Properties

Label 320.6.a.q.1.1
Level 320
Weight 6
Character 320.1
Self dual yes
Analytic conductor 51.323
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{129}) \)
Defining polynomial: \(x^{2} - x - 32\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(6.17891\) of defining polynomial
Character \(\chi\) \(=\) 320.1

$q$-expansion

\(f(q)\) \(=\) \(q-28.7156 q^{3} -25.0000 q^{5} +42.1469 q^{7} +581.588 q^{9} +O(q^{10})\) \(q-28.7156 q^{3} -25.0000 q^{5} +42.1469 q^{7} +581.588 q^{9} +416.294 q^{11} -966.588 q^{13} +717.891 q^{15} -1834.11 q^{17} +317.763 q^{19} -1210.27 q^{21} -1568.02 q^{23} +625.000 q^{25} -9722.76 q^{27} -7757.28 q^{29} -102.644 q^{31} -11954.1 q^{33} -1053.67 q^{35} -1936.58 q^{37} +27756.2 q^{39} +7994.36 q^{41} +16542.6 q^{43} -14539.7 q^{45} -18649.3 q^{47} -15030.6 q^{49} +52667.7 q^{51} +14972.4 q^{53} -10407.3 q^{55} -9124.76 q^{57} +19843.3 q^{59} +18024.1 q^{61} +24512.1 q^{63} +24164.7 q^{65} -55040.6 q^{67} +45026.8 q^{69} -11201.3 q^{71} -4013.95 q^{73} -17947.3 q^{75} +17545.5 q^{77} -24018.8 q^{79} +137869. q^{81} +70512.8 q^{83} +45852.8 q^{85} +222755. q^{87} -60765.7 q^{89} -40738.7 q^{91} +2947.49 q^{93} -7944.07 q^{95} -31112.2 q^{97} +242111. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 12q^{3} - 50q^{5} - 52q^{7} + 618q^{9} + O(q^{10}) \) \( 2q - 12q^{3} - 50q^{5} - 52q^{7} + 618q^{9} + 560q^{11} - 1388q^{13} + 300q^{15} + 148q^{17} - 1000q^{19} - 2784q^{21} + 2452q^{23} + 1250q^{25} - 13176q^{27} - 1340q^{29} + 2248q^{31} - 9552q^{33} + 1300q^{35} + 5940q^{37} + 20712q^{39} + 23076q^{41} + 17684q^{43} - 15450q^{45} + 2908q^{47} - 22974q^{49} + 85800q^{51} + 5412q^{53} - 14000q^{55} - 31152q^{57} + 62584q^{59} - 14108q^{61} + 21084q^{63} + 34700q^{65} - 85412q^{67} + 112224q^{69} - 47208q^{71} - 67452q^{73} - 7500q^{75} + 4016q^{77} + 65904q^{79} + 71298q^{81} + 108724q^{83} - 3700q^{85} + 330024q^{87} - 55020q^{89} - 1064q^{91} + 42240q^{93} + 25000q^{95} + 147668q^{97} + 247344q^{99} + O(q^{100}) \)

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 0
\(3\) −28.7156 −1.84211 −0.921054 0.389434i \(-0.872671\pi\)
−0.921054 + 0.389434i \(0.872671\pi\)
\(4\) 0 0
\(5\) −25.0000 −0.447214
\(6\) 0 0
\(7\) 42.1469 0.325103 0.162551 0.986700i \(-0.448028\pi\)
0.162551 + 0.986700i \(0.448028\pi\)
\(8\) 0 0
\(9\) 581.588 2.39336
\(10\) 0 0
\(11\) 416.294 1.03733 0.518667 0.854977i \(-0.326429\pi\)
0.518667 + 0.854977i \(0.326429\pi\)
\(12\) 0 0
\(13\) −966.588 −1.58629 −0.793145 0.609032i \(-0.791558\pi\)
−0.793145 + 0.609032i \(0.791558\pi\)
\(14\) 0 0
\(15\) 717.891 0.823816
\(16\) 0 0
\(17\) −1834.11 −1.53923 −0.769616 0.638508i \(-0.779552\pi\)
−0.769616 + 0.638508i \(0.779552\pi\)
\(18\) 0 0
\(19\) 317.763 0.201938 0.100969 0.994890i \(-0.467806\pi\)
0.100969 + 0.994890i \(0.467806\pi\)
\(20\) 0 0
\(21\) −1210.27 −0.598874
\(22\) 0 0
\(23\) −1568.02 −0.618063 −0.309032 0.951052i \(-0.600005\pi\)
−0.309032 + 0.951052i \(0.600005\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) 0 0
\(27\) −9722.76 −2.56673
\(28\) 0 0
\(29\) −7757.28 −1.71283 −0.856415 0.516288i \(-0.827313\pi\)
−0.856415 + 0.516288i \(0.827313\pi\)
\(30\) 0 0
\(31\) −102.644 −0.0191836 −0.00959180 0.999954i \(-0.503053\pi\)
−0.00959180 + 0.999954i \(0.503053\pi\)
\(32\) 0 0
\(33\) −11954.1 −1.91088
\(34\) 0 0
\(35\) −1053.67 −0.145390
\(36\) 0 0
\(37\) −1936.58 −0.232558 −0.116279 0.993217i \(-0.537097\pi\)
−0.116279 + 0.993217i \(0.537097\pi\)
\(38\) 0 0
\(39\) 27756.2 2.92212
\(40\) 0 0
\(41\) 7994.36 0.742718 0.371359 0.928489i \(-0.378892\pi\)
0.371359 + 0.928489i \(0.378892\pi\)
\(42\) 0 0
\(43\) 16542.6 1.36437 0.682186 0.731179i \(-0.261030\pi\)
0.682186 + 0.731179i \(0.261030\pi\)
\(44\) 0 0
\(45\) −14539.7 −1.07035
\(46\) 0 0
\(47\) −18649.3 −1.23146 −0.615728 0.787959i \(-0.711138\pi\)
−0.615728 + 0.787959i \(0.711138\pi\)
\(48\) 0 0
\(49\) −15030.6 −0.894308
\(50\) 0 0
\(51\) 52667.7 2.83543
\(52\) 0 0
\(53\) 14972.4 0.732155 0.366077 0.930584i \(-0.380701\pi\)
0.366077 + 0.930584i \(0.380701\pi\)
\(54\) 0 0
\(55\) −10407.3 −0.463909
\(56\) 0 0
\(57\) −9124.76 −0.371993
\(58\) 0 0
\(59\) 19843.3 0.742137 0.371069 0.928605i \(-0.378991\pi\)
0.371069 + 0.928605i \(0.378991\pi\)
\(60\) 0 0
\(61\) 18024.1 0.620195 0.310097 0.950705i \(-0.399638\pi\)
0.310097 + 0.950705i \(0.399638\pi\)
\(62\) 0 0
