Properties

Label 9200.2.a.cw.1.2
Level $9200$
Weight $2$
Character 9200.1
Self dual yes
Analytic conductor $73.462$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(73.4623698596\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.791953.1
Defining polynomial: \(x^{5} - 2 x^{4} - 7 x^{3} + 7 x^{2} + 9 x + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4600)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.72457\) of defining polynomial
Character \(\chi\) \(=\) 9200.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.724570 q^{3} -2.33840 q^{7} -2.47500 q^{9} +O(q^{10})\) \(q-0.724570 q^{3} -2.33840 q^{7} -2.47500 q^{9} +2.62783 q^{11} +4.29631 q^{13} -6.85404 q^{17} -2.87303 q^{19} +1.69433 q^{21} +1.00000 q^{23} +3.96702 q^{27} -5.03711 q^{29} +7.31504 q^{31} -1.90405 q^{33} +9.24142 q^{37} -3.11297 q^{39} -6.95078 q^{41} +7.01520 q^{43} +4.74690 q^{47} -1.53188 q^{49} +4.96623 q^{51} +12.3849 q^{53} +2.08171 q^{57} +2.14073 q^{59} -2.30567 q^{61} +5.78754 q^{63} -1.98376 q^{67} -0.724570 q^{69} +6.87990 q^{71} -13.2518 q^{73} -6.14492 q^{77} -16.6578 q^{79} +4.55062 q^{81} +9.09450 q^{83} +3.64974 q^{87} +0.676383 q^{89} -10.0465 q^{91} -5.30026 q^{93} -15.0759 q^{97} -6.50388 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 3 q^{3} - q^{7} + 4 q^{9} + O(q^{10}) \) \( 5 q + 3 q^{3} - q^{7} + 4 q^{9} + 4 q^{11} + q^{13} - 5 q^{17} - 4 q^{19} - 6 q^{21} + 5 q^{23} + 6 q^{27} - 11 q^{29} - 4 q^{31} - 13 q^{33} - 6 q^{37} - 31 q^{39} - 8 q^{41} + 3 q^{43} + 2 q^{47} - 2 q^{49} + 5 q^{51} - 18 q^{53} - 27 q^{57} - 23 q^{59} - 26 q^{61} + 5 q^{63} + 3 q^{67} + 3 q^{69} + 2 q^{71} - 4 q^{73} - 15 q^{77} - 43 q^{79} - 3 q^{81} + 30 q^{83} + 27 q^{87} + 15 q^{89} + 19 q^{91} + 15 q^{93} - 8 q^{97} - 37 q^{99} + O(q^{100}) \)

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
\(3\) −0.724570 −0.418331 −0.209165 0.977880i \(-0.567075\pi\)
−0.209165 + 0.977880i \(0.567075\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.33840 −0.883833 −0.441916 0.897056i \(-0.645701\pi\)
−0.441916 + 0.897056i \(0.645701\pi\)
\(8\) 0 0
\(9\) −2.47500 −0.825000
\(10\) 0 0
\(11\) 2.62783 0.792321 0.396160 0.918181i \(-0.370342\pi\)
0.396160 + 0.918181i \(0.370342\pi\)
\(12\) 0 0
\(13\) 4.29631 1.19158 0.595791 0.803140i \(-0.296839\pi\)
0.595791 + 0.803140i \(0.296839\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.85404 −1.66235 −0.831175 0.556011i \(-0.812331\pi\)
−0.831175 + 0.556011i \(0.812331\pi\)
\(18\) 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
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.96702 0.763453
\(28\) 0 0
\(29\) −5.03711 −0.935368 −0.467684 0.883896i \(-0.654912\pi\)
−0.467684 + 0.883896i \(0.654912\pi\)
\(30\) 0 0
\(31\) 7.31504 1.31382 0.656910 0.753969i \(-0.271863\pi\)
0.656910 + 0.753969i \(0.271863\pi\)
\(32\) 0 0
\(33\) −1.90405 −0.331452
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9.24142 1.51928 0.759640 0.650344i \(-0.225375\pi\)
0.759640 + 0.650344i \(0.225375\pi\)
\(38\) 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
\(43\) 7.01520 1.06981 0.534904 0.844913i \(-0.320348\pi\)
0.534904 + 0.844913i \(0.320348\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.74690 0.692406 0.346203 0.938160i \(-0.387471\pi\)
0.346203 + 0.938160i \(0.387471\pi\)
\(48\) 0 0
\(49\) −1.53188 −0.218840
\(50\) 0 0
\(51\) 4.96623 0.695412
\(52\) 0 0
\(53\) 12.3849 1.70120 0.850598 0.525817i \(-0.176240\pi\)
0.850598 + 0.525817i \(0.176240\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.08171 0.275729
\(58\) 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
\(63\) 5.78754 0.729162
