Properties

Label 462.6.a.a.1.1
Level $462$
Weight $6$
Character 462.1
Self dual yes
Analytic conductor $74.097$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 462.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(74.0973247536\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 462.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -78.0000 q^{5} +36.0000 q^{6} +49.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -78.0000 q^{5} +36.0000 q^{6} +49.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +312.000 q^{10} +121.000 q^{11} -144.000 q^{12} -502.000 q^{13} -196.000 q^{14} +702.000 q^{15} +256.000 q^{16} -642.000 q^{17} -324.000 q^{18} -520.000 q^{19} -1248.00 q^{20} -441.000 q^{21} -484.000 q^{22} +1020.00 q^{23} +576.000 q^{24} +2959.00 q^{25} +2008.00 q^{26} -729.000 q^{27} +784.000 q^{28} +4818.00 q^{29} -2808.00 q^{30} +1784.00 q^{31} -1024.00 q^{32} -1089.00 q^{33} +2568.00 q^{34} -3822.00 q^{35} +1296.00 q^{36} +7958.00 q^{37} +2080.00 q^{38} +4518.00 q^{39} +4992.00 q^{40} +2430.00 q^{41} +1764.00 q^{42} +22904.0 q^{43} +1936.00 q^{44} -6318.00 q^{45} -4080.00 q^{46} -11316.0 q^{47} -2304.00 q^{48} +2401.00 q^{49} -11836.0 q^{50} +5778.00 q^{51} -8032.00 q^{52} -12222.0 q^{53} +2916.00 q^{54} -9438.00 q^{55} -3136.00 q^{56} +4680.00 q^{57} -19272.0 q^{58} -15852.0 q^{59} +11232.0 q^{60} +46298.0 q^{61} -7136.00 q^{62} +3969.00 q^{63} +4096.00 q^{64} +39156.0 q^{65} +4356.00 q^{66} +19412.0 q^{67} -10272.0 q^{68} -9180.00 q^{69} +15288.0 q^{70} -17292.0 q^{71} -5184.00 q^{72} -30214.0 q^{73} -31832.0 q^{74} -26631.0 q^{75} -8320.00 q^{76} +5929.00 q^{77} -18072.0 q^{78} +35672.0 q^{79} -19968.0 q^{80} +6561.00 q^{81} -9720.00 q^{82} -43428.0 q^{83} -7056.00 q^{84} +50076.0 q^{85} -91616.0 q^{86} -43362.0 q^{87} -7744.00 q^{88} -14934.0 q^{89} +25272.0 q^{90} -24598.0 q^{91} +16320.0 q^{92} -16056.0 q^{93} +45264.0 q^{94} +40560.0 q^{95} +9216.00 q^{96} +85106.0 q^{97} -9604.00 q^{98} +9801.00 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) −9.00000 −0.577350
\(4\) 16.0000 0.500000
\(5\) −78.0000 −1.39531 −0.697653 0.716436i \(-0.745772\pi\)
−0.697653 + 0.716436i \(0.745772\pi\)
\(6\) 36.0000 0.408248
\(7\) 49.0000 0.377964
\(8\) −64.0000 −0.353553
\(9\) 81.0000 0.333333
\(10\) 312.000 0.986631
\(11\) 121.000 0.301511
\(12\) −144.000 −0.288675
\(13\) −502.000 −0.823845 −0.411922 0.911219i \(-0.635143\pi\)
−0.411922 + 0.911219i \(0.635143\pi\)
\(14\) −196.000 −0.267261
\(15\) 702.000 0.805581
\(16\) 256.000 0.250000
\(17\) −642.000 −0.538782 −0.269391 0.963031i \(-0.586822\pi\)
−0.269391 + 0.963031i \(0.586822\pi\)
\(18\) −324.000 −0.235702
\(19\) −520.000 −0.330460 −0.165230 0.986255i \(-0.552837\pi\)
−0.165230 + 0.986255i \(0.552837\pi\)
\(20\) −1248.00 −0.697653
\(21\) −441.000 −0.218218
\(22\) −484.000 −0.213201
\(23\) 1020.00 0.402050 0.201025 0.979586i \(-0.435573\pi\)
0.201025 + 0.979586i \(0.435573\pi\)
\(24\) 576.000 0.204124
\(25\) 2959.00 0.946880
\(26\) 2008.00 0.582546
\(27\) −729.000 −0.192450
\(28\) 784.000 0.188982
\(29\) 4818.00 1.06383 0.531914 0.846798i \(-0.321473\pi\)
0.531914 + 0.846798i \(0.321473\pi\)
\(30\) −2808.00 −0.569631
\(31\) 1784.00 0.333419 0.166710 0.986006i \(-0.446686\pi\)
0.166710 + 0.986006i \(0.446686\pi\)
\(32\) −1024.00 −0.176777
\(33\) −1089.00 −0.174078
\(34\) 2568.00 0.380976
\(35\) −3822.00 −0.527376
\(36\) 1296.00 0.166667
\(37\) 7958.00 0.955652 0.477826 0.878455i \(-0.341425\pi\)
0.477826 + 0.878455i \(0.341425\pi\)
\(38\) 2080.00 0.233671
\(39\) 4518.00 0.475647
\(40\) 4992.00 0.493315
\(41\) 2430.00 0.225760 0.112880 0.993609i \(-0.463992\pi\)
0.112880 + 0.993609i \(0.463992\pi\)
\(42\) 1764.00 0.154303
\(43\) 22904.0 1.88904 0.944518 0.328460i \(-0.106530\pi\)
0.944518 + 0.328460i \(0.106530\pi\)
\(44\) 1936.00 0.150756
\(45\) −6318.00 −0.465102
\(46\) −4080.00 −0.284293
\(47\) −11316.0 −0.747220 −0.373610 0.927586i \(-0.621880\pi\)
−0.373610 + 0.927586i \(0.621880\pi\)
\(48\) −2304.00 −0.144338
\(49\) 2401.00 0.142857
\(50\) −11836.0 −0.669545
\(51\) 5778.00 0.311066
\(52\) −8032.00 −0.411922
\(53\) −12222.0 −0.597658 −0.298829 0.954307i \(-0.596596\pi\)
−0.298829 + 0.954307i \(0.596596\pi\)
\(54\) 2916.00 0.136083
\(55\) −9438.00 −0.420701
\(56\) −3136.00 −0.133631
\(57\) 4680.00 0.190791
\(58\) −19272.0 −0.752241
\(59\) −15852.0 −0.592863 −0.296431 0.955054i \(-0.595797\pi\)
−0.296431 + 0.955054i \(0.595797\pi\)
\(60\) 11232.0 0.402790
\(61\) 46298.0 1.59308 0.796540 0.604586i \(-0.206661\pi\)
0.796540 + 0.604586i \(0.206661\pi\)
\(62\) −7136.00 −0.235763
\(63\) 3969.00 0.125988
\(64\) 4096.00 0.125000
\(65\) 39156.0 1.14952
\(66\) 4356.00 0.123091
\(67\) 19412.0 0.528303 0.264152 0.964481i \(-0.414908\pi\)
0.264152 + 0.964481i \(0.414908\pi\)
\(68\) −10272.0 −0.269391
\(69\) −9180.00 −0.232124
\(70\) 15288.0 0.372911
\(71\) −17292.0 −0.407098 −0.203549 0.979065i \(-0.565248\pi\)
−0.203549 + 0.979065i \(0.565248\pi\)
\(72\) −5184.00 −0.117851
\(73\) −30214.0 −0.663592 −0.331796 0.943351i \(-0.607655\pi\)
−0.331796 + 0.943351i \(0.607655\pi\)
\(74\) −31832.0 −0.675748
\(75\) −26631.0 −0.546681
\(76\) −8320.00 −0.165230
\(77\) 5929.00 0.113961
\(78\) −18072.0 −0.336333
