LMFDB - Embedded newform 1849.4.a.i.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,4,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1849.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1849.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:	$109.094531601$
Analytic rank:	$1$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.52844 q^{2} -3.06369 q^{3} +4.44988 q^{4} -14.8551 q^{5} +10.8100 q^{6} -18.1642 q^{7} +12.5264 q^{8} -17.6138 q^{9} +O(q^{10})$	\(q-3.52844 q^{2} -3.06369 q^{3} +4.44988 q^{4} -14.8551 q^{5} +10.8100 q^{6} -18.1642 q^{7} +12.5264 q^{8} -17.6138 q^{9} +52.4155 q^{10} -54.4158 q^{11} -13.6330 q^{12} -71.7442 q^{13} +64.0913 q^{14} +45.5115 q^{15} -79.7976 q^{16} -70.3013 q^{17} +62.1493 q^{18} -71.5106 q^{19} -66.1036 q^{20} +55.6494 q^{21} +192.003 q^{22} +107.841 q^{23} -38.3769 q^{24} +95.6754 q^{25} +253.145 q^{26} +136.683 q^{27} -80.8285 q^{28} +165.324 q^{29} -160.585 q^{30} -117.051 q^{31} +181.350 q^{32} +166.713 q^{33} +248.054 q^{34} +269.832 q^{35} -78.3794 q^{36} -37.0649 q^{37} +252.321 q^{38} +219.802 q^{39} -186.081 q^{40} +24.0387 q^{41} -196.356 q^{42} -242.144 q^{44} +261.656 q^{45} -380.509 q^{46} -454.480 q^{47} +244.475 q^{48} -13.0614 q^{49} -337.585 q^{50} +215.381 q^{51} -319.253 q^{52} -662.454 q^{53} -482.277 q^{54} +808.355 q^{55} -227.532 q^{56} +219.086 q^{57} -583.337 q^{58} +325.977 q^{59} +202.521 q^{60} -545.470 q^{61} +413.009 q^{62} +319.941 q^{63} -1.50096 q^{64} +1065.77 q^{65} -588.236 q^{66} +976.139 q^{67} -312.832 q^{68} -330.390 q^{69} -952.086 q^{70} -494.649 q^{71} -220.638 q^{72} +462.528 q^{73} +130.781 q^{74} -293.119 q^{75} -318.213 q^{76} +988.420 q^{77} -775.557 q^{78} +51.1561 q^{79} +1185.41 q^{80} +56.8202 q^{81} -84.8190 q^{82} +391.041 q^{83} +247.633 q^{84} +1044.34 q^{85} -506.502 q^{87} -681.633 q^{88} +1096.04 q^{89} -923.237 q^{90} +1303.18 q^{91} +479.877 q^{92} +358.609 q^{93} +1603.60 q^{94} +1062.30 q^{95} -555.599 q^{96} -685.682 q^{97} +46.0864 q^{98} +958.470 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9}+O(q^{10}) $ 50 * q - 10 * q^2 - 2 * q^3 + 186 * q^4 + 8 * q^5 - 51 * q^6 - 6 * q^7 - 138 * q^8 + 360 * q^9	\( 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9} - 137 q^{10} - 252 q^{11} - 48 q^{12} - 192 q^{13} - 272 q^{14} - 314 q^{15} + 542 q^{16} - 236 q^{17} - 386 q^{18} + 12 q^{19} + 108 q^{20} - 408 q^{21} + 1235 q^{22} - 630 q^{23} - 613 q^{24} + 1098 q^{25} + 1493 q^{26} + 10 q^{27} - 242 q^{28} + 208 q^{29} - 48 q^{30} - 932 q^{31} - 1124 q^{32} + 254 q^{33} + 765 q^{34} - 1452 q^{35} + 747 q^{36} - 90 q^{37} - 1213 q^{38} - 1610 q^{39} - 1693 q^{40} - 1354 q^{41} - 16 q^{42} - 2704 q^{44} + 4508 q^{45} + 233 q^{46} - 3484 q^{47} - 376 q^{48} + 1324 q^{49} - 408 q^{50} + 4054 q^{51} - 2176 q^{52} - 726 q^{53} - 6497 q^{54} - 3288 q^{55} - 7097 q^{56} - 870 q^{57} + 275 q^{58} - 4370 q^{59} - 3891 q^{60} + 1172 q^{61} - 1546 q^{62} - 3686 q^{63} + 606 q^{64} + 2610 q^{65} - 4697 q^{66} - 344 q^{67} - 3221 q^{68} + 136 q^{69} - 1310 q^{70} + 162 q^{71} - 5814 q^{72} + 746 q^{73} - 4332 q^{74} + 236 q^{75} + 1338 q^{76} + 2024 q^{77} - 2782 q^{78} - 2656 q^{79} - 5713 q^{80} - 86 q^{81} - 4168 q^{82} - 3514 q^{83} - 4269 q^{84} - 7558 q^{85} - 10278 q^{87} + 11692 q^{88} + 2640 q^{89} - 8286 q^{90} - 5946 q^{91} - 4271 q^{92} - 2 q^{93} + 9062 q^{94} - 12140 q^{95} - 700 q^{96} - 3864 q^{97} - 2826 q^{98} - 8174 q^{99}+O(q^{100}) \) 50 * q - 10 * q^2 - 2 * q^3 + 186 * q^4 + 8 * q^5 - 51 * q^6 - 6 * q^7 - 138 * q^8 + 360 * q^9 - 137 * q^10 - 252 * q^11 - 48 * q^12 - 192 * q^13 - 272 * q^14 - 314 * q^15 + 542 * q^16 - 236 * q^17 - 386 * q^18 + 12 * q^19 + 108 * q^20 - 408 * q^21 + 1235 * q^22 - 630 * q^23 - 613 * q^24 + 1098 * q^25 + 1493 * q^26 + 10 * q^27 - 242 * q^28 + 208 * q^29 - 48 * q^30 - 932 * q^31 - 1124 * q^32 + 254 * q^33 + 765 * q^34 - 1452 * q^35 + 747 * q^36 - 90 * q^37 - 1213 * q^38 - 1610 * q^39 - 1693 * q^40 - 1354 * q^41 - 16 * q^42 - 2704 * q^44 + 4508 * q^45 + 233 * q^46 - 3484 * q^47 - 376 * q^48 + 1324 * q^49 - 408 * q^50 + 4054 * q^51 - 2176 * q^52 - 726 * q^53 - 6497 * q^54 - 3288 * q^55 - 7097 * q^56 - 870 * q^57 + 275 * q^58 - 4370 * q^59 - 3891 * q^60 + 1172 * q^61 - 1546 * q^62 - 3686 * q^63 + 606 * q^64 + 2610 * q^65 - 4697 * q^66 - 344 * q^67 - 3221 * q^68 + 136 * q^69 - 1310 * q^70 + 162 * q^71 - 5814 * q^72 + 746 * q^73 - 4332 * q^74 + 236 * q^75 + 1338 * q^76 + 2024 * q^77 - 2782 * q^78 - 2656 * q^79 - 5713 * q^80 - 86 * q^81 - 4168 * q^82 - 3514 * q^83 - 4269 * q^84 - 7558 * q^85 - 10278 * q^87 + 11692 * q^88 + 2640 * q^89 - 8286 * q^90 - 5946 * q^91 - 4271 * q^92 - 2 * q^93 + 9062 * q^94 - 12140 * q^95 - 700 * q^96 - 3864 * q^97 - 2826 * q^98 - 8174 * 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 $

$2$	−3.52844	−1.24749	−0.623746	−	0.781627i	$-0.714390\pi$
$2$	−3.52844	−1.24749	−0.623746	+	0.781627i	$0.714390\pi$
$3$	−3.06369	−0.589607	−0.294803	−	0.955558i	$-0.595254\pi$
$3$	−3.06369	−0.589607	−0.294803	+	0.955558i	$0.595254\pi$
$4$	4.44988	0.556235
$5$	−14.8551	−1.32868	−0.664342	−	0.747428i	$-0.731288\pi$
$5$	−14.8551	−1.32868	−0.664342	+	0.747428i	$0.731288\pi$
$6$	10.8100	0.735529
$7$	−18.1642	−0.980775	−0.490388	−	0.871504i	$-0.663145\pi$
$7$	−18.1642	−0.980775	−0.490388	+	0.871504i	$0.663145\pi$
$8$	12.5264	0.553593
$9$	−17.6138	−0.652364
$10$	52.4155	1.65752
$11$	−54.4158	−1.49154	−0.745772	−	0.666201i	$-0.767919\pi$
$11$	−54.4158	−1.49154	−0.745772	+	0.666201i	$0.767919\pi$
$12$	−13.6330	−0.327960
$13$	−71.7442	−1.53064	−0.765318	−	0.643652i	$-0.777418\pi$
$13$	−71.7442	−1.53064	−0.765318	+	0.643652i	$0.777418\pi$
$14$	64.0913	1.22351
$15$	45.5115	0.783401
$16$	−79.7976	−1.24684
$17$	−70.3013	−1.00297	−0.501487	−	0.865165i	$-0.667214\pi$
$17$	−70.3013	−1.00297	−0.501487	+	0.865165i	$0.667214\pi$
$18$	62.1493	0.813818
$19$	−71.5106	−0.863456	−0.431728	−	0.902004i	$-0.642096\pi$
$19$	−71.5106	−0.863456	−0.431728	+	0.902004i	$0.642096\pi$
$20$	−66.1036	−0.739061
$21$	55.6494	0.578272
$22$	192.003	1.86069
$23$	107.841	0.977666	0.488833	−	0.872377i	$-0.337423\pi$
$23$	107.841	0.977666	0.488833	+	0.872377i	$0.337423\pi$