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.98376 −0.242355 −0.121178 0.992631i \(-0.538667\pi\)
−0.121178 + 0.992631i \(0.538667\pi\)
\(68\) 0 0
\(69\) −0.724570 −0.0872279
\(70\) 0 0
\(71\) 6.87990 0.816494 0.408247 0.912871i \(-0.366140\pi\)
0.408247 + 0.912871i \(0.366140\pi\)
\(72\) 0 0
\(73\) −13.2518 −1.55101 −0.775505 0.631342i \(-0.782504\pi\)
−0.775505 + 0.631342i \(0.782504\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.14492 −0.700279
\(78\) 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
\(83\) 9.09450 0.998251 0.499126 0.866530i \(-0.333655\pi\)
0.499126 + 0.866530i \(0.333655\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 3.64974 0.391293
\(88\) 0 0
\(89\) 0.676383 0.0716965 0.0358482 0.999357i \(-0.488587\pi\)
0.0358482 + 0.999357i \(0.488587\pi\)
\(90\) 0 0
\(91\) −10.0465 −1.05316
\(92\) 0 0
\(93\) −5.30026 −0.549611
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −15.0759 −1.53073 −0.765365 0.643597i \(-0.777441\pi\)
−0.765365 + 0.643597i \(0.777441\pi\)
\(98\) 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
\(103\) −16.4501 −1.62088 −0.810438 0.585824i \(-0.800771\pi\)
−0.810438 + 0.585824i \(0.800771\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.21143 0.697155 0.348578 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(108\) 0 0
\(109\) 18.6581 1.78712 0.893560 0.448943i \(-0.148199\pi\)
0.893560 + 0.448943i \(0.148199\pi\)
\(110\) 0 0
\(111\) −6.69605 −0.635561
\(112\) 0 0
\(113\) 0.722608 0.0679772 0.0339886 0.999422i \(-0.489179\pi\)
0.0339886 + 0.999422i \(0.489179\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −10.6334 −0.983054
\(118\) 0 0
\(119\) 16.0275 1.46924
\(120\) 0 0
\(121\) −4.09450 −0.372228
\(122\) 0 0
\(123\) 5.03633 0.454110
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 19.1029 1.69510 0.847552 0.530712i \(-0.178076\pi\)
0.847552 + 0.530712i \(0.178076\pi\)
\(128\) 0 0
\(129\) −5.08300 −0.447534
\(130\) 0 0
\(131\) −9.25213 −0.808362 −0.404181 0.914679i \(-0.632443\pi\)
−0.404181 + 0.914679i \(0.632443\pi\)
\(132\) 0 0
\(133\) 6.71829 0.582550
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −20.5484 −1.75557 −0.877783 0.479058i \(-0.840978\pi\)
−0.877783 + 0.479058i \(0.840978\pi\)
\(138\) 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
\(143\) 11.2900 0.944115
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.10995 0.0915473
\(148\) 0 0
\(149\) 5.51461 0.451775 0.225887 0.974153i \(-0.427472\pi\)
0.225887 + 0.974153i \(0.427472\pi\)
\(150\) 0 0
\(151\) −16.5363 −1.34570 −0.672851 0.739778i \(-0.734931\pi\)
−0.672851 + 0.739778i \(0.734931\pi\)
\(152\) 0 0
\(153\) 16.9638 1.37144
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −13.6888 −1.09249 −0.546244 0.837626i \(-0.683943\pi\)
−0.546244 + 0.837626i \(0.683943\pi\)
\(158\) 0 0
\(159\) −8.97372 −0.711662
\(160\) 0 0
\(161\) −2.33840 −0.184292
\(162\) 0 0
\(163\) 18.8266 1.47462 0.737308 0.675556i \(-0.236097\pi\)
0.737308 + 0.675556i \(0.236097\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.68830 0.285409 0.142705 0.989765i \(-0.454420\pi\)
0.142705 + 0.989765i \(0.454420\pi\)
\(168\) 0 0
\(169\) 5.45825 0.419866
\(170\) 0 0
\(171\) 7.11074 0.543772
\(172\) 0 0
\(173\) −7.94698 −0.604198 −0.302099 0.953277i \(-0.597687\pi\)
−0.302099 + 0.953277i \(0.597687\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1.55111 −0.116588
\(178\) 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
\(183\) 1.67062 0.123495
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −18.0113 −1.31711
\(188\) 0 0
\(189\) −9.27648 −0.674765
\(190\) 0 0
