Properties

Label 825.6.a.a.1.1
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(132.316651346\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -9.00000 q^{3} -31.0000 q^{4} +9.00000 q^{6} +26.0000 q^{7} +63.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -9.00000 q^{3} -31.0000 q^{4} +9.00000 q^{6} +26.0000 q^{7} +63.0000 q^{8} +81.0000 q^{9} +121.000 q^{11} +279.000 q^{12} +692.000 q^{13} -26.0000 q^{14} +929.000 q^{16} +1442.00 q^{17} -81.0000 q^{18} +2160.00 q^{19} -234.000 q^{21} -121.000 q^{22} +1582.00 q^{23} -567.000 q^{24} -692.000 q^{26} -729.000 q^{27} -806.000 q^{28} -5526.00 q^{29} +4792.00 q^{31} -2945.00 q^{32} -1089.00 q^{33} -1442.00 q^{34} -2511.00 q^{36} +10194.0 q^{37} -2160.00 q^{38} -6228.00 q^{39} -10622.0 q^{41} +234.000 q^{42} -8580.00 q^{43} -3751.00 q^{44} -1582.00 q^{46} +2362.00 q^{47} -8361.00 q^{48} -16131.0 q^{49} -12978.0 q^{51} -21452.0 q^{52} +30804.0 q^{53} +729.000 q^{54} +1638.00 q^{56} -19440.0 q^{57} +5526.00 q^{58} +6416.00 q^{59} +42096.0 q^{61} -4792.00 q^{62} +2106.00 q^{63} -26783.0 q^{64} +1089.00 q^{66} +28444.0 q^{67} -44702.0 q^{68} -14238.0 q^{69} +45690.0 q^{71} +5103.00 q^{72} +18374.0 q^{73} -10194.0 q^{74} -66960.0 q^{76} +3146.00 q^{77} +6228.00 q^{78} -105214. q^{79} +6561.00 q^{81} +10622.0 q^{82} -62292.0 q^{83} +7254.00 q^{84} +8580.00 q^{86} +49734.0 q^{87} +7623.00 q^{88} -72246.0 q^{89} +17992.0 q^{91} -49042.0 q^{92} -43128.0 q^{93} -2362.00 q^{94} +26505.0 q^{96} -79262.0 q^{97} +16131.0 q^{98} +9801.00 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.176777 −0.0883883 0.996086i \(-0.528172\pi\)
−0.0883883 + 0.996086i \(0.528172\pi\)
\(3\) −9.00000 −0.577350
\(4\) −31.0000 −0.968750
\(5\) 0 0
\(6\) 9.00000 0.102062
\(7\) 26.0000 0.200553 0.100276 0.994960i \(-0.468027\pi\)
0.100276 + 0.994960i \(0.468027\pi\)
\(8\) 63.0000 0.348029
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 279.000 0.559308
\(13\) 692.000 1.13566 0.567829 0.823146i \(-0.307783\pi\)
0.567829 + 0.823146i \(0.307783\pi\)
\(14\) −26.0000 −0.0354530
\(15\) 0 0
\(16\) 929.000 0.907227
\(17\) 1442.00 1.21016 0.605080 0.796165i \(-0.293141\pi\)
0.605080 + 0.796165i \(0.293141\pi\)
\(18\) −81.0000 −0.0589256
\(19\) 2160.00 1.37268 0.686341 0.727280i \(-0.259216\pi\)
0.686341 + 0.727280i \(0.259216\pi\)
\(20\) 0 0
\(21\) −234.000 −0.115789
\(22\) −121.000 −0.0533002
\(23\) 1582.00 0.623572 0.311786 0.950152i \(-0.399073\pi\)
0.311786 + 0.950152i \(0.399073\pi\)
\(24\) −567.000 −0.200935
\(25\) 0 0
\(26\) −692.000 −0.200758
\(27\) −729.000 −0.192450
\(28\) −806.000 −0.194285
\(29\) −5526.00 −1.22016 −0.610079 0.792341i \(-0.708862\pi\)
−0.610079 + 0.792341i \(0.708862\pi\)
\(30\) 0 0
\(31\) 4792.00 0.895597 0.447798 0.894135i \(-0.352208\pi\)
0.447798 + 0.894135i \(0.352208\pi\)
\(32\) −2945.00 −0.508406
\(33\) −1089.00 −0.174078
\(34\) −1442.00 −0.213928
\(35\) 0 0
\(36\) −2511.00 −0.322917
\(37\) 10194.0 1.22417 0.612083 0.790794i \(-0.290332\pi\)
0.612083 + 0.790794i \(0.290332\pi\)
\(38\) −2160.00 −0.242658
\(39\) −6228.00 −0.655673
\(40\) 0 0
\(41\) −10622.0 −0.986840 −0.493420 0.869791i \(-0.664253\pi\)
−0.493420 + 0.869791i \(0.664253\pi\)
\(42\) 234.000 0.0204688
\(43\) −8580.00 −0.707646 −0.353823 0.935312i \(-0.615119\pi\)
−0.353823 + 0.935312i \(0.615119\pi\)
\(44\) −3751.00 −0.292089
\(45\) 0 0
\(46\) −1582.00 −0.110233
\(47\) 2362.00 0.155968 0.0779840 0.996955i \(-0.475152\pi\)
0.0779840 + 0.996955i \(0.475152\pi\)
\(48\) −8361.00 −0.523788
\(49\) −16131.0 −0.959779
\(50\) 0 0
\(51\) −12978.0 −0.698686
\(52\) −21452.0 −1.10017
\(53\) 30804.0 1.50632 0.753160 0.657837i \(-0.228528\pi\)
0.753160 + 0.657837i \(0.228528\pi\)
\(54\) 729.000 0.0340207
\(55\) 0 0
\(56\) 1638.00 0.0697981
\(57\) −19440.0 −0.792518
\(58\) 5526.00 0.215695
\(59\) 6416.00 0.239957 0.119979 0.992776i \(-0.461717\pi\)
0.119979 + 0.992776i \(0.461717\pi\)
\(60\) 0 0
\(61\) 42096.0 1.44849 0.724246 0.689541i \(-0.242188\pi\)
0.724246 + 0.689541i \(0.242188\pi\)
\(62\) −4792.00 −0.158321
\(63\) 2106.00 0.0668509
\(64\) −26783.0 −0.817352
\(65\) 0 0
\(66\) 1089.00 0.0307729
\(67\) 28444.0 0.774112 0.387056 0.922056i \(-0.373492\pi\)
0.387056 + 0.922056i \(0.373492\pi\)
\(68\) −44702.0 −1.17234
\(69\) −14238.0 −0.360020
\(70\) 0 0
\(71\) 45690.0 1.07566 0.537830 0.843053i \(-0.319244\pi\)
0.537830 + 0.843053i \(0.319244\pi\)
\(72\) 5103.00 0.116010
\(73\) 18374.0 0.403549 0.201775 0.979432i \(-0.435329\pi\)
0.201775 + 0.979432i \(0.435329\pi\)
\(74\) −10194.0 −0.216404
\(75\) 0 0
\(76\) −66960.0 −1.32979
\(77\) 3146.00 0.0604689
\(78\) 6228.00 0.115908
\(79\) −105214. −1.89673 −0.948366 0.317179i \(-0.897264\pi\)
−0.948366 + 0.317179i \(0.897264\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 10622.0 0.174450
\(83\) −62292.0 −0.992515 −0.496257 0.868175i \(-0.665293\pi\)
