LMFDB - Embedded newform 1849.4.a.i.1.48

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			1.48
Character	$\chi$	$=$	1849.1

$q$-expansion

$f(q)$	$=$	$q+4.80413 q^{2} -2.56623 q^{3} +15.0797 q^{4} -6.34124 q^{5} -12.3285 q^{6} +6.17767 q^{7} +34.0117 q^{8} -20.4145 q^{9} +O(q^{10})$	\(q+4.80413 q^{2} -2.56623 q^{3} +15.0797 q^{4} -6.34124 q^{5} -12.3285 q^{6} +6.17767 q^{7} +34.0117 q^{8} -20.4145 q^{9} -30.4642 q^{10} +33.6300 q^{11} -38.6979 q^{12} -34.3998 q^{13} +29.6784 q^{14} +16.2731 q^{15} +42.7593 q^{16} +105.843 q^{17} -98.0737 q^{18} +14.3178 q^{19} -95.6239 q^{20} -15.8533 q^{21} +161.563 q^{22} -160.517 q^{23} -87.2819 q^{24} -84.7886 q^{25} -165.261 q^{26} +121.676 q^{27} +93.1573 q^{28} -42.2224 q^{29} +78.1781 q^{30} +0.100791 q^{31} -66.6725 q^{32} -86.3024 q^{33} +508.483 q^{34} -39.1741 q^{35} -307.843 q^{36} -369.747 q^{37} +68.7844 q^{38} +88.2778 q^{39} -215.676 q^{40} +450.083 q^{41} -76.1615 q^{42} +507.130 q^{44} +129.453 q^{45} -771.146 q^{46} -288.931 q^{47} -109.730 q^{48} -304.836 q^{49} -407.336 q^{50} -271.617 q^{51} -518.738 q^{52} +439.646 q^{53} +584.550 q^{54} -213.256 q^{55} +210.113 q^{56} -36.7427 q^{57} -202.842 q^{58} -16.1085 q^{59} +245.393 q^{60} -764.542 q^{61} +0.484211 q^{62} -126.114 q^{63} -662.378 q^{64} +218.137 q^{65} -414.608 q^{66} -415.572 q^{67} +1596.08 q^{68} +411.925 q^{69} -188.198 q^{70} +998.578 q^{71} -694.330 q^{72} +160.169 q^{73} -1776.31 q^{74} +217.587 q^{75} +215.907 q^{76} +207.755 q^{77} +424.098 q^{78} -1319.53 q^{79} -271.147 q^{80} +238.940 q^{81} +2162.26 q^{82} -374.027 q^{83} -239.063 q^{84} -671.175 q^{85} +108.352 q^{87} +1143.81 q^{88} -706.315 q^{89} +621.909 q^{90} -212.511 q^{91} -2420.55 q^{92} -0.258652 q^{93} -1388.06 q^{94} -90.7924 q^{95} +171.097 q^{96} -1260.72 q^{97} -1464.47 q^{98} -686.539 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} + O(q^{10}) $	\( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} - 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} - 386q^{18} + 12q^{19} + 108q^{20} - 408q^{21} + 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} + 1493q^{26} + 10q^{27} - 242q^{28} + 208q^{29} - 48q^{30} - 932q^{31} - 1124q^{32} + 254q^{33} + 765q^{34} - 1452q^{35} + 747q^{36} - 90q^{37} - 1213q^{38} - 1610q^{39} - 1693q^{40} - 1354q^{41} - 16q^{42} - 2704q^{44} + 4508q^{45} + 233q^{46} - 3484q^{47} - 376q^{48} + 1324q^{49} - 408q^{50} + 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} - 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} + 1172q^{61} - 1546q^{62} - 3686q^{63} + 606q^{64} + 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} + 136q^{69} - 1310q^{70} + 162q^{71} - 5814q^{72} + 746q^{73} - 4332q^{74} + 236q^{75} + 1338q^{76} + 2024q^{77} - 2782q^{78} - 2656q^{79} - 5713q^{80} - 86q^{81} - 4168q^{82} - 3514q^{83} - 4269q^{84} - 7558q^{85} - 10278q^{87} + 11692q^{88} + 2640q^{89} - 8286q^{90} - 5946q^{91} - 4271q^{92} - 2q^{93} + 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} - 2826q^{98} - 8174q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	4.80413	1.69852	0.849258	−	0.527977i	$-0.177049\pi$
$2$	4.80413	1.69852	0.849258	+	0.527977i	$0.177049\pi$
$3$	−2.56623	−0.493871	−0.246936	−	0.969032i	$-0.579424\pi$
$3$	−2.56623	−0.493871	−0.246936	+	0.969032i	$0.579424\pi$
$4$	15.0797	1.88496
$5$	−6.34124	−0.567178	−0.283589	−	0.958946i	$-0.591525\pi$
$5$	−6.34124	−0.567178	−0.283589	+	0.958946i	$0.591525\pi$
$6$	−12.3285	−0.838849
$7$	6.17767	0.333563	0.166782	−	0.985994i	$-0.446663\pi$
$7$	6.17767	0.333563	0.166782	+	0.985994i	$0.446663\pi$
$8$	34.0117	1.50312
$9$	−20.4145	−0.756091
$10$	−30.4642	−0.963361
$11$	33.6300	0.921803	0.460902	−	0.887451i	$-0.347526\pi$
$11$	33.6300	0.921803	0.460902	+	0.887451i	$0.347526\pi$
$12$	−38.6979	−0.930928
$13$	−34.3998	−0.733906	−0.366953	−	0.930239i	$-0.619599\pi$
$13$	−34.3998	−0.733906	−0.366953	+	0.930239i	$0.619599\pi$
$14$	29.6784	0.566562
$15$	16.2731	0.280113
$16$	42.7593	0.668114
$17$	105.843	1.51004	0.755020	−	0.655702i	$-0.227627\pi$
$17$	105.843	1.51004	0.755020	+	0.655702i	$0.227627\pi$
$18$	−98.0737	−1.28423
$19$	14.3178	0.172880	0.0864400	−	0.996257i	$-0.472451\pi$
$19$	14.3178	0.172880	0.0864400	+	0.996257i	$0.472451\pi$
$20$	−95.6239	−1.06911
$21$	−15.8533	−0.164737
$22$	161.563	1.56570
$23$	−160.517	−1.45522	−0.727612	−	0.685988i	$-0.759370\pi$
$23$	−160.517	−1.45522	−0.727612	+	0.685988i	$0.759370\pi$
$24$	−87.2819	−0.742348
$25$	−84.7886	−0.678309
$26$	−165.261	−1.24655
$27$	121.676	0.867283
$28$	93.1573	0.628753
$29$	−42.2224	−0.270362	−0.135181	−	0.990821i	$-0.543162\pi$
$29$	−42.2224	−0.270362	−0.135181	+	0.990821i	$0.543162\pi$
$30$	78.1781	0.475777
$31$	0.100791	0.000583952 0	0.000291976	−	1.00000i	$-0.499907\pi$
$31$	0.100791	0.000583952 0	0.000291976		1.00000i	$0.499907\pi$
$32$	−66.6725	−0.368317
$33$	−86.3024	−0.455252
$34$	508.483	2.56483
$35$	−39.1741	−0.189190
$36$	−307.843	−1.42520
$37$	−369.747	−1.64287	−0.821433	−	0.570305i	$-0.806825\pi$
$37$	−369.747	−1.64287	−0.821433	+	0.570305i	$0.806825\pi$
$38$	68.7844	0.293640
$39$	88.2778	0.362455
$40$	−215.676	−0.852536
$41$	450.083	1.71442	0.857210	−	0.514967i	$-0.172196\pi$
$41$	450.083	1.71442	0.857210	+	0.514967i	$0.172196\pi$
$42$	−76.1615	−0.279809
$43$	0	0
$43$	0	0
$44$	507.130	1.73756
$45$	129.453	0.428838
$46$	−771.146	−2.47172
$47$	−288.931	−0.896700	−0.448350	−	0.893858i	$-0.647988\pi$
$47$	−288.931	−0.896700	−0.448350	+	0.893858i	$0.647988\pi$
$48$	−109.730	−0.329962
$49$	−304.836	−0.888736
$50$	−407.336	−1.15212
$51$	−271.617	−0.745765
$52$	−518.738	−1.38338
$53$	439.646	1.13943	0.569717	−	0.821841i	$-0.307053\pi$
$53$	439.646	1.13943	0.569717	+	0.821841i	$0.307053\pi$
$54$	584.550	1.47310
$55$	−213.256	−0.522826
$56$	210.113	0.501385
