Properties

Label 585.2.a.n.1.1
Level $585$
Weight $2$
Character 585.1
Self dual yes
Analytic conductor $4.671$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 195)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.470683\) of defining polynomial
Character \(\chi\) \(=\) 585.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.77846 q^{2} +5.71982 q^{4} +1.00000 q^{5} -2.71982 q^{7} -10.3354 q^{8} +O(q^{10})\) \(q-2.77846 q^{2} +5.71982 q^{4} +1.00000 q^{5} -2.71982 q^{7} -10.3354 q^{8} -2.77846 q^{10} +2.71982 q^{11} +1.00000 q^{13} +7.55691 q^{14} +17.2767 q^{16} +2.83709 q^{17} -3.55691 q^{19} +5.71982 q^{20} -7.55691 q^{22} +4.83709 q^{23} +1.00000 q^{25} -2.77846 q^{26} -15.5569 q^{28} -6.00000 q^{29} +7.55691 q^{31} -27.3319 q^{32} -7.88273 q^{34} -2.71982 q^{35} -4.27674 q^{37} +9.88273 q^{38} -10.3354 q^{40} -2.83709 q^{41} +11.1138 q^{43} +15.5569 q^{44} -13.4396 q^{46} +11.5569 q^{47} +0.397442 q^{49} -2.77846 q^{50} +5.71982 q^{52} -1.16291 q^{53} +2.71982 q^{55} +28.1104 q^{56} +16.6707 q^{58} +2.11727 q^{59} +6.60256 q^{61} -20.9966 q^{62} +41.3871 q^{64} +1.00000 q^{65} +1.88273 q^{67} +16.2277 q^{68} +7.55691 q^{70} +6.71982 q^{71} +9.11383 q^{73} +11.8827 q^{74} -20.3449 q^{76} -7.39744 q^{77} +10.2767 q^{79} +17.2767 q^{80} +7.88273 q^{82} -2.11727 q^{83} +2.83709 q^{85} -30.8793 q^{86} -28.1104 q^{88} -1.16291 q^{89} -2.71982 q^{91} +27.6673 q^{92} -32.1104 q^{94} -3.55691 q^{95} -10.8371 q^{97} -1.10428 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 8 q^{4} + 3 q^{5} + q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 8 q^{4} + 3 q^{5} + q^{7} - 6 q^{8} - q^{11} + 3 q^{13} + 6 q^{14} + 26 q^{16} + q^{17} + 6 q^{19} + 8 q^{20} - 6 q^{22} + 7 q^{23} + 3 q^{25} - 30 q^{28} - 18 q^{29} + 6 q^{31} - 22 q^{32} - 22 q^{34} + q^{35} + 13 q^{37} + 28 q^{38} - 6 q^{40} - q^{41} + 30 q^{44} - 22 q^{46} + 18 q^{47} + 12 q^{49} + 8 q^{52} - 11 q^{53} - q^{55} + 16 q^{56} + 8 q^{59} + 9 q^{61} - 28 q^{62} + 30 q^{64} + 3 q^{65} + 4 q^{67} - 18 q^{68} + 6 q^{70} + 11 q^{71} - 6 q^{73} + 34 q^{74} + 4 q^{76} - 33 q^{77} + 5 q^{79} + 26 q^{80} + 22 q^{82} - 8 q^{83} + q^{85} - 56 q^{86} - 16 q^{88} - 11 q^{89} + q^{91} - 2 q^{92} - 28 q^{94} + 6 q^{95} - 25 q^{97} - 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.77846 −1.96467 −0.982333 0.187142i \(-0.940078\pi\)
−0.982333 + 0.187142i \(0.940078\pi\)
\(3\) 0 0
\(4\) 5.71982 2.85991
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.71982 −1.02800 −0.513998 0.857791i \(-0.671836\pi\)
−0.513998 + 0.857791i \(0.671836\pi\)
\(8\) −10.3354 −3.65411
\(9\) 0 0
\(10\) −2.77846 −0.878625
\(11\) 2.71982 0.820058 0.410029 0.912073i \(-0.365519\pi\)
0.410029 + 0.912073i \(0.365519\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 7.55691 2.01967
\(15\) 0 0
\(16\) 17.2767 4.31918
\(17\) 2.83709 0.688095 0.344048 0.938952i \(-0.388202\pi\)
0.344048 + 0.938952i \(0.388202\pi\)
\(18\) 0 0
\(19\) −3.55691 −0.816012 −0.408006 0.912979i \(-0.633776\pi\)
−0.408006 + 0.912979i \(0.633776\pi\)
\(20\) 5.71982 1.27899
\(21\) 0 0
\(22\) −7.55691 −1.61114
\(23\) 4.83709 1.00860 0.504302 0.863528i \(-0.331750\pi\)
0.504302 + 0.863528i \(0.331750\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.77846 −0.544900
\(27\) 0 0
\(28\) −15.5569 −2.93998
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 7.55691 1.35726 0.678631 0.734479i \(-0.262574\pi\)
0.678631 + 0.734479i \(0.262574\pi\)
\(32\) −27.3319 −4.83165
\(33\) 0 0
\(34\) −7.88273 −1.35188
\(35\) −2.71982 −0.459734
\(36\) 0 0
\(37\) −4.27674 −0.703091 −0.351546 0.936171i \(-0.614344\pi\)
−0.351546 + 0.936171i \(0.614344\pi\)
\(38\) 9.88273 1.60319
\(39\) 0 0
\(40\) −10.3354 −1.63417
\(41\) −2.83709 −0.443079 −0.221540 0.975151i \(-0.571108\pi\)
−0.221540 + 0.975151i \(0.571108\pi\)
\(42\) 0 0
\(43\) 11.1138 1.69484 0.847421 0.530921i \(-0.178154\pi\)
0.847421 + 0.530921i \(0.178154\pi\)
\(44\) 15.5569 2.34529
\(45\) 0 0
\(46\) −13.4396 −1.98157
\(47\) 11.5569 1.68575 0.842875 0.538110i \(-0.180862\pi\)
0.842875 + 0.538110i \(0.180862\pi\)
\(48\) 0 0
\(49\) 0.397442 0.0567775
\(50\) −2.77846 −0.392933
\(51\) 0 0
\(52\) 5.71982 0.793197
\(53\) −1.16291 −0.159738 −0.0798690 0.996805i \(-0.525450\pi\)
−0.0798690 + 0.996805i \(0.525450\pi\)
\(54\) 0 0
\(55\) 2.71982 0.366741
\(56\) 28.1104 3.75641
\(57\) 0 0
\(58\) 16.6707 2.18898
\(59\) 2.11727 0.275645 0.137822 0.990457i \(-0.455990\pi\)
0.137822 + 0.990457i \(0.455990\pi\)
\(60\) 0 0
\(61\) 6.60256 0.845371 0.422685 0.906276i \(-0.361088\pi\)
