Properties

Label 4600.2.e.v.4049.5
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 18 x^{8} + 103 x^{6} + 239 x^{4} + 197 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.5
Root \(-1.84717i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.v.4049.6

$q$-expansion

\(f(q)\) \(=\) \(q-0.724570i q^{3} +2.33840i q^{7} +2.47500 q^{9} +O(q^{10})\) \(q-0.724570i q^{3} +2.33840i q^{7} +2.47500 q^{9} -2.62783 q^{11} -4.29631i q^{13} -6.85404i q^{17} -2.87303 q^{19} +1.69433 q^{21} +1.00000i q^{23} -3.96702i q^{27} +5.03711 q^{29} -7.31504 q^{31} +1.90405i q^{33} +9.24142i q^{37} -3.11297 q^{39} -6.95078 q^{41} +7.01520i q^{43} -4.74690i q^{47} +1.53188 q^{49} -4.96623 q^{51} -12.3849i q^{53} +2.08171i q^{57} +2.14073 q^{59} -2.30567 q^{61} +5.78754i q^{63} +1.98376i q^{67} +0.724570 q^{69} -6.87990 q^{71} +13.2518i q^{73} -6.14492i q^{77} -16.6578 q^{79} +4.55062 q^{81} +9.09450i q^{83} -3.64974i q^{87} -0.676383 q^{89} +10.0465 q^{91} +5.30026i q^{93} -15.0759i q^{97} -6.50388 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 8 q^{9} + O(q^{10}) \) \( 10 q - 8 q^{9} - 8 q^{11} - 8 q^{19} - 12 q^{21} + 22 q^{29} + 8 q^{31} - 62 q^{39} - 16 q^{41} + 4 q^{49} - 10 q^{51} - 46 q^{59} - 52 q^{61} - 6 q^{69} - 4 q^{71} - 86 q^{79} - 6 q^{81} - 30 q^{89} - 38 q^{91} - 74 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 0.724570i − 0.418331i −0.977880 0.209165i \(-0.932925\pi\)
0.977880 0.209165i \(-0.0670747\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.33840i 0.883833i 0.897056 + 0.441916i \(0.145701\pi\)
−0.897056 + 0.441916i \(0.854299\pi\)
\(8\) 0 0
\(9\) 2.47500 0.825000
\(10\) 0 0
\(11\) −2.62783 −0.792321 −0.396160 0.918181i \(-0.629658\pi\)
−0.396160 + 0.918181i \(0.629658\pi\)
\(12\) 0 0
\(13\) − 4.29631i − 1.19158i −0.803140 0.595791i \(-0.796839\pi\)
0.803140 0.595791i \(-0.203161\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 6.85404i − 1.66235i −0.556011 0.831175i \(-0.687669\pi\)
0.556011 0.831175i \(-0.312331\pi\)
\(18\) 0 0
\(19\) −2.87303 −0.659118 −0.329559 0.944135i \(-0.606900\pi\)
−0.329559 + 0.944135i \(0.606900\pi\)
\(20\) 0 0
\(21\) 1.69433 0.369734
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 3.96702i − 0.763453i
\(28\) 0 0
\(29\) 5.03711 0.935368 0.467684 0.883896i \(-0.345088\pi\)
0.467684 + 0.883896i \(0.345088\pi\)
\(30\) 0 0
\(31\) −7.31504 −1.31382 −0.656910 0.753969i \(-0.728137\pi\)
−0.656910 + 0.753969i \(0.728137\pi\)
\(32\) 0 0
\(33\) 1.90405i 0.331452i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9.24142i 1.51928i 0.650344 + 0.759640i \(0.274625\pi\)
−0.650344 + 0.759640i \(0.725375\pi\)
\(38\) 0 0
\(39\) −3.11297 −0.498475
\(40\) 0 0
\(41\) −6.95078 −1.08553 −0.542765 0.839885i \(-0.682623\pi\)
−0.542765 + 0.839885i \(0.682623\pi\)
\(42\) 0 0
\(43\) 7.01520i 1.06981i 0.844913 + 0.534904i \(0.179652\pi\)
−0.844913 + 0.534904i \(0.820348\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 4.74690i − 0.692406i −0.938160 0.346203i \(-0.887471\pi\)
0.938160 0.346203i \(-0.112529\pi\)
\(48\) 0 0
\(49\) 1.53188 0.218840
\(50\) 0 0
\(51\) −4.96623 −0.695412
\(52\) 0 0
\(53\) − 12.3849i − 1.70120i −0.525817 0.850598i \(-0.676240\pi\)
0.525817 0.850598i \(-0.323760\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.08171i 0.275729i
\(58\) 0 0
\(59\) 2.14073 0.278699 0.139349 0.990243i \(-0.455499\pi\)
0.139349 + 0.990243i \(0.455499\pi\)
\(60\) 0 0
\(61\) −2.30567 −0.295210 −0.147605 0.989046i \(-0.547156\pi\)
−0.147605 + 0.989046i \(0.547156\pi\)
\(62\) 0 0
\(63\) 5.78754i 0.729162i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.98376i 0.242355i 0.992631 + 0.121178i \(0.0386671\pi\)
−0.992631 + 0.121178i \(0.961333\pi\)
\(68\) 0 0
\(69\) 0.724570 0.0872279
\(70\) 0 0
\(71\) −6.87990 −0.816494 −0.408247 0.912871i \(-0.633860\pi\)
−0.408247 + 0.912871i \(0.633860\pi\)
\(72\) 0 0
\(73\) 13.2518i 1.55101i 0.631342 + 0.775505i \(0.282504\pi\)
−0.631342 + 0.775505i \(0.717496\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 6.14492i − 0.700279i
\(78\) 0 0
\(79\) −16.6578 −1.87415 −0.937076 0.349126i \(-0.886478\pi\)
