Properties

Label 1089.2.d.g.1088.8
Level $1089$
Weight $2$
Character 1089.1088
Analytic conductor $8.696$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1089.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.69570878012\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 2 x^{14} - 16 x^{12} - 72 x^{10} + 26 x^{8} + 360 x^{6} + 725 x^{4} + 1000 x^{2} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1088.8
Root \(0.752864 + 0.902863i\) of defining polynomial
Character \(\chi\) \(=\) 1089.1088
Dual form 1089.2.d.g.1088.7

$q$-expansion

\(f(q)\) \(=\) \(q-0.253675 q^{2} -1.93565 q^{4} +2.92173i q^{5} +2.71893i q^{7} +0.998377 q^{8} +O(q^{10})\) \(q-0.253675 q^{2} -1.93565 q^{4} +2.92173i q^{5} +2.71893i q^{7} +0.998377 q^{8} -0.741170i q^{10} +2.85596i q^{13} -0.689726i q^{14} +3.61803 q^{16} +4.46218 q^{17} +1.91115i q^{19} -5.65544i q^{20} -0.816370i q^{23} -3.53649 q^{25} -0.724487i q^{26} -5.26290i q^{28} -9.54660 q^{29} -5.98605 q^{31} -2.91456 q^{32} -1.13195 q^{34} -7.94398 q^{35} +5.94628 q^{37} -0.484812i q^{38} +2.91698i q^{40} -8.43334 q^{41} +11.8763i q^{43} +0.207093i q^{46} -7.71816i q^{47} -0.392594 q^{49} +0.897120 q^{50} -5.52814i q^{52} -10.4556i q^{53} +2.71452i q^{56} +2.42174 q^{58} -0.0932996i q^{59} +2.14405i q^{61} +1.51851 q^{62} -6.49672 q^{64} -8.34434 q^{65} -1.40778 q^{67} -8.63722 q^{68} +2.01519 q^{70} +3.60916i q^{71} -1.75950i q^{73} -1.50842 q^{74} -3.69932i q^{76} +7.13289i q^{79} +10.5709i q^{80} +2.13933 q^{82} -7.83430 q^{83} +13.0373i q^{85} -3.01273i q^{86} +2.06830i q^{89} -7.76517 q^{91} +1.58021i q^{92} +1.95791i q^{94} -5.58387 q^{95} +1.67044 q^{97} +0.0995913 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4} + O(q^{10}) \) \( 16 q + 16 q^{4} + 40 q^{16} - 32 q^{25} + 16 q^{31} + 40 q^{34} + 8 q^{37} + 16 q^{49} + 32 q^{58} - 104 q^{64} + 96 q^{67} - 64 q^{70} + 88 q^{82} + 48 q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.253675 −0.179376 −0.0896878 0.995970i \(-0.528587\pi\)
−0.0896878 + 0.995970i \(0.528587\pi\)
\(3\) 0 0
\(4\) −1.93565 −0.967824
\(5\) 2.92173i 1.30664i 0.757084 + 0.653318i \(0.226624\pi\)
−0.757084 + 0.653318i \(0.773376\pi\)
\(6\) 0 0
\(7\) 2.71893i 1.02766i 0.857892 + 0.513830i \(0.171774\pi\)
−0.857892 + 0.513830i \(0.828226\pi\)
\(8\) 0.998377 0.352980
\(9\) 0 0
\(10\) − 0.741170i − 0.234379i
\(11\) 0 0
\(12\) 0 0
\(13\) 2.85596i 0.792102i 0.918229 + 0.396051i \(0.129620\pi\)
−0.918229 + 0.396051i \(0.870380\pi\)
\(14\) − 0.689726i − 0.184337i
\(15\) 0 0
\(16\) 3.61803 0.904508
\(17\) 4.46218 1.08224 0.541119 0.840946i \(-0.318001\pi\)
0.541119 + 0.840946i \(0.318001\pi\)
\(18\) 0 0
\(19\) 1.91115i 0.438449i 0.975674 + 0.219224i \(0.0703527\pi\)
−0.975674 + 0.219224i \(0.929647\pi\)
\(20\) − 5.65544i − 1.26459i
\(21\) 0 0
\(22\) 0 0
\(23\) − 0.816370i − 0.170225i −0.996371 0.0851124i \(-0.972875\pi\)
0.996371 0.0851124i \(-0.0271249\pi\)
\(24\) 0 0
\(25\) −3.53649 −0.707297
\(26\) − 0.724487i − 0.142084i
\(27\) 0 0
\(28\) − 5.26290i − 0.994594i
\(29\) −9.54660 −1.77276 −0.886380 0.462959i \(-0.846788\pi\)
−0.886380 + 0.462959i \(0.846788\pi\)
\(30\) 0 0
\(31\) −5.98605 −1.07513 −0.537563 0.843224i \(-0.680655\pi\)
−0.537563 + 0.843224i \(0.680655\pi\)
\(32\) −2.91456 −0.515226
\(33\) 0 0
\(34\) −1.13195 −0.194127
\(35\) −7.94398 −1.34278
\(36\) 0 0
\(37\) 5.94628 0.977562 0.488781 0.872407i \(-0.337442\pi\)
0.488781 + 0.872407i \(0.337442\pi\)
\(38\) − 0.484812i − 0.0786470i
\(39\) 0 0
\(40\) 2.91698i 0.461216i
\(41\) −8.43334 −1.31707 −0.658533 0.752552i \(-0.728823\pi\)
−0.658533 + 0.752552i \(0.728823\pi\)
\(42\) 0 0
\(43\) 11.8763i 1.81112i 0.424214 + 0.905562i \(0.360550\pi\)
−0.424214 + 0.905562i \(0.639450\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0.207093i 0.0305342i
\(47\) − 7.71816i − 1.12581i −0.826522 0.562905i \(-0.809684\pi\)
0.826522 0.562905i \(-0.190316\pi\)
\(48\) 0 0
\(49\) −0.392594 −0.0560848
\(50\) 0.897120 0.126872
\(51\) 0 0
\(52\) − 5.52814i − 0.766615i
\(53\) − 10.4556i − 1.43618i −0.695948 0.718092i \(-0.745016\pi\)
0.695948 0.718092i \(-0.254984\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.71452i 0.362743i
\(57\) 0 0
\(58\) 2.42174 0.317990
\(59\) − 0.0932996i − 0.0121466i −0.999982 0.00607329i \(-0.998067\pi\)
0.999982 0.00607329i \(-0.00193320\pi\)
\(60\) 0 0
\(61\) 2.14405i 0.274517i 0.990535 + 0.137259i \(0.0438291\pi\)
−0.990535 + 0.137259i \(0.956171\pi\)
\(62\) 1.51851 0.192851
\(63\) 0 0
\(64\) −6.49672 −0.812089
\(65\) −8.34434 −1.03499
\(66\) 0 0
\(67\) −1.40778 −0.171988 −0.0859941 0.996296i \(-0.527407\pi\)
−0.0859941 + 0.996296i \(0.527407\pi\)
\(68\) −8.63722 −1.04742
\(69\) 0 0
\(70\) 2.01519 0.240861
