Properties

Label 950.2.a.g.1.1
Level $950$
Weight $2$
Character 950.1
Self dual yes
Analytic conductor $7.586$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.58578819202\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 950.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.41421 q^{3} +1.00000 q^{4} -2.41421 q^{6} -1.58579 q^{7} +1.00000 q^{8} +2.82843 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.41421 q^{3} +1.00000 q^{4} -2.41421 q^{6} -1.58579 q^{7} +1.00000 q^{8} +2.82843 q^{9} +1.41421 q^{11} -2.41421 q^{12} -0.171573 q^{13} -1.58579 q^{14} +1.00000 q^{16} -1.00000 q^{17} +2.82843 q^{18} -1.00000 q^{19} +3.82843 q^{21} +1.41421 q^{22} -9.24264 q^{23} -2.41421 q^{24} -0.171573 q^{26} +0.414214 q^{27} -1.58579 q^{28} -5.82843 q^{29} -2.24264 q^{31} +1.00000 q^{32} -3.41421 q^{33} -1.00000 q^{34} +2.82843 q^{36} -8.48528 q^{37} -1.00000 q^{38} +0.414214 q^{39} +4.24264 q^{41} +3.82843 q^{42} -10.2426 q^{43} +1.41421 q^{44} -9.24264 q^{46} -2.41421 q^{48} -4.48528 q^{49} +2.41421 q^{51} -0.171573 q^{52} +11.4853 q^{53} +0.414214 q^{54} -1.58579 q^{56} +2.41421 q^{57} -5.82843 q^{58} -12.8995 q^{59} +5.75736 q^{61} -2.24264 q^{62} -4.48528 q^{63} +1.00000 q^{64} -3.41421 q^{66} -13.2426 q^{67} -1.00000 q^{68} +22.3137 q^{69} -10.5858 q^{71} +2.82843 q^{72} +5.48528 q^{73} -8.48528 q^{74} -1.00000 q^{76} -2.24264 q^{77} +0.414214 q^{78} +10.4853 q^{79} -9.48528 q^{81} +4.24264 q^{82} +2.48528 q^{83} +3.82843 q^{84} -10.2426 q^{86} +14.0711 q^{87} +1.41421 q^{88} -7.07107 q^{89} +0.272078 q^{91} -9.24264 q^{92} +5.41421 q^{93} -2.41421 q^{96} +11.6569 q^{97} -4.48528 q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} - 6q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} - 6q^{7} + 2q^{8} - 2q^{12} - 6q^{13} - 6q^{14} + 2q^{16} - 2q^{17} - 2q^{19} + 2q^{21} - 10q^{23} - 2q^{24} - 6q^{26} - 2q^{27} - 6q^{28} - 6q^{29} + 4q^{31} + 2q^{32} - 4q^{33} - 2q^{34} - 2q^{38} - 2q^{39} + 2q^{42} - 12q^{43} - 10q^{46} - 2q^{48} + 8q^{49} + 2q^{51} - 6q^{52} + 6q^{53} - 2q^{54} - 6q^{56} + 2q^{57} - 6q^{58} - 6q^{59} + 20q^{61} + 4q^{62} + 8q^{63} + 2q^{64} - 4q^{66} - 18q^{67} - 2q^{68} + 22q^{69} - 24q^{71} - 6q^{73} - 2q^{76} + 4q^{77} - 2q^{78} + 4q^{79} - 2q^{81} - 12q^{83} + 2q^{84} - 12q^{86} + 14q^{87} + 26q^{91} - 10q^{92} + 8q^{93} - 2q^{96} + 12q^{97} + 8q^{98} + 8q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −2.41421 −1.39385 −0.696923 0.717146i \(-0.745448\pi\)
−0.696923 + 0.717146i \(0.745448\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −2.41421 −0.985599
\(7\) −1.58579 −0.599371 −0.299685 0.954038i \(-0.596882\pi\)
−0.299685 + 0.954038i \(0.596882\pi\)
\(8\) 1.00000 0.353553
\(9\) 2.82843 0.942809
\(10\) 0 0
\(11\) 1.41421 0.426401 0.213201 0.977008i \(-0.431611\pi\)
0.213201 + 0.977008i \(0.431611\pi\)
\(12\) −2.41421 −0.696923
\(13\) −0.171573 −0.0475858 −0.0237929 0.999717i \(-0.507574\pi\)
−0.0237929 + 0.999717i \(0.507574\pi\)
\(14\) −1.58579 −0.423819
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 2.82843 0.666667
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 3.82843 0.835431
\(22\) 1.41421 0.301511
\(23\) −9.24264 −1.92722 −0.963612 0.267305i \(-0.913867\pi\)
−0.963612 + 0.267305i \(0.913867\pi\)
\(24\) −2.41421 −0.492799
\(25\) 0 0
\(26\) −0.171573 −0.0336482
\(27\) 0.414214 0.0797154
\(28\) −1.58579 −0.299685
\(29\) −5.82843 −1.08231 −0.541156 0.840922i \(-0.682013\pi\)
−0.541156 + 0.840922i \(0.682013\pi\)
\(30\) 0 0
\(31\) −2.24264 −0.402790 −0.201395 0.979510i \(-0.564548\pi\)
−0.201395 + 0.979510i \(0.564548\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.41421 −0.594338
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 2.82843 0.471405
\(37\) −8.48528 −1.39497 −0.697486 0.716599i \(-0.745698\pi\)
−0.697486 + 0.716599i \(0.745698\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0.414214 0.0663273
\(40\) 0 0
\(41\) 4.24264 0.662589 0.331295 0.943527i \(-0.392515\pi\)
0.331295 + 0.943527i \(0.392515\pi\)
\(42\) 3.82843 0.590739
\(43\) −10.2426 −1.56199 −0.780994 0.624538i \(-0.785287\pi\)
−0.780994 + 0.624538i \(0.785287\pi\)
\(44\) 1.41421 0.213201
\(45\) 0 0
\(46\) −9.24264 −1.36275
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −2.41421 −0.348462
\(49\) −4.48528 −0.640754
\(50\) 0 0
\(51\) 2.41421 0.338058
\(52\) −0.171573 −0.0237929
\(53\) 11.4853 1.57762 0.788812 0.614634i \(-0.210696\pi\)
0.788812 + 0.614634i \(0.210696\pi\)
\(54\) 0.414214 0.0563673
\(55\) 0 0
\(56\) −1.58579 −0.211910
\(57\) 2.41421 0.319770
\(58\) −5.82843 −0.765310
\(59\) −12.8995 −1.67937 −0.839686 0.543073i \(-0.817261\pi\)
−0.839686 + 0.543073i \(0.817261\pi\)
\(60\) 0 0
\(61\) 5.75736 0.737154 0.368577 0.929597i \(-0.379845\pi\)
0.368577 + 0.929597i \(0.379845\pi\)
