LMFDB - Embedded newform 75.22.a.j.1.3

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			1.3
Root			$89.2274$ 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)

$f(q)$	$=$	$q-340.910 q^{2} -59049.0 q^{3} -1.98093e6 q^{4} +2.01304e7 q^{6} +6.73159e8 q^{7} +1.39026e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$	\(q-340.910 q^{2} -59049.0 q^{3} -1.98093e6 q^{4} +2.01304e7 q^{6} +6.73159e8 q^{7} +1.39026e9 q^{8} +3.48678e9 q^{9} +8.81884e10 q^{11} +1.16972e11 q^{12} -4.35956e11 q^{13} -2.29486e11 q^{14} +3.68036e12 q^{16} +3.38833e12 q^{17} -1.18868e12 q^{18} -4.68698e13 q^{19} -3.97494e13 q^{21} -3.00643e13 q^{22} +2.77663e13 q^{23} -8.20934e13 q^{24} +1.48622e14 q^{26} -2.05891e14 q^{27} -1.33348e15 q^{28} -4.93794e14 q^{29} -1.99343e15 q^{31} -4.17025e15 q^{32} -5.20744e15 q^{33} -1.15512e15 q^{34} -6.90709e15 q^{36} -2.48383e16 q^{37} +1.59784e16 q^{38} +2.57428e16 q^{39} +1.53977e17 q^{41} +1.35509e16 q^{42} -8.38852e16 q^{43} -1.74695e17 q^{44} -9.46580e15 q^{46} +1.06134e16 q^{47} -2.17322e17 q^{48} -1.05402e17 q^{49} -2.00078e17 q^{51} +8.63600e17 q^{52} +2.83801e17 q^{53} +7.01903e16 q^{54} +9.35865e17 q^{56} +2.76761e18 q^{57} +1.68339e17 q^{58} +7.02994e18 q^{59} +6.54670e18 q^{61} +6.79580e17 q^{62} +2.34716e18 q^{63} -6.29660e18 q^{64} +1.77527e18 q^{66} +1.27152e18 q^{67} -6.71206e18 q^{68} -1.63957e18 q^{69} +9.18512e18 q^{71} +4.84753e18 q^{72} -3.67454e19 q^{73} +8.46760e18 q^{74} +9.28459e19 q^{76} +5.93649e19 q^{77} -8.77597e18 q^{78} -6.42331e19 q^{79} +1.21577e19 q^{81} -5.24921e19 q^{82} +2.62194e20 q^{83} +7.87409e19 q^{84} +2.85973e19 q^{86} +2.91580e19 q^{87} +1.22605e20 q^{88} -2.98217e20 q^{89} -2.93468e20 q^{91} -5.50032e19 q^{92} +1.17710e20 q^{93} -3.61820e18 q^{94} +2.46249e20 q^{96} +6.86712e20 q^{97} +3.59327e19 q^{98} +3.07494e20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−340.910	−0.235410	−0.117705	−	0.993049i	$-0.537554\pi$
$2$	−340.910	−0.235410	−0.117705	+	0.993049i	$0.537554\pi$
$3$	−59049.0	−0.577350
$3$	−59049.0	−0.577350
$4$	−1.98093e6	−0.944582
$5$	0	0
$5$	0	0
$6$	2.01304e7	0.135914
$7$	6.73159e8	0.900717	0.450359	−	0.892848i	$-0.351296\pi$
$7$	6.73159e8	0.900717	0.450359	+	0.892848i	$0.351296\pi$
$8$	1.39026e9	0.457773
$9$	3.48678e9	0.333333
$10$	0	0
$11$	8.81884e10	1.02515	0.512576	−	0.858642i	$-0.328691\pi$
$11$	8.81884e10	1.02515	0.512576	+	0.858642i	$0.328691\pi$
$12$	1.16972e11	0.545355
$13$	−4.35956e11	−0.877077	−0.438539	−	0.898712i	$-0.644504\pi$
$13$	−4.35956e11	−0.877077	−0.438539	+	0.898712i	$0.644504\pi$
$14$	−2.29486e11	−0.212038
$15$	0	0
$16$	3.68036e12	0.836818
$17$	3.38833e12	0.407636	0.203818	−	0.979009i	$-0.434665\pi$
$17$	3.38833e12	0.407636	0.203818	+	0.979009i	$0.434665\pi$
$18$	−1.18868e12	−0.0784699
$19$	−4.68698e13	−1.75380	−0.876900	−	0.480672i	$-0.840393\pi$
$19$	−4.68698e13	−1.75380	−0.876900	+	0.480672i	$0.840393\pi$
$20$	0	0
$21$	−3.97494e13	−0.520029
$22$	−3.00643e13	−0.241331
$23$	2.77663e13	0.139758	0.0698788	−	0.997555i	$-0.477739\pi$
$23$	2.77663e13	0.139758	0.0698788	+	0.997555i	$0.477739\pi$
$24$	−8.20934e13	−0.264296
$25$	0	0
$26$	1.48622e14	0.206472
$27$	−2.05891e14	−0.192450
$28$	−1.33348e15	−0.850802
$29$	−4.93794e14	−0.217955	−0.108977	−	0.994044i	$-0.534758\pi$
$29$	−4.93794e14	−0.217955	−0.108977	+	0.994044i	$0.534758\pi$
$30$	0	0
$31$	−1.99343e15	−0.436821	−0.218410	−	0.975857i	$-0.570087\pi$
$31$	−1.99343e15	−0.436821	−0.218410	+	0.975857i	$0.570087\pi$
$32$	−4.17025e15	−0.654768
$33$	−5.20744e15	−0.591872
$34$	−1.15512e15	−0.0959614
$35$	0	0
$36$	−6.90709e15	−0.314861
$37$	−2.48383e16	−0.849186	−0.424593	−	0.905384i	$-0.639583\pi$
$37$	−2.48383e16	−0.849186	−0.424593	+	0.905384i	$0.639583\pi$
$38$	1.59784e16	0.412862
$39$	2.57428e16	0.506381
$40$	0	0
$41$	1.53977e17	1.79153	0.895766	−	0.444526i	$-0.146628\pi$
$41$	1.53977e17	1.79153	0.895766	+	0.444526i	$0.146628\pi$
$42$	1.35509e16	0.122420
$43$	−8.38852e16	−0.591925	−0.295962	−	0.955200i	$-0.595640\pi$
$43$	−8.38852e16	−0.591925	−0.295962	+	0.955200i	$0.595640\pi$
$44$	−1.74695e17	−0.968341
$45$	0	0
$46$	−9.46580e15	−0.0329003
$47$	1.06134e16	0.0294324	0.0147162	−	0.999892i	$-0.495316\pi$
$47$	1.06134e16	0.0294324	0.0147162	+	0.999892i	$0.495316\pi$
$48$	−2.17322e17	−0.483137
$49$	−1.05402e17	−0.188708
$50$	0	0
$51$	−2.00078e17	−0.235349
$52$	8.63600e17	0.828472
$53$	2.83801e17	0.222904	0.111452	−	0.993770i	$-0.464450\pi$
$53$	2.83801e17	0.222904	0.111452	+	0.993770i	$0.464450\pi$
$54$	7.01903e16	0.0453046
$55$	0	0
$56$	9.35865e17	0.412324
$57$	2.76761e18	1.01256
$58$	1.68339e17	0.0513087
$59$	7.02994e18	1.79063	0.895314	−	0.445436i	$-0.146951\pi$
$59$	7.02994e18	1.79063	0.895314	+	0.445436i	$0.146951\pi$
$60$	0	0
$61$	6.54670e18	1.17506	0.587529	−	0.809203i	$-0.300101\pi$
$61$	6.54670e18	1.17506	0.587529	+	0.809203i	$0.300101\pi$
$62$	6.79580e17	0.102832
$63$	2.34716e18	0.300239
$64$	−6.29660e18	−0.682679
$65$	0	0
$66$	1.77527e18	0.139332
$67$	1.27152e18	0.0852192	0.0426096	−	0.999092i	$-0.486433\pi$
$67$	1.27152e18	0.0852192	0.0426096	+	0.999092i	$0.486433\pi$
$68$	−6.71206e18	−0.385046
$69$	−1.63957e18	−0.0806891
$70$	0	0
$71$	9.18512e18	0.334867	0.167433	−	0.985883i	$-0.446452\pi$
$71$	9.18512e18	0.334867	0.167433	+	0.985883i	$0.446452\pi$
$72$	4.84753e18	0.152591
$73$	−3.67454e19	−1.00072	−0.500361	−	0.865817i	$-0.666799\pi$
$73$	−3.67454e19	−1.00072	−0.500361	+	0.865817i	$0.666799\pi$
$74$	8.46760e18	0.199907
$75$	0	0
$76$	9.28459e19	1.65661
$77$	5.93649e19	0.923373
$78$	−8.77597e18	−0.119207
$79$	−6.42331e19	−0.763263	−0.381632	−	0.924315i	$-0.624638\pi$
