# Properties

 Label 1134.2.m.b Level $1134$ Weight $2$ Character orbit 1134.m Analytic conductor $9.055$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1134,2,Mod(377,1134)]

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([5, 3]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1134.377");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1134 = 2 \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1134.m (of order $$6$$, 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 $$\Q(\zeta_{6})$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 378) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{5} + (\zeta_{12}^{2} - 3) q^{7} + \zeta_{12}^{3} q^{8}+O(q^{10})$$ q + z * q^2 + z^2 * q^4 + (-z^3 - z) * q^5 + (z^2 - 3) * q^7 + z^3 * q^8 $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{5} + (\zeta_{12}^{2} - 3) q^{7} + \zeta_{12}^{3} q^{8} + ( - 2 \zeta_{12}^{2} + 1) q^{10} + 3 \zeta_{12} q^{11} + ( - 2 \zeta_{12}^{2} + 4) q^{13} + (\zeta_{12}^{3} - 3 \zeta_{12}) q^{14} + (\zeta_{12}^{2} - 1) q^{16} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}) q^{17} + (2 \zeta_{12}^{2} - 1) q^{19} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{20} + 3 \zeta_{12}^{2} q^{22} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{23} + ( - 2 \zeta_{12}^{2} + 2) q^{25} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{26} + ( - 2 \zeta_{12}^{2} - 1) q^{28} + 6 \zeta_{12} q^{29} + (3 \zeta_{12}^{2} - 6) q^{31} + (\zeta_{12}^{3} - \zeta_{12}) q^{32} + (4 \zeta_{12}^{2} + 4) q^{34} + (\zeta_{12}^{3} + 4 \zeta_{12}) q^{35} + 7 q^{37} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{38} + ( - \zeta_{12}^{2} + 2) q^{40} + (7 \zeta_{12}^{3} + 7 \zeta_{12}) q^{41} + ( - 2 \zeta_{12}^{2} + 2) q^{43} + 3 \zeta_{12}^{3} q^{44} - 3 q^{46} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}) q^{47} + ( - 5 \zeta_{12}^{2} + 8) q^{49} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{50} + (2 \zeta_{12}^{2} + 2) q^{52} + 12 \zeta_{12}^{3} q^{53} + ( - 6 \zeta_{12}^{2} + 3) q^{55} + ( - 2 \zeta_{12}^{3} - \zeta_{12}) q^{56} + 6 \zeta_{12}^{2} q^{58} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{59} + ( - 4 \zeta_{12}^{2} - 4) q^{61} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{62} - q^{64} - 6 \zeta_{12} q^{65} - 2 \zeta_{12}^{2} q^{67} + (4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{68} + (5 \zeta_{12}^{2} - 1) q^{70} + 3 \zeta_{12}^{3} q^{71} + ( - 4 \zeta_{12}^{2} + 2) q^{73} + 7 \zeta_{12} q^{74} + (\zeta_{12}^{2} - 2) q^{76} + (3 \zeta_{12}^{3} - 9 \zeta_{12}) q^{77} + ( - 10 \zeta_{12}^{2} + 