LMFDB - Embedded newform 471.4.a.c.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [471,4,Mod(1,471)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(471, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("471.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 471 = 3 \cdot 157 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	471.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:	$27.7898996127$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	471.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.36089 q^{2} -3.00000 q^{3} -6.14798 q^{4} +17.6223 q^{5} -4.08267 q^{6} +31.9947 q^{7} -19.2538 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+1.36089 q^{2} -3.00000 q^{3} -6.14798 q^{4} +17.6223 q^{5} -4.08267 q^{6} +31.9947 q^{7} -19.2538 q^{8} +9.00000 q^{9} +23.9820 q^{10} +10.4211 q^{11} +18.4439 q^{12} -12.2300 q^{13} +43.5413 q^{14} -52.8670 q^{15} +22.9815 q^{16} +89.2668 q^{17} +12.2480 q^{18} -114.164 q^{19} -108.342 q^{20} -95.9842 q^{21} +14.1819 q^{22} -62.2812 q^{23} +57.7615 q^{24} +185.546 q^{25} -16.6437 q^{26} -27.0000 q^{27} -196.703 q^{28} -42.4332 q^{29} -71.9461 q^{30} +273.472 q^{31} +185.306 q^{32} -31.2632 q^{33} +121.482 q^{34} +563.821 q^{35} -55.3318 q^{36} +94.0924 q^{37} -155.365 q^{38} +36.6900 q^{39} -339.297 q^{40} -506.022 q^{41} -130.624 q^{42} -77.1664 q^{43} -64.0685 q^{44} +158.601 q^{45} -84.7579 q^{46} +410.703 q^{47} -68.9444 q^{48} +680.662 q^{49} +252.508 q^{50} -267.800 q^{51} +75.1897 q^{52} +439.394 q^{53} -36.7440 q^{54} +183.643 q^{55} -616.021 q^{56} +342.492 q^{57} -57.7469 q^{58} +376.695 q^{59} +325.025 q^{60} +657.465 q^{61} +372.166 q^{62} +287.953 q^{63} +68.3294 q^{64} -215.521 q^{65} -42.5458 q^{66} -23.1459 q^{67} -548.810 q^{68} +186.844 q^{69} +767.299 q^{70} -572.643 q^{71} -173.285 q^{72} +649.680 q^{73} +128.049 q^{74} -556.638 q^{75} +701.878 q^{76} +333.419 q^{77} +49.9310 q^{78} +579.624 q^{79} +404.987 q^{80} +81.0000 q^{81} -688.640 q^{82} +232.454 q^{83} +590.109 q^{84} +1573.09 q^{85} -105.015 q^{86} +127.300 q^{87} -200.646 q^{88} -255.258 q^{89} +215.838 q^{90} -391.295 q^{91} +382.904 q^{92} -820.417 q^{93} +558.922 q^{94} -2011.84 q^{95} -555.918 q^{96} +893.990 q^{97} +926.307 q^{98} +93.7896 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9}+O(q^{10}) $ 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9} + 13 q^{10} + 61 q^{11} - 270 q^{12} + 4 q^{13} + 133 q^{14} - 96 q^{15} + 342 q^{16} + 308 q^{17} + 36 q^{18} + 32 q^{19} + 407 q^{20} + 12 q^{21} - 166 q^{22} + 53 q^{23} - 81 q^{24} + 746 q^{25} + 467 q^{26} - 594 q^{27} + 85 q^{28} + 634 q^{29} - 39 q^{30} - 163 q^{31} + 150 q^{32} - 183 q^{33} + 37 q^{34} + 782 q^{35} + 810 q^{36} - 2 q^{37} + 584 q^{38} - 12 q^{39} + 864 q^{40} + 1593 q^{41} - 399 q^{42} - 891 q^{43} + 2093 q^{44} + 288 q^{45} + 108 q^{46} + 1200 q^{47} - 1026 q^{48} + 2816 q^{49} + 4703 q^{50} - 924 q^{51} + 1866 q^{52} + 1182 q^{53} - 108 q^{54} + 970 q^{55} + 5362 q^{56} - 96 q^{57} + 1814 q^{58} + 2802 q^{59} - 1221 q^{60} + 2629 q^{61} + 2378 q^{62} - 36 q^{63} + 625 q^{64} + 2264 q^{65} + 498 q^{66} - 1074 q^{67} + 4383 q^{68} - 159 q^{69} + 4009 q^{70} + 3920 q^{71} + 243 q^{72} + 1086 q^{73} + 4904 q^{74} - 2238 q^{75} + 3750 q^{76} + 2966 q^{77} - 1401 q^{78} - 30 q^{79} + 7777 q^{80} + 1782 q^{81} + 2932 q^{82} + 1900 q^{83} - 255 q^{84} + 524 q^{85} + 3209 q^{86} - 1902 q^{87} - 100 q^{88} + 4488 q^{89} + 117 q^{90} - 818 q^{91} + 6210 q^{92} + 489 q^{93} + 3220 q^{94} + 3500 q^{95} - 450 q^{96} + 2178 q^{97} + 7629 q^{98} + 549 q^{99}+O(q^{100}) \) 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9 + 13 * q^10 + 61 * q^11 - 270 * q^12 + 4 * q^13 + 133 * q^14 - 96 * q^15 + 342 * q^16 + 308 * q^17 + 36 * q^18 + 32 * q^19 + 407 * q^20 + 12 * q^21 - 166 * q^22 + 53 * q^23 - 81 * q^24 + 746 * q^25 + 467 * q^26 - 594 * q^27 + 85 * q^28 + 634 * q^29 - 39 * q^30 - 163 * q^31 + 150 * q^32 - 183 * q^33 + 37 * q^34 + 782 * q^35 + 810 * q^36 - 2 * q^37 + 584 * q^38 - 12 * q^39 + 864 * q^40 + 1593 * q^41 - 399 * q^42 - 891 * q^43 + 2093 * q^44 + 288 * q^45 + 108 * q^46 + 1200 * q^47 - 1026 * q^48 + 2816 * q^49 + 4703 * q^50 - 924 * q^51 + 1866 * q^52 + 1182 * q^53 - 108 * q^54 + 970 * q^55 + 5362 * q^56 - 96 * q^57 + 1814 * q^58 + 2802 * q^59 - 1221 * q^60 + 2629 * q^61 + 2378 * q^62 - 36 * q^63 + 625 * q^64 + 2264 * q^65 + 498 * q^66 - 1074 * q^67 + 4383 * q^68 - 159 * q^69 + 4009 * q^70 + 3920 * q^71 + 243 * q^72 + 1086 * q^73 + 4904 * q^74 - 2238 * q^75 + 3750 * q^76 + 2966 * q^77 - 1401 * q^78 - 30 * q^79 + 7777 * q^80 + 1782 * q^81 + 2932 * q^82 + 1900 * q^83 - 255 * q^84 + 524 * q^85 + 3209 * q^86 - 1902 * q^87 - 100 * q^88 + 4488 * q^89 + 117 * q^90 - 818 * q^91 + 6210 * q^92 + 489 * q^93 + 3220 * q^94 + 3500 * q^95 - 450 * q^96 + 2178 * q^97 + 7629 * q^98 + 549 * 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 $

$2$	1.36089	0.481147	0.240574	−	0.970631i	$-0.422664\pi$
$2$	1.36089	0.481147	0.240574	+	0.970631i	$0.422664\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−6.14798	−0.768497
$5$	17.6223	1.57619	0.788094	−	0.615555i	$-0.211068\pi$
$5$	17.6223	1.57619	0.788094	+	0.615555i	$0.211068\pi$
$6$	−4.08267	−0.277791
$7$	31.9947	1.72755	0.863776	−	0.503875i	$-0.168093\pi$
$7$	31.9947	1.72755	0.863776	+	0.503875i	$0.168093\pi$
$8$	−19.2538	−0.850908
$9$	9.00000	0.333333
$10$	23.9820	0.758379
$11$	10.4211	0.285643	0.142821	−	0.989748i	$-0.454383\pi$
$11$	10.4211	0.285643	0.142821	+	0.989748i	$0.454383\pi$
$12$	18.4439	0.443692
$13$	−12.2300	−0.260922	−0.130461	−	0.991453i	$-0.541646\pi$
$13$	−12.2300	−0.260922	−0.130461	+	0.991453i	$0.541646\pi$
$14$	43.5413	0.831207
$15$	−52.8670	−0.910013
$16$	22.9815	0.359085
$17$	89.2668	1.27355	0.636776	−	0.771049i	$-0.280268\pi$
$17$	89.2668	1.27355	0.636776	+	0.771049i	$0.280268\pi$
$18$	12.2480	0.160382
$19$	−114.164	−1.37848	−0.689238	−	0.724535i	$-0.742054\pi$
$19$	−114.164	−1.37848	−0.689238	+	0.724535i	$0.742054\pi$
$20$	−108.342	−1.21130
$21$	−95.9842	−0.997403
$22$	14.1819	0.137436
$23$	−62.2812	−0.564632	−0.282316	−	0.959321i	$-0.591103\pi$
$23$	−62.2812	−0.564632	−0.282316	+	0.959321i	$0.591103\pi$
$24$	57.7615	0.491272
