Properties

Label 4014.2.a.r.1.1
Level 4014
Weight 2
Character 4014.1
Self dual yes
Analytic conductor 32.052
Analytic rank 1
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4014.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0519513713\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.356173.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1338)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.38363\)
Character \(\chi\) = 4014.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -4.35484 q^{5} +3.05676 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -4.35484 q^{5} +3.05676 q^{7} +1.00000 q^{8} -4.35484 q^{10} +0.734963 q^{11} -2.43689 q^{13} +3.05676 q^{14} +1.00000 q^{16} +0.681711 q^{17} -1.87811 q^{19} -4.35484 q^{20} +0.734963 q^{22} -6.91115 q^{23} +13.9646 q^{25} -2.43689 q^{26} +3.05676 q^{28} -0.113306 q^{29} +1.56736 q^{31} +1.00000 q^{32} +0.681711 q^{34} -13.3117 q^{35} -3.71884 q^{37} -1.87811 q^{38} -4.35484 q^{40} +11.1102 q^{41} -10.9512 q^{43} +0.734963 q^{44} -6.91115 q^{46} +6.22966 q^{47} +2.34379 q^{49} +13.9646 q^{50} -2.43689 q^{52} -6.47426 q^{53} -3.20064 q^{55} +3.05676 q^{56} -0.113306 q^{58} -7.94099 q^{59} -4.54537 q^{61} +1.56736 q^{62} +1.00000 q^{64} +10.6122 q^{65} -2.23117 q^{67} +0.681711 q^{68} -13.3117 q^{70} -10.5641 q^{71} -3.30232 q^{73} -3.71884 q^{74} -1.87811 q^{76} +2.24661 q^{77} -3.98486 q^{79} -4.35484 q^{80} +11.1102 q^{82} +9.77231 q^{83} -2.96874 q^{85} -10.9512 q^{86} +0.734963 q^{88} +3.23802 q^{89} -7.44898 q^{91} -6.91115 q^{92} +6.22966 q^{94} +8.17887 q^{95} -11.1019 q^{97} +2.34379 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{2} + 5q^{4} - 5q^{5} - q^{7} + 5q^{8} + O(q^{10}) \) \( 5q + 5q^{2} + 5q^{4} - 5q^{5} - q^{7} + 5q^{8} - 5q^{10} - 9q^{11} - q^{14} + 5q^{16} - 6q^{17} - 4q^{19} - 5q^{20} - 9q^{22} - 16q^{23} + 8q^{25} - q^{28} - 8q^{29} - q^{31} + 5q^{32} - 6q^{34} - 22q^{35} - 2q^{37} - 4q^{38} - 5q^{40} - 4q^{41} + 3q^{43} - 9q^{44} - 16q^{46} - 18q^{47} + 2q^{49} + 8q^{50} - 26q^{53} + q^{55} - q^{56} - 8q^{58} - 21q^{59} - 20q^{61} - q^{62} + 5q^{64} + 3q^{65} - 5q^{67} - 6q^{68} - 22q^{70} - 17q^{71} + 5q^{73} - 2q^{74} - 4q^{76} - 2q^{77} - 21q^{79} - 5q^{80} - 4q^{82} - 11q^{83} - 12q^{85} + 3q^{86} - 9q^{88} + 5q^{89} - 10q^{91} - 16q^{92} - 18q^{94} - 10q^{95} - 11q^{97} + 2q^{98} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −4.35484 −1.94754 −0.973772 0.227528i \(-0.926936\pi\)
−0.973772 + 0.227528i \(0.926936\pi\)
\(6\) 0 0
\(7\) 3.05676 1.15535 0.577674 0.816268i \(-0.303961\pi\)
0.577674 + 0.816268i \(0.303961\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −4.35484 −1.37712
\(11\) 0.734963 0.221600 0.110800 0.993843i \(-0.464659\pi\)
0.110800 + 0.993843i \(0.464659\pi\)
\(12\) 0 0
\(13\) −2.43689 −0.675871 −0.337935 0.941169i \(-0.609728\pi\)
−0.337935 + 0.941169i \(0.609728\pi\)
\(14\) 3.05676 0.816954
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.681711 0.165339 0.0826696 0.996577i \(-0.473655\pi\)
0.0826696 + 0.996577i \(0.473655\pi\)
\(18\) 0 0
\(19\) −1.87811 −0.430868 −0.215434 0.976518i \(-0.569117\pi\)
−0.215434 + 0.976518i \(0.569117\pi\)
\(20\) −4.35484 −0.973772
\(21\) 0 0
\(22\) 0.734963 0.156695
\(23\) −6.91115 −1.44107 −0.720537 0.693416i \(-0.756105\pi\)
−0.720537 + 0.693416i \(0.756105\pi\)
\(24\) 0 0
\(25\) 13.9646 2.79292
\(26\) −2.43689 −0.477913
\(27\) 0 0
\(28\) 3.05676 0.577674
\(29\) −0.113306 −0.0210404 −0.0105202 0.999945i \(-0.503349\pi\)
−0.0105202 + 0.999945i \(0.503349\pi\)
\(30\) 0 0
\(31\) 1.56736 0.281506 0.140753 0.990045i \(-0.455048\pi\)
0.140753 + 0.990045i \(0.455048\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0.681711 0.116912
\(35\) −13.3117 −2.25009
\(36\) 0 0
\(37\) −3.71884 −0.611374 −0.305687 0.952132i \(-0.598886\pi\)
−0.305687 + 0.952132i \(0.598886\pi\)
\(38\) −1.87811 −0.304670
\(39\) 0 0
\(40\) −4.35484 −0.688560
\(41\) 11.1102 1.73513 0.867563 0.497327i \(-0.165685\pi\)
0.867563 + 0.497327i \(0.165685\pi\)
\(42\) 0 0
\(43\) −10.9512 −1.67004 −0.835022 0.550217i \(-0.814545\pi\)
−0.835022 + 0.550217i \(0.814545\pi\)
\(44\) 0.734963 0.110800
\(45\) 0 0
\(46\) −6.91115 −1.01899
\(47\) 6.22966 0.908689 0.454344 0.890826i \(-0.349874\pi\)
0.454344 + 0.890826i \(0.349874\pi\)
\(48\) 0 0
\(49\) 2.34379 0.334827
\(50\) 13.9646 1.97490
\(51\) 0 0
\(52\) −2.43689 −0.337935
\(53\) −6.47426 −0.889309 −0.444654 0.895702i \(-0.646673\pi\)
−0.444654 + 0.895702i \(0.646673\pi\)
\(54\) 0 0
\(55\) −3.20064 −0.431575
\(56\) 3.05676 0.408477
\(57\) 0 0
