LMFDB - Embedded newform 1849.4.a.i.1.24

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			1.24
Character	$\chi$	$=$	1849.1

$q$-expansion

$f(q)$	$=$	$q-0.902811 q^{2} -9.58399 q^{3} -7.18493 q^{4} -2.03345 q^{5} +8.65253 q^{6} +26.8278 q^{7} +13.7091 q^{8} +64.8528 q^{9} +O(q^{10})$	\(q-0.902811 q^{2} -9.58399 q^{3} -7.18493 q^{4} -2.03345 q^{5} +8.65253 q^{6} +26.8278 q^{7} +13.7091 q^{8} +64.8528 q^{9} +1.83582 q^{10} +38.6799 q^{11} +68.8603 q^{12} -37.4866 q^{13} -24.2205 q^{14} +19.4886 q^{15} +45.1027 q^{16} -12.0881 q^{17} -58.5498 q^{18} -64.0712 q^{19} +14.6102 q^{20} -257.118 q^{21} -34.9206 q^{22} -85.2739 q^{23} -131.388 q^{24} -120.865 q^{25} +33.8433 q^{26} -362.781 q^{27} -192.756 q^{28} +129.469 q^{29} -17.5945 q^{30} +261.836 q^{31} -150.392 q^{32} -370.707 q^{33} +10.9133 q^{34} -54.5531 q^{35} -465.963 q^{36} +310.147 q^{37} +57.8442 q^{38} +359.271 q^{39} -27.8768 q^{40} -333.686 q^{41} +232.129 q^{42} -277.912 q^{44} -131.875 q^{45} +76.9862 q^{46} -475.663 q^{47} -432.264 q^{48} +376.733 q^{49} +109.118 q^{50} +115.852 q^{51} +269.339 q^{52} +14.1574 q^{53} +327.523 q^{54} -78.6536 q^{55} +367.786 q^{56} +614.057 q^{57} -116.886 q^{58} -652.202 q^{59} -140.024 q^{60} -233.602 q^{61} -236.388 q^{62} +1739.86 q^{63} -225.046 q^{64} +76.2271 q^{65} +334.679 q^{66} +963.472 q^{67} +86.8523 q^{68} +817.264 q^{69} +49.2511 q^{70} +887.185 q^{71} +889.075 q^{72} -345.150 q^{73} -280.004 q^{74} +1158.37 q^{75} +460.347 q^{76} +1037.70 q^{77} -324.354 q^{78} -396.269 q^{79} -91.7141 q^{80} +1725.86 q^{81} +301.255 q^{82} +539.877 q^{83} +1847.37 q^{84} +24.5806 q^{85} -1240.83 q^{87} +530.267 q^{88} +295.412 q^{89} +119.058 q^{90} -1005.68 q^{91} +612.687 q^{92} -2509.43 q^{93} +429.433 q^{94} +130.286 q^{95} +1441.36 q^{96} +191.963 q^{97} -340.119 q^{98} +2508.50 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.902811	−0.319192	−0.159596	−	0.987182i	$-0.551019\pi$
$2$	−0.902811	−0.319192	−0.159596	+	0.987182i	$0.551019\pi$
$3$	−9.58399	−1.84444	−0.922220	−	0.386667i	$-0.873626\pi$
$3$	−9.58399	−1.84444	−0.922220	+	0.386667i	$0.873626\pi$
$4$	−7.18493	−0.898117
$5$	−2.03345	−0.181877	−0.0909387	−	0.995856i	$-0.528987\pi$
$5$	−2.03345	−0.181877	−0.0909387	+	0.995856i	$0.528987\pi$
$6$	8.65253	0.588730
$7$	26.8278	1.44857	0.724284	−	0.689502i	$-0.242171\pi$
$7$	26.8278	1.44857	0.724284	+	0.689502i	$0.242171\pi$
$8$	13.7091	0.605863
$9$	64.8528	2.40196
$10$	1.83582	0.0580538
$11$	38.6799	1.06022	0.530110	−	0.847929i	$-0.322151\pi$
$11$	38.6799	1.06022	0.530110	+	0.847929i	$0.322151\pi$
$12$	68.8603	1.65652
$13$	−37.4866	−0.799762	−0.399881	−	0.916567i	$-0.630949\pi$
$13$	−37.4866	−0.799762	−0.399881	+	0.916567i	$0.630949\pi$
$14$	−24.2205	−0.462371
$15$	19.4886	0.335462
$16$	45.1027	0.704730
$17$	−12.0881	−0.172459	−0.0862294	−	0.996275i	$-0.527482\pi$
$17$	−12.0881	−0.172459	−0.0862294	+	0.996275i	$0.527482\pi$
$18$	−58.5498	−0.766685
$19$	−64.0712	−0.773628	−0.386814	−	0.922158i	$-0.626424\pi$
$19$	−64.0712	−0.773628	−0.386814	+	0.922158i	$0.626424\pi$
$20$	14.6102	0.163347
$21$	−257.118	−2.67179
$22$	−34.9206	−0.338414
$23$	−85.2739	−0.773080	−0.386540	−	0.922273i	$-0.626330\pi$
$23$	−85.2739	−0.773080	−0.386540	+	0.922273i	$0.626330\pi$
$24$	−131.388	−1.11748
$25$	−120.865	−0.966921
$26$	33.8433	0.255278
$27$	−362.781	−2.58582
$28$	−192.756	−1.30098
$29$	129.469	0.829029	0.414514	−	0.910043i	$-0.363951\pi$
$29$	129.469	0.829029	0.414514	+	0.910043i	$0.363951\pi$
$30$	−17.5945	−0.107077
$31$	261.836	1.51700	0.758501	−	0.651671i	$-0.225932\pi$
$31$	261.836	1.51700	0.758501	+	0.651671i	$0.225932\pi$
$32$	−150.392	−0.830808
$33$	−370.707	−1.95551
$34$	10.9133	0.0550474
$35$	−54.5531	−0.263462
$36$	−465.963	−2.15724
$37$	310.147	1.37805	0.689024	−	0.724739i	$-0.258039\pi$
$37$	310.147	1.37805	0.689024	+	0.724739i	$0.258039\pi$
$38$	57.8442	0.246936
$39$	359.271	1.47511
$40$	−27.8768	−0.110193
$41$	−333.686	−1.27105	−0.635524	−	0.772081i	$-0.719216\pi$
$41$	−333.686	−1.27105	−0.635524	+	0.772081i	$0.719216\pi$
$42$	232.129	0.852815
$43$	0	0
$43$	0	0
$44$	−277.912	−0.952201
$45$	−131.875	−0.436861
$46$	76.9862	0.246761
$47$	−475.663	−1.47622	−0.738112	−	0.674678i	$-0.764282\pi$
$47$	−475.663	−1.47622	−0.738112	+	0.674678i	$0.764282\pi$
$48$	−432.264	−1.29983
$49$	376.733	1.09835
$50$	109.118	0.308633
$51$	115.852	0.318090
$52$	269.339	0.718280
$53$	14.1574	0.0366919	0.0183460	−	0.999832i	$-0.494160\pi$
$53$	14.1574	0.0366919	0.0183460	+	0.999832i	$0.494160\pi$
$54$	327.523	0.825374
$55$	−78.6536	−0.192830
$56$	367.786	0.877634
$57$	614.057	1.42691
$58$	−116.886	−0.264619
$59$	−652.202	−1.43914	−0.719571	−	0.694419i	$-0.755662\pi$
$59$	−652.202	−1.43914	−0.719571	+	0.694419i	$0.755662\pi$
$60$	−140.024	−0.301284
$61$	−233.602	−0.490323	−0.245162	−	0.969482i	$-0.578841\pi$
$61$	−233.602	−0.490323	−0.245162	+	0.969482i	$0.578841\pi$
$62$	−236.388	−0.484215
$63$	1739.86	3.47939
$64$	−225.046	−0.439543
$65$	76.2271	0.145459
$66$	334.679	0.624183
$67$	963.472	1.75682	0.878409	−	0.477909i	$-0.158605\pi$
$67$	963.472	1.75682	0.878409	+	0.477909i	$0.158605\pi$
$68$	86.8523	0.154888
$69$	817.264	1.42590
$70$	49.2511	0.0840948
$71$	887.185	1.48295	0.741475	−	0.670980i	$-0.234126\pi$
$71$	887.185	1.48295	0.741475	+	0.670980i	$0.234126\pi$
$72$	889.075	1.45526
$73$	−345.150	−0.553380	−0.276690	−	0.960959i	$-0.589238\pi$
$73$	−345.150	−0.553380	−0.276690	+	0.960959i	$0.589238\pi$
$74$	−280.004	−0.439862
$75$	1158.37	1.78343
$76$	460.347	0.694808
$77$	1037.70	1.53580
$78$	−324.354	−0.470844
$79$	−396.269	−0.564351	−0.282176	−	0.959363i	$-0.591056\pi$
