# Properties

 Label 1134.2.e.t Level $1134$ Weight $2$ Character orbit 1134.e Analytic conductor $9.055$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 49$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + ( 1 + \beta_{1} + \beta_{2} ) q^{5} + \beta_{1} q^{7} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} + ( 1 + \beta_{1} + \beta_{2} ) q^{5} + \beta_{1} q^{7} + q^{8} + ( 1 + \beta_{1} + \beta_{2} ) q^{10} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{13} + \beta_{1} q^{14} + q^{16} + ( -1 - \beta_{1} - \beta_{2} ) q^{17} + 2 \beta_{2} q^{19} + ( 1 + \beta_{1} + \beta_{2} ) q^{20} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{22} + ( -4 + 2 \beta_{1} - 4 \beta_{2} ) q^{23} + ( 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{25} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{26} + \beta_{1} q^{28} + ( -5 + \beta_{1} - 5 \beta_{2} ) q^{29} + ( -2 + \beta_{3} ) q^{31} + q^{32} + ( -1 - \beta_{1} - \beta_{2} ) q^{34} + ( \beta_{1} + 7 \beta_{2} + \beta_{3} ) q^{35} + ( -3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{37} + 2 \beta_{2} q^{38} + ( 1 + \beta_{1} + \beta_{2} ) q^{40} + ( 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{41} + ( -5 - 5 \beta_{2} ) q^{43} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{44} + ( -4 + 2 \beta_{1} - 4 \beta_{2} ) q^{46} + ( -3 - 3 \beta_{3} ) q^{47} + 7 \beta_{2} q^{49} + ( 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{50} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{52} + ( -6 - 6 \beta_{2} ) q^{53} + ( -8 + 2 \beta_{3} ) q^{55} + \beta_{1} q^{56} + ( -5 + \beta_{1} - 5 \beta_{2} ) q^{58} + ( 11 + \beta_{3} ) q^{59} + ( 10 + \beta_{3} ) q^{61} + ( -2 + \beta_{3} ) q^{62} + q^{64} + ( 9 - 3 \beta_{3} ) q^{65} + ( 3 + 2 \beta_{3} ) q^{67} + ( -1 - \beta_{1} - \beta_{2} ) q^{68} + ( \beta_{1} + 7 \beta_{2} + \beta_{3} ) q^{70} + ( 13 - \beta_{3} ) q^{71} -4 \beta_{1} q^{73} + ( -3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{74} + 2 \beta_{2} q^{76} + ( -7 + \beta_{3} ) q^{77} + ( -2 - 5 \beta_{3} ) q^{79} + ( 1 + \beta_{1} + \beta_{2} ) q^{80} + ( 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{82} + ( 8 + 2 \beta_{1} + 8 \beta_{2} ) q^{83} + ( -2 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{85} + ( -5 - 5 \beta_{2} ) q^{86} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{88} + ( 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{89} + ( 7 - 2 \beta_{3} ) q^{91} + ( -4 + 2 \beta_{1} - 4 \beta_{2} ) q^{92} + ( -3 - 3 \beta_{3} ) q^{94} + ( -2 + 2 \beta_{3} ) q^{95} + ( -3 - 4 \beta_{1} - 3 \beta_{2} ) q^{97} + 7 \beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} + 4 q^{4} + 2 q^{5} + 4 q^{8} + O(q^{10})$$ $$4 q + 4 q^{2} + 4 q^{4} + 2 q^{5} + 4 q^{8} + 2 q^{10} - 2 q^{11} + 4 q^{13} + 4 q^{16} - 2 q^{17} - 4 q^{19} + 2 q^{20} - 2 q^{22} - 8 q^{23} - 6 q^{25} + 4 q^{26} - 10 q^{29} - 8 q^{31} + 4 q^{32} - 2 q^{34} - 14 q^{35} + 8 q^{37} - 4 q^{38} + 2 q^{40} - 6 q^{41} - 10 q^{43} - 2 q^{44} - 8 q^{46} - 12 q^{47} - 14 q^{49} - 6 q^{50} + 4 q^{52} - 12 q^{53} - 32 q^{55} - 10 q^{58} + 44 q^{59} + 40 q^{61} - 8 q^{62} + 4 q^{64} + 36 q^{65} + 12 q^{67} - 2 q^{68} - 14 q^{70} + 52 q^{71} + 8 q^{74} - 4 q^{76} - 28 q^{77} - 8 q^{79} + 2 q^{80} - 6 q^{82} + 16 q^{83} + 16 q^{85} - 10 q^{86} - 2 q^{88} + 6 q^{89} + 28 q^{91} - 8 q^{92} - 12 q^{94} - 8 q^{95} - 6 q^{97} - 14 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/7$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$7 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{3}$$

