# Properties

 Label 1134.2.f.c Level $1134$ Weight $2$ Character orbit 1134.f 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(379,1134)]

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([4, 0]))

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

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

chi := DirichletCharacter("1134.379");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1134 = 2 \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1134.f (of order $$3$$, degree $$2$$, not 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 378) 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} - \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} + q^{8} +O(q^{10})$$ q + (z - 1) * q^2 - z * q^4 - z * q^5 + (-z + 1) * q^7 + q^8 $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} - \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} + q^{8} + q^{10} + (5 \zeta_{6} - 5) q^{11} + \zeta_{6} q^{14} + (\zeta_{6} - 1) q^{16} + 2 q^{17} - q^{19} + (\zeta_{6} - 1) q^{20} - 5 \zeta_{6} q^{22} + \zeta_{6} q^{23} + ( - 4 \zeta_{6} + 4) q^{25} - q^{28} + (4 \zeta_{6} - 4) q^{29} + 9 \zeta_{6} q^{31} - \zeta_{6} q^{32} + (2 \zeta_{6} - 2) q^{34} - q^{35} + 5 q^{37} + ( - \zeta_{6} + 1) q^{38} - \zeta_{6} q^{40} + 9 \zeta_{6} q^{41} + ( - 10 \zeta_{6} + 10) q^{43} + 5 q^{44} - q^{46} + (6 \zeta_{6} - 6) q^{47} - \zeta_{6} q^{49} + 4 \zeta_{6} q^{50} + 12 q^{53} + 5 q^{55} + ( - \zeta_{6} + 1) q^{56} - 4 \zeta_{6} q^{58} + 14 \zeta_{6} q^{59} - 9 q^{62} + q^{64} + 8 \zeta_{6} q^{67} - 2 \zeta_{6} q^{68} + ( - \zeta_{6} + 1) q^{70} - 13 q^{71} - 2 q^{73} + (5 \zeta_{6} - 5) q^{74} + \zeta_{6} q^{76} + 5 \zeta_{6} q^{77} + (6 \zeta_{6} - 6) q^{79} + q^{80} - 9 q^{82} + ( - 4 \zeta_{6} + 4) q^{83} - 2 \zeta_{6} q^{85} + 10 \zeta_{6} q^{86} + (5 \zeta_{6} - 5) q^{88} - 9 q^{89} + ( - \zeta_{6} + 1) q^{92} - 6 \zeta_{6} q^{94} + \zeta_{6} q^{95} + (16 \zeta_{6} - 16) q^{97} + q^{98} +O(q^{100})$$ q + (z - 1) * q^2 - z * q^4 - z * q^5 + (-z + 1) * q^7 + q^8 + q^10 + (5*z - 5) * q^11 + z * q^14 + (z - 1) * q^16 + 2 * q^17 - q^19 + (z - 1) * q^20 - 5*z * q^22 + z * q^23 + (-4*z + 4) * q^25 - q^28 + (4*z - 4) * q^29 + 9*z * q^31 - z * q^32 + (2*z - 2) * q^34 - q^35 + 5 * q^37 + (-z + 1) * q^38 - z * q^40 + 9*z * q^41 + (-10*z + 10) * q^43 + 5 * q^44 - q^46 + (6*z - 6) * q^47 - z * q^49 + 4*z * q^50 + 12 * q^53 + 5 * q^55 + (-z + 1) * q^56 - 4*z * q^58 + 14*z * q^59 - 9 * q^62 + q^64 + 8*z * q^67 - 2*z * q^68 + (-z + 1) * q^70 - 13 * q^71 - 2 * q^73 + (5*z - 5) * q^74 + z * q^76 + 5*z * q^77 + (6*z - 6) * q^79 + q^80 - 9 * q^82 + (-4*z + 4) * q^83 - 2*z * q^85 + 10*z * q^86 + (5*z - 5) * q^88 - 9 * q^89 + (-z + 1) * q^92 - 6*z * q^94 + z * q^95 + (16*z - 16) * q^97 + q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} - q^{5} + q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q - q^2 - q^4 - q^5 + q^7 + 2 * q^8 $$2 q - q^{2} - q^{4} - q^{5} + q^{7} + 2 q^{8} + 2 q^{10} - 5 q^{11} + q^{14} - q^{16} + 4 q^{17} - 2 q^{19} - q^{20} - 5 q^{22} + q^{23} + 4 q^{25} - 2 q^{28} - 4 q^{29} + 9 q^{31} - q^{32} - 2 q^{34} - 2 q^{35} + 10 q^{37} + q^{38} - q^{40} + 9 q^{41} + 10 q^{43} + 10 q^{44} - 2 q^{46} - 6 q^{47} - q^{49} + 4 q^{50} + 24 q^{53} + 10 q^{55} + q^{56} - 4 q^{58} + 14 q^{59} - 18 q^{62} + 2 q^{64} + 8 q^{67} - 2 q^{68} + q^{70} - 26 q^{71} - 4 q^{73} - 5 q^{74} + q^{76} + 5 q^{77} - 6 q^{79} + 2 q^{80} - 18 q^{82} + 4 q^{83} - 2 q^{85} + 10 q^{86} - 5 q^{88} - 18 q^{89} + q^{92} - 6 q^{94} + q^{95} - 16 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q - q^2 - q^4 - q^5 + q^7 + 2 * q^8 + 2 * q^10 - 5 * q^11 + q^14 - q^16 + 4 * q^17 - 2 * q^19 - q^20 - 5 * q^22 + q^23 + 4 * q^25 - 2 * q^28 - 4 * q^29 + 9 * q^31 - q^32 - 2 * q^34 - 2 * q^35 + 10 * q^37 + q^38 - q^40 + 9 * q^41 + 10 * q^43 + 10 * q^44 - 2 * q^46 - 6 * q^47 - q^49 + 4 * q^50 + 24 * q^53 + 10 * q^55 + q^56 - 4 * q^58 + 14 * q^59 - 18 * q^62 + 2 * q^64 + 8 * q^67 - 2 * q^68 + q^70 - 26 * q^71 - 4 * q^73 - 5 * q^74 + q^76 + 5 * q^77 - 6 * q^79 + 2 * q^80 - 18 * q^82 + 4 * q^83 - 2 * q^85 + 10 * q^86 - 5 * q^88 - 18 * q^89 + q^92 - 6 * q^94 + q^95 - 16 * q^97 + 2 * q^98

