Properties

Label 8004.2.a.i.1.4
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} - 137 x^{8} - 121665 x^{7} + 60168 x^{6} + 172218 x^{5} - 138024 x^{4} - 17844 x^{3} + 14352 x^{2} + 72 x - 208\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.58465\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.58465 q^{5} -3.26725 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.58465 q^{5} -3.26725 q^{7} +1.00000 q^{9} -2.90115 q^{11} +5.43566 q^{13} +2.58465 q^{15} -2.76865 q^{17} -6.15044 q^{19} +3.26725 q^{21} +1.00000 q^{23} +1.68042 q^{25} -1.00000 q^{27} +1.00000 q^{29} -1.72826 q^{31} +2.90115 q^{33} +8.44471 q^{35} -4.97381 q^{37} -5.43566 q^{39} -6.02032 q^{41} -10.9734 q^{43} -2.58465 q^{45} -3.19760 q^{47} +3.67493 q^{49} +2.76865 q^{51} +5.41370 q^{53} +7.49845 q^{55} +6.15044 q^{57} -3.48319 q^{59} -3.77242 q^{61} -3.26725 q^{63} -14.0493 q^{65} -6.87084 q^{67} -1.00000 q^{69} +3.47332 q^{71} -5.99833 q^{73} -1.68042 q^{75} +9.47878 q^{77} +0.609370 q^{79} +1.00000 q^{81} -6.42111 q^{83} +7.15600 q^{85} -1.00000 q^{87} -15.0768 q^{89} -17.7597 q^{91} +1.72826 q^{93} +15.8967 q^{95} -6.60560 q^{97} -2.90115 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + O(q^{10}) \) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + 5q^{11} + 8q^{13} - 5q^{15} + 7q^{17} + q^{19} + 4q^{21} + 16q^{23} + 31q^{25} - 16q^{27} + 16q^{29} - 2q^{31} - 5q^{33} + 5q^{35} + 14q^{37} - 8q^{39} - q^{41} - 13q^{43} + 5q^{45} - 4q^{47} + 30q^{49} - 7q^{51} + 19q^{53} - 37q^{55} - q^{57} + 12q^{59} + 21q^{61} - 4q^{63} + 26q^{65} - 11q^{67} - 16q^{69} + 7q^{71} - 13q^{73} - 31q^{75} + 4q^{77} - 18q^{79} + 16q^{81} + 25q^{83} + 48q^{85} - 16q^{87} + 12q^{89} - 11q^{91} + 2q^{93} - q^{95} + 5q^{97} + 5q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −2.58465 −1.15589 −0.577946 0.816075i \(-0.696145\pi\)
−0.577946 + 0.816075i \(0.696145\pi\)
\(6\) 0 0
\(7\) −3.26725 −1.23491 −0.617453 0.786608i \(-0.711835\pi\)
−0.617453 + 0.786608i \(0.711835\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.90115 −0.874729 −0.437364 0.899284i \(-0.644088\pi\)
−0.437364 + 0.899284i \(0.644088\pi\)
\(12\) 0 0
\(13\) 5.43566 1.50758 0.753791 0.657114i \(-0.228223\pi\)
0.753791 + 0.657114i \(0.228223\pi\)
\(14\) 0 0
\(15\) 2.58465 0.667354
\(16\) 0 0
\(17\) −2.76865 −0.671497 −0.335748 0.941952i \(-0.608989\pi\)
−0.335748 + 0.941952i \(0.608989\pi\)
\(18\) 0 0
\(19\) −6.15044 −1.41101 −0.705504 0.708706i \(-0.749279\pi\)
−0.705504 + 0.708706i \(0.749279\pi\)
\(20\) 0 0
\(21\) 3.26725 0.712973
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.68042 0.336084
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −1.72826 −0.310404 −0.155202 0.987883i \(-0.549603\pi\)
−0.155202 + 0.987883i \(0.549603\pi\)
\(32\) 0 0
\(33\) 2.90115 0.505025
\(34\) 0 0
\(35\) 8.44471 1.42742
\(36\) 0 0
\(37\) −4.97381 −0.817689 −0.408845 0.912604i \(-0.634068\pi\)
−0.408845 + 0.912604i \(0.634068\pi\)
\(38\) 0 0
\(39\) −5.43566 −0.870403
\(40\) 0 0
\(41\) −6.02032 −0.940217 −0.470108 0.882609i \(-0.655785\pi\)
−0.470108 + 0.882609i \(0.655785\pi\)
\(42\) 0 0
\(43\) −10.9734 −1.67342 −0.836712 0.547643i \(-0.815525\pi\)
−0.836712 + 0.547643i \(0.815525\pi\)
\(44\) 0 0
\(45\) −2.58465 −0.385297
\(46\) 0 0
\(47\) −3.19760 −0.466419 −0.233209 0.972427i \(-0.574923\pi\)
−0.233209 + 0.972427i \(0.574923\pi\)
\(48\) 0 0
\(49\) 3.67493 0.524990
\(50\) 0 0
\(51\) 2.76865 0.387689
\(52\) 0 0
\(53\) 5.41370 0.743628 0.371814 0.928307i \(-0.378736\pi\)
0.371814 + 0.928307i \(0.378736\pi\)
\(54\) 0 0
\(55\) 7.49845 1.01109
\(56\) 0 0
\(57\) 6.15044 0.814646
\(58\) 0 0
\(59\) −3.48319 −0.453473 −0.226737 0.973956i \(-0.572806\pi\)
−0.226737 + 0.973956i \(0.572806\pi\)
\(60\) 0 0
\(61\) −3.77242 −0.483009 −0.241504 0.970400i \(-0.577641\pi\)
−0.241504 + 0.970400i \(0.577641\pi\)
\(62\) 0 0
\(63\) −3.26725 −0.411635
\(64\) 0 0
\(65\) −14.0493 −1.74260
\(66\) 0 0
\(67\) −6.87084 −0.839407 −0.419703 0.907661i \(-0.637866\pi\)
−0.419703 + 0.907661i \(0.637866\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 3.47332 0.412207 0.206104 0.978530i \(-0.433922\pi\)