\(63\) 24512.1 0.778089
\(64\) 0 0
\(65\) 24164.7 0.709411
\(66\) 0 0
\(67\) −55040.6 −1.49795 −0.748973 0.662601i \(-0.769453\pi\)
−0.748973 + 0.662601i \(0.769453\pi\)
\(68\) 0 0
\(69\) 45026.8 1.13854
\(70\) 0 0
\(71\) −11201.3 −0.263707 −0.131853 0.991269i \(-0.542093\pi\)
−0.131853 + 0.991269i \(0.542093\pi\)
\(72\) 0 0
\(73\) −4013.95 −0.0881587 −0.0440793 0.999028i \(-0.514035\pi\)
−0.0440793 + 0.999028i \(0.514035\pi\)
\(74\) 0 0
\(75\) −17947.3 −0.368422
\(76\) 0 0
\(77\) 17545.5 0.337240
\(78\) 0 0
\(79\) −24018.8 −0.432996 −0.216498 0.976283i \(-0.569463\pi\)
−0.216498 + 0.976283i \(0.569463\pi\)
\(80\) 0 0
\(81\) 137869. 2.33483
\(82\) 0 0
\(83\) 70512.8 1.12350 0.561750 0.827307i \(-0.310128\pi\)
0.561750 + 0.827307i \(0.310128\pi\)
\(84\) 0 0
\(85\) 45852.8 0.688365
\(86\) 0 0
\(87\) 222755. 3.15522
\(88\) 0 0
\(89\) −60765.7 −0.813174 −0.406587 0.913612i \(-0.633281\pi\)
−0.406587 + 0.913612i \(0.633281\pi\)
\(90\) 0 0
\(91\) −40738.7 −0.515707
\(92\) 0 0
\(93\) 2947.49 0.0353383
\(94\) 0 0
\(95\) −7944.07 −0.0903096
\(96\) 0 0
\(97\) −31112.2 −0.335739 −0.167869 0.985809i \(-0.553689\pi\)
−0.167869 + 0.985809i \(0.553689\pi\)
\(98\) 0 0
\(99\) 242111. 2.48272
\(100\) 0 0
\(101\) 49491.4 0.482755 0.241377 0.970431i \(-0.422401\pi\)
0.241377 + 0.970431i \(0.422401\pi\)
\(102\) 0 0
\(103\) 95117.9 0.883424 0.441712 0.897157i \(-0.354371\pi\)
0.441712 + 0.897157i \(0.354371\pi\)
\(104\) 0 0
\(105\) 30256.9 0.267825
\(106\) 0 0
\(107\) 88429.8 0.746688 0.373344 0.927693i \(-0.378211\pi\)
0.373344 + 0.927693i \(0.378211\pi\)
\(108\) 0 0
\(109\) 10598.9 0.0854464 0.0427232 0.999087i \(-0.486397\pi\)
0.0427232 + 0.999087i \(0.486397\pi\)
\(110\) 0 0
\(111\) 55610.0 0.428396
\(112\) 0 0
\(113\) 235124. 1.73221 0.866107 0.499859i \(-0.166615\pi\)
0.866107 + 0.499859i \(0.166615\pi\)
\(114\) 0 0
\(115\) 39200.6 0.276406
\(116\) 0 0
\(117\) −562155. −3.79657
\(118\) 0 0
\(119\) −77302.2 −0.500408
\(120\) 0 0
\(121\) 12249.5 0.0760599
\(122\) 0 0
\(123\) −229563. −1.36817
\(124\) 0 0
\(125\) −15625.0 −0.0894427
\(126\) 0 0
\(127\) 102533. 0.564098 0.282049 0.959400i \(-0.408986\pi\)
0.282049 + 0.959400i \(0.408986\pi\)
\(128\) 0 0
\(129\) −475031. −2.51332
\(130\) 0 0
\(131\) 50603.0 0.257631 0.128815 0.991669i \(-0.458883\pi\)
0.128815 + 0.991669i \(0.458883\pi\)
\(132\) 0 0
\(133\) 13392.7 0.0656507
\(134\) 0 0
\(135\) 243069. 1.14788
\(136\) 0 0
\(137\) 47811.7 0.217637 0.108818 0.994062i \(-0.465293\pi\)
0.108818 + 0.994062i \(0.465293\pi\)
\(138\) 0 0
\(139\) −220190. −0.966629 −0.483314 0.875447i \(-0.660567\pi\)
−0.483314 + 0.875447i \(0.660567\pi\)
\(140\) 0 0
\(141\) 535527. 2.26847
\(142\) 0 0
\(143\) −402384. −1.64551
\(144\) 0 0
\(145\) 193932. 0.766001
\(146\) 0 0
\(147\) 431614. 1.64741
\(148\) 0 0
\(149\) −154533. −0.570236 −0.285118 0.958492i \(-0.592033\pi\)
−0.285118 + 0.958492i \(0.592033\pi\)
\(150\) 0 0
\(151\) 455395. 1.62534 0.812672 0.582721i \(-0.198012\pi\)
0.812672 + 0.582721i \(0.198012\pi\)
\(152\) 0 0
\(153\) −1.06670e6 −3.68394
\(154\) 0 0
\(155\) 2566.11 0.00857917
\(156\) 0 0
\(157\) −361690. −1.17108 −0.585541 0.810643i \(-0.699118\pi\)
−0.585541 + 0.810643i \(0.699118\pi\)
\(158\) 0 0
\(159\) −429943. −1.34871
\(160\) 0 0
\(161\) −66087.3 −0.200934
\(162\) 0 0
\(163\) 490843. 1.44702 0.723508 0.690316i \(-0.242528\pi\)
0.723508 + 0.690316i \(0.242528\pi\)
\(164\) 0 0
\(165\) 298854. 0.854572
\(166\) 0 0
\(167\) 196273. 0.544591 0.272295 0.962214i \(-0.412217\pi\)
0.272295 + 0.962214i \(0.412217\pi\)
\(168\) 0 0
\(169\) 562999. 1.51632
\(170\) 0 0
\(171\) 184807. 0.483312
\(172\) 0 0
\(173\) −183395. −0.465877 −0.232938 0.972491i \(-0.574834\pi\)
−0.232938 + 0.972491i \(0.574834\pi\)
\(174\) 0 0
\(175\) 26341.8 0.0650205
\(176\) 0 0
\(177\) −569814. −1.36710
\(178\) 0 0
\(179\) −38660.0 −0.0901839 −0.0450920 0.998983i \(-0.514358\pi\)
−0.0450920 + 0.998983i \(0.514358\pi\)
\(180\) 0 0
\(181\) 561287. 1.27347 0.636735 0.771083i \(-0.280285\pi\)
0.636735 + 0.771083i \(0.280285\pi\)
\(182\) 0 0
\(183\) −517572. −1.14247
\(184\) 0 0
\(185\) 48414.4 0.104003
\(186\) 0 0
\(187\) −763530. −1.59670
\(188\) 0 0
\(189\) −409784. −0.834450
\(190\) 0 0
\(191\) −265393. −0.526387 −0.263194 0.964743i \(-0.584776\pi\)
−0.263194 + 0.964743i \(0.584776\pi\)
\(192\) 0 0
\(193\) 863148. 1.66798 0.833992 0.551776i \(-0.186050\pi\)
