# Properties

 Label 1134.2.f.o 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{7} - q^{8} +O(q^{10})$$ q + (-z + 1) * q^2 - z * q^4 + 3*z * q^5 + (z - 1) * q^7 - q^8 $$q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{7} - q^{8} + 3 q^{10} + (3 \zeta_{6} - 3) q^{11} + 4 \zeta_{6} q^{13} + \zeta_{6} q^{14} + (\zeta_{6} - 1) q^{16} - 6 q^{17} - 7 q^{19} + ( - 3 \zeta_{6} + 3) q^{20} + 3 \zeta_{6} q^{22} + 3 \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} + 4 q^{26} + q^{28} - 5 \zeta_{6} q^{31} + \zeta_{6} q^{32} + (6 \zeta_{6} - 6) q^{34} - 3 q^{35} - 7 q^{37} + (7 \zeta_{6} - 7) q^{38} - 3 \zeta_{6} q^{40} + 9 \zeta_{6} q^{41} + ( - 10 \zeta_{6} + 10) q^{43} + 3 q^{44} + 3 q^{46} + (6 \zeta_{6} - 6) q^{47} - \zeta_{6} q^{49} + 4 \zeta_{6} q^{50} + ( - 4 \zeta_{6} + 4) q^{52} + 12 q^{53} - 9 q^{55} + ( - \zeta_{6} + 1) q^{56} + 6 \zeta_{6} q^{59} + (8 \zeta_{6} - 8) q^{61} - 5 q^{62} + q^{64} + (12 \zeta_{6} - 12) q^{65} + 4 \zeta_{6} q^{67} + 6 \zeta_{6} q^{68} + (3 \zeta_{6} - 3) q^{70} + 9 q^{71} + 2 q^{73} + (7 \zeta_{6} - 7) q^{74} + 7 \zeta_{6} q^{76} - 3 \zeta_{6} q^{77} + ( - 10 \zeta_{6} + 10) q^{79} - 3 q^{80} + 9 q^{82} - 18 \zeta_{6} q^{85} - 10 \zeta_{6} q^{86} + ( - 3 \zeta_{6} + 3) q^{88} + 15 q^{89} - 4 q^{91} + ( - 3 \zeta_{6} + 3) q^{92} + 6 \zeta_{6} q^{94} - 21 \zeta_{6} q^{95} + (8 \zeta_{6} - 8) q^{97} - q^{98} +O(q^{100})$$ q + (-z + 1) * q^2 - z * q^4 + 3*z * q^5 + (z - 1) * q^7 - q^8 + 3 * q^10 + (3*z - 3) * q^11 + 4*z * q^13 + z * q^14 + (z - 1) * q^16 - 6 * q^17 - 7 * q^19 + (-3*z + 3) * q^20 + 3*z * q^22 + 3*z * q^23 + (4*z - 4) * q^25 + 4 * q^26 + q^28 - 5*z * q^31 + z * q^32 + (6*z - 6) * q^34 - 3 * q^35 - 7 * q^37 + (7*z - 7) * q^38 - 3*z * q^40 + 9*z * q^41 + (-10*z + 10) * q^43 + 3 * q^44 + 3 * q^46 + (6*z - 6) * q^47 - z * q^49 + 4*z * q^50 + (-4*z + 4) * q^52 + 12 * q^53 - 9 * q^55 + (-z + 1) * q^56 + 6*z * q^59 + (8*z - 8) * q^61 - 5 * q^62 + q^64 + (12*z - 12) * q^65 + 4*z * q^67 + 6*z * q^68 + (3*z - 3) * q^70 + 9 * q^71 + 2 * q^73 + (7*z - 7) * q^74 + 7*z * q^76 - 3*z * q^77 + (-10*z + 10) * q^79 - 3 * q^80 + 9 * q^82 - 18*z * q^85 - 10*z * q^86 + (-3*z + 3) * q^88 + 15 * q^89 - 4 * q^91 + (-3*z + 3) * q^92 + 6*z * q^94 - 21*z * q^95 + (8*z - 8) * q^97 - 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 $$2 q + q^{2} - q^{4} + 3 q^{5} - q^{7} - 2 q^{8} + 6 q^{10} - 3 q^{11} + 4 q^{13} + q^{14} - q^{16} - 12 q^{17} - 14 q^{19} + 3 q^{20} + 3 q^{22} + 3 q^{23} - 4 q^{25} + 8 q^{26} + 2 q^{28} - 5 q^{31} + q^{32} - 6 q^{34} - 6 q^{35} - 14 q^{37} - 7 q^{38} - 3 q^{40} + 9 q^{41} + 10 q^{43} + 6 q^{44} + 6 q^{46} - 6 q^{47} - q^{49} + 4 q^{50} + 4 q^{52} + 24 q^{53} - 18 q^{55} + q^{56} + 6 q^{59} - 8 q^{61} - 10 q^{62} + 2 q^{64} - 12 q^{65} + 4 q^{67} + 6 q^{68} - 3 q^{70} + 18 q^{71} + 4 q^{73} - 7 q^{74} + 7 q^{76} - 3 q^{77} + 10 q^{79} - 6 q^{80} + 18 q^{82} - 18 q^{85} - 10 q^{86} + 3 q^{88} + 30 q^{89} - 8 q^{91} + 3 q^{92} + 6 q^{94} - 21 q^{95} - 8 q^{97} - 2 q^{98}+O(q^{100})$$ 2 * q + q^2 - q^4 + 3 * q^5 - q^7 - 2 * q^8 + 6 * q^10 - 3 * q^11 + 4 * q^13 + q^14 - q^16 - 12 * q^17 - 14 * q^19 + 3 * q^20 + 3 * q^22 + 3 * q^23 - 4 * q^25 + 8 * q^26 + 2 * q^28 - 5 * q^31 + q^32 - 6 * q^34 - 6 * q^35 - 14 * q^37 - 7 * q^38 - 3 * q^40 + 9 * q^41 + 10 * q^43 + 6 * q^44 + 6 * q^46 - 6 * q^47 - q^49 + 4 * q^50 + 4 * q^52 + 24 * q^53 - 18 * q^55 + q^56 + 6 * q^59 - 8 * q^61 - 10 * q^62 + 2 * q^64 - 12 * q^65 + 4 * q^67 + 6 * q^68 - 3 * q^70 + 18 * q^71 + 4 * q^73 - 7 * q^74 + 7 * q^76 - 3 * q^77 + 10 * q^79 - 6 * q^80 + 18 * q^82 - 18 * q^85 - 10 * q^86 + 3 * q^88 + 30 * q^89 - 8 * q^91 + 3 * q^92 + 6 * q^94 - 21 * q^95 - 8 * 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 1.50000 + 2.59808i 0 −0.500000 + 0.866025i −1.00000 0 3.00000
757.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.50000 2.59808i 0 −0.500000 0.866025i −1.00000 0 3.00000
 $$n$$: 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.o 2
3.b odd 2 1 1134.2.f.b 2
9.c even 3 1 378.2.a.b 1
9.c even 3 1 inner 1134.2.f.o 2
9.d odd 6 1 378.2.a.g yes 1
9.d odd 6 1 1134.2.f.b 2
36.f odd 6 1 3024.2.a.c 1
36.h even 6 1 3024.2.a.bb 1
45.h odd 6 1 9450.2.a.h 1
45.j even 6 1 9450.2.a.cu 1
63.l odd 6 1 2646.2.a.n 1
63.o even 6 1 2646.2.a.q 1

