Properties

Label 6900.2.f.r.6349.5
Level $6900$
Weight $2$
Character 6900.6349
Analytic conductor $55.097$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6900.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(55.0967773947\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.158155776.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 24x^{3} + 81x^{2} + 54x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1380)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 6349.5
Root \(2.79911 + 2.79911i\) of defining polynomial
Character \(\chi\) \(=\) 6900.6349
Dual form 6900.2.f.r.6349.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} -1.52644i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} -1.52644i q^{7} -1.00000 q^{9} -3.59821 q^{11} +5.59821i q^{13} -4.07177i q^{17} +1.59821 q^{19} +1.52644 q^{21} +1.00000i q^{23} -1.00000i q^{27} +4.07177 q^{29} +2.47356 q^{31} -3.59821i q^{33} -9.66998i q^{37} -5.59821 q^{39} +5.01889 q^{41} +7.05288i q^{43} +4.54533i q^{47} +4.66998 q^{49} +4.07177 q^{51} +5.01889i q^{53} +1.59821i q^{57} -4.07177 q^{59} +6.54533 q^{61} +1.52644i q^{63} -2.47356i q^{67} -1.00000 q^{69} -5.01889 q^{71} -8.79463i q^{73} +5.49245i q^{77} -2.94712 q^{79} +1.00000 q^{81} +9.12465i q^{83} +4.07177i q^{87} -12.2493 q^{89} +8.54533 q^{91} +2.47356i q^{93} +13.0907i q^{97} +3.59821 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{9} + 8 q^{11} - 20 q^{19} + 4 q^{21} + 20 q^{31} - 4 q^{39} + 16 q^{41} - 26 q^{49} + 20 q^{61} - 6 q^{69} - 16 q^{71} - 28 q^{79} + 6 q^{81} - 4 q^{89} + 32 q^{91} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6900\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1201\) \(3451\) \(4601\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.52644i − 0.576940i −0.957489 0.288470i \(-0.906853\pi\)
0.957489 0.288470i \(-0.0931465\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −3.59821 −1.08490 −0.542451 0.840088i \(-0.682503\pi\)
−0.542451 + 0.840088i \(0.682503\pi\)
\(12\) 0 0
\(13\) 5.59821i 1.55266i 0.630324 + 0.776332i \(0.282922\pi\)
−0.630324 + 0.776332i \(0.717078\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.07177i − 0.987550i −0.869590 0.493775i \(-0.835617\pi\)
0.869590 0.493775i \(-0.164383\pi\)
\(18\) 0 0
\(19\) 1.59821 0.366655 0.183327 0.983052i \(-0.441313\pi\)
0.183327 + 0.983052i \(0.441313\pi\)
\(20\) 0 0
\(21\) 1.52644 0.333096
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) 4.07177 0.756109 0.378054 0.925783i \(-0.376593\pi\)
0.378054 + 0.925783i \(0.376593\pi\)
\(30\) 0 0
\(31\) 2.47356 0.444265 0.222132 0.975017i \(-0.428698\pi\)
0.222132 + 0.975017i \(0.428698\pi\)
\(32\) 0 0
\(33\) − 3.59821i − 0.626368i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 9.66998i − 1.58974i −0.606783 0.794868i \(-0.707540\pi\)
0.606783 0.794868i \(-0.292460\pi\)
\(38\) 0 0
\(39\) −5.59821 −0.896431
\(40\) 0 0
\(41\) 5.01889 0.783819 0.391910 0.920004i \(-0.371815\pi\)
0.391910 + 0.920004i \(0.371815\pi\)
\(42\) 0 0
\(43\) 7.05288i 1.07555i 0.843087 + 0.537777i \(0.180736\pi\)
−0.843087 + 0.537777i \(0.819264\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.54533i 0.663005i 0.943454 + 0.331502i \(0.107555\pi\)
−0.943454 + 0.331502i \(0.892445\pi\)
\(48\) 0 0
\(49\) 4.66998 0.667140
\(50\) 0 0
\(51\) 4.07177 0.570162
\(52\) 0 0
\(53\) 5.01889i 0.689398i 0.938713 + 0.344699i \(0.112019\pi\)
−0.938713 + 0.344699i \(0.887981\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.59821i 0.211688i
\(58\) 0 0
\(59\) −4.07177 −0.530099 −0.265050 0.964235i \(-0.585388\pi\)
−0.265050 + 0.964235i \(0.585388\pi\)
\(60\) 0 0
\(61\) 6.54533 0.838044 0.419022 0.907976i \(-0.362373\pi\)
0.419022 + 0.907976i \(0.362373\pi\)
\(62\) 0 0
\(63\) 1.52644i 0.192313i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 2.47356i − 0.302193i −0.988519 0.151097i \(-0.951720\pi\)
0.988519 0.151097i \(-0.0482805\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −5.01889 −0.595633 −0.297816 0.954623i \(-0.596258\pi\)
−0.297816 + 0.954623i \(0.596258\pi\)
\(72\) 0 0
\(73\) − 8.79463i − 1.02933i −0.857390 0.514667i \(-0.827916\pi\)