−0.937076 + 0.349126i \(0.886478\pi\)
\(80\) 0 0
\(81\) 4.55062 0.505624
\(82\) 0 0
\(83\) 9.09450i 0.998251i 0.866530 + 0.499126i \(0.166345\pi\)
−0.866530 + 0.499126i \(0.833655\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 3.64974i − 0.391293i
\(88\) 0 0
\(89\) −0.676383 −0.0716965 −0.0358482 0.999357i \(-0.511413\pi\)
−0.0358482 + 0.999357i \(0.511413\pi\)
\(90\) 0 0
\(91\) 10.0465 1.05316
\(92\) 0 0
\(93\) 5.30026i 0.549611i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 15.0759i − 1.53073i −0.643597 0.765365i \(-0.722559\pi\)
0.643597 0.765365i \(-0.277441\pi\)
\(98\) 0 0
\(99\) −6.50388 −0.653664
\(100\) 0 0
\(101\) −10.7713 −1.07178 −0.535892 0.844286i \(-0.680025\pi\)
−0.535892 + 0.844286i \(0.680025\pi\)
\(102\) 0 0
\(103\) − 16.4501i − 1.62088i −0.585824 0.810438i \(-0.699229\pi\)
0.585824 0.810438i \(-0.300771\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 7.21143i − 0.697155i −0.937280 0.348578i \(-0.886665\pi\)
0.937280 0.348578i \(-0.113335\pi\)
\(108\) 0 0
\(109\) −18.6581 −1.78712 −0.893560 0.448943i \(-0.851801\pi\)
−0.893560 + 0.448943i \(0.851801\pi\)
\(110\) 0 0
\(111\) 6.69605 0.635561
\(112\) 0 0
\(113\) − 0.722608i − 0.0679772i −0.999422 0.0339886i \(-0.989179\pi\)
0.999422 0.0339886i \(-0.0108210\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 10.6334i − 0.983054i
\(118\) 0 0
\(119\) 16.0275 1.46924
\(120\) 0 0
\(121\) −4.09450 −0.372228
\(122\) 0 0
\(123\) 5.03633i 0.454110i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 19.1029i − 1.69510i −0.530712 0.847552i \(-0.678076\pi\)
0.530712 0.847552i \(-0.321924\pi\)
\(128\) 0 0
\(129\) 5.08300 0.447534
\(130\) 0 0
\(131\) 9.25213 0.808362 0.404181 0.914679i \(-0.367557\pi\)
0.404181 + 0.914679i \(0.367557\pi\)
\(132\) 0 0
\(133\) − 6.71829i − 0.582550i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 20.5484i − 1.75557i −0.479058 0.877783i \(-0.659022\pi\)
0.479058 0.877783i \(-0.340978\pi\)
\(138\) 0 0
\(139\) 5.48875 0.465550 0.232775 0.972531i \(-0.425219\pi\)
0.232775 + 0.972531i \(0.425219\pi\)
\(140\) 0 0
\(141\) −3.43946 −0.289655
\(142\) 0 0
\(143\) 11.2900i 0.944115i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 1.10995i − 0.0915473i
\(148\) 0 0
\(149\) −5.51461 −0.451775 −0.225887 0.974153i \(-0.572528\pi\)
−0.225887 + 0.974153i \(0.572528\pi\)
\(150\) 0 0
\(151\) 16.5363 1.34570 0.672851 0.739778i \(-0.265069\pi\)
0.672851 + 0.739778i \(0.265069\pi\)
\(152\) 0 0
\(153\) − 16.9638i − 1.37144i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 13.6888i − 1.09249i −0.837626 0.546244i \(-0.816057\pi\)
0.837626 0.546244i \(-0.183943\pi\)
\(158\) 0 0
\(159\) −8.97372 −0.711662
\(160\) 0 0
\(161\) −2.33840 −0.184292
\(162\) 0 0
\(163\) 18.8266i 1.47462i 0.675556 + 0.737308i \(0.263903\pi\)
−0.675556 + 0.737308i \(0.736097\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 3.68830i − 0.285409i −0.989765 0.142705i \(-0.954420\pi\)
0.989765 0.142705i \(-0.0455799\pi\)
\(168\) 0 0
\(169\) −5.45825 −0.419866
\(170\) 0 0
\(171\) −7.11074 −0.543772
\(172\) 0 0
\(173\) 7.94698i 0.604198i 0.953277 + 0.302099i \(0.0976873\pi\)
−0.953277 + 0.302099i \(0.902313\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 1.55111i − 0.116588i
\(178\) 0 0
\(179\) −5.72373 −0.427811 −0.213906 0.976854i \(-0.568619\pi\)
−0.213906 + 0.976854i \(0.568619\pi\)
\(180\) 0 0
\(181\) −2.19348 −0.163040 −0.0815199 0.996672i \(-0.525977\pi\)
−0.0815199 + 0.996672i \(0.525977\pi\)
\(182\) 0 0
\(183\) 1.67062i 0.123495i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 18.0113i 1.31711i
\(188\) 0 0
\(189\) 9.27648 0.674765
\(190\) 0 0
\(191\) −14.4643 −1.04660 −0.523302 0.852148i \(-0.675300\pi\)
−0.523302 + 0.852148i \(0.675300\pi\)
\(192\) 0 0
\(193\) − 17.7400i − 1.27696i −0.769640 0.638478i \(-0.779564\pi\)
0.769640 0.638478i \(-0.220436\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 24.0847i 1.71596i 0.513679 + 0.857982i \(0.328282\pi\)
−0.513679 + 0.857982i \(0.671718\pi\)
\(198\) 0 0
\(199\) −7.78754 −0.552044 −0.276022 0.961151i \(-0.589016\pi\)
−0.276022 + 0.961151i \(0.589016\pi\)
\(200\) 0 0
\(201\) 1.43738 0.101385