−0.496257 + 0.868175i \(0.665293\pi\)
\(84\) 7254.00 0.112171
\(85\) 0 0
\(86\) 8580.00 0.125095
\(87\) 49734.0 0.704458
\(88\) 7623.00 0.104935
\(89\) −72246.0 −0.966805 −0.483402 0.875398i \(-0.660599\pi\)
−0.483402 + 0.875398i \(0.660599\pi\)
\(90\) 0 0
\(91\) 17992.0 0.227759
\(92\) −49042.0 −0.604086
\(93\) −43128.0 −0.517073
\(94\) −2362.00 −0.0275715
\(95\) 0 0
\(96\) 26505.0 0.293528
\(97\) −79262.0 −0.855334 −0.427667 0.903936i \(-0.640664\pi\)
−0.427667 + 0.903936i \(0.640664\pi\)
\(98\) 16131.0 0.169667
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) −24958.0 −0.243448 −0.121724 0.992564i \(-0.538842\pi\)
−0.121724 + 0.992564i \(0.538842\pi\)
\(102\) 12978.0 0.123511
\(103\) 56812.0 0.527651 0.263826 0.964570i \(-0.415016\pi\)
0.263826 + 0.964570i \(0.415016\pi\)
\(104\) 43596.0 0.395242
\(105\) 0 0
\(106\) −30804.0 −0.266282
\(107\) 12492.0 0.105481 0.0527403 0.998608i \(-0.483204\pi\)
0.0527403 + 0.998608i \(0.483204\pi\)
\(108\) 22599.0 0.186436
\(109\) 198748. 1.60227 0.801137 0.598482i \(-0.204229\pi\)
0.801137 + 0.598482i \(0.204229\pi\)
\(110\) 0 0
\(111\) −91746.0 −0.706773
\(112\) 24154.0 0.181947
\(113\) −166554. −1.22704 −0.613520 0.789679i \(-0.710247\pi\)
−0.613520 + 0.789679i \(0.710247\pi\)
\(114\) 19440.0 0.140099
\(115\) 0 0
\(116\) 171306. 1.18203
\(117\) 56052.0 0.378553
\(118\) −6416.00 −0.0424189
\(119\) 37492.0 0.242701
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −42096.0 −0.256060
\(123\) 95598.0 0.569752
\(124\) −148552. −0.867609
\(125\) 0 0
\(126\) −2106.00 −0.0118177
\(127\) −304226. −1.67374 −0.836868 0.547405i \(-0.815616\pi\)
−0.836868 + 0.547405i \(0.815616\pi\)
\(128\) 121023. 0.652894
\(129\) 77220.0 0.408560
\(130\) 0 0
\(131\) 274428. 1.39717 0.698586 0.715526i \(-0.253813\pi\)
0.698586 + 0.715526i \(0.253813\pi\)
\(132\) 33759.0 0.168638
\(133\) 56160.0 0.275295
\(134\) −28444.0 −0.136845
\(135\) 0 0
\(136\) 90846.0 0.421171
\(137\) 245458. 1.11732 0.558658 0.829398i \(-0.311317\pi\)
0.558658 + 0.829398i \(0.311317\pi\)
\(138\) 14238.0 0.0636431
\(139\) −59888.0 −0.262907 −0.131454 0.991322i \(-0.541964\pi\)
−0.131454 + 0.991322i \(0.541964\pi\)
\(140\) 0 0
\(141\) −21258.0 −0.0900481
\(142\) −45690.0 −0.190152
\(143\) 83732.0 0.342414
\(144\) 75249.0 0.302409
\(145\) 0 0
\(146\) −18374.0 −0.0713381
\(147\) 145179. 0.554128
\(148\) −316014. −1.18591
\(149\) 72038.0 0.265825 0.132913 0.991128i \(-0.457567\pi\)
0.132913 + 0.991128i \(0.457567\pi\)
\(150\) 0 0
\(151\) −323110. −1.15321 −0.576605 0.817023i \(-0.695623\pi\)
−0.576605 + 0.817023i \(0.695623\pi\)
\(152\) 136080. 0.477733
\(153\) 116802. 0.403387
\(154\) −3146.00 −0.0106895
\(155\) 0 0
\(156\) 193068. 0.635183
\(157\) 318766. 1.03210 0.516051 0.856558i \(-0.327401\pi\)
0.516051 + 0.856558i \(0.327401\pi\)
\(158\) 105214. 0.335298
\(159\) −277236. −0.869675
\(160\) 0 0
\(161\) 41132.0 0.125059
\(162\) −6561.00 −0.0196419
\(163\) 431996. 1.27353 0.636767 0.771056i \(-0.280271\pi\)
0.636767 + 0.771056i \(0.280271\pi\)
\(164\) 329282. 0.956001
\(165\) 0 0
\(166\) 62292.0 0.175454
\(167\) 251580. 0.698047 0.349024 0.937114i \(-0.386513\pi\)
0.349024 + 0.937114i \(0.386513\pi\)
\(168\) −14742.0 −0.0402980
\(169\) 107571. 0.289720
\(170\) 0 0
\(171\) 174960. 0.457560
\(172\) 265980. 0.685532
\(173\) −476634. −1.21079 −0.605396 0.795924i \(-0.706985\pi\)
−0.605396 + 0.795924i \(0.706985\pi\)
\(174\) −49734.0 −0.124532
\(175\) 0 0
\(176\) 112409. 0.273539
\(177\) −57744.0 −0.138540
\(178\) 72246.0 0.170909
\(179\) 90192.0 0.210395 0.105198 0.994451i \(-0.466453\pi\)
0.105198 + 0.994451i \(0.466453\pi\)
\(180\) 0 0
\(181\) 248002. 0.562676 0.281338 0.959609i \(-0.409222\pi\)
0.281338 + 0.959609i \(0.409222\pi\)
\(182\) −17992.0 −0.0402625
\(183\) −378864. −0.836288
\(184\) 99666.0 0.217021
\(185\) 0 0
\(186\) 43128.0 0.0914065
\(187\) 174482. 0.364877
\(188\) −73222.0 −0.151094
\(189\) −18954.0 −0.0385964
\(190\) 0 0
\(191\) −156802. −0.311006 −0.155503 0.987835i \(-0.549700\pi\)
−0.155503 + 0.987835i \(0.549700\pi\)
\(192\) 241047. 0.471899
\(193\) 431234. 0.833335 0.416668 0.909059i \(-0.363198\pi\)
0.416668 + 0.909059i \(0.363198\pi\)
\(194\) 79262.0 0.151203
\(195\) 0 0
\(196\) 500061. 0.929786
\(197\) 864974. 1.58795 0.793976 0.607949i \(-0.208007\pi\)
0.793976 + 0.607949i \(0.208007\pi\)
\(198\) −9801.00 −0.0177667
\(199\) −480060. −0.859336 −0.429668 0.902987i \(-0.641369\pi\)
−0.429668 + 0.902987i \(0.641369\pi\)
\(200\) 0 0
\(201\) −255996. −0.446934
\(202\) 24958.0 0.0430359
\(203\) −143676. −0.244706
\(204\) 402318. 0.676853
\(205\) 0 0
\(206\) −56812.0 −0.0932765
\(207\) 128142. 0.207857
\(208\) 642868. 1.03030
\(209\) 261360. 0.413879
\(210\) 0 0
\(211\) 525900. 0.813199 0.406600 0.913606i \(-0.366714\pi\)
0.406600 + 0.913606i \(0.366714\pi\)
