LMFDB - Embedded newform 50.26.a.k.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [50,26,Mod(1,50)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("50.1"); S:= CuspForms(chi, 26); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 26, names="a")

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 26 $
Character orbit:	$[\chi]$	$=$	50.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-24576,801416] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$197.998389976$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2 x^{5} - 27590779188 x^{4} + 26487255863952 x^{3} + \cdots - 30\!\cdots\!08 $ x^6 - 2x^5 - 27590779188x^4 + 26487255863952x^3 + 180405425727179733780x^2 - 1479960352440847053525960*x - 3018239644098422083113677808
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{4}\cdot 5^{18} $
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1691.05$ of defining polynomial
Character	$\chi$	$=$	50.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4096.00 q^{2} +116655. q^{3} +1.67772e7 q^{4} -4.77821e8 q^{6} -1.34370e10 q^{7} -6.87195e10 q^{8} -8.33680e11 q^{9} +4.67882e11 q^{11} +1.95715e12 q^{12} -9.38695e13 q^{13} +5.50381e13 q^{14} +2.81475e14 q^{16} +6.14184e14 q^{17} +3.41475e15 q^{18} -1.66661e16 q^{19} -1.56750e15 q^{21} -1.91644e15 q^{22} -1.60865e17 q^{23} -8.01650e15 q^{24} +3.84489e17 q^{26} -1.96094e17 q^{27} -2.25436e17 q^{28} -1.05009e18 q^{29} -3.00725e18 q^{31} -1.15292e18 q^{32} +5.45810e16 q^{33} -2.51570e18 q^{34} -1.39868e19 q^{36} -3.36937e19 q^{37} +6.82643e19 q^{38} -1.09504e19 q^{39} +5.09874e19 q^{41} +6.42050e18 q^{42} +2.92822e20 q^{43} +7.84976e18 q^{44} +6.58901e20 q^{46} -9.77356e20 q^{47} +3.28356e19 q^{48} -1.16051e21 q^{49} +7.16479e19 q^{51} -1.57487e21 q^{52} +5.26216e21 q^{53} +8.03202e20 q^{54} +9.23387e20 q^{56} -1.94419e21 q^{57} +4.30117e21 q^{58} +4.10935e20 q^{59} -5.44766e21 q^{61} +1.23177e22 q^{62} +1.12022e22 q^{63} +4.72237e21 q^{64} -2.23564e20 q^{66} +6.32396e22 q^{67} +1.03043e22 q^{68} -1.87657e22 q^{69} -1.81308e22 q^{71} +5.72901e22 q^{72} -3.48574e23 q^{73} +1.38010e23 q^{74} -2.79610e23 q^{76} -6.28695e21 q^{77} +4.48528e22 q^{78} +3.43193e23 q^{79} +6.83492e23 q^{81} -2.08844e23 q^{82} +6.85739e23 q^{83} -2.62984e22 q^{84} -1.19940e24 q^{86} -1.22499e23 q^{87} -3.21526e22 q^{88} -4.29595e24 q^{89} +1.26133e24 q^{91} -2.69886e24 q^{92} -3.50812e23 q^{93} +4.00325e24 q^{94} -1.34495e23 q^{96} -2.90151e24 q^{97} +4.75347e24 q^{98} -3.90064e23 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 24576 q^{2} + 801416 q^{3} + 100663296 q^{4} - 3282599936 q^{6} + 34007705352 q^{7} - 412316860416 q^{8} + 541468782118 q^{9} + 9861544614312 q^{11} + 13445529337856 q^{12} + 30787386783696 q^{13}+ \cdots - 20\!\cdots\!64 q^{99}+O(q^{100}) $ 6 * q - 24576 * q^2 + 801416 * q^3 + 100663296 * q^4 - 3282599936 * q^6 + 34007705352 * q^7 - 412316860416 * q^8 + 541468782118 * q^9 + 9861544614312 * q^11 + 13445529337856 * q^12 + 30787386783696 * q^13 - 139295561121792 * q^14 + 1688849860263936 * q^16 + 3432836939805312 * q^17 - 2217856131555328 * q^18 + 2007656488916520 * q^19 - 28181710291014728 * q^21 - 40392886740221952 * q^22 + 135711886432707576 * q^23 - 55072888167858176 * q^24 - 126105136266018816 * q^26 + 788054364552235760 * q^27 + 570554618354860032 * q^28 - 2926032031655504820 * q^29 + 79292935777014912 * q^31 - 6917529027641081856 * q^32 - 6844843638395003168 * q^33 - 14060900105442557952 * q^34 + 9084338714850623488 * q^36 + 3345885298190974032 * q^37 - 8223360978602065920 * q^38 + 124726636622561793456 * q^39 + 103229301719876244012 * q^41 + 115432285351996325888 * q^42 + 166390524239317503336 * q^43 + 165449264087949115392 * q^44 - 555875886828370231296 * q^46 + 405906243391034677992 * q^47 + 225578549935547088896 * q^48 + 1716280748139027694542 * q^49 - 1516225893661962013568 * q^51 + 516526638145613070336 * q^52 + 9772760301640799789616 * q^53 - 3227870677205957672960 * q^54 - 2336991716781506691072 * q^56 + 26174725126915157341920 * q^57 + 11985027201660947742720 * q^58 - 12499075130880780168840 * q^59 + 3687285610311118296612 * q^61 - 324783864942653079552 * q^62 + 93007818034972665339656 * q^63 + 28334198897217871282176 * q^64 + 28036479542865932976128 * q^66 + 16759935062266648886712 * q^67 + 57593446831892717371392 * q^68 + 185125325982378999854536 * q^69 + 74568876221253661649712 * q^71 - 37209451376028153806848 * q^72 + 381797761857106829638176 * q^73 - 13704746181390229635072 * q^74 + 33682886568354062008320 * q^76 - 80271541631956375339296 * q^77 - 510880303606013105995776 * q^78 - 400850316873811624799520 * q^79 - 1432110118616113828343714 * q^81 - 422827219844613095473152 * q^82 - 2423159733509985740001144 * q^83 - 472810640801776950837248 * q^84 - 681535587284244493664256 * q^86 - 1231328665433636473947120 * q^87 - 677680185704239576645632 * q^88 + 2982172528060341024616740 * q^89 - 7001711719121019241314768 * q^91 + 2276867632449004467388416 * q^92 - 8806417055581711481018368 * q^93 - 1662591972929678041055232 * q^94 - 923969740536000876118016 * q^96 - 4960469410756873985567808 * q^97 - 7029885944377457436844032 * q^98 - 2053720289282411723573464 * q^99

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−4096.00	−0.707107
$2$	−4096.00	−0.707107
$3$	116655.	0.126733	0.0633664	−	0.997990i	$-0.479816\pi$
$3$	116655.	0.126733	0.0633664	+	0.997990i	$0.479816\pi$
$4$	1.67772e7	0.500000
$5$	0	0
$5$	0	0
$6$	−4.77821e8	−0.0896137
$7$	−1.34370e10	−0.366926	−0.183463	−	0.983027i	$-0.558731\pi$
$7$	−1.34370e10	−0.366926	−0.183463	+	0.983027i	$0.558731\pi$
$8$	−6.87195e10	−0.353553
$9$	−8.33680e11	−0.983939
$10$	0	0
$11$	4.67882e11	0.0449498	0.0224749	−	0.999747i	$-0.492845\pi$
$11$	4.67882e11	0.0449498	0.0224749	+	0.999747i	$0.492845\pi$
$12$	1.95715e12	0.0633664
$13$	−9.38695e13	−1.11746	−0.558731	−	0.829349i	$-0.688711\pi$
$13$	−9.38695e13	−1.11746	−0.558731	+	0.829349i	$0.688711\pi$
$14$	5.50381e13	0.259456
$15$	0	0
$16$	2.81475e14	0.250000
$17$	6.14184e14	0.255674	0.127837	−	0.991795i	$-0.459197\pi$
$17$	6.14184e14	0.255674	0.127837	+	0.991795i	$0.459197\pi$
$18$	3.41475e15	0.695750
$19$	−1.66661e16	−1.72748	−0.863741	−	0.503936i	$-0.831885\pi$
$19$	−1.66661e16	−1.72748	−0.863741	+	0.503936i	$0.831885\pi$
$20$	0	0