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

## Hecke kernels

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

 $$T_{5}^{2} - 3T_{5} + 9$$ T5^2 - 3*T5 + 9 $$T_{11}^{2} + 3T_{11} + 9$$ T11^2 + 3*T11 + 9 $$T_{13}^{2} - 4T_{13} + 16$$ T13^2 - 4*T13 + 16 $$T_{17} + 6$$ T17 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$T^{2} + T + 1$$
$11$ $$T^{2} + 3T + 9$$
$13$ $$T^{2} - 4T + 16$$
$17$ $$(T + 6)^{2}$$
$19$ $$(T + 7)^{2}$$
$23$ $$T^{2} - 3T + 9$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 5T + 25$$
$37$ $$(T + 7)^{2}$$
$41$ $$T^{2} - 9T + 81$$
$43$ $$T^{2} - 10T + 100$$
$47$ $$T^{2} + 6T + 36$$
$53$ $$(T - 12)^{2}$$
$59$ $$T^{2} - 6T + 36$$
$61$ $$T^{2} + 8T + 64$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$(T - 9)^{2}$$
$73$ $$(T - 2)^{2}$$
$79$ $$T^{2} - 10T + 100$$
$83$ $$T^{2}$$
$89$ $$(T - 15)^{2}$$
$97$ $$T^{2} + 8T + 64$$