LMFDB - Embedded newform 75.22.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,22,Mod(1,75)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 22, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	75.a (trivial)

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:	yes
Analytic conductor:	$209.608008215$
Analytic rank:	$1$
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} - x^{5} - 530373x^{4} - 23850203x^{3} + 59940607490x^{2} + 2888057920800x - 684850637484000 $ x^6 - x^5 - 530373x^4 - 23850203x^3 + 59940607490x^2 + 2888057920800x - 684850637484000
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{6}\cdot 5^{7}\cdot 7^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$646.238$ of defining polynomial
Character	$\chi$	$=$	75.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2568.95 q^{2} -59049.0 q^{3} +4.50235e6 q^{4} +1.51694e8 q^{6} +5.58323e8 q^{7} -6.17885e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$	\(q-2568.95 q^{2} -59049.0 q^{3} +4.50235e6 q^{4} +1.51694e8 q^{6} +5.58323e8 q^{7} -6.17885e9 q^{8} +3.48678e9 q^{9} -3.70933e10 q^{11} -2.65860e11 q^{12} -7.75851e10 q^{13} -1.43430e12 q^{14} +6.43103e12 q^{16} +1.27282e13 q^{17} -8.95738e12 q^{18} +2.24955e12 q^{19} -3.29684e13 q^{21} +9.52909e13 q^{22} +2.53576e14 q^{23} +3.64855e14 q^{24} +1.99312e14 q^{26} -2.05891e14 q^{27} +2.51377e15 q^{28} -4.09365e15 q^{29} +2.85297e15 q^{31} -3.56301e15 q^{32} +2.19032e15 q^{33} -3.26982e16 q^{34} +1.56987e16 q^{36} +1.07806e16 q^{37} -5.77899e15 q^{38} +4.58132e15 q^{39} +4.19762e16 q^{41} +8.46942e16 q^{42} +6.32027e16 q^{43} -1.67007e17 q^{44} -6.51424e17 q^{46} -4.39281e17 q^{47} -3.79746e17 q^{48} -2.46822e17 q^{49} -7.51590e17 q^{51} -3.49316e17 q^{52} -1.77175e18 q^{53} +5.28924e17 q^{54} -3.44979e18 q^{56} -1.32834e17 q^{57} +1.05164e19 q^{58} -2.10312e18 q^{59} -4.93323e18 q^{61} -7.32914e18 q^{62} +1.94675e18 q^{63} -4.33365e18 q^{64} -5.62683e18 q^{66} -5.24702e18 q^{67} +5.73071e19 q^{68} -1.49734e19 q^{69} -4.41600e19 q^{71} -2.15443e19 q^{72} +2.66577e19 q^{73} -2.76949e19 q^{74} +1.01283e19 q^{76} -2.07100e19 q^{77} -1.17692e19 q^{78} +9.60972e19 q^{79} +1.21577e19 q^{81} -1.07835e20 q^{82} -1.56177e20 q^{83} -1.48435e20 q^{84} -1.62364e20 q^{86} +2.41726e20 q^{87} +2.29194e20 q^{88} -3.75254e20 q^{89} -4.33175e19 q^{91} +1.14169e21 q^{92} -1.68465e20 q^{93} +1.12849e21 q^{94} +2.10392e20 q^{96} +9.05983e20 q^{97} +6.34073e20 q^{98} -1.29336e20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 92 q^{2} - 354294 q^{3} + 4390448 q^{4} - 5432508 q^{6} + 1327143454 q^{7} - 4252271424 q^{8} + 20920706406 q^{9}+O(q^{10}) $ 6 * q + 92 * q^2 - 354294 * q^3 + 4390448 * q^4 - 5432508 * q^6 + 1327143454 * q^7 - 4252271424 * q^8 + 20920706406 * q^9	$ 6 q + 92 q^{2} - 354294 q^{3} + 4390448 q^{4} - 5432508 q^{6} + 1327143454 q^{7} - 4252271424 q^{8} + 20920706406 q^{9} - 78310112516 q^{11} - 259251563952 q^{12} + 116029746338 q^{13} - 313825730148 q^{14} + 1996701612800 q^{16} + 5382900513068 q^{17} + 320784164892 q^{18} + 6969638997622 q^{19} - 78366493815246 q^{21} - 19701817271864 q^{22} + 295754895311052 q^{23} + 251092375315776 q^{24} - 274314326263628 q^{26} - 12\!\cdots\!94 q^{27}+ \cdots - 27\!\cdots\!16 q^{99}+O(q^{100}) $ 6 * q + 92 * q^2 - 354294 * q^3 + 4390448 * q^4 - 5432508 * q^6 + 1327143454 * q^7 - 4252271424 * q^8 + 20920706406 * q^9 - 78310112516 * q^11 - 259251563952 * q^12 + 116029746338 * q^13 - 313825730148 * q^14 + 1996701612800 * q^16 + 5382900513068 * q^17 + 320784164892 * q^18 + 6969638997622 * q^19 - 78366493815246 * q^21 - 19701817271864 * q^22 + 295754895311052 * q^23 + 251092375315776 * q^24 - 274314326263628 * q^26 - 1235346792567894 * q^27 + 2674107576939568 * q^28 - 3895624305077596 * q^29 + 492586859482706 * q^31 - 10862288965692416 * q^32 + 4624133833957284 * q^33 - 30773279577490744 * q^34 + 15308545599801648 * q^36 - 26101348744076316 * q^37 - 8307793564414708 * q^38 - 6851440491512562 * q^39 + 112662882011424056 * q^41 + 18531095539509252 * q^42 - 13377804412633810 * q^43 - 23820753283797856 * q^44 - 156784693648581624 * q^46 - 243206274450163780 * q^47 - 117903233534227200 * q^48 + 327097566625336780 * q^49 - 317854892396152332 * q^51 - 192375894240694512 * q^52 + 554044170675798992 * q^53 - 18941984152707708 * q^54 - 5086364147554777920 * q^56 - 411550213170581478 * q^57 + 11350712056264661624 * q^58 - 7003944869710758212 * q^59 - 10235053877143753790 * q^61 - 1855763029836316572 * q^62 + 4627463093296461054 * q^63 - 10747451596339826688 * q^64 + 1163372608086297336 * q^66 + 8975831852618567158 * q^67 + 64266312753700360864 * q^68 - 17464030813222309548 * q^69 + 18009408229245489272 * q^71 - 14826753670021257024 * q^72 + 81172357549902895652 * q^73 - 103132266830355254968 * q^74 + 111484542579179194608 * q^76 - 40817017589776296372 * q^77 + 16197986651540969772 * q^78 - 79149540279186452080 * q^79 + 72945992754341572806 * q^81 + 17339452427244467168 * q^82 + 238715465076655746756 * q^83 - 157903378310704550832 * q^84 + 28767310268029844764 * q^86 + 230032719590526966204 * q^87 + 1023055046877663881856 * q^88 - 712331293797957179808 * q^89 + 670774308716312800426 * q^91 + 880736698470324218784 * q^92 - 29086761465594306594 * q^93 + 278898069330490980248 * q^94 + 641407301135171472384 * q^96 + 1498553500583421551178 * q^97 + 2662349783988420722392 * q^98 - 273050478761343662916 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−2568.95	−1.77395	−0.886974	−	0.461820i	$-0.847197\pi$
$2$	−2568.95	−1.77395	−0.886974	+	0.461820i	$0.847197\pi$
$3$	−59049.0	−0.577350
$3$	−59049.0	−0.577350
$4$	4.50235e6	2.14689
$5$	0	0
$5$	0	0
$6$	1.51694e8	1.02419
$7$	5.58323e8	0.747061	0.373530	−	0.927618i	$-0.378147\pi$
$7$	5.58323e8	0.747061	0.373530	+	0.927618i	$0.378147\pi$
$8$	−6.17885e9	−2.03452
$9$	3.48678e9	0.333333
$10$	0	0
$11$	−3.70933e10	−0.431194	−0.215597	−	0.976482i	$-0.569170\pi$
$11$	−3.70933e10	−0.431194	−0.215597	+	0.976482i	$0.569170\pi$
$12$	−2.65860e11	−1.23951
$13$	−7.75851e10	−0.156089	−0.0780446	−	0.996950i	$-0.524868\pi$
$13$	−7.75851e10	−0.156089	−0.0780446	+	0.996950i	$0.524868\pi$
$14$	−1.43430e12	−1.32525
$15$	0	0
$16$	6.43103e12	1.46225
$17$	1.27282e13	1.53128	0.765640	−	0.643269i	$-0.222422\pi$
$17$	1.27282e13	1.53128	0.765640	+	0.643269i	$0.222422\pi$
