Properties

Label 845.2.a.l.1.1
Level $845$
Weight $2$
Character 845.1
Self dual yes
Analytic conductor $6.747$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.74735897080\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4752.1
Defining polynomial: \(x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.49551\) of defining polynomial
Character \(\chi\) \(=\) 845.1

$q$-expansion

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.49551 −1.76459 −0.882295 0.470696i \(-0.844003\pi\)
−0.882295 + 0.470696i \(0.844003\pi\)
\(3\) −2.82684 −1.63208 −0.816038 0.577998i \(-0.803834\pi\)
−0.816038 + 0.577998i \(0.803834\pi\)
\(4\) 4.22756 2.11378
\(5\) 1.00000 0.447214
\(6\) 7.05440 2.87995
\(7\) −1.90521 −0.720103 −0.360051 0.932933i \(-0.617241\pi\)
−0.360051 + 0.932933i \(0.617241\pi\)
\(8\) −5.55889 −1.96536
\(9\) 4.99102 1.66367
\(10\) −2.49551 −0.789149
\(11\) −1.06939 −0.322433 −0.161217 0.986919i \(-0.551542\pi\)
−0.161217 + 0.986919i \(0.551542\pi\)
\(12\) −11.9506 −3.44985
\(13\) 0 0
\(14\) 4.75447 1.27069
\(15\) −2.82684 −0.729887
\(16\) 5.41713 1.35428
\(17\) 0.637263 0.154559 0.0772795 0.997009i \(-0.475377\pi\)
0.0772795 + 0.997009i \(0.475377\pi\)
\(18\) −12.4551 −2.93570
\(19\) −5.73205 −1.31502 −0.657511 0.753445i \(-0.728391\pi\)
−0.657511 + 0.753445i \(0.728391\pi\)
\(20\) 4.22756 0.945311
\(21\) 5.38573 1.17526
\(22\) 2.66867 0.568962
\(23\) 3.81785 0.796078 0.398039 0.917369i \(-0.369691\pi\)
0.398039 + 0.917369i \(0.369691\pi\)
\(24\) 15.7141 3.20762
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.62828 −1.08316
\(28\) −8.05440 −1.52214
\(29\) 9.45512 1.75577 0.877886 0.478870i \(-0.158954\pi\)
0.877886 + 0.478870i \(0.158954\pi\)
\(30\) 7.05440 1.28795
\(31\) 1.46410 0.262960 0.131480 0.991319i \(-0.458027\pi\)
0.131480 + 0.991319i \(0.458027\pi\)
\(32\) −2.40072 −0.424391
\(33\) 3.02299 0.526235
\(34\) −1.59030 −0.272733
\(35\) −1.90521 −0.322040
\(36\) 21.0998 3.51663
\(37\) −0.757449 −0.124524 −0.0622619 0.998060i \(-0.519831\pi\)
−0.0622619 + 0.998060i \(0.519831\pi\)
\(38\) 14.3044 2.32048
\(39\) 0 0
\(40\) −5.55889 −0.878938
\(41\) −0.267949 −0.0418466 −0.0209233 0.999781i \(-0.506661\pi\)
−0.0209233 + 0.999781i \(0.506661\pi\)
\(42\) −13.4401 −2.07386
\(43\) 0.637263 0.0971817 0.0485909 0.998819i \(-0.484527\pi\)
0.0485909 + 0.998819i \(0.484527\pi\)
\(44\) −4.52091 −0.681552
\(45\) 4.99102 0.744017
\(46\) −9.52748 −1.40475
\(47\) 9.44613 1.37786 0.688930 0.724828i \(-0.258081\pi\)
0.688930 + 0.724828i \(0.258081\pi\)
\(48\) −15.3134 −2.21029
\(49\) −3.37017 −0.481452
\(50\) −2.49551 −0.352918
\(51\) −1.80144 −0.252252
\(52\) 0 0
\(53\) −6.99102 −0.960290 −0.480145 0.877189i \(-0.659416\pi\)
−0.480145 + 0.877189i \(0.659416\pi\)
\(54\) 14.0454 1.91134
\(55\) −1.06939 −0.144196
\(56\) 10.5909 1.41526
\(57\) 16.2036 2.14622
\(58\) −23.5953 −3.09822
\(59\) −0.741035 −0.0964746 −0.0482373 0.998836i \(-0.515360\pi\)
−0.0482373 + 0.998836i \(0.515360\pi\)
\(60\) −11.9506 −1.54282
\(61\) 4.19856 0.537571 0.268785 0.963200i \(-0.413378\pi\)
0.268785 + 0.963200i \(0.413378\pi\)
\(62\) −3.65368 −0.464017
\(63\) −9.50894 −1.19801
\(64\) −4.84325 −0.605406
\(65\) 0 0
\(66\) −7.54390 −0.928589
\(67\) −8.09479 −0.988936 −0.494468 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(68\) 2.69407 0.326704
\(69\) −10.7925 −1.29926
\(70\) 4.75447 0.568268
\(71\) −9.76488 −1.15888 −0.579439 0.815016i \(-0.696728\pi\)
−0.579439 + 0.815016i \(0.696728\pi\)
\(72\) −27.7445 −3.26972
\(73\) 3.71649 0.434982 0.217491 0.976062i \(-0.430213\pi\)
0.217491 + 0.976062i \(0.430213\pi\)
\(74\) 1.89022 0.219734
\(75\) −2.82684 −0.326415
\(76\) −24.2326 −2.77967
\(77\) 2.03741 0.232185
\(78\) 0 0
\(79\) −9.31937 −1.04851 −0.524255 0.851561i \(-0.675656\pi\)
−0.524255 + 0.851561i \(0.675656\pi\)
\(80\) 5.41713 0.605654
\(81\) 0.937188 0.104132
\(82\) 0.668669 0.0738422
\(83\) −5.11778 −0.561749 −0.280875 0.959744i \(-0.590624\pi\)
−0.280875 + 0.959744i \(0.590624\pi\)
\(84\) 22.7685 2.48424
\(85\) 0.637263 0.0691209
\(86\) −1.59030 −0.171486
\(87\) −26.7281 −2.86555
\(88\) 5.94462 0.633698
\(89\) 12.5783 1.33330 0.666650 0.745371i \(-0.267727\pi\)
0.666650 + 0.745371i \(0.267727\pi\)
\(90\) −12.4551 −1.31288
\(91\) 0 0
\(92\) 16.1402 1.68273
\(93\) −4.13878 −0.429171
\(94\) −23.5729 −2.43136
\(95\) −5.73205 −0.588096
\(96\) 6.78645 0.692639
\(97\) 4.22155 0.428634 0.214317 0.976764i \(-0.431248\pi\)
0.214317 + 0.976764i \(0.431248\pi\)
\(98\) 8.41027 0.849566
\(99\) −5.33734 −0.536423
\(100\) 4.22756 0.422756
\(101\) −15.2476 −1.51719 −0.758595 0.651562i \(-0.774114\pi\)
−0.758595 + 0.651562i \(0.774114\pi\)
\(102\) 4.49551 0.445122
\(103\) −13.5269 −1.33285 −0.666423 0.745574i \(-0.732176\pi\)
−0.666423 + 0.745574i \(0.732176\pi\)
\(104\) 0 0
\(105\) 5.38573 0.525593
\(106\) 17.4461 1.69452
\(107\) −7.36274 −0.711783 −0.355891 0.934527i \(-0.615823\pi\)
−0.355891 + 0.934527i \(0.615823\pi\)
