LMFDB - Embedded newform 8018.2.a.j.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8018,2,Mod(1,8018)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8018, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8018.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8018 = 2 \cdot 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8018.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:	$64.0240523407$
Analytic rank:	$0$
Dimension:	$47$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8018.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.28341 q^{3} +1.00000 q^{4} +4.33431 q^{5} -2.28341 q^{6} -0.955796 q^{7} +1.00000 q^{8} +2.21398 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.28341 q^{3} +1.00000 q^{4} +4.33431 q^{5} -2.28341 q^{6} -0.955796 q^{7} +1.00000 q^{8} +2.21398 q^{9} +4.33431 q^{10} -5.13031 q^{11} -2.28341 q^{12} -0.223732 q^{13} -0.955796 q^{14} -9.89701 q^{15} +1.00000 q^{16} +2.89292 q^{17} +2.21398 q^{18} -1.00000 q^{19} +4.33431 q^{20} +2.18248 q^{21} -5.13031 q^{22} -3.30652 q^{23} -2.28341 q^{24} +13.7862 q^{25} -0.223732 q^{26} +1.79481 q^{27} -0.955796 q^{28} -0.0537792 q^{29} -9.89701 q^{30} -1.90761 q^{31} +1.00000 q^{32} +11.7146 q^{33} +2.89292 q^{34} -4.14271 q^{35} +2.21398 q^{36} -2.29313 q^{37} -1.00000 q^{38} +0.510873 q^{39} +4.33431 q^{40} +0.965538 q^{41} +2.18248 q^{42} -5.82516 q^{43} -5.13031 q^{44} +9.59606 q^{45} -3.30652 q^{46} +9.69579 q^{47} -2.28341 q^{48} -6.08645 q^{49} +13.7862 q^{50} -6.60574 q^{51} -0.223732 q^{52} +10.8683 q^{53} +1.79481 q^{54} -22.2363 q^{55} -0.955796 q^{56} +2.28341 q^{57} -0.0537792 q^{58} +4.27122 q^{59} -9.89701 q^{60} +1.64689 q^{61} -1.90761 q^{62} -2.11611 q^{63} +1.00000 q^{64} -0.969724 q^{65} +11.7146 q^{66} -11.7055 q^{67} +2.89292 q^{68} +7.55016 q^{69} -4.14271 q^{70} +6.06103 q^{71} +2.21398 q^{72} +3.78080 q^{73} -2.29313 q^{74} -31.4796 q^{75} -1.00000 q^{76} +4.90353 q^{77} +0.510873 q^{78} +9.19955 q^{79} +4.33431 q^{80} -10.7402 q^{81} +0.965538 q^{82} +16.2278 q^{83} +2.18248 q^{84} +12.5388 q^{85} -5.82516 q^{86} +0.122800 q^{87} -5.13031 q^{88} +15.5406 q^{89} +9.59606 q^{90} +0.213842 q^{91} -3.30652 q^{92} +4.35586 q^{93} +9.69579 q^{94} -4.33431 q^{95} -2.28341 q^{96} +12.7256 q^{97} -6.08645 q^{98} -11.3584 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 47 q + 47 q^{2} + 10 q^{3} + 47 q^{4} + 15 q^{5} + 10 q^{6} + 47 q^{8} + 69 q^{9}+O(q^{10}) $ 47 * q + 47 * q^2 + 10 * q^3 + 47 * q^4 + 15 * q^5 + 10 * q^6 + 47 * q^8 + 69 * q^9	\( 47 q + 47 q^{2} + 10 q^{3} + 47 q^{4} + 15 q^{5} + 10 q^{6} + 47 q^{8} + 69 q^{9} + 15 q^{10} + 17 q^{11} + 10 q^{12} + 27 q^{13} + 10 q^{15} + 47 q^{16} + 16 q^{17} + 69 q^{18} - 47 q^{19} + 15 q^{20} + 26 q^{21} + 17 q^{22} + 24 q^{23} + 10 q^{24} + 86 q^{25} + 27 q^{26} + 43 q^{27} + 58 q^{29} + 10 q^{30} + 21 q^{31} + 47 q^{32} + 18 q^{33} + 16 q^{34} + 24 q^{35} + 69 q^{36} + 78 q^{37} - 47 q^{38} - 4 q^{39} + 15 q^{40} + 53 q^{41} + 26 q^{42} + 47 q^{43} + 17 q^{44} + 23 q^{45} + 24 q^{46} + 16 q^{47} + 10 q^{48} + 93 q^{49} + 86 q^{50} + 28 q^{51} + 27 q^{52} + 43 q^{53} + 43 q^{54} + 23 q^{55} - 10 q^{57} + 58 q^{58} + 3 q^{59} + 10 q^{60} + 12 q^{61} + 21 q^{62} - 5 q^{63} + 47 q^{64} + 62 q^{65} + 18 q^{66} + 68 q^{67} + 16 q^{68} + 12 q^{69} + 24 q^{70} - 13 q^{71} + 69 q^{72} + 35 q^{73} + 78 q^{74} + 22 q^{75} - 47 q^{76} + 26 q^{77} - 4 q^{78} + 21 q^{79} + 15 q^{80} + 123 q^{81} + 53 q^{82} + 23 q^{83} + 26 q^{84} + 38 q^{85} + 47 q^{86} + 39 q^{87} + 17 q^{88} + 24 q^{89} + 23 q^{90} + 49 q^{91} + 24 q^{92} + 91 q^{93} + 16 q^{94} - 15 q^{95} + 10 q^{96} + 88 q^{97} + 93 q^{98} + 26 q^{99}+O(q^{100}) \) 47 * q + 47 * q^2 + 10 * q^3 + 47 * q^4 + 15 * q^5 + 10 * q^6 + 47 * q^8 + 69 * q^9 + 15 * q^10 + 17 * q^11 + 10 * q^12 + 27 * q^13 + 10 * q^15 + 47 * q^16 + 16 * q^17 + 69 * q^18 - 47 * q^19 + 15 * q^20 + 26 * q^21 + 17 * q^22 + 24 * q^23 + 10 * q^24 + 86 * q^25 + 27 * q^26 + 43 * q^27 + 58 * q^29 + 10 * q^30 + 21 * q^31 + 47 * q^32 + 18 * q^33 + 16 * q^34 + 24 * q^35 + 69 * q^36 + 78 * q^37 - 47 * q^38 - 4 * q^39 + 15 * q^40 + 53 * q^41 + 26 * q^42 + 47 * q^43 + 17 * q^44 + 23 * q^45 + 24 * q^46 + 16 * q^47 + 10 * q^48 + 93 * q^49 + 86 * q^50 + 28 * q^51 + 27 * q^52 + 43 * q^53 + 43 * q^54 + 23 * q^55 - 10 * q^57 + 58 * q^58 + 3 * q^59 + 10 * q^60 + 12 * q^61 + 21 * q^62 - 5 * q^63 + 47 * q^64 + 62 * q^65 + 18 * q^66 + 68 * q^67 + 16 * q^68 + 12 * q^69 + 24 * q^70 - 13 * q^71 + 69 * q^72 + 35 * q^73 + 78 * q^74 + 22 * q^75 - 47 * q^76 + 26 * q^77 - 4 * q^78 + 21 * q^79 + 15 * q^80 + 123 * q^81 + 53 * q^82 + 23 * q^83 + 26 * q^84 + 38 * q^85 + 47 * q^86 + 39 * q^87 + 17 * q^88 + 24 * q^89 + 23 * q^90 + 49 * q^91 + 24 * q^92 + 91 * q^93 + 16 * q^94 - 15 * q^95 + 10 * q^96 + 88 * q^97 + 93 * q^98 + 26 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.28341	−1.31833	−0.659165	−	0.751999i	$-0.729090\pi$
$3$	−2.28341	−1.31833	−0.659165	+	0.751999i	$0.729090\pi$
$4$	1.00000	0.500000
$5$	4.33431	1.93836	0.969180	−	0.246352i	$-0.0792321\pi$
$5$	4.33431	1.93836	0.969180	+	0.246352i	$0.0792321\pi$
$6$	−2.28341	−0.932200
$7$	−0.955796	−0.361257	−0.180628	−	0.983551i	$-0.557813\pi$
$7$	−0.955796	−0.361257	−0.180628	+	0.983551i	$0.557813\pi$
$8$	1.00000	0.353553
$9$	2.21398	0.737993
$10$	4.33431	1.37063
$11$	−5.13031	−1.54685	−0.773423	−	0.633890i	$-0.781457\pi$
$11$	−5.13031	−1.54685	−0.773423	+	0.633890i	$0.781457\pi$
$12$	−2.28341	−0.659165
$13$	−0.223732	−0.0620522	−0.0310261	−	0.999519i	$-0.509877\pi$
$13$	−0.223732	−0.0620522	−0.0310261	+	0.999519i	$0.509877\pi$
$14$	−0.955796	−0.255447
$15$	−9.89701	−2.55540
$16$	1.00000	0.250000
$17$	2.89292	0.701637	0.350819	−	0.936443i	$-0.385903\pi$
$17$	2.89292	0.701637	0.350819	+	0.936443i	$0.385903\pi$
$18$	2.21398	0.521840
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	4.33431	0.969180
$21$	2.18248	0.476256
$22$	−5.13031	−1.09379
$23$	−3.30652	−0.689458	−0.344729	−	0.938702i	$-0.612029\pi$
$23$	−3.30652	−0.689458	−0.344729	+	0.938702i	$0.612029\pi$
$24$	−2.28341	−0.466100
$25$	13.7862	2.75724
$26$	−0.223732	−0.0438775
$27$	1.79481	0.345412
$28$	−0.955796	−0.180628
$29$	−0.0537792	−0.00998654	−0.00499327	−	0.999988i	$-0.501589\pi$
