Properties

Label 4598.2.a.bc.1.1
Level $4598$
Weight $2$
Character 4598.1
Self dual yes
Analytic conductor $36.715$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7152148494\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 4598.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.30278 q^{3} +1.00000 q^{4} +1.30278 q^{5} -3.30278 q^{6} +2.30278 q^{7} +1.00000 q^{8} +7.90833 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -3.30278 q^{3} +1.00000 q^{4} +1.30278 q^{5} -3.30278 q^{6} +2.30278 q^{7} +1.00000 q^{8} +7.90833 q^{9} +1.30278 q^{10} -3.30278 q^{12} -0.302776 q^{13} +2.30278 q^{14} -4.30278 q^{15} +1.00000 q^{16} -2.60555 q^{17} +7.90833 q^{18} -1.00000 q^{19} +1.30278 q^{20} -7.60555 q^{21} -8.60555 q^{23} -3.30278 q^{24} -3.30278 q^{25} -0.302776 q^{26} -16.2111 q^{27} +2.30278 q^{28} +4.69722 q^{29} -4.30278 q^{30} +0.302776 q^{31} +1.00000 q^{32} -2.60555 q^{34} +3.00000 q^{35} +7.90833 q^{36} -9.21110 q^{37} -1.00000 q^{38} +1.00000 q^{39} +1.30278 q^{40} +6.90833 q^{41} -7.60555 q^{42} -11.9083 q^{43} +10.3028 q^{45} -8.60555 q^{46} -6.00000 q^{47} -3.30278 q^{48} -1.69722 q^{49} -3.30278 q^{50} +8.60555 q^{51} -0.302776 q^{52} -3.39445 q^{53} -16.2111 q^{54} +2.30278 q^{56} +3.30278 q^{57} +4.69722 q^{58} -4.30278 q^{60} +3.21110 q^{61} +0.302776 q^{62} +18.2111 q^{63} +1.00000 q^{64} -0.394449 q^{65} +11.5139 q^{67} -2.60555 q^{68} +28.4222 q^{69} +3.00000 q^{70} -13.3028 q^{71} +7.90833 q^{72} -4.60555 q^{73} -9.21110 q^{74} +10.9083 q^{75} -1.00000 q^{76} +1.00000 q^{78} +5.81665 q^{79} +1.30278 q^{80} +29.8167 q^{81} +6.90833 q^{82} -10.6972 q^{83} -7.60555 q^{84} -3.39445 q^{85} -11.9083 q^{86} -15.5139 q^{87} -2.60555 q^{89} +10.3028 q^{90} -0.697224 q^{91} -8.60555 q^{92} -1.00000 q^{93} -6.00000 q^{94} -1.30278 q^{95} -3.30278 q^{96} +8.00000 q^{97} -1.69722 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - q^{5} - 3 q^{6} + q^{7} + 2 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - q^{5} - 3 q^{6} + q^{7} + 2 q^{8} + 5 q^{9} - q^{10} - 3 q^{12} + 3 q^{13} + q^{14} - 5 q^{15} + 2 q^{16} + 2 q^{17} + 5 q^{18} - 2 q^{19} - q^{20} - 8 q^{21} - 10 q^{23} - 3 q^{24} - 3 q^{25} + 3 q^{26} - 18 q^{27} + q^{28} + 13 q^{29} - 5 q^{30} - 3 q^{31} + 2 q^{32} + 2 q^{34} + 6 q^{35} + 5 q^{36} - 4 q^{37} - 2 q^{38} + 2 q^{39} - q^{40} + 3 q^{41} - 8 q^{42} - 13 q^{43} + 17 q^{45} - 10 q^{46} - 12 q^{47} - 3 q^{48} - 7 q^{49} - 3 q^{50} + 10 q^{51} + 3 q^{52} - 14 q^{53} - 18 q^{54} + q^{56} + 3 q^{57} + 13 q^{58} - 5 q^{60} - 8 q^{61} - 3 q^{62} + 22 q^{63} + 2 q^{64} - 8 q^{65} + 5 q^{67} + 2 q^{68} + 28 q^{69} + 6 q^{70} - 23 q^{71} + 5 q^{72} - 2 q^{73} - 4 q^{74} + 11 q^{75} - 2 q^{76} + 2 q^{78} - 10 q^{79} - q^{80} + 38 q^{81} + 3 q^{82} - 25 q^{83} - 8 q^{84} - 14 q^{85} - 13 q^{86} - 13 q^{87} + 2 q^{89} + 17 q^{90} - 5 q^{91} - 10 q^{92} - 2 q^{93} - 12 q^{94} + q^{95} - 3 q^{96} + 16 q^{97} - 7 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −3.30278 −1.90686 −0.953429 0.301617i \(-0.902474\pi\)
−0.953429 + 0.301617i \(0.902474\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.30278 0.582619 0.291309 0.956629i \(-0.405909\pi\)
0.291309 + 0.956629i \(0.405909\pi\)
\(6\) −3.30278 −1.34835
\(7\) 2.30278 0.870367 0.435184 0.900342i \(-0.356683\pi\)
0.435184 + 0.900342i \(0.356683\pi\)
\(8\) 1.00000 0.353553
\(9\) 7.90833 2.63611
\(10\) 1.30278 0.411974
\(11\) 0 0
\(12\) −3.30278 −0.953429
\(13\) −0.302776 −0.0839749 −0.0419874 0.999118i \(-0.513369\pi\)
−0.0419874 + 0.999118i \(0.513369\pi\)
\(14\) 2.30278 0.615443
\(15\) −4.30278 −1.11097
\(16\) 1.00000 0.250000
\(17\) −2.60555 −0.631939 −0.315970 0.948769i \(-0.602330\pi\)
−0.315970 + 0.948769i \(0.602330\pi\)
\(18\) 7.90833 1.86401
\(19\) −1.00000 −0.229416
\(20\) 1.30278 0.291309
\(21\) −7.60555 −1.65967
\(22\) 0 0
\(23\) −8.60555 −1.79438 −0.897191 0.441643i \(-0.854396\pi\)
−0.897191 + 0.441643i \(0.854396\pi\)
\(24\) −3.30278 −0.674176
\(25\) −3.30278 −0.660555
\(26\) −0.302776 −0.0593792
\(27\) −16.2111 −3.11983
\(28\) 2.30278 0.435184
\(29\) 4.69722 0.872253 0.436126 0.899885i \(-0.356350\pi\)
0.436126 + 0.899885i \(0.356350\pi\)
\(30\) −4.30278 −0.785576
\(31\) 0.302776 0.0543801 0.0271901 0.999630i \(-0.491344\pi\)
0.0271901 + 0.999630i \(0.491344\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −2.60555 −0.446848
\(35\) 3.00000 0.507093
\(36\) 7.90833 1.31805
\(37\) −9.21110 −1.51430 −0.757148 0.653243i \(-0.773408\pi\)
−0.757148 + 0.653243i \(0.773408\pi\)
\(38\) −1.00000 −0.162221
\(39\) 1.00000 0.160128
\(40\) 1.30278 0.205987
\(41\) 6.90833 1.07890 0.539450 0.842018i \(-0.318632\pi\)
0.539450 + 0.842018i \(0.318632\pi\)
\(42\) −7.60555 −1.17356
\(43\) −11.9083 −1.81600 −0.908001 0.418967i \(-0.862392\pi\)
−0.908001 + 0.418967i \(0.862392\pi\)
\(44\) 0 0
\(45\) 10.3028 1.53585
\(46\) −8.60555 −1.26882
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) −3.30278 −0.476715
