LMFDB - Embedded newform 2013.4.a.d.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2013,4,Mod(1,2013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2013, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2013.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2013 = 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2013.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:	$118.770844842$
Analytic rank:	$1$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	2013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.250375 q^{2} -3.00000 q^{3} -7.93731 q^{4} +9.04215 q^{5} -0.751125 q^{6} +10.1810 q^{7} -3.99031 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+0.250375 q^{2} -3.00000 q^{3} -7.93731 q^{4} +9.04215 q^{5} -0.751125 q^{6} +10.1810 q^{7} -3.99031 q^{8} +9.00000 q^{9} +2.26393 q^{10} +11.0000 q^{11} +23.8119 q^{12} +22.7160 q^{13} +2.54908 q^{14} -27.1264 q^{15} +62.4994 q^{16} -80.4306 q^{17} +2.25338 q^{18} -19.3570 q^{19} -71.7703 q^{20} -30.5431 q^{21} +2.75413 q^{22} -96.2500 q^{23} +11.9709 q^{24} -43.2396 q^{25} +5.68751 q^{26} -27.0000 q^{27} -80.8101 q^{28} +202.485 q^{29} -6.79179 q^{30} -30.2038 q^{31} +47.5708 q^{32} -33.0000 q^{33} -20.1378 q^{34} +92.0585 q^{35} -71.4358 q^{36} +68.9040 q^{37} -4.84651 q^{38} -68.1479 q^{39} -36.0809 q^{40} +449.970 q^{41} -7.64724 q^{42} -298.000 q^{43} -87.3104 q^{44} +81.3793 q^{45} -24.0986 q^{46} -442.043 q^{47} -187.498 q^{48} -239.346 q^{49} -10.8261 q^{50} +241.292 q^{51} -180.304 q^{52} -125.008 q^{53} -6.76013 q^{54} +99.4636 q^{55} -40.6255 q^{56} +58.0709 q^{57} +50.6971 q^{58} -800.674 q^{59} +215.311 q^{60} -61.0000 q^{61} -7.56229 q^{62} +91.6294 q^{63} -488.085 q^{64} +205.401 q^{65} -8.26238 q^{66} +230.527 q^{67} +638.403 q^{68} +288.750 q^{69} +23.0491 q^{70} +617.923 q^{71} -35.9128 q^{72} +556.065 q^{73} +17.2518 q^{74} +129.719 q^{75} +153.642 q^{76} +111.991 q^{77} -17.0625 q^{78} +488.353 q^{79} +565.129 q^{80} +81.0000 q^{81} +112.661 q^{82} +1127.35 q^{83} +242.430 q^{84} -727.265 q^{85} -74.6117 q^{86} -607.454 q^{87} -43.8934 q^{88} +178.073 q^{89} +20.3754 q^{90} +231.272 q^{91} +763.966 q^{92} +90.6115 q^{93} -110.677 q^{94} -175.029 q^{95} -142.712 q^{96} -210.080 q^{97} -59.9264 q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q - 4 q^{2} - 111 q^{3} + 158 q^{4} - 15 q^{5} + 12 q^{6} - 77 q^{7} - 69 q^{8} + 333 q^{9}+O(q^{10}) $ 37 * q - 4 * q^2 - 111 * q^3 + 158 * q^4 - 15 * q^5 + 12 * q^6 - 77 * q^7 - 69 * q^8 + 333 * q^9	\( 37 q - 4 q^{2} - 111 q^{3} + 158 q^{4} - 15 q^{5} + 12 q^{6} - 77 q^{7} - 69 q^{8} + 333 q^{9} - 45 q^{10} + 407 q^{11} - 474 q^{12} - 169 q^{13} + 102 q^{14} + 45 q^{15} + 598 q^{16} - 338 q^{17} - 36 q^{18} - 235 q^{19} - 550 q^{20} + 231 q^{21} - 44 q^{22} - 53 q^{23} + 207 q^{24} + 750 q^{25} - 75 q^{26} - 999 q^{27} - 1378 q^{28} - 30 q^{29} + 135 q^{30} - 506 q^{31} - 841 q^{32} - 1221 q^{33} - 316 q^{34} - 822 q^{35} + 1422 q^{36} - 830 q^{37} - 371 q^{38} + 507 q^{39} - 613 q^{40} + 16 q^{41} - 306 q^{42} - 1137 q^{43} + 1738 q^{44} - 135 q^{45} - 659 q^{46} - 489 q^{47} - 1794 q^{48} + 2214 q^{49} + 1066 q^{50} + 1014 q^{51} - 2342 q^{52} + 731 q^{53} + 108 q^{54} - 165 q^{55} + 3051 q^{56} + 705 q^{57} - 611 q^{58} - 425 q^{59} + 1650 q^{60} - 2257 q^{61} + 453 q^{62} - 693 q^{63} + 4919 q^{64} + 1346 q^{65} + 132 q^{66} - 1907 q^{67} - 3236 q^{68} + 159 q^{69} - 1050 q^{70} - 561 q^{71} - 621 q^{72} - 2397 q^{73} - 1840 q^{74} - 2250 q^{75} - 3868 q^{76} - 847 q^{77} + 225 q^{78} + 393 q^{79} - 4031 q^{80} + 2997 q^{81} - 1946 q^{82} - 4191 q^{83} + 4134 q^{84} - 2667 q^{85} + 2405 q^{86} + 90 q^{87} - 759 q^{88} + 1437 q^{89} - 405 q^{90} - 5192 q^{91} - 737 q^{92} + 1518 q^{93} - 1960 q^{94} + 1356 q^{95} + 2523 q^{96} - 2368 q^{97} - 3014 q^{98} + 3663 q^{99}+O(q^{100}) \) 37 * q - 4 * q^2 - 111 * q^3 + 158 * q^4 - 15 * q^5 + 12 * q^6 - 77 * q^7 - 69 * q^8 + 333 * q^9 - 45 * q^10 + 407 * q^11 - 474 * q^12 - 169 * q^13 + 102 * q^14 + 45 * q^15 + 598 * q^16 - 338 * q^17 - 36 * q^18 - 235 * q^19 - 550 * q^20 + 231 * q^21 - 44 * q^22 - 53 * q^23 + 207 * q^24 + 750 * q^25 - 75 * q^26 - 999 * q^27 - 1378 * q^28 - 30 * q^29 + 135 * q^30 - 506 * q^31 - 841 * q^32 - 1221 * q^33 - 316 * q^34 - 822 * q^35 + 1422 * q^36 - 830 * q^37 - 371 * q^38 + 507 * q^39 - 613 * q^40 + 16 * q^41 - 306 * q^42 - 1137 * q^43 + 1738 * q^44 - 135 * q^45 - 659 * q^46 - 489 * q^47 - 1794 * q^48 + 2214 * q^49 + 1066 * q^50 + 1014 * q^51 - 2342 * q^52 + 731 * q^53 + 108 * q^54 - 165 * q^55 + 3051 * q^56 + 705 * q^57 - 611 * q^58 - 425 * q^59 + 1650 * q^60 - 2257 * q^61 + 453 * q^62 - 693 * q^63 + 4919 * q^64 + 1346 * q^65 + 132 * q^66 - 1907 * q^67 - 3236 * q^68 + 159 * q^69 - 1050 * q^70 - 561 * q^71 - 621 * q^72 - 2397 * q^73 - 1840 * q^74 - 2250 * q^75 - 3868 * q^76 - 847 * q^77 + 225 * q^78 + 393 * q^79 - 4031 * q^80 + 2997 * q^81 - 1946 * q^82 - 4191 * q^83 + 4134 * q^84 - 2667 * q^85 + 2405 * q^86 + 90 * q^87 - 759 * q^88 + 1437 * q^89 - 405 * q^90 - 5192 * q^91 - 737 * q^92 + 1518 * q^93 - 1960 * q^94 + 1356 * q^95 + 2523 * q^96 - 2368 * q^97 - 3014 * q^98 + 3663 * 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 $

$2$	0.250375	0.0885210	0.0442605	−	0.999020i	$-0.485907\pi$
$2$	0.250375	0.0885210	0.0442605	+	0.999020i	$0.485907\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.93731	−0.992164
$5$	9.04215	0.808754	0.404377	−	0.914592i	$-0.367488\pi$
$5$	9.04215	0.808754	0.404377	+	0.914592i	$0.367488\pi$
$6$	−0.751125	−0.0511076
$7$	10.1810	0.549724	0.274862	−	0.961484i	$-0.411368\pi$
$7$	10.1810	0.549724	0.274862	+	0.961484i	$0.411368\pi$
$8$	−3.99031	−0.176348
$9$	9.00000	0.333333
$10$	2.26393	0.0715917
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	23.8119	0.572826
$13$	22.7160	0.484637	0.242318	−	0.970197i	$-0.422092\pi$
$13$	22.7160	0.484637	0.242318	+	0.970197i	$0.422092\pi$
$14$	2.54908	0.0486621
$15$	−27.1264	−0.466934
$16$	62.4994	0.976554
$17$	−80.4306	−1.14749	−0.573744	−	0.819035i	$-0.694509\pi$
$17$	−80.4306	−1.14749	−0.573744	+	0.819035i	$0.694509\pi$
$18$	2.25338	0.0295070
$19$	−19.3570	−0.233726	−0.116863	−	0.993148i	$-0.537284\pi$
$19$	−19.3570	−0.233726	−0.116863	+	0.993148i	$0.537284\pi$
$20$	−71.7703	−0.802417
$21$	−30.5431	−0.317384
$22$	2.75413	0.0266901
