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Properties

Label 85.3.p.a
Level $85$
Weight $3$
Character orbit 85.p
Analytic conductor $2.316$
Analytic rank $0$
Dimension $128$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [85,3,Mod(14,85)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(85, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("85.14");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 85.p (of order \(16\), degree \(8\), 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: \(2.31608224706\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 16 q^{4} - 8 q^{5} - 16 q^{6} - 16 q^{9} + 16 q^{10} - 96 q^{11} - 16 q^{14} + 16 q^{15} - 16 q^{19} - 120 q^{20} + 224 q^{21} - 160 q^{24} - 240 q^{25} + 288 q^{26} - 176 q^{29} - 200 q^{30} + 48 q^{31}+ \cdots - 1664 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1 −1.45551 − 3.51390i −0.468685 − 0.0932273i −7.40057 + 7.40057i −0.936923 − 4.91143i 0.354582 + 1.78261i −6.09968 + 9.12882i 22.7209 + 9.41129i −8.10394 − 3.35676i −15.8946 + 10.4409i
14.2 −1.33424 − 3.22115i 4.93268 + 0.981171i −5.76715 + 5.76715i 4.22315 + 2.67675i −3.42090 − 17.1980i 3.95192 − 5.91447i 13.3870 + 5.54508i 15.0537 + 6.23546i 2.98750 − 17.1748i
14.3 −1.20149 − 2.90065i −3.75496 − 0.746907i −4.14176 + 4.14176i 0.501336 + 4.97480i 2.34502 + 11.7892i 0.580455 − 0.868712i 5.38745 + 2.23156i 5.22691 + 2.16506i 13.8278 − 7.43136i
14.4 −0.957956 − 2.31271i 1.86011 + 0.369999i −1.60252 + 1.60252i −4.85937 − 1.17749i −0.926203 − 4.65634i 5.79956 − 8.67965i −4.00954 − 1.66080i −4.99181 − 2.06767i 1.93188 + 12.3663i
14.5 −0.746854 − 1.80306i −1.51403 − 0.301160i 0.135175 − 0.135175i 4.28345 − 2.57915i 0.587751 + 2.95483i 1.25798 − 1.88271i −7.55695 − 3.13019i −6.11331 − 2.53222i −7.84949 − 5.79710i
14.6 −0.439826 − 1.06183i 5.22866 + 1.04004i 1.89438 − 1.89438i −1.35606 − 4.81260i −1.19534 − 6.00940i −5.63579 + 8.43455i −7.09205 − 2.93762i 17.9423 + 7.43192i −4.51375 + 3.55661i
14.7 −0.328307 − 0.792602i 2.04665 + 0.407105i 2.30799 − 2.30799i 3.60547 + 3.46419i −0.349258 − 1.75584i −2.59661 + 3.88610i −5.75746 − 2.38482i −4.29186 − 1.77774i 1.56202 − 3.99502i
14.8 −0.308672 − 0.745199i −3.19795 − 0.636111i 2.36838 − 2.36838i −4.85925 + 1.17800i 0.513086 + 2.57946i −3.07748 + 4.60577i −5.47677 − 2.26855i 1.50731 + 0.624350i 2.37776 + 3.25749i
14.9 0.308672 + 0.745199i 3.19795 + 0.636111i 2.36838 − 2.36838i −2.94789 + 4.03856i 0.513086 + 2.57946i 3.07748 − 4.60577i 5.47677 + 2.26855i 1.50731 + 0.624350i −3.91946 − 0.950175i
14.10 0.328307 + 0.792602i −2.04665 − 0.407105i 2.30799 − 2.30799i −1.82074 − 4.65671i −0.349258 − 1.75584i 2.59661 − 3.88610i 5.75746 + 2.38482i −4.29186 − 1.77774i 3.09315 − 2.97195i
14.11 0.439826 + 1.06183i −5.22866 − 1.04004i 1.89438 − 1.89438i 3.92732 + 3.09454i −1.19534 − 6.00940i 5.63579 − 8.43455i 7.09205 + 2.93762i 17.9423 + 7.43192i −1.55855 + 5.53121i
14.12 0.746854 + 1.80306i 1.51403 + 0.301160i 0.135175 − 0.135175i 4.02203 − 2.97040i 0.587751 + 2.95483i −1.25798 + 1.88271i 7.55695 + 3.13019i −6.11331 − 2.53222i 8.35969 + 5.03353i
14.13 0.957956 + 2.31271i −1.86011 − 0.369999i −1.60252 + 1.60252i −0.771745 + 4.94008i −0.926203 − 4.65634i −5.79956 + 8.67965i 4.00954 + 1.66080i −4.99181 − 2.06767i −12.1643 + 2.94756i
14.14 1.20149 + 2.90065i 3.75496 + 0.746907i −4.14176 + 4.14176i −4.40427 − 2.36695i 2.34502 + 11.7892i −0.580455 + 0.868712i −5.38745 − 2.23156i 5.22691 + 2.16506i 1.57401 − 15.6191i
14.15 1.33424 + 3.22115i −4.93268 − 0.981171i −5.76715 + 5.76715i −0.856866 − 4.92603i −3.42090 − 17.1980i −3.95192 + 5.91447i −13.3870 − 5.54508i 15.0537 + 6.23546i 14.7242 − 9.33261i
14.16 1.45551 + 3.51390i 0.468685 + 0.0932273i −7.40057 + 7.40057i 4.17903 + 2.74513i 0.354582 + 1.78261i 6.09968 − 9.12882i −22.7209 − 9.41129i −8.10394 − 3.35676i −3.56351 + 18.6802i
24.1 −3.41305 − 1.41373i −0.300774 − 0.200970i 6.82185 + 6.82185i −0.469678 − 4.97789i 0.742437 + 1.11114i 7.65847 + 1.52336i −7.98412 − 19.2754i −3.39408 − 8.19402i −5.43437 + 17.6538i
24.2 −2.93643 − 1.21631i 2.61324 + 1.74611i 4.31479 + 4.31479i −3.16949 + 3.86708i −5.54979 − 8.30585i −8.01084 − 1.59346i −2.55673 − 6.17250i 0.335974 + 0.811112i 14.0106 − 7.50035i
24.3 −2.69497 − 1.11629i −0.782295 − 0.522713i 3.18832 + 3.18832i 3.64794 + 3.41943i 1.52476 + 2.28196i 0.889448 + 0.176922i −0.568156 − 1.37165i −3.10539 − 7.49709i −6.01400 − 13.2874i
24.4 −2.05790 − 0.852408i 4.45190 + 2.97466i 0.679908 + 0.679908i 2.15905 − 4.50982i −6.62591 − 9.91638i 0.440630 + 0.0876467i 2.59001 + 6.25284i 7.52662 + 18.1709i −8.28731 + 7.44035i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
17.e odd 16 1 inner
85.p odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 85.3.p.a ✓ 128
5.b even 2 1 inner 85.3.p.a ✓ 128
5.c odd 4 2 425.3.u.f 128
17.e odd 16 1 inner 85.3.p.a ✓ 128
85.o even 16 1 425.3.u.f 128
85.p odd 16 1 inner 85.3.p.a ✓ 128
85.r even 16 1 425.3.u.f 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.3.p.a ✓ 128 1.a even 1 1 trivial
85.3.p.a ✓ 128 5.b even 2 1 inner
85.3.p.a ✓ 128 17.e odd 16 1 inner
85.3.p.a ✓ 128 85.p odd 16 1 inner
425.3.u.f 128 5.c odd 4 2
425.3.u.f 128 85.o even 16 1
425.3.u.f 128 85.r even 16 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(85, [\chi])\).