Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [320,2,Mod(3,320)]
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, 3, 12]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("320.3");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 320 = 2^{6} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 320.bj (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: | \(2.55521286468\) |
Analytic rank: | \(0\) |
Dimension: | \(368\) |
Relative dimension: | \(46\) 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.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −1.41250 | + | 0.0695938i | −0.910189 | − | 1.36219i | 1.99031 | − | 0.196602i | −0.381084 | + | 2.20336i | 1.38044 | + | 1.86076i | −1.16673 | − | 0.483276i | −2.79764 | + | 0.416214i | 0.120920 | − | 0.291928i | 0.384941 | − | 3.13876i |
3.2 | −1.41199 | − | 0.0792857i | 1.42562 | + | 2.13359i | 1.98743 | + | 0.223901i | −1.45784 | + | 1.69549i | −1.84380 | − | 3.12564i | −1.79842 | − | 0.744931i | −2.78847 | − | 0.473720i | −1.37176 | + | 3.31173i | 2.19289 | − | 2.27843i |
3.3 | −1.41105 | − | 0.0945208i | 0.0555723 | + | 0.0831698i | 1.98213 | + | 0.266747i | 1.63014 | − | 1.53057i | −0.0705541 | − | 0.122610i | 1.56812 | + | 0.649538i | −2.77168 | − | 0.563747i | 1.14422 | − | 2.76239i | −2.44488 | + | 2.00563i |
3.4 | −1.40259 | + | 0.180968i | −1.12830 | − | 1.68862i | 1.93450 | − | 0.507646i | −1.49135 | − | 1.66609i | 1.88812 | + | 2.16425i | −3.56811 | − | 1.47796i | −2.62144 | + | 1.06210i | −0.430327 | + | 1.03890i | 2.39326 | + | 2.06695i |
3.5 | −1.32650 | − | 0.490292i | 0.715650 | + | 1.07105i | 1.51923 | + | 1.30075i | −2.01260 | − | 0.974383i | −0.424187 | − | 1.77163i | 0.565906 | + | 0.234406i | −1.37751 | − | 2.47032i | 0.513065 | − | 1.23865i | 2.19200 | + | 2.27929i |
3.6 | −1.26744 | + | 0.627366i | 1.23558 | + | 1.84918i | 1.21282 | − | 1.59030i | 2.20390 | + | 0.377927i | −2.72615 | − | 1.56857i | −1.21576 | − | 0.503585i | −0.539487 | + | 2.77650i | −0.744758 | + | 1.79801i | −3.03042 | + | 0.903649i |
3.7 | −1.21873 | − | 0.717416i | −1.42561 | − | 2.13358i | 0.970629 | + | 1.74868i | −2.06596 | + | 0.855448i | 0.206779 | + | 3.62303i | 4.15639 | + | 1.72163i | 0.0715919 | − | 2.82752i | −1.37175 | + | 3.31170i | 3.13157 | + | 0.439592i |
3.8 | −1.19321 | + | 0.759115i | 0.0471430 | + | 0.0705545i | 0.847488 | − | 1.81156i | 0.727110 | + | 2.11455i | −0.109810 | − | 0.0483992i | 2.83007 | + | 1.17225i | 0.363957 | + | 2.80491i | 1.14529 | − | 2.76499i | −2.47278 | − | 1.97113i |
3.9 | −1.16043 | − | 0.808338i | −1.76630 | − | 2.64345i | 0.693179 | + | 1.87603i | 1.88676 | − | 1.20006i | −0.0871443 | + | 4.49529i | −2.23701 | − | 0.926602i | 0.712087 | − | 2.73732i | −2.71997 | + | 6.56659i | −3.15950 | − | 0.132556i |
3.10 | −1.14595 | + | 0.828734i | 1.03087 | + | 1.54280i | 0.626401 | − | 1.89937i | −1.21411 | − | 1.87775i | −2.45989 | − | 0.913659i | −2.97720 | − | 1.23320i | 0.856252 | + | 2.69571i | −0.169501 | + | 0.409211i | 2.94746 | + | 1.14564i |
