LMFDB - Embedded newform 1849.4.a.g.1.25

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:	$30$
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.25
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)

$f(q)$	$=$	$q+3.72135 q^{2} -5.79603 q^{3} +5.84847 q^{4} +14.8139 q^{5} -21.5691 q^{6} +12.2090 q^{7} -8.00659 q^{8} +6.59399 q^{9} +O(q^{10})$	\(q+3.72135 q^{2} -5.79603 q^{3} +5.84847 q^{4} +14.8139 q^{5} -21.5691 q^{6} +12.2090 q^{7} -8.00659 q^{8} +6.59399 q^{9} +55.1279 q^{10} +2.05254 q^{11} -33.8979 q^{12} +9.53050 q^{13} +45.4339 q^{14} -85.8621 q^{15} -76.5831 q^{16} -74.1396 q^{17} +24.5386 q^{18} +103.100 q^{19} +86.6389 q^{20} -70.7635 q^{21} +7.63821 q^{22} -146.983 q^{23} +46.4064 q^{24} +94.4528 q^{25} +35.4663 q^{26} +118.274 q^{27} +71.4038 q^{28} +102.214 q^{29} -319.523 q^{30} -308.185 q^{31} -220.940 q^{32} -11.8966 q^{33} -275.900 q^{34} +180.863 q^{35} +38.5648 q^{36} -434.968 q^{37} +383.670 q^{38} -55.2391 q^{39} -118.609 q^{40} +183.943 q^{41} -263.336 q^{42} +12.0042 q^{44} +97.6829 q^{45} -546.975 q^{46} +348.672 q^{47} +443.878 q^{48} -193.941 q^{49} +351.492 q^{50} +429.716 q^{51} +55.7389 q^{52} -199.807 q^{53} +440.139 q^{54} +30.4062 q^{55} -97.7522 q^{56} -597.568 q^{57} +380.375 q^{58} -156.936 q^{59} -502.162 q^{60} +261.234 q^{61} -1146.86 q^{62} +80.5057 q^{63} -209.532 q^{64} +141.184 q^{65} -44.2713 q^{66} -43.3366 q^{67} -433.604 q^{68} +851.917 q^{69} +673.055 q^{70} +654.799 q^{71} -52.7953 q^{72} -868.625 q^{73} -1618.67 q^{74} -547.452 q^{75} +602.975 q^{76} +25.0593 q^{77} -205.564 q^{78} -1354.05 q^{79} -1134.50 q^{80} -863.557 q^{81} +684.515 q^{82} +166.764 q^{83} -413.859 q^{84} -1098.30 q^{85} -592.436 q^{87} -16.4338 q^{88} +1189.13 q^{89} +363.513 q^{90} +116.357 q^{91} -859.625 q^{92} +1786.25 q^{93} +1297.53 q^{94} +1527.31 q^{95} +1280.58 q^{96} -398.059 q^{97} -721.724 q^{98} +13.5344 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 6 q^{2} - 2 q^{3} + 114 q^{4} - 27 q^{5} + 8 q^{6} - 48 q^{7} - 90 q^{8} + 216 q^{9}+O(q^{10}) $ 30 * q - 6 * q^2 - 2 * q^3 + 114 * q^4 - 27 * q^5 + 8 * q^6 - 48 * q^7 - 90 * q^8 + 216 * q^9	\( 30 q - 6 q^{2} - 2 q^{3} + 114 q^{4} - 27 q^{5} + 8 q^{6} - 48 q^{7} - 90 q^{8} + 216 q^{9} - 27 q^{10} + 80 q^{11} + 36 q^{12} - 13 q^{13} + 36 q^{14} + 16 q^{15} + 318 q^{16} + 66 q^{17} - 80 q^{18} - 254 q^{19} - 312 q^{20} - 548 q^{21} - 305 q^{22} - 105 q^{23} + 123 q^{24} + 523 q^{25} - 549 q^{26} + 10 q^{27} - 578 q^{28} - 793 q^{29} - 1560 q^{30} - 359 q^{31} - 676 q^{32} - 208 q^{33} - 1007 q^{34} - 514 q^{35} + 776 q^{36} - 510 q^{37} - 2066 q^{38} - 898 q^{39} - 1248 q^{40} - 270 q^{41} + 915 q^{42} + 3256 q^{44} - 807 q^{45} - 1960 q^{46} + 1421 q^{47} + 632 q^{48} + 386 q^{49} + 141 q^{50} - 209 q^{51} + 2825 q^{52} - 21 q^{53} + 2368 q^{54} - 2258 q^{55} + 2521 q^{56} - 1723 q^{57} - 347 q^{58} + 1752 q^{59} + 2711 q^{60} - 1759 q^{61} - 395 q^{62} - 2204 q^{63} + 222 q^{64} - 1151 q^{65} + 160 q^{66} - 3001 q^{67} + 1921 q^{68} - 1660 q^{69} - 1597 q^{70} - 727 q^{71} - 9100 q^{72} - 4623 q^{73} - 2649 q^{74} - 1027 q^{75} - 874 q^{76} - 3556 q^{77} - 4979 q^{78} + 546 q^{79} - 5809 q^{80} - 410 q^{81} + 4397 q^{82} - 492 q^{83} - 10611 q^{84} + 1723 q^{85} + 5937 q^{87} - 3974 q^{88} - 5218 q^{89} + 10492 q^{90} - 1104 q^{91} + 1060 q^{92} - 1997 q^{93} + 2134 q^{94} + 6346 q^{95} - 11984 q^{96} + 2590 q^{97} - 6270 q^{98} - 2693 q^{99}+O(q^{100}) \) 30 * q - 6 * q^2 - 2 * q^3 + 114 * q^4 - 27 * q^5 + 8 * q^6 - 48 * q^7 - 90 * q^8 + 216 * q^9 - 27 * q^10 + 80 * q^11 + 36 * q^12 - 13 * q^13 + 36 * q^14 + 16 * q^15 + 318 * q^16 + 66 * q^17 - 80 * q^18 - 254 * q^19 - 312 * q^20 - 548 * q^21 - 305 * q^22 - 105 * q^23 + 123 * q^24 + 523 * q^25 - 549 * q^26 + 10 * q^27 - 578 * q^28 - 793 * q^29 - 1560 * q^30 - 359 * q^31 - 676 * q^32 - 208 * q^33 - 1007 * q^34 - 514 * q^35 + 776 * q^36 - 510 * q^37 - 2066 * q^38 - 898 * q^39 - 1248 * q^40 - 270 * q^41 + 915 * q^42 + 3256 * q^44 - 807 * q^45 - 1960 * q^46 + 1421 * q^47 + 632 * q^48 + 386 * q^49 + 141 * q^50 - 209 * q^51 + 2825 * q^52 - 21 * q^53 + 2368 * q^54 - 2258 * q^55 + 2521 * q^56 - 1723 * q^57 - 347 * q^58 + 1752 * q^59 + 2711 * q^60 - 1759 * q^61 - 395 * q^62 - 2204 * q^63 + 222 * q^64 - 1151 * q^65 + 160 * q^66 - 3001 * q^67 + 1921 * q^68 - 1660 * q^69 - 1597 * q^70 - 727 * q^71 - 9100 * q^72 - 4623 * q^73 - 2649 * q^74 - 1027 * q^75 - 874 * q^76 - 3556 * q^77 - 4979 * q^78 + 546 * q^79 - 5809 * q^80 - 410 * q^81 + 4397 * q^82 - 492 * q^83 - 10611 * q^84 + 1723 * q^85 + 5937 * q^87 - 3974 * q^88 - 5218 * q^89 + 10492 * q^90 - 1104 * q^91 + 1060 * q^92 - 1997 * q^93 + 2134 * q^94 + 6346 * q^95 - 11984 * q^96 + 2590 * q^97 - 6270 * q^98 - 2693 * 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 $

$2$	3.72135	1.31570	0.657849	−	0.753150i	$-0.271467\pi$
$2$	3.72135	1.31570	0.657849	+	0.753150i	$0.271467\pi$
$3$	−5.79603	−1.11545	−0.557723	−	0.830027i	$-0.688325\pi$
$3$	−5.79603	−1.11545	−0.557723	+	0.830027i	$0.688325\pi$
$4$	5.84847	0.731059
$5$	14.8139	1.32500	0.662500	−	0.749062i	$-0.269496\pi$
$5$	14.8139	1.32500	0.662500	+	0.749062i	$0.269496\pi$
$6$	−21.5691	−1.46759
$7$	12.2090	0.659222	0.329611	−	0.944117i	$-0.393082\pi$
$7$	12.2090	0.659222	0.329611	+	0.944117i	$0.393082\pi$
$8$	−8.00659	−0.353845
$9$	6.59399	0.244222
$10$	55.1279	1.74330
$11$	2.05254	0.0562603	0.0281301	−	0.999604i	$-0.491045\pi$
$11$	2.05254	0.0562603	0.0281301	+	0.999604i	$0.491045\pi$
$12$	−33.8979	−0.815458
$13$	9.53050	0.203330	0.101665	−	0.994819i	$-0.467583\pi$
$13$	9.53050	0.203330	0.101665	+	0.994819i	$0.467583\pi$
$14$	45.4339	0.867337
$15$	−85.8621	−1.47797
$16$	−76.5831	−1.19661
$17$	−74.1396	−1.05774	−0.528868	−	0.848704i	$-0.677383\pi$
$17$	−74.1396	−1.05774	−0.528868	+	0.848704i	$0.677383\pi$
$18$	24.5386	0.321322
$19$	103.100	1.24488	0.622439	−	0.782669i	$-0.286142\pi$
$19$	103.100	1.24488	0.622439	+	0.782669i	$0.286142\pi$
$20$	86.6389	0.968653
$21$	−70.7635	−0.735327
$22$	7.63821	0.0740215
$23$	−146.983	−1.33252	−0.666261	−	0.745718i	$-0.732107\pi$
$23$	−146.983	−1.33252	−0.666261	+	0.745718i	$0.732107\pi$
$24$	46.4064	0.394695
$25$	94.4528	0.755623