$79$	−6.42331e19	−0.763263	−0.381632	+	0.924315i	$0.624638\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	−5.24921e19	−0.421744
$83$	2.62194e20	1.85482	0.927412	−	0.374042i	$-0.122028\pi$
$83$	2.62194e20	1.85482	0.927412	+	0.374042i	$0.122028\pi$
$84$	7.87409e19	0.491211
$85$	0	0
$86$	2.85973e19	0.139345
$87$	2.91580e19	0.125836
$88$	1.22605e20	0.469288
$89$	−2.98217e20	−1.01377	−0.506883	−	0.862015i	$-0.669202\pi$
$89$	−2.98217e20	−1.01377	−0.506883	+	0.862015i	$0.669202\pi$
$90$	0	0
$91$	−2.93468e20	−0.789999
$92$	−5.50032e19	−0.132013
$93$	1.17710e20	0.252199
$94$	−3.61820e18	−0.00692867
$95$	0	0
$96$	2.46249e20	0.378031
$97$	6.86712e20	0.945522	0.472761	−	0.881191i	$-0.343257\pi$
$97$	6.86712e20	0.945522	0.472761	+	0.881191i	$0.343257\pi$
$98$	3.59327e19	0.0444238
$99$	3.07494e20	0.341718
$100$	0	0
$101$	−1.72493e21	−1.55381	−0.776904	−	0.629619i	$-0.783211\pi$
$101$	−1.72493e21	−1.55381	−0.776904	+	0.629619i	$0.783211\pi$
$102$	6.82084e19	0.0554034
$103$	7.93905e20	0.582073	0.291037	−	0.956712i	$-0.406000\pi$
$103$	7.93905e20	0.582073	0.291037	+	0.956712i	$0.406000\pi$
$104$	−6.06092e20	−0.401503
$105$	0	0
$106$	−9.67505e19	−0.0524737
$107$	−2.41854e21	−1.18856	−0.594282	−	0.804257i	$-0.702564\pi$
$107$	−2.41854e21	−1.18856	−0.594282	+	0.804257i	$0.702564\pi$
$108$	4.07856e20	0.181785
$109$	−4.56052e20	−0.184517	−0.0922584	−	0.995735i	$-0.529409\pi$
$109$	−4.56052e20	−0.184517	−0.0922584	+	0.995735i	$0.529409\pi$
$110$	0	0
$111$	1.46667e21	0.490278
$112$	2.47747e21	0.753736
$113$	2.66253e21	0.737854	0.368927	−	0.929458i	$-0.379725\pi$
$113$	2.66253e21	0.737854	0.368927	+	0.929458i	$0.379725\pi$
$114$	−9.43506e20	−0.238366
$115$	0	0
$116$	9.78172e20	0.205876
$117$	−1.52009e21	−0.292359
$118$	−2.39657e21	−0.421531
$119$	2.28089e21	0.367165
$120$	0	0
$121$	3.76952e20	0.0509378
$122$	−2.23183e21	−0.276620
$123$	−9.09217e21	−1.03434
$124$	3.94885e21	0.412613
$125$	0	0
$126$	−8.00170e20	−0.0706792
$127$	3.90399e21	0.317372	0.158686	−	0.987329i	$-0.449274\pi$
$127$	3.90399e21	0.317372	0.158686	+	0.987329i	$0.449274\pi$
$128$	1.08922e22	0.815478
$129$	4.95334e21	0.341748
$130$	0	0
$131$	−1.19004e22	−0.698575	−0.349288	−	0.937016i	$-0.613576\pi$
$131$	−1.19004e22	−0.698575	−0.349288	+	0.937016i	$0.613576\pi$
$132$	1.03156e22	0.559072
$133$	−3.15508e22	−1.57968
$134$	−4.33473e20	−0.0200614
$135$	0	0
$136$	4.71066e21	0.186605
$137$	4.39883e22	1.61351	0.806754	−	0.590887i	$-0.201222\pi$
$137$	4.39883e22	1.61351	0.806754	+	0.590887i	$0.201222\pi$
$138$	5.58946e20	0.0189950
$139$	2.38509e22	0.751362	0.375681	−	0.926749i	$-0.377409\pi$
$139$	2.38509e22	0.751362	0.375681	+	0.926749i	$0.377409\pi$
$140$	0	0
$141$	−6.26709e20	−0.0169928
$142$	−3.13130e21	−0.0788309
$143$	−3.84463e22	−0.899138
$144$	1.28326e22	0.278939
$145$	0	0
$146$	1.25269e22	0.235579
$147$	6.22390e21	0.108951
$148$	4.92029e22	0.802126
$149$	9.91268e21	0.150569	0.0752844	−	0.997162i	$-0.476014\pi$
$149$	9.91268e21	0.150569	0.0752844	+	0.997162i	$0.476014\pi$
$150$	0	0
$151$	−1.80864e22	−0.238833	−0.119416	−	0.992844i	$-0.538102\pi$
$151$	−1.80864e22	−0.238833	−0.119416	+	0.992844i	$0.538102\pi$
$152$	−6.51611e22	−0.802843
$153$	1.18144e22	0.135879
$154$	−2.02381e22	−0.217371
$155$	0	0
$156$	−5.09947e22	−0.478318
$157$	−1.17597e23	−1.03146	−0.515729	−	0.856752i	$-0.672479\pi$
$157$	−1.17597e23	−1.03146	−0.515729	+	0.856752i	$0.672479\pi$
$158$	2.18977e22	0.179679
$159$	−1.67582e22	−0.128694
$160$	0	0
$161$	1.86911e22	0.125882
$162$	−4.14466e21	−0.0261566
$163$	−3.82771e22	−0.226448	−0.113224	−	0.993569i	$-0.536118\pi$
$163$	−3.82771e22	−0.226448	−0.113224	+	0.993569i	$0.536118\pi$
$164$	−3.05018e23	−1.69225
$165$	0	0
$166$	−8.93844e22	−0.436643
$167$	3.13595e23	1.43829	0.719145	−	0.694860i	$-0.244534\pi$
$167$	3.13595e23	1.43829	0.719145	+	0.694860i	$0.244534\pi$
$168$	−5.52619e22	−0.238056
$169$	−5.70065e22	−0.230735
$170$	0	0
$171$	−1.63425e23	−0.584600
$172$	1.66171e23	0.559121
$173$	4.81111e23	1.52322	0.761608	−	0.648039i	$-0.224410\pi$
$173$	4.81111e23	1.52322	0.761608	+	0.648039i	$0.224410\pi$
$174$	−9.94025e21	−0.0296231
$175$	0	0
$176$	3.24566e23	0.857866
$177$	−4.15111e23	−1.03382
$178$	1.01665e23	0.238650
$179$	−3.28060e22	−0.0726099	−0.0363050	−	0.999341i	$-0.511559\pi$
$179$	−3.28060e22	−0.0726099	−0.0363050	+	0.999341i	$0.511559\pi$
$180$	0	0
$181$	−5.50948e23	−1.08514	−0.542569	−	0.840011i	$-0.682548\pi$
$181$	−5.50948e23	−1.08514	−0.542569	+	0.840011i	$0.682548\pi$
$182$	1.00046e23	0.185973
$183$	−3.86576e23	−0.678420
$184$	3.86023e22	0.0639773
$185$	0	0
$186$	−4.01285e22	−0.0593700
$187$	2.98812e23	0.417889
$188$	−2.10244e22	−0.0278013
$189$	−1.38598e23	−0.173343
$190$	0	0
$191$	1.18267e24	1.32438	0.662191	−	0.749335i	$-0.269627\pi$
$191$	1.18267e24	1.32438	0.662191	+	0.749335i	$0.269627\pi$
$192$	3.71808e23	0.394145
$193$	−1.17187e24	−1.17632	−0.588161	−	0.808744i	$-0.700148\pi$
$193$	−1.17187e24	−1.17632	−0.588161	+	0.808744i	$0.700148\pi$
$194$	−2.34107e23	−0.222585
$195$	0	0
$196$	2.08795e23	0.178251
$197$	6.31695e23	0.511225	0.255613	−	0.966779i	$-0.417723\pi$
$197$	6.31695e23	0.511225	0.255613	+	0.966779i	$0.417723\pi$
$198$	−1.04828e23	−0.0804436
$199$	−1.80597e24	−1.31448	−0.657239	−	0.753682i	$-0.728276\pi$
$199$	−1.80597e24	−1.31448	−0.657239	+	0.753682i	$0.728276\pi$
$200$	0	0
$201$	−7.50820e22	−0.0492014
$202$	5.88046e23	0.365781
$203$	−3.32402e23	−0.196316
$204$	3.96340e23	0.222306
$205$	0	0
$206$	−2.70650e23	−0.137026
$207$	9.68151e22	0.0465859
$208$	−1.60448e24	−0.733954
$209$	−4.13338e24	−1.79791