\(79\) 35672.0 0.643072 0.321536 0.946897i \(-0.395801\pi\)
0.321536 + 0.946897i \(0.395801\pi\)
\(80\) −19968.0 −0.348827
\(81\) 6561.00 0.111111
\(82\) −9720.00 −0.159636
\(83\) −43428.0 −0.691950 −0.345975 0.938244i \(-0.612452\pi\)
−0.345975 + 0.938244i \(0.612452\pi\)
\(84\) −7056.00 −0.109109
\(85\) 50076.0 0.751765
\(86\) −91616.0 −1.33575
\(87\) −43362.0 −0.614202
\(88\) −7744.00 −0.106600
\(89\) −14934.0 −0.199849 −0.0999243 0.994995i \(-0.531860\pi\)
−0.0999243 + 0.994995i \(0.531860\pi\)
\(90\) 25272.0 0.328877
\(91\) −24598.0 −0.311384
\(92\) 16320.0 0.201025
\(93\) −16056.0 −0.192500
\(94\) 45264.0 0.528364
\(95\) 40560.0 0.461093
\(96\) 9216.00 0.102062
\(97\) 85106.0 0.918398 0.459199 0.888333i \(-0.348137\pi\)
0.459199 + 0.888333i \(0.348137\pi\)
\(98\) −9604.00 −0.101015
\(99\) 9801.00 0.100504
\(100\) 47344.0 0.473440
\(101\) −72150.0 −0.703773 −0.351887 0.936043i \(-0.614460\pi\)
−0.351887 + 0.936043i \(0.614460\pi\)
\(102\) −23112.0 −0.219957
\(103\) 11744.0 0.109074 0.0545372 0.998512i \(-0.482632\pi\)
0.0545372 + 0.998512i \(0.482632\pi\)
\(104\) 32128.0 0.291273
\(105\) 34398.0 0.304481
\(106\) 48888.0 0.422608
\(107\) −217932. −1.84019 −0.920093 0.391701i \(-0.871887\pi\)
−0.920093 + 0.391701i \(0.871887\pi\)
\(108\) −11664.0 −0.0962250
\(109\) 87674.0 0.706813 0.353407 0.935470i \(-0.385023\pi\)
0.353407 + 0.935470i \(0.385023\pi\)
\(110\) 37752.0 0.297480
\(111\) −71622.0 −0.551746
\(112\) 12544.0 0.0944911
\(113\) 145818. 1.07427 0.537137 0.843495i \(-0.319506\pi\)
0.537137 + 0.843495i \(0.319506\pi\)
\(114\) −18720.0 −0.134910
\(115\) −79560.0 −0.560984
\(116\) 77088.0 0.531914
\(117\) −40662.0 −0.274615
\(118\) 63408.0 0.419217
\(119\) −31458.0 −0.203640
\(120\) −44928.0 −0.284816
\(121\) 14641.0 0.0909091
\(122\) −185192. −1.12648
\(123\) −21870.0 −0.130342
\(124\) 28544.0 0.166710
\(125\) 12948.0 0.0741187
\(126\) −15876.0 −0.0890871
\(127\) −341752. −1.88019 −0.940095 0.340912i \(-0.889264\pi\)
−0.940095 + 0.340912i \(0.889264\pi\)
\(128\) −16384.0 −0.0883883
\(129\) −206136. −1.09064
\(130\) −156624. −0.812830
\(131\) −110460. −0.562376 −0.281188 0.959653i \(-0.590728\pi\)
−0.281188 + 0.959653i \(0.590728\pi\)
\(132\) −17424.0 −0.0870388
\(133\) −25480.0 −0.124902
\(134\) −77648.0 −0.373567
\(135\) 56862.0 0.268527
\(136\) 41088.0 0.190488
\(137\) −123510. −0.562213 −0.281106 0.959677i \(-0.590701\pi\)
−0.281106 + 0.959677i \(0.590701\pi\)
\(138\) 36720.0 0.164136
\(139\) 134720. 0.591419 0.295709 0.955278i \(-0.404444\pi\)
0.295709 + 0.955278i \(0.404444\pi\)
\(140\) −61152.0 −0.263688
\(141\) 101844. 0.431407
\(142\) 69168.0 0.287862
\(143\) −60742.0 −0.248399
\(144\) 20736.0 0.0833333
\(145\) −375804. −1.48437
\(146\) 120856. 0.469230
\(147\) −21609.0 −0.0824786
\(148\) 127328. 0.477826
\(149\) −387270. −1.42905 −0.714526 0.699609i \(-0.753358\pi\)
−0.714526 + 0.699609i \(0.753358\pi\)
\(150\) 106524. 0.386562
\(151\) −463792. −1.65532 −0.827658 0.561233i \(-0.810327\pi\)
−0.827658 + 0.561233i \(0.810327\pi\)
\(152\) 33280.0 0.116835
\(153\) −52002.0 −0.179594
\(154\) −23716.0 −0.0805823
\(155\) −139152. −0.465222
\(156\) 72288.0 0.237823
\(157\) −371122. −1.20162 −0.600811 0.799391i \(-0.705155\pi\)
−0.600811 + 0.799391i \(0.705155\pi\)
\(158\) −142688. −0.454721
\(159\) 109998. 0.345058
\(160\) 79872.0 0.246658
\(161\) 49980.0 0.151961
\(162\) −26244.0 −0.0785674
\(163\) −76156.0 −0.224510 −0.112255 0.993679i \(-0.535807\pi\)
−0.112255 + 0.993679i \(0.535807\pi\)
\(164\) 38880.0 0.112880
\(165\) 84942.0 0.242892
\(166\) 173712. 0.489282
\(167\) 172200. 0.477795 0.238898 0.971045i \(-0.423214\pi\)
0.238898 + 0.971045i \(0.423214\pi\)
\(168\) 28224.0 0.0771517
\(169\) −119289. −0.321280
\(170\) −200304. −0.531578
\(171\) −42120.0 −0.110153
\(172\) 366464. 0.944518
\(173\) −601134. −1.52706 −0.763530 0.645772i \(-0.776535\pi\)
−0.763530 + 0.645772i \(0.776535\pi\)
\(174\) 173448. 0.434306
\(175\) 144991. 0.357887
\(176\) 30976.0 0.0753778
\(177\) 142668. 0.342289
\(178\) 59736.0 0.141314
\(179\) 223572. 0.521537 0.260768 0.965401i \(-0.416024\pi\)
0.260768 + 0.965401i \(0.416024\pi\)
\(180\) −101088. −0.232551
\(181\) 181622. 0.412071 0.206035 0.978545i \(-0.433944\pi\)
0.206035 + 0.978545i \(0.433944\pi\)
\(182\) 98392.0 0.220182
\(183\) −416682. −0.919765
\(184\) −65280.0 −0.142146
\(185\) −620724. −1.33343
\(186\) 64224.0 0.136118
\(187\) −77682.0 −0.162449
\(188\) −181056. −0.373610
\(189\) −35721.0 −0.0727393
\(190\) −162240. −0.326042
\(191\) 117252. 0.232561 0.116280 0.993216i \(-0.462903\pi\)
0.116280 + 0.993216i \(0.462903\pi\)
\(192\) −36864.0 −0.0721688
\(193\) −410758. −0.793766 −0.396883 0.917869i \(-0.629908\pi\)
−0.396883 + 0.917869i \(0.629908\pi\)
\(194\) −340424. −0.649405
\(195\) −352404. −0.663673
\(196\) 38416.0 0.0714286
\(197\) 190506. 0.349738 0.174869 0.984592i \(-0.444050\pi\)
0.174869 + 0.984592i \(0.444050\pi\)
\(198\) −39204.0 −0.0710669
\(199\) −893560. −1.59952 −0.799762 0.600317i \(-0.795041\pi\)
−0.799762 + 0.600317i \(0.795041\pi\)
\(200\) −189376. −0.334773
\(201\) −174708. −0.305016
\(202\) 288600. 0.497643
\(203\) 236082. 0.402090
\(204\) 92448.0 0.155533
\(205\) −189540. −0.315004
\(206\) −46976.0 −0.0771273