\(191\) 14.4643 1.04660 0.523302 0.852148i \(-0.324700\pi\)
0.523302 + 0.852148i \(0.324700\pi\)
\(192\) 0 0
\(193\) 17.7400 1.27696 0.638478 0.769640i \(-0.279564\pi\)
0.638478 + 0.769640i \(0.279564\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 24.0847 1.71596 0.857982 0.513679i \(-0.171718\pi\)
0.857982 + 0.513679i \(0.171718\pi\)
\(198\) 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
\(203\) 11.7788 0.826709
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −2.47500 −0.172024
\(208\) 0 0
\(209\) −7.54983 −0.522233
\(210\) 0 0
\(211\) −16.5419 −1.13879 −0.569394 0.822065i \(-0.692822\pi\)
−0.569394 + 0.822065i \(0.692822\pi\)
\(212\) 0 0
\(213\) −4.98497 −0.341565
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −17.1055 −1.16120
\(218\) 0 0
\(219\) 9.60187 0.648834
\(220\) 0 0
\(221\) −29.4471 −1.98082
\(222\) 0 0
\(223\) −21.4323 −1.43521 −0.717606 0.696449i \(-0.754762\pi\)
−0.717606 + 0.696449i \(0.754762\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 18.6739 1.23943 0.619715 0.784827i \(-0.287248\pi\)
0.619715 + 0.784827i \(0.287248\pi\)
\(228\) 0 0
\(229\) −1.85981 −0.122900 −0.0614499 0.998110i \(-0.519572\pi\)
−0.0614499 + 0.998110i \(0.519572\pi\)
\(230\) 0 0
\(231\) 4.45243 0.292948
\(232\) 0 0
\(233\) 5.39803 0.353637 0.176818 0.984244i \(-0.443420\pi\)
0.176818 + 0.984244i \(0.443420\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 12.0698 0.784015
\(238\) 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
\(243\) −15.1983 −0.974971
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −12.3434 −0.785392
\(248\) 0 0
\(249\) −6.58960 −0.417599
\(250\) 0 0
\(251\) 14.0212 0.885013 0.442506 0.896765i \(-0.354089\pi\)
0.442506 + 0.896765i \(0.354089\pi\)
\(252\) 0 0
\(253\) 2.62783 0.165210
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7.14991 −0.445999 −0.222999 0.974819i \(-0.571585\pi\)
−0.222999 + 0.974819i \(0.571585\pi\)
\(258\) 0 0
\(259\) −21.6101 −1.34279
\(260\) 0 0
\(261\) 12.4668 0.771678
\(262\) 0 0
\(263\) 16.6723 1.02806 0.514030 0.857772i \(-0.328152\pi\)
0.514030 + 0.857772i \(0.328152\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −0.490087 −0.0299928
\(268\) 0 0
\(269\) −11.7884 −0.718752 −0.359376 0.933193i \(-0.617010\pi\)
−0.359376 + 0.933193i \(0.617010\pi\)
\(270\) 0 0
\(271\) 8.37717 0.508877 0.254438 0.967089i \(-0.418109\pi\)
0.254438 + 0.967089i \(0.418109\pi\)
\(272\) 0 0
\(273\) 7.27938 0.440568
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −3.99879 −0.240264 −0.120132 0.992758i \(-0.538332\pi\)
−0.120132 + 0.992758i \(0.538332\pi\)
\(278\) 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
\(283\) −10.6297 −0.631868 −0.315934 0.948781i \(-0.602318\pi\)
−0.315934 + 0.948781i \(0.602318\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 16.2537 0.959427
\(288\) 0 0
\(289\) 29.9779 1.76341
\(290\) 0 0
\(291\) 10.9236 0.640351
\(292\) 0 0
\(293\) −28.5108 −1.66562 −0.832810 0.553558i \(-0.813270\pi\)
−0.832810 + 0.553558i \(0.813270\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 10.4247 0.604900
\(298\) 0 0
\(299\) 4.29631 0.248462
\(300\) 0 0
\(301\) −16.4044 −0.945532
\(302\) 0 0
\(303\) 7.80456 0.448360
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −0.391652 −0.0223528 −0.0111764 0.999938i \(-0.503558\pi\)
−0.0111764 + 0.999938i \(0.503558\pi\)
\(308\) 0 0
\(309\) 11.9192 0.678062
\(310\) 0 0
\(311\) −22.5240 −1.27722 −0.638609 0.769531i \(-0.720490\pi\)
−0.638609 + 0.769531i \(0.720490\pi\)
\(312\) 0 0
\(313\) −22.0468 −1.24616 −0.623078 0.782160i \(-0.714118\pi\)