\(58\) −0.113306 −0.0148778
\(59\) −7.94099 −1.03383 −0.516915 0.856037i \(-0.672920\pi\)
−0.516915 + 0.856037i \(0.672920\pi\)
\(60\) 0 0
\(61\) −4.54537 −0.581974 −0.290987 0.956727i \(-0.593984\pi\)
−0.290987 + 0.956727i \(0.593984\pi\)
\(62\) 1.56736 0.199055
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 10.6122 1.31629
\(66\) 0 0
\(67\) −2.23117 −0.272581 −0.136290 0.990669i \(-0.543518\pi\)
−0.136290 + 0.990669i \(0.543518\pi\)
\(68\) 0.681711 0.0826696
\(69\) 0 0
\(70\) −13.3117 −1.59105
\(71\) −10.5641 −1.25372 −0.626862 0.779130i \(-0.715661\pi\)
−0.626862 + 0.779130i \(0.715661\pi\)
\(72\) 0 0
\(73\) −3.30232 −0.386507 −0.193254 0.981149i \(-0.561904\pi\)
−0.193254 + 0.981149i \(0.561904\pi\)
\(74\) −3.71884 −0.432307
\(75\) 0 0
\(76\) −1.87811 −0.215434
\(77\) 2.24661 0.256024
\(78\) 0 0
\(79\) −3.98486 −0.448332 −0.224166 0.974551i \(-0.571966\pi\)
−0.224166 + 0.974551i \(0.571966\pi\)
\(80\) −4.35484 −0.486886
\(81\) 0 0
\(82\) 11.1102 1.22692
\(83\) 9.77231 1.07265 0.536325 0.844011i \(-0.319812\pi\)
0.536325 + 0.844011i \(0.319812\pi\)
\(84\) 0 0
\(85\) −2.96874 −0.322005
\(86\) −10.9512 −1.18090
\(87\) 0 0
\(88\) 0.734963 0.0783473
\(89\) 3.23802 0.343230 0.171615 0.985164i \(-0.445102\pi\)
0.171615 + 0.985164i \(0.445102\pi\)
\(90\) 0 0
\(91\) −7.44898 −0.780865
\(92\) −6.91115 −0.720537
\(93\) 0 0
\(94\) 6.22966 0.642540
\(95\) 8.17887 0.839134
\(96\) 0 0
\(97\) −11.1019 −1.12723 −0.563613 0.826039i \(-0.690589\pi\)
−0.563613 + 0.826039i \(0.690589\pi\)
\(98\) 2.34379 0.236759
\(99\) 0 0
\(100\) 13.9646 1.39646
\(101\) −19.8419 −1.97434 −0.987171 0.159664i \(-0.948959\pi\)
−0.987171 + 0.159664i \(0.948959\pi\)
\(102\) 0 0
\(103\) −4.42428 −0.435937 −0.217968 0.975956i \(-0.569943\pi\)
−0.217968 + 0.975956i \(0.569943\pi\)
\(104\) −2.43689 −0.238956
\(105\) 0 0
\(106\) −6.47426 −0.628836
\(107\) −11.4126 −1.10330 −0.551651 0.834075i \(-0.686002\pi\)
−0.551651 + 0.834075i \(0.686002\pi\)
\(108\) 0 0
\(109\) −15.1912 −1.45506 −0.727528 0.686078i \(-0.759331\pi\)
−0.727528 + 0.686078i \(0.759331\pi\)
\(110\) −3.20064 −0.305169
\(111\) 0 0
\(112\) 3.05676 0.288837
\(113\) 0.723916 0.0681003 0.0340501 0.999420i \(-0.489159\pi\)
0.0340501 + 0.999420i \(0.489159\pi\)
\(114\) 0 0
\(115\) 30.0969 2.80655
\(116\) −0.113306 −0.0105202
\(117\) 0 0
\(118\) −7.94099 −0.731028
\(119\) 2.08383 0.191024
\(120\) 0 0
\(121\) −10.4598 −0.950894
\(122\) −4.54537 −0.411518
\(123\) 0 0
\(124\) 1.56736 0.140753
\(125\) −39.0395 −3.49180
\(126\) 0 0
\(127\) 17.7134 1.57181 0.785905 0.618347i \(-0.212197\pi\)
0.785905 + 0.618347i \(0.212197\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 10.6122 0.930755
\(131\) 4.62868 0.404409 0.202205 0.979343i \(-0.435189\pi\)
0.202205 + 0.979343i \(0.435189\pi\)
\(132\) 0 0
\(133\) −5.74094 −0.497802
\(134\) −2.23117 −0.192744
\(135\) 0 0
\(136\) 0.681711 0.0584562
\(137\) 6.95121 0.593882 0.296941 0.954896i \(-0.404034\pi\)
0.296941 + 0.954896i \(0.404034\pi\)
\(138\) 0 0
\(139\) 23.4964 1.99294 0.996468 0.0839681i \(-0.0267594\pi\)
0.996468 + 0.0839681i \(0.0267594\pi\)
\(140\) −13.3117 −1.12504
\(141\) 0 0
\(142\) −10.5641 −0.886517
\(143\) −1.79102 −0.149773
\(144\) 0 0
\(145\) 0.493430 0.0409772
\(146\) −3.30232 −0.273302
\(147\) 0 0
\(148\) −3.71884 −0.305687
\(149\) 8.25494 0.676271 0.338136 0.941097i \(-0.390204\pi\)
0.338136 + 0.941097i \(0.390204\pi\)
\(150\) 0 0
\(151\) −15.3701 −1.25080 −0.625402 0.780303i \(-0.715065\pi\)
−0.625402 + 0.780303i \(0.715065\pi\)
\(152\) −1.87811 −0.152335
\(153\) 0 0
\(154\) 2.24661 0.181037
\(155\) −6.82559 −0.548245
\(156\) 0 0
\(157\) 5.70675 0.455448 0.227724 0.973726i \(-0.426872\pi\)
0.227724 + 0.973726i \(0.426872\pi\)
\(158\) −3.98486 −0.317019
\(159\) 0 0
\(160\) −4.35484 −0.344280
\(161\) −21.1257 −1.66494
\(162\) 0 0
\(163\) 1.46144 0.114469 0.0572343 0.998361i \(-0.481772\pi\)
0.0572343 + 0.998361i \(0.481772\pi\)
\(164\) 11.1102 0.867563
\(165\) 0 0
\(166\) 9.77231 0.758478
\(167\) 0.335943 0.0259960 0.0129980 0.999916i \(-0.495862\pi\)
0.0129980 + 0.999916i \(0.495862\pi\)
\(168\) 0 0
\(169\) −7.06159 −0.543199
\(170\) −2.96874 −0.227692
\(171\) 0 0
\(172\) −10.9512 −0.835022
\(173\) −3.25577 −0.247532 −0.123766 0.992311i \(-0.539497\pi\)
−0.123766 + 0.992311i \(0.539497\pi\)
\(174\) 0 0
\(175\) 42.6865 3.22680
\(176\) 0.734963 0.0553999
\(177\) 0 0
\(178\) 3.23802 0.242700
\(179\) −2.01680 −0.150743 −0.0753715 0.997156i \(-0.524014\pi\)
−0.0753715 + 0.997156i \(0.524014\pi\)
\(180\) 0 0
\(181\) −0.718726 −0.0534225 −0.0267113 0.999643i \(-0.508503\pi\)
