Properties

Label 1849.4.a.i.1.23
Level $1849$
Weight $4$
Character 1849.1
Self dual yes
Analytic conductor $109.095$
Analytic rank $1$
Dimension $50$
CM no
Inner twists $1$

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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 1.23
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.928750 q^{2} +4.15887 q^{3} -7.13742 q^{4} -7.22084 q^{5} -3.86255 q^{6} -33.4889 q^{7} +14.0589 q^{8} -9.70384 q^{9} +O(q^{10})\) \(q-0.928750 q^{2} +4.15887 q^{3} -7.13742 q^{4} -7.22084 q^{5} -3.86255 q^{6} -33.4889 q^{7} +14.0589 q^{8} -9.70384 q^{9} +6.70635 q^{10} +21.8153 q^{11} -29.6836 q^{12} -37.7982 q^{13} +31.1028 q^{14} -30.0305 q^{15} +44.0422 q^{16} +87.4139 q^{17} +9.01244 q^{18} +115.186 q^{19} +51.5382 q^{20} -139.276 q^{21} -20.2609 q^{22} -3.50528 q^{23} +58.4690 q^{24} -72.8595 q^{25} +35.1051 q^{26} -152.646 q^{27} +239.025 q^{28} +149.824 q^{29} +27.8908 q^{30} +53.4626 q^{31} -153.375 q^{32} +90.7268 q^{33} -81.1857 q^{34} +241.818 q^{35} +69.2604 q^{36} +272.472 q^{37} -106.979 q^{38} -157.198 q^{39} -101.517 q^{40} -446.102 q^{41} +129.352 q^{42} -155.705 q^{44} +70.0699 q^{45} +3.25553 q^{46} +526.776 q^{47} +183.166 q^{48} +778.507 q^{49} +67.6683 q^{50} +363.543 q^{51} +269.782 q^{52} -262.810 q^{53} +141.770 q^{54} -157.525 q^{55} -470.817 q^{56} +479.044 q^{57} -139.149 q^{58} +575.151 q^{59} +214.340 q^{60} +112.448 q^{61} -49.6533 q^{62} +324.971 q^{63} -209.890 q^{64} +272.935 q^{65} -84.2625 q^{66} -6.38826 q^{67} -623.910 q^{68} -14.5780 q^{69} -224.588 q^{70} -570.402 q^{71} -136.425 q^{72} -424.131 q^{73} -253.059 q^{74} -303.013 q^{75} -822.133 q^{76} -730.570 q^{77} +145.997 q^{78} -395.839 q^{79} -318.022 q^{80} -372.832 q^{81} +414.317 q^{82} +472.458 q^{83} +994.071 q^{84} -631.202 q^{85} +623.098 q^{87} +306.698 q^{88} +836.438 q^{89} -65.0774 q^{90} +1265.82 q^{91} +25.0187 q^{92} +222.344 q^{93} -489.243 q^{94} -831.741 q^{95} -637.867 q^{96} -640.791 q^{97} -723.038 q^{98} -211.692 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} - 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} - 386q^{18} + 12q^{19} + 108q^{20} - 408q^{21} + 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} + 1493q^{26} + 10q^{27} - 242q^{28} + 208q^{29} - 48q^{30} - 932q^{31} - 1124q^{32} + 254q^{33} + 765q^{34} - 1452q^{35} + 747q^{36} - 90q^{37} - 1213q^{38} - 1610q^{39} - 1693q^{40} - 1354q^{41} - 16q^{42} - 2704q^{44} + 4508q^{45} + 233q^{46} - 3484q^{47} - 376q^{48} + 1324q^{49} - 408q^{50} + 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} - 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} + 1172q^{61} - 1546q^{62} - 3686q^{63} + 606q^{64} + 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} + 136q^{69} - 1310q^{70} + 162q^{71} - 5814q^{72} + 746q^{73} - 4332q^{74} + 236q^{75} + 1338q^{76} + 2024q^{77} - 2782q^{78} - 2656q^{79} - 5713q^{80} - 86q^{81} - 4168q^{82} - 3514q^{83} - 4269q^{84} - 7558q^{85} - 10278q^{87} + 11692q^{88} + 2640q^{89} - 8286q^{90} - 5946q^{91} - 4271q^{92} - 2q^{93} + 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} - 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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 \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.928750 −0.328363 −0.164181 0.986430i \(-0.552498\pi\)
−0.164181 + 0.986430i \(0.552498\pi\)
\(3\) 4.15887 0.800374 0.400187 0.916434i \(-0.368945\pi\)
0.400187 + 0.916434i \(0.368945\pi\)
\(4\) −7.13742 −0.892178
\(5\) −7.22084 −0.645851 −0.322926 0.946424i \(-0.604666\pi\)
−0.322926 + 0.946424i \(0.604666\pi\)
\(6\) −3.86255 −0.262813
\(7\) −33.4889 −1.80823 −0.904116 0.427288i \(-0.859469\pi\)
−0.904116 + 0.427288i \(0.859469\pi\)
\(8\) 14.0589 0.621321
\(9\) −9.70384 −0.359401
\(10\) 6.70635 0.212074
\(11\) 21.8153 0.597959 0.298980 0.954259i \(-0.403354\pi\)
0.298980 + 0.954259i \(0.403354\pi\)
\(12\) −29.6836 −0.714076
\(13\) −37.7982 −0.806411 −0.403206 0.915109i \(-0.632104\pi\)
−0.403206 + 0.915109i \(0.632104\pi\)
\(14\) 31.1028 0.593756
\(15\) −30.0305 −0.516923
\(16\) 44.0422 0.688159
\(17\) 87.4139 1.24712 0.623558 0.781777i \(-0.285686\pi\)
0.623558 + 0.781777i \(0.285686\pi\)
\(18\) 9.01244 0.118014
\(19\) 115.186 1.39082 0.695409 0.718614i \(-0.255223\pi\)
0.695409 + 0.718614i \(0.255223\pi\)
\(20\) 51.5382 0.576214
\(21\) −139.276 −1.44726
\(22\) −20.2609 −0.196348
\(23\) −3.50528 −0.0317783 −0.0158892 0.999874i \(-0.505058\pi\)
−0.0158892 + 0.999874i \(0.505058\pi\)
\(24\) 58.4690 0.497289
\(25\) −72.8595 −0.582876
\(26\) 35.1051 0.264795
\(27\) −152.646 −1.08803
\(28\) 239.025 1.61326
\(29\) 149.824 0.959367 0.479683 0.877442i \(-0.340752\pi\)
0.479683 + 0.877442i \(0.340752\pi\)
\(30\) 27.8908 0.169738
\(31\) 53.4626 0.309747 0.154874 0.987934i \(-0.450503\pi\)
0.154874 + 0.987934i \(0.450503\pi\)
\(32\) −153.375 −0.847287
\(33\) 90.7268 0.478591
\(34\) −81.1857 −0.409507
\(35\) 241.818 1.16785
\(36\) 69.2604 0.320650
\(37\) 272.472 1.21065 0.605327 0.795977i \(-0.293042\pi\)
0.605327 + 0.795977i \(0.293042\pi\)
\(38\) −106.979 −0.456693
\(39\) −157.198 −0.645431
\(40\) −101.517 −0.401281
\(41\) −446.102 −1.69925 −0.849626 0.527385i \(-0.823173\pi\)
