Properties

Label 2151.2.a.e.1.4
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1758214748\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.1767625.1
Defining polynomial: \(x^{6} - 7 x^{4} - x^{3} + 11 x^{2} + x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 717)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.39213\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.899709 q^{2} -1.19052 q^{4} -1.67380 q^{5} -1.75765 q^{7} -2.87054 q^{8} +O(q^{10})\) \(q+0.899709 q^{2} -1.19052 q^{4} -1.67380 q^{5} -1.75765 q^{7} -2.87054 q^{8} -1.50593 q^{10} +5.08858 q^{11} +6.43675 q^{13} -1.58137 q^{14} -0.201602 q^{16} +2.24001 q^{17} -3.96067 q^{19} +1.99270 q^{20} +4.57824 q^{22} -7.71459 q^{23} -2.19839 q^{25} +5.79120 q^{26} +2.09253 q^{28} -0.101311 q^{29} -4.63283 q^{31} +5.55970 q^{32} +2.01536 q^{34} +2.94196 q^{35} -7.78041 q^{37} -3.56345 q^{38} +4.80472 q^{40} -1.42586 q^{41} -3.65358 q^{43} -6.05808 q^{44} -6.94088 q^{46} +7.73754 q^{47} -3.91066 q^{49} -1.97791 q^{50} -7.66311 q^{52} -1.61074 q^{53} -8.51728 q^{55} +5.04542 q^{56} -0.0911499 q^{58} +2.46261 q^{59} -4.77088 q^{61} -4.16819 q^{62} +5.40531 q^{64} -10.7738 q^{65} -3.42357 q^{67} -2.66679 q^{68} +2.64691 q^{70} +13.1223 q^{71} -14.4582 q^{73} -7.00010 q^{74} +4.71528 q^{76} -8.94396 q^{77} -13.0644 q^{79} +0.337442 q^{80} -1.28286 q^{82} +0.423954 q^{83} -3.74933 q^{85} -3.28716 q^{86} -14.6070 q^{88} -17.1426 q^{89} -11.3136 q^{91} +9.18440 q^{92} +6.96153 q^{94} +6.62938 q^{95} +6.06424 q^{97} -3.51845 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{2} + 4q^{4} - 5q^{5} - 9q^{7} + 3q^{8} + O(q^{10}) \) \( 6q + 2q^{2} + 4q^{4} - 5q^{5} - 9q^{7} + 3q^{8} - 11q^{10} + 13q^{11} - q^{13} - 4q^{16} - 11q^{17} - 22q^{19} + q^{20} - 2q^{22} + 12q^{23} - q^{25} - 12q^{26} - 16q^{28} - 18q^{31} - 7q^{32} - 3q^{34} + 9q^{35} - 8q^{37} + 5q^{38} - 11q^{40} - 10q^{41} - 14q^{43} + 4q^{44} - 18q^{46} + 9q^{47} + 5q^{49} - 4q^{50} - 16q^{52} + 8q^{53} - 20q^{55} - 11q^{56} - 15q^{58} + 10q^{59} - 12q^{61} + 13q^{62} - 31q^{64} + 11q^{65} - 36q^{67} - 22q^{68} + q^{70} + 3q^{71} - 32q^{73} - 9q^{74} - 4q^{76} - 6q^{77} - q^{79} + 7q^{80} + 7q^{82} + 7q^{83} - 14q^{85} - 45q^{86} - 15q^{88} - 17q^{89} - 23q^{91} + 12q^{92} + 50q^{94} - 28q^{97} - 13q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.899709 0.636190 0.318095 0.948059i \(-0.396957\pi\)
0.318095 + 0.948059i \(0.396957\pi\)
\(3\) 0 0
\(4\) −1.19052 −0.595262
\(5\) −1.67380 −0.748547 −0.374273 0.927318i \(-0.622108\pi\)
−0.374273 + 0.927318i \(0.622108\pi\)
\(6\) 0 0
\(7\) −1.75765 −0.664330 −0.332165 0.943221i \(-0.607779\pi\)
−0.332165 + 0.943221i \(0.607779\pi\)
\(8\) −2.87054 −1.01489
\(9\) 0 0
\(10\) −1.50593 −0.476218
\(11\) 5.08858 1.53427 0.767133 0.641488i \(-0.221683\pi\)
0.767133 + 0.641488i \(0.221683\pi\)
\(12\) 0 0
\(13\) 6.43675 1.78523 0.892617 0.450816i \(-0.148867\pi\)
0.892617 + 0.450816i \(0.148867\pi\)
\(14\) −1.58137 −0.422640
\(15\) 0 0
\(16\) −0.201602 −0.0504005
\(17\) 2.24001 0.543283 0.271641 0.962399i \(-0.412434\pi\)
0.271641 + 0.962399i \(0.412434\pi\)
\(18\) 0 0
\(19\) −3.96067 −0.908641 −0.454320 0.890838i \(-0.650118\pi\)
−0.454320 + 0.890838i \(0.650118\pi\)
\(20\) 1.99270 0.445582
\(21\) 0 0
\(22\) 4.57824 0.976085
\(23\) −7.71459 −1.60860 −0.804301 0.594222i \(-0.797460\pi\)
−0.804301 + 0.594222i \(0.797460\pi\)
\(24\) 0 0
\(25\) −2.19839 −0.439678
\(26\) 5.79120 1.13575
\(27\) 0 0
\(28\) 2.09253 0.395451
\(29\) −0.101311 −0.0188129 −0.00940644 0.999956i \(-0.502994\pi\)
−0.00940644 + 0.999956i \(0.502994\pi\)
\(30\) 0 0
\(31\) −4.63283 −0.832080 −0.416040 0.909346i \(-0.636582\pi\)
−0.416040 + 0.909346i \(0.636582\pi\)
\(32\) 5.55970 0.982826
\(33\) 0 0
\(34\) 2.01536 0.345631
\(35\) 2.94196 0.497282
\(36\) 0 0
\(37\) −7.78041 −1.27909 −0.639546 0.768753i \(-0.720878\pi\)
−0.639546 + 0.768753i \(0.720878\pi\)
\(38\) −3.56345 −0.578068
\(39\) 0 0
\(40\) 4.80472 0.759692
\(41\) −1.42586 −0.222682 −0.111341 0.993782i \(-0.535515\pi\)
−0.111341 + 0.993782i \(0.535515\pi\)
\(42\) 0 0
\(43\) −3.65358 −0.557166 −0.278583 0.960412i \(-0.589865\pi\)
−0.278583 + 0.960412i \(0.589865\pi\)
\(44\) −6.05808 −0.913291
\(45\) 0 0
\(46\) −6.94088 −1.02338
\(47\) 7.73754 1.12864 0.564318 0.825558i \(-0.309139\pi\)
0.564318 + 0.825558i \(0.309139\pi\)
\(48\) 0 0
\(49\) −3.91066 −0.558665
\(50\) −1.97791 −0.279719
\(51\) 0 0
\(52\) −7.66311 −1.06268
\(53\) −1.61074 −0.221252 −0.110626 0.993862i \(-0.535285\pi\)
−0.110626 + 0.993862i \(0.535285\pi\)