$18$	−8.95738e12	−0.591316
$19$	2.24955e12	0.0841751	0.0420875	−	0.999114i	$-0.486599\pi$
$19$	2.24955e12	0.0841751	0.0420875	+	0.999114i	$0.486599\pi$
$20$	0	0
$21$	−3.29684e13	−0.431316
$22$	9.52909e13	0.764915
$23$	2.53576e14	1.27634	0.638169	−	0.769896i	$-0.279692\pi$
$23$	2.53576e14	1.27634	0.638169	+	0.769896i	$0.279692\pi$
$24$	3.64855e14	1.17463
$25$	0	0
$26$	1.99312e14	0.276894
$27$	−2.05891e14	−0.192450
$28$	2.51377e15	1.60386
$29$	−4.09365e15	−1.80689	−0.903444	−	0.428705i	$-0.858970\pi$
$29$	−4.09365e15	−1.80689	−0.903444	+	0.428705i	$0.858970\pi$
$30$	0	0
$31$	2.85297e15	0.625172	0.312586	−	0.949889i	$-0.398805\pi$
$31$	2.85297e15	0.625172	0.312586	+	0.949889i	$0.398805\pi$
$32$	−3.56301e15	−0.559425
$33$	2.19032e15	0.248950
$34$	−3.26982e16	−2.71641
$35$	0	0
$36$	1.56987e16	0.715630
$37$	1.07806e16	0.368575	0.184287	−	0.982872i	$-0.441002\pi$
$37$	1.07806e16	0.368575	0.184287	+	0.982872i	$0.441002\pi$
$38$	−5.77899e15	−0.149322
$39$	4.58132e15	0.0901182
$40$	0	0
$41$	4.19762e16	0.488396	0.244198	−	0.969725i	$-0.421475\pi$
$41$	4.19762e16	0.488396	0.244198	+	0.969725i	$0.421475\pi$
$42$	8.46942e16	0.765131
$43$	6.32027e16	0.445981	0.222991	−	0.974821i	$-0.428418\pi$
$43$	6.32027e16	0.445981	0.222991	+	0.974821i	$0.428418\pi$
$44$	−1.67007e17	−0.925726
$45$	0	0
$46$	−6.51424e17	−2.26416
$47$	−4.39281e17	−1.21819	−0.609095	−	0.793097i	$-0.708467\pi$
$47$	−4.39281e17	−1.21819	−0.609095	+	0.793097i	$0.708467\pi$
$48$	−3.79746e17	−0.844228
$49$	−2.46822e17	−0.441900
$50$	0	0
$51$	−7.51590e17	−0.884085
$52$	−3.49316e17	−0.335106
$53$	−1.77175e18	−1.39157	−0.695785	−	0.718250i	$-0.744943\pi$
$53$	−1.77175e18	−1.39157	−0.695785	+	0.718250i	$0.744943\pi$
$54$	5.28924e17	0.341396
$55$	0	0
$56$	−3.44979e18	−1.51991
$57$	−1.32834e17	−0.0485985
$58$	1.05164e19	3.20533
$59$	−2.10312e18	−0.535695	−0.267848	−	0.963461i	$-0.586312\pi$
$59$	−2.10312e18	−0.535695	−0.267848	+	0.963461i	$0.586312\pi$
$60$	0	0
$61$	−4.93323e18	−0.885457	−0.442729	−	0.896656i	$-0.645990\pi$
$61$	−4.93323e18	−0.885457	−0.442729	+	0.896656i	$0.645990\pi$
$62$	−7.32914e18	−1.10902
$63$	1.94675e18	0.249020
$64$	−4.33365e18	−0.469855
$65$	0	0
$66$	−5.62683e18	−0.441624
$67$	−5.24702e18	−0.351664	−0.175832	−	0.984420i	$-0.556261\pi$
$67$	−5.24702e18	−0.351664	−0.175832	+	0.984420i	$0.556261\pi$
$68$	5.73071e19	3.28749
$69$	−1.49734e19	−0.736894
$70$	0	0
$71$	−4.41600e19	−1.60996	−0.804982	−	0.593300i	$-0.797825\pi$
$71$	−4.41600e19	−1.60996	−0.804982	+	0.593300i	$0.797825\pi$
$72$	−2.15443e19	−0.678174
$73$	2.66577e19	0.725994	0.362997	−	0.931790i	$-0.381754\pi$
$73$	2.66577e19	0.725994	0.362997	+	0.931790i	$0.381754\pi$
$74$	−2.76949e19	−0.653832
$75$	0	0
$76$	1.01283e19	0.180715
$77$	−2.07100e19	−0.322128
$78$	−1.17692e19	−0.159865
$79$	9.60972e19	1.14189	0.570947	−	0.820987i	$-0.306576\pi$
$79$	9.60972e19	1.14189	0.570947	+	0.820987i	$0.306576\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	−1.07835e20	−0.866389
$83$	−1.56177e20	−1.10483	−0.552415	−	0.833569i	$-0.686294\pi$
$83$	−1.56177e20	−1.10483	−0.552415	+	0.833569i	$0.686294\pi$
$84$	−1.48435e20	−0.925987
$85$	0	0
$86$	−1.62364e20	−0.791147
$87$	2.41726e20	1.04321
$88$	2.29194e20	0.877273
$89$	−3.75254e20	−1.27564	−0.637822	−	0.770184i	$-0.720165\pi$
$89$	−3.75254e20	−1.27564	−0.637822	+	0.770184i	$0.720165\pi$
$90$	0	0
$91$	−4.33175e19	−0.116608
$92$	1.14169e21	2.74016
$93$	−1.68465e20	−0.360943
$94$	1.12849e21	2.16101
$95$	0	0
$96$	2.10392e20	0.322984
$97$	9.05983e20	1.24743	0.623716	−	0.781651i	$-0.285622\pi$
$97$	9.05983e20	1.24743	0.623716	+	0.781651i	$0.285622\pi$
$98$	6.34073e20	0.783908
$99$	−1.29336e20	−0.143731
$100$	0	0
$101$	1.86796e21	1.68265	0.841326	−	0.540528i	$-0.181776\pi$
$101$	1.86796e21	1.68265	0.841326	+	0.540528i	$0.181776\pi$
$102$	1.93080e21	1.56832
$103$	1.38607e21	1.01624	0.508118	−	0.861287i	$-0.330341\pi$
$103$	1.38607e21	1.01624	0.508118	+	0.861287i	$0.330341\pi$
$104$	4.79386e20	0.317567
$105$	0	0
$106$	4.55153e21	2.46857
$107$	2.83321e20	0.139235	0.0696176	−	0.997574i	$-0.477822\pi$
$107$	2.83321e20	0.139235	0.0696176	+	0.997574i	$0.477822\pi$
$108$	−9.26995e20	−0.413169
$109$	4.02076e21	1.62678	0.813391	−	0.581717i	$-0.197619\pi$
$109$	4.02076e21	1.62678	0.813391	+	0.581717i	$0.197619\pi$
$110$	0	0
$111$	−6.36584e20	−0.212797
$112$	3.59059e21	1.09239
$113$	−3.17848e21	−0.880838	−0.440419	−	0.897792i	$-0.645170\pi$
$113$	−3.17848e21	−0.880838	−0.440419	+	0.897792i	$0.645170\pi$
$114$	3.41244e20	0.0862112
$115$	0	0
$116$	−1.84311e22	−3.87919
$117$	−2.70522e20	−0.0520298
$118$	5.40281e21	0.950296
$119$	7.10647e21	1.14396
$120$	0	0
$121$	−6.02434e21	−0.814072
$122$	1.26732e22	1.57075
$123$	−2.47865e21	−0.281976
$124$	1.28451e22	1.34218
$125$	0	0
$126$	−5.00111e21	−0.441749
$127$	3.71270e21	0.301822	0.150911	−	0.988547i	$-0.451779\pi$
$127$	3.71270e21	0.301822	0.150911	+	0.988547i	$0.451779\pi$
$128$	1.86051e22	1.39292
$129$	−3.73205e21	−0.257487
$130$	0	0
$131$	−2.81704e22	−1.65365	−0.826827	−	0.562456i	$-0.809856\pi$
$131$	−2.81704e22	−1.65365	−0.826827	+	0.562456i	$0.809856\pi$
$132$	9.86161e21	0.534468
$133$	1.25598e21	0.0628839
$134$	1.34793e22	0.623833
$135$	0	0
$136$	−7.86458e22	−3.11542
$137$	9.97284e21	0.365808	0.182904	−	0.983131i	$-0.441450\pi$
$137$	9.97284e21	0.365808	0.182904	+	0.983131i	$0.441450\pi$
$138$	3.84659e22	1.30721
$139$	−4.59334e22	−1.44701	−0.723507	−	0.690317i	$-0.757471\pi$
$139$	−4.59334e22	−1.44701	−0.723507	+	0.690317i	$0.757471\pi$
$140$	0	0
$141$	2.59391e22	0.703323
$142$	1.13445e23	2.85599
$143$	2.87789e21	0.0673047
$144$	2.24236e22	0.487415
$145$	0	0
$146$	−6.84824e22	−1.28787
$147$	1.45746e22	0.255131
$148$	4.85381e22	0.791289
$149$	1.23009e23	1.86845	0.934227	−	0.356678i	$-0.116091\pi$
$149$	1.23009e23	1.86845	0.934227	+	0.356678i	$0.116091\pi$
$150$	0	0
$151$	−5.14951e22	−0.679999	−0.340000	−	0.940426i	$-0.610427\pi$
$151$	−5.14951e22	−0.679999	−0.340000	+	0.940426i	$0.610427\pi$
$152$	−1.38996e22	−0.171256
$153$	4.43806e22	0.510427
$154$	5.32031e22	0.571438
$155$	0	0
$156$	2.06267e22	0.193474
$157$	1.04915e23	0.920218	0.460109	−	0.887862i	$-0.347810\pi$
$157$	1.04915e23	0.920218	0.460109	+	0.887862i	$0.347810\pi$
$158$	−2.46869e23	−2.02566
$159$	1.04620e23	0.803423
$160$	0	0
$161$	1.41577e23	0.953502
$162$	−3.12324e22	−0.197105
