Properties

Label 6900.2.f.r.6349.3
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 6349.3
Root \(-0.376763 + 0.376763i\) of defining polynomial
Character \(\chi\) \(=\) 6900.6349
Dual form 6900.2.f.r.6349.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +4.20905i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +4.20905i q^{7} -1.00000 q^{9} +2.75353 q^{11} +0.753525i q^{13} -4.96257i q^{17} -4.75353 q^{19} +4.20905 q^{21} -1.00000i q^{23} +1.00000i q^{27} -4.96257 q^{29} -0.209050 q^{31} -2.75353i q^{33} -5.71610i q^{37} +0.753525 q^{39} -9.38067 q^{41} -12.4181i q^{43} +7.17162i q^{47} -10.7161 q^{49} -4.96257 q^{51} +9.38067i q^{53} +4.75353i q^{57} +4.96257 q^{59} -5.17162 q^{61} -4.20905i q^{63} -0.209050i q^{67} -1.00000 q^{69} +9.38067 q^{71} -10.2606i q^{73} +11.5897i q^{77} +2.41810 q^{79} +1.00000 q^{81} -5.45552i q^{83} +4.96257i q^{87} -4.91105 q^{89} -3.17162 q^{91} +0.209050i q^{93} +10.3432i q^{97} -2.75353 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9}+O(q^{10}) \) 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
\(7\) 4.20905i 1.59087i 0.606038 + 0.795436i \(0.292758\pi\)
−0.606038 + 0.795436i \(0.707242\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.75353 0.830219 0.415110 0.909771i \(-0.363743\pi\)
0.415110 + 0.909771i \(0.363743\pi\)
\(12\) 0 0
\(13\) 0.753525i 0.208990i 0.994525 + 0.104495i \(0.0333227\pi\)
−0.994525 + 0.104495i \(0.966677\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.96257i − 1.20360i −0.798646 0.601801i \(-0.794450\pi\)
0.798646 0.601801i \(-0.205550\pi\)
\(18\) 0 0
\(19\) −4.75353 −1.09053 −0.545267 0.838263i \(-0.683572\pi\)
−0.545267 + 0.838263i \(0.683572\pi\)
\(20\) 0 0
\(21\) 4.20905 0.918490
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −4.96257 −0.921527 −0.460764 0.887523i \(-0.652424\pi\)
−0.460764 + 0.887523i \(0.652424\pi\)
\(30\) 0 0
\(31\) −0.209050 −0.0375464 −0.0187732 0.999824i \(-0.505976\pi\)
−0.0187732 + 0.999824i \(0.505976\pi\)
\(32\) 0 0
\(33\) − 2.75353i − 0.479327i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 5.71610i − 0.939721i −0.882741 0.469861i \(-0.844304\pi\)
0.882741 0.469861i \(-0.155696\pi\)
\(38\) 0 0
\(39\) 0.753525 0.120661
\(40\) 0 0
\(41\) −9.38067 −1.46502 −0.732508 0.680759i \(-0.761650\pi\)
−0.732508 + 0.680759i \(0.761650\pi\)
\(42\) 0 0
\(43\) − 12.4181i − 1.89374i −0.321613 0.946871i \(-0.604225\pi\)
0.321613 0.946871i \(-0.395775\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.17162i 1.04609i 0.852305 + 0.523044i \(0.175204\pi\)
−0.852305 + 0.523044i \(0.824796\pi\)
\(48\) 0 0
\(49\) −10.7161 −1.53087
\(50\) 0 0
\(51\) −4.96257 −0.694899
\(52\) 0 0
\(53\) 9.38067i 1.28853i 0.764800 + 0.644267i \(0.222838\pi\)
−0.764800 + 0.644267i \(0.777162\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.75353i 0.629620i
\(58\) 0 0
\(59\) 4.96257 0.646072 0.323036 0.946387i \(-0.395296\pi\)
0.323036 + 0.946387i \(0.395296\pi\)
\(60\) 0 0
\(61\) −5.17162 −0.662159 −0.331079 0.943603i \(-0.607413\pi\)
−0.331079 + 0.943603i \(0.607413\pi\)
\(62\) 0 0
\(63\) − 4.20905i − 0.530290i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 0.209050i − 0.0255395i −0.999918 0.0127697i \(-0.995935\pi\)
0.999918 0.0127697i \(-0.00406485\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 9.38067 1.11328 0.556641 0.830753i \(-0.312090\pi\)
0.556641 + 0.830753i \(0.312090\pi\)
\(72\) 0 0
\(73\) − 10.2606i − 1.20091i −0.799659 0.600455i \(-0.794986\pi\)
0.799659 0.600455i \(-0.205014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 11.5897i 1.32077i
\(78\) 0 0
\(79\) 2.41810 0.272057 0.136029 0.990705i \(-0.456566\pi\)
0.136029 + 0.990705i \(0.456566\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 5.45552i − 0.598822i −0.954124 0.299411i \(-0.903210\pi\)
0.954124 0.299411i \(-0.0967901\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.96257i 0.532044i
\(88\) 0 0
\(89\) −4.91105 −0.520570 −0.260285 0.965532i \(-0.583817\pi\)
−0.260285 + 0.965532i \(0.583817\pi\)
\(90\) 0 0
\(91\) −3.17162 −0.332477