\(108\) −23.7939 −2.28957
\(109\) −10.0760 −0.965103 −0.482551 0.875868i \(-0.660290\pi\)
−0.482551 + 0.875868i \(0.660290\pi\)
\(110\) 2.66867 0.254448
\(111\) 2.14119 0.203232
\(112\) −10.3208 −0.975223
\(113\) −6.68806 −0.629160 −0.314580 0.949231i \(-0.601864\pi\)
−0.314580 + 0.949231i \(0.601864\pi\)
\(114\) −40.4362 −3.78719
\(115\) 3.81785 0.356017
\(116\) 39.9721 3.71131
\(117\) 0 0
\(118\) 1.84926 0.170238
\(119\) −1.21412 −0.111298
\(120\) 15.7141 1.43449
\(121\) −9.85641 −0.896037
\(122\) −10.4775 −0.948592
\(123\) 0.757449 0.0682969
\(124\) 6.18958 0.555840
\(125\) 1.00000 0.0894427
\(126\) 23.7296 2.11400
\(127\) 1.48950 0.132172 0.0660859 0.997814i \(-0.478949\pi\)
0.0660859 + 0.997814i \(0.478949\pi\)
\(128\) 16.8878 1.49269
\(129\) −1.80144 −0.158608
\(130\) 0 0
\(131\) 4.12676 0.360557 0.180278 0.983616i \(-0.442300\pi\)
0.180278 + 0.983616i \(0.442300\pi\)
\(132\) 12.7799 1.11234
\(133\) 10.9208 0.946951
\(134\) 20.2006 1.74507
\(135\) −5.62828 −0.484405
\(136\) −3.54248 −0.303765
\(137\) 20.1096 1.71808 0.859041 0.511906i \(-0.171060\pi\)
0.859041 + 0.511906i \(0.171060\pi\)
\(138\) 26.9327 2.29266
\(139\) 20.8253 1.76638 0.883189 0.469018i \(-0.155392\pi\)
0.883189 + 0.469018i \(0.155392\pi\)
\(140\) −8.05440 −0.680721
\(141\) −26.7027 −2.24877
\(142\) 24.3683 2.04494
\(143\) 0 0
\(144\) 27.0370 2.25308
\(145\) 9.45512 0.785205
\(146\) −9.27453 −0.767565
\(147\) 9.52691 0.785767
\(148\) −3.20216 −0.263216
\(149\) −13.3678 −1.09513 −0.547565 0.836763i \(-0.684445\pi\)
−0.547565 + 0.836763i \(0.684445\pi\)
\(150\) 7.05440 0.575989
\(151\) −18.2984 −1.48910 −0.744550 0.667567i \(-0.767336\pi\)
−0.744550 + 0.667567i \(0.767336\pi\)
\(152\) 31.8638 2.58450
\(153\) 3.18059 0.257136
\(154\) −5.08438 −0.409711
\(155\) 1.46410 0.117599
\(156\) 0 0
\(157\) 2.42229 0.193320 0.0966599 0.995317i \(-0.469184\pi\)
0.0966599 + 0.995317i \(0.469184\pi\)
\(158\) 23.2566 1.85019
\(159\) 19.7625 1.56727
\(160\) −2.40072 −0.189794
\(161\) −7.27382 −0.573258
\(162\) −2.33876 −0.183750
\(163\) −15.9829 −1.25188 −0.625938 0.779873i \(-0.715284\pi\)
−0.625938 + 0.779873i \(0.715284\pi\)
\(164\) −1.13277 −0.0884545
\(165\) 3.02299 0.235340
\(166\) 12.7715 0.991257
\(167\) 14.3932 1.11378 0.556888 0.830588i \(-0.311995\pi\)
0.556888 + 0.830588i \(0.311995\pi\)
\(168\) −29.9387 −2.30982
\(169\) 0 0
\(170\) −1.59030 −0.121970
\(171\) −28.6088 −2.18777
\(172\) 2.69407 0.205421
\(173\) −24.3489 −1.85122 −0.925608 0.378484i \(-0.876445\pi\)
−0.925608 + 0.378484i \(0.876445\pi\)
\(174\) 66.7001 5.05653
\(175\) −1.90521 −0.144021
\(176\) −5.79302 −0.436666
\(177\) 2.09479 0.157454
\(178\) −31.3893 −2.35273
\(179\) 3.78829 0.283150 0.141575 0.989928i \(-0.454783\pi\)
0.141575 + 0.989928i \(0.454783\pi\)
\(180\) 21.0998 1.57269
\(181\) −8.48794 −0.630904 −0.315452 0.948942i \(-0.602156\pi\)
−0.315452 + 0.948942i \(0.602156\pi\)
\(182\) 0 0
\(183\) −11.8687 −0.877356
\(184\) −21.2230 −1.56458
\(185\) −0.757449 −0.0556888
\(186\) 10.3284 0.757312
\(187\) −0.681482 −0.0498349
\(188\) 39.9341 2.91249
\(189\) 10.7231 0.779988
\(190\) 14.3044 1.03775
\(191\) −5.44310 −0.393849 −0.196924 0.980419i \(-0.563095\pi\)
−0.196924 + 0.980419i \(0.563095\pi\)
\(192\) 13.6911 0.988069
\(193\) −12.1576 −0.875123 −0.437562 0.899188i \(-0.644158\pi\)
−0.437562 + 0.899188i \(0.644158\pi\)
\(194\) −10.5349 −0.756363
\(195\) 0 0
\(196\) −14.2476 −1.01768
\(197\) 4.37830 0.311941 0.155970 0.987762i \(-0.450150\pi\)
0.155970 + 0.987762i \(0.450150\pi\)
\(198\) 13.3194 0.946566
\(199\) 20.8373 1.47712 0.738558 0.674189i \(-0.235507\pi\)
0.738558 + 0.674189i \(0.235507\pi\)
\(200\) −5.55889 −0.393073
\(201\) 22.8827 1.61402
\(202\) 38.0504 2.67722
\(203\) −18.0140 −1.26434
\(204\) −7.61569 −0.533205
\(205\) −0.267949 −0.0187144
\(206\) 33.7565 2.35193
\(207\) 19.0550 1.32441
\(208\) 0 0
\(209\) 6.12979 0.424007
\(210\) −13.4401 −0.927457
\(211\) −10.6537 −0.733429 −0.366715 0.930333i \(-0.619517\pi\)
−0.366715 + 0.930333i \(0.619517\pi\)
\(212\) −29.5549 −2.02984
\(213\) 27.6037 1.89138
\(214\) 18.3738 1.25600
\(215\) 0.637263 0.0434610
\(216\) 31.2870 2.12881
\(217\) −2.78942 −0.189358
\(218\) 25.1447 1.70301
\(219\) −10.5059 −0.709924
\(220\) −4.52091 −0.304799
\(221\) 0 0
\(222\) −5.34335 −0.358622
\(223\) −21.3393 −1.42899 −0.714494 0.699642i \(-0.753343\pi\)
−0.714494 + 0.699642i \(0.753343\pi\)
\(224\) 4.57388 0.305605
\(225\) 4.99102 0.332734
\(226\) 16.6901 1.11021
\(227\) −15.6857 −1.04109 −0.520547 0.853833i \(-0.674272\pi\)
−0.520547 + 0.853833i \(0.674272\pi\)
\(228\) 68.5016 4.53663
\(229\) 7.62085 0.503600 0.251800 0.967779i \(-0.418977\pi\)
0.251800 + 0.967779i \(0.418977\pi\)
\(230\) −9.52748 −0.628224
\(231\) −5.75944 −0.378943
\(232\) −52.5599 −3.45073
\(233\) −19.0550 −1.24833 −0.624166 0.781292i \(-0.714561\pi\)
−0.624166 + 0.781292i \(0.714561\pi\)
\(234\) 0 0
\(235\) 9.44613 0.616198