\(49\) −1.69722 −0.242461
\(50\) −3.30278 −0.467083
\(51\) 8.60555 1.20502
\(52\) −0.302776 −0.0419874
\(53\) −3.39445 −0.466263 −0.233132 0.972445i \(-0.574897\pi\)
−0.233132 + 0.972445i \(0.574897\pi\)
\(54\) −16.2111 −2.20605
\(55\) 0 0
\(56\) 2.30278 0.307721
\(57\) 3.30278 0.437463
\(58\) 4.69722 0.616776
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −4.30278 −0.555486
\(61\) 3.21110 0.411140 0.205570 0.978642i \(-0.434095\pi\)
0.205570 + 0.978642i \(0.434095\pi\)
\(62\) 0.302776 0.0384525
\(63\) 18.2111 2.29438
\(64\) 1.00000 0.125000
\(65\) −0.394449 −0.0489253
\(66\) 0 0
\(67\) 11.5139 1.40664 0.703322 0.710871i \(-0.251699\pi\)
0.703322 + 0.710871i \(0.251699\pi\)
\(68\) −2.60555 −0.315970
\(69\) 28.4222 3.42163
\(70\) 3.00000 0.358569
\(71\) −13.3028 −1.57875 −0.789375 0.613912i \(-0.789595\pi\)
−0.789375 + 0.613912i \(0.789595\pi\)
\(72\) 7.90833 0.932005
\(73\) −4.60555 −0.539039 −0.269520 0.962995i \(-0.586865\pi\)
−0.269520 + 0.962995i \(0.586865\pi\)
\(74\) −9.21110 −1.07077
\(75\) 10.9083 1.25959
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) 5.81665 0.654425 0.327212 0.944951i \(-0.393891\pi\)
0.327212 + 0.944951i \(0.393891\pi\)
\(80\) 1.30278 0.145655
\(81\) 29.8167 3.31296
\(82\) 6.90833 0.762897
\(83\) −10.6972 −1.17417 −0.587086 0.809524i \(-0.699725\pi\)
−0.587086 + 0.809524i \(0.699725\pi\)
\(84\) −7.60555 −0.829834
\(85\) −3.39445 −0.368180
\(86\) −11.9083 −1.28411
\(87\) −15.5139 −1.66326
\(88\) 0 0
\(89\) −2.60555 −0.276188 −0.138094 0.990419i \(-0.544098\pi\)
−0.138094 + 0.990419i \(0.544098\pi\)
\(90\) 10.3028 1.08601
\(91\) −0.697224 −0.0730890
\(92\) −8.60555 −0.897191
\(93\) −1.00000 −0.103695
\(94\) −6.00000 −0.618853
\(95\) −1.30278 −0.133662
\(96\) −3.30278 −0.337088
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −1.69722 −0.171446
\(99\) 0 0
\(100\) −3.30278 −0.330278
\(101\) 13.8167 1.37481 0.687404 0.726275i \(-0.258750\pi\)
0.687404 + 0.726275i \(0.258750\pi\)
\(102\) 8.60555 0.852077
\(103\) −5.30278 −0.522498 −0.261249 0.965271i \(-0.584134\pi\)
−0.261249 + 0.965271i \(0.584134\pi\)
\(104\) −0.302776 −0.0296896
\(105\) −9.90833 −0.966954
\(106\) −3.39445 −0.329698
\(107\) 11.2111 1.08382 0.541909 0.840437i \(-0.317702\pi\)
0.541909 + 0.840437i \(0.317702\pi\)
\(108\) −16.2111 −1.55991
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 30.4222 2.88755
\(112\) 2.30278 0.217592
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 3.30278 0.309333
\(115\) −11.2111 −1.04544
\(116\) 4.69722 0.436126
\(117\) −2.39445 −0.221367
\(118\) 0 0
\(119\) −6.00000 −0.550019
\(120\) −4.30278 −0.392788
\(121\) 0 0
\(122\) 3.21110 0.290720
\(123\) −22.8167 −2.05731
\(124\) 0.302776 0.0271901
\(125\) −10.8167 −0.967471
\(126\) 18.2111 1.62237
\(127\) 12.6056 1.11856 0.559281 0.828978i \(-0.311077\pi\)
0.559281 + 0.828978i \(0.311077\pi\)
\(128\) 1.00000 0.0883883
\(129\) 39.3305 3.46286
\(130\) −0.394449 −0.0345954
\(131\) 6.11943 0.534657 0.267329 0.963605i \(-0.413859\pi\)
0.267329 + 0.963605i \(0.413859\pi\)
\(132\) 0 0
\(133\) −2.30278 −0.199676
\(134\) 11.5139 0.994648
\(135\) −21.1194 −1.81767
\(136\) −2.60555 −0.223424
\(137\) −21.5139 −1.83805 −0.919027 0.394194i \(-0.871024\pi\)
−0.919027 + 0.394194i \(0.871024\pi\)
\(138\) 28.4222 2.41946
\(139\) 2.69722 0.228776 0.114388 0.993436i \(-0.463509\pi\)
0.114388 + 0.993436i \(0.463509\pi\)
\(140\) 3.00000 0.253546
\(141\) 19.8167 1.66886
\(142\) −13.3028 −1.11634
\(143\) 0 0
\(144\) 7.90833 0.659027
\(145\) 6.11943 0.508191
\(146\) −4.60555 −0.381158
\(147\) 5.60555 0.462338
\(148\) −9.21110 −0.757148
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 10.9083 0.890661
\(151\) −12.4222 −1.01090 −0.505452 0.862855i \(-0.668674\pi\)
−0.505452 + 0.862855i \(0.668674\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −20.6056 −1.66586
\(154\) 0 0
\(155\) 0.394449 0.0316829
\(156\) 1.00000 0.0800641
\(157\) 11.9083 0.950388 0.475194 0.879881i \(-0.342378\pi\)
0.475194 + 0.879881i \(0.342378\pi\)
\(158\) 5.81665 0.462748
\(159\) 11.2111 0.889098
\(160\) 1.30278 0.102993
\(161\) −19.8167 −1.56177
\(162\) 29.8167 2.34262
\(163\) −18.6056 −1.45730 −0.728650 0.684887i \(-0.759852\pi\)
−0.728650 + 0.684887i \(0.759852\pi\)
\(164\) 6.90833 0.539450
\(165\) 0 0
\(166\) −10.6972 −0.830266
\(167\) −16.4222 −1.27079 −0.635394 0.772188i \(-0.719162\pi\)
−0.635394 + 0.772188i \(0.719162\pi\)
\(168\) −7.60555 −0.586781
\(169\) −12.9083 −0.992948
\(170\) −3.39445 −0.260342
\(171\) −7.90833 −0.604765
\(172\) −11.9083 −0.908001
\(173\) −20.3305 −1.54570 −0.772851 0.634588i \(-0.781170\pi\)
−0.772851 + 0.634588i \(0.781170\pi\)
\(174\) −15.5139 −1.17610
\(175\) −7.60555 −0.574926
\(176\) 0 0
\(177\) 0 0
\(178\) −2.60555 −0.195294
\(179\) 12.1194 0.905849 0.452924 0.891549i \(-0.350381\pi\)
0.452924 + 0.891549i \(0.350381\pi\)
\(180\) 10.3028 0.767924
\(181\) 21.0278 1.56298 0.781490 0.623917i \(-0.214460\pi\)
