Properties

Label 5775.2.a.bz.1.2
Level 5775
Weight 2
Character 5775.1
Self dual yes
Analytic conductor 46.114
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5775.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(46.1136071673\)
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: no (minimal twist has level 1155)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.546295\) of defining polynomial
Character \(\chi\) \(=\) 5775.1

$q$-expansion

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

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 \(-0.561869\pi\)
−0.193144 + 0.981170i \(0.561869\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.70156 −0.850781
\(5\) 0 0
\(6\) 0.546295 0.223024
\(7\) 1.00000 0.377964
\(8\) 2.02214 0.714936
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 1.70156 0.491199
\(13\) 0.568438 0.157656 0.0788282 0.996888i \(-0.474882\pi\)
0.0788282 + 0.996888i \(0.474882\pi\)
\(14\) −0.546295 −0.146003
\(15\) 0 0
\(16\) 2.29844 0.574609
\(17\) −2.70156 −0.655225 −0.327613 0.944812i \(-0.606244\pi\)
−0.327613 + 0.944812i \(0.606244\pi\)
\(18\) −0.546295 −0.128763
\(19\) 6.62049 1.51885 0.759423 0.650598i \(-0.225482\pi\)
0.759423 + 0.650598i \(0.225482\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0.546295 0.116470
\(23\) −8.93103 −1.86225 −0.931124 0.364703i \(-0.881171\pi\)
−0.931124 + 0.364703i \(0.881171\pi\)
\(24\) −2.02214 −0.412768
\(25\) 0 0
\(26\) −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 0
\(31\) 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
\(35\) 0 0
\(36\) −1.70156 −0.283594
\(37\) 2.75362 0.452692 0.226346 0.974047i \(-0.427322\pi\)
0.226346 + 0.974047i \(0.427322\pi\)
\(38\) −3.61674 −0.586713
\(39\) −0.568438 −0.0910230
\(40\) 0 0
\(41\) −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 \(-0.498737\pi\)
0.00396916 + 0.999992i \(0.498737\pi\)
\(44\) 1.70156 0.256520
\(45\) 0 0
\(46\) 4.87897 0.719365
\(47\) 10.1567 1.48151 0.740756 0.671774i \(-0.234467\pi\)
0.740756 + 0.671774i \(0.234467\pi\)
\(48\) −2.29844 −0.331751
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.70156 0.378294
\(52\) −0.967233 −0.134131
\(53\) −1.56469 −0.214926 −0.107463 0.994209i \(-0.534273\pi\)
−0.107463 + 0.994209i \(0.534273\pi\)
\(54\) 0.546295 0.0743413
\(55\) 0 0
\(56\) 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 \(0.364072\pi\)
\(60\) 0 0
\(61\) −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 0
\(66\) −0.546295 −0.0672442
\(67\) 10.4146 1.27235 0.636176 0.771544i \(-0.280515\pi\)
0.636176 + 0.771544i \(0.280515\pi\)
\(68\) 4.59688 0.557453
\(69\) 8.93103 1.07517
\(70\) 0 0
\(71\) 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 \(-0.542340\pi\)
−0.132622 + 0.991167i \(0.542340\pi\)
\(74\) −1.50429 −0.174870
\(75\) 0 0
\(76\) −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 \(0.546153\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −17.1605 −1.88361 −0.941804 0.336161i \(-0.890871\pi\)
−0.941804 + 0.336161i \(0.890871\pi\)
\(84\) 1.70156 0.185656
\(85\) 0 0
\(86\) −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 0
\(91\) 0.568438 0.0595885
\(92\) 15.1967 1.58437
\(93\) −9.66103 −1.00180
\(94\) −5.54857 −0.572292
\(95\) 0 0
\(96\) 5.29991 0.540920
\(97\) −17.9952 −1.82713 −0.913567 0.406689i \(-0.866683\pi\)
−0.913567 + 0.406689i \(0.866683\pi\)
\(98\) −0.546295 −0.0551841
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 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 \(-0.674752\pi\)
−0.521833 + 0.853048i \(0.674752\pi\)
\(104\) 1.14946 0.112714
\(105\) 0 0