$25$	185.546	1.48437
$26$	−16.6437	−0.125542
$27$	−27.0000	−0.192450
$28$	−196.703	−1.32762
$29$	−42.4332	−0.271712	−0.135856	−	0.990729i	$-0.543378\pi$
$29$	−42.4332	−0.271712	−0.135856	+	0.990729i	$0.543378\pi$
$30$	−71.9461	−0.437850
$31$	273.472	1.58442	0.792211	−	0.610247i	$-0.208930\pi$
$31$	273.472	1.58442	0.792211	+	0.610247i	$0.208930\pi$
$32$	185.306	1.02368
$33$	−31.2632	−0.164916
$34$	121.482	0.612766
$35$	563.821	2.72295
$36$	−55.3318	−0.256166
$37$	94.0924	0.418073	0.209036	−	0.977908i	$-0.432967\pi$
$37$	94.0924	0.418073	0.209036	+	0.977908i	$0.432967\pi$
$38$	−155.365	−0.663250
$39$	36.6900	0.150644
$40$	−339.297	−1.34119
$41$	−506.022	−1.92749	−0.963747	−	0.266816i	$-0.914028\pi$
$41$	−506.022	−1.92749	−0.963747	+	0.266816i	$0.914028\pi$
$42$	−130.624	−0.479898
$43$	−77.1664	−0.273669	−0.136834	−	0.990594i	$-0.543693\pi$
$43$	−77.1664	−0.273669	−0.136834	+	0.990594i	$0.543693\pi$
$44$	−64.0685	−0.219516
$45$	158.601	0.525396
$46$	−84.7579	−0.271671
$47$	410.703	1.27462	0.637311	−	0.770606i	$-0.280047\pi$
$47$	410.703	1.27462	0.637311	+	0.770606i	$0.280047\pi$
$48$	−68.9444	−0.207318
$49$	680.662	1.98444
$50$	252.508	0.714200
$51$	−267.800	−0.735285
$52$	75.1897	0.200518
$53$	439.394	1.13878	0.569391	−	0.822067i	$-0.307179\pi$
$53$	439.394	1.13878	0.569391	+	0.822067i	$0.307179\pi$
$54$	−36.7440	−0.0925968
$55$	183.643	0.450227
$56$	−616.021	−1.46999
$57$	342.492	0.795863
$58$	−57.7469	−0.130733
$59$	376.695	0.831211	0.415606	−	0.909545i	$-0.363570\pi$
$59$	376.695	0.831211	0.415606	+	0.909545i	$0.363570\pi$
$60$	325.025	0.699342
$61$	657.465	1.38000	0.689999	−	0.723811i	$-0.257611\pi$
$61$	657.465	1.38000	0.689999	+	0.723811i	$0.257611\pi$
$62$	372.166	0.762340
$63$	287.953	0.575851
$64$	68.3294	0.133456
$65$	−215.521	−0.411263
$66$	−42.5458	−0.0793489
$67$	−23.1459	−0.0422048	−0.0211024	−	0.999777i	$-0.506718\pi$
$67$	−23.1459	−0.0422048	−0.0211024	+	0.999777i	$0.506718\pi$
$68$	−548.810	−0.978721
$69$	186.844	0.325990
$70$	767.299	1.31014
$71$	−572.643	−0.957186	−0.478593	−	0.878037i	$-0.658853\pi$
$71$	−572.643	−0.957186	−0.478593	+	0.878037i	$0.658853\pi$
$72$	−173.285	−0.283636
$73$	649.680	1.04163	0.520817	−	0.853668i	$-0.325627\pi$
$73$	649.680	1.04163	0.520817	+	0.853668i	$0.325627\pi$
$74$	128.049	0.201155
$75$	−556.638	−0.857001
$76$	701.878	1.05935
$77$	333.419	0.493463
$78$	49.9310	0.0724817
$79$	579.624	0.825479	0.412739	−	0.910849i	$-0.364572\pi$
$79$	579.624	0.825479	0.412739	+	0.910849i	$0.364572\pi$
$80$	404.987	0.565986
$81$	81.0000	0.111111
$82$	−688.640	−0.927409
$83$	232.454	0.307411	0.153706	−	0.988117i	$-0.450879\pi$
$83$	232.454	0.307411	0.153706	+	0.988117i	$0.450879\pi$
$84$	590.109	0.766502
$85$	1573.09	2.00736
$86$	−105.015	−0.131675
$87$	127.300	0.156873
$88$	−200.646	−0.243056
$89$	−255.258	−0.304015	−0.152007	−	0.988379i	$-0.548574\pi$
$89$	−255.258	−0.304015	−0.152007	+	0.988379i	$0.548574\pi$
$90$	215.838	0.252793
$91$	−391.295	−0.450757
$92$	382.904	0.433918
$93$	−820.417	−0.914767
$94$	558.922	0.613281
$95$	−2011.84	−2.17274
$96$	−555.918	−0.591022
$97$	893.990	0.935783	0.467891	−	0.883786i	$-0.345014\pi$
$97$	893.990	0.935783	0.467891	+	0.883786i	$0.345014\pi$
$98$	926.307	0.954807
$99$	93.7896	0.0952143
$100$	−1140.73	−1.14073
$101$	−899.639	−0.886311	−0.443156	−	0.896445i	$-0.646141\pi$
$101$	−899.639	−0.886311	−0.443156	+	0.896445i	$0.646141\pi$
$102$	−364.447	−0.353781
$103$	−1334.46	−1.27658	−0.638291	−	0.769795i	$-0.720358\pi$
$103$	−1334.46	−1.27658	−0.638291	+	0.769795i	$0.720358\pi$
$104$	235.474	0.222021
$105$	−1691.46	−1.57209
$106$	597.967	0.547922
$107$	−664.980	−0.600804	−0.300402	−	0.953813i	$-0.597121\pi$
$107$	−664.980	−0.600804	−0.300402	+	0.953813i	$0.597121\pi$
$108$	165.995	0.147897
$109$	−216.295	−0.190067	−0.0950334	−	0.995474i	$-0.530296\pi$
$109$	−216.295	−0.190067	−0.0950334	+	0.995474i	$0.530296\pi$
$110$	249.919	0.216625
$111$	−282.277	−0.241374
$112$	735.285	0.620339
$113$	2260.76	1.88208	0.941039	−	0.338299i	$-0.109852\pi$
$113$	2260.76	1.88208	0.941039	+	0.338299i	$0.109852\pi$
$114$	466.094	0.382927
$115$	−1097.54	−0.889966
$116$	260.878	0.208810
$117$	−110.070	−0.0869741
$118$	512.640	0.399935
$119$	2856.07	2.20013
$120$	1017.89	0.774337
$121$	−1222.40	−0.918408
$122$	894.738	0.663982
$123$	1518.06	1.11284
$124$	−1681.30	−1.21762
$125$	1066.96	0.763457
$126$	391.872	0.277069
$127$	−2716.78	−1.89823	−0.949116	−	0.314928i	$-0.898020\pi$
$127$	−2716.78	−1.89823	−0.949116	+	0.314928i	$0.898020\pi$
$128$	−1389.46	−0.959469
$129$	231.499	0.158003
$130$	−293.300	−0.197878
$131$	−1630.65	−1.08756	−0.543781	−	0.839227i	$-0.683008\pi$
$131$	−1630.65	−1.08756	−0.543781	+	0.839227i	$0.683008\pi$
$132$	192.206	0.126737
$133$	−3652.65	−2.38139
$134$	−31.4990	−0.0203067
$135$	−475.803	−0.303338
$136$	−1718.73	−1.08367
$137$	864.211	0.538938	0.269469	−	0.963009i	$-0.413152\pi$
$137$	864.211	0.538938	0.269469	+	0.963009i	$0.413152\pi$
$138$	254.274	0.156849
$139$	−1358.21	−0.828793	−0.414397	−	0.910096i	$-0.636007\pi$
$139$	−1358.21	−0.828793	−0.414397	+	0.910096i	$0.636007\pi$
$140$	−3466.36	−2.09258
$141$	−1232.11	−0.735904
$142$	−779.304	−0.460548
$143$	−127.450	−0.0745306
$144$	206.833	0.119695
$145$	−747.771	−0.428269
$146$	884.143	0.501179
$147$	−2041.99	−1.14572
$148$	−578.478	−0.321288
$149$	−474.290	−0.260774	−0.130387	−	0.991463i	$-0.541622\pi$
$149$	−474.290	−0.260774	−0.130387	+	0.991463i	$0.541622\pi$
$150$	−757.524	−0.412344
$151$	2116.13	1.14045	0.570225	−	0.821488i	$-0.306856\pi$
$151$	2116.13	1.14045	0.570225	+	0.821488i	$0.306856\pi$
$152$	2198.10	1.17296
$153$	803.401	0.424517
$154$	453.747	0.237428
$155$	4819.22	2.49735
$156$	−225.569	−0.115769
$157$	−157.000	−0.0798087
$157$	−157.000	−0.0798087
$158$	788.805	0.397177
$159$	−1318.18	−0.657476
$160$	3265.52	1.61351
$161$	−1992.67	−0.975431
$162$	110.232	0.0534608
$163$	−1513.92	−0.727482	−0.363741	−	0.931500i	$-0.618501\pi$
$163$	−1513.92	−0.727482	−0.363741	+	0.931500i	$0.618501\pi$
$164$	3111.01	1.48127
$165$	−550.930	−0.259939
$166$	316.344	0.147910
$167$	2702.53	1.25226	0.626132	−	0.779717i	$-0.284637\pi$
$167$	2702.53	1.25226	0.626132	+	0.779717i	$0.284637\pi$
$168$	1848.06	0.848698
$169$	−2047.43	−0.931920