−0.0267113 + 0.999643i \(0.508503\pi\)
\(182\) −7.44898 −0.552155
\(183\) 0 0
\(184\) −6.91115 −0.509497
\(185\) 16.1950 1.19068
\(186\) 0 0
\(187\) 0.501032 0.0366391
\(188\) 6.22966 0.454344
\(189\) 0 0
\(190\) 8.17887 0.593357
\(191\) −4.95760 −0.358719 −0.179360 0.983784i \(-0.557403\pi\)
−0.179360 + 0.983784i \(0.557403\pi\)
\(192\) 0 0
\(193\) −8.59130 −0.618415 −0.309208 0.950995i \(-0.600064\pi\)
−0.309208 + 0.950995i \(0.600064\pi\)
\(194\) −11.1019 −0.797070
\(195\) 0 0
\(196\) 2.34379 0.167414
\(197\) −17.0864 −1.21736 −0.608678 0.793417i \(-0.708300\pi\)
−0.608678 + 0.793417i \(0.708300\pi\)
\(198\) 0 0
\(199\) −0.350646 −0.0248566 −0.0124283 0.999923i \(-0.503956\pi\)
−0.0124283 + 0.999923i \(0.503956\pi\)
\(200\) 13.9646 0.987448
\(201\) 0 0
\(202\) −19.8419 −1.39607
\(203\) −0.346350 −0.0243090
\(204\) 0 0
\(205\) −48.3833 −3.37923
\(206\) −4.42428 −0.308254
\(207\) 0 0
\(208\) −2.43689 −0.168968
\(209\) −1.38034 −0.0954802
\(210\) 0 0
\(211\) −12.3103 −0.847475 −0.423738 0.905785i \(-0.639282\pi\)
−0.423738 + 0.905785i \(0.639282\pi\)
\(212\) −6.47426 −0.444654
\(213\) 0 0
\(214\) −11.4126 −0.780152
\(215\) 47.6907 3.25248
\(216\) 0 0
\(217\) 4.79104 0.325237
\(218\) −15.1912 −1.02888
\(219\) 0 0
\(220\) −3.20064 −0.215787
\(221\) −1.66125 −0.111748
\(222\) 0 0
\(223\) −1.00000 −0.0669650
\(224\) 3.05676 0.204238
\(225\) 0 0
\(226\) 0.723916 0.0481542
\(227\) −11.9814 −0.795233 −0.397616 0.917552i \(-0.630162\pi\)
−0.397616 + 0.917552i \(0.630162\pi\)
\(228\) 0 0
\(229\) −0.278694 −0.0184166 −0.00920831 0.999958i \(-0.502931\pi\)
−0.00920831 + 0.999958i \(0.502931\pi\)
\(230\) 30.0969 1.98453
\(231\) 0 0
\(232\) −0.113306 −0.00743892
\(233\) 12.2226 0.800732 0.400366 0.916355i \(-0.368883\pi\)
0.400366 + 0.916355i \(0.368883\pi\)
\(234\) 0 0
\(235\) −27.1291 −1.76971
\(236\) −7.94099 −0.516915
\(237\) 0 0
\(238\) 2.08383 0.135074
\(239\) −22.2407 −1.43863 −0.719316 0.694683i \(-0.755545\pi\)
−0.719316 + 0.694683i \(0.755545\pi\)
\(240\) 0 0
\(241\) −22.3491 −1.43963 −0.719814 0.694166i \(-0.755773\pi\)
−0.719814 + 0.694166i \(0.755773\pi\)
\(242\) −10.4598 −0.672383
\(243\) 0 0
\(244\) −4.54537 −0.290987
\(245\) −10.2068 −0.652091
\(246\) 0 0
\(247\) 4.57674 0.291211
\(248\) 1.56736 0.0995274
\(249\) 0 0
\(250\) −39.0395 −2.46907
\(251\) −19.7825 −1.24866 −0.624331 0.781160i \(-0.714628\pi\)
−0.624331 + 0.781160i \(0.714628\pi\)
\(252\) 0 0
\(253\) −5.07944 −0.319341
\(254\) 17.7134 1.11144
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 11.4401 0.713617 0.356808 0.934178i \(-0.383865\pi\)
0.356808 + 0.934178i \(0.383865\pi\)
\(258\) 0 0
\(259\) −11.3676 −0.706349
\(260\) 10.6122 0.658143
\(261\) 0 0
\(262\) 4.62868 0.285961
\(263\) −13.3295 −0.821929 −0.410965 0.911651i \(-0.634808\pi\)
−0.410965 + 0.911651i \(0.634808\pi\)
\(264\) 0 0
\(265\) 28.1944 1.73197
\(266\) −5.74094 −0.351999
\(267\) 0 0
\(268\) −2.23117 −0.136290
\(269\) −3.03055 −0.184776 −0.0923878 0.995723i \(-0.529450\pi\)
−0.0923878 + 0.995723i \(0.529450\pi\)
\(270\) 0 0
\(271\) 21.4604 1.30362 0.651812 0.758381i \(-0.274009\pi\)
0.651812 + 0.758381i \(0.274009\pi\)
\(272\) 0.681711 0.0413348
\(273\) 0 0
\(274\) 6.95121 0.419938
\(275\) 10.2635 0.618911
\(276\) 0 0
\(277\) 28.2695 1.69855 0.849276 0.527950i \(-0.177039\pi\)
0.849276 + 0.527950i \(0.177039\pi\)
\(278\) 23.4964 1.40922
\(279\) 0 0
\(280\) −13.3117 −0.795526
\(281\) 12.7637 0.761420 0.380710 0.924694i \(-0.375680\pi\)
0.380710 + 0.924694i \(0.375680\pi\)
\(282\) 0 0
\(283\) 32.5585 1.93540 0.967702 0.252098i \(-0.0811205\pi\)
0.967702 + 0.252098i \(0.0811205\pi\)
\(284\) −10.5641 −0.626862
\(285\) 0 0
\(286\) −1.79102 −0.105905
\(287\) 33.9613 2.00467
\(288\) 0 0
\(289\) −16.5353 −0.972663
\(290\) 0.493430 0.0289752
\(291\) 0 0
\(292\) −3.30232 −0.193254
\(293\) −28.3976 −1.65901 −0.829503 0.558503i \(-0.811376\pi\)
−0.829503 + 0.558503i \(0.811376\pi\)
\(294\) 0 0
\(295\) 34.5817 2.01343
\(296\) −3.71884 −0.216153
\(297\) 0 0
\(298\) 8.25494 0.478196
\(299\) 16.8417 0.973980
\(300\) 0 0
\(301\) −33.4752 −1.92948
\(302\) −15.3701 −0.884452
\(303\) 0 0
\(304\) −1.87811 −0.107717
\(305\) 19.7943 1.13342
\(306\) 0 0
\(307\) −27.9578 −1.59563 −0.797817 0.602900i \(-0.794012\pi\)
−0.797817 + 0.602900i \(0.794012\pi\)
\(308\) 2.24661 0.128012
\(309\) 0 0
\(310\) −6.82559 −0.387668
\(311\) 3.59459 0.203831 0.101915 0.994793i \(-0.467503\pi\)
0.101915 + 0.994793i \(0.467503\pi\)
\(312\) 0 0
\(313\) 11.4643 0.647998 0.323999 0.946057i \(-0.394972\pi\)
