Properties

Label 6900.2.a.x.1.2
Level $6900$
Weight $2$
Character 6900.1
Self dual yes
Analytic conductor $55.097$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.0967773947\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.3144.1
Defining polynomial: \( x^{3} - x^{2} - 16x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1380)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.526440\) of defining polynomial
Character \(\chi\) \(=\) 6900.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.52644 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.52644 q^{7} +1.00000 q^{9} -3.59821 q^{11} -5.59821 q^{13} -4.07177 q^{17} -1.59821 q^{19} +1.52644 q^{21} -1.00000 q^{23} -1.00000 q^{27} -4.07177 q^{29} +2.47356 q^{31} +3.59821 q^{33} -9.66998 q^{37} +5.59821 q^{39} +5.01889 q^{41} -7.05288 q^{43} +4.54533 q^{47} -4.66998 q^{49} +4.07177 q^{51} -5.01889 q^{53} +1.59821 q^{57} +4.07177 q^{59} +6.54533 q^{61} -1.52644 q^{63} -2.47356 q^{67} +1.00000 q^{69} -5.01889 q^{71} +8.79463 q^{73} +5.49245 q^{77} +2.94712 q^{79} +1.00000 q^{81} -9.12465 q^{83} +4.07177 q^{87} +12.2493 q^{89} +8.54533 q^{91} -2.47356 q^{93} +13.0907 q^{97} -3.59821 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} - 2 q^{7} + 3 q^{9} + 4 q^{11} - 2 q^{13} + 10 q^{19} + 2 q^{21} - 3 q^{23} - 3 q^{27} + 10 q^{31} - 4 q^{33} - 2 q^{37} + 2 q^{39} + 8 q^{41} - 16 q^{43} + 4 q^{47} + 13 q^{49} - 8 q^{53} - 10 q^{57} + 10 q^{61} - 2 q^{63} - 10 q^{67} + 3 q^{69} - 8 q^{71} - 18 q^{73} + 12 q^{77} + 14 q^{79} + 3 q^{81} - 10 q^{83} + 2 q^{89} + 16 q^{91} - 10 q^{93} + 20 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.52644 −0.576940 −0.288470 0.957489i \(-0.593147\pi\)
−0.288470 + 0.957489i \(0.593147\pi\)
\(8\) 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
\(13\) −5.59821 −1.55266 −0.776332 0.630324i \(-0.782922\pi\)
−0.776332 + 0.630324i \(0.782922\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.07177 −0.987550 −0.493775 0.869590i \(-0.664383\pi\)
−0.493775 + 0.869590i \(0.664383\pi\)
\(18\) 0 0
\(19\) −1.59821 −0.366655 −0.183327 0.983052i \(-0.558687\pi\)
−0.183327 + 0.983052i \(0.558687\pi\)
\(20\) 0 0
\(21\) 1.52644 0.333096
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −4.07177 −0.756109 −0.378054 0.925783i \(-0.623407\pi\)
−0.378054 + 0.925783i \(0.623407\pi\)
\(30\) 0 0
\(31\) 2.47356 0.444265 0.222132 0.975017i \(-0.428698\pi\)
0.222132 + 0.975017i \(0.428698\pi\)
\(32\) 0 0
\(33\) 3.59821 0.626368
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −9.66998 −1.58974 −0.794868 0.606783i \(-0.792460\pi\)
−0.794868 + 0.606783i \(0.792460\pi\)
\(38\) 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
\(43\) −7.05288 −1.07555 −0.537777 0.843087i \(-0.680736\pi\)
−0.537777 + 0.843087i \(0.680736\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.54533 0.663005 0.331502 0.943454i \(-0.392445\pi\)
0.331502 + 0.943454i \(0.392445\pi\)
\(48\) 0 0
\(49\) −4.66998 −0.667140
\(50\) 0 0
\(51\) 4.07177 0.570162
\(52\) 0 0
\(53\) −5.01889 −0.689398 −0.344699 0.938713i \(-0.612019\pi\)
−0.344699 + 0.938713i \(0.612019\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.59821 0.211688
\(58\) 0 0
\(59\) 4.07177 0.530099 0.265050 0.964235i \(-0.414612\pi\)
0.265050 + 0.964235i \(0.414612\pi\)
\(60\) 0 0
\(61\) 6.54533 0.838044 0.419022 0.907976i \(-0.362373\pi\)
0.419022 + 0.907976i \(0.362373\pi\)
\(62\) 0 0
\(63\) −1.52644 −0.192313
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.47356 −0.302193 −0.151097 0.988519i \(-0.548280\pi\)
−0.151097 + 0.988519i \(0.548280\pi\)
\(68\) 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