$163$	−2.23011e23	−1.31934	−0.659668	−	0.751557i	$-0.729303\pi$
$163$	−2.23011e23	−1.31934	−0.659668	+	0.751557i	$0.729303\pi$
$164$	1.88991e23	1.04853
$165$	0	0
$166$	4.01210e23	1.95991
$167$	9.52097e22	0.436675	0.218337	−	0.975873i	$-0.429937\pi$
$167$	9.52097e22	0.436675	0.218337	+	0.975873i	$0.429937\pi$
$168$	2.03707e23	0.877521
$169$	−2.41045e23	−0.975636
$170$	0	0
$171$	7.84371e21	0.0280584
$172$	2.84561e23	0.957472
$173$	−3.78983e23	−1.19987	−0.599937	−	0.800048i	$-0.704808\pi$
$173$	−3.78983e23	−1.19987	−0.599937	+	0.800048i	$0.704808\pi$
$174$	−6.20982e23	−1.85060
$175$	0	0
$176$	−2.38548e23	−0.630511
$177$	1.24187e23	0.309284
$178$	9.64008e23	2.26293
$179$	4.80425e23	1.06333	0.531666	−	0.846954i	$-0.321566\pi$
$179$	4.80425e23	1.06333	0.531666	+	0.846954i	$0.321566\pi$
$180$	0	0
$181$	7.81248e23	1.53874	0.769368	−	0.638806i	$-0.220571\pi$
$181$	7.81248e23	1.53874	0.769368	+	0.638806i	$0.220571\pi$
$182$	1.11281e23	0.206857
$183$	2.91302e23	0.511219
$184$	−1.56681e24	−2.59674
$185$	0	0
$186$	4.32778e23	0.640294
$187$	−4.72133e23	−0.660279
$188$	−1.97780e24	−2.61532
$189$	−1.14954e23	−0.143772
$190$	0	0
$191$	−6.69659e23	−0.749900	−0.374950	−	0.927045i	$-0.622340\pi$
$191$	−6.69659e23	−0.749900	−0.374950	+	0.927045i	$0.622340\pi$
$192$	2.55898e23	0.271271
$193$	4.74605e23	0.476410	0.238205	−	0.971215i	$-0.423441\pi$
$193$	4.74605e23	0.476410	0.238205	+	0.971215i	$0.423441\pi$
$194$	−2.32742e24	−2.21288
$195$	0	0
$196$	−1.11128e24	−0.948712
$197$	−2.28623e24	−1.85023	−0.925114	−	0.379689i	$-0.876031\pi$
$197$	−2.28623e24	−1.85023	−0.925114	+	0.379689i	$0.876031\pi$
$198$	3.32259e23	0.254972
$199$	4.11818e23	0.299743	0.149871	−	0.988706i	$-0.452114\pi$
$199$	4.11818e23	0.299743	0.149871	+	0.988706i	$0.452114\pi$
$200$	0	0
$201$	3.09831e23	0.203033
$202$	−4.79871e24	−2.98494
$203$	−2.28558e24	−1.34986
$204$	−3.38392e24	−1.89803
$205$	0	0
$206$	−3.56075e24	−1.80275
$207$	8.84165e23	0.425446
$208$	−4.98952e23	−0.228241
$209$	−8.34434e22	−0.0362958
$210$	0	0
$211$	2.31862e24	0.912565	0.456283	−	0.889835i	$-0.349181\pi$
$211$	2.31862e24	0.912565	0.456283	+	0.889835i	$0.349181\pi$
$212$	−7.97703e24	−2.98755
$213$	2.60760e24	0.929513
$214$	−7.27837e23	−0.246996
$215$	0	0
$216$	1.27217e24	0.391544
$217$	1.59288e24	0.467041
$218$	−1.03291e25	−2.88583
$219$	−1.57411e24	−0.419153
$220$	0	0
$221$	−9.87522e23	−0.239016
$222$	1.63535e24	0.377490
$223$	1.79125e24	0.394416	0.197208	−	0.980362i	$-0.436813\pi$
$223$	1.79125e24	0.394416	0.197208	+	0.980362i	$0.436813\pi$
$224$	−1.98931e24	−0.417925
$225$	0	0
$226$	8.16536e24	1.56256
$227$	7.37978e24	1.34826	0.674128	−	0.738615i	$-0.264520\pi$
$227$	7.37978e24	1.34826	0.674128	+	0.738615i	$0.264520\pi$
$228$	−5.98065e23	−0.104336
$229$	−9.84331e24	−1.64009	−0.820047	−	0.572297i	$-0.806053\pi$
$229$	−9.84331e24	−1.64009	−0.820047	+	0.572297i	$0.806053\pi$
$230$	0	0
$231$	1.22291e24	0.185981
$232$	2.52940e25	3.67616
$233$	−3.95232e23	−0.0549054	−0.0274527	−	0.999623i	$-0.508740\pi$
$233$	−3.95232e23	−0.0549054	−0.0274527	+	0.999623i	$0.508740\pi$
$234$	6.94959e23	0.0922981
$235$	0	0
$236$	−9.46899e24	−1.15008
$237$	−5.67444e24	−0.659273
$238$	−1.82562e25	−2.02932
$239$	8.45833e24	0.899718	0.449859	−	0.893100i	$-0.351474\pi$
$239$	8.45833e24	0.899718	0.449859	+	0.893100i	$0.351474\pi$
$240$	0	0
$241$	4.92775e24	0.480253	0.240126	−	0.970742i	$-0.422811\pi$
$241$	4.92775e24	0.480253	0.240126	+	0.970742i	$0.422811\pi$
$242$	1.54762e25	1.44412
$243$	−7.17898e23	−0.0641500
$244$	−2.22111e25	−1.90098
$245$	0	0
$246$	6.36753e24	0.500210
$247$	−1.74532e23	−0.0131388
$248$	−1.76281e25	−1.27193
$249$	9.22207e24	0.637874
$250$	0	0
$251$	−1.27655e25	−0.811825	−0.405912	−	0.913912i	$-0.633046\pi$
$251$	−1.27655e25	−0.811825	−0.405912	+	0.913912i	$0.633046\pi$
$252$	8.76496e24	0.534619
$253$	−9.40597e24	−0.550349
$254$	−9.53775e24	−0.535417
$255$	0	0
$256$	−3.87072e25	−2.00112
$257$	−2.25390e24	−0.111850	−0.0559252	−	0.998435i	$-0.517811\pi$
$257$	−2.25390e24	−0.111850	−0.0559252	+	0.998435i	$0.517811\pi$
$258$	9.58746e24	0.456769
$259$	6.01906e24	0.275348
$260$	0	0
$261$	−1.42737e25	−0.602296
$262$	7.23684e25	2.93350
$263$	2.95338e25	1.15023	0.575115	−	0.818073i	$-0.304957\pi$
$263$	2.95338e25	1.15023	0.575115	+	0.818073i	$0.304957\pi$
$264$	−1.35337e25	−0.506494
$265$	0	0
$266$	−3.22654e24	−0.111553
$267$	2.21583e25	0.736494
$268$	−2.36239e25	−0.754983
$269$	−8.12800e24	−0.249795	−0.124898	−	0.992170i	$-0.539860\pi$
$269$	−8.12800e24	−0.249795	−0.124898	+	0.992170i	$0.539860\pi$
$270$	0	0
$271$	−3.49730e25	−0.994388	−0.497194	−	0.867639i	$-0.665636\pi$
$271$	−3.49730e25	−0.994388	−0.497194	+	0.867639i	$0.665636\pi$
$272$	8.18557e25	2.23911
$273$	2.55786e24	0.0673237
$274$	−2.56197e25	−0.648924
$275$	0	0
$276$	−6.74156e25	−1.58203
$277$	−3.52529e25	−0.796448	−0.398224	−	0.917288i	$-0.630373\pi$
$277$	−3.52529e25	−0.796448	−0.398224	+	0.917288i	$0.630373\pi$
$278$	1.18001e26	2.56693
$279$	9.94769e24	0.208391
$280$	0	0
$281$	−1.13577e25	−0.220736	−0.110368	−	0.993891i	$-0.535203\pi$
$281$	−1.13577e25	−0.220736	−0.110368	+	0.993891i	$0.535203\pi$
$282$	−6.66363e25	−1.24766
$283$	7.93896e25	1.43221	0.716104	−	0.697994i	$-0.245924\pi$
$283$	7.93896e25	1.43221	0.716104	+	0.697994i	$0.245924\pi$
$284$	−1.98824e26	−3.45641
$285$	0	0
$286$	−7.39315e24	−0.119395
$287$	2.34362e25	0.364861
$288$	−1.24234e25	−0.186475
$289$	9.29162e25	1.34482
$290$	0	0
$291$	−5.34974e25	−0.720205
$292$	1.20022e26	1.55863
$293$	−1.37769e25	−0.172600	−0.0863001	−	0.996269i	$-0.527504\pi$
$293$	−1.37769e25	−0.172600	−0.0863001	+	0.996269i	$0.527504\pi$
$294$	−3.74414e25	−0.452590
$295$	0	0
$296$	−6.66117e25	−0.749873
$297$	7.63719e24	0.0829833
$298$	−3.16005e26	−3.31454
$299$	−1.96737e25	−0.199223
$300$	0	0
$301$	3.52875e25	0.333175
$302$	1.32288e26	1.20628
$303$	−1.10301e26	−0.971480
$304$	1.44669e25	0.123085
$305$	0	0
$306$	−1.14012e26	−0.905470
$307$	4.85445e24	0.0372552	0.0186276	−	0.999826i	$-0.494070\pi$
$307$	4.85445e24	0.0372552	0.0186276	+	0.999826i	$0.494070\pi$
$308$	−9.32439e25	−0.691573
$309$	−8.18461e25	−0.586724
$310$	0	0
$311$	4.06543e25	0.272347	0.136174	−	0.990685i	$-0.456520\pi$
$311$	4.06543e25	0.272347	0.136174	+	0.990685i	$0.456520\pi$
$312$	−2.83073e25	−0.183347
$313$	−1.76866e26	−1.10772	−0.553859	−	0.832611i	$-0.686845\pi$