\(212\) −954924. −1.45925
\(213\) −411210. −0.621033
\(214\) −12492.0 −0.0186465
\(215\) 0 0
\(216\) −45927.0 −0.0669782
\(217\) 124592. 0.179614
\(218\) −198748. −0.283245
\(219\) −165366. −0.232989
\(220\) 0 0
\(221\) 997864. 1.37433
\(222\) 91746.0 0.124941
\(223\) 245264. 0.330272 0.165136 0.986271i \(-0.447194\pi\)
0.165136 + 0.986271i \(0.447194\pi\)
\(224\) −76570.0 −0.101962
\(225\) 0 0
\(226\) 166554. 0.216912
\(227\) 799308. 1.02955 0.514777 0.857324i \(-0.327875\pi\)
0.514777 + 0.857324i \(0.327875\pi\)
\(228\) 602640. 0.767752
\(229\) −1.53989e6 −1.94045 −0.970224 0.242208i \(-0.922128\pi\)
−0.970224 + 0.242208i \(0.922128\pi\)
\(230\) 0 0
\(231\) −28314.0 −0.0349117
\(232\) −348138. −0.424650
\(233\) 721830. 0.871054 0.435527 0.900176i \(-0.356562\pi\)
0.435527 + 0.900176i \(0.356562\pi\)
\(234\) −56052.0 −0.0669193
\(235\) 0 0
\(236\) −198896. −0.232459
\(237\) 946926. 1.09508
\(238\) −37492.0 −0.0429038
\(239\) −638436. −0.722974 −0.361487 0.932377i \(-0.617731\pi\)
−0.361487 + 0.932377i \(0.617731\pi\)
\(240\) 0 0
\(241\) 220990. 0.245092 0.122546 0.992463i \(-0.460894\pi\)
0.122546 + 0.992463i \(0.460894\pi\)
\(242\) −14641.0 −0.0160706
\(243\) −59049.0 −0.0641500
\(244\) −1.30498e6 −1.40323
\(245\) 0 0
\(246\) −95598.0 −0.100719
\(247\) 1.49472e6 1.55890
\(248\) 301896. 0.311694
\(249\) 560628. 0.573029
\(250\) 0 0
\(251\) 627304. 0.628483 0.314242 0.949343i \(-0.398250\pi\)
0.314242 + 0.949343i \(0.398250\pi\)
\(252\) −65286.0 −0.0647618
\(253\) 191422. 0.188014
\(254\) 304226. 0.295878
\(255\) 0 0
\(256\) 736033. 0.701936
\(257\) 468014. 0.442004 0.221002 0.975273i \(-0.429067\pi\)
0.221002 + 0.975273i \(0.429067\pi\)
\(258\) −77220.0 −0.0722238
\(259\) 265044. 0.245510
\(260\) 0 0
\(261\) −447606. −0.406719
\(262\) −274428. −0.246988
\(263\) −1.54510e6 −1.37743 −0.688713 0.725034i \(-0.741824\pi\)
−0.688713 + 0.725034i \(0.741824\pi\)
\(264\) −68607.0 −0.0605841
\(265\) 0 0
\(266\) −56160.0 −0.0486657
\(267\) 650214. 0.558185
\(268\) −881764. −0.749921
\(269\) −1.07457e6 −0.905430 −0.452715 0.891655i \(-0.649544\pi\)
−0.452715 + 0.891655i \(0.649544\pi\)
\(270\) 0 0
\(271\) 1.58723e6 1.31285 0.656427 0.754389i \(-0.272067\pi\)
0.656427 + 0.754389i \(0.272067\pi\)
\(272\) 1.33962e6 1.09789
\(273\) −161928. −0.131497
\(274\) −245458. −0.197515
\(275\) 0 0
\(276\) 441378. 0.348769
\(277\) −692704. −0.542436 −0.271218 0.962518i \(-0.587426\pi\)
−0.271218 + 0.962518i \(0.587426\pi\)
\(278\) 59888.0 0.0464759
\(279\) 388152. 0.298532
\(280\) 0 0
\(281\) −567018. −0.428382 −0.214191 0.976792i \(-0.568711\pi\)
−0.214191 + 0.976792i \(0.568711\pi\)
\(282\) 21258.0 0.0159184
\(283\) −714916. −0.530626 −0.265313 0.964162i \(-0.585475\pi\)
−0.265313 + 0.964162i \(0.585475\pi\)
\(284\) −1.41639e6 −1.04205
\(285\) 0 0
\(286\) −83732.0 −0.0605308
\(287\) −276172. −0.197913
\(288\) −238545. −0.169469
\(289\) 659507. 0.464488
\(290\) 0 0
\(291\) 713358. 0.493827
\(292\) −569594. −0.390938
\(293\) −2.14409e6 −1.45907 −0.729533 0.683946i \(-0.760262\pi\)
−0.729533 + 0.683946i \(0.760262\pi\)
\(294\) −145179. −0.0979570
\(295\) 0 0
\(296\) 642222. 0.426045
\(297\) −88209.0 −0.0580259
\(298\) −72038.0 −0.0469917
\(299\) 1.09474e6 0.708165
\(300\) 0 0
\(301\) −223080. −0.141920
\(302\) 323110. 0.203860
\(303\) 224622. 0.140555
\(304\) 2.00664e6 1.24533
\(305\) 0 0
\(306\) −116802. −0.0713094
\(307\) 588808. 0.356556 0.178278 0.983980i \(-0.442947\pi\)
0.178278 + 0.983980i \(0.442947\pi\)
\(308\) −97526.0 −0.0585792
\(309\) −511308. −0.304640
\(310\) 0 0
\(311\) 2.51827e6 1.47639 0.738194 0.674588i \(-0.235679\pi\)
0.738194 + 0.674588i \(0.235679\pi\)
\(312\) −392364. −0.228193
\(313\) 2.23562e6 1.28985 0.644923 0.764248i \(-0.276890\pi\)
0.644923 + 0.764248i \(0.276890\pi\)
\(314\) −318766. −0.182452
\(315\) 0 0
\(316\) 3.26163e6 1.83746
\(317\) −1.06079e6 −0.592901 −0.296450 0.955048i \(-0.595803\pi\)
−0.296450 + 0.955048i \(0.595803\pi\)
\(318\) 277236. 0.153738
\(319\) −668646. −0.367891
\(320\) 0 0
\(321\) −112428. −0.0608992
\(322\) −41132.0 −0.0221075
\(323\) 3.11472e6 1.66116
\(324\) −203391. −0.107639
\(325\) 0 0
\(326\) −431996. −0.225131
\(327\) −1.78873e6 −0.925073
\(328\) −669186. −0.343449
\(329\) 61412.0 0.0312798
\(330\) 0 0
\(331\) −2.34566e6 −1.17678 −0.588390 0.808577i \(-0.700238\pi\)
−0.588390 + 0.808577i \(0.700238\pi\)
\(332\) 1.93105e6 0.961499
\(333\) 825714. 0.408055
\(334\) −251580. −0.123399
\(335\) 0 0
\(336\) −217386. −0.105047
\(337\) −839978. −0.402896 −0.201448 0.979499i \(-0.564565\pi\)
−0.201448 + 0.979499i \(0.564565\pi\)
\(338\) −107571. −0.0512157
\(339\) 1.49899e6 0.708432
\(340\) 0 0
\(341\) 579832. 0.270033
\(342\) −174960. −0.0808860
\(343\) −856388. −0.393039
\(344\) −540540. −0.246281
\(345\) 0 0
\(346\) 476634. 0.214040
