Properties

Label 975.2.a.o.1.1
Level $975$
Weight $2$
Character 975.1
Self dual yes
Analytic conductor $7.785$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.78541419707\)
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 \(\chi\) \(=\) 975.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.77846 q^{2} +1.00000 q^{3} +5.71982 q^{4} -2.77846 q^{6} +2.71982 q^{7} -10.3354 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.77846 q^{2} +1.00000 q^{3} +5.71982 q^{4} -2.77846 q^{6} +2.71982 q^{7} -10.3354 q^{8} +1.00000 q^{9} -2.71982 q^{11} +5.71982 q^{12} -1.00000 q^{13} -7.55691 q^{14} +17.2767 q^{16} +2.83709 q^{17} -2.77846 q^{18} -3.55691 q^{19} +2.71982 q^{21} +7.55691 q^{22} +4.83709 q^{23} -10.3354 q^{24} +2.77846 q^{26} +1.00000 q^{27} +15.5569 q^{28} +6.00000 q^{29} +7.55691 q^{31} -27.3319 q^{32} -2.71982 q^{33} -7.88273 q^{34} +5.71982 q^{36} +4.27674 q^{37} +9.88273 q^{38} -1.00000 q^{39} +2.83709 q^{41} -7.55691 q^{42} -11.1138 q^{43} -15.5569 q^{44} -13.4396 q^{46} +11.5569 q^{47} +17.2767 q^{48} +0.397442 q^{49} +2.83709 q^{51} -5.71982 q^{52} -1.16291 q^{53} -2.77846 q^{54} -28.1104 q^{56} -3.55691 q^{57} -16.6707 q^{58} -2.11727 q^{59} +6.60256 q^{61} -20.9966 q^{62} +2.71982 q^{63} +41.3871 q^{64} +7.55691 q^{66} -1.88273 q^{67} +16.2277 q^{68} +4.83709 q^{69} -6.71982 q^{71} -10.3354 q^{72} -9.11383 q^{73} -11.8827 q^{74} -20.3449 q^{76} -7.39744 q^{77} +2.77846 q^{78} +10.2767 q^{79} +1.00000 q^{81} -7.88273 q^{82} -2.11727 q^{83} +15.5569 q^{84} +30.8793 q^{86} +6.00000 q^{87} +28.1104 q^{88} +1.16291 q^{89} -2.71982 q^{91} +27.6673 q^{92} +7.55691 q^{93} -32.1104 q^{94} -27.3319 q^{96} +10.8371 q^{97} -1.10428 q^{98} -2.71982 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 8 q^{4} - q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 8 q^{4} - q^{7} - 6 q^{8} + 3 q^{9} + q^{11} + 8 q^{12} - 3 q^{13} - 6 q^{14} + 26 q^{16} + q^{17} + 6 q^{19} - q^{21} + 6 q^{22} + 7 q^{23} - 6 q^{24} + 3 q^{27} + 30 q^{28} + 18 q^{29} + 6 q^{31} - 22 q^{32} + q^{33} - 22 q^{34} + 8 q^{36} - 13 q^{37} + 28 q^{38} - 3 q^{39} + q^{41} - 6 q^{42} - 30 q^{44} - 22 q^{46} + 18 q^{47} + 26 q^{48} + 12 q^{49} + q^{51} - 8 q^{52} - 11 q^{53} - 16 q^{56} + 6 q^{57} - 8 q^{59} + 9 q^{61} - 28 q^{62} - q^{63} + 30 q^{64} + 6 q^{66} - 4 q^{67} - 18 q^{68} + 7 q^{69} - 11 q^{71} - 6 q^{72} + 6 q^{73} - 34 q^{74} + 4 q^{76} - 33 q^{77} + 5 q^{79} + 3 q^{81} - 22 q^{82} - 8 q^{83} + 30 q^{84} + 56 q^{86} + 18 q^{87} + 16 q^{88} + 11 q^{89} + q^{91} - 2 q^{92} + 6 q^{93} - 28 q^{94} - 22 q^{96} + 25 q^{97} - 10 q^{98} + q^{99}+O(q^{100}) \) 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 \(0.940078\pi\)
\(3\) 1.00000 0.577350
\(4\) 5.71982 2.85991
\(5\) 0 0
\(6\) −2.77846 −1.13430
\(7\) 2.71982 1.02800 0.513998 0.857791i \(-0.328164\pi\)
0.513998 + 0.857791i \(0.328164\pi\)
\(8\) −10.3354 −3.65411
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.71982 −0.820058 −0.410029 0.912073i \(-0.634481\pi\)
−0.410029 + 0.912073i \(0.634481\pi\)
\(12\) 5.71982 1.65117
\(13\) −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 \(0.388202\pi\)
\(18\) −2.77846 −0.654889
\(19\) −3.55691 −0.816012 −0.408006 0.912979i \(-0.633776\pi\)
−0.408006 + 0.912979i \(0.633776\pi\)
\(20\) 0 0
\(21\) 2.71982 0.593514
\(22\) 7.55691 1.61114
\(23\) 4.83709 1.00860 0.504302 0.863528i \(-0.331750\pi\)
0.504302 + 0.863528i \(0.331750\pi\)
\(24\) −10.3354 −2.10970
\(25\) 0 0
\(26\) 2.77846 0.544900
\(27\) 1.00000 0.192450
\(28\) 15.5569 2.93998
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 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
\(33\) −2.71982 −0.473461
\(34\) −7.88273 −1.35188
\(35\) 0 0
\(36\) 5.71982 0.953304
\(37\) 4.27674 0.703091 0.351546 0.936171i \(-0.385656\pi\)
0.351546 + 0.936171i \(0.385656\pi\)
\(38\) 9.88273 1.60319
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 2.83709 0.443079 0.221540 0.975151i \(-0.428892\pi\)
0.221540 + 0.975151i \(0.428892\pi\)
\(42\) −7.55691 −1.16606
\(43\) −11.1138 −1.69484 −0.847421 0.530921i \(-0.821846\pi\)
−0.847421 + 0.530921i \(0.821846\pi\)
\(44\) −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 \(0.180862\pi\)
\(48\) 17.2767 2.49368
\(49\) 0.397442 0.0567775
\(50\) 0 0
\(51\) 2.83709 0.397272
\(52\) −5.71982 −0.793197
\(53\) −1.16291 −0.159738 −0.0798690 0.996805i \(-0.525450\pi\)
