Properties

Label 1323.2.c.d.1322.9
Level $1323$
Weight $2$
Character 1323.1322
Analytic conductor $10.564$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 9 x^{10} + 59 x^{8} - 180 x^{6} + 403 x^{4} - 198 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1322.9
Root \(1.90412 - 1.09935i\) of defining polynomial
Character \(\chi\) \(=\) 1323.1322
Dual form 1323.2.c.d.1322.3

$q$-expansion

\(f(q)\) \(=\) \(q+1.83424i q^{2} -1.36445 q^{4} -3.80824 q^{5} +1.16576i q^{8} +O(q^{10})\) \(q+1.83424i q^{2} -1.36445 q^{4} -3.80824 q^{5} +1.16576i q^{8} -6.98525i q^{10} -1.16576i q^{11} -5.54030i q^{13} -4.86718 q^{16} +3.17700 q^{17} +0.631243i q^{19} +5.19615 q^{20} +2.13828 q^{22} -1.76183i q^{23} +9.50273 q^{25} +10.1622 q^{26} +4.83424i q^{29} -4.27781i q^{31} -6.59607i q^{32} +5.82739i q^{34} +4.23163 q^{37} -1.15785 q^{38} -4.43949i q^{40} +4.56491 q^{41} +7.23163 q^{43} +1.59061i q^{44} +3.23163 q^{46} -2.54576 q^{47} +17.4303i q^{50} +7.55945i q^{52} -0.0724126i q^{53} +4.43949i q^{55} -8.86718 q^{58} -6.98525 q^{59} -8.08606i q^{61} +7.84655 q^{62} +2.36445 q^{64} +21.0988i q^{65} +5.09880 q^{67} -4.33485 q^{68} -4.76183i q^{71} +5.59734i q^{73} +7.76183i q^{74} -0.861298i q^{76} +17.0988 q^{79} +18.5354 q^{80} +8.37315i q^{82} +10.1622 q^{83} -12.0988 q^{85} +13.2646i q^{86} +1.35899 q^{88} +11.5502 q^{89} +2.40393i q^{92} -4.66954i q^{94} -2.40393i q^{95} -0.688287i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 16q^{4} + O(q^{10}) \) \( 12q - 16q^{4} + 8q^{16} - 40q^{22} + 48q^{25} - 16q^{37} + 20q^{43} - 28q^{46} - 40q^{58} + 28q^{64} - 72q^{67} + 72q^{79} - 12q^{85} + 148q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.83424i 1.29701i 0.761212 + 0.648503i \(0.224605\pi\)
−0.761212 + 0.648503i \(0.775395\pi\)
\(3\) 0 0
\(4\) −1.36445 −0.682224
\(5\) −3.80824 −1.70310 −0.851549 0.524274i \(-0.824337\pi\)
−0.851549 + 0.524274i \(0.824337\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.16576i 0.412157i
\(9\) 0 0
\(10\) − 6.98525i − 2.20893i
\(11\) − 1.16576i − 0.351489i −0.984436 0.175744i \(-0.943767\pi\)
0.984436 0.175744i \(-0.0562332\pi\)
\(12\) 0 0
\(13\) − 5.54030i − 1.53660i −0.640089 0.768301i \(-0.721103\pi\)
0.640089 0.768301i \(-0.278897\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.86718 −1.21679
\(17\) 3.17700 0.770536 0.385268 0.922805i \(-0.374109\pi\)
0.385268 + 0.922805i \(0.374109\pi\)
\(18\) 0 0
\(19\) 0.631243i 0.144817i 0.997375 + 0.0724085i \(0.0230685\pi\)
−0.997375 + 0.0724085i \(0.976931\pi\)
\(20\) 5.19615 1.16190
\(21\) 0 0
\(22\) 2.13828 0.455883
\(23\) − 1.76183i − 0.367367i −0.982985 0.183684i \(-0.941198\pi\)
0.982985 0.183684i \(-0.0588022\pi\)
\(24\) 0 0
\(25\) 9.50273 1.90055
\(26\) 10.1622 1.99298
\(27\) 0 0
\(28\) 0 0
\(29\) 4.83424i 0.897696i 0.893608 + 0.448848i \(0.148166\pi\)
−0.893608 + 0.448848i \(0.851834\pi\)
\(30\) 0 0
\(31\) − 4.27781i − 0.768317i −0.923267 0.384159i \(-0.874492\pi\)
0.923267 0.384159i \(-0.125508\pi\)
\(32\) − 6.59607i − 1.16603i
\(33\) 0 0
\(34\) 5.82739i 0.999390i
\(35\) 0 0
\(36\) 0 0
\(37\) 4.23163 0.695675 0.347837 0.937555i \(-0.386916\pi\)
0.347837 + 0.937555i \(0.386916\pi\)
\(38\) −1.15785 −0.187828
\(39\) 0 0
\(40\) − 4.43949i − 0.701945i
\(41\) 4.56491 0.712919 0.356460 0.934311i \(-0.383984\pi\)
0.356460 + 0.934311i \(0.383984\pi\)
\(42\) 0 0
\(43\) 7.23163 1.10281 0.551406 0.834237i \(-0.314091\pi\)
0.551406 + 0.834237i \(0.314091\pi\)
\(44\) 1.59061i 0.239794i
\(45\) 0 0
\(46\) 3.23163 0.476477
\(47\) −2.54576 −0.371337 −0.185669 0.982612i \(-0.559445\pi\)
−0.185669 + 0.982612i \(0.559445\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 17.4303i 2.46502i
\(51\) 0 0
\(52\) 7.55945i 1.04831i
\(53\) − 0.0724126i − 0.00994664i −0.999988 0.00497332i \(-0.998417\pi\)
0.999988 0.00497332i \(-0.00158306\pi\)
\(54\) 0 0
\(55\) 4.43949i 0.598620i
\(56\) 0 0
\(57\) 0 0
\(58\) −8.86718 −1.16432
\(59\) −6.98525 −0.909402 −0.454701 0.890644i \(-0.650254\pi\)
−0.454701 + 0.890644i \(0.650254\pi\)
\(60\) 0 0
\(61\) − 8.08606i − 1.03531i −0.855588 0.517657i \(-0.826804\pi\)
0.855588 0.517657i \(-0.173196\pi\)
\(62\) 7.84655 0.996512
\(63\) 0 0
\(64\) 2.36445 0.295556
\(65\) 21.0988i 2.61698i
\(66\) 0 0
\(67\) 5.09880 0.622918 0.311459 0.950260i \(-0.399182\pi\)
0.311459 + 0.950260i \(0.399182\pi\)
\(68\) −4.33485 −0.525678
\(69\) 0 0
\(70\) 0 0
\(71\) − 4.76183i − 0.565125i −0.959249 0.282563i \(-0.908815\pi\)
0.959249 0.282563i \(-0.0911845\pi\)
\(72\) 0 0
\(73\) 5.59734i 0.655119i 0.944831 + 0.327560i \(0.106226\pi\)
−0.944831 + 0.327560i \(0.893774\pi\)