$210$	0	0
$211$	2.94629e23	0.115961	0.0579803	−	0.998318i	$-0.481534\pi$
$211$	2.94629e23	0.115961	0.0579803	+	0.998318i	$0.481534\pi$
$212$	−5.62191e23	−0.210551
$213$	−5.42372e23	−0.193336
$214$	8.24502e23	0.279800
$215$	0	0
$216$	−2.86242e23	−0.0880985
$217$	−1.34190e24	−0.393452
$218$	1.55472e23	0.0434371
$219$	2.16978e24	0.577767
$220$	0	0
$221$	−1.47717e24	−0.357528
$222$	−5.00003e23	−0.115416
$223$	−7.94458e24	−1.74932	−0.874662	−	0.484734i	$-0.838916\pi$
$223$	−7.94458e24	−1.74932	−0.874662	+	0.484734i	$0.838916\pi$
$224$	−2.80725e24	−0.589761
$225$	0	0
$226$	−9.07681e23	−0.173698
$227$	6.99177e24	1.27737	0.638684	−	0.769469i	$-0.279479\pi$
$227$	6.99177e24	1.27737	0.638684	+	0.769469i	$0.279479\pi$
$228$	−5.48246e24	−0.956444
$229$	−1.97790e24	−0.329558	−0.164779	−	0.986330i	$-0.552691\pi$
$229$	−1.97790e24	−0.329558	−0.164779	+	0.986330i	$0.552691\pi$
$230$	0	0
$231$	−3.50544e24	−0.533109
$232$	−6.86501e23	−0.0997740
$233$	5.40517e24	0.750883	0.375441	−	0.926846i	$-0.377491\pi$
$233$	5.40517e24	0.750883	0.375441	+	0.926846i	$0.377491\pi$
$234$	5.18212e23	0.0688242
$235$	0	0
$236$	−1.39258e25	−1.69140
$237$	3.79290e24	0.440670
$238$	−7.77577e23	−0.0864341
$239$	−1.53053e25	−1.62804	−0.814018	−	0.580840i	$-0.802724\pi$
$239$	−1.53053e25	−1.62804	−0.814018	+	0.580840i	$0.802724\pi$
$240$	0	0
$241$	−1.34112e25	−1.30704	−0.653519	−	0.756910i	$-0.726708\pi$
$241$	−1.34112e25	−1.30704	−0.653519	+	0.756910i	$0.726708\pi$
$242$	−1.28507e23	−0.0119912
$243$	−7.17898e23	−0.0641500
$244$	−1.29686e25	−1.10994
$245$	0	0
$246$	3.09961e24	0.243494
$247$	2.04332e25	1.53822
$248$	−2.77138e24	−0.199965
$249$	−1.54823e25	−1.07088
$250$	0	0
$251$	1.69729e24	0.107940	0.0539699	−	0.998543i	$-0.482812\pi$
$251$	1.69729e24	0.107940	0.0539699	+	0.998543i	$0.482812\pi$
$252$	−4.64957e24	−0.283601
$253$	2.44867e24	0.143273
$254$	−1.33091e24	−0.0747125
$255$	0	0
$256$	9.49167e24	0.490708
$257$	7.35666e24	0.365076	0.182538	−	0.983199i	$-0.441569\pi$
$257$	7.35666e24	0.365076	0.182538	+	0.983199i	$0.441569\pi$
$258$	−1.68864e24	−0.0804507
$259$	−1.67201e25	−0.764877
$260$	0	0
$261$	−1.72175e24	−0.0726516
$262$	4.05696e24	0.164451
$263$	−4.57145e25	−1.78040	−0.890202	−	0.455566i	$-0.849437\pi$
$263$	−4.57145e25	−1.78040	−0.890202	+	0.455566i	$0.849437\pi$
$264$	−7.23969e24	−0.270943
$265$	0	0
$266$	1.07560e25	0.371872
$267$	1.76094e25	0.585298
$268$	−2.51880e24	−0.0804966
$269$	−3.25178e25	−0.999360	−0.499680	−	0.866210i	$-0.666549\pi$
$269$	−3.25178e25	−0.999360	−0.499680	+	0.866210i	$0.666549\pi$
$270$	0	0
$271$	−1.68503e25	−0.479104	−0.239552	−	0.970884i	$-0.577001\pi$
$271$	−1.68503e25	−0.479104	−0.239552	+	0.970884i	$0.577001\pi$
$272$	1.24703e25	0.341117
$273$	1.73290e25	0.456106
$274$	−1.49960e25	−0.379835
$275$	0	0
$276$	3.24788e24	0.0762175
$277$	6.04936e25	1.36670	0.683348	−	0.730093i	$-0.260523\pi$
$277$	6.04936e25	1.36670	0.683348	+	0.730093i	$0.260523\pi$
$278$	−8.13100e24	−0.176878
$279$	−6.95067e24	−0.145607
$280$	0	0
$281$	9.79283e23	0.0190323	0.00951616	−	0.999955i	$-0.496971\pi$
$281$	9.79283e23	0.0190323	0.00951616	+	0.999955i	$0.496971\pi$
$282$	2.13651e23	0.00400027
$283$	−4.41068e25	−0.795697	−0.397849	−	0.917451i	$-0.630243\pi$
$283$	−4.41068e25	−0.795697	−0.397849	+	0.917451i	$0.630243\pi$
$284$	−1.81951e25	−0.316309
$285$	0	0
$286$	1.31067e25	0.211666
$287$	1.03651e26	1.61366
$288$	−1.45408e25	−0.218256
$289$	−5.76111e25	−0.833833
$290$	0	0
$291$	−4.05497e25	−0.545897
$292$	7.27902e25	0.945263
$293$	−8.77281e25	−1.09908	−0.549539	−	0.835468i	$-0.685197\pi$
$293$	−8.77281e25	−1.09908	−0.549539	+	0.835468i	$0.685197\pi$
$294$	−2.12179e24	−0.0256481
$295$	0	0
$296$	−3.45316e25	−0.388735
$297$	−1.81572e25	−0.197291
$298$	−3.37933e24	−0.0354454
$299$	−1.21049e25	−0.122578
$300$	0	0
$301$	−5.64681e25	−0.533157
$302$	6.16582e24	0.0562235
$303$	1.01855e26	0.897091
$304$	−1.72498e26	−1.46761
$305$	0	0
$306$	−4.02764e24	−0.0319871
$307$	−1.76055e26	−1.35113	−0.675563	−	0.737303i	$-0.736099\pi$
$307$	−1.76055e26	−1.35113	−0.675563	+	0.737303i	$0.736099\pi$
$308$	−1.17598e26	−0.872201
$309$	−4.68793e25	−0.336060
$310$	0	0
$311$	−2.72398e26	−1.82482	−0.912410	−	0.409278i	$-0.865781\pi$
$311$	−2.72398e26	−1.82482	−0.912410	+	0.409278i	$0.865781\pi$
$312$	3.57891e25	0.231808
$313$	−9.00000e25	−0.563672	−0.281836	−	0.959463i	$-0.590943\pi$
$313$	−9.00000e25	−0.563672	−0.281836	+	0.959463i	$0.590943\pi$
$314$	4.00901e25	0.242815
$315$	0	0
$316$	1.27241e26	0.720965
$317$	−8.30387e25	−0.455154	−0.227577	−	0.973760i	$-0.573080\pi$
$317$	−8.30387e25	−0.455154	−0.227577	+	0.973760i	$0.573080\pi$
$318$	5.71302e24	0.0302957
$319$	−4.35469e25	−0.223437
$320$	0	0
$321$	1.42812e26	0.686218
$322$	−6.37199e24	−0.0296339
$323$	−1.58811e26	−0.714912
$324$	−2.40835e25	−0.104954
$325$	0	0
$326$	1.30490e25	0.0533081
$327$	2.69294e25	0.106531
$328$	2.14067e26	0.820116
$329$	7.14449e24	0.0265103
$330$	0	0
$331$	−9.84201e25	−0.342681	−0.171340	−	0.985212i	$-0.554810\pi$
$331$	−9.84201e25	−0.342681	−0.171340	+	0.985212i	$0.554810\pi$
$332$	−5.19389e26	−1.75203
$333$	−8.66056e25	−0.283062
$334$	−1.06908e26	−0.338587
$335$	0	0
$336$	−1.46292e26	−0.435170
$337$	6.17471e26	1.78034	0.890169	−	0.455631i	$-0.150586\pi$
$337$	6.17471e26	1.78034	0.890169	+	0.455631i	$0.150586\pi$
$338$	1.94340e25	0.0543173
$339$	−1.57220e26	−0.426000
$340$	0	0
$341$	−1.75798e26	−0.447808
$342$	5.57131e25	0.137621
$343$	−4.46943e26	−1.07069
$344$	−1.16622e26	−0.270967
$345$	0	0
$346$	−1.64015e26	−0.358580
$347$	1.24054e26	0.263118	0.131559	−	0.991308i	$-0.458002\pi$
