LMFDB - Embedded newform 539.8.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [539,8,Mod(1,539)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("539.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(539, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 539 = 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	539.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-8,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$168.375528736$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{15}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 15 $ x^2 - 15
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.87298$ of defining polynomial
Character	$\chi$	$=$	539.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-11.7460 q^{2} -43.4758 q^{3} +9.96773 q^{4} +389.919 q^{5} +510.665 q^{6} +1386.40 q^{8} -296.855 q^{9} -4579.98 q^{10} +1331.00 q^{11} -433.355 q^{12} +3840.41 q^{13} -16952.1 q^{15} -17560.5 q^{16} -24162.7 q^{17} +3486.85 q^{18} -5458.47 q^{19} +3886.61 q^{20} -15633.9 q^{22} -63973.2 q^{23} -60275.0 q^{24} +73912.1 q^{25} -45109.3 q^{26} +107988. q^{27} +178351. q^{29} +199118. q^{30} +185129. q^{31} +28805.6 q^{32} -57866.3 q^{33} +283814. q^{34} -2958.97 q^{36} -409817. q^{37} +64115.0 q^{38} -166965. q^{39} +540585. q^{40} +675273. q^{41} +38903.5 q^{43} +13267.1 q^{44} -115749. q^{45} +751427. q^{46} -949397. q^{47} +763457. q^{48} -868169. q^{50} +1.05049e6 q^{51} +38280.2 q^{52} +294003. q^{53} -1.26842e6 q^{54} +518983. q^{55} +237311. q^{57} -2.09490e6 q^{58} +87803.7 q^{59} -168974. q^{60} +2.78573e6 q^{61} -2.17452e6 q^{62} +1.90940e6 q^{64} +1.49745e6 q^{65} +679696. q^{66} +2.95364e6 q^{67} -240847. q^{68} +2.78129e6 q^{69} -4.08380e6 q^{71} -411560. q^{72} -1.95525e6 q^{73} +4.81370e6 q^{74} -3.21339e6 q^{75} -54408.6 q^{76} +1.96116e6 q^{78} -608158. q^{79} -6.84718e6 q^{80} -4.04562e6 q^{81} -7.93173e6 q^{82} -214613. q^{83} -9.42151e6 q^{85} -456959. q^{86} -7.75394e6 q^{87} +1.84530e6 q^{88} +8.30366e6 q^{89} +1.35959e6 q^{90} -637668. q^{92} -8.04863e6 q^{93} +1.11516e7 q^{94} -2.12836e6 q^{95} -1.25235e6 q^{96} +1.38909e6 q^{97} -395114. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} + 6 q^{3} - 104 q^{4} + 470 q^{5} + 696 q^{6} + 480 q^{8} - 36 q^{9} - 4280 q^{10} + 2662 q^{11} - 6072 q^{12} - 344 q^{13} - 12990 q^{15} - 6368 q^{16} + 8468 q^{17} + 4464 q^{18} + 35280 q^{19}+ \cdots - 47916 q^{99}+O(q^{100}) $ 2 * q - 8 * q^2 + 6 * q^3 - 104 * q^4 + 470 * q^5 + 696 * q^6 + 480 * q^8 - 36 * q^9 - 4280 * q^10 + 2662 * q^11 - 6072 * q^12 - 344 * q^13 - 12990 * q^15 - 6368 * q^16 + 8468 * q^17 + 4464 * q^18 + 35280 * q^19 - 5240 * q^20 - 10648 * q^22 - 61486 * q^23 - 105120 * q^24 + 2200 * q^25 - 60784 * q^26 + 12690 * q^27 + 179040 * q^29 + 213960 * q^30 + 57166 * q^31 + 186752 * q^32 + 7986 * q^33 + 406048 * q^34 - 32688 * q^36 - 877698 * q^37 + 216720 * q^38 - 373992 * q^39 + 468000 * q^40 + 283616 * q^41 + 275484 * q^43 - 138424 * q^44 - 94860 * q^45 + 760744 * q^46 - 1662512 * q^47 + 1317216 * q^48 - 1136800 * q^50 + 2664924 * q^51 + 515168 * q^52 + 1616484 * q^53 - 1625400 * q^54 + 625570 * q^55 + 2252880 * q^57 - 2092320 * q^58 + 2454130 * q^59 - 620520 * q^60 + 6019176 * q^61 - 2653864 * q^62 + 1068416 * q^64 + 1162360 * q^65 + 926376 * q^66 - 174698 * q^67 - 3959696 * q^68 + 2904342 * q^69 - 1151466 * q^71 - 648000 * q^72 - 885944 * q^73 + 3061032 * q^74 - 6761400 * q^75 - 4697280 * q^76 + 1185648 * q^78 + 3801460 * q^79 - 5950880 * q^80 - 9331038 * q^81 - 9398864 * q^82 + 2282916 * q^83 - 6808420 * q^85 + 429264 * q^86 - 7719840 * q^87 + 638880 * q^88 + 13481970 * q^89 + 1437840 * q^90 - 921128 * q^92 - 14379702 * q^93 + 8480288 * q^94 + 1134000 * q^95 + 6562176 * q^96 + 68078 * q^97 - 47916 * q^99

Coefficient data

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

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

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

$2$	−11.7460	−1.03821	−0.519103	−	0.854712i	$-0.673734\pi$
$2$	−11.7460	−1.03821	−0.519103	+	0.854712i	$0.673734\pi$
$3$	−43.4758	−0.929658	−0.464829	−	0.885400i	$-0.653884\pi$
$3$	−43.4758	−0.929658	−0.464829	+	0.885400i	$0.653884\pi$
$4$	9.96773	0.0778729
$5$	389.919	1.39502	0.697509	−	0.716576i	$-0.254292\pi$
$5$	389.919	1.39502	0.697509	+	0.716576i	$0.254292\pi$
$6$	510.665	0.965177
$7$	0	0
$7$	0	0
$8$	1386.40	0.957358
$9$	−296.855	−0.135736
$10$	−4579.98	−1.44832
$11$	1331.00	0.301511
$11$	1331.00	0.301511
$12$	−433.355	−0.0723952
$13$	3840.41	0.484815	0.242407	−	0.970175i	$-0.422063\pi$
$13$	3840.41	0.484815	0.242407	+	0.970175i	$0.422063\pi$
$14$	0	0
$15$	−16952.1	−1.29689
$16$	−17560.5	−1.07181
$17$	−24162.7	−1.19282	−0.596409	−	0.802680i	$-0.703407\pi$
$17$	−24162.7	−1.19282	−0.596409	+	0.802680i	$0.703407\pi$
$18$	3486.85	0.140922
$19$	−5458.47	−0.182572	−0.0912859	−	0.995825i	$-0.529098\pi$
$19$	−5458.47	−0.182572	−0.0912859	+	0.995825i	$0.529098\pi$
$20$	3886.61	0.108634
$21$	0	0
$22$	−15633.9	−0.313031
$23$	−63973.2	−1.09635	−0.548177	−	0.836362i	$-0.684678\pi$
$23$	−63973.2	−1.09635	−0.548177	+	0.836362i	$0.684678\pi$
$24$	−60275.0	−0.890016
$25$	73912.1	0.946075
$26$	−45109.3	−0.503338
$27$	107988.	1.05585
$28$	0	0
$29$	178351.	1.35794	0.678972	−	0.734164i	$-0.262426\pi$
$29$	178351.	1.35794	0.678972	+	0.734164i	$0.262426\pi$
$30$	199118.	1.34644
$31$	185129.	1.11611	0.558057	−	0.829803i	$-0.311547\pi$
$31$	185129.	1.11611	0.558057	+	0.829803i	$0.311547\pi$
$32$	28805.6	0.155400
$33$	−57866.3	−0.280302
$34$	283814.	1.23839
$35$	0	0
$36$	−2958.97	−0.0105702
$37$	−409817.	−1.33010	−0.665050	−	0.746799i	$-0.731590\pi$
$37$	−409817.	−1.33010	−0.665050	+	0.746799i	$0.731590\pi$
$38$	64115.0	0.189547
$39$	−166965.	−0.450712
$40$	540585.	1.33553
$41$	675273.	1.53016	0.765078	−	0.643938i	$-0.222700\pi$
$41$	675273.	1.53016	0.765078	+	0.643938i	$0.222700\pi$
$42$	0	0
$43$	38903.5	0.0746189	0.0373094	−	0.999304i	$-0.488121\pi$
