Properties

Label 6003.2.a.e.1.2
Level 6003
Weight 2
Character 6003.1
Self dual yes
Analytic conductor 47.934
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.9341963334\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2001)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 6003.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{4} +2.44949 q^{5} +0.449490 q^{7} +O(q^{10})\) \(q-2.00000 q^{4} +2.44949 q^{5} +0.449490 q^{7} +1.00000 q^{13} +4.00000 q^{16} -3.44949 q^{17} -1.00000 q^{19} -4.89898 q^{20} +1.00000 q^{23} +1.00000 q^{25} -0.898979 q^{28} -1.00000 q^{29} -5.55051 q^{31} +1.10102 q^{35} +0.101021 q^{37} -5.34847 q^{41} -9.89898 q^{43} +7.55051 q^{47} -6.79796 q^{49} -2.00000 q^{52} +2.00000 q^{53} +5.44949 q^{59} -12.8990 q^{61} -8.00000 q^{64} +2.44949 q^{65} +0.898979 q^{67} +6.89898 q^{68} -2.34847 q^{71} +10.2474 q^{73} +2.00000 q^{76} -3.89898 q^{79} +9.79796 q^{80} +12.4495 q^{83} -8.44949 q^{85} -0.550510 q^{89} +0.449490 q^{91} -2.00000 q^{92} -2.44949 q^{95} +7.10102 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} - 4q^{7} + O(q^{10}) \) \( 2q - 4q^{4} - 4q^{7} + 2q^{13} + 8q^{16} - 2q^{17} - 2q^{19} + 2q^{23} + 2q^{25} + 8q^{28} - 2q^{29} - 16q^{31} + 12q^{35} + 10q^{37} + 4q^{41} - 10q^{43} + 20q^{47} + 6q^{49} - 4q^{52} + 4q^{53} + 6q^{59} - 16q^{61} - 16q^{64} - 8q^{67} + 4q^{68} + 10q^{71} - 4q^{73} + 4q^{76} + 2q^{79} + 20q^{83} - 12q^{85} - 6q^{89} - 4q^{91} - 4q^{92} + 24q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) 2.44949 1.09545 0.547723 0.836660i \(-0.315495\pi\)
0.547723 + 0.836660i \(0.315495\pi\)
\(6\) 0 0
\(7\) 0.449490 0.169891 0.0849456 0.996386i \(-0.472928\pi\)
0.0849456 + 0.996386i \(0.472928\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) −3.44949 −0.836624 −0.418312 0.908303i \(-0.637378\pi\)
−0.418312 + 0.908303i \(0.637378\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) −4.89898 −1.09545
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) −0.898979 −0.169891
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −5.55051 −0.996901 −0.498451 0.866918i \(-0.666097\pi\)
−0.498451 + 0.866918i \(0.666097\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.10102 0.186106
\(36\) 0 0
\(37\) 0.101021 0.0166077 0.00830384 0.999966i \(-0.497357\pi\)
0.00830384 + 0.999966i \(0.497357\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.34847 −0.835291 −0.417645 0.908610i \(-0.637145\pi\)
−0.417645 + 0.908610i \(0.637145\pi\)
\(42\) 0 0
\(43\) −9.89898 −1.50958 −0.754790 0.655966i \(-0.772261\pi\)
−0.754790 + 0.655966i \(0.772261\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.55051 1.10136 0.550678 0.834718i \(-0.314369\pi\)
0.550678 + 0.834718i \(0.314369\pi\)
\(48\) 0 0
\(49\) −6.79796 −0.971137
\(50\) 0 0
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.44949 0.709463 0.354732 0.934968i \(-0.384572\pi\)
0.354732 + 0.934968i \(0.384572\pi\)
\(60\) 0 0
\(61\) −12.8990 −1.65155 −0.825773 0.564003i \(-0.809261\pi\)
−0.825773 + 0.564003i \(0.809261\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 2.44949 0.303822
\(66\) 0 0
\(67\) 0.898979 0.109828 0.0549139 0.998491i \(-0.482512\pi\)
0.0549139 + 0.998491i \(0.482512\pi\)
\(68\) 6.89898 0.836624
\(69\) 0 0
\(70\) 0 0
\(71\) −2.34847 −0.278712 −0.139356 0.990242i \(-0.544503\pi\)
−0.139356 + 0.990242i \(0.544503\pi\)
\(72\) 0 0
\(73\) 10.2474 1.19937 0.599687 0.800235i \(-0.295292\pi\)
0.599687 + 0.800235i \(0.295292\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 0 0
\(79\) −3.89898 −0.438669 −0.219335 0.975650i \(-0.570389\pi\)
−0.219335 + 0.975650i \(0.570389\pi\)
\(80\) 9.79796 1.09545
\(81\) 0 0
\(82\) 0 0
\(83\) 12.4495 1.36651 0.683255 0.730180i \(-0.260564\pi\)
0.683255 + 0.730180i \(0.260564\pi\)
\(84\) 0 0
\(85\) −8.44949 −0.916476
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.550510 −0.0583540 −0.0291770 0.999574i \(-0.509289\pi\)
