Properties

Label 731.2.a.c.1.1
Level 731
Weight 2
Character 731.1
Self dual yes
Analytic conductor 5.837
Analytic rank 1
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 731 = 17 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 731.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.83706438776\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.2460365.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.08764\)
Character \(\chi\) = 731.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.35826 q^{2} +1.68213 q^{3} +3.56138 q^{4} +0.252087 q^{5} -3.96689 q^{6} -3.41152 q^{7} -3.68213 q^{8} -0.170443 q^{9} +O(q^{10})\) \(q-2.35826 q^{2} +1.68213 q^{3} +3.56138 q^{4} +0.252087 q^{5} -3.96689 q^{6} -3.41152 q^{7} -3.68213 q^{8} -0.170443 q^{9} -0.594485 q^{10} +0.120752 q^{11} +5.99069 q^{12} +3.34724 q^{13} +8.04523 q^{14} +0.424042 q^{15} +1.56065 q^{16} -1.00000 q^{17} +0.401948 q^{18} -8.40679 q^{19} +0.897775 q^{20} -5.73861 q^{21} -0.284764 q^{22} +0.438192 q^{23} -6.19381 q^{24} -4.93645 q^{25} -7.89366 q^{26} -5.33309 q^{27} -12.1497 q^{28} -4.34330 q^{29} -1.00000 q^{30} +7.74481 q^{31} +3.68384 q^{32} +0.203120 q^{33} +2.35826 q^{34} -0.859997 q^{35} -0.607011 q^{36} -8.36238 q^{37} +19.8254 q^{38} +5.63049 q^{39} -0.928215 q^{40} +5.90598 q^{41} +13.5331 q^{42} -1.00000 q^{43} +0.430042 q^{44} -0.0429663 q^{45} -1.03337 q^{46} +1.29175 q^{47} +2.62522 q^{48} +4.63844 q^{49} +11.6414 q^{50} -1.68213 q^{51} +11.9208 q^{52} -4.96574 q^{53} +12.5768 q^{54} +0.0304399 q^{55} +12.5616 q^{56} -14.1413 q^{57} +10.2426 q^{58} -4.88594 q^{59} +1.51017 q^{60} -12.9060 q^{61} -18.2643 q^{62} +0.581468 q^{63} -11.8087 q^{64} +0.843795 q^{65} -0.479009 q^{66} +6.34642 q^{67} -3.56138 q^{68} +0.737096 q^{69} +2.02809 q^{70} +11.5338 q^{71} +0.627593 q^{72} -10.1106 q^{73} +19.7206 q^{74} -8.30375 q^{75} -29.9398 q^{76} -0.411946 q^{77} -13.2782 q^{78} -11.3081 q^{79} +0.393419 q^{80} -8.45962 q^{81} -13.9278 q^{82} +13.3634 q^{83} -20.4373 q^{84} -0.252087 q^{85} +2.35826 q^{86} -7.30599 q^{87} -0.444623 q^{88} +8.52517 q^{89} +0.101326 q^{90} -11.4192 q^{91} +1.56057 q^{92} +13.0278 q^{93} -3.04627 q^{94} -2.11924 q^{95} +6.19669 q^{96} -6.27714 q^{97} -10.9386 q^{98} -0.0205813 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{2} - 3q^{3} + 5q^{4} + 3q^{5} - 7q^{6} - 7q^{7} - 9q^{8} - 3q^{9} + O(q^{10}) \) \( 6q - q^{2} - 3q^{3} + 5q^{4} + 3q^{5} - 7q^{6} - 7q^{7} - 9q^{8} - 3q^{9} - 4q^{10} + 4q^{11} + 11q^{12} - 10q^{13} - 7q^{14} + q^{15} - q^{16} - 6q^{17} + q^{18} - 20q^{19} + q^{20} - 8q^{21} + 2q^{22} - 3q^{23} - 9q^{24} - 7q^{25} - 3q^{26} - 6q^{27} - 11q^{28} - 15q^{29} - 6q^{30} + 12q^{31} + q^{32} - 2q^{33} + q^{34} - 9q^{35} - 16q^{36} - 14q^{37} + 27q^{38} + 5q^{39} - 7q^{40} - 2q^{41} + 19q^{42} - 6q^{43} - 12q^{44} - 18q^{45} + 14q^{46} - 11q^{47} - 6q^{48} + 3q^{49} + 7q^{50} + 3q^{51} - 5q^{52} + 3q^{53} + 25q^{54} + 6q^{55} + 22q^{56} + 11q^{57} - 21q^{58} + 2q^{59} - q^{60} - 20q^{61} + 3q^{62} + 23q^{63} - 39q^{64} - 34q^{65} + 7q^{66} - 2q^{67} - 5q^{68} - 17q^{69} - q^{70} + q^{71} + 21q^{72} + 13q^{73} + 28q^{74} - 5q^{75} - 29q^{76} - 11q^{77} - 26q^{79} + 12q^{80} + 2q^{81} - 9q^{82} + 10q^{83} - 3q^{84} - 3q^{85} + q^{86} + 12q^{87} - 6q^{88} - 15q^{89} + 15q^{90} + 8q^{91} - 9q^{92} - 11q^{93} - 33q^{94} - 21q^{95} + 25q^{96} - 22q^{97} - 3q^{98} - 5q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.35826 −1.66754 −0.833770 0.552112i \(-0.813822\pi\)
−0.833770 + 0.552112i \(0.813822\pi\)
\(3\) 1.68213 0.971177 0.485589 0.874187i \(-0.338605\pi\)
0.485589 + 0.874187i \(0.338605\pi\)
\(4\) 3.56138 1.78069
\(5\) 0.252087 0.112737 0.0563683 0.998410i \(-0.482048\pi\)
0.0563683 + 0.998410i \(0.482048\pi\)
\(6\) −3.96689 −1.61948
\(7\) −3.41152 −1.28943 −0.644716 0.764422i \(-0.723024\pi\)
−0.644716 + 0.764422i \(0.723024\pi\)
\(8\) −3.68213 −1.30183
\(9\) −0.170443 −0.0568143
\(10\) −0.594485 −0.187993
\(11\) 0.120752 0.0364080 0.0182040 0.999834i \(-0.494205\pi\)
0.0182040 + 0.999834i \(0.494205\pi\)
\(12\) 5.99069 1.72936
\(13\) 3.34724 0.928358 0.464179 0.885741i \(-0.346349\pi\)
0.464179 + 0.885741i \(0.346349\pi\)
\(14\) 8.04523 2.15018
\(15\) 0.424042 0.109487
\(16\) 1.56065 0.390163
\(17\) −1.00000 −0.242536
\(18\) 0.401948 0.0947401
\(19\) −8.40679 −1.92865 −0.964325 0.264720i \(-0.914720\pi\)
−0.964325 + 0.264720i \(0.914720\pi\)
\(20\) 0.897775 0.200749
\(21\) −5.73861 −1.25227
\(22\) −0.284764 −0.0607118
\(23\) 0.438192 0.0913694 0.0456847 0.998956i \(-0.485453\pi\)
0.0456847 + 0.998956i \(0.485453\pi\)
\(24\) −6.19381 −1.26431
\(25\) −4.93645 −0.987290
\(26\) −7.89366 −1.54807
\(27\) −5.33309 −1.02635
\(28\) −12.1497 −2.29608
\(29\) −4.34330 −0.806530 −0.403265 0.915083i \(-0.632125\pi\)
−0.403265 + 0.915083i \(0.632125\pi\)
\(30\) −1.00000 −0.182574
\(31\) 7.74481 1.39101 0.695505 0.718521i \(-0.255181\pi\)
0.695505 + 0.718521i \(0.255181\pi\)
\(32\) 3.68384 0.651217
\(33\) 0.203120 0.0353586
\(34\) 2.35826 0.404438
\(35\) −0.859997 −0.145366
\(36\) −0.607011 −0.101169
\(37\) −8.36238 −1.37477 −0.687383 0.726295i \(-0.741241\pi\)
−0.687383 + 0.726295i \(0.741241\pi\)
\(38\) 19.8254 3.21610
\(39\) 5.63049 0.901601
\(40\) −0.928215 −0.146764
\(41\) 5.90598 0.922359 0.461180 0.887307i \(-0.347426\pi\)