\(347\) 2.02560e6 0.903086 0.451543 0.892249i \(-0.350874\pi\)
0.451543 + 0.892249i \(0.350874\pi\)
\(348\) −1.54175e6 −0.682444
\(349\) −378924. −0.166528 −0.0832642 0.996528i \(-0.526535\pi\)
−0.0832642 + 0.996528i \(0.526535\pi\)
\(350\) 0 0
\(351\) −504468. −0.218558
\(352\) −356345. −0.153290
\(353\) 1.98730e6 0.848842 0.424421 0.905465i \(-0.360478\pi\)
0.424421 + 0.905465i \(0.360478\pi\)
\(354\) 57744.0 0.0244906
\(355\) 0 0
\(356\) 2.23963e6 0.936592
\(357\) −337428. −0.140123
\(358\) −90192.0 −0.0371929
\(359\) −3.43975e6 −1.40861 −0.704305 0.709898i \(-0.748741\pi\)
−0.704305 + 0.709898i \(0.748741\pi\)
\(360\) 0 0
\(361\) 2.18950e6 0.884254
\(362\) −248002. −0.0994681
\(363\) −131769. −0.0524864
\(364\) −557752. −0.220642
\(365\) 0 0
\(366\) 378864. 0.147836
\(367\) 1.79679e6 0.696358 0.348179 0.937428i \(-0.386800\pi\)
0.348179 + 0.937428i \(0.386800\pi\)
\(368\) 1.46968e6 0.565721
\(369\) −860382. −0.328947
\(370\) 0 0
\(371\) 800904. 0.302096
\(372\) 1.33697e6 0.500915
\(373\) 1.43541e6 0.534201 0.267100 0.963669i \(-0.413934\pi\)
0.267100 + 0.963669i \(0.413934\pi\)
\(374\) −174482. −0.0645018
\(375\) 0 0
\(376\) 148806. 0.0542814
\(377\) −3.82399e6 −1.38568
\(378\) 18954.0 0.00682294
\(379\) 2.66235e6 0.952065 0.476033 0.879428i \(-0.342074\pi\)
0.476033 + 0.879428i \(0.342074\pi\)
\(380\) 0 0
\(381\) 2.73803e6 0.966332
\(382\) 156802. 0.0549785
\(383\) −2.04091e6 −0.710932 −0.355466 0.934689i \(-0.615678\pi\)
−0.355466 + 0.934689i \(0.615678\pi\)
\(384\) −1.08921e6 −0.376949
\(385\) 0 0
\(386\) −431234. −0.147314
\(387\) −694980. −0.235882
\(388\) 2.45712e6 0.828605
\(389\) −4.29947e6 −1.44059 −0.720296 0.693667i \(-0.755994\pi\)
−0.720296 + 0.693667i \(0.755994\pi\)
\(390\) 0 0
\(391\) 2.28124e6 0.754623
\(392\) −1.01625e6 −0.334031
\(393\) −2.46985e6 −0.806658
\(394\) −864974. −0.280713
\(395\) 0 0
\(396\) −303831. −0.0973630
\(397\) −728818. −0.232083 −0.116041 0.993244i \(-0.537021\pi\)
−0.116041 + 0.993244i \(0.537021\pi\)
\(398\) 480060. 0.151911
\(399\) −505440. −0.158942
\(400\) 0 0
\(401\) −5.92515e6 −1.84009 −0.920044 0.391814i \(-0.871848\pi\)
−0.920044 + 0.391814i \(0.871848\pi\)
\(402\) 255996. 0.0790075
\(403\) 3.31606e6 1.01709
\(404\) 773698. 0.235840
\(405\) 0 0
\(406\) 143676. 0.0432583
\(407\) 1.23347e6 0.369100
\(408\) −817614. −0.243163
\(409\) 1.38212e6 0.408542 0.204271 0.978914i \(-0.434518\pi\)
0.204271 + 0.978914i \(0.434518\pi\)
\(410\) 0 0
\(411\) −2.20912e6 −0.645082
\(412\) −1.76117e6 −0.511162
\(413\) 166816. 0.0481241
\(414\) −128142. −0.0367444
\(415\) 0 0
\(416\) −2.03794e6 −0.577375
\(417\) 538992. 0.151790
\(418\) −261360. −0.0731642
\(419\) 5.47794e6 1.52434 0.762170 0.647377i \(-0.224134\pi\)
0.762170 + 0.647377i \(0.224134\pi\)
\(420\) 0 0
\(421\) 1.02873e6 0.282877 0.141439 0.989947i \(-0.454827\pi\)
0.141439 + 0.989947i \(0.454827\pi\)
\(422\) −525900. −0.143755
\(423\) 191322. 0.0519893
\(424\) 1.94065e6 0.524243
\(425\) 0 0
\(426\) 411210. 0.109784
\(427\) 1.09450e6 0.290499
\(428\) −387252. −0.102184
\(429\) −753588. −0.197693
\(430\) 0 0
\(431\) −5.14310e6 −1.33362 −0.666810 0.745228i \(-0.732341\pi\)
−0.666810 + 0.745228i \(0.732341\pi\)
\(432\) −677241. −0.174596
\(433\) −412954. −0.105848 −0.0529239 0.998599i \(-0.516854\pi\)
−0.0529239 + 0.998599i \(0.516854\pi\)
\(434\) −124592. −0.0317516
\(435\) 0 0
\(436\) −6.16119e6 −1.55220
\(437\) 3.41712e6 0.855966
\(438\) 165366. 0.0411871
\(439\) 5.96365e6 1.47690 0.738450 0.674309i \(-0.235558\pi\)
0.738450 + 0.674309i \(0.235558\pi\)
\(440\) 0 0
\(441\) −1.30661e6 −0.319926
\(442\) −997864. −0.242949
\(443\) −2.18433e6 −0.528821 −0.264410 0.964410i \(-0.585177\pi\)
−0.264410 + 0.964410i \(0.585177\pi\)
\(444\) 2.84413e6 0.684686
\(445\) 0 0
\(446\) −245264. −0.0583844
\(447\) −648342. −0.153474
\(448\) −696358. −0.163922
\(449\) −7858.00 −0.00183948 −0.000919742 1.00000i \(-0.500293\pi\)
−0.000919742 1.00000i \(0.500293\pi\)
\(450\) 0 0
\(451\) −1.28526e6 −0.297543
\(452\) 5.16317e6 1.18870
\(453\) 2.90799e6 0.665806
\(454\) −799308. −0.182001
\(455\) 0 0
\(456\) −1.22472e6 −0.275819
\(457\) 899922. 0.201565 0.100782 0.994908i \(-0.467865\pi\)
0.100782 + 0.994908i \(0.467865\pi\)
\(458\) 1.53989e6 0.343026
\(459\) −1.05122e6 −0.232895
\(460\) 0 0
\(461\) 1.13619e6 0.249000 0.124500 0.992220i \(-0.460267\pi\)
0.124500 + 0.992220i \(0.460267\pi\)
\(462\) 28314.0 0.00617158
\(463\) 7.38964e6 1.60203 0.801016 0.598643i \(-0.204293\pi\)
0.801016 + 0.598643i \(0.204293\pi\)
\(464\) −5.13365e6 −1.10696
\(465\) 0 0
\(466\) −721830. −0.153982
\(467\) −4.20851e6 −0.892968 −0.446484 0.894792i \(-0.647324\pi\)
−0.446484 + 0.894792i \(0.647324\pi\)
\(468\) −1.73761e6 −0.366723
\(469\) 739544. 0.155250
\(470\) 0 0
\(471\) −2.86889e6 −0.595885
\(472\) 404208. 0.0835122
\(473\) −1.03818e6 −0.213363