0.206104 + 0.978530i \(0.433922\pi\)
\(72\) 0 0
\(73\) −5.99833 −0.702052 −0.351026 0.936366i \(-0.614167\pi\)
−0.351026 + 0.936366i \(0.614167\pi\)
\(74\) 0 0
\(75\) −1.68042 −0.194038
\(76\) 0 0
\(77\) 9.47878 1.08021
\(78\) 0 0
\(79\) 0.609370 0.0685595 0.0342797 0.999412i \(-0.489086\pi\)
0.0342797 + 0.999412i \(0.489086\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.42111 −0.704809 −0.352404 0.935848i \(-0.614636\pi\)
−0.352404 + 0.935848i \(0.614636\pi\)
\(84\) 0 0
\(85\) 7.15600 0.776177
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) −15.0768 −1.59814 −0.799069 0.601239i \(-0.794674\pi\)
−0.799069 + 0.601239i \(0.794674\pi\)
\(90\) 0 0
\(91\) −17.7597 −1.86172
\(92\) 0 0
\(93\) 1.72826 0.179212
\(94\) 0 0
\(95\) 15.8967 1.63097
\(96\) 0 0
\(97\) −6.60560 −0.670697 −0.335349 0.942094i \(-0.608854\pi\)
−0.335349 + 0.942094i \(0.608854\pi\)
\(98\) 0 0
\(99\) −2.90115 −0.291576
\(100\) 0 0
\(101\) 15.5916 1.55142 0.775710 0.631090i \(-0.217392\pi\)
0.775710 + 0.631090i \(0.217392\pi\)
\(102\) 0 0
\(103\) −15.2178 −1.49945 −0.749727 0.661747i \(-0.769815\pi\)
−0.749727 + 0.661747i \(0.769815\pi\)
\(104\) 0 0
\(105\) −8.44471 −0.824119
\(106\) 0 0
\(107\) −3.87036 −0.374162 −0.187081 0.982345i \(-0.559903\pi\)
−0.187081 + 0.982345i \(0.559903\pi\)
\(108\) 0 0
\(109\) 9.19716 0.880928 0.440464 0.897770i \(-0.354814\pi\)
0.440464 + 0.897770i \(0.354814\pi\)
\(110\) 0 0
\(111\) 4.97381 0.472093
\(112\) 0 0
\(113\) 8.86530 0.833977 0.416988 0.908912i \(-0.363086\pi\)
0.416988 + 0.908912i \(0.363086\pi\)
\(114\) 0 0
\(115\) −2.58465 −0.241020
\(116\) 0 0
\(117\) 5.43566 0.502527
\(118\) 0 0
\(119\) 9.04588 0.829235
\(120\) 0 0
\(121\) −2.58335 −0.234850
\(122\) 0 0
\(123\) 6.02032 0.542834
\(124\) 0 0
\(125\) 8.57995 0.767414
\(126\) 0 0
\(127\) −0.0386548 −0.00343006 −0.00171503 0.999999i \(-0.500546\pi\)
−0.00171503 + 0.999999i \(0.500546\pi\)
\(128\) 0 0
\(129\) 10.9734 0.966152
\(130\) 0 0
\(131\) −18.2687 −1.59615 −0.798073 0.602561i \(-0.794147\pi\)
−0.798073 + 0.602561i \(0.794147\pi\)
\(132\) 0 0
\(133\) 20.0950 1.74246
\(134\) 0 0
\(135\) 2.58465 0.222451
\(136\) 0 0
\(137\) 3.12032 0.266587 0.133294 0.991077i \(-0.457445\pi\)
0.133294 + 0.991077i \(0.457445\pi\)
\(138\) 0 0
\(139\) −7.77344 −0.659335 −0.329667 0.944097i \(-0.606937\pi\)
−0.329667 + 0.944097i \(0.606937\pi\)
\(140\) 0 0
\(141\) 3.19760 0.269287
\(142\) 0 0
\(143\) −15.7697 −1.31872
\(144\) 0 0
\(145\) −2.58465 −0.214644
\(146\) 0 0
\(147\) −3.67493 −0.303103
\(148\) 0 0
\(149\) −16.7712 −1.37395 −0.686975 0.726681i \(-0.741062\pi\)
−0.686975 + 0.726681i \(0.741062\pi\)
\(150\) 0 0
\(151\) −12.6678 −1.03089 −0.515445 0.856923i \(-0.672373\pi\)
−0.515445 + 0.856923i \(0.672373\pi\)
\(152\) 0 0
\(153\) −2.76865 −0.223832
\(154\) 0 0
\(155\) 4.46695 0.358794
\(156\) 0 0
\(157\) −12.1569 −0.970229 −0.485115 0.874451i \(-0.661222\pi\)
−0.485115 + 0.874451i \(0.661222\pi\)
\(158\) 0 0
\(159\) −5.41370 −0.429334
\(160\) 0 0
\(161\) −3.26725 −0.257496
\(162\) 0 0
\(163\) −9.91818 −0.776852 −0.388426 0.921480i \(-0.626981\pi\)
−0.388426 + 0.921480i \(0.626981\pi\)
\(164\) 0 0
\(165\) −7.49845 −0.583754
\(166\) 0 0
\(167\) 16.7134 1.29332 0.646662 0.762776i \(-0.276164\pi\)
0.646662 + 0.762776i \(0.276164\pi\)
\(168\) 0 0
\(169\) 16.5464 1.27280
\(170\) 0 0
\(171\) −6.15044 −0.470336
\(172\) 0 0
\(173\) 4.59989 0.349723 0.174862 0.984593i \(-0.444052\pi\)
0.174862 + 0.984593i \(0.444052\pi\)
\(174\) 0 0
\(175\) −5.49036 −0.415032
\(176\) 0 0
\(177\) 3.48319 0.261813
\(178\) 0 0
\(179\) 20.7314 1.54953 0.774767 0.632247i \(-0.217867\pi\)
0.774767 + 0.632247i \(0.217867\pi\)
\(180\) 0 0
\(181\) 22.6705 1.68509 0.842543 0.538629i \(-0.181058\pi\)
0.842543 + 0.538629i \(0.181058\pi\)
\(182\) 0 0
\(183\) 3.77242 0.278865
\(184\) 0 0
\(185\) 12.8556 0.945160
\(186\) 0 0
\(187\) 8.03227 0.587377
\(188\) 0 0
\(189\) 3.26725 0.237658
\(190\) 0 0
\(191\) 2.86879 0.207578 0.103789 0.994599i \(-0.466903\pi\)
0.103789 + 0.994599i \(0.466903\pi\)
\(192\) 0 0
\(193\) −9.67593 −0.696488 −0.348244 0.937404i \(-0.613222\pi\)
−0.348244 + 0.937404i \(0.613222\pi\)
\(194\) 0 0