−0.0798690 + 0.996805i \(0.525450\pi\)
\(54\) −2.77846 −0.378100
\(55\) 0 0
\(56\) −28.1104 −3.75641
\(57\) −3.55691 −0.471125
\(58\) −16.6707 −2.18898
\(59\) −2.11727 −0.275645 −0.137822 0.990457i \(-0.544010\pi\)
−0.137822 + 0.990457i \(0.544010\pi\)
\(60\) 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
\(63\) 2.71982 0.342666
\(64\) 41.3871 5.17339
\(65\) 0 0
\(66\) 7.55691 0.930192
\(67\) −1.88273 −0.230013 −0.115006 0.993365i \(-0.536689\pi\)
−0.115006 + 0.993365i \(0.536689\pi\)
\(68\) 16.2277 1.96789
\(69\) 4.83709 0.582317
\(70\) 0 0
\(71\) −6.71982 −0.797496 −0.398748 0.917060i \(-0.630555\pi\)
−0.398748 + 0.917060i \(0.630555\pi\)
\(72\) −10.3354 −1.21804
\(73\) −9.11383 −1.06669 −0.533346 0.845897i \(-0.679066\pi\)
−0.533346 + 0.845897i \(0.679066\pi\)
\(74\) −11.8827 −1.38134
\(75\) 0 0
\(76\) −20.3449 −2.33372
\(77\) −7.39744 −0.843017
\(78\) 2.77846 0.314598
\(79\) 10.2767 1.15622 0.578112 0.815958i \(-0.303790\pi\)
0.578112 + 0.815958i \(0.303790\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −7.88273 −0.870502
\(83\) −2.11727 −0.232400 −0.116200 0.993226i \(-0.537071\pi\)
−0.116200 + 0.993226i \(0.537071\pi\)
\(84\) 15.5569 1.69740
\(85\) 0 0
\(86\) 30.8793 3.32980
\(87\) 6.00000 0.643268
\(88\) 28.1104 2.99658
\(89\) 1.16291 0.123268 0.0616341 0.998099i \(-0.480369\pi\)
0.0616341 + 0.998099i \(0.480369\pi\)
\(90\) 0 0
\(91\) −2.71982 −0.285115
\(92\) 27.6673 2.88452
\(93\) 7.55691 0.783616
\(94\) −32.1104 −3.31193
\(95\) 0 0
\(96\) −27.3319 −2.78955
\(97\) 10.8371 1.10034 0.550170 0.835053i \(-0.314563\pi\)
0.550170 + 0.835053i \(0.314563\pi\)
\(98\) −1.10428 −0.111549
\(99\) −2.71982 −0.273353
\(100\) 0 0
\(101\) 7.67418 0.763610 0.381805 0.924243i \(-0.375303\pi\)
0.381805 + 0.924243i \(0.375303\pi\)
\(102\) −7.88273 −0.780507
\(103\) −3.76547 −0.371023 −0.185511 0.982642i \(-0.559394\pi\)
−0.185511 + 0.982642i \(0.559394\pi\)
\(104\) 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 \(0.291504\pi\)
\(108\) 5.71982 0.550390
\(109\) 11.4396 1.09572 0.547860 0.836570i \(-0.315443\pi\)
0.547860 + 0.836570i \(0.315443\pi\)
\(110\) 0 0
\(111\) 4.27674 0.405930
\(112\) 46.9897 4.44011
\(113\) 13.1138 1.23365 0.616823 0.787102i \(-0.288420\pi\)
0.616823 + 0.787102i \(0.288420\pi\)
\(114\) 9.88273 0.925603
\(115\) 0 0
\(116\) 34.3189 3.18643
\(117\) −1.00000 −0.0924500
\(118\) 5.88273 0.541550
\(119\) 7.71639 0.707360
\(120\) 0 0
\(121\) −3.60256 −0.327505
\(122\) −18.3449 −1.66087
\(123\) 2.83709 0.255812
\(124\) 43.2242 3.88165
\(125\) 0 0
\(126\) −7.55691 −0.673223
\(127\) 13.4396 1.19258 0.596288 0.802771i \(-0.296642\pi\)
0.596288 + 0.802771i \(0.296642\pi\)
\(128\) −60.3285 −5.33234
\(129\) −11.1138 −0.978518
\(130\) 0 0
\(131\) −9.43965 −0.824746 −0.412373 0.911015i \(-0.635300\pi\)
−0.412373 + 0.911015i \(0.635300\pi\)
\(132\) −15.5569 −1.35406
\(133\) −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 \(0.524029\pi\)
\(138\) −13.4396 −1.14406
\(139\) −6.27674 −0.532386 −0.266193 0.963920i \(-0.585766\pi\)
−0.266193 + 0.963920i \(0.585766\pi\)
\(140\) 0 0
\(141\) 11.5569 0.973268
\(142\) 18.6707 1.56681
\(143\) 2.71982 0.227443
\(144\) 17.2767 1.43973
\(145\) 0 0
\(146\) 25.3224 2.09570
\(147\) 0.397442 0.0327805
\(148\) 24.4622 2.01078
\(149\) −20.8302 −1.70648 −0.853239 0.521520i \(-0.825365\pi\)
−0.853239 + 0.521520i \(0.825365\pi\)
\(150\) 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
\(153\) 2.83709 0.229365
\(154\) 20.5535 1.65625
\(155\) 0 0
\(156\) −5.71982 −0.457952
\(157\) −8.87930 −0.708645 −0.354322 0.935123i \(-0.615288\pi\)
−0.354322 + 0.935123i \(0.615288\pi\)
\(158\) −28.5535 −2.27159
\(159\) −1.16291 −0.0922247
\(160\) 0 0
\(161\) 13.1560 1.03684
\(162\) −2.77846 −0.218296
\(163\) 13.8337 1.08354 0.541768 0.840528i \(-0.317755\pi\)
0.541768 + 0.840528i \(0.317755\pi\)
\(164\) 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 \(0.375106\pi\)
\(168\) −28.1104 −2.16876
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −3.55691 −0.272004
\(172\) −63.5691 −4.84710
\(173\) −13.1138 −0.997026 −0.498513 0.866882i \(-0.666120\pi\)
−0.498513 + 0.866882i \(0.666120\pi\)
\(174\) −16.6707 −1.26381
\(175\) 0 0
\(176\) −46.9897 −3.54198
\(177\) −2.11727 −0.159143
\(178\) −3.23109 −0.242181
\(179\) 8.55348 0.639317 0.319658 0.947533i \(-0.396432\pi\)