0.857390 0.514667i \(-0.172084\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.49245i 0.625923i
\(78\) 0 0
\(79\) −2.94712 −0.331577 −0.165788 0.986161i \(-0.553017\pi\)
−0.165788 + 0.986161i \(0.553017\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.12465i 1.00156i 0.865574 + 0.500780i \(0.166954\pi\)
−0.865574 + 0.500780i \(0.833046\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.07177i 0.436540i
\(88\) 0 0
\(89\) −12.2493 −1.29842 −0.649212 0.760608i \(-0.724901\pi\)
−0.649212 + 0.760608i \(0.724901\pi\)
\(90\) 0 0
\(91\) 8.54533 0.895794
\(92\) 0 0
\(93\) 2.47356i 0.256496i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.0907i 1.32916i 0.747219 + 0.664578i \(0.231389\pi\)
−0.747219 + 0.664578i \(0.768611\pi\)
\(98\) 0 0
\(99\) 3.59821 0.361634
\(100\) 0 0
\(101\) −12.2153 −1.21547 −0.607735 0.794140i \(-0.707922\pi\)
−0.607735 + 0.794140i \(0.707922\pi\)
\(102\) 0 0
\(103\) − 0.143542i − 0.0141436i −0.999975 0.00707181i \(-0.997749\pi\)
0.999975 0.00707181i \(-0.00225105\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 5.26819i − 0.509295i −0.967034 0.254648i \(-0.918040\pi\)
0.967034 0.254648i \(-0.0819595\pi\)
\(108\) 0 0
\(109\) −12.7946 −1.22550 −0.612752 0.790275i \(-0.709937\pi\)
−0.612752 + 0.790275i \(0.709937\pi\)
\(110\) 0 0
\(111\) 9.66998 0.917834
\(112\) 0 0
\(113\) 5.01889i 0.472138i 0.971736 + 0.236069i \(0.0758591\pi\)
−0.971736 + 0.236069i \(0.924141\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 5.59821i − 0.517555i
\(118\) 0 0
\(119\) −6.21531 −0.569757
\(120\) 0 0
\(121\) 1.94712 0.177011
\(122\) 0 0
\(123\) 5.01889i 0.452538i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 17.7040i 1.57097i 0.618879 + 0.785487i \(0.287587\pi\)
−0.618879 + 0.785487i \(0.712413\pi\)
\(128\) 0 0
\(129\) −7.05288 −0.620971
\(130\) 0 0
\(131\) −6.94712 −0.606973 −0.303486 0.952836i \(-0.598151\pi\)
−0.303486 + 0.952836i \(0.598151\pi\)
\(132\) 0 0
\(133\) − 2.43957i − 0.211538i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.2493i 1.04653i 0.852171 + 0.523264i \(0.175286\pi\)
−0.852171 + 0.523264i \(0.824714\pi\)
\(138\) 0 0
\(139\) 6.61710 0.561255 0.280628 0.959817i \(-0.409457\pi\)
0.280628 + 0.959817i \(0.409457\pi\)
\(140\) 0 0
\(141\) −4.54533 −0.382786
\(142\) 0 0
\(143\) − 20.1435i − 1.68449i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.66998i 0.385174i
\(148\) 0 0
\(149\) −9.59821 −0.786316 −0.393158 0.919471i \(-0.628617\pi\)
−0.393158 + 0.919471i \(0.628617\pi\)
\(150\) 0 0
\(151\) −4.94712 −0.402591 −0.201295 0.979531i \(-0.564515\pi\)
−0.201295 + 0.979531i \(0.564515\pi\)
\(152\) 0 0
\(153\) 4.07177i 0.329183i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 7.63220i 0.609116i 0.952494 + 0.304558i \(0.0985087\pi\)
−0.952494 + 0.304558i \(0.901491\pi\)
\(158\) 0 0
\(159\) −5.01889 −0.398024
\(160\) 0 0
\(161\) 1.52644 0.120300
\(162\) 0 0
\(163\) 23.3400i 1.82813i 0.405571 + 0.914064i \(0.367073\pi\)
−0.405571 + 0.914064i \(0.632927\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.45467i 0.576860i 0.957501 + 0.288430i \(0.0931332\pi\)
−0.957501 + 0.288430i \(0.906867\pi\)
\(168\) 0 0
\(169\) −18.3400 −1.41077
\(170\) 0 0
\(171\) −1.59821 −0.122218
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 4.07177i − 0.306053i
\(178\) 0 0
\(179\) 22.2871 1.66581 0.832907 0.553412i \(-0.186675\pi\)
0.832907 + 0.553412i \(0.186675\pi\)
\(180\) 0 0
\(181\) 20.2493 1.50512 0.752559 0.658524i \(-0.228819\pi\)
0.752559 + 0.658524i \(0.228819\pi\)
\(182\) 0 0
\(183\) 6.54533i 0.483845i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 14.6511i 1.07139i
\(188\) 0 0
\(189\) −1.52644 −0.111032
\(190\) 0 0
\(191\) −21.8475 −1.58083 −0.790415 0.612571i \(-0.790135\pi\)
−0.790415 + 0.612571i \(0.790135\pi\)
\(192\) 0 0
\(193\) 12.0378i 0.866499i 0.901274 + 0.433249i \(0.142633\pi\)
−0.901274 + 0.433249i \(0.857367\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.2493i 1.30021i 0.759845 + 0.650104i \(0.225275\pi\)
−0.759845 + 0.650104i \(0.774725\pi\)
\(198\) 0 0
\(199\) 17.1964 1.21902 0.609511 0.792778i \(-0.291366\pi\)