\(92\) 0 0
\(93\) 0.209050i 0.0216775i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.3432i 1.05020i 0.851041 + 0.525099i \(0.175972\pi\)
−0.851041 + 0.525099i \(0.824028\pi\)
\(98\) 0 0
\(99\) −2.75353 −0.276740
\(100\) 0 0
\(101\) 14.8877 1.48138 0.740692 0.671845i \(-0.234498\pi\)
0.740692 + 0.671845i \(0.234498\pi\)
\(102\) 0 0
\(103\) − 17.9251i − 1.76622i −0.469168 0.883109i \(-0.655446\pi\)
0.469168 0.883109i \(-0.344554\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 16.4696i − 1.59218i −0.605179 0.796089i \(-0.706898\pi\)
0.605179 0.796089i \(-0.293102\pi\)
\(108\) 0 0
\(109\) 6.26058 0.599654 0.299827 0.953994i \(-0.403071\pi\)
0.299827 + 0.953994i \(0.403071\pi\)
\(110\) 0 0
\(111\) −5.71610 −0.542548
\(112\) 0 0
\(113\) 9.38067i 0.882460i 0.897394 + 0.441230i \(0.145458\pi\)
−0.897394 + 0.441230i \(0.854542\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 0.753525i − 0.0696634i
\(118\) 0 0
\(119\) 20.8877 1.91477
\(120\) 0 0
\(121\) −3.41810 −0.310736
\(122\) 0 0
\(123\) 9.38067i 0.845827i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 22.0827i − 1.95952i −0.200175 0.979760i \(-0.564151\pi\)
0.200175 0.979760i \(-0.435849\pi\)
\(128\) 0 0
\(129\) −12.4181 −1.09335
\(130\) 0 0
\(131\) −1.58190 −0.138211 −0.0691056 0.997609i \(-0.522015\pi\)
−0.0691056 + 0.997609i \(0.522015\pi\)
\(132\) 0 0
\(133\) − 20.0078i − 1.73490i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 4.91105i − 0.419579i −0.977747 0.209790i \(-0.932722\pi\)
0.977747 0.209790i \(-0.0672779\pi\)
\(138\) 0 0
\(139\) −14.1342 −1.19885 −0.599424 0.800432i \(-0.704603\pi\)
−0.599424 + 0.800432i \(0.704603\pi\)
\(140\) 0 0
\(141\) 7.17162 0.603960
\(142\) 0 0
\(143\) 2.07485i 0.173508i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 10.7161i 0.883849i
\(148\) 0 0
\(149\) −3.24647 −0.265962 −0.132981 0.991119i \(-0.542455\pi\)
−0.132981 + 0.991119i \(0.542455\pi\)
\(150\) 0 0
\(151\) 0.418100 0.0340245 0.0170122 0.999855i \(-0.494585\pi\)
0.0170122 + 0.999855i \(0.494585\pi\)
\(152\) 0 0
\(153\) 4.96257i 0.401200i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 21.0452i − 1.67959i −0.542901 0.839797i \(-0.682674\pi\)
0.542901 0.839797i \(-0.317326\pi\)
\(158\) 0 0
\(159\) 9.38067 0.743936
\(160\) 0 0
\(161\) 4.20905 0.331720
\(162\) 0 0
\(163\) 7.43220i 0.582135i 0.956703 + 0.291067i \(0.0940104\pi\)
−0.956703 + 0.291067i \(0.905990\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 19.1716i − 1.48354i −0.670652 0.741772i \(-0.733985\pi\)
0.670652 0.741772i \(-0.266015\pi\)
\(168\) 0 0
\(169\) 12.4322 0.956323
\(170\) 0 0
\(171\) 4.75353 0.363511
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 4.96257i − 0.373010i
\(178\) 0 0
\(179\) −13.8503 −1.03522 −0.517610 0.855617i \(-0.673178\pi\)
−0.517610 + 0.855617i \(0.673178\pi\)
\(180\) 0 0
\(181\) 12.9110 0.959671 0.479835 0.877359i \(-0.340696\pi\)
0.479835 + 0.877359i \(0.340696\pi\)
\(182\) 0 0
\(183\) 5.17162i 0.382297i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 13.6646i − 0.999253i
\(188\) 0 0
\(189\) −4.20905 −0.306163
\(190\) 0 0
\(191\) −8.15752 −0.590258 −0.295129 0.955457i \(-0.595363\pi\)
−0.295129 + 0.955457i \(0.595363\pi\)
\(192\) 0 0
\(193\) 16.7613i 1.20651i 0.797549 + 0.603254i \(0.206130\pi\)
−0.797549 + 0.603254i \(0.793870\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 10.9110i − 0.777380i −0.921369 0.388690i \(-0.872928\pi\)
0.921369 0.388690i \(-0.127072\pi\)
\(198\) 0 0
\(199\) 4.49295 0.318497 0.159248 0.987239i \(-0.449093\pi\)
0.159248 + 0.987239i \(0.449093\pi\)
\(200\) 0 0
\(201\) −0.209050 −0.0147452
\(202\) 0 0
\(203\) − 20.8877i − 1.46603i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.00000i 0.0695048i
\(208\) 0 0
\(209\) −13.0890 −0.905382
\(210\) 0 0
\(211\) 9.71610 0.668884 0.334442 0.942416i \(-0.391452\pi\)
0.334442 + 0.942416i \(0.391452\pi\)
\(212\) 0 0
\(213\) − 9.38067i − 0.642753i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 0.879901i − 0.0597316i
\(218\) 0 0
\(219\) −10.2606 −0.693345
\(220\) 0 0