$21$	−1.56750e15	−0.0465016
$22$	−1.91644e15	−0.0317843
$23$	−1.60865e17	−1.53060	−0.765301	−	0.643673i	$-0.777410\pi$
$23$	−1.60865e17	−1.53060	−0.765301	+	0.643673i	$0.777410\pi$
$24$	−8.01650e15	−0.0448068
$25$	0	0
$26$	3.84489e17	0.790165
$27$	−1.96094e17	−0.251430
$28$	−2.25436e17	−0.183463
$29$	−1.05009e18	−0.551126	−0.275563	−	0.961283i	$-0.588864\pi$
$29$	−1.05009e18	−0.551126	−0.275563	+	0.961283i	$0.588864\pi$
$30$	0	0
$31$	−3.00725e18	−0.685722	−0.342861	−	0.939386i	$-0.611396\pi$
$31$	−3.00725e18	−0.685722	−0.342861	+	0.939386i	$0.611396\pi$
$32$	−1.15292e18	−0.176777
$33$	5.45810e16	0.00569662
$34$	−2.51570e18	−0.180789
$35$	0	0
$36$	−1.39868e19	−0.491969
$37$	−3.36937e19	−0.841449	−0.420724	−	0.907189i	$-0.638224\pi$
$37$	−3.36937e19	−0.841449	−0.420724	+	0.907189i	$0.638224\pi$
$38$	6.82643e19	1.22151
$39$	−1.09504e19	−0.141619
$40$	0	0
$41$	5.09874e19	0.352911	0.176455	−	0.984309i	$-0.443537\pi$
$41$	5.09874e19	0.352911	0.176455	+	0.984309i	$0.443537\pi$
$42$	6.42050e18	0.0328816
$43$	2.92822e20	1.11750	0.558751	−	0.829335i	$-0.311281\pi$
$43$	2.92822e20	1.11750	0.558751	+	0.829335i	$0.311281\pi$
$44$	7.84976e18	0.0224749
$45$	0	0
$46$	6.58901e20	1.08230
$47$	−9.77356e20	−1.22696	−0.613479	−	0.789711i	$-0.710230\pi$
$47$	−9.77356e20	−1.22696	−0.613479	+	0.789711i	$0.710230\pi$
$48$	3.28356e19	0.0316832
$49$	−1.16051e21	−0.865365
$50$	0	0
$51$	7.16479e19	0.0324023
$52$	−1.57487e21	−0.558731
$53$	5.26216e21	1.47135	0.735675	−	0.677335i	$-0.236865\pi$
$53$	5.26216e21	1.47135	0.735675	+	0.677335i	$0.236865\pi$
$54$	8.03202e20	0.177788
$55$	0	0
$56$	9.23387e20	0.129728
$57$	−1.94419e21	−0.218929
$58$	4.30117e21	0.389705
$59$	4.10935e20	0.0300693	0.0150346	−	0.999887i	$-0.495214\pi$
$59$	4.10935e20	0.0300693	0.0150346	+	0.999887i	$0.495214\pi$
$60$	0	0
$61$	−5.44766e21	−0.262777	−0.131388	−	0.991331i	$-0.541943\pi$
$61$	−5.44766e21	−0.262777	−0.131388	+	0.991331i	$0.541943\pi$
$62$	1.23177e22	0.484879
$63$	1.12022e22	0.361032
$64$	4.72237e21	0.125000
$65$	0	0
$66$	−2.23564e20	−0.00402812
$67$	6.32396e22	0.944178	0.472089	−	0.881551i	$-0.343500\pi$
$67$	6.32396e22	0.944178	0.472089	+	0.881551i	$0.343500\pi$
$68$	1.03043e22	0.127837
$69$	−1.87657e22	−0.193978
$70$	0	0
$71$	−1.81308e22	−0.131126	−0.0655628	−	0.997848i	$-0.520884\pi$
$71$	−1.81308e22	−0.131126	−0.0655628	+	0.997848i	$0.520884\pi$
$72$	5.72901e22	0.347875
$73$	−3.48574e23	−1.78139	−0.890694	−	0.454603i	$-0.849781\pi$
$73$	−3.48574e23	−1.78139	−0.890694	+	0.454603i	$0.849781\pi$
$74$	1.38010e23	0.594994
$75$	0	0
$76$	−2.79610e23	−0.863741
$77$	−6.28695e21	−0.0164932
$78$	4.48528e22	0.100140
$79$	3.43193e23	0.653432	0.326716	−	0.945123i	$-0.394058\pi$
$79$	3.43193e23	0.653432	0.326716	+	0.945123i	$0.394058\pi$
$80$	0	0
$81$	6.83492e23	0.952074
$82$	−2.08844e23	−0.249545
$83$	6.85739e23	0.704178	0.352089	−	0.935967i	$-0.385471\pi$
$83$	6.85739e23	0.704178	0.352089	+	0.935967i	$0.385471\pi$
$84$	−2.62984e22	−0.0232508
$85$	0	0
$86$	−1.19940e24	−0.790194
$87$	−1.22499e23	−0.0698458
$88$	−3.21526e22	−0.0158922
$89$	−4.29595e24	−1.84367	−0.921837	−	0.387579i	$-0.873312\pi$
$89$	−4.29595e24	−1.84367	−0.921837	+	0.387579i	$0.873312\pi$
$90$	0	0
$91$	1.26133e24	0.410025
$92$	−2.69886e24	−0.765301
$93$	−3.50812e23	−0.0869035
$94$	4.00325e24	0.867590
$95$	0	0
$96$	−1.34495e23	−0.0224034
$97$	−2.90151e24	−0.424598	−0.212299	−	0.977205i	$-0.568095\pi$
$97$	−2.90151e24	−0.424598	−0.212299	+	0.977205i	$0.568095\pi$
$98$	4.75347e24	0.611906
$99$	−3.90064e23	−0.0442279
$100$	0	0
$101$	−8.46099e24	−0.747144	−0.373572	−	0.927601i	$-0.621867\pi$
$101$	−8.46099e24	−0.747144	−0.373572	+	0.927601i	$0.621867\pi$
$102$	−2.93470e23	−0.0229119
$103$	−1.06290e25	−0.734562	−0.367281	−	0.930110i	$-0.619711\pi$
$103$	−1.06290e25	−0.734562	−0.367281	+	0.930110i	$0.619711\pi$
$104$	6.45066e24	0.395082
$105$	0	0
$106$	−2.15538e25	−1.04040
$107$	2.28121e25	0.979193	0.489597	−	0.871949i	$-0.337144\pi$
$107$	2.28121e25	0.979193	0.489597	+	0.871949i	$0.337144\pi$
$108$	−3.28991e24	−0.125715
$109$	−3.17871e25	−1.08248	−0.541240	−	0.840868i	$-0.682045\pi$
$109$	−3.17871e25	−1.08248	−0.541240	+	0.840868i	$0.682045\pi$
$110$	0	0
$111$	−3.93056e24	−0.106639
$112$	−3.78219e24	−0.0917314
$113$	9.52597e24	0.206742	0.103371	−	0.994643i	$-0.467037\pi$
$113$	9.52597e24	0.206742	0.103371	+	0.994643i	$0.467037\pi$
$114$	7.96340e24	0.154806
$115$	0	0
$116$	−1.76176e25	−0.275563
$117$	7.82571e25	1.09951
$118$	−1.68319e24	−0.0212622
$119$	−8.25281e24	−0.0938134
$120$	0	0
$121$	−1.08128e26	−0.997980
$122$	2.23136e25	0.185811
$123$	5.94796e24	0.0447254
$124$	−5.04532e25	−0.342861
$125$	0	0
$126$	−4.58842e25	−0.255289
$127$	−2.95859e26	−1.49121	−0.745604	−	0.666390i	$-0.767839\pi$
$127$	−2.95859e26	−1.49121	−0.745604	+	0.666390i	$0.767839\pi$
$128$	−1.93428e25	−0.0883883
$129$	3.41593e25	0.141624
$130$	0	0
$131$	−2.19768e26	−0.751749	−0.375874	−	0.926671i	$-0.622658\pi$
$131$	−2.19768e26	−0.751749	−0.375874	+	0.926671i	$0.622658\pi$
$132$	9.15717e23	0.00284831
$133$	2.23943e26	0.633857
$134$	−2.59029e26	−0.667635
$135$	0	0
$136$	−4.22064e25	−0.0903945
$137$	−3.42423e25	−0.0669200	−0.0334600	−	0.999440i	$-0.510653\pi$
$137$	−3.42423e25	−0.0669200	−0.0334600	+	0.999440i	$0.510653\pi$
$138$	7.68644e25	0.137163
$139$	−2.79093e26	−0.455055	−0.227528	−	0.973772i	$-0.573064\pi$
$139$	−2.79093e26	−0.455055	−0.227528	+	0.973772i	$0.573064\pi$
$140$	0	0
$141$	−1.14014e26	−0.155496
$142$	7.42638e25	0.0927199
$143$	−4.39198e25	−0.0502297
$144$	−2.34660e26	−0.245985
$145$	0	0
$146$	1.42776e27	1.25963
$147$	−1.35380e26	−0.109670
$148$	−5.65287e26	−0.420724
$149$	1.42906e27	0.977739	0.488870	−	0.872357i	$-0.337409\pi$
$149$	1.42906e27	0.977739	0.488870	+	0.872357i	$0.337409\pi$
$150$	0	0
$151$	−2.51226e27	−1.45496	−0.727482	−	0.686127i	$-0.759310\pi$
$151$	−2.51226e27	−1.45496	−0.727482	+	0.686127i	$0.759310\pi$
$152$	1.14528e27	0.610757
$153$	−5.12033e26	−0.251568
$154$	2.57513e25	0.0116625
$155$	0	0
$156$	−1.83717e26	−0.0708096
$157$	3.34741e27	1.19114	0.595571	−	0.803302i	$-0.296926\pi$
$157$	3.34741e27	1.19114	0.595571	+	0.803302i	$0.296926\pi$
$158$	−1.40572e27	−0.462046
$159$	6.13860e26	0.186468
$160$	0	0
$161$	2.16154e27	0.561617
$162$	−2.79958e27	−0.673218
$163$	8.42668e26	0.187634	0.0938170	−	0.995589i	$-0.470093\pi$
