Properties

Label 1155.2.a.u.1.3
Level 1155
Weight 2
Character 1155.1
Self dual yes
Analytic conductor 9.223
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1155.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.13448.1
Defining polynomial: \(x^{4} - 7 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.546295\) of defining polynomial
Character \(\chi\) \(=\) 1155.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.546295 q^{2} +1.00000 q^{3} -1.70156 q^{4} -1.00000 q^{5} +0.546295 q^{6} -1.00000 q^{7} -2.02214 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.546295 q^{2} +1.00000 q^{3} -1.70156 q^{4} -1.00000 q^{5} +0.546295 q^{6} -1.00000 q^{7} -2.02214 q^{8} +1.00000 q^{9} -0.546295 q^{10} -1.00000 q^{11} -1.70156 q^{12} -0.568438 q^{13} -0.546295 q^{14} -1.00000 q^{15} +2.29844 q^{16} +2.70156 q^{17} +0.546295 q^{18} +6.62049 q^{19} +1.70156 q^{20} -1.00000 q^{21} -0.546295 q^{22} +8.93103 q^{23} -2.02214 q^{24} +1.00000 q^{25} -0.310535 q^{26} +1.00000 q^{27} +1.70156 q^{28} +5.27000 q^{29} -0.546295 q^{30} +9.66103 q^{31} +5.29991 q^{32} -1.00000 q^{33} +1.47585 q^{34} +1.00000 q^{35} -1.70156 q^{36} -2.75362 q^{37} +3.61674 q^{38} -0.568438 q^{39} +2.02214 q^{40} -10.9831 q^{41} -0.546295 q^{42} -0.0520550 q^{43} +1.70156 q^{44} -1.00000 q^{45} +4.87897 q^{46} -10.1567 q^{47} +2.29844 q^{48} +1.00000 q^{49} +0.546295 q^{50} +2.70156 q^{51} +0.967233 q^{52} +1.56469 q^{53} +0.546295 q^{54} +1.00000 q^{55} +2.02214 q^{56} +6.62049 q^{57} +2.87897 q^{58} +6.36259 q^{59} +1.70156 q^{60} -3.71308 q^{61} +5.27777 q^{62} -1.00000 q^{63} -1.70156 q^{64} +0.568438 q^{65} -0.546295 q^{66} -10.4146 q^{67} -4.59688 q^{68} +8.93103 q^{69} +0.546295 q^{70} +2.56844 q^{71} -2.02214 q^{72} +2.26625 q^{73} -1.50429 q^{74} +1.00000 q^{75} -11.2652 q^{76} +1.00000 q^{77} -0.310535 q^{78} -2.56844 q^{79} -2.29844 q^{80} +1.00000 q^{81} -6.00000 q^{82} +17.1605 q^{83} +1.70156 q^{84} -2.70156 q^{85} -0.0284374 q^{86} +5.27000 q^{87} +2.02214 q^{88} -10.7772 q^{89} -0.546295 q^{90} +0.568438 q^{91} -15.1967 q^{92} +9.66103 q^{93} -5.54857 q^{94} -6.62049 q^{95} +5.29991 q^{96} +17.9952 q^{97} +0.546295 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 6q^{4} - 4q^{5} - 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 6q^{4} - 4q^{5} - 4q^{7} + 4q^{9} - 4q^{11} + 6q^{12} + 8q^{13} - 4q^{15} + 22q^{16} - 2q^{17} + 10q^{19} - 6q^{20} - 4q^{21} - 2q^{23} + 4q^{25} + 20q^{26} + 4q^{27} - 6q^{28} - 2q^{29} + 24q^{31} - 4q^{33} + 4q^{35} + 6q^{36} + 8q^{37} + 16q^{38} + 8q^{39} + 6q^{43} - 6q^{44} - 4q^{45} - 12q^{46} + 4q^{47} + 22q^{48} + 4q^{49} - 2q^{51} + 12q^{52} + 14q^{53} + 4q^{55} + 10q^{57} - 20q^{58} - 2q^{59} - 6q^{60} + 6q^{61} + 8q^{62} - 4q^{63} + 6q^{64} - 8q^{65} - 8q^{67} - 44q^{68} - 2q^{69} + 4q^{73} - 36q^{74} + 4q^{75} + 56q^{76} + 4q^{77} + 20q^{78} - 22q^{80} + 4q^{81} - 24q^{82} + 6q^{83} - 6q^{84} + 2q^{85} - 36q^{86} - 2q^{87} + 18q^{89} - 8q^{91} - 44q^{92} + 24q^{93} - 36q^{94} - 10q^{95} - 6q^{97} - 4q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.546295 0.386289 0.193144 0.981170i \(-0.438131\pi\)
0.193144 + 0.981170i \(0.438131\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.70156 −0.850781
\(5\) −1.00000 −0.447214
\(6\) 0.546295 0.223024
\(7\) −1.00000 −0.377964
\(8\) −2.02214 −0.714936
\(9\) 1.00000 0.333333
\(10\) −0.546295 −0.172754
\(11\) −1.00000 −0.301511
\(12\) −1.70156 −0.491199
\(13\) −0.568438 −0.157656 −0.0788282 0.996888i \(-0.525118\pi\)
−0.0788282 + 0.996888i \(0.525118\pi\)
\(14\) −0.546295 −0.146003
\(15\) −1.00000 −0.258199
\(16\) 2.29844 0.574609
\(17\) 2.70156 0.655225 0.327613 0.944812i \(-0.393756\pi\)
0.327613 + 0.944812i \(0.393756\pi\)
\(18\) 0.546295 0.128763
\(19\) 6.62049 1.51885 0.759423 0.650598i \(-0.225482\pi\)
0.759423 + 0.650598i \(0.225482\pi\)
\(20\) 1.70156 0.380481
\(21\) −1.00000 −0.218218
\(22\) −0.546295 −0.116470
\(23\) 8.93103 1.86225 0.931124 0.364703i \(-0.118829\pi\)
0.931124 + 0.364703i \(0.118829\pi\)
\(24\) −2.02214 −0.412768
\(25\) 1.00000 0.200000
\(26\) −0.310535 −0.0609009
\(27\) 1.00000 0.192450
\(28\) 1.70156 0.321565
\(29\) 5.27000 0.978615 0.489307 0.872111i \(-0.337250\pi\)
0.489307 + 0.872111i \(0.337250\pi\)
\(30\) −0.546295 −0.0997393
\(31\) 9.66103 1.73517 0.867586 0.497287i \(-0.165671\pi\)
0.867586 + 0.497287i \(0.165671\pi\)
\(32\) 5.29991 0.936901
\(33\) −1.00000 −0.174078
\(34\) 1.47585 0.253106
\(35\) 1.00000 0.169031
\(36\) −1.70156 −0.283594
\(37\) −2.75362 −0.452692 −0.226346 0.974047i \(-0.572678\pi\)
−0.226346 + 0.974047i \(0.572678\pi\)
\(38\) 3.61674 0.586713
\(39\) −0.568438 −0.0910230
\(40\) 2.02214 0.319729
\(41\) −10.9831 −1.71527 −0.857635 0.514259i \(-0.828067\pi\)
−0.857635 + 0.514259i \(0.828067\pi\)
\(42\) −0.546295 −0.0842951
\(43\) −0.0520550 −0.00793831 −0.00396916 0.999992i \(-0.501263\pi\)
−0.00396916 + 0.999992i \(0.501263\pi\)
\(44\) 1.70156 0.256520
\(45\) −1.00000 −0.149071
\(46\) 4.87897 0.719365
\(47\) −10.1567 −1.48151 −0.740756 0.671774i \(-0.765533\pi\)
−0.740756 + 0.671774i \(0.765533\pi\)
\(48\) 2.29844 0.331751
\(49\) 1.00000 0.142857
\(50\) 0.546295 0.0772577
\(51\) 2.70156 0.378294
\(52\) 0.967233 0.134131
\(53\) 1.56469 0.214926 0.107463 0.994209i \(-0.465727\pi\)