$347$	1.24054e26	0.263118	0.131559	+	0.991308i	$0.458002\pi$
$348$	−5.77601e25	−0.118863
$349$	8.43585e26	1.68446	0.842232	−	0.539115i	$-0.181241\pi$
$349$	8.43585e26	1.68446	0.842232	+	0.539115i	$0.181241\pi$
$350$	0	0
$351$	8.97596e25	0.168794
$352$	−3.67768e26	−0.671238
$353$	8.92103e26	1.58045	0.790224	−	0.612818i	$-0.209964\pi$
$353$	8.92103e26	1.58045	0.790224	+	0.612818i	$0.209964\pi$
$354$	1.41515e26	0.243371
$355$	0	0
$356$	5.90748e26	0.957585
$357$	−1.34684e26	−0.211983
$358$	1.11839e25	0.0170931
$359$	−1.31171e26	−0.194692	−0.0973458	−	0.995251i	$-0.531035\pi$
$359$	−1.31171e26	−0.194692	−0.0973458	+	0.995251i	$0.531035\pi$
$360$	0	0
$361$	1.48257e27	2.07582
$362$	1.87823e26	0.255452
$363$	−2.22587e25	−0.0294090
$364$	5.81341e26	0.746219
$365$	0	0
$366$	1.31788e26	0.159707
$367$	6.63211e25	0.0781012	0.0390506	−	0.999237i	$-0.487567\pi$
$367$	6.63211e25	0.0781012	0.0390506	+	0.999237i	$0.487567\pi$
$368$	1.02190e26	0.116952
$369$	5.36884e26	0.597177
$370$	0	0
$371$	1.91043e26	0.200773
$372$	−2.33176e26	−0.238222
$373$	1.49862e27	1.48850	0.744252	−	0.667899i	$-0.232806\pi$
$373$	1.49862e27	1.48850	0.744252	+	0.667899i	$0.232806\pi$
$374$	−1.01868e26	−0.0983751
$375$	0	0
$376$	1.47553e25	0.0134734
$377$	2.15273e26	0.191163
$378$	4.72492e25	0.0408066
$379$	2.60984e26	0.219231	0.109615	−	0.993974i	$-0.465038\pi$
$379$	2.60984e26	0.219231	0.109615	+	0.993974i	$0.465038\pi$
$380$	0	0
$381$	−2.30527e26	−0.183235
$382$	−4.03184e26	−0.311772
$383$	6.73245e26	0.506508	0.253254	−	0.967400i	$-0.418499\pi$
$383$	6.73245e26	0.506508	0.253254	+	0.967400i	$0.418499\pi$
$384$	−6.43175e26	−0.470816
$385$	0	0
$386$	3.99501e26	0.276918
$387$	−2.92489e26	−0.197308
$388$	−1.36033e27	−0.893123
$389$	9.85802e26	0.629969	0.314984	−	0.949097i	$-0.398001\pi$
$389$	9.85802e26	0.629969	0.314984	+	0.949097i	$0.398001\pi$
$390$	0	0
$391$	9.40815e25	0.0569703
$392$	−1.46536e26	−0.0863857
$393$	7.02707e26	0.403323
$394$	−2.15351e26	−0.120347
$395$	0	0
$396$	−6.09125e26	−0.322780
$397$	−2.43807e27	−1.25819	−0.629093	−	0.777330i	$-0.716574\pi$
$397$	−2.43807e27	−1.25819	−0.629093	+	0.777330i	$0.716574\pi$
$398$	6.15673e26	0.309441
$399$	1.86305e27	0.912028
$400$	0	0
$401$	3.26497e26	0.151657	0.0758286	−	0.997121i	$-0.475840\pi$
$401$	3.26497e26	0.151657	0.0758286	+	0.997121i	$0.475840\pi$
$402$	2.55962e25	0.0115825
$403$	8.69049e26	0.383126
$404$	3.41697e27	1.46770
$405$	0	0
$406$	1.13319e26	0.0462146
$407$	−2.19045e27	−0.870546
$408$	−2.78160e26	−0.107736
$409$	4.25757e27	1.60719	0.803596	−	0.595176i	$-0.202918\pi$
$409$	4.25757e27	1.60719	0.803596	+	0.595176i	$0.202918\pi$
$410$	0	0
$411$	−2.59746e27	−0.931559
$412$	−1.57267e27	−0.549816
$413$	4.73227e27	1.61285
$414$	−3.30052e25	−0.0109668
$415$	0	0
$416$	1.81805e27	0.574283
$417$	−1.40837e27	−0.433799
$418$	1.40911e27	0.423246
$419$	−5.00076e26	−0.146483	−0.0732417	−	0.997314i	$-0.523334\pi$
$419$	−5.00076e26	−0.146483	−0.0732417	+	0.997314i	$0.523334\pi$
$420$	0	0
$421$	−4.63434e27	−1.29130	−0.645648	−	0.763635i	$-0.723413\pi$
$421$	−4.63434e27	−1.29130	−0.645648	+	0.763635i	$0.723413\pi$
$422$	−1.00442e26	−0.0272982
$423$	3.70065e25	0.00981080
$424$	3.94557e26	0.102039
$425$	0	0
$426$	1.84900e26	0.0455130
$427$	4.40697e27	1.05840
$428$	4.79096e27	1.12270
$429$	2.27022e27	0.519118
$430$	0	0
$431$	3.30727e27	0.720209	0.360105	−	0.932912i	$-0.382741\pi$
$431$	3.30727e27	0.720209	0.360105	+	0.932912i	$0.382741\pi$
$432$	−7.57754e26	−0.161046
$433$	−2.58162e27	−0.535512	−0.267756	−	0.963487i	$-0.586282\pi$
$433$	−2.58162e27	−0.535512	−0.267756	+	0.963487i	$0.586282\pi$
$434$	4.57466e26	0.0926224
$435$	0	0
$436$	9.03408e26	0.174291
$437$	−1.30140e27	−0.245107
$438$	−7.39699e26	−0.136012
$439$	6.51723e27	1.17000	0.585000	−	0.811034i	$-0.301095\pi$
$439$	6.51723e27	1.17000	0.585000	+	0.811034i	$0.301095\pi$
$440$	0	0
$441$	−3.67515e26	−0.0629028
$442$	5.03580e26	0.0841656
$443$	−1.02894e28	−1.67939	−0.839694	−	0.543060i	$-0.817266\pi$
$443$	−1.02894e28	−1.67939	−0.839694	+	0.543060i	$0.817266\pi$
$444$	−2.90538e27	−0.463108
$445$	0	0
$446$	2.70838e27	0.411808
$447$	−5.85334e26	−0.0869310
$448$	−4.23862e27	−0.614901
$449$	2.52487e27	0.357810	0.178905	−	0.983866i	$-0.442745\pi$
$449$	2.52487e27	0.357810	0.178905	+	0.983866i	$0.442745\pi$
$450$	0	0
$451$	1.35790e28	1.83659
$452$	−5.27429e27	−0.696964
$453$	1.06798e27	0.137890
$454$	−2.38356e27	−0.300705
$455$	0	0
$456$	3.84770e27	0.463522
$457$	−4.85166e27	−0.571176	−0.285588	−	0.958352i	$-0.592189\pi$
$457$	−4.85166e27	−0.571176	−0.285588	+	0.958352i	$0.592189\pi$
$458$	6.74286e26	0.0775812
$459$	−6.97628e26	−0.0784496
$460$	0	0
$461$	−1.01408e28	−1.08946	−0.544731	−	0.838611i	$-0.683368\pi$
$461$	−1.01408e28	−1.08946	−0.544731	+	0.838611i	$0.683368\pi$
$462$	1.19504e27	0.125499
$463$	9.15460e27	0.939808	0.469904	−	0.882718i	$-0.344289\pi$
$463$	9.15460e27	0.939808	0.469904	+	0.882718i	$0.344289\pi$
$464$	−1.81734e27	−0.182389
$465$	0	0
$466$	−1.84267e27	−0.176765
$467$	−3.58291e27	−0.336054	−0.168027	−	0.985782i	$-0.553740\pi$
$467$	−3.58291e27	−0.336054	−0.168027	+	0.985782i	$0.553740\pi$
$468$	3.01119e27	0.276157
$469$	8.55935e26	0.0767584
$470$	0	0
$471$	6.94401e27	0.595512
$472$	9.77343e27	0.819702
$473$	−7.39770e27	−0.606813
$474$	−1.29304e27	−0.103738
$475$	0	0
$476$	−4.51829e27	−0.346817
$477$	9.89554e26	0.0743013
$478$	5.21772e27	0.383255
$479$	−9.63095e27	−0.692064	−0.346032	−	0.938223i	$-0.612471\pi$
$479$	−9.63095e27	−0.692064	−0.346032	+	0.938223i	$0.612471\pi$
$480$	0	0
$481$	1.08284e28	0.744802
$482$	4.57200e27	0.307690