\(106\) 0.854779 0.0830235
\(107\) 13.7864 1.33278 0.666390 0.745603i \(-0.267839\pi\)
0.666390 + 0.745603i \(0.267839\pi\)
\(108\) 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 0
\(111\) −2.75362 −0.261362
\(112\) 2.29844 0.217182
\(113\) −6.43531 −0.605383 −0.302692 0.953089i \(-0.597885\pi\)
−0.302692 + 0.953089i \(0.597885\pi\)
\(114\) 3.61674 0.338739
\(115\) 0 0
\(116\) −8.96723 −0.832587
\(117\) 0.568438 0.0525521
\(118\) −3.47585 −0.319978
\(119\) −2.70156 −0.247652
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.02844 0.183646
\(123\) 10.9831 0.990311
\(124\) −16.4388 −1.47625
\(125\) 0 0
\(126\) −0.546295 −0.0486678
\(127\) −4.05206 −0.359562 −0.179781 0.983707i \(-0.557539\pi\)
−0.179781 + 0.983707i \(0.557539\pi\)
\(128\) 11.5294 1.01906
\(129\) −0.0520550 −0.00458319
\(130\) 0 0
\(131\) −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
\(135\) 0 0
\(136\) −5.46295 −0.468444
\(137\) 3.40312 0.290749 0.145374 0.989377i \(-0.453561\pi\)
0.145374 + 0.989377i \(0.453561\pi\)
\(138\) −4.87897 −0.415326
\(139\) −8.04724 −0.682558 −0.341279 0.939962i \(-0.610860\pi\)
−0.341279 + 0.939962i \(0.610860\pi\)
\(140\) 0 0
\(141\) −10.1567 −0.855352
\(142\) −1.40312 −0.117748
\(143\) −0.568438 −0.0475352
\(144\) 2.29844 0.191536
\(145\) 0 0
\(146\) 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 0
\(151\) 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
\(155\) 0 0
\(156\) 0.967233 0.0774406
\(157\) 3.35107 0.267444 0.133722 0.991019i \(-0.457307\pi\)
0.133722 + 0.991019i \(0.457307\pi\)
\(158\) 1.40312 0.111627
\(159\) 1.56469 0.124088
\(160\) 0 0
\(161\) −8.93103 −0.703864
\(162\) −0.546295 −0.0429210
\(163\) −10.9600 −0.858457 −0.429228 0.903196i \(-0.641214\pi\)
−0.429228 + 0.903196i \(0.641214\pi\)
\(164\) 18.6884 1.45932
\(165\) 0 0
\(166\) 9.37469 0.727617
\(167\) 7.58830 0.587201 0.293600 0.955928i \(-0.405147\pi\)
0.293600 + 0.955928i \(0.405147\pi\)
\(168\) −2.02214 −0.156012
\(169\) −12.6769 −0.975144
\(170\) 0 0
\(171\) 6.62049 0.506282
\(172\) −0.0885748 −0.00675377
\(173\) 18.2094 1.38443 0.692216 0.721690i \(-0.256634\pi\)
0.692216 + 0.721690i \(0.256634\pi\)
\(174\) 2.87897 0.218254
\(175\) 0 0
\(176\) −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 \(0.443150\pi\)
\(180\) 0 0
\(181\) 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
\(185\) 0 0
\(186\) 5.27777 0.386985
\(187\) 2.70156 0.197558
\(188\) −17.2823 −1.26044
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −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 \(-0.809477\pi\)
−0.826156 + 0.563442i \(0.809477\pi\)
\(194\) 9.83067 0.705801
\(195\) 0 0
\(196\) −1.70156 −0.121540
\(197\) 3.36634 0.239842 0.119921 0.992783i \(-0.461736\pi\)
0.119921 + 0.992783i \(0.461736\pi\)
\(198\) 0.546295 0.0388235
\(199\) 9.54402 0.676557 0.338279 0.941046i \(-0.390155\pi\)
0.338279 + 0.941046i \(0.390155\pi\)
\(200\) 0 0
\(201\) −10.4146 −0.734592
\(202\) −0.451426 −0.0317622
\(203\) 5.27000 0.369882
\(204\) −4.59688 −0.321846
\(205\) 0 0
\(206\) 5.78638 0.403156
\(207\) −8.93103 −0.620749
\(208\) 1.30652 0.0905909
\(209\) −6.62049 −0.457949
\(210\) 0 0
\(211\) −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 0
\(216\) −2.02214 −0.137589
\(217\) 9.66103 0.655833
\(218\) −10.8661 −0.735943
\(219\) 2.26625 0.153139
\(220\) 0 0
\(221\) −1.53567 −0.103300
\(222\) 1.50429 0.100961
\(223\) 13.9141 0.931758 0.465879 0.884848i \(-0.345738\pi\)
0.465879 + 0.884848i \(0.345738\pi\)
\(224\) −5.29991 −0.354115
\(225\) 0 0
\(226\) 3.51558 0.233853
\(227\) −0.597452 −0.0396543 −0.0198271 0.999803i \(-0.506312\pi\)
