Properties

Label 8085.2.a.bn.1.2
Level 8085
Weight 2
Character 8085.1
Self dual yes
Analytic conductor 64.559
Analytic rank 1
Dimension 4
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8085.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.5590500342\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 8085.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.546295 q^{2} -1.00000 q^{3} -1.70156 q^{4} +1.00000 q^{5} +0.546295 q^{6} +2.02214 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.546295 q^{2} -1.00000 q^{3} -1.70156 q^{4} +1.00000 q^{5} +0.546295 q^{6} +2.02214 q^{8} +1.00000 q^{9} -0.546295 q^{10} -1.00000 q^{11} +1.70156 q^{12} -4.56844 q^{13} -1.00000 q^{15} +2.29844 q^{16} -2.70156 q^{17} -0.546295 q^{18} +8.02362 q^{19} -1.70156 q^{20} +0.546295 q^{22} -3.52790 q^{23} -2.02214 q^{24} +1.00000 q^{25} +2.49571 q^{26} -1.00000 q^{27} +0.133124 q^{29} +0.546295 q^{30} -2.33897 q^{31} -5.29991 q^{32} +1.00000 q^{33} +1.47585 q^{34} -1.70156 q^{36} +6.75362 q^{37} -4.38326 q^{38} +4.56844 q^{39} +2.02214 q^{40} -10.9831 q^{41} +9.45518 q^{43} +1.70156 q^{44} +1.00000 q^{45} +1.92728 q^{46} +0.649507 q^{47} -2.29844 q^{48} -0.546295 q^{50} +2.70156 q^{51} +7.77348 q^{52} +11.8384 q^{53} +0.546295 q^{54} -1.00000 q^{55} -8.02362 q^{57} -0.0727248 q^{58} +0.959466 q^{59} +1.70156 q^{60} -13.1162 q^{61} +1.27777 q^{62} -1.70156 q^{64} -4.56844 q^{65} -0.546295 q^{66} +6.41464 q^{67} +4.59688 q^{68} +3.52790 q^{69} -2.56844 q^{71} +2.02214 q^{72} -12.5400 q^{73} -3.68947 q^{74} -1.00000 q^{75} -13.6527 q^{76} -2.49571 q^{78} +2.56844 q^{79} +2.29844 q^{80} +1.00000 q^{81} +6.00000 q^{82} +7.75737 q^{83} -2.70156 q^{85} -5.16531 q^{86} -0.133124 q^{87} -2.02214 q^{88} -13.3741 q^{89} -0.546295 q^{90} +6.00295 q^{92} +2.33897 q^{93} -0.354822 q^{94} +8.02362 q^{95} +5.29991 q^{96} +1.78581 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 6q^{4} + 4q^{5} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 6q^{4} + 4q^{5} + 4q^{9} - 4q^{11} - 6q^{12} - 8q^{13} - 4q^{15} + 22q^{16} + 2q^{17} - 10q^{19} + 6q^{20} - 2q^{23} + 4q^{25} - 20q^{26} - 4q^{27} - 2q^{29} - 24q^{31} + 4q^{33} + 6q^{36} + 8q^{37} - 16q^{38} + 8q^{39} + 6q^{43} - 6q^{44} + 4q^{45} - 12q^{46} - 4q^{47} - 22q^{48} - 2q^{51} - 12q^{52} + 14q^{53} - 4q^{55} + 10q^{57} - 20q^{58} + 2q^{59} - 6q^{60} - 6q^{61} - 8q^{62} + 6q^{64} - 8q^{65} - 8q^{67} + 44q^{68} + 2q^{69} - 4q^{73} - 36q^{74} - 4q^{75} - 56q^{76} + 20q^{78} + 22q^{80} + 4q^{81} + 24q^{82} - 6q^{83} + 2q^{85} - 36q^{86} + 2q^{87} - 18q^{89} - 44q^{92} + 24q^{93} + 36q^{94} - 10q^{95} + 6q^{97} - 4q^{99} + O(q^{100}) \)

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 \(-0.561869\pi\)