$79$	−396.269	−0.564351	−0.282176	+	0.959363i	$0.591056\pi$
$80$	−91.7141	−0.128174
$81$	1725.86	2.36744
$82$	301.255	0.405708
$83$	539.877	0.713966	0.356983	−	0.934111i	$-0.383805\pi$
$83$	539.877	0.713966	0.356983	+	0.934111i	$0.383805\pi$
$84$	1847.37	2.39958
$85$	24.5806	0.0313663
$86$	0	0
$87$	−1240.83	−1.52909
$88$	530.267	0.642349
$89$	295.412	0.351838	0.175919	−	0.984405i	$-0.443710\pi$
$89$	295.412	0.351838	0.175919	+	0.984405i	$0.443710\pi$
$90$	119.058	0.139443
$91$	−1005.68	−1.15851
$92$	612.687	0.694316
$93$	−2509.43	−2.79802
$94$	429.433	0.471199
$95$	130.286	0.140705
$96$	1441.36	1.53237
$97$	191.963	0.200937	0.100469	−	0.994940i	$-0.467966\pi$
$97$	191.963	0.200937	0.100469	+	0.994940i	$0.467966\pi$
$98$	−340.119	−0.350584
$99$	2508.50	2.54660
$100$	868.407	0.868407
$101$	−1717.20	−1.69176	−0.845878	−	0.533377i	$-0.820923\pi$
$101$	−1717.20	−1.69176	−0.845878	+	0.533377i	$0.820923\pi$
$102$	−104.593	−0.101532
$103$	−31.9437	−0.0305583	−0.0152792	−	0.999883i	$-0.504864\pi$
$103$	−31.9437	−0.0305583	−0.0152792	+	0.999883i	$0.504864\pi$
$104$	−513.908	−0.484547
$105$	522.836	0.485939
$106$	−12.7815	−0.0117118
$107$	1275.01	1.15196	0.575980	−	0.817463i	$-0.304620\pi$
$107$	1275.01	1.15196	0.575980	+	0.817463i	$0.304620\pi$
$108$	2606.56	2.32237
$109$	−1181.87	−1.03856	−0.519278	−	0.854606i	$-0.673799\pi$
$109$	−1181.87	−1.03856	−0.519278	+	0.854606i	$0.673799\pi$
$110$	71.0094	0.0615498
$111$	−2972.44	−2.54173
$112$	1210.01	1.02085
$113$	1007.21	0.838495	0.419248	−	0.907872i	$-0.362294\pi$
$113$	1007.21	0.838495	0.419248	+	0.907872i	$0.362294\pi$
$114$	−554.378	−0.455458
$115$	173.400	0.140606
$116$	−930.228	−0.744564
$117$	−2431.11	−1.92099
$118$	588.815	0.459363
$119$	−324.298	−0.249818
$120$	267.171	0.203244
$121$	165.132	0.124066
$122$	210.899	0.156507
$123$	3198.04	2.34437
$124$	−1881.27	−1.36245
$125$	499.955	0.357738
$126$	−1570.77	−1.11059
$127$	−694.035	−0.484926	−0.242463	−	0.970161i	$-0.577955\pi$
$127$	−694.035	−0.484926	−0.242463	+	0.970161i	$0.577955\pi$
$128$	1406.31	0.971106
$129$	0	0
$130$	−68.8187	−0.0464292
$131$	−1014.00	−0.676286	−0.338143	−	0.941095i	$-0.609799\pi$
$131$	−1014.00	−0.676286	−0.338143	+	0.941095i	$0.609799\pi$
$132$	2663.51	1.75628
$133$	−1718.89	−1.12065
$134$	−869.833	−0.560762
$135$	737.697	0.470303
$136$	−165.717	−0.104486
$137$	633.186	0.394867	0.197433	−	0.980316i	$-0.436739\pi$
$137$	633.186	0.394867	0.197433	+	0.980316i	$0.436739\pi$
$138$	−737.835	−0.455135
$139$	−437.269	−0.266825	−0.133412	−	0.991061i	$-0.542594\pi$
$139$	−437.269	−0.266825	−0.133412	+	0.991061i	$0.542594\pi$
$140$	391.960	0.236619
$141$	4558.74	2.72280
$142$	−800.961	−0.473346
$143$	−1449.98	−0.847924
$144$	2925.04	1.69273
$145$	−263.269	−0.150782
$146$	311.605	0.176635
$147$	−3610.60	−2.02583
$148$	−2228.38	−1.23765
$149$	2052.33	1.12841	0.564205	−	0.825635i	$-0.309183\pi$
$149$	2052.33	1.12841	0.564205	+	0.825635i	$0.309183\pi$
$150$	−1045.79	−0.569255
$151$	1021.62	0.550583	0.275292	−	0.961361i	$-0.411226\pi$
$151$	1021.62	0.550583	0.275292	+	0.961361i	$0.411226\pi$
$152$	−878.360	−0.468713
$153$	−783.948	−0.414238
$154$	−936.845	−0.490215
$155$	−532.430	−0.275909
$156$	−2581.34	−1.32482
$157$	183.529	0.0932942	0.0466471	−	0.998911i	$-0.485146\pi$
$157$	183.529	0.0932942	0.0466471	+	0.998911i	$0.485146\pi$
$158$	357.756	0.180136
$159$	−135.685	−0.0676760
$160$	305.815	0.151105
$161$	−2287.71	−1.11986
$162$	−1558.13	−0.755666
$163$	832.248	0.399918	0.199959	−	0.979804i	$-0.435919\pi$
$163$	832.248	0.399918	0.199959	+	0.979804i	$0.435919\pi$
$164$	2397.51	1.14155
$165$	753.815	0.355663
$166$	−487.407	−0.227892
$167$	−319.899	−0.148231	−0.0741153	−	0.997250i	$-0.523613\pi$
$167$	−319.899	−0.148231	−0.0741153	+	0.997250i	$0.523613\pi$
$168$	−3524.86	−1.61874
$169$	−791.756	−0.360380
$170$	−22.1916	−0.0100119
$171$	−4155.20	−1.85822
$172$	0	0
$173$	−3014.11	−1.32462	−0.662309	−	0.749231i	$-0.730423\pi$
$173$	−3014.11	−1.32462	−0.662309	+	0.749231i	$0.730423\pi$
$174$	1120.24	0.488074
$175$	−3242.55	−1.40065
$176$	1744.57	0.747169
$177$	6250.69	2.65441
$178$	−266.701	−0.112304
$179$	3477.58	1.45211	0.726053	−	0.687639i	$-0.241353\pi$
$179$	3477.58	1.45211	0.726053	+	0.687639i	$0.241353\pi$
$180$	947.513	0.392352
$181$	7.73635	0.00317701	0.00158850	−	0.999999i	$-0.499494\pi$
$181$	7.73635	0.00317701	0.00158850	+	0.999999i	$0.499494\pi$
$182$	907.943	0.369787
$183$	2238.84	0.904372
$184$	−1169.03	−0.468381
$185$	−630.668	−0.250636
$186$	2265.54	0.893105
$187$	−467.567	−0.182844
$188$	3417.60	1.32582
$189$	−9732.63	−3.74574
$190$	−117.623	−0.0449121
$191$	−1301.75	−0.493147	−0.246574	−	0.969124i	$-0.579305\pi$
$191$	−1301.75	−0.493147	−0.246574	+	0.969124i	$0.579305\pi$
$192$	2156.84	0.810710
$193$	−760.523	−0.283646	−0.141823	−	0.989892i	$-0.545296\pi$
$193$	−760.523	−0.283646	−0.141823	+	0.989892i	$0.545296\pi$
$194$	−173.306	−0.0641375
$195$	−730.560	−0.268290
$196$	−2706.80	−0.986444
$197$	−200.537	−0.0725263	−0.0362632	−	0.999342i	$-0.511545\pi$
$197$	−200.537	−0.0725263	−0.0362632	+	0.999342i	$0.511545\pi$
$198$	−2264.70	−0.812855
$199$	−1878.65	−0.669217	−0.334608	−	0.942357i	$-0.608604\pi$
$199$	−1878.65	−0.669217	−0.334608	+	0.942357i	$0.608604\pi$
$200$	−1656.95	−0.585822
$201$	−9233.90	−3.24034
$202$	1550.30	0.539995
$203$	3473.38	1.20090
$204$	−832.391	−0.285682
$205$	678.534	0.231175
$206$	28.8391	0.00975397
$207$	−5530.25	−1.85690
$208$	−1690.75	−0.563616
$209$	−2478.26	−0.820216
$210$	−472.022	−0.155108
$211$	−907.815	−0.296192	−0.148096	−	0.988973i	$-0.547315\pi$
$211$	−907.815	−0.296192	−0.148096	+	0.988973i	$0.547315\pi$