\(195\) 14.0493 1.00609
\(196\) 0 0
\(197\) 4.60951 0.328414 0.164207 0.986426i \(-0.447493\pi\)
0.164207 + 0.986426i \(0.447493\pi\)
\(198\) 0 0
\(199\) 17.3597 1.23059 0.615297 0.788296i \(-0.289036\pi\)
0.615297 + 0.788296i \(0.289036\pi\)
\(200\) 0 0
\(201\) 6.87084 0.484632
\(202\) 0 0
\(203\) −3.26725 −0.229316
\(204\) 0 0
\(205\) 15.5604 1.08679
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 17.8433 1.23425
\(210\) 0 0
\(211\) −6.77916 −0.466696 −0.233348 0.972393i \(-0.574968\pi\)
−0.233348 + 0.972393i \(0.574968\pi\)
\(212\) 0 0
\(213\) −3.47332 −0.237988
\(214\) 0 0
\(215\) 28.3624 1.93430
\(216\) 0 0
\(217\) 5.64666 0.383320
\(218\) 0 0
\(219\) 5.99833 0.405330
\(220\) 0 0
\(221\) −15.0495 −1.01234
\(222\) 0 0
\(223\) −12.4694 −0.835014 −0.417507 0.908674i \(-0.637096\pi\)
−0.417507 + 0.908674i \(0.637096\pi\)
\(224\) 0 0
\(225\) 1.68042 0.112028
\(226\) 0 0
\(227\) 29.2499 1.94139 0.970693 0.240321i \(-0.0772527\pi\)
0.970693 + 0.240321i \(0.0772527\pi\)
\(228\) 0 0
\(229\) −27.7176 −1.83163 −0.915817 0.401596i \(-0.868456\pi\)
−0.915817 + 0.401596i \(0.868456\pi\)
\(230\) 0 0
\(231\) −9.47878 −0.623658
\(232\) 0 0
\(233\) 20.4677 1.34089 0.670443 0.741961i \(-0.266104\pi\)
0.670443 + 0.741961i \(0.266104\pi\)
\(234\) 0 0
\(235\) 8.26469 0.539129
\(236\) 0 0
\(237\) −0.609370 −0.0395828
\(238\) 0 0
\(239\) 2.62599 0.169861 0.0849306 0.996387i \(-0.472933\pi\)
0.0849306 + 0.996387i \(0.472933\pi\)
\(240\) 0 0
\(241\) −8.14128 −0.524426 −0.262213 0.965010i \(-0.584452\pi\)
−0.262213 + 0.965010i \(0.584452\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −9.49842 −0.606832
\(246\) 0 0
\(247\) −33.4317 −2.12721
\(248\) 0 0
\(249\) 6.42111 0.406922
\(250\) 0 0
\(251\) 13.8908 0.876778 0.438389 0.898785i \(-0.355549\pi\)
0.438389 + 0.898785i \(0.355549\pi\)
\(252\) 0 0
\(253\) −2.90115 −0.182394
\(254\) 0 0
\(255\) −7.15600 −0.448126
\(256\) 0 0
\(257\) 22.3718 1.39551 0.697757 0.716334i \(-0.254181\pi\)
0.697757 + 0.716334i \(0.254181\pi\)
\(258\) 0 0
\(259\) 16.2507 1.00977
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) −24.8644 −1.53321 −0.766603 0.642121i \(-0.778055\pi\)
−0.766603 + 0.642121i \(0.778055\pi\)
\(264\) 0 0
\(265\) −13.9925 −0.859553
\(266\) 0 0
\(267\) 15.0768 0.922686
\(268\) 0 0
\(269\) −13.3151 −0.811838 −0.405919 0.913909i \(-0.633048\pi\)
−0.405919 + 0.913909i \(0.633048\pi\)
\(270\) 0 0
\(271\) −13.0537 −0.792959 −0.396479 0.918044i \(-0.629768\pi\)
−0.396479 + 0.918044i \(0.629768\pi\)
\(272\) 0 0
\(273\) 17.7597 1.07486
\(274\) 0 0
\(275\) −4.87515 −0.293983
\(276\) 0 0
\(277\) 31.3272 1.88227 0.941134 0.338033i \(-0.109762\pi\)
0.941134 + 0.338033i \(0.109762\pi\)
\(278\) 0 0
\(279\) −1.72826 −0.103468
\(280\) 0 0
\(281\) −8.97961 −0.535679 −0.267839 0.963464i \(-0.586310\pi\)
−0.267839 + 0.963464i \(0.586310\pi\)
\(282\) 0 0
\(283\) 2.83888 0.168754 0.0843769 0.996434i \(-0.473110\pi\)
0.0843769 + 0.996434i \(0.473110\pi\)
\(284\) 0 0
\(285\) −15.8967 −0.941642
\(286\) 0 0
\(287\) 19.6699 1.16108
\(288\) 0 0
\(289\) −9.33456 −0.549092
\(290\) 0 0
\(291\) 6.60560 0.387227
\(292\) 0 0
\(293\) −17.3913 −1.01601 −0.508005 0.861354i \(-0.669617\pi\)
−0.508005 + 0.861354i \(0.669617\pi\)
\(294\) 0 0
\(295\) 9.00284 0.524166
\(296\) 0 0
\(297\) 2.90115 0.168342
\(298\) 0 0
\(299\) 5.43566 0.314353
\(300\) 0 0
\(301\) 35.8528 2.06652
\(302\) 0 0
\(303\) −15.5916 −0.895712
\(304\) 0 0
\(305\) 9.75039 0.558306
\(306\) 0 0
\(307\) 18.8046 1.07323 0.536617 0.843826i \(-0.319702\pi\)
0.536617 + 0.843826i \(0.319702\pi\)
\(308\) 0 0
\(309\) 15.2178 0.865710
\(310\) 0 0
\(311\) −10.6986 −0.606661 −0.303330 0.952885i \(-0.598099\pi\)
−0.303330 + 0.952885i \(0.598099\pi\)
\(312\) 0 0
\(313\) 26.6605 1.50694 0.753471 0.657481i \(-0.228378\pi\)
0.753471 + 0.657481i \(0.228378\pi\)
\(314\) 0 0
\(315\) 8.44471 0.475805
\(316\) 0 0
\(317\) −24.9984 −1.40405 −0.702024 0.712153i \(-0.747720\pi\)
−0.702024 + 0.712153i \(0.747720\pi\)
\(318\) 0 0
\(319\) −2.90115 −0.162433
\(320\) 0 0
\(321\) 3.87036 0.216022
\(322\) 0 0
\(323\) 17.0284 0.947487
\(324\) 0 0
\(325\) 9.13421 0.506675
