LMFDB - Embedded newform 1849.4.a.j.1.6

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			1.6
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)

$f(q)$	$=$	$q-4.32699 q^{2} +4.31725 q^{3} +10.7228 q^{4} -7.58298 q^{5} -18.6807 q^{6} -15.1566 q^{7} -11.7817 q^{8} -8.36138 q^{9} +O(q^{10})$	\(q-4.32699 q^{2} +4.31725 q^{3} +10.7228 q^{4} -7.58298 q^{5} -18.6807 q^{6} -15.1566 q^{7} -11.7817 q^{8} -8.36138 q^{9} +32.8115 q^{10} -4.61811 q^{11} +46.2932 q^{12} +4.43823 q^{13} +65.5824 q^{14} -32.7376 q^{15} -34.8034 q^{16} -58.9409 q^{17} +36.1796 q^{18} +40.0098 q^{19} -81.3111 q^{20} -65.4347 q^{21} +19.9825 q^{22} +205.181 q^{23} -50.8646 q^{24} -67.4984 q^{25} -19.2042 q^{26} -152.664 q^{27} -162.522 q^{28} +248.436 q^{29} +141.655 q^{30} -63.1080 q^{31} +244.848 q^{32} -19.9375 q^{33} +255.037 q^{34} +114.932 q^{35} -89.6578 q^{36} +151.955 q^{37} -173.122 q^{38} +19.1609 q^{39} +89.3405 q^{40} -292.670 q^{41} +283.135 q^{42} -49.5192 q^{44} +63.4042 q^{45} -887.818 q^{46} -100.429 q^{47} -150.255 q^{48} -113.278 q^{49} +292.065 q^{50} -254.462 q^{51} +47.5905 q^{52} +566.570 q^{53} +660.575 q^{54} +35.0190 q^{55} +178.571 q^{56} +172.732 q^{57} -1074.98 q^{58} +730.172 q^{59} -351.040 q^{60} +928.090 q^{61} +273.068 q^{62} +126.730 q^{63} -781.026 q^{64} -33.6550 q^{65} +86.2694 q^{66} -61.7700 q^{67} -632.014 q^{68} +885.819 q^{69} -497.310 q^{70} +66.4400 q^{71} +98.5114 q^{72} +408.464 q^{73} -657.506 q^{74} -291.407 q^{75} +429.019 q^{76} +69.9947 q^{77} -82.9092 q^{78} +583.475 q^{79} +263.913 q^{80} -433.330 q^{81} +1266.38 q^{82} -1149.74 q^{83} -701.646 q^{84} +446.948 q^{85} +1072.56 q^{87} +54.4092 q^{88} -977.540 q^{89} -274.349 q^{90} -67.2684 q^{91} +2200.13 q^{92} -272.453 q^{93} +434.554 q^{94} -303.394 q^{95} +1057.07 q^{96} +254.707 q^{97} +490.153 q^{98} +38.6137 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−4.32699	−1.52982	−0.764911	−	0.644136i	$-0.777217\pi$
$2$	−4.32699	−1.52982	−0.764911	+	0.644136i	$0.777217\pi$
$3$	4.31725	0.830855	0.415427	−	0.909626i	$-0.363632\pi$
$3$	4.31725	0.830855	0.415427	+	0.909626i	$0.363632\pi$
$4$	10.7228	1.34036
$5$	−7.58298	−0.678242	−0.339121	−	0.940743i	$-0.610130\pi$
$5$	−7.58298	−0.678242	−0.339121	+	0.940743i	$0.610130\pi$
$6$	−18.6807	−1.27106
$7$	−15.1566	−0.818379	−0.409189	−	0.912450i	$-0.634188\pi$
$7$	−15.1566	−0.818379	−0.409189	+	0.912450i	$0.634188\pi$
$8$	−11.7817	−0.520683
$9$	−8.36138	−0.309681
$10$	32.8115	1.03759
$11$	−4.61811	−0.126583	−0.0632914	−	0.997995i	$-0.520160\pi$
$11$	−4.61811	−0.126583	−0.0632914	+	0.997995i	$0.520160\pi$
$12$	46.2932	1.11364
$13$	4.43823	0.0946880	0.0473440	−	0.998879i	$-0.484924\pi$
$13$	4.43823	0.0946880	0.0473440	+	0.998879i	$0.484924\pi$
$14$	65.5824	1.25197
$15$	−32.7376	−0.563521
$16$	−34.8034	−0.543803
$17$	−58.9409	−0.840898	−0.420449	−	0.907316i	$-0.638127\pi$
$17$	−58.9409	−0.840898	−0.420449	+	0.907316i	$0.638127\pi$
$18$	36.1796	0.473756
$19$	40.0098	0.483099	0.241550	−	0.970388i	$-0.422344\pi$
$19$	40.0098	0.483099	0.241550	+	0.970388i	$0.422344\pi$
$20$	−81.3111	−0.909086
$21$	−65.4347	−0.679953
$22$	19.9825	0.193649
$23$	205.181	1.86014	0.930071	−	0.367379i	$-0.119745\pi$
$23$	205.181	1.86014	0.930071	+	0.367379i	$0.119745\pi$
$24$	−50.8646	−0.432612
$25$	−67.4984	−0.539987
$26$	−19.2042	−0.144856
$27$	−152.664	−1.08815
$28$	−162.522	−1.09692
$29$	248.436	1.59081	0.795403	−	0.606081i	$-0.207259\pi$
$29$	248.436	1.59081	0.795403	+	0.606081i	$0.207259\pi$
$30$	141.655	0.862086
$31$	−63.1080	−0.365630	−0.182815	−	0.983147i	$-0.558521\pi$
$31$	−63.1080	−0.365630	−0.182815	+	0.983147i	$0.558521\pi$
$32$	244.848	1.35260
$33$	−19.9375	−0.105172
$34$	255.037	1.28642
$35$	114.932	0.555059
$36$	−89.6578	−0.415082
$37$	151.955	0.675167	0.337583	−	0.941296i	$-0.390390\pi$
$37$	151.955	0.675167	0.337583	+	0.941296i	$0.390390\pi$
$38$	−173.122	−0.739056
$39$	19.1609	0.0786720
$40$	89.3405	0.353150
$41$	−292.670	−1.11481	−0.557407	−	0.830239i	$-0.688204\pi$
$41$	−292.670	−1.11481	−0.557407	+	0.830239i	$0.688204\pi$
$42$	283.135	1.04021
$43$	0	0
$43$	0	0
$44$	−49.5192	−0.169666
$45$	63.4042	0.210039
$46$	−887.818	−2.84569
$47$	−100.429	−0.311682	−0.155841	−	0.987782i	$-0.549809\pi$
$47$	−100.429	−0.311682	−0.155841	+	0.987782i	$0.549809\pi$
$48$	−150.255	−0.451821
$49$	−113.278	−0.330257
$50$	292.065	0.826084
$51$	−254.462	−0.698664
$52$	47.5905	0.126916
$53$	566.570	1.46838	0.734192	−	0.678942i	$-0.237561\pi$
$53$	566.570	1.46838	0.734192	+	0.678942i	$0.237561\pi$
$54$	660.575	1.66468
$55$	35.0190	0.0858539
$56$	178.571	0.426116
$57$	172.732	0.401385
$58$	−1074.98	−2.43365
$59$	730.172	1.61119	0.805596	−	0.592466i	$-0.201846\pi$
$59$	730.172	1.61119	0.805596	+	0.592466i	$0.201846\pi$
$60$	−351.040	−0.755318
$61$	928.090	1.94803	0.974014	−	0.226486i	$-0.0727238\pi$
$61$	928.090	1.94803	0.974014	+	0.226486i	$0.0727238\pi$
$62$	273.068	0.559349
$63$	126.730	0.253436
$64$	−781.026	−1.52544
$65$	−33.6550	−0.0642214
$66$	86.2694	0.160894
$67$	−61.7700	−0.112633	−0.0563164	−	0.998413i	$-0.517936\pi$
$67$	−61.7700	−0.112633	−0.0563164	+	0.998413i	$0.517936\pi$
$68$	−632.014	−1.12710
$69$	885.819	1.54551
$70$	−497.310	−0.849141
$71$	66.4400	0.111056	0.0555280	−	0.998457i	$-0.482316\pi$
$71$	66.4400	0.111056	0.0555280	+	0.998457i	$0.482316\pi$
$72$	98.5114	0.161246
$73$	408.464	0.654892	0.327446	−	0.944870i	$-0.393812\pi$
$73$	408.464	0.654892	0.327446	+	0.944870i	$0.393812\pi$
$74$	−657.506	−1.03289