$212$	−101.720	−0.0329536
$213$	−8502.77	−2.73521
$214$	−1151.09	−0.367697
$215$	0	0
$216$	−4973.41	−1.56666
$217$	7024.49	2.19748
$218$	1067.00	0.331498
$219$	3307.91	1.02068
$220$	565.121	0.173184
$221$	453.142	0.137926
$222$	2683.55	0.811298
$223$	−2724.67	−0.818193	−0.409097	−	0.912491i	$-0.634156\pi$
$223$	−2724.67	−0.818193	−0.409097	+	0.912491i	$0.634156\pi$
$224$	−4034.70	−1.20348
$225$	−7838.44	−2.32250
$226$	−909.317	−0.267641
$227$	2297.13	0.671655	0.335828	−	0.941923i	$-0.390984\pi$
$227$	2297.13	0.671655	0.335828	+	0.941923i	$0.390984\pi$
$228$	−4411.96	−1.28153
$229$	3034.87	0.875763	0.437882	−	0.899033i	$-0.355729\pi$
$229$	3034.87	0.875763	0.437882	+	0.899033i	$0.355729\pi$
$230$	−156.548	−0.0448802
$231$	−9945.28	−2.83269
$232$	1774.91	0.502278
$233$	4994.74	1.40436	0.702180	−	0.711999i	$-0.252210\pi$
$233$	4994.74	1.40436	0.702180	+	0.711999i	$0.252210\pi$
$234$	2194.83	0.613166
$235$	967.237	0.268492
$236$	4686.02	1.29252
$237$	3797.84	1.04091
$238$	292.780	0.0797399
$239$	−2939.14	−0.795470	−0.397735	−	0.917500i	$-0.630204\pi$
$239$	−2939.14	−0.795470	−0.397735	+	0.917500i	$0.630204\pi$
$240$	878.987	0.236410
$241$	−4981.47	−1.33147	−0.665736	−	0.746187i	$-0.731882\pi$
$241$	−4981.47	−1.33147	−0.665736	+	0.746187i	$0.731882\pi$
$242$	−149.083	−0.0396010
$243$	−6745.54	−1.78077
$244$	1678.42	0.440368
$245$	−766.068	−0.199765
$246$	−2887.23	−0.748304
$247$	2401.81	0.618719
$248$	3589.54	0.919097
$249$	−5174.17	−1.31687
$250$	−451.365	−0.114187
$251$	5028.93	1.26463	0.632317	−	0.774709i	$-0.282104\pi$
$251$	5028.93	1.26463	0.632317	+	0.774709i	$0.282104\pi$
$252$	−12500.8	−3.12490
$253$	−3298.38	−0.819634
$254$	626.583	0.154785
$255$	−235.580	−0.0578533
$256$	530.733	0.129573
$257$	−846.260	−0.205402	−0.102701	−	0.994712i	$-0.532748\pi$
$257$	−846.260	−0.205402	−0.102701	+	0.994712i	$0.532748\pi$
$258$	0	0
$259$	8320.56	1.99620
$260$	−547.687	−0.130639
$261$	8396.44	1.99129
$262$	915.449	0.215865
$263$	−7725.74	−1.81137	−0.905684	−	0.423954i	$-0.860642\pi$
$263$	−7725.74	−1.81137	−0.905684	+	0.423954i	$0.860642\pi$
$264$	−5082.07	−1.18477
$265$	−28.7884	−0.00667343
$266$	1551.83	0.357703
$267$	−2831.23	−0.648944
$268$	−6922.48	−1.57783
$269$	−1281.93	−0.290559	−0.145279	−	0.989391i	$-0.546408\pi$
$269$	−1281.93	−0.290559	−0.145279	+	0.989391i	$0.546408\pi$
$270$	−666.001	−0.150117
$271$	1156.22	0.259171	0.129585	−	0.991568i	$-0.458635\pi$
$271$	1156.22	0.259171	0.129585	+	0.991568i	$0.458635\pi$
$272$	−545.207	−0.121537
$273$	9638.46	2.13680
$274$	−571.648	−0.126038
$275$	−4675.05	−1.02515
$276$	−5871.98	−1.28062
$277$	2086.27	0.452533	0.226266	−	0.974065i	$-0.427348\pi$
$277$	2086.27	0.452533	0.226266	+	0.974065i	$0.427348\pi$
$278$	394.771	0.0851684
$279$	16980.8	3.64377
$280$	−747.875	−0.159622
$281$	1154.08	0.245006	0.122503	−	0.992468i	$-0.460908\pi$
$281$	1154.08	0.245006	0.122503	+	0.992468i	$0.460908\pi$
$282$	−4115.68	−0.869097
$283$	5288.51	1.11085	0.555423	−	0.831568i	$-0.312556\pi$
$283$	5288.51	1.11085	0.555423	+	0.831568i	$0.312556\pi$
$284$	−6374.37	−1.33186
$285$	−1248.66	−0.259523
$286$	1309.05	0.270650
$287$	−8952.07	−1.84120
$288$	−9753.36	−1.99556
$289$	−4766.88	−0.970258
$290$	237.683	0.0481283
$291$	−1839.77	−0.370616
$292$	2479.88	0.497000
$293$	−3079.73	−0.614060	−0.307030	−	0.951700i	$-0.599335\pi$
$293$	−3079.73	−0.614060	−0.307030	+	0.951700i	$0.599335\pi$
$294$	3259.69	0.646630
$295$	1326.22	0.261747
$296$	4251.84	0.834909
$297$	−14032.3	−2.74154
$298$	−1852.86	−0.360179
$299$	3196.63	0.618280
$300$	−8322.80	−1.60172
$301$	0	0
$302$	−922.328	−0.175742
$303$	16457.6	3.12034
$304$	−2889.78	−0.545199
$305$	475.019	0.0891787
$306$	707.757	0.132222
$307$	−639.484	−0.118884	−0.0594419	−	0.998232i	$-0.518932\pi$
$307$	−639.484	−0.118884	−0.0594419	+	0.998232i	$0.518932\pi$
$308$	−7455.79	−1.37933
$309$	306.148	0.0563630
$310$	480.684	0.0880678
$311$	7068.39	1.28878	0.644391	−	0.764696i	$-0.277111\pi$
$311$	7068.39	1.28878	0.644391	+	0.764696i	$0.277111\pi$
$312$	4925.29	0.893717
$313$	5479.55	0.989528	0.494764	−	0.869027i	$-0.335254\pi$
$313$	5479.55	0.989528	0.494764	+	0.869027i	$0.335254\pi$
$314$	−165.692	−0.0297787
$315$	−3537.92	−0.632823
$316$	2847.17	0.506853
$317$	6514.68	1.15426	0.577131	−	0.816652i	$-0.304172\pi$
$317$	6514.68	1.15426	0.577131	+	0.816652i	$0.304172\pi$
$318$	122.498	0.0216016
$319$	5007.85	0.878953
$320$	457.620	0.0799429
$321$	−12219.7	−2.12472
$322$	2065.37	0.357450
$323$	774.500	0.133419
$324$	−12400.2	−2.12623
$325$	4530.82	0.773307
$326$	−751.363	−0.127651
$327$	11327.0	1.91555
$328$	−4574.54	−0.770081
$329$	−12761.0	−2.13841
$330$	−680.553	−0.113525
$331$	−3774.59	−0.626799	−0.313399	−	0.949621i	$-0.601468\pi$
$331$	−3774.59	−0.626799	−0.313399	+	0.949621i	$0.601468\pi$
$332$	−3878.98	−0.641225
$333$	20113.9	3.31001
$334$	288.808	0.0473140
$335$	−1959.17	−0.319526
$336$	−11596.7	−1.88289
$337$	11690.7	1.88972	0.944859	−	0.327478i	$-0.106199\pi$
$337$	11690.7	1.88972	0.944859	+	0.327478i	$0.106199\pi$
$338$	714.806	0.115031
$339$	−9653.05	−1.54655
$340$	−176.610	−0.0281706
$341$	10127.8	1.60836
$342$	3751.36	0.593129
$343$	904.985	0.142462
$344$	0	0
$345$	−1661.87	−0.259339
$346$	2721.18	0.422807
$347$	−11140.5	−1.72349	−0.861746	−	0.507340i	$-0.830629\pi$
$347$	−11140.5	−1.72349	−0.861746	+	0.507340i	$0.830629\pi$
$348$	8915.29	1.37330
$349$	1226.84	0.188170	0.0940852	−	0.995564i	$-0.470007\pi$
$349$	1226.84	0.188170	0.0940852	+	0.995564i	$0.470007\pi$
$350$	2927.41	0.447076
$351$	13599.4	2.06804
$352$	−5817.15	−0.880839
$353$	−1582.18	−0.238558	−0.119279	−	0.992861i	$-0.538058\pi$