0.319658 + 0.947533i \(0.396432\pi\)
\(180\) 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
\(183\) 6.60256 0.488075
\(184\) −49.9931 −3.68554
\(185\) 0 0
\(186\) −20.9966 −1.53954
\(187\) −7.71639 −0.564278
\(188\) 66.1035 4.82109
\(189\) 2.71982 0.197838
\(190\) 0 0
\(191\) −4.23453 −0.306400 −0.153200 0.988195i \(-0.548958\pi\)
−0.153200 + 0.988195i \(0.548958\pi\)
\(192\) 41.3871 2.98686
\(193\) 23.3906 1.68369 0.841845 0.539719i \(-0.181470\pi\)
0.841845 + 0.539719i \(0.181470\pi\)
\(194\) −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 \(0.673489\pi\)
\(198\) 7.55691 0.537047
\(199\) −15.1138 −1.07139 −0.535695 0.844411i \(-0.679950\pi\)
−0.535695 + 0.844411i \(0.679950\pi\)
\(200\) 0 0
\(201\) −1.88273 −0.132798
\(202\) −21.3224 −1.50024
\(203\) 16.3189 1.14537
\(204\) 16.2277 1.13616
\(205\) 0 0
\(206\) 10.4622 0.728935
\(207\) 4.83709 0.336201
\(208\) −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
\(213\) −6.71982 −0.460435
\(214\) −35.0157 −2.39362
\(215\) 0 0
\(216\) −10.3354 −0.703233
\(217\) 20.5535 1.39526
\(218\) −31.7846 −2.15272
\(219\) −9.11383 −0.615855
\(220\) 0 0
\(221\) −2.83709 −0.190843
\(222\) −11.8827 −0.797517
\(223\) −10.1173 −0.677502 −0.338751 0.940876i \(-0.610004\pi\)
−0.338751 + 0.940876i \(0.610004\pi\)
\(224\) −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 \(0.622614\pi\)
\(228\) −20.3449 −1.34738
\(229\) −6.23453 −0.411990 −0.205995 0.978553i \(-0.566043\pi\)
−0.205995 + 0.978553i \(0.566043\pi\)
\(230\) 0 0
\(231\) −7.39744 −0.486716
\(232\) −62.0122 −4.07130
\(233\) −6.83709 −0.447913 −0.223956 0.974599i \(-0.571897\pi\)
−0.223956 + 0.974599i \(0.571897\pi\)
\(234\) 2.77846 0.181633
\(235\) 0 0
\(236\) −12.1104 −0.788319
\(237\) 10.2767 0.667546
\(238\) −21.4396 −1.38973
\(239\) 1.28018 0.0828077 0.0414039 0.999142i \(-0.486817\pi\)
0.0414039 + 0.999142i \(0.486817\pi\)
\(240\) 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
\(243\) 1.00000 0.0641500
\(244\) 37.7655 2.41769
\(245\) 0 0
\(246\) −7.88273 −0.502585
\(247\) 3.55691 0.226321
\(248\) −78.1035 −4.95958
\(249\) −2.11727 −0.134176
\(250\) 0 0
\(251\) 18.2277 1.15052 0.575260 0.817971i \(-0.304901\pi\)
0.575260 + 0.817971i \(0.304901\pi\)
\(252\) 15.5569 0.979993
\(253\) −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 \(0.511060\pi\)
\(258\) 30.8793 1.92246
\(259\) 11.6320 0.722776
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 26.2277 1.62035
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 28.1104 1.73007
\(265\) 0 0
\(266\) 26.8793 1.64808
\(267\) 1.16291 0.0711689
\(268\) −10.7689 −0.657816
\(269\) 15.6742 0.955672 0.477836 0.878449i \(-0.341421\pi\)
0.477836 + 0.878449i \(0.341421\pi\)
\(270\) 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
\(273\) −2.71982 −0.164611
\(274\) 4.90528 0.296339
\(275\) 0 0
\(276\) 27.6673 1.66538
\(277\) 4.87930 0.293168 0.146584 0.989198i \(-0.453172\pi\)
0.146584 + 0.989198i \(0.453172\pi\)
\(278\) 17.4396 1.04596
\(279\) 7.55691 0.452421
\(280\) 0 0
\(281\) −9.11383 −0.543685 −0.271843 0.962342i \(-0.587633\pi\)
−0.271843 + 0.962342i \(0.587633\pi\)
\(282\) −32.1104 −1.91215
\(283\) −33.3415 −1.98195 −0.990973 0.134063i \(-0.957198\pi\)
−0.990973 + 0.134063i \(0.957198\pi\)
\(284\) −38.4362 −2.28077
\(285\) 0 0
\(286\) −7.55691 −0.446850
\(287\) 7.71639 0.455484
\(288\) −27.3319 −1.61055
\(289\) −8.95092 −0.526525
\(290\) 0 0
\(291\) 10.8371 0.635281
\(292\) −52.1295 −3.05065
\(293\) 29.4328 1.71948 0.859740 0.510731i \(-0.170625\pi\)
0.859740 + 0.510731i \(0.170625\pi\)
\(294\) −1.10428 −0.0644027
\(295\) 0 0
\(296\) −44.2017 −2.56917
\(297\) −2.71982 −0.157820
\(298\) 57.8759 3.35266
\(299\) −4.83709 −0.279736
\(300\) 0 0
\(301\) −30.2277 −1.74229
\(302\) −13.8827 −0.798862
\(303\) 7.67418 0.440870
\(304\) −61.4519 −3.52451
\(305\) 0 0
\(306\) −7.88273 −0.450626
\(307\) 21.8337 1.24611 0.623056 0.782177i \(-0.285891\pi\)
0.623056 + 0.782177i \(0.285891\pi\)
\(308\) −42.3121 −2.41095
\(309\) −3.76547 −0.214210
\(310\) 0 0
\(311\) 25.1070 1.42368 0.711842 0.702339i \(-0.247861\pi\)
0.711842 + 0.702339i \(0.247861\pi\)
\(312\) 10.3354 0.585125
\(313\) −8.22766 −0.465055 −0.232527 0.972590i \(-0.574700\pi\)
−0.232527 + 0.972590i \(0.574700\pi\)
\(314\) 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 \(0.783345\pi\)
\(318\) 3.23109 0.181191