$75$	−291.407	−0.448651
$76$	429.019	0.647525
$77$	69.9947	0.103593
$78$	−82.9092	−0.120354
$79$	583.475	0.830962	0.415481	−	0.909602i	$-0.363613\pi$
$79$	583.475	0.830962	0.415481	+	0.909602i	$0.363613\pi$
$80$	263.913	0.368830
$81$	−433.330	−0.594417
$82$	1266.38	1.70547
$83$	−1149.74	−1.52049	−0.760246	−	0.649635i	$-0.774922\pi$
$83$	−1149.74	−1.52049	−0.760246	+	0.649635i	$0.774922\pi$
$84$	−701.646	−0.911379
$85$	446.948	0.570333
$86$	0	0
$87$	1072.56	1.32173
$88$	54.4092	0.0659096
$89$	−977.540	−1.16426	−0.582129	−	0.813096i	$-0.697780\pi$
$89$	−977.540	−1.16426	−0.582129	+	0.813096i	$0.697780\pi$
$90$	−274.349	−0.321322
$91$	−67.2684	−0.0774906
$92$	2200.13	2.49325
$93$	−272.453	−0.303785
$94$	434.554	0.476818
$95$	−303.394	−0.327658
$96$	1057.07	1.12382
$97$	254.707	0.266614	0.133307	−	0.991075i	$-0.457440\pi$
$97$	254.707	0.266614	0.133307	+	0.991075i	$0.457440\pi$
$98$	490.153	0.505234
$99$	38.6137	0.0392003
$100$	−723.775	−0.723775
$101$	−1396.17	−1.37548	−0.687742	−	0.725955i	$-0.741398\pi$
$101$	−1396.17	−1.37548	−0.687742	+	0.725955i	$0.741398\pi$
$102$	1101.06	1.06883
$103$	−770.884	−0.737451	−0.368725	−	0.929538i	$-0.620206\pi$
$103$	−770.884	−0.737451	−0.368725	+	0.929538i	$0.620206\pi$
$104$	−52.2900	−0.0493025
$105$	496.190	0.461173
$106$	−2451.54	−2.24637
$107$	−559.189	−0.505223	−0.252611	−	0.967568i	$-0.581289\pi$
$107$	−559.189	−0.505223	−0.252611	+	0.967568i	$0.581289\pi$
$108$	−1636.99	−1.45851
$109$	309.168	0.271678	0.135839	−	0.990731i	$-0.456627\pi$
$109$	309.168	0.271678	0.135839	+	0.990731i	$0.456627\pi$
$110$	−151.527	−0.131341
$111$	656.025	0.560966
$112$	527.500	0.445036
$113$	−1649.23	−1.37298	−0.686490	−	0.727139i	$-0.740850\pi$
$113$	−1649.23	−1.37298	−0.686490	+	0.727139i	$0.740850\pi$
$114$	−747.411	−0.614048
$115$	−1555.89	−1.26163
$116$	2663.94	2.13225
$117$	−37.1098	−0.0293231
$118$	−3159.45	−2.46484
$119$	893.343	0.688173
$120$	385.705	0.293416
$121$	−1309.67	−0.983977
$122$	−4015.84	−2.98014
$123$	−1263.53	−0.926249
$124$	−676.697	−0.490074
$125$	1459.71	1.04448
$126$	−548.359	−0.387712
$127$	−1484.24	−1.03705	−0.518525	−	0.855062i	$-0.673519\pi$
$127$	−1484.24	−1.03705	−0.518525	+	0.855062i	$0.673519\pi$
$128$	1420.71	0.981050
$129$	0	0
$130$	145.625	0.0982474
$131$	−2344.14	−1.56343	−0.781713	−	0.623638i	$-0.785654\pi$
$131$	−2344.14	−1.56343	−0.781713	+	0.623638i	$0.785654\pi$
$132$	−213.787	−0.140968
$133$	−606.412	−0.395358
$134$	267.278	0.172308
$135$	1157.65	0.738032
$136$	694.425	0.437842
$137$	1016.32	0.633797	0.316899	−	0.948459i	$-0.397359\pi$
$137$	1016.32	0.633797	0.316899	+	0.948459i	$0.397359\pi$
$138$	−3832.93	−2.36435
$139$	1637.27	0.999073	0.499536	−	0.866293i	$-0.333504\pi$
$139$	1637.27	0.999073	0.499536	+	0.866293i	$0.333504\pi$
$140$	1232.40	0.743976
$141$	−433.576	−0.258962
$142$	−287.485	−0.169896
$143$	−20.4962	−0.0119859
$144$	291.004	0.168405
$145$	−1883.88	−1.07895
$146$	−1767.42	−1.00187
$147$	−489.049	−0.274395
$148$	1629.39	0.904964
$149$	2190.65	1.20446	0.602232	−	0.798321i	$-0.294278\pi$
$149$	2190.65	1.20446	0.602232	+	0.798321i	$0.294278\pi$
$150$	1260.92	0.686356
$151$	2993.00	1.61302	0.806512	−	0.591218i	$-0.201353\pi$
$151$	2993.00	1.61302	0.806512	+	0.591218i	$0.201353\pi$
$152$	−471.385	−0.251542
$153$	492.827	0.260410
$154$	−302.866	−0.158478
$155$	478.546	0.247986
$156$	205.460	0.105448
$157$	1276.61	0.648948	0.324474	−	0.945895i	$-0.394813\pi$
$157$	1276.61	0.648948	0.324474	+	0.945895i	$0.394813\pi$
$158$	−2524.69	−1.27122
$159$	2446.02	1.22001
$160$	−1856.67	−0.917394
$161$	−3109.85	−1.52230
$162$	1875.01	0.909352
$163$	1124.07	0.540148	0.270074	−	0.962840i	$-0.412952\pi$
$163$	1124.07	0.540148	0.270074	+	0.962840i	$0.412952\pi$
$164$	−3138.26	−1.49425
$165$	151.186	0.0713321
$166$	4974.93	2.32608
$167$	3171.73	1.46968	0.734838	−	0.678242i	$-0.237258\pi$
$167$	3171.73	1.46968	0.734838	+	0.678242i	$0.237258\pi$
$168$	770.933	0.354040
$169$	−2177.30	−0.991034
$170$	−1933.94	−0.872508
$171$	−334.537	−0.149607
$172$	0	0
$173$	−364.520	−0.160196	−0.0800982	−	0.996787i	$-0.525523\pi$
$173$	−364.520	−0.160196	−0.0800982	+	0.996787i	$0.525523\pi$
$174$	−4640.95	−2.02201
$175$	1023.05	0.441914
$176$	160.726	0.0688361
$177$	3152.33	1.33867
$178$	4229.81	1.78111
$179$	−2292.73	−0.957355	−0.478677	−	0.877991i	$-0.658884\pi$
$179$	−2292.73	−0.957355	−0.478677	+	0.877991i	$0.658884\pi$
$180$	679.873	0.281526
$181$	989.703	0.406431	0.203216	−	0.979134i	$-0.434861\pi$
$181$	989.703	0.406431	0.203216	+	0.979134i	$0.434861\pi$
$182$	291.070	0.118547
$183$	4006.79	1.61853
$184$	−2417.39	−0.968545
$185$	−1152.27	−0.457927
$186$	1178.90	0.464737
$187$	272.195	0.106443
$188$	−1076.88	−0.417764
$189$	2313.86	0.890522
$190$	1312.78	0.501259
$191$	−1523.76	−0.577254	−0.288627	−	0.957442i	$-0.593199\pi$
$191$	−1523.76	−0.577254	−0.288627	+	0.957442i	$0.593199\pi$
$192$	−3371.88	−1.26742
$193$	−794.523	−0.296326	−0.148163	−	0.988963i	$-0.547336\pi$
$193$	−794.523	−0.296326	−0.148163	+	0.988963i	$0.547336\pi$
$194$	−1102.12	−0.407872
$195$	−145.297	−0.0533587
$196$	−1214.66	−0.442661
$197$	−5363.39	−1.93973	−0.969863	−	0.243652i	$-0.921655\pi$
$197$	−5363.39	−1.93973	−0.969863	+	0.243652i	$0.921655\pi$
$198$	−167.081	−0.0599695
$199$	4058.33	1.44566	0.722832	−	0.691024i	$-0.242840\pi$
$199$	4058.33	1.44566	0.722832	+	0.691024i	$0.242840\pi$
$200$	795.247	0.281162
$201$	−266.676	−0.0935815
$202$	6041.21	2.10425
$203$	−3765.44	−1.30188
$204$	−2728.56	−0.936458
