Properties

Label 5445.2.a.m.1.2
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 5445.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} +4.82843 q^{7} -1.58579 q^{8} +O(q^{10})\) \(q+0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} +4.82843 q^{7} -1.58579 q^{8} +0.414214 q^{10} -5.65685 q^{13} +2.00000 q^{14} +3.00000 q^{16} -6.82843 q^{17} +1.17157 q^{19} -1.82843 q^{20} +4.00000 q^{23} +1.00000 q^{25} -2.34315 q^{26} -8.82843 q^{28} +0.828427 q^{29} +4.41421 q^{32} -2.82843 q^{34} +4.82843 q^{35} +0.343146 q^{37} +0.485281 q^{38} -1.58579 q^{40} -0.828427 q^{41} +3.17157 q^{43} +1.65685 q^{46} +4.00000 q^{47} +16.3137 q^{49} +0.414214 q^{50} +10.3431 q^{52} +13.3137 q^{53} -7.65685 q^{56} +0.343146 q^{58} +4.00000 q^{59} +0.343146 q^{61} -4.17157 q^{64} -5.65685 q^{65} +5.65685 q^{67} +12.4853 q^{68} +2.00000 q^{70} -13.6569 q^{71} +11.3137 q^{73} +0.142136 q^{74} -2.14214 q^{76} +8.48528 q^{79} +3.00000 q^{80} -0.343146 q^{82} -10.0000 q^{83} -6.82843 q^{85} +1.31371 q^{86} +7.65685 q^{89} -27.3137 q^{91} -7.31371 q^{92} +1.65685 q^{94} +1.17157 q^{95} +0.343146 q^{97} +6.75736 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} + 2q^{5} + 4q^{7} - 6q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} + 2q^{5} + 4q^{7} - 6q^{8} - 2q^{10} + 4q^{14} + 6q^{16} - 8q^{17} + 8q^{19} + 2q^{20} + 8q^{23} + 2q^{25} - 16q^{26} - 12q^{28} - 4q^{29} + 6q^{32} + 4q^{35} + 12q^{37} - 16q^{38} - 6q^{40} + 4q^{41} + 12q^{43} - 8q^{46} + 8q^{47} + 10q^{49} - 2q^{50} + 32q^{52} + 4q^{53} - 4q^{56} + 12q^{58} + 8q^{59} + 12q^{61} - 14q^{64} + 8q^{68} + 4q^{70} - 16q^{71} - 28q^{74} + 24q^{76} + 6q^{80} - 12q^{82} - 20q^{83} - 8q^{85} - 20q^{86} + 4q^{89} - 32q^{91} + 8q^{92} - 8q^{94} + 8q^{95} + 12q^{97} + 22q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214 0.292893 0.146447 0.989219i \(-0.453216\pi\)
0.146447 + 0.989219i \(0.453216\pi\)
\(3\) 0 0
\(4\) −1.82843 −0.914214
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.82843 1.82497 0.912487 0.409106i \(-0.134159\pi\)
0.912487 + 0.409106i \(0.134159\pi\)
\(8\) −1.58579 −0.560660
\(9\) 0 0
\(10\) 0.414214 0.130986
\(11\) 0 0
\(12\) 0 0
\(13\) −5.65685 −1.56893 −0.784465 0.620174i \(-0.787062\pi\)
−0.784465 + 0.620174i \(0.787062\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) −6.82843 −1.65614 −0.828068 0.560627i \(-0.810560\pi\)
−0.828068 + 0.560627i \(0.810560\pi\)
\(18\) 0 0
\(19\) 1.17157 0.268777 0.134389 0.990929i \(-0.457093\pi\)
0.134389 + 0.990929i \(0.457093\pi\)
\(20\) −1.82843 −0.408849
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.34315 −0.459529
\(27\) 0 0
\(28\) −8.82843 −1.66842
\(29\) 0.828427 0.153835 0.0769175 0.997037i \(-0.475492\pi\)
0.0769175 + 0.997037i \(0.475492\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 4.41421 0.780330
\(33\) 0 0
\(34\) −2.82843 −0.485071
\(35\) 4.82843 0.816153
\(36\) 0 0
\(37\) 0.343146 0.0564128 0.0282064 0.999602i \(-0.491020\pi\)
0.0282064 + 0.999602i \(0.491020\pi\)
\(38\) 0.485281 0.0787230
\(39\) 0 0
\(40\) −1.58579 −0.250735
\(41\) −0.828427 −0.129379 −0.0646893 0.997905i \(-0.520606\pi\)
−0.0646893 + 0.997905i \(0.520606\pi\)
\(42\) 0 0
\(43\) 3.17157 0.483660 0.241830 0.970319i \(-0.422252\pi\)
0.241830 + 0.970319i \(0.422252\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.65685 0.244290
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) 16.3137 2.33053
\(50\) 0.414214 0.0585786
\(51\) 0 0
\(52\) 10.3431 1.43434
\(53\) 13.3137 1.82878 0.914389 0.404836i \(-0.132671\pi\)
0.914389 + 0.404836i \(0.132671\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −7.65685 −1.02319
\(57\) 0 0
\(58\) 0.343146 0.0450572
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 0.343146 0.0439353 0.0219677 0.999759i \(-0.493007\pi\)
0.0219677 + 0.999759i \(0.493007\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −4.17157 −0.521447
\(65\) −5.65685 −0.701646
\(66\) 0 0
\(67\) 5.65685 0.691095 0.345547 0.938401i \(-0.387693\pi\)
0.345547 + 0.938401i \(0.387693\pi\)
\(68\) 12.4853 1.51406
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) −13.6569 −1.62077 −0.810385 0.585897i \(-0.800742\pi\)
−0.810385 + 0.585897i \(0.800742\pi\)
\(72\) 0 0
\(73\) 11.3137 1.32417 0.662085 0.749429i \(-0.269672\pi\)
0.662085 + 0.749429i \(0.269672\pi\)