−0.849626 + 0.527385i \(0.823173\pi\)
\(42\) 129.352 0.475227
\(43\) 0 0
\(44\) −155.705 −0.533486
\(45\) 70.0699 0.232120
\(46\) 3.25553 0.0104348
\(47\) 526.776 1.63485 0.817427 0.576032i \(-0.195400\pi\)
0.817427 + 0.576032i \(0.195400\pi\)
\(48\) 183.166 0.550785
\(49\) 778.507 2.26970
\(50\) 67.6683 0.191395
\(51\) 363.543 0.998160
\(52\) 269.782 0.719462
\(53\) −262.810 −0.681128 −0.340564 0.940221i \(-0.610618\pi\)
−0.340564 + 0.940221i \(0.610618\pi\)
\(54\) 141.770 0.357268
\(55\) −157.525 −0.386193
\(56\) −470.817 −1.12349
\(57\) 479.044 1.11317
\(58\) −139.149 −0.315020
\(59\) 575.151 1.26912 0.634562 0.772872i \(-0.281181\pi\)
0.634562 + 0.772872i \(0.281181\pi\)
\(60\) 214.340 0.461187
\(61\) 112.448 0.236024 0.118012 0.993012i \(-0.462348\pi\)
0.118012 + 0.993012i \(0.462348\pi\)
\(62\) −49.6533 −0.101709
\(63\) 324.971 0.649881
\(64\) −209.890 −0.409942
\(65\) 272.935 0.520822
\(66\) −84.2625 −0.157151
\(67\) −6.38826 −0.0116485 −0.00582425 0.999983i \(-0.501854\pi\)
−0.00582425 + 0.999983i \(0.501854\pi\)
\(68\) −623.910 −1.11265
\(69\) −14.5780 −0.0254346
\(70\) −224.588 −0.383478
\(71\) −570.402 −0.953440 −0.476720 0.879055i \(-0.658174\pi\)
−0.476720 + 0.879055i \(0.658174\pi\)
\(72\) −136.425 −0.223304
\(73\) −424.131 −0.680010 −0.340005 0.940424i \(-0.610429\pi\)
−0.340005 + 0.940424i \(0.610429\pi\)
\(74\) −253.059 −0.397534
\(75\) −303.013 −0.466519
\(76\) −822.133 −1.24086
\(77\) −730.570 −1.08125
\(78\) 145.997 0.211935
\(79\) −395.839 −0.563739 −0.281869 0.959453i \(-0.590955\pi\)
−0.281869 + 0.959453i \(0.590955\pi\)
\(80\) −318.022 −0.444449
\(81\) −372.832 −0.511429
\(82\) 414.317 0.557971
\(83\) 472.458 0.624807 0.312404 0.949949i \(-0.398866\pi\)
0.312404 + 0.949949i \(0.398866\pi\)
\(84\) 994.071 1.29121
\(85\) −631.202 −0.805452
\(86\) 0 0
\(87\) 623.098 0.767852
\(88\) 306.698 0.371525
\(89\) 836.438 0.996205 0.498102 0.867118i \(-0.334030\pi\)
0.498102 + 0.867118i \(0.334030\pi\)
\(90\) −65.0774 −0.0762195
\(91\) 1265.82 1.45818
\(92\) 25.0187 0.0283519
\(93\) 222.344 0.247913
\(94\) −489.243 −0.536825
\(95\) −831.741 −0.898262
\(96\) −637.867 −0.678146
\(97\) −640.791 −0.670747 −0.335374 0.942085i \(-0.608863\pi\)
−0.335374 + 0.942085i \(0.608863\pi\)
\(98\) −723.038 −0.745285
\(99\) −211.692 −0.214907
\(100\) 520.029 0.520029
\(101\) 1626.00 1.60191 0.800957 0.598722i \(-0.204324\pi\)
0.800957 + 0.598722i \(0.204324\pi\)
\(102\) −337.640 −0.327758
\(103\) −539.444 −0.516049 −0.258024 0.966138i \(-0.583071\pi\)
−0.258024 + 0.966138i \(0.583071\pi\)
\(104\) −531.401 −0.501040
\(105\) 1005.69 0.934716
\(106\) 244.085 0.223657
\(107\) 257.349 0.232513 0.116257 0.993219i \(-0.462911\pi\)
0.116257 + 0.993219i \(0.462911\pi\)
\(108\) 1089.50 0.970716
\(109\) −1923.21 −1.69000 −0.844998 0.534769i \(-0.820399\pi\)
−0.844998 + 0.534769i \(0.820399\pi\)
\(110\) 146.301 0.126811
\(111\) 1133.18 0.968976
\(112\) −1474.93 −1.24435
\(113\) 230.361 0.191774 0.0958872 0.995392i \(-0.469431\pi\)
0.0958872 + 0.995392i \(0.469431\pi\)
\(114\) −444.912 −0.365525
\(115\) 25.3111 0.0205241
\(116\) −1069.36 −0.855926
\(117\) 366.788 0.289825
\(118\) −534.171 −0.416733
\(119\) −2927.40 −2.25508
\(120\) −422.195 −0.321175
\(121\) −855.094 −0.642445
\(122\) −104.436 −0.0775014
\(123\) −1855.28 −1.36004
\(124\) −381.585 −0.276349
\(125\) 1428.71 1.02230
\(126\) −301.817 −0.213397
\(127\) −2386.40 −1.66739 −0.833695 0.552225i \(-0.813779\pi\)
−0.833695 + 0.552225i \(0.813779\pi\)
\(128\) 1421.94 0.981896
\(129\) 0 0
\(130\) −253.488 −0.171018
\(131\) 702.752 0.468700 0.234350 0.972152i \(-0.424704\pi\)
0.234350 + 0.972152i \(0.424704\pi\)
\(132\) −647.555 −0.426988
\(133\) −3857.46 −2.51492
\(134\) 5.93309 0.00382493
\(135\) 1102.23 0.702705
\(136\) 1228.94 0.774859
\(137\) −406.925 −0.253766 −0.126883 0.991918i \(-0.540497\pi\)
−0.126883 + 0.991918i \(0.540497\pi\)
\(138\) 13.5393 0.00835176
\(139\) −630.082 −0.384481 −0.192241 0.981348i \(-0.561575\pi\)
−0.192241 + 0.981348i \(0.561575\pi\)
\(140\) −1725.96 −1.04193
\(141\) 2190.79 1.30849
\(142\) 529.761 0.313074
\(143\) −824.579 −0.482201
\(144\) −427.378 −0.247326
\(145\) −1081.86 −0.619608
\(146\) 393.911 0.223290
\(147\) 3237.71 1.81661
\(148\) −1944.75 −1.08012
\(149\) −1284.41 −0.706193 −0.353097 0.935587i \(-0.614871\pi\)
−0.353097 + 0.935587i \(0.614871\pi\)
\(150\) 281.423 0.153187
\(151\) 909.515 0.490167 0.245084 0.969502i \(-0.421185\pi\)
0.245084 + 0.969502i \(0.421185\pi\)
\(152\) 1619.39 0.864144
\(153\) −848.251 −0.448216
\(154\) 678.517 0.355042
\(155\) −386.044 −0.200051
\(156\) 1121.99 0.575839
\(157\) −1974.99 −1.00396 −0.501979 0.864880i \(-0.667395\pi\)
−0.501979 + 0.864880i \(0.667395\pi\)
\(158\) 367.636 0.185111
\(159\) −1092.99 −0.545157
\(160\) 1107.50 0.547221
\(161\) 117.388 0.0574626
\(162\) 346.268 0.167934
\(163\) −1890.45 −0.908415 −0.454207 0.890896i \(-0.650077\pi\)
−0.454207 + 0.890896i \(0.650077\pi\)
\(164\) 3184.02 1.51604
\(165\) −655.123 −0.309099
\(166\) −438.795 −0.205163
\(167\) 3671.61 1.70130 0.850652 0.525729i \(-0.176208\pi\)
0.850652 + 0.525729i \(0.176208\pi\)
\(168\) −1958.06 −0.899213
\(169\) −768.293 −0.349701
\(170\) 586.229 0.264480
\(171\) −1117.75 −0.499862