\(71\) 3.60916i 0.428328i 0.976798 + 0.214164i \(0.0687028\pi\)
−0.976798 + 0.214164i \(0.931297\pi\)
\(72\) 0 0
\(73\) − 1.75950i − 0.205933i −0.994685 0.102967i \(-0.967167\pi\)
0.994685 0.102967i \(-0.0328335\pi\)
\(74\) −1.50842 −0.175351
\(75\) 0 0
\(76\) − 3.69932i − 0.424341i
\(77\) 0 0
\(78\) 0 0
\(79\) 7.13289i 0.802512i 0.915966 + 0.401256i \(0.131426\pi\)
−0.915966 + 0.401256i \(0.868574\pi\)
\(80\) 10.5709i 1.18186i
\(81\) 0 0
\(82\) 2.13933 0.236249
\(83\) −7.83430 −0.859926 −0.429963 0.902847i \(-0.641473\pi\)
−0.429963 + 0.902847i \(0.641473\pi\)
\(84\) 0 0
\(85\) 13.0373i 1.41409i
\(86\) − 3.01273i − 0.324871i
\(87\) 0 0
\(88\) 0 0
\(89\) 2.06830i 0.219240i 0.993974 + 0.109620i \(0.0349633\pi\)
−0.993974 + 0.109620i \(0.965037\pi\)
\(90\) 0 0
\(91\) −7.76517 −0.814011
\(92\) 1.58021i 0.164748i
\(93\) 0 0
\(94\) 1.95791i 0.201943i
\(95\) −5.58387 −0.572893
\(96\) 0 0
\(97\) 1.67044 0.169607 0.0848036 0.996398i \(-0.472974\pi\)
0.0848036 + 0.996398i \(0.472974\pi\)
\(98\) 0.0995913 0.0100602
\(99\) 0 0
\(100\) 6.84540 0.684540
\(101\) 9.04767 0.900277 0.450139 0.892959i \(-0.351375\pi\)
0.450139 + 0.892959i \(0.351375\pi\)
\(102\) 0 0
\(103\) −16.5729 −1.63298 −0.816490 0.577360i \(-0.804083\pi\)
−0.816490 + 0.577360i \(0.804083\pi\)
\(104\) 2.85133i 0.279596i
\(105\) 0 0
\(106\) 2.65232i 0.257616i
\(107\) 7.11298 0.687638 0.343819 0.939036i \(-0.388279\pi\)
0.343819 + 0.939036i \(0.388279\pi\)
\(108\) 0 0
\(109\) − 8.62045i − 0.825690i −0.910801 0.412845i \(-0.864535\pi\)
0.910801 0.412845i \(-0.135465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 9.83719i 0.929527i
\(113\) − 11.4079i − 1.07316i −0.843848 0.536582i \(-0.819715\pi\)
0.843848 0.536582i \(-0.180285\pi\)
\(114\) 0 0
\(115\) 2.38521 0.222422
\(116\) 18.4789 1.71572
\(117\) 0 0
\(118\) 0.0236678i 0.00217880i
\(119\) 12.1324i 1.11217i
\(120\) 0 0
\(121\) 0 0
\(122\) − 0.543892i − 0.0492417i
\(123\) 0 0
\(124\) 11.5869 1.04053
\(125\) 4.27598i 0.382456i
\(126\) 0 0
\(127\) − 2.88308i − 0.255832i −0.991785 0.127916i \(-0.959171\pi\)
0.991785 0.127916i \(-0.0408288\pi\)
\(128\) 7.47718 0.660895
\(129\) 0 0
\(130\) 2.11675 0.185652
\(131\) 2.10155 0.183613 0.0918067 0.995777i \(-0.470736\pi\)
0.0918067 + 0.995777i \(0.470736\pi\)
\(132\) 0 0
\(133\) −5.19630 −0.450576
\(134\) 0.357120 0.0308505
\(135\) 0 0
\(136\) 4.45494 0.382008
\(137\) 5.07404i 0.433504i 0.976227 + 0.216752i \(0.0695464\pi\)
−0.976227 + 0.216752i \(0.930454\pi\)
\(138\) 0 0
\(139\) 17.4839i 1.48297i 0.670970 + 0.741484i \(0.265878\pi\)
−0.670970 + 0.741484i \(0.734122\pi\)
\(140\) 15.3768 1.29957
\(141\) 0 0
\(142\) − 0.915555i − 0.0768317i
\(143\) 0 0
\(144\) 0 0
\(145\) − 27.8926i − 2.31635i
\(146\) 0.446341i 0.0369394i
\(147\) 0 0
\(148\) −11.5099 −0.946108
\(149\) −5.65654 −0.463402 −0.231701 0.972787i \(-0.574429\pi\)
−0.231701 + 0.972787i \(0.574429\pi\)
\(150\) 0 0
\(151\) − 9.17548i − 0.746690i −0.927693 0.373345i \(-0.878211\pi\)
0.927693 0.373345i \(-0.121789\pi\)
\(152\) 1.90805i 0.154763i
\(153\) 0 0
\(154\) 0 0
\(155\) − 17.4896i − 1.40480i
\(156\) 0 0
\(157\) −3.05372 −0.243714 −0.121857 0.992548i \(-0.538885\pi\)
−0.121857 + 0.992548i \(0.538885\pi\)
\(158\) − 1.80944i − 0.143951i
\(159\) 0 0
\(160\) − 8.51555i − 0.673213i
\(161\) 2.21965 0.174933
\(162\) 0 0
\(163\) 7.90450 0.619128 0.309564 0.950879i \(-0.399817\pi\)
0.309564 + 0.950879i \(0.399817\pi\)
\(164\) 16.3240 1.27469
\(165\) 0 0
\(166\) 1.98737 0.154250
\(167\) 16.4478 1.27277 0.636384 0.771372i \(-0.280429\pi\)
0.636384 + 0.771372i \(0.280429\pi\)
\(168\) 0 0
\(169\) 4.84347 0.372575
\(170\) − 3.30724i − 0.253653i
\(171\) 0 0
\(172\) − 22.9884i − 1.75285i
\(173\) −15.5224 −1.18015 −0.590074 0.807349i \(-0.700901\pi\)
−0.590074 + 0.807349i \(0.700901\pi\)
\(174\) 0 0
\(175\) − 9.61547i − 0.726861i
\(176\) 0 0
\(177\) 0 0
\(178\) − 0.524677i − 0.0393262i
\(179\) 5.18936i 0.387871i 0.981014 + 0.193936i \(0.0621253\pi\)
−0.981014 + 0.193936i \(0.937875\pi\)
\(180\) 0 0
\(181\) 5.31229 0.394859 0.197430 0.980317i \(-0.436741\pi\)
0.197430 + 0.980317i \(0.436741\pi\)
\(182\) 1.96983 0.146014
\(183\) 0 0
\(184\) − 0.815045i − 0.0600859i
\(185\) 17.3734i 1.27732i
\(186\) 0 0
\(187\) 0 0
\(188\) 14.9396i 1.08959i
\(189\) 0 0
\(190\) 1.41649 0.102763
\(191\) − 10.8996i − 0.788669i −0.918967 0.394335i \(-0.870975\pi\)
0.918967 0.394335i \(-0.129025\pi\)
\(192\) 0 0
\(193\) 15.3664i 1.10610i 0.833148 + 0.553050i \(0.186536\pi\)
−0.833148 + 0.553050i \(0.813464\pi\)
\(194\) −0.423749 −0.0304234
\(195\) 0 0
\(196\) 0.759923 0.0542802
\(197\) 7.24149 0.515935 0.257967 0.966154i \(-0.416947\pi\)
0.257967 + 0.966154i \(0.416947\pi\)
\(198\) 0 0
\(199\) −11.3726 −0.806181 −0.403090 0.915160i \(-0.632064\pi\)