0.107463 + 0.994209i \(0.465727\pi\)
\(54\) 0.546295 0.0743413
\(55\) 1.00000 0.134840
\(56\) 2.02214 0.270220
\(57\) 6.62049 0.876906
\(58\) 2.87897 0.378028
\(59\) 6.36259 0.828339 0.414169 0.910200i \(-0.364072\pi\)
0.414169 + 0.910200i \(0.364072\pi\)
\(60\) 1.70156 0.219671
\(61\) −3.71308 −0.475412 −0.237706 0.971337i \(-0.576395\pi\)
−0.237706 + 0.971337i \(0.576395\pi\)
\(62\) 5.27777 0.670277
\(63\) −1.00000 −0.125988
\(64\) −1.70156 −0.212695
\(65\) 0.568438 0.0705061
\(66\) −0.546295 −0.0672442
\(67\) −10.4146 −1.27235 −0.636176 0.771544i \(-0.719485\pi\)
−0.636176 + 0.771544i \(0.719485\pi\)
\(68\) −4.59688 −0.557453
\(69\) 8.93103 1.07517
\(70\) 0.546295 0.0652947
\(71\) 2.56844 0.304818 0.152409 0.988318i \(-0.451297\pi\)
0.152409 + 0.988318i \(0.451297\pi\)
\(72\) −2.02214 −0.238312
\(73\) 2.26625 0.265244 0.132622 0.991167i \(-0.457660\pi\)
0.132622 + 0.991167i \(0.457660\pi\)
\(74\) −1.50429 −0.174870
\(75\) 1.00000 0.115470
\(76\) −11.2652 −1.29220
\(77\) 1.00000 0.113961
\(78\) −0.310535 −0.0351612
\(79\) −2.56844 −0.288972 −0.144486 0.989507i \(-0.546153\pi\)
−0.144486 + 0.989507i \(0.546153\pi\)
\(80\) −2.29844 −0.256973
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 17.1605 1.88361 0.941804 0.336161i \(-0.109129\pi\)
0.941804 + 0.336161i \(0.109129\pi\)
\(84\) 1.70156 0.185656
\(85\) −2.70156 −0.293026
\(86\) −0.0284374 −0.00306648
\(87\) 5.27000 0.565003
\(88\) 2.02214 0.215561
\(89\) −10.7772 −1.14238 −0.571192 0.820816i \(-0.693519\pi\)
−0.571192 + 0.820816i \(0.693519\pi\)
\(90\) −0.546295 −0.0575845
\(91\) 0.568438 0.0595885
\(92\) −15.1967 −1.58437
\(93\) 9.66103 1.00180
\(94\) −5.54857 −0.572292
\(95\) −6.62049 −0.679248
\(96\) 5.29991 0.540920
\(97\) 17.9952 1.82713 0.913567 0.406689i \(-0.133317\pi\)
0.913567 + 0.406689i \(0.133317\pi\)
\(98\) 0.546295 0.0551841
\(99\) −1.00000 −0.100504
\(100\) −1.70156 −0.170156
\(101\) 0.826342 0.0822241 0.0411120 0.999155i \(-0.486910\pi\)
0.0411120 + 0.999155i \(0.486910\pi\)
\(102\) 1.47585 0.146131
\(103\) 10.5921 1.04367 0.521833 0.853048i \(-0.325248\pi\)
0.521833 + 0.853048i \(0.325248\pi\)
\(104\) 1.14946 0.112714
\(105\) 1.00000 0.0975900
\(106\) 0.854779 0.0830235
\(107\) −13.7864 −1.33278 −0.666390 0.745603i \(-0.732161\pi\)
−0.666390 + 0.745603i \(0.732161\pi\)
\(108\) −1.70156 −0.163733
\(109\) 19.8905 1.90516 0.952582 0.304282i \(-0.0984166\pi\)
0.952582 + 0.304282i \(0.0984166\pi\)
\(110\) 0.546295 0.0520872
\(111\) −2.75362 −0.261362
\(112\) −2.29844 −0.217182
\(113\) 6.43531 0.605383 0.302692 0.953089i \(-0.402115\pi\)
0.302692 + 0.953089i \(0.402115\pi\)
\(114\) 3.61674 0.338739
\(115\) −8.93103 −0.832823
\(116\) −8.96723 −0.832587
\(117\) −0.568438 −0.0525521
\(118\) 3.47585 0.319978
\(119\) −2.70156 −0.247652
\(120\) 2.02214 0.184596
\(121\) 1.00000 0.0909091
\(122\) −2.02844 −0.183646
\(123\) −10.9831 −0.990311
\(124\) −16.4388 −1.47625
\(125\) −1.00000 −0.0894427
\(126\) −0.546295 −0.0486678
\(127\) 4.05206 0.359562 0.179781 0.983707i \(-0.442461\pi\)
0.179781 + 0.983707i \(0.442461\pi\)
\(128\) −11.5294 −1.01906
\(129\) −0.0520550 −0.00458319
\(130\) 0.310535 0.0272357
\(131\) −5.69781 −0.497820 −0.248910 0.968527i \(-0.580072\pi\)
−0.248910 + 0.968527i \(0.580072\pi\)
\(132\) 1.70156 0.148102
\(133\) −6.62049 −0.574070
\(134\) −5.68947 −0.491495
\(135\) −1.00000 −0.0860663
\(136\) −5.46295 −0.468444
\(137\) −3.40312 −0.290749 −0.145374 0.989377i \(-0.546439\pi\)
−0.145374 + 0.989377i \(0.546439\pi\)
\(138\) 4.87897 0.415326
\(139\) −8.04724 −0.682558 −0.341279 0.939962i \(-0.610860\pi\)
−0.341279 + 0.939962i \(0.610860\pi\)
\(140\) −1.70156 −0.143808
\(141\) −10.1567 −0.855352
\(142\) 1.40312 0.117748
\(143\) 0.568438 0.0475352
\(144\) 2.29844 0.191536
\(145\) −5.27000 −0.437650
\(146\) 1.23804 0.102461
\(147\) 1.00000 0.0824786
\(148\) 4.68545 0.385142
\(149\) 7.50723 0.615017 0.307508 0.951545i \(-0.400505\pi\)
0.307508 + 0.951545i \(0.400505\pi\)
\(150\) 0.546295 0.0446048
\(151\) 3.21795 0.261873 0.130936 0.991391i \(-0.458202\pi\)
0.130936 + 0.991391i \(0.458202\pi\)
\(152\) −13.3876 −1.08588
\(153\) 2.70156 0.218408
\(154\) 0.546295 0.0440217
\(155\) −9.66103 −0.775992
\(156\) 0.967233 0.0774406
\(157\) −3.35107 −0.267444 −0.133722 0.991019i \(-0.542693\pi\)
−0.133722 + 0.991019i \(0.542693\pi\)
\(158\) −1.40312 −0.111627
\(159\) 1.56469 0.124088
\(160\) −5.29991 −0.418995
\(161\) −8.93103 −0.703864
\(162\) 0.546295 0.0429210
\(163\) 10.9600 0.858457 0.429228 0.903196i \(-0.358786\pi\)
0.429228 + 0.903196i \(0.358786\pi\)
\(164\) 18.6884 1.45932
\(165\) 1.00000 0.0778499
\(166\) 9.37469 0.727617
\(167\) −7.58830 −0.587201 −0.293600 0.955928i \(-0.594853\pi\)
−0.293600 + 0.955928i \(0.594853\pi\)
\(168\) 2.02214 0.156012
\(169\) −12.6769 −0.975144
\(170\) −1.47585 −0.113192
\(171\) 6.62049 0.506282
\(172\) 0.0885748 0.00675377
\(173\) −18.2094 −1.38443 −0.692216 0.721690i \(-0.743366\pi\)
−0.692216 + 0.721690i \(0.743366\pi\)
\(174\) 2.87897 0.218254
\(175\) −1.00000 −0.0755929
\(176\) −2.29844 −0.173251
\(177\) 6.36259 0.478242
\(178\) −5.88755 −0.441290
\(179\) 4.75362 0.355302 0.177651 0.984094i \(-0.443150\pi\)
0.177651 + 0.984094i \(0.443150\pi\)
\(180\) 1.70156 0.126827
\(181\) 25.2652 1.87795 0.938973 0.343991i \(-0.111779\pi\)