\(202\) 0 0
\(203\) 11.7788i 0.826709i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.47500i 0.172024i
\(208\) 0 0
\(209\) 7.54983 0.522233
\(210\) 0 0
\(211\) 16.5419 1.13879 0.569394 0.822065i \(-0.307178\pi\)
0.569394 + 0.822065i \(0.307178\pi\)
\(212\) 0 0
\(213\) 4.98497i 0.341565i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 17.1055i − 1.16120i
\(218\) 0 0
\(219\) 9.60187 0.648834
\(220\) 0 0
\(221\) −29.4471 −1.98082
\(222\) 0 0
\(223\) − 21.4323i − 1.43521i −0.696449 0.717606i \(-0.745238\pi\)
0.696449 0.717606i \(-0.254762\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 18.6739i − 1.23943i −0.784827 0.619715i \(-0.787248\pi\)
0.784827 0.619715i \(-0.212752\pi\)
\(228\) 0 0
\(229\) 1.85981 0.122900 0.0614499 0.998110i \(-0.480428\pi\)
0.0614499 + 0.998110i \(0.480428\pi\)
\(230\) 0 0
\(231\) −4.45243 −0.292948
\(232\) 0 0
\(233\) − 5.39803i − 0.353637i −0.984244 0.176818i \(-0.943420\pi\)
0.984244 0.176818i \(-0.0565805\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 12.0698i 0.784015i
\(238\) 0 0
\(239\) −21.1817 −1.37013 −0.685065 0.728482i \(-0.740226\pi\)
−0.685065 + 0.728482i \(0.740226\pi\)
\(240\) 0 0
\(241\) −29.8152 −1.92057 −0.960283 0.279028i \(-0.909988\pi\)
−0.960283 + 0.279028i \(0.909988\pi\)
\(242\) 0 0
\(243\) − 15.1983i − 0.974971i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 12.3434i 0.785392i
\(248\) 0 0
\(249\) 6.58960 0.417599
\(250\) 0 0
\(251\) −14.0212 −0.885013 −0.442506 0.896765i \(-0.645911\pi\)
−0.442506 + 0.896765i \(0.645911\pi\)
\(252\) 0 0
\(253\) − 2.62783i − 0.165210i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 7.14991i − 0.445999i −0.974819 0.222999i \(-0.928415\pi\)
0.974819 0.222999i \(-0.0715848\pi\)
\(258\) 0 0
\(259\) −21.6101 −1.34279
\(260\) 0 0
\(261\) 12.4668 0.771678
\(262\) 0 0
\(263\) 16.6723i 1.02806i 0.857772 + 0.514030i \(0.171848\pi\)
−0.857772 + 0.514030i \(0.828152\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0.490087i 0.0299928i
\(268\) 0 0
\(269\) 11.7884 0.718752 0.359376 0.933193i \(-0.382990\pi\)
0.359376 + 0.933193i \(0.382990\pi\)
\(270\) 0 0
\(271\) −8.37717 −0.508877 −0.254438 0.967089i \(-0.581891\pi\)
−0.254438 + 0.967089i \(0.581891\pi\)
\(272\) 0 0
\(273\) − 7.27938i − 0.440568i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 3.99879i − 0.240264i −0.992758 0.120132i \(-0.961668\pi\)
0.992758 0.120132i \(-0.0383318\pi\)
\(278\) 0 0
\(279\) −18.1047 −1.08390
\(280\) 0 0
\(281\) 21.9478 1.30930 0.654648 0.755934i \(-0.272817\pi\)
0.654648 + 0.755934i \(0.272817\pi\)
\(282\) 0 0
\(283\) − 10.6297i − 0.631868i −0.948781 0.315934i \(-0.897682\pi\)
0.948781 0.315934i \(-0.102318\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 16.2537i − 0.959427i
\(288\) 0 0
\(289\) −29.9779 −1.76341
\(290\) 0 0
\(291\) −10.9236 −0.640351
\(292\) 0 0
\(293\) 28.5108i 1.66562i 0.553558 + 0.832810i \(0.313270\pi\)
−0.553558 + 0.832810i \(0.686730\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 10.4247i 0.604900i
\(298\) 0 0
\(299\) 4.29631 0.248462
\(300\) 0 0
\(301\) −16.4044 −0.945532
\(302\) 0 0
\(303\) 7.80456i 0.448360i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0.391652i 0.0223528i 0.999938 + 0.0111764i \(0.00355763\pi\)
−0.999938 + 0.0111764i \(0.996442\pi\)
\(308\) 0 0
\(309\) −11.9192 −0.678062
\(310\) 0 0
\(311\) 22.5240 1.27722 0.638609 0.769531i \(-0.279510\pi\)
0.638609 + 0.769531i \(0.279510\pi\)
\(312\) 0 0
\(313\) 22.0468i 1.24616i 0.782160 + 0.623078i \(0.214118\pi\)
−0.782160 + 0.623078i \(0.785882\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 6.01192i − 0.337663i −0.985645 0.168831i \(-0.946001\pi\)
0.985645 0.168831i \(-0.0539994\pi\)
\(318\) 0 0
\(319\) −13.2367 −0.741112
\(320\) 0 0
\(321\) −5.22518 −0.291641
\(322\) 0 0
\(323\) 19.6919i 1.09568i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 13.5191i 0.747607i
\(328\) 0 0
\(329\) 11.1002 0.611971
\(330\) 0 0
\(331\) −27.5619 −1.51494 −0.757469 0.652871i \(-0.773564\pi\)
−0.757469 + 0.652871i \(0.773564\pi\)
\(332\) 0 0
\(333\) 22.8725i 1.25341i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 11.8095i − 0.643306i −0.946858 0.321653i \(-0.895761\pi\)