0.833992 + 0.551776i \(0.186050\pi\)
\(194\) 0 0
\(195\) −693904. −1.30681
\(196\) 0 0
\(197\) −281871. −0.517469 −0.258734 0.965949i \(-0.583305\pi\)
−0.258734 + 0.965949i \(0.583305\pi\)
\(198\) 0 0
\(199\) 192798. 0.345120 0.172560 0.984999i \(-0.444796\pi\)
0.172560 + 0.984999i \(0.444796\pi\)
\(200\) 0 0
\(201\) 1.58053e6 2.75938
\(202\) 0 0
\(203\) −326945. −0.556846
\(204\) 0 0
\(205\) −199859. −0.332154
\(206\) 0 0
\(207\) −911943. −1.47925
\(208\) 0 0
\(209\) 132283. 0.209477
\(210\) 0 0
\(211\) 44631.4 0.0690136 0.0345068 0.999404i \(-0.489014\pi\)
0.0345068 + 0.999404i \(0.489014\pi\)
\(212\) 0 0
\(213\) 321651. 0.485776
\(214\) 0 0
\(215\) −413565. −0.610165
\(216\) 0 0
\(217\) −4326.13 −0.00623664
\(218\) 0 0
\(219\) 115263. 0.162398
\(220\) 0 0
\(221\) 1.77283e6 2.44167
\(222\) 0 0
\(223\) 904469. 1.21796 0.608978 0.793187i \(-0.291580\pi\)
0.608978 + 0.793187i \(0.291580\pi\)
\(224\) 0 0
\(225\) 363492. 0.478673
\(226\) 0 0
\(227\) −1.02684e6 −1.32263 −0.661314 0.750109i \(-0.730001\pi\)
−0.661314 + 0.750109i \(0.730001\pi\)
\(228\) 0 0
\(229\) 101521. 0.127928 0.0639641 0.997952i \(-0.479626\pi\)
0.0639641 + 0.997952i \(0.479626\pi\)
\(230\) 0 0
\(231\) −503830. −0.621232
\(232\) 0 0
\(233\) −313713. −0.378567 −0.189283 0.981923i \(-0.560616\pi\)
−0.189283 + 0.981923i \(0.560616\pi\)
\(234\) 0 0
\(235\) 466233. 0.550724
\(236\) 0 0
\(237\) 689715. 0.797625
\(238\) 0 0
\(239\) 1.53093e6 1.73365 0.866825 0.498612i \(-0.166157\pi\)
0.866825 + 0.498612i \(0.166157\pi\)
\(240\) 0 0
\(241\) 506999. 0.562296 0.281148 0.959664i \(-0.409285\pi\)
0.281148 + 0.959664i \(0.409285\pi\)
\(242\) 0 0
\(243\) −1.59638e6 −1.73428
\(244\) 0 0
\(245\) 375766. 0.399947
\(246\) 0 0
\(247\) −307146. −0.320333
\(248\) 0 0
\(249\) −2.02482e6 −2.06961
\(250\) 0 0
\(251\) −511659. −0.512621 −0.256311 0.966594i \(-0.582507\pi\)
−0.256311 + 0.966594i \(0.582507\pi\)
\(252\) 0 0
\(253\) −652758. −0.641137
\(254\) 0 0
\(255\) −1.31669e6 −1.26804
\(256\) 0 0
\(257\) −256082. −0.241850 −0.120925 0.992662i \(-0.538586\pi\)
−0.120925 + 0.992662i \(0.538586\pi\)
\(258\) 0 0
\(259\) −81620.7 −0.0756051
\(260\) 0 0
\(261\) −4.51154e6 −4.09943
\(262\) 0 0
\(263\) −1.32451e6 −1.18077 −0.590386 0.807121i \(-0.701024\pi\)
−0.590386 + 0.807121i \(0.701024\pi\)
\(264\) 0 0
\(265\) −374311. −0.327430
\(266\) 0 0
\(267\) 1.74493e6 1.49795
\(268\) 0 0
\(269\) 1.42989e6 1.20482 0.602410 0.798187i \(-0.294207\pi\)
0.602410 + 0.798187i \(0.294207\pi\)
\(270\) 0 0
\(271\) 706426. 0.584310 0.292155 0.956371i \(-0.405628\pi\)
0.292155 + 0.956371i \(0.405628\pi\)
\(272\) 0 0
\(273\) 1.16984e6 0.949989
\(274\) 0 0
\(275\) 260184. 0.207467
\(276\) 0 0
\(277\) −314677. −0.246414 −0.123207 0.992381i \(-0.539318\pi\)
−0.123207 + 0.992381i \(0.539318\pi\)
\(278\) 0 0
\(279\) −59696.6 −0.0459134
\(280\) 0 0
\(281\) −437793. −0.330752 −0.165376 0.986231i \(-0.552884\pi\)
−0.165376 + 0.986231i \(0.552884\pi\)
\(282\) 0 0
\(283\) −2.08248e6 −1.54566 −0.772831 0.634612i \(-0.781160\pi\)
−0.772831 + 0.634612i \(0.781160\pi\)
\(284\) 0 0
\(285\) 228119. 0.166360
\(286\) 0 0
\(287\) 336938. 0.241460
\(288\) 0 0
\(289\) 1.94411e6 1.36923
\(290\) 0 0
\(291\) 893407. 0.618468
\(292\) 0 0
\(293\) −90716.3 −0.0617329 −0.0308664 0.999524i \(-0.509827\pi\)
−0.0308664 + 0.999524i \(0.509827\pi\)
\(294\) 0 0
\(295\) −496083. −0.331894
\(296\) 0 0
\(297\) −4.04752e6 −2.66255
\(298\) 0 0
\(299\) 1.51563e6 0.980428
\(300\) 0 0
\(301\) 697219. 0.443561
\(302\) 0 0
\(303\) −1.42118e6 −0.889286
\(304\) 0 0
\(305\) −450601. −0.277359
\(306\) 0 0
\(307\) −571699. −0.346196 −0.173098 0.984905i \(-0.555378\pi\)
−0.173098 + 0.984905i \(0.555378\pi\)
\(308\) 0 0
\(309\) −2.73137e6 −1.62736
\(310\) 0 0
\(311\) −2.59515e6 −1.52147 −0.760733 0.649065i \(-0.775160\pi\)
−0.760733 + 0.649065i \(0.775160\pi\)
\(312\) 0 0
\(313\) −510659. −0.294625 −0.147313 0.989090i \(-0.547062\pi\)
−0.147313 + 0.989090i \(0.547062\pi\)
\(314\) 0 0
\(315\) −612803. −0.347972
\(316\) 0 0
\(317\) 3.37030e6 1.88374 0.941868 0.335984i \(-0.109069\pi\)
0.941868 + 0.335984i \(0.109069\pi\)
\(318\) 0 0
\(319\) −3.22931e6 −1.77678
\(320\) 0 0
\(321\) −2.53932e6 −1.37548
\(322\) 0 0
\(323\) −582813. −0.310830
\(324\) 0 0
\(325\) −604117. −0.317258
\(326\) 0 0
\(327\) −304354. −0.157402
\(328\) 0 0
\(329\) −786012. −0.400349
