# Properties

 Label 1134.2.f.r Level $1134$ Weight $2$ Character orbit 1134.f Analytic conductor $9.055$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: yes 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_1 - 1) q^{4} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{5} + \beta_1 q^{7} + q^{8}+O(q^{10})$$ q - b1 * q^2 + (b1 - 1) * q^4 + (b3 - b2 - 2*b1 + 2) * q^5 + b1 * q^7 + q^8 $$q - \beta_1 q^{2} + (\beta_1 - 1) q^{4} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{5} + \beta_1 q^{7} + q^{8} + ( - \beta_{3} - 2) q^{10} + ( - 3 \beta_{2} + \beta_1) q^{11} + ( - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 3) q^{13} + ( - \beta_1 + 1) q^{14} - \beta_1 q^{16} - 7 q^{17} + ( - \beta_{3} - 1) q^{19} + (\beta_{2} + 2 \beta_1) q^{20} + ( - 3 \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{22} + (3 \beta_{3} - 3 \beta_{2} - \beta_1 + 1) q^{23} + ( - 4 \beta_{2} - 2 \beta_1) q^{25} + (2 \beta_{3} - 3) q^{26} - q^{28} + (2 \beta_{2} + 5 \beta_1) q^{29} + (3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 3) q^{31} + (\beta_1 - 1) q^{32} + 7 \beta_1 q^{34} + (\beta_{3} + 2) q^{35} + ( - 5 \beta_{3} + 2) q^{37} + (\beta_{2} + \beta_1) q^{38} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{40} + (2 \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 6) q^{41} + ( - 2 \beta_{2} - 2 \beta_1) q^{43} + (3 \beta_{3} - 1) q^{44} + ( - 3 \beta_{3} - 1) q^{46} + ( - \beta_{2} + 3 \beta_1) q^{47} + (\beta_1 - 1) q^{49} + ( - 4 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 2) q^{50} + ( - 2 \beta_{2} + 3 \beta_1) q^{52} + ( - 2 \beta_{3} + 6) q^{53} + ( - 5 \beta_{3} - 7) q^{55} + \beta_1 q^{56} + (2 \beta_{3} - 2 \beta_{2} - 5 \beta_1 + 5) q^{58} + ( - 3 \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{59} + (4 \beta_{2} + 3 \beta_1) q^{61} + ( - 3 \beta_{3} + 3) q^{62} + q^{64} + \beta_{2} q^{65} + ( - \beta_{3} + \beta_{2} - 5 \beta_1 + 5) q^{67} + ( - 7 \beta_1 + 7) q^{68} + ( - \beta_{2} - 2 \beta_1) q^{70} + ( - 2 \beta_{3} - 10) q^{71} + (\beta_{3} + 10) q^{73} + (5 \beta_{2} - 2 \beta_1) q^{74} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{76} + (3 \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{77} + ( - 7 \beta_{2} - 3 \beta_1) q^{79} + ( - \beta_{3} - 2) q^{80} + ( - 2 \beta_{3} - 6) q^{82} + ( - 9 \beta_{2} + \beta_1) q^{83} + ( - 7 \beta_{3} + 7 \beta_{2} + 14 \beta_1 - 14) q^{85} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{86} + ( - 3 \beta_{2} + \beta_1) q^{88} + (4 \beta_{3} - 3) q^{89} + ( - 2 \beta_{3} + 3) q^{91} + (3 \beta_{2} + \beta_1) q^{92} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 3) q^{94} + ( - 3 \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 5) q^{95} + (4 \beta_{2} - 4 \beta_1) q^{97} + q^{98}+O(q^{100})$$ q - b1 * q^2 + (b1 - 1) * q^4 + (b3 - b2 - 2*b1 + 2) * q^5 + b1 * q^7 + q^8 + (-b3 - 2) * q^10 + (-3*b2 + b1) * q^11 + (-2*b3 + 2*b2 - 3*b1 + 