\(236\) −3.13277 −0.203926
\(237\) 26.3444 1.71125
\(238\) 3.02985 0.196396
\(239\) −12.7535 −0.824954 −0.412477 0.910968i \(-0.635336\pi\)
−0.412477 + 0.910968i \(0.635336\pi\)
\(240\) −15.3134 −0.988473
\(241\) −25.9288 −1.67022 −0.835111 0.550081i \(-0.814597\pi\)
−0.835111 + 0.550081i \(0.814597\pi\)
\(242\) 24.5967 1.58114
\(243\) 14.2356 0.913211
\(244\) 17.7497 1.13631
\(245\) −3.37017 −0.215312
\(246\) −1.89022 −0.120516
\(247\) 0 0
\(248\) −8.13878 −0.516813
\(249\) 14.4671 0.916817
\(250\) −2.49551 −0.157830
\(251\) 7.61186 0.480457 0.240228 0.970716i \(-0.422778\pi\)
0.240228 + 0.970716i \(0.422778\pi\)
\(252\) −40.1996 −2.53234
\(253\) −4.08277 −0.256682
\(254\) −3.71706 −0.233229
\(255\) −1.80144 −0.112811
\(256\) −32.4572 −2.02857
\(257\) 0.335783 0.0209456 0.0104728 0.999945i \(-0.496666\pi\)
0.0104728 + 0.999945i \(0.496666\pi\)
\(258\) 4.49551 0.279878
\(259\) 1.44310 0.0896700
\(260\) 0 0
\(261\) 47.1906 2.92103
\(262\) −10.2984 −0.636235
\(263\) −5.37589 −0.331492 −0.165746 0.986169i \(-0.553003\pi\)
−0.165746 + 0.986169i \(0.553003\pi\)
\(264\) −16.8045 −1.03424
\(265\) −6.99102 −0.429455
\(266\) −27.2529 −1.67098
\(267\) −35.5569 −2.17605
\(268\) −34.2212 −2.09039
\(269\) −1.31038 −0.0798956 −0.0399478 0.999202i \(-0.512719\pi\)
−0.0399478 + 0.999202i \(0.512719\pi\)
\(270\) 14.0454 0.854777
\(271\) 11.6453 0.707403 0.353701 0.935358i \(-0.384923\pi\)
0.353701 + 0.935358i \(0.384923\pi\)
\(272\) 3.45214 0.209317
\(273\) 0 0
\(274\) −50.1838 −3.03171
\(275\) −1.06939 −0.0644866
\(276\) −45.6257 −2.74635
\(277\) −20.3161 −1.22068 −0.610338 0.792141i \(-0.708967\pi\)
−0.610338 + 0.792141i \(0.708967\pi\)
\(278\) −51.9697 −3.11693
\(279\) 7.30735 0.437480
\(280\) 10.5909 0.632925
\(281\) 11.8744 0.708366 0.354183 0.935176i \(-0.384759\pi\)
0.354183 + 0.935176i \(0.384759\pi\)
\(282\) 66.6368 3.96816
\(283\) −22.6521 −1.34653 −0.673264 0.739402i \(-0.735108\pi\)
−0.673264 + 0.739402i \(0.735108\pi\)
\(284\) −41.2816 −2.44961
\(285\) 16.2036 0.959817
\(286\) 0 0
\(287\) 0.510500 0.0301339
\(288\) −11.9820 −0.706048
\(289\) −16.5939 −0.976112
\(290\) −23.5953 −1.38556
\(291\) −11.9336 −0.699562
\(292\) 15.7117 0.919456
\(293\) −18.6127 −1.08737 −0.543683 0.839290i \(-0.682971\pi\)
−0.543683 + 0.839290i \(0.682971\pi\)
\(294\) −23.7745 −1.38656
\(295\) −0.741035 −0.0431448
\(296\) 4.21058 0.244735
\(297\) 6.01882 0.349247
\(298\) 33.3593 1.93245
\(299\) 0 0
\(300\) −11.9506 −0.689970
\(301\) −1.21412 −0.0699808
\(302\) 45.6637 2.62765
\(303\) 43.1024 2.47617
\(304\) −31.0513 −1.78091
\(305\) 4.19856 0.240409
\(306\) −7.93719 −0.453739
\(307\) −3.14776 −0.179652 −0.0898262 0.995957i \(-0.528631\pi\)
−0.0898262 + 0.995957i \(0.528631\pi\)
\(308\) 8.61329 0.490788
\(309\) 38.2384 2.17531
\(310\) −3.65368 −0.207515
\(311\) −3.18059 −0.180355 −0.0901774 0.995926i \(-0.528743\pi\)
−0.0901774 + 0.995926i \(0.528743\pi\)
\(312\) 0 0
\(313\) 35.3533 1.99829 0.999144 0.0413596i \(-0.0131689\pi\)
0.999144 + 0.0413596i \(0.0131689\pi\)
\(314\) −6.04484 −0.341130
\(315\) −9.50894 −0.535768
\(316\) −39.3982 −2.21632
\(317\) 13.6357 0.765858 0.382929 0.923778i \(-0.374915\pi\)
0.382929 + 0.923778i \(0.374915\pi\)
\(318\) −49.3174 −2.76558
\(319\) −10.1112 −0.566119
\(320\) −4.84325 −0.270746
\(321\) 20.8133 1.16168
\(322\) 18.1519 1.01156
\(323\) −3.65283 −0.203249
\(324\) 3.96202 0.220112
\(325\) 0 0
\(326\) 39.8854 2.20905
\(327\) 28.4831 1.57512
\(328\) 1.48950 0.0822439
\(329\) −17.9969 −0.992201
\(330\) −7.54390 −0.415278
\(331\) 28.7959 1.58277 0.791383 0.611320i \(-0.209361\pi\)
0.791383 + 0.611320i \(0.209361\pi\)
\(332\) −21.6357 −1.18741
\(333\) −3.78044 −0.207167
\(334\) −35.9182 −1.96536
\(335\) −8.09479 −0.442265
\(336\) 29.1752 1.59164
\(337\) 11.7493 0.640026 0.320013 0.947413i \(-0.396313\pi\)
0.320013 + 0.947413i \(0.396313\pi\)
\(338\) 0 0
\(339\) 18.9061 1.02684
\(340\) 2.69407 0.146106
\(341\) −1.56569 −0.0847871
\(342\) 71.3934 3.86051
\(343\) 19.7574 1.06680
\(344\) −3.54248 −0.190997
\(345\) −10.7925 −0.581046
\(346\) 60.7630 3.26664
\(347\) 1.89977 0.101985 0.0509926 0.998699i \(-0.483762\pi\)
0.0509926 + 0.998699i \(0.483762\pi\)
\(348\) −112.995 −6.05714
\(349\) 10.2691 0.549692 0.274846 0.961488i \(-0.411373\pi\)
0.274846 + 0.961488i \(0.411373\pi\)
\(350\) 4.75447 0.254137
\(351\) 0 0
\(352\) 2.56730 0.136838
\(353\) −0.800589 −0.0426110 −0.0213055 0.999773i \(-0.506782\pi\)
−0.0213055 + 0.999773i \(0.506782\pi\)
\(354\) −5.22756 −0.277842
\(355\) −9.76488 −0.518266
\(356\) 53.1756 2.81830
\(357\) 3.43213 0.181647
\(358\) −9.45370 −0.499643
\(359\) −8.13272 −0.429228 −0.214614 0.976699i \(-0.568849\pi\)
−0.214614 + 0.976699i \(0.568849\pi\)
\(360\) −27.7445 −1.46226
\(361\) 13.8564 0.729285
\(362\) 21.1817 1.11329
\(363\) 27.8625 1.46240
\(364\) 0 0
\(365\) 3.71649 0.194530
\(366\) 29.6183 1.54817
\(367\) −20.5265 −1.07147 −0.535737 0.844385i \(-0.679966\pi\)
