Properties

Label 845.2.c.g.506.1
Level $845$
Weight $2$
Character 845.506
Analytic conductor $6.747$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.22581504.2
Defining polynomial: \(x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 506.1
Root \(0.665665 + 1.24775i\) of defining polynomial
Character \(\chi\) \(=\) 845.506
Dual form 845.2.c.g.506.8

$q$-expansion

\(f(q)\) \(=\) \(q-2.49551i q^{2} -2.82684 q^{3} -4.22756 q^{4} +1.00000i q^{5} +7.05440i q^{6} +1.90521i q^{7} +5.55889i q^{8} +4.99102 q^{9} +O(q^{10})\) \(q-2.49551i q^{2} -2.82684 q^{3} -4.22756 q^{4} +1.00000i q^{5} +7.05440i q^{6} +1.90521i q^{7} +5.55889i q^{8} +4.99102 q^{9} +2.49551 q^{10} +1.06939i q^{11} +11.9506 q^{12} +4.75447 q^{14} -2.82684i q^{15} +5.41713 q^{16} -0.637263 q^{17} -12.4551i q^{18} -5.73205i q^{19} -4.22756i q^{20} -5.38573i q^{21} +2.66867 q^{22} -3.81785 q^{23} -15.7141i q^{24} -1.00000 q^{25} -5.62828 q^{27} -8.05440i q^{28} +9.45512 q^{29} -7.05440 q^{30} +1.46410i q^{31} -2.40072i q^{32} -3.02299i q^{33} +1.59030i q^{34} -1.90521 q^{35} -21.0998 q^{36} +0.757449i q^{37} -14.3044 q^{38} -5.55889 q^{40} -0.267949i q^{41} -13.4401 q^{42} -0.637263 q^{43} -4.52091i q^{44} +4.99102i q^{45} +9.52748i q^{46} -9.44613i q^{47} -15.3134 q^{48} +3.37017 q^{49} +2.49551i q^{50} +1.80144 q^{51} -6.99102 q^{53} +14.0454i q^{54} -1.06939 q^{55} -10.5909 q^{56} +16.2036i q^{57} -23.5953i q^{58} +0.741035i q^{59} +11.9506i q^{60} +4.19856 q^{61} +3.65368 q^{62} +9.50894i q^{63} +4.84325 q^{64} -7.54390 q^{66} -8.09479i q^{67} +2.69407 q^{68} +10.7925 q^{69} +4.75447i q^{70} -9.76488i q^{71} +27.7445i q^{72} -3.71649i q^{73} +1.89022 q^{74} +2.82684 q^{75} +24.2326i q^{76} -2.03741 q^{77} -9.31937 q^{79} +5.41713i q^{80} +0.937188 q^{81} -0.668669 q^{82} -5.11778i q^{83} +22.7685i q^{84} -0.637263i q^{85} +1.59030i q^{86} -26.7281 q^{87} -5.94462 q^{88} -12.5783i q^{89} +12.4551 q^{90} +16.1402 q^{92} -4.13878i q^{93} -23.5729 q^{94} +5.73205 q^{95} +6.78645i q^{96} +4.22155i q^{97} -8.41027i q^{98} +5.33734i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{3} - 4q^{4} + 8q^{9} + O(q^{10}) \) \( 8q - 4q^{3} - 4q^{4} + 8q^{9} + 4q^{10} + 20q^{12} + 4q^{14} + 4q^{16} + 4q^{17} + 24q^{22} + 20q^{23} - 8q^{25} - 4q^{27} + 16q^{29} - 8q^{30} - 20q^{35} - 40q^{36} - 16q^{38} - 12q^{40} - 8q^{42} + 4q^{43} - 56q^{48} - 24q^{49} - 8q^{51} - 24q^{53} - 24q^{56} + 56q^{61} - 8q^{62} - 8q^{64} + 12q^{66} + 28q^{68} + 32q^{69} - 20q^{74} + 4q^{75} - 36q^{77} - 16q^{79} - 16q^{81} - 8q^{82} - 44q^{87} + 36q^{88} + 40q^{90} + 44q^{92} - 64q^{94} + 32q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.49551i − 1.76459i −0.470696 0.882295i \(-0.655997\pi\)
0.470696 0.882295i \(-0.344003\pi\)
\(3\) −2.82684 −1.63208 −0.816038 0.577998i \(-0.803834\pi\)
−0.816038 + 0.577998i \(0.803834\pi\)
\(4\) −4.22756 −2.11378
\(5\) 1.00000i 0.447214i
\(6\) 7.05440i 2.87995i
\(7\) 1.90521i 0.720103i 0.932933 + 0.360051i \(0.117241\pi\)
−0.932933 + 0.360051i \(0.882759\pi\)
\(8\) 5.55889i 1.96536i
\(9\) 4.99102 1.66367
\(10\) 2.49551 0.789149
\(11\) 1.06939i 0.322433i 0.986919 + 0.161217i \(0.0515417\pi\)
−0.986919 + 0.161217i \(0.948458\pi\)
\(12\) 11.9506 3.44985
\(13\) 0 0
\(14\) 4.75447 1.27069
\(15\) − 2.82684i − 0.729887i
\(16\) 5.41713 1.35428
\(17\) −0.637263 −0.154559 −0.0772795 0.997009i \(-0.524623\pi\)
−0.0772795 + 0.997009i \(0.524623\pi\)
\(18\) − 12.4551i − 2.93570i
\(19\) − 5.73205i − 1.31502i −0.753445 0.657511i \(-0.771609\pi\)
0.753445 0.657511i \(-0.228391\pi\)
\(20\) − 4.22756i − 0.945311i
\(21\) − 5.38573i − 1.17526i
\(22\) 2.66867 0.568962
\(23\) −3.81785 −0.796078 −0.398039 0.917369i \(-0.630309\pi\)
−0.398039 + 0.917369i \(0.630309\pi\)
\(24\) − 15.7141i − 3.20762i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −5.62828 −1.08316
\(28\) − 8.05440i − 1.52214i
\(29\) 9.45512 1.75577 0.877886 0.478870i \(-0.158954\pi\)
0.877886 + 0.478870i \(0.158954\pi\)
\(30\) −7.05440 −1.28795
\(31\) 1.46410i 0.262960i 0.991319 + 0.131480i \(0.0419730\pi\)
−0.991319 + 0.131480i \(0.958027\pi\)
\(32\) − 2.40072i − 0.424391i
\(33\) − 3.02299i − 0.526235i
\(34\) 1.59030i 0.272733i
\(35\) −1.90521 −0.322040
\(36\) −21.0998 −3.51663
\(37\) 0.757449i 0.124524i 0.998060 + 0.0622619i \(0.0198314\pi\)
−0.998060 + 0.0622619i \(0.980169\pi\)
\(38\) −14.3044 −2.32048
\(39\) 0 0
\(40\) −5.55889 −0.878938
\(41\) − 0.267949i − 0.0418466i −0.999781 0.0209233i \(-0.993339\pi\)
0.999781 0.0209233i \(-0.00666058\pi\)
\(42\) −13.4401 −2.07386
\(43\) −0.637263 −0.0971817 −0.0485909 0.998819i \(-0.515473\pi\)
−0.0485909 + 0.998819i \(0.515473\pi\)
\(44\) − 4.52091i − 0.681552i
\(45\) 4.99102i 0.744017i
\(46\) 9.52748i 1.40475i
\(47\) − 9.44613i − 1.37786i −0.724828 0.688930i \(-0.758081\pi\)
0.724828 0.688930i \(-0.241919\pi\)
\(48\) −15.3134 −2.21029
\(49\) 3.37017 0.481452
\(50\) 2.49551i 0.352918i
\(51\) 1.80144 0.252252
\(52\) 0 0
\(53\) −6.99102 −0.960290 −0.480145 0.877189i \(-0.659416\pi\)
−0.480145 + 0.877189i \(0.659416\pi\)
\(54\) 14.0454i 1.91134i
\(55\) −1.06939 −0.144196
\(56\) −10.5909 −1.41526
\(57\) 16.2036i 2.14622i
\(58\) − 23.5953i − 3.09822i
\(59\) 0.741035i 0.0964746i 0.998836 + 0.0482373i \(0.0153604\pi\)
−0.998836 + 0.0482373i \(0.984640\pi\)
\(60\) 11.9506i 1.54282i
\(61\) 4.19856 0.537571 0.268785 0.963200i \(-0.413378\pi\)
0.268785 + 0.963200i \(0.413378\pi\)