\(172\) 0 0
\(173\) −2464.27 −1.08298 −0.541488 0.840709i \(-0.682139\pi\)
−0.541488 + 0.840709i \(0.682139\pi\)
\(174\) −578.703 −0.252134
\(175\) 2439.98 1.05397
\(176\) 960.793 0.411491
\(177\) 2391.98 1.01577
\(178\) −776.842 −0.327117
\(179\) −1155.83 −0.482629 −0.241314 0.970447i \(-0.577579\pi\)
−0.241314 + 0.970447i \(0.577579\pi\)
\(180\) −500.118 −0.207092
\(181\) −124.784 −0.0512437 −0.0256218 0.999672i \(-0.508157\pi\)
−0.0256218 + 0.999672i \(0.508157\pi\)
\(182\) −1175.63 −0.478811
\(183\) 467.655 0.188907
\(184\) −49.2804 −0.0197445
\(185\) −1967.48 −0.781903
\(186\) −206.502 −0.0814055
\(187\) 1906.96 0.745725
\(188\) −3759.82 −1.45858
\(189\) 5111.96 1.96741
\(190\) 772.480 0.294956
\(191\) 3922.64 1.48603 0.743016 0.669273i \(-0.233394\pi\)
0.743016 + 0.669273i \(0.233394\pi\)
\(192\) −872.906 −0.328107
\(193\) −3321.16 −1.23866 −0.619332 0.785129i \(-0.712596\pi\)
−0.619332 + 0.785129i \(0.712596\pi\)
\(194\) 595.135 0.220248
\(195\) 1135.10 0.416852
\(196\) −5556.53 −2.02498
\(197\) −1106.57 −0.400203 −0.200102 0.979775i \(-0.564127\pi\)
−0.200102 + 0.979775i \(0.564127\pi\)
\(198\) 196.609 0.0705676
\(199\) −3689.82 −1.31440 −0.657198 0.753718i \(-0.728258\pi\)
−0.657198 + 0.753718i \(0.728258\pi\)
\(200\) −1024.32 −0.362153
\(201\) −26.5679 −0.00932316
\(202\) −1510.15 −0.526009
\(203\) −5017.45 −1.73476
\(204\) −2594.76 −0.890536
\(205\) 3221.23 1.09746
\(206\) 501.009 0.169451
\(207\) 34.0147 0.0114212
\(208\) −1664.72 −0.554939
\(209\) 2512.82 0.831652
\(210\) −934.033 −0.306926
\(211\) −802.568 −0.261853 −0.130927 0.991392i \(-0.541795\pi\)
−0.130927 + 0.991392i \(0.541795\pi\)
\(212\) 1875.79 0.607687
\(213\) −2372.23 −0.763109
\(214\) −239.013 −0.0763486
\(215\) 0 0
\(216\) −2146.04 −0.676015
\(217\) −1790.40 −0.560094
\(218\) 1786.18 0.554932
\(219\) −1763.90 −0.544263
\(220\) 1124.32 0.344553
\(221\) −3304.09 −1.00569
\(222\) −1052.44 −0.318176
\(223\) 4705.21 1.41293 0.706467 0.707746i \(-0.250288\pi\)
0.706467 + 0.707746i \(0.250288\pi\)
\(224\) 5136.37 1.53209
\(225\) 707.017 0.209486
\(226\) −213.947 −0.0629716
\(227\) 1300.02 0.380112 0.190056 0.981773i \(-0.439133\pi\)
0.190056 + 0.981773i \(0.439133\pi\)
\(228\) −3419.14 −0.993149
\(229\) −4002.54 −1.15500 −0.577500 0.816391i \(-0.695972\pi\)
−0.577500 + 0.816391i \(0.695972\pi\)
\(230\) −23.5077 −0.00673935
\(231\) −3038.34 −0.865403
\(232\) 2106.36 0.596075
\(233\) 6770.78 1.90373 0.951864 0.306522i \(-0.0991654\pi\)
0.951864 + 0.306522i \(0.0991654\pi\)
\(234\) −340.654 −0.0951679
\(235\) −3803.76 −1.05587
\(236\) −4105.10 −1.13228
\(237\) −1646.24 −0.451202
\(238\) 2718.82 0.740483
\(239\) −6033.44 −1.63293 −0.816467 0.577392i \(-0.804070\pi\)
−0.816467 + 0.577392i \(0.804070\pi\)
\(240\) −1322.61 −0.355725
\(241\) 1721.55 0.460145 0.230073 0.973173i \(-0.426104\pi\)
0.230073 + 0.973173i \(0.426104\pi\)
\(242\) 794.168 0.210955
\(243\) 2570.89 0.678695
\(244\) −802.587 −0.210575
\(245\) −5621.47 −1.46589
\(246\) 1723.09 0.446586
\(247\) −4353.84 −1.12157
\(248\) 751.624 0.192452
\(249\) 1964.89 0.500079
\(250\) −1326.92 −0.335686
\(251\) 4782.06 1.20255 0.601277 0.799040i \(-0.294659\pi\)
0.601277 + 0.799040i \(0.294659\pi\)
\(252\) −2319.46 −0.579809
\(253\) −76.4687 −0.0190022
\(254\) 2216.37 0.547509
\(255\) −2625.08 −0.644663
\(256\) 358.498 0.0875239
\(257\) 6028.06 1.46311 0.731556 0.681781i \(-0.238794\pi\)
0.731556 + 0.681781i \(0.238794\pi\)
\(258\) 0 0
\(259\) −9124.80 −2.18914
\(260\) −1948.05 −0.464666
\(261\) −1453.87 −0.344798
\(262\) −652.681 −0.153904
\(263\) −1473.02 −0.345363 −0.172682 0.984978i \(-0.555243\pi\)
−0.172682 + 0.984978i \(0.555243\pi\)
\(264\) 1275.52 0.297359
\(265\) 1897.71 0.439907
\(266\) 3582.62 0.825806
\(267\) 3478.63 0.797337
\(268\) 45.5957 0.0103925
\(269\) 1629.48 0.369335 0.184668 0.982801i \(-0.440879\pi\)
0.184668 + 0.982801i \(0.440879\pi\)
\(270\) −1023.70 −0.230742
\(271\) −7123.49 −1.59676 −0.798379 0.602155i \(-0.794309\pi\)
−0.798379 + 0.602155i \(0.794309\pi\)
\(272\) 3849.90 0.858215
\(273\) 5264.38 1.16709
\(274\) 377.932 0.0833274
\(275\) −1589.45 −0.348536
\(276\) 104.049 0.0226922
\(277\) 4593.78 0.996438 0.498219 0.867051i \(-0.333988\pi\)
0.498219 + 0.867051i \(0.333988\pi\)
\(278\) 585.189 0.126249
\(279\) −518.792 −0.111324
\(280\) 3399.69 0.725609
\(281\) −3196.86 −0.678678 −0.339339 0.940664i \(-0.610203\pi\)
−0.339339 + 0.940664i \(0.610203\pi\)
\(282\) −2034.70 −0.429661
\(283\) 8652.58 1.81746 0.908732 0.417380i \(-0.137052\pi\)
0.908732 + 0.417380i \(0.137052\pi\)
\(284\) 4071.20 0.850638
\(285\) −3459.10 −0.718945
\(286\) 765.828 0.158337
\(287\) 14939.5 3.07264
\(288\) 1488.33 0.304516
\(289\) 2728.19 0.555300
\(290\) 1004.77 0.203456
\(291\) −2664.96 −0.536849
\(292\) 3027.20 0.606690
\(293\) 2526.24 0.503702 0.251851 0.967766i \(-0.418961\pi\)
0.251851 + 0.967766i \(0.418961\pi\)
\(294\) −3007.02 −0.596507
\(295\) −4153.07 −0.819665
\(296\) 3830.66 0.752204
\(297\) −3330.02 −0.650597
\(298\) 1192.89 0.231888
\(299\) 132.494 0.0256264
\(300\) 2162.73 0.416218
\(301\) 0 0
\(302\) −844.712 −0.160953
\(303\) 6762.33 1.28213
\(304\) 5073.06 0.957104
\(305\) −811.967 −0.152436
\(306\) 787.813 0.147177