0.461180 + 0.887307i \(0.347426\pi\)
\(42\) 13.5331 2.08820
\(43\) −1.00000 −0.152499
\(44\) 0.430042 0.0648313
\(45\) −0.0429663 −0.00640504
\(46\) −1.03337 −0.152362
\(47\) 1.29175 0.188421 0.0942104 0.995552i \(-0.469967\pi\)
0.0942104 + 0.995552i \(0.469967\pi\)
\(48\) 2.62522 0.378918
\(49\) 4.63844 0.662634
\(50\) 11.6414 1.64635
\(51\) −1.68213 −0.235545
\(52\) 11.9208 1.65312
\(53\) −4.96574 −0.682096 −0.341048 0.940046i \(-0.610782\pi\)
−0.341048 + 0.940046i \(0.610782\pi\)
\(54\) 12.5768 1.71149
\(55\) 0.0304399 0.00410451
\(56\) 12.5616 1.67862
\(57\) −14.1413 −1.87306
\(58\) 10.2426 1.34492
\(59\) −4.88594 −0.636095 −0.318048 0.948075i \(-0.603027\pi\)
−0.318048 + 0.948075i \(0.603027\pi\)
\(60\) 1.51017 0.194963
\(61\) −12.9060 −1.65244 −0.826219 0.563349i \(-0.809513\pi\)
−0.826219 + 0.563349i \(0.809513\pi\)
\(62\) −18.2643 −2.31956
\(63\) 0.581468 0.0732581
\(64\) −11.8087 −1.47609
\(65\) 0.843795 0.104660
\(66\) −0.479009 −0.0589619
\(67\) 6.34642 0.775339 0.387670 0.921798i \(-0.373280\pi\)
0.387670 + 0.921798i \(0.373280\pi\)
\(68\) −3.56138 −0.431880
\(69\) 0.737096 0.0887359
\(70\) 2.02809 0.242404
\(71\) 11.5338 1.36881 0.684407 0.729100i \(-0.260061\pi\)
0.684407 + 0.729100i \(0.260061\pi\)
\(72\) 0.627593 0.0739625
\(73\) −10.1106 −1.18335 −0.591677 0.806175i \(-0.701534\pi\)
−0.591677 + 0.806175i \(0.701534\pi\)
\(74\) 19.7206 2.29248
\(75\) −8.30375 −0.958834
\(76\) −29.9398 −3.43433
\(77\) −0.411946 −0.0469456
\(78\) −13.2782 −1.50345
\(79\) −11.3081 −1.27226 −0.636132 0.771580i \(-0.719467\pi\)
−0.636132 + 0.771580i \(0.719467\pi\)
\(80\) 0.393419 0.0439856
\(81\) −8.45962 −0.939958
\(82\) −13.9278 −1.53807
\(83\) 13.3634 1.46682 0.733412 0.679785i \(-0.237927\pi\)
0.733412 + 0.679785i \(0.237927\pi\)
\(84\) −20.4373 −2.22990
\(85\) −0.252087 −0.0273426
\(86\) 2.35826 0.254297
\(87\) −7.30599 −0.783284
\(88\) −0.444623 −0.0473970
\(89\) 8.52517 0.903666 0.451833 0.892102i \(-0.350770\pi\)
0.451833 + 0.892102i \(0.350770\pi\)
\(90\) 0.101326 0.0106807
\(91\) −11.4192 −1.19705
\(92\) 1.56057 0.162700
\(93\) 13.0278 1.35092
\(94\) −3.04627 −0.314199
\(95\) −2.11924 −0.217429
\(96\) 6.19669 0.632447
\(97\) −6.27714 −0.637347 −0.318674 0.947865i \(-0.603237\pi\)
−0.318674 + 0.947865i \(0.603237\pi\)
\(98\) −10.9386 −1.10497
\(99\) −0.0205813 −0.00206849
\(100\) −17.5806 −1.75806
\(101\) −19.3506 −1.92546 −0.962730 0.270465i \(-0.912823\pi\)
−0.962730 + 0.270465i \(0.912823\pi\)
\(102\) 3.96689 0.392781
\(103\) 1.40525 0.138464 0.0692318 0.997601i \(-0.477945\pi\)
0.0692318 + 0.997601i \(0.477945\pi\)
\(104\) −12.3250 −1.20856
\(105\) −1.44663 −0.141176
\(106\) 11.7105 1.13742
\(107\) −16.2108 −1.56716 −0.783579 0.621292i \(-0.786608\pi\)
−0.783579 + 0.621292i \(0.786608\pi\)
\(108\) −18.9932 −1.82762
\(109\) −1.93419 −0.185262 −0.0926308 0.995701i \(-0.529528\pi\)
−0.0926308 + 0.995701i \(0.529528\pi\)
\(110\) −0.0717850 −0.00684443
\(111\) −14.0666 −1.33514
\(112\) −5.32419 −0.503089
\(113\) 18.6189 1.75152 0.875758 0.482751i \(-0.160362\pi\)
0.875758 + 0.482751i \(0.160362\pi\)
\(114\) 33.3488 3.12341
\(115\) 0.110462 0.0103007
\(116\) −15.4681 −1.43618
\(117\) −0.570514 −0.0527440
\(118\) 11.5223 1.06071
\(119\) 3.41152 0.312733
\(120\) −1.56138 −0.142534
\(121\) −10.9854 −0.998674
\(122\) 30.4356 2.75551
\(123\) 9.93462 0.895775
\(124\) 27.5822 2.47696
\(125\) −2.50485 −0.224040
\(126\) −1.37125 −0.122161
\(127\) 15.4119 1.36759 0.683794 0.729675i \(-0.260329\pi\)
0.683794 + 0.729675i \(0.260329\pi\)
\(128\) 20.4804 1.81023
\(129\) −1.68213 −0.148103
\(130\) −1.98989 −0.174524
\(131\) −9.67701 −0.845484 −0.422742 0.906250i \(-0.638932\pi\)
−0.422742 + 0.906250i \(0.638932\pi\)
\(132\) 0.723386 0.0629627
\(133\) 28.6799 2.48686
\(134\) −14.9665 −1.29291
\(135\) −1.34440 −0.115708
\(136\) 3.68213 0.315740
\(137\) 7.35811 0.628646 0.314323 0.949316i \(-0.398223\pi\)
0.314323 + 0.949316i \(0.398223\pi\)
\(138\) −1.73826 −0.147971
\(139\) 2.42368 0.205574 0.102787 0.994703i \(-0.467224\pi\)
0.102787 + 0.994703i \(0.467224\pi\)
\(140\) −3.06277 −0.258852
\(141\) 2.17289 0.182990
\(142\) −27.1998 −2.28255
\(143\) 0.404185 0.0337997
\(144\) −0.266002 −0.0221668
\(145\) −1.09489 −0.0909254
\(146\) 23.8433 1.97329
\(147\) 7.80245 0.643535
\(148\) −29.7816 −2.44803
\(149\) −23.0185 −1.88575 −0.942876 0.333145i \(-0.891890\pi\)
−0.942876 + 0.333145i \(0.891890\pi\)
\(150\) 19.5824 1.59889
\(151\) −5.02639 −0.409041 −0.204521 0.978862i \(-0.565564\pi\)
−0.204521 + 0.978862i \(0.565564\pi\)
\(152\) 30.9549 2.51077
\(153\) 0.170443 0.0137795
\(154\) 0.971475 0.0782837
\(155\) 1.95236 0.156818
\(156\) 20.0523 1.60547
\(157\) −5.37370 −0.428868 −0.214434 0.976739i \(-0.568791\pi\)
−0.214434 + 0.976739i \(0.568791\pi\)
\(158\) 26.6675 2.12155
\(159\) −8.35301 −0.662437
\(160\) 0.928646 0.0734159
\(161\) −1.49490 −0.117815
\(162\) 19.9500 1.56742
\(163\) 8.85550 0.693616 0.346808 0.937936i \(-0.387266\pi\)
0.346808 + 0.937936i \(0.387266\pi\)
\(164\) 21.0334 1.64243
\(165\) 0.0512038 0.00398621
\(166\) −31.5143 −2.44599
\(167\) 6.72205 0.520168 0.260084 0.965586i \(-0.416250\pi\)
0.260084 + 0.965586i \(0.416250\pi\)
\(168\) 21.1303 1.63024
\(169\) −1.79596 −0.138151
\(170\) 0.594485 0.0455949
\(171\) 1.43288 0.109575