\(62\) 3.65368 0.464017
\(63\) 9.50894i 1.19801i
\(64\) 4.84325 0.605406
\(65\) 0 0
\(66\) −7.54390 −0.928589
\(67\) − 8.09479i − 0.988936i −0.869196 0.494468i \(-0.835363\pi\)
0.869196 0.494468i \(-0.164637\pi\)
\(68\) 2.69407 0.326704
\(69\) 10.7925 1.29926
\(70\) 4.75447i 0.568268i
\(71\) − 9.76488i − 1.15888i −0.815016 0.579439i \(-0.803272\pi\)
0.815016 0.579439i \(-0.196728\pi\)
\(72\) 27.7445i 3.26972i
\(73\) − 3.71649i − 0.434982i −0.976062 0.217491i \(-0.930213\pi\)
0.976062 0.217491i \(-0.0697873\pi\)
\(74\) 1.89022 0.219734
\(75\) 2.82684 0.326415
\(76\) 24.2326i 2.77967i
\(77\) −2.03741 −0.232185
\(78\) 0 0
\(79\) −9.31937 −1.04851 −0.524255 0.851561i \(-0.675656\pi\)
−0.524255 + 0.851561i \(0.675656\pi\)
\(80\) 5.41713i 0.605654i
\(81\) 0.937188 0.104132
\(82\) −0.668669 −0.0738422
\(83\) − 5.11778i − 0.561749i −0.959744 0.280875i \(-0.909376\pi\)
0.959744 0.280875i \(-0.0906245\pi\)
\(84\) 22.7685i 2.48424i
\(85\) − 0.637263i − 0.0691209i
\(86\) 1.59030i 0.171486i
\(87\) −26.7281 −2.86555
\(88\) −5.94462 −0.633698
\(89\) − 12.5783i − 1.33330i −0.745371 0.666650i \(-0.767727\pi\)
0.745371 0.666650i \(-0.232273\pi\)
\(90\) 12.4551 1.31288
\(91\) 0 0
\(92\) 16.1402 1.68273
\(93\) − 4.13878i − 0.429171i
\(94\) −23.5729 −2.43136
\(95\) 5.73205 0.588096
\(96\) 6.78645i 0.692639i
\(97\) 4.22155i 0.428634i 0.976764 + 0.214317i \(0.0687525\pi\)
−0.976764 + 0.214317i \(0.931248\pi\)
\(98\) − 8.41027i − 0.849566i
\(99\) 5.33734i 0.536423i
\(100\) 4.22756 0.422756
\(101\) 15.2476 1.51719 0.758595 0.651562i \(-0.225886\pi\)
0.758595 + 0.651562i \(0.225886\pi\)
\(102\) − 4.49551i − 0.445122i
\(103\) 13.5269 1.33285 0.666423 0.745574i \(-0.267824\pi\)
0.666423 + 0.745574i \(0.267824\pi\)
\(104\) 0 0
\(105\) 5.38573 0.525593
\(106\) 17.4461i 1.69452i
\(107\) −7.36274 −0.711783 −0.355891 0.934527i \(-0.615823\pi\)
−0.355891 + 0.934527i \(0.615823\pi\)
\(108\) 23.7939 2.28957
\(109\) − 10.0760i − 0.965103i −0.875868 0.482551i \(-0.839710\pi\)
0.875868 0.482551i \(-0.160290\pi\)
\(110\) 2.66867i 0.254448i
\(111\) − 2.14119i − 0.203232i
\(112\) 10.3208i 0.975223i
\(113\) −6.68806 −0.629160 −0.314580 0.949231i \(-0.601864\pi\)
−0.314580 + 0.949231i \(0.601864\pi\)
\(114\) 40.4362 3.78719
\(115\) − 3.81785i − 0.356017i
\(116\) −39.9721 −3.71131
\(117\) 0 0
\(118\) 1.84926 0.170238
\(119\) − 1.21412i − 0.111298i
\(120\) 15.7141 1.43449
\(121\) 9.85641 0.896037
\(122\) − 10.4775i − 0.948592i
\(123\) 0.757449i 0.0682969i
\(124\) − 6.18958i − 0.555840i
\(125\) − 1.00000i − 0.0894427i
\(126\) 23.7296 2.11400
\(127\) −1.48950 −0.132172 −0.0660859 0.997814i \(-0.521051\pi\)
−0.0660859 + 0.997814i \(0.521051\pi\)
\(128\) − 16.8878i − 1.49269i
\(129\) 1.80144 0.158608
\(130\) 0 0
\(131\) 4.12676 0.360557 0.180278 0.983616i \(-0.442300\pi\)
0.180278 + 0.983616i \(0.442300\pi\)
\(132\) 12.7799i 1.11234i
\(133\) 10.9208 0.946951
\(134\) −20.2006 −1.74507
\(135\) − 5.62828i − 0.484405i
\(136\) − 3.54248i − 0.303765i
\(137\) − 20.1096i − 1.71808i −0.511906 0.859041i \(-0.671060\pi\)
0.511906 0.859041i \(-0.328940\pi\)
\(138\) − 26.9327i − 2.29266i
\(139\) 20.8253 1.76638 0.883189 0.469018i \(-0.155392\pi\)
0.883189 + 0.469018i \(0.155392\pi\)
\(140\) 8.05440 0.680721
\(141\) 26.7027i 2.24877i
\(142\) −24.3683 −2.04494
\(143\) 0 0
\(144\) 27.0370 2.25308
\(145\) 9.45512i 0.785205i
\(146\) −9.27453 −0.767565
\(147\) −9.52691 −0.785767
\(148\) − 3.20216i − 0.263216i
\(149\) − 13.3678i − 1.09513i −0.836763 0.547565i \(-0.815555\pi\)
0.836763 0.547565i \(-0.184445\pi\)
\(150\) − 7.05440i − 0.575989i
\(151\) 18.2984i 1.48910i 0.667567 + 0.744550i \(0.267336\pi\)
−0.667567 + 0.744550i \(0.732664\pi\)
\(152\) 31.8638 2.58450
\(153\) −3.18059 −0.257136
\(154\) 5.08438i 0.409711i
\(155\) −1.46410 −0.117599
\(156\) 0 0
\(157\) 2.42229 0.193320 0.0966599 0.995317i \(-0.469184\pi\)
0.0966599 + 0.995317i \(0.469184\pi\)
\(158\) 23.2566i 1.85019i
\(159\) 19.7625 1.56727
\(160\) 2.40072 0.189794
\(161\) − 7.27382i − 0.573258i
\(162\) − 2.33876i − 0.183750i
\(163\) 15.9829i 1.25188i 0.779873 + 0.625938i \(0.215284\pi\)
−0.779873 + 0.625938i \(0.784716\pi\)
\(164\) 1.13277i 0.0884545i
\(165\) 3.02299 0.235340
\(166\) −12.7715 −0.991257
\(167\) − 14.3932i − 1.11378i −0.830588 0.556888i \(-0.811995\pi\)
0.830588 0.556888i \(-0.188005\pi\)
\(168\) 29.9387 2.30982
\(169\) 0 0
\(170\) −1.59030 −0.121970
\(171\) − 28.6088i − 2.18777i
\(172\) 2.69407 0.205421
\(173\) 24.3489 1.85122 0.925608 0.378484i \(-0.123555\pi\)
0.925608 + 0.378484i \(0.123555\pi\)
\(174\) 66.7001i 5.05653i
\(175\) − 1.90521i − 0.144021i
\(176\) 5.79302i 0.436666i
\(177\) − 2.09479i − 0.157454i
\(178\) −31.3893 −2.35273
\(179\) −3.78829 −0.283150 −0.141575 0.989928i \(-0.545217\pi\)
−0.141575 + 0.989928i \(0.545217\pi\)
\(180\) − 21.0998i − 1.57269i
\(181\) 8.48794 0.630904 0.315452 0.948942i \(-0.397844\pi\)
0.315452 + 0.948942i \(0.397844\pi\)
\(182\) 0 0
\(183\) −11.8687 −0.877356
\(184\) − 21.2230i − 1.56458i
\(185\) −0.757449 −0.0556888
\(186\) −10.3284 −0.757312
\(187\) − 0.681482i − 0.0498349i
\(188\) 39.9341i 2.91249i
\(189\) − 10.7231i − 0.779988i
\(190\) − 14.3044i − 1.03775i
\(191\) −5.44310 −0.393849 −0.196924 0.980419i \(-0.563095\pi\)
−0.196924 + 0.980419i \(0.563095\pi\)
\(192\) −13.6911 −0.988069
\(193\) 12.1576i 0.875123i 0.899188 + 0.437562i \(0.144158\pi\)