−0.623078 + 0.782160i \(0.714118\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.01192 −0.337663 −0.168831 0.985645i \(-0.553999\pi\)
−0.168831 + 0.985645i \(0.553999\pi\)
\(318\) 0 0
\(319\) −13.2367 −0.741112
\(320\) 0 0
\(321\) −5.22518 −0.291641
\(322\) 0 0
\(323\) 19.6919 1.09568
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −13.5191 −0.747607
\(328\) 0 0
\(329\) −11.1002 −0.611971
\(330\) 0 0
\(331\) 27.5619 1.51494 0.757469 0.652871i \(-0.226436\pi\)
0.757469 + 0.652871i \(0.226436\pi\)
\(332\) 0 0
\(333\) −22.8725 −1.25341
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −11.8095 −0.643306 −0.321653 0.946858i \(-0.604239\pi\)
−0.321653 + 0.946858i \(0.604239\pi\)
\(338\) 0 0
\(339\) −0.523580 −0.0284370
\(340\) 0 0
\(341\) 19.2227 1.04097
\(342\) 0 0
\(343\) 19.9510 1.07725
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −28.8288 −1.54761 −0.773806 0.633423i \(-0.781649\pi\)
−0.773806 + 0.633423i \(0.781649\pi\)
\(348\) 0 0
\(349\) −26.1403 −1.39926 −0.699629 0.714506i \(-0.746651\pi\)
−0.699629 + 0.714506i \(0.746651\pi\)
\(350\) 0 0
\(351\) 17.0435 0.909716
\(352\) 0 0
\(353\) 23.8004 1.26677 0.633383 0.773839i \(-0.281666\pi\)
0.633383 + 0.773839i \(0.281666\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −11.6130 −0.614628
\(358\) 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
\(363\) 2.96675 0.155714
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 2.84461 0.148487 0.0742437 0.997240i \(-0.476346\pi\)
0.0742437 + 0.997240i \(0.476346\pi\)
\(368\) 0 0
\(369\) 17.2032 0.895562
\(370\) 0 0
\(371\) −28.9609 −1.50357
\(372\) 0 0
\(373\) −4.65308 −0.240928 −0.120464 0.992718i \(-0.538438\pi\)
−0.120464 + 0.992718i \(0.538438\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −21.6410 −1.11457
\(378\) 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
\(383\) 10.7116 0.547337 0.273669 0.961824i \(-0.411763\pi\)
0.273669 + 0.961824i \(0.411763\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −17.3626 −0.882592
\(388\) 0 0
\(389\) 3.28523 0.166568 0.0832839 0.996526i \(-0.473459\pi\)
0.0832839 + 0.996526i \(0.473459\pi\)
\(390\) 0 0
\(391\) −6.85404 −0.346624
\(392\) 0 0
\(393\) 6.70381 0.338163
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −6.00319 −0.301292 −0.150646 0.988588i \(-0.548135\pi\)
−0.150646 + 0.988588i \(0.548135\pi\)
\(398\) 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
\(403\) 31.4277 1.56552
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 24.2849 1.20376
\(408\) 0 0
\(409\) −9.04425 −0.447210 −0.223605 0.974680i \(-0.571782\pi\)
−0.223605 + 0.974680i \(0.571782\pi\)
\(410\) 0 0
\(411\) 14.8887 0.734407
\(412\) 0 0
\(413\) −5.00588 −0.246323
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −3.97698 −0.194754
\(418\) 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
\(423\) −11.7486 −0.571235
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 5.39157 0.260916
\(428\) 0 0
\(429\) −8.18037 −0.394952
\(430\) 0 0
\(431\) −32.5005 −1.56549 −0.782747 0.622340i \(-0.786182\pi\)
−0.782747 + 0.622340i \(0.786182\pi\)
\(432\) 0 0
\(433\) −0.555560 −0.0266985 −0.0133492 0.999911i \(-0.504249\pi\)
−0.0133492 + 0.999911i \(0.504249\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.87303 −0.137436
\(438\) 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
\(443\) −30.2965 −1.43943 −0.719715 0.694270i \(-0.755728\pi\)
−0.719715 + 0.694270i \(0.755728\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −3.99572 −0.188991
\(448\) 0 0
\(449\) −31.6573 −1.49400 −0.747001 0.664823i \(-0.768507\pi\)
−0.747001 + 0.664823i \(0.768507\pi\)
\(450\) 0 0
\(451\) −18.2655 −0.860088
\(452\) 0 0
\(453\) 11.9817 0.562949