0.422685 + 0.906276i \(0.361088\pi\)
\(62\) −20.9966 −2.66657
\(63\) 0 0
\(64\) 41.3871 5.17339
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 1.88273 0.230013 0.115006 0.993365i \(-0.463311\pi\)
0.115006 + 0.993365i \(0.463311\pi\)
\(68\) 16.2277 1.96789
\(69\) 0 0
\(70\) 7.55691 0.903224
\(71\) 6.71982 0.797496 0.398748 0.917060i \(-0.369445\pi\)
0.398748 + 0.917060i \(0.369445\pi\)
\(72\) 0 0
\(73\) 9.11383 1.06669 0.533346 0.845897i \(-0.320934\pi\)
0.533346 + 0.845897i \(0.320934\pi\)
\(74\) 11.8827 1.38134
\(75\) 0 0
\(76\) −20.3449 −2.33372
\(77\) −7.39744 −0.843017
\(78\) 0 0
\(79\) 10.2767 1.15622 0.578112 0.815958i \(-0.303790\pi\)
0.578112 + 0.815958i \(0.303790\pi\)
\(80\) 17.2767 1.93160
\(81\) 0 0
\(82\) 7.88273 0.870502
\(83\) −2.11727 −0.232400 −0.116200 0.993226i \(-0.537071\pi\)
−0.116200 + 0.993226i \(0.537071\pi\)
\(84\) 0 0
\(85\) 2.83709 0.307726
\(86\) −30.8793 −3.32980
\(87\) 0 0
\(88\) −28.1104 −2.99658
\(89\) −1.16291 −0.123268 −0.0616341 0.998099i \(-0.519631\pi\)
−0.0616341 + 0.998099i \(0.519631\pi\)
\(90\) 0 0
\(91\) −2.71982 −0.285115
\(92\) 27.6673 2.88452
\(93\) 0 0
\(94\) −32.1104 −3.31193
\(95\) −3.55691 −0.364932
\(96\) 0 0
\(97\) −10.8371 −1.10034 −0.550170 0.835053i \(-0.685437\pi\)
−0.550170 + 0.835053i \(0.685437\pi\)
\(98\) −1.10428 −0.111549
\(99\) 0 0
\(100\) 5.71982 0.571982
\(101\) −7.67418 −0.763610 −0.381805 0.924243i \(-0.624697\pi\)
−0.381805 + 0.924243i \(0.624697\pi\)
\(102\) 0 0
\(103\) 3.76547 0.371023 0.185511 0.982642i \(-0.440606\pi\)
0.185511 + 0.982642i \(0.440606\pi\)
\(104\) −10.3354 −1.01347
\(105\) 0 0
\(106\) 3.23109 0.313832
\(107\) 12.6026 1.21834 0.609168 0.793041i \(-0.291504\pi\)
0.609168 + 0.793041i \(0.291504\pi\)
\(108\) 0 0
\(109\) 11.4396 1.09572 0.547860 0.836570i \(-0.315443\pi\)
0.547860 + 0.836570i \(0.315443\pi\)
\(110\) −7.55691 −0.720524
\(111\) 0 0
\(112\) −46.9897 −4.44011
\(113\) 13.1138 1.23365 0.616823 0.787102i \(-0.288420\pi\)
0.616823 + 0.787102i \(0.288420\pi\)
\(114\) 0 0
\(115\) 4.83709 0.451061
\(116\) −34.3189 −3.18643
\(117\) 0 0
\(118\) −5.88273 −0.541550
\(119\) −7.71639 −0.707360
\(120\) 0 0
\(121\) −3.60256 −0.327505
\(122\) −18.3449 −1.66087
\(123\) 0 0
\(124\) 43.2242 3.88165
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −13.4396 −1.19258 −0.596288 0.802771i \(-0.703358\pi\)
−0.596288 + 0.802771i \(0.703358\pi\)
\(128\) −60.3285 −5.33234
\(129\) 0 0
\(130\) −2.77846 −0.243687
\(131\) 9.43965 0.824746 0.412373 0.911015i \(-0.364700\pi\)
0.412373 + 0.911015i \(0.364700\pi\)
\(132\) 0 0
\(133\) 9.67418 0.838858
\(134\) −5.23109 −0.451898
\(135\) 0 0
\(136\) −29.3224 −2.51437
\(137\) −1.76547 −0.150834 −0.0754170 0.997152i \(-0.524029\pi\)
−0.0754170 + 0.997152i \(0.524029\pi\)
\(138\) 0 0
\(139\) −6.27674 −0.532386 −0.266193 0.963920i \(-0.585766\pi\)
−0.266193 + 0.963920i \(0.585766\pi\)
\(140\) −15.5569 −1.31480
\(141\) 0 0
\(142\) −18.6707 −1.56681
\(143\) 2.71982 0.227443
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) −25.3224 −2.09570
\(147\) 0 0
\(148\) −24.4622 −2.01078
\(149\) 20.8302 1.70648 0.853239 0.521520i \(-0.174635\pi\)
0.853239 + 0.521520i \(0.174635\pi\)
\(150\) 0 0
\(151\) 4.99656 0.406614 0.203307 0.979115i \(-0.434831\pi\)
0.203307 + 0.979115i \(0.434831\pi\)
\(152\) 36.7620 2.98179
\(153\) 0 0
\(154\) 20.5535 1.65625
\(155\) 7.55691 0.606986
\(156\) 0 0
\(157\) 8.87930 0.708645 0.354322 0.935123i \(-0.384712\pi\)
0.354322 + 0.935123i \(0.384712\pi\)
\(158\) −28.5535 −2.27159
\(159\) 0 0
\(160\) −27.3319 −2.16078
\(161\) −13.1560 −1.03684
\(162\) 0 0
\(163\) −13.8337 −1.08354 −0.541768 0.840528i \(-0.682245\pi\)
−0.541768 + 0.840528i \(0.682245\pi\)
\(164\) −16.2277 −1.26717
\(165\) 0 0
\(166\) 5.88273 0.456589
\(167\) 9.88273 0.764749 0.382374 0.924007i \(-0.375106\pi\)
0.382374 + 0.924007i \(0.375106\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −7.88273 −0.604578
\(171\) 0 0
\(172\) 63.5691 4.84710
\(173\) −13.1138 −0.997026 −0.498513 0.866882i \(-0.666120\pi\)
−0.498513 + 0.866882i \(0.666120\pi\)
\(174\) 0 0
\(175\) −2.71982 −0.205599
\(176\) 46.9897 3.54198
\(177\) 0 0
\(178\) 3.23109 0.242181
\(179\) −8.55348 −0.639317 −0.319658 0.947533i \(-0.603568\pi\)
−0.319658 + 0.947533i \(0.603568\pi\)
\(180\) 0 0
\(181\) 3.72326 0.276748 0.138374 0.990380i \(-0.455812\pi\)
0.138374 + 0.990380i \(0.455812\pi\)
\(182\) 7.55691 0.560156
\(183\) 0 0
\(184\) −49.9931 −3.68554
\(185\) −4.27674 −0.314432
\(186\) 0 0
\(187\) 7.71639 0.564278
\(188\) 66.1035 4.82109
\(189\) 0 0
\(190\) 9.88273 0.716969