\(221\) 3.73942 0.251541
\(222\) 0 0
\(223\) 19.4322i 1.30128i 0.759388 + 0.650638i \(0.225499\pi\)
−0.759388 + 0.650638i \(0.774501\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 0.985900i − 0.0654365i −0.999465 0.0327182i \(-0.989584\pi\)
0.999465 0.0327182i \(-0.0104164\pi\)
\(228\) 0 0
\(229\) −3.08895 −0.204124 −0.102062 0.994778i \(-0.532544\pi\)
−0.102062 + 0.994778i \(0.532544\pi\)
\(230\) 0 0
\(231\) 11.5897 0.762548
\(232\) 0 0
\(233\) − 8.83620i − 0.578879i −0.957196 0.289439i \(-0.906531\pi\)
0.957196 0.289439i \(-0.0934688\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 2.41810i − 0.157072i
\(238\) 0 0
\(239\) 21.3807 1.38300 0.691500 0.722376i \(-0.256950\pi\)
0.691500 + 0.722376i \(0.256950\pi\)
\(240\) 0 0
\(241\) 22.2606 1.43393 0.716965 0.697109i \(-0.245531\pi\)
0.716965 + 0.697109i \(0.245531\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 3.58190i − 0.227911i
\(248\) 0 0
\(249\) −5.45552 −0.345730
\(250\) 0 0
\(251\) −6.59600 −0.416336 −0.208168 0.978093i \(-0.566750\pi\)
−0.208168 + 0.978093i \(0.566750\pi\)
\(252\) 0 0
\(253\) − 2.75353i − 0.173113i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 18.1857i − 1.13439i −0.823582 0.567197i \(-0.808028\pi\)
0.823582 0.567197i \(-0.191972\pi\)
\(258\) 0 0
\(259\) 24.0593 1.49498
\(260\) 0 0
\(261\) 4.96257 0.307176
\(262\) 0 0
\(263\) 15.3807i 0.948413i 0.880414 + 0.474207i \(0.157265\pi\)
−0.880414 + 0.474207i \(0.842735\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 4.91105i 0.300551i
\(268\) 0 0
\(269\) −0.544475 −0.0331972 −0.0165986 0.999862i \(-0.505284\pi\)
−0.0165986 + 0.999862i \(0.505284\pi\)
\(270\) 0 0
\(271\) −12.2090 −0.741647 −0.370823 0.928703i \(-0.620925\pi\)
−0.370823 + 0.928703i \(0.620925\pi\)
\(272\) 0 0
\(273\) 3.17162i 0.191955i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 21.0141i − 1.26261i −0.775533 0.631307i \(-0.782519\pi\)
0.775533 0.631307i \(-0.217481\pi\)
\(278\) 0 0
\(279\) 0.209050 0.0125155
\(280\) 0 0
\(281\) −24.1857 −1.44280 −0.721400 0.692519i \(-0.756501\pi\)
−0.721400 + 0.692519i \(0.756501\pi\)
\(282\) 0 0
\(283\) − 15.1201i − 0.898797i −0.893331 0.449398i \(-0.851638\pi\)
0.893331 0.449398i \(-0.148362\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 39.4837i − 2.33065i
\(288\) 0 0
\(289\) −7.62715 −0.448656
\(290\) 0 0
\(291\) 10.3432 0.606332
\(292\) 0 0
\(293\) − 10.4696i − 0.611642i −0.952089 0.305821i \(-0.901069\pi\)
0.952089 0.305821i \(-0.0989309\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.75353i 0.159776i
\(298\) 0 0
\(299\) 0.753525 0.0435775
\(300\) 0 0
\(301\) 52.2684 3.01270
\(302\) 0 0
\(303\) − 14.8877i − 0.855277i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 12.1575i − 0.693867i −0.937890 0.346933i \(-0.887223\pi\)
0.937890 0.346933i \(-0.112777\pi\)
\(308\) 0 0
\(309\) −17.9251 −1.01973
\(310\) 0 0
\(311\) −34.4181 −1.95167 −0.975836 0.218506i \(-0.929882\pi\)
−0.975836 + 0.218506i \(0.929882\pi\)
\(312\) 0 0
\(313\) − 29.5664i − 1.67119i −0.549345 0.835596i \(-0.685123\pi\)
0.549345 0.835596i \(-0.314877\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.66457i 0.0934918i 0.998907 + 0.0467459i \(0.0148851\pi\)
−0.998907 + 0.0467459i \(0.985115\pi\)
\(318\) 0 0
\(319\) −13.6646 −0.765069
\(320\) 0 0
\(321\) −16.4696 −0.919245
\(322\) 0 0
\(323\) 23.5897i 1.31257i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 6.26058i − 0.346211i
\(328\) 0 0
\(329\) −30.1857 −1.66419
\(330\) 0 0
\(331\) 3.12010 0.171496 0.0857481 0.996317i \(-0.472672\pi\)
0.0857481 + 0.996317i \(0.472672\pi\)
\(332\) 0 0
\(333\) 5.71610i 0.313240i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 11.5819i − 0.630906i −0.948941 0.315453i \(-0.897843\pi\)
0.948941 0.315453i \(-0.102157\pi\)
\(338\) 0 0
\(339\) 9.38067 0.509488
\(340\) 0 0
\(341\) −0.575624 −0.0311718
\(342\) 0 0
\(343\) − 15.6412i − 0.844548i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 26.3432i 1.41418i 0.707124 + 0.707090i \(0.249992\pi\)
−0.707124 + 0.707090i \(0.750008\pi\)
\(348\) 0 0
