LMFDB - Embedded newform 1849.4.a.j.1.44

Newspace parameters

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

Newform invariants

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			1.44
Character	$\chi$	$=$	1849.1

$q$-expansion

$f(q)$	$=$	$q+4.78088 q^{2} -1.15757 q^{3} +14.8568 q^{4} +21.3015 q^{5} -5.53419 q^{6} -24.0536 q^{7} +32.7814 q^{8} -25.6600 q^{9} +O(q^{10})$	\(q+4.78088 q^{2} -1.15757 q^{3} +14.8568 q^{4} +21.3015 q^{5} -5.53419 q^{6} -24.0536 q^{7} +32.7814 q^{8} -25.6600 q^{9} +101.840 q^{10} +17.5943 q^{11} -17.1977 q^{12} -56.9423 q^{13} -114.997 q^{14} -24.6580 q^{15} +37.8697 q^{16} -74.3144 q^{17} -122.677 q^{18} -100.891 q^{19} +316.472 q^{20} +27.8436 q^{21} +84.1161 q^{22} -13.9922 q^{23} -37.9467 q^{24} +328.755 q^{25} -272.234 q^{26} +60.9576 q^{27} -357.359 q^{28} -87.1198 q^{29} -117.887 q^{30} -127.229 q^{31} -81.2011 q^{32} -20.3666 q^{33} -355.288 q^{34} -512.378 q^{35} -381.226 q^{36} -125.930 q^{37} -482.347 q^{38} +65.9146 q^{39} +698.294 q^{40} -82.7414 q^{41} +133.117 q^{42} +261.394 q^{44} -546.598 q^{45} -66.8948 q^{46} -71.5573 q^{47} -43.8367 q^{48} +235.575 q^{49} +1571.74 q^{50} +86.0239 q^{51} -845.980 q^{52} +152.062 q^{53} +291.431 q^{54} +374.785 q^{55} -788.511 q^{56} +116.788 q^{57} -416.509 q^{58} +319.919 q^{59} -366.338 q^{60} +353.020 q^{61} -608.268 q^{62} +617.216 q^{63} -691.170 q^{64} -1212.96 q^{65} -97.3701 q^{66} +37.2004 q^{67} -1104.07 q^{68} +16.1969 q^{69} -2449.62 q^{70} +781.921 q^{71} -841.173 q^{72} -1171.74 q^{73} -602.056 q^{74} -380.556 q^{75} -1498.91 q^{76} -423.206 q^{77} +315.130 q^{78} +841.963 q^{79} +806.682 q^{80} +622.259 q^{81} -395.577 q^{82} -1113.00 q^{83} +413.667 q^{84} -1583.01 q^{85} +100.847 q^{87} +576.766 q^{88} -308.616 q^{89} -2613.22 q^{90} +1369.67 q^{91} -207.879 q^{92} +147.277 q^{93} -342.106 q^{94} -2149.13 q^{95} +93.9957 q^{96} +1108.36 q^{97} +1126.25 q^{98} -451.470 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) $	\( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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 $

$2$	4.78088	1.69030	0.845148	−	0.534533i	$-0.179513\pi$
$2$	4.78088	1.69030	0.845148	+	0.534533i	$0.179513\pi$
$3$	−1.15757	−0.222774	−0.111387	−	0.993777i	$-0.535529\pi$
$3$	−1.15757	−0.222774	−0.111387	+	0.993777i	$0.535529\pi$
$4$	14.8568	1.85710
$5$	21.3015	1.90527	0.952633	−	0.304122i	$-0.0983630\pi$
$5$	21.3015	1.90527	0.952633	+	0.304122i	$0.0983630\pi$
$6$	−5.53419	−0.376554
$7$	−24.0536	−1.29877	−0.649386	−	0.760459i	$-0.724974\pi$
$7$	−24.0536	−1.29877	−0.649386	+	0.760459i	$0.724974\pi$
$8$	32.7814	1.44875
$9$	−25.6600	−0.950372
$10$	101.840	3.22046
$11$	17.5943	0.482262	0.241131	−	0.970493i	$-0.422482\pi$
$11$	17.5943	0.482262	0.241131	+	0.970493i	$0.422482\pi$
$12$	−17.1977	−0.413713
$13$	−56.9423	−1.21484	−0.607422	−	0.794380i	$-0.707796\pi$
$13$	−56.9423	−1.21484	−0.607422	+	0.794380i	$0.707796\pi$
$14$	−114.997	−2.19531
$15$	−24.6580	−0.424444
$16$	37.8697	0.591714
$17$	−74.3144	−1.06023	−0.530114	−	0.847926i	$-0.677851\pi$
$17$	−74.3144	−1.06023	−0.530114	+	0.847926i	$0.677851\pi$
$18$	−122.677	−1.60641
$19$	−100.891	−1.21821	−0.609104	−	0.793090i	$-0.708471\pi$
$19$	−100.891	−1.21821	−0.609104	+	0.793090i	$0.708471\pi$
$20$	316.472	3.53827
$21$	27.8436	0.289332
$22$	84.1161	0.815164
$23$	−13.9922	−0.126851	−0.0634254	−	0.997987i	$-0.520202\pi$
$23$	−13.9922	−0.126851	−0.0634254	+	0.997987i	$0.520202\pi$
$24$	−37.9467	−0.322743
$25$	328.755	2.63004
$26$	−272.234	−2.05344
$27$	60.9576	0.434492
$28$	−357.359	−2.41195
$29$	−87.1198	−0.557853	−0.278927	−	0.960312i	$-0.589979\pi$
$29$	−87.1198	−0.557853	−0.278927	+	0.960312i	$0.589979\pi$
$30$	−117.887	−0.717435
$31$	−127.229	−0.737131	−0.368566	−	0.929602i	$-0.620151\pi$
$31$	−127.229	−0.737131	−0.368566	+	0.929602i	$0.620151\pi$
$32$	−81.2011	−0.448577
$33$	−20.3666	−0.107435
$34$	−355.288	−1.79210
$35$	−512.378	−2.47451
$36$	−381.226	−1.76493
$37$	−125.930	−0.559535	−0.279767	−	0.960068i	$-0.590257\pi$
$37$	−125.930	−0.559535	−0.279767	+	0.960068i	$0.590257\pi$
$38$	−482.347	−2.05913
$39$	65.9146	0.270635
$40$	698.294	2.76025
$41$	−82.7414	−0.315172	−0.157586	−	0.987505i	$-0.550371\pi$
$41$	−82.7414	−0.315172	−0.157586	+	0.987505i	$0.550371\pi$
$42$	133.117	0.489057
$43$	0	0
$43$	0	0
$44$	261.394	0.895607
$45$	−546.598	−1.81071
$46$	−66.8948	−0.214415
$47$	−71.5573	−0.222079	−0.111039	−	0.993816i	$-0.535418\pi$
$47$	−71.5573	−0.222079	−0.111039	+	0.993816i	$0.535418\pi$
$48$	−43.8367	−0.131818
$49$	235.575	0.686807
$50$	1571.74	4.44554
$51$	86.0239	0.236191
$52$	−845.980	−2.25608
$53$	152.062	0.394101	0.197050	−	0.980393i	$-0.436864\pi$
$53$	152.062	0.394101	0.197050	+	0.980393i	$0.436864\pi$
$54$	291.431	0.734420
$55$	374.785	0.918837
$56$	−788.511	−1.88159
$57$	116.788	0.271385
$58$	−416.509	−0.942936
$59$	319.919	0.705931	0.352965	−	0.935636i	$-0.385173\pi$
$59$	319.919	0.705931	0.352965	+	0.935636i	$0.385173\pi$
$60$	−366.338	−0.788234
$61$	353.020	0.740977	0.370488	−	0.928837i	$-0.379190\pi$
$61$	353.020	0.740977	0.370488	+	0.928837i	$0.379190\pi$
$62$	−608.268	−1.24597
$63$	617.216	1.23432
$64$	−691.170	−1.34994
$65$	−1212.96	−2.31460
$66$	−97.3701	−0.181597
$67$	37.2004	0.0678320	0.0339160	−	0.999425i	$-0.489202\pi$
$67$	37.2004	0.0678320	0.0339160	+	0.999425i	$0.489202\pi$
$68$	−1104.07	−1.96895
$69$	16.1969	0.0282591
$70$	−2449.62	−4.18264
$71$	781.921	1.30700	0.653500	−	0.756927i	$-0.273300\pi$