0.781490 + 0.623917i \(0.214460\pi\)
\(182\) −0.697224 −0.0516817
\(183\) −10.6056 −0.783985
\(184\) −8.60555 −0.634410
\(185\) −12.0000 −0.882258
\(186\) −1.00000 −0.0733236
\(187\) 0 0
\(188\) −6.00000 −0.437595
\(189\) −37.3305 −2.71540
\(190\) −1.30278 −0.0945133
\(191\) −6.00000 −0.434145 −0.217072 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) −3.30278 −0.238357
\(193\) −7.72498 −0.556056 −0.278028 0.960573i \(-0.589681\pi\)
−0.278028 + 0.960573i \(0.589681\pi\)
\(194\) 8.00000 0.574367
\(195\) 1.30278 0.0932937
\(196\) −1.69722 −0.121230
\(197\) 21.6333 1.54131 0.770655 0.637253i \(-0.219929\pi\)
0.770655 + 0.637253i \(0.219929\pi\)
\(198\) 0 0
\(199\) −23.8167 −1.68832 −0.844159 0.536093i \(-0.819900\pi\)
−0.844159 + 0.536093i \(0.819900\pi\)
\(200\) −3.30278 −0.233542
\(201\) −38.0278 −2.68227
\(202\) 13.8167 0.972136
\(203\) 10.8167 0.759180
\(204\) 8.60555 0.602509
\(205\) 9.00000 0.628587
\(206\) −5.30278 −0.369462
\(207\) −68.0555 −4.73019
\(208\) −0.302776 −0.0209937
\(209\) 0 0
\(210\) −9.90833 −0.683740
\(211\) −17.3944 −1.19748 −0.598742 0.800942i \(-0.704332\pi\)
−0.598742 + 0.800942i \(0.704332\pi\)
\(212\) −3.39445 −0.233132
\(213\) 43.9361 3.01045
\(214\) 11.2111 0.766375
\(215\) −15.5139 −1.05804
\(216\) −16.2111 −1.10303
\(217\) 0.697224 0.0473307
\(218\) −2.00000 −0.135457
\(219\) 15.2111 1.02787
\(220\) 0 0
\(221\) 0.788897 0.0530670
\(222\) 30.4222 2.04180
\(223\) −10.7889 −0.722478 −0.361239 0.932473i \(-0.617646\pi\)
−0.361239 + 0.932473i \(0.617646\pi\)
\(224\) 2.30278 0.153861
\(225\) −26.1194 −1.74130
\(226\) 0 0
\(227\) 19.8167 1.31528 0.657639 0.753333i \(-0.271555\pi\)
0.657639 + 0.753333i \(0.271555\pi\)
\(228\) 3.30278 0.218732
\(229\) 1.09167 0.0721398 0.0360699 0.999349i \(-0.488516\pi\)
0.0360699 + 0.999349i \(0.488516\pi\)
\(230\) −11.2111 −0.739238
\(231\) 0 0
\(232\) 4.69722 0.308388
\(233\) 5.21110 0.341391 0.170695 0.985324i \(-0.445399\pi\)
0.170695 + 0.985324i \(0.445399\pi\)
\(234\) −2.39445 −0.156530
\(235\) −7.81665 −0.509902
\(236\) 0 0
\(237\) −19.2111 −1.24790
\(238\) −6.00000 −0.388922
\(239\) −22.9361 −1.48361 −0.741806 0.670615i \(-0.766030\pi\)
−0.741806 + 0.670615i \(0.766030\pi\)
\(240\) −4.30278 −0.277743
\(241\) −23.9083 −1.54007 −0.770035 0.638001i \(-0.779761\pi\)
−0.770035 + 0.638001i \(0.779761\pi\)
\(242\) 0 0
\(243\) −49.8444 −3.19752
\(244\) 3.21110 0.205570
\(245\) −2.21110 −0.141262
\(246\) −22.8167 −1.45474
\(247\) 0.302776 0.0192652
\(248\) 0.302776 0.0192263
\(249\) 35.3305 2.23898
\(250\) −10.8167 −0.684105
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 18.2111 1.14719
\(253\) 0 0
\(254\) 12.6056 0.790943
\(255\) 11.2111 0.702066
\(256\) 1.00000 0.0625000
\(257\) 2.60555 0.162530 0.0812649 0.996693i \(-0.474104\pi\)
0.0812649 + 0.996693i \(0.474104\pi\)
\(258\) 39.3305 2.44861
\(259\) −21.2111 −1.31799
\(260\) −0.394449 −0.0244627
\(261\) 37.1472 2.29935
\(262\) 6.11943 0.378060
\(263\) 16.6972 1.02959 0.514797 0.857312i \(-0.327867\pi\)
0.514797 + 0.857312i \(0.327867\pi\)
\(264\) 0 0
\(265\) −4.42221 −0.271654
\(266\) −2.30278 −0.141192
\(267\) 8.60555 0.526651
\(268\) 11.5139 0.703322
\(269\) 1.81665 0.110763 0.0553817 0.998465i \(-0.482362\pi\)
0.0553817 + 0.998465i \(0.482362\pi\)
\(270\) −21.1194 −1.28529
\(271\) −11.5139 −0.699418 −0.349709 0.936858i \(-0.613720\pi\)
−0.349709 + 0.936858i \(0.613720\pi\)
\(272\) −2.60555 −0.157985
\(273\) 2.30278 0.139370
\(274\) −21.5139 −1.29970
\(275\) 0 0
\(276\) 28.4222 1.71082
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 2.69722 0.161769
\(279\) 2.39445 0.143352
\(280\) 3.00000 0.179284
\(281\) 18.5139 1.10445 0.552223 0.833697i \(-0.313780\pi\)
0.552223 + 0.833697i \(0.313780\pi\)
\(282\) 19.8167 1.18006
\(283\) −27.9361 −1.66063 −0.830314 0.557296i \(-0.811839\pi\)
−0.830314 + 0.557296i \(0.811839\pi\)
\(284\) −13.3028 −0.789375
\(285\) 4.30278 0.254874
\(286\) 0 0
\(287\) 15.9083 0.939039
\(288\) 7.90833 0.466003
\(289\) −10.2111 −0.600653
\(290\) 6.11943 0.359345
\(291\) −26.4222 −1.54890
\(292\) −4.60555 −0.269520
\(293\) −28.5416 −1.66742 −0.833710 0.552202i \(-0.813788\pi\)
−0.833710 + 0.552202i \(0.813788\pi\)
\(294\) 5.60555 0.326922
\(295\) 0 0
\(296\) −9.21110 −0.535384
\(297\) 0 0
\(298\) 0 0
\(299\) 2.60555 0.150683
\(300\) 10.9083 0.629793
\(301\) −27.4222 −1.58059
\(302\) −12.4222 −0.714818
\(303\) −45.6333 −2.62157
\(304\) −1.00000 −0.0573539
\(305\) 4.18335 0.239538
\(306\) −20.6056 −1.17794
\(307\) 23.8167 1.35929 0.679644 0.733542i \(-0.262134\pi\)
0.679644 + 0.733542i \(0.262134\pi\)
\(308\) 0 0
\(309\) 17.5139 0.996330
\(310\) 0.394449 0.0224032
\(311\) 4.18335 0.237216 0.118608 0.992941i \(-0.462157\pi\)
0.118608 + 0.992941i \(0.462157\pi\)
\(312\) 1.00000 0.0566139
\(313\) −7.11943 −0.402414 −0.201207 0.979549i \(-0.564486\pi\)
−0.201207 + 0.979549i \(0.564486\pi\)
\(314\) 11.9083 0.672026
\(315\) 23.7250 1.33675
\(316\) 5.81665 0.327212