$313$	−1.76866e26	−1.10772	−0.553859	+	0.832611i	$0.686845\pi$
$314$	−2.69521e26	−1.63242
$315$	0	0
$316$	4.32664e26	2.45152
$317$	4.48172e25	0.245653	0.122827	−	0.992428i	$-0.460804\pi$
$317$	4.48172e25	0.245653	0.122827	+	0.992428i	$0.460804\pi$
$318$	−2.68763e26	−1.42523
$319$	1.51847e26	0.779119
$320$	0	0
$321$	−1.67298e25	−0.0803875
$322$	−3.63705e26	−1.69146
$323$	2.86329e25	0.128896
$324$	5.47381e25	0.238543
$325$	0	0
$326$	5.72903e26	2.34043
$327$	−2.37422e26	−0.939223
$328$	−2.59364e26	−0.993653
$329$	−2.45261e26	−0.910062
$330$	0	0
$331$	2.48182e26	0.864125	0.432063	−	0.901844i	$-0.357786\pi$
$331$	2.48182e26	0.864125	0.432063	+	0.901844i	$0.357786\pi$
$332$	−7.03162e26	−2.37195
$333$	3.75897e25	0.122858
$334$	−2.44589e26	−0.774638
$335$	0	0
$336$	−2.12021e26	−0.630689
$337$	5.22985e26	1.50791	0.753955	−	0.656926i	$-0.228144\pi$
$337$	5.22985e26	1.50791	0.753955	+	0.656926i	$0.228144\pi$
$338$	6.19233e26	1.73073
$339$	1.87686e26	0.508552
$340$	0	0
$341$	−1.05826e26	−0.269570
$342$	−2.01501e25	−0.0497740
$343$	−4.49655e26	−1.07719
$344$	−3.90519e26	−0.907359
$345$	0	0
$346$	9.73588e26	2.12851
$347$	−2.90390e26	−0.615917	−0.307959	−	0.951400i	$-0.599646\pi$
$347$	−2.90390e26	−0.615917	−0.307959	+	0.951400i	$0.599646\pi$
$348$	1.08834e27	2.23965
$349$	2.17227e26	0.433757	0.216878	−	0.976199i	$-0.430412\pi$
$349$	2.17227e26	0.433757	0.216878	+	0.976199i	$0.430412\pi$
$350$	0	0
$351$	1.59741e25	0.0300394
$352$	1.32164e26	0.241221
$353$	2.00928e26	0.355964	0.177982	−	0.984034i	$-0.443043\pi$
$353$	2.00928e26	0.355964	0.177982	+	0.984034i	$0.443043\pi$
$354$	−3.19031e26	−0.548653
$355$	0	0
$356$	−1.68952e27	−2.73867
$357$	−4.19630e26	−0.660465
$358$	−1.23419e27	−1.88629
$359$	5.85740e26	0.869388	0.434694	−	0.900578i	$-0.356857\pi$
$359$	5.85740e26	0.869388	0.434694	+	0.900578i	$0.356857\pi$
$360$	0	0
$361$	−7.09149e26	−0.992915
$362$	−2.00699e27	−2.72964
$363$	3.55731e26	0.470005
$364$	−1.95031e26	−0.250345
$365$	0	0
$366$	−7.48340e26	−0.906876
$367$	6.17342e26	0.726996	0.363498	−	0.931595i	$-0.381582\pi$
$367$	6.17342e26	0.726996	0.363498	+	0.931595i	$0.381582\pi$
$368$	1.63075e27	1.86632
$369$	1.46362e26	0.162799
$370$	0	0
$371$	−9.89206e26	−1.03959
$372$	−7.58489e26	−0.774905
$373$	2.81352e26	0.279452	0.139726	−	0.990190i	$-0.455378\pi$
$373$	2.81352e26	0.279452	0.139726	+	0.990190i	$0.455378\pi$
$374$	1.21289e27	1.17130
$375$	0	0
$376$	2.71425e27	2.47844
$377$	3.17606e26	0.282036
$378$	2.95310e26	0.255044
$379$	−5.21975e26	−0.438468	−0.219234	−	0.975672i	$-0.570356\pi$
$379$	−5.21975e26	−0.438468	−0.219234	+	0.975672i	$0.570356\pi$
$380$	0	0
$381$	−2.19232e26	−0.174257
$382$	1.72032e27	1.33028
$383$	−1.96111e27	−1.47542	−0.737709	−	0.675118i	$-0.764093\pi$
$383$	−1.96111e27	−1.47542	−0.737709	+	0.675118i	$0.764093\pi$
$384$	−1.09861e27	−0.804205
$385$	0	0
$386$	−1.21924e27	−0.845126
$387$	2.20374e26	0.148660
$388$	4.07905e27	2.67810
$389$	7.13182e25	0.0455753	0.0227876	−	0.999740i	$-0.492746\pi$
$389$	7.13182e25	0.0455753	0.0227876	+	0.999740i	$0.492746\pi$
$390$	0	0
$391$	3.22758e27	1.95443
$392$	1.52507e27	0.899056
$393$	1.66343e27	0.954737
$394$	5.87322e27	3.28221
$395$	0	0
$396$	−5.82318e26	−0.308575
$397$	1.96147e27	1.01224	0.506118	−	0.862464i	$-0.331080\pi$
$397$	1.96147e27	1.01224	0.506118	+	0.862464i	$0.331080\pi$
$398$	−1.05794e27	−0.531728
$399$	−7.41642e25	−0.0363060
$400$	0	0
$401$	−1.16289e27	−0.540158	−0.270079	−	0.962838i	$-0.587050\pi$
$401$	−1.16289e27	−0.540158	−0.270079	+	0.962838i	$0.587050\pi$
$402$	−7.95941e26	−0.360170
$403$	−2.21348e26	−0.0975826
$404$	8.41024e27	3.61247
$405$	0	0
$406$	5.87153e27	2.39457
$407$	−3.99889e26	−0.158927
$408$	4.64396e27	1.79869
$409$	3.42582e27	1.29321	0.646605	−	0.762825i	$-0.276188\pi$
$409$	3.42582e27	1.29321	0.646605	+	0.762825i	$0.276188\pi$
$410$	0	0
$411$	−5.88886e26	−0.211199
$412$	6.24058e27	2.18175
$413$	−1.17422e27	−0.400197
$414$	−2.27137e27	−0.754719
$415$	0	0
$416$	2.76436e26	0.0873203
$417$	2.71232e27	0.835434
$418$	2.14362e26	0.0643868
$419$	7.94594e26	0.232754	0.116377	−	0.993205i	$-0.462872\pi$
$419$	7.94594e26	0.232754	0.116377	+	0.993205i	$0.462872\pi$
$420$	0	0
$421$	−3.54391e27	−0.987462	−0.493731	−	0.869615i	$-0.664367\pi$
$421$	−3.54391e27	−0.987462	−0.493731	+	0.869615i	$0.664367\pi$
$422$	−5.95642e27	−1.61884
$423$	−1.53168e27	−0.406064
$424$	1.09473e28	2.83118
$425$	0	0
$426$	−6.69880e27	−1.64891
$427$	−2.75433e27	−0.661490
$428$	1.27561e27	0.298923
$429$	−1.69936e26	−0.0388584
$430$	0	0
$431$	−7.84829e27	−1.70909	−0.854543	−	0.519381i	$-0.826163\pi$
$431$	−7.84829e27	−1.70909	−0.854543	+	0.519381i	$0.826163\pi$
$432$	−1.32409e27	−0.281409
$433$	−9.01390e27	−1.86978	−0.934888	−	0.354942i	$-0.884501\pi$
$433$	−9.01390e27	−1.86978	−0.934888	+	0.354942i	$0.884501\pi$
$434$	−4.09202e27	−0.828507
$435$	0	0
$436$	1.81029e28	3.49252
$437$	5.70432e26	0.107436
$438$	4.04381e27	0.743555
$439$	4.74868e27	0.852503	0.426251	−	0.904605i	$-0.359834\pi$
$439$	4.74868e27	0.852503	0.426251	+	0.904605i	$0.359834\pi$
$440$	0	0
$441$	−8.60614e26	−0.147300
$442$	2.53689e27	0.424003
$443$	−3.11361e27	−0.508188	−0.254094	−	0.967179i	$-0.581777\pi$
$443$	−3.11361e27	−0.508188	−0.254094	+	0.967179i	$0.581777\pi$
$444$	−2.86613e27	−0.456851
$445$	0	0
$446$	−4.60163e27	−0.699673
$447$	−7.26358e27	−1.07875
$448$	−2.41957e27	−0.351010
$449$	2.79453e27	0.396025	0.198013	−	0.980199i	$-0.436551\pi$
$449$	2.79453e27	0.396025	0.198013	+	0.980199i	$0.436551\pi$
$450$	0	0
$451$	−1.55703e27	−0.210593
$452$	−1.43106e28	−1.89106
$453$	3.04074e27	0.392598
$454$	−1.89583e28	−2.39173
$455$	0	0
$456$	8.20760e26	0.0988747
$457$	−1.33559e28	−1.57236	−0.786180	−	0.617998i	$-0.787944\pi$
$457$	−1.33559e28	−1.57236	−0.786180	+	0.617998i	$0.787944\pi$
$458$	2.52870e28	2.90944
$459$	−2.62063e27	−0.294695
$460$	0	0
$461$	−7.84878e27	−0.843223	−0.421612	−	0.906777i	$-0.638535\pi$
$461$	−7.84878e27	−0.843223	−0.421612	+	0.906777i	$0.638535\pi$
$462$	−3.14159e27	−0.329920
$463$	7.86992e27	0.807922	0.403961	−	0.914776i	$-0.367633\pi$
$463$	7.86992e27	0.807922	0.403961	+	0.914776i	$0.367633\pi$
$464$	−2.63263e28	−2.64212
$465$	0	0
$466$	1.01533e27	0.0973992
$467$	−9.65288e26	−0.0905377	−0.0452689	−	0.998975i	$-0.514414\pi$
$467$	−9.65288e26	−0.0905377	−0.0452689	+	0.998975i	$0.514414\pi$
$468$	−1.21799e27	−0.111702