$353$	−1582.18	−0.238558	−0.119279	+	0.992861i	$0.538058\pi$
$354$	−5643.19	−0.847267
$355$	−1804.05	−0.269715
$356$	−2122.52	−0.315992
$357$	3108.07	0.460774
$358$	−3139.60	−0.463500
$359$	−613.266	−0.0901586	−0.0450793	−	0.998983i	$-0.514354\pi$
$359$	−613.266	−0.0901586	−0.0450793	+	0.998983i	$0.514354\pi$
$360$	−1807.89	−0.264678
$361$	−2753.88	−0.401499
$362$	−6.98446	−0.00101407
$363$	−1582.63	−0.228833
$364$	7225.77	1.04048
$365$	701.846	0.100647
$366$	−2021.25	−0.288668
$367$	2202.45	0.313261	0.156631	−	0.987657i	$-0.449937\pi$
$367$	2202.45	0.313261	0.156631	+	0.987657i	$0.449937\pi$
$368$	−3846.08	−0.544812
$369$	−21640.5	−3.05300
$370$	569.374	0.0800009
$371$	379.813	0.0531507
$372$	18030.1	2.51295
$373$	6268.78	0.870201	0.435101	−	0.900382i	$-0.356713\pi$
$373$	6268.78	0.870201	0.435101	+	0.900382i	$0.356713\pi$
$374$	422.124	0.0583624
$375$	−4791.56	−0.659827
$376$	−6520.92	−0.894390
$377$	−4853.36	−0.663026
$378$	8786.72	1.19561
$379$	3247.19	0.440097	0.220049	−	0.975489i	$-0.429378\pi$
$379$	3247.19	0.440097	0.220049	+	0.975489i	$0.429378\pi$
$380$	−936.093	−0.126370
$381$	6651.62	0.894417
$382$	1175.23	0.157409
$383$	−8938.80	−1.19256	−0.596281	−	0.802776i	$-0.703356\pi$
$383$	−8938.80	−1.19256	−0.596281	+	0.802776i	$0.703356\pi$
$384$	−13478.1	−1.79115
$385$	−2110.11	−0.279327
$386$	686.609	0.0905375
$387$	0	0
$388$	−1379.24	−0.180465
$389$	−4626.84	−0.603060	−0.301530	−	0.953457i	$-0.597497\pi$
$389$	−4626.84	−0.603060	−0.301530	+	0.953457i	$0.597497\pi$
$390$	659.558	0.0856359
$391$	1030.80	0.133324
$392$	5164.68	0.665448
$393$	9718.15	1.24737
$394$	181.047	0.0231498
$395$	805.794	0.102643
$396$	−18023.4	−2.28714
$397$	10840.9	1.37051	0.685253	−	0.728305i	$-0.259692\pi$
$397$	10840.9	1.37051	0.685253	+	0.728305i	$0.259692\pi$
$398$	1696.07	0.213609
$399$	16473.8	2.06698
$400$	−5451.34	−0.681418
$401$	−12391.8	−1.54319	−0.771594	−	0.636116i	$-0.780540\pi$
$401$	−12391.8	−1.54319	−0.771594	+	0.636116i	$0.780540\pi$
$402$	8336.47	1.03429
$403$	−9815.33	−1.21324
$404$	12337.9	1.51939
$405$	−3509.45	−0.430583
$406$	−3135.81	−0.383319
$407$	11996.4	1.46103
$408$	1588.23	0.192719
$409$	7910.71	0.956380	0.478190	−	0.878256i	$-0.341293\pi$
$409$	7910.71	0.956380	0.478190	+	0.878256i	$0.341293\pi$
$410$	−612.588	−0.0737891
$411$	−6068.45	−0.728308
$412$	229.513	0.0274449
$413$	−17497.2	−2.08469
$414$	4992.77	0.592708
$415$	−1097.81	−0.129854
$416$	5637.69	0.664448
$417$	4190.78	0.492142
$418$	2237.41	0.261806
$419$	−12901.8	−1.50428	−0.752142	−	0.659001i	$-0.770979\pi$
$419$	−12901.8	−1.50428	−0.752142	+	0.659001i	$0.770979\pi$
$420$	−3756.54	−0.436430
$421$	−16075.1	−1.86093	−0.930464	−	0.366382i	$-0.880596\pi$
$421$	−16075.1	−1.86093	−0.930464	+	0.366382i	$0.880596\pi$
$422$	819.586	0.0945422
$423$	−30848.1	−3.54582
$424$	194.086	0.0222303
$425$	1461.03	0.166754
$426$	7676.40	0.873058
$427$	−6267.05	−0.710266
$428$	−9160.86	−1.03460
$429$	13896.6	1.56394
$430$	0	0
$431$	−9277.47	−1.03684	−0.518422	−	0.855125i	$-0.673480\pi$
$431$	−9277.47	−1.03684	−0.518422	+	0.855125i	$0.673480\pi$
$432$	−16362.4	−1.82231
$433$	−182.499	−0.0202548	−0.0101274	−	0.999949i	$-0.503224\pi$
$433$	−182.499	−0.0202548	−0.0101274	+	0.999949i	$0.503224\pi$
$434$	−6341.79	−0.701418
$435$	2523.17	0.278107
$436$	8491.65	0.932744
$437$	5463.60	0.598076
$438$	−2986.42	−0.325792
$439$	−5806.09	−0.631229	−0.315614	−	0.948888i	$-0.602211\pi$
$439$	−5806.09	−0.631229	−0.315614	+	0.948888i	$0.602211\pi$
$440$	−1078.27	−0.116829
$441$	24432.2	2.63818
$442$	−409.102	−0.0440249
$443$	13300.1	1.42643	0.713214	−	0.700946i	$-0.247239\pi$
$443$	13300.1	1.42643	0.713214	+	0.700946i	$0.247239\pi$
$444$	21356.8	2.28277
$445$	−600.706	−0.0639914
$446$	2459.86	0.261161
$447$	−19669.5	−2.08128
$448$	−6037.49	−0.636707
$449$	7189.18	0.755631	0.377815	−	0.925881i	$-0.376675\pi$
$449$	7189.18	0.755631	0.377815	+	0.925881i	$0.376675\pi$
$450$	7076.63	0.741323
$451$	−12906.9	−1.34759
$452$	−7236.71	−0.753066
$453$	−9791.17	−1.01552
$454$	−2073.87	−0.214387
$455$	2045.01	0.210707
$456$	8418.19	0.864513
$457$	−4199.15	−0.429820	−0.214910	−	0.976634i	$-0.568946\pi$
$457$	−4199.15	−0.429820	−0.214910	+	0.976634i	$0.568946\pi$
$458$	−2739.91	−0.279537
$459$	4385.34	0.445948
$460$	−1245.87	−0.126280
$461$	−1185.60	−0.119781	−0.0598905	−	0.998205i	$-0.519075\pi$
$461$	−1185.60	−0.119781	−0.0598905	+	0.998205i	$0.519075\pi$
$462$	8978.71	0.904172
$463$	−2522.44	−0.253192	−0.126596	−	0.991954i	$-0.540405\pi$
$463$	−2522.44	−0.253192	−0.126596	+	0.991954i	$0.540405\pi$
$464$	5839.41	0.584241
$465$	5102.80	0.508897
$466$	−4509.30	−0.448261
$467$	9629.56	0.954182	0.477091	−	0.878854i	$-0.341691\pi$
$467$	9629.56	0.954182	0.477091	+	0.878854i	$0.341691\pi$
$468$	17467.4	1.72528
$469$	25847.9	2.54487
$470$	−873.232	−0.0857004
$471$	−1758.94	−0.172075
$472$	−8941.11	−0.871924
$473$	0	0
$474$	−3428.73	−0.332251
$475$	7743.97	0.748037
$476$	2330.06	0.224366
$477$	918.149	0.0881324
$478$	2653.49	0.253908
$479$	12358.5	1.17886	0.589431	−	0.807819i	$-0.299352\pi$
$479$	12358.5	1.17886	0.589431	+	0.807819i	$0.299352\pi$
$480$	−2930.93	−0.278704
$481$	−11626.3	−1.10211
$482$	4497.33	0.424995
$483$	21925.4	2.06551
$484$	−1186.47	−0.111426
$485$	−390.348	−0.0365459
$486$	6089.95	0.568407
$487$	2723.31	0.253398	0.126699	−	0.991941i	$-0.459562\pi$
$487$	2723.31	0.253398	0.126699	+	0.991941i	$0.459562\pi$
$488$	−3202.49	−0.297069
$489$	−7976.25	−0.737625
$490$	691.615	0.0637632
$491$	−4199.46	−0.385986	−0.192993	−	0.981200i	$-0.561819\pi$
$491$	−4199.46	−0.385986	−0.192993	+	0.981200i	$0.561819\pi$
$492$	−22977.7	−2.10552