$24$	−38.3769	−0.326402
$25$	95.6754	0.765403
$26$	253.145	1.90946
$27$	136.683	0.974245
$28$	−80.8285	−0.545541
$29$	165.324	1.05862	0.529310	−	0.848429i	$-0.322451\pi$
$29$	165.324	1.05862	0.529310	+	0.848429i	$0.322451\pi$
$30$	−160.585	−0.977286
$31$	−117.051	−0.678163	−0.339082	−	0.940757i	$-0.610116\pi$
$31$	−117.051	−0.678163	−0.339082	+	0.940757i	$0.610116\pi$
$32$	181.350	1.00183
$33$	166.713	0.879424
$34$	248.054	1.25120
$35$	269.832	1.30314
$36$	−78.3794	−0.362868
$37$	−37.0649	−0.164687	−0.0823437	−	0.996604i	$-0.526241\pi$
$37$	−37.0649	−0.164687	−0.0823437	+	0.996604i	$0.526241\pi$
$38$	252.321	1.07715
$39$	219.802	0.902473
$40$	−186.081	−0.735551
$41$	24.0387	0.0915662	0.0457831	−	0.998951i	$-0.485422\pi$
$41$	24.0387	0.0915662	0.0457831	+	0.998951i	$0.485422\pi$
$42$	−196.356	−0.721389
$43$	0	0
$43$	0	0
$44$	−242.144	−0.829648
$45$	261.656	0.866786
$46$	−380.509	−1.21963
$47$	−454.480	−1.41048	−0.705241	−	0.708968i	$-0.749161\pi$
$47$	−454.480	−1.41048	−0.705241	+	0.708968i	$0.749161\pi$
$48$	244.475	0.735144
$49$	−13.0614	−0.0380799
$50$	−337.585	−0.954834
$51$	215.381	0.591361
$52$	−319.253	−0.851393
$53$	−662.454	−1.71689	−0.858444	−	0.512907i	$-0.828569\pi$
$53$	−662.454	−1.71689	−0.858444	+	0.512907i	$0.828569\pi$
$54$	−482.277	−1.21536
$55$	808.355	1.98179
$56$	−227.532	−0.542951
$57$	219.086	0.509099
$58$	−583.337	−1.32062
$59$	325.977	0.719299	0.359650	−	0.933087i	$-0.382896\pi$
$59$	325.977	0.719299	0.359650	+	0.933087i	$0.382896\pi$
$60$	202.521	0.435755
$61$	−545.470	−1.14492	−0.572461	−	0.819932i	$-0.694011\pi$
$61$	−545.470	−1.14492	−0.572461	+	0.819932i	$0.694011\pi$
$62$	413.009	0.846003
$63$	319.941	0.639822
$64$	−1.50096	−0.00293156
$65$	1065.77	2.03373
$66$	−588.236	−1.09707
$67$	976.139	1.77992	0.889958	−	0.456042i	$-0.150733\pi$
$67$	976.139	1.77992	0.889958	+	0.456042i	$0.150733\pi$
$68$	−312.832	−0.557889
$69$	−330.390	−0.576438
$70$	−952.086	−1.62566
$71$	−494.649	−0.826818	−0.413409	−	0.910546i	$-0.635662\pi$
$71$	−494.649	−0.826818	−0.413409	+	0.910546i	$0.635662\pi$
$72$	−220.638	−0.361144
$73$	462.528	0.741573	0.370787	−	0.928718i	$-0.379088\pi$
$73$	462.528	0.741573	0.370787	+	0.928718i	$0.379088\pi$
$74$	130.781	0.205446
$75$	−293.119	−0.451287
$76$	−318.213	−0.480284
$77$	988.420	1.46287
$78$	−775.557	−1.12583
$79$	51.1561	0.0728546	0.0364273	−	0.999336i	$-0.488402\pi$
$79$	51.1561	0.0728546	0.0364273	+	0.999336i	$0.488402\pi$
$80$	1185.41	1.65665
$81$	56.8202	0.0779427
$82$	−84.8190	−0.114228
$83$	391.041	0.517137	0.258568	−	0.965993i	$-0.416749\pi$
$83$	391.041	0.517137	0.258568	+	0.965993i	$0.416749\pi$
$84$	247.633	0.321655
$85$	1044.34	1.33264
$86$	0	0
$87$	−506.502	−0.624169
$88$	−681.633	−0.825709
$89$	1096.04	1.30540	0.652698	−	0.757618i	$-0.273637\pi$
$89$	1096.04	1.30540	0.652698	+	0.757618i	$0.273637\pi$
$90$	−923.237	−1.08131
$91$	1303.18	1.50121
$92$	479.877	0.543812
$93$	358.609	0.399850
$94$	1603.60	1.75956
$95$	1062.30	1.14726
$96$	−555.599	−0.590683
$97$	−685.682	−0.717737	−0.358869	−	0.933388i	$-0.616837\pi$
$97$	−685.682	−0.717737	−0.358869	+	0.933388i	$0.616837\pi$
$98$	46.0864	0.0475044
$99$	958.470	0.973029
$100$	425.744	0.425744
$101$	991.235	0.976550	0.488275	−	0.872690i	$-0.337626\pi$
$101$	991.235	0.976550	0.488275	+	0.872690i	$0.337626\pi$
$102$	−759.959	−0.737717
$103$	−1673.10	−1.60054	−0.800270	−	0.599639i	$-0.795311\pi$
$103$	−1673.10	−1.60054	−0.800270	+	0.599639i	$0.795311\pi$
$104$	−898.696	−0.847350
$105$	−826.681	−0.768341
$106$	2337.43	2.14180
$107$	511.415	0.462059	0.231030	−	0.972947i	$-0.425791\pi$
$107$	511.415	0.462059	0.231030	+	0.972947i	$0.425791\pi$
$108$	608.222	0.541909
$109$	−112.112	−0.0985176	−0.0492588	−	0.998786i	$-0.515686\pi$
$109$	−112.112	−0.0985176	−0.0492588	+	0.998786i	$0.515686\pi$
$110$	−2852.23	−2.47227
$111$	113.555	0.0971007
$112$	1449.46	1.22287
$113$	−1391.69	−1.15858	−0.579290	−	0.815122i	$-0.696670\pi$
$113$	−1391.69	−1.15858	−0.579290	+	0.815122i	$0.696670\pi$
$114$	−773.032	−0.635097
$115$	−1601.99	−1.29901
$116$	735.674	0.588841
$117$	1263.69	0.998532
$118$	−1150.19	−0.897319
$119$	1276.97	0.983693
$120$	570.095	0.433686
$121$	1630.08	1.22470
$122$	1924.66	1.42828
$123$	−73.6470	−0.0539880
$124$	−520.865	−0.377218
$125$	435.622	0.311706
$126$	−1128.89	−0.798173
$127$	2009.58	1.40410	0.702052	−	0.712126i	$-0.252268\pi$
$127$	2009.58	1.40410	0.702052	+	0.712126i	$0.252268\pi$
$128$	−1445.50	−0.998169
$129$	0	0
$130$	−3760.51	−2.53706
$131$	990.242	0.660442	0.330221	−	0.943904i	$-0.392877\pi$
$131$	990.242	0.660442	0.330221	+	0.943904i	$0.392877\pi$
$132$	741.852	0.489166
$133$	1298.93	0.846856
$134$	−3444.25	−2.22043
$135$	−2030.44	−1.29446
$136$	−880.621	−0.555240
$137$	−1662.89	−1.03701	−0.518505	−	0.855075i	$-0.673511\pi$
$137$	−1662.89	−1.03701	−0.518505	+	0.855075i	$0.673511\pi$
$138$	1165.76	0.719102
$139$	867.074	0.529095	0.264548	−	0.964373i	$-0.414777\pi$
$139$	867.074	0.529095	0.264548	+	0.964373i	$0.414777\pi$
$140$	1200.72	0.724852
$141$	1392.38	0.831630
$142$	1745.34	1.03145
$143$	3904.02	2.28301
$144$	1405.54	0.813392
$145$	−2455.92	−1.40657
$146$	−1632.00	−0.925107
$147$	40.0161	0.0224522
$148$	−164.934	−0.0916048
$149$	1411.40	0.776014	0.388007	−	0.921656i	$-0.373164\pi$
$149$	1411.40	0.776014	0.388007	+	0.921656i	$0.373164\pi$
$150$	1034.25	0.562976
$151$	1638.02	0.882781	0.441390	−	0.897315i	$-0.354485\pi$
$151$	1638.02	0.882781	0.441390	+	0.897315i	$0.354485\pi$
$152$	−895.769	−0.478003
$153$	1238.27	0.654304
$154$	−3487.58	−1.82492
$155$	1738.82	0.901065
$156$	978.091	0.501987
$157$	1244.53	0.632639	0.316319	−	0.948653i	$-0.397553\pi$
$157$	1244.53	0.632639	0.316319	+	0.948653i	$0.397553\pi$
$158$	−180.501	−0.0908855
$159$	2029.55	1.01229
$160$	−2693.98	−1.33111
$161$	−1958.84	−0.958870
$162$	−200.487	−0.0972328
$163$	−1756.05	−0.843831	−0.421915	−	0.906635i	$-0.638642\pi$
$163$	−1756.05	−0.843831	−0.421915	+	0.906635i	$0.638642\pi$
$164$	106.969	0.0509323
$165$	−2476.55	−1.16848
$166$	−1379.77	−0.645124
$167$	179.987	0.0834001	0.0417001	−	0.999130i	$-0.486723\pi$
$167$	179.987	0.0834001	0.0417001	+	0.999130i	$0.486723\pi$
$168$	697.086	0.320127
