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Properties

Label 206.2.e.c
Level $206$
Weight $2$
Character orbit 206.e
Analytic conductor $1.645$
Analytic rank $0$
Dimension $80$
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] = [206,2,Mod(9,206)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(206, base_ring=CyclotomicField(34))
 
chi = DirichletCharacter(H, H._module([26]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("206.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 206 = 2 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 206.e (of order \(17\), degree \(16\), 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: \(1.64491828164\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(5\) over \(\Q(\zeta_{17})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{17}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 5 q^{2} - 6 q^{3} - 5 q^{4} - 6 q^{5} - 6 q^{6} - 10 q^{7} - 5 q^{8} - 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 5 q^{2} - 6 q^{3} - 5 q^{4} - 6 q^{5} - 6 q^{6} - 10 q^{7} - 5 q^{8} - 15 q^{9} + 11 q^{10} - 16 q^{11} - 6 q^{12} - 14 q^{13} + 7 q^{14} - 3 q^{15} - 5 q^{16} - 4 q^{17} - 15 q^{18} - 3 q^{19} + 11 q^{20} + 14 q^{21} + q^{22} - 9 q^{23} + 11 q^{24} + 9 q^{25} + 20 q^{26} - 30 q^{27} - 10 q^{28} - 13 q^{29} + 14 q^{30} - 40 q^{31} - 5 q^{32} - 32 q^{33} - 4 q^{34} + 30 q^{35} + 19 q^{36} + 37 q^{37} - 3 q^{38} - 30 q^{39} - 6 q^{40} + 10 q^{41} + 14 q^{42} - 48 q^{43} - 16 q^{44} - 17 q^{45} - 26 q^{46} + 20 q^{47} + 11 q^{48} - 33 q^{49} + 26 q^{50} - 28 q^{51} - 14 q^{52} + 103 q^{53} - 30 q^{54} + 90 q^{55} - 10 q^{56} + 38 q^{57} + 4 q^{58} + 119 q^{59} - 20 q^{60} + 27 q^{61} + 11 q^{62} - 63 q^{63} - 5 q^{64} + 12 q^{65} + 36 q^{66} - 35 q^{67} - 4 q^{68} - 16 q^{69} - 21 q^{70} + 26 q^{71} - 15 q^{72} - 52 q^{73} - 31 q^{74} - 6 q^{75} - 20 q^{76} + 40 q^{77} - 64 q^{78} - 98 q^{79} - 6 q^{80} - 101 q^{81} - 24 q^{82} - 79 q^{83} + 31 q^{84} - 84 q^{85} + 37 q^{86} - 86 q^{87} + q^{88} - 43 q^{89} + 17 q^{90} + 106 q^{91} - 9 q^{92} + 140 q^{93} + 3 q^{94} + 73 q^{95} - 6 q^{96} + 44 q^{97} + 18 q^{98} - 95 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0.445738 − 0.895163i −2.38175 + 1.47472i −0.602635 − 0.798017i 1.99011 − 1.81422i 0.258475 + 2.78939i 0.330560 + 0.0617924i −0.982973 + 0.183750i 2.16073 − 4.33932i −0.736959 − 2.59014i
9.2 0.445738 − 0.895163i −1.50572 + 0.932303i −0.602635 − 0.798017i −1.89493 + 1.72746i 0.163406 + 1.76343i 4.14833 + 0.775457i −0.982973 + 0.183750i 0.0607927 − 0.122088i 0.701714 + 2.46627i
9.3 0.445738 − 0.895163i −1.00377 + 0.621506i −0.602635 − 0.798017i −2.09986 + 1.91428i 0.108932 + 1.17556i −4.58742 − 0.857538i −0.982973 + 0.183750i −0.715938 + 1.43780i 0.777601 + 2.73299i
9.4 0.445738 − 0.895163i 0.936341 − 0.579758i −0.602635 − 0.798017i 2.06232 − 1.88005i −0.101615 − 1.09660i −0.172475 − 0.0322412i −0.982973 + 0.183750i −0.796599 + 1.59979i −0.763699 − 2.68413i
9.5 0.445738 − 0.895163i 1.46317 − 0.905958i −0.602635 − 0.798017i −0.436947 + 0.398330i −0.158788 − 1.71360i 2.79414 + 0.522315i −0.982973 + 0.183750i −0.0171027 + 0.0343468i 0.161806 + 0.568690i
13.1 −0.982973 − 0.183750i −0.206838 + 2.23214i 0.932472 + 0.361242i −1.86254 + 2.46641i 0.613471 − 2.15613i 0.928603 − 0.574967i −0.850217 − 0.526432i −1.99075 − 0.372135i 2.28403 − 2.08217i