0.323999 + 0.946057i \(0.394972\pi\)
\(314\) 5.70675 0.322050
\(315\) 0 0
\(316\) −3.98486 −0.224166
\(317\) 33.3632 1.87386 0.936931 0.349514i \(-0.113653\pi\)
0.936931 + 0.349514i \(0.113653\pi\)
\(318\) 0 0
\(319\) −0.0832759 −0.00466255
\(320\) −4.35484 −0.243443
\(321\) 0 0
\(322\) −21.1257 −1.17729
\(323\) −1.28033 −0.0712394
\(324\) 0 0
\(325\) −34.0302 −1.88765
\(326\) 1.46144 0.0809415
\(327\) 0 0
\(328\) 11.1102 0.613460
\(329\) 19.0426 1.04985
\(330\) 0 0
\(331\) −11.6186 −0.638615 −0.319308 0.947651i \(-0.603450\pi\)
−0.319308 + 0.947651i \(0.603450\pi\)
\(332\) 9.77231 0.536325
\(333\) 0 0
\(334\) 0.335943 0.0183820
\(335\) 9.71638 0.530862
\(336\) 0 0
\(337\) −7.91263 −0.431029 −0.215514 0.976501i \(-0.569143\pi\)
−0.215514 + 0.976501i \(0.569143\pi\)
\(338\) −7.06159 −0.384100
\(339\) 0 0
\(340\) −2.96874 −0.161003
\(341\) 1.15195 0.0623816
\(342\) 0 0
\(343\) −14.2329 −0.768505
\(344\) −10.9512 −0.590450
\(345\) 0 0
\(346\) −3.25577 −0.175031
\(347\) 21.1718 1.13656 0.568280 0.822835i \(-0.307609\pi\)
0.568280 + 0.822835i \(0.307609\pi\)
\(348\) 0 0
\(349\) −11.3838 −0.609360 −0.304680 0.952455i \(-0.598550\pi\)
−0.304680 + 0.952455i \(0.598550\pi\)
\(350\) 42.6865 2.28169
\(351\) 0 0
\(352\) 0.734963 0.0391736
\(353\) 20.6961 1.10154 0.550772 0.834656i \(-0.314333\pi\)
0.550772 + 0.834656i \(0.314333\pi\)
\(354\) 0 0
\(355\) 46.0048 2.44168
\(356\) 3.23802 0.171615
\(357\) 0 0
\(358\) −2.01680 −0.106591
\(359\) 13.4560 0.710183 0.355092 0.934832i \(-0.384450\pi\)
0.355092 + 0.934832i \(0.384450\pi\)
\(360\) 0 0
\(361\) −15.4727 −0.814353
\(362\) −0.718726 −0.0377754
\(363\) 0 0
\(364\) −7.44898 −0.390433
\(365\) 14.3811 0.752740
\(366\) 0 0
\(367\) 8.80612 0.459675 0.229838 0.973229i \(-0.426180\pi\)
0.229838 + 0.973229i \(0.426180\pi\)
\(368\) −6.91115 −0.360269
\(369\) 0 0
\(370\) 16.1950 0.841936
\(371\) −19.7903 −1.02746
\(372\) 0 0
\(373\) 5.86346 0.303598 0.151799 0.988411i \(-0.451493\pi\)
0.151799 + 0.988411i \(0.451493\pi\)
\(374\) 0.501032 0.0259078
\(375\) 0 0
\(376\) 6.22966 0.321270
\(377\) 0.276114 0.0142206
\(378\) 0 0
\(379\) −20.7423 −1.06546 −0.532729 0.846286i \(-0.678833\pi\)
−0.532729 + 0.846286i \(0.678833\pi\)
\(380\) 8.17887 0.419567
\(381\) 0 0
\(382\) −4.95760 −0.253653
\(383\) −25.8637 −1.32158 −0.660788 0.750573i \(-0.729778\pi\)
−0.660788 + 0.750573i \(0.729778\pi\)
\(384\) 0 0
\(385\) −9.78361 −0.498619
\(386\) −8.59130 −0.437286
\(387\) 0 0
\(388\) −11.1019 −0.563613
\(389\) −2.44929 −0.124184 −0.0620920 0.998070i \(-0.519777\pi\)
−0.0620920 + 0.998070i \(0.519777\pi\)
\(390\) 0 0
\(391\) −4.71141 −0.238266
\(392\) 2.34379 0.118379
\(393\) 0 0
\(394\) −17.0864 −0.860801
\(395\) 17.3534 0.873146
\(396\) 0 0
\(397\) 19.7131 0.989370 0.494685 0.869072i \(-0.335283\pi\)
0.494685 + 0.869072i \(0.335283\pi\)
\(398\) −0.350646 −0.0175763
\(399\) 0 0
\(400\) 13.9646 0.698231
\(401\) −11.7643 −0.587483 −0.293742 0.955885i \(-0.594900\pi\)
−0.293742 + 0.955885i \(0.594900\pi\)
\(402\) 0 0
\(403\) −3.81947 −0.190262
\(404\) −19.8419 −0.987171
\(405\) 0 0
\(406\) −0.346350 −0.0171891
\(407\) −2.73321 −0.135480
\(408\) 0 0
\(409\) 29.9196 1.47943 0.739715 0.672920i \(-0.234960\pi\)
0.739715 + 0.672920i \(0.234960\pi\)
\(410\) −48.3833 −2.38948
\(411\) 0 0
\(412\) −4.42428 −0.217968
\(413\) −24.2737 −1.19443
\(414\) 0 0
\(415\) −42.5568 −2.08903
\(416\) −2.43689 −0.119478
\(417\) 0 0
\(418\) −1.38034 −0.0675147
\(419\) 36.6787 1.79187 0.895935 0.444185i \(-0.146507\pi\)
0.895935 + 0.444185i \(0.146507\pi\)
\(420\) 0 0
\(421\) 30.9616 1.50898 0.754488 0.656313i \(-0.227885\pi\)
0.754488 + 0.656313i \(0.227885\pi\)
\(422\) −12.3103 −0.599255
\(423\) 0 0
\(424\) −6.47426 −0.314418
\(425\) 9.51983 0.461780
\(426\) 0 0
\(427\) −13.8941 −0.672383
\(428\) −11.4126 −0.551651
\(429\) 0 0
\(430\) 47.6907 2.29985
\(431\) 32.4852 1.56476 0.782378 0.622804i \(-0.214006\pi\)
0.782378 + 0.622804i \(0.214006\pi\)
\(432\) 0 0
\(433\) −24.9126 −1.19722 −0.598612 0.801039i \(-0.704281\pi\)
−0.598612 + 0.801039i \(0.704281\pi\)
\(434\) 4.79104 0.229977
\(435\) 0 0
\(436\) −15.1912 −0.727528
\(437\) 12.9799 0.620913
\(438\) 0 0
\(439\) 31.8189 1.51863 0.759317 0.650721i \(-0.225533\pi\)
0.759317 + 0.650721i \(0.225533\pi\)
\(440\) −3.20064 −0.152585
\(441\) 0 0
\(442\) −1.66125 −0.0790177
\(443\) 22.4064 1.06456 0.532281 0.846568i \(-0.321335\pi\)
0.532281 + 0.846568i \(0.321335\pi\)
\(444\) 0 0
\(445\) −14.1011 −0.668455
\(446\) −1.00000 −0.0473514
\(447\) 0 0
\(448\) 3.05676 0.144418
\(449\) −27.5968 −1.30237 −0.651187 0.758917i \(-0.725729\pi\)