$469$	−2.92953e27	−0.262714
$470$	0	0
$471$	−6.19512e27	−0.531288
$472$	1.29949e28	1.08988
$473$	−2.34440e27	−0.192304
$474$	1.45774e28	1.16952
$475$	0	0
$476$	3.19958e28	2.45595
$477$	−6.17770e27	−0.463856
$478$	−2.17290e28	−1.59605
$479$	−7.09688e27	−0.509970	−0.254985	−	0.966945i	$-0.582070\pi$
$479$	−7.09688e27	−0.509970	−0.254985	+	0.966945i	$0.582070\pi$
$480$	0	0
$481$	−8.36415e26	−0.0575305
$482$	−1.26591e28	−0.851943
$483$	−8.35999e27	−0.550505
$484$	−2.71237e28	−1.74772
$485$	0	0
$486$	1.84424e27	0.113799
$487$	−5.40175e27	−0.326198	−0.163099	−	0.986610i	$-0.552149\pi$
$487$	−5.40175e27	−0.326198	−0.163099	+	0.986610i	$0.552149\pi$
$488$	3.04816e28	1.80148
$489$	1.31686e28	0.761720
$490$	0	0
$491$	1.37622e28	0.762665	0.381332	−	0.924438i	$-0.375465\pi$
$491$	1.37622e28	0.762665	0.381332	+	0.924438i	$0.375465\pi$
$492$	−1.11598e28	−0.605370
$493$	−5.21049e28	−2.76685
$494$	4.48363e26	0.0233076
$495$	0	0
$496$	1.83475e28	0.914155
$497$	−2.46555e28	−1.20274
$498$	−2.36910e28	−1.13156
$499$	2.91157e28	1.36167	0.680834	−	0.732437i	$-0.261617\pi$
$499$	2.91157e28	1.36167	0.680834	+	0.732437i	$0.261617\pi$
$500$	0	0
$501$	−5.62204e27	−0.252114
$502$	3.27938e28	1.44013
$503$	−2.54685e28	−1.09532	−0.547659	−	0.836702i	$-0.684481\pi$
$503$	−2.54685e28	−1.09532	−0.547659	+	0.836702i	$0.684481\pi$
$504$	−1.20287e28	−0.506637
$505$	0	0
$506$	2.41635e28	0.976290
$507$	1.42335e28	0.563284
$508$	1.67159e28	0.647979
$509$	−1.73800e28	−0.659954	−0.329977	−	0.943989i	$-0.607041\pi$
$509$	−1.73800e28	−0.659954	−0.329977	+	0.943989i	$0.607041\pi$
$510$	0	0
$511$	1.48836e28	0.542361
$512$	6.04193e28	2.15695
$513$	−4.63163e26	−0.0161995
$514$	5.79016e27	0.198417
$515$	0	0
$516$	−1.68030e28	−0.552797
$517$	1.62944e28	0.525276
$518$	−1.54627e28	−0.488452
$519$	2.23786e28	0.692747
$520$	0	0
$521$	−1.00200e28	−0.297901	−0.148950	−	0.988845i	$-0.547589\pi$
$521$	−1.00200e28	−0.297901	−0.148950	+	0.988845i	$0.547589\pi$
$522$	3.66683e28	1.06844
$523$	−3.19105e28	−0.911309	−0.455654	−	0.890157i	$-0.650595\pi$
$523$	−3.19105e28	−0.911309	−0.455654	+	0.890157i	$0.650595\pi$
$524$	−1.26833e29	−3.55021
$525$	0	0
$526$	−7.58709e28	−2.04045
$527$	3.63133e28	0.957314
$528$	1.40860e28	0.364026
$529$	2.48292e28	0.629039
$530$	0	0
$531$	−7.33313e27	−0.178565
$532$	5.65485e27	0.135005
$533$	−3.25672e27	−0.0762334
$534$	−5.69237e28	−1.30650
$535$	0	0
$536$	3.24205e28	0.715467
$537$	−2.83686e28	−0.613915
$538$	2.08804e28	0.443124
$539$	9.15544e27	0.190545
$540$	0	0
$541$	−5.44256e28	−1.08951	−0.544755	−	0.838595i	$-0.683378\pi$
$541$	−5.44256e28	−1.08951	−0.544755	+	0.838595i	$0.683378\pi$
$542$	8.98440e28	1.76399
$543$	−4.61319e28	−0.888390
$544$	−4.53508e28	−0.856637
$545$	0	0
$546$	−6.57100e27	−0.119429
$547$	−1.01579e29	−1.81107	−0.905536	−	0.424270i	$-0.860531\pi$
$547$	−1.01579e29	−1.81107	−0.905536	+	0.424270i	$0.860531\pi$
$548$	4.49013e28	0.785349
$549$	−1.72011e28	−0.295152
$550$	0	0
$551$	−9.20888e27	−0.152095
$552$	9.25183e28	1.49923
$553$	5.36532e28	0.853065
$554$	9.05630e28	1.41286
$555$	0	0
$556$	−2.06808e29	−3.10658
$557$	2.16249e28	0.318767	0.159384	−	0.987217i	$-0.449049\pi$
$557$	2.16249e28	0.318767	0.159384	+	0.987217i	$0.449049\pi$
$558$	−2.55551e28	−0.369674
$559$	−4.90358e27	−0.0696129
$560$	0	0
$561$	2.78790e28	0.381212
$562$	2.91774e28	0.391575
$563$	−1.31464e29	−1.73168	−0.865840	−	0.500320i	$-0.833216\pi$
$563$	−1.31464e29	−1.73168	−0.865840	+	0.500320i	$0.833216\pi$
$564$	1.16787e29	1.50996
$565$	0	0
$566$	−2.03948e29	−2.54066
$567$	6.78790e27	0.0830067
$568$	2.72858e29	3.27551
$569$	1.59225e29	1.87643	0.938215	−	0.346053i	$-0.112478\pi$
$569$	1.59225e29	1.87643	0.938215	+	0.346053i	$0.112478\pi$
$570$	0	0
$571$	−1.20057e29	−1.36367	−0.681833	−	0.731508i	$-0.738817\pi$
$571$	−1.20057e29	−1.36367	−0.681833	+	0.731508i	$0.738817\pi$
$572$	1.29573e28	0.144496
$573$	3.95427e28	0.432955
$574$	−6.02065e28	−0.647245
$575$	0	0
$576$	−1.51105e28	−0.156618
$577$	8.22193e28	0.836812	0.418406	−	0.908260i	$-0.362589\pi$
$577$	8.22193e28	0.836812	0.418406	+	0.908260i	$0.362589\pi$
$578$	−2.38697e29	−2.38564
$579$	−2.80249e28	−0.275055
$580$	0	0
$581$	−8.71969e28	−0.825376
$582$	1.37432e29	1.27761
$583$	6.57199e28	0.600036
$584$	−1.64714e29	−1.47705
$585$	0	0
$586$	3.53921e28	0.306184
$587$	−2.00399e29	−1.70292	−0.851461	−	0.524417i	$-0.824283\pi$
$587$	−2.00399e29	−1.70292	−0.851461	+	0.524417i	$0.824283\pi$
$588$	6.56199e28	0.547739
$589$	6.41791e27	0.0526239
$590$	0	0
$591$	1.35000e29	1.06823
$592$	6.93304e28	0.538947
$593$	−6.29550e28	−0.480791	−0.240395	−	0.970675i	$-0.577277\pi$
$593$	−6.29550e28	−0.480791	−0.240395	+	0.970675i	$0.577277\pi$
$594$	−1.96196e28	−0.147208
$595$	0	0
$596$	5.53832e29	4.01137
$597$	−2.43175e28	−0.173056
$598$	5.05408e28	0.353411
$599$	6.47634e28	0.444988	0.222494	−	0.974934i	$-0.428580\pi$
$599$	6.47634e28	0.444988	0.222494	+	0.974934i	$0.428580\pi$
$600$	0	0
$601$	8.96436e26	0.00594754	0.00297377	−	0.999996i	$-0.499053\pi$
$601$	8.96436e26	0.00594754	0.00297377	+	0.999996i	$0.499053\pi$
$602$	−9.06518e28	−0.591035
$603$	−1.82952e28	−0.117221
$604$	−2.31849e29	−1.45988
$605$	0	0
$606$	2.83359e29	1.72335
$607$	−2.88429e28	−0.172408	−0.0862042	−	0.996277i	$-0.527474\pi$
$607$	−2.88429e28	−0.172408	−0.0862042	+	0.996277i	$0.527474\pi$
$608$	−8.01518e27	−0.0470897
$609$	1.34961e29	0.779339
$610$	0	0
$611$	3.40817e28	0.190146
$612$	1.99817e29	1.09583
$613$	−2.23267e29	−1.20362	−0.601811	−	0.798638i	$-0.705554\pi$
$613$	−2.23267e29	−1.20362	−0.601811	+	0.798638i	$0.705554\pi$
$614$	−1.24708e28	−0.0660887
$615$	0	0
$616$	1.27964e29	0.655376
$617$	−8.48914e28	−0.427435	−0.213717	−	0.976896i	$-0.568557\pi$
$617$	−8.48914e28	−0.427435	−0.213717	+	0.976896i	$0.568557\pi$
$618$	2.10259e29	1.04082
$619$	−2.50545e29	−1.21937	−0.609684	−	0.792645i	$-0.708704\pi$
$619$	−2.50545e29	−1.21937	−0.609684	+	0.792645i	$0.708704\pi$
$620$	0	0
$621$	−5.22090e28	−0.245631
$622$	−1.04439e29	−0.483129
$623$	−2.09513e29	−0.952984
$624$	2.94626e28	0.131775
$625$	0	0
$626$	4.54360e29	1.96503
$627$	4.92725e27	0.0209554
$628$	4.72364e29	1.97561
$629$	1.37218e29	0.564391
$630$	0	0
$631$	−3.77252e29	−1.50080	−0.750400	−	0.660984i	$-0.770139\pi$
$631$	−3.77252e29	−1.50080	−0.750400	+	0.660984i	$0.770139\pi$
$632$	−5.93770e29	−2.32321
$633$	−1.36912e29	−0.526870