−0.193144 + 0.981170i \(0.561869\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.70156 −0.850781
\(5\) 1.00000 0.447214
\(6\) 0.546295 0.223024
\(7\) 0 0
\(8\) 2.02214 0.714936
\(9\) 1.00000 0.333333
\(10\) −0.546295 −0.172754
\(11\) −1.00000 −0.301511
\(12\) 1.70156 0.491199
\(13\) −4.56844 −1.26706 −0.633528 0.773719i \(-0.718394\pi\)
−0.633528 + 0.773719i \(0.718394\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 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
\(19\) 8.02362 1.84074 0.920372 0.391044i \(-0.127886\pi\)
0.920372 + 0.391044i \(0.127886\pi\)
\(20\) −1.70156 −0.380481
\(21\) 0 0
\(22\) 0.546295 0.116470
\(23\) −3.52790 −0.735619 −0.367809 0.929901i \(-0.619892\pi\)
−0.367809 + 0.929901i \(0.619892\pi\)
\(24\) −2.02214 −0.412768
\(25\) 1.00000 0.200000
\(26\) 2.49571 0.489450
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 0.133124 0.0247205 0.0123602 0.999924i \(-0.496066\pi\)
0.0123602 + 0.999924i \(0.496066\pi\)
\(30\) 0.546295 0.0997393
\(31\) −2.33897 −0.420092 −0.210046 0.977692i \(-0.567361\pi\)
−0.210046 + 0.977692i \(0.567361\pi\)
\(32\) −5.29991 −0.936901
\(33\) 1.00000 0.174078
\(34\) 1.47585 0.253106
\(35\) 0 0
\(36\) −1.70156 −0.283594
\(37\) 6.75362 1.11029 0.555144 0.831754i \(-0.312663\pi\)
0.555144 + 0.831754i \(0.312663\pi\)
\(38\) −4.38326 −0.711059
\(39\) 4.56844 0.731536
\(40\) 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 0
\(43\) 9.45518 1.44190 0.720951 0.692986i \(-0.243705\pi\)
0.720951 + 0.692986i \(0.243705\pi\)
\(44\) 1.70156 0.256520
\(45\) 1.00000 0.149071
\(46\) 1.92728 0.284161
\(47\) 0.649507 0.0947403 0.0473702 0.998877i \(-0.484916\pi\)
0.0473702 + 0.998877i \(0.484916\pi\)
\(48\) −2.29844 −0.331751
\(49\) 0 0
\(50\) −0.546295 −0.0772577
\(51\) 2.70156 0.378294
\(52\) 7.77348 1.07799
\(53\) 11.8384 1.62613 0.813067 0.582170i \(-0.197796\pi\)
0.813067 + 0.582170i \(0.197796\pi\)
\(54\) 0.546295 0.0743413
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) −8.02362 −1.06275
\(58\) −0.0727248 −0.00954923
\(59\) 0.959466 0.124912 0.0624559 0.998048i \(-0.480107\pi\)
0.0624559 + 0.998048i \(0.480107\pi\)
\(60\) 1.70156 0.219671
\(61\) −13.1162 −1.67936 −0.839679 0.543083i \(-0.817257\pi\)
−0.839679 + 0.543083i \(0.817257\pi\)
\(62\) 1.27777 0.162277
\(63\) 0 0
\(64\) −1.70156 −0.212695
\(65\) −4.56844 −0.566645
\(66\) −0.546295 −0.0672442
\(67\) 6.41464 0.783674 0.391837 0.920035i \(-0.371840\pi\)
0.391837 + 0.920035i \(0.371840\pi\)
\(68\) 4.59688 0.557453
\(69\) 3.52790 0.424710
\(70\) 0 0
\(71\) −2.56844 −0.304818 −0.152409 0.988318i \(-0.548703\pi\)
−0.152409 + 0.988318i \(0.548703\pi\)
\(72\) 2.02214 0.238312
\(73\) −12.5400 −1.46770 −0.733848 0.679314i \(-0.762278\pi\)
−0.733848 + 0.679314i \(0.762278\pi\)
\(74\) −3.68947 −0.428892
\(75\) −1.00000 −0.115470
