Properties

Label 162.10.c.k
Level 162162
Weight 1010
Character orbit 162.c
Analytic conductor 83.43683.436
Analytic rank 00
Dimension 44
Inner twists 22

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [162,10,Mod(55,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 10, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("162.55"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Level: N N == 162=234 162 = 2 \cdot 3^{4}
Weight: k k == 10 10
Character orbit: [χ][\chi] == 162.c (of order 33, degree 22, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-32,0,-512,-1704] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 83.435805458583.4358054585
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q(3,301)\Q(\sqrt{-3}, \sqrt{301})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x4x3+76x2+75x+5625 x^{4} - x^{3} + 76x^{2} + 75x + 5625 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 2636 2^{6}\cdot 3^{6}
Twist minimal: no (minimal twist has level 54)
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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) == q+(16β116)q2256β1q4+(β2852β1)q5+(4β3+4β2+3269)q7+4096q8+(16β3+13632)q10+(23β323β2+7284)q11++(418432β3+424105248)q98+O(q100) q + (16 \beta_1 - 16) q^{2} - 256 \beta_1 q^{4} + ( - \beta_{2} - 852 \beta_1) q^{5} + (4 \beta_{3} + 4 \beta_{2} + \cdots - 3269) q^{7} + 4096 q^{8} + ( - 16 \beta_{3} + 13632) q^{10} + ( - 23 \beta_{3} - 23 \beta_{2} + \cdots - 7284) q^{11}+ \cdots + ( - 418432 \beta_{3} + 424105248) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q32q2512q41704q56538q7+16384q8+54528q1014568q1137138q13104608q14131072q16175824q17+1418300q19436224q20233088q22++1696420992q98+O(q100) 4 q - 32 q^{2} - 512 q^{4} - 1704 q^{5} - 6538 q^{7} + 16384 q^{8} + 54528 q^{10} - 14568 q^{11} - 37138 q^{13} - 104608 q^{14} - 131072 q^{16} - 175824 q^{17} + 1418300 q^{19} - 436224 q^{20} - 233088 q^{22}+ \cdots + 1696420992 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x4x3+76x2+75x+5625 x^{4} - x^{3} + 76x^{2} + 75x + 5625 : Copy content Toggle raw display

