Properties

Label 4225.2.a.bl.1.4
Level $4225$
Weight $2$
Character 4225.1
Self dual yes
Analytic conductor $33.737$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4225 = 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.7367948540\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 4225.1

$q$-expansion

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

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 \(-0.155997\pi\)
0.882295 + 0.470696i \(0.155997\pi\)
\(3\) 2.82684 1.63208 0.816038 0.577998i \(-0.196166\pi\)
0.816038 + 0.577998i \(0.196166\pi\)
\(4\) 4.22756 2.11378
\(5\) 0 0
\(6\) 7.05440 2.87995
\(7\) 1.90521 0.720103 0.360051 0.932933i \(-0.382759\pi\)
0.360051 + 0.932933i \(0.382759\pi\)
\(8\) 5.55889 1.96536
\(9\) 4.99102 1.66367
\(10\) 0 0
\(11\) −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
\(15\) 0 0
\(16\) 5.41713 1.35428
\(17\) −0.637263 −0.154559 −0.0772795 0.997009i \(-0.524623\pi\)
−0.0772795 + 0.997009i \(0.524623\pi\)
\(18\) 12.4551 2.93570
\(19\) −5.73205 −1.31502 −0.657511 0.753445i \(-0.728391\pi\)
−0.657511 + 0.753445i \(0.728391\pi\)
\(20\) 0 0
\(21\) 5.38573 1.17526
\(22\) −2.66867 −0.568962
\(23\) −3.81785 −0.796078 −0.398039 0.917369i \(-0.630309\pi\)
−0.398039 + 0.917369i \(0.630309\pi\)
\(24\) 15.7141 3.20762
\(25\) 0 0
\(26\) 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 \(0.158954\pi\)
\(30\) 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
\(33\) −3.02299 −0.526235
\(34\) −1.59030 −0.272733
\(35\) 0 0
\(36\) 21.0998 3.51663
\(37\) 0.757449 0.124524 0.0622619 0.998060i \(-0.480169\pi\)
0.0622619 + 0.998060i \(0.480169\pi\)
\(38\) −14.3044 −2.32048
\(39\) 0 0
\(40\) 0 0
\(41\) −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 \(-0.515473\pi\)
−0.0485909 + 0.998819i \(0.515473\pi\)
\(44\) −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 \(0.741919\pi\)
\(48\) 15.3134 2.21029
\(49\) −3.37017 −0.481452
\(50\) 0 0
\(51\) −1.80144 −0.252252
\(52\) 0 0
\(53\) 6.99102 0.960290 0.480145 0.877189i \(-0.340584\pi\)
0.480145 + 0.877189i \(0.340584\pi\)
\(54\) 14.0454 1.91134
\(55\) 0 0
\(56\) 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 \(0.515360\pi\)
\(60\) 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
\(63\) 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 \(-0.335363\pi\)
0.494468 + 0.869196i \(0.335363\pi\)
\(68\) −2.69407 −0.326704
\(69\) −10.7925 −1.29926
\(70\) 0 0
\(71\) −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 \(-0.569787\pi\)
−0.217491 + 0.976062i \(0.569787\pi\)
\(74\) 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 \(0.675656\pi\)
\(80\) 0 0
\(81\) 0.937188 0.104132
\(82\) −0.668669 −0.0738422
\(83\) 5.11778 0.561749 0.280875 0.959744i \(-0.409376\pi\)
0.280875 + 0.959744i \(0.409376\pi\)
\(84\) 22.7685 2.48424
\(85\) 0 0
\(86\) −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 \(0.267727\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −16.1402 −1.68273
\(93\) 4.13878 0.429171
\(94\) −23.5729 −2.43136
\(95\) 0 0
\(96\) 6.78645 0.692639
\(97\) −4.22155 −0.428634 −0.214317 0.976764i \(-0.568752\pi\)
−0.214317 + 0.976764i \(0.568752\pi\)
\(98\) −8.41027 −0.849566
\(99\) −5.33734 −0.536423
\(100\) 0 0
\(101\) −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 \(-0.267824\pi\)