0.609511 + 0.792778i \(0.291366\pi\)
\(200\) 0 0
\(201\) 2.47356 0.174471
\(202\) 0 0
\(203\) − 6.21531i − 0.436229i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 1.00000i − 0.0695048i
\(208\) 0 0
\(209\) −5.75070 −0.397784
\(210\) 0 0
\(211\) −5.66998 −0.390338 −0.195169 0.980770i \(-0.562525\pi\)
−0.195169 + 0.980770i \(0.562525\pi\)
\(212\) 0 0
\(213\) − 5.01889i − 0.343889i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 3.77574i − 0.256314i
\(218\) 0 0
\(219\) 8.79463 0.594286
\(220\) 0 0
\(221\) 22.7946 1.53333
\(222\) 0 0
\(223\) 11.3400i 0.759380i 0.925114 + 0.379690i \(0.123969\pi\)
−0.925114 + 0.379690i \(0.876031\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 26.3928i 1.75175i 0.482534 + 0.875877i \(0.339716\pi\)
−0.482534 + 0.875877i \(0.660284\pi\)
\(228\) 0 0
\(229\) 4.24930 0.280802 0.140401 0.990095i \(-0.455161\pi\)
0.140401 + 0.990095i \(0.455161\pi\)
\(230\) 0 0
\(231\) −5.49245 −0.361377
\(232\) 0 0
\(233\) − 1.89424i − 0.124096i −0.998073 0.0620479i \(-0.980237\pi\)
0.998073 0.0620479i \(-0.0197632\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 2.94712i − 0.191436i
\(238\) 0 0
\(239\) 6.98111 0.451570 0.225785 0.974177i \(-0.427505\pi\)
0.225785 + 0.974177i \(0.427505\pi\)
\(240\) 0 0
\(241\) 3.20537 0.206476 0.103238 0.994657i \(-0.467080\pi\)
0.103238 + 0.994657i \(0.467080\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 8.94712i 0.569292i
\(248\) 0 0
\(249\) −9.12465 −0.578251
\(250\) 0 0
\(251\) 13.4457 0.848686 0.424343 0.905501i \(-0.360505\pi\)
0.424343 + 0.905501i \(0.360505\pi\)
\(252\) 0 0
\(253\) − 3.59821i − 0.226218i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 18.9382i − 1.18133i −0.806917 0.590665i \(-0.798865\pi\)
0.806917 0.590665i \(-0.201135\pi\)
\(258\) 0 0
\(259\) −14.7606 −0.917182
\(260\) 0 0
\(261\) −4.07177 −0.252036
\(262\) 0 0
\(263\) − 0.981108i − 0.0604977i −0.999542 0.0302489i \(-0.990370\pi\)
0.999542 0.0302489i \(-0.00962998\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 12.2493i − 0.749645i
\(268\) 0 0
\(269\) 3.12465 0.190513 0.0952567 0.995453i \(-0.469633\pi\)
0.0952567 + 0.995453i \(0.469633\pi\)
\(270\) 0 0
\(271\) −9.52644 −0.578690 −0.289345 0.957225i \(-0.593437\pi\)
−0.289345 + 0.957225i \(0.593437\pi\)
\(272\) 0 0
\(273\) 8.54533i 0.517187i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 4.39284i − 0.263940i −0.991254 0.131970i \(-0.957870\pi\)
0.991254 0.131970i \(-0.0421303\pi\)
\(278\) 0 0
\(279\) −2.47356 −0.148088
\(280\) 0 0
\(281\) 12.9382 0.771827 0.385913 0.922535i \(-0.373886\pi\)
0.385913 + 0.922535i \(0.373886\pi\)
\(282\) 0 0
\(283\) 19.7757i 1.17555i 0.809026 + 0.587773i \(0.199995\pi\)
−0.809026 + 0.587773i \(0.800005\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 7.66104i − 0.452217i
\(288\) 0 0
\(289\) 0.420681 0.0247459
\(290\) 0 0
\(291\) −13.0907 −0.767388
\(292\) 0 0
\(293\) − 11.2682i − 0.658295i −0.944279 0.329147i \(-0.893239\pi\)
0.944279 0.329147i \(-0.106761\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.59821i 0.208789i
\(298\) 0 0
\(299\) −5.59821 −0.323753
\(300\) 0 0
\(301\) 10.7658 0.620530
\(302\) 0 0
\(303\) − 12.2153i − 0.701751i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 25.8475i 1.47520i 0.675240 + 0.737598i \(0.264040\pi\)
−0.675240 + 0.737598i \(0.735960\pi\)
\(308\) 0 0
\(309\) 0.143542 0.00816582
\(310\) 0 0
\(311\) −29.0529 −1.64744 −0.823719 0.566998i \(-0.808105\pi\)
−0.823719 + 0.566998i \(0.808105\pi\)
\(312\) 0 0
\(313\) − 21.9571i − 1.24109i −0.784172 0.620543i \(-0.786912\pi\)
0.784172 0.620543i \(-0.213088\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 2.65109i − 0.148900i −0.997225 0.0744500i \(-0.976280\pi\)
0.997225 0.0744500i \(-0.0237201\pi\)
\(318\) 0 0
\(319\) −14.6511 −0.820304
\(320\) 0 0
\(321\) 5.26819 0.294042
\(322\) 0 0
\(323\) − 6.50755i − 0.362090i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 12.7946i − 0.707545i
\(328\) 0 0
\(329\) 6.93817 0.382514
\(330\) 0 0
\(331\) 7.77574 0.427393 0.213697 0.976900i \(-0.431450\pi\)
0.213697 + 0.976900i \(0.431450\pi\)
\(332\) 0 0