\(74\) 7.76183i 0.902294i
\(75\) 0 0
\(76\) − 0.861298i − 0.0987976i
\(77\) 0 0
\(78\) 0 0
\(79\) 17.0988 1.92377 0.961883 0.273462i \(-0.0881687\pi\)
0.961883 + 0.273462i \(0.0881687\pi\)
\(80\) 18.5354 2.07232
\(81\) 0 0
\(82\) 8.37315i 0.924660i
\(83\) 10.1622 1.11545 0.557726 0.830025i \(-0.311674\pi\)
0.557726 + 0.830025i \(0.311674\pi\)
\(84\) 0 0
\(85\) −12.0988 −1.31230
\(86\) 13.2646i 1.43035i
\(87\) 0 0
\(88\) 1.35899 0.144869
\(89\) 11.5502 1.22431 0.612157 0.790736i \(-0.290302\pi\)
0.612157 + 0.790736i \(0.290302\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.40393i 0.250627i
\(93\) 0 0
\(94\) − 4.66954i − 0.481627i
\(95\) − 2.40393i − 0.246638i
\(96\) 0 0
\(97\) − 0.688287i − 0.0698849i −0.999389 0.0349425i \(-0.988875\pi\)
0.999389 0.0349425i \(-0.0111248\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −12.9660 −1.29660
\(101\) −6.58406 −0.655138 −0.327569 0.944827i \(-0.606229\pi\)
−0.327569 + 0.944827i \(0.606229\pi\)
\(102\) 0 0
\(103\) 9.34854i 0.921139i 0.887624 + 0.460570i \(0.152355\pi\)
−0.887624 + 0.460570i \(0.847645\pi\)
\(104\) 6.45864 0.633322
\(105\) 0 0
\(106\) 0.132822 0.0129009
\(107\) − 15.7673i − 1.52428i −0.647411 0.762141i \(-0.724148\pi\)
0.647411 0.762141i \(-0.275852\pi\)
\(108\) 0 0
\(109\) −14.4633 −1.38533 −0.692664 0.721260i \(-0.743563\pi\)
−0.692664 + 0.721260i \(0.743563\pi\)
\(110\) −8.14310 −0.776414
\(111\) 0 0
\(112\) 0 0
\(113\) − 19.8606i − 1.86833i −0.356840 0.934166i \(-0.616146\pi\)
0.356840 0.934166i \(-0.383854\pi\)
\(114\) 0 0
\(115\) 6.70948i 0.625662i
\(116\) − 6.59607i − 0.612430i
\(117\) 0 0
\(118\) − 12.8126i − 1.17950i
\(119\) 0 0
\(120\) 0 0
\(121\) 9.64101 0.876456
\(122\) 14.8318 1.34281
\(123\) 0 0
\(124\) 5.83685i 0.524165i
\(125\) −17.1475 −1.53372
\(126\) 0 0
\(127\) −8.90666 −0.790338 −0.395169 0.918608i \(-0.629314\pi\)
−0.395169 + 0.918608i \(0.629314\pi\)
\(128\) − 8.85517i − 0.782694i
\(129\) 0 0
\(130\) −38.7003 −3.39424
\(131\) 6.75519 0.590204 0.295102 0.955466i \(-0.404646\pi\)
0.295102 + 0.955466i \(0.404646\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 9.35245i 0.807928i
\(135\) 0 0
\(136\) 3.70361i 0.317582i
\(137\) − 16.8606i − 1.44050i −0.693714 0.720251i \(-0.744027\pi\)
0.693714 0.720251i \(-0.255973\pi\)
\(138\) 0 0
\(139\) − 0.813709i − 0.0690179i −0.999404 0.0345089i \(-0.989013\pi\)
0.999404 0.0345089i \(-0.0109867\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.73436 0.732971
\(143\) −6.45864 −0.540098
\(144\) 0 0
\(145\) − 18.4100i − 1.52887i
\(146\) −10.2669 −0.849693
\(147\) 0 0
\(148\) −5.77383 −0.474606
\(149\) 8.43032i 0.690638i 0.938485 + 0.345319i \(0.112229\pi\)
−0.938485 + 0.345319i \(0.887771\pi\)
\(150\) 0 0
\(151\) 3.03948 0.247349 0.123675 0.992323i \(-0.460532\pi\)
0.123675 + 0.992323i \(0.460532\pi\)
\(152\) −0.735875 −0.0596874
\(153\) 0 0
\(154\) 0 0
\(155\) 16.2910i 1.30852i
\(156\) 0 0
\(157\) − 3.80824i − 0.303931i −0.988386 0.151966i \(-0.951440\pi\)
0.988386 0.151966i \(-0.0485603\pi\)
\(158\) 31.3634i 2.49514i
\(159\) 0 0
\(160\) 25.1195i 1.98587i
\(161\) 0 0
\(162\) 0 0
\(163\) −11.0933 −0.868898 −0.434449 0.900696i \(-0.643057\pi\)
−0.434449 + 0.900696i \(0.643057\pi\)
\(164\) −6.22858 −0.486371
\(165\) 0 0
\(166\) 18.6400i 1.44675i
\(167\) 3.70361 0.286594 0.143297 0.989680i \(-0.454230\pi\)
0.143297 + 0.989680i \(0.454230\pi\)
\(168\) 0 0
\(169\) −17.6949 −1.36114
\(170\) − 22.1921i − 1.70206i
\(171\) 0 0
\(172\) −9.86718 −0.752365
\(173\) −9.63564 −0.732584 −0.366292 0.930500i \(-0.619373\pi\)
−0.366292 + 0.930500i \(0.619373\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 5.67395i 0.427690i
\(177\) 0 0
\(178\) 21.1858i 1.58794i
\(179\) − 18.4512i − 1.37911i −0.724233 0.689556i \(-0.757806\pi\)
0.724233 0.689556i \(-0.242194\pi\)
\(180\) 0 0
\(181\) − 16.6209i − 1.23542i −0.786406 0.617710i \(-0.788060\pi\)
0.786406 0.617710i \(-0.211940\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 2.05387 0.151413
\(185\) −16.1151 −1.18480
\(186\) 0 0
\(187\) − 3.70361i − 0.270835i
\(188\) 3.47356 0.253335
\(189\) 0 0
\(190\) 4.40939 0.319890
\(191\) − 7.16576i − 0.518496i −0.965811 0.259248i \(-0.916525\pi\)
0.965811 0.259248i \(-0.0834747\pi\)
\(192\) 0 0
\(193\) −3.17776 −0.228740 −0.114370 0.993438i \(-0.536485\pi\)
−0.114370 + 0.993438i \(0.536485\pi\)
\(194\) 1.26249 0.0906411
\(195\) 0 0
\(196\) 0 0
\(197\) 1.93305i 0.137724i 0.997626 + 0.0688619i \(0.0219368\pi\)
−0.997626 + 0.0688619i \(0.978063\pi\)
\(198\) 0 0
\(199\) − 8.66025i − 0.613909i −0.951724 0.306955i \(-0.900690\pi\)