\(62\) −2.24264 −0.284816
\(63\) −4.48528 −0.565092
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.41421 −0.420261
\(67\) −13.2426 −1.61785 −0.808923 0.587915i \(-0.799949\pi\)
−0.808923 + 0.587915i \(0.799949\pi\)
\(68\) −1.00000 −0.121268
\(69\) 22.3137 2.68625
\(70\) 0 0
\(71\) −10.5858 −1.25630 −0.628151 0.778092i \(-0.716188\pi\)
−0.628151 + 0.778092i \(0.716188\pi\)
\(72\) 2.82843 0.333333
\(73\) 5.48528 0.642004 0.321002 0.947079i \(-0.395980\pi\)
0.321002 + 0.947079i \(0.395980\pi\)
\(74\) −8.48528 −0.986394
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −2.24264 −0.255573
\(78\) 0.414214 0.0469005
\(79\) 10.4853 1.17969 0.589843 0.807518i \(-0.299190\pi\)
0.589843 + 0.807518i \(0.299190\pi\)
\(80\) 0 0
\(81\) −9.48528 −1.05392
\(82\) 4.24264 0.468521
\(83\) 2.48528 0.272795 0.136398 0.990654i \(-0.456448\pi\)
0.136398 + 0.990654i \(0.456448\pi\)
\(84\) 3.82843 0.417716
\(85\) 0 0
\(86\) −10.2426 −1.10449
\(87\) 14.0711 1.50858
\(88\) 1.41421 0.150756
\(89\) −7.07107 −0.749532 −0.374766 0.927119i \(-0.622277\pi\)
−0.374766 + 0.927119i \(0.622277\pi\)
\(90\) 0 0
\(91\) 0.272078 0.0285215
\(92\) −9.24264 −0.963612
\(93\) 5.41421 0.561428
\(94\) 0 0
\(95\) 0 0
\(96\) −2.41421 −0.246400
\(97\) 11.6569 1.18357 0.591787 0.806094i \(-0.298423\pi\)
0.591787 + 0.806094i \(0.298423\pi\)
\(98\) −4.48528 −0.453082
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) −1.07107 −0.106575 −0.0532876 0.998579i \(-0.516970\pi\)
−0.0532876 + 0.998579i \(0.516970\pi\)
\(102\) 2.41421 0.239043
\(103\) 4.24264 0.418040 0.209020 0.977911i \(-0.432973\pi\)
0.209020 + 0.977911i \(0.432973\pi\)
\(104\) −0.171573 −0.0168241
\(105\) 0 0
\(106\) 11.4853 1.11555
\(107\) 5.72792 0.553739 0.276870 0.960908i \(-0.410703\pi\)
0.276870 + 0.960908i \(0.410703\pi\)
\(108\) 0.414214 0.0398577
\(109\) 15.9706 1.52970 0.764851 0.644207i \(-0.222812\pi\)
0.764851 + 0.644207i \(0.222812\pi\)
\(110\) 0 0
\(111\) 20.4853 1.94438
\(112\) −1.58579 −0.149843
\(113\) 1.75736 0.165318 0.0826592 0.996578i \(-0.473659\pi\)
0.0826592 + 0.996578i \(0.473659\pi\)
\(114\) 2.41421 0.226112
\(115\) 0 0
\(116\) −5.82843 −0.541156
\(117\) −0.485281 −0.0448643
\(118\) −12.8995 −1.18749
\(119\) 1.58579 0.145369
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 5.75736 0.521247
\(123\) −10.2426 −0.923548
\(124\) −2.24264 −0.201395
\(125\) 0 0
\(126\) −4.48528 −0.399581
\(127\) −14.4853 −1.28536 −0.642680 0.766134i \(-0.722178\pi\)
−0.642680 + 0.766134i \(0.722178\pi\)
\(128\) 1.00000 0.0883883
\(129\) 24.7279 2.17717
\(130\) 0 0
\(131\) 16.9706 1.48272 0.741362 0.671105i \(-0.234180\pi\)
0.741362 + 0.671105i \(0.234180\pi\)
\(132\) −3.41421 −0.297169
\(133\) 1.58579 0.137505
\(134\) −13.2426 −1.14399
\(135\) 0 0
\(136\) −1.00000 −0.0857493
\(137\) −13.0000 −1.11066 −0.555332 0.831628i \(-0.687409\pi\)
−0.555332 + 0.831628i \(0.687409\pi\)
\(138\) 22.3137 1.89947
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −10.5858 −0.888339
\(143\) −0.242641 −0.0202906
\(144\) 2.82843 0.235702
\(145\) 0 0
\(146\) 5.48528 0.453965
\(147\) 10.8284 0.893114
\(148\) −8.48528 −0.697486
\(149\) 6.34315 0.519651 0.259825 0.965656i \(-0.416335\pi\)
0.259825 + 0.965656i \(0.416335\pi\)
\(150\) 0 0
\(151\) 6.48528 0.527765 0.263882 0.964555i \(-0.414997\pi\)
0.263882 + 0.964555i \(0.414997\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −2.82843 −0.228665
\(154\) −2.24264 −0.180717
\(155\) 0 0
\(156\) 0.414214 0.0331636
\(157\) −0.343146 −0.0273860 −0.0136930 0.999906i \(-0.504359\pi\)
−0.0136930 + 0.999906i \(0.504359\pi\)
\(158\) 10.4853 0.834164
\(159\) −27.7279 −2.19897
\(160\) 0 0
\(161\) 14.6569 1.15512
\(162\) −9.48528 −0.745234
\(163\) 1.75736 0.137647 0.0688235 0.997629i \(-0.478075\pi\)
0.0688235 + 0.997629i \(0.478075\pi\)
\(164\) 4.24264 0.331295
\(165\) 0 0
\(166\) 2.48528 0.192895
\(167\) 9.75736 0.755047 0.377524 0.926000i \(-0.376776\pi\)
0.377524 + 0.926000i \(0.376776\pi\)
\(168\) 3.82843 0.295370
\(169\) −12.9706 −0.997736
\(170\) 0 0
\(171\) −2.82843 −0.216295
\(172\) −10.2426 −0.780994
\(173\) 16.4853 1.25335 0.626676 0.779280i \(-0.284415\pi\)
0.626676 + 0.779280i \(0.284415\pi\)
\(174\) 14.0711 1.06672
\(175\) 0 0
\(176\) 1.41421 0.106600
\(177\) 31.1421 2.34079
\(178\) −7.07107 −0.529999
\(179\) −0.343146 −0.0256479 −0.0128240 0.999918i \(-0.504082\pi\)
−0.0128240 + 0.999918i \(0.504082\pi\)
\(180\) 0 0
\(181\) 8.48528 0.630706 0.315353 0.948974i \(-0.397877\pi\)
0.315353 + 0.948974i \(0.397877\pi\)
\(182\) 0.272078 0.0201678
\(183\) −13.8995 −1.02748
\(184\) −9.24264 −0.681377
\(185\) 0 0
\(186\) 5.41421 0.396989
\(187\) −1.41421 −0.103418
\(188\) 0 0
\(189\) −0.656854 −0.0477791
\(190\) 0 0
\(191\) −18.5563 −1.34269 −0.671345 0.741145i \(-0.734283\pi\)
−0.671345 + 0.741145i \(0.734283\pi\)