\(319\) −16.3189 −0.913685
\(320\) 0 0
\(321\) 12.6026 0.703406
\(322\) −36.5535 −2.03705
\(323\) −10.0913 −0.561494
\(324\) 5.71982 0.317768
\(325\) 0 0
\(326\) −38.4362 −2.12878
\(327\) 11.4396 0.632614
\(328\) −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
\(333\) 4.27674 0.234364
\(334\) −27.4588 −1.50248
\(335\) 0 0
\(336\) 46.9897 2.56350
\(337\) 4.32582 0.235642 0.117821 0.993035i \(-0.462409\pi\)
0.117821 + 0.993035i \(0.462409\pi\)
\(338\) −2.77846 −0.151128
\(339\) 13.1138 0.712245
\(340\) 0 0
\(341\) −20.5535 −1.11303
\(342\) 9.88273 0.534397
\(343\) −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 \(0.553885\pi\)
\(348\) 34.3189 1.83969
\(349\) 17.6673 0.945709 0.472855 0.881140i \(-0.343224\pi\)
0.472855 + 0.881140i \(0.343224\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 74.3380 3.96223
\(353\) 13.7655 0.732662 0.366331 0.930485i \(-0.380614\pi\)
0.366331 + 0.930485i \(0.380614\pi\)
\(354\) 5.88273 0.312664
\(355\) 0 0
\(356\) 6.65164 0.352536
\(357\) 7.71639 0.408394
\(358\) −23.7655 −1.25604
\(359\) 0.996562 0.0525965 0.0262983 0.999654i \(-0.491628\pi\)
0.0262983 + 0.999654i \(0.491628\pi\)
\(360\) 0 0
\(361\) −6.34836 −0.334124
\(362\) −10.3449 −0.543717
\(363\) −3.60256 −0.189085
\(364\) −15.5569 −0.815404
\(365\) 0 0
\(366\) −18.3449 −0.958905
\(367\) −14.2277 −0.742678 −0.371339 0.928497i \(-0.621101\pi\)
−0.371339 + 0.928497i \(0.621101\pi\)
\(368\) 83.5691 4.35634
\(369\) 2.83709 0.147693
\(370\) 0 0
\(371\) −3.16291 −0.164210
\(372\) 43.2242 2.24107
\(373\) −15.6742 −0.811578 −0.405789 0.913967i \(-0.633003\pi\)
−0.405789 + 0.913967i \(0.633003\pi\)
\(374\) 21.4396 1.10862
\(375\) 0 0
\(376\) −119.445 −6.15991
\(377\) −6.00000 −0.309016
\(378\) −7.55691 −0.388686
\(379\) 26.2017 1.34589 0.672945 0.739693i \(-0.265029\pi\)
0.672945 + 0.739693i \(0.265029\pi\)
\(380\) 0 0
\(381\) 13.4396 0.688534
\(382\) 11.7655 0.601974
\(383\) −22.4362 −1.14644 −0.573218 0.819403i \(-0.694305\pi\)
−0.573218 + 0.819403i \(0.694305\pi\)
\(384\) −60.3285 −3.07863
\(385\) 0 0
\(386\) −64.9897 −3.30789
\(387\) −11.1138 −0.564948
\(388\) 61.9862 3.14687
\(389\) 31.6742 1.60594 0.802972 0.596016i \(-0.203251\pi\)
0.802972 + 0.596016i \(0.203251\pi\)
\(390\) 0 0
\(391\) 13.7233 0.694015
\(392\) −4.10771 −0.207471
\(393\) −9.43965 −0.476167
\(394\) 40.4362 2.03715
\(395\) 0 0
\(396\) −15.5569 −0.781764
\(397\) −17.9509 −0.900931 −0.450465 0.892794i \(-0.648742\pi\)
−0.450465 + 0.892794i \(0.648742\pi\)
\(398\) 41.9931 2.10493
\(399\) −9.67418 −0.484315
\(400\) 0 0
\(401\) 13.5829 0.678297 0.339149 0.940733i \(-0.389861\pi\)
0.339149 + 0.940733i \(0.389861\pi\)
\(402\) 5.23109 0.260903
\(403\) −7.55691 −0.376437
\(404\) 43.8950 2.18386
\(405\) 0 0
\(406\) −45.3415 −2.25026
\(407\) −11.6320 −0.576576
\(408\) −29.3224 −1.45167
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) −1.76547 −0.0870841
\(412\) −21.5378 −1.06109
\(413\) −5.75859 −0.283362
\(414\) −13.4396 −0.660523
\(415\) 0 0
\(416\) 27.3319 1.34006
\(417\) −6.27674 −0.307373
\(418\) −26.8793 −1.31471
\(419\) 12.3189 0.601820 0.300910 0.953653i \(-0.402710\pi\)
0.300910 + 0.953653i \(0.402710\pi\)
\(420\) 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
\(423\) 11.5569 0.561916
\(424\) 12.0191 0.583699
\(425\) 0 0
\(426\) 18.6707 0.904600
\(427\) 17.9578 0.869039
\(428\) 72.0844 3.48433
\(429\) 2.71982 0.131314
\(430\) 0 0
\(431\) −8.99656 −0.433349 −0.216675 0.976244i \(-0.569521\pi\)
−0.216675 + 0.976244i \(0.569521\pi\)
\(432\) 17.2767 0.831227
\(433\) 20.3258 0.976797 0.488398 0.872621i \(-0.337581\pi\)
0.488398 + 0.872621i \(0.337581\pi\)
\(434\) −57.1070 −2.74122
\(435\) 0 0
\(436\) 65.4328 3.13366
\(437\) −17.2051 −0.823032
\(438\) 25.3224 1.20995
\(439\) −25.3906 −1.21183 −0.605913 0.795531i \(-0.707192\pi\)
−0.605913 + 0.795531i \(0.707192\pi\)
\(440\) 0 0
\(441\) 0.397442 0.0189258
\(442\) 7.88273 0.374943
\(443\) −10.9284 −0.519223 −0.259611 0.965713i \(-0.583594\pi\)
−0.259611 + 0.965713i \(0.583594\pi\)
\(444\) 24.4622 1.16092
\(445\) 0 0
\(446\) 28.1104 1.33107
\(447\) −20.8302 −0.985235
\(448\) 112.566 5.31823
\(449\) 2.83709 0.133891 0.0669453 0.997757i \(-0.478675\pi\)
0.0669453 + 0.997757i \(0.478675\pi\)
\(450\) 0 0
\(451\) −7.71639 −0.363350
\(452\) 75.0088 3.52812
\(453\) 4.99656 0.234759
\(454\) 31.4588 1.47643
\(455\) 0 0
\(456\) 36.7620 1.72154