0.938973 + 0.343991i \(0.111779\pi\)
\(182\) 0.310535 0.0230184
\(183\) −3.71308 −0.274479
\(184\) −18.0598 −1.33139
\(185\) 2.75362 0.202450
\(186\) 5.27777 0.386985
\(187\) −2.70156 −0.197558
\(188\) 17.2823 1.26044
\(189\) −1.00000 −0.0727393
\(190\) −3.61674 −0.262386
\(191\) −17.4504 −1.26266 −0.631332 0.775513i \(-0.717491\pi\)
−0.631332 + 0.775513i \(0.717491\pi\)
\(192\) −1.70156 −0.122800
\(193\) 22.9546 1.65231 0.826156 0.563442i \(-0.190523\pi\)
0.826156 + 0.563442i \(0.190523\pi\)
\(194\) 9.83067 0.705801
\(195\) 0.568438 0.0407067
\(196\) −1.70156 −0.121540
\(197\) −3.36634 −0.239842 −0.119921 0.992783i \(-0.538264\pi\)
−0.119921 + 0.992783i \(0.538264\pi\)
\(198\) −0.546295 −0.0388235
\(199\) 9.54402 0.676557 0.338279 0.941046i \(-0.390155\pi\)
0.338279 + 0.941046i \(0.390155\pi\)
\(200\) −2.02214 −0.142987
\(201\) −10.4146 −0.734592
\(202\) 0.451426 0.0317622
\(203\) −5.27000 −0.369882
\(204\) −4.59688 −0.321846
\(205\) 10.9831 0.767092
\(206\) 5.78638 0.403156
\(207\) 8.93103 0.620749
\(208\) −1.30652 −0.0905909
\(209\) −6.62049 −0.457949
\(210\) 0.546295 0.0376979
\(211\) −7.73375 −0.532413 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(212\) −2.66241 −0.182855
\(213\) 2.56844 0.175986
\(214\) −7.53143 −0.514838
\(215\) 0.0520550 0.00355012
\(216\) −2.02214 −0.137589
\(217\) −9.66103 −0.655833
\(218\) 10.8661 0.735943
\(219\) 2.26625 0.153139
\(220\) −1.70156 −0.114719
\(221\) −1.53567 −0.103300
\(222\) −1.50429 −0.100961
\(223\) −13.9141 −0.931758 −0.465879 0.884848i \(-0.654262\pi\)
−0.465879 + 0.884848i \(0.654262\pi\)
\(224\) −5.29991 −0.354115
\(225\) 1.00000 0.0666667
\(226\) 3.51558 0.233853
\(227\) 0.597452 0.0396543 0.0198271 0.999803i \(-0.493688\pi\)
0.0198271 + 0.999803i \(0.493688\pi\)
\(228\) −11.2652 −0.746055
\(229\) 21.1895 1.40024 0.700121 0.714024i \(-0.253129\pi\)
0.700121 + 0.714024i \(0.253129\pi\)
\(230\) −4.87897 −0.321710
\(231\) 1.00000 0.0657952
\(232\) −10.6567 −0.699647
\(233\) −20.2821 −1.32872 −0.664362 0.747411i \(-0.731297\pi\)
−0.664362 + 0.747411i \(0.731297\pi\)
\(234\) −0.310535 −0.0203003
\(235\) 10.1567 0.662553
\(236\) −10.8263 −0.704735
\(237\) −2.56844 −0.166838
\(238\) −1.47585 −0.0956651
\(239\) 2.23723 0.144715 0.0723573 0.997379i \(-0.476948\pi\)
0.0723573 + 0.997379i \(0.476948\pi\)
\(240\) −2.29844 −0.148364
\(241\) 8.93045 0.575261 0.287630 0.957741i \(-0.407133\pi\)
0.287630 + 0.957741i \(0.407133\pi\)
\(242\) 0.546295 0.0351172
\(243\) 1.00000 0.0641500
\(244\) 6.31804 0.404471
\(245\) −1.00000 −0.0638877
\(246\) −6.00000 −0.382546
\(247\) −3.76334 −0.239456
\(248\) −19.5360 −1.24054
\(249\) 17.1605 1.08750
\(250\) −0.546295 −0.0345507
\(251\) 3.35884 0.212008 0.106004 0.994366i \(-0.466194\pi\)
0.106004 + 0.994366i \(0.466194\pi\)
\(252\) 1.70156 0.107188
\(253\) −8.93103 −0.561489
\(254\) 2.21362 0.138895
\(255\) −2.70156 −0.169178
\(256\) −2.89531 −0.180957
\(257\) 5.80943 0.362382 0.181191 0.983448i \(-0.442005\pi\)
0.181191 + 0.983448i \(0.442005\pi\)
\(258\) −0.0284374 −0.00177043
\(259\) 2.75362 0.171101
\(260\) −0.967233 −0.0599853
\(261\) 5.27000 0.326205
\(262\) −3.11268 −0.192302
\(263\) −23.3221 −1.43810 −0.719050 0.694959i \(-0.755423\pi\)
−0.719050 + 0.694959i \(0.755423\pi\)
\(264\) 2.02214 0.124454
\(265\) −1.56469 −0.0961179
\(266\) −3.61674 −0.221757
\(267\) −10.7772 −0.659556
\(268\) 17.7212 1.08249
\(269\) 29.1246 1.77576 0.887878 0.460080i \(-0.152179\pi\)
0.887878 + 0.460080i \(0.152179\pi\)
\(270\) −0.546295 −0.0332464
\(271\) 31.2416 1.89779 0.948895 0.315592i \(-0.102203\pi\)
0.948895 + 0.315592i \(0.102203\pi\)
\(272\) 6.20937 0.376499
\(273\) 0.568438 0.0344035
\(274\) −1.85911 −0.112313
\(275\) −1.00000 −0.0603023
\(276\) −15.1967 −0.914734
\(277\) −15.8988 −0.955269 −0.477634 0.878559i \(-0.658506\pi\)
−0.477634 + 0.878559i \(0.658506\pi\)
\(278\) −4.39616 −0.263664
\(279\) 9.66103 0.578391
\(280\) −2.02214 −0.120846
\(281\) −14.1041 −0.841381 −0.420690 0.907204i \(-0.638212\pi\)
−0.420690 + 0.907204i \(0.638212\pi\)
\(282\) −5.54857 −0.330413
\(283\) −1.32745 −0.0789088 −0.0394544 0.999221i \(-0.512562\pi\)
−0.0394544 + 0.999221i \(0.512562\pi\)
\(284\) −4.37036 −0.259333
\(285\) −6.62049 −0.392164
\(286\) 0.310535 0.0183623
\(287\) 10.9831 0.648311
\(288\) 5.29991 0.312300
\(289\) −9.70156 −0.570680
\(290\) −2.87897 −0.169059
\(291\) 17.9952 1.05490
\(292\) −3.85616 −0.225665
\(293\) 18.3784 1.07368 0.536840 0.843684i \(-0.319618\pi\)
0.536840 + 0.843684i \(0.319618\pi\)
\(294\) 0.546295 0.0318606
\(295\) −6.36259 −0.370444
\(296\) 5.56821 0.323646
\(297\) −1.00000 −0.0580259
\(298\) 4.10116 0.237574
\(299\) −5.07674 −0.293595
\(300\) −1.70156 −0.0982397
\(301\) 0.0520550 0.00300040
\(302\) 1.75795 0.101158
\(303\) 0.826342 0.0474721
\(304\) 15.2168 0.872743
\(305\) 3.71308 0.212611
\(306\) 1.47585 0.0843687
\(307\) 6.45143 0.368202 0.184101 0.982907i \(-0.441063\pi\)
0.184101 + 0.982907i \(0.441063\pi\)
\(308\) −1.70156 −0.0969555
\(309\) 10.5921 0.602561
\(310\) −5.27777 −0.299757
\(311\) −29.4136 −1.66789 −0.833946 0.551847i \(-0.813923\pi\)
−0.833946 + 0.551847i \(0.813923\pi\)
\(312\) 1.14946 0.0650756
\(313\) −24.7542 −1.39919 −0.699595 0.714540i \(-0.746636\pi\)