\(74\) 0.142136 0.0165229
\(75\) 0 0
\(76\) −2.14214 −0.245720
\(77\) 0 0
\(78\) 0 0
\(79\) 8.48528 0.954669 0.477334 0.878722i \(-0.341603\pi\)
0.477334 + 0.878722i \(0.341603\pi\)
\(80\) 3.00000 0.335410
\(81\) 0 0
\(82\) −0.343146 −0.0378941
\(83\) −10.0000 −1.09764 −0.548821 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(84\) 0 0
\(85\) −6.82843 −0.740647
\(86\) 1.31371 0.141661
\(87\) 0 0
\(88\) 0 0
\(89\) 7.65685 0.811625 0.405812 0.913956i \(-0.366989\pi\)
0.405812 + 0.913956i \(0.366989\pi\)
\(90\) 0 0
\(91\) −27.3137 −2.86325
\(92\) −7.31371 −0.762507
\(93\) 0 0
\(94\) 1.65685 0.170891
\(95\) 1.17157 0.120201
\(96\) 0 0
\(97\) 0.343146 0.0348412 0.0174206 0.999848i \(-0.494455\pi\)
0.0174206 + 0.999848i \(0.494455\pi\)
\(98\) 6.75736 0.682596
\(99\) 0 0
\(100\) −1.82843 −0.182843
\(101\) 4.82843 0.480446 0.240223 0.970718i \(-0.422779\pi\)
0.240223 + 0.970718i \(0.422779\pi\)
\(102\) 0 0
\(103\) 19.3137 1.90304 0.951518 0.307593i \(-0.0995234\pi\)
0.951518 + 0.307593i \(0.0995234\pi\)
\(104\) 8.97056 0.879636
\(105\) 0 0
\(106\) 5.51472 0.535637
\(107\) −5.31371 −0.513696 −0.256848 0.966452i \(-0.582684\pi\)
−0.256848 + 0.966452i \(0.582684\pi\)
\(108\) 0 0
\(109\) 5.31371 0.508961 0.254480 0.967078i \(-0.418096\pi\)
0.254480 + 0.967078i \(0.418096\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 14.4853 1.36873
\(113\) −14.9706 −1.40831 −0.704156 0.710045i \(-0.748674\pi\)
−0.704156 + 0.710045i \(0.748674\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) −1.51472 −0.140638
\(117\) 0 0
\(118\) 1.65685 0.152526
\(119\) −32.9706 −3.02241
\(120\) 0 0
\(121\) 0 0
\(122\) 0.142136 0.0128684
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −2.48528 −0.220533 −0.110267 0.993902i \(-0.535170\pi\)
−0.110267 + 0.993902i \(0.535170\pi\)
\(128\) −10.5563 −0.933058
\(129\) 0 0
\(130\) −2.34315 −0.205507
\(131\) −19.3137 −1.68745 −0.843723 0.536778i \(-0.819641\pi\)
−0.843723 + 0.536778i \(0.819641\pi\)
\(132\) 0 0
\(133\) 5.65685 0.490511
\(134\) 2.34315 0.202417
\(135\) 0 0
\(136\) 10.8284 0.928530
\(137\) −9.31371 −0.795724 −0.397862 0.917445i \(-0.630248\pi\)
−0.397862 + 0.917445i \(0.630248\pi\)
\(138\) 0 0
\(139\) 16.4853 1.39826 0.699132 0.714993i \(-0.253570\pi\)
0.699132 + 0.714993i \(0.253570\pi\)
\(140\) −8.82843 −0.746138
\(141\) 0 0
\(142\) −5.65685 −0.474713
\(143\) 0 0
\(144\) 0 0
\(145\) 0.828427 0.0687971
\(146\) 4.68629 0.387840
\(147\) 0 0
\(148\) −0.627417 −0.0515734
\(149\) 18.4853 1.51437 0.757187 0.653199i \(-0.226573\pi\)
0.757187 + 0.653199i \(0.226573\pi\)
\(150\) 0 0
\(151\) 0.485281 0.0394916 0.0197458 0.999805i \(-0.493714\pi\)
0.0197458 + 0.999805i \(0.493714\pi\)
\(152\) −1.85786 −0.150693
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 3.51472 0.279616
\(159\) 0 0
\(160\) 4.41421 0.348974
\(161\) 19.3137 1.52213
\(162\) 0 0
\(163\) 15.3137 1.19946 0.599731 0.800202i \(-0.295274\pi\)
0.599731 + 0.800202i \(0.295274\pi\)
\(164\) 1.51472 0.118280
\(165\) 0 0
\(166\) −4.14214 −0.321492
\(167\) 9.31371 0.720716 0.360358 0.932814i \(-0.382654\pi\)
0.360358 + 0.932814i \(0.382654\pi\)
\(168\) 0 0
\(169\) 19.0000 1.46154
\(170\) −2.82843 −0.216930
\(171\) 0 0
\(172\) −5.79899 −0.442169
\(173\) −2.82843 −0.215041 −0.107521 0.994203i \(-0.534291\pi\)
−0.107521 + 0.994203i \(0.534291\pi\)
\(174\) 0 0
\(175\) 4.82843 0.364995
\(176\) 0 0
\(177\) 0 0
\(178\) 3.17157 0.237719
\(179\) 6.34315 0.474109 0.237054 0.971496i \(-0.423818\pi\)
0.237054 + 0.971496i \(0.423818\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) −11.3137 −0.838628
\(183\) 0 0
\(184\) −6.34315 −0.467623
\(185\) 0.343146 0.0252286
\(186\) 0 0
\(187\) 0 0
\(188\) −7.31371 −0.533407
\(189\) 0 0
\(190\) 0.485281 0.0352060
\(191\) 5.65685 0.409316 0.204658 0.978834i \(-0.434392\pi\)
0.204658 + 0.978834i \(0.434392\pi\)
\(192\) 0 0
\(193\) −2.34315 −0.168663 −0.0843317 0.996438i \(-0.526876\pi\)
−0.0843317 + 0.996438i \(0.526876\pi\)
\(194\) 0.142136 0.0102047
\(195\) 0 0
\(196\) −29.8284 −2.13060
\(197\) 8.48528 0.604551 0.302276 0.953221i \(-0.402254\pi\)
0.302276 + 0.953221i \(0.402254\pi\)
\(198\) 0 0
\(199\) −10.3431 −0.733206 −0.366603 0.930377i \(-0.619479\pi\)
−0.366603 + 0.930377i \(0.619479\pi\)
\(200\) −1.58579 −0.112132
\(201\) 0 0
\(202\) 2.00000 0.140720
\(203\) 4.00000 0.280745
