Properties

Label 5445.2.a.ca.1.3
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5445.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.4785439006\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 4 x^{7} - 3 x^{6} + 22 x^{5} - 3 x^{4} - 32 x^{3} + 9 x^{2} + 8 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 495)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.226007\) of defining polynomial
Character \(\chi\) \(=\) 5445.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.22601 q^{2} -0.496906 q^{4} -1.00000 q^{5} -0.451695 q^{7} +3.06123 q^{8} +O(q^{10})\) \(q-1.22601 q^{2} -0.496906 q^{4} -1.00000 q^{5} -0.451695 q^{7} +3.06123 q^{8} +1.22601 q^{10} -4.84034 q^{13} +0.553781 q^{14} -2.75927 q^{16} +0.740078 q^{17} +6.80375 q^{19} +0.496906 q^{20} -0.00634166 q^{23} +1.00000 q^{25} +5.93429 q^{26} +0.224450 q^{28} -0.323900 q^{29} -5.60583 q^{31} -2.73956 q^{32} -0.907341 q^{34} +0.451695 q^{35} +7.36894 q^{37} -8.34144 q^{38} -3.06123 q^{40} -10.9705 q^{41} -1.80668 q^{43} +0.00777492 q^{46} -1.67948 q^{47} -6.79597 q^{49} -1.22601 q^{50} +2.40520 q^{52} +8.63948 q^{53} -1.38274 q^{56} +0.397104 q^{58} -1.50054 q^{59} +13.0454 q^{61} +6.87279 q^{62} +8.87727 q^{64} +4.84034 q^{65} +9.60773 q^{67} -0.367750 q^{68} -0.553781 q^{70} +11.4126 q^{71} +10.2399 q^{73} -9.03438 q^{74} -3.38083 q^{76} -2.09965 q^{79} +2.75927 q^{80} +13.4499 q^{82} -15.8945 q^{83} -0.740078 q^{85} +2.21500 q^{86} -9.36925 q^{89} +2.18636 q^{91} +0.00315121 q^{92} +2.05906 q^{94} -6.80375 q^{95} +5.87983 q^{97} +8.33191 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 6 q^{4} - 8 q^{5} + 8 q^{7} - 12 q^{8} + O(q^{10}) \) \( 8 q - 4 q^{2} + 6 q^{4} - 8 q^{5} + 8 q^{7} - 12 q^{8} + 4 q^{10} + 6 q^{13} - 14 q^{14} + 14 q^{16} - 8 q^{17} - 2 q^{19} - 6 q^{20} - 4 q^{23} + 8 q^{25} + 2 q^{26} + 24 q^{28} - 22 q^{29} + 10 q^{31} - 28 q^{32} - 2 q^{34} - 8 q^{35} - 14 q^{37} + 20 q^{38} + 12 q^{40} - 22 q^{41} + 14 q^{43} - 2 q^{46} - 10 q^{47} - 4 q^{50} - 10 q^{52} + 18 q^{53} - 34 q^{56} + 12 q^{58} - 2 q^{59} - 14 q^{61} - 30 q^{62} + 30 q^{64} - 6 q^{65} + 10 q^{67} - 6 q^{68} + 14 q^{70} + 2 q^{71} + 16 q^{73} - 24 q^{74} - 22 q^{76} - 16 q^{79} - 14 q^{80} + 10 q^{82} - 46 q^{83} + 8 q^{85} + 28 q^{86} - 38 q^{89} + 8 q^{91} + 24 q^{92} - 10 q^{94} + 2 q^{95} - 4 q^{97} - 4 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.22601 −0.866918 −0.433459 0.901173i \(-0.642707\pi\)
−0.433459 + 0.901173i \(0.642707\pi\)
\(3\) 0 0
\(4\) −0.496906 −0.248453
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.451695 −0.170725 −0.0853623 0.996350i \(-0.527205\pi\)
−0.0853623 + 0.996350i \(0.527205\pi\)
\(8\) 3.06123 1.08231
\(9\) 0 0
\(10\) 1.22601 0.387698
\(11\) 0 0
\(12\) 0 0
\(13\) −4.84034 −1.34247 −0.671235 0.741245i \(-0.734236\pi\)
−0.671235 + 0.741245i \(0.734236\pi\)
\(14\) 0.553781 0.148004
\(15\) 0 0
\(16\) −2.75927 −0.689818
\(17\) 0.740078 0.179495 0.0897477 0.995965i \(-0.471394\pi\)
0.0897477 + 0.995965i \(0.471394\pi\)
\(18\) 0 0
\(19\) 6.80375 1.56089 0.780443 0.625227i \(-0.214993\pi\)
0.780443 + 0.625227i \(0.214993\pi\)
\(20\) 0.496906 0.111112
\(21\) 0 0
\(22\) 0 0
\(23\) −0.00634166 −0.00132233 −0.000661164 1.00000i \(-0.500210\pi\)
−0.000661164 1.00000i \(0.500210\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 5.93429 1.16381
\(27\) 0 0
\(28\) 0.224450 0.0424171
\(29\) −0.323900 −0.0601468 −0.0300734 0.999548i \(-0.509574\pi\)
−0.0300734 + 0.999548i \(0.509574\pi\)
\(30\) 0 0
\(31\) −5.60583 −1.00684 −0.503419 0.864043i \(-0.667925\pi\)
−0.503419 + 0.864043i \(0.667925\pi\)
\(32\) −2.73956 −0.484291
\(33\) 0 0
\(34\) −0.907341 −0.155608
\(35\) 0.451695 0.0763504
\(36\) 0 0
\(37\) 7.36894 1.21145 0.605723 0.795675i \(-0.292884\pi\)
0.605723 + 0.795675i \(0.292884\pi\)
\(38\) −8.34144 −1.35316
\(39\) 0 0
\(40\) −3.06123 −0.484022
\(41\) −10.9705 −1.71330 −0.856649 0.515900i \(-0.827457\pi\)
−0.856649 + 0.515900i \(0.827457\pi\)
\(42\) 0 0
\(43\) −1.80668 −0.275516 −0.137758 0.990466i \(-0.543990\pi\)
−0.137758 + 0.990466i \(0.543990\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0.00777492 0.00114635
\(47\) −1.67948 −0.244978 −0.122489 0.992470i \(-0.539088\pi\)
−0.122489 + 0.992470i \(0.539088\pi\)
\(48\) 0 0
\(49\) −6.79597 −0.970853
\(50\) −1.22601 −0.173384
\(51\) 0 0
\(52\) 2.40520 0.333541
\(53\) 8.63948 1.18672 0.593362 0.804936i \(-0.297800\pi\)
0.593362 + 0.804936i \(0.297800\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.38274 −0.184776
\(57\) 0 0
\(58\) 0.397104 0.0521423