$163$	8.42668e26	0.187634	0.0938170	+	0.995589i	$0.470093\pi$
$164$	8.55426e26	0.176455
$165$	0	0
$166$	−2.80879e27	−0.497929
$167$	3.08789e27	0.507815	0.253907	−	0.967229i	$-0.418284\pi$
$167$	3.08789e27	0.507815	0.253907	+	0.967229i	$0.418284\pi$
$168$	1.07718e26	0.0164408
$169$	1.75507e27	0.248720
$170$	0	0
$171$	1.38942e28	1.69974
$172$	4.91274e27	0.558751
$173$	9.76292e27	1.03277	0.516385	−	0.856357i	$-0.327277\pi$
$173$	9.76292e27	1.03277	0.516385	+	0.856357i	$0.327277\pi$
$174$	5.01754e26	0.0493885
$175$	0	0
$176$	1.31697e26	0.0112375
$177$	4.79378e25	0.00381077
$178$	1.75962e28	1.30367
$179$	−1.48035e28	−1.02259	−0.511294	−	0.859406i	$-0.670834\pi$
$179$	−1.48035e28	−1.02259	−0.511294	+	0.859406i	$0.670834\pi$
$180$	0	0
$181$	1.20192e28	0.722591	0.361295	−	0.932451i	$-0.382335\pi$
$181$	1.20192e28	0.722591	0.361295	+	0.932451i	$0.382335\pi$
$182$	−5.16640e27	−0.289932
$183$	−6.35499e26	−0.0333025
$184$	1.10545e28	0.541150
$185$	0	0
$186$	1.43693e27	0.0614501
$187$	2.87366e26	0.0114925
$188$	−1.63973e28	−0.613479
$189$	2.63493e27	0.0922563
$190$	0	0
$191$	8.66884e27	0.266099	0.133050	−	0.991109i	$-0.457523\pi$
$191$	8.66884e27	0.266099	0.133050	+	0.991109i	$0.457523\pi$
$192$	5.50890e26	0.0158416
$193$	−1.60225e28	−0.431782	−0.215891	−	0.976417i	$-0.569266\pi$
$193$	−1.60225e28	−0.431782	−0.215891	+	0.976417i	$0.569266\pi$
$194$	1.18846e28	0.300236
$195$	0	0
$196$	−1.94702e28	−0.432683
$197$	6.69730e28	1.39660	0.698300	−	0.715805i	$-0.253940\pi$
$197$	6.69730e28	1.39660	0.698300	+	0.715805i	$0.253940\pi$
$198$	1.59770e27	0.0312738
$199$	1.81836e28	0.334208	0.167104	−	0.985939i	$-0.446558\pi$
$199$	1.81836e28	0.334208	0.167104	+	0.985939i	$0.446558\pi$
$200$	0	0
$201$	7.37724e27	0.119658
$202$	3.46562e28	0.528310
$203$	1.41101e28	0.202222
$204$	1.20205e27	0.0162012
$205$	0	0
$206$	4.35365e28	0.519414
$207$	1.34110e29	1.50602
$208$	−2.64219e28	−0.279365
$209$	−7.79776e27	−0.0776500
$210$	0	0
$211$	−1.59042e29	−1.40598	−0.702992	−	0.711197i	$-0.748153\pi$
$211$	−1.59042e29	−1.40598	−0.702992	+	0.711197i	$0.748153\pi$
$212$	8.82844e28	0.735675
$213$	−2.11506e27	−0.0166179
$214$	−9.34384e28	−0.692394
$215$	0	0
$216$	1.34755e28	0.0888940
$217$	4.04085e28	0.251609
$218$	1.30200e29	0.765428
$219$	−4.06630e28	−0.225761
$220$	0	0
$221$	−5.76531e28	−0.285706
$222$	1.60996e28	0.0754053
$223$	3.74438e29	1.65794	0.828970	−	0.559293i	$-0.188927\pi$
$223$	3.74438e29	1.65794	0.828970	+	0.559293i	$0.188927\pi$
$224$	1.54919e28	0.0648639
$225$	0	0
$226$	−3.90184e28	−0.146189
$227$	3.09492e29	1.09730	0.548651	−	0.836052i	$-0.315142\pi$
$227$	3.09492e29	1.09730	0.548651	+	0.836052i	$0.315142\pi$
$228$	−3.26181e28	−0.109464
$229$	4.36225e29	1.38601	0.693006	−	0.720932i	$-0.256286\pi$
$229$	4.36225e29	1.38601	0.693006	+	0.720932i	$0.256286\pi$
$230$	0	0
$231$	−7.33407e26	−0.00209024
$232$	7.21616e28	0.194853
$233$	2.03884e29	0.521715	0.260858	−	0.965377i	$-0.415995\pi$
$233$	2.03884e29	0.521715	0.260858	+	0.965377i	$0.415995\pi$
$234$	−3.20541e29	−0.777474
$235$	0	0
$236$	6.89435e27	0.0150346
$237$	4.00354e28	0.0828113
$238$	3.38035e28	0.0663361
$239$	−7.19381e29	−1.33963	−0.669816	−	0.742527i	$-0.733627\pi$
$239$	−7.19381e29	−1.33963	−0.669816	+	0.742527i	$0.733627\pi$
$240$	0	0
$241$	−5.94932e28	−0.0998285	−0.0499142	−	0.998754i	$-0.515895\pi$
$241$	−5.94932e28	−0.0998285	−0.0499142	+	0.998754i	$0.515895\pi$
$242$	4.42893e29	0.705678
$243$	2.45881e29	0.372089
$244$	−9.13966e28	−0.131388
$245$	0	0
$246$	−2.43628e28	−0.0316256
$247$	1.56444e30	1.93039
$248$	2.06656e29	0.242439
$249$	7.99951e28	0.0892425
$250$	0	0
$251$	−1.11692e30	−1.12745	−0.563727	−	0.825961i	$-0.690633\pi$
$251$	−1.11692e30	−1.12745	−0.563727	+	0.825961i	$0.690633\pi$
$252$	1.87942e29	0.180516
$253$	−7.52656e28	−0.0688003
$254$	1.21184e30	1.05444
$255$	0	0
$256$	7.92282e28	0.0625000
$257$	2.23332e30	1.67798	0.838991	−	0.544146i	$-0.183146\pi$
$257$	2.23332e30	1.67798	0.838991	+	0.544146i	$0.183146\pi$
$258$	−1.39917e29	−0.100144
$259$	4.52744e29	0.308749
$260$	0	0
$261$	8.75438e29	0.542275
$262$	9.00168e29	0.531566
$263$	−1.67388e30	−0.942489	−0.471244	−	0.882003i	$-0.656195\pi$
$263$	−1.67388e30	−0.942489	−0.471244	+	0.882003i	$0.656195\pi$
$264$	−3.75078e27	−0.00201406
$265$	0	0
$266$	−9.17270e29	−0.448205
$267$	−5.01146e29	−0.233654
$268$	1.06098e30	0.472089
$269$	−1.51842e30	−0.644895	−0.322448	−	0.946587i	$-0.604506\pi$
$269$	−1.51842e30	−0.644895	−0.322448	+	0.946587i	$0.604506\pi$
$270$	0	0
$271$	3.73589e30	1.44637	0.723183	−	0.690657i	$-0.242678\pi$
$271$	3.73589e30	1.44637	0.723183	+	0.690657i	$0.242678\pi$
$272$	1.72877e29	0.0639185
$273$	1.47141e29	0.0519637
$274$	1.40256e29	0.0473196
$275$	0	0
$276$	−3.14837e29	−0.0969888
$277$	4.88475e30	1.43829	0.719143	−	0.694863i	$-0.244535\pi$
$277$	4.88475e30	1.43829	0.719143	+	0.694863i	$0.244535\pi$
$278$	1.14316e30	0.321772
$279$	2.50708e30	0.674708
$280$	0	0
$281$	4.07868e30	1.00390	0.501950	−	0.864896i	$-0.332616\pi$
$281$	4.07868e30	1.00390	0.501950	+	0.864896i	$0.332616\pi$
$282$	4.67001e29	0.109952
$283$	8.77909e30	1.97751	0.988756	−	0.149537i	$-0.0477782\pi$
$283$	8.77909e30	1.97751	0.988756	+	0.149537i	$0.0477782\pi$
$284$	−3.04184e29	−0.0655628
$285$	0	0
$286$	1.79896e29	0.0355177
$287$	−6.85120e29	−0.129492
$288$	9.61168e29	0.173937
$289$	−5.39341e30	−0.934631
$290$	0	0
$291$	−3.38477e29	−0.0538105
$292$	−5.84809e30	−0.890694
$293$	4.02063e30	0.586744	0.293372	−	0.955998i	$-0.405222\pi$
$293$	4.02063e30	0.586744	0.293372	+	0.955998i	$0.405222\pi$
$294$	5.54518e29	0.0775486
$295$	0	0
$296$	2.31542e30	0.297497
$297$	−9.17489e28	−0.0113017
$298$	−5.85344e30	−0.691366
$299$	1.51003e31	1.71039
$300$	0	0
$301$	−3.93467e30	−0.410040
$302$	1.02902e31	1.02882
$303$	−9.87021e29	−0.0946877
$304$	−4.69108e30	−0.431870
$305$	0	0
$306$	2.09729e30	0.177885
$307$	3.20834e29	0.0261247	0.0130623	−	0.999915i	$-0.495842\pi$
$307$	3.20834e29	0.0261247	0.0130623	+	0.999915i	$0.495842\pi$
$308$	−1.05478e29	−0.00824662
$309$	−1.23993e30	−0.0930932
$310$	0	0
$311$	−1.26721e31	−0.877692	−0.438846	−	0.898562i	$-0.644613\pi$
$311$	−1.26721e31	−0.877692	−0.438846	+	0.898562i	$0.644613\pi$
$312$	7.52505e29	0.0500699
$313$	−2.76579e31	−1.76813	−0.884066	−	0.467361i	$-0.845205\pi$
$313$	−2.76579e31	−1.76813	−0.884066	+	0.467361i	$0.845205\pi$