\(191\) 4.23453 0.306400 0.153200 0.988195i \(-0.451042\pi\)
0.153200 + 0.988195i \(0.451042\pi\)
\(192\) 0 0
\(193\) −23.3906 −1.68369 −0.841845 0.539719i \(-0.818530\pi\)
−0.841845 + 0.539719i \(0.818530\pi\)
\(194\) 30.1104 2.16180
\(195\) 0 0
\(196\) 2.27330 0.162379
\(197\) −14.5535 −1.03689 −0.518446 0.855110i \(-0.673489\pi\)
−0.518446 + 0.855110i \(0.673489\pi\)
\(198\) 0 0
\(199\) −15.1138 −1.07139 −0.535695 0.844411i \(-0.679950\pi\)
−0.535695 + 0.844411i \(0.679950\pi\)
\(200\) −10.3354 −0.730821
\(201\) 0 0
\(202\) 21.3224 1.50024
\(203\) 16.3189 1.14537
\(204\) 0 0
\(205\) −2.83709 −0.198151
\(206\) −10.4622 −0.728935
\(207\) 0 0
\(208\) 17.2767 1.19793
\(209\) −9.67418 −0.669177
\(210\) 0 0
\(211\) −18.2277 −1.25484 −0.627422 0.778680i \(-0.715890\pi\)
−0.627422 + 0.778680i \(0.715890\pi\)
\(212\) −6.65164 −0.456836
\(213\) 0 0
\(214\) −35.0157 −2.39362
\(215\) 11.1138 0.757957
\(216\) 0 0
\(217\) −20.5535 −1.39526
\(218\) −31.7846 −2.15272
\(219\) 0 0
\(220\) 15.5569 1.04885
\(221\) 2.83709 0.190843
\(222\) 0 0
\(223\) 10.1173 0.677502 0.338751 0.940876i \(-0.389996\pi\)
0.338751 + 0.940876i \(0.389996\pi\)
\(224\) 74.3380 4.96692
\(225\) 0 0
\(226\) −36.4362 −2.42370
\(227\) −11.3224 −0.751493 −0.375746 0.926723i \(-0.622614\pi\)
−0.375746 + 0.926723i \(0.622614\pi\)
\(228\) 0 0
\(229\) −6.23453 −0.411990 −0.205995 0.978553i \(-0.566043\pi\)
−0.205995 + 0.978553i \(0.566043\pi\)
\(230\) −13.4396 −0.886184
\(231\) 0 0
\(232\) 62.0122 4.07130
\(233\) −6.83709 −0.447913 −0.223956 0.974599i \(-0.571897\pi\)
−0.223956 + 0.974599i \(0.571897\pi\)
\(234\) 0 0
\(235\) 11.5569 0.753890
\(236\) 12.1104 0.788319
\(237\) 0 0
\(238\) 21.4396 1.38973
\(239\) −1.28018 −0.0828077 −0.0414039 0.999142i \(-0.513183\pi\)
−0.0414039 + 0.999142i \(0.513183\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 10.0096 0.643438
\(243\) 0 0
\(244\) 37.7655 2.41769
\(245\) 0.397442 0.0253917
\(246\) 0 0
\(247\) −3.55691 −0.226321
\(248\) −78.1035 −4.95958
\(249\) 0 0
\(250\) −2.77846 −0.175725
\(251\) −18.2277 −1.15052 −0.575260 0.817971i \(-0.695099\pi\)
−0.575260 + 0.817971i \(0.695099\pi\)
\(252\) 0 0
\(253\) 13.1560 0.827113
\(254\) 37.3415 2.34301
\(255\) 0 0
\(256\) 84.8459 5.30287
\(257\) −1.11383 −0.0694787 −0.0347394 0.999396i \(-0.511060\pi\)
−0.0347394 + 0.999396i \(0.511060\pi\)
\(258\) 0 0
\(259\) 11.6320 0.722776
\(260\) 5.71982 0.354728
\(261\) 0 0
\(262\) −26.2277 −1.62035
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 0 0
\(265\) −1.16291 −0.0714370
\(266\) −26.8793 −1.64808
\(267\) 0 0
\(268\) 10.7689 0.657816
\(269\) −15.6742 −0.955672 −0.477836 0.878449i \(-0.658579\pi\)
−0.477836 + 0.878449i \(0.658579\pi\)
\(270\) 0 0
\(271\) 0.443086 0.0269155 0.0134578 0.999909i \(-0.495716\pi\)
0.0134578 + 0.999909i \(0.495716\pi\)
\(272\) 49.0157 2.97201
\(273\) 0 0
\(274\) 4.90528 0.296339
\(275\) 2.71982 0.164012
\(276\) 0 0
\(277\) −4.87930 −0.293168 −0.146584 0.989198i \(-0.546828\pi\)
−0.146584 + 0.989198i \(0.546828\pi\)
\(278\) 17.4396 1.04596
\(279\) 0 0
\(280\) 28.1104 1.67992
\(281\) 9.11383 0.543685 0.271843 0.962342i \(-0.412367\pi\)
0.271843 + 0.962342i \(0.412367\pi\)
\(282\) 0 0
\(283\) 33.3415 1.98195 0.990973 0.134063i \(-0.0428025\pi\)
0.990973 + 0.134063i \(0.0428025\pi\)
\(284\) 38.4362 2.28077
\(285\) 0 0
\(286\) −7.55691 −0.446850
\(287\) 7.71639 0.455484
\(288\) 0 0
\(289\) −8.95092 −0.526525
\(290\) 16.6707 0.978940
\(291\) 0 0
\(292\) 52.1295 3.05065
\(293\) 29.4328 1.71948 0.859740 0.510731i \(-0.170625\pi\)
0.859740 + 0.510731i \(0.170625\pi\)
\(294\) 0 0
\(295\) 2.11727 0.123272
\(296\) 44.2017 2.56917
\(297\) 0 0
\(298\) −57.8759 −3.35266
\(299\) 4.83709 0.279736
\(300\) 0 0
\(301\) −30.2277 −1.74229
\(302\) −13.8827 −0.798862
\(303\) 0 0
\(304\) −61.4519 −3.52451
\(305\) 6.60256 0.378061
\(306\) 0 0
\(307\) −21.8337 −1.24611 −0.623056 0.782177i \(-0.714109\pi\)
−0.623056 + 0.782177i \(0.714109\pi\)
\(308\) −42.3121 −2.41095
\(309\) 0 0
\(310\) −20.9966 −1.19252
\(311\) −25.1070 −1.42368 −0.711842 0.702339i \(-0.752139\pi\)
−0.711842 + 0.702339i \(0.752139\pi\)
\(312\) 0 0
\(313\) 8.22766 0.465055 0.232527 0.972590i \(-0.425300\pi\)
0.232527 + 0.972590i \(0.425300\pi\)
\(314\) −24.6707 −1.39225
\(315\) 0 0
\(316\) 58.7811 3.30670
\(317\) −27.6742 −1.55434 −0.777168 0.629293i \(-0.783345\pi\)
−0.777168 + 0.629293i \(0.783345\pi\)
\(318\) 0 0
\(319\) −16.3189 −0.913685
\(320\) 41.3871 2.31361
\(321\) 0 0
\(322\) 36.5535 2.03705
\(323\) −10.0913 −0.561494