$493$	−1565.04	−0.142973
$494$	−2168.38	−0.197490
$495$	−5100.91	−0.463169
$496$	11809.5	1.06908
$497$	23801.3	2.14815
$498$	4671.30	0.420333
$499$	19985.3	1.79292	0.896458	−	0.443129i	$-0.146132\pi$
$499$	19985.3	1.79292	0.896458	+	0.443129i	$0.146132\pi$
$500$	−3592.14	−0.321291
$501$	3065.90	0.273402
$502$	−4540.18	−0.403661
$503$	520.717	0.0461583	0.0230792	−	0.999734i	$-0.492653\pi$
$503$	520.717	0.0461583	0.0230792	+	0.999734i	$0.492653\pi$
$504$	23852.0	2.10804
$505$	3491.83	0.307692
$506$	2977.82	0.261621
$507$	7588.18	0.664700
$508$	4986.60	0.435520
$509$	1746.69	0.152103	0.0760517	−	0.997104i	$-0.475769\pi$
$509$	1746.69	0.152103	0.0760517	+	0.997104i	$0.475769\pi$
$510$	212.684	0.0184663
$511$	−9259.63	−0.801609
$512$	−11729.6	−1.01246
$513$	23243.8	2.00046
$514$	764.013	0.0655626
$515$	64.9560	0.00555787
$516$	0	0
$517$	−18398.6	−1.56512
$518$	−7511.90	−0.637169
$519$	28887.2	2.44318
$520$	1045.01	0.0881281
$521$	−12223.8	−1.02790	−0.513948	−	0.857821i	$-0.671818\pi$
$521$	−12223.8	−1.02790	−0.513948	+	0.857821i	$0.671818\pi$
$522$	−7580.40	−0.635604
$523$	−12923.8	−1.08053	−0.540265	−	0.841495i	$-0.681676\pi$
$523$	−12923.8	−1.08053	−0.540265	+	0.841495i	$0.681676\pi$
$524$	7285.51	0.607383
$525$	31076.5	2.58341
$526$	6974.88	0.578174
$527$	−3165.10	−0.261620
$528$	−16719.9	−1.37811
$529$	−4895.37	−0.402348
$530$	25.9905	0.00213011
$531$	−42297.1	−3.45676
$532$	12350.1	1.00648
$533$	12508.7	1.01654
$534$	2556.06	0.207138
$535$	−2592.67	−0.209516
$536$	13208.4	1.06439
$537$	−33329.1	−2.67832
$538$	1157.34	0.0927441
$539$	14572.0	1.16449
$540$	−5300.30	−0.422387
$541$	−10061.3	−0.799574	−0.399787	−	0.916608i	$-0.630916\pi$
$541$	−10061.3	−0.799574	−0.399787	+	0.916608i	$0.630916\pi$
$542$	−1043.85	−0.0827252
$543$	−74.1451	−0.00585980
$544$	1817.96	0.143280
$545$	2403.27	0.188890
$546$	−8701.71	−0.682049
$547$	−23951.3	−1.87218	−0.936090	−	0.351761i	$-0.885583\pi$
$547$	−23951.3	−1.87218	−0.936090	+	0.351761i	$0.885583\pi$
$548$	−4549.40	−0.354636
$549$	−15149.8	−1.17774
$550$	4220.68	0.327219
$551$	−8295.25	−0.641360
$552$	11204.0	0.863900
$553$	−10631.0	−0.817501
$554$	−1883.50	−0.144445
$555$	6044.31	0.462282
$556$	3141.75	0.239640
$557$	8803.19	0.669665	0.334832	−	0.942278i	$-0.391320\pi$
$557$	8803.19	0.669665	0.334832	+	0.942278i	$0.391320\pi$
$558$	−15330.4	−1.16306
$559$	0	0
$560$	−2460.49	−0.185669
$561$	4481.15	0.337245
$562$	−1041.92	−0.0782038
$563$	−21876.5	−1.63762	−0.818812	−	0.574061i	$-0.805367\pi$
$563$	−21876.5	−1.63762	−0.818812	+	0.574061i	$0.805367\pi$
$564$	−32754.3	−2.44540
$565$	−2048.10	−0.152503
$566$	−4774.53	−0.354573
$567$	46301.1	3.42939
$568$	12162.5	0.898466
$569$	−22198.4	−1.63551	−0.817756	−	0.575565i	$-0.804782\pi$
$569$	−22198.4	−1.63551	−0.817756	+	0.575565i	$0.804782\pi$
$570$	1127.30	0.0828376
$571$	−13270.8	−0.972619	−0.486310	−	0.873787i	$-0.661657\pi$
$571$	−13270.8	−0.972619	−0.486310	+	0.873787i	$0.661657\pi$
$572$	10418.0	0.761534
$573$	12475.9	0.909580
$574$	8082.03	0.587696
$575$	10306.6	0.747507
$576$	−14594.9	−1.05576
$577$	−15373.6	−1.10921	−0.554603	−	0.832115i	$-0.687130\pi$
$577$	−15373.6	−1.10921	−0.554603	+	0.832115i	$0.687130\pi$
$578$	4303.59	0.309699
$579$	7288.85	0.523168
$580$	1891.57	0.135419
$581$	14483.7	1.03423
$582$	1660.97	0.118298
$583$	547.607	0.0389015
$584$	−4731.71	−0.335273
$585$	4943.54	0.349385
$586$	2780.41	0.196003
$587$	863.918	0.0607457	0.0303728	−	0.999539i	$-0.490331\pi$
$587$	863.918	0.0607457	0.0303728	+	0.999539i	$0.490331\pi$
$588$	25942.0	1.81944
$589$	−16776.1	−1.17360
$590$	−1197.33	−0.0835477
$591$	1921.95	0.133770
$592$	13988.4	0.971151
$593$	−1387.83	−0.0961071	−0.0480536	−	0.998845i	$-0.515302\pi$
$593$	−1387.83	−0.0961071	−0.0480536	+	0.998845i	$0.515302\pi$
$594$	12668.5	0.875078
$595$	659.444	0.0454363
$596$	−14745.8	−1.01344
$597$	18005.0	1.23433
$598$	−2885.95	−0.197350
$599$	−23119.8	−1.57704	−0.788521	−	0.615008i	$-0.789153\pi$
$599$	−23119.8	−1.57704	−0.788521	+	0.615008i	$0.789153\pi$
$600$	15880.2	1.08051
$601$	29375.9	1.99379	0.996897	−	0.0787233i	$-0.0250844\pi$
$601$	29375.9	1.99379	0.996897	+	0.0787233i	$0.0250844\pi$
$602$	0	0
$603$	62483.9	4.21980
$604$	−7340.26	−0.494488
$605$	−335.789	−0.0225649
$606$	−14858.1	−0.995987
$607$	−10354.5	−0.692383	−0.346191	−	0.938164i	$-0.612525\pi$
$607$	−10354.5	−0.692383	−0.346191	+	0.938164i	$0.612525\pi$
$608$	9635.81	0.642736
$609$	−33288.8	−2.21499
$610$	−428.853	−0.0284651
$611$	17831.0	1.18063
$612$	5632.61	0.372034
$613$	−9960.96	−0.656312	−0.328156	−	0.944624i	$-0.606427\pi$
$613$	−9960.96	−0.656312	−0.328156	+	0.944624i	$0.606427\pi$
$614$	577.334	0.0379467
$615$	−6503.06	−0.426388
$616$	14225.9	0.930485
$617$	−13796.7	−0.900218	−0.450109	−	0.892974i	$-0.648615\pi$
$617$	−13796.7	−0.900218	−0.450109	+	0.892974i	$0.648615\pi$
$618$	−276.394	−0.0179906
$619$	25631.5	1.66432	0.832162	−	0.554533i	$-0.187103\pi$
$619$	25631.5	1.66432	0.832162	+	0.554533i	$0.187103\pi$
$620$	3825.47	0.247798
$621$	30935.7	1.99905
$622$	−6381.42	−0.411369
$623$	7925.27	0.509662
$624$	16204.1	1.03956
$625$	14091.5	0.901856
$626$	−4947.00	−0.315849
$627$	23751.7	1.51284
$628$	−1318.64	−0.0837890
$629$	−3749.09	−0.237656
$630$	3194.08	0.201992
$631$	−17655.1	−1.11385	−0.556925	−	0.830563i	$-0.688019\pi$
$631$	−17655.1	−1.11385	−0.556925	+	0.830563i	$0.688019\pi$
$632$	−5432.50	−0.341920
$633$	8700.49	0.546309
$634$	−5881.53	−0.368431
$635$	1411.29	0.0881972
$636$	974.884	0.0607809
$637$	−14122.4	−0.878417
$638$	−4521.15	−0.280555
$639$	57536.5	3.56198
$640$	−2859.67	−0.176622
$641$	374.249	0.0230608	0.0115304	−	0.999934i	$-0.496330\pi$