\(317\) 7.81665 0.439027 0.219514 0.975609i \(-0.429553\pi\)
0.219514 + 0.975609i \(0.429553\pi\)
\(318\) 11.2111 0.628687
\(319\) 0 0
\(320\) 1.30278 0.0728274
\(321\) −37.0278 −2.06669
\(322\) −19.8167 −1.10434
\(323\) 2.60555 0.144977
\(324\) 29.8167 1.65648
\(325\) 1.00000 0.0554700
\(326\) −18.6056 −1.03047
\(327\) 6.60555 0.365288
\(328\) 6.90833 0.381449
\(329\) −13.8167 −0.761737
\(330\) 0 0
\(331\) −9.72498 −0.534533 −0.267267 0.963623i \(-0.586120\pi\)
−0.267267 + 0.963623i \(0.586120\pi\)
\(332\) −10.6972 −0.587086
\(333\) −72.8444 −3.99185
\(334\) −16.4222 −0.898583
\(335\) 15.0000 0.819538
\(336\) −7.60555 −0.414917
\(337\) 5.69722 0.310348 0.155174 0.987887i \(-0.450406\pi\)
0.155174 + 0.987887i \(0.450406\pi\)
\(338\) −12.9083 −0.702120
\(339\) 0 0
\(340\) −3.39445 −0.184090
\(341\) 0 0
\(342\) −7.90833 −0.427633
\(343\) −20.0278 −1.08140
\(344\) −11.9083 −0.642054
\(345\) 37.0278 1.99351
\(346\) −20.3305 −1.09298
\(347\) −17.2111 −0.923940 −0.461970 0.886895i \(-0.652857\pi\)
−0.461970 + 0.886895i \(0.652857\pi\)
\(348\) −15.5139 −0.831631
\(349\) −3.57779 −0.191515 −0.0957575 0.995405i \(-0.530527\pi\)
−0.0957575 + 0.995405i \(0.530527\pi\)
\(350\) −7.60555 −0.406534
\(351\) 4.90833 0.261987
\(352\) 0 0
\(353\) −0.788897 −0.0419888 −0.0209944 0.999780i \(-0.506683\pi\)
−0.0209944 + 0.999780i \(0.506683\pi\)
\(354\) 0 0
\(355\) −17.3305 −0.919809
\(356\) −2.60555 −0.138094
\(357\) 19.8167 1.04881
\(358\) 12.1194 0.640532
\(359\) 27.5139 1.45213 0.726063 0.687628i \(-0.241348\pi\)
0.726063 + 0.687628i \(0.241348\pi\)
\(360\) 10.3028 0.543004
\(361\) 1.00000 0.0526316
\(362\) 21.0278 1.10519
\(363\) 0 0
\(364\) −0.697224 −0.0365445
\(365\) −6.00000 −0.314054
\(366\) −10.6056 −0.554361
\(367\) 7.21110 0.376416 0.188208 0.982129i \(-0.439732\pi\)
0.188208 + 0.982129i \(0.439732\pi\)
\(368\) −8.60555 −0.448595
\(369\) 54.6333 2.84410
\(370\) −12.0000 −0.623850
\(371\) −7.81665 −0.405820
\(372\) −1.00000 −0.0518476
\(373\) −17.9083 −0.927258 −0.463629 0.886029i \(-0.653453\pi\)
−0.463629 + 0.886029i \(0.653453\pi\)
\(374\) 0 0
\(375\) 35.7250 1.84483
\(376\) −6.00000 −0.309426
\(377\) −1.42221 −0.0732473
\(378\) −37.3305 −1.92008
\(379\) −31.5139 −1.61876 −0.809380 0.587286i \(-0.800196\pi\)
−0.809380 + 0.587286i \(0.800196\pi\)
\(380\) −1.30278 −0.0668310
\(381\) −41.6333 −2.13294
\(382\) −6.00000 −0.306987
\(383\) 21.5139 1.09931 0.549654 0.835392i \(-0.314760\pi\)
0.549654 + 0.835392i \(0.314760\pi\)
\(384\) −3.30278 −0.168544
\(385\) 0 0
\(386\) −7.72498 −0.393191
\(387\) −94.1749 −4.78718
\(388\) 8.00000 0.406138
\(389\) −19.5416 −0.990800 −0.495400 0.868665i \(-0.664979\pi\)
−0.495400 + 0.868665i \(0.664979\pi\)
\(390\) 1.30278 0.0659686
\(391\) 22.4222 1.13394
\(392\) −1.69722 −0.0857228
\(393\) −20.2111 −1.01952
\(394\) 21.6333 1.08987
\(395\) 7.57779 0.381280
\(396\) 0 0
\(397\) 35.7527 1.79438 0.897189 0.441646i \(-0.145605\pi\)
0.897189 + 0.441646i \(0.145605\pi\)
\(398\) −23.8167 −1.19382
\(399\) 7.60555 0.380754
\(400\) −3.30278 −0.165139
\(401\) 15.3944 0.768762 0.384381 0.923175i \(-0.374415\pi\)
0.384381 + 0.923175i \(0.374415\pi\)
\(402\) −38.0278 −1.89665
\(403\) −0.0916731 −0.00456656
\(404\) 13.8167 0.687404
\(405\) 38.8444 1.93019
\(406\) 10.8167 0.536822
\(407\) 0 0
\(408\) 8.60555 0.426038
\(409\) 38.1472 1.88626 0.943128 0.332428i \(-0.107868\pi\)
0.943128 + 0.332428i \(0.107868\pi\)
\(410\) 9.00000 0.444478
\(411\) 71.0555 3.50491
\(412\) −5.30278 −0.261249
\(413\) 0 0
\(414\) −68.0555 −3.34475
\(415\) −13.9361 −0.684095
\(416\) −0.302776 −0.0148448
\(417\) −8.90833 −0.436243
\(418\) 0 0
\(419\) 23.2111 1.13394 0.566968 0.823740i \(-0.308116\pi\)
0.566968 + 0.823740i \(0.308116\pi\)
\(420\) −9.90833 −0.483477
\(421\) 20.2389 0.986382 0.493191 0.869921i \(-0.335830\pi\)
0.493191 + 0.869921i \(0.335830\pi\)
\(422\) −17.3944 −0.846749
\(423\) −47.4500 −2.30710
\(424\) −3.39445 −0.164849
\(425\) 8.60555 0.417431
\(426\) 43.9361 2.12871
\(427\) 7.39445 0.357842
\(428\) 11.2111 0.541909
\(429\) 0 0
\(430\) −15.5139 −0.748146
\(431\) −17.2111 −0.829030 −0.414515 0.910043i \(-0.636049\pi\)
−0.414515 + 0.910043i \(0.636049\pi\)
\(432\) −16.2111 −0.779957
\(433\) 26.2389 1.26096 0.630480 0.776206i \(-0.282858\pi\)
0.630480 + 0.776206i \(0.282858\pi\)
\(434\) 0.697224 0.0334678
\(435\) −20.2111 −0.969048
\(436\) −2.00000 −0.0957826
\(437\) 8.60555 0.411659
\(438\) 15.2111 0.726815
\(439\) 10.7889 0.514926 0.257463 0.966288i \(-0.417113\pi\)
0.257463 + 0.966288i \(0.417113\pi\)
\(440\) 0 0
\(441\) −13.4222 −0.639153
\(442\) 0.788897 0.0375240
\(443\) −3.63331 −0.172624 −0.0863118 0.996268i \(-0.527508\pi\)
−0.0863118 + 0.996268i \(0.527508\pi\)
\(444\) 30.4222 1.44377
\(445\) −3.39445 −0.160912
\(446\) −10.7889 −0.510869
\(447\) 0 0
\(448\) 2.30278 0.108796
\(449\) −10.1833 −0.480582 −0.240291 0.970701i \(-0.577243\pi\)
−0.240291 + 0.970701i \(0.577243\pi\)