$43$	38903.5	0.0746189	0.0373094	+	0.999304i	$0.488121\pi$
$44$	13267.1	0.0234796
$45$	−115749.	−0.189354
$46$	751427.	1.13824
$47$	−949397.	−1.33385	−0.666923	−	0.745127i	$-0.732389\pi$
$47$	−949397.	−1.33385	−0.666923	+	0.745127i	$0.732389\pi$
$48$	763457.	0.996416
$49$	0	0
$50$	−868169.	−0.982221
$51$	1.05049e6	1.10891
$52$	38280.2	0.0377539
$53$	294003.	0.271260	0.135630	−	0.990760i	$-0.456694\pi$
$53$	294003.	0.271260	0.135630	+	0.990760i	$0.456694\pi$
$54$	−1.26842e6	−1.09619
$55$	518983.	0.420614
$56$	0	0
$57$	237311.	0.169729
$58$	−2.09490e6	−1.40983
$59$	87803.7	0.0556584	0.0278292	−	0.999613i	$-0.491141\pi$
$59$	87803.7	0.0556584	0.0278292	+	0.999613i	$0.491141\pi$
$60$	−168974.	−0.100993
$61$	2.78573e6	1.57139	0.785696	−	0.618613i	$-0.212305\pi$
$61$	2.78573e6	1.57139	0.785696	+	0.618613i	$0.212305\pi$
$62$	−2.17452e6	−1.15876
$63$	0	0
$64$	1.90940e6	0.910471
$65$	1.49745e6	0.676325
$66$	679696.	0.291012
$67$	2.95364e6	1.19976	0.599882	−	0.800089i	$-0.295214\pi$
$67$	2.95364e6	1.19976	0.599882	+	0.800089i	$0.295214\pi$
$68$	−240847.	−0.0928883
$69$	2.78129e6	1.01923
$70$	0	0
$71$	−4.08380e6	−1.35413	−0.677065	−	0.735923i	$-0.736749\pi$
$71$	−4.08380e6	−1.35413	−0.677065	+	0.735923i	$0.736749\pi$
$72$	−411560.	−0.129948
$73$	−1.95525e6	−0.588263	−0.294132	−	0.955765i	$-0.595030\pi$
$73$	−1.95525e6	−0.588263	−0.294132	+	0.955765i	$0.595030\pi$
$74$	4.81370e6	1.38092
$75$	−3.21339e6	−0.879526
$76$	−54408.6	−0.0142174
$77$	0	0
$78$	1.96116e6	0.467932
$79$	−608158.	−0.138778	−0.0693891	−	0.997590i	$-0.522105\pi$
$79$	−608158.	−0.138778	−0.0693891	+	0.997590i	$0.522105\pi$
$80$	−6.84718e6	−1.49519
$81$	−4.04562e6	−0.845840
$82$	−7.93173e6	−1.58862
$83$	−214613.	−0.0411986	−0.0205993	−	0.999788i	$-0.506557\pi$
$83$	−214613.	−0.0411986	−0.0205993	+	0.999788i	$0.506557\pi$
$84$	0	0
$85$	−9.42151e6	−1.66400
$86$	−456959.	−0.0774698
$87$	−7.75394e6	−1.26242
$88$	1.84530e6	0.288654
$89$	8.30366e6	1.24855	0.624273	−	0.781206i	$-0.285395\pi$
$89$	8.30366e6	1.24855	0.624273	+	0.781206i	$0.285395\pi$
$90$	1.35959e6	0.196589
$91$	0	0
$92$	−637668.	−0.0853763
$93$	−8.04863e6	−1.03760
$94$	1.11516e7	1.38481
$95$	−2.12836e6	−0.254691
$96$	−1.25235e6	−0.144469
$97$	1.38909e6	0.154536	0.0772678	−	0.997010i	$-0.475380\pi$
$97$	1.38909e6	0.154536	0.0772678	+	0.997010i	$0.475380\pi$
$98$	0	0
$99$	−395114.	−0.0409260
$100$	736736.	0.0736736
$101$	−1.22974e7	−1.18765	−0.593825	−	0.804595i	$-0.702383\pi$
$101$	−1.22974e7	−1.18765	−0.593825	+	0.804595i	$0.702383\pi$
$102$	−1.23391e7	−1.15128
$103$	−2.14543e7	−1.93457	−0.967283	−	0.253701i	$-0.918352\pi$
$103$	−2.14543e7	−1.93457	−0.967283	+	0.253701i	$0.918352\pi$
$104$	5.32436e6	0.464142
$105$	0	0
$106$	−3.45335e6	−0.281624
$107$	4.25610e6	0.335868	0.167934	−	0.985798i	$-0.446290\pi$
$107$	4.25610e6	0.335868	0.167934	+	0.985798i	$0.446290\pi$
$108$	1.07639e6	0.0822218
$109$	−6.75437e6	−0.499565	−0.249783	−	0.968302i	$-0.580359\pi$
$109$	−6.75437e6	−0.499565	−0.249783	+	0.968302i	$0.580359\pi$
$110$	−6.09595e6	−0.436684
$111$	1.78171e7	1.23654
$112$	0	0
$113$	2.26593e6	0.147731	0.0738656	−	0.997268i	$-0.476466\pi$
$113$	2.26593e6	0.147731	0.0738656	+	0.997268i	$0.476466\pi$
$114$	−2.78745e6	−0.176214
$115$	−2.49444e7	−1.52943
$116$	1.77775e6	0.105747
$117$	−1.14004e6	−0.0658069
$118$	−1.03134e6	−0.0577849
$119$	0	0
$120$	−2.35024e7	−1.24159
$121$	1.77156e6	0.0909091
$122$	−3.27211e7	−1.63143
$123$	−2.93580e7	−1.42252
$124$	1.84532e6	0.0869150
$125$	−1.64270e6	−0.0752267
$126$	0	0
$127$	1.10495e7	0.478661	0.239331	−	0.970938i	$-0.423072\pi$
$127$	1.10495e7	0.478661	0.239331	+	0.970938i	$0.423072\pi$
$128$	−2.61148e7	−1.10066
$129$	−1.69136e6	−0.0693700
$130$	−1.75890e7	−0.702165
$131$	1.05834e7	0.411316	0.205658	−	0.978624i	$-0.434067\pi$
$131$	1.05834e7	0.411316	0.205658	+	0.978624i	$0.434067\pi$
$132$	−576796.	−0.0218280
$133$	0	0
$134$	−3.46934e7	−1.24560
$135$	4.21064e7	1.47292
$136$	−3.34993e7	−1.14196
$137$	−3.39909e7	−1.12938	−0.564691	−	0.825302i	$-0.691005\pi$
$137$	−3.39909e7	−1.12938	−0.564691	+	0.825302i	$0.691005\pi$
$138$	−3.26689e7	−1.05818
$139$	−2.60529e6	−0.0822818	−0.0411409	−	0.999153i	$-0.513099\pi$
$139$	−2.60529e6	−0.0822818	−0.0411409	+	0.999153i	$0.513099\pi$
$140$	0	0
$141$	4.12758e7	1.24002
$142$	4.79682e7	1.40587
$143$	5.11159e6	0.146177
$144$	5.21292e6	0.145483
$145$	6.95424e7	1.89436
$146$	2.29663e7	0.610739
$147$	0	0
$148$	−4.08495e6	−0.103579
$149$	9.39464e6	0.232663	0.116332	−	0.993210i	$-0.462886\pi$
$149$	9.39464e6	0.232663	0.116332	+	0.993210i	$0.462886\pi$
$150$	3.77443e7	0.913130
$151$	−2.58693e7	−0.611457	−0.305729	−	0.952119i	$-0.598900\pi$
$151$	−2.58693e7	−0.611457	−0.305729	+	0.952119i	$0.598900\pi$
$152$	−7.56764e6	−0.174787
$153$	7.17282e6	0.161909
$154$	0	0
$155$	7.21854e7	1.55700
$156$	−1.66426e6	−0.0350983
$157$	4.17516e7	0.861043	0.430521	−	0.902580i	$-0.358330\pi$
$157$	4.17516e7	0.861043	0.430521	+	0.902580i	$0.358330\pi$
$158$	7.14340e6	0.144080
$159$	−1.27820e7	−0.252179
$160$	1.12319e7	0.216786
$161$	0	0
$162$	4.75198e7	0.878156
$163$	−2.45070e7	−0.443235	−0.221618	−	0.975134i	$-0.571134\pi$
$163$	−2.45070e7	−0.443235	−0.221618	+	0.975134i	$0.571134\pi$
$164$	6.73094e6	0.119158
$165$	−2.25632e7	−0.391027
$166$	2.52084e6	0.0427727
$167$	6.73433e7	1.11889	0.559444	−	0.828868i	$-0.311015\pi$
$167$	6.73433e7	1.11889	0.559444	+	0.828868i	$0.311015\pi$
$168$	0	0
$169$	−4.79998e7	−0.764955
$170$	1.10665e8	1.72758
$171$	1.62037e6	0.0247816
$172$	387779.	0.00581079
$173$	−1.29430e7	−0.190052	−0.0950259	−	0.995475i	$-0.530293\pi$
$173$	−1.29430e7	−0.190052	−0.0950259	+	0.995475i	$0.530293\pi$
$174$	9.10775e7	1.31066
$175$	0	0
$176$	−2.33730e7	−0.323162
$177$	−3.81734e6	−0.0517433
$178$	−9.75345e7	−1.29625
$179$	6.35496e6	0.0828185	0.0414093	−	0.999142i	$-0.486815\pi$
$179$	6.35496e6	0.0828185	0.0414093	+	0.999142i	$0.486815\pi$
$180$	−1.15376e6	−0.0147456
$181$	3.46385e7	0.434194	0.217097	−	0.976150i	$-0.430341\pi$
