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Properties

Label 639.2.e.b
Level $639$
Weight $2$
Character orbit 639.e
Analytic conductor $5.102$
Analytic rank $0$
Dimension $56$
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: \(56\)
Relative dimension: \(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - q^{2} - q^{3} - 21 q^{4} - 2 q^{5} + 3 q^{6} + 11 q^{7} + 6 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - q^{2} - q^{3} - 21 q^{4} - 2 q^{5} + 3 q^{6} + 11 q^{7} + 6 q^{8} - 5 q^{9} - 32 q^{10} + q^{11} + 6 q^{12} + 19 q^{13} + 2 q^{14} - 7 q^{16} - 6 q^{17} + 2 q^{18} - 60 q^{19} - 13 q^{20} - 6 q^{21} + 17 q^{22} + 3 q^{23} - 3 q^{24} - 16 q^{26} - 7 q^{27} - 64 q^{28} + 3 q^{29} + 26 q^{30} + 28 q^{31} + 3 q^{32} + 20 q^{33} + 33 q^{34} - 10 q^{35} - 73 q^{36} - 46 q^{37} + 67 q^{38} + 15 q^{39} + 42 q^{40} + 3 q^{41} + 32 q^{42} + 19 q^{43} + 4 q^{44} - 11 q^{45} - 72 q^{46} - 10 q^{47} + 34 q^{48} + 33 q^{49} - 14 q^{50} + 24 q^{51} + 46 q^{52} + 20 q^{53} - 117 q^{54} - 52 q^{55} + 5 q^{57} + 22 q^{58} + 7 q^{59} + 81 q^{60} + 67 q^{61} + 44 q^{62} - 36 q^{63} - 30 q^{64} + 21 q^{65} + 18 q^{66} + 22 q^{67} - 6 q^{68} + 28 q^{70} - 56 q^{71} - 105 q^{72} - 164 q^{73} - 67 q^{74} - 9 q^{75} + 83 q^{76} + 3 q^{77} + 122 q^{78} + 37 q^{79} + 2 q^{80} - 53 q^{81} - 46 q^{82} - 15 q^{83} + 80 q^{84} + 44 q^{85} - 15 q^{86} + 52 q^{87} + 51 q^{88} - 2 q^{89} - 94 q^{90} - 74 q^{91} + 61 q^{92} - 4 q^{93} + 58 q^{94} - 14 q^{95} + 119 q^{96} + 74 q^{97} + 24 q^{98} - 8 q^{99}+O(q^{100}) \) 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 −1.38189 + 2.39351i −1.45058 + 0.946482i −2.81925 − 4.88309i 0.884609 + 1.53219i −0.260873 − 4.77990i 1.49853 − 2.59554i 10.0560 1.20834 − 2.74589i −4.88974
214.2 −1.21161 + 2.09857i −0.579181 − 1.63234i −1.93599 − 3.35322i 0.189277 + 0.327837i 4.12732 + 0.762311i −1.02077 + 1.76803i 4.53619 −2.32910 + 1.89085i −0.917317
214.3 −1.11686 + 1.93446i 1.56939 + 0.732820i −1.49476 − 2.58900i −0.451233 − 0.781558i −3.17040 + 2.21746i 1.47032 − 2.54666i 2.21033 1.92595 + 2.30016i 2.01586
214.4 −1.09527 + 1.89706i 0.189015 + 1.72171i −1.39921 − 2.42351i 0.0632411 + 0.109537i −3.47319 − 1.52715i −0.0160222 + 0.0277512i 1.74898 −2.92855 + 0.650857i −0.277063
214.5 −0.946446 + 1.63929i 0.966725 − 1.43717i −0.791518 − 1.37095i −1.78324 − 3.08865i 1.44098 + 2.94494i 2.28098 − 3.95078i −0.789266 −1.13089 − 2.77869i 6.75094
214.6 −0.927694 + 1.60681i 1.68586 − 0.397344i −0.721233 − 1.24921i 0.850825 + 1.47367i −0.925502 + 3.07747i −1.29390 + 2.24111i −1.03444 2.68423 − 1.33973i −3.15722
214.7 −0.879866 + 1.52397i −0.913890 + 1.47133i −0.548327 − 0.949730i 2.17616 + 3.76921i −1.43816 − 2.68731i −0.815176 + 1.41193i −1.58965 −1.32961 − 2.68926i −7.65890