$57$	−36.7427	−0.0853805
$58$	−202.842	−0.459215
$59$	−16.1085	−0.0355450	−0.0177725	−	0.999842i	$-0.505657\pi$
$59$	−16.1085	−0.0355450	−0.0177725	+	0.999842i	$0.505657\pi$
$60$	245.393	0.528002
$61$	−764.542	−1.60475	−0.802374	−	0.596822i	$-0.796430\pi$
$61$	−764.542	−1.60475	−0.802374	+	0.596822i	$0.796430\pi$
$62$	0.484211	0.000991853 0
$63$	−126.114	−0.252204
$64$	−662.378	−1.29371
$65$	218.137	0.416256
$66$	−414.608	−0.773254
$67$	−415.572	−0.757764	−0.378882	−	0.925445i	$-0.623691\pi$
$67$	−415.572	−0.757764	−0.378882	+	0.925445i	$0.623691\pi$
$68$	1596.08	2.84636
$69$	411.925	0.718694
$70$	−188.198	−0.321342
$71$	998.578	1.66915	0.834573	−	0.550897i	$-0.185714\pi$
$71$	998.578	1.66915	0.834573	+	0.550897i	$0.185714\pi$
$72$	−694.330	−1.13649
$73$	160.169	0.256799	0.128400	−	0.991723i	$-0.459016\pi$
$73$	160.169	0.256799	0.128400	+	0.991723i	$0.459016\pi$
$74$	−1776.31	−2.79044
$75$	217.587	0.334998
$76$	215.907	0.325872
$77$	207.755	0.307479
$78$	424.098	0.615637
$79$	−1319.53	−1.87922	−0.939609	−	0.342251i	$-0.888811\pi$
$79$	−1319.53	−1.87922	−0.939609	+	0.342251i	$0.888811\pi$
$80$	−271.147	−0.378939
$81$	238.940	0.327765
$82$	2162.26	2.91197
$83$	−374.027	−0.494636	−0.247318	−	0.968934i	$-0.579549\pi$
$83$	−374.027	−0.494636	−0.247318	+	0.968934i	$0.579549\pi$
$84$	−239.063	−0.310523
$85$	−671.175	−0.856461
$86$	0	0
$87$	108.352	0.133524
$88$	1143.81	1.38558
$89$	−706.315	−0.841227	−0.420614	−	0.907240i	$-0.638185\pi$
$89$	−706.315	−0.841227	−0.420614	+	0.907240i	$0.638185\pi$
$90$	621.909	0.728389
$91$	−212.511	−0.244804
$92$	−2420.55	−2.74304
$93$	−0.258652	−0.000288397 0
$94$	−1388.06	−1.52306
$95$	−90.7924	−0.0980538
$96$	171.097	0.181901
$97$	−1260.72	−1.31966	−0.659829	−	0.751416i	$-0.729371\pi$
$97$	−1260.72	−1.31966	−0.659829	+	0.751416i	$0.729371\pi$
$98$	−1464.47	−1.50953
$99$	−686.539	−0.696967
$100$	−1278.59	−1.27859
$101$	−617.433	−0.608286	−0.304143	−	0.952626i	$-0.598370\pi$
$101$	−617.433	−0.608286	−0.304143	+	0.952626i	$0.598370\pi$
$102$	−1304.89	−1.26669
$103$	−217.291	−0.207867	−0.103934	−	0.994584i	$-0.533143\pi$
$103$	−217.291	−0.207867	−0.103934	+	0.994584i	$0.533143\pi$
$104$	−1170.00	−1.10315
$105$	100.530	0.0934353
$106$	2112.12	1.93535
$107$	322.466	0.291346	0.145673	−	0.989333i	$-0.453465\pi$
$107$	322.466	0.291346	0.145673	+	0.989333i	$0.453465\pi$
$108$	1834.84	1.63479
$109$	74.8798	0.0657999	0.0328999	−	0.999459i	$-0.489526\pi$
$109$	74.8798	0.0657999	0.0328999	+	0.999459i	$0.489526\pi$
$110$	−1024.51	−0.888030
$111$	948.856	0.811365
$112$	264.153	0.222858
$113$	−1937.66	−1.61309	−0.806545	−	0.591172i	$-0.798665\pi$
$113$	−1937.66	−1.61309	−0.806545	+	0.591172i	$0.798665\pi$
$114$	−176.517	−0.145020
$115$	1017.88	0.825371
$116$	−636.700	−0.509622
$117$	702.253	0.554900
$118$	−77.3876	−0.0603737
$119$	653.863	0.503693
$120$	553.476	0.421043
$121$	−200.021	−0.150279
$122$	−3672.96	−2.72569
$123$	−1155.02	−0.846703
$124$	1.51989	0.00110073
$125$	1330.32	0.951900
$126$	−605.867	−0.428373
$127$	−85.5015	−0.0597404	−0.0298702	−	0.999554i	$-0.509509\pi$
$127$	−85.5015	−0.0597404	−0.0298702	+	0.999554i	$0.509509\pi$
$128$	−2648.77	−1.82906
$129$	0	0
$130$	1047.96	0.707017
$131$	−1370.38	−0.913975	−0.456988	−	0.889473i	$-0.651072\pi$
$131$	−1370.38	−0.913975	−0.456988	+	0.889473i	$0.651072\pi$
$132$	−1301.41	−0.858132
$133$	88.4505	0.0576664
$134$	−1996.46	−1.28708
$135$	−771.580	−0.491904
$136$	3599.90	2.26977
$137$	−355.929	−0.221964	−0.110982	−	0.993822i	$-0.535400\pi$
$137$	−355.929	−0.221964	−0.110982	+	0.993822i	$0.535400\pi$
$138$	1978.94	1.22071
$139$	−2301.37	−1.40431	−0.702156	−	0.712024i	$-0.747779\pi$
$139$	−2301.37	−1.40431	−0.702156	+	0.712024i	$0.747779\pi$
$140$	−590.733	−0.356615
$141$	741.463	0.442855
$142$	4797.30	2.83507
$143$	−1156.87	−0.676517
$144$	−872.907	−0.505155
$145$	267.743	0.153343
$146$	769.472	0.436178
$147$	782.281	0.438921
$148$	−5575.67	−3.09674
$149$	1231.68	0.677202	0.338601	−	0.940930i	$-0.390046\pi$
$149$	1231.68	0.677202	0.338601	+	0.940930i	$0.390046\pi$
$150$	1045.32	0.568999
$151$	1504.88	0.811032	0.405516	−	0.914088i	$-0.367092\pi$
$151$	1504.88	0.811032	0.405516	+	0.914088i	$0.367092\pi$
$152$	486.972	0.259859
$153$	−2160.72	−1.14173
$154$	998.084	0.522259
$155$	−0.639138	−0.000331205 0
$156$	1331.20	0.683214
$157$	−1740.89	−0.884955	−0.442477	−	0.896780i	$-0.645900\pi$
$157$	−1740.89	−0.884955	−0.442477	+	0.896780i	$0.645900\pi$
$158$	−6339.17	−3.19188
$159$	−1128.23	−0.562734
$160$	422.786	0.208901
$161$	−991.623	−0.485409
$162$	1147.90	0.556714
$163$	1171.60	0.562986	0.281493	−	0.959563i	$-0.409170\pi$
$163$	1171.60	0.562986	0.281493	+	0.959563i	$0.409170\pi$
$164$	6787.11	3.23161
$165$	547.265	0.258209
$166$	−1796.88	−0.840148
$167$	1555.44	0.720741	0.360370	−	0.932809i	$-0.382650\pi$
$167$	1555.44	0.720741	0.360370	+	0.932809i	$0.382650\pi$
$168$	−539.199	−0.247620
$169$	−1013.65	−0.461381
$170$	−3224.41	−1.45471
$171$	−292.289	−0.130713
$172$	0	0
$173$	2048.52	0.900268	0.450134	−	0.892961i	$-0.351376\pi$
$173$	2048.52	0.900268	0.450134	+	0.892961i	$0.351376\pi$
$174$	520.539	0.226793
$175$	−523.797	−0.226259
$176$	1438.00	0.615869
$177$	41.3382	0.0175547
$178$	−3393.23	−1.42884
$179$	4392.22	1.83402	0.917011	−	0.398863i	$-0.130595\pi$
$179$	4392.22	1.83402	0.917011	+	0.398863i	$0.130595\pi$
$180$	1952.11	0.808343
$181$	2563.35	1.05266	0.526332	−	0.850279i	$-0.323567\pi$
$181$	2563.35	1.05266	0.526332	+	0.850279i	$0.323567\pi$
$182$	−1020.93	−0.415804
$183$	1961.99	0.792539
$184$	−5459.47	−2.18738
$185$	2344.66	0.931797
$186$	−1.24260	−0.000489848 0
$187$	3559.50	1.39196
$188$	−4356.98	−1.69024
$189$	751.677	0.289294
$190$	−436.179	−0.166546
$191$	1139.09	0.431528	0.215764	−	0.976446i	$-0.430776\pi$
$191$	1139.09	0.431528	0.215764	+	0.976446i	$0.430776\pi$