$181$	3.46385e7	0.434194	0.217097	+	0.976150i	$0.430341\pi$
$182$	0	0
$183$	−1.21112e8	−1.46086
$184$	−8.86926e7	−1.04960
$185$	−1.59796e8	−1.85551
$186$	9.45389e7	1.07725
$187$	−3.21606e7	−0.359648
$188$	−9.46334e6	−0.103870
$189$	0	0
$190$	2.49997e7	0.264422
$191$	8.18197e7	0.849652	0.424826	−	0.905275i	$-0.360335\pi$
$191$	8.18197e7	0.849652	0.424826	+	0.905275i	$0.360335\pi$
$192$	−8.30125e7	−0.846427
$193$	−6.82998e7	−0.683863	−0.341931	−	0.939725i	$-0.611081\pi$
$193$	−6.82998e7	−0.683863	−0.341931	+	0.939725i	$0.611081\pi$
$194$	−1.63162e7	−0.160440
$195$	−6.51029e7	−0.628751
$196$	0	0
$197$	4.70586e7	0.438538	0.219269	−	0.975664i	$-0.429633\pi$
$197$	4.70586e7	0.438538	0.219269	+	0.975664i	$0.429633\pi$
$198$	4.64099e6	0.0424896
$199$	9.87712e7	0.888474	0.444237	−	0.895909i	$-0.353475\pi$
$199$	9.87712e7	0.888474	0.444237	+	0.895909i	$0.353475\pi$
$200$	1.02472e8	0.905733
$201$	−1.28412e8	−1.11537
$202$	1.44445e8	1.23303
$203$	0	0
$204$	1.04710e7	0.0863543
$205$	2.63302e8	2.13459
$206$	2.52001e8	2.00848
$207$	1.89908e7	0.148815
$208$	−6.74396e7	−0.519629
$209$	−7.26523e6	−0.0550474
$210$	0	0
$211$	−1.37863e8	−1.01032	−0.505160	−	0.863026i	$-0.668566\pi$
$211$	−1.37863e8	−1.01032	−0.505160	+	0.863026i	$0.668566\pi$
$212$	2.93054e6	0.0211238
$213$	1.77547e8	1.25888
$214$	−4.99920e7	−0.348700
$215$	1.51692e7	0.104095
$216$	1.49714e8	1.01082
$217$	0	0
$218$	7.93366e7	0.518652
$219$	8.50060e7	0.546884
$220$	5.17308e6	0.0327544
$221$	−9.27947e7	−0.578296
$222$	−2.09279e8	−1.28378
$223$	−2.67373e8	−1.61454	−0.807272	−	0.590180i	$-0.799057\pi$
$223$	−2.67373e8	−1.61454	−0.807272	+	0.590180i	$0.799057\pi$
$224$	0	0
$225$	−2.19412e7	−0.128416
$226$	−2.66155e7	−0.153375
$227$	5.68774e7	0.322737	0.161369	−	0.986894i	$-0.448409\pi$
$227$	5.68774e7	0.322737	0.161369	+	0.986894i	$0.448409\pi$
$228$	2.36546e6	0.0132173
$229$	3.00661e8	1.65445	0.827224	−	0.561872i	$-0.189919\pi$
$229$	3.00661e8	1.65445	0.827224	+	0.561872i	$0.189919\pi$
$230$	2.92996e8	1.58787
$231$	0	0
$232$	2.47266e8	1.30004
$233$	3.70266e8	1.91765	0.958823	−	0.284006i	$-0.0916635\pi$
$233$	3.70266e8	1.91765	0.958823	+	0.284006i	$0.0916635\pi$
$234$	1.33909e7	0.0683211
$235$	−3.70188e8	−1.86074
$236$	875204.	0.00433428
$237$	2.64401e7	0.129016
$238$	0	0
$239$	3.00432e8	1.42349	0.711743	−	0.702440i	$-0.247906\pi$
$239$	3.00432e8	1.42349	0.711743	+	0.702440i	$0.247906\pi$
$240$	2.97687e8	1.39002
$241$	−2.75534e7	−0.126799	−0.0633994	−	0.997988i	$-0.520194\pi$
$241$	−2.75534e7	−0.126799	−0.0633994	+	0.997988i	$0.520194\pi$
$242$	−2.08087e7	−0.0943824
$243$	−6.02821e7	−0.269505
$244$	2.77674e7	0.122369
$245$	0	0
$246$	3.44838e8	1.47687
$247$	−2.09628e7	−0.0885135
$248$	2.56663e8	1.06852
$249$	9.33047e6	0.0383006
$250$	1.92951e7	0.0781009
$251$	1.17084e8	0.467347	0.233673	−	0.972315i	$-0.424925\pi$
$251$	1.17084e8	0.467347	0.233673	+	0.972315i	$0.424925\pi$
$252$	0	0
$253$	−8.51483e7	−0.330563
$254$	−1.29787e8	−0.496949
$255$	4.09608e8	1.54695
$256$	6.23411e7	0.232239
$257$	1.05899e8	0.389157	0.194579	−	0.980887i	$-0.437666\pi$
$257$	1.05899e8	0.389157	0.194579	+	0.980887i	$0.437666\pi$
$258$	1.98666e7	0.0720204
$259$	0	0
$260$	1.49262e7	0.0526674
$261$	−5.29443e7	−0.184322
$262$	−1.24312e8	−0.427031
$263$	2.69446e8	0.913328	0.456664	−	0.889639i	$-0.349044\pi$
$263$	2.69446e8	0.913328	0.456664	+	0.889639i	$0.349044\pi$
$264$	−8.02260e7	−0.268350
$265$	1.14637e8	0.378413
$266$	0	0
$267$	−3.61008e8	−1.16072
$268$	2.94411e7	0.0934291
$269$	6.89849e7	0.216083	0.108042	−	0.994146i	$-0.465542\pi$
$269$	6.89849e7	0.216083	0.108042	+	0.994146i	$0.465542\pi$
$270$	−4.94581e8	−1.52920
$271$	−6.86311e7	−0.209473	−0.104737	−	0.994500i	$-0.533400\pi$
$271$	−6.86311e7	−0.209473	−0.104737	+	0.994500i	$0.533400\pi$
$272$	4.24310e8	1.27847
$273$	0	0
$274$	3.99256e8	1.17253
$275$	9.83770e7	0.285252
$276$	2.77231e7	0.0793707
$277$	−6.22102e8	−1.75866	−0.879330	−	0.476212i	$-0.842009\pi$
$277$	−6.22102e8	−1.75866	−0.879330	+	0.476212i	$0.842009\pi$
$278$	3.06016e7	0.0854255
$279$	−5.49564e7	−0.151497
$280$	0	0
$281$	−8.07383e7	−0.217074	−0.108537	−	0.994092i	$-0.534617\pi$
$281$	−8.07383e7	−0.217074	−0.108537	+	0.994092i	$0.534617\pi$
$282$	−4.84824e8	−1.28740
$283$	3.19850e8	0.838868	0.419434	−	0.907786i	$-0.362229\pi$
$283$	3.19850e8	0.838868	0.419434	+	0.907786i	$0.362229\pi$
$284$	−4.07063e7	−0.105450
$285$	9.25323e7	0.236775
$286$	−6.00405e7	−0.151762
$287$	0	0
$288$	−8.55109e6	−0.0210935
$289$	1.73498e8	0.422817
$290$	−8.16843e8	−1.96673
$291$	−6.03917e7	−0.143665
$292$	−1.94894e7	−0.0458098
$293$	4.26992e8	0.991707	0.495854	−	0.868406i	$-0.334855\pi$
$293$	4.26992e8	0.991707	0.495854	+	0.868406i	$0.334855\pi$
$294$	0	0
$295$	3.42364e7	0.0776445
$296$	−5.68172e8	−1.27338
$297$	1.43731e8	0.318350
$298$	−1.10349e8	−0.241553
$299$	−2.45683e8	−0.531529
$300$	−3.20302e7	−0.0684912
$301$	0	0
$302$	3.03860e8	0.634819
$303$	5.34639e8	1.10411
$304$	9.58536e7	0.195682
$305$	1.08621e9	2.19212
$306$	−8.42517e7	−0.168095
$307$	−1.63077e8	−0.321669	−0.160834	−	0.986981i	$-0.551418\pi$
$307$	−1.63077e8	−0.321669	−0.160834	+	0.986981i	$0.551418\pi$
$308$	0	0
$309$	9.32741e8	1.79848
$310$	−8.47887e8	−1.61649
$311$	−1.22234e8	−0.230426	−0.115213	−	0.993341i	$-0.536755\pi$
$311$	−1.22234e8	−0.230426	−0.115213	+	0.993341i	$0.536755\pi$
$312$	−2.31481e8	−0.431493
$313$	−7.58549e7	−0.139823	−0.0699115	−	0.997553i	$-0.522272\pi$
$313$	−7.58549e7	−0.139823	−0.0699115	+	0.997553i	$0.522272\pi$
$314$	−4.90413e8	−0.893940
$315$	0	0
$316$	−6.06195e6	−0.0108071
$317$	−9.48844e8	−1.67297	−0.836483	−	0.547992i	$-0.815392\pi$
$317$	−9.48844e8	−1.67297	−0.836483	+	0.547992i	$0.815392\pi$
$318$	1.50137e8	0.261814
$319$	2.37385e8	0.409436
$320$	7.44510e8	1.27012
$321$	−1.85037e8	−0.312242
$322$	0	0
$323$	1.31892e8	0.217775
$324$	−4.03257e7	−0.0658680
$325$	2.83853e8	0.458671
$326$	2.87859e8	0.460170
$327$	2.93652e8	0.464425
$328$	9.36200e8	1.46491
$329$	0	0
$330$	2.65026e8	0.405967