$71$	781.921	1.30700	0.653500	+	0.756927i	$0.273300\pi$
$72$	−841.173	−1.37685
$73$	−1171.74	−1.87866	−0.939331	−	0.343013i	$-0.888552\pi$
$73$	−1171.74	−1.87866	−0.939331	+	0.343013i	$0.888552\pi$
$74$	−602.056	−0.945779
$75$	−380.556	−0.585905
$76$	−1498.91	−2.26233
$77$	−423.206	−0.626347
$78$	315.130	0.457454
$79$	841.963	1.19909	0.599546	−	0.800340i	$-0.295348\pi$
$79$	841.963	1.19909	0.599546	+	0.800340i	$0.295348\pi$
$80$	806.682	1.12737
$81$	622.259	0.853578
$82$	−395.577	−0.532733
$83$	−1113.00	−1.47190	−0.735950	−	0.677036i	$-0.763264\pi$
$83$	−1113.00	−1.47190	−0.735950	+	0.677036i	$0.763264\pi$
$84$	413.667	0.537319
$85$	−1583.01	−2.02002
$86$	0	0
$87$	100.847	0.124275
$88$	576.766	0.698675
$89$	−308.616	−0.367564	−0.183782	−	0.982967i	$-0.558834\pi$
$89$	−308.616	−0.367564	−0.183782	+	0.982967i	$0.558834\pi$
$90$	−2613.22	−3.06064
$91$	1369.67	1.57780
$92$	−207.879	−0.235574
$93$	147.277	0.164214
$94$	−342.106	−0.375379
$95$	−2149.13	−2.32101
$96$	93.9957	0.0999312
$97$	1108.36	1.16017	0.580085	−	0.814556i	$-0.303019\pi$
$97$	1108.36	1.16017	0.580085	+	0.814556i	$0.303019\pi$
$98$	1126.25	1.16091
$99$	−451.470	−0.458328
$100$	4884.24	4.88424
$101$	−774.120	−0.762652	−0.381326	−	0.924441i	$-0.624532\pi$
$101$	−774.120	−0.762652	−0.381326	+	0.924441i	$0.624532\pi$
$102$	411.270	0.399233
$103$	317.377	0.303612	0.151806	−	0.988410i	$-0.451491\pi$
$103$	317.377	0.303612	0.151806	+	0.988410i	$0.451491\pi$
$104$	−1866.65	−1.76000
$105$	593.112	0.551255
$106$	726.990	0.666146
$107$	648.591	0.585997	0.292998	−	0.956113i	$-0.405347\pi$
$107$	648.591	0.585997	0.292998	+	0.956113i	$0.405347\pi$
$108$	905.633	0.806894
$109$	575.465	0.505684	0.252842	−	0.967508i	$-0.418635\pi$
$109$	575.465	0.505684	0.252842	+	0.967508i	$0.418635\pi$
$110$	1791.80	1.55311
$111$	145.773	0.124650
$112$	−910.902	−0.768501
$113$	−509.868	−0.424463	−0.212232	−	0.977219i	$-0.568073\pi$
$113$	−509.868	−0.424463	−0.212232	+	0.977219i	$0.568073\pi$
$114$	558.349	0.458721
$115$	−298.055	−0.241685
$116$	−1294.32	−1.03599
$117$	1461.14	1.15455
$118$	1529.49	1.19323
$119$	1787.53	1.37699
$120$	−808.323	−0.614912
$121$	−1021.44	−0.767424
$122$	1687.75	1.25247
$123$	95.7788	0.0702121
$124$	−1890.22	−1.36892
$125$	4340.29	3.10566
$126$	2950.83	2.08636
$127$	2623.19	1.83284	0.916421	−	0.400217i	$-0.131065\pi$
$127$	2623.19	1.83284	0.916421	+	0.400217i	$0.131065\pi$
$128$	−2654.79	−1.83322
$129$	0	0
$130$	−5799.01	−3.91236
$131$	−2604.53	−1.73709	−0.868544	−	0.495611i	$-0.834944\pi$
$131$	−2604.53	−1.73709	−0.868544	+	0.495611i	$0.834944\pi$
$132$	−302.582	−0.199518
$133$	2426.79	1.58217
$134$	177.850	0.114656
$135$	1298.49	0.827823
$136$	−2436.13	−1.53600
$137$	665.268	0.414873	0.207437	−	0.978248i	$-0.433488\pi$
$137$	665.268	0.414873	0.207437	+	0.978248i	$0.433488\pi$
$138$	77.4353	0.0477662
$139$	−876.154	−0.534636	−0.267318	−	0.963608i	$-0.586137\pi$
$139$	−876.154	−0.534636	−0.267318	+	0.963608i	$0.586137\pi$
$140$	−7612.29	−4.59540
$141$	82.8324	0.0494734
$142$	3738.27	2.20921
$143$	−1001.86	−0.585872
$144$	−971.738	−0.562348
$145$	−1855.78	−1.06286
$146$	−5601.96	−3.17549
$147$	−272.694	−0.153003
$148$	−1870.92	−1.03911
$149$	1157.27	0.636289	0.318144	−	0.948042i	$-0.396940\pi$
$149$	1157.27	0.636289	0.318144	+	0.948042i	$0.396940\pi$
$150$	−1819.39	−0.990352
$151$	1380.36	0.743922	0.371961	−	0.928248i	$-0.378686\pi$
$151$	1380.36	0.743922	0.371961	+	0.928248i	$0.378686\pi$
$152$	−3307.35	−1.76488
$153$	1906.91	1.00761
$154$	−2023.29	−1.05871
$155$	−2710.18	−1.40443
$156$	979.279	0.502596
$157$	2859.38	1.45352	0.726762	−	0.686890i	$-0.241024\pi$
$157$	2859.38	1.45352	0.726762	+	0.686890i	$0.241024\pi$
$158$	4025.32	2.02682
$159$	−176.022	−0.0877954
$160$	−1729.71	−0.854658
$161$	336.562	0.164750
$162$	2974.94	1.44280
$163$	1990.78	0.956627	0.478314	−	0.878189i	$-0.341248\pi$
$163$	1990.78	0.956627	0.478314	+	0.878189i	$0.341248\pi$
$164$	−1229.27	−0.585305
$165$	−433.839	−0.204693
$166$	−5321.12	−2.48794
$167$	−1753.53	−0.812527	−0.406263	−	0.913756i	$-0.633168\pi$
$167$	−1753.53	−0.812527	−0.406263	+	0.913756i	$0.633168\pi$
$168$	912.754	0.419170
$169$	1045.43	0.475844
$170$	−7568.17	−3.41443
$171$	2588.86	1.15775
$172$	0	0
$173$	1857.37	0.816263	0.408131	−	0.912923i	$-0.366180\pi$
$173$	1857.37	0.816263	0.408131	+	0.912923i	$0.366180\pi$
$174$	482.137	0.210062
$175$	−7907.73	−3.41582
$176$	666.290	0.285361
$177$	−370.328	−0.157263
$178$	−1475.45	−0.621291
$179$	−3915.43	−1.63493	−0.817467	−	0.575975i	$-0.804623\pi$
$179$	−3915.43	−1.63493	−0.817467	+	0.575975i	$0.804623\pi$
$180$	−8120.69	−3.36267
$181$	3009.44	1.23586	0.617928	−	0.786234i	$-0.287972\pi$
$181$	3009.44	1.23586	0.617928	+	0.786234i	$0.287972\pi$
$182$	6548.21	2.66695
$183$	−408.645	−0.165070
$184$	−458.683	−0.183775
$185$	−2682.50	−1.06606
$186$	704.111	0.277570
$187$	−1307.51	−0.511308
$188$	−1063.11	−0.412422
$189$	−1466.25	−0.564306
$190$	−10274.7	−3.92319
$191$	−393.305	−0.148998	−0.0744989	−	0.997221i	$-0.523736\pi$
$191$	−393.305	−0.148998	−0.0744989	+	0.997221i	$0.523736\pi$
$192$	800.076	0.300732
$193$	1329.16	0.495725	0.247862	−	0.968795i	$-0.420272\pi$
$193$	1329.16	0.495725	0.247862	+	0.968795i	$0.420272\pi$
$194$	5298.91	1.96103
$195$	1404.08	0.515633
$196$	3499.88	1.27547
$197$	−533.591	−0.192978	−0.0964892	−	0.995334i	$-0.530761\pi$
$197$	−533.591	−0.192978	−0.0964892	+	0.995334i	$0.530761\pi$
$198$	−2158.42	−0.774709
$199$	−2186.74	−0.778964	−0.389482	−	0.921034i	$-0.627346\pi$