0.951724 0.306955i \(-0.0993100\pi\)
\(200\) 11.0779i 0.783324i
\(201\) 0 0
\(202\) − 12.0768i − 0.849718i
\(203\) 0 0
\(204\) 0 0
\(205\) −17.3843 −1.21417
\(206\) −17.1475 −1.19472
\(207\) 0 0
\(208\) 26.9656i 1.86973i
\(209\) 0.735875 0.0509016
\(210\) 0 0
\(211\) 12.4183 0.854912 0.427456 0.904036i \(-0.359410\pi\)
0.427456 + 0.904036i \(0.359410\pi\)
\(212\) 0.0988033i 0.00678584i
\(213\) 0 0
\(214\) 28.9210 1.97700
\(215\) −27.5398 −1.87820
\(216\) 0 0
\(217\) 0 0
\(218\) − 26.5291i − 1.79678i
\(219\) 0 0
\(220\) − 6.05745i − 0.408393i
\(221\) − 17.6015i − 1.18401i
\(222\) 0 0
\(223\) 18.9366i 1.26809i 0.773297 + 0.634044i \(0.218606\pi\)
−0.773297 + 0.634044i \(0.781394\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 36.4292 2.42324
\(227\) −5.72276 −0.379833 −0.189917 0.981800i \(-0.560822\pi\)
−0.189917 + 0.981800i \(0.560822\pi\)
\(228\) 0 0
\(229\) − 11.1376i − 0.735996i −0.929827 0.367998i \(-0.880043\pi\)
0.929827 0.367998i \(-0.119957\pi\)
\(230\) −12.3068 −0.811488
\(231\) 0 0
\(232\) −5.63555 −0.369992
\(233\) 10.3370i 0.677198i 0.940931 + 0.338599i \(0.109953\pi\)
−0.940931 + 0.338599i \(0.890047\pi\)
\(234\) 0 0
\(235\) 9.69488 0.632424
\(236\) 9.53101 0.620416
\(237\) 0 0
\(238\) 0 0
\(239\) − 16.5806i − 1.07251i −0.844056 0.536255i \(-0.819839\pi\)
0.844056 0.536255i \(-0.180161\pi\)
\(240\) 0 0
\(241\) 23.8932i 1.53910i 0.638587 + 0.769549i \(0.279519\pi\)
−0.638587 + 0.769549i \(0.720481\pi\)
\(242\) 17.6840i 1.13677i
\(243\) 0 0
\(244\) 11.0330i 0.706316i
\(245\) 0 0
\(246\) 0 0
\(247\) 3.49727 0.222526
\(248\) 4.98689 0.316668
\(249\) 0 0
\(250\) − 31.4527i − 1.98924i
\(251\) −19.9233 −1.25755 −0.628774 0.777588i \(-0.716443\pi\)
−0.628774 + 0.777588i \(0.716443\pi\)
\(252\) 0 0
\(253\) −2.05387 −0.129125
\(254\) − 16.3370i − 1.02507i
\(255\) 0 0
\(256\) 20.9714 1.31072
\(257\) 23.2049 1.44748 0.723742 0.690070i \(-0.242420\pi\)
0.723742 + 0.690070i \(0.242420\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) − 28.7882i − 1.78537i
\(261\) 0 0
\(262\) 12.3907i 0.765498i
\(263\) 9.62246i 0.593347i 0.954979 + 0.296673i \(0.0958772\pi\)
−0.954979 + 0.296673i \(0.904123\pi\)
\(264\) 0 0
\(265\) 0.275765i 0.0169401i
\(266\) 0 0
\(267\) 0 0
\(268\) −6.95705 −0.424970
\(269\) 20.8303 1.27005 0.635024 0.772493i \(-0.280990\pi\)
0.635024 + 0.772493i \(0.280990\pi\)
\(270\) 0 0
\(271\) − 19.6838i − 1.19571i −0.801606 0.597853i \(-0.796021\pi\)
0.801606 0.597853i \(-0.203979\pi\)
\(272\) −15.4630 −0.937584
\(273\) 0 0
\(274\) 30.9265 1.86834
\(275\) − 11.0779i − 0.668021i
\(276\) 0 0
\(277\) 8.31604 0.499662 0.249831 0.968289i \(-0.419625\pi\)
0.249831 + 0.968289i \(0.419625\pi\)
\(278\) 1.49254 0.0895166
\(279\) 0 0
\(280\) 0 0
\(281\) − 22.7618i − 1.35786i −0.734204 0.678928i \(-0.762445\pi\)
0.734204 0.678928i \(-0.237555\pi\)
\(282\) 0 0
\(283\) − 17.0334i − 1.01253i −0.862378 0.506266i \(-0.831026\pi\)
0.862378 0.506266i \(-0.168974\pi\)
\(284\) 6.49727i 0.385542i
\(285\) 0 0
\(286\) − 11.8467i − 0.700511i
\(287\) 0 0
\(288\) 0 0
\(289\) −6.90666 −0.406274
\(290\) 33.7684 1.98295
\(291\) 0 0
\(292\) − 7.63728i − 0.446938i
\(293\) −21.4159 −1.25113 −0.625564 0.780173i \(-0.715131\pi\)
−0.625564 + 0.780173i \(0.715131\pi\)
\(294\) 0 0
\(295\) 26.6015 1.54880
\(296\) 4.93305i 0.286728i
\(297\) 0 0
\(298\) −15.4633 −0.895762
\(299\) −9.76106 −0.564497
\(300\) 0 0
\(301\) 0 0
\(302\) 5.57514i 0.320813i
\(303\) 0 0
\(304\) − 3.07237i − 0.176212i
\(305\) 30.7937i 1.76324i
\(306\) 0 0
\(307\) 19.4537i 1.11028i 0.831756 + 0.555142i \(0.187336\pi\)
−0.831756 + 0.555142i \(0.812664\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −29.8816 −1.69716
\(311\) −6.81412 −0.386393 −0.193197 0.981160i \(-0.561885\pi\)
−0.193197 + 0.981160i \(0.561885\pi\)
\(312\) 0 0
\(313\) 16.6417i 0.940643i 0.882495 + 0.470322i \(0.155862\pi\)
−0.882495 + 0.470322i \(0.844138\pi\)
\(314\) 6.98525 0.394200
\(315\) 0 0
\(316\) −23.3304 −1.31244
\(317\) − 5.07787i − 0.285202i −0.989780 0.142601i \(-0.954453\pi\)
0.989780 0.142601i \(-0.0455465\pi\)
\(318\) 0 0
\(319\) 5.63555 0.315530
\(320\) −9.00440 −0.503361
\(321\) 0 0
\(322\) 0 0
\(323\) 2.00546i 0.111587i
\(324\) 0 0
\(325\) − 52.6479i − 2.92038i
\(326\) − 20.3479i − 1.12697i
\(327\) 0 0
\(328\) 5.32157i 0.293835i
\(329\) 0 0
\(330\) 0 0
\(331\) 16.3304 0.897602 0.448801 0.893632i \(-0.351851\pi\)
0.448801 + 0.893632i \(0.351851\pi\)
\(332\) −13.8659 −0.760988
\(333\) 0 0
\(334\) 6.79333i 0.371714i
\(335\) −19.4175 −1.06089
\(336\) 0 0
\(337\) −3.95506 −0.215446 −0.107723 0.994181i \(-0.534356\pi\)