$29$	−0.0537792	−0.00998654	−0.00499327	+	0.999988i	$0.501589\pi$
$30$	−9.89701	−1.80694
$31$	−1.90761	−0.342617	−0.171308	−	0.985217i	$-0.554799\pi$
$31$	−1.90761	−0.342617	−0.171308	+	0.985217i	$0.554799\pi$
$32$	1.00000	0.176777
$33$	11.7146	2.03925
$34$	2.89292	0.496132
$35$	−4.14271	−0.700246
$36$	2.21398	0.368996
$37$	−2.29313	−0.376989	−0.188494	−	0.982074i	$-0.560361\pi$
$37$	−2.29313	−0.376989	−0.188494	+	0.982074i	$0.560361\pi$
$38$	−1.00000	−0.162221
$39$	0.510873	0.0818052
$40$	4.33431	0.685314
$41$	0.965538	0.150792	0.0753958	−	0.997154i	$-0.475978\pi$
$41$	0.965538	0.150792	0.0753958	+	0.997154i	$0.475978\pi$
$42$	2.18248	0.336764
$43$	−5.82516	−0.888328	−0.444164	−	0.895945i	$-0.646499\pi$
$43$	−5.82516	−0.888328	−0.444164	+	0.895945i	$0.646499\pi$
$44$	−5.13031	−0.773423
$45$	9.59606	1.43050
$46$	−3.30652	−0.487520
$47$	9.69579	1.41428	0.707138	−	0.707076i	$-0.249986\pi$
$47$	9.69579	1.41428	0.707138	+	0.707076i	$0.249986\pi$
$48$	−2.28341	−0.329582
$49$	−6.08645	−0.869493
$50$	13.7862	1.94966
$51$	−6.60574	−0.924989
$52$	−0.223732	−0.0310261
$53$	10.8683	1.49287	0.746437	−	0.665456i	$-0.231763\pi$
$53$	10.8683	1.49287	0.746437	+	0.665456i	$0.231763\pi$
$54$	1.79481	0.244243
$55$	−22.2363	−2.99835
$56$	−0.955796	−0.127724
$57$	2.28341	0.302446
$58$	−0.0537792	−0.00706155
$59$	4.27122	0.556066	0.278033	−	0.960572i	$-0.410318\pi$
$59$	4.27122	0.556066	0.278033	+	0.960572i	$0.410318\pi$
$60$	−9.89701	−1.27770
$61$	1.64689	0.210862	0.105431	−	0.994427i	$-0.466378\pi$
$61$	1.64689	0.210862	0.105431	+	0.994427i	$0.466378\pi$
$62$	−1.90761	−0.242267
$63$	−2.11611	−0.266605
$64$	1.00000	0.125000
$65$	−0.969724	−0.120280
$66$	11.7146	1.44197
$67$	−11.7055	−1.43006	−0.715028	−	0.699096i	$-0.753586\pi$
$67$	−11.7055	−1.43006	−0.715028	+	0.699096i	$0.753586\pi$
$68$	2.89292	0.350819
$69$	7.55016	0.908933
$70$	−4.14271	−0.495149
$71$	6.06103	0.719312	0.359656	−	0.933085i	$-0.382894\pi$
$71$	6.06103	0.719312	0.359656	+	0.933085i	$0.382894\pi$
$72$	2.21398	0.260920
$73$	3.78080	0.442509	0.221255	−	0.975216i	$-0.428985\pi$
$73$	3.78080	0.442509	0.221255	+	0.975216i	$0.428985\pi$
$74$	−2.29313	−0.266571
$75$	−31.4796	−3.63495
$76$	−1.00000	−0.114708
$77$	4.90353	0.558809
$78$	0.510873	0.0578450
$79$	9.19955	1.03503	0.517515	−	0.855674i	$-0.326857\pi$
$79$	9.19955	1.03503	0.517515	+	0.855674i	$0.326857\pi$
$80$	4.33431	0.484590
$81$	−10.7402	−1.19336
$82$	0.965538	0.106626
$83$	16.2278	1.78123	0.890616	−	0.454756i	$-0.150273\pi$
$83$	16.2278	1.78123	0.890616	+	0.454756i	$0.150273\pi$
$84$	2.18248	0.238128
$85$	12.5388	1.36003
$86$	−5.82516	−0.628143
$87$	0.122800	0.0131656
$88$	−5.13031	−0.546893
$89$	15.5406	1.64730	0.823652	−	0.567095i	$-0.191933\pi$
$89$	15.5406	1.64730	0.823652	+	0.567095i	$0.191933\pi$
$90$	9.59606	1.01151
$91$	0.213842	0.0224168
$92$	−3.30652	−0.344729
$93$	4.35586	0.451682
$94$	9.69579	1.00004
$95$	−4.33431	−0.444690
$96$	−2.28341	−0.233050
$97$	12.7256	1.29209	0.646045	−	0.763299i	$-0.276422\pi$
$97$	12.7256	1.29209	0.646045	+	0.763299i	$0.276422\pi$
$98$	−6.08645	−0.614825
$99$	−11.3584	−1.14156
$100$	13.7862	1.37862
$101$	17.0356	1.69510	0.847551	−	0.530714i	$-0.178076\pi$
$101$	17.0356	1.69510	0.847551	+	0.530714i	$0.178076\pi$
$102$	−6.60574	−0.654066
$103$	−10.5631	−1.04081	−0.520407	−	0.853918i	$-0.674220\pi$
$103$	−10.5631	−1.04081	−0.520407	+	0.853918i	$0.674220\pi$
$104$	−0.223732	−0.0219388
$105$	9.45952	0.923155
$106$	10.8683	1.05562
$107$	−3.59047	−0.347104	−0.173552	−	0.984825i	$-0.555524\pi$
$107$	−3.59047	−0.347104	−0.173552	+	0.984825i	$0.555524\pi$
$108$	1.79481	0.172706
$109$	4.87950	0.467372	0.233686	−	0.972312i	$-0.424921\pi$
$109$	4.87950	0.467372	0.233686	+	0.972312i	$0.424921\pi$
$110$	−22.2363	−2.12015
$111$	5.23617	0.496996
$112$	−0.955796	−0.0903142
$113$	−8.29563	−0.780388	−0.390194	−	0.920733i	$-0.627592\pi$
$113$	−8.29563	−0.780388	−0.390194	+	0.920733i	$0.627592\pi$
$114$	2.28341	0.213861
$115$	−14.3315	−1.33642
$116$	−0.0537792	−0.00499327
$117$	−0.495339	−0.0457941
$118$	4.27122	0.393198
$119$	−2.76504	−0.253471
$120$	−9.89701	−0.903470
$121$	15.3201	1.39273
$122$	1.64689	0.149102
$123$	−2.20472	−0.198793
$124$	−1.90761	−0.171308
$125$	38.0821	3.40617
$126$	−2.11611	−0.188518
$127$	19.3439	1.71649	0.858247	−	0.513237i	$-0.171554\pi$
$127$	19.3439	1.71649	0.858247	+	0.513237i	$0.171554\pi$
$128$	1.00000	0.0883883
$129$	13.3012	1.17111
$130$	−0.969724	−0.0850505
$131$	9.20602	0.804334	0.402167	−	0.915566i	$-0.368257\pi$
$131$	9.20602	0.804334	0.402167	+	0.915566i	$0.368257\pi$
$132$	11.7146	1.01963
$133$	0.955796	0.0828780
$134$	−11.7055	−1.01120
$135$	7.77926	0.669532
$136$	2.89292	0.248066
$137$	−16.3335	−1.39547	−0.697734	−	0.716357i	$-0.745808\pi$
$137$	−16.3335	−1.39547	−0.697734	+	0.716357i	$0.745808\pi$
$138$	7.55016	0.642713
$139$	9.81016	0.832087	0.416043	−	0.909345i	$-0.363416\pi$
$139$	9.81016	0.832087	0.416043	+	0.909345i	$0.363416\pi$
$140$	−4.14271	−0.350123
$141$	−22.1395	−1.86448
$142$	6.06103	0.508631
$143$	1.14782	0.0959852
$144$	2.21398	0.184498
$145$	−0.233095	−0.0193575
$146$	3.78080	0.312901
$147$	13.8979	1.14628
$148$	−2.29313	−0.188494
$149$	0.990937	0.0811808	0.0405904	−	0.999176i	$-0.487076\pi$
$149$	0.990937	0.0811808	0.0405904	+	0.999176i	$0.487076\pi$
$150$	−31.4796	−2.57030
$151$	10.4306	0.848827	0.424414	−	0.905468i	$-0.360480\pi$
$151$	10.4306	0.848827	0.424414	+	0.905468i	$0.360480\pi$
$152$	−1.00000	−0.0811107
$153$	6.40487	0.517803
$154$	4.90353	0.395137
$155$	−8.26816	−0.664115
$156$	0.510873	0.0409026
$157$	−8.84405	−0.705832	−0.352916	−	0.935655i	$-0.614810\pi$
$157$	−8.84405	−0.705832	−0.352916	+	0.935655i	$0.614810\pi$
$158$	9.19955	0.731877
$159$	−24.8168	−1.96810
$160$	4.33431	0.342657
$161$	3.16036	0.249071
$162$	−10.7402	−0.843832
$163$	0.889463	0.0696681	0.0348340	−	0.999393i	$-0.488910\pi$
$163$	0.889463	0.0696681	0.0348340	+	0.999393i	$0.488910\pi$
$164$	0.965538	0.0753958
$165$	50.7747	3.95281
$166$	16.2278	1.25952
$167$	7.07473	0.547459	0.273729	−	0.961807i	$-0.411743\pi$
$167$	7.07473	0.547459	0.273729	+	0.961807i	$0.411743\pi$
$168$	2.18248	0.168382
$169$	−12.9499	−0.996150
$170$	12.5388	0.961683
$171$	−2.21398	−0.169307