$199$	−2186.74	−0.778964	−0.389482	+	0.921034i	$0.627346\pi$
$200$	10777.1	3.81026
$201$	−43.0619	−0.0151112
$202$	−3700.97	−1.28911
$203$	2095.54	0.724524
$204$	1278.04	0.438630
$205$	−1762.52	−0.600486
$206$	1517.34	0.513195
$207$	359.040	0.120555
$208$	−2156.39	−0.718840
$209$	−1775.10	−0.587495
$210$	2835.60	0.931784
$211$	−5062.24	−1.65165	−0.825827	−	0.563923i	$-0.809291\pi$
$211$	−5062.24	−1.65165	−0.825827	+	0.563923i	$0.809291\pi$
$212$	2259.15	0.731883
$213$	−905.127	−0.291165
$214$	3100.83	0.990508
$215$	0	0
$216$	1998.28	0.629470
$217$	3060.32	0.957365
$218$	2751.23	0.854755
$219$	1356.37	0.418517
$220$	5568.10	1.70637
$221$	4231.63	1.28801
$222$	696.921	0.210695
$223$	−5423.05	−1.62850	−0.814248	−	0.580518i	$-0.802850\pi$
$223$	−5423.05	−1.62850	−0.814248	+	0.580518i	$0.802850\pi$
$224$	1953.18	0.582599
$225$	−8435.87	−2.49952
$226$	−2437.62	−0.717468
$227$	−1368.39	−0.400103	−0.200052	−	0.979785i	$-0.564111\pi$
$227$	−1368.39	−0.400103	−0.200052	+	0.979785i	$0.564111\pi$
$228$	1735.09	0.503988
$229$	3079.65	0.888685	0.444342	−	0.895857i	$-0.353437\pi$
$229$	3079.65	0.888685	0.444342	+	0.895857i	$0.353437\pi$
$230$	−1424.96	−0.408518
$231$	489.889	0.139534
$232$	−2855.91	−0.808189
$233$	−1874.11	−0.526939	−0.263469	−	0.964668i	$-0.584867\pi$
$233$	−1874.11	−0.526939	−0.263469	+	0.964668i	$0.584867\pi$
$234$	6985.54	1.95153
$235$	−1524.28	−0.423119
$236$	4752.97	1.31098
$237$	−974.629	−0.267126
$238$	8545.95	2.32753
$239$	−2112.85	−0.571837	−0.285919	−	0.958254i	$-0.592299\pi$
$239$	−2112.85	−0.571837	−0.285919	+	0.958254i	$0.592299\pi$
$240$	−933.789	−0.251149
$241$	242.170	0.0647285	0.0323642	−	0.999476i	$-0.489696\pi$
$241$	242.170	0.0647285	0.0323642	+	0.999476i	$0.489696\pi$
$242$	−4883.38	−1.29717
$243$	−2366.16	−0.624647
$244$	5244.74	1.37607
$245$	5018.10	1.30855
$246$	457.907	0.118679
$247$	5744.96	1.47993
$248$	−4170.76	−1.06792
$249$	1288.37	0.327901
$250$	20750.4	5.24948
$251$	−2026.35	−0.509569	−0.254784	−	0.966998i	$-0.582005\pi$
$251$	−2026.35	−0.509569	−0.254784	+	0.966998i	$0.582005\pi$
$252$	9169.84	2.29224
$253$	−246.182	−0.0611753
$254$	12541.2	3.09804
$255$	1832.44	0.450007
$256$	−7162.86	−1.74875
$257$	−2709.06	−0.657534	−0.328767	−	0.944411i	$-0.606633\pi$
$257$	−2709.06	−0.657534	−0.328767	+	0.944411i	$0.606633\pi$
$258$	0	0
$259$	3029.07	0.726707
$260$	−18020.7	−4.29844
$261$	2235.50	0.530168
$262$	−12451.9	−2.93619
$263$	−1614.38	−0.378507	−0.189253	−	0.981928i	$-0.560607\pi$
$263$	−1614.38	−0.378507	−0.189253	+	0.981928i	$0.560607\pi$
$264$	−667.645	−0.155647
$265$	3239.15	0.750867
$266$	11602.2	2.67434
$267$	357.243	0.0818837
$268$	552.678	0.125971
$269$	−3571.10	−0.809420	−0.404710	−	0.914445i	$-0.632627\pi$
$269$	−3571.10	−0.809420	−0.404710	+	0.914445i	$0.632627\pi$
$270$	6207.92	1.39927
$271$	−7251.79	−1.62552	−0.812758	−	0.582602i	$-0.802035\pi$
$271$	−7251.79	−1.62552	−0.812758	+	0.582602i	$0.802035\pi$
$272$	−2814.26	−0.627352
$273$	−1585.48	−0.351494
$274$	3180.56	0.701258
$275$	5784.21	1.26837
$276$	240.634	0.0524799
$277$	4897.85	1.06239	0.531197	−	0.847248i	$-0.321742\pi$
$277$	4897.85	1.06239	0.531197	+	0.847248i	$0.321742\pi$
$278$	−4188.78	−0.903693
$279$	3264.71	0.700549
$280$	−16796.5	−3.58493
$281$	−3332.97	−0.707573	−0.353787	−	0.935326i	$-0.615106\pi$
$281$	−3332.97	−0.707573	−0.353787	+	0.935326i	$0.615106\pi$
$282$	396.011	0.0836246
$283$	−3141.03	−0.659770	−0.329885	−	0.944021i	$-0.607010\pi$
$283$	−3141.03	−0.659770	−0.329885	+	0.944021i	$0.607010\pi$
$284$	11616.8	2.42723
$285$	2487.76	0.517061
$286$	−4789.77	−0.990297
$287$	1990.23	0.409336
$288$	2083.62	0.426315
$289$	609.628	0.124085
$290$	−8872.28	−1.79654
$291$	−1283.00	−0.258456
$292$	−17408.3	−3.48886
$293$	−4342.75	−0.865891	−0.432945	−	0.901420i	$-0.642526\pi$
$293$	−4342.75	−0.865891	−0.432945	+	0.901420i	$0.642526\pi$
$294$	−1303.71	−0.258620
$295$	6814.76	1.34499
$296$	−4128.17	−0.810625
$297$	1072.50	0.209539
$298$	5532.75	1.07552
$299$	796.747	0.154104
$300$	−5653.84	−1.08808
$301$	0	0
$302$	6599.34	1.25745
$303$	896.096	0.169899
$304$	−3820.70	−0.720830
$305$	7519.87	1.41176
$306$	9116.70	1.70316
$307$	−8180.43	−1.52079	−0.760394	−	0.649462i	$-0.774994\pi$
$307$	−8180.43	−1.52079	−0.760394	+	0.649462i	$0.774994\pi$
$308$	−6287.47	−1.16319
$309$	−367.385	−0.0676370
$310$	−12957.0	−2.37390
$311$	−4932.66	−0.899375	−0.449687	−	0.893186i	$-0.648465\pi$
$311$	−4932.66	−0.899375	−0.449687	+	0.893186i	$0.648465\pi$
$312$	2160.77	0.392083
$313$	4008.91	0.723952	0.361976	−	0.932187i	$-0.382102\pi$
$313$	4008.91	0.723952	0.361976	+	0.932187i	$0.382102\pi$
$314$	13670.3	2.45688
$315$	13147.6	2.35170
$316$	12508.9	2.22683
$317$	1568.08	0.277831	0.138915	−	0.990304i	$-0.455638\pi$
$317$	1568.08	0.277831	0.138915	+	0.990304i	$0.455638\pi$
$318$	−841.540	−0.148400
$319$	−1532.81	−0.269031
$320$	−14723.0	−2.57200
$321$	−750.788	−0.130545
$322$	1609.06	0.278477
$323$	7497.64	1.29158
$324$	9244.76	1.58518
$325$	−18720.1	−3.19509
$326$	9517.69	1.61698
$327$	−666.140	−0.112653
$328$	−2712.38	−0.456604
$329$	1721.21	0.288429
$330$	−2074.13	−0.345991
$331$	−2987.88	−0.496160	−0.248080	−	0.968740i	$-0.579800\pi$
$331$	−2987.88	−0.496160	−0.248080	+	0.968740i	$0.579800\pi$
$332$	−16535.6	−2.73346
$333$	3231.37	0.531766
$334$	−8383.39	−1.37341
$335$	792.424	0.129238
$336$	1054.43	0.171202
$337$	−5177.88	−0.836965	−0.418483	−	0.908225i	$-0.637438\pi$
$337$	−5177.88	−0.836965	−0.418483	+	0.908225i	$0.637438\pi$