$205$	2219.31	0.756114
$206$	3335.61	1.12817
$207$	−1715.60	−0.576050
$208$	−154.465	−0.0514916
$209$	−184.770	−0.0611521
$210$	−2147.01	−0.705513
$211$	−1338.34	−0.436658	−0.218329	−	0.975875i	$-0.570061\pi$
$211$	−1338.34	−0.436658	−0.218329	+	0.975875i	$0.570061\pi$
$212$	6075.24	1.96816
$213$	286.838	0.0922715
$214$	2419.60	0.772901
$215$	0	0
$216$	1798.64	0.566584
$217$	956.501	0.299224
$218$	−1337.77	−0.415619
$219$	1763.44	0.544120
$220$	375.503	0.115075
$221$	−261.593	−0.0796230
$222$	−2838.62	−0.858177
$223$	334.702	0.100508	0.0502540	−	0.998736i	$-0.483997\pi$
$223$	334.702	0.100508	0.0502540	+	0.998736i	$0.483997\pi$
$224$	−3711.05	−1.10694
$225$	564.380	0.167224
$226$	7136.22	2.10042
$227$	−1321.17	−0.386294	−0.193147	−	0.981170i	$-0.561869\pi$
$227$	−1321.17	−0.386294	−0.193147	+	0.981170i	$0.561869\pi$
$228$	1852.18	0.537999
$229$	−1838.72	−0.530594	−0.265297	−	0.964167i	$-0.585470\pi$
$229$	−1838.72	−0.530594	−0.265297	+	0.964167i	$0.585470\pi$
$230$	6732.31	1.93007
$231$	302.184	0.0860705
$232$	−2927.00	−0.828306
$233$	−2469.71	−0.694403	−0.347202	−	0.937791i	$-0.612868\pi$
$233$	−2469.71	−0.694403	−0.347202	+	0.937791i	$0.612868\pi$
$234$	160.574	0.0448591
$235$	761.550	0.211396
$236$	7829.52	2.15957
$237$	2519.00	0.690409
$238$	−3865.48	−1.05278
$239$	−6493.53	−1.75746	−0.878728	−	0.477324i	$-0.841607\pi$
$239$	−6493.53	−1.75746	−0.878728	+	0.477324i	$0.841607\pi$
$240$	1139.38	0.306444
$241$	546.111	0.145967	0.0729836	−	0.997333i	$-0.476748\pi$
$241$	546.111	0.145967	0.0729836	+	0.997333i	$0.476748\pi$
$242$	5666.94	1.50531
$243$	2251.13	0.594280
$244$	9951.76	2.61105
$245$	858.985	0.223994
$246$	5467.28	1.41700
$247$	177.573	0.0457437
$248$	743.520	0.190377
$249$	−4963.73	−1.26331
$250$	−6316.16	−1.59788
$251$	3433.75	0.863493	0.431746	−	0.901995i	$-0.357898\pi$
$251$	3433.75	0.863493	0.431746	+	0.901995i	$0.357898\pi$
$252$	1358.91	0.339694
$253$	−947.550	−0.235462
$254$	6422.31	1.58650
$255$	1929.58	0.473864
$256$	100.803	0.0246101
$257$	485.127	0.117749	0.0588743	−	0.998265i	$-0.481249\pi$
$257$	485.127	0.117749	0.0588743	+	0.998265i	$0.481249\pi$
$258$	0	0
$259$	−2303.11	−0.552542
$260$	−360.878	−0.0860795
$261$	−2077.27	−0.492642
$262$	10143.1	2.39176
$263$	1318.47	0.309126	0.154563	−	0.987983i	$-0.450603\pi$
$263$	1318.47	0.309126	0.154563	+	0.987983i	$0.450603\pi$
$264$	234.898	0.0547613
$265$	−4296.29	−0.995920
$266$	2623.94	0.604827
$267$	−4220.28	−0.967330
$268$	−662.350	−0.150968
$269$	−1988.06	−0.450610	−0.225305	−	0.974288i	$-0.572338\pi$
$269$	−1988.06	−0.450610	−0.225305	+	0.974288i	$0.572338\pi$
$270$	−5009.13	−1.12906
$271$	7179.43	1.60930	0.804648	−	0.593752i	$-0.202354\pi$
$271$	7179.43	1.60930	0.804648	+	0.593752i	$0.202354\pi$
$272$	2051.34	0.457283
$273$	−290.414	−0.0643834
$274$	−4397.61	−0.969597
$275$	311.715	0.0683531
$276$	9498.50	2.07153
$277$	−3471.68	−0.753043	−0.376521	−	0.926408i	$-0.622880\pi$
$277$	−3471.68	−0.753043	−0.376521	+	0.926408i	$0.622880\pi$
$278$	−7084.44	−1.52840
$279$	527.670	0.113229
$280$	−1354.10	−0.289010
$281$	2360.71	0.501168	0.250584	−	0.968095i	$-0.419377\pi$
$281$	2360.71	0.501168	0.250584	+	0.968095i	$0.419377\pi$
$282$	1876.08	0.396166
$283$	−3584.19	−0.752854	−0.376427	−	0.926446i	$-0.622847\pi$
$283$	−3584.19	−0.752854	−0.376427	+	0.926446i	$0.622847\pi$
$284$	712.426	0.148855
$285$	−1309.83	−0.272236
$286$	88.6870	0.0183363
$287$	4435.88	0.912340
$288$	−2047.26	−0.418876
$289$	−1438.97	−0.292890
$290$	8151.55	1.65060
$291$	1099.63	0.221518
$292$	4379.90	0.877788
$293$	−218.871	−0.0436403	−0.0218201	−	0.999762i	$-0.506946\pi$
$293$	−218.871	−0.0436403	−0.0218201	+	0.999762i	$0.506946\pi$
$294$	2116.11	0.419776
$295$	−5536.88	−1.09278
$296$	−1790.29	−0.351548
$297$	705.018	0.137742
$298$	−9478.93	−1.84262
$299$	910.643	0.176133
$300$	−3124.72	−0.601352
$301$	0	0
$302$	−12950.7	−2.46764
$303$	−6027.60	−1.14283
$304$	−1392.48	−0.262711
$305$	−7037.69	−1.32124
$306$	−2132.46	−0.398381
$307$	−2450.11	−0.455488	−0.227744	−	0.973721i	$-0.573135\pi$
$307$	−2450.11	−0.455488	−0.227744	+	0.973721i	$0.573135\pi$
$308$	750.542	0.138851
$309$	−3328.09	−0.612714
$310$	−2070.67	−0.379374
$311$	−818.907	−0.149312	−0.0746559	−	0.997209i	$-0.523786\pi$
$311$	−818.907	−0.149312	−0.0746559	+	0.997209i	$0.523786\pi$
$312$	−225.749	−0.0409632
$313$	1988.86	0.359159	0.179579	−	0.983743i	$-0.442526\pi$
$313$	1988.86	0.359159	0.179579	+	0.983743i	$0.442526\pi$
$314$	−5523.89	−0.992775
$315$	−960.991	−0.171891
$316$	6256.51	1.11379
$317$	−4046.09	−0.716880	−0.358440	−	0.933553i	$-0.616691\pi$
$317$	−4046.09	−0.716880	−0.358440	+	0.933553i	$0.616691\pi$
$318$	−10583.9	−1.86640
$319$	−1147.30	−0.201369
$320$	5922.51	1.03462
$321$	−2414.16	−0.419767
$322$	13456.3	2.32885
$323$	−2358.22	−0.406237
$324$	−4646.53	−0.796730
$325$	−299.574	−0.0511303
$326$	−4863.85	−0.826330
$327$	1334.75	0.225725
$328$	3448.16	0.580465
$329$	1522.16	0.255074
$330$	−654.179	−0.109125
$331$	−11197.4	−1.85940	−0.929702	−	0.368313i	$-0.879935\pi$
$331$	−11197.4	−1.85940	−0.929702	+	0.368313i	$0.879935\pi$
$332$	−12328.5	−2.03800
$333$	−1270.55	−0.209086
$334$	−13724.1	−2.24834
$335$	468.400	0.0763924
$336$	2277.35	0.369760
$337$	9070.66	1.46620	0.733102	−	0.680119i	$-0.238072\pi$
$337$	9070.66	1.46620	0.733102	+	0.680119i	$0.238072\pi$
$338$	9421.16	1.51611
$339$	−7120.15	−1.14075
$340$	4792.55	0.764449
$341$	291.439	0.0462825
$342$	1447.54	0.228871
$343$	6915.62	1.08865
$344$	0	0
