# Properties

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

# Learn more

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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} + (\zeta_{6} - 1) q^{7} - q^{8} +O(q^{10})$$ q + (-z + 1) * q^2 - z * q^4 + (z - 1) * q^7 - q^8 $$q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} + (\zeta_{6} - 1) q^{7} - q^{8} - 5 \zeta_{6} q^{13} + \zeta_{6} q^{14} + (\zeta_{6} - 1) q^{16} - 3 q^{17} + 2 q^{19} - 9 \zeta_{6} q^{23} + ( - 5 \zeta_{6} + 5) q^{25} - 5 q^{26} + q^{28} + (3 \zeta_{6} - 3) q^{29} - 5 \zeta_{6} q^{31} + \zeta_{6} q^{32} + (3 \zeta_{6} - 3) q^{34} + 2 q^{37} + ( - 2 \zeta_{6} + 2) q^{38} - 6 \zeta_{6} q^{41} + ( - \zeta_{6} + 1) q^{43} - 9 q^{46} + (6 \zeta_{6} - 6) q^{47} - \zeta_{6} q^{49} - 5 \zeta_{6} q^{50} + (5 \zeta_{6} - 5) q^{52} - 3 q^{53} + ( - \zeta_{6} + 1) q^{56} + 3 \zeta_{6} q^{58} - 3 \zeta_{6} q^{59} + ( - 10 \zeta_{6} + 10) q^{61} - 5 q^{62} + q^{64} + 13 \zeta_{6} q^{67} + 3 \zeta_{6} q^{68} - 9 q^{71} + 2 q^{73} + ( - 2 \zeta_{6} + 2) q^{74} - 2 \zeta_{6} q^{76} + ( - 10 \zeta_{6} + 10) q^{79} - 6 q^{82} + (12 \zeta_{6} - 12) q^{83} - \zeta_{6} q^{86} - 15 q^{89} + 5 q^{91} + (9 \zeta_{6} - 9) q^{92} + 6 \zeta_{6} q^{94} + (8 \zeta_{6} - 8) q^{97} - q^{98} +O(q^{100})$$ q + (-z + 1) * q^2 - z * q^4 + (z - 1) * q^7 - q^8 - 5*z * q^13 + z * q^14 + (z - 1) * q^16 - 3 * q^17 + 2 * q^19 - 9*z * q^23 + (-5*z + 5) * q^25 - 5 * q^26 + q^28 + (3*z - 3) * q^29 - 5*z * q^31 + z * q^32 + (3*z - 3) * q^34 + 2 * q^37 + (-2*z + 2) * q^38 - 6*z * q^41 + (-z + 1) * q^43 - 9 * q^46 + (6*z - 6) * q^47 - z * q^49 - 5*z * q^50 + (5*z - 5) * q^52 - 3 * q^53 + (-z + 1) * q^56 + 3*z * q^58 - 3*z * q^59 + (-10*z + 10) * q^61 - 5 * q^62 + q^64 + 13*z * q^67 + 3*z * q^68 - 9 * q^71 + 2 * q^73 + (-2*z + 2) * q^74 - 2*z * q^76 + (-10*z + 10) * q^79 - 6 * q^82 + (12*z - 12) * q^83 - z * q^86 - 15 * q^89 + 5 * q^91 + (9*z - 9) * q^92 + 6*z * q^94 + (8*z - 8) * q^97 - q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} - q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q + q^2 - q^4 - q^7 - 2 * q^8 $$2 q + q^{2} - q^{4} - q^{7} - 2 q^{8} - 5 q^{13} + q^{14} - q^{16} - 6 q^{17} + 4 q^{19} - 9 q^{23} + 5 q^{25} - 10 q^{26} + 2 q^{28} - 3 q^{29} - 5 q^{31} + q^{32} - 3 q^{34} + 4 q^{37} + 2 q^{38} - 6 q^{41} + q^{43} - 18 q^{46} - 6 q^{47} - q^{49} - 5 q^{50} - 5 q^{52} - 6 q^{53} + q^{56} + 3 q^{58} - 3 q^{59} + 10 q^{61} - 10 q^{62} + 2 q^{64} + 13 q^{67} + 3 q^{68} - 18 