LMFDB - Embedded newform 1849.4.a.j.1.2

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			1.2
Character	$\chi$	$=$	1849.1

$q$-expansion

$f(q)$	$=$	$q-5.20801 q^{2} -5.00842 q^{3} +19.1234 q^{4} +13.8519 q^{5} +26.0839 q^{6} +26.6225 q^{7} -57.9307 q^{8} -1.91570 q^{9} +O(q^{10})$	\(q-5.20801 q^{2} -5.00842 q^{3} +19.1234 q^{4} +13.8519 q^{5} +26.0839 q^{6} +26.6225 q^{7} -57.9307 q^{8} -1.91570 q^{9} -72.1409 q^{10} +47.0765 q^{11} -95.7780 q^{12} -60.3088 q^{13} -138.650 q^{14} -69.3762 q^{15} +148.717 q^{16} +69.2105 q^{17} +9.97699 q^{18} +9.96884 q^{19} +264.895 q^{20} -133.337 q^{21} -245.175 q^{22} -26.9731 q^{23} +290.142 q^{24} +66.8755 q^{25} +314.089 q^{26} +144.822 q^{27} +509.112 q^{28} +158.022 q^{29} +361.312 q^{30} -129.716 q^{31} -311.073 q^{32} -235.779 q^{33} -360.449 q^{34} +368.773 q^{35} -36.6347 q^{36} +59.9333 q^{37} -51.9178 q^{38} +302.052 q^{39} -802.451 q^{40} -415.611 q^{41} +694.419 q^{42} +900.263 q^{44} -26.5361 q^{45} +140.476 q^{46} -398.299 q^{47} -744.836 q^{48} +365.758 q^{49} -348.289 q^{50} -346.636 q^{51} -1153.31 q^{52} -280.790 q^{53} -754.235 q^{54} +652.100 q^{55} -1542.26 q^{56} -49.9282 q^{57} -822.981 q^{58} +143.280 q^{59} -1326.71 q^{60} -791.993 q^{61} +675.562 q^{62} -51.0008 q^{63} +430.337 q^{64} -835.392 q^{65} +1227.94 q^{66} -635.977 q^{67} +1323.54 q^{68} +135.093 q^{69} -1920.57 q^{70} -707.876 q^{71} +110.978 q^{72} +168.798 q^{73} -312.133 q^{74} -334.941 q^{75} +190.638 q^{76} +1253.30 q^{77} -1573.09 q^{78} -631.664 q^{79} +2060.01 q^{80} -673.606 q^{81} +2164.50 q^{82} +516.463 q^{83} -2549.85 q^{84} +958.698 q^{85} -791.441 q^{87} -2727.18 q^{88} -1272.00 q^{89} +138.200 q^{90} -1605.57 q^{91} -515.817 q^{92} +649.672 q^{93} +2074.34 q^{94} +138.088 q^{95} +1557.98 q^{96} -298.811 q^{97} -1904.87 q^{98} -90.1846 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−5.20801	−1.84131	−0.920655	−	0.390377i	$-0.872345\pi$
$2$	−5.20801	−1.84131	−0.920655	+	0.390377i	$0.872345\pi$
$3$	−5.00842	−0.963871	−0.481936	−	0.876207i	$-0.660066\pi$
$3$	−5.00842	−0.963871	−0.481936	+	0.876207i	$0.660066\pi$
$4$	19.1234	2.39042
$5$	13.8519	1.23895	0.619476	−	0.785015i	$-0.287345\pi$
$5$	13.8519	1.23895	0.619476	+	0.785015i	$0.287345\pi$
$6$	26.0839	1.77479
$7$	26.6225	1.43748	0.718740	−	0.695279i	$-0.244719\pi$
$7$	26.6225	1.43748	0.718740	+	0.695279i	$0.244719\pi$
$8$	−57.9307	−2.56020
$9$	−1.91570	−0.0709519
$10$	−72.1409	−2.28130
$11$	47.0765	1.29037	0.645187	−	0.764025i	$-0.276780\pi$
$11$	47.0765	1.29037	0.645187	+	0.764025i	$0.276780\pi$
$12$	−95.7780	−2.30406
$13$	−60.3088	−1.28667	−0.643333	−	0.765587i	$-0.722449\pi$
$13$	−60.3088	−1.28667	−0.643333	+	0.765587i	$0.722449\pi$
$14$	−138.650	−2.64685
$15$	−69.3762	−1.19419
$16$	148.717	2.32370
$17$	69.2105	0.987413	0.493706	−	0.869629i	$-0.335642\pi$
$17$	69.2105	0.987413	0.493706	+	0.869629i	$0.335642\pi$
$18$	9.97699	0.130644
$19$	9.96884	0.120369	0.0601845	−	0.998187i	$-0.480831\pi$
$19$	9.96884	0.120369	0.0601845	+	0.998187i	$0.480831\pi$
$20$	264.895	2.96162
$21$	−133.337	−1.38555
$22$	−245.175	−2.37598
$23$	−26.9731	−0.244534	−0.122267	−	0.992497i	$-0.539016\pi$
$23$	−26.9731	−0.244534	−0.122267	+	0.992497i	$0.539016\pi$
$24$	290.142	2.46770
$25$	66.8755	0.535004
$26$	314.089	2.36915
$27$	144.822	1.03226
$28$	509.112	3.43619
$29$	158.022	1.01186	0.505930	−	0.862574i	$-0.331149\pi$
$29$	158.022	1.01186	0.505930	+	0.862574i	$0.331149\pi$
$30$	361.312	2.19888
$31$	−129.716	−0.751537	−0.375769	−	0.926713i	$-0.622621\pi$
$31$	−129.716	−0.751537	−0.375769	+	0.926713i	$0.622621\pi$
$32$	−311.073	−1.71845
$33$	−235.779	−1.24375
$34$	−360.449	−1.81813
$35$	368.773	1.78097
$36$	−36.6347	−0.169605
$37$	59.9333	0.266296	0.133148	−	0.991096i	$-0.457491\pi$
$37$	59.9333	0.266296	0.133148	+	0.991096i	$0.457491\pi$
$38$	−51.9178	−0.221636
$39$	302.052	1.24018
$40$	−802.451	−3.17197
$41$	−415.611	−1.58311	−0.791554	−	0.611099i	$-0.790728\pi$
$41$	−415.611	−1.58311	−0.791554	+	0.611099i	$0.790728\pi$
$42$	694.419	2.55122
$43$	0	0
$43$	0	0
$44$	900.263	3.08454
$45$	−26.5361	−0.0879061
$46$	140.476	0.450263
$47$	−398.299	−1.23612	−0.618062	−	0.786129i	$-0.712082\pi$
$47$	−398.299	−1.23612	−0.618062	+	0.786129i	$0.712082\pi$
$48$	−744.836	−2.23975
$49$	365.758	1.06635
$50$	−348.289	−0.985109
$51$	−346.636	−0.951739
$52$	−1153.31	−3.07567
$53$	−280.790	−0.727725	−0.363863	−	0.931453i	$-0.618542\pi$
$53$	−280.790	−0.727725	−0.363863	+	0.931453i	$0.618542\pi$
$54$	−754.235	−1.90071
$55$	652.100	1.59871
$56$	−1542.26	−3.68024
$57$	−49.9282	−0.116020
$58$	−822.981	−1.86315
$59$	143.280	0.316159	0.158080	−	0.987426i	$-0.449470\pi$
$59$	143.280	0.316159	0.158080	+	0.987426i	$0.449470\pi$
$60$	−1326.71	−2.85462
$61$	−791.993	−1.66237	−0.831183	−	0.555999i	$-0.812336\pi$
$61$	−791.993	−1.66237	−0.831183	+	0.555999i	$0.812336\pi$
$62$	675.562	1.38381
$63$	−51.0008	−0.101992
$64$	430.337	0.840502
$65$	−835.392	−1.59412
$66$	1227.94	2.29014
$67$	−635.977	−1.15966	−0.579828	−	0.814739i	$-0.696880\pi$
$67$	−635.977	−1.15966	−0.579828	+	0.814739i	$0.696880\pi$
$68$	1323.54	2.36033
$69$	135.093	0.235699
$70$	−1920.57	−3.27932
$71$	−707.876	−1.18323	−0.591615	−	0.806220i	$-0.701510\pi$
$71$	−707.876	−1.18323	−0.591615	+	0.806220i	$0.701510\pi$
$72$	110.978	0.181651
$73$	168.798	0.270634	0.135317	−	0.990802i	$-0.456795\pi$
$73$	168.798	0.270634	0.135317	+	0.990802i	$0.456795\pi$
$74$	−312.133	−0.490334