$26$	35.4663	0.267520
$27$	118.274	0.843031
$28$	71.4038	0.481930
$29$	102.214	0.654506	0.327253	−	0.944937i	$-0.393877\pi$
$29$	102.214	0.654506	0.327253	+	0.944937i	$0.393877\pi$
$30$	−319.523	−1.94456
$31$	−308.185	−1.78554	−0.892768	−	0.450516i	$-0.851240\pi$
$31$	−308.185	−1.78554	−0.892768	+	0.450516i	$0.851240\pi$
$32$	−220.940	−1.22053
$33$	−11.8966	−0.0627553
$34$	−275.900	−1.39166
$35$	180.863	0.873469
$36$	38.5648	0.178541
$37$	−434.968	−1.93266	−0.966329	−	0.257312i	$-0.917163\pi$
$37$	−434.968	−1.93266	−0.966329	+	0.257312i	$0.917163\pi$
$38$	383.670	1.63788
$39$	−55.2391	−0.226803
$40$	−118.609	−0.468844
$41$	183.943	0.700659	0.350329	−	0.936627i	$-0.386070\pi$
$41$	183.943	0.700659	0.350329	+	0.936627i	$0.386070\pi$
$42$	−263.336	−0.967468
$43$	0	0
$43$	0	0
$44$	12.0042	0.0411296
$45$	97.6829	0.323594
$46$	−546.975	−1.75320
$47$	348.672	1.08211	0.541053	−	0.840988i	$-0.318026\pi$
$47$	348.672	1.08211	0.541053	+	0.840988i	$0.318026\pi$
$48$	443.878	1.33476
$49$	−193.941	−0.565426
$50$	351.492	0.994171
$51$	429.716	1.17985
$52$	55.7389	0.148646
$53$	−199.807	−0.517842	−0.258921	−	0.965898i	$-0.583367\pi$
$53$	−199.807	−0.517842	−0.258921	+	0.965898i	$0.583367\pi$
$54$	440.139	1.10917
$55$	30.4062	0.0745448
$56$	−97.7522	−0.233262
$57$	−597.568	−1.38859
$58$	380.375	0.861132
$59$	−156.936	−0.346294	−0.173147	−	0.984896i	$-0.555394\pi$
$59$	−156.936	−0.346294	−0.173147	+	0.984896i	$0.555394\pi$
$60$	−502.162	−1.08048
$61$	261.234	0.548321	0.274160	−	0.961684i	$-0.411600\pi$
$61$	261.234	0.548321	0.274160	+	0.961684i	$0.411600\pi$
$62$	−1146.86	−2.34923
$63$	80.5057	0.160996
$64$	−209.532	−0.409242
$65$	141.184	0.269411
$66$	−44.2713	−0.0825670
$67$	−43.3366	−0.0790209	−0.0395105	−	0.999219i	$-0.512580\pi$
$67$	−43.3366	−0.0790209	−0.0395105	+	0.999219i	$0.512580\pi$
$68$	−433.604	−0.773267
$69$	851.917	1.48636
$70$	673.055	1.14922
$71$	654.799	1.09451	0.547256	−	0.836965i	$-0.315672\pi$
$71$	654.799	1.09451	0.547256	+	0.836965i	$0.315672\pi$
$72$	−52.7953	−0.0864165
$73$	−868.625	−1.39267	−0.696335	−	0.717717i	$-0.745187\pi$
$73$	−868.625	−1.39267	−0.696335	+	0.717717i	$0.745187\pi$
$74$	−1618.67	−2.54279
$75$	−547.452	−0.842857
$76$	602.975	0.910079
$77$	25.0593	0.0370880
$78$	−205.564	−0.298404
$79$	−1354.05	−1.92838	−0.964190	−	0.265213i	$-0.914558\pi$
$79$	−1354.05	−1.92838	−0.964190	+	0.265213i	$0.914558\pi$
$80$	−1134.50	−1.58551
$81$	−863.557	−1.18458
$82$	684.515	0.921855
$83$	166.764	0.220539	0.110270	−	0.993902i	$-0.464829\pi$
$83$	166.764	0.220539	0.110270	+	0.993902i	$0.464829\pi$
$84$	−413.859	−0.537568
$85$	−1098.30	−1.40150
$86$	0	0
$87$	−592.436	−0.730067
$88$	−16.4338	−0.0199074
$89$	1189.13	1.41627	0.708134	−	0.706078i	$-0.249537\pi$
$89$	1189.13	1.41627	0.708134	+	0.706078i	$0.249537\pi$
$90$	363.513	0.425751
$91$	116.357	0.134039
$92$	−859.625	−0.974153
$93$	1786.25	1.99167
$94$	1297.53	1.42373
$95$	1527.31	1.64946
$96$	1280.58	1.36144
$97$	−398.059	−0.416668	−0.208334	−	0.978058i	$-0.566804\pi$
$97$	−398.059	−0.416668	−0.208334	+	0.978058i	$0.566804\pi$
$98$	−721.724	−0.743930
$99$	13.5344	0.0137400
$100$	552.405	0.552405
$101$	592.846	0.584063	0.292032	−	0.956409i	$-0.405669\pi$
$101$	592.846	0.584063	0.292032	+	0.956409i	$0.405669\pi$
$102$	1599.12	1.55232
$103$	644.170	0.616233	0.308116	−	0.951349i	$-0.400301\pi$
$103$	644.170	0.616233	0.308116	+	0.951349i	$0.400301\pi$
$104$	−76.3068	−0.0719471
$105$	−1048.29	−0.974308
$106$	−743.553	−0.681323
$107$	215.065	0.194309	0.0971547	−	0.995269i	$-0.469026\pi$
$107$	215.065	0.194309	0.0971547	+	0.995269i	$0.469026\pi$
$108$	691.722	0.616305
$109$	−1496.74	−1.31524	−0.657621	−	0.753349i	$-0.728437\pi$
$109$	−1496.74	−1.31524	−0.657621	+	0.753349i	$0.728437\pi$
$110$	113.152	0.0980784
$111$	2521.09	2.15578
$112$	−935.001	−0.788833
$113$	436.886	0.363706	0.181853	−	0.983326i	$-0.441791\pi$
$113$	436.886	0.363706	0.181853	+	0.983326i	$0.441791\pi$
$114$	−2223.76	−1.82697
$115$	−2177.39	−1.76559
$116$	597.797	0.478483
$117$	62.8440	0.0496575
$118$	−584.016	−0.455619
$119$	−905.168	−0.697283
$120$	687.462	0.522970
$121$	−1326.79	−0.996835
$122$	972.143	0.721424
$123$	−1066.14	−0.781547
$124$	−1802.41	−1.30533
$125$	−452.524	−0.323800
$126$	299.590	0.211822
$127$	−1556.68	−1.08766	−0.543830	−	0.839196i	$-0.683026\pi$
$127$	−1556.68	−1.08766	−0.543830	+	0.839196i	$0.683026\pi$
$128$	987.780	0.682096
$129$	0	0
$130$	525.396	0.354464
$131$	278.172	0.185527	0.0927633	−	0.995688i	$-0.470430\pi$
$131$	278.172	0.185527	0.0927633	+	0.995688i	$0.470430\pi$
$132$	−69.5768	−0.0458779
$133$	1258.74	0.820651
$134$	−161.271	−0.103968
$135$	1752.10	1.11701
$136$	593.606	0.374274
$137$	1770.68	1.10423	0.552114	−	0.833769i	$-0.313821\pi$
$137$	1770.68	1.10423	0.552114	+	0.833769i	$0.313821\pi$
$138$	3170.28	1.95560
$139$	−1023.01	−0.624249	−0.312125	−	0.950041i	$-0.601041\pi$
$139$	−1023.01	−0.624249	−0.312125	+	0.950041i	$0.601041\pi$
$140$	1057.77	0.638557
$141$	−2020.91	−1.20703
$142$	2436.74	1.44005
$143$	19.5617	0.0114394
$144$	−504.988	−0.292239
$145$	1514.19	0.867220
$146$	−3232.46	−1.83233
$147$	1124.09	0.630703
$148$	−2543.90	−1.41289
$149$	−2224.57	−1.22311	−0.611556	−	0.791201i	$-0.709456\pi$
$149$	−2224.57	−1.22311	−0.611556	+	0.791201i	$0.709456\pi$
$150$	−2037.26	−1.10894
$151$	753.417	0.406041	0.203020	−	0.979174i	$-0.434924\pi$
$151$	753.417	0.406041	0.203020	+	0.979174i	$0.434924\pi$
$152$	−825.476	−0.440493
$153$	−488.876	−0.258322
$154$	93.2547	0.0487966
$155$	−4565.43	−2.36583
$156$	−323.064	−0.165807
$157$	−297.824	−0.151395	−0.0756973	−	0.997131i	$-0.524118\pi$
$157$	−297.824	−0.151395	−0.0756973	+	0.997131i	$0.524118\pi$
$158$	−5038.88	−2.53716
$159$	1158.09	0.577625
$160$	−3273.00	−1.61721
$161$	−1794.51	−0.878429
$162$	−3213.60	−1.55855
$163$	−2993.34	−1.43838	−0.719191	−	0.694812i	$-0.755487\pi$
$163$	−2993.34	−1.43838	−0.719191	+	0.694812i	$0.755487\pi$
$164$	1075.78	0.512223
$165$	−176.235	−0.0831508
$166$	620.589	0.290163
$167$	−300.196	−0.139101	−0.0695504	−	0.997578i	$-0.522156\pi$
$167$	−300.196	−0.139101	−0.0695504	+	0.997578i	$0.522156\pi$
$168$	566.575	0.260192
$169$	−2106.17	−0.958657
$170$	−4087.16	−1.84395
$171$	679.837	0.304026