q^{71} + 4 q^{73} + 2 q^{74} - 2 q^{76} + 10 q^{79} - 12 q^{82} - 12 q^{83} - q^{86} - 30 q^{89} + 10 q^{91} - 9 q^{92} + 6 q^{94} - 8 q^{97} - 2 q^{98}+O(q^{100})$$ 2 * q + q^2 - q^4 - q^7 - 2 * q^8 - 5 * q^13 + q^14 - q^16 - 6 * q^17 + 4 * q^19 - 9 * q^23 + 5 * q^25 - 10 * q^26 + 2 * q^28 - 3 * q^29 - 5 * q^31 + q^32 - 3 * q^34 + 4 * q^37 + 2 * q^38 - 6 * q^41 + q^43 - 18 * q^46 - 6 * q^47 - q^49 - 5 * q^50 - 5 * q^52 - 6 * q^53 + q^56 + 3 * q^58 - 3 * q^59 + 10 * q^61 - 10 * q^62 + 2 * q^64 + 13 * q^67 + 3 * q^68 - 18 * q^71 + 4 * q^73 + 2 * q^74 - 2 * q^76 + 10 * q^79 - 12 * q^82 - 12 * q^83 - q^86 - 30 * q^89 + 10 * q^91 - 9 * q^92 + 6 * q^94 - 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 0 0 −0.500000 + 0.866025i −1.00000 0 0
757.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 −0.500000 0.866025i −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
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.k 2
3.b odd 2 1 1134.2.f.e 2
9.c even 3 1 378.2.a.d 1
9.c even 3 1 inner 1134.2.f.k 2
9.d odd 6 1 378.2.a.e yes 1
9.d odd 6 1 1134.2.f.e 2
36.f odd 6 1 3024.2.a.o 1
36.h even 6 1 3024.2.a.p 1
45.h odd 6 1 9450.2.a.l 1
45.j even 6 1 9450.2.a.cl 1
63.l odd 6 1 2646.2.a.f 1
63.o even 6 1 2646.2.a.y 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.d 1 9.c even 3 1
378.2.a.e yes 1 9.d odd 6 1
1134.2.f.e 2 3.b odd 2 1
1134.2.f.e 2 9.d odd 6 1
1134.2.f.k 2 1.a even 1 1 trivial
1134.2.f.k 2 9.c even 3 1 inner
2646.2.a.f 1 63.l odd 6 1
2646.2.a.y 1 63.o even 6 1
3024.2.a.o 1 36.f odd 6 1
3024.2.a.p 1 36.h even 6 1
9450.2.a.l 1 45.h odd 6 1
9450.2.a.cl 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 $$S_{2}^{\mathrm{new}}(1134, [\chi])$$:

 $$T_{5}$$ T5 $$T_{11}$$ T11 $$T_{13}^{2} + 5T_{13} + 25$$ T13^2 + 5*T13 + 25 $$T_{17} + 3$$ T17 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + T + 1$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 5T + 25$$
$17$ $$(T + 3)^{2}$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2} + 9T + 81$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$T^{2} + 5T + 25$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2} + 6T + 36$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2} + 6T + 36$$
$53$ $$(T + 3)^{2}$$
$59$ $$T^{2} + 3T + 9$$
$61$ $$T^{2} - 10T + 100$$
$67$ $$T^{2} - 13T + 169$$
$71$ $$(T + 9)^{2}$$
$73$ $$(T - 2)^{2}$$
$79$ $$T^{2} - 10T + 100$$
$83$ $$T^{2} + 12T + 144$$
$89$ $$(T + 15)^{2}$$
$97$ $$T^{2} + 8T + 64$$
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