Properties

Label 162.4.c.d
Level 162162
Weight 44
Character orbit 162.c
Analytic conductor 9.5589.558
Analytic rank 00
Dimension 22
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.55");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 162=234 162 = 2 \cdot 3^{4}
Weight: k k == 4 4
Character orbit: [χ][\chi] == 162.c (of order 33, degree 22, 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: 9.558309420939.55830942093
Analytic rank: 00
Dimension: 22
Coefficient field: Q(3)\Q(\sqrt{-3})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2x+1 x^{2} - x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

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 primitive root of unity ζ6\zeta_{6}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(2ζ62)q24ζ6q4+21ζ6q5+(8ζ68)q7+8q842q10+(36ζ6+36)q11+49ζ6q1316ζ6q14+(16ζ616)q16+558q98+O(q100) q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + 21 \zeta_{6} q^{5} + (8 \zeta_{6} - 8) q^{7} + 8 q^{8} - 42 q^{10} + ( - 36 \zeta_{6} + 36) q^{11} + 49 \zeta_{6} q^{13} - 16 \zeta_{6} q^{14} + (16 \zeta_{6} - 16) q^{16} + \cdots - 558 q^{98} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q2q24q4+21q58q7+16q884q10+36q11+49q1316q1416q1642q17224q19+84q20+72q22+180q23316q25196q26+1116q98+O(q100) 2 q - 2 q^{2} - 4 q^{4} + 21 q^{5} - 8 q^{7} + 16 q^{8} - 84 q^{10} + 36 q^{11} + 49 q^{13} - 16 q^{14} - 16 q^{16} - 42 q^{17} - 224 q^{19} + 84 q^{20} + 72 q^{22} + 180 q^{23} - 316 q^{25} - 196 q^{26}+ \cdots - 1116 q^{98}+O(q^{100}) 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) ζ6-\zeta_{6}

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}
55.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.00000 + 1.73205i 0 −2.00000 3.46410i 10.5000 + 18.1865i 0 −4.00000 + 6.92820i 8.00000 0 −42.0000
109.1 −1.00000 1.73205i 0 −2.00000 + 3.46410i 10.5000 18.1865i 0 −4.00000 6.92820i 8.00000 0 −42.0000
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

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 162.4.c.d 2
3.b odd 2 1 162.4.c.e 2
9.c even 3 1 162.4.a.c yes 1
9.c even 3 1 inner 162.4.c.d 2
9.d odd 6 1 162.4.a.b 1
9.d odd 6 1 162.4.c.e 2
36.f odd 6 1 1296.4.a.a 1
36.h even 6 1 1296.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.4.a.b 1 9.d odd 6 1
162.4.a.c yes 1 9.c even 3 1
162.4.c.d 2 1.a even 1 1 trivial
162.4.c.d 2 9.c even 3 1 inner
162.4.c.e 2 3.b odd 2 1
162.4.c.e 2 9.d odd 6 1
1296.4.a.a 1 36.f odd 6 1
1296.4.a.h 1 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5221T5+441 T_{5}^{2} - 21T_{5} + 441 acting on S4new(162,[χ])S_{4}^{\mathrm{new}}(162, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2+2T+4 T^{2} + 2T + 4 Copy content Toggle raw display
33 T2 T^{2} Copy content Toggle raw display
55 T221T+441 T^{2} - 21T + 441 Copy content Toggle raw display
77 T2+8T+64 T^{2} + 8T + 64 Copy content Toggle raw display
1111 T236T+1296 T^{2} - 36T + 1296 Copy content Toggle raw display
1313 T249T+2401 T^{2} - 49T + 2401 Copy content Toggle raw display
1717 (T+21)2 (T + 21)^{2} Copy content Toggle raw display
1919 (T+112)2 (T + 112)^{2} Copy content Toggle raw display
2323 T2180T+32400 T^{2} - 180T + 32400 Copy content Toggle raw display
2929 T2+135T+18225 T^{2} + 135T + 18225 Copy content Toggle raw display
3131 T2+308T+94864 T^{2} + 308T + 94864 Copy content Toggle raw display
3737 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
4141 T2+42T+1764 T^{2} + 42T + 1764 Copy content Toggle raw display
4343 T2+20T+400 T^{2} + 20T + 400 Copy content Toggle raw display
4747 T284T+7056 T^{2} - 84T + 7056 Copy content Toggle raw display
5353 (T174)2 (T - 174)^{2} Copy content Toggle raw display
5959 T2504T+254016 T^{2} - 504T + 254016 Copy content Toggle raw display
6161 T2385T+148225 T^{2} - 385T + 148225 Copy content Toggle raw display
6767 T2+272T+73984 T^{2} + 272T + 73984 Copy content Toggle raw display
7171 (T888)2 (T - 888)^{2} Copy content Toggle raw display
7373 (T371)2 (T - 371)^{2} Copy content Toggle raw display
7979 T2652T+425104 T^{2} - 652T + 425104 Copy content Toggle raw display
8383 T284T+7056 T^{2} - 84T + 7056 Copy content Toggle raw display
8989 (T+21)2 (T + 21)^{2} Copy content Toggle raw display
9797 T21246T+1552516 T^{2} - 1246 T + 1552516 Copy content Toggle raw display
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