# Properties

 Label 1134.2.f.f 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 14) 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 $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + (\zeta_{6} - 1) q^{7} + q^{8} + 4 \zeta_{6} q^{13} - \zeta_{6} q^{14} + (\zeta_{6} - 1) q^{16} - 6 q^{17} + 2 q^{19} + ( - 5 \zeta_{6} + 5) q^{25} - 4 q^{26} + q^{28} + (6 \zeta_{6} - 6) q^{29} + 4 \zeta_{6} q^{31} - \zeta_{6} q^{32} + ( - 6 \zeta_{6} + 6) q^{34} + 2 q^{37} + (2 \zeta_{6} - 2) q^{38} + 6 \zeta_{6} q^{41} + (8 \zeta_{6} - 8) q^{43} + (12 \zeta_{6} - 12) q^{47} - \zeta_{6} q^{49} + 5 \zeta_{6} q^{50} + ( - 4 \zeta_{6} + 4) q^{52} - 6 q^{53} + (\zeta_{6} - 1) q^{56} - 6 \zeta_{6} q^{58} - 6 \zeta_{6} q^{59} + (8 \zeta_{6} - 8) q^{61} - 4 q^{62} + q^{64} + 4 \zeta_{6} q^{67} + 6 \zeta_{6} q^{68} + 2 q^{73} + (2 \zeta_{6} - 2) q^{74} - 2 \zeta_{6} q^{76} + (8 \zeta_{6} - 8) q^{79} - 6 q^{82} + (6 \zeta_{6} - 6) q^{83} - 8 \zeta_{6} q^{86} + 6 q^{89} - 4 q^{91} - 12 \zeta_{6} q^{94} + ( - 10 \zeta_{6} + 10) q^{97} + q^{98} +O(q^{100})$$ q + (z - 1) * q^2 - z * q^4 + (z - 1) * q^7 + q^8 + 4*z * q^13 - z * q^14 + (z - 1) * q^16 - 6 * q^17 + 2 * q^19 + (-5*z + 5) * q^25 - 4 * q^26 + q^28 + (6*z - 6) * q^29 + 4*z * q^31 - z * q^32 + (-6*z + 6) * q^34 + 2 * q^37 + (2*z - 2) * q^38 + 6*z * q^41 + (8*z - 8) * q^43 + (12*z - 12) * q^47 - z * q^49 + 5*z * q^50 + (-4*z + 4) * q^52 - 6 * q^53 + (z - 1) * q^56 - 6*z * q^58 - 6*z * q^59 + (8*z - 8) * q^61 - 4 * q^62 + q^64 + 4*z * q^67 + 6*z * q^68 + 2 * q^73 + (2*z - 2) * q^74 - 2*z * q^76 + (8*z - 8) * q^79 - 6 * q^82 + (6*z - 6) * q^83 - 8*z * q^86 + 6 * q^89 - 4 * q^91 - 12*z * q^94 + (-10*z + 10) * 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 $$2 q - q^{2} - q^{4} - q^{7} + 2 q^{8} + 4 q^{13} - q^{14} - q^{16} - 12 q^{17} + 4 q^{19} + 5 q^{25} - 8 q^{26} + 2 q^{28} - 6 q^{29} + 4 q^{31} - q^{32} + 6 q^{34} + 4 q^{37} - 2 q^{38} + 6 q^{41} - 8 q^{43} - 12 q^{47} - q^{49} + 5 q^{50} + 4 q^{52} - 12 q^{53} - q^{56} - 6 q^{58} - 6 q^{59} - 8 q^{61} - 8 q^{62} + 2 q^{64} + 4 q^{67} + 6 q^{68} + 4 q^{73} - 2 q^{74} - 2 q^{76} - 8 q^{79} - 12 q^{82} - 6 q^{83} - 8 q^{86} + 12 q^{89} - 8 q^{91} - 12 q^{94} + 10 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q - q^2 - q^4 - q^7 + 2 * q^8 + 4 * q^13 - q^14 - q^16 - 12 * q^17 + 4 * q^19 + 5 * q^25 - 8 * q^26 + 2 * q^28 - 6 * q^29 + 4 * q^31 - q^32 + 6 * q^34 + 4 * q^37 - 2 * q^38 + 6 * q^41 - 8 * q^43 - 12 * q^47 - q^49 + 5 * q^50 + 4 * q^52 - 12 * q^53 - q^56 - 6 * q^58 - 6 * q^59 - 8 * q^61 - 8 * q^62 + 2 * q^64 + 4 * q^67 + 6 * q^68 + 4 * q^73 - 2 * q^74 - 2 * q^76 - 8 * q^79 - 12 * q^82 - 6 * q^83 - 8 * q^86 + 12 * q^89 - 8 * q^91 - 12 * q^94 + 10 * 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.f 2
3.b odd 2 1 1134.2.f.l 2
9.c even 3 1 126.2.a.b 1
9.c even 3 1 inner 1134.2.f.f 2
9.d odd 6 1 14.2.a.a 1
9.d odd 6 1 1134.2.f.l 2
36.f odd 6 1 1008.2.a.h 1
36.h even 6 1 112.2.a.c 1