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 38.4362 2.12878
\(327\) 0 0
\(328\) 29.3224 1.61906
\(329\) −31.4328 −1.73294
\(330\) 0 0
\(331\) −13.2311 −0.727247 −0.363623 0.931546i \(-0.618460\pi\)
−0.363623 + 0.931546i \(0.618460\pi\)
\(332\) −12.1104 −0.664644
\(333\) 0 0
\(334\) −27.4588 −1.50248
\(335\) 1.88273 0.102865
\(336\) 0 0
\(337\) −4.32582 −0.235642 −0.117821 0.993035i \(-0.537591\pi\)
−0.117821 + 0.993035i \(0.537591\pi\)
\(338\) −2.77846 −0.151128
\(339\) 0 0
\(340\) 16.2277 0.880068
\(341\) 20.5535 1.11303
\(342\) 0 0
\(343\) 17.9578 0.969630
\(344\) −114.866 −6.19314
\(345\) 0 0
\(346\) 36.4362 1.95882
\(347\) −6.27674 −0.336953 −0.168476 0.985706i \(-0.553885\pi\)
−0.168476 + 0.985706i \(0.553885\pi\)
\(348\) 0 0
\(349\) 17.6673 0.945709 0.472855 0.881140i \(-0.343224\pi\)
0.472855 + 0.881140i \(0.343224\pi\)
\(350\) 7.55691 0.403934
\(351\) 0 0
\(352\) −74.3380 −3.96223
\(353\) 13.7655 0.732662 0.366331 0.930485i \(-0.380614\pi\)
0.366331 + 0.930485i \(0.380614\pi\)
\(354\) 0 0
\(355\) 6.71982 0.356651
\(356\) −6.65164 −0.352536
\(357\) 0 0
\(358\) 23.7655 1.25604
\(359\) −0.996562 −0.0525965 −0.0262983 0.999654i \(-0.508372\pi\)
−0.0262983 + 0.999654i \(0.508372\pi\)
\(360\) 0 0
\(361\) −6.34836 −0.334124
\(362\) −10.3449 −0.543717
\(363\) 0 0
\(364\) −15.5569 −0.815404
\(365\) 9.11383 0.477040
\(366\) 0 0
\(367\) 14.2277 0.742678 0.371339 0.928497i \(-0.378899\pi\)
0.371339 + 0.928497i \(0.378899\pi\)
\(368\) 83.5691 4.35634
\(369\) 0 0
\(370\) 11.8827 0.617754
\(371\) 3.16291 0.164210
\(372\) 0 0
\(373\) 15.6742 0.811578 0.405789 0.913967i \(-0.366997\pi\)
0.405789 + 0.913967i \(0.366997\pi\)
\(374\) −21.4396 −1.10862
\(375\) 0 0
\(376\) −119.445 −6.15991
\(377\) −6.00000 −0.309016
\(378\) 0 0
\(379\) 26.2017 1.34589 0.672945 0.739693i \(-0.265029\pi\)
0.672945 + 0.739693i \(0.265029\pi\)
\(380\) −20.3449 −1.04367
\(381\) 0 0
\(382\) −11.7655 −0.601974
\(383\) −22.4362 −1.14644 −0.573218 0.819403i \(-0.694305\pi\)
−0.573218 + 0.819403i \(0.694305\pi\)
\(384\) 0 0
\(385\) −7.39744 −0.377009
\(386\) 64.9897 3.30789
\(387\) 0 0
\(388\) −61.9862 −3.14687
\(389\) −31.6742 −1.60594 −0.802972 0.596016i \(-0.796749\pi\)
−0.802972 + 0.596016i \(0.796749\pi\)
\(390\) 0 0
\(391\) 13.7233 0.694015
\(392\) −4.10771 −0.207471
\(393\) 0 0
\(394\) 40.4362 2.03715
\(395\) 10.2767 0.517079
\(396\) 0 0
\(397\) 17.9509 0.900931 0.450465 0.892794i \(-0.351258\pi\)
0.450465 + 0.892794i \(0.351258\pi\)
\(398\) 41.9931 2.10493
\(399\) 0 0
\(400\) 17.2767 0.863837
\(401\) −13.5829 −0.678297 −0.339149 0.940733i \(-0.610139\pi\)
−0.339149 + 0.940733i \(0.610139\pi\)
\(402\) 0 0
\(403\) 7.55691 0.376437
\(404\) −43.8950 −2.18386
\(405\) 0 0
\(406\) −45.3415 −2.25026
\(407\) −11.6320 −0.576576
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 7.88273 0.389300
\(411\) 0 0
\(412\) 21.5378 1.06109
\(413\) −5.75859 −0.283362
\(414\) 0 0
\(415\) −2.11727 −0.103933
\(416\) −27.3319 −1.34006
\(417\) 0 0
\(418\) 26.8793 1.31471
\(419\) −12.3189 −0.601820 −0.300910 0.953653i \(-0.597290\pi\)
−0.300910 + 0.953653i \(0.597290\pi\)
\(420\) 0 0
\(421\) 22.7880 1.11062 0.555310 0.831644i \(-0.312600\pi\)
0.555310 + 0.831644i \(0.312600\pi\)
\(422\) 50.6448 2.46535
\(423\) 0 0
\(424\) 12.0191 0.583699
\(425\) 2.83709 0.137619
\(426\) 0 0
\(427\) −17.9578 −0.869039
\(428\) 72.0844 3.48433
\(429\) 0 0
\(430\) −30.8793 −1.48913
\(431\) 8.99656 0.433349 0.216675 0.976244i \(-0.430479\pi\)
0.216675 + 0.976244i \(0.430479\pi\)
\(432\) 0 0
\(433\) −20.3258 −0.976797 −0.488398 0.872621i \(-0.662419\pi\)
−0.488398 + 0.872621i \(0.662419\pi\)
\(434\) 57.1070 2.74122
\(435\) 0 0
\(436\) 65.4328 3.13366
\(437\) −17.2051 −0.823032
\(438\) 0 0
\(439\) −25.3906 −1.21183 −0.605913 0.795531i \(-0.707192\pi\)
−0.605913 + 0.795531i \(0.707192\pi\)
\(440\) −28.1104 −1.34011
\(441\) 0 0
\(442\) −7.88273 −0.374943
\(443\) −10.9284 −0.519223 −0.259611 0.965713i \(-0.583594\pi\)
−0.259611 + 0.965713i \(0.583594\pi\)
\(444\) 0 0
\(445\) −1.16291 −0.0551272
\(446\) −28.1104 −1.33107
\(447\) 0 0
\(448\) −112.566 −5.31823
\(449\) −2.83709 −0.133891 −0.0669453 0.997757i \(-0.521325\pi\)
−0.0669453 + 0.997757i \(0.521325\pi\)
\(450\) 0 0
\(451\) −7.71639 −0.363350
\(452\) 75.0088 3.52812
\(453\) 0 0
\(454\) 31.4588 1.47643
\(455\) −2.71982 −0.127507
\(456\) 0 0
\(457\) −13.7164 −0.641625 −0.320813 0.947143i \(-0.603956\pi\)
−0.320813 + 0.947143i \(0.603956\pi\)
\(458\) 17.3224 0.809422
\(459\) 0 0
\(460\) 27.6673 1.28999
\(461\) 19.6251 0.914032 0.457016 0.889458i \(-0.348918\pi\)