−0.107723 + 0.994181i \(0.534356\pi\)
\(338\) − 32.4567i − 1.76541i
\(339\) 0 0
\(340\) 16.5082 0.895282
\(341\) −4.98689 −0.270055
\(342\) 0 0
\(343\) 0 0
\(344\) 8.43032i 0.454532i
\(345\) 0 0
\(346\) − 17.6741i − 0.950166i
\(347\) 7.33697i 0.393869i 0.980417 + 0.196935i \(0.0630987\pi\)
−0.980417 + 0.196935i \(0.936901\pi\)
\(348\) 0 0
\(349\) − 25.5093i − 1.36548i −0.730660 0.682741i \(-0.760788\pi\)
0.730660 0.682741i \(-0.239212\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −7.68942 −0.409847
\(353\) 0.0207896 0.00110652 0.000553260 1.00000i \(-0.499824\pi\)
0.000553260 1.00000i \(0.499824\pi\)
\(354\) 0 0
\(355\) 18.1342i 0.962464i
\(356\) −15.7596 −0.835257
\(357\) 0 0
\(358\) 33.8441 1.78872
\(359\) 36.2240i 1.91183i 0.293644 + 0.955915i \(0.405132\pi\)
−0.293644 + 0.955915i \(0.594868\pi\)
\(360\) 0 0
\(361\) 18.6015 0.979028
\(362\) 30.4867 1.60235
\(363\) 0 0
\(364\) 0 0
\(365\) − 21.3160i − 1.11573i
\(366\) 0 0
\(367\) − 8.87897i − 0.463479i −0.972778 0.231739i \(-0.925558\pi\)
0.972778 0.231739i \(-0.0744416\pi\)
\(368\) 8.57514i 0.447010i
\(369\) 0 0
\(370\) − 29.5590i − 1.53670i
\(371\) 0 0
\(372\) 0 0
\(373\) −12.8188 −0.663731 −0.331865 0.943327i \(-0.607678\pi\)
−0.331865 + 0.943327i \(0.607678\pi\)
\(374\) 6.79333 0.351274
\(375\) 0 0
\(376\) − 2.96774i − 0.153049i
\(377\) 26.7831 1.37940
\(378\) 0 0
\(379\) 3.13828 0.161203 0.0806013 0.996746i \(-0.474316\pi\)
0.0806013 + 0.996746i \(0.474316\pi\)
\(380\) 3.28003i 0.168262i
\(381\) 0 0
\(382\) 13.1437 0.672492
\(383\) −27.5190 −1.40616 −0.703078 0.711113i \(-0.748192\pi\)
−0.703078 + 0.711113i \(0.748192\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 5.82878i − 0.296677i
\(387\) 0 0
\(388\) 0.939131i 0.0476772i
\(389\) 18.2436i 0.924989i 0.886622 + 0.462494i \(0.153045\pi\)
−0.886622 + 0.462494i \(0.846955\pi\)
\(390\) 0 0
\(391\) − 5.59734i − 0.283070i
\(392\) 0 0
\(393\) 0 0
\(394\) −3.54568 −0.178629
\(395\) −65.1164 −3.27636
\(396\) 0 0
\(397\) − 1.16919i − 0.0586798i −0.999569 0.0293399i \(-0.990659\pi\)
0.999569 0.0293399i \(-0.00934052\pi\)
\(398\) 15.8850 0.796244
\(399\) 0 0
\(400\) −46.2515 −2.31257
\(401\) − 13.9594i − 0.697101i −0.937290 0.348551i \(-0.886674\pi\)
0.937290 0.348551i \(-0.113326\pi\)
\(402\) 0 0
\(403\) −23.7003 −1.18060
\(404\) 8.98361 0.446951
\(405\) 0 0
\(406\) 0 0
\(407\) − 4.93305i − 0.244522i
\(408\) 0 0
\(409\) 20.3721i 1.00733i 0.863898 + 0.503667i \(0.168016\pi\)
−0.863898 + 0.503667i \(0.831984\pi\)
\(410\) − 31.8870i − 1.57479i
\(411\) 0 0
\(412\) − 12.7556i − 0.628423i
\(413\) 0 0
\(414\) 0 0
\(415\) −38.7003 −1.89972
\(416\) −36.5442 −1.79173
\(417\) 0 0
\(418\) 1.34977i 0.0660196i
\(419\) 25.5207 1.24677 0.623383 0.781917i \(-0.285758\pi\)
0.623383 + 0.781917i \(0.285758\pi\)
\(420\) 0 0
\(421\) 0.0933442 0.00454932 0.00227466 0.999997i \(-0.499276\pi\)
0.00227466 + 0.999997i \(0.499276\pi\)
\(422\) 22.7782i 1.10883i
\(423\) 0 0
\(424\) 0.0844155 0.00409958
\(425\) 30.1902 1.46444
\(426\) 0 0
\(427\) 0 0
\(428\) 21.5136i 1.03990i
\(429\) 0 0
\(430\) − 50.5147i − 2.43603i
\(431\) − 33.1712i − 1.59780i −0.601463 0.798901i \(-0.705415\pi\)
0.601463 0.798901i \(-0.294585\pi\)
\(432\) 0 0
\(433\) 17.9518i 0.862706i 0.902183 + 0.431353i \(0.141964\pi\)
−0.902183 + 0.431353i \(0.858036\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 19.7344 0.945104
\(437\) 1.11214 0.0532010
\(438\) 0 0
\(439\) 10.0368i 0.479032i 0.970892 + 0.239516i \(0.0769887\pi\)
−0.970892 + 0.239516i \(0.923011\pi\)
\(440\) −5.17536 −0.246726
\(441\) 0 0
\(442\) 32.2855 1.53566
\(443\) − 26.3579i − 1.25230i −0.779702 0.626151i \(-0.784629\pi\)
0.779702 0.626151i \(-0.215371\pi\)
\(444\) 0 0
\(445\) −43.9858 −2.08513
\(446\) −34.7343 −1.64472
\(447\) 0 0
\(448\) 0 0
\(449\) 24.0264i 1.13388i 0.823761 + 0.566938i \(0.191872\pi\)
−0.823761 + 0.566938i \(0.808128\pi\)
\(450\) 0 0
\(451\) − 5.32157i − 0.250583i
\(452\) 27.0988i 1.27462i
\(453\) 0 0
\(454\) − 10.4969i − 0.492646i
\(455\) 0 0
\(456\) 0 0
\(457\) 20.8222 0.974023 0.487012 0.873395i \(-0.338087\pi\)
0.487012 + 0.873395i \(0.338087\pi\)
\(458\) 20.4291 0.954591
\(459\) 0 0
\(460\) − 9.15474i − 0.426842i
\(461\) −8.35237 −0.389008 −0.194504 0.980902i \(-0.562310\pi\)
−0.194504 + 0.980902i \(0.562310\pi\)
\(462\) 0 0
\(463\) 11.8133 0.549011 0.274506 0.961586i \(-0.411486\pi\)
0.274506 + 0.961586i \(0.411486\pi\)
\(464\) − 23.5291i − 1.09231i
\(465\) 0 0
\(466\) −18.9605 −0.878329
\(467\) −29.1578 −1.34926 −0.674630 0.738156i \(-0.735697\pi\)