−0.535737 + 0.844385i \(0.679966\pi\)
\(368\) 20.6818 1.07811
\(369\) −1.33734 −0.0696191
\(370\) 1.89022 0.0982679
\(371\) 13.3194 0.691507
\(372\) −17.4969 −0.907173
\(373\) −17.8058 −0.921951 −0.460976 0.887413i \(-0.652500\pi\)
−0.460976 + 0.887413i \(0.652500\pi\)
\(374\) 1.70064 0.0879382
\(375\) −2.82684 −0.145977
\(376\) −52.5100 −2.70800
\(377\) 0 0
\(378\) −26.7595 −1.37636
\(379\) −2.04555 −0.105073 −0.0525363 0.998619i \(-0.516731\pi\)
−0.0525363 + 0.998619i \(0.516731\pi\)
\(380\) −24.2326 −1.24311
\(381\) −4.21058 −0.215714
\(382\) 13.5833 0.694982
\(383\) 7.90521 0.403937 0.201969 0.979392i \(-0.435266\pi\)
0.201969 + 0.979392i \(0.435266\pi\)
\(384\) −47.7391 −2.43618
\(385\) 2.03741 0.103836
\(386\) 30.3394 1.54423
\(387\) 3.18059 0.161679
\(388\) 17.8469 0.906037
\(389\) −9.21171 −0.467052 −0.233526 0.972351i \(-0.575026\pi\)
−0.233526 + 0.972351i \(0.575026\pi\)
\(390\) 0 0
\(391\) 2.43298 0.123041
\(392\) 18.7344 0.946229
\(393\) −11.6657 −0.588456
\(394\) −10.9261 −0.550448
\(395\) −9.31937 −0.468908
\(396\) −22.5639 −1.13388
\(397\) 6.35438 0.318917 0.159458 0.987205i \(-0.449025\pi\)
0.159458 + 0.987205i \(0.449025\pi\)
\(398\) −51.9996 −2.60651
\(399\) −30.8713 −1.54550
\(400\) 5.41713 0.270857
\(401\) −4.16920 −0.208200 −0.104100 0.994567i \(-0.533196\pi\)
−0.104100 + 0.994567i \(0.533196\pi\)
\(402\) −57.1038 −2.84808
\(403\) 0 0
\(404\) −64.4600 −3.20701
\(405\) 0.937188 0.0465692
\(406\) 44.9541 2.23103
\(407\) 0.810008 0.0401506
\(408\) 10.0140 0.495767
\(409\) −10.1681 −0.502778 −0.251389 0.967886i \(-0.580887\pi\)
−0.251389 + 0.967886i \(0.580887\pi\)
\(410\) 0.668669 0.0330232
\(411\) −56.8467 −2.80404
\(412\) −57.1858 −2.81734
\(413\) 1.41183 0.0694716
\(414\) −47.5518 −2.33704
\(415\) −5.11778 −0.251222
\(416\) 0 0
\(417\) −58.8697 −2.88286
\(418\) −15.2969 −0.748198
\(419\) 28.5909 1.39676 0.698378 0.715730i \(-0.253906\pi\)
0.698378 + 0.715730i \(0.253906\pi\)
\(420\) 22.7685 1.11099
\(421\) −2.01797 −0.0983498 −0.0491749 0.998790i \(-0.515659\pi\)
−0.0491749 + 0.998790i \(0.515659\pi\)
\(422\) 26.5863 1.29420
\(423\) 47.1458 2.29231
\(424\) 38.8623 1.88732
\(425\) 0.637263 0.0309118
\(426\) −68.8853 −3.33750
\(427\) −7.99915 −0.387106
\(428\) −31.1264 −1.50455
\(429\) 0 0
\(430\) −1.59030 −0.0766908
\(431\) 20.6123 0.992860 0.496430 0.868077i \(-0.334644\pi\)
0.496430 + 0.868077i \(0.334644\pi\)
\(432\) −30.4891 −1.46691
\(433\) 29.4356 1.41458 0.707292 0.706921i \(-0.249917\pi\)
0.707292 + 0.706921i \(0.249917\pi\)
\(434\) 6.96103 0.334140
\(435\) −26.7281 −1.28151
\(436\) −42.5967 −2.04001
\(437\) −21.8841 −1.04686
\(438\) 26.2176 1.25272
\(439\) 16.9520 0.809077 0.404538 0.914521i \(-0.367432\pi\)
0.404538 + 0.914521i \(0.367432\pi\)
\(440\) 5.94462 0.283398
\(441\) −16.8205 −0.800978
\(442\) 0 0
\(443\) −24.1399 −1.14692 −0.573461 0.819233i \(-0.694400\pi\)
−0.573461 + 0.819233i \(0.694400\pi\)
\(444\) 9.05199 0.429588
\(445\) 12.5783 0.596270
\(446\) 53.2525 2.52158
\(447\) 37.7885 1.78733
\(448\) 9.22742 0.435955
\(449\) 20.8630 0.984585 0.492293 0.870430i \(-0.336159\pi\)
0.492293 + 0.870430i \(0.336159\pi\)
\(450\) −12.4551 −0.587140
\(451\) 0.286542 0.0134927
\(452\) −28.2742 −1.32990
\(453\) 51.7265 2.43032
\(454\) 39.1437 1.83710
\(455\) 0 0
\(456\) −90.0739 −4.21810
\(457\) 30.5659 1.42981 0.714906 0.699220i \(-0.246469\pi\)
0.714906 + 0.699220i \(0.246469\pi\)
\(458\) −19.0179 −0.888648
\(459\) −3.58669 −0.167413
\(460\) 16.1402 0.752541
\(461\) −4.67822 −0.217887 −0.108943 0.994048i \(-0.534747\pi\)
−0.108943 + 0.994048i \(0.534747\pi\)
\(462\) 14.3727 0.668680
\(463\) 14.0011 0.650688 0.325344 0.945596i \(-0.394520\pi\)
0.325344 + 0.945596i \(0.394520\pi\)
\(464\) 51.2196 2.37781
\(465\) −4.13878 −0.191931
\(466\) 47.5518 2.20280
\(467\) −6.98506 −0.323230 −0.161615 0.986854i \(-0.551670\pi\)
−0.161615 + 0.986854i \(0.551670\pi\)
\(468\) 0 0
\(469\) 15.4223 0.712135
\(470\) −23.5729 −1.08734
\(471\) −6.84742 −0.315513
\(472\) 4.11933 0.189608
\(473\) −0.681482 −0.0313346
\(474\) −65.7425 −3.01965
\(475\) −5.73205 −0.263005
\(476\) −5.13277 −0.235260
\(477\) −34.8923 −1.59761
\(478\) 31.8264 1.45571
\(479\) 16.2888 0.744252 0.372126 0.928182i \(-0.378629\pi\)
0.372126 + 0.928182i \(0.378629\pi\)
\(480\) 6.78645 0.309758
\(481\) 0 0
\(482\) 64.7056 2.94726
\(483\) 20.5619 0.935600
\(484\) −41.6685 −1.89402
\(485\) 4.22155 0.191691
\(486\) −35.5249 −1.61144
\(487\) −20.0409 −0.908139 −0.454069 0.890966i \(-0.650028\pi\)
−0.454069 + 0.890966i \(0.650028\pi\)
\(488\) −23.3393 −1.05652
\(489\) 45.1810 2.04316
\(490\) 8.41027 0.379937
\(491\) −15.7983 −0.712969 −0.356484 0.934301i \(-0.616025\pi\)
−0.356484 + 0.934301i \(0.616025\pi\)
\(492\) 3.20216 0.144365
\(493\) 6.02540 0.271370
\(494\) 0 0
\(495\) −5.33734 −0.239896
\(496\) 7.93123 0.356123
\(497\) 18.6042 0.834511
\(498\) −36.1028 −1.61781
\(499\) −1.24651 −0.0558016 −0.0279008 0.999611i \(-0.508882\pi\)