3.11 | −1.12995 | + | 0.850415i | −1.38917 | − | 2.07904i | 0.553590 | − | 1.92186i | −0.180810 | − | 2.22875i | 3.33774 | + | 1.16785i | 4.18944 | + | 1.73532i | 1.00885 | + | 2.64239i | −1.24456 | + | 3.00463i | 2.09967 | + | 2.36462i |
3.12 | −1.11439 | − | 0.870707i | 1.42151 | + | 2.12744i | 0.483739 | + | 1.94062i | 1.98658 | + | 1.02640i | 0.268257 | − | 3.60852i | 2.97150 | + | 1.23083i | 1.15063 | − | 2.58380i | −1.35726 | + | 3.27671i | −1.32014 | − | 2.87354i |
3.13 | −1.06436 | − | 0.931203i | −0.000769131 | − | 0.00115109i | 0.265723 | + | 1.98227i | 1.37433 | + | 1.76387i | −0.000253262 | 0.00194139i | −3.49768 | − | 1.44879i | 1.56307 | − | 2.35729i | 1.14805 | − | 2.77164i | 0.179742 | − | 3.15717i | |
3.14 | −0.822872 | + | 1.15017i | −1.02225 | − | 1.52991i | −0.645765 | − | 1.89288i | 2.22448 | + | 0.227309i | 2.60083 | + | 0.0831583i | −2.61953 | − | 1.08505i | 2.70851 | + | 0.814859i | −0.147570 | + | 0.356266i | −2.09191 | + | 2.37148i |
3.15 | −0.790504 | + | 1.17265i | −0.936632 | − | 1.40177i | −0.750206 | − | 1.85397i | −2.14343 | + | 0.636966i | 2.38419 | + | 0.00976493i | −0.275234 | − | 0.114006i | 2.76709 | + | 0.585841i | 0.0603745 | − | 0.145757i | 0.947451 | − | 3.01701i |
3.16 | −0.779137 | − | 1.18023i | 0.0599818 | + | 0.0897691i | −0.785890 | + | 1.83912i | −1.62511 | + | 1.53591i | 0.0592142 | − | 0.140735i | −0.748850 | − | 0.310184i | 2.78291 | − | 0.505399i | 1.14359 | − | 2.76087i | 3.07891 | + | 0.721319i |
3.17 | −0.691783 | + | 1.23347i | 1.72903 | + | 2.58768i | −1.04287 | − | 1.70658i | −1.72169 | + | 1.42680i | −4.38793 | + | 0.342588i | 3.92678 | + | 1.62653i | 2.82645 | − | 0.105762i | −2.55849 | + | 6.17673i | −0.568874 | − | 3.11069i |
3.18 | −0.592295 | + | 1.28421i | 0.658738 | + | 0.985871i | −1.29837 | − | 1.52126i | 0.0564951 | − | 2.23535i | −1.65623 | + | 0.262029i | 0.804125 | + | 0.333079i | 2.72263 | − | 0.766346i | 0.610044 | − | 1.47278i | 2.83719 | + | 1.39654i |
3.19 | −0.544145 | − | 1.30534i | −0.783307 | − | 1.17230i | −1.40781 | + | 1.42059i | 2.13877 | − | 0.652412i | −1.10402 | + | 1.66038i | 3.71308 | + | 1.53801i | 2.62040 | + | 1.06467i | 0.387328 | − | 0.935094i | −2.01542 | − | 2.43682i |
3.20 | −0.530381 | − | 1.31099i | 1.41093 | + | 2.11161i | −1.43739 | + | 1.39065i | −1.73973 | − | 1.40476i | 2.01997 | − | 2.96967i | 2.37843 | + | 0.985179i | 2.58549 | + | 1.14683i | −1.32011 | + | 3.18703i | −0.918903 | + | 3.02582i |
See next 80 embeddings (of 368 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
320.bj | even | 16 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 320.2.bj.a | yes | 368 |
5.c | odd | 4 | 1 | 320.2.bd.a | ✓ | 368 | |
64.j | odd | 16 | 1 | 320.2.bd.a | ✓ | 368 | |
320.bj | even | 16 | 1 | inner | 320.2.bj.a | yes | 368 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
320.2.bd.a | ✓ | 368 | 5.c | odd | 4 | 1 | |
320.2.bd.a | ✓ | 368 | 64.j | odd | 16 | 1 | |
320.2.bj.a | yes | 368 | 1.a | even | 1 | 1 | trivial |
320.2.bj.a | yes | 368 | 320.bj | even | 16 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(320, [\chi])\).