\(333\) 9.66998i 0.529912i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 16.9471i 0.923168i 0.887096 + 0.461584i \(0.152719\pi\)
−0.887096 + 0.461584i \(0.847281\pi\)
\(338\) 0 0
\(339\) −5.01889 −0.272589
\(340\) 0 0
\(341\) −8.90039 −0.481983
\(342\) 0 0
\(343\) − 17.8135i − 0.961840i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 2.90934i − 0.156181i −0.996946 0.0780907i \(-0.975118\pi\)
0.996946 0.0780907i \(-0.0248824\pi\)
\(348\) 0 0
\(349\) 18.8664 1.00990 0.504948 0.863150i \(-0.331512\pi\)
0.504948 + 0.863150i \(0.331512\pi\)
\(350\) 0 0
\(351\) 5.59821 0.298810
\(352\) 0 0
\(353\) 18.9382i 1.00798i 0.863710 + 0.503989i \(0.168135\pi\)
−0.863710 + 0.503989i \(0.831865\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 6.21531i − 0.328949i
\(358\) 0 0
\(359\) 4.54533 0.239893 0.119947 0.992780i \(-0.461728\pi\)
0.119947 + 0.992780i \(0.461728\pi\)
\(360\) 0 0
\(361\) −16.4457 −0.865564
\(362\) 0 0
\(363\) 1.94712i 0.102197i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 33.9571i 1.77254i 0.463165 + 0.886272i \(0.346714\pi\)
−0.463165 + 0.886272i \(0.653286\pi\)
\(368\) 0 0
\(369\) −5.01889 −0.261273
\(370\) 0 0
\(371\) 7.66104 0.397741
\(372\) 0 0
\(373\) − 12.2115i − 0.632288i −0.948711 0.316144i \(-0.897612\pi\)
0.948711 0.316144i \(-0.102388\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 22.7946i 1.17398i
\(378\) 0 0
\(379\) −13.0529 −0.670481 −0.335241 0.942133i \(-0.608818\pi\)
−0.335241 + 0.942133i \(0.608818\pi\)
\(380\) 0 0
\(381\) −17.7040 −0.907002
\(382\) 0 0
\(383\) − 26.4268i − 1.35035i −0.737659 0.675174i \(-0.764069\pi\)
0.737659 0.675174i \(-0.235931\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 7.05288i − 0.358518i
\(388\) 0 0
\(389\) 15.5893 0.790407 0.395204 0.918594i \(-0.370674\pi\)
0.395204 + 0.918594i \(0.370674\pi\)
\(390\) 0 0
\(391\) 4.07177 0.205918
\(392\) 0 0
\(393\) − 6.94712i − 0.350436i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 35.4457i 1.77897i 0.456965 + 0.889485i \(0.348937\pi\)
−0.456965 + 0.889485i \(0.651063\pi\)
\(398\) 0 0
\(399\) 2.43957 0.122131
\(400\) 0 0
\(401\) 27.5893 1.37774 0.688871 0.724884i \(-0.258107\pi\)
0.688871 + 0.724884i \(0.258107\pi\)
\(402\) 0 0
\(403\) 13.8475i 0.689794i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 34.7946i 1.72471i
\(408\) 0 0
\(409\) −12.7606 −0.630973 −0.315487 0.948930i \(-0.602168\pi\)
−0.315487 + 0.948930i \(0.602168\pi\)
\(410\) 0 0
\(411\) −12.2493 −0.604213
\(412\) 0 0
\(413\) 6.21531i 0.305836i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 6.61710i 0.324041i
\(418\) 0 0
\(419\) 22.7946 1.11359 0.556795 0.830650i \(-0.312031\pi\)
0.556795 + 0.830650i \(0.312031\pi\)
\(420\) 0 0
\(421\) 26.6889 1.30074 0.650368 0.759619i \(-0.274615\pi\)
0.650368 + 0.759619i \(0.274615\pi\)
\(422\) 0 0
\(423\) − 4.54533i − 0.221002i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 9.99105i − 0.483501i
\(428\) 0 0
\(429\) 20.1435 0.972539
\(430\) 0 0
\(431\) −27.4079 −1.32019 −0.660097 0.751180i \(-0.729485\pi\)
−0.660097 + 0.751180i \(0.729485\pi\)
\(432\) 0 0
\(433\) − 8.57932i − 0.412296i −0.978521 0.206148i \(-0.933907\pi\)
0.978521 0.206148i \(-0.0660928\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.59821i 0.0764528i
\(438\) 0 0
\(439\) −9.96222 −0.475471 −0.237735 0.971330i \(-0.576405\pi\)
−0.237735 + 0.971330i \(0.576405\pi\)
\(440\) 0 0
\(441\) −4.66998 −0.222380
\(442\) 0 0
\(443\) − 1.63599i − 0.0777284i −0.999245 0.0388642i \(-0.987626\pi\)
0.999245 0.0388642i \(-0.0123740\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 9.59821i − 0.453980i
\(448\) 0 0
\(449\) −38.6082 −1.82203 −0.911016 0.412372i \(-0.864701\pi\)
−0.911016 + 0.412372i \(0.864701\pi\)
\(450\) 0 0
\(451\) −18.0590 −0.850366
\(452\) 0 0
\(453\) − 4.94712i − 0.232436i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 17.6700i 0.826567i 0.910602 + 0.413283i \(0.135618\pi\)
−0.910602 + 0.413283i \(0.864382\pi\)
\(458\) 0 0
\(459\) −4.07177 −0.190054
\(460\) 0 0
\(461\) −15.0907 −0.702842 −0.351421 0.936217i \(-0.614301\pi\)
−0.351421 + 0.936217i \(0.614301\pi\)
\(462\) 0 0
\(463\) 22.1346i 1.02868i 0.857586 + 0.514341i \(0.171963\pi\)