$641$	374.249	0.0230608	0.0115304	+	0.999934i	$0.496330\pi$
$642$	11032.1	0.678194
$643$	−18264.0	−1.12016	−0.560080	−	0.828438i	$-0.689230\pi$
$643$	−18264.0	−1.12016	−0.560080	+	0.828438i	$0.689230\pi$
$644$	16437.1	1.00576
$645$	0	0
$646$	−699.227	−0.0425863
$647$	−1121.95	−0.0681735	−0.0340868	−	0.999419i	$-0.510852\pi$
$647$	−1121.95	−0.0681735	−0.0340868	+	0.999419i	$0.510852\pi$
$648$	23660.0	1.43434
$649$	−25227.1	−1.52581
$650$	−4090.47	−0.246833
$651$	−67322.6	−4.05312
$652$	−5979.65	−0.359173
$653$	−14019.2	−0.840142	−0.420071	−	0.907491i	$-0.637995\pi$
$653$	−14019.2	−0.840142	−0.420071	+	0.907491i	$0.637995\pi$
$654$	−10226.2	−0.611429
$655$	2061.92	0.123001
$656$	−15050.1	−0.895745
$657$	−22384.0	−1.32920
$658$	11520.8	0.682563
$659$	−26897.3	−1.58994	−0.794969	−	0.606650i	$-0.792513\pi$
$659$	−26897.3	−1.58994	−0.794969	+	0.606650i	$0.792513\pi$
$660$	−5416.11	−0.319427
$661$	4456.60	0.262241	0.131121	−	0.991366i	$-0.458142\pi$
$661$	4456.60	0.262241	0.131121	+	0.991366i	$0.458142\pi$
$662$	3407.74	0.200069
$663$	−4342.91	−0.254396
$664$	7401.24	0.432566
$665$	3495.28	0.203821
$666$	−18159.0	−1.05653
$667$	−11040.3	−0.640905
$668$	2298.45	0.133128
$669$	26113.2	1.50911
$670$	1768.76	0.101990
$671$	−9035.71	−0.519851
$672$	38668.5	2.21975
$673$	12270.1	0.702787	0.351394	−	0.936228i	$-0.385708\pi$
$673$	12270.1	0.702787	0.351394	+	0.936228i	$0.385708\pi$
$674$	−10554.5	−0.603183
$675$	43847.5	2.50028
$676$	5688.71	0.323664
$677$	5587.24	0.317186	0.158593	−	0.987344i	$-0.449304\pi$
$677$	5587.24	0.317186	0.158593	+	0.987344i	$0.449304\pi$
$678$	8714.88	0.493647
$679$	5149.96	0.291071
$680$	336.978	0.0190037
$681$	−22015.7	−1.23883
$682$	−9143.47	−0.513375
$683$	−16419.2	−0.919860	−0.459930	−	0.887955i	$-0.652126\pi$
$683$	−16419.2	−0.919860	−0.459930	+	0.887955i	$0.652126\pi$
$684$	29854.8	1.66890
$685$	−1287.55	−0.0718173
$686$	−817.031	−0.0454729
$687$	−29086.1	−1.61529
$688$	0	0
$689$	−530.714	−0.0293448
$690$	1500.35	0.0827788
$691$	−21390.2	−1.17760	−0.588801	−	0.808278i	$-0.700400\pi$
$691$	−21390.2	−1.17760	−0.588801	+	0.808278i	$0.700400\pi$
$692$	21656.2	1.18966
$693$	67297.6	3.68892
$694$	10057.7	0.550125
$695$	889.165	0.0485294
$696$	−17010.7	−0.926422
$697$	4033.63	0.219203
$698$	−1107.61	−0.0600625
$699$	−47869.5	−2.59026
$700$	23297.5	1.25795
$701$	2029.82	0.109365	0.0546827	−	0.998504i	$-0.482585\pi$
$701$	2029.82	0.109365	0.0546827	+	0.998504i	$0.482585\pi$
$702$	−12277.7	−0.660103
$703$	−19871.5	−1.06610
$704$	−8704.74	−0.466012
$705$	−9269.98	−0.495217
$706$	1428.41	0.0761459
$707$	−46068.6	−2.45062
$708$	−44910.8	−2.38397
$709$	17269.2	0.914751	0.457376	−	0.889274i	$-0.348789\pi$
$709$	17269.2	0.914751	0.457376	+	0.889274i	$0.348789\pi$
$710$	1628.71	0.0860909
$711$	−25699.2	−1.35555
$712$	4049.84	0.213166
$713$	−22327.7	−1.17276
$714$	−2806.00	−0.147075
$715$	2948.46	0.154218
$716$	−24986.2	−1.30416
$717$	28168.7	1.46720
$718$	553.663	0.0287779
$719$	13317.0	0.690739	0.345369	−	0.938467i	$-0.387754\pi$
$719$	13317.0	0.690739	0.345369	+	0.938467i	$0.387754\pi$
$720$	−5947.92	−0.307869
$721$	−856.981	−0.0442658
$722$	2486.24	0.128155
$723$	47742.4	2.45582
$724$	−55.5851	−0.00285332
$725$	−15648.3	−0.801605
$726$	1428.81	0.0730416
$727$	−22136.1	−1.12927	−0.564636	−	0.825340i	$-0.690983\pi$
$727$	−22136.1	−1.12927	−0.564636	+	0.825340i	$0.690983\pi$
$728$	−13787.1	−0.701899
$729$	18051.0	0.917084
$730$	−633.634	−0.0321258
$731$	0	0
$732$	−16085.9	−0.812231
$733$	−17545.9	−0.884139	−0.442069	−	0.896981i	$-0.645756\pi$
$733$	−17545.9	−0.884139	−0.442069	+	0.896981i	$0.645756\pi$
$734$	−1988.39	−0.0999904
$735$	7341.99	0.368453
$736$	12824.5	0.642280
$737$	37267.0	1.86261
$738$	19537.2	0.974493
$739$	−13702.8	−0.682094	−0.341047	−	0.940046i	$-0.610781\pi$
$739$	−13702.8	−0.682094	−0.341047	+	0.940046i	$0.610781\pi$
$740$	4531.31	0.225100
$741$	−23018.9	−1.14119
$742$	−342.900	−0.0169653
$743$	−12371.8	−0.610871	−0.305435	−	0.952213i	$-0.598802\pi$
$743$	−12371.8	−0.610871	−0.305435	+	0.952213i	$0.598802\pi$
$744$	−34402.1	−1.69522
$745$	−4173.30	−0.205232
$746$	−5659.52	−0.277761
$747$	35012.5	1.71492
$748$	3359.44	0.164215
$749$	34205.7	1.66869
$750$	4325.87	0.210611
$751$	9938.30	0.482895	0.241447	−	0.970414i	$-0.422378\pi$
$751$	9938.30	0.482895	0.241447	+	0.970414i	$0.422378\pi$
$752$	−21453.7	−1.04034
$753$	−48197.2	−2.33254
$754$	4381.67	0.211633
$755$	−2077.41	−0.100139
$756$	69928.3	3.36411
$757$	−10311.6	−0.495088	−0.247544	−	0.968877i	$-0.579623\pi$
$757$	−10311.6	−0.495088	−0.247544	+	0.968877i	$0.579623\pi$
$758$	−2931.60	−0.140475
$759$	31611.7	1.51177
$760$	1786.10	0.0852483
$761$	19234.7	0.916238	0.458119	−	0.888891i	$-0.348523\pi$
$761$	19234.7	0.916238	0.458119	+	0.888891i	$0.348523\pi$
$762$	−6005.16	−0.285491
$763$	−31707.0	−1.50442
$764$	9352.97	0.442904
$765$	1594.12	0.0753406
$766$	8070.05	0.380656
$767$	24448.8	1.15097
$768$	−5086.54	−0.238990
$769$	13634.6	0.639370	0.319685	−	0.947524i	$-0.396423\pi$
$769$	13634.6	0.639370	0.319685	+	0.947524i	$0.396423\pi$
$770$	1905.03	0.0891590
$771$	8110.55	0.378851
$772$	5464.31	0.254747
$773$	11204.3	0.521335	0.260667	−	0.965429i	$-0.416057\pi$
$773$	11204.3	0.521335	0.260667	+	0.965429i	$0.416057\pi$
$774$	0	0
$775$	−31646.8	−1.46682
$776$	2631.65	0.121740
$777$	−79744.2	−3.68186
$778$	4177.17	0.192492
$779$	21379.6	0.983318
$780$	5249.02	0.240955
$781$	34316.2	1.57225
$782$	−930.618	−0.0425561
$783$	−46969.0	−2.14372
$784$	16991.7	0.774038
$785$	−373.197	−0.0169681
$786$	−8773.65	−0.398150
$787$	−40995.2	−1.85683	−0.928413	−	0.371551i	$-0.878826\pi$