−0.674630 + 0.738156i \(0.735697\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 17.7828i 0.820258i
\(471\) 0 0
\(472\) − 8.14310i − 0.374817i
\(473\) − 8.43032i − 0.387626i
\(474\) 0 0
\(475\) 5.99853i 0.275231i
\(476\) 0 0
\(477\) 0 0
\(478\) 30.4129 1.39105
\(479\) 40.5236 1.85157 0.925785 0.378051i \(-0.123406\pi\)
0.925785 + 0.378051i \(0.123406\pi\)
\(480\) 0 0
\(481\) − 23.4445i − 1.06898i
\(482\) −43.8260 −1.99622
\(483\) 0 0
\(484\) −13.1547 −0.597939
\(485\) 2.62116i 0.119021i
\(486\) 0 0
\(487\) 12.4183 0.562728 0.281364 0.959601i \(-0.409213\pi\)
0.281364 + 0.959601i \(0.409213\pi\)
\(488\) 9.42637 0.426712
\(489\) 0 0
\(490\) 0 0
\(491\) − 6.17122i − 0.278503i −0.990257 0.139252i \(-0.955530\pi\)
0.990257 0.139252i \(-0.0444697\pi\)
\(492\) 0 0
\(493\) 15.3584i 0.691708i
\(494\) 6.41484i 0.288617i
\(495\) 0 0
\(496\) 20.8209i 0.934884i
\(497\) 0 0
\(498\) 0 0
\(499\) −2.96052 −0.132531 −0.0662656 0.997802i \(-0.521108\pi\)
−0.0662656 + 0.997802i \(0.521108\pi\)
\(500\) 23.3969 1.04634
\(501\) 0 0
\(502\) − 36.5442i − 1.63105i
\(503\) −4.33485 −0.193282 −0.0966408 0.995319i \(-0.530810\pi\)
−0.0966408 + 0.995319i \(0.530810\pi\)
\(504\) 0 0
\(505\) 25.0737 1.11577
\(506\) − 3.76729i − 0.167476i
\(507\) 0 0
\(508\) 12.1527 0.539188
\(509\) 28.9485 1.28312 0.641560 0.767073i \(-0.278288\pi\)
0.641560 + 0.767073i \(0.278288\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 20.7564i 0.917311i
\(513\) 0 0
\(514\) 42.5635i 1.87740i
\(515\) − 35.6015i − 1.56879i
\(516\) 0 0
\(517\) 2.96774i 0.130521i
\(518\) 0 0
\(519\) 0 0
\(520\) −24.5961 −1.07861
\(521\) 12.1606 0.532766 0.266383 0.963867i \(-0.414171\pi\)
0.266383 + 0.963867i \(0.414171\pi\)
\(522\) 0 0
\(523\) 38.2857i 1.67412i 0.547113 + 0.837059i \(0.315727\pi\)
−0.547113 + 0.837059i \(0.684273\pi\)
\(524\) −9.21711 −0.402651
\(525\) 0 0
\(526\) −17.6499 −0.769574
\(527\) − 13.5906i − 0.592016i
\(528\) 0 0
\(529\) 19.8960 0.865041
\(530\) −0.505820 −0.0219714
\(531\) 0 0
\(532\) 0 0
\(533\) − 25.2910i − 1.09547i
\(534\) 0 0
\(535\) 60.0457i 2.59600i
\(536\) 5.94396i 0.256740i
\(537\) 0 0
\(538\) 38.2079i 1.64726i
\(539\) 0 0
\(540\) 0 0
\(541\) 0.508189 0.0218487 0.0109244 0.999940i \(-0.496523\pi\)
0.0109244 + 0.999940i \(0.496523\pi\)
\(542\) 36.1049 1.55084
\(543\) 0 0
\(544\) − 20.9557i − 0.898470i
\(545\) 55.0796 2.35935
\(546\) 0 0
\(547\) −40.2964 −1.72295 −0.861475 0.507800i \(-0.830459\pi\)
−0.861475 + 0.507800i \(0.830459\pi\)
\(548\) 23.0055i 0.982745i
\(549\) 0 0
\(550\) 20.3195 0.866427
\(551\) −3.05158 −0.130002
\(552\) 0 0
\(553\) 0 0
\(554\) 15.2536i 0.648065i
\(555\) 0 0
\(556\) 1.11026i 0.0470857i
\(557\) 43.4358i 1.84043i 0.391410 + 0.920216i \(0.371987\pi\)
−0.391410 + 0.920216i \(0.628013\pi\)
\(558\) 0 0
\(559\) − 40.0653i − 1.69458i
\(560\) 0 0
\(561\) 0 0
\(562\) 41.7507 1.76115
\(563\) 6.33322 0.266913 0.133457 0.991055i \(-0.457392\pi\)
0.133457 + 0.991055i \(0.457392\pi\)
\(564\) 0 0
\(565\) 75.6342i 3.18195i
\(566\) 31.2434 1.31326
\(567\) 0 0
\(568\) 5.55114 0.232921
\(569\) 8.23271i 0.345133i 0.984998 + 0.172567i \(0.0552060\pi\)
−0.984998 + 0.172567i \(0.944794\pi\)
\(570\) 0 0
\(571\) 8.49727 0.355600 0.177800 0.984067i \(-0.443102\pi\)
0.177800 + 0.984067i \(0.443102\pi\)
\(572\) 8.81248 0.368468
\(573\) 0 0
\(574\) 0 0
\(575\) − 16.7422i − 0.698198i
\(576\) 0 0
\(577\) 39.4798i 1.64357i 0.569801 + 0.821783i \(0.307020\pi\)
−0.569801 + 0.821783i \(0.692980\pi\)
\(578\) − 12.6685i − 0.526940i
\(579\) 0 0
\(580\) 25.1195i 1.04303i
\(581\) 0 0
\(582\) 0 0
\(583\) −0.0844155 −0.00349613
\(584\) −6.52514 −0.270012
\(585\) 0 0
\(586\) − 39.2819i − 1.62272i
\(587\) −24.1120 −0.995207 −0.497603 0.867405i \(-0.665787\pi\)
−0.497603 + 0.867405i \(0.665787\pi\)
\(588\) 0 0
\(589\) 2.70034 0.111265
\(590\) 48.7937i 2.00880i
\(591\) 0 0
\(592\) −20.5961 −0.846493
\(593\) −18.9366 −0.777633 −0.388816 0.921315i \(-0.627116\pi\)
−0.388816 + 0.921315i \(0.627116\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) − 11.5027i − 0.471170i
\(597\) 0 0
\(598\) − 17.9042i − 0.732156i
\(599\) − 34.6794i − 1.41696i −0.705730 0.708481i \(-0.749381\pi\)
0.705730 0.708481i \(-0.250619\pi\)
\(600\) 0 0
\(601\) 13.2709i 0.541330i 0.962674 + 0.270665i \(0.0872436\pi\)
−0.962674 + 0.270665i \(0.912756\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −4.14721 −0.168748
\(605\) −36.7153 −1.49269
\(606\) 0 0
\(607\) 25.0148i 1.01532i 0.861557 + 0.507660i \(0.169490\pi\)
−0.861557 + 0.507660i \(0.830510\pi\)
\(608\) 4.16372 0.168861
\(609\) 0 0
\(610\) −56.4831 −2.28693
\(611\) 14.1043i 0.570597i
\(612\) 0 0