3) * q^13 + (-b1 + 1) * q^14 - b1 * q^16 - 7 * q^17 + (-b3 - 1) * q^19 + (b2 + 2*b1) * q^20 + (-3*b3 + 3*b2 - b1 + 1) * q^22 + (3*b3 - 3*b2 - b1 + 1) * q^23 + (-4*b2 - 2*b1) * q^25 + (2*b3 - 3) * q^26 - q^28 + (2*b2 + 5*b1) * q^29 + (3*b3 - 3*b2 + 3*b1 - 3) * q^31 + (b1 - 1) * q^32 + 7*b1 * q^34 + (b3 + 2) * q^35 + (-5*b3 + 2) * q^37 + (b2 + b1) * q^38 + (b3 - b2 - 2*b1 + 2) * q^40 + (2*b3 - 2*b2 - 6*b1 + 6) * q^41 + (-2*b2 - 2*b1) * q^43 + (3*b3 - 1) * q^44 + (-3*b3 - 1) * q^46 + (-b2 + 3*b1) * q^47 + (b1 - 1) * q^49 + (-4*b3 + 4*b2 + 2*b1 - 2) * q^50 + (-2*b2 + 3*b1) * q^52 + (-2*b3 + 6) * q^53 + (-5*b3 - 7) * q^55 + b1 * q^56 + (2*b3 - 2*b2 - 5*b1 + 5) * q^58 + (-3*b3 + 3*b2 + b1 - 1) * q^59 + (4*b2 + 3*b1) * q^61 + (-3*b3 + 3) * q^62 + q^64 + b2 * q^65 + (-b3 + b2 - 5*b1 + 5) * q^67 + (-7*b1 + 7) * q^68 + (-b2 - 2*b1) * q^70 + (-2*b3 - 10) * q^71 + (b3 + 10) * q^73 + (5*b2 - 2*b1) * q^74 + (b3 - b2 - b1 + 1) * q^76 + (3*b3 - 3*b2 + b1 - 1) * q^77 + (-7*b2 - 3*b1) * q^79 + (-b3 - 2) * q^80 + (-2*b3 - 6) * q^82 + (-9*b2 + b1) * q^83 + (-7*b3 + 7*b2 + 14*b1 - 14) * q^85 + (-2*b3 + 2*b2 + 2*b1 - 2) * q^86 + (-3*b2 + b1) * q^88 + (4*b3 - 3) * q^89 + (-2*b3 + 3) * q^91 + (3*b2 + b1) * q^92 + (-b3 + b2 - 3*b1 + 3) * q^94 + (-3*b3 + 3*b2 + 5*b1 - 5) * q^95 + (4*b2 - 4*b1) * q^97 + q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 2 q^{4} + 4 q^{5} + 2 q^{7} + 4 q^{8}+O(q^{10})$$ 4 * q - 2 * q^2 - 2 * q^4 + 4 * q^5 + 2 * q^7 + 4 * q^8 $$4 q - 2 q^{2} - 2 q^{4} + 4 q^{5} + 2 q^{7} + 4 q^{8} - 8 q^{10} + 2 q^{11} + 6 q^{13} + 2 q^{14} - 2 q^{16} - 28 q^{17} - 4 q^{19} + 4 q^{20} + 2 q^{22} + 2 q^{23} - 4 q^{25} - 12 q^{26} - 4 q^{28} + 10 q^{29} - 6 q^{31} - 2 q^{32} + 14 q^{34} + 8 q^{35} + 8 q^{37} + 2 q^{38} + 4 q^{40} + 12 q^{41} - 4 q^{43} - 4 q^{44} - 4 q^{46} + 6 q^{47} - 2 q^{49} - 4 q^{50} + 6 q^{52} + 24 q^{53} - 28 q^{55} + 2 q^{56} + 10 q^{58} - 2 q^{59} + 6 q^{61} + 12 q^{62} + 4 q^{64} + 10 q^{67} + 14 q^{68} - 4 q^{70} - 40 q^{71} + 40 q^{73} - 4 q^{74} + 2 q^{76} - 2 q^{77} - 6 q^{79} - 8 q^{80} - 24 q^{82} + 2 q^{83} - 28 q^{85} - 4 q^{86} + 2 q^{88} - 12 q^{89} + 12 q^{91} + 2 q^{92} + 6 q^{94} - 10 q^{95} - 8 q^{97} + 4 q^{98}+O(q^{100})$$ 4 * q - 2 * q^2 - 2 * q^4 + 4 * q^5 + 2 * q^7 + 4 * q^8 - 8 * q^10 + 2 * q^11 + 6 * q^13 + 2 * q^14 - 2 * q^16 - 28 * q^17 - 4 * q^19 + 4 * q^20 + 2 * q^22 + 2 * q^23 - 4 * q^25 - 12 * q^26 - 4 * q^28 + 10 * q^29 - 6 * q^31 - 2 * q^32 + 14 * q^34 + 8 * q^35 + 8 * q^37 + 2 * q^38 + 4 * q^40 + 12 * q^41 - 4 * q^43 - 4 * q^44 - 4 * q^46 + 6 * q^47 - 2 * q^49 - 4 * q^50 + 6 * q^52 + 24 * q^53 - 28 * q^55 + 2 * q^56 + 10 * q^58 - 2 * q^59 + 6 * q^61 + 12 * q^62 + 4 * q^64 + 10 * q^67 + 14 * q^68 - 4 * q^70 - 40 * q^71 + 40 * q^73 - 4 * q^74 + 2 * q^76 - 2 * q^77 - 6 * q^79 - 8 * q^80 - 24 * q^82 + 2 * q^83 - 28 * q^85 - 4 * q^86 + 2 * q^88 - 12 * q^89 + 12 * q^91 + 2 * q^92 + 6 * q^94 - 10 * q^95 - 8 * q^97 + 4 * q^98