−0.857586 + 0.514341i \(0.828037\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 29.2682i − 1.35437i −0.735813 0.677185i \(-0.763200\pi\)
0.735813 0.677185i \(-0.236800\pi\)
\(468\) 0 0
\(469\) −3.77574 −0.174348
\(470\) 0 0
\(471\) −7.63220 −0.351673
\(472\) 0 0
\(473\) − 25.3777i − 1.16687i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 5.01889i − 0.229799i
\(478\) 0 0
\(479\) −18.9382 −0.865307 −0.432654 0.901560i \(-0.642423\pi\)
−0.432654 + 0.901560i \(0.642423\pi\)
\(480\) 0 0
\(481\) 54.1346 2.46833
\(482\) 0 0
\(483\) 1.52644i 0.0694554i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 5.20537i 0.235878i 0.993021 + 0.117939i \(0.0376287\pi\)
−0.993021 + 0.117939i \(0.962371\pi\)
\(488\) 0 0
\(489\) −23.3400 −1.05547
\(490\) 0 0
\(491\) −12.2153 −0.551269 −0.275635 0.961262i \(-0.588888\pi\)
−0.275635 + 0.961262i \(0.588888\pi\)
\(492\) 0 0
\(493\) − 16.5793i − 0.746695i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.66104i 0.343644i
\(498\) 0 0
\(499\) −35.1157 −1.57199 −0.785997 0.618230i \(-0.787850\pi\)
−0.785997 + 0.618230i \(0.787850\pi\)
\(500\) 0 0
\(501\) −7.45467 −0.333050
\(502\) 0 0
\(503\) − 6.21531i − 0.277127i −0.990354 0.138564i \(-0.955751\pi\)
0.990354 0.138564i \(-0.0442485\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 18.3400i − 0.814506i
\(508\) 0 0
\(509\) −9.85646 −0.436880 −0.218440 0.975850i \(-0.570097\pi\)
−0.218440 + 0.975850i \(0.570097\pi\)
\(510\) 0 0
\(511\) −13.4245 −0.593864
\(512\) 0 0
\(513\) − 1.59821i − 0.0705627i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 16.3551i − 0.719295i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −41.2253 −1.80611 −0.903056 0.429524i \(-0.858682\pi\)
−0.903056 + 0.429524i \(0.858682\pi\)
\(522\) 0 0
\(523\) − 22.6799i − 0.991724i −0.868401 0.495862i \(-0.834852\pi\)
0.868401 0.495862i \(-0.165148\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 10.0718i − 0.438733i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 4.07177 0.176700
\(532\) 0 0
\(533\) 28.0968i 1.21701i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 22.2871i 0.961759i
\(538\) 0 0
\(539\) −16.8036 −0.723781
\(540\) 0 0
\(541\) 22.1435 0.952025 0.476013 0.879438i \(-0.342082\pi\)
0.476013 + 0.879438i \(0.342082\pi\)
\(542\) 0 0
\(543\) 20.2493i 0.868981i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 28.0000i 1.19719i 0.801050 + 0.598597i \(0.204275\pi\)
−0.801050 + 0.598597i \(0.795725\pi\)
\(548\) 0 0
\(549\) −6.54533 −0.279348
\(550\) 0 0
\(551\) 6.50755 0.277231
\(552\) 0 0
\(553\) 4.49860i 0.191300i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.37395i 0.397187i 0.980082 + 0.198594i \(0.0636374\pi\)
−0.980082 + 0.198594i \(0.936363\pi\)
\(558\) 0 0
\(559\) −39.4835 −1.66997
\(560\) 0 0
\(561\) −14.6511 −0.618570
\(562\) 0 0
\(563\) 27.3060i 1.15081i 0.817869 + 0.575405i \(0.195155\pi\)
−0.817869 + 0.575405i \(0.804845\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 1.52644i − 0.0641044i
\(568\) 0 0
\(569\) −23.3022 −0.976878 −0.488439 0.872598i \(-0.662434\pi\)
−0.488439 + 0.872598i \(0.662434\pi\)
\(570\) 0 0
\(571\) 32.9382 1.37842 0.689210 0.724562i \(-0.257958\pi\)
0.689210 + 0.724562i \(0.257958\pi\)
\(572\) 0 0
\(573\) − 21.8475i − 0.912693i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 41.3400i − 1.72101i −0.509446 0.860503i \(-0.670150\pi\)
0.509446 0.860503i \(-0.329850\pi\)
\(578\) 0 0
\(579\) −12.0378 −0.500273
\(580\) 0 0
\(581\) 13.9282 0.577840
\(582\) 0 0
\(583\) − 18.0590i − 0.747929i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 23.0529i − 0.951494i −0.879582 0.475747i \(-0.842178\pi\)
0.879582 0.475747i \(-0.157822\pi\)
\(588\) 0 0
\(589\) 3.95327 0.162892
\(590\) 0 0
\(591\) −18.2493 −0.750676
\(592\) 0 0
\(593\) 31.9533i 1.31216i 0.754690 + 0.656082i \(0.227787\pi\)
−0.754690 + 0.656082i \(0.772213\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 17.1964i 0.703803i
\(598\) 0 0
\(599\) −26.6421 −1.08857 −0.544284 0.838901i \(-0.683199\pi\)
−0.544284 + 0.838901i \(0.683199\pi\)
\(600\) 0 0
\(601\) 7.45846 0.304237 0.152119 0.988362i \(-0.451390\pi\)