−0.403090 + 0.915160i \(0.632064\pi\)
\(200\) −3.53075 −0.249662
\(201\) 0 0
\(202\) −2.29517 −0.161488
\(203\) − 25.9566i − 1.82179i
\(204\) 0 0
\(205\) − 24.6399i − 1.72093i
\(206\) 4.20414 0.292917
\(207\) 0 0
\(208\) 10.3330i 0.716463i
\(209\) 0 0
\(210\) 0 0
\(211\) 3.69822i 0.254596i 0.991864 + 0.127298i \(0.0406305\pi\)
−0.991864 + 0.127298i \(0.959369\pi\)
\(212\) 20.2383i 1.38997i
\(213\) 0 0
\(214\) −1.80439 −0.123345
\(215\) −34.6994 −2.36648
\(216\) 0 0
\(217\) − 16.2757i − 1.10486i
\(218\) 2.18680i 0.148109i
\(219\) 0 0
\(220\) 0 0
\(221\) 12.7438i 0.857243i
\(222\) 0 0
\(223\) −0.283223 −0.0189660 −0.00948302 0.999955i \(-0.503019\pi\)
−0.00948302 + 0.999955i \(0.503019\pi\)
\(224\) − 7.92449i − 0.529477i
\(225\) 0 0
\(226\) 2.89390i 0.192499i
\(227\) 18.2560 1.21169 0.605846 0.795582i \(-0.292835\pi\)
0.605846 + 0.795582i \(0.292835\pi\)
\(228\) 0 0
\(229\) 26.3758 1.74296 0.871482 0.490428i \(-0.163160\pi\)
0.871482 + 0.490428i \(0.163160\pi\)
\(230\) −0.605069 −0.0398971
\(231\) 0 0
\(232\) −9.53111 −0.625748
\(233\) 10.8022 0.707675 0.353838 0.935307i \(-0.384877\pi\)
0.353838 + 0.935307i \(0.384877\pi\)
\(234\) 0 0
\(235\) 22.5504 1.47102
\(236\) 0.180595i 0.0117558i
\(237\) 0 0
\(238\) − 3.07768i − 0.199497i
\(239\) −7.42332 −0.480175 −0.240087 0.970751i \(-0.577176\pi\)
−0.240087 + 0.970751i \(0.577176\pi\)
\(240\) 0 0
\(241\) − 2.66873i − 0.171908i −0.996299 0.0859539i \(-0.972606\pi\)
0.996299 0.0859539i \(-0.0273938\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) − 4.15012i − 0.265684i
\(245\) − 1.14705i − 0.0732824i
\(246\) 0 0
\(247\) −5.45818 −0.347296
\(248\) −5.97633 −0.379498
\(249\) 0 0
\(250\) − 1.08471i − 0.0686032i
\(251\) 24.2656i 1.53163i 0.643060 + 0.765816i \(0.277665\pi\)
−0.643060 + 0.765816i \(0.722335\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0.731367i 0.0458901i
\(255\) 0 0
\(256\) 11.0967 0.693541
\(257\) 16.7561i 1.04522i 0.852573 + 0.522608i \(0.175041\pi\)
−0.852573 + 0.522608i \(0.824959\pi\)
\(258\) 0 0
\(259\) 16.1675i 1.00460i
\(260\) 16.1517 1.00169
\(261\) 0 0
\(262\) −0.533112 −0.0329357
\(263\) −10.9146 −0.673021 −0.336511 0.941680i \(-0.609247\pi\)
−0.336511 + 0.941680i \(0.609247\pi\)
\(264\) 0 0
\(265\) 30.5484 1.87657
\(266\) 1.31817 0.0808223
\(267\) 0 0
\(268\) 2.72498 0.166454
\(269\) − 15.6867i − 0.956435i −0.878241 0.478218i \(-0.841283\pi\)
0.878241 0.478218i \(-0.158717\pi\)
\(270\) 0 0
\(271\) − 2.90507i − 0.176471i −0.996100 0.0882353i \(-0.971877\pi\)
0.996100 0.0882353i \(-0.0281227\pi\)
\(272\) 16.1443 0.978894
\(273\) 0 0
\(274\) − 1.28716i − 0.0777601i
\(275\) 0 0
\(276\) 0 0
\(277\) − 21.1150i − 1.26868i −0.773056 0.634338i \(-0.781273\pi\)
0.773056 0.634338i \(-0.218727\pi\)
\(278\) − 4.43524i − 0.266008i
\(279\) 0 0
\(280\) −7.93109 −0.473973
\(281\) −3.09036 −0.184355 −0.0921777 0.995743i \(-0.529383\pi\)
−0.0921777 + 0.995743i \(0.529383\pi\)
\(282\) 0 0
\(283\) 22.4247i 1.33301i 0.745500 + 0.666505i \(0.232211\pi\)
−0.745500 + 0.666505i \(0.767789\pi\)
\(284\) − 6.98607i − 0.414547i
\(285\) 0 0
\(286\) 0 0
\(287\) − 22.9297i − 1.35350i
\(288\) 0 0
\(289\) 2.91107 0.171239
\(290\) 7.07565i 0.415497i
\(291\) 0 0
\(292\) 3.40576i 0.199307i
\(293\) 20.7409 1.21170 0.605849 0.795579i \(-0.292833\pi\)
0.605849 + 0.795579i \(0.292833\pi\)
\(294\) 0 0
\(295\) 0.272596 0.0158712
\(296\) 5.93663 0.345059
\(297\) 0 0
\(298\) 1.43492 0.0831229
\(299\) 2.33152 0.134835
\(300\) 0 0
\(301\) −32.2909 −1.86122
\(302\) 2.32759i 0.133938i
\(303\) 0 0
\(304\) 6.91462i 0.396581i
\(305\) −6.26432 −0.358694
\(306\) 0 0
\(307\) 15.0077i 0.856533i 0.903652 + 0.428267i \(0.140876\pi\)
−0.903652 + 0.428267i \(0.859124\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 4.43668i 0.251986i
\(311\) 2.05901i 0.116756i 0.998295 + 0.0583780i \(0.0185929\pi\)
−0.998295 + 0.0583780i \(0.981407\pi\)
\(312\) 0 0
\(313\) −24.8725 −1.40588 −0.702939 0.711250i \(-0.748130\pi\)
−0.702939 + 0.711250i \(0.748130\pi\)
\(314\) 0.774654 0.0437163
\(315\) 0 0
\(316\) − 13.8068i − 0.776691i
\(317\) − 0.432728i − 0.0243044i −0.999926 0.0121522i \(-0.996132\pi\)
0.999926 0.0121522i \(-0.00386827\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) − 18.9816i − 1.06111i
\(321\) 0 0
\(322\) −0.563072 −0.0313787
\(323\) 8.52791i 0.474506i
\(324\) 0 0
\(325\) − 10.1001i − 0.560251i
\(326\) −2.00518 −0.111056
\(327\) 0 0
\(328\) −8.41965 −0.464897
\(329\) 20.9852 1.15695
\(330\) 0 0
\(331\) −17.3090 −0.951391 −0.475696 0.879610i \(-0.657804\pi\)
−0.475696 + 0.879610i \(0.657804\pi\)
\(332\) 15.1644 0.832257
\(333\) 0 0
\(334\) −4.17240 −0.228304
\(335\) − 4.11316i − 0.224726i
\(336\) 0 0