$75$	−334.941	−0.515675
$76$	190.638	0.287733
$77$	1253.30	1.85489
$78$	−1573.09	−2.28356
$79$	−631.664	−0.899591	−0.449796	−	0.893131i	$-0.648503\pi$
$79$	−631.664	−0.899591	−0.449796	+	0.893131i	$0.648503\pi$
$80$	2060.01	2.87895
$81$	−673.606	−0.924014
$82$	2164.50	2.91499
$83$	516.463	0.683002	0.341501	−	0.939881i	$-0.389065\pi$
$83$	516.463	0.683002	0.341501	+	0.939881i	$0.389065\pi$
$84$	−2549.85	−3.31204
$85$	958.698	1.22336
$86$	0	0
$87$	−791.441	−0.975303
$88$	−2727.18	−3.30361
$89$	−1272.00	−1.51496	−0.757481	−	0.652857i	$-0.773570\pi$
$89$	−1272.00	−1.51496	−0.757481	+	0.652857i	$0.773570\pi$
$90$	138.200	0.161862
$91$	−1605.57	−1.84956
$92$	−515.817	−0.584540
$93$	649.672	0.724385
$94$	2074.34	2.27609
$95$	138.088	0.149131
$96$	1557.98	1.65637
$97$	−298.811	−0.312780	−0.156390	−	0.987695i	$-0.549986\pi$
$97$	−298.811	−0.312780	−0.156390	+	0.987695i	$0.549986\pi$
$98$	−1904.87	−1.96348
$99$	−90.1846	−0.0915545
$100$	1278.89	1.27889
$101$	−842.187	−0.829711	−0.414855	−	0.909887i	$-0.636168\pi$
$101$	−842.187	−0.829711	−0.414855	+	0.909887i	$0.636168\pi$
$102$	1805.28	1.75245
$103$	330.701	0.316359	0.158179	−	0.987410i	$-0.449438\pi$
$103$	330.701	0.316359	0.158179	+	0.987410i	$0.449438\pi$
$104$	3493.73	3.29412
$105$	−1846.97	−1.71663
$106$	1462.36	1.33997
$107$	−691.923	−0.625147	−0.312573	−	0.949894i	$-0.601191\pi$
$107$	−691.923	−0.625147	−0.312573	+	0.949894i	$0.601191\pi$
$108$	2769.49	2.46754
$109$	−1114.65	−0.979483	−0.489742	−	0.871868i	$-0.662909\pi$
$109$	−1114.65	−0.979483	−0.489742	+	0.871868i	$0.662909\pi$
$110$	−3396.15	−2.94373
$111$	−300.171	−0.256675
$112$	3959.21	3.34027
$113$	−1732.15	−1.44201	−0.721005	−	0.692929i	$-0.756320\pi$
$113$	−1732.15	−1.44201	−0.721005	+	0.692929i	$0.756320\pi$
$114$	260.027	0.213629
$115$	−373.629	−0.302966
$116$	3021.92	2.41877
$117$	115.534	0.0912913
$118$	−746.201	−0.582148
$119$	1842.56	1.41939
$120$	4019.02	3.05737
$121$	885.201	0.665064
$122$	4124.71	3.06093
$123$	2081.55	1.52591
$124$	−2480.61	−1.79649
$125$	−805.135	−0.576108
$126$	265.613	0.187799
$127$	2426.06	1.69510	0.847550	−	0.530715i	$-0.178077\pi$
$127$	2426.06	1.69510	0.847550	+	0.530715i	$0.178077\pi$
$128$	247.383	0.170826
$129$	0	0
$130$	4350.73	2.93527
$131$	−260.318	−0.173619	−0.0868094	−	0.996225i	$-0.527667\pi$
$131$	−260.318	−0.173619	−0.0868094	+	0.996225i	$0.527667\pi$
$132$	−4508.90	−2.97310
$133$	265.396	0.173028
$134$	3312.18	2.13529
$135$	2006.06	1.27892
$136$	−4009.42	−2.52797
$137$	−209.330	−0.130542	−0.0652711	−	0.997868i	$-0.520791\pi$
$137$	−209.330	−0.130542	−0.0652711	+	0.997868i	$0.520791\pi$
$138$	−703.565	−0.433996
$139$	253.533	0.154708	0.0773540	−	0.997004i	$-0.475353\pi$
$139$	253.533	0.154708	0.0773540	+	0.997004i	$0.475353\pi$
$140$	7052.18	4.25727
$141$	1994.85	1.19146
$142$	3686.62	2.17869
$143$	−2839.13	−1.66028
$144$	−284.897	−0.164871
$145$	2188.91	1.25365
$146$	−879.100	−0.498321
$147$	−1831.87	−1.02782
$148$	1146.13	0.636561
$149$	−2506.92	−1.37836	−0.689178	−	0.724592i	$-0.742028\pi$
$149$	−2506.92	−1.37836	−0.689178	+	0.724592i	$0.742028\pi$
$150$	1744.38	0.949518
$151$	1804.46	0.972482	0.486241	−	0.873825i	$-0.338368\pi$
$151$	1804.46	0.972482	0.486241	+	0.873825i	$0.338368\pi$
$152$	−577.502	−0.308168
$153$	−132.587	−0.0700588
$154$	−6527.18	−3.41542
$155$	−1796.81	−0.931120
$156$	5776.25	2.96455
$157$	676.565	0.343922	0.171961	−	0.985104i	$-0.444990\pi$
$157$	676.565	0.343922	0.171961	+	0.985104i	$0.444990\pi$
$158$	3289.71	1.65643
$159$	1406.31	0.701433
$160$	−4308.95	−2.12908
$161$	−718.092	−0.351513
$162$	3508.15	1.70140
$163$	−509.421	−0.244791	−0.122396	−	0.992481i	$-0.539058\pi$
$163$	−509.421	−0.244791	−0.122396	+	0.992481i	$0.539058\pi$
$164$	−7947.88	−3.78430
$165$	−3265.99	−1.54095
$166$	−2689.74	−1.25762
$167$	−1635.97	−0.758057	−0.379028	−	0.925385i	$-0.623742\pi$
$167$	−1635.97	−0.758057	−0.379028	+	0.925385i	$0.623742\pi$
$168$	7724.29	3.54727
$169$	1440.15	0.655507
$170$	−4992.91	−2.25258
$171$	−19.0973	−0.00854040
$172$	0	0
$173$	−336.002	−0.147663	−0.0738317	−	0.997271i	$-0.523523\pi$
$173$	−336.002	−0.147663	−0.0738317	+	0.997271i	$0.523523\pi$
$174$	4121.83	1.79584
$175$	1780.39	0.769058
$176$	7001.07	2.99844
$177$	−717.604	−0.304737
$178$	6624.59	2.78952
$179$	−885.817	−0.369883	−0.184941	−	0.982750i	$-0.559210\pi$
$179$	−885.817	−0.369883	−0.184941	+	0.982750i	$0.559210\pi$
$180$	−507.461	−0.210133
$181$	−3866.02	−1.58762	−0.793808	−	0.608168i	$-0.791905\pi$
$181$	−3866.02	−1.58762	−0.793808	+	0.608168i	$0.791905\pi$
$182$	8361.83	3.40561
$183$	3966.64	1.60231
$184$	1562.57	0.626056
$185$	830.190	0.329929
$186$	−3383.50	−1.33382
$187$	3258.19	1.27413
$188$	−7616.82	−2.95486
$189$	3855.53	1.48385
$190$	−719.162	−0.274597
$191$	3601.46	1.36436	0.682180	−	0.731184i	$-0.261032\pi$
$191$	3601.46	1.36436	0.682180	+	0.731184i	$0.261032\pi$
$192$	−2155.31	−0.810136
$193$	−907.399	−0.338425	−0.169213	−	0.985580i	$-0.554122\pi$
$193$	−907.399	−0.338425	−0.169213	+	0.985580i	$0.554122\pi$
$194$	1556.21	0.575926
$195$	4184.00	1.53652
$196$	6994.52	2.54902
$197$	3133.94	1.13342	0.566711	−	0.823917i	$-0.308216\pi$
$197$	3133.94	1.13342	0.566711	+	0.823917i	$0.308216\pi$
$198$	469.682	0.168580
$199$	702.456	0.250230	0.125115	−	0.992142i	$-0.460070\pi$
$199$	702.456	0.250230	0.125115	+	0.992142i	$0.460070\pi$
$200$	−3874.15	−1.36972
$201$	3185.24	1.11776
$202$	4386.12	1.52775
$203$	4206.94	1.45453
