Properties

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

$q$-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 \(-0.707242\pi\)
0.606038 0.795436i \(-0.292758\pi\)
\(8\) 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 \(-0.966677\pi\)
0.994525 0.104495i \(-0.0333227\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.96257i 1.20360i 0.798646 + 0.601801i \(0.205550\pi\)
−0.798646 + 0.601801i \(0.794450\pi\)
\(18\) 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 \(0.155696\pi\)
−0.882741 + 0.469861i \(0.844304\pi\)
\(38\) 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 \(0.395775\pi\)
−0.321613 + 0.946871i \(0.604225\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 7.17162i − 1.04609i −0.852305 0.523044i \(-0.824796\pi\)
0.852305 0.523044i \(-0.175204\pi\)
\(48\) 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 \(-0.777162\pi\)
0.764800 0.644267i \(-0.222838\pi\)
\(54\) 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 \(0.00406485\pi\)
−0.999918 + 0.0127697i \(0.995935\pi\)
\(68\) 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 \(0.205014\pi\)
−0.799659 + 0.600455i \(0.794986\pi\)
\(74\) 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 \(0.0967901\pi\)
−0.954124 + 0.299411i \(0.903210\pi\)
\(84\) 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 \(-0.824028\pi\)
0.851041 0.525099i \(-0.175972\pi\)
\(98\) 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 \(0.344554\pi\)
−0.469168 + 0.883109i \(0.655446\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.4696i 1.59218i 0.605179 + 0.796089i \(0.293102\pi\)
−0.605179 + 0.796089i \(0.706898\pi\)
\(108\) 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 \(-0.854542\pi\)
0.897394 0.441230i \(-0.145458\pi\)
\(114\) 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 \(0.435849\pi\)
−0.200175 + 0.979760i \(0.564151\pi\)
\(128\) 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 \(0.0672779\pi\)
−0.977747 + 0.209790i \(0.932722\pi\)
\(138\) 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 \(0.317326\pi\)
−0.542901 + 0.839797i \(0.682674\pi\)
\(158\) 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 \(-0.905990\pi\)
0.956703 0.291067i \(-0.0940104\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 19.1716i 1.48354i 0.670652 + 0.741772i \(0.266015\pi\)
−0.670652 + 0.741772i \(0.733985\pi\)
\(168\) 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 \(-0.793870\pi\)
0.797549 0.603254i \(-0.206130\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.9110i 0.777380i 0.921369 + 0.388690i \(0.127072\pi\)
−0.921369 + 0.388690i \(0.872928\pi\)
\(198\) 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 \(-0.774501\pi\)
0.759388 0.650638i \(-0.225499\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0.985900i 0.0654365i 0.999465 + 0.0327182i \(0.0104164\pi\)
−0.999465 + 0.0327182i \(0.989584\pi\)
\(228\) 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 \(0.0934688\pi\)
−0.957196 + 0.289439i \(0.906531\pi\)
\(234\) 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 \(0.191972\pi\)
−0.823582 + 0.567197i \(0.808028\pi\)
\(258\) 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 \(-0.842735\pi\)
0.880414 0.474207i \(-0.157265\pi\)
\(264\) 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 \(0.217481\pi\)
−0.775533 + 0.631307i \(0.782519\pi\)
\(278\) 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 \(0.148362\pi\)
−0.893331 + 0.449398i \(0.851638\pi\)
\(284\) 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 \(0.0989309\pi\)
−0.952089 + 0.305821i \(0.901069\pi\)
\(294\) 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 \(0.112777\pi\)
−0.937890 + 0.346933i \(0.887223\pi\)
\(308\) 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 \(0.314877\pi\)
−0.549345 + 0.835596i \(0.685123\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 1.66457i − 0.0934918i −0.998907 0.0467459i \(-0.985115\pi\)
0.998907 0.0467459i \(-0.0148851\pi\)
\(318\) 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 \(0.102157\pi\)
−0.948941 + 0.315453i \(0.897843\pi\)
\(338\) 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 \(-0.750008\pi\)
0.707124 0.707090i \(-0.249992\pi\)
\(348\) 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 \(-0.839196\pi\)
0.875088 0.483964i \(-0.160804\pi\)
\(354\) 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 \(-0.848394\pi\)
0.888705 0.458479i \(-0.151606\pi\)
\(368\) 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 \(-0.662988\pi\)
0.489959 0.871745i \(-0.337012\pi\)
\(374\) 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 \(-0.821808\pi\)
0.847357 0.531024i \(-0.178192\pi\)
\(384\) 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 \(0.126334\pi\)
−0.922267 + 0.386552i \(0.873666\pi\)
\(398\) 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 \(-0.869175\pi\)
0.916722 0.399525i \(-0.130825\pi\)
\(434\) 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 \(0.293144\pi\)
−0.605073 + 0.796170i \(0.706856\pi\)
\(444\) 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 \(0.0170116\pi\)
−0.998572 + 0.0534182i \(0.982988\pi\)
\(458\) 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 \(-0.777483\pi\)
0.765449 0.643496i \(-0.222517\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 7.53037i − 0.348464i −0.984705 0.174232i \(-0.944256\pi\)
0.984705 0.174232i \(-0.0557443\pi\)
