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Properties

Label 63.3.r.a.29.2
Level $63$
Weight $3$
Character 63.29
Analytic conductor $1.717$
Analytic rank $0$
Dimension $24$
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] = [63,3,Mod(29,63)] 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("63.29"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 63.r (of order \(6\), 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: \(1.71662566547\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 29.2
Character \(\chi\) \(=\) 63.29
Dual form 63.3.r.a.50.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.95047 - 1.70346i) q^{2} +(2.25172 - 1.98236i) q^{3} +(3.80352 + 6.58789i) q^{4} +(6.84828 - 3.95386i) q^{5} +(-10.0205 + 2.01321i) q^{6} +(-1.32288 + 2.29129i) q^{7} -12.2888i q^{8} +(1.14047 - 8.92745i) q^{9} -26.9409 q^{10} +(-13.1415 - 7.58726i) q^{11} +(21.6240 + 7.29410i) q^{12} +(2.78232 + 4.81912i) q^{13} +(7.80621 - 4.50692i) q^{14} +(7.58241 - 22.4788i) q^{15} +(-5.71943 + 9.90634i) q^{16} +18.1612i q^{17} +(-18.5724 + 24.3974i) q^{18} +14.4333 q^{19} +(52.0951 + 30.0771i) q^{20} +(1.56342 + 7.78175i) q^{21} +(25.8491 + 44.7720i) q^{22} +(-12.4647 + 7.19650i) q^{23} +(-24.3610 - 27.6710i) q^{24} +(18.7660 - 32.5036i) q^{25} -18.9582i q^{26} +(-15.1294 - 22.3629i) q^{27} -20.1263 q^{28} +(12.4370 + 7.18052i) q^{29} +(-60.6633 + 53.4066i) q^{30} +(7.54257 + 13.0641i) q^{31} +(-8.81981 + 5.09212i) q^{32} +(-44.6317 + 8.96691i) q^{33} +(30.9369 - 53.5842i) q^{34} +20.9219i q^{35} +(63.1508 - 26.4425i) q^{36} +15.6901 q^{37} +(-42.5850 - 24.5865i) q^{38} +(15.8182 + 5.33573i) q^{39} +(-48.5884 - 84.1575i) q^{40} +(26.5137 - 15.3077i) q^{41} +(8.64303 - 25.6231i) q^{42} +(-23.5824 + 40.8459i) q^{43} -115.433i q^{44} +(-27.4876 - 65.6469i) q^{45} +49.0356 q^{46} +(54.9423 + 31.7209i) q^{47} +(6.75944 + 33.6443i) q^{48} +(-3.50000 - 6.06218i) q^{49} +(-110.737 + 63.9340i) q^{50} +(36.0022 + 40.8940i) q^{51} +(-21.1652 + 36.6592i) q^{52} +55.0381i q^{53} +(6.54478 + 91.7534i) q^{54} -119.996 q^{55} +(28.1573 + 16.2566i) q^{56} +(32.4997 - 28.6120i) q^{57} +(-24.4634 - 42.3719i) q^{58} +(-58.5495 + 33.8036i) q^{59} +(176.927 - 35.5463i) q^{60} +(-7.27466 + 12.6001i) q^{61} -51.3937i q^{62} +(18.9467 + 14.4230i) q^{63} +80.4522 q^{64} +(38.1082 + 22.0018i) q^{65} +(146.959 + 49.5715i) q^{66} +(-1.84090 - 3.18854i) q^{67} +(-119.644 + 69.0766i) q^{68} +(-13.8009 + 40.9141i) q^{69} +(35.6394 - 61.7293i) q^{70} -88.8477i q^{71} +(-109.708 - 14.0150i) q^{72} +41.4417 q^{73} +(-46.2932 - 26.7274i) q^{74} +(-22.1784 - 110.390i) q^{75} +(54.8973 + 95.0849i) q^{76} +(34.7692 - 