$192$	1699.81	0.638924
$193$	2118.90	0.790267	0.395133	−	0.918624i	$-0.370698\pi$
$193$	2118.90	0.790267	0.395133	+	0.918624i	$0.370698\pi$
$194$	−6056.67	−2.24146
$195$	−559.791	−0.205577
$196$	−4596.83	−1.67523
$197$	−4044.27	−1.46265	−0.731326	−	0.682028i	$-0.761098\pi$
$197$	−4044.27	−1.46265	−0.731326	+	0.682028i	$0.761098\pi$
$198$	−3298.22	−1.18381
$199$	−1717.52	−0.611817	−0.305908	−	0.952061i	$-0.598960\pi$
$199$	−1717.52	−0.611817	−0.305908	+	0.952061i	$0.598960\pi$
$200$	−2883.81	−1.01958
$201$	1066.45	0.374238
$202$	−2966.23	−1.03318
$203$	−260.836	−0.0901828
$204$	−4095.90	−1.40574
$205$	−2854.09	−0.972381
$206$	−1043.90	−0.353066
$207$	3276.87	1.10028
$208$	−1470.91	−0.490333
$209$	481.507	0.159361
$210$	482.959	0.158702
$211$	−3943.80	−1.28674	−0.643370	−	0.765555i	$-0.722464\pi$
$211$	−3943.80	−1.28674	−0.643370	+	0.765555i	$0.722464\pi$
$212$	6629.71	2.14779
$213$	−2562.58	−0.824344
$214$	1549.17	0.494856
$215$	0	0
$216$	4138.42	1.30363
$217$	0.622651	0.000194785 0
$218$	359.732	0.111762
$219$	−411.030	−0.126826
$220$	−3215.83	−0.985507
$221$	−3640.97	−1.10823
$222$	4558.43	1.37812
$223$	1798.16	0.539971	0.269985	−	0.962864i	$-0.412981\pi$
$223$	1798.16	0.539971	0.269985	+	0.962864i	$0.412981\pi$
$224$	−411.881	−0.122857
$225$	1730.91	0.512863
$226$	−9308.75	−2.73986
$227$	6340.20	1.85381	0.926903	−	0.375301i	$-0.122461\pi$
$227$	6340.20	1.85381	0.926903	+	0.375301i	$0.122461\pi$
$228$	−554.068	−0.160939
$229$	−1568.74	−0.452686	−0.226343	−	0.974048i	$-0.572677\pi$
$229$	−1568.74	−0.452686	−0.226343	+	0.974048i	$0.572677\pi$
$230$	4890.03	1.40191
$231$	−533.148	−0.151855
$232$	−1436.06	−0.406387
$233$	2409.61	0.677505	0.338752	−	0.940876i	$-0.389995\pi$
$233$	2409.61	0.677505	0.338752	+	0.940876i	$0.389995\pi$
$234$	3373.72	0.942507
$235$	1832.18	0.508588
$236$	−242.912	−0.0670009
$237$	3386.21	0.928092
$238$	3141.24	0.855531
$239$	2111.13	0.571371	0.285685	−	0.958324i	$-0.407779\pi$
$239$	2111.13	0.571371	0.285685	+	0.958324i	$0.407779\pi$
$240$	695.826	0.187147
$241$	−3837.49	−1.02570	−0.512852	−	0.858477i	$-0.671411\pi$
$241$	−3837.49	−1.02570	−0.512852	+	0.858477i	$0.671411\pi$
$242$	−960.929	−0.255252
$243$	−3898.44	−1.02916
$244$	−11529.0	−3.02488
$245$	1933.04	0.504071
$246$	−5548.86	−1.43814
$247$	−492.528	−0.126878
$248$	3.42806	0.000877750 0
$249$	959.840	0.244287
$250$	6391.04	1.61682
$251$	−7394.81	−1.85959	−0.929793	−	0.368083i	$-0.880014\pi$
$251$	−7394.81	−1.85959	−0.929793	+	0.368083i	$0.880014\pi$
$252$	−1901.76	−0.475394
$253$	−5398.20	−1.34143
$254$	−410.760	−0.101470
$255$	1722.39	0.422982
$256$	−7426.01	−1.81299
$257$	2496.57	0.605959	0.302980	−	0.952997i	$-0.402019\pi$
$257$	2496.57	0.605959	0.302980	+	0.952997i	$0.402019\pi$
$258$	0	0
$259$	−2284.18	−0.547999
$260$	3289.44	0.784625
$261$	861.947	0.204418
$262$	−6583.49	−1.55240
$263$	1815.08	0.425562	0.212781	−	0.977100i	$-0.431748\pi$
$263$	1815.08	0.425562	0.212781	+	0.977100i	$0.431748\pi$
$264$	−2935.29	−0.684298
$265$	−2787.90	−0.646261
$266$	424.928	0.0979473
$267$	1812.57	0.415458
$268$	−6266.69	−1.42836
$269$	−673.864	−0.152737	−0.0763684	−	0.997080i	$-0.524333\pi$
$269$	−673.864	−0.152737	−0.0763684	+	0.997080i	$0.524333\pi$
$270$	−3706.77	−0.835507
$271$	2582.80	0.578944	0.289472	−	0.957186i	$-0.406520\pi$
$271$	2582.80	0.578944	0.289472	+	0.957186i	$0.406520\pi$
$272$	4525.76	1.00888
$273$	545.351	0.120902
$274$	−1709.93	−0.377009
$275$	−2851.44	−0.625267
$276$	6211.69	1.35471
$277$	−2042.43	−0.443024	−0.221512	−	0.975158i	$-0.571099\pi$
$277$	−2042.43	−0.443024	−0.221512	+	0.975158i	$0.571099\pi$
$278$	−11056.1	−2.38525
$279$	−2.05759	−0.000441521 0
$280$	−1332.38	−0.284375
$281$	5529.75	1.17394	0.586970	−	0.809608i	$-0.300321\pi$
$281$	5529.75	1.17394	0.586970	+	0.809608i	$0.300321\pi$
$282$	3562.09	0.752196
$283$	213.594	0.0448653	0.0224326	−	0.999748i	$-0.492859\pi$
$283$	213.594	0.0448653	0.0224326	+	0.999748i	$0.492859\pi$
$284$	15058.2	3.14627
$285$	232.994	0.0484260
$286$	−5557.73	−1.14908
$287$	2780.47	0.571867
$288$	1361.08	0.278481
$289$	6289.71	1.28022
$290$	1286.27	0.260456
$291$	3235.30	0.651741
$292$	2415.29	0.484056
$293$	−6029.67	−1.20224	−0.601121	−	0.799158i	$-0.705279\pi$
$293$	−6029.67	−1.20224	−0.601121	+	0.799158i	$0.705279\pi$
$294$	3758.18	0.745515
$295$	102.148	0.0201603
$296$	−12575.7	−2.46942
$297$	4091.98	0.799464
$298$	5917.15	1.15024
$299$	5521.76	1.06800
$300$	3281.15	0.631457
$301$	0	0
$302$	7229.66	1.37755
$303$	1584.47	0.300415
$304$	612.217	0.115504
$305$	4848.15	0.910177
$306$	−10380.4	−1.93924
$307$	9957.66	1.85118	0.925592	−	0.378523i	$-0.123568\pi$
$307$	9957.66	1.85118	0.925592	+	0.378523i	$0.123568\pi$
$308$	3132.88	0.579586
$309$	557.619	0.102660
$310$	−3.07050	−0.000562557 0
$311$	9950.53	1.81429	0.907143	−	0.420824i	$-0.138259\pi$
$311$	9950.53	1.81429	0.907143	+	0.420824i	$0.138259\pi$
$312$	3002.48	0.544814
$313$	−1906.47	−0.344281	−0.172140	−	0.985072i	$-0.555068\pi$
$313$	−1906.47	−0.344281	−0.172140	+	0.985072i	$0.555068\pi$
$314$	−8363.45	−1.50311
$315$	799.718	0.143045
$316$	−19898.0	−3.54225
$317$	−2985.80	−0.529019	−0.264510	−	0.964383i	$-0.585210\pi$
$317$	−2985.80	−0.529019	−0.264510	+	0.964383i	$0.585210\pi$
$318$	−5420.18	−0.955812
$319$	−1419.94	−0.249221
$320$	4200.30	0.733762
$321$	−827.523	−0.143887
$322$	−4763.89	−0.824476
$323$	1515.43	0.261056
$324$	3603.14	0.617823
$325$	2916.71	0.497815
$326$	5628.52	0.956242
$327$	−192.159	−0.0324967
$328$	15308.1	2.57698
$329$	−1784.92	−0.299106
$330$	2629.13	0.438572
$331$	842.438	0.139893	0.0699465	−	0.997551i	$-0.477717\pi$
$331$	842.438	0.139893	0.0699465	+	0.997551i	$0.477717\pi$
$332$	−5640.21	−0.932370
$333$	7548.18	1.24216
$334$	7472.55	1.22419
$335$	2635.24	0.429787
$336$	−677.877	−0.110063
