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Properties

Label 206.2.e.b
Level $206$
Weight $2$
Character orbit 206.e
Analytic conductor $1.645$
Analytic rank $0$
Dimension $64$
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: \(64\)
Relative dimension: \(4\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 4 q^{2} + 3 q^{3} - 4 q^{4} - 15 q^{5} - 3 q^{6} + 5 q^{7} + 4 q^{8} - 43 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 4 q^{2} + 3 q^{3} - 4 q^{4} - 15 q^{5} - 3 q^{6} + 5 q^{7} + 4 q^{8} - 43 q^{9} - 2 q^{10} - 2 q^{11} + 3 q^{12} - 28 q^{13} + 29 q^{14} - 36 q^{15} - 4 q^{16} + 11 q^{17} + 26 q^{18} - 27 q^{19} + 2 q^{20} + 9 q^{21} - 32 q^{22} + 22 q^{23} - 20 q^{24} + 25 q^{25} - 23 q^{26} + 15 q^{27} + 5 q^{28} - 5 q^{29} + 70 q^{30} - 20 q^{31} + 4 q^{32} + 8 q^{33} + 6 q^{34} - 66 q^{35} + 25 q^{36} + 19 q^{37} + 27 q^{38} + 81 q^{39} - 2 q^{40} + q^{41} - 9 q^{42} + 7 q^{43} - 19 q^{44} + 101 q^{45} - 22 q^{46} - 26 q^{47} - 14 q^{48} - 29 q^{49} - 8 q^{50} + 37 q^{51} - 28 q^{52} - 57 q^{53} - 15 q^{54} - 99 q^{55} + 12 q^{56} - 20 q^{57} - 12 q^{58} - 128 q^{59} - 19 q^{60} - 111 q^{61} - 48 q^{62} - 37 q^{63} - 4 q^{64} - 81 q^{65} + 43 q^{66} + 69 q^{67} - 6 q^{68} + 30 q^{69} - 87 q^{70} + 37 q^{71} + 9 q^{72} - 4 q^{73} - 19 q^{74} - 86 q^{75} + 7 q^{76} + 46 q^{77} - 64 q^{78} - 58 q^{79} + 2 q^{80} - 45 q^{81} - 35 q^{82} + 52 q^{83} + 9 q^{84} + 28 q^{85} - 41 q^{86} + 139 q^{87} + 19 q^{88} + 53 q^{89} + 52 q^{90} + 37 q^{91} + 22 q^{92} + 258 q^{93} + 43 q^{94} + 245 q^{95} - 3 q^{96} + 35 q^{97} + 12 q^{98} + 21 q^{99}+O(q^{100}) \) 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 −2.38957 + 1.47956i −0.602635 − 0.798017i −2.31917 + 2.11420i −0.259324 − 2.79855i 0.496161 + 0.0927485i 0.982973 − 0.183750i 2.18373 − 4.38551i −0.858812 − 3.01841i
9.2 −0.445738 + 0.895163i −0.379399 + 0.234914i −0.602635 − 0.798017i −0.474886 + 0.432915i −0.0411737 − 0.444335i −4.11414 − 0.769067i 0.982973 − 0.183750i −1.24846 + 2.50724i −0.175855 − 0.618067i
9.3 −0.445738 + 0.895163i 1.32727 − 0.821812i −0.602635 − 0.798017i −0.693515 + 0.632223i 0.144040 + 1.55444i 2.60915 + 0.487734i 0.982973 − 0.183750i −0.250938 + 0.503952i −0.256816 − 0.902616i
9.4 −0.445738 + 0.895163i 1.32788 − 0.822187i −0.602635 − 0.798017i 2.39530 − 2.18360i 0.144106 + 1.55515i −2.16747 − 0.405170i 0.982973 − 0.183750i −0.249947 + 0.501962i 0.887005 + 3.11750i
13.1 0.982973 + 0.183750i −0.276165 + 2.98029i 0.932472 + 0.361242i −0.602025 + 0.797210i −0.819090 + 2.87880i 2.01843 − 1.24976i 0.850217 + 0.526432i −5.85695 − 1.09485i −0.738261 + 0.673014i
13.2 0.982973 + 0.183750i −0.136551 + 1.47362i 0.932472 + 0.361242i 2.14368 − 2.83869i −0.405002 + 1.42344i −0.652470 + 0.403992i 0.850217 + 0.526432i 0.796015 + 0.148801i 2.62879 − 2.39646i
13.3 0.982973 + 0.183750i 0.0269455 − 0.290788i 0.932472 + 0.361242i −2.65137 + 3.51098i 0.0799188 − 0.280885i 3.64912 − 2.25944i 0.850217 + 0.526432i 2.86509 + 0.535578i −3.25137 + 2.96402i