\(172\) −3.56138 −0.271552
\(173\) 11.0675 0.841446 0.420723 0.907189i \(-0.361776\pi\)
0.420723 + 0.907189i \(0.361776\pi\)
\(174\) 17.2294 1.30616
\(175\) 16.8408 1.27304
\(176\) 0.188451 0.0142051
\(177\) −8.21878 −0.617762
\(178\) −20.1045 −1.50690
\(179\) 11.5790 0.865455 0.432728 0.901525i \(-0.357551\pi\)
0.432728 + 0.901525i \(0.357551\pi\)
\(180\) −0.153019 −0.0114054
\(181\) −7.43861 −0.552908 −0.276454 0.961027i \(-0.589159\pi\)
−0.276454 + 0.961027i \(0.589159\pi\)
\(182\) 26.9293 1.99614
\(183\) −21.7095 −1.60481
\(184\) −1.61348 −0.118947
\(185\) −2.10804 −0.154986
\(186\) −30.7228 −2.25271
\(187\) −0.120752 −0.00883024
\(188\) 4.60040 0.335519
\(189\) 18.1939 1.32341
\(190\) 4.99771 0.362572
\(191\) −5.57474 −0.403374 −0.201687 0.979450i \(-0.564642\pi\)
−0.201687 + 0.979450i \(0.564642\pi\)
\(192\) −19.8638 −1.43355
\(193\) 22.1403 1.59369 0.796846 0.604183i \(-0.206500\pi\)
0.796846 + 0.604183i \(0.206500\pi\)
\(194\) 14.8031 1.06280
\(195\) 1.41937 0.101643
\(196\) 16.5192 1.17994
\(197\) 9.02222 0.642806 0.321403 0.946942i \(-0.395846\pi\)
0.321403 + 0.946942i \(0.395846\pi\)
\(198\) 0.0485359 0.00344930
\(199\) −1.63539 −0.115930 −0.0579649 0.998319i \(-0.518461\pi\)
−0.0579649 + 0.998319i \(0.518461\pi\)
\(200\) 18.1767 1.28528
\(201\) 10.6755 0.752992
\(202\) 45.6338 3.21078
\(203\) 14.8172 1.03997
\(204\) −5.99069 −0.419433
\(205\) 1.48882 0.103984
\(206\) −3.31395 −0.230894
\(207\) −0.0746867 −0.00519109
\(208\) 5.22388 0.362211
\(209\) −1.01513 −0.0702183
\(210\) 3.41152 0.235417
\(211\) 14.4367 0.993864 0.496932 0.867789i \(-0.334460\pi\)
0.496932 + 0.867789i \(0.334460\pi\)
\(212\) −17.6849 −1.21460
\(213\) 19.4014 1.32936
\(214\) 38.2293 2.61330
\(215\) −0.252087 −0.0171922
\(216\) 19.6371 1.33614
\(217\) −26.4216 −1.79361
\(218\) 4.56131 0.308931
\(219\) −17.0073 −1.14925
\(220\) 0.108408 0.00730886
\(221\) −3.34724 −0.225160
\(222\) 33.1727 2.22640
\(223\) 12.9197 0.865165 0.432583 0.901594i \(-0.357602\pi\)
0.432583 + 0.901594i \(0.357602\pi\)
\(224\) −12.5675 −0.839699
\(225\) 0.841383 0.0560922
\(226\) −43.9081 −2.92072
\(227\) −17.1004 −1.13500 −0.567498 0.823375i \(-0.692088\pi\)
−0.567498 + 0.823375i \(0.692088\pi\)
\(228\) −50.3625 −3.33534
\(229\) 27.8932 1.84323 0.921617 0.388100i \(-0.126869\pi\)
0.921617 + 0.388100i \(0.126869\pi\)
\(230\) −0.260499 −0.0171768
\(231\) −0.692947 −0.0455925
\(232\) 15.9926 1.04996
\(233\) 11.5099 0.754036 0.377018 0.926206i \(-0.376949\pi\)
0.377018 + 0.926206i \(0.376949\pi\)
\(234\) 1.34542 0.0879527
\(235\) 0.325632 0.0212419
\(236\) −17.4007 −1.13269
\(237\) −19.0217 −1.23559
\(238\) −8.04523 −0.521495
\(239\) −20.1863 −1.30574 −0.652871 0.757469i \(-0.726436\pi\)
−0.652871 + 0.757469i \(0.726436\pi\)
\(240\) 0.661782 0.0427179
\(241\) −4.99186 −0.321554 −0.160777 0.986991i \(-0.551400\pi\)
−0.160777 + 0.986991i \(0.551400\pi\)
\(242\) 25.9064 1.66533
\(243\) 1.76911 0.113488
\(244\) −45.9630 −2.94248
\(245\) 1.16929 0.0747030
\(246\) −23.4284 −1.49374
\(247\) −28.1396 −1.79048
\(248\) −28.5174 −1.81086
\(249\) 22.4790 1.42455
\(250\) 5.90707 0.373596
\(251\) 15.8937 1.00320 0.501602 0.865099i \(-0.332744\pi\)
0.501602 + 0.865099i \(0.332744\pi\)
\(252\) 2.07083 0.130450
\(253\) 0.0529124 0.00332658
\(254\) −36.3453 −2.28051
\(255\) −0.424042 −0.0265545
\(256\) −24.6805 −1.54253
\(257\) 4.98203 0.310770 0.155385 0.987854i \(-0.450338\pi\)
0.155385 + 0.987854i \(0.450338\pi\)
\(258\) 3.96689 0.246968
\(259\) 28.5284 1.77267
\(260\) 3.00507 0.186367
\(261\) 0.740284 0.0458224
\(262\) 22.8209 1.40988
\(263\) 0.361153 0.0222696 0.0111348 0.999938i \(-0.496456\pi\)
0.0111348 + 0.999938i \(0.496456\pi\)
\(264\) −0.747913 −0.0460309
\(265\) −1.25180 −0.0768972
\(266\) −67.6346 −4.14694
\(267\) 14.3404 0.877620
\(268\) 22.6020 1.38064
\(269\) 14.4925 0.883621 0.441811 0.897108i \(-0.354336\pi\)
0.441811 + 0.897108i \(0.354336\pi\)
\(270\) 3.17044 0.192947
\(271\) 7.85002 0.476855 0.238427 0.971160i \(-0.423368\pi\)
0.238427 + 0.971160i \(0.423368\pi\)
\(272\) −1.56065 −0.0946285
\(273\) −19.2085 −1.16255
\(274\) −17.3523 −1.04829
\(275\) −0.596085 −0.0359453
\(276\) 2.62508 0.158011
\(277\) 1.83631 0.110333 0.0551667 0.998477i \(-0.482431\pi\)
0.0551667 + 0.998477i \(0.482431\pi\)
\(278\) −5.71567 −0.342803
\(279\) −1.32005 −0.0790292
\(280\) 3.16662 0.189242
\(281\) 13.0247 0.776987 0.388493 0.921451i \(-0.372996\pi\)
0.388493 + 0.921451i \(0.372996\pi\)
\(282\) −5.12422 −0.305143
\(283\) −9.08423 −0.540001 −0.270001 0.962860i \(-0.587024\pi\)
−0.270001 + 0.962860i \(0.587024\pi\)
\(284\) 41.0763 2.43743
\(285\) −3.56483 −0.211162
\(286\) −0.953173 −0.0563623
\(287\) −20.1483 −1.18932
\(288\) −0.627884 −0.0369984
\(289\) 1.00000 0.0588235
\(290\) 2.58202 0.151622
\(291\) −10.5590 −0.618977
\(292\) −36.0076 −2.10718
\(293\) −20.5237 −1.19901 −0.599503 0.800372i \(-0.704635\pi\)
−0.599503 + 0.800372i \(0.704635\pi\)
\(294\) −18.4002 −1.07312
\(295\) −1.23168 −0.0717112
\(296\) 30.7914 1.78971
\(297\) −0.643980 −0.0373675
\(298\) 54.2836 3.14457
\(299\) 1.46674 0.0848235
\(300\) −29.5728 −1.70739
\(301\) 3.41152 0.196636
\(302\) 11.8535 0.682093
\(303\) −32.5503 −1.86996
\(304\) −13.1201 −0.752488
\(305\) −3.25342 −0.186290
\(306\) −0.401948 −0.0229778
