Properties

Label 320.3.bh.a
Level $320$
Weight $3$
Character orbit 320.bh
Analytic conductor $8.719$
Analytic rank $0$
Dimension $752$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,3,Mod(19,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 7, 8]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 320.bh (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: \(8.71936845953\)
Analytic rank: \(0\)
Dimension: \(752\)
Relative dimension: \(94\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 752 q - 16 q^{4} - 8 q^{5} - 16 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 752 q - 16 q^{4} - 8 q^{5} - 16 q^{6} - 16 q^{9} - 8 q^{10} - 16 q^{11} - 16 q^{14} - 8 q^{15} - 16 q^{16} - 16 q^{19} - 8 q^{20} - 16 q^{21} - 16 q^{24} - 8 q^{25} + 384 q^{26} - 16 q^{29} - 88 q^{30} - 32 q^{31} - 16 q^{34} - 8 q^{35} - 816 q^{36} - 16 q^{39} + 352 q^{40} - 16 q^{41} - 16 q^{44} - 8 q^{45} - 16 q^{46} - 16 q^{49} - 320 q^{50} - 400 q^{51} - 592 q^{54} + 504 q^{55} + 656 q^{56} + 240 q^{59} - 296 q^{60} - 16 q^{61} + 80 q^{64} - 16 q^{65} - 368 q^{66} - 16 q^{69} + 664 q^{70} + 496 q^{71} + 1040 q^{74} - 776 q^{75} - 848 q^{76} - 528 q^{79} + 400 q^{80} - 16 q^{81} + 1216 q^{84} - 8 q^{85} - 16 q^{86} - 16 q^{89} - 8 q^{90} - 16 q^{91} + 160 q^{94} - 16 q^{95} - 16 q^{96} - 160 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.99980 + 0.0281854i 2.55665 + 3.82630i 3.99841 0.112730i −4.32079 2.51611i −5.22064 7.57978i −1.57357 3.79893i −7.99285 + 0.338135i −4.65995 + 11.2501i 8.71164 + 4.90993i
19.2 −1.99976 + 0.0312769i −0.222450 0.332920i 3.99804 0.125092i −2.11099 4.53252i 0.455259 + 0.658801i 3.34064 + 8.06503i −7.99120 + 0.375200i 3.38280 8.16680i 4.36322 + 8.99791i
19.3 −1.99936 0.0504630i −1.71781 2.57088i 3.99491 + 0.201788i 3.28925 + 3.76575i 3.30479 + 5.22681i 1.24880 + 3.01488i −7.97709 0.605042i −0.214417 + 0.517649i −6.38637 7.69508i
19.4 −1.99775 0.0947843i 0.132332 + 0.198048i 3.98203 + 0.378711i 3.01166 3.99123i −0.245594 0.408194i −1.16058 2.80188i −7.91922 1.13401i 3.42244 8.26250i −6.39485 + 7.68804i
19.5 −1.99464 0.146372i 2.90773 + 4.35173i 3.95715 + 0.583917i 3.94474 + 3.07230i −5.16290 9.10572i −4.28936 10.3554i −7.80761 1.74392i −7.03847 + 16.9924i −7.41863 6.70551i
19.6 −1.98966 + 0.203107i −2.41543 3.61494i 3.91750 0.808226i 4.97198 + 0.528631i 5.54010 + 6.70192i −3.66182 8.84042i −7.63033 + 2.40376i −3.78936 + 9.14833i −9.99991 0.0419546i
19.7 −1.94208 0.477854i −0.278697 0.417100i 3.54331 + 1.85606i −0.765800 + 4.94101i 0.341938 + 0.943216i −0.143376 0.346141i −5.99445 5.29779i 3.34785 8.08243i 3.84832 9.22987i
19.8 −1.93592 + 0.502193i 2.05543 + 3.07617i 3.49560 1.94442i −2.13995 + 4.51892i −5.52399 4.92300i 3.65857 + 8.83258i −5.79075 + 5.51971i −1.79387 + 4.33078i 1.87341 9.82295i