\(474\) −946926. −0.193584
\(475\) 0 0
\(476\) −1.16225e6 −0.235116
\(477\) 2.49512e6 0.502107
\(478\) 638436. 0.127805
\(479\) 7.39441e6 1.47253 0.736266 0.676692i \(-0.236587\pi\)
0.736266 + 0.676692i \(0.236587\pi\)
\(480\) 0 0
\(481\) 7.05425e6 1.39023
\(482\) −220990. −0.0433266
\(483\) −370188. −0.0722029
\(484\) −453871. −0.0880682
\(485\) 0 0
\(486\) 59049.0 0.0113402
\(487\) 3.81644e6 0.729181 0.364591 0.931168i \(-0.381209\pi\)
0.364591 + 0.931168i \(0.381209\pi\)
\(488\) 2.65205e6 0.504118
\(489\) −3.88796e6 −0.735275
\(490\) 0 0
\(491\) 1.69716e6 0.317702 0.158851 0.987303i \(-0.449221\pi\)
0.158851 + 0.987303i \(0.449221\pi\)
\(492\) −2.96354e6 −0.551947
\(493\) −7.96849e6 −1.47659
\(494\) −1.49472e6 −0.275577
\(495\) 0 0
\(496\) 4.45177e6 0.812509
\(497\) 1.18794e6 0.215727
\(498\) −560628. −0.101298
\(499\) 6.95160e6 1.24978 0.624889 0.780713i \(-0.285144\pi\)
0.624889 + 0.780713i \(0.285144\pi\)
\(500\) 0 0
\(501\) −2.26422e6 −0.403018
\(502\) −627304. −0.111101
\(503\) −6.01023e6 −1.05918 −0.529591 0.848253i \(-0.677655\pi\)
−0.529591 + 0.848253i \(0.677655\pi\)
\(504\) 132678. 0.0232660
\(505\) 0 0
\(506\) −191422. −0.0332365
\(507\) −968139. −0.167270
\(508\) 9.43101e6 1.62143
\(509\) 624660. 0.106868 0.0534342 0.998571i \(-0.482983\pi\)
0.0534342 + 0.998571i \(0.482983\pi\)
\(510\) 0 0
\(511\) 477724. 0.0809328
\(512\) −4.60877e6 −0.776980
\(513\) −1.57464e6 −0.264173
\(514\) −468014. −0.0781360
\(515\) 0 0
\(516\) −2.39382e6 −0.395792
\(517\) 285802. 0.0470261
\(518\) −265044. −0.0434004
\(519\) 4.28971e6 0.699051
\(520\) 0 0
\(521\) −647490. −0.104505 −0.0522527 0.998634i \(-0.516640\pi\)
−0.0522527 + 0.998634i \(0.516640\pi\)
\(522\) 447606. 0.0718985
\(523\) 114676. 0.0183324 0.00916618 0.999958i \(-0.497082\pi\)
0.00916618 + 0.999958i \(0.497082\pi\)
\(524\) −8.50727e6 −1.35351
\(525\) 0 0
\(526\) 1.54510e6 0.243497
\(527\) 6.91006e6 1.08382
\(528\) −1.01168e6 −0.157928
\(529\) −3.93362e6 −0.611157
\(530\) 0 0
\(531\) 519696. 0.0799858
\(532\) −1.74096e6 −0.266692
\(533\) −7.35042e6 −1.12071
\(534\) −650214. −0.0986741
\(535\) 0 0
\(536\) 1.79197e6 0.269413
\(537\) −811728. −0.121472
\(538\) 1.07457e6 0.160059
\(539\) −1.95185e6 −0.289384
\(540\) 0 0
\(541\) −2.12404e6 −0.312011 −0.156006 0.987756i \(-0.549862\pi\)
−0.156006 + 0.987756i \(0.549862\pi\)
\(542\) −1.58723e6 −0.232082
\(543\) −2.23202e6 −0.324861
\(544\) −4.24669e6 −0.615252
\(545\) 0 0
\(546\) 161928. 0.0232456
\(547\) −1.22672e7 −1.75299 −0.876494 0.481413i \(-0.840124\pi\)
−0.876494 + 0.481413i \(0.840124\pi\)
\(548\) −7.60920e6 −1.08240
\(549\) 3.40978e6 0.482831
\(550\) 0 0
\(551\) −1.19362e7 −1.67489
\(552\) −896994. −0.125297
\(553\) −2.73556e6 −0.380394
\(554\) 692704. 0.0958900
\(555\) 0 0
\(556\) 1.85653e6 0.254692
\(557\) −1.10980e7 −1.51568 −0.757839 0.652442i \(-0.773745\pi\)
−0.757839 + 0.652442i \(0.773745\pi\)
\(558\) −388152. −0.0527736
\(559\) −5.93736e6 −0.803644
\(560\) 0 0
\(561\) −1.57034e6 −0.210662
\(562\) 567018. 0.0757279
\(563\) −4.61984e6 −0.614265 −0.307132 0.951667i \(-0.599369\pi\)
−0.307132 + 0.951667i \(0.599369\pi\)
\(564\) 658998. 0.0872341
\(565\) 0 0
\(566\) 714916. 0.0938024
\(567\) 170586. 0.0222836
\(568\) 2.87847e6 0.374361
\(569\) 1.01716e7 1.31707 0.658537 0.752548i \(-0.271176\pi\)
0.658537 + 0.752548i \(0.271176\pi\)
\(570\) 0 0
\(571\) −9.36866e6 −1.20251 −0.601253 0.799059i \(-0.705331\pi\)
−0.601253 + 0.799059i \(0.705331\pi\)
\(572\) −2.59569e6 −0.331713
\(573\) 1.41122e6 0.179559
\(574\) 276172. 0.0349865
\(575\) 0 0
\(576\) −2.16942e6 −0.272451
\(577\) 6.14973e6 0.768983 0.384491 0.923129i \(-0.374377\pi\)
0.384491 + 0.923129i \(0.374377\pi\)
\(578\) −659507. −0.0821107
\(579\) −3.88111e6 −0.481126
\(580\) 0 0
\(581\) −1.61959e6 −0.199051
\(582\) −713358. −0.0872972
\(583\) 3.72728e6 0.454173
\(584\) 1.15756e6 0.140447
\(585\) 0 0
\(586\) 2.14409e6 0.257929
\(587\) −1.04649e6 −0.125354 −0.0626771 0.998034i \(-0.519964\pi\)
−0.0626771 + 0.998034i \(0.519964\pi\)
\(588\) −4.50055e6 −0.536812
\(589\) 1.03507e7 1.22937
\(590\) 0 0
\(591\) −7.78477e6 −0.916805
\(592\) 9.47023e6 1.11060
\(593\) 3.31784e6 0.387453 0.193726 0.981056i \(-0.437943\pi\)
0.193726 + 0.981056i \(0.437943\pi\)
\(594\) 88209.0 0.0102576
\(595\) 0 0
\(596\) −2.23318e6 −0.257518
\(597\) 4.32054e6 0.496138
\(598\) −1.09474e6 −0.125187
\(599\) −1.73991e7 −1.98134 −0.990670 0.136280i \(-0.956485\pi\)
−0.990670 + 0.136280i \(0.956485\pi\)
\(600\) 0 0
\(601\) 7.13163e6 0.805383 0.402691 0.915336i \(-0.368075\pi\)
0.402691 + 0.915336i \(0.368075\pi\)
\(602\) 223080. 0.0250882
\(603\) 2.30396e6 0.258037
\(604\) 1.00164e7 1.11717
\(605\) 0 0
\(606\) −224622. −0.0248468
\(607\) 9.64617e6 1.06263 0.531317 0.847173i \(-0.321697\pi\)
0.531317 + 0.847173i \(0.321697\pi\)
\(608\) −6.36120e6 −0.697879