13.2 −0.982973 − 0.183750i −0.0839719 + 0.906201i 0.932472 + 0.361242i 0.366347 − 0.485122i 0.249056 − 0.875341i −0.232843 + 0.144170i −0.850217 − 0.526432i 2.13477 + 0.399058i −0.449250 + 0.409546i
13.3 −0.982973 − 0.183750i 0.0461059 − 0.497561i 0.932472 + 0.361242i 2.02935 − 2.68730i −0.136747 + 0.480618i −1.33690 + 0.827773i −0.850217 − 0.526432i 2.70348 + 0.505368i −2.48859 + 2.26865i
13.4 −0.982973 − 0.183750i 0.222092 − 2.39676i 0.932472 + 0.361242i −2.30884 + 3.05739i −0.658713 + 2.31514i −3.61214 + 2.23654i −0.850217 − 0.526432i −2.74619 − 0.513352i 2.83132 − 2.58109i
13.5 −0.982973 − 0.183750i 0.232315 − 2.50708i 0.932472 + 0.361242i 0.704447 − 0.932838i −0.689034 + 2.42170i 2.06128 − 1.27629i −0.850217 − 0.526432i −3.28255 − 0.613614i −0.863861 + 0.787513i
23.1 0.445738 + 0.895163i −2.38175 − 1.47472i −0.602635 + 0.798017i 1.99011 + 1.81422i 0.258475 − 2.78939i 0.330560 − 0.0617924i −0.982973 − 0.183750i 2.16073 + 4.33932i −0.736959 + 2.59014i
23.2 0.445738 + 0.895163i −1.50572 − 0.932303i −0.602635 + 0.798017i −1.89493 − 1.72746i 0.163406 − 1.76343i 4.14833 − 0.775457i −0.982973 − 0.183750i 0.0607927 + 0.122088i 0.701714 − 2.46627i
23.3 0.445738 + 0.895163i −1.00377 − 0.621506i −0.602635 + 0.798017i −2.09986 − 1.91428i 0.108932 − 1.17556i −4.58742 + 0.857538i −0.982973 − 0.183750i −0.715938 − 1.43780i 0.777601 − 2.73299i
23.4 0.445738 + 0.895163i 0.936341 + 0.579758i −0.602635 + 0.798017i 2.06232 + 1.88005i −0.101615 + 1.09660i −0.172475 + 0.0322412i −0.982973 − 0.183750i −0.796599 − 1.59979i −0.763699 + 2.68413i
23.5 0.445738 + 0.895163i 1.46317 + 0.905958i −0.602635 + 0.798017i −0.436947 − 0.398330i −0.158788 + 1.71360i 2.79414 − 0.522315i −0.982973 − 0.183750i −0.0171027 − 0.0343468i 0.161806 − 0.568690i
61.1 0.0922684 − 0.995734i −1.40176 + 1.27788i −0.982973 − 0.183750i 0.478639 + 0.961236i 1.14309 + 1.51369i −0.0843891 − 0.296597i −0.273663 + 0.961826i 0.0551677 − 0.595354i 1.00130 − 0.387905i
61.2 0.0922684 − 0.995734i −0.0613685 + 0.0559448i −0.982973 − 0.183750i 1.23120 + 2.47258i 0.0500437 + 0.0662686i 1.25007 + 4.39353i −0.273663 + 0.961826i −0.276169 + 2.98034i 2.57563 − 0.997804i
61.3 0.0922684 − 0.995734i 0.717345 − 0.653946i −0.982973 − 0.183750i −0.629557 − 1.26432i −0.584969 − 0.774623i −0.767849 − 2.69871i −0.273663 + 0.961826i −0.189867 + 2.04899i −1.31702 + 0.510215i
61.4 0.0922684 − 0.995734i 2.07805 − 1.89439i −0.982973 − 0.183750i 1.83324 + 3.68163i −1.69457 − 2.24398i −0.959753 − 3.37318i −0.273663 + 0.961826i 0.452761 − 4.88607i 3.83508 − 1.48572i
61.5 0.0922684 − 0.995734i 2.18278 − 1.98986i −0.982973 − 0.183750i −1.43472 − 2.88131i −1.77997 − 2.35707i 1.08616 + 3.81745i −0.273663 + 0.961826i 0.528150 − 5.69964i −3.00140 + 1.16275i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.5
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
103.e even 17 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 206.2.e.c ✓ 80
103.e even 17 1 inner 206.2.e.c ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
206.2.e.c ✓ 80 1.a even 1 1 trivial
206.2.e.c ✓ 80 103.e even 17 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{80} + 6 T_{3}^{79} + 33 T_{3}^{78} + 148 T_{3}^{77} + 650 T_{3}^{76} + 2630 T_{3}^{75} + \cdots + 871546299373369 \) acting on \(S_{2}^{\mathrm{new}}(206, [\chi])\). Copy content Toggle raw display