−0.0291770 + 0.999574i \(0.509289\pi\)
\(90\) 0 0
\(91\) 0.449490 0.0471193
\(92\) −2.00000 −0.208514
\(93\) 0 0
\(94\) 0 0
\(95\) −2.44949 −0.251312
\(96\) 0 0
\(97\) 7.10102 0.720999 0.360500 0.932759i \(-0.382606\pi\)
0.360500 + 0.932759i \(0.382606\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −2.00000 −0.200000
\(101\) −5.55051 −0.552296 −0.276148 0.961115i \(-0.589058\pi\)
−0.276148 + 0.961115i \(0.589058\pi\)
\(102\) 0 0
\(103\) −17.7980 −1.75369 −0.876843 0.480778i \(-0.840354\pi\)
−0.876843 + 0.480778i \(0.840354\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −10.2474 −0.990658 −0.495329 0.868705i \(-0.664953\pi\)
−0.495329 + 0.868705i \(0.664953\pi\)
\(108\) 0 0
\(109\) −1.55051 −0.148512 −0.0742560 0.997239i \(-0.523658\pi\)
−0.0742560 + 0.997239i \(0.523658\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.79796 0.169891
\(113\) 7.10102 0.668008 0.334004 0.942572i \(-0.391600\pi\)
0.334004 + 0.942572i \(0.391600\pi\)
\(114\) 0 0
\(115\) 2.44949 0.228416
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) 0 0
\(119\) −1.55051 −0.142135
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 11.1010 0.996901
\(125\) −9.79796 −0.876356
\(126\) 0 0
\(127\) −14.2474 −1.26426 −0.632128 0.774864i \(-0.717818\pi\)
−0.632128 + 0.774864i \(0.717818\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 10.6969 0.934596 0.467298 0.884100i \(-0.345228\pi\)
0.467298 + 0.884100i \(0.345228\pi\)
\(132\) 0 0
\(133\) −0.449490 −0.0389757
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 14.1464 1.20861 0.604305 0.796753i \(-0.293451\pi\)
0.604305 + 0.796753i \(0.293451\pi\)
\(138\) 0 0
\(139\) 6.00000 0.508913 0.254457 0.967084i \(-0.418103\pi\)
0.254457 + 0.967084i \(0.418103\pi\)
\(140\) −2.20204 −0.186106
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −2.44949 −0.203419
\(146\) 0 0
\(147\) 0 0
\(148\) −0.202041 −0.0166077
\(149\) −8.89898 −0.729033 −0.364516 0.931197i \(-0.618766\pi\)
−0.364516 + 0.931197i \(0.618766\pi\)
\(150\) 0 0
\(151\) −20.5959 −1.67607 −0.838036 0.545615i \(-0.816296\pi\)
−0.838036 + 0.545615i \(0.816296\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −13.5959 −1.09205
\(156\) 0 0
\(157\) 18.5959 1.48412 0.742058 0.670336i \(-0.233850\pi\)
0.742058 + 0.670336i \(0.233850\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0.449490 0.0354248
\(162\) 0 0
\(163\) 3.10102 0.242891 0.121445 0.992598i \(-0.461247\pi\)
0.121445 + 0.992598i \(0.461247\pi\)
\(164\) 10.6969 0.835291
\(165\) 0 0
\(166\) 0 0
\(167\) 8.34847 0.646024 0.323012 0.946395i \(-0.395305\pi\)
0.323012 + 0.946395i \(0.395305\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 19.7980 1.50958
\(173\) 1.24745 0.0948418 0.0474209 0.998875i \(-0.484900\pi\)
0.0474209 + 0.998875i \(0.484900\pi\)
\(174\) 0 0
\(175\) 0.449490 0.0339782
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −16.3485 −1.22194 −0.610971 0.791653i \(-0.709221\pi\)
−0.610971 + 0.791653i \(0.709221\pi\)
\(180\) 0 0
\(181\) 3.79796 0.282300 0.141150 0.989988i \(-0.454920\pi\)
0.141150 + 0.989988i \(0.454920\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.247449 0.0181928
\(186\) 0 0
\(187\) 0 0
\(188\) −15.1010 −1.10136
\(189\) 0 0
\(190\) 0 0
\(191\) −12.1464 −0.878885 −0.439442 0.898271i \(-0.644824\pi\)
−0.439442 + 0.898271i \(0.644824\pi\)
\(192\) 0 0
\(193\) 3.79796 0.273383 0.136692 0.990614i \(-0.456353\pi\)
0.136692 + 0.990614i \(0.456353\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 13.5959 0.971137
\(197\) 10.1464 0.722903 0.361452 0.932391i \(-0.382281\pi\)
0.361452 + 0.932391i \(0.382281\pi\)
\(198\) 0 0
\(199\) 4.24745 0.301094 0.150547 0.988603i \(-0.451897\pi\)
0.150547 + 0.988603i \(0.451897\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −0.449490 −0.0315480
\(204\) 0 0
\(205\) −13.1010 −0.915015
\(206\) 0 0
\(207\) 0 0
\(208\) 4.00000 0.277350
\(209\) 0 0
\(210\) 0 0
\(211\) 26.0454 1.79304 0.896520 0.443003i \(-0.146087\pi\)