$172$	−5.82516	−0.444164
$173$	−2.39066	−0.181758	−0.0908792	−	0.995862i	$-0.528968\pi$
$173$	−2.39066	−0.181758	−0.0908792	+	0.995862i	$0.528968\pi$
$174$	0.122800	0.00930945
$175$	−13.1768	−0.996073
$176$	−5.13031	−0.386712
$177$	−9.75297	−0.733078
$178$	15.5406	1.16482
$179$	8.08589	0.604368	0.302184	−	0.953250i	$-0.402284\pi$
$179$	8.08589	0.604368	0.302184	+	0.953250i	$0.402284\pi$
$180$	9.59606	0.715248
$181$	0.291984	0.0217030	0.0108515	−	0.999941i	$-0.496546\pi$
$181$	0.291984	0.0217030	0.0108515	+	0.999941i	$0.496546\pi$
$182$	0.213842	0.0158511
$183$	−3.76053	−0.277986
$184$	−3.30652	−0.243760
$185$	−9.93915	−0.730741
$186$	4.35586	0.319387
$187$	−14.8416	−1.08532
$188$	9.69579	0.707138
$189$	−1.71547	−0.124782
$190$	−4.33431	−0.314444
$191$	−16.3318	−1.18172	−0.590862	−	0.806772i	$-0.701212\pi$
$191$	−16.3318	−1.18172	−0.590862	+	0.806772i	$0.701212\pi$
$192$	−2.28341	−0.164791
$193$	−22.2848	−1.60410	−0.802048	−	0.597260i	$-0.796256\pi$
$193$	−22.2848	−1.60410	−0.802048	+	0.597260i	$0.796256\pi$
$194$	12.7256	0.913646
$195$	2.21428	0.158568
$196$	−6.08645	−0.434747
$197$	−7.01787	−0.500003	−0.250001	−	0.968245i	$-0.580431\pi$
$197$	−7.01787	−0.500003	−0.250001	+	0.968245i	$0.580431\pi$
$198$	−11.3584	−0.807206
$199$	−8.26753	−0.586069	−0.293035	−	0.956102i	$-0.594665\pi$
$199$	−8.26753	−0.586069	−0.293035	+	0.956102i	$0.594665\pi$
$200$	13.7862	0.974832
$201$	26.7285	1.88529
$202$	17.0356	1.19862
$203$	0.0514019	0.00360771
$204$	−6.60574	−0.462494
$205$	4.18494	0.292289
$206$	−10.5631	−0.735967
$207$	−7.32058	−0.508815
$208$	−0.223732	−0.0155130
$209$	5.13031	0.354871
$210$	9.45952	0.652769
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	10.8683	0.746437
$213$	−13.8398	−0.948291
$214$	−3.59047	−0.245439
$215$	−25.2480	−1.72190
$216$	1.79481	0.122121
$217$	1.82329	0.123773
$218$	4.87950	0.330482
$219$	−8.63313	−0.583373
$220$	−22.2363	−1.49917
$221$	−0.647241	−0.0435381
$222$	5.23617	0.351429
$223$	20.6508	1.38288	0.691441	−	0.722433i	$-0.256976\pi$
$223$	20.6508	1.38288	0.691441	+	0.722433i	$0.256976\pi$
$224$	−0.955796	−0.0638618
$225$	30.5224	2.03483
$226$	−8.29563	−0.551817
$227$	7.68360	0.509979	0.254989	−	0.966944i	$-0.417928\pi$
$227$	7.68360	0.509979	0.254989	+	0.966944i	$0.417928\pi$
$228$	2.28341	0.151223
$229$	−29.6585	−1.95989	−0.979945	−	0.199269i	$-0.936143\pi$
$229$	−29.6585	−1.95989	−0.979945	+	0.199269i	$0.936143\pi$
$230$	−14.3315	−0.944990
$231$	−11.1968	−0.736694
$232$	−0.0537792	−0.00353078
$233$	2.77318	0.181677	0.0908385	−	0.995866i	$-0.471045\pi$
$233$	2.77318	0.181677	0.0908385	+	0.995866i	$0.471045\pi$
$234$	−0.495339	−0.0323813
$235$	42.0245	2.74138
$236$	4.27122	0.278033
$237$	−21.0064	−1.36451
$238$	−2.76504	−0.179231
$239$	−5.22141	−0.337745	−0.168872	−	0.985638i	$-0.554013\pi$
$239$	−5.22141	−0.337745	−0.168872	+	0.985638i	$0.554013\pi$
$240$	−9.89701	−0.638850
$241$	−2.31966	−0.149422	−0.0747111	−	0.997205i	$-0.523803\pi$
$241$	−2.31966	−0.149422	−0.0747111	+	0.997205i	$0.523803\pi$
$242$	15.3201	0.984811
$243$	19.1400	1.22783
$244$	1.64689	0.105431
$245$	−26.3806	−1.68539
$246$	−2.20472	−0.140568
$247$	0.223732	0.0142357
$248$	−1.90761	−0.121133
$249$	−37.0548	−2.34825
$250$	38.0821	2.40852
$251$	17.6215	1.11226	0.556130	−	0.831095i	$-0.312286\pi$
$251$	17.6215	1.11226	0.556130	+	0.831095i	$0.312286\pi$
$252$	−2.11611	−0.133303
$253$	16.9635	1.06649
$254$	19.3439	1.21374
$255$	−28.6313	−1.79296
$256$	1.00000	0.0625000
$257$	−18.0907	−1.12847	−0.564233	−	0.825616i	$-0.690828\pi$
$257$	−18.0907	−1.12847	−0.564233	+	0.825616i	$0.690828\pi$
$258$	13.3012	0.828099
$259$	2.19177	0.136190
$260$	−0.969724	−0.0601398
$261$	−0.119066	−0.00737000
$262$	9.20602	0.568750
$263$	−5.52366	−0.340603	−0.170302	−	0.985392i	$-0.554474\pi$
$263$	−5.52366	−0.340603	−0.170302	+	0.985392i	$0.554474\pi$
$264$	11.7146	0.720985
$265$	47.1065	2.89373
$266$	0.955796	0.0586036
$267$	−35.4857	−2.17169
$268$	−11.7055	−0.715028
$269$	22.3892	1.36509	0.682546	−	0.730843i	$-0.260873\pi$
$269$	22.3892	1.36509	0.682546	+	0.730843i	$0.260873\pi$
$270$	7.77926	0.473431
$271$	21.1723	1.28613	0.643063	−	0.765813i	$-0.277663\pi$
$271$	21.1723	1.28613	0.643063	+	0.765813i	$0.277663\pi$
$272$	2.89292	0.175409
$273$	−0.488291	−0.0295527
$274$	−16.3335	−0.986745
$275$	−70.7275	−4.26503
$276$	7.55016	0.454466
$277$	30.5060	1.83293	0.916464	−	0.400117i	$-0.131031\pi$
$277$	30.5060	1.83293	0.916464	+	0.400117i	$0.131031\pi$
$278$	9.81016	0.588374
$279$	−4.22341	−0.252849
$280$	−4.14271	−0.247574
$281$	0.299543	0.0178692	0.00893461	−	0.999960i	$-0.497156\pi$
$281$	0.299543	0.0178692	0.00893461	+	0.999960i	$0.497156\pi$
$282$	−22.1395	−1.31839
$283$	6.89607	0.409929	0.204964	−	0.978769i	$-0.434292\pi$
$283$	6.89607	0.409929	0.204964	+	0.978769i	$0.434292\pi$
$284$	6.06103	0.359656
$285$	9.89701	0.586249
$286$	1.14782	0.0678718
$287$	−0.922857	−0.0544745
$288$	2.21398	0.130460
$289$	−8.63099	−0.507705
$290$	−0.233095	−0.0136878
$291$	−29.0579	−1.70340
$292$	3.78080	0.221255
$293$	10.4173	0.608586	0.304293	−	0.952578i	$-0.401580\pi$
$293$	10.4173	0.608586	0.304293	+	0.952578i	$0.401580\pi$
$294$	13.8979	0.810542
$295$	18.5128	1.07786
$296$	−2.29313	−0.133286
$297$	−9.20794	−0.534299
$298$	0.990937	0.0574035
$299$	0.739776	0.0427824
$300$	−31.4796	−1.81748
$301$	5.56766	0.320915
$302$	10.4306	0.600212
$303$	−38.8992	−2.23470
$304$	−1.00000	−0.0573539
$305$	7.13812	0.408727
$306$	6.40487	0.366142
$307$	−24.1824	−1.38016	−0.690080	−	0.723733i	$-0.742425\pi$
$307$	−24.1824	−1.38016	−0.690080	+	0.723733i	$0.742425\pi$
$308$	4.90353	0.279404
$309$	24.1199	1.37214
$310$	−8.26816	−0.469600
$311$	4.73300	0.268384	0.134192	−	0.990955i	$-0.457156\pi$
$311$	4.73300	0.268384	0.134192	+	0.990955i	$0.457156\pi$
$312$	0.510873	0.0289225
$313$	9.38214	0.530310	0.265155	−	0.964206i	$-0.414577\pi$
$313$	9.38214	0.530310	0.265155	+	0.964206i	$0.414577\pi$
$314$	−8.84405	−0.499099
$315$	−9.17188	−0.516777
$316$	9.19955	0.517515
$317$	16.9015	0.949282	0.474641	−	0.880179i	$-0.342578\pi$
$317$	16.9015	0.949282	0.474641	+	0.880179i	$0.342578\pi$
$318$	−24.8168	−1.39166
$319$	0.275904	0.0154476
$320$	4.33431	0.242295
$321$	8.19853	0.457597