\(609\) 1.29308e6 0.141281
\(610\) 0 0
\(611\) 1.63450e6 0.177126
\(612\) −3.62086e6 −0.390781
\(613\) −3.68170e6 −0.395729 −0.197864 0.980229i \(-0.563401\pi\)
−0.197864 + 0.980229i \(0.563401\pi\)
\(614\) −588808. −0.0630308
\(615\) 0 0
\(616\) 198198. 0.0210449
\(617\) −1.83190e7 −1.93727 −0.968635 0.248489i \(-0.920066\pi\)
−0.968635 + 0.248489i \(0.920066\pi\)
\(618\) 511308. 0.0538532
\(619\) 1.09660e6 0.115033 0.0575166 0.998345i \(-0.481682\pi\)
0.0575166 + 0.998345i \(0.481682\pi\)
\(620\) 0 0
\(621\) −1.15328e6 −0.120007
\(622\) −2.51827e6 −0.260991
\(623\) −1.87840e6 −0.193895
\(624\) −5.78581e6 −0.594844
\(625\) 0 0
\(626\) −2.23562e6 −0.228015
\(627\) −2.35224e6 −0.238953
\(628\) −9.88175e6 −0.999849
\(629\) 1.46997e7 1.48144
\(630\) 0 0
\(631\) −9.58030e6 −0.957869 −0.478934 0.877851i \(-0.658977\pi\)
−0.478934 + 0.877851i \(0.658977\pi\)
\(632\) −6.62848e6 −0.660118
\(633\) −4.73310e6 −0.469501
\(634\) 1.06079e6 0.104811
\(635\) 0 0
\(636\) 8.59432e6 0.842497
\(637\) −1.11627e7 −1.08998
\(638\) 668646. 0.0650346
\(639\) 3.70089e6 0.358554
\(640\) 0 0
\(641\) −1.18062e7 −1.13492 −0.567462 0.823400i \(-0.692075\pi\)
−0.567462 + 0.823400i \(0.692075\pi\)
\(642\) 112428. 0.0107656
\(643\) 5.88298e6 0.561138 0.280569 0.959834i \(-0.409477\pi\)
0.280569 + 0.959834i \(0.409477\pi\)
\(644\) −1.27509e6 −0.121151
\(645\) 0 0
\(646\) −3.11472e6 −0.293655
\(647\) 3.62822e6 0.340748 0.170374 0.985379i \(-0.445502\pi\)
0.170374 + 0.985379i \(0.445502\pi\)
\(648\) 413343. 0.0386699
\(649\) 776336. 0.0723499
\(650\) 0 0
\(651\) −1.12133e6 −0.103700
\(652\) −1.33919e7 −1.23374
\(653\) 5.70795e6 0.523838 0.261919 0.965090i \(-0.415645\pi\)
0.261919 + 0.965090i \(0.415645\pi\)
\(654\) 1.78873e6 0.163531
\(655\) 0 0
\(656\) −9.86784e6 −0.895287
\(657\) 1.48829e6 0.134516
\(658\) −61412.0 −0.00552953
\(659\) 1.08205e7 0.970588 0.485294 0.874351i \(-0.338713\pi\)
0.485294 + 0.874351i \(0.338713\pi\)
\(660\) 0 0
\(661\) 1.14311e7 1.01762 0.508809 0.860879i \(-0.330086\pi\)
0.508809 + 0.860879i \(0.330086\pi\)
\(662\) 2.34566e6 0.208027
\(663\) −8.98078e6 −0.793469
\(664\) −3.92440e6 −0.345424
\(665\) 0 0
\(666\) −825714. −0.0721347
\(667\) −8.74213e6 −0.760857
\(668\) −7.79898e6 −0.676233
\(669\) −2.20738e6 −0.190683
\(670\) 0 0
\(671\) 5.09362e6 0.436737
\(672\) 689130. 0.0588678
\(673\) 2.03858e7 1.73496 0.867482 0.497468i \(-0.165737\pi\)
0.867482 + 0.497468i \(0.165737\pi\)
\(674\) 839978. 0.0712227
\(675\) 0 0
\(676\) −3.33470e6 −0.280666
\(677\) −6.09278e6 −0.510909 −0.255455 0.966821i \(-0.582225\pi\)
−0.255455 + 0.966821i \(0.582225\pi\)
\(678\) −1.49899e6 −0.125234
\(679\) −2.06081e6 −0.171539
\(680\) 0 0
\(681\) −7.19377e6 −0.594414
\(682\) −579832. −0.0477355
\(683\) −1.44978e7 −1.18918 −0.594592 0.804027i \(-0.702686\pi\)
−0.594592 + 0.804027i \(0.702686\pi\)
\(684\) −5.42376e6 −0.443262
\(685\) 0 0
\(686\) 856388. 0.0694801
\(687\) 1.38590e7 1.12032
\(688\) −7.97082e6 −0.641995
\(689\) 2.13164e7 1.71067
\(690\) 0 0
\(691\) 9.87069e6 0.786416 0.393208 0.919449i \(-0.371365\pi\)
0.393208 + 0.919449i \(0.371365\pi\)
\(692\) 1.47757e7 1.17296
\(693\) 254826. 0.0201563
\(694\) −2.02560e6 −0.159645
\(695\) 0 0
\(696\) 3.13324e6 0.245172
\(697\) −1.53169e7 −1.19423
\(698\) 378924. 0.0294384
\(699\) −6.49647e6 −0.502903
\(700\) 0 0
\(701\) 6.35411e6 0.488382 0.244191 0.969727i \(-0.421478\pi\)
0.244191 + 0.969727i \(0.421478\pi\)
\(702\) 504468. 0.0386359
\(703\) 2.20190e7 1.68039
\(704\) −3.24074e6 −0.246441
\(705\) 0 0
\(706\) −1.98730e6 −0.150056
\(707\) −648908. −0.0488241
\(708\) 1.79006e6 0.134210
\(709\) −411382. −0.0307348 −0.0153674 0.999882i \(-0.504892\pi\)
−0.0153674 + 0.999882i \(0.504892\pi\)
\(710\) 0 0
\(711\) −8.52233e6 −0.632244
\(712\) −4.55150e6 −0.336476
\(713\) 7.58094e6 0.558470
\(714\) 337428. 0.0247705
\(715\) 0 0
\(716\) −2.79595e6 −0.203820
\(717\) 5.74592e6 0.417409
\(718\) 3.43975e6 0.249009
\(719\) 6.29795e6 0.454336 0.227168 0.973856i \(-0.427053\pi\)
0.227168 + 0.973856i \(0.427053\pi\)
\(720\) 0 0
\(721\) 1.47711e6 0.105822
\(722\) −2.18950e6 −0.156316
\(723\) −1.98891e6 −0.141504
\(724\) −7.68806e6 −0.545093
\(725\) 0 0
\(726\) 131769. 0.00927837
\(727\) −1.14699e7 −0.804866 −0.402433 0.915449i \(-0.631835\pi\)
−0.402433 + 0.915449i \(0.631835\pi\)
\(728\) 1.13350e6 0.0792668
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −1.23724e7 −0.856365
\(732\) 1.17448e7 0.810154
\(733\) 1.87547e7 1.28929 0.644646 0.764481i \(-0.277005\pi\)
0.644646 + 0.764481i \(0.277005\pi\)
\(734\) −1.79679e6 −0.123100
\(735\) 0 0
\(736\) −4.65899e6 −0.317028
\(737\) 3.44172e6 0.233403
\(738\) 860382. 0.0581501
\(739\) 2.79727e6 0.188418 0.0942091 0.995552i \(-0.469968\pi\)
0.0942091 + 0.995552i \(0.469968\pi\)
\(740\) 0 0
\(741\) −1.34525e7 −0.900030
\(742\) −800904. −0.0534036