\(326\) 0 0
\(327\) −9.19716 −0.508604
\(328\) 0 0
\(329\) 10.4474 0.575983
\(330\) 0 0
\(331\) 35.7206 1.96338 0.981692 0.190476i \(-0.0610030\pi\)
0.981692 + 0.190476i \(0.0610030\pi\)
\(332\) 0 0
\(333\) −4.97381 −0.272563
\(334\) 0 0
\(335\) 17.7587 0.970263
\(336\) 0 0
\(337\) 13.0895 0.713028 0.356514 0.934290i \(-0.383965\pi\)
0.356514 + 0.934290i \(0.383965\pi\)
\(338\) 0 0
\(339\) −8.86530 −0.481497
\(340\) 0 0
\(341\) 5.01393 0.271520
\(342\) 0 0
\(343\) 10.8638 0.586592
\(344\) 0 0
\(345\) 2.58465 0.139153
\(346\) 0 0
\(347\) 1.35007 0.0724758 0.0362379 0.999343i \(-0.488463\pi\)
0.0362379 + 0.999343i \(0.488463\pi\)
\(348\) 0 0
\(349\) 3.78246 0.202470 0.101235 0.994863i \(-0.467721\pi\)
0.101235 + 0.994863i \(0.467721\pi\)
\(350\) 0 0
\(351\) −5.43566 −0.290134
\(352\) 0 0
\(353\) −27.7802 −1.47859 −0.739295 0.673381i \(-0.764841\pi\)
−0.739295 + 0.673381i \(0.764841\pi\)
\(354\) 0 0
\(355\) −8.97732 −0.476467
\(356\) 0 0
\(357\) −9.04588 −0.478759
\(358\) 0 0
\(359\) 33.7405 1.78075 0.890377 0.455224i \(-0.150441\pi\)
0.890377 + 0.455224i \(0.150441\pi\)
\(360\) 0 0
\(361\) 18.8279 0.990944
\(362\) 0 0
\(363\) 2.58335 0.135591
\(364\) 0 0
\(365\) 15.5036 0.811496
\(366\) 0 0
\(367\) −1.27364 −0.0664836 −0.0332418 0.999447i \(-0.510583\pi\)
−0.0332418 + 0.999447i \(0.510583\pi\)
\(368\) 0 0
\(369\) −6.02032 −0.313406
\(370\) 0 0
\(371\) −17.6879 −0.918310
\(372\) 0 0
\(373\) 19.2000 0.994137 0.497069 0.867711i \(-0.334410\pi\)
0.497069 + 0.867711i \(0.334410\pi\)
\(374\) 0 0
\(375\) −8.57995 −0.443067
\(376\) 0 0
\(377\) 5.43566 0.279951
\(378\) 0 0
\(379\) 0.333289 0.0171199 0.00855995 0.999963i \(-0.497275\pi\)
0.00855995 + 0.999963i \(0.497275\pi\)
\(380\) 0 0
\(381\) 0.0386548 0.00198035
\(382\) 0 0
\(383\) 1.46228 0.0747188 0.0373594 0.999302i \(-0.488105\pi\)
0.0373594 + 0.999302i \(0.488105\pi\)
\(384\) 0 0
\(385\) −24.4993 −1.24860
\(386\) 0 0
\(387\) −10.9734 −0.557808
\(388\) 0 0
\(389\) 18.3472 0.930241 0.465120 0.885247i \(-0.346011\pi\)
0.465120 + 0.885247i \(0.346011\pi\)
\(390\) 0 0
\(391\) −2.76865 −0.140017
\(392\) 0 0
\(393\) 18.2687 0.921535
\(394\) 0 0
\(395\) −1.57501 −0.0792473
\(396\) 0 0
\(397\) 7.65239 0.384062 0.192031 0.981389i \(-0.438493\pi\)
0.192031 + 0.981389i \(0.438493\pi\)
\(398\) 0 0
\(399\) −20.0950 −1.00601
\(400\) 0 0
\(401\) 16.3401 0.815986 0.407993 0.912985i \(-0.366229\pi\)
0.407993 + 0.912985i \(0.366229\pi\)
\(402\) 0 0
\(403\) −9.39423 −0.467960
\(404\) 0 0
\(405\) −2.58465 −0.128432
\(406\) 0 0
\(407\) 14.4298 0.715256
\(408\) 0 0
\(409\) 0.101086 0.00499837 0.00249919 0.999997i \(-0.499204\pi\)
0.00249919 + 0.999997i \(0.499204\pi\)
\(410\) 0 0
\(411\) −3.12032 −0.153914
\(412\) 0 0
\(413\) 11.3805 0.559996
\(414\) 0 0
\(415\) 16.5963 0.814682
\(416\) 0 0
\(417\) 7.77344 0.380667
\(418\) 0 0
\(419\) 10.3659 0.506409 0.253204 0.967413i \(-0.418516\pi\)
0.253204 + 0.967413i \(0.418516\pi\)
\(420\) 0 0
\(421\) 5.97694 0.291298 0.145649 0.989336i \(-0.453473\pi\)
0.145649 + 0.989336i \(0.453473\pi\)
\(422\) 0 0
\(423\) −3.19760 −0.155473
\(424\) 0 0
\(425\) −4.65250 −0.225680
\(426\) 0 0
\(427\) 12.3254 0.596470
\(428\) 0 0
\(429\) 15.7697 0.761366
\(430\) 0 0
\(431\) −7.85116 −0.378177 −0.189088 0.981960i \(-0.560553\pi\)
−0.189088 + 0.981960i \(0.560553\pi\)
\(432\) 0 0
\(433\) 12.7222 0.611389 0.305694 0.952130i \(-0.401111\pi\)
0.305694 + 0.952130i \(0.401111\pi\)
\(434\) 0 0
\(435\) 2.58465 0.123925
\(436\) 0 0
\(437\) −6.15044 −0.294216
\(438\) 0 0
\(439\) 23.4129 1.11744 0.558718 0.829357i \(-0.311293\pi\)
0.558718 + 0.829357i \(0.311293\pi\)
\(440\) 0 0
\(441\) 3.67493 0.174997
\(442\) 0 0
\(443\) −5.14557 −0.244474 −0.122237 0.992501i \(-0.539007\pi\)
−0.122237 + 0.992501i \(0.539007\pi\)
\(444\) 0 0
\(445\) 38.9683 1.84727
\(446\) 0 0
\(447\) 16.7712 0.793250
\(448\) 0 0
\(449\) −16.5786 −0.782391 −0.391195 0.920308i \(-0.627938\pi\)
−0.391195 + 0.920308i \(0.627938\pi\)
\(450\) 0 0
\(451\) 17.4658 0.822434
\(452\) 0 0
\(453\) 12.6678 0.595184
\(454\) 0 0
\(455\) 45.9026 2.15195
\(456\) 0 0
\(457\) −11.3367 −0.530307 −0.265154 0.964206i \(-0.585423\pi\)