10) q^{79} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{80} + (14 \zeta_{12}^{2} - 7) q^{82} + ( - 20 \zeta_{12}^{3} + 10 \zeta_{12}) q^{83} - 12 \zeta_{12}^{2} q^{85} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{86} + (3 \zeta_{12}^{2} - 3) q^{88} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{89} + (8 \zeta_{12}^{2} - 10) q^{91} - 3 \zeta_{12} q^{92} + ( - 2 \zeta_{12}^{2} + 4) q^{94} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{95} + ( - 8 \zeta_{12}^{2} - 8) q^{97} + ( - 5 \zeta_{12}^{3} + 8 \zeta_{12}) q^{98} +O(q^{100})$$ q + z * q^2 + z^2 * q^4 + (-z^3 - z) * q^5 + (z^2 - 3) * q^7 + z^3 * q^8 + (-2*z^2 + 1) * q^10 + 3*z * q^11 + (-2*z^2 + 4) * q^13 + (z^3 - 3*z) * q^14 + (z^2 - 1) * q^16 + (-4*z^3 + 8*z) * q^17 + (2*z^2 - 1) * q^19 + (-2*z^3 + z) * q^20 + 3*z^2 * q^22 + (3*z^3 - 3*z) * q^23 + (-2*z^2 + 2) * q^25 + (-2*z^3 + 4*z) * q^26 + (-2*z^2 - 1) * q^28 + 6*z * q^29 + (3*z^2 - 6) * q^31 + (z^3 - z) * q^32 + (4*z^2 + 4) * q^34 + (z^3 + 4*z) * q^35 + 7 * q^37 + (2*z^3 - z) * q^38 + (-z^2 + 2) * q^40 + (7*z^3 + 7*z) * q^41 + (-2*z^2 + 2) * q^43 + 3*z^3 * q^44 - 3 * q^46 + (-4*z^3 + 2*z) * q^47 + (-5*z^2 + 8) * q^49 + (-2*z^3 + 2*z) * q^50 + (2*z^2 + 2) * q^52 + 12*z^3 * q^53 + (-6*z^2 + 3) * q^55 + (-2*z^3 - z) * q^56 + 6*z^2 * q^58 + (-2*z^3 - 2*z) * q^59 + (-4*z^2 - 4) * q^61 + (3*z^3 - 6*z) * q^62 - q^64 - 6*z * q^65 - 2*z^2 * q^67 + (4*z^3 + 4*z) * q^68 + (5*z^2 - 1) * q^70 + 3*z^3 * q^71 + (-4*z^2 + 2) * q^73 + 7*z * q^74 + (z^2 - 2) * q^76 + (3*z^3 - 9*z) * q^77 + (-10*z^2 + 10) * q^79 + (-z^3 + 2*z) * q^80 + (14*z^2 - 7) * q^82 + (-20*z^3 + 10*z) * q^83 - 12*z^2 * q^85 + (-2*z^3 + 2*z) * q^86 + (3*z^2 - 3) * q^88 + (3*z^3 - 6*z) * q^89 + (8*z^2 - 10) * q^91 - 3*z * q^92 + (-2*z^2 + 4) * q^94 + (-3*z^3 + 3*z) * q^95 + (-8*z^2 - 8) * q^97 + (-5*z^3 + 8*z) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} - 10 q^{7}+O(q^{10})$$ 4 * q + 2 * q^4 - 10 * q^7 $$4 q + 2 q^{4} - 10 q^{7} + 12 q^{13} - 2 q^{16} + 6 q^{22} + 4 q^{25} - 8 q^{28} - 18 q^{31} + 24 q^{34} + 28 q^{37} + 6 q^{40} + 4 q^{43} - 12 q^{46} + 22 q^{49} + 12 q^{52} + 12 q^{58} - 24 q^{61} - 4 q^{64} - 4 q^{67} + 6 q^{70} - 6 q^{76} + 20 q^{79} - 24 q^{85} - 6 q^{88} - 24 q^{91} + 12 q^{94} - 48 q^{97}+O(q^{100})$$ 4 * q + 2 * q^4 - 10 * q^7 + 12 * q^13 - 2 * q^16 + 6 * q^22 + 4 * q^25 - 8 * q^28 - 18 * q^31 + 24 * q^34 + 28 * q^37 + 6 * q^40 + 4 * q^43 - 12 * q^46 + 22 * q^49 + 12 * q^52 + 12 * q^58 - 24 * q^61 - 4 * q^64 - 4 * q^67 + 6 * q^70 - 6 * q^76 + 20 * q^79 - 24 * q^85 - 6 * q^88 - 24 * q^91 + 12 * q^94 - 48 * q^97