20.0740i) q^{77} +(-37.5821 - 42.6886i) q^{78} +(8.98464 - 15.5619i) q^{79} +90.4552i q^{80} +(-78.3987 - 20.3629i) q^{81} -104.304 q^{82} +(-102.353 - 59.0934i) q^{83} +(-45.3188 + 39.8977i) q^{84} +(71.8070 + 124.373i) q^{85} +(139.158 - 80.3431i) q^{86} +(42.2391 - 8.48621i) q^{87} +(-93.2387 + 161.494i) q^{88} -11.3142i q^{89} +(-30.7251 + 240.513i) q^{90} -14.7227 q^{91} +(-94.8194 - 54.7440i) q^{92} +(42.8816 + 14.4646i) q^{93} +(-108.070 - 187.183i) q^{94} +(98.8433 - 57.0672i) q^{95} +(-9.76529 + 28.9501i) q^{96} +(25.9398 - 44.9290i) q^{97} +23.8484i q^{98} +(-82.7223 + 108.667i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 2 q^{3} + 24 q^{4} - 18 q^{5} - 14 q^{6} + 26 q^{9} - 18 q^{11} + 4 q^{12} - 10 q^{15} - 48 q^{16} - 62 q^{18} - 24 q^{19} - 18 q^{20} - 14 q^{21} - 24 q^{22} + 72 q^{23} + 54 q^{24} + 54 q^{25}+ \cdots + 296 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.95047 − 1.70346i −1.47524 − 0.851728i −0.475625 − 0.879648i \(-0.657778\pi\)
−0.999610 + 0.0279205i \(0.991111\pi\)
\(3\) 2.25172 − 1.98236i 0.750573 − 0.660788i
\(4\) 3.80352 + 6.58789i 0.950880 + 1.64697i
\(5\) 6.84828 − 3.95386i 1.36966 − 0.790772i 0.378773 − 0.925490i \(-0.376346\pi\)
0.990884 + 0.134718i \(0.0430129\pi\)
\(6\) −10.0205 + 2.01321i −1.67008 + 0.335535i
\(7\) −1.32288 + 2.29129i −0.188982 + 0.327327i
\(8\) − 12.2888i − 1.53611i
\(9\) 1.14047 − 8.92745i 0.126718 − 0.991939i
\(10\) −26.9409 −2.69409
\(11\) −13.1415 − 7.58726i −1.19468 − 0.689751i −0.235318 − 0.971918i \(-0.575613\pi\)
−0.959365 + 0.282168i \(0.908947\pi\)
\(12\) 21.6240 + 7.29410i 1.80200 + 0.607842i
\(13\) 2.78232 + 4.81912i 0.214025 + 0.370701i 0.952970 − 0.303063i \(-0.0980094\pi\)
−0.738946 + 0.673765i \(0.764676\pi\)
\(14\) 7.80621 − 4.50692i 0.557587 − 0.321923i
\(15\) 7.58241 − 22.4788i 0.505494 − 1.49858i
\(16\) −5.71943 + 9.90634i −0.357464 + 0.619146i
\(17\) 18.1612i 1.06831i 0.845387 + 0.534154i \(0.179370\pi\)
−0.845387 + 0.534154i \(0.820630\pi\)
\(18\) −18.5724 + 24.3974i −1.03180 + 1.35541i
\(19\) 14.4333 0.759647 0.379824 − 0.925059i \(-0.375985\pi\)
0.379824 + 0.925059i \(0.375985\pi\)
\(20\) 52.0951 + 30.0771i 2.60476 + 1.50386i
\(21\) 1.56342 + 7.78175i 0.0744488 + 0.370560i
\(22\) 25.8491 + 44.7720i 1.17496 + 2.03509i
\(23\) −12.4647 + 7.19650i −0.541943 + 0.312891i −0.745866 − 0.666096i \(-0.767964\pi\)
0.203923 + 0.978987i \(0.434631\pi\)
\(24\) −24.3610 − 27.6710i −1.01504 − 1.15296i
\(25\) 18.7660 − 32.5036i 0.750640 − 1.30015i
\(26\) − 18.9582i − 0.729163i
\(27\) −15.1294 − 22.3629i −0.560350 − 0.828256i
\(28\) −20.1263 −0.718797
\(29\) 12.4370 + 7.18052i 0.428863 + 0.247604i 0.698862 − 0.715256i \(-0.253690\pi\)