\(192\) −2.41421 −0.174231
\(193\) 11.6569 0.839079 0.419539 0.907737i \(-0.362192\pi\)
0.419539 + 0.907737i \(0.362192\pi\)
\(194\) 11.6569 0.836913
\(195\) 0 0
\(196\) −4.48528 −0.320377
\(197\) 20.2426 1.44223 0.721114 0.692816i \(-0.243630\pi\)
0.721114 + 0.692816i \(0.243630\pi\)
\(198\) 4.00000 0.284268
\(199\) 0.757359 0.0536878 0.0268439 0.999640i \(-0.491454\pi\)
0.0268439 + 0.999640i \(0.491454\pi\)
\(200\) 0 0
\(201\) 31.9706 2.25503
\(202\) −1.07107 −0.0753601
\(203\) 9.24264 0.648706
\(204\) 2.41421 0.169029
\(205\) 0 0
\(206\) 4.24264 0.295599
\(207\) −26.1421 −1.81700
\(208\) −0.171573 −0.0118964
\(209\) −1.41421 −0.0978232
\(210\) 0 0
\(211\) 5.72792 0.394326 0.197163 0.980371i \(-0.436827\pi\)
0.197163 + 0.980371i \(0.436827\pi\)
\(212\) 11.4853 0.788812
\(213\) 25.5563 1.75109
\(214\) 5.72792 0.391553
\(215\) 0 0
\(216\) 0.414214 0.0281837
\(217\) 3.55635 0.241421
\(218\) 15.9706 1.08166
\(219\) −13.2426 −0.894855
\(220\) 0 0
\(221\) 0.171573 0.0115412
\(222\) 20.4853 1.37488
\(223\) 20.8284 1.39477 0.697387 0.716694i \(-0.254346\pi\)
0.697387 + 0.716694i \(0.254346\pi\)
\(224\) −1.58579 −0.105955
\(225\) 0 0
\(226\) 1.75736 0.116898
\(227\) 25.2426 1.67541 0.837706 0.546121i \(-0.183896\pi\)
0.837706 + 0.546121i \(0.183896\pi\)
\(228\) 2.41421 0.159885
\(229\) −18.9706 −1.25361 −0.626805 0.779176i \(-0.715638\pi\)
−0.626805 + 0.779176i \(0.715638\pi\)
\(230\) 0 0
\(231\) 5.41421 0.356229
\(232\) −5.82843 −0.382655
\(233\) −8.97056 −0.587681 −0.293841 0.955854i \(-0.594933\pi\)
−0.293841 + 0.955854i \(0.594933\pi\)
\(234\) −0.485281 −0.0317238
\(235\) 0 0
\(236\) −12.8995 −0.839686
\(237\) −25.3137 −1.64430
\(238\) 1.58579 0.102791
\(239\) 12.8995 0.834399 0.417199 0.908815i \(-0.363012\pi\)
0.417199 + 0.908815i \(0.363012\pi\)
\(240\) 0 0
\(241\) −24.9706 −1.60850 −0.804248 0.594294i \(-0.797431\pi\)
−0.804248 + 0.594294i \(0.797431\pi\)
\(242\) −9.00000 −0.578542
\(243\) 21.6569 1.38929
\(244\) 5.75736 0.368577
\(245\) 0 0
\(246\) −10.2426 −0.653047
\(247\) 0.171573 0.0109169
\(248\) −2.24264 −0.142408
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) −27.5563 −1.73934 −0.869671 0.493632i \(-0.835669\pi\)
−0.869671 + 0.493632i \(0.835669\pi\)
\(252\) −4.48528 −0.282546
\(253\) −13.0711 −0.821771
\(254\) −14.4853 −0.908887
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 4.72792 0.294920 0.147460 0.989068i \(-0.452890\pi\)
0.147460 + 0.989068i \(0.452890\pi\)
\(258\) 24.7279 1.53949
\(259\) 13.4558 0.836105
\(260\) 0 0
\(261\) −16.4853 −1.02041
\(262\) 16.9706 1.04844
\(263\) −6.97056 −0.429823 −0.214912 0.976633i \(-0.568946\pi\)
−0.214912 + 0.976633i \(0.568946\pi\)
\(264\) −3.41421 −0.210130
\(265\) 0 0
\(266\) 1.58579 0.0972308
\(267\) 17.0711 1.04473
\(268\) −13.2426 −0.808923
\(269\) 28.6274 1.74544 0.872722 0.488217i \(-0.162353\pi\)
0.872722 + 0.488217i \(0.162353\pi\)
\(270\) 0 0
\(271\) −18.7574 −1.13943 −0.569714 0.821843i \(-0.692946\pi\)
−0.569714 + 0.821843i \(0.692946\pi\)
\(272\) −1.00000 −0.0606339
\(273\) −0.656854 −0.0397546
\(274\) −13.0000 −0.785359
\(275\) 0 0
\(276\) 22.3137 1.34313
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) −12.0000 −0.719712
\(279\) −6.34315 −0.379754
\(280\) 0 0
\(281\) −4.24264 −0.253095 −0.126547 0.991961i \(-0.540390\pi\)
−0.126547 + 0.991961i \(0.540390\pi\)
\(282\) 0 0
\(283\) −3.85786 −0.229326 −0.114663 0.993404i \(-0.536579\pi\)
−0.114663 + 0.993404i \(0.536579\pi\)
\(284\) −10.5858 −0.628151
\(285\) 0 0
\(286\) −0.242641 −0.0143476
\(287\) −6.72792 −0.397137
\(288\) 2.82843 0.166667
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) −28.1421 −1.64972
\(292\) 5.48528 0.321002
\(293\) −11.4853 −0.670977 −0.335489 0.942044i \(-0.608901\pi\)
−0.335489 + 0.942044i \(0.608901\pi\)
\(294\) 10.8284 0.631527
\(295\) 0 0
\(296\) −8.48528 −0.493197
\(297\) 0.585786 0.0339908
\(298\) 6.34315 0.367449
\(299\) 1.58579 0.0917084
\(300\) 0 0
\(301\) 16.2426 0.936210
\(302\) 6.48528 0.373186
\(303\) 2.58579 0.148550
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) −2.82843 −0.161690
\(307\) 6.34315 0.362022 0.181011 0.983481i \(-0.442063\pi\)
0.181011 + 0.983481i \(0.442063\pi\)
\(308\) −2.24264 −0.127786
\(309\) −10.2426 −0.582683
\(310\) 0 0
\(311\) −13.2426 −0.750921 −0.375461 0.926838i \(-0.622515\pi\)
−0.375461 + 0.926838i \(0.622515\pi\)
\(312\) 0.414214 0.0234502
\(313\) −25.9706 −1.46794 −0.733971 0.679180i \(-0.762335\pi\)
−0.733971 + 0.679180i \(0.762335\pi\)
\(314\) −0.343146 −0.0193648
\(315\) 0 0
\(316\) 10.4853 0.589843
\(317\) −7.48528 −0.420415 −0.210208 0.977657i \(-0.567414\pi\)
−0.210208 + 0.977657i \(0.567414\pi\)
\(318\) −27.7279 −1.55490
\(319\) −8.24264 −0.461499
\(320\) 0 0
\(321\) −13.8284 −0.771828
\(322\) 14.6569 0.816795
\(323\) 1.00000 0.0556415