\(337\) 25.0649i 1.36537i 0.730712 + 0.682686i \(0.239188\pi\)
−0.730712 + 0.682686i \(0.760812\pi\)
\(338\) −1.22867 −0.0668308
\(339\) 0 0
\(340\) − 25.2356i − 1.36859i
\(341\) 0 0
\(342\) 0 0
\(343\) 17.9651i 0.970024i
\(344\) 11.8571i 0.639290i
\(345\) 0 0
\(346\) 3.93765 0.211690
\(347\) 20.8730 1.12052 0.560260 0.828317i \(-0.310701\pi\)
0.560260 + 0.828317i \(0.310701\pi\)
\(348\) 0 0
\(349\) 18.0205i 0.964617i 0.876002 + 0.482308i \(0.160202\pi\)
−0.876002 + 0.482308i \(0.839798\pi\)
\(350\) 2.43921i 0.130381i
\(351\) 0 0
\(352\) 0 0
\(353\) 16.2953i 0.867313i 0.901078 + 0.433656i \(0.142777\pi\)
−0.901078 + 0.433656i \(0.857223\pi\)
\(354\) 0 0
\(355\) −10.5450 −0.559669
\(356\) − 4.00351i − 0.212185i
\(357\) 0 0
\(358\) − 1.31641i − 0.0695746i
\(359\) −24.7842 −1.30806 −0.654031 0.756468i \(-0.726923\pi\)
−0.654031 + 0.756468i \(0.726923\pi\)
\(360\) 0 0
\(361\) 15.3475 0.807763
\(362\) −1.34760 −0.0708281
\(363\) 0 0
\(364\) 15.0306 0.787820
\(365\) 5.14076 0.269080
\(366\) 0 0
\(367\) 8.95366 0.467377 0.233689 0.972311i \(-0.424920\pi\)
0.233689 + 0.972311i \(0.424920\pi\)
\(368\) − 2.95365i − 0.153970i
\(369\) 0 0
\(370\) − 4.40720i − 0.229120i
\(371\) 28.4280 1.47591
\(372\) 0 0
\(373\) − 16.9007i − 0.875088i −0.899197 0.437544i \(-0.855848\pi\)
0.899197 0.437544i \(-0.144152\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) − 7.70564i − 0.397388i
\(377\) − 27.2647i − 1.40421i
\(378\) 0 0
\(379\) 19.7308 1.01350 0.506751 0.862092i \(-0.330846\pi\)
0.506751 + 0.862092i \(0.330846\pi\)
\(380\) 10.8084 0.554460
\(381\) 0 0
\(382\) 2.76497i 0.141468i
\(383\) − 4.99993i − 0.255485i −0.991807 0.127742i \(-0.959227\pi\)
0.991807 0.127742i \(-0.0407730\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 3.89808i − 0.198407i
\(387\) 0 0
\(388\) −3.23338 −0.164150
\(389\) − 9.85341i − 0.499588i −0.968299 0.249794i \(-0.919637\pi\)
0.968299 0.249794i \(-0.0803629\pi\)
\(390\) 0 0
\(391\) − 3.64279i − 0.184224i
\(392\) −0.391956 −0.0197968
\(393\) 0 0
\(394\) −1.83699 −0.0925461
\(395\) −20.8403 −1.04859
\(396\) 0 0
\(397\) −19.0245 −0.954812 −0.477406 0.878683i \(-0.658423\pi\)
−0.477406 + 0.878683i \(0.658423\pi\)
\(398\) 2.88494 0.144609
\(399\) 0 0
\(400\) −12.7951 −0.639757
\(401\) 30.7055i 1.53336i 0.642030 + 0.766680i \(0.278093\pi\)
−0.642030 + 0.766680i \(0.721907\pi\)
\(402\) 0 0
\(403\) − 17.0959i − 0.851609i
\(404\) −17.5131 −0.871310
\(405\) 0 0
\(406\) 6.58454i 0.326785i
\(407\) 0 0
\(408\) 0 0
\(409\) 11.8735i 0.587105i 0.955943 + 0.293552i \(0.0948375\pi\)
−0.955943 + 0.293552i \(0.905162\pi\)
\(410\) 6.25054i 0.308692i
\(411\) 0 0
\(412\) 32.0794 1.58044
\(413\) 0.253675 0.0124825
\(414\) 0 0
\(415\) − 22.8897i − 1.12361i
\(416\) − 8.32388i − 0.408112i
\(417\) 0 0
\(418\) 0 0
\(419\) 10.0629i 0.491606i 0.969320 + 0.245803i \(0.0790516\pi\)
−0.969320 + 0.245803i \(0.920948\pi\)
\(420\) 0 0
\(421\) −6.25651 −0.304923 −0.152462 0.988309i \(-0.548720\pi\)
−0.152462 + 0.988309i \(0.548720\pi\)
\(422\) − 0.938148i − 0.0456684i
\(423\) 0 0
\(424\) − 10.4386i − 0.506944i
\(425\) −15.7804 −0.765464
\(426\) 0 0
\(427\) −5.82952 −0.282110
\(428\) −13.7682 −0.665512
\(429\) 0 0
\(430\) 8.80238 0.424489
\(431\) −19.6392 −0.945988 −0.472994 0.881066i \(-0.656827\pi\)
−0.472994 + 0.881066i \(0.656827\pi\)
\(432\) 0 0
\(433\) 31.2889 1.50365 0.751825 0.659362i \(-0.229174\pi\)
0.751825 + 0.659362i \(0.229174\pi\)
\(434\) 4.12873i 0.198186i
\(435\) 0 0
\(436\) 16.6862i 0.799123i
\(437\) 1.56021 0.0746349
\(438\) 0 0
\(439\) 20.1935i 0.963782i 0.876231 + 0.481891i \(0.160050\pi\)
−0.876231 + 0.481891i \(0.839950\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 3.23279i − 0.153768i
\(443\) 24.3679i 1.15775i 0.815415 + 0.578876i \(0.196509\pi\)
−0.815415 + 0.578876i \(0.803491\pi\)
\(444\) 0 0
\(445\) −6.04301 −0.286466
\(446\) 0.0718468 0.00340205
\(447\) 0 0
\(448\) − 17.6641i − 0.834552i
\(449\) 3.30192i 0.155827i 0.996960 + 0.0779136i \(0.0248258\pi\)
−0.996960 + 0.0779136i \(0.975174\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 22.0817i 1.03863i
\(453\) 0 0
\(454\) −4.63109 −0.217348
\(455\) − 22.6877i − 1.06362i
\(456\) 0 0
\(457\) − 15.6174i − 0.730551i −0.930899 0.365276i \(-0.880975\pi\)
0.930899 0.365276i \(-0.119025\pi\)
\(458\) −6.69090 −0.312645
\(459\) 0 0
\(460\) −4.61693 −0.215265
\(461\) 36.5685 1.70316 0.851582 0.524221i \(-0.175644\pi\)
0.851582 + 0.524221i \(0.175644\pi\)
\(462\) 0 0
\(463\) 4.45531 0.207056 0.103528 0.994627i \(-0.466987\pi\)
0.103528 + 0.994627i \(0.466987\pi\)
\(464\) −34.5399 −1.60348
\(465\) 0 0
\(466\) −2.74025 −0.126940
\(467\) 41.6268i 1.92626i 0.269037 + 0.963130i \(0.413294\pi\)
−0.269037 + 0.963130i \(0.586706\pi\)