$787$	−40995.2	−1.85683	−0.928413	+	0.371551i	$0.878826\pi$
$788$	1440.85	0.0651371
$789$	74043.4	3.34096
$790$	−727.480	−0.0327627
$791$	27021.2	1.21462
$792$	34389.3	1.54289
$793$	8756.96	0.392142
$794$	−9787.31	−0.437454
$795$	275.908	0.0123087
$796$	13498.0	0.601034
$797$	−1380.09	−0.0613368	−0.0306684	−	0.999530i	$-0.509764\pi$
$797$	−1380.09	−0.0613368	−0.0306684	+	0.999530i	$0.509764\pi$
$798$	−14872.8	−0.659762
$799$	5749.86	0.254588
$800$	18177.2	0.803325
$801$	19158.3	0.845100
$802$	11187.5	0.492573
$803$	−13350.4	−0.586705
$804$	66345.0	2.91021
$805$	4651.95	0.203677
$806$	8861.39	0.387257
$807$	12286.0	0.535918
$808$	−23541.2	−1.02497
$809$	−18158.2	−0.789131	−0.394566	−	0.918868i	$-0.629105\pi$
$809$	−18158.2	−0.789131	−0.394566	+	0.918868i	$0.629105\pi$
$810$	3168.37	0.137439
$811$	−478.154	−0.0207031	−0.0103516	−	0.999946i	$-0.503295\pi$
$811$	−478.154	−0.0207031	−0.0103516	+	0.999946i	$0.503295\pi$
$812$	−24956.0	−1.07855
$813$	−11081.2	−0.478024
$814$	−10830.5	−0.466350
$815$	−1692.34	−0.0727361
$816$	5225.25	0.224167
$817$	0	0
$818$	−7141.88	−0.305269
$819$	−65221.4	−2.78269
$820$	−4875.22	−0.207622
$821$	7722.38	0.328274	0.164137	−	0.986438i	$-0.447516\pi$
$821$	7722.38	0.328274	0.164137	+	0.986438i	$0.447516\pi$
$822$	5478.66	0.232470
$823$	−1765.67	−0.0747842	−0.0373921	−	0.999301i	$-0.511905\pi$
$823$	−1765.67	−0.0747842	−0.0373921	+	0.999301i	$0.511905\pi$
$824$	−437.920	−0.0185142
$825$	44805.6	1.89082
$826$	15796.6	0.665418
$827$	−19472.6	−0.818776	−0.409388	−	0.912360i	$-0.634258\pi$
$827$	−19472.6	−0.818776	−0.409388	+	0.912360i	$0.634258\pi$
$828$	39734.5	1.66772
$829$	26318.0	1.10261	0.551304	−	0.834304i	$-0.314130\pi$
$829$	26318.0	1.10261	0.551304	+	0.834304i	$0.314130\pi$
$830$	991.118	0.0414485
$831$	−19994.7	−0.834669
$832$	8436.20	0.351530
$833$	−4553.99	−0.189420
$834$	−3783.48	−0.157088
$835$	650.498	0.0269598
$836$	17806.2	0.736650
$837$	−94989.0	−3.92270
$838$	11647.9	0.480156
$839$	6566.60	0.270208	0.135104	−	0.990831i	$-0.456863\pi$
$839$	6566.60	0.270208	0.135104	+	0.990831i	$0.456863\pi$
$840$	7167.63	0.294413
$841$	−7626.71	−0.312711
$842$	14512.8	0.593994
$843$	−11060.7	−0.451898
$844$	6522.59	0.266015
$845$	1610.00	0.0655450
$846$	27850.0	1.13180
$847$	4430.15	0.179719
$848$	638.538	0.0258579
$849$	−50685.0	−2.04889
$850$	−1319.04	−0.0532265
$851$	−26447.4	−1.06534
$852$	61091.8	2.45654
$853$	39036.9	1.56694	0.783468	−	0.621432i	$-0.213449\pi$
$853$	39036.9	1.56694	0.783468	+	0.621432i	$0.213449\pi$
$854$	5657.96	0.226711
$855$	8449.39	0.337968
$856$	17479.3	0.697931
$857$	−6060.17	−0.241554	−0.120777	−	0.992680i	$-0.538539\pi$
$857$	−6060.17	−0.241554	−0.120777	+	0.992680i	$0.538539\pi$
$858$	−12546.0	−0.499198
$859$	13664.8	0.542767	0.271383	−	0.962471i	$-0.412519\pi$
$859$	13664.8	0.542767	0.271383	+	0.962471i	$0.412519\pi$
$860$	0	0
$861$	85796.5	3.39598
$862$	8375.80	0.330952
$863$	−8805.11	−0.347311	−0.173655	−	0.984806i	$-0.555558\pi$
$863$	−8805.11	−0.347311	−0.173655	+	0.984806i	$0.555558\pi$
$864$	54559.4	2.14832
$865$	6129.05	0.240918
$866$	164.762	0.00646517
$867$	45685.7	1.78958
$868$	−50470.5	−1.97359
$869$	−15327.6	−0.598337
$870$	−2277.95	−0.0887697
$871$	−36117.3	−1.40504
$872$	−16202.4	−0.629223
$873$	12449.3	0.482642
$874$	−4932.60	−0.190901
$875$	13412.7	0.518208
$876$	−23767.1	−0.916686
$877$	24442.4	0.941120	0.470560	−	0.882368i	$-0.344052\pi$
$877$	24442.4	0.941120	0.470560	+	0.882368i	$0.344052\pi$
$878$	5241.80	0.201483
$879$	29516.1	1.13260
$880$	−3547.49	−0.135893
$881$	−21839.8	−0.835190	−0.417595	−	0.908633i	$-0.637127\pi$
$881$	−21839.8	−0.835190	−0.417595	+	0.908633i	$0.637127\pi$
$882$	−22057.7	−0.842086
$883$	17757.9	0.676785	0.338393	−	0.941005i	$-0.390117\pi$
$883$	17757.9	0.676785	0.338393	+	0.941005i	$0.390117\pi$
$884$	−3255.80	−0.123874
$885$	−12710.5	−0.482777
$886$	−12007.5	−0.455304
$887$	−42106.9	−1.59392	−0.796962	−	0.604030i	$-0.793561\pi$
$887$	−42106.9	−1.59392	−0.796962	+	0.604030i	$0.793561\pi$
$888$	−40749.6	−1.53994
$889$	−18619.5	−0.702449
$890$	542.324	0.0204256
$891$	66756.1	2.51000
$892$	19576.5	0.734833
$893$	30476.3	1.14205
$894$	17757.8	0.664329
$895$	−7071.50	−0.264105
$896$	37728.3	1.40671
$897$	−30636.4	−1.14038
$898$	−6490.47	−0.241191
$899$	33899.7	1.25764
$900$	56318.7	2.08588
$901$	−171.137	−0.00632784
$902$	11652.5	0.430140
$903$	0	0
$904$	13807.9	0.508014
$905$	−15.7315	−0.000577826 0
$906$	8839.58	0.324145
$907$	1377.33	0.0504227	0.0252114	−	0.999682i	$-0.491974\pi$
$907$	1377.33	0.0504227	0.0252114	+	0.999682i	$0.491974\pi$
$908$	−16504.7	−0.603225
$909$	−111365.	−4.06352
$910$	−1846.26	−0.0672559
$911$	−39864.1	−1.44979	−0.724895	−	0.688860i	$-0.758112\pi$
$911$	−39864.1	−1.44979	−0.724895	+	0.688860i	$0.758112\pi$
$912$	27695.6	1.00559
$913$	20882.4	0.756961
$914$	3791.04	0.137195
$915$	−4552.58	−0.164485
$916$	−21805.3	−0.786537
$917$	−27203.4	−0.979646
$918$	−3959.13	−0.142343
$919$	5830.88	0.209296	0.104648	−	0.994509i	$-0.466628\pi$
$919$	5830.88	0.209296	0.104648	+	0.994509i	$0.466628\pi$
$920$	2377.17	0.0851879
$921$	6128.81	0.219274
$922$	1070.38	0.0382331
$923$	−33257.5	−1.18601
$924$	71456.2	2.54409
$925$	−37485.9	−1.33246
$926$	2277.29	0.0808169
$927$	−2071.64	−0.0733998
$928$	−19471.2	−0.688763
$929$	−4121.82	−0.145568	−0.0727840	−	0.997348i	$-0.523188\pi$
$929$	−4121.82	−0.145568	−0.0727840	+	0.997348i	$0.523188\pi$
$930$	−4606.87	−0.162436
$931$	−24137.7	−0.849712
$932$	−35886.8	−1.26128
$933$	−67743.3	−2.37708
$934$	−8693.67	−0.304567
$935$	950.774	0.0332552
$936$	−33328.4	−1.16386
$937$	33608.5	1.17176	0.585881	−	0.810397i	$-0.300749\pi$