−0.0198271 + 0.999803i \(0.506312\pi\)
\(228\) 11.2652 0.746055
\(229\) 21.1895 1.40024 0.700121 0.714024i \(-0.253129\pi\)
0.700121 + 0.714024i \(0.253129\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) 10.6567 0.699647
\(233\) 20.2821 1.32872 0.664362 0.747411i \(-0.268703\pi\)
0.664362 + 0.747411i \(0.268703\pi\)
\(234\) −0.310535 −0.0203003
\(235\) 0 0
\(236\) −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 \(0.476948\pi\)
\(240\) 0 0
\(241\) 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
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 3.76334 0.239456
\(248\) 19.5360 1.24054
\(249\) 17.1605 1.08750
\(250\) 0 0
\(251\) 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
\(255\) 0 0
\(256\) −2.89531 −0.180957
\(257\) −5.80943 −0.362382 −0.181191 0.983448i \(-0.557995\pi\)
−0.181191 + 0.983448i \(0.557995\pi\)
\(258\) 0.0284374 0.00177043
\(259\) 2.75362 0.171101
\(260\) 0 0
\(261\) 5.27000 0.326205
\(262\) 3.11268 0.192302
\(263\) 23.3221 1.43810 0.719050 0.694959i \(-0.244577\pi\)
0.719050 + 0.694959i \(0.244577\pi\)
\(264\) 2.02214 0.124454
\(265\) 0 0
\(266\) −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 0
\(271\) 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
\(275\) 0 0
\(276\) −15.1967 −0.914734
\(277\) 15.8988 0.955269 0.477634 0.878559i \(-0.341494\pi\)
0.477634 + 0.878559i \(0.341494\pi\)
\(278\) 4.39616 0.263664
\(279\) 9.66103 0.578391
\(280\) 0 0
\(281\) −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 \(-0.487438\pi\)
0.0394544 + 0.999221i \(0.487438\pi\)
\(284\) −4.37036 −0.259333
\(285\) 0 0
\(286\) 0.310535 0.0183623
\(287\) −10.9831 −0.648311
\(288\) −5.29991 −0.312300
\(289\) −9.70156 −0.570680
\(290\) 0 0
\(291\) 17.9952 1.05490
\(292\) 3.85616 0.225665
\(293\) −18.3784 −1.07368 −0.536840 0.843684i \(-0.680382\pi\)
−0.536840 + 0.843684i \(0.680382\pi\)
\(294\) 0.546295 0.0318606
\(295\) 0 0
\(296\) 5.56821 0.323646
\(297\) 1.00000 0.0580259
\(298\) −4.10116 −0.237574
\(299\) −5.07674 −0.293595
\(300\) 0 0
\(301\) 0.0520550 0.00300040
\(302\) −1.75795 −0.101158
\(303\) −0.826342 −0.0474721
\(304\) 15.2168 0.872743
\(305\) 0 0
\(306\) 1.47585 0.0843687
\(307\) −6.45143 −0.368202 −0.184101 0.982907i \(-0.558937\pi\)
−0.184101 + 0.982907i \(0.558937\pi\)
\(308\) 1.70156 0.0969555
\(309\) 10.5921 0.602561
\(310\) 0 0
\(311\) −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 \(-0.253364\pi\)
0.699595 + 0.714540i \(0.253364\pi\)
\(314\) −1.83067 −0.103311
\(315\) 0 0
\(316\) 4.37036 0.245852
\(317\) 15.0955 0.847850 0.423925 0.905697i \(-0.360652\pi\)
0.423925 + 0.905697i \(0.360652\pi\)
\(318\) −0.854779 −0.0479336
\(319\) −5.27000 −0.295063
\(320\) 0 0
\(321\) −13.7864 −0.769481
\(322\) 4.87897 0.271895
\(323\) −17.8857 −0.995186
\(324\) −1.70156 −0.0945312
\(325\) 0 0
\(326\) 5.98741 0.331612
\(327\) −19.8905 −1.09995
\(328\) −22.2094 −1.22631
\(329\) 10.1567 0.559959
\(330\) 0 0
\(331\) 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
\(335\) 0 0
\(336\) −2.29844 −0.125390
\(337\) 31.9509 1.74048 0.870238 0.492631i \(-0.163965\pi\)
0.870238 + 0.492631i \(0.163965\pi\)
\(338\) 6.92531 0.376687
\(339\) 6.43531 0.349518
\(340\) 0 0
\(341\) −9.66103 −0.523174
\(342\) −3.61674 −0.195571
\(343\) 1.00000 0.0539949
\(344\) 0.105263 0.00567538
\(345\) 0 0
\(346\) −9.94768 −0.534791
\(347\) 13.6694 0.733810 0.366905 0.930258i \(-0.380417\pi\)
0.366905 + 0.930258i \(0.380417\pi\)
\(348\) 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 0
\(351\) −0.568438 −0.0303410
\(352\) 5.29991 0.282486