$23$	−96.2500	−0.872587	−0.436294	−	0.899804i	$-0.643709\pi$
$23$	−96.2500	−0.872587	−0.436294	+	0.899804i	$0.643709\pi$
$24$	11.9709	0.101815
$25$	−43.2396	−0.345917
$26$	5.68751	0.0429005
$27$	−27.0000	−0.192450
$28$	−80.8101	−0.545417
$29$	202.485	1.29657	0.648284	−	0.761399i	$-0.275487\pi$
$29$	202.485	1.29657	0.648284	+	0.761399i	$0.275487\pi$
$30$	−6.79179	−0.0413335
$31$	−30.2038	−0.174993	−0.0874963	−	0.996165i	$-0.527887\pi$
$31$	−30.2038	−0.174993	−0.0874963	+	0.996165i	$0.527887\pi$
$32$	47.5708	0.262794
$33$	−33.0000	−0.174078
$34$	−20.1378	−0.101577
$35$	92.0585	0.444592
$36$	−71.4358	−0.330721
$37$	68.9040	0.306155	0.153078	−	0.988214i	$-0.451082\pi$
$37$	68.9040	0.306155	0.153078	+	0.988214i	$0.451082\pi$
$38$	−4.84651	−0.0206897
$39$	−68.1479	−0.279805
$40$	−36.0809	−0.142622
$41$	449.970	1.71399	0.856994	−	0.515327i	$-0.172329\pi$
$41$	449.970	1.71399	0.856994	+	0.515327i	$0.172329\pi$
$42$	−7.64724	−0.0280951
$43$	−298.000	−1.05685	−0.528425	−	0.848980i	$-0.677217\pi$
$43$	−298.000	−1.05685	−0.528425	+	0.848980i	$0.677217\pi$
$44$	−87.3104	−0.299149
$45$	81.3793	0.269585
$46$	−24.0986	−0.0772423
$47$	−442.043	−1.37189	−0.685943	−	0.727655i	$-0.740610\pi$
$47$	−442.043	−1.37189	−0.685943	+	0.727655i	$0.740610\pi$
$48$	−187.498	−0.563813
$49$	−239.346	−0.697803
$50$	−10.8261	−0.0306209
$51$	241.292	0.662502
$52$	−180.304	−0.480839
$53$	−125.008	−0.323985	−0.161992	−	0.986792i	$-0.551792\pi$
$53$	−125.008	−0.323985	−0.161992	+	0.986792i	$0.551792\pi$
$54$	−6.76013	−0.0170359
$55$	99.4636	0.243849
$56$	−40.6255	−0.0969430
$57$	58.0709	0.134942
$58$	50.6971	0.114773
$59$	−800.674	−1.76676	−0.883380	−	0.468658i	$-0.844738\pi$
$59$	−800.674	−1.76676	−0.883380	+	0.468658i	$0.844738\pi$
$60$	215.311	0.463276
$61$	−61.0000	−0.128037
$61$	−61.0000	−0.128037
$62$	−7.56229	−0.0154905
$63$	91.6294	0.183241
$64$	−488.085	−0.953291
$65$	205.401	0.391952
$66$	−8.26238	−0.0154095
$67$	230.527	0.420349	0.210175	−	0.977664i	$-0.432597\pi$
$67$	230.527	0.420349	0.210175	+	0.977664i	$0.432597\pi$
$68$	638.403	1.13850
$69$	288.750	0.503788
$70$	23.0491	0.0393557
$71$	617.923	1.03287	0.516437	−	0.856325i	$-0.327258\pi$
$71$	617.923	1.03287	0.516437	+	0.856325i	$0.327258\pi$
$72$	−35.9128	−0.0587828
$73$	556.065	0.891540	0.445770	−	0.895147i	$-0.352930\pi$
$73$	556.065	0.891540	0.445770	+	0.895147i	$0.352930\pi$
$74$	17.2518	0.0271012
$75$	129.719	0.199715
$76$	153.642	0.231895
$77$	111.991	0.165748
$78$	−17.0625	−0.0247686
$79$	488.353	0.695494	0.347747	−	0.937588i	$-0.386947\pi$
$79$	488.353	0.695494	0.347747	+	0.937588i	$0.386947\pi$
$80$	565.129	0.789792
$81$	81.0000	0.111111
$82$	112.661	0.151724
$83$	1127.35	1.49088	0.745438	−	0.666575i	$-0.232240\pi$
$83$	1127.35	1.49088	0.745438	+	0.666575i	$0.232240\pi$
$84$	242.430	0.314897
$85$	−727.265	−0.928035
$86$	−74.6117	−0.0935534
$87$	−607.454	−0.748573
$88$	−43.8934	−0.0531710
$89$	178.073	0.212087	0.106043	−	0.994362i	$-0.466182\pi$
$89$	178.073	0.212087	0.106043	+	0.994362i	$0.466182\pi$
$90$	20.3754	0.0238639
$91$	231.272	0.266417
$92$	763.966	0.865750
$93$	90.6115	0.101032
$94$	−110.677	−0.121441
$95$	−175.029	−0.189027
$96$	−142.712	−0.151724
$97$	−210.080	−0.219901	−0.109950	−	0.993937i	$-0.535069\pi$
$97$	−210.080	−0.219901	−0.109950	+	0.993937i	$0.535069\pi$
$98$	−59.9264	−0.0617702
$99$	99.0000	0.100504
$100$	343.206	0.343206
$101$	1805.57	1.77882	0.889408	−	0.457114i	$-0.151117\pi$
$101$	1805.57	1.77882	0.889408	+	0.457114i	$0.151117\pi$
$102$	60.4135	0.0586453
$103$	−786.732	−0.752612	−0.376306	−	0.926495i	$-0.622806\pi$
$103$	−786.732	−0.752612	−0.376306	+	0.926495i	$0.622806\pi$
$104$	−90.6437	−0.0854649
$105$	−276.175	−0.256685
$106$	−31.2989	−0.0286794
$107$	−1300.07	−1.17460	−0.587301	−	0.809368i	$-0.699810\pi$
$107$	−1300.07	−1.17460	−0.587301	+	0.809368i	$0.699810\pi$
$108$	214.307	0.190942
$109$	1216.70	1.06916	0.534579	−	0.845118i	$-0.320470\pi$
$109$	1216.70	1.06916	0.534579	+	0.845118i	$0.320470\pi$
$110$	24.9032	0.0215857
$111$	−206.712	−0.176759
$112$	636.309	0.536835
$113$	312.692	0.260315	0.130158	−	0.991493i	$-0.458452\pi$
$113$	312.692	0.260315	0.130158	+	0.991493i	$0.458452\pi$
$114$	14.5395	0.0119452
$115$	−870.306	−0.705709
$116$	−1607.18	−1.28641
$117$	204.444	0.161546
$118$	−200.469	−0.156395
$119$	−818.867	−0.630802
$120$	108.243	0.0823431
$121$	121.000	0.0909091
$122$	−15.2729	−0.0113339
$123$	−1349.91	−0.989571
$124$	239.737	0.173621
$125$	−1521.25	−1.08852
$126$	22.9417	0.0162207
$127$	−308.982	−0.215887	−0.107944	−	0.994157i	$-0.534427\pi$
$127$	−308.982	−0.215887	−0.107944	+	0.994157i	$0.534427\pi$
$128$	−502.770	−0.347180
$129$	893.999	0.610173
$130$	51.4273	0.0346960
$131$	−286.009	−0.190754	−0.0953769	−	0.995441i	$-0.530406\pi$
$131$	−286.009	−0.190754	−0.0953769	+	0.995441i	$0.530406\pi$
$132$	261.931	0.172714
$133$	−197.074	−0.128485
$134$	57.7183	0.0372097
$135$	−244.138	−0.155645
$136$	320.943	0.202357
$137$	−2229.87	−1.39059	−0.695293	−	0.718726i	$-0.744725\pi$
$137$	−2229.87	−1.39059	−0.695293	+	0.718726i	$0.744725\pi$
$138$	72.2958	0.0445958
$139$	1392.28	0.849580	0.424790	−	0.905292i	$-0.360348\pi$
$139$	1392.28	0.849580	0.424790	+	0.905292i	$0.360348\pi$
$140$	−730.697	−0.441108
$141$	1326.13	0.792059
$142$	154.713	0.0914309
$143$	249.876	0.146123
$144$	562.495	0.325518
$145$	1830.90	1.04860
$146$	139.225	0.0789200
$147$	718.039	0.402877
$148$	−546.912	−0.303756
$149$	−2058.19	−1.13163	−0.565817	−	0.824531i	$-0.691439\pi$
$149$	−2058.19	−1.13163	−0.565817	+	0.824531i	$0.691439\pi$
$150$	32.4783	0.0176790
$151$	2033.26	1.09579	0.547897	−	0.836546i	$-0.315429\pi$
$151$	2033.26	1.09579	0.547897	+	0.836546i	$0.315429\pi$
$152$	77.2403	0.0412172
$153$	−723.875	−0.382496
$154$	28.0399	0.0146722
$155$	−273.108	−0.141526
$156$	540.911	0.277613
$157$	−465.161	−0.236458	−0.118229	−	0.992986i	$-0.537722\pi$
$157$	−465.161	−0.236458	−0.118229	+	0.992986i	$0.537722\pi$
$158$	122.272	0.0615658
$159$	375.024	0.187053
$160$	430.142	0.212536
$161$	−979.925	−0.479683
$162$	20.2804	0.00983566
$163$	−2865.81	−1.37710	−0.688552	−	0.725187i	$-0.741753\pi$
$163$	−2865.81	−1.37710	−0.688552	+	0.725187i	$0.741753\pi$
$164$	−3571.55	−1.70056
$165$	−298.391	−0.140786
$166$	282.260	0.131974
$167$	−3359.34	−1.55661	−0.778304	−	0.627887i	$-0.783920\pi$
$167$	−3359.34	−1.55661	−0.778304	+	0.627887i	$0.783920\pi$