\(204\) 0 0
\(205\) −0.828427 −0.0578599
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) −16.9706 −1.17670
\(209\) 0 0
\(210\) 0 0
\(211\) −6.82843 −0.470088 −0.235044 0.971985i \(-0.575523\pi\)
−0.235044 + 0.971985i \(0.575523\pi\)
\(212\) −24.3431 −1.67189
\(213\) 0 0
\(214\) −2.20101 −0.150458
\(215\) 3.17157 0.216299
\(216\) 0 0
\(217\) 0 0
\(218\) 2.20101 0.149071
\(219\) 0 0
\(220\) 0 0
\(221\) 38.6274 2.59836
\(222\) 0 0
\(223\) −17.6569 −1.18239 −0.591195 0.806529i \(-0.701344\pi\)
−0.591195 + 0.806529i \(0.701344\pi\)
\(224\) 21.3137 1.42408
\(225\) 0 0
\(226\) −6.20101 −0.412485
\(227\) −14.0000 −0.929213 −0.464606 0.885517i \(-0.653804\pi\)
−0.464606 + 0.885517i \(0.653804\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 1.65685 0.109250
\(231\) 0 0
\(232\) −1.31371 −0.0862492
\(233\) −13.1716 −0.862898 −0.431449 0.902137i \(-0.641998\pi\)
−0.431449 + 0.902137i \(0.641998\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) −7.31371 −0.476082
\(237\) 0 0
\(238\) −13.6569 −0.885242
\(239\) 6.34315 0.410304 0.205152 0.978730i \(-0.434231\pi\)
0.205152 + 0.978730i \(0.434231\pi\)
\(240\) 0 0
\(241\) 23.6569 1.52387 0.761936 0.647652i \(-0.224249\pi\)
0.761936 + 0.647652i \(0.224249\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −0.627417 −0.0401663
\(245\) 16.3137 1.04224
\(246\) 0 0
\(247\) −6.62742 −0.421692
\(248\) 0 0
\(249\) 0 0
\(250\) 0.414214 0.0261972
\(251\) 12.9706 0.818695 0.409347 0.912379i \(-0.365756\pi\)
0.409347 + 0.912379i \(0.365756\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −1.02944 −0.0645926
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) 27.6569 1.72519 0.862594 0.505898i \(-0.168839\pi\)
0.862594 + 0.505898i \(0.168839\pi\)
\(258\) 0 0
\(259\) 1.65685 0.102952
\(260\) 10.3431 0.641455
\(261\) 0 0
\(262\) −8.00000 −0.494242
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) 13.3137 0.817855
\(266\) 2.34315 0.143667
\(267\) 0 0
\(268\) −10.3431 −0.631808
\(269\) 24.6274 1.50156 0.750780 0.660552i \(-0.229678\pi\)
0.750780 + 0.660552i \(0.229678\pi\)
\(270\) 0 0
\(271\) −27.7990 −1.68867 −0.844334 0.535817i \(-0.820004\pi\)
−0.844334 + 0.535817i \(0.820004\pi\)
\(272\) −20.4853 −1.24210
\(273\) 0 0
\(274\) −3.85786 −0.233062
\(275\) 0 0
\(276\) 0 0
\(277\) 13.6569 0.820561 0.410280 0.911959i \(-0.365431\pi\)
0.410280 + 0.911959i \(0.365431\pi\)
\(278\) 6.82843 0.409542
\(279\) 0 0
\(280\) −7.65685 −0.457585
\(281\) −16.8284 −1.00390 −0.501950 0.864897i \(-0.667384\pi\)
−0.501950 + 0.864897i \(0.667384\pi\)
\(282\) 0 0
\(283\) 3.17157 0.188530 0.0942652 0.995547i \(-0.469950\pi\)
0.0942652 + 0.995547i \(0.469950\pi\)
\(284\) 24.9706 1.48173
\(285\) 0 0
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) 0 0
\(289\) 29.6274 1.74279
\(290\) 0.343146 0.0201502
\(291\) 0 0
\(292\) −20.6863 −1.21057
\(293\) −1.17157 −0.0684440 −0.0342220 0.999414i \(-0.510895\pi\)
−0.0342220 + 0.999414i \(0.510895\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) −0.544156 −0.0316284
\(297\) 0 0
\(298\) 7.65685 0.443550
\(299\) −22.6274 −1.30858
\(300\) 0 0
\(301\) 15.3137 0.882667
\(302\) 0.201010 0.0115668
\(303\) 0 0
\(304\) 3.51472 0.201583
\(305\) 0.343146 0.0196485
\(306\) 0 0
\(307\) 8.82843 0.503865 0.251932 0.967745i \(-0.418934\pi\)
0.251932 + 0.967745i \(0.418934\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −19.3137 −1.09518 −0.547590 0.836747i \(-0.684455\pi\)
−0.547590 + 0.836747i \(0.684455\pi\)
\(312\) 0 0
\(313\) 4.34315 0.245489 0.122745 0.992438i \(-0.460830\pi\)
0.122745 + 0.992438i \(0.460830\pi\)
\(314\) 7.45584 0.420758
\(315\) 0 0
\(316\) −15.5147 −0.872771
\(317\) −30.2843 −1.70093 −0.850467 0.526028i \(-0.823681\pi\)
−0.850467 + 0.526028i \(0.823681\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −4.17157 −0.233198
\(321\) 0 0
\(322\) 8.00000 0.445823
\(323\) −8.00000 −0.445132
\(324\) 0 0
\(325\) −5.65685 −0.313786
\(326\) 6.34315 0.351314
\(327\) 0 0
\(328\) 1.31371 0.0725374
\(329\) 19.3137 1.06480
\(330\) 0 0
\(331\) 17.6569 0.970508 0.485254 0.874373i \(-0.338727\pi\)
0.485254 + 0.874373i \(0.338727\pi\)
\(332\) 18.2843 1.00348
\(333\) 0 0
\(334\) 3.85786 0.211093
\(335\) 5.65685 0.309067
\(336\) 0 0
\(337\) 19.3137 1.05208 0.526042 0.850458i \(-0.323675\pi\)
0.526042 + 0.850458i \(0.323675\pi\)