\(349\) −9.22315 −0.493704 −0.246852 0.969053i \(-0.579396\pi\)
−0.246852 + 0.969053i \(0.579396\pi\)
\(350\) 0 0
\(351\) −0.753525 −0.0402202
\(352\) 0 0
\(353\) 18.1857i 0.967928i 0.875088 + 0.483964i \(0.160804\pi\)
−0.875088 + 0.483964i \(0.839196\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 20.8877i − 1.10550i
\(358\) 0 0
\(359\) −7.17162 −0.378504 −0.189252 0.981929i \(-0.560606\pi\)
−0.189252 + 0.981929i \(0.560606\pi\)
\(360\) 0 0
\(361\) 3.59600 0.189263
\(362\) 0 0
\(363\) 3.41810i 0.179404i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 17.5664i 0.916959i 0.888705 + 0.458479i \(0.151606\pi\)
−0.888705 + 0.458479i \(0.848394\pi\)
\(368\) 0 0
\(369\) 9.38067 0.488338
\(370\) 0 0
\(371\) −39.4837 −2.04989
\(372\) 0 0
\(373\) 33.6724i 1.74349i 0.489959 + 0.871745i \(0.337012\pi\)
−0.489959 + 0.871745i \(0.662988\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 3.73942i − 0.192590i
\(378\) 0 0
\(379\) −18.4181 −0.946074 −0.473037 0.881043i \(-0.656842\pi\)
−0.473037 + 0.881043i \(0.656842\pi\)
\(380\) 0 0
\(381\) −22.0827 −1.13133
\(382\) 0 0
\(383\) 20.7847i 1.06205i 0.847357 + 0.531024i \(0.178192\pi\)
−0.847357 + 0.531024i \(0.821808\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 12.4181i 0.631247i
\(388\) 0 0
\(389\) −22.5212 −1.14187 −0.570934 0.820996i \(-0.693419\pi\)
−0.570934 + 0.820996i \(0.693419\pi\)
\(390\) 0 0
\(391\) −4.96257 −0.250968
\(392\) 0 0
\(393\) 1.58190i 0.0797963i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 15.4040i − 0.773105i −0.922267 0.386552i \(-0.873666\pi\)
0.922267 0.386552i \(-0.126334\pi\)
\(398\) 0 0
\(399\) −20.0078 −1.00164
\(400\) 0 0
\(401\) −10.5212 −0.525401 −0.262701 0.964877i \(-0.584613\pi\)
−0.262701 + 0.964877i \(0.584613\pi\)
\(402\) 0 0
\(403\) − 0.157524i − 0.00784684i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 15.7394i − 0.780174i
\(408\) 0 0
\(409\) 26.0593 1.28855 0.644276 0.764793i \(-0.277159\pi\)
0.644276 + 0.764793i \(0.277159\pi\)
\(410\) 0 0
\(411\) −4.91105 −0.242244
\(412\) 0 0
\(413\) 20.8877i 1.02782i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 14.1342i 0.692155i
\(418\) 0 0
\(419\) 3.73942 0.182683 0.0913414 0.995820i \(-0.470885\pi\)
0.0913414 + 0.995820i \(0.470885\pi\)
\(420\) 0 0
\(421\) −3.09677 −0.150928 −0.0754638 0.997149i \(-0.524044\pi\)
−0.0754638 + 0.997149i \(0.524044\pi\)
\(422\) 0 0
\(423\) − 7.17162i − 0.348696i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 21.7676i − 1.05341i
\(428\) 0 0
\(429\) 2.07485 0.100175
\(430\) 0 0
\(431\) −36.1653 −1.74202 −0.871012 0.491262i \(-0.836536\pi\)
−0.871012 + 0.491262i \(0.836536\pi\)
\(432\) 0 0
\(433\) 16.6271i 0.799050i 0.916722 + 0.399525i \(0.130825\pi\)
−0.916722 + 0.399525i \(0.869175\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.75353i 0.227392i
\(438\) 0 0
\(439\) −38.7613 −1.84998 −0.924989 0.379994i \(-0.875926\pi\)
−0.924989 + 0.379994i \(0.875926\pi\)
\(440\) 0 0
\(441\) 10.7161 0.510290
\(442\) 0 0
\(443\) − 33.5149i − 1.59234i −0.605073 0.796170i \(-0.706856\pi\)
0.605073 0.796170i \(-0.293144\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 3.24647i 0.153553i
\(448\) 0 0
\(449\) 13.9018 0.656068 0.328034 0.944666i \(-0.393614\pi\)
0.328034 + 0.944666i \(0.393614\pi\)
\(450\) 0 0
\(451\) −25.8299 −1.21628
\(452\) 0 0
\(453\) − 0.418100i − 0.0196440i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 2.28390i − 0.106836i −0.998572 0.0534182i \(-0.982988\pi\)
0.998572 0.0534182i \(-0.0170116\pi\)
\(458\) 0 0
\(459\) 4.96257 0.231633
\(460\) 0 0
\(461\) 8.34325 0.388584 0.194292 0.980944i \(-0.437759\pi\)
0.194292 + 0.980944i \(0.437759\pi\)
\(462\) 0 0
\(463\) 27.6928i 1.28699i 0.765449 + 0.643496i \(0.222517\pi\)
−0.765449 + 0.643496i \(0.777483\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7.53037i 0.348464i 0.984705 + 0.174232i \(0.0557443\pi\)
−0.984705 + 0.174232i \(0.944256\pi\)
\(468\) 0 0
\(469\) 0.879901 0.0406300
\(470\) 0 0
\(471\) −21.0452 −0.969714
\(472\) 0 0