−0.651187 + 0.758917i \(0.725729\pi\)
\(450\) 0 0
\(451\) 8.16560 0.384503
\(452\) 0.723916 0.0340501
\(453\) 0 0
\(454\) −11.9814 −0.562314
\(455\) 32.4391 1.52077
\(456\) 0 0
\(457\) 6.26270 0.292957 0.146478 0.989214i \(-0.453206\pi\)
0.146478 + 0.989214i \(0.453206\pi\)
\(458\) −0.278694 −0.0130225
\(459\) 0 0
\(460\) 30.0969 1.40328
\(461\) 24.4624 1.13933 0.569664 0.821878i \(-0.307074\pi\)
0.569664 + 0.821878i \(0.307074\pi\)
\(462\) 0 0
\(463\) −10.5467 −0.490146 −0.245073 0.969505i \(-0.578812\pi\)
−0.245073 + 0.969505i \(0.578812\pi\)
\(464\) −0.113306 −0.00526011
\(465\) 0 0
\(466\) 12.2226 0.566203
\(467\) 27.0318 1.25088 0.625442 0.780271i \(-0.284919\pi\)
0.625442 + 0.780271i \(0.284919\pi\)
\(468\) 0 0
\(469\) −6.82015 −0.314925
\(470\) −27.1291 −1.25137
\(471\) 0 0
\(472\) −7.94099 −0.365514
\(473\) −8.04873 −0.370081
\(474\) 0 0
\(475\) −26.2271 −1.20338
\(476\) 2.08383 0.0955121
\(477\) 0 0
\(478\) −22.2407 −1.01727
\(479\) −23.9419 −1.09393 −0.546967 0.837154i \(-0.684218\pi\)
−0.546967 + 0.837154i \(0.684218\pi\)
\(480\) 0 0
\(481\) 9.06239 0.413210
\(482\) −22.3491 −1.01797
\(483\) 0 0
\(484\) −10.4598 −0.475447
\(485\) 48.3470 2.19532
\(486\) 0 0
\(487\) −2.04679 −0.0927490 −0.0463745 0.998924i \(-0.514767\pi\)
−0.0463745 + 0.998924i \(0.514767\pi\)
\(488\) −4.54537 −0.205759
\(489\) 0 0
\(490\) −10.2068 −0.461098
\(491\) 30.6746 1.38433 0.692163 0.721741i \(-0.256658\pi\)
0.692163 + 0.721741i \(0.256658\pi\)
\(492\) 0 0
\(493\) −0.0772421 −0.00347881
\(494\) 4.57674 0.205917
\(495\) 0 0
\(496\) 1.56736 0.0703765
\(497\) −32.2918 −1.44849
\(498\) 0 0
\(499\) 29.0813 1.30186 0.650929 0.759138i \(-0.274379\pi\)
0.650929 + 0.759138i \(0.274379\pi\)
\(500\) −39.0395 −1.74590
\(501\) 0 0
\(502\) −19.7825 −0.882937
\(503\) −32.4064 −1.44493 −0.722466 0.691407i \(-0.756991\pi\)
−0.722466 + 0.691407i \(0.756991\pi\)
\(504\) 0 0
\(505\) 86.4083 3.84512
\(506\) −5.07944 −0.225809
\(507\) 0 0
\(508\) 17.7134 0.785905
\(509\) 20.3134 0.900376 0.450188 0.892934i \(-0.351357\pi\)
0.450188 + 0.892934i \(0.351357\pi\)
\(510\) 0 0
\(511\) −10.0944 −0.446550
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 11.4401 0.504603
\(515\) 19.2670 0.849006
\(516\) 0 0
\(517\) 4.57856 0.201365
\(518\) −11.3676 −0.499464
\(519\) 0 0
\(520\) 10.6122 0.465378
\(521\) 23.8912 1.04669 0.523346 0.852120i \(-0.324683\pi\)
0.523346 + 0.852120i \(0.324683\pi\)
\(522\) 0 0
\(523\) 26.0276 1.13811 0.569054 0.822300i \(-0.307309\pi\)
0.569054 + 0.822300i \(0.307309\pi\)
\(524\) 4.62868 0.202205
\(525\) 0 0
\(526\) −13.3295 −0.581192
\(527\) 1.06849 0.0465440
\(528\) 0 0
\(529\) 24.7640 1.07670
\(530\) 28.1944 1.22469
\(531\) 0 0
\(532\) −5.74094 −0.248901
\(533\) −27.0744 −1.17272
\(534\) 0 0
\(535\) 49.7002 2.14873
\(536\) −2.23117 −0.0963718
\(537\) 0 0
\(538\) −3.03055 −0.130656
\(539\) 1.72260 0.0741976
\(540\) 0 0
\(541\) −5.46873 −0.235119 −0.117559 0.993066i \(-0.537507\pi\)
−0.117559 + 0.993066i \(0.537507\pi\)
\(542\) 21.4604 0.921801
\(543\) 0 0
\(544\) 0.681711 0.0292281
\(545\) 66.1554 2.83378
\(546\) 0 0
\(547\) 0.600980 0.0256961 0.0128480 0.999917i \(-0.495910\pi\)
0.0128480 + 0.999917i \(0.495910\pi\)
\(548\) 6.95121 0.296941
\(549\) 0 0
\(550\) 10.2635 0.437636
\(551\) 0.212802 0.00906565
\(552\) 0 0
\(553\) −12.1808 −0.517979
\(554\) 28.2695 1.20106
\(555\) 0 0
\(556\) 23.4964 0.996468
\(557\) 33.5046 1.41963 0.709817 0.704386i \(-0.248778\pi\)
0.709817 + 0.704386i \(0.248778\pi\)
\(558\) 0 0
\(559\) 26.6868 1.12873
\(560\) −13.3117 −0.562522
\(561\) 0 0
\(562\) 12.7637 0.538405
\(563\) 0.0190630 0.000803411 0 0.000401705 1.00000i \(-0.499872\pi\)
0.000401705 1.00000i \(0.499872\pi\)
\(564\) 0 0
\(565\) −3.15254 −0.132628
\(566\) 32.5585 1.36854
\(567\) 0 0
\(568\) −10.5641 −0.443258
\(569\) −17.4587 −0.731908 −0.365954 0.930633i \(-0.619257\pi\)
−0.365954 + 0.930633i \(0.619257\pi\)
\(570\) 0 0
\(571\) 18.0879 0.756955 0.378477 0.925611i \(-0.376448\pi\)
0.378477 + 0.925611i \(0.376448\pi\)
\(572\) −1.79102 −0.0748863
\(573\) 0 0
\(574\) 33.9613 1.41752
\(575\) −96.5116 −4.02481
\(576\) 0 0
\(577\) 35.8487 1.49240 0.746200 0.665722i \(-0.231876\pi\)
0.746200 + 0.665722i \(0.231876\pi\)
\(578\) −16.5353 −0.687777
\(579\) 0 0
\(580\) 0.493430 0.0204886
\(581\) 29.8716 1.23928
\(582\) 0 0
\(583\) −4.75834 −0.197070
\(584\) −3.30232 −0.136651
\(585\) 0 0
\(586\) −28.3976 −1.17309
\(587\) −2.69524 −0.111244 −0.0556222 0.998452i \(-0.517714\pi\)
−0.0556222 + 0.998452i \(0.517714\pi\)
\(588\) 0 0
\(589\) −2.94367 −0.121292