0.666423 + 0.745574i \(0.267824\pi\)
\(104\) 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 \(0.384177\pi\)
\(108\) 23.7939 2.28957
\(109\) −10.0760 −0.965103 −0.482551 0.875868i \(-0.660290\pi\)
−0.482551 + 0.875868i \(0.660290\pi\)
\(110\) 0 0
\(111\) 2.14119 0.203232
\(112\) 10.3208 0.975223
\(113\) 6.68806 0.629160 0.314580 0.949231i \(-0.398136\pi\)
0.314580 + 0.949231i \(0.398136\pi\)
\(114\) −40.4362 −3.78719
\(115\) 0 0
\(116\) 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
\(123\) −0.757449 −0.0682969
\(124\) 6.18958 0.555840
\(125\) 0 0
\(126\) 23.7296 2.11400
\(127\) −1.48950 −0.132172 −0.0660859 0.997814i \(-0.521051\pi\)
−0.0660859 + 0.997814i \(0.521051\pi\)
\(128\) −16.8878 −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
\(135\) 0 0
\(136\) −3.54248 −0.303765
\(137\) −20.1096 −1.71808 −0.859041 0.511906i \(-0.828940\pi\)
−0.859041 + 0.511906i \(0.828940\pi\)
\(138\) −26.9327 −2.29266
\(139\) 20.8253 1.76638 0.883189 0.469018i \(-0.155392\pi\)
0.883189 + 0.469018i \(0.155392\pi\)
\(140\) 0 0
\(141\) −26.7027 −2.24877
\(142\) −24.3683 −2.04494
\(143\) 0 0
\(144\) 27.0370 2.25308
\(145\) 0 0
\(146\) −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 \(0.684445\pi\)
\(150\) 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
\(153\) −3.18059 −0.257136
\(154\) −5.08438 −0.409711
\(155\) 0 0
\(156\) 0 0
\(157\) −2.42229 −0.193320 −0.0966599 0.995317i \(-0.530816\pi\)
−0.0966599 + 0.995317i \(0.530816\pi\)
\(158\) −23.2566 −1.85019
\(159\) 19.7625 1.56727
\(160\) 0 0
\(161\) −7.27382 −0.573258
\(162\) 2.33876 0.183750
\(163\) 15.9829 1.25188 0.625938 0.779873i \(-0.284716\pi\)
0.625938 + 0.779873i \(0.284716\pi\)
\(164\) −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 \(0.688005\pi\)
\(168\) 29.9387 2.30982
\(169\) 0 0
\(170\) 0 0
\(171\) −28.6088 −2.18777
\(172\) −2.69407 −0.205421
\(173\) 24.3489 1.85122 0.925608 0.378484i \(-0.123555\pi\)
0.925608 + 0.378484i \(0.123555\pi\)
\(174\) 66.7001 5.05653
\(175\) 0 0
\(176\) −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 \(0.454783\pi\)
\(180\) 0 0
\(181\) −8.48794 −0.630904 −0.315452 0.948942i \(-0.602156\pi\)
−0.315452 + 0.948942i \(0.602156\pi\)
\(182\) 0 0
\(183\) 11.8687 0.877356
\(184\) −21.2230 −1.56458
\(185\) 0 0
\(186\) 10.3284 0.757312
\(187\) 0.681482 0.0498349
\(188\) −39.9341 −2.91249
\(189\) 10.7231 0.779988
\(190\) 0 0
\(191\) −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 \(-0.355842\pi\)
0.437562 + 0.899188i \(0.355842\pi\)
\(194\) −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 \(0.549850\pi\)
\(198\) −13.3194 −0.946566
\(199\) 20.8373 1.47712 0.738558 0.674189i \(-0.235507\pi\)
0.738558 + 0.674189i \(0.235507\pi\)
\(200\) 0 0
\(201\) 22.8827 1.61402
\(202\) −38.0504 −2.67722
\(203\) 18.0140 1.26434
\(204\) −7.61569 −0.533205
\(205\) 0 0
\(206\) 33.7565 2.35193
\(207\) −19.0550 −1.32441
\(208\) 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
\(213\) −27.6037 −1.89138
\(214\) 18.3738 1.25600
\(215\) 0 0
\(216\) 31.2870 2.12881
\(217\) 2.78942 0.189358
\(218\) −25.1447 −1.70301
\(219\) −10.5059 −0.709924
\(220\) 0 0
\(221\) 0 0
\(222\) 5.34335 0.358622
\(223\) 21.3393 1.42899 0.714494 0.699642i \(-0.246657\pi\)
0.714494 + 0.699642i \(0.246657\pi\)
\(224\) 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 \(0.325728\pi\)
