Properties

Label 320.2.bj.a
Level $320$
Weight $2$
Character orbit 320.bj
Analytic conductor $2.555$
Analytic rank $0$
Dimension $368$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

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.

\(\operatorname{Tr}(f)(q) = \) \( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{10} - 16 q^{11} - 40 q^{12} - 8 q^{13} - 32 q^{14} - 8 q^{15} - 16 q^{16} - 16 q^{17} - 8 q^{18} - 8 q^{20} - 16 q^{21} + 24 q^{22} - 8 q^{23} + 16 q^{24} - 8 q^{25} - 16 q^{26} - 8 q^{27} - 8 q^{28} - 104 q^{30} - 32 q^{31} - 8 q^{32} - 32 q^{34} - 8 q^{35} - 16 q^{36} - 8 q^{37} + 48 q^{38} + 16 q^{40} - 16 q^{41} - 8 q^{42} - 8 q^{43} + 16 q^{45} - 16 q^{46} - 112 q^{48} - 112 q^{50} - 48 q^{51} - 8 q^{52} - 8 q^{53} - 8 q^{55} + 80 q^{56} - 8 q^{57} + 56 q^{58} + 48 q^{60} - 16 q^{61} - 24 q^{62} - 16 q^{65} + 80 q^{66} - 8 q^{67} - 96 q^{68} + 64 q^{69} - 8 q^{70} - 80 q^{71} + 112 q^{72} - 8 q^{73} - 8 q^{75} + 48 q^{76} - 8 q^{77} + 144 q^{78} - 32 q^{79} - 8 q^{80} - 16 q^{81} - 168 q^{82} - 8 q^{83} - 48 q^{85} - 16 q^{86} + 104 q^{87} - 96 q^{88} - 8 q^{90} - 16 q^{91} - 88 q^{92} - 32 q^{93} + 32 q^{94} - 16 q^{95} - 16 q^{96} + 32 q^{98} + 128 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.46
Significant digits:
Format:

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 \(S_{2}^{\mathrm{new}}(320, [\chi])\).