$337$	−3058.48	−0.494380	−0.247190	−	0.968967i	$-0.579507\pi$
$337$	−3058.48	−0.494380	−0.247190	+	0.968967i	$0.579507\pi$
$338$	−4869.73	−0.783664
$339$	4972.47	0.796659
$340$	−10121.1	−1.61439
$341$	3.38959	0.000538289 0
$342$	−1404.20	−0.222018
$343$	−4002.12	−0.630012
$344$	0	0
$345$	−2612.11	−0.407627
$346$	9841.37	1.52912
$347$	11415.0	1.76597	0.882985	−	0.469402i	$-0.155530\pi$
$347$	11415.0	1.76597	0.882985	+	0.469402i	$0.155530\pi$
$348$	1633.92	0.251688
$349$	−2729.17	−0.418594	−0.209297	−	0.977852i	$-0.567118\pi$
$349$	−2729.17	−0.418594	−0.209297	+	0.977852i	$0.567118\pi$
$350$	−2516.39	−0.384304
$351$	−4185.64	−0.636505
$352$	−2242.20	−0.339516
$353$	−11443.4	−1.72541	−0.862703	−	0.505710i	$-0.831230\pi$
$353$	−11443.4	−1.72541	−0.862703	+	0.505710i	$0.831230\pi$
$354$	198.594	0.0298169
$355$	−6332.23	−0.946703
$356$	−10651.0	−1.58568
$357$	−1677.96	−0.248760
$358$	21100.8	3.11512
$359$	−3938.99	−0.579086	−0.289543	−	0.957165i	$-0.593503\pi$
$359$	−3938.99	−0.579086	−0.289543	+	0.957165i	$0.593503\pi$
$360$	4402.92	0.644595
$361$	−6654.00	−0.970112
$362$	12314.7	1.78797
$363$	513.301	0.0742185
$364$	−3204.59	−0.461446
$365$	−1015.67	−0.145651
$366$	9425.67	1.34614
$367$	−7300.33	−1.03835	−0.519175	−	0.854668i	$-0.673761\pi$
$367$	−7300.33	−1.03835	−0.519175	+	0.854668i	$0.673761\pi$
$368$	−6863.60	−0.972256
$369$	−9188.21	−1.29626
$370$	11264.0	1.58267
$371$	2715.99	0.380073
$372$	−3.90039	−0.000543618 0
$373$	9652.92	1.33997	0.669985	−	0.742374i	$-0.266300\pi$
$373$	9652.92	1.33997	0.669985	+	0.742374i	$0.266300\pi$
$374$	17100.3	2.36427
$375$	−3413.91	−0.470116
$376$	−9827.03	−1.34785
$377$	1452.44	0.198421
$378$	3611.16	0.491370
$379$	−1436.71	−0.194720	−0.0973601	−	0.995249i	$-0.531040\pi$
$379$	−1436.71	−0.194720	−0.0973601	+	0.995249i	$0.531040\pi$
$380$	−1369.12	−0.184827
$381$	219.417	0.0295041
$382$	5472.35	0.732958
$383$	9742.88	1.29984	0.649919	−	0.760003i	$-0.274803\pi$
$383$	9742.88	1.29984	0.649919	+	0.760003i	$0.274803\pi$
$384$	6797.35	0.903323
$385$	−1317.43	−0.174396
$386$	10179.5	1.34228
$387$	0	0
$388$	−19011.3	−2.48750
$389$	853.522	0.111248	0.0556238	−	0.998452i	$-0.482285\pi$
$389$	853.522	0.111248	0.0556238	+	0.998452i	$0.482285\pi$
$390$	−2689.31	−0.349176
$391$	−16989.6	−2.19745
$392$	−10368.0	−1.33588
$393$	3516.71	0.451386
$394$	−19429.2	−2.48434
$395$	8367.43	1.06585
$396$	−10352.8	−1.31375
$397$	7093.70	0.896782	0.448391	−	0.893837i	$-0.351997\pi$
$397$	7093.70	0.896782	0.448391	+	0.893837i	$0.351997\pi$
$398$	−8251.18	−1.03918
$399$	−226.984	−0.0284798
$400$	−3625.50	−0.453188
$401$	−2816.61	−0.350761	−0.175380	−	0.984501i	$-0.556115\pi$
$401$	−2816.61	−0.350761	−0.175380	+	0.984501i	$0.556115\pi$
$402$	5123.38	0.635650
$403$	−3.46718	−0.000428567 0
$404$	−9310.69	−1.14659
$405$	−1515.18	−0.185901
$406$	−1253.09	−0.153177
$407$	−12434.6	−1.51440
$408$	−9238.17	−1.12097
$409$	11950.0	1.44472	0.722358	−	0.691520i	$-0.243058\pi$
$409$	11950.0	1.44472	0.722358	+	0.691520i	$0.243058\pi$
$410$	−13711.4	−1.65161
$411$	913.395	0.109622
$412$	−3276.68	−0.391822
$413$	−99.5133	−0.0118565
$414$	15742.5	1.86885
$415$	2371.80	0.280547
$416$	2293.52	0.270310
$417$	5905.84	0.693549
$418$	2313.22	0.270678
$419$	−2847.25	−0.331974	−0.165987	−	0.986128i	$-0.553081\pi$
$419$	−2847.25	−0.331974	−0.165987	+	0.986128i	$0.553081\pi$
$420$	1515.96	0.176122
$421$	10205.8	1.18147	0.590737	−	0.806864i	$-0.298837\pi$
$421$	10205.8	1.18147	0.590737	+	0.806864i	$0.298837\pi$
$422$	−18946.5	−2.18555
$423$	5898.37	0.677987
$424$	14953.1	1.71270
$425$	−8974.27	−1.02427
$426$	−12311.0	−1.40016
$427$	−4723.09	−0.535284
$428$	4862.69	0.549175
$429$	2968.78	0.334113
$430$	0	0
$431$	−13829.3	−1.54555	−0.772775	−	0.634680i	$-0.781132\pi$
$431$	−13829.3	−1.54555	−0.772775	+	0.634680i	$0.781132\pi$
$432$	5202.80	0.579444
$433$	−8717.43	−0.967512	−0.483756	−	0.875203i	$-0.660728\pi$
$433$	−8717.43	−0.967512	−0.483756	+	0.875203i	$0.660728\pi$
$434$	2.99130	0.000330846 0
$435$	−687.089	−0.0757320
$436$	1129.16	0.124030
$437$	−2298.25	−0.251579
$438$	−1974.64	−0.215416
$439$	−14904.6	−1.62041	−0.810203	−	0.586149i	$-0.800643\pi$
$439$	−14904.6	−1.62041	−0.810203	+	0.586149i	$0.800643\pi$
$440$	−7253.21	−0.785870
$441$	6223.07	0.671965
$442$	−17491.7	−1.88234
$443$	16289.2	1.74700	0.873502	−	0.486820i	$-0.161843\pi$
$443$	16289.2	1.74700	0.873502	+	0.486820i	$0.161843\pi$
$444$	14308.4	1.52939
$445$	4478.91	0.477126
$446$	8638.58	0.917149
$447$	−3160.77	−0.334451
$448$	−4091.95	−0.431532
$449$	12947.9	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$449$	12947.9	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$450$	8315.54	0.871107
$451$	15136.3	1.58036
$452$	−29219.2	−3.04061
$453$	−3861.88	−0.400545
$454$	30459.2	3.14872
$455$	1347.58	0.138847
$456$	−1249.68	−0.128337
$457$	1521.06	0.155694	0.0778469	−	0.996965i	$-0.475195\pi$
$457$	1521.06	0.155694	0.0778469	+	0.996965i	$0.475195\pi$
$458$	−7536.42	−0.768895
$459$	12878.6	1.30963
$460$	15349.3	1.55579
$461$	−4500.11	−0.454644	−0.227322	−	0.973820i	$-0.572997\pi$
$461$	−4500.11	−0.454644	−0.227322	+	0.973820i	$0.572997\pi$
$462$	−2561.31	−0.257929
$463$	−5096.27	−0.511542	−0.255771	−	0.966737i	$-0.582329\pi$
$463$	−5096.27	−0.511542	−0.255771	+	0.966737i	$0.582329\pi$
$464$	−1805.40	−0.180633
$465$	1.64018	0.000163573 0
$466$	11576.1	1.15075
$467$	−5931.03	−0.587699	−0.293850	−	0.955852i	$-0.594936\pi$
$467$	−5931.03	−0.587699	−0.293850	+	0.955852i	$0.594936\pi$
$468$	10589.7	1.04596
$469$	−2567.27	−0.252762
$470$	8802.04	0.863846
$471$	4467.52	0.437054
$472$	−547.879	−0.0534283
$473$	0	0
$474$	16267.8	1.57638
$475$	−1213.98	−0.117266
$476$	9860.04	0.949441
$477$	−8975.13	−0.861515
$478$	10142.1	0.970482
$479$	−13403.9	−1.27858	−0.639292	−	0.768964i	$-0.720772\pi$