−0.699595 + 0.714540i \(0.746636\pi\)
\(314\) −1.83067 −0.103311
\(315\) 1.00000 0.0563436
\(316\) 4.37036 0.245852
\(317\) −15.0955 −0.847850 −0.423925 0.905697i \(-0.639348\pi\)
−0.423925 + 0.905697i \(0.639348\pi\)
\(318\) 0.854779 0.0479336
\(319\) −5.27000 −0.295063
\(320\) 1.70156 0.0951202
\(321\) −13.7864 −0.769481
\(322\) −4.87897 −0.271895
\(323\) 17.8857 0.995186
\(324\) −1.70156 −0.0945312
\(325\) −0.568438 −0.0315313
\(326\) 5.98741 0.331612
\(327\) 19.8905 1.09995
\(328\) 22.2094 1.22631
\(329\) 10.1567 0.559959
\(330\) 0.546295 0.0300725
\(331\) 6.78263 0.372807 0.186404 0.982473i \(-0.440317\pi\)
0.186404 + 0.982473i \(0.440317\pi\)
\(332\) −29.1996 −1.60254
\(333\) −2.75362 −0.150897
\(334\) −4.14545 −0.226829
\(335\) 10.4146 0.569013
\(336\) −2.29844 −0.125390
\(337\) −31.9509 −1.74048 −0.870238 0.492631i \(-0.836035\pi\)
−0.870238 + 0.492631i \(0.836035\pi\)
\(338\) −6.92531 −0.376687
\(339\) 6.43531 0.349518
\(340\) 4.59688 0.249301
\(341\) −9.66103 −0.523174
\(342\) 3.61674 0.195571
\(343\) −1.00000 −0.0539949
\(344\) 0.105263 0.00567538
\(345\) −8.93103 −0.480830
\(346\) −9.94768 −0.534791
\(347\) −13.6694 −0.733810 −0.366905 0.930258i \(-0.619583\pi\)
−0.366905 + 0.930258i \(0.619583\pi\)
\(348\) −8.96723 −0.480694
\(349\) 16.1720 0.865668 0.432834 0.901474i \(-0.357514\pi\)
0.432834 + 0.901474i \(0.357514\pi\)
\(350\) −0.546295 −0.0292007
\(351\) −0.568438 −0.0303410
\(352\) −5.29991 −0.282486
\(353\) −9.94313 −0.529219 −0.264610 0.964356i \(-0.585243\pi\)
−0.264610 + 0.964356i \(0.585243\pi\)
\(354\) 3.47585 0.184739
\(355\) −2.56844 −0.136319
\(356\) 18.3381 0.971919
\(357\) −2.70156 −0.142982
\(358\) 2.59688 0.137249
\(359\) −12.5845 −0.664187 −0.332094 0.943246i \(-0.607755\pi\)
−0.332094 + 0.943246i \(0.607755\pi\)
\(360\) 2.02214 0.106576
\(361\) 24.8309 1.30689
\(362\) 13.8022 0.725429
\(363\) 1.00000 0.0524864
\(364\) −0.967233 −0.0506968
\(365\) −2.26625 −0.118621
\(366\) −2.02844 −0.106028
\(367\) 36.9699 1.92981 0.964907 0.262592i \(-0.0845772\pi\)
0.964907 + 0.262592i \(0.0845772\pi\)
\(368\) 20.5274 1.07007
\(369\) −10.9831 −0.571756
\(370\) 1.50429 0.0782041
\(371\) −1.56469 −0.0812344
\(372\) −16.4388 −0.852314
\(373\) −13.4109 −0.694390 −0.347195 0.937793i \(-0.612866\pi\)
−0.347195 + 0.937793i \(0.612866\pi\)
\(374\) −1.47585 −0.0763143
\(375\) −1.00000 −0.0516398
\(376\) 20.5384 1.05919
\(377\) −2.99567 −0.154285
\(378\) −0.546295 −0.0280984
\(379\) −0.805672 −0.0413846 −0.0206923 0.999786i \(-0.506587\pi\)
−0.0206923 + 0.999786i \(0.506587\pi\)
\(380\) 11.2652 0.577892
\(381\) 4.05206 0.207593
\(382\) −9.53304 −0.487753
\(383\) −22.3494 −1.14200 −0.571001 0.820949i \(-0.693445\pi\)
−0.571001 + 0.820949i \(0.693445\pi\)
\(384\) −11.5294 −0.588356
\(385\) −1.00000 −0.0509647
\(386\) 12.5400 0.638269
\(387\) −0.0520550 −0.00264610
\(388\) −30.6199 −1.55449
\(389\) −29.4863 −1.49501 −0.747507 0.664253i \(-0.768750\pi\)
−0.747507 + 0.664253i \(0.768750\pi\)
\(390\) 0.310535 0.0157245
\(391\) 24.1277 1.22019
\(392\) −2.02214 −0.102134
\(393\) −5.69781 −0.287416
\(394\) −1.83902 −0.0926482
\(395\) 2.56844 0.129232
\(396\) 1.70156 0.0855067
\(397\) 4.45143 0.223411 0.111705 0.993741i \(-0.464369\pi\)
0.111705 + 0.993741i \(0.464369\pi\)
\(398\) 5.21384 0.261346
\(399\) −6.62049 −0.331439
\(400\) 2.29844 0.114922
\(401\) 22.7123 1.13420 0.567099 0.823650i \(-0.308066\pi\)
0.567099 + 0.823650i \(0.308066\pi\)
\(402\) −5.68947 −0.283765
\(403\) −5.49170 −0.273561
\(404\) −1.40607 −0.0699547
\(405\) −1.00000 −0.0496904
\(406\) −2.87897 −0.142881
\(407\) 2.75362 0.136492
\(408\) −5.46295 −0.270456
\(409\) 5.09259 0.251812 0.125906 0.992042i \(-0.459816\pi\)
0.125906 + 0.992042i \(0.459816\pi\)
\(410\) 6.00000 0.296319
\(411\) −3.40312 −0.167864
\(412\) −18.0230 −0.887932
\(413\) −6.36259 −0.313083
\(414\) 4.87897 0.239788
\(415\) −17.1605 −0.842376
\(416\) −3.01267 −0.147708
\(417\) −8.04724 −0.394075
\(418\) −3.61674 −0.176901
\(419\) −2.86286 −0.139860 −0.0699300 0.997552i \(-0.522278\pi\)
−0.0699300 + 0.997552i \(0.522278\pi\)
\(420\) −1.70156 −0.0830277
\(421\) −16.1047 −0.784894 −0.392447 0.919775i \(-0.628371\pi\)
−0.392447 + 0.919775i \(0.628371\pi\)
\(422\) −4.22491 −0.205665
\(423\) −10.1567 −0.493838
\(424\) −3.16402 −0.153658
\(425\) 2.70156 0.131045
\(426\) 1.40312 0.0679816
\(427\) 3.71308 0.179689
\(428\) 23.4584 1.13390
\(429\) 0.568438 0.0274445
\(430\) 0.0284374 0.00137137
\(431\) −4.67794 −0.225329 −0.112664 0.993633i \(-0.535938\pi\)
−0.112664 + 0.993633i \(0.535938\pi\)
\(432\) 2.29844 0.110584
\(433\) −27.4337 −1.31838 −0.659189 0.751977i \(-0.729100\pi\)
−0.659189 + 0.751977i \(0.729100\pi\)
\(434\) −5.27777 −0.253341
\(435\) −5.27000 −0.252677
\(436\) −33.8449 −1.62088
\(437\) 59.1278 2.82847
\(438\) 1.23804 0.0591558
\(439\) 5.73433 0.273685 0.136842 0.990593i \(-0.456305\pi\)
0.136842 + 0.990593i \(0.456305\pi\)
\(440\) −2.02214 −0.0964019
\(441\) 1.00000 0.0476190
\(442\) −0.838929 −0.0399038
\(443\) −9.58536 −0.455414 −0.227707 0.973730i \(-0.573123\pi\)
−0.227707 + 0.973730i \(0.573123\pi\)
\(444\) 4.68545 0.222362
\(445\) 10.7772 0.510890
\(446\) −7.60121 −0.359927
\(447\) 7.50723 0.355080
\(448\) 1.70156 0.0803913
\(449\) 30.9463 1.46045 0.730223 0.683209i \(-0.239416\pi\)