\(338\) 7.87006 0.428075
\(339\) 0 0
\(340\) 12.4853 0.677109
\(341\) 0 0
\(342\) 0 0
\(343\) 44.9706 2.42818
\(344\) −5.02944 −0.271169
\(345\) 0 0
\(346\) −1.17157 −0.0629841
\(347\) −6.68629 −0.358939 −0.179469 0.983764i \(-0.557438\pi\)
−0.179469 + 0.983764i \(0.557438\pi\)
\(348\) 0 0
\(349\) 22.9706 1.22959 0.614793 0.788688i \(-0.289240\pi\)
0.614793 + 0.788688i \(0.289240\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) 0 0
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 0 0
\(355\) −13.6569 −0.724831
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) 2.62742 0.138863
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) −17.6274 −0.927759
\(362\) −5.79899 −0.304788
\(363\) 0 0
\(364\) 49.9411 2.61763
\(365\) 11.3137 0.592187
\(366\) 0 0
\(367\) −1.65685 −0.0864871 −0.0432435 0.999065i \(-0.513769\pi\)
−0.0432435 + 0.999065i \(0.513769\pi\)
\(368\) 12.0000 0.625543
\(369\) 0 0
\(370\) 0.142136 0.00738928
\(371\) 64.2843 3.33747
\(372\) 0 0
\(373\) 34.6274 1.79294 0.896470 0.443105i \(-0.146123\pi\)
0.896470 + 0.443105i \(0.146123\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −6.34315 −0.327123
\(377\) −4.68629 −0.241356
\(378\) 0 0
\(379\) −0.686292 −0.0352524 −0.0176262 0.999845i \(-0.505611\pi\)
−0.0176262 + 0.999845i \(0.505611\pi\)
\(380\) −2.14214 −0.109889
\(381\) 0 0
\(382\) 2.34315 0.119886
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −0.970563 −0.0494003
\(387\) 0 0
\(388\) −0.627417 −0.0318523
\(389\) 12.3431 0.625822 0.312911 0.949782i \(-0.398696\pi\)
0.312911 + 0.949782i \(0.398696\pi\)
\(390\) 0 0
\(391\) −27.3137 −1.38131
\(392\) −25.8701 −1.30664
\(393\) 0 0
\(394\) 3.51472 0.177069
\(395\) 8.48528 0.426941
\(396\) 0 0
\(397\) 18.9706 0.952105 0.476053 0.879417i \(-0.342067\pi\)
0.476053 + 0.879417i \(0.342067\pi\)
\(398\) −4.28427 −0.214751
\(399\) 0 0
\(400\) 3.00000 0.150000
\(401\) 29.3137 1.46386 0.731928 0.681382i \(-0.238621\pi\)
0.731928 + 0.681382i \(0.238621\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −8.82843 −0.439231
\(405\) 0 0
\(406\) 1.65685 0.0822283
\(407\) 0 0
\(408\) 0 0
\(409\) 8.34315 0.412542 0.206271 0.978495i \(-0.433867\pi\)
0.206271 + 0.978495i \(0.433867\pi\)
\(410\) −0.343146 −0.0169468
\(411\) 0 0
\(412\) −35.3137 −1.73978
\(413\) 19.3137 0.950365
\(414\) 0 0
\(415\) −10.0000 −0.490881
\(416\) −24.9706 −1.22428
\(417\) 0 0
\(418\) 0 0
\(419\) 3.02944 0.147998 0.0739988 0.997258i \(-0.476424\pi\)
0.0739988 + 0.997258i \(0.476424\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) −2.82843 −0.137686
\(423\) 0 0
\(424\) −21.1127 −1.02532
\(425\) −6.82843 −0.331227
\(426\) 0 0
\(427\) 1.65685 0.0801808
\(428\) 9.71573 0.469627
\(429\) 0 0
\(430\) 1.31371 0.0633526
\(431\) 10.3431 0.498212 0.249106 0.968476i \(-0.419863\pi\)
0.249106 + 0.968476i \(0.419863\pi\)
\(432\) 0 0
\(433\) −4.34315 −0.208718 −0.104359 0.994540i \(-0.533279\pi\)
−0.104359 + 0.994540i \(0.533279\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −9.71573 −0.465299
\(437\) 4.68629 0.224176
\(438\) 0 0
\(439\) 3.51472 0.167748 0.0838742 0.996476i \(-0.473271\pi\)
0.0838742 + 0.996476i \(0.473271\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 16.0000 0.761042
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) 7.65685 0.362970
\(446\) −7.31371 −0.346314
\(447\) 0 0
\(448\) −20.1421 −0.951626
\(449\) 2.97056 0.140190 0.0700948 0.997540i \(-0.477670\pi\)
0.0700948 + 0.997540i \(0.477670\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 27.3726 1.28750
\(453\) 0 0
\(454\) −5.79899 −0.272160
\(455\) −27.3137 −1.28049
\(456\) 0 0
\(457\) 0.686292 0.0321034 0.0160517 0.999871i \(-0.494890\pi\)
0.0160517 + 0.999871i \(0.494890\pi\)
\(458\) −0.828427 −0.0387099
\(459\) 0 0
\(460\) −7.31371 −0.341003
\(461\) −28.1421 −1.31071 −0.655355 0.755321i \(-0.727481\pi\)
−0.655355 + 0.755321i \(0.727481\pi\)
\(462\) 0 0
\(463\) −28.9706 −1.34638 −0.673188 0.739471i \(-0.735076\pi\)
−0.673188 + 0.739471i \(0.735076\pi\)
\(464\) 2.48528 0.115376
\(465\) 0 0
\(466\) −5.45584 −0.252737
\(467\) −22.6274 −1.04707 −0.523536 0.852004i \(-0.675387\pi\)
−0.523536 + 0.852004i \(0.675387\pi\)
\(468\) 0 0
\(469\) 27.3137 1.26123
\(470\) 1.65685 0.0764250
\(471\) 0 0
\(472\) −6.34315 −0.291967
\(473\) 0 0
\(474\) 0 0
\(475\) 1.17157 0.0537555
