LMFDB - Embedded newform 471.4.a.c.1.11

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			1.11
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)

$f(q)$	$=$	$q+0.196646 q^{2} -3.00000 q^{3} -7.96133 q^{4} -2.23494 q^{5} -0.589937 q^{6} -24.0345 q^{7} -3.13873 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+0.196646 q^{2} -3.00000 q^{3} -7.96133 q^{4} -2.23494 q^{5} -0.589937 q^{6} -24.0345 q^{7} -3.13873 q^{8} +9.00000 q^{9} -0.439492 q^{10} -21.1995 q^{11} +23.8840 q^{12} -15.5713 q^{13} -4.72628 q^{14} +6.70483 q^{15} +63.0734 q^{16} -123.126 q^{17} +1.76981 q^{18} -128.154 q^{19} +17.7931 q^{20} +72.1035 q^{21} -4.16880 q^{22} -91.3050 q^{23} +9.41618 q^{24} -120.005 q^{25} -3.06202 q^{26} -27.0000 q^{27} +191.347 q^{28} +240.484 q^{29} +1.31848 q^{30} +115.339 q^{31} +37.5129 q^{32} +63.5986 q^{33} -24.2121 q^{34} +53.7158 q^{35} -71.6520 q^{36} +349.560 q^{37} -25.2010 q^{38} +46.7138 q^{39} +7.01487 q^{40} -235.828 q^{41} +14.1788 q^{42} +348.195 q^{43} +168.777 q^{44} -20.1145 q^{45} -17.9547 q^{46} -221.080 q^{47} -189.220 q^{48} +234.657 q^{49} -23.5985 q^{50} +369.377 q^{51} +123.968 q^{52} -341.722 q^{53} -5.30943 q^{54} +47.3798 q^{55} +75.4377 q^{56} +384.463 q^{57} +47.2902 q^{58} +679.517 q^{59} -53.3794 q^{60} +444.531 q^{61} +22.6809 q^{62} -216.310 q^{63} -497.211 q^{64} +34.8009 q^{65} +12.5064 q^{66} -597.295 q^{67} +980.245 q^{68} +273.915 q^{69} +10.5630 q^{70} -207.433 q^{71} -28.2485 q^{72} +289.889 q^{73} +68.7394 q^{74} +360.015 q^{75} +1020.28 q^{76} +509.520 q^{77} +9.18606 q^{78} -977.803 q^{79} -140.966 q^{80} +81.0000 q^{81} -46.3745 q^{82} -153.530 q^{83} -574.040 q^{84} +275.179 q^{85} +68.4710 q^{86} -721.453 q^{87} +66.5395 q^{88} +1194.50 q^{89} -3.95543 q^{90} +374.247 q^{91} +726.910 q^{92} -346.017 q^{93} -43.4743 q^{94} +286.418 q^{95} -112.539 q^{96} -654.329 q^{97} +46.1443 q^{98} -190.796 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.196646	0.0695247	0.0347624	−	0.999396i	$-0.488933\pi$
$2$	0.196646	0.0695247	0.0347624	+	0.999396i	$0.488933\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.96133	−0.995166
$5$	−2.23494	−0.199899	−0.0999497	−	0.994992i	$-0.531868\pi$
$5$	−2.23494	−0.199899	−0.0999497	+	0.994992i	$0.531868\pi$
$6$	−0.589937	−0.0401401
$7$	−24.0345	−1.29774	−0.648870	−	0.760899i	$-0.724758\pi$
$7$	−24.0345	−1.29774	−0.648870	+	0.760899i	$0.724758\pi$
$8$	−3.13873	−0.138713
$9$	9.00000	0.333333
$10$	−0.439492	−0.0138980
$11$	−21.1995	−0.581082	−0.290541	−	0.956863i	$-0.593835\pi$
$11$	−21.1995	−0.581082	−0.290541	+	0.956863i	$0.593835\pi$
$12$	23.8840	0.574560
$13$	−15.5713	−0.332207	−0.166103	−	0.986108i	$-0.553119\pi$
$13$	−15.5713	−0.332207	−0.166103	+	0.986108i	$0.553119\pi$
$14$	−4.72628	−0.0902251
$15$	6.70483	0.115412
$16$	63.0734	0.985522
$17$	−123.126	−1.75661	−0.878305	−	0.478100i	$-0.841326\pi$
$17$	−123.126	−1.75661	−0.878305	+	0.478100i	$0.841326\pi$
$18$	1.76981	0.0231749
$19$	−128.154	−1.54740	−0.773700	−	0.633552i	$-0.781596\pi$
$19$	−128.154	−1.54740	−0.773700	+	0.633552i	$0.781596\pi$
$20$	17.7931	0.198933
$21$	72.1035	0.749251
$22$	−4.16880	−0.0403996
$23$	−91.3050	−0.827757	−0.413879	−	0.910332i	$-0.635826\pi$
$23$	−91.3050	−0.827757	−0.413879	+	0.910332i	$0.635826\pi$
$24$	9.41618	0.0800862
$25$	−120.005	−0.960040
$26$	−3.06202	−0.0230966
$27$	−27.0000	−0.192450
$28$	191.347	1.29147
$29$	240.484	1.53989	0.769945	−	0.638110i	$-0.220283\pi$
$29$	240.484	1.53989	0.769945	+	0.638110i	$0.220283\pi$
$30$	1.31848	0.00802399
$31$	115.339	0.668242	0.334121	−	0.942530i	$-0.391561\pi$
$31$	115.339	0.668242	0.334121	+	0.942530i	$0.391561\pi$
$32$	37.5129	0.207232
$33$	63.5986	0.335488
$34$	−24.2121	−0.122128
$35$	53.7158	0.259418
$36$	−71.6520	−0.331722
$37$	349.560	1.55317	0.776585	−	0.630013i	$-0.216950\pi$
$37$	349.560	1.55317	0.776585	+	0.630013i	$0.216950\pi$
$38$	−25.2010	−0.107583
$39$	46.7138	0.191800
$40$	7.01487	0.0277287
$41$	−235.828	−0.898296	−0.449148	−	0.893457i	$-0.648272\pi$
$41$	−235.828	−0.898296	−0.449148	+	0.893457i	$0.648272\pi$
$42$	14.1788	0.0520915
$43$	348.195	1.23487	0.617433	−	0.786623i	$-0.288173\pi$
$43$	348.195	1.23487	0.617433	+	0.786623i	$0.288173\pi$
$44$	168.777	0.578273
$45$	−20.1145	−0.0666331
$46$	−17.9547	−0.0575496
$47$	−221.080	−0.686123	−0.343061	−	0.939313i	$-0.611464\pi$
$47$	−221.080	−0.686123	−0.343061	+	0.939313i	$0.611464\pi$
$48$	−189.220	−0.568992
$49$	234.657	0.684132
$50$	−23.5985	−0.0667465
$51$	369.377	1.01418
$52$	123.968	0.330601
$53$	−341.722	−0.885645	−0.442822	−	0.896609i	$-0.646023\pi$
$53$	−341.722	−0.885645	−0.442822	+	0.896609i	$0.646023\pi$
$54$	−5.30943	−0.0133800
$55$	47.3798	0.116158
$56$	75.4377	0.180014
$57$	384.463	0.893392
$58$	47.2902	0.107060
$59$	679.517	1.49942	0.749708	−	0.661769i	$-0.230194\pi$
$59$	679.517	1.49942	0.749708	+	0.661769i	$0.230194\pi$
$60$	−53.3794	−0.114854
$61$	444.531	0.933054	0.466527	−	0.884507i	$-0.345505\pi$
$61$	444.531	0.933054	0.466527	+	0.884507i	$0.345505\pi$
$62$	22.6809	0.0464593
$63$	−216.310	−0.432580
$64$	−497.211	−0.971115
$65$	34.8009	0.0664080
$66$	12.5064	0.0233247
$67$	−597.295	−1.08912	−0.544561	−	0.838721i	$-0.683304\pi$
$67$	−597.295	−1.08912	−0.544561	+	0.838721i	$0.683304\pi$
$68$	980.245	1.74812
$69$	273.915	0.477906
$70$	10.5630	0.0180359
$71$	−207.433	−0.346729	−0.173365	−	0.984858i	$-0.555464\pi$
$71$	−207.433	−0.346729	−0.173365	+	0.984858i	$0.555464\pi$
$72$	−28.2485	−0.0462378
$73$	289.889	0.464780	0.232390	−	0.972623i	$-0.425345\pi$
$73$	289.889	0.464780	0.232390	+	0.972623i	$0.425345\pi$