\(457\) 13.7164 0.641625 0.320813 0.947143i \(-0.396044\pi\)
0.320813 + 0.947143i \(0.396044\pi\)
\(458\) 17.3224 0.809422
\(459\) 2.83709 0.132424
\(460\) 0 0
\(461\) −19.6251 −0.914032 −0.457016 0.889458i \(-0.651082\pi\)
−0.457016 + 0.889458i \(0.651082\pi\)
\(462\) 20.5535 0.956234
\(463\) −27.0388 −1.25660 −0.628299 0.777972i \(-0.716249\pi\)
−0.628299 + 0.777972i \(0.716249\pi\)
\(464\) 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 \(0.733348\pi\)
\(468\) −5.71982 −0.264399
\(469\) −5.12070 −0.236452
\(470\) 0 0
\(471\) −8.87930 −0.409136
\(472\) 21.8827 1.00723
\(473\) 30.2277 1.38987
\(474\) −28.5535 −1.31150
\(475\) 0 0
\(476\) 44.1364 2.02299
\(477\) −1.16291 −0.0532460
\(478\) −3.55691 −0.162689
\(479\) −12.1595 −0.555580 −0.277790 0.960642i \(-0.589602\pi\)
−0.277790 + 0.960642i \(0.589602\pi\)
\(480\) 0 0
\(481\) −4.27674 −0.195002
\(482\) 16.6707 0.759332
\(483\) 13.1560 0.598620
\(484\) −20.6060 −0.936636
\(485\) 0 0
\(486\) −2.77846 −0.126033
\(487\) 0.159472 0.00722636 0.00361318 0.999993i \(-0.498850\pi\)
0.00361318 + 0.999993i \(0.498850\pi\)
\(488\) −68.2399 −3.08907
\(489\) 13.8337 0.625579
\(490\) 0 0
\(491\) −42.2277 −1.90571 −0.952854 0.303430i \(-0.901868\pi\)
−0.952854 + 0.303430i \(0.901868\pi\)
\(492\) 16.2277 0.731599
\(493\) 17.0225 0.766657
\(494\) −9.88273 −0.444645
\(495\) 0 0
\(496\) 130.559 5.86226
\(497\) −18.2767 −0.819824
\(498\) 5.88273 0.263612
\(499\) 7.79145 0.348793 0.174397 0.984676i \(-0.444203\pi\)
0.174397 + 0.984676i \(0.444203\pi\)
\(500\) 0 0
\(501\) 9.88273 0.441528
\(502\) −50.6448 −2.26039
\(503\) −27.3484 −1.21940 −0.609702 0.792631i \(-0.708711\pi\)
−0.609702 + 0.792631i \(0.708711\pi\)
\(504\) −28.1104 −1.25214
\(505\) 0 0
\(506\) 36.5535 1.62500
\(507\) 1.00000 0.0444116
\(508\) 76.8724 3.41066
\(509\) −33.4819 −1.48406 −0.742029 0.670368i \(-0.766136\pi\)
−0.742029 + 0.670368i \(0.766136\pi\)
\(510\) 0 0
\(511\) −24.7880 −1.09656
\(512\) −115.084 −5.08603
\(513\) −3.55691 −0.157042
\(514\) 3.09472 0.136502
\(515\) 0 0
\(516\) −63.5691 −2.79848
\(517\) −31.4328 −1.38241
\(518\) −32.3189 −1.42001
\(519\) −13.1138 −0.575633
\(520\) 0 0
\(521\) −17.3484 −0.760045 −0.380023 0.924977i \(-0.624084\pi\)
−0.380023 + 0.924977i \(0.624084\pi\)
\(522\) −16.6707 −0.729659
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) −53.9931 −2.35870
\(525\) 0 0
\(526\) 22.2277 0.969172
\(527\) 21.4396 0.933926
\(528\) −46.9897 −2.04496
\(529\) 0.397442 0.0172801
\(530\) 0 0
\(531\) −2.11727 −0.0918815
\(532\) −55.3346 −2.39906
\(533\) −2.83709 −0.122888
\(534\) −3.23109 −0.139823
\(535\) 0 0
\(536\) 19.4588 0.840490
\(537\) 8.55348 0.369110
\(538\) −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
\(543\) 3.72326 0.159780
\(544\) −77.5432 −3.32464
\(545\) 0 0
\(546\) 7.55691 0.323406
\(547\) −34.2277 −1.46347 −0.731734 0.681590i \(-0.761289\pi\)
−0.731734 + 0.681590i \(0.761289\pi\)
\(548\) −10.0982 −0.431372
\(549\) 6.60256 0.281790
\(550\) 0 0
\(551\) −21.3415 −0.909178
\(552\) −49.9931 −2.12785
\(553\) 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 \(0.545006\pi\)
\(558\) −20.9966 −0.888855
\(559\) 11.1138 0.470065
\(560\) 0 0
\(561\) −7.71639 −0.325786
\(562\) 25.3224 1.06816
\(563\) −40.2699 −1.69717 −0.848586 0.529057i \(-0.822546\pi\)
−0.848586 + 0.529057i \(0.822546\pi\)
\(564\) 66.1035 2.78346
\(565\) 0 0
\(566\) 92.6379 3.89386
\(567\) 2.71982 0.114222
\(568\) 69.4519 2.91414
\(569\) −13.4328 −0.563131 −0.281566 0.959542i \(-0.590854\pi\)
−0.281566 + 0.959542i \(0.590854\pi\)
\(570\) 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
\(573\) −4.23453 −0.176900
\(574\) −21.4396 −0.894874
\(575\) 0 0
\(576\) 41.3871 1.72446
\(577\) 13.7164 0.571021 0.285510 0.958376i \(-0.407837\pi\)
0.285510 + 0.958376i \(0.407837\pi\)
\(578\) 24.8697 1.03444
\(579\) 23.3906 0.972079
\(580\) 0 0
\(581\) −5.75859 −0.238907
\(582\) −30.1104 −1.24812
\(583\) 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 \(0.718160\pi\)
\(588\) 2.27330 0.0937493
\(589\) −26.8793 −1.10754
\(590\) 0 0
\(591\) −14.5535 −0.598650
\(592\) 73.8881 3.03678
\(593\) 45.6673 1.87533 0.937666 0.347538i \(-0.112982\pi\)
0.937666 + 0.347538i \(0.112982\pi\)
\(594\) 7.55691 0.310064
\(595\) 0 0
\(596\) −119.145 −4.88038
\(597\) −15.1138 −0.618568
\(598\) 13.4396 0.549588