\(473\) − 34.1935i − 1.57222i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 9.38067i − 0.429512i
\(478\) 0 0
\(479\) 18.1857 0.830927 0.415463 0.909610i \(-0.363619\pi\)
0.415463 + 0.909610i \(0.363619\pi\)
\(480\) 0 0
\(481\) 4.30722 0.196393
\(482\) 0 0
\(483\) − 4.20905i − 0.191518i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 24.2606i − 1.09935i −0.835378 0.549676i \(-0.814751\pi\)
0.835378 0.549676i \(-0.185249\pi\)
\(488\) 0 0
\(489\) 7.43220 0.336096
\(490\) 0 0
\(491\) 14.8877 0.671874 0.335937 0.941885i \(-0.390947\pi\)
0.335937 + 0.941885i \(0.390947\pi\)
\(492\) 0 0
\(493\) 24.6271i 1.10915i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 39.4837i 1.77109i
\(498\) 0 0
\(499\) 0.312101 0.0139716 0.00698578 0.999976i \(-0.497776\pi\)
0.00698578 + 0.999976i \(0.497776\pi\)
\(500\) 0 0
\(501\) −19.1716 −0.856525
\(502\) 0 0
\(503\) − 20.8877i − 0.931338i −0.884959 0.465669i \(-0.845814\pi\)
0.884959 0.465669i \(-0.154186\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 12.4322i − 0.552133i
\(508\) 0 0
\(509\) −27.9251 −1.23776 −0.618880 0.785485i \(-0.712413\pi\)
−0.618880 + 0.785485i \(0.712413\pi\)
\(510\) 0 0
\(511\) 43.1873 1.91049
\(512\) 0 0
\(513\) − 4.75353i − 0.209873i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 19.7472i 0.868483i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 32.0360 1.40352 0.701762 0.712412i \(-0.252397\pi\)
0.701762 + 0.712412i \(0.252397\pi\)
\(522\) 0 0
\(523\) − 38.8644i − 1.69942i −0.527249 0.849711i \(-0.676777\pi\)
0.527249 0.849711i \(-0.323223\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.03743i 0.0451909i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −4.96257 −0.215357
\(532\) 0 0
\(533\) − 7.06857i − 0.306174i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 13.8503i 0.597685i
\(538\) 0 0
\(539\) −29.5071 −1.27096
\(540\) 0 0
\(541\) 4.07485 0.175191 0.0875957 0.996156i \(-0.472082\pi\)
0.0875957 + 0.996156i \(0.472082\pi\)
\(542\) 0 0
\(543\) − 12.9110i − 0.554066i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 28.0000i − 1.19719i −0.801050 0.598597i \(-0.795725\pi\)
0.801050 0.598597i \(-0.204275\pi\)
\(548\) 0 0
\(549\) 5.17162 0.220720
\(550\) 0 0
\(551\) 23.5897 1.00496
\(552\) 0 0
\(553\) 10.1779i 0.432808i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.63343i 0.0692105i 0.999401 + 0.0346052i \(0.0110174\pi\)
−0.999401 + 0.0346052i \(0.988983\pi\)
\(558\) 0 0
\(559\) 9.35735 0.395774
\(560\) 0 0
\(561\) −13.6646 −0.576919
\(562\) 0 0
\(563\) 23.2310i 0.979069i 0.871984 + 0.489534i \(0.162833\pi\)
−0.871984 + 0.489534i \(0.837167\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 4.20905i 0.176763i
\(568\) 0 0
\(569\) −21.3291 −0.894164 −0.447082 0.894493i \(-0.647537\pi\)
−0.447082 + 0.894493i \(0.647537\pi\)
\(570\) 0 0
\(571\) −4.18573 −0.175167 −0.0875836 0.996157i \(-0.527914\pi\)
−0.0875836 + 0.996157i \(0.527914\pi\)
\(572\) 0 0
\(573\) 8.15752i 0.340785i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 10.5678i 0.439943i 0.975506 + 0.219972i \(0.0705965\pi\)
−0.975506 + 0.219972i \(0.929404\pi\)
\(578\) 0 0
\(579\) 16.7613 0.696578
\(580\) 0 0
\(581\) 22.9626 0.952648
\(582\) 0 0
\(583\) 25.8299i 1.06977i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 28.4181i 1.17294i 0.809971 + 0.586470i \(0.199483\pi\)
−0.809971 + 0.586470i \(0.800517\pi\)
\(588\) 0 0
\(589\) 0.993723 0.0409457
\(590\) 0 0
\(591\) −10.9110 −0.448821
\(592\) 0 0
\(593\) − 28.9937i − 1.19063i −0.803493 0.595315i \(-0.797027\pi\)
0.803493 0.595315i \(-0.202973\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 4.49295i − 0.183884i
\(598\) 0 0
\(599\) 6.10305 0.249364 0.124682 0.992197i \(-0.460209\pi\)
0.124682 + 0.992197i \(0.460209\pi\)
\(600\) 0 0
\(601\) −29.3885 −1.19878 −0.599391 0.800456i \(-0.704590\pi\)
−0.599391 + 0.800456i \(0.704590\pi\)
\(602\) 0 0
\(603\) 0.209050i 0.00851316i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 3.58972i − 0.145702i −0.997343 0.0728512i \(-0.976790\pi\)
0.997343 0.0728512i \(-0.0232098\pi\)
\(608\) 0 0
\(609\) −20.8877 −0.846413