0.152119 + 0.988362i \(0.451390\pi\)
\(602\) 0 0
\(603\) 2.47356i 0.100731i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 13.4925i − 0.547642i −0.961781 0.273821i \(-0.911712\pi\)
0.961781 0.273821i \(-0.0882875\pi\)
\(608\) 0 0
\(609\) 6.21531 0.251857
\(610\) 0 0
\(611\) −25.4457 −1.02942
\(612\) 0 0
\(613\) − 41.4457i − 1.67398i −0.547221 0.836988i \(-0.684314\pi\)
0.547221 0.836988i \(-0.315686\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 25.2304i − 1.01574i −0.861434 0.507869i \(-0.830433\pi\)
0.861434 0.507869i \(-0.169567\pi\)
\(618\) 0 0
\(619\) 5.12845 0.206130 0.103065 0.994675i \(-0.467135\pi\)
0.103065 + 0.994675i \(0.467135\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 18.6978i 0.749112i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 5.75070i − 0.229661i
\(628\) 0 0
\(629\) −39.3740 −1.56994
\(630\) 0 0
\(631\) −11.6360 −0.463222 −0.231611 0.972809i \(-0.574400\pi\)
−0.231611 + 0.972809i \(0.574400\pi\)
\(632\) 0 0
\(633\) − 5.66998i − 0.225362i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 26.1435i 1.03584i
\(638\) 0 0
\(639\) 5.01889 0.198544
\(640\) 0 0
\(641\) 17.7418 0.700757 0.350379 0.936608i \(-0.386053\pi\)
0.350379 + 0.936608i \(0.386053\pi\)
\(642\) 0 0
\(643\) 31.7757i 1.25311i 0.779376 + 0.626556i \(0.215536\pi\)
−0.779376 + 0.626556i \(0.784464\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 46.7946i 1.83969i 0.392286 + 0.919843i \(0.371684\pi\)
−0.392286 + 0.919843i \(0.628316\pi\)
\(648\) 0 0
\(649\) 14.6511 0.575106
\(650\) 0 0
\(651\) 3.77574 0.147983
\(652\) 0 0
\(653\) 3.59821i 0.140809i 0.997519 + 0.0704044i \(0.0224290\pi\)
−0.997519 + 0.0704044i \(0.977571\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 8.79463i 0.343111i
\(658\) 0 0
\(659\) 5.49245 0.213956 0.106978 0.994261i \(-0.465883\pi\)
0.106978 + 0.994261i \(0.465883\pi\)
\(660\) 0 0
\(661\) 4.90934 0.190951 0.0954755 0.995432i \(-0.469563\pi\)
0.0954755 + 0.995432i \(0.469563\pi\)
\(662\) 0 0
\(663\) 22.7946i 0.885270i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.07177i 0.157660i
\(668\) 0 0
\(669\) −11.3400 −0.438428
\(670\) 0 0
\(671\) −23.5515 −0.909195
\(672\) 0 0
\(673\) 25.2253i 0.972362i 0.873858 + 0.486181i \(0.161610\pi\)
−0.873858 + 0.486181i \(0.838390\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 49.6610i − 1.90863i −0.298805 0.954314i \(-0.596588\pi\)
0.298805 0.954314i \(-0.403412\pi\)
\(678\) 0 0
\(679\) 19.9821 0.766843
\(680\) 0 0
\(681\) −26.3928 −1.01138
\(682\) 0 0
\(683\) 9.84751i 0.376805i 0.982092 + 0.188402i \(0.0603309\pi\)
−0.982092 + 0.188402i \(0.939669\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 4.24930i 0.162121i
\(688\) 0 0
\(689\) −28.0968 −1.07040
\(690\) 0 0
\(691\) −17.4457 −0.663667 −0.331833 0.943338i \(-0.607667\pi\)
−0.331833 + 0.943338i \(0.607667\pi\)
\(692\) 0 0
\(693\) − 5.49245i − 0.208641i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 20.4358i − 0.774060i
\(698\) 0 0
\(699\) 1.89424 0.0716468
\(700\) 0 0
\(701\) 24.5075 0.925637 0.462819 0.886453i \(-0.346838\pi\)
0.462819 + 0.886453i \(0.346838\pi\)
\(702\) 0 0
\(703\) − 15.4547i − 0.582884i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 18.6459i 0.701253i
\(708\) 0 0
\(709\) 41.4547 1.55686 0.778431 0.627730i \(-0.216016\pi\)
0.778431 + 0.627730i \(0.216016\pi\)
\(710\) 0 0
\(711\) 2.94712 0.110526
\(712\) 0 0
\(713\) 2.47356i 0.0926356i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.98111i 0.260714i
\(718\) 0 0
\(719\) −30.9811 −1.15540 −0.577700 0.816249i \(-0.696050\pi\)
−0.577700 + 0.816249i \(0.696050\pi\)
\(720\) 0 0
\(721\) −0.219108 −0.00816002
\(722\) 0 0
\(723\) 3.20537i 0.119209i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 15.8513i − 0.587892i −0.955822 0.293946i \(-0.905031\pi\)
0.955822 0.293946i \(-0.0949687\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 28.7177 1.06216
\(732\) 0 0
\(733\) 0.0807173i 0.00298136i 0.999999 + 0.00149068i \(0.000474499\pi\)
−0.999999 + 0.00149068i \(0.999526\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.90039i 0.327850i
\(738\) 0 0
