Properties

Label 208.2.bj.a
Level $208$
Weight $2$
Character orbit 208.bj
Analytic conductor $1.661$
Analytic rank $0$
Dimension $104$
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] = [208,2,Mod(29,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 9, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 208.bj (of order \(12\), degree \(4\), 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: \(1.66088836204\)
Analytic rank: \(0\)
Dimension: \(104\)
Relative dimension: \(26\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 104 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 8 q^{5} + 4 q^{6} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 104 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 8 q^{5} + 4 q^{6} - 8 q^{8} - 2 q^{10} - 2 q^{11} - 36 q^{12} - 4 q^{13} - 4 q^{15} - 2 q^{16} - 4 q^{17} - 16 q^{18} - 2 q^{19} - 10 q^{20} - 20 q^{21} + 18 q^{22} + 14 q^{24} - 40 q^{26} + 4 q^{27} - 16 q^{28} - 2 q^{29} - 2 q^{30} - 16 q^{31} - 2 q^{32} - 4 q^{33} + 20 q^{34} + 8 q^{35} - 24 q^{36} - 2 q^{37} + 12 q^{38} - 28 q^{40} + 14 q^{42} - 18 q^{43} - 24 q^{44} + 20 q^{45} + 32 q^{46} - 16 q^{47} - 48 q^{48} + 24 q^{49} + 64 q^{50} + 4 q^{51} - 42 q^{52} - 8 q^{53} - 26 q^{54} + 32 q^{56} + 34 q^{58} - 42 q^{59} + 56 q^{60} - 2 q^{61} + 18 q^{62} - 60 q^{63} - 44 q^{64} - 16 q^{65} - 24 q^{66} - 2 q^{67} + 32 q^{68} - 14 q^{69} - 112 q^{70} - 60 q^{72} + 6 q^{74} + 10 q^{75} + 46 q^{76} - 36 q^{77} - 116 q^{78} + 64 q^{79} + 30 q^{80} + 16 q^{81} + 68 q^{82} - 48 q^{83} + 88 q^{84} - 12 q^{85} - 48 q^{86} - 44 q^{88} + 220 q^{90} + 38 q^{91} - 72 q^{92} - 56 q^{93} - 2 q^{94} + 60 q^{95} + 168 q^{96} - 4 q^{97} + 38 q^{98} - 160 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −1.41044 + 0.103276i −0.623697 + 2.32767i 1.97867 0.291328i −2.33843 + 2.33843i 0.639294 3.34744i −3.16277 1.82602i −2.76070 + 0.615248i −2.43097 1.40352i 3.05670 3.53971i
29.2 −1.39507 + 0.231899i 0.0845055 0.315379i 1.89245 0.647031i 2.63109 2.63109i −0.0447552 + 0.459573i −3.34168 1.92932i −2.49005 + 1.34151i 2.50575 + 1.44670i −3.06041 + 4.28070i
29.3 −1.39445 0.235600i 0.156066 0.582446i 1.88899 + 0.657065i −0.921546 + 0.921546i −0.354850 + 0.775422i 1.44176 + 0.832400i −2.47929 1.36129i 2.28319 + 1.31820i 1.50217 1.06793i
29.4 −1.34109 0.448861i −0.762293 + 2.84492i 1.59705 + 1.20393i 2.03642 2.03642i 2.29928 3.47313i 3.62835 + 2.09483i −1.60139 2.33143i −4.91439 2.83732i −3.64509 + 1.81695i
29.5 −1.28175 + 0.597591i 0.722632 2.69690i 1.28577 1.53193i 0.609482 0.609482i 0.685410 + 3.88859i 3.93274 + 2.27057i −0.732570 + 2.73191i −4.15300 2.39773i −0.416983 + 1.14542i
29.6 −0.979428 1.02016i 0.598940 2.23527i −0.0814433 + 1.99834i 2.39927 2.39927i −2.86695 + 1.57828i −0.442240 0.255327i 2.11839 1.87415i −2.03964 1.17759i −4.79754 0.0977225i
29.7 −0.967244 + 1.03172i 0.371119 1.38503i −0.128880 1.99584i −2.99717 + 2.99717i 1.07000 + 1.72255i −1.68939 0.975368i 2.18380 + 1.79750i 0.817487 + 0.471976i −0.193237 5.99123i
