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Properties

Label 639.2.w.a
Level $639$
Weight $2$
Character orbit 639.w
Analytic conductor $5.102$
Analytic rank $0$
Dimension $840$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [639,2,Mod(23,639)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(639, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([35, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("639.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 639 = 3^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 639.w (of order \(42\), degree \(12\), 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: \(840\)
Relative dimension: \(70\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 840 q - 15 q^{2} - 5 q^{3} - 73 q^{4} - 42 q^{5} - 7 q^{7} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 840 q - 15 q^{2} - 5 q^{3} - 73 q^{4} - 42 q^{5} - 7 q^{7} - 13 q^{9} - 40 q^{10} - 21 q^{11} - 17 q^{12} - 7 q^{13} - 7 q^{15} + 55 q^{16} - 2 q^{18} - 20 q^{19} - 6 q^{20} - 56 q^{21} - 7 q^{22} - 21 q^{23} + 25 q^{24} + 366 q^{25} - 68 q^{27} - 15 q^{29} - 28 q^{30} + 14 q^{31} - 75 q^{32} - 14 q^{33} - 21 q^{34} + 4 q^{36} - 32 q^{37} - 3 q^{38} + 14 q^{39} + 36 q^{40} - 21 q^{41} + 7 q^{42} - 5 q^{43} + 35 q^{45} - 21 q^{47} - 91 q^{48} - 63 q^{49} - 114 q^{50} + 14 q^{51} - 7 q^{52} + 2 q^{54} - 98 q^{55} - 21 q^{56} + 102 q^{57} + 21 q^{58} - 126 q^{59} + 85 q^{60} - 7 q^{61} - 126 q^{63} + 48 q^{64} - 63 q^{65} - 7 q^{67} - 105 q^{68} + 7 q^{69} + 142 q^{72} - 20 q^{73} + 24 q^{74} + 48 q^{75} + 4 q^{76} + 117 q^{77} - 126 q^{78} - 17 q^{79} + 19 q^{81} - 28 q^{82} - 99 q^{83} + 105 q^{84} + 255 q^{86} - 7 q^{87} - 75 q^{90} + 32 q^{91} + 21 q^{92} - 21 q^{93} - 7 q^{94} - 78 q^{95} + 18 q^{96} - 7 q^{97} - 70 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −2.56356 − 1.00612i −0.168046 − 1.72388i 4.09346 + 3.79817i 1.82620 + 1.05436i −1.30364 + 4.58835i 0.714683 − 4.74161i −4.28263 − 8.89296i −2.94352 + 0.579382i −3.62076 − 4.54029i
23.2 −2.55520 − 1.00284i −0.301520 + 1.70560i 4.05726 + 3.76458i 0.937698 + 0.541380i 2.48090 − 4.05578i −0.576123 + 3.82233i −4.20984 − 8.74183i −2.81817 − 1.02855i −1.85309 − 2.32370i
23.3 −2.54766 − 0.999884i −1.72529 − 0.152936i 4.02471 + 3.73439i −1.84928 − 1.06768i 4.24253 + 2.11472i −0.172972 + 1.14759i −4.14471 − 8.60657i 2.95322 + 0.527717i 3.64379 + 4.56916i
23.4 −2.44674 − 0.960273i 0.517113 + 1.65306i 3.59828 + 3.33872i −3.45862 − 1.99683i 0.322145 − 4.54116i 0.666523 − 4.42209i −3.31710 − 6.88803i −2.46519 + 1.70963i 6.54481 + 8.20694i
23.5 −2.33115 − 0.914909i 1.49730 − 0.870678i 3.13110 + 2.90524i −1.36481 − 0.787971i −4.28703 + 0.659785i 0.120506 − 0.799502i −2.46793 − 5.12471i 1.48384 − 2.60734i 2.46065 + 3.08555i
23.6 −2.27920 − 0.894518i −0.342698 − 1.69781i 2.92846 + 2.71722i 0.278948 + 0.161051i −0.737647 + 4.17619i −0.450904 + 2.99156i −2.11926 − 4.40068i −2.76512 + 1.16367i −0.491714 − 0.616590i
23.7 −2.21200 − 0.868144i 1.55252 − 0.767909i 2.67315 + 2.48032i 3.45974 + 1.99748i −4.10082 + 0.350802i −0.445701 + 2.95703i −1.69768 − 3.52526i 1.82063 − 2.38439i −5.91882 − 7.42197i