$169$	2950.23	1.34285
$170$	−3684.88	−1.66245
$171$	1259.58	0.563287
$172$	0	0
$173$	−3964.61	−1.74233	−0.871167	−	0.490987i	$-0.836636\pi$
$173$	−3964.61	−1.74233	−0.871167	+	0.490987i	$0.836636\pi$
$174$	1787.16	0.778646
$175$	−1737.87	−0.750688
$176$	4342.25	1.85971
$177$	−998.693	−0.424104
$178$	−3867.32	−1.62847
$179$	−3188.70	−1.33148	−0.665739	−	0.746185i	$-0.731884\pi$
$179$	−3188.70	−1.33148	−0.665739	+	0.746185i	$0.731884\pi$
$180$	1164.34	0.482137
$181$	−53.3936	−0.0219266	−0.0109633	−	0.999940i	$-0.503490\pi$
$181$	−53.3936	−0.0219266	−0.0109633	+	0.999940i	$0.503490\pi$
$182$	−4598.18	−1.87275
$183$	1671.15	0.675054
$184$	1350.85	0.541229
$185$	550.604	0.218818
$186$	−1265.33	−0.498809
$187$	3825.50	1.49598
$188$	−2022.38	−0.784559
$189$	−2482.73	−0.955515
$190$	−3748.26	−1.43120
$191$	222.120	0.0841466	0.0420733	−	0.999115i	$-0.486604\pi$
$191$	222.120	0.0841466	0.0420733	+	0.999115i	$0.486604\pi$
$192$	4.59846	0.00172847
$193$	5074.95	1.89276	0.946380	−	0.323056i	$-0.104710\pi$
$193$	5074.95	1.89276	0.946380	+	0.323056i	$0.104710\pi$
$194$	2419.39	0.895371
$195$	−3265.19	−1.19910
$196$	−58.1217	−0.0211814
$197$	−2931.10	−1.06006	−0.530032	−	0.847978i	$-0.677820\pi$
$197$	−2931.10	−1.06006	−0.530032	+	0.847978i	$0.677820\pi$
$198$	−3381.90	−1.21385
$199$	−2008.74	−0.715556	−0.357778	−	0.933807i	$-0.616465\pi$
$199$	−2008.74	−0.715556	−0.357778	+	0.933807i	$0.616465\pi$
$200$	1198.47	0.423722
$201$	−2990.58	−1.04945
$202$	−3497.51	−1.21824
$203$	−3002.99	−1.03827
$204$	958.420	0.328935
$205$	−357.098	−0.121663
$206$	5903.44	1.99666
$207$	−1899.49	−0.637794
$208$	5725.02	1.90845
$209$	3891.31	1.28788
$210$	2916.89	0.958498
$211$	−823.929	−0.268823	−0.134411	−	0.990926i	$-0.542914\pi$
$211$	−823.929	−0.268823	−0.134411	+	0.990926i	$0.542914\pi$
$212$	−2947.84	−0.954993
$213$	1515.45	0.487497
$214$	−1804.50	−0.576415
$215$	0	0
$216$	1712.14	0.539335
$217$	2126.15	0.665126
$218$	395.582	0.122900
$219$	−1417.04	−0.437237
$220$	3597.08	1.10234
$221$	5043.71	1.53519
$222$	−400.673	−0.121132
$223$	2138.39	0.642139	0.321070	−	0.947056i	$-0.395958\pi$
$223$	2138.39	0.642139	0.321070	+	0.947056i	$0.395958\pi$
$224$	−3294.08	−0.982566
$225$	−1685.21	−0.499321
$226$	4910.51	1.44532
$227$	−448.136	−0.131030	−0.0655151	−	0.997852i	$-0.520869\pi$
$227$	−448.136	−0.131030	−0.0655151	+	0.997852i	$0.520869\pi$
$228$	974.906	0.283179
$229$	−5716.16	−1.64950	−0.824748	−	0.565501i	$-0.808683\pi$
$229$	−5716.16	−1.64950	−0.824748	+	0.565501i	$0.808683\pi$
$230$	5652.51	1.62050
$231$	−3028.21	−0.862517
$232$	2070.92	0.586045
$233$	6035.33	1.69694	0.848471	−	0.529242i	$-0.177524\pi$
$233$	6035.33	1.69694	0.848471	+	0.529242i	$0.177524\pi$
$234$	−4458.85	−1.24566
$235$	6751.36	1.87409
$236$	1450.56	0.400099
$237$	−156.726	−0.0429556
$238$	−4505.70	−1.22715
$239$	2662.70	0.720653	0.360326	−	0.932826i	$-0.382665\pi$
$239$	2662.70	0.720653	0.360326	+	0.932826i	$0.382665\pi$
$240$	−3631.71	−0.976774
$241$	81.7963	0.0218629	0.0109315	−	0.999940i	$-0.496520\pi$
$241$	81.7963	0.0218629	0.0109315	+	0.999940i	$0.496520\pi$
$242$	−5751.63	−1.52781
$243$	−3864.51	−1.02020
$244$	−2427.27	−0.636845
$245$	194.029	0.0505962
$246$	259.859	0.0673496
$247$	5130.47	1.32164
$248$	−1466.23	−0.375427
$249$	−1198.03	−0.304907
$250$	−1537.06	−0.388850
$251$	−3638.89	−0.915078	−0.457539	−	0.889190i	$-0.651269\pi$
$251$	−3638.89	−0.915078	−0.457539	+	0.889190i	$0.651269\pi$
$252$	1423.70	0.355891
$253$	−5868.23	−1.45823
$254$	−7090.67	−1.75161
$255$	−3199.52	−0.785732
$256$	5112.38	1.24814
$257$	−2962.31	−0.719003	−0.359502	−	0.933144i	$-0.617053\pi$
$257$	−2962.31	−0.719003	−0.359502	+	0.933144i	$0.617053\pi$
$258$	0	0
$259$	673.255	0.161521
$260$	4742.55	1.13123
$261$	−2912.00	−0.690605
$262$	−3494.01	−0.823895
$263$	−113.737	−0.0266666	−0.0133333	−	0.999911i	$-0.504244\pi$
$263$	−113.737	−0.0266666	−0.0133333	+	0.999911i	$0.504244\pi$
$264$	2088.31	0.486843
$265$	9840.86	2.28120
$266$	−4583.21	−1.05645
$267$	−3357.93	−0.769671
$268$	4343.70	0.990051
$269$	−6938.53	−1.57267	−0.786337	−	0.617797i	$-0.788025\pi$
$269$	−6938.53	−1.57267	−0.786337	+	0.617797i	$0.788025\pi$
$270$	7164.29	1.61483
$271$	589.833	0.132213	0.0661066	−	0.997813i	$-0.478942\pi$
$271$	589.833	0.132213	0.0661066	+	0.997813i	$0.478942\pi$
$272$	5609.88	1.25055
$273$	−3992.53	−0.885123
$274$	5867.41	1.29366
$275$	−5206.25	−1.14163
$276$	−1470.19	−0.320635
$277$	5117.30	1.11000	0.554998	−	0.831852i	$-0.312719\pi$
$277$	5117.30	1.11000	0.554998	+	0.831852i	$0.312719\pi$
$278$	−3059.42	−0.660042
$279$	2061.72	0.442409
$280$	3380.02	0.721410
$281$	−157.427	−0.0334210	−0.0167105	−	0.999860i	$-0.505319\pi$
$281$	−157.427	−0.0334210	−0.0167105	+	0.999860i	$0.505319\pi$
$282$	−4912.94	−1.03745
$283$	3584.31	0.752879	0.376440	−	0.926441i	$-0.377148\pi$
$283$	3584.31	0.752879	0.376440	+	0.926441i	$0.377148\pi$
$284$	−2201.13	−0.459905
$285$	−3254.55	−0.676432
$286$	−13775.1	−2.84804
$287$	−436.644	−0.0898058
$288$	−3194.27	−0.653555
$289$	29.2718	0.00595802
$290$	8665.56	1.75469
$291$	2100.72	0.423183
$292$	2058.20	0.412489
$293$	−6808.92	−1.35761	−0.678807	−	0.734316i	$-0.737503\pi$
$293$	−6808.92	−1.35761	−0.678807	+	0.734316i	$0.737503\pi$
$294$	−141.194	−0.0280089
$295$	−4842.44	−0.955722
$296$	−464.289	−0.0911698
$297$	−7437.70	−1.45313
$298$	−4980.03	−0.968071
$299$	−7736.94	−1.49645
$300$	−1304.35	−0.251021
$301$	0	0
$302$	−5779.64	−1.10126
$303$	−3036.83	−0.575781
$304$	5706.37	1.07659
$305$	8103.03	1.52124
$306$	−4369.18	−0.816239
$307$	7195.92	1.33776	0.668880	−	0.743370i	$-0.266774\pi$
$307$	7195.92	1.33776	0.668880	+	0.743370i	$0.266774\pi$
$308$	4398.35	0.813699
$309$	5125.86	0.943689
$310$	−6135.31	−1.12407
$311$	−5076.42	−0.925586	−0.462793	−	0.886466i	$-0.653153\pi$
$311$	−5076.42	−0.925586	−0.462793	+	0.886466i	$0.653153\pi$
$312$	2753.32	0.499603
$313$	5157.09	0.931297	0.465649	−	0.884970i	$-0.345821\pi$
$313$	5157.09	0.931297	0.465649	+	0.884970i	$0.345821\pi$
$314$	−4391.25	−0.789212
$315$	−4752.77	−0.850122
$316$	227.639	0.0405243
$317$	−6858.21	−1.21513	−0.607564	−	0.794271i	$-0.707853\pi$
$317$	−6858.21	−1.21513	−0.607564	+	0.794271i	$0.707853\pi$
