Properties

Label 7605.2.a.cj.1.4
Level $7605$
Weight $2$
Character 7605.1
Self dual yes
Analytic conductor $60.726$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7605.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.7262307372\)
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 1.4
Root \(2.49551\) of defining polynomial
Character \(\chi\) \(=\) 7605.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.49551 q^{2} +4.22756 q^{4} -1.00000 q^{5} -1.90521 q^{7} +5.55889 q^{8} +O(q^{10})\) \(q+2.49551 q^{2} +4.22756 q^{4} -1.00000 q^{5} -1.90521 q^{7} +5.55889 q^{8} -2.49551 q^{10} +1.06939 q^{11} -4.75447 q^{14} +5.41713 q^{16} -0.637263 q^{17} -5.73205 q^{19} -4.22756 q^{20} +2.66867 q^{22} -3.81785 q^{23} +1.00000 q^{25} -8.05440 q^{28} -9.45512 q^{29} +1.46410 q^{31} +2.40072 q^{32} -1.59030 q^{34} +1.90521 q^{35} -0.757449 q^{37} -14.3044 q^{38} -5.55889 q^{40} +0.267949 q^{41} +0.637263 q^{43} +4.52091 q^{44} -9.52748 q^{46} -9.44613 q^{47} -3.37017 q^{49} +2.49551 q^{50} +6.99102 q^{53} -1.06939 q^{55} -10.5909 q^{56} -23.5953 q^{58} +0.741035 q^{59} +4.19856 q^{61} +3.65368 q^{62} -4.84325 q^{64} -8.09479 q^{67} -2.69407 q^{68} +4.75447 q^{70} +9.76488 q^{71} +3.71649 q^{73} -1.89022 q^{74} -24.2326 q^{76} -2.03741 q^{77} -9.31937 q^{79} -5.41713 q^{80} +0.668669 q^{82} +5.11778 q^{83} +0.637263 q^{85} +1.59030 q^{86} +5.94462 q^{88} -12.5783 q^{89} -16.1402 q^{92} -23.5729 q^{94} +5.73205 q^{95} +4.22155 q^{97} -8.41027 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} + 2q^{4} - 4q^{5} - 10q^{7} + 6q^{8} + O(q^{10}) \) \( 4q + 2q^{2} + 2q^{4} - 4q^{5} - 10q^{7} + 6q^{8} - 2q^{10} - 2q^{14} + 2q^{16} + 2q^{17} - 16q^{19} - 2q^{20} + 12q^{22} + 10q^{23} + 4q^{25} - 8q^{28} - 8q^{29} - 8q^{31} + 4q^{32} + 4q^{34} + 10q^{35} + 2q^{37} - 8q^{38} - 6q^{40} + 8q^{41} - 2q^{43} + 12q^{44} - 16q^{46} + 8q^{47} + 12q^{49} + 2q^{50} + 12q^{53} - 12q^{56} - 22q^{58} + 12q^{59} + 28q^{61} - 4q^{62} + 4q^{64} - 30q^{67} - 14q^{68} + 2q^{70} + 4q^{71} + 8q^{73} + 10q^{74} - 20q^{76} - 18q^{77} - 8q^{79} - 2q^{80} + 4q^{82} - 12q^{83} - 2q^{85} - 4q^{86} - 18q^{88} - 12q^{89} - 22q^{92} - 32q^{94} + 16q^{95} - 2q^{97} - 24q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.49551 1.76459 0.882295 0.470696i \(-0.155997\pi\)
0.882295 + 0.470696i \(0.155997\pi\)
\(3\) 0 0
\(4\) 4.22756 2.11378
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.90521 −0.720103 −0.360051 0.932933i \(-0.617241\pi\)
−0.360051 + 0.932933i \(0.617241\pi\)
\(8\) 5.55889 1.96536
\(9\) 0 0
\(10\) −2.49551 −0.789149
\(11\) 1.06939 0.322433 0.161217 0.986919i \(-0.448458\pi\)
0.161217 + 0.986919i \(0.448458\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −4.75447 −1.27069
\(15\) 0 0
\(16\) 5.41713 1.35428
\(17\) −0.637263 −0.154559 −0.0772795 0.997009i \(-0.524623\pi\)
−0.0772795 + 0.997009i \(0.524623\pi\)
\(18\) 0 0
\(19\) −5.73205 −1.31502 −0.657511 0.753445i \(-0.728391\pi\)
−0.657511 + 0.753445i \(0.728391\pi\)
\(20\) −4.22756 −0.945311
\(21\) 0 0
\(22\) 2.66867 0.568962
\(23\) −3.81785 −0.796078 −0.398039 0.917369i \(-0.630309\pi\)
−0.398039 + 0.917369i \(0.630309\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) −8.05440 −1.52214
\(29\) −9.45512 −1.75577 −0.877886 0.478870i \(-0.841046\pi\)
−0.877886 + 0.478870i \(0.841046\pi\)
\(30\) 0 0
\(31\) 1.46410 0.262960 0.131480 0.991319i \(-0.458027\pi\)
0.131480 + 0.991319i \(0.458027\pi\)
\(32\) 2.40072 0.424391
\(33\) 0 0
\(34\) −1.59030 −0.272733
\(35\) 1.90521 0.322040
\(36\) 0 0
\(37\) −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 \(-0.493339\pi\)
0.0209233 + 0.999781i \(0.493339\pi\)
\(42\) 0 0
\(43\) 0.637263 0.0971817 0.0485909 0.998819i \(-0.484527\pi\)
0.0485909 + 0.998819i \(0.484527\pi\)
\(44\) 4.52091 0.681552
\(45\) 0 0
\(46\) −9.52748 −1.40475
\(47\) −9.44613 −1.37786 −0.688930 0.724828i \(-0.741919\pi\)
−0.688930 + 0.724828i \(0.741919\pi\)
\(48\) 0 0
\(49\) −3.37017 −0.481452
\(50\) 2.49551 0.352918
\(51\) 0 0
\(52\) 0 0
\(53\) 6.99102 0.960290 0.480145 0.877189i \(-0.340584\pi\)
0.480145 + 0.877189i \(0.340584\pi\)
\(54\) 0 0
\(55\) −1.06939 −0.144196
\(56\) −10.5909 −1.41526
\(57\) 0 0
\(58\) −23.5953 −3.09822
\(59\) 0.741035 0.0964746 0.0482373 0.998836i \(-0.484640\pi\)
0.0482373 + 0.998836i \(0.484640\pi\)
\(60\) 0 0