$634$	−1.15133e29	−0.435776
$635$	0	0
$636$	4.71036e29	1.72486
$637$	1.91497e28	0.0689759
$638$	−3.90087e29	−1.38212
$639$	−1.53976e29	−0.536654
$640$	0	0
$641$	−2.07562e29	−0.700064	−0.350032	−	0.936738i	$-0.613829\pi$
$641$	−2.07562e29	−0.700064	−0.350032	+	0.936738i	$0.613829\pi$
$642$	4.29781e28	0.142603
$643$	−2.75939e29	−0.900738	−0.450369	−	0.892843i	$-0.648708\pi$
$643$	−2.75939e29	−0.900738	−0.450369	+	0.892843i	$0.648708\pi$
$644$	6.37431e29	2.04706
$645$	0	0
$646$	−7.35564e28	−0.228654
$647$	−1.30012e29	−0.397638	−0.198819	−	0.980036i	$-0.563711\pi$
$647$	−1.30012e29	−0.397638	−0.198819	+	0.980036i	$0.563711\pi$
$648$	−7.51203e28	−0.226058
$649$	7.80117e28	0.230989
$650$	0	0
$651$	−9.40578e28	−0.269646
$652$	−1.00407e30	−2.83247
$653$	−1.20201e29	−0.333672	−0.166836	−	0.985985i	$-0.553355\pi$
$653$	−1.20201e29	−0.333672	−0.166836	+	0.985985i	$0.553355\pi$
$654$	6.09924e29	1.66613
$655$	0	0
$656$	2.69950e29	0.714155
$657$	9.29497e28	0.241998
$658$	6.30063e29	1.61440
$659$	−3.25641e29	−0.821186	−0.410593	−	0.911819i	$-0.634678\pi$
$659$	−3.25641e29	−0.821186	−0.410593	+	0.911819i	$0.634678\pi$
$660$	0	0
$661$	−3.48891e28	−0.0852264	−0.0426132	−	0.999092i	$-0.513568\pi$
$661$	−3.48891e28	−0.0852264	−0.0426132	+	0.999092i	$0.513568\pi$
$662$	−6.37568e29	−1.53291
$663$	5.83122e28	0.137996
$664$	9.64991e29	2.24780
$665$	0	0
$666$	−9.65660e28	−0.217944
$667$	−1.03805e30	−2.30620
$668$	4.28668e29	0.937493
$669$	−1.05771e29	−0.227716
$670$	0	0
$671$	1.82990e29	0.381804
$672$	1.17467e29	0.241289
$673$	9.26788e29	1.87423	0.937113	−	0.349025i	$-0.113487\pi$
$673$	9.26788e29	1.87423	0.937113	+	0.349025i	$0.113487\pi$
$674$	−1.34352e30	−2.67495
$675$	0	0
$676$	−1.08527e30	−2.09458
$677$	3.95810e29	0.752152	0.376076	−	0.926589i	$-0.377273\pi$
$677$	3.95810e29	0.752152	0.376076	+	0.926589i	$0.377273\pi$
$678$	−4.82156e29	−0.902145
$679$	5.05831e29	0.931907
$680$	0	0
$681$	−4.35769e29	−0.778416
$682$	2.71862e29	0.478203
$683$	8.69191e29	1.50556	0.752779	−	0.658273i	$-0.228713\pi$
$683$	8.69191e29	1.50556	0.752779	+	0.658273i	$0.228713\pi$
$684$	3.53151e28	0.0602382
$685$	0	0
$686$	1.15514e30	1.91087
$687$	5.81238e29	0.946908
$688$	4.06458e29	0.652134
$689$	1.37461e29	0.217209
$690$	0	0
$691$	5.25953e29	0.806172	0.403086	−	0.915162i	$-0.367938\pi$
$691$	5.25953e29	0.806172	0.403086	+	0.915162i	$0.367938\pi$
$692$	−1.70631e30	−2.57600
$693$	−7.22114e28	−0.107376
$694$	7.45998e29	1.09260
$695$	0	0
$696$	−1.49359e30	−2.12243
$697$	5.34283e29	0.747871
$698$	−5.58045e29	−0.769462
$699$	2.33380e28	0.0316996
$700$	0	0
$701$	−7.49111e29	−0.987431	−0.493716	−	0.869623i	$-0.664362\pi$
$701$	−7.49111e29	−0.987431	−0.493716	+	0.869623i	$0.664362\pi$
$702$	−4.10366e28	−0.0532883
$703$	2.42516e28	0.0310248
$704$	1.60749e29	0.202599
$705$	0	0
$706$	−5.16174e29	−0.631461
$707$	1.04293e30	1.25704
$708$	5.59135e29	0.663999
$709$	−4.05321e29	−0.474257	−0.237128	−	0.971478i	$-0.576206\pi$
$709$	−4.05321e29	−0.474257	−0.237128	+	0.971478i	$0.576206\pi$
$710$	0	0
$711$	3.35070e29	0.380632
$712$	2.31863e30	2.59533
$713$	7.23444e29	0.797931
$714$	1.07801e30	1.17163
$715$	0	0
$716$	2.16304e30	2.28286
$717$	−4.99456e29	−0.519453
$718$	−1.50474e30	−1.54225
$719$	1.84533e28	0.0186389	0.00931945	−	0.999957i	$-0.497033\pi$
$719$	1.84533e28	0.0186389	0.00931945	+	0.999957i	$0.497033\pi$
$720$	0	0
$721$	7.73875e29	0.759190
$722$	1.82177e30	1.76138
$723$	−2.90979e29	−0.277274
$724$	3.51746e30	3.30350
$725$	0	0
$726$	−9.13855e29	−0.833764
$727$	5.77932e29	0.519715	0.259858	−	0.965647i	$-0.416324\pi$
$727$	5.77932e29	0.519715	0.259858	+	0.965647i	$0.416324\pi$
$728$	2.67652e29	0.237242
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	8.04459e29	0.682922
$732$	1.31154e30	1.09753
$733$	−1.71345e30	−1.41344	−0.706722	−	0.707491i	$-0.749827\pi$
$733$	−1.71345e30	−1.41344	−0.706722	+	0.707491i	$0.749827\pi$
$734$	−1.58592e30	−1.28965
$735$	0	0
$736$	−9.03493e29	−0.714016
$737$	1.94629e29	0.151635
$738$	−3.75996e29	−0.288796
$739$	1.47473e29	0.111672	0.0558362	−	0.998440i	$-0.482218\pi$
$739$	1.47473e29	0.111672	0.0558362	+	0.998440i	$0.482218\pi$
$740$	0	0
$741$	1.03059e28	0.00758570
$742$	2.54122e30	1.84417
$743$	1.29857e30	0.929144	0.464572	−	0.885535i	$-0.346208\pi$
$743$	1.29857e30	0.929144	0.464572	+	0.885535i	$0.346208\pi$
$744$	1.04092e30	0.734347
$745$	0	0
$746$	−7.22778e29	−0.495733
$747$	−5.44554e29	−0.368277
$748$	−2.12571e30	−1.41755
$749$	1.58184e29	0.104017
$750$	0	0
$751$	−2.64216e30	−1.68943	−0.844713	−	0.535220i	$-0.820229\pi$
$751$	−2.64216e30	−1.68943	−0.844713	+	0.535220i	$0.820229\pi$
$752$	−2.82503e30	−1.78129
$753$	7.53788e29	0.468707
$754$	−8.15914e29	−0.500317
$755$	0	0
$756$	−5.17562e29	−0.308662
$757$	1.84730e30	1.08650	0.543252	−	0.839570i	$-0.317193\pi$
$757$	1.84730e30	1.08650	0.543252	+	0.839570i	$0.317193\pi$
$758$	1.34093e30	0.777819
$759$	5.55413e29	0.317744
$760$	0	0
$761$	2.82617e30	1.57275	0.786376	−	0.617749i	$-0.211955\pi$
$761$	2.82617e30	1.57275	0.786376	+	0.617749i	$0.211955\pi$
$762$	5.63195e29	0.309123
$763$	2.24488e30	1.21530
$764$	−3.01504e30	−1.60995
$765$	0	0
$766$	5.03800e30	2.61732
$767$	1.63171e29	0.0836163
$768$	2.28562e30	1.15535
$769$	−1.57374e30	−0.784705	−0.392352	−	0.919815i	$-0.628339\pi$
$769$	−1.57374e30	−0.784705	−0.392352	+	0.919815i	$0.628339\pi$
$770$	0	0
$771$	1.33091e29	0.0645768
$772$	2.13684e30	1.02280
$773$	3.55240e30	1.67740	0.838702	−	0.544591i	$-0.183315\pi$
$773$	3.55240e30	1.67740	0.838702	+	0.544591i	$0.183315\pi$
$774$	−5.66130e29	−0.263716
$775$	0	0
$776$	−5.59793e30	−2.53793
$777$	−3.55420e29	−0.158972
$778$	−1.83213e29	−0.0808481
$779$	9.44276e28	0.0411108
$780$	0	0
$781$	1.63804e30	0.694206
$782$	−8.29148e30	−3.46706
$783$	8.42846e29	0.347736
$784$	−1.58732e30	−0.646167
$785$	0	0
$786$	−4.27328e30	−1.69365
$787$	1.46623e30	0.573414	0.286707	−	0.958018i	$-0.407439\pi$
$787$	1.46623e30	0.573414	0.286707	+	0.958018i	$0.407439\pi$
$788$	−1.02934e31	−3.97224
$789$	−1.74394e30	−0.664085
$790$	0	0
$791$	−1.77462e30	−0.658039
$792$	7.99150e29	0.292424
$793$	3.82745e29	0.138210
$794$	−5.03892e30	−1.79565
$795$	0	0
$796$	1.85415e30	0.643514
$797$	−3.89960e30	−1.33569	−0.667847	−	0.744298i	$-0.732784\pi$
$797$	−3.89960e30	−1.33569	−0.667847	+	0.744298i	$0.732784\pi$
$798$	1.90524e29	0.0644050
$799$	−5.59128e30	−1.86539
$800$	0	0
