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Properties

Label 85.4.o.a
Level $85$
Weight $4$
Character orbit 85.o
Analytic conductor $5.015$
Analytic rank $0$
Dimension $200$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [85,4,Mod(3,85)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("85.3"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(85, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([12, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 85.o (of order \(16\), degree \(8\), 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: \(5.01516235049\)
Analytic rank: \(0\)
Dimension: \(200\)
Relative dimension: \(25\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 200 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} + 56 q^{8} + 216 q^{10} - 16 q^{11} - 224 q^{12} - 416 q^{14} - 344 q^{15} - 8 q^{17} - 16 q^{18} + 96 q^{19} + 408 q^{20} - 16 q^{21} - 8 q^{22}+ \cdots - 16320 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −2.09828 + 5.06569i −0.0758006 − 0.113444i −15.6016 − 15.6016i −10.7272 − 3.15086i 0.733721 − 0.145946i 3.63545 + 18.2767i 71.2440 − 29.5102i 10.3253 − 24.9275i 38.4699 − 47.7292i
3.2 −1.91456 + 4.62217i −4.14558 − 6.20430i −12.0420 − 12.0420i 10.0375 + 4.92425i 36.6143 − 7.28303i −0.687966 − 3.45864i 41.7380 − 17.2885i −10.9750 + 26.4961i −41.9782 + 36.9673i
3.3 −1.67848 + 4.05222i 0.154007 + 0.230488i −7.94631 − 7.94631i 6.80547 − 8.87049i −1.19249 + 0.237201i −4.73603 − 23.8096i 13.1202 − 5.43457i 10.3030 − 24.8738i 24.5223 + 42.4662i
3.4 −1.59940 + 3.86130i 4.95060 + 7.40910i −6.69468 − 6.69468i 8.22651 − 7.57130i −36.5267 + 7.26562i 5.58918 + 28.0987i 5.66723 − 2.34744i −20.0538 + 48.4142i 16.0776 + 43.8745i
3.5 −1.53952 + 3.71672i 4.53296 + 6.78405i −5.78705 − 5.78705i −11.1058 + 1.28868i −32.1930 + 6.40358i −5.74925 − 28.9034i 0.684338 − 0.283462i −15.1432 + 36.5589i 12.3079 − 43.2612i
3.6 −1.44980 + 3.50012i 1.69943 + 2.54337i −4.49204 − 4.49204i 3.31010 + 10.6791i −11.3659 + 2.26082i 1.92612 + 9.68326i −5.76570 + 2.38823i 6.75176 − 16.3002i −42.1771 − 3.89677i
3.7 −1.32628 + 3.20192i −2.60296 − 3.89560i −2.83643 − 2.83643i −5.71326 + 9.61034i 15.9257 − 3.16781i −0.609781 − 3.06558i −12.7714 + 5.29010i 1.93213 − 4.66458i −23.1942 − 31.0394i
3.8 −1.20151 + 2.90071i −4.46209 − 6.67799i −1.31364 − 1.31364i −1.19905 − 11.1159i 24.7322 − 4.91954i 5.81526 + 29.2353i −17.8168 + 7.37998i −14.3528 + 34.6508i 33.6846 + 9.87776i
3.9 −0.772500 + 1.86498i −1.32654 − 1.98530i 2.77546 + 2.77546i −6.06235 − 9.39404i 4.72731 − 0.940320i −3.48570 − 17.5238i −22.2401 + 9.21214i 8.15072 − 19.6776i 22.2029 − 4.04926i
3.10 −0.464568 + 1.12157i 1.00690 + 1.50693i 4.61477 + 4.61477i 11.1199 − 1.16107i −2.15790 + 0.429232i 0.0769207 + 0.386707i −16.2922 + 6.74844i 9.07546 − 21.9101i −3.86372 + 13.0111i
3.11 −0.436760 + 1.05443i 2.57186 + 3.84906i 4.73579 + 4.73579i −10.7688 − 3.00558i −5.18186 + 1.03074i 4.80914 + 24.1772i −15.4974 + 6.41924i 2.13165 − 5.14626i 7.87255 − 10.0422i
3.12 −0.219976 + 0.531069i −3.44939 − 5.16237i 5.42321 + 5.42321i −7.28583 + 8.48037i 3.50036 − 0.696264i −0.486076 − 2.44367i −8.32162 + 3.44693i −4.41937 + 10.6693i −2.90095 − 5.73475i
3.13 −0.209437 + 0.505625i 4.08200 + 6.10915i 5.44506 + 5.44506i 2.92018 + 10.7922i −3.94386 + 0.784482i −3.02159 − 15.1905i −7.93855 + 3.28826i −10.3265 + 24.9304i −6.06842 − 0.783778i
3.14 −0.138966 + 0.335493i −5.02050 − 7.51371i 5.56361 + 5.56361i 11.1778 + 0.239064i 3.21848 − 0.640195i −4.01014 − 20.1604i −5.32365 + 2.20513i −20.9180 + 50.5004i −1.63354 + 3.71685i
3.15 0.494498 − 1.19382i 4.95318 + 7.41295i 4.47617 + 4.47617i 2.98585 − 10.7743i 11.2991 − 2.24753i −2.60018 − 13.0720i 17.1078 − 7.08629i −20.0854 + 48.4905i −11.3861 − 8.89243i
3.16 0.603439 − 1.45683i −2.21091 − 3.30886i 3.89864 + 3.89864i 4.74223 + 10.1248i −6.15459 + 1.22422i 5.98431 + 30.0852i 19.6869 − 8.15457i 4.27203 − 10.3136i 17.6117 − 0.798929i
3.17 0.702898 − 1.69694i −0.566470 − 0.847783i 3.27130 + 3.27130i 3.95442 − 10.4577i −1.83681 + 0.365364i 3.10353 + 15.6025i 21.4262 − 8.87501i 9.93461 − 23.9843i −14.9665 − 14.0611i
3.18 0.767564 − 1.85306i −0.509513 − 0.762540i 2.81217 + 2.81217i −10.8989 − 2.49266i −1.80412 + 0.358862i −5.01892 − 25.2318i 22.1941 − 9.19311i 10.0106 − 24.1677i −12.9847 + 18.2831i
3.19 1.09807 − 2.65098i −5.07792 − 7.59965i −0.165092 − 0.165092i −9.49705 − 5.89967i −25.7225 + 5.11652i 0.404943 + 2.03578i 20.5889 − 8.52821i −21.6369 + 52.2361i −26.0684 + 18.6983i
3.20 1.12361 − 2.71264i 3.41643 + 5.11305i −0.439064 − 0.439064i −8.39860 + 7.37994i 17.7086 − 3.52246i 2.32607 + 11.6940i 20.0168 − 8.29122i −4.13882 + 9.99200i 10.5824 + 31.0746i
See next 80 embeddings (of 200 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.25
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.o even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 85.4.o.a ✓ 200
5.c odd 4 1 85.4.r.a yes 200
17.e odd 16 1 85.4.r.a yes 200
85.o even 16 1 inner 85.4.o.a ✓ 200
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.4.o.a ✓ 200 1.a even 1 1 trivial
85.4.o.a ✓ 200 85.o even 16 1 inner
85.4.r.a yes 200 5.c odd 4 1
85.4.r.a yes 200 17.e odd 16 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(85, [\chi])\).