\(61\) 4.19856 0.537571 0.268785 0.963200i \(-0.413378\pi\)
0.268785 + 0.963200i \(0.413378\pi\)
\(62\) 3.65368 0.464017
\(63\) 0 0
\(64\) −4.84325 −0.605406
\(65\) 0 0
\(66\) 0 0
\(67\) −8.09479 −0.988936 −0.494468 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(68\) −2.69407 −0.326704
\(69\) 0 0
\(70\) 4.75447 0.568268
\(71\) 9.76488 1.15888 0.579439 0.815016i \(-0.303272\pi\)
0.579439 + 0.815016i \(0.303272\pi\)
\(72\) 0 0
\(73\) 3.71649 0.434982 0.217491 0.976062i \(-0.430213\pi\)
0.217491 + 0.976062i \(0.430213\pi\)
\(74\) −1.89022 −0.219734
\(75\) 0 0
\(76\) −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 0
\(82\) 0.668669 0.0738422
\(83\) 5.11778 0.561749 0.280875 0.959744i \(-0.409376\pi\)
0.280875 + 0.959744i \(0.409376\pi\)
\(84\) 0 0
\(85\) 0.637263 0.0691209
\(86\) 1.59030 0.171486
\(87\) 0 0
\(88\) 5.94462 0.633698
\(89\) −12.5783 −1.33330 −0.666650 0.745371i \(-0.732273\pi\)
−0.666650 + 0.745371i \(0.732273\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −16.1402 −1.68273
\(93\) 0 0
\(94\) −23.5729 −2.43136
\(95\) 5.73205 0.588096
\(96\) 0 0
\(97\) 4.22155 0.428634 0.214317 0.976764i \(-0.431248\pi\)
0.214317 + 0.976764i \(0.431248\pi\)
\(98\) −8.41027 −0.849566
\(99\) 0 0
\(100\) 4.22756 0.422756
\(101\) 15.2476 1.51719 0.758595 0.651562i \(-0.225886\pi\)
0.758595 + 0.651562i \(0.225886\pi\)
\(102\) 0 0
\(103\) −13.5269 −1.33285 −0.666423 0.745574i \(-0.732176\pi\)
−0.666423 + 0.745574i \(0.732176\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 17.4461 1.69452
\(107\) 7.36274 0.711783 0.355891 0.934527i \(-0.384177\pi\)
0.355891 + 0.934527i \(0.384177\pi\)
\(108\) 0 0
\(109\) −10.0760 −0.965103 −0.482551 0.875868i \(-0.660290\pi\)
−0.482551 + 0.875868i \(0.660290\pi\)
\(110\) −2.66867 −0.254448
\(111\) 0 0
\(112\) −10.3208 −0.975223
\(113\) 6.68806 0.629160 0.314580 0.949231i \(-0.398136\pi\)
0.314580 + 0.949231i \(0.398136\pi\)
\(114\) 0 0
\(115\) 3.81785 0.356017
\(116\) −39.9721 −3.71131
\(117\) 0 0
\(118\) 1.84926 0.170238
\(119\) 1.21412 0.111298
\(120\) 0 0
\(121\) −9.85641 −0.896037
\(122\) 10.4775 0.948592
\(123\) 0 0
\(124\) 6.18958 0.555840
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 1.48950 0.132172 0.0660859 0.997814i \(-0.478949\pi\)
0.0660859 + 0.997814i \(0.478949\pi\)
\(128\) −16.8878 −1.49269
\(129\) 0 0
\(130\) 0 0
\(131\) −4.12676 −0.360557 −0.180278 0.983616i \(-0.557700\pi\)
−0.180278 + 0.983616i \(0.557700\pi\)
\(132\) 0 0
\(133\) 10.9208 0.946951
\(134\) −20.2006 −1.74507
\(135\) 0 0
\(136\) −3.54248 −0.303765
\(137\) −20.1096 −1.71808 −0.859041 0.511906i \(-0.828940\pi\)
−0.859041 + 0.511906i \(0.828940\pi\)
\(138\) 0 0
\(139\) 20.8253 1.76638 0.883189 0.469018i \(-0.155392\pi\)
0.883189 + 0.469018i \(0.155392\pi\)
\(140\) 8.05440 0.680721
\(141\) 0 0
\(142\) 24.3683 2.04494
\(143\) 0 0
\(144\) 0 0
\(145\) 9.45512 0.785205
\(146\) 9.27453 0.767565
\(147\) 0 0
\(148\) −3.20216 −0.263216
\(149\) 13.3678 1.09513 0.547565 0.836763i \(-0.315555\pi\)
0.547565 + 0.836763i \(0.315555\pi\)
\(150\) 0 0
\(151\) −18.2984 −1.48910 −0.744550 0.667567i \(-0.767336\pi\)
−0.744550 + 0.667567i \(0.767336\pi\)
\(152\) −31.8638 −2.58450
\(153\) 0 0
\(154\) −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
\(159\) 0 0
\(160\) −2.40072 −0.189794
\(161\) 7.27382 0.573258
\(162\) 0 0
\(163\) −15.9829 −1.25188 −0.625938 0.779873i \(-0.715284\pi\)
−0.625938 + 0.779873i \(0.715284\pi\)
\(164\) 1.13277 0.0884545
\(165\) 0 0
\(166\) 12.7715 0.991257
\(167\) −14.3932 −1.11378 −0.556888 0.830588i \(-0.688005\pi\)
−0.556888 + 0.830588i \(0.688005\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 1.59030 0.121970
\(171\) 0 0
\(172\) 2.69407 0.205421
\(173\) 24.3489 1.85122 0.925608 0.378484i \(-0.123555\pi\)
0.925608 + 0.378484i \(0.123555\pi\)
\(174\) 0 0
\(175\) −1.90521 −0.144021
\(176\) 5.79302 0.436666
\(177\) 0 0
\(178\) −31.3893 −2.35273
\(179\) −3.78829 −0.283150 −0.141575 0.989928i \(-0.545217\pi\)
−0.141575 + 0.989928i \(0.545217\pi\)
\(180\) 0 0
\(181\) −8.48794 −0.630904 −0.315452 0.948942i \(-0.602156\pi\)
−0.315452 + 0.948942i \(0.602156\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −21.2230 −1.56458
\(185\) 0.757449 0.0556888
\(186\) 0 0
\(187\) −0.681482 −0.0498349
\(188\) −39.9341 −2.91249
\(189\) 0 0
\(190\) 14.3044 1.03775