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −7.34455 −0.343564 −0.171782 0.985135i \(-0.554952\pi\)
−0.171782 + 0.985135i \(0.554952\pi\)
\(458\) 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
\(463\) 1.82421 0.0847782 0.0423891 0.999101i \(-0.486503\pi\)
0.0423891 + 0.999101i \(0.486503\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.43722 −0.0665067 −0.0332533 0.999447i \(-0.510587\pi\)
−0.0332533 + 0.999447i \(0.510587\pi\)
\(468\) 0 0
\(469\) 4.63884 0.214202
\(470\) 0 0
\(471\) 9.91852 0.457021
\(472\) 0 0
\(473\) 18.4348 0.847632
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −30.6526 −1.40349
\(478\) 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
\(483\) 1.69433 0.0770949
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −33.8112 −1.53213 −0.766066 0.642762i \(-0.777788\pi\)
−0.766066 + 0.642762i \(0.777788\pi\)
\(488\) 0 0
\(489\) −13.6412 −0.616877
\(490\) 0 0
\(491\) −22.4820 −1.01460 −0.507298 0.861770i \(-0.669356\pi\)
−0.507298 + 0.861770i \(0.669356\pi\)
\(492\) 0 0
\(493\) 34.5246 1.55491
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −16.0880 −0.721644
\(498\) 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
\(503\) −16.3142 −0.727414 −0.363707 0.931513i \(-0.618489\pi\)
−0.363707 + 0.931513i \(0.618489\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −3.95488 −0.175643
\(508\) 0 0
\(509\) −8.01935 −0.355451 −0.177726 0.984080i \(-0.556874\pi\)
−0.177726 + 0.984080i \(0.556874\pi\)
\(510\) 0 0
\(511\) 30.9881 1.37083
\(512\) 0 0
\(513\) −11.3973 −0.503205
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 12.4740 0.548608
\(518\) 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
\(523\) 1.91940 0.0839294 0.0419647 0.999119i \(-0.486638\pi\)
0.0419647 + 0.999119i \(0.486638\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −50.1376 −2.18403
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −5.29830 −0.229927
\(532\) 0 0
\(533\) −29.8627 −1.29350
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 4.14724 0.178967
\(538\) 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
\(543\) 1.58933 0.0682045
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 36.0750 1.54246 0.771228 0.636559i \(-0.219643\pi\)
0.771228 + 0.636559i \(0.219643\pi\)
\(548\) 0 0
\(549\) 5.70652 0.243548
\(550\) 0 0
\(551\) 14.4718 0.616518
\(552\) 0 0
\(553\) 38.9527 1.65644
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −31.9500 −1.35377 −0.676883 0.736091i \(-0.736670\pi\)
−0.676883 + 0.736091i \(0.736670\pi\)
\(558\) 0 0
\(559\) 30.1395 1.27476
\(560\) 0 0
\(561\) 13.0504 0.550989
\(562\) 0 0
\(563\) 6.26575 0.264070 0.132035 0.991245i \(-0.457849\pi\)
0.132035 + 0.991245i \(0.457849\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −10.6412 −0.446887
\(568\) 0 0
\(569\) −12.7997 −0.536593 −0.268296 0.963336i \(-0.586461\pi\)
−0.268296 + 0.963336i \(0.586461\pi\)
\(570\) 0 0
\(571\) −13.7690 −0.576217 −0.288108 0.957598i \(-0.593026\pi\)
−0.288108 + 0.957598i \(0.593026\pi\)
\(572\) 0 0
\(573\) −10.4804 −0.437826
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 13.5830 0.565470 0.282735 0.959198i \(-0.408758\pi\)
0.282735 + 0.959198i \(0.408758\pi\)
\(578\) 0 0
\(579\) −12.8539 −0.534190
\(580\) 0 0
\(581\) −21.2666 −0.882287
\(582\) 0 0
\(583\) 32.5454 1.34789
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −19.3233 −0.797557 −0.398778 0.917047i \(-0.630566\pi\)
−0.398778 + 0.917047i \(0.630566\pi\)
\(588\) 0 0
\(589\) −21.0163 −0.865962
\(590\) 0 0
\(591\) −17.4511 −0.717840
\(592\) 0 0
\(593\) −25.1344 −1.03215 −0.516073 0.856545i \(-0.672607\pi\)
