Properties

Label 171.4.y.a
Level 171171
Weight 44
Character orbit 171.y
Analytic conductor 10.08910.089
Analytic rank 00
Dimension 120120
Inner twists 44

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [171,4,Mod(53,171)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 11])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("171.53"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 171=3219 171 = 3^{2} \cdot 19
Weight: k k == 4 4
Character orbit: [χ][\chi] == 171.y (of order 1818, degree 66, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 10.089326611010.0893266110
Analytic rank: 00
Dimension: 120120
Relative dimension: 2020 over Q(ζ18)\Q(\zeta_{18})
Twist minimal: yes
Sato-Tate group: SU(2)[C18]\mathrm{SU}(2)[C_{18}]

qq-expansion

The algebraic qq-expansion of this newform has not been computed, but we have computed the trace expansion.

Tr(f)(q)=\operatorname{Tr}(f)(q) = 120q36q4180q10156q13+180q16+924q19+432q22360q25624q28+324q34+1440q40+1524q43+3888q463228q496000q524464q55++5904q97+O(q100) 120 q - 36 q^{4} - 180 q^{10} - 156 q^{13} + 180 q^{16} + 924 q^{19} + 432 q^{22} - 360 q^{25} - 624 q^{28} + 324 q^{34} + 1440 q^{40} + 1524 q^{43} + 3888 q^{46} - 3228 q^{49} - 6000 q^{52} - 4464 q^{55}+ \cdots + 5904 q^{97}+O(q^{100}) Copy content Toggle raw display

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   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}
53.1 −4.90666 1.78588i 0 14.7576 + 12.3831i −3.22388 3.84207i 0 −0.347063 + 0.601131i −29.4095 50.9388i 0 8.95701 + 24.6092i
53.2 −4.79619 1.74567i 0 13.8277 + 11.6028i 0.869736 + 1.03651i 0 −17.0305 + 29.4978i −25.6497 44.4266i 0 −2.36201 6.48958i
53.3 −4.20988 1.53227i 0 9.24691 + 7.75908i −12.4048 14.7835i 0 14.3815 24.9094i −9.11914 15.7948i 0 29.5705 + 81.2444i
53.4 −3.41758 1.24390i 0 4.00419 + 3.35992i 8.24027 + 9.82037i 0 8.78095 15.2090i 5.04239 + 8.73367i 0 −15.9462 43.8119i
53.5 −3.36331 1.22414i 0 3.68495 + 3.09204i 13.3019 + 15.8526i 0 −7.77429 + 13.4655i 5.70811 + 9.88673i 0 −25.3325 69.6004i
53.6 −3.13604 1.14142i 0 2.40352 + 2.01679i −0.417489 0.497545i 0 11.8643 20.5495i 8.11369 + 14.0533i 0 0.741353 + 2.03685i
53.7 −2.30065 0.837369i 0 −1.53654 1.28931i −10.9536 13.0540i 0 −16.1646 + 27.9979i 12.2486 + 21.2153i 0 14.2695 + 39.2050i
53.8 −1.91636 0.697498i 0 −2.94242 2.46899i 0.792095 + 0.943982i 0 −7.99477 + 13.8473i 12.0740 + 20.9128i 0 −0.859514 2.36149i
53.9 −1.02676 0.373709i 0 −5.21378 4.37488i −4.49187 5.35320i 0 9.55306 16.5464i 8.08896 + 14.0105i 0 2.61152 + 7.17509i
53.10 −0.393398 0.143185i 0 −5.99410 5.02964i −6.91842 8.24506i 0 −0.354982 + 0.614847i 3.31248 + 5.73738i 0 1.54113 + 4.23421i
53.11 0.393398 + 0.143185i 0 −5.99410 5.02964i 6.91842 + 8.24506i 0 −0.354982 + 0.614847i −3.31248 5.73738i 0 1.54113 + 4.23421i
53.12 1.02676 + 0.373709i 0 −5.21378 4.37488i 4.49187 + 5.35320i 0 9.55306 16.5464i −8.08896 14.0105i 0 2.61152 + 7.17509i
53.13 1.91636 + 0.697498i 0 −2.94242 2.46899i −0.792095 0.943982i 0 −7.99477 + 13.8473i −12.0740 20.9128i 0 −0.859514 2.36149i
53.14 2.30065 + 0.837369i 0 −1.53654 1.28931i 10.9536 + 13.0540i 0 −16.1646 + 27.9979i −12.2486 21.2153i 0 14.2695 + 39.2050i
53.15 3.13604 + 1.14142i 0 2.40352 + 2.01679i 0.417489 + 0.497545i 0 11.8643 20.5495i −8.11369 14.0533i 0 0.741353 + 2.03685i
53.16 3.36331 + 1.22414i 0 3.68495 + 3.09204i −13.3019 15.8526i 0 −7.77429 + 13.4655i −5.70811 9.88673i 0 −25.3325 69.6004i
53.17 3.41758 + 1.24390i 0 4.00419 + 3.35992i −8.24027 9.82037i 0 8.78095 15.2090i −5.04239 8.73367i 0 −15.9462 43.8119i
53.18 4.20988 + 1.53227i 0 9.24691 + 7.75908i 12.4048 + 14.7835i 0 14.3815 24.9094i 9.11914 + 15.7948i 0 29.5705 + 81.2444i
53.19 4.79619 + 1.74567i 0 13.8277 + 11.6028i −0.869736 1.03651i 0 −17.0305 + 29.4978i 25.6497 + 44.4266i 0 −2.36201 6.48958i
53.20 4.90666 + 1.78588i 0 14.7576 + 12.3831i 3.22388 + 3.84207i 0 −0.347063 + 0.601131i 29.4095 + 50.9388i 0 8.95701 + 24.6092i
See next 80 embeddings (of 120 total)
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.f odd 18 1 inner
57.j even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.4.y.a 120
3.b odd 2 1 inner 171.4.y.a 120
19.f odd 18 1 inner 171.4.y.a 120
57.j even 18 1 inner 171.4.y.a 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.4.y.a 120 1.a even 1 1 trivial
171.4.y.a 120 3.b odd 2 1 inner
171.4.y.a 120 19.f odd 18 1 inner
171.4.y.a 120 57.j even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace S4new(171,[χ])S_{4}^{\mathrm{new}}(171, [\chi]).