$338$	4998.07	0.804317
$339$	590.207	0.0945594
$340$	−23518.4	−3.75137
$341$	−2238.51	−0.355490
$342$	12377.0	1.95694
$343$	2583.96	0.406766
$344$	0	0
$345$	345.018	0.0538411
$346$	8879.87	1.37973
$347$	8237.38	1.27437	0.637184	−	0.770712i	$-0.280099\pi$
$347$	8237.38	1.27437	0.637184	+	0.770712i	$0.280099\pi$
$348$	1498.26	0.230791
$349$	−2750.91	−0.421928	−0.210964	−	0.977494i	$-0.567660\pi$
$349$	−2750.91	−0.421928	−0.210964	+	0.977494i	$0.567660\pi$
$350$	−37805.9	−5.77374
$351$	−3471.07	−0.527840
$352$	−1428.67	−0.216331
$353$	−109.272	−0.0164758	−0.00823789	−	0.999966i	$-0.502622\pi$
$353$	−109.272	−0.0164758	−0.00823789	+	0.999966i	$0.502622\pi$
$354$	−1770.49	−0.265821
$355$	16656.1	2.49018
$356$	−4585.03	−0.682602
$357$	−2069.18	−0.306759
$358$	−18719.2	−2.76352
$359$	−11539.3	−1.69643	−0.848216	−	0.529650i	$-0.822323\pi$
$359$	−11539.3	−1.69643	−0.848216	+	0.529650i	$0.822323\pi$
$360$	−17918.3	−2.62326
$361$	3319.96	0.484030
$362$	14387.8	2.08896
$363$	1182.39	0.170962
$364$	20348.8	2.93013
$365$	−24959.9	−3.57935
$366$	−1953.68	−0.279018
$367$	−13092.1	−1.86213	−0.931064	−	0.364857i	$-0.881118\pi$
$367$	−13092.1	−1.86213	−0.931064	+	0.364857i	$0.881118\pi$
$368$	−529.879	−0.0750594
$369$	2123.15	0.299530
$370$	−12824.7	−1.80196
$371$	−3657.64	−0.511847
$372$	2188.06	0.304961
$373$	9972.25	1.38430	0.692150	−	0.721754i	$-0.256664\pi$
$373$	9972.25	1.38430	0.692150	+	0.721754i	$0.256664\pi$
$374$	−6251.04	−0.864261
$375$	−5024.18	−0.691860
$376$	−2345.75	−0.321736
$377$	4960.80	0.677704
$378$	−7009.95	−0.953843
$379$	1702.20	0.230702	0.115351	−	0.993325i	$-0.463201\pi$
$379$	1702.20	0.230702	0.115351	+	0.993325i	$0.463201\pi$
$380$	−31929.1	−4.31034
$381$	−3036.52	−0.408309
$382$	−1880.34	−0.251850
$383$	−3035.45	−0.404972	−0.202486	−	0.979285i	$-0.564902\pi$
$383$	−3035.45	−0.404972	−0.202486	+	0.979285i	$0.564902\pi$
$384$	3073.10	0.408394
$385$	−9014.92	−1.19336
$386$	6354.54	0.837921
$387$	0	0
$388$	16466.6	2.15455
$389$	9435.36	1.22980	0.614900	−	0.788605i	$-0.289196\pi$
$389$	9435.36	1.22980	0.614900	+	0.788605i	$0.289196\pi$
$390$	6712.74	0.871571
$391$	1039.82	0.134491
$392$	7722.47	0.995010
$393$	3014.92	0.386978
$394$	−2551.03	−0.326191
$395$	17935.1	2.28459
$396$	−6707.39	−0.851159
$397$	4943.82	0.624995	0.312498	−	0.949919i	$-0.398834\pi$
$397$	4943.82	0.624995	0.312498	+	0.949919i	$0.398834\pi$
$398$	−10454.5	−1.31668
$399$	−2809.17	−0.352467
$400$	12449.8	1.55623
$401$	−10139.0	−1.26264	−0.631318	−	0.775524i	$-0.717486\pi$
$401$	−10139.0	−1.26264	−0.631318	+	0.775524i	$0.717486\pi$
$402$	−205.874	−0.0255424
$403$	7244.74	0.895499
$404$	−11500.9	−1.41632
$405$	13255.1	1.62629
$406$	10018.5	1.22466
$407$	−2215.65	−0.269842
$408$	2819.99	0.342182
$409$	−740.411	−0.0895134	−0.0447567	−	0.998998i	$-0.514251\pi$
$409$	−740.411	−0.0895134	−0.0447567	+	0.998998i	$0.514251\pi$
$410$	−8426.39	−1.01500
$411$	−770.092	−0.0924230
$412$	4715.20	0.563838
$413$	−7695.20	−0.916843
$414$	1716.52	0.203774
$415$	−23708.6	−2.80436
$416$	4623.78	0.544950
$417$	1014.21	0.119103
$418$	−8486.55	−0.993040
$419$	12031.8	1.40284	0.701421	−	0.712747i	$-0.252549\pi$
$419$	12031.8	1.40284	0.701421	+	0.712747i	$0.252549\pi$
$420$	8811.74	1.02374
$421$	−1281.44	−0.148346	−0.0741731	−	0.997245i	$-0.523632\pi$
$421$	−1281.44	−0.148346	−0.0741731	+	0.997245i	$0.523632\pi$
$422$	−24202.0	−2.79178
$423$	1836.16	0.211057
$424$	4984.81	0.570952
$425$	−24431.2	−2.78844
$426$	−4327.30	−0.492156
$427$	−8491.40	−0.962360
$428$	9635.98	1.08825
$429$	1159.72	0.130517
$430$	0	0
$431$	−5537.45	−0.618862	−0.309431	−	0.950922i	$-0.600139\pi$
$431$	−5537.45	−0.618862	−0.309431	+	0.950922i	$0.600139\pi$
$432$	2308.44	0.257095
$433$	10109.8	1.12205	0.561023	−	0.827800i	$-0.310408\pi$
$433$	10109.8	1.12205	0.561023	+	0.827800i	$0.310408\pi$
$434$	14631.0	1.61823
$435$	2148.20	0.236777
$436$	8549.56	0.939104
$437$	1411.68	0.154531
$438$	6484.65	0.707417
$439$	13005.7	1.41396	0.706981	−	0.707233i	$-0.250057\pi$
$439$	13005.7	1.41396	0.706981	+	0.707233i	$0.250057\pi$
$440$	12286.0	1.33116
$441$	−6044.86	−0.652722
$442$	20230.9	2.17712
$443$	12636.5	1.35526	0.677630	−	0.735403i	$-0.263007\pi$
$443$	12636.5	1.35526	0.677630	+	0.735403i	$0.263007\pi$
$444$	2165.71	0.231487
$445$	−6573.98	−0.700307
$446$	−25926.9	−2.75264
$447$	−1339.61	−0.141749
$448$	16625.1	1.75326
$449$	1602.63	0.168447	0.0842237	−	0.996447i	$-0.473159\pi$
$449$	1602.63	0.168447	0.0842237	+	0.996447i	$0.473159\pi$
$450$	−40330.8	−4.22492
$451$	−1455.78	−0.151995
$452$	−7575.00	−0.788269
$453$	−1597.86	−0.165726
$454$	−6542.12	−0.676292
$455$	29176.0	3.00614
$456$	3828.48	0.393168
$457$	−12386.5	−1.26786	−0.633932	−	0.773389i	$-0.718560\pi$
$457$	−12386.5	−1.26786	−0.633932	+	0.773389i	$0.718560\pi$
$458$	14723.4	1.50214
$459$	−4530.02	−0.460661
$460$	−4428.13	−0.448832
$461$	7814.88	0.789534	0.394767	−	0.918781i	$-0.370825\pi$
$461$	7814.88	0.789534	0.394767	+	0.918781i	$0.370825\pi$
$462$	2342.10	0.235854
$463$	−7828.20	−0.785761	−0.392881	−	0.919589i	$-0.628521\pi$
$463$	−7828.20	−0.785761	−0.392881	+	0.919589i	$0.628521\pi$
$464$	−3299.20	−0.330089
$465$	3137.22	0.312871
$466$	−8959.87	−0.890682
$467$	−372.814	−0.0369417	−0.0184709	−	0.999829i	$-0.505880\pi$
$467$	−372.814	−0.0369417	−0.0184709	+	0.999829i	$0.505880\pi$
$468$	21707.9	2.14412
$469$	−894.802	−0.0880983
$470$	−7287.39	−0.715196
$471$	−3309.92	−0.323807
$472$	10487.4	1.02272
$473$	0	0
$474$	−4659.58	−0.451522
$475$	−33168.4	−3.20393