$331$	−5.20078e8	−0.788262	−0.394131	−	0.919054i	$-0.628954\pi$
$331$	−5.20078e8	−0.788262	−0.394131	+	0.919054i	$0.628954\pi$
$332$	−2.13920e6	−0.00320826
$333$	1.21656e8	0.180542
$334$	−7.91013e8	−1.16164
$335$	1.15168e9	1.67369
$336$	0	0
$337$	6.53022e8	0.929443	0.464722	−	0.885457i	$-0.346154\pi$
$337$	6.53022e8	0.929443	0.464722	+	0.885457i	$0.346154\pi$
$338$	5.63804e8	0.794181
$339$	−9.85131e7	−0.137339
$340$	−9.39111e7	−0.129581
$341$	2.46407e8	0.336521
$342$	−1.90329e7	−0.0257284
$343$	0	0
$344$	5.39359e7	0.0714370
$345$	1.08448e9	1.42185
$346$	1.52028e8	0.197313
$347$	4.63419e8	0.595416	0.297708	−	0.954657i	$-0.403778\pi$
$347$	4.63419e8	0.595416	0.297708	+	0.954657i	$0.403778\pi$
$348$	−7.72892e7	−0.0983086
$349$	−2.76638e8	−0.348355	−0.174178	−	0.984714i	$-0.555727\pi$
$349$	−2.76638e8	−0.348355	−0.174178	+	0.984714i	$0.555727\pi$
$350$	0	0
$351$	4.14717e8	0.511890
$352$	3.83403e7	0.0468550
$353$	5.47428e8	0.662393	0.331197	−	0.943562i	$-0.392548\pi$
$353$	5.47428e8	0.662393	0.331197	+	0.943562i	$0.392548\pi$
$354$	4.48383e7	0.0537202
$355$	−1.59235e9	−1.88904
$356$	8.27686e7	0.0972279
$357$	0	0
$358$	−7.46452e7	−0.0859827
$359$	−4.51664e8	−0.515211	−0.257605	−	0.966250i	$-0.582933\pi$
$359$	−4.51664e8	−0.515211	−0.257605	+	0.966250i	$0.582933\pi$
$360$	−1.60475e8	−0.181280
$361$	−8.64077e8	−0.966668
$362$	−4.06863e8	−0.450783
$363$	−7.70200e7	−0.0845144
$364$	0	0
$365$	−7.62389e8	−0.820638
$366$	1.42258e9	1.51667
$367$	8.85429e8	0.935024	0.467512	−	0.883987i	$-0.345151\pi$
$367$	8.85429e8	0.935024	0.467512	+	0.883987i	$0.345151\pi$
$368$	1.12340e9	1.17508
$369$	−2.00458e8	−0.207697
$370$	1.87695e9	1.92640
$371$	0	0
$372$	−8.02266e7	−0.0808013
$373$	8.02566e7	0.0800755	0.0400377	−	0.999198i	$-0.487252\pi$
$373$	8.02566e7	0.0800755	0.0400377	+	0.999198i	$0.487252\pi$
$374$	3.77757e8	0.373389
$375$	7.14175e7	0.0699351
$376$	−1.31625e9	−1.27697
$377$	6.84940e8	0.658352
$378$	0	0
$379$	1.14669e9	1.08196	0.540978	−	0.841036i	$-0.318054\pi$
$379$	1.14669e9	1.08196	0.540978	+	0.841036i	$0.318054\pi$
$380$	−2.12150e7	−0.0198335
$381$	−4.80384e8	−0.444991
$382$	−9.61051e8	−0.882114
$383$	−5.01359e8	−0.455987	−0.227994	−	0.973663i	$-0.573217\pi$
$383$	−5.01359e8	−0.455987	−0.227994	+	0.973663i	$0.573217\pi$
$384$	1.13536e9	1.02323
$385$	0	0
$386$	8.02247e8	0.709991
$387$	−1.15487e7	−0.0101285
$388$	1.38461e7	0.0120341
$389$	6.92770e8	0.596713	0.298357	−	0.954454i	$-0.403562\pi$
$389$	6.92770e8	0.596713	0.298357	+	0.954454i	$0.403562\pi$
$390$	7.64696e8	0.652774
$391$	1.54577e9	1.30775
$392$	0	0
$393$	−4.60121e8	−0.382383
$394$	−5.52748e8	−0.455293
$395$	−2.37132e8	−0.193598
$396$	−3.93839e6	−0.00318702
$397$	1.51646e9	1.21637	0.608184	−	0.793796i	$-0.291898\pi$
$397$	1.51646e9	1.21637	0.608184	+	0.793796i	$0.291898\pi$
$398$	−1.16016e9	−0.922419
$399$	0	0
$400$	−1.29793e9	−1.01401
$401$	1.51899e9	1.17639	0.588193	−	0.808721i	$-0.299840\pi$
$401$	1.51899e9	1.17639	0.588193	+	0.808721i	$0.299840\pi$
$402$	1.50832e9	1.15798
$403$	7.10971e8	0.541109
$404$	−1.22577e8	−0.0924857
$405$	−1.57747e9	−1.17996
$406$	0	0
$407$	−5.45467e8	−0.401040
$408$	1.45641e9	1.06163
$409$	6.06955e7	0.0438657	0.0219329	−	0.999759i	$-0.493018\pi$
$409$	6.06955e7	0.0438657	0.0219329	+	0.999759i	$0.493018\pi$
$410$	−3.09274e9	−2.21615
$411$	1.47778e9	1.04994
$412$	−2.13850e8	−0.150650
$413$	0	0
$414$	−2.23065e8	−0.154500
$415$	−8.36817e7	−0.0574728
$416$	1.10625e8	0.0753405
$417$	1.13267e8	0.0764939
$418$	8.53371e7	0.0571506
$419$	4.38581e8	0.291274	0.145637	−	0.989338i	$-0.453477\pi$
$419$	4.38581e8	0.291274	0.145637	+	0.989338i	$0.453477\pi$
$420$	0	0
$421$	−1.88412e9	−1.23061	−0.615305	−	0.788289i	$-0.710967\pi$
$421$	−1.88412e9	−1.23061	−0.615305	+	0.788289i	$0.710967\pi$
$422$	1.61933e9	1.04892
$423$	2.81833e8	0.181051
$424$	4.07606e8	0.259693
$425$	−1.78592e9	−1.12850
$426$	−2.08546e9	−1.30698
$427$	0	0
$428$	4.24237e7	0.0261550
$429$	−2.22230e8	−0.135895
$430$	−1.78177e8	−0.108072
$431$	7.36561e8	0.443137	0.221569	−	0.975145i	$-0.428882\pi$
$431$	7.36561e8	0.443137	0.221569	+	0.975145i	$0.428882\pi$
$432$	−1.89632e9	−1.13167
$433$	−7.99028e8	−0.472992	−0.236496	−	0.971632i	$-0.575999\pi$
$433$	−7.99028e8	−0.472992	−0.236496	+	0.971632i	$0.575999\pi$
$434$	0	0
$435$	−3.02341e9	−1.76110
$436$	−6.73258e7	−0.0389026
$437$	3.49196e8	0.200163
$438$	−9.98477e8	−0.567778
$439$	−9.89554e8	−0.558230	−0.279115	−	0.960258i	$-0.590041\pi$
$439$	−9.89554e8	−0.558230	−0.279115	+	0.960258i	$0.590041\pi$
$440$	7.19519e8	0.402678
$441$	0	0
$442$	1.08996e9	0.600391
$443$	−7.19768e8	−0.393350	−0.196675	−	0.980469i	$-0.563014\pi$
$443$	−7.19768e8	−0.393350	−0.196675	+	0.980469i	$0.563014\pi$
$444$	1.77596e8	0.0962928
$445$	3.23776e9	1.74174
$446$	3.14055e9	1.67623
$447$	−4.08439e8	−0.216297
$448$	0	0
$449$	1.35412e8	0.0705987	0.0352993	−	0.999377i	$-0.488762\pi$
$449$	1.35412e8	0.0705987	0.0352993	+	0.999377i	$0.488762\pi$
$450$	2.57720e8	0.133323
$451$	8.98788e8	0.461359
$452$	2.25862e7	0.0115043
$453$	1.12469e9	0.568446
$454$	−6.68080e8	−0.335068
$455$	0	0
$456$	3.29009e8	0.162492
$457$	3.03772e9	1.48882	0.744409	−	0.667724i	$-0.232731\pi$
$457$	3.03772e9	1.48882	0.744409	+	0.667724i	$0.232731\pi$
$458$	−3.53156e9	−1.71766
$459$	−2.60927e9	−1.25943
$460$	−2.48639e8	−0.119101
$461$	1.35818e9	0.645660	0.322830	−	0.946457i	$-0.395366\pi$
$461$	1.35818e9	0.645660	0.322830	+	0.946457i	$0.395366\pi$
$462$	0	0
$463$	−1.31208e9	−0.614366	−0.307183	−	0.951650i	$-0.599386\pi$
$463$	−1.31208e9	−0.614366	−0.307183	+	0.951650i	$0.599386\pi$
$464$	−3.13193e9	−1.45546
$465$	−3.13832e9	−1.44748
$466$	−4.34913e9	−1.99091
$467$	−1.37898e8	−0.0626542	−0.0313271	−	0.999509i	$-0.509973\pi$
$467$	−1.37898e8	−0.0626542	−0.0313271	+	0.999509i	$0.509973\pi$
$468$	−1.13637e7	−0.00512457
$469$	0	0
$470$	4.34822e9	1.93183
$471$	−1.81519e9	−0.800475
$472$	1.21731e8	0.0532851
$473$	5.17805e7	0.0224984
$474$	−3.10565e8	−0.133946
$475$	−4.03447e8	−0.172726