\(450\) −26.1194 −1.23128
\(451\) 0 0
\(452\) 0 0
\(453\) 41.0278 1.92765
\(454\) 19.8167 0.930042
\(455\) −0.908327 −0.0425830
\(456\) 3.30278 0.154667
\(457\) −0.183346 −0.00857657 −0.00428829 0.999991i \(-0.501365\pi\)
−0.00428829 + 0.999991i \(0.501365\pi\)
\(458\) 1.09167 0.0510105
\(459\) 42.2389 1.97154
\(460\) −11.2111 −0.522720
\(461\) 22.4222 1.04431 0.522153 0.852852i \(-0.325129\pi\)
0.522153 + 0.852852i \(0.325129\pi\)
\(462\) 0 0
\(463\) 13.2111 0.613972 0.306986 0.951714i \(-0.400680\pi\)
0.306986 + 0.951714i \(0.400680\pi\)
\(464\) 4.69722 0.218063
\(465\) −1.30278 −0.0604148
\(466\) 5.21110 0.241400
\(467\) −15.3944 −0.712370 −0.356185 0.934415i \(-0.615923\pi\)
−0.356185 + 0.934415i \(0.615923\pi\)
\(468\) −2.39445 −0.110683
\(469\) 26.5139 1.22430
\(470\) −7.81665 −0.360555
\(471\) −39.3305 −1.81226
\(472\) 0 0
\(473\) 0 0
\(474\) −19.2111 −0.882395
\(475\) 3.30278 0.151542
\(476\) −6.00000 −0.275010
\(477\) −26.8444 −1.22912
\(478\) −22.9361 −1.04907
\(479\) −3.27502 −0.149639 −0.0748197 0.997197i \(-0.523838\pi\)
−0.0748197 + 0.997197i \(0.523838\pi\)
\(480\) −4.30278 −0.196394
\(481\) 2.78890 0.127163
\(482\) −23.9083 −1.08899
\(483\) 65.4500 2.97808
\(484\) 0 0
\(485\) 10.4222 0.473248
\(486\) −49.8444 −2.26099
\(487\) 10.3305 0.468121 0.234061 0.972222i \(-0.424799\pi\)
0.234061 + 0.972222i \(0.424799\pi\)
\(488\) 3.21110 0.145360
\(489\) 61.4500 2.77886
\(490\) −2.21110 −0.0998874
\(491\) −31.9361 −1.44126 −0.720628 0.693322i \(-0.756146\pi\)
−0.720628 + 0.693322i \(0.756146\pi\)
\(492\) −22.8167 −1.02865
\(493\) −12.2389 −0.551210
\(494\) 0.302776 0.0136225
\(495\) 0 0
\(496\) 0.302776 0.0135950
\(497\) −30.6333 −1.37409
\(498\) 35.3305 1.58320
\(499\) −34.0000 −1.52205 −0.761025 0.648723i \(-0.775303\pi\)
−0.761025 + 0.648723i \(0.775303\pi\)
\(500\) −10.8167 −0.483735
\(501\) 54.2389 2.42321
\(502\) 24.0000 1.07117
\(503\) 3.11943 0.139088 0.0695442 0.997579i \(-0.477845\pi\)
0.0695442 + 0.997579i \(0.477845\pi\)
\(504\) 18.2111 0.811187
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) 42.6333 1.89341
\(508\) 12.6056 0.559281
\(509\) 10.4222 0.461956 0.230978 0.972959i \(-0.425807\pi\)
0.230978 + 0.972959i \(0.425807\pi\)
\(510\) 11.2111 0.496436
\(511\) −10.6056 −0.469162
\(512\) 1.00000 0.0441942
\(513\) 16.2111 0.715738
\(514\) 2.60555 0.114926
\(515\) −6.90833 −0.304417
\(516\) 39.3305 1.73143
\(517\) 0 0
\(518\) −21.2111 −0.931962
\(519\) 67.1472 2.94743
\(520\) −0.394449 −0.0172977
\(521\) 7.57779 0.331989 0.165995 0.986127i \(-0.446917\pi\)
0.165995 + 0.986127i \(0.446917\pi\)
\(522\) 37.1472 1.62589
\(523\) 17.0278 0.744572 0.372286 0.928118i \(-0.378574\pi\)
0.372286 + 0.928118i \(0.378574\pi\)
\(524\) 6.11943 0.267329
\(525\) 25.1194 1.09630
\(526\) 16.6972 0.728034
\(527\) −0.788897 −0.0343649
\(528\) 0 0
\(529\) 51.0555 2.21980
\(530\) −4.42221 −0.192088
\(531\) 0 0
\(532\) −2.30278 −0.0998380
\(533\) −2.09167 −0.0906004
\(534\) 8.60555 0.372399
\(535\) 14.6056 0.631453
\(536\) 11.5139 0.497324
\(537\) −40.0278 −1.72733
\(538\) 1.81665 0.0783215
\(539\) 0 0
\(540\) −21.1194 −0.908836
\(541\) 17.8167 0.765998 0.382999 0.923749i \(-0.374891\pi\)
0.382999 + 0.923749i \(0.374891\pi\)
\(542\) −11.5139 −0.494563
\(543\) −69.4500 −2.98038
\(544\) −2.60555 −0.111712
\(545\) −2.60555 −0.111610
\(546\) 2.30278 0.0985497
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) −21.5139 −0.919027
\(549\) 25.3944 1.08381
\(550\) 0 0
\(551\) −4.69722 −0.200108
\(552\) 28.4222 1.20973
\(553\) 13.3944 0.569590
\(554\) 10.0000 0.424859
\(555\) 39.6333 1.68234
\(556\) 2.69722 0.114388
\(557\) −15.6333 −0.662405 −0.331202 0.943560i \(-0.607454\pi\)
−0.331202 + 0.943560i \(0.607454\pi\)
\(558\) 2.39445 0.101365
\(559\) 3.60555 0.152499
\(560\) 3.00000 0.126773
\(561\) 0 0
\(562\) 18.5139 0.780961
\(563\) 21.6333 0.911735 0.455868 0.890048i \(-0.349329\pi\)
0.455868 + 0.890048i \(0.349329\pi\)
\(564\) 19.8167 0.834432
\(565\) 0 0
\(566\) −27.9361 −1.17424
\(567\) 68.6611 2.88349
\(568\) −13.3028 −0.558172
\(569\) −44.7250 −1.87497 −0.937484 0.348027i \(-0.886852\pi\)
−0.937484 + 0.348027i \(0.886852\pi\)
\(570\) 4.30278 0.180223
\(571\) −39.9361 −1.67127 −0.835637 0.549283i \(-0.814901\pi\)
−0.835637 + 0.549283i \(0.814901\pi\)
\(572\) 0 0
\(573\) 19.8167 0.827853
\(574\) 15.9083 0.664001
\(575\) 28.4222 1.18529
\(576\) 7.90833 0.329514
\(577\) −28.9083 −1.20347 −0.601735 0.798696i \(-0.705524\pi\)
−0.601735 + 0.798696i \(0.705524\pi\)
\(578\) −10.2111 −0.424726
\(579\) 25.5139 1.06032
\(580\) 6.11943 0.254095
\(581\) −24.6333 −1.02196
\(582\) −26.4222 −1.09524
\(583\) 0 0
\(584\) −4.60555 −0.190579
\(585\) −3.11943 −0.128973
\(586\) −28.5416 −1.17904
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 5.60555 0.231169
\(589\) −0.302776 −0.0124757
\(590\) 0 0
\(591\) −71.4500 −2.93906
\(592\) −9.21110 −0.378574
\(593\) 42.0000 1.72473 0.862367 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) 0 0