$168$	121.876	0.0559700
$169$	−1680.98	−0.765127
$170$	−182.089	−0.0821506
$171$	−174.213	−0.0779087
$172$	2365.32	1.04857
$173$	−2886.80	−1.26867	−0.634334	−	0.773059i	$-0.718725\pi$
$173$	−2886.80	−1.26867	−0.634334	+	0.773059i	$0.718725\pi$
$174$	−152.091	−0.0662645
$175$	−440.224	−0.190159
$176$	687.494	0.294442
$177$	2402.02	1.02004
$178$	44.5851	0.0187741
$179$	−483.081	−0.201716	−0.100858	−	0.994901i	$-0.532159\pi$
$179$	−483.081	−0.201716	−0.100858	+	0.994901i	$0.532159\pi$
$180$	−645.933	−0.267472
$181$	−930.622	−0.382169	−0.191084	−	0.981574i	$-0.561200\pi$
$181$	−930.622	−0.382169	−0.191084	+	0.981574i	$0.561200\pi$
$182$	57.9048	0.0235835
$183$	183.000	0.0739221
$184$	384.067	0.153879
$185$	623.040	0.247604
$186$	22.6869	0.00894345
$187$	−884.737	−0.345980
$188$	3508.64	1.36114
$189$	−274.888	−0.105795
$190$	−43.8228	−0.0167328
$191$	1388.20	0.525900	0.262950	−	0.964810i	$-0.415305\pi$
$191$	1388.20	0.525900	0.262950	+	0.964810i	$0.415305\pi$
$192$	1464.25	0.550383
$193$	3370.21	1.25696	0.628480	−	0.777826i	$-0.283677\pi$
$193$	3370.21	1.25696	0.628480	+	0.777826i	$0.283677\pi$
$194$	−52.5988	−0.0194658
$195$	−616.204	−0.226294
$196$	1899.77	0.692335
$197$	−4021.99	−1.45459	−0.727296	−	0.686324i	$-0.759223\pi$
$197$	−4021.99	−1.45459	−0.727296	+	0.686324i	$0.759223\pi$
$198$	24.7871	0.00889669
$199$	−3885.06	−1.38394	−0.691972	−	0.721925i	$-0.743258\pi$
$199$	−3885.06	−1.38394	−0.691972	+	0.721925i	$0.743258\pi$
$200$	172.539	0.0610018
$201$	−691.582	−0.242689
$202$	452.069	0.157463
$203$	2061.50	0.712755
$204$	−1915.21	−0.657311
$205$	4068.69	1.38619
$206$	−196.978	−0.0666219
$207$	−866.250	−0.290862
$208$	1419.74	0.473274
$209$	−212.927	−0.0704711
$210$	−69.1474	−0.0227220
$211$	−4437.76	−1.44790	−0.723952	−	0.689850i	$-0.757676\pi$
$211$	−4437.76	−1.44790	−0.723952	+	0.689850i	$0.757676\pi$
$212$	992.228	0.321446
$213$	−1853.77	−0.596330
$214$	−325.505	−0.103977
$215$	−2694.56	−0.854732
$216$	107.738	0.0339382
$217$	−307.506	−0.0961977
$218$	304.630	0.0946429
$219$	−1668.19	−0.514731
$220$	−789.474	−0.241938
$221$	−1827.06	−0.556115
$222$	−51.7555	−0.0156469
$223$	−1744.24	−0.523779	−0.261890	−	0.965098i	$-0.584346\pi$
$223$	−1744.24	−0.523779	−0.261890	+	0.965098i	$0.584346\pi$
$224$	484.320	0.144464
$225$	−389.156	−0.115306
$226$	78.2904	0.0230434
$227$	−3595.01	−1.05114	−0.525571	−	0.850750i	$-0.676148\pi$
$227$	−3595.01	−1.05114	−0.525571	+	0.850750i	$0.676148\pi$
$228$	−460.927	−0.133884
$229$	5226.18	1.50810	0.754052	−	0.656815i	$-0.228097\pi$
$229$	5226.18	1.50810	0.754052	+	0.656815i	$0.228097\pi$
$230$	−217.903	−0.0624700
$231$	−335.974	−0.0956947
$232$	−807.976	−0.228647
$233$	−653.593	−0.183769	−0.0918847	−	0.995770i	$-0.529289\pi$
$233$	−653.593	−0.183769	−0.0918847	+	0.995770i	$0.529289\pi$
$234$	51.1876	0.0143002
$235$	−3997.02	−1.10952
$236$	6355.20	1.75292
$237$	−1465.06	−0.401544
$238$	−205.024	−0.0558392
$239$	5136.37	1.39014	0.695071	−	0.718941i	$-0.255373\pi$
$239$	5136.37	1.39014	0.695071	+	0.718941i	$0.255373\pi$
$240$	−1695.39	−0.455987
$241$	−5223.95	−1.39628	−0.698141	−	0.715960i	$-0.745989\pi$
$241$	−5223.95	−1.39628	−0.698141	+	0.715960i	$0.745989\pi$
$242$	30.2954	0.00804736
$243$	−243.000	−0.0641500
$244$	484.176	0.127034
$245$	−2164.21	−0.564351
$246$	−337.984	−0.0875978
$247$	−439.713	−0.113272
$248$	120.523	0.0308596
$249$	−3382.05	−0.860757
$250$	−380.882	−0.0963565
$251$	898.404	0.225923	0.112962	−	0.993599i	$-0.463966\pi$
$251$	898.404	0.225923	0.112962	+	0.993599i	$0.463966\pi$
$252$	−727.291	−0.181806
$253$	−1058.75	−0.263095
$254$	−77.3614	−0.0191106
$255$	2181.80	0.535801
$256$	3778.80	0.922558
$257$	−6998.01	−1.69854	−0.849269	−	0.527961i	$-0.822957\pi$
$257$	−6998.01	−1.69854	−0.849269	+	0.527961i	$0.822957\pi$
$258$	223.835	0.0540131
$259$	701.514	0.168301
$260$	−1630.33	−0.388881
$261$	1822.36	0.432189
$262$	−71.6096	−0.0168857
$263$	−2798.27	−0.656079	−0.328039	−	0.944664i	$-0.606388\pi$
$263$	−2798.27	−0.656079	−0.328039	+	0.944664i	$0.606388\pi$
$264$	131.680	0.0306983
$265$	−1130.34	−0.262024
$266$	−49.3425	−0.0113736
$267$	−534.219	−0.122448
$268$	−1829.77	−0.417055
$269$	4215.58	0.955496	0.477748	−	0.878497i	$-0.341453\pi$
$269$	4215.58	0.955496	0.477748	+	0.878497i	$0.341453\pi$
$270$	−61.1261	−0.0137778
$271$	−515.330	−0.115513	−0.0577566	−	0.998331i	$-0.518395\pi$
$271$	−515.330	−0.115513	−0.0577566	+	0.998331i	$0.518395\pi$
$272$	−5026.87	−1.12058
$273$	−693.817	−0.153816
$274$	−558.303	−0.123096
$275$	−475.635	−0.104298
$276$	−2291.90	−0.499841
$277$	−4992.26	−1.08287	−0.541436	−	0.840742i	$-0.682119\pi$
$277$	−4992.26	−1.08287	−0.541436	+	0.840742i	$0.682119\pi$
$278$	348.592	0.0752057
$279$	−271.835	−0.0583309
$280$	−367.341	−0.0784030
$281$	−562.029	−0.119316	−0.0596581	−	0.998219i	$-0.519001\pi$
$281$	−562.029	−0.119316	−0.0596581	+	0.998219i	$0.519001\pi$
$282$	332.030	0.0701138
$283$	1943.95	0.408324	0.204162	−	0.978937i	$-0.434553\pi$
$283$	1943.95	0.408324	0.204162	+	0.978937i	$0.434553\pi$
$284$	−4904.65	−1.02478
$285$	525.086	0.109135
$286$	62.5627	0.0129350
$287$	4581.16	0.942221
$288$	428.137	0.0875979
$289$	1556.08	0.316727
$290$	458.411	0.0928235
$291$	630.240	0.126960
$292$	−4413.66	−0.884554
$293$	5703.96	1.13730	0.568650	−	0.822580i	$-0.307466\pi$
$293$	5703.96	1.13730	0.568650	+	0.822580i	$0.307466\pi$
$294$	179.779	0.0356630
$295$	−7239.81	−1.42887
$296$	−274.948	−0.0539899
$297$	−297.000	−0.0580259
$298$	−515.320	−0.100173
$299$	−2186.41	−0.422888
$300$	−1029.62	−0.198150
$301$	−3033.95	−0.580976
$302$	509.079	0.0970007
$303$	−5416.70	−1.02700
$304$	−1209.80	−0.228246
$305$	−551.571	−0.103550
$306$	−181.240	−0.0338589
$307$	−1.80713	−0.000335956 0	−0.000167978	−	1.00000i	$-0.500053\pi$
$307$	−1.80713	−0.000335956 0	−0.000167978		1.00000i	$0.500053\pi$
$308$	−888.911	−0.164449
$309$	2360.20	0.434521
$310$	−68.3793	−0.0125280
$311$	−4285.99	−0.781466	−0.390733	−	0.920504i	$-0.627778\pi$
$311$	−4285.99	−0.781466	−0.390733	+	0.920504i	$0.627778\pi$
$312$	271.931	0.0493432
$313$	1823.38	0.329277	0.164638	−	0.986354i	$-0.447354\pi$
$313$	1823.38	0.329277	0.164638	+	0.986354i	$0.447354\pi$
$314$	−116.465	−0.0209315
$315$	828.526	0.148197
$316$	−3876.21	−0.690045
$317$	−4427.61	−0.784477	−0.392238	−	0.919864i	$-0.628299\pi$