$937$	33608.5	1.17176	0.585881	+	0.810397i	$0.300749\pi$
$938$	−23335.7	−0.812302
$939$	−52515.9	−1.82512
$940$	−6949.53	−0.241137
$941$	−30075.8	−1.04192	−0.520958	−	0.853582i	$-0.674425\pi$
$941$	−30075.8	−1.04192	−0.520958	+	0.853582i	$0.674425\pi$
$942$	1587.99	0.0549251
$943$	28454.7	0.982621
$944$	−29416.1	−1.01421
$945$	19790.8	0.681265
$946$	0	0
$947$	8945.08	0.306944	0.153472	−	0.988153i	$-0.450955\pi$
$947$	8945.08	0.306944	0.153472	+	0.988153i	$0.450955\pi$
$948$	−27287.2	−0.934860
$949$	12938.5	0.442573
$950$	−6991.34	−0.238767
$951$	−62436.6	−2.12897
$952$	−4445.84	−0.151356
$953$	−51373.5	−1.74622	−0.873112	−	0.487519i	$-0.837902\pi$
$953$	−51373.5	−1.74622	−0.873112	+	0.487519i	$0.837902\pi$
$954$	−828.915	−0.0281311
$955$	2647.04	0.0896924
$956$	21117.5	0.714425
$957$	−47995.2	−1.62118
$958$	−11157.4	−0.376283
$959$	16987.0	0.571991
$960$	−4385.82	−0.147450
$961$	38767.0	1.30130
$962$	10496.4	0.351785
$963$	82687.9	2.76696
$964$	35791.5	1.19582
$965$	1546.49	0.0515888
$966$	−19794.5	−0.659294
$967$	22479.1	0.747549	0.373775	−	0.927520i	$-0.378063\pi$
$967$	22479.1	0.747549	0.373775	+	0.927520i	$0.378063\pi$
$968$	2263.82	0.0751673
$969$	−7422.80	−0.246083
$970$	352.410	0.0116652
$971$	45003.4	1.48736	0.743680	−	0.668536i	$-0.233079\pi$
$971$	45003.4	1.48736	0.743680	+	0.668536i	$0.233079\pi$
$972$	48466.3	1.59934
$973$	−11731.0	−0.386514
$974$	−2458.63	−0.0808826
$975$	−43423.3	−1.42632
$976$	−10536.1	−0.345545
$977$	−21946.4	−0.718657	−0.359329	−	0.933211i	$-0.616994\pi$
$977$	−21946.4	−0.718657	−0.359329	+	0.933211i	$0.616994\pi$
$978$	7201.05	0.235444
$979$	11426.5	0.373026
$980$	5504.15	0.179412
$981$	−76647.5	−2.49456
$982$	3791.32	0.123204
$983$	950.865	0.0308524	0.0154262	−	0.999881i	$-0.495089\pi$
$983$	950.865	0.0308524	0.0154262	+	0.999881i	$0.495089\pi$
$984$	43842.3	1.42037
$985$	407.783	0.0131909
$986$	1412.93	0.0456359
$987$	122301.	3.94417
$988$	−17256.8	−0.555681
$989$	0	0
$990$	4605.16	0.147840
$991$	51696.8	1.65712	0.828559	−	0.559901i	$-0.189161\pi$
$991$	51696.8	1.65712	0.828559	+	0.559901i	$0.189161\pi$
$992$	−39378.1	−1.26034
$993$	36175.6	1.15609
$994$	−21488.0	−0.685674
$995$	3820.15	0.121715
$996$	37176.1	1.18270
$997$	44474.6	1.41276	0.706381	−	0.707832i	$-0.250326\pi$
$997$	44474.6	1.41276	0.706381	+	0.707832i	$0.250326\pi$
$998$	−18042.9	−0.572284
$999$	−112515.	−3.56339

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.i.1.24	✓	50
43.42	odd	2		1849.4.a.j.1.27	yes	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.24	✓	50	1.1	even	1	trivial
1849.4.a.j.1.27	yes	50	43.42	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.902811 q^{2} -9.58399 q^{3} -7.18493 q^{4} -2.03345 q^{5} +8.65253 q^{6} +26.8278 q^{7} +13.7091 q^{8} +64.8528 q^{9} +O(q^{10})\)	\(q-0.902811 q^{2} -9.58399 q^{3} -7.18493 q^{4} -2.03345 q^{5} +8.65253 q^{6} +26.8278 q^{7} +13.7091 q^{8} +64.8528 q^{9} +1.83582 q^{10} +38.6799 q^{11} +68.8603 q^{12} -37.4866 q^{13} -24.2205 q^{14} +19.4886 q^{15} +45.1027 q^{16} -12.0881 q^{17} -58.5498 q^{18} -64.0712 q^{19} +14.6102 q^{20} -257.118 q^{21} -34.9206 q^{22} -85.2739 q^{23} -131.388 q^{24} -120.865 q^{25} +33.8433 q^{26} -362.781 q^{27} -192.756 q^{28} +129.469 q^{29} -17.5945 q^{30} +261.836 q^{31} -150.392 q^{32} -370.707 q^{33} +10.9133 q^{34} -54.5531 q^{35} -465.963 q^{36} +310.147 q^{37} +57.8442 q^{38} +359.271 q^{39} -27.8768 q^{40} -333.686 q^{41} +232.129 q^{42} -277.912 q^{44} -131.875 q^{45} +76.9862 q^{46} -475.663 q^{47} -432.264 q^{48} +376.733 q^{49} +109.118 q^{50} +115.852 q^{51} +269.339 q^{52} +14.1574 q^{53} +327.523 q^{54} -78.6536 q^{55} +367.786 q^{56} +614.057 q^{57} -116.886 q^{58} -652.202 q^{59} -140.024 q^{60} -233.602 q^{61} -236.388 q^{62} +1739.86 q^{63} -225.046 q^{64} +76.2271 q^{65} +334.679 q^{66} +963.472 q^{67} +86.8523 q^{68} +817.264 q^{69} +49.2511 q^{70} +887.185 q^{71} +889.075 q^{72} -345.150 q^{73} -280.004 q^{74} +1158.37 q^{75} +460.347 q^{76} +1037.70 q^{77} -324.354 q^{78} -396.269 q^{79} -91.7141 q^{80} +1725.86 q^{81} +301.255 q^{82} +539.877 q^{83} +1847.37 q^{84} +24.5806 q^{85} -1240.83 q^{87} +530.267 q^{88} +295.412 q^{89} +119.058 q^{90} -1005.68 q^{91} +612.687 q^{92} -2509.43 q^{93} +429.433 q^{94} +130.286 q^{95} +1441.36 q^{96} +191.963 q^{97} -340.119 q^{98} +2508.50 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} + O(q^{10}) \)	\( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} - 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} - 386q^{18} + 12q^{19} + 108q^{20} - 408q^{21} + 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} + 1493q^{26} + 10q^{27} - 242q^{28} + 208q^{29} - 48q^{30} - 932q^{31} - 1124q^{32} + 254q^{33} + 765q^{34} - 1452q^{35} + 747q^{36} - 90q^{37} - 1213q^{38} - 1610q^{39} - 1693q^{40} - 1354q^{41} - 16q^{42} - 2704q^{44} + 4508q^{45} + 233q^{46} - 3484q^{47} - 376q^{48} + 1324q^{49} - 408q^{50} + 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} - 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} + 1172q^{61} - 1546q^{62} - 3686q^{63} + 606q^{64} + 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} + 136q^{69} - 1310q^{70} + 162q^{71} - 5814q^{72} + 746q^{73} - 4332q^{74} + 236q^{75} + 1338q^{76} + 2024q^{77} - 2782q^{78} - 2656q^{79} - 5713q^{80} - 86q^{81} - 4168q^{82} - 3514q^{83} - 4269q^{84} - 7558q^{85} - 10278q^{87} + 11692q^{88} + 2640q^{89} - 8286q^{90} - 5946q^{91} - 4271q^{92} - 2q^{93} + 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} - 2826q^{98} - 8174q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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