0.896520 + 0.443003i \(0.146087\pi\)
\(212\) −4.00000 −0.274721
\(213\) 0 0
\(214\) 0 0
\(215\) −24.2474 −1.65366
\(216\) 0 0
\(217\) −2.49490 −0.169365
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −3.44949 −0.232038
\(222\) 0 0
\(223\) −12.1010 −0.810344 −0.405172 0.914240i \(-0.632788\pi\)
−0.405172 + 0.914240i \(0.632788\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −14.0000 −0.929213 −0.464606 0.885517i \(-0.653804\pi\)
−0.464606 + 0.885517i \(0.653804\pi\)
\(228\) 0 0
\(229\) −26.7980 −1.77086 −0.885429 0.464774i \(-0.846136\pi\)
−0.885429 + 0.464774i \(0.846136\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −22.1464 −1.45086 −0.725430 0.688296i \(-0.758359\pi\)
−0.725430 + 0.688296i \(0.758359\pi\)
\(234\) 0 0
\(235\) 18.4949 1.20647
\(236\) −10.8990 −0.709463
\(237\) 0 0
\(238\) 0 0
\(239\) −13.4495 −0.869975 −0.434988 0.900436i \(-0.643247\pi\)
−0.434988 + 0.900436i \(0.643247\pi\)
\(240\) 0 0
\(241\) −8.69694 −0.560219 −0.280110 0.959968i \(-0.590371\pi\)
−0.280110 + 0.959968i \(0.590371\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 25.7980 1.65155
\(245\) −16.6515 −1.06383
\(246\) 0 0
\(247\) −1.00000 −0.0636285
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 13.4495 0.848924 0.424462 0.905446i \(-0.360463\pi\)
0.424462 + 0.905446i \(0.360463\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −10.5505 −0.658123 −0.329061 0.944309i \(-0.606732\pi\)
−0.329061 + 0.944309i \(0.606732\pi\)
\(258\) 0 0
\(259\) 0.0454077 0.00282150
\(260\) −4.89898 −0.303822
\(261\) 0 0
\(262\) 0 0
\(263\) 7.10102 0.437868 0.218934 0.975740i \(-0.429742\pi\)
0.218934 + 0.975740i \(0.429742\pi\)
\(264\) 0 0
\(265\) 4.89898 0.300942
\(266\) 0 0
\(267\) 0 0
\(268\) −1.79796 −0.109828
\(269\) −15.5959 −0.950900 −0.475450 0.879743i \(-0.657715\pi\)
−0.475450 + 0.879743i \(0.657715\pi\)
\(270\) 0 0
\(271\) 16.8990 1.02654 0.513270 0.858227i \(-0.328434\pi\)
0.513270 + 0.858227i \(0.328434\pi\)
\(272\) −13.7980 −0.836624
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −7.00000 −0.420589 −0.210295 0.977638i \(-0.567442\pi\)
−0.210295 + 0.977638i \(0.567442\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 24.4949 1.46124 0.730622 0.682783i \(-0.239230\pi\)
0.730622 + 0.682783i \(0.239230\pi\)
\(282\) 0 0
\(283\) −22.0454 −1.31046 −0.655232 0.755428i \(-0.727429\pi\)
−0.655232 + 0.755428i \(0.727429\pi\)
\(284\) 4.69694 0.278712
\(285\) 0 0
\(286\) 0 0
\(287\) −2.40408 −0.141908
\(288\) 0 0
\(289\) −5.10102 −0.300060
\(290\) 0 0
\(291\) 0 0
\(292\) −20.4949 −1.19937
\(293\) 27.2474 1.59181 0.795906 0.605420i \(-0.206995\pi\)
0.795906 + 0.605420i \(0.206995\pi\)
\(294\) 0 0
\(295\) 13.3485 0.777178
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.00000 0.0578315
\(300\) 0 0
\(301\) −4.44949 −0.256464
\(302\) 0 0
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) −31.5959 −1.80918
\(306\) 0 0
\(307\) 12.6515 0.722061 0.361030 0.932554i \(-0.382425\pi\)
0.361030 + 0.932554i \(0.382425\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.79796 0.555591 0.277796 0.960640i \(-0.410396\pi\)
0.277796 + 0.960640i \(0.410396\pi\)
\(312\) 0 0
\(313\) 0.651531 0.0368267 0.0184133 0.999830i \(-0.494139\pi\)
0.0184133 + 0.999830i \(0.494139\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 7.79796 0.438669
\(317\) −18.2474 −1.02488 −0.512439 0.858723i \(-0.671258\pi\)
−0.512439 + 0.858723i \(0.671258\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −19.5959 −1.09545
\(321\) 0 0
\(322\) 0 0
\(323\) 3.44949 0.191935
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.39388 0.187110
\(330\) 0 0
\(331\) −5.59592 −0.307579 −0.153790 0.988104i \(-0.549148\pi\)
−0.153790 + 0.988104i \(0.549148\pi\)
\(332\) −24.8990 −1.36651
\(333\) 0 0
\(334\) 0 0
\(335\) 2.20204 0.120310
\(336\) 0 0
\(337\) −12.7980 −0.697149 −0.348575 0.937281i \(-0.613334\pi\)