−0.0279008 + 0.999611i \(0.508882\pi\)
\(500\) 4.22756 0.189062
\(501\) −40.6871 −1.81777
\(502\) −18.9955 −0.847809
\(503\) −7.65345 −0.341250 −0.170625 0.985336i \(-0.554579\pi\)
−0.170625 + 0.985336i \(0.554579\pi\)
\(504\) 52.8592 2.35453
\(505\) −15.2476 −0.678508
\(506\) 10.1886 0.452938
\(507\) 0 0
\(508\) 6.29695 0.279382
\(509\) −25.7241 −1.14020 −0.570101 0.821575i \(-0.693096\pi\)
−0.570101 + 0.821575i \(0.693096\pi\)
\(510\) 4.49551 0.199064
\(511\) −7.08070 −0.313232
\(512\) 47.2215 2.08691
\(513\) 32.2616 1.42438
\(514\) −0.837948 −0.0369603
\(515\) −13.5269 −0.596067
\(516\) −7.61569 −0.335262
\(517\) −10.1016 −0.444268
\(518\) −3.60127 −0.158231
\(519\) 68.8305 3.02132
\(520\) 0 0
\(521\) −30.1519 −1.32098 −0.660490 0.750835i \(-0.729651\pi\)
−0.660490 + 0.750835i \(0.729651\pi\)
\(522\) −117.765 −5.15442
\(523\) 3.93752 0.172176 0.0860880 0.996288i \(-0.472563\pi\)
0.0860880 + 0.996288i \(0.472563\pi\)
\(524\) 17.4461 0.762138
\(525\) 5.38573 0.235052
\(526\) 13.4156 0.584947
\(527\) 0.933018 0.0406429
\(528\) 16.3759 0.712672
\(529\) −8.42399 −0.366261
\(530\) 17.4461 0.757812
\(531\) −3.69852 −0.160502
\(532\) 46.1682 2.00165
\(533\) 0 0
\(534\) 88.7326 3.83983
\(535\) −7.36274 −0.318319
\(536\) 44.9980 1.94362
\(537\) −10.7089 −0.462122
\(538\) 3.27007 0.140983
\(539\) 3.60402 0.155236
\(540\) −23.7939 −1.02393
\(541\) 15.8881 0.683083 0.341541 0.939867i \(-0.389051\pi\)
0.341541 + 0.939867i \(0.389051\pi\)
\(542\) −29.0610 −1.24828
\(543\) 23.9940 1.02968
\(544\) −1.52989 −0.0655935
\(545\) −10.0760 −0.431607
\(546\) 0 0
\(547\) −6.56107 −0.280531 −0.140266 0.990114i \(-0.544796\pi\)
−0.140266 + 0.990114i \(0.544796\pi\)
\(548\) 85.0147 3.63165
\(549\) 20.9551 0.894341
\(550\) 2.66867 0.113792
\(551\) −54.1972 −2.30888
\(552\) 59.9941 2.55352
\(553\) 17.7554 0.755035
\(554\) 50.6990 2.15399
\(555\) 2.14119 0.0908883
\(556\) 88.0401 3.73373
\(557\) −7.85006 −0.332618 −0.166309 0.986074i \(-0.553185\pi\)
−0.166309 + 0.986074i \(0.553185\pi\)
\(558\) −18.2356 −0.771973
\(559\) 0 0
\(560\) −10.3208 −0.436133
\(561\) 1.92644 0.0813344
\(562\) −29.6326 −1.24998
\(563\) −15.5595 −0.655755 −0.327878 0.944720i \(-0.606333\pi\)
−0.327878 + 0.944720i \(0.606333\pi\)
\(564\) −112.887 −4.75341
\(565\) −6.68806 −0.281369
\(566\) 56.5285 2.37607
\(567\) −1.78554 −0.0749857
\(568\) 54.2819 2.27762
\(569\) 3.47915 0.145853 0.0729267 0.997337i \(-0.476766\pi\)
0.0729267 + 0.997337i \(0.476766\pi\)
\(570\) −40.4362 −1.69368
\(571\) −21.5118 −0.900240 −0.450120 0.892968i \(-0.648619\pi\)
−0.450120 + 0.892968i \(0.648619\pi\)
\(572\) 0 0
\(573\) 15.3868 0.642791
\(574\) −1.27396 −0.0531739
\(575\) 3.81785 0.159216
\(576\) −24.1727 −1.00720
\(577\) −9.97608 −0.415310 −0.207655 0.978202i \(-0.566583\pi\)
−0.207655 + 0.978202i \(0.566583\pi\)
\(578\) 41.4102 1.72244
\(579\) 34.3676 1.42827
\(580\) 39.9721 1.65975
\(581\) 9.75045 0.404517
\(582\) 29.7805 1.23444
\(583\) 7.47612 0.309629
\(584\) −20.6595 −0.854898
\(585\) 0 0
\(586\) 46.4482 1.91876
\(587\) 24.0571 0.992945 0.496472 0.868053i \(-0.334628\pi\)
0.496472 + 0.868053i \(0.334628\pi\)
\(588\) 40.2756 1.66094
\(589\) −8.39230 −0.345799
\(590\) 1.84926 0.0761328
\(591\) −12.3767 −0.509111
\(592\) −4.10320 −0.168641
\(593\) −0.940219 −0.0386102 −0.0193051 0.999814i \(-0.506145\pi\)
−0.0193051 + 0.999814i \(0.506145\pi\)
\(594\) −15.0200 −0.616279
\(595\) −1.21412 −0.0497741
\(596\) −56.5130 −2.31486
\(597\) −58.9037 −2.41077
\(598\) 0 0
\(599\) −11.4270 −0.466896 −0.233448 0.972369i \(-0.575001\pi\)
−0.233448 + 0.972369i \(0.575001\pi\)
\(600\) 15.7141 0.641525
\(601\) −36.0431 −1.47023 −0.735114 0.677944i \(-0.762871\pi\)
−0.735114 + 0.677944i \(0.762871\pi\)
\(602\) 3.02985 0.123487
\(603\) −40.4012 −1.64526
\(604\) −77.3574 −3.14763
\(605\) −9.85641 −0.400720
\(606\) −107.562 −4.36942
\(607\) 39.8907 1.61911 0.809557 0.587041i \(-0.199707\pi\)
0.809557 + 0.587041i \(0.199707\pi\)
\(608\) 13.7610 0.558084
\(609\) 50.9227 2.06349
\(610\) −10.4775 −0.424223
\(611\) 0 0
\(612\) 13.4461 0.543528
\(613\) 0.345472 0.0139535 0.00697673 0.999976i \(-0.497779\pi\)
0.00697673 + 0.999976i \(0.497779\pi\)
\(614\) 7.85527 0.317013
\(615\) 0.757449 0.0305433
\(616\) −11.3258 −0.456328
\(617\) −38.6850 −1.55740 −0.778700 0.627397i \(-0.784121\pi\)
−0.778700 + 0.627397i \(0.784121\pi\)
\(618\) −95.4242 −3.83852
\(619\) −14.8971 −0.598764 −0.299382 0.954133i \(-0.596781\pi\)
−0.299382 + 0.954133i \(0.596781\pi\)
\(620\) 6.18958 0.248579
\(621\) −21.4879 −0.862281
\(622\) 7.93719 0.318252
\(623\) −23.9644 −0.960113
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −88.2245 −3.52616
\(627\) −17.3279 −0.692011
\(628\) 10.2404 0.408635
\(629\) −0.482694 −0.0192463
\(630\) 23.7296 0.945412
\(631\) 38.8450 1.54640 0.773198 0.634165i \(-0.218656\pi\)
0.773198 + 0.634165i \(0.218656\pi\)
\(632\) 51.8053 2.06071
\(633\) 30.1162 1.19701
