# Properties

 Label 1134.2.h.r Level $1134$ Weight $2$ Character orbit 1134.h 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.h (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: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{1} ) q^{2} + \beta_{1} q^{4} + ( 1 + \beta_{2} ) q^{7} + q^{8} +O(q^{10})$$ $$q + ( -1 - \beta_{1} ) q^{2} + \beta_{1} q^{4} + ( 1 + \beta_{2} ) q^{7} + q^{8} + ( \beta_{2} - 2 \beta_{3} ) q^{11} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{13} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{14} + ( -1 - \beta_{1} ) q^{16} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{19} + ( \beta_{2} + \beta_{3} ) q^{22} + ( 3 + \beta_{2} - 2 \beta_{3} ) q^{23} -5 q^{25} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{26} + ( \beta_{1} - \beta_{3} ) q^{28} + ( 2 \beta_{2} - \beta_{3} ) q^{29} + ( 5 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{31} + \beta_{1} q^{32} -4 \beta_{1} q^{37} + ( 2 + \beta_{2} - 2 \beta_{3} ) q^{38} + ( -3 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{41} + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{43} + ( -2 \beta_{2} + \beta_{3} ) q^{44} + ( -3 - 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{46} + ( -9 - 9 \beta_{1} + \beta_{2} + \beta_{3} ) q^{47} + ( -5 + 2 \beta_{2} ) q^{49} + ( 5 + 5 \beta_{1} ) q^{50} + ( 2 + \beta_{2} - 2 \beta_{3} ) q^{52} + ( \beta_{2} + \beta_{3} ) q^{53} + ( 1 + \beta_{2} ) q^{56} + ( -\beta_{2} + 2 \beta_{3} ) q^{58} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{61} + ( 5 - \beta_{2} + 2 \beta_{3} ) q^{62} + q^{64} + ( -4 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{67} + ( -3 - \beta_{2} + 2 \beta_{3} ) q^{71} + ( 7 + 7 \beta_{1} ) q^{73} -4 q^{74} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{76} + ( -6 - 12 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{77} + ( -5 - 5 \beta_{1} + \beta_{2} + \beta_{3} ) q^{79} + ( 3 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{82} + ( 12 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{83} + ( 2 - 2 \beta_{2} + 4 \beta_{3} ) q^{86} + ( \beta_{2} - 2 \beta_{3} ) q^{88} + ( 3 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{89} + ( -8 + 4 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{91} + ( 3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{92} + ( 9 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{94} + ( -4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{97} + ( 5 + 5 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - 2q^{4} + 4q^{7} + 4q^{8} + O(q^{10})$$ $$4q - 2q^{2} - 2q^{4} + 4q^{7} + 4q^{8} - 4q^{13} - 2q^{14} - 2q^{16} - 4q^{19} + 12q^{23} - 20q^{25} - 4q^{26} - 2q^{28} - 10q^{31} - 2q^{32} + 8q^{37} + 8q^{38} - 6q^{41} - 4q^{43} - 6q^{46} - 18q^{47} - 20q^{49} + 10q^{50} + 8q^{52} + 4q^{56} - 4q^{61} + 20q^{62} + 4q^{64} + 8q^{67} - 12q^{71} + 14q^{73} - 16q^{74} - 4q^{76} - 10q^{79} - 6q^{82} - 24q^{83} + 8q^{86} - 6q^{89} - 40q^{91} - 6q^{92} - 18q^{94} + 8q^{97} + 10q^{98} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{3} + 2 \beta_{2}$$$$)/3$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 0.707107 − 1.22474i −0.707107 + 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 1.00000 2.44949i 1.00000 0 0
109.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 1.00000 + 2.44949i 1.00000 0 0
541.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 1.00000 2.44949i 1.00000 0 0
541.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 1.00000 + 2.44949i 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.r 4
3.b odd 2 1 1134.2.h.s 4
7.c even 3 1 1134.2.e.s 4
9.c even 3 1 1134.2.e.s 4
9.c even 3 1 1134.2.g.i 4
9.d odd 6 1 1134.2.e.r 4
9.d odd 6 1 1134.2.g.j yes 4
21.h odd 6 1 1134.2.e.r 4
63.g even 3 1 inner 1134.2.h.r 4
63.g even 3 1 7938.2.a.bq 2
63.h even 3 1 1134.2.g.i 4
63.j odd 6 1 1134.2.g.j yes 4
63.k odd 6 1 7938.2.a.bp 2
63.n odd 6 1 1134.2.h.s 4
63.n odd 6 1 7938.2.a.bk 2
63.s even 6 1 7938.2.a.bj 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.e.r 4 9.d odd 6 1
1134.2.e.r 4 21.h odd 6 1
1134.2.e.s 4 7.c even 3 1
1134.2.e.s 4 9.c even 3 1
1134.2.g.i 4 9.c even 3 1
1134.2.g.i 4 63.h even 3 1
1134.2.g.j yes 4 9.d odd 6 1
1134.2.g.j yes 4 63.j odd 6 1
1134.2.h.r 4 1.a even 1 1 trivial
1134.2.h.r 4 63.g even 3 1 inner
1134.2.h.s 4 3.b odd 2 1
1134.2.h.s 4 63.n odd 6 1
7938.2.a.bj 2 63.s even 6 1
7938.2.a.bk 2 63.n odd 6 1
7938.2.a.bp 2 63.k odd 6 1
7938.2.a.bq 2 63.g even 3 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{11}^{2} - 18$$ $$T_{17}$$ $$T_{23}^{2} - 6 T_{23} - 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 7 - 2 T + T^{2} )^{2}$$
$11$ $$( -18 + T^{2} )^{2}$$
$13$ $$196 - 56 T + 30 T^{2} + 4 T^{3} + T^{4}$$
$17$ $$T^{4}$$
$19$ $$196 - 56 T + 30 T^{2} + 4 T^{3} + T^{4}$$
$23$ $$( -9 - 6 T + T^{2} )^{2}$$
$29$ $$324 + 18 T^{2} + T^{4}$$
$31$ $$49 + 70 T + 93 T^{2} + 10 T^{3} + T^{4}$$
$37$ $$( 16 - 4 T + T^{2} )^{2}$$
$41$ $$3969 - 378 T + 99 T^{2} + 6 T^{3} + T^{4}$$
$43$ $$4624 - 272 T + 84 T^{2} + 4 T^{3} + T^{4}$$
$47$ $$3969 + 1134 T + 261 T^{2} + 18 T^{3} + T^{4}$$
$53$ $$324 + 18 T^{2} + T^{4}$$
$59$ $$T^{4}$$
$61$ $$196 - 56 T + 30 T^{2} + 4 T^{3} + T^{4}$$
$67$ $$4 + 16 T + 66 T^{2} - 8 T^{3} + T^{4}$$
$71$ $$( -9 + 6 T + T^{2} )^{2}$$
$73$ $$( 49 - 7 T + T^{2} )^{2}$$
$79$ $$49 + 70 T + 93 T^{2} + 10 T^{3} + T^{4}$$
$83$ $$15876 + 3024 T + 450 T^{2} + 24 T^{3} + T^{4}$$
$89$ $$3969 - 378 T + 99 T^{2} + 6 T^{3} + T^{4}$$
$97$ $$3136 + 448 T + 120 T^{2} - 8 T^{3} + T^{4}$$