\(610\) 0 0
\(611\) −5.40400 −0.218622
\(612\) 0 0
\(613\) 21.4040i 0.864499i 0.901754 + 0.432250i \(0.142280\pi\)
−0.901754 + 0.432250i \(0.857720\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 32.2917i 1.30002i 0.759927 + 0.650008i \(0.225234\pi\)
−0.759927 + 0.650008i \(0.774766\pi\)
\(618\) 0 0
\(619\) −47.1046 −1.89329 −0.946647 0.322273i \(-0.895553\pi\)
−0.946647 + 0.322273i \(0.895553\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) − 20.6709i − 0.828160i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 13.0890i 0.522722i
\(628\) 0 0
\(629\) −28.3666 −1.13105
\(630\) 0 0
\(631\) 23.5149 0.936112 0.468056 0.883699i \(-0.344954\pi\)
0.468056 + 0.883699i \(0.344954\pi\)
\(632\) 0 0
\(633\) − 9.71610i − 0.386180i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 8.07485i − 0.319937i
\(638\) 0 0
\(639\) −9.38067 −0.371094
\(640\) 0 0
\(641\) −6.67867 −0.263792 −0.131896 0.991264i \(-0.542106\pi\)
−0.131896 + 0.991264i \(0.542106\pi\)
\(642\) 0 0
\(643\) − 27.1201i − 1.06951i −0.845006 0.534756i \(-0.820403\pi\)
0.845006 0.534756i \(-0.179597\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 27.7394i − 1.09055i −0.838257 0.545275i \(-0.816425\pi\)
0.838257 0.545275i \(-0.183575\pi\)
\(648\) 0 0
\(649\) 13.6646 0.536381
\(650\) 0 0
\(651\) −0.879901 −0.0344860
\(652\) 0 0
\(653\) 2.75353i 0.107754i 0.998548 + 0.0538769i \(0.0171579\pi\)
−0.998548 + 0.0538769i \(0.982842\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 10.2606i 0.400303i
\(658\) 0 0
\(659\) −11.5897 −0.451472 −0.225736 0.974189i \(-0.572479\pi\)
−0.225736 + 0.974189i \(0.572479\pi\)
\(660\) 0 0
\(661\) 28.3432 1.10242 0.551212 0.834365i \(-0.314165\pi\)
0.551212 + 0.834365i \(0.314165\pi\)
\(662\) 0 0
\(663\) − 3.73942i − 0.145227i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.96257i 0.192152i
\(668\) 0 0
\(669\) 19.4322 0.751292
\(670\) 0 0
\(671\) −14.2402 −0.549737
\(672\) 0 0
\(673\) 48.0360i 1.85165i 0.377949 + 0.925826i \(0.376629\pi\)
−0.377949 + 0.925826i \(0.623371\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2.51627i 0.0967083i 0.998830 + 0.0483541i \(0.0153976\pi\)
−0.998830 + 0.0483541i \(0.984602\pi\)
\(678\) 0 0
\(679\) −43.5353 −1.67073
\(680\) 0 0
\(681\) −0.985900 −0.0377798
\(682\) 0 0
\(683\) 3.84248i 0.147028i 0.997294 + 0.0735141i \(0.0234214\pi\)
−0.997294 + 0.0735141i \(0.976579\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 3.08895i 0.117851i
\(688\) 0 0
\(689\) −7.06857 −0.269291
\(690\) 0 0
\(691\) 2.59600 0.0987565 0.0493783 0.998780i \(-0.484276\pi\)
0.0493783 + 0.998780i \(0.484276\pi\)
\(692\) 0 0
\(693\) − 11.5897i − 0.440257i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 46.5523i 1.76329i
\(698\) 0 0
\(699\) −8.83620 −0.334216
\(700\) 0 0
\(701\) 41.5897 1.57082 0.785411 0.618974i \(-0.212452\pi\)
0.785411 + 0.618974i \(0.212452\pi\)
\(702\) 0 0
\(703\) 27.1716i 1.02480i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 62.6632i 2.35669i
\(708\) 0 0
\(709\) 53.1716 1.99690 0.998451 0.0556356i \(-0.0177185\pi\)
0.998451 + 0.0556356i \(0.0177185\pi\)
\(710\) 0 0
\(711\) −2.41810 −0.0906858
\(712\) 0 0
\(713\) 0.209050i 0.00782898i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 21.3807i − 0.798476i
\(718\) 0 0
\(719\) −45.3807 −1.69241 −0.846207 0.532855i \(-0.821119\pi\)
−0.846207 + 0.532855i \(0.821119\pi\)
\(720\) 0 0
\(721\) 75.4478 2.80982
\(722\) 0 0
\(723\) − 22.2606i − 0.827880i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 46.4026i − 1.72098i −0.509470 0.860489i \(-0.670158\pi\)
0.509470 0.860489i \(-0.329842\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −61.6257 −2.27931
\(732\) 0 0
\(733\) − 22.8051i − 0.842324i −0.906985 0.421162i \(-0.861622\pi\)
0.906985 0.421162i \(-0.138378\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 0.575624i − 0.0212034i
\(738\) 0 0
\(739\) 16.5241 0.607849 0.303924 0.952696i \(-0.401703\pi\)
0.303924 + 0.952696i \(0.401703\pi\)
\(740\) 0 0
\(741\) −3.58190 −0.131584
\(742\) 0 0