\(599\) −40.2208 −1.64338 −0.821688 0.569937i \(-0.806968\pi\)
−0.821688 + 0.569937i \(0.806968\pi\)
\(600\) 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
\(603\) −1.88273 −0.0766708
\(604\) 28.5795 1.16288
\(605\) 0 0
\(606\) −21.3224 −0.866163
\(607\) 14.2277 0.577483 0.288741 0.957407i \(-0.406763\pi\)
0.288741 + 0.957407i \(0.406763\pi\)
\(608\) 97.2173 3.94268
\(609\) 16.3189 0.661277
\(610\) 0 0
\(611\) −11.5569 −0.467543
\(612\) 16.2277 0.655964
\(613\) 40.8302 1.64912 0.824558 0.565777i \(-0.191424\pi\)
0.824558 + 0.565777i \(0.191424\pi\)
\(614\) −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 \(0.532824\pi\)
\(618\) 10.4622 0.420851
\(619\) 11.5569 0.464512 0.232256 0.972655i \(-0.425389\pi\)
0.232256 + 0.972655i \(0.425389\pi\)
\(620\) 0 0
\(621\) 4.83709 0.194106
\(622\) −69.7586 −2.79706
\(623\) 3.16291 0.126719
\(624\) −17.2767 −0.691623
\(625\) 0 0
\(626\) 22.8602 0.913677
\(627\) 9.67418 0.386350
\(628\) −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
\(633\) −18.2277 −0.724484
\(634\) 76.8915 3.05375
\(635\) 0 0
\(636\) −6.65164 −0.263755
\(637\) −0.397442 −0.0157472
\(638\) 45.3415 1.79509
\(639\) −6.71982 −0.265832
\(640\) 0 0
\(641\) −21.9018 −0.865071 −0.432535 0.901617i \(-0.642381\pi\)
−0.432535 + 0.901617i \(0.642381\pi\)
\(642\) −35.0157 −1.38196
\(643\) −7.50783 −0.296080 −0.148040 0.988981i \(-0.547296\pi\)
−0.148040 + 0.988981i \(0.547296\pi\)
\(644\) 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 \(0.410982\pi\)
\(648\) −10.3354 −0.406012
\(649\) 5.75859 0.226044
\(650\) 0 0
\(651\) 20.5535 0.805554
\(652\) 79.1261 3.09882
\(653\) −7.99312 −0.312795 −0.156398 0.987694i \(-0.549988\pi\)
−0.156398 + 0.987694i \(0.549988\pi\)
\(654\) −31.7846 −1.24288
\(655\) 0 0
\(656\) 49.0157 1.91374
\(657\) −9.11383 −0.355564
\(658\) −87.3346 −3.40466
\(659\) −25.3415 −0.987164 −0.493582 0.869699i \(-0.664313\pi\)
−0.493582 + 0.869699i \(0.664313\pi\)
\(660\) 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
\(663\) −2.83709 −0.110183
\(664\) 21.8827 0.849215
\(665\) 0 0
\(666\) −11.8827 −0.460447
\(667\) 29.0225 1.12376
\(668\) 56.5275 2.18711
\(669\) −10.1173 −0.391156
\(670\) 0 0
\(671\) −17.9578 −0.693253
\(672\) −74.3380 −2.86765
\(673\) −27.1070 −1.04490 −0.522448 0.852671i \(-0.674981\pi\)
−0.522448 + 0.852671i \(0.674981\pi\)
\(674\) −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 \(0.251735\pi\)
\(678\) −36.4362 −1.39932
\(679\) 29.4750 1.13115
\(680\) 0 0
\(681\) −11.3224 −0.433875
\(682\) 57.1070 2.18674
\(683\) 13.4656 0.515248 0.257624 0.966245i \(-0.417060\pi\)
0.257624 + 0.966245i \(0.417060\pi\)
\(684\) −20.3449 −0.777908
\(685\) 0 0
\(686\) 49.8950 1.90500
\(687\) −6.23453 −0.237862
\(688\) −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
\(693\) −7.39744 −0.281006
\(694\) 17.4396 0.662000
\(695\) 0 0
\(696\) −62.0122 −2.35057
\(697\) 8.04908 0.304881
\(698\) −49.0878 −1.85800
\(699\) −6.83709 −0.258603
\(700\) 0 0
\(701\) 43.6604 1.64903 0.824516 0.565839i \(-0.191448\pi\)
0.824516 + 0.565839i \(0.191448\pi\)
\(702\) 2.77846 0.104866
\(703\) −15.2120 −0.573731
\(704\) −112.566 −4.24248
\(705\) 0 0
\(706\) −38.2468 −1.43944
\(707\) 20.8724 0.784988
\(708\) −12.1104 −0.455136
\(709\) −26.7880 −1.00604 −0.503022 0.864273i \(-0.667779\pi\)
−0.503022 + 0.864273i \(0.667779\pi\)
\(710\) 0 0
\(711\) 10.2767 0.385408
\(712\) −12.0191 −0.450435
\(713\) 36.5535 1.36894
\(714\) −21.4396 −0.802359
\(715\) 0 0
\(716\) 48.9244 1.82839
\(717\) 1.28018 0.0478091
\(718\) −2.76891 −0.103335
\(719\) −34.8793 −1.30078 −0.650389 0.759601i \(-0.725394\pi\)
−0.650389 + 0.759601i \(0.725394\pi\)
\(720\) 0 0
\(721\) −10.2414 −0.381410
\(722\) 17.6386 0.656443
\(723\) −6.00000 −0.223142
\(724\) 21.2964 0.791475
\(725\) 0 0
\(726\) 10.0096 0.371489
\(727\) −37.4396 −1.38856 −0.694280 0.719705i \(-0.744277\pi\)
−0.694280 + 0.719705i \(0.744277\pi\)
\(728\) 28.1104 1.04184
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −31.5309 −1.16621
\(732\) 37.7655 1.39585
\(733\) 47.1560 1.74175 0.870874 0.491506i \(-0.163554\pi\)
0.870874 + 0.491506i \(0.163554\pi\)
\(734\) 39.5309 1.45911
\(735\) 0 0
\(736\) −132.207 −4.87322
\(737\) 5.12070 0.188624
\(738\) −7.88273 −0.290167
\(739\) −31.7914 −1.16947 −0.584734 0.811225i \(-0.698801\pi\)
−0.584734 + 0.811225i \(0.698801\pi\)
\(740\) 0 0
\(741\) 3.55691 0.130667