\(73\) 8.79463 1.02933 0.514667 0.857390i \(-0.327916\pi\)
0.514667 + 0.857390i \(0.327916\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.49245 0.625923
\(78\) 0 0
\(79\) 2.94712 0.331577 0.165788 0.986161i \(-0.446983\pi\)
0.165788 + 0.986161i \(0.446983\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −9.12465 −1.00156 −0.500780 0.865574i \(-0.666954\pi\)
−0.500780 + 0.865574i \(0.666954\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.07177 0.436540
\(88\) 0 0
\(89\) 12.2493 1.29842 0.649212 0.760608i \(-0.275099\pi\)
0.649212 + 0.760608i \(0.275099\pi\)
\(90\) 0 0
\(91\) 8.54533 0.895794
\(92\) 0 0
\(93\) −2.47356 −0.256496
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.0907 1.32916 0.664578 0.747219i \(-0.268611\pi\)
0.664578 + 0.747219i \(0.268611\pi\)
\(98\) 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
\(103\) 0.143542 0.0141436 0.00707181 0.999975i \(-0.497749\pi\)
0.00707181 + 0.999975i \(0.497749\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.26819 −0.509295 −0.254648 0.967034i \(-0.581960\pi\)
−0.254648 + 0.967034i \(0.581960\pi\)
\(108\) 0 0
\(109\) 12.7946 1.22550 0.612752 0.790275i \(-0.290063\pi\)
0.612752 + 0.790275i \(0.290063\pi\)
\(110\) 0 0
\(111\) 9.66998 0.917834
\(112\) 0 0
\(113\) −5.01889 −0.472138 −0.236069 0.971736i \(-0.575859\pi\)
−0.236069 + 0.971736i \(0.575859\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −5.59821 −0.517555
\(118\) 0 0
\(119\) 6.21531 0.569757
\(120\) 0 0
\(121\) 1.94712 0.177011
\(122\) 0 0
\(123\) −5.01889 −0.452538
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 17.7040 1.57097 0.785487 0.618879i \(-0.212413\pi\)
0.785487 + 0.618879i \(0.212413\pi\)
\(128\) 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
\(133\) 2.43957 0.211538
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.2493 1.04653 0.523264 0.852171i \(-0.324714\pi\)
0.523264 + 0.852171i \(0.324714\pi\)
\(138\) 0 0
\(139\) −6.61710 −0.561255 −0.280628 0.959817i \(-0.590543\pi\)
−0.280628 + 0.959817i \(0.590543\pi\)
\(140\) 0 0
\(141\) −4.54533 −0.382786
\(142\) 0 0
\(143\) 20.1435 1.68449
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.66998 0.385174
\(148\) 0 0
\(149\) 9.59821 0.786316 0.393158 0.919471i \(-0.371383\pi\)
0.393158 + 0.919471i \(0.371383\pi\)
\(150\) 0 0
\(151\) −4.94712 −0.402591 −0.201295 0.979531i \(-0.564515\pi\)
−0.201295 + 0.979531i \(0.564515\pi\)
\(152\) 0 0
\(153\) −4.07177 −0.329183
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 7.63220 0.609116 0.304558 0.952494i \(-0.401491\pi\)
0.304558 + 0.952494i \(0.401491\pi\)
\(158\) 0 0
\(159\) 5.01889 0.398024
\(160\) 0 0
\(161\) 1.52644 0.120300
\(162\) 0 0
\(163\) −23.3400 −1.82813 −0.914064 0.405571i \(-0.867073\pi\)
−0.914064 + 0.405571i \(0.867073\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.45467 0.576860 0.288430 0.957501i \(-0.406867\pi\)
0.288430 + 0.957501i \(0.406867\pi\)
\(168\) 0 0
\(169\) 18.3400 1.41077
\(170\) 0 0
\(171\) −1.59821 −0.122218
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −4.07177 −0.306053
\(178\) 0 0
\(179\) −22.2871 −1.66581 −0.832907 0.553412i \(-0.813325\pi\)
−0.832907 + 0.553412i \(0.813325\pi\)
\(180\) 0 0
\(181\) 20.2493 1.50512 0.752559 0.658524i \(-0.228819\pi\)
0.752559 + 0.658524i \(0.228819\pi\)
\(182\) 0 0
\(183\) −6.54533 −0.483845
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 14.6511 1.07139
\(188\) 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
\(193\) −12.0378 −0.866499 −0.433249 0.901274i \(-0.642633\pi\)
−0.433249 + 0.901274i \(0.642633\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.2493 1.30021 0.650104 0.759845i \(-0.274725\pi\)