\(59\) −1.50054 −0.195353 −0.0976766 0.995218i \(-0.531141\pi\)
−0.0976766 + 0.995218i \(0.531141\pi\)
\(60\) 0 0
\(61\) 13.0454 1.67029 0.835147 0.550027i \(-0.185383\pi\)
0.835147 + 0.550027i \(0.185383\pi\)
\(62\) 6.87279 0.872845
\(63\) 0 0
\(64\) 8.87727 1.10966
\(65\) 4.84034 0.600371
\(66\) 0 0
\(67\) 9.60773 1.17377 0.586885 0.809670i \(-0.300354\pi\)
0.586885 + 0.809670i \(0.300354\pi\)
\(68\) −0.367750 −0.0445962
\(69\) 0 0
\(70\) −0.553781 −0.0661895
\(71\) 11.4126 1.35442 0.677211 0.735789i \(-0.263188\pi\)
0.677211 + 0.735789i \(0.263188\pi\)
\(72\) 0 0
\(73\) 10.2399 1.19849 0.599243 0.800567i \(-0.295468\pi\)
0.599243 + 0.800567i \(0.295468\pi\)
\(74\) −9.03438 −1.05023
\(75\) 0 0
\(76\) −3.38083 −0.387807
\(77\) 0 0
\(78\) 0 0
\(79\) −2.09965 −0.236229 −0.118115 0.993000i \(-0.537685\pi\)
−0.118115 + 0.993000i \(0.537685\pi\)
\(80\) 2.75927 0.308496
\(81\) 0 0
\(82\) 13.4499 1.48529
\(83\) −15.8945 −1.74465 −0.872323 0.488929i \(-0.837388\pi\)
−0.872323 + 0.488929i \(0.837388\pi\)
\(84\) 0 0
\(85\) −0.740078 −0.0802727
\(86\) 2.21500 0.238850
\(87\) 0 0
\(88\) 0 0
\(89\) −9.36925 −0.993138 −0.496569 0.867997i \(-0.665407\pi\)
−0.496569 + 0.867997i \(0.665407\pi\)
\(90\) 0 0
\(91\) 2.18636 0.229193
\(92\) 0.00315121 0.000328536 0
\(93\) 0 0
\(94\) 2.05906 0.212375
\(95\) −6.80375 −0.698050
\(96\) 0 0
\(97\) 5.87983 0.597007 0.298503 0.954409i \(-0.403513\pi\)
0.298503 + 0.954409i \(0.403513\pi\)
\(98\) 8.33191 0.841650
\(99\) 0 0
\(100\) −0.496906 −0.0496906
\(101\) −2.20797 −0.219701 −0.109851 0.993948i \(-0.535037\pi\)
−0.109851 + 0.993948i \(0.535037\pi\)
\(102\) 0 0
\(103\) 9.67943 0.953743 0.476871 0.878973i \(-0.341771\pi\)
0.476871 + 0.878973i \(0.341771\pi\)
\(104\) −14.8174 −1.45296
\(105\) 0 0
\(106\) −10.5921 −1.02879
\(107\) −11.7564 −1.13654 −0.568269 0.822843i \(-0.692387\pi\)
−0.568269 + 0.822843i \(0.692387\pi\)
\(108\) 0 0
\(109\) −12.5091 −1.19816 −0.599078 0.800691i \(-0.704466\pi\)
−0.599078 + 0.800691i \(0.704466\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.24635 0.117769
\(113\) 9.90608 0.931885 0.465943 0.884815i \(-0.345715\pi\)
0.465943 + 0.884815i \(0.345715\pi\)
\(114\) 0 0
\(115\) 0.00634166 0.000591363 0
\(116\) 0.160948 0.0149437
\(117\) 0 0
\(118\) 1.83967 0.169355
\(119\) −0.334290 −0.0306443
\(120\) 0 0
\(121\) 0 0
\(122\) −15.9938 −1.44801
\(123\) 0 0
\(124\) 2.78557 0.250152
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −0.561561 −0.0498304 −0.0249152 0.999690i \(-0.507932\pi\)
−0.0249152 + 0.999690i \(0.507932\pi\)
\(128\) −5.40447 −0.477692
\(129\) 0 0
\(130\) −5.93429 −0.520472
\(131\) 3.03500 0.265170 0.132585 0.991172i \(-0.457672\pi\)
0.132585 + 0.991172i \(0.457672\pi\)
\(132\) 0 0
\(133\) −3.07322 −0.266482
\(134\) −11.7791 −1.01756
\(135\) 0 0
\(136\) 2.26555 0.194269
\(137\) −20.7004 −1.76855 −0.884275 0.466966i \(-0.845347\pi\)
−0.884275 + 0.466966i \(0.845347\pi\)
\(138\) 0 0
\(139\) 3.54873 0.300999 0.150500 0.988610i \(-0.451912\pi\)
0.150500 + 0.988610i \(0.451912\pi\)
\(140\) −0.224450 −0.0189695
\(141\) 0 0
\(142\) −13.9919 −1.17417
\(143\) 0 0
\(144\) 0 0
\(145\) 0.323900 0.0268985
\(146\) −12.5542 −1.03899
\(147\) 0 0
\(148\) −3.66168 −0.300988
\(149\) 6.30459 0.516492 0.258246 0.966079i \(-0.416855\pi\)
0.258246 + 0.966079i \(0.416855\pi\)
\(150\) 0 0
\(151\) −3.68432 −0.299826 −0.149913 0.988699i \(-0.547899\pi\)
−0.149913 + 0.988699i \(0.547899\pi\)
\(152\) 20.8278 1.68936
\(153\) 0 0
\(154\) 0 0
\(155\) 5.60583 0.450271
\(156\) 0 0
\(157\) 2.82869 0.225754 0.112877 0.993609i \(-0.463993\pi\)
0.112877 + 0.993609i \(0.463993\pi\)
\(158\) 2.57419 0.204792
\(159\) 0 0
\(160\) 2.73956 0.216582
\(161\) 0.00286450 0.000225754 0
\(162\) 0 0
\(163\) 8.85141 0.693296 0.346648 0.937995i \(-0.387320\pi\)
0.346648 + 0.937995i \(0.387320\pi\)
\(164\) 5.45129 0.425674
\(165\) 0 0
\(166\) 19.4868 1.51247
\(167\) −19.6842 −1.52321 −0.761605 0.648042i \(-0.775588\pi\)
−0.761605 + 0.648042i \(0.775588\pi\)
\(168\) 0 0
\(169\) 10.4289 0.802224
\(170\) 0.907341 0.0695899
\(171\) 0 0
\(172\) 0.897750 0.0684528
\(173\) −2.91136 −0.221347 −0.110673 0.993857i \(-0.535301\pi\)
−0.110673 + 0.993857i \(0.535301\pi\)
\(174\) 0 0
\(175\) −0.451695 −0.0341449
\(176\) 0 0
\(177\) 0 0
\(178\) 11.4868 0.860969
\(179\) 6.29309 0.470368 0.235184 0.971951i \(-0.424431\pi\)
0.235184 + 0.971951i \(0.424431\pi\)
\(180\) 0 0
\(181\) −11.1642 −0.829826 −0.414913 0.909861i \(-0.636188\pi\)
−0.414913 + 0.909861i \(0.636188\pi\)
\(182\) −2.68049 −0.198691
\(183\) 0 0