$172$	0	0
$173$	−1441.64	−0.633559	−0.316780	−	0.948499i	$-0.602602\pi$
$173$	−1441.64	−0.633559	−0.316780	+	0.948499i	$0.602602\pi$
$174$	−2204.66	−0.960547
$175$	1153.17	0.498123
$176$	−157.190	−0.0673217
$177$	909.608	0.386273
$178$	4425.19	1.86338
$179$	−4500.30	−1.87915	−0.939576	−	0.342340i	$-0.888781\pi$
$179$	−4500.30	−1.87915	−0.939576	+	0.342340i	$0.888781\pi$
$180$	571.296	0.236566
$181$	3180.09	1.30593	0.652967	−	0.757386i	$-0.273524\pi$
$181$	3180.09	1.30593	0.652967	+	0.757386i	$0.273524\pi$
$182$	433.007	0.176355
$183$	−1514.12	−0.611623
$184$	1176.83	0.471506
$185$	−6443.59	−2.56077
$186$	6647.27	2.62044
$187$	−152.174	−0.0595085
$188$	2039.20	0.791084
$189$	1444.00	0.555744
$190$	5683.67	2.17019
$191$	−3654.44	−1.38443	−0.692215	−	0.721691i	$-0.743365\pi$
$191$	−3654.44	−1.38443	−0.692215	+	0.721691i	$0.743365\pi$
$192$	1214.45	0.456487
$193$	2259.01	0.842522	0.421261	−	0.906939i	$-0.361588\pi$
$193$	2259.01	0.842522	0.421261	+	0.906939i	$0.361588\pi$
$194$	−1481.32	−0.548209
$195$	−818.308	−0.300514
$196$	−1134.26	−0.413360
$197$	350.103	0.126618	0.0633091	−	0.997994i	$-0.479835\pi$
$197$	350.103	0.126618	0.0633091	+	0.997994i	$0.479835\pi$
$198$	50.3663	0.0180777
$199$	−3583.53	−1.27653	−0.638265	−	0.769817i	$-0.720348\pi$
$199$	−3583.53	−1.27653	−0.638265	+	0.769817i	$0.720348\pi$
$200$	−756.245	−0.267373
$201$	251.180	0.0881437
$202$	2206.19	0.768451
$203$	1247.93	0.431465
$204$	2513.18	0.862539
$205$	2724.91	0.928372
$206$	2397.18	0.810776
$207$	−969.202	−0.325431
$208$	−729.875	−0.243307
$209$	211.616	0.0700371
$210$	−3901.05	−1.28189
$211$	−653.378	−0.213177	−0.106589	−	0.994303i	$-0.533993\pi$
$211$	−653.378	−0.213177	−0.106589	+	0.994303i	$0.533993\pi$
$212$	−1168.57	−0.378573
$213$	−3795.24	−1.22087
$214$	800.332	0.255652
$215$	0	0
$216$	−946.971	−0.298302
$217$	−3762.62	−1.17707
$218$	−5569.88	−1.73046
$219$	5034.58	1.55345
$220$	177.830	0.0544967
$221$	−706.588	−0.215069
$222$	9381.86	2.83635
$223$	−304.058	−0.0913061	−0.0456530	−	0.998957i	$-0.514537\pi$
$223$	−304.058	−0.0913061	−0.0456530	+	0.998957i	$0.514537\pi$
$224$	−2697.45	−0.804603
$225$	622.821	0.184539
$226$	1625.81	0.478527
$227$	1310.85	0.383277	0.191638	−	0.981466i	$-0.438620\pi$
$227$	1310.85	0.383277	0.191638	+	0.981466i	$0.438620\pi$
$228$	−3494.86	−1.01514
$229$	−4790.56	−1.38240	−0.691199	−	0.722664i	$-0.742917\pi$
$229$	−4790.56	−1.38240	−0.691199	+	0.722664i	$0.742917\pi$
$230$	−8102.85	−2.32298
$231$	−145.245	−0.0413697
$232$	−818.386	−0.231594
$233$	−374.148	−0.105199	−0.0525993	−	0.998616i	$-0.516751\pi$
$233$	−374.148	−0.105199	−0.0525993	+	0.998616i	$0.516751\pi$
$234$	233.865	0.0653342
$235$	5165.20	1.43379
$236$	−917.838	−0.253162
$237$	7848.09	2.15101
$238$	−3368.45	−0.917413
$239$	−4096.57	−1.10872	−0.554362	−	0.832276i	$-0.687038\pi$
$239$	−4096.57	−1.10872	−0.554362	+	0.832276i	$0.687038\pi$
$240$	6575.59	1.76855
$241$	6255.68	1.67205	0.836024	−	0.548693i	$-0.184874\pi$
$241$	6255.68	1.67205	0.836024	+	0.548693i	$0.184874\pi$
$242$	−4937.44	−1.31153
$243$	1811.81	0.478303
$244$	1527.82	0.400855
$245$	−2873.03	−0.749189
$246$	−3967.47	−1.02828
$247$	982.590	0.253120
$248$	2467.51	0.631802
$249$	−966.572	−0.246000
$250$	−1684.00	−0.426022
$251$	−404.137	−0.101629	−0.0508146	−	0.998708i	$-0.516182\pi$
$251$	−404.137	−0.101629	−0.0508146	+	0.998708i	$0.516182\pi$
$252$	470.836	0.117698
$253$	−301.687	−0.0749681
$254$	−5792.95	−1.43103
$255$	6365.78	1.56330
$256$	5352.13	1.30667
$257$	2899.07	0.703655	0.351827	−	0.936065i	$-0.385560\pi$
$257$	2899.07	0.703655	0.351827	+	0.936065i	$0.385560\pi$
$258$	0	0
$259$	−5310.51	−1.27405
$260$	825.712	0.196956
$261$	673.998	0.159845
$262$	1035.18	0.244097
$263$	−5855.49	−1.37287	−0.686435	−	0.727191i	$-0.740825\pi$
$263$	−5855.49	−1.37287	−0.686435	+	0.727191i	$0.740825\pi$
$264$	95.2509	0.0222056
$265$	−2959.93	−0.686140
$266$	4684.21	1.07973
$267$	−6892.26	−1.57977
$268$	−253.453	−0.0577690
$269$	6624.95	1.50160	0.750800	−	0.660530i	$-0.229668\pi$
$269$	6624.95	1.50160	0.750800	+	0.660530i	$0.229668\pi$
$270$	6520.19	1.46965
$271$	−3887.82	−0.871470	−0.435735	−	0.900075i	$-0.643512\pi$
$271$	−3887.82	−0.871470	−0.435735	+	0.900075i	$0.643512\pi$
$272$	5677.85	1.26570
$273$	−674.412	−0.149514
$274$	6589.32	1.45283
$275$	193.868	0.0425115
$276$	4982.41	1.08662
$277$	−1269.13	−0.275287	−0.137644	−	0.990482i	$-0.543953\pi$
$277$	−1269.13	−0.275287	−0.137644	+	0.990482i	$0.543953\pi$
$278$	−3806.98	−0.821323
$279$	−2032.17	−0.436067
$280$	−1448.09	−0.309072
$281$	−1292.70	−0.274434	−0.137217	−	0.990541i	$-0.543816\pi$
$281$	−1292.70	−0.274434	−0.137217	+	0.990541i	$0.543816\pi$
$282$	−7520.53	−1.58809
$283$	−291.953	−0.0613243	−0.0306622	−	0.999530i	$-0.509762\pi$
$283$	−291.953	−0.0613243	−0.0306622	+	0.999530i	$0.509762\pi$
$284$	3829.58	0.800153
$285$	−8852.34	−1.83989
$286$	72.7960	0.0150508
$287$	2245.75	0.461890
$288$	−1456.88	−0.298081
$289$	583.686	0.118804
$290$	5634.85	1.14100
$291$	2307.17	0.464771
$292$	−5080.13	−1.01812
$293$	4659.82	0.929111	0.464555	−	0.885544i	$-0.346214\pi$
$293$	4659.82	0.929111	0.464555	+	0.885544i	$0.346214\pi$
$294$	4183.13	0.829814
$295$	−2324.85	−0.458840
$296$	3482.61	0.683860
$297$	242.762	0.0474291
$298$	−8278.40	−1.60924
$299$	−1400.82	−0.270941
$300$	−3201.76	−0.616178
$301$	0	0
$302$	2803.73	0.534227
$303$	−3436.16	−0.651492
$304$	−7895.69	−1.48963
$305$	3869.90	0.726524
$306$	−1819.28	−0.339873
$307$	4104.01	0.762959	0.381479	−	0.924377i	$-0.375415\pi$
$307$	4104.01	0.762959	0.381479	+	0.924377i	$0.375415\pi$
$308$	146.559	0.0271135
$309$	−3733.63	−0.687375
$310$	−16989.6	−3.11272
$311$	5349.77	0.975426	0.487713	−	0.873004i	$-0.337831\pi$
$311$	5349.77	0.975426	0.487713	+	0.873004i	$0.337831\pi$
$312$	442.277	0.0802531
$313$	2013.28	0.363569	0.181784	−	0.983338i	$-0.441813\pi$
$313$	2013.28	0.363569	0.181784	+	0.983338i	$0.441813\pi$
$314$	−1108.31	−0.199189
$315$	1192.61	0.213320
$316$	−7919.10	−1.40976
$317$	−4266.99	−0.756019	−0.378010	−	0.925802i	$-0.623391\pi$
$317$	−4266.99	−0.756019	−0.378010	+	0.925802i	$0.623391\pi$
$318$	4309.66	0.759980
$319$	209.798	0.0368227
$320$	−3103.99	−0.542245
$321$	−1246.52	−0.216742