\(191\) 5.44310 0.393849 0.196924 0.980419i \(-0.436905\pi\)
0.196924 + 0.980419i \(0.436905\pi\)
\(192\) 0 0
\(193\) −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 \(-0.549850\pi\)
−0.155970 + 0.987762i \(0.549850\pi\)
\(198\) 0 0
\(199\) 20.8373 1.47712 0.738558 0.674189i \(-0.235507\pi\)
0.738558 + 0.674189i \(0.235507\pi\)
\(200\) 5.55889 0.393073
\(201\) 0 0
\(202\) 38.0504 2.67722
\(203\) 18.0140 1.26434
\(204\) 0 0
\(205\) −0.267949 −0.0187144
\(206\) −33.7565 −2.35193
\(207\) 0 0
\(208\) 0 0
\(209\) −6.12979 −0.424007
\(210\) 0 0
\(211\) −10.6537 −0.733429 −0.366715 0.930333i \(-0.619517\pi\)
−0.366715 + 0.930333i \(0.619517\pi\)
\(212\) 29.5549 2.02984
\(213\) 0 0
\(214\) 18.3738 1.25600
\(215\) −0.637263 −0.0434610
\(216\) 0 0
\(217\) −2.78942 −0.189358
\(218\) −25.1447 −1.70301
\(219\) 0 0
\(220\) −4.52091 −0.304799
\(221\) 0 0
\(222\) 0 0
\(223\) −21.3393 −1.42899 −0.714494 0.699642i \(-0.753343\pi\)
−0.714494 + 0.699642i \(0.753343\pi\)
\(224\) −4.57388 −0.305605
\(225\) 0 0
\(226\) 16.6901 1.11021
\(227\) 15.6857 1.04109 0.520547 0.853833i \(-0.325728\pi\)
0.520547 + 0.853833i \(0.325728\pi\)
\(228\) 0 0
\(229\) 7.62085 0.503600 0.251800 0.967779i \(-0.418977\pi\)
0.251800 + 0.967779i \(0.418977\pi\)
\(230\) 9.52748 0.628224
\(231\) 0 0
\(232\) −52.5599 −3.45073
\(233\) 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
\(236\) 3.13277 0.203926
\(237\) 0 0
\(238\) 3.02985 0.196396
\(239\) 12.7535 0.824954 0.412477 0.910968i \(-0.364664\pi\)
0.412477 + 0.910968i \(0.364664\pi\)
\(240\) 0 0
\(241\) −25.9288 −1.67022 −0.835111 0.550081i \(-0.814597\pi\)
−0.835111 + 0.550081i \(0.814597\pi\)
\(242\) −24.5967 −1.58114
\(243\) 0 0
\(244\) 17.7497 1.13631
\(245\) 3.37017 0.215312
\(246\) 0 0
\(247\) 0 0
\(248\) 8.13878 0.516813
\(249\) 0 0
\(250\) −2.49551 −0.157830
\(251\) −7.61186 −0.480457 −0.240228 0.970716i \(-0.577222\pi\)
−0.240228 + 0.970716i \(0.577222\pi\)
\(252\) 0 0
\(253\) −4.08277 −0.256682
\(254\) 3.71706 0.233229
\(255\) 0 0
\(256\) −32.4572 −2.02857
\(257\) −0.335783 −0.0209456 −0.0104728 0.999945i \(-0.503334\pi\)
−0.0104728 + 0.999945i \(0.503334\pi\)
\(258\) 0 0
\(259\) 1.44310 0.0896700
\(260\) 0 0
\(261\) 0 0
\(262\) −10.2984 −0.636235
\(263\) 5.37589 0.331492 0.165746 0.986169i \(-0.446997\pi\)
0.165746 + 0.986169i \(0.446997\pi\)
\(264\) 0 0
\(265\) −6.99102 −0.429455
\(266\) 27.2529 1.67098
\(267\) 0 0
\(268\) −34.2212 −2.09039
\(269\) 1.31038 0.0798956 0.0399478 0.999202i \(-0.487281\pi\)
0.0399478 + 0.999202i \(0.487281\pi\)
\(270\) 0 0
\(271\) 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
\(276\) 0 0
\(277\) −20.3161 −1.22068 −0.610338 0.792141i \(-0.708967\pi\)
−0.610338 + 0.792141i \(0.708967\pi\)
\(278\) 51.9697 3.11693
\(279\) 0 0
\(280\) 10.5909 0.632925
\(281\) −11.8744 −0.708366 −0.354183 0.935176i \(-0.615241\pi\)
−0.354183 + 0.935176i \(0.615241\pi\)
\(282\) 0 0
\(283\) −22.6521 −1.34653 −0.673264 0.739402i \(-0.735108\pi\)
−0.673264 + 0.739402i \(0.735108\pi\)
\(284\) 41.2816 2.44961
\(285\) 0 0
\(286\) 0 0
\(287\) −0.510500 −0.0301339
\(288\) 0 0
\(289\) −16.5939 −0.976112
\(290\) 23.5953 1.38556
\(291\) 0 0
\(292\) 15.7117 0.919456
\(293\) 18.6127 1.08737 0.543683 0.839290i \(-0.317029\pi\)
0.543683 + 0.839290i \(0.317029\pi\)
\(294\) 0 0
\(295\) −0.741035 −0.0431448
\(296\) −4.21058 −0.244735
\(297\) 0 0
\(298\) 33.3593 1.93245
\(299\) 0 0
\(300\) 0 0
\(301\) −1.21412 −0.0699808
\(302\) −45.6637 −2.62765
\(303\) 0 0
\(304\) −31.0513 −1.78091
\(305\) −4.19856 −0.240409
\(306\) 0 0
\(307\) −3.14776 −0.179652 −0.0898262 0.995957i \(-0.528631\pi\)
−0.0898262 + 0.995957i \(0.528631\pi\)
\(308\) −8.61329 −0.490788
\(309\) 0 0
\(310\) −3.65368 −0.207515
\(311\) 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 \(0.0131689\pi\)
\(314\) 6.04484 0.341130
\(315\) 0 0
\(316\) −39.3982 −2.21632
\(317\) −13.6357 −0.765858 −0.382929 0.923778i \(-0.625085\pi\)
−0.382929 + 0.923778i \(0.625085\pi\)
\(318\) 0 0
\(319\) −10.1112 −0.566119
\(320\) 4.84325 0.270746
\(321\) 0 0
\(322\) 18.1519 1.01156
\(323\) 3.65283 0.203249
\(324\) 0 0
\(325\) 0 0
\(326\) −39.8854 −2.20905
\(327\) 0 0
\(328\) 1.48950 0.0822439
\(329\) 17.9969 0.992201
\(330\) 0 0
\(331\) 28.7959 1.58277 0.791383 0.611320i \(-0.209361\pi\)
0.791383 + 0.611320i \(0.209361\pi\)
\(332\) 21.6357 1.18741
\(333\) 0 0