$476$	26556.9	2.55721
$477$	−3901.92	−0.374542
$478$	−10101.3	−0.966574
$479$	6085.36	0.580474	0.290237	−	0.956955i	$-0.406266\pi$
$479$	6085.36	0.580474	0.290237	+	0.956955i	$0.406266\pi$
$480$	2002.25	0.190396
$481$	7170.75	0.679747
$482$	1157.79	0.109410
$483$	−389.593	−0.0367021
$484$	−15175.3	−1.42518
$485$	23609.7	2.21043
$486$	−11312.3	−1.05584
$487$	−1655.22	−0.154015	−0.0770075	−	0.997031i	$-0.524537\pi$
$487$	−1655.22	−0.154015	−0.0770075	+	0.997031i	$0.524537\pi$
$488$	11572.5	1.07349
$489$	−2304.47	−0.213112
$490$	23990.9	2.21184
$491$	−14763.9	−1.35700	−0.678498	−	0.734602i	$-0.737369\pi$
$491$	−14763.9	−1.35700	−0.678498	+	0.734602i	$0.737369\pi$
$492$	1422.97	0.130391
$493$	6474.25	0.591452
$494$	27465.9	2.50152
$495$	−9617.00	−0.873236
$496$	−4818.14	−0.436171
$497$	−18808.0	−1.69749
$498$	6159.55	0.554249
$499$	5889.43	0.528350	0.264175	−	0.964475i	$-0.414900\pi$
$499$	5889.43	0.528350	0.264175	+	0.964475i	$0.414900\pi$
$500$	64482.8	5.76751
$501$	2029.83	0.181010
$502$	−9687.71	−0.861322
$503$	18376.1	1.62893	0.814463	−	0.580215i	$-0.197032\pi$
$503$	18376.1	1.62893	0.814463	+	0.580215i	$0.197032\pi$
$504$	20233.2	1.78821
$505$	−16489.9	−1.45305
$506$	−1176.97	−0.103404
$507$	−1210.16	−0.106006
$508$	38972.2	3.40376
$509$	8355.89	0.727639	0.363820	−	0.931469i	$-0.381472\pi$
$509$	8355.89	0.727639	0.363820	+	0.931469i	$0.381472\pi$
$510$	8760.67	0.760645
$511$	28184.6	2.43995
$512$	−13006.4	−1.12267
$513$	−6150.06	−0.529302
$514$	−12951.7	−1.11143
$515$	6760.61	0.578463
$516$	0	0
$517$	−1259.00	−0.107100
$518$	14481.6	1.22835
$519$	−2150.04	−0.181842
$520$	−39762.5	−3.35327
$521$	19044.7	1.60146	0.800732	−	0.599022i	$-0.204444\pi$
$521$	19044.7	1.60146	0.800732	+	0.599022i	$0.204444\pi$
$522$	10687.6	0.896140
$523$	−5907.61	−0.493923	−0.246961	−	0.969025i	$-0.579432\pi$
$523$	−5907.61	−0.493923	−0.246961	+	0.969025i	$0.579432\pi$
$524$	−38694.9	−3.22594
$525$	9153.74	0.760956
$526$	−7718.18	−0.639788
$527$	9454.97	0.781528
$528$	−771.276	−0.0635710
$529$	−11971.2	−0.983909
$530$	15486.0	1.26919
$531$	−8209.14	−0.670897
$532$	36054.2	2.93825
$533$	4711.49	0.382884
$534$	1707.94	0.138408
$535$	13816.0	1.11648
$536$	1219.48	0.0982715
$537$	4532.38	0.364221
$538$	−17073.0	−1.36816
$539$	4144.77	0.331221
$540$	19291.4	1.53735
$541$	−22011.7	−1.74927	−0.874636	−	0.484780i	$-0.838900\pi$
$541$	−22011.7	−1.74927	−0.874636	+	0.484780i	$0.838900\pi$
$542$	−34669.9	−2.74760
$543$	−3483.63	−0.275317
$544$	6034.41	0.475594
$545$	12258.3	0.963463
$546$	−7579.99	−0.594128
$547$	−19182.2	−1.49940	−0.749700	−	0.661777i	$-0.769802\pi$
$547$	−19182.2	−1.49940	−0.749700	+	0.661777i	$0.769802\pi$
$548$	9883.74	0.770460
$549$	−9058.51	−0.704204
$550$	27653.6	2.14391
$551$	8789.59	0.679581
$552$	530.957	0.0409403
$553$	−20252.2	−1.55735
$554$	23416.0	1.79576
$555$	3105.18	0.237491
$556$	−13016.8	−0.992872
$557$	12163.8	0.925306	0.462653	−	0.886539i	$-0.346898\pi$
$557$	12163.8	0.925306	0.462653	+	0.886539i	$0.346898\pi$
$558$	15608.2	1.18413
$559$	0	0
$560$	−19403.6	−1.46420
$561$	1513.53	0.113906
$562$	−15934.5	−1.19601
$563$	9583.99	0.717437	0.358718	−	0.933446i	$-0.383214\pi$
$563$	9583.99	0.717437	0.358718	+	0.933446i	$0.383214\pi$
$564$	1230.62	0.0918769
$565$	−10861.0	−0.808715
$566$	−15016.9	−1.11521
$567$	−14967.5	−1.10860
$568$	25632.5	1.89351
$569$	7956.37	0.586201	0.293100	−	0.956082i	$-0.405313\pi$
$569$	7956.37	0.586201	0.293100	+	0.956082i	$0.405313\pi$
$570$	11893.7	0.873985
$571$	4971.46	0.364360	0.182180	−	0.983265i	$-0.441685\pi$
$571$	4971.46	0.364360	0.182180	+	0.983265i	$0.441685\pi$
$572$	−14884.4	−1.08802
$573$	455.277	0.0331928
$574$	9515.03	0.691899
$575$	−4600.00	−0.333623
$576$	17735.4	1.28295
$577$	23376.7	1.68663	0.843315	−	0.537420i	$-0.180601\pi$
$577$	23376.7	1.68663	0.843315	+	0.537420i	$0.180601\pi$
$578$	2914.56	0.209740
$579$	−1538.59	−0.110435
$580$	−27571.0	−1.97383
$581$	26771.6	1.91166
$582$	−6133.85	−0.436866
$583$	2675.42	0.190060
$584$	−38411.4	−2.72171
$585$	31124.6	2.19973
$586$	−20762.1	−1.46361
$587$	−13790.5	−0.969670	−0.484835	−	0.874606i	$-0.661120\pi$
$587$	−13790.5	−0.969670	−0.484835	+	0.874606i	$0.661120\pi$
$588$	−4051.35	−0.284141
$589$	12836.3	0.897979
$590$	32580.5	2.27342
$591$	617.667	0.0429906
$592$	−4768.93	−0.331084
$593$	22432.3	1.55343	0.776716	−	0.629851i	$-0.216884\pi$
$593$	22432.3	1.55343	0.776716	+	0.629851i	$0.216884\pi$
$594$	5127.51	0.354182
$595$	38077.1	2.62354
$596$	17193.3	1.18165
$597$	2531.30	0.173533
$598$	3809.15	0.260481
$599$	2181.92	0.148833	0.0744165	−	0.997227i	$-0.476291\pi$
$599$	2181.92	0.148833	0.0744165	+	0.997227i	$0.476291\pi$
$600$	−12475.2	−0.848828
$601$	−10415.4	−0.706907	−0.353453	−	0.935452i	$-0.614993\pi$
$601$	−10415.4	−0.706907	−0.353453	+	0.935452i	$0.614993\pi$
$602$	0	0
$603$	−954.562	−0.0644657
$604$	20507.7	1.38154
$605$	−21758.3	−1.46215
$606$	4284.12	0.287179
$607$	6163.42	0.412134	0.206067	−	0.978538i	$-0.433933\pi$
$607$	6163.42	0.412134	0.206067	+	0.978538i	$0.433933\pi$
$608$	8192.44	0.546460
$609$	−2425.73	−0.161405
$610$	35951.6	2.38629
$611$	4074.64	0.269791
$612$	28330.5	1.87123
$613$	6705.56	0.441819	0.220910	−	0.975294i	$-0.429097\pi$
$613$	6705.56	0.441819	0.220910	+	0.975294i	$0.429097\pi$
$614$	−39109.6	−2.57058
$615$	2040.24	0.133773
$616$	−13873.3	−0.907420
$617$	−1560.14	−0.101797	−0.0508985	−	0.998704i	$-0.516209\pi$
$617$	−1560.14	−0.101797	−0.0508985	+	0.998704i	$0.516209\pi$
$618$	−1756.42	−0.114326