$322$	−6678.00	−1.15575
$323$	−7643.77	−1.31675
$324$	−5050.49	−0.865996
$325$	900.182	0.153640
$326$	−11139.3	−1.89248
$327$	8675.13	1.46708
$328$	−1472.75	−0.247924
$329$	4256.92	0.713349
$330$	−655.833	−0.109401
$331$	−815.461	−0.135413	−0.0677066	−	0.997705i	$-0.521568\pi$
$331$	−815.461	−0.135413	−0.0677066	+	0.997705i	$0.521568\pi$
$332$	975.317	0.161227
$333$	−2868.17	−0.471997
$334$	−1117.13	−0.183014
$335$	−641.985	−0.104703
$336$	5419.29	0.879901
$337$	8384.84	1.35534	0.677672	−	0.735364i	$-0.262989\pi$
$337$	8384.84	1.35534	0.677672	+	0.735364i	$0.262989\pi$
$338$	−7837.80	−1.26130
$339$	−2532.20	−0.405695
$340$	−6423.38	−1.02458
$341$	−632.561	−0.100455
$342$	2529.91	0.400006
$343$	−6555.50	−1.03196
$344$	0	0
$345$	12620.2	1.96942
$346$	−5364.85	−0.833572
$347$	−7627.48	−1.18001	−0.590006	−	0.807399i	$-0.700875\pi$
$347$	−7627.48	−1.18001	−0.590006	+	0.807399i	$0.700875\pi$
$348$	−3464.85	−0.533722
$349$	1526.25	0.234093	0.117046	−	0.993126i	$-0.462657\pi$
$349$	1526.25	0.234093	0.117046	+	0.993126i	$0.462657\pi$
$350$	4291.36	0.655379
$351$	1127.21	0.171413
$352$	−453.488	−0.0686676
$353$	11239.7	1.69471	0.847353	−	0.531031i	$-0.178195\pi$
$353$	11239.7	1.69471	0.847353	+	0.531031i	$0.178195\pi$
$354$	3384.97	0.508218
$355$	9700.15	1.45023
$356$	6954.62	1.03538
$357$	5246.38	0.777782
$358$	−16747.2	−2.47240
$359$	3923.43	0.576799	0.288400	−	0.957510i	$-0.406877\pi$
$359$	3923.43	0.576799	0.288400	+	0.957510i	$0.406877\pi$
$360$	−782.107	−0.114502
$361$	3770.52	0.549719
$362$	11834.2	1.71821
$363$	7690.10	1.11192
$364$	680.514	0.0979907
$365$	−12867.8	−1.84529
$366$	−5634.57	−0.804710
$367$	1953.59	0.277866	0.138933	−	0.990302i	$-0.455633\pi$
$367$	1953.59	0.277866	0.138933	+	0.990302i	$0.455633\pi$
$368$	11256.4	1.59451
$369$	1212.91	0.171116
$370$	−23978.9	−3.36920
$371$	−2439.44	−0.341373
$372$	10446.8	1.45603
$373$	2889.05	0.401043	0.200522	−	0.979689i	$-0.435736\pi$
$373$	2889.05	0.401043	0.200522	+	0.979689i	$0.435736\pi$
$374$	−566.294	−0.0782952
$375$	2622.84	0.361181
$376$	−2791.67	−0.382898
$377$	974.151	0.133080
$378$	5373.64	0.731191
$379$	2202.12	0.298457	0.149229	−	0.988803i	$-0.452321\pi$
$379$	2202.12	0.298457	0.149229	+	0.988803i	$0.452321\pi$
$380$	8932.44	1.20585
$381$	9022.55	1.21323
$382$	−13599.5	−1.82149
$383$	−13559.5	−1.80903	−0.904515	−	0.426442i	$-0.859767\pi$
$383$	−13559.5	−1.80903	−0.904515	+	0.426442i	$0.859767\pi$
$384$	−5725.21	−0.760842
$385$	371.228	0.0491416
$386$	8406.56	1.10850
$387$	0	0
$388$	−2328.04	−0.304609
$389$	−6665.93	−0.868833	−0.434417	−	0.900712i	$-0.643045\pi$
$389$	−6665.93	−0.868833	−0.434417	+	0.900712i	$0.643045\pi$
$390$	−3045.21	−0.395386
$391$	10897.2	1.40946
$392$	1552.81	0.200073
$393$	−1612.29	−0.206945
$394$	1302.86	0.166591
$395$	−20058.7	−2.55510
$396$	79.1556	0.0100447
$397$	2959.76	0.374172	0.187086	−	0.982344i	$-0.440096\pi$
$397$	2959.76	0.374172	0.187086	+	0.982344i	$0.440096\pi$
$398$	−13335.6	−1.67953
$399$	−7295.69	−0.915392
$400$	−7233.50	−0.904187
$401$	516.898	0.0643707	0.0321854	−	0.999482i	$-0.489753\pi$
$401$	516.898	0.0643707	0.0321854	+	0.999482i	$0.489753\pi$
$402$	934.730	0.115970
$403$	−2937.15	−0.363052
$404$	3467.25	0.426985
$405$	−12792.7	−1.56956
$406$	4643.98	0.567677
$407$	−892.788	−0.108732
$408$	−3440.56	−0.417483
$409$	−10919.9	−1.32018	−0.660091	−	0.751186i	$-0.729482\pi$
$409$	−10919.9	−1.32018	−0.660091	+	0.751186i	$0.729482\pi$
$410$	10140.4	1.22146
$411$	−10262.9	−1.23171
$412$	3767.41	0.450503
$413$	−1916.03	−0.228285
$414$	−3606.74	−0.428169
$415$	2470.44	0.292214
$416$	−2105.67	−0.248171
$417$	5929.40	0.696317
$418$	787.497	0.0921477
$419$	9649.13	1.12504	0.562519	−	0.826784i	$-0.309832\pi$
$419$	9649.13	1.12504	0.562519	+	0.826784i	$0.309832\pi$
$420$	−6130.88	−0.712277
$421$	−14758.4	−1.70850	−0.854250	−	0.519862i	$-0.825983\pi$
$421$	−14758.4	−1.70850	−0.854250	+	0.519862i	$0.825983\pi$
$422$	−2431.45	−0.280477
$423$	2299.14	0.264274
$424$	1599.77	0.183236
$425$	−7002.70	−0.799249
$426$	−14123.4	−1.60630
$427$	3189.39	0.361465
$428$	1257.80	0.142052
$429$	−113.380	−0.0127600
$430$	0	0
$431$	11145.4	1.24560	0.622799	−	0.782382i	$-0.285995\pi$
$431$	11145.4	1.24560	0.622799	+	0.782382i	$0.285995\pi$
$432$	−9057.79	−1.00878
$433$	14713.2	1.63296	0.816479	−	0.577375i	$-0.195923\pi$
$433$	14713.2	1.63296	0.816479	+	0.577375i	$0.195923\pi$
$434$	−14002.0	−1.54866
$435$	−8776.32	−0.967338
$436$	−8753.62	−0.961519
$437$	−15153.9	−1.65883
$438$	18735.5	2.04387
$439$	−7024.07	−0.763646	−0.381823	−	0.924235i	$-0.624704\pi$
$439$	−7024.07	−0.763646	−0.381823	+	0.924235i	$0.624704\pi$
$440$	−243.450	−0.0263773
$441$	−1278.85	−0.138089
$442$	−2629.46	−0.282966
$443$	6017.83	0.645408	0.322704	−	0.946500i	$-0.395408\pi$
$443$	6017.83	0.645408	0.322704	+	0.946500i	$0.395408\pi$
$444$	14744.5	1.57600
$445$	17615.8	1.87656
$446$	−1131.51	−0.120131
$447$	12893.7	1.36432
$448$	−2558.17	−0.269781
$449$	11364.2	1.19446	0.597229	−	0.802071i	$-0.296268\pi$
$449$	11364.2	1.19446	0.597229	+	0.802071i	$0.296268\pi$
$450$	2317.74	0.242798
$451$	377.549	0.0394192
$452$	2555.12	0.265890
$453$	−4366.83	−0.452917
$454$	4878.12	0.504276
$455$	1723.71	0.177602
$456$	4784.49	0.491347
$457$	9100.15	0.931482	0.465741	−	0.884921i	$-0.345788\pi$
$457$	9100.15	0.931482	0.465741	+	0.884921i	$0.345788\pi$
$458$	−17827.4	−1.81882
$459$	−8768.78	−0.891703
$460$	−12734.4	−1.29075
$461$	−5401.64	−0.545725	−0.272863	−	0.962053i	$-0.587970\pi$
$461$	−5401.64	−0.545725	−0.272863	+	0.962053i	$0.587970\pi$
$462$	−540.507	−0.0544300
$463$	11462.7	1.15057	0.575286	−	0.817953i	$-0.304891\pi$
$463$	11462.7	1.15057	0.575286	+	0.817953i	$0.304891\pi$
$464$	−7827.88	−0.783190
$465$	26461.4	2.63896
$466$	−1392.34	−0.138410
$467$	5843.46	0.579022	0.289511	−	0.957175i	$-0.406507\pi$
$467$	5843.46	0.579022	0.289511	+	0.957175i	$0.406507\pi$
$468$	367.541	0.0363026
$469$	−529.094	−0.0520924
$470$	19221.6	1.88643
$471$	1726.20	0.168873
$472$	1256.53	0.122534
$473$	0	0
$474$	29205.5	2.83007
$475$	9738.05	0.940658
$476$	−5293.85	−0.509755
$477$	−1317.53	−0.126468
$478$	−15244.8	−1.45874
$479$	−7333.33	−0.699516	−0.349758	−	0.936840i	$-0.613736\pi$