\(228\) −68.5016 −4.53663
\(229\) 7.62085 0.503600 0.251800 0.967779i \(-0.418977\pi\)
0.251800 + 0.967779i \(0.418977\pi\)
\(230\) 0 0
\(231\) −5.75944 −0.378943
\(232\) 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
\(235\) 0 0
\(236\) −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 \(0.635336\pi\)
\(240\) 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
\(243\) −14.2356 −0.913211
\(244\) 17.7497 1.13631
\(245\) 0 0
\(246\) −1.89022 −0.120516
\(247\) 0 0
\(248\) 8.13878 0.516813
\(249\) 14.4671 0.916817
\(250\) 0 0
\(251\) 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
\(255\) 0 0
\(256\) −32.4572 −2.02857
\(257\) −0.335783 −0.0209456 −0.0104728 0.999945i \(-0.503334\pi\)
−0.0104728 + 0.999945i \(0.503334\pi\)
\(258\) −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 \(-0.446997\pi\)
0.165746 + 0.986169i \(0.446997\pi\)
\(264\) −16.8045 −1.03424
\(265\) 0 0
\(266\) −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 \(0.512719\pi\)
\(270\) 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
\(275\) 0 0
\(276\) −45.6257 −2.74635
\(277\) 20.3161 1.22068 0.610338 0.792141i \(-0.291033\pi\)
0.610338 + 0.792141i \(0.291033\pi\)
\(278\) 51.9697 3.11693
\(279\) 7.30735 0.437480
\(280\) 0 0
\(281\) 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 \(-0.264892\pi\)
0.673264 + 0.739402i \(0.264892\pi\)
\(284\) −41.2816 −2.44961
\(285\) 0 0
\(286\) 0 0
\(287\) −0.510500 −0.0301339
\(288\) 11.9820 0.706048
\(289\) −16.5939 −0.976112
\(290\) 0 0
\(291\) −11.9336 −0.699562
\(292\) −15.7117 −0.919456
\(293\) 18.6127 1.08737 0.543683 0.839290i \(-0.317029\pi\)
0.543683 + 0.839290i \(0.317029\pi\)
\(294\) −23.7745 −1.38656
\(295\) 0 0
\(296\) 4.21058 0.244735
\(297\) −6.01882 −0.349247
\(298\) −33.3593 −1.93245
\(299\) 0 0
\(300\) 0 0
\(301\) −1.21412 −0.0699808
\(302\) −45.6637 −2.62765
\(303\) −43.1024 −2.47617
\(304\) −31.0513 −1.78091
\(305\) 0 0
\(306\) −7.93719 −0.453739
\(307\) 3.14776 0.179652 0.0898262 0.995957i \(-0.471369\pi\)
0.0898262 + 0.995957i \(0.471369\pi\)
\(308\) −8.61329 −0.490788
\(309\) 38.2384 2.17531
\(310\) 0 0
\(311\) −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 \(-0.986831\pi\)
−0.999144 + 0.0413596i \(0.986831\pi\)
\(314\) −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 \(0.625085\pi\)
\(318\) 49.3174 2.76558
\(319\) −10.1112 −0.566119
\(320\) 0 0
\(321\) 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
\(330\) 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
\(333\) 3.78044 0.207167
\(334\) −35.9182 −1.96536
\(335\) 0 0
\(336\) 29.1752 1.59164
\(337\) −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
\(340\) 0 0
\(341\) −1.56569 −0.0847871
\(342\) −71.3934 −3.86051
\(343\) −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 \(0.516238\pi\)
\(348\) 112.995 6.05714
\(349\) 10.2691 0.549692 0.274846 0.961488i \(-0.411373\pi\)
0.274846 + 0.961488i \(0.411373\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.56730 −0.136838
\(353\) 0.800589 0.0426110 0.0213055 0.999773i \(-0.493218\pi\)
0.0213055 + 0.999773i \(0.493218\pi\)
\(354\) −5.22756 −0.277842
\(355\) 0 0
\(356\) 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 \(0.568849\pi\)
\(360\) 0 0
\(361\) 13.8564 0.729285
\(362\) −21.1817 −1.11329
\(363\) −27.8625 −1.46240