\(307\) 2682.30 0.498654 0.249327 0.968419i \(-0.419791\pi\)
0.249327 + 0.968419i \(0.419791\pi\)
\(308\) 5214.39 0.964666
\(309\) −2243.48 −0.413032
\(310\) 358.539 0.0656891
\(311\) −3757.40 −0.685089 −0.342544 0.939502i \(-0.611289\pi\)
−0.342544 + 0.939502i \(0.611289\pi\)
\(312\) −2210.03 −0.401019
\(313\) 4548.20 0.821341 0.410670 0.911784i \(-0.365295\pi\)
0.410670 + 0.911784i \(0.365295\pi\)
\(314\) 1834.27 0.329663
\(315\) −2346.56 −0.419727
\(316\) 2825.27 0.502955
\(317\) −9653.47 −1.71039 −0.855193 0.518309i \(-0.826562\pi\)
−0.855193 + 0.518309i \(0.826562\pi\)
\(318\) 1015.12 0.179009
\(319\) 3268.45 0.573662
\(320\) 1515.58 0.264762
\(321\) 1070.28 0.186097
\(322\) −109.024 −0.0188686
\(323\) 10068.9 1.73451
\(324\) 2661.06 0.456286
\(325\) 2753.96 0.470038
\(326\) 1755.76 0.298290
\(327\) −7998.35 −1.35263
\(328\) −6271.69 −1.05578
\(329\) −17641.1 −2.95619
\(330\) 608.446 0.101497
\(331\) 5200.02 0.863501 0.431751 0.901993i \(-0.357896\pi\)
0.431751 + 0.901993i \(0.357896\pi\)
\(332\) −3372.13 −0.557439
\(333\) −2644.03 −0.435111
\(334\) −3410.01 −0.558645
\(335\) 46.1286 0.00752320
\(336\) −6134.01 −0.995946
\(337\) −10242.7 −1.65566 −0.827829 0.560980i \(-0.810424\pi\)
−0.827829 + 0.560980i \(0.810424\pi\)
\(338\) 713.552 0.114829
\(339\) 958.039 0.153491
\(340\) 4505.15 0.718607
\(341\) 1166.30 0.185216
\(342\) 1038.11 0.164136
\(343\) −14584.7 −2.29591
\(344\) 0 0
\(345\) 105.265 0.0164269
\(346\) 2288.69 0.355609
\(347\) −5275.43 −0.816137 −0.408069 0.912951i \(-0.633798\pi\)
−0.408069 + 0.912951i \(0.633798\pi\)
\(348\) −4447.32 −0.685061
\(349\) 76.7692 0.0117747 0.00588733 0.999983i \(-0.498126\pi\)
0.00588733 + 0.999983i \(0.498126\pi\)
\(350\) −2266.14 −0.346086
\(351\) 5769.76 0.877399
\(352\) −3345.92 −0.506643
\(353\) −1230.16 −0.185480 −0.0927402 0.995690i \(-0.529563\pi\)
−0.0927402 + 0.995690i \(0.529563\pi\)
\(354\) −2221.55 −0.333542
\(355\) 4118.78 0.615781
\(356\) −5970.01 −0.888792
\(357\) −12174.6 −1.80490
\(358\) 1073.47 0.158477
\(359\) 498.064 0.0732223 0.0366112 0.999330i \(-0.488344\pi\)
0.0366112 + 0.999330i \(0.488344\pi\)
\(360\) 985.104 0.144221
\(361\) 6408.87 0.934374
\(362\) 115.893 0.0168265
\(363\) −3556.22 −0.514196
\(364\) −9034.71 −1.30095
\(365\) 3062.58 0.439186
\(366\) −434.334 −0.0620301
\(367\) −8123.76 −1.15547 −0.577734 0.816225i \(-0.696063\pi\)
−0.577734 + 0.816225i \(0.696063\pi\)
\(368\) −154.380 −0.0218686
\(369\) 4328.90 0.610714
\(370\) 1827.30 0.256748
\(371\) 8801.23 1.23164
\(372\) −1586.96 −0.221183
\(373\) 4460.89 0.619238 0.309619 0.950861i \(-0.399798\pi\)
0.309619 + 0.950861i \(0.399798\pi\)
\(374\) −1771.09 −0.244868
\(375\) 5941.82 0.818224
\(376\) 7405.88 1.01577
\(377\) −5663.09 −0.773644
\(378\) −4747.73 −0.646024
\(379\) −3176.29 −0.430488 −0.215244 0.976560i \(-0.569055\pi\)
−0.215244 + 0.976560i \(0.569055\pi\)
\(380\) 5936.49 0.801409
\(381\) −9924.70 −1.33454
\(382\) −3643.15 −0.487958
\(383\) 9741.88 1.29970 0.649852 0.760061i \(-0.274831\pi\)
0.649852 + 0.760061i \(0.274831\pi\)
\(384\) 5913.65 0.785884
\(385\) 5275.33 0.698326
\(386\) 3084.52 0.406731
\(387\) 0 0
\(388\) 4573.60 0.598426
\(389\) −5989.93 −0.780723 −0.390362 0.920662i \(-0.627650\pi\)
−0.390362 + 0.920662i \(0.627650\pi\)
\(390\) −1054.22 −0.136879
\(391\) −306.410 −0.0396313
\(392\) 10944.9 1.41021
\(393\) 2922.65 0.375135
\(394\) 1027.73 0.131412
\(395\) 2858.29 0.364092
\(396\) 1510.93 0.191736
\(397\) −6376.95 −0.806171 −0.403086 0.915162i \(-0.632062\pi\)
−0.403086 + 0.915162i \(0.632062\pi\)
\(398\) 3426.93 0.431599
\(399\) −16042.7 −2.01288
\(400\) −3208.89 −0.401112
\(401\) 11130.6 1.38613 0.693065 0.720875i \(-0.256260\pi\)
0.693065 + 0.720875i \(0.256260\pi\)
\(402\) 24.6749 0.00306138
\(403\) −2020.79 −0.249784
\(404\) −11605.5 −1.42919
\(405\) 2692.16 0.330307
\(406\) 4659.95 0.569630
\(407\) 5944.06 0.723922
\(408\) 5111.00 0.620177
\(409\) −4881.98 −0.590216 −0.295108 0.955464i \(-0.595356\pi\)
−0.295108 + 0.955464i \(0.595356\pi\)
\(410\) −2991.72 −0.360367
\(411\) −1692.35 −0.203108
\(412\) 3850.24 0.460407
\(413\) −19261.2 −2.29487
\(414\) −31.5912 −0.00375029
\(415\) −3411.54 −0.403533
\(416\) 5797.32 0.683261
\(417\) −2620.43 −0.307729
\(418\) −2333.78 −0.273084
\(419\) −14222.9 −1.65831 −0.829156 0.559017i \(-0.811179\pi\)
−0.829156 + 0.559017i \(0.811179\pi\)
\(420\) −7178.02 −0.833933
\(421\) −11362.9 −1.31543 −0.657713 0.753269i \(-0.728476\pi\)
−0.657713 + 0.753269i \(0.728476\pi\)
\(422\) 745.385 0.0859829
\(423\) −5111.75 −0.587569
\(424\) −3694.82 −0.423199
\(425\) −6368.93 −0.726914
\(426\) 2203.20 0.250576
\(427\) −3765.75 −0.426786
\(428\) −1836.81 −0.207443
\(429\) −3429.31 −0.385941
\(430\) 0 0
\(431\) −8523.94 −0.952631 −0.476315 0.879275i \(-0.658028\pi\)
−0.476315 + 0.879275i \(0.658028\pi\)
\(432\) −6722.88 −0.748738
\(433\) 2382.07 0.264376 0.132188 0.991225i \(-0.457800\pi\)
0.132188 + 0.991225i \(0.457800\pi\)
\(434\) 1662.84 0.183914
\(435\) −4499.29 −0.495918
\(436\) 13726.7 1.50778
\(437\) −403.760 −0.0441979
\(438\) 1638.22 0.178716
\(439\) 7841.44 0.852509 0.426255 0.904603i \(-0.359833\pi\)
0.426255 + 0.904603i \(0.359833\pi\)
\(440\) −2214.62 −0.239950
\(441\) −7554.51 −0.815733
