chi(n)$$ $$1 - \zeta_{12}^{2}$$ $$1 - \zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
215.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
1.00000i 0 −1.00000 0 0 2.00000 1.73205i 1.00000i 0 0
215.2 1.00000i 0 −1.00000 0 0 2.00000 1.73205i 1.00000i 0 0
269.1 1.00000i 0 −1.00000 0 0 2.00000 + 1.73205i 1.00000i 0 0
269.2 1.00000i 0 −1.00000 0 0 2.00000 + 1.73205i 1.00000i 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
3.b odd 2 1 inner
63.i even 6 1 inner
63.t odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.2.l.d 4
3.b odd 2 1 inner 1134.2.l.d 4
7.d odd 6 1 1134.2.t.a 4
9.c even 3 1 378.2.k.a 4
9.c even 3 1 1134.2.t.a 4
9.d odd 6 1 378.2.k.a 4
9.d odd 6 1 1134.2.t.a 4
21.g even 6 1 1134.2.t.a 4
63.h even 3 1 2646.2.d.c 4
63.i even 6 1 inner 1134.2.l.d 4
63.i even 6 1 2646.2.d.c 4
63.j odd 6 1 2646.2.d.c 4
63.k odd 6 1 378.2.k.a 4
63.s even 6 1 378.2.k.a 4
63.t odd 6 1 inner 1134.2.l.d 4
63.t odd 6 1 2646.2.d.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.k.a 4 9.c even 3 1
378.2.k.a 4 9.d odd 6 1
378.2.k.a 4 63.k odd 6 1
378.2.k.a 4 63.s even 6 1
1134.2.l.d 4 1.a even 1 1 trivial
1134.2.l.d 4 3.b odd 2 1 inner
1134.2.l.d 4 63.i even 6 1 inner
1134.2.l.d 4 63.t odd 6 1 inner
1134.2.t.a 4 7.d odd 6 1
1134.2.t.a 4 9.c even 3 1
1134.2.t.a 4 9.d odd 6 1
1134.2.t.a 4 21.g even 6 1
2646.2.d.c 4 63.h even 3 1
2646.2.d.c 4 63.i even 6 1
2646.2.d.c 4 63.j odd 6 1
2646.2.d.c 4 63.t odd 6 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{11}^{4} - 9 T_{11}^{2} + 81$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 7 - 4 T + T^{2} )^{2}$$
$11$ $$81 - 9 T^{2} + T^{4}$$
$13$ $$( 12 + 6 T + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$6561 - 81 T^{2} + T^{4}$$
$31$ $$( 3 + T^{2} )^{2}$$
$37$ $$( 64 - 8 T + T^{2} )^{2}$$
$41$ $$11664 + 108 T^{2} + T^{4}$$
$43$ $$( 16 + 4 T + T^{2} )^{2}$$
$47$ $$( -108 + T^{2} )^{2}$$
$53$ $$1296 - 36 T^{2} + T^{4}$$
$59$ $$( -27 + T^{2} )^{2}$$
$61$ $$( 192 + T^{2} )^{2}$$
$67$ $$( -2 + T )^{4}$$
$71$ $$( 144 + T^{2} )^{2}$$
$73$ $$( 27 - 9 T + T^{2} )^{2}$$
$79$ $$( 13 + T )^{4}$$
$83$ $$729 + 27 T^{2} + T^{4}$$
$89$ $$11664 + 108 T^{2} + T^{4}$$
$97$ $$( 75 + 15 T + T^{2} )^{2}$$