−0.348575 + 0.937281i \(0.613334\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 16.8990 0.916476
\(341\) 0 0
\(342\) 0 0
\(343\) −6.20204 −0.334879
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −22.1464 −1.18888 −0.594441 0.804139i \(-0.702627\pi\)
−0.594441 + 0.804139i \(0.702627\pi\)
\(348\) 0 0
\(349\) −11.6969 −0.626123 −0.313061 0.949733i \(-0.601355\pi\)
−0.313061 + 0.949733i \(0.601355\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.898979 0.0478479 0.0239239 0.999714i \(-0.492384\pi\)
0.0239239 + 0.999714i \(0.492384\pi\)
\(354\) 0 0
\(355\) −5.75255 −0.305314
\(356\) 1.10102 0.0583540
\(357\) 0 0
\(358\) 0 0
\(359\) 29.0454 1.53296 0.766479 0.642269i \(-0.222007\pi\)
0.766479 + 0.642269i \(0.222007\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 0 0
\(364\) −0.898979 −0.0471193
\(365\) 25.1010 1.31385
\(366\) 0 0
\(367\) −14.0000 −0.730794 −0.365397 0.930852i \(-0.619067\pi\)
−0.365397 + 0.930852i \(0.619067\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) 0 0
\(371\) 0.898979 0.0466727
\(372\) 0 0
\(373\) −19.3485 −1.00183 −0.500913 0.865498i \(-0.667002\pi\)
−0.500913 + 0.865498i \(0.667002\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.00000 −0.0515026
\(378\) 0 0
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 4.89898 0.251312
\(381\) 0 0
\(382\) 0 0
\(383\) 3.79796 0.194067 0.0970333 0.995281i \(-0.469065\pi\)
0.0970333 + 0.995281i \(0.469065\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −14.2020 −0.720999
\(389\) −15.6515 −0.793564 −0.396782 0.917913i \(-0.629873\pi\)
−0.396782 + 0.917913i \(0.629873\pi\)
\(390\) 0 0
\(391\) −3.44949 −0.174448
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −9.55051 −0.480538
\(396\) 0 0
\(397\) −22.6969 −1.13913 −0.569563 0.821947i \(-0.692888\pi\)
−0.569563 + 0.821947i \(0.692888\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) −16.6969 −0.833805 −0.416903 0.908951i \(-0.636884\pi\)
−0.416903 + 0.908951i \(0.636884\pi\)
\(402\) 0 0
\(403\) −5.55051 −0.276491
\(404\) 11.1010 0.552296
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −11.1464 −0.551155 −0.275578 0.961279i \(-0.588869\pi\)
−0.275578 + 0.961279i \(0.588869\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 35.5959 1.75369
\(413\) 2.44949 0.120532
\(414\) 0 0
\(415\) 30.4949 1.49694
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −31.5959 −1.54356 −0.771781 0.635889i \(-0.780634\pi\)
−0.771781 + 0.635889i \(0.780634\pi\)
\(420\) 0 0
\(421\) 19.6969 0.959970 0.479985 0.877277i \(-0.340642\pi\)
0.479985 + 0.877277i \(0.340642\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.44949 −0.167325
\(426\) 0 0
\(427\) −5.79796 −0.280583
\(428\) 20.4949 0.990658
\(429\) 0 0
\(430\) 0 0
\(431\) −14.4949 −0.698195 −0.349097 0.937086i \(-0.613512\pi\)
−0.349097 + 0.937086i \(0.613512\pi\)
\(432\) 0 0
\(433\) −6.00000 −0.288342 −0.144171 0.989553i \(-0.546051\pi\)
−0.144171 + 0.989553i \(0.546051\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 3.10102 0.148512
\(437\) −1.00000 −0.0478365
\(438\) 0 0
\(439\) 37.6969 1.79918 0.899588 0.436739i \(-0.143867\pi\)
0.899588 + 0.436739i \(0.143867\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 23.1464 1.09972 0.549860 0.835257i \(-0.314681\pi\)
0.549860 + 0.835257i \(0.314681\pi\)
\(444\) 0 0
\(445\) −1.34847 −0.0639236
\(446\) 0 0
\(447\) 0 0
\(448\) −3.59592 −0.169891
\(449\) −33.7980 −1.59502 −0.797512 0.603303i \(-0.793851\pi\)
−0.797512 + 0.603303i \(0.793851\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −14.2020 −0.668008
\(453\) 0 0
\(454\) 0 0
\(455\) 1.10102 0.0516166
\(456\) 0 0
\(457\) 11.5505 0.540310 0.270155 0.962817i \(-0.412925\pi\)
0.270155 + 0.962817i \(0.412925\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −4.89898 −0.228416
\(461\) −4.00000 −0.186299 −0.0931493 0.995652i \(-0.529693\pi\)
−0.0931493 + 0.995652i \(0.529693\pi\)
\(462\) 0 0
\(463\) 13.8990 0.645940 0.322970 0.946409i \(-0.395319\pi\)