\(54\) 0 0
\(55\) −8.51728 −1.14847
\(56\) 5.04542 0.674222
\(57\) 0 0
\(58\) −0.0911499 −0.0119686
\(59\) 2.46261 0.320604 0.160302 0.987068i \(-0.448753\pi\)
0.160302 + 0.987068i \(0.448753\pi\)
\(60\) 0 0
\(61\) −4.77088 −0.610849 −0.305424 0.952216i \(-0.598798\pi\)
−0.305424 + 0.952216i \(0.598798\pi\)
\(62\) −4.16819 −0.529361
\(63\) 0 0
\(64\) 5.40531 0.675664
\(65\) −10.7738 −1.33633
\(66\) 0 0
\(67\) −3.42357 −0.418255 −0.209128 0.977888i \(-0.567062\pi\)
−0.209128 + 0.977888i \(0.567062\pi\)
\(68\) −2.66679 −0.323396
\(69\) 0 0
\(70\) 2.64691 0.316366
\(71\) 13.1223 1.55733 0.778665 0.627440i \(-0.215897\pi\)
0.778665 + 0.627440i \(0.215897\pi\)
\(72\) 0 0
\(73\) −14.4582 −1.69220 −0.846100 0.533025i \(-0.821055\pi\)
−0.846100 + 0.533025i \(0.821055\pi\)
\(74\) −7.00010 −0.813745
\(75\) 0 0
\(76\) 4.71528 0.540880
\(77\) −8.94396 −1.01926
\(78\) 0 0
\(79\) −13.0644 −1.46986 −0.734930 0.678143i \(-0.762785\pi\)
−0.734930 + 0.678143i \(0.762785\pi\)
\(80\) 0.337442 0.0377271
\(81\) 0 0
\(82\) −1.28286 −0.141668
\(83\) 0.423954 0.0465350 0.0232675 0.999729i \(-0.492593\pi\)
0.0232675 + 0.999729i \(0.492593\pi\)
\(84\) 0 0
\(85\) −3.74933 −0.406672
\(86\) −3.28716 −0.354463
\(87\) 0 0
\(88\) −14.6070 −1.55711
\(89\) −17.1426 −1.81711 −0.908556 0.417763i \(-0.862814\pi\)
−0.908556 + 0.417763i \(0.862814\pi\)
\(90\) 0 0
\(91\) −11.3136 −1.18598
\(92\) 9.18440 0.957540
\(93\) 0 0
\(94\) 6.96153 0.718027
\(95\) 6.62938 0.680160
\(96\) 0 0
\(97\) 6.06424 0.615730 0.307865 0.951430i \(-0.400386\pi\)
0.307865 + 0.951430i \(0.400386\pi\)
\(98\) −3.51845 −0.355417
\(99\) 0 0
\(100\) 2.61724 0.261724
\(101\) −10.1166 −1.00664 −0.503320 0.864100i \(-0.667888\pi\)
−0.503320 + 0.864100i \(0.667888\pi\)
\(102\) 0 0
\(103\) 2.53914 0.250189 0.125094 0.992145i \(-0.460077\pi\)
0.125094 + 0.992145i \(0.460077\pi\)
\(104\) −18.4770 −1.81182
\(105\) 0 0
\(106\) −1.44919 −0.140758
\(107\) −9.82497 −0.949816 −0.474908 0.880035i \(-0.657519\pi\)
−0.474908 + 0.880035i \(0.657519\pi\)
\(108\) 0 0
\(109\) −15.1110 −1.44738 −0.723688 0.690127i \(-0.757555\pi\)
−0.723688 + 0.690127i \(0.757555\pi\)
\(110\) −7.66307 −0.730645
\(111\) 0 0
\(112\) 0.354346 0.0334826
\(113\) −2.58916 −0.243568 −0.121784 0.992557i \(-0.538861\pi\)
−0.121784 + 0.992557i \(0.538861\pi\)
\(114\) 0 0
\(115\) 12.9127 1.20411
\(116\) 0.120613 0.0111986
\(117\) 0 0
\(118\) 2.21563 0.203965
\(119\) −3.93716 −0.360919
\(120\) 0 0
\(121\) 14.8937 1.35397
\(122\) −4.29240 −0.388616
\(123\) 0 0
\(124\) 5.51549 0.495306
\(125\) 12.0487 1.07767
\(126\) 0 0
\(127\) 7.24277 0.642692 0.321346 0.946962i \(-0.395865\pi\)
0.321346 + 0.946962i \(0.395865\pi\)
\(128\) −6.25619 −0.552975
\(129\) 0 0
\(130\) −9.69332 −0.850160
\(131\) −2.14842 −0.187709 −0.0938543 0.995586i \(-0.529919\pi\)
−0.0938543 + 0.995586i \(0.529919\pi\)
\(132\) 0 0
\(133\) 6.96149 0.603637
\(134\) −3.08021 −0.266090
\(135\) 0 0
\(136\) −6.43005 −0.551372
\(137\) −6.53872 −0.558640 −0.279320 0.960198i \(-0.590109\pi\)
−0.279320 + 0.960198i \(0.590109\pi\)
\(138\) 0 0
\(139\) 9.78192 0.829691 0.414846 0.909892i \(-0.363836\pi\)
0.414846 + 0.909892i \(0.363836\pi\)
\(140\) −3.50248 −0.296013
\(141\) 0 0
\(142\) 11.8062 0.990758
\(143\) 32.7540 2.73902
\(144\) 0 0
\(145\) 0.169574 0.0140823
\(146\) −13.0081 −1.07656
\(147\) 0 0
\(148\) 9.26277 0.761395
\(149\) 12.0860 0.990120 0.495060 0.868859i \(-0.335146\pi\)
0.495060 + 0.868859i \(0.335146\pi\)
\(150\) 0 0
\(151\) 12.6923 1.03288 0.516442 0.856322i \(-0.327256\pi\)
0.516442 + 0.856322i \(0.327256\pi\)
\(152\) 11.3693 0.922170
\(153\) 0 0
\(154\) −8.04696 −0.648442
\(155\) 7.75443 0.622851
\(156\) 0 0
\(157\) −20.3326 −1.62272 −0.811359 0.584548i \(-0.801272\pi\)
−0.811359 + 0.584548i \(0.801272\pi\)
\(158\) −11.7542 −0.935110
\(159\) 0 0
\(160\) −9.30583 −0.735691
\(161\) 13.5596 1.06864
\(162\) 0 0
\(163\) −14.3750 −1.12593 −0.562967 0.826480i \(-0.690340\pi\)
−0.562967 + 0.826480i \(0.690340\pi\)
\(164\) 1.69752 0.132554
\(165\) 0 0
\(166\) 0.381435 0.0296051
\(167\) 18.7294 1.44933 0.724663 0.689103i \(-0.241995\pi\)
0.724663 + 0.689103i \(0.241995\pi\)
\(168\) 0 0
\(169\) 28.4318 2.18706
\(170\) −3.37331 −0.258721
\(171\) 0 0
\(172\) 4.34968 0.331660
\(173\) 11.7428 0.892786 0.446393 0.894837i \(-0.352708\pi\)
0.446393 + 0.894837i \(0.352708\pi\)
\(174\) 0 0
\(175\) 3.86401 0.292091
\(176\) −1.02587 −0.0773278
\(177\) 0 0
\(178\) −15.4233 −1.15603
\(179\) −24.5193 −1.83266 −0.916328 0.400429i \(-0.868861\pi\)
−0.916328 + 0.400429i \(0.868861\pi\)