\(468\) 0 0
\(469\) − 3.82767i − 0.176745i
\(470\) −5.72047 −0.263866
\(471\) 0 0
\(472\) − 0.0931482i − 0.00428749i
\(473\) 0 0
\(474\) 0 0
\(475\) − 6.75877i − 0.310114i
\(476\) − 23.4840i − 1.07639i
\(477\) 0 0
\(478\) 1.88311 0.0861316
\(479\) 3.52150 0.160902 0.0804508 0.996759i \(-0.474364\pi\)
0.0804508 + 0.996759i \(0.474364\pi\)
\(480\) 0 0
\(481\) 16.9823i 0.774328i
\(482\) 0.676990i 0.0308360i
\(483\) 0 0
\(484\) 0 0
\(485\) 4.88056i 0.221615i
\(486\) 0 0
\(487\) 31.9004 1.44554 0.722772 0.691087i \(-0.242868\pi\)
0.722772 + 0.691087i \(0.242868\pi\)
\(488\) 2.14057i 0.0968990i
\(489\) 0 0
\(490\) 0.290979i 0.0131451i
\(491\) −24.4192 −1.10202 −0.551012 0.834497i \(-0.685758\pi\)
−0.551012 + 0.834497i \(0.685758\pi\)
\(492\) 0 0
\(493\) −42.5987 −1.91855
\(494\) 1.38461 0.0622964
\(495\) 0 0
\(496\) −21.6577 −0.972460
\(497\) −9.81306 −0.440176
\(498\) 0 0
\(499\) −23.2692 −1.04167 −0.520836 0.853657i \(-0.674380\pi\)
−0.520836 + 0.853657i \(0.674380\pi\)
\(500\) − 8.27680i − 0.370150i
\(501\) 0 0
\(502\) − 6.15559i − 0.274737i
\(503\) −36.5209 −1.62839 −0.814193 0.580595i \(-0.802820\pi\)
−0.814193 + 0.580595i \(0.802820\pi\)
\(504\) 0 0
\(505\) 26.4348i 1.17633i
\(506\) 0 0
\(507\) 0 0
\(508\) 5.58064i 0.247601i
\(509\) − 24.8510i − 1.10150i −0.834670 0.550751i \(-0.814341\pi\)
0.834670 0.550751i \(-0.185659\pi\)
\(510\) 0 0
\(511\) 4.78395 0.211629
\(512\) −17.7693 −0.785300
\(513\) 0 0
\(514\) − 4.25060i − 0.187486i
\(515\) − 48.4216i − 2.13371i
\(516\) 0 0
\(517\) 0 0
\(518\) − 4.10130i − 0.180201i
\(519\) 0 0
\(520\) −8.33080 −0.365330
\(521\) − 9.74005i − 0.426719i −0.976974 0.213360i \(-0.931559\pi\)
0.976974 0.213360i \(-0.0684406\pi\)
\(522\) 0 0
\(523\) 33.2649i 1.45457i 0.686335 + 0.727285i \(0.259218\pi\)
−0.686335 + 0.727285i \(0.740782\pi\)
\(524\) −4.06786 −0.177705
\(525\) 0 0
\(526\) 2.76876 0.120724
\(527\) −26.7108 −1.16354
\(528\) 0 0
\(529\) 22.3335 0.971023
\(530\) −7.74936 −0.336611
\(531\) 0 0
\(532\) 10.0582 0.436079
\(533\) − 24.0853i − 1.04325i
\(534\) 0 0
\(535\) 20.7822i 0.898492i
\(536\) −1.40550 −0.0607084
\(537\) 0 0
\(538\) 3.97933i 0.171561i
\(539\) 0 0
\(540\) 0 0
\(541\) 11.6552i 0.501096i 0.968104 + 0.250548i \(0.0806108\pi\)
−0.968104 + 0.250548i \(0.919389\pi\)
\(542\) 0.736945i 0.0316545i
\(543\) 0 0
\(544\) −13.0053 −0.557598
\(545\) 25.1866 1.07888
\(546\) 0 0
\(547\) − 0.812785i − 0.0347522i −0.999849 0.0173761i \(-0.994469\pi\)
0.999849 0.0173761i \(-0.00553126\pi\)
\(548\) − 9.82156i − 0.419556i
\(549\) 0 0
\(550\) 0 0
\(551\) − 18.2450i − 0.777264i
\(552\) 0 0
\(553\) −19.3938 −0.824710
\(554\) 5.35635i 0.227569i
\(555\) 0 0
\(556\) − 33.8428i − 1.43525i
\(557\) −6.95247 −0.294586 −0.147293 0.989093i \(-0.547056\pi\)
−0.147293 + 0.989093i \(0.547056\pi\)
\(558\) 0 0
\(559\) −33.9184 −1.43459
\(560\) −28.7416 −1.21455
\(561\) 0 0
\(562\) 0.783948 0.0330689
\(563\) −28.7576 −1.21199 −0.605994 0.795469i \(-0.707224\pi\)
−0.605994 + 0.795469i \(0.707224\pi\)
\(564\) 0 0
\(565\) 33.3307 1.40223
\(566\) − 5.68859i − 0.239109i
\(567\) 0 0
\(568\) 3.60330i 0.151191i
\(569\) 10.4489 0.438042 0.219021 0.975720i \(-0.429714\pi\)
0.219021 + 0.975720i \(0.429714\pi\)
\(570\) 0 0
\(571\) − 21.2530i − 0.889409i −0.895677 0.444704i \(-0.853309\pi\)
0.895677 0.444704i \(-0.146691\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 5.81669i 0.242784i
\(575\) 2.88708i 0.120400i
\(576\) 0 0
\(577\) 15.6803 0.652779 0.326390 0.945235i \(-0.394168\pi\)
0.326390 + 0.945235i \(0.394168\pi\)
\(578\) −0.738466 −0.0307162
\(579\) 0 0
\(580\) 53.9902i 2.24182i
\(581\) − 21.3009i − 0.883711i
\(582\) 0 0
\(583\) 0 0
\(584\) − 1.75664i − 0.0726903i
\(585\) 0 0
\(586\) −5.26146 −0.217349
\(587\) 20.1556i 0.831912i 0.909385 + 0.415956i \(0.136553\pi\)
−0.909385 + 0.415956i \(0.863447\pi\)
\(588\) 0 0
\(589\) − 11.4403i − 0.471387i
\(590\) −0.0691509 −0.00284690
\(591\) 0 0
\(592\) 21.5138 0.884213
\(593\) −40.5694 −1.66599 −0.832993 0.553284i \(-0.813374\pi\)
−0.832993 + 0.553284i \(0.813374\pi\)
\(594\) 0 0
\(595\) −35.4475 −1.45320
\(596\) 10.9491 0.448491
\(597\) 0 0
\(598\) −0.591450 −0.0241862
\(599\) 10.9375i 0.446894i 0.974716 + 0.223447i \(0.0717309\pi\)
−0.974716 + 0.223447i \(0.928269\pi\)
\(600\) 0 0
\(601\) 16.3562i 0.667182i 0.942718 + 0.333591i \(0.108260\pi\)
−0.942718 + 0.333591i \(0.891740\pi\)
\(602\) 8.19142 0.333857
\(603\) 0 0
\(604\) 17.7605i 0.722665i
\(605\) 0 0
\(606\) 0 0
\(607\) 23.8646i 0.968633i 0.874893 + 0.484317i \(0.160932\pi\)
−0.874893 + 0.484317i \(0.839068\pi\)
\(608\) − 5.57017i − 0.225900i
\(609\) 0 0
\(610\) 1.58910 0.0643410
\(611\) 22.0428 0.891756
\(612\) 0 0
\(613\) − 0.0485285i − 0.00196005i −1.00000 0.000980023i \(-0.999688\pi\)
