# Properties

 Label 162.2.c.a Level $162$ Weight $2$ Character orbit 162.c Analytic conductor $1.294$ 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] = [162,2,Mod(55,162)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

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

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

chi := DirichletCharacter("162.55");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 162.c (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: $$1.29357651274$$ 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: 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$$.

 $$n$$ $$83$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −1.50000 2.59808i 0 2.00000 3.46410i 1.00000 0 3.00000
109.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −1.50000 + 2.59808i 0 2.00000 + 3.46410i 1.00000 0 3.00000
 $$n$$: 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 162.2.c.a 2
3.b odd 2 1 162.2.c.d 2
4.b odd 2 1 1296.2.i.b 2
9.c even 3 1 162.2.a.d yes 1
9.c even 3 1 inner 162.2.c.a 2
9.d odd 6 1 162.2.a.a 1
9.d odd 6 1 162.2.c.d 2
12.b even 2 1 1296.2.i.n 2
36.f odd 6 1 1296.2.a.l 1
36.f odd 6 1 1296.2.i.b 2
36.h even 6 1 1296.2.a.c 1
36.h even 6 1 1296.2.i.n 2
45.h odd 6 1 4050.2.a.bh 1
45.j even 6 1 4050.2.a.r 1
45.k odd 12 2 4050.2.c.n 2
45.l even 12 2 4050.2.c.g 2
63.l odd 6 1 7938.2.a.s 1
63.o even 6 1 7938.2.a.n 1
72.j odd 6 1 5184.2.a.y 1
72.l even 6 1 5184.2.a.bd 1
72.n even 6 1 5184.2.a.c 1
72.p odd 6 1 5184.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.2.a.a 1 9.d odd 6 1
162.2.a.d yes 1 9.c even 3 1
162.2.c.a 2 1.a even 1 1 trivial
162.2.c.a 2 9.c even 3 1 inner
162.2.c.d 2 3.b odd 2 1
162.2.c.d 2 9.d odd 6 1
1296.2.a.c 1 36.h even 6 1
1296.2.a.l 1 36.f odd 6 1
1296.2.i.b 2 4.b odd 2 1
1296.2.i.b 2 36.f odd 6 1
1296.2.i.n 2 12.b even 2 1
1296.2.i.n 2 36.h even 6 1
4050.2.a.r 1 45.j even 6 1
4050.2.a.bh 1 45.h odd 6 1
4050.2.c.g 2 45.l even 12 2
4050.2.c.n 2 45.k odd 12 2
5184.2.a.c 1 72.n even 6 1
5184.2.a.h 1 72.p odd 6 1
5184.2.a.y 1 72.j odd 6 1
5184.2.a.bd 1 72.l even 6 1
7938.2.a.n 1 63.o even 6 1
7938.2.a.s 1 63.l odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(162, [\chi])$$:

 $$T_{5}^{2} + 3T_{5} + 9$$ T5^2 + 3*T5 + 9 $$T_{7}^{2} - 4T_{7} + 16$$ T7^2 - 4*T7 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 3T + 9$$
$7$ $$T^{2} - 4T + 16$$
$11$ $$T^{2}$$
$13$ $$T^{2} - T + 1$$
$17$ $$(T - 3)^{2}$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} - 9T + 81$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$(T + 1)^{2}$$
$41$ $$T^{2} - 6T + 36$$
$43$ $$T^{2} + 8T + 64$$
$47$ $$T^{2} + 12T + 144$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$(T - 12)^{2}$$
$73$ $$(T - 11)^{2}$$
$79$ $$T^{2} - 16T + 256$$
$83$ $$T^{2} + 12T + 144$$
$89$ $$(T - 3)^{2}$$
$97$ $$T^{2} + 2T + 4$$