$170$	2140.80	0.965834
$171$	−1027.48	−0.459492
$172$	474.417	0.210314
$173$	1722.62	0.757042	0.378521	−	0.925593i	$-0.376433\pi$
$173$	1722.62	0.757042	0.378521	+	0.925593i	$0.376433\pi$
$174$	173.241	0.0754790
$175$	5936.50	2.56433
$176$	239.491	0.102570
$177$	−1130.08	−0.479900
$178$	−347.379	−0.146276
$179$	−2805.53	−1.17148	−0.585740	−	0.810499i	$-0.699196\pi$
$179$	−2805.53	−1.17148	−0.585740	+	0.810499i	$0.699196\pi$
$180$	−975.075	−0.403765
$181$	−3810.01	−1.56462	−0.782308	−	0.622891i	$-0.785958\pi$
$181$	−3810.01	−1.56462	−0.782308	+	0.622891i	$0.785958\pi$
$182$	−532.510	−0.216881
$183$	−1972.40	−0.796742
$184$	1199.15	0.480450
$185$	1658.13	0.658961
$186$	−1116.50	−0.440137
$187$	930.256	0.363781
$188$	−2525.00	−0.979544
$189$	−863.858	−0.332468
$190$	−2737.89	−1.04541
$191$	3691.91	1.39862	0.699311	−	0.714817i	$-0.253490\pi$
$191$	3691.91	1.39862	0.699311	+	0.714817i	$0.253490\pi$
$192$	−204.988	−0.0770508
$193$	2326.63	0.867745	0.433873	−	0.900974i	$-0.357147\pi$
$193$	2326.63	0.867745	0.433873	+	0.900974i	$0.357147\pi$
$194$	1216.62	0.450249
$195$	646.562	0.237443
$196$	−4184.70	−1.52504
$197$	3374.98	1.22060	0.610298	−	0.792172i	$-0.291050\pi$
$197$	3374.98	1.22060	0.610298	+	0.792172i	$0.291050\pi$
$198$	127.637	0.0458121
$199$	2557.49	0.911034	0.455517	−	0.890227i	$-0.349454\pi$
$199$	2557.49	0.911034	0.455517	+	0.890227i	$0.349454\pi$
$200$	−3572.48	−1.26306
$201$	69.4377	0.0243670
$202$	−1224.31	−0.426446
$203$	−1357.64	−0.469397
$204$	1646.43	0.565065
$205$	−8917.27	−3.03809
$206$	−1816.05	−0.614224
$207$	−560.531	−0.188211
$208$	−281.063	−0.0936933
$209$	−1189.71	−0.393752
$210$	−2301.90	−0.756409
$211$	2112.94	0.689389	0.344694	−	0.938715i	$-0.387983\pi$
$211$	2112.94	0.689389	0.344694	+	0.938715i	$0.387983\pi$
$212$	−2701.38	−0.875150
$213$	1717.93	0.552632
$214$	−904.965	−0.289075
$215$	−1359.85	−0.431354
$216$	519.854	0.163757
$217$	8749.67	2.73717
$218$	−294.354	−0.0914502
$219$	−1949.04	−0.601388
$220$	−1129.04	−0.345998
$221$	−1091.73	−0.332298
$222$	−384.148	−0.116137
$223$	−5735.38	−1.72228	−0.861142	−	0.508364i	$-0.830250\pi$
$223$	−5735.38	−1.72228	−0.861142	+	0.508364i	$0.830250\pi$
$224$	5928.81	1.76846
$225$	1669.92	0.494790
$226$	3076.65	0.905556
$227$	5584.66	1.63289	0.816447	−	0.577421i	$-0.195941\pi$
$227$	5584.66	1.63289	0.816447	+	0.577421i	$0.195941\pi$
$228$	−2105.63	−0.611619
$229$	−4995.69	−1.44159	−0.720795	−	0.693148i	$-0.756223\pi$
$229$	−4995.69	−1.44159	−0.720795	+	0.693148i	$0.756223\pi$
$230$	−1493.63	−0.428205
$231$	−1000.26	−0.284901
$232$	817.002	0.231202
$233$	1098.49	0.308861	0.154431	−	0.988004i	$-0.450646\pi$
$233$	1098.49	0.308861	0.154431	+	0.988004i	$0.450646\pi$
$234$	−149.793	−0.0418473
$235$	7237.55	2.00905
$236$	−2315.91	−0.638784
$237$	−1738.87	−0.476590
$238$	3886.79	1.05859
$239$	1981.42	0.536267	0.268133	−	0.963382i	$-0.413593\pi$
$239$	1981.42	0.536267	0.268133	+	0.963382i	$0.413593\pi$
$240$	−1214.96	−0.326772
$241$	−1518.49	−0.405869	−0.202935	−	0.979192i	$-0.565048\pi$
$241$	−1518.49	−0.405869	−0.202935	+	0.979192i	$0.565048\pi$
$242$	−1663.55	−0.441890
$243$	−243.000	−0.0641500
$244$	−4042.08	−1.06052
$245$	11994.9	3.12785
$246$	2065.92	0.535440
$247$	1396.23	0.359675
$248$	−5265.39	−1.34820
$249$	−697.362	−0.177484
$250$	1452.02	0.367335
$251$	−5236.76	−1.31690	−0.658449	−	0.752626i	$-0.728787\pi$
$251$	−5236.76	−1.31690	−0.658449	+	0.752626i	$0.728787\pi$
$252$	−1770.33	−0.442540
$253$	−649.037	−0.161283
$254$	−3697.24	−0.913329
$255$	−4719.26	−1.15895
$256$	−2437.54	−0.595102
$257$	675.389	0.163929	0.0819643	−	0.996635i	$-0.473881\pi$
$257$	675.389	0.163929	0.0819643	+	0.996635i	$0.473881\pi$
$258$	315.045	0.0760226
$259$	3010.46	0.722243
$260$	1325.02	0.316054
$261$	−381.899	−0.0905706
$262$	−2219.13	−0.523277
$263$	−4929.13	−1.15568	−0.577838	−	0.816151i	$-0.696104\pi$
$263$	−4929.13	−1.15568	−0.577838	+	0.816151i	$0.696104\pi$
$264$	601.937	0.140328
$265$	7743.14	1.79493
$266$	−4970.85	−1.14580
$267$	765.775	0.175523
$268$	142.301	0.0324343
$269$	−2798.37	−0.634273	−0.317136	−	0.948380i	$-0.602721\pi$
$269$	−2798.37	−0.634273	−0.317136	+	0.948380i	$0.602721\pi$
$270$	−647.515	−0.145950
$271$	−993.864	−0.222778	−0.111389	−	0.993777i	$-0.535530\pi$
$271$	−993.864	−0.222778	−0.111389	+	0.993777i	$0.535530\pi$
$272$	2051.48	0.457314
$273$	1173.89	0.260245
$274$	1176.10	0.259309
$275$	1933.59	0.423999
$276$	−1148.71	−0.250523
$277$	964.450	0.209199	0.104600	−	0.994514i	$-0.466644\pi$
$277$	964.450	0.209199	0.104600	+	0.994514i	$0.466644\pi$
$278$	−1848.38	−0.398772
$279$	2461.25	0.528141
$280$	−10855.7	−2.31698
$281$	473.926	0.100612	0.0503061	−	0.998734i	$-0.483980\pi$
$281$	473.926	0.100612	0.0503061	+	0.998734i	$0.483980\pi$
$282$	−1676.77	−0.354078
$283$	−1442.08	−0.302907	−0.151454	−	0.988464i	$-0.548395\pi$
$283$	−1442.08	−0.302907	−0.151454	+	0.988464i	$0.548395\pi$
$284$	3520.60	0.735595
$285$	6035.51	1.25443
$286$	−173.445	−0.0358602
$287$	−16190.0	−3.32985
$288$	1667.75	0.341227
$289$	3055.56	0.621934
$290$	−1017.63	−0.206061
$291$	−2681.97	−0.540274
$292$	−3994.22	−0.800493
$293$	−4369.94	−0.871314	−0.435657	−	0.900113i	$-0.643484\pi$
$293$	−4369.94	−0.871314	−0.435657	+	0.900113i	$0.643484\pi$
$294$	−2778.92	−0.551258
$295$	6638.23	1.31015
$296$	−1811.64	−0.355741
$297$	−281.369	−0.0549720
$298$	−645.456	−0.125471
$299$	761.699	0.147325
$300$	3422.20	0.658603
$301$	−2468.92	−0.472777
$302$	2879.82	0.548725
$303$	2698.92	0.511712
$304$	−2623.66	−0.494990
$305$	11586.1	2.17514
$306$	1093.34	0.204255
$307$	−6412.39	−1.19210	−0.596049	−	0.802948i	$-0.703264\pi$
$307$	−6412.39	−1.19210	−0.596049	+	0.802948i	$0.703264\pi$
$308$	−2049.85	−0.379225
$309$	4003.37	0.737035
$310$	6558.43	1.20159
$311$	−4616.84	−0.841791	−0.420896	−	0.907109i	$-0.638284\pi$
$311$	−4616.84	−0.841791	−0.420896	+	0.907109i	$0.638284\pi$
$312$	−706.423	−0.128184
$313$	−1365.09	−0.246515	−0.123258	−	0.992375i	$-0.539334\pi$
$313$	−1365.09	−0.246515	−0.123258	+	0.992375i	$0.539334\pi$
$314$	−213.660	−0.0383997
$315$	5074.39	0.907649
$316$	−3563.52	−0.634378
$317$	7832.53	1.38776	0.693878	−	0.720092i	$-0.255901\pi$
$317$	7832.53	1.38776	0.693878	+	0.720092i	$0.255901\pi$
