# Properties

 Label 1134.2.f.j 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 42) 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} - 2 \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} - q^{8} +O(q^{10})$$ q + (-z + 1) * q^2 - z * q^4 - 2*z * q^5 + (-z + 1) * q^7 - q^8 $$q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} - 2 \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} - q^{8} - 2 q^{10} + (4 \zeta_{6} - 4) q^{11} - 6 \zeta_{6} q^{13} - \zeta_{6} q^{14} + (\zeta_{6} - 1) q^{16} - 2 q^{17} - 4 q^{19} + (2 \zeta_{6} - 2) q^{20} + 4 \zeta_{6} q^{22} + 8 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} - 6 q^{26} - q^{28} + (2 \zeta_{6} - 2) q^{29} + \zeta_{6} q^{32} + (2 \zeta_{6} - 2) q^{34} - 2 q^{35} - 10 q^{37} + (4 \zeta_{6} - 4) q^{38} + 2 \zeta_{6} q^{40} - 6 \zeta_{6} q^{41} + ( - 4 \zeta_{6} + 4) q^{43} + 4 q^{44} + 8 q^{46} - \zeta_{6} q^{49} - \zeta_{6} q^{50} + (6 \zeta_{6} - 6) q^{52} - 6 q^{53} + 8 q^{55} + (\zeta_{6} - 1) q^{56} + 2 \zeta_{6} q^{58} + 4 \zeta_{6} q^{59} + (6 \zeta_{6} - 6) q^{61} + q^{64} + (12 \zeta_{6} - 12) q^{65} - 4 \zeta_{6} q^{67} + 2 \zeta_{6} q^{68} + (2 \zeta_{6} - 2) q^{70} - 8 q^{71} + 10 q^{73} + (10 \zeta_{6} - 10) q^{74} + 4 \zeta_{6} q^{76} + 4 \zeta_{6} q^{77} + 2 q^{80} - 6 q^{82} + (4 \zeta_{6} - 4) q^{83} + 4 \zeta_{6} q^{85} - 4 \zeta_{6} q^{86} + ( - 4 \zeta_{6} + 4) q^{88} + 6 q^{89} - 6 q^{91} + ( - 8 \zeta_{6} + 8) q^{92} + 8 \zeta_{6} q^{95} + ( - 14 \zeta_{6} + 14) q^{97} - q^{98} +O(q^{100})$$ q + (-z + 1) * q^2 - z * q^4 - 2*z * q^5 + (-z + 1) * q^7 - q^8 - 2 * q^10 + (4*z - 4) * q^11 - 6*z * q^13 - z * q^14 + (z - 1) * q^16 - 2 * q^17 - 4 * q^19 + (2*z - 2) * q^20 + 4*z * q^22 + 8*z * q^23 + (-z + 1) * q^25 - 6 * q^26 - q^28 + (2*z - 2) * q^29 + z * q^32 + (2*z - 2) * q^34 - 2 * q^35 - 10 * q^37 + (4*z - 4) * q^38 + 2*z * q^40 - 6*z * q^41 + (-4*z + 4) * q^43 + 4 * q^44 + 8 * q^46 - z * q^49 - z * q^50 + (6*z - 6) * q^52 - 6 * q^53 + 8 * q^55 + (z - 1) * q^56 + 2*z * q^58 + 4*z * q^59 + (6*z - 6) * q^61 + q^64 + (12*z - 12) * q^65 - 4*z * q^67 + 2*z * q^68 + (2*z - 2) * q^70 - 8 * q^71 + 10 * q^73 + (10*z - 10) * q^74 + 4*z * q^76 + 4*z * q^77 + 2 * q^80 - 6 * q^82 + (4*z - 4) * q^83 + 4*z * q^85 - 4*z * q^86 + (-4*z + 4) * q^88 + 6 * q^89 - 6 * q^91 + (-8*z + 8) * q^92 + 8*z * q^95 + (-14*z + 14) * q^97 - q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} - 2 q^{5} + q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q + q^2 - q^4 - 2 * q^5 + q^7 - 2 * q^8 $$2 q + q^{2} - q^{4} - 2 q^{5} + q^{7} - 2 q^{8} - 4 q^{10} - 4 q^{11} - 6 q^{13} - q^{14} - q^{16} - 4 q^{17} - 8 q^{19} - 2 q^{20} + 4 q^{22} + 8 q^{23} + q^{25} - 12 q^{26} - 2 q^{28} - 2 q^{29} + q^{32} - 2 q^{34} - 4 q^{35} - 20 q^{37} - 4 q^{38} + 2 q^{40} - 6 q^{41} + 4 q^{43} + 8 q^{44} + 16 q^{46} - q^{49} - q^{50} - 6 q^{52} - 12 q^{53} + 16 q^{55} - q^{56} + 2 q^{58} + 4 q^{59} - 6 q^{61} + 2 q^{64} - 12 q^{65} - 4 q^{67} + 2 q^{68} - 2 q^{70} - 16 q^{71} + 20 q^{73} - 10 q^{74} + 4 q^{76} + 4 q^{77} + 4 q^{80} - 12 q^{82} - 4 q^{83} + 4 q^{85} - 4 q^{86} + 4 q^{88} + 12 q^{89} - 12 q^{91} + 8 q^{92} + 8 q^{95} + 14 q^{97} - 2 q^{98}+O(q^{100})$$ 2 * q + q^2 - q^4 - 2 * q^5 + q^7 - 2 * q^8 - 4 * q^10 - 4 * q^11 - 6 * q^13 - q^14 - q^16 - 4 * q^17 - 8 * q^19 - 2 * q^20 + 4 * q^22 + 8 * q^23 + q^25 - 12 * q^26 - 2 * q^28 - 2 * q^29 + q^32 - 2 * q^34 - 4 * q^35 - 20 * q^37 - 4 * q^38 + 2 * q^40 - 6 * q^41 + 4 * q^43 + 8 * q^44 + 16 * q^46 - q^49 - q^50 - 6 * q^52 - 12 * q^53 + 16 * q^55 - q^56 + 2 * q^58 + 4 * q^59 - 6 * q^61 + 2 * q^64 - 12 * q^65 - 4 * q^67 + 2 * q^68 - 2 * q^70 - 16 * q^71 + 20 * q^73 - 10 * q^74 + 4 * q^76 + 4 * q^77 + 4 * q^80 - 12 * q^82 - 4 * q^83 + 4 * q^85 - 4 * q^86 + 4 * q^88 + 12 * q^89 - 12 * q^91 + 8 * q^92 + 8 * q^95 + 14 * 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 −1.00000 1.73205i 0 0.500000 0.866025i −1.00000 0 −2.00000
757.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.00000 + 1.73205i 0 0.500000 + 0.866025i −1.00000 0 −2.00000
 $$n$$: 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.j 2
3.b odd 2 1 1134.2.f.g 2
9.c even 3 1 126.2.a.a 1
9.c even 3 1 inner 1134.2.f.j 2
9.d odd 6 1 42.2.a.a 1
9.d odd 6 1 1134.2.f.g 2
36.f odd 6 1 1008.2.a.j 1
36.h even 6 1 336.2.a.d 1
45.h odd 6 1 1050.2.a.i 1
45.j even 6 1 3150.2.a.bo 1
45.k odd 12 2 3150.2.g.r 2
45.l even 12 2 1050.2.g.a 2
63.g even 3 1 882.2.g.h 2
63.h even 3 1 882.2.g.h 2
63.i even 6 1 294.2.e.a 2
63.j odd 6 1 294.2.e.c 2
63.k odd 6 1 882.2.g.j 2
63.l odd 6 1 882.2.a.b 1
63.n odd 6 1 294.2.e.c 2
63.o even 6 1 294.2.a.g 1
63.s even 6 1 294.2.e.a 2
63.t odd 6 1 882.2.g.j 2
72.j odd 6 1 1344.2.a.q 1
72.l even 6 1 1344.2.a.i 1
72.n even 6 1 4032.2.a.e 1
72.p odd 6 1 4032.2.a.m 1
99.g even 6 1 5082.2.a.d 1
117.n odd 6 1 7098.2.a.f 1
144.u even 12 2 5376.2.c.e 2
144.w odd 12 2 5376.2.c.bc 2
180.n even 6 1 8400.2.a.k 1
252.o even 6 1 2352.2.q.i 2
252.r odd 6 1 2352.2.q.n 2
252.s odd 6 1 2352.2.a.l 1
252.bb even 6 1 2352.2.q.i 2
252.bi even 6 1 7056.2.a.k 1
252.bn odd 6 1 2352.2.q.n 2
315.z even 6 1 7350.2.a.f 1
504.cc even 6 1 9408.2.a.n 1
504.co odd 6 1 9408.2.a.bw 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.a.a 1 9.d odd 6 1
126.2.a.a 1 9.c even 3 1
294.2.a.g 1 63.o even 6 1
294.2.e.a 2 63.i even 6 1
294.2.e.a 2 63.s even 6 1
294.2.e.c 2 63.j odd 6 1
294.2.e.c 2 63.n odd 6 1
336.2.a.d 1 36.h even 6 1
882.2.a.b 1 63.l odd 6 1
882.2.g.h 2 63.g even 3 1
882.2.g.h 2 63.h even 3 1
882.2.g.j 2 63.k odd 6 1
882.2.g.j 2 63.t odd 6 1
1008.2.a.j 1 36.f odd 6 1
1050.2.a.i 1 45.h odd 6 1
1050.2.g.a 2 45.l even 12 2
1134.2.f.g 2 3.b odd 2 1
1134.2.f.g 2 9.d odd 6 1
1134.2.f.j 2 1.a even 1 1 trivial
1134.2.f.j 2 9.c even 3 1 inner
1344.2.a.i 1 72.l even 6 1
1344.2.a.q 1 72.j odd 6 1
2352.2.a.l 1 252.s odd 6 1
2352.2.q.i 2 252.o even 6 1
2352.2.q.i 2 252.bb even 6 1
2352.2.q.n 2 252.r odd 6 1
2352.2.q.n 2 252.bn odd 6 1
3150.2.a.bo 1 45.j even 6 1
3150.2.g.r 2 45.k odd 12 2
4032.2.a.e 1 72.n even 6 1
4032.2.a.m 1 72.p odd 6 1
5082.2.a.d 1 99.g even 6 1
5376.2.c.e 2 144.u even 12 2
5376.2.c.bc 2 144.w odd 12 2
7056.2.a.k 1 252.bi even 6 1
7098.2.a.f 1 117.n odd 6 1
7350.2.a.f 1 315.z even 6 1
8400.2.a.k 1 180.n even 6 1
9408.2.a.n 1 504.cc even 6 1
9408.2.a.bw 1 504.co odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} + 2T_{5} + 4$$ T5^2 + 2*T5 + 4 $$T_{11}^{2} + 4T_{11} + 16$$ T11^2 + 4*T11 + 16 $$T_{13}^{2} + 6T_{13} + 36$$ T13^2 + 6*T13 + 36 $$T_{17} + 2$$ T17 + 2

## Hecke characteristic polynomials

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