$74$	68.7394	0.107984
$75$	360.015	0.554279
$76$	1020.28	1.53992
$77$	509.520	0.754094
$78$	9.18606	0.0133348
$79$	−977.803	−1.39255	−0.696275	−	0.717775i	$-0.745161\pi$
$79$	−977.803	−1.39255	−0.696275	+	0.717775i	$0.745161\pi$
$80$	−140.966	−0.197005
$81$	81.0000	0.111111
$82$	−46.3745	−0.0624538
$83$	−153.530	−0.203037	−0.101519	−	0.994834i	$-0.532370\pi$
$83$	−153.530	−0.203037	−0.101519	+	0.994834i	$0.532370\pi$
$84$	−574.040	−0.745629
$85$	275.179	0.351146
$86$	68.4710	0.0858538
$87$	−721.453	−0.889056
$88$	66.5395	0.0806039
$89$	1194.50	1.42265	0.711327	−	0.702861i	$-0.248094\pi$
$89$	1194.50	1.42265	0.711327	+	0.702861i	$0.248094\pi$
$90$	−3.95543	−0.00463265
$91$	374.247	0.431118
$92$	726.910	0.823756
$93$	−346.017	−0.385809
$94$	−43.4743	−0.0477025
$95$	286.418	0.309324
$96$	−112.539	−0.119645
$97$	−654.329	−0.684918	−0.342459	−	0.939533i	$-0.611260\pi$
$97$	−654.329	−0.684918	−0.342459	+	0.939533i	$0.611260\pi$
$98$	46.1443	0.0475641
$99$	−190.796	−0.193694
$100$	955.400	0.955400
$101$	1136.95	1.12010	0.560052	−	0.828457i	$-0.310781\pi$
$101$	1136.95	1.12010	0.560052	+	0.828457i	$0.310781\pi$
$102$	72.6364	0.0705106
$103$	476.638	0.455966	0.227983	−	0.973665i	$-0.426787\pi$
$103$	476.638	0.455966	0.227983	+	0.973665i	$0.426787\pi$
$104$	48.8739	0.0460815
$105$	−161.147	−0.149775
$106$	−67.1982	−0.0615742
$107$	−873.816	−0.789486	−0.394743	−	0.918792i	$-0.629166\pi$
$107$	−873.816	−0.789486	−0.394743	+	0.918792i	$0.629166\pi$
$108$	214.956	0.191520
$109$	−777.780	−0.683466	−0.341733	−	0.939797i	$-0.611014\pi$
$109$	−777.780	−0.683466	−0.341733	+	0.939797i	$0.611014\pi$
$110$	9.31703	0.00807585
$111$	−1048.68	−0.896723
$112$	−1515.94	−1.27895
$113$	52.3138	0.0435511	0.0217755	−	0.999763i	$-0.493068\pi$
$113$	52.3138	0.0435511	0.0217755	+	0.999763i	$0.493068\pi$
$114$	75.6029	0.0621128
$115$	204.062	0.165468
$116$	−1914.58	−1.53245
$117$	−140.141	−0.110736
$118$	133.624	0.104246
$119$	2959.27	2.27963
$120$	−21.0446	−0.0160092
$121$	−881.579	−0.662344
$122$	87.4150	0.0648703
$123$	707.484	0.518631
$124$	−918.252	−0.665012
$125$	547.572	0.391811
$126$	−42.5365	−0.0300750
$127$	−350.764	−0.245081	−0.122540	−	0.992464i	$-0.539104\pi$
$127$	−350.764	−0.245081	−0.122540	+	0.992464i	$0.539104\pi$
$128$	−397.878	−0.274748
$129$	−1044.59	−0.712951
$130$	6.84344	0.00461699
$131$	1151.55	0.768027	0.384013	−	0.923328i	$-0.374542\pi$
$131$	1151.55	0.768027	0.384013	+	0.923328i	$0.374542\pi$
$132$	−506.330	−0.333866
$133$	3080.12	2.00812
$134$	−117.455	−0.0757209
$135$	60.3435	0.0384707
$136$	386.458	0.243665
$137$	−2528.80	−1.57701	−0.788503	−	0.615031i	$-0.789143\pi$
$137$	−2528.80	−1.57701	−0.788503	+	0.615031i	$0.789143\pi$
$138$	53.8642	0.0332263
$139$	−221.866	−0.135385	−0.0676923	−	0.997706i	$-0.521564\pi$
$139$	−221.866	−0.135385	−0.0676923	+	0.997706i	$0.521564\pi$
$140$	−427.649	−0.258164
$141$	663.239	0.396133
$142$	−40.7908	−0.0241063
$143$	330.103	0.193039
$144$	567.661	0.328507
$145$	−537.469	−0.307823
$146$	57.0054	0.0323137
$147$	−703.971	−0.394984
$148$	−2782.96	−1.54566
$149$	269.389	0.148115	0.0740577	−	0.997254i	$-0.476405\pi$
$149$	269.389	0.148115	0.0740577	+	0.997254i	$0.476405\pi$
$150$	70.7954	0.0385361
$151$	−1304.07	−0.702805	−0.351402	−	0.936225i	$-0.614295\pi$
$151$	−1304.07	−0.702805	−0.351402	+	0.936225i	$0.614295\pi$
$152$	402.241	0.214645
$153$	−1108.13	−0.585537
$154$	100.195	0.0524282
$155$	−257.776	−0.133581
$156$	−371.904	−0.190873
$157$	−157.000	−0.0798087
$157$	−157.000	−0.0798087
$158$	−192.281	−0.0968167
$159$	1025.17	0.511327
$160$	−83.8393	−0.0414255
$161$	2194.47	1.07421
$162$	15.9283	0.00772497
$163$	−2212.99	−1.06340	−0.531702	−	0.846931i	$-0.678447\pi$
$163$	−2212.99	−1.06340	−0.531702	+	0.846931i	$0.678447\pi$
$164$	1877.50	0.893954
$165$	−142.139	−0.0670638
$166$	−30.1910	−0.0141161
$167$	−2782.64	−1.28938	−0.644692	−	0.764443i	$-0.723014\pi$
$167$	−2782.64	−1.28938	−0.644692	+	0.764443i	$0.723014\pi$
$168$	−226.313	−0.103931
$169$	−1954.54	−0.889639
$170$	54.1128	0.0244133
$171$	−1153.39	−0.515800
$172$	−2772.10	−1.22890
$173$	1676.78	0.736896	0.368448	−	0.929648i	$-0.379889\pi$
$173$	1676.78	0.736896	0.368448	+	0.929648i	$0.379889\pi$
$174$	−141.871	−0.0618114
$175$	2884.26	1.24588
$176$	−1337.13	−0.572669
$177$	−2038.55	−0.865688
$178$	234.892	0.0989096
$179$	184.628	0.0770936	0.0385468	−	0.999257i	$-0.487727\pi$
$179$	184.628	0.0770936	0.0385468	+	0.999257i	$0.487727\pi$
$180$	160.138	0.0663111
$181$	1122.42	0.460931	0.230466	−	0.973080i	$-0.425975\pi$
$181$	1122.42	0.460931	0.230466	+	0.973080i	$0.425975\pi$
$182$	73.5941	0.0299734
$183$	−1333.59	−0.538699
$184$	286.581	0.114821
$185$	−781.246	−0.310478
$186$	−68.0427	−0.0268233
$187$	2610.21	1.02074
$188$	1760.09	0.682806
$189$	648.931	0.249750
$190$	56.3227	0.0215057
$191$	4377.33	1.65829	0.829143	−	0.559036i	$-0.188829\pi$
$191$	4377.33	1.65829	0.829143	+	0.559036i	$0.188829\pi$
$192$	1491.63	0.560673
$193$	4584.67	1.70991	0.854953	−	0.518705i	$-0.173586\pi$
$193$	4584.67	1.70991	0.854953	+	0.518705i	$0.173586\pi$
$194$	−128.671	−0.0476187
$195$	−104.403	−0.0383407
$196$	−1868.18	−0.680825
$197$	5516.46	1.99508	0.997542	−	0.0700754i	$-0.0223240\pi$
$197$	5516.46	1.99508	0.997542	+	0.0700754i	$0.0223240\pi$
$198$	−37.5192	−0.0134665
$199$	−1651.58	−0.588328	−0.294164	−	0.955755i	$-0.595041\pi$
$199$	−1651.58	−0.588328	−0.294164	+	0.955755i	$0.595041\pi$
$200$	376.663	0.133170
$201$	1791.89	0.628805
$202$	223.576	0.0778750
$203$	−5779.92	−1.99838
$204$	−2940.73	−1.00928
$205$	527.062	0.179569
$206$	93.7288	0.0317009