\(184\) −0.0194132 −0.00143116
\(185\) −7.36894 −0.541776
\(186\) 0 0
\(187\) 0 0
\(188\) 0.834545 0.0608655
\(189\) 0 0
\(190\) 8.34144 0.605152
\(191\) 20.3946 1.47570 0.737851 0.674964i \(-0.235841\pi\)
0.737851 + 0.674964i \(0.235841\pi\)
\(192\) 0 0
\(193\) −2.97433 −0.214097 −0.107049 0.994254i \(-0.534140\pi\)
−0.107049 + 0.994254i \(0.534140\pi\)
\(194\) −7.20872 −0.517556
\(195\) 0 0
\(196\) 3.37696 0.241212
\(197\) −5.20127 −0.370575 −0.185288 0.982684i \(-0.559322\pi\)
−0.185288 + 0.982684i \(0.559322\pi\)
\(198\) 0 0
\(199\) 8.10264 0.574381 0.287191 0.957873i \(-0.407279\pi\)
0.287191 + 0.957873i \(0.407279\pi\)
\(200\) 3.06123 0.216461
\(201\) 0 0
\(202\) 2.70699 0.190463
\(203\) 0.146304 0.0102685
\(204\) 0 0
\(205\) 10.9705 0.766210
\(206\) −11.8671 −0.826817
\(207\) 0 0
\(208\) 13.3558 0.926059
\(209\) 0 0
\(210\) 0 0
\(211\) −1.31335 −0.0904146 −0.0452073 0.998978i \(-0.514395\pi\)
−0.0452073 + 0.998978i \(0.514395\pi\)
\(212\) −4.29301 −0.294845
\(213\) 0 0
\(214\) 14.4135 0.985286
\(215\) 1.80668 0.123214
\(216\) 0 0
\(217\) 2.53213 0.171892
\(218\) 15.3363 1.03870
\(219\) 0 0
\(220\) 0 0
\(221\) −3.58223 −0.240967
\(222\) 0 0
\(223\) 28.6347 1.91752 0.958761 0.284214i \(-0.0917325\pi\)
0.958761 + 0.284214i \(0.0917325\pi\)
\(224\) 1.23745 0.0826804
\(225\) 0 0
\(226\) −12.1449 −0.807868
\(227\) 7.86603 0.522087 0.261043 0.965327i \(-0.415933\pi\)
0.261043 + 0.965327i \(0.415933\pi\)
\(228\) 0 0
\(229\) −16.1510 −1.06729 −0.533644 0.845710i \(-0.679178\pi\)
−0.533644 + 0.845710i \(0.679178\pi\)
\(230\) −0.00777492 −0.000512663 0
\(231\) 0 0
\(232\) −0.991532 −0.0650973
\(233\) −18.4345 −1.20768 −0.603841 0.797105i \(-0.706364\pi\)
−0.603841 + 0.797105i \(0.706364\pi\)
\(234\) 0 0
\(235\) 1.67948 0.109557
\(236\) 0.745626 0.0485361
\(237\) 0 0
\(238\) 0.409841 0.0265661
\(239\) −22.3638 −1.44660 −0.723298 0.690536i \(-0.757375\pi\)
−0.723298 + 0.690536i \(0.757375\pi\)
\(240\) 0 0
\(241\) −13.2213 −0.851662 −0.425831 0.904803i \(-0.640018\pi\)
−0.425831 + 0.904803i \(0.640018\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −6.48235 −0.414990
\(245\) 6.79597 0.434179
\(246\) 0 0
\(247\) −32.9325 −2.09544
\(248\) −17.1607 −1.08971
\(249\) 0 0
\(250\) 1.22601 0.0775395
\(251\) 21.0032 1.32571 0.662854 0.748749i \(-0.269345\pi\)
0.662854 + 0.748749i \(0.269345\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0.688477 0.0431989
\(255\) 0 0
\(256\) −11.1286 −0.695539
\(257\) 3.36772 0.210073 0.105036 0.994468i \(-0.466504\pi\)
0.105036 + 0.994468i \(0.466504\pi\)
\(258\) 0 0
\(259\) −3.32852 −0.206824
\(260\) −2.40520 −0.149164
\(261\) 0 0
\(262\) −3.72094 −0.229880
\(263\) −21.0450 −1.29769 −0.648845 0.760920i \(-0.724748\pi\)
−0.648845 + 0.760920i \(0.724748\pi\)
\(264\) 0 0
\(265\) −8.63948 −0.530719
\(266\) 3.76779 0.231018
\(267\) 0 0
\(268\) −4.77414 −0.291627
\(269\) −23.3816 −1.42560 −0.712800 0.701367i \(-0.752573\pi\)
−0.712800 + 0.701367i \(0.752573\pi\)
\(270\) 0 0
\(271\) −20.5474 −1.24817 −0.624084 0.781357i \(-0.714528\pi\)
−0.624084 + 0.781357i \(0.714528\pi\)
\(272\) −2.04208 −0.123819
\(273\) 0 0
\(274\) 25.3788 1.53319
\(275\) 0 0
\(276\) 0 0
\(277\) −19.5818 −1.17656 −0.588280 0.808658i \(-0.700195\pi\)
−0.588280 + 0.808658i \(0.700195\pi\)
\(278\) −4.35077 −0.260942
\(279\) 0 0
\(280\) 1.38274 0.0826345
\(281\) −13.7926 −0.822800 −0.411400 0.911455i \(-0.634960\pi\)
−0.411400 + 0.911455i \(0.634960\pi\)
\(282\) 0 0
\(283\) 31.5177 1.87353 0.936765 0.349958i \(-0.113804\pi\)
0.936765 + 0.349958i \(0.113804\pi\)
\(284\) −5.67098 −0.336511
\(285\) 0 0
\(286\) 0 0
\(287\) 4.95530 0.292502
\(288\) 0 0
\(289\) −16.4523 −0.967781
\(290\) −0.397104 −0.0233188
\(291\) 0 0
\(292\) −5.08826 −0.297768
\(293\) 23.3131 1.36196 0.680982 0.732300i \(-0.261553\pi\)
0.680982 + 0.732300i \(0.261553\pi\)
\(294\) 0 0
\(295\) 1.50054 0.0873646
\(296\) 22.5580 1.31116
\(297\) 0 0
\(298\) −7.72948 −0.447757
\(299\) 0.0306958 0.00177518
\(300\) 0 0
\(301\) 0.816068 0.0470374
\(302\) 4.51700 0.259924
\(303\) 0 0
\(304\) −18.7734 −1.07673
\(305\) −13.0454 −0.746978
\(306\) 0 0
\(307\) −10.0938 −0.576083 −0.288042 0.957618i \(-0.593004\pi\)
−0.288042 + 0.957618i \(0.593004\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −6.87279 −0.390348
\(311\) 13.3254 0.755614 0.377807 0.925884i \(-0.376678\pi\)
0.377807 + 0.925884i \(0.376678\pi\)
\(312\) 0 0
\(313\) 23.1682 1.30955 0.654773 0.755825i \(-0.272764\pi\)
0.654773 + 0.755825i \(0.272764\pi\)
\(314\) −3.46799 −0.195710
\(315\) 0 0
\(316\) 1.04333 0.0586920