$479$	−13403.9	−1.27858	−0.639292	+	0.768964i	$0.720772\pi$
$480$	−1084.97	−0.103170
$481$	12719.2	1.20571
$482$	−18435.8	−1.74218
$483$	2544.74	0.239730
$484$	−3016.26	−0.283270
$485$	7994.54	0.748481
$486$	−18728.6	−1.74804
$487$	9526.78	0.886447	0.443223	−	0.896411i	$-0.353835\pi$
$487$	9526.78	0.886447	0.443223	+	0.896411i	$0.353835\pi$
$488$	−26003.4	−2.41213
$489$	−3006.60	−0.278043
$490$	9286.58	0.856174
$491$	6851.57	0.629750	0.314875	−	0.949133i	$-0.398037\pi$
$491$	6851.57	0.629750	0.314875	+	0.949133i	$0.398037\pi$
$492$	−17417.3	−1.59600
$493$	−4468.94	−0.408257
$494$	−2366.17	−0.215504
$495$	4353.51	0.395304
$496$	4.30973	0.000390147 0
$497$	6168.89	0.556766
$498$	4611.20	0.414925
$499$	−3033.89	−0.272176	−0.136088	−	0.990697i	$-0.543453\pi$
$499$	−3033.89	−0.272176	−0.136088	+	0.990697i	$0.543453\pi$
$500$	20060.8	1.79429
$501$	−3991.62	−0.355953
$502$	−35525.6	−3.15854
$503$	−6789.54	−0.601850	−0.300925	−	0.953648i	$-0.597295\pi$
$503$	−6789.54	−0.601850	−0.300925	+	0.953648i	$0.597295\pi$
$504$	−4289.35	−0.379093
$505$	3915.29	0.345006
$506$	−25933.7	−2.27844
$507$	2601.27	0.227863
$508$	−1289.33	−0.112608
$509$	−2197.89	−0.191395	−0.0956974	−	0.995410i	$-0.530508\pi$
$509$	−2197.89	−0.191395	−0.0956974	+	0.995410i	$0.530508\pi$
$510$	8274.59	0.718441
$511$	989.470	0.0856587
$512$	−14485.4	−1.25033
$513$	1742.14	0.149936
$514$	11993.8	1.02923
$515$	1377.90	0.117898
$516$	0	0
$517$	−9716.75	−0.826581
$518$	−10973.5	−0.930786
$519$	−5256.98	−0.444617
$520$	7419.22	0.625682
$521$	17320.1	1.45644	0.728222	−	0.685341i	$-0.240347\pi$
$521$	17320.1	1.45644	0.728222	+	0.685341i	$0.240347\pi$
$522$	4140.91	0.347208
$523$	11154.5	0.932601	0.466300	−	0.884626i	$-0.345587\pi$
$523$	11154.5	0.932601	0.466300	+	0.884626i	$0.345587\pi$
$524$	−20664.9	−1.72281
$525$	1344.18	0.111743
$526$	8719.90	0.722824
$527$	10.6680	0.000881791 0
$528$	−3690.23	−0.304160
$529$	13598.8	1.11768
$530$	−13393.4	−1.09769
$531$	328.847	0.0268752
$532$	1333.80	0.108699
$533$	−15482.8	−1.25822
$534$	8707.81	0.705663
$535$	−2044.84	−0.165245
$536$	−14134.3	−1.13901
$537$	−11271.4	−0.905771
$538$	−3237.33	−0.259426
$539$	−10251.7	−0.819239
$540$	−11635.2	−0.927219
$541$	−18613.0	−1.47918	−0.739591	−	0.673057i	$-0.764981\pi$
$541$	−18613.0	−1.47918	−0.739591	+	0.673057i	$0.764981\pi$
$542$	12408.1	0.983347
$543$	−6578.14	−0.519881
$544$	−7056.81	−0.556173
$545$	−474.831	−0.0373202
$546$	2619.94	0.205354
$547$	19378.5	1.51474	0.757371	−	0.652985i	$-0.226484\pi$
$547$	19378.5	1.51474	0.757371	+	0.652985i	$0.226484\pi$
$548$	−5367.29	−0.418393
$549$	15607.7	1.21333
$550$	−13698.7	−1.06203
$551$	−604.531	−0.0467402
$552$	14010.3	1.08028
$553$	−8151.59	−0.626837
$554$	−9812.11	−0.752485
$555$	−6016.93	−0.460188
$556$	−34703.8	−2.64707
$557$	−8962.10	−0.681753	−0.340877	−	0.940108i	$-0.610724\pi$
$557$	−8962.10	−0.681753	−0.340877	+	0.940108i	$0.610724\pi$
$558$	−9.88491	−0.000749931 0
$559$	0	0
$560$	−1675.06	−0.126400
$561$	−9134.50	−0.687449
$562$	26565.7	1.99396
$563$	12298.5	0.920640	0.460320	−	0.887753i	$-0.347735\pi$
$563$	12298.5	0.920640	0.460320	+	0.887753i	$0.347735\pi$
$564$	11181.0	0.834763
$565$	12287.1	0.914910
$566$	1026.14	0.0762044
$567$	1476.10	0.109330
$568$	33963.3	2.50893
$569$	5282.29	0.389183	0.194592	−	0.980884i	$-0.437662\pi$
$569$	5282.29	0.389183	0.194592	+	0.980884i	$0.437662\pi$
$570$	1119.34	0.0822523
$571$	20315.8	1.48895	0.744475	−	0.667650i	$-0.232700\pi$
$571$	20315.8	1.48895	0.744475	+	0.667650i	$0.232700\pi$
$572$	−17445.2	−1.27521
$573$	−2923.18	−0.213119
$574$	13357.7	0.971326
$575$	13610.0	0.987092
$576$	13522.1	0.978160
$577$	12757.9	0.920484	0.460242	−	0.887793i	$-0.347763\pi$
$577$	12757.9	0.920484	0.460242	+	0.887793i	$0.347763\pi$
$578$	30216.6	2.17447
$579$	−5437.58	−0.390290
$580$	4037.47	0.289046
$581$	−2310.62	−0.164992
$582$	15542.8	1.10699
$583$	14785.3	1.05033
$584$	5447.61	0.386000
$585$	−4453.16	−0.314727
$586$	−28967.3	−2.04203
$587$	10685.7	0.751354	0.375677	−	0.926751i	$-0.377410\pi$
$587$	10685.7	0.751354	0.375677	+	0.926751i	$0.377410\pi$
$588$	11796.5	0.827349
$589$	1.44310	0.000100954 0
$590$	490.733	0.0342427
$591$	10378.5	0.722362
$592$	−15810.1	−1.09762
$593$	2637.21	0.182626	0.0913130	−	0.995822i	$-0.470894\pi$
$593$	2637.21	0.182626	0.0913130	+	0.995822i	$0.470894\pi$
$594$	19658.4	1.35790
$595$	−4146.30	−0.285684
$596$	18573.3	1.27650
$597$	4407.55	0.302159
$598$	26527.3	1.81401
$599$	20472.5	1.39647	0.698234	−	0.715869i	$-0.253969\pi$
$599$	20472.5	1.39647	0.698234	+	0.715869i	$0.253969\pi$
$600$	7400.51	0.503541
$601$	−848.655	−0.0575996	−0.0287998	−	0.999585i	$-0.509169\pi$
$601$	−848.655	−0.0575996	−0.0287998	+	0.999585i	$0.509169\pi$
$602$	0	0
$603$	8483.68	0.572939
$604$	22693.2	1.52876
$605$	1268.38	0.0852350
$606$	7612.03	0.510260
$607$	−19963.6	−1.33492	−0.667461	−	0.744645i	$-0.732619\pi$
$607$	−19963.6	−1.33492	−0.667461	+	0.744645i	$0.732619\pi$
$608$	−954.601	−0.0636746
$609$	669.366	0.0445387
$610$	23291.1	1.54595
$611$	9939.16	0.658094
$612$	−32583.0	−2.15211
$613$	−7530.10	−0.496147	−0.248073	−	0.968741i	$-0.579797\pi$
$613$	−7530.10	−0.496147	−0.248073	+	0.968741i	$0.579797\pi$
$614$	47837.9	3.14427
$615$	7324.25	0.480231
$616$	7066.11	0.462178
$617$	10468.8	0.683075	0.341538	−	0.939868i	$-0.389052\pi$
$617$	10468.8	0.683075	0.341538	+	0.939868i	$0.389052\pi$
$618$	2678.88	0.174369
$619$	5756.07	0.373757	0.186879	−	0.982383i	$-0.440163\pi$
$619$	5756.07	0.373757	0.186879	+	0.982383i	$0.440163\pi$
$620$	−9.63799	−0.000624308 0
$621$	−19531.2	−1.26209
$622$	47803.6	3.08159
$623$	−4363.38	−0.280602
$624$	3774.69	0.242161
$625$	2162.69	0.138412
$626$	−9158.91	−0.584766
$627$	−1235.66	−0.0787040
$628$	−26252.0	−1.66810
$629$	−39135.1	−2.48079