−0.899188 + 0.437562i \(0.855842\pi\)
\(194\) 10.5349 0.756363
\(195\) 0 0
\(196\) −14.2476 −1.01768
\(197\) 4.37830i 0.311941i 0.987762 + 0.155970i \(0.0498505\pi\)
−0.987762 + 0.155970i \(0.950150\pi\)
\(198\) 13.3194 0.946566
\(199\) −20.8373 −1.47712 −0.738558 0.674189i \(-0.764493\pi\)
−0.738558 + 0.674189i \(0.764493\pi\)
\(200\) − 5.55889i − 0.393073i
\(201\) 22.8827i 1.61402i
\(202\) − 38.0504i − 2.67722i
\(203\) 18.0140i 1.26434i
\(204\) −7.61569 −0.533205
\(205\) 0.267949 0.0187144
\(206\) − 33.7565i − 2.35193i
\(207\) −19.0550 −1.32441
\(208\) 0 0
\(209\) 6.12979 0.424007
\(210\) − 13.4401i − 0.927457i
\(211\) −10.6537 −0.733429 −0.366715 0.930333i \(-0.619517\pi\)
−0.366715 + 0.930333i \(0.619517\pi\)
\(212\) 29.5549 2.02984
\(213\) 27.6037i 1.89138i
\(214\) 18.3738i 1.25600i
\(215\) − 0.637263i − 0.0434610i
\(216\) − 31.2870i − 2.12881i
\(217\) −2.78942 −0.189358
\(218\) −25.1447 −1.70301
\(219\) 10.5059i 0.709924i
\(220\) 4.52091 0.304799
\(221\) 0 0
\(222\) −5.34335 −0.358622
\(223\) − 21.3393i − 1.42899i −0.699642 0.714494i \(-0.746657\pi\)
0.699642 0.714494i \(-0.253343\pi\)
\(224\) 4.57388 0.305605
\(225\) −4.99102 −0.332734
\(226\) 16.6901i 1.11021i
\(227\) − 15.6857i − 1.04109i −0.853833 0.520547i \(-0.825728\pi\)
0.853833 0.520547i \(-0.174272\pi\)
\(228\) − 68.5016i − 4.53663i
\(229\) − 7.62085i − 0.503600i −0.967779 0.251800i \(-0.918977\pi\)
0.967779 0.251800i \(-0.0810225\pi\)
\(230\) −9.52748 −0.628224
\(231\) 5.75944 0.378943
\(232\) 52.5599i 3.45073i
\(233\) 19.0550 1.24833 0.624166 0.781292i \(-0.285439\pi\)
0.624166 + 0.781292i \(0.285439\pi\)
\(234\) 0 0
\(235\) 9.44613 0.616198
\(236\) − 3.13277i − 0.203926i
\(237\) 26.3444 1.71125
\(238\) −3.02985 −0.196396
\(239\) − 12.7535i − 0.824954i −0.910968 0.412477i \(-0.864664\pi\)
0.910968 0.412477i \(-0.135336\pi\)
\(240\) − 15.3134i − 0.988473i
\(241\) 25.9288i 1.67022i 0.550081 + 0.835111i \(0.314597\pi\)
−0.550081 + 0.835111i \(0.685403\pi\)
\(242\) − 24.5967i − 1.58114i
\(243\) 14.2356 0.913211
\(244\) −17.7497 −1.13631
\(245\) 3.37017i 0.215312i
\(246\) 1.89022 0.120516
\(247\) 0 0
\(248\) −8.13878 −0.516813
\(249\) 14.4671i 0.916817i
\(250\) −2.49551 −0.157830
\(251\) −7.61186 −0.480457 −0.240228 0.970716i \(-0.577222\pi\)
−0.240228 + 0.970716i \(0.577222\pi\)
\(252\) − 40.1996i − 2.53234i
\(253\) − 4.08277i − 0.256682i
\(254\) 3.71706i 0.233229i
\(255\) 1.80144i 0.112811i
\(256\) −32.4572 −2.02857
\(257\) −0.335783 −0.0209456 −0.0104728 0.999945i \(-0.503334\pi\)
−0.0104728 + 0.999945i \(0.503334\pi\)
\(258\) − 4.49551i − 0.279878i
\(259\) −1.44310 −0.0896700
\(260\) 0 0
\(261\) 47.1906 2.92103
\(262\) − 10.2984i − 0.636235i
\(263\) −5.37589 −0.331492 −0.165746 0.986169i \(-0.553003\pi\)
−0.165746 + 0.986169i \(0.553003\pi\)
\(264\) 16.8045 1.03424
\(265\) − 6.99102i − 0.429455i
\(266\) − 27.2529i − 1.67098i
\(267\) 35.5569i 2.17605i
\(268\) 34.2212i 2.09039i
\(269\) −1.31038 −0.0798956 −0.0399478 0.999202i \(-0.512719\pi\)
−0.0399478 + 0.999202i \(0.512719\pi\)
\(270\) −14.0454 −0.854777
\(271\) − 11.6453i − 0.707403i −0.935358 0.353701i \(-0.884923\pi\)
0.935358 0.353701i \(-0.115077\pi\)
\(272\) −3.45214 −0.209317
\(273\) 0 0
\(274\) −50.1838 −3.03171
\(275\) − 1.06939i − 0.0644866i
\(276\) −45.6257 −2.74635
\(277\) 20.3161 1.22068 0.610338 0.792141i \(-0.291033\pi\)
0.610338 + 0.792141i \(0.291033\pi\)
\(278\) − 51.9697i − 3.11693i
\(279\) 7.30735i 0.437480i
\(280\) − 10.5909i − 0.632925i
\(281\) − 11.8744i − 0.708366i −0.935176 0.354183i \(-0.884759\pi\)
0.935176 0.354183i \(-0.115241\pi\)
\(282\) 66.6368 3.96816
\(283\) 22.6521 1.34653 0.673264 0.739402i \(-0.264892\pi\)
0.673264 + 0.739402i \(0.264892\pi\)
\(284\) 41.2816i 2.44961i
\(285\) −16.2036 −0.959817
\(286\) 0 0
\(287\) 0.510500 0.0301339
\(288\) − 11.9820i − 0.706048i
\(289\) −16.5939 −0.976112
\(290\) 23.5953 1.38556
\(291\) − 11.9336i − 0.699562i
\(292\) 15.7117i 0.919456i
\(293\) 18.6127i 1.08737i 0.839290 + 0.543683i \(0.182971\pi\)
−0.839290 + 0.543683i \(0.817029\pi\)
\(294\) 23.7745i 1.38656i
\(295\) −0.741035 −0.0431448
\(296\) −4.21058 −0.244735
\(297\) − 6.01882i − 0.349247i
\(298\) −33.3593 −1.93245
\(299\) 0 0
\(300\) −11.9506 −0.689970
\(301\) − 1.21412i − 0.0699808i
\(302\) 45.6637 2.62765
\(303\) −43.1024 −2.47617
\(304\) − 31.0513i − 1.78091i
\(305\) 4.19856i 0.240409i
\(306\) 7.93719i 0.453739i
\(307\) 3.14776i 0.179652i 0.995957 + 0.0898262i \(0.0286311\pi\)
−0.995957 + 0.0898262i \(0.971369\pi\)
\(308\) 8.61329 0.490788
\(309\) −38.2384 −2.17531
\(310\) 3.65368i 0.207515i
\(311\) 3.18059 0.180355 0.0901774 0.995926i \(-0.471257\pi\)
0.0901774 + 0.995926i \(0.471257\pi\)
\(312\) 0 0
\(313\) 35.3533 1.99829 0.999144 0.0413596i \(-0.0131689\pi\)
0.999144 + 0.0413596i \(0.0131689\pi\)
\(314\) − 6.04484i − 0.341130i
\(315\) −9.50894 −0.535768
\(316\) 39.3982 2.21632
\(317\) 13.6357i 0.765858i 0.923778 + 0.382929i \(0.125085\pi\)
−0.923778 + 0.382929i \(0.874915\pi\)
\(318\) − 49.3174i − 2.76558i
\(319\) 10.1112i 0.566119i
\(320\) 4.84325i 0.270746i
\(321\) 20.8133 1.16168
\(322\) −18.1519 −1.01156
\(323\) 3.65283i 0.203249i
\(324\) −3.96202 −0.220112
\(325\) 0 0
\(326\) 39.8854 2.20905
\(327\) 28.4831i 1.57512i
\(328\) 1.48950 0.0822439
\(329\) 17.9969 0.992201
\(330\) − 7.54390i − 0.415278i
\(331\) 28.7959i 1.58277i 0.611320 + 0.791383i \(0.290639\pi\)