0.946858 0.321653i \(-0.104239\pi\)
\(338\) 0 0
\(339\) −0.523580 −0.0284370
\(340\) 0 0
\(341\) 19.2227 1.04097
\(342\) 0 0
\(343\) 19.9510i 1.07725i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 28.8288i 1.54761i 0.633423 + 0.773806i \(0.281649\pi\)
−0.633423 + 0.773806i \(0.718351\pi\)
\(348\) 0 0
\(349\) 26.1403 1.39926 0.699629 0.714506i \(-0.253349\pi\)
0.699629 + 0.714506i \(0.253349\pi\)
\(350\) 0 0
\(351\) −17.0435 −0.909716
\(352\) 0 0
\(353\) − 23.8004i − 1.26677i −0.773839 0.633383i \(-0.781666\pi\)
0.773839 0.633383i \(-0.218334\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 11.6130i − 0.614628i
\(358\) 0 0
\(359\) −29.0102 −1.53110 −0.765551 0.643375i \(-0.777534\pi\)
−0.765551 + 0.643375i \(0.777534\pi\)
\(360\) 0 0
\(361\) −10.7457 −0.565564
\(362\) 0 0
\(363\) 2.96675i 0.155714i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 2.84461i − 0.148487i −0.997240 0.0742437i \(-0.976346\pi\)
0.997240 0.0742437i \(-0.0236543\pi\)
\(368\) 0 0
\(369\) −17.2032 −0.895562
\(370\) 0 0
\(371\) 28.9609 1.50357
\(372\) 0 0
\(373\) 4.65308i 0.240928i 0.992718 + 0.120464i \(0.0384382\pi\)
−0.992718 + 0.120464i \(0.961562\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 21.6410i − 1.11457i
\(378\) 0 0
\(379\) −20.8607 −1.07154 −0.535772 0.844363i \(-0.679979\pi\)
−0.535772 + 0.844363i \(0.679979\pi\)
\(380\) 0 0
\(381\) −13.8413 −0.709114
\(382\) 0 0
\(383\) 10.7116i 0.547337i 0.961824 + 0.273669i \(0.0882372\pi\)
−0.961824 + 0.273669i \(0.911763\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 17.3626i 0.882592i
\(388\) 0 0
\(389\) −3.28523 −0.166568 −0.0832839 0.996526i \(-0.526541\pi\)
−0.0832839 + 0.996526i \(0.526541\pi\)
\(390\) 0 0
\(391\) 6.85404 0.346624
\(392\) 0 0
\(393\) − 6.70381i − 0.338163i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 6.00319i − 0.301292i −0.988588 0.150646i \(-0.951865\pi\)
0.988588 0.150646i \(-0.0481353\pi\)
\(398\) 0 0
\(399\) −4.86787 −0.243698
\(400\) 0 0
\(401\) −22.9420 −1.14567 −0.572835 0.819671i \(-0.694156\pi\)
−0.572835 + 0.819671i \(0.694156\pi\)
\(402\) 0 0
\(403\) 31.4277i 1.56552i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 24.2849i − 1.20376i
\(408\) 0 0
\(409\) 9.04425 0.447210 0.223605 0.974680i \(-0.428218\pi\)
0.223605 + 0.974680i \(0.428218\pi\)
\(410\) 0 0
\(411\) −14.8887 −0.734407
\(412\) 0 0
\(413\) 5.00588i 0.246323i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 3.97698i − 0.194754i
\(418\) 0 0
\(419\) 28.1921 1.37727 0.688637 0.725106i \(-0.258209\pi\)
0.688637 + 0.725106i \(0.258209\pi\)
\(420\) 0 0
\(421\) −9.14897 −0.445893 −0.222947 0.974831i \(-0.571568\pi\)
−0.222947 + 0.974831i \(0.571568\pi\)
\(422\) 0 0
\(423\) − 11.7486i − 0.571235i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 5.39157i − 0.260916i
\(428\) 0 0
\(429\) 8.18037 0.394952
\(430\) 0 0
\(431\) 32.5005 1.56549 0.782747 0.622340i \(-0.213818\pi\)
0.782747 + 0.622340i \(0.213818\pi\)
\(432\) 0 0
\(433\) 0.555560i 0.0266985i 0.999911 + 0.0133492i \(0.00424932\pi\)
−0.999911 + 0.0133492i \(0.995751\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 2.87303i − 0.137436i
\(438\) 0 0
\(439\) 25.8680 1.23461 0.617305 0.786724i \(-0.288224\pi\)
0.617305 + 0.786724i \(0.288224\pi\)
\(440\) 0 0
\(441\) 3.79140 0.180543
\(442\) 0 0
\(443\) − 30.2965i − 1.43943i −0.694270 0.719715i \(-0.744272\pi\)
0.694270 0.719715i \(-0.255728\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 3.99572i 0.188991i
\(448\) 0 0
\(449\) 31.6573 1.49400 0.747001 0.664823i \(-0.231493\pi\)
0.747001 + 0.664823i \(0.231493\pi\)
\(450\) 0 0
\(451\) 18.2655 0.860088
\(452\) 0 0
\(453\) − 11.9817i − 0.562949i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 7.34455i − 0.343564i −0.985135 0.171782i \(-0.945048\pi\)
0.985135 0.171782i \(-0.0549524\pi\)
\(458\) 0 0
\(459\) −27.1901 −1.26913
\(460\) 0 0
\(461\) 15.6987 0.731163 0.365581 0.930779i \(-0.380870\pi\)
0.365581 + 0.930779i \(0.380870\pi\)
\(462\) 0 0
\(463\) 1.82421i 0.0847782i 0.999101 + 0.0423891i \(0.0134969\pi\)