$317$	−4427.61	−0.784477	−0.392238	+	0.919864i	$0.628299\pi$
$318$	93.8968	0.0165581
$319$	2227.33	0.390930
$320$	−4413.34	−0.770978
$321$	3900.21	0.678157
$322$	−245.349	−0.0424620
$323$	1556.89	0.268198
$324$	−642.922	−0.110240
$325$	−982.229	−0.167644
$326$	−717.528	−0.121902
$327$	−3650.09	−0.617279
$328$	−1795.52	−0.302259
$329$	−4500.46	−0.754159
$330$	−74.7096	−0.0124625
$331$	−2226.35	−0.369701	−0.184850	−	0.982767i	$-0.559180\pi$
$331$	−2226.35	−0.369701	−0.184850	+	0.982767i	$0.559180\pi$
$332$	−8948.12	−1.47919
$333$	620.136	0.102052
$334$	−841.095	−0.137792
$335$	2084.46	0.339959
$336$	−1908.93	−0.309942
$337$	−5159.58	−0.834006	−0.417003	−	0.908905i	$-0.636920\pi$
$337$	−5159.58	−0.834006	−0.417003	+	0.908905i	$0.636920\pi$
$338$	−420.877	−0.0677298
$339$	−938.077	−0.150293
$340$	5772.53	0.920763
$341$	−332.242	−0.0527622
$342$	−43.6186	−0.00689655
$343$	−5928.89	−0.933324
$344$	1189.11	0.186374
$345$	2610.92	0.407441
$346$	−722.783	−0.112304
$347$	6649.43	1.02870	0.514352	−	0.857579i	$-0.328033\pi$
$347$	6649.43	1.02870	0.514352	+	0.857579i	$0.328033\pi$
$348$	4821.55	0.742708
$349$	−983.869	−0.150903	−0.0754517	−	0.997149i	$-0.524040\pi$
$349$	−983.869	−0.150903	−0.0754517	+	0.997149i	$0.524040\pi$
$350$	−110.221	−0.0168330
$351$	−613.331	−0.0932684
$352$	523.278	0.0792353
$353$	−8783.99	−1.32443	−0.662216	−	0.749313i	$-0.730384\pi$
$353$	−8783.99	−1.32443	−0.662216	+	0.749313i	$0.730384\pi$
$354$	601.406	0.0902949
$355$	5587.35	0.835341
$356$	−1413.42	−0.210425
$357$	2456.60	0.364194
$358$	−120.952	−0.0178561
$359$	11576.6	1.70192	0.850959	−	0.525232i	$-0.176021\pi$
$359$	11576.6	1.70192	0.850959	+	0.525232i	$0.176021\pi$
$360$	−324.728	−0.0475408
$361$	−6484.31	−0.945372
$362$	−233.005	−0.0338300
$363$	−363.000	−0.0524864
$364$	−1835.68	−0.264329
$365$	5028.02	0.721037
$366$	45.8186	0.00654366
$367$	−11711.2	−1.66572	−0.832861	−	0.553482i	$-0.813299\pi$
$367$	−11711.2	−1.66572	−0.832861	+	0.553482i	$0.813299\pi$
$368$	−6015.57	−0.852128
$369$	4049.73	0.571329
$370$	155.994	0.0219182
$371$	−1272.71	−0.178102
$372$	−719.212	−0.100240
$373$	−4930.72	−0.684459	−0.342229	−	0.939616i	$-0.611182\pi$
$373$	−4930.72	−0.684459	−0.342229	+	0.939616i	$0.611182\pi$
$374$	−221.516	−0.0306265
$375$	4563.74	0.628455
$376$	1763.89	0.241930
$377$	4599.64	0.628364
$378$	−68.8251	−0.00936503
$379$	9014.70	1.22178	0.610889	−	0.791716i	$-0.290812\pi$
$379$	9014.70	1.22178	0.610889	+	0.791716i	$0.290812\pi$
$380$	1389.26	0.187546
$381$	926.946	0.124643
$382$	347.571	0.0465531
$383$	−10650.2	−1.42089	−0.710444	−	0.703754i	$-0.751506\pi$
$383$	−10650.2	−1.42089	−0.710444	+	0.703754i	$0.751506\pi$
$384$	1508.31	0.200444
$385$	1012.64	0.134050
$386$	843.817	0.111267
$387$	−2682.00	−0.352283
$388$	1667.47	0.218178
$389$	9234.72	1.20365	0.601824	−	0.798629i	$-0.294441\pi$
$389$	9234.72	1.20365	0.601824	+	0.798629i	$0.294441\pi$
$390$	−154.282	−0.0200317
$391$	7741.44	1.00128
$392$	955.066	0.123056
$393$	858.028	0.110132
$394$	−1007.01	−0.128762
$395$	4415.76	0.562484
$396$	−785.794	−0.0997162
$397$	−4730.03	−0.597968	−0.298984	−	0.954258i	$-0.596648\pi$
$397$	−4730.03	−0.597968	−0.298984	+	0.954258i	$0.596648\pi$
$398$	−972.723	−0.122508
$399$	591.222	0.0741808
$400$	−2702.45	−0.337806
$401$	−11142.3	−1.38758	−0.693792	−	0.720175i	$-0.744061\pi$
$401$	−11142.3	−1.38758	−0.693792	+	0.720175i	$0.744061\pi$
$402$	−173.155	−0.0214830
$403$	−686.110	−0.0848078
$404$	−14331.3	−1.76488
$405$	732.414	0.0898616
$406$	516.149	0.0630937
$407$	757.944	0.0923093
$408$	−962.828	−0.116831
$409$	107.977	0.0130541	0.00652705	−	0.999979i	$-0.497922\pi$
$409$	107.977	0.0130541	0.00652705	+	0.999979i	$0.497922\pi$
$410$	1018.70	0.122707
$411$	6689.60	0.802855
$412$	6244.54	0.746714
$413$	−8151.69	−0.971231
$414$	−216.887	−0.0257474
$415$	10193.7	1.20575
$416$	1080.62	0.127360
$417$	−4176.84	−0.490505
$418$	−53.3116	−0.00623817
$419$	2104.27	0.245347	0.122674	−	0.992447i	$-0.460853\pi$
$419$	2104.27	0.245347	0.122674	+	0.992447i	$0.460853\pi$
$420$	2192.09	0.254674
$421$	128.433	0.0148680	0.00743400	−	0.999972i	$-0.497634\pi$
$421$	128.433	0.0148680	0.00743400	+	0.999972i	$0.497634\pi$
$422$	−1111.10	−0.128170
$423$	−3978.39	−0.457295
$424$	498.821	0.0571341
$425$	3477.78	0.396935
$426$	−464.138	−0.0527877
$427$	−621.043	−0.0703850
$428$	10319.1	1.16540
$429$	−749.627	−0.0843644
$430$	−674.650	−0.0756617
$431$	−2726.80	−0.304746	−0.152373	−	0.988323i	$-0.548691\pi$
$431$	−2726.80	−0.304746	−0.152373	+	0.988323i	$0.548691\pi$
$432$	−1687.48	−0.187938
$433$	−3875.34	−0.430109	−0.215054	−	0.976602i	$-0.568993\pi$
$433$	−3875.34	−0.430109	−0.215054	+	0.976602i	$0.568993\pi$
$434$	−76.9920	−0.00851551
$435$	−5492.69	−0.605412
$436$	−9657.29	−1.06078
$437$	1863.11	0.203946
$438$	−417.674	−0.0455645
$439$	−6975.52	−0.758368	−0.379184	−	0.925321i	$-0.623795\pi$
$439$	−6975.52	−0.758368	−0.379184	+	0.925321i	$0.623795\pi$
$440$	−396.890	−0.0430023
$441$	−2154.12	−0.232601
$442$	−457.450	−0.0492278
$443$	16083.1	1.72490	0.862452	−	0.506138i	$-0.168927\pi$
$443$	16083.1	1.72490	0.862452	+	0.506138i	$0.168927\pi$
$444$	1640.74	0.175374
$445$	1610.16	0.171526
$446$	−436.714	−0.0463654
$447$	6174.57	0.653349
$448$	−4969.21	−0.524047
$449$	6261.13	0.658086	0.329043	−	0.944315i	$-0.393274\pi$
$449$	6261.13	0.658086	0.329043	+	0.944315i	$0.393274\pi$
$450$	−97.4350	−0.0102070
$451$	4949.67	0.516787
$452$	−2481.94	−0.258275
$453$	−6099.79	−0.632656
$454$	−900.101	−0.0930481
$455$	2091.20	0.215466
$456$	−231.721	−0.0237968
$457$	−7397.06	−0.757155	−0.378577	−	0.925570i	$-0.623587\pi$
$457$	−7397.06	−0.757155	−0.378577	+	0.925570i	$0.623587\pi$
$458$	1308.51	0.133499
$459$	2171.63	0.220834
$460$	6907.89	0.700179
$461$	−8655.85	−0.874497	−0.437249	−	0.899341i	$-0.644047\pi$
$461$	−8655.85	−0.874497	−0.437249	+	0.899341i	$0.644047\pi$
$462$	−84.1196	−0.00847099
$463$	−15113.8	−1.51706	−0.758528	−	0.651640i	$-0.774081\pi$
$463$	−15113.8	−1.51706	−0.758528	+	0.651640i	$0.774081\pi$
$464$	12655.2	1.26617
$465$	819.323	0.0817101
$466$	−163.643	−0.0162675
$467$	−508.178	−0.0503547	−0.0251774	−	0.999683i	$-0.508015\pi$
$467$	−508.178	−0.0503547	−0.0251774	+	0.999683i	$0.508015\pi$
$468$	−1622.73	−0.160280
$469$	2347.01	0.231076
$470$	−1000.75	−0.0982157
$471$	1395.48	0.136519
$472$	3194.93	0.311565