−0.516073 + 0.856545i \(0.672607\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5.64262 0.230937
\(598\) 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
\(603\) 4.90982 0.199943
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 3.50465 0.142249 0.0711246 0.997467i \(-0.477341\pi\)
0.0711246 + 0.997467i \(0.477341\pi\)
\(608\) 0 0
\(609\) −8.53455 −0.345838
\(610\) 0 0
\(611\) 20.3941 0.825058
\(612\) 0 0
\(613\) 7.59433 0.306732 0.153366 0.988169i \(-0.450989\pi\)
0.153366 + 0.988169i \(0.450989\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 13.4277 0.540579 0.270290 0.962779i \(-0.412881\pi\)
0.270290 + 0.962779i \(0.412881\pi\)
\(618\) 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
\(623\) −1.58166 −0.0633677
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 5.47038 0.218466
\(628\) 0 0
\(629\) −63.3411 −2.52557
\(630\) 0 0
\(631\) 14.9030 0.593280 0.296640 0.954989i \(-0.404134\pi\)
0.296640 + 0.954989i \(0.404134\pi\)
\(632\) 0 0
\(633\) 11.9857 0.476390
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −6.58142 −0.260765
\(638\) 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
\(643\) −18.8777 −0.744465 −0.372232 0.928140i \(-0.621408\pi\)
−0.372232 + 0.928140i \(0.621408\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 20.5272 0.807009 0.403505 0.914978i \(-0.367792\pi\)
0.403505 + 0.914978i \(0.367792\pi\)
\(648\) 0 0
\(649\) 5.62547 0.220819
\(650\) 0 0
\(651\) 12.3941 0.485764
\(652\) 0 0
\(653\) 3.07302 0.120256 0.0601282 0.998191i \(-0.480849\pi\)
0.0601282 + 0.998191i \(0.480849\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 32.7983 1.27958
\(658\) 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
\(663\) 21.3365 0.828639
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −5.03711 −0.195038
\(668\) 0 0
\(669\) 15.5292 0.600393
\(670\) 0 0
\(671\) −6.05890 −0.233901
\(672\) 0 0
\(673\) 45.0529 1.73666 0.868331 0.495984i \(-0.165193\pi\)
0.868331 + 0.495984i \(0.165193\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 16.5395 0.635665 0.317833 0.948147i \(-0.397045\pi\)
0.317833 + 0.948147i \(0.397045\pi\)
\(678\) 0 0
\(679\) 35.2536 1.35291
\(680\) 0 0
\(681\) −13.5305 −0.518492
\(682\) 0 0
\(683\) −26.9328 −1.03055 −0.515277 0.857024i \(-0.672311\pi\)
−0.515277 + 0.857024i \(0.672311\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1.34756 0.0514128
\(688\) 0 0
\(689\) 53.2093 2.02711
\(690\) 0 0
\(691\) −44.1583 −1.67986 −0.839931 0.542693i \(-0.817405\pi\)
−0.839931 + 0.542693i \(0.817405\pi\)
\(692\) 0 0
\(693\) 15.2087 0.577730
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 47.6410 1.80453
\(698\) 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
\(703\) −26.5508 −1.00138
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 25.1876 0.947279
\(708\) 0 0
\(709\) 8.79068 0.330141 0.165070 0.986282i \(-0.447215\pi\)
0.165070 + 0.986282i \(0.447215\pi\)
\(710\) 0 0
\(711\) 41.2281 1.54617
\(712\) 0 0
\(713\) 7.31504 0.273951
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 15.3476 0.573167
\(718\) 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
\(723\) 21.6032 0.803431
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −16.1156 −0.597695 −0.298848 0.954301i \(-0.596602\pi\)
−0.298848 + 0.954301i \(0.596602\pi\)
\(728\) 0 0
\(729\) −2.63962 −0.0977638
\(730\) 0 0
\(731\) −48.0825 −1.77840
\(732\) 0 0
\(733\) −9.66306 −0.356913 −0.178457 0.983948i \(-0.557110\pi\)
−0.178457 + 0.983948i \(0.557110\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5.21300 −0.192023
\(738\) 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