\(317\) −6.93189 −0.389334 −0.194667 0.980869i \(-0.562363\pi\)
−0.194667 + 0.980869i \(0.562363\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −8.87727 −0.496254
\(321\) 0 0
\(322\) −0.00351189 −0.000195710 0
\(323\) 5.03530 0.280172
\(324\) 0 0
\(325\) −4.84034 −0.268494
\(326\) −10.8519 −0.601031
\(327\) 0 0
\(328\) −33.5830 −1.85431
\(329\) 0.758613 0.0418237
\(330\) 0 0
\(331\) −10.5717 −0.581075 −0.290538 0.956864i \(-0.593834\pi\)
−0.290538 + 0.956864i \(0.593834\pi\)
\(332\) 7.89807 0.433463
\(333\) 0 0
\(334\) 24.1330 1.32050
\(335\) −9.60773 −0.524926
\(336\) 0 0
\(337\) −6.34668 −0.345726 −0.172863 0.984946i \(-0.555302\pi\)
−0.172863 + 0.984946i \(0.555302\pi\)
\(338\) −12.7859 −0.695463
\(339\) 0 0
\(340\) 0.367750 0.0199440
\(341\) 0 0
\(342\) 0 0
\(343\) 6.23157 0.336473
\(344\) −5.53065 −0.298193
\(345\) 0 0
\(346\) 3.56935 0.191890
\(347\) −25.2417 −1.35505 −0.677524 0.735501i \(-0.736947\pi\)
−0.677524 + 0.735501i \(0.736947\pi\)
\(348\) 0 0
\(349\) −11.1035 −0.594359 −0.297179 0.954822i \(-0.596046\pi\)
−0.297179 + 0.954822i \(0.596046\pi\)
\(350\) 0.553781 0.0296009
\(351\) 0 0
\(352\) 0 0
\(353\) −18.8552 −1.00356 −0.501780 0.864996i \(-0.667321\pi\)
−0.501780 + 0.864996i \(0.667321\pi\)
\(354\) 0 0
\(355\) −11.4126 −0.605716
\(356\) 4.65564 0.246748
\(357\) 0 0
\(358\) −7.71537 −0.407770
\(359\) −21.2928 −1.12379 −0.561895 0.827208i \(-0.689928\pi\)
−0.561895 + 0.827208i \(0.689928\pi\)
\(360\) 0 0
\(361\) 27.2910 1.43637
\(362\) 13.6873 0.719391
\(363\) 0 0
\(364\) −1.08642 −0.0569437
\(365\) −10.2399 −0.535980
\(366\) 0 0
\(367\) −31.2857 −1.63310 −0.816550 0.577275i \(-0.804116\pi\)
−0.816550 + 0.577275i \(0.804116\pi\)
\(368\) 0.0174984 0.000912165 0
\(369\) 0 0
\(370\) 9.03438 0.469675
\(371\) −3.90241 −0.202603
\(372\) 0 0
\(373\) 9.39368 0.486386 0.243193 0.969978i \(-0.421805\pi\)
0.243193 + 0.969978i \(0.421805\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −5.14127 −0.265141
\(377\) 1.56779 0.0807452
\(378\) 0 0
\(379\) −8.19727 −0.421065 −0.210533 0.977587i \(-0.567520\pi\)
−0.210533 + 0.977587i \(0.567520\pi\)
\(380\) 3.38083 0.173433
\(381\) 0 0
\(382\) −25.0039 −1.27931
\(383\) 33.6830 1.72112 0.860560 0.509349i \(-0.170114\pi\)
0.860560 + 0.509349i \(0.170114\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3.64656 0.185605
\(387\) 0 0
\(388\) −2.92173 −0.148328
\(389\) −32.9338 −1.66981 −0.834905 0.550395i \(-0.814477\pi\)
−0.834905 + 0.550395i \(0.814477\pi\)
\(390\) 0 0
\(391\) −0.00469332 −0.000237352 0
\(392\) −20.8040 −1.05076
\(393\) 0 0
\(394\) 6.37680 0.321259
\(395\) 2.09965 0.105645
\(396\) 0 0
\(397\) −15.3865 −0.772228 −0.386114 0.922451i \(-0.626183\pi\)
−0.386114 + 0.922451i \(0.626183\pi\)
\(398\) −9.93390 −0.497941
\(399\) 0 0
\(400\) −2.75927 −0.137964
\(401\) −6.64295 −0.331733 −0.165866 0.986148i \(-0.553042\pi\)
−0.165866 + 0.986148i \(0.553042\pi\)
\(402\) 0 0
\(403\) 27.1341 1.35165
\(404\) 1.09715 0.0545855
\(405\) 0 0
\(406\) −0.179370 −0.00890198
\(407\) 0 0
\(408\) 0 0
\(409\) 24.4850 1.21071 0.605353 0.795957i \(-0.293032\pi\)
0.605353 + 0.795957i \(0.293032\pi\)
\(410\) −13.4499 −0.664241
\(411\) 0 0
\(412\) −4.80977 −0.236960
\(413\) 0.677784 0.0333516
\(414\) 0 0
\(415\) 15.8945 0.780230
\(416\) 13.2604 0.650146
\(417\) 0 0
\(418\) 0 0
\(419\) −20.8656 −1.01935 −0.509676 0.860367i \(-0.670235\pi\)
−0.509676 + 0.860367i \(0.670235\pi\)
\(420\) 0 0
\(421\) −16.9311 −0.825172 −0.412586 0.910919i \(-0.635374\pi\)
−0.412586 + 0.910919i \(0.635374\pi\)
\(422\) 1.61017 0.0783821
\(423\) 0 0
\(424\) 26.4474 1.28440
\(425\) 0.740078 0.0358991
\(426\) 0 0
\(427\) −5.89255 −0.285160
\(428\) 5.84186 0.282377
\(429\) 0 0
\(430\) −2.21500 −0.106817
\(431\) −26.1647 −1.26031 −0.630154 0.776470i \(-0.717008\pi\)
−0.630154 + 0.776470i \(0.717008\pi\)
\(432\) 0 0
\(433\) −25.0593 −1.20427 −0.602137 0.798393i \(-0.705684\pi\)
−0.602137 + 0.798393i \(0.705684\pi\)
\(434\) −3.10440 −0.149016
\(435\) 0 0
\(436\) 6.21586 0.297686
\(437\) −0.0431470 −0.00206400
\(438\) 0 0
\(439\) 5.11234 0.243999 0.121999 0.992530i \(-0.461069\pi\)
0.121999 + 0.992530i \(0.461069\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 4.39184 0.208899
\(443\) −28.0773 −1.33399 −0.666995 0.745062i \(-0.732420\pi\)
−0.666995 + 0.745062i \(0.732420\pi\)
\(444\) 0 0
\(445\) 9.36925 0.444145
\(446\) −35.1064 −1.66233
\(447\) 0 0
\(448\) −4.00982 −0.189446
\(449\) −36.9951 −1.74591 −0.872953 0.487804i \(-0.837798\pi\)
−0.872953 + 0.487804i \(0.837798\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −4.92239 −0.231530