$479$	−7333.33	−0.699516	−0.349758	+	0.936840i	$0.613736\pi$
$480$	18970.4	1.80391
$481$	−4145.46	−0.392966
$482$	23279.6	2.19991
$483$	10401.0	0.979840
$484$	−7759.68	−0.728745
$485$	−5896.83	−0.552085
$486$	6742.38	0.629301
$487$	−9898.48	−0.921032	−0.460516	−	0.887651i	$-0.652336\pi$
$487$	−9898.48	−0.921032	−0.460516	+	0.887651i	$0.652336\pi$
$488$	−2091.59	−0.194020
$489$	17349.5	1.60444
$490$	−10691.6	−0.985706
$491$	−1867.36	−0.171635	−0.0858176	−	0.996311i	$-0.527350\pi$
$491$	−1867.36	−0.171635	−0.0858176	+	0.996311i	$0.527350\pi$
$492$	−6235.27	−0.571357
$493$	−7578.12	−0.692295
$494$	3656.57	0.333030
$495$	200.498	0.0182055
$496$	23601.8	2.13659
$497$	7994.42	0.721526
$498$	−3596.95	−0.323661
$499$	8422.05	0.755556	0.377778	−	0.925896i	$-0.376688\pi$
$499$	8422.05	0.755556	0.377778	+	0.925896i	$0.376688\pi$
$500$	−2646.57	−0.236717
$501$	1739.94	0.155160
$502$	−1503.94	−0.133713
$503$	13588.9	1.20457	0.602286	−	0.798280i	$-0.294257\pi$
$503$	13588.9	1.20457	0.602286	+	0.798280i	$0.294257\pi$
$504$	−644.576	−0.0569677
$505$	8782.39	0.773884
$506$	−1122.69	−0.0986353
$507$	12207.4	1.06933
$508$	−9104.18	−0.795143
$509$	−1287.12	−0.112084	−0.0560420	−	0.998428i	$-0.517848\pi$
$509$	−1287.12	−0.112084	−0.0560420	+	0.998428i	$0.517848\pi$
$510$	23689.3	2.05683
$511$	−10605.0	−0.918079
$512$	12014.9	1.03709
$513$	12194.0	1.04947
$514$	10788.5	0.925797
$515$	9542.70	0.816508
$516$	0	0
$517$	715.662	0.0608796
$518$	−19762.3	−1.67626
$519$	8355.78	0.706701
$520$	−1130.40	−0.0953298
$521$	18347.4	1.54283	0.771415	−	0.636333i	$-0.219549\pi$
$521$	18347.4	1.54283	0.771415	+	0.636333i	$0.219549\pi$
$522$	2508.19	0.210307
$523$	10556.8	0.882635	0.441318	−	0.897351i	$-0.354511\pi$
$523$	10556.8	0.882635	0.441318	+	0.897351i	$0.354511\pi$
$524$	1626.88	0.135631
$525$	−6683.82	−0.555630
$526$	−21790.3	−1.80628
$527$	22848.7	1.88863
$528$	911.076	0.0750938
$529$	9436.94	0.775617
$530$	−11015.0	−0.902753
$531$	−1034.84	−0.0845726
$532$	7361.70	0.599944
$533$	1753.06	0.142465
$534$	−25648.5	−2.07850
$535$	3185.96	0.257460
$536$	346.978	0.0279611
$537$	26083.9	2.09609
$538$	24653.8	1.97565
$539$	−398.071	−0.0318110
$540$	10247.1	0.816604
$541$	−18649.2	−1.48205	−0.741027	−	0.671475i	$-0.765661\pi$
$541$	−18649.2	−1.48205	−0.741027	+	0.671475i	$0.765661\pi$
$542$	−14468.0	−1.14659
$543$	−18431.9	−1.45670
$544$	16380.4	1.29100
$545$	−22172.6	−1.74269
$546$	−2509.72	−0.196715
$547$	12594.9	0.984498	0.492249	−	0.870454i	$-0.336175\pi$
$547$	12594.9	0.984498	0.492249	+	0.870454i	$0.336175\pi$
$548$	10355.8	0.807256
$549$	1722.57	0.133912
$550$	721.451	0.0559323
$551$	10538.2	0.814780
$552$	−6820.95	−0.525940
$553$	−16531.5	−1.27123
$554$	−4722.87	−0.362194
$555$	37347.3	2.85640
$556$	−5983.05	−0.456363
$557$	−2619.40	−0.199259	−0.0996297	−	0.995025i	$-0.531766\pi$
$557$	−2619.40	−0.199259	−0.0996297	+	0.995025i	$0.531766\pi$
$558$	−7562.41	−0.573732
$559$	0	0
$560$	−13851.0	−1.04520
$561$	882.007	0.0663786
$562$	−4810.58	−0.361071
$563$	3061.94	0.229211	0.114605	−	0.993411i	$-0.463440\pi$
$563$	3061.94	0.229211	0.114605	+	0.993411i	$0.463440\pi$
$564$	−11819.3	−0.882412
$565$	6472.00	0.481910
$566$	−1086.46	−0.0806842
$567$	−10543.1	−0.780900
$568$	−5242.71	−0.387287
$569$	−7153.09	−0.527017	−0.263509	−	0.964657i	$-0.584880\pi$
$569$	−7153.09	−0.527017	−0.263509	+	0.964657i	$0.584880\pi$
$570$	−32942.7	−2.42073
$571$	−439.404	−0.0322040	−0.0161020	−	0.999870i	$-0.505126\pi$
$571$	−439.404	−0.0322040	−0.0161020	+	0.999870i	$0.505126\pi$
$572$	114.406	0.00836286
$573$	21181.3	1.54426
$574$	8357.22	0.607707
$575$	−13882.9	−1.00688
$576$	−1381.65	−0.0999457
$577$	12722.8	0.917951	0.458975	−	0.888449i	$-0.348217\pi$
$577$	12722.8	0.917951	0.458975	+	0.888449i	$0.348217\pi$
$578$	2172.10	0.156311
$579$	−13093.3	−0.939789
$580$	8855.72	0.633989
$581$	2036.02	0.145384
$582$	8585.78	0.611498
$583$	−410.112	−0.0291339
$584$	6954.73	0.492789
$585$	930.967	0.0657961
$586$	17340.8	1.22243
$587$	−1310.20	−0.0921256	−0.0460628	−	0.998939i	$-0.514667\pi$
$587$	−1310.20	−0.0921256	−0.0460628	+	0.998939i	$0.514667\pi$
$588$	6574.21	0.461081
$589$	−31773.7	−2.22277
$590$	−8651.57	−0.603694
$591$	−2029.21	−0.141236
$592$	33311.2	2.31264
$593$	−8523.59	−0.590256	−0.295128	−	0.955458i	$-0.595362\pi$
$593$	−8523.59	−0.590256	−0.295128	+	0.955458i	$0.595362\pi$
$594$	903.401	0.0624024
$595$	−13409.1	−0.923899
$596$	−13010.3	−0.894167
$597$	20770.2	1.42390
$598$	−5212.94	−0.356477
$599$	−21107.4	−1.43977	−0.719887	−	0.694091i	$-0.755806\pi$
$599$	−21107.4	−1.43977	−0.719887	+	0.694091i	$0.755806\pi$
$600$	4383.22	0.298240
$601$	27867.5	1.89141	0.945705	−	0.325026i	$-0.105373\pi$
$601$	27867.5	1.89141	0.945705	+	0.325026i	$0.105373\pi$
$602$	0	0
$603$	−285.761	−0.0192986
$604$	4406.34	0.296840
$605$	−19654.9	−1.32081
$606$	−12787.2	−0.857166
$607$	21396.3	1.43072	0.715361	−	0.698755i	$-0.246262\pi$
$607$	21396.3	1.43072	0.715361	+	0.698755i	$0.246262\pi$
$608$	−22778.8	−1.51942
$609$	−7233.03	−0.481276
$610$	14401.3	0.955886
$611$	3323.02	0.220024
$612$	−2859.18	−0.188849
$613$	21983.4	1.44845	0.724226	−	0.689563i	$-0.242197\pi$
$613$	21983.4	1.44845	0.724226	+	0.689563i	$0.242197\pi$
$614$	15272.5	1.00382
$615$	−15793.7	−1.03555
$616$	−200.640	−0.0131234
$617$	11090.6	0.723647	0.361824	−	0.932247i	$-0.382154\pi$
$617$	11090.6	0.723647	0.361824	+	0.932247i	$0.382154\pi$
$618$	−13894.2	−0.904377
$619$	−14015.0	−0.910030	−0.455015	−	0.890484i	$-0.650366\pi$
$619$	−14015.0	−0.910030	−0.455015	+	0.890484i	$0.650366\pi$
$620$	−26700.8	−1.72957
$621$	−17384.2	−1.12336
$622$	19908.4	1.28337
$623$	14518.1	0.933636
$624$	4230.38	0.271395
$625$	−18510.3	−1.18466
$626$	7492.11	0.478346
$627$	−1226.53	−0.0781227
$628$	−1741.82	−0.110678
$629$	32248.4	2.04424
$630$	4438.11	0.280665
$631$	20579.0	1.29832	0.649158	−	0.760654i	$-0.275122\pi$
$631$	20579.0	1.29832	0.649158	+	0.760654i	$0.275122\pi$
$632$	10841.3	0.682347
$633$	3787.00	0.237788
$634$	−15879.0	−0.994693
$635$	−23060.5	−1.44115
$636$	6773.05	0.422278
$637$	−1848.36	−0.114968
$638$	780.733	0.0484475
$639$	4317.74	0.267304
$640$	14632.9	0.903777
$641$	21587.1	1.33017	0.665087	−	0.746766i	$-0.268395\pi$