$619$	−17652.2	−1.14621	−0.573104	−	0.819483i	$-0.694261\pi$
$619$	−17652.2	−1.14621	−0.573104	+	0.819483i	$0.694261\pi$
$620$	−40264.5	−2.60817
$621$	−852.929	−0.0551157
$622$	−23582.4	−1.52021
$623$	7423.31	0.477381
$624$	2496.17	0.160139
$625$	51360.5	3.28707
$626$	19166.1	1.22369
$627$	2054.80	0.130879
$628$	42481.1	2.69933
$629$	9358.42	0.593235
$630$	62857.2	3.97507
$631$	31138.2	1.96449	0.982245	−	0.187601i	$-0.0600712\pi$
$631$	31138.2	1.96449	0.982245	+	0.187601i	$0.0600712\pi$
$632$	27600.7	1.73718
$633$	5859.89	0.367946
$634$	7496.81	0.469616
$635$	55878.0	3.49205
$636$	−2615.12	−0.163045
$637$	−13414.2	−0.834363
$638$	−7328.18	−0.454742
$639$	−20064.1	−1.24214
$640$	−56551.1	−3.49278
$641$	17820.0	1.09805	0.549023	−	0.835807i	$-0.315000\pi$
$641$	17820.0	1.09805	0.549023	+	0.835807i	$0.315000\pi$
$642$	−3589.43	−0.220659
$643$	−3961.12	−0.242941	−0.121471	−	0.992595i	$-0.538761\pi$
$643$	−3961.12	−0.242941	−0.121471	+	0.992595i	$0.538761\pi$
$644$	5000.23	0.305957
$645$	0	0
$646$	35845.3	2.18315
$647$	1807.07	0.109804	0.0549019	−	0.998492i	$-0.482515\pi$
$647$	1807.07	0.109804	0.0549019	+	0.998492i	$0.482515\pi$
$648$	20398.5	1.23662
$649$	5628.75	0.340443
$650$	−89498.4	−5.40064
$651$	−3542.53	−0.213276
$652$	29576.6	1.77655
$653$	−20608.3	−1.23502	−0.617509	−	0.786564i	$-0.711858\pi$
$653$	−20608.3	−1.23502	−0.617509	+	0.786564i	$0.711858\pi$
$654$	−3184.73	−0.190417
$655$	−55480.4	−3.30962
$656$	−3133.39	−0.186492
$657$	30067.0	1.78543
$658$	8228.89	0.487531
$659$	−5125.22	−0.302959	−0.151480	−	0.988460i	$-0.548404\pi$
$659$	−5125.22	−0.302959	−0.151480	+	0.988460i	$0.548404\pi$
$660$	−6445.45	−0.380135
$661$	−19830.2	−1.16688	−0.583439	−	0.812157i	$-0.698293\pi$
$661$	−19830.2	−1.16688	−0.583439	+	0.812157i	$0.698293\pi$
$662$	−14284.7	−0.838656
$663$	−4898.40	−0.286935
$664$	−36485.7	−2.13241
$665$	51694.2	3.01446
$666$	15448.8	0.898841
$667$	1219.00	0.0707641
$668$	−26051.8	−1.50894
$669$	6277.55	0.362786
$670$	3788.48	0.218451
$671$	6211.14	0.357345
$672$	−2260.93	−0.129788
$673$	−20904.7	−1.19735	−0.598675	−	0.800992i	$-0.704306\pi$
$673$	−20904.7	−1.19735	−0.598675	+	0.800992i	$0.704306\pi$
$674$	−24754.8	−1.41472
$675$	20040.1	1.14273
$676$	15531.7	0.883689
$677$	−8266.41	−0.469282	−0.234641	−	0.972082i	$-0.575392\pi$
$677$	−8266.41	−0.469282	−0.234641	+	0.972082i	$0.575392\pi$
$678$	2821.70	0.159833
$679$	−26659.9	−1.50680
$680$	−51893.3	−2.92650
$681$	1584.01	0.0891326
$682$	−10702.0	−0.600883
$683$	−27099.4	−1.51820	−0.759098	−	0.650976i	$-0.774360\pi$
$683$	−27099.4	−1.51820	−0.759098	+	0.650976i	$0.774360\pi$
$684$	38462.2	2.15006
$685$	14171.2	0.790444
$686$	12353.6	0.687555
$687$	−3564.90	−0.197976
$688$	0	0
$689$	−8658.77	−0.478770
$690$	1649.49	0.0910073
$691$	−18247.5	−1.00458	−0.502292	−	0.864698i	$-0.667510\pi$
$691$	−18247.5	−1.00458	−0.502292	+	0.864698i	$0.667510\pi$
$692$	27594.6	1.51588
$693$	10859.5	0.595263
$694$	39381.9	2.15406
$695$	−18663.4	−1.01862
$696$	3305.91	0.180043
$697$	6148.88	0.334154
$698$	−13151.8	−0.713183
$699$	2169.40	0.117388
$700$	−117483.	−6.34351
$701$	27373.3	1.47486	0.737430	−	0.675424i	$-0.236039\pi$
$701$	27373.3	1.47486	0.737430	+	0.675424i	$0.236039\pi$
$702$	−16594.7	−0.892205
$703$	12705.2	0.681629
$704$	−12160.6	−0.651025
$705$	1764.46	0.0942599
$706$	−522.415	−0.0278489
$707$	18620.4	0.990510
$708$	−5501.88	−0.292053
$709$	−12752.9	−0.675524	−0.337762	−	0.941232i	$-0.609670\pi$
$709$	−12752.9	−0.675524	−0.337762	+	0.941232i	$0.609670\pi$
$710$	79630.8	4.20914
$711$	−21604.8	−1.13958
$712$	−10116.9	−0.532507
$713$	1780.21	0.0935057
$714$	−9892.51	−0.518513
$715$	−21341.1	−1.11624
$716$	−58170.7	−3.03623
$717$	2445.77	0.127391
$718$	−55167.8	−2.86747
$719$	−26662.5	−1.38295	−0.691477	−	0.722398i	$-0.743040\pi$
$719$	−26662.5	−1.38295	−0.691477	+	0.722398i	$0.743040\pi$
$720$	−20699.5	−1.07142
$721$	−7634.05	−0.394323
$722$	15872.3	0.818154
$723$	−280.328	−0.0144198
$724$	44710.6	2.29511
$725$	−28641.1	−1.46718
$726$	5652.85	0.288976
$727$	27767.2	1.41655	0.708273	−	0.705939i	$-0.249475\pi$
$727$	27767.2	1.41655	0.708273	+	0.705939i	$0.249475\pi$
$728$	44899.6	2.28584
$729$	−14062.0	−0.714423
$730$	−119330.	−6.05016
$731$	0	0
$732$	−6071.14	−0.306552
$733$	5819.92	0.293265	0.146633	−	0.989191i	$-0.453156\pi$
$733$	5819.92	0.293265	0.146633	+	0.989191i	$0.453156\pi$
$734$	−62591.6	−3.14754
$735$	−5808.79	−0.291511
$736$	1136.18	0.0569023
$737$	654.514	0.0327128
$738$	10150.5	0.506295
$739$	−4469.69	−0.222490	−0.111245	−	0.993793i	$-0.535484\pi$
$739$	−4469.69	−0.222490	−0.111245	+	0.993793i	$0.535484\pi$
$740$	−39853.4	−1.97978
$741$	−6650.18	−0.329690
$742$	−17486.7	−0.865172
$743$	−31260.1	−1.54350	−0.771752	−	0.635924i	$-0.780619\pi$
$743$	−31260.1	−1.54350	−0.771752	+	0.635924i	$0.780619\pi$
$744$	4827.94	0.237904
$745$	24651.6	1.21230
$746$	47676.1	2.33987
$747$	28559.6	1.39885
$748$	−19425.4	−0.949548
$749$	−15600.9	−0.761076
$750$	−24020.0	−1.16945
$751$	9943.65	0.483154	0.241577	−	0.970382i	$-0.422335\pi$
$751$	9943.65	0.483154	0.241577	+	0.970382i	$0.422335\pi$
$752$	−2709.85	−0.131407
$753$	2345.63	0.113519
$754$	23717.0	1.14552
$755$	29403.8	1.41737
$756$	−21783.7	−1.04797
$757$	−11664.7	−0.560056	−0.280028	−	0.959992i	$-0.590344\pi$
$757$	−11664.7	−0.560056	−0.280028	+	0.959992i	$0.590344\pi$
$758$	8138.02	0.389955
$759$	284.973	0.0136283
$760$	−70451.5	−3.36256
$761$	−23261.8	−1.10807	−0.554033	−	0.832495i	$-0.686912\pi$