0.650104 + 0.759845i \(0.274725\pi\)
\(198\) 0 0
\(199\) −17.1964 −1.21902 −0.609511 0.792778i \(-0.708634\pi\)
−0.609511 + 0.792778i \(0.708634\pi\)
\(200\) 0 0
\(201\) 2.47356 0.174471
\(202\) 0 0
\(203\) 6.21531 0.436229
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 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
\(213\) 5.01889 0.343889
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −3.77574 −0.256314
\(218\) 0 0
\(219\) −8.79463 −0.594286
\(220\) 0 0
\(221\) 22.7946 1.53333
\(222\) 0 0
\(223\) −11.3400 −0.759380 −0.379690 0.925114i \(-0.623969\pi\)
−0.379690 + 0.925114i \(0.623969\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 26.3928 1.75175 0.875877 0.482534i \(-0.160284\pi\)
0.875877 + 0.482534i \(0.160284\pi\)
\(228\) 0 0
\(229\) −4.24930 −0.280802 −0.140401 0.990095i \(-0.544839\pi\)
−0.140401 + 0.990095i \(0.544839\pi\)
\(230\) 0 0
\(231\) −5.49245 −0.361377
\(232\) 0 0
\(233\) 1.89424 0.124096 0.0620479 0.998073i \(-0.480237\pi\)
0.0620479 + 0.998073i \(0.480237\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −2.94712 −0.191436
\(238\) 0 0
\(239\) −6.98111 −0.451570 −0.225785 0.974177i \(-0.572495\pi\)
−0.225785 + 0.974177i \(0.572495\pi\)
\(240\) 0 0
\(241\) 3.20537 0.206476 0.103238 0.994657i \(-0.467080\pi\)
0.103238 + 0.994657i \(0.467080\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 8.94712 0.569292
\(248\) 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
\(253\) 3.59821 0.226218
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −18.9382 −1.18133 −0.590665 0.806917i \(-0.701135\pi\)
−0.590665 + 0.806917i \(0.701135\pi\)
\(258\) 0 0
\(259\) 14.7606 0.917182
\(260\) 0 0
\(261\) −4.07177 −0.252036
\(262\) 0 0
\(263\) 0.981108 0.0604977 0.0302489 0.999542i \(-0.490370\pi\)
0.0302489 + 0.999542i \(0.490370\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −12.2493 −0.749645
\(268\) 0 0
\(269\) −3.12465 −0.190513 −0.0952567 0.995453i \(-0.530367\pi\)
−0.0952567 + 0.995453i \(0.530367\pi\)
\(270\) 0 0
\(271\) −9.52644 −0.578690 −0.289345 0.957225i \(-0.593437\pi\)
−0.289345 + 0.957225i \(0.593437\pi\)
\(272\) 0 0
\(273\) −8.54533 −0.517187
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −4.39284 −0.263940 −0.131970 0.991254i \(-0.542130\pi\)
−0.131970 + 0.991254i \(0.542130\pi\)
\(278\) 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
\(283\) −19.7757 −1.17555 −0.587773 0.809026i \(-0.699995\pi\)
−0.587773 + 0.809026i \(0.699995\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −7.66104 −0.452217
\(288\) 0 0
\(289\) −0.420681 −0.0247459
\(290\) 0 0
\(291\) −13.0907 −0.767388
\(292\) 0 0
\(293\) 11.2682 0.658295 0.329147 0.944279i \(-0.393239\pi\)
0.329147 + 0.944279i \(0.393239\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.59821 0.208789
\(298\) 0 0
\(299\) 5.59821 0.323753
\(300\) 0 0
\(301\) 10.7658 0.620530
\(302\) 0 0
\(303\) 12.2153 0.701751
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 25.8475 1.47520 0.737598 0.675240i \(-0.235960\pi\)
0.737598 + 0.675240i \(0.235960\pi\)
\(308\) 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
\(313\) 21.9571 1.24109 0.620543 0.784172i \(-0.286912\pi\)
0.620543 + 0.784172i \(0.286912\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.65109 −0.148900 −0.0744500 0.997225i \(-0.523720\pi\)
−0.0744500 + 0.997225i \(0.523720\pi\)
\(318\) 0 0
\(319\) 14.6511 0.820304
\(320\) 0 0
\(321\) 5.26819 0.294042
\(322\) 0 0
\(323\) 6.50755 0.362090
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −12.7946 −0.707545
\(328\) 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
\(333\) −9.66998 −0.529912