$204$	−6628.85	−2.27506
$205$	−5757.00	−1.96140
$206$	−1722.30	−0.582515
$207$	51.6724	0.0173502
$208$	−8968.93	−2.98982
$209$	469.299	0.155321
$210$	9619.04	3.16084
$211$	−2634.32	−0.859497	−0.429749	−	0.902949i	$-0.641398\pi$
$211$	−2634.32	−0.859497	−0.429749	+	0.902949i	$0.641398\pi$
$212$	−5369.65	−1.73957
$213$	3545.34	1.14048
$214$	3603.54	1.15109
$215$	0	0
$216$	−8389.65	−2.64279
$217$	−3453.36	−1.08032
$218$	5805.09	1.80353
$219$	−845.410	−0.260856
$220$	12470.4	3.82160
$221$	−4174.00	−1.27047
$222$	1563.29	0.472619
$223$	5852.83	1.75755	0.878776	−	0.477234i	$-0.158361\pi$
$223$	5852.83	1.75755	0.878776	+	0.477234i	$0.158361\pi$
$224$	−8281.54	−2.47024
$225$	−128.114	−0.0379596
$226$	9021.07	2.65519
$227$	−2690.65	−0.786718	−0.393359	−	0.919385i	$-0.628687\pi$
$227$	−2690.65	−0.786718	−0.393359	+	0.919385i	$0.628687\pi$
$228$	−954.796	−0.277337
$229$	909.642	0.262493	0.131246	−	0.991350i	$-0.458102\pi$
$229$	909.642	0.262493	0.131246	+	0.991350i	$0.458102\pi$
$230$	1945.87	0.557855
$231$	−6277.03	−1.78787
$232$	−9154.33	−2.59057
$233$	−2711.93	−0.762509	−0.381255	−	0.924470i	$-0.624508\pi$
$233$	−2711.93	−0.762509	−0.381255	+	0.924470i	$0.624508\pi$
$234$	−601.700	−0.168096
$235$	−5517.20	−1.53150
$236$	2739.99	0.755755
$237$	3163.64	0.867090
$238$	−9596.06	−2.61353
$239$	2199.65	0.595329	0.297665	−	0.954671i	$-0.403792\pi$
$239$	2199.65	0.595329	0.297665	+	0.954671i	$0.403792\pi$
$240$	−10317.4	−2.77494
$241$	2700.40	0.721776	0.360888	−	0.932609i	$-0.382474\pi$
$241$	2700.40	0.721776	0.360888	+	0.932609i	$0.382474\pi$
$242$	−4610.14	−1.22459
$243$	−536.491	−0.141629
$244$	−15145.6	−3.97376
$245$	5066.44	1.32116
$246$	−10840.8	−2.80968
$247$	−601.209	−0.154874
$248$	7514.53	1.92409
$249$	−2586.66	−0.658326
$250$	4193.15	1.06079
$251$	−5537.81	−1.39260	−0.696302	−	0.717749i	$-0.745173\pi$
$251$	−5537.81	−1.39260	−0.696302	+	0.717749i	$0.745173\pi$
$252$	−975.307	−0.243804
$253$	−1269.80	−0.315540
$254$	−12634.9	−3.12121
$255$	−4801.57	−1.17916
$256$	−4731.07	−1.15505
$257$	−468.256	−0.113654	−0.0568268	−	0.998384i	$-0.518098\pi$
$257$	−468.256	−0.113654	−0.0568268	+	0.998384i	$0.518098\pi$
$258$	0	0
$259$	1595.57	0.382796
$260$	−15975.5	−3.81062
$261$	−302.723	−0.0717934
$262$	1355.74	0.319686
$263$	−6435.62	−1.50889	−0.754443	−	0.656365i	$-0.772093\pi$
$263$	−6435.62	−1.50889	−0.754443	+	0.656365i	$0.772093\pi$
$264$	13658.9	3.18426
$265$	−3889.47	−0.901617
$266$	−1382.18	−0.318598
$267$	6370.71	1.46023
$268$	−12162.0	−2.77207
$269$	−882.896	−0.200116	−0.100058	−	0.994982i	$-0.531903\pi$
$269$	−882.896	−0.200116	−0.100058	+	0.994982i	$0.531903\pi$
$270$	−10447.6	−2.35489
$271$	6424.80	1.44014	0.720071	−	0.693900i	$-0.244109\pi$
$271$	6424.80	1.44014	0.720071	+	0.693900i	$0.244109\pi$
$272$	10292.8	2.29445
$273$	8041.38	1.78273
$274$	1090.19	0.240369
$275$	3148.27	0.690356
$276$	2583.43	0.563421
$277$	1236.99	0.268315	0.134158	−	0.990960i	$-0.457167\pi$
$277$	1236.99	0.268315	0.134158	+	0.990960i	$0.457167\pi$
$278$	−1320.41	−0.284866
$279$	248.497	0.0533230
$280$	−21363.3	−4.55964
$281$	1461.25	0.310217	0.155109	−	0.987897i	$-0.450427\pi$
$281$	1461.25	0.310217	0.155109	+	0.987897i	$0.450427\pi$
$282$	−10389.2	−2.19386
$283$	1698.07	0.356678	0.178339	−	0.983969i	$-0.442928\pi$
$283$	1698.07	0.356678	0.178339	+	0.983969i	$0.442928\pi$
$284$	−13537.0	−2.82842
$285$	−691.601	−0.143743
$286$	14786.2	3.05709
$287$	−11064.6	−2.27569
$288$	595.923	0.121927
$289$	−122.902	−0.0250158
$290$	−11399.9	−2.30835
$291$	1496.57	0.301480
$292$	3227.98	0.646929
$293$	3559.87	0.709795	0.354897	−	0.934905i	$-0.384516\pi$
$293$	3559.87	0.709795	0.354897	+	0.934905i	$0.384516\pi$
$294$	9540.39	1.89254
$295$	1984.70	0.391707
$296$	−3471.98	−0.681772
$297$	6817.72	1.33200
$298$	13056.1	2.53798
$299$	1626.72	0.314634
$300$	−6405.21	−1.23268
$301$	0	0
$302$	−9397.64	−1.79064
$303$	4218.03	0.799734
$304$	1482.53	0.279701
$305$	−10970.6	−2.05959
$306$	690.513	0.129000
$307$	−5590.48	−1.03930	−0.519651	−	0.854379i	$-0.673938\pi$
$307$	−5590.48	−1.03930	−0.519651	+	0.854379i	$0.673938\pi$
$308$	23967.2	4.43396
$309$	−1656.29	−0.304929
$310$	9357.83	1.71448
$311$	2708.95	0.493924	0.246962	−	0.969025i	$-0.420568\pi$
$311$	2708.95	0.493924	0.246962	+	0.969025i	$0.420568\pi$
$312$	−17498.1	−3.17511
$313$	6730.92	1.21551	0.607754	−	0.794125i	$-0.292071\pi$
$313$	6730.92	1.21551	0.607754	+	0.794125i	$0.292071\pi$
$314$	−3523.56	−0.633267
$315$	−706.458	−0.126363
$316$	−12079.5	−2.15040
$317$	5984.74	1.06037	0.530184	−	0.847883i	$-0.322123\pi$
$317$	5984.74	1.06037	0.530184	+	0.847883i	$0.322123\pi$
$318$	−7324.10	−1.29156
$319$	7439.13	1.30568
$320$	5960.99	1.04134
$321$	3465.44	0.602561
$322$	3739.83	0.647244
$323$	689.949	0.118854
$324$	−12881.6	−2.20878
$325$	−4033.18	−0.688372
$326$	2653.07	0.450737
$327$	5582.62	0.944096
$328$	24076.6	4.05308
$329$	−10603.7	−1.77690
$330$	17009.3	2.83737
$331$	−2114.28	−0.351092	−0.175546	−	0.984471i	$-0.556169\pi$
$331$	−2114.28	−0.351092	−0.175546	+	0.984471i	$0.556169\pi$
$332$	9876.51	1.63266
$333$	−114.814	−0.0188942
$334$	8520.17	1.39582
$335$	−8809.51	−1.43676
$336$	−19829.4	−3.21959
$337$	10476.9	1.69351	0.846756	−	0.531982i	$-0.178552\pi$
$337$	10476.9	1.69351	0.846756	+	0.531982i	$0.178552\pi$
$338$	−7500.32	−1.20699
$339$	8675.35	1.38991
$340$	18333.6	2.92434
$341$	−6106.58	−0.969764
$342$	99.4591	0.0157255
$343$	605.864	0.0953749
$344$	0	0
$345$	1871.29	0.292021