\(364\) 0 0
\(365\) 0 0
\(366\) 29.6183 1.54817
\(367\) 20.5265 1.07147 0.535737 0.844385i \(-0.320034\pi\)
0.535737 + 0.844385i \(0.320034\pi\)
\(368\) −20.6818 −1.07811
\(369\) −1.33734 −0.0696191
\(370\) 0 0
\(371\) 13.3194 0.691507
\(372\) 17.4969 0.907173
\(373\) 17.8058 0.921951 0.460976 0.887413i \(-0.347500\pi\)
0.460976 + 0.887413i \(0.347500\pi\)
\(374\) 1.70064 0.0879382
\(375\) 0 0
\(376\) −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 \(0.516731\pi\)
\(380\) 0 0
\(381\) −4.21058 −0.215714
\(382\) −13.5833 −0.694982
\(383\) −7.90521 −0.403937 −0.201969 0.979392i \(-0.564734\pi\)
−0.201969 + 0.979392i \(0.564734\pi\)
\(384\) −47.7391 −2.43618
\(385\) 0 0
\(386\) 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
\(395\) 0 0
\(396\) −22.5639 −1.13388
\(397\) −6.35438 −0.318917 −0.159458 0.987205i \(-0.550975\pi\)
−0.159458 + 0.987205i \(0.550975\pi\)
\(398\) 51.9996 2.60651
\(399\) −30.8713 −1.54550
\(400\) 0 0
\(401\) −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 0
\(406\) 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 0
\(411\) −56.8467 −2.80404
\(412\) 57.1858 2.81734
\(413\) −1.41183 −0.0694716
\(414\) −47.5518 −2.33704
\(415\) 0 0
\(416\) 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 \(0.253906\pi\)
\(420\) 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
\(423\) −47.1458 −2.29231
\(424\) 38.8623 1.88732
\(425\) 0 0
\(426\) −68.8853 −3.33750
\(427\) 7.99915 0.387106
\(428\) 31.1264 1.50455
\(429\) 0 0
\(430\) 0 0
\(431\) 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 \(-0.750083\pi\)
−0.707292 + 0.706921i \(0.750083\pi\)
\(434\) 6.96103 0.334140
\(435\) 0 0
\(436\) −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 \(0.367432\pi\)
\(440\) 0 0
\(441\) −16.8205 −0.800978
\(442\) 0 0
\(443\) 24.1399 1.14692 0.573461 0.819233i \(-0.305600\pi\)
0.573461 + 0.819233i \(0.305600\pi\)
\(444\) 9.05199 0.429588
\(445\) 0 0
\(446\) 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 \(0.336159\pi\)
\(450\) 0 0
\(451\) 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 \(-0.753531\pi\)
−0.714906 + 0.699220i \(0.753531\pi\)
\(458\) 19.0179 0.888648
\(459\) −3.58669 −0.167413
\(460\) 0 0
\(461\) −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 \(-0.605480\pi\)
−0.325344 + 0.945596i \(0.605480\pi\)
\(464\) 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
\(470\) 0 0
\(471\) −6.84742 −0.315513
\(472\) −4.11933 −0.189608
\(473\) 0.681482 0.0313346
\(474\) −65.7425 −3.01965
\(475\) 0 0
\(476\) −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 \(0.378629\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −64.7056 −2.94726
\(483\) −20.5619 −0.935600
\(484\) −41.6685 −1.89402
\(485\) 0 0
\(486\) −35.5249 −1.61144
\(487\) 20.0409 0.908139 0.454069 0.890966i \(-0.349972\pi\)
0.454069 + 0.890966i \(0.349972\pi\)
\(488\) 23.3393 1.05652
\(489\) 45.1810 2.04316
\(490\) 0 0
\(491\) −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
\(495\) 0 0
\(496\) 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 \(0.508882\pi\)
\(500\) 0 0
\(501\) −40.6871 −1.81777
\(502\) 18.9955 0.847809
\(503\) 7.65345 0.341250 0.170625 0.985336i \(-0.445421\pi\)
0.170625 + 0.985336i \(0.445421\pi\)
\(504\) 52.8592 2.35453
\(505\) 0 0
\(506\) 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 \(0.693096\pi\)