$318$	−7161.15	−1.26282
$319$	−8996.26	−1.57898
$320$	22.2969	0.00389511
$321$	−1566.81	−0.272433
$322$	6911.64	1.19618
$323$	5027.29	0.866024
$324$	252.843	0.0433544
$325$	−6864.15	−1.17155
$326$	6196.11	1.05267
$327$	343.477	0.0580866
$328$	301.118	0.0506904
$329$	8255.26	1.38337
$330$	8738.34	1.45767
$331$	−7748.36	−1.28667	−0.643336	−	0.765584i	$-0.722450\pi$
$331$	−7748.36	−1.28667	−0.643336	+	0.765584i	$0.722450\pi$
$332$	1740.09	0.287649
$333$	652.855	0.107436
$334$	−635.074	−0.104041
$335$	−14500.7	−2.36495
$336$	−4440.69	−0.721011
$337$	−3882.30	−0.627544	−0.313772	−	0.949498i	$-0.601593\pi$
$337$	−3882.30	−0.627544	−0.313772	+	0.949498i	$0.601593\pi$
$338$	−10409.7	−1.67519
$339$	4263.71	0.683107
$340$	4647.17	0.741259
$341$	6369.45	1.01151
$342$	−4444.33	−0.702696
$343$	6467.57	1.01812
$344$	0	0
$345$	4907.99	0.765905
$346$	13988.9	2.17355
$347$	−109.552	−0.0169484	−0.00847418	−	0.999964i	$-0.502697\pi$
$347$	−109.552	−0.0169484	−0.00847418	+	0.999964i	$0.502697\pi$
$348$	−2253.87	−0.347185
$349$	6182.65	0.948280	0.474140	−	0.880450i	$-0.342759\pi$
$349$	6182.65	0.948280	0.474140	+	0.880450i	$0.342759\pi$
$350$	6131.96	0.936477
$351$	−9806.20	−1.49121
$352$	−9868.30	−1.49427
$353$	−8396.83	−1.26606	−0.633029	−	0.774128i	$-0.718188\pi$
$353$	−8396.83	−1.26606	−0.633029	+	0.774128i	$0.718188\pi$
$354$	3523.83	0.529066
$355$	7348.09	1.09858
$356$	4877.26	0.726107
$357$	−3912.23	−0.579992
$358$	11251.1	1.66101
$359$	−9437.09	−1.38738	−0.693692	−	0.720272i	$-0.744017\pi$
$359$	−9437.09	−1.38738	−0.693692	+	0.720272i	$0.744017\pi$
$360$	3277.60	0.479847
$361$	−1745.23	−0.254444
$362$	188.396	0.0273532
$363$	−4994.05	−0.722093
$364$	5798.98	0.835025
$365$	−6870.93	−0.985317
$366$	−5896.54	−0.842124
$367$	9389.93	1.33556	0.667780	−	0.744359i	$-0.267245\pi$
$367$	9389.93	1.33556	0.667780	+	0.744359i	$0.267245\pi$
$368$	−8605.42	−1.21899
$369$	−423.413	−0.0597345
$370$	−1942.77	−0.272973
$371$	12033.0	1.68388
$372$	1595.77	0.222410
$373$	443.328	0.0615406	0.0307703	−	0.999526i	$-0.490204\pi$
$373$	443.328	0.0615406	0.0307703	+	0.999526i	$0.490204\pi$
$374$	−13498.0	−1.86622
$375$	−1334.61	−0.183784
$376$	−5692.99	−0.780833
$377$	−11861.1	−1.62036
$378$	8760.18	1.19200
$379$	−4688.25	−0.635407	−0.317704	−	0.948190i	$-0.602912\pi$
$379$	−4688.25	−0.635407	−0.317704	+	0.948190i	$0.602912\pi$
$380$	4727.11	0.638146
$381$	−6156.71	−0.827869
$382$	−783.735	−0.104972
$383$	3374.31	0.450180	0.225090	−	0.974338i	$-0.427732\pi$
$383$	3374.31	0.450180	0.225090	+	0.974338i	$0.427732\pi$
$384$	4428.57	0.588527
$385$	−14683.1	−1.94369
$386$	−17906.6	−2.36120
$387$	0	0
$388$	−3051.20	−0.399230
$389$	13756.1	1.79297	0.896483	−	0.443078i	$-0.146114\pi$
$389$	13756.1	1.79297	0.896483	+	0.443078i	$0.146114\pi$
$390$	11521.0	1.49587
$391$	−7581.33	−0.980574
$392$	−163.612	−0.0210808
$393$	−3033.79	−0.389401
$394$	10342.2	1.32242
$395$	−759.932	−0.0968008
$396$	4265.08	0.541233
$397$	6440.24	0.814173	0.407086	−	0.913390i	$-0.366545\pi$
$397$	6440.24	0.814173	0.407086	+	0.913390i	$0.366545\pi$
$398$	7087.70	0.892650
$399$	−3979.52	−0.499312
$400$	−7634.67	−0.954333
$401$	4498.42	0.560200	0.280100	−	0.959971i	$-0.409632\pi$
$401$	4498.42	0.560200	0.280100	+	0.959971i	$0.409632\pi$
$402$	10552.1	1.30918
$403$	8397.76	1.03802
$404$	4410.88	0.543191
$405$	−844.073	−0.103561
$406$	10595.9	1.29523
$407$	2016.92	0.245638
$408$	2697.95	0.327373
$409$	749.470	0.0906086	0.0453043	−	0.998973i	$-0.485574\pi$
$409$	749.470	0.0906086	0.0453043	+	0.998973i	$0.485574\pi$
$410$	1260.00	0.151773
$411$	5094.58	0.611428
$412$	−7445.10	−0.890276
$413$	−5921.12	−0.705471
$414$	6702.22	0.795642
$415$	−5808.98	−0.687112
$416$	−13010.8	−1.53343
$417$	−2656.44	−0.311958
$418$	−13730.2	−1.60662
$419$	9940.58	1.15902	0.579510	−	0.814965i	$-0.303244\pi$
$419$	9940.58	1.15902	0.579510	+	0.814965i	$0.303244\pi$
$420$	−3678.63	−0.427378
$421$	−3626.40	−0.419810	−0.209905	−	0.977722i	$-0.567315\pi$
$421$	−3626.40	−0.419810	−0.209905	+	0.977722i	$0.567315\pi$
$422$	2907.18	0.335354
$423$	8005.12	0.920148
$424$	−8298.16	−0.950458
$425$	−6726.10	−0.767680
$426$	−5347.17	−0.608149
$427$	9908.03	1.12291
$428$	2275.73	0.257013
$429$	−11960.7	−1.34608
$430$	0	0
$431$	10247.8	1.14529	0.572644	−	0.819804i	$-0.305918\pi$
$431$	10247.8	1.14529	0.572644	+	0.819804i	$0.305918\pi$
$432$	−10907.0	−1.21473
$433$	−474.905	−0.0527078	−0.0263539	−	0.999653i	$-0.508390\pi$
$433$	−474.905	−0.0527078	−0.0263539	+	0.999653i	$0.508390\pi$
$434$	−7501.98	−0.829739
$435$	7524.16	0.829324
$436$	−498.886	−0.0547989
$437$	−7711.74	−0.844171
$438$	4999.95	0.545449
$439$	−5591.28	−0.607875	−0.303938	−	0.952692i	$-0.598301\pi$
$439$	−5591.28	−0.607875	−0.303938	+	0.952692i	$0.598301\pi$
$440$	10125.8	1.09711
$441$	230.062	0.0248420
$442$	−17796.4	−1.91513
$443$	8177.66	0.877048	0.438524	−	0.898719i	$-0.355501\pi$
$443$	8177.66	0.877048	0.438524	+	0.898719i	$0.355501\pi$
$444$	505.307	0.0540108
$445$	−16281.9	−1.73446
$446$	−7545.17	−0.801063
$447$	−4324.08	−0.457543
$448$	27.2637	0.00287520
$449$	−15836.2	−1.66450	−0.832248	−	0.554404i	$-0.812946\pi$
$449$	−15836.2	−1.66450	−0.832248	+	0.554404i	$0.812946\pi$
$450$	5946.16	0.622899
$451$	−1308.08	−0.136575
$452$	−6192.87	−0.644442
$453$	−5018.37	−0.520494
$454$	1581.22	0.163459
$455$	−19358.9	−1.99463
$456$	2744.36	0.281834
$457$	−10990.1	−1.12493	−0.562465	−	0.826821i	$-0.690147\pi$
$457$	−10990.1	−1.12493	−0.562465	+	0.826821i	$0.690147\pi$
$458$	20169.1	2.05773
$459$	−9608.98	−0.977143
$460$	−7128.65	−0.722554
$461$	−6790.50	−0.686041	−0.343021	−	0.939328i	$-0.611450\pi$
$461$	−6790.50	−0.686041	−0.343021	+	0.939328i	$0.611450\pi$
$462$	10684.8	1.07598
$463$	−9388.34	−0.942361	−0.471180	−	0.882037i	$-0.656172\pi$
$463$	−9388.34	−0.942361	−0.471180	+	0.882037i	$0.656172\pi$
$464$	−13192.5	−1.31993
$465$	−5327.19	−0.531274
$466$	−21295.3	−2.11692
$467$	14261.3	1.41314	0.706569	−	0.707644i	$-0.250242\pi$
$467$	14261.3	1.41314	0.706569	+	0.707644i	$0.250242\pi$
$468$	5623.27	0.555418
$469$	−17730.8	−1.74570
$470$	−23821.8	−2.33791
$471$	−3812.85	−0.373008
$472$	4083.32	0.398199
$473$	0	0
$474$	552.999	0.0535867
$475$	−6841.80	−0.660891