$801$	−1.30843e30	−0.425215
$802$	2.98740e30	0.958213
$803$	−9.88823e29	−0.313044
$804$	1.39497e30	0.435890
$805$	0	0
$806$	5.68632e29	0.173106
$807$	4.79950e29	0.144219
$808$	−1.15419e31	−3.42339
$809$	−6.55857e30	−1.92021	−0.960107	−	0.279633i	$-0.909787\pi$
$809$	−6.55857e30	−1.92021	−0.960107	+	0.279633i	$0.909787\pi$
$810$	0	0
$811$	5.52522e28	0.0157627	0.00788135	−	0.999969i	$-0.497491\pi$
$811$	5.52522e28	0.0157627	0.00788135	+	0.999969i	$0.497491\pi$
$812$	−1.02905e31	−2.89799
$813$	2.06512e30	0.574110
$814$	1.02729e30	0.281928
$815$	0	0
$816$	−4.83349e30	−1.29275
$817$	1.42178e29	0.0375405
$818$	−8.80075e30	−2.29409
$819$	−1.51039e29	−0.0388694
$820$	0	0
$821$	−5.51600e30	−1.38363	−0.691817	−	0.722073i	$-0.743189\pi$
$821$	−5.51600e30	−1.38363	−0.691817	+	0.722073i	$0.743189\pi$
$822$	1.51282e30	0.374656
$823$	4.67949e30	1.14419	0.572097	−	0.820186i	$-0.306130\pi$
$823$	4.67949e30	1.14419	0.572097	+	0.820186i	$0.306130\pi$
$824$	−8.56432e30	−2.06755
$825$	0	0
$826$	3.01651e30	0.709928
$827$	−4.73472e30	−1.10024	−0.550119	−	0.835086i	$-0.685418\pi$
$827$	−4.73472e30	−1.10024	−0.550119	+	0.835086i	$0.685418\pi$
$828$	3.98082e30	0.913386
$829$	7.07617e30	1.60316	0.801578	−	0.597890i	$-0.203994\pi$
$829$	7.07617e30	1.60316	0.801578	+	0.597890i	$0.203994\pi$
$830$	0	0
$831$	2.08165e30	0.459829
$832$	3.36226e29	0.0733393
$833$	−3.14161e30	−0.676674
$834$	−6.96781e30	−1.48202
$835$	0	0
$836$	−3.75692e29	−0.0779230
$837$	−5.87401e29	−0.120314
$838$	−2.04127e30	−0.412894
$839$	5.63558e30	1.12574	0.562870	−	0.826545i	$-0.309697\pi$
$839$	5.63558e30	1.12574	0.562870	+	0.826545i	$0.309697\pi$
$840$	0	0
$841$	1.16251e31	2.26485
$842$	9.10413e30	1.75171
$843$	6.70661e29	0.127442
$844$	1.04392e31	1.95918
$845$	0	0
$846$	3.93481e30	0.720335
$847$	−3.36352e30	−0.608161
$848$	−1.13941e31	−2.03482
$849$	−4.68787e30	−0.826885
$850$	0	0
$851$	2.73370e30	0.470426
$852$	1.17403e31	1.99556
$853$	3.93728e30	0.661045	0.330523	−	0.943798i	$-0.392775\pi$
$853$	3.93728e30	0.661045	0.330523	+	0.943798i	$0.392775\pi$
$854$	7.07574e30	1.17345
$855$	0	0
$856$	−1.75060e30	−0.283277
$857$	3.31208e30	0.529422	0.264711	−	0.964328i	$-0.414723\pi$
$857$	3.31208e30	0.529422	0.264711	+	0.964328i	$0.414723\pi$
$858$	4.36558e29	0.0689328
$859$	−7.48047e30	−1.16681	−0.583406	−	0.812181i	$-0.698280\pi$
$859$	−7.48047e30	−1.16681	−0.583406	+	0.812181i	$0.698280\pi$
$860$	0	0
$861$	−1.38389e30	−0.210653
$862$	2.01619e31	3.03183
$863$	8.74392e30	1.29895	0.649476	−	0.760382i	$-0.274988\pi$
$863$	8.74392e30	1.29895	0.649476	+	0.760382i	$0.274988\pi$
$864$	7.33592e29	0.107661
$865$	0	0
$866$	2.31563e31	3.31689
$867$	−5.48661e30	−0.776432
$868$	7.17170e30	1.00269
$869$	−3.56456e30	−0.492378
$870$	0	0
$871$	4.07091e29	0.0548909
$872$	−2.48436e31	−3.30972
$873$	3.15897e30	0.415810
$874$	−1.46541e30	−0.190586
$875$	0	0
$876$	−7.08721e30	−0.899875
$877$	−6.82497e30	−0.856259	−0.428130	−	0.903717i	$-0.640827\pi$
$877$	−6.82497e30	−0.856259	−0.428130	+	0.903717i	$0.640827\pi$
$878$	−1.21991e31	−1.51229
$879$	8.13512e29	0.0996507
$880$	0	0
$881$	−1.33104e31	−1.59200	−0.796002	−	0.605294i	$-0.793056\pi$
$881$	−1.33104e31	−1.59200	−0.796002	+	0.605294i	$0.793056\pi$
$882$	2.21087e30	0.261303
$883$	2.94784e30	0.344284	0.172142	−	0.985072i	$-0.444931\pi$
$883$	2.94784e30	0.344284	0.172142	+	0.985072i	$0.444931\pi$
$884$	−4.44617e30	−0.513142
$885$	0	0
$886$	7.99870e30	0.901499
$887$	1.22822e31	1.36797	0.683987	−	0.729494i	$-0.260244\pi$
$887$	1.22822e31	1.36797	0.683987	+	0.729494i	$0.260244\pi$
$888$	3.93336e30	0.432939
$889$	2.07289e30	0.225480
$890$	0	0
$891$	−4.50968e29	−0.0479104
$892$	8.06483e30	0.846768
$893$	−9.88187e29	−0.102541
$894$	1.86598e31	1.91365
$895$	0	0
$896$	1.03876e31	1.04060
$897$	1.16171e30	0.115021
$898$	−7.17902e30	−0.702528
$899$	−1.16791e31	−1.12962
$900$	0	0
$901$	−2.25512e31	−2.13088
$902$	3.99995e30	0.373582
$903$	−2.08369e30	−0.192359
$904$	1.96393e31	1.79208
$905$	0	0
$906$	−7.81150e30	−0.696448
$907$	9.64171e30	0.849724	0.424862	−	0.905258i	$-0.360323\pi$
$907$	9.64171e30	0.849724	0.424862	+	0.905258i	$0.360323\pi$
$908$	3.32264e31	2.89456
$909$	6.51319e30	0.560884
$910$	0	0
$911$	−1.47519e30	−0.124138	−0.0620690	−	0.998072i	$-0.519770\pi$
$911$	−1.47519e30	−0.124138	−0.0620690	+	0.998072i	$0.519770\pi$
$912$	−8.54258e29	−0.0710630
$913$	5.79311e30	0.476396
$914$	3.43106e31	2.78928
$915$	0	0
$916$	−4.43181e31	−3.52110
$917$	−1.57282e31	−1.23538
$918$	6.73227e30	0.522774
$919$	−1.21235e31	−0.930714	−0.465357	−	0.885123i	$-0.654074\pi$
$919$	−1.21235e31	−0.930714	−0.465357	+	0.885123i	$0.654074\pi$
$920$	0	0
$921$	−2.86650e29	−0.0215093
$922$	2.01631e31	1.49583
$923$	3.42615e30	0.251298
$924$	5.50596e30	0.399280
$925$	0	0
$926$	−2.02174e31	−1.43321
$927$	4.83293e30	0.338745
$928$	1.45857e31	1.01082
$929$	−1.25133e31	−0.857447	−0.428724	−	0.903436i	$-0.641037\pi$
$929$	−1.25133e31	−0.857447	−0.428724	+	0.903436i	$0.641037\pi$
$930$	0	0
$931$	−5.55239e29	−0.0371970
$932$	−1.77947e30	−0.117876
$933$	−2.40060e30	−0.157240
$934$	2.47978e30	0.160609
$935$	0	0
$936$	1.67152e30	0.105856
$937$	−6.38356e30	−0.399758	−0.199879	−	0.979821i	$-0.564055\pi$
$937$	−6.38356e30	−0.399758	−0.199879	+	0.979821i	$0.564055\pi$
$938$	7.52582e30	0.466041
$939$	1.04438e31	0.639541
$940$	0	0
$941$	2.28359e31	1.36750	0.683751	−	0.729716i	$-0.260348\pi$
$941$	2.28359e31	1.36750	0.683751	+	0.729716i	$0.260348\pi$
$942$	1.59150e31	0.942478
$943$	1.06441e31	0.623358
$944$	−1.35252e31	−0.783319
$945$	0	0
$946$	6.02264e30	0.341138
$947$	−6.01017e30	−0.336676	−0.168338	−	0.985729i	$-0.553840\pi$
$947$	−6.01017e30	−0.336676	−0.168338	+	0.985729i	$0.553840\pi$
$948$	−2.55483e31	−1.41539
$949$	−2.06824e30	−0.113320
$950$	0	0
$951$	−2.64641e30	−0.141828
$952$	−4.39098e31	−2.32741
$953$	3.54334e31	1.85754	0.928769	−	0.370660i	$-0.120868\pi$
$953$	3.54334e31	1.85754	0.928769	+	0.370660i	$0.120868\pi$
$954$	1.58702e31	0.822857
$955$	0	0
$956$	3.80824e31	1.93160
$957$	−8.96641e30	−0.449825
$958$	1.82315e31	0.904660
$959$	5.56806e30	0.273281
$960$	0	0
$961$	−1.26861e31	−0.609160
$962$	2.14871e30	0.102056
$963$	9.87879e29	0.0464117
$964$	2.21865e31	1.03105
$965$	0	0
$966$	2.14764e31	0.976566
$967$	−2.99312e31	−1.34631	−0.673157	−	0.739499i	$-0.735062\pi$
$967$	−2.99312e31	−1.34631	−0.673157	+	0.739499i	$0.735062\pi$