$314$	−1.37110e31	−0.842265
$315$	0	0
$316$	5.75783e30	0.326716
$317$	−8.12742e30	−0.443315	−0.221657	−	0.975125i	$-0.571147\pi$
$317$	−8.12742e30	−0.443315	−0.221657	+	0.975125i	$0.571147\pi$
$318$	−2.51437e30	−0.131853
$319$	−4.91318e29	−0.0247730
$320$	0	0
$321$	2.66116e30	0.124096
$322$	−8.85368e30	−0.397123
$323$	−1.02360e31	−0.441672
$324$	1.14671e31	0.476037
$325$	0	0
$326$	−3.45157e30	−0.132677
$327$	−3.70814e30	−0.137186
$328$	−3.50383e30	−0.124773
$329$	1.31328e31	0.450203
$330$	0	0
$331$	−3.24680e31	−1.03182	−0.515912	−	0.856642i	$-0.672547\pi$
$331$	−3.24680e31	−1.03182	−0.515912	+	0.856642i	$0.672547\pi$
$332$	1.15048e31	0.352089
$333$	2.80898e31	0.827934
$334$	−1.26480e31	−0.359079
$335$	0	0
$336$	−4.41213e29	−0.0116254
$337$	−6.16105e31	−1.56416	−0.782080	−	0.623178i	$-0.785841\pi$
$337$	−6.16105e31	−1.56416	−0.782080	+	0.623178i	$0.785841\pi$
$338$	−7.18877e30	−0.175872
$339$	1.11126e30	0.0262010
$340$	0	0
$341$	−1.40704e30	−0.0308231
$342$	−5.69106e31	−1.20189
$343$	3.36139e31	0.684451
$344$	−2.01226e31	−0.395097
$345$	0	0
$346$	−3.99889e31	−0.730279
$347$	−8.34408e31	−1.46981	−0.734903	−	0.678172i	$-0.762772\pi$
$347$	−8.34408e31	−1.46981	−0.734903	+	0.678172i	$0.762772\pi$
$348$	−2.05519e30	−0.0349229
$349$	−1.07311e32	−1.75924	−0.879621	−	0.475676i	$-0.842204\pi$
$349$	−1.07311e32	−1.75924	−0.879621	+	0.475676i	$0.842204\pi$
$350$	0	0
$351$	1.84073e31	0.280964
$352$	−5.39431e29	−0.00794608
$353$	−1.05139e32	−1.49479	−0.747394	−	0.664381i	$-0.768695\pi$
$353$	−1.05139e32	−1.49479	−0.747394	+	0.664381i	$0.768695\pi$
$354$	−1.96353e29	−0.00269462
$355$	0	0
$356$	−7.20741e31	−0.921837
$357$	−9.62736e29	−0.0118892
$358$	6.06350e31	0.723079
$359$	1.15543e32	1.33064	0.665321	−	0.746557i	$-0.268295\pi$
$359$	1.15543e32	1.33064	0.665321	+	0.746557i	$0.268295\pi$
$360$	0	0
$361$	1.84682e32	1.98419
$362$	−4.92305e31	−0.510949
$363$	−1.26137e31	−0.126477
$364$	2.11616e31	0.205013
$365$	0	0
$366$	2.60300e30	0.0235484
$367$	−5.65157e31	−0.494133	−0.247066	−	0.968999i	$-0.579467\pi$
$367$	−5.65157e31	−0.494133	−0.247066	+	0.968999i	$0.579467\pi$
$368$	−4.52793e31	−0.382651
$369$	−4.25072e31	−0.347242
$370$	0	0
$371$	−7.07079e31	−0.539876
$372$	−5.88565e30	−0.0434518
$373$	−1.41123e31	−0.100748	−0.0503741	−	0.998730i	$-0.516041\pi$
$373$	−1.41123e31	−0.100748	−0.0503741	+	0.998730i	$0.516041\pi$
$374$	−1.17705e30	−0.00812643
$375$	0	0
$376$	6.71634e31	0.433795
$377$	9.85713e31	0.615862
$378$	−1.07927e31	−0.0652350
$379$	2.60316e32	1.52234	0.761169	−	0.648554i	$-0.224626\pi$
$379$	2.60316e32	1.52234	0.761169	+	0.648554i	$0.224626\pi$
$380$	0	0
$381$	−3.45135e31	−0.188985
$382$	−3.55076e31	−0.188161
$383$	−2.22460e32	−1.14095	−0.570475	−	0.821315i	$-0.693241\pi$
$383$	−2.22460e32	−1.14095	−0.570475	+	0.821315i	$0.693241\pi$
$384$	−2.25644e30	−0.0112017
$385$	0	0
$386$	6.56283e31	0.305316
$387$	−2.44120e32	−1.09955
$388$	−4.86793e31	−0.212299
$389$	2.83579e31	0.119758	0.0598789	−	0.998206i	$-0.480929\pi$
$389$	2.83579e31	0.119758	0.0598789	+	0.998206i	$0.480929\pi$
$390$	0	0
$391$	−9.88004e31	−0.391335
$392$	7.97499e31	0.305953
$393$	−2.56371e31	−0.0952713
$394$	−2.74322e32	−0.987545
$395$	0	0
$396$	−6.54419e30	−0.0221139
$397$	−1.97865e32	−0.647870	−0.323935	−	0.946079i	$-0.605006\pi$
$397$	−1.97865e32	−0.647870	−0.323935	+	0.946079i	$0.605006\pi$
$398$	−7.44802e31	−0.236321
$399$	2.61242e31	0.0803306
$400$	0	0
$401$	5.91837e32	1.70961	0.854806	−	0.518947i	$-0.173676\pi$
$401$	5.91837e32	1.70961	0.854806	+	0.518947i	$0.173676\pi$
$402$	−3.02172e31	−0.0846113
$403$	2.82289e32	0.766268
$404$	−1.41952e32	−0.373572
$405$	0	0
$406$	−5.77949e31	−0.142993
$407$	−1.57647e31	−0.0378229
$408$	−4.92361e30	−0.0114560
$409$	5.10098e32	1.15110	0.575549	−	0.817767i	$-0.304788\pi$
$409$	5.10098e32	1.15110	0.575549	+	0.817767i	$0.304788\pi$
$410$	0	0
$411$	−3.99455e30	−0.00848097
$412$	−1.78326e32	−0.367281
$413$	−5.52175e30	−0.0110332
$414$	−5.49313e32	−1.06492
$415$	0	0
$416$	1.08224e32	0.197541
$417$	−3.25577e31	−0.0576704
$418$	3.19396e31	0.0549068
$419$	3.45646e32	0.576709	0.288354	−	0.957524i	$-0.406892\pi$
$419$	3.45646e32	0.576709	0.288354	+	0.957524i	$0.406892\pi$
$420$	0	0
$421$	1.15256e31	0.0181191	0.00905955	−	0.999959i	$-0.497116\pi$
$421$	1.15256e31	0.0181191	0.00905955	+	0.999959i	$0.497116\pi$
$422$	6.51435e32	0.994181
$423$	8.14803e32	1.20725
$424$	−3.61613e32	−0.520201
$425$	0	0
$426$	8.66328e30	0.0117507
$427$	7.32004e31	0.0964196
$428$	3.82724e32	0.489597
$429$	−5.12349e30	−0.00636575
$430$	0	0
$431$	−1.12581e33	−1.31978	−0.659888	−	0.751364i	$-0.729396\pi$
$431$	−1.12581e33	−1.31978	−0.659888	+	0.751364i	$0.729396\pi$
$432$	−5.51956e31	−0.0628576
$433$	5.31109e32	0.587604	0.293802	−	0.955866i	$-0.405079\pi$
$433$	5.31109e32	0.587604	0.293802	+	0.955866i	$0.405079\pi$
$434$	−1.65513e32	−0.177914
$435$	0	0
$436$	−5.33299e32	−0.541240
$437$	2.68098e33	2.64409
$438$	1.66556e32	0.159637
$439$	7.89847e32	0.735760	0.367880	−	0.929873i	$-0.380084\pi$
$439$	7.89847e32	0.735760	0.367880	+	0.929873i	$0.380084\pi$
$440$	0	0
$441$	9.67498e32	0.851467
$442$	2.36147e32	0.202025
$443$	−1.82758e33	−1.51995	−0.759974	−	0.649953i	$-0.774788\pi$
$443$	−1.82758e33	−1.51995	−0.759974	+	0.649953i	$0.774788\pi$
$444$	−6.59438e31	−0.0533196
$445$	0	0
$446$	−1.53370e33	−1.17234
$447$	1.66708e32	0.123912
$448$	−6.34546e31	−0.0458657
$449$	−8.32692e31	−0.0585336	−0.0292668	−	0.999572i	$-0.509317\pi$
$449$	−8.32692e31	−0.0585336	−0.0292668	+	0.999572i	$0.509317\pi$
$450$	0	0
$451$	2.38561e31	0.0158633
$452$	1.59819e32	0.103371
$453$	−2.93069e32	−0.184392
$454$	−1.26768e33	−0.775909
$455$	0	0
$456$	1.33604e32	0.0774030
$457$	2.02366e33	1.14074	0.570368	−	0.821389i	$-0.306801\pi$
$457$	2.02366e33	1.14074	0.570368	+	0.821389i	$0.306801\pi$
$458$	−1.78678e33	−0.980058
$459$	−1.20438e32	−0.0642842
$460$	0	0
$461$	−8.37646e32	−0.423447	−0.211724	−	0.977330i	$-0.567908\pi$
$461$	−8.37646e32	−0.423447	−0.211724	+	0.977330i	$0.567908\pi$
$462$	3.00404e30	0.00147802
$463$	3.37029e33	1.61401	0.807003	−	0.590547i	$-0.201088\pi$
$463$	3.37029e33	1.61401	0.807003	+	0.590547i	$0.201088\pi$
$464$	−2.95574e32	−0.137782
$465$	0	0
$466$	−8.35108e32	−0.368909
$467$	2.79564e33	1.20232	0.601161	−	0.799128i	$-0.294705\pi$