0.322970 + 0.946409i \(0.395319\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) 0 0
\(467\) −4.14643 −0.191874 −0.0959369 0.995387i \(-0.530585\pi\)
−0.0959369 + 0.995387i \(0.530585\pi\)
\(468\) 0 0
\(469\) 0.404082 0.0186588
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 3.10102 0.142135
\(477\) 0 0
\(478\) 0 0
\(479\) −7.10102 −0.324454 −0.162227 0.986753i \(-0.551868\pi\)
−0.162227 + 0.986753i \(0.551868\pi\)
\(480\) 0 0
\(481\) 0.101021 0.00460614
\(482\) 0 0
\(483\) 0 0
\(484\) 22.0000 1.00000
\(485\) 17.3939 0.789815
\(486\) 0 0
\(487\) −4.69694 −0.212839 −0.106419 0.994321i \(-0.533939\pi\)
−0.106419 + 0.994321i \(0.533939\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 20.4495 0.922873 0.461436 0.887173i \(-0.347334\pi\)
0.461436 + 0.887173i \(0.347334\pi\)
\(492\) 0 0
\(493\) 3.44949 0.155357
\(494\) 0 0
\(495\) 0 0
\(496\) −22.2020 −0.996901
\(497\) −1.05561 −0.0473507
\(498\) 0 0
\(499\) −37.8990 −1.69659 −0.848296 0.529523i \(-0.822371\pi\)
−0.848296 + 0.529523i \(0.822371\pi\)
\(500\) 19.5959 0.876356
\(501\) 0 0
\(502\) 0 0
\(503\) 25.2474 1.12573 0.562864 0.826549i \(-0.309699\pi\)
0.562864 + 0.826549i \(0.309699\pi\)
\(504\) 0 0
\(505\) −13.5959 −0.605010
\(506\) 0 0
\(507\) 0 0
\(508\) 28.4949 1.26426
\(509\) −36.1464 −1.60216 −0.801081 0.598556i \(-0.795741\pi\)
−0.801081 + 0.598556i \(0.795741\pi\)
\(510\) 0 0
\(511\) 4.60612 0.203763
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −43.5959 −1.92107
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −18.0454 −0.790584 −0.395292 0.918556i \(-0.629357\pi\)
−0.395292 + 0.918556i \(0.629357\pi\)
\(522\) 0 0
\(523\) 13.3485 0.583688 0.291844 0.956466i \(-0.405731\pi\)
0.291844 + 0.956466i \(0.405731\pi\)
\(524\) −21.3939 −0.934596
\(525\) 0 0
\(526\) 0 0
\(527\) 19.1464 0.834032
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0.898979 0.0389757
\(533\) −5.34847 −0.231668
\(534\) 0 0
\(535\) −25.1010 −1.08521
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −7.10102 −0.305297 −0.152648 0.988281i \(-0.548780\pi\)
−0.152648 + 0.988281i \(0.548780\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −3.79796 −0.162687
\(546\) 0 0
\(547\) 15.6969 0.671153 0.335576 0.942013i \(-0.391069\pi\)
0.335576 + 0.942013i \(0.391069\pi\)
\(548\) −28.2929 −1.20861
\(549\) 0 0
\(550\) 0 0
\(551\) 1.00000 0.0426014
\(552\) 0 0
\(553\) −1.75255 −0.0745261
\(554\) 0 0
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) 0.853572 0.0361670 0.0180835 0.999836i \(-0.494244\pi\)
0.0180835 + 0.999836i \(0.494244\pi\)
\(558\) 0 0
\(559\) −9.89898 −0.418682
\(560\) 4.40408 0.186106
\(561\) 0 0
\(562\) 0 0
\(563\) −15.2474 −0.642603 −0.321302 0.946977i \(-0.604120\pi\)
−0.321302 + 0.946977i \(0.604120\pi\)
\(564\) 0 0
\(565\) 17.3939 0.731766
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 35.7980 1.50073 0.750364 0.661025i \(-0.229878\pi\)
0.750364 + 0.661025i \(0.229878\pi\)
\(570\) 0 0
\(571\) −10.0454 −0.420387 −0.210194 0.977660i \(-0.567409\pi\)
−0.210194 + 0.977660i \(0.567409\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 3.75255 0.156221 0.0781104 0.996945i \(-0.475111\pi\)
0.0781104 + 0.996945i \(0.475111\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 4.89898 0.203419
\(581\) 5.59592 0.232158
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −23.0454 −0.951186 −0.475593 0.879666i \(-0.657766\pi\)
−0.475593 + 0.879666i \(0.657766\pi\)
\(588\) 0 0
\(589\) 5.55051 0.228705
\(590\) 0 0
\(591\) 0 0
\(592\) 0.404082 0.0166077
\(593\) −14.7526 −0.605815 −0.302907 0.953020i \(-0.597957\pi\)
−0.302907 + 0.953020i \(0.597957\pi\)
\(594\) 0 0
\(595\) −3.79796 −0.155701
\(596\) 17.7980 0.729033
\(597\) 0 0
\(598\) 0 0
\(599\) 1.34847 0.0550970 0.0275485 0.999620i \(-0.491230\pi\)
0.0275485 + 0.999620i \(0.491230\pi\)
\(600\) 0 0
\(601\) 30.6969 1.25215 0.626077 0.779761i \(-0.284660\pi\)