$473$	−3278.00	−0.318652
$474$	−366.815	−0.0355451
$475$	836.987	0.0808497
$476$	6499.60	0.625859
$477$	−1125.07	−0.107995
$478$	1286.02	0.123057
$479$	−5389.50	−0.514097	−0.257048	−	0.966399i	$-0.582750\pi$
$479$	−5389.50	−0.514097	−0.257048	+	0.966399i	$0.582750\pi$
$480$	−1290.43	−0.122707
$481$	1565.22	0.148374
$482$	−1307.95	−0.123600
$483$	2939.77	0.276945
$484$	−960.415	−0.0901967
$485$	−1899.57	−0.177846
$486$	−60.8412	−0.00567862
$487$	14509.6	1.35009	0.675045	−	0.737777i	$-0.264124\pi$
$487$	14509.6	1.35009	0.675045	+	0.737777i	$0.264124\pi$
$488$	243.409	0.0225791
$489$	8597.44	0.795071
$490$	−541.863	−0.0499569
$491$	−15135.7	−1.39117	−0.695584	−	0.718445i	$-0.744854\pi$
$491$	−15135.7	−1.39117	−0.695584	+	0.718445i	$0.744854\pi$
$492$	10714.7	0.981817
$493$	−16286.0	−1.48779
$494$	−110.093	−0.0100270
$495$	895.173	0.0812829
$496$	−1887.72	−0.170890
$497$	6291.10	0.567796
$498$	−846.781	−0.0761951
$499$	9752.89	0.874948	0.437474	−	0.899231i	$-0.355873\pi$
$499$	9752.89	0.874948	0.437474	+	0.899231i	$0.355873\pi$
$500$	12074.6	1.07999
$501$	10078.0	0.898708
$502$	224.938	0.0199989
$503$	13103.1	1.16151	0.580755	−	0.814079i	$-0.302758\pi$
$503$	13103.1	1.16151	0.580755	+	0.814079i	$0.302758\pi$
$504$	−365.629	−0.0323143
$505$	16326.2	1.43863
$506$	−265.085	−0.0232894
$507$	5042.95	0.441746
$508$	2452.49	0.214196
$509$	−17620.1	−1.53438	−0.767190	−	0.641420i	$-0.778346\pi$
$509$	−17620.1	−1.53438	−0.767190	+	0.641420i	$0.778346\pi$
$510$	546.267	0.0474297
$511$	5661.32	0.490101
$512$	4968.28	0.428846
$513$	522.638	0.0449806
$514$	−1752.13	−0.150356
$515$	−7113.75	−0.608678
$516$	−7095.95	−0.605391
$517$	−4862.48	−0.413639
$518$	175.642	0.0148982
$519$	8660.40	0.732465
$520$	−819.614	−0.0691201
$521$	5690.07	0.478477	0.239239	−	0.970961i	$-0.423102\pi$
$521$	5690.07	0.478477	0.239239	+	0.970961i	$0.423102\pi$
$522$	456.274	0.0382578
$523$	15837.5	1.32414	0.662071	−	0.749441i	$-0.269678\pi$
$523$	15837.5	1.32414	0.662071	+	0.749441i	$0.269678\pi$
$524$	2270.14	0.189259
$525$	1320.67	0.109788
$526$	−700.617	−0.0580767
$527$	2429.31	0.200802
$528$	−2062.48	−0.169996
$529$	−2902.94	−0.238592
$530$	−283.009	−0.0231946
$531$	−7206.06	−0.588920
$532$	1564.24	0.127478
$533$	10221.5	0.830661
$534$	−133.755	−0.0108392
$535$	−11755.4	−0.949965
$536$	−919.874	−0.0741278
$537$	1449.24	0.116461
$538$	1055.48	0.0845814
$539$	−2632.81	−0.210396
$540$	1937.80	0.154425
$541$	17493.0	1.39017	0.695087	−	0.718926i	$-0.255366\pi$
$541$	17493.0	1.39017	0.695087	+	0.718926i	$0.255366\pi$
$542$	−129.026	−0.0102253
$543$	2791.87	0.220645
$544$	−3826.14	−0.301553
$545$	11001.5	0.864687
$546$	−173.714	−0.0136159
$547$	−12704.3	−0.993050	−0.496525	−	0.868022i	$-0.665391\pi$
$547$	−12704.3	−0.993050	−0.496525	+	0.868022i	$0.665391\pi$
$548$	17699.1	1.37969
$549$	−549.000	−0.0426790
$550$	−119.087	−0.00923254
$551$	−3919.49	−0.303042
$552$	−1152.20	−0.0888422
$553$	4971.95	0.382330
$554$	−1249.94	−0.0958569
$555$	−1869.12	−0.142954
$556$	−11051.0	−0.842923
$557$	19731.7	1.50101	0.750503	−	0.660867i	$-0.229811\pi$
$557$	19731.7	1.50101	0.750503	+	0.660867i	$0.229811\pi$
$558$	−68.0606	−0.00516350
$559$	−6769.36	−0.512188
$560$	5753.60	0.434168
$561$	2654.21	0.199752
$562$	−140.718	−0.0105620
$563$	−13631.1	−1.02039	−0.510196	−	0.860058i	$-0.670427\pi$
$563$	−13631.1	−1.02039	−0.510196	+	0.860058i	$0.670427\pi$
$564$	−10525.9	−0.785852
$565$	2827.41	0.210531
$566$	486.716	0.0361452
$567$	824.664	0.0610805
$568$	−2465.70	−0.182145
$569$	−20872.0	−1.53779	−0.768893	−	0.639378i	$-0.779192\pi$
$569$	−20872.0	−1.53779	−0.768893	+	0.639378i	$0.779192\pi$
$570$	131.468	0.00966072
$571$	−4328.08	−0.317206	−0.158603	−	0.987342i	$-0.550699\pi$
$571$	−4328.08	−0.317206	−0.158603	+	0.987342i	$0.550699\pi$
$572$	−1983.34	−0.144978
$573$	−4164.61	−0.303628
$574$	1147.01	0.0834063
$575$	4161.81	0.301842
$576$	−4392.76	−0.317764
$577$	5951.33	0.429389	0.214694	−	0.976681i	$-0.431124\pi$
$577$	5951.33	0.429389	0.214694	+	0.976681i	$0.431124\pi$
$578$	389.604	0.0280370
$579$	−10110.6	−0.725706
$580$	−14532.4	−1.04039
$581$	11477.6	0.819571
$582$	157.796	0.0112386
$583$	−1375.09	−0.0976850
$584$	−2218.87	−0.157222
$585$	1848.61	0.130651
$586$	1428.13	0.100675
$587$	4638.28	0.326137	0.163068	−	0.986615i	$-0.447861\pi$
$587$	4638.28	0.326137	0.163068	+	0.986615i	$0.447861\pi$
$588$	−5699.30	−0.399720
$589$	584.655	0.0409003
$590$	−1812.67	−0.126485
$591$	12066.0	0.839809
$592$	4306.46	0.298977
$593$	24763.5	1.71486	0.857431	−	0.514598i	$-0.172059\pi$
$593$	24763.5	1.71486	0.857431	+	0.514598i	$0.172059\pi$
$594$	−74.3614	−0.00513651
$595$	−7404.32	−0.510164
$596$	16336.5	1.12277
$597$	11655.2	0.799020
$598$	−547.423	−0.0374344
$599$	−15863.9	−1.08211	−0.541053	−	0.840989i	$-0.681974\pi$
$599$	−15863.9	−1.08211	−0.541053	+	0.840989i	$0.681974\pi$
$600$	−517.617	−0.0352194
$601$	8588.40	0.582908	0.291454	−	0.956585i	$-0.405861\pi$
$601$	8588.40	0.582908	0.291454	+	0.956585i	$0.405861\pi$
$602$	−759.625	−0.0514286
$603$	2074.75	0.140116
$604$	−16138.7	−1.08721
$605$	1094.10	0.0735231
$606$	−1356.21	−0.0909110
$607$	−8657.01	−0.578875	−0.289438	−	0.957197i	$-0.593468\pi$
$607$	−8657.01	−0.578875	−0.289438	+	0.957197i	$0.593468\pi$
$608$	−920.826	−0.0614218
$609$	−6184.51	−0.411509
$610$	−138.100	−0.00916638
$611$	−10041.4	−0.664866
$612$	5745.63	0.379499
$613$	−3569.35	−0.235179	−0.117589	−	0.993062i	$-0.537517\pi$
$613$	−3569.35	−0.235179	−0.117589	+	0.993062i	$0.537517\pi$
$614$	−0.452461	−2.97392e−5 0
$615$	−12206.1	−0.800320
$616$	−446.880	−0.0292294
$617$	1564.45	0.102078	0.0510392	−	0.998697i	$-0.483747\pi$
$617$	1564.45	0.102078	0.0510392	+	0.998697i	$0.483747\pi$
$618$	590.934	0.0384642
$619$	8269.46	0.536959	0.268480	−	0.963285i	$-0.413479\pi$
$619$	8269.46	0.536959	0.268480	+	0.963285i	$0.413479\pi$
$620$	2167.74	0.140417
$621$	2598.75	0.167929
$622$	−1073.10	−0.0691761
$623$	1812.97	0.116589
$624$	−4259.21	−0.273245
$625$	−8350.39	−0.534425
$626$	456.529	0.0291479
$627$	638.780	0.0406865
$628$	3692.13	0.234605
$629$	−5541.99	−0.351309
$630$	207.442	0.0131186
$631$	−29242.0	−1.84486	−0.922429	−	0.386166i	$-0.873799\pi$
$631$	−29242.0	−1.84486	−0.922429	+	0.386166i	$0.873799\pi$
$632$	−1948.68	−0.122649
$633$	13313.3	0.835948
$634$	−1108.56	−0.0694426
$635$	−2793.86	−0.174600