\(634\) −34.0280 −1.35143
\(635\) 1.48950 0.0591090
\(636\) 83.5470 3.31285
\(637\) 0 0
\(638\) 25.2326 0.998967
\(639\) −48.7367 −1.92799
\(640\) 16.8878 0.667549
\(641\) 37.1816 1.46859 0.734293 0.678832i \(-0.237514\pi\)
0.734293 + 0.678832i \(0.237514\pi\)
\(642\) −51.9397 −2.04990
\(643\) −9.10377 −0.359018 −0.179509 0.983756i \(-0.557451\pi\)
−0.179509 + 0.983756i \(0.557451\pi\)
\(644\) −30.7505 −1.21174
\(645\) −1.80144 −0.0709316
\(646\) 9.11565 0.358651
\(647\) −19.1224 −0.751778 −0.375889 0.926665i \(-0.622663\pi\)
−0.375889 + 0.926665i \(0.622663\pi\)
\(648\) −5.20972 −0.204657
\(649\) 0.792455 0.0311066
\(650\) 0 0
\(651\) 7.88525 0.309047
\(652\) −67.5686 −2.64619
\(653\) 34.6324 1.35527 0.677636 0.735397i \(-0.263004\pi\)
0.677636 + 0.735397i \(0.263004\pi\)
\(654\) −71.0799 −2.77944
\(655\) 4.12676 0.161246
\(656\) −1.45152 −0.0566722
\(657\) 18.5491 0.723667
\(658\) 44.9114 1.75083
\(659\) −6.69852 −0.260937 −0.130469 0.991452i \(-0.541648\pi\)
−0.130469 + 0.991452i \(0.541648\pi\)
\(660\) 12.7799 0.497456
\(661\) −6.02758 −0.234446 −0.117223 0.993106i \(-0.537399\pi\)
−0.117223 + 0.993106i \(0.537399\pi\)
\(662\) −71.8604 −2.79294
\(663\) 0 0
\(664\) 28.4492 1.10404
\(665\) 10.9208 0.423489
\(666\) 9.43412 0.365565
\(667\) 36.0983 1.39773
\(668\) 60.8479 2.35428
\(669\) 60.3228 2.33222
\(670\) 20.2006 0.780417
\(671\) −4.48990 −0.173330
\(672\) −12.9296 −0.498771
\(673\) 23.3568 0.900338 0.450169 0.892943i \(-0.351364\pi\)
0.450169 + 0.892943i \(0.351364\pi\)
\(674\) −29.3205 −1.12938
\(675\) −5.62828 −0.216633
\(676\) 0 0
\(677\) −45.4042 −1.74503 −0.872513 0.488590i \(-0.837511\pi\)
−0.872513 + 0.488590i \(0.837511\pi\)
\(678\) −47.1802 −1.81195
\(679\) −8.04295 −0.308660
\(680\) −3.54248 −0.135848
\(681\) 44.3408 1.69914
\(682\) 3.90720 0.149615
\(683\) 25.4978 0.975645 0.487823 0.872943i \(-0.337791\pi\)
0.487823 + 0.872943i \(0.337791\pi\)
\(684\) −120.945 −4.62445
\(685\) 20.1096 0.768350
\(686\) −49.3047 −1.88246
\(687\) −21.5429 −0.821913
\(688\) 3.45214 0.131612
\(689\) 0 0
\(690\) 26.9327 1.02531
\(691\) 6.59630 0.250935 0.125468 0.992098i \(-0.459957\pi\)
0.125468 + 0.992098i \(0.459957\pi\)
\(692\) −102.937 −3.91306
\(693\) 10.1688 0.386279
\(694\) −4.74090 −0.179962
\(695\) 20.8253 0.789948
\(696\) 148.578 5.63185
\(697\) −0.170754 −0.00646778
\(698\) −25.6266 −0.969981
\(699\) 53.8653 2.03737
\(700\) −8.05440 −0.304428
\(701\) 29.2474 1.10466 0.552329 0.833626i \(-0.313739\pi\)
0.552329 + 0.833626i \(0.313739\pi\)
\(702\) 0 0
\(703\) 4.34174 0.163752
\(704\) 5.17932 0.195203
\(705\) −26.7027 −1.00568
\(706\) 1.99787 0.0751910
\(707\) 29.0499 1.09253
\(708\) 8.85584 0.332823
\(709\) −10.9335 −0.410614 −0.205307 0.978698i \(-0.565819\pi\)
−0.205307 + 0.978698i \(0.565819\pi\)
\(710\) 24.3683 0.914527
\(711\) −46.5131 −1.74438
\(712\) −69.9216 −2.62042
\(713\) 5.58973 0.209337
\(714\) −8.56490 −0.320533
\(715\) 0 0
\(716\) 16.0152 0.598516
\(717\) 36.0520 1.34639
\(718\) 20.2953 0.757412
\(719\) 16.0598 0.598929 0.299464 0.954107i \(-0.403192\pi\)
0.299464 + 0.954107i \(0.403192\pi\)
\(720\) 27.0370 1.00761
\(721\) 25.7716 0.959786
\(722\) −34.5788 −1.28689
\(723\) 73.2966 2.72593
\(724\) −35.8833 −1.33359
\(725\) 9.45512 0.351154
\(726\) −69.5310 −2.58054
\(727\) 51.3754 1.90541 0.952704 0.303900i \(-0.0982889\pi\)
0.952704 + 0.303900i \(0.0982889\pi\)
\(728\) 0 0
\(729\) −43.0532 −1.59456
\(730\) −9.27453 −0.343266
\(731\) 0.406104 0.0150203
\(732\) −50.1754 −1.85454
\(733\) 9.82358 0.362842 0.181421 0.983406i \(-0.441930\pi\)
0.181421 + 0.983406i \(0.441930\pi\)
\(734\) 51.2240 1.89071
\(735\) 9.52691 0.351406
\(736\) −9.16560 −0.337848
\(737\) 8.65648 0.318866
\(738\) 3.33734 0.122849
\(739\) −49.0842 −1.80559 −0.902797 0.430068i \(-0.858490\pi\)
−0.902797 + 0.430068i \(0.858490\pi\)
\(740\) −3.20216 −0.117714
\(741\) 0 0
\(742\) −33.2386 −1.22023
\(743\) −40.8375 −1.49818 −0.749091 0.662467i \(-0.769510\pi\)
−0.749091 + 0.662467i \(0.769510\pi\)
\(744\) 23.0070 0.843478
\(745\) −13.3678 −0.489757
\(746\) 44.4346 1.62687
\(747\) −25.5429 −0.934566
\(748\) −2.88101 −0.105340
\(749\) 14.0276 0.512557
\(750\) 7.05440 0.257590
\(751\) −2.72680 −0.0995024 −0.0497512 0.998762i \(-0.515843\pi\)
−0.0497512 + 0.998762i \(0.515843\pi\)
\(752\) 51.1710 1.86601
\(753\) −21.5175 −0.784142
\(754\) 0 0
\(755\) −18.2984 −0.665946
\(756\) 45.3324 1.64872
\(757\) 14.8060 0.538134 0.269067 0.963121i \(-0.413285\pi\)
0.269067 + 0.963121i \(0.413285\pi\)
\(758\) 5.10468 0.185410
\(759\) 11.5413 0.418924
\(760\) 31.8638 1.15582
\(761\) 11.3689 0.412122 0.206061 0.978539i \(-0.433935\pi\)
0.206061 + 0.978539i \(0.433935\pi\)
\(762\) 10.5075 0.380647
\(763\) 19.1969 0.694973
\(764\) −23.0110 −0.832510
\(765\) 3.18059 0.114994
\(766\) −19.7275 −0.712784
\(767\) 0 0
\(768\) 91.7512 3.31078
\(769\) 21.0562 0.759307 0.379654 0.925129i \(-0.376043\pi\)
0.379654 + 0.925129i \(0.376043\pi\)