\(742\) 8.78801 0.322618
\(743\) −16.6776 −0.611842 −0.305921 0.952057i \(-0.598964\pi\)
−0.305921 + 0.952057i \(0.598964\pi\)
\(744\) −78.1035 −2.86341
\(745\) 0 0
\(746\) 43.5500 1.59448
\(747\) −2.11727 −0.0774667
\(748\) −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
\(753\) 18.2277 0.664253
\(754\) 16.6707 0.607113
\(755\) 0 0
\(756\) 15.5569 0.565800
\(757\) −12.3258 −0.447990 −0.223995 0.974590i \(-0.571910\pi\)
−0.223995 + 0.974590i \(0.571910\pi\)
\(758\) −72.8002 −2.64422
\(759\) −13.1560 −0.477534
\(760\) 0 0
\(761\) −0.00687569 −0.000249244 0 −0.000124622 1.00000i \(-0.500040\pi\)
−0.000124622 1.00000i \(0.500040\pi\)
\(762\) −37.3415 −1.35274
\(763\) 31.1138 1.12640
\(764\) −24.2208 −0.876277
\(765\) 0 0
\(766\) 62.3380 2.25237
\(767\) 2.11727 0.0764501
\(768\) 84.8459 3.06161
\(769\) −20.3258 −0.732968 −0.366484 0.930424i \(-0.619439\pi\)
−0.366484 + 0.930424i \(0.619439\pi\)
\(770\) 0 0
\(771\) −1.11383 −0.0401136
\(772\) 133.790 4.81520
\(773\) −9.90184 −0.356144 −0.178072 0.984017i \(-0.556986\pi\)
−0.178072 + 0.984017i \(0.556986\pi\)
\(774\) 30.8793 1.10993
\(775\) 0 0
\(776\) −112.005 −4.02076
\(777\) 11.6320 0.417295
\(778\) −88.0054 −3.15514
\(779\) −10.0913 −0.361558
\(780\) 0 0
\(781\) 18.2767 0.653993
\(782\) −38.1295 −1.36351
\(783\) 6.00000 0.214423
\(784\) 6.86651 0.245232
\(785\) 0 0
\(786\) 26.2277 0.935510
\(787\) −36.3449 −1.29556 −0.647778 0.761829i \(-0.724302\pi\)
−0.647778 + 0.761829i \(0.724302\pi\)
\(788\) −83.2433 −2.96542
\(789\) −8.00000 −0.284808
\(790\) 0 0
\(791\) 35.6673 1.26818
\(792\) 28.1104 0.998859
\(793\) −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 \(0.608271\pi\)
\(798\) 26.8793 0.951517
\(799\) 32.7880 1.15996
\(800\) 0 0
\(801\) 1.16291 0.0410894
\(802\) −37.7395 −1.33263
\(803\) 24.7880 0.874750
\(804\) −10.7689 −0.379790
\(805\) 0 0
\(806\) 20.9966 0.739572
\(807\) 15.6742 0.551757
\(808\) −79.3155 −2.79031
\(809\) 32.2277 1.13306 0.566532 0.824040i \(-0.308285\pi\)
0.566532 + 0.824040i \(0.308285\pi\)
\(810\) 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
\(813\) 0.443086 0.0155397
\(814\) 32.3189 1.13278
\(815\) 0 0
\(816\) 49.0157 1.71589
\(817\) 39.5309 1.38301
\(818\) 38.8984 1.36005
\(819\) −2.71982 −0.0950383
\(820\) 0 0
\(821\) 49.9372 1.74282 0.871410 0.490556i \(-0.163206\pi\)
0.871410 + 0.490556i \(0.163206\pi\)
\(822\) 4.90528 0.171091
\(823\) −28.2345 −0.984194 −0.492097 0.870540i \(-0.663769\pi\)
−0.492097 + 0.870540i \(0.663769\pi\)
\(824\) 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 \(0.446901\pi\)
\(828\) 27.6673 0.961505
\(829\) −37.9862 −1.31932 −0.659658 0.751565i \(-0.729299\pi\)
−0.659658 + 0.751565i \(0.729299\pi\)
\(830\) 0 0
\(831\) 4.87930 0.169261
\(832\) −41.3871 −1.43484
\(833\) 1.12758 0.0390683
\(834\) 17.4396 0.603886
\(835\) 0 0
\(836\) 55.3346 1.91379
\(837\) 7.55691 0.261205
\(838\) −34.2277 −1.18237
\(839\) −4.72670 −0.163184 −0.0815919 0.996666i \(-0.526000\pi\)
−0.0815919 + 0.996666i \(0.526000\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −63.3155 −2.18200
\(843\) −9.11383 −0.313897
\(844\) −104.259 −3.58874
\(845\) 0 0
\(846\) −32.1104 −1.10398
\(847\) −9.79832 −0.336674
\(848\) −20.0913 −0.689938
\(849\) −33.3415 −1.14428
\(850\) 0 0
\(851\) 20.6870 0.709140
\(852\) −38.4362 −1.31680
\(853\) 48.3611 1.65585 0.827927 0.560836i \(-0.189520\pi\)
0.827927 + 0.560836i \(0.189520\pi\)
\(854\) −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 \(0.537256\pi\)
\(858\) −7.55691 −0.257989
\(859\) 12.6026 0.429994 0.214997 0.976615i \(-0.431026\pi\)
0.214997 + 0.976615i \(0.431026\pi\)
\(860\) 0 0
\(861\) 7.71639 0.262974
\(862\) 24.9966 0.851386
\(863\) −8.20855 −0.279422 −0.139711 0.990192i \(-0.544617\pi\)
−0.139711 + 0.990192i \(0.544617\pi\)
\(864\) −27.3319 −0.929851
\(865\) 0 0
\(866\) −56.4744 −1.91908
\(867\) −8.95092 −0.303989
\(868\) 117.562 3.99032
\(869\) −27.9509 −0.948170
\(870\) 0 0
\(871\) 1.88273 0.0637940
\(872\) −118.233 −4.00387
\(873\) 10.8371 0.366780
\(874\) 47.8037 1.61698
\(875\) 0 0
\(876\) −52.1295 −1.76129
\(877\) −13.5309 −0.456907 −0.228454 0.973555i \(-0.573367\pi\)
−0.228454 + 0.973555i \(0.573367\pi\)
\(878\) 70.5466 2.38083
\(879\) 29.4328 0.992743
\(880\) 0 0
\(881\) −9.34836 −0.314954 −0.157477 0.987523i \(-0.550336\pi\)
−0.157477 + 0.987523i \(0.550336\pi\)
\(882\) −1.10428 −0.0371829