\(442\) 3068.68 0.330231
\(443\) 14488.4 1.55387 0.776934 0.629582i \(-0.216774\pi\)
0.776934 + 0.629582i \(0.216774\pi\)
\(444\) −8087.96 −0.864499
\(445\) −6039.78 −0.643400
\(446\) −4369.96 −0.463955
\(447\) −5341.68 −0.565219
\(448\) 7029.00 0.741270
\(449\) −4280.14 −0.449871 −0.224936 0.974374i \(-0.572217\pi\)
−0.224936 + 0.974374i \(0.572217\pi\)
\(450\) −656.642 −0.0687876
\(451\) −9731.83 −1.01608
\(452\) −1644.18 −0.171097
\(453\) 3782.55 0.392317
\(454\) −1207.39 −0.124815
\(455\) −9140.30 −0.941766
\(456\) 6734.82 0.691638
\(457\) −5468.42 −0.559742 −0.279871 0.960038i \(-0.590292\pi\)
−0.279871 + 0.960038i \(0.590292\pi\)
\(458\) 3717.36 0.379259
\(459\) −13343.4 −1.35690
\(460\) −180.656 −0.0183111
\(461\) −7987.57 −0.806981 −0.403490 0.914984i \(-0.632203\pi\)
−0.403490 + 0.914984i \(0.632203\pi\)
\(462\) 2821.86 0.284166
\(463\) −2657.37 −0.266735 −0.133368 0.991067i \(-0.542579\pi\)
−0.133368 + 0.991067i \(0.542579\pi\)
\(464\) 6598.58 0.660197
\(465\) −1605.51 −0.160115
\(466\) −6288.36 −0.625113
\(467\) 2485.15 0.246250 0.123125 0.992391i \(-0.460708\pi\)
0.123125 + 0.992391i \(0.460708\pi\)
\(468\) −2617.92 −0.258576
\(469\) 213.936 0.0210632
\(470\) 3532.74 0.346709
\(471\) −8213.72 −0.803542
\(472\) 8085.98 0.788533
\(473\) 0 0
\(474\) 1528.95 0.148158
\(475\) −8392.41 −0.810674
\(476\) 20894.1 2.01193
\(477\) 2550.27 0.244798
\(478\) 5603.56 0.536194
\(479\) −536.798 −0.0512044 −0.0256022 0.999672i \(-0.508150\pi\)
−0.0256022 + 0.999672i \(0.508150\pi\)
\(480\) 4605.93 0.437982
\(481\) −10299.0 −0.976285
\(482\) −1598.89 −0.151095
\(483\) 488.201 0.0459916
\(484\) 6103.17 0.573175
\(485\) 4627.05 0.433203
\(486\) −2387.72 −0.222858
\(487\) −8983.76 −0.835920 −0.417960 0.908465i \(-0.637255\pi\)
−0.417960 + 0.908465i \(0.637255\pi\)
\(488\) 1580.89 0.146646
\(489\) −7862.13 −0.727071
\(490\) 5220.94 0.481343
\(491\) −15775.6 −1.44998 −0.724992 0.688757i \(-0.758157\pi\)
−0.724992 + 0.688757i \(0.758157\pi\)
\(492\) 13241.9 1.21340
\(493\) 13096.7 1.19644
\(494\) 4043.63 0.368282
\(495\) 1528.59 0.138798
\(496\) 2354.61 0.213155
\(497\) 19102.1 1.72404
\(498\) −1824.89 −0.164207
\(499\) 9664.44 0.867014 0.433507 0.901150i \(-0.357276\pi\)
0.433507 + 0.901150i \(0.357276\pi\)
\(500\) −10197.3 −0.912076
\(501\) 15269.7 1.36168
\(502\) −4441.34 −0.394874
\(503\) −5422.87 −0.480704 −0.240352 0.970686i \(-0.577263\pi\)
−0.240352 + 0.970686i \(0.577263\pi\)
\(504\) 4568.73 0.403784
\(505\) −11741.1 −1.03460
\(506\) 71.0203 0.00623960
\(507\) −3195.23 −0.279891
\(508\) 17032.7 1.48761
\(509\) 18023.2 1.56948 0.784738 0.619828i \(-0.212798\pi\)
0.784738 + 0.619828i \(0.212798\pi\)
\(510\) 2438.05 0.211683
\(511\) 14203.7 1.22962
\(512\) −11708.5 −1.01064
\(513\) −17582.8 −1.51325
\(514\) −5598.56 −0.480432
\(515\) 3895.24 0.333291
\(516\) 0 0
\(517\) 11491.8 0.977576
\(518\) 8474.66 0.718833
\(519\) −10248.6 −0.866785
\(520\) 3837.16 0.323597
\(521\) 1580.04 0.132865 0.0664325 0.997791i \(-0.478838\pi\)
0.0664325 + 0.997791i \(0.478838\pi\)
\(522\) 1350.28 0.113219
\(523\) −12441.2 −1.04019 −0.520093 0.854110i \(-0.674103\pi\)
−0.520093 + 0.854110i \(0.674103\pi\)
\(524\) −5015.84 −0.418164
\(525\) 10147.6 0.843574
\(526\) 1368.07 0.113404
\(527\) 4673.37 0.386291
\(528\) 3995.81 0.329347
\(529\) −12154.7 −0.998990
\(530\) −1762.50 −0.144449
\(531\) −5581.17 −0.456125
\(532\) 27532.3 2.24376
\(533\) 16861.9 1.37030
\(534\) −3230.78 −0.261816
\(535\) −1858.28 −0.150169
\(536\) −89.8117 −0.00723746
\(537\) −4806.93 −0.386284
\(538\) −1513.38 −0.121276
\(539\) 16983.3 1.35719
\(540\) −7867.11 −0.626938
\(541\) −14128.8 −1.12282 −0.561408 0.827539i \(-0.689740\pi\)
−0.561408 + 0.827539i \(0.689740\pi\)
\(542\) 6615.94 0.524316
\(543\) −518.959 −0.0410141
\(544\) −13407.1 −1.05667
\(545\) 13887.2 1.09149
\(546\) −4889.30 −0.383228
\(547\) 8410.19 0.657393 0.328696 0.944436i \(-0.393391\pi\)
0.328696 + 0.944436i \(0.393391\pi\)
\(548\) 2904.40 0.226405
\(549\) −1091.17 −0.0848273
\(550\) 1476.20 0.114446
\(551\) 17257.7 1.33430
\(552\) −204.950 −0.0158030
\(553\) 13256.2 1.01937
\(554\) −4266.47 −0.327193
\(555\) −8182.48 −0.625814
\(556\) 4497.16 0.343025
\(557\) −7911.88 −0.601862 −0.300931 0.953646i \(-0.597297\pi\)
−0.300931 + 0.953646i \(0.597297\pi\)
\(558\) 481.828 0.0365545
\(559\) 0 0
\(560\) 10650.2 0.803666
\(561\) 7930.78 0.596859
\(562\) 2969.08 0.222853
\(563\) −2289.48 −0.171386 −0.0856929 0.996322i \(-0.527310\pi\)
−0.0856929 + 0.996322i \(0.527310\pi\)
\(564\) −15636.6 −1.16741
\(565\) −1663.40 −0.123858
\(566\) −8036.08 −0.596787
\(567\) 12485.7 0.924782
\(568\) −8019.21 −0.592392
\(569\) 8732.61 0.643392 0.321696 0.946843i \(-0.395747\pi\)
0.321696 + 0.946843i \(0.395747\pi\)
\(570\) 3212.64 0.236075
\(571\) 9031.46 0.661918 0.330959 0.943645i \(-0.392628\pi\)
0.330959 + 0.943645i \(0.392628\pi\)
\(572\) 5885.37 0.430209
\(573\) 16313.7 1.18938
\(574\) −13875.0 −1.00894
\(575\) 255.393 0.0185228
\(576\) 2036.74 0.147334
\(577\) 1205.57 0.0869821 0.0434910 0.999054i \(-0.486152\pi\)
0.0434910 + 0.999054i \(0.486152\pi\)
\(578\) −2533.81 −0.182340
\(579\) −13812.2 −0.991394
\(580\) 7721.66 0.552801
\(581\) −15822.1 −1.12980
\(582\) 2475.09 0.176281
