Properties

Label 162.3.b.b
Level 162162
Weight 33
Character orbit 162.b
Analytic conductor 4.4144.414
Analytic rank 00
Dimension 44
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,3,Mod(161,162)]
 
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([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: N N == 162=234 162 = 2 \cdot 3^{4}
Weight: k k == 3 3
Character orbit: [χ][\chi] == 162.b (of order 22, degree 11, 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: 4.414180282644.41418028264
Analytic rank: 00
Dimension: 44
Coefficient field: Q(2,3)\Q(\sqrt{-2}, \sqrt{3})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4+4x2+1 x^{4} + 4x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 32 3^{2}
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,β2,β31,\beta_1,\beta_2,\beta_3 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == qβ1q22q4+(β2+β1)q5+(2β3+2)q7+2β1q8+(β3+3)q10+(4β22β1)q11+(β316)q13+(4β24β1)q14++(16β271β1)q98+O(q100) q - \beta_1 q^{2} - 2 q^{4} + ( - \beta_{2} + \beta_1) q^{5} + (2 \beta_{3} + 2) q^{7} + 2 \beta_1 q^{8} + ( - \beta_{3} + 3) q^{10} + ( - 4 \beta_{2} - 2 \beta_1) q^{11} + (\beta_{3} - 16) q^{13} + ( - 4 \beta_{2} - 4 \beta_1) q^{14}+ \cdots + ( - 16 \beta_{2} - 71 \beta_1) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q8q4+8q7+12q1064q13+16q16+56q19+28q2516q28+32q3160q3476q3724q4088q43+252q49+128q52216q55+180q58++32q97+O(q100) 4 q - 8 q^{4} + 8 q^{7} + 12 q^{10} - 64 q^{13} + 16 q^{16} + 56 q^{19} + 28 q^{25} - 16 q^{28} + 32 q^{31} - 60 q^{34} - 76 q^{37} - 24 q^{40} - 88 q^{43} + 252 q^{49} + 128 q^{52} - 216 q^{55} + 180 q^{58}+ \cdots + 32 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x4+4x2+1 x^{4} + 4x^{2} + 1 : Copy content Toggle raw display

β1\beta_{1}== ν3+3ν \nu^{3} + 3\nu Copy content Toggle raw display
β2\beta_{2}== ν3+6ν \nu^{3} + 6\nu Copy content Toggle raw display
β3\beta_{3}== 3ν2+6 3\nu^{2} + 6 Copy content Toggle raw display
ν\nu== (β2β1)/3 ( \beta_{2} - \beta_1 ) / 3 Copy content Toggle raw display
ν2\nu^{2}== (β36)/3 ( \beta_{3} - 6 ) / 3 Copy content Toggle raw display
ν3\nu^{3}== β2+2β1 -\beta_{2} + 2\beta_1 Copy content Toggle raw display

Character values

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

nn 8383
χ(n)\chi(n) 1-1

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
161.1
0.517638i
1.93185i
1.93185i
0.517638i
1.41421i 0 −2.00000 1.55291i 0 12.3923 2.82843i 0 −2.19615
161.2 1.41421i 0 −2.00000 5.79555i 0 −8.39230 2.82843i 0 8.19615
161.3 1.41421i 0 −2.00000 5.79555i 0 −8.39230 2.82843i 0 8.19615
161.4 1.41421i 0 −2.00000 1.55291i 0 12.3923 2.82843i 0 −2.19615
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.3.b.b 4
3.b odd 2 1 inner 162.3.b.b 4
4.b odd 2 1 1296.3.e.d 4
9.c even 3 2 162.3.d.c 8
9.d odd 6 2 162.3.d.c 8
12.b even 2 1 1296.3.e.d 4
36.f odd 6 2 1296.3.q.o 8
36.h even 6 2 1296.3.q.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.3.b.b 4 1.a even 1 1 trivial
162.3.b.b 4 3.b odd 2 1 inner
162.3.d.c 8 9.c even 3 2
162.3.d.c 8 9.d odd 6 2
1296.3.e.d 4 4.b odd 2 1
1296.3.e.d 4 12.b even 2 1
1296.3.q.o 8 36.f odd 6 2
1296.3.q.o 8 36.h even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T54+36T52+81 T_{5}^{4} + 36T_{5}^{2} + 81 acting on S3new(162,[χ])S_{3}^{\mathrm{new}}(162, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T2+2)2 (T^{2} + 2)^{2} Copy content Toggle raw display
33 T4 T^{4} Copy content Toggle raw display
55 T4+36T2+81 T^{4} + 36T^{2} + 81 Copy content Toggle raw display
77 (T24T104)2 (T^{2} - 4 T - 104)^{2} Copy content Toggle raw display
1111 (T2+216)2 (T^{2} + 216)^{2} Copy content Toggle raw display
1313 (T2+32T+229)2 (T^{2} + 32 T + 229)^{2} Copy content Toggle raw display
1717 T4+900T2+50625 T^{4} + 900 T^{2} + 50625 Copy content Toggle raw display
1919 (T228T+88)2 (T^{2} - 28 T + 88)^{2} Copy content Toggle raw display
2323 (T2+216)2 (T^{2} + 216)^{2} Copy content Toggle raw display
2929 T4+2052T2+998001 T^{4} + 2052 T^{2} + 998001 Copy content Toggle raw display
3131 (T8)4 (T - 8)^{4} Copy content Toggle raw display
3737 (T2+38T1367)2 (T^{2} + 38 T - 1367)^{2} Copy content Toggle raw display
4141 T4+1764T2+715716 T^{4} + 1764 T^{2} + 715716 Copy content Toggle raw display
4343 (T2+44T488)2 (T^{2} + 44 T - 488)^{2} Copy content Toggle raw display
4747 (T2+288)2 (T^{2} + 288)^{2} Copy content Toggle raw display
5353 T4+7812T2+4743684 T^{4} + 7812 T^{2} + 4743684 Copy content Toggle raw display
5959 T4+12096T2+31539456 T^{4} + 12096 T^{2} + 31539456 Copy content Toggle raw display
6161 (T+13)4 (T + 13)^{4} Copy content Toggle raw display
6767 (T2+20T872)2 (T^{2} + 20 T - 872)^{2} Copy content Toggle raw display
7171 T4+10512T2+2742336 T^{4} + 10512 T^{2} + 2742336 Copy content Toggle raw display
7373 (T2112T+2893)2 (T^{2} - 112 T + 2893)^{2} Copy content Toggle raw display
7979 (T252T4616)2 (T^{2} - 52 T - 4616)^{2} Copy content Toggle raw display
8383 T4+10944T2+2509056 T^{4} + 10944 T^{2} + 2509056 Copy content Toggle raw display
8989 T4+24228T2+112211649 T^{4} + 24228 T^{2} + 112211649 Copy content Toggle raw display
9797 (T216T6848)2 (T^{2} - 16 T - 6848)^{2} Copy content Toggle raw display
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