$467$	2.79564e33	1.20232	0.601161	+	0.799128i	$0.294705\pi$
$468$	1.31294e33	0.549757
$469$	−8.49753e32	−0.346443
$470$	0	0
$471$	3.90494e32	0.150957
$472$	−2.82392e31	−0.0106311
$473$	1.37006e32	0.0502315
$474$	−1.63985e32	−0.0585564
$475$	0	0
$476$	−1.38459e32	−0.0469067
$477$	−4.38696e33	−1.44772
$478$	2.94659e33	0.947263
$479$	6.03192e33	1.88913	0.944565	−	0.328324i	$-0.106484\pi$
$479$	6.03192e33	1.88913	0.944565	+	0.328324i	$0.106484\pi$
$480$	0	0
$481$	3.16281e33	0.940286
$482$	2.43684e32	0.0705894
$483$	2.52156e32	0.0711754
$484$	−1.81409e33	−0.498990
$485$	0	0
$486$	−1.00713e33	−0.263107
$487$	2.63729e33	0.671500	0.335750	−	0.941951i	$-0.391010\pi$
$487$	2.63729e33	0.671500	0.335750	+	0.941951i	$0.391010\pi$
$488$	3.74360e32	0.0929056
$489$	9.83018e31	0.0237794
$490$	0	0
$491$	4.59365e33	1.05594	0.527970	−	0.849263i	$-0.322953\pi$
$491$	4.59365e33	1.05594	0.527970	+	0.849263i	$0.322953\pi$
$492$	9.97902e31	0.0223627
$493$	−6.44948e32	−0.140909
$494$	−6.40793e33	−1.36499
$495$	0	0
$496$	−8.46465e32	−0.171430
$497$	2.43624e32	0.0481134
$498$	−3.27660e32	−0.0631040
$499$	4.28804e33	0.805382	0.402691	−	0.915336i	$-0.368075\pi$
$499$	4.28804e33	0.805382	0.402691	+	0.915336i	$0.368075\pi$
$500$	0	0
$501$	3.60219e32	0.0643569
$502$	4.57488e33	0.797230
$503$	−1.27163e33	−0.216152	−0.108076	−	0.994143i	$-0.534469\pi$
$503$	−1.27163e33	−0.216152	−0.108076	+	0.994143i	$0.534469\pi$
$504$	−7.69809e32	−0.127644
$505$	0	0
$506$	3.08288e32	0.0486491
$507$	2.04739e32	0.0315210
$508$	−4.96369e33	−0.745604
$509$	8.49642e33	1.24527	0.622635	−	0.782512i	$-0.286062\pi$
$509$	8.49642e33	1.24527	0.622635	+	0.782512i	$0.286062\pi$
$510$	0	0
$511$	4.68380e33	0.653637
$512$	−3.24519e32	−0.0441942
$513$	3.26812e33	0.434341
$514$	−9.14768e33	−1.18651
$515$	0	0
$516$	5.73098e32	0.0708122
$517$	−4.57287e32	−0.0551515
$518$	−1.85444e33	−0.218319
$519$	1.13890e33	0.130886
$520$	0	0
$521$	−1.16997e34	−1.28146	−0.640730	−	0.767767i	$-0.721368\pi$
$521$	−1.16997e34	−1.28146	−0.640730	+	0.767767i	$0.721368\pi$
$522$	−3.58580e33	−0.383446
$523$	−2.58251e33	−0.269631	−0.134816	−	0.990871i	$-0.543044\pi$
$523$	−2.58251e33	−0.269631	−0.134816	+	0.990871i	$0.543044\pi$
$524$	−3.68709e33	−0.375874
$525$	0	0
$526$	6.85619e33	0.666440
$527$	−1.84700e33	−0.175321
$528$	1.53632e31	0.00142415
$529$	1.48316e34	1.34274
$530$	0	0
$531$	−3.42588e32	−0.0295863
$532$	3.75714e33	0.316929
$533$	−4.78616e33	−0.394364
$534$	2.05269e33	0.165218
$535$	0	0
$536$	−4.34579e33	−0.333817
$537$	−1.72691e33	−0.129596
$538$	6.21946e33	0.456010
$539$	−5.42984e32	−0.0388980
$540$	0	0
$541$	8.08434e33	0.552941	0.276470	−	0.961022i	$-0.410835\pi$
$541$	8.08434e33	0.552941	0.276470	+	0.961022i	$0.410835\pi$
$542$	−1.53022e34	−1.02274
$543$	1.40210e33	0.0915760
$544$	−7.08106e32	−0.0451972
$545$	0	0
$546$	−6.02689e32	−0.0367439
$547$	−3.07411e34	−1.83180	−0.915900	−	0.401408i	$-0.868521\pi$
$547$	−3.07411e34	−1.83180	−0.915900	+	0.401408i	$0.868521\pi$
$548$	−5.74491e32	−0.0334600
$549$	4.54160e33	0.258556
$550$	0	0
$551$	1.75009e34	0.952061
$552$	1.28957e33	0.0685814
$553$	−4.61150e33	−0.239761
$554$	−2.00079e34	−1.01702
$555$	0	0
$556$	−4.68240e33	−0.227528
$557$	−2.42852e34	−1.15385	−0.576927	−	0.816796i	$-0.695748\pi$
$557$	−2.42852e34	−1.15385	−0.576927	+	0.816796i	$0.695748\pi$
$558$	−1.02690e34	−0.477091
$559$	−2.74871e34	−1.24877
$560$	0	0
$561$	3.35228e31	0.00145648
$562$	−1.67063e34	−0.709865
$563$	−2.54352e34	−1.05701	−0.528507	−	0.848929i	$-0.677248\pi$
$563$	−2.54352e34	−1.05701	−0.528507	+	0.848929i	$0.677248\pi$
$564$	−1.91284e33	−0.0777480
$565$	0	0
$566$	−3.59592e34	−1.39831
$567$	−9.18411e33	−0.349341
$568$	1.24594e33	0.0463599
$569$	9.11823e33	0.331900	0.165950	−	0.986134i	$-0.446931\pi$
$569$	9.11823e33	0.331900	0.165950	+	0.986134i	$0.446931\pi$
$570$	0	0
$571$	−7.94351e33	−0.276733	−0.138366	−	0.990381i	$-0.544185\pi$
$571$	−7.94351e33	−0.276733	−0.138366	+	0.990381i	$0.544185\pi$
$572$	−7.36853e32	−0.0251148
$573$	1.01127e33	0.0337236
$574$	2.80625e33	0.0915646
$575$	0	0
$576$	−3.93694e33	−0.122992
$577$	3.99407e33	0.122101	0.0610503	−	0.998135i	$-0.480555\pi$
$577$	3.99407e33	0.122101	0.0610503	+	0.998135i	$0.480555\pi$
$578$	2.20914e34	0.660884
$579$	−1.86912e33	−0.0547210
$580$	0	0
$581$	−9.21430e33	−0.258381
$582$	1.38640e33	0.0380498
$583$	2.46207e33	0.0661369
$584$	2.39538e34	0.629816
$585$	0	0
$586$	−1.64685e34	−0.414891
$587$	−1.28546e34	−0.317017	−0.158508	−	0.987358i	$-0.550668\pi$
$587$	−1.28546e34	−0.317017	−0.158508	+	0.987358i	$0.550668\pi$
$588$	−2.27131e33	−0.0548351
$589$	5.01190e34	1.18457
$590$	0	0
$591$	7.81277e33	0.176995
$592$	−9.48394e33	−0.210362
$593$	1.35249e34	0.293732	0.146866	−	0.989156i	$-0.453081\pi$
$593$	1.35249e34	0.293732	0.146866	+	0.989156i	$0.453081\pi$
$594$	3.75804e32	0.00799154
$595$	0	0
$596$	2.39757e34	0.488870
$597$	2.12122e33	0.0423552
$598$	−6.18507e34	−1.20943
$599$	−4.11263e34	−0.787561	−0.393781	−	0.919204i	$-0.628833\pi$
$599$	−4.11263e34	−0.787561	−0.393781	+	0.919204i	$0.628833\pi$
$600$	0	0
$601$	−8.66879e34	−1.59231	−0.796155	−	0.605092i	$-0.793136\pi$
$601$	−8.66879e34	−1.59231	−0.796155	+	0.605092i	$0.793136\pi$
$602$	1.61164e34	0.289942
$603$	−5.27216e34	−0.929013
$604$	−4.21487e34	−0.727482
$605$	0	0
$606$	4.04284e33	0.0669543
$607$	1.06263e35	1.72394	0.861972	−	0.506956i	$-0.169229\pi$
$607$	1.06263e35	1.72394	0.861972	+	0.506956i	$0.169229\pi$
$608$	1.92147e34	0.305378
$609$	1.64602e33	0.0256282
$610$	0	0
$611$	9.17439e34	1.37108
$612$	−8.59049e33	−0.125784
$613$	3.14879e34	0.451740	0.225870	−	0.974157i	$-0.427478\pi$
$613$	3.14879e34	0.451740	0.225870	+	0.974157i	$0.427478\pi$
$614$	−1.31414e33	−0.0184729
$615$	0	0
$616$	4.32036e32	0.00583124
$617$	−1.34862e35	−1.78371	−0.891856	−	0.452320i	$-0.850597\pi$
$617$	−1.34862e35	−1.78371	−0.891856	+	0.452320i	$0.850597\pi$
$618$	5.07877e33	0.0658268
$619$	2.48732e34	0.315936	0.157968	−	0.987444i	$-0.449506\pi$
$619$	2.48732e34	0.315936	0.157968	+	0.987444i	$0.449506\pi$
$620$	0	0
$621$	3.15446e34	0.384840
$622$	5.19048e34	0.620622
$623$	5.77248e34	0.676491
$624$	−3.08226e33	−0.0354048
$625$	0	0
$626$	1.13287e35	1.25026
$627$	−9.09651e32	−0.00984080
$628$	5.61603e34	0.595571
$629$	−2.06941e34	−0.215137
$630$	0	0
$631$	−9.95340e34	−0.994499	−0.497250	−	0.867607i	$-0.665657\pi$