−0.611320 + 0.791383i \(0.709361\pi\)
\(332\) 21.6357i 1.18741i
\(333\) 3.78044i 0.207167i
\(334\) −35.9182 −1.96536
\(335\) 8.09479 0.442265
\(336\) − 29.1752i − 1.59164i
\(337\) −11.7493 −0.640026 −0.320013 0.947413i \(-0.603687\pi\)
−0.320013 + 0.947413i \(0.603687\pi\)
\(338\) 0 0
\(339\) 18.9061 1.02684
\(340\) 2.69407i 0.146106i
\(341\) −1.56569 −0.0847871
\(342\) −71.3934 −3.86051
\(343\) 19.7574i 1.06680i
\(344\) − 3.54248i − 0.190997i
\(345\) 10.7925i 0.581046i
\(346\) − 60.7630i − 3.26664i
\(347\) 1.89977 0.101985 0.0509926 0.998699i \(-0.483762\pi\)
0.0509926 + 0.998699i \(0.483762\pi\)
\(348\) 112.995 6.05714
\(349\) − 10.2691i − 0.549692i −0.961488 0.274846i \(-0.911373\pi\)
0.961488 0.274846i \(-0.0886268\pi\)
\(350\) −4.75447 −0.254137
\(351\) 0 0
\(352\) 2.56730 0.136838
\(353\) − 0.800589i − 0.0426110i −0.999773 0.0213055i \(-0.993218\pi\)
0.999773 0.0213055i \(-0.00678227\pi\)
\(354\) −5.22756 −0.277842
\(355\) 9.76488 0.518266
\(356\) 53.1756i 2.81830i
\(357\) 3.43213i 0.181647i
\(358\) 9.45370i 0.499643i
\(359\) 8.13272i 0.429228i 0.976699 + 0.214614i \(0.0688494\pi\)
−0.976699 + 0.214614i \(0.931151\pi\)
\(360\) −27.7445 −1.46226
\(361\) −13.8564 −0.729285
\(362\) − 21.1817i − 1.11329i
\(363\) −27.8625 −1.46240
\(364\) 0 0
\(365\) 3.71649 0.194530
\(366\) 29.6183i 1.54817i
\(367\) −20.5265 −1.07147 −0.535737 0.844385i \(-0.679966\pi\)
−0.535737 + 0.844385i \(0.679966\pi\)
\(368\) −20.6818 −1.07811
\(369\) − 1.33734i − 0.0696191i
\(370\) 1.89022i 0.0982679i
\(371\) − 13.3194i − 0.691507i
\(372\) 17.4969i 0.907173i
\(373\) −17.8058 −0.921951 −0.460976 0.887413i \(-0.652500\pi\)
−0.460976 + 0.887413i \(0.652500\pi\)
\(374\) −1.70064 −0.0879382
\(375\) 2.82684i 0.145977i
\(376\) 52.5100 2.70800
\(377\) 0 0
\(378\) −26.7595 −1.37636
\(379\) − 2.04555i − 0.105073i −0.998619 0.0525363i \(-0.983269\pi\)
0.998619 0.0525363i \(-0.0167305\pi\)
\(380\) −24.2326 −1.24311
\(381\) 4.21058 0.215714
\(382\) 13.5833i 0.694982i
\(383\) 7.90521i 0.403937i 0.979392 + 0.201969i \(0.0647339\pi\)
−0.979392 + 0.201969i \(0.935266\pi\)
\(384\) 47.7391i 2.43618i
\(385\) − 2.03741i − 0.103836i
\(386\) 30.3394 1.54423
\(387\) −3.18059 −0.161679
\(388\) − 17.8469i − 0.906037i
\(389\) 9.21171 0.467052 0.233526 0.972351i \(-0.424974\pi\)
0.233526 + 0.972351i \(0.424974\pi\)
\(390\) 0 0
\(391\) 2.43298 0.123041
\(392\) 18.7344i 0.946229i
\(393\) −11.6657 −0.588456
\(394\) 10.9261 0.550448
\(395\) − 9.31937i − 0.468908i
\(396\) − 22.5639i − 1.13388i
\(397\) − 6.35438i − 0.318917i −0.987205 0.159458i \(-0.949025\pi\)
0.987205 0.159458i \(-0.0509748\pi\)
\(398\) 51.9996i 2.60651i
\(399\) −30.8713 −1.54550
\(400\) −5.41713 −0.270857
\(401\) 4.16920i 0.208200i 0.994567 + 0.104100i \(0.0331962\pi\)
−0.994567 + 0.104100i \(0.966804\pi\)
\(402\) 57.1038 2.84808
\(403\) 0 0
\(404\) −64.4600 −3.20701
\(405\) 0.937188i 0.0465692i
\(406\) 44.9541 2.23103
\(407\) −0.810008 −0.0401506
\(408\) 10.0140i 0.495767i
\(409\) − 10.1681i − 0.502778i −0.967886 0.251389i \(-0.919113\pi\)
0.967886 0.251389i \(-0.0808874\pi\)
\(410\) − 0.668669i − 0.0330232i
\(411\) 56.8467i 2.80404i
\(412\) −57.1858 −2.81734
\(413\) −1.41183 −0.0694716
\(414\) 47.5518i 2.33704i
\(415\) 5.11778 0.251222
\(416\) 0 0
\(417\) −58.8697 −2.88286
\(418\) − 15.2969i − 0.748198i
\(419\) 28.5909 1.39676 0.698378 0.715730i \(-0.253906\pi\)
0.698378 + 0.715730i \(0.253906\pi\)
\(420\) −22.7685 −1.11099
\(421\) − 2.01797i − 0.0983498i −0.998790 0.0491749i \(-0.984341\pi\)
0.998790 0.0491749i \(-0.0156592\pi\)
\(422\) 26.5863i 1.29420i
\(423\) − 47.1458i − 2.29231i
\(424\) − 38.8623i − 1.88732i
\(425\) 0.637263 0.0309118
\(426\) 68.8853 3.33750
\(427\) 7.99915i 0.387106i
\(428\) 31.1264 1.50455
\(429\) 0 0
\(430\) −1.59030 −0.0766908
\(431\) 20.6123i 0.992860i 0.868077 + 0.496430i \(0.165356\pi\)
−0.868077 + 0.496430i \(0.834644\pi\)
\(432\) −30.4891 −1.46691
\(433\) −29.4356 −1.41458 −0.707292 0.706921i \(-0.750083\pi\)
−0.707292 + 0.706921i \(0.750083\pi\)
\(434\) 6.96103i 0.334140i
\(435\) − 26.7281i − 1.28151i
\(436\) 42.5967i 2.04001i
\(437\) 21.8841i 1.04686i
\(438\) 26.2176 1.25272
\(439\) −16.9520 −0.809077 −0.404538 0.914521i \(-0.632568\pi\)
−0.404538 + 0.914521i \(0.632568\pi\)
\(440\) − 5.94462i − 0.283398i
\(441\) 16.8205 0.800978
\(442\) 0 0
\(443\) −24.1399 −1.14692 −0.573461 0.819233i \(-0.694400\pi\)
−0.573461 + 0.819233i \(0.694400\pi\)
\(444\) 9.05199i 0.429588i
\(445\) 12.5783 0.596270
\(446\) −53.2525 −2.52158
\(447\) 37.7885i 1.78733i
\(448\) 9.22742i 0.435955i
\(449\) − 20.8630i − 0.984585i −0.870430 0.492293i \(-0.836159\pi\)
0.870430 0.492293i \(-0.163841\pi\)
\(450\) 12.4551i 0.587140i
\(451\) 0.286542 0.0134927
\(452\) 28.2742 1.32990
\(453\) − 51.7265i − 2.43032i
\(454\) −39.1437 −1.83710
\(455\) 0 0
\(456\) −90.0739 −4.21810
\(457\) 30.5659i 1.42981i 0.699220 + 0.714906i \(0.253531\pi\)
−0.699220 + 0.714906i \(0.746469\pi\)
\(458\) −19.0179 −0.888648
\(459\) 3.58669 0.167413
\(460\) 16.1402i 0.752541i
\(461\) − 4.67822i − 0.217887i −0.994048 0.108943i \(-0.965253\pi\)
0.994048 0.108943i \(-0.0347467\pi\)
\(462\) − 14.3727i − 0.668680i
\(463\) − 14.0011i − 0.650688i −0.945596 0.325344i \(-0.894520\pi\)
0.945596 0.325344i \(-0.105480\pi\)
\(464\) 51.2196 2.37781
\(465\) 4.13878 0.191931
\(466\) − 47.5518i − 2.20280i
\(467\) 6.98506 0.323230 0.161615 0.986854i \(-0.448330\pi\)