\(207\) 82620.0 0.134017
\(208\) −128512. −0.205961
\(209\) −62920.0 −0.0996375
\(210\) −137592. −0.215300
\(211\) 294392. 0.455218 0.227609 0.973753i \(-0.426909\pi\)
0.227609 + 0.973753i \(0.426909\pi\)
\(212\) −195552. −0.298829
\(213\) 155628. 0.235038
\(214\) 871728. 1.30121
\(215\) −1.78651e6 −2.63578
\(216\) 46656.0 0.0680414
\(217\) 87416.0 0.126021
\(218\) −350696. −0.499792
\(219\) 271926. 0.383125
\(220\) −151008. −0.210350
\(221\) 322284. 0.443872
\(222\) 286488. 0.390143
\(223\) −399040. −0.537346 −0.268673 0.963231i \(-0.586585\pi\)
−0.268673 + 0.963231i \(0.586585\pi\)
\(224\) −50176.0 −0.0668153
\(225\) 239679. 0.315627
\(226\) −583272. −0.759626
\(227\) 86100.0 0.110902 0.0554509 0.998461i \(-0.482340\pi\)
0.0554509 + 0.998461i \(0.482340\pi\)
\(228\) 74880.0 0.0953957
\(229\) 371126. 0.467663 0.233831 0.972277i \(-0.424874\pi\)
0.233831 + 0.972277i \(0.424874\pi\)
\(230\) 318240. 0.396675
\(231\) −53361.0 −0.0657952
\(232\) −308352. −0.376120
\(233\) −57786.0 −0.0697321 −0.0348661 0.999392i \(-0.511100\pi\)
−0.0348661 + 0.999392i \(0.511100\pi\)
\(234\) 162648. 0.194182
\(235\) 882648. 1.04260
\(236\) −253632. −0.296431
\(237\) −321048. −0.371278
\(238\) 125832. 0.143995
\(239\) 725616. 0.821698 0.410849 0.911703i \(-0.365232\pi\)
0.410849 + 0.911703i \(0.365232\pi\)
\(240\) 179712. 0.201395
\(241\) 625370. 0.693577 0.346788 0.937943i \(-0.387272\pi\)
0.346788 + 0.937943i \(0.387272\pi\)
\(242\) −58564.0 −0.0642824
\(243\) −59049.0 −0.0641500
\(244\) 740768. 0.796540
\(245\) −187278. −0.199329
\(246\) 87480.0 0.0921661
\(247\) 261040. 0.272248
\(248\) −114176. −0.117881
\(249\) 390852. 0.399497
\(250\) −51792.0 −0.0524098
\(251\) 267084. 0.267586 0.133793 0.991009i \(-0.457284\pi\)
0.133793 + 0.991009i \(0.457284\pi\)
\(252\) 63504.0 0.0629941
\(253\) 123420. 0.121223
\(254\) 1.36701e6 1.32950
\(255\) −450684. −0.434032
\(256\) 65536.0 0.0625000
\(257\) −63726.0 −0.0601844 −0.0300922 0.999547i \(-0.509580\pi\)
−0.0300922 + 0.999547i \(0.509580\pi\)
\(258\) 824544. 0.771196
\(259\) 389942. 0.361202
\(260\) 626496. 0.574758
\(261\) 390258. 0.354610
\(262\) 441840. 0.397660
\(263\) −153888. −0.137188 −0.0685939 0.997645i \(-0.521851\pi\)
−0.0685939 + 0.997645i \(0.521851\pi\)
\(264\) 69696.0 0.0615457
\(265\) 953316. 0.833916
\(266\) 101920. 0.0883192
\(267\) 134406. 0.115383
\(268\) 310592. 0.264152
\(269\) 1.79997e6 1.51665 0.758324 0.651878i \(-0.226019\pi\)
0.758324 + 0.651878i \(0.226019\pi\)
\(270\) −227448. −0.189877
\(271\) −1.05129e6 −0.869558 −0.434779 0.900537i \(-0.643173\pi\)
−0.434779 + 0.900537i \(0.643173\pi\)
\(272\) −164352. −0.134695
\(273\) 221382. 0.179778
\(274\) 494040. 0.397544
\(275\) 358039. 0.285495
\(276\) −146880. −0.116062
\(277\) 2.14313e6 1.67822 0.839110 0.543961i \(-0.183076\pi\)
0.839110 + 0.543961i \(0.183076\pi\)
\(278\) −538880. −0.418196
\(279\) 144504. 0.111140
\(280\) 244608. 0.186456
\(281\) 893910. 0.675349 0.337674 0.941263i \(-0.390360\pi\)
0.337674 + 0.941263i \(0.390360\pi\)
\(282\) −407376. −0.305051
\(283\) 665216. 0.493738 0.246869 0.969049i \(-0.420598\pi\)
0.246869 + 0.969049i \(0.420598\pi\)
\(284\) −276672. −0.203549
\(285\) −365040. −0.266212
\(286\) 242968. 0.175644
\(287\) 119070. 0.0853292
\(288\) −82944.0 −0.0589256
\(289\) −1.00769e6 −0.709714
\(290\) 1.50322e6 1.04961
\(291\) −765954. −0.530237
\(292\) −483424. −0.331796
\(293\) −2.20009e6 −1.49717 −0.748584 0.663040i \(-0.769266\pi\)
−0.748584 + 0.663040i \(0.769266\pi\)
\(294\) 86436.0 0.0583212
\(295\) 1.23646e6 0.827225
\(296\) −509312. −0.337874
\(297\) −88209.0 −0.0580259
\(298\) 1.54908e6 1.01049
\(299\) −512040. −0.331227
\(300\) −426096. −0.273341
\(301\) 1.12230e6 0.713988
\(302\) 1.85517e6 1.17049
\(303\) 649350. 0.406324
\(304\) −133120. −0.0826151
\(305\) −3.61124e6 −2.22284
\(306\) 208008. 0.126992
\(307\) −491272. −0.297493 −0.148746 0.988875i \(-0.547524\pi\)
−0.148746 + 0.988875i \(0.547524\pi\)
\(308\) 94864.0 0.0569803
\(309\) −105696. −0.0629742
\(310\) 556608. 0.328962
\(311\) −105180. −0.0616641 −0.0308320 0.999525i \(-0.509816\pi\)
−0.0308320 + 0.999525i \(0.509816\pi\)
\(312\) −289152. −0.168167
\(313\) −1.89175e6 −1.09145 −0.545724 0.837965i \(-0.683745\pi\)
−0.545724 + 0.837965i \(0.683745\pi\)
\(314\) 1.48449e6 0.849674
\(315\) −309582. −0.175792
\(316\) 570752. 0.321536
\(317\) −672198. −0.375707 −0.187853 0.982197i \(-0.560153\pi\)
−0.187853 + 0.982197i \(0.560153\pi\)
\(318\) −439992. −0.243993
\(319\) 582978. 0.320756
\(320\) −319488. −0.174413
\(321\) 1.96139e6 1.06243
\(322\) −199920. −0.107453
\(323\) 333840. 0.178046
\(324\) 104976. 0.0555556
\(325\) −1.48542e6 −0.780082
\(326\) 304624. 0.158752
\(327\) −789066. −0.408079
\(328\) −155520. −0.0798181
\(329\) −554484. −0.282422
\(330\) −339768. −0.171750
\(331\) −1.07290e6 −0.538256 −0.269128 0.963104i \(-0.586736\pi\)
−0.269128 + 0.963104i \(0.586736\pi\)
\(332\) −694848. −0.345975
\(333\) 644598. 0.318551
\(334\) −688800. −0.337852
\(335\) −1.51414e6 −0.737145
\(336\) −112896. −0.0545545
\(337\) 1.03582e6 0.496831 0.248416 0.968654i \(-0.420090\pi\)
0.248416 + 0.968654i \(0.420090\pi\)
\(338\) 477156. 0.227179
\(339\) −1.31236e6 −0.620232