0.626077 + 0.779761i \(0.284660\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 41.1918 1.67607
\(605\) −26.9444 −1.09545
\(606\) 0 0
\(607\) −9.34847 −0.379443 −0.189721 0.981838i \(-0.560758\pi\)
−0.189721 + 0.981838i \(0.560758\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7.55051 0.305461
\(612\) 0 0
\(613\) 10.0000 0.403896 0.201948 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −27.7980 −1.11910 −0.559552 0.828795i \(-0.689027\pi\)
−0.559552 + 0.828795i \(0.689027\pi\)
\(618\) 0 0
\(619\) −8.79796 −0.353620 −0.176810 0.984245i \(-0.556578\pi\)
−0.176810 + 0.984245i \(0.556578\pi\)
\(620\) 27.1918 1.09205
\(621\) 0 0
\(622\) 0 0
\(623\) −0.247449 −0.00991382
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) −37.1918 −1.48412
\(629\) −0.348469 −0.0138944
\(630\) 0 0
\(631\) −3.30306 −0.131493 −0.0657464 0.997836i \(-0.520943\pi\)
−0.0657464 + 0.997836i \(0.520943\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −34.8990 −1.38492
\(636\) 0 0
\(637\) −6.79796 −0.269345
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −11.0454 −0.436267 −0.218134 0.975919i \(-0.569997\pi\)
−0.218134 + 0.975919i \(0.569997\pi\)
\(642\) 0 0
\(643\) −12.8990 −0.508686 −0.254343 0.967114i \(-0.581859\pi\)
−0.254343 + 0.967114i \(0.581859\pi\)
\(644\) −0.898979 −0.0354248
\(645\) 0 0
\(646\) 0 0
\(647\) −23.2474 −0.913952 −0.456976 0.889479i \(-0.651067\pi\)
−0.456976 + 0.889479i \(0.651067\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −6.20204 −0.242891
\(653\) 30.7423 1.20304 0.601520 0.798857i \(-0.294562\pi\)
0.601520 + 0.798857i \(0.294562\pi\)
\(654\) 0 0
\(655\) 26.2020 1.02380
\(656\) −21.3939 −0.835291
\(657\) 0 0
\(658\) 0 0
\(659\) 4.55051 0.177263 0.0886314 0.996064i \(-0.471751\pi\)
0.0886314 + 0.996064i \(0.471751\pi\)
\(660\) 0 0
\(661\) 7.34847 0.285822 0.142911 0.989736i \(-0.454354\pi\)
0.142911 + 0.989736i \(0.454354\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.10102 −0.0426957
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) −16.6969 −0.646024
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −7.89898 −0.304483 −0.152242 0.988343i \(-0.548649\pi\)
−0.152242 + 0.988343i \(0.548649\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 24.0000 0.923077
\(677\) −4.20204 −0.161498 −0.0807488 0.996734i \(-0.525731\pi\)
−0.0807488 + 0.996734i \(0.525731\pi\)
\(678\) 0 0
\(679\) 3.19184 0.122491
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 2.69694 0.103195 0.0515977 0.998668i \(-0.483569\pi\)
0.0515977 + 0.998668i \(0.483569\pi\)
\(684\) 0 0
\(685\) 34.6515 1.32397
\(686\) 0 0
\(687\) 0 0
\(688\) −39.5959 −1.50958
\(689\) 2.00000 0.0761939
\(690\) 0 0
\(691\) 38.8990 1.47979 0.739893 0.672724i \(-0.234876\pi\)
0.739893 + 0.672724i \(0.234876\pi\)
\(692\) −2.49490 −0.0948418
\(693\) 0 0
\(694\) 0 0
\(695\) 14.6969 0.557487
\(696\) 0 0
\(697\) 18.4495 0.698824
\(698\) 0 0
\(699\) 0 0
\(700\) −0.898979 −0.0339782
\(701\) 42.2474 1.59566 0.797832 0.602880i \(-0.205980\pi\)
0.797832 + 0.602880i \(0.205980\pi\)
\(702\) 0 0
\(703\) −0.101021 −0.00381006
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2.49490 −0.0938303
\(708\) 0 0
\(709\) 0.651531 0.0244688 0.0122344 0.999925i \(-0.496106\pi\)
0.0122344 + 0.999925i \(0.496106\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −5.55051 −0.207868
\(714\) 0 0
\(715\) 0 0
\(716\) 32.6969 1.22194
\(717\) 0 0
\(718\) 0 0
\(719\) 51.7423 1.92966 0.964832 0.262867i \(-0.0846682\pi\)
0.964832 + 0.262867i \(0.0846682\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 0 0
\(724\) −7.59592 −0.282300
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) 3.59592 0.133365 0.0666826 0.997774i \(-0.478759\pi\)
0.0666826 + 0.997774i \(0.478759\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 34.1464 1.26295
\(732\) 0 0
\(733\) −31.1010 −1.14874 −0.574371 0.818595i \(-0.694753\pi\)