−0.265154 + 0.964206i \(0.585423\pi\)
\(458\) 0 0
\(459\) 2.76865 0.129230
\(460\) 0 0
\(461\) −35.1134 −1.63539 −0.817697 0.575649i \(-0.804750\pi\)
−0.817697 + 0.575649i \(0.804750\pi\)
\(462\) 0 0
\(463\) −28.4019 −1.31995 −0.659975 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(464\) 0 0
\(465\) −4.46695 −0.207150
\(466\) 0 0
\(467\) 12.3209 0.570143 0.285072 0.958506i \(-0.407983\pi\)
0.285072 + 0.958506i \(0.407983\pi\)
\(468\) 0 0
\(469\) 22.4488 1.03659
\(470\) 0 0
\(471\) 12.1569 0.560162
\(472\) 0 0
\(473\) 31.8354 1.46379
\(474\) 0 0
\(475\) −10.3353 −0.474218
\(476\) 0 0
\(477\) 5.41370 0.247876
\(478\) 0 0
\(479\) −1.37261 −0.0627162 −0.0313581 0.999508i \(-0.509983\pi\)
−0.0313581 + 0.999508i \(0.509983\pi\)
\(480\) 0 0
\(481\) −27.0360 −1.23273
\(482\) 0 0
\(483\) 3.26725 0.148665
\(484\) 0 0
\(485\) 17.0732 0.775253
\(486\) 0 0
\(487\) 19.9010 0.901803 0.450901 0.892574i \(-0.351103\pi\)
0.450901 + 0.892574i \(0.351103\pi\)
\(488\) 0 0
\(489\) 9.91818 0.448516
\(490\) 0 0
\(491\) −0.0888982 −0.00401192 −0.00200596 0.999998i \(-0.500639\pi\)
−0.00200596 + 0.999998i \(0.500639\pi\)
\(492\) 0 0
\(493\) −2.76865 −0.124694
\(494\) 0 0
\(495\) 7.49845 0.337030
\(496\) 0 0
\(497\) −11.3482 −0.509037
\(498\) 0 0
\(499\) 16.6820 0.746790 0.373395 0.927672i \(-0.378194\pi\)
0.373395 + 0.927672i \(0.378194\pi\)
\(500\) 0 0
\(501\) −16.7134 −0.746701
\(502\) 0 0
\(503\) −3.99675 −0.178206 −0.0891032 0.996022i \(-0.528400\pi\)
−0.0891032 + 0.996022i \(0.528400\pi\)
\(504\) 0 0
\(505\) −40.2988 −1.79327
\(506\) 0 0
\(507\) −16.5464 −0.734853
\(508\) 0 0
\(509\) 9.87300 0.437613 0.218807 0.975768i \(-0.429784\pi\)
0.218807 + 0.975768i \(0.429784\pi\)
\(510\) 0 0
\(511\) 19.5981 0.866967
\(512\) 0 0
\(513\) 6.15044 0.271549
\(514\) 0 0
\(515\) 39.3327 1.73321
\(516\) 0 0
\(517\) 9.27672 0.407990
\(518\) 0 0
\(519\) −4.59989 −0.201913
\(520\) 0 0
\(521\) −25.7062 −1.12621 −0.563104 0.826386i \(-0.690393\pi\)
−0.563104 + 0.826386i \(0.690393\pi\)
\(522\) 0 0
\(523\) −20.9459 −0.915902 −0.457951 0.888978i \(-0.651416\pi\)
−0.457951 + 0.888978i \(0.651416\pi\)
\(524\) 0 0
\(525\) 5.49036 0.239619
\(526\) 0 0
\(527\) 4.78495 0.208436
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −3.48319 −0.151158
\(532\) 0 0
\(533\) −32.7245 −1.41745
\(534\) 0 0
\(535\) 10.0035 0.432490
\(536\) 0 0
\(537\) −20.7314 −0.894624
\(538\) 0 0
\(539\) −10.6615 −0.459224
\(540\) 0 0
\(541\) −19.9499 −0.857715 −0.428857 0.903372i \(-0.641084\pi\)
−0.428857 + 0.903372i \(0.641084\pi\)
\(542\) 0 0
\(543\) −22.6705 −0.972885
\(544\) 0 0
\(545\) −23.7714 −1.01826
\(546\) 0 0
\(547\) −17.5884 −0.752026 −0.376013 0.926614i \(-0.622705\pi\)
−0.376013 + 0.926614i \(0.622705\pi\)
\(548\) 0 0
\(549\) −3.77242 −0.161003
\(550\) 0 0
\(551\) −6.15044 −0.262018
\(552\) 0 0
\(553\) −1.99096 −0.0846644
\(554\) 0 0
\(555\) −12.8556 −0.545688
\(556\) 0 0
\(557\) 39.4366 1.67098 0.835492 0.549503i \(-0.185183\pi\)
0.835492 + 0.549503i \(0.185183\pi\)
\(558\) 0 0
\(559\) −59.6476 −2.52282
\(560\) 0 0
\(561\) −8.03227 −0.339123
\(562\) 0 0
\(563\) 29.4911 1.24290 0.621452 0.783453i \(-0.286543\pi\)
0.621452 + 0.783453i \(0.286543\pi\)
\(564\) 0 0
\(565\) −22.9137 −0.963987
\(566\) 0 0
\(567\) −3.26725 −0.137212
\(568\) 0 0
\(569\) 3.32457 0.139373 0.0696866 0.997569i \(-0.477800\pi\)
0.0696866 + 0.997569i \(0.477800\pi\)
\(570\) 0 0
\(571\) −36.9084 −1.54457 −0.772284 0.635278i \(-0.780886\pi\)
−0.772284 + 0.635278i \(0.780886\pi\)
\(572\) 0 0
\(573\) −2.86879 −0.119845
\(574\) 0 0
\(575\) 1.68042 0.0700784
\(576\) 0 0
\(577\) 25.8580 1.07648 0.538242 0.842790i \(-0.319089\pi\)
0.538242 + 0.842790i \(0.319089\pi\)
\(578\) 0 0
\(579\) 9.67593 0.402118
\(580\) 0 0
\(581\) 20.9794 0.870372
\(582\) 0 0
\(583\) −15.7059 −0.650473
\(584\) 0 0
\(585\) −14.0493 −0.580867
\(586\) 0 0
\(587\) −38.2916 −1.58046 −0.790232 0.612808i \(-0.790040\pi\)
−0.790232 + 0.612808i \(0.790040\pi\)
\(588\) 0 0
\(589\) 10.6296 0.437983
\(590\) 0 0
\(591\) −4.60951 −0.189610
\(592\) 0 0
\(593\) −9.36472 −0.384563 −0.192281 0.981340i \(-0.561589\pi\)
−0.192281 + 0.981340i \(0.561589\pi\)