\(595\) −7.81665 −0.320452
\(596\) 0 0
\(597\) 78.6611 3.21938
\(598\) 2.60555 0.106549
\(599\) −45.1194 −1.84353 −0.921765 0.387749i \(-0.873253\pi\)
−0.921765 + 0.387749i \(0.873253\pi\)
\(600\) 10.9083 0.445331
\(601\) −14.9083 −0.608123 −0.304062 0.952652i \(-0.598343\pi\)
−0.304062 + 0.952652i \(0.598343\pi\)
\(602\) −27.4222 −1.11765
\(603\) 91.0555 3.70807
\(604\) −12.4222 −0.505452
\(605\) 0 0
\(606\) −45.6333 −1.85373
\(607\) 17.8167 0.723156 0.361578 0.932342i \(-0.382238\pi\)
0.361578 + 0.932342i \(0.382238\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −35.7250 −1.44765
\(610\) 4.18335 0.169379
\(611\) 1.81665 0.0734939
\(612\) −20.6056 −0.832930
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 23.8167 0.961162
\(615\) −29.7250 −1.19863
\(616\) 0 0
\(617\) 12.9083 0.519670 0.259835 0.965653i \(-0.416332\pi\)
0.259835 + 0.965653i \(0.416332\pi\)
\(618\) 17.5139 0.704512
\(619\) 22.8444 0.918194 0.459097 0.888386i \(-0.348173\pi\)
0.459097 + 0.888386i \(0.348173\pi\)
\(620\) 0.394449 0.0158414
\(621\) 139.505 5.59816
\(622\) 4.18335 0.167737
\(623\) −6.00000 −0.240385
\(624\) 1.00000 0.0400320
\(625\) 2.42221 0.0968882
\(626\) −7.11943 −0.284550
\(627\) 0 0
\(628\) 11.9083 0.475194
\(629\) 24.0000 0.956943
\(630\) 23.7250 0.945226
\(631\) −8.97224 −0.357179 −0.178590 0.983924i \(-0.557153\pi\)
−0.178590 + 0.983924i \(0.557153\pi\)
\(632\) 5.81665 0.231374
\(633\) 57.4500 2.28343
\(634\) 7.81665 0.310439
\(635\) 16.4222 0.651695
\(636\) 11.2111 0.444549
\(637\) 0.513878 0.0203606
\(638\) 0 0
\(639\) −105.203 −4.16175
\(640\) 1.30278 0.0514967
\(641\) −30.2389 −1.19436 −0.597182 0.802106i \(-0.703713\pi\)
−0.597182 + 0.802106i \(0.703713\pi\)
\(642\) −37.0278 −1.46137
\(643\) −2.42221 −0.0955224 −0.0477612 0.998859i \(-0.515209\pi\)
−0.0477612 + 0.998859i \(0.515209\pi\)
\(644\) −19.8167 −0.780886
\(645\) 51.2389 2.01753
\(646\) 2.60555 0.102514
\(647\) 4.18335 0.164464 0.0822322 0.996613i \(-0.473795\pi\)
0.0822322 + 0.996613i \(0.473795\pi\)
\(648\) 29.8167 1.17131
\(649\) 0 0
\(650\) 1.00000 0.0392232
\(651\) −2.30278 −0.0902529
\(652\) −18.6056 −0.728650
\(653\) −17.3305 −0.678196 −0.339098 0.940751i \(-0.610122\pi\)
−0.339098 + 0.940751i \(0.610122\pi\)
\(654\) 6.60555 0.258297
\(655\) 7.97224 0.311501
\(656\) 6.90833 0.269725
\(657\) −36.4222 −1.42097
\(658\) −13.8167 −0.538629
\(659\) 7.02776 0.273763 0.136881 0.990587i \(-0.456292\pi\)
0.136881 + 0.990587i \(0.456292\pi\)
\(660\) 0 0
\(661\) −19.6333 −0.763647 −0.381824 0.924235i \(-0.624704\pi\)
−0.381824 + 0.924235i \(0.624704\pi\)
\(662\) −9.72498 −0.377972
\(663\) −2.60555 −0.101191
\(664\) −10.6972 −0.415133
\(665\) −3.00000 −0.116335
\(666\) −72.8444 −2.82266
\(667\) −40.4222 −1.56515
\(668\) −16.4222 −0.635394
\(669\) 35.6333 1.37766
\(670\) 15.0000 0.579501
\(671\) 0 0
\(672\) −7.60555 −0.293391
\(673\) 2.30278 0.0887655 0.0443827 0.999015i \(-0.485868\pi\)
0.0443827 + 0.999015i \(0.485868\pi\)
\(674\) 5.69722 0.219449
\(675\) 53.5416 2.06082
\(676\) −12.9083 −0.496474
\(677\) 4.69722 0.180529 0.0902645 0.995918i \(-0.471229\pi\)
0.0902645 + 0.995918i \(0.471229\pi\)
\(678\) 0 0
\(679\) 18.4222 0.706979
\(680\) −3.39445 −0.130171
\(681\) −65.4500 −2.50805
\(682\) 0 0
\(683\) −46.4222 −1.77630 −0.888148 0.459557i \(-0.848008\pi\)
−0.888148 + 0.459557i \(0.848008\pi\)
\(684\) −7.90833 −0.302382
\(685\) −28.0278 −1.07089
\(686\) −20.0278 −0.764663
\(687\) −3.60555 −0.137560
\(688\) −11.9083 −0.454001
\(689\) 1.02776 0.0391544
\(690\) 37.0278 1.40962
\(691\) 3.57779 0.136106 0.0680529 0.997682i \(-0.478321\pi\)
0.0680529 + 0.997682i \(0.478321\pi\)
\(692\) −20.3305 −0.772851
\(693\) 0 0
\(694\) −17.2111 −0.653325
\(695\) 3.51388 0.133289
\(696\) −15.5139 −0.588052
\(697\) −18.0000 −0.681799
\(698\) −3.57779 −0.135422
\(699\) −17.2111 −0.650984
\(700\) −7.60555 −0.287463
\(701\) −15.6333 −0.590462 −0.295231 0.955426i \(-0.595397\pi\)
−0.295231 + 0.955426i \(0.595397\pi\)
\(702\) 4.90833 0.185253
\(703\) 9.21110 0.347403
\(704\) 0 0
\(705\) 25.8167 0.972311
\(706\) −0.788897 −0.0296905
\(707\) 31.8167 1.19659
\(708\) 0 0
\(709\) 18.3028 0.687375 0.343688 0.939084i \(-0.388324\pi\)
0.343688 + 0.939084i \(0.388324\pi\)
\(710\) −17.3305 −0.650403
\(711\) 46.0000 1.72513
\(712\) −2.60555 −0.0976472
\(713\) −2.60555 −0.0975787
\(714\) 19.8167 0.741620
\(715\) 0 0
\(716\) 12.1194 0.452924
\(717\) 75.7527 2.82904
\(718\) 27.5139 1.02681
\(719\) −4.18335 −0.156012 −0.0780062 0.996953i \(-0.524855\pi\)
−0.0780062 + 0.996953i \(0.524855\pi\)
\(720\) 10.3028 0.383962
\(721\) −12.2111 −0.454765
\(722\) 1.00000 0.0372161
\(723\) 78.9638 2.93670
\(724\) 21.0278 0.781490
\(725\) −15.5139 −0.576171
\(726\) 0 0
\(727\) −43.6333 −1.61827 −0.809135 0.587623i \(-0.800064\pi\)
−0.809135 + 0.587623i \(0.800064\pi\)
\(728\) −0.697224 −0.0258409
\(729\) 75.1749 2.78426
\(730\) −6.00000 −0.222070
\(731\) 31.0278 1.14760