\(353\) 9.94313 0.529219 0.264610 0.964356i \(-0.414757\pi\)
0.264610 + 0.964356i \(0.414757\pi\)
\(354\) 3.47585 0.184739
\(355\) 0 0
\(356\) 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 \(0.607755\pi\)
\(360\) 0 0
\(361\) 24.8309 1.30689
\(362\) −13.8022 −0.725429
\(363\) −1.00000 −0.0524864
\(364\) −0.967233 −0.0506968
\(365\) 0 0
\(366\) −2.02844 −0.106028
\(367\) −36.9699 −1.92981 −0.964907 0.262592i \(-0.915423\pi\)
−0.964907 + 0.262592i \(0.915423\pi\)
\(368\) −20.5274 −1.07007
\(369\) −10.9831 −0.571756
\(370\) 0 0
\(371\) −1.56469 −0.0812344
\(372\) 16.4388 0.852314
\(373\) 13.4109 0.694390 0.347195 0.937793i \(-0.387134\pi\)
0.347195 + 0.937793i \(0.387134\pi\)
\(374\) −1.47585 −0.0763143
\(375\) 0 0
\(376\) 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 \(0.506587\pi\)
\(380\) 0 0
\(381\) 4.05206 0.207593
\(382\) 9.53304 0.487753
\(383\) 22.3494 1.14200 0.571001 0.820949i \(-0.306555\pi\)
0.571001 + 0.820949i \(0.306555\pi\)
\(384\) −11.5294 −0.588356
\(385\) 0 0
\(386\) 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 0
\(391\) 24.1277 1.22019
\(392\) 2.02214 0.102134
\(393\) 5.69781 0.287416
\(394\) −1.83902 −0.0926482
\(395\) 0 0
\(396\) 1.70156 0.0855067
\(397\) −4.45143 −0.223411 −0.111705 0.993741i \(-0.535631\pi\)
−0.111705 + 0.993741i \(0.535631\pi\)
\(398\) −5.21384 −0.261346
\(399\) −6.62049 −0.331439
\(400\) 0 0
\(401\) 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
\(405\) 0 0
\(406\) −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 \(0.459816\pi\)
\(410\) 0 0
\(411\) −3.40312 −0.167864
\(412\) 18.0230 0.887932
\(413\) 6.36259 0.313083
\(414\) 4.87897 0.239788
\(415\) 0 0
\(416\) −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 \(0.522278\pi\)
\(420\) 0 0
\(421\) −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
\(425\) 0 0
\(426\) 1.40312 0.0679816
\(427\) −3.71308 −0.179689
\(428\) −23.4584 −1.13390
\(429\) 0.568438 0.0274445
\(430\) 0 0
\(431\) −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 \(-0.270900\pi\)
0.659189 + 0.751977i \(0.270900\pi\)
\(434\) −5.27777 −0.253341
\(435\) 0 0
\(436\) −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 \(0.456305\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0.838929 0.0399038
\(443\) 9.58536 0.455414 0.227707 0.973730i \(-0.426877\pi\)
0.227707 + 0.973730i \(0.426877\pi\)
\(444\) 4.68545 0.222362
\(445\) 0 0
\(446\) −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 0
\(451\) 10.9831 0.517173
\(452\) 10.9501 0.515049
\(453\) −3.21795 −0.151192
\(454\) 0.326385 0.0153180
\(455\) 0 0
\(456\) −13.3876 −0.626931
\(457\) 42.2246 1.97519 0.987593 0.157036i \(-0.0501940\pi\)
0.987593 + 0.157036i \(0.0501940\pi\)
\(458\) −11.5757 −0.540898
\(459\) 2.70156 0.126098
\(460\) 0 0
\(461\) 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 \(-0.518947\pi\)
−0.0594888 + 0.998229i \(0.518947\pi\)
\(464\) 12.1128 0.562321
\(465\) 0 0
\(466\) −11.0800 −0.513271
\(467\) 19.6924 0.911256 0.455628 0.890170i \(-0.349415\pi\)
0.455628 + 0.890170i \(0.349415\pi\)
\(468\) −0.967233 −0.0447104
\(469\) 10.4146 0.480904
\(470\) 0 0
\(471\) −3.35107 −0.154409
\(472\) 12.8661 0.592209
\(473\) −0.0520550 −0.00239349
\(474\) −1.40312 −0.0644476
\(475\) 0 0
\(476\) 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 \(0.385817\pi\)
\(480\) 0 0
\(481\) 1.56526 0.0713698
\(482\) −4.87866 −0.222217
\(483\) 8.93103 0.406376
\(484\) −1.70156 −0.0773437
\(485\) 0 0
\(486\) 0.546295 0.0247804
\(487\) 1.52848 0.0692621 0.0346310 0.999400i \(-0.488974\pi\)
0.0346310 + 0.999400i \(0.488974\pi\)