$476$	5682.35	0.547164
$477$	11668.4	1.12004
$478$	−9395.19	−0.899008
$479$	−3859.13	−0.368117	−0.184058	−	0.982915i	$-0.558924\pi$
$479$	−3859.13	−0.368117	−0.184058	+	0.982915i	$0.558924\pi$
$480$	8253.51	0.784832
$481$	2659.19	0.252076
$482$	−288.613	−0.0272738
$483$	6001.27	0.565356
$484$	7253.65	0.681222
$485$	10185.9	0.953646
$486$	13635.7	1.27269
$487$	−7036.51	−0.654732	−0.327366	−	0.944898i	$-0.606161\pi$
$487$	−7036.51	−0.654732	−0.327366	+	0.944898i	$0.606161\pi$
$488$	−6832.77	−0.633821
$489$	5379.98	0.497528
$490$	−684.620	−0.0631184
$491$	−15125.5	−1.39023	−0.695116	−	0.718898i	$-0.744647\pi$
$491$	−15125.5	−1.39023	−0.695116	+	0.718898i	$0.744647\pi$
$492$	−327.720	−0.0300300
$493$	−11622.5	−1.06177
$494$	−18102.6	−1.64873
$495$	−14238.2	−1.29285
$496$	9340.43	0.845559
$497$	8984.91	0.810922
$498$	4227.17	0.380369
$499$	−5619.20	−0.504108	−0.252054	−	0.967713i	$-0.581106\pi$
$499$	−5619.20	−0.504108	−0.252054	+	0.967713i	$0.581106\pi$
$500$	1938.46	0.173381
$501$	−551.424	−0.0491733
$502$	12839.6	1.14155
$503$	−1050.56	−0.0931258	−0.0465629	−	0.998915i	$-0.514827\pi$
$503$	−1050.56	−0.0931258	−0.0465629	+	0.998915i	$0.514827\pi$
$504$	4007.71	0.354201
$505$	−14724.9	−1.29753
$506$	20705.7	1.81913
$507$	−9038.59	−0.791751
$508$	8942.37	0.781011
$509$	2928.22	0.254993	0.127496	−	0.991839i	$-0.459306\pi$
$509$	2928.22	0.254993	0.127496	+	0.991839i	$0.459306\pi$
$510$	11289.3	0.980193
$511$	−8401.47	−0.727317
$512$	−6474.68	−0.558873
$513$	−9774.27	−0.841217
$514$	10452.3	0.896951
$515$	24854.2	2.12661
$516$	0	0
$517$	24730.9	2.10380
$518$	−2375.54	−0.201496
$519$	12146.3	1.02729
$520$	13350.3	1.12586
$521$	−14601.6	−1.22785	−0.613925	−	0.789364i	$-0.710410\pi$
$521$	−14601.6	−1.22785	−0.613925	+	0.789364i	$0.710410\pi$
$522$	10274.8	0.861524
$523$	−6271.99	−0.524388	−0.262194	−	0.965015i	$-0.584446\pi$
$523$	−6271.99	−0.524388	−0.262194	+	0.965015i	$0.584446\pi$
$524$	4406.46	0.367361
$525$	5324.28	0.442611
$526$	401.313	0.0332663
$527$	8228.87	0.680180
$528$	−13303.3	−1.09650
$529$	−537.410	−0.0441695
$530$	−34722.9	−2.84578
$531$	−5741.71	−0.469245
$532$	5780.10	0.471051
$533$	−1724.64	−0.140154
$534$	11848.3	0.960157
$535$	−7597.14	−0.613931
$536$	12227.5	0.985350
$537$	9769.17	0.785048
$538$	24482.2	1.96190
$539$	710.748	0.0567979
$540$	−9035.22	−0.720026
$541$	3413.67	0.271285	0.135643	−	0.990758i	$-0.456690\pi$
$541$	3413.67	0.271285	0.135643	+	0.990758i	$0.456690\pi$
$542$	−2081.19	−0.164935
$543$	163.581	0.0129281
$544$	−12749.1	−1.00481
$545$	1665.45	0.130899
$546$	14087.4	1.10418
$547$	−258.591	−0.0202131	−0.0101065	−	0.999949i	$-0.503217\pi$
$547$	−258.591	−0.0202131	−0.0101065	+	0.999949i	$0.503217\pi$
$548$	−7399.66	−0.576821
$549$	9607.81	0.746906
$550$	18369.9	1.42418
$551$	−11822.4	−0.914071
$552$	−4138.59	−0.319112
$553$	−929.211	−0.0714540
$554$	−18056.1	−1.38471
$555$	−1686.88	−0.129016
$556$	3858.37	0.294301
$557$	8465.22	0.643955	0.321978	−	0.946747i	$-0.395652\pi$
$557$	8465.22	0.643955	0.321978	+	0.946747i	$0.395652\pi$
$558$	−7274.67	−0.551902
$559$	0	0
$560$	−21532.0	−1.62481
$561$	−11720.1	−0.882040
$562$	555.471	0.0416924
$563$	−622.915	−0.0466301	−0.0233150	−	0.999728i	$-0.507422\pi$
$563$	−622.915	−0.0466301	−0.0233150	+	0.999728i	$0.507422\pi$
$564$	6195.93	0.462581
$565$	20673.8	1.53939
$566$	−12647.0	−0.939210
$567$	−1032.09	−0.0764443
$568$	−6196.17	−0.457721
$569$	6933.34	0.510827	0.255413	−	0.966832i	$-0.417788\pi$
$569$	6933.34	0.510827	0.255413	+	0.966832i	$0.417788\pi$
$570$	11483.5	0.843843
$571$	18877.4	1.38353	0.691764	−	0.722123i	$-0.256834\pi$
$571$	18877.4	1.38353	0.691764	+	0.722123i	$0.256834\pi$
$572$	17372.4	1.26989
$573$	−680.505	−0.0496134
$574$	1540.67	0.112032
$575$	10317.7	0.748308
$576$	26.4376	0.00191244
$577$	11798.0	0.851227	0.425613	−	0.904905i	$-0.360058\pi$
$577$	11798.0	0.851227	0.425613	+	0.904905i	$0.360058\pi$
$578$	−103.284	−0.00743258
$579$	−15548.0	−1.11598
$580$	−10928.5	−0.782384
$581$	−7102.96	−0.507195
$582$	−7412.25	−0.527917
$583$	36048.0	2.56081
$584$	5793.81	0.410530
$585$	−18772.3	−1.32673
$586$	24024.8	1.69361
$587$	663.788	0.0466737	0.0233368	−	0.999728i	$-0.492571\pi$
$587$	663.788	0.0466737	0.0233368	+	0.999728i	$0.492571\pi$
$588$	178.067	0.0124887
$589$	8370.42	0.585564
$590$	17086.3	1.19225
$591$	8979.98	0.625021
$592$	2957.69	0.205338
$593$	−3823.87	−0.264802	−0.132401	−	0.991196i	$-0.542269\pi$
$593$	−3823.87	−0.264802	−0.132401	+	0.991196i	$0.542269\pi$
$594$	26243.5	1.81277
$595$	−18969.5	−1.30702
$596$	6280.54	0.431646
$597$	6154.14	0.421896
$598$	27299.3	1.86681
$599$	−4835.69	−0.329851	−0.164926	−	0.986306i	$-0.552738\pi$
$599$	−4835.69	−0.329851	−0.164926	+	0.986306i	$0.552738\pi$
$600$	−3671.73	−0.249829
$601$	−1776.64	−0.120584	−0.0602918	−	0.998181i	$-0.519203\pi$
$601$	−1776.64	−0.120584	−0.0602918	+	0.998181i	$0.519203\pi$
$602$	0	0
$603$	−17193.5	−1.16115
$604$	7288.98	0.491033
$605$	−24215.1	−1.62724
$606$	10715.3	0.718281
$607$	−17302.1	−1.15695	−0.578477	−	0.815699i	$-0.696353\pi$
$607$	−17302.1	−1.15695	−0.578477	+	0.815699i	$0.696353\pi$
$608$	−12968.4	−0.865032
$609$	9200.21	0.612170
$610$	−28591.1	−1.89773
$611$	32606.3	2.15893
$612$	5510.17	0.363947
$613$	−18526.0	−1.22065	−0.610323	−	0.792152i	$-0.708960\pi$
$613$	−18526.0	−1.22065	−0.610323	+	0.792152i	$0.708960\pi$
$614$	−25390.3	−1.66885
$615$	1094.04	0.0717331
$616$	12381.3	0.809835
$617$	29791.8	1.94387	0.971937	−	0.235240i	$-0.0755875\pi$
$617$	29791.8	1.94387	0.971937	+	0.235240i	$0.0755875\pi$
$618$	−18086.3	−1.17724
$619$	11505.2	0.747065	0.373533	−	0.927617i	$-0.378146\pi$
$619$	11505.2	0.747065	0.373533	+	0.927617i	$0.378146\pi$
$620$	7737.52	0.501204
$621$	14739.9	0.952486
$622$	17911.8	1.15466
$623$	−19908.8	−1.28030
$624$	−17539.7	−1.12524
$625$	−18430.6	−1.17956
$626$	−18196.5	−1.16179
$627$	−11921.7	−0.759344
$628$	5538.01	0.351896
$629$	2605.71	0.165177
$630$	16769.9	1.06052
$631$	30684.7	1.93588	0.967939	−	0.251184i	$-0.0808198\pi$
$631$	30684.7	1.93588	0.967939	+	0.251184i	$0.0808198\pi$
$632$	640.801	0.0403318
$633$	2524.26	0.158500
$634$	24198.8	1.51586
$635$	−29852.6	−1.86561
$636$	9031.26	0.563070