\(594\) 0 0
\(595\) −23.3805 −0.958505
\(596\) 0 0
\(597\) −17.3597 −0.710483
\(598\) 0 0
\(599\) −5.91152 −0.241538 −0.120769 0.992681i \(-0.538536\pi\)
−0.120769 + 0.992681i \(0.538536\pi\)
\(600\) 0 0
\(601\) 18.1326 0.739643 0.369821 0.929103i \(-0.379419\pi\)
0.369821 + 0.929103i \(0.379419\pi\)
\(602\) 0 0
\(603\) −6.87084 −0.279802
\(604\) 0 0
\(605\) 6.67706 0.271461
\(606\) 0 0
\(607\) −47.2698 −1.91862 −0.959311 0.282353i \(-0.908885\pi\)
−0.959311 + 0.282353i \(0.908885\pi\)
\(608\) 0 0
\(609\) 3.26725 0.132396
\(610\) 0 0
\(611\) −17.3811 −0.703164
\(612\) 0 0
\(613\) −41.3828 −1.67144 −0.835718 0.549159i \(-0.814948\pi\)
−0.835718 + 0.549159i \(0.814948\pi\)
\(614\) 0 0
\(615\) −15.5604 −0.627457
\(616\) 0 0
\(617\) 5.22450 0.210331 0.105165 0.994455i \(-0.466463\pi\)
0.105165 + 0.994455i \(0.466463\pi\)
\(618\) 0 0
\(619\) −2.08761 −0.0839080 −0.0419540 0.999120i \(-0.513358\pi\)
−0.0419540 + 0.999120i \(0.513358\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 49.2597 1.97355
\(624\) 0 0
\(625\) −30.5783 −1.22313
\(626\) 0 0
\(627\) −17.8433 −0.712594
\(628\) 0 0
\(629\) 13.7708 0.549076
\(630\) 0 0
\(631\) 0.758471 0.0301943 0.0150971 0.999886i \(-0.495194\pi\)
0.0150971 + 0.999886i \(0.495194\pi\)
\(632\) 0 0
\(633\) 6.77916 0.269447
\(634\) 0 0
\(635\) 0.0999093 0.00396478
\(636\) 0 0
\(637\) 19.9757 0.791466
\(638\) 0 0
\(639\) 3.47332 0.137402
\(640\) 0 0
\(641\) 9.04373 0.357206 0.178603 0.983921i \(-0.442842\pi\)
0.178603 + 0.983921i \(0.442842\pi\)
\(642\) 0 0
\(643\) 37.8286 1.49181 0.745907 0.666050i \(-0.232016\pi\)
0.745907 + 0.666050i \(0.232016\pi\)
\(644\) 0 0
\(645\) −28.3624 −1.11677
\(646\) 0 0
\(647\) 9.03911 0.355364 0.177682 0.984088i \(-0.443140\pi\)
0.177682 + 0.984088i \(0.443140\pi\)
\(648\) 0 0
\(649\) 10.1053 0.396666
\(650\) 0 0
\(651\) −5.64666 −0.221310
\(652\) 0 0
\(653\) −3.81709 −0.149374 −0.0746871 0.997207i \(-0.523796\pi\)
−0.0746871 + 0.997207i \(0.523796\pi\)
\(654\) 0 0
\(655\) 47.2183 1.84497
\(656\) 0 0
\(657\) −5.99833 −0.234017
\(658\) 0 0
\(659\) −9.64692 −0.375791 −0.187895 0.982189i \(-0.560167\pi\)
−0.187895 + 0.982189i \(0.560167\pi\)
\(660\) 0 0
\(661\) −19.7624 −0.768667 −0.384334 0.923194i \(-0.625569\pi\)
−0.384334 + 0.923194i \(0.625569\pi\)
\(662\) 0 0
\(663\) 15.0495 0.584473
\(664\) 0 0
\(665\) −51.9387 −2.01410
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) 0 0
\(669\) 12.4694 0.482095
\(670\) 0 0
\(671\) 10.9443 0.422502
\(672\) 0 0
\(673\) −29.6832 −1.14420 −0.572101 0.820183i \(-0.693871\pi\)
−0.572101 + 0.820183i \(0.693871\pi\)
\(674\) 0 0
\(675\) −1.68042 −0.0646795
\(676\) 0 0
\(677\) 19.7488 0.759008 0.379504 0.925190i \(-0.376095\pi\)
0.379504 + 0.925190i \(0.376095\pi\)
\(678\) 0 0
\(679\) 21.5822 0.828247
\(680\) 0 0
\(681\) −29.2499 −1.12086
\(682\) 0 0
\(683\) 13.3720 0.511667 0.255834 0.966721i \(-0.417650\pi\)
0.255834 + 0.966721i \(0.417650\pi\)
\(684\) 0 0
\(685\) −8.06495 −0.308146
\(686\) 0 0
\(687\) 27.7176 1.05749
\(688\) 0 0
\(689\) 29.4270 1.12108
\(690\) 0 0
\(691\) −49.0254 −1.86501 −0.932507 0.361153i \(-0.882383\pi\)
−0.932507 + 0.361153i \(0.882383\pi\)
\(692\) 0 0
\(693\) 9.47878 0.360069
\(694\) 0 0
\(695\) 20.0916 0.762119
\(696\) 0 0
\(697\) 16.6682 0.631353
\(698\) 0 0
\(699\) −20.4677 −0.774161
\(700\) 0 0
\(701\) 0.916175 0.0346035 0.0173017 0.999850i \(-0.494492\pi\)
0.0173017 + 0.999850i \(0.494492\pi\)
\(702\) 0 0
\(703\) 30.5911 1.15377
\(704\) 0 0
\(705\) −8.26469 −0.311266
\(706\) 0 0
\(707\) −50.9416 −1.91586
\(708\) 0 0
\(709\) −4.11514 −0.154547 −0.0772736 0.997010i \(-0.524621\pi\)
−0.0772736 + 0.997010i \(0.524621\pi\)
\(710\) 0 0
\(711\) 0.609370 0.0228532
\(712\) 0 0
\(713\) −1.72826 −0.0647238
\(714\) 0 0
\(715\) 40.7591 1.52430
\(716\) 0 0
\(717\) −2.62599 −0.0980694
\(718\) 0 0
\(719\) −43.0447 −1.60530 −0.802648 0.596453i \(-0.796576\pi\)
−0.802648 + 0.596453i \(0.796576\pi\)
\(720\) 0 0
\(721\) 49.7204 1.85168
\(722\) 0 0
\(723\) 8.14128 0.302778
\(724\) 0 0
\(725\) 1.68042 0.0624093
\(726\) 0 0
\(727\) −8.96336 −0.332433 −0.166216 0.986089i \(-0.553155\pi\)