\(340\) 801216. 0.375883
\(341\) 215864. 0.100530
\(342\) 168480. 0.0778902
\(343\) 117649. 0.0539949
\(344\) −1.46586e6 −0.667875
\(345\) 716040. 0.323884
\(346\) 2.40454e6 1.07979
\(347\) −1.52821e6 −0.681334 −0.340667 0.940184i \(-0.610653\pi\)
−0.340667 + 0.940184i \(0.610653\pi\)
\(348\) −693792. −0.307101
\(349\) −1.45205e6 −0.638141 −0.319071 0.947731i \(-0.603371\pi\)
−0.319071 + 0.947731i \(0.603371\pi\)
\(350\) −579964. −0.253064
\(351\) 365958. 0.158549
\(352\) −123904. −0.0533002
\(353\) −184926. −0.0789880 −0.0394940 0.999220i \(-0.512575\pi\)
−0.0394940 + 0.999220i \(0.512575\pi\)
\(354\) −570672. −0.242035
\(355\) 1.34878e6 0.568027
\(356\) −238944. −0.0999243
\(357\) 283122. 0.117572
\(358\) −894288. −0.368782
\(359\) 1.09661e6 0.449071 0.224536 0.974466i \(-0.427913\pi\)
0.224536 + 0.974466i \(0.427913\pi\)
\(360\) 404352. 0.164438
\(361\) −2.20570e6 −0.890796
\(362\) −726488. −0.291378
\(363\) −131769. −0.0524864
\(364\) −393568. −0.155692
\(365\) 2.35669e6 0.925914
\(366\) 1.66673e6 0.650372
\(367\) 3.73587e6 1.44786 0.723930 0.689873i \(-0.242334\pi\)
0.723930 + 0.689873i \(0.242334\pi\)
\(368\) 261120. 0.100513
\(369\) 196830. 0.0752533
\(370\) 2.48290e6 0.942875
\(371\) −598878. −0.225893
\(372\) −256896. −0.0962498
\(373\) 4.10724e6 1.52854 0.764272 0.644893i \(-0.223098\pi\)
0.764272 + 0.644893i \(0.223098\pi\)
\(374\) 310728. 0.114869
\(375\) −116532. −0.0427924
\(376\) 724224. 0.264182
\(377\) −2.41864e6 −0.876430
\(378\) 142884. 0.0514344
\(379\) 3.18638e6 1.13946 0.569731 0.821832i \(-0.307048\pi\)
0.569731 + 0.821832i \(0.307048\pi\)
\(380\) 648960. 0.230547
\(381\) 3.07577e6 1.08553
\(382\) −469008. −0.164445
\(383\) −4.20383e6 −1.46436 −0.732180 0.681111i \(-0.761497\pi\)
−0.732180 + 0.681111i \(0.761497\pi\)
\(384\) 147456. 0.0510310
\(385\) −462462. −0.159010
\(386\) 1.64303e6 0.561278
\(387\) 1.85522e6 0.629679
\(388\) 1.36170e6 0.459199
\(389\) −3.32540e6 −1.11422 −0.557108 0.830440i \(-0.688089\pi\)
−0.557108 + 0.830440i \(0.688089\pi\)
\(390\) 1.40962e6 0.469288
\(391\) −654840. −0.216617
\(392\) −153664. −0.0505076
\(393\) 994140. 0.324688
\(394\) −762024. −0.247302
\(395\) −2.78242e6 −0.897283
\(396\) 156816. 0.0502519
\(397\) 5.34467e6 1.70194 0.850971 0.525213i \(-0.176014\pi\)
0.850971 + 0.525213i \(0.176014\pi\)
\(398\) 3.57424e6 1.13103
\(399\) 229320. 0.0721124
\(400\) 757504. 0.236720
\(401\) −17262.0 −0.00536081 −0.00268040 0.999996i \(-0.500853\pi\)
−0.00268040 + 0.999996i \(0.500853\pi\)
\(402\) 698832. 0.215679
\(403\) −895568. −0.274686
\(404\) −1.15440e6 −0.351887
\(405\) −511758. −0.155034
\(406\) −944328. −0.284320
\(407\) 962918. 0.288140
\(408\) −369792. −0.109978
\(409\) −591886. −0.174956 −0.0874782 0.996166i \(-0.527881\pi\)
−0.0874782 + 0.996166i \(0.527881\pi\)
\(410\) 758160. 0.222742
\(411\) 1.11159e6 0.324594
\(412\) 187904. 0.0545372
\(413\) −776748. −0.224081
\(414\) −330480. −0.0947642
\(415\) 3.38738e6 0.965482
\(416\) 514048. 0.145637
\(417\) −1.21248e6 −0.341456
\(418\) 251680. 0.0704544
\(419\) 1.64303e6 0.457204 0.228602 0.973520i \(-0.426585\pi\)
0.228602 + 0.973520i \(0.426585\pi\)
\(420\) 550368. 0.152240
\(421\) −2.62575e6 −0.722019 −0.361010 0.932562i \(-0.617568\pi\)
−0.361010 + 0.932562i \(0.617568\pi\)
\(422\) −1.17757e6 −0.321888
\(423\) −916596. −0.249073
\(424\) 782208. 0.211304
\(425\) −1.89968e6 −0.510161
\(426\) −622512. −0.166197
\(427\) 2.26860e6 0.602128
\(428\) −3.48691e6 −0.920093
\(429\) 546678. 0.143413
\(430\) 7.14605e6 1.86378
\(431\) −1.79868e6 −0.466402 −0.233201 0.972429i \(-0.574920\pi\)
−0.233201 + 0.972429i \(0.574920\pi\)
\(432\) −186624. −0.0481125
\(433\) −2.84174e6 −0.728392 −0.364196 0.931322i \(-0.618656\pi\)
−0.364196 + 0.931322i \(0.618656\pi\)
\(434\) −349664. −0.0891100
\(435\) 3.38224e6 0.857000
\(436\) 1.40278e6 0.353407
\(437\) −530400. −0.132862
\(438\) −1.08770e6 −0.270910
\(439\) 6.94278e6 1.71938 0.859690 0.510816i \(-0.170657\pi\)
0.859690 + 0.510816i \(0.170657\pi\)
\(440\) 604032. 0.148740
\(441\) 194481. 0.0476190
\(442\) −1.28914e6 −0.313865
\(443\) 887508. 0.214864 0.107432 0.994212i \(-0.465737\pi\)
0.107432 + 0.994212i \(0.465737\pi\)
\(444\) −1.14595e6 −0.275873
\(445\) 1.16485e6 0.278850
\(446\) 1.59616e6 0.379961
\(447\) 3.48543e6 0.825064
\(448\) 200704. 0.0472456
\(449\) −6.17391e6 −1.44525 −0.722627 0.691238i \(-0.757066\pi\)
−0.722627 + 0.691238i \(0.757066\pi\)
\(450\) −958716. −0.223182
\(451\) 294030. 0.0680691
\(452\) 2.33309e6 0.537137
\(453\) 4.17413e6 0.955697
\(454\) −344400. −0.0784194
\(455\) 1.91864e6 0.434476
\(456\) −299520. −0.0674549
\(457\) −6.54564e6 −1.46609 −0.733046 0.680179i \(-0.761902\pi\)
−0.733046 + 0.680179i \(0.761902\pi\)
\(458\) −1.48450e6 −0.330687
\(459\) 468018. 0.103689
\(460\) −1.27296e6 −0.280492
\(461\) 3.59248e6 0.787304 0.393652 0.919260i \(-0.371212\pi\)
0.393652 + 0.919260i \(0.371212\pi\)
\(462\) 213444. 0.0465242
\(463\) −1.51974e6 −0.329472 −0.164736 0.986338i \(-0.552677\pi\)
−0.164736 + 0.986338i \(0.552677\pi\)
\(464\) 1.23341e6 0.265957
\(465\) 1.25237e6 0.268596
\(466\) 231144. 0.0493081
\(467\) 1.33352e6 0.282949 0.141475 0.989942i \(-0.454816\pi\)
0.141475 + 0.989942i \(0.454816\pi\)