\(180\) 0 0
\(181\) 2.13689 0.158833 0.0794167 0.996842i \(-0.474694\pi\)
0.0794167 + 0.996842i \(0.474694\pi\)
\(182\) −10.1789 −0.754512
\(183\) 0 0
\(184\) 22.1450 1.63255
\(185\) 13.0229 0.957460
\(186\) 0 0
\(187\) 11.3985 0.833540
\(188\) −9.21173 −0.671834
\(189\) 0 0
\(190\) 5.96451 0.432711
\(191\) −11.6936 −0.846118 −0.423059 0.906102i \(-0.639044\pi\)
−0.423059 + 0.906102i \(0.639044\pi\)
\(192\) 0 0
\(193\) 2.15548 0.155155 0.0775774 0.996986i \(-0.475282\pi\)
0.0775774 + 0.996986i \(0.475282\pi\)
\(194\) 5.45605 0.391721
\(195\) 0 0
\(196\) 4.65573 0.332552
\(197\) −6.46675 −0.460737 −0.230368 0.973104i \(-0.573993\pi\)
−0.230368 + 0.973104i \(0.573993\pi\)
\(198\) 0 0
\(199\) 21.8975 1.55228 0.776138 0.630563i \(-0.217176\pi\)
0.776138 + 0.630563i \(0.217176\pi\)
\(200\) 6.31057 0.446225
\(201\) 0 0
\(202\) −9.10200 −0.640415
\(203\) 0.178069 0.0124980
\(204\) 0 0
\(205\) 2.38661 0.166688
\(206\) 2.28449 0.159168
\(207\) 0 0
\(208\) −1.29766 −0.0899767
\(209\) −20.1542 −1.39410
\(210\) 0 0
\(211\) 11.2465 0.774243 0.387122 0.922029i \(-0.373469\pi\)
0.387122 + 0.922029i \(0.373469\pi\)
\(212\) 1.91762 0.131703
\(213\) 0 0
\(214\) −8.83961 −0.604263
\(215\) 6.11537 0.417065
\(216\) 0 0
\(217\) 8.14290 0.552776
\(218\) −13.5955 −0.920806
\(219\) 0 0
\(220\) 10.1400 0.683641
\(221\) 14.4184 0.969887
\(222\) 0 0
\(223\) −19.8849 −1.33159 −0.665797 0.746133i \(-0.731908\pi\)
−0.665797 + 0.746133i \(0.731908\pi\)
\(224\) −9.77202 −0.652921
\(225\) 0 0
\(226\) −2.32949 −0.154955
\(227\) 6.20415 0.411784 0.205892 0.978575i \(-0.433990\pi\)
0.205892 + 0.978575i \(0.433990\pi\)
\(228\) 0 0
\(229\) −6.29386 −0.415910 −0.207955 0.978138i \(-0.566681\pi\)
−0.207955 + 0.978138i \(0.566681\pi\)
\(230\) 11.6177 0.766045
\(231\) 0 0
\(232\) 0.290816 0.0190930
\(233\) 1.37434 0.0900362 0.0450181 0.998986i \(-0.485665\pi\)
0.0450181 + 0.998986i \(0.485665\pi\)
\(234\) 0 0
\(235\) −12.9511 −0.844836
\(236\) −2.93179 −0.190843
\(237\) 0 0
\(238\) −3.54230 −0.229613
\(239\) 1.00000 0.0646846
\(240\) 0 0
\(241\) −16.6512 −1.07260 −0.536299 0.844028i \(-0.680178\pi\)
−0.536299 + 0.844028i \(0.680178\pi\)
\(242\) 13.4000 0.861383
\(243\) 0 0
\(244\) 5.67985 0.363615
\(245\) 6.54566 0.418187
\(246\) 0 0
\(247\) −25.4939 −1.62214
\(248\) 13.2987 0.844470
\(249\) 0 0
\(250\) 10.8403 0.685600
\(251\) −8.70369 −0.549372 −0.274686 0.961534i \(-0.588574\pi\)
−0.274686 + 0.961534i \(0.588574\pi\)
\(252\) 0 0
\(253\) −39.2563 −2.46802
\(254\) 6.51638 0.408874
\(255\) 0 0
\(256\) −16.4394 −1.02746
\(257\) 17.8087 1.11087 0.555437 0.831559i \(-0.312551\pi\)
0.555437 + 0.831559i \(0.312551\pi\)
\(258\) 0 0
\(259\) 13.6753 0.849739
\(260\) 12.8265 0.795467
\(261\) 0 0
\(262\) −1.93295 −0.119418
\(263\) −21.2660 −1.31132 −0.655658 0.755058i \(-0.727609\pi\)
−0.655658 + 0.755058i \(0.727609\pi\)
\(264\) 0 0
\(265\) 2.69605 0.165617
\(266\) 6.26331 0.384028
\(267\) 0 0
\(268\) 4.07584 0.248971
\(269\) 17.1887 1.04801 0.524005 0.851715i \(-0.324437\pi\)
0.524005 + 0.851715i \(0.324437\pi\)
\(270\) 0 0
\(271\) 19.1030 1.16043 0.580213 0.814465i \(-0.302969\pi\)
0.580213 + 0.814465i \(0.302969\pi\)
\(272\) −0.451591 −0.0273817
\(273\) 0 0
\(274\) −5.88294 −0.355401
\(275\) −11.1867 −0.674583
\(276\) 0 0
\(277\) 19.3315 1.16152 0.580760 0.814075i \(-0.302755\pi\)
0.580760 + 0.814075i \(0.302755\pi\)
\(278\) 8.80087 0.527841
\(279\) 0 0
\(280\) −8.44502 −0.504687
\(281\) 4.05588 0.241954 0.120977 0.992655i \(-0.461397\pi\)
0.120977 + 0.992655i \(0.461397\pi\)
\(282\) 0 0
\(283\) −10.8007 −0.642035 −0.321018 0.947073i \(-0.604025\pi\)
−0.321018 + 0.947073i \(0.604025\pi\)
\(284\) −15.6224 −0.927020
\(285\) 0 0
\(286\) 29.4690 1.74254
\(287\) 2.50617 0.147934
\(288\) 0 0
\(289\) −11.9823 −0.704844
\(290\) 0.152567 0.00895903
\(291\) 0 0
\(292\) 17.2128 1.00730
\(293\) −5.01609 −0.293043 −0.146522 0.989207i \(-0.546808\pi\)
−0.146522 + 0.989207i \(0.546808\pi\)
\(294\) 0 0
\(295\) −4.12191 −0.239987
\(296\) 22.3340 1.29814
\(297\) 0 0
\(298\) 10.8738 0.629904
\(299\) −49.6569 −2.87173
\(300\) 0 0
\(301\) 6.42173 0.370142
\(302\) 11.4194 0.657110
\(303\) 0 0
\(304\) 0.798480 0.0457959
\(305\) 7.98550 0.457249
\(306\) 0 0
\(307\) −9.70591 −0.553945 −0.276973 0.960878i \(-0.589331\pi\)
−0.276973 + 0.960878i \(0.589331\pi\)
\(308\) 10.6480 0.606727
\(309\) 0 0
\(310\) 6.97673 0.396251
\(311\) 10.3411 0.586389 0.293195 0.956053i \(-0.405282\pi\)
0.293195 + 0.956053i \(0.405282\pi\)
\(312\) 0 0
\(313\) −23.8987 −1.35084 −0.675418 0.737435i \(-0.736037\pi\)