−0.166216 + 0.986089i \(0.553155\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 30.3815 1.12370
\(732\) 0 0
\(733\) −10.5056 −0.388034 −0.194017 0.980998i \(-0.562152\pi\)
−0.194017 + 0.980998i \(0.562152\pi\)
\(734\) 0 0
\(735\) 9.49842 0.350355
\(736\) 0 0
\(737\) 19.9333 0.734253
\(738\) 0 0
\(739\) 52.9028 1.94606 0.973030 0.230679i \(-0.0740946\pi\)
0.973030 + 0.230679i \(0.0740946\pi\)
\(740\) 0 0
\(741\) 33.4317 1.22815
\(742\) 0 0
\(743\) −11.0250 −0.404469 −0.202234 0.979337i \(-0.564820\pi\)
−0.202234 + 0.979337i \(0.564820\pi\)
\(744\) 0 0
\(745\) 43.3477 1.58814
\(746\) 0 0
\(747\) −6.42111 −0.234936
\(748\) 0 0
\(749\) 12.6454 0.462054
\(750\) 0 0
\(751\) −45.1829 −1.64875 −0.824374 0.566046i \(-0.808473\pi\)
−0.824374 + 0.566046i \(0.808473\pi\)
\(752\) 0 0
\(753\) −13.8908 −0.506208
\(754\) 0 0
\(755\) 32.7418 1.19160
\(756\) 0 0
\(757\) 5.11999 0.186089 0.0930446 0.995662i \(-0.470340\pi\)
0.0930446 + 0.995662i \(0.470340\pi\)
\(758\) 0 0
\(759\) 2.90115 0.105305
\(760\) 0 0
\(761\) 47.5479 1.72361 0.861805 0.507240i \(-0.169334\pi\)
0.861805 + 0.507240i \(0.169334\pi\)
\(762\) 0 0
\(763\) −30.0494 −1.08786
\(764\) 0 0
\(765\) 7.15600 0.258726
\(766\) 0 0
\(767\) −18.9335 −0.683648
\(768\) 0 0
\(769\) −32.5122 −1.17242 −0.586210 0.810159i \(-0.699381\pi\)
−0.586210 + 0.810159i \(0.699381\pi\)
\(770\) 0 0
\(771\) −22.3718 −0.805701
\(772\) 0 0
\(773\) −23.6598 −0.850984 −0.425492 0.904962i \(-0.639899\pi\)
−0.425492 + 0.904962i \(0.639899\pi\)
\(774\) 0 0
\(775\) −2.90420 −0.104322
\(776\) 0 0
\(777\) −16.2507 −0.582990
\(778\) 0 0
\(779\) 37.0277 1.32665
\(780\) 0 0
\(781\) −10.0766 −0.360569
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 31.4214 1.12148
\(786\) 0 0
\(787\) −35.3648 −1.26062 −0.630310 0.776344i \(-0.717072\pi\)
−0.630310 + 0.776344i \(0.717072\pi\)
\(788\) 0 0
\(789\) 24.8644 0.885197
\(790\) 0 0
\(791\) −28.9652 −1.02988
\(792\) 0 0
\(793\) −20.5056 −0.728175
\(794\) 0 0
\(795\) 13.9925 0.496263
\(796\) 0 0
\(797\) 4.02243 0.142482 0.0712408 0.997459i \(-0.477304\pi\)
0.0712408 + 0.997459i \(0.477304\pi\)
\(798\) 0 0
\(799\) 8.85306 0.313199
\(800\) 0 0
\(801\) −15.0768 −0.532713
\(802\) 0 0
\(803\) 17.4020 0.614105
\(804\) 0 0
\(805\) 8.44471 0.297637
\(806\) 0 0
\(807\) 13.3151 0.468715
\(808\) 0 0
\(809\) 31.0035 1.09003 0.545013 0.838428i \(-0.316525\pi\)
0.545013 + 0.838428i \(0.316525\pi\)
\(810\) 0 0
\(811\) 9.52266 0.334386 0.167193 0.985924i \(-0.446530\pi\)
0.167193 + 0.985924i \(0.446530\pi\)
\(812\) 0 0
\(813\) 13.0537 0.457815
\(814\) 0 0
\(815\) 25.6350 0.897956
\(816\) 0 0
\(817\) 67.4911 2.36122
\(818\) 0 0
\(819\) −17.7597 −0.620574
\(820\) 0 0
\(821\) 1.87751 0.0655257 0.0327628 0.999463i \(-0.489569\pi\)
0.0327628 + 0.999463i \(0.489569\pi\)
\(822\) 0 0
\(823\) −26.7916 −0.933897 −0.466948 0.884285i \(-0.654647\pi\)
−0.466948 + 0.884285i \(0.654647\pi\)
\(824\) 0 0
\(825\) 4.87515 0.169731
\(826\) 0 0
\(827\) 25.2439 0.877818 0.438909 0.898532i \(-0.355365\pi\)
0.438909 + 0.898532i \(0.355365\pi\)
\(828\) 0 0
\(829\) 43.8248 1.52210 0.761050 0.648693i \(-0.224684\pi\)
0.761050 + 0.648693i \(0.224684\pi\)
\(830\) 0 0
\(831\) −31.3272 −1.08673
\(832\) 0 0
\(833\) −10.1746 −0.352529
\(834\) 0 0
\(835\) −43.1984 −1.49494
\(836\) 0 0
\(837\) 1.72826 0.0597374
\(838\) 0 0
\(839\) 50.4257 1.74089 0.870445 0.492266i \(-0.163831\pi\)
0.870445 + 0.492266i \(0.163831\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 8.97961 0.309274
\(844\) 0 0
\(845\) −42.7668 −1.47122
\(846\) 0 0
\(847\) 8.44045 0.290017
\(848\) 0 0
\(849\) −2.83888 −0.0974300
\(850\) 0 0
\(851\) −4.97381 −0.170500
\(852\) 0 0
\(853\) −28.0675 −0.961014 −0.480507 0.876991i \(-0.659547\pi\)
−0.480507 + 0.876991i \(0.659547\pi\)
\(854\) 0 0
\(855\) 15.8967 0.543657
\(856\) 0 0
\(857\) −10.2561 −0.350341 −0.175170 0.984538i \(-0.556048\pi\)
−0.175170 + 0.984538i \(0.556048\pi\)
\(858\) 0 0
\(859\) 41.7755 1.42536 0.712681 0.701488i \(-0.247481\pi\)
0.712681 + 0.701488i \(0.247481\pi\)
\(860\) 0 0
\(861\) −19.6699 −0.670349
\(862\) 0 0
\(863\) −13.2911 −0.452433 −0.226217 0.974077i \(-0.572636\pi\)
−0.226217 + 0.974077i \(0.572636\pi\)