$637$	937.081	0.0582865
$638$	31742.8	1.96976
$639$	8712.67	0.539386
$640$	21473.2	1.32625
$641$	12353.5	0.761209	0.380605	−	0.924738i	$-0.375716\pi$
$641$	12353.5	0.761209	0.380605	+	0.924738i	$0.375716\pi$
$642$	5528.41	0.339858
$643$	8901.37	0.545934	0.272967	−	0.962023i	$-0.411995\pi$
$643$	8901.37	0.545934	0.272967	+	0.962023i	$0.411995\pi$
$644$	−8716.60	−0.533357
$645$	0	0
$646$	−17738.5	−1.08036
$647$	−15209.5	−0.924186	−0.462093	−	0.886831i	$-0.652901\pi$
$647$	−15209.5	−0.924186	−0.462093	+	0.886831i	$0.652901\pi$
$648$	711.752	0.0431486
$649$	−17738.3	−1.07287
$650$	24219.7	1.46150
$651$	−6513.85	−0.392163
$652$	−7814.21	−0.469368
$653$	11591.2	0.694638	0.347319	−	0.937747i	$-0.387092\pi$
$653$	11591.2	0.694638	0.347319	+	0.937747i	$0.387092\pi$
$654$	−1211.94	−0.0724626
$655$	−14710.2	−0.877519
$656$	−1918.23	−0.114168
$657$	−8146.90	−0.483776
$658$	−29128.2	−1.72574
$659$	15290.8	0.903863	0.451932	−	0.892053i	$-0.350735\pi$
$659$	15290.8	0.903863	0.451932	+	0.892053i	$0.350735\pi$
$660$	−11020.3	−0.649948
$661$	787.114	0.0463165	0.0231582	−	0.999732i	$-0.492628\pi$
$661$	787.114	0.0463165	0.0231582	+	0.999732i	$0.492628\pi$
$662$	27339.6	1.60511
$663$	−15452.3	−0.905158
$664$	4898.33	0.286283
$665$	−19295.8	−1.12520
$666$	−2303.56	−0.134026
$667$	17828.7	1.03498
$668$	800.921	0.0463901
$669$	−6551.35	−0.378610
$670$	51164.8	2.95025
$671$	29682.2	1.70770
$672$	10092.0	0.579328
$673$	10873.2	0.622783	0.311391	−	0.950282i	$-0.399205\pi$
$673$	10873.2	0.622783	0.311391	+	0.950282i	$0.399205\pi$
$674$	13698.4	0.782855
$675$	13077.2	0.745690
$676$	13128.2	0.746938
$677$	−29398.0	−1.66891	−0.834457	−	0.551073i	$-0.814219\pi$
$677$	−29398.0	−1.66891	−0.834457	+	0.551073i	$0.814219\pi$
$678$	−15044.2	−0.852170
$679$	12454.9	0.703939
$680$	13081.8	0.737739
$681$	1372.95	0.0772563
$682$	−22474.2	−1.26185
$683$	23535.9	1.31856	0.659279	−	0.751899i	$-0.270862\pi$
$683$	23535.9	1.31856	0.659279	+	0.751899i	$0.270862\pi$
$684$	5604.96	0.313320
$685$	24702.5	1.37786
$686$	−22820.4	−1.27010
$687$	17512.5	0.972554
$688$	0	0
$689$	47527.3	2.62793
$690$	−17317.5	−0.955460
$691$	−15755.6	−0.867397	−0.433698	−	0.901058i	$-0.642792\pi$
$691$	−15755.6	−0.867397	−0.433698	+	0.901058i	$0.642792\pi$
$692$	−17642.0	−0.969147
$693$	−17409.9	−0.954323
$694$	386.549	0.0211429
$695$	−12880.5	−0.703001
$696$	−6344.64	−0.345536
$697$	−1689.95	−0.0918385
$698$	−21815.1	−1.18297
$699$	−18490.4	−1.00053
$700$	−7733.30	−0.417559
$701$	−11016.5	−0.593564	−0.296782	−	0.954945i	$-0.595913\pi$
$701$	−11016.5	−0.593564	−0.296782	+	0.954945i	$0.595913\pi$
$702$	34600.6	1.86028
$703$	2650.53	0.142200
$704$	81.6758	0.00437254
$705$	−20684.0	−1.10497
$706$	29627.7	1.57940
$707$	−18005.0	−0.957776
$708$	−4444.06	−0.235901
$709$	26713.3	1.41500	0.707502	−	0.706711i	$-0.249822\pi$
$709$	26713.3	1.41500	0.707502	+	0.706711i	$0.249822\pi$
$710$	−25927.3	−1.37047
$711$	−901.055	−0.0475277
$712$	13729.5	0.722659
$713$	−12622.9	−0.663017
$714$	13804.1	0.723535
$715$	−57994.8	−3.03340
$716$	−14189.3	−0.740614
$717$	−8157.69	−0.424902
$718$	33298.2	1.73075
$719$	−28318.6	−1.46885	−0.734427	−	0.678688i	$-0.762549\pi$
$719$	−28318.6	−1.46885	−0.734427	+	0.678688i	$0.762549\pi$
$720$	−20879.5	−1.08074
$721$	30390.6	1.56977
$722$	6157.95	0.317417
$723$	−250.598	−0.0128905
$724$	−237.595	−0.0121963
$725$	15817.5	0.810271
$726$	17621.2	0.900805
$727$	2736.88	0.139622	0.0698111	−	0.997560i	$-0.477760\pi$
$727$	2736.88	0.139622	0.0698111	+	0.997560i	$0.477760\pi$
$728$	16324.1	0.831060
$729$	10305.5	0.523574
$730$	24243.6	1.22917
$731$	0	0
$732$	7436.41	0.375488
$733$	26362.0	1.32838	0.664189	−	0.747565i	$-0.268777\pi$
$733$	26362.0	1.32838	0.664189	+	0.747565i	$0.268777\pi$
$734$	−33131.8	−1.66610
$735$	−594.445	−0.0298319
$736$	19556.9	0.979451
$737$	−53117.4	−2.65482
$738$	1493.99	0.0745182
$739$	4576.23	0.227793	0.113897	−	0.993493i	$-0.463667\pi$
$739$	4576.23	0.227793	0.113897	+	0.993493i	$0.463667\pi$
$740$	2450.12	0.121714
$741$	−15718.2	−0.779245
$742$	−42457.6	−2.10063
$743$	11690.3	0.577219	0.288610	−	0.957447i	$-0.406807\pi$
$743$	11690.3	0.577219	0.288610	+	0.957447i	$0.406807\pi$
$744$	4492.07	0.221354
$745$	−20966.5	−1.03108
$746$	−1564.25	−0.0767713
$747$	−6887.73	−0.337361
$748$	17023.0	0.832116
$749$	−9289.45	−0.453176
$750$	4709.08	0.229269
$751$	715.623	0.0347716	0.0173858	−	0.999849i	$-0.494466\pi$
$751$	715.623	0.0347716	0.0173858	+	0.999849i	$0.494466\pi$
$752$	36266.4	1.75864
$753$	11148.4	0.539536
$754$	41851.1	2.02139
$755$	−24333.0	−1.17294
$756$	−11047.9	−0.531491
$757$	32317.1	1.55163	0.775815	−	0.630960i	$-0.217339\pi$
$757$	32317.1	1.55163	0.775815	+	0.630960i	$0.217339\pi$
$758$	16542.2	0.792665
$759$	17978.4	0.859783
$760$	13306.8	0.635116
$761$	11740.3	0.559247	0.279623	−	0.960110i	$-0.409790\pi$
$761$	11740.3	0.559247	0.279623	+	0.960110i	$0.409790\pi$
$762$	21723.6	1.03276
$763$	2036.43	0.0966236
$764$	988.405	0.0468053
$765$	−18394.8	−0.869364
$766$	−11906.0	−0.561596
$767$	−23387.0	−1.10098
$768$	−15662.7	−0.735911
$769$	−18488.7	−0.866995	−0.433498	−	0.901155i	$-0.642721\pi$
$769$	−18488.7	−0.866995	−0.433498	+	0.901155i	$0.642721\pi$
$770$	51808.5	2.42474
$771$	9075.59	0.423929
$772$	22582.9	1.05282
$773$	28213.0	1.31274	0.656372	−	0.754437i	$-0.272090\pi$
$773$	28213.0	1.31274	0.656372	+	0.754437i	$0.272090\pi$
$774$	0	0
$775$	−11198.9	−0.519068
$776$	−8589.12	−0.397334
$777$	−2062.64	−0.0952340
$778$	−48537.7	−2.23671
$779$	−1719.02	−0.0790633
$780$	−14529.7	−0.666982
$781$	26916.7	1.23323
$782$	26750.3	1.22326
$783$	22597.0	1.03136
$784$	1042.27	0.0474795
$785$	−18487.7	−0.840578
$786$	10704.5	0.485774
$787$	−30281.3	−1.37155	−0.685776	−	0.727812i	$-0.740537\pi$
$787$	−30281.3	−1.37155	−0.685776	+	0.727812i	$0.740537\pi$
$788$	−13043.1	−0.589644
$789$	348.454	0.0157228
$790$	2681.37	0.120758
$791$	25279.0	1.13631
$792$	12006.2	0.538663
$793$	39134.3	1.75246
$794$	−22724.0	−1.01567
$795$	−30149.3	−1.34501
$796$	−8938.63	−0.398017
$797$	15491.1	0.688484	0.344242	−	0.938881i	$-0.388136\pi$
$797$	15491.1	0.688484	0.344242	+	0.938881i	$0.388136\pi$
$798$	14041.5	0.622887
$799$	31950.5	1.41468