−0.574371 + 0.818595i \(0.694753\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 16.2020 0.596002 0.298001 0.954566i \(-0.403680\pi\)
0.298001 + 0.954566i \(0.403680\pi\)
\(740\) −0.494897 −0.0181928
\(741\) 0 0
\(742\) 0 0
\(743\) −41.2474 −1.51322 −0.756611 0.653865i \(-0.773146\pi\)
−0.756611 + 0.653865i \(0.773146\pi\)
\(744\) 0 0
\(745\) −21.7980 −0.798615
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −4.60612 −0.168304
\(750\) 0 0
\(751\) 37.4949 1.36821 0.684104 0.729384i \(-0.260193\pi\)
0.684104 + 0.729384i \(0.260193\pi\)
\(752\) 30.2020 1.10136
\(753\) 0 0
\(754\) 0 0
\(755\) −50.4495 −1.83604
\(756\) 0 0
\(757\) 21.8990 0.795932 0.397966 0.917400i \(-0.369716\pi\)
0.397966 + 0.917400i \(0.369716\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −21.2474 −0.770219 −0.385110 0.922871i \(-0.625836\pi\)
−0.385110 + 0.922871i \(0.625836\pi\)
\(762\) 0 0
\(763\) −0.696938 −0.0252309
\(764\) 24.2929 0.878885
\(765\) 0 0
\(766\) 0 0
\(767\) 5.44949 0.196770
\(768\) 0 0
\(769\) −51.8990 −1.87153 −0.935763 0.352631i \(-0.885287\pi\)
−0.935763 + 0.352631i \(0.885287\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −7.59592 −0.273383
\(773\) 7.30306 0.262673 0.131336 0.991338i \(-0.458073\pi\)
0.131336 + 0.991338i \(0.458073\pi\)
\(774\) 0 0
\(775\) −5.55051 −0.199380
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.34847 0.191629
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −27.1918 −0.971137
\(785\) 45.5505 1.62577
\(786\) 0 0
\(787\) 28.2929 1.00853 0.504266 0.863549i \(-0.331763\pi\)
0.504266 + 0.863549i \(0.331763\pi\)
\(788\) −20.2929 −0.722903
\(789\) 0 0
\(790\) 0 0
\(791\) 3.19184 0.113489
\(792\) 0 0
\(793\) −12.8990 −0.458056
\(794\) 0 0
\(795\) 0 0
\(796\) −8.49490 −0.301094
\(797\) −3.30306 −0.117000 −0.0585002 0.998287i \(-0.518632\pi\)
−0.0585002 + 0.998287i \(0.518632\pi\)
\(798\) 0 0
\(799\) −26.0454 −0.921420
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 1.10102 0.0388059
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −30.8990 −1.08635 −0.543175 0.839619i \(-0.682778\pi\)
−0.543175 + 0.839619i \(0.682778\pi\)
\(810\) 0 0
\(811\) 48.6969 1.70998 0.854990 0.518644i \(-0.173563\pi\)
0.854990 + 0.518644i \(0.173563\pi\)
\(812\) 0.898979 0.0315480
\(813\) 0 0
\(814\) 0 0
\(815\) 7.59592 0.266073
\(816\) 0 0
\(817\) 9.89898 0.346321
\(818\) 0 0
\(819\) 0 0
\(820\) 26.2020 0.915015
\(821\) 9.65153 0.336841 0.168420 0.985715i \(-0.446133\pi\)
0.168420 + 0.985715i \(0.446133\pi\)
\(822\) 0 0
\(823\) −51.1464 −1.78285 −0.891426 0.453166i \(-0.850295\pi\)
−0.891426 + 0.453166i \(0.850295\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 7.65153 0.266070 0.133035 0.991111i \(-0.457528\pi\)
0.133035 + 0.991111i \(0.457528\pi\)
\(828\) 0 0
\(829\) 18.2929 0.635337 0.317669 0.948202i \(-0.397100\pi\)
0.317669 + 0.948202i \(0.397100\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −8.00000 −0.277350
\(833\) 23.4495 0.812477
\(834\) 0 0
\(835\) 20.4495 0.707684
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 7.39388 0.255265 0.127632 0.991822i \(-0.459262\pi\)
0.127632 + 0.991822i \(0.459262\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 0 0
\(844\) −52.0908 −1.79304
\(845\) −29.3939 −1.01118
\(846\) 0 0
\(847\) −4.94439 −0.169891
\(848\) 8.00000 0.274721
\(849\) 0 0
\(850\) 0 0
\(851\) 0.101021 0.00346294
\(852\) 0 0
\(853\) 35.3939 1.21186 0.605932 0.795517i \(-0.292800\pi\)
0.605932 + 0.795517i \(0.292800\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 28.2929 0.966466 0.483233 0.875492i \(-0.339462\pi\)
0.483233 + 0.875492i \(0.339462\pi\)
\(858\) 0 0
\(859\) −27.3485 −0.933118 −0.466559 0.884490i \(-0.654506\pi\)
−0.466559 + 0.884490i \(0.654506\pi\)
\(860\) 48.4949 1.65366
\(861\) 0 0
\(862\) 0 0
\(863\) 38.2929 1.30350 0.651752 0.758432i \(-0.274034\pi\)
0.651752 + 0.758432i \(0.274034\pi\)
\(864\) 0 0
\(865\) 3.05561 0.103894
\(866\) 0 0
\(867\) 0 0
\(868\) 4.98979 0.169365
\(869\) 0 0