\(488\) −7.50839 −0.339889
\(489\) 10.9600 0.495630
\(490\) 0 0
\(491\) −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
\(495\) 0 0
\(496\) 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 \(0.650388\pi\)
\(500\) 0 0
\(501\) −7.58830 −0.339020
\(502\) −1.83491 −0.0818963
\(503\) 23.5952 1.05206 0.526030 0.850466i \(-0.323680\pi\)
0.526030 + 0.850466i \(0.323680\pi\)
\(504\) 2.02214 0.0900734
\(505\) 0 0
\(506\) −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 \(0.179981\pi\)
\(510\) 0 0
\(511\) −2.26625 −0.100253
\(512\) −21.4771 −0.949161
\(513\) −6.62049 −0.292302
\(514\) 3.17366 0.139984
\(515\) 0 0
\(516\) 0.0885748 0.00389929
\(517\) −10.1567 −0.446693
\(518\) −1.50429 −0.0660945
\(519\) −18.2094 −0.799303
\(520\) 0 0
\(521\) 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 \(-0.576965\pi\)
−0.239442 + 0.970911i \(0.576965\pi\)
\(524\) 9.69518 0.423536
\(525\) 0 0
\(526\) −12.7407 −0.555522
\(527\) −26.0999 −1.13693
\(528\) 2.29844 0.100027
\(529\) 56.7633 2.46797
\(530\) 0 0
\(531\) 6.36259 0.276113
\(532\) −11.2652 −0.488408
\(533\) −6.24321 −0.270423
\(534\) −5.88755 −0.254779
\(535\) 0 0
\(536\) 21.0599 0.909650
\(537\) −4.75362 −0.205134
\(538\) −15.9106 −0.685954
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 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
\(545\) 0 0
\(546\) 0.310535 0.0132897
\(547\) 42.0182 1.79657 0.898285 0.439414i \(-0.144814\pi\)
0.898285 + 0.439414i \(0.144814\pi\)
\(548\) −5.79063 −0.247363
\(549\) −3.71308 −0.158471
\(550\) 0 0
\(551\) 34.8900 1.48636
\(552\) 18.0598 0.768677
\(553\) −2.56844 −0.109221
\(554\) −8.68545 −0.369009
\(555\) 0 0
\(556\) 13.6929 0.580707
\(557\) 27.9461 1.18411 0.592057 0.805896i \(-0.298316\pi\)
0.592057 + 0.805896i \(0.298316\pi\)
\(558\) −5.27777 −0.223426
\(559\) 0.0295901 0.00125153
\(560\) 0 0
\(561\) −2.70156 −0.114060
\(562\) 7.70500 0.325016
\(563\) 0.266247 0.0112210 0.00561050 0.999984i \(-0.498214\pi\)
0.00561050 + 0.999984i \(0.498214\pi\)
\(564\) 17.2823 0.727717
\(565\) 0 0
\(566\) −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 \(0.766116\pi\)
\(570\) 0 0
\(571\) −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
\(575\) 0 0
\(576\) −1.70156 −0.0708984
\(577\) 25.1030 1.04505 0.522527 0.852623i \(-0.324990\pi\)
0.522527 + 0.852623i \(0.324990\pi\)
\(578\) 5.29991 0.220447
\(579\) 22.9546 0.953963
\(580\) 0 0
\(581\) −17.1605 −0.711937
\(582\) −9.83067 −0.407494
\(583\) 1.56469 0.0648026
\(584\) −4.58268 −0.189633
\(585\) 0 0
\(586\) 10.0400 0.414750
\(587\) −3.24638 −0.133993 −0.0669963 0.997753i \(-0.521342\pi\)
−0.0669963 + 0.997753i \(0.521342\pi\)
\(588\) 1.70156 0.0701712
\(589\) 63.9608 2.63546
\(590\) 0 0
\(591\) −3.36634 −0.138473
\(592\) 6.32902 0.260121
\(593\) −10.7252 −0.440430 −0.220215 0.975451i \(-0.570676\pi\)
−0.220215 + 0.975451i \(0.570676\pi\)
\(594\) −0.546295 −0.0224147
\(595\) 0 0
\(596\) −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 \(0.233483\pi\)
\(600\) 0 0
\(601\) −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
\(605\) 0 0
\(606\) 0.451426 0.0183379
\(607\) 9.91893 0.402597 0.201299 0.979530i \(-0.435484\pi\)
0.201299 + 0.979530i \(0.435484\pi\)
\(608\) −35.0880 −1.42301
\(609\) −5.27000 −0.213551
\(610\) 0 0
\(611\) 5.77348 0.233570
\(612\) 4.59688 0.185818
\(613\) −39.2681 −1.58602 −0.793012 0.609206i \(-0.791488\pi\)
−0.793012 + 0.609206i \(0.791488\pi\)
\(614\) 3.52438 0.142232
\(615\) 0 0
\(616\) −2.02214 −0.0814745
\(617\) −31.3462 −1.26195 −0.630976 0.775802i \(-0.717345\pi\)
−0.630976 + 0.775802i \(0.717345\pi\)