\(307\) 21.1819 1.20891 0.604456 0.796638i \(-0.293390\pi\)
0.604456 + 0.796638i \(0.293390\pi\)
\(308\) −1.46710 −0.0835955
\(309\) 2.36382 0.134473
\(310\) −4.60417 −0.261500
\(311\) −2.94814 −0.167173 −0.0835867 0.996501i \(-0.526638\pi\)
−0.0835867 + 0.996501i \(0.526638\pi\)
\(312\) −20.7322 −1.17373
\(313\) −23.0727 −1.30414 −0.652072 0.758157i \(-0.726100\pi\)
−0.652072 + 0.758157i \(0.726100\pi\)
\(314\) 12.6726 0.715154
\(315\) 0.146580 0.00825887
\(316\) −40.2725 −2.26550
\(317\) 15.3858 0.864155 0.432077 0.901837i \(-0.357781\pi\)
0.432077 + 0.901837i \(0.357781\pi\)
\(318\) 19.6985 1.10464
\(319\) −0.524460 −0.0293641
\(320\) −2.97682 −0.166410
\(321\) −27.2687 −1.52199
\(322\) 3.52536 0.196460
\(323\) 8.40679 0.467766
\(324\) −30.1279 −1.67377
\(325\) −16.5235 −0.916559
\(326\) −20.8835 −1.15663
\(327\) −3.25355 −0.179922
\(328\) −21.7466 −1.20075
\(329\) −4.40682 −0.242956
\(330\) −0.120752 −0.00664716
\(331\) −10.4726 −0.575626 −0.287813 0.957687i \(-0.592928\pi\)
−0.287813 + 0.957687i \(0.592928\pi\)
\(332\) 47.5921 2.61196
\(333\) 1.42531 0.0781064
\(334\) −15.8523 −0.867400
\(335\) 1.59985 0.0874090
\(336\) −8.95597 −0.488588
\(337\) −19.2658 −1.04947 −0.524737 0.851264i \(-0.675836\pi\)
−0.524737 + 0.851264i \(0.675836\pi\)
\(338\) 4.23534 0.230372
\(339\) 31.3193 1.70103
\(340\) −0.897775 −0.0486887
\(341\) 0.935199 0.0506439
\(342\) −3.37909 −0.182721
\(343\) 8.05651 0.435010
\(344\) 3.68213 0.198527
\(345\) 0.185812 0.0100038
\(346\) −26.1000 −1.40314
\(347\) 28.0162 1.50399 0.751995 0.659169i \(-0.229092\pi\)
0.751995 + 0.659169i \(0.229092\pi\)
\(348\) −26.0194 −1.39478
\(349\) −18.1695 −0.972589 −0.486294 0.873795i \(-0.661652\pi\)
−0.486294 + 0.873795i \(0.661652\pi\)
\(350\) −39.7149 −2.12285
\(351\) −17.8512 −0.952824
\(352\) 0.444830 0.0237095
\(353\) −10.2567 −0.545907 −0.272954 0.962027i \(-0.588001\pi\)
−0.272954 + 0.962027i \(0.588001\pi\)
\(354\) 19.3820 1.03014
\(355\) 2.90753 0.154315
\(356\) 30.3613 1.60915
\(357\) 5.73861 0.303719
\(358\) −27.3063 −1.44318
\(359\) −18.6942 −0.986641 −0.493320 0.869848i \(-0.664217\pi\)
−0.493320 + 0.869848i \(0.664217\pi\)
\(360\) 0.158208 0.00833827
\(361\) 51.6742 2.71969
\(362\) 17.5422 0.921996
\(363\) −18.4789 −0.969890
\(364\) −40.6680 −2.13158
\(365\) −2.54874 −0.133407
\(366\) 51.1965 2.67609
\(367\) 15.8306 0.826349 0.413174 0.910652i \(-0.364420\pi\)
0.413174 + 0.910652i \(0.364420\pi\)
\(368\) 0.683866 0.0356490
\(369\) −1.00663 −0.0524032
\(370\) 4.97131 0.258446
\(371\) 16.9407 0.879517
\(372\) 46.3968 2.40556
\(373\) 14.1309 0.731671 0.365836 0.930679i \(-0.380783\pi\)
0.365836 + 0.930679i \(0.380783\pi\)
\(374\) 0.284764 0.0147248
\(375\) −4.21347 −0.217583
\(376\) −4.75638 −0.245292
\(377\) −14.5381 −0.748749
\(378\) −42.9060 −2.20684
\(379\) 19.9764 1.02612 0.513060 0.858353i \(-0.328512\pi\)
0.513060 + 0.858353i \(0.328512\pi\)
\(380\) −7.54741 −0.387174
\(381\) 25.9249 1.32817
\(382\) 13.1467 0.672642
\(383\) 5.17491 0.264425 0.132213 0.991221i \(-0.457792\pi\)
0.132213 + 0.991221i \(0.457792\pi\)
\(384\) 34.4506 1.75805
\(385\) −0.103846 −0.00529249
\(386\) −52.2125 −2.65754
\(387\) 0.170443 0.00866410
\(388\) −22.3553 −1.13492
\(389\) −10.7768 −0.546408 −0.273204 0.961956i \(-0.588083\pi\)
−0.273204 + 0.961956i \(0.588083\pi\)
\(390\) −3.34724 −0.169494
\(391\) −0.438192 −0.0221603
\(392\) −17.0793 −0.862636
\(393\) −16.2780 −0.821115
\(394\) −21.2767 −1.07191
\(395\) −2.85063 −0.143431
\(396\) −0.0732976 −0.00368334
\(397\) −15.3644 −0.771117 −0.385558 0.922683i \(-0.625991\pi\)
−0.385558 + 0.922683i \(0.625991\pi\)
\(398\) 3.85667 0.193318
\(399\) 48.2433 2.41519
\(400\) −7.70409 −0.385204
\(401\) 9.16323 0.457590 0.228795 0.973475i \(-0.426521\pi\)
0.228795 + 0.973475i \(0.426521\pi\)
\(402\) −25.1756 −1.25564
\(403\) 25.9238 1.29136
\(404\) −68.9149 −3.42864
\(405\) −2.13256 −0.105968
\(406\) −34.9428 −1.73418
\(407\) −1.00977 −0.0500525
\(408\) 6.19381 0.306640
\(409\) −11.1488 −0.551271 −0.275635 0.961262i \(-0.588888\pi\)
−0.275635 + 0.961262i \(0.588888\pi\)
\(410\) −3.51102 −0.173397
\(411\) 12.3773 0.610527
\(412\) 5.00464 0.246561
\(413\) 16.6685 0.820202
\(414\) 0.176131 0.00865634
\(415\) 3.36873 0.165365
\(416\) 12.3307 0.604562
\(417\) 4.07695 0.199649
\(418\) 2.39395 0.117092
\(419\) −20.6208 −1.00739 −0.503696 0.863881i \(-0.668027\pi\)
−0.503696 + 0.863881i \(0.668027\pi\)
\(420\) −5.15198 −0.251391
\(421\) −27.0578 −1.31872 −0.659359 0.751829i \(-0.729172\pi\)
−0.659359 + 0.751829i \(0.729172\pi\)
\(422\) −34.0455 −1.65731
\(423\) −0.220169 −0.0107050
\(424\) 18.2845 0.887973
\(425\) 4.93645 0.239453
\(426\) −45.7535 −2.21676
\(427\) 44.0289 2.13071
\(428\) −57.7328 −2.79062
\(429\) 0.679892 0.0328255
\(430\) 0.594485 0.0286686
\(431\) 3.83029 0.184499 0.0922493 0.995736i \(-0.470594\pi\)
0.0922493 + 0.995736i \(0.470594\pi\)
\(432\) −8.32311 −0.400446
\(433\) −5.46875 −0.262811 −0.131406 0.991329i \(-0.541949\pi\)
−0.131406 + 0.991329i \(0.541949\pi\)
\(434\) 62.3088 2.99092
\(435\) −1.84174 −0.0883047
\(436\) −6.88837 −0.329893
\(437\) −3.68379 −0.176220
\(438\) 40.1076 1.91641
\(439\) 17.1856 0.820222 0.410111 0.912036i \(-0.365490\pi\)
0.410111 + 0.912036i \(0.365490\pi\)
\(440\) −0.112084 −0.00534337
\(441\) −0.790589 −0.0376471