$346$	1749.90	0.271894
$347$	−11253.4	−1.74096	−0.870482	−	0.492199i	$-0.836193\pi$
$347$	−11253.4	−1.74096	−0.870482	+	0.492199i	$0.836193\pi$
$348$	−15135.0	−2.33139
$349$	1170.22	0.179486	0.0897428	−	0.995965i	$-0.471396\pi$
$349$	1170.22	0.179486	0.0897428	+	0.995965i	$0.471396\pi$
$350$	−9272.31	−1.41607
$351$	−8734.04	−1.32817
$352$	−14644.2	−2.21744
$353$	−804.464	−0.121295	−0.0606477	−	0.998159i	$-0.519317\pi$
$353$	−804.464	−0.121295	−0.0606477	+	0.998159i	$0.519317\pi$
$354$	3737.29	0.561115
$355$	−9805.43	−1.46597
$356$	−24324.9	−3.62140
$357$	−9228.31	−1.36811
$358$	4613.34	0.681069
$359$	9746.71	1.43290	0.716451	−	0.697638i	$-0.245765\pi$
$359$	9746.71	1.43290	0.716451	+	0.697638i	$0.245765\pi$
$360$	1537.26	0.225057
$361$	−6759.62	−0.985511
$362$	20134.3	2.92329
$363$	−4433.46	−0.641037
$364$	−30703.9	−4.42122
$365$	2338.17	0.335303
$366$	−20658.3	−2.95034
$367$	−3663.53	−0.521075	−0.260538	−	0.965464i	$-0.583900\pi$
$367$	−3663.53	−0.521075	−0.260538	+	0.965464i	$0.583900\pi$
$368$	−4011.36	−0.568224
$369$	796.186	0.112325
$370$	−4323.64	−0.607501
$371$	−7475.32	−1.04609
$372$	12423.9	1.73159
$373$	13468.9	1.86969	0.934846	−	0.355053i	$-0.115537\pi$
$373$	13468.9	1.86969	0.934846	+	0.355053i	$0.115537\pi$
$374$	−16968.7	−2.34607
$375$	4032.46	0.555294
$376$	23073.7	3.16472
$377$	−9530.12	−1.30193
$378$	−20079.6	−2.73223
$379$	2326.69	0.315341	0.157670	−	0.987492i	$-0.449602\pi$
$379$	2326.69	0.315341	0.157670	+	0.987492i	$0.449602\pi$
$380$	2640.70	0.356487
$381$	−12150.7	−1.63386
$382$	−18756.5	−2.51221
$383$	−1329.44	−0.177366	−0.0886831	−	0.996060i	$-0.528266\pi$
$383$	−1329.44	−0.177366	−0.0886831	+	0.996060i	$0.528266\pi$
$384$	−1239.00	−0.164655
$385$	17360.5	2.29812
$386$	4725.75	0.623145
$387$	0	0
$388$	−5714.28	−0.747677
$389$	−10163.9	−1.32475	−0.662375	−	0.749172i	$-0.730451\pi$
$389$	−10163.9	−1.32475	−0.662375	+	0.749172i	$0.730451\pi$
$390$	−21790.3	−2.82922
$391$	−1866.82	−0.241456
$392$	−21188.6	−2.73007
$393$	1303.78	0.167346
$394$	−16321.6	−2.08698
$395$	−8749.75	−1.11455
$396$	−1724.63	−0.218854
$397$	−13306.5	−1.68221	−0.841103	−	0.540875i	$-0.818093\pi$
$397$	−13306.5	−1.68221	−0.841103	+	0.540875i	$0.818093\pi$
$398$	−3658.40	−0.460751
$399$	−1329.21	−0.166777
$400$	9945.51	1.24319
$401$	15275.3	1.90228	0.951140	−	0.308759i	$-0.0999137\pi$
$401$	15275.3	1.90228	0.951140	+	0.308759i	$0.0999137\pi$
$402$	−16588.8	−2.05814
$403$	7823.01	0.966977
$404$	−16105.5	−1.98336
$405$	−9330.74	−1.14481
$406$	−21909.8	−2.67824
$407$	2821.45	0.343622
$408$	20080.8	2.43664
$409$	16043.3	1.93958	0.969792	−	0.243931i	$-0.0784372\pi$
$409$	16043.3	1.93958	0.969792	+	0.243931i	$0.0784372\pi$
$410$	29982.5	3.61154
$411$	1048.41	0.125826
$412$	6324.13	0.756232
$413$	3814.46	0.454473
$414$	−269.111	−0.0319470
$415$	7154.00	0.846207
$416$	18760.4	2.21107
$417$	−1269.80	−0.149119
$418$	−2444.11	−0.285994
$419$	8887.02	1.03618	0.518090	−	0.855326i	$-0.326643\pi$
$419$	8887.02	1.03618	0.518090	+	0.855326i	$0.326643\pi$
$420$	−35320.3	−4.10346
$421$	600.862	0.0695587	0.0347794	−	0.999395i	$-0.488927\pi$
$421$	600.862	0.0695587	0.0347794	+	0.999395i	$0.488927\pi$
$422$	13719.6	1.58260
$423$	763.021	0.0877053
$424$	16266.3	1.86312
$425$	4628.49	0.528270
$426$	−18464.2	−2.09998
$427$	−21084.8	−2.38962
$428$	−13231.9	−1.49436
$429$	14219.6	1.60030
$430$	0	0
$431$	−16834.7	−1.88143	−0.940717	−	0.339193i	$-0.889846\pi$
$431$	−16834.7	−1.88143	−0.940717	+	0.339193i	$0.889846\pi$
$432$	21537.5	2.39866
$433$	−1169.19	−0.129764	−0.0648820	−	0.997893i	$-0.520667\pi$
$433$	−1169.19	−0.129764	−0.0648820	+	0.997893i	$0.520667\pi$
$434$	17985.1	1.98920
$435$	−10963.0	−1.20836
$436$	−21315.8	−2.34138
$437$	−268.891	−0.0294343
$438$	4402.91	0.480317
$439$	404.634	0.0439912	0.0219956	−	0.999758i	$-0.492998\pi$
$439$	404.634	0.0439912	0.0219956	+	0.999758i	$0.492998\pi$
$440$	−37776.6	−4.09302
$441$	−700.682	−0.0756595
$442$	21738.3	2.33933
$443$	8522.87	0.914071	0.457036	−	0.889448i	$-0.348911\pi$
$443$	8522.87	0.914071	0.457036	+	0.889448i	$0.348911\pi$
$444$	−5740.29	−0.613563
$445$	−17619.6	−1.87697
$446$	−30481.6	−3.23620
$447$	12555.7	1.32856
$448$	11456.6	1.20820
$449$	2477.35	0.260386	0.130193	−	0.991489i	$-0.458440\pi$
$449$	2477.35	0.260386	0.130193	+	0.991489i	$0.458440\pi$
$450$	667.217	0.0698954
$451$	−19565.5	−2.04280
$452$	−33124.6	−3.44702
$453$	−9037.49	−0.937347
$454$	14013.0	1.44859
$455$	−22240.2	−2.29151
$456$	2892.37	0.297035
$457$	4605.88	0.471453	0.235727	−	0.971819i	$-0.424253\pi$
$457$	4605.88	0.471453	0.235727	+	0.971819i	$0.424253\pi$
$458$	−4737.42	−0.483330
$459$	10023.2	1.01927
$460$	−7145.06	−0.724218
$461$	−11285.7	−1.14019	−0.570093	−	0.821580i	$-0.693093\pi$
$461$	−11285.7	−1.14019	−0.570093	+	0.821580i	$0.693093\pi$
$462$	32690.9	3.29203
$463$	508.870	0.0510782	0.0255391	−	0.999674i	$-0.491870\pi$
$463$	508.870	0.0510782	0.0255391	+	0.999674i	$0.491870\pi$
$464$	23500.5	2.35126
$465$	8999.20	0.897480
$466$	14123.8	1.40402
$467$	−568.788	−0.0563606	−0.0281803	−	0.999603i	$-0.508971\pi$
$467$	−568.788	−0.0563606	−0.0281803	+	0.999603i	$0.508971\pi$
$468$	2209.39	0.218225
$469$	−16931.3	−1.66698
$470$	28733.6	2.81997
$471$	−3388.52	−0.331497
$472$	−8300.29	−0.809431
$473$	0	0
$474$	−16476.3	−1.59658
$475$	666.672	0.0643979
$476$	35235.9	3.39293
$477$	537.909	0.0516335
$478$	−11455.8	−1.09619
$479$	−3278.52	−0.312734	−0.156367	−	0.987699i	$-0.549978\pi$