\(743\) 32.0702 1.17654 0.588271 0.808664i \(-0.299809\pi\)
0.588271 + 0.808664i \(0.299809\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −22.5089 −0.823557
\(748\) 0 0
\(749\) −16.8632 −0.616168
\(750\) 0 0
\(751\) −1.88705 −0.0688596 −0.0344298 0.999407i \(-0.510962\pi\)
−0.0344298 + 0.999407i \(0.510962\pi\)
\(752\) 0 0
\(753\) −10.1594 −0.370228
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −5.13762 −0.186730 −0.0933651 0.995632i \(-0.529762\pi\)
−0.0933651 + 0.995632i \(0.529762\pi\)
\(758\) 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
\(763\) −43.6301 −1.57952
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.19722 0.332092
\(768\) 0 0
\(769\) −1.32121 −0.0476439 −0.0238220 0.999716i \(-0.507583\pi\)
−0.0238220 + 0.999716i \(0.507583\pi\)
\(770\) 0 0
\(771\) 5.18061 0.186575
\(772\) 0 0
\(773\) 33.7473 1.21381 0.606903 0.794776i \(-0.292412\pi\)
0.606903 + 0.794776i \(0.292412\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 15.6581 0.561730
\(778\) 0 0
\(779\) 19.9698 0.715492
\(780\) 0 0
\(781\) 18.0792 0.646926
\(782\) 0 0
\(783\) −19.9823 −0.714110
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 13.1800 0.469816 0.234908 0.972018i \(-0.424521\pi\)
0.234908 + 0.972018i \(0.424521\pi\)
\(788\) 0 0
\(789\) −12.0803 −0.430069
\(790\) 0 0
\(791\) −1.68975 −0.0600805
\(792\) 0 0
\(793\) −9.90584 −0.351767
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 21.0017 0.743919 0.371959 0.928249i \(-0.378686\pi\)
0.371959 + 0.928249i \(0.378686\pi\)
\(798\) 0 0
\(799\) −32.5354 −1.15102
\(800\) 0 0
\(801\) −1.67405 −0.0591496
\(802\) 0 0
\(803\) −34.8236 −1.22890
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 8.54153 0.300676
\(808\) 0 0
\(809\) 4.45831 0.156746 0.0783729 0.996924i \(-0.475028\pi\)
0.0783729 + 0.996924i \(0.475028\pi\)
\(810\) 0 0
\(811\) −27.9804 −0.982525 −0.491263 0.871012i \(-0.663464\pi\)
−0.491263 + 0.871012i \(0.663464\pi\)
\(812\) 0 0
\(813\) −6.06984 −0.212879
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −20.1549 −0.705130
\(818\) 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
\(823\) 6.83168 0.238138 0.119069 0.992886i \(-0.462009\pi\)
0.119069 + 0.992886i \(0.462009\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −31.5165 −1.09594 −0.547968 0.836499i \(-0.684598\pi\)
−0.547968 + 0.836499i \(0.684598\pi\)
\(828\) 0 0
\(829\) −22.9202 −0.796052 −0.398026 0.917374i \(-0.630305\pi\)
−0.398026 + 0.917374i \(0.630305\pi\)
\(830\) 0 0
\(831\) 2.89741 0.100510
\(832\) 0 0
\(833\) 10.4996 0.363788
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 29.0189 1.00304
\(838\) 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
\(843\) −15.9027 −0.547718
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 9.57459 0.328987
\(848\) 0 0
\(849\) 7.70193 0.264330
\(850\) 0 0
\(851\) 9.24142 0.316792
\(852\) 0 0
\(853\) −9.76951 −0.334502 −0.167251 0.985914i \(-0.553489\pi\)
−0.167251 + 0.985914i \(0.553489\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −26.5308 −0.906273 −0.453137 0.891441i \(-0.649695\pi\)
−0.453137 + 0.891441i \(0.649695\pi\)
\(858\) 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
\(863\) −6.15457 −0.209504 −0.104752 0.994498i \(-0.533405\pi\)
−0.104752 + 0.994498i \(0.533405\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −21.7211 −0.737687
\(868\) 0 0
\(869\) −43.7739 −1.48493
\(870\) 0 0
\(871\) −8.52286 −0.288786
\(872\) 0 0
\(873\) 37.3129 1.26285
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 22.0440 0.744373 0.372186 0.928158i \(-0.378608\pi\)
0.372186 + 0.928158i \(0.378608\pi\)