\(510\) 0 0
\(511\) −7.08070 −0.313232
\(512\) −47.2215 −2.08691
\(513\) −32.2616 −1.42438
\(514\) −0.837948 −0.0369603
\(515\) 0 0
\(516\) −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 \(-0.527437\pi\)
−0.0860880 + 0.996288i \(0.527437\pi\)
\(524\) 17.4461 0.762138
\(525\) 0 0
\(526\) 13.4156 0.584947
\(527\) −0.933018 −0.0406429
\(528\) −16.3759 −0.712672
\(529\) −8.42399 −0.366261
\(530\) 0 0
\(531\) −3.69852 −0.160502
\(532\) −46.1682 −2.00165
\(533\) 0 0
\(534\) 88.7326 3.83983
\(535\) 0 0
\(536\) 44.9980 1.94362
\(537\) 10.7089 0.462122
\(538\) −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
\(543\) −23.9940 −1.02968
\(544\) −1.52989 −0.0655935
\(545\) 0 0
\(546\) 0 0
\(547\) 6.56107 0.280531 0.140266 0.990114i \(-0.455204\pi\)
0.140266 + 0.990114i \(0.455204\pi\)
\(548\) −85.0147 −3.63165
\(549\) 20.9551 0.894341
\(550\) 0 0
\(551\) −54.1972 −2.30888
\(552\) −59.9941 −2.55352
\(553\) −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 \(0.446815\pi\)
\(558\) 18.2356 0.771973
\(559\) 0 0
\(560\) 0 0
\(561\) 1.92644 0.0813344
\(562\) 29.6326 1.24998
\(563\) 15.5595 0.655755 0.327878 0.944720i \(-0.393667\pi\)
0.327878 + 0.944720i \(0.393667\pi\)
\(564\) −112.887 −4.75341
\(565\) 0 0
\(566\) 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 \(0.476766\pi\)
\(570\) 0 0
\(571\) −21.5118 −0.900240 −0.450120 0.892968i \(-0.648619\pi\)
−0.450120 + 0.892968i \(0.648619\pi\)
\(572\) 0 0
\(573\) −15.3868 −0.642791
\(574\) −1.27396 −0.0531739
\(575\) 0 0
\(576\) −24.1727 −1.00720
\(577\) 9.97608 0.415310 0.207655 0.978202i \(-0.433417\pi\)
0.207655 + 0.978202i \(0.433417\pi\)
\(578\) −41.4102 −1.72244
\(579\) 34.3676 1.42827
\(580\) 0 0
\(581\) 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 \(-0.665372\pi\)
−0.496472 + 0.868053i \(0.665372\pi\)
\(588\) −40.2756 −1.66094
\(589\) −8.39230 −0.345799
\(590\) 0 0
\(591\) −12.3767 −0.509111
\(592\) 4.10320 0.168641
\(593\) 0.940219 0.0386102 0.0193051 0.999814i \(-0.493855\pi\)
0.0193051 + 0.999814i \(0.493855\pi\)
\(594\) −15.0200 −0.616279
\(595\) 0 0
\(596\) −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 \(0.575001\pi\)
\(600\) 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
\(603\) 40.4012 1.64526
\(604\) −77.3574 −3.14763
\(605\) 0 0
\(606\) −107.562 −4.36942
\(607\) −39.8907 −1.61911 −0.809557 0.587041i \(-0.800293\pi\)
−0.809557 + 0.587041i \(0.800293\pi\)
\(608\) −13.7610 −0.558084
\(609\) 50.9227 2.06349
\(610\) 0 0
\(611\) 0 0
\(612\) −13.4461 −0.543528
\(613\) −0.345472 −0.0139535 −0.00697673 0.999976i \(-0.502221\pi\)
−0.00697673 + 0.999976i \(0.502221\pi\)
\(614\) 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 \(0.215879\pi\)
\(618\) 95.4242 3.83852
\(619\) −14.8971 −0.598764 −0.299382 0.954133i \(-0.596781\pi\)
−0.299382 + 0.954133i \(0.596781\pi\)
\(620\) 0 0
\(621\) −21.4879 −0.862281
\(622\) −7.93719 −0.318252
\(623\) 23.9644 0.960113
\(624\) 0 0
\(625\) 0 0
\(626\) −88.2245 −3.52616
\(627\) 17.3279 0.692011
\(628\) −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
\(633\) −30.1162 −1.19701
\(634\) −34.0280 −1.35143
\(635\) 0 0
\(636\) 83.5470 3.31285
\(637\) 0 0
\(638\) −25.2326 −0.998967
\(639\) −48.7367 −1.92799
\(640\) 0 0
\(641\) 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 \(-0.442549\pi\)