\(743\) 2.25651e7 1.49956 0.749781 0.661686i \(-0.230159\pi\)
0.749781 + 0.661686i \(0.230159\pi\)
\(744\) −2.71706e6 −0.179956
\(745\) 0 0
\(746\) −1.43541e6 −0.0944342
\(747\) −5.04565e6 −0.330838
\(748\) −5.40894e6 −0.353475
\(749\) 324792. 0.0211544
\(750\) 0 0
\(751\) −7.49233e6 −0.484749 −0.242375 0.970183i \(-0.577926\pi\)
−0.242375 + 0.970183i \(0.577926\pi\)
\(752\) 2.19430e6 0.141498
\(753\) −5.64574e6 −0.362855
\(754\) 3.82399e6 0.244956
\(755\) 0 0
\(756\) 587574. 0.0373902
\(757\) −2.88492e7 −1.82976 −0.914880 0.403727i \(-0.867715\pi\)
−0.914880 + 0.403727i \(0.867715\pi\)
\(758\) −2.66235e6 −0.168303
\(759\) −1.72280e6 −0.108550
\(760\) 0 0
\(761\) −9.56279e6 −0.598581 −0.299291 0.954162i \(-0.596750\pi\)
−0.299291 + 0.954162i \(0.596750\pi\)
\(762\) −2.73803e6 −0.170825
\(763\) 5.16745e6 0.321340
\(764\) 4.86086e6 0.301287
\(765\) 0 0
\(766\) 2.04091e6 0.125676
\(767\) 4.43987e6 0.272510
\(768\) −6.62430e6 −0.405263
\(769\) −744898. −0.0454235 −0.0227118 0.999742i \(-0.507230\pi\)
−0.0227118 + 0.999742i \(0.507230\pi\)
\(770\) 0 0
\(771\) −4.21213e6 −0.255191
\(772\) −1.33683e7 −0.807293
\(773\) −6.07336e6 −0.365578 −0.182789 0.983152i \(-0.558513\pi\)
−0.182789 + 0.983152i \(0.558513\pi\)
\(774\) 694980. 0.0416984
\(775\) 0 0
\(776\) −4.99351e6 −0.297681
\(777\) −2.38540e6 −0.141745
\(778\) 4.29947e6 0.254663
\(779\) −2.29435e7 −1.35462
\(780\) 0 0
\(781\) 5.52849e6 0.324324
\(782\) −2.28124e6 −0.133400
\(783\) 4.02845e6 0.234819
\(784\) −1.49857e7 −0.870737
\(785\) 0 0
\(786\) 2.46985e6 0.142598
\(787\) 1.47512e7 0.848966 0.424483 0.905436i \(-0.360456\pi\)
0.424483 + 0.905436i \(0.360456\pi\)
\(788\) −2.68142e7 −1.53833
\(789\) 1.39059e7 0.795257
\(790\) 0 0
\(791\) −4.33040e6 −0.246086
\(792\) 617463. 0.0349782
\(793\) 2.91304e7 1.64499
\(794\) 728818. 0.0410268
\(795\) 0 0
\(796\) 1.48819e7 0.832481
\(797\) 2.78359e7 1.55224 0.776121 0.630584i \(-0.217185\pi\)
0.776121 + 0.630584i \(0.217185\pi\)
\(798\) 505440. 0.0280972
\(799\) 3.40600e6 0.188746
\(800\) 0 0
\(801\) −5.85193e6 −0.322268
\(802\) 5.92515e6 0.325285
\(803\) 2.22325e6 0.121675
\(804\) 7.93588e6 0.432967
\(805\) 0 0
\(806\) −3.31606e6 −0.179798
\(807\) 9.67115e6 0.522750
\(808\) −1.57235e6 −0.0847270
\(809\) 2.54767e7 1.36859 0.684293 0.729207i \(-0.260111\pi\)
0.684293 + 0.729207i \(0.260111\pi\)
\(810\) 0 0
\(811\) −1.91915e7 −1.02460 −0.512302 0.858805i \(-0.671207\pi\)
−0.512302 + 0.858805i \(0.671207\pi\)
\(812\) 4.45396e6 0.237059
\(813\) −1.42851e7 −0.757977
\(814\) −1.23347e6 −0.0652483
\(815\) 0 0
\(816\) −1.20566e7 −0.633867
\(817\) −1.85328e7 −0.971373
\(818\) −1.38212e6 −0.0722207
\(819\) 1.45735e6 0.0759197
\(820\) 0 0
\(821\) 3.27107e6 0.169368 0.0846840 0.996408i \(-0.473012\pi\)
0.0846840 + 0.996408i \(0.473012\pi\)
\(822\) 2.20912e6 0.114036
\(823\) 3.19195e7 1.64269 0.821347 0.570430i \(-0.193223\pi\)
0.821347 + 0.570430i \(0.193223\pi\)
\(824\) 3.57916e6 0.183638
\(825\) 0 0
\(826\) −166816. −0.00850722
\(827\) −2.45556e7 −1.24850 −0.624248 0.781226i \(-0.714595\pi\)
−0.624248 + 0.781226i \(0.714595\pi\)
\(828\) −3.97240e6 −0.201362
\(829\) −1.40969e7 −0.712421 −0.356211 0.934406i \(-0.615931\pi\)
−0.356211 + 0.934406i \(0.615931\pi\)
\(830\) 0 0
\(831\) 6.23434e6 0.313175
\(832\) −1.85338e7 −0.928233
\(833\) −2.32609e7 −1.16149
\(834\) −538992. −0.0268329
\(835\) 0 0
\(836\) −8.10216e6 −0.400945
\(837\) −3.49337e6 −0.172358
\(838\) −5.47794e6 −0.269468
\(839\) −3.01443e6 −0.147843 −0.0739213 0.997264i \(-0.523551\pi\)
−0.0739213 + 0.997264i \(0.523551\pi\)
\(840\) 0 0
\(841\) 1.00255e7 0.488784
\(842\) −1.02873e6 −0.0500061
\(843\) 5.10316e6 0.247326
\(844\) −1.63029e7 −0.787787
\(845\) 0 0
\(846\) −191322. −0.00919050
\(847\) 380666. 0.0182321
\(848\) 2.86169e7 1.36657
\(849\) 6.43424e6 0.306357
\(850\) 0 0
\(851\) 1.61269e7 0.763356
\(852\) 1.27475e7 0.601626
\(853\) 1.67201e7 0.786806 0.393403 0.919366i \(-0.371298\pi\)
0.393403 + 0.919366i \(0.371298\pi\)
\(854\) −1.09450e6 −0.0513534
\(855\) 0 0
\(856\) 786996. 0.0367103
\(857\) 9.15871e6 0.425973 0.212987 0.977055i \(-0.431681\pi\)
0.212987 + 0.977055i \(0.431681\pi\)
\(858\) 753588. 0.0349475
\(859\) 1.51068e7 0.698536 0.349268 0.937023i \(-0.386430\pi\)
0.349268 + 0.937023i \(0.386430\pi\)
\(860\) 0 0
\(861\) 2.48555e6 0.114265
\(862\) 5.14310e6 0.235753
\(863\) −5.11568e6 −0.233817 −0.116909 0.993143i \(-0.537298\pi\)
−0.116909 + 0.993143i \(0.537298\pi\)
\(864\) 2.14690e6 0.0978427
\(865\) 0 0
\(866\) 412954. 0.0187114
\(867\) −5.93556e6 −0.268172
\(868\) −3.86235e6 −0.174001
\(869\) −1.27309e7 −0.571886
\(870\) 0 0
\(871\) 1.96832e7 0.879127
\(872\) 1.25211e7 0.557638
\(873\) −6.42022e6 −0.285111
\(874\) −3.41712e6 −0.151315
\(875\) 0 0
\(876\) 5.12635e6 0.225708
\(877\) 1.26998e7 0.557568 0.278784 0.960354i \(-0.410069\pi\)