$800$	17350.7	0.766801
$801$	−19305.5	−0.851594
$802$	−15872.4	−0.698845
$803$	−25168.9	−1.10609
$804$	−13307.7	−0.583741
$805$	29098.8	1.27404
$806$	−29631.0	−1.29492
$807$	21257.5	0.927259
$808$	12416.6	0.540612
$809$	−16916.7	−0.735179	−0.367590	−	0.929988i	$-0.619817\pi$
$809$	−16916.7	−0.735179	−0.367590	+	0.929988i	$0.619817\pi$
$810$	2978.26	0.129192
$811$	42803.7	1.85332	0.926659	−	0.375903i	$-0.122667\pi$
$811$	42803.7	1.85332	0.926659	+	0.375903i	$0.122667\pi$
$812$	−13362.9	−0.577521
$813$	−1807.06	−0.0779538
$814$	−7116.56	−0.306432
$815$	26086.4	1.12118
$816$	−17186.9	−0.737331
$817$	0	0
$818$	−2644.46	−0.113033
$819$	−22953.9	−0.979335
$820$	−1589.04	−0.0676729
$821$	14953.9	0.635683	0.317842	−	0.948144i	$-0.397042\pi$
$821$	14953.9	0.635683	0.317842	+	0.948144i	$0.397042\pi$
$822$	−17975.9	−0.762751
$823$	−6802.39	−0.288112	−0.144056	−	0.989570i	$-0.546015\pi$
$823$	−6802.39	−0.288112	−0.144056	+	0.989570i	$0.546015\pi$
$824$	−20957.9	−0.886049
$825$	15950.3	0.673114
$826$	20892.3	0.880069
$827$	−43.8240	−0.00184270	−0.000921348	−	1.00000i	$-0.500293\pi$
$827$	−43.8240	−0.00184270	−0.000921348		1.00000i	$0.500293\pi$
$828$	−8452.48	−0.354763
$829$	16106.3	0.674781	0.337391	−	0.941365i	$-0.390456\pi$
$829$	16106.3	0.674781	0.337391	+	0.941365i	$0.390456\pi$
$830$	20496.6	0.857166
$831$	−15677.8	−0.654461
$832$	107.685	0.00448715
$833$	918.235	0.0381932
$834$	9373.10	0.389165
$835$	−2673.74	−0.110812
$836$	17315.8	0.716365
$837$	−15998.9	−0.660697
$838$	−35074.7	−1.44587
$839$	−4269.60	−0.175689	−0.0878444	−	0.996134i	$-0.527998\pi$
$839$	−4269.60	−0.175689	−0.0878444	+	0.996134i	$0.527998\pi$
$840$	−10355.3	−0.425348
$841$	2943.17	0.120676
$842$	12795.5	0.523709
$843$	482.306	0.0197052
$844$	−3666.38	−0.149529
$845$	−43826.1	−1.78422
$846$	−28245.6	−1.14788
$847$	−29609.1	−1.20116
$848$	52862.3	2.14068
$849$	−10981.2	−0.443903
$850$	23732.6	0.957674
$851$	−3997.10	−0.161009
$852$	6743.57	0.271163
$853$	35651.3	1.43104	0.715521	−	0.698591i	$-0.246189\pi$
$853$	35651.3	1.43104	0.715521	+	0.698591i	$0.246189\pi$
$854$	−34959.9	−1.40082
$855$	−18711.2	−0.748431
$856$	6406.18	0.255793
$857$	11015.6	0.439072	0.219536	−	0.975604i	$-0.429546\pi$
$857$	11015.6	0.439072	0.219536	+	0.975604i	$0.429546\pi$
$858$	42202.6	1.67922
$859$	−10268.1	−0.407851	−0.203925	−	0.978986i	$-0.565370\pi$
$859$	−10268.1	−0.407851	−0.203925	+	0.978986i	$0.565370\pi$
$860$	0	0
$861$	1337.74	0.0529501
$862$	−36158.7	−1.42874
$863$	3943.51	0.155549	0.0777743	−	0.996971i	$-0.475219\pi$
$863$	3943.51	0.155549	0.0777743	+	0.996971i	$0.475219\pi$
$864$	24787.4	0.976024
$865$	58894.9	2.31501
$866$	1675.67	0.0657526
$867$	−89.6795	−0.00351289
$868$	9461.10	0.369966
$869$	−2783.70	−0.108666
$870$	−26548.6	−1.03457
$871$	−70032.4	−2.72440
$872$	−1404.36	−0.0545387
$873$	12077.5	0.468226
$874$	27210.4	1.05310
$875$	−7912.72	−0.305713
$876$	−6305.66	−0.243206
$877$	48677.1	1.87424	0.937120	−	0.349007i	$-0.113481\pi$
$877$	48677.1	1.87424	0.937120	+	0.349007i	$0.113481\pi$
$878$	19728.5	0.758319
$879$	20860.4	0.800459
$880$	−64504.8	−2.47097
$881$	4875.25	0.186437	0.0932186	−	0.995646i	$-0.470284\pi$
$881$	4875.25	0.186437	0.0932186	+	0.995646i	$0.470284\pi$
$882$	−811.758	−0.0309902
$883$	−13952.6	−0.531759	−0.265880	−	0.964006i	$-0.585662\pi$
$883$	−13952.6	−0.531759	−0.265880	+	0.964006i	$0.585662\pi$
$884$	22443.9	0.853925
$885$	14835.7	0.563500
$886$	−28854.4	−1.09411
$887$	10494.9	0.397275	0.198638	−	0.980073i	$-0.436348\pi$
$887$	10494.9	0.397275	0.198638	+	0.980073i	$0.436348\pi$
$888$	1422.44	0.0537543
$889$	−36502.4	−1.37711
$890$	57449.6	2.16372
$891$	−3091.92	−0.116255
$892$	9515.56	0.357180
$893$	32500.1	1.21789
$894$	15257.2	0.570781
$895$	47368.6	1.76911
$896$	26256.4	0.978979
$897$	23703.5	0.882317
$898$	55877.2	2.07644
$899$	−19351.5	−0.717917
$900$	−7498.98	−0.277740
$901$	46571.4	1.72200
$902$	4615.50	0.170376
$903$	0	0
$904$	−17432.9	−0.641382
$905$	793.169	0.0291335
$906$	17707.0	0.649311
$907$	−1846.24	−0.0675892	−0.0337946	−	0.999429i	$-0.510759\pi$
$907$	−1846.24	−0.0675892	−0.0337946	+	0.999429i	$0.510759\pi$
$908$	−1994.15	−0.0728836
$909$	−17459.4	−0.637066
$910$	68306.6	2.48829
$911$	26662.6	0.969671	0.484835	−	0.874605i	$-0.338879\pi$
$911$	26662.6	0.969671	0.484835	+	0.874605i	$0.338879\pi$
$912$	−17482.5	−0.634764
$913$	−21278.8	−0.771332
$914$	38777.7	1.40334
$915$	−24825.2	−0.896934
$916$	−25436.2	−0.917507
$917$	−17987.0	−0.647745
$918$	33904.7	1.21898
$919$	26028.5	0.934279	0.467139	−	0.884184i	$-0.345285\pi$
$919$	26028.5	0.934279	0.467139	+	0.884184i	$0.345285\pi$
$920$	−20067.1	−0.719123
$921$	−22046.0	−0.788753
$922$	23959.9	0.855831
$923$	35488.2	1.26556
$924$	−13475.2	−0.479762
$925$	−3546.20	−0.126052
$926$	33126.2	1.17559
$927$	29469.7	1.04414
$928$	29981.6	1.06055
$929$	−34867.3	−1.23139	−0.615693	−	0.787986i	$-0.711124\pi$
$929$	−34867.3	−1.23139	−0.615693	+	0.787986i	$0.711124\pi$
$930$	18796.7	0.662760
$931$	934.030	0.0328803
$932$	26856.5	0.943898
$933$	15552.6	0.545732
$934$	−50320.2	−1.76288
$935$	−56828.4	−1.98769
$936$	15829.5	0.552780
$937$	13748.9	0.479357	0.239678	−	0.970852i	$-0.422958\pi$
$937$	13748.9	0.479357	0.239678	+	0.970852i	$0.422958\pi$
$938$	62562.0	2.17774
$939$	−15799.7	−0.549099
$940$	30042.7	1.04243
$941$	−42210.4	−1.46230	−0.731148	−	0.682219i	$-0.761015\pi$
$941$	−42210.4	−1.46230	−0.731148	+	0.682219i	$0.761015\pi$
$942$	13453.4	0.465324
$943$	2592.35	0.0895211
$944$	−26012.2	−0.896849
$945$	36881.4	1.26958
$946$	0	0
$947$	−12216.7	−0.419208	−0.209604	−	0.977786i	$-0.567217\pi$
$947$	−12216.7	−0.419208	−0.209604	+	0.977786i	$0.567217\pi$
$948$	−697.413	−0.0238934
$949$	−33183.7	−1.13508
$950$	24140.9	0.824456
$951$	21011.4	0.716447
$952$	15995.8	0.544566
$953$	−37883.5	−1.28769	−0.643845	−	0.765156i	$-0.722662\pi$
$953$	−37883.5	−1.28769	−0.643845	+	0.765156i	$0.722662\pi$
$954$	−41171.1	−1.39724
$955$	−3299.62	−0.111804
$956$	11848.7	0.400852
$957$	27561.7	0.930976
$958$	13616.7	0.459223
$959$	30205.1	1.01707
$960$	−68.3108	−0.00229659
$961$	−16090.0	−0.540095
$962$	−9382.79	−0.314463
$963$	−9007.97	−0.301431
$964$	363.983	0.0121609
$965$	−75389.1	−2.51488