−0.675418 + 0.737435i \(0.736037\pi\)
\(314\) −18.2934 −1.03236
\(315\) 0 0
\(316\) 15.5535 0.874952
\(317\) 1.48619 0.0834729 0.0417365 0.999129i \(-0.486711\pi\)
0.0417365 + 0.999129i \(0.486711\pi\)
\(318\) 0 0
\(319\) −0.515527 −0.0288640
\(320\) −9.04742 −0.505766
\(321\) 0 0
\(322\) 12.1997 0.679860
\(323\) −8.87196 −0.493649
\(324\) 0 0
\(325\) −14.1505 −0.784928
\(326\) −12.9333 −0.716307
\(327\) 0 0
\(328\) 4.09299 0.225998
\(329\) −13.5999 −0.749787
\(330\) 0 0
\(331\) −17.6038 −0.967593 −0.483797 0.875180i \(-0.660743\pi\)
−0.483797 + 0.875180i \(0.660743\pi\)
\(332\) −0.504728 −0.0277006
\(333\) 0 0
\(334\) 16.8510 0.922047
\(335\) 5.73037 0.313083
\(336\) 0 0
\(337\) 18.9294 1.03115 0.515576 0.856844i \(-0.327578\pi\)
0.515576 + 0.856844i \(0.327578\pi\)
\(338\) 25.5803 1.39139
\(339\) 0 0
\(340\) 4.46368 0.242077
\(341\) −23.5745 −1.27663
\(342\) 0 0
\(343\) 19.1771 1.03547
\(344\) 10.4878 0.565462
\(345\) 0 0
\(346\) 10.5651 0.567982
\(347\) 1.63077 0.0875444 0.0437722 0.999042i \(-0.486062\pi\)
0.0437722 + 0.999042i \(0.486062\pi\)
\(348\) 0 0
\(349\) 8.00213 0.428344 0.214172 0.976796i \(-0.431295\pi\)
0.214172 + 0.976796i \(0.431295\pi\)
\(350\) 3.47648 0.185826
\(351\) 0 0
\(352\) 28.2910 1.50792
\(353\) −9.60124 −0.511022 −0.255511 0.966806i \(-0.582244\pi\)
−0.255511 + 0.966806i \(0.582244\pi\)
\(354\) 0 0
\(355\) −21.9641 −1.16573
\(356\) 20.4087 1.08166
\(357\) 0 0
\(358\) −22.0602 −1.16592
\(359\) 31.4338 1.65901 0.829506 0.558498i \(-0.188622\pi\)
0.829506 + 0.558498i \(0.188622\pi\)
\(360\) 0 0
\(361\) −3.31307 −0.174372
\(362\) 1.92257 0.101048
\(363\) 0 0
\(364\) 13.4691 0.705972
\(365\) 24.2001 1.26669
\(366\) 0 0
\(367\) −9.76744 −0.509856 −0.254928 0.966960i \(-0.582052\pi\)
−0.254928 + 0.966960i \(0.582052\pi\)
\(368\) 1.55528 0.0810744
\(369\) 0 0
\(370\) 11.7168 0.609126
\(371\) 2.83111 0.146984
\(372\) 0 0
\(373\) 18.4876 0.957252 0.478626 0.878019i \(-0.341135\pi\)
0.478626 + 0.878019i \(0.341135\pi\)
\(374\) 10.2553 0.530290
\(375\) 0 0
\(376\) −22.2109 −1.14544
\(377\) −0.652111 −0.0335854
\(378\) 0 0
\(379\) −1.80626 −0.0927813 −0.0463907 0.998923i \(-0.514772\pi\)
−0.0463907 + 0.998923i \(0.514772\pi\)
\(380\) −7.89244 −0.404874
\(381\) 0 0
\(382\) −10.5208 −0.538292
\(383\) 25.2177 1.28856 0.644281 0.764789i \(-0.277157\pi\)
0.644281 + 0.764789i \(0.277157\pi\)
\(384\) 0 0
\(385\) 14.9704 0.762963
\(386\) 1.93930 0.0987079
\(387\) 0 0
\(388\) −7.21962 −0.366521
\(389\) −29.1698 −1.47897 −0.739485 0.673173i \(-0.764931\pi\)
−0.739485 + 0.673173i \(0.764931\pi\)
\(390\) 0 0
\(391\) −17.2808 −0.873926
\(392\) 11.2257 0.566984
\(393\) 0 0
\(394\) −5.81819 −0.293116
\(395\) 21.8672 1.10026
\(396\) 0 0
\(397\) 37.3463 1.87435 0.937177 0.348853i \(-0.113429\pi\)
0.937177 + 0.348853i \(0.113429\pi\)
\(398\) 19.7014 0.987542
\(399\) 0 0
\(400\) 0.443200 0.0221600
\(401\) −5.50916 −0.275114 −0.137557 0.990494i \(-0.543925\pi\)
−0.137557 + 0.990494i \(0.543925\pi\)
\(402\) 0 0
\(403\) −29.8204 −1.48546
\(404\) 12.0441 0.599215
\(405\) 0 0
\(406\) 0.160210 0.00795108
\(407\) −39.5913 −1.96247
\(408\) 0 0
\(409\) −28.2723 −1.39797 −0.698987 0.715134i \(-0.746366\pi\)
−0.698987 + 0.715134i \(0.746366\pi\)
\(410\) 2.14725 0.106045
\(411\) 0 0
\(412\) −3.02291 −0.148928
\(413\) −4.32840 −0.212987
\(414\) 0 0
\(415\) −0.709615 −0.0348337
\(416\) 35.7864 1.75457
\(417\) 0 0
\(418\) −18.1329 −0.886910
\(419\) −12.0357 −0.587983 −0.293992 0.955808i \(-0.594984\pi\)
−0.293992 + 0.955808i \(0.594984\pi\)
\(420\) 0 0
\(421\) −29.9983 −1.46203 −0.731015 0.682362i \(-0.760953\pi\)
−0.731015 + 0.682362i \(0.760953\pi\)
\(422\) 10.1186 0.492566
\(423\) 0 0
\(424\) 4.62368 0.224546
\(425\) −4.92442 −0.238869
\(426\) 0 0
\(427\) 8.38555 0.405805
\(428\) 11.6969 0.565390
\(429\) 0 0
\(430\) 5.50205 0.265332
\(431\) 17.8098 0.857870 0.428935 0.903335i \(-0.358889\pi\)
0.428935 + 0.903335i \(0.358889\pi\)
\(432\) 0 0
\(433\) 20.2488 0.973094 0.486547 0.873654i \(-0.338256\pi\)
0.486547 + 0.873654i \(0.338256\pi\)
\(434\) 7.32623 0.351670
\(435\) 0 0
\(436\) 17.9901 0.861568
\(437\) 30.5550 1.46164
\(438\) 0 0
\(439\) −34.7608 −1.65904 −0.829521 0.558475i \(-0.811387\pi\)
−0.829521 + 0.558475i \(0.811387\pi\)
\(440\) 24.4492 1.16557
\(441\) 0 0
\(442\) 12.9724 0.617032
\(443\) 25.7595 1.22387 0.611935 0.790908i \(-0.290392\pi\)
0.611935 + 0.790908i \(0.290392\pi\)
\(444\) 0 0
\(445\) 28.6933 1.36019
\(446\) −17.8907 −0.847147
\(447\) 0 0
\(448\) −9.50066 −0.448864
\(449\) 30.2122 1.42580 0.712901 0.701265i \(-0.247381\pi\)