$483$	−1.10369e27	−0.0726781
$484$	−7.46717e26	−0.0481149
$485$	0	0
$486$	2.44738e26	0.0151015
$487$	1.93192e27	0.116664	0.0583318	−	0.998297i	$-0.481422\pi$
$487$	1.93192e27	0.116664	0.0583318	+	0.998297i	$0.481422\pi$
$488$	9.10161e27	0.537910
$489$	2.26022e27	0.130740
$490$	0	0
$491$	−8.62908e27	−0.478199	−0.239100	−	0.970995i	$-0.576852\pi$
$491$	−8.62908e27	−0.478199	−0.239100	+	0.970995i	$0.576852\pi$
$492$	1.80110e28	0.977021
$493$	−1.67314e27	−0.0888463
$494$	−6.96587e27	−0.362112
$495$	0	0
$496$	−7.33655e27	−0.365540
$497$	6.18305e27	0.301620
$498$	5.27806e27	0.252096
$499$	−3.45711e28	−1.61681	−0.808403	−	0.588629i	$-0.799668\pi$
$499$	−3.45711e28	−1.61681	−0.808403	+	0.588629i	$0.799668\pi$
$500$	0	0
$501$	−1.85175e28	−0.830397
$502$	−5.78622e26	−0.0254101
$503$	−2.28934e28	−0.984569	−0.492285	−	0.870434i	$-0.663838\pi$
$503$	−2.28934e28	−0.984569	−0.492285	+	0.870434i	$0.663838\pi$
$504$	3.26316e27	0.137441
$505$	0	0
$506$	−8.34774e26	−0.0337278
$507$	3.36617e27	0.133215
$508$	−7.73353e27	−0.299784
$509$	−3.01380e28	−1.14440	−0.572199	−	0.820115i	$-0.693910\pi$
$509$	−3.01380e28	−1.14440	−0.572199	+	0.820115i	$0.693910\pi$
$510$	0	0
$511$	−2.47355e28	−0.901367
$512$	−2.60785e28	−0.930995
$513$	9.65008e27	0.337519
$514$	−2.50796e27	−0.0859423
$515$	0	0
$516$	−9.81222e27	−0.322809
$517$	9.35977e26	0.0301727
$518$	5.70004e27	0.180059
$519$	−2.84091e28	−0.879429
$520$	0	0
$521$	−7.99529e27	−0.237705	−0.118853	−	0.992912i	$-0.537922\pi$
$521$	−7.99529e27	−0.237705	−0.118853	+	0.992912i	$0.537922\pi$
$522$	5.86962e26	0.0171029
$523$	−2.08661e28	−0.595901	−0.297950	−	0.954581i	$-0.596303\pi$
$523$	−2.08661e28	−0.595901	−0.297950	+	0.954581i	$0.596303\pi$
$524$	2.35739e28	0.659862
$525$	0	0
$526$	1.55845e28	0.419124
$527$	−6.75441e27	−0.178064
$528$	−1.91653e28	−0.495289
$529$	−3.87006e28	−0.980468
$530$	0	0
$531$	2.45119e28	0.596876
$532$	6.25001e28	1.49214
$533$	−6.71272e28	−1.57131
$534$	−6.00322e27	−0.137785
$535$	0	0
$536$	1.76774e27	0.0390111
$537$	1.93716e27	0.0419214
$538$	1.10856e28	0.235259
$539$	−9.29526e27	−0.193455
$540$	0	0
$541$	−6.81219e28	−1.36369	−0.681844	−	0.731497i	$-0.738822\pi$
$541$	−6.81219e28	−1.36369	−0.681844	+	0.731497i	$0.738822\pi$
$542$	5.74442e27	0.112786
$543$	3.25329e28	0.626505
$544$	−1.41302e28	−0.266907
$545$	0	0
$546$	−5.90762e27	−0.107372
$547$	5.16259e27	0.0920450	0.0460225	−	0.998940i	$-0.485345\pi$
$547$	5.16259e27	0.0920450	0.0460225	+	0.998940i	$0.485345\pi$
$548$	−8.71379e28	−1.52409
$549$	2.28269e28	0.391686
$550$	0	0
$551$	2.31440e28	0.382250
$552$	−2.27943e27	−0.0369373
$553$	−4.32391e28	−0.687484
$554$	−2.06228e28	−0.321733
$555$	0	0
$556$	−4.72470e28	−0.709724
$557$	8.97767e28	1.32338	0.661689	−	0.749778i	$-0.269840\pi$
$557$	8.97767e28	1.32338	0.661689	+	0.749778i	$0.269840\pi$
$558$	2.36955e27	0.0342773
$559$	3.65703e28	0.519164
$560$	0	0
$561$	−1.76445e28	−0.241268
$562$	−3.33847e26	−0.00448039
$563$	−7.85492e26	−0.0103467	−0.00517337	−	0.999987i	$-0.501647\pi$
$563$	−7.85492e26	−0.0103467	−0.00517337	+	0.999987i	$0.501647\pi$
$564$	1.24147e27	0.0160511
$565$	0	0
$566$	1.50364e28	0.187315
$567$	8.18405e27	0.100080
$568$	1.27697e28	0.153293
$569$	−9.58086e28	−1.12908	−0.564541	−	0.825405i	$-0.690947\pi$
$569$	−9.58086e28	−1.12908	−0.564541	+	0.825405i	$0.690947\pi$
$570$	0	0
$571$	−6.50183e26	−0.00738511	−0.00369255	−	0.999993i	$-0.501175\pi$
$571$	−6.50183e26	−0.00738511	−0.00369255	+	0.999993i	$0.501175\pi$
$572$	7.61596e28	0.849310
$573$	−6.98355e28	−0.764632
$574$	−3.53356e28	−0.379872
$575$	0	0
$576$	−2.19549e28	−0.227560
$577$	1.67471e29	1.70448	0.852242	−	0.523148i	$-0.175242\pi$
$577$	1.67471e29	1.70448	0.852242	+	0.523148i	$0.175242\pi$
$578$	1.96402e28	0.196292
$579$	6.91975e28	0.679150
$580$	0	0
$581$	1.76498e29	1.67067
$582$	1.38238e28	0.128510
$583$	2.50280e28	0.228510
$584$	−5.10856e28	−0.458104
$585$	0	0
$586$	2.99074e28	0.258734
$587$	−2.23014e29	−1.89510	−0.947551	−	0.319603i	$-0.896450\pi$
$587$	−2.23014e29	−1.89510	−0.947551	+	0.319603i	$0.896450\pi$
$588$	−1.23291e28	−0.102913
$589$	9.34317e28	0.766097
$590$	0	0
$591$	−3.73010e28	−0.295156
$592$	−9.14138e28	−0.710614
$593$	1.71015e29	1.30605	0.653024	−	0.757337i	$-0.273500\pi$
$593$	1.71015e29	1.30605	0.653024	+	0.757337i	$0.273500\pi$
$594$	6.18997e27	0.0464441
$595$	0	0
$596$	−1.96363e28	−0.142225
$597$	1.06641e29	0.758914
$598$	4.12668e27	0.0288561
$599$	−1.56747e29	−1.07701	−0.538504	−	0.842623i	$-0.681010\pi$
$599$	−1.56747e29	−1.07701	−0.538504	+	0.842623i	$0.681010\pi$
$600$	0	0
$601$	−1.53561e29	−1.01883	−0.509413	−	0.860522i	$-0.670137\pi$
$601$	−1.53561e29	−1.01883	−0.509413	+	0.860522i	$0.670137\pi$
$602$	1.92505e28	0.125510
$603$	4.43352e27	0.0284064
$604$	3.58279e28	0.225597
$605$	0	0
$606$	−3.47235e28	−0.211184
$607$	−2.86065e29	−1.70995	−0.854976	−	0.518667i	$-0.826428\pi$
$607$	−2.86065e29	−1.70995	−0.854976	+	0.518667i	$0.826428\pi$
$608$	1.95459e29	1.14833
$609$	1.96280e28	0.113343
$610$	0	0
$611$	−4.62697e27	−0.0258145
$612$	−2.34035e28	−0.128349
$613$	−4.90169e28	−0.264247	−0.132124	−	0.991233i	$-0.542180\pi$
$613$	−4.90169e28	−0.264247	−0.132124	+	0.991233i	$0.542180\pi$
$614$	6.00189e28	0.318068
$615$	0	0
$616$	8.25325e28	0.422695
$617$	−2.77126e29	−1.39535	−0.697676	−	0.716413i	$-0.745783\pi$
$617$	−2.77126e29	−1.39535	−0.697676	+	0.716413i	$0.745783\pi$
$618$	1.59816e28	0.0791118
$619$	−8.92512e28	−0.434373	−0.217186	−	0.976130i	$-0.569688\pi$
$619$	−8.92512e28	−0.434373	−0.217186	+	0.976130i	$0.569688\pi$
$620$	0	0
$621$	−5.71683e27	−0.0268964
$622$	9.28631e28	0.429580