\(324\) −9.48528 −0.526960
\(325\) 0 0
\(326\) 1.75736 0.0973311
\(327\) −38.5563 −2.13217
\(328\) 4.24264 0.234261
\(329\) 0 0
\(330\) 0 0
\(331\) −10.7574 −0.591278 −0.295639 0.955300i \(-0.595533\pi\)
−0.295639 + 0.955300i \(0.595533\pi\)
\(332\) 2.48528 0.136398
\(333\) −24.0000 −1.31519
\(334\) 9.75736 0.533899
\(335\) 0 0
\(336\) 3.82843 0.208858
\(337\) −33.8995 −1.84662 −0.923312 0.384052i \(-0.874528\pi\)
−0.923312 + 0.384052i \(0.874528\pi\)
\(338\) −12.9706 −0.705506
\(339\) −4.24264 −0.230429
\(340\) 0 0
\(341\) −3.17157 −0.171750
\(342\) −2.82843 −0.152944
\(343\) 18.2132 0.983421
\(344\) −10.2426 −0.552246
\(345\) 0 0
\(346\) 16.4853 0.886254
\(347\) 22.4853 1.20707 0.603537 0.797335i \(-0.293758\pi\)
0.603537 + 0.797335i \(0.293758\pi\)
\(348\) 14.0711 0.754288
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 0 0
\(351\) −0.0710678 −0.00379332
\(352\) 1.41421 0.0753778
\(353\) 19.4853 1.03710 0.518548 0.855048i \(-0.326473\pi\)
0.518548 + 0.855048i \(0.326473\pi\)
\(354\) 31.1421 1.65519
\(355\) 0 0
\(356\) −7.07107 −0.374766
\(357\) −3.82843 −0.202622
\(358\) −0.343146 −0.0181358
\(359\) −31.2426 −1.64892 −0.824462 0.565918i \(-0.808522\pi\)
−0.824462 + 0.565918i \(0.808522\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 8.48528 0.445976
\(363\) 21.7279 1.14042
\(364\) 0.272078 0.0142608
\(365\) 0 0
\(366\) −13.8995 −0.726538
\(367\) −25.4558 −1.32878 −0.664392 0.747384i \(-0.731309\pi\)
−0.664392 + 0.747384i \(0.731309\pi\)
\(368\) −9.24264 −0.481806
\(369\) 12.0000 0.624695
\(370\) 0 0
\(371\) −18.2132 −0.945582
\(372\) 5.41421 0.280714
\(373\) −9.00000 −0.466002 −0.233001 0.972476i \(-0.574855\pi\)
−0.233001 + 0.972476i \(0.574855\pi\)
\(374\) −1.41421 −0.0731272
\(375\) 0 0
\(376\) 0 0
\(377\) 1.00000 0.0515026
\(378\) −0.656854 −0.0337849
\(379\) 2.75736 0.141636 0.0708180 0.997489i \(-0.477439\pi\)
0.0708180 + 0.997489i \(0.477439\pi\)
\(380\) 0 0
\(381\) 34.9706 1.79160
\(382\) −18.5563 −0.949425
\(383\) −3.75736 −0.191992 −0.0959960 0.995382i \(-0.530604\pi\)
−0.0959960 + 0.995382i \(0.530604\pi\)
\(384\) −2.41421 −0.123200
\(385\) 0 0
\(386\) 11.6569 0.593318
\(387\) −28.9706 −1.47266
\(388\) 11.6569 0.591787
\(389\) 37.0711 1.87958 0.939789 0.341756i \(-0.111021\pi\)
0.939789 + 0.341756i \(0.111021\pi\)
\(390\) 0 0
\(391\) 9.24264 0.467420
\(392\) −4.48528 −0.226541
\(393\) −40.9706 −2.06669
\(394\) 20.2426 1.01981
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) −24.0000 −1.20453 −0.602263 0.798298i \(-0.705734\pi\)
−0.602263 + 0.798298i \(0.705734\pi\)
\(398\) 0.757359 0.0379630
\(399\) −3.82843 −0.191661
\(400\) 0 0
\(401\) −22.5858 −1.12788 −0.563940 0.825816i \(-0.690715\pi\)
−0.563940 + 0.825816i \(0.690715\pi\)
\(402\) 31.9706 1.59455
\(403\) 0.384776 0.0191671
\(404\) −1.07107 −0.0532876
\(405\) 0 0
\(406\) 9.24264 0.458705
\(407\) −12.0000 −0.594818
\(408\) 2.41421 0.119521
\(409\) −17.2132 −0.851138 −0.425569 0.904926i \(-0.639926\pi\)
−0.425569 + 0.904926i \(0.639926\pi\)
\(410\) 0 0
\(411\) 31.3848 1.54810
\(412\) 4.24264 0.209020
\(413\) 20.4558 1.00657
\(414\) −26.1421 −1.28482
\(415\) 0 0
\(416\) −0.171573 −0.00841205
\(417\) 28.9706 1.41869
\(418\) −1.41421 −0.0691714
\(419\) 16.5858 0.810269 0.405134 0.914257i \(-0.367225\pi\)
0.405134 + 0.914257i \(0.367225\pi\)
\(420\) 0 0
\(421\) 2.51472 0.122560 0.0612799 0.998121i \(-0.480482\pi\)
0.0612799 + 0.998121i \(0.480482\pi\)
\(422\) 5.72792 0.278831
\(423\) 0 0
\(424\) 11.4853 0.557775
\(425\) 0 0
\(426\) 25.5563 1.23821
\(427\) −9.12994 −0.441829
\(428\) 5.72792 0.276870
\(429\) 0.585786 0.0282820
\(430\) 0 0
\(431\) 30.3848 1.46358 0.731792 0.681528i \(-0.238684\pi\)
0.731792 + 0.681528i \(0.238684\pi\)
\(432\) 0.414214 0.0199289
\(433\) −21.5563 −1.03593 −0.517966 0.855401i \(-0.673311\pi\)
−0.517966 + 0.855401i \(0.673311\pi\)
\(434\) 3.55635 0.170710
\(435\) 0 0
\(436\) 15.9706 0.764851
\(437\) 9.24264 0.442135
\(438\) −13.2426 −0.632758
\(439\) −14.2426 −0.679764 −0.339882 0.940468i \(-0.610387\pi\)
−0.339882 + 0.940468i \(0.610387\pi\)
\(440\) 0 0
\(441\) −12.6863 −0.604109
\(442\) 0.171573 0.00816089
\(443\) 4.24264 0.201574 0.100787 0.994908i \(-0.467864\pi\)
0.100787 + 0.994908i \(0.467864\pi\)
\(444\) 20.4853 0.972188
\(445\) 0 0
\(446\) 20.8284 0.986255
\(447\) −15.3137 −0.724314
\(448\) −1.58579 −0.0749214
\(449\) −5.31371 −0.250769 −0.125385 0.992108i \(-0.540017\pi\)
−0.125385 + 0.992108i \(0.540017\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) 1.75736 0.0826592
\(453\) −15.6569 −0.735623
\(454\) 25.2426 1.18470
\(455\) 0 0
\(456\) 2.41421 0.113056
\(457\) 3.00000 0.140334 0.0701670 0.997535i \(-0.477647\pi\)
0.0701670 + 0.997535i \(0.477647\pi\)
\(458\) −18.9706 −0.886436
\(459\) −0.414214 −0.0193338
\(460\) 0 0