0.712901 + 0.701265i \(0.247381\pi\)
\(450\) 0 0
\(451\) −7.25561 −0.341653
\(452\) 3.08246 0.144987
\(453\) 0 0
\(454\) 5.58193 0.261973
\(455\) 18.9367 0.887765
\(456\) 0 0
\(457\) 1.33578 0.0624850 0.0312425 0.999512i \(-0.490054\pi\)
0.0312425 + 0.999512i \(0.490054\pi\)
\(458\) −5.66264 −0.264598
\(459\) 0 0
\(460\) −15.3729 −0.716764
\(461\) −16.6243 −0.774272 −0.387136 0.922023i \(-0.626536\pi\)
−0.387136 + 0.922023i \(0.626536\pi\)
\(462\) 0 0
\(463\) 29.4246 1.36748 0.683739 0.729726i \(-0.260353\pi\)
0.683739 + 0.729726i \(0.260353\pi\)
\(464\) 0.0204244 0.000948179 0
\(465\) 0 0
\(466\) 1.23651 0.0572801
\(467\) 2.53660 0.117380 0.0586899 0.998276i \(-0.481308\pi\)
0.0586899 + 0.998276i \(0.481308\pi\)
\(468\) 0 0
\(469\) 6.01744 0.277859
\(470\) −11.6522 −0.537476
\(471\) 0 0
\(472\) −7.06901 −0.325378
\(473\) −18.5916 −0.854841
\(474\) 0 0
\(475\) 8.70710 0.399509
\(476\) 4.68729 0.214842
\(477\) 0 0
\(478\) 0.899709 0.0411517
\(479\) −38.6620 −1.76651 −0.883255 0.468893i \(-0.844653\pi\)
−0.883255 + 0.468893i \(0.844653\pi\)
\(480\) 0 0
\(481\) −50.0806 −2.28348
\(482\) −14.9812 −0.682376
\(483\) 0 0
\(484\) −17.7313 −0.805968
\(485\) −10.1503 −0.460903
\(486\) 0 0
\(487\) 9.29522 0.421207 0.210603 0.977572i \(-0.432457\pi\)
0.210603 + 0.977572i \(0.432457\pi\)
\(488\) 13.6950 0.619944
\(489\) 0 0
\(490\) 5.88919 0.266046
\(491\) −11.3475 −0.512105 −0.256052 0.966663i \(-0.582422\pi\)
−0.256052 + 0.966663i \(0.582422\pi\)
\(492\) 0 0
\(493\) −0.226937 −0.0102207
\(494\) −22.9371 −1.03199
\(495\) 0 0
\(496\) 0.933987 0.0419372
\(497\) −23.0644 −1.03458
\(498\) 0 0
\(499\) 24.8745 1.11353 0.556767 0.830669i \(-0.312042\pi\)
0.556767 + 0.830669i \(0.312042\pi\)
\(500\) −14.3442 −0.641494
\(501\) 0 0
\(502\) −7.83078 −0.349505
\(503\) −13.2786 −0.592064 −0.296032 0.955178i \(-0.595663\pi\)
−0.296032 + 0.955178i \(0.595663\pi\)
\(504\) 0 0
\(505\) 16.9332 0.753517
\(506\) −35.3192 −1.57013
\(507\) 0 0
\(508\) −8.62270 −0.382570
\(509\) −33.4053 −1.48067 −0.740333 0.672241i \(-0.765332\pi\)
−0.740333 + 0.672241i \(0.765332\pi\)
\(510\) 0 0
\(511\) 25.4124 1.12418
\(512\) −2.27826 −0.100686
\(513\) 0 0
\(514\) 16.0226 0.706727
\(515\) −4.25001 −0.187278
\(516\) 0 0
\(517\) 39.3731 1.73163
\(518\) 12.3037 0.540596
\(519\) 0 0
\(520\) 30.9268 1.35623
\(521\) −33.4430 −1.46516 −0.732582 0.680678i \(-0.761685\pi\)
−0.732582 + 0.680678i \(0.761685\pi\)
\(522\) 0 0
\(523\) −25.7941 −1.12790 −0.563948 0.825810i \(-0.690718\pi\)
−0.563948 + 0.825810i \(0.690718\pi\)
\(524\) 2.55775 0.111736
\(525\) 0 0
\(526\) −19.1332 −0.834246
\(527\) −10.3776 −0.452055
\(528\) 0 0
\(529\) 36.5148 1.58760
\(530\) 2.42566 0.105364
\(531\) 0 0
\(532\) −8.28782 −0.359323
\(533\) −9.17791 −0.397540
\(534\) 0 0
\(535\) 16.4451 0.710982
\(536\) 9.82749 0.424483
\(537\) 0 0
\(538\) 15.4648 0.666734
\(539\) −19.8997 −0.857141
\(540\) 0 0
\(541\) 4.49430 0.193225 0.0966126 0.995322i \(-0.469199\pi\)
0.0966126 + 0.995322i \(0.469199\pi\)
\(542\) 17.1871 0.738251
\(543\) 0 0
\(544\) 12.4538 0.533952
\(545\) 25.2929 1.08343
\(546\) 0 0
\(547\) −24.5160 −1.04823 −0.524114 0.851648i \(-0.675603\pi\)
−0.524114 + 0.851648i \(0.675603\pi\)
\(548\) 7.78450 0.332537
\(549\) 0 0
\(550\) −10.0648 −0.429163
\(551\) 0.401258 0.0170942
\(552\) 0 0
\(553\) 22.9627 0.976472
\(554\) 17.3928 0.738947
\(555\) 0 0
\(556\) −11.6456 −0.493884
\(557\) −25.5502 −1.08260 −0.541299 0.840830i \(-0.682067\pi\)
−0.541299 + 0.840830i \(0.682067\pi\)
\(558\) 0 0
\(559\) −23.5172 −0.994672
\(560\) −0.593105 −0.0250633
\(561\) 0 0
\(562\) 3.64911 0.153929
\(563\) 13.4776 0.568011 0.284006 0.958823i \(-0.408337\pi\)
0.284006 + 0.958823i \(0.408337\pi\)
\(564\) 0 0
\(565\) 4.33374 0.182322
\(566\) −9.71749 −0.408456
\(567\) 0 0
\(568\) −37.6681 −1.58052
\(569\) 6.39428 0.268062 0.134031 0.990977i \(-0.457208\pi\)
0.134031 + 0.990977i \(0.457208\pi\)
\(570\) 0 0
\(571\) −17.9506 −0.751210 −0.375605 0.926780i \(-0.622565\pi\)
−0.375605 + 0.926780i \(0.622565\pi\)
\(572\) −38.9944 −1.63044
\(573\) 0 0
\(574\) 2.25482 0.0941144
\(575\) 16.9597 0.707267
\(576\) 0 0
\(577\) −37.1291 −1.54571 −0.772853 0.634585i \(-0.781171\pi\)
−0.772853 + 0.634585i \(0.781171\pi\)
\(578\) −10.7806 −0.448415
\(579\) 0 0
\(580\) −0.201882 −0.00838268
\(581\) −0.745165 −0.0309146
\(582\) 0 0
\(583\) −8.19636 −0.339459
\(584\) 41.5027 1.71740
\(585\) 0 0
\(586\) −4.51302 −0.186431
\(587\) 36.6232 1.51160 0.755800 0.654803i \(-0.227248\pi\)
0.755800 + 0.654803i \(0.227248\pi\)