$636$	−2976.69	−0.185587
$637$	−5436.99	−0.338181
$638$	557.668	0.0346055
$639$	5561.31	0.344291
$640$	−4546.12	−0.280783
$641$	−9608.70	−0.592076	−0.296038	−	0.955176i	$-0.595665\pi$
$641$	−9608.70	−0.592076	−0.296038	+	0.955176i	$0.595665\pi$
$642$	976.515	0.0600311
$643$	−18022.8	−1.10536	−0.552682	−	0.833392i	$-0.686396\pi$
$643$	−18022.8	−1.10536	−0.552682	+	0.833392i	$0.686396\pi$
$644$	7777.97	0.475924
$645$	8083.67	0.493480
$646$	389.807	0.0237411
$647$	20929.2	1.27173	0.635866	−	0.771800i	$-0.280643\pi$
$647$	20929.2	1.27173	0.635866	+	0.771800i	$0.280643\pi$
$648$	−323.215	−0.0195943
$649$	−8807.41	−0.532698
$650$	−245.926	−0.0148400
$651$	922.519	0.0555398
$652$	22746.9	1.36631
$653$	19965.3	1.19648	0.598242	−	0.801316i	$-0.295866\pi$
$653$	19965.3	1.19648	0.598242	+	0.801316i	$0.295866\pi$
$654$	−913.891	−0.0546421
$655$	−2586.14	−0.154273
$656$	28122.9	1.67380
$657$	5004.58	0.297180
$658$	−1126.80	−0.0667589
$659$	2238.26	0.132307	0.0661534	−	0.997809i	$-0.478927\pi$
$659$	2238.26	0.132307	0.0661534	+	0.997809i	$0.478927\pi$
$660$	2368.42	0.139683
$661$	6705.98	0.394603	0.197301	−	0.980343i	$-0.436782\pi$
$661$	6705.98	0.394603	0.197301	+	0.980343i	$0.436782\pi$
$662$	−557.422	−0.0327263
$663$	5481.18	0.321073
$664$	−4498.47	−0.262913
$665$	−1781.97	−0.103913
$666$	155.267	0.00903372
$667$	−19489.1	−1.13137
$668$	26664.1	1.54441
$669$	5232.71	0.302404
$670$	521.897	0.0300935
$671$	−671.000	−0.0386046
$672$	−1452.96	−0.0834064
$673$	17901.5	1.02534	0.512668	−	0.858587i	$-0.328657\pi$
$673$	17901.5	1.02534	0.512668	+	0.858587i	$0.328657\pi$
$674$	−1291.83	−0.0738270
$675$	1167.47	0.0665717
$676$	13342.5	0.759132
$677$	15776.5	0.895627	0.447814	−	0.894127i	$-0.352203\pi$
$677$	15776.5	0.895627	0.447814	+	0.894127i	$0.352203\pi$
$678$	−234.871	−0.0133041
$679$	−2138.83	−0.120885
$680$	2902.01	0.163657
$681$	10785.0	0.606877
$682$	−83.1852	−0.00467057
$683$	−15685.0	−0.878723	−0.439362	−	0.898310i	$-0.644795\pi$
$683$	−15685.0	−0.878723	−0.439362	+	0.898310i	$0.644795\pi$
$684$	1382.78	0.0772982
$685$	−20162.8	−1.12464
$686$	−1484.45	−0.0826187
$687$	−15678.5	−0.870704
$688$	−18624.8	−1.03207
$689$	−2839.68	−0.157015
$690$	653.709	0.0360671
$691$	−30938.7	−1.70328	−0.851639	−	0.524129i	$-0.824391\pi$
$691$	−30938.7	−1.70328	−0.851639	+	0.524129i	$0.824391\pi$
$692$	22913.4	1.25873
$693$	1007.92	0.0552494
$694$	1664.85	0.0910618
$695$	12589.2	0.687102
$696$	2423.93	0.132010
$697$	−36191.4	−1.96678
$698$	−246.336	−0.0133581
$699$	1960.78	0.106099
$700$	3494.19	0.188669
$701$	25269.8	1.36152	0.680761	−	0.732506i	$-0.261649\pi$
$701$	25269.8	1.36152	0.680761	+	0.732506i	$0.261649\pi$
$702$	−153.563	−0.00825621
$703$	−1333.77	−0.0715564
$704$	−5368.93	−0.287428
$705$	11991.1	0.640581
$706$	−2199.29	−0.117240
$707$	18382.5	0.977859
$708$	−19065.6	−1.01205
$709$	28591.2	1.51448	0.757240	−	0.653136i	$-0.226547\pi$
$709$	28591.2	1.51448	0.757240	+	0.653136i	$0.226547\pi$
$710$	1398.93	0.0739452
$711$	4395.18	0.231831
$712$	−710.566	−0.0374011
$713$	2907.12	0.152696
$714$	615.072	0.0322388
$715$	2259.41	0.118178
$716$	3834.37	0.200136
$717$	−15409.1	−0.802599
$718$	2898.49	0.150655
$719$	−28920.6	−1.50008	−0.750040	−	0.661392i	$-0.769966\pi$
$719$	−28920.6	−1.50008	−0.750040	+	0.661392i	$0.769966\pi$
$720$	5086.16	0.263264
$721$	−8009.75	−0.413729
$722$	−1623.51	−0.0836853
$723$	15671.8	0.806144
$724$	7386.63	0.379174
$725$	−8755.35	−0.448504
$726$	−90.8862	−0.00464615
$727$	−24680.0	−1.25905	−0.629526	−	0.776979i	$-0.716751\pi$
$727$	−24680.0	−1.25905	−0.629526	+	0.776979i	$0.716751\pi$
$728$	−922.847	−0.0469821
$729$	729.000	0.0370370
$730$	1258.89	0.0638269
$731$	23968.3	1.21272
$732$	−1452.53	−0.0733429
$733$	16318.4	0.822285	0.411142	−	0.911571i	$-0.365130\pi$
$733$	16318.4	0.822285	0.411142	+	0.911571i	$0.365130\pi$
$734$	−2932.19	−0.147451
$735$	6492.62	0.325828
$736$	−4578.68	−0.229310
$737$	2535.80	0.126740
$738$	1013.95	0.0505746
$739$	−22447.5	−1.11738	−0.558690	−	0.829376i	$-0.688696\pi$
$739$	−22447.5	−1.11738	−0.558690	+	0.829376i	$0.688696\pi$
$740$	−4945.26	−0.245664
$741$	1319.14	0.0653978
$742$	−318.656	−0.0157658
$743$	−6282.96	−0.310228	−0.155114	−	0.987897i	$-0.549574\pi$
$743$	−6282.96	−0.310228	−0.155114	+	0.987897i	$0.549574\pi$
$744$	−361.568	−0.0178168
$745$	−18610.5	−0.915214
$746$	−1234.53	−0.0605890
$747$	10146.1	0.496959
$748$	7022.43	0.343269
$749$	−13236.1	−0.645708
$750$	1142.65	0.0556314
$751$	4817.06	0.234057	0.117029	−	0.993129i	$-0.462663\pi$
$751$	4817.06	0.234057	0.117029	+	0.993129i	$0.462663\pi$
$752$	−27627.5	−1.33972
$753$	−2695.21	−0.130437
$754$	1151.63	0.0556234
$755$	18385.1	0.886227
$756$	2181.87	0.104966
$757$	−2224.96	−0.106826	−0.0534132	−	0.998572i	$-0.517010\pi$
$757$	−2224.96	−0.106826	−0.0534132	+	0.998572i	$0.517010\pi$
$758$	2257.06	0.108153
$759$	3176.25	0.151898
$760$	698.418	0.0333346
$761$	−5908.44	−0.281447	−0.140723	−	0.990049i	$-0.544943\pi$
$761$	−5908.44	−0.281447	−0.140723	+	0.990049i	$0.544943\pi$
$762$	232.084	0.0110335
$763$	12387.2	0.587743
$764$	−11018.6	−0.521779
$765$	−6545.39	−0.309345
$766$	−2666.55	−0.125778
$767$	−18188.1	−0.856237
$768$	−11336.4	−0.532639
$769$	−37740.1	−1.76976	−0.884879	−	0.465821i	$-0.845759\pi$
$769$	−37740.1	−1.76976	−0.884879	+	0.465821i	$0.845759\pi$
$770$	253.541	0.0118662
$771$	20994.0	0.980651
$772$	−26750.4	−1.24711
$773$	−9149.43	−0.425721	−0.212860	−	0.977083i	$-0.568278\pi$
$773$	−9149.43	−0.425721	−0.212860	+	0.977083i	$0.568278\pi$
$774$	−671.506	−0.0311845
$775$	1306.00	0.0605328
$776$	838.283	0.0387791
$777$	−2104.54	−0.0971686
$778$	2312.15	0.106548
$779$	−8710.06	−0.400604
$780$	4891.00	0.224520
$781$	6797.16	0.311423
$782$	1938.26	0.0886345
$783$	−5467.09	−0.249524
$784$	−14959.0	−0.681442
$785$	−4206.06	−0.191237
$786$	214.829	0.00974897
$787$	33086.1	1.49859	0.749296	−	0.662236i	$-0.230392\pi$
$787$	33086.1	1.49859	0.749296	+	0.662236i	$0.230392\pi$
$788$	31923.8	1.44319
$789$	8394.81	0.378787
$790$	1105.60	0.0497916
$791$	3183.53	0.143102
$792$	−395.040	−0.0177237
$793$	−1385.67	−0.0620514
$794$	−1184.28	−0.0529327
$795$	3391.03	0.151280
$796$	30837.0	1.37310
$797$	22085.0	0.981545	0.490772	−	0.871288i	$-0.336715\pi$
$797$	22085.0	0.981545	0.490772	+	0.871288i	$0.336715\pi$
$798$	148.027	0.00656656
$799$	35553.8	1.57422