1.00000 0.000980023i \(-0.000311951\pi\)
\(614\) − 3.80708i − 0.153641i
\(615\) 0 0
\(616\) 0 0
\(617\) 8.24561i 0.331956i 0.986129 + 0.165978i \(0.0530780\pi\)
−0.986129 + 0.165978i \(0.946922\pi\)
\(618\) 0 0
\(619\) −1.73927 −0.0699072 −0.0349536 0.999389i \(-0.511128\pi\)
−0.0349536 + 0.999389i \(0.511128\pi\)
\(620\) 33.8537i 1.35960i
\(621\) 0 0
\(622\) − 0.522321i − 0.0209432i
\(623\) −5.62357 −0.225304
\(624\) 0 0
\(625\) −30.1757 −1.20703
\(626\) 6.30955 0.252180
\(627\) 0 0
\(628\) 5.91094 0.235872
\(629\) 26.5334 1.05795
\(630\) 0 0
\(631\) −26.3475 −1.04888 −0.524439 0.851448i \(-0.675725\pi\)
−0.524439 + 0.851448i \(0.675725\pi\)
\(632\) 7.12131i 0.283270i
\(633\) 0 0
\(634\) 0.109772i 0.00435962i
\(635\) 8.42358 0.334280
\(636\) 0 0
\(637\) − 1.12123i − 0.0444249i
\(638\) 0 0
\(639\) 0 0
\(640\) 21.8463i 0.863550i
\(641\) − 8.69577i − 0.343462i −0.985144 0.171731i \(-0.945064\pi\)
0.985144 0.171731i \(-0.0549361\pi\)
\(642\) 0 0
\(643\) −21.6159 −0.852450 −0.426225 0.904617i \(-0.640157\pi\)
−0.426225 + 0.904617i \(0.640157\pi\)
\(644\) −4.29647 −0.169305
\(645\) 0 0
\(646\) − 2.16332i − 0.0851147i
\(647\) − 29.3673i − 1.15455i −0.816551 0.577273i \(-0.804117\pi\)
0.816551 0.577273i \(-0.195883\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 2.56214i 0.100495i
\(651\) 0 0
\(652\) −15.3003 −0.599207
\(653\) 23.6935i 0.927199i 0.886045 + 0.463599i \(0.153442\pi\)
−0.886045 + 0.463599i \(0.846558\pi\)
\(654\) 0 0
\(655\) 6.14016i 0.239916i
\(656\) −30.5121 −1.19130
\(657\) 0 0
\(658\) −5.32342 −0.207528
\(659\) −34.5941 −1.34759 −0.673797 0.738916i \(-0.735338\pi\)
−0.673797 + 0.738916i \(0.735338\pi\)
\(660\) 0 0
\(661\) 35.4913 1.38045 0.690225 0.723595i \(-0.257512\pi\)
0.690225 + 0.723595i \(0.257512\pi\)
\(662\) 4.39088 0.170656
\(663\) 0 0
\(664\) −7.82158 −0.303536
\(665\) − 15.1822i − 0.588739i
\(666\) 0 0
\(667\) 7.79356i 0.301768i
\(668\) −31.8372 −1.23182
\(669\) 0 0
\(670\) 1.04341i 0.0403104i
\(671\) 0 0
\(672\) 0 0
\(673\) 20.9013i 0.805684i 0.915270 + 0.402842i \(0.131978\pi\)
−0.915270 + 0.402842i \(0.868022\pi\)
\(674\) − 6.35835i − 0.244914i
\(675\) 0 0
\(676\) −9.37527 −0.360587
\(677\) 24.2380 0.931543 0.465772 0.884905i \(-0.345777\pi\)
0.465772 + 0.884905i \(0.345777\pi\)
\(678\) 0 0
\(679\) 4.54181i 0.174299i
\(680\) 13.0161i 0.499145i
\(681\) 0 0
\(682\) 0 0
\(683\) − 17.6311i − 0.674636i −0.941391 0.337318i \(-0.890480\pi\)
0.941391 0.337318i \(-0.109520\pi\)
\(684\) 0 0
\(685\) −14.8250 −0.566433
\(686\) − 4.55730i − 0.173999i
\(687\) 0 0
\(688\) 42.9690i 1.63818i
\(689\) 29.8608 1.13760
\(690\) 0 0
\(691\) 29.1363 1.10840 0.554199 0.832384i \(-0.313024\pi\)
0.554199 + 0.832384i \(0.313024\pi\)
\(692\) 30.0459 1.14218
\(693\) 0 0
\(694\) −5.29496 −0.200994
\(695\) −51.0833 −1.93770
\(696\) 0 0
\(697\) −37.6311 −1.42538
\(698\) − 4.57136i − 0.173029i
\(699\) 0 0
\(700\) 18.6122i 0.703474i
\(701\) 14.7015 0.555268 0.277634 0.960687i \(-0.410450\pi\)
0.277634 + 0.960687i \(0.410450\pi\)
\(702\) 0 0
\(703\) 11.3642i 0.428611i
\(704\) 0 0
\(705\) 0 0
\(706\) − 4.13372i − 0.155575i
\(707\) 24.6000i 0.925179i
\(708\) 0 0
\(709\) 10.7824 0.404940 0.202470 0.979288i \(-0.435103\pi\)
0.202470 + 0.979288i \(0.435103\pi\)
\(710\) 2.67500 0.100391
\(711\) 0 0
\(712\) 2.06495i 0.0773871i
\(713\) 4.88683i 0.183013i
\(714\) 0 0
\(715\) 0 0
\(716\) − 10.0448i − 0.375391i
\(717\) 0 0
\(718\) 6.28715 0.234634
\(719\) − 31.7500i − 1.18407i −0.805911 0.592037i \(-0.798324\pi\)
0.805911 0.592037i \(-0.201676\pi\)
\(720\) 0 0
\(721\) − 45.0607i − 1.67815i
\(722\) −3.89328 −0.144893
\(723\) 0 0
\(724\) −10.2827 −0.382154
\(725\) 33.7614 1.25387
\(726\) 0 0
\(727\) 23.6418 0.876828 0.438414 0.898773i \(-0.355540\pi\)
0.438414 + 0.898773i \(0.355540\pi\)
\(728\) −7.75257 −0.287329
\(729\) 0 0
\(730\) −1.30409 −0.0482664
\(731\) 52.9944i 1.96007i
\(732\) 0 0
\(733\) 16.4244i 0.606651i 0.952887 + 0.303325i \(0.0980969\pi\)
−0.952887 + 0.303325i \(0.901903\pi\)
\(734\) −2.27132 −0.0838361
\(735\) 0 0
\(736\) 2.37936i 0.0877043i
\(737\) 0 0
\(738\) 0 0
\(739\) − 34.8449i − 1.28179i −0.767629 0.640895i \(-0.778563\pi\)
0.767629 0.640895i \(-0.221437\pi\)
\(740\) − 33.6288i − 1.23622i
\(741\) 0 0
\(742\) −7.21149 −0.264742
\(743\) −52.7926 −1.93677 −0.968386 0.249458i \(-0.919748\pi\)
−0.968386 + 0.249458i \(0.919748\pi\)
\(744\) 0 0
\(745\) − 16.5269i − 0.605497i
\(746\) 4.28730i 0.156969i
\(747\) 0 0
\(748\) 0 0
\(749\) 19.3397i 0.706657i
\(750\) 0 0
\(751\) 51.3777 1.87480 0.937400 0.348255i \(-0.113226\pi\)
0.937400 + 0.348255i \(0.113226\pi\)
\(752\) − 27.9246i − 1.01830i
\(753\) 0 0
\(754\) 6.91639i 0.251880i
\(755\) 26.8083 0.975652