\(588\) 0 0
\(589\) 18.3491 0.756062
\(590\) −3.70852 −0.152677
\(591\) 0 0
\(592\) 1.56855 0.0644669
\(593\) −38.1568 −1.56691 −0.783456 0.621447i \(-0.786545\pi\)
−0.783456 + 0.621447i \(0.786545\pi\)
\(594\) 0 0
\(595\) 6.59003 0.270165
\(596\) −14.3886 −0.589381
\(597\) 0 0
\(598\) −44.6767 −1.82697
\(599\) −48.4161 −1.97823 −0.989115 0.147148i \(-0.952991\pi\)
−0.989115 + 0.147148i \(0.952991\pi\)
\(600\) 0 0
\(601\) 22.2469 0.907472 0.453736 0.891136i \(-0.350091\pi\)
0.453736 + 0.891136i \(0.350091\pi\)
\(602\) 5.77768 0.235481
\(603\) 0 0
\(604\) −15.1105 −0.614837
\(605\) −24.9291 −1.01351
\(606\) 0 0
\(607\) 40.3888 1.63933 0.819665 0.572842i \(-0.194159\pi\)
0.819665 + 0.572842i \(0.194159\pi\)
\(608\) −22.0202 −0.893035
\(609\) 0 0
\(610\) 7.18463 0.290897
\(611\) 49.8046 2.01488
\(612\) 0 0
\(613\) −16.3156 −0.658979 −0.329490 0.944159i \(-0.606877\pi\)
−0.329490 + 0.944159i \(0.606877\pi\)
\(614\) −8.73249 −0.352415
\(615\) 0 0
\(616\) 25.6740 1.03444
\(617\) −5.46752 −0.220114 −0.110057 0.993925i \(-0.535103\pi\)
−0.110057 + 0.993925i \(0.535103\pi\)
\(618\) 0 0
\(619\) 43.6271 1.75352 0.876761 0.480926i \(-0.159700\pi\)
0.876761 + 0.480926i \(0.159700\pi\)
\(620\) −9.23184 −0.370760
\(621\) 0 0
\(622\) 9.30397 0.373055
\(623\) 30.1307 1.20716
\(624\) 0 0
\(625\) −9.17513 −0.367005
\(626\) −21.5019 −0.859389
\(627\) 0 0
\(628\) 24.2065 0.965943
\(629\) −17.4282 −0.694908
\(630\) 0 0
\(631\) 25.7573 1.02538 0.512692 0.858573i \(-0.328648\pi\)
0.512692 + 0.858573i \(0.328648\pi\)
\(632\) 37.5019 1.49175
\(633\) 0 0
\(634\) 1.33714 0.0531046
\(635\) −12.1230 −0.481085
\(636\) 0 0
\(637\) −25.1719 −0.997349
\(638\) −0.463824 −0.0183630
\(639\) 0 0
\(640\) 10.4716 0.413927
\(641\) −2.85693 −0.112842 −0.0564210 0.998407i \(-0.517969\pi\)
−0.0564210 + 0.998407i \(0.517969\pi\)
\(642\) 0 0
\(643\) 10.7053 0.422176 0.211088 0.977467i \(-0.432299\pi\)
0.211088 + 0.977467i \(0.432299\pi\)
\(644\) −16.1430 −0.636123
\(645\) 0 0
\(646\) −7.98217 −0.314054
\(647\) −10.9862 −0.431913 −0.215956 0.976403i \(-0.569287\pi\)
−0.215956 + 0.976403i \(0.569287\pi\)
\(648\) 0 0
\(649\) 12.5312 0.491892
\(650\) −12.7313 −0.499363
\(651\) 0 0
\(652\) 17.1137 0.670226
\(653\) 41.1768 1.61137 0.805687 0.592342i \(-0.201797\pi\)
0.805687 + 0.592342i \(0.201797\pi\)
\(654\) 0 0
\(655\) 3.59603 0.140509
\(656\) 0.287456 0.0112233
\(657\) 0 0
\(658\) −12.2359 −0.477007
\(659\) −1.39962 −0.0545216 −0.0272608 0.999628i \(-0.508678\pi\)
−0.0272608 + 0.999628i \(0.508678\pi\)
\(660\) 0 0
\(661\) 26.9112 1.04673 0.523363 0.852110i \(-0.324677\pi\)
0.523363 + 0.852110i \(0.324677\pi\)
\(662\) −15.8383 −0.615573
\(663\) 0 0
\(664\) −1.21698 −0.0472280
\(665\) −11.6521 −0.451851
\(666\) 0 0
\(667\) 0.781569 0.0302625
\(668\) −22.2978 −0.862729
\(669\) 0 0
\(670\) 5.15566 0.199181
\(671\) −24.2770 −0.937204
\(672\) 0 0
\(673\) 35.4943 1.36821 0.684103 0.729386i \(-0.260194\pi\)
0.684103 + 0.729386i \(0.260194\pi\)
\(674\) 17.0310 0.656009
\(675\) 0 0
\(676\) −33.8487 −1.30187
\(677\) −7.66192 −0.294471 −0.147236 0.989101i \(-0.547038\pi\)
−0.147236 + 0.989101i \(0.547038\pi\)
\(678\) 0 0
\(679\) −10.6588 −0.409048
\(680\) 10.7626 0.412728
\(681\) 0 0
\(682\) −21.2102 −0.812180
\(683\) 11.5533 0.442074 0.221037 0.975265i \(-0.429056\pi\)
0.221037 + 0.975265i \(0.429056\pi\)
\(684\) 0 0
\(685\) 10.9445 0.418168
\(686\) 17.2538 0.658755
\(687\) 0 0
\(688\) 0.736569 0.0280814
\(689\) −10.3679 −0.394986
\(690\) 0 0
\(691\) 9.48943 0.360995 0.180498 0.983575i \(-0.442229\pi\)
0.180498 + 0.983575i \(0.442229\pi\)
\(692\) −13.9801 −0.531442
\(693\) 0 0
\(694\) 1.46722 0.0556949
\(695\) −16.3730 −0.621063
\(696\) 0 0
\(697\) −3.19395 −0.120979
\(698\) 7.19959 0.272508
\(699\) 0 0
\(700\) −4.60019 −0.173871
\(701\) 33.6461 1.27079 0.635397 0.772186i \(-0.280836\pi\)
0.635397 + 0.772186i \(0.280836\pi\)
\(702\) 0 0
\(703\) 30.8157 1.16223
\(704\) 27.5054 1.03665
\(705\) 0 0
\(706\) −8.63831 −0.325107
\(707\) 17.7815 0.668742
\(708\) 0 0
\(709\) −27.7519 −1.04224 −0.521122 0.853482i \(-0.674486\pi\)
−0.521122 + 0.853482i \(0.674486\pi\)
\(710\) −19.7613 −0.741629
\(711\) 0 0
\(712\) 49.2085 1.84417
\(713\) 35.7403 1.33849
\(714\) 0 0
\(715\) −54.8236 −2.05029
\(716\) 29.1908 1.09091
\(717\) 0 0
\(718\) 28.2812 1.05545
\(719\) 28.9767 1.08065 0.540324 0.841457i \(-0.318302\pi\)
0.540324 + 0.841457i \(0.318302\pi\)
\(720\) 0 0
\(721\) −4.46292 −0.166208
\(722\) −2.98080 −0.110934
\(723\) 0 0
\(724\) −2.54402 −0.0945476
\(725\) 0.222720 0.00827161
\(726\) 0 0