\(468\) −650592. −0.137307
\(469\) 951188. 0.199680
\(470\) −3.53059e6 −0.737230
\(471\) 3.34010e6 0.693756
\(472\) 1.01453e6 0.209609
\(473\) 2.77138e6 0.569566
\(474\) 1.28419e6 0.262533
\(475\) −1.53868e6 −0.312906
\(476\) −503328. −0.101820
\(477\) −989982. −0.199219
\(478\) −2.90246e6 −0.581028
\(479\) 5.15705e6 1.02698 0.513490 0.858095i \(-0.328352\pi\)
0.513490 + 0.858095i \(0.328352\pi\)
\(480\) −718848. −0.142408
\(481\) −3.99492e6 −0.787309
\(482\) −2.50148e6 −0.490433
\(483\) −449820. −0.0877346
\(484\) 234256. 0.0454545
\(485\) −6.63827e6 −1.28145
\(486\) 236196. 0.0453609
\(487\) −3.42899e6 −0.655155 −0.327578 0.944824i \(-0.606232\pi\)
−0.327578 + 0.944824i \(0.606232\pi\)
\(488\) −2.96307e6 −0.563239
\(489\) 685404. 0.129621
\(490\) 749112. 0.140947
\(491\) 9.44840e6 1.76870 0.884351 0.466822i \(-0.154601\pi\)
0.884351 + 0.466822i \(0.154601\pi\)
\(492\) −349920. −0.0651712
\(493\) −3.09316e6 −0.573171
\(494\) −1.04416e6 −0.192508
\(495\) −764478. −0.140234
\(496\) 456704. 0.0833548
\(497\) −847308. −0.153869
\(498\) −1.56341e6 −0.282487
\(499\) −4.07717e6 −0.733006 −0.366503 0.930417i \(-0.619445\pi\)
−0.366503 + 0.930417i \(0.619445\pi\)
\(500\) 207168. 0.0370593
\(501\) −1.54980e6 −0.275855
\(502\) −1.06834e6 −0.189212
\(503\) −4.32338e6 −0.761910 −0.380955 0.924594i \(-0.624405\pi\)
−0.380955 + 0.924594i \(0.624405\pi\)
\(504\) −254016. −0.0445435
\(505\) 5.62770e6 0.981980
\(506\) −493680. −0.0857174
\(507\) 1.07360e6 0.185491
\(508\) −5.46803e6 −0.940095
\(509\) −6.35043e6 −1.08645 −0.543224 0.839588i \(-0.682796\pi\)
−0.543224 + 0.839588i \(0.682796\pi\)
\(510\) 1.80274e6 0.306907
\(511\) −1.48049e6 −0.250814
\(512\) −262144. −0.0441942
\(513\) 379080. 0.0635971
\(514\) 254904. 0.0425568
\(515\) −916032. −0.152192
\(516\) −3.29818e6 −0.545318
\(517\) −1.36924e6 −0.225295
\(518\) −1.55977e6 −0.255409
\(519\) 5.41021e6 0.881648
\(520\) −2.50598e6 −0.406415
\(521\) 3.12777e6 0.504825 0.252412 0.967620i \(-0.418776\pi\)
0.252412 + 0.967620i \(0.418776\pi\)
\(522\) −1.56103e6 −0.250747
\(523\) −6.76898e6 −1.08210 −0.541052 0.840989i \(-0.681974\pi\)
−0.541052 + 0.840989i \(0.681974\pi\)
\(524\) −1.76736e6 −0.281188
\(525\) −1.30492e6 −0.206626
\(526\) 615552. 0.0970064
\(527\) −1.14533e6 −0.179640
\(528\) −278784. −0.0435194
\(529\) −5.39594e6 −0.838355
\(530\) −3.81326e6 −0.589668
\(531\) −1.28401e6 −0.197621
\(532\) −407680. −0.0624511
\(533\) −1.21986e6 −0.185991
\(534\) −537624. −0.0815879
\(535\) 1.69987e7 2.56762
\(536\) −1.24237e6 −0.186783
\(537\) −2.01215e6 −0.301109
\(538\) −7.19988e6 −1.07243
\(539\) 290521. 0.0430730
\(540\) 909792. 0.134263
\(541\) −7.42759e6 −1.09108 −0.545538 0.838086i \(-0.683675\pi\)
−0.545538 + 0.838086i \(0.683675\pi\)
\(542\) 4.20515e6 0.614870
\(543\) −1.63460e6 −0.237909
\(544\) 657408. 0.0952440
\(545\) −6.83857e6 −0.986221
\(546\) −885528. −0.127122
\(547\) −1.14050e7 −1.62977 −0.814885 0.579623i \(-0.803200\pi\)
−0.814885 + 0.579623i \(0.803200\pi\)
\(548\) −1.97616e6 −0.281106
\(549\) 3.75014e6 0.531027
\(550\) −1.43216e6 −0.201875
\(551\) −2.50536e6 −0.351553
\(552\) 587520. 0.0820682
\(553\) 1.74793e6 0.243058
\(554\) −8.57252e6 −1.18668
\(555\) 5.58652e6 0.769854
\(556\) 2.15552e6 0.295709
\(557\) −7.50733e6 −1.02529 −0.512646 0.858600i \(-0.671335\pi\)
−0.512646 + 0.858600i \(0.671335\pi\)
\(558\) −578016. −0.0785877
\(559\) −1.14978e7 −1.55627
\(560\) −978432. −0.131844
\(561\) 699138. 0.0937898
\(562\) −3.57564e6 −0.477544
\(563\) 3.67782e6 0.489012 0.244506 0.969648i \(-0.421374\pi\)
0.244506 + 0.969648i \(0.421374\pi\)
\(564\) 1.62950e6 0.215704
\(565\) −1.13738e7 −1.49894
\(566\) −2.66086e6 −0.349126
\(567\) 321489. 0.0419961
\(568\) 1.10669e6 0.143931
\(569\) 330558. 0.0428023 0.0214011 0.999771i \(-0.493187\pi\)
0.0214011 + 0.999771i \(0.493187\pi\)
\(570\) 1.46016e6 0.188241
\(571\) −3.91658e6 −0.502709 −0.251354 0.967895i \(-0.580876\pi\)
−0.251354 + 0.967895i \(0.580876\pi\)
\(572\) −971872. −0.124199
\(573\) −1.05527e6 −0.134269
\(574\) −476280. −0.0603368
\(575\) 3.01818e6 0.380694
\(576\) 331776. 0.0416667
\(577\) −1.64371e6 −0.205535 −0.102767 0.994705i \(-0.532770\pi\)
−0.102767 + 0.994705i \(0.532770\pi\)
\(578\) 4.03077e6 0.501844
\(579\) 3.69682e6 0.458281
\(580\) −6.01286e6 −0.742184
\(581\) −2.12797e6 −0.261532
\(582\) 3.06382e6 0.374934
\(583\) −1.47886e6 −0.180201
\(584\) 1.93370e6 0.234615
\(585\) 3.17164e6 0.383172
\(586\) 8.80034e6 1.05866
\(587\) 1.43860e7 1.72323 0.861616 0.507560i \(-0.169452\pi\)
0.861616 + 0.507560i \(0.169452\pi\)
\(588\) −345744. −0.0412393
\(589\) −927680. −0.110182
\(590\) −4.94582e6 −0.584936
\(591\) −1.71455e6 −0.201921
\(592\) 2.03725e6 0.238913
\(593\) −1.49626e7 −1.74731 −0.873653 0.486549i \(-0.838255\pi\)
−0.873653 + 0.486549i \(0.838255\pi\)
\(594\) 352836. 0.0410305
\(595\) 2.45372e6 0.284141
\(596\) −6.19632e6 −0.714526
\(597\) 8.04204e6 0.923486
\(598\) 2.04816e6 0.234213
\(599\) 904956. 0.103053 0.0515265 0.998672i \(-0.483591\pi\)
0.0515265 + 0.998672i \(0.483591\pi\)
\(600\) 1.70438e6 0.193281
\(601\) 6.11184e6 0.690217 0.345109 0.938563i \(-0.387842\pi\)
0.345109 + 0.938563i \(0.387842\pi\)
\(602\) −4.48918e6 −0.504866
\(603\) 1.57237e6 0.176101