$207$	−821.745	−0.275919
$208$	−982.132	−0.327397
$209$	2716.81	0.899166
$210$	−31.6889	−0.0104131
$211$	−4353.33	−1.42036	−0.710179	−	0.704021i	$-0.751386\pi$
$211$	−4353.33	−1.42036	−0.710179	+	0.704021i	$0.751386\pi$
$212$	2720.56	0.881364
$213$	622.299	0.200184
$214$	−171.832	−0.0548888
$215$	−778.197	−0.246849
$216$	84.7456	0.0266954
$217$	−2772.11	−0.867205
$218$	−152.947	−0.0475178
$219$	−869.667	−0.268341
$220$	−377.206	−0.115597
$221$	1917.22	0.583558
$222$	−206.218	−0.0623444
$223$	−3019.45	−0.906715	−0.453358	−	0.891329i	$-0.649774\pi$
$223$	−3019.45	−0.906715	−0.453358	+	0.891329i	$0.649774\pi$
$224$	−901.604	−0.268933
$225$	−1080.05	−0.320013
$226$	10.2873	0.00302788
$227$	−867.319	−0.253595	−0.126797	−	0.991929i	$-0.540470\pi$
$227$	−867.319	−0.253595	−0.126797	+	0.991929i	$0.540470\pi$
$228$	−3060.83	−0.889073
$229$	−785.274	−0.226604	−0.113302	−	0.993561i	$-0.536143\pi$
$229$	−785.274	−0.226604	−0.113302	+	0.993561i	$0.536143\pi$
$230$	40.1278	0.0115041
$231$	−1528.56	−0.435376
$232$	−754.815	−0.213603
$233$	−5521.32	−1.55242	−0.776210	−	0.630475i	$-0.782860\pi$
$233$	−5521.32	−1.55242	−0.776210	+	0.630475i	$0.782860\pi$
$234$	−27.5582	−0.00769886
$235$	494.100	0.137156
$236$	−5409.86	−1.49217
$237$	2933.41	0.803989
$238$	581.927	0.158490
$239$	828.113	0.224126	0.112063	−	0.993701i	$-0.464254\pi$
$239$	828.113	0.224126	0.112063	+	0.993701i	$0.464254\pi$
$240$	422.897	0.113741
$241$	−5237.81	−1.39999	−0.699994	−	0.714149i	$-0.746814\pi$
$241$	−5237.81	−1.39999	−0.699994	+	0.714149i	$0.746814\pi$
$242$	−173.359	−0.0460493
$243$	−243.000	−0.0641500
$244$	−3539.06	−0.928544
$245$	−524.445	−0.136758
$246$	139.124	0.0360577
$247$	1995.52	0.514057
$248$	−362.017	−0.0926941
$249$	460.589	0.117224
$250$	107.678	0.0272405
$251$	1318.88	0.331661	0.165830	−	0.986154i	$-0.446970\pi$
$251$	1318.88	0.331661	0.165830	+	0.986154i	$0.446970\pi$
$252$	1722.12	0.430489
$253$	1935.63	0.480995
$254$	−68.9762	−0.0170392
$255$	−825.537	−0.202734
$256$	3899.44	0.952013
$257$	2459.82	0.597040	0.298520	−	0.954403i	$-0.403507\pi$
$257$	2459.82	0.597040	0.298520	+	0.954403i	$0.403507\pi$
$258$	−205.413	−0.0495677
$259$	−8401.49	−2.01561
$260$	−277.061	−0.0660870
$261$	2164.36	0.513297
$262$	226.448	0.0533969
$263$	855.130	0.200493	0.100246	−	0.994963i	$-0.468037\pi$
$263$	855.130	0.200493	0.100246	+	0.994963i	$0.468037\pi$
$264$	−199.619	−0.0465367
$265$	763.730	0.177040
$266$	605.693	0.139614
$267$	−3583.49	−0.821370
$268$	4755.26	1.08386
$269$	2503.20	0.567370	0.283685	−	0.958918i	$-0.408443\pi$
$269$	2503.20	0.567370	0.283685	+	0.958918i	$0.408443\pi$
$270$	11.8663	0.00267466
$271$	5557.01	1.24563	0.622813	−	0.782371i	$-0.285990\pi$
$271$	5557.01	1.24563	0.622813	+	0.782371i	$0.285990\pi$
$272$	−7765.96	−1.73118
$273$	−1122.74	−0.248906
$274$	−497.277	−0.109641
$275$	2544.05	0.557862
$276$	−2180.73	−0.475596
$277$	−2153.86	−0.467195	−0.233598	−	0.972333i	$-0.575050\pi$
$277$	−2153.86	−0.467195	−0.233598	+	0.972333i	$0.575050\pi$
$278$	−43.6291	−0.00941258
$279$	1038.05	0.222747
$280$	−168.599	−0.0359847
$281$	−6228.83	−1.32235	−0.661176	−	0.750231i	$-0.729942\pi$
$281$	−6228.83	−1.32235	−0.661176	+	0.750231i	$0.729942\pi$
$282$	130.423	0.0275411
$283$	−8653.33	−1.81762	−0.908811	−	0.417209i	$-0.863008\pi$
$283$	−8653.33	−1.81762	−0.908811	+	0.417209i	$0.863008\pi$
$284$	1651.44	0.345053
$285$	−859.253	−0.178589
$286$	64.9134	0.0134210
$287$	5668.00	1.16576
$288$	337.616	0.0690772
$289$	10247.0	2.08568
$290$	−105.691	−0.0214013
$291$	1962.99	0.395438
$292$	−2307.90	−0.462533
$293$	3165.92	0.631246	0.315623	−	0.948885i	$-0.397787\pi$
$293$	3165.92	0.631246	0.315623	+	0.948885i	$0.397787\pi$
$294$	−138.433	−0.0274611
$295$	−1518.68	−0.299732
$296$	−1097.17	−0.215445
$297$	572.388	0.111829
$298$	52.9741	0.0102977
$299$	1421.73	0.274987
$300$	−2866.20	−0.551600
$301$	−8368.70	−1.60254
$302$	−256.439	−0.0488623
$303$	−3410.85	−0.646693
$304$	−8083.13	−1.52500
$305$	−993.501	−0.186517
$306$	−217.909	−0.0407093
$307$	3839.72	0.713826	0.356913	−	0.934138i	$-0.383829\pi$
$307$	3839.72	0.713826	0.356913	+	0.934138i	$0.383829\pi$
$308$	−4056.46	−0.750449
$309$	−1429.91	−0.263252
$310$	−50.6905	−0.00928719
$311$	−13.0396	−0.00237752	−0.00118876	−	0.999999i	$-0.500378\pi$
$311$	−13.0396	−0.00237752	−0.00118876	+	0.999999i	$0.500378\pi$
$312$	−146.622	−0.0266052
$313$	409.283	0.0739107	0.0369553	−	0.999317i	$-0.488234\pi$
$313$	409.283	0.0739107	0.0369553	+	0.999317i	$0.488234\pi$
$314$	−30.8734	−0.00554868
$315$	483.442	0.0864726
$316$	7784.62	1.38582
$317$	4888.92	0.866212	0.433106	−	0.901343i	$-0.357418\pi$
$317$	4888.92	0.866212	0.433106	+	0.901343i	$0.357418\pi$
$318$	201.595	0.0355499
$319$	−5098.16	−0.894803
$320$	1111.24	0.194125
$321$	2621.45	0.455810
$322$	431.533	0.0746845
$323$	15779.1	2.71818
$324$	−644.868	−0.110574
$325$	1868.63	0.318932
$326$	−435.175	−0.0739329
$327$	2333.34	0.394599
$328$	740.199	0.124606
$329$	5313.54	0.890410
$330$	−27.9511	−0.00466259
$331$	−6822.61	−1.13294	−0.566472	−	0.824081i	$-0.691692\pi$
$331$	−6822.61	−1.13294	−0.566472	+	0.824081i	$0.691692\pi$
$332$	1222.30	0.202056
$333$	3146.04	0.517723
$334$	−547.194	−0.0896440
$335$	1334.92	0.217715
$336$	4547.81	0.738404
$337$	12054.2	1.94847	0.974234	−	0.225538i	$-0.0724141\pi$
$337$	12054.2	1.94847	0.974234	+	0.225538i	$0.0724141\pi$
$338$	−384.351	−0.0618519
$339$	−156.941	−0.0251442
$340$	−2190.79	−0.349448
$341$	−2445.13	−0.388303
$342$	−226.809	−0.0358608
$343$	2603.97	0.409915
$344$	−1092.89	−0.171293
$345$	−612.185	−0.0955331