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 16.9471 0.923168 0.461584 0.887096i \(-0.347281\pi\)
0.461584 + 0.887096i \(0.347281\pi\)
\(338\) 0 0
\(339\) 5.01889 0.272589
\(340\) 0 0
\(341\) −8.90039 −0.481983
\(342\) 0 0
\(343\) 17.8135 0.961840
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.90934 −0.156181 −0.0780907 0.996946i \(-0.524882\pi\)
−0.0780907 + 0.996946i \(0.524882\pi\)
\(348\) 0 0
\(349\) −18.8664 −1.00990 −0.504948 0.863150i \(-0.668488\pi\)
−0.504948 + 0.863150i \(0.668488\pi\)
\(350\) 0 0
\(351\) 5.59821 0.298810
\(352\) 0 0
\(353\) −18.9382 −1.00798 −0.503989 0.863710i \(-0.668135\pi\)
−0.503989 + 0.863710i \(0.668135\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −6.21531 −0.328949
\(358\) 0 0
\(359\) −4.54533 −0.239893 −0.119947 0.992780i \(-0.538272\pi\)
−0.119947 + 0.992780i \(0.538272\pi\)
\(360\) 0 0
\(361\) −16.4457 −0.865564
\(362\) 0 0
\(363\) −1.94712 −0.102197
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 33.9571 1.77254 0.886272 0.463165i \(-0.153286\pi\)
0.886272 + 0.463165i \(0.153286\pi\)
\(368\) 0 0
\(369\) 5.01889 0.261273
\(370\) 0 0
\(371\) 7.66104 0.397741
\(372\) 0 0
\(373\) 12.2115 0.632288 0.316144 0.948711i \(-0.397612\pi\)
0.316144 + 0.948711i \(0.397612\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 22.7946 1.17398
\(378\) 0 0
\(379\) 13.0529 0.670481 0.335241 0.942133i \(-0.391182\pi\)
0.335241 + 0.942133i \(0.391182\pi\)
\(380\) 0 0
\(381\) −17.7040 −0.907002
\(382\) 0 0
\(383\) 26.4268 1.35035 0.675174 0.737659i \(-0.264069\pi\)
0.675174 + 0.737659i \(0.264069\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −7.05288 −0.358518
\(388\) 0 0
\(389\) −15.5893 −0.790407 −0.395204 0.918594i \(-0.629326\pi\)
−0.395204 + 0.918594i \(0.629326\pi\)
\(390\) 0 0
\(391\) 4.07177 0.205918
\(392\) 0 0
\(393\) 6.94712 0.350436
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 35.4457 1.77897 0.889485 0.456965i \(-0.151063\pi\)
0.889485 + 0.456965i \(0.151063\pi\)
\(398\) 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
\(403\) −13.8475 −0.689794
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 34.7946 1.72471
\(408\) 0 0
\(409\) 12.7606 0.630973 0.315487 0.948930i \(-0.397832\pi\)
0.315487 + 0.948930i \(0.397832\pi\)
\(410\) 0 0
\(411\) −12.2493 −0.604213
\(412\) 0 0
\(413\) −6.21531 −0.305836
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 6.61710 0.324041
\(418\) 0 0
\(419\) −22.7946 −1.11359 −0.556795 0.830650i \(-0.687969\pi\)
−0.556795 + 0.830650i \(0.687969\pi\)
\(420\) 0 0
\(421\) 26.6889 1.30074 0.650368 0.759619i \(-0.274615\pi\)
0.650368 + 0.759619i \(0.274615\pi\)
\(422\) 0 0
\(423\) 4.54533 0.221002
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −9.99105 −0.483501
\(428\) 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
\(433\) 8.57932 0.412296 0.206148 0.978521i \(-0.433907\pi\)
0.206148 + 0.978521i \(0.433907\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.59821 0.0764528
\(438\) 0 0
\(439\) 9.96222 0.475471 0.237735 0.971330i \(-0.423595\pi\)
0.237735 + 0.971330i \(0.423595\pi\)
\(440\) 0 0
\(441\) −4.66998 −0.222380
\(442\) 0 0
\(443\) 1.63599 0.0777284 0.0388642 0.999245i \(-0.487626\pi\)
0.0388642 + 0.999245i \(0.487626\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −9.59821 −0.453980
\(448\) 0 0
\(449\) 38.6082 1.82203 0.911016 0.412372i \(-0.135299\pi\)
0.911016 + 0.412372i \(0.135299\pi\)
\(450\) 0 0
\(451\) −18.0590 −0.850366
\(452\) 0 0
\(453\) 4.94712 0.232436
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 17.6700 0.826567 0.413283 0.910602i \(-0.364382\pi\)
0.413283 + 0.910602i \(0.364382\pi\)
\(458\) 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