## Character values

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

 $$n$$ $$325$$ $$407$$ $$\chi(n)$$ $$-1$$ $$\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
377.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i 0.866025 1.50000i 0 −2.50000 0.866025i 1.00000i 0 1.73205i
377.2 0.866025 0.500000i 0 0.500000 0.866025i −0.866025 + 1.50000i 0 −2.50000 0.866025i 1.00000i 0 1.73205i
755.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.866025 + 1.50000i 0 −2.50000 + 0.866025i 1.00000i 0 1.73205i
755.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.866025 1.50000i 0 −2.50000 + 0.866025i 1.00000i 0 1.73205i
 $$n$$: 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
3.b odd 2 1 inner
63.l odd 6 1 inner
63.o even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.2.m.b 4
3.b odd 2 1 inner 1134.2.m.b 4
7.b odd 2 1 1134.2.m.c 4
9.c even 3 1 378.2.d.b 4
9.c even 3 1 1134.2.m.c 4
9.d odd 6 1 378.2.d.b 4
9.d odd 6 1 1134.2.m.c 4
21.c even 2 1 1134.2.m.c 4
36.f odd 6 1 3024.2.k.e 4
36.h even 6 1 3024.2.k.e 4
63.l odd 6 1 378.2.d.b 4
63.l odd 6 1 inner 1134.2.m.b 4
63.o even 6 1 378.2.d.b 4
63.o even 6 1 inner 1134.2.m.b 4
252.s odd 6 1 3024.2.k.e 4
252.bi even 6 1 3024.2.k.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.d.b 4 9.c even 3 1
378.2.d.b 4 9.d odd 6 1
378.2.d.b 4 63.l odd 6 1
378.2.d.b 4 63.o even 6 1
1134.2.m.b 4 1.a even 1 1 trivial
1134.2.m.b 4 3.b odd 2 1 inner
1134.2.m.b 4 63.l odd 6 1 inner
1134.2.m.b 4 63.o even 6 1 inner
1134.2.m.c 4 7.b odd 2 1
1134.2.m.c 4 9.c even 3 1
1134.2.m.c 4 9.d odd 6 1
1134.2.m.c 4 21.c even 2 1
3024.2.k.e 4 36.f odd 6 1
3024.2.k.e 4 36.h even 6 1
3024.2.k.e 4 252.s odd 6 1
3024.2.k.e 4 252.bi even 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 3T_{5}^{2} + 9$$ T5^4 + 3*T5^2 + 9 $$T_{11}^{4} - 9T_{11}^{2} + 81$$ T11^4 - 9*T11^2 + 81 $$T_{13}^{2} - 6T_{13} + 12$$ T13^2 - 6*T13 + 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 3T^{2} + 9$$
$7$ $$(T^{2} + 5 T + 7)^{2}$$
$11$ $$T^{4} - 9T^{2} + 81$$
$13$ $$(T^{2} - 6 T + 12)^{2}$$
$17$ $$(T^{2} - 48)^{2}$$
$19$ $$(T^{2} + 3)^{2}$$
$23$ $$T^{4} - 9T^{2} + 81$$
$29$ $$T^{4} - 36T^{2} + 1296$$
$31$ $$(T^{2} + 9 T + 27)^{2}$$
$37$ $$(T - 7)^{4}$$
$41$ $$T^{4} + 147 T^{2} + 21609$$
$43$ $$(T^{2} - 2 T + 4)^{2}$$
$47$ $$T^{4} + 12T^{2} + 144$$
$53$ $$(T^{2} + 144)^{2}$$
$59$ $$T^{4} + 12T^{2} + 144$$
$61$ $$(T^{2} + 12 T + 48)^{2}$$
$67$ $$(T^{2} + 2 T + 4)^{2}$$
$71$ $$(T^{2} + 9)^{2}$$
$73$ $$(T^{2} + 12)^{2}$$
$79$ $$(T^{2} - 10 T + 100)^{2}$$
$83$ $$T^{4} + 300 T^{2} + 90000$$
$89$ $$(T^{2} - 27)^{2}$$
$97$ $$(T^{2} + 24 T + 192)^{2}$$