\(618\) −5.78638 −0.232762
\(619\) −12.3916 −0.498061 −0.249030 0.968496i \(-0.580112\pi\)
−0.249030 + 0.968496i \(0.580112\pi\)
\(620\) 0 0
\(621\) 8.93103 0.358390
\(622\) 16.0685 0.644287
\(623\) −10.7772 −0.431781
\(624\) −1.30652 −0.0523027
\(625\) 0 0
\(626\) −13.5231 −0.540491
\(627\) 6.62049 0.264397
\(628\) −5.70205 −0.227537
\(629\) −7.43907 −0.296615
\(630\) 0 0
\(631\) 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
\(635\) 0 0
\(636\) −2.66241 −0.105571
\(637\) 0.568438 0.0225223
\(638\) 2.87897 0.113980
\(639\) 2.56844 0.101606
\(640\) 0 0
\(641\) 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 \(-0.463355\pi\)
0.114868 + 0.993381i \(0.463355\pi\)
\(644\) 15.1967 0.598834
\(645\) 0 0
\(646\) 9.77085 0.384429
\(647\) −8.30759 −0.326605 −0.163302 0.986576i \(-0.552215\pi\)
−0.163302 + 0.986576i \(0.552215\pi\)
\(648\) 2.02214 0.0794373
\(649\) −6.36259 −0.249753
\(650\) 0 0
\(651\) −9.66103 −0.378646
\(652\) 18.6492 0.730359
\(653\) −44.4182 −1.73822 −0.869109 0.494621i \(-0.835307\pi\)
−0.869109 + 0.494621i \(0.835307\pi\)
\(654\) 10.8661 0.424897
\(655\) 0 0
\(656\) −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 \(0.584671\pi\)
\(660\) 0 0
\(661\) 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
\(665\) 0 0
\(666\) −1.50429 −0.0582899
\(667\) −47.0665 −1.82242
\(668\) −12.9120 −0.499579
\(669\) −13.9141 −0.537951
\(670\) 0 0
\(671\) 3.71308 0.143342
\(672\) 5.29991 0.204449
\(673\) −5.81295 −0.224073 −0.112036 0.993704i \(-0.535737\pi\)
−0.112036 + 0.993704i \(0.535737\pi\)
\(674\) −17.4546 −0.672326
\(675\) 0 0
\(676\) 21.5705 0.829634
\(677\) 23.3699 0.898177 0.449088 0.893487i \(-0.351749\pi\)
0.449088 + 0.893487i \(0.351749\pi\)
\(678\) −3.51558 −0.135015
\(679\) −17.9952 −0.690592
\(680\) 0 0
\(681\) 0.597452 0.0228944
\(682\) 5.27777 0.202096
\(683\) −13.4705 −0.515433 −0.257716 0.966221i \(-0.582970\pi\)
−0.257716 + 0.966221i \(0.582970\pi\)
\(684\) −11.2652 −0.430735
\(685\) 0 0
\(686\) −0.546295 −0.0208576
\(687\) −21.1895 −0.808430
\(688\) 0.119645 0.00456143
\(689\) −0.889427 −0.0338845
\(690\) 0 0
\(691\) 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
\(695\) 0 0
\(696\) −10.6567 −0.403941
\(697\) 29.6715 1.12389
\(698\) −8.83469 −0.334398
\(699\) −20.2821 −0.767139
\(700\) 0 0
\(701\) −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
\(705\) 0 0
\(706\) −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 \(0.622483\pi\)
\(710\) 0 0
\(711\) −2.56844 −0.0963240
\(712\) −21.7931 −0.816732
\(713\) −86.2829 −3.23132
\(714\) −1.47585 −0.0552323
\(715\) 0 0
\(716\) −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 \(0.666004\pi\)
\(720\) 0 0
\(721\) −10.5921 −0.394469
\(722\) −13.5650 −0.504837
\(723\) −8.93045 −0.332127
\(724\) −42.9903 −1.59772
\(725\) 0 0
\(726\) 0.546295 0.0202749
\(727\) −16.4955 −0.611782 −0.305891 0.952066i \(-0.598954\pi\)
−0.305891 + 0.952066i \(0.598954\pi\)
\(728\) 1.14946 0.0426020
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −0.140630 −0.00520138
\(732\) −6.31804 −0.233522
\(733\) 22.6838 0.837847 0.418923 0.908022i \(-0.362408\pi\)
0.418923 + 0.908022i \(0.362408\pi\)
\(734\) 20.1965 0.745465
\(735\) 0 0
\(736\) 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 \(0.778393\pi\)
\(740\) 0 0
\(741\) −3.76334 −0.138250
\(742\) 0.854779 0.0313799
\(743\) 26.4664 0.970959 0.485480 0.874248i \(-0.338645\pi\)
0.485480 + 0.874248i \(0.338645\pi\)
\(744\) −19.5360 −0.716224
\(745\) 0 0
\(746\) −7.32630 −0.268235
\(747\) −17.1605 −0.627870
\(748\) −4.59688 −0.168078
\(749\) 13.7864 0.503744