$968$	3.72234e31	1.65625
$969$	−1.69074e30	−0.0744179
$970$	0	0
$971$	−1.42001e31	−0.611632	−0.305816	−	0.952091i	$-0.598929\pi$
$971$	−1.42001e31	−0.611632	−0.305816	+	0.952091i	$0.598929\pi$
$972$	−3.23223e30	−0.137723
$973$	−2.56456e31	−1.08101
$974$	1.38768e31	0.578657
$975$	0	0
$976$	−3.17257e31	−1.29476
$977$	2.56243e31	1.03457	0.517283	−	0.855814i	$-0.326943\pi$
$977$	2.56243e31	1.03457	0.517283	+	0.855814i	$0.326943\pi$
$978$	−3.38294e31	−1.35125
$979$	1.39194e31	0.550050
$980$	0	0
$981$	1.40195e31	0.542261
$982$	−3.53545e31	−1.35293
$983$	−3.29184e31	−1.24631	−0.623156	−	0.782098i	$-0.714150\pi$
$983$	−3.29184e31	−1.24631	−0.623156	+	0.782098i	$0.714150\pi$
$984$	1.53152e31	0.573686
$985$	0	0
$986$	1.33855e32	4.90825
$987$	1.44824e31	0.525425
$988$	−7.85804e29	−0.0282076
$989$	1.60267e31	0.569223
$990$	0	0
$991$	−2.22756e31	−0.774560	−0.387280	−	0.921962i	$-0.626585\pi$
$991$	−2.22756e31	−0.774560	−0.387280	+	0.921962i	$0.626585\pi$
$992$	−1.01652e31	−0.349737
$993$	−1.46549e31	−0.498903
$994$	6.33388e31	2.13360
$995$	0	0
$996$	4.15210e31	1.36945
$997$	−2.49754e31	−0.815105	−0.407553	−	0.913182i	$-0.633618\pi$
$997$	−2.49754e31	−0.815105	−0.407553	+	0.913182i	$0.633618\pi$
$998$	−7.47968e31	−2.41553
$999$	−2.21963e30	−0.0709322

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.j.1.1	yes	6
5.2	odd	4		75.22.b.i.49.1		12
5.3	odd	4		75.22.b.i.49.12		12
5.4	even	2		75.22.a.i.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.22.a.i.1.6	✓	6	5.4	even	2
75.22.a.j.1.1	yes	6	1.1	even	1	trivial
75.22.b.i.49.1		12	5.2	odd	4
75.22.b.i.49.12		12	5.3	odd	4

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q-2568.95 q^{2} -59049.0 q^{3} +4.50235e6 q^{4} +1.51694e8 q^{6} +5.58323e8 q^{7} -6.17885e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\)	\(q-2568.95 q^{2} -59049.0 q^{3} +4.50235e6 q^{4} +1.51694e8 q^{6} +5.58323e8 q^{7} -6.17885e9 q^{8} +3.48678e9 q^{9} -3.70933e10 q^{11} -2.65860e11 q^{12} -7.75851e10 q^{13} -1.43430e12 q^{14} +6.43103e12 q^{16} +1.27282e13 q^{17} -8.95738e12 q^{18} +2.24955e12 q^{19} -3.29684e13 q^{21} +9.52909e13 q^{22} +2.53576e14 q^{23} +3.64855e14 q^{24} +1.99312e14 q^{26} -2.05891e14 q^{27} +2.51377e15 q^{28} -4.09365e15 q^{29} +2.85297e15 q^{31} -3.56301e15 q^{32} +2.19032e15 q^{33} -3.26982e16 q^{34} +1.56987e16 q^{36} +1.07806e16 q^{37} -5.77899e15 q^{38} +4.58132e15 q^{39} +4.19762e16 q^{41} +8.46942e16 q^{42} +6.32027e16 q^{43} -1.67007e17 q^{44} -6.51424e17 q^{46} -4.39281e17 q^{47} -3.79746e17 q^{48} -2.46822e17 q^{49} -7.51590e17 q^{51} -3.49316e17 q^{52} -1.77175e18 q^{53} +5.28924e17 q^{54} -3.44979e18 q^{56} -1.32834e17 q^{57} +1.05164e19 q^{58} -2.10312e18 q^{59} -4.93323e18 q^{61} -7.32914e18 q^{62} +1.94675e18 q^{63} -4.33365e18 q^{64} -5.62683e18 q^{66} -5.24702e18 q^{67} +5.73071e19 q^{68} -1.49734e19 q^{69} -4.41600e19 q^{71} -2.15443e19 q^{72} +2.66577e19 q^{73} -2.76949e19 q^{74} +1.01283e19 q^{76} -2.07100e19 q^{77} -1.17692e19 q^{78} +9.60972e19 q^{79} +1.21577e19 q^{81} -1.07835e20 q^{82} -1.56177e20 q^{83} -1.48435e20 q^{84} -1.62364e20 q^{86} +2.41726e20 q^{87} +2.29194e20 q^{88} -3.75254e20 q^{89} -4.33175e19 q^{91} +1.14169e21 q^{92} -1.68465e20 q^{93} +1.12849e21 q^{94} +2.10392e20 q^{96} +9.05983e20 q^{97} +6.34073e20 q^{98} -1.29336e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 92 q^{2} - 354294 q^{3} + 4390448 q^{4} - 5432508 q^{6} + 1327143454 q^{7} - 4252271424 q^{8} + 20920706406 q^{9}+O(q^{10}) \) 6 * q + 92 * q^2 - 354294 * q^3 + 4390448 * q^4 - 5432508 * q^6 + 1327143454 * q^7 - 4252271424 * q^8 + 20920706406 * q^9	\( 6 q + 92 q^{2} - 354294 q^{3} + 4390448 q^{4} - 5432508 q^{6} + 1327143454 q^{7} - 4252271424 q^{8} + 20920706406 q^{9} - 78310112516 q^{11} - 259251563952 q^{12} + 116029746338 q^{13} - 313825730148 q^{14} + 1996701612800 q^{16} + 5382900513068 q^{17} + 320784164892 q^{18} + 6969638997622 q^{19} - 78366493815246 q^{21} - 19701817271864 q^{22} + 295754895311052 q^{23} + 251092375315776 q^{24} - 274314326263628 q^{26} - 12\!\cdots\!94 q^{27}+ \cdots - 27\!\cdots\!16 q^{99}+O(q^{100}) \) 6 * q + 92 * q^2 - 354294 * q^3 + 4390448 * q^4 - 5432508 * q^6 + 1327143454 * q^7 - 4252271424 * q^8 + 20920706406 * q^9 - 78310112516 * q^11 - 259251563952 * q^12 + 116029746338 * q^13 - 313825730148 * q^14 + 1996701612800 * q^16 + 5382900513068 * q^17 + 320784164892 * q^18 + 6969638997622 * q^19 - 78366493815246 * q^21 - 19701817271864 * q^22 + 295754895311052 * q^23 + 251092375315776 * q^24 - 274314326263628 * q^26 - 1235346792567894 * q^27 + 2674107576939568 * q^28 - 3895624305077596 * q^29 + 492586859482706 * q^31 - 10862288965692416 * q^32 + 4624133833957284 * q^33 - 30773279577490744 * q^34 + 15308545599801648 * q^36 - 26101348744076316 * q^37 - 8307793564414708 * q^38 - 6851440491512562 * q^39 + 112662882011424056 * q^41 + 18531095539509252 * q^42 - 13377804412633810 * q^43 - 23820753283797856 * q^44 - 156784693648581624 * q^46 - 243206274450163780 * q^47 - 117903233534227200 * q^48 + 327097566625336780 * q^49 - 317854892396152332 * q^51 - 192375894240694512 * q^52 + 554044170675798992 * q^53 - 18941984152707708 * q^54 - 5086364147554777920 * q^56 - 411550213170581478 * q^57 + 11350712056264661624 * q^58 - 7003944869710758212 * q^59 - 10235053877143753790 * q^61 - 1855763029836316572 * q^62 + 4627463093296461054 * q^63 - 10747451596339826688 * q^64 + 1163372608086297336 * q^66 + 8975831852618567158 * q^67 + 64266312753700360864 * q^68 - 17464030813222309548 * q^69 + 18009408229245489272 * q^71 - 14826753670021257024 * q^72 + 81172357549902895652 * q^73 - 103132266830355254968 * q^74 + 111484542579179194608 * q^76 - 40817017589776296372 * q^77 + 16197986651540969772 * q^78 - 79149540279186452080 * q^79 + 72945992754341572806 * q^81 + 17339452427244467168 * q^82 + 238715465076655746756 * q^83 - 157903378310704550832 * q^84 + 28767310268029844764 * q^86 + 230032719590526966204 * q^87 + 1023055046877663881856 * q^88 - 712331293797957179808 * q^89 + 670774308716312800426 * q^91 + 880736698470324218784 * q^92 - 29086761465594306594 * q^93 + 278898069330490980248 * q^94 + 641407301135171472384 * q^96 + 1498553500583421551178 * q^97 + 2662349783988420722392 * q^98 - 273050478761343662916 * q^99

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