0.278784 + 0.960354i \(0.410069\pi\)
\(878\) −5.96365e6 −0.261081
\(879\) 1.92968e7 0.842392
\(880\) 0 0
\(881\) −8.38173e6 −0.363826 −0.181913 0.983315i \(-0.558229\pi\)
−0.181913 + 0.983315i \(0.558229\pi\)
\(882\) 1.30661e6 0.0565555
\(883\) 1.69529e7 0.731715 0.365858 0.930671i \(-0.380776\pi\)
0.365858 + 0.930671i \(0.380776\pi\)
\(884\) −3.09338e7 −1.33138
\(885\) 0 0
\(886\) 2.18433e6 0.0934832
\(887\) 1.05143e7 0.448717 0.224359 0.974507i \(-0.427971\pi\)
0.224359 + 0.974507i \(0.427971\pi\)
\(888\) −5.78000e6 −0.245977
\(889\) −7.90988e6 −0.335672
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) −7.60318e6 −0.319951
\(893\) 5.10192e6 0.214094
\(894\) 648342. 0.0271307
\(895\) 0 0
\(896\) 3.14660e6 0.130940
\(897\) −9.85270e6 −0.408859
\(898\) 7858.00 0.000325178 0
\(899\) −2.64806e7 −1.09277
\(900\) 0 0
\(901\) 4.44194e7 1.82289
\(902\) 1.28526e6 0.0525987
\(903\) 2.00772e6 0.0819377
\(904\) −1.04929e7 −0.427046
\(905\) 0 0
\(906\) −2.90799e6 −0.117699
\(907\) 1.53747e7 0.620569 0.310284 0.950644i \(-0.399576\pi\)
0.310284 + 0.950644i \(0.399576\pi\)
\(908\) −2.47785e7 −0.997381
\(909\) −2.02160e6 −0.0811494
\(910\) 0 0
\(911\) 1.25424e7 0.500708 0.250354 0.968154i \(-0.419453\pi\)
0.250354 + 0.968154i \(0.419453\pi\)
\(912\) −1.80598e7 −0.718993
\(913\) −7.53733e6 −0.299255
\(914\) −899922. −0.0356319
\(915\) 0 0
\(916\) 4.77367e7 1.87981
\(917\) 7.13513e6 0.280207
\(918\) 1.05122e6 0.0411705
\(919\) 3.31432e7 1.29451 0.647256 0.762273i \(-0.275916\pi\)
0.647256 + 0.762273i \(0.275916\pi\)
\(920\) 0 0
\(921\) −5.29927e6 −0.205858
\(922\) −1.13619e6 −0.0440173
\(923\) 3.16175e7 1.22158
\(924\) 877734. 0.0338207
\(925\) 0 0
\(926\) −7.38964e6 −0.283202
\(927\) 4.60177e6 0.175884
\(928\) 1.62741e7 0.620335
\(929\) 3.10442e7 1.18016 0.590080 0.807345i \(-0.299096\pi\)
0.590080 + 0.807345i \(0.299096\pi\)
\(930\) 0 0
\(931\) −3.48430e7 −1.31747
\(932\) −2.23767e7 −0.843834
\(933\) −2.26644e7 −0.852393
\(934\) 4.20851e6 0.157856
\(935\) 0 0
\(936\) 3.53128e6 0.131747
\(937\) 3.10737e7 1.15623 0.578115 0.815955i \(-0.303788\pi\)
0.578115 + 0.815955i \(0.303788\pi\)
\(938\) −739544. −0.0274446
\(939\) −2.01206e7 −0.744692
\(940\) 0 0
\(941\) −2.50349e7 −0.921664 −0.460832 0.887488i \(-0.652449\pi\)
−0.460832 + 0.887488i \(0.652449\pi\)
\(942\) 2.86889e6 0.105339
\(943\) −1.68040e7 −0.615366
\(944\) 5.96046e6 0.217696
\(945\) 0 0
\(946\) 1.03818e6 0.0377177
\(947\) 5.37383e6 0.194719 0.0973596 0.995249i \(-0.468960\pi\)
0.0973596 + 0.995249i \(0.468960\pi\)
\(948\) −2.93547e7 −1.06086
\(949\) 1.27148e7 0.458294
\(950\) 0 0
\(951\) 9.54713e6 0.342311
\(952\) 2.36200e6 0.0844669
\(953\) −7.26908e6 −0.259267 −0.129634 0.991562i \(-0.541380\pi\)
−0.129634 + 0.991562i \(0.541380\pi\)
\(954\) −2.49512e6 −0.0887608
\(955\) 0 0
\(956\) 1.97915e7 0.700381
\(957\) 6.01781e6 0.212402
\(958\) −7.39441e6 −0.260309
\(959\) 6.38191e6 0.224080
\(960\) 0 0
\(961\) −5.66589e6 −0.197906
\(962\) −7.05425e6 −0.245761
\(963\) 1.01185e6 0.0351602
\(964\) −6.85069e6 −0.237433
\(965\) 0 0
\(966\) 370188. 0.0127638
\(967\) 2.54428e7 0.874983 0.437491 0.899223i \(-0.355867\pi\)
0.437491 + 0.899223i \(0.355867\pi\)
\(968\) 922383. 0.0316390
\(969\) −2.80325e7 −0.959074
\(970\) 0 0
\(971\) 9.88213e6 0.336358 0.168179 0.985756i \(-0.446211\pi\)
0.168179 + 0.985756i \(0.446211\pi\)
\(972\) 1.83052e6 0.0621453
\(973\) −1.55709e6 −0.0527268
\(974\) −3.81644e6 −0.128902
\(975\) 0 0
\(976\) 3.91072e7 1.31411
\(977\) −2.22197e6 −0.0744736 −0.0372368 0.999306i \(-0.511856\pi\)
−0.0372368 + 0.999306i \(0.511856\pi\)
\(978\) 3.88796e6 0.129980
\(979\) −8.74177e6 −0.291503
\(980\) 0 0
\(981\) 1.60986e7 0.534091
\(982\) −1.69716e6 −0.0561623
\(983\) −2.53706e7 −0.837428 −0.418714 0.908118i \(-0.637519\pi\)
−0.418714 + 0.908118i \(0.637519\pi\)
\(984\) 6.02267e6 0.198290
\(985\) 0 0
\(986\) 7.96849e6 0.261026
\(987\) −552708. −0.0180594
\(988\) −4.63363e7 −1.51018
\(989\) −1.35736e7 −0.441269
\(990\) 0 0
\(991\) 3.24132e7 1.04843 0.524214 0.851587i \(-0.324359\pi\)
0.524214 + 0.851587i \(0.324359\pi\)
\(992\) −1.41124e7 −0.455327
\(993\) 2.11109e7 0.679414
\(994\) −1.18794e6 −0.0381354
\(995\) 0 0
\(996\) −1.73795e7 −0.555122
\(997\) 1.55048e6 0.0494000 0.0247000 0.999695i \(-0.492137\pi\)
0.0247000 + 0.999695i \(0.492137\pi\)
\(998\) −6.95160e6 −0.220932
\(999\) −7.43143e6 −0.235591
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 825.6.a.a.1.1 1
5.4 even 2 33.6.a.b.1.1 1
15.14 odd 2 99.6.a.a.1.1 1
20.19 odd 2 528.6.a.a.1.1 1
55.54 odd 2 363.6.a.b.1.1 1
165.164 even 2 1089.6.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.6.a.b.1.1 1 5.4 even 2
99.6.a.a.1.1 1 15.14 odd 2
363.6.a.b.1.1 1 55.54 odd 2
528.6.a.a.1.1 1 20.19 odd 2
825.6.a.a.1.1 1 1.1 even 1 trivial
1089.6.a.h.1.1 1 165.164 even 2