0.730223 + 0.683209i \(0.239416\pi\)
\(450\) 0.546295 0.0257526
\(451\) 10.9831 0.517173
\(452\) −10.9501 −0.515049
\(453\) 3.21795 0.151192
\(454\) 0.326385 0.0153180
\(455\) −0.568438 −0.0266488
\(456\) −13.3876 −0.626931
\(457\) −42.2246 −1.97519 −0.987593 0.157036i \(-0.949806\pi\)
−0.987593 + 0.157036i \(0.949806\pi\)
\(458\) 11.5757 0.540898
\(459\) 2.70156 0.126098
\(460\) 15.1967 0.708550
\(461\) 9.87464 0.459908 0.229954 0.973201i \(-0.426142\pi\)
0.229954 + 0.973201i \(0.426142\pi\)
\(462\) 0.546295 0.0254159
\(463\) 2.56009 0.118978 0.0594888 0.998229i \(-0.481053\pi\)
0.0594888 + 0.998229i \(0.481053\pi\)
\(464\) 12.1128 0.562321
\(465\) −9.66103 −0.448019
\(466\) −11.0800 −0.513271
\(467\) −19.6924 −0.911256 −0.455628 0.890170i \(-0.650585\pi\)
−0.455628 + 0.890170i \(0.650585\pi\)
\(468\) 0.967233 0.0447104
\(469\) 10.4146 0.480904
\(470\) 5.54857 0.255937
\(471\) −3.35107 −0.154409
\(472\) −12.8661 −0.592209
\(473\) 0.0520550 0.00239349
\(474\) −1.40312 −0.0644476
\(475\) 6.62049 0.303769
\(476\) 4.59688 0.210697
\(477\) 1.56469 0.0716420
\(478\) 1.22219 0.0559016
\(479\) 15.3672 0.702144 0.351072 0.936348i \(-0.385817\pi\)
0.351072 + 0.936348i \(0.385817\pi\)
\(480\) −5.29991 −0.241907
\(481\) 1.56526 0.0713698
\(482\) 4.87866 0.222217
\(483\) −8.93103 −0.406376
\(484\) −1.70156 −0.0773437
\(485\) −17.9952 −0.817119
\(486\) 0.546295 0.0247804
\(487\) −1.52848 −0.0692621 −0.0346310 0.999400i \(-0.511026\pi\)
−0.0346310 + 0.999400i \(0.511026\pi\)
\(488\) 7.50839 0.339889
\(489\) 10.9600 0.495630
\(490\) −0.546295 −0.0246791
\(491\) −40.7359 −1.83839 −0.919193 0.393808i \(-0.871157\pi\)
−0.919193 + 0.393808i \(0.871157\pi\)
\(492\) 18.6884 0.842538
\(493\) 14.2372 0.641213
\(494\) −2.05589 −0.0924990
\(495\) 1.00000 0.0449467
\(496\) 22.2053 0.997046
\(497\) −2.56844 −0.115210
\(498\) 9.37469 0.420090
\(499\) −20.3312 −0.910150 −0.455075 0.890453i \(-0.650388\pi\)
−0.455075 + 0.890453i \(0.650388\pi\)
\(500\) 1.70156 0.0760962
\(501\) −7.58830 −0.339020
\(502\) 1.83491 0.0818963
\(503\) −23.5952 −1.05206 −0.526030 0.850466i \(-0.676320\pi\)
−0.526030 + 0.850466i \(0.676320\pi\)
\(504\) 2.02214 0.0900734
\(505\) −0.826342 −0.0367717
\(506\) −4.87897 −0.216897
\(507\) −12.6769 −0.563000
\(508\) −6.89482 −0.305908
\(509\) 38.0993 1.68872 0.844361 0.535775i \(-0.179981\pi\)
0.844361 + 0.535775i \(0.179981\pi\)
\(510\) −1.47585 −0.0653517
\(511\) −2.26625 −0.100253
\(512\) 21.4771 0.949161
\(513\) 6.62049 0.292302
\(514\) 3.17366 0.139984
\(515\) −10.5921 −0.466742
\(516\) 0.0885748 0.00389929
\(517\) 10.1567 0.446693
\(518\) 1.50429 0.0660945
\(519\) −18.2094 −0.799303
\(520\) −1.14946 −0.0504073
\(521\) 33.9952 1.48936 0.744678 0.667424i \(-0.232603\pi\)
0.744678 + 0.667424i \(0.232603\pi\)
\(522\) 2.87897 0.126009
\(523\) 10.9517 0.478884 0.239442 0.970911i \(-0.423035\pi\)
0.239442 + 0.970911i \(0.423035\pi\)
\(524\) 9.69518 0.423536
\(525\) −1.00000 −0.0436436
\(526\) −12.7407 −0.555522
\(527\) 26.0999 1.13693
\(528\) −2.29844 −0.100027
\(529\) 56.7633 2.46797
\(530\) −0.854779 −0.0371292
\(531\) 6.36259 0.276113
\(532\) 11.2652 0.488408
\(533\) 6.24321 0.270423
\(534\) −5.88755 −0.254779
\(535\) 13.7864 0.596037
\(536\) 21.0599 0.909650
\(537\) 4.75362 0.205134
\(538\) 15.9106 0.685954
\(539\) −1.00000 −0.0430730
\(540\) 1.70156 0.0732236
\(541\) 13.4971 0.580285 0.290143 0.956983i \(-0.406297\pi\)
0.290143 + 0.956983i \(0.406297\pi\)
\(542\) 17.0671 0.733095
\(543\) 25.2652 1.08423
\(544\) 14.3180 0.613881
\(545\) −19.8905 −0.852015
\(546\) 0.310535 0.0132897
\(547\) −42.0182 −1.79657 −0.898285 0.439414i \(-0.855186\pi\)
−0.898285 + 0.439414i \(0.855186\pi\)
\(548\) 5.79063 0.247363
\(549\) −3.71308 −0.158471
\(550\) −0.546295 −0.0232941
\(551\) 34.8900 1.48636
\(552\) −18.0598 −0.768677
\(553\) 2.56844 0.109221
\(554\) −8.68545 −0.369009
\(555\) 2.75362 0.116885
\(556\) 13.6929 0.580707
\(557\) −27.9461 −1.18411 −0.592057 0.805896i \(-0.701684\pi\)
−0.592057 + 0.805896i \(0.701684\pi\)
\(558\) 5.27777 0.223426
\(559\) 0.0295901 0.00125153
\(560\) 2.29844 0.0971267
\(561\) −2.70156 −0.114060
\(562\) −7.70500 −0.325016
\(563\) −0.266247 −0.0112210 −0.00561050 0.999984i \(-0.501786\pi\)
−0.00561050 + 0.999984i \(0.501786\pi\)
\(564\) 17.2823 0.727717
\(565\) −6.43531 −0.270736
\(566\) −0.725180 −0.0304816
\(567\) −1.00000 −0.0419961
\(568\) −5.19375 −0.217925
\(569\) −35.3983 −1.48397 −0.741987 0.670414i \(-0.766116\pi\)
−0.741987 + 0.670414i \(0.766116\pi\)
\(570\) −3.61674 −0.151489
\(571\) −0.440134 −0.0184191 −0.00920953 0.999958i \(-0.502932\pi\)
−0.00920953 + 0.999958i \(0.502932\pi\)
\(572\) −0.967233 −0.0404421
\(573\) −17.4504 −0.728999
\(574\) 6.00000 0.250435
\(575\) 8.93103 0.372450
\(576\) −1.70156 −0.0708984
\(577\) −25.1030 −1.04505 −0.522527 0.852623i \(-0.675010\pi\)
−0.522527 + 0.852623i \(0.675010\pi\)
\(578\) −5.29991 −0.220447
\(579\) 22.9546 0.953963
\(580\) 8.96723 0.372344
\(581\) −17.1605 −0.711937
\(582\) 9.83067 0.407494
\(583\) −1.56469 −0.0648026
\(584\) −4.58268 −0.189633
\(585\) 0.568438 0.0235020
\(586\) 10.0400 0.414750
\(587\) 3.24638 0.133993 0.0669963 0.997753i \(-0.478658\pi\)
0.0669963 + 0.997753i \(0.478658\pi\)
\(588\) −1.70156 −0.0701712
\(589\) 63.9608 2.63546