β1\beta_{1}== (ν3+76ν276ν+5625)/5700 ( -\nu^{3} + 76\nu^{2} - 76\nu + 5625 ) / 5700 Copy content Toggle raw display
β2\beta_{2}== (9ν3684ν2+103284ν50625)/475 ( 9\nu^{3} - 684\nu^{2} + 103284\nu - 50625 ) / 475 Copy content Toggle raw display
β3\beta_{3}== (54ν3+6102)/19 ( 54\nu^{3} + 6102 ) / 19 Copy content Toggle raw display
ν\nu== (β2+108β1)/216 ( \beta_{2} + 108\beta_1 ) / 216 Copy content Toggle raw display
ν2\nu^{2}== (β3+β2+16308β116308)/216 ( \beta_{3} + \beta_{2} + 16308\beta _1 - 16308 ) / 216 Copy content Toggle raw display
ν3\nu^{3}== (19β36102)/54 ( 19\beta_{3} - 6102 ) / 54 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) β1-\beta_{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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
4.58734 + 7.94550i
−4.08734 7.07948i
4.58734 7.94550i
−4.08734 + 7.07948i
−8.00000 + 13.8564i 0 −128.000 221.703i −1362.86 2360.55i 0 −5381.96 + 9321.83i 4096.00 0 43611.7
55.2 −8.00000 + 13.8564i 0 −128.000 221.703i 510.865 + 884.844i 0 2112.96 3659.75i 4096.00 0 −16347.7
109.1 −8.00000 13.8564i 0 −128.000 + 221.703i −1362.86 + 2360.55i 0 −5381.96 9321.83i 4096.00 0 43611.7
109.2 −8.00000 13.8564i 0 −128.000 + 221.703i 510.865 884.844i 0 2112.96 + 3659.75i 4096.00 0 −16347.7
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 162.10.c.k 4
3.b odd 2 1 162.10.c.p 4
9.c even 3 1 54.10.a.h yes 2
9.c even 3 1 inner 162.10.c.k 4
9.d odd 6 1 54.10.a.e 2
9.d odd 6 1 162.10.c.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.10.a.e 2 9.d odd 6 1
54.10.a.h yes 2 9.c even 3 1
162.10.c.k 4 1.a even 1 1 trivial
162.10.c.k 4 9.c even 3 1 inner
162.10.c.p 4 3.b odd 2 1
162.10.c.p 4 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T54+1704T53+5688576T524745571840T5+7756002201600 T_{5}^{4} + 1704T_{5}^{3} + 5688576T_{5}^{2} - 4745571840T_{5} + 7756002201600 acting on S10new(162,[χ])S_{10}^{\mathrm{new}}(162, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T2+16T+256)2 (T^{2} + 16 T + 256)^{2} Copy content Toggle raw display
33 T4 T^{4} Copy content Toggle raw display
55 T4++7756002201600 T^{4} + \cdots + 7756002201600 Copy content Toggle raw display
77 T4++20 ⁣ ⁣69 T^{4} + \cdots + 20\!\cdots\!69 Copy content Toggle raw display
1111 T4++32 ⁣ ⁣00 T^{4} + \cdots + 32\!\cdots\!00 Copy content Toggle raw display
1313 T4++83 ⁣ ⁣25 T^{4} + \cdots + 83\!\cdots\!25 Copy content Toggle raw display
1717 (T2+87912T49470429888)2 (T^{2} + 87912 T - 49470429888)^{2} Copy content Toggle raw display
1919 (T2709150T+81683152609)2 (T^{2} - 709150 T + 81683152609)^{2} Copy content Toggle raw display
2323 T4++92 ⁣ ⁣76 T^{4} + \cdots + 92\!\cdots\!76 Copy content Toggle raw display
2929 T4++67 ⁣ ⁣44 T^{4} + \cdots + 67\!\cdots\!44 Copy content Toggle raw display
3131 T4++10 ⁣ ⁣00 T^{4} + \cdots + 10\!\cdots\!00 Copy content Toggle raw display
3737 (T2++204253599323425)2 (T^{2} + \cdots + 204253599323425)^{2} Copy content Toggle raw display
4141 T4++53 ⁣ ⁣00 T^{4} + \cdots + 53\!\cdots\!00 Copy content Toggle raw display
4343 T4++49 ⁣ ⁣36 T^{4} + \cdots + 49\!\cdots\!36 Copy content Toggle raw display
4747 T4++22 ⁣ ⁣64 T^{4} + \cdots + 22\!\cdots\!64 Copy content Toggle raw display
5353 (T2++16 ⁣ ⁣00)2 (T^{2} + \cdots + 16\!\cdots\!00)^{2} Copy content Toggle raw display
5959 T4++19 ⁣ ⁣04 T^{4} + \cdots + 19\!\cdots\!04 Copy content Toggle raw display
6161 T4++20 ⁣ ⁣81 T^{4} + \cdots + 20\!\cdots\!81 Copy content Toggle raw display
6767 T4++20 ⁣ ⁣41 T^{4} + \cdots + 20\!\cdots\!41 Copy content Toggle raw display
7171 (T2+10 ⁣ ⁣12)2 (T^{2} + \cdots - 10\!\cdots\!12)^{2} Copy content Toggle raw display
7373 (T2++74 ⁣ ⁣45)2 (T^{2} + \cdots + 74\!\cdots\!45)^{2} Copy content Toggle raw display
7979 T4++33 ⁣ ⁣49 T^{4} + \cdots + 33\!\cdots\!49 Copy content Toggle raw display
8383 T4++16 ⁣ ⁣76 T^{4} + \cdots + 16\!\cdots\!76 Copy content Toggle raw display
8989 (T2+79 ⁣ ⁣92)2 (T^{2} + \cdots - 79\!\cdots\!92)^{2} Copy content Toggle raw display
9797 T4++25 ⁣ ⁣25 T^{4} + \cdots + 25\!\cdots\!25 Copy content Toggle raw display
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