$345$	−6717.15	−1.04823
$346$	1577.28	0.245072
$347$	2409.25	0.372724	0.186362	−	0.982481i	$-0.440330\pi$
$347$	2409.25	0.372724	0.186362	+	0.982481i	$0.440330\pi$
$348$	11500.9	1.77159
$349$	−10646.2	−1.63289	−0.816443	−	0.577426i	$-0.804057\pi$
$349$	−10646.2	−1.63289	−0.816443	+	0.577426i	$0.804057\pi$
$350$	−4426.71	−0.676050
$351$	−677.557	−0.103035
$352$	−1130.73	−0.171217
$353$	6413.78	0.967056	0.483528	−	0.875329i	$-0.339355\pi$
$353$	6413.78	0.967056	0.483528	+	0.875329i	$0.339355\pi$
$354$	−13640.1	−2.04792
$355$	−503.814	−0.0753229
$356$	−10482.0	−1.56052
$357$	3856.78	0.571772
$358$	9920.61	1.46458
$359$	3428.56	0.504046	0.252023	−	0.967721i	$-0.418904\pi$
$359$	3428.56	0.504046	0.252023	+	0.967721i	$0.418904\pi$
$360$	−747.010	−0.109364
$361$	−5258.21	−0.766615
$362$	−4282.44	−0.621768
$363$	−5654.18	−0.817542
$364$	−721.309	−0.103865
$365$	−3097.38	−0.444175
$366$	−17337.4	−2.47606
$367$	−10004.5	−1.42297	−0.711483	−	0.702703i	$-0.751976\pi$
$367$	−10004.5	−1.42297	−0.711483	+	0.702703i	$0.751976\pi$
$368$	−7141.00	−1.01155
$369$	2447.13	0.345237
$370$	4985.85	0.700547
$371$	−8587.26	−1.20169
$372$	−2921.47	−0.407180
$373$	6168.67	0.856305	0.428153	−	0.903706i	$-0.359165\pi$
$373$	6168.67	0.856305	0.428153	+	0.903706i	$0.359165\pi$
$374$	−1177.79	−0.162839
$375$	6301.94	0.867815
$376$	1183.22	0.162288
$377$	1102.62	0.150630
$378$	−10012.1	−1.36234
$379$	12794.7	1.73409	0.867045	−	0.498230i	$-0.166016\pi$
$379$	12794.7	1.73409	0.867045	+	0.498230i	$0.166016\pi$
$380$	−3253.24	−0.439179
$381$	−6407.85	−0.861638
$382$	6593.30	0.883096
$383$	−3461.90	−0.461867	−0.230933	−	0.972970i	$-0.574178\pi$
$383$	−3461.90	−0.461867	−0.230933	+	0.972970i	$0.574178\pi$
$384$	6133.56	0.815110
$385$	−530.768	−0.0702610
$386$	3437.89	0.453327
$387$	0	0
$388$	2731.18	0.357358
$389$	−13493.8	−1.75877	−0.879384	−	0.476113i	$-0.842045\pi$
$389$	−13493.8	−1.75877	−0.879384	+	0.476113i	$0.842045\pi$
$390$	628.699	0.0816293
$391$	−12093.6	−1.56419
$392$	1334.61	0.171959
$393$	−10120.2	−1.29898
$394$	23207.3	2.96743
$395$	−4424.48	−0.563594
$396$	414.049	0.0525423
$397$	−8336.42	−1.05389	−0.526943	−	0.849900i	$-0.676662\pi$
$397$	−8336.42	−1.05389	−0.526943	+	0.849900i	$0.676662\pi$
$398$	−17560.3	−2.21161
$399$	−2618.03	−0.328485
$400$	2349.17	0.293647
$401$	−1635.58	−0.203684	−0.101842	−	0.994801i	$-0.532474\pi$
$401$	−1635.58	−0.203684	−0.101842	+	0.994801i	$0.532474\pi$
$402$	1153.91	0.143163
$403$	−280.088	−0.0346208
$404$	−14970.9	−1.84364
$405$	3285.93	0.403159
$406$	16293.0	1.99165
$407$	−701.742	−0.0854646
$408$	2998.00	0.363783
$409$	−13688.6	−1.65490	−0.827452	−	0.561536i	$-0.810211\pi$
$409$	−13688.6	−1.65490	−0.827452	+	0.561536i	$0.810211\pi$
$410$	−9602.94	−1.15672
$411$	4387.71	0.526593
$412$	−8266.07	−0.988446
$413$	−11066.9	−1.31856
$414$	7423.38	0.881255
$415$	8718.49	1.03126
$416$	1086.69	0.128075
$417$	7068.48	0.830084
$418$	799.496	0.0935518
$419$	253.682	0.0295780	0.0147890	−	0.999891i	$-0.495292\pi$
$419$	253.682	0.0295780	0.0147890	+	0.999891i	$0.495292\pi$
$420$	5320.57	0.618136
$421$	−11417.5	−1.32174	−0.660872	−	0.750498i	$-0.729813\pi$
$421$	−11417.5	−1.32174	−0.660872	+	0.750498i	$0.729813\pi$
$422$	5790.96	0.668009
$423$	839.724	0.0965219
$424$	−6675.17	−0.764563
$425$	3978.42	0.454074
$426$	−1241.15	−0.141159
$427$	−14066.7	−1.59422
$428$	−5996.10	−0.677178
$429$	−88.4873	−0.00995852
$430$	0	0
$431$	−4723.09	−0.527850	−0.263925	−	0.964543i	$-0.585017\pi$
$431$	−4723.09	−0.527850	−0.263925	+	0.964543i	$0.585017\pi$
$432$	5313.21	0.591741
$433$	−5231.32	−0.580604	−0.290302	−	0.956935i	$-0.593756\pi$
$433$	−5231.32	−0.580604	−0.290302	+	0.956935i	$0.593756\pi$
$434$	−4138.77	−0.457759
$435$	−8133.19	−0.896452
$436$	3315.16	0.364145
$437$	8209.27	0.898633
$438$	−7630.39	−0.832407
$439$	12258.1	1.33268	0.666342	−	0.745646i	$-0.267859\pi$
$439$	12258.1	1.33268	0.666342	+	0.745646i	$0.267859\pi$
$440$	−412.584	−0.0447027
$441$	947.161	0.102274
$442$	1131.91	0.121809
$443$	−15064.2	−1.61563	−0.807813	−	0.589439i	$-0.799349\pi$
$443$	−15064.2	−1.61563	−0.807813	+	0.589439i	$0.799349\pi$
$444$	7034.46	0.751893
$445$	7412.67	0.789649
$446$	−1448.25	−0.153759
$447$	9457.59	1.00073
$448$	11837.7	1.24839
$449$	−1007.95	−0.105943	−0.0529713	−	0.998596i	$-0.516869\pi$
$449$	−1007.95	−0.105943	−0.0529713	+	0.998596i	$0.516869\pi$
$450$	−2442.07	−0.255822
$451$	1351.58	0.141116
$452$	−17684.5	−1.84028
$453$	12921.5	1.34019
$454$	5716.67	0.590962
$455$	510.095	0.0525574
$456$	−2035.08	−0.208995
$457$	2569.85	0.263047	0.131523	−	0.991313i	$-0.458013\pi$
$457$	2569.85	0.263047	0.131523	+	0.991313i	$0.458013\pi$
$458$	7956.13	0.811715
$459$	8998.14	0.915027
$460$	−16683.5	−1.69103
$461$	−2634.42	−0.266155	−0.133077	−	0.991106i	$-0.542486\pi$
$461$	−2634.42	−0.266155	−0.133077	+	0.991106i	$0.542486\pi$
$462$	−1307.55	−0.131672
$463$	−9459.74	−0.949528	−0.474764	−	0.880113i	$-0.657467\pi$
$463$	−9459.74	−0.949528	−0.474764	+	0.880113i	$0.657467\pi$
$464$	−8646.40	−0.865085
$465$	2066.00	0.206040
$466$	10686.4	1.06231
$467$	−4529.63	−0.448835	−0.224418	−	0.974493i	$-0.572048\pi$
$467$	−4529.63	−0.448835	−0.224418	+	0.974493i	$0.572048\pi$
$468$	−397.922	−0.0393033
$469$	936.221	0.0921763
$470$	−3295.22	−0.323398
$471$	5511.45	0.539181
$472$	−8602.68	−0.838920
$473$	0	0
$474$	−10899.7	−1.05620
$475$	−2700.60	−0.260867
$476$	9579.17	0.922396
$477$	−4737.31	−0.454730
$478$	28097.5	2.68859
$479$	−14764.9	−1.40841	−0.704203	−	0.709999i	$-0.748695\pi$