\(334\) −35.9182 −1.96536
\(335\) 8.09479 0.442265
\(336\) 0 0
\(337\) 11.7493 0.640026 0.320013 0.947413i \(-0.396313\pi\)
0.320013 + 0.947413i \(0.396313\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 2.69407 0.146106
\(341\) 1.56569 0.0847871
\(342\) 0 0
\(343\) 19.7574 1.06680
\(344\) 3.54248 0.190997
\(345\) 0 0
\(346\) 60.7630 3.26664
\(347\) −1.89977 −0.101985 −0.0509926 0.998699i \(-0.516238\pi\)
−0.0509926 + 0.998699i \(0.516238\pi\)
\(348\) 0 0
\(349\) 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 \(-0.493218\pi\)
0.0213055 + 0.999773i \(0.493218\pi\)
\(354\) 0 0
\(355\) −9.76488 −0.518266
\(356\) −53.1756 −2.81830
\(357\) 0 0
\(358\) −9.45370 −0.499643
\(359\) 8.13272 0.429228 0.214614 0.976699i \(-0.431151\pi\)
0.214614 + 0.976699i \(0.431151\pi\)
\(360\) 0 0
\(361\) 13.8564 0.729285
\(362\) −21.1817 −1.11329
\(363\) 0 0
\(364\) 0 0
\(365\) −3.71649 −0.194530
\(366\) 0 0
\(367\) −20.5265 −1.07147 −0.535737 0.844385i \(-0.679966\pi\)
−0.535737 + 0.844385i \(0.679966\pi\)
\(368\) −20.6818 −1.07811
\(369\) 0 0
\(370\) 1.89022 0.0982679
\(371\) −13.3194 −0.691507
\(372\) 0 0
\(373\) −17.8058 −0.921951 −0.460976 0.887413i \(-0.652500\pi\)
−0.460976 + 0.887413i \(0.652500\pi\)
\(374\) −1.70064 −0.0879382
\(375\) 0 0
\(376\) −52.5100 −2.70800
\(377\) 0 0
\(378\) 0 0
\(379\) −2.04555 −0.105073 −0.0525363 0.998619i \(-0.516731\pi\)
−0.0525363 + 0.998619i \(0.516731\pi\)
\(380\) 24.2326 1.24311
\(381\) 0 0
\(382\) 13.5833 0.694982
\(383\) −7.90521 −0.403937 −0.201969 0.979392i \(-0.564734\pi\)
−0.201969 + 0.979392i \(0.564734\pi\)
\(384\) 0 0
\(385\) 2.03741 0.103836
\(386\) −30.3394 −1.54423
\(387\) 0 0
\(388\) 17.8469 0.906037
\(389\) 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
\(392\) −18.7344 −0.946229
\(393\) 0 0
\(394\) −10.9261 −0.550448
\(395\) 9.31937 0.468908
\(396\) 0 0
\(397\) 6.35438 0.318917 0.159458 0.987205i \(-0.449025\pi\)
0.159458 + 0.987205i \(0.449025\pi\)
\(398\) 51.9996 2.60651
\(399\) 0 0
\(400\) 5.41713 0.270857
\(401\) 4.16920 0.208200 0.104100 0.994567i \(-0.466804\pi\)
0.104100 + 0.994567i \(0.466804\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 64.4600 3.20701
\(405\) 0 0
\(406\) 44.9541 2.23103
\(407\) −0.810008 −0.0401506
\(408\) 0 0
\(409\) −10.1681 −0.502778 −0.251389 0.967886i \(-0.580887\pi\)
−0.251389 + 0.967886i \(0.580887\pi\)
\(410\) −0.668669 −0.0330232
\(411\) 0 0
\(412\) −57.1858 −2.81734
\(413\) −1.41183 −0.0694716
\(414\) 0 0
\(415\) −5.11778 −0.251222
\(416\) 0 0
\(417\) 0 0
\(418\) −15.2969 −0.748198
\(419\) −28.5909 −1.39676 −0.698378 0.715730i \(-0.746094\pi\)
−0.698378 + 0.715730i \(0.746094\pi\)
\(420\) 0 0
\(421\) −2.01797 −0.0983498 −0.0491749 0.998790i \(-0.515659\pi\)
−0.0491749 + 0.998790i \(0.515659\pi\)
\(422\) −26.5863 −1.29420
\(423\) 0 0
\(424\) 38.8623 1.88732
\(425\) −0.637263 −0.0309118
\(426\) 0 0
\(427\) −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 \(-0.665356\pi\)
−0.496430 + 0.868077i \(0.665356\pi\)
\(432\) 0 0
\(433\) 29.4356 1.41458 0.707292 0.706921i \(-0.249917\pi\)
0.707292 + 0.706921i \(0.249917\pi\)
\(434\) −6.96103 −0.334140
\(435\) 0 0
\(436\) −42.5967 −2.04001
\(437\) 21.8841 1.04686
\(438\) 0 0
\(439\) 16.9520 0.809077 0.404538 0.914521i \(-0.367432\pi\)
0.404538 + 0.914521i \(0.367432\pi\)
\(440\) −5.94462 −0.283398
\(441\) 0 0
\(442\) 0 0
\(443\) 24.1399 1.14692 0.573461 0.819233i \(-0.305600\pi\)
0.573461 + 0.819233i \(0.305600\pi\)
\(444\) 0 0
\(445\) 12.5783 0.596270
\(446\) −53.2525 −2.52158
\(447\) 0 0
\(448\) 9.22742 0.435955
\(449\) −20.8630 −0.984585 −0.492293 0.870430i \(-0.663841\pi\)
−0.492293 + 0.870430i \(0.663841\pi\)
\(450\) 0 0
\(451\) 0.286542 0.0134927
\(452\) 28.2742 1.32990
\(453\) 0 0
\(454\) 39.1437 1.83710
\(455\) 0 0
\(456\) 0 0
\(457\) 30.5659 1.42981 0.714906 0.699220i \(-0.246469\pi\)
0.714906 + 0.699220i \(0.246469\pi\)
\(458\) 19.0179 0.888648
\(459\) 0 0
\(460\) 16.1402 0.752541
\(461\) 4.67822 0.217887 0.108943 0.994048i \(-0.465253\pi\)
0.108943 + 0.994048i \(0.465253\pi\)
\(462\) 0 0
\(463\) 14.0011 0.650688 0.325344 0.945596i \(-0.394520\pi\)
0.325344 + 0.945596i \(0.394520\pi\)
\(464\) −51.2196 −2.37781
\(465\) 0 0
\(466\) 47.5518 2.20280
\(467\) 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
\(470\) 23.5729 1.08734
\(471\) 0 0
\(472\) 4.11933 0.189608