0.457016 + 0.889458i \(0.348918\pi\)
\(462\) 0 0
\(463\) 27.0388 1.25660 0.628299 0.777972i \(-0.283751\pi\)
0.628299 + 0.777972i \(0.283751\pi\)
\(464\) −103.660 −4.81231
\(465\) 0 0
\(466\) 18.9966 0.879999
\(467\) −28.9215 −1.33833 −0.669164 0.743115i \(-0.733348\pi\)
−0.669164 + 0.743115i \(0.733348\pi\)
\(468\) 0 0
\(469\) −5.12070 −0.236452
\(470\) −32.1104 −1.48114
\(471\) 0 0
\(472\) −21.8827 −1.00723
\(473\) 30.2277 1.38987
\(474\) 0 0
\(475\) −3.55691 −0.163202
\(476\) −44.1364 −2.02299
\(477\) 0 0
\(478\) 3.55691 0.162689
\(479\) 12.1595 0.555580 0.277790 0.960642i \(-0.410398\pi\)
0.277790 + 0.960642i \(0.410398\pi\)
\(480\) 0 0
\(481\) −4.27674 −0.195002
\(482\) 16.6707 0.759332
\(483\) 0 0
\(484\) −20.6060 −0.936636
\(485\) −10.8371 −0.492087
\(486\) 0 0
\(487\) −0.159472 −0.00722636 −0.00361318 0.999993i \(-0.501150\pi\)
−0.00361318 + 0.999993i \(0.501150\pi\)
\(488\) −68.2399 −3.08907
\(489\) 0 0
\(490\) −1.10428 −0.0498861
\(491\) 42.2277 1.90571 0.952854 0.303430i \(-0.0981318\pi\)
0.952854 + 0.303430i \(0.0981318\pi\)
\(492\) 0 0
\(493\) −17.0225 −0.766657
\(494\) 9.88273 0.444645
\(495\) 0 0
\(496\) 130.559 5.86226
\(497\) −18.2767 −0.819824
\(498\) 0 0
\(499\) 7.79145 0.348793 0.174397 0.984676i \(-0.444203\pi\)
0.174397 + 0.984676i \(0.444203\pi\)
\(500\) 5.71982 0.255798
\(501\) 0 0
\(502\) 50.6448 2.26039
\(503\) −27.3484 −1.21940 −0.609702 0.792631i \(-0.708711\pi\)
−0.609702 + 0.792631i \(0.708711\pi\)
\(504\) 0 0
\(505\) −7.67418 −0.341497
\(506\) −36.5535 −1.62500
\(507\) 0 0
\(508\) −76.8724 −3.41066
\(509\) 33.4819 1.48406 0.742029 0.670368i \(-0.233864\pi\)
0.742029 + 0.670368i \(0.233864\pi\)
\(510\) 0 0
\(511\) −24.7880 −1.09656
\(512\) −115.084 −5.08603
\(513\) 0 0
\(514\) 3.09472 0.136502
\(515\) 3.76547 0.165926
\(516\) 0 0
\(517\) 31.4328 1.38241
\(518\) −32.3189 −1.42001
\(519\) 0 0
\(520\) −10.3354 −0.453236
\(521\) 17.3484 0.760045 0.380023 0.924977i \(-0.375916\pi\)
0.380023 + 0.924977i \(0.375916\pi\)
\(522\) 0 0
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 53.9931 2.35870
\(525\) 0 0
\(526\) 22.2277 0.969172
\(527\) 21.4396 0.933926
\(528\) 0 0
\(529\) 0.397442 0.0172801
\(530\) 3.23109 0.140350
\(531\) 0 0
\(532\) 55.3346 2.39906
\(533\) −2.83709 −0.122888
\(534\) 0 0
\(535\) 12.6026 0.544856
\(536\) −19.4588 −0.840490
\(537\) 0 0
\(538\) 43.5500 1.87758
\(539\) 1.08097 0.0465608
\(540\) 0 0
\(541\) −32.6448 −1.40351 −0.701754 0.712419i \(-0.747599\pi\)
−0.701754 + 0.712419i \(0.747599\pi\)
\(542\) −1.23109 −0.0528800
\(543\) 0 0
\(544\) −77.5432 −3.32464
\(545\) 11.4396 0.490021
\(546\) 0 0
\(547\) 34.2277 1.46347 0.731734 0.681590i \(-0.238711\pi\)
0.731734 + 0.681590i \(0.238711\pi\)
\(548\) −10.0982 −0.431372
\(549\) 0 0
\(550\) −7.55691 −0.322228
\(551\) 21.3415 0.909178
\(552\) 0 0
\(553\) −27.9509 −1.18859
\(554\) 13.5569 0.575978
\(555\) 0 0
\(556\) −35.9018 −1.52258
\(557\) −6.65164 −0.281839 −0.140919 0.990021i \(-0.545006\pi\)
−0.140919 + 0.990021i \(0.545006\pi\)
\(558\) 0 0
\(559\) 11.1138 0.470065
\(560\) −46.9897 −1.98568
\(561\) 0 0
\(562\) −25.3224 −1.06816
\(563\) −40.2699 −1.69717 −0.848586 0.529057i \(-0.822546\pi\)
−0.848586 + 0.529057i \(0.822546\pi\)
\(564\) 0 0
\(565\) 13.1138 0.551703
\(566\) −92.6379 −3.89386
\(567\) 0 0
\(568\) −69.4519 −2.91414
\(569\) 13.4328 0.563131 0.281566 0.959542i \(-0.409146\pi\)
0.281566 + 0.959542i \(0.409146\pi\)
\(570\) 0 0
\(571\) −35.7164 −1.49468 −0.747342 0.664439i \(-0.768670\pi\)
−0.747342 + 0.664439i \(0.768670\pi\)
\(572\) 15.5569 0.650467
\(573\) 0 0
\(574\) −21.4396 −0.894874
\(575\) 4.83709 0.201721
\(576\) 0 0
\(577\) −13.7164 −0.571021 −0.285510 0.958376i \(-0.592163\pi\)
−0.285510 + 0.958376i \(0.592163\pi\)
\(578\) 24.8697 1.03444
\(579\) 0 0
\(580\) −34.3189 −1.42502
\(581\) 5.75859 0.238907
\(582\) 0 0
\(583\) −3.16291 −0.130994
\(584\) −94.1948 −3.89781
\(585\) 0 0
\(586\) −81.7777 −3.37821
\(587\) −30.6707 −1.26592 −0.632959 0.774186i \(-0.718160\pi\)
−0.632959 + 0.774186i \(0.718160\pi\)
\(588\) 0 0
\(589\) −26.8793 −1.10754
\(590\) −5.88273 −0.242188
\(591\) 0 0
\(592\) −73.8881 −3.03678
\(593\) 45.6673 1.87533 0.937666 0.347538i \(-0.112982\pi\)
0.937666 + 0.347538i \(0.112982\pi\)
\(594\) 0 0
\(595\) −7.71639 −0.316341
\(596\) 119.145 4.88038
\(597\) 0 0
\(598\) −13.4396 −0.549588
\(599\) 40.2208 1.64338 0.821688 0.569937i \(-0.193032\pi\)
0.821688 + 0.569937i \(0.193032\pi\)
\(600\) 0 0
\(601\) 17.3974 0.709656 0.354828 0.934932i \(-0.384539\pi\)
0.354828 + 0.934932i \(0.384539\pi\)
\(602\) 83.9862 3.42302