\(590\) 34.5817 1.42371
\(591\) 0 0
\(592\) −3.71884 −0.152843
\(593\) 31.1847 1.28060 0.640302 0.768124i \(-0.278809\pi\)
0.640302 + 0.768124i \(0.278809\pi\)
\(594\) 0 0
\(595\) −9.07473 −0.372028
\(596\) 8.25494 0.338136
\(597\) 0 0
\(598\) 16.8417 0.688708
\(599\) 5.49303 0.224439 0.112220 0.993683i \(-0.464204\pi\)
0.112220 + 0.993683i \(0.464204\pi\)
\(600\) 0 0
\(601\) −25.3653 −1.03467 −0.517335 0.855783i \(-0.673076\pi\)
−0.517335 + 0.855783i \(0.673076\pi\)
\(602\) −33.4752 −1.36435
\(603\) 0 0
\(604\) −15.3701 −0.625402
\(605\) 45.5509 1.85191
\(606\) 0 0
\(607\) −12.3964 −0.503154 −0.251577 0.967837i \(-0.580949\pi\)
−0.251577 + 0.967837i \(0.580949\pi\)
\(608\) −1.87811 −0.0761674
\(609\) 0 0
\(610\) 19.7943 0.801449
\(611\) −15.1810 −0.614156
\(612\) 0 0
\(613\) 11.0514 0.446363 0.223181 0.974777i \(-0.428356\pi\)
0.223181 + 0.974777i \(0.428356\pi\)
\(614\) −27.9578 −1.12828
\(615\) 0 0
\(616\) 2.24661 0.0905183
\(617\) −24.7403 −0.996006 −0.498003 0.867175i \(-0.665933\pi\)
−0.498003 + 0.867175i \(0.665933\pi\)
\(618\) 0 0
\(619\) −4.37667 −0.175913 −0.0879566 0.996124i \(-0.528034\pi\)
−0.0879566 + 0.996124i \(0.528034\pi\)
\(620\) −6.82559 −0.274122
\(621\) 0 0
\(622\) 3.59459 0.144130
\(623\) 9.89786 0.396550
\(624\) 0 0
\(625\) 100.187 4.00750
\(626\) 11.4643 0.458204
\(627\) 0 0
\(628\) 5.70675 0.227724
\(629\) −2.53518 −0.101084
\(630\) 0 0
\(631\) −39.8325 −1.58571 −0.792854 0.609411i \(-0.791406\pi\)
−0.792854 + 0.609411i \(0.791406\pi\)
\(632\) −3.98486 −0.158509
\(633\) 0 0
\(634\) 33.3632 1.32502
\(635\) −77.1390 −3.06117
\(636\) 0 0
\(637\) −5.71155 −0.226300
\(638\) −0.0832759 −0.00329692
\(639\) 0 0
\(640\) −4.35484 −0.172140
\(641\) 44.5598 1.76000 0.880002 0.474969i \(-0.157541\pi\)
0.880002 + 0.474969i \(0.157541\pi\)
\(642\) 0 0
\(643\) 33.6455 1.32685 0.663425 0.748243i \(-0.269102\pi\)
0.663425 + 0.748243i \(0.269102\pi\)
\(644\) −21.1257 −0.832471
\(645\) 0 0
\(646\) −1.28033 −0.0503738
\(647\) 7.69538 0.302537 0.151268 0.988493i \(-0.451664\pi\)
0.151268 + 0.988493i \(0.451664\pi\)
\(648\) 0 0
\(649\) −5.83633 −0.229096
\(650\) −34.0302 −1.33477
\(651\) 0 0
\(652\) 1.46144 0.0572343
\(653\) −11.4671 −0.448742 −0.224371 0.974504i \(-0.572033\pi\)
−0.224371 + 0.974504i \(0.572033\pi\)
\(654\) 0 0
\(655\) −20.1571 −0.787605
\(656\) 11.1102 0.433782
\(657\) 0 0
\(658\) 19.0426 0.742357
\(659\) −35.9351 −1.39983 −0.699916 0.714225i \(-0.746779\pi\)
−0.699916 + 0.714225i \(0.746779\pi\)
\(660\) 0 0
\(661\) −45.1850 −1.75749 −0.878747 0.477288i \(-0.841620\pi\)
−0.878747 + 0.477288i \(0.841620\pi\)
\(662\) −11.6186 −0.451569
\(663\) 0 0
\(664\) 9.77231 0.379239
\(665\) 25.0009 0.969491
\(666\) 0 0
\(667\) 0.783076 0.0303208
\(668\) 0.335943 0.0129980
\(669\) 0 0
\(670\) 9.71638 0.375376
\(671\) −3.34067 −0.128965
\(672\) 0 0
\(673\) 1.54949 0.0597284 0.0298642 0.999554i \(-0.490493\pi\)
0.0298642 + 0.999554i \(0.490493\pi\)
\(674\) −7.91263 −0.304783
\(675\) 0 0
\(676\) −7.06159 −0.271600
\(677\) 18.2028 0.699590 0.349795 0.936826i \(-0.386251\pi\)
0.349795 + 0.936826i \(0.386251\pi\)
\(678\) 0 0
\(679\) −33.9358 −1.30234
\(680\) −2.96874 −0.113846
\(681\) 0 0
\(682\) 1.15195 0.0441104
\(683\) −32.0454 −1.22618 −0.613092 0.790012i \(-0.710074\pi\)
−0.613092 + 0.790012i \(0.710074\pi\)
\(684\) 0 0
\(685\) −30.2714 −1.15661
\(686\) −14.2329 −0.543415
\(687\) 0 0
\(688\) −10.9512 −0.417511
\(689\) 15.7770 0.601057
\(690\) 0 0
\(691\) 8.17206 0.310880 0.155440 0.987845i \(-0.450321\pi\)
0.155440 + 0.987845i \(0.450321\pi\)
\(692\) −3.25577 −0.123766
\(693\) 0 0
\(694\) 21.1718 0.803669
\(695\) −102.323 −3.88133
\(696\) 0 0
\(697\) 7.57397 0.286884
\(698\) −11.3838 −0.430883
\(699\) 0 0
\(700\) 42.6865 1.61340
\(701\) 25.3718 0.958281 0.479140 0.877738i \(-0.340948\pi\)
0.479140 + 0.877738i \(0.340948\pi\)
\(702\) 0 0
\(703\) 6.98440 0.263422
\(704\) 0.734963 0.0276999
\(705\) 0 0
\(706\) 20.6961 0.778910
\(707\) −60.6520 −2.28105
\(708\) 0 0
\(709\) 7.95024 0.298578 0.149289 0.988794i \(-0.452302\pi\)
0.149289 + 0.988794i \(0.452302\pi\)
\(710\) 46.0048 1.72653
\(711\) 0 0
\(712\) 3.23802 0.121350
\(713\) −10.8322 −0.405671
\(714\) 0 0
\(715\) 7.79960 0.291689
\(716\) −2.01680 −0.0753715
\(717\) 0 0
\(718\) 13.4560 0.502175
\(719\) 31.5123 1.17521 0.587606 0.809147i \(-0.300071\pi\)
0.587606 + 0.809147i \(0.300071\pi\)
\(720\) 0 0
\(721\) −13.5240 −0.503658
\(722\) −15.4727 −0.575834
\(723\) 0 0
\(724\) −0.718726 −0.0267113
\(725\) −1.58228 −0.0587643
\(726\) 0 0