## Character values

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

 $$n$$ $$325$$ $$407$$ $$\chi(n)$$ $$\beta_{2}$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
865.1
 −1.32288 + 2.29129i 1.32288 − 2.29129i −1.32288 − 2.29129i 1.32288 + 2.29129i
1.00000 0 1.00000 −0.822876 + 1.42526i 0 −1.32288 + 2.29129i 1.00000 0 −0.822876 + 1.42526i
865.2 1.00000 0 1.00000 1.82288 3.15731i 0 1.32288 2.29129i 1.00000 0 1.82288 3.15731i
919.1 1.00000 0 1.00000 −0.822876 1.42526i 0 −1.32288 2.29129i 1.00000 0 −0.822876 1.42526i
919.2 1.00000 0 1.00000 1.82288 + 3.15731i 0 1.32288 + 2.29129i 1.00000 0 1.82288 + 3.15731i
 $$n$$: 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.t 4
3.b odd 2 1 1134.2.e.q 4
7.c even 3 1 1134.2.h.q 4
9.c even 3 1 378.2.g.g 4
9.c even 3 1 1134.2.h.q 4
9.d odd 6 1 378.2.g.h yes 4
9.d odd 6 1 1134.2.h.t 4
21.h odd 6 1 1134.2.h.t 4
63.g even 3 1 378.2.g.g 4
63.h even 3 1 inner 1134.2.e.t 4
63.h even 3 1 2646.2.a.bl 2
63.i even 6 1 2646.2.a.bf 2
63.j odd 6 1 1134.2.e.q 4
63.j odd 6 1 2646.2.a.bi 2
63.n odd 6 1 378.2.g.h yes 4
63.t odd 6 1 2646.2.a.bo 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.g.g 4 9.c even 3 1
378.2.g.g 4 63.g even 3 1
378.2.g.h yes 4 9.d odd 6 1
378.2.g.h yes 4 63.n odd 6 1
1134.2.e.q 4 3.b odd 2 1
1134.2.e.q 4 63.j odd 6 1
1134.2.e.t 4 1.a even 1 1 trivial
1134.2.e.t 4 63.h even 3 1 inner
1134.2.h.q 4 7.c even 3 1
1134.2.h.q 4 9.c even 3 1
1134.2.h.t 4 9.d odd 6 1
1134.2.h.t 4 21.h odd 6 1
2646.2.a.bf 2 63.i even 6 1
2646.2.a.bi 2 63.j odd 6 1
2646.2.a.bl 2 63.h even 3 1
2646.2.a.bo 2 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 $$S_{2}^{\mathrm{new}}(1134, [\chi])$$:

 $$T_{5}^{4} - 2 T_{5}^{3} + 10 T_{5}^{2} + 12 T_{5} + 36$$ $$T_{11}^{4} + 2 T_{11}^{3} + 10 T_{11}^{2} - 12 T_{11} + 36$$ $$T_{17}^{4} + 2 T_{17}^{3} + 10 T_{17}^{2} - 12 T_{17} + 36$$ $$T_{23}^{4} + 8 T_{23}^{3} + 76 T_{23}^{2} - 96 T_{23} + 144$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{4}$$
$3$ $$T^{4}$$
$5$ $$36 + 12 T + 10 T^{2} - 2 T^{3} + T^{4}$$
$7$ $$49 + 7 T^{2} + T^{4}$$
$11$ $$36 - 12 T + 10 T^{2} + 2 T^{3} + T^{4}$$
$13$ $$9 + 12 T + 19 T^{2} - 4 T^{3} + T^{4}$$
$17$ $$36 - 12 T + 10 T^{2} + 2 T^{3} + T^{4}$$
$19$ $$( 4 + 2 T + T^{2} )^{2}$$
$23$ $$144 - 96 T + 76 T^{2} + 8 T^{3} + T^{4}$$
$29$ $$324 + 180 T + 82 T^{2} + 10 T^{3} + T^{4}$$
$31$ $$( -3 + 4 T + T^{2} )^{2}$$
$37$ $$2209 + 376 T + 111 T^{2} - 8 T^{3} + T^{4}$$
$41$ $$2916 - 324 T + 90 T^{2} + 6 T^{3} + T^{4}$$
$43$ $$( 25 + 5 T + T^{2} )^{2}$$
$47$ $$( -54 + 6 T + T^{2} )^{2}$$
$53$ $$( 36 + 6 T + T^{2} )^{2}$$
$59$ $$( 114 - 22 T + T^{2} )^{2}$$
$61$ $$( 93 - 20 T + T^{2} )^{2}$$
$67$ $$( -19 - 6 T + T^{2} )^{2}$$
$71$ $$( 162 - 26 T + T^{2} )^{2}$$
$73$ $$12544 + 112 T^{2} + T^{4}$$
$79$ $$( -171 + 4 T + T^{2} )^{2}$$
$83$ $$1296 - 576 T + 220 T^{2} - 16 T^{3} + T^{4}$$
$89$ $$2916 + 324 T + 90 T^{2} - 6 T^{3} + T^{4}$$
$97$ $$10609 - 618 T + 139 T^{2} + 6 T^{3} + T^{4}$$