\(613\) −44.1097 −1.78157 −0.890787 0.454420i \(-0.849846\pi\)
−0.890787 + 0.454420i \(0.849846\pi\)
\(614\) −35.6829 −1.44004
\(615\) 0 0
\(616\) 0 0
\(617\) 19.3370i 0.778477i 0.921137 + 0.389239i \(0.127262\pi\)
−0.921137 + 0.389239i \(0.872738\pi\)
\(618\) 0 0
\(619\) 11.0598i 0.444531i 0.974986 + 0.222265i \(0.0713452\pi\)
−0.974986 + 0.222265i \(0.928655\pi\)
\(620\) − 22.2282i − 0.892704i
\(621\) 0 0
\(622\) − 12.4987i − 0.501154i
\(623\) 0 0
\(624\) 0 0
\(625\) 17.7882 0.711529
\(626\) −30.5249 −1.22002
\(627\) 0 0
\(628\) 5.19615i 0.207349i
\(629\) 13.4439 0.536043
\(630\) 0 0
\(631\) −27.6015 −1.09880 −0.549400 0.835560i \(-0.685144\pi\)
−0.549400 + 0.835560i \(0.685144\pi\)
\(632\) 19.9330i 0.792894i
\(633\) 0 0
\(634\) 9.31405 0.369908
\(635\) 33.9187 1.34602
\(636\) 0 0
\(637\) 0 0
\(638\) 10.3370i 0.409245i
\(639\) 0 0
\(640\) 33.7227i 1.33301i
\(641\) − 6.07241i − 0.239846i −0.992783 0.119923i \(-0.961735\pi\)
0.992783 0.119923i \(-0.0382648\pi\)
\(642\) 0 0
\(643\) − 13.2709i − 0.523352i −0.965156 0.261676i \(-0.915725\pi\)
0.965156 0.261676i \(-0.0842752\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −3.67850 −0.144729
\(647\) 30.7376 1.20842 0.604210 0.796825i \(-0.293489\pi\)
0.604210 + 0.796825i \(0.293489\pi\)
\(648\) 0 0
\(649\) 8.14310i 0.319645i
\(650\) 96.5691 3.78775
\(651\) 0 0
\(652\) 15.1363 0.592783
\(653\) 37.2185i 1.45647i 0.685325 + 0.728237i \(0.259660\pi\)
−0.685325 + 0.728237i \(0.740340\pi\)
\(654\) 0 0
\(655\) −25.7254 −1.00518
\(656\) −22.2182 −0.867476
\(657\) 0 0
\(658\) 0 0
\(659\) 46.2545i 1.80182i 0.434005 + 0.900911i \(0.357100\pi\)
−0.434005 + 0.900911i \(0.642900\pi\)
\(660\) 0 0
\(661\) 13.1889i 0.512989i 0.966546 + 0.256495i \(0.0825676\pi\)
−0.966546 + 0.256495i \(0.917432\pi\)
\(662\) 29.9540i 1.16419i
\(663\) 0 0
\(664\) 11.8467i 0.459742i
\(665\) 0 0
\(666\) 0 0
\(667\) 8.51712 0.329784
\(668\) −5.05339 −0.195521
\(669\) 0 0
\(670\) − 35.6164i − 1.37598i
\(671\) −9.42637 −0.363901
\(672\) 0 0
\(673\) 40.9265 1.57760 0.788800 0.614649i \(-0.210703\pi\)
0.788800 + 0.614649i \(0.210703\pi\)
\(674\) − 7.25455i − 0.279435i
\(675\) 0 0
\(676\) 24.1437 0.928605
\(677\) 1.61796 0.0621834 0.0310917 0.999517i \(-0.490102\pi\)
0.0310917 + 0.999517i \(0.490102\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) − 14.1043i − 0.540874i
\(681\) 0 0
\(682\) − 9.14716i − 0.350263i
\(683\) − 26.5555i − 1.01612i −0.861322 0.508059i \(-0.830363\pi\)
0.861322 0.508059i \(-0.169637\pi\)
\(684\) 0 0
\(685\) 64.2094i 2.45332i
\(686\) 0 0
\(687\) 0 0
\(688\) −35.1976 −1.34190
\(689\) −0.401187 −0.0152840
\(690\) 0 0
\(691\) − 37.6337i − 1.43165i −0.698278 0.715826i \(-0.746050\pi\)
0.698278 0.715826i \(-0.253950\pi\)
\(692\) 13.1473 0.499787
\(693\) 0 0
\(694\) −13.4578 −0.510851
\(695\) 3.09880i 0.117544i
\(696\) 0 0
\(697\) 14.5027 0.549330
\(698\) 46.7903 1.77104
\(699\) 0 0
\(700\) 0 0
\(701\) 24.3228i 0.918659i 0.888266 + 0.459330i \(0.151910\pi\)
−0.888266 + 0.459330i \(0.848090\pi\)
\(702\) 0 0
\(703\) 2.67118i 0.100746i
\(704\) − 2.75637i − 0.103885i
\(705\) 0 0
\(706\) 0.0381332i 0.00143516i
\(707\) 0 0
\(708\) 0 0
\(709\) 1.03402 0.0388334 0.0194167 0.999811i \(-0.493819\pi\)
0.0194167 + 0.999811i \(0.493819\pi\)
\(710\) −33.2626 −1.24832
\(711\) 0 0
\(712\) 13.4647i 0.504610i
\(713\) −7.53678 −0.282255
\(714\) 0 0
\(715\) 24.5961 0.919841
\(716\) 25.1758i 0.940863i
\(717\) 0 0
\(718\) −66.4436 −2.47965
\(719\) 16.7463 0.624532 0.312266 0.949995i \(-0.398912\pi\)
0.312266 + 0.949995i \(0.398912\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 34.1197i 1.26981i
\(723\) 0 0
\(724\) 22.6783i 0.842834i
\(725\) 45.9385i 1.70611i
\(726\) 0 0
\(727\) − 32.8976i − 1.22011i −0.792361 0.610053i \(-0.791148\pi\)
0.792361 0.610053i \(-0.208852\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 39.0988 1.44711
\(731\) 22.9749 0.849757
\(732\) 0 0
\(733\) 19.9120i 0.735466i 0.929931 + 0.367733i \(0.119866\pi\)
−0.929931 + 0.367733i \(0.880134\pi\)
\(734\) 16.2862 0.601135
\(735\) 0 0
\(736\) −11.6212 −0.428362
\(737\) − 5.94396i − 0.218949i
\(738\) 0 0
\(739\) −50.6698 −1.86392 −0.931959 0.362563i \(-0.881902\pi\)
−0.931959 + 0.362563i \(0.881902\pi\)
\(740\) 21.9882 0.808301
\(741\) 0 0
\(742\) 0 0
\(743\) 44.2491i 1.62334i 0.584115 + 0.811671i \(0.301442\pi\)
−0.584115 + 0.811671i \(0.698558\pi\)
\(744\) 0 0
\(745\) − 32.1047i − 1.17623i
\(746\) − 23.5127i − 0.860863i
\(747\) 0 0
\(748\) 5.05339i 0.184770i
\(749\) 0 0
\(750\) 0 0
\(751\) 2.34460 0.0855557 0.0427779 0.999085i \(-0.486379\pi\)