\(476\) 60.2843 2.76313
\(477\) 0 0
\(478\) 2.62742 0.120175
\(479\) 3.02944 0.138419 0.0692093 0.997602i \(-0.477952\pi\)
0.0692093 + 0.997602i \(0.477952\pi\)
\(480\) 0 0
\(481\) −1.94113 −0.0885077
\(482\) 9.79899 0.446332
\(483\) 0 0
\(484\) 0 0
\(485\) 0.343146 0.0155814
\(486\) 0 0
\(487\) 20.9706 0.950267 0.475133 0.879914i \(-0.342400\pi\)
0.475133 + 0.879914i \(0.342400\pi\)
\(488\) −0.544156 −0.0246328
\(489\) 0 0
\(490\) 6.75736 0.305266
\(491\) 25.6569 1.15788 0.578939 0.815371i \(-0.303467\pi\)
0.578939 + 0.815371i \(0.303467\pi\)
\(492\) 0 0
\(493\) −5.65685 −0.254772
\(494\) −2.74517 −0.123511
\(495\) 0 0
\(496\) 0 0
\(497\) −65.9411 −2.95786
\(498\) 0 0
\(499\) −33.6569 −1.50669 −0.753344 0.657627i \(-0.771560\pi\)
−0.753344 + 0.657627i \(0.771560\pi\)
\(500\) −1.82843 −0.0817697
\(501\) 0 0
\(502\) 5.37258 0.239790
\(503\) −5.31371 −0.236927 −0.118463 0.992958i \(-0.537797\pi\)
−0.118463 + 0.992958i \(0.537797\pi\)
\(504\) 0 0
\(505\) 4.82843 0.214862
\(506\) 0 0
\(507\) 0 0
\(508\) 4.54416 0.201614
\(509\) −41.3137 −1.83120 −0.915599 0.402093i \(-0.868283\pi\)
−0.915599 + 0.402093i \(0.868283\pi\)
\(510\) 0 0
\(511\) 54.6274 2.41657
\(512\) 22.7574 1.00574
\(513\) 0 0
\(514\) 11.4558 0.505296
\(515\) 19.3137 0.851064
\(516\) 0 0
\(517\) 0 0
\(518\) 0.686292 0.0301539
\(519\) 0 0
\(520\) 8.97056 0.393385
\(521\) −12.6274 −0.553217 −0.276609 0.960983i \(-0.589211\pi\)
−0.276609 + 0.960983i \(0.589211\pi\)
\(522\) 0 0
\(523\) 26.4853 1.15812 0.579060 0.815285i \(-0.303420\pi\)
0.579060 + 0.815285i \(0.303420\pi\)
\(524\) 35.3137 1.54269
\(525\) 0 0
\(526\) 7.45584 0.325090
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 5.51472 0.239544
\(531\) 0 0
\(532\) −10.3431 −0.448432
\(533\) 4.68629 0.202986
\(534\) 0 0
\(535\) −5.31371 −0.229732
\(536\) −8.97056 −0.387469
\(537\) 0 0
\(538\) 10.2010 0.439797
\(539\) 0 0
\(540\) 0 0
\(541\) 5.31371 0.228454 0.114227 0.993455i \(-0.463561\pi\)
0.114227 + 0.993455i \(0.463561\pi\)
\(542\) −11.5147 −0.494600
\(543\) 0 0
\(544\) −30.1421 −1.29233
\(545\) 5.31371 0.227614
\(546\) 0 0
\(547\) −20.1421 −0.861216 −0.430608 0.902539i \(-0.641701\pi\)
−0.430608 + 0.902539i \(0.641701\pi\)
\(548\) 17.0294 0.727462
\(549\) 0 0
\(550\) 0 0
\(551\) 0.970563 0.0413474
\(552\) 0 0
\(553\) 40.9706 1.74225
\(554\) 5.65685 0.240337
\(555\) 0 0
\(556\) −30.1421 −1.27831
\(557\) 10.8284 0.458815 0.229408 0.973330i \(-0.426321\pi\)
0.229408 + 0.973330i \(0.426321\pi\)
\(558\) 0 0
\(559\) −17.9411 −0.758829
\(560\) 14.4853 0.612115
\(561\) 0 0
\(562\) −6.97056 −0.294035
\(563\) 20.3431 0.857361 0.428681 0.903456i \(-0.358979\pi\)
0.428681 + 0.903456i \(0.358979\pi\)
\(564\) 0 0
\(565\) −14.9706 −0.629816
\(566\) 1.31371 0.0552193
\(567\) 0 0
\(568\) 21.6569 0.908701
\(569\) 15.4558 0.647943 0.323971 0.946067i \(-0.394982\pi\)
0.323971 + 0.946067i \(0.394982\pi\)
\(570\) 0 0
\(571\) −0.485281 −0.0203084 −0.0101542 0.999948i \(-0.503232\pi\)
−0.0101542 + 0.999948i \(0.503232\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −1.65685 −0.0691558
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 12.2721 0.510451
\(579\) 0 0
\(580\) −1.51472 −0.0628953
\(581\) −48.2843 −2.00317
\(582\) 0 0
\(583\) 0 0
\(584\) −17.9411 −0.742409
\(585\) 0 0
\(586\) −0.485281 −0.0200468
\(587\) 30.6274 1.26413 0.632064 0.774916i \(-0.282208\pi\)
0.632064 + 0.774916i \(0.282208\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 1.65685 0.0682116
\(591\) 0 0
\(592\) 1.02944 0.0423096
\(593\) −17.1716 −0.705152 −0.352576 0.935783i \(-0.614694\pi\)
−0.352576 + 0.935783i \(0.614694\pi\)
\(594\) 0 0
\(595\) −32.9706 −1.35166
\(596\) −33.7990 −1.38446
\(597\) 0 0
\(598\) −9.37258 −0.383273
\(599\) −4.68629 −0.191477 −0.0957383 0.995407i \(-0.530521\pi\)
−0.0957383 + 0.995407i \(0.530521\pi\)
\(600\) 0 0
\(601\) −17.3137 −0.706241 −0.353120 0.935578i \(-0.614879\pi\)
−0.353120 + 0.935578i \(0.614879\pi\)
\(602\) 6.34315 0.258527
\(603\) 0 0
\(604\) −0.887302 −0.0361038
\(605\) 0 0
\(606\) 0 0
\(607\) −18.4853 −0.750294 −0.375147 0.926965i \(-0.622408\pi\)
−0.375147 + 0.926965i \(0.622408\pi\)
\(608\) 5.17157 0.209735
\(609\) 0 0
\(610\) 0.142136 0.00575490
\(611\) −22.6274 −0.915407
\(612\) 0 0
\(613\) −21.9411 −0.886194 −0.443097 0.896474i \(-0.646120\pi\)
−0.443097 + 0.896474i \(0.646120\pi\)