$479$	−14764.9	−1.40841	−0.704203	+	0.709999i	$0.748695\pi$
$480$	−8015.72	−0.762221
$481$	674.410	0.0639302
$482$	−2363.02	−0.223304
$483$	−13426.0	−1.26481
$484$	−14043.4	−1.31888
$485$	−1931.44	−0.180829
$486$	−9740.62	−0.909143
$487$	−6246.07	−0.581184	−0.290592	−	0.956847i	$-0.593852\pi$
$487$	−6246.07	−0.581184	−0.290592	+	0.956847i	$0.593852\pi$
$488$	−10934.5	−1.01431
$489$	4852.89	0.448784
$490$	−3716.82	−0.342671
$491$	−9471.49	−0.870555	−0.435277	−	0.900296i	$-0.643350\pi$
$491$	−9471.49	−0.870555	−0.435277	+	0.900296i	$0.643350\pi$
$492$	−13548.6	−1.24150
$493$	−14643.0	−1.33771
$494$	−768.356	−0.0699797
$495$	−292.807	−0.0265873
$496$	2196.37	0.198830
$497$	−1007.00	−0.0908859
$498$	21478.0	1.93264
$499$	9708.83	0.870996	0.435498	−	0.900190i	$-0.356572\pi$
$499$	9708.83	0.870996	0.435498	+	0.900190i	$0.356572\pi$
$500$	15652.3	1.39998
$501$	13693.2	1.22109
$502$	−14857.8	−1.32099
$503$	5971.16	0.529306	0.264653	−	0.964344i	$-0.414743\pi$
$503$	5971.16	0.529306	0.264653	+	0.964344i	$0.414743\pi$
$504$	−1493.10	−0.131960
$505$	10587.1	0.932912
$506$	4100.04	0.360215
$507$	−9399.95	−0.823405
$508$	−15915.3	−1.39002
$509$	−10850.7	−0.944887	−0.472443	−	0.881361i	$-0.656628\pi$
$509$	−10850.7	−0.944887	−0.472443	+	0.881361i	$0.656628\pi$
$510$	−8349.29	−0.724927
$511$	−6190.92	−0.535949
$512$	−11801.9	−1.01870
$513$	−6108.05	−0.525686
$514$	−2099.14	−0.180134
$515$	5845.60	0.500170
$516$	0	0
$517$	463.791	0.0394536
$518$	9965.54	0.845291
$519$	−1573.72	−0.133100
$520$	396.514	0.0334390
$521$	−13234.7	−1.11290	−0.556450	−	0.830881i	$-0.687837\pi$
$521$	−13234.7	−1.11290	−0.556450	+	0.830881i	$0.687837\pi$
$522$	8988.31	0.753655
$523$	−15760.1	−1.31767	−0.658836	−	0.752286i	$-0.728951\pi$
$523$	−15760.1	−1.31767	−0.658836	+	0.752286i	$0.728951\pi$
$524$	−25135.9	−2.09555
$525$	4416.74	0.367166
$526$	−5705.00	−0.472908
$527$	3719.64	0.307457
$528$	693.892	0.0571928
$529$	29932.4	2.46013
$530$	18590.0	1.52358
$531$	−6105.24	−0.498955
$532$	−6502.46	−0.529920
$533$	−1298.94	−0.105560
$534$	18261.1	1.47984
$535$	4240.32	0.342663
$536$	727.756	0.0586461
$537$	−9898.27	−0.795423
$538$	8602.31	0.689353
$539$	523.130	0.0418048
$540$	12413.3	0.989226
$541$	−8717.14	−0.692753	−0.346376	−	0.938096i	$-0.612588\pi$
$541$	−8717.14	−0.692753	−0.346376	+	0.938096i	$0.612588\pi$
$542$	−31065.3	−2.46194
$543$	4272.79	0.337685
$544$	−14431.5	−1.13740
$545$	−2344.41	−0.184263
$546$	1256.62	0.0984952
$547$	13375.8	1.04553	0.522766	−	0.852476i	$-0.324900\pi$
$547$	13375.8	1.04553	0.522766	+	0.852476i	$0.324900\pi$
$548$	10897.9	0.849514
$549$	−7760.11	−0.603267
$550$	−1348.79	−0.104568
$551$	9939.88	0.768517
$552$	−10436.5	−0.804720
$553$	−8843.48	−0.680042
$554$	15021.9	1.15202
$555$	−4974.63	−0.380471
$556$	17556.2	1.33911
$557$	18892.9	1.43720	0.718598	−	0.695426i	$-0.244784\pi$
$557$	18892.9	1.43720	0.718598	+	0.695426i	$0.244784\pi$
$558$	−2283.22	−0.173220
$559$	0	0
$560$	−4000.02	−0.301843
$561$	1175.13	0.0884389
$562$	−10214.8	−0.766698
$563$	−24531.2	−1.83635	−0.918177	−	0.396170i	$-0.870339\pi$
$563$	−24531.2	−1.83635	−0.918177	+	0.396170i	$0.870339\pi$
$564$	−4649.17	−0.347102
$565$	12506.1	0.931214
$566$	15508.7	1.15173
$567$	6567.80	0.486458
$568$	−782.778	−0.0578251
$569$	18497.1	1.36281	0.681403	−	0.731908i	$-0.261370\pi$
$569$	18497.1	1.36281	0.681403	+	0.731908i	$0.261370\pi$
$570$	5667.60	0.416473
$571$	−4775.61	−0.350006	−0.175003	−	0.984568i	$-0.555993\pi$
$571$	−4775.61	−0.350006	−0.175003	+	0.984568i	$0.555993\pi$
$572$	−219.778	−0.0160653
$573$	−6578.46	−0.479614
$574$	−19194.0	−1.39572
$575$	−13849.4	−1.00445
$576$	6530.46	0.472400
$577$	−17822.1	−1.28586	−0.642932	−	0.765923i	$-0.722282\pi$
$577$	−17822.1	−1.28586	−0.642932	+	0.765923i	$0.722282\pi$
$578$	6226.41	0.448070
$579$	−3430.15	−0.246204
$580$	−20200.6	−1.44618
$581$	17426.2	1.24434
$582$	−4758.10	−0.338883
$583$	−2616.48	−0.185872
$584$	−4812.41	−0.340991
$585$	281.403	0.0198881
$586$	947.054	0.0667618
$587$	−8344.83	−0.586760	−0.293380	−	0.955996i	$-0.594780\pi$
$587$	−8344.83	−0.586760	−0.293380	+	0.955996i	$0.594780\pi$
$588$	−5244.00	−0.367787
$589$	−2524.94	−0.176635
$590$	23958.0	1.67176
$591$	−23155.1	−1.61163
$592$	−5288.53	−0.367158
$593$	12261.2	0.849084	0.424542	−	0.905408i	$-0.360435\pi$
$593$	12261.2	0.849084	0.424542	+	0.905408i	$0.360435\pi$
$594$	−3050.60	−0.210720
$595$	−6774.20	−0.466748
$596$	23490.0	1.61441
$597$	17520.8	1.20114
$598$	−3940.34	−0.269452
$599$	20681.9	1.41075	0.705374	−	0.708835i	$-0.250779\pi$
$599$	20681.9	1.41075	0.705374	+	0.708835i	$0.250779\pi$
$600$	3433.28	0.233605
$601$	−7757.58	−0.526519	−0.263260	−	0.964725i	$-0.584798\pi$
$601$	−7757.58	−0.526519	−0.263260	+	0.964725i	$0.584798\pi$
$602$	0	0
$603$	516.482	0.0348802
$604$	32093.4	2.16203
$605$	9931.22	0.667375
$606$	26081.4	1.74832
$607$	−1766.42	−0.118116	−0.0590582	−	0.998255i	$-0.518810\pi$
$607$	−1766.42	−0.118116	−0.0590582	+	0.998255i	$0.518810\pi$
$608$	9796.31	0.653442
$609$	−16256.3	−1.08167
$610$	30452.0	2.02126
$611$	−445.726	−0.0295125
$612$	5284.51	0.349042
$613$	10071.3	0.663584	0.331792	−	0.943352i	$-0.392347\pi$
$613$	10071.3	0.663584	0.331792	+	0.943352i	$0.392347\pi$
$614$	10601.6	0.696816
$615$	9581.32	0.628221
$616$	−824.658	−0.0539390
$617$	21039.4	1.37280	0.686399	−	0.727225i	$-0.259191\pi$
$617$	21039.4	1.37280	0.686399	+	0.727225i	$0.259191\pi$
$618$	14400.6	0.937344
$619$	9690.78	0.629249	0.314625	−	0.949216i	$-0.398121\pi$