\(739\) 41.2215 1.51636 0.758178 0.652048i \(-0.226090\pi\)
0.758178 + 0.652048i \(0.226090\pi\)
\(740\) 0 0
\(741\) −8.94712 −0.328681
\(742\) 0 0
\(743\) 7.71292i 0.282959i 0.989941 + 0.141480i \(0.0451860\pi\)
−0.989941 + 0.141480i \(0.954814\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 9.12465i − 0.333854i
\(748\) 0 0
\(749\) −8.04158 −0.293833
\(750\) 0 0
\(751\) 24.7946 0.904769 0.452384 0.891823i \(-0.350573\pi\)
0.452384 + 0.891823i \(0.350573\pi\)
\(752\) 0 0
\(753\) 13.4457i 0.489989i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 30.6171i 1.11280i 0.830915 + 0.556399i \(0.187817\pi\)
−0.830915 + 0.556399i \(0.812183\pi\)
\(758\) 0 0
\(759\) 3.59821 0.130607
\(760\) 0 0
\(761\) 37.1624 1.34714 0.673569 0.739125i \(-0.264761\pi\)
0.673569 + 0.739125i \(0.264761\pi\)
\(762\) 0 0
\(763\) 19.5302i 0.707042i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 22.7946i − 0.823066i
\(768\) 0 0
\(769\) 40.4217 1.45764 0.728822 0.684704i \(-0.240068\pi\)
0.728822 + 0.684704i \(0.240068\pi\)
\(770\) 0 0
\(771\) 18.9382 0.682042
\(772\) 0 0
\(773\) 25.1964i 0.906252i 0.891446 + 0.453126i \(0.149691\pi\)
−0.891446 + 0.453126i \(0.850309\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 14.7606i − 0.529535i
\(778\) 0 0
\(779\) 8.02125 0.287391
\(780\) 0 0
\(781\) 18.0590 0.646203
\(782\) 0 0
\(783\) − 4.07177i − 0.145513i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 9.66998i − 0.344698i −0.985036 0.172349i \(-0.944864\pi\)
0.985036 0.172349i \(-0.0551356\pi\)
\(788\) 0 0
\(789\) 0.981108 0.0349284
\(790\) 0 0
\(791\) 7.66104 0.272395
\(792\) 0 0
\(793\) 36.6421i 1.30120i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 37.6610i − 1.33402i −0.745048 0.667011i \(-0.767573\pi\)
0.745048 0.667011i \(-0.232427\pi\)
\(798\) 0 0
\(799\) 18.5075 0.654750
\(800\) 0 0
\(801\) 12.2493 0.432808
\(802\) 0 0
\(803\) 31.6449i 1.11673i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 3.12465i 0.109993i
\(808\) 0 0
\(809\) 11.2002 0.393779 0.196889 0.980426i \(-0.436916\pi\)
0.196889 + 0.980426i \(0.436916\pi\)
\(810\) 0 0
\(811\) 37.0099 1.29959 0.649797 0.760107i \(-0.274854\pi\)
0.649797 + 0.760107i \(0.274854\pi\)
\(812\) 0 0
\(813\) − 9.52644i − 0.334107i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 11.2720i 0.394357i
\(818\) 0 0
\(819\) −8.54533 −0.298598
\(820\) 0 0
\(821\) −12.2493 −0.427504 −0.213752 0.976888i \(-0.568568\pi\)
−0.213752 + 0.976888i \(0.568568\pi\)
\(822\) 0 0
\(823\) − 48.5742i − 1.69319i −0.532238 0.846595i \(-0.678649\pi\)
0.532238 0.846595i \(-0.321351\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 10.5704i 0.367568i 0.982967 + 0.183784i \(0.0588347\pi\)
−0.982967 + 0.183784i \(0.941165\pi\)
\(828\) 0 0
\(829\) 12.1865 0.423254 0.211627 0.977351i \(-0.432124\pi\)
0.211627 + 0.977351i \(0.432124\pi\)
\(830\) 0 0
\(831\) 4.39284 0.152386
\(832\) 0 0
\(833\) − 19.0151i − 0.658834i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 2.47356i − 0.0854988i
\(838\) 0 0
\(839\) 25.4457 0.878484 0.439242 0.898369i \(-0.355247\pi\)
0.439242 + 0.898369i \(0.355247\pi\)
\(840\) 0 0
\(841\) −12.4207 −0.428299
\(842\) 0 0
\(843\) 12.9382i 0.445614i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 2.97216i − 0.102125i
\(848\) 0 0
\(849\) −19.7757 −0.678702
\(850\) 0 0
\(851\) 9.66998 0.331483
\(852\) 0 0
\(853\) 17.2720i 0.591382i 0.955284 + 0.295691i \(0.0955498\pi\)
−0.955284 + 0.295691i \(0.904450\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 41.7328i − 1.42557i −0.701385 0.712783i \(-0.747435\pi\)
0.701385 0.712783i \(-0.252565\pi\)
\(858\) 0 0
\(859\) −34.6171 −1.18112 −0.590560 0.806994i \(-0.701093\pi\)
−0.590560 + 0.806994i \(0.701093\pi\)
\(860\) 0 0
\(861\) 7.66104 0.261087
\(862\) 0 0
\(863\) 38.4608i 1.30922i 0.755966 + 0.654611i \(0.227167\pi\)
−0.755966 + 0.654611i \(0.772833\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0.420681i 0.0142871i
\(868\) 0 0
\(869\) 10.6044 0.359728
\(870\) 0 0
\(871\) 13.8475 0.469205
\(872\) 0 0
\(873\) − 13.0907i − 0.443052i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 27.3022i 0.921929i 0.887419 + 0.460965i \(0.152496\pi\)