0.161615 + 0.986854i \(0.448330\pi\)
\(468\) 0 0
\(469\) 15.4223 0.712135
\(470\) − 23.5729i − 1.08734i
\(471\) −6.84742 −0.315513
\(472\) −4.11933 −0.189608
\(473\) − 0.681482i − 0.0313346i
\(474\) − 65.7425i − 3.01965i
\(475\) 5.73205i 0.263005i
\(476\) 5.13277i 0.235260i
\(477\) −34.8923 −1.59761
\(478\) −31.8264 −1.45571
\(479\) − 16.2888i − 0.744252i −0.928182 0.372126i \(-0.878629\pi\)
0.928182 0.372126i \(-0.121371\pi\)
\(480\) −6.78645 −0.309758
\(481\) 0 0
\(482\) 64.7056 2.94726
\(483\) 20.5619i 0.935600i
\(484\) −41.6685 −1.89402
\(485\) −4.22155 −0.191691
\(486\) − 35.5249i − 1.61144i
\(487\) − 20.0409i − 0.908139i −0.890966 0.454069i \(-0.849972\pi\)
0.890966 0.454069i \(-0.150028\pi\)
\(488\) 23.3393i 1.05652i
\(489\) − 45.1810i − 2.04316i
\(490\) 8.41027 0.379937
\(491\) 15.7983 0.712969 0.356484 0.934301i \(-0.383975\pi\)
0.356484 + 0.934301i \(0.383975\pi\)
\(492\) − 3.20216i − 0.144365i
\(493\) −6.02540 −0.271370
\(494\) 0 0
\(495\) −5.33734 −0.239896
\(496\) 7.93123i 0.356123i
\(497\) 18.6042 0.834511
\(498\) 36.1028 1.61781
\(499\) − 1.24651i − 0.0558016i −0.999611 0.0279008i \(-0.991118\pi\)
0.999611 0.0279008i \(-0.00888226\pi\)
\(500\) 4.22756i 0.189062i
\(501\) 40.6871i 1.81777i
\(502\) 18.9955i 0.847809i
\(503\) −7.65345 −0.341250 −0.170625 0.985336i \(-0.554579\pi\)
−0.170625 + 0.985336i \(0.554579\pi\)
\(504\) −52.8592 −2.35453
\(505\) 15.2476i 0.678508i
\(506\) −10.1886 −0.452938
\(507\) 0 0
\(508\) 6.29695 0.279382
\(509\) − 25.7241i − 1.14020i −0.821575 0.570101i \(-0.806904\pi\)
0.821575 0.570101i \(-0.193096\pi\)
\(510\) 4.49551 0.199064
\(511\) 7.08070 0.313232
\(512\) 47.2215i 2.08691i
\(513\) 32.2616i 1.42438i
\(514\) 0.837948i 0.0369603i
\(515\) 13.5269i 0.596067i
\(516\) −7.61569 −0.335262
\(517\) 10.1016 0.444268
\(518\) 3.60127i 0.158231i
\(519\) −68.8305 −3.02132
\(520\) 0 0
\(521\) −30.1519 −1.32098 −0.660490 0.750835i \(-0.729651\pi\)
−0.660490 + 0.750835i \(0.729651\pi\)
\(522\) − 117.765i − 5.15442i
\(523\) 3.93752 0.172176 0.0860880 0.996288i \(-0.472563\pi\)
0.0860880 + 0.996288i \(0.472563\pi\)
\(524\) −17.4461 −0.762138
\(525\) 5.38573i 0.235052i
\(526\) 13.4156i 0.584947i
\(527\) − 0.933018i − 0.0406429i
\(528\) − 16.3759i − 0.712672i
\(529\) −8.42399 −0.366261
\(530\) −17.4461 −0.757812
\(531\) 3.69852i 0.160502i
\(532\) −46.1682 −2.00165
\(533\) 0 0
\(534\) 88.7326 3.83983
\(535\) − 7.36274i − 0.318319i
\(536\) 44.9980 1.94362
\(537\) 10.7089 0.462122
\(538\) 3.27007i 0.140983i
\(539\) 3.60402i 0.155236i
\(540\) 23.7939i 1.02393i
\(541\) − 15.8881i − 0.683083i −0.939867 0.341541i \(-0.889051\pi\)
0.939867 0.341541i \(-0.110949\pi\)
\(542\) −29.0610 −1.24828
\(543\) −23.9940 −1.02968
\(544\) 1.52989i 0.0655935i
\(545\) 10.0760 0.431607
\(546\) 0 0
\(547\) −6.56107 −0.280531 −0.140266 0.990114i \(-0.544796\pi\)
−0.140266 + 0.990114i \(0.544796\pi\)
\(548\) 85.0147i 3.63165i
\(549\) 20.9551 0.894341
\(550\) −2.66867 −0.113792
\(551\) − 54.1972i − 2.30888i
\(552\) 59.9941i 2.55352i
\(553\) − 17.7554i − 0.755035i
\(554\) − 50.6990i − 2.15399i
\(555\) 2.14119 0.0908883
\(556\) −88.0401 −3.73373
\(557\) 7.85006i 0.332618i 0.986074 + 0.166309i \(0.0531849\pi\)
−0.986074 + 0.166309i \(0.946815\pi\)
\(558\) 18.2356 0.771973
\(559\) 0 0
\(560\) −10.3208 −0.436133
\(561\) 1.92644i 0.0813344i
\(562\) −29.6326 −1.24998
\(563\) 15.5595 0.655755 0.327878 0.944720i \(-0.393667\pi\)
0.327878 + 0.944720i \(0.393667\pi\)
\(564\) − 112.887i − 4.75341i
\(565\) − 6.68806i − 0.281369i
\(566\) − 56.5285i − 2.37607i
\(567\) 1.78554i 0.0749857i
\(568\) 54.2819 2.27762
\(569\) −3.47915 −0.145853 −0.0729267 0.997337i \(-0.523234\pi\)
−0.0729267 + 0.997337i \(0.523234\pi\)
\(570\) 40.4362i 1.69368i
\(571\) 21.5118 0.900240 0.450120 0.892968i \(-0.351381\pi\)
0.450120 + 0.892968i \(0.351381\pi\)
\(572\) 0 0
\(573\) 15.3868 0.642791
\(574\) − 1.27396i − 0.0531739i
\(575\) 3.81785 0.159216
\(576\) 24.1727 1.00720
\(577\) − 9.97608i − 0.415310i −0.978202 0.207655i \(-0.933417\pi\)
0.978202 0.207655i \(-0.0665831\pi\)
\(578\) 41.4102i 1.72244i
\(579\) − 34.3676i − 1.42827i
\(580\) − 39.9721i − 1.65975i
\(581\) 9.75045 0.404517
\(582\) −29.7805 −1.23444
\(583\) − 7.47612i − 0.309629i
\(584\) 20.6595 0.854898
\(585\) 0 0
\(586\) 46.4482 1.91876
\(587\) 24.0571i 0.992945i 0.868053 + 0.496472i \(0.165372\pi\)
−0.868053 + 0.496472i \(0.834628\pi\)
\(588\) 40.2756 1.66094
\(589\) 8.39230 0.345799
\(590\) 1.84926i 0.0761328i
\(591\) − 12.3767i − 0.509111i
\(592\) 4.10320i 0.168641i
\(593\) 0.940219i 0.0386102i 0.999814 + 0.0193051i \(0.00614538\pi\)
−0.999814 + 0.0193051i \(0.993855\pi\)
\(594\) −15.0200 −0.616279
\(595\) 1.21412 0.0497741
\(596\) 56.5130i 2.31486i
\(597\) 58.9037 2.41077
\(598\) 0 0
\(599\) −11.4270 −0.466896 −0.233448 0.972369i \(-0.575001\pi\)
−0.233448 + 0.972369i \(0.575001\pi\)
\(600\) 15.7141i 0.641525i
\(601\) −36.0431 −1.47023 −0.735114 0.677944i \(-0.762871\pi\)
−0.735114 + 0.677944i \(0.762871\pi\)
\(602\) −3.02985 −0.123487
\(603\) − 40.4012i − 1.64526i
\(604\) − 77.3574i − 3.14763i
\(605\) 9.85641i 0.400720i
\(606\) 107.562i 4.36942i
\(607\) 39.8907 1.61911 0.809557 0.587041i \(-0.199707\pi\)
0.809557 + 0.587041i \(0.199707\pi\)
\(608\) −13.7610 −0.558084
\(609\) − 50.9227i − 2.06349i
\(610\) 10.4775 0.424223
\(611\) 0 0
\(612\) 13.4461 0.543528
\(613\) 0.345472i 0.0139535i 0.999976 + 0.00697673i \(0.00222078\pi\)
