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Properties

Label 639.2.e.c
Level $639$
Weight $2$
Character orbit 639.e
Analytic conductor $5.102$
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] = [639,2,Mod(214,639)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(639, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("639.214");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 639 = 3^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 639.e (of order \(3\), degree \(2\), 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: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 2 q^{2} - q^{3} - 46 q^{4} + 2 q^{5} + 12 q^{6} - 17 q^{7} - 6 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 2 q^{2} - q^{3} - 46 q^{4} + 2 q^{5} + 12 q^{6} - 17 q^{7} - 6 q^{8} + 7 q^{9} + 10 q^{10} + 5 q^{11} - 3 q^{12} - 31 q^{13} + q^{14} - 12 q^{15} - 50 q^{16} - 10 q^{17} + 7 q^{18} + 56 q^{19} + 29 q^{20} - 28 q^{21} - 10 q^{22} + 3 q^{23} - 38 q^{24} - 62 q^{25} + 14 q^{26} + 17 q^{27} + 62 q^{28} + 9 q^{29} - 37 q^{30} - 52 q^{31} + 18 q^{32} - 2 q^{33} - 31 q^{34} - 14 q^{35} + 78 q^{36} + 70 q^{37} - 57 q^{38} + 17 q^{39} - 22 q^{40} + q^{41} - 43 q^{42} - 39 q^{43} + 22 q^{44} + 57 q^{45} + 42 q^{46} + 20 q^{47} - 18 q^{48} - 89 q^{49} + 18 q^{50} + 4 q^{51} - 69 q^{52} - 16 q^{53} + 112 q^{54} + 80 q^{55} + 8 q^{56} - 23 q^{57} - 35 q^{58} + 5 q^{59} + 39 q^{60} - 71 q^{61} + 102 q^{62} + 84 q^{63} + 186 q^{64} - 35 q^{65} - 79 q^{66} - 44 q^{67} + 12 q^{68} - 24 q^{69} - 25 q^{70} + 80 q^{71} + 109 q^{72} + 116 q^{73} + 28 q^{74} + 7 q^{75} - 73 q^{76} - 15 q^{77} - 104 q^{78} - 59 q^{79} - 164 q^{80} - 57 q^{81} + 108 q^{82} + 9 q^{83} + 4 q^{84} - 54 q^{85} - 42 q^{86} - 22 q^{87} - 41 q^{88} - 22 q^{89} + 182 q^{90} + 78 q^{91} + 10 q^{92} + 36 q^{93} - 31 q^{94} - 8 q^{95} - 144 q^{96} - 104 q^{97} + 100 q^{98} + 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
214.1 −1.36005 + 2.35568i 1.23903 + 1.21029i −2.69948 − 4.67564i 1.62873 + 2.82105i −4.53620 + 1.27269i −1.95659 + 3.38892i 9.24552 0.0703777 + 2.99917i −8.86065
214.2 −1.30843 + 2.26626i −1.28395 − 1.16253i −2.42397 − 4.19843i −1.69881 − 2.94243i 4.31455 − 1.38869i 1.90309 − 3.29624i 7.45263 0.297069 + 2.98526i 8.89111
214.3 −1.28366 + 2.22337i 0.441492 − 1.67484i −2.29559 − 3.97608i −0.468290 − 0.811102i 3.15706 + 3.13153i −1.13616 + 1.96788i 6.65241 −2.61017 − 1.47885i 2.40451
214.4 −1.26234 + 2.18644i 1.38222 − 1.04378i −2.18701 − 3.78801i 1.38662 + 2.40169i 0.537326 + 4.33974i 1.47513 − 2.55501i 5.99362 0.821054 − 2.88546i −7.00153
214.5 −1.19772 + 2.07451i −1.42543 + 0.983940i −1.86906 − 3.23730i −0.248174 − 0.429850i −0.333925 − 4.13556i −2.60510 + 4.51216i 4.16353 1.06372 − 2.80508i 1.18897
214.6 −1.15761 + 2.00503i −1.62526 − 0.598763i −1.68010 − 2.91002i −0.309840 − 0.536659i 3.08195 − 2.56557i −0.401024 + 0.694594i 3.14914 2.28296 + 1.94630i 1.43469