\(743\) − 43.8503i − 1.60871i −0.594148 0.804356i \(-0.702511\pi\)
0.594148 0.804356i \(-0.297489\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 5.45552i 0.199607i
\(748\) 0 0
\(749\) 69.3215 2.53295
\(750\) 0 0
\(751\) 5.73942 0.209435 0.104717 0.994502i \(-0.466606\pi\)
0.104717 + 0.994502i \(0.466606\pi\)
\(752\) 0 0
\(753\) 6.59600i 0.240372i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 9.86580i − 0.358579i −0.983796 0.179289i \(-0.942620\pi\)
0.983796 0.179289i \(-0.0573798\pi\)
\(758\) 0 0
\(759\) −2.75353 −0.0999466
\(760\) 0 0
\(761\) 4.69418 0.170164 0.0850819 0.996374i \(-0.472885\pi\)
0.0850819 + 0.996374i \(0.472885\pi\)
\(762\) 0 0
\(763\) 26.3511i 0.953973i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.73942i 0.135023i
\(768\) 0 0
\(769\) −45.5431 −1.64233 −0.821163 0.570694i \(-0.806674\pi\)
−0.821163 + 0.570694i \(0.806674\pi\)
\(770\) 0 0
\(771\) −18.1857 −0.654943
\(772\) 0 0
\(773\) − 12.4929i − 0.449340i −0.974435 0.224670i \(-0.927870\pi\)
0.974435 0.224670i \(-0.0721304\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 24.0593i − 0.863124i
\(778\) 0 0
\(779\) 44.5913 1.59765
\(780\) 0 0
\(781\) 25.8299 0.924267
\(782\) 0 0
\(783\) − 4.96257i − 0.177348i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 5.71610i − 0.203757i −0.994797 0.101878i \(-0.967515\pi\)
0.994797 0.101878i \(-0.0324853\pi\)
\(788\) 0 0
\(789\) 15.3807 0.547567
\(790\) 0 0
\(791\) −39.4837 −1.40388
\(792\) 0 0
\(793\) − 3.89695i − 0.138385i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 9.48373i − 0.335931i −0.985793 0.167965i \(-0.946280\pi\)
0.985793 0.167965i \(-0.0537197\pi\)
\(798\) 0 0
\(799\) 35.5897 1.25907
\(800\) 0 0
\(801\) 4.91105 0.173523
\(802\) 0 0
\(803\) − 28.2528i − 0.997018i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0.544475i 0.0191664i
\(808\) 0 0
\(809\) −50.0672 −1.76027 −0.880134 0.474725i \(-0.842547\pi\)
−0.880134 + 0.474725i \(0.842547\pi\)
\(810\) 0 0
\(811\) −9.14830 −0.321240 −0.160620 0.987016i \(-0.551349\pi\)
−0.160620 + 0.987016i \(0.551349\pi\)
\(812\) 0 0
\(813\) 12.2090i 0.428190i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 59.0297i 2.06519i
\(818\) 0 0
\(819\) 3.17162 0.110826
\(820\) 0 0
\(821\) −4.91105 −0.171397 −0.0856984 0.996321i \(-0.527312\pi\)
−0.0856984 + 0.996321i \(0.527312\pi\)
\(822\) 0 0
\(823\) − 23.7006i − 0.826151i −0.910697 0.413075i \(-0.864455\pi\)
0.910697 0.413075i \(-0.135545\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 13.1405i 0.456939i 0.973551 + 0.228470i \(0.0733721\pi\)
−0.973551 + 0.228470i \(0.926628\pi\)
\(828\) 0 0
\(829\) 45.6412 1.58519 0.792593 0.609751i \(-0.208731\pi\)
0.792593 + 0.609751i \(0.208731\pi\)
\(830\) 0 0
\(831\) −21.0141 −0.728971
\(832\) 0 0
\(833\) 53.1794i 1.84256i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 0.209050i − 0.00722582i
\(838\) 0 0
\(839\) 5.40400 0.186567 0.0932834 0.995640i \(-0.470264\pi\)
0.0932834 + 0.995640i \(0.470264\pi\)
\(840\) 0 0
\(841\) −4.37285 −0.150788
\(842\) 0 0
\(843\) 24.1857i 0.833001i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 14.3870i − 0.494341i
\(848\) 0 0
\(849\) −15.1201 −0.518920
\(850\) 0 0
\(851\) −5.71610 −0.195945
\(852\) 0 0
\(853\) 53.0297i 1.81570i 0.419291 + 0.907852i \(0.362279\pi\)
−0.419291 + 0.907852i \(0.637721\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 14.4463i − 0.493476i −0.969082 0.246738i \(-0.920641\pi\)
0.969082 0.246738i \(-0.0793587\pi\)
\(858\) 0 0
\(859\) −13.8658 −0.473095 −0.236548 0.971620i \(-0.576016\pi\)
−0.236548 + 0.971620i \(0.576016\pi\)
\(860\) 0 0
\(861\) −39.4837 −1.34560
\(862\) 0 0
\(863\) − 52.5834i − 1.78996i −0.446105 0.894981i \(-0.647189\pi\)
0.446105 0.894981i \(-0.352811\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 7.62715i 0.259032i
\(868\) 0 0
\(869\) 6.65830 0.225867
\(870\) 0 0
\(871\) 0.157524 0.00533751
\(872\) 0 0
\(873\) − 10.3432i − 0.350066i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 25.3291i − 0.855305i −0.903943 0.427652i \(-0.859341\pi\)
0.903943 0.427652i \(-0.140659\pi\)