\(330\) 0 0
\(331\) 3.80172e6 1.90726 0.953632 0.300976i \(-0.0973125\pi\)
0.953632 + 0.300976i \(0.0973125\pi\)
\(332\) 0 0
\(333\) −1.12629e6 −0.556595
\(334\) 0 0
\(335\) 1.37601e6 0.669902
\(336\) 0 0
\(337\) 2.06627e6 0.991088 0.495544 0.868583i \(-0.334969\pi\)
0.495544 + 0.868583i \(0.334969\pi\)
\(338\) 0 0
\(339\) −6.75174e6 −3.19093
\(340\) 0 0
\(341\) −42730.1 −0.0198998
\(342\) 0 0
\(343\) −1.34186e6 −0.615845
\(344\) 0 0
\(345\) −1.12567e6 −0.509170
\(346\) 0 0
\(347\) 2.35066e6 1.04801 0.524005 0.851715i \(-0.324437\pi\)
0.524005 + 0.851715i \(0.324437\pi\)
\(348\) 0 0
\(349\) −1.67220e6 −0.734896 −0.367448 0.930044i \(-0.619768\pi\)
−0.367448 + 0.930044i \(0.619768\pi\)
\(350\) 0 0
\(351\) 9.39790e6 4.07158
\(352\) 0 0
\(353\) 1.71355e6 0.731914 0.365957 0.930632i \(-0.380742\pi\)
0.365957 + 0.930632i \(0.380742\pi\)
\(354\) 0 0
\(355\) 280032. 0.117933
\(356\) 0 0
\(357\) 2.21978e6 0.921806
\(358\) 0 0
\(359\) 3.85773e6 1.57978 0.789888 0.613251i \(-0.210139\pi\)
0.789888 + 0.613251i \(0.210139\pi\)
\(360\) 0 0
\(361\) −2.37513e6 −0.959221
\(362\) 0 0
\(363\) −351753. −0.140111
\(364\) 0 0
\(365\) 100349. 0.0394258
\(366\) 0 0
\(367\) 3.18109e6 1.23285 0.616427 0.787412i \(-0.288580\pi\)
0.616427 + 0.787412i \(0.288580\pi\)
\(368\) 0 0
\(369\) 4.64942e6 1.77760
\(370\) 0 0
\(371\) 631042. 0.238026
\(372\) 0 0
\(373\) 4.79295e6 1.78374 0.891868 0.452295i \(-0.149395\pi\)
0.891868 + 0.452295i \(0.149395\pi\)
\(374\) 0 0
\(375\) 448682. 0.164763
\(376\) 0 0
\(377\) 7.49809e6 2.71705
\(378\) 0 0
\(379\) 7018.36 0.00250979 0.00125490 0.999999i \(-0.499601\pi\)
0.00125490 + 0.999999i \(0.499601\pi\)
\(380\) 0 0
\(381\) −2.94430e6 −1.03913
\(382\) 0 0
\(383\) 694105. 0.241784 0.120892 0.992666i \(-0.461424\pi\)
0.120892 + 0.992666i \(0.461424\pi\)
\(384\) 0 0
\(385\) −438637. −0.150818
\(386\) 0 0
\(387\) 9.62097e6 3.26544
\(388\) 0 0
\(389\) 53514.1 0.0179306 0.00896529 0.999960i \(-0.497146\pi\)
0.00896529 + 0.999960i \(0.497146\pi\)
\(390\) 0 0
\(391\) 2.87593e6 0.951342
\(392\) 0 0
\(393\) −1.45310e6 −0.474584
\(394\) 0 0
\(395\) 600470. 0.193642
\(396\) 0 0
\(397\) −907937. −0.289121 −0.144560 0.989496i \(-0.546177\pi\)
−0.144560 + 0.989496i \(0.546177\pi\)
\(398\) 0 0
\(399\) −384580. −0.120936
\(400\) 0 0
\(401\) −514404. −0.159751 −0.0798755 0.996805i \(-0.525452\pi\)
−0.0798755 + 0.996805i \(0.525452\pi\)
\(402\) 0 0
\(403\) 99214.6 0.0304308
\(404\) 0 0
\(405\) −3.44673e6 −1.04417
\(406\) 0 0
\(407\) −806185. −0.241240
\(408\) 0 0
\(409\) 5.61814e6 1.66067 0.830337 0.557262i \(-0.188148\pi\)
0.830337 + 0.557262i \(0.188148\pi\)
\(410\) 0 0
\(411\) −1.37294e6 −0.400911
\(412\) 0 0
\(413\) 836334. 0.241271
\(414\) 0 0
\(415\) −1.76282e6 −0.502444
\(416\) 0 0
\(417\) 6.32288e6 1.78064
\(418\) 0 0
\(419\) 708382. 0.197121 0.0985605 0.995131i \(-0.468576\pi\)
0.0985605 + 0.995131i \(0.468576\pi\)
\(420\) 0 0
\(421\) 3.91741e6 1.07719 0.538597 0.842563i \(-0.318954\pi\)
0.538597 + 0.842563i \(0.318954\pi\)
\(422\) 0 0
\(423\) −1.08462e7 −2.94732
\(424\) 0 0
\(425\) −1.14632e6 −0.307846
\(426\) 0 0
\(427\) 759658. 0.201627
\(428\) 0 0
\(429\) 1.15547e7 3.03121
\(430\) 0 0
\(431\) −4.42095e6 −1.14636 −0.573181 0.819429i \(-0.694291\pi\)
−0.573181 + 0.819429i \(0.694291\pi\)
\(432\) 0 0
\(433\) −4.01914e6 −1.03018 −0.515090 0.857136i \(-0.672242\pi\)
−0.515090 + 0.857136i \(0.672242\pi\)
\(434\) 0 0
\(435\) −5.56888e6 −1.41106
\(436\) 0 0
\(437\) −498259. −0.124811
\(438\) 0 0
\(439\) 5.41795e6 1.34176 0.670879 0.741567i \(-0.265917\pi\)
0.670879 + 0.741567i \(0.265917\pi\)
\(440\) 0 0
\(441\) −8.74163e6 −2.14041
\(442\) 0 0
\(443\) −7.21518e6 −1.74678 −0.873389 0.487023i \(-0.838083\pi\)
−0.873389 + 0.487023i \(0.838083\pi\)
\(444\) 0 0
\(445\) 1.51914e6 0.363662
\(446\) 0 0
\(447\) 4.43751e6 1.05044
\(448\) 0 0
\(449\) −203792. −0.0477057 −0.0238529 0.999715i \(-0.507593\pi\)
−0.0238529 + 0.999715i \(0.507593\pi\)
\(450\) 0 0
\(451\) 3.32800e6 0.770446
\(452\) 0 0
\(453\) −1.30769e7 −2.99406
\(454\) 0 0
\(455\) 1.01847e6 0.230631
\(456\) 0 0
\(457\) −3.45642e6 −0.774169 −0.387085 0.922044i \(-0.626518\pi\)
−0.387085 + 0.922044i \(0.626518\pi\)
\(458\) 0 0
\(459\) 1.78326e7 3.95079
\(460\) 0 0
\(461\) −6.40596e6 −1.40389 −0.701944 0.712233i \(-0.747684\pi\)
−0.701944 + 0.712233i \(0.747684\pi\)
\(462\) 0 0