\(770\) −5.08438 −0.183228
\(771\) −0.949203 −0.0341847
\(772\) −51.3970 −1.84982
\(773\) −14.0829 −0.506526 −0.253263 0.967397i \(-0.581504\pi\)
−0.253263 + 0.967397i \(0.581504\pi\)
\(774\) −7.93719 −0.285296
\(775\) 1.46410 0.0525921
\(776\) −23.4671 −0.842421
\(777\) −4.07941 −0.146348
\(778\) 22.9879 0.824156
\(779\) 1.53590 0.0550293
\(780\) 0 0
\(781\) 10.4425 0.373660
\(782\) −6.07151 −0.217117
\(783\) −53.2160 −1.90179
\(784\) −18.2566 −0.652023
\(785\) 2.42229 0.0864552
\(786\) 29.1118 1.03838
\(787\) −33.0242 −1.17719 −0.588593 0.808429i \(-0.700318\pi\)
−0.588593 + 0.808429i \(0.700318\pi\)
\(788\) 18.5095 0.659374
\(789\) 15.1968 0.541020
\(790\) 23.2566 0.827431
\(791\) 12.7422 0.453060
\(792\) 29.6697 1.05427
\(793\) 0 0
\(794\) −15.8574 −0.562758
\(795\) 19.7625 0.700903
\(796\) 88.0909 3.12230
\(797\) −16.9416 −0.600102 −0.300051 0.953923i \(-0.597004\pi\)
−0.300051 + 0.953923i \(0.597004\pi\)
\(798\) 77.0395 2.72717
\(799\) 6.01967 0.212961
\(800\) −2.40072 −0.0848783
\(801\) 62.7787 2.21817
\(802\) 10.4043 0.367387
\(803\) −3.97437 −0.140253
\(804\) 96.7378 3.41168
\(805\) −7.27382 −0.256369
\(806\) 0 0
\(807\) 3.70425 0.130396
\(808\) 84.7596 2.98183
\(809\) −51.7635 −1.81991 −0.909954 0.414708i \(-0.863884\pi\)
−0.909954 + 0.414708i \(0.863884\pi\)
\(810\) −2.33876 −0.0821756
\(811\) −22.6699 −0.796047 −0.398023 0.917375i \(-0.630304\pi\)
−0.398023 + 0.917375i \(0.630304\pi\)
\(812\) −76.1553 −2.67253
\(813\) −32.9194 −1.15453
\(814\) −2.02138 −0.0708494
\(815\) −15.9829 −0.559856
\(816\) −9.75864 −0.341621
\(817\) −3.65283 −0.127796
\(818\) 25.3745 0.887198
\(819\) 0 0
\(820\) −1.13277 −0.0395581
\(821\) −28.6631 −1.00035 −0.500174 0.865925i \(-0.666731\pi\)
−0.500174 + 0.865925i \(0.666731\pi\)
\(822\) 141.861 4.94799
\(823\) −25.8327 −0.900472 −0.450236 0.892910i \(-0.648660\pi\)
−0.450236 + 0.892910i \(0.648660\pi\)
\(824\) 75.1946 2.61953
\(825\) 3.02299 0.105247
\(826\) −3.52323 −0.122589
\(827\) 16.0820 0.559227 0.279613 0.960113i \(-0.409794\pi\)
0.279613 + 0.960113i \(0.409794\pi\)
\(828\) 80.5560 2.79951
\(829\) −22.5818 −0.784298 −0.392149 0.919902i \(-0.628268\pi\)
−0.392149 + 0.919902i \(0.628268\pi\)
\(830\) 12.7715 0.443304
\(831\) 57.4304 1.99224
\(832\) 0 0
\(833\) −2.14768 −0.0744128
\(834\) 146.910 5.08707
\(835\) 14.3932 0.498096
\(836\) 25.9141 0.896257
\(837\) −8.24037 −0.284829
\(838\) −71.3487 −2.46470
\(839\) 17.8440 0.616042 0.308021 0.951380i \(-0.400333\pi\)
0.308021 + 0.951380i \(0.400333\pi\)
\(840\) −29.9387 −1.03298
\(841\) 60.3992 2.08273
\(842\) 5.03586 0.173547
\(843\) −33.5669 −1.15611
\(844\) −45.0390 −1.55031
\(845\) 0 0
\(846\) −117.653 −4.04498
\(847\) 18.7785 0.645239
\(848\) −37.8713 −1.30050
\(849\) 64.0339 2.19764
\(850\) −1.59030 −0.0545467
\(851\) −2.89183 −0.0991306
\(852\) 116.696 3.99795
\(853\) −19.7936 −0.677720 −0.338860 0.940837i \(-0.610041\pi\)
−0.338860 + 0.940837i \(0.610041\pi\)
\(854\) 19.9619 0.683083
\(855\) −28.6088 −0.978399
\(856\) 40.9286 1.39891
\(857\) −11.7302 −0.400696 −0.200348 0.979725i \(-0.564207\pi\)
−0.200348 + 0.979725i \(0.564207\pi\)
\(858\) 0 0
\(859\) 5.37452 0.183376 0.0916882 0.995788i \(-0.470774\pi\)
0.0916882 + 0.995788i \(0.470774\pi\)
\(860\) 2.69407 0.0918669
\(861\) −1.44310 −0.0491808
\(862\) −51.4382 −1.75199
\(863\) 25.3234 0.862017 0.431008 0.902348i \(-0.358158\pi\)
0.431008 + 0.902348i \(0.358158\pi\)
\(864\) 13.5119 0.459685
\(865\) −24.3489 −0.827889
\(866\) −73.4567 −2.49616
\(867\) 46.9083 1.59309
\(868\) −11.7925 −0.400262
\(869\) 9.96603 0.338075
\(870\) 66.7001 2.26135
\(871\) 0 0
\(872\) 56.0112 1.89678
\(873\) 21.0698 0.713106
\(874\) 54.6120 1.84728
\(875\) −1.90521 −0.0644079
\(876\) −44.4144 −1.50062
\(877\) −20.6915 −0.698703 −0.349352 0.936992i \(-0.613598\pi\)
−0.349352 + 0.936992i \(0.613598\pi\)
\(878\) −42.3040 −1.42769
\(879\) 52.6151 1.77466
\(880\) −5.79302 −0.195283
\(881\) 48.3993 1.63061 0.815307 0.579029i \(-0.196568\pi\)
0.815307 + 0.579029i \(0.196568\pi\)
\(882\) 41.9758 1.41340
\(883\) 45.8550 1.54314 0.771572 0.636142i \(-0.219471\pi\)
0.771572 + 0.636142i \(0.219471\pi\)
\(884\) 0 0
\(885\) 2.09479 0.0704155
\(886\) 60.2413 2.02385
\(887\) −1.08234 −0.0363413 −0.0181707 0.999835i \(-0.505784\pi\)
−0.0181707 + 0.999835i \(0.505784\pi\)
\(888\) −11.9026 −0.399426
\(889\) −2.83781 −0.0951772
\(890\) −31.3893 −1.05217
\(891\) −1.00222 −0.0335756
\(892\) −90.2133 −3.02056
\(893\) −54.1457 −1.81192
\(894\) −94.3015 −3.15391
\(895\) 3.78829 0.126628
\(896\) −32.1749 −1.07489
\(897\) 0 0
\(898\) −52.0637 −1.73739
\(899\) 13.8433 0.461698
\(900\) 21.0998 0.703327
\(901\) −4.45512 −0.148421
\(902\) −0.715068 −0.0238092
\(903\) 3.43213 0.114214
\(904\) 37.1782 1.23653
\(905\) −8.48794 −0.282149
\(906\) −129.084 −4.28853
\(907\) −45.5307 −1.51182 −0.755910 0.654675i \(-0.772805\pi\)
−0.755910 + 0.654675i \(0.772805\pi\)
\(908\) −66.3120 −2.20064