$631$	−9.95340e34	−0.994499	−0.497250	+	0.867607i	$0.665657\pi$
$632$	−2.35841e34	−0.231023
$633$	−1.85531e34	−0.178184
$634$	3.32899e34	0.313471
$635$	0	0
$636$	1.02989e34	0.0932342
$637$	1.08937e35	0.967013
$638$	2.01244e33	0.0175172
$639$	1.51153e34	0.129020
$640$	0	0
$641$	2.11836e33	0.0173890	0.00869452	−	0.999962i	$-0.497232\pi$
$641$	2.11836e33	0.0173890	0.00869452	+	0.999962i	$0.497232\pi$
$642$	−1.09001e34	−0.0877491
$643$	5.48861e34	0.433336	0.216668	−	0.976245i	$-0.430481\pi$
$643$	5.48861e34	0.433336	0.216668	+	0.976245i	$0.430481\pi$
$644$	3.62647e34	0.280809
$645$	0	0
$646$	4.19268e34	0.312310
$647$	−8.23657e34	−0.601787	−0.300894	−	0.953658i	$-0.597285\pi$
$647$	−8.23657e34	−0.601787	−0.300894	+	0.953658i	$0.597285\pi$
$648$	−4.69692e34	−0.336609
$649$	1.92269e32	0.00135161
$650$	0	0
$651$	4.71387e33	0.0318871
$652$	1.41376e34	0.0938170
$653$	2.06557e35	1.34470	0.672349	−	0.740235i	$-0.265286\pi$
$653$	2.06557e35	1.34470	0.672349	+	0.740235i	$0.265286\pi$
$654$	1.51885e34	0.0970050
$655$	0	0
$656$	1.43517e34	0.0882276
$657$	2.90599e35	1.75278
$658$	−5.37919e34	−0.318341
$659$	−1.48751e35	−0.863756	−0.431878	−	0.901932i	$-0.642149\pi$
$659$	−1.48751e35	−0.863756	−0.431878	+	0.901932i	$0.642149\pi$
$660$	0	0
$661$	1.93614e35	1.08248	0.541238	−	0.840869i	$-0.317956\pi$
$661$	1.93614e35	1.08248	0.541238	+	0.840869i	$0.317956\pi$
$662$	1.32989e35	0.729610
$663$	−6.72555e33	−0.0362084
$664$	−4.71236e34	−0.248965
$665$	0	0
$666$	−1.15056e35	−0.585438
$667$	1.68922e35	0.843555
$668$	5.18061e34	0.253907
$669$	4.36802e34	0.210116
$670$	0	0
$671$	−2.54886e33	−0.0118118
$672$	1.80721e33	0.00822039
$673$	−2.68004e35	−1.19661	−0.598305	−	0.801268i	$-0.704159\pi$
$673$	−2.68004e35	−1.19661	−0.598305	+	0.801268i	$0.704159\pi$
$674$	2.52357e35	1.10603
$675$	0	0
$676$	2.94452e34	0.124360
$677$	2.37470e35	0.984578	0.492289	−	0.870432i	$-0.336160\pi$
$677$	2.37470e35	0.984578	0.492289	+	0.870432i	$0.336160\pi$
$678$	−4.55170e33	−0.0185269
$679$	3.89878e34	0.155796
$680$	0	0
$681$	3.61039e34	0.139064
$682$	5.76322e33	0.0217952
$683$	−2.30743e35	−0.856779	−0.428390	−	0.903594i	$-0.640919\pi$
$683$	−2.30743e35	−0.856779	−0.428390	+	0.903594i	$0.640919\pi$
$684$	2.33106e35	0.849868
$685$	0	0
$686$	−1.37682e35	−0.483980
$687$	5.08880e34	0.175653
$688$	8.24222e34	0.279376
$689$	−4.93957e35	−1.64418
$690$	0	0
$691$	−1.65717e35	−0.531974	−0.265987	−	0.963977i	$-0.585698\pi$
$691$	−1.65717e35	−0.531974	−0.265987	+	0.963977i	$0.585698\pi$
$692$	1.63795e35	0.516385
$693$	5.24131e33	0.0162283
$694$	3.41774e35	1.03931
$695$	0	0
$696$	8.41804e33	0.0246942
$697$	3.13156e34	0.0902301
$698$	4.39545e35	1.24397
$699$	2.37842e34	0.0661185
$700$	0	0
$701$	−4.91864e35	−1.31938	−0.659690	−	0.751538i	$-0.729312\pi$
$701$	−4.91864e35	−1.31938	−0.659690	+	0.751538i	$0.729312\pi$
$702$	−7.53961e34	−0.198671
$703$	5.61542e35	1.45359
$704$	2.20951e33	0.00561873
$705$	0	0
$706$	4.30649e35	1.05697
$707$	1.13691e35	0.274146
$708$	8.04263e32	0.00190538
$709$	−7.85100e35	−1.82746	−0.913728	−	0.406326i	$-0.866810\pi$
$709$	−7.85100e35	−1.82746	−0.913728	+	0.406326i	$0.866810\pi$
$710$	0	0
$711$	−2.86114e35	−0.642937
$712$	2.95215e35	0.651837
$713$	4.83759e35	1.04957
$714$	3.94337e33	0.00840697
$715$	0	0
$716$	−2.48361e35	−0.511294
$717$	−8.39197e34	−0.169775
$718$	−4.73262e35	−0.940906
$719$	2.69785e33	0.00527118	0.00263559	−	0.999997i	$-0.499161\pi$
$719$	2.69785e33	0.00527118	0.00263559	+	0.999997i	$0.499161\pi$
$720$	0	0
$721$	1.42823e35	0.269530
$722$	−7.56456e35	−1.40304
$723$	−6.94021e33	−0.0126516
$724$	2.01648e35	0.361295
$725$	0	0
$726$	5.16659e34	0.0894326
$727$	−5.83647e35	−0.993048	−0.496524	−	0.868023i	$-0.665391\pi$
$727$	−5.83647e35	−0.993048	−0.496524	+	0.868023i	$0.665391\pi$
$728$	−8.66778e34	−0.144966
$729$	−5.50432e35	−0.904918
$730$	0	0
$731$	1.79847e35	0.285717
$732$	−1.06619e34	−0.0166512
$733$	−6.31572e35	−0.969668	−0.484834	−	0.874606i	$-0.661120\pi$
$733$	−6.31572e35	−0.969668	−0.484834	+	0.874606i	$0.661120\pi$
$734$	2.31488e35	0.349405
$735$	0	0
$736$	1.85464e35	0.270575
$737$	2.95887e34	0.0424406
$738$	1.74109e35	0.245537
$739$	6.44973e35	0.894305	0.447153	−	0.894458i	$-0.352438\pi$
$739$	6.44973e35	0.894305	0.447153	+	0.894458i	$0.352438\pi$
$740$	0	0
$741$	1.82500e35	0.244644
$742$	2.89620e35	0.381750
$743$	−5.26299e35	−0.682138	−0.341069	−	0.940038i	$-0.610789\pi$
$743$	−5.26299e35	−0.682138	−0.341069	+	0.940038i	$0.610789\pi$
$744$	2.41076e34	0.0307250
$745$	0	0
$746$	5.78040e34	0.0712398
$747$	−5.71687e35	−0.692868
$748$	4.82119e33	0.00574625
$749$	−3.06527e35	−0.359291
$750$	0	0
$751$	−1.61425e36	−1.83009	−0.915044	−	0.403355i	$-0.867844\pi$
$751$	−1.61425e36	−1.83009	−0.915044	+	0.403355i	$0.867844\pi$
$752$	−2.75101e35	−0.306740
$753$	−1.30294e35	−0.142885
$754$	−4.03748e35	−0.435481
$755$	0	0
$756$	4.42067e34	0.0461281
$757$	−6.23656e35	−0.640098	−0.320049	−	0.947401i	$-0.603699\pi$
$757$	−6.23656e35	−0.640098	−0.320049	+	0.947401i	$0.603699\pi$
$758$	−1.06625e36	−1.07645
$759$	−8.78014e33	−0.00871926
$760$	0	0
$761$	4.69451e35	0.451110	0.225555	−	0.974230i	$-0.427580\pi$
$761$	4.69451e35	0.451110	0.225555	+	0.974230i	$0.427580\pi$
$762$	1.41367e35	0.133633
$763$	4.27125e35	0.397190
$764$	1.45439e35	0.133050
$765$	0	0
$766$	9.11195e35	0.806774
$767$	−3.85743e34	−0.0336013
$768$	9.24240e33	0.00792081
$769$	1.07024e36	0.902402	0.451201	−	0.892422i	$-0.350996\pi$
$769$	1.07024e36	0.902402	0.451201	+	0.892422i	$0.350996\pi$
$770$	0	0
$771$	2.60529e35	0.212655
$772$	−2.68814e35	−0.215891
$773$	−1.40143e36	−1.10746	−0.553732	−	0.832695i	$-0.686797\pi$
$773$	−1.40143e36	−1.10746	−0.553732	+	0.832695i	$0.686797\pi$
$774$	9.99916e35	0.777502
$775$	0	0
$776$	1.99390e35	0.150118
$777$	5.28151e34	0.0391287
$778$	−1.16154e35	−0.0846816
$779$	−8.49760e35	−0.609646
$780$	0	0
$781$	−8.48308e33	−0.00589407
$782$	4.04686e35	0.276716
$783$	2.05916e35	0.138570
$784$	−3.26656e35	−0.216341
$785$	0	0
$786$	1.05010e35	0.0673670
$787$	−2.04262e36	−1.28974	−0.644870	−	0.764292i	$-0.723089\pi$
$787$	−2.04262e36	−1.28974	−0.644870	+	0.764292i	$0.723089\pi$
$788$	1.12362e36	0.698300
$789$	−1.95267e35	−0.119444
$790$	0	0
$791$	−1.28001e35	−0.0758589
$792$	2.68050e34	0.0156369
$793$	5.11369e35	0.293643
$794$	8.10456e35	0.458114
$795$	0	0