−0.999976 + 0.00697673i \(0.997779\pi\)
\(614\) 7.85527 0.317013
\(615\) −0.757449 −0.0305433
\(616\) − 11.3258i − 0.456328i
\(617\) − 38.6850i − 1.55740i −0.627397 0.778700i \(-0.715879\pi\)
0.627397 0.778700i \(-0.284121\pi\)
\(618\) 95.4242i 3.83852i
\(619\) 14.8971i 0.598764i 0.954133 + 0.299382i \(0.0967805\pi\)
−0.954133 + 0.299382i \(0.903219\pi\)
\(620\) 6.18958 0.248579
\(621\) 21.4879 0.862281
\(622\) − 7.93719i − 0.318252i
\(623\) 23.9644 0.960113
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) − 88.2245i − 3.52616i
\(627\) −17.3279 −0.692011
\(628\) −10.2404 −0.408635
\(629\) − 0.482694i − 0.0192463i
\(630\) 23.7296i 0.945412i
\(631\) − 38.8450i − 1.54640i −0.634165 0.773198i \(-0.718656\pi\)
0.634165 0.773198i \(-0.281344\pi\)
\(632\) − 51.8053i − 2.06071i
\(633\) 30.1162 1.19701
\(634\) 34.0280 1.35143
\(635\) − 1.48950i − 0.0591090i
\(636\) −83.5470 −3.31285
\(637\) 0 0
\(638\) 25.2326 0.998967
\(639\) − 48.7367i − 1.92799i
\(640\) 16.8878 0.667549
\(641\) −37.1816 −1.46859 −0.734293 0.678832i \(-0.762486\pi\)
−0.734293 + 0.678832i \(0.762486\pi\)
\(642\) − 51.9397i − 2.04990i
\(643\) − 9.10377i − 0.359018i −0.983756 0.179509i \(-0.942549\pi\)
0.983756 0.179509i \(-0.0574509\pi\)
\(644\) 30.7505i 1.21174i
\(645\) 1.80144i 0.0709316i
\(646\) 9.11565 0.358651
\(647\) 19.1224 0.751778 0.375889 0.926665i \(-0.377337\pi\)
0.375889 + 0.926665i \(0.377337\pi\)
\(648\) 5.20972i 0.204657i
\(649\) −0.792455 −0.0311066
\(650\) 0 0
\(651\) 7.88525 0.309047
\(652\) − 67.5686i − 2.64619i
\(653\) 34.6324 1.35527 0.677636 0.735397i \(-0.263004\pi\)
0.677636 + 0.735397i \(0.263004\pi\)
\(654\) 71.0799 2.77944
\(655\) 4.12676i 0.161246i
\(656\) − 1.45152i − 0.0566722i
\(657\) − 18.5491i − 0.723667i
\(658\) − 44.9114i − 1.75083i
\(659\) −6.69852 −0.260937 −0.130469 0.991452i \(-0.541648\pi\)
−0.130469 + 0.991452i \(0.541648\pi\)
\(660\) −12.7799 −0.497456
\(661\) 6.02758i 0.234446i 0.993106 + 0.117223i \(0.0373992\pi\)
−0.993106 + 0.117223i \(0.962601\pi\)
\(662\) 71.8604 2.79294
\(663\) 0 0
\(664\) 28.4492 1.10404
\(665\) 10.9208i 0.423489i
\(666\) 9.43412 0.365565
\(667\) −36.0983 −1.39773
\(668\) 60.8479i 2.35428i
\(669\) 60.3228i 2.33222i
\(670\) − 20.2006i − 0.780417i
\(671\) 4.48990i 0.173330i
\(672\) −12.9296 −0.498771
\(673\) −23.3568 −0.900338 −0.450169 0.892943i \(-0.648636\pi\)
−0.450169 + 0.892943i \(0.648636\pi\)
\(674\) 29.3205i 1.12938i
\(675\) 5.62828 0.216633
\(676\) 0 0
\(677\) −45.4042 −1.74503 −0.872513 0.488590i \(-0.837511\pi\)
−0.872513 + 0.488590i \(0.837511\pi\)
\(678\) − 47.1802i − 1.81195i
\(679\) −8.04295 −0.308660
\(680\) 3.54248 0.135848
\(681\) 44.3408i 1.69914i
\(682\) 3.90720i 0.149615i
\(683\) − 25.4978i − 0.975645i −0.872943 0.487823i \(-0.837791\pi\)
0.872943 0.487823i \(-0.162209\pi\)
\(684\) 120.945i 4.62445i
\(685\) 20.1096 0.768350
\(686\) 49.3047 1.88246
\(687\) 21.5429i 0.821913i
\(688\) −3.45214 −0.131612
\(689\) 0 0
\(690\) 26.9327 1.02531
\(691\) 6.59630i 0.250935i 0.992098 + 0.125468i \(0.0400431\pi\)
−0.992098 + 0.125468i \(0.959957\pi\)
\(692\) −102.937 −3.91306
\(693\) −10.1688 −0.386279
\(694\) − 4.74090i − 0.179962i
\(695\) 20.8253i 0.789948i
\(696\) − 148.578i − 5.63185i
\(697\) 0.170754i 0.00646778i
\(698\) −25.6266 −0.969981
\(699\) −53.8653 −2.03737
\(700\) 8.05440i 0.304428i
\(701\) −29.2474 −1.10466 −0.552329 0.833626i \(-0.686261\pi\)
−0.552329 + 0.833626i \(0.686261\pi\)
\(702\) 0 0
\(703\) 4.34174 0.163752
\(704\) 5.17932i 0.195203i
\(705\) −26.7027 −1.00568
\(706\) −1.99787 −0.0751910
\(707\) 29.0499i 1.09253i
\(708\) 8.85584i 0.332823i
\(709\) 10.9335i 0.410614i 0.978698 + 0.205307i \(0.0658194\pi\)
−0.978698 + 0.205307i \(0.934181\pi\)
\(710\) − 24.3683i − 0.914527i
\(711\) −46.5131 −1.74438
\(712\) 69.9216 2.62042
\(713\) − 5.58973i − 0.209337i
\(714\) 8.56490 0.320533
\(715\) 0 0
\(716\) 16.0152 0.598516
\(717\) 36.0520i 1.34639i
\(718\) 20.2953 0.757412
\(719\) −16.0598 −0.598929 −0.299464 0.954107i \(-0.596808\pi\)
−0.299464 + 0.954107i \(0.596808\pi\)
\(720\) 27.0370i 1.00761i
\(721\) 25.7716i 0.959786i
\(722\) 34.5788i 1.28689i
\(723\) − 73.2966i − 2.72593i
\(724\) −35.8833 −1.33359
\(725\) −9.45512 −0.351154
\(726\) 69.5310i 2.58054i
\(727\) −51.3754 −1.90541 −0.952704 0.303900i \(-0.901711\pi\)
−0.952704 + 0.303900i \(0.901711\pi\)
\(728\) 0 0
\(729\) −43.0532 −1.59456
\(730\) − 9.27453i − 0.343266i
\(731\) 0.406104 0.0150203
\(732\) 50.1754 1.85454
\(733\) 9.82358i 0.362842i 0.983406 + 0.181421i \(0.0580697\pi\)
−0.983406 + 0.181421i \(0.941930\pi\)
\(734\) 51.2240i 1.89071i
\(735\) − 9.52691i − 0.351406i
\(736\) 9.16560i 0.337848i
\(737\) 8.65648 0.318866
\(738\) −3.33734 −0.122849
\(739\) 49.0842i 1.80559i 0.430068 + 0.902797i \(0.358490\pi\)
−0.430068 + 0.902797i \(0.641510\pi\)
\(740\) 3.20216 0.117714
\(741\) 0 0
\(742\) −33.2386 −1.22023
\(743\) − 40.8375i − 1.49818i −0.662467 0.749091i \(-0.730490\pi\)
0.662467 0.749091i \(-0.269510\pi\)
\(744\) 23.0070 0.843478
\(745\) 13.3678 0.489757
\(746\) 44.4346i 1.62687i
\(747\) − 25.5429i − 0.934566i
\(748\) 2.88101i 0.105340i
\(749\) − 14.0276i − 0.512557i
\(750\) 7.05440 0.257590
\(751\) 2.72680 0.0995024 0.0497512 0.998762i \(-0.484157\pi\)
0.0497512 + 0.998762i \(0.484157\pi\)
\(752\) − 51.1710i − 1.86601i
\(753\) 21.5175 0.784142
\(754\) 0 0
\(755\) −18.2984 −0.665946