0.0427779 + 0.999085i \(0.486379\pi\)
\(752\) 12.3907 0.451841
\(753\) 0 0
\(754\) 49.1268i 1.78909i
\(755\) −11.5751 −0.421260
\(756\) 0 0
\(757\) −24.8530 −0.903298 −0.451649 0.892196i \(-0.649164\pi\)
−0.451649 + 0.892196i \(0.649164\pi\)
\(758\) 5.75637i 0.209081i
\(759\) 0 0
\(760\) 2.80239 0.101653
\(761\) −0.882087 −0.0319756 −0.0159878 0.999872i \(-0.505089\pi\)
−0.0159878 + 0.999872i \(0.505089\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 9.77730i 0.353730i
\(765\) 0 0
\(766\) − 50.4766i − 1.82379i
\(767\) 38.7003i 1.39739i
\(768\) 0 0
\(769\) 20.7049i 0.746638i 0.927703 + 0.373319i \(0.121780\pi\)
−0.927703 + 0.373319i \(0.878220\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.33589 0.156052
\(773\) 9.40558 0.338295 0.169148 0.985591i \(-0.445899\pi\)
0.169148 + 0.985591i \(0.445899\pi\)
\(774\) 0 0
\(775\) − 40.6509i − 1.46022i
\(776\) 0.802375 0.0288036
\(777\) 0 0
\(778\) −33.4633 −1.19972
\(779\) 2.88157i 0.103243i
\(780\) 0 0
\(781\) −5.55114 −0.198635
\(782\) 10.2669 0.367143
\(783\) 0 0
\(784\) 0 0
\(785\) 14.5027i 0.517625i
\(786\) 0 0
\(787\) − 25.1557i − 0.896705i −0.893857 0.448352i \(-0.852011\pi\)
0.893857 0.448352i \(-0.147989\pi\)
\(788\) − 2.63754i − 0.0939585i
\(789\) 0 0
\(790\) − 119.439i − 4.24946i
\(791\) 0 0
\(792\) 0 0
\(793\) −44.7991 −1.59086
\(794\) 2.14457 0.0761080
\(795\) 0 0
\(796\) 11.8165i 0.418824i
\(797\) 30.7168 1.08805 0.544023 0.839071i \(-0.316901\pi\)
0.544023 + 0.839071i \(0.316901\pi\)
\(798\) 0 0
\(799\) −8.08789 −0.286129
\(800\) − 62.6807i − 2.21610i
\(801\) 0 0
\(802\) 25.6050 0.904144
\(803\) 6.52514 0.230267
\(804\) 0 0
\(805\) 0 0
\(806\) − 43.4722i − 1.53124i
\(807\) 0 0
\(808\) − 7.67541i − 0.270020i
\(809\) − 14.2755i − 0.501899i −0.968000 0.250950i \(-0.919257\pi\)
0.968000 0.250950i \(-0.0807428\pi\)
\(810\) 0 0
\(811\) − 10.0160i − 0.351711i −0.984416 0.175855i \(-0.943731\pi\)
0.984416 0.175855i \(-0.0562691\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 9.04841 0.317146
\(815\) 42.2462 1.47982
\(816\) 0 0
\(817\) 4.56491i 0.159706i
\(818\) −37.3674 −1.30652
\(819\) 0 0
\(820\) 23.7200 0.828337
\(821\) 12.4609i 0.434887i 0.976073 + 0.217444i \(0.0697718\pi\)
−0.976073 + 0.217444i \(0.930228\pi\)
\(822\) 0 0
\(823\) −4.59607 −0.160209 −0.0801045 0.996786i \(-0.525525\pi\)
−0.0801045 + 0.996786i \(0.525525\pi\)
\(824\) −10.8981 −0.379654
\(825\) 0 0
\(826\) 0 0
\(827\) 37.9330i 1.31906i 0.751678 + 0.659531i \(0.229245\pi\)
−0.751678 + 0.659531i \(0.770755\pi\)
\(828\) 0 0
\(829\) − 36.7992i − 1.27809i −0.769170 0.639044i \(-0.779330\pi\)
0.769170 0.639044i \(-0.220670\pi\)
\(830\) − 70.9858i − 2.46395i
\(831\) 0 0
\(832\) − 13.0997i − 0.454152i
\(833\) 0 0
\(834\) 0 0
\(835\) −14.1043 −0.488098
\(836\) −1.00406 −0.0347263
\(837\) 0 0
\(838\) 46.8111i 1.61706i
\(839\) −21.3569 −0.737323 −0.368662 0.929564i \(-0.620184\pi\)
−0.368662 + 0.929564i \(0.620184\pi\)
\(840\) 0 0
\(841\) 5.63009 0.194141
\(842\) 0.171216i 0.00590049i
\(843\) 0 0
\(844\) −16.9441 −0.583242
\(845\) 67.3864 2.31816
\(846\) 0 0
\(847\) 0 0
\(848\) 0.352445i 0.0121030i
\(849\) 0 0
\(850\) 55.3762i 1.89939i
\(851\) − 7.45541i − 0.255568i
\(852\) 0 0
\(853\) − 49.5031i − 1.69495i −0.530833 0.847476i \(-0.678121\pi\)
0.530833 0.847476i \(-0.321879\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 18.3808 0.628244
\(857\) −14.0162 −0.478784 −0.239392 0.970923i \(-0.576948\pi\)
−0.239392 + 0.970923i \(0.576948\pi\)
\(858\) 0 0
\(859\) − 1.21490i − 0.0414517i −0.999785 0.0207259i \(-0.993402\pi\)
0.999785 0.0207259i \(-0.00659772\pi\)
\(860\) 37.5766 1.28135
\(861\) 0 0
\(862\) 60.8441 2.07236
\(863\) − 36.0724i − 1.22792i −0.789337 0.613960i \(-0.789576\pi\)
0.789337 0.613960i \(-0.210424\pi\)
\(864\) 0 0
\(865\) 36.6949 1.24766
\(866\) −32.9279 −1.11893
\(867\) 0 0
\(868\) 0 0
\(869\) − 19.9330i − 0.676182i
\(870\) 0 0
\(871\) − 28.2489i − 0.957177i
\(872\) − 16.8606i − 0.570973i
\(873\) 0 0
\(874\) 2.03994i 0.0690020i
\(875\) 0 0
\(876\) 0 0
\(877\) −11.1921 −0.377932 −0.188966 0.981984i \(-0.560514\pi\)
−0.188966 + 0.981984i \(0.560514\pi\)
\(878\) −18.4100 −0.621307
\(879\) 0 0
\(880\) − 21.6078i − 0.728398i
\(881\) −4.16372 −0.140279 −0.0701397 0.997537i \(-0.522345\pi\)
−0.0701397 + 0.997537i \(0.522345\pi\)
\(882\) 0 0
\(883\) −38.7433 −1.30382 −0.651908 0.758298i \(-0.726031\pi\)
−0.651908 + 0.758298i \(0.726031\pi\)
\(884\) 24.0164i 0.807758i
\(885\) 0 0
\(886\) 48.3468 1.62424
\(887\) −38.0824 −1.27868 −0.639342 0.768923i \(-0.720793\pi\)
−0.639342 + 0.768923i \(0.720793\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) − 80.6807i − 2.70442i