\(756\) 0 0
\(757\) −7.06878 −0.256919 −0.128460 0.991715i \(-0.541003\pi\)
−0.128460 + 0.991715i \(0.541003\pi\)
\(758\) −5.00521 −0.181797
\(759\) 0 0
\(760\) −5.57481 −0.202219
\(761\) 29.1327 1.05606 0.528030 0.849226i \(-0.322931\pi\)
0.528030 + 0.849226i \(0.322931\pi\)
\(762\) 0 0
\(763\) 23.4384 0.848528
\(764\) 21.0979i 0.763294i
\(765\) 0 0
\(766\) 1.26836i 0.0458277i
\(767\) 0.266460 0.00962132
\(768\) 0 0
\(769\) − 44.2408i − 1.59536i −0.603079 0.797682i \(-0.706060\pi\)
0.603079 0.797682i \(-0.293940\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 29.7440i − 1.07051i
\(773\) 40.0382i 1.44007i 0.693936 + 0.720037i \(0.255875\pi\)
−0.693936 + 0.720037i \(0.744125\pi\)
\(774\) 0 0
\(775\) 21.1696 0.760434
\(776\) 1.66773 0.0598679
\(777\) 0 0
\(778\) 2.49957i 0.0896139i
\(779\) − 16.1174i − 0.577466i
\(780\) 0 0
\(781\) 0 0
\(782\) 0.924086i 0.0330453i
\(783\) 0 0
\(784\) −1.42042 −0.0507292
\(785\) − 8.92215i − 0.318445i
\(786\) 0 0
\(787\) − 28.3641i − 1.01107i −0.862806 0.505535i \(-0.831295\pi\)
0.862806 0.505535i \(-0.168705\pi\)
\(788\) −14.0170 −0.499334
\(789\) 0 0
\(790\) 5.28668 0.188092
\(791\) 31.0173 1.10285
\(792\) 0 0
\(793\) −6.12332 −0.217446
\(794\) 4.82605 0.171270
\(795\) 0 0
\(796\) 22.0133 0.780241
\(797\) − 14.9437i − 0.529334i −0.964340 0.264667i \(-0.914738\pi\)
0.964340 0.264667i \(-0.0852621\pi\)
\(798\) 0 0
\(799\) − 34.4398i − 1.21839i
\(800\) 10.3073 0.364418
\(801\) 0 0
\(802\) − 7.78923i − 0.275047i
\(803\) 0 0
\(804\) 0 0
\(805\) 6.48522i 0.228574i
\(806\) 4.33682i 0.152758i
\(807\) 0 0
\(808\) 9.03299 0.317779
\(809\) 29.7759 1.04687 0.523433 0.852067i \(-0.324651\pi\)
0.523433 + 0.852067i \(0.324651\pi\)
\(810\) 0 0
\(811\) − 44.1463i − 1.55019i −0.631846 0.775094i \(-0.717703\pi\)
0.631846 0.775094i \(-0.282297\pi\)
\(812\) 50.2428i 1.76318i
\(813\) 0 0
\(814\) 0 0
\(815\) 23.0948i 0.808975i
\(816\) 0 0
\(817\) −22.6975 −0.794085
\(818\) − 3.01200i − 0.105312i
\(819\) 0 0
\(820\) 47.6942i 1.66555i
\(821\) 4.00495 0.139774 0.0698868 0.997555i \(-0.477736\pi\)
0.0698868 + 0.997555i \(0.477736\pi\)
\(822\) 0 0
\(823\) 42.1823 1.47038 0.735191 0.677860i \(-0.237093\pi\)
0.735191 + 0.677860i \(0.237093\pi\)
\(824\) −16.5460 −0.576409
\(825\) 0 0
\(826\) −0.0643512 −0.00223906
\(827\) −26.5981 −0.924907 −0.462454 0.886643i \(-0.653031\pi\)
−0.462454 + 0.886643i \(0.653031\pi\)
\(828\) 0 0
\(829\) 12.2316 0.424819 0.212410 0.977181i \(-0.431869\pi\)
0.212410 + 0.977181i \(0.431869\pi\)
\(830\) 5.80655i 0.201548i
\(831\) 0 0
\(832\) − 18.5544i − 0.643257i
\(833\) −1.75182 −0.0606971
\(834\) 0 0
\(835\) 48.0560i 1.66305i
\(836\) 0 0
\(837\) 0 0
\(838\) − 2.55271i − 0.0881820i
\(839\) − 9.64647i − 0.333033i −0.986039 0.166517i \(-0.946748\pi\)
0.986039 0.166517i \(-0.0532519\pi\)
\(840\) 0 0
\(841\) 62.1376 2.14268
\(842\) 1.58712 0.0546958
\(843\) 0 0
\(844\) − 7.15846i − 0.246405i
\(845\) 14.1513i 0.486820i
\(846\) 0 0
\(847\) 0 0
\(848\) − 37.8287i − 1.29904i
\(849\) 0 0
\(850\) 4.00311 0.137306
\(851\) − 4.85436i − 0.166405i
\(852\) 0 0
\(853\) − 11.7386i − 0.401921i −0.979599 0.200960i \(-0.935594\pi\)
0.979599 0.200960i \(-0.0644063\pi\)
\(854\) 1.47881 0.0506037
\(855\) 0 0
\(856\) 7.10143 0.242722
\(857\) 48.7738 1.66608 0.833041 0.553212i \(-0.186598\pi\)
0.833041 + 0.553212i \(0.186598\pi\)
\(858\) 0 0
\(859\) −31.0880 −1.06071 −0.530355 0.847776i \(-0.677941\pi\)
−0.530355 + 0.847776i \(0.677941\pi\)
\(860\) 67.1659 2.29034
\(861\) 0 0
\(862\) 4.98199 0.169687
\(863\) − 52.1380i − 1.77480i −0.461002 0.887399i \(-0.652510\pi\)
0.461002 0.887399i \(-0.347490\pi\)
\(864\) 0 0
\(865\) − 45.3522i − 1.54202i
\(866\) −7.93723 −0.269718
\(867\) 0 0
\(868\) 31.5040i 1.06931i
\(869\) 0 0
\(870\) 0 0
\(871\) − 4.02058i − 0.136232i
\(872\) − 8.60646i − 0.291452i
\(873\) 0 0
\(874\) −0.395786 −0.0133877
\(875\) −11.6261 −0.393034
\(876\) 0 0
\(877\) − 16.9878i − 0.573637i −0.957985 0.286819i \(-0.907402\pi\)
0.957985 0.286819i \(-0.0925977\pi\)
\(878\) − 5.12259i − 0.172879i
\(879\) 0 0
\(880\) 0 0
\(881\) 50.3115i 1.69504i 0.530765 + 0.847519i \(0.321905\pi\)
−0.530765 + 0.847519i \(0.678095\pi\)
\(882\) 0 0
\(883\) 14.3693 0.483567 0.241783 0.970330i \(-0.422268\pi\)
0.241783 + 0.970330i \(0.422268\pi\)
\(884\) − 24.6676i − 0.829660i
\(885\) 0 0
\(886\) − 6.18153i − 0.207673i
\(887\) −41.3003 −1.38673 −0.693365 0.720587i \(-0.743873\pi\)
−0.693365 + 0.720587i \(0.743873\pi\)
\(888\) 0 0
\(889\) 7.83891 0.262909
\(890\) 1.53296 0.0513851
\(891\) 0 0
\(892\) 0.548221 0.0183558
\(893\) 14.7506 0.493610
\(894\) 0 0
\(895\) −15.1619 −0.506807
\(896\) 20.3299i 0.679176i
\(897\) 0 0
\(898\) − 0.837616i − 0.0279516i
\(899\) 57.1464 1.90594