$476$	0	0
$477$	−8.72761e7	−0.0368198
$478$	−3.52886e9	−1.47787
$479$	−2.23257e9	−0.928178	−0.464089	−	0.885788i	$-0.653618\pi$
$479$	−2.23257e9	−0.928178	−0.464089	+	0.885788i	$0.653618\pi$
$480$	−4.88314e8	−0.201537
$481$	−1.57387e9	−0.644852
$482$	3.23641e8	0.131643
$483$	0	0
$484$	1.76584e7	0.00707936
$485$	5.41632e8	0.215580
$486$	7.08071e8	0.279801
$487$	3.51454e9	1.37885	0.689425	−	0.724357i	$-0.257863\pi$
$487$	3.51454e9	1.37885	0.689425	+	0.724357i	$0.257863\pi$
$488$	3.86214e9	1.50439
$489$	1.06546e9	0.412057
$490$	0	0
$491$	−1.57552e9	−0.600672	−0.300336	−	0.953833i	$-0.597099\pi$
$491$	−1.57552e9	−0.600672	−0.300336	+	0.953833i	$0.597099\pi$
$492$	−2.92633e8	−0.110776
$493$	−4.30944e9	−1.61978
$494$	2.46228e8	0.0918953
$495$	−1.54062e8	−0.0570925
$496$	−3.25096e9	−1.19626
$497$	0	0
$498$	−1.09595e8	−0.0397639
$499$	4.77854e9	1.72164	0.860822	−	0.508907i	$-0.169950\pi$
$499$	4.77854e9	1.72164	0.860822	+	0.508907i	$0.169950\pi$
$500$	−1.63740e7	−0.00585813
$501$	−2.92781e9	−1.04018
$502$	−1.37526e9	−0.485202
$503$	−1.99980e8	−0.0700646	−0.0350323	−	0.999386i	$-0.511153\pi$
$503$	−1.99980e8	−0.0700646	−0.0350323	+	0.999386i	$0.511153\pi$
$504$	0	0
$505$	−4.79499e9	−1.65679
$506$	1.00015e9	0.343193
$507$	2.08683e9	0.711146
$508$	1.10138e8	0.0372747
$509$	5.40122e9	1.81543	0.907715	−	0.419587i	$-0.137825\pi$
$509$	5.40122e9	1.81543	0.907715	+	0.419587i	$0.137825\pi$
$510$	−4.81124e9	−1.60606
$511$	0	0
$512$	2.61044e9	0.859546
$513$	−5.89447e8	−0.192768
$514$	−1.24388e9	−0.404026
$515$	−8.36543e9	−2.69875
$516$	−1.68590e7	−0.00540205
$517$	−1.26365e9	−0.402170
$518$	0	0
$519$	5.62705e8	0.176683
$520$	2.07607e9	0.647486
$521$	−4.76502e9	−1.47616	−0.738079	−	0.674715i	$-0.764267\pi$
$521$	−4.76502e9	−1.47616	−0.738079	+	0.674715i	$0.764267\pi$
$522$	6.21882e8	0.191364
$523$	4.61783e9	1.41151	0.705753	−	0.708458i	$-0.250609\pi$
$523$	4.61783e9	1.41151	0.705753	+	0.708458i	$0.250609\pi$
$524$	1.05492e8	0.0320304
$525$	0	0
$526$	−3.16490e9	−0.948223
$527$	−4.47322e9	−1.33132
$528$	1.01616e9	0.300431
$529$	6.87745e8	0.201991
$530$	−1.34653e9	−0.392870
$531$	−2.60650e7	−0.00755485
$532$	0	0
$533$	2.59332e9	0.741842
$534$	4.24039e9	1.20507
$535$	1.65954e9	0.468542
$536$	4.09494e9	1.14860
$537$	−2.76287e8	−0.0769929
$538$	−8.10294e8	−0.224339
$539$	0	0
$540$	4.19706e8	0.114701
$541$	−4.03282e9	−1.09501	−0.547506	−	0.836802i	$-0.684423\pi$
$541$	−4.03282e9	−1.09501	−0.547506	+	0.836802i	$0.684423\pi$
$542$	8.06139e8	0.217476
$543$	−1.50594e9	−0.403652
$544$	−6.96022e8	−0.185365
$545$	−2.63366e9	−0.696902
$546$	0	0
$547$	−3.23762e9	−0.845804	−0.422902	−	0.906175i	$-0.638989\pi$
$547$	−3.23762e9	−0.845804	−0.422902	+	0.906175i	$0.638989\pi$
$548$	−3.38813e8	−0.0879483
$549$	−8.26957e8	−0.213295
$550$	−1.15553e9	−0.296151
$551$	−9.73523e8	−0.247922
$552$	3.85598e9	0.975772
$553$	0	0
$554$	7.30718e9	1.82585
$555$	6.94724e9	1.72499
$556$	−2.59688e7	−0.00640752
$557$	6.03647e9	1.48009	0.740047	−	0.672555i	$-0.234803\pi$
$557$	6.03647e9	1.48009	0.740047	+	0.672555i	$0.234803\pi$
$558$	6.45516e8	0.157285
$559$	1.49405e8	0.0361763
$560$	0	0
$561$	1.39821e9	0.334350
$562$	9.48349e8	0.225367
$563$	−7.03032e9	−1.66033	−0.830167	−	0.557515i	$-0.811755\pi$
$563$	−7.03032e9	−1.66033	−0.830167	+	0.557515i	$0.811755\pi$
$564$	4.11426e8	0.0965640
$565$	8.83530e8	0.206088
$566$	−3.75695e9	−0.870918
$567$	0	0
$568$	−5.66180e9	−1.29639
$569$	−4.25341e9	−0.967930	−0.483965	−	0.875087i	$-0.660804\pi$
$569$	−4.25341e9	−0.967930	−0.483965	+	0.875087i	$0.660804\pi$
$570$	−1.08688e9	−0.245822
$571$	3.26097e9	0.733029	0.366514	−	0.930412i	$-0.380551\pi$
$571$	3.26097e9	0.733029	0.366514	+	0.930412i	$0.380551\pi$
$572$	5.09509e7	0.0113832
$573$	−3.55718e9	−0.789886
$574$	0	0
$575$	−4.72839e9	−1.03723
$576$	−5.66813e8	−0.123584
$577$	7.44043e9	1.61244	0.806219	−	0.591617i	$-0.201510\pi$
$577$	7.44043e9	1.61244	0.806219	+	0.591617i	$0.201510\pi$
$578$	−2.03790e9	−0.438971
$579$	2.96939e9	0.635758
$580$	6.93180e8	0.147519
$581$	0	0
$582$	7.09359e8	0.149154
$583$	3.91318e8	0.0817880
$584$	−2.71076e9	−0.563179
$585$	−4.44525e8	−0.0918018
$586$	−5.01544e9	−1.02960
$587$	−4.66637e9	−0.952238	−0.476119	−	0.879381i	$-0.657957\pi$
$587$	−4.66637e9	−0.952238	−0.476119	+	0.879381i	$0.657957\pi$
$588$	0	0
$589$	−1.01052e9	−0.203771
$590$	−4.02139e8	−0.0806110
$591$	−2.04591e9	−0.407690
$592$	7.19660e9	1.42561
$593$	−5.22317e9	−1.02859	−0.514296	−	0.857613i	$-0.671947\pi$
$593$	−5.22317e9	−1.02859	−0.514296	+	0.857613i	$0.671947\pi$
$594$	−1.68826e9	−0.330513
$595$	0	0
$596$	9.36433e7	0.0181182
$597$	−4.29416e9	−0.825977
$598$	2.88579e9	0.551836
$599$	3.32573e9	0.632257	0.316128	−	0.948716i	$-0.397617\pi$
$599$	3.32573e9	0.632257	0.316128	+	0.948716i	$0.397617\pi$
$600$	−4.45505e9	−0.842022
$601$	9.44629e9	1.77501	0.887504	−	0.460799i	$-0.152437\pi$
$601$	9.44629e9	1.77501	0.887504	+	0.460799i	$0.152437\pi$
$602$	0	0
$603$	−8.76802e8	−0.162851
$604$	−2.57859e8	−0.0476160
$605$	6.90766e8	0.126820
$606$	−6.27985e9	−1.14629
$607$	4.62671e9	0.839676	0.419838	−	0.907599i	$-0.362087\pi$
$607$	4.62671e9	0.839676	0.419838	+	0.907599i	$0.362087\pi$
$608$	−1.57235e8	−0.0283717
$609$	0	0
$610$	−1.27586e10	−2.27587
$611$	−3.64608e9	−0.646668
$612$	7.14967e7	0.0126083
$613$	8.77518e9	1.53867	0.769333	−	0.638848i	$-0.220589\pi$
$613$	8.77518e9	1.53867	0.769333	+	0.638848i	$0.220589\pi$
$614$	1.91550e9	0.333959
$615$	−1.14473e10	−1.98444
$616$	0	0
$617$	−3.17693e9	−0.544514	−0.272257	−	0.962225i	$-0.587770\pi$
$617$	−3.17693e9	−0.544514	−0.272257	+	0.962225i	$0.587770\pi$
$618$	−1.09559e10	−1.86720
$619$	5.77076e9	0.977948	0.488974	−	0.872298i	$-0.337371\pi$
$619$	5.77076e9	0.977948	0.488974	+	0.872298i	$0.337371\pi$
$620$	7.19525e8	0.121248
$621$	−6.90831e9	−1.15758
$622$	1.43576e9	0.239230
$623$	0	0
$624$	2.93199e9	0.483077
$625$	−6.41490e9	−1.05102
$626$	8.90989e8	0.145165
$627$	3.15862e8	0.0511753
$628$	4.16169e8	0.0670519
$629$	9.90229e9	1.58657