\(442\) 7.89366 0.375463
\(443\) −2.32735 −0.110576 −0.0552880 0.998470i \(-0.517608\pi\)
−0.0552880 + 0.998470i \(0.517608\pi\)
\(444\) −50.0965 −2.37747
\(445\) 2.14908 0.101876
\(446\) −30.4679 −1.44270
\(447\) −38.7201 −1.83140
\(448\) 40.2857 1.90332
\(449\) 11.2166 0.529346 0.264673 0.964338i \(-0.414736\pi\)
0.264673 + 0.964338i \(0.414736\pi\)
\(450\) −1.98420 −0.0935360
\(451\) 0.713157 0.0335813
\(452\) 66.3088 3.11890
\(453\) −8.45503 −0.397252
\(454\) 40.3272 1.89265
\(455\) −2.87862 −0.134952
\(456\) 52.0701 2.43841
\(457\) 31.7186 1.48373 0.741867 0.670547i \(-0.233941\pi\)
0.741867 + 0.670547i \(0.233941\pi\)
\(458\) −65.7793 −3.07367
\(459\) 5.33309 0.248927
\(460\) 0.393398 0.0183423
\(461\) −25.9508 −1.20865 −0.604326 0.796738i \(-0.706557\pi\)
−0.604326 + 0.796738i \(0.706557\pi\)
\(462\) 1.63415 0.0760274
\(463\) −20.8000 −0.966660 −0.483330 0.875438i \(-0.660573\pi\)
−0.483330 + 0.875438i \(0.660573\pi\)
\(464\) −6.77838 −0.314678
\(465\) 3.28413 0.152298
\(466\) −27.1432 −1.25739
\(467\) 17.6282 0.815737 0.407868 0.913041i \(-0.366272\pi\)
0.407868 + 0.913041i \(0.366272\pi\)
\(468\) −2.03181 −0.0939207
\(469\) −21.6509 −0.999747
\(470\) −0.767925 −0.0354217
\(471\) −9.03925 −0.416507
\(472\) 17.9907 0.828088
\(473\) −0.120752 −0.00555217
\(474\) 44.8581 2.06040
\(475\) 41.4997 1.90414
\(476\) 12.1497 0.556880
\(477\) 0.846374 0.0387528
\(478\) 47.6044 2.17738
\(479\) −26.4045 −1.20645 −0.603227 0.797569i \(-0.706119\pi\)
−0.603227 + 0.797569i \(0.706119\pi\)
\(480\) 1.56210 0.0712999
\(481\) −27.9909 −1.27628
\(482\) 11.7721 0.536204
\(483\) −2.51461 −0.114419
\(484\) −39.1232 −1.77833
\(485\) −1.58238 −0.0718523
\(486\) −4.17201 −0.189246
\(487\) −40.2114 −1.82215 −0.911076 0.412239i \(-0.864747\pi\)
−0.911076 + 0.412239i \(0.864747\pi\)
\(488\) 47.5214 2.15119
\(489\) 14.8961 0.673624
\(490\) −2.75748 −0.124570
\(491\) −23.6521 −1.06740 −0.533702 0.845673i \(-0.679199\pi\)
−0.533702 + 0.845673i \(0.679199\pi\)
\(492\) 35.3809 1.59510
\(493\) 4.34330 0.195612
\(494\) 66.3604 2.98569
\(495\) −0.00518826 −0.000233195 0
\(496\) 12.0870 0.542721
\(497\) −39.3479 −1.76499
\(498\) −53.0112 −2.37549
\(499\) 5.80912 0.260052 0.130026 0.991511i \(-0.458494\pi\)
0.130026 + 0.991511i \(0.458494\pi\)
\(500\) −8.92070 −0.398946
\(501\) 11.3073 0.505175
\(502\) −37.4815 −1.67288
\(503\) −7.51699 −0.335166 −0.167583 0.985858i \(-0.553596\pi\)
−0.167583 + 0.985858i \(0.553596\pi\)
\(504\) −2.14104 −0.0953696
\(505\) −4.87803 −0.217070
\(506\) −0.124781 −0.00554720
\(507\) −3.02104 −0.134169
\(508\) 54.8877 2.43525
\(509\) −4.08998 −0.181285 −0.0906426 0.995883i \(-0.528892\pi\)
−0.0906426 + 0.995883i \(0.528892\pi\)
\(510\) 1.00000 0.0442807
\(511\) 34.4924 1.52585
\(512\) 17.2422 0.762006
\(513\) 44.8342 1.97948
\(514\) −11.7489 −0.518222
\(515\) 0.354245 0.0156099
\(516\) −5.99069 −0.263726
\(517\) 0.155981 0.00686002
\(518\) −67.2773 −2.95599
\(519\) 18.6170 0.817194
\(520\) −3.10696 −0.136249
\(521\) 43.5895 1.90969 0.954844 0.297106i \(-0.0960216\pi\)
0.954844 + 0.297106i \(0.0960216\pi\)
\(522\) −1.74578 −0.0764107
\(523\) 10.6045 0.463702 0.231851 0.972751i \(-0.425522\pi\)
0.231851 + 0.972751i \(0.425522\pi\)
\(524\) −34.4635 −1.50554
\(525\) 28.3284 1.23635
\(526\) −0.851691 −0.0371355
\(527\) −7.74481 −0.337369
\(528\) 0.317000 0.0137956
\(529\) −22.8080 −0.991652
\(530\) 2.95206 0.128229
\(531\) 0.832774 0.0361393
\(532\) 102.140 4.42833
\(533\) 19.7688 0.856280
\(534\) −33.8184 −1.46347
\(535\) −4.08653 −0.176676
\(536\) −23.3683 −1.00936
\(537\) 19.4774 0.840511
\(538\) −34.1770 −1.47347
\(539\) 0.560099 0.0241252
\(540\) −4.78792 −0.206039
\(541\) −36.4433 −1.56682 −0.783409 0.621506i \(-0.786521\pi\)
−0.783409 + 0.621506i \(0.786521\pi\)
\(542\) −18.5124 −0.795175
\(543\) −12.5127 −0.536972
\(544\) −3.68384 −0.157943
\(545\) −0.487583 −0.0208857
\(546\) 45.2986 1.93860
\(547\) −4.31868 −0.184653 −0.0923267 0.995729i \(-0.529430\pi\)
−0.0923267 + 0.995729i \(0.529430\pi\)
\(548\) 26.2050 1.11942
\(549\) 2.19973 0.0938821
\(550\) 1.40572 0.0599402
\(551\) 36.5132 1.55551
\(552\) −2.71408 −0.115519
\(553\) 38.5778 1.64050
\(554\) −4.33050 −0.183985
\(555\) −3.54600 −0.150519
\(556\) 8.63165 0.366064
\(557\) 10.0168 0.424424 0.212212 0.977224i \(-0.431933\pi\)
0.212212 + 0.977224i \(0.431933\pi\)
\(558\) 3.11301 0.131784
\(559\) −3.34724 −0.141573
\(560\) −1.34216 −0.0567165
\(561\) −0.203120 −0.00857573
\(562\) −30.7155 −1.29566
\(563\) −33.3798 −1.40679 −0.703395 0.710799i \(-0.748334\pi\)
−0.703395 + 0.710799i \(0.748334\pi\)
\(564\) 7.73847 0.325848
\(565\) 4.69357 0.197460
\(566\) 21.4229 0.900474
\(567\) 28.8601 1.21201
\(568\) −42.4691 −1.78196
\(569\) 40.5491 1.69991 0.849953 0.526859i \(-0.176630\pi\)
0.849953 + 0.526859i \(0.176630\pi\)
\(570\) 8.40679 0.352122
\(571\) −17.2176 −0.720534 −0.360267 0.932849i \(-0.617314\pi\)
−0.360267 + 0.932849i \(0.617314\pi\)
\(572\) 1.43946 0.0601867
\(573\) −9.37743 −0.391748
\(574\) 47.5150 1.98324
\(575\) −2.16311 −0.0902081
\(576\) 2.01272 0.0838632
\(577\) 3.69189 0.153695 0.0768477 0.997043i \(-0.475514\pi\)
0.0768477 + 0.997043i \(0.475514\pi\)
\(578\) −2.35826 −0.0980906
\(579\) 37.2428 1.54776
\(580\) −3.89930 −0.161910
\(581\) −45.5894 −1.89137