\(727\) −41.5993 −1.54283 −0.771416 0.636332i \(-0.780451\pi\)
−0.771416 + 0.636332i \(0.780451\pi\)
\(728\) −7.44898 −0.276078
\(729\) 0 0
\(730\) 14.3811 0.532267
\(731\) −7.46556 −0.276124
\(732\) 0 0
\(733\) −0.280816 −0.0103722 −0.00518608 0.999987i \(-0.501651\pi\)
−0.00518608 + 0.999987i \(0.501651\pi\)
\(734\) 8.80612 0.325040
\(735\) 0 0
\(736\) −6.91115 −0.254748
\(737\) −1.63983 −0.0604038
\(738\) 0 0
\(739\) 30.0973 1.10715 0.553574 0.832800i \(-0.313264\pi\)
0.553574 + 0.832800i \(0.313264\pi\)
\(740\) 16.1950 0.595339
\(741\) 0 0
\(742\) −19.7903 −0.726524
\(743\) −1.49835 −0.0549692 −0.0274846 0.999622i \(-0.508750\pi\)
−0.0274846 + 0.999622i \(0.508750\pi\)
\(744\) 0 0
\(745\) −35.9489 −1.31707
\(746\) 5.86346 0.214676
\(747\) 0 0
\(748\) 0.501032 0.0183195
\(749\) −34.8857 −1.27470
\(750\) 0 0
\(751\) 19.0975 0.696878 0.348439 0.937331i \(-0.386712\pi\)
0.348439 + 0.937331i \(0.386712\pi\)
\(752\) 6.22966 0.227172
\(753\) 0 0
\(754\) 0.276114 0.0100555
\(755\) 66.9344 2.43599
\(756\) 0 0
\(757\) −29.6151 −1.07638 −0.538189 0.842824i \(-0.680891\pi\)
−0.538189 + 0.842824i \(0.680891\pi\)
\(758\) −20.7423 −0.753392
\(759\) 0 0
\(760\) 8.17887 0.296679
\(761\) −14.8382 −0.537886 −0.268943 0.963156i \(-0.586674\pi\)
−0.268943 + 0.963156i \(0.586674\pi\)
\(762\) 0 0
\(763\) −46.4360 −1.68110
\(764\) −4.95760 −0.179360
\(765\) 0 0
\(766\) −25.8637 −0.934495
\(767\) 19.3513 0.698735
\(768\) 0 0
\(769\) −23.8346 −0.859498 −0.429749 0.902948i \(-0.641398\pi\)
−0.429749 + 0.902948i \(0.641398\pi\)
\(770\) −9.78361 −0.352577
\(771\) 0 0
\(772\) −8.59130 −0.309208
\(773\) −55.4521 −1.99447 −0.997237 0.0742897i \(-0.976331\pi\)
−0.997237 + 0.0742897i \(0.976331\pi\)
\(774\) 0 0
\(775\) 21.8876 0.786224
\(776\) −11.1019 −0.398535
\(777\) 0 0
\(778\) −2.44929 −0.0878113
\(779\) −20.8662 −0.747611
\(780\) 0 0
\(781\) −7.76419 −0.277825
\(782\) −4.71141 −0.168480
\(783\) 0 0
\(784\) 2.34379 0.0837068
\(785\) −24.8520 −0.887005
\(786\) 0 0
\(787\) 33.1602 1.18203 0.591017 0.806659i \(-0.298726\pi\)
0.591017 + 0.806659i \(0.298726\pi\)
\(788\) −17.0864 −0.608678
\(789\) 0 0
\(790\) 17.3534 0.617407
\(791\) 2.21284 0.0786795
\(792\) 0 0
\(793\) 11.0765 0.393339
\(794\) 19.7131 0.699590
\(795\) 0 0
\(796\) −0.350646 −0.0124283
\(797\) −15.2713 −0.540936 −0.270468 0.962729i \(-0.587179\pi\)
−0.270468 + 0.962729i \(0.587179\pi\)
\(798\) 0 0
\(799\) 4.24682 0.150242
\(800\) 13.9646 0.493724
\(801\) 0 0
\(802\) −11.7643 −0.415413
\(803\) −2.42708 −0.0856499
\(804\) 0 0
\(805\) 91.9992 3.24255
\(806\) −3.81947 −0.134535
\(807\) 0 0
\(808\) −19.8419 −0.698036
\(809\) −39.8830 −1.40221 −0.701106 0.713057i \(-0.747310\pi\)
−0.701106 + 0.713057i \(0.747310\pi\)
\(810\) 0 0
\(811\) −21.4838 −0.754400 −0.377200 0.926132i \(-0.623113\pi\)
−0.377200 + 0.926132i \(0.623113\pi\)
\(812\) −0.346350 −0.0121545
\(813\) 0 0
\(814\) −2.73321 −0.0957990
\(815\) −6.36432 −0.222932
\(816\) 0 0
\(817\) 20.5676 0.719569
\(818\) 29.9196 1.04611
\(819\) 0 0
\(820\) −48.3833 −1.68962
\(821\) −14.5566 −0.508029 −0.254014 0.967200i \(-0.581751\pi\)
−0.254014 + 0.967200i \(0.581751\pi\)
\(822\) 0 0
\(823\) 3.79452 0.132269 0.0661344 0.997811i \(-0.478933\pi\)
0.0661344 + 0.997811i \(0.478933\pi\)
\(824\) −4.42428 −0.154127
\(825\) 0 0
\(826\) −24.2737 −0.844591
\(827\) −56.9287 −1.97960 −0.989802 0.142452i \(-0.954501\pi\)
−0.989802 + 0.142452i \(0.954501\pi\)
\(828\) 0 0
\(829\) −44.0562 −1.53013 −0.765067 0.643951i \(-0.777294\pi\)
−0.765067 + 0.643951i \(0.777294\pi\)
\(830\) −42.5568 −1.47717
\(831\) 0 0
\(832\) −2.43689 −0.0844838
\(833\) 1.59779 0.0553601
\(834\) 0 0
\(835\) −1.46298 −0.0506284
\(836\) −1.38034 −0.0477401
\(837\) 0 0
\(838\) 36.6787 1.26704
\(839\) −2.03615 −0.0702957 −0.0351479 0.999382i \(-0.511190\pi\)
−0.0351479 + 0.999382i \(0.511190\pi\)
\(840\) 0 0
\(841\) −28.9872 −0.999557
\(842\) 30.9616 1.06701
\(843\) 0 0
\(844\) −12.3103 −0.423738
\(845\) 30.7521 1.05790
\(846\) 0 0
\(847\) −31.9732 −1.09861
\(848\) −6.47426 −0.222327
\(849\) 0 0
\(850\) 9.51983 0.326528
\(851\) 25.7015 0.881035
\(852\) 0 0
\(853\) 45.1825 1.54702 0.773510 0.633785i \(-0.218499\pi\)
0.773510 + 0.633785i \(0.218499\pi\)
\(854\) −13.8941 −0.475446
\(855\) 0 0
\(856\) −11.4126 −0.390076
\(857\) 13.9020 0.474883 0.237442 0.971402i \(-0.423691\pi\)
0.237442 + 0.971402i \(0.423691\pi\)
\(858\) 0 0
\(859\) 25.9715 0.886134 0.443067 0.896488i \(-0.353890\pi\)
0.443067 + 0.896488i \(0.353890\pi\)
\(860\) 47.6907 1.62624
\(861\) 0 0
\(862\) 32.4852 1.10645
\(863\) −24.8717 −0.846642 −0.423321 0.905980i \(-0.639136\pi\)