\(473\) 0.681482 0.0313346
\(474\) 0 0
\(475\) −5.73205 −0.263005
\(476\) 5.13277 0.235260
\(477\) 0 0
\(478\) 31.8264 1.45571
\(479\) −16.2888 −0.744252 −0.372126 0.928182i \(-0.621371\pi\)
−0.372126 + 0.928182i \(0.621371\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −64.7056 −2.94726
\(483\) 0 0
\(484\) −41.6685 −1.89402
\(485\) −4.22155 −0.191691
\(486\) 0 0
\(487\) −20.0409 −0.908139 −0.454069 0.890966i \(-0.650028\pi\)
−0.454069 + 0.890966i \(0.650028\pi\)
\(488\) 23.3393 1.05652
\(489\) 0 0
\(490\) 8.41027 0.379937
\(491\) 15.7983 0.712969 0.356484 0.934301i \(-0.383975\pi\)
0.356484 + 0.934301i \(0.383975\pi\)
\(492\) 0 0
\(493\) 6.02540 0.271370
\(494\) 0 0
\(495\) 0 0
\(496\) 7.93123 0.356123
\(497\) −18.6042 −0.834511
\(498\) 0 0
\(499\) −1.24651 −0.0558016 −0.0279008 0.999611i \(-0.508882\pi\)
−0.0279008 + 0.999611i \(0.508882\pi\)
\(500\) −4.22756 −0.189062
\(501\) 0 0
\(502\) −18.9955 −0.847809
\(503\) 7.65345 0.341250 0.170625 0.985336i \(-0.445421\pi\)
0.170625 + 0.985336i \(0.445421\pi\)
\(504\) 0 0
\(505\) −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 \(-0.306904\pi\)
0.570101 + 0.821575i \(0.306904\pi\)
\(510\) 0 0
\(511\) −7.08070 −0.313232
\(512\) −47.2215 −2.08691
\(513\) 0 0
\(514\) −0.837948 −0.0369603
\(515\) 13.5269 0.596067
\(516\) 0 0
\(517\) −10.1016 −0.444268
\(518\) 3.60127 0.158231
\(519\) 0 0
\(520\) 0 0
\(521\) 30.1519 1.32098 0.660490 0.750835i \(-0.270349\pi\)
0.660490 + 0.750835i \(0.270349\pi\)
\(522\) 0 0
\(523\) 3.93752 0.172176 0.0860880 0.996288i \(-0.472563\pi\)
0.0860880 + 0.996288i \(0.472563\pi\)
\(524\) −17.4461 −0.762138
\(525\) 0 0
\(526\) 13.4156 0.584947
\(527\) −0.933018 −0.0406429
\(528\) 0 0
\(529\) −8.42399 −0.366261
\(530\) −17.4461 −0.757812
\(531\) 0 0
\(532\) 46.1682 2.00165
\(533\) 0 0
\(534\) 0 0
\(535\) −7.36274 −0.318319
\(536\) −44.9980 −1.94362
\(537\) 0 0
\(538\) 3.27007 0.140983
\(539\) −3.60402 −0.155236
\(540\) 0 0
\(541\) 15.8881 0.683083 0.341541 0.939867i \(-0.389051\pi\)
0.341541 + 0.939867i \(0.389051\pi\)
\(542\) 29.0610 1.24828
\(543\) 0 0
\(544\) −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
\(549\) 0 0
\(550\) 2.66867 0.113792
\(551\) 54.1972 2.30888
\(552\) 0 0
\(553\) 17.7554 0.755035
\(554\) −50.6990 −2.15399
\(555\) 0 0
\(556\) 88.0401 3.73373
\(557\) 7.85006 0.332618 0.166309 0.986074i \(-0.446815\pi\)
0.166309 + 0.986074i \(0.446815\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 10.3208 0.436133
\(561\) 0 0
\(562\) −29.6326 −1.24998
\(563\) 15.5595 0.655755 0.327878 0.944720i \(-0.393667\pi\)
0.327878 + 0.944720i \(0.393667\pi\)
\(564\) 0 0
\(565\) −6.68806 −0.281369
\(566\) −56.5285 −2.37607
\(567\) 0 0
\(568\) 54.2819 2.27762
\(569\) −3.47915 −0.145853 −0.0729267 0.997337i \(-0.523234\pi\)
−0.0729267 + 0.997337i \(0.523234\pi\)
\(570\) 0 0
\(571\) −21.5118 −0.900240 −0.450120 0.892968i \(-0.648619\pi\)
−0.450120 + 0.892968i \(0.648619\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −1.27396 −0.0531739
\(575\) −3.81785 −0.159216
\(576\) 0 0
\(577\) −9.97608 −0.415310 −0.207655 0.978202i \(-0.566583\pi\)
−0.207655 + 0.978202i \(0.566583\pi\)
\(578\) −41.4102 −1.72244
\(579\) 0 0
\(580\) 39.9721 1.65975
\(581\) −9.75045 −0.404517
\(582\) 0 0
\(583\) 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 \(-0.665372\pi\)
−0.496472 + 0.868053i \(0.665372\pi\)
\(588\) 0 0
\(589\) −8.39230 −0.345799
\(590\) −1.84926 −0.0761328
\(591\) 0 0
\(592\) −4.10320 −0.168641
\(593\) 0.940219 0.0386102 0.0193051 0.999814i \(-0.493855\pi\)
0.0193051 + 0.999814i \(0.493855\pi\)
\(594\) 0 0
\(595\) −1.21412 −0.0497741
\(596\) 56.5130 2.31486
\(597\) 0 0
\(598\) 0 0
\(599\) 11.4270 0.466896 0.233448 0.972369i \(-0.424999\pi\)
0.233448 + 0.972369i \(0.424999\pi\)
\(600\) 0 0
\(601\) −36.0431 −1.47023 −0.735114 0.677944i \(-0.762871\pi\)
−0.735114 + 0.677944i \(0.762871\pi\)
\(602\) −3.02985 −0.123487
\(603\) 0 0
\(604\) −77.3574 −3.14763
\(605\) 9.85641 0.400720
\(606\) 0 0
\(607\) 39.8907 1.61911 0.809557 0.587041i \(-0.199707\pi\)
0.809557 + 0.587041i \(0.199707\pi\)
\(608\) −13.7610 −0.558084
\(609\) 0 0
\(610\) −10.4775 −0.424223
\(611\) 0 0
\(612\) 0 0
\(613\) 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 0
\(616\) −11.3258 −0.456328
\(617\) 38.6850 1.55740 0.778700 0.627397i \(-0.215879\pi\)
