# Properties

 Label 1134.2.m.d Level $1134$ Weight $2$ Character orbit 1134.m 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.m (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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{5} + ( -1 + 3 \zeta_{12}^{2} ) q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10})$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{5} + ( -1 + 3 \zeta_{12}^{2} ) q^{7} + \zeta_{12}^{3} q^{8} + ( 2 - 4 \zeta_{12}^{2} ) q^{10} -6 \zeta_{12} q^{11} + ( 2 - \zeta_{12}^{2} ) q^{13} + ( -\zeta_{12} + 3 \zeta_{12}^{3} ) q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{17} + ( 4 - 8 \zeta_{12}^{2} ) q^{19} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{20} -6 \zeta_{12}^{2} q^{22} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{23} + ( -7 + 7 \zeta_{12}^{2} ) q^{25} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{26} + ( -3 + 2 \zeta_{12}^{2} ) q^{28} -3 \zeta_{12} q^{29} + ( -6 + 3 \zeta_{12}^{2} ) q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} + ( -1 - \zeta_{12}^{2} ) q^{34} + ( 8 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{35} -2 q^{37} + ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{38} + ( 4 - 2 \zeta_{12}^{2} ) q^{40} + ( -4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{41} + ( 11 - 11 \zeta_{12}^{2} ) q^{43} -6 \zeta_{12}^{3} q^{44} -3 q^{46} + ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{47} + ( -8 + 3 \zeta_{12}^{2} ) q^{49} + ( -7 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{50} + ( 1 + \zeta_{12}^{2} ) q^{52} + 3 \zeta_{12}^{3} q^{53} + ( -12 + 24 \zeta_{12}^{2} ) q^{55} + ( -3 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{56} -3 \zeta_{12}^{2} q^{58} + ( 5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{59} + ( -8 - 8 \zeta_{12}^{2} ) q^{61} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{62} - q^{64} -6 \zeta_{12} q^{65} + 7 \zeta_{12}^{2} q^{67} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{68} + ( 10 - 2 \zeta_{12}^{2} ) q^{70} + 3 \zeta_{12}^{3} q^{71} + ( 4 - 8 \zeta_{12}^{2} ) q^{73} -2 \zeta_{12} q^{74} + ( 8 - 4 \zeta_{12}^{2} ) q^{76} + ( 6 \zeta_{12} - 18 \zeta_{12}^{3} ) q^{77} + ( -8 + 8 \zeta_{12}^{2} ) q^{79} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{80} + ( 4 - 8 \zeta_{12}^{2} ) q^{82} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{83} + 6 \zeta_{12}^{2} q^{85} + ( 11 \zeta_{12} - 11 \zeta_{12}^{3} ) q^{86} + ( 6 - 6 \zeta_{12}^{2} ) q^{88} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{89} + ( 1 + 4 \zeta_{12}^{2} ) q^{91} -3 \zeta_{12} q^{92} + ( 8 - 4 \zeta_{12}^{2} ) q^{94} + ( -24 \zeta_{12} + 24 \zeta_{12}^{3} ) q^{95} + ( -4 - 4 \zeta_{12}^{2} ) q^{97} + ( -8 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + 2q^{7} + O(q^{10})$$ $$4q + 2q^{4} + 2q^{7} + 6q^{13} - 2q^{16} - 12q^{22} - 14q^{25} - 8q^{28} - 18q^{31} - 6q^{34} - 8q^{37} + 12q^{40} + 22q^{43} - 12q^{46} - 26q^{49} + 6q^{52} - 6q^{58} - 48q^{61} - 4q^{64} + 14q^{67} + 36q^{70} + 24q^{76} - 16q^{79} + 12q^{85} + 12q^{88} + 12q^{91} + 24q^{94} - 24q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
377.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i 1.73205 3.00000i 0 0.500000 2.59808i 1.00000i 0 3.46410i
377.2 0.866025 0.500000i 0 0.500000 0.866025i −1.73205 + 3.00000i 0 0.500000 2.59808i 1.00000i 0 3.46410i
755.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.73205 + 3.00000i 0 0.500000 + 2.59808i 1.00000i 0 3.46410i
755.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.73205 3.00000i 0 0.500000 + 2.59808i 1.00000i 0 3.46410i
 $$n$$: 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.l odd 6 1 inner
63.o even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.2.m.d 4
3.b odd 2 1 inner 1134.2.m.d 4
7.b odd 2 1 1134.2.m.a 4
9.c even 3 1 378.2.d.c 4
9.c even 3 1 1134.2.m.a 4
9.d odd 6 1 378.2.d.c 4
9.d odd 6 1 1134.2.m.a 4
21.c even 2 1 1134.2.m.a 4
36.f odd 6 1 3024.2.k.f 4
36.h even 6 1 3024.2.k.f 4
63.l odd 6 1 378.2.d.c 4
63.l odd 6 1 inner 1134.2.m.d 4
63.o even 6 1 378.2.d.c 4
63.o even 6 1 inner 1134.2.m.d 4
252.s odd 6 1 3024.2.k.f 4
252.bi even 6 1 3024.2.k.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.d.c 4 9.c even 3 1
378.2.d.c 4 9.d odd 6 1
378.2.d.c 4 63.l odd 6 1
378.2.d.c 4 63.o even 6 1
1134.2.m.a 4 7.b odd 2 1
1134.2.m.a 4 9.c even 3 1
1134.2.m.a 4 9.d odd 6 1
1134.2.m.a 4 21.c even 2 1
1134.2.m.d 4 1.a even 1 1 trivial
1134.2.m.d 4 3.b odd 2 1 inner
1134.2.m.d 4 63.l odd 6 1 inner
1134.2.m.d 4 63.o even 6 1 inner
3024.2.k.f 4 36.f odd 6 1
3024.2.k.f 4 36.h even 6 1
3024.2.k.f 4 252.s odd 6 1
3024.2.k.f 4 252.bi even 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 12 T_{5}^{2} + 144$$ $$T_{11}^{4} - 36 T_{11}^{2} + 1296$$ $$T_{13}^{2} - 3 T_{13} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ 1
$5$ $$1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8}$$
$7$ $$( 1 - T + 7 T^{2} )^{2}$$
$11$ $$1 - 14 T^{2} + 75 T^{4} - 1694 T^{6} + 14641 T^{8}$$
$13$ $$( 1 - 5 T + 13 T^{2} )^{2}( 1 + 2 T + 13 T^{2} )^{2}$$
$17$ $$( 1 + 31 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 + 10 T^{2} + 361 T^{4} )^{2}$$
$23$ $$1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8}$$
$29$ $$1 + 49 T^{2} + 1560 T^{4} + 41209 T^{6} + 707281 T^{8}$$
$31$ $$( 1 + 9 T + 58 T^{2} + 279 T^{3} + 961 T^{4} )^{2}$$
$37$ $$( 1 + 2 T + 37 T^{2} )^{4}$$
$41$ $$1 - 34 T^{2} - 525 T^{4} - 57154 T^{6} + 2825761 T^{8}$$
$43$ $$( 1 - 11 T + 78 T^{2} - 473 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$1 - 46 T^{2} - 93 T^{4} - 101614 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 - 97 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$1 - 43 T^{2} - 1632 T^{4} - 149683 T^{6} + 12117361 T^{8}$$
$61$ $$( 1 + 24 T + 253 T^{2} + 1464 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 133 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 98 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$1 - 154 T^{2} + 16827 T^{4} - 1060906 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 + 151 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 12 T + 145 T^{2} + 1164 T^{3} + 9409 T^{4} )^{2}$$