\(614\) 3.65685 0.147579
\(615\) 0 0
\(616\) 0 0
\(617\) −11.6569 −0.469287 −0.234644 0.972081i \(-0.575392\pi\)
−0.234644 + 0.972081i \(0.575392\pi\)
\(618\) 0 0
\(619\) −25.6569 −1.03124 −0.515618 0.856819i \(-0.672438\pi\)
−0.515618 + 0.856819i \(0.672438\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −8.00000 −0.320771
\(623\) 36.9706 1.48119
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 1.79899 0.0719021
\(627\) 0 0
\(628\) −32.9117 −1.31332
\(629\) −2.34315 −0.0934273
\(630\) 0 0
\(631\) 34.3431 1.36718 0.683590 0.729867i \(-0.260418\pi\)
0.683590 + 0.729867i \(0.260418\pi\)
\(632\) −13.4558 −0.535245
\(633\) 0 0
\(634\) −12.5442 −0.498192
\(635\) −2.48528 −0.0986254
\(636\) 0 0
\(637\) −92.2843 −3.65644
\(638\) 0 0
\(639\) 0 0
\(640\) −10.5563 −0.417276
\(641\) 26.9706 1.06527 0.532637 0.846344i \(-0.321201\pi\)
0.532637 + 0.846344i \(0.321201\pi\)
\(642\) 0 0
\(643\) −29.9411 −1.18076 −0.590381 0.807124i \(-0.701023\pi\)
−0.590381 + 0.807124i \(0.701023\pi\)
\(644\) −35.3137 −1.39156
\(645\) 0 0
\(646\) −3.31371 −0.130376
\(647\) −27.3137 −1.07381 −0.536906 0.843642i \(-0.680407\pi\)
−0.536906 + 0.843642i \(0.680407\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −2.34315 −0.0919057
\(651\) 0 0
\(652\) −28.0000 −1.09656
\(653\) 26.9706 1.05544 0.527720 0.849418i \(-0.323047\pi\)
0.527720 + 0.849418i \(0.323047\pi\)
\(654\) 0 0
\(655\) −19.3137 −0.754649
\(656\) −2.48528 −0.0970339
\(657\) 0 0
\(658\) 8.00000 0.311872
\(659\) 7.31371 0.284902 0.142451 0.989802i \(-0.454502\pi\)
0.142451 + 0.989802i \(0.454502\pi\)
\(660\) 0 0
\(661\) −13.3137 −0.517843 −0.258922 0.965898i \(-0.583367\pi\)
−0.258922 + 0.965898i \(0.583367\pi\)
\(662\) 7.31371 0.284255
\(663\) 0 0
\(664\) 15.8579 0.615404
\(665\) 5.65685 0.219363
\(666\) 0 0
\(667\) 3.31371 0.128307
\(668\) −17.0294 −0.658889
\(669\) 0 0
\(670\) 2.34315 0.0905236
\(671\) 0 0
\(672\) 0 0
\(673\) −29.6569 −1.14319 −0.571594 0.820537i \(-0.693675\pi\)
−0.571594 + 0.820537i \(0.693675\pi\)
\(674\) 8.00000 0.308148
\(675\) 0 0
\(676\) −34.7401 −1.33616
\(677\) −21.4558 −0.824615 −0.412308 0.911045i \(-0.635277\pi\)
−0.412308 + 0.911045i \(0.635277\pi\)
\(678\) 0 0
\(679\) 1.65685 0.0635842
\(680\) 10.8284 0.415251
\(681\) 0 0
\(682\) 0 0
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 0 0
\(685\) −9.31371 −0.355859
\(686\) 18.6274 0.711198
\(687\) 0 0
\(688\) 9.51472 0.362745
\(689\) −75.3137 −2.86922
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 5.17157 0.196594
\(693\) 0 0
\(694\) −2.76955 −0.105131
\(695\) 16.4853 0.625322
\(696\) 0 0
\(697\) 5.65685 0.214269
\(698\) 9.51472 0.360137
\(699\) 0 0
\(700\) −8.82843 −0.333683
\(701\) 7.85786 0.296787 0.148394 0.988928i \(-0.452590\pi\)
0.148394 + 0.988928i \(0.452590\pi\)
\(702\) 0 0
\(703\) 0.402020 0.0151625
\(704\) 0 0
\(705\) 0 0
\(706\) −10.7696 −0.405317
\(707\) 23.3137 0.876802
\(708\) 0 0
\(709\) 29.3137 1.10090 0.550450 0.834868i \(-0.314456\pi\)
0.550450 + 0.834868i \(0.314456\pi\)
\(710\) −5.65685 −0.212298
\(711\) 0 0
\(712\) −12.1421 −0.455046
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −11.5980 −0.433437
\(717\) 0 0
\(718\) 4.97056 0.185500
\(719\) −31.5980 −1.17841 −0.589203 0.807985i \(-0.700558\pi\)
−0.589203 + 0.807985i \(0.700558\pi\)
\(720\) 0 0
\(721\) 93.2548 3.47299
\(722\) −7.30152 −0.271734
\(723\) 0 0
\(724\) 25.5980 0.951341
\(725\) 0.828427 0.0307670
\(726\) 0 0
\(727\) −33.9411 −1.25881 −0.629403 0.777079i \(-0.716701\pi\)
−0.629403 + 0.777079i \(0.716701\pi\)
\(728\) 43.3137 1.60531
\(729\) 0 0
\(730\) 4.68629 0.173447
\(731\) −21.6569 −0.801008
\(732\) 0 0
\(733\) 17.6569 0.652171 0.326085 0.945340i \(-0.394270\pi\)
0.326085 + 0.945340i \(0.394270\pi\)
\(734\) −0.686292 −0.0253315
\(735\) 0 0
\(736\) 17.6569 0.650840
\(737\) 0 0
\(738\) 0 0
\(739\) −47.1127 −1.73307 −0.866534 0.499118i \(-0.833658\pi\)
−0.866534 + 0.499118i \(0.833658\pi\)
\(740\) −0.627417 −0.0230643
\(741\) 0 0
\(742\) 26.6274 0.977523
\(743\) 47.6569 1.74836 0.874180 0.485602i \(-0.161399\pi\)
0.874180 + 0.485602i \(0.161399\pi\)
\(744\) 0 0
\(745\) 18.4853 0.677248
\(746\) 14.3431 0.525140
\(747\) 0 0
\(748\) 0 0
\(749\) −25.6569 −0.937481
\(750\) 0 0
\(751\) −36.2843 −1.32403 −0.662016 0.749490i \(-0.730299\pi\)
−0.662016 + 0.749490i \(0.730299\pi\)