0.778700 + 0.627397i \(0.215879\pi\)
\(618\) 0 0
\(619\) −14.8971 −0.598764 −0.299382 0.954133i \(-0.596781\pi\)
−0.299382 + 0.954133i \(0.596781\pi\)
\(620\) −6.18958 −0.248579
\(621\) 0 0
\(622\) 7.93719 0.318252
\(623\) 23.9644 0.960113
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 88.2245 3.52616
\(627\) 0 0
\(628\) 10.2404 0.408635
\(629\) 0.482694 0.0192463
\(630\) 0 0
\(631\) 38.8450 1.54640 0.773198 0.634165i \(-0.218656\pi\)
0.773198 + 0.634165i \(0.218656\pi\)
\(632\) −51.8053 −2.06071
\(633\) 0 0
\(634\) −34.0280 −1.35143
\(635\) −1.48950 −0.0591090
\(636\) 0 0
\(637\) 0 0
\(638\) −25.2326 −0.998967
\(639\) 0 0
\(640\) 16.8878 0.667549
\(641\) −37.1816 −1.46859 −0.734293 0.678832i \(-0.762486\pi\)
−0.734293 + 0.678832i \(0.762486\pi\)
\(642\) 0 0
\(643\) −9.10377 −0.359018 −0.179509 0.983756i \(-0.557451\pi\)
−0.179509 + 0.983756i \(0.557451\pi\)
\(644\) 30.7505 1.21174
\(645\) 0 0
\(646\) 9.11565 0.358651
\(647\) 19.1224 0.751778 0.375889 0.926665i \(-0.377337\pi\)
0.375889 + 0.926665i \(0.377337\pi\)
\(648\) 0 0
\(649\) 0.792455 0.0311066
\(650\) 0 0
\(651\) 0 0
\(652\) −67.5686 −2.64619
\(653\) −34.6324 −1.35527 −0.677636 0.735397i \(-0.736996\pi\)
−0.677636 + 0.735397i \(0.736996\pi\)
\(654\) 0 0
\(655\) 4.12676 0.161246
\(656\) 1.45152 0.0566722
\(657\) 0 0
\(658\) 44.9114 1.75083
\(659\) 6.69852 0.260937 0.130469 0.991452i \(-0.458352\pi\)
0.130469 + 0.991452i \(0.458352\pi\)
\(660\) 0 0
\(661\) −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
\(666\) 0 0
\(667\) 36.0983 1.39773
\(668\) −60.8479 −2.35428
\(669\) 0 0
\(670\) 20.2006 0.780417
\(671\) 4.48990 0.173330
\(672\) 0 0
\(673\) 23.3568 0.900338 0.450169 0.892943i \(-0.351364\pi\)
0.450169 + 0.892943i \(0.351364\pi\)
\(674\) 29.3205 1.12938
\(675\) 0 0
\(676\) 0 0
\(677\) 45.4042 1.74503 0.872513 0.488590i \(-0.162489\pi\)
0.872513 + 0.488590i \(0.162489\pi\)
\(678\) 0 0
\(679\) −8.04295 −0.308660
\(680\) 3.54248 0.135848
\(681\) 0 0
\(682\) 3.90720 0.149615
\(683\) −25.4978 −0.975645 −0.487823 0.872943i \(-0.662209\pi\)
−0.487823 + 0.872943i \(0.662209\pi\)
\(684\) 0 0
\(685\) 20.1096 0.768350
\(686\) 49.3047 1.88246
\(687\) 0 0
\(688\) 3.45214 0.131612
\(689\) 0 0
\(690\) 0 0
\(691\) 6.59630 0.250935 0.125468 0.992098i \(-0.459957\pi\)
0.125468 + 0.992098i \(0.459957\pi\)
\(692\) 102.937 3.91306
\(693\) 0 0
\(694\) −4.74090 −0.179962
\(695\) −20.8253 −0.789948
\(696\) 0 0
\(697\) −0.170754 −0.00646778
\(698\) 25.6266 0.969981
\(699\) 0 0
\(700\) −8.05440 −0.304428
\(701\) −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
\(704\) −5.17932 −0.195203
\(705\) 0 0
\(706\) 1.99787 0.0751910
\(707\) −29.0499 −1.09253
\(708\) 0 0
\(709\) −10.9335 −0.410614 −0.205307 0.978698i \(-0.565819\pi\)
−0.205307 + 0.978698i \(0.565819\pi\)
\(710\) −24.3683 −0.914527
\(711\) 0 0
\(712\) −69.9216 −2.62042
\(713\) −5.58973 −0.209337
\(714\) 0 0
\(715\) 0 0
\(716\) −16.0152 −0.598516
\(717\) 0 0
\(718\) 20.2953 0.757412
\(719\) −16.0598 −0.598929 −0.299464 0.954107i \(-0.596808\pi\)
−0.299464 + 0.954107i \(0.596808\pi\)
\(720\) 0 0
\(721\) 25.7716 0.959786
\(722\) 34.5788 1.28689
\(723\) 0 0
\(724\) −35.8833 −1.33359
\(725\) −9.45512 −0.351154
\(726\) 0 0
\(727\) 51.3754 1.90541 0.952704 0.303900i \(-0.0982889\pi\)
0.952704 + 0.303900i \(0.0982889\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −9.27453 −0.343266
\(731\) −0.406104 −0.0150203
\(732\) 0 0
\(733\) 9.82358 0.362842 0.181421 0.983406i \(-0.441930\pi\)
0.181421 + 0.983406i \(0.441930\pi\)
\(734\) −51.2240 −1.89071
\(735\) 0 0
\(736\) −9.16560 −0.337848
\(737\) −8.65648 −0.318866
\(738\) 0 0
\(739\) −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 \(-0.230490\pi\)
0.749091 + 0.662467i \(0.230490\pi\)
\(744\) 0 0
\(745\) −13.3678 −0.489757
\(746\) −44.4346 −1.62687
\(747\) 0 0
\(748\) −2.88101 −0.105340
\(749\) −14.0276 −0.512557
\(750\) 0 0
\(751\) −2.72680 −0.0995024 −0.0497512 0.998762i \(-0.515843\pi\)
−0.0497512 + 0.998762i \(0.515843\pi\)
\(752\) −51.1710 −1.86601
\(753\) 0 0
\(754\) 0 0
\(755\) 18.2984 0.665946
\(756\) 0 0
\(757\) 14.8060 0.538134 0.269067 0.963121i \(-0.413285\pi\)
0.269067 + 0.963121i \(0.413285\pi\)
\(758\) −5.10468 −0.185410
\(759\) 0 0
\(760\) 31.8638 1.15582
\(761\) −11.3689 −0.412122 −0.206061 0.978539i \(-0.566065\pi\)