\(461\) 27.5563 1.28343 0.641714 0.766944i \(-0.278224\pi\)
0.641714 + 0.766944i \(0.278224\pi\)
\(462\) 5.41421 0.251892
\(463\) 14.1421 0.657241 0.328620 0.944462i \(-0.393416\pi\)
0.328620 + 0.944462i \(0.393416\pi\)
\(464\) −5.82843 −0.270578
\(465\) 0 0
\(466\) −8.97056 −0.415553
\(467\) 24.7279 1.14427 0.572136 0.820159i \(-0.306115\pi\)
0.572136 + 0.820159i \(0.306115\pi\)
\(468\) −0.485281 −0.0224321
\(469\) 21.0000 0.969690
\(470\) 0 0
\(471\) 0.828427 0.0381719
\(472\) −12.8995 −0.593747
\(473\) −14.4853 −0.666034
\(474\) −25.3137 −1.16270
\(475\) 0 0
\(476\) 1.58579 0.0726844
\(477\) 32.4853 1.48740
\(478\) 12.8995 0.590009
\(479\) 31.1127 1.42158 0.710788 0.703407i \(-0.248339\pi\)
0.710788 + 0.703407i \(0.248339\pi\)
\(480\) 0 0
\(481\) 1.45584 0.0663808
\(482\) −24.9706 −1.13738
\(483\) −35.3848 −1.61006
\(484\) −9.00000 −0.409091
\(485\) 0 0
\(486\) 21.6569 0.982375
\(487\) 1.79899 0.0815200 0.0407600 0.999169i \(-0.487022\pi\)
0.0407600 + 0.999169i \(0.487022\pi\)
\(488\) 5.75736 0.260623
\(489\) −4.24264 −0.191859
\(490\) 0 0
\(491\) −2.44365 −0.110280 −0.0551402 0.998479i \(-0.517561\pi\)
−0.0551402 + 0.998479i \(0.517561\pi\)
\(492\) −10.2426 −0.461774
\(493\) 5.82843 0.262499
\(494\) 0.171573 0.00771943
\(495\) 0 0
\(496\) −2.24264 −0.100698
\(497\) 16.7868 0.752991
\(498\) −6.00000 −0.268866
\(499\) −34.2426 −1.53291 −0.766456 0.642297i \(-0.777981\pi\)
−0.766456 + 0.642297i \(0.777981\pi\)
\(500\) 0 0
\(501\) −23.5563 −1.05242
\(502\) −27.5563 −1.22990
\(503\) −39.7279 −1.77138 −0.885690 0.464277i \(-0.846314\pi\)
−0.885690 + 0.464277i \(0.846314\pi\)
\(504\) −4.48528 −0.199790
\(505\) 0 0
\(506\) −13.0711 −0.581080
\(507\) 31.3137 1.39069
\(508\) −14.4853 −0.642680
\(509\) −4.97056 −0.220316 −0.110158 0.993914i \(-0.535136\pi\)
−0.110158 + 0.993914i \(0.535136\pi\)
\(510\) 0 0
\(511\) −8.69848 −0.384798
\(512\) 1.00000 0.0441942
\(513\) −0.414214 −0.0182880
\(514\) 4.72792 0.208540
\(515\) 0 0
\(516\) 24.7279 1.08859
\(517\) 0 0
\(518\) 13.4558 0.591216
\(519\) −39.7990 −1.74698
\(520\) 0 0
\(521\) 0.686292 0.0300670 0.0150335 0.999887i \(-0.495215\pi\)
0.0150335 + 0.999887i \(0.495215\pi\)
\(522\) −16.4853 −0.721541
\(523\) 27.7279 1.21246 0.606229 0.795290i \(-0.292682\pi\)
0.606229 + 0.795290i \(0.292682\pi\)
\(524\) 16.9706 0.741362
\(525\) 0 0
\(526\) −6.97056 −0.303931
\(527\) 2.24264 0.0976910
\(528\) −3.41421 −0.148585
\(529\) 62.4264 2.71419
\(530\) 0 0
\(531\) −36.4853 −1.58333
\(532\) 1.58579 0.0687526
\(533\) −0.727922 −0.0315298
\(534\) 17.0711 0.738737
\(535\) 0 0
\(536\) −13.2426 −0.571995
\(537\) 0.828427 0.0357493
\(538\) 28.6274 1.23422
\(539\) −6.34315 −0.273219
\(540\) 0 0
\(541\) 18.2426 0.784312 0.392156 0.919899i \(-0.371729\pi\)
0.392156 + 0.919899i \(0.371729\pi\)
\(542\) −18.7574 −0.805698
\(543\) −20.4853 −0.879108
\(544\) −1.00000 −0.0428746
\(545\) 0 0
\(546\) −0.656854 −0.0281108
\(547\) −5.31371 −0.227198 −0.113599 0.993527i \(-0.536238\pi\)
−0.113599 + 0.993527i \(0.536238\pi\)
\(548\) −13.0000 −0.555332
\(549\) 16.2843 0.694996
\(550\) 0 0
\(551\) 5.82843 0.248299
\(552\) 22.3137 0.949735
\(553\) −16.6274 −0.707070
\(554\) 0 0
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) −16.0000 −0.677942 −0.338971 0.940797i \(-0.610079\pi\)
−0.338971 + 0.940797i \(0.610079\pi\)
\(558\) −6.34315 −0.268527
\(559\) 1.75736 0.0743284
\(560\) 0 0
\(561\) 3.41421 0.144148
\(562\) −4.24264 −0.178965
\(563\) 28.9706 1.22096 0.610482 0.792030i \(-0.290976\pi\)
0.610482 + 0.792030i \(0.290976\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −3.85786 −0.162158
\(567\) 15.0416 0.631689
\(568\) −10.5858 −0.444170
\(569\) 28.2843 1.18574 0.592869 0.805299i \(-0.297995\pi\)
0.592869 + 0.805299i \(0.297995\pi\)
\(570\) 0 0
\(571\) 6.24264 0.261246 0.130623 0.991432i \(-0.458302\pi\)
0.130623 + 0.991432i \(0.458302\pi\)
\(572\) −0.242641 −0.0101453
\(573\) 44.7990 1.87150
\(574\) −6.72792 −0.280818
\(575\) 0 0
\(576\) 2.82843 0.117851
\(577\) −2.31371 −0.0963209 −0.0481605 0.998840i \(-0.515336\pi\)
−0.0481605 + 0.998840i \(0.515336\pi\)
\(578\) −16.0000 −0.665512
\(579\) −28.1421 −1.16955
\(580\) 0 0
\(581\) −3.94113 −0.163505
\(582\) −28.1421 −1.16653
\(583\) 16.2426 0.672701
\(584\) 5.48528 0.226983
\(585\) 0 0
\(586\) −11.4853 −0.474453
\(587\) −21.7574 −0.898022 −0.449011 0.893526i \(-0.648224\pi\)
−0.449011 + 0.893526i \(0.648224\pi\)
\(588\) 10.8284 0.446557
\(589\) 2.24264 0.0924064
\(590\) 0 0
\(591\) −48.8701 −2.01025
\(592\) −8.48528 −0.348743
\(593\) −22.0000 −0.903432 −0.451716 0.892162i \(-0.649188\pi\)
−0.451716 + 0.892162i \(0.649188\pi\)
\(594\) 0.585786 0.0240351
\(595\) 0 0
\(596\) 6.34315 0.259825
\(597\) −1.82843 −0.0748325
\(598\) 1.58579 0.0648476
\(599\) 27.2132 1.11190 0.555951 0.831215i \(-0.312354\pi\)