\(878\) 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
\(883\) 33.8860 1.14035 0.570177 0.821522i \(-0.306875\pi\)
0.570177 + 0.821522i \(0.306875\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −3.89387 −0.130743 −0.0653717 0.997861i \(-0.520823\pi\)
−0.0653717 + 0.997861i \(0.520823\pi\)
\(888\) 0 0
\(889\) −44.6701 −1.49819
\(890\) 0 0
\(891\) 11.9582 0.400616
\(892\) 0 0
\(893\) −13.6380 −0.456377
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −3.11297 −0.103939
\(898\) 0 0
\(899\) −36.8467 −1.22891
\(900\) 0 0
\(901\) −84.8866 −2.82798
\(902\) 0 0
\(903\) 11.8861 0.395545
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −30.5048 −1.01289 −0.506447 0.862271i \(-0.669041\pi\)
−0.506447 + 0.862271i \(0.669041\pi\)
\(908\) 0 0
\(909\) 26.6590 0.884222
\(910\) 0 0
\(911\) 15.8042 0.523616 0.261808 0.965120i \(-0.415681\pi\)
0.261808 + 0.965120i \(0.415681\pi\)
\(912\) 0 0
\(913\) 23.8988 0.790935
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 21.6352 0.714457
\(918\) 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
\(923\) 29.5582 0.972919
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 40.7140 1.33722
\(928\) 0 0
\(929\) 13.3600 0.438326 0.219163 0.975688i \(-0.429667\pi\)
0.219163 + 0.975688i \(0.429667\pi\)
\(930\) 0 0
\(931\) 4.40113 0.144241
\(932\) 0 0
\(933\) 16.3202 0.534299
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 16.6980 0.545501 0.272751 0.962085i \(-0.412067\pi\)
0.272751 + 0.962085i \(0.412067\pi\)
\(938\) 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
\(943\) −6.95078 −0.226349
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 7.88894 0.256356 0.128178 0.991751i \(-0.459087\pi\)
0.128178 + 0.991751i \(0.459087\pi\)
\(948\) 0 0
\(949\) −56.9339 −1.84815
\(950\) 0 0
\(951\) 4.35605 0.141255
\(952\) 0 0
\(953\) −2.26143 −0.0732549 −0.0366275 0.999329i \(-0.511661\pi\)
−0.0366275 + 0.999329i \(0.511661\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 9.59090 0.310030
\(958\) 0 0
\(959\) 48.0504 1.55163
\(960\) 0 0
\(961\) 22.5099 0.726125
\(962\) 0 0
\(963\) −17.8483 −0.575153
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 23.1416 0.744182 0.372091 0.928196i \(-0.378641\pi\)
0.372091 + 0.928196i \(0.378641\pi\)
\(968\) 0 0
\(969\) −14.2681 −0.458358
\(970\) 0 0
\(971\) 27.9414 0.896683 0.448341 0.893862i \(-0.352015\pi\)
0.448341 + 0.893862i \(0.352015\pi\)
\(972\) 0 0
\(973\) −12.8349 −0.411468
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −9.62964 −0.308079 −0.154040 0.988065i \(-0.549228\pi\)
−0.154040 + 0.988065i \(0.549228\pi\)
\(978\) 0 0
\(979\) 1.77742 0.0568066
\(980\) 0 0
\(981\) −46.1787 −1.47437
\(982\) 0 0
\(983\) 23.3104 0.743487 0.371743 0.928336i \(-0.378760\pi\)
0.371743 + 0.928336i \(0.378760\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 8.04283 0.256006
\(988\) 0 0
\(989\) 7.01520 0.223071
\(990\) 0 0
\(991\) 2.62769 0.0834712 0.0417356 0.999129i \(-0.486711\pi\)
0.0417356 + 0.999129i \(0.486711\pi\)
\(992\) 0 0
\(993\) −19.9705 −0.633745
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −23.4056 −0.741264 −0.370632 0.928780i \(-0.620859\pi\)
−0.370632 + 0.928780i \(0.620859\pi\)
\(998\) 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 9200.2.a.cw.1.2 5
4.3 odd 2 4600.2.a.bc.1.4 5
5.4 even 2 9200.2.a.cs.1.4 5
20.3 even 4 4600.2.e.v.4049.6 10
20.7 even 4 4600.2.e.v.4049.5 10
20.19 odd 2 4600.2.a.bg.1.2 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bc.1.4 5 4.3 odd 2
4600.2.a.bg.1.2 yes 5 20.19 odd 2
4600.2.e.v.4049.5 10 20.7 even 4
4600.2.e.v.4049.6 10 20.3 even 4
9200.2.a.cs.1.4 5 5.4 even 2
9200.2.a.cw.1.2 5 1.1 even 1 trivial