$318$	−1793.90	−0.316343
$319$	−442.199	−0.0776126
$320$	1204.12	0.210352
$321$	1994.94	0.346874
$322$	−2711.81	−0.469326
$323$	−10191.1	−1.75556
$324$	−497.986	−0.0853886
$325$	−2269.23	−0.387305
$326$	−2060.28	−0.350026
$327$	648.885	0.109735
$328$	9742.86	1.64012
$329$	13140.3	2.20198
$330$	−749.756	−0.125069
$331$	−10935.0	−1.81583	−0.907914	−	0.419156i	$-0.862326\pi$
$331$	−10935.0	−1.81583	−0.907914	+	0.419156i	$0.862326\pi$
$332$	−1429.12	−0.236245
$333$	846.831	0.139358
$334$	3677.85	0.602524
$335$	−407.885	−0.0665227
$336$	−2205.86	−0.358153
$337$	−265.987	−0.0429948	−0.0214974	−	0.999769i	$-0.506843\pi$
$337$	−265.987	−0.0429948	−0.0214974	+	0.999769i	$0.506843\pi$
$338$	−2786.32	−0.448391
$339$	−6782.29	−1.08662
$340$	−9671.31	−1.54265
$341$	2849.87	0.452579
$342$	−1398.28	−0.221083
$343$	10803.4	1.70067
$344$	1485.75	0.232867
$345$	3292.62	0.513822
$346$	2344.30	0.364249
$347$	−4917.49	−0.760763	−0.380382	−	0.924830i	$-0.624207\pi$
$347$	−4917.49	−0.760763	−0.380382	+	0.924830i	$0.624207\pi$
$348$	−782.635	−0.120556
$349$	8250.25	1.26540	0.632702	−	0.774395i	$-0.281946\pi$
$349$	8250.25	1.26540	0.632702	+	0.774395i	$0.281946\pi$
$350$	8078.92	1.23382
$351$	330.210	0.0502145
$352$	1931.09	0.292407
$353$	−7954.24	−1.19932	−0.599662	−	0.800253i	$-0.704698\pi$
$353$	−7954.24	−1.19932	−0.599662	+	0.800253i	$0.704698\pi$
$354$	−1537.92	−0.230903
$355$	−10091.3	−1.50871
$356$	1569.32	0.233635
$357$	−8568.20	−1.27024
$358$	−3818.02	−0.563655
$359$	397.801	0.0584822	0.0292411	−	0.999572i	$-0.490691\pi$
$359$	397.801	0.0584822	0.0292411	+	0.999572i	$0.490691\pi$
$360$	−3053.68	−0.447064
$361$	6174.43	0.900195
$362$	−5185.00	−0.752811
$363$	3667.20	0.530243
$364$	2405.67	0.346406
$365$	11448.9	1.64181
$366$	−2684.21	−0.383350
$367$	3819.64	0.543280	0.271640	−	0.962399i	$-0.412434\pi$
$367$	3819.64	0.543280	0.271640	+	0.962399i	$0.412434\pi$
$368$	−1431.31	−0.202751
$369$	−4554.19	−0.642498
$370$	2256.53	0.317057
$371$	14058.3	1.96730
$372$	5043.91	0.702996
$373$	4108.50	0.570322	0.285161	−	0.958480i	$-0.407953\pi$
$373$	4108.50	0.570322	0.285161	+	0.958480i	$0.407953\pi$
$374$	1265.98	0.175032
$375$	−3200.89	−0.440782
$376$	−7907.62	−1.08459
$377$	518.958	0.0708957
$378$	−1175.62	−0.159966
$379$	−14348.8	−1.94471	−0.972357	−	0.233500i	$-0.924982\pi$
$379$	−14348.8	−1.94471	−0.972357	+	0.233500i	$0.924982\pi$
$380$	12368.7	1.66974
$381$	8150.34	1.09594
$382$	5024.28	0.672944
$383$	8927.14	1.19101	0.595503	−	0.803353i	$-0.296953\pi$
$383$	8927.14	1.19101	0.595503	+	0.803353i	$0.296953\pi$
$384$	4168.38	0.553950
$385$	5875.62	0.777791
$386$	3166.29	0.417513
$387$	−694.497	−0.0912229
$388$	−5496.23	−0.719147
$389$	11409.6	1.48712	0.743558	−	0.668671i	$-0.233137\pi$
$389$	11409.6	1.48712	0.743558	+	0.668671i	$0.233137\pi$
$390$	879.900	0.114245
$391$	−5559.65	−0.719088
$392$	−13105.4	−1.68857
$393$	4891.95	0.627904
$394$	4592.98	0.587287
$395$	10214.3	1.30111
$396$	−576.617	−0.0731719
$397$	−10526.4	−1.33074	−0.665370	−	0.746514i	$-0.731726\pi$
$397$	−10526.4	−1.33074	−0.665370	+	0.746514i	$0.731726\pi$
$398$	3480.46	0.438342
$399$	10957.9	1.37490
$400$	4264.12	0.533015
$401$	−8559.87	−1.06598	−0.532992	−	0.846120i	$-0.678932\pi$
$401$	−8559.87	−1.06598	−0.532992	+	0.846120i	$0.678932\pi$
$402$	94.4971	0.0117241
$403$	−3344.56	−0.413411
$404$	5530.96	0.681128
$405$	1427.41	0.175132
$406$	−1847.60	−0.225849
$407$	980.543	0.119419
$408$	5156.19	0.625660
$409$	−11829.1	−1.43010	−0.715048	−	0.699075i	$-0.753595\pi$
$409$	−11829.1	−1.43010	−0.715048	+	0.699075i	$0.753595\pi$
$410$	−12135.4	−1.46177
$411$	−2592.63	−0.311156
$412$	8204.21	0.981050
$413$	12052.2	1.43596
$414$	−762.821	−0.0905570
$415$	4096.38	0.484538
$416$	−2266.29	−0.267101
$417$	4074.64	0.478504
$418$	−1619.07	−0.189453
$419$	−12126.4	−1.41387	−0.706937	−	0.707277i	$-0.749924\pi$
$419$	−12126.4	−1.41387	−0.706937	+	0.707277i	$0.749924\pi$
$420$	10399.1	1.20815
$421$	−7542.97	−0.873211	−0.436606	−	0.899653i	$-0.643819\pi$
$421$	−7542.97	−0.873211	−0.436606	+	0.899653i	$0.643819\pi$
$422$	2875.48	0.331698
$423$	3696.33	0.424874
$424$	−8460.02	−0.968998
$425$	16563.1	1.89042
$426$	2337.91	0.265897
$427$	21035.4	2.38402
$428$	4088.28	0.461716
$429$	382.349	0.0430302
$430$	−1850.61	−0.207545
$431$	−6162.90	−0.688762	−0.344381	−	0.938830i	$-0.611911\pi$
$431$	−6162.90	−0.688762	−0.344381	+	0.938830i	$0.611911\pi$
$432$	−620.499	−0.0691060
$433$	−3554.10	−0.394456	−0.197228	−	0.980358i	$-0.563194\pi$
$433$	−3554.10	−0.394456	−0.197228	+	0.980358i	$0.563194\pi$
$434$	11907.3	1.31698
$435$	2243.31	0.247261
$436$	1329.78	0.146066
$437$	7110.28	0.778331
$438$	−2652.43	−0.289356
$439$	3827.94	0.416168	0.208084	−	0.978111i	$-0.433277\pi$
$439$	3827.94	0.416168	0.208084	+	0.978111i	$0.433277\pi$
$440$	−3535.84	−0.383102
$441$	6125.96	0.661480
$442$	−1485.73	−0.159884
$443$	12152.4	1.30334	0.651669	−	0.758503i	$-0.274069\pi$
$443$	12152.4	1.30334	0.651669	+	0.758503i	$0.274069\pi$
$444$	1735.43	0.185496
$445$	−4498.24	−0.479185
$446$	−7805.22	−0.828673
$447$	1422.87	0.150558
$448$	2186.18	0.230552
$449$	−5212.02	−0.547818	−0.273909	−	0.961756i	$-0.588317\pi$
$449$	−5212.02	−0.547818	−0.273909	+	0.961756i	$0.588317\pi$
$450$	2272.57	0.238067
$451$	−5273.29	−0.550575
$452$	−13899.1	−1.44637
$453$	−6348.38	−0.658439
$454$	7600.11	0.785662
$455$	−6895.53	−0.710478
$456$	−6594.29	−0.677206
$457$	15868.3	1.62427	0.812133	−	0.583472i	$-0.198306\pi$
$457$	15868.3	1.62427	0.812133	+	0.583472i	$0.198306\pi$
$458$	−6798.58	−0.693617
$459$	−2410.20	−0.245095
$460$	6747.65	0.683936
$461$	6334.43	0.639965	0.319983	−	0.947423i	$-0.396323\pi$
$461$	6334.43	0.639965	0.319983	+	0.947423i	$0.396323\pi$
$462$	−1361.24	−0.137079
$463$	9139.61	0.917394	0.458697	−	0.888593i	$-0.348316\pi$
$463$	9139.61	0.917394	0.458697	+	0.888593i	$0.348316\pi$
$464$	−975.177	−0.0975678
$465$	−14457.7	−1.44184
$466$	1494.93	0.148608
$467$	6935.51	0.687232	0.343616	−	0.939110i	$-0.388348\pi$
$467$	6935.51	0.687232	0.343616	+	0.939110i	$0.388348\pi$
$468$	676.707	0.0668393
$469$	−740.547	−0.0729111
$470$	9849.51	0.966647
$471$	471.000	0.0460776
$472$	−7252.82	−0.707284
$473$	−804.156	−0.0781715
$474$	−2366.42	−0.229310