\(463\) −22.1346 −1.02868 −0.514341 0.857586i \(-0.671963\pi\)
−0.514341 + 0.857586i \(0.671963\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −29.2682 −1.35437 −0.677185 0.735813i \(-0.736800\pi\)
−0.677185 + 0.735813i \(0.736800\pi\)
\(468\) 0 0
\(469\) 3.77574 0.174348
\(470\) 0 0
\(471\) −7.63220 −0.351673
\(472\) 0 0
\(473\) 25.3777 1.16687
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −5.01889 −0.229799
\(478\) 0 0
\(479\) 18.9382 0.865307 0.432654 0.901560i \(-0.357577\pi\)
0.432654 + 0.901560i \(0.357577\pi\)
\(480\) 0 0
\(481\) 54.1346 2.46833
\(482\) 0 0
\(483\) −1.52644 −0.0694554
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 5.20537 0.235878 0.117939 0.993021i \(-0.462371\pi\)
0.117939 + 0.993021i \(0.462371\pi\)
\(488\) 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
\(493\) 16.5793 0.746695
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.66104 0.343644
\(498\) 0 0
\(499\) 35.1157 1.57199 0.785997 0.618230i \(-0.212150\pi\)
0.785997 + 0.618230i \(0.212150\pi\)
\(500\) 0 0
\(501\) −7.45467 −0.333050
\(502\) 0 0
\(503\) 6.21531 0.277127 0.138564 0.990354i \(-0.455751\pi\)
0.138564 + 0.990354i \(0.455751\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −18.3400 −0.814506
\(508\) 0 0
\(509\) 9.85646 0.436880 0.218440 0.975850i \(-0.429903\pi\)
0.218440 + 0.975850i \(0.429903\pi\)
\(510\) 0 0
\(511\) −13.4245 −0.593864
\(512\) 0 0
\(513\) 1.59821 0.0705627
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −16.3551 −0.719295
\(518\) 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
\(523\) 22.6799 0.991724 0.495862 0.868401i \(-0.334852\pi\)
0.495862 + 0.868401i \(0.334852\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10.0718 −0.438733
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 4.07177 0.176700
\(532\) 0 0
\(533\) −28.0968 −1.21701
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 22.2871 0.961759
\(538\) 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
\(543\) −20.2493 −0.868981
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 0 0
\(549\) 6.54533 0.279348
\(550\) 0 0
\(551\) 6.50755 0.277231
\(552\) 0 0
\(553\) −4.49860 −0.191300
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.37395 0.397187 0.198594 0.980082i \(-0.436363\pi\)
0.198594 + 0.980082i \(0.436363\pi\)
\(558\) 0 0
\(559\) 39.4835 1.66997
\(560\) 0 0
\(561\) −14.6511 −0.618570
\(562\) 0 0
\(563\) −27.3060 −1.15081 −0.575405 0.817869i \(-0.695155\pi\)
−0.575405 + 0.817869i \(0.695155\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.52644 −0.0641044
\(568\) 0 0
\(569\) 23.3022 0.976878 0.488439 0.872598i \(-0.337566\pi\)
0.488439 + 0.872598i \(0.337566\pi\)
\(570\) 0 0
\(571\) 32.9382 1.37842 0.689210 0.724562i \(-0.257958\pi\)
0.689210 + 0.724562i \(0.257958\pi\)
\(572\) 0 0
\(573\) 21.8475 0.912693
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −41.3400 −1.72101 −0.860503 0.509446i \(-0.829850\pi\)
−0.860503 + 0.509446i \(0.829850\pi\)
\(578\) 0 0
\(579\) 12.0378 0.500273
\(580\) 0 0
\(581\) 13.9282 0.577840
\(582\) 0 0
\(583\) 18.0590 0.747929
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −23.0529 −0.951494 −0.475747 0.879582i \(-0.657822\pi\)
−0.475747 + 0.879582i \(0.657822\pi\)
\(588\) 0 0
\(589\) −3.95327 −0.162892
\(590\) 0 0
\(591\) −18.2493 −0.750676
\(592\) 0 0
\(593\) −31.9533 −1.31216 −0.656082 0.754690i \(-0.727787\pi\)
−0.656082 + 0.754690i \(0.727787\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 17.1964 0.703803
\(598\) 0 0
\(599\) 26.6421 1.08857 0.544284 0.838901i \(-0.316801\pi\)
0.544284 + 0.838901i \(0.316801\pi\)
\(600\) 0 0
\(601\) 7.45846 0.304237 0.152119 0.988362i \(-0.451390\pi\)