\(582\) 24.9007 1.03217
\(583\) −0.599621 −0.0248338
\(584\) 37.2284 1.54052
\(585\) −0.143819 −0.00594618
\(586\) 48.4001 1.99939
\(587\) 0.209448 0.00864484 0.00432242 0.999991i \(-0.498624\pi\)
0.00432242 + 0.999991i \(0.498624\pi\)
\(588\) 27.7875 1.14594
\(589\) −65.1091 −2.68277
\(590\) 2.90462 0.119581
\(591\) 15.1765 0.624279
\(592\) −13.0508 −0.536383
\(593\) −45.8332 −1.88215 −0.941073 0.338204i \(-0.890181\pi\)
−0.941073 + 0.338204i \(0.890181\pi\)
\(594\) 1.51867 0.0623118
\(595\) 0.859997 0.0352564
\(596\) −81.9777 −3.35794
\(597\) −2.75094 −0.112588
\(598\) −3.45894 −0.141447
\(599\) −23.5892 −0.963830 −0.481915 0.876218i \(-0.660059\pi\)
−0.481915 + 0.876218i \(0.660059\pi\)
\(600\) 30.5755 1.24824
\(601\) 19.5597 0.797857 0.398928 0.916982i \(-0.369382\pi\)
0.398928 + 0.916982i \(0.369382\pi\)
\(602\) −8.04523 −0.327899
\(603\) −1.08170 −0.0440503
\(604\) −17.9009 −0.728375
\(605\) −2.76928 −0.112587
\(606\) 76.7619 3.11824
\(607\) 35.6132 1.44550 0.722748 0.691111i \(-0.242879\pi\)
0.722748 + 0.691111i \(0.242879\pi\)
\(608\) −30.9693 −1.25597
\(609\) 24.9245 1.00999
\(610\) 7.67239 0.310646
\(611\) 4.32379 0.174922
\(612\) 0.607011 0.0245370
\(613\) 2.61160 0.105482 0.0527408 0.998608i \(-0.483204\pi\)
0.0527408 + 0.998608i \(0.483204\pi\)
\(614\) −49.9523 −2.01591
\(615\) 2.50438 0.100986
\(616\) 1.51684 0.0611152
\(617\) −25.7789 −1.03782 −0.518911 0.854829i \(-0.673662\pi\)
−0.518911 + 0.854829i \(0.673662\pi\)
\(618\) −5.57449 −0.224239
\(619\) 17.8472 0.717338 0.358669 0.933465i \(-0.383231\pi\)
0.358669 + 0.933465i \(0.383231\pi\)
\(620\) 6.95310 0.279243
\(621\) −2.33692 −0.0937774
\(622\) 6.95246 0.278768
\(623\) −29.0838 −1.16522
\(624\) 8.78725 0.351771
\(625\) 24.0508 0.962033
\(626\) 54.4113 2.17471
\(627\) −1.70759 −0.0681944
\(628\) −19.1378 −0.763680
\(629\) 8.36238 0.333430
\(630\) −0.345674 −0.0137720
\(631\) 3.33820 0.132892 0.0664458 0.997790i \(-0.478834\pi\)
0.0664458 + 0.997790i \(0.478834\pi\)
\(632\) 41.6380 1.65627
\(633\) 24.2844 0.965219
\(634\) −36.2838 −1.44101
\(635\) 3.88514 0.154177
\(636\) −29.7482 −1.17959
\(637\) 15.5260 0.615162
\(638\) 1.23681 0.0489659
\(639\) −1.96586 −0.0777682
\(640\) 5.16283 0.204079
\(641\) −13.9216 −0.549869 −0.274935 0.961463i \(-0.588656\pi\)
−0.274935 + 0.961463i \(0.588656\pi\)
\(642\) 64.3066 2.53798
\(643\) −5.61656 −0.221496 −0.110748 0.993849i \(-0.535325\pi\)
−0.110748 + 0.993849i \(0.535325\pi\)
\(644\) −5.32390 −0.209791
\(645\) −0.424042 −0.0166966
\(646\) −19.8254 −0.780019
\(647\) −12.8755 −0.506190 −0.253095 0.967441i \(-0.581448\pi\)
−0.253095 + 0.967441i \(0.581448\pi\)
\(648\) 31.1494 1.22366
\(649\) −0.589986 −0.0231590
\(650\) 38.9667 1.52840
\(651\) −44.4445 −1.74192
\(652\) 31.5378 1.23511
\(653\) −49.8385 −1.95033 −0.975166 0.221477i \(-0.928912\pi\)
−0.975166 + 0.221477i \(0.928912\pi\)
\(654\) 7.67271 0.300027
\(655\) −2.43944 −0.0953169
\(656\) 9.21718 0.359871
\(657\) 1.72328 0.0672314
\(658\) 10.3924 0.405138
\(659\) 27.6176 1.07583 0.537914 0.843000i \(-0.319212\pi\)
0.537914 + 0.843000i \(0.319212\pi\)
\(660\) 0.182356 0.00709820
\(661\) 2.49646 0.0971010 0.0485505 0.998821i \(-0.484540\pi\)
0.0485505 + 0.998821i \(0.484540\pi\)
\(662\) 24.6971 0.959878
\(663\) −5.63049 −0.218670
\(664\) −49.2057 −1.90955
\(665\) 7.22982 0.280360
\(666\) −3.36124 −0.130245
\(667\) −1.90320 −0.0736922
\(668\) 23.9397 0.926256
\(669\) 21.7325 0.840229
\(670\) −3.77285 −0.145758
\(671\) −1.55842 −0.0601620
\(672\) −21.1401 −0.815497
\(673\) −46.0809 −1.77629 −0.888143 0.459566i \(-0.848005\pi\)
−0.888143 + 0.459566i \(0.848005\pi\)
\(674\) 45.4337 1.75004
\(675\) 26.3266 1.01331
\(676\) −6.39610 −0.246004
\(677\) −13.9244 −0.535158 −0.267579 0.963536i \(-0.586224\pi\)
−0.267579 + 0.963536i \(0.586224\pi\)
\(678\) −73.8590 −2.83654
\(679\) 21.4146 0.821816
\(680\) 0.928215 0.0355954
\(681\) −28.7651 −1.10228
\(682\) −2.20544 −0.0844507
\(683\) −13.3748 −0.511771 −0.255885 0.966707i \(-0.582367\pi\)
−0.255885 + 0.966707i \(0.582367\pi\)
\(684\) 5.10302 0.195119
\(685\) 1.85488 0.0708713
\(686\) −18.9993 −0.725397
\(687\) 46.9199 1.79011
\(688\) −1.56065 −0.0594993
\(689\) −16.6215 −0.633230
\(690\) −0.438192 −0.0166817
\(691\) −13.5757 −0.516442 −0.258221 0.966086i \(-0.583136\pi\)
−0.258221 + 0.966086i \(0.583136\pi\)
\(692\) 39.4155 1.49835
\(693\) 0.0702133 0.00266718
\(694\) −66.0694 −2.50796
\(695\) 0.610978 0.0231757
\(696\) 26.9016 1.01970
\(697\) −5.90598 −0.223705
\(698\) 42.8483 1.62183
\(699\) 19.3611 0.732303
\(700\) 59.9764 2.26689
\(701\) 2.05978 0.0777969 0.0388984 0.999243i \(-0.487615\pi\)
0.0388984 + 0.999243i \(0.487615\pi\)
\(702\) 42.0976 1.58887
\(703\) 70.3008 2.65144
\(704\) −1.42593 −0.0537416
\(705\) 0.547755 0.0206297
\(706\) 24.1879 0.910322
\(707\) 66.0150 2.48275
\(708\) −29.2702 −1.10004
\(709\) −19.3260 −0.725802 −0.362901 0.931828i \(-0.618214\pi\)
−0.362901 + 0.931828i \(0.618214\pi\)
\(710\) −6.85669 −0.257327
\(711\) 1.92739 0.0722827
\(712\) −31.3908 −1.17642
\(713\) 3.39372 0.127096
\(714\) −13.5331 −0.506464
\(715\) 0.101890 0.00381046
\(716\) 41.2372 1.54111
\(717\) −33.9559 −1.26811
\(718\) 44.0857 1.64526
\(719\) −30.9365 −1.15374 −0.576869 0.816837i \(-0.695726\pi\)
−0.576869 + 0.816837i \(0.695726\pi\)