$623$	−2.00748e29	−0.913116
$624$	9.47429e28	0.423749
$625$	0	0
$626$	3.06818e28	0.132694
$627$	2.44072e29	1.03803
$628$	2.32952e29	0.974296
$629$	−8.41603e28	−0.346159
$630$	0	0
$631$	7.08364e28	0.281805	0.140902	−	0.990024i	$-0.455000\pi$
$631$	7.08364e28	0.281805	0.140902	+	0.990024i	$0.455000\pi$
$632$	−8.93006e28	−0.349402
$633$	−1.73976e28	−0.0669499
$634$	2.83087e28	0.107148
$635$	0	0
$636$	3.31968e28	0.121562
$637$	4.59508e28	0.165512
$638$	1.48456e28	0.0525992
$639$	3.20265e28	0.111622
$640$	0	0
$641$	2.59855e28	0.0876440	0.0438220	−	0.999039i	$-0.486047\pi$
$641$	2.59855e28	0.0876440	0.0438220	+	0.999039i	$0.486047\pi$
$642$	−4.86860e28	−0.161542
$643$	1.38976e29	0.453652	0.226826	−	0.973935i	$-0.427165\pi$
$643$	1.38976e29	0.453652	0.226826	+	0.973935i	$0.427165\pi$
$644$	−3.70259e28	−0.118906
$645$	0	0
$646$	5.41400e28	0.168297
$647$	−4.60493e29	−1.40841	−0.704203	−	0.709999i	$-0.748695\pi$
$647$	−4.60493e29	−1.40841	−0.704203	+	0.709999i	$0.748695\pi$
$648$	1.69023e28	0.0508637
$649$	6.19959e29	1.83567
$650$	0	0
$651$	7.92377e28	0.227160
$652$	7.58243e28	0.213899
$653$	6.22853e29	1.72901	0.864504	−	0.502625i	$-0.167632\pi$
$653$	6.22853e29	1.72901	0.864504	+	0.502625i	$0.167632\pi$
$654$	−9.18049e27	−0.0250784
$655$	0	0
$656$	5.66691e29	1.49919
$657$	−1.28123e29	−0.333574
$658$	−2.43562e27	−0.00624077
$659$	6.30181e29	1.58916	0.794582	−	0.607157i	$-0.207690\pi$
$659$	6.30181e29	1.58916	0.794582	+	0.607157i	$0.207690\pi$
$660$	0	0
$661$	−3.89533e29	−0.951545	−0.475773	−	0.879568i	$-0.657832\pi$
$661$	−3.89533e29	−0.951545	−0.475773	+	0.879568i	$0.657832\pi$
$662$	3.35524e28	0.0806704
$663$	8.72252e28	0.206419
$664$	3.64517e29	0.849089
$665$	0	0
$666$	2.95247e28	0.0666356
$667$	−1.37108e28	−0.0304609
$668$	−6.21211e29	−1.35858
$669$	4.69120e29	1.00997
$670$	0	0
$671$	5.77344e29	1.20461
$672$	1.65765e29	0.340499
$673$	2.70633e29	0.547297	0.273648	−	0.961830i	$-0.411770\pi$
$673$	2.70633e29	0.547297	0.273648	+	0.961830i	$0.411770\pi$
$674$	−2.10502e29	−0.419109
$675$	0	0
$676$	1.12926e29	0.217948
$677$	−4.23921e29	−0.805572	−0.402786	−	0.915294i	$-0.631958\pi$
$677$	−4.23921e29	−0.805572	−0.402786	+	0.915294i	$0.631958\pi$
$678$	5.35977e28	0.100285
$679$	4.62267e29	0.851648
$680$	0	0
$681$	−4.12857e29	−0.737488
$682$	5.99311e28	0.105418
$683$	−9.87881e29	−1.71115	−0.855573	−	0.517682i	$-0.826795\pi$
$683$	−9.87881e29	−1.71115	−0.855573	+	0.517682i	$0.826795\pi$
$684$	3.23734e29	0.552203
$685$	0	0
$686$	1.52367e29	0.252051
$687$	1.16793e29	0.190271
$688$	−3.08728e29	−0.495333
$689$	−1.23725e29	−0.195504
$690$	0	0
$691$	−1.20248e29	−0.184314	−0.0921569	−	0.995744i	$-0.529376\pi$
$691$	−1.20248e29	−0.184314	−0.0921569	+	0.995744i	$0.529376\pi$
$692$	−9.53049e29	−1.43880
$693$	2.06993e29	0.307791
$694$	−4.22911e28	−0.0619405
$695$	0	0
$696$	4.05372e28	0.0576045
$697$	5.21725e29	0.730293
$698$	−2.87586e29	−0.396539
$699$	−3.19170e29	−0.433522
$700$	0	0
$701$	3.01199e29	0.397022	0.198511	−	0.980099i	$-0.436389\pi$
$701$	3.01199e29	0.397022	0.198511	+	0.980099i	$0.436389\pi$
$702$	−3.05999e28	−0.0397356
$703$	1.16416e30	1.48930
$704$	−5.55288e29	−0.699850
$705$	0	0
$706$	−3.04126e29	−0.372053
$707$	−1.16115e30	−1.39954
$708$	8.22306e29	0.976527
$709$	−4.42994e29	−0.518337	−0.259168	−	0.965832i	$-0.583448\pi$
$709$	−4.42994e29	−0.518337	−0.259168	+	0.965832i	$0.583448\pi$
$710$	0	0
$711$	−2.23967e29	−0.254421
$712$	−4.14599e29	−0.464075
$713$	−5.53502e28	−0.0610491
$714$	4.59151e28	0.0499028
$715$	0	0
$716$	6.49864e28	0.0685861
$717$	9.03763e29	0.939947
$718$	4.47176e28	0.0458323
$719$	1.48484e30	1.49978	0.749888	−	0.661565i	$-0.230108\pi$
$719$	1.48484e30	1.49978	0.749888	+	0.661565i	$0.230108\pi$
$720$	0	0
$721$	5.34425e29	0.524283
$722$	−5.05422e29	−0.488668
$723$	7.91918e29	0.754619
$724$	1.09139e30	1.02500
$725$	0	0
$726$	7.58819e27	0.00692315
$727$	1.41765e30	1.27484	0.637421	−	0.770516i	$-0.280001\pi$
$727$	1.41765e30	1.27484	0.637421	+	0.770516i	$0.280001\pi$
$728$	−4.07997e29	−0.361640
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	−2.84231e29	−0.241290
$732$	7.65782e29	0.640824
$733$	1.26744e30	1.04553	0.522764	−	0.852477i	$-0.324901\pi$
$733$	1.26744e30	1.04553	0.522764	+	0.852477i	$0.324901\pi$
$734$	−2.26095e28	−0.0183858
$735$	0	0
$736$	−1.15793e29	−0.0915089
$737$	1.12133e29	0.0873627
$738$	−1.83029e29	−0.140581
$739$	−1.86110e30	−1.40930	−0.704650	−	0.709555i	$-0.748896\pi$
$739$	−1.86110e30	−1.40930	−0.704650	+	0.709555i	$0.748896\pi$
$740$	0	0
$741$	−1.20656e30	−0.888091
$742$	−6.51285e28	−0.0472640
$743$	5.15013e29	0.368499	0.184249	−	0.982880i	$-0.441015\pi$
$743$	5.15013e29	0.368499	0.184249	+	0.982880i	$0.441015\pi$
$744$	1.63647e29	0.115450
$745$	0	0
$746$	−5.10895e29	−0.350408
$747$	9.14214e29	0.618275
$748$	−5.91926e29	−0.394731
$749$	−1.62806e30	−1.07056
$750$	0	0
$751$	−1.60839e30	−1.02842	−0.514211	−	0.857663i	$-0.671915\pi$
$751$	−1.60839e30	−1.02842	−0.514211	+	0.857663i	$0.671915\pi$
$752$	3.90611e28	0.0246296
$753$	−1.00223e29	−0.0623191
$754$	−7.33885e28	−0.0450017
$755$	0	0
$756$	2.74552e29	0.163737
$757$	−2.14104e29	−0.125927	−0.0629634	−	0.998016i	$-0.520055\pi$
$757$	−2.14104e29	−0.125927	−0.0629634	+	0.998016i	$0.520055\pi$
$758$	−8.89718e28	−0.0516090
$759$	−1.44591e29	−0.0827187
$760$	0	0
$761$	1.25293e30	0.697250	0.348625	−	0.937262i	$-0.386649\pi$
$761$	1.25293e30	0.697250	0.348625	+	0.937262i	$0.386649\pi$
$762$	7.85887e28	0.0431353
$763$	−3.06996e29	−0.166198
$764$	−2.34279e30	−1.25099
$765$	0	0
$766$	−2.29516e29	−0.119237
$767$	−3.06475e30	−1.57052
$768$	−5.60474e29	−0.283310