\(750\) 0 0
\(751\) −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
\(755\) 0 0
\(756\) 1.70156 0.0618852
\(757\) −27.3451 −0.993874 −0.496937 0.867786i \(-0.665542\pi\)
−0.496937 + 0.867786i \(0.665542\pi\)
\(758\) 0.440134 0.0159864
\(759\) −8.93103 −0.324176
\(760\) 0 0
\(761\) −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
\(765\) 0 0
\(766\) −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 0
\(771\) 5.80943 0.209221
\(772\) 39.0588 1.40576
\(773\) −6.60438 −0.237543 −0.118772 0.992922i \(-0.537896\pi\)
−0.118772 + 0.992922i \(0.537896\pi\)
\(774\) −0.0284374 −0.00102216
\(775\) 0 0
\(776\) −36.3888 −1.30628
\(777\) −2.75362 −0.0987855
\(778\) 16.1082 0.577507
\(779\) −72.7134 −2.60523
\(780\) 0 0
\(781\) −2.56844 −0.0919060
\(782\) −13.1808 −0.471346
\(783\) −5.27000 −0.188334
\(784\) 2.29844 0.0820871
\(785\) 0 0
\(786\) −3.11268 −0.111026
\(787\) 23.1916 0.826692 0.413346 0.910574i \(-0.364360\pi\)
0.413346 + 0.910574i \(0.364360\pi\)
\(788\) −5.72804 −0.204053
\(789\) −23.3221 −0.830287
\(790\) 0 0
\(791\) −6.43531 −0.228813
\(792\) −2.02214 −0.0718537
\(793\) −2.11066 −0.0749517
\(794\) 2.43179 0.0863010
\(795\) 0 0
\(796\) −16.2397 −0.575602
\(797\) −32.5872 −1.15430 −0.577150 0.816638i \(-0.695835\pi\)
−0.577150 + 0.816638i \(0.695835\pi\)
\(798\) 3.61674 0.128031
\(799\) −27.4391 −0.970724
\(800\) 0 0
\(801\) −10.7772 −0.380795
\(802\) −12.4076 −0.438127
\(803\) 2.26625 0.0799741
\(804\) 17.7212 0.624977
\(805\) 0 0
\(806\) −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 0
\(811\) 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
\(815\) 0 0
\(816\) 6.20937 0.217372
\(817\) 0.344630 0.0120571
\(818\) −2.78205 −0.0972723
\(819\) 0.568438 0.0198628
\(820\) 0 0
\(821\) 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 \(-0.518171\pi\)
−0.0570539 + 0.998371i \(0.518171\pi\)
\(824\) −21.4187 −0.746154
\(825\) 0 0
\(826\) −3.47585 −0.120940
\(827\) 36.5217 1.26998 0.634992 0.772519i \(-0.281003\pi\)
0.634992 + 0.772519i \(0.281003\pi\)
\(828\) 15.1967 0.528122
\(829\) 21.2335 0.737469 0.368735 0.929535i \(-0.379791\pi\)
0.368735 + 0.929535i \(0.379791\pi\)
\(830\) 0 0
\(831\) −15.8988 −0.551525
\(832\) −0.967233 −0.0335328
\(833\) −2.70156 −0.0936036
\(834\) −4.39616 −0.152227
\(835\) 0 0
\(836\) 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 \(0.383354\pi\)
\(840\) 0 0
\(841\) −1.22709 −0.0423136
\(842\) 8.79790 0.303196
\(843\) 14.1041 0.485771
\(844\) 13.1595 0.452967
\(845\) 0 0
\(846\) −5.54857 −0.190764
\(847\) 1.00000 0.0343604
\(848\) −3.59633 −0.123499
\(849\) −1.32745 −0.0455580
\(850\) 0 0
\(851\) −24.5926 −0.843025
\(852\) 4.37036 0.149726
\(853\) 31.3333 1.07283 0.536417 0.843953i \(-0.319778\pi\)
0.536417 + 0.843953i \(0.319778\pi\)
\(854\) 2.02844 0.0694117
\(855\) 0 0
\(856\) 27.8780 0.952852
\(857\) −13.5411 −0.462554 −0.231277 0.972888i \(-0.574290\pi\)
−0.231277 + 0.972888i \(0.574290\pi\)
\(858\) −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 0
\(861\) 10.9831 0.374302
\(862\) 2.55554 0.0870419
\(863\) 8.91434 0.303448 0.151724 0.988423i \(-0.451518\pi\)
0.151724 + 0.988423i \(0.451518\pi\)
\(864\) 5.29991 0.180307
\(865\) 0 0
\(866\) −14.9869 −0.509275
\(867\) 9.70156 0.329482
\(868\) −16.4388 −0.557971
\(869\) 2.56844 0.0871283
\(870\) 0 0
\(871\) 5.92008 0.200594
\(872\) 40.2214 1.36207
\(873\) −17.9952 −0.609045
\(874\) 32.3012 1.09260
\(875\) 0 0
\(876\) −3.85616 −0.130288
\(877\) −6.17626 −0.208557 −0.104279 0.994548i \(-0.533253\pi\)
−0.104279 + 0.994548i \(0.533253\pi\)
\(878\) −3.13263 −0.105721