\(604\) −7.42067e6 −0.827658
\(605\) −1.14200e6 −0.126846
\(606\) −2.59740e6 −0.287314
\(607\) −1.03513e7 −1.14031 −0.570155 0.821537i \(-0.693117\pi\)
−0.570155 + 0.821537i \(0.693117\pi\)
\(608\) 532480. 0.0584177
\(609\) −2.12474e6 −0.232147
\(610\) 1.44450e7 1.57178
\(611\) 5.68063e6 0.615593
\(612\) −832032. −0.0897969
\(613\) 4.54354e6 0.488363 0.244182 0.969730i \(-0.421481\pi\)
0.244182 + 0.969730i \(0.421481\pi\)
\(614\) 1.96509e6 0.210359
\(615\) 1.70586e6 0.181868
\(616\) −379456. −0.0402911
\(617\) −8.34929e6 −0.882951 −0.441476 0.897273i \(-0.645545\pi\)
−0.441476 + 0.897273i \(0.645545\pi\)
\(618\) 422784. 0.0445295
\(619\) 5.15954e6 0.541233 0.270617 0.962687i \(-0.412772\pi\)
0.270617 + 0.962687i \(0.412772\pi\)
\(620\) −2.22643e6 −0.232611
\(621\) −743580. −0.0773747
\(622\) 420720. 0.0436031
\(623\) −731766. −0.0755357
\(624\) 1.15661e6 0.118912
\(625\) −1.02568e7 −1.05030
\(626\) 7.56700e6 0.771770
\(627\) 566280. 0.0575258
\(628\) −5.93795e6 −0.600811
\(629\) −5.10904e6 −0.514888
\(630\) 1.23833e6 0.124304
\(631\) 3.33812e6 0.333756 0.166878 0.985978i \(-0.446631\pi\)
0.166878 + 0.985978i \(0.446631\pi\)
\(632\) −2.28301e6 −0.227360
\(633\) −2.64953e6 −0.262821
\(634\) 2.68879e6 0.265665
\(635\) 2.66567e7 2.62344
\(636\) 1.75997e6 0.172529
\(637\) −1.20530e6 −0.117692
\(638\) −2.33191e6 −0.226809
\(639\) −1.40065e6 −0.135699
\(640\) 1.27795e6 0.123329
\(641\) −1.51004e7 −1.45159 −0.725795 0.687911i \(-0.758528\pi\)
−0.725795 + 0.687911i \(0.758528\pi\)
\(642\) −7.84555e6 −0.751253
\(643\) −1.71050e7 −1.63153 −0.815766 0.578382i \(-0.803684\pi\)
−0.815766 + 0.578382i \(0.803684\pi\)
\(644\) 799680. 0.0759804
\(645\) 1.60786e7 1.52177
\(646\) −1.33536e6 −0.125897
\(647\) −8.36243e6 −0.785365 −0.392683 0.919674i \(-0.628453\pi\)
−0.392683 + 0.919674i \(0.628453\pi\)
\(648\) −419904. −0.0392837
\(649\) −1.91809e6 −0.178755
\(650\) 5.94167e6 0.551601
\(651\) −786744. −0.0727580
\(652\) −1.21850e6 −0.112255
\(653\) 1.29485e7 1.18832 0.594162 0.804345i \(-0.297484\pi\)
0.594162 + 0.804345i \(0.297484\pi\)
\(654\) 3.15626e6 0.288555
\(655\) 8.61588e6 0.784687
\(656\) 622080. 0.0564400
\(657\) −2.44733e6 −0.221197
\(658\) 2.21794e6 0.199703
\(659\) −5.39982e6 −0.484357 −0.242179 0.970232i \(-0.577862\pi\)
−0.242179 + 0.970232i \(0.577862\pi\)
\(660\) 1.35907e6 0.121446
\(661\) −3.63908e6 −0.323958 −0.161979 0.986794i \(-0.551788\pi\)
−0.161979 + 0.986794i \(0.551788\pi\)
\(662\) 4.29160e6 0.380605
\(663\) −2.90056e6 −0.256270
\(664\) 2.77939e6 0.244641
\(665\) 1.98744e6 0.174277
\(666\) −2.57839e6 −0.225249
\(667\) 4.91436e6 0.427713
\(668\) 2.75520e6 0.238898
\(669\) 3.59136e6 0.310237
\(670\) 6.05654e6 0.521240
\(671\) 5.60206e6 0.480332
\(672\) 451584. 0.0385758
\(673\) 5.82483e6 0.495730 0.247865 0.968795i \(-0.420271\pi\)
0.247865 + 0.968795i \(0.420271\pi\)
\(674\) −4.14327e6 −0.351313
\(675\) −2.15711e6 −0.182227
\(676\) −1.90862e6 −0.160640
\(677\) −1.36657e6 −0.114593 −0.0572966 0.998357i \(-0.518248\pi\)
−0.0572966 + 0.998357i \(0.518248\pi\)
\(678\) 5.24945e6 0.438570
\(679\) 4.17019e6 0.347122
\(680\) −3.20486e6 −0.265789
\(681\) −774900. −0.0640292
\(682\) −863456. −0.0710852
\(683\) 5.67865e6 0.465794 0.232897 0.972501i \(-0.425180\pi\)
0.232897 + 0.972501i \(0.425180\pi\)
\(684\) −673920. −0.0550767
\(685\) 9.63378e6 0.784459
\(686\) −470596. −0.0381802
\(687\) −3.34013e6 −0.270005
\(688\) 5.86342e6 0.472259
\(689\) 6.13544e6 0.492377
\(690\) −2.86416e6 −0.229021
\(691\) 2.89483e6 0.230636 0.115318 0.993329i \(-0.463211\pi\)
0.115318 + 0.993329i \(0.463211\pi\)
\(692\) −9.61814e6 −0.763530
\(693\) 480249. 0.0379869
\(694\) 6.11285e6 0.481776
\(695\) −1.05082e7 −0.825210
\(696\) 2.77517e6 0.217153
\(697\) −1.56006e6 −0.121635
\(698\) 5.80818e6 0.451234
\(699\) 520074. 0.0402599
\(700\) 2.31986e6 0.178944
\(701\) 1.75699e7 1.35044 0.675218 0.737618i \(-0.264050\pi\)
0.675218 + 0.737618i \(0.264050\pi\)
\(702\) −1.46383e6 −0.112111
\(703\) −4.13816e6 −0.315805
\(704\) 495616. 0.0376889
\(705\) −7.94383e6 −0.601946
\(706\) 739704. 0.0558530
\(707\) −3.53535e6 −0.266001
\(708\) 2.28269e6 0.171145
\(709\) −1.92613e7 −1.43903 −0.719515 0.694477i \(-0.755636\pi\)
−0.719515 + 0.694477i \(0.755636\pi\)
\(710\) −5.39510e6 −0.401656
\(711\) 2.88943e6 0.214357
\(712\) 955776. 0.0706572
\(713\) 1.81968e6 0.134051
\(714\) −1.13249e6 −0.0831358
\(715\) 4.73788e6 0.346592
\(716\) 3.57715e6 0.260768
\(717\) −6.53054e6 −0.474407
\(718\) −4.38643e6 −0.317541
\(719\) −2.88337e6 −0.208007 −0.104004 0.994577i \(-0.533165\pi\)
−0.104004 + 0.994577i \(0.533165\pi\)
\(720\) −1.61741e6 −0.116276
\(721\) 575456. 0.0412263
\(722\) 8.82280e6 0.629888
\(723\) −5.62833e6 −0.400437
\(724\) 2.90595e6 0.206035
\(725\) 1.42565e7 1.00732
\(726\) 527076. 0.0371135
\(727\) 5.01217e6 0.351714 0.175857 0.984416i \(-0.443730\pi\)
0.175857 + 0.984416i \(0.443730\pi\)
\(728\) 1.57427e6 0.110091
\(729\) 531441. 0.0370370
\(730\) −9.42677e6 −0.654720
\(731\) −1.47044e7 −1.01778
\(732\) −6.66691e6 −0.459883
\(733\) 416426. 0.0286271 0.0143136 0.999898i \(-0.495444\pi\)
0.0143136 + 0.999898i \(0.495444\pi\)
\(734\) −1.49435e7 −1.02379
\(735\) 1.68550e6 0.115083