$322$	3.16036	0.176120
$323$	−2.89292	−0.160967
$324$	−10.7402	−0.596680
$325$	−3.08442	−0.171093
$326$	0.889463	0.0492628
$327$	−11.1419	−0.616150
$328$	0.965538	0.0533129
$329$	−9.26719	−0.510917
$330$	50.7747	2.79506
$331$	3.06898	0.168686	0.0843432	−	0.996437i	$-0.473121\pi$
$331$	3.06898	0.168686	0.0843432	+	0.996437i	$0.473121\pi$
$332$	16.2278	0.890616
$333$	−5.07695	−0.278215
$334$	7.07473	0.387112
$335$	−50.7353	−2.77197
$336$	2.18248	0.119064
$337$	12.0774	0.657898	0.328949	−	0.944348i	$-0.393306\pi$
$337$	12.0774	0.657898	0.328949	+	0.944348i	$0.393306\pi$
$338$	−12.9499	−0.704384
$339$	18.9424	1.02881
$340$	12.5388	0.680013
$341$	9.78662	0.529975
$342$	−2.21398	−0.119718
$343$	12.5080	0.675367
$344$	−5.82516	−0.314071
$345$	32.7247	1.76184
$346$	−2.39066	−0.128523
$347$	−30.2646	−1.62469	−0.812345	−	0.583177i	$-0.801809\pi$
$347$	−30.2646	−1.62469	−0.812345	+	0.583177i	$0.801809\pi$
$348$	0.122800	0.00658278
$349$	−0.577196	−0.0308966	−0.0154483	−	0.999881i	$-0.504918\pi$
$349$	−0.577196	−0.0308966	−0.0154483	+	0.999881i	$0.504918\pi$
$350$	−13.1768	−0.704330
$351$	−0.401557	−0.0214335
$352$	−5.13031	−0.273446
$353$	−32.8742	−1.74972	−0.874860	−	0.484376i	$-0.839047\pi$
$353$	−32.8742	−1.74972	−0.874860	+	0.484376i	$0.839047\pi$
$354$	−9.75297	−0.518365
$355$	26.2704	1.39429
$356$	15.5406	0.823652
$357$	6.31374	0.334159
$358$	8.08589	0.427353
$359$	−22.0424	−1.16336	−0.581678	−	0.813419i	$-0.697604\pi$
$359$	−22.0424	−1.16336	−0.581678	+	0.813419i	$0.697604\pi$
$360$	9.59606	0.505757
$361$	1.00000	0.0526316
$362$	0.291984	0.0153463
$363$	−34.9820	−1.83608
$364$	0.213842	0.0112084
$365$	16.3872	0.857743
$366$	−3.76053	−0.196566
$367$	28.0980	1.46670	0.733352	−	0.679849i	$-0.237955\pi$
$367$	28.0980	1.46670	0.733352	+	0.679849i	$0.237955\pi$
$368$	−3.30652	−0.172365
$369$	2.13768	0.111283
$370$	−9.93915	−0.516712
$371$	−10.3879	−0.539311
$372$	4.35586	0.225841
$373$	32.0311	1.65851	0.829255	−	0.558871i	$-0.188765\pi$
$373$	32.0311	1.65851	0.829255	+	0.558871i	$0.188765\pi$
$374$	−14.8416	−0.767440
$375$	−86.9572	−4.49045
$376$	9.69579	0.500022
$377$	0.0120321	0.000619687 0
$378$	−1.71547	−0.0882344
$379$	−37.9753	−1.95066	−0.975331	−	0.220749i	$-0.929150\pi$
$379$	−37.9753	−1.95066	−0.975331	+	0.220749i	$0.929150\pi$
$380$	−4.33431	−0.222345
$381$	−44.1701	−2.26291
$382$	−16.3318	−0.835606
$383$	7.68475	0.392672	0.196336	−	0.980537i	$-0.437096\pi$
$383$	7.68475	0.392672	0.196336	+	0.980537i	$0.437096\pi$
$384$	−2.28341	−0.116525
$385$	21.2534	1.08317
$386$	−22.2848	−1.13427
$387$	−12.8968	−0.655580
$388$	12.7256	0.646045
$389$	30.6879	1.55594	0.777968	−	0.628304i	$-0.216251\pi$
$389$	30.6879	1.55594	0.777968	+	0.628304i	$0.216251\pi$
$390$	2.21428	0.112125
$391$	−9.56552	−0.483749
$392$	−6.08645	−0.307412
$393$	−21.0212	−1.06038
$394$	−7.01787	−0.353555
$395$	39.8737	2.00626
$396$	−11.3584	−0.570781
$397$	−11.9920	−0.601860	−0.300930	−	0.953646i	$-0.597297\pi$
$397$	−11.9920	−0.601860	−0.300930	+	0.953646i	$0.597297\pi$
$398$	−8.26753	−0.414414
$399$	−2.18248	−0.109261
$400$	13.7862	0.689310
$401$	−3.28872	−0.164231	−0.0821154	−	0.996623i	$-0.526168\pi$
$401$	−3.28872	−0.164231	−0.0821154	+	0.996623i	$0.526168\pi$
$402$	26.7285	1.33310
$403$	0.426794	0.0212601
$404$	17.0356	0.847551
$405$	−46.5515	−2.31316
$406$	0.0514019	0.00255103
$407$	11.7645	0.583144
$408$	−6.60574	−0.327033
$409$	29.2267	1.44517	0.722584	−	0.691283i	$-0.242954\pi$
$409$	29.2267	1.44517	0.722584	+	0.691283i	$0.242954\pi$
$410$	4.18494	0.206679
$411$	37.2962	1.83969
$412$	−10.5631	−0.520407
$413$	−4.08242	−0.200883
$414$	−7.32058	−0.359787
$415$	70.3363	3.45267
$416$	−0.223732	−0.0109694
$417$	−22.4007	−1.09696
$418$	5.13031	0.250932
$419$	35.5644	1.73744	0.868718	−	0.495307i	$-0.164945\pi$
$419$	35.5644	1.73744	0.868718	+	0.495307i	$0.164945\pi$
$420$	9.45952	0.461578
$421$	−18.5289	−0.903045	−0.451523	−	0.892260i	$-0.649119\pi$
$421$	−18.5289	−0.903045	−0.451523	+	0.892260i	$0.649119\pi$
$422$	−1.00000	−0.0486792
$423$	21.4663	1.04373
$424$	10.8683	0.527811
$425$	39.8825	1.93458
$426$	−13.8398	−0.670543
$427$	−1.57409	−0.0761755
$428$	−3.59047	−0.173552
$429$	−2.62094	−0.126540
$430$	−25.2480	−1.21757
$431$	24.6150	1.18566	0.592830	−	0.805327i	$-0.298010\pi$
$431$	24.6150	1.18566	0.592830	+	0.805327i	$0.298010\pi$
$432$	1.79481	0.0863529
$433$	−38.2251	−1.83698	−0.918491	−	0.395441i	$-0.870592\pi$
$433$	−38.2251	−1.83698	−0.918491	+	0.395441i	$0.870592\pi$
$434$	1.82329	0.0875205
$435$	0.532253	0.0255196
$436$	4.87950	0.233686
$437$	3.30652	0.158173
$438$	−8.63313	−0.412507
$439$	4.87452	0.232648	0.116324	−	0.993211i	$-0.462889\pi$
$439$	4.87452	0.232648	0.116324	+	0.993211i	$0.462889\pi$
$440$	−22.2363	−1.06008
$441$	−13.4753	−0.641680
$442$	−0.647241	−0.0307861
$443$	36.7284	1.74502	0.872509	−	0.488598i	$-0.162492\pi$
$443$	36.7284	1.74502	0.872509	+	0.488598i	$0.162492\pi$
$444$	5.23617	0.248498
$445$	67.3579	3.19307
$446$	20.6508	0.977845
$447$	−2.26272	−0.107023
$448$	−0.955796	−0.0451571
$449$	14.8606	0.701316	0.350658	−	0.936504i	$-0.385958\pi$
$449$	14.8606	0.701316	0.350658	+	0.936504i	$0.385958\pi$
$450$	30.5224	1.43884
$451$	−4.95351	−0.233252
$452$	−8.29563	−0.390194
$453$	−23.8173	−1.11903
$454$	7.68360	0.360609
$455$	0.926858	0.0434518
$456$	2.28341	0.106931
$457$	−34.4174	−1.60998	−0.804990	−	0.593288i	$-0.797829\pi$
$457$	−34.4174	−1.60998	−0.804990	+	0.593288i	$0.797829\pi$
$458$	−29.6585	−1.38585
$459$	5.19225	0.242354
$460$	−14.3315	−0.668209
$461$	−12.0352	−0.560533	−0.280267	−	0.959922i	$-0.590423\pi$
$461$	−12.0352	−0.560533	−0.280267	+	0.959922i	$0.590423\pi$
$462$	−11.1968	−0.520921
$463$	13.7270	0.637949	0.318974	−	0.947763i	$-0.396662\pi$
$463$	13.7270	0.637949	0.318974	+	0.947763i	$0.396662\pi$
$464$	−0.0537792	−0.00249664
$465$	18.8796	0.875522
$466$	2.77318	0.128465
$467$	16.6043	0.768355	0.384178	−	0.923259i	$-0.374485\pi$
$467$	16.6043	0.768355	0.384178	+	0.923259i	$0.374485\pi$
$468$	−0.495339	−0.0228970
$469$	11.1881	0.516618
$470$	42.0245	1.93845
$471$	20.1946	0.930519
$472$	4.27122	0.196599
$473$	29.8848	1.37411
$474$	−21.0064	−0.964855
$475$	−13.7862	−0.632555
$476$	−2.76504	−0.126736
$477$	24.0622	1.10173
$478$	−5.22141	−0.238822
