# Properties

 Label 162.2.a.d Level $162$ Weight $2$ Character orbit 162.a Self dual yes Analytic conductor $1.294$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,2,Mod(1,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

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

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

chi := DirichletCharacter("162.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 162.a (trivial)

## 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: yes Analytic conductor: $$1.29357651274$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 0 1.00000 3.00000 0 −4.00000 1.00000 0 3.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.2.a.d yes 1
3.b odd 2 1 162.2.a.a 1
4.b odd 2 1 1296.2.a.l 1
5.b even 2 1 4050.2.a.r 1
5.c odd 4 2 4050.2.c.n 2
7.b odd 2 1 7938.2.a.s 1
8.b even 2 1 5184.2.a.c 1
8.d odd 2 1 5184.2.a.h 1
9.c even 3 2 162.2.c.a 2
9.d odd 6 2 162.2.c.d 2
12.b even 2 1 1296.2.a.c 1
15.d odd 2 1 4050.2.a.bh 1
15.e even 4 2 4050.2.c.g 2
21.c even 2 1 7938.2.a.n 1
24.f even 2 1 5184.2.a.bd 1
24.h odd 2 1 5184.2.a.y 1
36.f odd 6 2 1296.2.i.b 2
36.h even 6 2 1296.2.i.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.2.a.a 1 3.b odd 2 1
162.2.a.d yes 1 1.a even 1 1 trivial
162.2.c.a 2 9.c even 3 2
162.2.c.d 2 9.d odd 6 2
1296.2.a.c 1 12.b even 2 1
1296.2.a.l 1 4.b odd 2 1
1296.2.i.b 2 36.f odd 6 2
1296.2.i.n 2 36.h even 6 2
4050.2.a.r 1 5.b even 2 1
4050.2.a.bh 1 15.d odd 2 1
4050.2.c.g 2 15.e even 4 2
4050.2.c.n 2 5.c odd 4 2
5184.2.a.c 1 8.b even 2 1
5184.2.a.h 1 8.d odd 2 1
5184.2.a.y 1 24.h odd 2 1
5184.2.a.bd 1 24.f even 2 1
7938.2.a.n 1 21.c even 2 1
7938.2.a.s 1 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(162))$$:

 $$T_{5} - 3$$ T5 - 3 $$T_{11}$$ T11

## Hecke characteristic polynomials

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