\(468\) 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 \(0.185249\pi\)
−0.835378 + 0.549676i \(0.814751\pi\)
\(488\) 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 \(0.154186\pi\)
−0.884959 + 0.465669i \(0.845814\pi\)
\(504\) 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 \(0.323223\pi\)
−0.527249 + 0.849711i \(0.676777\pi\)
\(524\) 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 \(0.204275\pi\)
−0.801050 + 0.598597i \(0.795725\pi\)
\(548\) 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 \(-0.988983\pi\)
0.999401 0.0346052i \(-0.0110174\pi\)
\(558\) 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 \(-0.837167\pi\)
0.871984 0.489534i \(-0.162833\pi\)
\(564\) 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 \(-0.929404\pi\)
0.975506 0.219972i \(-0.0705965\pi\)
\(578\) 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 \(-0.800517\pi\)
0.809971 0.586470i \(-0.199483\pi\)
\(588\) 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 \(0.202973\pi\)
−0.803493 + 0.595315i \(0.797027\pi\)
\(594\) 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 \(0.0232098\pi\)
−0.997343 + 0.0728512i \(0.976790\pi\)
\(608\) 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 \(-0.857720\pi\)
0.901754 0.432250i \(-0.142280\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 32.2917i − 1.30002i −0.759927 0.650008i \(-0.774766\pi\)
0.759927 0.650008i \(-0.225234\pi\)
\(618\) 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 \(0.179597\pi\)
−0.845006 + 0.534756i \(0.820403\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 27.7394i 1.09055i 0.838257 + 0.545275i \(0.183575\pi\)
−0.838257 + 0.545275i \(0.816425\pi\)
\(648\) 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 \(-0.982842\pi\)
0.998548 0.0538769i \(-0.0171579\pi\)
\(654\) 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 \(-0.623371\pi\)
0.377949 0.925826i \(-0.376629\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 2.51627i − 0.0967083i −0.998830 0.0483541i \(-0.984602\pi\)
0.998830 0.0483541i \(-0.0153976\pi\)
\(678\) 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 \(-0.976579\pi\)
0.997294 0.0735141i \(-0.0234214\pi\)
\(684\) 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 \(0.329842\pi\)
−0.509470 + 0.860489i \(0.670158\pi\)
\(728\) 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 \(0.138378\pi\)
−0.906985 + 0.421162i \(0.861622\pi\)
\(734\) 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 \(0.297489\pi\)
−0.594148 + 0.804356i \(0.702511\pi\)
\(744\) 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 \(0.0573798\pi\)
−0.983796 + 0.179289i \(0.942620\pi\)
\(758\) 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 \(0.0721304\pi\)
−0.974435 + 0.224670i \(0.927870\pi\)
\(774\) 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 \(0.0324853\pi\)
−0.994797 + 0.101878i \(0.967515\pi\)
\(788\) 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 \(0.0537197\pi\)
−0.985793 + 0.167965i \(0.946280\pi\)
\(798\) 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 \(0.135545\pi\)
−0.910697 + 0.413075i \(0.864455\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 13.1405i − 0.456939i −0.973551 0.228470i \(-0.926628\pi\)
0.973551 0.228470i \(-0.0733721\pi\)
\(828\) 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 \(-0.637721\pi\)
0.419291 0.907852i \(-0.362279\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 14.4463i 0.493476i 0.969082 + 0.246738i \(0.0793587\pi\)
−0.969082 + 0.246738i \(0.920641\pi\)
\(858\) 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 \(0.352811\pi\)
−0.446105 + 0.894981i \(0.647189\pi\)
\(864\) 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 \(0.140659\pi\)
−0.903943 + 0.427652i \(0.859341\pi\)
\(878\) 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 \(0.239625\pi\)
−0.729774 + 0.683688i \(0.760375\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 19.9534i 0.669968i 0.942224 + 0.334984i \(0.108731\pi\)
−0.942224 + 0.334984i \(0.891269\pi\)
\(888\) 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 \(0.165235\pi\)
−0.868266 + 0.496099i \(0.834765\pi\)
\(908\) 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 \(0.236091\pi\)
−0.737321 + 0.675543i \(0.763909\pi\)
\(938\) 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 \(-0.971480\pi\)
0.995989 0.0894775i \(-0.0285197\pi\)
\(948\) 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 \(0.00445659\pi\)
−0.999902 + 0.0140003i \(0.995543\pi\)
\(954\) 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 \(-0.944685\pi\)
0.984939 0.172905i \(-0.0553154\pi\)
\(968\) 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 \(-0.917737\pi\)
0.966791 0.255570i \(-0.0822630\pi\)
\(978\) 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 \(-0.633649\pi\)
0.407642 0.913142i \(-0.366351\pi\)
\(984\) 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 \(0.216884\pi\)
−0.776716 + 0.629851i \(0.783116\pi\)
\(998\) 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.6349.4 6
5.2 odd 4 1380.2.a.j.1.3 3
5.3 odd 4 6900.2.a.x.1.1 3
5.4 even 2 inner 6900.2.f.r.6349.3 6
15.2 even 4 4140.2.a.s.1.3 3
20.7 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 5.2 odd 4
4140.2.a.s.1.3 3 15.2 even 4
5520.2.a.bv.1.1 3 20.7 even 4
6900.2.a.x.1.1 3 5.3 odd 4
6900.2.f.r.6349.3 6 5.4 even 2 inner
6900.2.f.r.6349.4 6 1.1 even 1 trivial