0.179509 + 0.983756i \(0.442549\pi\)
\(644\) −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 \(0.377337\pi\)
\(648\) 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 \(-0.736996\pi\)
−0.677636 + 0.735397i \(0.736996\pi\)
\(654\) −71.0799 −2.77944
\(655\) 0 0
\(656\) −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 \(0.541648\pi\)
\(660\) 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
\(665\) 0 0
\(666\) 9.43412 0.365565
\(667\) −36.0983 −1.39773
\(668\) −60.8479 −2.35428
\(669\) 60.3228 2.33222
\(670\) 0 0
\(671\) −4.48990 −0.173330
\(672\) 12.9296 0.498771
\(673\) −23.3568 −0.900338 −0.450169 0.892943i \(-0.648636\pi\)
−0.450169 + 0.892943i \(0.648636\pi\)
\(674\) −29.3205 −1.12938
\(675\) 0 0
\(676\) 0 0
\(677\) 45.4042 1.74503 0.872513 0.488590i \(-0.162489\pi\)
0.872513 + 0.488590i \(0.162489\pi\)
\(678\) 47.1802 1.81195
\(679\) −8.04295 −0.308660
\(680\) 0 0
\(681\) 44.3408 1.69914
\(682\) −3.90720 −0.149615
\(683\) −25.4978 −0.975645 −0.487823 0.872943i \(-0.662209\pi\)
−0.487823 + 0.872943i \(0.662209\pi\)
\(684\) −120.945 −4.62445
\(685\) 0 0
\(686\) −49.3047 −1.88246
\(687\) 21.5429 0.821913
\(688\) −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
\(693\) −10.1688 −0.386279
\(694\) −4.74090 −0.179962
\(695\) 0 0
\(696\) 148.578 5.63185
\(697\) 0.170754 0.00646778
\(698\) 25.6266 0.969981
\(699\) 53.8653 2.03737
\(700\) 0 0
\(701\) 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
\(705\) 0 0
\(706\) 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 \(0.565819\pi\)
\(710\) 0 0
\(711\) −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 \(0.403192\pi\)
\(720\) 0 0
\(721\) 25.7716 0.959786
\(722\) 34.5788 1.28689
\(723\) −73.2966 −2.72593
\(724\) −35.8833 −1.33359
\(725\) 0 0
\(726\) −69.5310 −2.58054
\(727\) −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
\(730\) 0 0
\(731\) 0.406104 0.0150203
\(732\) 50.1754 1.85454
\(733\) −9.82358 −0.362842 −0.181421 0.983406i \(-0.558070\pi\)
−0.181421 + 0.983406i \(0.558070\pi\)
\(734\) 51.2240 1.89071
\(735\) 0 0
\(736\) −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 \(0.858490\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 33.2386 1.22023
\(743\) 40.8375 1.49818 0.749091 0.662467i \(-0.230490\pi\)
0.749091 + 0.662467i \(0.230490\pi\)
\(744\) 23.0070 0.843478
\(745\) 0 0
\(746\) 44.4346 1.62687
\(747\) 25.5429 0.934566
\(748\) 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
\(753\) 21.5175 0.784142
\(754\) 0 0
\(755\) 0 0
\(756\) 45.3324 1.64872
\(757\) −14.8060 −0.538134 −0.269067 0.963121i \(-0.586715\pi\)
−0.269067 + 0.963121i \(0.586715\pi\)
\(758\) −5.10468 −0.185410
\(759\) 11.5413 0.418924
\(760\) 0 0
\(761\) 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
\(765\) 0 0
\(766\) −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 \(0.376043\pi\)
\(770\) 0 0
\(771\) −0.949203 −0.0341847
\(772\) 51.3970 1.84982
\(773\) 14.0829 0.506526 0.253263 0.967397i \(-0.418496\pi\)
0.253263 + 0.967397i \(0.418496\pi\)
\(774\) −7.93719 −0.285296
\(775\) 0 0
\(776\) −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
\(785\) 0 0
\(786\) 29.1118 1.03838
\(787\) 33.0242 1.17719 0.588593 0.808429i \(-0.299682\pi\)
0.588593 + 0.808429i \(0.299682\pi\)
\(788\) −18.5095 −0.659374
\(789\) 15.1968 0.541020