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
379.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −0.500000 0.866025i 0 0.500000 0.866025i 1.00000 0 1.00000
757.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.500000 + 0.866025i 1.00000 0 1.00000
 $$n$$: 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
9.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.f.c 2
3.b odd 2 1 1134.2.f.n 2
9.c even 3 1 378.2.a.f yes 1
9.c even 3 1 inner 1134.2.f.c 2
9.d odd 6 1 378.2.a.c 1
9.d odd 6 1 1134.2.f.n 2
36.f odd 6 1 3024.2.a.t 1
36.h even 6 1 3024.2.a.m 1
45.h odd 6 1 9450.2.a.dc 1
45.j even 6 1 9450.2.a.bx 1
63.l odd 6 1 2646.2.a.v 1
63.o even 6 1 2646.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.c 1 9.d odd 6 1
378.2.a.f yes 1 9.c even 3 1
1134.2.f.c 2 1.a even 1 1 trivial
1134.2.f.c 2 9.c even 3 1 inner
1134.2.f.n 2 3.b odd 2 1
1134.2.f.n 2 9.d odd 6 1
2646.2.a.i 1 63.o even 6 1
2646.2.a.v 1 63.l odd 6 1
3024.2.a.m 1 36.h even 6 1
3024.2.a.t 1 36.f odd 6 1
9450.2.a.bx 1 45.j even 6 1
9450.2.a.dc 1 45.h odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} + T_{5} + 1$$ T5^2 + T5 + 1 $$T_{11}^{2} + 5T_{11} + 25$$ T11^2 + 5*T11 + 25 $$T_{13}$$ T13 $$T_{17} - 2$$ T17 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2} - T + 1$$
$11$ $$T^{2} + 5T + 25$$
$13$ $$T^{2}$$
$17$ $$(T - 2)^{2}$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} - T + 1$$
$29$ $$T^{2} + 4T + 16$$
$31$ $$T^{2} - 9T + 81$$
$37$ $$(T - 5)^{2}$$
$41$ $$T^{2} - 9T + 81$$
$43$ $$T^{2} - 10T + 100$$
$47$ $$T^{2} + 6T + 36$$
$53$ $$(T - 12)^{2}$$
$59$ $$T^{2} - 14T + 196$$
$61$ $$T^{2}$$
$67$ $$T^{2} - 8T + 64$$
$71$ $$(T + 13)^{2}$$
$73$ $$(T + 2)^{2}$$
$79$ $$T^{2} + 6T + 36$$
$83$ $$T^{2} - 4T + 16$$
$89$ $$(T + 9)^{2}$$
$97$ $$T^{2} + 16T + 256$$