\(736\) −1.04448e6 −0.0710732
\(737\) 2.34885e6 0.159289
\(738\) −787320. −0.0532121
\(739\) 2.26842e7 1.52796 0.763979 0.645241i \(-0.223243\pi\)
0.763979 + 0.645241i \(0.223243\pi\)
\(740\) −9.93158e6 −0.666713
\(741\) −2.34936e6 −0.157182
\(742\) 2.39551e6 0.159731
\(743\) 2.74750e7 1.82585 0.912925 0.408128i \(-0.133818\pi\)
0.912925 + 0.408128i \(0.133818\pi\)
\(744\) 1.02758e6 0.0680589
\(745\) 3.02071e7 1.99397
\(746\) −1.64290e7 −1.08084
\(747\) −3.51767e6 −0.230650
\(748\) −1.24291e6 −0.0812244
\(749\) −1.06787e7 −0.695525
\(750\) 466128. 0.0302588
\(751\) −2.96805e6 −0.192031 −0.0960154 0.995380i \(-0.530610\pi\)
−0.0960154 + 0.995380i \(0.530610\pi\)
\(752\) −2.89690e6 −0.186805
\(753\) −2.40376e6 −0.154491
\(754\) 9.67454e6 0.619729
\(755\) 3.61758e7 2.30967
\(756\) −571536. −0.0363696
\(757\) 1.75593e7 1.11370 0.556849 0.830614i \(-0.312010\pi\)
0.556849 + 0.830614i \(0.312010\pi\)
\(758\) −1.27455e7 −0.805721
\(759\) −1.11078e6 −0.0699880
\(760\) −2.59584e6 −0.163021
\(761\) −2.23450e7 −1.39868 −0.699341 0.714789i \(-0.746523\pi\)
−0.699341 + 0.714789i \(0.746523\pi\)
\(762\) −1.23031e7 −0.767584
\(763\) 4.29603e6 0.267150
\(764\) 1.87603e6 0.116280
\(765\) 4.05616e6 0.250588
\(766\) 1.68153e7 1.03546
\(767\) 7.95770e6 0.488427
\(768\) −589824. −0.0360844
\(769\) −1.11300e7 −0.678705 −0.339353 0.940659i \(-0.610208\pi\)
−0.339353 + 0.940659i \(0.610208\pi\)
\(770\) 1.84985e6 0.112437
\(771\) 573534. 0.0347475
\(772\) −6.57213e6 −0.396883
\(773\) 1.37300e7 0.826461 0.413230 0.910627i \(-0.364400\pi\)
0.413230 + 0.910627i \(0.364400\pi\)
\(774\) −7.42090e6 −0.445250
\(775\) 5.27886e6 0.315708
\(776\) −5.44678e6 −0.324703
\(777\) −3.50948e6 −0.208540
\(778\) 1.33016e7 0.787870
\(779\) −1.26360e6 −0.0746047
\(780\) −5.63846e6 −0.331837
\(781\) −2.09233e6 −0.122745
\(782\) 2.61936e6 0.153172
\(783\) −3.51232e6 −0.204734
\(784\) 614656. 0.0357143
\(785\) 2.89475e7 1.67663
\(786\) −3.97656e6 −0.229589
\(787\) −1.86313e7 −1.07227 −0.536137 0.844131i \(-0.680117\pi\)
−0.536137 + 0.844131i \(0.680117\pi\)
\(788\) 3.04810e6 0.174869
\(789\) 1.38499e6 0.0792054
\(790\) 1.11297e7 0.634475
\(791\) 7.14508e6 0.406037
\(792\) −627264. −0.0355335
\(793\) −2.32416e7 −1.31245
\(794\) −2.13787e7 −1.20345
\(795\) −8.57984e6 −0.481462
\(796\) −1.42970e7 −0.799762
\(797\) 5.06039e6 0.282188 0.141094 0.989996i \(-0.454938\pi\)
0.141094 + 0.989996i \(0.454938\pi\)
\(798\) −917280. −0.0509911
\(799\) 7.26487e6 0.402588
\(800\) −3.03002e6 −0.167386
\(801\) −1.20965e6 −0.0666162
\(802\) 69048.0 0.00379066
\(803\) −3.65589e6 −0.200080
\(804\) −2.79533e6 −0.152508
\(805\) −3.89844e6 −0.212032
\(806\) 3.58227e6 0.194232
\(807\) −1.61997e7 −0.875637
\(808\) 4.61760e6 0.248822
\(809\) −1.19684e7 −0.642933 −0.321467 0.946921i \(-0.604176\pi\)
−0.321467 + 0.946921i \(0.604176\pi\)
\(810\) 2.04703e6 0.109626
\(811\) −2.69460e7 −1.43861 −0.719305 0.694695i \(-0.755539\pi\)
−0.719305 + 0.694695i \(0.755539\pi\)
\(812\) 3.77731e6 0.201045
\(813\) 9.46159e6 0.502039
\(814\) −3.85167e6 −0.203746
\(815\) 5.94017e6 0.313260
\(816\) 1.47917e6 0.0777664
\(817\) −1.19101e7 −0.624251
\(818\) 2.36754e6 0.123713
\(819\) −1.99244e6 −0.103795
\(820\) −3.03264e6 −0.157502
\(821\) −9.46724e6 −0.490191 −0.245096 0.969499i \(-0.578819\pi\)
−0.245096 + 0.969499i \(0.578819\pi\)
\(822\) −4.44636e6 −0.229522
\(823\) −2.20562e7 −1.13509 −0.567546 0.823342i \(-0.692107\pi\)
−0.567546 + 0.823342i \(0.692107\pi\)
\(824\) −751616. −0.0385636
\(825\) −3.22235e6 −0.164831
\(826\) 3.10699e6 0.158449
\(827\) 1.22724e7 0.623971 0.311986 0.950087i \(-0.399006\pi\)
0.311986 + 0.950087i \(0.399006\pi\)
\(828\) 1.32192e6 0.0670084
\(829\) 6.34453e6 0.320637 0.160318 0.987065i \(-0.448748\pi\)
0.160318 + 0.987065i \(0.448748\pi\)
\(830\) −1.35495e7 −0.682699
\(831\) −1.92882e7 −0.968921
\(832\) −2.05619e6 −0.102981
\(833\) −1.54144e6 −0.0769688
\(834\) 4.84992e6 0.241446
\(835\) −1.34316e7 −0.666671
\(836\) −1.00672e6 −0.0498188
\(837\) −1.30054e6 −0.0641666
\(838\) −6.57211e6 −0.323292
\(839\) 1.82048e7 0.892857 0.446428 0.894819i \(-0.352696\pi\)
0.446428 + 0.894819i \(0.352696\pi\)
\(840\) −2.20147e6 −0.107650
\(841\) 2.70198e6 0.131732
\(842\) 1.05030e7 0.510545
\(843\) −8.04519e6 −0.389913
\(844\) 4.71027e6 0.227609
\(845\) 9.30454e6 0.448284
\(846\) 3.66638e6 0.176121
\(847\) 717409. 0.0343604
\(848\) −3.12883e6 −0.149414
\(849\) −5.98694e6 −0.285060
\(850\) 7.59871e6 0.360739
\(851\) 8.11716e6 0.384220
\(852\) 2.49005e6 0.117519
\(853\) −595294. −0.0280130 −0.0140065 0.999902i \(-0.504459\pi\)
−0.0140065 + 0.999902i \(0.504459\pi\)
\(854\) −9.07441e6 −0.425769
\(855\) 3.28536e6 0.153698
\(856\) 1.39476e7 0.650604
\(857\) −1.98720e7 −0.924250 −0.462125 0.886815i \(-0.652913\pi\)
−0.462125 + 0.886815i \(0.652913\pi\)
\(858\) −2.18671e6 −0.101408
\(859\) −4.44092e6 −0.205348 −0.102674 0.994715i \(-0.532740\pi\)
−0.102674 + 0.994715i \(0.532740\pi\)
\(860\) −2.85842e7 −1.31789
\(861\) −1.07163e6 −0.0492648
\(862\) 7.19472e6 0.329796
\(863\) −3.04580e7 −1.39211 −0.696055 0.717988i \(-0.745063\pi\)
−0.696055 + 0.717988i \(0.745063\pi\)
\(864\) 746496. 0.0340207
\(865\) 4.68885e7 2.13072
\(866\) 1.13670e7 0.515051
\(867\) 9.06924e6 0.409754