\(583\) −5733.28 −0.407287
\(584\) −5962.80 −0.422504
\(585\) −2648.52 −0.187184
\(586\) −2346.25 −0.165397
\(587\) −3929.14 −0.276274 −0.138137 0.990413i \(-0.544111\pi\)
−0.138137 + 0.990413i \(0.544111\pi\)
\(588\) −23108.9 −1.62074
\(589\) 6158.15 0.430802
\(590\) 3857.17 0.269147
\(591\) −4602.09 −0.320312
\(592\) 12000.3 0.833123
\(593\) −12857.6 −0.890386 −0.445193 0.895435i \(-0.646865\pi\)
−0.445193 + 0.895435i \(0.646865\pi\)
\(594\) 3092.76 0.213632
\(595\) 21138.3 1.45644
\(596\) 9167.36 0.630050
\(597\) −15345.5 −1.05201
\(598\) −123.053 −0.00841476
\(599\) −16097.0 −1.09801 −0.549004 0.835820i \(-0.684993\pi\)
−0.549004 + 0.835820i \(0.684993\pi\)
\(600\) −4260.02 −0.289858
\(601\) 242.345 0.0164483 0.00822416 0.999966i \(-0.497382\pi\)
0.00822416 + 0.999966i \(0.497382\pi\)
\(602\) 0 0
\(603\) 61.9906 0.00418649
\(604\) −6491.59 −0.437316
\(605\) 6174.49 0.414924
\(606\) −6280.51 −0.421004
\(607\) 2132.21 0.142576 0.0712882 0.997456i \(-0.477289\pi\)
0.0712882 + 0.997456i \(0.477289\pi\)
\(608\) −17666.7 −1.17842
\(609\) −20866.9 −1.38845
\(610\) 754.114 0.0500544
\(611\) −19911.2 −1.31836
\(612\) 6054.32 0.399888
\(613\) −26948.2 −1.77557 −0.887787 0.460254i \(-0.847758\pi\)
−0.887787 + 0.460254i \(0.847758\pi\)
\(614\) −2491.19 −0.163740
\(615\) 13396.7 0.878382
\(616\) −10271.0 −0.671802
\(617\) −4081.93 −0.266341 −0.133170 0.991093i \(-0.542516\pi\)
−0.133170 + 0.991093i \(0.542516\pi\)
\(618\) 2083.63 0.135624
\(619\) −3662.59 −0.237822 −0.118911 0.992905i \(-0.537940\pi\)
−0.118911 + 0.992905i \(0.537940\pi\)
\(620\) 2755.36 0.178481
\(621\) 535.069 0.0345758
\(622\) 3489.69 0.224958
\(623\) −28011.4 −1.80137
\(624\) −6923.34 −0.444159
\(625\) −1209.06 −0.0773797
\(626\) −4224.14 −0.269698
\(627\) 10450.5 0.665633
\(628\) 14096.3 0.895710
\(629\) 23817.9 1.50983
\(630\) 2179.37 0.137823
\(631\) −15666.2 −0.988367 −0.494184 0.869358i \(-0.664533\pi\)
−0.494184 + 0.869358i \(0.664533\pi\)
\(632\) −5565.05 −0.350263
\(633\) −3337.77 −0.209581
\(634\) 8965.66 0.561627
\(635\) 17231.8 1.07689
\(636\) 7801.15 0.486377
\(637\) −29426.2 −1.83031
\(638\) −3035.58 −0.188369
\(639\) 5535.09 0.342668
\(640\) −10267.6 −0.634159
\(641\) −10576.0 −0.651679 −0.325839 0.945425i \(-0.605647\pi\)
−0.325839 + 0.945425i \(0.605647\pi\)
\(642\) −994.024 −0.0611075
\(643\) 3630.64 0.222672 0.111336 0.993783i \(-0.464487\pi\)
0.111336 + 0.993783i \(0.464487\pi\)
\(644\) −837.849 −0.0512669
\(645\) 0 0
\(646\) −9351.47 −0.569549
\(647\) 20208.7 1.22795 0.613977 0.789324i \(-0.289569\pi\)
0.613977 + 0.789324i \(0.289569\pi\)
\(648\) −5241.60 −0.317761
\(649\) 12547.1 0.758884
\(650\) −2557.74 −0.154343
\(651\) −7446.04 −0.448285
\(652\) 13493.0 0.810468
\(653\) −13988.7 −0.838316 −0.419158 0.907913i \(-0.637675\pi\)
−0.419158 + 0.907913i \(0.637675\pi\)
\(654\) 7428.47 0.444153
\(655\) −5074.46 −0.302711
\(656\) −19647.3 −1.16936
\(657\) 4115.70 0.244397
\(658\) 16384.2 0.970704
\(659\) −9951.46 −0.588245 −0.294123 0.955768i \(-0.595027\pi\)
−0.294123 + 0.955768i \(0.595027\pi\)
\(660\) 4675.89 0.275771
\(661\) 11708.5 0.688969 0.344484 0.938792i \(-0.388054\pi\)
0.344484 + 0.938792i \(0.388054\pi\)
\(662\) −4829.52 −0.283542
\(663\) −13741.3 −0.804927
\(664\) 6642.23 0.388206
\(665\) 27854.1 1.62426
\(666\) 2455.64 0.142874
\(667\) −525.176 −0.0304871
\(668\) −26205.8 −1.51787
\(669\) 19568.3 1.13088
\(670\) −42.8419 −0.00247034
\(671\) 2453.08 0.141133
\(672\) 21361.5 1.22624
\(673\) 29959.1 1.71595 0.857977 0.513688i \(-0.171721\pi\)
0.857977 + 0.513688i \(0.171721\pi\)
\(674\) 9512.93 0.543657
\(675\) 11121.7 0.634186
\(676\) 5483.63 0.311995
\(677\) 6509.81 0.369560 0.184780 0.982780i \(-0.440843\pi\)
0.184780 + 0.982780i \(0.440843\pi\)
\(678\) −889.779 −0.0504008
\(679\) 21459.4 1.21287
\(680\) −8873.99 −0.500444
\(681\) 5406.61 0.304232
\(682\) −1083.20 −0.0608181
\(683\) 18770.4 1.05158 0.525791 0.850614i \(-0.323770\pi\)
0.525791 + 0.850614i \(0.323770\pi\)
\(684\) 7977.85 0.445966
\(685\) 2938.34 0.163895
\(686\) 13545.5 0.753891
\(687\) −16646.0 −0.924432
\(688\) 0 0
\(689\) 9933.77 0.549269
\(690\) −97.7652 −0.00539400
\(691\) −23467.6 −1.29197 −0.645985 0.763350i \(-0.723553\pi\)
−0.645985 + 0.763350i \(0.723553\pi\)
\(692\) 17588.5 0.966206
\(693\) 7089.33 0.388602
\(694\) 4899.55 0.267989
\(695\) 4549.72 0.248318
\(696\) 8760.07 0.477083
\(697\) −38995.5 −2.11917
\(698\) −71.2994 −0.00386636
\(699\) 28158.8 1.52369
\(700\) −17415.2 −0.940333
\(701\) 12471.1 0.671938 0.335969 0.941873i \(-0.390936\pi\)
0.335969 + 0.941873i \(0.390936\pi\)
\(702\) −5358.67 −0.288105
\(703\) 31385.1 1.68380
\(704\) −4578.81 −0.245129
\(705\) −15819.3 −0.845093
\(706\) 1142.51 0.0609048
\(707\) −54453.1 −2.89663
\(708\) −17072.5 −0.906250
\(709\) 29195.7 1.54650 0.773250 0.634102i \(-0.218630\pi\)
0.773250 + 0.634102i \(0.218630\pi\)
\(710\) −3825.32 −0.202199
\(711\) 3841.16 0.202609
\(712\) 11759.4 0.618963
\(713\) −187.401 −0.00984325
\(714\) 11307.2 0.592663
\(715\) 5954.15 0.311430
\(716\) 8249.63 0.430591
\(717\) −25092.3 −1.30696
\(718\) −462.577 −0.0240435
\(719\) −14395.4 −0.746670 −0.373335 0.927697i \(-0.621786\pi\)
−0.373335 + 0.927697i \(0.621786\pi\)