$346$	329.731	0.0512325
$347$	723.201	0.111883	0.0559416	−	0.998434i	$-0.482184\pi$
$347$	723.201	0.111883	0.0559416	+	0.998434i	$0.482184\pi$
$348$	5743.73	0.884759
$349$	2652.94	0.406901	0.203450	−	0.979085i	$-0.434784\pi$
$349$	2652.94	0.406901	0.203450	+	0.979085i	$0.434784\pi$
$350$	567.177	0.0866197
$351$	420.424	0.0639332
$352$	−795.257	−0.120419
$353$	608.790	0.0917921	0.0458960	−	0.998946i	$-0.485386\pi$
$353$	608.790	0.0917921	0.0458960	+	0.998946i	$0.485386\pi$
$354$	−400.872	−0.0601867
$355$	463.601	0.0693110
$356$	−9509.77	−1.41578
$357$	−8877.80	−1.31614
$358$	36.3063	0.00535991
$359$	8144.48	1.19735	0.598676	−	0.800991i	$-0.295694\pi$
$359$	8144.48	1.19735	0.598676	+	0.800991i	$0.295694\pi$
$360$	63.1339	0.00924291
$361$	9564.51	1.39445
$362$	220.718	0.0320461
$363$	2644.74	0.382404
$364$	−2979.51	−0.429034
$365$	−647.885	−0.0929092
$366$	−262.245	−0.0374529
$367$	−11138.3	−1.58424	−0.792119	−	0.610367i	$-0.791022\pi$
$367$	−11138.3	−1.58424	−0.792119	+	0.610367i	$0.791022\pi$
$368$	−5758.92	−0.815773
$369$	−2122.45	−0.299432
$370$	−153.629	−0.0215859
$371$	8213.13	1.14934
$372$	2754.75	0.383945
$373$	549.932	0.0763389	0.0381694	−	0.999271i	$-0.487847\pi$
$373$	549.932	0.0763389	0.0381694	+	0.999271i	$0.487847\pi$
$374$	513.286	0.0709663
$375$	−1642.72	−0.226212
$376$	693.908	0.0951744
$377$	−3744.64	−0.511562
$378$	127.610	0.0173638
$379$	5742.22	0.778254	0.389127	−	0.921184i	$-0.372777\pi$
$379$	5742.22	0.778254	0.389127	+	0.921184i	$0.372777\pi$
$380$	−2280.26	−0.307829
$381$	1052.29	0.141498
$382$	860.783	0.115292
$383$	13131.8	1.75196	0.875982	−	0.482343i	$-0.160214\pi$
$383$	13131.8	1.75196	0.875982	+	0.482343i	$0.160214\pi$
$384$	1193.63	0.158626
$385$	−1138.75	−0.150743
$386$	901.556	0.118881
$387$	3133.76	0.411622
$388$	5209.33	0.681607
$389$	−3641.36	−0.474613	−0.237306	−	0.971435i	$-0.576265\pi$
$389$	−3641.36	−0.474613	−0.237306	+	0.971435i	$0.576265\pi$
$390$	−20.5303	−0.00266562
$391$	11242.0	1.45405
$392$	−736.524	−0.0948982
$393$	−3454.65	−0.443421
$394$	1084.79	0.138708
$395$	2185.34	0.278370
$396$	1518.99	0.192758
$397$	4537.94	0.573684	0.286842	−	0.957978i	$-0.407394\pi$
$397$	4537.94	0.573684	0.286842	+	0.957978i	$0.407394\pi$
$398$	−324.776	−0.0409034
$399$	−9240.37	−1.15939
$400$	−7569.13	−0.946141
$401$	−8702.07	−1.08369	−0.541846	−	0.840478i	$-0.682275\pi$
$401$	−8702.07	−1.08369	−0.541846	+	0.840478i	$0.682275\pi$
$402$	352.366	0.0437175
$403$	−1795.97	−0.221994
$404$	−9051.62	−1.11469
$405$	−181.030	−0.0222110
$406$	−1136.60	−0.138937
$407$	−7410.51	−0.902519
$408$	−1159.37	−0.140680
$409$	11254.5	1.36063	0.680315	−	0.732920i	$-0.261843\pi$
$409$	11254.5	1.36063	0.680315	+	0.732920i	$0.261843\pi$
$410$	103.644	0.0124845
$411$	7586.39	0.910485
$412$	−3794.67	−0.453762
$413$	−16331.8	−1.94585
$414$	−161.593	−0.0191832
$415$	343.130	0.0405870
$416$	−584.123	−0.0688437
$417$	665.599	0.0781644
$418$	534.249	0.0625143
$419$	−11848.5	−1.38147	−0.690737	−	0.723106i	$-0.742714\pi$
$419$	−11848.5	−1.38147	−0.690737	+	0.723106i	$0.742714\pi$
$420$	1282.95	0.149051
$421$	625.594	0.0724219	0.0362109	−	0.999344i	$-0.488471\pi$
$421$	625.594	0.0724219	0.0362109	+	0.999344i	$0.488471\pi$
$422$	−856.063	−0.0987499
$423$	−1989.72	−0.228708
$424$	1072.57	0.122851
$425$	14775.7	1.68642
$426$	122.372	0.0139178
$427$	−10684.1	−1.21086
$428$	6956.74	0.785670
$429$	−990.310	−0.111451
$430$	−153.029	−0.0171621
$431$	−12524.3	−1.39971	−0.699857	−	0.714283i	$-0.746753\pi$
$431$	−12524.3	−1.39971	−0.699857	+	0.714283i	$0.746753\pi$
$432$	−1702.98	−0.189664
$433$	665.267	0.0738354	0.0369177	−	0.999318i	$-0.488246\pi$
$433$	665.267	0.0738354	0.0369177	+	0.999318i	$0.488246\pi$
$434$	−545.124	−0.0602922
$435$	1612.41	0.177722
$436$	6192.16	0.680162
$437$	11701.1	1.28087
$438$	−171.016	−0.0186563
$439$	13529.0	1.47086	0.735428	−	0.677603i	$-0.236981\pi$
$439$	13529.0	1.47086	0.735428	+	0.677603i	$0.236981\pi$
$440$	−148.712	−0.0161127
$441$	2111.91	0.228044
$442$	377.013	0.0405717
$443$	−3146.50	−0.337459	−0.168730	−	0.985662i	$-0.553967\pi$
$443$	−3146.50	−0.337459	−0.168730	+	0.985662i	$0.553967\pi$
$444$	8348.88	0.892388
$445$	−2669.63	−0.284388
$446$	−593.762	−0.0630391
$447$	−808.166	−0.0855145
$448$	11950.2	1.26026
$449$	14569.0	1.53130	0.765649	−	0.643258i	$-0.222418\pi$
$449$	14569.0	1.53130	0.765649	+	0.643258i	$0.222418\pi$
$450$	−212.386	−0.0222488
$451$	4999.44	0.521983
$452$	−416.488	−0.0433406
$453$	3912.20	0.405765
$454$	−170.554	−0.0176311
$455$	−836.422	−0.0861803
$456$	−1206.72	−0.123925
$457$	−7756.44	−0.793941	−0.396971	−	0.917831i	$-0.629939\pi$
$457$	−7756.44	−0.793941	−0.396971	+	0.917831i	$0.629939\pi$
$458$	−154.421	−0.0157546
$459$	3324.40	0.338060
$460$	−1624.60	−0.164668
$461$	3248.41	0.328186	0.164093	−	0.986445i	$-0.447530\pi$
$461$	3248.41	0.328186	0.164093	+	0.986445i	$0.447530\pi$
$462$	−300.585	−0.0302694
$463$	9274.12	0.930896	0.465448	−	0.885075i	$-0.345893\pi$
$463$	9274.12	0.930896	0.465448	+	0.885075i	$0.345893\pi$
$464$	15168.2	1.51760
$465$	773.328	0.0771231
$466$	−1085.74	−0.107932
$467$	−4476.69	−0.443590	−0.221795	−	0.975093i	$-0.571192\pi$
$467$	−4476.69	−0.443590	−0.221795	+	0.975093i	$0.571192\pi$
$468$	1115.71	0.110200
$469$	14355.7	1.41340
$470$	97.1627	0.00953570
$471$	471.000	0.0460776
$472$	−2132.82	−0.207989
$473$	−7381.58	−0.717559
$474$	576.842	0.0558971
$475$	15379.2	1.48557
$476$	−23559.7	−2.26861
$477$	−3075.50	−0.295215
$478$	162.845	0.0155823
$479$	11597.3	1.10625	0.553127	−	0.833097i	$-0.313435\pi$
$479$	11597.3	1.10625	0.553127	+	0.833097i	$0.313435\pi$