\(870\) 0 0
\(871\) 0.898979 0.0304608
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −4.40408 −0.148885
\(876\) 0 0
\(877\) −7.40408 −0.250018 −0.125009 0.992156i \(-0.539896\pi\)
−0.125009 + 0.992156i \(0.539896\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 14.3485 0.483412 0.241706 0.970350i \(-0.422293\pi\)
0.241706 + 0.970350i \(0.422293\pi\)
\(882\) 0 0
\(883\) 53.0908 1.78665 0.893324 0.449413i \(-0.148367\pi\)
0.893324 + 0.449413i \(0.148367\pi\)
\(884\) 6.89898 0.232038
\(885\) 0 0
\(886\) 0 0
\(887\) 50.5403 1.69698 0.848489 0.529214i \(-0.177513\pi\)
0.848489 + 0.529214i \(0.177513\pi\)
\(888\) 0 0
\(889\) −6.40408 −0.214786
\(890\) 0 0
\(891\) 0 0
\(892\) 24.2020 0.810344
\(893\) −7.55051 −0.252668
\(894\) 0 0
\(895\) −40.0454 −1.33857
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 5.55051 0.185120
\(900\) 0 0
\(901\) −6.89898 −0.229838
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 9.30306 0.309244
\(906\) 0 0
\(907\) 12.5959 0.418241 0.209120 0.977890i \(-0.432940\pi\)
0.209120 + 0.977890i \(0.432940\pi\)
\(908\) 28.0000 0.929213
\(909\) 0 0
\(910\) 0 0
\(911\) 42.5505 1.40976 0.704881 0.709326i \(-0.251001\pi\)
0.704881 + 0.709326i \(0.251001\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 53.5959 1.77086
\(917\) 4.80816 0.158780
\(918\) 0 0
\(919\) 0.696938 0.0229899 0.0114949 0.999934i \(-0.496341\pi\)
0.0114949 + 0.999934i \(0.496341\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −2.34847 −0.0773008
\(924\) 0 0
\(925\) 0.101021 0.00332153
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −58.8434 −1.93059 −0.965294 0.261165i \(-0.915893\pi\)
−0.965294 + 0.261165i \(0.915893\pi\)
\(930\) 0 0
\(931\) 6.79796 0.222794
\(932\) 44.2929 1.45086
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −49.1918 −1.60703 −0.803514 0.595286i \(-0.797039\pi\)
−0.803514 + 0.595286i \(0.797039\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −36.9898 −1.20647
\(941\) −16.4949 −0.537718 −0.268859 0.963180i \(-0.586647\pi\)
−0.268859 + 0.963180i \(0.586647\pi\)
\(942\) 0 0
\(943\) −5.34847 −0.174170
\(944\) 21.7980 0.709463
\(945\) 0 0
\(946\) 0 0
\(947\) −22.6969 −0.737551 −0.368776 0.929518i \(-0.620223\pi\)
−0.368776 + 0.929518i \(0.620223\pi\)
\(948\) 0 0
\(949\) 10.2474 0.332646
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 41.3485 1.33941 0.669704 0.742628i \(-0.266421\pi\)
0.669704 + 0.742628i \(0.266421\pi\)
\(954\) 0 0
\(955\) −29.7526 −0.962770
\(956\) 26.8990 0.869975
\(957\) 0 0
\(958\) 0 0
\(959\) 6.35867 0.205332
\(960\) 0 0
\(961\) −0.191836 −0.00618825
\(962\) 0 0
\(963\) 0 0
\(964\) 17.3939 0.560219
\(965\) 9.30306 0.299476
\(966\) 0 0
\(967\) −26.4495 −0.850558 −0.425279 0.905062i \(-0.639824\pi\)
−0.425279 + 0.905062i \(0.639824\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 46.2929 1.48561 0.742804 0.669509i \(-0.233495\pi\)
0.742804 + 0.669509i \(0.233495\pi\)
\(972\) 0 0
\(973\) 2.69694 0.0864599
\(974\) 0 0
\(975\) 0 0
\(976\) −51.5959 −1.65155
\(977\) −21.5505 −0.689462 −0.344731 0.938702i \(-0.612030\pi\)
−0.344731 + 0.938702i \(0.612030\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 33.3031 1.06383
\(981\) 0 0
\(982\) 0 0
\(983\) 61.7423 1.96928 0.984638 0.174611i \(-0.0558668\pi\)
0.984638 + 0.174611i \(0.0558668\pi\)
\(984\) 0 0
\(985\) 24.8536 0.791901
\(986\) 0 0
\(987\) 0 0
\(988\) 2.00000 0.0636285
\(989\) −9.89898 −0.314769
\(990\) 0 0
\(991\) −12.4041 −0.394029 −0.197014 0.980401i \(-0.563125\pi\)
−0.197014 + 0.980401i \(0.563125\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 10.4041 0.329832
\(996\) 0 0
\(997\) 36.9898 1.17148 0.585739 0.810500i \(-0.300804\pi\)
0.585739 + 0.810500i \(0.300804\pi\)
\(998\) 0 0
\(999\) 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 6003.2.a.e.1.2 2
3.2 odd 2 2001.2.a.e.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.e.1.1 2 3.2 odd 2
6003.2.a.e.1.2 2 1.1 even 1 trivial