$479$	−35.6304	−1.62800	−0.813998	−	0.580867i	$-0.802714\pi$
$479$	−35.6304	−1.62800	−0.813998	+	0.580867i	$0.802714\pi$
$480$	−9.89701	−0.451735
$481$	0.513048	0.0233930
$482$	−2.31966	−0.105657
$483$	−7.21641	−0.328358
$484$	15.3201	0.696366
$485$	55.1567	2.50454
$486$	19.1400	0.868206
$487$	−27.0689	−1.22661	−0.613304	−	0.789847i	$-0.710160\pi$
$487$	−27.0689	−1.22661	−0.613304	+	0.789847i	$0.710160\pi$
$488$	1.64689	0.0745511
$489$	−2.03101	−0.0918455
$490$	−26.3806	−1.19175
$491$	20.6028	0.929791	0.464896	−	0.885365i	$-0.346092\pi$
$491$	20.6028	0.929791	0.464896	+	0.885365i	$0.346092\pi$
$492$	−2.20472	−0.0993966
$493$	−0.155579	−0.00700693
$494$	0.223732	0.0100662
$495$	−49.2308	−2.21276
$496$	−1.90761	−0.0856542
$497$	−5.79311	−0.259857
$498$	−37.0548	−1.66046
$499$	19.1318	0.856459	0.428229	−	0.903670i	$-0.359138\pi$
$499$	19.1318	0.856459	0.428229	+	0.903670i	$0.359138\pi$
$500$	38.0821	1.70308
$501$	−16.1545	−0.721731
$502$	17.6215	0.786487
$503$	19.4783	0.868493	0.434246	−	0.900794i	$-0.357015\pi$
$503$	19.4783	0.868493	0.434246	+	0.900794i	$0.357015\pi$
$504$	−2.11611	−0.0942591
$505$	73.8373	3.28572
$506$	16.9635	0.754119
$507$	29.5701	1.31325
$508$	19.3439	0.858247
$509$	15.6613	0.694174	0.347087	−	0.937833i	$-0.387171\pi$
$509$	15.6613	0.694174	0.347087	+	0.937833i	$0.387171\pi$
$510$	−28.6313	−1.26782
$511$	−3.61367	−0.159860
$512$	1.00000	0.0441942
$513$	−1.79481	−0.0792429
$514$	−18.0907	−0.797945
$515$	−45.7837	−2.01747
$516$	13.3012	0.585555
$517$	−49.7424	−2.18767
$518$	2.19177	0.0963008
$519$	5.45887	0.239618
$520$	−0.969724	−0.0425252
$521$	24.7153	1.08280	0.541398	−	0.840766i	$-0.317895\pi$
$521$	24.7153	1.08280	0.541398	+	0.840766i	$0.317895\pi$
$522$	−0.119066	−0.00521137
$523$	31.4808	1.37656	0.688280	−	0.725445i	$-0.258366\pi$
$523$	31.4808	1.37656	0.688280	+	0.725445i	$0.258366\pi$
$524$	9.20602	0.402167
$525$	30.0881	1.31315
$526$	−5.52366	−0.240843
$527$	−5.51857	−0.240393
$528$	11.7146	0.509813
$529$	−12.0669	−0.524648
$530$	47.1065	2.04618
$531$	9.45640	0.410373
$532$	0.955796	0.0414390
$533$	−0.216022	−0.00935695
$534$	−35.4857	−1.53562
$535$	−15.5622	−0.672812
$536$	−11.7055	−0.505601
$537$	−18.4634	−0.796756
$538$	22.3892	0.965266
$539$	31.2254	1.34497
$540$	7.77926	0.334766
$541$	−16.1161	−0.692887	−0.346444	−	0.938071i	$-0.612611\pi$
$541$	−16.1161	−0.692887	−0.346444	+	0.938071i	$0.612611\pi$
$542$	21.1723	0.909429
$543$	−0.666719	−0.0286117
$544$	2.89292	0.124033
$545$	21.1493	0.905935
$546$	−0.488291	−0.0208969
$547$	−16.4297	−0.702485	−0.351242	−	0.936285i	$-0.614241\pi$
$547$	−16.4297	−0.702485	−0.351242	+	0.936285i	$0.614241\pi$
$548$	−16.3335	−0.697734
$549$	3.64618	0.155615
$550$	−70.7275	−3.01583
$551$	0.0537792	0.00229107
$552$	7.55016	0.321356
$553$	−8.79289	−0.373912
$554$	30.5060	1.29608
$555$	22.6952	0.963357
$556$	9.81016	0.416043
$557$	−40.4682	−1.71469	−0.857347	−	0.514739i	$-0.827889\pi$
$557$	−40.4682	−1.71469	−0.857347	+	0.514739i	$0.827889\pi$
$558$	−4.22341	−0.178791
$559$	1.30328	0.0551227
$560$	−4.14271	−0.175062
$561$	33.8895	1.43082
$562$	0.299543	0.0126354
$563$	8.20347	0.345735	0.172868	−	0.984945i	$-0.444697\pi$
$563$	8.20347	0.345735	0.172868	+	0.984945i	$0.444697\pi$
$564$	−22.1395	−0.932241
$565$	−35.9558	−1.51267
$566$	6.89607	0.289864
$567$	10.2655	0.431109
$568$	6.06103	0.254315
$569$	22.1725	0.929520	0.464760	−	0.885437i	$-0.346141\pi$
$569$	22.1725	0.929520	0.464760	+	0.885437i	$0.346141\pi$
$570$	9.89701	0.414540
$571$	−36.4729	−1.52634	−0.763172	−	0.646195i	$-0.776359\pi$
$571$	−36.4729	−1.52634	−0.763172	+	0.646195i	$0.776359\pi$
$572$	1.14782	0.0479926
$573$	37.2922	1.55790
$574$	−0.922857	−0.0385193
$575$	−45.5844	−1.90100
$576$	2.21398	0.0922491
$577$	4.67388	0.194576	0.0972881	−	0.995256i	$-0.468983\pi$
$577$	4.67388	0.194576	0.0972881	+	0.995256i	$0.468983\pi$
$578$	−8.63099	−0.359002
$579$	50.8855	2.11473
$580$	−0.233095	−0.00967876
$581$	−15.5105	−0.643482
$582$	−29.0579	−1.20449
$583$	−55.7577	−2.30925
$584$	3.78080	0.156451
$585$	−2.14695	−0.0887654
$586$	10.4173	0.430335
$587$	−0.250896	−0.0103556	−0.00517780	−	0.999987i	$-0.501648\pi$
$587$	−0.250896	−0.0103556	−0.00517780	+	0.999987i	$0.501648\pi$
$588$	13.8979	0.573140
$589$	1.90761	0.0786017
$590$	18.5128	0.762160
$591$	16.0247	0.659169
$592$	−2.29313	−0.0942472
$593$	32.7873	1.34641	0.673207	−	0.739454i	$-0.264916\pi$
$593$	32.7873	1.34641	0.673207	+	0.739454i	$0.264916\pi$
$594$	−9.20794	−0.377806
$595$	−11.9845	−0.491319
$596$	0.990937	0.0405904
$597$	18.8782	0.772633
$598$	0.739776	0.0302517
$599$	−15.4877	−0.632812	−0.316406	−	0.948624i	$-0.602476\pi$
$599$	−15.4877	−0.632812	−0.316406	+	0.948624i	$0.602476\pi$
$600$	−31.4796	−1.28515
$601$	21.1575	0.863031	0.431516	−	0.902106i	$-0.357979\pi$
$601$	21.1575	0.863031	0.431516	+	0.902106i	$0.357979\pi$
$602$	5.56766	0.226921
$603$	−25.9158	−1.05537
$604$	10.4306	0.424414
$605$	66.4018	2.69962
$606$	−38.8992	−1.58017
$607$	3.06005	0.124204	0.0621018	−	0.998070i	$-0.480220\pi$
$607$	3.06005	0.124204	0.0621018	+	0.998070i	$0.480220\pi$
$608$	−1.00000	−0.0405554
$609$	−0.117372	−0.00475615
$610$	7.13812	0.289014
$611$	−2.16926	−0.0877589
$612$	6.40487	0.258902
$613$	−18.4730	−0.746117	−0.373059	−	0.927808i	$-0.621691\pi$
$613$	−18.4730	−0.746117	−0.373059	+	0.927808i	$0.621691\pi$
$614$	−24.1824	−0.975921
$615$	−9.55594	−0.385333
$616$	4.90353	0.197569
$617$	18.4812	0.744026	0.372013	−	0.928227i	$-0.378668\pi$
$617$	18.4812	0.744026	0.372013	+	0.928227i	$0.378668\pi$
$618$	24.1199	0.970247
$619$	−19.8455	−0.797659	−0.398829	−	0.917025i	$-0.630583\pi$
$619$	−19.8455	−0.797659	−0.398829	+	0.917025i	$0.630583\pi$
$620$	−8.26816	−0.332057
$621$	−5.93459	−0.238147
$622$	4.73300	0.189776
$623$	−14.8537	−0.595100
$624$	0.510873	0.0204513
$625$	96.1285	3.84514
$626$	9.38214	0.374986
$627$	−11.7146	−0.467837
$628$	−8.84405	−0.352916
$629$	−6.63386	−0.264509
$630$	−9.17188	−0.365416
$631$	0.596367	0.0237410	0.0118705	−	0.999930i	$-0.496221\pi$
$631$	0.596367	0.0237410	0.0118705	+	0.999930i	$0.496221\pi$
$632$	9.19955	0.365939
$633$	2.28341	0.0907576
$634$	16.9015	0.671244
$635$	83.8424	3.32718
$636$	−24.8168	−0.984050
$637$	1.36174	0.0539540