$480$	251.518	0.0239170
$481$	−5443.08	−0.515974
$482$	−1029.99	−0.0973338
$483$	−6583.41	−0.620198
$484$	7018.54	0.659142
$485$	1462.39	0.136915
$486$	−47.7849	−0.00446001
$487$	−18048.4	−1.67937	−0.839684	−	0.543075i	$-0.817260\pi$
$487$	−18048.4	−1.67937	−0.839684	+	0.543075i	$0.817260\pi$
$488$	−1395.26	−0.129427
$489$	6638.97	0.613957
$490$	−103.130	−0.00950803
$491$	16451.1	1.51207	0.756036	−	0.654531i	$-0.227134\pi$
$491$	16451.1	1.51207	0.756036	+	0.654531i	$0.227134\pi$
$492$	−5632.51	−0.516124
$493$	−29609.8	−2.70499
$494$	392.411	0.0357397
$495$	426.418	0.0387193
$496$	7274.82	0.658567
$497$	4985.55	0.449965
$498$	90.5729	0.00814993
$499$	8178.01	0.733664	0.366832	−	0.930287i	$-0.380442\pi$
$499$	8178.01	0.733664	0.366832	+	0.930287i	$0.380442\pi$
$500$	−4359.41	−0.389917
$501$	8347.92	0.744426
$502$	259.352	0.0230586
$503$	17671.9	1.56651	0.783253	−	0.621703i	$-0.213559\pi$
$503$	17671.9	1.56651	0.783253	+	0.621703i	$0.213559\pi$
$504$	678.939	0.0600047
$505$	−2541.02	−0.223908
$506$	380.632	0.0334410
$507$	5863.61	0.513633
$508$	2792.55	0.243896
$509$	2531.66	0.220459	0.110230	−	0.993906i	$-0.464841\pi$
$509$	2531.66	0.220459	0.110230	+	0.993906i	$0.464841\pi$
$510$	−162.338	−0.0140950
$511$	−6967.33	−0.603164
$512$	3949.83	0.340936
$513$	3460.16	0.297797
$514$	483.712	0.0415090
$515$	−1065.26	−0.0911474
$516$	8316.29	0.709504
$517$	4686.79	0.398694
$518$	−1652.12	−0.140135
$519$	−5030.33	−0.425447
$520$	−109.230	−0.00921167
$521$	−1812.71	−0.152431	−0.0762153	−	0.997091i	$-0.524284\pi$
$521$	−1812.71	−0.152431	−0.0762153	+	0.997091i	$0.524284\pi$
$522$	425.612	0.0356868
$523$	14364.0	1.20095	0.600474	−	0.799644i	$-0.294979\pi$
$523$	14364.0	1.20095	0.600474	+	0.799644i	$0.294979\pi$
$524$	−9167.88	−0.764314
$525$	−8652.78	−0.719311
$526$	168.158	0.0139392
$527$	−14201.2	−1.17384
$528$	4011.38	0.330631
$529$	−3830.39	−0.314818
$530$	150.184	0.0123086
$531$	6115.65	0.499805
$532$	−24521.9	−1.99842
$533$	3672.14	0.298420
$534$	−704.677	−0.0571055
$535$	1952.93	0.157818
$536$	1874.75	0.151076
$537$	−553.885	−0.0445100
$538$	492.242	0.0394462
$539$	−4974.62	−0.397537
$540$	−480.414	−0.0382847
$541$	−21624.0	−1.71846	−0.859232	−	0.511587i	$-0.829058\pi$
$541$	−21624.0	−1.71846	−0.859232	+	0.511587i	$0.829058\pi$
$542$	1092.76	0.0866017
$543$	−3367.25	−0.266119
$544$	−4618.81	−0.364025
$545$	1738.29	0.136624
$546$	−220.782	−0.0173051
$547$	−16710.3	−1.30618	−0.653089	−	0.757282i	$-0.726527\pi$
$547$	−16710.3	−1.30618	−0.653089	+	0.757282i	$0.726527\pi$
$548$	20132.6	1.56938
$549$	4000.78	0.311018
$550$	500.277	0.0387852
$551$	−30819.1	−2.38283
$552$	−859.744	−0.0662919
$553$	23501.0	1.80717
$554$	−423.548	−0.0324816
$555$	2343.74	0.179254
$556$	1766.35	0.134730
$557$	−706.615	−0.0537527	−0.0268764	−	0.999639i	$-0.508556\pi$
$557$	−706.615	−0.0537527	−0.0268764	+	0.999639i	$0.508556\pi$
$558$	204.128	0.0154864
$559$	−5421.83	−0.410231
$560$	3388.04	0.255662
$561$	−7830.63	−0.589322
$562$	−1224.87	−0.0919361
$563$	21027.6	1.57408	0.787039	−	0.616903i	$-0.211613\pi$
$563$	21027.6	1.57408	0.787039	+	0.616903i	$0.211613\pi$
$564$	−5280.26	−0.394218
$565$	−116.918	−0.00870584
$566$	−1701.64	−0.126370
$567$	−1946.79	−0.144193
$568$	651.076	0.0480960
$569$	14442.5	1.06408	0.532039	−	0.846720i	$-0.321426\pi$
$569$	14442.5	1.06408	0.532039	+	0.846720i	$0.321426\pi$
$570$	−168.968	−0.0124163
$571$	−20040.3	−1.46876	−0.734380	−	0.678739i	$-0.762527\pi$
$571$	−20040.3	−1.46876	−0.734380	+	0.678739i	$0.762527\pi$
$572$	−2628.06	−0.192106
$573$	−13132.0	−0.957412
$574$	1114.59	0.0810488
$575$	10957.1	0.794680
$576$	−4474.90	−0.323705
$577$	−9041.79	−0.652365	−0.326183	−	0.945307i	$-0.605762\pi$
$577$	−9041.79	−0.652365	−0.326183	+	0.945307i	$0.605762\pi$
$578$	2015.02	0.145006
$579$	−13754.0	−0.987215
$580$	4278.97	0.306335
$581$	3690.01	0.263490
$582$	386.013	0.0274927
$583$	7244.36	0.514632
$584$	−909.881	−0.0644712
$585$	313.208	0.0221360
$586$	622.564	0.0438872
$587$	−16166.6	−1.13674	−0.568369	−	0.822774i	$-0.692425\pi$
$587$	−16166.6	−1.13674	−0.568369	+	0.822774i	$0.692425\pi$
$588$	5604.55	0.393074
$589$	−14781.2	−1.03404
$590$	−298.642	−0.0208388
$591$	−16549.4	−1.15186
$592$	22047.9	1.53068
$593$	−1062.37	−0.0735690	−0.0367845	−	0.999323i	$-0.511712\pi$
$593$	−1062.37	−0.0735690	−0.0367845	+	0.999323i	$0.511712\pi$
$594$	112.558	0.00777490
$595$	−6613.79	−0.455696
$596$	−2144.69	−0.147399
$597$	4954.74	0.339672
$598$	279.578	0.0191184
$599$	1982.94	0.135260	0.0676299	−	0.997710i	$-0.478456\pi$
$599$	1982.94	0.135260	0.0676299	+	0.997710i	$0.478456\pi$
$600$	−1129.99	−0.0768860
$601$	21063.3	1.42960	0.714799	−	0.699330i	$-0.246518\pi$
$601$	21063.3	1.42960	0.714799	+	0.699330i	$0.246518\pi$
$602$	−1645.67	−0.111416
$603$	−5375.66	−0.363041
$604$	10382.1	0.699408
$605$	1970.28	0.132402
$606$	−670.728	−0.0449611
$607$	17515.6	1.17123	0.585613	−	0.810591i	$-0.300854\pi$
$607$	17515.6	1.17123	0.585613	+	0.810591i	$0.300854\pi$
$608$	−4807.44	−0.320670
$609$	17339.8	1.15376
$610$	−195.368	−0.0129675
$611$	3442.49	0.227935
$612$	8822.20	0.582707
$613$	−4623.88	−0.304660	−0.152330	−	0.988330i	$-0.548678\pi$
$613$	−4623.88	−0.304660	−0.152330	+	0.988330i	$0.548678\pi$
$614$	755.065	0.0496286
$615$	−1581.19	−0.103674
$616$	−1599.24	−0.104603
$617$	−18254.4	−1.19108	−0.595538	−	0.803327i	$-0.703061\pi$
$617$	−18254.4	−1.19108	−0.595538	+	0.803327i	$0.703061\pi$
$618$	−281.186	−0.0183025
$619$	45.9974	0.00298674	0.00149337	−	0.999999i	$-0.499525\pi$
$619$	45.9974	0.00298674	0.00149337	+	0.999999i	$0.499525\pi$