$630$	0	0
$631$	9.29360e9	1.47259	0.736294	−	0.676662i	$-0.236574\pi$
$631$	9.29360e9	1.47259	0.736294	+	0.676662i	$0.236574\pi$
$632$	−8.43152e8	−0.132861
$633$	5.99370e9	0.939251
$634$	1.11451e10	1.73689
$635$	4.30840e9	0.667741
$636$	−1.27408e8	−0.0196379
$637$	0	0
$638$	−2.78831e9	−0.425079
$639$	1.21230e9	0.183804
$640$	−1.01827e10	−1.53544
$641$	6.12597e9	0.918696	0.459348	−	0.888256i	$-0.348083\pi$
$641$	6.12597e9	0.918696	0.459348	+	0.888256i	$0.348083\pi$
$642$	2.17344e9	0.324172
$643$	−7.62998e8	−0.113184	−0.0565920	−	0.998397i	$-0.518023\pi$
$643$	−7.62998e8	−0.113184	−0.0565920	+	0.998397i	$0.518023\pi$
$644$	0	0
$645$	−6.59494e8	−0.0967724
$646$	−1.54919e9	−0.226095
$647$	1.50623e9	0.218638	0.109319	−	0.994007i	$-0.465133\pi$
$647$	1.50623e9	0.218638	0.109319	+	0.994007i	$0.465133\pi$
$648$	−5.60887e9	−0.809772
$649$	1.16867e8	0.0167816
$650$	−3.33413e9	−0.476195
$651$	0	0
$652$	−2.44280e8	−0.0345160
$653$	−3.12302e9	−0.438913	−0.219456	−	0.975622i	$-0.570428\pi$
$653$	−3.12302e9	−0.438913	−0.219456	+	0.975622i	$0.570428\pi$
$654$	−3.44922e9	−0.482169
$655$	4.12667e9	0.573793
$656$	−1.18581e10	−1.64003
$657$	5.80425e8	0.0798486
$658$	0	0
$659$	−1.41329e10	−1.92368	−0.961840	−	0.273613i	$-0.911781\pi$
$659$	−1.41329e10	−1.92368	−0.961840	+	0.273613i	$0.911781\pi$
$660$	−2.24904e8	−0.0304504
$661$	5.09870e9	0.686680	0.343340	−	0.939211i	$-0.388442\pi$
$661$	5.09870e9	0.686680	0.343340	+	0.939211i	$0.388442\pi$
$662$	6.10882e9	0.818379
$663$	4.03433e9	0.537618
$664$	−2.97540e8	−0.0394418
$665$	0	0
$666$	−1.42897e9	−0.187440
$667$	−1.14097e10	−1.48879
$668$	6.71260e8	0.0871312
$669$	1.16242e10	1.50097
$670$	−1.35276e10	−1.73764
$671$	3.70781e9	0.473793
$672$	0	0
$673$	1.45170e10	1.83580	0.917901	−	0.396811i	$-0.129883\pi$
$673$	1.45170e10	1.83580	0.917901	+	0.396811i	$0.129883\pi$
$674$	−7.67037e9	−0.964954
$675$	7.98159e9	0.998909
$676$	−4.78449e8	−0.0595692
$677$	7.52045e9	0.931501	0.465751	−	0.884916i	$-0.345784\pi$
$677$	7.52045e9	0.931501	0.465751	+	0.884916i	$0.345784\pi$
$678$	1.15713e9	0.142587
$679$	0	0
$680$	−1.30620e10	−1.59305
$681$	−2.47279e9	−0.300035
$682$	−2.89428e9	−0.349378
$683$	−2.67143e9	−0.320828	−0.160414	−	0.987050i	$-0.551283\pi$
$683$	−2.67143e9	−0.320828	−0.160414	+	0.987050i	$0.551283\pi$
$684$	1.61515e7	0.00192981
$685$	−1.32537e10	−1.57551
$686$	0	0
$687$	−1.30715e10	−1.53807
$688$	−6.83165e8	−0.0799772
$689$	1.12909e9	0.131511
$690$	−1.27382e10	−1.47617
$691$	8.80524e8	0.101524	0.0507619	−	0.998711i	$-0.483835\pi$
$691$	8.80524e8	0.101524	0.0507619	+	0.998711i	$0.483835\pi$
$692$	−1.29012e8	−0.0147999
$693$	0	0
$694$	−5.44330e9	−0.618165
$695$	−1.01585e9	−0.114785
$696$	−1.07501e10	−1.20859
$697$	−1.63164e10	−1.82520
$698$	3.24938e9	0.361665
$699$	−1.60976e10	−1.78275
$700$	0	0
$701$	1.16510e10	1.27747	0.638733	−	0.769429i	$-0.279459\pi$
$701$	1.16510e10	1.27747	0.638733	+	0.769429i	$0.279459\pi$
$702$	−4.87125e9	−0.531447
$703$	2.23698e9	0.242839
$704$	2.54141e9	0.274517
$705$	1.60942e10	1.72985
$706$	−6.43007e9	−0.687701
$707$	0	0
$708$	−3.80502e7	−0.00402940
$709$	1.07976e10	1.13779	0.568897	−	0.822409i	$-0.307370\pi$
$709$	1.07976e10	1.13779	0.568897	+	0.822409i	$0.307370\pi$
$710$	1.87037e10	1.96121
$711$	1.80535e8	0.0188372
$712$	1.15122e10	1.19531
$713$	−1.18433e10	−1.22366
$714$	0	0
$715$	1.99311e9	0.203920
$716$	6.33446e7	0.00644932
$717$	−1.30615e10	−1.32335
$718$	5.30523e9	0.534895
$719$	2.39680e9	0.240481	0.120241	−	0.992745i	$-0.461633\pi$
$719$	2.39680e9	0.240481	0.120241	+	0.992745i	$0.461633\pi$
$720$	2.03262e9	0.202952
$721$	0	0
$722$	1.01494e10	1.00360
$723$	1.19791e9	0.117880
$724$	3.45267e8	0.0338120
$725$	1.31823e10	1.28472
$726$	9.04675e8	0.0877434
$727$	−7.10027e9	−0.685337	−0.342669	−	0.939456i	$-0.611331\pi$
$727$	−7.10027e9	−0.685337	−0.342669	+	0.939456i	$0.611331\pi$
$728$	0	0
$729$	1.14686e10	1.09639
$730$	8.95500e9	0.851992
$731$	−9.40013e8	−0.0890068
$732$	−1.20721e9	−0.113761
$733$	1.02464e10	0.960964	0.480482	−	0.877005i	$-0.340462\pi$
$733$	1.02464e10	0.960964	0.480482	+	0.877005i	$0.340462\pi$
$734$	−1.04002e10	−0.970748
$735$	0	0
$736$	−1.84279e9	−0.170374
$737$	3.93129e9	0.361742
$738$	2.35457e9	0.215633
$739$	1.01080e10	0.921318	0.460659	−	0.887577i	$-0.347613\pi$
$739$	1.01080e10	0.921318	0.460659	+	0.887577i	$0.347613\pi$
$740$	−1.59280e9	−0.144494
$741$	9.11374e8	0.0822873
$742$	0	0
$743$	1.79087e10	1.60178	0.800892	−	0.598809i	$-0.204359\pi$
$743$	1.79087e10	1.60178	0.800892	+	0.598809i	$0.204359\pi$
$744$	−1.11586e10	−0.993359
$745$	3.66315e9	0.324570
$746$	−9.42691e8	−0.0831349
$747$	6.37089e7	0.00559214
$748$	−3.20568e8	−0.0280069
$749$	0	0
$750$	−8.38868e8	−0.0726071
$751$	1.14603e10	0.987313	0.493657	−	0.869657i	$-0.335660\pi$
$751$	1.14603e10	0.987313	0.493657	+	0.869657i	$0.335660\pi$
$752$	1.66719e10	1.42963
$753$	−5.09032e9	−0.434473
$754$	−8.04528e9	−0.683505
$755$	−1.00870e10	−0.852994
$756$	0	0
$757$	1.69859e10	1.42315	0.711577	−	0.702608i	$-0.247981\pi$
$757$	1.69859e10	1.42315	0.711577	+	0.702608i	$0.247981\pi$
$758$	−1.34690e10	−1.12329
$759$	3.70189e9	0.307311
$760$	−2.95077e9	−0.243830
$761$	1.67046e10	1.37401	0.687004	−	0.726654i	$-0.258926\pi$
$761$	1.67046e10	1.37401	0.687004	+	0.726654i	$0.258926\pi$
$762$	5.64258e9	0.461993
$763$	0	0
$764$	8.15557e8	0.0661649
$765$	2.79682e9	0.225865
$766$	5.88894e9	0.473409
$767$	3.37202e8	0.0269840
$768$	−2.71033e9	−0.215903
$769$	1.27067e10	1.00761	0.503803	−	0.863818i	$-0.331934\pi$
$769$	1.27067e10	1.00761	0.503803	+	0.863818i	$0.331934\pi$
$770$	0	0
$771$	−4.60404e9	−0.361783
$772$	−6.80794e8	−0.0532544
$773$	8.89505e9	0.692660	0.346330	−	0.938113i	$-0.387428\pi$
$773$	8.89505e9	0.692660	0.346330	+	0.938113i	$0.387428\pi$
$774$	1.35650e8	0.0105154
$775$	1.36833e10	1.05593
$776$	1.92584e9	0.147946
$777$	0	0
$778$	−8.13726e9	−0.619512
$779$	−3.68596e9	−0.279363
$780$	−6.48928e8	−0.0489627
$781$	−5.43554e9	−0.408286
$782$	−1.81565e10	−1.35772