$638$	0.275904	0.0109231
$639$	13.4190	0.530847
$640$	4.33431	0.171328
$641$	−16.1306	−0.637121	−0.318560	−	0.947903i	$-0.603199\pi$
$641$	−16.1306	−0.637121	−0.318560	+	0.947903i	$0.603199\pi$
$642$	8.19853	0.323570
$643$	39.1153	1.54256	0.771278	−	0.636498i	$-0.219618\pi$
$643$	39.1153	1.54256	0.771278	+	0.636498i	$0.219618\pi$
$644$	3.16036	0.124536
$645$	57.6517	2.27003
$646$	−2.89292	−0.113821
$647$	−19.0893	−0.750478	−0.375239	−	0.926928i	$-0.622439\pi$
$647$	−19.0893	−0.750478	−0.375239	+	0.926928i	$0.622439\pi$
$648$	−10.7402	−0.421916
$649$	−21.9127	−0.860149
$650$	−3.08442	−0.120981
$651$	−4.16331	−0.163173
$652$	0.889463	0.0348340
$653$	21.4411	0.839056	0.419528	−	0.907742i	$-0.362196\pi$
$653$	21.4411	0.839056	0.419528	+	0.907742i	$0.362196\pi$
$654$	−11.1419	−0.435684
$655$	39.9017	1.55909
$656$	0.965538	0.0376979
$657$	8.37061	0.326569
$658$	−9.26719	−0.361273
$659$	4.82406	0.187919	0.0939593	−	0.995576i	$-0.470048\pi$
$659$	4.82406	0.187919	0.0939593	+	0.995576i	$0.470048\pi$
$660$	50.7747	1.97640
$661$	20.1301	0.782969	0.391484	−	0.920185i	$-0.371962\pi$
$661$	20.1301	0.782969	0.391484	+	0.920185i	$0.371962\pi$
$662$	3.06898	0.119279
$663$	1.47792	0.0573976
$664$	16.2278	0.629761
$665$	4.14271	0.160647
$666$	−5.07695	−0.196728
$667$	0.177822	0.00688530
$668$	7.07473	0.273729
$669$	−47.1544	−1.82309
$670$	−50.7353	−1.96008
$671$	−8.44904	−0.326172
$672$	2.18248	0.0841909
$673$	18.7453	0.722577	0.361288	−	0.932454i	$-0.382337\pi$
$673$	18.7453	0.722577	0.361288	+	0.932454i	$0.382337\pi$
$674$	12.0774	0.465204
$675$	24.7436	0.952383
$676$	−12.9499	−0.498075
$677$	29.3705	1.12880	0.564399	−	0.825502i	$-0.309108\pi$
$677$	29.3705	1.12880	0.564399	+	0.825502i	$0.309108\pi$
$678$	18.9424	0.727477
$679$	−12.1631	−0.466777
$680$	12.5388	0.480842
$681$	−17.5448	−0.672320
$682$	9.78662	0.374749
$683$	−10.0911	−0.386125	−0.193062	−	0.981186i	$-0.561842\pi$
$683$	−10.0911	−0.386125	−0.193062	+	0.981186i	$0.561842\pi$
$684$	−2.21398	−0.0846536
$685$	−70.7945	−2.70492
$686$	12.5080	0.477557
$687$	67.7227	2.58378
$688$	−5.82516	−0.222082
$689$	−2.43159	−0.0926361
$690$	32.7247	1.24581
$691$	0.478655	0.0182089	0.00910445	−	0.999959i	$-0.497102\pi$
$691$	0.478655	0.0182089	0.00910445	+	0.999959i	$0.497102\pi$
$692$	−2.39066	−0.0908792
$693$	10.8563	0.412397
$694$	−30.2646	−1.14883
$695$	42.5202	1.61288
$696$	0.122800	0.00465473
$697$	2.79323	0.105801
$698$	−0.577196	−0.0218472
$699$	−6.33231	−0.239510
$700$	−13.1768	−0.498036
$701$	−51.6805	−1.95194	−0.975972	−	0.217895i	$-0.930081\pi$
$701$	−51.6805	−1.95194	−0.975972	+	0.217895i	$0.930081\pi$
$702$	−0.401557	−0.0151558
$703$	2.29313	0.0864872
$704$	−5.13031	−0.193356
$705$	−95.9593	−3.61404
$706$	−32.8742	−1.23724
$707$	−16.2825	−0.612367
$708$	−9.75297	−0.366539
$709$	−34.8782	−1.30988	−0.654940	−	0.755681i	$-0.727306\pi$
$709$	−34.8782	−1.30988	−0.654940	+	0.755681i	$0.727306\pi$
$710$	26.2704	0.985910
$711$	20.3676	0.763845
$712$	15.5406	0.582410
$713$	6.30756	0.236220
$714$	6.31374	0.236286
$715$	4.97498	0.186054
$716$	8.08589	0.302184
$717$	11.9226	0.445259
$718$	−22.0424	−0.822617
$719$	−44.3798	−1.65509	−0.827544	−	0.561400i	$-0.810263\pi$
$719$	−44.3798	−1.65509	−0.827544	+	0.561400i	$0.810263\pi$
$720$	9.59606	0.357624
$721$	10.0962	0.376001
$722$	1.00000	0.0372161
$723$	5.29673	0.196988
$724$	0.291984	0.0108515
$725$	−0.741411	−0.0275353
$726$	−34.9820	−1.29831
$727$	−28.5440	−1.05864	−0.529319	−	0.848423i	$-0.677552\pi$
$727$	−28.5440	−1.05864	−0.529319	+	0.848423i	$0.677552\pi$
$728$	0.213842	0.00792553
$729$	−11.4838	−0.425324
$730$	16.3872	0.606516
$731$	−16.8517	−0.623284
$732$	−3.76053	−0.138993
$733$	−38.2904	−1.41429	−0.707143	−	0.707070i	$-0.750016\pi$
$733$	−38.2904	−1.41429	−0.707143	+	0.707070i	$0.750016\pi$
$734$	28.0980	1.03712
$735$	60.2377	2.22190
$736$	−3.30652	−0.121880
$737$	60.0529	2.21208
$738$	2.13768	0.0786891
$739$	−4.86293	−0.178886	−0.0894429	−	0.995992i	$-0.528509\pi$
$739$	−4.86293	−0.178886	−0.0894429	+	0.995992i	$0.528509\pi$
$740$	−9.93915	−0.365370
$741$	−0.510873	−0.0187674
$742$	−10.3879	−0.381350
$743$	−0.793285	−0.0291028	−0.0145514	−	0.999894i	$-0.504632\pi$
$743$	−0.793285	−0.0291028	−0.0145514	+	0.999894i	$0.504632\pi$
$744$	4.35586	0.159694
$745$	4.29503	0.157358
$746$	32.0311	1.17274
$747$	35.9280	1.31454
$748$	−14.8416	−0.542662
$749$	3.43176	0.125394
$750$	−86.9572	−3.17523
$751$	4.93314	0.180013	0.0900065	−	0.995941i	$-0.471311\pi$
$751$	4.93314	0.180013	0.0900065	+	0.995941i	$0.471311\pi$
$752$	9.69579	0.353569
$753$	−40.2372	−1.46633
$754$	0.0120321	0.000438185 0
$755$	45.2093	1.64533
$756$	−1.71547	−0.0623912
$757$	−39.2225	−1.42557	−0.712783	−	0.701385i	$-0.752566\pi$
$757$	−39.2225	−1.42557	−0.712783	+	0.701385i	$0.752566\pi$
$758$	−37.9753	−1.37933
$759$	−38.7347	−1.40598
$760$	−4.33431	−0.157222
$761$	35.6627	1.29277	0.646387	−	0.763010i	$-0.276279\pi$
$761$	35.6627	1.29277	0.646387	+	0.763010i	$0.276279\pi$
$762$	−44.1701	−1.60012
$763$	−4.66381	−0.168841
$764$	−16.3318	−0.590862
$765$	27.7607	1.00369
$766$	7.68475	0.277661
$767$	−0.955611	−0.0345051
$768$	−2.28341	−0.0823956
$769$	−32.0463	−1.15562	−0.577810	−	0.816172i	$-0.696092\pi$
$769$	−32.0463	−1.15562	−0.577810	+	0.816172i	$0.696092\pi$
$770$	21.2534	0.765919
$771$	41.3085	1.48769
$772$	−22.2848	−0.802048
$773$	−36.0321	−1.29598	−0.647992	−	0.761647i	$-0.724391\pi$
$773$	−36.0321	−1.29598	−0.647992	+	0.761647i	$0.724391\pi$
$774$	−12.8968	−0.463565
$775$	−26.2987	−0.944677
$776$	12.7256	0.456823
$777$	−5.00471	−0.179543
$778$	30.6879	1.10021
$779$	−0.965538	−0.0345940
$780$	2.21428	0.0792840
$781$	−31.0950	−1.11267
$782$	−9.56552	−0.342062
$783$	−0.0965235	−0.00344947
$784$	−6.08645	−0.217373
$785$	−38.3328	−1.36816
$786$	−21.0212	−0.749800
$787$	−15.8706	−0.565726	−0.282863	−	0.959160i	$-0.591284\pi$
$787$	−15.8706	−0.565726	−0.282863	+	0.959160i	$0.591284\pi$
$788$	−7.01787	−0.250001
$789$	12.6128	0.449027
$790$	39.8737	1.41864
$791$	7.92893	0.281920
$792$	−11.3584	−0.403603
$793$	−0.368462	−0.0130845
$794$	−11.9920	−0.425579
$795$	−107.564	−3.81489
$796$	−8.26753	−0.293035
$797$	−20.3226	−0.719862	−0.359931	−	0.932979i	$-0.617200\pi$