$620$	2052.24	0.132935
$621$	2465.24	0.159302
$622$	−2.56418	−0.000165296 0
$623$	−28709.1	−1.84624
$624$	2946.40	0.189023
$625$	13776.8	0.881717
$626$	80.4837	0.00513862
$627$	−8150.43	−0.519134
$628$	1249.93	0.0794229
$629$	−43039.8	−2.72831
$630$	95.0667	0.00601198
$631$	5406.55	0.341095	0.170548	−	0.985349i	$-0.445446\pi$
$631$	5406.55	0.341095	0.170548	+	0.985349i	$0.445446\pi$
$632$	3069.06	0.193165
$633$	13060.0	0.820044
$634$	961.385	0.0602231
$635$	783.938	0.0489915
$636$	−8161.69	−0.508856
$637$	−3653.91	−0.227273
$638$	−1002.53	−0.0622109
$639$	−1866.90	−0.115576
$640$	889.234	0.0549220
$641$	−6467.68	−0.398530	−0.199265	−	0.979946i	$-0.563856\pi$
$641$	−6467.68	−0.398530	−0.199265	+	0.979946i	$0.563856\pi$
$642$	515.496	0.0316900
$643$	−7147.41	−0.438361	−0.219181	−	0.975684i	$-0.570338\pi$
$643$	−7147.41	−0.438361	−0.219181	+	0.975684i	$0.570338\pi$
$644$	−17470.9	−1.06902
$645$	2334.59	0.142518
$646$	3102.89	0.188981
$647$	−29650.0	−1.80164	−0.900821	−	0.434191i	$-0.857034\pi$
$647$	−29650.0	−1.80164	−0.900821	+	0.434191i	$0.857034\pi$
$648$	−254.237	−0.0154126
$649$	−14405.4	−0.871284
$650$	367.458	0.0221737
$651$	8316.34	0.500681
$652$	17618.4	1.05826
$653$	−17005.7	−1.01912	−0.509560	−	0.860435i	$-0.670192\pi$
$653$	−17005.7	−1.01912	−0.509560	+	0.860435i	$0.670192\pi$
$654$	458.841	0.0274344
$655$	−2573.65	−0.153528
$656$	−14874.5	−0.885290
$657$	2609.00	0.154927
$658$	1044.88	0.0619055
$659$	−3743.84	−0.221304	−0.110652	−	0.993859i	$-0.535294\pi$
$659$	−3743.84	−0.221304	−0.110652	+	0.993859i	$0.535294\pi$
$660$	1131.62	0.0667397
$661$	−10291.6	−0.605591	−0.302795	−	0.953056i	$-0.597920\pi$
$661$	−10291.6	−0.605591	−0.302795	+	0.953056i	$0.597920\pi$
$662$	−1341.64	−0.0787677
$663$	−5751.67	−0.336917
$664$	481.888	0.0281640
$665$	−6883.90	−0.401423
$666$	618.655	0.0359946
$667$	−21957.4	−1.27466
$668$	22153.5	1.28315
$669$	9058.36	0.523492
$670$	262.506	0.0151366
$671$	−9423.85	−0.542181
$672$	2704.81	0.155268
$673$	−12580.5	−0.720569	−0.360284	−	0.932842i	$-0.617320\pi$
$673$	−12580.5	−0.720569	−0.360284	+	0.932842i	$0.617320\pi$
$674$	2370.40	0.135467
$675$	3240.14	0.184760
$676$	15560.7	0.885338
$677$	4032.36	0.228916	0.114458	−	0.993428i	$-0.463487\pi$
$677$	4032.36	0.228916	0.114458	+	0.993428i	$0.463487\pi$
$678$	−30.8619	−0.00174815
$679$	15726.5	0.888846
$680$	−863.712	−0.0487086
$681$	2601.96	0.146413
$682$	−480.825	−0.0269967
$683$	12147.2	0.680527	0.340263	−	0.940330i	$-0.389484\pi$
$683$	12147.2	0.680527	0.340263	+	0.940330i	$0.389484\pi$
$684$	9182.50	0.513307
$685$	5651.72	0.315243
$686$	512.059	0.0284993
$687$	2355.82	0.130830
$688$	21961.9	1.21699
$689$	5321.05	0.294217
$690$	−120.383	−0.00664191
$691$	−28274.9	−1.55663	−0.778313	−	0.627877i	$-0.783924\pi$
$691$	−28274.9	−1.55663	−0.778313	+	0.627877i	$0.783924\pi$
$692$	−13349.4	−0.733334
$693$	4585.68	0.251365
$694$	142.214	0.00777865
$695$	495.859	0.0270633
$696$	2264.44	0.123324
$697$	29036.5	1.57796
$698$	521.688	0.0282897
$699$	16564.0	0.896290
$700$	−22962.6	−1.23986
$701$	−16720.1	−0.900870	−0.450435	−	0.892809i	$-0.648731\pi$
$701$	−16720.1	−0.900870	−0.450435	+	0.892809i	$0.648731\pi$
$702$	82.6745	0.00444494
$703$	−44797.6	−2.40337
$704$	10540.6	0.564297
$705$	−1482.30	−0.0791868
$706$	119.716	0.00638182
$707$	−27326.0	−1.45361
$708$	16229.6	0.861504
$709$	16617.1	0.880207	0.440103	−	0.897947i	$-0.354942\pi$
$709$	16617.1	0.880207	0.440103	+	0.897947i	$0.354942\pi$
$710$	91.1652	0.00481883
$711$	−8800.23	−0.464183
$712$	−3749.19	−0.197341
$713$	−10531.0	−0.553142
$714$	−1745.78	−0.0915044
$715$	−737.763	−0.0385885
$716$	−1469.89	−0.0767210
$717$	−2484.34	−0.129399
$718$	1601.58	0.0832455
$719$	−13434.6	−0.696837	−0.348419	−	0.937339i	$-0.613281\pi$
$719$	−13434.6	−0.696837	−0.348419	+	0.937339i	$0.613281\pi$
$720$	−1268.69	−0.0656685
$721$	−11455.8	−0.591726
$722$	1880.82	0.0969485
$723$	15713.4	0.808283
$724$	−8935.93	−0.458703
$725$	−28859.3	−1.47836
$726$	520.076	0.0265866
$727$	−18414.5	−0.939418	−0.469709	−	0.882821i	$-0.655641\pi$
$727$	−18414.5	−0.939418	−0.469709	+	0.882821i	$0.655641\pi$
$728$	−1174.66	−0.0598019
$729$	729.000	0.0370370
$730$	−127.404	−0.00645949
$731$	−42871.8	−2.16918
$732$	10617.2	0.536095
$733$	30637.6	1.54383	0.771913	−	0.635729i	$-0.219300\pi$
$733$	30637.6	1.54383	0.771913	+	0.635729i	$0.219300\pi$
$734$	−2190.30	−0.110144
$735$	1573.34	0.0789570
$736$	−3425.12	−0.171537
$737$	12662.4	0.632869
$738$	−417.371	−0.0208179
$739$	5167.36	0.257218	0.128609	−	0.991695i	$-0.458949\pi$
$739$	5167.36	0.257218	0.128609	+	0.991695i	$0.458949\pi$
$740$	6219.76	0.308977
$741$	−5986.57	−0.296791
$742$	1615.08	0.0799074
$743$	22993.0	1.13530	0.567652	−	0.823269i	$-0.307852\pi$
$743$	22993.0	1.13530	0.567652	+	0.823269i	$0.307852\pi$
$744$	1086.05	0.0535169
$745$	−602.069	−0.0296082
$746$	108.142	0.00530744
$747$	−1381.77	−0.0676790
$748$	−20780.7	−1.01580
$749$	21001.7	1.02455
$750$	−323.033	−0.0157273
$751$	2484.02	0.120696	0.0603482	−	0.998177i	$-0.480779\pi$
$751$	2484.02	0.120696	0.0603482	+	0.998177i	$0.480779\pi$
$752$	−13944.2	−0.676189
$753$	−3956.64	−0.191485
$754$	−736.368	−0.0355662
$755$	2914.52	0.140490
$756$	−5166.36	−0.248543
$757$	25694.3	1.23365	0.616827	−	0.787099i	$-0.288418\pi$
$757$	25694.3	1.23365	0.616827	+	0.787099i	$0.288418\pi$
$758$	1129.18	0.0541079
$759$	−5806.88	−0.277703
$760$	−898.986	−0.0429074
$761$	17403.7	0.829021	0.414511	−	0.910044i	$-0.363953\pi$
$761$	17403.7	0.829021	0.414511	+	0.910044i	$0.363953\pi$
$762$	206.929	0.00983758
$763$	18693.5	0.886962