$641$	21587.1	1.33017	0.665087	+	0.746766i	$0.268395\pi$
$642$	−4638.75	−0.285166
$643$	6576.06	0.403320	0.201660	−	0.979456i	$-0.435366\pi$
$643$	6576.06	0.403320	0.201660	+	0.979456i	$0.435366\pi$
$644$	−10495.1	−0.642183
$645$	0	0
$646$	−28445.2	−1.73245
$647$	−10311.6	−0.626571	−0.313285	−	0.949659i	$-0.601430\pi$
$647$	−10311.6	−0.626571	−0.313285	+	0.949659i	$0.601430\pi$
$648$	6914.15	0.419156
$649$	−322.118	−0.0194826
$650$	3349.90	0.202144
$651$	21808.3	1.31295
$652$	−17506.5	−1.05154
$653$	−10297.6	−0.617116	−0.308558	−	0.951206i	$-0.599846\pi$
$653$	−10297.6	−0.617116	−0.308558	+	0.951206i	$0.599846\pi$
$654$	32283.2	1.93024
$655$	4120.82	0.245823
$656$	−14086.9	−0.838416
$657$	−5727.70	−0.340120
$658$	15841.5	0.938551
$659$	12986.6	0.767658	0.383829	−	0.923404i	$-0.374605\pi$
$659$	12986.6	0.767658	0.383829	+	0.923404i	$0.374605\pi$
$660$	−1030.71	−0.0607881
$661$	6720.12	0.395435	0.197717	−	0.980259i	$-0.436647\pi$
$661$	6720.12	0.395435	0.197717	+	0.980259i	$0.436647\pi$
$662$	−3034.62	−0.178163
$663$	4095.40	0.239898
$664$	−1335.21	−0.0780367
$665$	18646.9	1.08736
$666$	−10673.5	−0.621005
$667$	−15023.7	−0.872145
$668$	−1755.69	−0.101691
$669$	1762.33	0.101847
$670$	−2389.05	−0.137757
$671$	536.192	0.0308487
$672$	15634.5	0.897492
$673$	5865.01	0.335928	0.167964	−	0.985793i	$-0.446281\pi$
$673$	5865.01	0.335928	0.167964	+	0.985793i	$0.446281\pi$
$674$	31202.9	1.78322
$675$	11171.3	0.637013
$676$	−12317.9	−0.700835
$677$	−22538.1	−1.27948	−0.639740	−	0.768591i	$-0.720958\pi$
$677$	−22538.1	−1.27948	−0.639740	+	0.768591i	$0.720958\pi$
$678$	−9423.23	−0.533771
$679$	−4859.89	−0.274677
$680$	8793.64	0.495913
$681$	−7597.70	−0.427525
$682$	−2353.98	−0.132168
$683$	15142.2	0.848318	0.424159	−	0.905588i	$-0.360570\pi$
$683$	15142.2	0.848318	0.424159	+	0.905588i	$0.360570\pi$
$684$	3976.01	0.222261
$685$	26230.7	1.46310
$686$	−24395.3	−1.35775
$687$	27766.3	1.54199
$688$	0	0
$689$	−1904.26	−0.105293
$690$	46964.4	2.59117
$691$	9967.42	0.548739	0.274369	−	0.961624i	$-0.411531\pi$
$691$	9967.42	0.548739	0.274369	+	0.961624i	$0.411531\pi$
$692$	−8431.38	−0.463169
$693$	165.241	0.00905770
$694$	−28384.5	−1.55254
$695$	−15154.8	−0.827129
$696$	4743.39	0.258330
$697$	−13637.4	−0.741111
$698$	5679.72	0.307995
$699$	2168.58	0.117343
$700$	6744.29	0.364158
$701$	14945.7	0.805267	0.402634	−	0.915361i	$-0.368095\pi$
$701$	14945.7	0.805267	0.402634	+	0.915361i	$0.368095\pi$
$702$	4194.74	0.225528
$703$	−44845.0	−2.40592
$704$	−430.071	−0.0230240
$705$	−29937.7	−1.59932
$706$	41827.0	2.22972
$707$	7238.04	0.385027
$708$	5319.82	0.282389
$709$	−16453.5	−0.871541	−0.435770	−	0.900058i	$-0.643524\pi$
$709$	−16453.5	−0.871541	−0.435770	+	0.900058i	$0.643524\pi$
$710$	36097.7	1.90806
$711$	−8928.55	−0.470952
$712$	−9520.91	−0.501139
$713$	45297.9	2.37927
$714$	19523.6	1.02333
$715$	289.786	0.0151572
$716$	−26319.9	−1.37377
$717$	23743.8	1.23672
$718$	14600.5	0.758893
$719$	17776.5	0.922048	0.461024	−	0.887388i	$-0.347482\pi$
$719$	17776.5	0.922048	0.461024	+	0.887388i	$0.347482\pi$
$720$	−7480.86	−0.387216
$721$	7864.65	0.406234
$722$	14031.5	0.723264
$723$	−36258.1	−1.86508
$724$	18598.7	0.954716
$725$	9654.41	0.494560
$726$	28617.6	1.46295
$727$	782.314	0.0399098	0.0199549	−	0.999801i	$-0.493648\pi$
$727$	782.314	0.0399098	0.0199549	+	0.999801i	$0.493648\pi$
$728$	−931.627	−0.0474291
$729$	12814.7	0.651056
$730$	−47885.5	−2.42784
$731$	0	0
$732$	−8855.29	−0.447132
$733$	38475.2	1.93876	0.969381	−	0.245560i	$-0.0789720\pi$
$733$	38475.2	1.93876	0.969381	+	0.245560i	$0.0789720\pi$
$734$	7270.02	0.365588
$735$	16652.2	0.835681
$736$	32474.4	1.62639
$737$	−88.9499	−0.00444574
$738$	4513.68	0.225137
$739$	−12300.1	−0.612266	−0.306133	−	0.951989i	$-0.599035\pi$
$739$	−12300.1	−0.612266	−0.306133	+	0.951989i	$0.599035\pi$
$740$	−37685.2	−1.87207
$741$	−5695.12	−0.282342
$742$	−9078.02	−0.449143
$743$	−38280.9	−1.89016	−0.945080	−	0.326840i	$-0.894016\pi$
$743$	−38280.9	−1.89016	−0.945080	+	0.326840i	$0.894016\pi$
$744$	−14301.8	−0.704742
$745$	−32954.6	−1.62062
$746$	10751.2	0.527652
$747$	1099.64	0.0538605
$748$	−889.987	−0.0435042
$749$	2625.72	0.128093
$750$	9760.53	0.475205
$751$	−12564.6	−0.610506	−0.305253	−	0.952271i	$-0.598741\pi$
$751$	−12564.6	−0.610506	−0.305253	+	0.952271i	$0.598741\pi$
$752$	−26702.4	−1.29486
$753$	2342.39	0.113362
$754$	3625.16	0.175094
$755$	11161.1	0.538004
$756$	8445.21	0.406282
$757$	12863.3	0.617603	0.308801	−	0.951127i	$-0.400072\pi$
$757$	12863.3	0.617603	0.308801	+	0.951127i	$0.400072\pi$
$758$	8194.86	0.392679
$759$	1748.59	0.0836229
$760$	−12228.6	−0.583653
$761$	3597.42	0.171362	0.0856810	−	0.996323i	$-0.472693\pi$
$761$	3597.42	0.171362	0.0856810	+	0.996323i	$0.472693\pi$
$762$	33576.1	1.59624
$763$	−18273.6	−0.867036
$764$	−21372.9	−1.01210
$765$	−7242.18	−0.342276
$766$	−50459.7	−2.38014
$767$	−1495.68	−0.0704119
$768$	−31021.1	−1.45752
$769$	18986.1	0.890318	0.445159	−	0.895452i	$-0.353147\pi$
$769$	18986.1	0.890318	0.445159	+	0.895452i	$0.353147\pi$
$770$	1381.47	0.0646555
$771$	−16803.1	−0.784890
$772$	13211.7	0.615934
$773$	−23805.7	−1.10767	−0.553836	−	0.832626i	$-0.686836\pi$
$773$	−23805.7	−1.10767	−0.553836	+	0.832626i	$0.686836\pi$
$774$	0	0
$775$	−29108.9	−1.34919
$776$	3187.10	0.147436
$777$	30779.9	1.42114
$778$	−24806.3	−1.14312
$779$	18964.4	0.872234
$780$	−4785.85	−0.219694
$781$	1344.00	0.0615775
$782$	40552.5	1.85442
$783$	12089.3	0.551769
$784$	14852.6	0.676596
$785$	−4411.95	−0.200598
$786$	−5999.91	−0.272277
$787$	5921.07	0.268187	0.134094	−	0.990969i	$-0.457188\pi$
$787$	5921.07	0.268187	0.134094	+	0.990969i	$0.457188\pi$
$788$	2047.57	0.0925655
$789$	33938.6	1.53136
$790$	−74645.7	−3.36174
$791$	5333.92	0.239763
$792$	−108.364	−0.00486182
$793$	2489.69	0.111490
$794$	11014.3	0.492297
$795$	17155.9	0.765353
$796$	−20958.2	−0.933219
$797$	−31492.1	−1.39963	−0.699817	−	0.714322i	$-0.746735\pi$
$797$	−31492.1	−1.39963	−0.699817	+	0.714322i	$0.746735\pi$
$798$	−27149.9	−1.20438
$799$	−25850.4	−1.14458
$800$	−20868.4	−0.922263
$801$	7841.13	0.345884
$802$	1923.56	0.0846924
$803$	−1782.89	−0.0783520
$804$	1469.02	0.0644382
$805$	−26583.7	−1.16392
$806$	−10930.2	−0.477667
$807$	−38398.4	−1.67495