\(891\) 0 0
\(892\) − 25.8380i − 0.865120i
\(893\) − 1.60699i − 0.0537759i
\(894\) 0 0
\(895\) 70.2669i 2.34876i
\(896\) 0 0
\(897\) 0 0
\(898\) −44.0702 −1.47064
\(899\) 20.6800 0.689716
\(900\) 0 0
\(901\) − 0.230055i − 0.00766425i
\(902\) 9.76106 0.325008
\(903\) 0 0
\(904\) 23.1527 0.770046
\(905\) 63.2964i 2.10404i
\(906\) 0 0
\(907\) 1.54767 0.0513894 0.0256947 0.999670i \(-0.491820\pi\)
0.0256947 + 0.999670i \(0.491820\pi\)
\(908\) 7.80841 0.259131
\(909\) 0 0
\(910\) 0 0
\(911\) − 38.9704i − 1.29115i −0.763699 0.645573i \(-0.776619\pi\)
0.763699 0.645573i \(-0.223381\pi\)
\(912\) 0 0
\(913\) − 11.8467i − 0.392069i
\(914\) 38.1931i 1.26331i
\(915\) 0 0
\(916\) 15.1967i 0.502114i
\(917\) 0 0
\(918\) 0 0
\(919\) 11.7398 0.387261 0.193630 0.981075i \(-0.437974\pi\)
0.193630 + 0.981075i \(0.437974\pi\)
\(920\) −7.82162 −0.257871
\(921\) 0 0
\(922\) − 15.3203i − 0.504546i
\(923\) −26.3819 −0.868372
\(924\) 0 0
\(925\) 40.2120 1.32216
\(926\) 21.6685i 0.712071i
\(927\) 0 0
\(928\) 31.8870 1.04674
\(929\) −44.6284 −1.46421 −0.732105 0.681192i \(-0.761462\pi\)
−0.732105 + 0.681192i \(0.761462\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 14.1043i − 0.462000i
\(933\) 0 0
\(934\) − 53.4824i − 1.75000i
\(935\) 14.1043i 0.461259i
\(936\) 0 0
\(937\) 33.3351i 1.08901i 0.838758 + 0.544505i \(0.183282\pi\)
−0.838758 + 0.544505i \(0.816718\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −13.2282 −0.431455
\(941\) −33.9984 −1.10832 −0.554159 0.832411i \(-0.686960\pi\)
−0.554159 + 0.832411i \(0.686960\pi\)
\(942\) 0 0
\(943\) − 8.04260i − 0.261903i
\(944\) 33.9984 1.10655
\(945\) 0 0
\(946\) 15.4633 0.502754
\(947\) − 46.1461i − 1.49955i −0.661694 0.749774i \(-0.730162\pi\)
0.661694 0.749774i \(-0.269838\pi\)
\(948\) 0 0
\(949\) 31.0109 1.00666
\(950\) −11.0028 −0.356977
\(951\) 0 0
\(952\) 0 0
\(953\) 35.1143i 1.13746i 0.822523 + 0.568731i \(0.192566\pi\)
−0.822523 + 0.568731i \(0.807434\pi\)
\(954\) 0 0
\(955\) 27.2890i 0.883050i
\(956\) 22.6234i 0.731692i
\(957\) 0 0
\(958\) 74.3301i 2.40150i
\(959\) 0 0
\(960\) 0 0
\(961\) 12.7003 0.409688
\(962\) 43.0028 1.38647
\(963\) 0 0
\(964\) − 32.6011i − 1.05001i
\(965\) 12.1017 0.389567
\(966\) 0 0
\(967\) −13.4148 −0.431392 −0.215696 0.976461i \(-0.569202\pi\)
−0.215696 + 0.976461i \(0.569202\pi\)
\(968\) 11.2391i 0.361238i
\(969\) 0 0
\(970\) −4.80785 −0.154371
\(971\) −18.3892 −0.590137 −0.295069 0.955476i \(-0.595343\pi\)
−0.295069 + 0.955476i \(0.595343\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 22.7782i 0.729861i
\(975\) 0 0
\(976\) 39.3563i 1.25976i
\(977\) 22.6530i 0.724734i 0.932035 + 0.362367i \(0.118031\pi\)
−0.932035 + 0.362367i \(0.881969\pi\)
\(978\) 0 0
\(979\) − 13.4647i − 0.430333i
\(980\) 0 0
\(981\) 0 0
\(982\) 11.3195 0.361220
\(983\) −39.3408 −1.25478 −0.627388 0.778707i \(-0.715876\pi\)
−0.627388 + 0.778707i \(0.715876\pi\)
\(984\) 0 0
\(985\) − 7.36151i − 0.234557i
\(986\) −28.1710 −0.897149
\(987\) 0 0
\(988\) −4.77184 −0.151813
\(989\) − 12.7409i − 0.405137i
\(990\) 0 0
\(991\) 0.951593 0.0302284 0.0151142 0.999886i \(-0.495189\pi\)
0.0151142 + 0.999886i \(0.495189\pi\)
\(992\) −28.2168 −0.895883
\(993\) 0 0
\(994\) 0 0
\(995\) 32.9804i 1.04555i
\(996\) 0 0
\(997\) − 31.4640i − 0.996475i −0.867041 0.498238i \(-0.833981\pi\)
0.867041 0.498238i \(-0.166019\pi\)
\(998\) − 5.43032i − 0.171894i
\(999\) 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 1323.2.c.d.1322.9 12
3.2 odd 2 inner 1323.2.c.d.1322.4 12
7.4 even 3 189.2.p.d.26.2 12
7.5 odd 6 189.2.p.d.80.5 yes 12
7.6 odd 2 inner 1323.2.c.d.1322.10 12
21.5 even 6 189.2.p.d.80.2 yes 12
21.11 odd 6 189.2.p.d.26.5 yes 12
21.20 even 2 inner 1323.2.c.d.1322.3 12
63.4 even 3 567.2.s.f.26.5 12
63.5 even 6 567.2.i.f.269.2 12
63.11 odd 6 567.2.i.f.215.2 12
63.25 even 3 567.2.i.f.215.5 12
63.32 odd 6 567.2.s.f.26.2 12
63.40 odd 6 567.2.i.f.269.5 12
63.47 even 6 567.2.s.f.458.5 12
63.61 odd 6 567.2.s.f.458.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.p.d.26.2 12 7.4 even 3
189.2.p.d.26.5 yes 12 21.11 odd 6
189.2.p.d.80.2 yes 12 21.5 even 6
189.2.p.d.80.5 yes 12 7.5 odd 6
567.2.i.f.215.2 12 63.11 odd 6
567.2.i.f.215.5 12 63.25 even 3
567.2.i.f.269.2 12 63.5 even 6
567.2.i.f.269.5 12 63.40 odd 6
567.2.s.f.26.2 12 63.32 odd 6
567.2.s.f.26.5 12 63.4 even 3
567.2.s.f.458.2 12 63.61 odd 6
567.2.s.f.458.5 12 63.47 even 6
1323.2.c.d.1322.3 12 21.20 even 2 inner
1323.2.c.d.1322.4 12 3.2 odd 2 inner
1323.2.c.d.1322.9 12 1.1 even 1 trivial
1323.2.c.d.1322.10 12 7.6 odd 2 inner