214.7 −1.01342 + 1.75530i −0.454420 − 1.67138i −1.05405 − 1.82566i 1.98751 + 3.44248i 3.39428 + 0.896168i 0.900865 − 1.56034i 0.219082 −2.58701 + 1.51901i −8.05675
214.8 −0.952775 + 1.65026i 1.67500 − 0.440892i −0.815562 − 1.41259i −1.71559 − 2.97149i −0.868311 + 3.18424i −1.98604 + 3.43991i −0.702912 2.61123 − 1.47699i 6.53829
214.9 −0.866095 + 1.50012i −1.02522 + 1.39604i −0.500241 − 0.866443i 0.446083 + 0.772638i −1.20629 − 2.74706i 1.85339 − 3.21017i −1.73135 −0.897844 − 2.86249i −1.54540
214.10 −0.825837 + 1.43039i −1.18875 + 1.25971i −0.364014 − 0.630491i −1.10416 − 1.91246i −0.820168 − 2.74070i 0.668525 − 1.15792i −2.10088 −0.173748 − 2.99496i 3.64742
214.11 −0.773700 + 1.34009i 1.01250 + 1.40529i −0.197225 − 0.341603i 0.0584390 + 0.101219i −2.66659 + 0.269559i −1.15194 + 1.99522i −2.48443 −0.949701 + 2.84571i −0.180857
214.12 −0.697611 + 1.20830i −1.72145 − 0.191349i 0.0266787 + 0.0462089i 0.625279 + 1.08301i 1.43211 − 1.94653i 2.13046 − 3.69007i −2.86489 2.92677 + 0.658796i −1.74481
214.13 −0.587656 + 1.01785i 1.14258 + 1.30173i 0.309320 + 0.535759i −1.91279 − 3.31305i −1.99641 + 0.398009i −0.398852 + 0.690833i −3.07772 −0.389005 + 2.97467i 4.49625
214.14 −0.575128 + 0.996150i −1.15856 − 1.28753i 0.338456 + 0.586223i 0.880940 + 1.52583i 1.94890 − 0.413604i −2.52643 + 4.37591i −3.07913 −0.315479 + 2.98337i −2.02661
214.15 −0.471484 + 0.816634i 1.07893 − 1.35495i 0.555406 + 0.961991i 1.40492 + 2.43340i 0.597802 + 1.51993i −0.0661639 + 0.114599i −2.93340 −0.671805 − 2.92381i −2.64959
214.16 −0.408974 + 0.708363i 0.538260 + 1.64629i 0.665481 + 1.15265i 1.30084 + 2.25311i −1.38631 − 0.292006i −0.373723 + 0.647308i −2.72455 −2.42055 + 1.77227i −2.12803
214.17 −0.257079 + 0.445273i 1.72325 − 0.174379i 0.867821 + 1.50311i −1.16991 − 2.02634i −0.365365 + 0.812146i 1.56398 − 2.70890i −1.92071 2.93918 − 0.600997i 1.20303
214.18 −0.157476 + 0.272757i 0.574136 − 1.63413i 0.950402 + 1.64615i −1.21380 − 2.10236i 0.355307 + 0.413936i −2.47252 + 4.28253i −1.22857 −2.34074 − 1.87642i 0.764577
214.19 −0.0750929 + 0.130065i 1.59972 − 0.663993i 0.988722 + 1.71252i −0.786266 − 1.36185i −0.0337658 + 0.257929i 0.292078 − 0.505893i −0.597356 2.11823 − 2.12441i 0.236172
214.20 0.0383906 − 0.0664944i −1.62826 + 0.590554i 0.997052 + 1.72695i 0.217271 + 0.376325i −0.0232415 + 0.130942i 0.974662 − 1.68816i 0.306672 2.30249 − 1.92316i 0.0333647
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 214.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 639.2.e.c ✓ 80
9.c even 3 1 inner 639.2.e.c ✓ 80
9.c even 3 1 5751.2.a.k 40
9.d odd 6 1 5751.2.a.l 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
639.2.e.c ✓ 80 1.a even 1 1 trivial
639.2.e.c ✓ 80 9.c even 3 1 inner
5751.2.a.k 40 9.c even 3 1
5751.2.a.l 40 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{80} - 2 T_{2}^{79} + 65 T_{2}^{78} - 120 T_{2}^{77} + 2258 T_{2}^{76} - 3916 T_{2}^{75} + \cdots + 3272481 \) acting on \(S_{2}^{\mathrm{new}}(639, [\chi])\). Copy content Toggle raw display