\(603\) 0 0
\(604\) 28.5795 1.16288
\(605\) −3.60256 −0.146465
\(606\) 0 0
\(607\) −14.2277 −0.577483 −0.288741 0.957407i \(-0.593237\pi\)
−0.288741 + 0.957407i \(0.593237\pi\)
\(608\) 97.2173 3.94268
\(609\) 0 0
\(610\) −18.3449 −0.742764
\(611\) 11.5569 0.467543
\(612\) 0 0
\(613\) −40.8302 −1.64912 −0.824558 0.565777i \(-0.808576\pi\)
−0.824558 + 0.565777i \(0.808576\pi\)
\(614\) 60.6639 2.44819
\(615\) 0 0
\(616\) 76.4553 3.08047
\(617\) −5.11383 −0.205875 −0.102937 0.994688i \(-0.532824\pi\)
−0.102937 + 0.994688i \(0.532824\pi\)
\(618\) 0 0
\(619\) 11.5569 0.464512 0.232256 0.972655i \(-0.425389\pi\)
0.232256 + 0.972655i \(0.425389\pi\)
\(620\) 43.2242 1.73593
\(621\) 0 0
\(622\) 69.7586 2.79706
\(623\) 3.16291 0.126719
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −22.8602 −0.913677
\(627\) 0 0
\(628\) 50.7880 2.02666
\(629\) −12.1335 −0.483794
\(630\) 0 0
\(631\) 35.2242 1.40225 0.701127 0.713036i \(-0.252681\pi\)
0.701127 + 0.713036i \(0.252681\pi\)
\(632\) −106.214 −4.22496
\(633\) 0 0
\(634\) 76.8915 3.05375
\(635\) −13.4396 −0.533336
\(636\) 0 0
\(637\) 0.397442 0.0157472
\(638\) 45.3415 1.79509
\(639\) 0 0
\(640\) −60.3285 −2.38469
\(641\) 21.9018 0.865071 0.432535 0.901617i \(-0.357619\pi\)
0.432535 + 0.901617i \(0.357619\pi\)
\(642\) 0 0
\(643\) 7.50783 0.296080 0.148040 0.988981i \(-0.452704\pi\)
0.148040 + 0.988981i \(0.452704\pi\)
\(644\) −75.2502 −2.96527
\(645\) 0 0
\(646\) 28.0382 1.10315
\(647\) 14.0422 0.552056 0.276028 0.961150i \(-0.410982\pi\)
0.276028 + 0.961150i \(0.410982\pi\)
\(648\) 0 0
\(649\) 5.75859 0.226044
\(650\) −2.77846 −0.108980
\(651\) 0 0
\(652\) −79.1261 −3.09882
\(653\) −7.99312 −0.312795 −0.156398 0.987694i \(-0.549988\pi\)
−0.156398 + 0.987694i \(0.549988\pi\)
\(654\) 0 0
\(655\) 9.43965 0.368838
\(656\) −49.0157 −1.91374
\(657\) 0 0
\(658\) 87.3346 3.40466
\(659\) 25.3415 0.987164 0.493582 0.869699i \(-0.335687\pi\)
0.493582 + 0.869699i \(0.335687\pi\)
\(660\) 0 0
\(661\) 27.4396 1.06728 0.533639 0.845712i \(-0.320824\pi\)
0.533639 + 0.845712i \(0.320824\pi\)
\(662\) 36.7620 1.42880
\(663\) 0 0
\(664\) 21.8827 0.849215
\(665\) 9.67418 0.375149
\(666\) 0 0
\(667\) −29.0225 −1.12376
\(668\) 56.5275 2.18711
\(669\) 0 0
\(670\) −5.23109 −0.202095
\(671\) 17.9578 0.693253
\(672\) 0 0
\(673\) 27.1070 1.04490 0.522448 0.852671i \(-0.325019\pi\)
0.522448 + 0.852671i \(0.325019\pi\)
\(674\) 12.0191 0.462959
\(675\) 0 0
\(676\) 5.71982 0.219993
\(677\) 36.5957 1.40649 0.703243 0.710949i \(-0.251735\pi\)
0.703243 + 0.710949i \(0.251735\pi\)
\(678\) 0 0
\(679\) 29.4750 1.13115
\(680\) −29.3224 −1.12446
\(681\) 0 0
\(682\) −57.1070 −2.18674
\(683\) 13.4656 0.515248 0.257624 0.966245i \(-0.417060\pi\)
0.257624 + 0.966245i \(0.417060\pi\)
\(684\) 0 0
\(685\) −1.76547 −0.0674550
\(686\) −49.8950 −1.90500
\(687\) 0 0
\(688\) 192.011 7.32034
\(689\) −1.16291 −0.0443033
\(690\) 0 0
\(691\) 29.5500 1.12414 0.562068 0.827091i \(-0.310006\pi\)
0.562068 + 0.827091i \(0.310006\pi\)
\(692\) −75.0088 −2.85141
\(693\) 0 0
\(694\) 17.4396 0.662000
\(695\) −6.27674 −0.238090
\(696\) 0 0
\(697\) −8.04908 −0.304881
\(698\) −49.0878 −1.85800
\(699\) 0 0
\(700\) −15.5569 −0.587996
\(701\) −43.6604 −1.64903 −0.824516 0.565839i \(-0.808552\pi\)
−0.824516 + 0.565839i \(0.808552\pi\)
\(702\) 0 0
\(703\) 15.2120 0.573731
\(704\) 112.566 4.24248
\(705\) 0 0
\(706\) −38.2468 −1.43944
\(707\) 20.8724 0.784988
\(708\) 0 0
\(709\) −26.7880 −1.00604 −0.503022 0.864273i \(-0.667779\pi\)
−0.503022 + 0.864273i \(0.667779\pi\)
\(710\) −18.6707 −0.700700
\(711\) 0 0
\(712\) 12.0191 0.450435
\(713\) 36.5535 1.36894
\(714\) 0 0
\(715\) 2.71982 0.101716
\(716\) −48.9244 −1.82839
\(717\) 0 0
\(718\) 2.76891 0.103335
\(719\) 34.8793 1.30078 0.650389 0.759601i \(-0.274606\pi\)
0.650389 + 0.759601i \(0.274606\pi\)
\(720\) 0 0
\(721\) −10.2414 −0.381410
\(722\) 17.6386 0.656443
\(723\) 0 0
\(724\) 21.2964 0.791475
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 37.4396 1.38856 0.694280 0.719705i \(-0.255723\pi\)
0.694280 + 0.719705i \(0.255723\pi\)
\(728\) 28.1104 1.04184
\(729\) 0 0
\(730\) −25.3224 −0.937223
\(731\) 31.5309 1.16621
\(732\) 0 0
\(733\) −47.1560 −1.74175 −0.870874 0.491506i \(-0.836446\pi\)
−0.870874 + 0.491506i \(0.836446\pi\)
\(734\) −39.5309 −1.45911
\(735\) 0 0
\(736\) −132.207 −4.87322
\(737\) 5.12070 0.188624
\(738\) 0 0
\(739\) −31.7914 −1.16947 −0.584734 0.811225i \(-0.698801\pi\)
−0.584734 + 0.811225i \(0.698801\pi\)
\(740\) −24.4622 −0.899248
\(741\) 0 0
\(742\) −8.78801 −0.322618