$479$	−3278.52	−0.312734	−0.156367	+	0.987699i	$0.549978\pi$
$480$	21581.1	2.05216
$481$	−3614.50	−0.342634
$482$	−14063.7	−1.32901
$483$	3596.51	0.338813
$484$	16928.0	1.58979
$485$	−4139.11	−0.387520
$486$	2794.05	0.260783
$487$	3439.30	0.320020	0.160010	−	0.987115i	$-0.448847\pi$
$487$	3439.30	0.320020	0.160010	+	0.987115i	$0.448847\pi$
$488$	45880.7	4.25599
$489$	2551.40	0.235947
$490$	−26386.1	−2.43266
$491$	3116.14	0.286414	0.143207	−	0.989693i	$-0.454259\pi$
$491$	3116.14	0.286414	0.143207	+	0.989693i	$0.454259\pi$
$492$	39806.3	3.64758
$493$	10936.8	0.999124
$494$	3131.10	0.285172
$495$	−1249.23	−0.113432
$496$	−19290.9	−1.74635
$497$	−18845.4	−1.70087
$498$	13471.4	1.21218
$499$	16962.6	1.52175	0.760874	−	0.648899i	$-0.224770\pi$
$499$	16962.6	1.52175	0.760874	+	0.648899i	$0.224770\pi$
$500$	−15396.9	−1.37714
$501$	8193.65	0.730669
$502$	28841.0	2.56422
$503$	17725.1	1.57122	0.785610	−	0.618722i	$-0.212349\pi$
$503$	17725.1	1.57122	0.785610	+	0.618722i	$0.212349\pi$
$504$	2954.51	0.261120
$505$	−11665.9	−1.02797
$506$	6613.14	0.581008
$507$	−7212.88	−0.631825
$508$	46394.4	4.05201
$509$	3044.24	0.265095	0.132548	−	0.991177i	$-0.457684\pi$
$509$	3044.24	0.265095	0.132548	+	0.991177i	$0.457684\pi$
$510$	25006.6	2.17120
$511$	4493.82	0.389031
$512$	22660.4	1.95597
$513$	1443.71	0.124252
$514$	2438.68	0.209272
$515$	4580.85	0.391954
$516$	0	0
$517$	−18750.5	−1.59506
$518$	−8309.76	−0.704846
$519$	1682.84	0.142329
$520$	48394.9	4.08126
$521$	12593.9	1.05902	0.529509	−	0.848304i	$-0.322376\pi$
$521$	12593.9	1.05902	0.529509	+	0.848304i	$0.322376\pi$
$522$	1576.59	0.132194
$523$	15100.1	1.26249	0.631243	−	0.775585i	$-0.282545\pi$
$523$	15100.1	1.26249	0.631243	+	0.775585i	$0.282545\pi$
$524$	−4978.16	−0.415023
$525$	−8916.97	−0.741273
$526$	33516.8	2.77833
$527$	−8977.71	−0.742078
$528$	−35064.3	−2.89011
$529$	−11439.5	−0.940203
$530$	20256.4	1.66016
$531$	−274.481	−0.0224321
$532$	5075.26	0.413610
$533$	25065.0	2.03693
$534$	−33178.7	−2.68873
$535$	−9584.45	−0.774527
$536$	36842.6	2.96895
$537$	4436.54	0.356520
$538$	4598.13	0.368475
$539$	17218.6	1.37599
$540$	38362.7	3.05716
$541$	−11318.5	−0.899485	−0.449743	−	0.893158i	$-0.648484\pi$
$541$	−11318.5	−0.899485	−0.449743	+	0.893158i	$0.648484\pi$
$542$	−33460.4	−2.65175
$543$	19362.6	1.53026
$544$	−21529.5	−1.69682
$545$	−15440.0	−1.21353
$546$	−41879.6	−3.28257
$547$	−4299.81	−0.336100	−0.168050	−	0.985778i	$-0.553747\pi$
$547$	−4299.81	−0.336100	−0.168050	+	0.985778i	$0.553747\pi$
$548$	−4003.10	−0.312051
$549$	1517.22	0.117948
$550$	−16396.2	−1.27116
$551$	1575.30	0.121797
$552$	−7826.02	−0.603438
$553$	−16816.5	−1.29314
$554$	−6442.24	−0.494052
$555$	−4157.94	−0.318009
$556$	4848.42	0.369818
$557$	−8356.80	−0.635707	−0.317854	−	0.948140i	$-0.602962\pi$
$557$	−8356.80	−0.635707	−0.317854	+	0.948140i	$0.602962\pi$
$558$	−1294.17	−0.0981842
$559$	0	0
$560$	54842.7	4.13844
$561$	−16318.4	−1.22810
$562$	−7610.22	−0.571206
$563$	−17758.8	−1.32938	−0.664692	−	0.747118i	$-0.731437\pi$
$563$	−17758.8	−1.32938	−0.664692	+	0.747118i	$0.731437\pi$
$564$	38148.2	2.84810
$565$	−23993.6	−1.78658
$566$	−8843.57	−0.656754
$567$	−17933.1	−1.32825
$568$	41007.7	3.02931
$569$	−6954.03	−0.512352	−0.256176	−	0.966630i	$-0.582463\pi$
$569$	−6954.03	−0.512352	−0.256176	+	0.966630i	$0.582463\pi$
$570$	3601.87	0.264676
$571$	20289.4	1.48701	0.743505	−	0.668730i	$-0.233162\pi$
$571$	20289.4	1.48701	0.743505	+	0.668730i	$0.233162\pi$
$572$	−54293.8	−3.96877
$573$	−18037.6	−1.31507
$574$	57624.5	4.19025
$575$	−1803.84	−0.130827
$576$	−824.397	−0.0596352
$577$	−5613.49	−0.405013	−0.202507	−	0.979281i	$-0.564909\pi$
$577$	−5613.49	−0.405013	−0.202507	+	0.979281i	$0.564909\pi$
$578$	640.078	0.0460618
$579$	4544.64	0.326198
$580$	41859.3	2.99675
$581$	13749.5	0.981801
$582$	−7794.17	−0.555118
$583$	−13218.6	−0.939037
$584$	−9778.57	−0.692877
$585$	1600.36	0.113106
$586$	−18539.8	−1.30695
$587$	−8010.42	−0.563246	−0.281623	−	0.959525i	$-0.590873\pi$
$587$	−8010.42	−0.563246	−0.281623	+	0.959525i	$0.590873\pi$
$588$	−35031.5	−2.45693
$589$	−1293.12	−0.0904617
$590$	−10336.3	−0.721253
$591$	−15696.1	−1.09247
$592$	8913.08	0.618793
$593$	−6472.49	−0.448218	−0.224109	−	0.974564i	$-0.571947\pi$
$593$	−6472.49	−0.448218	−0.224109	+	0.974564i	$0.571947\pi$
$594$	−35506.8	−2.45263
$595$	25522.9	1.75855
$596$	−47940.8	−3.29485
$597$	−3518.20	−0.241190
$598$	−8471.96	−0.579338
$599$	−25455.3	−1.73635	−0.868176	−	0.496257i	$-0.834708\pi$
$599$	−25455.3	−1.73635	−0.868176	+	0.496257i	$0.834708\pi$
$600$	19403.4	1.32023
$601$	−13600.1	−0.923064	−0.461532	−	0.887124i	$-0.652700\pi$
$601$	−13600.1	−0.923064	−0.461532	+	0.887124i	$0.652700\pi$
$602$	0	0
$603$	1218.34	0.0822799
$604$	34507.4	2.32464
$605$	12261.7	0.823983
$606$	−21967.5	−1.47256
$607$	−8144.66	−0.544616	−0.272308	−	0.962210i	$-0.587787\pi$
$607$	−8144.66	−0.544616	−0.272308	+	0.962210i	$0.587787\pi$
$608$	−3101.04	−0.206848
$609$	−21070.1	−1.40198
$610$	57135.1	3.79235
$611$	24020.9	1.59048
$612$	−2535.51	−0.167470
$613$	21750.5	1.43311	0.716554	−	0.697531i	$-0.245718\pi$
$613$	21750.5	1.43311	0.716554	+	0.697531i	$0.245718\pi$
$614$	29115.3	1.91368
$615$	28833.5	1.89053
$616$	−72604.3	−4.74888
$617$	11659.4	0.760758	0.380379	−	0.924831i	$-0.375793\pi$
$617$	11659.4	0.760758	0.380379	+	0.924831i	$0.375793\pi$
$618$	8625.99	0.561469
$619$	17054.3	1.10738	0.553692	−	0.832722i	$-0.313219\pi$