\(756\) 45.3324i 1.64872i
\(757\) 14.8060 0.538134 0.269067 0.963121i \(-0.413285\pi\)
0.269067 + 0.963121i \(0.413285\pi\)
\(758\) −5.10468 −0.185410
\(759\) 11.5413i 0.418924i
\(760\) 31.8638i 1.15582i
\(761\) − 11.3689i − 0.412122i −0.978539 0.206061i \(-0.933935\pi\)
0.978539 0.206061i \(-0.0660646\pi\)
\(762\) − 10.5075i − 0.380647i
\(763\) 19.1969 0.694973
\(764\) 23.0110 0.832510
\(765\) − 3.18059i − 0.114994i
\(766\) 19.7275 0.712784
\(767\) 0 0
\(768\) 91.7512 3.31078
\(769\) 21.0562i 0.759307i 0.925129 + 0.379654i \(0.123957\pi\)
−0.925129 + 0.379654i \(0.876043\pi\)
\(770\) −5.08438 −0.183228
\(771\) 0.949203 0.0341847
\(772\) − 51.3970i − 1.84982i
\(773\) − 14.0829i − 0.506526i −0.967397 0.253263i \(-0.918496\pi\)
0.967397 0.253263i \(-0.0815038\pi\)
\(774\) 7.93719i 0.285296i
\(775\) − 1.46410i − 0.0525921i
\(776\) −23.4671 −0.842421
\(777\) 4.07941 0.146348
\(778\) − 22.9879i − 0.824156i
\(779\) −1.53590 −0.0550293
\(780\) 0 0
\(781\) 10.4425 0.373660
\(782\) − 6.07151i − 0.217117i
\(783\) −53.2160 −1.90179
\(784\) 18.2566 0.652023
\(785\) 2.42229i 0.0864552i
\(786\) 29.1118i 1.03838i
\(787\) 33.0242i 1.17719i 0.808429 + 0.588593i \(0.200318\pi\)
−0.808429 + 0.588593i \(0.799682\pi\)
\(788\) − 18.5095i − 0.659374i
\(789\) 15.1968 0.541020
\(790\) −23.2566 −0.827431
\(791\) − 12.7422i − 0.453060i
\(792\) −29.6697 −1.05427
\(793\) 0 0
\(794\) −15.8574 −0.562758
\(795\) 19.7625i 0.700903i
\(796\) 88.0909 3.12230
\(797\) 16.9416 0.600102 0.300051 0.953923i \(-0.402996\pi\)
0.300051 + 0.953923i \(0.402996\pi\)
\(798\) 77.0395i 2.72717i
\(799\) 6.01967i 0.212961i
\(800\) 2.40072i 0.0848783i
\(801\) − 62.7787i − 2.21817i
\(802\) 10.4043 0.367387
\(803\) 3.97437 0.140253
\(804\) − 96.7378i − 3.41168i
\(805\) 7.27382 0.256369
\(806\) 0 0
\(807\) 3.70425 0.130396
\(808\) 84.7596i 2.98183i
\(809\) −51.7635 −1.81991 −0.909954 0.414708i \(-0.863884\pi\)
−0.909954 + 0.414708i \(0.863884\pi\)
\(810\) 2.33876 0.0821756
\(811\) − 22.6699i − 0.796047i −0.917375 0.398023i \(-0.869696\pi\)
0.917375 0.398023i \(-0.130304\pi\)
\(812\) − 76.1553i − 2.67253i
\(813\) 32.9194i 1.15453i
\(814\) 2.02138i 0.0708494i
\(815\) −15.9829 −0.559856
\(816\) 9.75864 0.341621
\(817\) 3.65283i 0.127796i
\(818\) −25.3745 −0.887198
\(819\) 0 0
\(820\) −1.13277 −0.0395581
\(821\) − 28.6631i − 1.00035i −0.865925 0.500174i \(-0.833269\pi\)
0.865925 0.500174i \(-0.166731\pi\)
\(822\) 141.861 4.94799
\(823\) 25.8327 0.900472 0.450236 0.892910i \(-0.351340\pi\)
0.450236 + 0.892910i \(0.351340\pi\)
\(824\) 75.1946i 2.61953i
\(825\) 3.02299i 0.105247i
\(826\) 3.52323i 0.122589i
\(827\) − 16.0820i − 0.559227i −0.960113 0.279613i \(-0.909794\pi\)
0.960113 0.279613i \(-0.0902063\pi\)
\(828\) 80.5560 2.79951
\(829\) 22.5818 0.784298 0.392149 0.919902i \(-0.371732\pi\)
0.392149 + 0.919902i \(0.371732\pi\)
\(830\) − 12.7715i − 0.443304i
\(831\) −57.4304 −1.99224
\(832\) 0 0
\(833\) −2.14768 −0.0744128
\(834\) 146.910i 5.08707i
\(835\) 14.3932 0.498096
\(836\) −25.9141 −0.896257
\(837\) − 8.24037i − 0.284829i
\(838\) − 71.3487i − 2.46470i
\(839\) − 17.8440i − 0.616042i −0.951380 0.308021i \(-0.900333\pi\)
0.951380 0.308021i \(-0.0996667\pi\)
\(840\) 29.9387i 1.03298i
\(841\) 60.3992 2.08273
\(842\) −5.03586 −0.173547
\(843\) 33.5669i 1.15611i
\(844\) 45.0390 1.55031
\(845\) 0 0
\(846\) −117.653 −4.04498
\(847\) 18.7785i 0.645239i
\(848\) −37.8713 −1.30050
\(849\) −64.0339 −2.19764
\(850\) − 1.59030i − 0.0545467i
\(851\) − 2.89183i − 0.0991306i
\(852\) − 116.696i − 3.99795i
\(853\) 19.7936i 0.677720i 0.940837 + 0.338860i \(0.110041\pi\)
−0.940837 + 0.338860i \(0.889959\pi\)
\(854\) 19.9619 0.683083
\(855\) 28.6088 0.978399
\(856\) − 40.9286i − 1.39891i
\(857\) 11.7302 0.400696 0.200348 0.979725i \(-0.435793\pi\)
0.200348 + 0.979725i \(0.435793\pi\)
\(858\) 0 0
\(859\) 5.37452 0.183376 0.0916882 0.995788i \(-0.470774\pi\)
0.0916882 + 0.995788i \(0.470774\pi\)
\(860\) 2.69407i 0.0918669i
\(861\) −1.44310 −0.0491808
\(862\) 51.4382 1.75199
\(863\) 25.3234i 0.862017i 0.902348 + 0.431008i \(0.141842\pi\)
−0.902348 + 0.431008i \(0.858158\pi\)
\(864\) 13.5119i 0.459685i
\(865\) 24.3489i 0.827889i
\(866\) 73.4567i 2.49616i
\(867\) 46.9083 1.59309
\(868\) 11.7925 0.400262
\(869\) − 9.96603i − 0.338075i
\(870\) −66.7001 −2.26135
\(871\) 0 0
\(872\) 56.0112 1.89678
\(873\) 21.0698i 0.713106i
\(874\) 54.6120 1.84728
\(875\) 1.90521 0.0644079
\(876\) − 44.4144i − 1.50062i
\(877\) − 20.6915i − 0.698703i −0.936992 0.349352i \(-0.886402\pi\)
0.936992 0.349352i \(-0.113598\pi\)
\(878\) 42.3040i 1.42769i
\(879\) − 52.6151i − 1.77466i
\(880\) −5.79302 −0.195283
\(881\) −48.3993 −1.63061 −0.815307 0.579029i \(-0.803432\pi\)
−0.815307 + 0.579029i \(0.803432\pi\)
\(882\) − 41.9758i − 1.41340i
\(883\) −45.8550 −1.54314 −0.771572 0.636142i \(-0.780529\pi\)
−0.771572 + 0.636142i \(0.780529\pi\)
\(884\) 0 0
\(885\) 2.09479 0.0704155
\(886\) 60.2413i 2.02385i
\(887\) −1.08234 −0.0363413 −0.0181707 0.999835i \(-0.505784\pi\)
−0.0181707 + 0.999835i \(0.505784\pi\)
\(888\) 11.9026 0.399426
\(889\) − 2.83781i − 0.0951772i
\(890\) − 31.3893i − 1.05217i
\(891\) 1.00222i 0.0335756i
\(892\) 90.2133i 3.02056i
\(893\) −54.1457 −1.81192
\(894\) 94.3015 3.15391
\(895\) − 3.78829i − 0.126628i
\(896\) 32.1749 1.07489
\(897\) 0 0
\(898\) −52.0637 −1.73739
\(899\) 13.8433i 0.461698i
\(900\) 21.0998 0.703327