0.152119 + 0.988362i \(0.451390\pi\)
\(602\) 0 0
\(603\) −2.47356 −0.100731
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −13.4925 −0.547642 −0.273821 0.961781i \(-0.588288\pi\)
−0.273821 + 0.961781i \(0.588288\pi\)
\(608\) 0 0
\(609\) −6.21531 −0.251857
\(610\) 0 0
\(611\) −25.4457 −1.02942
\(612\) 0 0
\(613\) 41.4457 1.67398 0.836988 0.547221i \(-0.184314\pi\)
0.836988 + 0.547221i \(0.184314\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −25.2304 −1.01574 −0.507869 0.861434i \(-0.669567\pi\)
−0.507869 + 0.861434i \(0.669567\pi\)
\(618\) 0 0
\(619\) −5.12845 −0.206130 −0.103065 0.994675i \(-0.532865\pi\)
−0.103065 + 0.994675i \(0.532865\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −18.6978 −0.749112
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −5.75070 −0.229661
\(628\) 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
\(633\) 5.66998 0.225362
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 26.1435 1.03584
\(638\) 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
\(643\) −31.7757 −1.25311 −0.626556 0.779376i \(-0.715536\pi\)
−0.626556 + 0.779376i \(0.715536\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 46.7946 1.83969 0.919843 0.392286i \(-0.128316\pi\)
0.919843 + 0.392286i \(0.128316\pi\)
\(648\) 0 0
\(649\) −14.6511 −0.575106
\(650\) 0 0
\(651\) 3.77574 0.147983
\(652\) 0 0
\(653\) −3.59821 −0.140809 −0.0704044 0.997519i \(-0.522429\pi\)
−0.0704044 + 0.997519i \(0.522429\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 8.79463 0.343111
\(658\) 0 0
\(659\) −5.49245 −0.213956 −0.106978 0.994261i \(-0.534117\pi\)
−0.106978 + 0.994261i \(0.534117\pi\)
\(660\) 0 0
\(661\) 4.90934 0.190951 0.0954755 0.995432i \(-0.469563\pi\)
0.0954755 + 0.995432i \(0.469563\pi\)
\(662\) 0 0
\(663\) −22.7946 −0.885270
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.07177 0.157660
\(668\) 0 0
\(669\) 11.3400 0.438428
\(670\) 0 0
\(671\) −23.5515 −0.909195
\(672\) 0 0
\(673\) −25.2253 −0.972362 −0.486181 0.873858i \(-0.661610\pi\)
−0.486181 + 0.873858i \(0.661610\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −49.6610 −1.90863 −0.954314 0.298805i \(-0.903412\pi\)
−0.954314 + 0.298805i \(0.903412\pi\)
\(678\) 0 0
\(679\) −19.9821 −0.766843
\(680\) 0 0
\(681\) −26.3928 −1.01138
\(682\) 0 0
\(683\) −9.84751 −0.376805 −0.188402 0.982092i \(-0.560331\pi\)
−0.188402 + 0.982092i \(0.560331\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 4.24930 0.162121
\(688\) 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
\(693\) 5.49245 0.208641
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −20.4358 −0.774060
\(698\) 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
\(703\) 15.4547 0.582884
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 18.6459 0.701253
\(708\) 0 0
\(709\) −41.4547 −1.55686 −0.778431 0.627730i \(-0.783984\pi\)
−0.778431 + 0.627730i \(0.783984\pi\)
\(710\) 0 0
\(711\) 2.94712 0.110526
\(712\) 0 0
\(713\) −2.47356 −0.0926356
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.98111 0.260714
\(718\) 0 0
\(719\) 30.9811 1.15540 0.577700 0.816249i \(-0.303950\pi\)
0.577700 + 0.816249i \(0.303950\pi\)
\(720\) 0 0
\(721\) −0.219108 −0.00816002
\(722\) 0 0
\(723\) −3.20537 −0.119209
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −15.8513 −0.587892 −0.293946 0.955822i \(-0.594969\pi\)
−0.293946 + 0.955822i \(0.594969\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 28.7177 1.06216
\(732\) 0 0
\(733\) −0.0807173 −0.00298136 −0.00149068 0.999999i \(-0.500474\pi\)
−0.00149068 + 0.999999i \(0.500474\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.90039 0.327850