$808$	−4746.68	−0.206668
$809$	−8959.91	−0.389386	−0.194693	−	0.980864i	$-0.562371\pi$
$809$	−8959.91	−0.389386	−0.194693	+	0.980864i	$0.562371\pi$
$810$	−47606.1	−2.06507
$811$	−15879.3	−0.687543	−0.343772	−	0.939053i	$-0.611705\pi$
$811$	−15879.3	−0.687543	−0.343772	+	0.939053i	$0.611705\pi$
$812$	7298.48	0.315427
$813$	22533.9	0.972079
$814$	−3322.38	−0.143058
$815$	−44343.1	−1.90586
$816$	−32909.0	−1.41182
$817$	0	0
$818$	−40636.8	−1.73696
$819$	767.260	0.0327353
$820$	15936.6	0.678695
$821$	1143.25	0.0485990	0.0242995	−	0.999705i	$-0.492264\pi$
$821$	1143.25	0.0485990	0.0242995	+	0.999705i	$0.492264\pi$
$822$	−38191.9	−1.62055
$823$	6089.64	0.257924	0.128962	−	0.991650i	$-0.458835\pi$
$823$	6089.64	0.257924	0.128962	+	0.991650i	$0.458835\pi$
$824$	−5157.61	−0.218051
$825$	−1123.66	−0.0474194
$826$	−7130.23	−0.300354
$827$	−347.605	−0.0146160	−0.00730798	−	0.999973i	$-0.502326\pi$
$827$	−347.605	−0.0146160	−0.00730798	+	0.999973i	$0.502326\pi$
$828$	−5668.35	−0.237909
$829$	10417.2	0.436437	0.218218	−	0.975900i	$-0.429976\pi$
$829$	10417.2	0.436437	0.218218	+	0.975900i	$0.429976\pi$
$830$	9193.37	0.384466
$831$	7355.91	0.307068
$832$	−1996.94	−0.0832109
$833$	14378.7	0.598071
$834$	22065.4	0.916142
$835$	−4447.08	−0.184308
$836$	1237.63	0.0512013
$837$	−36450.2	−1.50526
$838$	35907.8	1.48021
$839$	−27147.7	−1.11710	−0.558548	−	0.829472i	$-0.688641\pi$
$839$	−27147.7	−1.11710	−0.558548	+	0.829472i	$0.688641\pi$
$840$	8393.20	0.344754
$841$	−13941.3	−0.571621
$842$	−54921.1	−2.24787
$843$	7492.51	0.306116
$844$	−3821.26	−0.155845
$845$	−31200.7	−1.27022
$846$	8555.90	0.347705
$847$	−16198.7	−0.657136
$848$	15301.9	0.619656
$849$	1692.17	0.0684040
$850$	−26059.5	−1.05157
$851$	63932.8	2.57531
$852$	−22196.3	−0.892528
$853$	−28796.5	−1.15589	−0.577945	−	0.816076i	$-0.696145\pi$
$853$	−28796.5	−1.15589	−0.577945	+	0.816076i	$0.696145\pi$
$854$	11868.9	0.475579
$855$	10071.1	0.402834
$856$	−1721.94	−0.0687553
$857$	183.644	0.00731990	0.00365995	−	0.999993i	$-0.498835\pi$
$857$	183.644	0.00731990	0.00365995	+	0.999993i	$0.498835\pi$
$858$	−421.928	−0.0167883
$859$	−29667.8	−1.17841	−0.589203	−	0.807985i	$-0.700558\pi$
$859$	−29667.8	−1.17841	−0.589203	+	0.807985i	$0.700558\pi$
$860$	0	0
$861$	−13016.4	−0.515213
$862$	41475.8	1.63883
$863$	37230.7	1.46854	0.734268	−	0.678860i	$-0.237526\pi$
$863$	37230.7	1.46854	0.734268	+	0.678860i	$0.237526\pi$
$864$	−26131.5	−1.02895
$865$	−21356.3	−0.839465
$866$	54753.0	2.14848
$867$	−3383.06	−0.132520
$868$	−22005.6	−0.860504
$869$	−2779.23	−0.108491
$870$	−32659.8	−1.27272
$871$	−413.019	−0.0160673
$872$	11983.8	0.465391
$873$	−2624.80	−0.101759
$874$	−56392.9	−2.18251
$875$	−5524.85	−0.213456
$876$	29444.6	1.13566
$877$	−9915.97	−0.381800	−0.190900	−	0.981610i	$-0.561141\pi$
$877$	−9915.97	−0.381800	−0.190900	+	0.981610i	$0.561141\pi$
$878$	−26139.1	−1.00473
$879$	−27008.4	−1.03637
$880$	−2328.60	−0.0892012
$881$	31781.6	1.21538	0.607691	−	0.794174i	$-0.292096\pi$
$881$	31781.6	1.21538	0.607691	+	0.794174i	$0.292096\pi$
$882$	−4759.04	−0.181684
$883$	−30414.3	−1.15914	−0.579572	−	0.814921i	$-0.696780\pi$
$883$	−30414.3	−1.15914	−0.579572	+	0.814921i	$0.696780\pi$
$884$	−4132.46	−0.157228
$885$	13474.9	0.511812
$886$	22394.5	0.849161
$887$	−32756.2	−1.23996	−0.619981	−	0.784617i	$-0.712860\pi$
$887$	−32756.2	−1.23996	−0.619981	+	0.784617i	$0.712860\pi$
$888$	−20185.3	−0.762810
$889$	−19005.4	−0.717009
$890$	65554.5	2.46898
$891$	−1772.48	−0.0666447
$892$	−1778.28	−0.0667501
$893$	35947.9	1.34709
$894$	47981.9	1.79503
$895$	−66667.2	−2.48988
$896$	12059.8	0.449653
$897$	8119.19	0.302221
$898$	42290.4	1.57155
$899$	−31500.8	−1.16865
$900$	3642.55	0.134909
$901$	14813.6	0.547740
$902$	1404.99	0.0518638
$903$	0	0
$904$	−3497.97	−0.128695
$905$	47109.7	1.73036
$906$	−16250.5	−0.595902
$907$	11089.6	0.405981	0.202991	−	0.979181i	$-0.434934\pi$
$907$	11089.6	0.405981	0.202991	+	0.979181i	$0.434934\pi$
$908$	7666.45	0.280198
$909$	3909.22	0.142641
$910$	6414.55	0.233670
$911$	−27247.3	−0.990938	−0.495469	−	0.868626i	$-0.665004\pi$
$911$	−27247.3	−0.990938	−0.495469	+	0.868626i	$0.665004\pi$
$912$	45763.7	1.66161
$913$	342.290	0.0124076
$914$	33864.9	1.22555
$915$	−22430.1	−0.810399
$916$	−28017.5	−1.01062
$917$	3396.19	0.122303
$918$	−32631.8	−1.17321
$919$	−3098.95	−0.111235	−0.0556174	−	0.998452i	$-0.517713\pi$
$919$	−3098.95	−0.111235	−0.0556174	+	0.998452i	$0.517713\pi$
$920$	17433.5	0.624745
$921$	−23787.0	−0.851040
$922$	−20101.4	−0.718009
$923$	6240.56	0.222547
$924$	−849.460	−0.0302437
$925$	−41084.0	−1.46036
$926$	42656.6	1.51380
$927$	4247.65	0.150497
$928$	−22583.2	−0.798847
$929$	−15691.2	−0.554158	−0.277079	−	0.960847i	$-0.589366\pi$
$929$	−15691.2	−0.554158	−0.277079	+	0.960847i	$0.589366\pi$
$930$	98472.2	3.47208
$931$	−19995.3	−0.703886
$932$	−2188.20	−0.0769064
$933$	−31007.4	−1.08804
$934$	21745.6	0.761817
$935$	−2254.30	−0.0788487
$936$	−503.166	−0.0175710
$937$	30261.1	1.05505	0.527527	−	0.849538i	$-0.323119\pi$
$937$	30261.1	1.05505	0.527527	+	0.849538i	$0.323119\pi$
$938$	−1968.95	−0.0685378
$939$	−11669.0	−0.405542
$940$	30208.6	1.04819
$941$	−51902.6	−1.79806	−0.899032	−	0.437884i	$-0.855728\pi$
$941$	−51902.6	−1.79806	−0.899032	+	0.437884i	$0.855728\pi$
$942$	6423.79	0.222185
$943$	−27036.4	−0.933644
$944$	12018.7	0.414380
$945$	21391.4	0.736361
$946$	0	0
$947$	−18308.0	−0.628225	−0.314112	−	0.949386i	$-0.601707\pi$
$947$	−18308.0	−0.628225	−0.314112	+	0.949386i	$0.601707\pi$
$948$	45899.3	1.57251
$949$	−8278.43	−0.283171
$950$	36238.7	1.23762
$951$	24731.6	0.843299
$952$	7247.31	0.246730
$953$	1809.03	0.0614904	0.0307452	−	0.999527i	$-0.490212\pi$
$953$	1809.03	0.0614904	0.0307452	+	0.999527i	$0.490212\pi$
$954$	−4902.98	−0.166394
$955$	−54136.7	−1.83437
$956$	−23958.7	−0.810543
$957$	−1216.00	−0.0410738
$958$	−27289.9	−0.920351
$959$	21618.1	0.727931
$960$	17990.8	0.604845
$961$	65186.9	2.18814
$962$	−15426.7	−0.517025
$963$	1418.13	0.0474546
$964$	36586.2	1.22237
$965$	33464.8	1.11634
$966$	38705.9	1.28917
$967$	−9570.15	−0.318258	−0.159129	−	0.987258i	$-0.550869\pi$
$967$	−9570.15	−0.318258	−0.159129	+	0.987258i	$0.550869\pi$
$968$	10623.0	0.352725