$769$	2.53517e30	1.26410	0.632048	−	0.774929i	$-0.282215\pi$
$769$	2.53517e30	1.26410	0.632048	+	0.774929i	$0.282215\pi$
$770$	0	0
$771$	−4.34403e29	−0.210777
$772$	2.32139e30	1.11113
$773$	−1.70099e30	−0.803187	−0.401594	−	0.915818i	$-0.631544\pi$
$773$	−1.70099e30	−0.803187	−0.401594	+	0.915818i	$0.631544\pi$
$774$	9.97125e28	0.0464483
$775$	0	0
$776$	9.54707e29	0.432835
$777$	9.87306e29	0.441602
$778$	−3.36069e29	−0.148301
$779$	−7.21686e30	−3.14199
$780$	0	0
$781$	8.10022e29	0.343290
$782$	−3.20733e28	−0.0134113
$783$	1.01668e29	0.0419454
$784$	−3.87919e29	−0.157915
$785$	0	0
$786$	−2.39560e29	−0.0949460
$787$	4.98615e30	1.94998	0.974991	−	0.222246i	$-0.0713388\pi$
$787$	4.98615e30	1.94998	0.974991	+	0.222246i	$0.0713388\pi$
$788$	−1.25135e30	−0.482894
$789$	2.69940e30	1.02792
$790$	0	0
$791$	1.79231e30	0.664598
$792$	4.27496e29	0.156429
$793$	−2.85408e30	−1.03062
$794$	8.31160e29	0.296189
$795$	0	0
$796$	3.57751e30	1.24163
$797$	−4.48685e30	−1.53684	−0.768420	−	0.639946i	$-0.778957\pi$
$797$	−4.48685e30	−1.53684	−0.768420	+	0.639946i	$0.778957\pi$
$798$	−6.35130e29	−0.214700
$799$	3.59616e28	0.0119977
$800$	0	0
$801$	−1.03982e30	−0.337922
$802$	−1.11306e29	−0.0357016
$803$	−3.24052e30	−1.02589
$804$	1.48732e29	0.0464747
$805$	0	0
$806$	−2.96267e29	−0.0901915
$807$	1.92014e30	0.576981
$808$	−2.39810e30	−0.711292
$809$	−5.22233e30	−1.52899	−0.764496	−	0.644629i	$-0.777012\pi$
$809$	−5.22233e30	−1.52899	−0.764496	+	0.644629i	$0.777012\pi$
$810$	0	0
$811$	−6.36541e30	−1.81597	−0.907983	−	0.419007i	$-0.862378\pi$
$811$	−6.36541e30	−1.81597	−0.907983	+	0.419007i	$0.862378\pi$
$812$	6.58466e29	0.185436
$813$	9.94992e29	0.276611
$814$	7.46744e29	0.204935
$815$	0	0
$816$	−7.36359e29	−0.196944
$817$	3.93168e30	1.03812
$818$	−1.45145e30	−0.378348
$819$	−1.02326e30	−0.263333
$820$	0	0
$821$	−7.95104e29	−0.199444	−0.0997220	−	0.995015i	$-0.531795\pi$
$821$	−7.95104e29	−0.199444	−0.0997220	+	0.995015i	$0.531795\pi$
$822$	8.85501e29	0.219298
$823$	1.00191e30	0.244980	0.122490	−	0.992470i	$-0.460912\pi$
$823$	1.00191e30	0.244980	0.122490	+	0.992470i	$0.460912\pi$
$824$	1.10373e30	0.266458
$825$	0	0
$826$	−1.61328e30	−0.379680
$827$	−4.61386e30	−1.07215	−0.536076	−	0.844170i	$-0.680094\pi$
$827$	−4.61386e30	−1.07215	−0.536076	+	0.844170i	$0.680094\pi$
$828$	−1.91784e29	−0.0440042
$829$	−2.84827e30	−0.645294	−0.322647	−	0.946519i	$-0.604573\pi$
$829$	−2.84827e30	−0.645294	−0.322647	+	0.946519i	$0.604573\pi$
$830$	0	0
$831$	−3.57209e30	−0.789062
$832$	2.74505e30	0.598763
$833$	−3.57138e29	−0.0769243
$834$	4.80127e29	0.102121
$835$	0	0
$836$	8.18794e30	1.69828
$837$	4.10430e29	0.0840662
$838$	1.70481e29	0.0344836
$839$	−6.35489e30	−1.26943	−0.634713	−	0.772748i	$-0.718882\pi$
$839$	−6.35489e30	−1.26943	−0.634713	+	0.772748i	$0.718882\pi$
$840$	0	0
$841$	−4.88901e30	−0.952496
$842$	1.57989e30	0.303984
$843$	−5.78257e28	−0.0109883
$844$	−5.83641e29	−0.109534
$845$	0	0
$846$	−1.26159e28	−0.00230956
$847$	2.53749e29	0.0458806
$848$	1.04449e30	0.186530
$849$	2.60446e30	0.459396
$850$	0	0
$851$	−6.89666e29	−0.118680
$852$	1.07440e30	0.182621
$853$	−7.05513e30	−1.18451	−0.592257	−	0.805749i	$-0.701763\pi$
$853$	−7.05513e30	−1.18451	−0.592257	+	0.805749i	$0.701763\pi$
$854$	−1.50238e30	−0.249156
$855$	0	0
$856$	−3.36239e30	−0.544093
$857$	−3.70327e30	−0.591953	−0.295976	−	0.955195i	$-0.595645\pi$
$857$	−3.70327e30	−0.591953	−0.295976	+	0.955195i	$0.595645\pi$
$858$	−7.73939e29	−0.122205
$859$	−5.45289e29	−0.0850548	−0.0425274	−	0.999095i	$-0.513541\pi$
$859$	−5.45289e29	−0.0850548	−0.0425274	+	0.999095i	$0.513541\pi$
$860$	0	0
$861$	−6.12048e30	−0.931649
$862$	−1.12748e30	−0.169544
$863$	7.07308e30	1.05074	0.525370	−	0.850874i	$-0.323927\pi$
$863$	7.07308e30	1.05074	0.525370	+	0.850874i	$0.323927\pi$
$864$	8.58618e29	0.126010
$865$	0	0
$866$	8.80098e29	0.126065
$867$	3.40188e30	0.481414
$868$	2.65821e30	0.371648
$869$	−5.66462e30	−0.782461
$870$	0	0
$871$	−5.54327e29	−0.0747439
$872$	−6.34030e29	−0.0844669
$873$	2.39442e30	0.315174
$874$	4.43660e29	0.0577006
$875$	0	0
$876$	−4.29819e30	−0.545748
$877$	−1.58002e31	−1.98229	−0.991146	−	0.132779i	$-0.957610\pi$
$877$	−1.58002e31	−1.98229	−0.991146	+	0.132779i	$0.957610\pi$
$878$	−2.22179e30	−0.275429
$879$	5.18026e30	0.634553
$880$	0	0
$881$	9.64321e30	1.15339	0.576693	−	0.816961i	$-0.304343\pi$
$881$	9.64321e30	1.15339	0.576693	+	0.816961i	$0.304343\pi$
$882$	1.25289e29	0.0148079
$883$	7.91697e30	0.924638	0.462319	−	0.886714i	$-0.347018\pi$
$883$	7.91697e30	0.924638	0.462319	+	0.886714i	$0.347018\pi$
$884$	2.92617e30	0.337715
$885$	0	0
$886$	3.50776e30	0.395344
$887$	−7.76439e30	−0.864787	−0.432394	−	0.901685i	$-0.642331\pi$
$887$	−7.76439e30	−0.864787	−0.432394	+	0.901685i	$0.642331\pi$
$888$	2.03906e30	0.224436
$889$	2.62801e30	0.285863
$890$	0	0
$891$	1.07217e30	0.113906
$892$	1.57377e31	1.65238
$893$	−4.97447e29	−0.0516186
$894$	1.99546e29	0.0204644
$895$	0	0
$896$	7.33221e30	0.734515
$897$	7.14782e29	0.0707706
$898$	−8.60752e29	−0.0842318
$899$	9.84344e29	0.0952073
$900$	0	0
$901$	9.61613e29	0.0908636
$902$	−4.62920e30	−0.432352
$903$	3.33438e30	0.307818
$904$	3.70160e30	0.337770
$905$	0	0
$906$	−3.64086e29	−0.0324607
$907$	1.25327e31	1.10451	0.552253	−	0.833677i	$-0.313768\pi$
$907$	1.25327e31	1.10451	0.552253	+	0.833677i	$0.313768\pi$
$908$	−1.38502e31	−1.20658
$909$	−6.01446e30	−0.517936
$910$	0	0
$911$	9.93009e30	0.835623	0.417811	−	0.908534i	$-0.362797\pi$
$911$	9.93009e30	0.835623	0.417811	+	0.908534i	$0.362797\pi$
$912$	1.01858e31	0.847326
$913$	2.31225e31	1.90148