$619$	9690.78	0.629249	0.314625	+	0.949216i	$0.398121\pi$
$620$	5131.38	0.332389
$621$	−31323.8	−2.02412
$622$	3543.40	0.228420
$623$	14816.2	0.952804
$624$	−666.865	−0.0427820
$625$	−2631.66	−0.168426
$626$	−8605.76	−0.549449
$627$	−797.696	−0.0508085
$628$	13688.9	0.869821
$629$	−8956.34	−0.567747
$630$	4158.20	0.262963
$631$	20307.8	1.28121	0.640603	−	0.767872i	$-0.278684\pi$
$631$	20307.8	1.28121	0.640603	+	0.767872i	$0.278684\pi$
$632$	−6874.34	−0.432668
$633$	−5777.92	−0.362799
$634$	17507.4	1.09670
$635$	11255.0	0.703372
$636$	26228.3	1.63525
$637$	−502.754	−0.0312713
$638$	4964.37	0.308058
$639$	−555.531	−0.0343919
$640$	−10773.2	−0.665390
$641$	−29060.8	−1.79069	−0.895344	−	0.445374i	$-0.853071\pi$
$641$	−29060.8	−1.79069	−0.895344	+	0.445374i	$0.853071\pi$
$642$	10446.0	0.642168
$643$	13231.3	0.811494	0.405747	−	0.913985i	$-0.367011\pi$
$643$	13231.3	0.811494	0.405747	+	0.913985i	$0.367011\pi$
$644$	−33346.4	−2.04042
$645$	0	0
$646$	10204.0	0.621471
$647$	−25154.9	−1.52850	−0.764252	−	0.644918i	$-0.776891\pi$
$647$	−25154.9	−1.52850	−0.764252	+	0.644918i	$0.776891\pi$
$648$	5105.37	0.309503
$649$	−3372.01	−0.203949
$650$	1296.25	0.0782203
$651$	4129.45	0.248611
$652$	12053.2	0.723990
$653$	6734.12	0.403563	0.201781	−	0.979431i	$-0.435327\pi$
$653$	6734.12	0.403563	0.201781	+	0.979431i	$0.435327\pi$
$654$	−5775.46	−0.345319
$655$	17775.6	1.06038
$656$	10185.9	0.606239
$657$	−3415.32	−0.202807
$658$	−6586.36	−0.390217
$659$	−3685.41	−0.217850	−0.108925	−	0.994050i	$-0.534741\pi$
$659$	−3685.41	−0.217850	−0.108925	+	0.994050i	$0.534741\pi$
$660$	1621.14	0.0956103
$661$	3938.81	0.231773	0.115887	−	0.993262i	$-0.463029\pi$
$661$	3938.81	0.231773	0.115887	+	0.993262i	$0.463029\pi$
$662$	48450.9	2.84456
$663$	−1129.36	−0.0661551
$664$	13546.0	0.791695
$665$	4598.41	0.268149
$666$	5497.66	0.319865
$667$	50974.4	2.95913
$668$	34010.0	1.96989
$669$	1444.99	0.0835076
$670$	−2026.76	−0.116867
$671$	−4286.02	−0.246587
$672$	−16021.5	−0.919708
$673$	−31975.1	−1.83143	−0.915714	−	0.401831i	$-0.868374\pi$
$673$	−31975.1	−1.83143	−0.915714	+	0.401831i	$0.868374\pi$
$674$	−39248.7	−2.24303
$675$	10304.6	0.587589
$676$	−23346.9	−1.32834
$677$	−10683.3	−0.606487	−0.303244	−	0.952913i	$-0.598070\pi$
$677$	−10683.3	−0.606487	−0.303244	+	0.952913i	$0.598070\pi$
$678$	30808.8	1.74514
$679$	−3860.49	−0.218191
$680$	−5265.81	−0.296963
$681$	−5703.80	−0.320954
$682$	−1261.06	−0.0708039
$683$	−15614.9	−0.874800	−0.437400	−	0.899267i	$-0.644101\pi$
$683$	−15614.9	−0.874800	−0.437400	+	0.899267i	$0.644101\pi$
$684$	−3587.19	−0.200526
$685$	−7706.75	−0.429868
$686$	−29923.8	−1.66545
$687$	−7938.21	−0.440847
$688$	0	0
$689$	2514.57	0.139038
$690$	29065.0	1.60360
$691$	−25222.6	−1.38858	−0.694292	−	0.719693i	$-0.744283\pi$
$691$	−25222.6	−1.38858	−0.694292	+	0.719693i	$0.744283\pi$
$692$	−3908.70	−0.214720
$693$	−585.252	−0.0320807
$694$	−10424.8	−0.570201
$695$	−12415.4	−0.677614
$696$	−12636.6	−0.688202
$697$	17250.2	0.937445
$698$	46065.9	2.49802
$699$	−10662.3	−0.576948
$700$	10970.0	0.592322
$701$	−4178.65	−0.225143	−0.112572	−	0.993644i	$-0.535909\pi$
$701$	−4178.65	−0.225143	−0.112572	+	0.993644i	$0.535909\pi$
$702$	2931.78	0.157625
$703$	6079.68	0.326173
$704$	3606.86	0.193095
$705$	3287.80	0.175639
$706$	−27752.3	−1.47942
$707$	21161.1	1.12567
$708$	33802.0	1.79429
$709$	−27765.1	−1.47072	−0.735360	−	0.677677i	$-0.762987\pi$
$709$	−27765.1	−1.47072	−0.735360	+	0.677677i	$0.762987\pi$
$710$	2180.00	0.115231
$711$	−4878.65	−0.257333
$712$	11517.1	0.606210
$713$	−12948.6	−0.680124
$714$	−16688.3	−0.874709
$715$	155.423	0.00812933
$716$	−24584.6	−1.28320
$717$	−28034.2	−1.46019
$718$	−14835.4	−0.771101
$719$	−25705.2	−1.33330	−0.666649	−	0.745372i	$-0.732272\pi$
$719$	−25705.2	−1.33330	−0.666649	+	0.745372i	$0.732272\pi$
$720$	−2206.68	−0.114220
$721$	11684.0	0.603514
$722$	22752.2	1.17278
$723$	2357.69	0.121277
$724$	10612.4	0.544763
$725$	−16769.0	−0.859015
$726$	24465.6	1.25069
$727$	27877.2	1.42216	0.711080	−	0.703111i	$-0.248207\pi$
$727$	27877.2	1.42216	0.711080	+	0.703111i	$0.248207\pi$
$728$	792.538	0.0403481
$729$	21418.6	1.08818
$730$	13402.3	0.679509
$731$	0	0
$732$	42964.2	2.16940
$733$	1914.75	0.0964841	0.0482420	−	0.998836i	$-0.484638\pi$
$733$	1914.75	0.0964841	0.0482420	+	0.998836i	$0.484638\pi$
$734$	43289.2	2.17689
$735$	3708.45	0.186106
$736$	50238.2	2.51604
$737$	285.260	0.0142574
$738$	−10588.7	−0.528151
$739$	−397.742	−0.0197986	−0.00989932	−	0.999951i	$-0.503151\pi$
$739$	−397.742	−0.0197986	−0.00989932	+	0.999951i	$0.503151\pi$
$740$	−12355.6	−0.613785
$741$	766.626	0.0380064
$742$	37157.0	1.83838
$743$	9023.68	0.445554	0.222777	−	0.974869i	$-0.428488\pi$
$743$	9023.68	0.445554	0.222777	+	0.974869i	$0.428488\pi$
$744$	3209.96	0.158176
$745$	−16611.7	−0.816919
$746$	−26691.8	−1.30999
$747$	9613.45	0.470867
$748$	2918.71	0.142672
$749$	8475.39	0.413463
$750$	−27268.4	−1.32760
$751$	19915.2	0.967665	0.483832	−	0.875161i	$-0.339244\pi$
$751$	19915.2	0.967665	0.483832	+	0.875161i	$0.339244\pi$
$752$	3495.26	0.169493
$753$	14824.4	0.717437
$754$	−4771.01	−0.230438
$755$	−22695.8	−1.09402
$756$	24811.2	1.19362
$757$	24897.3	1.19539	0.597694	−	0.801724i	$-0.296084\pi$
$757$	24897.3	1.19539	0.597694	+	0.801724i	$0.296084\pi$
$758$	−55362.6	−2.65285
$759$	−4090.81	−0.195635
$760$	3574.50	0.170606
$761$	34554.9	1.64601	0.823006	−	0.568033i	$-0.192295\pi$
$761$	34554.9	1.64601	0.823006	+	0.568033i	$0.192295\pi$