−0.206061 + 0.978539i \(0.566065\pi\)
\(762\) 0 0
\(763\) 19.1969 0.694973
\(764\) 23.0110 0.832510
\(765\) 0 0
\(766\) −19.7275 −0.712784
\(767\) 0 0
\(768\) 0 0
\(769\) 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 0
\(772\) −51.3970 −1.84982
\(773\) 14.0829 0.506526 0.253263 0.967397i \(-0.418496\pi\)
0.253263 + 0.967397i \(0.418496\pi\)
\(774\) 0 0
\(775\) 1.46410 0.0525921
\(776\) 23.4671 0.842421
\(777\) 0 0
\(778\) 22.9879 0.824156
\(779\) −1.53590 −0.0550293
\(780\) 0 0
\(781\) 10.4425 0.373660
\(782\) 6.07151 0.217117
\(783\) 0 0
\(784\) −18.2566 −0.652023
\(785\) −2.42229 −0.0864552
\(786\) 0 0
\(787\) −33.0242 −1.17719 −0.588593 0.808429i \(-0.700318\pi\)
−0.588593 + 0.808429i \(0.700318\pi\)
\(788\) −18.5095 −0.659374
\(789\) 0 0
\(790\) 23.2566 0.827431
\(791\) −12.7422 −0.453060
\(792\) 0 0
\(793\) 0 0
\(794\) 15.8574 0.562758
\(795\) 0 0
\(796\) 88.0909 3.12230
\(797\) 16.9416 0.600102 0.300051 0.953923i \(-0.402996\pi\)
0.300051 + 0.953923i \(0.402996\pi\)
\(798\) 0 0
\(799\) 6.01967 0.212961
\(800\) 2.40072 0.0848783
\(801\) 0 0
\(802\) 10.4043 0.367387
\(803\) 3.97437 0.140253
\(804\) 0 0
\(805\) −7.27382 −0.256369
\(806\) 0 0
\(807\) 0 0
\(808\) 84.7596 2.98183
\(809\) 51.7635 1.81991 0.909954 0.414708i \(-0.136116\pi\)
0.909954 + 0.414708i \(0.136116\pi\)
\(810\) 0 0
\(811\) −22.6699 −0.796047 −0.398023 0.917375i \(-0.630304\pi\)
−0.398023 + 0.917375i \(0.630304\pi\)
\(812\) 76.1553 2.67253
\(813\) 0 0
\(814\) −2.02138 −0.0708494
\(815\) 15.9829 0.559856
\(816\) 0 0
\(817\) −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 \(-0.333269\pi\)
0.500174 + 0.865925i \(0.333269\pi\)
\(822\) 0 0
\(823\) −25.8327 −0.900472 −0.450236 0.892910i \(-0.648660\pi\)
−0.450236 + 0.892910i \(0.648660\pi\)
\(824\) −75.1946 −2.61953
\(825\) 0 0
\(826\) −3.52323 −0.122589
\(827\) −16.0820 −0.559227 −0.279613 0.960113i \(-0.590206\pi\)
−0.279613 + 0.960113i \(0.590206\pi\)
\(828\) 0 0
\(829\) −22.5818 −0.784298 −0.392149 0.919902i \(-0.628268\pi\)
−0.392149 + 0.919902i \(0.628268\pi\)
\(830\) −12.7715 −0.443304
\(831\) 0 0
\(832\) 0 0
\(833\) 2.14768 0.0744128
\(834\) 0 0
\(835\) 14.3932 0.498096
\(836\) −25.9141 −0.896257
\(837\) 0 0
\(838\) −71.3487 −2.46470
\(839\) −17.8440 −0.616042 −0.308021 0.951380i \(-0.599667\pi\)
−0.308021 + 0.951380i \(0.599667\pi\)
\(840\) 0 0
\(841\) 60.3992 2.08273
\(842\) −5.03586 −0.173547
\(843\) 0 0
\(844\) −45.0390 −1.55031
\(845\) 0 0
\(846\) 0 0
\(847\) 18.7785 0.645239
\(848\) 37.8713 1.30050
\(849\) 0 0
\(850\) −1.59030 −0.0545467
\(851\) 2.89183 0.0991306
\(852\) 0 0
\(853\) −19.7936 −0.677720 −0.338860 0.940837i \(-0.610041\pi\)
−0.338860 + 0.940837i \(0.610041\pi\)
\(854\) −19.9619 −0.683083
\(855\) 0 0
\(856\) 40.9286 1.39891
\(857\) 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 \(0.470774\pi\)
\(860\) −2.69407 −0.0918669
\(861\) 0 0
\(862\) −51.4382 −1.75199
\(863\) −25.3234 −0.862017 −0.431008 0.902348i \(-0.641842\pi\)
−0.431008 + 0.902348i \(0.641842\pi\)
\(864\) 0 0
\(865\) −24.3489 −0.827889
\(866\) 73.4567 2.49616
\(867\) 0 0
\(868\) −11.7925 −0.400262
\(869\) −9.96603 −0.338075
\(870\) 0 0
\(871\) 0 0
\(872\) −56.0112 −1.89678
\(873\) 0 0
\(874\) 54.6120 1.84728
\(875\) 1.90521 0.0644079
\(876\) 0 0
\(877\) −20.6915 −0.698703 −0.349352 0.936992i \(-0.613598\pi\)
−0.349352 + 0.936992i \(0.613598\pi\)
\(878\) 42.3040 1.42769
\(879\) 0 0
\(880\) −5.79302 −0.195283
\(881\) −48.3993 −1.63061 −0.815307 0.579029i \(-0.803432\pi\)
−0.815307 + 0.579029i \(0.803432\pi\)
\(882\) 0 0
\(883\) 45.8550 1.54314 0.771572 0.636142i \(-0.219471\pi\)
0.771572 + 0.636142i \(0.219471\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 60.2413 2.02385
\(887\) 1.08234 0.0363413 0.0181707 0.999835i \(-0.494216\pi\)
0.0181707 + 0.999835i \(0.494216\pi\)
\(888\) 0 0
\(889\) −2.83781 −0.0951772
\(890\) 31.3893 1.05217
\(891\) 0 0
\(892\) −90.2133 −3.02056
\(893\) 54.1457 1.81192
\(894\) 0 0
\(895\) 3.78829 0.126628
\(896\) 32.1749 1.07489
\(897\) 0 0
\(898\) −52.0637 −1.73739
\(899\) −13.8433 −0.461698
\(900\) 0 0
\(901\) −4.45512 −0.148421
\(902\) 0.715068 0.0238092
\(903\) 0 0
\(904\) 37.1782 1.23653
\(905\) 8.48794 0.282149
\(906\) 0 0
\(907\) −45.5307 −1.51182 −0.755910 0.654675i \(-0.772805\pi\)
−0.755910 + 0.654675i \(0.772805\pi\)
\(908\) 66.3120 2.20064
\(909\) 0 0