$969$	44303.5	1.46877
$970$	−21944.2	−0.726377
$971$	24689.1	0.815973	0.407987	−	0.912988i	$-0.366231\pi$
$971$	24689.1	0.815973	0.407987	+	0.912988i	$0.366231\pi$
$972$	10596.3	0.349668
$973$	−12489.9	−0.411519
$974$	−36835.7	−1.21180
$975$	−5217.49	−0.171378
$976$	−20006.1	−0.656127
$977$	−57659.0	−1.88810	−0.944051	−	0.329801i	$-0.893018\pi$
$977$	−57659.0	−1.88810	−0.944051	+	0.329801i	$0.893018\pi$
$978$	64563.6	2.11096
$979$	2440.74	0.0796797
$980$	−16802.9	−0.547702
$981$	−9869.46	−0.321210
$982$	−6949.11	−0.225820
$983$	−22540.1	−0.731351	−0.365676	−	0.930742i	$-0.619162\pi$
$983$	−22540.1	−0.731351	−0.365676	+	0.930742i	$0.619162\pi$
$984$	8536.12	0.276546
$985$	5186.40	0.167769
$986$	−28200.9	−0.910850
$987$	−24673.3	−0.795703
$988$	5746.65	0.185046
$989$	0	0
$990$	746.123	0.0239529
$991$	−19732.0	−0.632499	−0.316249	−	0.948676i	$-0.602424\pi$
$991$	−19732.0	−0.632499	−0.316249	+	0.948676i	$0.602424\pi$
$992$	68090.4	2.17931
$993$	4726.44	0.151046
$994$	29750.1	0.949310
$995$	−53086.1	−1.69140
$996$	−5652.97	−0.179841
$997$	34498.1	1.09585	0.547927	−	0.836526i	$-0.315417\pi$
$997$	34498.1	1.09585	0.547927	+	0.836526i	$0.315417\pi$
$998$	31341.4	0.994084
$999$	−51445.4	−1.62929

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.g.1.25		30
43.2	odd	14		43.4.e.a.4.2	✓	60
43.22	odd	14		43.4.e.a.11.2	yes	60
43.42	odd	2		1849.4.a.h.1.6		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.e.a.4.2	✓	60	43.2	odd	14
43.4.e.a.11.2	yes	60	43.22	odd	14
1849.4.a.g.1.25		30	1.1	even	1	trivial
1849.4.a.h.1.6		30	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)

\(f(q)\)	\(=\)	\(q+3.72135 q^{2} -5.79603 q^{3} +5.84847 q^{4} +14.8139 q^{5} -21.5691 q^{6} +12.2090 q^{7} -8.00659 q^{8} +6.59399 q^{9} +O(q^{10})\)	\(q+3.72135 q^{2} -5.79603 q^{3} +5.84847 q^{4} +14.8139 q^{5} -21.5691 q^{6} +12.2090 q^{7} -8.00659 q^{8} +6.59399 q^{9} +55.1279 q^{10} +2.05254 q^{11} -33.8979 q^{12} +9.53050 q^{13} +45.4339 q^{14} -85.8621 q^{15} -76.5831 q^{16} -74.1396 q^{17} +24.5386 q^{18} +103.100 q^{19} +86.6389 q^{20} -70.7635 q^{21} +7.63821 q^{22} -146.983 q^{23} +46.4064 q^{24} +94.4528 q^{25} +35.4663 q^{26} +118.274 q^{27} +71.4038 q^{28} +102.214 q^{29} -319.523 q^{30} -308.185 q^{31} -220.940 q^{32} -11.8966 q^{33} -275.900 q^{34} +180.863 q^{35} +38.5648 q^{36} -434.968 q^{37} +383.670 q^{38} -55.2391 q^{39} -118.609 q^{40} +183.943 q^{41} -263.336 q^{42} +12.0042 q^{44} +97.6829 q^{45} -546.975 q^{46} +348.672 q^{47} +443.878 q^{48} -193.941 q^{49} +351.492 q^{50} +429.716 q^{51} +55.7389 q^{52} -199.807 q^{53} +440.139 q^{54} +30.4062 q^{55} -97.7522 q^{56} -597.568 q^{57} +380.375 q^{58} -156.936 q^{59} -502.162 q^{60} +261.234 q^{61} -1146.86 q^{62} +80.5057 q^{63} -209.532 q^{64} +141.184 q^{65} -44.2713 q^{66} -43.3366 q^{67} -433.604 q^{68} +851.917 q^{69} +673.055 q^{70} +654.799 q^{71} -52.7953 q^{72} -868.625 q^{73} -1618.67 q^{74} -547.452 q^{75} +602.975 q^{76} +25.0593 q^{77} -205.564 q^{78} -1354.05 q^{79} -1134.50 q^{80} -863.557 q^{81} +684.515 q^{82} +166.764 q^{83} -413.859 q^{84} -1098.30 q^{85} -592.436 q^{87} -16.4338 q^{88} +1189.13 q^{89} +363.513 q^{90} +116.357 q^{91} -859.625 q^{92} +1786.25 q^{93} +1297.53 q^{94} +1527.31 q^{95} +1280.58 q^{96} -398.059 q^{97} -721.724 q^{98} +13.5344 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 6 q^{2} - 2 q^{3} + 114 q^{4} - 27 q^{5} + 8 q^{6} - 48 q^{7} - 90 q^{8} + 216 q^{9}+O(q^{10}) \) 30 * q - 6 * q^2 - 2 * q^3 + 114 * q^4 - 27 * q^5 + 8 * q^6 - 48 * q^7 - 90 * q^8 + 216 * q^9	\( 30 q - 6 q^{2} - 2 q^{3} + 114 q^{4} - 27 q^{5} + 8 q^{6} - 48 q^{7} - 90 q^{8} + 216 q^{9} - 27 q^{10} + 80 q^{11} + 36 q^{12} - 13 q^{13} + 36 q^{14} + 16 q^{15} + 318 q^{16} + 66 q^{17} - 80 q^{18} - 254 q^{19} - 312 q^{20} - 548 q^{21} - 305 q^{22} - 105 q^{23} + 123 q^{24} + 523 q^{25} - 549 q^{26} + 10 q^{27} - 578 q^{28} - 793 q^{29} - 1560 q^{30} - 359 q^{31} - 676 q^{32} - 208 q^{33} - 1007 q^{34} - 514 q^{35} + 776 q^{36} - 510 q^{37} - 2066 q^{38} - 898 q^{39} - 1248 q^{40} - 270 q^{41} + 915 q^{42} + 3256 q^{44} - 807 q^{45} - 1960 q^{46} + 1421 q^{47} + 632 q^{48} + 386 q^{49} + 141 q^{50} - 209 q^{51} + 2825 q^{52} - 21 q^{53} + 2368 q^{54} - 2258 q^{55} + 2521 q^{56} - 1723 q^{57} - 347 q^{58} + 1752 q^{59} + 2711 q^{60} - 1759 q^{61} - 395 q^{62} - 2204 q^{63} + 222 q^{64} - 1151 q^{65} + 160 q^{66} - 3001 q^{67} + 1921 q^{68} - 1660 q^{69} - 1597 q^{70} - 727 q^{71} - 9100 q^{72} - 4623 q^{73} - 2649 q^{74} - 1027 q^{75} - 874 q^{76} - 3556 q^{77} - 4979 q^{78} + 546 q^{79} - 5809 q^{80} - 410 q^{81} + 4397 q^{82} - 492 q^{83} - 10611 q^{84} + 1723 q^{85} + 5937 q^{87} - 3974 q^{88} - 5218 q^{89} + 10492 q^{90} - 1104 q^{91} + 1060 q^{92} - 1997 q^{93} + 2134 q^{94} + 6346 q^{95} - 11984 q^{96} + 2590 q^{97} - 6270 q^{98} - 2693 q^{99}+O(q^{100}) \) 30 * q - 6 * q^2 - 2 * q^3 + 114 * q^4 - 27 * q^5 + 8 * q^6 - 48 * q^7 - 90 * q^8 + 216 * q^9 - 27 * q^10 + 80 * q^11 + 36 * q^12 - 13 * q^13 + 36 * q^14 + 16 * q^15 + 318 * q^16 + 66 * q^17 - 80 * q^18 - 254 * q^19 - 312 * q^20 - 548 * q^21 - 305 * q^22 - 105 * q^23 + 123 * q^24 + 523 * q^25 - 549 * q^26 + 10 * q^27 - 578 * q^28 - 793 * q^29 - 1560 * q^30 - 359 * q^31 - 676 * q^32 - 208 * q^33 - 1007 * q^34 - 514 * q^35 + 776 * q^36 - 510 * q^37 - 2066 * q^38 - 898 * q^39 - 1248 * q^40 - 270 * q^41 + 915 * q^42 + 3256 * q^44 - 807 * q^45 - 1960 * q^46 + 1421 * q^47 + 632 * q^48 + 386 * q^49 + 141 * q^50 - 209 * q^51 + 2825 * q^52 - 21 * q^53 + 2368 * q^54 - 2258 * q^55 + 2521 * q^56 - 1723 * q^57 - 347 * q^58 + 1752 * q^59 + 2711 * q^60 - 1759 * q^61 - 395 * q^62 - 2204 * q^63 + 222 * q^64 - 1151 * q^65 + 160 * q^66 - 3001 * q^67 + 1921 * q^68 - 1660 * q^69 - 1597 * q^70 - 727 * q^71 - 9100 * q^72 - 4623 * q^73 - 2649 * q^74 - 1027 * q^75 - 874 * q^76 - 3556 * q^77 - 4979 * q^78 + 546 * q^79 - 5809 * q^80 - 410 * q^81 + 4397 * q^82 - 492 * q^83 - 10611 * q^84 + 1723 * q^85 + 5937 * q^87 - 3974 * q^88 - 5218 * q^89 + 10492 * q^90 - 1104 * q^91 + 1060 * q^92 - 1997 * q^93 + 2134 * q^94 + 6346 * q^95 - 11984 * q^96 + 2590 * q^97 - 6270 * q^98 - 2693 * q^99

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