23.8 −2.18001 − 0.855591i −1.71904 − 0.211907i 2.55431 + 2.37005i 3.62776 + 2.09449i 3.56622 + 1.93275i −0.0152420 + 0.101124i −1.50840 − 3.13222i 2.91019 + 0.728552i −6.11653 − 7.66988i
23.9 −2.16970 − 0.851545i −0.888366 + 1.48688i 2.51637 + 2.33485i 2.32238 + 1.34082i 3.19363 − 2.46960i 0.325319 − 2.15835i −1.44893 − 3.00873i −1.42161 − 2.64178i −3.89709 − 4.88680i
23.10 −2.08803 − 0.819493i 1.44716 + 0.951698i 2.22221 + 2.06191i −2.34379 − 1.35319i −2.24181 − 3.17311i −0.769278 + 5.10383i −1.00385 − 2.08452i 1.18854 + 2.75452i 3.78498 + 4.74621i
23.11 −2.03394 − 0.798263i −0.584267 − 1.63053i 2.03359 + 1.88690i −3.09808 − 1.78868i −0.113228 + 3.78280i −0.205552 + 1.36375i −0.733909 − 1.52398i −2.31726 + 1.90533i 4.87349 + 6.11116i
23.12 −2.01359 − 0.790278i 1.64036 + 0.556070i 1.96392 + 1.82225i 0.608339 + 0.351225i −2.86357 − 2.41604i 0.192116 − 1.27461i −0.637367 − 1.32351i 2.38157 + 1.82431i −0.947384 − 1.18798i
23.13 −1.99172 − 0.781693i −1.33915 + 1.09848i 1.88980 + 1.75348i −0.779278 − 0.449916i 3.52590 − 1.14107i 0.123543 − 0.819655i −0.536581 − 1.11422i 0.586662 − 2.94208i 1.20041 + 1.50526i
23.14 −1.94977 − 0.765227i 1.02901 + 1.39324i 1.74991 + 1.62368i 1.44563 + 0.834634i −0.940190 − 3.50393i 0.359858 − 2.38750i −0.351846 − 0.730615i −0.882257 + 2.86734i −2.17995 − 2.73357i
23.15 −1.67822 − 0.658653i −1.73119 + 0.0545628i 0.916498 + 0.850386i −1.64942 − 0.952293i 2.94126 + 1.04869i 0.646913 − 4.29199i 0.586474 + 1.21782i 2.99405 − 0.188917i 2.14086 + 2.68455i
23.16 −1.58255 − 0.621103i 0.817927 − 1.52676i 0.652577 + 0.605503i −1.43787 − 0.830152i −2.24268 + 1.90815i 0.265903 − 1.76415i 0.818608 + 1.69986i −1.66199 − 2.49756i 1.75988 + 2.20682i
23.17 −1.50561 − 0.590909i −1.03678 − 1.38748i 0.451589 + 0.419013i 0.508369 + 0.293507i 0.741112 + 2.70164i 0.426688 − 2.83089i 0.971223 + 2.01677i −0.850184 + 2.87701i −0.591970 − 0.742307i
23.18 −1.47759 − 0.579911i −0.215903 + 1.71854i 0.380868 + 0.353394i −2.27745 − 1.31489i 1.31562 − 2.41409i −0.0880423 + 0.584123i 1.01959 + 2.11720i −2.90677 − 0.742077i 2.60262 + 3.26358i
23.19 −1.39904 − 0.549081i 0.807382 − 1.53236i 0.189706 + 0.176022i 0.675728 + 0.390132i −1.97095 + 1.70051i −0.332193 + 2.20395i 1.13544 + 2.35776i −1.69627 − 2.47440i −0.731154 − 0.916838i
23.20 −1.36738 − 0.536658i −1.61323 + 0.630468i 0.115629 + 0.107288i 0.590487 + 0.340918i 2.54425 + 0.00366166i −0.673681 + 4.46958i 1.17415 + 2.43815i 2.20502 − 2.03418i −0.624465 − 0.783055i
See next 80 embeddings (of 840 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.70
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
71.f odd 14 1 inner
639.w even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 639.2.w.a ✓ 840
9.d odd 6 1 inner 639.2.w.a ✓ 840
71.f odd 14 1 inner 639.2.w.a ✓ 840
639.w even 42 1 inner 639.2.w.a ✓ 840
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
639.2.w.a ✓ 840 1.a even 1 1 trivial
639.2.w.a ✓ 840 9.d odd 6 1 inner
639.2.w.a ✓ 840 71.f odd 14 1 inner
639.2.w.a ✓ 840 639.w even 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(639, [\chi])\).