$764$	−34849.4	−1.65027
$765$	2476.61	0.117049
$766$	2582.31	0.121805
$767$	−10580.9	−0.498116
$768$	−11698.3	−0.549645
$769$	−1624.71	−0.0761879	−0.0380940	−	0.999274i	$-0.512129\pi$
$769$	−1624.71	−0.0761879	−0.0380940	+	0.999274i	$0.512129\pi$
$770$	−223.930	−0.0104804
$771$	−7379.45	−0.344701
$772$	−36500.1	−1.70164
$773$	39022.6	1.81571	0.907855	−	0.419284i	$-0.137719\pi$
$773$	39022.6	1.81571	0.907855	+	0.419284i	$0.137719\pi$
$774$	616.239	0.0286179
$775$	−13841.3	−0.641539
$776$	2053.76	0.0950073
$777$	25204.5	1.16371
$778$	−716.058	−0.0329973
$779$	30222.3	1.39002
$780$	831.184	0.0381553
$781$	4397.49	0.201478
$782$	2210.69	0.101092
$783$	−6493.08	−0.296352
$784$	14800.6	0.674227
$785$	350.886	0.0159537
$786$	−679.343	−0.0308287
$787$	31902.8	1.44499	0.722497	−	0.691374i	$-0.242994\pi$
$787$	31902.8	1.44499	0.722497	+	0.691374i	$0.242994\pi$
$788$	−43918.4	−1.98544
$789$	−2565.39	−0.115755
$790$	429.737	0.0193536
$791$	−1257.34	−0.0565180
$792$	598.856	0.0268680
$793$	−6921.90	−0.309967
$794$	892.366	0.0398852
$795$	−2291.19	−0.102214
$796$	13148.8	0.585485
$797$	−932.841	−0.0414591	−0.0207296	−	0.999785i	$-0.506599\pi$
$797$	−932.841	−0.0414591	−0.0207296	+	0.999785i	$0.506599\pi$
$798$	−1817.08	−0.0806063
$799$	27220.6	1.20525
$800$	−4501.74	−0.198951
$801$	10750.5	0.474218
$802$	−1711.22	−0.0753434
$803$	−6145.51	−0.270075
$804$	−14265.8	−0.625766
$805$	−4904.52	−0.214735
$806$	−353.170	−0.0154341
$807$	−7509.59	−0.327571
$808$	−3568.57	−0.155374
$809$	25237.0	1.09677	0.548385	−	0.836226i	$-0.315243\pi$
$809$	25237.0	1.09677	0.548385	+	0.836226i	$0.315243\pi$
$810$	−35.5988	−0.00154422
$811$	−9733.19	−0.421429	−0.210714	−	0.977548i	$-0.567579\pi$
$811$	−9733.19	−0.421429	−0.210714	+	0.977548i	$0.567579\pi$
$812$	46015.9	1.98872
$813$	−16671.0	−0.719162
$814$	−1457.24	−0.0627474
$815$	4945.91	0.212574
$816$	23297.9	0.999497
$817$	−44622.7	−1.91083
$818$	2213.14	0.0945974
$819$	3368.23	0.143706
$820$	−4196.11	−0.178701
$821$	9418.02	0.400355	0.200177	−	0.979760i	$-0.435848\pi$
$821$	9418.02	0.400355	0.200177	+	0.979760i	$0.435848\pi$
$822$	1491.83	0.0633012
$823$	−20264.3	−0.858287	−0.429143	−	0.903236i	$-0.641184\pi$
$823$	−20264.3	−0.858287	−0.429143	+	0.903236i	$0.641184\pi$
$824$	−1496.04	−0.0632486
$825$	−7632.16	−0.322082
$826$	−3211.58	−0.135285
$827$	4383.63	0.184321	0.0921607	−	0.995744i	$-0.470623\pi$
$827$	4383.63	0.184321	0.0921607	+	0.995744i	$0.470623\pi$
$828$	6542.19	0.274585
$829$	41152.8	1.72412	0.862060	−	0.506806i	$-0.169174\pi$
$829$	41152.8	1.72412	0.862060	+	0.506806i	$0.169174\pi$
$830$	67.4751	0.00282180
$831$	6461.59	0.269735
$832$	7742.19	0.322611
$833$	−28892.3	−1.20175
$834$	130.887	0.00543436
$835$	6219.04	0.257747
$836$	−21629.4	−0.894820
$837$	−3114.15	−0.128603
$838$	−2329.96	−0.0960467
$839$	−34613.6	−1.42431	−0.712153	−	0.702024i	$-0.752280\pi$
$839$	−34613.6	−1.42431	−0.712153	+	0.702024i	$0.752280\pi$
$840$	505.797	0.0207758
$841$	33443.7	1.37126
$842$	123.020	0.00503511
$843$	18686.5	0.763460
$844$	34658.3	1.41349
$845$	4368.28	0.177838
$846$	−391.269	−0.0159008
$847$	21188.3	0.859550
$848$	−21553.6	−0.872823
$849$	25960.0	1.04940
$850$	2905.58	0.117248
$851$	−31916.6	−1.28565
$852$	−4954.33	−0.199217
$853$	−45666.8	−1.83306	−0.916530	−	0.399965i	$-0.869022\pi$
$853$	−45666.8	−1.83306	−0.916530	+	0.399965i	$0.869022\pi$
$854$	−2100.98	−0.0841849
$855$	2577.76	0.103108
$856$	2742.67	0.109512
$857$	5476.54	0.218291	0.109145	−	0.994026i	$-0.465189\pi$
$857$	5476.54	0.218291	0.109145	+	0.994026i	$0.465189\pi$
$858$	−194.740	−0.00774862
$859$	−22178.6	−0.880936	−0.440468	−	0.897768i	$-0.645187\pi$
$859$	−22178.6	−0.880936	−0.440468	+	0.897768i	$0.645187\pi$
$860$	6195.48	0.245656
$861$	−17004.0	−0.673049
$862$	−2462.86	−0.0973147
$863$	−31399.3	−1.23852	−0.619260	−	0.785186i	$-0.712567\pi$
$863$	−31399.3	−1.23852	−0.619260	+	0.785186i	$0.712567\pi$
$864$	−1012.85	−0.0398817
$865$	−3747.50	−0.147305
$866$	130.822	0.00513338
$867$	−30740.9	−1.20417
$868$	22069.7	0.863013
$869$	20729.0	0.809186
$870$	317.073	0.0123561
$871$	9300.63	0.361814
$872$	2441.24	0.0948059
$873$	−5888.96	−0.228306
$874$	2300.98	0.0890522
$875$	−13160.6	−0.508469
$876$	6923.70	0.267044
$877$	41447.8	1.59589	0.797943	−	0.602732i	$-0.205921\pi$
$877$	41447.8	1.59589	0.797943	+	0.602732i	$0.205921\pi$
$878$	2660.43	0.102261
$879$	−9497.76	−0.364450
$880$	2988.41	0.114476
$881$	−8629.72	−0.330014	−0.165007	−	0.986292i	$-0.552765\pi$
$881$	−8629.72	−0.330014	−0.165007	+	0.986292i	$0.552765\pi$
$882$	415.299	0.0158547
$883$	−6132.27	−0.233711	−0.116856	−	0.993149i	$-0.537282\pi$
$883$	−6132.27	−0.233711	−0.116856	+	0.993149i	$0.537282\pi$
$884$	−15263.6	−0.580737
$885$	4556.04	0.173051
$886$	−618.745	−0.0234618
$887$	−17562.8	−0.664828	−0.332414	−	0.943134i	$-0.607863\pi$
$887$	−17562.8	−0.664828	−0.332414	+	0.943134i	$0.607863\pi$
$888$	3291.52	0.124387
$889$	8430.44	0.318052
$890$	−524.971	−0.0197720
$891$	−1717.16	−0.0645647
$892$	24038.9	0.902332
$893$	28332.3	1.06171
$894$	−158.922	−0.00594537
$895$	−412.634	−0.0154110
$896$	9562.79	0.356552
$897$	−4265.20	−0.158764
$898$	2864.93	0.106463
$899$	27737.2	1.02902
$900$	8598.60	0.318467
$901$	42074.8	1.55573
$902$	983.118	0.0362908
$903$	25106.1	0.925225
$904$	−164.199	−0.00604112
$905$	−2508.54	−0.0921399
$906$	769.318	0.0282107
$907$	21607.9	0.791045	0.395522	−	0.918456i	$-0.370564\pi$
$907$	21607.9	0.791045	0.395522	+	0.918456i	$0.370564\pi$
$908$	6905.01	0.252369
$909$	10232.5	0.373368
$910$	−164.479	−0.00599166
$911$	710.841	0.0258520	0.0129260	−	0.999916i	$-0.495885\pi$
