# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1134,2,Mod(163,1134)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1134, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1134.163");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1134 = 2 \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1134.g (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$9.05503558921$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} + 3 \zeta_{6} q^{5} + (3 \zeta_{6} - 2) q^{7} + q^{8} +O(q^{10})$$ q - z * q^2 + (z - 1) * q^4 + 3*z * q^5 + (3*z - 2) * q^7 + q^8 $$q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} + 3 \zeta_{6} q^{5} + (3 \zeta_{6} - 2) q^{7} + q^{8} + ( - 3 \zeta_{6} + 3) q^{10} + (3 \zeta_{6} - 3) q^{11} - q^{13} + ( - \zeta_{6} + 3) q^{14} - \zeta_{6} q^{16} + ( - 3 \zeta_{6} + 3) q^{17} + 7 \zeta_{6} q^{19} - 3 q^{20} + 3 q^{22} - 9 \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} + \zeta_{6} q^{26} + ( - 2 \zeta_{6} - 1) q^{28} - 3 q^{29} + (8 \zeta_{6} - 8) q^{31} + (\zeta_{6} - 1) q^{32} - 3 q^{34} + (3 \zeta_{6} - 9) q^{35} + \zeta_{6} q^{37} + ( - 7 \zeta_{6} + 7) q^{38} + 3 \zeta_{6} q^{40} - 3 q^{41} - q^{43} - 3 \zeta_{6} q^{44} + (9 \zeta_{6} - 9) q^{46} + ( - 3 \zeta_{6} - 5) q^{49} + 4 q^{50} + ( - \zeta_{6} + 1) q^{52} + ( - 3 \zeta_{6} + 3) q^{53} - 9 q^{55} + (3 \zeta_{6} - 2) q^{56} + 3 \zeta_{6} q^{58} - 2 \zeta_{6} q^{61} + 8 q^{62} + q^{64} - 3 \zeta_{6} q^{65} + ( - 4 \zeta_{6} + 4) q^{67} + 3 \zeta_{6} q^{68} + (6 \zeta_{6} + 3) q^{70} - 12 q^{71} + (11 \zeta_{6} - 11) q^{73} + ( - \zeta_{6} + 1) q^{74} - 7 q^{76} + ( - 6 \zeta_{6} - 3) q^{77} + 16 \zeta_{6} q^{79} + ( - 3 \zeta_{6} + 3) q^{80} + 3 \zeta_{6} q^{82} + 9 q^{83} + 9 q^{85} + \zeta_{6} q^{86} + (3 \zeta_{6} - 3) q^{88} + 3 \zeta_{6} q^{89} + ( - 3 \zeta_{6} + 2) q^{91} + 9 q^{92} + (21 \zeta_{6} - 21) q^{95} - q^{97} + (8 \zeta_{6} - 3) q^{98} +O(q^{100})$$ q - z * q^2 + (z - 1) * q^4 + 3*z * q^5 + (3*z - 2) * q^7 + q^8 + (-3*z + 3) * q^10 + (3*z - 3) * q^11 - q^13 + (-z + 3) * q^14 - z * q^16 + (-3*z + 3) * q^17 + 7*z * q^19 - 3 * q^20 + 3 * q^22 - 9*z * q^23 + (4*z - 4) * q^25 + z * q^26 + (-2*z - 1) * q^28 - 3 * q^29 + (8*z - 8) * q^31 + (z - 1) * q^32 - 3 * q^34 + (3*z - 9) * q^35 + z * q^37 + (-7*z + 7) * q^38 + 3*z * q^40 - 3 * q^41 - q^43 - 3*z * q^44 + (9*z - 9) * q^46 + (-3*z - 5) * q^49 + 4 * q^50 + (-z + 1) * q^52 + (-3*z + 3) * q^53 - 9 * q^55 + (3*z - 2) * q^56 + 3*z * q^58 - 2*z * q^61 + 8 * q^62 + q^64 - 3*z * q^65 + (-4*z + 4) * q^67 + 3*z * q^68 + (6*z + 3) * q^70 - 12 * q^71 + (11*z - 11) * q^73 + (-z + 1) * q^74 - 7 * q^76 + (-6*z - 3) * q^77 + 16*z * q^79 + (-3*z + 3) * q^80 + 3*z * q^82 + 9 * q^83 + 9 * q^85 + z * q^86 + (3*z - 3) * q^88 + 3*z * q^89 + (-3*z + 2) * q^91 + 9 * q^92 + (21*z - 21) * q^95 - q^97 + (8*z - 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} + 3 q^{5} - q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q - q^2 - q^4 + 3 * q^5 - q^7 + 2 * q^8 $$2 q - q^{2} - q^{4} + 3 q^{5} - q^{7} + 2 q^{8} + 3 q^{10} - 3 q^{11} - 2 q^{13} + 5 q^{14} - q^{16} + 3 q^{17} + 7 q^{19} - 6 q^{20} + 6 q^{22} - 9 q^{23} - 4 q^{25} + q^{26} - 4 q^{28} - 6 q^{29} - 8 q^{31} - q^{32} - 6 q^{34} - 15 q^{35} + q^{37} + 7 q^{38} + 3 q^{40} - 6 q^{41} - 2 q^{43} - 3 q^{44} - 9 q^{46} - 13 q^{49} + 8 q^{50} + q^{52} + 3 q^{53} - 18 q^{55} - q^{56} + 3 q^{58} - 2 q^{61} + 16 q^{62} + 2 q^{64} - 3 q^{65} + 4 q^{67} + 3 q^{68} + 12 q^{70} - 24 q^{71} - 11 q^{73} + q^{74} - 14 q^{76} - 12 q^{77} + 16 q^{79} + 3 q^{80} + 3 q^{82} + 18 q^{83} + 18 q^{85} + q^{86} - 3 q^{88} + 3 q^{89} + q^{91} + 18 q^{92} - 21 q^{95} - 2 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q - q^2 - q^4 + 3 * q^5 - q^7 + 2 * q^8 + 3 * q^10 - 3 * q^11 - 2 * q^13 + 5 * q^14 - q^16 + 3 * q^17 + 7 * q^19 - 6 * q^20 + 6 * q^22 - 9 * q^23 - 4 * q^25 + q^26 - 4 * q^28 - 6 * q^29 - 8 * q^31 - q^32 - 6 * q^34 - 15 * q^35 + q^37 + 7 * q^38 + 3 * q^40 - 6 * q^41 - 2 * q^43 - 3 * q^44 - 9 * q^46 - 13 * q^49 + 8 * q^50 + q^52 + 3 * q^53 - 18 * q^55 - q^56 + 3 * q^58 - 2 * q^61 + 16 * q^62 + 2 * q^64 - 3 * q^65 + 4 * q^67 + 3 * q^68 + 12 * q^70 - 24 * q^71 - 11 * q^73 + q^74 - 14 * q^76 - 12 * q^77 + 16 * q^79 + 3 * q^80 + 3 * q^82 + 18 * q^83 + 18 * q^85 + q^86 - 3 * q^88 + 3 * q^89 + q^91 + 18 * q^92 - 21 * q^95 - 2 * q^97 + 2 * q^98

## Character values

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

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
163.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 1.50000 2.59808i 0 −0.500000 2.59808i 1.00000 0 1.50000 + 2.59808i
487.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.50000 + 2.59808i 0 −0.500000 + 2.59808i 1.00000 0 1.50000 2.59808i
 $$n$$: 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
7.c 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.g.c 2
3.b odd 2 1 1134.2.g.e 2
7.c even 3 1 inner 1134.2.g.c 2
7.c even 3 1 7938.2.a.t 1
7.d odd 6 1 7938.2.a.be 1
9.c even 3 1 378.2.e.b 2
9.c even 3 1 378.2.h.a 2
9.d odd 6 1 126.2.e.a 2
9.d odd 6 1 126.2.h.b yes 2
21.g even 6 1 7938.2.a.b 1
21.h odd 6 1 1134.2.g.e 2
21.h odd 6 1 7938.2.a.m 1
36.f odd 6 1 3024.2.q.f 2
36.f odd 6 1 3024.2.t.a 2