\(790\) 0 0
\(791\) 12.7422 0.453060
\(792\) −29.6697 −1.05427
\(793\) 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 \(0.402996\pi\)
\(798\) −77.0395 −2.72717
\(799\) 6.01967 0.212961
\(800\) 0 0
\(801\) 62.7787 2.21817
\(802\) −10.4043 −0.367387
\(803\) 3.97437 0.140253
\(804\) 96.7378 3.41168
\(805\) 0 0
\(806\) 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 \(0.863884\pi\)
\(810\) 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
\(813\) 32.9194 1.15453
\(814\) −2.02138 −0.0708494
\(815\) 0 0
\(816\) −9.75864 −0.341621
\(817\) 3.65283 0.127796
\(818\) −25.3745 −0.887198
\(819\) 0 0
\(820\) 0 0
\(821\) −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 \(-0.351340\pi\)
0.450236 + 0.892910i \(0.351340\pi\)
\(824\) 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 \(0.590206\pi\)
\(828\) −80.5560 −2.79951
\(829\) −22.5818 −0.784298 −0.392149 0.919902i \(-0.628268\pi\)
−0.392149 + 0.919902i \(0.628268\pi\)
\(830\) 0 0
\(831\) 57.4304 1.99224
\(832\) 0 0
\(833\) 2.14768 0.0744128
\(834\) 146.910 5.08707
\(835\) 0 0
\(836\) 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 \(0.400333\pi\)
\(840\) 0 0
\(841\) 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
\(850\) 0 0
\(851\) −2.89183 −0.0991306
\(852\) −116.696 −3.99795
\(853\) 19.7936 0.677720 0.338860 0.940837i \(-0.389959\pi\)
0.338860 + 0.940837i \(0.389959\pi\)
\(854\) 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 \(0.470774\pi\)
\(860\) 0 0
\(861\) −1.44310 −0.0491808
\(862\) 51.4382 1.75199
\(863\) −25.3234 −0.862017 −0.431008 0.902348i \(-0.641842\pi\)
−0.431008 + 0.902348i \(0.641842\pi\)
\(864\) 13.5119 0.459685
\(865\) 0 0
\(866\) −73.4567 −2.49616
\(867\) −46.9083 −1.59309
\(868\) 11.7925 0.400262
\(869\) 9.96603 0.338075
\(870\) 0 0
\(871\) 0 0
\(872\) −56.0112 −1.89678
\(873\) −21.0698 −0.713106
\(874\) 54.6120 1.84728
\(875\) 0 0
\(876\) −44.4144 −1.50062
\(877\) 20.6915 0.698703 0.349352 0.936992i \(-0.386402\pi\)
0.349352 + 0.936992i \(0.386402\pi\)
\(878\) 42.3040 1.42769
\(879\) 52.6151 1.77466
\(880\) 0 0
\(881\) 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 \(-0.780529\pi\)
−0.771572 + 0.636142i \(0.780529\pi\)
\(884\) 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 \(0.494216\pi\)
\(888\) 11.9026 0.399426
\(889\) −2.83781 −0.0951772
\(890\) 0 0
\(891\) −1.00222 −0.0335756
\(892\) 90.2133 3.02056
\(893\) 54.1457 1.81192
\(894\) −94.3015 −3.15391
\(895\) 0 0
\(896\) −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
\(903\) −3.43213 −0.114214
\(904\) 37.1782 1.23653
\(905\) 0 0
\(906\) −129.084 −4.28853
\(907\) 45.5307 1.51182 0.755910 0.654675i \(-0.227195\pi\)
0.755910 + 0.654675i \(0.227195\pi\)
\(908\) 66.3120 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
\(915\) 0 0
\(916\) 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 \(0.217704\pi\)
\(920\) 0 0
\(921\) 8.89822 0.293206
\(922\) −11.6745 −0.384481
\(923\) 0 0
\(924\) −24.3484 −0.801002
\(925\) 0 0
\(926\) −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 \(0.419669\pi\)
\(930\) 0 0
\(931\) 19.3180 0.633121
\(932\) 80.5560 2.63870
\(933\) −8.99102 −0.294353
\(934\) 17.4313 0.570369
\(935\) 0 0
\(936\) 0 0
\(937\) −6.07285 −0.198392 −0.0991958 0.995068i \(-0.531627\pi\)
−0.0991958 + 0.995068i \(0.531627\pi\)