\(878\) 0 0
\(879\) −10.4696 −0.353132
\(880\) 0 0
\(881\) 14.2606 0.480451 0.240225 0.970717i \(-0.422779\pi\)
0.240225 + 0.970717i \(0.422779\pi\)
\(882\) 0 0
\(883\) − 40.6320i − 1.36738i −0.729774 0.683688i \(-0.760375\pi\)
0.729774 0.683688i \(-0.239625\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 19.9534i − 0.669968i −0.942224 0.334984i \(-0.891269\pi\)
0.942224 0.334984i \(-0.108731\pi\)
\(888\) 0 0
\(889\) 92.9471 3.11734
\(890\) 0 0
\(891\) 2.75353 0.0922466
\(892\) 0 0
\(893\) − 34.0905i − 1.14080i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 0.753525i − 0.0251595i
\(898\) 0 0
\(899\) 1.03743 0.0346001
\(900\) 0 0
\(901\) 46.5523 1.55088
\(902\) 0 0
\(903\) − 52.2684i − 1.73938i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 29.8814i − 0.992197i −0.868266 0.496099i \(-0.834765\pi\)
0.868266 0.496099i \(-0.165235\pi\)
\(908\) 0 0
\(909\) −14.8877 −0.493795
\(910\) 0 0
\(911\) 18.8644 0.625005 0.312503 0.949917i \(-0.398833\pi\)
0.312503 + 0.949917i \(0.398833\pi\)
\(912\) 0 0
\(913\) − 15.0219i − 0.497153i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 6.65830i − 0.219876i
\(918\) 0 0
\(919\) −16.2402 −0.535715 −0.267857 0.963459i \(-0.586316\pi\)
−0.267857 + 0.963459i \(0.586316\pi\)
\(920\) 0 0
\(921\) −12.1575 −0.400604
\(922\) 0 0
\(923\) 7.06857i 0.232665i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 17.9251i 0.588739i
\(928\) 0 0
\(929\) −29.3340 −0.962418 −0.481209 0.876606i \(-0.659802\pi\)
−0.481209 + 0.876606i \(0.659802\pi\)
\(930\) 0 0
\(931\) 50.9393 1.66947
\(932\) 0 0
\(933\) 34.4181i 1.12680i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 41.3573i − 1.35109i −0.737321 0.675543i \(-0.763909\pi\)
0.737321 0.675543i \(-0.236091\pi\)
\(938\) 0 0
\(939\) −29.5664 −0.964863
\(940\) 0 0
\(941\) 50.4259 1.64384 0.821919 0.569604i \(-0.192904\pi\)
0.821919 + 0.569604i \(0.192904\pi\)
\(942\) 0 0
\(943\) 9.38067i 0.305477i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 5.50705i 0.178955i 0.995989 + 0.0894775i \(0.0285197\pi\)
−0.995989 + 0.0894775i \(0.971480\pi\)
\(948\) 0 0
\(949\) 7.73160 0.250978
\(950\) 0 0
\(951\) 1.66457 0.0539775
\(952\) 0 0
\(953\) − 0.864400i − 0.0280007i −0.999902 0.0140003i \(-0.995543\pi\)
0.999902 0.0140003i \(-0.00445659\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 13.6646i 0.441713i
\(958\) 0 0
\(959\) 20.6709 0.667497
\(960\) 0 0
\(961\) −30.9563 −0.998590
\(962\) 0 0
\(963\) 16.4696i 0.530726i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 10.7535i 0.345810i 0.984939 + 0.172905i \(0.0553154\pi\)
−0.984939 + 0.172905i \(0.944685\pi\)
\(968\) 0 0
\(969\) 23.5897 0.757811
\(970\) 0 0
\(971\) 35.1794 1.12896 0.564481 0.825446i \(-0.309076\pi\)
0.564481 + 0.825446i \(0.309076\pi\)
\(972\) 0 0
\(973\) − 59.4915i − 1.90721i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 15.9767i 0.511139i 0.966791 + 0.255570i \(0.0822630\pi\)
−0.966791 + 0.255570i \(0.917737\pi\)
\(978\) 0 0
\(979\) −13.5227 −0.432187
\(980\) 0 0
\(981\) −6.26058 −0.199885
\(982\) 0 0
\(983\) 57.2592i 1.82628i 0.407642 + 0.913142i \(0.366351\pi\)
−0.407642 + 0.913142i \(0.633649\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 30.1857i 0.960822i
\(988\) 0 0
\(989\) −12.4181 −0.394873
\(990\) 0 0
\(991\) 12.8799 0.409144 0.204572 0.978852i \(-0.434420\pi\)
0.204572 + 0.978852i \(0.434420\pi\)
\(992\) 0 0
\(993\) − 3.12010i − 0.0990134i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 39.7754i − 1.25970i −0.776716 0.629851i \(-0.783116\pi\)
0.776716 0.629851i \(-0.216884\pi\)
\(998\) 0 0
\(999\) 5.71610 0.180849
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.6349.3 6
5.2 odd 4 6900.2.a.x.1.1 3
5.3 odd 4 1380.2.a.j.1.3 3
5.4 even 2 inner 6900.2.f.r.6349.4 6
15.8 even 4 4140.2.a.s.1.3 3
20.3 even 4 5520.2.a.bv.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.a.j.1.3 3 5.3 odd 4
4140.2.a.s.1.3 3 15.8 even 4
5520.2.a.bv.1.1 3 20.3 even 4
6900.2.a.x.1.1 3 5.2 odd 4
6900.2.f.r.6349.3 6 1.1 even 1 trivial
6900.2.f.r.6349.4 6 5.4 even 2 inner