\(743\) −16.6776 −0.611842 −0.305921 0.952057i \(-0.598964\pi\)
−0.305921 + 0.952057i \(0.598964\pi\)
\(744\) 0 0
\(745\) 20.8302 0.763160
\(746\) −43.5500 −1.59448
\(747\) 0 0
\(748\) 44.1364 1.61379
\(749\) −34.2767 −1.25244
\(750\) 0 0
\(751\) 16.1855 0.590616 0.295308 0.955402i \(-0.404578\pi\)
0.295308 + 0.955402i \(0.404578\pi\)
\(752\) 199.666 7.28106
\(753\) 0 0
\(754\) 16.6707 0.607113
\(755\) 4.99656 0.181844
\(756\) 0 0
\(757\) 12.3258 0.447990 0.223995 0.974590i \(-0.428090\pi\)
0.223995 + 0.974590i \(0.428090\pi\)
\(758\) −72.8002 −2.64422
\(759\) 0 0
\(760\) 36.7620 1.33350
\(761\) 0.00687569 0.000249244 0 0.000124622 1.00000i \(-0.499960\pi\)
0.000124622 1.00000i \(0.499960\pi\)
\(762\) 0 0
\(763\) −31.1138 −1.12640
\(764\) 24.2208 0.876277
\(765\) 0 0
\(766\) 62.3380 2.25237
\(767\) 2.11727 0.0764501
\(768\) 0 0
\(769\) −20.3258 −0.732968 −0.366484 0.930424i \(-0.619439\pi\)
−0.366484 + 0.930424i \(0.619439\pi\)
\(770\) 20.5535 0.740696
\(771\) 0 0
\(772\) −133.790 −4.81520
\(773\) −9.90184 −0.356144 −0.178072 0.984017i \(-0.556986\pi\)
−0.178072 + 0.984017i \(0.556986\pi\)
\(774\) 0 0
\(775\) 7.55691 0.271452
\(776\) 112.005 4.02076
\(777\) 0 0
\(778\) 88.0054 3.15514
\(779\) 10.0913 0.361558
\(780\) 0 0
\(781\) 18.2767 0.653993
\(782\) −38.1295 −1.36351
\(783\) 0 0
\(784\) 6.86651 0.245232
\(785\) 8.87930 0.316916
\(786\) 0 0
\(787\) 36.3449 1.29556 0.647778 0.761829i \(-0.275698\pi\)
0.647778 + 0.761829i \(0.275698\pi\)
\(788\) −83.2433 −2.96542
\(789\) 0 0
\(790\) −28.5535 −1.01589
\(791\) −35.6673 −1.26818
\(792\) 0 0
\(793\) 6.60256 0.234464
\(794\) −49.8759 −1.77003
\(795\) 0 0
\(796\) −86.4484 −3.06408
\(797\) −18.8371 −0.667244 −0.333622 0.942707i \(-0.608271\pi\)
−0.333622 + 0.942707i \(0.608271\pi\)
\(798\) 0 0
\(799\) 32.7880 1.15996
\(800\) −27.3319 −0.966330
\(801\) 0 0
\(802\) 37.7395 1.33263
\(803\) 24.7880 0.874750
\(804\) 0 0
\(805\) −13.1560 −0.463689
\(806\) −20.9966 −0.739572
\(807\) 0 0
\(808\) 79.3155 2.79031
\(809\) −32.2277 −1.13306 −0.566532 0.824040i \(-0.691715\pi\)
−0.566532 + 0.824040i \(0.691715\pi\)
\(810\) 0 0
\(811\) −23.0034 −0.807760 −0.403880 0.914812i \(-0.632339\pi\)
−0.403880 + 0.914812i \(0.632339\pi\)
\(812\) 93.3415 3.27564
\(813\) 0 0
\(814\) 32.3189 1.13278
\(815\) −13.8337 −0.484572
\(816\) 0 0
\(817\) −39.5309 −1.38301
\(818\) 38.8984 1.36005
\(819\) 0 0
\(820\) −16.2277 −0.566694
\(821\) −49.9372 −1.74282 −0.871410 0.490556i \(-0.836794\pi\)
−0.871410 + 0.490556i \(0.836794\pi\)
\(822\) 0 0
\(823\) 28.2345 0.984194 0.492097 0.870540i \(-0.336231\pi\)
0.492097 + 0.870540i \(0.336231\pi\)
\(824\) −38.9175 −1.35576
\(825\) 0 0
\(826\) 16.0000 0.556711
\(827\) 9.55004 0.332087 0.166044 0.986118i \(-0.446901\pi\)
0.166044 + 0.986118i \(0.446901\pi\)
\(828\) 0 0
\(829\) −37.9862 −1.31932 −0.659658 0.751565i \(-0.729299\pi\)
−0.659658 + 0.751565i \(0.729299\pi\)
\(830\) 5.88273 0.204193
\(831\) 0 0
\(832\) 41.3871 1.43484
\(833\) 1.12758 0.0390683
\(834\) 0 0
\(835\) 9.88273 0.342006
\(836\) −55.3346 −1.91379
\(837\) 0 0
\(838\) 34.2277 1.18237
\(839\) 4.72670 0.163184 0.0815919 0.996666i \(-0.474000\pi\)
0.0815919 + 0.996666i \(0.474000\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −63.3155 −2.18200
\(843\) 0 0
\(844\) −104.259 −3.58874
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 9.79832 0.336674
\(848\) −20.0913 −0.689938
\(849\) 0 0
\(850\) −7.88273 −0.270376
\(851\) −20.6870 −0.709140
\(852\) 0 0
\(853\) −48.3611 −1.65585 −0.827927 0.560836i \(-0.810480\pi\)
−0.827927 + 0.560836i \(0.810480\pi\)
\(854\) 49.8950 1.70737
\(855\) 0 0
\(856\) −130.252 −4.45193
\(857\) −6.83709 −0.233551 −0.116775 0.993158i \(-0.537256\pi\)
−0.116775 + 0.993158i \(0.537256\pi\)
\(858\) 0 0
\(859\) 12.6026 0.429994 0.214997 0.976615i \(-0.431026\pi\)
0.214997 + 0.976615i \(0.431026\pi\)
\(860\) 63.5691 2.16769
\(861\) 0 0
\(862\) −24.9966 −0.851386
\(863\) −8.20855 −0.279422 −0.139711 0.990192i \(-0.544617\pi\)
−0.139711 + 0.990192i \(0.544617\pi\)
\(864\) 0 0
\(865\) −13.1138 −0.445884
\(866\) 56.4744 1.91908
\(867\) 0 0
\(868\) −117.562 −3.99032
\(869\) 27.9509 0.948170
\(870\) 0 0
\(871\) 1.88273 0.0637940
\(872\) −118.233 −4.00387
\(873\) 0 0
\(874\) 47.8037 1.61698
\(875\) −2.71982 −0.0919468
\(876\) 0 0
\(877\) 13.5309 0.456907 0.228454 0.973555i \(-0.426633\pi\)
0.228454 + 0.973555i \(0.426633\pi\)
\(878\) 70.5466 2.38083
\(879\) 0 0
\(880\) 46.9897 1.58402
\(881\) 9.34836 0.314954 0.157477 0.987523i \(-0.449664\pi\)
0.157477 + 0.987523i \(0.449664\pi\)
\(882\) 0 0