$475$	−21182.7	−2.04617
$476$	−17559.0	−1.69079
$477$	3954.55	0.379594
$478$	2696.50	0.258023
$479$	−5528.81	−0.527386	−0.263693	−	0.964607i	$-0.584941\pi$
$479$	−5528.81	−0.527386	−0.263693	+	0.964607i	$0.584941\pi$
$480$	−9796.56	−0.931562
$481$	−1150.75	−0.109084
$482$	−2066.50	−0.195283
$483$	5978.01	0.563166
$484$	7515.30	0.705794
$485$	15754.2	1.47497
$486$	−330.696	−0.0308656
$487$	−14213.7	−1.32256	−0.661278	−	0.750141i	$-0.729986\pi$
$487$	−14213.7	−1.32256	−0.661278	+	0.750141i	$0.729986\pi$
$488$	−12658.7	−1.17425
$489$	4541.77	0.420012
$490$	16323.7	1.50496
$491$	2043.13	0.187790	0.0938952	−	0.995582i	$-0.470068\pi$
$491$	2043.13	0.187790	0.0938952	+	0.995582i	$0.470068\pi$
$492$	−9333.03	−0.855214
$493$	−3787.87	−0.346039
$494$	1900.11	0.173057
$495$	1652.79	0.150076
$496$	6284.79	0.568943
$497$	−18321.6	−1.65359
$498$	−949.033	−0.0853959
$499$	−11937.9	−1.07097	−0.535484	−	0.844545i	$-0.679871\pi$
$499$	−11937.9	−1.07097	−0.535484	+	0.844545i	$0.679871\pi$
$500$	−6559.67	−0.586715
$501$	−8107.60	−0.722995
$502$	−7126.65	−0.633622
$503$	−12570.1	−1.11426	−0.557129	−	0.830426i	$-0.688097\pi$
$503$	−12570.1	−1.11426	−0.557129	+	0.830426i	$0.688097\pi$
$504$	−5544.19	−0.489996
$505$	−15853.7	−1.39699
$506$	−883.268	−0.0776009
$507$	6142.28	0.538044
$508$	16702.7	1.45879
$509$	−663.140	−0.0577469	−0.0288735	−	0.999583i	$-0.509192\pi$
$509$	−663.140	−0.0577469	−0.0288735	+	0.999583i	$0.509192\pi$
$510$	−6422.40	−0.557625
$511$	20786.3	1.79948
$512$	7798.45	0.673137
$513$	3082.43	0.265288
$514$	919.131	0.0788738
$515$	−23516.2	−2.01213
$516$	−1423.25	−0.121425
$517$	4279.97	0.364087
$518$	4096.90	0.347505
$519$	−5167.86	−0.437079
$520$	4149.60	0.349947
$521$	−7036.30	−0.591682	−0.295841	−	0.955237i	$-0.595600\pi$
$521$	−7036.30	−0.591682	−0.295841	+	0.955237i	$0.595600\pi$
$522$	−519.722	−0.0435778
$523$	−6075.58	−0.507966	−0.253983	−	0.967209i	$-0.581741\pi$
$523$	−6075.58	−0.507966	−0.253983	+	0.967209i	$0.581741\pi$
$524$	10025.2	0.835788
$525$	−17809.5	−1.48051
$526$	−6708.00	−0.556051
$527$	24412.0	2.01784
$528$	−718.474	−0.0592189
$529$	−8288.05	−0.681191
$530$	10537.6	0.863628
$531$	3390.25	0.277070
$532$	22456.4	1.83009
$533$	6188.64	0.502926
$534$	1042.14	0.0844525
$535$	−11718.5	−0.946981
$536$	445.648	0.0359124
$537$	8416.58	0.676355
$538$	−3808.27	−0.305179
$539$	7093.23	0.566841
$540$	2925.22	0.233114
$541$	10676.3	0.848451	0.424225	−	0.905557i	$-0.360546\pi$
$541$	10676.3	0.848451	0.424225	+	0.905557i	$0.360546\pi$
$542$	−1352.54	−0.107189
$543$	11430.0	0.903332
$544$	16541.7	1.30371
$545$	−3811.62	−0.299581
$546$	1597.53	0.125216
$547$	2515.62	0.196637	0.0983183	−	0.995155i	$-0.468654\pi$
$547$	2515.62	0.196637	0.0983183	+	0.995155i	$0.468654\pi$
$548$	−5313.15	−0.414172
$549$	5917.19	0.459999
$550$	2631.40	0.204006
$551$	4844.35	0.374548
$552$	−3597.46	−0.277388
$553$	18544.9	1.42606
$554$	1312.51	0.100656
$555$	−4974.38	−0.380451
$556$	8350.28	0.636925
$557$	12043.7	0.916175	0.458088	−	0.888907i	$-0.348535\pi$
$557$	12043.7	0.916175	0.458088	+	0.888907i	$0.348535\pi$
$558$	3349.49	0.254113
$559$	943.744	0.0714063
$560$	12957.4	0.977771
$561$	−2790.77	−0.210029
$562$	644.961	0.0484093
$563$	−12449.6	−0.931949	−0.465974	−	0.884798i	$-0.654296\pi$
$563$	−12449.6	−0.931949	−0.465974	+	0.884798i	$0.654296\pi$
$564$	7574.99	0.565540
$565$	39839.9	2.96651
$566$	−1962.51	−0.145743
$567$	2591.57	0.191950
$568$	11025.6	0.814477
$569$	−5213.38	−0.384106	−0.192053	−	0.981385i	$-0.561515\pi$
$569$	−5213.38	−0.384106	−0.192053	+	0.981385i	$0.561515\pi$
$570$	8213.66	0.603566
$571$	−15372.7	−1.12667	−0.563333	−	0.826230i	$-0.690481\pi$
$571$	−15372.7	−1.12667	−0.563333	+	0.826230i	$0.690481\pi$
$572$	783.557	0.0572765
$573$	−11075.7	−0.807495
$574$	−22032.8	−1.60215
$575$	−11556.0	−0.838122
$576$	614.965	0.0444853
$577$	11656.1	0.840985	0.420493	−	0.907296i	$-0.361857\pi$
$577$	11656.1	0.840985	0.420493	+	0.907296i	$0.361857\pi$
$578$	4158.28	0.299242
$579$	−6979.90	−0.500993
$580$	4597.28	0.329124
$581$	7437.30	0.531069
$582$	−3649.87	−0.259952
$583$	4578.96	0.325285
$584$	−12508.8	−0.886334
$585$	−1939.69	−0.137088
$586$	−5947.01	−0.419230
$587$	−27212.9	−1.91345	−0.956725	−	0.290994i	$-0.906014\pi$
$587$	−27212.9	−1.91345	−0.956725	+	0.290994i	$0.906014\pi$
$588$	12554.1	0.880480
$589$	−31220.7	−2.18409
$590$	9033.91	0.630373
$591$	−10124.9	−0.704712
$592$	2162.38	0.150124
$593$	3078.05	0.213154	0.106577	−	0.994304i	$-0.466011\pi$
$593$	3078.05	0.213154	0.106577	+	0.994304i	$0.466011\pi$
$594$	−382.912	−0.0264496
$595$	50330.5	3.46782
$596$	2915.92	0.200404
$597$	−7672.47	−0.525986
$598$	1036.59	0.0708850
$599$	−33.3703	−0.00227625	−0.00113812	−	0.999999i	$-0.500362\pi$
$599$	−33.3703	−0.00227625	−0.00113812	+	0.999999i	$0.500362\pi$
$600$	10717.4	0.729229
$601$	8966.87	0.608596	0.304298	−	0.952577i	$-0.401578\pi$
$601$	8966.87	0.608596	0.304298	+	0.952577i	$0.401578\pi$
$602$	−3359.92	−0.227476
$603$	−208.313	−0.0140683
$604$	−13009.9	−0.876433
$605$	−21541.5	−1.44758
$606$	3672.93	0.246209
$607$	−26079.4	−1.74387	−0.871937	−	0.489618i	$-0.837136\pi$
$607$	−26079.4	−1.74387	−0.871937	+	0.489618i	$0.837136\pi$
$608$	−21155.3	−1.41112
$609$	4072.91	0.271006
$610$	15767.4	1.04656
$611$	−5022.90	−0.332577
$612$	−4939.29	−0.326240
$613$	5862.99	0.386304	0.193152	−	0.981169i	$-0.438129\pi$
$613$	5862.99	0.386304	0.193152	+	0.981169i	$0.438129\pi$
$614$	−8726.55	−0.573575
$615$	26751.8	1.75404
$616$	−6419.60	−0.419892
$617$	−2097.15	−0.136836	−0.0684181	−	0.997657i	$-0.521795\pi$
$617$	−2097.15	−0.136836	−0.0684181	+	0.997657i	$0.521795\pi$
$618$	5448.15	0.354622
$619$	15962.2	1.03647	0.518234	−	0.855239i	$-0.326590\pi$
$619$	15962.2	1.03647	0.518234	+	0.855239i	$0.326590\pi$
$620$	−29628.4	−1.91920
$621$	1681.59	0.108663
$622$	−6283.01	−0.405026
$623$	−8166.92	−0.525202
$624$	843.189	0.0540939
$625$	−4390.90	−0.281017
$626$	−1857.73	−0.118610
$627$	3569.14	0.227333
$628$	965.233	0.0613328
$629$	8399.32	0.532437
$630$	6905.69	0.436713
$631$	6824.75	0.430569	0.215285	−	0.976551i	$-0.430932\pi$
$631$	6824.75	0.430569	0.215285	+	0.976551i	$0.430932\pi$
$632$	−11160.0	−0.702406
$633$	−6338.83	−0.398019
$634$	10659.2	0.667715
$635$	−47876.0	−2.99197