\(453\) 0 0
\(454\) −9.64381 −0.452606
\(455\) −2.18636 −0.102498
\(456\) 0 0
\(457\) 36.8462 1.72359 0.861795 0.507256i \(-0.169340\pi\)
0.861795 + 0.507256i \(0.169340\pi\)
\(458\) 19.8012 0.925250
\(459\) 0 0
\(460\) −0.00315121 −0.000146926 0
\(461\) −23.0013 −1.07128 −0.535640 0.844447i \(-0.679929\pi\)
−0.535640 + 0.844447i \(0.679929\pi\)
\(462\) 0 0
\(463\) −36.1571 −1.68036 −0.840181 0.542306i \(-0.817551\pi\)
−0.840181 + 0.542306i \(0.817551\pi\)
\(464\) 0.893729 0.0414903
\(465\) 0 0
\(466\) 22.6008 1.04696
\(467\) 2.26498 0.104811 0.0524053 0.998626i \(-0.483311\pi\)
0.0524053 + 0.998626i \(0.483311\pi\)
\(468\) 0 0
\(469\) −4.33976 −0.200392
\(470\) −2.05906 −0.0949772
\(471\) 0 0
\(472\) −4.59348 −0.211432
\(473\) 0 0
\(474\) 0 0
\(475\) 6.80375 0.312177
\(476\) 0.166111 0.00761367
\(477\) 0 0
\(478\) 27.4182 1.25408
\(479\) 30.5308 1.39499 0.697495 0.716590i \(-0.254298\pi\)
0.697495 + 0.716590i \(0.254298\pi\)
\(480\) 0 0
\(481\) −35.6682 −1.62633
\(482\) 16.2095 0.738321
\(483\) 0 0
\(484\) 0 0
\(485\) −5.87983 −0.266989
\(486\) 0 0
\(487\) −16.1543 −0.732021 −0.366010 0.930611i \(-0.619277\pi\)
−0.366010 + 0.930611i \(0.619277\pi\)
\(488\) 39.9349 1.80777
\(489\) 0 0
\(490\) −8.33191 −0.376397
\(491\) 11.6766 0.526960 0.263480 0.964665i \(-0.415130\pi\)
0.263480 + 0.964665i \(0.415130\pi\)
\(492\) 0 0
\(493\) −0.239712 −0.0107961
\(494\) 40.3754 1.81658
\(495\) 0 0
\(496\) 15.4680 0.694534
\(497\) −5.15500 −0.231233
\(498\) 0 0
\(499\) −40.9366 −1.83257 −0.916287 0.400522i \(-0.868829\pi\)
−0.916287 + 0.400522i \(0.868829\pi\)
\(500\) 0.496906 0.0222223
\(501\) 0 0
\(502\) −25.7500 −1.14928
\(503\) 0.424478 0.0189266 0.00946328 0.999955i \(-0.496988\pi\)
0.00946328 + 0.999955i \(0.496988\pi\)
\(504\) 0 0
\(505\) 2.20797 0.0982533
\(506\) 0 0
\(507\) 0 0
\(508\) 0.279043 0.0123805
\(509\) 19.8053 0.877854 0.438927 0.898523i \(-0.355359\pi\)
0.438927 + 0.898523i \(0.355359\pi\)
\(510\) 0 0
\(511\) −4.62530 −0.204611
\(512\) 24.4527 1.08067
\(513\) 0 0
\(514\) −4.12885 −0.182116
\(515\) −9.67943 −0.426527
\(516\) 0 0
\(517\) 0 0
\(518\) 4.08078 0.179299
\(519\) 0 0
\(520\) 14.8174 0.649785
\(521\) −36.8095 −1.61265 −0.806327 0.591470i \(-0.798548\pi\)
−0.806327 + 0.591470i \(0.798548\pi\)
\(522\) 0 0
\(523\) −10.5015 −0.459198 −0.229599 0.973285i \(-0.573742\pi\)
−0.229599 + 0.973285i \(0.573742\pi\)
\(524\) −1.50811 −0.0658822
\(525\) 0 0
\(526\) 25.8013 1.12499
\(527\) −4.14875 −0.180723
\(528\) 0 0
\(529\) −23.0000 −0.999998
\(530\) 10.5921 0.460090
\(531\) 0 0
\(532\) 1.52710 0.0662083
\(533\) 53.1008 2.30005
\(534\) 0 0
\(535\) 11.7564 0.508276
\(536\) 29.4114 1.27038
\(537\) 0 0
\(538\) 28.6660 1.23588
\(539\) 0 0
\(540\) 0 0
\(541\) 0.654613 0.0281440 0.0140720 0.999901i \(-0.495521\pi\)
0.0140720 + 0.999901i \(0.495521\pi\)
\(542\) 25.1913 1.08206
\(543\) 0 0
\(544\) −2.02749 −0.0869280
\(545\) 12.5091 0.535832
\(546\) 0 0
\(547\) −13.4159 −0.573621 −0.286810 0.957987i \(-0.592595\pi\)
−0.286810 + 0.957987i \(0.592595\pi\)
\(548\) 10.2861 0.439402
\(549\) 0 0
\(550\) 0 0
\(551\) −2.20374 −0.0938823
\(552\) 0 0
\(553\) 0.948403 0.0403302
\(554\) 24.0075 1.01998
\(555\) 0 0
\(556\) −1.76339 −0.0747843
\(557\) 13.2421 0.561087 0.280544 0.959841i \(-0.409485\pi\)
0.280544 + 0.959841i \(0.409485\pi\)
\(558\) 0 0
\(559\) 8.74494 0.369872
\(560\) −1.24635 −0.0526679
\(561\) 0 0
\(562\) 16.9099 0.713300
\(563\) 21.3619 0.900297 0.450149 0.892954i \(-0.351371\pi\)
0.450149 + 0.892954i \(0.351371\pi\)
\(564\) 0 0
\(565\) −9.90608 −0.416752
\(566\) −38.6409 −1.62420
\(567\) 0 0
\(568\) 34.9364 1.46590
\(569\) −45.3375 −1.90065 −0.950323 0.311267i \(-0.899247\pi\)
−0.950323 + 0.311267i \(0.899247\pi\)
\(570\) 0 0
\(571\) −25.1544 −1.05268 −0.526339 0.850275i \(-0.676436\pi\)
−0.526339 + 0.850275i \(0.676436\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −6.07523 −0.253575
\(575\) −0.00634166 −0.000264465 0
\(576\) 0 0
\(577\) 7.62653 0.317496 0.158748 0.987319i \(-0.449254\pi\)
0.158748 + 0.987319i \(0.449254\pi\)
\(578\) 20.1706 0.838987
\(579\) 0 0
\(580\) −0.160948 −0.00668301
\(581\) 7.17946 0.297854
\(582\) 0 0
\(583\) 0 0
\(584\) 31.3466 1.29713
\(585\) 0 0
\(586\) −28.5820 −1.18071
\(587\) 7.88825 0.325583 0.162791 0.986660i \(-0.447950\pi\)
0.162791 + 0.986660i \(0.447950\pi\)
\(588\) 0 0
\(589\) −38.1407 −1.57156
\(590\) −1.83967 −0.0757379
\(591\) 0 0
\(592\) −20.3329 −0.835678
\(593\) −8.68507 −0.356653 −0.178327 0.983971i \(-0.557068\pi\)
−0.178327 + 0.983971i \(0.557068\pi\)