\(910\) 0 0
\(911\) −39.7417 −1.31670 −0.658350 0.752712i \(-0.728745\pi\)
−0.658350 + 0.752712i \(0.728745\pi\)
\(912\) 0 0
\(913\) 5.47290 0.181126
\(914\) 76.2774 2.52303
\(915\) 0 0
\(916\) 32.2176 1.06450
\(917\) 7.86236 0.259638
\(918\) 0 0
\(919\) 46.9938 1.55018 0.775091 0.631850i \(-0.217704\pi\)
0.775091 + 0.631850i \(0.217704\pi\)
\(920\) 21.2230 0.699702
\(921\) 0 0
\(922\) 11.6745 0.384481
\(923\) 0 0
\(924\) 0 0
\(925\) −0.757449 −0.0249048
\(926\) 34.9399 1.14820
\(927\) 0 0
\(928\) −22.6991 −0.745134
\(929\) −15.2213 −0.499395 −0.249698 0.968324i \(-0.580331\pi\)
−0.249698 + 0.968324i \(0.580331\pi\)
\(930\) 0 0
\(931\) 19.3180 0.633121
\(932\) 80.5560 2.63870
\(933\) 0 0
\(934\) 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
\(939\) 0 0
\(940\) 39.9341 1.30251
\(941\) −0.0496576 −0.00161879 −0.000809396 1.00000i \(-0.500258\pi\)
−0.000809396 1.00000i \(0.500258\pi\)
\(942\) 0 0
\(943\) −1.02299 −0.0333132
\(944\) 4.01429 0.130654
\(945\) 0 0
\(946\) 1.70064 0.0552927
\(947\) −18.6581 −0.606308 −0.303154 0.952942i \(-0.598040\pi\)
−0.303154 + 0.952942i \(0.598040\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −14.3044 −0.464095
\(951\) 0 0
\(952\) 6.74917 0.218742
\(953\) 1.52953 0.0495463 0.0247731 0.999693i \(-0.492114\pi\)
0.0247731 + 0.999693i \(0.492114\pi\)
\(954\) 0 0
\(955\) −5.44310 −0.176135
\(956\) 53.9161 1.74377
\(957\) 0 0
\(958\) −40.6487 −1.31330
\(959\) 38.3131 1.23720
\(960\) 0 0
\(961\) −28.8564 −0.930852
\(962\) 0 0
\(963\) 0 0
\(964\) −109.616 −3.53048
\(965\) 12.1576 0.391367
\(966\) 0 0
\(967\) −32.1716 −1.03457 −0.517285 0.855813i \(-0.673057\pi\)
−0.517285 + 0.855813i \(0.673057\pi\)
\(968\) −54.7907 −1.76104
\(969\) 0 0
\(970\) −10.5349 −0.338256
\(971\) 17.2541 0.553710 0.276855 0.960912i \(-0.410708\pi\)
0.276855 + 0.960912i \(0.410708\pi\)
\(972\) 0 0
\(973\) −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 \(-0.419073\pi\)
0.251509 + 0.967855i \(0.419073\pi\)
\(978\) 0 0
\(979\) −13.4511 −0.429900
\(980\) 14.2476 0.455122
\(981\) 0 0
\(982\) 39.4248 1.25810
\(983\) 38.5356 1.22910 0.614548 0.788880i \(-0.289338\pi\)
0.614548 + 0.788880i \(0.289338\pi\)
\(984\) 0 0
\(985\) 4.37830 0.139504
\(986\) 15.0364 0.478857
\(987\) 0 0
\(988\) 0 0
\(989\) −2.43298 −0.0773642
\(990\) 0 0
\(991\) 8.59143 0.272916 0.136458 0.990646i \(-0.456428\pi\)
0.136458 + 0.990646i \(0.456428\pi\)
\(992\) 3.51490 0.111598
\(993\) 0 0
\(994\) −46.4268 −1.47257
\(995\) −20.8373 −0.660587
\(996\) 0 0
\(997\) −20.5374 −0.650425 −0.325213 0.945641i \(-0.605436\pi\)
−0.325213 + 0.945641i \(0.605436\pi\)
\(998\) −3.11069 −0.0984670
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7605.2.a.cj.1.4 4
3.2 odd 2 845.2.a.l.1.1 4
13.2 odd 12 585.2.bu.c.316.4 8
13.7 odd 12 585.2.bu.c.361.4 8
13.12 even 2 7605.2.a.cf.1.1 4
15.14 odd 2 4225.2.a.bl.1.4 4
39.2 even 12 65.2.m.a.56.1 yes 8
39.5 even 4 845.2.c.g.506.8 8
39.8 even 4 845.2.c.g.506.1 8
39.11 even 12 845.2.m.g.316.4 8
39.17 odd 6 845.2.e.m.146.1 8
39.20 even 12 65.2.m.a.36.1 8
39.23 odd 6 845.2.e.m.191.1 8
39.29 odd 6 845.2.e.n.191.4 8
39.32 even 12 845.2.m.g.361.4 8
39.35 odd 6 845.2.e.n.146.4 8
39.38 odd 2 845.2.a.m.1.4 4
156.59 odd 12 1040.2.da.b.881.1 8
156.119 odd 12 1040.2.da.b.641.1 8
195.2 odd 12 325.2.m.c.199.4 8
195.59 even 12 325.2.n.d.101.4 8
195.98 odd 12 325.2.m.c.49.4 8
195.119 even 12 325.2.n.d.251.4 8
195.137 odd 12 325.2.m.b.49.1 8
195.158 odd 12 325.2.m.b.199.1 8
195.194 odd 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 39.20 even 12
65.2.m.a.56.1 yes 8 39.2 even 12
325.2.m.b.49.1 8 195.137 odd 12
325.2.m.b.199.1 8 195.158 odd 12
325.2.m.c.49.4 8 195.98 odd 12
325.2.m.c.199.4 8 195.2 odd 12
325.2.n.d.101.4 8 195.59 even 12
325.2.n.d.251.4 8 195.119 even 12
585.2.bu.c.316.4 8 13.2 odd 12
585.2.bu.c.361.4 8 13.7 odd 12
845.2.a.l.1.1 4 3.2 odd 2
845.2.a.m.1.4 4 39.38 odd 2
845.2.c.g.506.1 8 39.8 even 4
845.2.c.g.506.8 8 39.5 even 4
845.2.e.m.146.1 8 39.17 odd 6
845.2.e.m.191.1 8 39.23 odd 6
845.2.e.n.146.4 8 39.35 odd 6
845.2.e.n.191.4 8 39.29 odd 6
845.2.m.g.316.4 8 39.11 even 12
845.2.m.g.361.4 8 39.32 even 12
1040.2.da.b.641.1 8 156.119 odd 12
1040.2.da.b.881.1 8 156.59 odd 12
4225.2.a.bi.1.1 4 195.194 odd 2
4225.2.a.bl.1.4 4 15.14 odd 2
7605.2.a.cf.1.1 4 13.12 even 2
7605.2.a.cj.1.4 4 1.1 even 1 trivial