\(463\) 7.59550e6 1.64666 0.823330 0.567563i \(-0.192113\pi\)
0.823330 + 0.567563i \(0.192113\pi\)
\(464\) 0 0
\(465\) −73687.3 −0.0158038
\(466\) 0 0
\(467\) −4.58166e6 −0.972145 −0.486072 0.873919i \(-0.661571\pi\)
−0.486072 + 0.873919i \(0.661571\pi\)
\(468\) 0 0
\(469\) −2.31979e6 −0.486986
\(470\) 0 0
\(471\) 1.03862e7 2.15726
\(472\) 0 0
\(473\) 6.88658e6 1.41531
\(474\) 0 0
\(475\) 198602. 0.0403877
\(476\) 0 0
\(477\) 8.70779e6 1.75231
\(478\) 0 0
\(479\) −5.01195e6 −0.998085 −0.499043 0.866577i \(-0.666315\pi\)
−0.499043 + 0.866577i \(0.666315\pi\)
\(480\) 0 0
\(481\) 1.87187e6 0.368904
\(482\) 0 0
\(483\) 1.89774e6 0.370142
\(484\) 0 0
\(485\) 777806. 0.150147
\(486\) 0 0
\(487\) 2.10670e6 0.402514 0.201257 0.979539i \(-0.435497\pi\)
0.201257 + 0.979539i \(0.435497\pi\)
\(488\) 0 0
\(489\) −1.40949e7 −2.66556
\(490\) 0 0
\(491\) 5.43322e6 1.01708 0.508538 0.861040i \(-0.330186\pi\)
0.508538 + 0.861040i \(0.330186\pi\)
\(492\) 0 0
\(493\) 1.42277e7 2.63644
\(494\) 0 0
\(495\) −6.05278e6 −1.11030
\(496\) 0 0
\(497\) −472099. −0.0857318
\(498\) 0 0
\(499\) 9.65183e6 1.73523 0.867617 0.497233i \(-0.165650\pi\)
0.867617 + 0.497233i \(0.165650\pi\)
\(500\) 0 0
\(501\) −5.63611e6 −1.00319
\(502\) 0 0
\(503\) −1.33954e6 −0.236067 −0.118034 0.993010i \(-0.537659\pi\)
−0.118034 + 0.993010i \(0.537659\pi\)
\(504\) 0 0
\(505\) −1.23729e6 −0.215894
\(506\) 0 0
\(507\) −1.61669e7 −2.79322
\(508\) 0 0
\(509\) −4.60771e6 −0.788299 −0.394150 0.919046i \(-0.628961\pi\)
−0.394150 + 0.919046i \(0.628961\pi\)
\(510\) 0 0
\(511\) −169176. −0.0286606
\(512\) 0 0
\(513\) −3.08953e6 −0.518321
\(514\) 0 0
\(515\) −2.37795e6 −0.395079
\(516\) 0 0
\(517\) −7.76360e6 −1.27743
\(518\) 0 0
\(519\) 5.26629e6 0.858196
\(520\) 0 0
\(521\) −5.80941e6 −0.937644 −0.468822 0.883293i \(-0.655321\pi\)
−0.468822 + 0.883293i \(0.655321\pi\)
\(522\) 0 0
\(523\) −3.83877e6 −0.613674 −0.306837 0.951762i \(-0.599271\pi\)
−0.306837 + 0.951762i \(0.599271\pi\)
\(524\) 0 0
\(525\) −756422. −0.119775
\(526\) 0 0
\(527\) 188261. 0.0295280
\(528\) 0 0
\(529\) −3.97765e6 −0.617998
\(530\) 0 0
\(531\) 1.15406e7 1.77621
\(532\) 0 0
\(533\) −7.72725e6 −1.17817
\(534\) 0 0
\(535\) −2.21075e6 −0.333929
\(536\) 0 0
\(537\) 1.11015e6 0.166129
\(538\) 0 0
\(539\) −6.25716e6 −0.927696
\(540\) 0 0
\(541\) −6.28830e6 −0.923720 −0.461860 0.886953i \(-0.652818\pi\)
−0.461860 + 0.886953i \(0.652818\pi\)
\(542\) 0 0
\(543\) −1.61177e7 −2.34587
\(544\) 0 0
\(545\) −264972. −0.0382128
\(546\) 0 0
\(547\) 4.54365e6 0.649287 0.324644 0.945836i \(-0.394756\pi\)
0.324644 + 0.945836i \(0.394756\pi\)
\(548\) 0 0
\(549\) 1.04826e7 1.48435
\(550\) 0 0
\(551\) −2.46497e6 −0.345886
\(552\) 0 0
\(553\) −1.01232e6 −0.140768
\(554\) 0 0
\(555\) −1.39025e6 −0.191585
\(556\) 0 0
\(557\) −1.20198e7 −1.64157 −0.820783 0.571239i \(-0.806463\pi\)
−0.820783 + 0.571239i \(0.806463\pi\)
\(558\) 0 0
\(559\) −1.59899e7 −2.16429
\(560\) 0 0
\(561\) 2.19252e7 2.94129
\(562\) 0 0
\(563\) 3.65741e6 0.486298 0.243149 0.969989i \(-0.421820\pi\)
0.243149 + 0.969989i \(0.421820\pi\)
\(564\) 0 0
\(565\) −5.87811e6 −0.774670
\(566\) 0 0
\(567\) 5.81077e6 0.759059
\(568\) 0 0
\(569\) −2.80214e6 −0.362835 −0.181418 0.983406i \(-0.558069\pi\)
−0.181418 + 0.983406i \(0.558069\pi\)
\(570\) 0 0
\(571\) −6.78472e6 −0.870846 −0.435423 0.900226i \(-0.643401\pi\)
−0.435423 + 0.900226i \(0.643401\pi\)
\(572\) 0 0
\(573\) 7.62091e6 0.969662
\(574\) 0 0
\(575\) −980014. −0.123613
\(576\) 0 0
\(577\) −1.25416e6 −0.156825 −0.0784123 0.996921i \(-0.524985\pi\)
−0.0784123 + 0.996921i \(0.524985\pi\)
\(578\) 0 0
\(579\) −2.47858e7 −3.07261
\(580\) 0 0
\(581\) 2.97190e6 0.365253
\(582\) 0 0
\(583\) 6.23293e6 0.759488
\(584\) 0 0
\(585\) 1.40539e7 1.69788
\(586\) 0 0
\(587\) 726476. 0.0870213 0.0435107 0.999053i \(-0.486146\pi\)
0.0435107 + 0.999053i \(0.486146\pi\)
\(588\) 0 0
\(589\) −32616.5 −0.00387391
\(590\) 0 0
\(591\) 8.09409e6 0.953234
\(592\) 0 0
\(593\) 933494. 0.109012 0.0545060 0.998513i \(-0.482642\pi\)
0.0545060 + 0.998513i \(0.482642\pi\)
\(594\) 0 0
\(595\) 1.93255e6 0.223789
\(596\) 0 0
\(597\) −5.53632e6 −0.635749
\(598\) 0 0
\(599\) 1.27354e7 1.45025 0.725127 0.688615i \(-0.241781\pi\)
0.725127 + 0.688615i \(0.241781\pi\)
\(600\) 0 0
\(601\) −6.87190e6 −0.776052 −0.388026 0.921648i \(-0.626843\pi\)
−0.388026 + 0.921648i \(0.626843\pi\)