$761$	−23261.8	−1.10807	−0.554033	+	0.832495i	$0.686912\pi$
$762$	−14517.2	−0.690163
$763$	−13842.0	−0.656768
$764$	−5843.25	−0.276703
$765$	40620.1	1.91977
$766$	−14512.1	−0.684522
$767$	−18216.9	−0.857595
$768$	8291.50	0.389575
$769$	−12725.0	−0.596717	−0.298358	−	0.954454i	$-0.596439\pi$
$769$	−12725.0	−0.596717	−0.298358	+	0.954454i	$0.596439\pi$
$770$	−43099.2	−2.01713
$771$	3135.92	0.146482
$772$	19747.0	0.920609
$773$	31896.0	1.48411	0.742056	−	0.670337i	$-0.233851\pi$
$773$	31896.0	1.48411	0.742056	+	0.670337i	$0.233851\pi$
$774$	0	0
$775$	−41827.3	−1.93868
$776$	36333.5	1.68079
$777$	−3506.35	−0.161892
$778$	45109.3	2.07872
$779$	8347.85	0.383945
$780$	20860.1	0.957580
$781$	13757.3	0.630316
$782$	4971.25	0.227329
$783$	−5310.61	−0.242383
$784$	8921.14	0.406393
$785$	60909.1	2.76935
$786$	14413.9	0.654107
$787$	−3554.63	−0.161003	−0.0805013	−	0.996755i	$-0.525652\pi$
$787$	−3554.63	−0.161003	−0.0805013	+	0.996755i	$0.525652\pi$
$788$	−7927.44	−0.358380
$789$	1868.76	0.0843214
$790$	85745.5	3.86163
$791$	12264.1	0.551281
$792$	−14799.8	−0.664001
$793$	−20101.8	−0.900171
$794$	23635.8	1.05643
$795$	−3749.54	−0.167274
$796$	−32487.9	−1.44661
$797$	15944.6	0.708640	0.354320	−	0.935124i	$-0.384712\pi$
$797$	15944.6	0.708640	0.354320	+	0.935124i	$0.384712\pi$
$798$	−13430.3	−0.595773
$799$	5317.73	0.235454
$800$	−26695.3	−1.17977
$801$	7919.09	0.349322
$802$	−48473.2	−2.13423
$803$	−20616.0	−0.906006
$804$	−639.762	−0.0280630
$805$	7169.28	0.313893
$806$	34636.2	1.51366
$807$	4133.79	0.180318
$808$	−25376.7	−1.10489
$809$	12758.5	0.554467	0.277234	−	0.960803i	$-0.410582\pi$
$809$	12758.5	0.554467	0.277234	+	0.960803i	$0.410582\pi$
$810$	63370.8	2.74892
$811$	33232.7	1.43891	0.719457	−	0.694537i	$-0.244391\pi$
$811$	33232.7	1.43891	0.719457	+	0.694537i	$0.244391\pi$
$812$	31133.0	1.34551
$813$	8394.44	0.362123
$814$	−10592.8	−0.456113
$815$	42406.7	1.82263
$816$	3257.70	0.139758
$817$	0	0
$818$	−3539.81	−0.151304
$819$	−35145.7	−1.49950
$820$	−26185.4	−1.11516
$821$	1216.94	0.0517314	0.0258657	−	0.999665i	$-0.491766\pi$
$821$	1216.94	0.0517314	0.0258657	+	0.999665i	$0.491766\pi$
$822$	−3681.72	−0.156222
$823$	4475.87	0.189574	0.0947869	−	0.995498i	$-0.469783\pi$
$823$	4475.87	0.189574	0.0947869	+	0.995498i	$0.469783\pi$
$824$	10404.1	0.439858
$825$	−6695.61	−0.282559
$826$	−36789.8	−1.54973
$827$	38299.6	1.61041	0.805205	−	0.592997i	$-0.202055\pi$
$827$	38299.6	1.61041	0.805205	+	0.592997i	$0.202055\pi$
$828$	5334.17	0.223883
$829$	14373.6	0.602192	0.301096	−	0.953594i	$-0.402648\pi$
$829$	14373.6	0.602192	0.301096	+	0.953594i	$0.402648\pi$
$830$	−113348.	−4.74020
$831$	−5669.59	−0.236674
$832$	39356.8	1.63997
$833$	−17506.6	−0.728172
$834$	4848.80	0.201319
$835$	−37352.8	−1.54808
$836$	−26372.3	−1.09104
$837$	−7755.59	−0.320278
$838$	57522.5	2.37122
$839$	20770.8	0.854691	0.427346	−	0.904088i	$-0.359449\pi$
$839$	20770.8	0.854691	0.427346	+	0.904088i	$0.359449\pi$
$840$	19443.1	0.798630
$841$	−16799.1	−0.688800
$842$	−6126.43	−0.250749
$843$	3858.13	0.157629
$844$	−75208.6	−3.06728
$845$	22269.2	0.906609
$846$	8778.47	0.356749
$847$	24569.3	0.996708
$848$	5758.54	0.233195
$849$	3635.96	0.146980
$850$	−116803.	−4.71329
$851$	1762.04	0.0709774
$852$	−13447.3	−0.540723
$853$	21536.0	0.864455	0.432228	−	0.901765i	$-0.357728\pi$
$853$	21536.0	0.864455	0.432228	+	0.901765i	$0.357728\pi$
$854$	−40596.3	−1.62667
$855$	55146.7	2.20582
$856$	21261.7	0.848962
$857$	34255.5	1.36540	0.682699	−	0.730700i	$-0.260806\pi$
$857$	34255.5	1.36540	0.682699	+	0.730700i	$0.260806\pi$
$858$	5544.48	0.220612
$859$	39691.7	1.57656	0.788279	−	0.615317i	$-0.210972\pi$
$859$	39691.7	1.57656	0.788279	+	0.615317i	$0.210972\pi$
$860$	0	0
$861$	−2303.82	−0.0911894
$862$	−26473.9	−1.04606
$863$	10554.4	0.416310	0.208155	−	0.978096i	$-0.433254\pi$
$863$	10554.4	0.416310	0.208155	+	0.978096i	$0.433254\pi$
$864$	−4949.82	−0.194903
$865$	39564.9	1.55520
$866$	48333.7	1.89659
$867$	−705.686	−0.0276429
$868$	45466.5	1.77792
$869$	14813.7	0.578276
$870$	10270.3	0.400224
$871$	−2118.27	−0.0824053
$872$	18864.6	0.732608
$873$	−28440.5	−1.10259
$874$	6749.08	0.261202
$875$	−104400.	−4.03354
$876$	20151.3	0.777227
$877$	−2628.97	−0.101224	−0.0506122	−	0.998718i	$-0.516117\pi$
$877$	−2628.97	−0.101224	−0.0506122	+	0.998718i	$0.516117\pi$
$878$	62178.7	2.39001
$879$	5027.02	0.192898
$880$	14193.0	0.543688
$881$	−44531.4	−1.70295	−0.851476	−	0.524394i	$-0.824292\pi$
$881$	−44531.4	−1.70295	−0.851476	+	0.524394i	$0.824292\pi$
$882$	−28899.7	−1.10329
$883$	12638.0	0.481657	0.240828	−	0.970568i	$-0.422581\pi$
$883$	12638.0	0.481657	0.240828	+	0.970568i	$0.422581\pi$
$884$	62868.5	2.39196
$885$	−7888.55	−0.299628
$886$	60413.8	2.29079
$887$	−47366.5	−1.79302	−0.896511	−	0.443022i	$-0.853906\pi$
$887$	−47366.5	−1.79302	−0.896511	+	0.443022i	$0.853906\pi$
$888$	4778.63	0.180586
$889$	−63097.2	−2.38044
$890$	−31429.4	−1.18373
$891$	10948.2	0.411648
$892$	−80569.1	−3.02427
$893$	7219.47	0.270538
$894$	−6404.53	−0.239597
$895$	−83404.7	−3.11499
$896$	63857.2	2.38094
$897$	−922.288	−0.0343303
$898$	7661.99	0.284726
$899$	11084.2	0.411211
$900$	−125330.	−4.64184
$901$	−11300.4	−0.417837
$902$	−6959.89	−0.256917
$903$	0	0
$904$	−16714.2	−0.614940
$905$	64105.7	2.35464
$906$	−7639.18	−0.280127
$907$	3162.85	0.115789	0.0578946	−	0.998323i	$-0.481561\pi$
$907$	3162.85	0.115789	0.0578946	+	0.998323i	$0.481561\pi$
$908$	−20329.9	−0.743031
$909$	19863.9	0.724802
$910$	139487.	5.08126