−0.423321 + 0.905980i \(0.639136\pi\)
\(864\) 0 0
\(865\) 14.1784 0.482079
\(866\) −24.9126 −0.846566
\(867\) 0 0
\(868\) 4.79104 0.162619
\(869\) −2.92872 −0.0993502
\(870\) 0 0
\(871\) 5.43710 0.184229
\(872\) −15.1912 −0.514440
\(873\) 0 0
\(874\) 12.9799 0.439052
\(875\) −119.334 −4.03424
\(876\) 0 0
\(877\) −41.9092 −1.41517 −0.707586 0.706627i \(-0.750216\pi\)
−0.707586 + 0.706627i \(0.750216\pi\)
\(878\) 31.8189 1.07384
\(879\) 0 0
\(880\) −3.20064 −0.107894
\(881\) −0.208232 −0.00701552 −0.00350776 0.999994i \(-0.501117\pi\)
−0.00350776 + 0.999994i \(0.501117\pi\)
\(882\) 0 0
\(883\) −4.62099 −0.155509 −0.0777544 0.996973i \(-0.524775\pi\)
−0.0777544 + 0.996973i \(0.524775\pi\)
\(884\) −1.66125 −0.0558739
\(885\) 0 0
\(886\) 22.4064 0.752759
\(887\) 11.1644 0.374865 0.187433 0.982277i \(-0.439983\pi\)
0.187433 + 0.982277i \(0.439983\pi\)
\(888\) 0 0
\(889\) 54.1457 1.81599
\(890\) −14.1011 −0.472669
\(891\) 0 0
\(892\) −1.00000 −0.0334825
\(893\) −11.7000 −0.391525
\(894\) 0 0
\(895\) 8.78285 0.293578
\(896\) 3.05676 0.102119
\(897\) 0 0
\(898\) −27.5968 −0.920918
\(899\) −0.177591 −0.00592301
\(900\) 0 0
\(901\) −4.41358 −0.147038
\(902\) 8.16560 0.271885
\(903\) 0 0
\(904\) 0.723916 0.0240771
\(905\) 3.12994 0.104043
\(906\) 0 0
\(907\) 0.749208 0.0248770 0.0124385 0.999923i \(-0.496041\pi\)
0.0124385 + 0.999923i \(0.496041\pi\)
\(908\) −11.9814 −0.397616
\(909\) 0 0
\(910\) 32.4391 1.07535
\(911\) 45.7170 1.51467 0.757336 0.653026i \(-0.226501\pi\)
0.757336 + 0.653026i \(0.226501\pi\)
\(912\) 0 0
\(913\) 7.18228 0.237699
\(914\) 6.26270 0.207152
\(915\) 0 0
\(916\) −0.278694 −0.00920831
\(917\) 14.1488 0.467233
\(918\) 0 0
\(919\) −4.43703 −0.146364 −0.0731821 0.997319i \(-0.523315\pi\)
−0.0731821 + 0.997319i \(0.523315\pi\)
\(920\) 30.0969 0.992267
\(921\) 0 0
\(922\) 24.4624 0.805626
\(923\) 25.7434 0.847355
\(924\) 0 0
\(925\) −51.9322 −1.70752
\(926\) −10.5467 −0.346585
\(927\) 0 0
\(928\) −0.113306 −0.00371946
\(929\) −0.789965 −0.0259179 −0.0129590 0.999916i \(-0.504125\pi\)
−0.0129590 + 0.999916i \(0.504125\pi\)
\(930\) 0 0
\(931\) −4.40190 −0.144266
\(932\) 12.2226 0.400366
\(933\) 0 0
\(934\) 27.0318 0.884508
\(935\) −2.18191 −0.0713562
\(936\) 0 0
\(937\) 11.0887 0.362252 0.181126 0.983460i \(-0.442026\pi\)
0.181126 + 0.983460i \(0.442026\pi\)
\(938\) −6.82015 −0.222686
\(939\) 0 0
\(940\) −27.1291 −0.884855
\(941\) 49.8346 1.62456 0.812281 0.583267i \(-0.198226\pi\)
0.812281 + 0.583267i \(0.198226\pi\)
\(942\) 0 0
\(943\) −76.7845 −2.50045
\(944\) −7.94099 −0.258457
\(945\) 0 0
\(946\) −8.04873 −0.261687
\(947\) −3.22300 −0.104733 −0.0523667 0.998628i \(-0.516676\pi\)
−0.0523667 + 0.998628i \(0.516676\pi\)
\(948\) 0 0
\(949\) 8.04738 0.261229
\(950\) −26.2271 −0.850919
\(951\) 0 0
\(952\) 2.08383 0.0675372
\(953\) −35.4866 −1.14952 −0.574762 0.818320i \(-0.694905\pi\)
−0.574762 + 0.818320i \(0.694905\pi\)
\(954\) 0 0
\(955\) 21.5895 0.698621
\(956\) −22.2407 −0.719316
\(957\) 0 0
\(958\) −23.9419 −0.773528
\(959\) 21.2482 0.686140
\(960\) 0 0
\(961\) −28.5434 −0.920754
\(962\) 9.06239 0.292183
\(963\) 0 0
\(964\) −22.3491 −0.719814
\(965\) 37.4137 1.20439
\(966\) 0 0
\(967\) −4.41806 −0.142075 −0.0710376 0.997474i \(-0.522631\pi\)
−0.0710376 + 0.997474i \(0.522631\pi\)
\(968\) −10.4598 −0.336192
\(969\) 0 0
\(970\) 48.3470 1.55233
\(971\) −12.8435 −0.412169 −0.206084 0.978534i \(-0.566072\pi\)
−0.206084 + 0.978534i \(0.566072\pi\)
\(972\) 0 0
\(973\) 71.8228 2.30253
\(974\) −2.04679 −0.0655834
\(975\) 0 0
\(976\) −4.54537 −0.145494
\(977\) −14.9739 −0.479057 −0.239528 0.970889i \(-0.576993\pi\)
−0.239528 + 0.970889i \(0.576993\pi\)
\(978\) 0 0
\(979\) 2.37983 0.0760596
\(980\) −10.2068 −0.326045
\(981\) 0 0
\(982\) 30.6746 0.978866
\(983\) 22.0615 0.703652 0.351826 0.936065i \(-0.385561\pi\)
0.351826 + 0.936065i \(0.385561\pi\)
\(984\) 0 0
\(985\) 74.4086 2.37085
\(986\) −0.0772421 −0.00245989
\(987\) 0 0
\(988\) 4.57674 0.145606
\(989\) 75.6855 2.40666
\(990\) 0 0
\(991\) −8.96101 −0.284656 −0.142328 0.989820i \(-0.545459\pi\)
−0.142328 + 0.989820i \(0.545459\pi\)
\(992\) 1.56736 0.0497637
\(993\) 0 0
\(994\) −32.2918 −1.02423
\(995\) 1.52701 0.0484093
\(996\) 0 0
\(997\) 51.5423 1.63236 0.816180 0.577798i \(-0.196088\pi\)
0.816180 + 0.577798i \(0.196088\pi\)
\(998\) 29.0813 0.920553
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4014.2.a.r.1.1 5
3.2 odd 2 1338.2.a.h.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1338.2.a.h.1.5 5 3.2 odd 2
4014.2.a.r.1.1 5 1.1 even 1 trivial