$797$	−20.3226	−0.719862	−0.359931	+	0.932979i	$0.617200\pi$
$798$	−2.18248	−0.0772589
$799$	28.0492	0.992308
$800$	13.7862	0.487416
$801$	34.4067	1.21570
$802$	−3.28872	−0.116129
$803$	−19.3967	−0.684494
$804$	26.7285	0.942643
$805$	13.6980	0.482790
$806$	0.426794	0.0150332
$807$	−51.1238	−1.79964
$808$	17.0356	0.599309
$809$	−19.0189	−0.668669	−0.334334	−	0.942454i	$-0.608512\pi$
$809$	−19.0189	−0.668669	−0.334334	+	0.942454i	$0.608512\pi$
$810$	−46.5515	−1.63565
$811$	−14.8294	−0.520730	−0.260365	−	0.965510i	$-0.583843\pi$
$811$	−14.8294	−0.520730	−0.260365	+	0.965510i	$0.583843\pi$
$812$	0.0514019	0.00180385
$813$	−48.3452	−1.69554
$814$	11.7645	0.412345
$815$	3.85520	0.135042
$816$	−6.60574	−0.231247
$817$	5.82516	0.203796
$818$	29.2267	1.02189
$819$	0.473443	0.0165434
$820$	4.18494	0.146144
$821$	14.0275	0.489562	0.244781	−	0.969578i	$-0.421284\pi$
$821$	14.0275	0.489562	0.244781	+	0.969578i	$0.421284\pi$
$822$	37.2962	1.30086
$823$	28.2974	0.986386	0.493193	−	0.869920i	$-0.335830\pi$
$823$	28.2974	0.986386	0.493193	+	0.869920i	$0.335830\pi$
$824$	−10.5631	−0.367983
$825$	161.500	5.62271
$826$	−4.08242	−0.142046
$827$	14.4380	0.502057	0.251029	−	0.967980i	$-0.419231\pi$
$827$	14.4380	0.502057	0.251029	+	0.967980i	$0.419231\pi$
$828$	−7.32058	−0.254408
$829$	44.2391	1.53649	0.768244	−	0.640157i	$-0.221131\pi$
$829$	44.2391	1.53649	0.768244	+	0.640157i	$0.221131\pi$
$830$	70.3363	2.44141
$831$	−69.6578	−2.41640
$832$	−0.223732	−0.00775652
$833$	−17.6076	−0.610069
$834$	−22.4007	−0.775671
$835$	30.6640	1.06117
$836$	5.13031	0.177435
$837$	−3.42380	−0.118344
$838$	35.5644	1.22855
$839$	−8.61764	−0.297514	−0.148757	−	0.988874i	$-0.547527\pi$
$839$	−8.61764	−0.297514	−0.148757	+	0.988874i	$0.547527\pi$
$840$	9.45952	0.326385
$841$	−28.9971	−0.999900
$842$	−18.5289	−0.638549
$843$	−0.683980	−0.0235575
$844$	−1.00000	−0.0344214
$845$	−56.1290	−1.93090
$846$	21.4663	0.738025
$847$	−14.6428	−0.503134
$848$	10.8683	0.373219
$849$	−15.7466	−0.540421
$850$	39.8825	1.36796
$851$	7.58230	0.259918
$852$	−13.8398	−0.474145
$853$	43.5815	1.49220	0.746101	−	0.665833i	$-0.231924\pi$
$853$	43.5815	1.49220	0.746101	+	0.665833i	$0.231924\pi$
$854$	−1.57409	−0.0538642
$855$	−9.59606	−0.328178
$856$	−3.59047	−0.122720
$857$	7.04546	0.240668	0.120334	−	0.992733i	$-0.461603\pi$
$857$	7.04546	0.240668	0.120334	+	0.992733i	$0.461603\pi$
$858$	−2.62094	−0.0894774
$859$	−14.2081	−0.484774	−0.242387	−	0.970180i	$-0.577930\pi$
$859$	−14.2081	−0.484774	−0.242387	+	0.970180i	$0.577930\pi$
$860$	−25.2480	−0.860950
$861$	2.10726	0.0718154
$862$	24.6150	0.838389
$863$	−22.7307	−0.773763	−0.386882	−	0.922129i	$-0.626448\pi$
$863$	−22.7307	−0.773763	−0.386882	+	0.922129i	$0.626448\pi$
$864$	1.79481	0.0610607
$865$	−10.3619	−0.352313
$866$	−38.2251	−1.29894
$867$	19.7081	0.669323
$868$	1.82329	0.0618863
$869$	−47.1965	−1.60103
$870$	0.532253	0.0180451
$871$	2.61890	0.0887381
$872$	4.87950	0.165241
$873$	28.1743	0.953554
$874$	3.30652	0.111845
$875$	−36.3987	−1.23050
$876$	−8.63313	−0.291687
$877$	35.8319	1.20996	0.604979	−	0.796241i	$-0.293182\pi$
$877$	35.8319	1.20996	0.604979	+	0.796241i	$0.293182\pi$
$878$	4.87452	0.164507
$879$	−23.7871	−0.802317
$880$	−22.2363	−0.749586
$881$	8.59293	0.289503	0.144752	−	0.989468i	$-0.453762\pi$
$881$	8.59293	0.289503	0.144752	+	0.989468i	$0.453762\pi$
$882$	−13.4753	−0.453736
$883$	20.6165	0.693799	0.346899	−	0.937902i	$-0.387235\pi$
$883$	20.6165	0.693799	0.346899	+	0.937902i	$0.387235\pi$
$884$	−0.647241	−0.0217691
$885$	−42.2724	−1.42097
$886$	36.7284	1.23391
$887$	47.8875	1.60790	0.803952	−	0.594694i	$-0.202727\pi$
$887$	47.8875	1.60790	0.803952	+	0.594694i	$0.202727\pi$
$888$	5.23617	0.175715
$889$	−18.4888	−0.620095
$890$	67.3579	2.25784
$891$	55.1007	1.84594
$892$	20.6508	0.691441
$893$	−9.69579	−0.324457
$894$	−2.26272	−0.0756767
$895$	35.0467	1.17148
$896$	−0.955796	−0.0319309
$897$	−1.68922	−0.0564013
$898$	14.8606	0.495906
$899$	0.102590	0.00342156
$900$	30.5224	1.01741
$901$	31.4411	1.04746
$902$	−4.95351	−0.164934
$903$	−12.7133	−0.423071
$904$	−8.29563	−0.275909
$905$	1.26555	0.0420682
$906$	−23.8173	−0.791277
$907$	−7.85610	−0.260857	−0.130429	−	0.991458i	$-0.541635\pi$
$907$	−7.85610	−0.260857	−0.130429	+	0.991458i	$0.541635\pi$
$908$	7.68360	0.254989
$909$	37.7164	1.25097
$910$	0.926858	0.0307251
$911$	31.2631	1.03579	0.517897	−	0.855443i	$-0.326715\pi$
$911$	31.2631	1.03579	0.517897	+	0.855443i	$0.326715\pi$
$912$	2.28341	0.0756114
$913$	−83.2536	−2.75529
$914$	−34.4174	−1.13843
$915$	−16.2993	−0.538837
$916$	−29.6585	−0.979945
$917$	−8.79908	−0.290571
$918$	5.19225	0.171370
$919$	−34.9245	−1.15205	−0.576027	−	0.817431i	$-0.695398\pi$
$919$	−34.9245	−1.15205	−0.576027	+	0.817431i	$0.695398\pi$
$920$	−14.3315	−0.472495
$921$	55.2184	1.81951
$922$	−12.0352	−0.396357
$923$	−1.35605	−0.0446349
$924$	−11.1968	−0.368347
$925$	−31.6136	−1.03945
$926$	13.7270	0.451098
$927$	−23.3865	−0.768113
$928$	−0.0537792	−0.00176539
$929$	40.7402	1.33664	0.668321	−	0.743873i	$-0.267013\pi$
$929$	40.7402	1.33664	0.668321	+	0.743873i	$0.267013\pi$
$930$	18.8796	0.619088
$931$	6.08645	0.199475
$932$	2.77318	0.0908385
$933$	−10.8074	−0.353818
$934$	16.6043	0.543309
$935$	−64.3280	−2.10375
$936$	−0.495339	−0.0161907
$937$	26.8862	0.878333	0.439167	−	0.898406i	$-0.355274\pi$
$937$	26.8862	0.878333	0.439167	+	0.898406i	$0.355274\pi$
$938$	11.1881	0.365304
$939$	−21.4233	−0.699123
$940$	42.0245	1.37069
$941$	−22.0354	−0.718334	−0.359167	−	0.933273i	$-0.616939\pi$
$941$	−22.0354	−0.718334	−0.359167	+	0.933273i	$0.616939\pi$
$942$	20.1946	0.657976
$943$	−3.19257	−0.103965
$944$	4.27122	0.139017
$945$	−7.43539	−0.241873
$946$	29.8848	0.971640
$947$	−18.9954	−0.617266	−0.308633	−	0.951181i	$-0.599871\pi$
$947$	−18.9954	−0.617266	−0.308633	+	0.951181i	$0.599871\pi$
$948$	−21.0064	−0.682256
$949$	−0.845887	−0.0274587
$950$	−13.7862	−0.447284
$951$	−38.5931	−1.25147
$952$	−2.76504	−0.0896156
$953$	−16.2966	−0.527899	−0.263949	−	0.964537i	$-0.585025\pi$
$953$	−16.2966	−0.527899	−0.263949	+	0.964537i	$0.585025\pi$
$954$	24.0622	0.779041
$955$	−70.7869	−2.29061
$956$	−5.22141	−0.168872
$957$	−0.630002	−0.0203651
$958$	−35.6304	−1.15117
$959$	15.6115	0.504122
$960$	−9.89701	−0.319425