$800$	−2056.94	−0.0909047
$801$	1602.66	0.0706955
$802$	−2789.76	−0.122830
$803$	6116.71	0.268809
$804$	5489.30	0.240787
$805$	−8860.62	−0.387945
$806$	−171.785	−0.00750727
$807$	−12646.7	−0.551656
$808$	−7204.76	−0.313691
$809$	23759.7	1.03257	0.516283	−	0.856418i	$-0.327315\pi$
$809$	23759.7	1.03257	0.516283	+	0.856418i	$0.327315\pi$
$810$	183.378	0.00795463
$811$	−4636.56	−0.200754	−0.100377	−	0.994949i	$-0.532005\pi$
$811$	−4636.56	−0.200754	−0.100377	+	0.994949i	$0.532005\pi$
$812$	−16362.8	−0.707170
$813$	1545.99	0.0666916
$814$	189.770	0.00817131
$815$	−25913.1	−1.11374
$816$	15080.6	0.646969
$817$	5768.38	0.247013
$818$	27.0348	0.00115556
$819$	2081.45	0.0888056
$820$	−32294.5	−1.37533
$821$	−31004.9	−1.31800	−0.659000	−	0.752143i	$-0.729020\pi$
$821$	−31004.9	−1.31800	−0.659000	+	0.752143i	$0.729020\pi$
$822$	1674.91	0.0710695
$823$	−6002.48	−0.254232	−0.127116	−	0.991888i	$-0.540572\pi$
$823$	−6002.48	−0.254232	−0.127116	+	0.991888i	$0.540572\pi$
$824$	3139.30	0.132722
$825$	1426.91	0.0602163
$826$	−2040.98	−0.0859743
$827$	40315.2	1.69516	0.847579	−	0.530669i	$-0.178059\pi$
$827$	40315.2	1.69516	0.847579	+	0.530669i	$0.178059\pi$
$828$	6875.69	0.288583
$829$	−15502.0	−0.649467	−0.324733	−	0.945806i	$-0.605275\pi$
$829$	−15502.0	−0.649467	−0.324733	+	0.945806i	$0.605275\pi$
$830$	2552.24	0.106734
$831$	14976.8	0.625197
$832$	−11087.3	−0.462000
$833$	19250.8	0.800720
$834$	−1045.78	−0.0434200
$835$	−30375.7	−1.25891
$836$	1690.07	0.0699189
$837$	815.504	0.0336773
$838$	526.858	0.0217184
$839$	44990.6	1.85131	0.925655	−	0.378369i	$-0.123515\pi$
$839$	44990.6	1.85131	0.925655	+	0.378369i	$0.123515\pi$
$840$	1102.02	0.0452660
$841$	16611.0	0.681087
$842$	32.1563	0.00131613
$843$	1686.09	0.0688872
$844$	35223.9	1.43656
$845$	−15199.7	−0.618800
$846$	−996.090	−0.0404802
$847$	1231.91	0.0499749
$848$	−7812.93	−0.316388
$849$	−5831.84	−0.235746
$850$	870.751	0.0351371
$851$	−6632.00	−0.267147
$852$	14714.0	0.591657
$853$	−11194.0	−0.449324	−0.224662	−	0.974437i	$-0.572128\pi$
$853$	−11194.0	−0.449324	−0.224662	+	0.974437i	$0.572128\pi$
$854$	−155.494	−0.00623055
$855$	−1575.26	−0.0630090
$856$	5187.68	0.207139
$857$	3840.97	0.153098	0.0765489	−	0.997066i	$-0.475610\pi$
$857$	3840.97	0.153098	0.0765489	+	0.997066i	$0.475610\pi$
$858$	−187.688	−0.00746802
$859$	6728.75	0.267266	0.133633	−	0.991031i	$-0.457336\pi$
$859$	6728.75	0.267266	0.133633	+	0.991031i	$0.457336\pi$
$860$	21387.6	0.848034
$861$	−13743.5	−0.543992
$862$	−682.723	−0.0269764
$863$	−37893.9	−1.49470	−0.747349	−	0.664432i	$-0.768674\pi$
$863$	−37893.9	−1.49470	−0.747349	+	0.664432i	$0.768674\pi$
$864$	−1284.41	−0.0505747
$865$	−26102.9	−1.02604
$866$	−970.289	−0.0380736
$867$	−4668.24	−0.182863
$868$	2440.77	0.0954439
$869$	5371.89	0.209699
$870$	−1375.23	−0.0535917
$871$	5236.65	0.203717
$872$	−4854.99	−0.188544
$873$	−1890.72	−0.0733003
$874$	466.476	0.0180535
$875$	−15487.9	−0.598384
$876$	13241.0	0.510698
$877$	−15552.8	−0.598837	−0.299418	−	0.954122i	$-0.596793\pi$
$877$	−15552.8	−0.598837	−0.299418	+	0.954122i	$0.596793\pi$
$878$	−1746.50	−0.0671315
$879$	−17111.9	−0.656620
$880$	6216.42	0.238131
$881$	−7381.08	−0.282265	−0.141132	−	0.989991i	$-0.545074\pi$
$881$	−7381.08	−0.282265	−0.141132	+	0.989991i	$0.545074\pi$
$882$	−539.338	−0.0205901
$883$	28903.2	1.10155	0.550776	−	0.834653i	$-0.314332\pi$
$883$	28903.2	1.10155	0.550776	+	0.834653i	$0.314332\pi$
$884$	14501.9	0.551757
$885$	21719.4	0.824961
$886$	4026.82	0.152690
$887$	−10942.2	−0.414210	−0.207105	−	0.978319i	$-0.566404\pi$
$887$	−10942.2	−0.414210	−0.207105	+	0.978319i	$0.566404\pi$
$888$	824.844	0.0311711
$889$	−3145.76	−0.118679
$890$	403.145	0.0151836
$891$	891.000	0.0335013
$892$	13844.6	0.519675
$893$	8556.62	0.320646
$894$	1545.96	0.0578351
$895$	−4368.09	−0.163139
$896$	−5118.72	−0.190853
$897$	6559.24	0.244154
$898$	1567.63	0.0582544
$899$	−6115.81	−0.226890
$900$	3088.85	0.114402
$901$	10054.5	0.371768
$902$	1239.27	0.0457465
$903$	9101.84	0.335427
$904$	−1247.74	−0.0459062
$905$	−8414.82	−0.309081
$906$	−1527.24	−0.0560034
$907$	−42480.9	−1.55519	−0.777595	−	0.628766i	$-0.783560\pi$
$907$	−42480.9	−1.55519	−0.777595	+	0.628766i	$0.783560\pi$
$908$	28534.7	1.04290
$909$	16250.1	0.592939
$910$	523.584	0.0190732
$911$	12977.3	0.471960	0.235980	−	0.971758i	$-0.424170\pi$
$911$	12977.3	0.471960	0.235980	+	0.971758i	$0.424170\pi$
$912$	3629.40	0.131778
$913$	12400.8	0.449516
$914$	−1852.04	−0.0670241
$915$	1654.71	0.0597848
$916$	−41481.8	−1.49629
$917$	−2911.87	−0.104862
$918$	543.721	0.0195484
$919$	−28700.0	−1.03017	−0.515085	−	0.857139i	$-0.672240\pi$
$919$	−28700.0	−1.03017	−0.515085	+	0.857139i	$0.672240\pi$
$920$	3472.79	0.124451
$921$	5.42140	0.000193964 0
$922$	−2167.21	−0.0774113
$923$	14036.7	0.500568
$924$	2666.73	0.0949449
$925$	−2979.38	−0.105904
$926$	−3784.12	−0.134291
$927$	−7080.59	−0.250871
$928$	9632.35	0.340730
$929$	22430.6	0.792169	0.396084	−	0.918214i	$-0.370369\pi$
$929$	22430.6	0.792169	0.396084	+	0.918214i	$0.370369\pi$
$930$	205.138	0.00723305
$931$	4633.02	0.163095
$932$	5187.77	0.182329
$933$	12858.0	0.451180
$934$	−127.235	−0.00445745
$935$	−7999.92	−0.279813
$936$	−815.793	−0.0284883
$937$	15667.6	0.546252	0.273126	−	0.961978i	$-0.411942\pi$
$937$	15667.6	0.546252	0.273126	+	0.961978i	$0.411942\pi$
$938$	587.632	0.0204551
$939$	−5470.14	−0.190108
$940$	31725.6	1.10082
$941$	35239.5	1.22080	0.610401	−	0.792093i	$-0.291008\pi$
$941$	35239.5	1.22080	0.610401	+	0.792093i	$0.291008\pi$
$942$	349.395	0.0120848
$943$	−43309.6	−1.49560
$944$	−50041.6	−1.72534
$945$	−2485.58	−0.0855618
$946$	−820.729	−0.0282074
$947$	12778.9	0.438500	0.219250	−	0.975669i	$-0.429639\pi$
$947$	12778.9	0.438500	0.219250	+	0.975669i	$0.429639\pi$
$948$	11628.6	0.398397
$949$	12631.5	0.432073
$950$	209.561	0.00715689
$951$	13282.8	0.452918
$952$	3267.53	0.111241
$953$	3803.59	0.129287	0.0646435	−	0.997908i	$-0.479409\pi$
$953$	3803.59	0.129287	0.0646435	+	0.997908i	$0.479409\pi$
$954$	−281.690	−0.00955981
$955$	12552.3	0.425324
$956$	−40768.9	−1.37925
$957$	−6681.99	−0.225703
$958$	−1349.40	−0.0455084
$959$	−22702.4	−0.764439
$960$	13240.0	0.445124
$961$	−28878.7	−0.969378
$962$	391.892	0.0131342
$963$	−11700.6	−0.391534
$964$	41464.1	1.38534
$965$	30474.0	1.01657
$966$	736.046	0.0245154