\(732\) −10.6056 −0.391992
\(733\) −7.21110 −0.266348 −0.133174 0.991093i \(-0.542517\pi\)
−0.133174 + 0.991093i \(0.542517\pi\)
\(734\) 7.21110 0.266167
\(735\) 7.30278 0.269367
\(736\) −8.60555 −0.317205
\(737\) 0 0
\(738\) 54.6333 2.01108
\(739\) −35.3583 −1.30068 −0.650338 0.759645i \(-0.725373\pi\)
−0.650338 + 0.759645i \(0.725373\pi\)
\(740\) −12.0000 −0.441129
\(741\) −1.00000 −0.0367359
\(742\) −7.81665 −0.286958
\(743\) −45.6333 −1.67412 −0.837062 0.547108i \(-0.815729\pi\)
−0.837062 + 0.547108i \(0.815729\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 0 0
\(746\) −17.9083 −0.655670
\(747\) −84.5971 −3.09525
\(748\) 0 0
\(749\) 25.8167 0.943320
\(750\) 35.7250 1.30449
\(751\) 16.8444 0.614661 0.307331 0.951603i \(-0.400564\pi\)
0.307331 + 0.951603i \(0.400564\pi\)
\(752\) −6.00000 −0.218797
\(753\) −79.2666 −2.88864
\(754\) −1.42221 −0.0517937
\(755\) −16.1833 −0.588972
\(756\) −37.3305 −1.35770
\(757\) −23.5416 −0.855635 −0.427818 0.903865i \(-0.640717\pi\)
−0.427818 + 0.903865i \(0.640717\pi\)
\(758\) −31.5139 −1.14464
\(759\) 0 0
\(760\) −1.30278 −0.0472566
\(761\) 30.2389 1.09616 0.548079 0.836427i \(-0.315359\pi\)
0.548079 + 0.836427i \(0.315359\pi\)
\(762\) −41.6333 −1.50822
\(763\) −4.60555 −0.166732
\(764\) −6.00000 −0.217072
\(765\) −26.8444 −0.970562
\(766\) 21.5139 0.777328
\(767\) 0 0
\(768\) −3.30278 −0.119179
\(769\) −5.63331 −0.203142 −0.101571 0.994828i \(-0.532387\pi\)
−0.101571 + 0.994828i \(0.532387\pi\)
\(770\) 0 0
\(771\) −8.60555 −0.309921
\(772\) −7.72498 −0.278028
\(773\) 19.5778 0.704164 0.352082 0.935969i \(-0.385474\pi\)
0.352082 + 0.935969i \(0.385474\pi\)
\(774\) −94.1749 −3.38505
\(775\) −1.00000 −0.0359211
\(776\) 8.00000 0.287183
\(777\) 70.0555 2.51323
\(778\) −19.5416 −0.700602
\(779\) −6.90833 −0.247516
\(780\) 1.30278 0.0466469
\(781\) 0 0
\(782\) 22.4222 0.801816
\(783\) −76.1472 −2.72128
\(784\) −1.69722 −0.0606152
\(785\) 15.5139 0.553714
\(786\) −20.2111 −0.720906
\(787\) −21.0278 −0.749559 −0.374779 0.927114i \(-0.622282\pi\)
−0.374779 + 0.927114i \(0.622282\pi\)
\(788\) 21.6333 0.770655
\(789\) −55.1472 −1.96329
\(790\) 7.57779 0.269606
\(791\) 0 0
\(792\) 0 0
\(793\) −0.972244 −0.0345254
\(794\) 35.7527 1.26882
\(795\) 14.6056 0.518006
\(796\) −23.8167 −0.844159
\(797\) 9.39445 0.332768 0.166384 0.986061i \(-0.446791\pi\)
0.166384 + 0.986061i \(0.446791\pi\)
\(798\) 7.60555 0.269234
\(799\) 15.6333 0.553067
\(800\) −3.30278 −0.116771
\(801\) −20.6056 −0.728061
\(802\) 15.3944 0.543597
\(803\) 0 0
\(804\) −38.0278 −1.34114
\(805\) −25.8167 −0.909917
\(806\) −0.0916731 −0.00322905
\(807\) −6.00000 −0.211210
\(808\) 13.8167 0.486068
\(809\) −18.7889 −0.660582 −0.330291 0.943879i \(-0.607147\pi\)
−0.330291 + 0.943879i \(0.607147\pi\)
\(810\) 38.8444 1.36485
\(811\) 13.6333 0.478730 0.239365 0.970930i \(-0.423061\pi\)
0.239365 + 0.970930i \(0.423061\pi\)
\(812\) 10.8167 0.379590
\(813\) 38.0278 1.33369
\(814\) 0 0
\(815\) −24.2389 −0.849050
\(816\) 8.60555 0.301255
\(817\) 11.9083 0.416620
\(818\) 38.1472 1.33379
\(819\) −5.51388 −0.192670
\(820\) 9.00000 0.314294
\(821\) 27.3944 0.956073 0.478036 0.878340i \(-0.341349\pi\)
0.478036 + 0.878340i \(0.341349\pi\)
\(822\) 71.0555 2.47835
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) −5.30278 −0.184731
\(825\) 0 0
\(826\) 0 0
\(827\) 50.0555 1.74060 0.870300 0.492521i \(-0.163925\pi\)
0.870300 + 0.492521i \(0.163925\pi\)
\(828\) −68.0555 −2.36509
\(829\) −56.4222 −1.95962 −0.979812 0.199921i \(-0.935932\pi\)
−0.979812 + 0.199921i \(0.935932\pi\)
\(830\) −13.9361 −0.483729
\(831\) −33.0278 −1.14572
\(832\) −0.302776 −0.0104969
\(833\) 4.42221 0.153220
\(834\) −8.90833 −0.308470
\(835\) −21.3944 −0.740385
\(836\) 0 0
\(837\) −4.90833 −0.169657
\(838\) 23.2111 0.801814
\(839\) 45.1194 1.55770 0.778848 0.627213i \(-0.215804\pi\)
0.778848 + 0.627213i \(0.215804\pi\)
\(840\) −9.90833 −0.341870
\(841\) −6.93608 −0.239175
\(842\) 20.2389 0.697477
\(843\) −61.1472 −2.10602
\(844\) −17.3944 −0.598742
\(845\) −16.8167 −0.578510
\(846\) −47.4500 −1.63136
\(847\) 0 0
\(848\) −3.39445 −0.116566
\(849\) 92.2666 3.16658
\(850\) 8.60555 0.295168
\(851\) 79.2666 2.71722
\(852\) 43.9361 1.50523
\(853\) 17.5778 0.601852 0.300926 0.953647i \(-0.402704\pi\)
0.300926 + 0.953647i \(0.402704\pi\)
\(854\) 7.39445 0.253033
\(855\) −10.3028 −0.352347
\(856\) 11.2111 0.383188
\(857\) −10.5416 −0.360095 −0.180048 0.983658i \(-0.557625\pi\)
−0.180048 + 0.983658i \(0.557625\pi\)
\(858\) 0 0
\(859\) −48.8444 −1.66655 −0.833275 0.552859i \(-0.813537\pi\)
−0.833275 + 0.552859i \(0.813537\pi\)
\(860\) −15.5139 −0.529019
\(861\) −52.5416 −1.79061
\(862\) −17.2111 −0.586212
\(863\) 14.7250 0.501244 0.250622 0.968085i \(-0.419365\pi\)
0.250622 + 0.968085i \(0.419365\pi\)
\(864\) −16.2111 −0.551513
\(865\) −26.4861 −0.900555
\(866\) 26.2389 0.891633
\(867\) 33.7250 1.14536
\(868\) 0.697224 0.0236653
\(869\) 0 0