$619$	17054.3	1.10738	0.553692	+	0.832722i	$0.313219\pi$
$620$	−34361.2	−2.22577
$621$	−3906.30	−0.252423
$622$	−14108.2	−0.909467
$623$	−33863.8	−2.17773
$624$	44920.2	2.88180
$625$	−19512.1	−1.24877
$626$	−35054.7	−2.23813
$627$	−2350.45	−0.149709
$628$	12938.2	0.822119
$629$	4148.01	0.262944
$630$	3679.24	0.232674
$631$	−15049.7	−0.949475	−0.474737	−	0.880128i	$-0.657457\pi$
$631$	−15049.7	−0.949475	−0.474737	+	0.880128i	$0.657457\pi$
$632$	36592.7	2.30313
$633$	13193.8	0.828445
$634$	−31168.6	−1.95247
$635$	33605.5	2.10015
$636$	26893.5	1.67672
$637$	−22058.4	−1.37203
$638$	−38743.1	−2.40416
$639$	1356.08	0.0839525
$640$	3426.73	0.211646
$641$	22558.3	1.39002	0.695008	−	0.719002i	$-0.255401\pi$
$641$	22558.3	1.39002	0.695008	+	0.719002i	$0.255401\pi$
$642$	−18048.1	−1.10950
$643$	16170.3	0.991748	0.495874	−	0.868395i	$-0.334848\pi$
$643$	16170.3	0.991748	0.495874	+	0.868395i	$0.334848\pi$
$644$	−13732.3	−0.840265
$645$	0	0
$646$	−3593.26	−0.218847
$647$	−16223.2	−0.985778	−0.492889	−	0.870092i	$-0.664059\pi$
$647$	−16223.2	−0.985778	−0.492889	+	0.870092i	$0.664059\pi$
$648$	39022.5	2.36566
$649$	6745.10	0.407964
$650$	21004.9	1.26751
$651$	17295.9	1.04129
$652$	−9741.86	−0.585155
$653$	6142.50	0.368108	0.184054	−	0.982916i	$-0.441078\pi$
$653$	6142.50	0.368108	0.184054	+	0.982916i	$0.441078\pi$
$654$	−29074.3	−1.73837
$655$	−3605.90	−0.215106
$656$	−61808.3	−3.67867
$657$	−323.366	−0.0192020
$658$	55224.2	3.27183
$659$	−14009.5	−0.828123	−0.414061	−	0.910249i	$-0.635890\pi$
$659$	−14009.5	−0.828123	−0.414061	+	0.910249i	$0.635890\pi$
$660$	−62456.9	−3.68353
$661$	−26667.6	−1.56921	−0.784605	−	0.619996i	$-0.787134\pi$
$661$	−26667.6	−1.56921	−0.784605	+	0.619996i	$0.787134\pi$
$662$	11011.2	0.646470
$663$	20905.2	1.22457
$664$	−29919.0	−1.74862
$665$	3676.24	0.214373
$666$	597.954	0.0347901
$667$	−4262.35	−0.247434
$668$	−31285.4	−1.81208
$669$	−29313.4	−1.69405
$670$	45880.0	2.64552
$671$	−37284.3	−2.14507
$672$	41477.4	2.38099
$673$	26390.1	1.51153	0.755767	−	0.654841i	$-0.227264\pi$
$673$	26390.1	1.51153	0.755767	+	0.654841i	$0.227264\pi$
$674$	−54563.8	−3.11828
$675$	9685.05	0.552264
$676$	27540.5	1.56694
$677$	10340.1	0.587006	0.293503	−	0.955958i	$-0.405179\pi$
$677$	10340.1	0.587006	0.293503	+	0.955958i	$0.405179\pi$
$678$	−45181.3	−2.55926
$679$	−7955.10	−0.449615
$680$	−55538.1	−3.13204
$681$	13475.9	0.758295
$682$	31803.1	1.78564
$683$	3929.56	0.220147	0.110074	−	0.993923i	$-0.464891\pi$
$683$	3929.56	0.220147	0.110074	+	0.993923i	$0.464891\pi$
$684$	−365.205	−0.0204152
$685$	−2899.62	−0.161736
$686$	−3155.35	−0.175615
$687$	−4555.87	−0.253009
$688$	0	0
$689$	16934.1	0.936339
$690$	−9745.72	−0.537700
$691$	−2660.45	−0.146466	−0.0732332	−	0.997315i	$-0.523332\pi$
$691$	−2660.45	−0.146466	−0.0732332	+	0.997315i	$0.523332\pi$
$692$	−6425.50	−0.352978
$693$	−2400.94	−0.131608
$694$	58607.9	3.20566
$695$	3511.92	0.191676
$696$	45848.8	2.49697
$697$	−28764.6	−1.56318
$698$	−6094.52	−0.330489
$699$	13582.5	0.734961
$700$	34047.2	1.83837
$701$	27310.9	1.47150	0.735749	−	0.677255i	$-0.236830\pi$
$701$	27310.9	1.47150	0.735749	+	0.677255i	$0.236830\pi$
$702$	45487.0	2.44558
$703$	597.465	0.0320538
$704$	20258.8	1.08456
$705$	27632.5	1.47617
$706$	4189.66	0.223342
$707$	−22421.1	−1.19269
$708$	−13723.0	−0.728450
$709$	2281.77	0.120866	0.0604328	−	0.998172i	$-0.480752\pi$
$709$	2281.77	0.120866	0.0604328	+	0.998172i	$0.480752\pi$
$710$	51066.8	2.69930
$711$	1210.08	0.0638277
$712$	73687.8	3.87861
$713$	3498.84	0.183777
$714$	48061.1	2.51911
$715$	−39327.4	−2.05701
$716$	−16939.8	−0.884177
$717$	−11016.8	−0.573821
$718$	−50761.0	−2.63842
$719$	−7427.41	−0.385251	−0.192626	−	0.981272i	$-0.561700\pi$
$719$	−7427.41	−0.385251	−0.192626	+	0.981272i	$0.561700\pi$
$720$	−3946.37	−0.204267
$721$	8804.10	0.454760
$722$	35204.2	1.81463
$723$	−13524.7	−0.695699
$724$	−73931.3	−3.79508
$725$	10567.8	0.541350
$726$	23089.5	1.18035
$727$	31960.7	1.63048	0.815240	−	0.579124i	$-0.196605\pi$
$727$	31960.7	1.63048	0.815240	+	0.579124i	$0.196605\pi$
$728$	93011.9	4.73523
$729$	20874.3	1.06053
$730$	−12177.2	−0.617396
$731$	0	0
$732$	75855.5	3.83019
$733$	3379.50	0.170293	0.0851463	−	0.996368i	$-0.472864\pi$
$733$	3379.50	0.170293	0.0851463	+	0.996368i	$0.472864\pi$
$734$	19079.7	0.959462
$735$	−25374.9	−1.27342
$736$	8390.61	0.420220
$737$	−29939.6	−1.49639
$738$	−4146.54	−0.206824
$739$	−32194.7	−1.60257	−0.801285	−	0.598283i	$-0.795850\pi$
$739$	−32194.7	−1.60257	−0.801285	+	0.598283i	$0.795850\pi$
$740$	15876.0	0.788669
$741$	3011.11	0.149279
$742$	38931.6	1.92618
$743$	−8237.10	−0.406716	−0.203358	−	0.979104i	$-0.565185\pi$
$743$	−8237.10	−0.406716	−0.203358	+	0.979104i	$0.565185\pi$
$744$	−37636.0	−1.85457
$745$	−34725.6	−1.70772
$746$	−70146.4	−3.44268
$747$	−989.388	−0.0484603
$748$	62307.7	3.04571
$749$	−18420.7	−0.898636
$750$	−21001.1	−1.02247
$751$	−33321.1	−1.61904	−0.809522	−	0.587089i	$-0.800274\pi$
$751$	−33321.1	−1.61904	−0.809522	+	0.587089i	$0.800274\pi$
$752$	−59233.7	−2.87238
$753$	27735.7	1.34229
$754$	49633.0	2.39725
$755$	24995.2	1.20486
$756$	73730.7	3.54704
$757$	40005.7	1.92078	0.960391	−	0.278656i	$-0.0898890\pi$
$757$	40005.7	1.92078	0.960391	+	0.278656i	$0.0898890\pi$
$758$	−12117.4	−0.580640
$759$	6359.70	0.304140
$760$	−7999.51	−0.381806
$761$	−1344.71	−0.0640549	−0.0320274	−	0.999487i	$-0.510196\pi$
$761$	−1344.71	−0.0640549	−0.0320274	+	0.999487i	$0.510196\pi$
$762$	63281.1	3.00844