19.9 −1.92098 + 0.556639i −3.12248 4.67312i 3.38031 2.13858i −2.04138 4.56429i 8.59945 + 7.23886i 0.911578 + 2.20074i −5.30307 + 5.98978i −8.64401 + 20.8685i 6.46211 + 7.63159i
19.10 −1.89436 + 0.641417i −0.368816 0.551972i 3.17717 2.43014i −4.94122 0.764408i 1.05271 + 0.809067i −4.69689 11.3393i −4.45996 + 6.64144i 3.27550 7.90776i 9.85074 1.72132i
19.11 −1.88434 0.670266i −2.53228 3.78983i 3.10149 + 2.52602i −4.77579 + 1.48047i 2.23149 + 8.83864i 1.82456 + 4.40488i −4.15115 6.83871i −4.50621 + 10.8789i 9.99154 + 0.411344i
19.12 −1.83592 0.793342i 1.45741 + 2.18116i 2.74122 + 2.91303i 4.71962 + 1.65082i −0.945275 5.16066i 3.79732 + 9.16754i −2.72163 7.52281i 0.810715 1.95724i −7.35518 6.77506i
19.13 −1.80115 + 0.869401i −1.52115 2.27657i 2.48828 3.13184i −3.98138 + 3.02466i 4.71908 + 2.77795i 1.81341 + 4.37797i −1.75895 + 7.80424i 0.575297 1.38889i 4.54143 8.90929i
19.14 −1.78511 + 0.901887i 1.57535 + 2.35768i 2.37320 3.21993i 4.15406 2.78277i −4.93853 2.78792i −0.244084 0.589270i −1.33240 + 7.88826i 0.367225 0.886559i −4.90569 + 8.71403i
19.15 −1.78397 0.904121i 3.00507 + 4.49741i 2.36513 + 3.22586i 1.29786 4.82862i −1.29477 10.7402i 3.88041 + 9.36813i −1.30277 7.89321i −7.75206 + 18.7151i −6.68101 + 7.44071i
19.16 −1.77717 0.917423i 1.50868 + 2.25790i 2.31667 + 3.26084i −4.54717 + 2.07923i −0.609733 5.39678i −1.69013 4.08034i −1.12555 7.92043i 0.622145 1.50199i 9.98864 + 0.476530i
19.17 −1.70689 1.04237i −1.63844 2.45209i 1.82692 + 3.55842i −2.16920 4.50495i 0.240632 + 5.89330i −4.03762 9.74768i 0.590847 7.97815i 0.115865 0.279723i −0.993257 + 9.95055i
19.18 −1.68479 + 1.07772i 0.795457 + 1.19049i 1.67702 3.63147i 1.75943 + 4.68021i −2.62319 1.14843i −3.20466 7.73674i 1.08829 + 7.92563i 2.65965 6.42096i −8.00825 5.98899i
19.19 −1.66317 1.11079i −2.54647 3.81106i 1.53229 + 3.69487i 3.64742 3.41999i 0.00193507 + 9.16703i 1.01842 + 2.45869i 1.55575 7.84727i −4.59550 + 11.0945i −9.86518 + 1.63651i
19.20 −1.63387 + 1.15346i −1.28831 1.92809i 1.33908 3.76920i 4.99552 + 0.211605i 4.32891 + 1.66425i 4.87580 + 11.7712i 2.15971 + 7.70296i 1.38635 3.34695i −8.40612 + 5.41637i
See next 80 embeddings (of 752 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.94
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
64.j odd 16 1 inner
320.bh odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.3.bh.a 752
5.b even 2 1 inner 320.3.bh.a 752
64.j odd 16 1 inner 320.3.bh.a 752
320.bh odd 16 1 inner 320.3.bh.a 752
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.3.bh.a 752 1.a even 1 1 trivial
320.3.bh.a 752 5.b even 2 1 inner
320.3.bh.a 752 64.j odd 16 1 inner
320.3.bh.a 752 320.bh odd 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(320, [\chi])\).