\(720\) 3086.03 0.159736
\(721\) 18065.4 0.933135
\(722\) −5952.24 −0.306814
\(723\) 7159.71 0.368288
\(724\) 890.635 0.0457185
\(725\) −10916.1 −0.559192
\(726\) 3302.84 0.168843
\(727\) 19221.5 0.980587 0.490293 0.871557i \(-0.336890\pi\)
0.490293 + 0.871557i \(0.336890\pi\)
\(728\) 17796.0 0.905996
\(729\) 20758.5 1.05464
\(730\) −2844.37 −0.144212
\(731\) 0 0
\(732\) −3337.85 −0.168539
\(733\) −1107.28 −0.0557958 −0.0278979 0.999611i \(-0.508881\pi\)
−0.0278979 + 0.999611i \(0.508881\pi\)
\(734\) 7544.94 0.379412
\(735\) −23378.9 −1.17326
\(736\) 537.624 0.0269254
\(737\) −139.362 −0.00696533
\(738\) −4020.47 −0.200536
\(739\) −31591.9 −1.57257 −0.786283 0.617867i \(-0.787997\pi\)
−0.786283 + 0.617867i \(0.787997\pi\)
\(740\) 14042.7 0.697596
\(741\) −18107.0 −0.897676
\(742\) −8174.14 −0.404424
\(743\) 29466.4 1.45494 0.727468 0.686142i \(-0.240697\pi\)
0.727468 + 0.686142i \(0.240697\pi\)
\(744\) 3125.90 0.154034
\(745\) 9274.50 0.456096
\(746\) −4143.05 −0.203335
\(747\) −4584.66 −0.224557
\(748\) −13610.8 −0.665319
\(749\) −8618.35 −0.420437
\(750\) −5518.46 −0.268674
\(751\) −5615.56 −0.272856 −0.136428 0.990650i \(-0.543562\pi\)
−0.136428 + 0.990650i \(0.543562\pi\)
\(752\) 23200.4 1.12504
\(753\) 19888.0 0.962494
\(754\) 5259.59 0.254036
\(755\) −6567.46 −0.316575
\(756\) −36486.2 −1.75528
\(757\) −18192.9 −0.873488 −0.436744 0.899586i \(-0.643869\pi\)
−0.436744 + 0.899586i \(0.643869\pi\)
\(758\) 2949.98 0.141356
\(759\) −318.023 −0.0152088
\(760\) −11693.4 −0.558109
\(761\) 27160.4 1.29378 0.646889 0.762584i \(-0.276070\pi\)
0.646889 + 0.762584i \(0.276070\pi\)
\(762\) 9217.57 0.438212
\(763\) 64406.0 3.05590
\(764\) −27997.5 −1.32581
\(765\) 6125.08 0.289481
\(766\) −9047.77 −0.426774
\(767\) −21739.7 −1.02344
\(768\) 1490.94 0.0700518
\(769\) −31551.5 −1.47955 −0.739776 0.672853i \(-0.765069\pi\)
−0.739776 + 0.672853i \(0.765069\pi\)
\(770\) −4899.46 −0.229304
\(771\) 25069.9 1.17104
\(772\) 23704.5 1.10511
\(773\) 19613.7 0.912619 0.456309 0.889821i \(-0.349171\pi\)
0.456309 + 0.889821i \(0.349171\pi\)
\(774\) 0 0
\(775\) −3895.25 −0.180544
\(776\) −9008.81 −0.416749
\(777\) −37948.8 −1.75213
\(778\) 5563.14 0.256360
\(779\) −51384.8 −2.36335
\(780\) −8101.69 −0.371906
\(781\) −12443.5 −0.570119
\(782\) 284.579 0.0130134
\(783\) −22870.1 −1.04382
\(784\) 34287.2 1.56192
\(785\) 14261.1 0.648408
\(786\) −2714.41 −0.123180
\(787\) 19355.7 0.876693 0.438347 0.898806i \(-0.355564\pi\)
0.438347 + 0.898806i \(0.355564\pi\)
\(788\) 7898.08 0.357052
\(789\) −6126.11 −0.276420
\(790\) −2654.64 −0.119554
\(791\) −7714.53 −0.346772
\(792\) −2976.15 −0.133526
\(793\) −4250.33 −0.190332
\(794\) 5922.59 0.264717
\(795\) 7892.32 0.352090
\(796\) 26335.8 1.17267
\(797\) −15317.5 −0.680771 −0.340386 0.940286i \(-0.610558\pi\)
−0.340386 + 0.940286i \(0.610558\pi\)
\(798\) 14899.6 0.660954
\(799\) 46047.5 2.03885
\(800\) 11174.8 0.493863
\(801\) −8116.66 −0.358038
\(802\) −10337.6 −0.455153
\(803\) −9252.53 −0.406618
\(804\) 189.626 0.00831792
\(805\) −847.641 −0.0371123
\(806\) 1876.81 0.0820196
\(807\) 6776.79 0.295606
\(808\) 22859.8 0.995303
\(809\) −39005.9 −1.69515 −0.847573 0.530678i \(-0.821937\pi\)
−0.847573 + 0.530678i \(0.821937\pi\)
\(810\) −2500.34 −0.108461
\(811\) 28344.6 1.22727 0.613633 0.789591i \(-0.289707\pi\)
0.613633 + 0.789591i \(0.289707\pi\)
\(812\) 35811.6 1.54771
\(813\) −29625.6 −1.27800
\(814\) −5520.55 −0.237709
\(815\) 13650.6 0.586701
\(816\) 16011.2 0.686893
\(817\) 0 0
\(818\) 4534.14 0.193805
\(819\) −12283.3 −0.524071
\(820\) −22991.3 −0.979134
\(821\) −17104.4 −0.727096 −0.363548 0.931575i \(-0.618435\pi\)
−0.363548 + 0.931575i \(0.618435\pi\)
\(822\) 1571.77 0.0666931
\(823\) 5202.60 0.220354 0.110177 0.993912i \(-0.464858\pi\)
0.110177 + 0.993912i \(0.464858\pi\)
\(824\) −7583.98 −0.320632
\(825\) −6610.31 −0.278959
\(826\) 17888.8 0.753549
\(827\) 31811.8 1.33761 0.668805 0.743438i \(-0.266806\pi\)
0.668805 + 0.743438i \(0.266806\pi\)
\(828\) −242.777 −0.0101897
\(829\) 2820.60 0.118171 0.0590854 0.998253i \(-0.481182\pi\)
0.0590854 + 0.998253i \(0.481182\pi\)
\(830\) 3168.47 0.132505
\(831\) 19104.9 0.797523
\(832\) 7933.49 0.330582
\(833\) 68052.3 2.83058
\(834\) 2433.72 0.101047
\(835\) −26512.1 −1.09879
\(836\) −17935.1 −0.741982
\(837\) −8160.86 −0.337014
\(838\) 13209.5 0.544528
\(839\) −33523.4 −1.37945 −0.689723 0.724074i \(-0.742268\pi\)
−0.689723 + 0.724074i \(0.742268\pi\)
\(840\) 14138.9 0.580758
\(841\) −1941.73 −0.0796152
\(842\) 10553.3 0.431937
\(843\) −13295.3 −0.543196
\(844\) 5728.27 0.233620
\(845\) 5547.72 0.225855
\(846\) 4747.53 0.192936
\(847\) 28636.2 1.16169
\(848\) −11574.7 −0.468725
\(849\) 35984.9 1.45465
\(850\) 5915.15 0.238692
\(851\) −955.093 −0.0384726
\(852\) 16931.6 0.680829
\(853\) −4185.92 −0.168023 −0.0840113 0.996465i \(-0.526773\pi\)
−0.0840113 + 0.996465i \(0.526773\pi\)
\(854\) 3497.44 0.140140
\(855\) 8071.08 0.322837
\(856\) 3618.04 0.144465
\(857\) 32627.5 1.30051 0.650254 0.759717i \(-0.274663\pi\)
0.650254 + 0.759717i \(0.274663\pi\)
\(858\) 3184.97 0.126729
\(859\) −23475.1 −0.932433 −0.466217 0.884671i \(-0.654383\pi\)
−0.466217 + 0.884671i \(0.654383\pi\)
\(860\) 0 0
\(861\) 62131.2 2.45926