$762$	27726.7	1.31815
$763$	−4685.93	−0.222335
$764$	−16339.1	−0.773726
$765$	−3737.10	−0.176621
$766$	14979.6	0.706574
$767$	3240.67	0.152560
$768$	435.191	0.0204474
$769$	−12520.7	−0.587138	−0.293569	−	0.955938i	$-0.594843\pi$
$769$	−12520.7	−0.587138	−0.293569	+	0.955938i	$0.594843\pi$
$770$	2296.63	0.107487
$771$	2094.41	0.0978319
$772$	−8519.54	−0.397183
$773$	23875.0	1.11090	0.555450	−	0.831550i	$-0.312546\pi$
$773$	23875.0	1.11090	0.555450	+	0.831550i	$0.312546\pi$
$774$	0	0
$775$	4259.69	0.197435
$776$	−3000.89	−0.138822
$777$	−9943.10	−0.459082
$778$	58387.4	2.69060
$779$	−11709.7	−0.538566
$780$	−1558.00	−0.0715196
$781$	−306.827	−0.0140578
$782$	52328.8	2.39293
$783$	−37927.2	−1.73104
$784$	3942.46	0.179594
$785$	−9680.53	−0.440144
$786$	43790.2	1.98721
$787$	3612.43	0.163620	0.0818102	−	0.996648i	$-0.473930\pi$
$787$	3612.43	0.163620	0.0818102	+	0.996648i	$0.473930\pi$
$788$	−57510.8	−2.59992
$789$	5692.15	0.256839
$790$	19144.7	0.862198
$791$	24996.7	1.12362
$792$	−454.936	−0.0204109
$793$	4119.08	0.184455
$794$	36071.6	1.61226
$795$	−18548.1	−0.827465
$796$	43516.8	1.93770
$797$	14565.8	0.647363	0.323682	−	0.946166i	$-0.395079\pi$
$797$	14565.8	0.647363	0.323682	+	0.946166i	$0.395079\pi$
$798$	11328.2	0.502524
$799$	5919.36	0.262093
$800$	−16526.8	−0.730389
$801$	8173.58	0.360549
$802$	7077.16	0.311600
$803$	−1886.33	−0.0828981
$804$	−2859.53	−0.125433
$805$	23581.9	1.03249
$806$	1211.94	0.0529636
$807$	−8582.94	−0.374391
$808$	16449.3	0.716192
$809$	−22200.5	−0.964806	−0.482403	−	0.875949i	$-0.660236\pi$
$809$	−22200.5	−0.964806	−0.482403	+	0.875949i	$0.660236\pi$
$810$	−14218.2	−0.616761
$811$	−805.377	−0.0348713	−0.0174356	−	0.999848i	$-0.505550\pi$
$811$	−805.377	−0.0348713	−0.0174356	+	0.999848i	$0.505550\pi$
$812$	−40376.2	−1.74498
$813$	30995.4	1.33709
$814$	3036.43	0.130746
$815$	−8523.81	−0.366351
$816$	8856.15	0.379935
$817$	0	0
$818$	59230.3	2.53171
$819$	562.457	0.0239974
$820$	23797.3	1.01346
$821$	−30715.5	−1.30570	−0.652848	−	0.757489i	$-0.726426\pi$
$821$	−30715.5	−1.30570	−0.652848	+	0.757489i	$0.726426\pi$
$822$	−18985.6	−0.805594
$823$	−10776.3	−0.456424	−0.228212	−	0.973611i	$-0.573288\pi$
$823$	−10776.3	−0.456424	−0.228212	+	0.973611i	$0.573288\pi$
$824$	9082.34	0.383978
$825$	1345.75	0.0567915
$826$	47886.4	2.01717
$827$	−26112.2	−1.09796	−0.548978	−	0.835837i	$-0.684983\pi$
$827$	−26112.2	−1.09796	−0.548978	+	0.835837i	$0.684983\pi$
$828$	−18396.1	−0.772112
$829$	44126.9	1.84872	0.924362	−	0.381518i	$-0.124598\pi$
$829$	44126.9	1.84872	0.924362	+	0.381518i	$0.124598\pi$
$830$	−37724.8	−1.57765
$831$	−14988.1	−0.625669
$832$	−3466.38	−0.144441
$833$	6676.71	0.277712
$834$	−30585.3	−1.26988
$835$	−24051.2	−0.996797
$836$	−1981.26	−0.0819655
$837$	9634.30	0.397862
$838$	−1097.68	−0.0452491
$839$	2366.99	0.0973986	0.0486993	−	0.998813i	$-0.484492\pi$
$839$	2366.99	0.0973986	0.0486993	+	0.998813i	$0.484492\pi$
$840$	−5845.97	−0.240125
$841$	37331.4	1.53066
$842$	49403.4	2.02203
$843$	10191.8	0.416398
$844$	−14350.8	−0.585277
$845$	16510.4	0.672161
$846$	−3633.48	−0.147661
$847$	19850.2	0.805265
$848$	−19718.5	−0.798511
$849$	−15473.8	−0.625512
$850$	−17214.6	−0.694653
$851$	31178.3	1.25591
$852$	3075.72	0.123677
$853$	−24301.3	−0.975452	−0.487726	−	0.872997i	$-0.662173\pi$
$853$	−24301.3	−0.975452	−0.487726	+	0.872997i	$0.662173\pi$
$854$	60866.4	2.43888
$855$	2536.79	0.101469
$856$	6588.21	0.263061
$857$	−40357.3	−1.60861	−0.804304	−	0.594218i	$-0.797462\pi$
$857$	−40357.3	−1.60861	−0.804304	+	0.594218i	$0.797462\pi$
$858$	382.884	0.0152348
$859$	10352.8	0.411212	0.205606	−	0.978635i	$-0.434083\pi$
$859$	10352.8	0.411212	0.205606	+	0.978635i	$0.434083\pi$
$860$	0	0
$861$	19150.8	0.758022
$862$	20436.8	0.807516
$863$	15149.6	0.597563	0.298781	−	0.954322i	$-0.403420\pi$
$863$	15149.6	0.597563	0.298781	+	0.954322i	$0.403420\pi$
$864$	−37379.4	−1.47184
$865$	2764.15	0.108652
$866$	22635.9	0.888220
$867$	−6212.39	−0.243349
$868$	10256.4	0.401066
$869$	−2694.55	−0.105186
$870$	35192.2	1.37141
$871$	−274.149	−0.0106650
$872$	−3642.53	−0.141458
$873$	−2129.70	−0.0825653
$874$	−35521.4	−1.37475
$875$	−22124.2	−0.854784
$876$	18909.1	0.729314
$877$	19411.3	0.747405	0.373703	−	0.927549i	$-0.378088\pi$
$877$	19411.3	0.747405	0.373703	+	0.927549i	$0.378088\pi$
$878$	−53040.8	−2.03877
$879$	−944.921	−0.0362587
$880$	−1218.78	−0.0466876
$881$	−44659.5	−1.70785	−0.853926	−	0.520394i	$-0.825785\pi$
$881$	−44659.5	−1.70785	−0.853926	+	0.520394i	$0.825785\pi$
$882$	−4098.36	−0.156461
$883$	21280.2	0.811024	0.405512	−	0.914090i	$-0.367093\pi$
$883$	21280.2	0.811024	0.405512	+	0.914090i	$0.367093\pi$
$884$	−2805.03	−0.106723
$885$	−23904.1	−0.907940
$886$	65182.7	2.47162
$887$	−12849.9	−0.486423	−0.243211	−	0.969973i	$-0.578201\pi$
$887$	−12849.9	−0.486423	−0.243211	+	0.969973i	$0.578201\pi$
$888$	−7729.11	−0.292085
$889$	22496.1	0.848700
$890$	−32074.5	−1.20802
$891$	2001.16	0.0752430
$892$	3588.96	0.134717
$893$	−4018.14	−0.150573
$894$	−40922.9	−1.53095
$895$	17385.7	0.649319
$896$	−21533.1	−0.802870
$897$	3931.47	0.146341
$898$	4361.40	0.162073
$899$	−15678.3	−0.581646
$900$	6051.76	0.224139
$901$	−33394.1	−1.23476
$902$	−5848.28	−0.215883
$903$	0	0
$904$	19430.8	0.714888
$905$	−7504.90	−0.275659
$906$	−55911.2	−2.05025
$907$	38267.7	1.40094	0.700472	−	0.713680i	$-0.252973\pi$
$907$	38267.7	1.40094	0.700472	+	0.713680i	$0.252973\pi$
$908$	−14166.6	−0.517772
$909$	11673.9	0.425961
$910$	−2207.18	−0.0804035