\(602\) 0 0
\(603\) −3.20109e7 −3.58513
\(604\) 0 0
\(605\) −306238. −0.0340150
\(606\) 0 0
\(607\) 3.87130e6 0.426467 0.213233 0.977001i \(-0.431601\pi\)
0.213233 + 0.977001i \(0.431601\pi\)
\(608\) 0 0
\(609\) 9.38844e6 1.02577
\(610\) 0 0
\(611\) 1.80262e7 1.95345
\(612\) 0 0
\(613\) 2.38824e6 0.256701 0.128350 0.991729i \(-0.459032\pi\)
0.128350 + 0.991729i \(0.459032\pi\)
\(614\) 0 0
\(615\) 5.73908e6 0.611863
\(616\) 0 0
\(617\) −3.07299e6 −0.324974 −0.162487 0.986711i \(-0.551952\pi\)
−0.162487 + 0.986711i \(0.551952\pi\)
\(618\) 0 0
\(619\) 8.52257e6 0.894013 0.447007 0.894531i \(-0.352490\pi\)
0.447007 + 0.894531i \(0.352490\pi\)
\(620\) 0 0
\(621\) 1.52455e7 1.58640
\(622\) 0 0
\(623\) −2.56109e6 −0.264365
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 0 0
\(627\) −3.79858e6 −0.385880
\(628\) 0 0
\(629\) 3.55190e6 0.357960
\(630\) 0 0
\(631\) −8.54170e6 −0.854026 −0.427013 0.904246i \(-0.640434\pi\)
−0.427013 + 0.904246i \(0.640434\pi\)
\(632\) 0 0
\(633\) −1.28162e6 −0.127131
\(634\) 0 0
\(635\) −2.56333e6 −0.252272
\(636\) 0 0
\(637\) 1.45284e7 1.41863
\(638\) 0 0
\(639\) −6.51452e6 −0.631146
\(640\) 0 0
\(641\) 3.81006e6 0.366257 0.183129 0.983089i \(-0.441377\pi\)
0.183129 + 0.983089i \(0.441377\pi\)
\(642\) 0 0
\(643\) 1.40516e7 1.34029 0.670144 0.742231i \(-0.266232\pi\)
0.670144 + 0.742231i \(0.266232\pi\)
\(644\) 0 0
\(645\) 1.18758e7 1.12399
\(646\) 0 0
\(647\) −962366. −0.0903815 −0.0451908 0.998978i \(-0.514390\pi\)
−0.0451908 + 0.998978i \(0.514390\pi\)
\(648\) 0 0
\(649\) 8.26065e6 0.769844
\(650\) 0 0
\(651\) 124228. 0.0114886
\(652\) 0 0
\(653\) 413989. 0.0379932 0.0189966 0.999820i \(-0.493953\pi\)
0.0189966 + 0.999820i \(0.493953\pi\)
\(654\) 0 0
\(655\) −1.26508e6 −0.115216
\(656\) 0 0
\(657\) −2.33446e6 −0.210996
\(658\) 0 0
\(659\) −1.92401e7 −1.72581 −0.862905 0.505367i \(-0.831357\pi\)
−0.862905 + 0.505367i \(0.831357\pi\)
\(660\) 0 0
\(661\) 2.04652e7 1.82185 0.910924 0.412574i \(-0.135370\pi\)
0.910924 + 0.412574i \(0.135370\pi\)
\(662\) 0 0
\(663\) −5.09080e7 −4.49782
\(664\) 0 0
\(665\) −334818. −0.0293599
\(666\) 0 0
\(667\) 1.21636e7 1.05864
\(668\) 0 0
\(669\) −2.59724e7 −2.24361
\(670\) 0 0
\(671\) 7.50330e6 0.643348
\(672\) 0 0
\(673\) −9.12086e6 −0.776244 −0.388122 0.921608i \(-0.626876\pi\)
−0.388122 + 0.921608i \(0.626876\pi\)
\(674\) 0 0
\(675\) −6.07672e6 −0.513346
\(676\) 0 0
\(677\) 1.29457e7 1.08556 0.542778 0.839876i \(-0.317372\pi\)
0.542778 + 0.839876i \(0.317372\pi\)
\(678\) 0 0
\(679\) −1.31128e6 −0.109150
\(680\) 0 0
\(681\) 2.94863e7 2.43642
\(682\) 0 0
\(683\) −8.56637e6 −0.702660 −0.351330 0.936252i \(-0.614270\pi\)
−0.351330 + 0.936252i \(0.614270\pi\)
\(684\) 0 0
\(685\) −1.19529e6 −0.0973302
\(686\) 0 0
\(687\) −2.91523e6 −0.235658
\(688\) 0 0
\(689\) −1.44722e7 −1.16141
\(690\) 0 0
\(691\) −1.01163e7 −0.805984 −0.402992 0.915204i \(-0.632030\pi\)
−0.402992 + 0.915204i \(0.632030\pi\)
\(692\) 0 0
\(693\) 1.02042e7 0.807138
\(694\) 0 0
\(695\) 5.50474e6 0.432289
\(696\) 0 0
\(697\) −1.46626e7 −1.14322
\(698\) 0 0
\(699\) 9.00846e6 0.697361
\(700\) 0 0
\(701\) 1.95732e7 1.50441 0.752207 0.658927i \(-0.228989\pi\)
0.752207 + 0.658927i \(0.228989\pi\)
\(702\) 0 0
\(703\) −615372. −0.0469623
\(704\) 0 0
\(705\) −1.33882e7 −1.01449
\(706\) 0 0
\(707\) 2.08591e6 0.156945
\(708\) 0 0
\(709\) −2.36252e7 −1.76506 −0.882531 0.470254i \(-0.844162\pi\)
−0.882531 + 0.470254i \(0.844162\pi\)
\(710\) 0 0
\(711\) −1.39690e7 −1.03632
\(712\) 0 0
\(713\) 160948. 0.0118567
\(714\) 0 0
\(715\) 1.00596e7 0.735895
\(716\) 0 0
\(717\) −4.39617e7 −3.19357
\(718\) 0 0
\(719\) −2.44994e7 −1.76740 −0.883698 0.468058i \(-0.844954\pi\)
−0.883698 + 0.468058i \(0.844954\pi\)
\(720\) 0 0
\(721\) 4.00892e6 0.287204
\(722\) 0 0
\(723\) −1.45588e7 −1.03581
\(724\) 0 0
\(725\) −4.84830e6 −0.342566
\(726\) 0 0
\(727\) 1.75199e7 1.22941 0.614703 0.788759i \(-0.289276\pi\)
0.614703 + 0.788759i \(0.289276\pi\)
\(728\) 0 0
\(729\) 1.23387e7 0.859904
\(730\) 0 0
\(731\) −3.03410e7 −2.10008
\(732\) 0 0
\(733\) −2.29025e7 −1.57443 −0.787215 0.616679i \(-0.788478\pi\)
−0.787215 + 0.616679i \(0.788478\pi\)
\(734\) 0 0
\(735\) −1.07904e7 −0.736746
\(736\) 0 0
\(737\) −2.29131e7 −1.55387
\(738\) 0 0
\(739\) 1.31983e7 0.889010 0.444505 0.895776i \(-0.353380\pi\)
0.444505 + 0.895776i \(0.353380\pi\)