−0.269999 + 0.962861i \(0.587023\pi\)
\(30\) −60.6633 + 53.4066i −2.02211 + 1.78022i
\(31\) 7.54257 + 13.0641i 0.243309 + 0.421423i 0.961655 − 0.274263i \(-0.0884339\pi\)
−0.718346 + 0.695686i \(0.755101\pi\)
\(32\) −8.81981 + 5.09212i −0.275619 + 0.159129i
\(33\) −44.6317 + 8.96691i −1.35248 + 0.271724i
\(34\) 30.9369 − 53.5842i 0.909908 − 1.57601i
\(35\) 20.9219i 0.597767i
\(36\) 63.1508 − 26.4425i 1.75419 − 0.734513i
\(37\) 15.6901 0.424057 0.212029 − 0.977263i \(-0.431993\pi\)
0.212029 + 0.977263i \(0.431993\pi\)
\(38\) −42.5850 − 24.5865i −1.12066 − 0.647012i
\(39\) 15.8182 + 5.33573i 0.405596 + 0.136813i
\(40\) −48.5884 − 84.1575i −1.21471 − 2.10394i
\(41\) 26.5137 − 15.3077i 0.646676 − 0.373359i −0.140506 − 0.990080i \(-0.544873\pi\)
0.787182 + 0.616721i \(0.211539\pi\)
\(42\) 8.64303 − 25.6231i 0.205786 − 0.610073i
\(43\) −23.5824 + 40.8459i −0.548428 + 0.949905i 0.449955 + 0.893051i \(0.351440\pi\)
−0.998383 + 0.0568536i \(0.981893\pi\)
\(44\) − 115.433i − 2.62348i
\(45\) −27.4876 − 65.6469i −0.610836 − 1.45882i
\(46\) 49.0356 1.06599
\(47\) 54.9423 + 31.7209i 1.16898 + 0.674914i 0.953441 − 0.301581i \(-0.0975142\pi\)
0.215544 + 0.976494i \(0.430848\pi\)
\(48\) 6.75944 + 33.6443i 0.140822 + 0.700922i
\(49\) −3.50000 − 6.06218i −0.0714286 − 0.123718i
\(50\) −110.737 + 63.9340i −2.21474 + 1.27868i
\(51\) 36.0022 + 40.8940i 0.705926 + 0.801843i
\(52\) −21.1652 + 36.6592i −0.407023 + 0.704985i
\(53\) 55.0381i 1.03845i 0.854636 + 0.519227i \(0.173780\pi\)
−0.854636 + 0.519227i \(0.826220\pi\)
\(54\) 6.54478 + 91.7534i 0.121200 + 1.69914i
\(55\) −119.996 −2.18174
\(56\) 28.1573 + 16.2566i 0.502809 + 0.290297i
\(57\) 32.4997 − 28.6120i 0.570170 − 0.501966i
\(58\) −24.4634 − 42.3719i −0.421783 − 0.730549i
\(59\) −58.5495 + 33.8036i −0.992365 + 0.572942i −0.905980 − 0.423320i \(-0.860865\pi\)
−0.0863845 + 0.996262i \(0.527531\pi\)
\(60\) 176.927 − 35.5463i 2.94879 − 0.592438i
\(61\) −7.27466 + 12.6001i −0.119257 + 0.206559i −0.919473 − 0.393152i \(-0.871384\pi\)
0.800217 + 0.599711i \(0.204718\pi\)
\(62\) − 51.3937i − 0.828931i
\(63\) 18.9467 + 14.4230i 0.300741 + 0.228937i
\(64\) 80.4522 1.25707
\(65\) 38.1082 + 22.0018i 0.586280 + 0.338489i
\(66\) 146.959 + 49.5715i 2.22666 + 0.751083i
\(67\) −1.84090 − 3.18854i −0.0274762 − 0.0475901i 0.851960 − 0.523606i \(-0.175414\pi\)
−0.879437 + 0.476016i \(0.842080\pi\)
\(68\) −119.644 + 69.0766i −1.75947 + 1.01583i
\(69\) −13.8009 + 40.9141i −0.200013 + 0.592957i
\(70\) 35.6394 − 61.7293i 0.509135 − 0.881847i
\(71\) − 88.8477i − 1.25138i −0.780073 − 0.625688i \(-0.784818\pi\)
0.780073 − 0.625688i \(-0.215182\pi\)
\(72\) −109.708 − 14.0150i −1.52372 − 0.194653i