$796$	3.05071e35	0.167104
$797$	−4.60347e35	−0.248231	−0.124115	−	0.992268i	$-0.539609\pi$
$797$	−4.60347e35	−0.248231	−0.124115	+	0.992268i	$0.539609\pi$
$798$	−1.07005e35	−0.0568023
$799$	−6.00276e35	−0.313701
$800$	0	0
$801$	3.58145e36	1.81406
$802$	−2.42417e36	−1.20888
$803$	−1.63091e35	−0.0800731
$804$	1.23770e35	0.0598292
$805$	0	0
$806$	−1.15625e36	−0.541833
$807$	−1.77132e35	−0.0817294
$808$	5.81435e35	0.264155
$809$	2.81919e35	0.126115	0.0630576	−	0.998010i	$-0.479915\pi$
$809$	2.81919e35	0.126115	0.0630576	+	0.998010i	$0.479915\pi$
$810$	0	0
$811$	8.66756e35	0.375955	0.187978	−	0.982173i	$-0.439807\pi$
$811$	8.66756e35	0.375955	0.187978	+	0.982173i	$0.439807\pi$
$812$	2.36728e35	0.101111
$813$	4.35812e35	0.183302
$814$	6.45722e34	0.0267449
$815$	0	0
$816$	2.01671e34	0.00810058
$817$	−4.88020e36	−1.93047
$818$	−2.08936e36	−0.813950
$819$	−1.05154e36	−0.403440
$820$	0	0
$821$	−2.61490e36	−0.973120	−0.486560	−	0.873647i	$-0.661748\pi$
$821$	−2.61490e36	−0.973120	−0.486560	+	0.873647i	$0.661748\pi$
$822$	1.63617e34	0.00599695
$823$	3.63703e35	0.131295	0.0656477	−	0.997843i	$-0.479089\pi$
$823$	3.63703e35	0.131295	0.0656477	+	0.997843i	$0.479089\pi$
$824$	7.30422e35	0.259707
$825$	0	0
$826$	2.26171e34	0.00780165
$827$	3.87599e36	1.31693	0.658467	−	0.752610i	$-0.271205\pi$
$827$	3.87599e36	1.31693	0.658467	+	0.752610i	$0.271205\pi$
$828$	2.24998e36	0.753009
$829$	1.44009e36	0.474743	0.237371	−	0.971419i	$-0.423714\pi$
$829$	1.44009e36	0.474743	0.237371	+	0.971419i	$0.423714\pi$
$830$	0	0
$831$	5.69832e35	0.182278
$832$	−4.43286e35	−0.139683
$833$	−7.12769e35	−0.221252
$834$	1.33356e35	0.0407792
$835$	0	0
$836$	−1.30825e35	−0.0388250
$837$	5.89704e35	0.172411
$838$	−1.41577e36	−0.407795
$839$	−2.94352e36	−0.835300	−0.417650	−	0.908608i	$-0.637146\pi$
$839$	−2.94352e36	−0.835300	−0.417650	+	0.908608i	$0.637146\pi$
$840$	0	0
$841$	−2.52767e36	−0.696260
$842$	−4.72088e34	−0.0128121
$843$	4.75800e35	0.127227
$844$	−2.66828e36	−0.702992
$845$	0	0
$846$	−3.33743e36	−0.853656
$847$	1.45292e36	0.366184
$848$	1.48117e36	0.367837
$849$	1.02413e36	0.250616
$850$	0	0
$851$	5.42013e36	1.28792
$852$	−3.54848e34	−0.00830897
$853$	−1.60861e36	−0.371183	−0.185591	−	0.982627i	$-0.559420\pi$
$853$	−1.60861e36	−0.371183	−0.185591	+	0.982627i	$0.559420\pi$
$854$	−2.99829e35	−0.0681789
$855$	0	0
$856$	−1.56764e36	−0.346197
$857$	7.04998e36	1.53436	0.767180	−	0.641432i	$-0.221659\pi$
$857$	7.04998e36	1.53436	0.767180	+	0.641432i	$0.221659\pi$
$858$	2.09858e34	0.00450127
$859$	6.59563e36	1.39425	0.697126	−	0.716948i	$-0.254462\pi$
$859$	6.59563e36	1.39425	0.697126	+	0.716948i	$0.254462\pi$
$860$	0	0
$861$	−7.99230e34	−0.0164109
$862$	4.61133e36	0.933222
$863$	−9.41312e36	−1.87758	−0.938791	−	0.344487i	$-0.888053\pi$
$863$	−9.41312e36	−1.87758	−0.938791	+	0.344487i	$0.888053\pi$
$864$	2.26081e35	0.0444470
$865$	0	0
$866$	−2.17542e36	−0.415499
$867$	−6.29170e35	−0.118448
$868$	6.77942e35	0.125805
$869$	1.60574e35	0.0293716
$870$	0	0
$871$	−5.93627e36	−1.05508
$872$	2.18439e36	0.382714
$873$	2.41893e36	0.417778
$874$	−1.09813e37	−1.86965
$875$	0	0
$876$	−6.82212e35	−0.112880
$877$	4.23262e36	0.690422	0.345211	−	0.938525i	$-0.387807\pi$
$877$	4.23262e36	0.690422	0.345211	+	0.938525i	$0.387807\pi$
$878$	−3.23521e36	−0.520261
$879$	4.69029e35	0.0743598
$880$	0	0
$881$	−9.23339e36	−1.42286	−0.711431	−	0.702756i	$-0.751952\pi$
$881$	−9.23339e36	−1.42286	−0.711431	+	0.702756i	$0.751952\pi$
$882$	−3.96287e36	−0.602078
$883$	9.76831e36	1.46322	0.731612	−	0.681721i	$-0.238768\pi$
$883$	9.76831e36	1.46322	0.731612	+	0.681721i	$0.238768\pi$
$884$	−9.67259e35	−0.142853
$885$	0	0
$886$	7.48575e36	1.07477
$887$	−1.62985e36	−0.230729	−0.115365	−	0.993323i	$-0.536804\pi$
$887$	−1.62985e36	−0.230729	−0.115365	+	0.993323i	$0.536804\pi$
$888$	2.70106e35	0.0377027
$889$	3.97547e36	0.547162
$890$	0	0
$891$	3.19794e35	0.0427955
$892$	6.28202e36	0.828970
$893$	1.62887e37	2.11955
$894$	−6.82836e35	−0.0876188
$895$	0	0
$896$	2.59910e35	0.0324320
$897$	1.76153e36	0.216763
$898$	3.41071e35	0.0413895
$899$	3.15788e36	0.377919
$900$	0	0
$901$	3.23194e36	0.376186
$902$	−9.77145e34	−0.0112170
$903$	−4.59000e35	−0.0519656
$904$	−6.54619e35	−0.0730943
$905$	0	0
$906$	1.20041e36	0.130385
$907$	4.47773e36	0.479697	0.239848	−	0.970810i	$-0.422902\pi$
$907$	4.47773e36	0.479697	0.239848	+	0.970810i	$0.422902\pi$
$908$	5.19241e36	0.548651
$909$	7.05376e36	0.735144
$910$	0	0
$911$	9.15716e36	0.928498	0.464249	−	0.885705i	$-0.346324\pi$
$911$	9.15716e36	0.928498	0.464249	+	0.885705i	$0.346324\pi$
$912$	−5.47241e35	−0.0547322
$913$	3.20845e35	0.0316527
$914$	−8.28891e36	−0.806622
$915$	0	0
$916$	7.31864e36	0.693006
$917$	2.95303e36	0.275836
$918$	4.93313e35	0.0454558
$919$	1.60416e36	0.145816	0.0729078	−	0.997339i	$-0.476772\pi$
$919$	1.60416e36	0.145816	0.0729078	+	0.997339i	$0.476772\pi$
$920$	0	0
$921$	3.74270e34	0.00331085
$922$	3.43100e36	0.299422
$923$	1.70193e36	0.146528
$924$	−1.23045e34	−0.00104512
$925$	0	0
$926$	−1.38047e37	−1.14127
$927$	8.86121e36	0.722764
$928$	1.21067e36	0.0974263
$929$	−8.64983e35	−0.0686770	−0.0343385	−	0.999410i	$-0.510932\pi$
$929$	−8.64983e35	−0.0686770	−0.0343385	+	0.999410i	$0.510932\pi$
$930$	0	0
$931$	1.93412e37	1.49490
$932$	3.42060e36	0.260858
$933$	−1.47826e36	−0.111232
$934$	−1.14510e37	−0.850170
$935$	0	0
$936$	−5.37779e36	−0.388737
$937$	−7.42933e36	−0.529913	−0.264956	−	0.964260i	$-0.585358\pi$
$937$	−7.42933e36	−0.529913	−0.264956	+	0.964260i	$0.585358\pi$
$938$	3.48059e36	0.244972
$939$	−3.22644e36	−0.224081
$940$	0	0
$941$	6.99775e36	0.473247	0.236623	−	0.971601i	$-0.423959\pi$
$941$	6.99775e36	0.473247	0.236623	+	0.971601i	$0.423959\pi$
$942$	−1.59946e36	−0.106743
$943$	−8.20206e36	−0.540166
$944$	1.15668e35	0.00751732
$945$	0	0
$946$	−5.61178e35	−0.0355190
$947$	9.94818e36	0.621397	0.310698	−	0.950509i	$-0.399437\pi$
$947$	9.94818e36	0.621397	0.310698	+	0.950509i	$0.399437\pi$
$948$	6.71682e35	0.0414057
$949$	3.27204e37	1.99063
$950$	0	0
$951$	−9.48108e35	−0.0561825
$952$	5.67129e35	0.0331681
$953$	−1.89836e37	−1.09577	−0.547883	−	0.836555i	$-0.684566\pi$
$953$	−1.89836e37	−1.09577	−0.547883	+	0.836555i	$0.684566\pi$
$954$	1.79690e37	1.02369
$955$	0	0
$956$	−1.20692e37	−0.669816
$957$	−5.73149e34	−0.00313956
$958$	−2.47068e37	−1.33582