\(740\) 0 0
\(741\) 8.81988e6 0.590089
\(742\) 0 0
\(743\) 1.89399e6 0.125865 0.0629326 0.998018i \(-0.479955\pi\)
0.0629326 + 0.998018i \(0.479955\pi\)
\(744\) 0 0
\(745\) 3.86332e6 0.255017
\(746\) 0 0
\(747\) 4.10094e7 2.68894
\(748\) 0 0
\(749\) 3.72704e6 0.242750
\(750\) 0 0
\(751\) 1.71988e7 1.11275 0.556376 0.830930i \(-0.312191\pi\)
0.556376 + 0.830930i \(0.312191\pi\)
\(752\) 0 0
\(753\) 1.46926e7 0.944304
\(754\) 0 0
\(755\) −1.13849e7 −0.726876
\(756\) 0 0
\(757\) −2.02697e7 −1.28561 −0.642804 0.766031i \(-0.722229\pi\)
−0.642804 + 0.766031i \(0.722229\pi\)
\(758\) 0 0
\(759\) 1.87444e7 1.18104
\(760\) 0 0
\(761\) −6.54894e6 −0.409929 −0.204965 0.978769i \(-0.565708\pi\)
−0.204965 + 0.978769i \(0.565708\pi\)
\(762\) 0 0
\(763\) 446710. 0.0277789
\(764\) 0 0
\(765\) 2.66674e7 1.64751
\(766\) 0 0
\(767\) −1.91803e7 −1.17725
\(768\) 0 0
\(769\) −2.48043e7 −1.51256 −0.756278 0.654251i \(-0.772984\pi\)
−0.756278 + 0.654251i \(0.772984\pi\)
\(770\) 0 0
\(771\) 7.35357e6 0.445515
\(772\) 0 0
\(773\) −2.27616e7 −1.37010 −0.685052 0.728495i \(-0.740220\pi\)
−0.685052 + 0.728495i \(0.740220\pi\)
\(774\) 0 0
\(775\) −64152.6 −0.00383672
\(776\) 0 0
\(777\) 2.34379e6 0.139273
\(778\) 0 0
\(779\) 2.54031e6 0.149983
\(780\) 0 0
\(781\) −4.66302e6 −0.273552
\(782\) 0 0
\(783\) 7.54221e7 4.39637
\(784\) 0 0
\(785\) 9.04225e6 0.523724
\(786\) 0 0
\(787\) 1.56494e7 0.900658 0.450329 0.892863i \(-0.351307\pi\)
0.450329 + 0.892863i \(0.351307\pi\)
\(788\) 0 0
\(789\) 3.80341e7 2.17511
\(790\) 0 0
\(791\) 9.90976e6 0.563147
\(792\) 0 0
\(793\) −1.74218e7 −0.983809
\(794\) 0 0
\(795\) 1.07486e7 0.603161
\(796\) 0 0
\(797\) 2.26205e7 1.26141 0.630704 0.776023i \(-0.282766\pi\)
0.630704 + 0.776023i \(0.282766\pi\)
\(798\) 0 0
\(799\) 3.42050e7 1.89549
\(800\) 0 0
\(801\) −3.53406e7 −1.94622
\(802\) 0 0
\(803\) −1.67098e6 −0.0914499
\(804\) 0 0
\(805\) 1.65218e6 0.0898604
\(806\) 0 0
\(807\) −4.10602e7 −2.21941
\(808\) 0 0
\(809\) −2.25367e7 −1.21065 −0.605326 0.795977i \(-0.706957\pi\)
−0.605326 + 0.795977i \(0.706957\pi\)
\(810\) 0 0
\(811\) −410196. −0.0218997 −0.0109499 0.999940i \(-0.503486\pi\)
−0.0109499 + 0.999940i \(0.503486\pi\)
\(812\) 0 0
\(813\) −2.02855e7 −1.07636
\(814\) 0 0
\(815\) −1.22711e7 −0.647126
\(816\) 0 0
\(817\) 5.25662e6 0.275519
\(818\) 0 0
\(819\) −2.36931e7 −1.23428
\(820\) 0 0
\(821\) −1.62749e7 −0.842678 −0.421339 0.906903i \(-0.638440\pi\)
−0.421339 + 0.906903i \(0.638440\pi\)
\(822\) 0 0
\(823\) 628881. 0.0323645 0.0161823 0.999869i \(-0.494849\pi\)
0.0161823 + 0.999869i \(0.494849\pi\)
\(824\) 0 0
\(825\) −7.47134e6 −0.382176
\(826\) 0 0
\(827\) −1.94403e7 −0.988417 −0.494209 0.869343i \(-0.664542\pi\)
−0.494209 + 0.869343i \(0.664542\pi\)
\(828\) 0 0
\(829\) −3.39088e7 −1.71366 −0.856832 0.515595i \(-0.827571\pi\)
−0.856832 + 0.515595i \(0.827571\pi\)
\(830\) 0 0
\(831\) 9.03614e6 0.453921
\(832\) 0 0
\(833\) 2.75679e7 1.37655
\(834\) 0 0
\(835\) −4.90683e6 −0.243548
\(836\) 0 0
\(837\) 997985. 0.0492391
\(838\) 0 0
\(839\) −6.84583e6 −0.335754 −0.167877 0.985808i \(-0.553691\pi\)
−0.167877 + 0.985808i \(0.553691\pi\)
\(840\) 0 0
\(841\) 3.96642e7 1.93379
\(842\) 0 0
\(843\) 1.25715e7 0.609282
\(844\) 0 0
\(845\) −1.40750e7 −0.678118
\(846\) 0 0
\(847\) 516280. 0.0247273
\(848\) 0 0
\(849\) 5.97997e7 2.84728
\(850\) 0 0
\(851\) 3.03660e6 0.143735
\(852\) 0 0
\(853\) 2.67852e7 1.26044 0.630221 0.776416i \(-0.282964\pi\)
0.630221 + 0.776416i \(0.282964\pi\)
\(854\) 0 0
\(855\) −4.62017e6 −0.216144
\(856\) 0 0
\(857\) 9.32339e6 0.433632 0.216816 0.976212i \(-0.430433\pi\)
0.216816 + 0.976212i \(0.430433\pi\)
\(858\) 0 0
\(859\) 3.73055e7 1.72500 0.862502 0.506054i \(-0.168896\pi\)
0.862502 + 0.506054i \(0.168896\pi\)
\(860\) 0 0
\(861\) −9.67537e6 −0.444795
\(862\) 0 0
\(863\) 5.18636e6 0.237048 0.118524 0.992951i \(-0.462184\pi\)
0.118524 + 0.992951i \(0.462184\pi\)
\(864\) 0 0
\(865\) 4.58486e6 0.208346
\(866\) 0 0
\(867\) −5.58265e7 −2.52228
\(868\) 0 0
\(869\) −9.99888e6 −0.449161
\(870\) 0 0
\(871\) 5.32016e7 2.37618
\(872\) 0 0
\(873\) −1.80945e7 −0.803546
\(874\) 0 0
\(875\) −658545. −0.0290781
\(876\) 0 0
\(877\) 5.63396e6 0.247352 0.123676 0.992323i \(-0.460532\pi\)
0.123676 + 0.992323i \(0.460532\pi\)
\(878\) 0 0
\(879\) 2.60498e6 0.113719
\(880\) 0 0
\(881\) −5.76880e6 −0.250407 −0.125203 0.992131i \(-0.539958\pi\)