\(73\) 41.4417 0.567695 0.283847 − 0.958869i \(-0.408389\pi\)
0.283847 + 0.958869i \(0.408389\pi\)
\(74\) −46.2932 − 26.7274i −0.625584 − 0.361181i
\(75\) −22.1784 − 110.390i −0.295711 − 1.47187i
\(76\) 54.8973 + 95.0849i 0.722333 + 1.25112i
\(77\) 34.7692 − 20.0740i 0.451548 − 0.260701i
\(78\) −37.5821 − 42.6886i −0.481822 − 0.547289i
\(79\) 8.98464 − 15.5619i 0.113730 − 0.196986i −0.803542 − 0.595249i \(-0.797054\pi\)
0.917271 + 0.398263i \(0.130387\pi\)
\(80\) 90.4552i 1.13069i
\(81\) −78.3987 − 20.3629i −0.967885 − 0.251394i
\(82\) −104.304 −1.27200
\(83\) −102.353 − 59.0934i −1.23317 − 0.711969i −0.265478 − 0.964117i \(-0.585530\pi\)
−0.967689 + 0.252148i \(0.918863\pi\)
\(84\) −45.3188 + 39.8977i −0.539510 + 0.474973i
\(85\) 71.8070 + 124.373i 0.844788 + 1.46322i
\(86\) 139.158 − 80.3431i 1.61812 − 0.934222i
\(87\) 42.2391 − 8.48621i 0.485507 − 0.0975427i
\(88\) −93.2387 + 161.494i −1.05953 + 1.83516i
\(89\) − 11.3142i − 0.127126i −0.997978 − 0.0635629i \(-0.979754\pi\)
0.997978 − 0.0635629i \(-0.0202464\pi\)
\(90\) −30.7251 + 240.513i −0.341390 + 2.67237i
\(91\) −14.7227 −0.161787
\(92\) −94.8194 − 54.7440i −1.03065 − 0.595044i
\(93\) 42.8816 + 14.4646i 0.461092 + 0.155533i
\(94\) −108.070 − 187.183i −1.14968 − 1.99131i
\(95\) 98.8433 − 57.0672i 1.04046 − 0.600707i
\(96\) −9.76529 + 28.9501i −0.101722 + 0.301563i
\(97\) 25.9398 − 44.9290i 0.267420 − 0.463186i −0.700774 − 0.713383i \(-0.747162\pi\)
0.968195 + 0.250197i \(0.0804954\pi\)
\(98\) 23.8484i 0.243351i
\(99\) −82.7223 + 108.667i −0.835579 + 1.09765i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 63.3.r.a.29.2 ✓ 24
3.2 odd 2 189.3.r.a.8.11 24
7.2 even 3 441.3.j.h.263.11 24
7.3 odd 6 441.3.n.h.128.11 24
7.4 even 3 441.3.n.g.128.11 24
7.5 odd 6 441.3.j.g.263.11 24
7.6 odd 2 441.3.r.h.344.2 24
9.2 odd 6 567.3.b.a.323.3 24
9.4 even 3 189.3.r.a.71.11 24
9.5 odd 6 inner 63.3.r.a.50.2 yes 24
9.7 even 3 567.3.b.a.323.22 24
63.5 even 6 441.3.n.h.410.11 24
63.23 odd 6 441.3.n.g.410.11 24
63.32 odd 6 441.3.j.h.275.2 24
63.41 even 6 441.3.r.h.50.2 24
63.59 even 6 441.3.j.g.275.2 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.3.r.a.29.2 ✓ 24 1.1 even 1 trivial
63.3.r.a.50.2 yes 24 9.5 odd 6 inner
189.3.r.a.8.11 24 3.2 odd 2
189.3.r.a.71.11 24 9.4 even 3
441.3.j.g.263.11 24 7.5 odd 6
441.3.j.g.275.2 24 63.59 even 6
441.3.j.h.263.11 24 7.2 even 3
441.3.j.h.275.2 24 63.32 odd 6
441.3.n.g.128.11 24 7.4 even 3
441.3.n.g.410.11 24 63.23 odd 6
441.3.n.h.128.11 24 7.3 odd 6
441.3.n.h.410.11 24 63.5 even 6
441.3.r.h.50.2 24 63.41 even 6
441.3.r.h.344.2 24 7.6 odd 2
567.3.b.a.323.3 24 9.2 odd 6
567.3.b.a.323.22 24 9.7 even 3