\(720\) −0.0670555 −0.00249901
\(721\) −4.79404 −0.178539
\(722\) −121.861 −4.53520
\(723\) −8.39695 −0.312286
\(724\) −26.4917 −0.984556
\(725\) 21.4405 0.796279
\(726\) 43.5780 1.61733
\(727\) −30.6157 −1.13547 −0.567737 0.823210i \(-0.692181\pi\)
−0.567737 + 0.823210i \(0.692181\pi\)
\(728\) 42.0469 1.55836
\(729\) 28.3547 1.05018
\(730\) 6.01058 0.222462
\(731\) 1.00000 0.0369863
\(732\) −77.3156 −2.85767
\(733\) 9.54484 0.352547 0.176273 0.984341i \(-0.443596\pi\)
0.176273 + 0.984341i \(0.443596\pi\)
\(734\) −37.3325 −1.37797
\(735\) 1.96689 0.0725499
\(736\) 1.61423 0.0595013
\(737\) 0.766341 0.0282285
\(738\) 2.37390 0.0873844
\(739\) 32.6005 1.19923 0.599614 0.800289i \(-0.295321\pi\)
0.599614 + 0.800289i \(0.295321\pi\)
\(740\) −7.50753 −0.275982
\(741\) −47.3344 −1.73887
\(742\) −39.9505 −1.46663
\(743\) 14.1350 0.518562 0.259281 0.965802i \(-0.416514\pi\)
0.259281 + 0.965802i \(0.416514\pi\)
\(744\) −47.9699 −1.75866
\(745\) −5.80266 −0.212593
\(746\) −33.3243 −1.22009
\(747\) −2.27770 −0.0833365
\(748\) −0.430042 −0.0157239
\(749\) 55.3034 2.02074
\(750\) 9.93645 0.362828
\(751\) −30.7397 −1.12171 −0.560854 0.827915i \(-0.689527\pi\)
−0.560854 + 0.827915i \(0.689527\pi\)
\(752\) 2.01597 0.0735148
\(753\) 26.7353 0.974288
\(754\) 34.2845 1.24857
\(755\) −1.26708 −0.0461139
\(756\) 64.7954 2.35659
\(757\) −52.8011 −1.91909 −0.959544 0.281557i \(-0.909149\pi\)
−0.959544 + 0.281557i \(0.909149\pi\)
\(758\) −47.1095 −1.71109
\(759\) 0.0890055 0.00323070
\(760\) 7.80331 0.283056
\(761\) −18.1300 −0.657211 −0.328606 0.944467i \(-0.606579\pi\)
−0.328606 + 0.944467i \(0.606579\pi\)
\(762\) −61.1375 −2.21478
\(763\) 6.59851 0.238882
\(764\) −19.8537 −0.718283
\(765\) 0.0429663 0.00155345
\(766\) −12.2038 −0.440940
\(767\) −16.3544 −0.590524
\(768\) −41.5158 −1.49807
\(769\) 14.9904 0.540567 0.270283 0.962781i \(-0.412883\pi\)
0.270283 + 0.962781i \(0.412883\pi\)
\(770\) 0.244896 0.00882543
\(771\) 8.38041 0.301813
\(772\) 78.8499 2.83787
\(773\) −54.9568 −1.97666 −0.988329 0.152332i \(-0.951322\pi\)
−0.988329 + 0.152332i \(0.951322\pi\)
\(774\) −0.401948 −0.0144477
\(775\) −38.2319 −1.37333
\(776\) 23.1132 0.829717
\(777\) 47.9884 1.72157
\(778\) 25.4146 0.911157
\(779\) −49.6504 −1.77891
\(780\) 5.05492 0.180995
\(781\) 1.39273 0.0498358
\(782\) 1.03337 0.0369532
\(783\) 23.1632 0.827786
\(784\) 7.23899 0.258535
\(785\) −1.35464 −0.0483490
\(786\) 38.3876 1.36924
\(787\) −2.55479 −0.0910684 −0.0455342 0.998963i \(-0.514499\pi\)
−0.0455342 + 0.998963i \(0.514499\pi\)
\(788\) 32.1315 1.14464
\(789\) 0.607506 0.0216278
\(790\) 6.72251 0.239176
\(791\) −63.5186 −2.25846
\(792\) 0.0757829 0.00269283
\(793\) −43.1994 −1.53405
\(794\) 36.2332 1.28587
\(795\) −2.10568 −0.0746808
\(796\) −5.82424 −0.206435
\(797\) 9.28439 0.328870 0.164435 0.986388i \(-0.447420\pi\)
0.164435 + 0.986388i \(0.447420\pi\)
\(798\) −113.770 −4.02742
\(799\) −1.29175 −0.0456988
\(800\) −18.1851 −0.642940
\(801\) −1.45305 −0.0513412
\(802\) −21.6093 −0.763050
\(803\) −1.22087 −0.0430835
\(804\) 38.0195 1.34084
\(805\) −0.376844 −0.0132820
\(806\) −61.1349 −2.15339
\(807\) 24.3782 0.858153
\(808\) 71.2515 2.50662
\(809\) −49.4945 −1.74013 −0.870067 0.492934i \(-0.835924\pi\)
−0.870067 + 0.492934i \(0.835924\pi\)
\(810\) 5.02912 0.176705
\(811\) −7.79500 −0.273720 −0.136860 0.990590i \(-0.543701\pi\)
−0.136860 + 0.990590i \(0.543701\pi\)
\(812\) 52.7697 1.85185
\(813\) 13.2047 0.463111
\(814\) 2.38130 0.0834645
\(815\) 2.23235 0.0781959
\(816\) −2.62522 −0.0919010
\(817\) 8.40679 0.294116
\(818\) 26.2916 0.919266
\(819\) 1.94632 0.0680098
\(820\) 5.30224 0.185162
\(821\) 13.7637 0.480357 0.240179 0.970729i \(-0.422794\pi\)
0.240179 + 0.970729i \(0.422794\pi\)
\(822\) −29.1888 −1.01808
\(823\) 13.8886 0.484126 0.242063 0.970260i \(-0.422176\pi\)
0.242063 + 0.970260i \(0.422176\pi\)
\(824\) −5.17432 −0.180256
\(825\) −1.00269 −0.0349092
\(826\) −39.3085 −1.36772
\(827\) 39.7097 1.38084 0.690421 0.723408i \(-0.257425\pi\)
0.690421 + 0.723408i \(0.257425\pi\)
\(828\) −0.265988 −0.00924371
\(829\) −55.3677 −1.92300 −0.961500 0.274804i \(-0.911387\pi\)
−0.961500 + 0.274804i \(0.911387\pi\)
\(830\) −7.94434 −0.275752
\(831\) 3.08892 0.107153
\(832\) −39.5267 −1.37034
\(833\) −4.63844 −0.160712
\(834\) −9.61450 −0.332923
\(835\) 1.69454 0.0586419
\(836\) −3.61528 −0.125037
\(837\) −41.3038 −1.42767
\(838\) 48.6292 1.67987
\(839\) −10.8918 −0.376028 −0.188014 0.982166i \(-0.560205\pi\)
−0.188014 + 0.982166i \(0.560205\pi\)
\(840\) 5.32666 0.183787
\(841\) −10.1358 −0.349509
\(842\) 63.8093 2.19901
\(843\) 21.9092 0.754592
\(844\) 51.4146 1.76976
\(845\) −0.452738 −0.0155747
\(846\) 0.519216 0.0178510
\(847\) 37.4769 1.28772
\(848\) −7.74979 −0.266129
\(849\) −15.2808 −0.524437
\(850\) −11.6414 −0.399298
\(851\) −3.66433 −0.125612
\(852\) 69.0957 2.36718
\(853\) −32.6496 −1.11790 −0.558950 0.829201i \(-0.688796\pi\)
−0.558950 + 0.829201i \(0.688796\pi\)
\(854\) −103.831 −3.55304
\(855\) 0.361209 0.0123531
\(856\) 59.6903 2.04017
\(857\) 32.8424 1.12188 0.560938 0.827858i \(-0.310441\pi\)
0.560938 + 0.827858i \(0.310441\pi\)
\(858\) −1.60336 −0.0547378
\(859\) −38.5322 −1.31470 −0.657351 0.753585i \(-0.728323\pi\)
−0.657351 + 0.753585i \(0.728323\pi\)
\(860\) −0.897775 −0.0306139