$630$	3841.95	0.242964
$631$	27752.2	1.75087	0.875433	−	0.483339i	$-0.160576\pi$
$631$	27752.2	1.75087	0.875433	+	0.483339i	$0.160576\pi$
$632$	−44879.3	−2.82469
$633$	10120.7	0.635484
$634$	−14344.2	−0.898548
$635$	542.186	0.0338834
$636$	−17013.4	−1.06073
$637$	10486.3	0.652249
$638$	−6821.58	−0.423306
$639$	−20385.4	−1.26203
$640$	16796.5	1.03741
$641$	−2219.51	−0.136763	−0.0683816	−	0.997659i	$-0.521784\pi$
$641$	−2219.51	−0.136763	−0.0683816	+	0.997659i	$0.521784\pi$
$642$	−3975.53	−0.244395
$643$	−20580.1	−1.26221	−0.631104	−	0.775698i	$-0.717398\pi$
$643$	−20580.1	−1.26221	−0.631104	+	0.775698i	$0.717398\pi$
$644$	−14953.4	−0.914977
$645$	0	0
$646$	7280.34	0.443407
$647$	16963.7	1.03078	0.515389	−	0.856957i	$-0.327648\pi$
$647$	16963.7	1.03078	0.515389	+	0.856957i	$0.327648\pi$
$648$	8126.77	0.492669
$649$	−541.731	−0.0327655
$650$	14012.3	0.845548
$651$	−1.59787	−9.61987e−5 0
$652$	17667.3	1.06121
$653$	5184.83	0.310717	0.155358	−	0.987858i	$-0.450347\pi$
$653$	5184.83	0.310717	0.155358	+	0.987858i	$0.450347\pi$
$654$	−923.157	−0.0551961
$655$	8689.92	0.518387
$656$	19245.2	1.14543
$657$	−3269.76	−0.194163
$658$	−8574.99	−0.508037
$659$	−30887.9	−1.82583	−0.912915	−	0.408149i	$-0.866174\pi$
$659$	−30887.9	−1.82583	−0.912915	+	0.408149i	$0.866174\pi$
$660$	8252.57	0.486714
$661$	−5445.52	−0.320433	−0.160216	−	0.987082i	$-0.551219\pi$
$661$	−5445.52	−0.320433	−0.160216	+	0.987082i	$0.551219\pi$
$662$	4047.18	0.237611
$663$	9343.58	0.547322
$664$	−12721.3	−0.743497
$665$	−560.886	−0.0327071
$666$	36262.5	2.10982
$667$	6777.43	0.393438
$668$	23455.6	1.35857
$669$	−4614.48	−0.266676
$670$	12660.1	0.730001
$671$	−25711.6	−1.47926
$672$	1056.98	0.0606755
$673$	−12480.7	−0.714850	−0.357425	−	0.933942i	$-0.616345\pi$
$673$	−12480.7	−0.714850	−0.357425	+	0.933942i	$0.616345\pi$
$674$	−14693.3	−0.839712
$675$	−10316.8	−0.588286
$676$	−15285.6	−0.869685
$677$	30422.7	1.72709	0.863546	−	0.504270i	$-0.168238\pi$
$677$	30422.7	1.72709	0.863546	+	0.504270i	$0.168238\pi$
$678$	23888.4	1.35314
$679$	−7788.32	−0.440189
$680$	−22827.8	−1.28736
$681$	−16270.4	−0.915542
$682$	16.2840	0.000914293 0
$683$	−9567.17	−0.535985	−0.267992	−	0.963421i	$-0.586360\pi$
$683$	−9567.17	−0.535985	−0.267992	+	0.963421i	$0.586360\pi$
$684$	−4407.63	−0.246389
$685$	2257.03	0.125893
$686$	−19226.7	−1.07009
$687$	4025.74	0.223569
$688$	0	0
$689$	−15123.7	−0.836237
$690$	−12548.9	−0.692362
$691$	−14041.5	−0.773031	−0.386516	−	0.922283i	$-0.626321\pi$
$691$	−14041.5	−0.773031	−0.386516	+	0.922283i	$0.626321\pi$
$692$	30891.1	1.69697
$693$	−4241.21	−0.232482
$694$	54839.3	2.99953
$695$	14593.5	0.796494
$696$	3685.25	0.200703
$697$	47638.1	2.58884
$698$	−13111.3	−0.710989
$699$	−6183.61	−0.334600
$700$	−7898.68	−0.426489
$701$	4386.14	0.236323	0.118161	−	0.992994i	$-0.462300\pi$
$701$	4386.14	0.236323	0.118161	+	0.992994i	$0.462300\pi$
$702$	−20108.4	−1.08111
$703$	−5293.95	−0.284019
$704$	−22275.8	−1.19254
$705$	−4701.80	−0.251177
$706$	−54975.4	−2.93063
$707$	−3814.30	−0.202902
$708$	623.368	0.0330898
$709$	−2852.53	−0.151099	−0.0755494	−	0.997142i	$-0.524071\pi$
$709$	−2852.53	−0.151099	−0.0755494	+	0.997142i	$0.524071\pi$
$710$	−30420.8	−1.60799
$711$	26937.4	1.42086
$712$	−24023.0	−1.26447
$713$	−16.1786	−0.000849782 0
$714$	−8061.15	−0.422523
$715$	7335.97	0.383706
$716$	66233.2	3.45706
$717$	−5417.64	−0.282184
$718$	−18923.4	−0.983588
$719$	608.604	0.0315676	0.0157838	−	0.999875i	$-0.494976\pi$
$719$	608.604	0.0315676	0.0157838	+	0.999875i	$0.494976\pi$
$720$	5535.32	0.286513
$721$	−1342.35	−0.0693369
$722$	−31966.7	−1.64775
$723$	9847.90	0.506566
$724$	38654.5	1.98423
$725$	3579.98	0.183389
$726$	2465.97	0.126061
$727$	38172.7	1.94738	0.973691	−	0.227872i	$-0.0731767\pi$
$727$	38172.7	1.94738	0.973691	+	0.227872i	$0.0731767\pi$
$728$	−7227.85	−0.367970
$729$	3552.91	0.180507
$730$	−4879.41	−0.247390
$731$	0	0
$732$	29586.2	1.49390
$733$	−31896.8	−1.60728	−0.803639	−	0.595117i	$-0.797106\pi$
$733$	−31896.8	−1.60728	−0.803639	+	0.595117i	$0.797106\pi$
$734$	−35071.8	−1.76365
$735$	−4960.63	−0.248946
$736$	10702.1	0.535984
$737$	−13975.7	−0.698509
$738$	−44141.4	−2.20172
$739$	3826.00	0.190449	0.0952245	−	0.995456i	$-0.469643\pi$
$739$	3826.00	0.190449	0.0952245	+	0.995456i	$0.469643\pi$
$740$	35356.7	1.75640
$741$	1263.94	0.0626613
$742$	13048.0	0.645560
$743$	17425.8	0.860418	0.430209	−	0.902729i	$-0.358440\pi$
$743$	17425.8	0.860418	0.430209	+	0.902729i	$0.358440\pi$
$744$	−8.79720	−0.000433496 0
$745$	−7810.38	−0.384094
$746$	46373.9	2.27596
$747$	7635.56	0.373990
$748$	53676.1	2.62379
$749$	1992.09	0.0971822
$750$	−16400.9	−0.798500
$751$	15398.9	0.748223	0.374112	−	0.927384i	$-0.377948\pi$
$751$	15398.9	0.748223	0.374112	+	0.927384i	$0.377948\pi$
$752$	−12354.5	−0.599097
$753$	18976.8	0.918396
$754$	6977.72	0.337021
$755$	−9542.84	−0.459999
$756$	11335.1	0.545307
$757$	−15476.4	−0.743065	−0.371532	−	0.928420i	$-0.621168\pi$
$757$	−15476.4	−0.743065	−0.371532	+	0.928420i	$0.621168\pi$
$758$	−6902.16	−0.330736
$759$	13853.0	0.662494
$760$	−3088.01	−0.147386
$761$	−19666.4	−0.936802	−0.468401	−	0.883516i	$-0.655170\pi$
$761$	−19666.4	−0.936802	−0.468401	+	0.883516i	$0.655170\pi$
$762$	1054.11	0.0501132
$763$	462.583	0.0219484
$764$	17177.2	0.813413
$765$	13701.7	0.647562
$766$	46806.1	2.20780
$767$	554.130	0.0260867
$768$	19056.9	0.895385
$769$	18122.5	0.849821	0.424911	−	0.905235i	$-0.360306\pi$
$769$	18122.5	0.849821	0.424911	+	0.905235i	$0.360306\pi$
$770$	−6329.09	−0.296214
$771$	−6406.77	−0.299266
$772$	31952.3	1.48962
$773$	−20770.5	−0.966444	−0.483222	−	0.875498i	$-0.660534\pi$
$773$	−20770.5	−0.966444	−0.483222	+	0.875498i	$0.660534\pi$
$774$	0	0
$775$	−8.54590	−0.000396100 0
$776$	−42879.3	−1.98360
$777$	5861.72	0.270641
$778$	4100.43	0.188956