\(862\) 7916.61 0.312808
\(863\) −2337.41 −0.0921975 −0.0460987 0.998937i \(-0.514679\pi\)
−0.0460987 + 0.998937i \(0.514679\pi\)
\(864\) 23412.2 0.921873
\(865\) 17794.1 0.699441
\(866\) −2212.35 −0.0868113
\(867\) 11346.2 0.444448
\(868\) 12778.9 0.499704
\(869\) −8635.34 −0.337093
\(870\) 4178.72 0.162841
\(871\) 241.465 0.00939348
\(872\) −27038.1 −1.05003
\(873\) 6218.14 0.241068
\(874\) 374.992 0.0145129
\(875\) −47846.0 −1.84856
\(876\) 12589.7 0.485579
\(877\) −31546.5 −1.21465 −0.607326 0.794453i \(-0.707758\pi\)
−0.607326 + 0.794453i \(0.707758\pi\)
\(878\) −7282.74 −0.279932
\(879\) 10506.3 0.403150
\(880\) −6937.73 −0.265762
\(881\) −8467.25 −0.323801 −0.161901 0.986807i \(-0.551762\pi\)
−0.161901 + 0.986807i \(0.551762\pi\)
\(882\) 7016.25 0.267856
\(883\) −33603.1 −1.28067 −0.640336 0.768095i \(-0.721205\pi\)
−0.640336 + 0.768095i \(0.721205\pi\)
\(884\) 23582.7 0.897254
\(885\) −17272.1 −0.656039
\(886\) −13456.1 −0.510232
\(887\) −17129.2 −0.648415 −0.324207 0.945986i \(-0.605098\pi\)
−0.324207 + 0.945986i \(0.605098\pi\)
\(888\) 15931.2 0.602045
\(889\) 79917.8 3.01503
\(890\) 5609.45 0.211269
\(891\) −8133.43 −0.305814
\(892\) −33583.1 −1.26059
\(893\) 60677.3 2.27378
\(894\) 4961.09 0.185597
\(895\) 8346.04 0.311707
\(896\) −47619.1 −1.77550
\(897\) 551.023 0.0205107
\(898\) 3975.18 0.147721
\(899\) 8009.98 0.297161
\(900\) −5046.28 −0.186899
\(901\) −22973.3 −0.849446
\(902\) 9038.44 0.333644
\(903\) 0 0
\(904\) 3238.61 0.119153
\(905\) 901.043 0.0330958
\(906\) −3513.04 −0.128822
\(907\) 20927.5 0.766138 0.383069 0.923720i \(-0.374867\pi\)
0.383069 + 0.923720i \(0.374867\pi\)
\(908\) −9278.80 −0.339127
\(909\) −15778.5 −0.575730
\(910\) 8489.05 0.309241
\(911\) 46040.5 1.67441 0.837206 0.546888i \(-0.184188\pi\)
0.837206 + 0.546888i \(0.184188\pi\)
\(912\) 21098.2 0.766041
\(913\) 10306.8 0.373609
\(914\) 5078.80 0.183798
\(915\) −3376.86 −0.122006
\(916\) 28567.8 1.03047
\(917\) −23534.4 −0.847518
\(918\) 12392.7 0.445555
\(919\) −21510.4 −0.772104 −0.386052 0.922477i \(-0.626162\pi\)
−0.386052 + 0.922477i \(0.626162\pi\)
\(920\) 355.846 0.0127520
\(921\) 11155.3 0.399110
\(922\) 7418.46 0.264982
\(923\) 21560.2 0.768865
\(924\) 21685.9 0.772094
\(925\) −19852.2 −0.705661
\(926\) 2468.03 0.0875860
\(927\) 5234.68 0.185469
\(928\) −22979.3 −0.812859
\(929\) 6327.56 0.223467 0.111733 0.993738i \(-0.464360\pi\)
0.111733 + 0.993738i \(0.464360\pi\)
\(930\) 1491.11 0.0525759
\(931\) 89673.3 3.15674
\(932\) −48325.9 −1.69846
\(933\) −15626.5 −0.548327
\(934\) −2308.08 −0.0808594
\(935\) −13769.8 −0.481628
\(936\) 5156.63 0.180075
\(937\) −18761.8 −0.654133 −0.327066 0.945001i \(-0.606060\pi\)
−0.327066 + 0.945001i \(0.606060\pi\)
\(938\) −198.693 −0.00691636
\(939\) 18915.4 0.657380
\(940\) 27149.1 0.942026
\(941\) 638.717 0.0221271 0.0110635 0.999939i \(-0.496478\pi\)
0.0110635 + 0.999939i \(0.496478\pi\)
\(942\) 7628.50 0.263853
\(943\) 1563.71 0.0539995
\(944\) 25330.9 0.873359
\(945\) −36912.6 −1.27065
\(946\) 0 0
\(947\) −11356.8 −0.389699 −0.194850 0.980833i \(-0.562422\pi\)
−0.194850 + 0.980833i \(0.562422\pi\)
\(948\) 11749.9 0.402552
\(949\) 16031.4 0.548368
\(950\) 7794.45 0.266195
\(951\) −40147.5 −1.36895
\(952\) −41155.9 −1.40112
\(953\) 18470.0 0.627809 0.313904 0.949455i \(-0.398363\pi\)
0.313904 + 0.949455i \(0.398363\pi\)
\(954\) −2368.56 −0.0803827
\(955\) −28324.7 −0.959756
\(956\) 43063.2 1.45687
\(957\) 13593.1 0.459144
\(958\) 498.551 0.0168136
\(959\) 13627.5 0.458868
\(960\) 6303.11 0.211908
\(961\) −26932.8 −0.904057
\(962\) 9565.18 0.320576
\(963\) −2497.28 −0.0835655
\(964\) −12287.5 −0.410531
\(965\) 23981.5 0.799993
\(966\) −453.417 −0.0151019
\(967\) −29051.7 −0.966120 −0.483060 0.875587i \(-0.660475\pi\)
−0.483060 + 0.875587i \(0.660475\pi\)
\(968\) −12021.7 −0.399164
\(969\) 41875.1 1.38826
\(970\) −4297.37 −0.142248
\(971\) −10491.2 −0.346734 −0.173367 0.984857i \(-0.555465\pi\)
−0.173367 + 0.984857i \(0.555465\pi\)
\(972\) −18349.6 −0.605517
\(973\) 21100.8 0.695230
\(974\) 8343.67 0.274485
\(975\) 11453.4 0.376206
\(976\) 4952.44 0.162422
\(977\) −1951.61 −0.0639075 −0.0319537 0.999489i \(-0.510173\pi\)
−0.0319537 + 0.999489i \(0.510173\pi\)
\(978\) 7301.96 0.238743
\(979\) 18247.1 0.595690
\(980\) 40122.8 1.30783
\(981\) 18662.5 0.607387
\(982\) 14651.6 0.476121
\(983\) −58691.5 −1.90434 −0.952172 0.305563i \(-0.901155\pi\)
−0.952172 + 0.305563i \(0.901155\pi\)
\(984\) −26083.1 −0.845020
\(985\) 7990.38 0.258472
\(986\) −12163.6 −0.392867
\(987\) −73367.1 −2.36606
\(988\) 31075.2 1.00064
\(989\) 0 0
\(990\) −1419.68 −0.0455762
\(991\) −7723.20 −0.247564 −0.123782 0.992309i \(-0.539502\pi\)
−0.123782 + 0.992309i \(0.539502\pi\)
\(992\) −8199.83 −0.262445
\(993\) 21626.2 0.691124
\(994\) −17741.1 −0.566111
\(995\) 26643.6 0.848904
\(996\) −14024.2 −0.446160
\(997\) 51198.5 1.62635 0.813176 0.582018i \(-0.197737\pi\)
0.813176 + 0.582018i \(0.197737\pi\)
\(998\) −8975.85 −0.284695
\(999\) −41591.9 −1.31723
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1849.4.a.i.1.23 50
43.42 odd 2 1849.4.a.j.1.28 yes 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.23 50 1.1 even 1 trivial
1849.4.a.j.1.28 yes 50 43.42 odd 2