45.h odd 6 1 350.2.a.f 1
45.j even 6 1 3150.2.a.i 1
45.k odd 12 2 3150.2.g.j 2
45.l even 12 2 350.2.c.d 2
63.g even 3 1 882.2.g.c 2
63.h even 3 1 882.2.g.c 2
63.i even 6 1 98.2.c.a 2
63.j odd 6 1 98.2.c.b 2
63.k odd 6 1 882.2.g.d 2
63.l odd 6 1 882.2.a.i 1
63.n odd 6 1 98.2.c.b 2
63.o even 6 1 98.2.a.a 1
63.s even 6 1 98.2.c.a 2
63.t odd 6 1 882.2.g.d 2
72.j odd 6 1 448.2.a.g 1
72.l even 6 1 448.2.a.a 1
72.n even 6 1 4032.2.a.w 1
72.p odd 6 1 4032.2.a.r 1
99.g even 6 1 1694.2.a.e 1
117.n odd 6 1 2366.2.a.j 1
117.z even 12 2 2366.2.d.b 2
144.u even 12 2 1792.2.b.g 2
144.w odd 12 2 1792.2.b.c 2
153.i odd 6 1 4046.2.a.f 1
171.l even 6 1 5054.2.a.c 1
180.n even 6 1 2800.2.a.g 1
180.v odd 12 2 2800.2.g.h 2
207.g even 6 1 7406.2.a.a 1
252.o even 6 1 784.2.i.c 2
252.r odd 6 1 784.2.i.i 2
252.s odd 6 1 784.2.a.b 1
252.bb even 6 1 784.2.i.c 2
252.bi even 6 1 7056.2.a.bd 1
252.bn odd 6 1 784.2.i.i 2
315.z even 6 1 2450.2.a.t 1
315.cf odd 12 2 2450.2.c.c 2
504.cc even 6 1 3136.2.a.e 1
504.co odd 6 1 3136.2.a.z 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.2.a.a 1 9.d odd 6 1
98.2.a.a 1 63.o even 6 1
98.2.c.a 2 63.i even 6 1
98.2.c.a 2 63.s even 6 1
98.2.c.b 2 63.j odd 6 1
98.2.c.b 2 63.n odd 6 1
112.2.a.c 1 36.h even 6 1
126.2.a.b 1 9.c even 3 1
350.2.a.f 1 45.h odd 6 1
350.2.c.d 2 45.l even 12 2
448.2.a.a 1 72.l even 6 1
448.2.a.g 1 72.j odd 6 1
784.2.a.b 1 252.s odd 6 1
784.2.i.c 2 252.o even 6 1
784.2.i.c 2 252.bb even 6 1
784.2.i.i 2 252.r odd 6 1
784.2.i.i 2 252.bn odd 6 1
882.2.a.i 1 63.l odd 6 1
882.2.g.c 2 63.g even 3 1
882.2.g.c 2 63.h even 3 1
882.2.g.d 2 63.k odd 6 1
882.2.g.d 2 63.t odd 6 1
1008.2.a.h 1 36.f odd 6 1
1134.2.f.f 2 1.a even 1 1 trivial
1134.2.f.f 2 9.c even 3 1 inner
1134.2.f.l 2 3.b odd 2 1
1134.2.f.l 2 9.d odd 6 1
1694.2.a.e 1 99.g even 6 1
1792.2.b.c 2 144.w odd 12 2
1792.2.b.g 2 144.u even 12 2
2366.2.a.j 1 117.n odd 6 1
2366.2.d.b 2 117.z even 12 2
2450.2.a.t 1 315.z even 6 1
2450.2.c.c 2 315.cf odd 12 2
2800.2.a.g 1 180.n even 6 1
2800.2.g.h 2 180.v odd 12 2
3136.2.a.e 1 504.cc even 6 1
3136.2.a.z 1 504.co odd 6 1
3150.2.a.i 1 45.j even 6 1
3150.2.g.j 2 45.k odd 12 2
4032.2.a.r 1 72.p odd 6 1
4032.2.a.w 1 72.n even 6 1
4046.2.a.f 1 153.i odd 6 1
5054.2.a.c 1 171.l even 6 1
7056.2.a.bd 1 252.bi even 6 1
7406.2.a.a 1 207.g 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 $$T_{13}^{2} - 4T_{13} + 16$$ T13^2 - 4*T13 + 16 $$T_{17} + 6$$ T17 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + T + 1$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 4T + 16$$
$17$ $$(T + 6)^{2}$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2} - 6T + 36$$
$43$ $$T^{2} + 8T + 64$$
$47$ $$T^{2} + 12T + 144$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} + 6T + 36$$
$61$ $$T^{2} + 8T + 64$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$T^{2}$$
$73$ $$(T - 2)^{2}$$
$79$ $$T^{2} + 8T + 64$$
$83$ $$T^{2} + 6T + 36$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} - 10T + 100$$