$636$	8104.15	0.505268
$637$	−8324.50	−0.517784
$638$	−601.785	−0.0373431
$639$	−5153.79	−0.319062
$640$	−24485.5	−1.51230
$641$	5934.11	0.365652	0.182826	−	0.983145i	$-0.441475\pi$
$641$	5934.11	0.365652	0.182826	+	0.983145i	$0.441475\pi$
$642$	2714.89	0.166898
$643$	25564.0	1.56788	0.783939	−	0.620838i	$-0.213208\pi$
$643$	25564.0	1.56788	0.783939	+	0.620838i	$0.213208\pi$
$644$	12250.9	0.749616
$645$	4079.55	0.249042
$646$	−13868.9	−0.844683
$647$	544.649	0.0330948	0.0165474	−	0.999863i	$-0.494733\pi$
$647$	544.649	0.0330948	0.0165474	+	0.999863i	$0.494733\pi$
$648$	−1559.56	−0.0945453
$649$	3925.56	0.237430
$650$	−3088.17	−0.186351
$651$	−26249.0	−1.58031
$652$	9307.56	0.559068
$653$	27028.2	1.61975	0.809874	−	0.586603i	$-0.199535\pi$
$653$	27028.2	1.61975	0.809874	+	0.586603i	$0.199535\pi$
$654$	883.061	0.0527988
$655$	−28735.8	−1.71420
$656$	−11629.1	−0.692135
$657$	5847.12	0.347211
$658$	17882.6	1.05948
$659$	−17584.5	−1.03944	−0.519722	−	0.854335i	$-0.673964\pi$
$659$	−17584.5	−1.03944	−0.519722	+	0.854335i	$0.673964\pi$
$660$	3387.11	0.199762
$661$	13782.4	0.811003	0.405502	−	0.914094i	$-0.367097\pi$
$661$	13782.4	0.811003	0.405502	+	0.914094i	$0.367097\pi$
$662$	−14881.3	−0.873681
$663$	3275.20	0.191852
$664$	−4475.63	−0.261579
$665$	−64368.1	−3.75352
$666$	1152.44	0.0670515
$667$	2642.79	0.153417
$668$	−16615.1	−0.962362
$669$	17206.1	0.994362
$670$	−555.086	−0.0320072
$671$	6851.49	0.394186
$672$	−17786.4	−1.02102
$673$	−22519.6	−1.28985	−0.644924	−	0.764247i	$-0.723111\pi$
$673$	−22519.6	−1.28985	−0.644924	+	0.764247i	$0.723111\pi$
$674$	−361.979	−0.0206868
$675$	−5009.75	−0.285667
$676$	12587.5	0.716178
$677$	10247.2	0.581733	0.290867	−	0.956764i	$-0.406056\pi$
$677$	10247.2	0.581733	0.290867	+	0.956764i	$0.406056\pi$
$678$	−9229.95	−0.522823
$679$	28603.0	1.61661
$680$	−30288.0	−1.70808
$681$	−16754.0	−0.942751
$682$	3878.37	0.217757
$683$	9901.98	0.554742	0.277371	−	0.960763i	$-0.410537\pi$
$683$	9901.98	0.554742	0.277371	+	0.960763i	$0.410537\pi$
$684$	6316.90	0.353118
$685$	15229.4	0.849467
$686$	14702.3	0.818273
$687$	14987.1	0.832303
$688$	−1773.40	−0.0982704
$689$	−5373.79	−0.297133
$690$	4480.89	0.247224
$691$	−29762.4	−1.63852	−0.819259	−	0.573424i	$-0.805615\pi$
$691$	−29762.4	−1.63852	−0.819259	+	0.573424i	$0.805615\pi$
$692$	−10590.6	−0.581785
$693$	3000.77	0.164488
$694$	−6692.17	−0.366039
$695$	−23934.9	−1.30633
$696$	−2451.01	−0.133484
$697$	−45170.9	−2.45476
$698$	11227.7	0.608846
$699$	−3295.48	−0.178321
$700$	−36497.5	−1.97068
$701$	28645.9	1.54343	0.771713	−	0.635971i	$-0.219400\pi$
$701$	28645.9	1.54343	0.771713	+	0.635971i	$0.219400\pi$
$702$	449.379	0.0241606
$703$	−10742.0	−0.576303
$704$	712.066	0.0381207
$705$	−21712.6	−1.15992
$706$	−10824.8	−0.577051
$707$	−28783.7	−1.53115
$708$	6947.73	0.368802
$709$	−24775.3	−1.31235	−0.656175	−	0.754609i	$-0.727827\pi$
$709$	−24775.3	−1.31235	−0.656175	+	0.754609i	$0.727827\pi$
$710$	−13733.1	−0.725910
$711$	5216.62	0.275160
$712$	4914.70	0.258689
$713$	−17032.2	−0.894615
$714$	−11660.4	−0.611175
$715$	−2245.96	−0.117474
$716$	17248.3	0.900280
$717$	−5944.27	−0.309614
$718$	541.363	0.0281386
$719$	13251.5	0.687342	0.343671	−	0.939090i	$-0.388329\pi$
$719$	13251.5	0.687342	0.343671	+	0.939090i	$0.388329\pi$
$720$	3644.88	0.188662
$721$	−42695.6	−2.20536
$722$	8402.73	0.433126
$723$	4555.47	0.234329
$724$	23423.8	1.20240
$725$	−7873.31	−0.403321
$726$	4990.66	0.255125
$727$	−30242.2	−1.54281	−0.771404	−	0.636346i	$-0.780445\pi$
$727$	−30242.2	−1.54281	−0.771404	+	0.636346i	$0.780445\pi$
$728$	7533.94	0.383553
$729$	729.000	0.0370370
$730$	15580.6	0.789953
$731$	−6888.39	−0.348531
$732$	12126.3	0.612294
$733$	−6532.42	−0.329169	−0.164584	−	0.986363i	$-0.552628\pi$
$733$	−6532.42	−0.329169	−0.164584	+	0.986363i	$0.552628\pi$
$734$	5198.11	0.261398
$735$	−35984.6	−1.80586
$736$	−11541.1	−0.578003
$737$	−241.205	−0.0120555
$738$	−6197.76	−0.309136
$739$	11957.9	0.595237	0.297618	−	0.954685i	$-0.403808\pi$
$739$	11957.9	0.595237	0.297618	+	0.954685i	$0.403808\pi$
$740$	−10194.1	−0.506410
$741$	−4188.68	−0.207658
$742$	19131.8	0.946564
$743$	−23341.8	−1.15253	−0.576263	−	0.817264i	$-0.695490\pi$
$743$	−23341.8	−1.15253	−0.576263	+	0.817264i	$0.695490\pi$
$744$	15796.2	0.778382
$745$	−8358.08	−0.411029
$746$	5591.21	0.274409
$747$	2092.09	0.102470
$748$	−5719.19	−0.279565
$749$	−21275.9	−1.03792
$750$	−4356.06	−0.212081
$751$	−4901.52	−0.238161	−0.119081	−	0.992885i	$-0.537995\pi$
$751$	−4901.52	−0.238161	−0.119081	+	0.992885i	$0.537995\pi$
$752$	9438.56	0.457698
$753$	15710.3	0.760311
$754$	706.244	0.0341113
$755$	37291.1	1.79756
$756$	5310.98	0.255501
$757$	−25740.2	−1.23586	−0.617928	−	0.786235i	$-0.712028\pi$
$757$	−25740.2	−1.23586	−0.617928	+	0.786235i	$0.712028\pi$
$758$	−19527.1	−0.935694
$759$	1947.11	0.0931168
$760$	38735.6	1.84880
$761$	−18351.9	−0.874186	−0.437093	−	0.899416i	$-0.643992\pi$
$761$	−18351.9	−0.874186	−0.437093	+	0.899416i	$0.643992\pi$
$762$	11091.7	0.527311
$763$	−6920.30	−0.328351
$764$	−22697.8	−1.07484
$765$	14157.8	0.669119
$766$	12148.9	0.573050
$767$	−4606.97	−0.216882
$768$	7312.61	0.343582
$769$	11223.1	0.526289	0.263144	−	0.964756i	$-0.415240\pi$
$769$	11223.1	0.526289	0.263144	+	0.964756i	$0.415240\pi$
$770$	7996.08	0.374232
$771$	−2026.17	−0.0946442
$772$	−14304.1	−0.666860
$773$	10688.1	0.497316	0.248658	−	0.968591i	$-0.420011\pi$
$773$	10688.1	0.497316	0.248658	+	0.968591i	$0.420011\pi$
$774$	−945.134	−0.0438917
$775$	50741.7	2.35187
$776$	−17212.7	−0.796265
$777$	−9031.38	−0.416987
$778$	15527.2	0.715522
$779$	57769.5	2.65700
$780$	−3975.05	−0.182474
$781$	−5967.55	−0.273413
$782$	−7566.07	−0.345987
$783$	1145.70	0.0522910
$784$	15642.6	0.712583
$785$	−2766.70	−0.125794
$786$	6657.40	0.302114
$787$	−22399.7	−1.01457	−0.507283	−	0.861780i	$-0.669350\pi$
$787$	−22399.7	−1.01457	−0.507283	+	0.861780i	$0.669350\pi$
$788$	−20749.3	−0.938025
$789$	14787.4	0.667230
$790$	13900.6	0.626026
$791$	72332.5	3.25139
$792$	−1805.81	−0.0810186
$793$	−8040.80	−0.360072
$794$	−14325.2	−0.640282
$795$	−23229.4	−1.03631
$796$	−15723.4	−0.700127
$797$	41999.4	1.86662	0.933309	−	0.359074i	$-0.116908\pi$
$797$	41999.4	1.86662	0.933309	+	0.359074i	$0.116908\pi$