\(738\) 0 0
\(739\) −41.2215 −1.51636 −0.758178 0.652048i \(-0.773910\pi\)
−0.758178 + 0.652048i \(0.773910\pi\)
\(740\) 0 0
\(741\) −8.94712 −0.328681
\(742\) 0 0
\(743\) −7.71292 −0.282959 −0.141480 0.989941i \(-0.545186\pi\)
−0.141480 + 0.989941i \(0.545186\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −9.12465 −0.333854
\(748\) 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
\(753\) −13.4457 −0.489989
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 30.6171 1.11280 0.556399 0.830915i \(-0.312183\pi\)
0.556399 + 0.830915i \(0.312183\pi\)
\(758\) 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
\(763\) −19.5302 −0.707042
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −22.7946 −0.823066
\(768\) 0 0
\(769\) −40.4217 −1.45764 −0.728822 0.684704i \(-0.759932\pi\)
−0.728822 + 0.684704i \(0.759932\pi\)
\(770\) 0 0
\(771\) 18.9382 0.682042
\(772\) 0 0
\(773\) −25.1964 −0.906252 −0.453126 0.891446i \(-0.649691\pi\)
−0.453126 + 0.891446i \(0.649691\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −14.7606 −0.529535
\(778\) 0 0
\(779\) −8.02125 −0.287391
\(780\) 0 0
\(781\) 18.0590 0.646203
\(782\) 0 0
\(783\) 4.07177 0.145513
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −9.66998 −0.344698 −0.172349 0.985036i \(-0.555136\pi\)
−0.172349 + 0.985036i \(0.555136\pi\)
\(788\) 0 0
\(789\) −0.981108 −0.0349284
\(790\) 0 0
\(791\) 7.66104 0.272395
\(792\) 0 0
\(793\) −36.6421 −1.30120
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −37.6610 −1.33402 −0.667011 0.745048i \(-0.732427\pi\)
−0.667011 + 0.745048i \(0.732427\pi\)
\(798\) 0 0
\(799\) −18.5075 −0.654750
\(800\) 0 0
\(801\) 12.2493 0.432808
\(802\) 0 0
\(803\) −31.6449 −1.11673
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 3.12465 0.109993
\(808\) 0 0
\(809\) −11.2002 −0.393779 −0.196889 0.980426i \(-0.563084\pi\)
−0.196889 + 0.980426i \(0.563084\pi\)
\(810\) 0 0
\(811\) 37.0099 1.29959 0.649797 0.760107i \(-0.274854\pi\)
0.649797 + 0.760107i \(0.274854\pi\)
\(812\) 0 0
\(813\) 9.52644 0.334107
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 11.2720 0.394357
\(818\) 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
\(823\) 48.5742 1.69319 0.846595 0.532238i \(-0.178649\pi\)
0.846595 + 0.532238i \(0.178649\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 10.5704 0.367568 0.183784 0.982967i \(-0.441165\pi\)
0.183784 + 0.982967i \(0.441165\pi\)
\(828\) 0 0
\(829\) −12.1865 −0.423254 −0.211627 0.977351i \(-0.567876\pi\)
−0.211627 + 0.977351i \(0.567876\pi\)
\(830\) 0 0
\(831\) 4.39284 0.152386
\(832\) 0 0
\(833\) 19.0151 0.658834
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −2.47356 −0.0854988
\(838\) 0 0
\(839\) −25.4457 −0.878484 −0.439242 0.898369i \(-0.644753\pi\)
−0.439242 + 0.898369i \(0.644753\pi\)
\(840\) 0 0
\(841\) −12.4207 −0.428299
\(842\) 0 0
\(843\) −12.9382 −0.445614
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −2.97216 −0.102125
\(848\) 0 0
\(849\) 19.7757 0.678702
\(850\) 0 0
\(851\) 9.66998 0.331483
\(852\) 0 0
\(853\) −17.2720 −0.591382 −0.295691 0.955284i \(-0.595550\pi\)
−0.295691 + 0.955284i \(0.595550\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −41.7328 −1.42557 −0.712783 0.701385i \(-0.752565\pi\)
−0.712783 + 0.701385i \(0.752565\pi\)
\(858\) 0 0
\(859\) 34.6171 1.18112 0.590560 0.806994i \(-0.298907\pi\)
0.590560 + 0.806994i \(0.298907\pi\)
\(860\) 0 0
\(861\) 7.66104 0.261087
\(862\) 0 0
\(863\) −38.4608 −1.30922 −0.654611 0.755966i \(-0.727167\pi\)
−0.654611 + 0.755966i \(0.727167\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0.420681 0.0142871
\(868\) 0 0
\(869\) −10.6044 −0.359728
\(870\) 0 0