\(883\) 55.1001 1.85427 0.927133 0.374733i \(-0.122266\pi\)
0.927133 + 0.374733i \(0.122266\pi\)
\(884\) −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 \(0.500713\pi\)
\(888\) −44.2017 −1.48331
\(889\) 36.5535 1.22596
\(890\) 0 0
\(891\) −2.71982 −0.0911175
\(892\) −57.8690 −1.93760
\(893\) −41.1070 −1.37559
\(894\) 57.8759 1.93566
\(895\) 0 0
\(896\) −164.083 −5.48162
\(897\) −4.83709 −0.161506
\(898\) −7.88273 −0.263050
\(899\) 45.3415 1.51222
\(900\) 0 0
\(901\) −3.29928 −0.109915
\(902\) 21.4396 0.713862
\(903\) −30.2277 −1.00591
\(904\) −135.536 −4.50787
\(905\) 0 0
\(906\) −13.8827 −0.461223
\(907\) 58.5466 1.94401 0.972004 0.234964i \(-0.0754973\pi\)
0.972004 + 0.234964i \(0.0754973\pi\)
\(908\) −64.7620 −2.14920
\(909\) 7.67418 0.254537
\(910\) 0 0
\(911\) 50.4622 1.67189 0.835943 0.548816i \(-0.184921\pi\)
0.835943 + 0.548816i \(0.184921\pi\)
\(912\) −61.4519 −2.03487
\(913\) 5.75859 0.190582
\(914\) −38.1104 −1.26058
\(915\) 0 0
\(916\) −35.6604 −1.17825
\(917\) −25.6742 −0.847836
\(918\) −7.88273 −0.260169
\(919\) 56.9735 1.87938 0.939691 0.342026i \(-0.111113\pi\)
0.939691 + 0.342026i \(0.111113\pi\)
\(920\) 0 0
\(921\) 21.8337 0.719443
\(922\) 54.5275 1.79577
\(923\) 6.71982 0.221186
\(924\) −42.3121 −1.39196
\(925\) 0 0
\(926\) 75.1261 2.46880
\(927\) −3.76547 −0.123674
\(928\) −163.992 −5.38329
\(929\) −36.5957 −1.20067 −0.600333 0.799750i \(-0.704965\pi\)
−0.600333 + 0.799750i \(0.704965\pi\)
\(930\) 0 0
\(931\) −1.41367 −0.0463311
\(932\) −39.1070 −1.28099
\(933\) 25.1070 0.821965
\(934\) 80.3572 2.62937
\(935\) 0 0
\(936\) 10.3354 0.337822
\(937\) 47.1070 1.53892 0.769459 0.638697i \(-0.220526\pi\)
0.769459 + 0.638697i \(0.220526\pi\)
\(938\) 14.2277 0.464549
\(939\) −8.22766 −0.268499
\(940\) 0 0
\(941\) −36.3611 −1.18534 −0.592670 0.805446i \(-0.701926\pi\)
−0.592670 + 0.805446i \(0.701926\pi\)
\(942\) 24.6707 0.803816
\(943\) 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 \(0.757846\pi\)
\(948\) 58.7811 1.90912
\(949\) 9.11383 0.295847
\(950\) 0 0
\(951\) −27.6742 −0.897397
\(952\) −79.7517 −2.58477
\(953\) −19.8596 −0.643317 −0.321658 0.946856i \(-0.604240\pi\)
−0.321658 + 0.946856i \(0.604240\pi\)
\(954\) 3.23109 0.104611
\(955\) 0 0
\(956\) 7.32238 0.236823
\(957\) −16.3189 −0.527517
\(958\) 33.7846 1.09153
\(959\) −4.80176 −0.155057
\(960\) 0 0
\(961\) 26.1070 0.842160
\(962\) 11.8827 0.383115
\(963\) 12.6026 0.406112
\(964\) −34.3189 −1.10534
\(965\) 0 0
\(966\) −36.5535 −1.17609
\(967\) −47.4068 −1.52450 −0.762250 0.647283i \(-0.775905\pi\)
−0.762250 + 0.647283i \(0.775905\pi\)
\(968\) 37.2338 1.19674
\(969\) −10.0913 −0.324179
\(970\) 0 0
\(971\) −10.6448 −0.341607 −0.170803 0.985305i \(-0.554636\pi\)
−0.170803 + 0.985305i \(0.554636\pi\)
\(972\) 5.71982 0.183463
\(973\) −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 \(0.493828\pi\)
\(978\) −38.4362 −1.22905
\(979\) −3.16291 −0.101087
\(980\) 0 0
\(981\) 11.4396 0.365240
\(982\) 117.328 3.74408
\(983\) −51.8759 −1.65458 −0.827291 0.561773i \(-0.810119\pi\)
−0.827291 + 0.561773i \(0.810119\pi\)
\(984\) −29.3224 −0.934763
\(985\) 0 0
\(986\) −47.2964 −1.50622
\(987\) 31.4328 1.00052
\(988\) 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
\(993\) −13.2311 −0.419876
\(994\) 50.7811 1.61068
\(995\) 0 0
\(996\) −12.1104 −0.383732
\(997\) −23.2051 −0.734913 −0.367457 0.930041i \(-0.619771\pi\)
−0.367457 + 0.930041i \(0.619771\pi\)
\(998\) −21.6482 −0.685262
\(999\) 4.27674 0.135310
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 975.2.a.o.1.1 3
3.2 odd 2 2925.2.a.bh.1.3 3
5.2 odd 4 975.2.c.i.274.1 6
5.3 odd 4 975.2.c.i.274.6 6
5.4 even 2 195.2.a.e.1.3 3
15.2 even 4 2925.2.c.w.2224.6 6
15.8 even 4 2925.2.c.w.2224.1 6
15.14 odd 2 585.2.a.n.1.1 3
20.19 odd 2 3120.2.a.bj.1.3 3
35.34 odd 2 9555.2.a.bq.1.3 3
60.59 even 2 9360.2.a.dd.1.3 3
65.64 even 2 2535.2.a.bc.1.1 3
195.194 odd 2 7605.2.a.bx.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.e.1.3 3 5.4 even 2
585.2.a.n.1.1 3 15.14 odd 2
975.2.a.o.1.1 3 1.1 even 1 trivial
975.2.c.i.274.1 6 5.2 odd 4
975.2.c.i.274.6 6 5.3 odd 4
2535.2.a.bc.1.1 3 65.64 even 2
2925.2.a.bh.1.3 3 3.2 odd 2
2925.2.c.w.2224.1 6 15.8 even 4
2925.2.c.w.2224.6 6 15.2 even 4
3120.2.a.bj.1.3 3 20.19 odd 2
7605.2.a.bx.1.3 3 195.194 odd 2
9360.2.a.dd.1.3 3 60.59 even 2
9555.2.a.bq.1.3 3 35.34 odd 2