\(594\) 0 0
\(595\) 0.334290 0.0137045
\(596\) −3.13279 −0.128324
\(597\) 0 0
\(598\) −0.0376333 −0.00153894
\(599\) −14.0016 −0.572088 −0.286044 0.958216i \(-0.592340\pi\)
−0.286044 + 0.958216i \(0.592340\pi\)
\(600\) 0 0
\(601\) −10.2936 −0.419886 −0.209943 0.977714i \(-0.567328\pi\)
−0.209943 + 0.977714i \(0.567328\pi\)
\(602\) −1.00050 −0.0407775
\(603\) 0 0
\(604\) 1.83076 0.0744927
\(605\) 0 0
\(606\) 0 0
\(607\) 27.4756 1.11520 0.557599 0.830110i \(-0.311723\pi\)
0.557599 + 0.830110i \(0.311723\pi\)
\(608\) −18.6393 −0.755923
\(609\) 0 0
\(610\) 15.9938 0.647569
\(611\) 8.12926 0.328875
\(612\) 0 0
\(613\) −10.8580 −0.438550 −0.219275 0.975663i \(-0.570369\pi\)
−0.219275 + 0.975663i \(0.570369\pi\)
\(614\) 12.3751 0.499417
\(615\) 0 0
\(616\) 0 0
\(617\) 18.2404 0.734330 0.367165 0.930156i \(-0.380329\pi\)
0.367165 + 0.930156i \(0.380329\pi\)
\(618\) 0 0
\(619\) 19.6799 0.791004 0.395502 0.918465i \(-0.370571\pi\)
0.395502 + 0.918465i \(0.370571\pi\)
\(620\) −2.78557 −0.111871
\(621\) 0 0
\(622\) −16.3370 −0.655056
\(623\) 4.23204 0.169553
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −28.4044 −1.13527
\(627\) 0 0
\(628\) −1.40559 −0.0560893
\(629\) 5.45359 0.217449
\(630\) 0 0
\(631\) −30.4249 −1.21120 −0.605598 0.795771i \(-0.707066\pi\)
−0.605598 + 0.795771i \(0.707066\pi\)
\(632\) −6.42751 −0.255673
\(633\) 0 0
\(634\) 8.49854 0.337520
\(635\) 0.561561 0.0222849
\(636\) 0 0
\(637\) 32.8948 1.30334
\(638\) 0 0
\(639\) 0 0
\(640\) 5.40447 0.213630
\(641\) 37.3528 1.47535 0.737674 0.675157i \(-0.235924\pi\)
0.737674 + 0.675157i \(0.235924\pi\)
\(642\) 0 0
\(643\) 1.06494 0.0419971 0.0209986 0.999780i \(-0.493315\pi\)
0.0209986 + 0.999780i \(0.493315\pi\)
\(644\) −0.00142339 −5.60893e−5 0
\(645\) 0 0
\(646\) −6.17332 −0.242886
\(647\) −15.8049 −0.621354 −0.310677 0.950516i \(-0.600556\pi\)
−0.310677 + 0.950516i \(0.600556\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 5.93429 0.232762
\(651\) 0 0
\(652\) −4.39832 −0.172252
\(653\) 35.3513 1.38340 0.691701 0.722184i \(-0.256862\pi\)
0.691701 + 0.722184i \(0.256862\pi\)
\(654\) 0 0
\(655\) −3.03500 −0.118587
\(656\) 30.2705 1.18186
\(657\) 0 0
\(658\) −0.930065 −0.0362577
\(659\) −28.4474 −1.10815 −0.554077 0.832465i \(-0.686929\pi\)
−0.554077 + 0.832465i \(0.686929\pi\)
\(660\) 0 0
\(661\) 39.5989 1.54022 0.770110 0.637911i \(-0.220201\pi\)
0.770110 + 0.637911i \(0.220201\pi\)
\(662\) 12.9610 0.503745
\(663\) 0 0
\(664\) −48.6566 −1.88824
\(665\) 3.07322 0.119174
\(666\) 0 0
\(667\) 0.00205407 7.95337e−5 0
\(668\) 9.78121 0.378446
\(669\) 0 0
\(670\) 11.7791 0.455068
\(671\) 0 0
\(672\) 0 0
\(673\) −17.5621 −0.676969 −0.338484 0.940972i \(-0.609914\pi\)
−0.338484 + 0.940972i \(0.609914\pi\)
\(674\) 7.78108 0.299716
\(675\) 0 0
\(676\) −5.18220 −0.199315
\(677\) −45.0690 −1.73214 −0.866071 0.499922i \(-0.833362\pi\)
−0.866071 + 0.499922i \(0.833362\pi\)
\(678\) 0 0
\(679\) −2.65589 −0.101924
\(680\) −2.26555 −0.0868797
\(681\) 0 0
\(682\) 0 0
\(683\) 15.7677 0.603334 0.301667 0.953413i \(-0.402457\pi\)
0.301667 + 0.953413i \(0.402457\pi\)
\(684\) 0 0
\(685\) 20.7004 0.790920
\(686\) −7.63995 −0.291695
\(687\) 0 0
\(688\) 4.98512 0.190056
\(689\) −41.8180 −1.59314
\(690\) 0 0
\(691\) 2.96619 0.112839 0.0564197 0.998407i \(-0.482032\pi\)
0.0564197 + 0.998407i \(0.482032\pi\)
\(692\) 1.44668 0.0549944
\(693\) 0 0
\(694\) 30.9465 1.17471
\(695\) −3.54873 −0.134611
\(696\) 0 0
\(697\) −8.11899 −0.307529
\(698\) 13.6130 0.515260
\(699\) 0 0
\(700\) 0.224450 0.00848342
\(701\) 37.1301 1.40238 0.701192 0.712973i \(-0.252652\pi\)
0.701192 + 0.712973i \(0.252652\pi\)
\(702\) 0 0
\(703\) 50.1364 1.89093
\(704\) 0 0
\(705\) 0 0
\(706\) 23.1166 0.870003
\(707\) 0.997329 0.0375084
\(708\) 0 0
\(709\) −31.7505 −1.19241 −0.596207 0.802831i \(-0.703326\pi\)
−0.596207 + 0.802831i \(0.703326\pi\)
\(710\) 13.9919 0.525106
\(711\) 0 0
\(712\) −28.6814 −1.07488
\(713\) 0.0355503 0.00133137
\(714\) 0 0
\(715\) 0 0
\(716\) −3.12708 −0.116864
\(717\) 0 0
\(718\) 26.1051 0.974234
\(719\) −31.8772 −1.18882 −0.594409 0.804163i \(-0.702614\pi\)
−0.594409 + 0.804163i \(0.702614\pi\)
\(720\) 0 0
\(721\) −4.37215 −0.162827
\(722\) −33.4589 −1.24521
\(723\) 0 0
\(724\) 5.54755 0.206173
\(725\) −0.323900 −0.0120294
\(726\) 0 0
\(727\) 41.4129 1.53592 0.767959 0.640499i \(-0.221272\pi\)
0.767959 + 0.640499i \(0.221272\pi\)
\(728\) 6.69294 0.248057
\(729\) 0 0
\(730\) 12.5542 0.464650
\(731\) −1.33708 −0.0494538
\(732\) 0 0
\(733\) 47.6920 1.76155 0.880773 0.473540i \(-0.157024\pi\)