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{3} + \zeta_{12}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\zeta_{12}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{12}^{3}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} ) / 3$$ (-b3 + 2*b2) / 3

## Character values

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

 $$n$$ $$325$$ $$407$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
379.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0.133975 + 0.232051i 0 0.500000 0.866025i 1.00000 0 −0.267949
379.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 1.86603 + 3.23205i 0 0.500000 0.866025i 1.00000 0 −3.73205
757.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0.133975 0.232051i 0 0.500000 + 0.866025i 1.00000 0 −0.267949
757.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.86603 3.23205i 0 0.500000 + 0.866025i 1.00000 0 −3.73205
 $$n$$: 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.r 4
3.b odd 2 1 1134.2.f.s 4
9.c even 3 1 1134.2.a.m yes 2
9.c even 3 1 inner 1134.2.f.r 4
9.d odd 6 1 1134.2.a.l 2
9.d odd 6 1 1134.2.f.s 4
36.f odd 6 1 9072.2.a.y 2
36.h even 6 1 9072.2.a.bp 2
63.l odd 6 1 7938.2.a.bt 2
63.o even 6 1 7938.2.a.bg 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.a.l 2 9.d odd 6 1
1134.2.a.m yes 2 9.c even 3 1
1134.2.f.r 4 1.a even 1 1 trivial
1134.2.f.r 4 9.c even 3 1 inner
1134.2.f.s 4 3.b odd 2 1
1134.2.f.s 4 9.d odd 6 1
7938.2.a.bg 2 63.o even 6 1
7938.2.a.bt 2 63.l odd 6 1
9072.2.a.y 2 36.f odd 6 1
9072.2.a.bp 2 36.h even 6 1

## Hecke kernels

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

 $$T_{5}^{4} - 4T_{5}^{3} + 15T_{5}^{2} - 4T_{5} + 1$$ T5^4 - 4*T5^3 + 15*T5^2 - 4*T5 + 1 $$T_{11}^{4} - 2T_{11}^{3} + 30T_{11}^{2} + 52T_{11} + 676$$ T11^4 - 2*T11^3 + 30*T11^2 + 52*T11 + 676 $$T_{13}^{4} - 6T_{13}^{3} + 39T_{13}^{2} + 18T_{13} + 9$$ T13^4 - 6*T13^3 + 39*T13^2 + 18*T13 + 9 $$T_{17} + 7$$ T17 + 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 4 T^{3} + 15 T^{2} - 4 T + 1$$
$7$ $$(T^{2} - T + 1)^{2}$$
$11$ $$T^{4} - 2 T^{3} + 30 T^{2} + 52 T + 676$$
$13$ $$T^{4} - 6 T^{3} + 39 T^{2} + 18 T + 9$$
$17$ $$(T + 7)^{4}$$
$19$ $$(T^{2} + 2 T - 2)^{2}$$
$23$ $$T^{4} - 2 T^{3} + 30 T^{2} + 52 T + 676$$
$29$ $$T^{4} - 10 T^{3} + 87 T^{2} + \cdots + 169$$
$31$ $$T^{4} + 6 T^{3} + 54 T^{2} - 108 T + 324$$
$37$ $$(T^{2} - 4 T - 71)^{2}$$
$41$ $$T^{4} - 12 T^{3} + 120 T^{2} + \cdots + 576$$
$43$ $$T^{4} + 4 T^{3} + 24 T^{2} - 32 T + 64$$
$47$ $$T^{4} - 6 T^{3} + 30 T^{2} - 36 T + 36$$
$53$ $$(T^{2} - 12 T + 24)^{2}$$
$59$ $$T^{4} + 2 T^{3} + 30 T^{2} - 52 T + 676$$
$61$ $$T^{4} - 6 T^{3} + 75 T^{2} + \cdots + 1521$$
$67$ $$T^{4} - 10 T^{3} + 78 T^{2} + \cdots + 484$$
$71$ $$(T^{2} + 20 T + 88)^{2}$$
$73$ $$(T^{2} - 20 T + 97)^{2}$$
$79$ $$T^{4} + 6 T^{3} + 174 T^{2} + \cdots + 19044$$
$83$ $$T^{4} - 2 T^{3} + 246 T^{2} + \cdots + 58564$$
$89$ $$(T^{2} + 6 T - 39)^{2}$$
$97$ $$T^{4} + 8 T^{3} + 96 T^{2} + \cdots + 1024$$