−0.125203 + 0.992131i \(0.539958\pi\)
\(882\) 0 0
\(883\) 2.60630e7 1.12492 0.562462 0.826823i \(-0.309854\pi\)
0.562462 + 0.826823i \(0.309854\pi\)
\(884\) 0 0
\(885\) 1.42453e7 0.611385
\(886\) 0 0
\(887\) −3.24889e7 −1.38652 −0.693259 0.720688i \(-0.743826\pi\)
−0.693259 + 0.720688i \(0.743826\pi\)
\(888\) 0 0
\(889\) 4.32145e6 0.183390
\(890\) 0 0
\(891\) 5.73942e7 2.42200
\(892\) 0 0
\(893\) −5.92607e6 −0.248678
\(894\) 0 0
\(895\) 966499. 0.0403315
\(896\) 0 0
\(897\) −4.35223e7 −1.80605
\(898\) 0 0
\(899\) 796240. 0.0328583
\(900\) 0 0
\(901\) −2.74612e7 −1.12696
\(902\) 0 0
\(903\) −2.00211e7 −0.817087
\(904\) 0 0
\(905\) −1.40322e7 −0.569513
\(906\) 0 0
\(907\) −3.11745e7 −1.25829 −0.629145 0.777288i \(-0.716595\pi\)
−0.629145 + 0.777288i \(0.716595\pi\)
\(908\) 0 0
\(909\) 2.87836e7 1.15541
\(910\) 0 0
\(911\) −3.58254e7 −1.43019 −0.715097 0.699025i \(-0.753618\pi\)
−0.715097 + 0.699025i \(0.753618\pi\)
\(912\) 0 0
\(913\) 2.93540e7 1.16544
\(914\) 0 0
\(915\) 1.29393e7 0.510926
\(916\) 0 0
\(917\) 2.13276e6 0.0837565
\(918\) 0 0
\(919\) 3.14710e7 1.22920 0.614598 0.788840i \(-0.289318\pi\)
0.614598 + 0.788840i \(0.289318\pi\)
\(920\) 0 0
\(921\) 1.64167e7 0.637730
\(922\) 0 0
\(923\) 1.08270e7 0.418316
\(924\) 0 0
\(925\) −1.21036e6 −0.0465115
\(926\) 0 0
\(927\) 5.53194e7 2.11436
\(928\) 0 0
\(929\) 532600. 0.0202471 0.0101235 0.999949i \(-0.496778\pi\)
0.0101235 + 0.999949i \(0.496778\pi\)
\(930\) 0 0
\(931\) −4.77618e6 −0.180595
\(932\) 0 0
\(933\) 7.45215e7 2.80270
\(934\) 0 0
\(935\) 1.90882e7 0.714064
\(936\) 0 0
\(937\) 1.97320e7 0.734214 0.367107 0.930179i \(-0.380348\pi\)
0.367107 + 0.930179i \(0.380348\pi\)
\(938\) 0 0
\(939\) 1.46639e7 0.542732
\(940\) 0 0
\(941\) 2.12060e7 0.780702 0.390351 0.920666i \(-0.372354\pi\)
0.390351 + 0.920666i \(0.372354\pi\)
\(942\) 0 0
\(943\) −1.25353e7 −0.459047
\(944\) 0 0
\(945\) 1.02446e7 0.373178
\(946\) 0 0
\(947\) 4.60224e7 1.66761 0.833804 0.552060i \(-0.186158\pi\)
0.833804 + 0.552060i \(0.186158\pi\)
\(948\) 0 0
\(949\) 3.87984e6 0.139845
\(950\) 0 0
\(951\) −9.67802e7 −3.47004
\(952\) 0 0
\(953\) 4.76525e7 1.69963 0.849813 0.527085i \(-0.176715\pi\)
0.849813 + 0.527085i \(0.176715\pi\)
\(954\) 0 0
\(955\) 6.63481e6 0.235407
\(956\) 0 0
\(957\) 9.27316e7 3.27301
\(958\) 0 0
\(959\) 2.01511e6 0.0707543
\(960\) 0 0
\(961\) −2.86186e7 −0.999632
\(962\) 0 0
\(963\) 5.14297e7 1.78710
\(964\) 0 0
\(965\) −2.15787e7 −0.745946
\(966\) 0 0
\(967\) 2.29686e7 0.789893 0.394946 0.918704i \(-0.370763\pi\)
0.394946 + 0.918704i \(0.370763\pi\)
\(968\) 0 0
\(969\) 1.67358e7 0.572583
\(970\) 0 0
\(971\) 2.22231e7 0.756408 0.378204 0.925722i \(-0.376542\pi\)
0.378204 + 0.925722i \(0.376542\pi\)
\(972\) 0 0
\(973\) −9.28031e6 −0.314254
\(974\) 0 0
\(975\) 1.73476e7 0.584424
\(976\) 0 0
\(977\) −3.68581e7 −1.23537 −0.617684 0.786426i \(-0.711929\pi\)
−0.617684 + 0.786426i \(0.711929\pi\)
\(978\) 0 0
\(979\) −2.52964e7 −0.843532
\(980\) 0 0
\(981\) 6.16418e6 0.204504
\(982\) 0 0
\(983\) 4.53136e7 1.49570 0.747850 0.663868i \(-0.231086\pi\)
0.747850 + 0.663868i \(0.231086\pi\)
\(984\) 0 0
\(985\) 7.04676e6 0.231419
\(986\) 0 0
\(987\) 2.25708e7 0.737487
\(988\) 0 0
\(989\) −2.59392e7 −0.843268
\(990\) 0 0
\(991\) 3.68582e7 1.19220 0.596102 0.802909i \(-0.296716\pi\)
0.596102 + 0.802909i \(0.296716\pi\)
\(992\) 0 0
\(993\) −1.09169e8 −3.51339
\(994\) 0 0
\(995\) −4.81995e6 −0.154342
\(996\) 0 0
\(997\) 4.19245e7 1.33577 0.667883 0.744266i \(-0.267201\pi\)
0.667883 + 0.744266i \(0.267201\pi\)
\(998\) 0 0
\(999\) 1.88289e7 0.596912
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 320.6.a.q.1.1 2
4.3 odd 2 320.6.a.w.1.2 2
8.3 odd 2 40.6.a.d.1.1 2
8.5 even 2 80.6.a.i.1.2 2
24.5 odd 2 720.6.a.z.1.2 2
24.11 even 2 360.6.a.l.1.1 2
40.3 even 4 200.6.c.e.49.1 4
40.13 odd 4 400.6.c.l.49.4 4
40.19 odd 2 200.6.a.g.1.2 2
40.27 even 4 200.6.c.e.49.4 4
40.29 even 2 400.6.a.q.1.1 2
40.37 odd 4 400.6.c.l.49.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.6.a.d.1.1 2 8.3 odd 2
80.6.a.i.1.2 2 8.5 even 2
200.6.a.g.1.2 2 40.19 odd 2
200.6.c.e.49.1 4 40.3 even 4
200.6.c.e.49.4 4 40.27 even 4
320.6.a.q.1.1 2 1.1 even 1 trivial
320.6.a.w.1.2 2 4.3 odd 2
360.6.a.l.1.1 2 24.11 even 2
400.6.a.q.1.1 2 40.29 even 2
400.6.c.l.49.1 4 40.37 odd 4
400.6.c.l.49.4 4 40.13 odd 4
720.6.a.z.1.2 2 24.5 odd 2