\(870\) −20.2111 −0.685221
\(871\) −3.48612 −0.118123
\(872\) −2.00000 −0.0677285
\(873\) 63.2666 2.14125
\(874\) 8.60555 0.291087
\(875\) −24.9083 −0.842055
\(876\) 15.2111 0.513936
\(877\) 42.5694 1.43747 0.718733 0.695286i \(-0.244722\pi\)
0.718733 + 0.695286i \(0.244722\pi\)
\(878\) 10.7889 0.364108
\(879\) 94.2666 3.17953
\(880\) 0 0
\(881\) −23.0917 −0.777978 −0.388989 0.921242i \(-0.627176\pi\)
−0.388989 + 0.921242i \(0.627176\pi\)
\(882\) −13.4222 −0.451949
\(883\) −2.97224 −0.100024 −0.0500120 0.998749i \(-0.515926\pi\)
−0.0500120 + 0.998749i \(0.515926\pi\)
\(884\) 0.788897 0.0265335
\(885\) 0 0
\(886\) −3.63331 −0.122063
\(887\) 32.8444 1.10281 0.551404 0.834239i \(-0.314092\pi\)
0.551404 + 0.834239i \(0.314092\pi\)
\(888\) 30.4222 1.02090
\(889\) 29.0278 0.973560
\(890\) −3.39445 −0.113782
\(891\) 0 0
\(892\) −10.7889 −0.361239
\(893\) 6.00000 0.200782
\(894\) 0 0
\(895\) 15.7889 0.527765
\(896\) 2.30278 0.0769303
\(897\) −8.60555 −0.287331
\(898\) −10.1833 −0.339823
\(899\) 1.42221 0.0474332
\(900\) −26.1194 −0.870648
\(901\) 8.84441 0.294650
\(902\) 0 0
\(903\) 90.5694 3.01396
\(904\) 0 0
\(905\) 27.3944 0.910622
\(906\) 41.0278 1.36306
\(907\) 52.8444 1.75467 0.877335 0.479879i \(-0.159319\pi\)
0.877335 + 0.479879i \(0.159319\pi\)
\(908\) 19.8167 0.657639
\(909\) 109.267 3.62414
\(910\) −0.908327 −0.0301107
\(911\) −15.6333 −0.517955 −0.258977 0.965883i \(-0.583385\pi\)
−0.258977 + 0.965883i \(0.583385\pi\)
\(912\) 3.30278 0.109366
\(913\) 0 0
\(914\) −0.183346 −0.00606455
\(915\) −13.8167 −0.456764
\(916\) 1.09167 0.0360699
\(917\) 14.0917 0.465348
\(918\) 42.2389 1.39409
\(919\) 19.1194 0.630692 0.315346 0.948977i \(-0.397879\pi\)
0.315346 + 0.948977i \(0.397879\pi\)
\(920\) −11.2111 −0.369619
\(921\) −78.6611 −2.59197
\(922\) 22.4222 0.738436
\(923\) 4.02776 0.132575
\(924\) 0 0
\(925\) 30.4222 1.00028
\(926\) 13.2111 0.434144
\(927\) −41.9361 −1.37736
\(928\) 4.69722 0.154194
\(929\) 9.51388 0.312140 0.156070 0.987746i \(-0.450117\pi\)
0.156070 + 0.987746i \(0.450117\pi\)
\(930\) −1.30278 −0.0427197
\(931\) 1.69722 0.0556243
\(932\) 5.21110 0.170695
\(933\) −13.8167 −0.452337
\(934\) −15.3944 −0.503722
\(935\) 0 0
\(936\) −2.39445 −0.0782650
\(937\) −58.8444 −1.92236 −0.961182 0.275917i \(-0.911019\pi\)
−0.961182 + 0.275917i \(0.911019\pi\)
\(938\) 26.5139 0.865709
\(939\) 23.5139 0.767346
\(940\) −7.81665 −0.254951
\(941\) 14.3667 0.468341 0.234170 0.972196i \(-0.424763\pi\)
0.234170 + 0.972196i \(0.424763\pi\)
\(942\) −39.3305 −1.28146
\(943\) −59.4500 −1.93596
\(944\) 0 0
\(945\) −48.6333 −1.58204
\(946\) 0 0
\(947\) 4.18335 0.135940 0.0679702 0.997687i \(-0.478348\pi\)
0.0679702 + 0.997687i \(0.478348\pi\)
\(948\) −19.2111 −0.623948
\(949\) 1.39445 0.0452657
\(950\) 3.30278 0.107156
\(951\) −25.8167 −0.837162
\(952\) −6.00000 −0.194461
\(953\) −21.6333 −0.700772 −0.350386 0.936605i \(-0.613950\pi\)
−0.350386 + 0.936605i \(0.613950\pi\)
\(954\) −26.8444 −0.869120
\(955\) −7.81665 −0.252941
\(956\) −22.9361 −0.741806
\(957\) 0 0
\(958\) −3.27502 −0.105811
\(959\) −49.5416 −1.59978
\(960\) −4.30278 −0.138871
\(961\) −30.9083 −0.997043
\(962\) 2.78890 0.0899177
\(963\) 88.6611 2.85706
\(964\) −23.9083 −0.770035
\(965\) −10.0639 −0.323969
\(966\) 65.4500 2.10582
\(967\) 19.6333 0.631365 0.315682 0.948865i \(-0.397767\pi\)
0.315682 + 0.948865i \(0.397767\pi\)
\(968\) 0 0
\(969\) −8.60555 −0.276450
\(970\) 10.4222 0.334637
\(971\) 22.9361 0.736054 0.368027 0.929815i \(-0.380033\pi\)
0.368027 + 0.929815i \(0.380033\pi\)
\(972\) −49.8444 −1.59876
\(973\) 6.21110 0.199119
\(974\) 10.3305 0.331012
\(975\) −3.30278 −0.105773
\(976\) 3.21110 0.102785
\(977\) −15.6333 −0.500154 −0.250077 0.968226i \(-0.580456\pi\)
−0.250077 + 0.968226i \(0.580456\pi\)
\(978\) 61.4500 1.96495
\(979\) 0 0
\(980\) −2.21110 −0.0706311
\(981\) −15.8167 −0.504987
\(982\) −31.9361 −1.01912
\(983\) 19.6972 0.628244 0.314122 0.949383i \(-0.398290\pi\)
0.314122 + 0.949383i \(0.398290\pi\)
\(984\) −22.8167 −0.727368
\(985\) 28.1833 0.897996
\(986\) −12.2389 −0.389765
\(987\) 45.6333 1.45252
\(988\) 0.302776 0.00963258
\(989\) 102.478 3.25860
\(990\) 0 0
\(991\) −15.0917 −0.479403 −0.239701 0.970847i \(-0.577050\pi\)
−0.239701 + 0.970847i \(0.577050\pi\)
\(992\) 0.302776 0.00961314
\(993\) 32.1194 1.01928
\(994\) −30.6333 −0.971630
\(995\) −31.0278 −0.983646
\(996\) 35.3305 1.11949
\(997\) −21.8167 −0.690940 −0.345470 0.938430i \(-0.612281\pi\)
−0.345470 + 0.938430i \(0.612281\pi\)
\(998\) −34.0000 −1.07625
\(999\) 149.322 4.72434
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 4598.2.a.bc.1.1 2
11.10 odd 2 418.2.a.d.1.1 2
33.32 even 2 3762.2.a.bb.1.1 2
44.43 even 2 3344.2.a.o.1.2 2
209.208 even 2 7942.2.a.bb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.d.1.1 2 11.10 odd 2
3344.2.a.o.1.2 2 44.43 even 2
3762.2.a.bb.1.1 2 33.32 even 2
4598.2.a.bc.1.1 2 1.1 even 1 trivial
7942.2.a.bb.1.2 2 209.208 even 2