\(727\) −17.6229 −0.653599 −0.326799 0.945094i \(-0.605970\pi\)
−0.326799 + 0.945094i \(0.605970\pi\)
\(728\) 32.4761 1.20364
\(729\) 0 0
\(730\) 21.7730 0.805855
\(731\) −8.18407 −0.302699
\(732\) 0 0
\(733\) 10.2978 0.380359 0.190180 0.981749i \(-0.439093\pi\)
0.190180 + 0.981749i \(0.439093\pi\)
\(734\) −8.78785 −0.324366
\(735\) 0 0
\(736\) −42.8908 −1.58098
\(737\) −17.4211 −0.641714
\(738\) 0 0
\(739\) 13.2229 0.486411 0.243206 0.969975i \(-0.421801\pi\)
0.243206 + 0.969975i \(0.421801\pi\)
\(740\) −15.5040 −0.569940
\(741\) 0 0
\(742\) 2.54718 0.0935098
\(743\) −7.80394 −0.286299 −0.143149 0.989701i \(-0.545723\pi\)
−0.143149 + 0.989701i \(0.545723\pi\)
\(744\) 0 0
\(745\) −20.2295 −0.741151
\(746\) 16.6335 0.608994
\(747\) 0 0
\(748\) −13.5702 −0.496175
\(749\) 17.2689 0.630991
\(750\) 0 0
\(751\) 50.3886 1.83871 0.919353 0.393435i \(-0.128713\pi\)
0.919353 + 0.393435i \(0.128713\pi\)
\(752\) −1.55990 −0.0568838
\(753\) 0 0
\(754\) −0.586710 −0.0213667
\(755\) −21.2444 −0.773162
\(756\) 0 0
\(757\) −46.6796 −1.69660 −0.848299 0.529517i \(-0.822373\pi\)
−0.848299 + 0.529517i \(0.822373\pi\)
\(758\) −1.62511 −0.0590265
\(759\) 0 0
\(760\) −19.0299 −0.690287
\(761\) −24.2370 −0.878591 −0.439296 0.898343i \(-0.644772\pi\)
−0.439296 + 0.898343i \(0.644772\pi\)
\(762\) 0 0
\(763\) 26.5600 0.961536
\(764\) 13.9215 0.503662
\(765\) 0 0
\(766\) 22.6886 0.819771
\(767\) 15.8512 0.572353
\(768\) 0 0
\(769\) 23.9656 0.864222 0.432111 0.901820i \(-0.357769\pi\)
0.432111 + 0.901820i \(0.357769\pi\)
\(770\) 13.4690 0.485389
\(771\) 0 0
\(772\) −2.56615 −0.0923578
\(773\) −1.40235 −0.0504391 −0.0252195 0.999682i \(-0.508028\pi\)
−0.0252195 + 0.999682i \(0.508028\pi\)
\(774\) 0 0
\(775\) 10.1848 0.365847
\(776\) −17.4076 −0.624898
\(777\) 0 0
\(778\) −26.2444 −0.940906
\(779\) 5.64737 0.202338
\(780\) 0 0
\(781\) 66.7739 2.38936
\(782\) −15.5477 −0.555983
\(783\) 0 0
\(784\) 0.788396 0.0281570
\(785\) 34.0327 1.21468
\(786\) 0 0
\(787\) −22.5141 −0.802540 −0.401270 0.915960i \(-0.631431\pi\)
−0.401270 + 0.915960i \(0.631431\pi\)
\(788\) 7.69882 0.274259
\(789\) 0 0
\(790\) 19.6741 0.699974
\(791\) 4.55085 0.161809
\(792\) 0 0
\(793\) −30.7090 −1.09051
\(794\) 33.6007 1.19245
\(795\) 0 0
\(796\) −26.0696 −0.924011
\(797\) 35.7937 1.26788 0.633939 0.773383i \(-0.281437\pi\)
0.633939 + 0.773383i \(0.281437\pi\)
\(798\) 0 0
\(799\) 17.3322 0.613168
\(800\) −12.2224 −0.432127
\(801\) 0 0
\(802\) −4.95664 −0.175025
\(803\) −73.5715 −2.59628
\(804\) 0 0
\(805\) −22.6960 −0.799929
\(806\) −26.8296 −0.945033
\(807\) 0 0
\(808\) 29.0402 1.02163
\(809\) 22.3480 0.785712 0.392856 0.919600i \(-0.371487\pi\)
0.392856 + 0.919600i \(0.371487\pi\)
\(810\) 0 0
\(811\) 12.5070 0.439182 0.219591 0.975592i \(-0.429528\pi\)
0.219591 + 0.975592i \(0.429528\pi\)
\(812\) −0.211995 −0.00743957
\(813\) 0 0
\(814\) −35.6206 −1.24850
\(815\) 24.0608 0.842813
\(816\) 0 0
\(817\) 14.4706 0.506264
\(818\) −25.4368 −0.889377
\(819\) 0 0
\(820\) −2.84131 −0.0992230
\(821\) 38.7711 1.35312 0.676560 0.736387i \(-0.263470\pi\)
0.676560 + 0.736387i \(0.263470\pi\)
\(822\) 0 0
\(823\) 51.3284 1.78920 0.894598 0.446872i \(-0.147462\pi\)
0.894598 + 0.446872i \(0.147462\pi\)
\(824\) −7.28871 −0.253914
\(825\) 0 0
\(826\) −3.89430 −0.135500
\(827\) 15.0423 0.523072 0.261536 0.965194i \(-0.415771\pi\)
0.261536 + 0.965194i \(0.415771\pi\)
\(828\) 0 0
\(829\) −24.9052 −0.864992 −0.432496 0.901636i \(-0.642367\pi\)
−0.432496 + 0.901636i \(0.642367\pi\)
\(830\) −0.638447 −0.0221608
\(831\) 0 0
\(832\) 34.7927 1.20622
\(833\) −8.75992 −0.303513
\(834\) 0 0
\(835\) −31.3493 −1.08489
\(836\) 23.9941 0.829853
\(837\) 0 0
\(838\) −10.8286 −0.374069
\(839\) −14.5488 −0.502279 −0.251139 0.967951i \(-0.580805\pi\)
−0.251139 + 0.967951i \(0.580805\pi\)
\(840\) 0 0
\(841\) −28.9897 −0.999646
\(842\) −26.9898 −0.930129
\(843\) 0 0
\(844\) −13.3893 −0.460878
\(845\) −47.5892 −1.63712
\(846\) 0 0
\(847\) −26.1779 −0.899484
\(848\) 0.324727 0.0111512
\(849\) 0 0
\(850\) −4.43054 −0.151966
\(851\) 60.0226 2.05755
\(852\) 0 0
\(853\) −33.7550 −1.15575 −0.577875 0.816125i \(-0.696118\pi\)
−0.577875 + 0.816125i \(0.696118\pi\)
\(854\) 7.54455 0.258169
\(855\) 0 0
\(856\) 28.2030 0.963959
\(857\) −47.0497 −1.60719 −0.803593 0.595179i \(-0.797081\pi\)
−0.803593 + 0.595179i \(0.797081\pi\)
\(858\) 0 0
\(859\) −47.7888 −1.63053 −0.815267 0.579085i \(-0.803410\pi\)
−0.815267 + 0.579085i \(0.803410\pi\)
\(860\) −7.28050 −0.248263
\(861\) 0 0
\(862\) 16.0237 0.545768