214.8 −0.763857 + 1.32304i −1.72660 − 0.137352i −0.166956 − 0.289176i −1.63625 − 2.83406i 1.50059 − 2.17944i −0.600451 + 1.04001i −2.54531 2.96227 + 0.474301i 4.99943
214.9 −0.717606 + 1.24293i 0.519374 − 1.65235i −0.0299153 − 0.0518148i −0.169516 − 0.293611i 1.68104 + 1.83128i 0.112211 − 0.194354i −2.78455 −2.46050 − 1.71637i 0.486583
214.10 −0.504378 + 0.873609i −0.977673 − 1.42974i 0.491205 + 0.850792i −0.596072 − 1.03243i 1.74215 − 0.132974i 0.633594 − 1.09742i −3.00853 −1.08831 + 2.79564i 1.20258
214.11 −0.470582 + 0.815072i 1.66105 + 0.490836i 0.557105 + 0.964935i 1.52971 + 2.64953i −1.18173 + 1.12289i 1.64284 − 2.84549i −2.93098 2.51816 + 1.63060i −2.87941
214.12 −0.201428 + 0.348883i −1.63894 + 0.560253i 0.918854 + 1.59150i 0.787203 + 1.36348i 0.134665 − 0.684649i −1.21256 + 2.10021i −1.54604 2.37223 − 1.83644i −0.634258
214.13 −0.195185 + 0.338071i −0.916340 + 1.46980i 0.923805 + 1.60008i −1.34895 − 2.33645i −0.318041 − 0.596672i −0.544114 + 0.942433i −1.50199 −1.32064 − 2.69368i 1.05318
214.14 −0.0346993 + 0.0601010i −1.40903 − 1.00729i 0.997592 + 1.72788i 1.89001 + 3.27359i 0.109431 − 0.0497320i 0.766858 − 1.32824i −0.277260 0.970746 + 2.83860i −0.262328
214.15 0.101714 − 0.176173i 1.54377 + 0.785341i 0.979309 + 1.69621i −0.416772 − 0.721871i 0.295379 − 0.192092i −1.63638 + 2.83429i 0.805292 1.76648 + 2.42478i −0.169566
214.16 0.191602 − 0.331865i 0.366522 − 1.69283i 0.926577 + 1.60488i 0.493408 + 0.854607i −0.491563 − 0.445985i 2.22595 − 3.85546i 1.47654 −2.73132 − 1.24092i 0.378152
214.17 0.262319 − 0.454349i 0.931957 + 1.45995i 0.862378 + 1.49368i −1.60462 − 2.77928i 0.907797 − 0.0404617i 0.214045 − 0.370737i 1.95415 −1.26291 + 2.72122i −1.68369
214.18 0.324748 − 0.562481i −1.51565 − 0.838338i 0.789077 + 1.36672i −0.392744 − 0.680253i −0.963753 + 0.580274i −0.323055 + 0.559549i 2.32400 1.59438 + 2.54125i −0.510172
214.19 0.451075 − 0.781284i 1.05726 + 1.37193i 0.593063 + 1.02722i 1.23020 + 2.13077i 1.54877 − 0.207178i 1.11596 − 1.93290i 2.87436 −0.764398 + 2.90098i 2.21965
214.20 0.576984 − 0.999366i −0.474205 + 1.66587i 0.334179 + 0.578815i −1.25468 − 2.17317i 1.39121 + 1.43509i 1.38071 − 2.39147i 3.07920 −2.55026 − 1.57993i −2.89572
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 214.28
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.b ✓ 56
9.c even 3 1 inner 639.2.e.b ✓ 56
9.c even 3 1 5751.2.a.h 28
9.d odd 6 1 5751.2.a.g 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
639.2.e.b ✓ 56 1.a even 1 1 trivial
639.2.e.b ✓ 56 9.c even 3 1 inner
5751.2.a.g 28 9.d odd 6 1
5751.2.a.h 28 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} + T_{2}^{55} + 39 T_{2}^{54} + 34 T_{2}^{53} + 845 T_{2}^{52} + 663 T_{2}^{51} + 12559 T_{2}^{50} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(639, [\chi])\). Copy content Toggle raw display