0.880773 + 0.473540i \(0.157024\pi\)
\(734\) 38.3565 1.41576
\(735\) 0 0
\(736\) 0.0173734 0.000640391 0
\(737\) 0 0
\(738\) 0 0
\(739\) −12.4567 −0.458228 −0.229114 0.973400i \(-0.573583\pi\)
−0.229114 + 0.973400i \(0.573583\pi\)
\(740\) 3.66168 0.134606
\(741\) 0 0
\(742\) 4.78438 0.175640
\(743\) −27.3723 −1.00419 −0.502095 0.864812i \(-0.667437\pi\)
−0.502095 + 0.864812i \(0.667437\pi\)
\(744\) 0 0
\(745\) −6.30459 −0.230982
\(746\) −11.5167 −0.421657
\(747\) 0 0
\(748\) 0 0
\(749\) 5.31033 0.194035
\(750\) 0 0
\(751\) −41.8725 −1.52795 −0.763975 0.645246i \(-0.776755\pi\)
−0.763975 + 0.645246i \(0.776755\pi\)
\(752\) 4.63414 0.168990
\(753\) 0 0
\(754\) −1.92212 −0.0699995
\(755\) 3.68432 0.134086
\(756\) 0 0
\(757\) 16.1192 0.585863 0.292932 0.956133i \(-0.405369\pi\)
0.292932 + 0.956133i \(0.405369\pi\)
\(758\) 10.0499 0.365029
\(759\) 0 0
\(760\) −20.8278 −0.755504
\(761\) 10.9239 0.395990 0.197995 0.980203i \(-0.436557\pi\)
0.197995 + 0.980203i \(0.436557\pi\)
\(762\) 0 0
\(763\) 5.65030 0.204555
\(764\) −10.1342 −0.366643
\(765\) 0 0
\(766\) −41.2956 −1.49207
\(767\) 7.26311 0.262256
\(768\) 0 0
\(769\) −34.2074 −1.23355 −0.616775 0.787139i \(-0.711561\pi\)
−0.616775 + 0.787139i \(0.711561\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.47797 0.0531932
\(773\) −8.63688 −0.310647 −0.155323 0.987864i \(-0.549642\pi\)
−0.155323 + 0.987864i \(0.549642\pi\)
\(774\) 0 0
\(775\) −5.60583 −0.201367
\(776\) 17.9995 0.646144
\(777\) 0 0
\(778\) 40.3771 1.44759
\(779\) −74.6402 −2.67426
\(780\) 0 0
\(781\) 0 0
\(782\) 0.00575405 0.000205764 0
\(783\) 0 0
\(784\) 18.7519 0.669712
\(785\) −2.82869 −0.100960
\(786\) 0 0
\(787\) −23.1043 −0.823581 −0.411790 0.911279i \(-0.635096\pi\)
−0.411790 + 0.911279i \(0.635096\pi\)
\(788\) 2.58455 0.0920707
\(789\) 0 0
\(790\) −2.57419 −0.0915856
\(791\) −4.47453 −0.159096
\(792\) 0 0
\(793\) −63.1443 −2.24232
\(794\) 18.8640 0.669458
\(795\) 0 0
\(796\) −4.02626 −0.142707
\(797\) 39.7333 1.40743 0.703714 0.710484i \(-0.251524\pi\)
0.703714 + 0.710484i \(0.251524\pi\)
\(798\) 0 0
\(799\) −1.24295 −0.0439723
\(800\) −2.73956 −0.0968582
\(801\) 0 0
\(802\) 8.14430 0.287585
\(803\) 0 0
\(804\) 0 0
\(805\) −0.00286450 −0.000100960 0
\(806\) −33.2667 −1.17177
\(807\) 0 0
\(808\) −6.75909 −0.237784
\(809\) 31.0140 1.09039 0.545197 0.838308i \(-0.316455\pi\)
0.545197 + 0.838308i \(0.316455\pi\)
\(810\) 0 0
\(811\) 7.45922 0.261929 0.130964 0.991387i \(-0.458193\pi\)
0.130964 + 0.991387i \(0.458193\pi\)
\(812\) −0.0726995 −0.00255125
\(813\) 0 0
\(814\) 0 0
\(815\) −8.85141 −0.310051
\(816\) 0 0
\(817\) −12.2922 −0.430049
\(818\) −30.0188 −1.04958
\(819\) 0 0
\(820\) −5.45129 −0.190367
\(821\) 12.6556 0.441683 0.220841 0.975310i \(-0.429120\pi\)
0.220841 + 0.975310i \(0.429120\pi\)
\(822\) 0 0
\(823\) −27.2890 −0.951234 −0.475617 0.879652i \(-0.657775\pi\)
−0.475617 + 0.879652i \(0.657775\pi\)
\(824\) 29.6309 1.03224
\(825\) 0 0
\(826\) −0.830969 −0.0289131
\(827\) 16.9544 0.589563 0.294781 0.955565i \(-0.404753\pi\)
0.294781 + 0.955565i \(0.404753\pi\)
\(828\) 0 0
\(829\) 19.7794 0.686967 0.343483 0.939159i \(-0.388393\pi\)
0.343483 + 0.939159i \(0.388393\pi\)
\(830\) −19.4868 −0.676395
\(831\) 0 0
\(832\) −42.9690 −1.48968
\(833\) −5.02955 −0.174264
\(834\) 0 0
\(835\) 19.6842 0.681200
\(836\) 0 0
\(837\) 0 0
\(838\) 25.5814 0.883694
\(839\) 4.97974 0.171920 0.0859598 0.996299i \(-0.472604\pi\)
0.0859598 + 0.996299i \(0.472604\pi\)
\(840\) 0 0
\(841\) −28.8951 −0.996382
\(842\) 20.7577 0.715357
\(843\) 0 0
\(844\) 0.652611 0.0224638
\(845\) −10.4289 −0.358766
\(846\) 0 0
\(847\) 0 0
\(848\) −23.8387 −0.818623
\(849\) 0 0
\(850\) −0.907341 −0.0311215
\(851\) −0.0467313 −0.00160193
\(852\) 0 0
\(853\) −23.2464 −0.795943 −0.397971 0.917398i \(-0.630286\pi\)
−0.397971 + 0.917398i \(0.630286\pi\)
\(854\) 7.22430 0.247211
\(855\) 0 0
\(856\) −35.9891 −1.23008
\(857\) 30.6976 1.04861 0.524306 0.851530i \(-0.324325\pi\)
0.524306 + 0.851530i \(0.324325\pi\)
\(858\) 0 0
\(859\) −11.1830 −0.381560 −0.190780 0.981633i \(-0.561102\pi\)
−0.190780 + 0.981633i \(0.561102\pi\)
\(860\) −0.897750 −0.0306130
\(861\) 0 0
\(862\) 32.0781 1.09258
\(863\) 3.66335 0.124702 0.0623510 0.998054i \(-0.480140\pi\)
0.0623510 + 0.998054i \(0.480140\pi\)
\(864\) 0 0
\(865\) 2.91136 0.0989894
\(866\) 30.7229 1.04401
\(867\) 0 0
\(868\) −1.25823 −0.0427071
\(869\) 0 0
\(870\) 0 0
\(871\) −46.5047 −1.57575
\(872\) −38.2932 −1.29677
\(873\) 0 0
\(874\) 0.0528986 0.00178932
\(875\) 0.451695 0.0152701
\(876\) 0 0