\(590\) −3.47585 −0.143098
\(591\) −3.36634 −0.138473
\(592\) −6.32902 −0.260121
\(593\) 10.7252 0.440430 0.220215 0.975451i \(-0.429324\pi\)
0.220215 + 0.975451i \(0.429324\pi\)
\(594\) −0.546295 −0.0224147
\(595\) 2.70156 0.110753
\(596\) −12.7740 −0.523244
\(597\) 9.54402 0.390611
\(598\) −2.77340 −0.113413
\(599\) 36.3607 1.48566 0.742829 0.669481i \(-0.233483\pi\)
0.742829 + 0.669481i \(0.233483\pi\)
\(600\) −2.02214 −0.0825537
\(601\) −39.7034 −1.61954 −0.809769 0.586749i \(-0.800407\pi\)
−0.809769 + 0.586749i \(0.800407\pi\)
\(602\) 0.0284374 0.00115902
\(603\) −10.4146 −0.424117
\(604\) −5.47553 −0.222796
\(605\) −1.00000 −0.0406558
\(606\) 0.451426 0.0183379
\(607\) −9.91893 −0.402597 −0.201299 0.979530i \(-0.564516\pi\)
−0.201299 + 0.979530i \(0.564516\pi\)
\(608\) 35.0880 1.42301
\(609\) −5.27000 −0.213551
\(610\) 2.02844 0.0821290
\(611\) 5.77348 0.233570
\(612\) −4.59688 −0.185818
\(613\) 39.2681 1.58602 0.793012 0.609206i \(-0.208512\pi\)
0.793012 + 0.609206i \(0.208512\pi\)
\(614\) 3.52438 0.142232
\(615\) 10.9831 0.442881
\(616\) −2.02214 −0.0814745
\(617\) 31.3462 1.26195 0.630976 0.775802i \(-0.282655\pi\)
0.630976 + 0.775802i \(0.282655\pi\)
\(618\) 5.78638 0.232762
\(619\) −12.3916 −0.498061 −0.249030 0.968496i \(-0.580112\pi\)
−0.249030 + 0.968496i \(0.580112\pi\)
\(620\) 16.4388 0.660200
\(621\) 8.93103 0.358390
\(622\) −16.0685 −0.644287
\(623\) 10.7772 0.431781
\(624\) −1.30652 −0.0523027
\(625\) 1.00000 0.0400000
\(626\) −13.5231 −0.540491
\(627\) −6.62049 −0.264397
\(628\) 5.70205 0.227537
\(629\) −7.43907 −0.296615
\(630\) 0.546295 0.0217649
\(631\) 33.2233 1.32260 0.661299 0.750123i \(-0.270006\pi\)
0.661299 + 0.750123i \(0.270006\pi\)
\(632\) 5.19375 0.206596
\(633\) −7.73375 −0.307389
\(634\) −8.24661 −0.327515
\(635\) −4.05206 −0.160801
\(636\) −2.66241 −0.105571
\(637\) −0.568438 −0.0225223
\(638\) −2.87897 −0.113980
\(639\) 2.56844 0.101606
\(640\) 11.5294 0.455739
\(641\) 18.9179 0.747211 0.373605 0.927588i \(-0.378121\pi\)
0.373605 + 0.927588i \(0.378121\pi\)
\(642\) −7.53143 −0.297242
\(643\) −5.82554 −0.229737 −0.114868 0.993381i \(-0.536645\pi\)
−0.114868 + 0.993381i \(0.536645\pi\)
\(644\) 15.1967 0.598834
\(645\) 0.0520550 0.00204966
\(646\) 9.77085 0.384429
\(647\) 8.30759 0.326605 0.163302 0.986576i \(-0.447785\pi\)
0.163302 + 0.986576i \(0.447785\pi\)
\(648\) −2.02214 −0.0794373
\(649\) −6.36259 −0.249753
\(650\) −0.310535 −0.0121802
\(651\) −9.66103 −0.378646
\(652\) −18.6492 −0.730359
\(653\) 44.4182 1.73822 0.869109 0.494621i \(-0.164693\pi\)
0.869109 + 0.494621i \(0.164693\pi\)
\(654\) 10.8661 0.424897
\(655\) 5.69781 0.222632
\(656\) −25.2439 −0.985610
\(657\) 2.26625 0.0884147
\(658\) 5.54857 0.216306
\(659\) −13.4965 −0.525750 −0.262875 0.964830i \(-0.584671\pi\)
−0.262875 + 0.964830i \(0.584671\pi\)
\(660\) −1.70156 −0.0662332
\(661\) 31.7450 1.23474 0.617370 0.786673i \(-0.288198\pi\)
0.617370 + 0.786673i \(0.288198\pi\)
\(662\) 3.70532 0.144011
\(663\) −1.53567 −0.0596405
\(664\) −34.7010 −1.34666
\(665\) 6.62049 0.256732
\(666\) −1.50429 −0.0582899
\(667\) 47.0665 1.82242
\(668\) 12.9120 0.499579
\(669\) −13.9141 −0.537951
\(670\) 5.68947 0.219803
\(671\) 3.71308 0.143342
\(672\) −5.29991 −0.204449
\(673\) 5.81295 0.224073 0.112036 0.993704i \(-0.464263\pi\)
0.112036 + 0.993704i \(0.464263\pi\)
\(674\) −17.4546 −0.672326
\(675\) 1.00000 0.0384900
\(676\) 21.5705 0.829634
\(677\) −23.3699 −0.898177 −0.449088 0.893487i \(-0.648251\pi\)
−0.449088 + 0.893487i \(0.648251\pi\)
\(678\) 3.51558 0.135015
\(679\) −17.9952 −0.690592
\(680\) 5.46295 0.209494
\(681\) 0.597452 0.0228944
\(682\) −5.27777 −0.202096
\(683\) 13.4705 0.515433 0.257716 0.966221i \(-0.417030\pi\)
0.257716 + 0.966221i \(0.417030\pi\)
\(684\) −11.2652 −0.430735
\(685\) 3.40312 0.130027
\(686\) −0.546295 −0.0208576
\(687\) 21.1895 0.808430
\(688\) −0.119645 −0.00456143
\(689\) −0.889427 −0.0338845
\(690\) −4.87897 −0.185739
\(691\) 9.79358 0.372565 0.186283 0.982496i \(-0.440356\pi\)
0.186283 + 0.982496i \(0.440356\pi\)
\(692\) 30.9844 1.17785
\(693\) 1.00000 0.0379869
\(694\) −7.46751 −0.283463
\(695\) 8.04724 0.305249
\(696\) −10.6567 −0.403941
\(697\) −29.6715 −1.12389
\(698\) 8.83469 0.334398
\(699\) −20.2821 −0.767139
\(700\) 1.70156 0.0643130
\(701\) −9.27000 −0.350123 −0.175062 0.984557i \(-0.556012\pi\)
−0.175062 + 0.984557i \(0.556012\pi\)
\(702\) −0.310535 −0.0117204
\(703\) −18.2303 −0.687569
\(704\) 1.70156 0.0641300
\(705\) 10.1567 0.382525
\(706\) −5.43188 −0.204431
\(707\) −0.826342 −0.0310778
\(708\) −10.8263 −0.406879
\(709\) −19.9898 −0.750732 −0.375366 0.926877i \(-0.622483\pi\)
−0.375366 + 0.926877i \(0.622483\pi\)
\(710\) −1.40312 −0.0526583
\(711\) −2.56844 −0.0963240
\(712\) 21.7931 0.816732
\(713\) 86.2829 3.23132
\(714\) −1.47585 −0.0552323
\(715\) −0.568438 −0.0212584
\(716\) −8.08857 −0.302284
\(717\) 2.23723 0.0835510
\(718\) −6.87487 −0.256568
\(719\) −26.7174 −0.996391 −0.498196 0.867065i \(-0.666004\pi\)
−0.498196 + 0.867065i \(0.666004\pi\)
\(720\) −2.29844 −0.0856577
\(721\) −10.5921 −0.394469
\(722\) 13.5650 0.504837
\(723\) 8.93045 0.332127
\(724\) −42.9903 −1.59772
\(725\) 5.27000 0.195723
\(726\) 0.546295 0.0202749
\(727\) 16.4955 0.611782 0.305891 0.952066i \(-0.401046\pi\)