0.555951 + 0.831215i \(0.312354\pi\)
\(600\) 0 0
\(601\) −11.7574 −0.479593 −0.239796 0.970823i \(-0.577081\pi\)
−0.239796 + 0.970823i \(0.577081\pi\)
\(602\) 16.2426 0.662001
\(603\) −37.4558 −1.52532
\(604\) 6.48528 0.263882
\(605\) 0 0
\(606\) 2.58579 0.105040
\(607\) −8.82843 −0.358335 −0.179167 0.983819i \(-0.557340\pi\)
−0.179167 + 0.983819i \(0.557340\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −22.3137 −0.904197
\(610\) 0 0
\(611\) 0 0
\(612\) −2.82843 −0.114332
\(613\) −47.6985 −1.92652 −0.963262 0.268564i \(-0.913451\pi\)
−0.963262 + 0.268564i \(0.913451\pi\)
\(614\) 6.34315 0.255989
\(615\) 0 0
\(616\) −2.24264 −0.0903586
\(617\) −12.4853 −0.502639 −0.251319 0.967904i \(-0.580864\pi\)
−0.251319 + 0.967904i \(0.580864\pi\)
\(618\) −10.2426 −0.412019
\(619\) 16.2426 0.652847 0.326423 0.945224i \(-0.394156\pi\)
0.326423 + 0.945224i \(0.394156\pi\)
\(620\) 0 0
\(621\) −3.82843 −0.153629
\(622\) −13.2426 −0.530982
\(623\) 11.2132 0.449248
\(624\) 0.414214 0.0165818
\(625\) 0 0
\(626\) −25.9706 −1.03799
\(627\) 3.41421 0.136351
\(628\) −0.343146 −0.0136930
\(629\) 8.48528 0.338330
\(630\) 0 0
\(631\) −5.02944 −0.200219 −0.100109 0.994976i \(-0.531919\pi\)
−0.100109 + 0.994976i \(0.531919\pi\)
\(632\) 10.4853 0.417082
\(633\) −13.8284 −0.549631
\(634\) −7.48528 −0.297279
\(635\) 0 0
\(636\) −27.7279 −1.09948
\(637\) 0.769553 0.0304908
\(638\) −8.24264 −0.326329
\(639\) −29.9411 −1.18445
\(640\) 0 0
\(641\) 30.0416 1.18657 0.593287 0.804991i \(-0.297830\pi\)
0.593287 + 0.804991i \(0.297830\pi\)
\(642\) −13.8284 −0.545764
\(643\) −2.48528 −0.0980099 −0.0490050 0.998799i \(-0.515605\pi\)
−0.0490050 + 0.998799i \(0.515605\pi\)
\(644\) 14.6569 0.577561
\(645\) 0 0
\(646\) 1.00000 0.0393445
\(647\) 27.2426 1.07102 0.535509 0.844529i \(-0.320120\pi\)
0.535509 + 0.844529i \(0.320120\pi\)
\(648\) −9.48528 −0.372617
\(649\) −18.2426 −0.716086
\(650\) 0 0
\(651\) −8.58579 −0.336504
\(652\) 1.75736 0.0688235
\(653\) −4.97056 −0.194513 −0.0972566 0.995259i \(-0.531007\pi\)
−0.0972566 + 0.995259i \(0.531007\pi\)
\(654\) −38.5563 −1.50767
\(655\) 0 0
\(656\) 4.24264 0.165647
\(657\) 15.5147 0.605287
\(658\) 0 0
\(659\) −0.899495 −0.0350393 −0.0175197 0.999847i \(-0.505577\pi\)
−0.0175197 + 0.999847i \(0.505577\pi\)
\(660\) 0 0
\(661\) 32.4558 1.26239 0.631193 0.775626i \(-0.282566\pi\)
0.631193 + 0.775626i \(0.282566\pi\)
\(662\) −10.7574 −0.418097
\(663\) −0.414214 −0.0160867
\(664\) 2.48528 0.0964476
\(665\) 0 0
\(666\) −24.0000 −0.929981
\(667\) 53.8701 2.08586
\(668\) 9.75736 0.377524
\(669\) −50.2843 −1.94410
\(670\) 0 0
\(671\) 8.14214 0.314324
\(672\) 3.82843 0.147685
\(673\) 12.0000 0.462566 0.231283 0.972887i \(-0.425708\pi\)
0.231283 + 0.972887i \(0.425708\pi\)
\(674\) −33.8995 −1.30576
\(675\) 0 0
\(676\) −12.9706 −0.498868
\(677\) −26.9411 −1.03543 −0.517716 0.855553i \(-0.673218\pi\)
−0.517716 + 0.855553i \(0.673218\pi\)
\(678\) −4.24264 −0.162938
\(679\) −18.4853 −0.709400
\(680\) 0 0
\(681\) −60.9411 −2.33527
\(682\) −3.17157 −0.121446
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) −2.82843 −0.108148
\(685\) 0 0
\(686\) 18.2132 0.695383
\(687\) 45.7990 1.74734
\(688\) −10.2426 −0.390497
\(689\) −1.97056 −0.0750725
\(690\) 0 0
\(691\) 5.51472 0.209790 0.104895 0.994483i \(-0.466549\pi\)
0.104895 + 0.994483i \(0.466549\pi\)
\(692\) 16.4853 0.626676
\(693\) −6.34315 −0.240956
\(694\) 22.4853 0.853530
\(695\) 0 0
\(696\) 14.0711 0.533362
\(697\) −4.24264 −0.160701
\(698\) 6.00000 0.227103
\(699\) 21.6569 0.819137
\(700\) 0 0
\(701\) −16.9706 −0.640969 −0.320485 0.947254i \(-0.603846\pi\)
−0.320485 + 0.947254i \(0.603846\pi\)
\(702\) −0.0710678 −0.00268228
\(703\) 8.48528 0.320028
\(704\) 1.41421 0.0533002
\(705\) 0 0
\(706\) 19.4853 0.733338
\(707\) 1.69848 0.0638781
\(708\) 31.1421 1.17039
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 0 0
\(711\) 29.6569 1.11222
\(712\) −7.07107 −0.264999
\(713\) 20.7279 0.776267
\(714\) −3.82843 −0.143275
\(715\) 0 0
\(716\) −0.343146 −0.0128240
\(717\) −31.1421 −1.16302
\(718\) −31.2426 −1.16596
\(719\) −24.8995 −0.928594 −0.464297 0.885679i \(-0.653693\pi\)
−0.464297 + 0.885679i \(0.653693\pi\)
\(720\) 0 0
\(721\) −6.72792 −0.250561
\(722\) 1.00000 0.0372161
\(723\) 60.2843 2.24200
\(724\) 8.48528 0.315353
\(725\) 0 0
\(726\) 21.7279 0.806399
\(727\) 15.7279 0.583316 0.291658 0.956523i \(-0.405793\pi\)
0.291658 + 0.956523i \(0.405793\pi\)
\(728\) 0.272078 0.0100839
\(729\) −23.8284 −0.882534
\(730\) 0 0
\(731\) 10.2426 0.378838
\(732\) −13.8995 −0.513740
\(733\) −12.0000 −0.443230 −0.221615 0.975134i \(-0.571133\pi\)
−0.221615 + 0.975134i \(0.571133\pi\)
\(734\) −25.4558 −0.939592
\(735\) 0 0
\(736\) −9.24264 −0.340688
\(737\) −18.7279 −0.689852
\(738\) 12.0000 0.441726