$783$	1.92597e10	1.43378
$784$	0	0
$785$	1.62798e10	1.20117
$786$	5.40457e9	0.396992
$787$	−6.24086e9	−0.456386	−0.228193	−	0.973616i	$-0.573282\pi$
$787$	−6.24086e9	−0.456386	−0.228193	+	0.973616i	$0.573282\pi$
$788$	4.69067e8	0.0341502
$789$	−1.17144e10	−0.849082
$790$	2.78535e9	0.200995
$791$	0	0
$792$	−5.47787e8	−0.0391808
$793$	1.06983e10	0.761834
$794$	−1.78123e10	−1.26284
$795$	−4.98395e9	−0.351794
$796$	9.84525e8	0.0691880
$797$	7.50982e9	0.525442	0.262721	−	0.964872i	$-0.415380\pi$
$797$	7.50982e9	0.525442	0.262721	+	0.964872i	$0.415380\pi$
$798$	0	0
$799$	2.29400e10	1.59104
$800$	2.12908e9	0.147020
$801$	−2.46498e9	−0.169473
$802$	−1.78420e10	−1.22133
$803$	−2.60244e9	−0.177368
$804$	−1.27998e9	−0.0868571
$805$	0	0
$806$	−8.35105e9	−0.561783
$807$	−2.99917e9	−0.200883
$808$	−1.70491e10	−1.13701
$809$	5.76686e9	0.382930	0.191465	−	0.981499i	$-0.438676\pi$
$809$	5.76686e9	0.382930	0.191465	+	0.981499i	$0.438676\pi$
$810$	1.85289e10	1.22504
$811$	1.72046e10	1.13259	0.566293	−	0.824204i	$-0.308377\pi$
$811$	1.72046e10	1.13259	0.566293	+	0.824204i	$0.308377\pi$
$812$	0	0
$813$	2.98379e9	0.194738
$814$	6.40703e9	0.416362
$815$	−9.55577e9	−0.618321
$816$	−1.84472e10	−1.18854
$817$	−2.12354e8	−0.0136233
$818$	−7.12928e8	−0.0455417
$819$	0	0
$820$	2.62452e9	0.166227
$821$	−5.07112e9	−0.319818	−0.159909	−	0.987132i	$-0.551120\pi$
$821$	−5.07112e9	−0.319818	−0.159909	+	0.987132i	$0.551120\pi$
$822$	−1.73580e10	−1.09005
$823$	−2.75808e10	−1.72467	−0.862337	−	0.506335i	$-0.831000\pi$
$823$	−2.75808e10	−1.72467	−0.862337	+	0.506335i	$0.831000\pi$
$824$	−2.97443e10	−1.85207
$825$	−4.27702e9	−0.265187
$826$	0	0
$827$	−1.37072e10	−0.842710	−0.421355	−	0.906896i	$-0.638445\pi$
$827$	−1.37072e10	−0.842710	−0.421355	+	0.906896i	$0.638445\pi$
$828$	1.89295e8	0.0115886
$829$	5.39751e9	0.329043	0.164521	−	0.986374i	$-0.447392\pi$
$829$	5.39751e9	0.329043	0.164521	+	0.986374i	$0.447392\pi$
$830$	9.82923e8	0.0596686
$831$	2.70464e10	1.63495
$832$	7.33286e9	0.441410
$833$	0	0
$834$	−1.33043e9	−0.0794165
$835$	2.62585e10	1.56087
$836$	−7.24178e7	−0.00428671
$837$	1.99916e10	1.17844
$838$	−5.15156e9	−0.302402
$839$	−7.66291e9	−0.447948	−0.223974	−	0.974595i	$-0.571903\pi$
$839$	−7.66291e9	−0.447948	−0.223974	+	0.974595i	$0.571903\pi$
$840$	0	0
$841$	1.45591e10	0.844013
$842$	2.21308e10	1.27763
$843$	3.51016e9	0.201804
$844$	−1.37418e9	−0.0786765
$845$	−1.87160e10	−1.06713
$846$	−3.31040e9	−0.187968
$847$	0	0
$848$	−5.16284e9	−0.290739
$849$	−1.39057e10	−0.779860
$850$	2.09773e10	1.17161
$851$	2.62173e10	1.45826
$852$	1.76974e9	0.0980325
$853$	−9.11744e9	−0.502981	−0.251490	−	0.967860i	$-0.580921\pi$
$853$	−9.11744e9	−0.502981	−0.251490	+	0.967860i	$0.580921\pi$
$854$	0	0
$855$	6.31815e8	0.0345707
$856$	5.90067e9	0.321546
$857$	2.27356e10	1.23388	0.616942	−	0.787009i	$-0.288371\pi$
$857$	2.27356e10	1.23388	0.616942	+	0.787009i	$0.288371\pi$
$858$	2.61031e9	0.141087
$859$	4.15338e9	0.223576	0.111788	−	0.993732i	$-0.464342\pi$
$859$	4.15338e9	0.223576	0.111788	+	0.993732i	$0.464342\pi$
$860$	1.51203e8	0.00810615
$861$	0	0
$862$	−8.65162e9	−0.460068
$863$	−1.94331e10	−1.02921	−0.514606	−	0.857427i	$-0.672062\pi$
$863$	−1.94331e10	−1.02921	−0.514606	+	0.857427i	$0.672062\pi$
$864$	3.11065e9	0.164079
$865$	−5.04671e9	−0.265126
$866$	9.38535e9	0.491064
$867$	−7.54297e9	−0.393075
$868$	0	0
$869$	−8.09458e8	−0.0418432
$870$	3.55129e10	1.82839
$871$	1.13432e10	0.581663
$872$	−9.36428e9	−0.478263
$873$	−4.12357e8	−0.0209761
$874$	−4.10164e9	−0.207811
$875$	0	0
$876$	8.47317e8	0.0425874
$877$	−1.61113e10	−0.806550	−0.403275	−	0.915079i	$-0.632128\pi$
$877$	−1.61113e10	−0.806550	−0.403275	+	0.915079i	$0.632128\pi$
$878$	1.16233e10	0.579559
$879$	−1.85638e10	−0.921948
$880$	−9.11360e9	−0.450817
$881$	1.85288e10	0.912916	0.456458	−	0.889745i	$-0.349118\pi$
$881$	1.85288e10	0.912916	0.456458	+	0.889745i	$0.349118\pi$
$882$	0	0
$883$	−1.99740e10	−0.976345	−0.488172	−	0.872747i	$-0.662336\pi$
$883$	−1.99740e10	−0.976345	−0.488172	+	0.872747i	$0.662336\pi$
$884$	−9.24953e8	−0.0450336
$885$	−1.48845e9	−0.0721828
$886$	8.45437e9	0.408379
$887$	5.34255e9	0.257049	0.128525	−	0.991706i	$-0.458976\pi$
$887$	5.34255e9	0.257049	0.128525	+	0.991706i	$0.458976\pi$
$888$	2.47017e10	1.18381
$889$	0	0
$890$	−3.80306e10	−1.80829
$891$	−5.38473e9	−0.255030
$892$	−2.66510e9	−0.125729
$893$	5.18226e9	0.243523
$894$	4.79752e9	0.224561
$895$	2.47792e9	0.115533
$896$	0	0
$897$	1.06813e10	0.494140
$898$	−1.59055e9	−0.0732960
$899$	3.30179e10	1.51562
$900$	−2.18704e8	−0.0100002
$901$	−7.10390e9	−0.323564
$902$	−1.05571e10	−0.478986
$903$	0	0
$904$	3.14149e9	0.141432
$905$	1.35062e10	0.605709
$906$	−1.32106e10	−0.590165
$907$	7.80696e9	0.347421	0.173711	−	0.984797i	$-0.444424\pi$
$907$	7.80696e9	0.347421	0.173711	+	0.984797i	$0.444424\pi$
$908$	5.66939e8	0.0251325
$909$	3.65054e9	0.161207
$910$	0	0
$911$	−2.92975e10	−1.28386	−0.641928	−	0.766765i	$-0.721865\pi$
$911$	−2.92975e10	−1.28386	−0.641928	+	0.766765i	$0.721865\pi$
$912$	−4.16731e9	−0.181917
$913$	−2.85650e8	−0.0124218
$914$	−3.56810e10	−1.54570
$915$	−4.72238e10	−2.03792
$916$	2.99691e9	0.128837
$917$	0	0
$918$	3.06484e10	1.30755
$919$	4.50916e10	1.91642	0.958212	−	0.286061i	$-0.0923459\pi$
$919$	4.50916e10	1.91642	0.958212	+	0.286061i	$0.0923459\pi$
$920$	−3.45830e10	−1.46422
$921$	7.08991e9	0.299042
$922$	−1.59531e10	−0.670328
$923$	−1.56835e10	−0.656503
$924$	0	0
$925$	−3.02904e10	−1.25837
$926$	1.54117e10	0.637839
$927$	6.36880e9	0.262590
$928$	5.13750e9	0.211025
$929$	3.94849e10	1.61576	0.807878	−	0.589350i	$-0.200616\pi$
$929$	3.94849e10	1.61576	0.807878	+	0.589350i	$0.200616\pi$
$930$	3.68626e10	1.50278
$931$	0	0
$932$	3.69071e9	0.149333
$933$	5.31424e9	0.214218
$934$	1.61975e9	0.0650480
$935$	−1.25400e10	−0.501716
$936$	−1.58056e9	−0.0630008
$937$	2.59072e10	1.02880	0.514402	−	0.857549i	$-0.328014\pi$
$937$	2.59072e10	1.02880	0.514402	+	0.857549i	$0.328014\pi$
$938$	0	0
$939$	3.29785e9	0.129987
