Properties

Label 133.2.bf.a
Level $133$
Weight $2$
Character orbit 133.bf
Analytic conductor $1.062$
Analytic rank $0$
Dimension $66$
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] = [133,2,Mod(10,133)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(133, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([3, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("133.10");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 133 = 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 133.bf (of order \(18\), degree \(6\), 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: \(1.06201034688\)
Analytic rank: \(0\)
Dimension: \(66\)
Relative dimension: \(11\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 66 q - 3 q^{2} - 9 q^{3} - 3 q^{4} - 9 q^{5} + 3 q^{7} - 18 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q - 3 q^{2} - 9 q^{3} - 3 q^{4} - 9 q^{5} + 3 q^{7} - 18 q^{8} - 3 q^{9} - 9 q^{10} - 12 q^{11} + 6 q^{12} - 30 q^{13} - 15 q^{14} + 9 q^{15} - 15 q^{16} + 18 q^{17} + 36 q^{18} + 12 q^{19} - 30 q^{21} - 3 q^{23} - 36 q^{24} - 27 q^{25} + 12 q^{27} - 33 q^{28} - 6 q^{29} + 3 q^{30} - 9 q^{31} + 60 q^{32} - 9 q^{33} - 36 q^{34} + 9 q^{35} + 27 q^{36} - 36 q^{37} + 18 q^{38} + 12 q^{39} + 9 q^{40} + 54 q^{41} - 9 q^{42} + 12 q^{43} + 18 q^{44} - 27 q^{45} + 45 q^{47} + 63 q^{48} - 45 q^{49} - 63 q^{50} - 3 q^{51} + 57 q^{52} + 27 q^{53} - 9 q^{54} - 45 q^{55} - 54 q^{56} - 54 q^{57} + 30 q^{58} + 36 q^{59} - 78 q^{60} - 42 q^{61} - 45 q^{62} + 57 q^{63} - 36 q^{64} + 45 q^{65} + 9 q^{66} + 30 q^{67} - 9 q^{68} + 69 q^{70} - 6 q^{71} - 6 q^{72} + 60 q^{73} + 9 q^{74} - 21 q^{75} + 54 q^{76} - 18 q^{77} + 3 q^{78} + 27 q^{79} - 45 q^{80} + 24 q^{81} - 9 q^{82} + 36 q^{83} + 99 q^{84} - 48 q^{85} - 48 q^{86} - 9 q^{88} - 9 q^{89} - 18 q^{90} + 24 q^{91} + 48 q^{92} - 3 q^{93} + 90 q^{94} - 75 q^{95} + 63 q^{96} - 27 q^{97} + 96 q^{98} + 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −1.61995 1.93058i 0.0453294 + 0.257076i −0.755612 + 4.28529i 4.11821 0.726151i 0.422875 0.503963i −0.270204 + 2.63192i 5.13205 2.96299i 2.75504 1.00275i −8.07319 6.77421i
10.2 −1.52009 1.81157i −0.439106 2.49029i −0.623824 + 3.53788i −1.04499 + 0.184260i −3.84386 + 4.58094i −0.591482 2.57879i 3.26138 1.88296i −3.18966 + 1.16094i 1.92228 + 1.61298i
10.3 −1.27330 1.51746i 0.476887 + 2.70456i −0.334098 + 1.89477i −2.20568 + 0.388921i 3.49685 4.16738i −2.50884 + 0.840077i −0.130386 + 0.0752786i −4.26815 + 1.55348i 3.39868 + 2.85183i
10.4 −0.790508 0.942090i 0.0284469 + 0.161330i 0.0846645 0.480156i 0.850790 0.150017i 0.129500 0.154333i −0.990946 2.45317i −2.64937 + 1.52962i 2.79386 1.01688i −0.813886 0.682931i
10.5 −0.432162 0.515031i −0.514014 2.91512i 0.268804 1.52446i 0.960858 0.169425i −1.27924 + 1.52454i −0.595102 + 2.57796i −2.06581 + 1.19270i −5.41462 + 1.97076i −0.502506 0.421652i
10.6 −0.373593 0.445230i 0.289267 + 1.64052i 0.288638 1.63695i 0.294690 0.0519618i 0.622340 0.741676i 2.63474 + 0.241108i −1.84333 + 1.06425i 0.211456 0.0769638i −0.133229 0.111792i
10.7 0.414484 + 0.493962i −0.315845 1.79125i 0.275094 1.56014i −3.56788 + 0.629114i 0.753896 0.898458i 2.01676 1.71251i 2.00154 1.15559i −0.289728 + 0.105452i −1.78959 1.50164i