\(739\) 26.7279 0.983203 0.491601 0.870820i \(-0.336412\pi\)
0.491601 + 0.870820i \(0.336412\pi\)
\(740\) 0 0
\(741\) −0.414214 −0.0152165
\(742\) −18.2132 −0.668628
\(743\) 18.7279 0.687061 0.343530 0.939142i \(-0.388377\pi\)
0.343530 + 0.939142i \(0.388377\pi\)
\(744\) 5.41421 0.198495
\(745\) 0 0
\(746\) −9.00000 −0.329513
\(747\) 7.02944 0.257194
\(748\) −1.41421 −0.0517088
\(749\) −9.08326 −0.331895
\(750\) 0 0
\(751\) −1.27208 −0.0464188 −0.0232094 0.999731i \(-0.507388\pi\)
−0.0232094 + 0.999731i \(0.507388\pi\)
\(752\) 0 0
\(753\) 66.5269 2.42438
\(754\) 1.00000 0.0364179
\(755\) 0 0
\(756\) −0.656854 −0.0238896
\(757\) −23.6569 −0.859823 −0.429911 0.902871i \(-0.641455\pi\)
−0.429911 + 0.902871i \(0.641455\pi\)
\(758\) 2.75736 0.100152
\(759\) 31.5563 1.14542
\(760\) 0 0
\(761\) 10.0294 0.363567 0.181783 0.983339i \(-0.441813\pi\)
0.181783 + 0.983339i \(0.441813\pi\)
\(762\) 34.9706 1.26685
\(763\) −25.3259 −0.916859
\(764\) −18.5563 −0.671345
\(765\) 0 0
\(766\) −3.75736 −0.135759
\(767\) 2.21320 0.0799141
\(768\) −2.41421 −0.0871154
\(769\) 14.4558 0.521291 0.260646 0.965435i \(-0.416065\pi\)
0.260646 + 0.965435i \(0.416065\pi\)
\(770\) 0 0
\(771\) −11.4142 −0.411073
\(772\) 11.6569 0.419539
\(773\) 19.9706 0.718291 0.359146 0.933282i \(-0.383068\pi\)
0.359146 + 0.933282i \(0.383068\pi\)
\(774\) −28.9706 −1.04133
\(775\) 0 0
\(776\) 11.6569 0.418457
\(777\) −32.4853 −1.16540
\(778\) 37.0711 1.32906
\(779\) −4.24264 −0.152008
\(780\) 0 0
\(781\) −14.9706 −0.535689
\(782\) 9.24264 0.330516
\(783\) −2.41421 −0.0862770
\(784\) −4.48528 −0.160189
\(785\) 0 0
\(786\) −40.9706 −1.46137
\(787\) −35.1838 −1.25417 −0.627083 0.778953i \(-0.715751\pi\)
−0.627083 + 0.778953i \(0.715751\pi\)
\(788\) 20.2426 0.721114
\(789\) 16.8284 0.599108
\(790\) 0 0
\(791\) −2.78680 −0.0990871
\(792\) 4.00000 0.142134
\(793\) −0.987807 −0.0350780
\(794\) −24.0000 −0.851728
\(795\) 0 0
\(796\) 0.757359 0.0268439
\(797\) 30.5147 1.08089 0.540443 0.841380i \(-0.318257\pi\)
0.540443 + 0.841380i \(0.318257\pi\)
\(798\) −3.82843 −0.135525
\(799\) 0 0
\(800\) 0 0
\(801\) −20.0000 −0.706665
\(802\) −22.5858 −0.797532
\(803\) 7.75736 0.273751
\(804\) 31.9706 1.12751
\(805\) 0 0
\(806\) 0.384776 0.0135532
\(807\) −69.1127 −2.43288
\(808\) −1.07107 −0.0376800
\(809\) 10.7990 0.379672 0.189836 0.981816i \(-0.439204\pi\)
0.189836 + 0.981816i \(0.439204\pi\)
\(810\) 0 0
\(811\) −6.69848 −0.235216 −0.117608 0.993060i \(-0.537523\pi\)
−0.117608 + 0.993060i \(0.537523\pi\)
\(812\) 9.24264 0.324353
\(813\) 45.2843 1.58819
\(814\) −12.0000 −0.420600
\(815\) 0 0
\(816\) 2.41421 0.0845144
\(817\) 10.2426 0.358345
\(818\) −17.2132 −0.601846
\(819\) 0.769553 0.0268903
\(820\) 0 0
\(821\) 17.6569 0.616228 0.308114 0.951349i \(-0.400302\pi\)
0.308114 + 0.951349i \(0.400302\pi\)
\(822\) 31.3848 1.09467
\(823\) −9.72792 −0.339094 −0.169547 0.985522i \(-0.554230\pi\)
−0.169547 + 0.985522i \(0.554230\pi\)
\(824\) 4.24264 0.147799
\(825\) 0 0
\(826\) 20.4558 0.711750
\(827\) −38.6985 −1.34568 −0.672839 0.739789i \(-0.734925\pi\)
−0.672839 + 0.739789i \(0.734925\pi\)
\(828\) −26.1421 −0.908502
\(829\) −40.4558 −1.40509 −0.702545 0.711640i \(-0.747953\pi\)
−0.702545 + 0.711640i \(0.747953\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −0.171573 −0.00594822
\(833\) 4.48528 0.155406
\(834\) 28.9706 1.00317
\(835\) 0 0
\(836\) −1.41421 −0.0489116
\(837\) −0.928932 −0.0321086
\(838\) 16.5858 0.572946
\(839\) −22.6274 −0.781185 −0.390593 0.920564i \(-0.627730\pi\)
−0.390593 + 0.920564i \(0.627730\pi\)
\(840\) 0 0
\(841\) 4.97056 0.171399
\(842\) 2.51472 0.0866629
\(843\) 10.2426 0.352775
\(844\) 5.72792 0.197163
\(845\) 0 0
\(846\) 0 0
\(847\) 14.2721 0.490394
\(848\) 11.4853 0.394406
\(849\) 9.31371 0.319646
\(850\) 0 0
\(851\) 78.4264 2.68842
\(852\) 25.5563 0.875546
\(853\) −39.9411 −1.36756 −0.683779 0.729689i \(-0.739665\pi\)
−0.683779 + 0.729689i \(0.739665\pi\)
\(854\) −9.12994 −0.312420
\(855\) 0 0
\(856\) 5.72792 0.195776
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0.585786 0.0199984
\(859\) −20.7279 −0.707228 −0.353614 0.935392i \(-0.615047\pi\)
−0.353614 + 0.935392i \(0.615047\pi\)
\(860\) 0 0
\(861\) 16.2426 0.553548
\(862\) 30.3848 1.03491
\(863\) 24.7279 0.841748 0.420874 0.907119i \(-0.361723\pi\)
0.420874 + 0.907119i \(0.361723\pi\)
\(864\) 0.414214 0.0140918
\(865\) 0 0
\(866\) −21.5563 −0.732515
\(867\) 38.6274 1.31186
\(868\) 3.55635 0.120710
\(869\) 14.8284 0.503020
\(870\) 0 0
\(871\) 2.27208 0.0769864
\(872\) 15.9706 0.540831
\(873\) 32.9706 1.11588
\(874\) 9.24264 0.312637
\(875\) 0 0
\(876\) −13.2426 −0.447427
\(877\) 47.1421 1.59188 0.795938 0.605378i \(-0.206978\pi\)
0.795938 + 0.605378i \(0.206978\pi\)
\(878\) −14.2426 −0.480666
\(879\) 27.7279 0.935240
\(880\) 0 0