\(752\) 12.0000 0.437595
\(753\) 0 0
\(754\) −1.94113 −0.0706916
\(755\) 0.485281 0.0176612
\(756\) 0 0
\(757\) 8.62742 0.313569 0.156784 0.987633i \(-0.449887\pi\)
0.156784 + 0.987633i \(0.449887\pi\)
\(758\) −0.284271 −0.0103252
\(759\) 0 0
\(760\) −1.85786 −0.0673918
\(761\) −23.1716 −0.839969 −0.419984 0.907531i \(-0.637964\pi\)
−0.419984 + 0.907531i \(0.637964\pi\)
\(762\) 0 0
\(763\) 25.6569 0.928840
\(764\) −10.3431 −0.374202
\(765\) 0 0
\(766\) 3.31371 0.119729
\(767\) −22.6274 −0.817029
\(768\) 0 0
\(769\) −33.3137 −1.20132 −0.600662 0.799503i \(-0.705096\pi\)
−0.600662 + 0.799503i \(0.705096\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.28427 0.154194
\(773\) 7.65685 0.275398 0.137699 0.990474i \(-0.456029\pi\)
0.137699 + 0.990474i \(0.456029\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −0.544156 −0.0195341
\(777\) 0 0
\(778\) 5.11270 0.183299
\(779\) −0.970563 −0.0347740
\(780\) 0 0
\(781\) 0 0
\(782\) −11.3137 −0.404577
\(783\) 0 0
\(784\) 48.9411 1.74790
\(785\) 18.0000 0.642448
\(786\) 0 0
\(787\) −8.14214 −0.290236 −0.145118 0.989414i \(-0.546356\pi\)
−0.145118 + 0.989414i \(0.546356\pi\)
\(788\) −15.5147 −0.552689
\(789\) 0 0
\(790\) 3.51472 0.125048
\(791\) −72.2843 −2.57013
\(792\) 0 0
\(793\) −1.94113 −0.0689314
\(794\) 7.85786 0.278865
\(795\) 0 0
\(796\) 18.9117 0.670307
\(797\) −1.02944 −0.0364645 −0.0182323 0.999834i \(-0.505804\pi\)
−0.0182323 + 0.999834i \(0.505804\pi\)
\(798\) 0 0
\(799\) −27.3137 −0.966290
\(800\) 4.41421 0.156066
\(801\) 0 0
\(802\) 12.1421 0.428754
\(803\) 0 0
\(804\) 0 0
\(805\) 19.3137 0.680719
\(806\) 0 0
\(807\) 0 0
\(808\) −7.65685 −0.269367
\(809\) −56.4264 −1.98385 −0.991923 0.126838i \(-0.959517\pi\)
−0.991923 + 0.126838i \(0.959517\pi\)
\(810\) 0 0
\(811\) 16.4853 0.578877 0.289438 0.957197i \(-0.406532\pi\)
0.289438 + 0.957197i \(0.406532\pi\)
\(812\) −7.31371 −0.256661
\(813\) 0 0
\(814\) 0 0
\(815\) 15.3137 0.536416
\(816\) 0 0
\(817\) 3.71573 0.129997
\(818\) 3.45584 0.120831
\(819\) 0 0
\(820\) 1.51472 0.0528963
\(821\) −7.17157 −0.250290 −0.125145 0.992138i \(-0.539940\pi\)
−0.125145 + 0.992138i \(0.539940\pi\)
\(822\) 0 0
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) −30.6274 −1.06696
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) 18.6863 0.649786 0.324893 0.945751i \(-0.394672\pi\)
0.324893 + 0.945751i \(0.394672\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) −4.14214 −0.143776
\(831\) 0 0
\(832\) 23.5980 0.818113
\(833\) −111.397 −3.85968
\(834\) 0 0
\(835\) 9.31371 0.322314
\(836\) 0 0
\(837\) 0 0
\(838\) 1.25483 0.0433475
\(839\) −22.6274 −0.781185 −0.390593 0.920564i \(-0.627730\pi\)
−0.390593 + 0.920564i \(0.627730\pi\)
\(840\) 0 0
\(841\) −28.3137 −0.976335
\(842\) −2.48528 −0.0856485
\(843\) 0 0
\(844\) 12.4853 0.429761
\(845\) 19.0000 0.653620
\(846\) 0 0
\(847\) 0 0
\(848\) 39.9411 1.37158
\(849\) 0 0
\(850\) −2.82843 −0.0970143
\(851\) 1.37258 0.0470515
\(852\) 0 0
\(853\) 31.3137 1.07216 0.536080 0.844167i \(-0.319904\pi\)
0.536080 + 0.844167i \(0.319904\pi\)
\(854\) 0.686292 0.0234844
\(855\) 0 0
\(856\) 8.42641 0.288009
\(857\) −11.5147 −0.393335 −0.196668 0.980470i \(-0.563012\pi\)
−0.196668 + 0.980470i \(0.563012\pi\)
\(858\) 0 0
\(859\) −19.0294 −0.649276 −0.324638 0.945838i \(-0.605242\pi\)
−0.324638 + 0.945838i \(0.605242\pi\)
\(860\) −5.79899 −0.197744
\(861\) 0 0
\(862\) 4.28427 0.145923
\(863\) 43.3137 1.47442 0.737208 0.675666i \(-0.236144\pi\)
0.737208 + 0.675666i \(0.236144\pi\)
\(864\) 0 0
\(865\) −2.82843 −0.0961694
\(866\) −1.79899 −0.0611322
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −32.0000 −1.08428
\(872\) −8.42641 −0.285354
\(873\) 0 0
\(874\) 1.94113 0.0656595
\(875\) 4.82843 0.163231
\(876\) 0 0
\(877\) 42.6274 1.43943 0.719713 0.694272i \(-0.244273\pi\)
0.719713 + 0.694272i \(0.244273\pi\)
\(878\) 1.45584 0.0491324
\(879\) 0 0
\(880\) 0 0
\(881\) 13.0294 0.438973 0.219486 0.975616i \(-0.429562\pi\)
0.219486 + 0.975616i \(0.429562\pi\)
\(882\) 0 0
\(883\) −50.6274 −1.70375 −0.851874 0.523747i \(-0.824534\pi\)
−0.851874 + 0.523747i \(0.824534\pi\)
\(884\) −70.6274 −2.37546
\(885\) 0 0
\(886\) −4.97056 −0.166989
\(887\) −4.34315 −0.145829 −0.0729143 0.997338i \(-0.523230\pi\)
−0.0729143 + 0.997338i \(0.523230\pi\)
\(888\) 0 0