−0.999101 + 0.0423891i \(0.986503\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.43722i 0.0665067i 0.999447 + 0.0332533i \(0.0105868\pi\)
−0.999447 + 0.0332533i \(0.989413\pi\)
\(468\) 0 0
\(469\) −4.63884 −0.214202
\(470\) 0 0
\(471\) −9.91852 −0.457021
\(472\) 0 0
\(473\) − 18.4348i − 0.847632i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 30.6526i − 1.40349i
\(478\) 0 0
\(479\) −26.5858 −1.21474 −0.607368 0.794420i \(-0.707775\pi\)
−0.607368 + 0.794420i \(0.707775\pi\)
\(480\) 0 0
\(481\) 39.7040 1.81035
\(482\) 0 0
\(483\) 1.69433i 0.0770949i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 33.8112i 1.53213i 0.642762 + 0.766066i \(0.277788\pi\)
−0.642762 + 0.766066i \(0.722212\pi\)
\(488\) 0 0
\(489\) 13.6412 0.616877
\(490\) 0 0
\(491\) 22.4820 1.01460 0.507298 0.861770i \(-0.330644\pi\)
0.507298 + 0.861770i \(0.330644\pi\)
\(492\) 0 0
\(493\) − 34.5246i − 1.55491i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 16.0880i − 0.721644i
\(498\) 0 0
\(499\) −22.6152 −1.01240 −0.506198 0.862417i \(-0.668949\pi\)
−0.506198 + 0.862417i \(0.668949\pi\)
\(500\) 0 0
\(501\) −2.67243 −0.119395
\(502\) 0 0
\(503\) − 16.3142i − 0.727414i −0.931513 0.363707i \(-0.881511\pi\)
0.931513 0.363707i \(-0.118489\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 3.95488i 0.175643i
\(508\) 0 0
\(509\) 8.01935 0.355451 0.177726 0.984080i \(-0.443126\pi\)
0.177726 + 0.984080i \(0.443126\pi\)
\(510\) 0 0
\(511\) −30.9881 −1.37083
\(512\) 0 0
\(513\) 11.3973i 0.503205i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 12.4740i 0.548608i
\(518\) 0 0
\(519\) 5.75814 0.252754
\(520\) 0 0
\(521\) 18.0130 0.789162 0.394581 0.918861i \(-0.370890\pi\)
0.394581 + 0.918861i \(0.370890\pi\)
\(522\) 0 0
\(523\) 1.91940i 0.0839294i 0.999119 + 0.0419647i \(0.0133617\pi\)
−0.999119 + 0.0419647i \(0.986638\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 50.1376i 2.18403i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 5.29830 0.229927
\(532\) 0 0
\(533\) 29.8627i 1.29350i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 4.14724i 0.178967i
\(538\) 0 0
\(539\) −4.02552 −0.173391
\(540\) 0 0
\(541\) −42.0297 −1.80700 −0.903500 0.428589i \(-0.859011\pi\)
−0.903500 + 0.428589i \(0.859011\pi\)
\(542\) 0 0
\(543\) 1.58933i 0.0682045i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 36.0750i − 1.54246i −0.636559 0.771228i \(-0.719643\pi\)
0.636559 0.771228i \(-0.280357\pi\)
\(548\) 0 0
\(549\) −5.70652 −0.243548
\(550\) 0 0
\(551\) −14.4718 −0.616518
\(552\) 0 0
\(553\) − 38.9527i − 1.65644i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 31.9500i − 1.35377i −0.736091 0.676883i \(-0.763330\pi\)
0.736091 0.676883i \(-0.236670\pi\)
\(558\) 0 0
\(559\) 30.1395 1.27476
\(560\) 0 0
\(561\) 13.0504 0.550989
\(562\) 0 0
\(563\) 6.26575i 0.264070i 0.991245 + 0.132035i \(0.0421511\pi\)
−0.991245 + 0.132035i \(0.957849\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 10.6412i 0.446887i
\(568\) 0 0
\(569\) 12.7997 0.536593 0.268296 0.963336i \(-0.413539\pi\)
0.268296 + 0.963336i \(0.413539\pi\)
\(570\) 0 0
\(571\) 13.7690 0.576217 0.288108 0.957598i \(-0.406974\pi\)
0.288108 + 0.957598i \(0.406974\pi\)
\(572\) 0 0
\(573\) 10.4804i 0.437826i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 13.5830i 0.565470i 0.959198 + 0.282735i \(0.0912416\pi\)
−0.959198 + 0.282735i \(0.908758\pi\)
\(578\) 0 0
\(579\) −12.8539 −0.534190
\(580\) 0 0
\(581\) −21.2666 −0.882287
\(582\) 0 0
\(583\) 32.5454i 1.34789i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 19.3233i 0.797557i 0.917047 + 0.398778i \(0.130566\pi\)
−0.917047 + 0.398778i \(0.869434\pi\)
\(588\) 0 0
\(589\) 21.0163 0.865962
\(590\) 0 0
\(591\) 17.4511 0.717840
\(592\) 0 0
\(593\) 25.1344i 1.03215i 0.856545 + 0.516073i \(0.172607\pi\)
−0.856545 + 0.516073i \(0.827393\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5.64262i 0.230937i
\(598\) 0 0
\(599\) −1.88678 −0.0770918 −0.0385459 0.999257i \(-0.512273\pi\)
−0.0385459 + 0.999257i \(0.512273\pi\)
\(600\) 0 0
\(601\) 13.2105 0.538866 0.269433 0.963019i \(-0.413164\pi\)
0.269433 + 0.963019i \(0.413164\pi\)
\(602\) 0 0
