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Properties

Label 63.6.g.a
Level $63$
Weight $6$
Character orbit 63.g
Analytic conductor $10.104$
Analytic rank $0$
Dimension $76$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [63,6,Mod(4,63)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("63.4"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 63.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.1041806482\)
Analytic rank: \(0\)
Dimension: \(76\)
Relative dimension: \(38\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 76 q + q^{2} - q^{3} - 575 q^{4} - 202 q^{5} - 116 q^{6} + 28 q^{7} - 72 q^{8} + 299 q^{9} - 66 q^{10} - 382 q^{11} + 1639 q^{12} + 179 q^{13} - 907 q^{14} - 1828 q^{15} - 8159 q^{16} + 2043 q^{17} - 2171 q^{18}+ \cdots + 54410 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −5.45858 + 9.45455i 6.02055 + 14.3789i −43.5923 − 75.5041i −81.8996 −168.810 − 21.5669i 116.686 − 56.4925i 602.459 −170.506 + 173.138i 447.056 − 774.324i
4.2 −5.19274 + 8.99409i −14.3357 + 6.12265i −37.9291 − 65.6951i 78.9178 19.3741 − 160.730i 78.2959 + 103.328i 455.487 168.026 − 175.545i −409.800 + 709.794i
4.3 −4.90850 + 8.50178i −9.78720 − 12.1330i −32.1868 − 55.7492i −0.906572 151.193 − 23.6535i −27.1127 − 126.775i 317.812 −51.4216 + 237.497i 4.44991 − 7.70748i
4.4 −4.86454 + 8.42562i 13.8209 − 7.20991i −31.3274 − 54.2607i −33.1194 −6.48422 + 151.523i −126.153 + 29.8716i 298.243 139.034 − 199.295i 161.111 − 279.052i
4.5 −4.57252 + 7.91984i 10.9649 + 11.0802i −25.8159 − 44.7144i 83.7927 −137.891 + 36.1755i −129.140 + 11.3996i 179.533 −2.54290 + 242.987i −383.144 + 663.624i
4.6 −4.40700 + 7.63314i 3.38767 − 15.2159i −22.8433 − 39.5657i 11.3542 101.216 + 92.9150i 85.1554 + 97.7525i 120.633 −220.047 − 103.093i −50.0378 + 86.6681i
4.7 −4.16732 + 7.21801i −15.3204 + 2.87825i −18.7331 − 32.4467i −101.394 43.0699 − 122.578i −84.9872 + 97.8988i 45.5588 226.431 − 88.1920i 422.542 − 731.864i
4.8 −3.53515 + 6.12306i 15.2582 + 3.19159i −8.99455 − 15.5790i 32.9744 −73.4824 + 82.1443i 129.546 − 4.97106i −99.0612 222.628 + 97.3960i −116.569 + 201.904i
4.9 −3.47017 + 6.01050i −8.53482 + 13.0444i −8.08411 − 14.0021i 11.3731 −48.7863 − 96.5649i −20.3156 − 128.040i −109.878 −97.3138 − 222.663i −39.4664 + 68.3578i
4.10 −2.80913 + 4.86555i 0.935686 + 15.5604i 0.217629 + 0.376944i −19.0252 −78.3381 − 39.1584i 30.5840 + 125.983i −182.229 −241.249 + 29.1192i 53.4442 − 92.5680i
4.11 −2.66412 + 4.61439i 14.7925 − 4.91762i 1.80491 + 3.12620i −76.4115 −16.7171 + 81.3594i 27.6739 − 126.654i −189.738 194.634 − 145.487i 203.570 − 352.593i
4.12 −2.45815 + 4.25764i −13.1905 − 8.30722i 3.91502 + 6.78102i 42.3225 67.7934 − 35.7401i −115.362 + 59.1492i −195.816 104.980 + 219.153i −104.035 + 180.194i
4.13 −2.27973 + 3.94860i 6.66259 − 14.0929i 5.60571 + 9.70937i 107.131 40.4584 + 58.4358i −22.2750 − 127.714i −197.020 −154.220 − 187.790i −244.230 + 423.019i
4.14 −2.18387 + 3.78258i −7.05203 − 13.9021i 6.46140 + 11.1915i −87.8756 67.9866 + 3.68562i 129.631 − 1.69202i −196.211 −143.538 + 196.076i 191.909 − 332.396i
4.15 −1.46936 + 2.54501i −15.5712 − 0.733322i 11.6820 + 20.2337i 35.4936 24.7460 − 38.5513i 128.007 − 20.5232i −162.699 241.924 + 22.8374i −52.1529 + 90.3315i
4.16 −1.34728 + 2.33355i 10.7103 + 11.3265i 12.3697 + 21.4249i −54.9827 −40.8607 + 9.73323i −126.245 − 29.4838i −152.887 −13.5776 + 242.620i 74.0768 − 128.305i
4.17 −0.992584 + 1.71921i 5.06511 − 14.7426i 14.0296 + 24.2999i −33.8405 20.3180 + 23.3413i −94.9899 + 88.2265i −119.227 −191.689 − 149.346i 33.5895 − 58.1787i
4.18 −0.487864 + 0.845006i 15.4052 − 2.38330i 15.5240 + 26.8883i 41.3255 −5.50174 + 14.1802i 3.17928 + 129.603i −61.5177 231.640 − 73.4303i −20.1612 + 34.9203i
4.19 0.372053 − 0.644415i −7.56275 + 13.6310i 15.7232 + 27.2333i 97.3099 5.97029 + 9.94501i −107.248 + 72.8345i 47.2108 −128.610 − 206.176i 36.2044 − 62.7079i
4.20 0.482480 − 0.835680i −14.7836 + 4.94422i 15.5344 + 26.9064i −67.7724 −3.00101 + 14.7398i −77.6500 − 103.815i 60.8589 194.109 − 146.187i −32.6989 + 56.6361i
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.38
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.6.g.a ✓ 76
3.b odd 2 1 189.6.g.a 76
7.c even 3 1 63.6.h.a yes 76
9.c even 3 1 63.6.h.a yes 76
9.d odd 6 1 189.6.h.a 76
21.h odd 6 1 189.6.h.a 76
63.g even 3 1 inner 63.6.g.a ✓ 76
63.n odd 6 1 189.6.g.a 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.6.g.a ✓ 76 1.a even 1 1 trivial
63.6.g.a ✓ 76 63.g even 3 1 inner
63.6.h.a yes 76 7.c even 3 1
63.6.h.a yes 76 9.c even 3 1
189.6.g.a 76 3.b odd 2 1
189.6.g.a 76 63.n odd 6 1
189.6.h.a 76 9.d odd 6 1
189.6.h.a 76 21.h odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(63, [\chi])\).