\(889\) −12.0000 −0.402467
\(890\) 3.17157 0.106311
\(891\) 0 0
\(892\) 32.2843 1.08096
\(893\) 4.68629 0.156821
\(894\) 0 0
\(895\) 6.34315 0.212028
\(896\) −50.9706 −1.70281
\(897\) 0 0
\(898\) 1.23045 0.0410606
\(899\) 0 0
\(900\) 0 0
\(901\) −90.9117 −3.02871
\(902\) 0 0
\(903\) 0 0
\(904\) 23.7401 0.789584
\(905\) −14.0000 −0.465376
\(906\) 0 0
\(907\) 7.02944 0.233409 0.116704 0.993167i \(-0.462767\pi\)
0.116704 + 0.993167i \(0.462767\pi\)
\(908\) 25.5980 0.849499
\(909\) 0 0
\(910\) −11.3137 −0.375046
\(911\) 15.0294 0.497947 0.248974 0.968510i \(-0.419907\pi\)
0.248974 + 0.968510i \(0.419907\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0.284271 0.00940286
\(915\) 0 0
\(916\) 3.65685 0.120826
\(917\) −93.2548 −3.07955
\(918\) 0 0
\(919\) −28.4853 −0.939643 −0.469821 0.882762i \(-0.655682\pi\)
−0.469821 + 0.882762i \(0.655682\pi\)
\(920\) −6.34315 −0.209127
\(921\) 0 0
\(922\) −11.6569 −0.383898
\(923\) 77.2548 2.54287
\(924\) 0 0
\(925\) 0.343146 0.0112826
\(926\) −12.0000 −0.394344
\(927\) 0 0
\(928\) 3.65685 0.120042
\(929\) −33.5980 −1.10231 −0.551157 0.834402i \(-0.685813\pi\)
−0.551157 + 0.834402i \(0.685813\pi\)
\(930\) 0 0
\(931\) 19.1127 0.626393
\(932\) 24.0833 0.788873
\(933\) 0 0
\(934\) −9.37258 −0.306680
\(935\) 0 0
\(936\) 0 0
\(937\) −44.9706 −1.46912 −0.734562 0.678541i \(-0.762612\pi\)
−0.734562 + 0.678541i \(0.762612\pi\)
\(938\) 11.3137 0.369406
\(939\) 0 0
\(940\) −7.31371 −0.238547
\(941\) 38.7696 1.26385 0.631926 0.775029i \(-0.282265\pi\)
0.631926 + 0.775029i \(0.282265\pi\)
\(942\) 0 0
\(943\) −3.31371 −0.107909
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) 38.6274 1.25522 0.627611 0.778527i \(-0.284033\pi\)
0.627611 + 0.778527i \(0.284033\pi\)
\(948\) 0 0
\(949\) −64.0000 −2.07753
\(950\) 0.485281 0.0157446
\(951\) 0 0
\(952\) 52.2843 1.69454
\(953\) 27.7990 0.900498 0.450249 0.892903i \(-0.351335\pi\)
0.450249 + 0.892903i \(0.351335\pi\)
\(954\) 0 0
\(955\) 5.65685 0.183052
\(956\) −11.5980 −0.375105
\(957\) 0 0
\(958\) 1.25483 0.0405418
\(959\) −44.9706 −1.45218
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −0.804041 −0.0259233
\(963\) 0 0
\(964\) −43.2548 −1.39314
\(965\) −2.34315 −0.0754285
\(966\) 0 0
\(967\) 39.4558 1.26881 0.634407 0.772999i \(-0.281244\pi\)
0.634407 + 0.772999i \(0.281244\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0.142136 0.00456370
\(971\) −10.6274 −0.341050 −0.170525 0.985353i \(-0.554546\pi\)
−0.170525 + 0.985353i \(0.554546\pi\)
\(972\) 0 0
\(973\) 79.5980 2.55179
\(974\) 8.68629 0.278327
\(975\) 0 0
\(976\) 1.02944 0.0329515
\(977\) −25.3137 −0.809857 −0.404929 0.914348i \(-0.632704\pi\)
−0.404929 + 0.914348i \(0.632704\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −29.8284 −0.952834
\(981\) 0 0
\(982\) 10.6274 0.339135
\(983\) −14.6274 −0.466542 −0.233271 0.972412i \(-0.574943\pi\)
−0.233271 + 0.972412i \(0.574943\pi\)
\(984\) 0 0
\(985\) 8.48528 0.270364
\(986\) −2.34315 −0.0746210
\(987\) 0 0
\(988\) 12.1177 0.385517
\(989\) 12.6863 0.403401
\(990\) 0 0
\(991\) 14.6274 0.464655 0.232328 0.972638i \(-0.425366\pi\)
0.232328 + 0.972638i \(0.425366\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −27.3137 −0.866338
\(995\) −10.3431 −0.327900
\(996\) 0 0
\(997\) 16.6863 0.528460 0.264230 0.964460i \(-0.414882\pi\)
0.264230 + 0.964460i \(0.414882\pi\)
\(998\) −13.9411 −0.441299
\(999\) 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.m.1.2 2
3.2 odd 2 1815.2.a.k.1.1 2
11.10 odd 2 495.2.a.d.1.1 2
15.14 odd 2 9075.2.a.v.1.2 2
33.32 even 2 165.2.a.a.1.2 2
44.43 even 2 7920.2.a.cg.1.2 2
55.32 even 4 2475.2.c.m.199.2 4
55.43 even 4 2475.2.c.m.199.3 4
55.54 odd 2 2475.2.a.m.1.2 2
132.131 odd 2 2640.2.a.bb.1.2 2
165.32 odd 4 825.2.c.e.199.3 4
165.98 odd 4 825.2.c.e.199.2 4
165.164 even 2 825.2.a.g.1.1 2
231.230 odd 2 8085.2.a.ba.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.a.1.2 2 33.32 even 2
495.2.a.d.1.1 2 11.10 odd 2
825.2.a.g.1.1 2 165.164 even 2
825.2.c.e.199.2 4 165.98 odd 4
825.2.c.e.199.3 4 165.32 odd 4
1815.2.a.k.1.1 2 3.2 odd 2
2475.2.a.m.1.2 2 55.54 odd 2
2475.2.c.m.199.2 4 55.32 even 4
2475.2.c.m.199.3 4 55.43 even 4
2640.2.a.bb.1.2 2 132.131 odd 2
5445.2.a.m.1.2 2 1.1 even 1 trivial
7920.2.a.cg.1.2 2 44.43 even 2
8085.2.a.ba.1.2 2 231.230 odd 2
9075.2.a.v.1.2 2 15.14 odd 2