\(861\) −33.8921 −1.15504
\(862\) −9.03281 −0.307659
\(863\) −3.59644 −0.122424 −0.0612121 0.998125i \(-0.519497\pi\)
−0.0612121 + 0.998125i \(0.519497\pi\)
\(864\) −19.6462 −0.668379
\(865\) 2.78997 0.0948617
\(866\) 12.8967 0.438248
\(867\) 1.68213 0.0571281
\(868\) −94.0971 −3.19386
\(869\) −1.36548 −0.0463206
\(870\) 4.34330 0.147252
\(871\) 21.2430 0.719792
\(872\) 7.12193 0.241179
\(873\) 1.06989 0.0362104
\(874\) 8.68733 0.293853
\(875\) 8.54532 0.288885
\(876\) −60.5694 −2.04645
\(877\) −20.5421 −0.693658 −0.346829 0.937928i \(-0.612742\pi\)
−0.346829 + 0.937928i \(0.612742\pi\)
\(878\) −40.5280 −1.36775
\(879\) −34.5235 −1.16445
\(880\) 0.0475061 0.00160143
\(881\) −4.49143 −0.151320 −0.0756601 0.997134i \(-0.524106\pi\)
−0.0756601 + 0.997134i \(0.524106\pi\)
\(882\) 1.86441 0.0627780
\(883\) 15.9912 0.538145 0.269073 0.963120i \(-0.413283\pi\)
0.269073 + 0.963120i \(0.413283\pi\)
\(884\) −11.9208 −0.400940
\(885\) −2.07184 −0.0696443
\(886\) 5.48850 0.184390
\(887\) 53.9426 1.81122 0.905608 0.424115i \(-0.139415\pi\)
0.905608 + 0.424115i \(0.139415\pi\)
\(888\) 51.7950 1.73813
\(889\) −52.5780 −1.76341
\(890\) −5.06808 −0.169883
\(891\) −1.02151 −0.0342220
\(892\) 46.0118 1.54059
\(893\) −10.8595 −0.363398
\(894\) 91.3120 3.05393
\(895\) 2.91891 0.0975684
\(896\) −69.8691 −2.33416
\(897\) 2.46724 0.0823787
\(898\) −26.4517 −0.882705
\(899\) −33.6380 −1.12189
\(900\) 2.99648 0.0998827
\(901\) 4.96574 0.165433
\(902\) −1.68181 −0.0559981
\(903\) 5.73861 0.190969
\(904\) −68.5571 −2.28017
\(905\) −1.87517 −0.0623329
\(906\) 19.9391 0.662433
\(907\) 18.8282 0.625179 0.312589 0.949888i \(-0.398804\pi\)
0.312589 + 0.949888i \(0.398804\pi\)
\(908\) −60.9011 −2.02107
\(909\) 3.29818 0.109394
\(910\) 6.78852 0.225037
\(911\) 34.2713 1.13546 0.567730 0.823215i \(-0.307822\pi\)
0.567730 + 0.823215i \(0.307822\pi\)
\(912\) −22.0697 −0.730800
\(913\) 1.61365 0.0534041
\(914\) −74.8006 −2.47418
\(915\) −5.47267 −0.180921
\(916\) 99.3382 3.28223
\(917\) 33.0133 1.09019
\(918\) −12.5768 −0.415096
\(919\) −13.6058 −0.448815 −0.224407 0.974495i \(-0.572045\pi\)
−0.224407 + 0.974495i \(0.572045\pi\)
\(920\) −0.406737 −0.0134097
\(921\) 35.6306 1.17407
\(922\) 61.1988 2.01547
\(923\) 38.6066 1.27075
\(924\) −2.46784 −0.0811861
\(925\) 41.2805 1.35729
\(926\) 49.0519 1.61194
\(927\) −0.239515 −0.00786672
\(928\) −16.0000 −0.525226
\(929\) 28.8100 0.945224 0.472612 0.881271i \(-0.343311\pi\)
0.472612 + 0.881271i \(0.343311\pi\)
\(930\) −7.74481 −0.253962
\(931\) −38.9944 −1.27799
\(932\) 40.9910 1.34270
\(933\) −4.95914 −0.162355
\(934\) −41.5719 −1.36027
\(935\) −0.0304399 −0.000995490 0
\(936\) 2.10070 0.0686637
\(937\) −44.8103 −1.46389 −0.731945 0.681364i \(-0.761387\pi\)
−0.731945 + 0.681364i \(0.761387\pi\)
\(938\) 51.0584 1.66712
\(939\) −38.8112 −1.26656
\(940\) 1.15970 0.0378252
\(941\) −15.2691 −0.497760 −0.248880 0.968534i \(-0.580062\pi\)
−0.248880 + 0.968534i \(0.580062\pi\)
\(942\) 21.3169 0.694541
\(943\) 2.58795 0.0842754
\(944\) −7.62526 −0.248181
\(945\) 4.58644 0.149197
\(946\) 0.284764 0.00925846
\(947\) −1.13590 −0.0369118 −0.0184559 0.999830i \(-0.505875\pi\)
−0.0184559 + 0.999830i \(0.505875\pi\)
\(948\) −67.7435 −2.20021
\(949\) −33.8426 −1.09858
\(950\) −97.8670 −3.17523
\(951\) 25.8810 0.839248
\(952\) −12.5616 −0.407125
\(953\) 16.6629 0.539764 0.269882 0.962893i \(-0.413015\pi\)
0.269882 + 0.962893i \(0.413015\pi\)
\(954\) −1.99597 −0.0646219
\(955\) −1.40532 −0.0454750
\(956\) −71.8910 −2.32512
\(957\) −0.882210 −0.0285178
\(958\) 62.2687 2.01181
\(959\) −25.1023 −0.810596
\(960\) −5.00740 −0.161613
\(961\) 28.9822 0.934908
\(962\) 66.0098 2.12824
\(963\) 2.76302 0.0890370
\(964\) −17.7779 −0.572587
\(965\) 5.58126 0.179667
\(966\) 5.93011 0.190798
\(967\) −26.7181 −0.859197 −0.429599 0.903020i \(-0.641345\pi\)
−0.429599 + 0.903020i \(0.641345\pi\)
\(968\) 40.4497 1.30010
\(969\) 14.1413 0.454284
\(970\) 3.73167 0.119817
\(971\) 44.7350 1.43561 0.717807 0.696242i \(-0.245146\pi\)
0.717807 + 0.696242i \(0.245146\pi\)
\(972\) 6.30046 0.202087
\(973\) −8.26844 −0.265074
\(974\) 94.8288 3.03851
\(975\) −27.7947 −0.890142
\(976\) −20.1417 −0.644720
\(977\) −16.1109 −0.515435 −0.257717 0.966220i \(-0.582970\pi\)
−0.257717 + 0.966220i \(0.582970\pi\)
\(978\) −35.1288 −1.12330
\(979\) 1.02943 0.0329007
\(980\) 4.16427 0.133023
\(981\) 0.329668 0.0105255
\(982\) 55.7777 1.77994
\(983\) −47.2877 −1.50824 −0.754122 0.656735i \(-0.771937\pi\)
−0.754122 + 0.656735i \(0.771937\pi\)
\(984\) −36.5806 −1.16615
\(985\) 2.27438 0.0724677
\(986\) −10.2426 −0.326191
\(987\) −7.41284 −0.235953
\(988\) −100.216 −3.18828
\(989\) −0.438192 −0.0139337
\(990\) 0.0122352 0.000388862 0
\(991\) 29.9314 0.950803 0.475402 0.879769i \(-0.342303\pi\)
0.475402 + 0.879769i \(0.342303\pi\)
\(992\) 28.5306 0.905849
\(993\) −17.6162 −0.559035
\(994\) 92.7924 2.94320
\(995\) −0.412260 −0.0130695
\(996\) 80.0560 2.53667
\(997\) 51.3858 1.62740 0.813702 0.581282i \(-0.197449\pi\)
0.813702 + 0.581282i \(0.197449\pi\)
\(998\) −13.6994 −0.433647
\(999\) 44.5973 1.41100
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 731.2.a.c.1.1 6
3.2 odd 2 6579.2.a.j.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
731.2.a.c.1.1 6 1.1 even 1 trivial
6579.2.a.j.1.6 6 3.2 odd 2