$911$	−33801.3	−1.22929	−0.614646	−	0.788803i	$-0.710701\pi$
$911$	−33801.3	−1.22929	−0.614646	+	0.788803i	$0.710701\pi$
$912$	−6011.67	−0.218274
$913$	5309.64	0.192468
$914$	−11119.7	−0.402415
$915$	−30383.4	−1.09775
$916$	−19716.3	−0.711185
$917$	35529.2	1.27947
$918$	−38934.9	−1.39983
$919$	18924.2	0.679274	0.339637	−	0.940557i	$-0.389696\pi$
$919$	18924.2	0.679274	0.339637	+	0.940557i	$0.389696\pi$
$920$	18331.0	0.656908
$921$	−10577.7	−0.378445
$922$	11399.1	0.407170
$923$	294.876	0.0105157
$924$	3240.28	0.115365
$925$	−10256.7	−0.364582
$926$	40932.2	1.45261
$927$	6445.65	0.228374
$928$	60828.9	2.15173
$929$	47171.0	1.66591	0.832955	−	0.553340i	$-0.186647\pi$
$929$	47171.0	1.66591	0.832955	+	0.553340i	$0.186647\pi$
$930$	−8939.58	−0.315205
$931$	−4532.23	−0.159547
$932$	−26482.3	−0.930747
$933$	−3535.42	−0.124056
$934$	19599.6	0.686638
$935$	−2064.05	−0.0721944
$936$	437.217	0.0152680
$937$	39747.2	1.38579	0.692894	−	0.721039i	$-0.256335\pi$
$937$	39747.2	1.38579	0.692894	+	0.721039i	$0.256335\pi$
$938$	−4051.02	−0.141013
$939$	8586.38	0.298409
$940$	8165.98	0.283346
$941$	9781.20	0.338850	0.169425	−	0.985543i	$-0.445809\pi$
$941$	9781.20	0.338850	0.169425	+	0.985543i	$0.445809\pi$
$942$	−23848.0	−0.824852
$943$	−60050.5	−2.07371
$944$	−25412.4	−0.876170
$945$	−17546.0	−0.603990
$946$	0	0
$947$	33077.2	1.13502	0.567511	−	0.823366i	$-0.307906\pi$
$947$	33077.2	1.13502	0.567511	+	0.823366i	$0.307906\pi$
$948$	27010.9	0.925393
$949$	1812.86	0.0620104
$950$	11685.5	0.399081
$951$	−17468.0	−0.595623
$952$	−10525.1	−0.358320
$953$	−4221.07	−0.143477	−0.0717386	−	0.997423i	$-0.522855\pi$
$953$	−4221.07	−0.143477	−0.0717386	+	0.997423i	$0.522855\pi$
$954$	20498.3	0.695657
$955$	11554.7	0.391518
$956$	−69629.2	−2.35561
$957$	−4953.19	−0.167308
$958$	63887.6	2.15461
$959$	−15404.0	−0.518686
$960$	25568.9	0.859618
$961$	−25808.4	−0.866315
$962$	−2918.16	−0.0978019
$963$	4675.59	0.156458
$964$	5855.86	0.195648
$965$	6024.85	0.200981
$966$	58094.1	1.93493
$967$	17701.1	0.588655	0.294328	−	0.955705i	$-0.404904\pi$
$967$	17701.1	0.588655	0.294328	+	0.955705i	$0.404904\pi$
$968$	15430.2	0.512340
$969$	−10181.0	−0.337524
$970$	8357.32	0.276636
$971$	22881.9	0.756248	0.378124	−	0.925755i	$-0.376569\pi$
$971$	22881.9	0.756248	0.378124	+	0.925755i	$0.376569\pi$
$972$	24138.5	0.796547
$973$	−24815.4	−0.817620
$974$	27026.7	0.889108
$975$	−1293.33	−0.0424819
$976$	−32300.7	−1.05934
$977$	27596.4	0.903671	0.451835	−	0.892101i	$-0.350769\pi$
$977$	27596.4	0.903671	0.451835	+	0.892101i	$0.350769\pi$
$978$	−20998.4	−0.686560
$979$	4514.38	0.147375
$980$	9210.76	0.300232
$981$	−2585.07	−0.0841334
$982$	40983.0	1.33179
$983$	−13288.7	−0.431175	−0.215587	−	0.976485i	$-0.569167\pi$
$983$	−13288.7	−0.431175	−0.215587	+	0.976485i	$0.569167\pi$
$984$	14886.5	0.482282
$985$	40670.5	1.31560
$986$	63360.3	2.04645
$987$	6571.53	0.211929
$988$	1904.09	0.0613128
$989$	0	0
$990$	1266.97	0.0406738
$991$	−43350.5	−1.38958	−0.694789	−	0.719213i	$-0.744502\pi$
$991$	−43350.5	−1.38958	−0.694789	+	0.719213i	$0.744502\pi$
$992$	−15451.8	−0.494553
$993$	−48341.8	−1.54489
$994$	4357.30	0.139039
$995$	−30774.2	−0.980510
$996$	−53225.3	−1.69328
$997$	−45404.7	−1.44231	−0.721154	−	0.692775i	$-0.756388\pi$
$997$	−45404.7	−1.44231	−0.721154	+	0.692775i	$0.756388\pi$
$998$	−42010.0	−1.33247
$999$	−23198.0	−0.734686

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.j.1.6	yes	50
43.42	odd	2		1849.4.a.i.1.45	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.45	✓	50	43.42	odd	2
1849.4.a.j.1.6	yes	50	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q-4.32699 q^{2} +4.31725 q^{3} +10.7228 q^{4} -7.58298 q^{5} -18.6807 q^{6} -15.1566 q^{7} -11.7817 q^{8} -8.36138 q^{9} +O(q^{10})\)	\(q-4.32699 q^{2} +4.31725 q^{3} +10.7228 q^{4} -7.58298 q^{5} -18.6807 q^{6} -15.1566 q^{7} -11.7817 q^{8} -8.36138 q^{9} +32.8115 q^{10} -4.61811 q^{11} +46.2932 q^{12} +4.43823 q^{13} +65.5824 q^{14} -32.7376 q^{15} -34.8034 q^{16} -58.9409 q^{17} +36.1796 q^{18} +40.0098 q^{19} -81.3111 q^{20} -65.4347 q^{21} +19.9825 q^{22} +205.181 q^{23} -50.8646 q^{24} -67.4984 q^{25} -19.2042 q^{26} -152.664 q^{27} -162.522 q^{28} +248.436 q^{29} +141.655 q^{30} -63.1080 q^{31} +244.848 q^{32} -19.9375 q^{33} +255.037 q^{34} +114.932 q^{35} -89.6578 q^{36} +151.955 q^{37} -173.122 q^{38} +19.1609 q^{39} +89.3405 q^{40} -292.670 q^{41} +283.135 q^{42} -49.5192 q^{44} +63.4042 q^{45} -887.818 q^{46} -100.429 q^{47} -150.255 q^{48} -113.278 q^{49} +292.065 q^{50} -254.462 q^{51} +47.5905 q^{52} +566.570 q^{53} +660.575 q^{54} +35.0190 q^{55} +178.571 q^{56} +172.732 q^{57} -1074.98 q^{58} +730.172 q^{59} -351.040 q^{60} +928.090 q^{61} +273.068 q^{62} +126.730 q^{63} -781.026 q^{64} -33.6550 q^{65} +86.2694 q^{66} -61.7700 q^{67} -632.014 q^{68} +885.819 q^{69} -497.310 q^{70} +66.4400 q^{71} +98.5114 q^{72} +408.464 q^{73} -657.506 q^{74} -291.407 q^{75} +429.019 q^{76} +69.9947 q^{77} -82.9092 q^{78} +583.475 q^{79} +263.913 q^{80} -433.330 q^{81} +1266.38 q^{82} -1149.74 q^{83} -701.646 q^{84} +446.948 q^{85} +1072.56 q^{87} +54.4092 q^{88} -977.540 q^{89} -274.349 q^{90} -67.2684 q^{91} +2200.13 q^{92} -272.453 q^{93} +434.554 q^{94} -303.394 q^{95} +1057.07 q^{96} +254.707 q^{97} +490.153 q^{98} +38.6137 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9}+O(q^{10}) \) 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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