36.h even 6 1 1008.2.q.a 2
36.h even 6 1 1008.2.t.f 2
63.g even 3 1 378.2.e.b 2
63.g even 3 1 2646.2.f.d 2
63.h even 3 1 378.2.h.a 2
63.h even 3 1 2646.2.f.d 2
63.i even 6 1 882.2.f.g 2
63.i even 6 1 882.2.h.i 2
63.j odd 6 1 126.2.h.b yes 2
63.j odd 6 1 882.2.f.i 2
63.k odd 6 1 2646.2.e.g 2
63.k odd 6 1 2646.2.f.a 2
63.l odd 6 1 2646.2.e.g 2
63.l odd 6 1 2646.2.h.d 2
63.n odd 6 1 126.2.e.a 2
63.n odd 6 1 882.2.f.i 2
63.o even 6 1 882.2.e.c 2
63.o even 6 1 882.2.h.i 2
63.s even 6 1 882.2.e.c 2
63.s even 6 1 882.2.f.g 2
63.t odd 6 1 2646.2.f.a 2
63.t odd 6 1 2646.2.h.d 2
252.o even 6 1 1008.2.q.a 2
252.u odd 6 1 3024.2.t.a 2
252.bb even 6 1 1008.2.t.f 2
252.bl odd 6 1 3024.2.q.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.a 2 9.d odd 6 1
126.2.e.a 2 63.n odd 6 1
126.2.h.b yes 2 9.d odd 6 1
126.2.h.b yes 2 63.j odd 6 1
378.2.e.b 2 9.c even 3 1
378.2.e.b 2 63.g even 3 1
378.2.h.a 2 9.c even 3 1
378.2.h.a 2 63.h even 3 1
882.2.e.c 2 63.o even 6 1
882.2.e.c 2 63.s even 6 1
882.2.f.g 2 63.i even 6 1
882.2.f.g 2 63.s even 6 1
882.2.f.i 2 63.j odd 6 1
882.2.f.i 2 63.n odd 6 1
882.2.h.i 2 63.i even 6 1
882.2.h.i 2 63.o even 6 1
1008.2.q.a 2 36.h even 6 1
1008.2.q.a 2 252.o even 6 1
1008.2.t.f 2 36.h even 6 1
1008.2.t.f 2 252.bb even 6 1
1134.2.g.c 2 1.a even 1 1 trivial
1134.2.g.c 2 7.c even 3 1 inner
1134.2.g.e 2 3.b odd 2 1
1134.2.g.e 2 21.h odd 6 1
2646.2.e.g 2 63.k odd 6 1
2646.2.e.g 2 63.l odd 6 1
2646.2.f.a 2 63.k odd 6 1
2646.2.f.a 2 63.t odd 6 1
2646.2.f.d 2 63.g even 3 1
2646.2.f.d 2 63.h even 3 1
2646.2.h.d 2 63.l odd 6 1
2646.2.h.d 2 63.t odd 6 1
3024.2.q.f 2 36.f odd 6 1
3024.2.q.f 2 252.bl odd 6 1
3024.2.t.a 2 36.f odd 6 1
3024.2.t.a 2 252.u odd 6 1
7938.2.a.b 1 21.g even 6 1
7938.2.a.m 1 21.h odd 6 1
7938.2.a.t 1 7.c even 3 1
7938.2.a.be 1 7.d odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 3T_{5} + 9$$ T5^2 - 3*T5 + 9 $$T_{11}^{2} + 3T_{11} + 9$$ T11^2 + 3*T11 + 9 $$T_{17}^{2} - 3T_{17} + 9$$ T17^2 - 3*T17 + 9 $$T_{23}^{2} + 9T_{23} + 81$$ T23^2 + 9*T23 + 81

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$T^{2} + T + 7$$
$11$ $$T^{2} + 3T + 9$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2} - 7T + 49$$
$23$ $$T^{2} + 9T + 81$$
$29$ $$(T + 3)^{2}$$
$31$ $$T^{2} + 8T + 64$$
$37$ $$T^{2} - T + 1$$
$41$ $$(T + 3)^{2}$$
$43$ $$(T + 1)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 3T + 9$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 2T + 4$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} + 11T + 121$$
$79$ $$T^{2} - 16T + 256$$
$83$ $$(T - 9)^{2}$$
$89$ $$T^{2} - 3T + 9$$
$97$ $$(T + 1)^{2}$$