\(864\) 0 0
\(865\) −11.8891 −0.404242
\(866\) 0 0
\(867\) 9.33456 0.317018
\(868\) 0 0
\(869\) −1.76787 −0.0599709
\(870\) 0 0
\(871\) −37.3476 −1.26547
\(872\) 0 0
\(873\) −6.60560 −0.223566
\(874\) 0 0
\(875\) −28.0329 −0.947684
\(876\) 0 0
\(877\) −28.6871 −0.968694 −0.484347 0.874876i \(-0.660943\pi\)
−0.484347 + 0.874876i \(0.660943\pi\)
\(878\) 0 0
\(879\) 17.3913 0.586594
\(880\) 0 0
\(881\) −13.6373 −0.459451 −0.229726 0.973255i \(-0.573783\pi\)
−0.229726 + 0.973255i \(0.573783\pi\)
\(882\) 0 0
\(883\) −30.8554 −1.03837 −0.519183 0.854663i \(-0.673764\pi\)
−0.519183 + 0.854663i \(0.673764\pi\)
\(884\) 0 0
\(885\) −9.00284 −0.302627
\(886\) 0 0
\(887\) 20.2891 0.681242 0.340621 0.940201i \(-0.389363\pi\)
0.340621 + 0.940201i \(0.389363\pi\)
\(888\) 0 0
\(889\) 0.126295 0.00423580
\(890\) 0 0
\(891\) −2.90115 −0.0971921
\(892\) 0 0
\(893\) 19.6667 0.658120
\(894\) 0 0
\(895\) −53.5833 −1.79109
\(896\) 0 0
\(897\) −5.43566 −0.181492
\(898\) 0 0
\(899\) −1.72826 −0.0576407
\(900\) 0 0
\(901\) −14.9886 −0.499344
\(902\) 0 0
\(903\) −35.8528 −1.19311
\(904\) 0 0
\(905\) −58.5954 −1.94778
\(906\) 0 0
\(907\) −7.37111 −0.244754 −0.122377 0.992484i \(-0.539052\pi\)
−0.122377 + 0.992484i \(0.539052\pi\)
\(908\) 0 0
\(909\) 15.5916 0.517140
\(910\) 0 0
\(911\) 29.3900 0.973735 0.486867 0.873476i \(-0.338140\pi\)
0.486867 + 0.873476i \(0.338140\pi\)
\(912\) 0 0
\(913\) 18.6286 0.616516
\(914\) 0 0
\(915\) −9.75039 −0.322338
\(916\) 0 0
\(917\) 59.6885 1.97109
\(918\) 0 0
\(919\) −47.9846 −1.58287 −0.791433 0.611256i \(-0.790665\pi\)
−0.791433 + 0.611256i \(0.790665\pi\)
\(920\) 0 0
\(921\) −18.8046 −0.619632
\(922\) 0 0
\(923\) 18.8798 0.621436
\(924\) 0 0
\(925\) −8.35810 −0.274813
\(926\) 0 0
\(927\) −15.2178 −0.499818
\(928\) 0 0
\(929\) −11.9401 −0.391741 −0.195871 0.980630i \(-0.562753\pi\)
−0.195871 + 0.980630i \(0.562753\pi\)
\(930\) 0 0
\(931\) −22.6025 −0.740766
\(932\) 0 0
\(933\) 10.6986 0.350256
\(934\) 0 0
\(935\) −20.7606 −0.678944
\(936\) 0 0
\(937\) 6.48683 0.211915 0.105958 0.994371i \(-0.466209\pi\)
0.105958 + 0.994371i \(0.466209\pi\)
\(938\) 0 0
\(939\) −26.6605 −0.870034
\(940\) 0 0
\(941\) 53.7386 1.75183 0.875915 0.482466i \(-0.160259\pi\)
0.875915 + 0.482466i \(0.160259\pi\)
\(942\) 0 0
\(943\) −6.02032 −0.196049
\(944\) 0 0
\(945\) −8.44471 −0.274706
\(946\) 0 0
\(947\) 1.43493 0.0466291 0.0233145 0.999728i \(-0.492578\pi\)
0.0233145 + 0.999728i \(0.492578\pi\)
\(948\) 0 0
\(949\) −32.6049 −1.05840
\(950\) 0 0
\(951\) 24.9984 0.810628
\(952\) 0 0
\(953\) −47.4944 −1.53850 −0.769248 0.638950i \(-0.779369\pi\)
−0.769248 + 0.638950i \(0.779369\pi\)
\(954\) 0 0
\(955\) −7.41482 −0.239938
\(956\) 0 0
\(957\) 2.90115 0.0937807
\(958\) 0 0
\(959\) −10.1949 −0.329210
\(960\) 0 0
\(961\) −28.0131 −0.903649
\(962\) 0 0
\(963\) −3.87036 −0.124721
\(964\) 0 0
\(965\) 25.0089 0.805065
\(966\) 0 0
\(967\) 9.45128 0.303933 0.151966 0.988386i \(-0.451439\pi\)
0.151966 + 0.988386i \(0.451439\pi\)
\(968\) 0 0
\(969\) −17.0284 −0.547032
\(970\) 0 0
\(971\) 2.14235 0.0687514 0.0343757 0.999409i \(-0.489056\pi\)
0.0343757 + 0.999409i \(0.489056\pi\)
\(972\) 0 0
\(973\) 25.3978 0.814216
\(974\) 0 0
\(975\) −9.13421 −0.292529
\(976\) 0 0
\(977\) 58.8448 1.88261 0.941306 0.337555i \(-0.109600\pi\)
0.941306 + 0.337555i \(0.109600\pi\)
\(978\) 0 0
\(979\) 43.7400 1.39794
\(980\) 0 0
\(981\) 9.19716 0.293643
\(982\) 0 0
\(983\) −55.4775 −1.76946 −0.884730 0.466105i \(-0.845657\pi\)
−0.884730 + 0.466105i \(0.845657\pi\)
\(984\) 0 0
\(985\) −11.9140 −0.379611
\(986\) 0 0
\(987\) −10.4474 −0.332544
\(988\) 0 0
\(989\) −10.9734 −0.348933
\(990\) 0 0
\(991\) 2.28446 0.0725683 0.0362841 0.999342i \(-0.488448\pi\)
0.0362841 + 0.999342i \(0.488448\pi\)
\(992\) 0 0
\(993\) −35.7206 −1.13356
\(994\) 0 0
\(995\) −44.8687 −1.42243
\(996\) 0 0
\(997\) 36.7888 1.16511 0.582557 0.812790i \(-0.302052\pi\)
0.582557 + 0.812790i \(0.302052\pi\)
\(998\) 0 0
\(999\) 4.97381 0.157364
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 8004.2.a.i.1.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.i.1.4 16 1.1 even 1 trivial