$959$	4.60115e35	0.0245547
$960$	0	0
$961$	−1.01893e37	−0.529785
$962$	−1.29549e37	−0.664883
$963$	−1.90180e37	−0.963466
$964$	−9.98131e35	−0.0499142
$965$	0	0
$966$	−1.03283e36	−0.0503286
$967$	1.99094e37	0.957693	0.478846	−	0.877899i	$-0.341055\pi$
$967$	1.99094e37	0.957693	0.478846	+	0.877899i	$0.341055\pi$
$968$	7.43051e36	0.352839
$969$	−1.19409e36	−0.0559744
$970$	0	0
$971$	1.95878e37	0.894841	0.447421	−	0.894324i	$-0.352343\pi$
$971$	1.95878e37	0.894841	0.447421	+	0.894324i	$0.352343\pi$
$972$	4.12521e36	0.186045
$973$	3.75018e36	0.166971
$974$	−1.08023e37	−0.474822
$975$	0	0
$976$	−1.53338e36	−0.0656942
$977$	−2.97587e37	−1.25873	−0.629365	−	0.777110i	$-0.716685\pi$
$977$	−2.97587e37	−1.25873	−0.629365	+	0.777110i	$0.716685\pi$
$978$	−4.02644e35	−0.0168146
$979$	−2.01000e36	−0.0828727
$980$	0	0
$981$	2.65003e37	1.06509
$982$	−1.88156e37	−0.746663
$983$	4.05260e37	1.58787	0.793934	−	0.608004i	$-0.208030\pi$
$983$	4.05260e37	1.58787	0.793934	+	0.608004i	$0.208030\pi$
$984$	−4.08741e35	−0.0158128
$985$	0	0
$986$	2.64171e36	0.0996376
$987$	1.53201e36	0.0570555
$988$	2.62469e37	0.965197
$989$	−4.71047e37	−1.71045
$990$	0	0
$991$	−3.58276e37	−1.26852	−0.634260	−	0.773120i	$-0.718695\pi$
$991$	−3.58276e37	−1.26852	−0.634260	+	0.773120i	$0.718695\pi$
$992$	3.46712e36	0.121220
$993$	−3.78757e36	−0.130766
$994$	−9.97886e35	−0.0340213
$995$	0	0
$996$	1.34210e36	0.0446213
$997$	−9.64459e36	−0.316661	−0.158330	−	0.987386i	$-0.550611\pi$
$997$	−9.64459e36	−0.316661	−0.158330	+	0.987386i	$0.550611\pi$
$998$	−1.75638e37	−0.569491
$999$	6.60714e36	0.211566

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.26.a.k.1.3		6
5.2	odd	4		10.26.b.a.9.4	✓	12
5.3	odd	4		10.26.b.a.9.9	yes	12
5.4	even	2		50.26.a.l.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.26.b.a.9.4	✓	12	5.2	odd	4
10.26.b.a.9.9	yes	12	5.3	odd	4
50.26.a.k.1.3		6	1.1	even	1	trivial
50.26.a.l.1.4		6	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 26 \)
Character orbit:	\([\chi]\)	\(=\)	50.a (trivial)

\(f(q)\)	\(=\)	\(q-4096.00 q^{2} +116655. q^{3} +1.67772e7 q^{4} -4.77821e8 q^{6} -1.34370e10 q^{7} -6.87195e10 q^{8} -8.33680e11 q^{9} +4.67882e11 q^{11} +1.95715e12 q^{12} -9.38695e13 q^{13} +5.50381e13 q^{14} +2.81475e14 q^{16} +6.14184e14 q^{17} +3.41475e15 q^{18} -1.66661e16 q^{19} -1.56750e15 q^{21} -1.91644e15 q^{22} -1.60865e17 q^{23} -8.01650e15 q^{24} +3.84489e17 q^{26} -1.96094e17 q^{27} -2.25436e17 q^{28} -1.05009e18 q^{29} -3.00725e18 q^{31} -1.15292e18 q^{32} +5.45810e16 q^{33} -2.51570e18 q^{34} -1.39868e19 q^{36} -3.36937e19 q^{37} +6.82643e19 q^{38} -1.09504e19 q^{39} +5.09874e19 q^{41} +6.42050e18 q^{42} +2.92822e20 q^{43} +7.84976e18 q^{44} +6.58901e20 q^{46} -9.77356e20 q^{47} +3.28356e19 q^{48} -1.16051e21 q^{49} +7.16479e19 q^{51} -1.57487e21 q^{52} +5.26216e21 q^{53} +8.03202e20 q^{54} +9.23387e20 q^{56} -1.94419e21 q^{57} +4.30117e21 q^{58} +4.10935e20 q^{59} -5.44766e21 q^{61} +1.23177e22 q^{62} +1.12022e22 q^{63} +4.72237e21 q^{64} -2.23564e20 q^{66} +6.32396e22 q^{67} +1.03043e22 q^{68} -1.87657e22 q^{69} -1.81308e22 q^{71} +5.72901e22 q^{72} -3.48574e23 q^{73} +1.38010e23 q^{74} -2.79610e23 q^{76} -6.28695e21 q^{77} +4.48528e22 q^{78} +3.43193e23 q^{79} +6.83492e23 q^{81} -2.08844e23 q^{82} +6.85739e23 q^{83} -2.62984e22 q^{84} -1.19940e24 q^{86} -1.22499e23 q^{87} -3.21526e22 q^{88} -4.29595e24 q^{89} +1.26133e24 q^{91} -2.69886e24 q^{92} -3.50812e23 q^{93} +4.00325e24 q^{94} -1.34495e23 q^{96} -2.90151e24 q^{97} +4.75347e24 q^{98} -3.90064e23 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 24576 q^{2} + 801416 q^{3} + 100663296 q^{4} - 3282599936 q^{6} + 34007705352 q^{7} - 412316860416 q^{8} + 541468782118 q^{9} + 9861544614312 q^{11} + 13445529337856 q^{12} + 30787386783696 q^{13}+ \cdots - 20\!\cdots\!64 q^{99}+O(q^{100}) \) 6 * q - 24576 * q^2 + 801416 * q^3 + 100663296 * q^4 - 3282599936 * q^6 + 34007705352 * q^7 - 412316860416 * q^8 + 541468782118 * q^9 + 9861544614312 * q^11 + 13445529337856 * q^12 + 30787386783696 * q^13 - 139295561121792 * q^14 + 1688849860263936 * q^16 + 3432836939805312 * q^17 - 2217856131555328 * q^18 + 2007656488916520 * q^19 - 28181710291014728 * q^21 - 40392886740221952 * q^22 + 135711886432707576 * q^23 - 55072888167858176 * q^24 - 126105136266018816 * q^26 + 788054364552235760 * q^27 + 570554618354860032 * q^28 - 2926032031655504820 * q^29 + 79292935777014912 * q^31 - 6917529027641081856 * q^32 - 6844843638395003168 * q^33 - 14060900105442557952 * q^34 + 9084338714850623488 * q^36 + 3345885298190974032 * q^37 - 8223360978602065920 * q^38 + 124726636622561793456 * q^39 + 103229301719876244012 * q^41 + 115432285351996325888 * q^42 + 166390524239317503336 * q^43 + 165449264087949115392 * q^44 - 555875886828370231296 * q^46 + 405906243391034677992 * q^47 + 225578549935547088896 * q^48 + 1716280748139027694542 * q^49 - 1516225893661962013568 * q^51 + 516526638145613070336 * q^52 + 9772760301640799789616 * q^53 - 3227870677205957672960 * q^54 - 2336991716781506691072 * q^56 + 26174725126915157341920 * q^57 + 11985027201660947742720 * q^58 - 12499075130880780168840 * q^59 + 3687285610311118296612 * q^61 - 324783864942653079552 * q^62 + 93007818034972665339656 * q^63 + 28334198897217871282176 * q^64 + 28036479542865932976128 * q^66 + 16759935062266648886712 * q^67 + 57593446831892717371392 * q^68 + 185125325982378999854536 * q^69 + 74568876221253661649712 * q^71 - 37209451376028153806848 * q^72 + 381797761857106829638176 * q^73 - 13704746181390229635072 * q^74 + 33682886568354062008320 * q^76 - 80271541631956375339296 * q^77 - 510880303606013105995776 * q^78 - 400850316873811624799520 * q^79 - 1432110118616113828343714 * q^81 - 422827219844613095473152 * q^82 - 2423159733509985740001144 * q^83 - 472810640801776950837248 * q^84 - 681535587284244493664256 * q^86 - 1231328665433636473947120 * q^87 - 677680185704239576645632 * q^88 + 2982172528060341024616740 * q^89 - 7001711719121019241314768 * q^91 + 2276867632449004467388416 * q^92 - 8806417055581711481018368 * q^93 - 1662591972929678041055232 * q^94 - 923969740536000876118016 * q^96 - 4960469410756873985567808 * q^97 - 7029885944377457436844032 * q^98 - 2053720289282411723573464 * q^99

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