$966$	−21175.1	−0.705277
$967$	9175.62	0.305138	0.152569	−	0.988293i	$-0.451245\pi$
$967$	9175.62	0.305138	0.152569	+	0.988293i	$0.451245\pi$
$968$	20419.0	0.677987
$969$	−15402.0	−0.510614
$970$	−35940.4	−1.18967
$971$	−27782.1	−0.918199	−0.459099	−	0.888385i	$-0.651828\pi$
$971$	−27782.1	−0.918199	−0.459099	+	0.888385i	$0.651828\pi$
$972$	−17196.6	−0.567471
$973$	−15749.7	−0.518924
$974$	24827.9	0.816773
$975$	21029.6	0.690756
$976$	43527.2	1.42753
$977$	8808.24	0.288435	0.144217	−	0.989546i	$-0.453934\pi$
$977$	8808.24	0.288435	0.144217	+	0.989546i	$0.453934\pi$
$978$	−18982.9	−0.620662
$979$	−59642.0	−1.94706
$980$	863.407	0.0281434
$981$	1974.73	0.0642693
$982$	53369.4	1.73430
$983$	−45190.1	−1.46627	−0.733134	−	0.680085i	$-0.761943\pi$
$983$	−45190.1	−1.46627	−0.733134	+	0.680085i	$0.761943\pi$
$984$	−922.531	−0.0298874
$985$	43542.0	1.40849
$986$	41009.4	1.32455
$987$	−25291.5	−0.815642
$988$	22830.0	0.735140
$989$	0	0
$990$	50238.7	1.61282
$991$	−29048.5	−0.931135	−0.465567	−	0.885012i	$-0.654150\pi$
$991$	−29048.5	−0.931135	−0.465567	+	0.885012i	$0.654150\pi$
$992$	−21227.3	−0.679401
$993$	23738.5	0.758630
$994$	−31702.7	−1.01162
$995$	29840.1	0.950748
$996$	−5331.08	−0.169600
$997$	20346.6	0.646322	0.323161	−	0.946344i	$-0.395255\pi$
$997$	20346.6	0.646322	0.323161	+	0.946344i	$0.395255\pi$
$998$	19827.0	0.628870
$999$	−5066.13	−0.160446

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.i.1.12	✓	50
43.42	odd	2		1849.4.a.j.1.39	yes	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.12	✓	50	1.1	even	1	trivial
1849.4.a.j.1.39	yes	50	43.42	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:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q-3.52844 q^{2} -3.06369 q^{3} +4.44988 q^{4} -14.8551 q^{5} +10.8100 q^{6} -18.1642 q^{7} +12.5264 q^{8} -17.6138 q^{9} +O(q^{10})\)	\(q-3.52844 q^{2} -3.06369 q^{3} +4.44988 q^{4} -14.8551 q^{5} +10.8100 q^{6} -18.1642 q^{7} +12.5264 q^{8} -17.6138 q^{9} +52.4155 q^{10} -54.4158 q^{11} -13.6330 q^{12} -71.7442 q^{13} +64.0913 q^{14} +45.5115 q^{15} -79.7976 q^{16} -70.3013 q^{17} +62.1493 q^{18} -71.5106 q^{19} -66.1036 q^{20} +55.6494 q^{21} +192.003 q^{22} +107.841 q^{23} -38.3769 q^{24} +95.6754 q^{25} +253.145 q^{26} +136.683 q^{27} -80.8285 q^{28} +165.324 q^{29} -160.585 q^{30} -117.051 q^{31} +181.350 q^{32} +166.713 q^{33} +248.054 q^{34} +269.832 q^{35} -78.3794 q^{36} -37.0649 q^{37} +252.321 q^{38} +219.802 q^{39} -186.081 q^{40} +24.0387 q^{41} -196.356 q^{42} -242.144 q^{44} +261.656 q^{45} -380.509 q^{46} -454.480 q^{47} +244.475 q^{48} -13.0614 q^{49} -337.585 q^{50} +215.381 q^{51} -319.253 q^{52} -662.454 q^{53} -482.277 q^{54} +808.355 q^{55} -227.532 q^{56} +219.086 q^{57} -583.337 q^{58} +325.977 q^{59} +202.521 q^{60} -545.470 q^{61} +413.009 q^{62} +319.941 q^{63} -1.50096 q^{64} +1065.77 q^{65} -588.236 q^{66} +976.139 q^{67} -312.832 q^{68} -330.390 q^{69} -952.086 q^{70} -494.649 q^{71} -220.638 q^{72} +462.528 q^{73} +130.781 q^{74} -293.119 q^{75} -318.213 q^{76} +988.420 q^{77} -775.557 q^{78} +51.1561 q^{79} +1185.41 q^{80} +56.8202 q^{81} -84.8190 q^{82} +391.041 q^{83} +247.633 q^{84} +1044.34 q^{85} -506.502 q^{87} -681.633 q^{88} +1096.04 q^{89} -923.237 q^{90} +1303.18 q^{91} +479.877 q^{92} +358.609 q^{93} +1603.60 q^{94} +1062.30 q^{95} -555.599 q^{96} -685.682 q^{97} +46.0864 q^{98} +958.470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9}+O(q^{10}) \) 50 * q - 10 * q^2 - 2 * q^3 + 186 * q^4 + 8 * q^5 - 51 * q^6 - 6 * q^7 - 138 * q^8 + 360 * q^9	\( 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9} - 137 q^{10} - 252 q^{11} - 48 q^{12} - 192 q^{13} - 272 q^{14} - 314 q^{15} + 542 q^{16} - 236 q^{17} - 386 q^{18} + 12 q^{19} + 108 q^{20} - 408 q^{21} + 1235 q^{22} - 630 q^{23} - 613 q^{24} + 1098 q^{25} + 1493 q^{26} + 10 q^{27} - 242 q^{28} + 208 q^{29} - 48 q^{30} - 932 q^{31} - 1124 q^{32} + 254 q^{33} + 765 q^{34} - 1452 q^{35} + 747 q^{36} - 90 q^{37} - 1213 q^{38} - 1610 q^{39} - 1693 q^{40} - 1354 q^{41} - 16 q^{42} - 2704 q^{44} + 4508 q^{45} + 233 q^{46} - 3484 q^{47} - 376 q^{48} + 1324 q^{49} - 408 q^{50} + 4054 q^{51} - 2176 q^{52} - 726 q^{53} - 6497 q^{54} - 3288 q^{55} - 7097 q^{56} - 870 q^{57} + 275 q^{58} - 4370 q^{59} - 3891 q^{60} + 1172 q^{61} - 1546 q^{62} - 3686 q^{63} + 606 q^{64} + 2610 q^{65} - 4697 q^{66} - 344 q^{67} - 3221 q^{68} + 136 q^{69} - 1310 q^{70} + 162 q^{71} - 5814 q^{72} + 746 q^{73} - 4332 q^{74} + 236 q^{75} + 1338 q^{76} + 2024 q^{77} - 2782 q^{78} - 2656 q^{79} - 5713 q^{80} - 86 q^{81} - 4168 q^{82} - 3514 q^{83} - 4269 q^{84} - 7558 q^{85} - 10278 q^{87} + 11692 q^{88} + 2640 q^{89} - 8286 q^{90} - 5946 q^{91} - 4271 q^{92} - 2 q^{93} + 9062 q^{94} - 12140 q^{95} - 700 q^{96} - 3864 q^{97} - 2826 q^{98} - 8174 q^{99}+O(q^{100}) \) 50 * q - 10 * q^2 - 2 * q^3 + 186 * q^4 + 8 * q^5 - 51 * q^6 - 6 * q^7 - 138 * q^8 + 360 * q^9 - 137 * q^10 - 252 * q^11 - 48 * q^12 - 192 * q^13 - 272 * q^14 - 314 * q^15 + 542 * q^16 - 236 * q^17 - 386 * q^18 + 12 * q^19 + 108 * q^20 - 408 * q^21 + 1235 * q^22 - 630 * q^23 - 613 * q^24 + 1098 * q^25 + 1493 * q^26 + 10 * q^27 - 242 * q^28 + 208 * q^29 - 48 * q^30 - 932 * q^31 - 1124 * q^32 + 254 * q^33 + 765 * q^34 - 1452 * q^35 + 747 * q^36 - 90 * q^37 - 1213 * q^38 - 1610 * q^39 - 1693 * q^40 - 1354 * q^41 - 16 * q^42 - 2704 * q^44 + 4508 * q^45 + 233 * q^46 - 3484 * q^47 - 376 * q^48 + 1324 * q^49 - 408 * q^50 + 4054 * q^51 - 2176 * q^52 - 726 * q^53 - 6497 * q^54 - 3288 * q^55 - 7097 * q^56 - 870 * q^57 + 275 * q^58 - 4370 * q^59 - 3891 * q^60 + 1172 * q^61 - 1546 * q^62 - 3686 * q^63 + 606 * q^64 + 2610 * q^65 - 4697 * q^66 - 344 * q^67 - 3221 * q^68 + 136 * q^69 - 1310 * q^70 + 162 * q^71 - 5814 * q^72 + 746 * q^73 - 4332 * q^74 + 236 * q^75 + 1338 * q^76 + 2024 * q^77 - 2782 * q^78 - 2656 * q^79 - 5713 * q^80 - 86 * q^81 - 4168 * q^82 - 3514 * q^83 - 4269 * q^84 - 7558 * q^85 - 10278 * q^87 + 11692 * q^88 + 2640 * q^89 - 8286 * q^90 - 5946 * q^91 - 4271 * q^92 - 2 * q^93 + 9062 * q^94 - 12140 * q^95 - 700 * q^96 - 3864 * q^97 - 2826 * q^98 - 8174 * q^99

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