\(603\) 4.90982i 0.199943i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 3.50465i − 0.142249i −0.997467 0.0711246i \(-0.977341\pi\)
0.997467 0.0711246i \(-0.0226588\pi\)
\(608\) 0 0
\(609\) 8.53455 0.345838
\(610\) 0 0
\(611\) −20.3941 −0.825058
\(612\) 0 0
\(613\) − 7.59433i − 0.306732i −0.988169 0.153366i \(-0.950989\pi\)
0.988169 0.153366i \(-0.0490114\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 13.4277i 0.540579i 0.962779 + 0.270290i \(0.0871195\pi\)
−0.962779 + 0.270290i \(0.912881\pi\)
\(618\) 0 0
\(619\) 45.0584 1.81105 0.905525 0.424292i \(-0.139477\pi\)
0.905525 + 0.424292i \(0.139477\pi\)
\(620\) 0 0
\(621\) 3.96702 0.159191
\(622\) 0 0
\(623\) − 1.58166i − 0.0633677i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 5.47038i − 0.218466i
\(628\) 0 0
\(629\) 63.3411 2.52557
\(630\) 0 0
\(631\) −14.9030 −0.593280 −0.296640 0.954989i \(-0.595866\pi\)
−0.296640 + 0.954989i \(0.595866\pi\)
\(632\) 0 0
\(633\) − 11.9857i − 0.476390i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 6.58142i − 0.260765i
\(638\) 0 0
\(639\) −17.0278 −0.673608
\(640\) 0 0
\(641\) 37.7852 1.49243 0.746213 0.665707i \(-0.231870\pi\)
0.746213 + 0.665707i \(0.231870\pi\)
\(642\) 0 0
\(643\) − 18.8777i − 0.744465i −0.928140 0.372232i \(-0.878592\pi\)
0.928140 0.372232i \(-0.121408\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 20.5272i − 0.807009i −0.914978 0.403505i \(-0.867792\pi\)
0.914978 0.403505i \(-0.132208\pi\)
\(648\) 0 0
\(649\) −5.62547 −0.220819
\(650\) 0 0
\(651\) −12.3941 −0.485764
\(652\) 0 0
\(653\) − 3.07302i − 0.120256i −0.998191 0.0601282i \(-0.980849\pi\)
0.998191 0.0601282i \(-0.0191509\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 32.7983i 1.27958i
\(658\) 0 0
\(659\) 20.4014 0.794725 0.397363 0.917662i \(-0.369925\pi\)
0.397363 + 0.917662i \(0.369925\pi\)
\(660\) 0 0
\(661\) −4.17682 −0.162460 −0.0812298 0.996695i \(-0.525885\pi\)
−0.0812298 + 0.996695i \(0.525885\pi\)
\(662\) 0 0
\(663\) 21.3365i 0.828639i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 5.03711i 0.195038i
\(668\) 0 0
\(669\) −15.5292 −0.600393
\(670\) 0 0
\(671\) 6.05890 0.233901
\(672\) 0 0
\(673\) − 45.0529i − 1.73666i −0.495984 0.868331i \(-0.665193\pi\)
0.495984 0.868331i \(-0.334807\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 16.5395i 0.635665i 0.948147 + 0.317833i \(0.102955\pi\)
−0.948147 + 0.317833i \(0.897045\pi\)
\(678\) 0 0
\(679\) 35.2536 1.35291
\(680\) 0 0
\(681\) −13.5305 −0.518492
\(682\) 0 0
\(683\) − 26.9328i − 1.03055i −0.857024 0.515277i \(-0.827689\pi\)
0.857024 0.515277i \(-0.172311\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 1.34756i − 0.0514128i
\(688\) 0 0
\(689\) −53.2093 −2.02711
\(690\) 0 0
\(691\) 44.1583 1.67986 0.839931 0.542693i \(-0.182595\pi\)
0.839931 + 0.542693i \(0.182595\pi\)
\(692\) 0 0
\(693\) − 15.2087i − 0.577730i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 47.6410i 1.80453i
\(698\) 0 0
\(699\) −3.91125 −0.147937
\(700\) 0 0
\(701\) −15.6476 −0.591002 −0.295501 0.955342i \(-0.595487\pi\)
−0.295501 + 0.955342i \(0.595487\pi\)
\(702\) 0 0
\(703\) − 26.5508i − 1.00138i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 25.1876i − 0.947279i
\(708\) 0 0
\(709\) −8.79068 −0.330141 −0.165070 0.986282i \(-0.552785\pi\)
−0.165070 + 0.986282i \(0.552785\pi\)
\(710\) 0 0
\(711\) −41.2281 −1.54617
\(712\) 0 0
\(713\) − 7.31504i − 0.273951i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 15.3476i 0.573167i
\(718\) 0 0
\(719\) −39.3081 −1.46594 −0.732972 0.680259i \(-0.761867\pi\)
−0.732972 + 0.680259i \(0.761867\pi\)
\(720\) 0 0
\(721\) 38.4669 1.43258
\(722\) 0 0
\(723\) 21.6032i 0.803431i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 16.1156i 0.597695i 0.954301 + 0.298848i \(0.0966022\pi\)
−0.954301 + 0.298848i \(0.903398\pi\)
\(728\) 0 0
\(729\) 2.63962 0.0977638
\(730\) 0 0
\(731\) 48.0825 1.77840
\(732\) 0 0
\(733\) 9.66306i 0.356913i 0.983948 + 0.178457i \(0.0571104\pi\)
−0.983948 + 0.178457i \(0.942890\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 5.21300i − 0.192023i
\(738\) 0 0
\(739\) −24.3518 −0.895796 −0.447898 0.894085i \(-0.647827\pi\)