10.8 0.480891 + 0.573103i −0.0906786 0.514264i 0.250105 1.41842i 1.59609 0.281434i 0.251120 0.299273i −2.47659 + 0.930864i 2.22898 1.28690i 2.56283 0.932795i 0.928835 + 0.779385i
10.9 0.818304 + 0.975217i 0.347752 + 1.97220i 0.0658701 0.373568i −2.34435 + 0.413372i −1.63876 + 1.95299i 0.0571233 + 2.64513i 2.62321 1.51451i −0.949566 + 0.345614i −2.32152 1.94798i
10.10 1.48693 + 1.77205i −0.429930 2.43826i −0.581918 + 3.30022i 0.765719 0.135017i 3.68144 4.38737i 1.90031 + 1.84088i −2.70677 + 1.56275i −2.94117 + 1.07050i 1.37783 + 1.15613i
10.11 1.54295 + 1.83882i 0.0415830 + 0.235829i −0.653260 + 3.70482i −0.923449 + 0.162829i −0.369487 + 0.440337i −1.91360 1.82706i −3.66282 + 2.11473i 2.76519 1.00645i −1.72425 1.44682i
33.1 −2.42618 0.427801i 1.81066 0.659027i 3.82394 + 1.39180i −0.965340 2.65225i −4.67492 + 0.824314i 2.39627 + 1.12156i −4.41506 2.54904i 0.546042 0.458184i 1.20745 + 6.84780i
33.2 −1.99215 0.351269i −0.662476 + 0.241122i 1.96587 + 0.715519i 0.0536790 + 0.147482i 1.40445 0.247642i −1.17382 2.37111i −0.161239 0.0930914i −1.91740 + 1.60889i −0.0551306 0.312661i
33.3 −1.84540 0.325394i −2.57736 + 0.938083i 1.42023 + 0.516922i −0.854536 2.34782i 5.06151 0.892480i −1.55266 + 2.14225i 0.792944 + 0.457807i 3.46466 2.90719i 0.812995 + 4.61072i
33.4 −0.928599 0.163737i −0.187541 + 0.0682594i −1.04390 0.379948i 0.803754 + 2.20830i 0.185327 0.0326782i 2.64571 0.0147547i 2.54034 + 1.46667i −2.26762 + 1.90276i −0.384786 2.18223i
33.5 −0.697238 0.122942i 2.27698 0.828751i −1.40836 0.512601i −0.581955 1.59891i −1.68948 + 0.297901i −0.813635 2.51754i 2.14522 + 1.23854i 2.19966 1.84573i 0.209188 + 1.18637i
33.6 −0.0226089 0.00398656i −1.47792 + 0.537920i −1.87889 0.683860i 0.513048 + 1.40959i 0.0355586 0.00626995i −1.84251 + 1.89873i 0.0795172 + 0.0459093i −0.403236 + 0.338355i −0.00598004 0.0339145i
33.7 0.397723 + 0.0701294i −1.42173 + 0.517468i −1.72612 0.628256i −1.40124 3.84988i −0.601745 + 0.106104i 2.51181 0.831138i −1.34196 0.774783i −0.544588 + 0.456963i −0.287317 1.62946i
33.8 1.19145 + 0.210084i 1.79234 0.652360i −0.503976 0.183432i −0.404136 1.11035i 2.27253 0.400709i 0.366724 + 2.62021i −2.65741 1.53425i 0.488789 0.410143i −0.248238 1.40783i
33.9 1.20110 + 0.211786i 1.44898 0.527386i −0.481603 0.175289i 1.20912 + 3.32203i 1.85206 0.326568i −1.34413 2.27889i −2.65378 1.53216i −0.476721 + 0.400016i 0.748713 + 4.24616i
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.bf even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 133.2.bf.a yes 66
7.b odd 2 1 931.2.bj.a 66
7.c even 3 1 931.2.be.a 66
7.c even 3 1 931.2.bf.a 66
7.d odd 6 1 133.2.bb.a 66
7.d odd 6 1 931.2.be.b 66
19.f odd 18 1 133.2.bb.a 66
133.ba even 18 1 931.2.bf.a 66
133.bb even 18 1 931.2.be.a 66
133.bd odd 18 1 931.2.be.b 66
133.be odd 18 1 931.2.bj.a 66
133.bf even 18 1 inner 133.2.bf.a yes 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.bb.a 66 7.d odd 6 1
133.2.bb.a 66 19.f odd 18 1
133.2.bf.a yes 66 1.a even 1 1 trivial
133.2.bf.a yes 66 133.bf even 18 1 inner
931.2.be.a 66 7.c even 3 1
931.2.be.a 66 133.bb even 18 1
931.2.be.b 66 7.d odd 6 1
931.2.be.b 66 133.bd odd 18 1
931.2.bf.a 66 7.c even 3 1
931.2.bf.a 66 133.ba even 18 1
931.2.bj.a 66 7.b odd 2 1
931.2.bj.a 66 133.be odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(133, [\chi])\).