$763$	−29674.6	−1.40799
$764$	68872.1	3.26140
$765$	−1836.58	−0.0867996
$766$	6923.74	0.326586
$767$	−8641.02	−0.406791
$768$	23695.2	1.11332
$769$	−41407.9	−1.94175	−0.970875	−	0.239586i	$-0.922988\pi$
$769$	−41407.9	−1.94175	−0.970875	+	0.239586i	$0.922988\pi$
$770$	−90413.9	−4.23155
$771$	2345.22	0.109547
$772$	−17352.5	−0.808979
$773$	10328.8	0.480598	0.240299	−	0.970699i	$-0.422755\pi$
$773$	10328.8	0.480598	0.240299	+	0.970699i	$0.422755\pi$
$774$	0	0
$775$	−8674.82	−0.402076
$776$	17310.4	0.800780
$777$	−7991.30	−0.368966
$778$	52933.5	2.43928
$779$	−4143.16	−0.190557
$780$	80012.2	3.67294
$781$	−33324.3	−1.52681
$782$	9722.44	0.444596
$783$	22885.1	1.04450
$784$	54394.3	2.47787
$785$	9371.72	0.426103
$786$	−6790.11	−0.308136
$787$	−35833.2	−1.62302	−0.811509	−	0.584341i	$-0.801353\pi$
$787$	−35833.2	−1.62302	−0.811509	+	0.584341i	$0.801353\pi$
$788$	59931.5	2.70936
$789$	32232.3	1.45437
$790$	45568.8	2.05223
$791$	−46114.2	−2.07286
$792$	5224.46	0.234398
$793$	47764.2	2.13891
$794$	69300.6	3.09746
$795$	19480.1	0.869043
$796$	13433.3	0.598156
$797$	9312.94	0.413904	0.206952	−	0.978351i	$-0.433646\pi$
$797$	9312.94	0.413904	0.206952	+	0.978351i	$0.433646\pi$
$798$	6922.56	0.307088
$799$	−27566.5	−1.22056
$800$	−20803.2	−0.919379
$801$	2436.77	0.107489
$802$	−79554.2	−3.50269
$803$	7946.41	0.349219
$804$	60912.6	2.67192
$805$	−9946.95	−0.435508
$806$	−40742.3	−1.78050
$807$	4421.92	0.192886
$808$	48788.5	2.12422
$809$	29114.8	1.26529	0.632646	−	0.774441i	$-0.281969\pi$
$809$	29114.8	1.26529	0.632646	+	0.774441i	$0.281969\pi$
$810$	48594.6	2.10795
$811$	−41592.3	−1.80087	−0.900434	−	0.434993i	$-0.856751\pi$
$811$	−41592.3	−1.80087	−0.900434	+	0.434993i	$0.856751\pi$
$812$	80451.0	3.47694
$813$	−32178.1	−1.38811
$814$	−14694.1	−0.632714
$815$	−7056.46	−0.303285
$816$	−51550.5	−2.21156
$817$	0	0
$818$	−83553.7	−3.57138
$819$	3075.79	0.131229
$820$	−110093.	−4.68857
$821$	−39650.4	−1.68551	−0.842757	−	0.538294i	$-0.819069\pi$
$821$	−39650.4	−1.68551	−0.842757	+	0.538294i	$0.819069\pi$
$822$	−5460.15	−0.231684
$823$	−18879.4	−0.799629	−0.399815	−	0.916596i	$-0.630925\pi$
$823$	−18879.4	−0.799629	−0.399815	+	0.916596i	$0.630925\pi$
$824$	−19157.8	−0.809942
$825$	−15767.9	−0.665414
$826$	−19865.8	−0.836825
$827$	19740.1	0.830024	0.415012	−	0.909816i	$-0.363777\pi$
$827$	19740.1	0.830024	0.415012	+	0.909816i	$0.363777\pi$
$828$	988.152	0.0414742
$829$	−26241.2	−1.09939	−0.549695	−	0.835365i	$-0.685256\pi$
$829$	−26241.2	−1.09939	−0.549695	+	0.835365i	$0.685256\pi$
$830$	−37258.1	−1.55813
$831$	−6195.35	−0.258621
$832$	−25953.1	−1.08144
$833$	25314.3	1.05293
$834$	6613.15	0.274574
$835$	−22661.4	−0.939197
$836$	8974.58	0.371283
$837$	−18785.7	−0.775782
$838$	−46283.7	−1.90793
$839$	−9598.19	−0.394954	−0.197477	−	0.980308i	$-0.563275\pi$
$839$	−9598.19	−0.394954	−0.197477	+	0.980308i	$0.563275\pi$
$840$	106996.	4.39491
$841$	581.966	0.0238618
$842$	−3129.30	−0.128079
$843$	−7318.57	−0.299009
$844$	−50377.1	−2.05456
$845$	19948.8	0.812143
$846$	−3973.82	−0.161493
$847$	23566.3	0.956017
$848$	−41758.1	−1.69101
$849$	−8504.65	−0.343791
$850$	−24105.2	−0.972709
$851$	−1616.59	−0.0651186
$852$	67798.9	2.72623
$853$	−13278.1	−0.532981	−0.266490	−	0.963838i	$-0.585864\pi$
$853$	−13278.1	−0.532981	−0.266490	+	0.963838i	$0.585864\pi$
$854$	109810.	4.40003
$855$	−264.534	−0.0105812
$856$	40083.6	1.60050
$857$	−18835.6	−0.750773	−0.375386	−	0.926868i	$-0.622490\pi$
$857$	−18835.6	−0.750773	−0.375386	+	0.926868i	$0.622490\pi$
$858$	−74055.6	−2.94664
$859$	−8470.49	−0.336449	−0.168224	−	0.985749i	$-0.553803\pi$
$859$	−8470.49	−0.336449	−0.168224	+	0.985749i	$0.553803\pi$
$860$	0	0
$861$	55416.2	2.19347
$862$	87675.2	3.46430
$863$	−14654.0	−0.578017	−0.289008	−	0.957327i	$-0.593326\pi$
$863$	−14654.0	−0.578017	−0.289008	+	0.957327i	$0.593326\pi$
$864$	−45050.2	−1.77389
$865$	−4654.27	−0.182948
$866$	6089.17	0.238936
$867$	615.548	0.0241120
$868$	−66040.0	−2.58242
$869$	−29736.5	−1.16081
$870$	57095.3	2.22496
$871$	38355.0	1.49209
$872$	64572.2	2.50767
$873$	572.433	0.0221924
$874$	1400.39	0.0541977
$875$	−21434.7	−0.828143
$876$	−16167.1	−0.623557
$877$	31034.9	1.19495	0.597477	−	0.801886i	$-0.296170\pi$
$877$	31034.9	1.19495	0.597477	+	0.801886i	$0.296170\pi$
$878$	−2107.34	−0.0810015
$879$	−17829.3	−0.684151
$880$	96978.2	3.71493
$881$	−2331.10	−0.0891449	−0.0445725	−	0.999006i	$-0.514193\pi$
$881$	−2331.10	−0.0891449	−0.0445725	+	0.999006i	$0.514193\pi$
$882$	3649.16	0.139313
$883$	38070.6	1.45094	0.725469	−	0.688255i	$-0.241623\pi$
$883$	38070.6	1.45094	0.725469	+	0.688255i	$0.241623\pi$
$884$	−79821.1	−3.03696
$885$	−9940.20	−0.377555
$886$	−44387.2	−1.68309
$887$	19683.4	0.745102	0.372551	−	0.928012i	$-0.378483\pi$
$887$	19683.4	0.745102	0.372551	+	0.928012i	$0.378483\pi$
$888$	17389.1	0.657141
$889$	64587.7	2.43667
$890$	91763.2	3.45608
$891$	−31711.0	−1.19232
$892$	111926.	4.20129
$893$	−3970.58	−0.148791
$894$	−65390.3	−2.44629
$895$	−12270.3	−0.458268
$896$	6585.95	0.245559
$897$	−8147.28	−0.303266
$898$	−12902.1	−0.479451
$899$	−20498.0	−0.760451
$900$	−2449.97	−0.0907394
$901$	−19433.6	−0.718565
$902$	101897.	3.76143
$903$	0	0
$904$	100345.	3.69184
$905$	−53551.7	−1.96698
$906$	47067.4	1.72595
$907$	−3322.49	−0.121633	−0.0608167	−	0.998149i	$-0.519371\pi$
$907$	−3322.49	−0.121633	−0.0608167	+	0.998149i	$0.519371\pi$
$908$	−51454.4	−1.88059
$909$	1613.38	0.0588695
$910$	115827.	4.21938