$779$	6444.19	0.296389
$780$	−8441.47	−0.387504
$781$	33582.2	1.53862
$782$	−81620.3	−3.73240
$783$	−5137.47	−0.234481
$784$	−13034.6	−0.593776
$785$	11039.4	0.501927
$786$	16894.8	0.766687
$787$	10239.3	0.463776	0.231888	−	0.972743i	$-0.425510\pi$
$787$	10239.3	0.463776	0.231888	+	0.972743i	$0.425510\pi$
$788$	−60986.3	−2.75704
$789$	−4657.92	−0.210173
$790$	40198.2	1.81037
$791$	−11970.2	−0.538067
$792$	−23350.4	−1.04762
$793$	26300.1	1.17773
$794$	34079.1	1.52320
$795$	7154.39	0.319170
$796$	−25899.6	−1.15325
$797$	−25982.1	−1.15475	−0.577373	−	0.816481i	$-0.695922\pi$
$797$	−25982.1	−1.15475	−0.577373	+	0.816481i	$0.695922\pi$
$798$	−1090.46	−0.0483734
$799$	−30581.3	−1.35405
$800$	5653.07	0.249833
$801$	14419.0	0.636044
$802$	−13531.4	−0.595773
$803$	5386.48	0.236718
$804$	16081.8	0.705424
$805$	6288.12	0.275313
$806$	−16.6568	−0.000727927 0
$807$	1729.29	0.0754324
$808$	−20999.9	−0.914326
$809$	−22187.4	−0.964236	−0.482118	−	0.876106i	$-0.660132\pi$
$809$	−22187.4	−0.964236	−0.482118	+	0.876106i	$0.660132\pi$
$810$	−7279.12	−0.315756
$811$	−8163.38	−0.353459	−0.176729	−	0.984259i	$-0.556552\pi$
$811$	−8163.38	−0.353459	−0.176729	+	0.984259i	$0.556552\pi$
$812$	−3933.33	−0.169991
$813$	−6628.06	−0.285924
$814$	−59737.5	−2.57223
$815$	−7429.40	−0.319313
$816$	−11614.2	−0.498256
$817$	0	0
$818$	57409.3	2.45387
$819$	4338.29	0.185094
$820$	−43038.7	−1.83290
$821$	−1562.98	−0.0664415	−0.0332208	−	0.999448i	$-0.510576\pi$
$821$	−1562.98	−0.0664415	−0.0332208	+	0.999448i	$0.510576\pi$
$822$	4388.07	0.186194
$823$	−35023.3	−1.48340	−0.741699	−	0.670733i	$-0.765980\pi$
$823$	−35023.3	−1.48340	−0.741699	+	0.670733i	$0.765980\pi$
$824$	−7390.44	−0.312449
$825$	7317.47	0.308802
$826$	−478.075	−0.0201384
$827$	19516.9	0.820638	0.410319	−	0.911942i	$-0.365417\pi$
$827$	19516.9	0.820638	0.410319	+	0.911942i	$0.365417\pi$
$828$	49414.2	2.07399
$829$	−17033.7	−0.713635	−0.356818	−	0.934174i	$-0.616138\pi$
$829$	−17033.7	−0.713635	−0.356818	+	0.934174i	$0.616138\pi$
$830$	11394.4	0.476514
$831$	5241.35	0.218797
$832$	22785.6	0.949459
$833$	−32264.8	−1.34203
$834$	28372.4	1.17800
$835$	−9863.43	−0.408788
$836$	7260.97	0.300390
$837$	12.2638	0.000506452 0
$838$	−13678.6	−0.563864
$839$	7296.99	0.300262	0.150131	−	0.988666i	$-0.452030\pi$
$839$	7296.99	0.300262	0.150131	+	0.988666i	$0.452030\pi$
$840$	3419.19	0.140444
$841$	−22606.3	−0.926904
$842$	49030.1	2.00675
$843$	−14190.6	−0.579776
$844$	−59471.2	−2.42545
$845$	6427.83	0.261685
$846$	28336.5	1.15157
$847$	−1235.67	−0.0501275
$848$	18798.9	0.761271
$849$	−548.133	−0.0221577
$850$	−43113.6	−1.73975
$851$	59350.8	2.39074
$852$	−38642.9	−1.55386
$853$	−9689.81	−0.388948	−0.194474	−	0.980908i	$-0.562300\pi$
$853$	−9689.81	−0.388948	−0.194474	+	0.980908i	$0.562300\pi$
$854$	−22690.4	−0.909189
$855$	1853.48	0.0741376
$856$	10967.6	0.437927
$857$	25619.7	1.02118	0.510590	−	0.859824i	$-0.329427\pi$
$857$	25619.7	1.02118	0.510590	+	0.859824i	$0.329427\pi$
$858$	14262.4	0.567496
$859$	−33986.1	−1.34993	−0.674966	−	0.737849i	$-0.735842\pi$
$859$	−33986.1	−1.34993	−0.674966	+	0.737849i	$0.735842\pi$
$860$	0	0
$861$	−7135.32	−0.282429
$862$	−66437.6	−2.62514
$863$	10162.6	0.400855	0.200427	−	0.979709i	$-0.435767\pi$
$863$	10162.6	0.400855	0.200427	+	0.979709i	$0.435767\pi$
$864$	−8112.47	−0.319435
$865$	−12990.2	−0.510612
$866$	−41879.7	−1.64334
$867$	−16140.9	−0.632263
$868$	9.38938	0.000367162 0
$869$	−44375.7	−1.73227
$870$	−3300.87	−0.128632
$871$	14295.6	0.556128
$872$	2546.79	0.0989050
$873$	25736.9	0.997781
$874$	−11041.1	−0.427312
$875$	8218.29	0.317519
$876$	−6198.20	−0.239061
$877$	33751.4	1.29955	0.649774	−	0.760127i	$-0.274863\pi$
$877$	33751.4	1.29955	0.649774	+	0.760127i	$0.274863\pi$
$878$	−71603.7	−2.75229
$879$	15473.5	0.593753
$880$	−9118.68	−0.349307
$881$	18540.8	0.709030	0.354515	−	0.935050i	$-0.384646\pi$
$881$	18540.8	0.709030	0.354515	+	0.935050i	$0.384646\pi$
$882$	29896.4	1.14134
$883$	−33606.6	−1.28081	−0.640403	−	0.768039i	$-0.721233\pi$
$883$	−33606.6	−1.28081	−0.640403	+	0.768039i	$0.721233\pi$
$884$	−54904.7	−2.08896
$885$	−262.136	−0.00995661
$886$	78255.4	2.96732
$887$	−36259.0	−1.37256	−0.686279	−	0.727339i	$-0.740757\pi$
$887$	−36259.0	−1.37256	−0.686279	+	0.727339i	$0.740757\pi$
$888$	32272.2	1.21958
$889$	−528.200	−0.0199272
$890$	21517.3	0.810406
$891$	8035.57	0.302134
$892$	27115.6	1.01782
$893$	−4136.84	−0.155022
$894$	−15184.8	−0.568070
$895$	−27852.1	−1.04022
$896$	−16363.2	−0.610108
$897$	−14170.1	−0.527454
$898$	62203.6	2.31154
$899$	−4.25562	−0.000157879 0
$900$	26101.6	0.966727
$901$	46533.3	1.72059
$902$	72716.8	2.68426
$903$	0	0
$904$	−65903.0	−2.42467
$905$	−16254.8	−0.597048
$906$	−18553.0	−0.680333
$907$	2435.81	0.0891730	0.0445865	−	0.999006i	$-0.485803\pi$
$907$	2435.81	0.0891730	0.0445865	+	0.999006i	$0.485803\pi$
$908$	95608.2	3.49435
$909$	12604.6	0.459919
$910$	6473.96	0.235835
$911$	−26035.1	−0.946851	−0.473426	−	0.880834i	$-0.656983\pi$
$911$	−26035.1	−0.946851	−0.473426	+	0.880834i	$0.656983\pi$
$912$	−1571.09	−0.0570439
$913$	−12578.5	−0.455957
$914$	7307.36	0.264449
$915$	−12441.5	−0.449511
$916$	−23656.1	−0.853295
$917$	−8465.76	−0.304868
$918$	61870.4	2.22443
$919$	−37812.0	−1.35724	−0.678620	−	0.734490i	$-0.737422\pi$
$919$	−37812.0	−1.35724	−0.678620	+	0.734490i	$0.737422\pi$
$920$	34619.8	1.24063
$921$	−25553.7	−0.914247
$922$	−21619.1	−0.772221
$923$	−34350.9	−1.22500
$924$	−8039.70	−0.286241
$925$	31350.3	1.11437
$926$	−24483.2	−0.868862
$927$	4435.88	0.157167
$928$	2815.07	0.0995790
$929$	−27043.9	−0.955093	−0.477546	−	0.878607i	$-0.658474\pi$
$929$	−27043.9	−0.955093	−0.477546	+	0.878607i	$0.658474\pi$
$930$	7.87962	0.000277831 0
$931$	−4364.58	−0.153645
$932$	36336.1	1.27707
$933$	−25535.4	−0.896024
$934$	−28493.5	−0.998217