$798$	14912.6	0.661527
$799$	36662.2	1.62330
$800$	34382.8	1.51952
$801$	−2297.32	−0.101338
$802$	−11649.0	−0.512896
$803$	6770.36	0.297535
$804$	−426.902	−0.0187259
$805$	−35115.5	−1.53746
$806$	−4551.59	−0.198912
$807$	8395.10	0.366198
$808$	17321.5	0.754169
$809$	8486.68	0.368821	0.184410	−	0.982849i	$-0.440962\pi$
$809$	8486.68	0.368821	0.184410	+	0.982849i	$0.440962\pi$
$810$	1942.55	0.0842643
$811$	5897.10	0.255333	0.127667	−	0.991817i	$-0.459251\pi$
$811$	5897.10	0.255333	0.127667	+	0.991817i	$0.459251\pi$
$812$	8346.73	0.360730
$813$	2981.59	0.128621
$814$	1334.41	0.0574584
$815$	−26678.8	−1.14665
$816$	−6154.44	−0.264030
$817$	8809.62	0.377246
$818$	−16098.1	−0.688087
$819$	−3521.66	−0.150252
$820$	54823.2	2.33477
$821$	28076.6	1.19352	0.596761	−	0.802419i	$-0.296454\pi$
$821$	28076.6	1.19352	0.596761	+	0.802419i	$0.296454\pi$
$822$	−3528.29	−0.149712
$823$	−3050.40	−0.129198	−0.0645992	−	0.997911i	$-0.520577\pi$
$823$	−3050.40	−0.129198	−0.0645992	+	0.997911i	$0.520577\pi$
$824$	25693.4	1.08625
$825$	−5800.77	−0.244796
$826$	16401.8	0.690909
$827$	−24319.6	−1.02258	−0.511290	−	0.859408i	$-0.670832\pi$
$827$	−24319.6	−1.02258	−0.511290	+	0.859408i	$0.670832\pi$
$828$	3446.13	0.144639
$829$	4515.95	0.189198	0.0945992	−	0.995515i	$-0.469843\pi$
$829$	4515.95	0.189198	0.0945992	+	0.995515i	$0.469843\pi$
$830$	5574.72	0.233134
$831$	−2893.35	−0.120781
$832$	−835.668	−0.0348216
$833$	60760.6	2.52728
$834$	5545.14	0.230231
$835$	47624.9	1.97380
$836$	7314.32	0.302597
$837$	−7383.75	−0.304922
$838$	−16502.7	−0.680282
$839$	−19734.4	−0.812045	−0.406023	−	0.913863i	$-0.633085\pi$
$839$	−19734.4	−0.812045	−0.406023	+	0.913863i	$0.633085\pi$
$840$	32567.2	1.33771
$841$	−22588.4	−0.926173
$842$	−10265.2	−0.420143
$843$	−1421.78	−0.0580885
$844$	−12990.3	−0.529793
$845$	−36080.4	−1.46888
$846$	5030.30	0.204427
$847$	−39110.4	−1.58660
$848$	10097.9	0.408920
$849$	4326.24	0.174884
$850$	22540.6	0.909571
$851$	−5860.19	−0.236057
$852$	−10561.8	−0.424696
$853$	−3269.82	−0.131250	−0.0656251	−	0.997844i	$-0.520904\pi$
$853$	−3269.82	−0.131250	−0.0656251	+	0.997844i	$0.520904\pi$
$854$	28626.9	1.14706
$855$	−18106.5	−0.724246
$856$	12803.4	0.511229
$857$	39317.1	1.56715	0.783574	−	0.621298i	$-0.213395\pi$
$857$	39317.1	1.56715	0.783574	+	0.621298i	$0.213395\pi$
$858$	520.335	0.0207039
$859$	24522.8	0.974048	0.487024	−	0.873389i	$-0.338082\pi$
$859$	24522.8	0.974048	0.487024	+	0.873389i	$0.338082\pi$
$860$	8360.33	0.331494
$861$	48570.1	1.92249
$862$	−8387.03	−0.331396
$863$	35031.9	1.38181	0.690903	−	0.722947i	$-0.257213\pi$
$863$	35031.9	1.38181	0.690903	+	0.722947i	$0.257213\pi$
$864$	−5003.26	−0.197007
$865$	30356.6	1.19324
$866$	−4836.74	−0.189791
$867$	−9166.68	−0.359074
$868$	−53792.8	−2.10351
$869$	6040.31	0.235792
$870$	3052.90	0.118969
$871$	283.074	0.0110122
$872$	4164.51	0.161729
$873$	8045.91	0.311928
$874$	9676.31	0.374492
$875$	34137.2	1.31891
$876$	11982.7	0.462165
$877$	28047.0	1.07991	0.539953	−	0.841695i	$-0.318442\pi$
$877$	28047.0	1.07991	0.539953	+	0.841695i	$0.318442\pi$
$878$	5209.40	0.200238
$879$	13109.8	0.503053
$880$	4220.39	0.161670
$881$	38614.4	1.47668	0.738339	−	0.674430i	$-0.235610\pi$
$881$	38614.4	1.47668	0.738339	+	0.674430i	$0.235610\pi$
$882$	8336.76	0.318269
$883$	−24035.6	−0.916040	−0.458020	−	0.888942i	$-0.651441\pi$
$883$	−24035.6	−0.916040	−0.458020	+	0.888942i	$0.651441\pi$
$884$	6711.95	0.255370
$885$	−19914.7	−0.756413
$886$	16538.1	0.627098
$887$	7276.57	0.275449	0.137725	−	0.990471i	$-0.456021\pi$
$887$	7276.57	0.275449	0.137725	+	0.990471i	$0.456021\pi$
$888$	5434.92	0.205387
$889$	−86922.7	−3.27929
$890$	−6121.62	−0.230558
$891$	844.107	0.0317381
$892$	35261.0	1.32357
$893$	−46887.6	−1.75704
$894$	1936.37	0.0724405
$895$	−49439.9	−1.84647
$896$	−44455.4	−1.65753
$897$	−2285.10	−0.0850581
$898$	−7092.99	−0.263581
$899$	−11604.3	−0.430506
$900$	−10266.6	−0.380245
$901$	39223.3	1.45030
$902$	−7176.36	−0.264908
$903$	7406.75	0.272958
$904$	−43528.4	−1.60147
$905$	−67141.2	−2.46613
$906$	−8639.45	−0.316806
$907$	−46662.2	−1.70826	−0.854130	−	0.520059i	$-0.825910\pi$
$907$	−46662.2	−1.70826	−0.854130	+	0.520059i	$0.825910\pi$
$908$	−34334.4	−1.25487
$909$	−8096.75	−0.295437
$910$	−9384.06	−0.341845
$911$	11830.9	0.430271	0.215135	−	0.976584i	$-0.430981\pi$
$911$	11830.9	0.430271	0.215135	+	0.976584i	$0.430981\pi$
$912$	7870.97	0.285783
$913$	2422.42	0.0878098
$914$	21595.1	0.781512
$915$	−34758.2	−1.25581
$916$	30713.4	1.10786
$917$	−52172.2	−1.87882
$918$	−3280.02	−0.117927
$919$	45201.0	1.62246	0.811232	−	0.584724i	$-0.198797\pi$
$919$	45201.0	1.62246	0.811232	+	0.584724i	$0.198797\pi$
$920$	21131.9	0.757279
$921$	19237.2	0.688258
$922$	8620.47	0.307918
$923$	7003.42	0.249751
$924$	6149.56	0.218946
$925$	17458.5	0.620574
$926$	12438.0	0.441402
$927$	−12010.1	−0.425527
$928$	−7863.12	−0.278146
$929$	12140.6	0.428764	0.214382	−	0.976750i	$-0.431226\pi$
$929$	12140.6	0.428764	0.214382	+	0.976750i	$0.431226\pi$
$930$	−19675.3	−0.693739
$931$	−77707.2	−2.73550
$932$	−6753.51	−0.237359
$933$	13850.5	0.486008
$934$	9438.47	0.330660
$935$	16393.3	0.573387
$936$	2119.27	0.0740069
$937$	9243.05	0.322260	0.161130	−	0.986933i	$-0.448486\pi$
$937$	9243.05	0.322260	0.161130	+	0.986933i	$0.448486\pi$
$938$	−1007.80	−0.0350810
$939$	4095.26	0.142326
$940$	−44496.3	−1.54395
$941$	45463.0	1.57498	0.787488	−	0.616330i	$-0.211381\pi$
$941$	45463.0	1.57498	0.787488	+	0.616330i	$0.211381\pi$
$942$	640.979	0.0221701
$943$	31515.6	1.08833
$944$	8656.99	0.298476
$945$	−15223.2	−0.524032
$946$	−1094.37	−0.0376120
$947$	6301.52	0.216232	0.108116	−	0.994138i	$-0.465518\pi$
$947$	6301.52	0.216232	0.108116	+	0.994138i	$0.465518\pi$
$948$	10690.6	0.366258
$949$	−7945.58	−0.271785
$950$	−28827.3	−0.984507
$951$	−23497.6	−0.801222
$952$	−54990.3	−1.87211
$953$	−23434.6	−0.796559	−0.398280	−	0.917264i	$-0.630393\pi$
$953$	−23434.6	−0.796559	−0.398280	+	0.917264i	$0.630393\pi$
$954$	5381.70	0.182641
$955$	65059.9	2.20449
$956$	−12181.8	−0.412119
$957$	1326.60	0.0448096
$958$	−7524.11	−0.253750
$959$	27650.2	0.931044
$960$	−3612.37	−0.121447
$961$	44996.1	1.51039
$962$	−1566.04	−0.0524857
$963$	−5984.82	−0.200268