−0.447898 + 0.894085i \(0.647827\pi\)
\(740\) 0 0
\(741\) 8.94366 0.328553
\(742\) 0 0
\(743\) 32.0702i 1.17654i 0.808664 + 0.588271i \(0.200191\pi\)
−0.808664 + 0.588271i \(0.799809\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 22.5089i 0.823557i
\(748\) 0 0
\(749\) 16.8632 0.616168
\(750\) 0 0
\(751\) 1.88705 0.0688596 0.0344298 0.999407i \(-0.489038\pi\)
0.0344298 + 0.999407i \(0.489038\pi\)
\(752\) 0 0
\(753\) 10.1594i 0.370228i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 5.13762i − 0.186730i −0.995632 0.0933651i \(-0.970238\pi\)
0.995632 0.0933651i \(-0.0297624\pi\)
\(758\) 0 0
\(759\) −1.90405 −0.0691125
\(760\) 0 0
\(761\) 39.4855 1.43135 0.715674 0.698435i \(-0.246120\pi\)
0.715674 + 0.698435i \(0.246120\pi\)
\(762\) 0 0
\(763\) − 43.6301i − 1.57952i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 9.19722i − 0.332092i
\(768\) 0 0
\(769\) 1.32121 0.0476439 0.0238220 0.999716i \(-0.492417\pi\)
0.0238220 + 0.999716i \(0.492417\pi\)
\(770\) 0 0
\(771\) −5.18061 −0.186575
\(772\) 0 0
\(773\) − 33.7473i − 1.21381i −0.794776 0.606903i \(-0.792412\pi\)
0.794776 0.606903i \(-0.207588\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 15.6581i 0.561730i
\(778\) 0 0
\(779\) 19.9698 0.715492
\(780\) 0 0
\(781\) 18.0792 0.646926
\(782\) 0 0
\(783\) − 19.9823i − 0.714110i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 13.1800i − 0.469816i −0.972018 0.234908i \(-0.924521\pi\)
0.972018 0.234908i \(-0.0754789\pi\)
\(788\) 0 0
\(789\) 12.0803 0.430069
\(790\) 0 0
\(791\) 1.68975 0.0600805
\(792\) 0 0
\(793\) 9.90584i 0.351767i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 21.0017i 0.743919i 0.928249 + 0.371959i \(0.121314\pi\)
−0.928249 + 0.371959i \(0.878686\pi\)
\(798\) 0 0
\(799\) −32.5354 −1.15102
\(800\) 0 0
\(801\) −1.67405 −0.0591496
\(802\) 0 0
\(803\) − 34.8236i − 1.22890i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 8.54153i − 0.300676i
\(808\) 0 0
\(809\) −4.45831 −0.156746 −0.0783729 0.996924i \(-0.524972\pi\)
−0.0783729 + 0.996924i \(0.524972\pi\)
\(810\) 0 0
\(811\) 27.9804 0.982525 0.491263 0.871012i \(-0.336536\pi\)
0.491263 + 0.871012i \(0.336536\pi\)
\(812\) 0 0
\(813\) 6.06984i 0.212879i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 20.1549i − 0.705130i
\(818\) 0 0
\(819\) 24.8651 0.868855
\(820\) 0 0
\(821\) 29.0587 1.01416 0.507078 0.861900i \(-0.330726\pi\)
0.507078 + 0.861900i \(0.330726\pi\)
\(822\) 0 0
\(823\) 6.83168i 0.238138i 0.992886 + 0.119069i \(0.0379909\pi\)
−0.992886 + 0.119069i \(0.962009\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 31.5165i 1.09594i 0.836499 + 0.547968i \(0.184598\pi\)
−0.836499 + 0.547968i \(0.815402\pi\)
\(828\) 0 0
\(829\) 22.9202 0.796052 0.398026 0.917374i \(-0.369695\pi\)
0.398026 + 0.917374i \(0.369695\pi\)
\(830\) 0 0
\(831\) −2.89741 −0.100510
\(832\) 0 0
\(833\) − 10.4996i − 0.363788i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 29.0189i 1.00304i
\(838\) 0 0
\(839\) −19.6003 −0.676676 −0.338338 0.941025i \(-0.609865\pi\)
−0.338338 + 0.941025i \(0.609865\pi\)
\(840\) 0 0
\(841\) −3.62750 −0.125086
\(842\) 0 0
\(843\) − 15.9027i − 0.547718i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 9.57459i − 0.328987i
\(848\) 0 0
\(849\) −7.70193 −0.264330
\(850\) 0 0
\(851\) −9.24142 −0.316792
\(852\) 0 0
\(853\) 9.76951i 0.334502i 0.985914 + 0.167251i \(0.0534890\pi\)
−0.985914 + 0.167251i \(0.946511\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 26.5308i − 0.906273i −0.891441 0.453137i \(-0.850305\pi\)
0.891441 0.453137i \(-0.149695\pi\)
\(858\) 0 0
\(859\) −37.1341 −1.26700 −0.633499 0.773744i \(-0.718382\pi\)
−0.633499 + 0.773744i \(0.718382\pi\)
\(860\) 0 0
\(861\) −11.7770 −0.401358
\(862\) 0 0
\(863\) − 6.15457i − 0.209504i −0.994498 0.104752i \(-0.966595\pi\)
0.994498 0.104752i \(-0.0334049\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 21.7211i 0.737687i
\(868\) 0 0
\(869\) 43.7739 1.48493
\(870\) 0 0
\(871\) 8.52286 0.288786
\(872\) 0 0
\(873\) − 37.3129i − 1.26285i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 22.0440i 0.744373i 0.928158 + 0.372186i \(0.121392\pi\)