\(881\) 39.5980 1.33409 0.667045 0.745018i \(-0.267559\pi\)
0.667045 + 0.745018i \(0.267559\pi\)
\(882\) −12.6863 −0.427170
\(883\) 2.48528 0.0836364 0.0418182 0.999125i \(-0.486685\pi\)
0.0418182 + 0.999125i \(0.486685\pi\)
\(884\) 0.171573 0.00577062
\(885\) 0 0
\(886\) 4.24264 0.142534
\(887\) 2.78680 0.0935715 0.0467857 0.998905i \(-0.485102\pi\)
0.0467857 + 0.998905i \(0.485102\pi\)
\(888\) 20.4853 0.687441
\(889\) 22.9706 0.770408
\(890\) 0 0
\(891\) −13.4142 −0.449393
\(892\) 20.8284 0.697387
\(893\) 0 0
\(894\) −15.3137 −0.512167
\(895\) 0 0
\(896\) −1.58579 −0.0529774
\(897\) −3.82843 −0.127827
\(898\) −5.31371 −0.177321
\(899\) 13.0711 0.435945
\(900\) 0 0
\(901\) −11.4853 −0.382630
\(902\) 6.00000 0.199778
\(903\) −39.2132 −1.30493
\(904\) 1.75736 0.0584489
\(905\) 0 0
\(906\) −15.6569 −0.520164
\(907\) 20.6985 0.687282 0.343641 0.939101i \(-0.388340\pi\)
0.343641 + 0.939101i \(0.388340\pi\)
\(908\) 25.2426 0.837706
\(909\) −3.02944 −0.100480
\(910\) 0 0
\(911\) −16.2843 −0.539522 −0.269761 0.962927i \(-0.586945\pi\)
−0.269761 + 0.962927i \(0.586945\pi\)
\(912\) 2.41421 0.0799426
\(913\) 3.51472 0.116320
\(914\) 3.00000 0.0992312
\(915\) 0 0
\(916\) −18.9706 −0.626805
\(917\) −26.9117 −0.888702
\(918\) −0.414214 −0.0136711
\(919\) 36.2132 1.19456 0.597282 0.802032i \(-0.296247\pi\)
0.597282 + 0.802032i \(0.296247\pi\)
\(920\) 0 0
\(921\) −15.3137 −0.504604
\(922\) 27.5563 0.907520
\(923\) 1.81623 0.0597821
\(924\) 5.41421 0.178115
\(925\) 0 0
\(926\) 14.1421 0.464739
\(927\) 12.0000 0.394132
\(928\) −5.82843 −0.191327
\(929\) −17.8284 −0.584932 −0.292466 0.956276i \(-0.594476\pi\)
−0.292466 + 0.956276i \(0.594476\pi\)
\(930\) 0 0
\(931\) 4.48528 0.146999
\(932\) −8.97056 −0.293841
\(933\) 31.9706 1.04667
\(934\) 24.7279 0.809122
\(935\) 0 0
\(936\) −0.485281 −0.0158619
\(937\) −27.0000 −0.882052 −0.441026 0.897494i \(-0.645385\pi\)
−0.441026 + 0.897494i \(0.645385\pi\)
\(938\) 21.0000 0.685674
\(939\) 62.6985 2.04609
\(940\) 0 0
\(941\) 24.8579 0.810343 0.405172 0.914241i \(-0.367212\pi\)
0.405172 + 0.914241i \(0.367212\pi\)
\(942\) 0.828427 0.0269916
\(943\) −39.2132 −1.27696
\(944\) −12.8995 −0.419843
\(945\) 0 0
\(946\) −14.4853 −0.470957
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −25.3137 −0.822151
\(949\) −0.941125 −0.0305502
\(950\) 0 0
\(951\) 18.0711 0.585995
\(952\) 1.58579 0.0513956
\(953\) 6.72792 0.217939 0.108969 0.994045i \(-0.465245\pi\)
0.108969 + 0.994045i \(0.465245\pi\)
\(954\) 32.4853 1.05175
\(955\) 0 0
\(956\) 12.8995 0.417199
\(957\) 19.8995 0.643259
\(958\) 31.1127 1.00521
\(959\) 20.6152 0.665700
\(960\) 0 0
\(961\) −25.9706 −0.837760
\(962\) 1.45584 0.0469383
\(963\) 16.2010 0.522070
\(964\) −24.9706 −0.804248
\(965\) 0 0
\(966\) −35.3848 −1.13849
\(967\) 19.1127 0.614623 0.307311 0.951609i \(-0.400571\pi\)
0.307311 + 0.951609i \(0.400571\pi\)
\(968\) −9.00000 −0.289271
\(969\) −2.41421 −0.0775557
\(970\) 0 0
\(971\) 30.3431 0.973758 0.486879 0.873469i \(-0.338135\pi\)
0.486879 + 0.873469i \(0.338135\pi\)
\(972\) 21.6569 0.694644
\(973\) 19.0294 0.610056
\(974\) 1.79899 0.0576434
\(975\) 0 0
\(976\) 5.75736 0.184289
\(977\) 8.24264 0.263705 0.131853 0.991269i \(-0.457907\pi\)
0.131853 + 0.991269i \(0.457907\pi\)
\(978\) −4.24264 −0.135665
\(979\) −10.0000 −0.319601
\(980\) 0 0
\(981\) 45.1716 1.44222
\(982\) −2.44365 −0.0779800
\(983\) 0.544156 0.0173559 0.00867794 0.999962i \(-0.497238\pi\)
0.00867794 + 0.999962i \(0.497238\pi\)
\(984\) −10.2426 −0.326523
\(985\) 0 0
\(986\) 5.82843 0.185615
\(987\) 0 0
\(988\) 0.171573 0.00545846
\(989\) 94.6690 3.01030
\(990\) 0 0
\(991\) 22.2426 0.706561 0.353280 0.935517i \(-0.385066\pi\)
0.353280 + 0.935517i \(0.385066\pi\)
\(992\) −2.24264 −0.0712039
\(993\) 25.9706 0.824151
\(994\) 16.7868 0.532445
\(995\) 0 0
\(996\) −6.00000 −0.190117
\(997\) −23.2721 −0.737034 −0.368517 0.929621i \(-0.620134\pi\)
−0.368517 + 0.929621i \(0.620134\pi\)
\(998\) −34.2426 −1.08393
\(999\) −3.51472 −0.111201
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 950.2.a.g.1.1 2
3.2 odd 2 8550.2.a.bn.1.2 2
4.3 odd 2 7600.2.a.bg.1.2 2
5.2 odd 4 190.2.b.a.39.4 yes 4
5.3 odd 4 190.2.b.a.39.1 4
5.4 even 2 950.2.a.f.1.2 2
15.2 even 4 1710.2.d.c.1369.2 4
15.8 even 4 1710.2.d.c.1369.4 4
15.14 odd 2 8550.2.a.cb.1.1 2
20.3 even 4 1520.2.d.e.609.4 4
20.7 even 4 1520.2.d.e.609.1 4
20.19 odd 2 7600.2.a.v.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.b.a.39.1 4 5.3 odd 4
190.2.b.a.39.4 yes 4 5.2 odd 4
950.2.a.f.1.2 2 5.4 even 2
950.2.a.g.1.1 2 1.1 even 1 trivial
1520.2.d.e.609.1 4 20.7 even 4
1520.2.d.e.609.4 4 20.3 even 4
1710.2.d.c.1369.2 4 15.2 even 4
1710.2.d.c.1369.4 4 15.8 even 4
7600.2.a.v.1.1 2 20.19 odd 2
7600.2.a.bg.1.2 2 4.3 odd 2
8550.2.a.bn.1.2 2 3.2 odd 2
8550.2.a.cb.1.1 2 15.14 odd 2