$961$	−27.3610	−0.882614
$962$	0.513048	0.0165413
$963$	−7.94922	−0.256160
$964$	−2.31966	−0.0747111
$965$	−96.5892	−3.10932
$966$	−7.21641	−0.232184
$967$	2.68153	0.0862323	0.0431161	−	0.999070i	$-0.486271\pi$
$967$	2.68153	0.0862323	0.0431161	+	0.999070i	$0.486271\pi$
$968$	15.3201	0.492405
$969$	6.60574	0.212207
$970$	55.1567	1.77098
$971$	47.4001	1.52114	0.760571	−	0.649255i	$-0.224919\pi$
$971$	47.4001	1.52114	0.760571	+	0.649255i	$0.224919\pi$
$972$	19.1400	0.613915
$973$	−9.37651	−0.300597
$974$	−27.0689	−0.867343
$975$	7.04301	0.225557
$976$	1.64689	0.0527156
$977$	−42.5792	−1.36223	−0.681114	−	0.732177i	$-0.738504\pi$
$977$	−42.5792	−1.36223	−0.681114	+	0.732177i	$0.738504\pi$
$978$	−2.03101	−0.0649446
$979$	−79.7283	−2.54813
$980$	−26.3806	−0.842696
$981$	10.8031	0.344917
$982$	20.6028	0.657462
$983$	−27.2983	−0.870681	−0.435340	−	0.900266i	$-0.643372\pi$
$983$	−27.2983	−0.870681	−0.435340	+	0.900266i	$0.643372\pi$
$984$	−2.20472	−0.0702840
$985$	−30.4176	−0.969186
$986$	−0.155579	−0.00495465
$987$	21.1608	0.673557
$988$	0.223732	0.00711787
$989$	19.2610	0.612465
$990$	−49.2308	−1.56466
$991$	−60.9292	−1.93548	−0.967740	−	0.251951i	$-0.918928\pi$
$991$	−60.9292	−1.93548	−0.967740	+	0.251951i	$0.918928\pi$
$992$	−1.90761	−0.0605667
$993$	−7.00776	−0.222384
$994$	−5.79311	−0.183746
$995$	−35.8340	−1.13601
$996$	−37.0548	−1.17413
$997$	−3.25619	−0.103125	−0.0515623	−	0.998670i	$-0.516420\pi$
$997$	−3.25619	−0.103125	−0.0515623	+	0.998670i	$0.516420\pi$
$998$	19.1318	0.605608
$999$	−4.11574	−0.130216

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8018.2.a.j.1.8	✓	47

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8018.2.a.j.1.8	✓	47	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:	\( N \)	\(=\)	\( 8018 = 2 \cdot 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8018.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.28341 q^{3} +1.00000 q^{4} +4.33431 q^{5} -2.28341 q^{6} -0.955796 q^{7} +1.00000 q^{8} +2.21398 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.28341 q^{3} +1.00000 q^{4} +4.33431 q^{5} -2.28341 q^{6} -0.955796 q^{7} +1.00000 q^{8} +2.21398 q^{9} +4.33431 q^{10} -5.13031 q^{11} -2.28341 q^{12} -0.223732 q^{13} -0.955796 q^{14} -9.89701 q^{15} +1.00000 q^{16} +2.89292 q^{17} +2.21398 q^{18} -1.00000 q^{19} +4.33431 q^{20} +2.18248 q^{21} -5.13031 q^{22} -3.30652 q^{23} -2.28341 q^{24} +13.7862 q^{25} -0.223732 q^{26} +1.79481 q^{27} -0.955796 q^{28} -0.0537792 q^{29} -9.89701 q^{30} -1.90761 q^{31} +1.00000 q^{32} +11.7146 q^{33} +2.89292 q^{34} -4.14271 q^{35} +2.21398 q^{36} -2.29313 q^{37} -1.00000 q^{38} +0.510873 q^{39} +4.33431 q^{40} +0.965538 q^{41} +2.18248 q^{42} -5.82516 q^{43} -5.13031 q^{44} +9.59606 q^{45} -3.30652 q^{46} +9.69579 q^{47} -2.28341 q^{48} -6.08645 q^{49} +13.7862 q^{50} -6.60574 q^{51} -0.223732 q^{52} +10.8683 q^{53} +1.79481 q^{54} -22.2363 q^{55} -0.955796 q^{56} +2.28341 q^{57} -0.0537792 q^{58} +4.27122 q^{59} -9.89701 q^{60} +1.64689 q^{61} -1.90761 q^{62} -2.11611 q^{63} +1.00000 q^{64} -0.969724 q^{65} +11.7146 q^{66} -11.7055 q^{67} +2.89292 q^{68} +7.55016 q^{69} -4.14271 q^{70} +6.06103 q^{71} +2.21398 q^{72} +3.78080 q^{73} -2.29313 q^{74} -31.4796 q^{75} -1.00000 q^{76} +4.90353 q^{77} +0.510873 q^{78} +9.19955 q^{79} +4.33431 q^{80} -10.7402 q^{81} +0.965538 q^{82} +16.2278 q^{83} +2.18248 q^{84} +12.5388 q^{85} -5.82516 q^{86} +0.122800 q^{87} -5.13031 q^{88} +15.5406 q^{89} +9.59606 q^{90} +0.213842 q^{91} -3.30652 q^{92} +4.35586 q^{93} +9.69579 q^{94} -4.33431 q^{95} -2.28341 q^{96} +12.7256 q^{97} -6.08645 q^{98} -11.3584 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 47 q + 47 q^{2} + 10 q^{3} + 47 q^{4} + 15 q^{5} + 10 q^{6} + 47 q^{8} + 69 q^{9}+O(q^{10}) \) 47 * q + 47 * q^2 + 10 * q^3 + 47 * q^4 + 15 * q^5 + 10 * q^6 + 47 * q^8 + 69 * q^9	\( 47 q + 47 q^{2} + 10 q^{3} + 47 q^{4} + 15 q^{5} + 10 q^{6} + 47 q^{8} + 69 q^{9} + 15 q^{10} + 17 q^{11} + 10 q^{12} + 27 q^{13} + 10 q^{15} + 47 q^{16} + 16 q^{17} + 69 q^{18} - 47 q^{19} + 15 q^{20} + 26 q^{21} + 17 q^{22} + 24 q^{23} + 10 q^{24} + 86 q^{25} + 27 q^{26} + 43 q^{27} + 58 q^{29} + 10 q^{30} + 21 q^{31} + 47 q^{32} + 18 q^{33} + 16 q^{34} + 24 q^{35} + 69 q^{36} + 78 q^{37} - 47 q^{38} - 4 q^{39} + 15 q^{40} + 53 q^{41} + 26 q^{42} + 47 q^{43} + 17 q^{44} + 23 q^{45} + 24 q^{46} + 16 q^{47} + 10 q^{48} + 93 q^{49} + 86 q^{50} + 28 q^{51} + 27 q^{52} + 43 q^{53} + 43 q^{54} + 23 q^{55} - 10 q^{57} + 58 q^{58} + 3 q^{59} + 10 q^{60} + 12 q^{61} + 21 q^{62} - 5 q^{63} + 47 q^{64} + 62 q^{65} + 18 q^{66} + 68 q^{67} + 16 q^{68} + 12 q^{69} + 24 q^{70} - 13 q^{71} + 69 q^{72} + 35 q^{73} + 78 q^{74} + 22 q^{75} - 47 q^{76} + 26 q^{77} - 4 q^{78} + 21 q^{79} + 15 q^{80} + 123 q^{81} + 53 q^{82} + 23 q^{83} + 26 q^{84} + 38 q^{85} + 47 q^{86} + 39 q^{87} + 17 q^{88} + 24 q^{89} + 23 q^{90} + 49 q^{91} + 24 q^{92} + 91 q^{93} + 16 q^{94} - 15 q^{95} + 10 q^{96} + 88 q^{97} + 93 q^{98} + 26 q^{99}+O(q^{100}) \) 47 * q + 47 * q^2 + 10 * q^3 + 47 * q^4 + 15 * q^5 + 10 * q^6 + 47 * q^8 + 69 * q^9 + 15 * q^10 + 17 * q^11 + 10 * q^12 + 27 * q^13 + 10 * q^15 + 47 * q^16 + 16 * q^17 + 69 * q^18 - 47 * q^19 + 15 * q^20 + 26 * q^21 + 17 * q^22 + 24 * q^23 + 10 * q^24 + 86 * q^25 + 27 * q^26 + 43 * q^27 + 58 * q^29 + 10 * q^30 + 21 * q^31 + 47 * q^32 + 18 * q^33 + 16 * q^34 + 24 * q^35 + 69 * q^36 + 78 * q^37 - 47 * q^38 - 4 * q^39 + 15 * q^40 + 53 * q^41 + 26 * q^42 + 47 * q^43 + 17 * q^44 + 23 * q^45 + 24 * q^46 + 16 * q^47 + 10 * q^48 + 93 * q^49 + 86 * q^50 + 28 * q^51 + 27 * q^52 + 43 * q^53 + 43 * q^54 + 23 * q^55 - 10 * q^57 + 58 * q^58 + 3 * q^59 + 10 * q^60 + 12 * q^61 + 21 * q^62 - 5 * q^63 + 47 * q^64 + 62 * q^65 + 18 * q^66 + 68 * q^67 + 16 * q^68 + 12 * q^69 + 24 * q^70 - 13 * q^71 + 69 * q^72 + 35 * q^73 + 78 * q^74 + 22 * q^75 - 47 * q^76 + 26 * q^77 - 4 * q^78 + 21 * q^79 + 15 * q^80 + 123 * q^81 + 53 * q^82 + 23 * q^83 + 26 * q^84 + 38 * q^85 + 47 * q^86 + 39 * q^87 + 17 * q^88 + 24 * q^89 + 23 * q^90 + 49 * q^91 + 24 * q^92 + 91 * q^93 + 16 * q^94 - 15 * q^95 + 10 * q^96 + 88 * q^97 + 93 * q^98 + 26 * q^99

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