\(883\) −55.1001 −1.85427 −0.927133 0.374733i \(-0.877734\pi\)
−0.927133 + 0.374733i \(0.877734\pi\)
\(884\) 16.2277 0.545795
\(885\) 0 0
\(886\) 30.3640 1.02010
\(887\) −0.133492 −0.00448223 −0.00224112 0.999997i \(-0.500713\pi\)
−0.00224112 + 0.999997i \(0.500713\pi\)
\(888\) 0 0
\(889\) 36.5535 1.22596
\(890\) 3.23109 0.108307
\(891\) 0 0
\(892\) 57.8690 1.93760
\(893\) −41.1070 −1.37559
\(894\) 0 0
\(895\) −8.55348 −0.285911
\(896\) 164.083 5.48162
\(897\) 0 0
\(898\) 7.88273 0.263050
\(899\) −45.3415 −1.51222
\(900\) 0 0
\(901\) −3.29928 −0.109915
\(902\) 21.4396 0.713862
\(903\) 0 0
\(904\) −135.536 −4.50787
\(905\) 3.72326 0.123765
\(906\) 0 0
\(907\) −58.5466 −1.94401 −0.972004 0.234964i \(-0.924503\pi\)
−0.972004 + 0.234964i \(0.924503\pi\)
\(908\) −64.7620 −2.14920
\(909\) 0 0
\(910\) 7.55691 0.250509
\(911\) −50.4622 −1.67189 −0.835943 0.548816i \(-0.815079\pi\)
−0.835943 + 0.548816i \(0.815079\pi\)
\(912\) 0 0
\(913\) −5.75859 −0.190582
\(914\) 38.1104 1.26058
\(915\) 0 0
\(916\) −35.6604 −1.17825
\(917\) −25.6742 −0.847836
\(918\) 0 0
\(919\) 56.9735 1.87938 0.939691 0.342026i \(-0.111113\pi\)
0.939691 + 0.342026i \(0.111113\pi\)
\(920\) −49.9931 −1.64822
\(921\) 0 0
\(922\) −54.5275 −1.79577
\(923\) 6.71982 0.221186
\(924\) 0 0
\(925\) −4.27674 −0.140618
\(926\) −75.1261 −2.46880
\(927\) 0 0
\(928\) 163.992 5.38329
\(929\) 36.5957 1.20067 0.600333 0.799750i \(-0.295035\pi\)
0.600333 + 0.799750i \(0.295035\pi\)
\(930\) 0 0
\(931\) −1.41367 −0.0463311
\(932\) −39.1070 −1.28099
\(933\) 0 0
\(934\) 80.3572 2.62937
\(935\) 7.71639 0.252353
\(936\) 0 0
\(937\) −47.1070 −1.53892 −0.769459 0.638697i \(-0.779474\pi\)
−0.769459 + 0.638697i \(0.779474\pi\)
\(938\) 14.2277 0.464549
\(939\) 0 0
\(940\) 66.1035 2.15606
\(941\) 36.3611 1.18534 0.592670 0.805446i \(-0.298074\pi\)
0.592670 + 0.805446i \(0.298074\pi\)
\(942\) 0 0
\(943\) −13.7233 −0.446891
\(944\) 36.5795 1.19056
\(945\) 0 0
\(946\) −83.9862 −2.73063
\(947\) −44.5795 −1.44864 −0.724319 0.689465i \(-0.757846\pi\)
−0.724319 + 0.689465i \(0.757846\pi\)
\(948\) 0 0
\(949\) 9.11383 0.295847
\(950\) 9.88273 0.320638
\(951\) 0 0
\(952\) 79.7517 2.58477
\(953\) −19.8596 −0.643317 −0.321658 0.946856i \(-0.604240\pi\)
−0.321658 + 0.946856i \(0.604240\pi\)
\(954\) 0 0
\(955\) 4.23453 0.137026
\(956\) −7.32238 −0.236823
\(957\) 0 0
\(958\) −33.7846 −1.09153
\(959\) 4.80176 0.155057
\(960\) 0 0
\(961\) 26.1070 0.842160
\(962\) 11.8827 0.383115
\(963\) 0 0
\(964\) −34.3189 −1.10534
\(965\) −23.3906 −0.752969
\(966\) 0 0
\(967\) 47.4068 1.52450 0.762250 0.647283i \(-0.224095\pi\)
0.762250 + 0.647283i \(0.224095\pi\)
\(968\) 37.2338 1.19674
\(969\) 0 0
\(970\) 30.1104 0.966786
\(971\) 10.6448 0.341607 0.170803 0.985305i \(-0.445364\pi\)
0.170803 + 0.985305i \(0.445364\pi\)
\(972\) 0 0
\(973\) 17.0716 0.547291
\(974\) 0.443086 0.0141974
\(975\) 0 0
\(976\) 114.071 3.65131
\(977\) 1.21199 0.0387750 0.0193875 0.999812i \(-0.493828\pi\)
0.0193875 + 0.999812i \(0.493828\pi\)
\(978\) 0 0
\(979\) −3.16291 −0.101087
\(980\) 2.27330 0.0726179
\(981\) 0 0
\(982\) −117.328 −3.74408
\(983\) −51.8759 −1.65458 −0.827291 0.561773i \(-0.810119\pi\)
−0.827291 + 0.561773i \(0.810119\pi\)
\(984\) 0 0
\(985\) −14.5535 −0.463712
\(986\) 47.2964 1.50622
\(987\) 0 0
\(988\) −20.3449 −0.647258
\(989\) 53.7586 1.70942
\(990\) 0 0
\(991\) −21.6251 −0.686944 −0.343472 0.939163i \(-0.611603\pi\)
−0.343472 + 0.939163i \(0.611603\pi\)
\(992\) −206.545 −6.55781
\(993\) 0 0
\(994\) 50.7811 1.61068
\(995\) −15.1138 −0.479141
\(996\) 0 0
\(997\) 23.2051 0.734913 0.367457 0.930041i \(-0.380229\pi\)
0.367457 + 0.930041i \(0.380229\pi\)
\(998\) −21.6482 −0.685262
\(999\) 0 0
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 585.2.a.n.1.1 3
3.2 odd 2 195.2.a.e.1.3 3
4.3 odd 2 9360.2.a.dd.1.3 3
5.2 odd 4 2925.2.c.w.2224.1 6
5.3 odd 4 2925.2.c.w.2224.6 6
5.4 even 2 2925.2.a.bh.1.3 3
12.11 even 2 3120.2.a.bj.1.3 3
13.12 even 2 7605.2.a.bx.1.3 3
15.2 even 4 975.2.c.i.274.6 6
15.8 even 4 975.2.c.i.274.1 6
15.14 odd 2 975.2.a.o.1.1 3
21.20 even 2 9555.2.a.bq.1.3 3
39.38 odd 2 2535.2.a.bc.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.e.1.3 3 3.2 odd 2
585.2.a.n.1.1 3 1.1 even 1 trivial
975.2.a.o.1.1 3 15.14 odd 2
975.2.c.i.274.1 6 15.8 even 4
975.2.c.i.274.6 6 15.2 even 4
2535.2.a.bc.1.1 3 39.38 odd 2
2925.2.a.bh.1.3 3 5.4 even 2
2925.2.c.w.2224.1 6 5.2 odd 4
2925.2.c.w.2224.6 6 5.3 odd 4
3120.2.a.bj.1.3 3 12.11 even 2
7605.2.a.bx.1.3 3 13.12 even 2
9360.2.a.dd.1.3 3 4.3 odd 2
9555.2.a.bq.1.3 3 21.20 even 2