\(938\) 38.4864 1.25663
\(939\) −99.9382 −3.26136
\(940\) 0 0
\(941\) 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
\(945\) 0 0
\(946\) 1.70064 0.0552927
\(947\) −18.6581 −0.606308 −0.303154 0.952942i \(-0.598040\pi\)
−0.303154 + 0.952942i \(0.598040\pi\)
\(948\) −111.372 −3.61720
\(949\) 0 0
\(950\) 0 0
\(951\) −38.5459 −1.24994
\(952\) −6.74917 −0.218742
\(953\) 1.52953 0.0495463 0.0247731 0.999693i \(-0.492114\pi\)
0.0247731 + 0.999693i \(0.492114\pi\)
\(954\) 87.0739 2.81912
\(955\) 0 0
\(956\) −53.9161 −1.74377
\(957\) −28.5827 −0.923948
\(958\) 40.6487 1.31330
\(959\) −38.3131 −1.23720
\(960\) 0 0
\(961\) −28.8564 −0.930852
\(962\) 0 0
\(963\) 36.7475 1.18417
\(964\) −109.616 −3.53048
\(965\) 0 0
\(966\) −51.3124 −1.65095
\(967\) 32.1716 1.03457 0.517285 0.855813i \(-0.326943\pi\)
0.517285 + 0.855813i \(0.326943\pi\)
\(968\) −54.7907 −1.76104
\(969\) 10.3259 0.331717
\(970\) 0 0
\(971\) −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 \(-0.419073\pi\)
0.251509 + 0.967855i \(0.419073\pi\)
\(978\) 112.750 3.60533
\(979\) −13.4511 −0.429900
\(980\) 0 0
\(981\) −50.2893 −1.60561
\(982\) −39.4248 −1.25810
\(983\) 38.5356 1.22910 0.614548 0.788880i \(-0.289338\pi\)
0.614548 + 0.788880i \(0.289338\pi\)
\(984\) −4.21058 −0.134228
\(985\) 0 0
\(986\) −15.0364 −0.478857
\(987\) −50.8743 −1.61935
\(988\) 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
\(993\) 81.4014 2.58320
\(994\) −46.4268 −1.47257
\(995\) 0 0
\(996\) 61.1606 1.93795
\(997\) 20.5374 0.650425 0.325213 0.945641i \(-0.394564\pi\)
0.325213 + 0.945641i \(0.394564\pi\)
\(998\) −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 4225.2.a.bl.1.4 4
5.4 even 2 845.2.a.l.1.1 4
13.2 odd 12 325.2.n.d.251.4 8
13.7 odd 12 325.2.n.d.101.4 8
13.12 even 2 4225.2.a.bi.1.1 4
15.14 odd 2 7605.2.a.cj.1.4 4
65.2 even 12 325.2.m.b.199.1 8
65.4 even 6 845.2.e.m.146.1 8
65.7 even 12 325.2.m.c.49.4 8
65.9 even 6 845.2.e.n.146.4 8
65.19 odd 12 845.2.m.g.361.4 8
65.24 odd 12 845.2.m.g.316.4 8
65.28 even 12 325.2.m.c.199.4 8
65.29 even 6 845.2.e.n.191.4 8
65.33 even 12 325.2.m.b.49.1 8
65.34 odd 4 845.2.c.g.506.1 8
65.44 odd 4 845.2.c.g.506.8 8
65.49 even 6 845.2.e.m.191.1 8
65.54 odd 12 65.2.m.a.56.1 yes 8
65.59 odd 12 65.2.m.a.36.1 8
65.64 even 2 845.2.a.m.1.4 4
195.59 even 12 585.2.bu.c.361.4 8
195.119 even 12 585.2.bu.c.316.4 8
195.194 odd 2 7605.2.a.cf.1.1 4
260.59 even 12 1040.2.da.b.881.1 8
260.119 even 12 1040.2.da.b.641.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.m.a.36.1 8 65.59 odd 12
65.2.m.a.56.1 yes 8 65.54 odd 12
325.2.m.b.49.1 8 65.33 even 12
325.2.m.b.199.1 8 65.2 even 12
325.2.m.c.49.4 8 65.7 even 12
325.2.m.c.199.4 8 65.28 even 12
325.2.n.d.101.4 8 13.7 odd 12
325.2.n.d.251.4 8 13.2 odd 12
585.2.bu.c.316.4 8 195.119 even 12
585.2.bu.c.361.4 8 195.59 even 12
845.2.a.l.1.1 4 5.4 even 2
845.2.a.m.1.4 4 65.64 even 2
845.2.c.g.506.1 8 65.34 odd 4
845.2.c.g.506.8 8 65.44 odd 4
845.2.e.m.146.1 8 65.4 even 6
845.2.e.m.191.1 8 65.49 even 6
845.2.e.n.146.4 8 65.9 even 6
845.2.e.n.191.4 8 65.29 even 6
845.2.m.g.316.4 8 65.24 odd 12
845.2.m.g.361.4 8 65.19 odd 12
1040.2.da.b.641.1 8 260.119 even 12
1040.2.da.b.881.1 8 260.59 even 12
4225.2.a.bi.1.1 4 13.12 even 2
4225.2.a.bl.1.4 4 1.1 even 1 trivial
7605.2.a.cf.1.1 4 195.194 odd 2
7605.2.a.cj.1.4 4 15.14 odd 2