$940$	−3.68994e9	−0.144901
$941$	−4.37803e10	−1.71283	−0.856417	−	0.516285i	$-0.827315\pi$
$941$	−4.37803e10	−1.71283	−0.856417	+	0.516285i	$0.827315\pi$
$942$	2.13211e10	0.831059
$943$	−4.31994e10	−1.67759
$944$	−1.54188e9	−0.0596552
$945$	0	0
$946$	−6.08212e8	−0.0233580
$947$	2.38242e10	0.911578	0.455789	−	0.890088i	$-0.349357\pi$
$947$	2.38242e10	0.911578	0.455789	+	0.890088i	$0.349357\pi$
$948$	2.63548e8	0.0100469
$949$	−7.50896e9	−0.285199
$950$	4.73888e9	0.179326
$951$	4.12517e10	1.55529
$952$	0	0
$953$	−3.79309e9	−0.141961	−0.0709803	−	0.997478i	$-0.522613\pi$
$953$	−3.79309e9	−0.141961	−0.0709803	+	0.997478i	$0.522613\pi$
$954$	1.02514e9	0.0382265
$955$	3.19031e10	1.18528
$956$	2.99462e9	0.110851
$957$	−1.03205e10	−0.380635
$958$	2.62237e10	0.963641
$959$	0	0
$960$	−3.23682e10	−1.18078
$961$	6.76013e9	0.245710
$962$	1.84866e10	0.669490
$963$	−1.26344e9	−0.0455894
$964$	−2.74645e8	−0.00987419
$965$	−2.66314e10	−0.954000
$966$	0	0
$967$	5.24034e10	1.86366	0.931830	−	0.362895i	$-0.118212\pi$
$967$	5.24034e10	1.86366	0.931830	+	0.362895i	$0.118212\pi$
$968$	2.45610e9	0.0870326
$969$	−5.73409e9	−0.202456
$970$	−6.36199e9	−0.223816
$971$	−2.89908e10	−1.01623	−0.508116	−	0.861288i	$-0.669658\pi$
$971$	−2.89908e10	−1.01623	−0.508116	+	0.861288i	$0.669658\pi$
$972$	−6.00875e8	−0.0209871
$973$	0	0
$974$	−4.12817e10	−1.43153
$975$	−1.23407e10	−0.426407
$976$	−4.89188e10	−1.68423
$977$	5.13449e10	1.76143	0.880717	−	0.473643i	$-0.157061\pi$
$977$	5.13449e10	1.76143	0.880717	+	0.473643i	$0.157061\pi$
$978$	−1.25149e10	−0.427800
$979$	1.10522e10	0.376451
$980$	0	0
$981$	2.00507e9	0.0678090
$982$	1.85060e10	0.623622
$983$	4.63039e10	1.55482	0.777410	−	0.628994i	$-0.216533\pi$
$983$	4.63039e10	1.55482	0.777410	+	0.628994i	$0.216533\pi$
$984$	−4.07021e10	−1.36186
$985$	1.83490e10	0.611768
$986$	5.06185e10	1.68167
$987$	0	0
$988$	−2.08951e8	−0.00689280
$989$	−2.48878e9	−0.0818087
$990$	1.80961e9	0.0592738
$991$	1.61581e10	0.527389	0.263695	−	0.964606i	$-0.415059\pi$
$991$	1.61581e10	0.527389	0.263695	+	0.964606i	$0.415059\pi$
$992$	5.33276e9	0.173445
$993$	2.26108e10	0.732814
$994$	0	0
$995$	3.85128e10	1.23944
$996$	9.30036e7	0.00298258
$997$	−5.17011e10	−1.65222	−0.826108	−	0.563512i	$-0.809450\pi$
$997$	−5.17011e10	−1.65222	−0.826108	+	0.563512i	$0.809450\pi$
$998$	−5.61285e10	−1.78742
$999$	−4.42552e10	−1.40438

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.8.a.a.1.1		2
7.6	odd	2		11.8.a.a.1.1	✓	2
21.20	even	2		99.8.a.c.1.2		2
28.27	even	2		176.8.a.d.1.1		2
35.34	odd	2		275.8.a.a.1.2		2
77.76	even	2		121.8.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.8.a.a.1.1	✓	2	7.6	odd	2
99.8.a.c.1.2		2	21.20	even	2
121.8.a.b.1.2		2	77.76	even	2
176.8.a.d.1.1		2	28.27	even	2
275.8.a.a.1.2		2	35.34	odd	2
539.8.a.a.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	539.a (trivial)

\(f(q)\)	\(=\)	\(q-11.7460 q^{2} -43.4758 q^{3} +9.96773 q^{4} +389.919 q^{5} +510.665 q^{6} +1386.40 q^{8} -296.855 q^{9} -4579.98 q^{10} +1331.00 q^{11} -433.355 q^{12} +3840.41 q^{13} -16952.1 q^{15} -17560.5 q^{16} -24162.7 q^{17} +3486.85 q^{18} -5458.47 q^{19} +3886.61 q^{20} -15633.9 q^{22} -63973.2 q^{23} -60275.0 q^{24} +73912.1 q^{25} -45109.3 q^{26} +107988. q^{27} +178351. q^{29} +199118. q^{30} +185129. q^{31} +28805.6 q^{32} -57866.3 q^{33} +283814. q^{34} -2958.97 q^{36} -409817. q^{37} +64115.0 q^{38} -166965. q^{39} +540585. q^{40} +675273. q^{41} +38903.5 q^{43} +13267.1 q^{44} -115749. q^{45} +751427. q^{46} -949397. q^{47} +763457. q^{48} -868169. q^{50} +1.05049e6 q^{51} +38280.2 q^{52} +294003. q^{53} -1.26842e6 q^{54} +518983. q^{55} +237311. q^{57} -2.09490e6 q^{58} +87803.7 q^{59} -168974. q^{60} +2.78573e6 q^{61} -2.17452e6 q^{62} +1.90940e6 q^{64} +1.49745e6 q^{65} +679696. q^{66} +2.95364e6 q^{67} -240847. q^{68} +2.78129e6 q^{69} -4.08380e6 q^{71} -411560. q^{72} -1.95525e6 q^{73} +4.81370e6 q^{74} -3.21339e6 q^{75} -54408.6 q^{76} +1.96116e6 q^{78} -608158. q^{79} -6.84718e6 q^{80} -4.04562e6 q^{81} -7.93173e6 q^{82} -214613. q^{83} -9.42151e6 q^{85} -456959. q^{86} -7.75394e6 q^{87} +1.84530e6 q^{88} +8.30366e6 q^{89} +1.35959e6 q^{90} -637668. q^{92} -8.04863e6 q^{93} +1.11516e7 q^{94} -2.12836e6 q^{95} -1.25235e6 q^{96} +1.38909e6 q^{97} -395114. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 6 q^{3} - 104 q^{4} + 470 q^{5} + 696 q^{6} + 480 q^{8} - 36 q^{9} - 4280 q^{10} + 2662 q^{11} - 6072 q^{12} - 344 q^{13} - 12990 q^{15} - 6368 q^{16} + 8468 q^{17} + 4464 q^{18} + 35280 q^{19}+ \cdots - 47916 q^{99}+O(q^{100}) \) 2 * q - 8 * q^2 + 6 * q^3 - 104 * q^4 + 470 * q^5 + 696 * q^6 + 480 * q^8 - 36 * q^9 - 4280 * q^10 + 2662 * q^11 - 6072 * q^12 - 344 * q^13 - 12990 * q^15 - 6368 * q^16 + 8468 * q^17 + 4464 * q^18 + 35280 * q^19 - 5240 * q^20 - 10648 * q^22 - 61486 * q^23 - 105120 * q^24 + 2200 * q^25 - 60784 * q^26 + 12690 * q^27 + 179040 * q^29 + 213960 * q^30 + 57166 * q^31 + 186752 * q^32 + 7986 * q^33 + 406048 * q^34 - 32688 * q^36 - 877698 * q^37 + 216720 * q^38 - 373992 * q^39 + 468000 * q^40 + 283616 * q^41 + 275484 * q^43 - 138424 * q^44 - 94860 * q^45 + 760744 * q^46 - 1662512 * q^47 + 1317216 * q^48 - 1136800 * q^50 + 2664924 * q^51 + 515168 * q^52 + 1616484 * q^53 - 1625400 * q^54 + 625570 * q^55 + 2252880 * q^57 - 2092320 * q^58 + 2454130 * q^59 - 620520 * q^60 + 6019176 * q^61 - 2653864 * q^62 + 1068416 * q^64 + 1162360 * q^65 + 926376 * q^66 - 174698 * q^67 - 3959696 * q^68 + 2904342 * q^69 - 1151466 * q^71 - 648000 * q^72 - 885944 * q^73 + 3061032 * q^74 - 6761400 * q^75 - 4697280 * q^76 + 1185648 * q^78 + 3801460 * q^79 - 5950880 * q^80 - 9331038 * q^81 - 9398864 * q^82 + 2282916 * q^83 - 6808420 * q^85 + 429264 * q^86 - 7719840 * q^87 + 638880 * q^88 + 13481970 * q^89 + 1437840 * q^90 - 921128 * q^92 - 14379702 * q^93 + 8480288 * q^94 + 1134000 * q^95 + 6562176 * q^96 + 68078 * q^97 - 47916 * q^99

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