−0.928158 + 0.372186i \(0.878608\pi\)
\(878\) 0 0
\(879\) 20.6581 0.696780
\(880\) 0 0
\(881\) 11.0196 0.371260 0.185630 0.982620i \(-0.440567\pi\)
0.185630 + 0.982620i \(0.440567\pi\)
\(882\) 0 0
\(883\) 33.8860i 1.14035i 0.821522 + 0.570177i \(0.193125\pi\)
−0.821522 + 0.570177i \(0.806875\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 3.89387i 0.130743i 0.997861 + 0.0653717i \(0.0208233\pi\)
−0.997861 + 0.0653717i \(0.979177\pi\)
\(888\) 0 0
\(889\) 44.6701 1.49819
\(890\) 0 0
\(891\) −11.9582 −0.400616
\(892\) 0 0
\(893\) 13.6380i 0.456377i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 3.11297i − 0.103939i
\(898\) 0 0
\(899\) −36.8467 −1.22891
\(900\) 0 0
\(901\) −84.8866 −2.82798
\(902\) 0 0
\(903\) 11.8861i 0.395545i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 30.5048i 1.01289i 0.862271 + 0.506447i \(0.169041\pi\)
−0.862271 + 0.506447i \(0.830959\pi\)
\(908\) 0 0
\(909\) −26.6590 −0.884222
\(910\) 0 0
\(911\) −15.8042 −0.523616 −0.261808 0.965120i \(-0.584319\pi\)
−0.261808 + 0.965120i \(0.584319\pi\)
\(912\) 0 0
\(913\) − 23.8988i − 0.790935i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 21.6352i 0.714457i
\(918\) 0 0
\(919\) 41.2196 1.35971 0.679855 0.733347i \(-0.262043\pi\)
0.679855 + 0.733347i \(0.262043\pi\)
\(920\) 0 0
\(921\) 0.283779 0.00935085
\(922\) 0 0
\(923\) 29.5582i 0.972919i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 40.7140i − 1.33722i
\(928\) 0 0
\(929\) −13.3600 −0.438326 −0.219163 0.975688i \(-0.570333\pi\)
−0.219163 + 0.975688i \(0.570333\pi\)
\(930\) 0 0
\(931\) −4.40113 −0.144241
\(932\) 0 0
\(933\) − 16.3202i − 0.534299i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 16.6980i 0.545501i 0.962085 + 0.272751i \(0.0879334\pi\)
−0.962085 + 0.272751i \(0.912067\pi\)
\(938\) 0 0
\(939\) 15.9744 0.521305
\(940\) 0 0
\(941\) 16.3650 0.533482 0.266741 0.963768i \(-0.414053\pi\)
0.266741 + 0.963768i \(0.414053\pi\)
\(942\) 0 0
\(943\) − 6.95078i − 0.226349i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 7.88894i − 0.256356i −0.991751 0.128178i \(-0.959087\pi\)
0.991751 0.128178i \(-0.0409129\pi\)
\(948\) 0 0
\(949\) 56.9339 1.84815
\(950\) 0 0
\(951\) −4.35605 −0.141255
\(952\) 0 0
\(953\) 2.26143i 0.0732549i 0.999329 + 0.0366275i \(0.0116615\pi\)
−0.999329 + 0.0366275i \(0.988339\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 9.59090i 0.310030i
\(958\) 0 0
\(959\) 48.0504 1.55163
\(960\) 0 0
\(961\) 22.5099 0.726125
\(962\) 0 0
\(963\) − 17.8483i − 0.575153i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 23.1416i − 0.744182i −0.928196 0.372091i \(-0.878641\pi\)
0.928196 0.372091i \(-0.121359\pi\)
\(968\) 0 0
\(969\) 14.2681 0.458358
\(970\) 0 0
\(971\) −27.9414 −0.896683 −0.448341 0.893862i \(-0.647985\pi\)
−0.448341 + 0.893862i \(0.647985\pi\)
\(972\) 0 0
\(973\) 12.8349i 0.411468i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 9.62964i − 0.308079i −0.988065 0.154040i \(-0.950772\pi\)
0.988065 0.154040i \(-0.0492284\pi\)
\(978\) 0 0
\(979\) 1.77742 0.0568066
\(980\) 0 0
\(981\) −46.1787 −1.47437
\(982\) 0 0
\(983\) 23.3104i 0.743487i 0.928336 + 0.371743i \(0.121240\pi\)
−0.928336 + 0.371743i \(0.878760\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 8.04283i − 0.256006i
\(988\) 0 0
\(989\) −7.01520 −0.223071
\(990\) 0 0
\(991\) −2.62769 −0.0834712 −0.0417356 0.999129i \(-0.513289\pi\)
−0.0417356 + 0.999129i \(0.513289\pi\)
\(992\) 0 0
\(993\) 19.9705i 0.633745i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 23.4056i − 0.741264i −0.928780 0.370632i \(-0.879141\pi\)
0.928780 0.370632i \(-0.120859\pi\)
\(998\) 0 0
\(999\) 36.6609 1.15990
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 4600.2.e.v.4049.5 10
5.2 odd 4 4600.2.a.bg.1.2 yes 5
5.3 odd 4 4600.2.a.bc.1.4 5
5.4 even 2 inner 4600.2.e.v.4049.6 10
20.3 even 4 9200.2.a.cw.1.2 5
20.7 even 4 9200.2.a.cs.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bc.1.4 5 5.3 odd 4
4600.2.a.bg.1.2 yes 5 5.2 odd 4
4600.2.e.v.4049.5 10 1.1 even 1 trivial
4600.2.e.v.4049.6 10 5.4 even 2 inner
9200.2.a.cs.1.4 5 20.7 even 4
9200.2.a.cw.1.2 5 20.3 even 4