$914$	1.65398e30	0.134460
$915$	0	0
$916$	3.91809e30	0.311295
$917$	−8.01087e30	−0.629219
$918$	2.37828e29	0.0184678
$919$	2.22319e31	1.70673	0.853364	−	0.521316i	$-0.174559\pi$
$919$	2.22319e31	1.70673	0.853364	+	0.521316i	$0.174559\pi$
$920$	0	0
$921$	1.03959e31	0.780072
$922$	3.45709e30	0.256470
$923$	−4.00431e30	−0.293704
$924$	6.94403e30	0.503566
$925$	0	0
$926$	−3.12089e30	−0.221240
$927$	2.76818e30	0.194024
$928$	2.05925e30	0.142710
$929$	−1.62508e31	−1.11355	−0.556773	−	0.830664i	$-0.687961\pi$
$929$	−1.62508e31	−1.11355	−0.556773	+	0.830664i	$0.687961\pi$
$930$	0	0
$931$	4.94018e30	0.330957
$932$	−1.07073e31	−0.709271
$933$	1.60848e31	1.05356
$934$	1.22145e30	0.0791103
$935$	0	0
$936$	−2.11331e30	−0.133834
$937$	1.81129e30	0.113428	0.0567142	−	0.998390i	$-0.481938\pi$
$937$	1.81129e30	0.113428	0.0567142	+	0.998390i	$0.481938\pi$
$938$	−2.91797e29	−0.0180697
$939$	5.31441e30	0.325436
$940$	0	0
$941$	−3.25431e31	−1.94880	−0.974402	−	0.224812i	$-0.927823\pi$
$941$	−3.25431e31	−1.94880	−0.974402	+	0.224812i	$0.927823\pi$
$942$	−2.36728e30	−0.140189
$943$	4.27536e30	0.250380
$944$	2.58727e31	1.49843
$945$	0	0
$946$	2.52195e30	0.142850
$947$	−1.36434e31	−0.764270	−0.382135	−	0.924107i	$-0.624811\pi$
$947$	−1.36434e31	−0.764270	−0.382135	+	0.924107i	$0.624811\pi$
$948$	−7.51348e30	−0.416249
$949$	1.60194e31	0.877710
$950$	0	0
$951$	4.90335e30	0.262783
$952$	3.17102e30	0.168078
$953$	−1.03136e31	−0.540673	−0.270337	−	0.962766i	$-0.587135\pi$
$953$	−1.03136e31	−0.540673	−0.270337	+	0.962766i	$0.587135\pi$
$954$	−3.37348e29	−0.0174912
$955$	0	0
$956$	3.03188e31	1.53781
$957$	2.57140e30	0.129001
$958$	3.28328e30	0.162918
$959$	2.96111e31	1.45331
$960$	0	0
$961$	−1.68517e31	−0.809187
$962$	−3.69151e30	−0.175334
$963$	−8.43291e30	−0.396188
$964$	2.65667e31	1.23461
$965$	0	0
$966$	3.76260e29	0.0171091
$967$	−3.33709e31	−1.50103	−0.750517	−	0.660851i	$-0.770196\pi$
$967$	−3.33709e31	−1.50103	−0.750517	+	0.660851i	$0.770196\pi$
$968$	5.24061e29	0.0233180
$969$	9.37760e30	0.412755
$970$	0	0
$971$	2.13037e30	0.0917598	0.0458799	−	0.998947i	$-0.485391\pi$
$971$	2.13037e30	0.0917598	0.0458799	+	0.998947i	$0.485391\pi$
$972$	1.42211e30	0.0605950
$973$	1.60555e31	0.676765
$974$	−6.58611e29	−0.0274637
$975$	0	0
$976$	2.40943e31	0.983310
$977$	−4.34834e31	−1.75562	−0.877811	−	0.479008i	$-0.840997\pi$
$977$	−4.34834e31	−1.75562	−0.877811	+	0.479008i	$0.840997\pi$
$978$	−7.70532e29	−0.0307774
$979$	−2.62993e31	−1.03926
$980$	0	0
$981$	−1.59015e30	−0.0615056
$982$	2.94174e30	0.112573
$983$	4.41457e31	1.67138	0.835692	−	0.549198i	$-0.185067\pi$
$983$	4.41457e31	1.67138	0.835692	+	0.549198i	$0.185067\pi$
$984$	−1.26405e31	−0.473494
$985$	0	0
$986$	5.70389e29	0.0209153
$987$	−4.21875e29	−0.0153057
$988$	−4.04768e31	−1.45297
$989$	−2.32918e30	−0.0827260
$990$	0	0
$991$	−3.77164e31	−1.31147	−0.655734	−	0.754992i	$-0.727641\pi$
$991$	−3.77164e31	−1.31147	−0.655734	+	0.754992i	$0.727641\pi$
$992$	8.31311e30	0.286017
$993$	5.81161e30	0.197847
$994$	−2.10786e30	−0.0710044
$995$	0	0
$996$	3.06694e31	1.01154
$997$	−2.89594e31	−0.945126	−0.472563	−	0.881297i	$-0.656671\pi$
$997$	−2.89594e31	−0.945126	−0.472563	+	0.881297i	$0.656671\pi$
$998$	1.17856e31	0.380612
$999$	5.11398e30	0.163426

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.j.1.3	yes	6
5.2	odd	4		75.22.b.i.49.6		12
5.3	odd	4		75.22.b.i.49.7		12
5.4	even	2		75.22.a.i.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.22.a.i.1.4	✓	6	5.4	even	2
75.22.a.j.1.3	yes	6	1.1	even	1	trivial
75.22.b.i.49.6		12	5.2	odd	4
75.22.b.i.49.7		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)

\(f(q)\)	\(=\)	\(q-340.910 q^{2} -59049.0 q^{3} -1.98093e6 q^{4} +2.01304e7 q^{6} +6.73159e8 q^{7} +1.39026e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\)	\(q-340.910 q^{2} -59049.0 q^{3} -1.98093e6 q^{4} +2.01304e7 q^{6} +6.73159e8 q^{7} +1.39026e9 q^{8} +3.48678e9 q^{9} +8.81884e10 q^{11} +1.16972e11 q^{12} -4.35956e11 q^{13} -2.29486e11 q^{14} +3.68036e12 q^{16} +3.38833e12 q^{17} -1.18868e12 q^{18} -4.68698e13 q^{19} -3.97494e13 q^{21} -3.00643e13 q^{22} +2.77663e13 q^{23} -8.20934e13 q^{24} +1.48622e14 q^{26} -2.05891e14 q^{27} -1.33348e15 q^{28} -4.93794e14 q^{29} -1.99343e15 q^{31} -4.17025e15 q^{32} -5.20744e15 q^{33} -1.15512e15 q^{34} -6.90709e15 q^{36} -2.48383e16 q^{37} +1.59784e16 q^{38} +2.57428e16 q^{39} +1.53977e17 q^{41} +1.35509e16 q^{42} -8.38852e16 q^{43} -1.74695e17 q^{44} -9.46580e15 q^{46} +1.06134e16 q^{47} -2.17322e17 q^{48} -1.05402e17 q^{49} -2.00078e17 q^{51} +8.63600e17 q^{52} +2.83801e17 q^{53} +7.01903e16 q^{54} +9.35865e17 q^{56} +2.76761e18 q^{57} +1.68339e17 q^{58} +7.02994e18 q^{59} +6.54670e18 q^{61} +6.79580e17 q^{62} +2.34716e18 q^{63} -6.29660e18 q^{64} +1.77527e18 q^{66} +1.27152e18 q^{67} -6.71206e18 q^{68} -1.63957e18 q^{69} +9.18512e18 q^{71} +4.84753e18 q^{72} -3.67454e19 q^{73} +8.46760e18 q^{74} +9.28459e19 q^{76} +5.93649e19 q^{77} -8.77597e18 q^{78} -6.42331e19 q^{79} +1.21577e19 q^{81} -5.24921e19 q^{82} +2.62194e20 q^{83} +7.87409e19 q^{84} +2.85973e19 q^{86} +2.91580e19 q^{87} +1.22605e20 q^{88} -2.98217e20 q^{89} -2.93468e20 q^{91} -5.50032e19 q^{92} +1.17710e20 q^{93} -3.61820e18 q^{94} +2.46249e20 q^{96} +6.86712e20 q^{97} +3.59327e19 q^{98} +3.07494e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 92 q^{2} - 354294 q^{3} + 4390448 q^{4} - 5432508 q^{6} + 1327143454 q^{7} - 4252271424 q^{8} + 20920706406 q^{9}+O(q^{10}) \) 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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