13.4 0.982973 + 0.183750i 0.227314 − 2.45311i 0.932472 + 0.361242i 0.383381 − 0.507678i 0.674201 − 2.36957i −0.557809 + 0.345381i 0.850217 + 0.526432i −3.01714 − 0.564002i 0.470138 − 0.428588i
23.1 −0.445738 − 0.895163i −2.38957 − 1.47956i −0.602635 + 0.798017i −2.31917 − 2.11420i −0.259324 + 2.79855i 0.496161 − 0.0927485i 0.982973 + 0.183750i 2.18373 + 4.38551i −0.858812 + 3.01841i
23.2 −0.445738 − 0.895163i −0.379399 − 0.234914i −0.602635 + 0.798017i −0.474886 − 0.432915i −0.0411737 + 0.444335i −4.11414 + 0.769067i 0.982973 + 0.183750i −1.24846 − 2.50724i −0.175855 + 0.618067i
23.3 −0.445738 − 0.895163i 1.32727 + 0.821812i −0.602635 + 0.798017i −0.693515 − 0.632223i 0.144040 − 1.55444i 2.60915 − 0.487734i 0.982973 + 0.183750i −0.250938 − 0.503952i −0.256816 + 0.902616i
23.4 −0.445738 − 0.895163i 1.32788 + 0.822187i −0.602635 + 0.798017i 2.39530 + 2.18360i 0.144106 − 1.55515i −2.16747 + 0.405170i 0.982973 + 0.183750i −0.249947 − 0.501962i 0.887005 − 3.11750i
61.1 −0.0922684 + 0.995734i −2.28041 + 2.07887i −0.982973 − 0.183750i 1.84955 + 3.71439i −1.85959 − 2.46250i −0.0654303 − 0.229964i 0.273663 − 0.961826i 0.601772 − 6.49416i −3.86920 + 1.49894i
61.2 −0.0922684 + 0.995734i −1.15018 + 1.04853i −0.982973 − 0.183750i −1.15566 − 2.32088i −0.937929 − 1.24202i −1.34878 − 4.74045i 0.273663 − 0.961826i −0.0533016 + 0.575216i 2.41761 − 0.936586i
61.3 −0.0922684 + 0.995734i −0.220408 + 0.200928i −0.982973 − 0.183750i −0.404003 − 0.811347i −0.179735 − 0.238007i 0.974975 + 3.42668i 0.273663 − 0.961826i −0.268598 + 2.89863i 0.845163 − 0.327418i
61.4 −0.0922684 + 0.995734i 2.05621 − 1.87449i −0.982973 − 0.183750i −0.687248 − 1.38018i 1.67677 + 2.22040i −0.695526 − 2.44452i 0.273663 − 0.961826i 0.437511 − 4.72149i 1.43770 − 0.556970i
79.1 −0.739009 + 0.673696i −2.41027 + 0.933745i 0.0922684 − 0.995734i 3.52750 + 2.18413i 1.15215 − 2.31384i 0.253744 − 0.336012i 0.602635 + 0.798017i 2.72051 − 2.48007i −4.07829 + 0.762365i
79.2 −0.739009 + 0.673696i −1.65726 + 0.642028i 0.0922684 − 0.995734i −1.15824 − 0.717150i 0.792202 − 1.59096i 1.33477 − 1.76753i 0.602635 + 0.798017i 0.117300 − 0.106933i 1.33909 − 0.250319i
79.3 −0.739009 + 0.673696i −0.325518 + 0.126106i 0.0922684 − 0.995734i −1.12247 − 0.695006i 0.155603 − 0.312493i −1.95461 + 2.58832i 0.602635 + 0.798017i −2.12697 + 1.93899i 1.29774 − 0.242590i
79.4 −0.739009 + 0.673696i 2.32995 − 0.902629i 0.0922684 − 0.995734i −2.69252 − 1.66714i −1.11376 + 2.23673i 1.31156 − 1.73679i 0.602635 + 0.798017i 2.39692 − 2.18508i 3.11294 − 0.581910i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.4
Significant digits:
Format:

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.b ✓ 64
103.e even 17 1 inner 206.2.e.b ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
206.2.e.b ✓ 64 1.a even 1 1 trivial
206.2.e.b ✓ 64 103.e even 17 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{64} - 3 T_{3}^{63} + 32 T_{3}^{62} - 98 T_{3}^{61} + 626 T_{3}^{60} - 1940 T_{3}^{59} + \cdots + 75151561 \) acting on \(S_{2}^{\mathrm{new}}(206, [\chi])\). Copy content Toggle raw display