\(877\) 7.36271 0.248621 0.124310 0.992243i \(-0.460328\pi\)
0.124310 + 0.992243i \(0.460328\pi\)
\(878\) −6.26776 −0.211527
\(879\) 0 0
\(880\) 0 0
\(881\) −6.07165 −0.204559 −0.102280 0.994756i \(-0.532614\pi\)
−0.102280 + 0.994756i \(0.532614\pi\)
\(882\) 0 0
\(883\) 31.1767 1.04918 0.524589 0.851356i \(-0.324219\pi\)
0.524589 + 0.851356i \(0.324219\pi\)
\(884\) 1.78003 0.0598690
\(885\) 0 0
\(886\) 34.4229 1.15646
\(887\) 5.68999 0.191051 0.0955255 0.995427i \(-0.469547\pi\)
0.0955255 + 0.995427i \(0.469547\pi\)
\(888\) 0 0
\(889\) 0.253654 0.00850729
\(890\) −11.4868 −0.385037
\(891\) 0 0
\(892\) −14.2288 −0.476414
\(893\) −11.4268 −0.382382
\(894\) 0 0
\(895\) −6.29309 −0.210355
\(896\) 2.44117 0.0815538
\(897\) 0 0
\(898\) 45.3563 1.51356
\(899\) 1.81573 0.0605580
\(900\) 0 0
\(901\) 6.39389 0.213011
\(902\) 0 0
\(903\) 0 0
\(904\) 30.3247 1.00859
\(905\) 11.1642 0.371109
\(906\) 0 0
\(907\) 42.4917 1.41091 0.705456 0.708754i \(-0.250742\pi\)
0.705456 + 0.708754i \(0.250742\pi\)
\(908\) −3.90868 −0.129714
\(909\) 0 0
\(910\) 2.68049 0.0888574
\(911\) 2.14405 0.0710356 0.0355178 0.999369i \(-0.488692\pi\)
0.0355178 + 0.999369i \(0.488692\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −45.1737 −1.49421
\(915\) 0 0
\(916\) 8.02553 0.265171
\(917\) −1.37090 −0.0452710
\(918\) 0 0
\(919\) 16.0683 0.530044 0.265022 0.964242i \(-0.414621\pi\)
0.265022 + 0.964242i \(0.414621\pi\)
\(920\) 0.0194132 0.000640036 0
\(921\) 0 0
\(922\) 28.1998 0.928711
\(923\) −55.2407 −1.81827
\(924\) 0 0
\(925\) 7.36894 0.242289
\(926\) 44.3289 1.45674
\(927\) 0 0
\(928\) 0.887346 0.0291286
\(929\) 37.4770 1.22958 0.614791 0.788690i \(-0.289240\pi\)
0.614791 + 0.788690i \(0.289240\pi\)
\(930\) 0 0
\(931\) −46.2381 −1.51539
\(932\) 9.16021 0.300053
\(933\) 0 0
\(934\) −2.77688 −0.0908621
\(935\) 0 0
\(936\) 0 0
\(937\) −26.0054 −0.849560 −0.424780 0.905297i \(-0.639649\pi\)
−0.424780 + 0.905297i \(0.639649\pi\)
\(938\) 5.32058 0.173723
\(939\) 0 0
\(940\) −0.834545 −0.0272199
\(941\) −8.58482 −0.279857 −0.139929 0.990162i \(-0.544687\pi\)
−0.139929 + 0.990162i \(0.544687\pi\)
\(942\) 0 0
\(943\) 0.0695709 0.00226554
\(944\) 4.14038 0.134758
\(945\) 0 0
\(946\) 0 0
\(947\) 36.0304 1.17083 0.585416 0.810733i \(-0.300931\pi\)
0.585416 + 0.810733i \(0.300931\pi\)
\(948\) 0 0
\(949\) −49.5645 −1.60893
\(950\) −8.34144 −0.270632
\(951\) 0 0
\(952\) −1.02334 −0.0331665
\(953\) 43.8525 1.42052 0.710261 0.703938i \(-0.248577\pi\)
0.710261 + 0.703938i \(0.248577\pi\)
\(954\) 0 0
\(955\) −20.3946 −0.659954
\(956\) 11.1127 0.359412
\(957\) 0 0
\(958\) −37.4310 −1.20934
\(959\) 9.35025 0.301935
\(960\) 0 0
\(961\) 0.425347 0.0137209
\(962\) 43.7295 1.40990
\(963\) 0 0
\(964\) 6.56977 0.211598
\(965\) 2.97433 0.0957472
\(966\) 0 0
\(967\) −48.1271 −1.54766 −0.773831 0.633392i \(-0.781662\pi\)
−0.773831 + 0.633392i \(0.781662\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 7.20872 0.231458
\(971\) −31.7334 −1.01837 −0.509187 0.860656i \(-0.670054\pi\)
−0.509187 + 0.860656i \(0.670054\pi\)
\(972\) 0 0
\(973\) −1.60294 −0.0513880
\(974\) 19.8053 0.634602
\(975\) 0 0
\(976\) −35.9958 −1.15220
\(977\) −43.1280 −1.37979 −0.689893 0.723911i \(-0.742342\pi\)
−0.689893 + 0.723911i \(0.742342\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −3.37696 −0.107873
\(981\) 0 0
\(982\) −14.3156 −0.456831
\(983\) −44.6922 −1.42546 −0.712729 0.701439i \(-0.752541\pi\)
−0.712729 + 0.701439i \(0.752541\pi\)
\(984\) 0 0
\(985\) 5.20127 0.165726
\(986\) 0.293888 0.00935931
\(987\) 0 0
\(988\) 16.3644 0.520619
\(989\) 0.0114573 0.000364322 0
\(990\) 0 0
\(991\) 13.7657 0.437282 0.218641 0.975805i \(-0.429838\pi\)
0.218641 + 0.975805i \(0.429838\pi\)
\(992\) 15.3575 0.487602
\(993\) 0 0
\(994\) 6.32007 0.200460
\(995\) −8.10264 −0.256871
\(996\) 0 0
\(997\) 4.89609 0.155061 0.0775303 0.996990i \(-0.475297\pi\)
0.0775303 + 0.996990i \(0.475297\pi\)
\(998\) 50.1886 1.58869
\(999\) 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 5445.2.a.ca.1.3 8
3.2 odd 2 5445.2.a.cd.1.6 8
11.5 even 5 495.2.n.h.91.3 yes 16
11.9 even 5 495.2.n.h.136.3 yes 16
11.10 odd 2 5445.2.a.cc.1.6 8
33.5 odd 10 495.2.n.g.91.2 16
33.20 odd 10 495.2.n.g.136.2 yes 16
33.32 even 2 5445.2.a.cb.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
495.2.n.g.91.2 16 33.5 odd 10
495.2.n.g.136.2 yes 16 33.20 odd 10
495.2.n.h.91.3 yes 16 11.5 even 5
495.2.n.h.136.3 yes 16 11.9 even 5
5445.2.a.ca.1.3 8 1.1 even 1 trivial
5445.2.a.cb.1.3 8 33.32 even 2
5445.2.a.cc.1.6 8 11.10 odd 2
5445.2.a.cd.1.6 8 3.2 odd 2