\(868\) 1.39866e6 0.0630103
\(869\) 4.31631e6 0.193894
\(870\) −1.35289e7 −0.605990
\(871\) −9.74482e6 −0.435240
\(872\) −5.61114e6 −0.249896
\(873\) 6.89359e6 0.306133
\(874\) 2.12160e6 0.0939474
\(875\) 634452. 0.0280142
\(876\) 4.35082e6 0.191562
\(877\) 4.11013e7 1.80450 0.902249 0.431215i \(-0.141915\pi\)
0.902249 + 0.431215i \(0.141915\pi\)
\(878\) −2.77711e7 −1.21579
\(879\) 1.98008e7 0.864390
\(880\) −2.41613e6 −0.105175
\(881\) 2.45200e7 1.06434 0.532171 0.846637i \(-0.321376\pi\)
0.532171 + 0.846637i \(0.321376\pi\)
\(882\) −777924. −0.0336718
\(883\) −3.08283e6 −0.133060 −0.0665300 0.997784i \(-0.521193\pi\)
−0.0665300 + 0.997784i \(0.521193\pi\)
\(884\) 5.15654e6 0.221936
\(885\) −1.11281e7 −0.477599
\(886\) −3.55003e6 −0.151932
\(887\) 2.37525e7 1.01368 0.506839 0.862041i \(-0.330814\pi\)
0.506839 + 0.862041i \(0.330814\pi\)
\(888\) 4.58381e6 0.195072
\(889\) −1.67458e7 −0.710645
\(890\) −4.65941e6 −0.197177
\(891\) 793881. 0.0335013
\(892\) −6.38464e6 −0.268673
\(893\) 5.88432e6 0.246926
\(894\) −1.39417e7 −0.583408
\(895\) −1.74386e7 −0.727703
\(896\) −802816. −0.0334077
\(897\) 4.60836e6 0.191234
\(898\) 2.46956e7 1.02195
\(899\) 8.59531e6 0.354701
\(900\) 3.83486e6 0.157813
\(901\) 7.84652e6 0.322007
\(902\) −1.17612e6 −0.0481322
\(903\) −1.01007e7 −0.412221
\(904\) −9.33235e6 −0.379813
\(905\) −1.41665e7 −0.574965
\(906\) −1.66965e7 −0.675780
\(907\) 2.50973e7 1.01300 0.506500 0.862240i \(-0.330939\pi\)
0.506500 + 0.862240i \(0.330939\pi\)
\(908\) 1.37760e6 0.0554509
\(909\) −5.84415e6 −0.234591
\(910\) −7.67458e6 −0.307221
\(911\) 2.39629e7 0.956627 0.478314 0.878189i \(-0.341248\pi\)
0.478314 + 0.878189i \(0.341248\pi\)
\(912\) 1.19808e6 0.0476978
\(913\) −5.25479e6 −0.208631
\(914\) 2.61826e7 1.03668
\(915\) 3.25012e7 1.28335
\(916\) 5.93802e6 0.233831
\(917\) −5.41254e6 −0.212558
\(918\) −1.87207e6 −0.0733189
\(919\) 9.36056e6 0.365606 0.182803 0.983150i \(-0.441483\pi\)
0.182803 + 0.983150i \(0.441483\pi\)
\(920\) 5.09184e6 0.198338
\(921\) 4.42145e6 0.171757
\(922\) −1.43699e7 −0.556708
\(923\) 8.68058e6 0.335386
\(924\) −853776. −0.0328976
\(925\) 2.35477e7 0.904887
\(926\) 6.07898e6 0.232972
\(927\) 951264. 0.0363581
\(928\) −4.93363e6 −0.188060
\(929\) 2.56159e6 0.0973800 0.0486900 0.998814i \(-0.484495\pi\)
0.0486900 + 0.998814i \(0.484495\pi\)
\(930\) −5.00947e6 −0.189926
\(931\) −1.24852e6 −0.0472086
\(932\) −924576. −0.0348661
\(933\) 946620. 0.0356018
\(934\) −5.33410e6 −0.200075
\(935\) 6.05920e6 0.226666
\(936\) 2.60237e6 0.0970910
\(937\) 1.22738e7 0.456701 0.228350 0.973579i \(-0.426667\pi\)
0.228350 + 0.973579i \(0.426667\pi\)
\(938\) −3.80475e6 −0.141195
\(939\) 1.70258e7 0.630148
\(940\) 1.41224e7 0.521300
\(941\) 3.76486e7 1.38604 0.693019 0.720919i \(-0.256280\pi\)
0.693019 + 0.720919i \(0.256280\pi\)
\(942\) −1.33604e7 −0.490560
\(943\) 2.47860e6 0.0907668
\(944\) −4.05811e6 −0.148216
\(945\) 2.78624e6 0.101494
\(946\) −1.10855e7 −0.402744
\(947\) −2.06302e7 −0.747530 −0.373765 0.927524i \(-0.621933\pi\)
−0.373765 + 0.927524i \(0.621933\pi\)
\(948\) −5.13677e6 −0.185639
\(949\) 1.51674e7 0.546697
\(950\) 6.15472e6 0.221258
\(951\) 6.04978e6 0.216914
\(952\) 2.01331e6 0.0719977
\(953\) −2.79681e7 −0.997540 −0.498770 0.866734i \(-0.666215\pi\)
−0.498770 + 0.866734i \(0.666215\pi\)
\(954\) 3.95993e6 0.140869
\(955\) −9.14566e6 −0.324494
\(956\) 1.16099e7 0.410849
\(957\) −5.24680e6 −0.185189
\(958\) −2.06282e7 −0.726185
\(959\) −6.05199e6 −0.212496
\(960\) 2.87539e6 0.100698
\(961\) −2.54465e7 −0.888832
\(962\) 1.59797e7 0.556711
\(963\) −1.76525e7 −0.613395
\(964\) 1.00059e7 0.346788
\(965\) 3.20391e7 1.10755
\(966\) 1.79928e6 0.0620377
\(967\) 2.74786e7 0.944994 0.472497 0.881332i \(-0.343353\pi\)
0.472497 + 0.881332i \(0.343353\pi\)
\(968\) −937024. −0.0321412
\(969\) −3.00456e6 −0.102795
\(970\) 2.65531e7 0.906119
\(971\) −3.26904e7 −1.11269 −0.556343 0.830953i \(-0.687796\pi\)
−0.556343 + 0.830953i \(0.687796\pi\)
\(972\) −944784. −0.0320750
\(973\) 6.60128e6 0.223535
\(974\) 1.37160e7 0.463265
\(975\) 1.33688e7 0.450381
\(976\) 1.18523e7 0.398270
\(977\) −4.94406e7 −1.65709 −0.828547 0.559920i \(-0.810832\pi\)
−0.828547 + 0.559920i \(0.810832\pi\)
\(978\) −2.74162e6 −0.0916557
\(979\) −1.80701e6 −0.0602566
\(980\) −2.99645e6 −0.0996647
\(981\) 7.10159e6 0.235604
\(982\) −3.77936e7 −1.25066
\(983\) 7.23892e6 0.238940 0.119470 0.992838i \(-0.461880\pi\)
0.119470 + 0.992838i \(0.461880\pi\)
\(984\) 1.39968e6 0.0460830
\(985\) −1.48595e7 −0.487992
\(986\) 1.23726e7 0.405293
\(987\) 4.99036e6 0.163057
\(988\) 4.17664e6 0.136124
\(989\) 2.33621e7 0.759488
\(990\) 3.05791e6 0.0991601
\(991\) 3.59078e7 1.16146 0.580730 0.814096i \(-0.302767\pi\)
0.580730 + 0.814096i \(0.302767\pi\)
\(992\) −1.82682e6 −0.0589407
\(993\) 9.65610e6 0.310763
\(994\) 3.38923e6 0.108802
\(995\) 6.96977e7 2.23183
\(996\) 6.25363e6 0.199749
\(997\) 351914. 0.0112124 0.00560620 0.999984i \(-0.498215\pi\)
0.00560620 + 0.999984i \(0.498215\pi\)
\(998\) 1.63087e7 0.518314
\(999\) −5.80138e6 −0.183915
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 462.6.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.6.a.a.1.1 1 1.1 even 1 trivial