\(879\) 18.3784 0.619889
\(880\) 0 0
\(881\) 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 \(-0.660988\pi\)
−0.484473 + 0.874806i \(0.660988\pi\)
\(884\) 2.61304 0.0878861
\(885\) 0 0
\(886\) −5.23643 −0.175921
\(887\) 43.4970 1.46049 0.730243 0.683187i \(-0.239407\pi\)
0.730243 + 0.683187i \(0.239407\pi\)
\(888\) −5.56821 −0.186857
\(889\) −4.05206 −0.135902
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) −23.6757 −0.792722
\(893\) 67.2426 2.25019
\(894\) 4.10116 0.137163
\(895\) 0 0
\(896\) 11.5294 0.385169
\(897\) 5.07674 0.169507
\(898\) −16.9058 −0.564154
\(899\) 50.9136 1.69806
\(900\) 0 0
\(901\) 4.22709 0.140825
\(902\) −6.00000 −0.199778
\(903\) −0.0520550 −0.00173228
\(904\) −13.0131 −0.432810
\(905\) 0 0
\(906\) 1.75795 0.0584039
\(907\) 9.20210 0.305551 0.152775 0.988261i \(-0.451179\pi\)
0.152775 + 0.988261i \(0.451179\pi\)
\(908\) 1.01660 0.0337371
\(909\) 0.826342 0.0274080
\(910\) 0 0
\(911\) 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
\(915\) 0 0
\(916\) −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 \(0.705093\pi\)
\(920\) 0 0
\(921\) 6.45143 0.212582
\(922\) −5.39447 −0.177657
\(923\) 1.46000 0.0480565
\(924\) −1.70156 −0.0559773
\(925\) 0 0
\(926\) 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 \(0.494430\pi\)
\(930\) 0 0
\(931\) 6.62049 0.216978
\(932\) −34.5112 −1.13045
\(933\) 29.4136 0.962957
\(934\) −10.7579 −0.352008
\(935\) 0 0
\(936\) 1.14946 0.0375714
\(937\) 4.74611 0.155049 0.0775243 0.996990i \(-0.475298\pi\)
0.0775243 + 0.996990i \(0.475298\pi\)
\(938\) −5.68947 −0.185768
\(939\) −24.7542 −0.807823
\(940\) 0 0
\(941\) 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
\(945\) 0 0
\(946\) 0.0284374 0.000924579 0
\(947\) −6.30361 −0.204840 −0.102420 0.994741i \(-0.532659\pi\)
−0.102420 + 0.994741i \(0.532659\pi\)
\(948\) −4.37036 −0.141943
\(949\) −1.28822 −0.0418175
\(950\) 0 0
\(951\) −15.0955 −0.489506
\(952\) −5.46295 −0.177055
\(953\) 39.4705 1.27857 0.639287 0.768968i \(-0.279230\pi\)
0.639287 + 0.768968i \(0.279230\pi\)
\(954\) 0.854779 0.0276745
\(955\) 0 0
\(956\) −3.80679 −0.123120
\(957\) 5.27000 0.170355
\(958\) −8.39501 −0.271230
\(959\) 3.40312 0.109893
\(960\) 0 0
\(961\) 62.3355 2.01082
\(962\) −0.855094 −0.0275693
\(963\) 13.7864 0.444260
\(964\) −15.1957 −0.489421
\(965\) 0 0
\(966\) −4.87897 −0.156978
\(967\) 20.7192 0.666285 0.333142 0.942877i \(-0.391891\pi\)
0.333142 + 0.942877i \(0.391891\pi\)
\(968\) 2.02214 0.0649942
\(969\) 17.8857 0.574571
\(970\) 0 0
\(971\) 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 0
\(976\) −8.53429 −0.273176
\(977\) −53.1267 −1.69967 −0.849836 0.527047i \(-0.823299\pi\)
−0.849836 + 0.527047i \(0.823299\pi\)
\(978\) −5.98741 −0.191456
\(979\) 10.7772 0.344442
\(980\) 0 0
\(981\) 19.8905 0.635055
\(982\) 22.2538 0.710147
\(983\) 47.2072 1.50567 0.752837 0.658207i \(-0.228685\pi\)
0.752837 + 0.658207i \(0.228685\pi\)
\(984\) 22.2094 0.708009
\(985\) 0 0
\(986\) 7.77773 0.247693
\(987\) −10.1567 −0.323293
\(988\) −6.40356 −0.203724
\(989\) −0.464905 −0.0147831
\(990\) 0 0
\(991\) −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
\(995\) 0 0
\(996\) −29.1996 −0.925226
\(997\) −6.62107 −0.209691 −0.104846 0.994489i \(-0.533435\pi\)
−0.104846 + 0.994489i \(0.533435\pi\)
\(998\) 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 5775.2.a.bz.1.2 4
5.4 even 2 1155.2.a.u.1.3 4
15.14 odd 2 3465.2.a.bl.1.2 4
35.34 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 5.4 even 2
3465.2.a.bl.1.2 4 15.14 odd 2
5775.2.a.bz.1.2 4 1.1 even 1 trivial
8085.2.a.bn.1.3 4 35.34 odd 2