$911$	36251.9	1.31842	0.659209	−	0.751959i	$-0.270891\pi$
$911$	36251.9	1.31842	0.659209	+	0.751959i	$0.270891\pi$
$912$	−7425.16	−0.269596
$913$	24313.3	0.881327
$914$	−23987.5	−0.868092
$915$	54945.5	1.98518
$916$	17395.4	0.627468
$917$	−6930.31	−0.249574
$918$	−52201.0	−1.87679
$919$	20127.9	0.722477	0.361239	−	0.932473i	$-0.382354\pi$
$919$	20127.9	0.722477	0.361239	+	0.932473i	$0.382354\pi$
$920$	21644.6	0.775654
$921$	27999.5	1.00175
$922$	58775.9	2.09944
$923$	42691.1	1.52242
$924$	−120038.	−4.27377
$925$	4008.07	0.142470
$926$	−2650.20	−0.0940508
$927$	−633.525	−0.0224463
$928$	−49156.4	−1.73883
$929$	−39220.1	−1.38511	−0.692557	−	0.721364i	$-0.743516\pi$
$929$	−39220.1	−1.38511	−0.692557	+	0.721364i	$0.743516\pi$
$930$	−46867.9	−1.65254
$931$	3646.18	0.128355
$932$	−51861.4	−1.82272
$933$	−13567.5	−0.476079
$934$	2962.26	0.103777
$935$	45132.2	1.57859
$936$	−6692.94	−0.233724
$937$	12798.4	0.446218	0.223109	−	0.974794i	$-0.428379\pi$
$937$	12798.4	0.446218	0.223109	+	0.974794i	$0.428379\pi$
$938$	88178.5	3.06943
$939$	−33711.3	−1.17159
$940$	−105507.	−3.66093
$941$	−45827.7	−1.58761	−0.793805	−	0.608172i	$-0.791903\pi$
$941$	−45827.7	−1.58761	−0.793805	+	0.608172i	$0.791903\pi$
$942$	17647.5	0.610388
$943$	11210.3	0.387124
$944$	21308.1	0.734659
$945$	53406.4	1.83842
$946$	0	0
$947$	8956.69	0.307342	0.153671	−	0.988122i	$-0.450890\pi$
$947$	8956.69	0.307342	0.153671	+	0.988122i	$0.450890\pi$
$948$	60499.5	2.07271
$949$	−10180.0	−0.348215
$950$	−3472.03	−0.118576
$951$	−29974.1	−1.02206
$952$	−106741.	−3.63391
$953$	−32038.7	−1.08902	−0.544510	−	0.838754i	$-0.683285\pi$
$953$	−32038.7	−1.08902	−0.544510	+	0.838754i	$0.683285\pi$
$954$	−2801.44	−0.0950732
$955$	49887.2	1.69038
$956$	42064.8	1.42309
$957$	−37258.3	−1.25851
$958$	17074.6	0.575840
$959$	−5572.89	−0.187652
$960$	−29855.2	−1.00372
$961$	−12964.8	−0.435191
$962$	18824.4	0.630896
$963$	1325.52	0.0443553
$964$	51640.8	1.72535
$965$	−12569.2	−0.419293
$966$	−18730.7	−0.623860
$967$	−33896.0	−1.12722	−0.563610	−	0.826041i	$-0.690588\pi$
$967$	−33896.0	−1.12722	−0.563610	+	0.826041i	$0.690588\pi$
$968$	−51280.3	−1.70270
$969$	−3455.56	−0.114560
$970$	21556.5	0.713545
$971$	−37446.8	−1.23762	−0.618808	−	0.785542i	$-0.712384\pi$
$971$	−37446.8	−1.23762	−0.618808	+	0.785542i	$0.712384\pi$
$972$	−10259.5	−0.338554
$973$	6749.69	0.222390
$974$	−17911.9	−0.589256
$975$	20199.9	0.663502
$976$	−117783.	−3.86284
$977$	23276.6	0.762214	0.381107	−	0.924531i	$-0.375543\pi$
$977$	23276.6	0.762214	0.381107	+	0.924531i	$0.375543\pi$
$978$	−13287.7	−0.434452
$979$	−59881.3	−1.95487
$980$	96887.5	3.15812
$981$	2135.33	0.0694962
$982$	−16228.9	−0.527377
$983$	20647.7	0.669947	0.334974	−	0.942228i	$-0.391273\pi$
$983$	20647.7	0.669947	0.334974	+	0.942228i	$0.391273\pi$
$984$	−120586.	−3.90664
$985$	43411.1	1.40426
$986$	−56958.9	−1.83970
$987$	53107.8	1.71271
$988$	−11497.1	−0.370216
$989$	0	0
$990$	6506.00	0.208863
$991$	−20706.8	−0.663745	−0.331873	−	0.943324i	$-0.607680\pi$
$991$	−20706.8	−0.663745	−0.331873	+	0.943324i	$0.607680\pi$
$992$	40351.1	1.29148
$993$	10589.2	0.338408
$994$	98147.2	3.13183
$995$	9730.36	0.310023
$996$	−49465.7	−1.57368
$997$	55958.1	1.77754	0.888771	−	0.458351i	$-0.151560\pi$
$997$	55958.1	1.77754	0.888771	+	0.458351i	$0.151560\pi$
$998$	−88341.7	−2.80201
$999$	8679.66	0.274887

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.j.1.2	yes	50
43.42	odd	2		1849.4.a.i.1.49	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.49	✓	50	43.42	odd	2
1849.4.a.j.1.2	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)

\(f(q)\)	\(=\)	\(q-5.20801 q^{2} -5.00842 q^{3} +19.1234 q^{4} +13.8519 q^{5} +26.0839 q^{6} +26.6225 q^{7} -57.9307 q^{8} -1.91570 q^{9} +O(q^{10})\)	\(q-5.20801 q^{2} -5.00842 q^{3} +19.1234 q^{4} +13.8519 q^{5} +26.0839 q^{6} +26.6225 q^{7} -57.9307 q^{8} -1.91570 q^{9} -72.1409 q^{10} +47.0765 q^{11} -95.7780 q^{12} -60.3088 q^{13} -138.650 q^{14} -69.3762 q^{15} +148.717 q^{16} +69.2105 q^{17} +9.97699 q^{18} +9.96884 q^{19} +264.895 q^{20} -133.337 q^{21} -245.175 q^{22} -26.9731 q^{23} +290.142 q^{24} +66.8755 q^{25} +314.089 q^{26} +144.822 q^{27} +509.112 q^{28} +158.022 q^{29} +361.312 q^{30} -129.716 q^{31} -311.073 q^{32} -235.779 q^{33} -360.449 q^{34} +368.773 q^{35} -36.6347 q^{36} +59.9333 q^{37} -51.9178 q^{38} +302.052 q^{39} -802.451 q^{40} -415.611 q^{41} +694.419 q^{42} +900.263 q^{44} -26.5361 q^{45} +140.476 q^{46} -398.299 q^{47} -744.836 q^{48} +365.758 q^{49} -348.289 q^{50} -346.636 q^{51} -1153.31 q^{52} -280.790 q^{53} -754.235 q^{54} +652.100 q^{55} -1542.26 q^{56} -49.9282 q^{57} -822.981 q^{58} +143.280 q^{59} -1326.71 q^{60} -791.993 q^{61} +675.562 q^{62} -51.0008 q^{63} +430.337 q^{64} -835.392 q^{65} +1227.94 q^{66} -635.977 q^{67} +1323.54 q^{68} +135.093 q^{69} -1920.57 q^{70} -707.876 q^{71} +110.978 q^{72} +168.798 q^{73} -312.133 q^{74} -334.941 q^{75} +190.638 q^{76} +1253.30 q^{77} -1573.09 q^{78} -631.664 q^{79} +2060.01 q^{80} -673.606 q^{81} +2164.50 q^{82} +516.463 q^{83} -2549.85 q^{84} +958.698 q^{85} -791.441 q^{87} -2727.18 q^{88} -1272.00 q^{89} +138.200 q^{90} -1605.57 q^{91} -515.817 q^{92} +649.672 q^{93} +2074.34 q^{94} +138.088 q^{95} +1557.98 q^{96} -298.811 q^{97} -1904.87 q^{98} -90.1846 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \)	\( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

Downloads

Learn more about