0.305891 + 0.952066i \(0.401046\pi\)
\(728\) −1.14946 −0.0426020
\(729\) 1.00000 0.0370370
\(730\) −1.23804 −0.0458219
\(731\) −0.140630 −0.00520138
\(732\) 6.31804 0.233522
\(733\) −22.6838 −0.837847 −0.418923 0.908022i \(-0.637592\pi\)
−0.418923 + 0.908022i \(0.637592\pi\)
\(734\) 20.1965 0.745465
\(735\) −1.00000 −0.0368856
\(736\) 47.3337 1.74474
\(737\) 10.4146 0.383628
\(738\) −6.00000 −0.220863
\(739\) −41.7166 −1.53457 −0.767285 0.641306i \(-0.778393\pi\)
−0.767285 + 0.641306i \(0.778393\pi\)
\(740\) −4.68545 −0.172241
\(741\) −3.76334 −0.138250
\(742\) −0.854779 −0.0313799
\(743\) −26.4664 −0.970959 −0.485480 0.874248i \(-0.661355\pi\)
−0.485480 + 0.874248i \(0.661355\pi\)
\(744\) −19.5360 −0.716224
\(745\) −7.50723 −0.275044
\(746\) −7.32630 −0.268235
\(747\) 17.1605 0.627870
\(748\) 4.59688 0.168078
\(749\) 13.7864 0.503744
\(750\) −0.546295 −0.0199479
\(751\) −36.2802 −1.32388 −0.661942 0.749555i \(-0.730268\pi\)
−0.661942 + 0.749555i \(0.730268\pi\)
\(752\) −23.3446 −0.851291
\(753\) 3.35884 0.122403
\(754\) −1.63652 −0.0595985
\(755\) −3.21795 −0.117113
\(756\) 1.70156 0.0618852
\(757\) 27.3451 0.993874 0.496937 0.867786i \(-0.334458\pi\)
0.496937 + 0.867786i \(0.334458\pi\)
\(758\) −0.440134 −0.0159864
\(759\) −8.93103 −0.324176
\(760\) 13.3876 0.485619
\(761\) −16.2537 −0.589195 −0.294597 0.955621i \(-0.595186\pi\)
−0.294597 + 0.955621i \(0.595186\pi\)
\(762\) 2.21362 0.0801909
\(763\) −19.8905 −0.720084
\(764\) 29.6929 1.07425
\(765\) −2.70156 −0.0976752
\(766\) −12.2094 −0.441143
\(767\) −3.61674 −0.130593
\(768\) −2.89531 −0.104476
\(769\) −38.2698 −1.38004 −0.690022 0.723789i \(-0.742399\pi\)
−0.690022 + 0.723789i \(0.742399\pi\)
\(770\) −0.546295 −0.0196871
\(771\) 5.80943 0.209221
\(772\) −39.0588 −1.40576
\(773\) 6.60438 0.237543 0.118772 0.992922i \(-0.462104\pi\)
0.118772 + 0.992922i \(0.462104\pi\)
\(774\) −0.0284374 −0.00102216
\(775\) 9.66103 0.347034
\(776\) −36.3888 −1.30628
\(777\) 2.75362 0.0987855
\(778\) −16.1082 −0.577507
\(779\) −72.7134 −2.60523
\(780\) −0.967233 −0.0346325
\(781\) −2.56844 −0.0919060
\(782\) 13.1808 0.471346
\(783\) 5.27000 0.188334
\(784\) 2.29844 0.0820871
\(785\) 3.35107 0.119605
\(786\) −3.11268 −0.111026
\(787\) −23.1916 −0.826692 −0.413346 0.910574i \(-0.635640\pi\)
−0.413346 + 0.910574i \(0.635640\pi\)
\(788\) 5.72804 0.204053
\(789\) −23.3221 −0.830287
\(790\) 1.40312 0.0499209
\(791\) −6.43531 −0.228813
\(792\) 2.02214 0.0718537
\(793\) 2.11066 0.0749517
\(794\) 2.43179 0.0863010
\(795\) −1.56469 −0.0554937
\(796\) −16.2397 −0.575602
\(797\) 32.5872 1.15430 0.577150 0.816638i \(-0.304165\pi\)
0.577150 + 0.816638i \(0.304165\pi\)
\(798\) −3.61674 −0.128031
\(799\) −27.4391 −0.970724
\(800\) 5.29991 0.187380
\(801\) −10.7772 −0.380795
\(802\) 12.4076 0.438127
\(803\) −2.26625 −0.0799741
\(804\) 17.7212 0.624977
\(805\) 8.93103 0.314777
\(806\) −3.00009 −0.105674
\(807\) 29.1246 1.02523
\(808\) −1.67098 −0.0587849
\(809\) 13.5303 0.475699 0.237850 0.971302i \(-0.423557\pi\)
0.237850 + 0.971302i \(0.423557\pi\)
\(810\) −0.546295 −0.0191948
\(811\) 27.7241 0.973525 0.486763 0.873534i \(-0.338178\pi\)
0.486763 + 0.873534i \(0.338178\pi\)
\(812\) 8.96723 0.314688
\(813\) 31.2416 1.09569
\(814\) 1.50429 0.0527252
\(815\) −10.9600 −0.383914
\(816\) 6.20937 0.217372
\(817\) −0.344630 −0.0120571
\(818\) 2.78205 0.0972723
\(819\) 0.568438 0.0198628
\(820\) −18.6884 −0.652627
\(821\) 40.9774 1.43012 0.715061 0.699062i \(-0.246399\pi\)
0.715061 + 0.699062i \(0.246399\pi\)
\(822\) −1.85911 −0.0648439
\(823\) 3.27352 0.114108 0.0570539 0.998371i \(-0.481829\pi\)
0.0570539 + 0.998371i \(0.481829\pi\)
\(824\) −21.4187 −0.746154
\(825\) −1.00000 −0.0348155
\(826\) −3.47585 −0.120940
\(827\) −36.5217 −1.26998 −0.634992 0.772519i \(-0.718997\pi\)
−0.634992 + 0.772519i \(0.718997\pi\)
\(828\) −15.1967 −0.528122
\(829\) 21.2335 0.737469 0.368735 0.929535i \(-0.379791\pi\)
0.368735 + 0.929535i \(0.379791\pi\)
\(830\) −9.37469 −0.325400
\(831\) −15.8988 −0.551525
\(832\) 0.967233 0.0335328
\(833\) 2.70156 0.0936036
\(834\) −4.39616 −0.152227
\(835\) 7.58830 0.262604
\(836\) 11.2652 0.389614
\(837\) 9.66103 0.333934
\(838\) −1.56397 −0.0540263
\(839\) 20.7571 0.716616 0.358308 0.933603i \(-0.383354\pi\)
0.358308 + 0.933603i \(0.383354\pi\)
\(840\) −2.02214 −0.0697706
\(841\) −1.22709 −0.0423136
\(842\) −8.79790 −0.303196
\(843\) −14.1041 −0.485771
\(844\) 13.1595 0.452967
\(845\) 12.6769 0.436098
\(846\) −5.54857 −0.190764
\(847\) −1.00000 −0.0343604
\(848\) 3.59633 0.123499
\(849\) −1.32745 −0.0455580
\(850\) 1.47585 0.0506212
\(851\) −24.5926 −0.843025
\(852\) −4.37036 −0.149726
\(853\) −31.3333 −1.07283 −0.536417 0.843953i \(-0.680222\pi\)
−0.536417 + 0.843953i \(0.680222\pi\)
\(854\) 2.02844 0.0694117
\(855\) −6.62049 −0.226416
\(856\) 27.8780 0.952852
\(857\) 13.5411 0.462554 0.231277 0.972888i \(-0.425710\pi\)
0.231277 + 0.972888i \(0.425710\pi\)
\(858\) 0.310535 0.0106015
\(859\) −19.5644 −0.667530 −0.333765 0.942656i \(-0.608319\pi\)
−0.333765 + 0.942656i \(0.608319\pi\)
\(860\) −0.0885748 −0.00302038
\(861\) 10.9831 0.374302
\(862\) −2.55554 −0.0870419
\(863\) −8.91434 −0.303448 −0.151724 0.988423i \(-0.548482\pi\)
−0.151724 + 0.988423i \(0.548482\pi\)