$911$	710.841	0.0258520	0.0129260	+	0.999916i	$0.495885\pi$
$912$	24249.4	0.880458
$913$	3254.76	0.117981
$914$	−1525.27	−0.0551985
$915$	2980.50	0.107686
$916$	6251.82	0.225509
$917$	−27677.0	−0.996700
$918$	653.728	0.0235035
$919$	−24699.8	−0.886585	−0.443293	−	0.896377i	$-0.646190\pi$
$919$	−24699.8	−0.886585	−0.443293	+	0.896377i	$0.646190\pi$
$920$	−640.493	−0.0229527
$921$	−11519.2	−0.412128
$922$	638.786	0.0228170
$923$	3229.99	0.115186
$924$	12169.4	0.433272
$925$	−41948.9	−1.49111
$926$	1823.71	0.0647203
$927$	4289.74	0.151989
$928$	9021.27	0.319114
$929$	42500.4	1.50096	0.750481	−	0.660892i	$-0.229822\pi$
$929$	42500.4	1.50096	0.750481	+	0.660892i	$0.229822\pi$
$930$	152.072	0.00536196
$931$	−30072.3	−1.05863
$932$	43957.0	1.54492
$933$	39.1188	0.00137266
$934$	−880.322	−0.0308405
$935$	−5833.67	−0.204044
$936$	439.865	0.0153605
$937$	29745.3	1.03707	0.518537	−	0.855055i	$-0.326477\pi$
$937$	29745.3	1.03707	0.518537	+	0.855055i	$0.326477\pi$
$938$	2822.98	0.0982662
$939$	−1227.85	−0.0426724
$940$	−3933.70	−0.136493
$941$	9917.92	0.343587	0.171793	−	0.985133i	$-0.445044\pi$
$941$	9917.92	0.343587	0.171793	+	0.985133i	$0.445044\pi$
$942$	92.6201	0.00320353
$943$	21532.3	0.743571
$944$	42859.4	1.47771
$945$	−1450.33	−0.0499250
$946$	−1451.55	−0.0498881
$947$	30940.4	1.06170	0.530849	−	0.847466i	$-0.321873\pi$
$947$	30940.4	1.06170	0.530849	+	0.847466i	$0.321873\pi$
$948$	−23353.8	−0.800103
$949$	−4513.93	−0.154403
$950$	3024.24	0.103284
$951$	−14666.8	−0.500108
$952$	−9288.32	−0.316215
$953$	710.287	0.0241432	0.0120716	−	0.999927i	$-0.496157\pi$
$953$	710.287	0.0241432	0.0120716	+	0.999927i	$0.496157\pi$
$954$	−604.784	−0.0205247
$955$	−9783.09	−0.331491
$956$	−6592.88	−0.223043
$957$	15294.5	0.516615
$958$	2280.56	0.0769119
$959$	60778.4	2.04654
$960$	−3333.71	−0.112078
$961$	−16487.9	−0.553453
$962$	−1070.36	−0.0358729
$963$	−7864.34	−0.263162
$964$	41699.9	1.39322
$965$	−10246.5	−0.341809
$966$	−1294.60	−0.0431191
$967$	−45687.9	−1.51936	−0.759682	−	0.650295i	$-0.774645\pi$
$967$	−45687.9	−1.51936	−0.759682	+	0.650295i	$0.774645\pi$
$968$	2767.04	0.0918759
$969$	−47337.3	−1.56934
$970$	287.572	0.00951896
$971$	53715.3	1.77529	0.887644	−	0.460530i	$-0.152341\pi$
$971$	53715.3	1.77529	0.887644	+	0.460530i	$0.152341\pi$
$972$	1934.60	0.0638399
$973$	5332.45	0.175694
$974$	−3549.14	−0.116758
$975$	−5605.89	−0.184135
$976$	28038.1	0.919546
$977$	−6522.58	−0.213588	−0.106794	−	0.994281i	$-0.534059\pi$
$977$	−6522.58	−0.213588	−0.106794	+	0.994281i	$0.534059\pi$
$978$	1305.53	0.0426852
$979$	−25322.8	−0.826679
$980$	4175.28	0.136096
$981$	−7000.02	−0.227822
$982$	3235.03	0.105126
$983$	24039.0	0.779986	0.389993	−	0.920818i	$-0.372477\pi$
$983$	24039.0	0.779986	0.389993	+	0.920818i	$0.372477\pi$
$984$	−2220.60	−0.0719411
$985$	−12329.0	−0.398816
$986$	−5822.64	−0.188064
$987$	−15940.6	−0.514078
$988$	−15887.0	−0.511572
$989$	−31792.0	−1.02217
$990$	83.8532	0.00269195
$991$	5752.55	0.184395	0.0921977	−	0.995741i	$-0.470611\pi$
$991$	5752.55	0.184395	0.0921977	+	0.995741i	$0.470611\pi$
$992$	4326.70	0.138481
$993$	20467.8	0.654106
$994$	980.387	0.0312837
$995$	3691.19	0.117607
$996$	−3666.90	−0.116657
$997$	7196.58	0.228604	0.114302	−	0.993446i	$-0.463537\pi$
$997$	7196.58	0.228604	0.114302	+	0.993446i	$0.463537\pi$
$998$	1608.17	0.0510078
$999$	−9438.11	−0.298908

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.4.a.c.1.11	✓	22
3.2	odd	2		1413.4.a.e.1.12		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.4.a.c.1.11	✓	22	1.1	even	1	trivial
1413.4.a.e.1.12		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)

\(f(q)\)	\(=\)	\(q+0.196646 q^{2} -3.00000 q^{3} -7.96133 q^{4} -2.23494 q^{5} -0.589937 q^{6} -24.0345 q^{7} -3.13873 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+0.196646 q^{2} -3.00000 q^{3} -7.96133 q^{4} -2.23494 q^{5} -0.589937 q^{6} -24.0345 q^{7} -3.13873 q^{8} +9.00000 q^{9} -0.439492 q^{10} -21.1995 q^{11} +23.8840 q^{12} -15.5713 q^{13} -4.72628 q^{14} +6.70483 q^{15} +63.0734 q^{16} -123.126 q^{17} +1.76981 q^{18} -128.154 q^{19} +17.7931 q^{20} +72.1035 q^{21} -4.16880 q^{22} -91.3050 q^{23} +9.41618 q^{24} -120.005 q^{25} -3.06202 q^{26} -27.0000 q^{27} +191.347 q^{28} +240.484 q^{29} +1.31848 q^{30} +115.339 q^{31} +37.5129 q^{32} +63.5986 q^{33} -24.2121 q^{34} +53.7158 q^{35} -71.6520 q^{36} +349.560 q^{37} -25.2010 q^{38} +46.7138 q^{39} +7.01487 q^{40} -235.828 q^{41} +14.1788 q^{42} +348.195 q^{43} +168.777 q^{44} -20.1145 q^{45} -17.9547 q^{46} -221.080 q^{47} -189.220 q^{48} +234.657 q^{49} -23.5985 q^{50} +369.377 q^{51} +123.968 q^{52} -341.722 q^{53} -5.30943 q^{54} +47.3798 q^{55} +75.4377 q^{56} +384.463 q^{57} +47.2902 q^{58} +679.517 q^{59} -53.3794 q^{60} +444.531 q^{61} +22.6809 q^{62} -216.310 q^{63} -497.211 q^{64} +34.8009 q^{65} +12.5064 q^{66} -597.295 q^{67} +980.245 q^{68} +273.915 q^{69} +10.5630 q^{70} -207.433 q^{71} -28.2485 q^{72} +289.889 q^{73} +68.7394 q^{74} +360.015 q^{75} +1020.28 q^{76} +509.520 q^{77} +9.18606 q^{78} -977.803 q^{79} -140.966 q^{80} +81.0000 q^{81} -46.3745 q^{82} -153.530 q^{83} -574.040 q^{84} +275.179 q^{85} +68.4710 q^{86} -721.453 q^{87} +66.5395 q^{88} +1194.50 q^{89} -3.95543 q^{90} +374.247 q^{91} +726.910 q^{92} -346.017 q^{93} -43.4743 q^{94} +286.418 q^{95} -112.539 q^{96} -654.329 q^{97} +46.1443 q^{98} -190.796 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9}+O(q^{10}) \) 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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