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Properties

Label 639.2.v.a
Level $639$
Weight $2$
Character orbit 639.v
Analytic conductor $5.102$
Analytic rank $0$
Dimension $120$
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] = [639,2,Mod(10,639)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(639, base_ring=CyclotomicField(70))
 
chi = DirichletCharacter(H, H._module([0, 34]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("639.10");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 639 = 3^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 639.v (of order \(35\), degree \(24\), 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: \(5.10244068916\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(5\) over \(\Q(\zeta_{35})\)
Twist minimal: no (minimal twist has level 71)
Sato-Tate group: $\mathrm{SU}(2)[C_{35}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 22 q^{2} - 18 q^{4} + 20 q^{5} - 27 q^{7} + 27 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 22 q^{2} - 18 q^{4} + 20 q^{5} - 27 q^{7} + 27 q^{8} - 8 q^{10} + 27 q^{11} - 31 q^{13} - 2 q^{14} + 30 q^{16} - 9 q^{17} - 31 q^{19} - 72 q^{20} - 24 q^{22} + 6 q^{23} - 42 q^{25} - 77 q^{26} - 18 q^{28} - q^{29} + 15 q^{31} + 10 q^{32} + 20 q^{34} - 74 q^{35} - 22 q^{37} + 39 q^{38} - 30 q^{40} - 39 q^{41} + 33 q^{43} + 23 q^{44} + 124 q^{46} - 6 q^{47} - 22 q^{49} + 15 q^{50} + 83 q^{52} + 30 q^{53} + 32 q^{55} - 11 q^{56} + 12 q^{58} + 80 q^{59} - 12 q^{61} + 68 q^{62} - 117 q^{64} + 89 q^{65} - 74 q^{67} + 67 q^{68} - 64 q^{70} + 115 q^{71} - 14 q^{73} + 70 q^{74} + 8 q^{76} - 47 q^{77} - 25 q^{79} + 103 q^{80} - 88 q^{82} + 20 q^{83} + 22 q^{85} - 159 q^{86} + 43 q^{88} - 33 q^{89} + 60 q^{91} + 2 q^{92} + 174 q^{94} - 20 q^{95} + 121 q^{97} - 128 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −1.59197 + 0.439356i 0 0.624443 − 0.373087i −0.346904 + 1.06766i 0 −0.160876 − 3.58218i 1.45238 − 1.51908i 0 0.0831780 − 1.85210i
10.2 −0.931498 + 0.257077i 0 −0.915297 + 0.546865i 1.09157 − 3.35950i 0 −0.0991455 − 2.20765i 2.04759 − 2.14161i 0 −0.153143 + 3.40999i
10.3 −0.440194 + 0.121486i 0 −1.53789 + 0.918844i −1.24312 + 3.82593i 0 0.213032 + 4.74353i 1.19649 − 1.25143i 0 0.0824179 − 1.83518i
10.4 0.713225 − 0.196838i 0 −1.24695 + 0.745020i 0.623421 − 1.91869i 0 0.0146541 + 0.326298i −1.76533 + 1.84639i 0 0.0669687 − 1.49117i
10.5 2.65702 − 0.733292i 0 4.80515 − 2.87095i 0.451356 − 1.38913i 0 0.118039 + 2.62834i 6.85254 − 7.16719i 0 0.180625 − 4.02193i
19.1 −0.871177 − 2.03822i 0 −2.01326 + 2.10571i 0.768083 + 2.36392i 0 −1.65970 − 1.45003i 1.89530 + 0.711319i 0 4.14905 − 3.62491i
19.2 −0.682008 − 1.59564i 0 −0.698796 + 0.730883i −0.984762 − 3.03079i 0 −1.95579 − 1.70872i −1.60645 − 0.602910i 0 −4.16442 + 3.63834i
19.3 −0.0758189 − 0.177387i 0 1.35641 − 1.41869i 0.0139966 + 0.0430772i 0 1.74206 + 1.52199i −0.715719 − 0.268614i 0 0.00658013 − 0.00574888i
19.4 0.569315 + 1.33198i 0 −0.0679233 + 0.0710422i −0.862898 − 2.65573i 0 −2.26960 − 1.98289i 2.57906 + 0.967939i 0 3.04611 − 2.66131i
19.5 0.996141 + 2.33059i 0 −3.05722 + 3.19760i 1.11904 + 3.44404i 0 1.50266 + 1.31283i −5.75184 − 2.15870i 0 −6.91193 + 6.03877i
64.1 −1.59197 − 0.439356i 0 0.624443 + 0.373087i −0.346904 − 1.06766i 0 −0.160876 + 3.58218i 1.45238 + 1.51908i 0 0.0831780 + 1.85210i
64.2 −0.931498 − 0.257077i 0 −0.915297 − 0.546865i 1.09157 + 3.35950i 0 −0.0991455 + 2.20765i 2.04759 + 2.14161i 0 −0.153143 − 3.40999i
64.3 −0.440194 − 0.121486i 0 −1.53789 − 0.918844i −1.24312 − 3.82593i 0 0.213032 − 4.74353i 1.19649 + 1.25143i 0 0.0824179 + 1.83518i
64.4 0.713225 + 0.196838i 0 −1.24695 − 0.745020i 0.623421 + 1.91869i 0 0.0146541 − 0.326298i −1.76533 − 1.84639i 0 0.0669687 + 1.49117i
64.5 2.65702 + 0.733292i 0 4.80515 + 2.87095i 0.451356 + 1.38913i 0 0.118039 − 2.62834i 6.85254 + 7.16719i 0 0.180625 + 4.02193i
73.1 −0.119211 + 2.65443i 0 −5.03985 − 0.453595i 0.171217 + 0.526952i 0 −1.94286 − 0.536196i 1.09150 − 8.05775i 0 −1.41917 + 0.391666i
73.2 −0.0417847 + 0.930409i 0 1.12803 + 0.101525i −0.322385 − 0.992198i 0 −2.48776 − 0.686577i −0.391630 + 2.89113i 0 0.936621 − 0.258491i
73.3 0.0135643 − 0.302033i 0 1.90091 + 0.171085i 1.01225 + 3.11539i 0 2.21035 + 0.610018i 0.158625 − 1.17102i 0 0.954681 − 0.263475i
73.4 0.0669265 − 1.49023i 0 −0.224368 − 0.0201935i −0.774775 − 2.38451i 0 −1.36859 − 0.377708i 0.355372 − 2.62346i 0 −3.60533 + 0.995008i
73.5 0.119796 − 2.66747i 0 −5.10911 − 0.459828i 0.490011 + 1.50810i 0 3.65834 + 1.00964i −1.12178 + 8.28134i 0 4.08151 − 1.12643i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.g even 35 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 639.2.v.a 120
3.b odd 2 1 71.2.g.a ✓ 120
71.g even 35 1 inner 639.2.v.a 120
213.n even 70 1 5041.2.a.t 60
213.o odd 70 1 71.2.g.a ✓ 120
213.o odd 70 1 5041.2.a.s 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.2.g.a ✓ 120 3.b odd 2 1
71.2.g.a ✓ 120 213.o odd 70 1
639.2.v.a 120 1.a even 1 1 trivial
639.2.v.a 120 71.g even 35 1 inner
5041.2.a.s 60 213.o odd 70 1
5041.2.a.t 60 213.n even 70 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{120} - 22 T_{2}^{119} + 246 T_{2}^{118} - 1877 T_{2}^{117} + 11024 T_{2}^{116} - 53104 T_{2}^{115} + \cdots + 54037201 \) acting on \(S_{2}^{\mathrm{new}}(639, [\chi])\). Copy content Toggle raw display