\(871\) 13.8475 0.469205
\(872\) 0 0
\(873\) 13.0907 0.443052
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 27.3022 0.921929 0.460965 0.887419i \(-0.347504\pi\)
0.460965 + 0.887419i \(0.347504\pi\)
\(878\) 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
\(883\) 52.6710 1.77252 0.886260 0.463188i \(-0.153295\pi\)
0.886260 + 0.463188i \(0.153295\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −48.9292 −1.64288 −0.821441 0.570293i \(-0.806830\pi\)
−0.821441 + 0.570293i \(0.806830\pi\)
\(888\) 0 0
\(889\) −27.0240 −0.906357
\(890\) 0 0
\(891\) −3.59821 −0.120545
\(892\) 0 0
\(893\) −7.26440 −0.243094
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −5.59821 −0.186919
\(898\) 0 0
\(899\) −10.0718 −0.335912
\(900\) 0 0
\(901\) 20.4358 0.680814
\(902\) 0 0
\(903\) −10.7658 −0.358263
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 5.73796 0.190526 0.0952629 0.995452i \(-0.469631\pi\)
0.0952629 + 0.995452i \(0.469631\pi\)
\(908\) 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
\(913\) 32.8324 1.08659
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 10.6044 0.350187
\(918\) 0 0
\(919\) 25.5515 0.842866 0.421433 0.906860i \(-0.361527\pi\)
0.421433 + 0.906860i \(0.361527\pi\)
\(920\) 0 0
\(921\) −25.8475 −0.851704
\(922\) 0 0
\(923\) 28.0968 0.924818
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0.143542 0.00471454
\(928\) 0 0
\(929\) −53.9481 −1.76998 −0.884990 0.465610i \(-0.845835\pi\)
−0.884990 + 0.465610i \(0.845835\pi\)
\(930\) 0 0
\(931\) 7.46362 0.244610
\(932\) 0 0
\(933\) 29.0529 0.951149
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −7.48351 −0.244475 −0.122238 0.992501i \(-0.539007\pi\)
−0.122238 + 0.992501i \(0.539007\pi\)
\(938\) 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
\(943\) −5.01889 −0.163438
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 7.19642 0.233852 0.116926 0.993141i \(-0.462696\pi\)
0.116926 + 0.993141i \(0.462696\pi\)
\(948\) 0 0
\(949\) −49.2342 −1.59821
\(950\) 0 0
\(951\) 2.65109 0.0859675
\(952\) 0 0
\(953\) 60.6799 1.96562 0.982808 0.184631i \(-0.0591092\pi\)
0.982808 + 0.184631i \(0.0591092\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −14.6511 −0.473602
\(958\) 0 0
\(959\) −18.6978 −0.603784
\(960\) 0 0
\(961\) −24.8815 −0.802629
\(962\) 0 0
\(963\) −5.26819 −0.169765
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −4.40179 −0.141552 −0.0707760 0.997492i \(-0.522548\pi\)
−0.0707760 + 0.997492i \(0.522548\pi\)
\(968\) 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
\(973\) 10.1006 0.323811
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18.4646 0.590735 0.295368 0.955384i \(-0.404558\pi\)
0.295368 + 0.955384i \(0.404558\pi\)
\(978\) 0 0
\(979\) −44.0756 −1.40866
\(980\) 0 0
\(981\) 12.7946 0.408501
\(982\) 0 0
\(983\) −44.0917 −1.40631 −0.703153 0.711039i \(-0.748225\pi\)
−0.703153 + 0.711039i \(0.748225\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 6.93817 0.220845
\(988\) 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
\(993\) −7.77574 −0.246756
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −14.4306 −0.457023 −0.228511 0.973541i \(-0.573386\pi\)
−0.228511 + 0.973541i \(0.573386\pi\)
\(998\) 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.a.x.1.2 3
5.2 odd 4 6900.2.f.r.6349.5 6
5.3 odd 4 6900.2.f.r.6349.2 6
5.4 even 2 1380.2.a.j.1.2 3
15.14 odd 2 4140.2.a.s.1.2 3
20.19 odd 2 5520.2.a.bv.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.a.j.1.2 3 5.4 even 2
4140.2.a.s.1.2 3 15.14 odd 2
5520.2.a.bv.1.2 3 20.19 odd 2
6900.2.a.x.1.2 3 1.1 even 1 trivial
6900.2.f.r.6349.2 6 5.3 odd 4
6900.2.f.r.6349.5 6 5.2 odd 4