$935$	−22571.6	−0.789488
$936$	23884.8	0.834081
$937$	−35620.5	−1.24191	−0.620956	−	0.783845i	$-0.713255\pi$
$937$	−35620.5	−1.24191	−0.620956	+	0.783845i	$0.713255\pi$
$938$	−12333.5	−0.429321
$939$	4892.43	0.170030
$940$	27628.7	0.958669
$941$	−54098.1	−1.87412	−0.937060	−	0.349168i	$-0.886464\pi$
$941$	−54098.1	−1.87412	−0.937060	+	0.349168i	$0.886464\pi$
$942$	21462.5	0.742343
$943$	−72246.2	−2.49487
$944$	−688.790	−0.0237481
$945$	−4766.57	−0.164081
$946$	0	0
$947$	−1896.92	−0.0650914	−0.0325457	−	0.999470i	$-0.510361\pi$
$947$	−1896.92	−0.0650914	−0.0325457	+	0.999470i	$0.510361\pi$
$948$	51062.9	1.74942
$949$	−5509.77	−0.188467
$950$	−5832.14	−0.199178
$951$	7662.25	0.261268
$952$	22239.0	0.757111
$953$	−38430.6	−1.30629	−0.653143	−	0.757235i	$-0.726550\pi$
$953$	−38430.6	−1.30629	−0.653143	+	0.757235i	$0.726550\pi$
$954$	−43117.7	−1.46330
$955$	−7223.27	−0.244753
$956$	31835.1	1.07701
$957$	3643.90	0.123083
$958$	−64394.3	−2.17170
$959$	−2198.81	−0.0740389
$960$	−10778.9	−0.362384
$961$	−29791.0	−1.00000
$962$	61104.8	2.04792
$963$	−6582.98	−0.220284
$964$	−57868.2	−1.93341
$965$	−13436.4	−0.448222
$966$	12225.2	0.407185
$967$	−43807.6	−1.45683	−0.728417	−	0.685135i	$-0.759743\pi$
$967$	−43807.6	−1.45683	−0.728417	+	0.685135i	$0.759743\pi$
$968$	−6803.07	−0.225887
$969$	−3888.95	−0.128928
$970$	38406.8	1.27131
$971$	13880.3	0.458742	0.229371	−	0.973339i	$-0.426333\pi$
$971$	13880.3	0.458742	0.229371	+	0.973339i	$0.426333\pi$
$972$	−58787.2	−1.93992
$973$	−14217.1	−0.468426
$974$	45767.9	1.50564
$975$	−7484.96	−0.245857
$976$	−32691.3	−1.07215
$977$	16948.3	0.554990	0.277495	−	0.960727i	$-0.410496\pi$
$977$	16948.3	0.554990	0.277495	+	0.960727i	$0.410496\pi$
$978$	−14444.1	−0.472260
$979$	−23753.4	−0.775446
$980$	29149.6	0.950154
$981$	−1528.63	−0.0497507
$982$	32915.8	1.06964
$983$	−6232.10	−0.202211	−0.101105	−	0.994876i	$-0.532238\pi$
$983$	−6232.10	−0.202211	−0.101105	+	0.994876i	$0.532238\pi$
$984$	−39284.1	−1.27270
$985$	25645.7	0.829584
$986$	−21469.4	−0.693432
$987$	4580.52	0.147720
$988$	−7427.17	−0.239160
$989$	0	0
$990$	20914.8	0.671431
$991$	41002.7	1.31432	0.657161	−	0.753751i	$-0.271757\pi$
$991$	41002.7	1.31432	0.657161	+	0.753751i	$0.271757\pi$
$992$	−6.71996	−0.000215080 0
$993$	−2161.89	−0.0690892
$994$	29636.2	0.945676
$995$	10891.2	0.347009
$996$	14474.1	0.460471
$997$	4141.03	0.131542	0.0657712	−	0.997835i	$-0.479049\pi$
$997$	4141.03	0.131542	0.0657712	+	0.997835i	$0.479049\pi$
$998$	−14575.2	−0.462295
$999$	−44989.5	−1.42483

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.i.1.48	✓	50
43.42	odd	2		1849.4.a.j.1.3	yes	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.48	✓	50	1.1	even	1	trivial
1849.4.a.j.1.3	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)

\(f(q)\)	\(=\)	\(q+4.80413 q^{2} -2.56623 q^{3} +15.0797 q^{4} -6.34124 q^{5} -12.3285 q^{6} +6.17767 q^{7} +34.0117 q^{8} -20.4145 q^{9} +O(q^{10})\)	\(q+4.80413 q^{2} -2.56623 q^{3} +15.0797 q^{4} -6.34124 q^{5} -12.3285 q^{6} +6.17767 q^{7} +34.0117 q^{8} -20.4145 q^{9} -30.4642 q^{10} +33.6300 q^{11} -38.6979 q^{12} -34.3998 q^{13} +29.6784 q^{14} +16.2731 q^{15} +42.7593 q^{16} +105.843 q^{17} -98.0737 q^{18} +14.3178 q^{19} -95.6239 q^{20} -15.8533 q^{21} +161.563 q^{22} -160.517 q^{23} -87.2819 q^{24} -84.7886 q^{25} -165.261 q^{26} +121.676 q^{27} +93.1573 q^{28} -42.2224 q^{29} +78.1781 q^{30} +0.100791 q^{31} -66.6725 q^{32} -86.3024 q^{33} +508.483 q^{34} -39.1741 q^{35} -307.843 q^{36} -369.747 q^{37} +68.7844 q^{38} +88.2778 q^{39} -215.676 q^{40} +450.083 q^{41} -76.1615 q^{42} +507.130 q^{44} +129.453 q^{45} -771.146 q^{46} -288.931 q^{47} -109.730 q^{48} -304.836 q^{49} -407.336 q^{50} -271.617 q^{51} -518.738 q^{52} +439.646 q^{53} +584.550 q^{54} -213.256 q^{55} +210.113 q^{56} -36.7427 q^{57} -202.842 q^{58} -16.1085 q^{59} +245.393 q^{60} -764.542 q^{61} +0.484211 q^{62} -126.114 q^{63} -662.378 q^{64} +218.137 q^{65} -414.608 q^{66} -415.572 q^{67} +1596.08 q^{68} +411.925 q^{69} -188.198 q^{70} +998.578 q^{71} -694.330 q^{72} +160.169 q^{73} -1776.31 q^{74} +217.587 q^{75} +215.907 q^{76} +207.755 q^{77} +424.098 q^{78} -1319.53 q^{79} -271.147 q^{80} +238.940 q^{81} +2162.26 q^{82} -374.027 q^{83} -239.063 q^{84} -671.175 q^{85} +108.352 q^{87} +1143.81 q^{88} -706.315 q^{89} +621.909 q^{90} -212.511 q^{91} -2420.55 q^{92} -0.258652 q^{93} -1388.06 q^{94} -90.7924 q^{95} +171.097 q^{96} -1260.72 q^{97} -1464.47 q^{98} -686.539 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} + O(q^{10}) \)	\( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} - 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} - 386q^{18} + 12q^{19} + 108q^{20} - 408q^{21} + 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} + 1493q^{26} + 10q^{27} - 242q^{28} + 208q^{29} - 48q^{30} - 932q^{31} - 1124q^{32} + 254q^{33} + 765q^{34} - 1452q^{35} + 747q^{36} - 90q^{37} - 1213q^{38} - 1610q^{39} - 1693q^{40} - 1354q^{41} - 16q^{42} - 2704q^{44} + 4508q^{45} + 233q^{46} - 3484q^{47} - 376q^{48} + 1324q^{49} - 408q^{50} + 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} - 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} + 1172q^{61} - 1546q^{62} - 3686q^{63} + 606q^{64} + 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} + 136q^{69} - 1310q^{70} + 162q^{71} - 5814q^{72} + 746q^{73} - 4332q^{74} + 236q^{75} + 1338q^{76} + 2024q^{77} - 2782q^{78} - 2656q^{79} - 5713q^{80} - 86q^{81} - 4168q^{82} - 3514q^{83} - 4269q^{84} - 7558q^{85} - 10278q^{87} + 11692q^{88} + 2640q^{89} - 8286q^{90} - 5946q^{91} - 4271q^{92} - 2q^{93} + 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} - 2826q^{98} - 8174q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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