\(901\) 4.45512 0.148421
\(902\) − 0.715068i − 0.0238092i
\(903\) 3.43213i 0.114214i
\(904\) − 37.1782i − 1.23653i
\(905\) 8.48794i 0.282149i
\(906\) −129.084 −4.28853
\(907\) 45.5307 1.51182 0.755910 0.654675i \(-0.227195\pi\)
0.755910 + 0.654675i \(0.227195\pi\)
\(908\) 66.3120i 2.20064i
\(909\) 76.1009 2.52411
\(910\) 0 0
\(911\) 39.7417 1.31670 0.658350 0.752712i \(-0.271255\pi\)
0.658350 + 0.752712i \(0.271255\pi\)
\(912\) 87.7770i 2.90659i
\(913\) 5.47290 0.181126
\(914\) 76.2774 2.52303
\(915\) − 11.8687i − 0.392365i
\(916\) 32.2176i 1.06450i
\(917\) 7.86236i 0.259638i
\(918\) − 8.95062i − 0.295415i
\(919\) 46.9938 1.55018 0.775091 0.631850i \(-0.217704\pi\)
0.775091 + 0.631850i \(0.217704\pi\)
\(920\) 21.2230 0.699702
\(921\) − 8.89822i − 0.293206i
\(922\) −11.6745 −0.384481
\(923\) 0 0
\(924\) −24.3484 −0.801002
\(925\) − 0.757449i − 0.0249048i
\(926\) −34.9399 −1.14820
\(927\) 67.5130 2.21742
\(928\) − 22.6991i − 0.745134i
\(929\) 15.2213i 0.499395i 0.968324 + 0.249698i \(0.0803312\pi\)
−0.968324 + 0.249698i \(0.919669\pi\)
\(930\) − 10.3284i − 0.338680i
\(931\) − 19.3180i − 0.633121i
\(932\) −80.5560 −2.63870
\(933\) −8.99102 −0.294353
\(934\) − 17.4313i − 0.570369i
\(935\) 0.681482 0.0222869
\(936\) 0 0
\(937\) 6.07285 0.198392 0.0991958 0.995068i \(-0.468373\pi\)
0.0991958 + 0.995068i \(0.468373\pi\)
\(938\) − 38.4864i − 1.25663i
\(939\) −99.9382 −3.26136
\(940\) −39.9341 −1.30251
\(941\) 0.0496576i 0.00161879i 1.00000 0.000809396i \(0.000257639\pi\)
−1.00000 0.000809396i \(0.999742\pi\)
\(942\) 17.0878i 0.556750i
\(943\) 1.02299i 0.0333132i
\(944\) 4.01429i 0.130654i
\(945\) 10.7231 0.348821
\(946\) −1.70064 −0.0552927
\(947\) − 18.6581i − 0.606308i −0.952942 0.303154i \(-0.901960\pi\)
0.952942 0.303154i \(-0.0980396\pi\)
\(948\) −111.372 −3.61720
\(949\) 0 0
\(950\) 14.3044 0.464095
\(951\) − 38.5459i − 1.24994i
\(952\) 6.74917 0.218742
\(953\) 1.52953 0.0495463 0.0247731 0.999693i \(-0.492114\pi\)
0.0247731 + 0.999693i \(0.492114\pi\)
\(954\) 87.0739i 2.81912i
\(955\) − 5.44310i − 0.176135i
\(956\) 53.9161i 1.74377i
\(957\) − 28.5827i − 0.923948i
\(958\) −40.6487 −1.31330
\(959\) 38.3131 1.23720
\(960\) − 13.6911i − 0.441878i
\(961\) 28.8564 0.930852
\(962\) 0 0
\(963\) −36.7475 −1.18417
\(964\) − 109.616i − 3.53048i
\(965\) −12.1576 −0.391367
\(966\) 51.3124 1.65095
\(967\) − 32.1716i − 1.03457i −0.855813 0.517285i \(-0.826943\pi\)
0.855813 0.517285i \(-0.173057\pi\)
\(968\) 54.7907i 1.76104i
\(969\) − 10.3259i − 0.331717i
\(970\) 10.5349i 0.338256i
\(971\) −17.2541 −0.553710 −0.276855 0.960912i \(-0.589292\pi\)
−0.276855 + 0.960912i \(0.589292\pi\)
\(972\) −60.1816 −1.93033
\(973\) 39.6766i 1.27197i
\(974\) −50.0122 −1.60249
\(975\) 0 0
\(976\) 22.7442 0.728023
\(977\) − 15.7228i − 0.503018i −0.967855 0.251509i \(-0.919073\pi\)
0.967855 0.251509i \(-0.0809268\pi\)
\(978\) −112.750 −3.60533
\(979\) 13.4511 0.429900
\(980\) − 14.2476i − 0.455122i
\(981\) − 50.2893i − 1.60561i
\(982\) − 39.4248i − 1.25810i
\(983\) 38.5356i 1.22910i 0.788880 + 0.614548i \(0.210662\pi\)
−0.788880 + 0.614548i \(0.789338\pi\)
\(984\) −4.21058 −0.134228
\(985\) −4.37830 −0.139504
\(986\) 15.0364i 0.478857i
\(987\) −50.8743 −1.61935
\(988\) 0 0
\(989\) 2.43298 0.0773642
\(990\) 13.3194i 0.423317i
\(991\) 8.59143 0.272916 0.136458 0.990646i \(-0.456428\pi\)
0.136458 + 0.990646i \(0.456428\pi\)
\(992\) 3.51490 0.111598
\(993\) − 81.4014i − 2.58320i
\(994\) − 46.4268i − 1.47257i
\(995\) − 20.8373i − 0.660587i
\(996\) − 61.1606i − 1.93795i
\(997\) −20.5374 −0.650425 −0.325213 0.945641i \(-0.605436\pi\)
−0.325213 + 0.945641i \(0.605436\pi\)
\(998\) −3.11069 −0.0984670
\(999\) − 4.26313i − 0.134880i
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 845.2.c.g.506.1 8
13.2 odd 12 845.2.e.n.191.4 8
13.3 even 3 845.2.m.g.316.4 8
13.4 even 6 845.2.m.g.361.4 8
13.5 odd 4 845.2.a.l.1.1 4
13.6 odd 12 845.2.e.n.146.4 8
13.7 odd 12 845.2.e.m.146.1 8
13.8 odd 4 845.2.a.m.1.4 4
13.9 even 3 65.2.m.a.36.1 8
13.10 even 6 65.2.m.a.56.1 yes 8
13.11 odd 12 845.2.e.m.191.1 8
13.12 even 2 inner 845.2.c.g.506.8 8
39.5 even 4 7605.2.a.cj.1.4 4
39.8 even 4 7605.2.a.cf.1.1 4
39.23 odd 6 585.2.bu.c.316.4 8
39.35 odd 6 585.2.bu.c.361.4 8
52.23 odd 6 1040.2.da.b.641.1 8
52.35 odd 6 1040.2.da.b.881.1 8
65.9 even 6 325.2.n.d.101.4 8
65.22 odd 12 325.2.m.b.49.1 8
65.23 odd 12 325.2.m.b.199.1 8
65.34 odd 4 4225.2.a.bi.1.1 4
65.44 odd 4 4225.2.a.bl.1.4 4
65.48 odd 12 325.2.m.c.49.4 8
65.49 even 6 325.2.n.d.251.4 8
65.62 odd 12 325.2.m.c.199.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.m.a.36.1 8 13.9 even 3
65.2.m.a.56.1 yes 8 13.10 even 6
325.2.m.b.49.1 8 65.22 odd 12
325.2.m.b.199.1 8 65.23 odd 12
325.2.m.c.49.4 8 65.48 odd 12
325.2.m.c.199.4 8 65.62 odd 12
325.2.n.d.101.4 8 65.9 even 6
325.2.n.d.251.4 8 65.49 even 6
585.2.bu.c.316.4 8 39.23 odd 6
585.2.bu.c.361.4 8 39.35 odd 6
845.2.a.l.1.1 4 13.5 odd 4
845.2.a.m.1.4 4 13.8 odd 4
845.2.c.g.506.1 8 1.1 even 1 trivial
845.2.c.g.506.8 8 13.12 even 2 inner
845.2.e.m.146.1 8 13.7 odd 12
845.2.e.m.191.1 8 13.11 odd 12
845.2.e.n.146.4 8 13.6 odd 12
845.2.e.n.191.4 8 13.2 odd 12
845.2.m.g.316.4 8 13.3 even 3
845.2.m.g.361.4 8 13.4 even 6
1040.2.da.b.641.1 8 52.23 odd 6
1040.2.da.b.881.1 8 52.35 odd 6
4225.2.a.bi.1.1 4 65.34 odd 4
4225.2.a.bl.1.4 4 65.44 odd 4
7605.2.a.cf.1.1 4 39.8 even 4
7605.2.a.cj.1.4 4 39.5 even 4