−0.887419 + 0.460965i \(0.847504\pi\)
\(878\) 0 0
\(879\) 11.2682 0.380067
\(880\) 0 0
\(881\) −4.79463 −0.161535 −0.0807676 0.996733i \(-0.525737\pi\)
−0.0807676 + 0.996733i \(0.525737\pi\)
\(882\) 0 0
\(883\) − 52.6710i − 1.77252i −0.463188 0.886260i \(-0.653295\pi\)
0.463188 0.886260i \(-0.346705\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 48.9292i − 1.64288i −0.570293 0.821441i \(-0.693170\pi\)
0.570293 0.821441i \(-0.306830\pi\)
\(888\) 0 0
\(889\) 27.0240 0.906357
\(890\) 0 0
\(891\) −3.59821 −0.120545
\(892\) 0 0
\(893\) 7.26440i 0.243094i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 5.59821i − 0.186919i
\(898\) 0 0
\(899\) 10.0718 0.335912
\(900\) 0 0
\(901\) 20.4358 0.680814
\(902\) 0 0
\(903\) 10.7658i 0.358263i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 5.73796i 0.190526i 0.995452 + 0.0952629i \(0.0303692\pi\)
−0.995452 + 0.0952629i \(0.969631\pi\)
\(908\) 0 0
\(909\) 12.2153 0.405156
\(910\) 0 0
\(911\) −42.6799 −1.41405 −0.707025 0.707189i \(-0.749963\pi\)
−0.707025 + 0.707189i \(0.749963\pi\)
\(912\) 0 0
\(913\) − 32.8324i − 1.08659i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 10.6044i 0.350187i
\(918\) 0 0
\(919\) −25.5515 −0.842866 −0.421433 0.906860i \(-0.638473\pi\)
−0.421433 + 0.906860i \(0.638473\pi\)
\(920\) 0 0
\(921\) −25.8475 −0.851704
\(922\) 0 0
\(923\) − 28.0968i − 0.924818i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0.143542i 0.00471454i
\(928\) 0 0
\(929\) 53.9481 1.76998 0.884990 0.465610i \(-0.154165\pi\)
0.884990 + 0.465610i \(0.154165\pi\)
\(930\) 0 0
\(931\) 7.46362 0.244610
\(932\) 0 0
\(933\) − 29.0529i − 0.951149i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 7.48351i − 0.244475i −0.992501 0.122238i \(-0.960993\pi\)
0.992501 0.122238i \(-0.0390070\pi\)
\(938\) 0 0
\(939\) 21.9571 0.716542
\(940\) 0 0
\(941\) 22.6133 0.737173 0.368586 0.929594i \(-0.379842\pi\)
0.368586 + 0.929594i \(0.379842\pi\)
\(942\) 0 0
\(943\) 5.01889i 0.163438i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 7.19642i 0.233852i 0.993141 + 0.116926i \(0.0373041\pi\)
−0.993141 + 0.116926i \(0.962696\pi\)
\(948\) 0 0
\(949\) 49.2342 1.59821
\(950\) 0 0
\(951\) 2.65109 0.0859675
\(952\) 0 0
\(953\) − 60.6799i − 1.96562i −0.184631 0.982808i \(-0.559109\pi\)
0.184631 0.982808i \(-0.440891\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 14.6511i − 0.473602i
\(958\) 0 0
\(959\) 18.6978 0.603784
\(960\) 0 0
\(961\) −24.8815 −0.802629
\(962\) 0 0
\(963\) 5.26819i 0.169765i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 4.40179i − 0.141552i −0.997492 0.0707760i \(-0.977452\pi\)
0.997492 0.0707760i \(-0.0225476\pi\)
\(968\) 0 0
\(969\) 6.50755 0.209053
\(970\) 0 0
\(971\) 1.01510 0.0325760 0.0162880 0.999867i \(-0.494815\pi\)
0.0162880 + 0.999867i \(0.494815\pi\)
\(972\) 0 0
\(973\) − 10.1006i − 0.323811i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18.4646i 0.590735i 0.955384 + 0.295368i \(0.0954421\pi\)
−0.955384 + 0.295368i \(0.904558\pi\)
\(978\) 0 0
\(979\) 44.0756 1.40866
\(980\) 0 0
\(981\) 12.7946 0.408501
\(982\) 0 0
\(983\) 44.0917i 1.40631i 0.711039 + 0.703153i \(0.248225\pi\)
−0.711039 + 0.703153i \(0.751775\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 6.93817i 0.220845i
\(988\) 0 0
\(989\) −7.05288 −0.224269
\(990\) 0 0
\(991\) 8.22426 0.261252 0.130626 0.991432i \(-0.458301\pi\)
0.130626 + 0.991432i \(0.458301\pi\)
\(992\) 0 0
\(993\) 7.77574i 0.246756i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 14.4306i − 0.457023i −0.973541 0.228511i \(-0.926614\pi\)
0.973541 0.228511i \(-0.0733858\pi\)
\(998\) 0 0
\(999\) −9.66998 −0.305945
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 6900.2.f.r.6349.5 6
5.2 odd 4 1380.2.a.j.1.2 3
5.3 odd 4 6900.2.a.x.1.2 3
5.4 even 2 inner 6900.2.f.r.6349.2 6
15.2 even 4 4140.2.a.s.1.2 3
20.7 even 4 5520.2.a.bv.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.a.j.1.2 3 5.2 odd 4
4140.2.a.s.1.2 3 15.2 even 4
5520.2.a.bv.1.2 3 20.7 even 4
6900.2.a.x.1.2 3 5.3 odd 4
6900.2.f.r.6349.2 6 5.4 even 2 inner
6900.2.f.r.6349.5 6 1.1 even 1 trivial