$964$	9335.64	0.311909
$965$	41000.7	1.36773
$966$	8135.42	0.270966
$967$	−2934.80	−0.0975974	−0.0487987	−	0.998809i	$-0.515539\pi$
$967$	−2934.80	−0.0975974	−0.0487987	+	0.998809i	$0.515539\pi$
$968$	23535.9	0.781481
$969$	30573.2	1.01357
$970$	21439.7	0.709678
$971$	−59049.0	−1.95157	−0.975784	−	0.218737i	$-0.929806\pi$
$971$	−59049.0	−1.95157	−0.975784	+	0.218737i	$0.929806\pi$
$972$	1493.96	0.0492991
$973$	−43455.7	−1.43178
$974$	−19343.3	−0.636345
$975$	6807.68	0.223611
$976$	15109.5	0.495537
$977$	−53462.4	−1.75068	−0.875339	−	0.483509i	$-0.839362\pi$
$977$	−53462.4	−1.75068	−0.875339	+	0.483509i	$0.839362\pi$
$978$	6180.84	0.202088
$979$	−2660.07	−0.0868397
$980$	−73744.1	−2.40374
$981$	−1946.65	−0.0633556
$982$	2780.47	0.0903549
$983$	17440.6	0.565890	0.282945	−	0.959136i	$-0.408689\pi$
$983$	17440.6	0.565890	0.282945	+	0.959136i	$0.408689\pi$
$984$	−29228.6	−0.946924
$985$	59475.0	1.92389
$986$	−5154.88	−0.166496
$987$	−39421.0	−1.27131
$988$	−8583.96	−0.276409
$989$	4806.01	0.154522
$990$	2249.27	0.0722085
$991$	45972.0	1.47361	0.736805	−	0.676105i	$-0.236333\pi$
$991$	45972.0	1.47361	0.736805	+	0.676105i	$0.236333\pi$
$992$	50676.1	1.62194
$993$	32804.9	1.04837
$994$	−24933.6	−0.795620
$995$	45068.9	1.43596
$996$	4287.36	0.136396
$997$	40978.1	1.30169	0.650847	−	0.759209i	$-0.274414\pi$
$997$	40978.1	1.30169	0.650847	+	0.759209i	$0.274414\pi$
$998$	−16246.2	−0.515294
$999$	−2540.49	−0.0804581

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.4.a.c.1.13	✓	22
3.2	odd	2		1413.4.a.e.1.10		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.4.a.c.1.13	✓	22	1.1	even	1	trivial
1413.4.a.e.1.10		22	3.2	odd	2

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:	\( N \)	\(=\)	\( 471 = 3 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	471.a (trivial)

\(f(q)\)	\(=\)	\(q+1.36089 q^{2} -3.00000 q^{3} -6.14798 q^{4} +17.6223 q^{5} -4.08267 q^{6} +31.9947 q^{7} -19.2538 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+1.36089 q^{2} -3.00000 q^{3} -6.14798 q^{4} +17.6223 q^{5} -4.08267 q^{6} +31.9947 q^{7} -19.2538 q^{8} +9.00000 q^{9} +23.9820 q^{10} +10.4211 q^{11} +18.4439 q^{12} -12.2300 q^{13} +43.5413 q^{14} -52.8670 q^{15} +22.9815 q^{16} +89.2668 q^{17} +12.2480 q^{18} -114.164 q^{19} -108.342 q^{20} -95.9842 q^{21} +14.1819 q^{22} -62.2812 q^{23} +57.7615 q^{24} +185.546 q^{25} -16.6437 q^{26} -27.0000 q^{27} -196.703 q^{28} -42.4332 q^{29} -71.9461 q^{30} +273.472 q^{31} +185.306 q^{32} -31.2632 q^{33} +121.482 q^{34} +563.821 q^{35} -55.3318 q^{36} +94.0924 q^{37} -155.365 q^{38} +36.6900 q^{39} -339.297 q^{40} -506.022 q^{41} -130.624 q^{42} -77.1664 q^{43} -64.0685 q^{44} +158.601 q^{45} -84.7579 q^{46} +410.703 q^{47} -68.9444 q^{48} +680.662 q^{49} +252.508 q^{50} -267.800 q^{51} +75.1897 q^{52} +439.394 q^{53} -36.7440 q^{54} +183.643 q^{55} -616.021 q^{56} +342.492 q^{57} -57.7469 q^{58} +376.695 q^{59} +325.025 q^{60} +657.465 q^{61} +372.166 q^{62} +287.953 q^{63} +68.3294 q^{64} -215.521 q^{65} -42.5458 q^{66} -23.1459 q^{67} -548.810 q^{68} +186.844 q^{69} +767.299 q^{70} -572.643 q^{71} -173.285 q^{72} +649.680 q^{73} +128.049 q^{74} -556.638 q^{75} +701.878 q^{76} +333.419 q^{77} +49.9310 q^{78} +579.624 q^{79} +404.987 q^{80} +81.0000 q^{81} -688.640 q^{82} +232.454 q^{83} +590.109 q^{84} +1573.09 q^{85} -105.015 q^{86} +127.300 q^{87} -200.646 q^{88} -255.258 q^{89} +215.838 q^{90} -391.295 q^{91} +382.904 q^{92} -820.417 q^{93} +558.922 q^{94} -2011.84 q^{95} -555.918 q^{96} +893.990 q^{97} +926.307 q^{98} +93.7896 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9}+O(q^{10}) \) 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9} + 13 q^{10} + 61 q^{11} - 270 q^{12} + 4 q^{13} + 133 q^{14} - 96 q^{15} + 342 q^{16} + 308 q^{17} + 36 q^{18} + 32 q^{19} + 407 q^{20} + 12 q^{21} - 166 q^{22} + 53 q^{23} - 81 q^{24} + 746 q^{25} + 467 q^{26} - 594 q^{27} + 85 q^{28} + 634 q^{29} - 39 q^{30} - 163 q^{31} + 150 q^{32} - 183 q^{33} + 37 q^{34} + 782 q^{35} + 810 q^{36} - 2 q^{37} + 584 q^{38} - 12 q^{39} + 864 q^{40} + 1593 q^{41} - 399 q^{42} - 891 q^{43} + 2093 q^{44} + 288 q^{45} + 108 q^{46} + 1200 q^{47} - 1026 q^{48} + 2816 q^{49} + 4703 q^{50} - 924 q^{51} + 1866 q^{52} + 1182 q^{53} - 108 q^{54} + 970 q^{55} + 5362 q^{56} - 96 q^{57} + 1814 q^{58} + 2802 q^{59} - 1221 q^{60} + 2629 q^{61} + 2378 q^{62} - 36 q^{63} + 625 q^{64} + 2264 q^{65} + 498 q^{66} - 1074 q^{67} + 4383 q^{68} - 159 q^{69} + 4009 q^{70} + 3920 q^{71} + 243 q^{72} + 1086 q^{73} + 4904 q^{74} - 2238 q^{75} + 3750 q^{76} + 2966 q^{77} - 1401 q^{78} - 30 q^{79} + 7777 q^{80} + 1782 q^{81} + 2932 q^{82} + 1900 q^{83} - 255 q^{84} + 524 q^{85} + 3209 q^{86} - 1902 q^{87} - 100 q^{88} + 4488 q^{89} + 117 q^{90} - 818 q^{91} + 6210 q^{92} + 489 q^{93} + 3220 q^{94} + 3500 q^{95} - 450 q^{96} + 2178 q^{97} + 7629 q^{98} + 549 q^{99}+O(q^{100}) \) 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9 + 13 * q^10 + 61 * q^11 - 270 * q^12 + 4 * q^13 + 133 * q^14 - 96 * q^15 + 342 * q^16 + 308 * q^17 + 36 * q^18 + 32 * q^19 + 407 * q^20 + 12 * q^21 - 166 * q^22 + 53 * q^23 - 81 * q^24 + 746 * q^25 + 467 * q^26 - 594 * q^27 + 85 * q^28 + 634 * q^29 - 39 * q^30 - 163 * q^31 + 150 * q^32 - 183 * q^33 + 37 * q^34 + 782 * q^35 + 810 * q^36 - 2 * q^37 + 584 * q^38 - 12 * q^39 + 864 * q^40 + 1593 * q^41 - 399 * q^42 - 891 * q^43 + 2093 * q^44 + 288 * q^45 + 108 * q^46 + 1200 * q^47 - 1026 * q^48 + 2816 * q^49 + 4703 * q^50 - 924 * q^51 + 1866 * q^52 + 1182 * q^53 - 108 * q^54 + 970 * q^55 + 5362 * q^56 - 96 * q^57 + 1814 * q^58 + 2802 * q^59 - 1221 * q^60 + 2629 * q^61 + 2378 * q^62 - 36 * q^63 + 625 * q^64 + 2264 * q^65 + 498 * q^66 - 1074 * q^67 + 4383 * q^68 - 159 * q^69 + 4009 * q^70 + 3920 * q^71 + 243 * q^72 + 1086 * q^73 + 4904 * q^74 - 2238 * q^75 + 3750 * q^76 + 2966 * q^77 - 1401 * q^78 - 30 * q^79 + 7777 * q^80 + 1782 * q^81 + 2932 * q^82 + 1900 * q^83 - 255 * q^84 + 524 * q^85 + 3209 * q^86 - 1902 * q^87 - 100 * q^88 + 4488 * q^89 + 117 * q^90 - 818 * q^91 + 6210 * q^92 + 489 * q^93 + 3220 * q^94 + 3500 * q^95 - 450 * q^96 + 2178 * q^97 + 7629 * q^98 + 549 * q^99

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