29.8 −0.890578 1.09858i 0.100590 0.375407i −0.413740 + 1.95674i −1.52134 + 1.52134i −0.501996 + 0.223823i 0.798611 + 0.461078i 2.51809 1.28810i 2.46726 + 1.42448i 3.02617 + 0.316435i
29.9 −0.851616 1.12905i −0.520271 + 1.94168i −0.549502 + 1.92303i −0.499375 + 0.499375i 2.63532 1.06615i −2.35238 1.35814i 2.63916 1.01727i −0.901359 0.520400i 0.989095 + 0.138543i
29.10 −0.672039 + 1.24433i 0.0577913 0.215680i −1.09673 1.67248i 0.668900 0.668900i 0.229540 + 0.216857i −0.616743 0.356077i 2.81816 0.240721i 2.55490 + 1.47507i 0.382807 + 1.28186i
29.11 −0.237188 + 1.39418i −0.623936 + 2.32856i −1.88748 0.661365i −1.66259 + 1.66259i −3.09845 1.42219i 2.78550 + 1.60821i 1.36975 2.47463i −2.43482 1.40575i −1.92361 2.71230i
29.12 −0.176994 1.40309i −0.465351 + 1.73671i −1.93735 + 0.496679i 1.82441 1.82441i 2.51914 + 0.345544i 0.405225 + 0.233957i 1.03979 + 2.63037i −0.201550 0.116365i −2.88273 2.23691i
29.13 −0.0791950 + 1.41199i 0.815998 3.04535i −1.98746 0.223646i 0.626691 0.626691i 4.23539 + 1.39336i −1.94682 1.12400i 0.473183 2.78857i −6.01021 3.47000i 0.835254 + 0.934515i
29.14 −0.0263250 1.41397i 0.465695 1.73800i −1.99861 + 0.0744453i −0.627944 + 0.627944i −2.46973 0.612725i −4.03746 2.33103i 0.157877 + 2.82402i −0.205684 0.118752i 0.904424 + 0.871363i
29.15 0.266053 + 1.38896i −0.0165565 + 0.0617898i −1.85843 + 0.739075i 2.60178 2.60178i −0.0902286 0.00655704i 2.68608 + 1.55081i −1.52099 2.38466i 2.59453 + 1.49795i 4.30598 + 2.92156i
29.16 0.298771 1.38229i −0.354336 + 1.32240i −1.82147 0.825978i −2.17499 + 2.17499i 1.72208 + 0.884891i 3.13079 + 1.80756i −1.68595 + 2.27103i 0.974889 + 0.562852i 2.35665 + 3.65630i
29.17 0.443624 1.34283i 0.502442 1.87514i −1.60640 1.19142i 1.07638 1.07638i −2.29510 1.50655i 2.22572 + 1.28502i −2.31252 + 1.62858i −0.665623 0.384298i −0.967891 1.92291i
29.18 0.586716 + 1.28676i −0.356544 + 1.33064i −1.31153 + 1.50993i −1.01509 + 1.01509i −1.92141 + 0.321920i −3.02737 1.74785i −2.71242 0.801727i 0.954598 + 0.551137i −1.90175 0.710612i
29.19 0.884002 + 1.10388i 0.301612 1.12563i −0.437081 + 1.95166i −2.02015 + 2.02015i 1.50918 0.662118i 2.08993 + 1.20662i −2.54077 + 1.24278i 1.42200 + 0.820993i −4.01580 0.444176i
29.20 0.995696 1.00429i −0.356961 + 1.33220i −0.0171782 1.99993i 1.82137 1.82137i 0.982481 + 1.68495i −1.32362 0.764192i −2.02560 1.97407i 0.950750 + 0.548916i −0.0156441 3.64270i
See next 80 embeddings (of 104 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
16.e even 4 1 inner
208.bj even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.2.bj.a 104
4.b odd 2 1 832.2.br.a 104
13.c even 3 1 inner 208.2.bj.a 104
16.e even 4 1 inner 208.2.bj.a 104
16.f odd 4 1 832.2.br.a 104
52.j odd 6 1 832.2.br.a 104
208.bg odd 12 1 832.2.br.a 104
208.bj even 12 1 inner 208.2.bj.a 104
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
208.2.bj.a 104 1.a even 1 1 trivial
208.2.bj.a 104 13.c even 3 1 inner
208.2.bj.a 104 16.e even 4 1 inner
208.2.bj.a 104 208.bj even 12 1 inner
832.2.br.a 104 4.b odd 2 1
832.2.br.a 104 16.f odd 4 1
832.2.br.a 104 52.j odd 6 1
832.2.br.a 104 208.bg odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(208, [\chi])\).