\(863\) −30.8008 −1.04847 −0.524236 0.851573i \(-0.675649\pi\)
−0.524236 + 0.851573i \(0.675649\pi\)
\(864\) 0 0
\(865\) −19.6551 −0.668292
\(866\) 18.2180 0.619072
\(867\) 0 0
\(868\) −9.69432 −0.329047
\(869\) −66.4793 −2.25516
\(870\) 0 0
\(871\) −22.0366 −0.746683
\(872\) 43.3769 1.46893
\(873\) 0 0
\(874\) 27.4906 0.929882
\(875\) −21.1774 −0.715926
\(876\) 0 0
\(877\) −43.0660 −1.45424 −0.727118 0.686512i \(-0.759141\pi\)
−0.727118 + 0.686512i \(0.759141\pi\)
\(878\) −31.2746 −1.05547
\(879\) 0 0
\(880\) 1.71710 0.0578834
\(881\) −30.2543 −1.01929 −0.509647 0.860384i \(-0.670224\pi\)
−0.509647 + 0.860384i \(0.670224\pi\)
\(882\) 0 0
\(883\) 33.4773 1.12660 0.563300 0.826252i \(-0.309531\pi\)
0.563300 + 0.826252i \(0.309531\pi\)
\(884\) −17.1655 −0.577337
\(885\) 0 0
\(886\) 23.1760 0.778613
\(887\) 34.7429 1.16655 0.583277 0.812274i \(-0.301770\pi\)
0.583277 + 0.812274i \(0.301770\pi\)
\(888\) 0 0
\(889\) −12.7303 −0.426960
\(890\) 25.8156 0.865341
\(891\) 0 0
\(892\) 23.6735 0.792648
\(893\) −30.6458 −1.02552
\(894\) 0 0
\(895\) 41.0404 1.37183
\(896\) 10.9962 0.367358
\(897\) 0 0
\(898\) 27.1822 0.907081
\(899\) 0.469354 0.0156538
\(900\) 0 0
\(901\) −3.60807 −0.120202
\(902\) −6.52794 −0.217356
\(903\) 0 0
\(904\) 7.43230 0.247194
\(905\) −3.57672 −0.118894
\(906\) 0 0
\(907\) −18.9277 −0.628484 −0.314242 0.949343i \(-0.601750\pi\)
−0.314242 + 0.949343i \(0.601750\pi\)
\(908\) −7.38620 −0.245120
\(909\) 0 0
\(910\) 17.0375 0.564787
\(911\) 53.6024 1.77593 0.887964 0.459914i \(-0.152120\pi\)
0.887964 + 0.459914i \(0.152120\pi\)
\(912\) 0 0
\(913\) 2.15733 0.0713971
\(914\) 1.20181 0.0397524
\(915\) 0 0
\(916\) 7.49300 0.247576
\(917\) 3.77618 0.124701
\(918\) 0 0
\(919\) −38.1249 −1.25762 −0.628812 0.777558i \(-0.716458\pi\)
−0.628812 + 0.777558i \(0.716458\pi\)
\(920\) −37.0664 −1.22204
\(921\) 0 0
\(922\) −14.9571 −0.492584
\(923\) 84.4650 2.78020
\(924\) 0 0
\(925\) 17.1044 0.562388
\(926\) 26.4736 0.869976
\(927\) 0 0
\(928\) −0.563256 −0.0184898
\(929\) −23.1631 −0.759957 −0.379978 0.924995i \(-0.624069\pi\)
−0.379978 + 0.924995i \(0.624069\pi\)
\(930\) 0 0
\(931\) 15.4888 0.507626
\(932\) −1.63619 −0.0535952
\(933\) 0 0
\(934\) 2.28220 0.0746758
\(935\) −19.0788 −0.623944
\(936\) 0 0
\(937\) 3.99000 0.130348 0.0651738 0.997874i \(-0.479240\pi\)
0.0651738 + 0.997874i \(0.479240\pi\)
\(938\) 5.41394 0.176771
\(939\) 0 0
\(940\) 15.4186 0.502899
\(941\) −15.0860 −0.491790 −0.245895 0.969296i \(-0.579082\pi\)
−0.245895 + 0.969296i \(0.579082\pi\)
\(942\) 0 0
\(943\) 10.9999 0.358207
\(944\) −0.496466 −0.0161586
\(945\) 0 0
\(946\) −16.7270 −0.543841
\(947\) 55.4188 1.80087 0.900435 0.434991i \(-0.143248\pi\)
0.900435 + 0.434991i \(0.143248\pi\)
\(948\) 0 0
\(949\) −93.0636 −3.02097
\(950\) 7.83385 0.254164
\(951\) 0 0
\(952\) 11.3018 0.366293
\(953\) 46.8160 1.51652 0.758259 0.651953i \(-0.226050\pi\)
0.758259 + 0.651953i \(0.226050\pi\)
\(954\) 0 0
\(955\) 19.5727 0.633359
\(956\) −1.19052 −0.0385043
\(957\) 0 0
\(958\) −34.7845 −1.12384
\(959\) 11.4928 0.371122
\(960\) 0 0
\(961\) −9.53693 −0.307643
\(962\) −45.0579 −1.45273
\(963\) 0 0
\(964\) 19.8237 0.638477
\(965\) −3.60784 −0.116141
\(966\) 0 0
\(967\) −32.6031 −1.04844 −0.524222 0.851582i \(-0.675644\pi\)
−0.524222 + 0.851582i \(0.675644\pi\)
\(968\) −42.7530 −1.37413
\(969\) 0 0
\(970\) −9.13233 −0.293222
\(971\) 22.8194 0.732310 0.366155 0.930554i \(-0.380674\pi\)
0.366155 + 0.930554i \(0.380674\pi\)
\(972\) 0 0
\(973\) −17.1932 −0.551189
\(974\) 8.36299 0.267967
\(975\) 0 0
\(976\) 0.961819 0.0307871
\(977\) 13.1827 0.421751 0.210875 0.977513i \(-0.432369\pi\)
0.210875 + 0.977513i \(0.432369\pi\)
\(978\) 0 0
\(979\) −87.2315 −2.78793
\(980\) −7.79277 −0.248931
\(981\) 0 0
\(982\) −10.2094 −0.325796
\(983\) 26.5649 0.847290 0.423645 0.905828i \(-0.360750\pi\)
0.423645 + 0.905828i \(0.360750\pi\)
\(984\) 0 0
\(985\) 10.8240 0.344883
\(986\) −0.204177 −0.00650232
\(987\) 0 0
\(988\) 30.3511 0.965597
\(989\) 28.1859 0.896259
\(990\) 0 0
\(991\) 29.8250 0.947423 0.473711 0.880680i \(-0.342914\pi\)
0.473711 + 0.880680i \(0.342914\pi\)
\(992\) −25.7571 −0.817790
\(993\) 0 0
\(994\) −20.7513 −0.658191
\(995\) −36.6521 −1.16195
\(996\) 0 0
\(997\) 21.3442 0.675976 0.337988 0.941150i \(-0.390254\pi\)
0.337988 + 0.941150i \(0.390254\pi\)
\(998\) 22.3798 0.708419
\(999\) 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 2151.2.a.e.1.4 6
3.2 odd 2 717.2.a.d.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
717.2.a.d.1.3 6 3.2 odd 2
2151.2.a.e.1.4 6 1.1 even 1 trivial