\(909\) −76.1009 −2.52411
\(910\) 0 0
\(911\) 39.7417 1.31670 0.658350 0.752712i \(-0.271255\pi\)
0.658350 + 0.752712i \(0.271255\pi\)
\(912\) 87.7770 2.90659
\(913\) 5.47290 0.181126
\(914\) −76.2774 −2.52303
\(915\) −11.8687 −0.392365
\(916\) 32.2176 1.06450
\(917\) −7.86236 −0.259638
\(918\) 8.95062 0.295415
\(919\) 46.9938 1.55018 0.775091 0.631850i \(-0.217704\pi\)
0.775091 + 0.631850i \(0.217704\pi\)
\(920\) −21.2230 −0.699702
\(921\) 8.89822 0.293206
\(922\) 11.6745 0.384481
\(923\) 0 0
\(924\) −24.3484 −0.801002
\(925\) −0.757449 −0.0249048
\(926\) −34.9399 −1.14820
\(927\) −67.5130 −2.21742
\(928\) −22.6991 −0.745134
\(929\) 15.2213 0.499395 0.249698 0.968324i \(-0.419669\pi\)
0.249698 + 0.968324i \(0.419669\pi\)
\(930\) 10.3284 0.338680
\(931\) 19.3180 0.633121
\(932\) −80.5560 −2.63870
\(933\) 8.99102 0.294353
\(934\) 17.4313 0.570369
\(935\) −0.681482 −0.0222869
\(936\) 0 0
\(937\) 6.07285 0.198392 0.0991958 0.995068i \(-0.468373\pi\)
0.0991958 + 0.995068i \(0.468373\pi\)
\(938\) −38.4864 −1.25663
\(939\) −99.9382 −3.26136
\(940\) 39.9341 1.30251
\(941\) 0.0496576 0.00161879 0.000809396 1.00000i \(-0.499742\pi\)
0.000809396 1.00000i \(0.499742\pi\)
\(942\) 17.0878 0.556750
\(943\) −1.02299 −0.0333132
\(944\) −4.01429 −0.130654
\(945\) 10.7231 0.348821
\(946\) 1.70064 0.0552927
\(947\) 18.6581 0.606308 0.303154 0.952942i \(-0.401960\pi\)
0.303154 + 0.952942i \(0.401960\pi\)
\(948\) 111.372 3.61720
\(949\) 0 0
\(950\) 14.3044 0.464095
\(951\) −38.5459 −1.24994
\(952\) 6.74917 0.218742
\(953\) −1.52953 −0.0495463 −0.0247731 0.999693i \(-0.507886\pi\)
−0.0247731 + 0.999693i \(0.507886\pi\)
\(954\) 87.0739 2.81912
\(955\) −5.44310 −0.176135
\(956\) −53.9161 −1.74377
\(957\) 28.5827 0.923948
\(958\) −40.6487 −1.31330
\(959\) −38.3131 −1.23720
\(960\) 13.6911 0.441878
\(961\) −28.8564 −0.930852
\(962\) 0 0
\(963\) −36.7475 −1.18417
\(964\) −109.616 −3.53048
\(965\) −12.1576 −0.391367
\(966\) −51.3124 −1.65095
\(967\) −32.1716 −1.03457 −0.517285 0.855813i \(-0.673057\pi\)
−0.517285 + 0.855813i \(0.673057\pi\)
\(968\) 54.7907 1.76104
\(969\) 10.3259 0.331717
\(970\) −10.5349 −0.338256
\(971\) −17.2541 −0.553710 −0.276855 0.960912i \(-0.589292\pi\)
−0.276855 + 0.960912i \(0.589292\pi\)
\(972\) 60.1816 1.93033
\(973\) −39.6766 −1.27197
\(974\) 50.0122 1.60249
\(975\) 0 0
\(976\) 22.7442 0.728023
\(977\) −15.7228 −0.503018 −0.251509 0.967855i \(-0.580927\pi\)
−0.251509 + 0.967855i \(0.580927\pi\)
\(978\) −112.750 −3.60533
\(979\) −13.4511 −0.429900
\(980\) −14.2476 −0.455122
\(981\) −50.2893 −1.60561
\(982\) 39.4248 1.25810
\(983\) −38.5356 −1.22910 −0.614548 0.788880i \(-0.710662\pi\)
−0.614548 + 0.788880i \(0.710662\pi\)
\(984\) −4.21058 −0.134228
\(985\) 4.37830 0.139504
\(986\) −15.0364 −0.478857
\(987\) 50.8743 1.61935
\(988\) 0 0
\(989\) 2.43298 0.0773642
\(990\) 13.3194 0.423317
\(991\) 8.59143 0.272916 0.136458 0.990646i \(-0.456428\pi\)
0.136458 + 0.990646i \(0.456428\pi\)
\(992\) −3.51490 −0.111598
\(993\) −81.4014 −2.58320
\(994\) −46.4268 −1.47257
\(995\) 20.8373 0.660587
\(996\) 61.1606 1.93795
\(997\) −20.5374 −0.650425 −0.325213 0.945641i \(-0.605436\pi\)
−0.325213 + 0.945641i \(0.605436\pi\)
\(998\) 3.11069 0.0984670
\(999\) 4.26313 0.134880
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.a.l.1.1 4
3.2 odd 2 7605.2.a.cj.1.4 4
5.4 even 2 4225.2.a.bl.1.4 4
13.2 odd 12 65.2.m.a.56.1 yes 8
13.3 even 3 845.2.e.n.191.4 8
13.4 even 6 845.2.e.m.146.1 8
13.5 odd 4 845.2.c.g.506.8 8
13.6 odd 12 845.2.m.g.361.4 8
13.7 odd 12 65.2.m.a.36.1 8
13.8 odd 4 845.2.c.g.506.1 8
13.9 even 3 845.2.e.n.146.4 8
13.10 even 6 845.2.e.m.191.1 8
13.11 odd 12 845.2.m.g.316.4 8
13.12 even 2 845.2.a.m.1.4 4
39.2 even 12 585.2.bu.c.316.4 8
39.20 even 12 585.2.bu.c.361.4 8
39.38 odd 2 7605.2.a.cf.1.1 4
52.7 even 12 1040.2.da.b.881.1 8
52.15 even 12 1040.2.da.b.641.1 8
65.2 even 12 325.2.m.c.199.4 8
65.7 even 12 325.2.m.b.49.1 8
65.28 even 12 325.2.m.b.199.1 8
65.33 even 12 325.2.m.c.49.4 8
65.54 odd 12 325.2.n.d.251.4 8
65.59 odd 12 325.2.n.d.101.4 8
65.64 even 2 4225.2.a.bi.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.m.a.36.1 8 13.7 odd 12
65.2.m.a.56.1 yes 8 13.2 odd 12
325.2.m.b.49.1 8 65.7 even 12
325.2.m.b.199.1 8 65.28 even 12
325.2.m.c.49.4 8 65.33 even 12
325.2.m.c.199.4 8 65.2 even 12
325.2.n.d.101.4 8 65.59 odd 12
325.2.n.d.251.4 8 65.54 odd 12
585.2.bu.c.316.4 8 39.2 even 12
585.2.bu.c.361.4 8 39.20 even 12
845.2.a.l.1.1 4 1.1 even 1 trivial
845.2.a.m.1.4 4 13.12 even 2
845.2.c.g.506.1 8 13.8 odd 4
845.2.c.g.506.8 8 13.5 odd 4
845.2.e.m.146.1 8 13.4 even 6
845.2.e.m.191.1 8 13.10 even 6
845.2.e.n.146.4 8 13.9 even 3
845.2.e.n.191.4 8 13.3 even 3
845.2.m.g.316.4 8 13.11 odd 12
845.2.m.g.361.4 8 13.6 odd 12
1040.2.da.b.641.1 8 52.15 even 12
1040.2.da.b.881.1 8 52.7 even 12
4225.2.a.bi.1.1 4 65.64 even 2
4225.2.a.bl.1.4 4 5.4 even 2
7605.2.a.cf.1.1 4 39.38 odd 2
7605.2.a.cj.1.4 4 3.2 odd 2