\(864\) 5.29991 0.180307
\(865\) 18.2094 0.619137
\(866\) −14.9869 −0.509275
\(867\) −9.70156 −0.329482
\(868\) 16.4388 0.557971
\(869\) 2.56844 0.0871283
\(870\) −2.87897 −0.0976063
\(871\) 5.92008 0.200594
\(872\) −40.2214 −1.36207
\(873\) 17.9952 0.609045
\(874\) 32.3012 1.09260
\(875\) 1.00000 0.0338062
\(876\) −3.85616 −0.130288
\(877\) 6.17626 0.208557 0.104279 0.994548i \(-0.466747\pi\)
0.104279 + 0.994548i \(0.466747\pi\)
\(878\) 3.13263 0.105721
\(879\) 18.3784 0.619889
\(880\) 2.29844 0.0774803
\(881\) 19.3511 0.651954 0.325977 0.945378i \(-0.394307\pi\)
0.325977 + 0.945378i \(0.394307\pi\)
\(882\) 0.546295 0.0183947
\(883\) 28.7925 0.968945 0.484473 0.874806i \(-0.339012\pi\)
0.484473 + 0.874806i \(0.339012\pi\)
\(884\) 2.61304 0.0878861
\(885\) −6.36259 −0.213876
\(886\) −5.23643 −0.175921
\(887\) −43.4970 −1.46049 −0.730243 0.683187i \(-0.760593\pi\)
−0.730243 + 0.683187i \(0.760593\pi\)
\(888\) 5.56821 0.186857
\(889\) −4.05206 −0.135902
\(890\) 5.88755 0.197351
\(891\) −1.00000 −0.0335013
\(892\) 23.6757 0.792722
\(893\) −67.2426 −2.25019
\(894\) 4.10116 0.137163
\(895\) −4.75362 −0.158896
\(896\) 11.5294 0.385169
\(897\) −5.07674 −0.169507
\(898\) 16.9058 0.564154
\(899\) 50.9136 1.69806
\(900\) −1.70156 −0.0567187
\(901\) 4.22709 0.140825
\(902\) 6.00000 0.199778
\(903\) 0.0520550 0.00173228
\(904\) −13.0131 −0.432810
\(905\) −25.2652 −0.839843
\(906\) 1.75795 0.0584039
\(907\) −9.20210 −0.305551 −0.152775 0.988261i \(-0.548821\pi\)
−0.152775 + 0.988261i \(0.548821\pi\)
\(908\) −1.01660 −0.0337371
\(909\) 0.826342 0.0274080
\(910\) −0.310535 −0.0102941
\(911\) 7.98331 0.264499 0.132249 0.991216i \(-0.457780\pi\)
0.132249 + 0.991216i \(0.457780\pi\)
\(912\) 15.2168 0.503878
\(913\) −17.1605 −0.567929
\(914\) −23.0671 −0.762992
\(915\) 3.71308 0.122751
\(916\) −36.0553 −1.19130
\(917\) 5.69781 0.188158
\(918\) 1.47585 0.0487103
\(919\) −36.4176 −1.20131 −0.600653 0.799510i \(-0.705093\pi\)
−0.600653 + 0.799510i \(0.705093\pi\)
\(920\) 18.0598 0.595415
\(921\) 6.45143 0.212582
\(922\) 5.39447 0.177657
\(923\) −1.46000 −0.0480565
\(924\) −1.70156 −0.0559773
\(925\) −2.75362 −0.0905384
\(926\) 1.39857 0.0459597
\(927\) 10.5921 0.347889
\(928\) 27.9305 0.916865
\(929\) 1.06660 0.0349940 0.0174970 0.999847i \(-0.494430\pi\)
0.0174970 + 0.999847i \(0.494430\pi\)
\(930\) −5.27777 −0.173065
\(931\) 6.62049 0.216978
\(932\) 34.5112 1.13045
\(933\) −29.4136 −0.962957
\(934\) −10.7579 −0.352008
\(935\) 2.70156 0.0883505
\(936\) 1.14946 0.0375714
\(937\) −4.74611 −0.155049 −0.0775243 0.996990i \(-0.524702\pi\)
−0.0775243 + 0.996990i \(0.524702\pi\)
\(938\) 5.68947 0.185768
\(939\) −24.7542 −0.807823
\(940\) −17.2823 −0.563687
\(941\) 33.1988 1.08225 0.541125 0.840942i \(-0.317999\pi\)
0.541125 + 0.840942i \(0.317999\pi\)
\(942\) −1.83067 −0.0596465
\(943\) −98.0902 −3.19426
\(944\) 14.6240 0.475971
\(945\) 1.00000 0.0325300
\(946\) 0.0284374 0.000924579 0
\(947\) 6.30361 0.204840 0.102420 0.994741i \(-0.467341\pi\)
0.102420 + 0.994741i \(0.467341\pi\)
\(948\) 4.37036 0.141943
\(949\) −1.28822 −0.0418175
\(950\) 3.61674 0.117343
\(951\) −15.0955 −0.489506
\(952\) 5.46295 0.177055
\(953\) −39.4705 −1.27857 −0.639287 0.768968i \(-0.720770\pi\)
−0.639287 + 0.768968i \(0.720770\pi\)
\(954\) 0.854779 0.0276745
\(955\) 17.4504 0.564680
\(956\) −3.80679 −0.123120
\(957\) −5.27000 −0.170355
\(958\) 8.39501 0.271230
\(959\) 3.40312 0.109893
\(960\) 1.70156 0.0549177
\(961\) 62.3355 2.01082
\(962\) 0.855094 0.0275693
\(963\) −13.7864 −0.444260
\(964\) −15.1957 −0.489421
\(965\) −22.9546 −0.738936
\(966\) −4.87897 −0.156978
\(967\) −20.7192 −0.666285 −0.333142 0.942877i \(-0.608109\pi\)
−0.333142 + 0.942877i \(0.608109\pi\)
\(968\) −2.02214 −0.0649942
\(969\) 17.8857 0.574571
\(970\) −9.83067 −0.315644
\(971\) 0.281521 0.00903444 0.00451722 0.999990i \(-0.498562\pi\)
0.00451722 + 0.999990i \(0.498562\pi\)
\(972\) −1.70156 −0.0545776
\(973\) 8.04724 0.257983
\(974\) −0.835001 −0.0267551
\(975\) −0.568438 −0.0182046
\(976\) −8.53429 −0.273176
\(977\) 53.1267 1.69967 0.849836 0.527047i \(-0.176701\pi\)
0.849836 + 0.527047i \(0.176701\pi\)
\(978\) 5.98741 0.191456
\(979\) 10.7772 0.344442
\(980\) 1.70156 0.0543544
\(981\) 19.8905 0.635055
\(982\) −22.2538 −0.710147
\(983\) −47.2072 −1.50567 −0.752837 0.658207i \(-0.771315\pi\)
−0.752837 + 0.658207i \(0.771315\pi\)
\(984\) 22.2094 0.708009
\(985\) 3.36634 0.107261
\(986\) 7.77773 0.247693
\(987\) 10.1567 0.323293
\(988\) 6.40356 0.203724
\(989\) −0.464905 −0.0147831
\(990\) 0.546295 0.0173624
\(991\) −30.4901 −0.968549 −0.484274 0.874916i \(-0.660916\pi\)
−0.484274 + 0.874916i \(0.660916\pi\)
\(992\) 51.2026 1.62568
\(993\) 6.78263 0.215240
\(994\) −1.40312 −0.0445044
\(995\) −9.54402 −0.302566
\(996\) −29.1996 −0.925226
\(997\) 6.62107 0.209691 0.104846 0.994489i \(-0.466565\pi\)
0.104846 + 0.994489i \(0.466565\pi\)
\(998\) −11.1068 −0.351581
\(999\) −2.75362 −0.0871206
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 1155.2.a.u.1.3 4
3.2 odd 2 3465.2.a.bl.1.2 4
5.4 even 2 5775.2.a.bz.1.2 4
7.6 odd 2 8085.2.a.bn.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.u.1.3 4 1.1 even 1 trivial
3465.2.a.bl.1.2 4 3.2 odd 2
5775.2.a.bz.1.2 4 5.4 even 2
8085.2.a.bn.1.3 4 7.6 odd 2