$967$	28169.9	0.936797	0.468399	−	0.883517i	$-0.344831\pi$
$967$	28169.9	0.936797	0.468399	+	0.883517i	$0.344831\pi$
$968$	−482.827	−0.0160317
$969$	−4670.68	−0.154844
$970$	−475.606	−0.0157431
$971$	10072.6	0.332899	0.166450	−	0.986050i	$-0.446770\pi$
$971$	10072.6	0.332899	0.166450	+	0.986050i	$0.446770\pi$
$972$	1928.77	0.0636474
$973$	14174.9	0.467035
$974$	3632.85	0.119511
$975$	2946.69	0.0967892
$976$	−3812.46	−0.125035
$977$	24295.4	0.795578	0.397789	−	0.917477i	$-0.369778\pi$
$977$	24295.4	0.795578	0.397789	+	0.917477i	$0.369778\pi$
$978$	2152.58	0.0703804
$979$	1958.80	0.0639465
$980$	17178.0	0.559929
$981$	10950.3	0.356386
$982$	−3789.60	−0.123148
$983$	19818.8	0.643054	0.321527	−	0.946900i	$-0.395804\pi$
$983$	19818.8	0.643054	0.321527	+	0.946900i	$0.395804\pi$
$984$	5386.55	0.174509
$985$	−36367.4	−1.17641
$986$	−4077.60	−0.131701
$987$	13501.4	0.435414
$988$	3490.14	0.112385
$989$	28682.5	0.922194
$990$	224.129	0.00719524
$991$	−48232.5	−1.54607	−0.773035	−	0.634363i	$-0.781262\pi$
$991$	−48232.5	−1.54607	−0.773035	+	0.634363i	$0.781262\pi$
$992$	−1436.82	−0.0459870
$993$	6679.04	0.213447
$994$	1575.14	0.0502618
$995$	−35129.3	−1.11927
$996$	26844.4	0.854013
$997$	−55390.0	−1.75950	−0.879748	−	0.475440i	$-0.842289\pi$
$997$	−55390.0	−1.75950	−0.879748	+	0.475440i	$0.842289\pi$
$998$	2441.88	0.0774513
$999$	−1860.41	−0.0589196

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2013.4.a.d.1.20	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.4.a.d.1.20	✓	37	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2013 = 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2013.a (trivial)

\(f(q)\)	\(=\)	\(q+0.250375 q^{2} -3.00000 q^{3} -7.93731 q^{4} +9.04215 q^{5} -0.751125 q^{6} +10.1810 q^{7} -3.99031 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+0.250375 q^{2} -3.00000 q^{3} -7.93731 q^{4} +9.04215 q^{5} -0.751125 q^{6} +10.1810 q^{7} -3.99031 q^{8} +9.00000 q^{9} +2.26393 q^{10} +11.0000 q^{11} +23.8119 q^{12} +22.7160 q^{13} +2.54908 q^{14} -27.1264 q^{15} +62.4994 q^{16} -80.4306 q^{17} +2.25338 q^{18} -19.3570 q^{19} -71.7703 q^{20} -30.5431 q^{21} +2.75413 q^{22} -96.2500 q^{23} +11.9709 q^{24} -43.2396 q^{25} +5.68751 q^{26} -27.0000 q^{27} -80.8101 q^{28} +202.485 q^{29} -6.79179 q^{30} -30.2038 q^{31} +47.5708 q^{32} -33.0000 q^{33} -20.1378 q^{34} +92.0585 q^{35} -71.4358 q^{36} +68.9040 q^{37} -4.84651 q^{38} -68.1479 q^{39} -36.0809 q^{40} +449.970 q^{41} -7.64724 q^{42} -298.000 q^{43} -87.3104 q^{44} +81.3793 q^{45} -24.0986 q^{46} -442.043 q^{47} -187.498 q^{48} -239.346 q^{49} -10.8261 q^{50} +241.292 q^{51} -180.304 q^{52} -125.008 q^{53} -6.76013 q^{54} +99.4636 q^{55} -40.6255 q^{56} +58.0709 q^{57} +50.6971 q^{58} -800.674 q^{59} +215.311 q^{60} -61.0000 q^{61} -7.56229 q^{62} +91.6294 q^{63} -488.085 q^{64} +205.401 q^{65} -8.26238 q^{66} +230.527 q^{67} +638.403 q^{68} +288.750 q^{69} +23.0491 q^{70} +617.923 q^{71} -35.9128 q^{72} +556.065 q^{73} +17.2518 q^{74} +129.719 q^{75} +153.642 q^{76} +111.991 q^{77} -17.0625 q^{78} +488.353 q^{79} +565.129 q^{80} +81.0000 q^{81} +112.661 q^{82} +1127.35 q^{83} +242.430 q^{84} -727.265 q^{85} -74.6117 q^{86} -607.454 q^{87} -43.8934 q^{88} +178.073 q^{89} +20.3754 q^{90} +231.272 q^{91} +763.966 q^{92} +90.6115 q^{93} -110.677 q^{94} -175.029 q^{95} -142.712 q^{96} -210.080 q^{97} -59.9264 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q - 4 q^{2} - 111 q^{3} + 158 q^{4} - 15 q^{5} + 12 q^{6} - 77 q^{7} - 69 q^{8} + 333 q^{9}+O(q^{10}) \) 37 * q - 4 * q^2 - 111 * q^3 + 158 * q^4 - 15 * q^5 + 12 * q^6 - 77 * q^7 - 69 * q^8 + 333 * q^9	\( 37 q - 4 q^{2} - 111 q^{3} + 158 q^{4} - 15 q^{5} + 12 q^{6} - 77 q^{7} - 69 q^{8} + 333 q^{9} - 45 q^{10} + 407 q^{11} - 474 q^{12} - 169 q^{13} + 102 q^{14} + 45 q^{15} + 598 q^{16} - 338 q^{17} - 36 q^{18} - 235 q^{19} - 550 q^{20} + 231 q^{21} - 44 q^{22} - 53 q^{23} + 207 q^{24} + 750 q^{25} - 75 q^{26} - 999 q^{27} - 1378 q^{28} - 30 q^{29} + 135 q^{30} - 506 q^{31} - 841 q^{32} - 1221 q^{33} - 316 q^{34} - 822 q^{35} + 1422 q^{36} - 830 q^{37} - 371 q^{38} + 507 q^{39} - 613 q^{40} + 16 q^{41} - 306 q^{42} - 1137 q^{43} + 1738 q^{44} - 135 q^{45} - 659 q^{46} - 489 q^{47} - 1794 q^{48} + 2214 q^{49} + 1066 q^{50} + 1014 q^{51} - 2342 q^{52} + 731 q^{53} + 108 q^{54} - 165 q^{55} + 3051 q^{56} + 705 q^{57} - 611 q^{58} - 425 q^{59} + 1650 q^{60} - 2257 q^{61} + 453 q^{62} - 693 q^{63} + 4919 q^{64} + 1346 q^{65} + 132 q^{66} - 1907 q^{67} - 3236 q^{68} + 159 q^{69} - 1050 q^{70} - 561 q^{71} - 621 q^{72} - 2397 q^{73} - 1840 q^{74} - 2250 q^{75} - 3868 q^{76} - 847 q^{77} + 225 q^{78} + 393 q^{79} - 4031 q^{80} + 2997 q^{81} - 1946 q^{82} - 4191 q^{83} + 4134 q^{84} - 2667 q^{85} + 2405 q^{86} + 90 q^{87} - 759 q^{88} + 1437 q^{89} - 405 q^{90} - 5192 q^{91} - 737 q^{92} + 1518 q^{93} - 1960 q^{94} + 1356 q^{95} + 2523 q^{96} - 2368 q^{97} - 3014 q^{98} + 3663 q^{99}+O(q^{100}) \) 37 * q - 4 * q^2 - 111 * q^3 + 158 * q^4 - 15 * q^5 + 12 * q^6 - 77 * q^7 - 69 * q^8 + 333 * q^9 - 45 * q^10 + 407 * q^11 - 474 * q^12 - 169 * q^13 + 102 * q^14 + 45 * q^15 + 598 * q^16 - 338 * q^17 - 36 * q^18 - 235 * q^19 - 550 * q^20 + 231 * q^21 - 44 * q^22 - 53 * q^23 + 207 * q^24 + 750 * q^25 - 75 * q^26 - 999 * q^27 - 1378 * q^28 - 30 * q^29 + 135 * q^30 - 506 * q^31 - 841 * q^32 - 1221 * q^33 - 316 * q^34 - 822 * q^35 + 1422 * q^36 - 830 * q^37 - 371 * q^38 + 507 * q^39 - 613 * q^40 + 16 * q^41 - 306 * q^42 - 1137 * q^43 + 1738 * q^44 - 135 * q^45 - 659 * q^46 - 489 * q^47 - 1794 * q^48 + 2214 * q^49 + 1066 * q^50 + 1014 * q^51 - 2342 * q^52 + 731 * q^53 + 108 * q^54 - 165 * q^55 + 3051 * q^56 + 705 * q^57 - 611 * q^58 - 425 * q^59 + 1650 * q^60 - 2257 * q^61 + 453 * q^62 - 693 * q^63 + 4919 * q^64 + 1346 * q^65 + 132 * q^66 - 1907 * q^67 - 3236 * q^68 + 159 * q^69 - 1050 * q^70 - 561 * q^71 - 621 * q^72 - 2397 * q^73 - 1840 * q^74 - 2250 * q^75 - 3868 * q^76 - 847 * q^77 + 225 * q^78 + 393 * q^79 - 4031 * q^80 + 2997 * q^81 - 1946 * q^82 - 4191 * q^83 + 4134 * q^84 - 2667 * q^85 + 2405 * q^86 + 90 * q^87 - 759 * q^88 + 1437 * q^89 - 405 * q^90 - 5192 * q^91 - 737 * q^92 + 1518 * q^93 - 1960 * q^94 + 1356 * q^95 + 2523 * q^96 - 2368 * q^97 - 3014 * q^98 + 3663 * q^99

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