\(900\) 0 0
\(901\) − 46.6547i − 1.55429i
\(902\) 0 0
\(903\) 0 0
\(904\) − 11.3894i − 0.378805i
\(905\) 15.5210i 0.515937i
\(906\) 0 0
\(907\) −26.1167 −0.867190 −0.433595 0.901108i \(-0.642755\pi\)
−0.433595 + 0.901108i \(0.642755\pi\)
\(908\) −35.3371 −1.17270
\(909\) 0 0
\(910\) 5.75531i 0.190787i
\(911\) 45.4127i 1.50459i 0.658826 + 0.752296i \(0.271054\pi\)
−0.658826 + 0.752296i \(0.728946\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 3.96175i 0.131043i
\(915\) 0 0
\(916\) −51.0543 −1.68688
\(917\) 5.71398i 0.188692i
\(918\) 0 0
\(919\) − 27.8566i − 0.918905i −0.888202 0.459453i \(-0.848046\pi\)
0.888202 0.459453i \(-0.151954\pi\)
\(920\) 2.38134 0.0785104
\(921\) 0 0
\(922\) −9.27652 −0.305506
\(923\) −10.3076 −0.339280
\(924\) 0 0
\(925\) −21.0289 −0.691427
\(926\) −1.13020 −0.0371408
\(927\) 0 0
\(928\) 27.8241 0.913372
\(929\) − 49.7545i − 1.63239i −0.577774 0.816197i \(-0.696079\pi\)
0.577774 0.816197i \(-0.303921\pi\)
\(930\) 0 0
\(931\) − 0.750306i − 0.0245903i
\(932\) −20.9093 −0.684905
\(933\) 0 0
\(934\) − 10.5597i − 0.345524i
\(935\) 0 0
\(936\) 0 0
\(937\) − 20.4059i − 0.666633i −0.942815 0.333316i \(-0.891832\pi\)
0.942815 0.333316i \(-0.108168\pi\)
\(938\) 0.970986i 0.0317038i
\(939\) 0 0
\(940\) −43.6496 −1.42369
\(941\) 17.5434 0.571898 0.285949 0.958245i \(-0.407691\pi\)
0.285949 + 0.958245i \(0.407691\pi\)
\(942\) 0 0
\(943\) 6.88472i 0.224197i
\(944\) − 0.337561i − 0.0109867i
\(945\) 0 0
\(946\) 0 0
\(947\) − 47.0672i − 1.52948i −0.644340 0.764739i \(-0.722868\pi\)
0.644340 0.764739i \(-0.277132\pi\)
\(948\) 0 0
\(949\) 5.02505 0.163120
\(950\) 1.71453i 0.0556268i
\(951\) 0 0
\(952\) 12.1127i 0.392574i
\(953\) 51.6298 1.67245 0.836227 0.548384i \(-0.184757\pi\)
0.836227 + 0.548384i \(0.184757\pi\)
\(954\) 0 0
\(955\) 31.8457 1.03050
\(956\) 14.3689 0.464725
\(957\) 0 0
\(958\) −0.893318 −0.0288618
\(959\) −13.7960 −0.445495
\(960\) 0 0
\(961\) 4.83277 0.155896
\(962\) − 4.30800i − 0.138896i
\(963\) 0 0
\(964\) 5.16572i 0.166376i
\(965\) −44.8965 −1.44527
\(966\) 0 0
\(967\) − 33.6964i − 1.08360i −0.840506 0.541802i \(-0.817742\pi\)
0.840506 0.541802i \(-0.182258\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) − 1.23808i − 0.0397523i
\(971\) 55.8818i 1.79333i 0.442708 + 0.896666i \(0.354018\pi\)
−0.442708 + 0.896666i \(0.645982\pi\)
\(972\) 0 0
\(973\) −47.5376 −1.52399
\(974\) −8.09233 −0.259295
\(975\) 0 0
\(976\) 7.75724i 0.248303i
\(977\) 20.6593i 0.660949i 0.943815 + 0.330475i \(0.107209\pi\)
−0.943815 + 0.330475i \(0.892791\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 2.22029i 0.0709245i
\(981\) 0 0
\(982\) 6.19455 0.197676
\(983\) 50.1274i 1.59882i 0.600789 + 0.799408i \(0.294853\pi\)
−0.600789 + 0.799408i \(0.705147\pi\)
\(984\) 0 0
\(985\) 21.1577i 0.674139i
\(986\) 10.8062 0.344141
\(987\) 0 0
\(988\) 10.5651 0.336121
\(989\) 9.69548 0.308298
\(990\) 0 0
\(991\) 45.3068 1.43922 0.719609 0.694379i \(-0.244321\pi\)
0.719609 + 0.694379i \(0.244321\pi\)
\(992\) 17.4467 0.553933
\(993\) 0 0
\(994\) 2.48933 0.0789568
\(995\) − 33.2276i − 1.05338i
\(996\) 0 0
\(997\) 10.2799i 0.325567i 0.986662 + 0.162784i \(0.0520473\pi\)
−0.986662 + 0.162784i \(0.947953\pi\)
\(998\) 5.90282 0.186850
\(999\) 0 0
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 1089.2.d.g.1088.8 16
3.2 odd 2 inner 1089.2.d.g.1088.9 16
11.2 odd 10 99.2.j.a.62.2 yes 16
11.5 even 5 99.2.j.a.8.3 yes 16
11.10 odd 2 inner 1089.2.d.g.1088.10 16
33.2 even 10 99.2.j.a.62.3 yes 16
33.5 odd 10 99.2.j.a.8.2 16
33.32 even 2 inner 1089.2.d.g.1088.7 16
44.27 odd 10 1584.2.cd.c.305.3 16
44.35 even 10 1584.2.cd.c.161.2 16
99.2 even 30 891.2.u.c.458.3 32
99.5 odd 30 891.2.u.c.107.2 32
99.13 odd 30 891.2.u.c.755.3 32
99.16 even 15 891.2.u.c.701.2 32
99.38 odd 30 891.2.u.c.701.3 32
99.49 even 15 891.2.u.c.107.3 32
99.68 even 30 891.2.u.c.755.2 32
99.79 odd 30 891.2.u.c.458.2 32
132.35 odd 10 1584.2.cd.c.161.3 16
132.71 even 10 1584.2.cd.c.305.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.j.a.8.2 16 33.5 odd 10
99.2.j.a.8.3 yes 16 11.5 even 5
99.2.j.a.62.2 yes 16 11.2 odd 10
99.2.j.a.62.3 yes 16 33.2 even 10
891.2.u.c.107.2 32 99.5 odd 30
891.2.u.c.107.3 32 99.49 even 15
891.2.u.c.458.2 32 99.79 odd 30
891.2.u.c.458.3 32 99.2 even 30
891.2.u.c.701.2 32 99.16 even 15
891.2.u.c.701.3 32 99.38 odd 30
891.2.u.c.755.2 32 99.68 even 30
891.2.u.c.755.3 32 99.13 odd 30
1089.2.d.g.1088.7 16 33.32 even 2 inner
1089.2.d.g.1088.8 16 1.1 even 1 trivial
1089.2.d.g.1088.9 16 3.2 odd 2 inner
1089.2.d.g.1088.10 16 11.10 odd 2 inner
1584.2.cd.c.161.2 16 44.35 even 10
1584.2.cd.c.161.3 16 132.35 odd 10
1584.2.cd.c.305.2 16 132.71 even 10
1584.2.cd.c.305.3 16 44.27 odd 10