$911$	−13963.5	−0.507829	−0.253914	−	0.967227i	$-0.581718\pi$
$911$	−13963.5	−0.507829	−0.253914	+	0.967227i	$0.581718\pi$
$912$	4422.72	0.160582
$913$	−19582.4	−0.709840
$914$	−59218.1	−2.14306
$915$	−8704.75	−0.314503
$916$	45753.7	1.65037
$917$	62648.2	2.25608
$918$	−21657.5	−0.778653
$919$	5006.75	0.179714	0.0898572	−	0.995955i	$-0.471359\pi$
$919$	5006.75	0.179714	0.0898572	+	0.995955i	$0.471359\pi$
$920$	−9770.65	−0.350140
$921$	9469.40	0.338792
$922$	37362.0	1.33455
$923$	−44524.4	−1.58780
$924$	7278.18	0.259128
$925$	−41400.1	−1.47160
$926$	−37425.7	−1.32817
$927$	−8143.91	−0.288545
$928$	7074.22	0.250240
$929$	−35356.7	−1.24867	−0.624336	−	0.781156i	$-0.714630\pi$
$929$	−35356.7	−1.24867	−0.624336	+	0.781156i	$0.714630\pi$
$930$	14998.6	0.528844
$931$	−23767.3	−0.836673
$932$	−27843.2	−0.978577
$933$	5709.89	0.200357
$934$	−1782.38	−0.0624425
$935$	−27851.9	−0.974177
$936$	47898.3	1.67266
$937$	−42579.5	−1.48454	−0.742269	−	0.670103i	$-0.766250\pi$
$937$	−42579.5	−1.48454	−0.742269	+	0.670103i	$0.766250\pi$
$938$	−4277.94	−0.148912
$939$	−4640.58	−0.161278
$940$	−22645.9	−0.785774
$941$	−24475.5	−0.847905	−0.423952	−	0.905685i	$-0.639358\pi$
$941$	−24475.5	−0.847905	−0.423952	+	0.905685i	$0.639358\pi$
$942$	−15824.3	−0.547330
$943$	1157.73	0.0399798
$944$	12115.2	0.417709
$945$	−31233.3	−1.07515
$946$	0	0
$947$	5042.58	0.173032	0.0865162	−	0.996250i	$-0.472427\pi$
$947$	5042.58	0.173032	0.0865162	+	0.996250i	$0.472427\pi$
$948$	−14479.9	−0.496080
$949$	66721.8	2.28228
$950$	−158574.	−5.41560
$951$	−1815.16	−0.0618934
$952$	58597.7	1.99492
$953$	22662.7	0.770323	0.385161	−	0.922849i	$-0.374146\pi$
$953$	22662.7	0.770323	0.385161	+	0.922849i	$0.374146\pi$
$954$	−18654.6	−0.633087
$955$	−8378.00	−0.283880
$956$	−31390.2	−1.06196
$957$	1774.33	0.0599331
$958$	29093.4	0.981173
$959$	−16002.1	−0.538826
$960$	17042.8	0.572974
$961$	−13603.7	−0.456638
$962$	34282.5	1.14897
$963$	−16642.9	−0.556915
$964$	3597.87	0.120207
$965$	28313.1	0.944488
$966$	−1862.60	−0.0620373
$967$	20319.2	0.675720	0.337860	−	0.941196i	$-0.390297\pi$
$967$	20319.2	0.675720	0.337860	+	0.941196i	$0.390297\pi$
$968$	−33484.3	−1.11180
$969$	−8679.03	−0.287730
$970$	112875.	3.73628
$971$	2458.32	0.0812474	0.0406237	−	0.999175i	$-0.487066\pi$
$971$	2458.32	0.0812474	0.0406237	+	0.999175i	$0.487066\pi$
$972$	−35153.5	−1.16003
$973$	21074.6	0.694370
$974$	−7913.41	−0.260331
$975$	21669.8	0.711782
$976$	13368.8	0.438446
$977$	56845.4	1.86146	0.930729	−	0.365709i	$-0.119174\pi$
$977$	56845.4	1.86146	0.930729	+	0.365709i	$0.119174\pi$
$978$	−11017.4	−0.360222
$979$	−5429.87	−0.177262
$980$	74552.8	2.43010
$981$	−14766.5	−0.480588
$982$	−70584.4	−2.29372
$983$	9749.84	0.316350	0.158175	−	0.987411i	$-0.449439\pi$
$983$	9749.84	0.316350	0.158175	+	0.987411i	$0.449439\pi$
$984$	3139.77	0.101720
$985$	−11366.3	−0.367675
$986$	30952.6	0.999728
$987$	−1992.42	−0.0642546
$988$	85351.6	2.74838
$989$	0	0
$990$	−45977.7	−1.47603
$991$	−44926.5	−1.44010	−0.720050	−	0.693922i	$-0.755881\pi$
$991$	−44926.5	−1.44010	−0.720050	+	0.693922i	$0.755881\pi$
$992$	10331.2	0.330660
$993$	3458.68	0.110531
$994$	−89918.7	−2.86926
$995$	−46580.9	−1.48413
$996$	19141.1	0.608944
$997$	−1536.04	−0.0487934	−0.0243967	−	0.999702i	$-0.507766\pi$
$997$	−1536.04	−0.0487934	−0.0243967	+	0.999702i	$0.507766\pi$
$998$	28156.6	0.893068
$999$	−7676.39	−0.243113

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.j.1.44	yes	50
43.42	odd	2		1849.4.a.i.1.7	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.7	✓	50	43.42	odd	2
1849.4.a.j.1.44	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)

\(f(q)\)	\(=\)	\(q+4.78088 q^{2} -1.15757 q^{3} +14.8568 q^{4} +21.3015 q^{5} -5.53419 q^{6} -24.0536 q^{7} +32.7814 q^{8} -25.6600 q^{9} +O(q^{10})\)	\(q+4.78088 q^{2} -1.15757 q^{3} +14.8568 q^{4} +21.3015 q^{5} -5.53419 q^{6} -24.0536 q^{7} +32.7814 q^{8} -25.6600 q^{9} +101.840 q^{10} +17.5943 q^{11} -17.1977 q^{12} -56.9423 q^{13} -114.997 q^{14} -24.6580 q^{15} +37.8697 q^{16} -74.3144 q^{17} -122.677 q^{18} -100.891 q^{19} +316.472 q^{20} +27.8436 q^{21} +84.1161 q^{22} -13.9922 q^{23} -37.9467 q^{24} +328.755 q^{25} -272.234 q^{26} +60.9576 q^{27} -357.359 q^{28} -87.1198 q^{29} -117.887 q^{30} -127.229 q^{31} -81.2011 q^{32} -20.3666 q^{33} -355.288 q^{34} -512.378 q^{35} -381.226 q^{36} -125.930 q^{37} -482.347 q^{38} +65.9146 q^{39} +698.294 q^{40} -82.7414 q^{41} +133.117 q^{42} +261.394 q^{44} -546.598 q^{45} -66.8948 q^{46} -71.5573 q^{47} -43.8367 q^{48} +235.575 q^{49} +1571.74 q^{50} +86.0239 q^{51} -845.980 q^{52} +152.062 q^{53} +291.431 q^{54} +374.785 q^{55} -788.511 q^{56} +116.788 q^{57} -416.509 q^{58} +319.919 q^{59} -366.338 q^{60} +353.020 q^{61} -608.268 q^{62} +617.216 q^{63} -691.170 q^{64} -1212.96 q^{65} -97.3701 q^{66} +37.2004 q^{67} -1104.07 q^{68} +16.1969 q^{69} -2449.62 q^{70} +781.921 q^{71} -841.173 q^{72} -1171.74 q^{73} -602.056 q^{74} -380.556 q^{75} -1498.91 q^{76} -423.206 q^{77} +315.130 q^{78} +841.963 q^{79} +806.682 q^{80} +622.259 q^{81} -395.577 q^{82} -1113.00 q^{83} +413.667 q^{84} -1583.01 q^{85} +100.847 q^{87} +576.766 q^{88} -308.616 q^{89} -2613.22 q^{90} +1369.67 q^{91} -207.879 q^{92} +147.277 q^{93} -342.106 q^{94} -2149.13 q^{95} +93.9957 q^{96} +1108.36 q^{97} +1126.25 q^{98} -451.470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \)	\( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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