\(76\) −13.6527 −1.56607
\(77\) 0 0
\(78\) −2.49571 −0.282584
\(79\) 2.56844 0.288972 0.144486 0.989507i \(-0.453847\pi\)
0.144486 + 0.989507i \(0.453847\pi\)
\(80\) 2.29844 0.256973
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 7.75737 0.851482 0.425741 0.904845i \(-0.360013\pi\)
0.425741 + 0.904845i \(0.360013\pi\)
\(84\) 0 0
\(85\) −2.70156 −0.293026
\(86\) −5.16531 −0.556990
\(87\) −0.133124 −0.0142724
\(88\) −2.02214 −0.215561
\(89\) −13.3741 −1.41765 −0.708826 0.705383i \(-0.750775\pi\)
−0.708826 + 0.705383i \(0.750775\pi\)
\(90\) −0.546295 −0.0575845
\(91\) 0 0
\(92\) 6.00295 0.625851
\(93\) 2.33897 0.242540
\(94\) −0.354822 −0.0365971
\(95\) 8.02362 0.823206
\(96\) 5.29991 0.540920
\(97\) 1.78581 0.181321 0.0906606 0.995882i \(-0.471102\pi\)
0.0906606 + 0.995882i \(0.471102\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) −1.70156 −0.170156
\(101\) 11.6326 1.15749 0.578743 0.815510i \(-0.303543\pi\)
0.578743 + 0.815510i \(0.303543\pi\)
\(102\) −1.47585 −0.146131
\(103\) 9.18893 0.905412 0.452706 0.891660i \(-0.350459\pi\)
0.452706 + 0.891660i \(0.350459\pi\)
\(104\) −9.23804 −0.905864
\(105\) 0 0
\(106\) −6.46728 −0.628157
\(107\) −13.0199 −1.25868 −0.629339 0.777131i \(-0.716674\pi\)
−0.629339 + 0.777131i \(0.716674\pi\)
\(108\) 1.70156 0.163733
\(109\) 0.109506 0.0104888 0.00524439 0.999986i \(-0.498331\pi\)
0.00524439 + 0.999986i \(0.498331\pi\)
\(110\) 0.546295 0.0520872
\(111\) −6.75362 −0.641025
\(112\) 0 0
\(113\) −3.83844 −0.361090 −0.180545 0.983567i \(-0.557786\pi\)
−0.180545 + 0.983567i \(0.557786\pi\)
\(114\) 4.38326 0.410530
\(115\) −3.52790 −0.328979
\(116\) −0.226518 −0.0210317
\(117\) −4.56844 −0.422352
\(118\) −0.524151 −0.0482520
\(119\) 0 0
\(120\) −2.02214 −0.184596
\(121\) 1.00000 0.0909091
\(122\) 7.16531 0.648717
\(123\) 10.9831 0.990311
\(124\) 3.97991 0.357406
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −5.45518 −0.484069 −0.242034 0.970268i \(-0.577815\pi\)
−0.242034 + 0.970268i \(0.577815\pi\)
\(128\) 11.5294 1.01906
\(129\) −9.45518 −0.832482
\(130\) 2.49571 0.218889
\(131\) 21.1084 1.84425 0.922126 0.386889i \(-0.126450\pi\)
0.922126 + 0.386889i \(0.126450\pi\)
\(132\) −1.70156 −0.148102
\(133\) 0 0
\(134\) −3.50429 −0.302724
\(135\) −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 \(0.546439\pi\)
\(138\) −1.92728 −0.164061
\(139\) −21.2410 −1.80164 −0.900818 0.434196i \(-0.857033\pi\)
−0.900818 + 0.434196i \(0.857033\pi\)
\(140\) 0 0
\(141\) −0.649507 −0.0546984
\(142\) 1.40312 0.117748
\(143\) 4.56844 0.382032
\(144\) 2.29844 0.191536
\(145\) 0.133124 0.0110553
\(146\) 6.85054 0.566954
\(147\) 0 0
\(148\) −11.4917 −0.944612
\(149\) −11.5072 −0.942709 −0.471355 0.881944i \(-0.656235\pi\)
−0.471355 + 0.881944i \(0.656235\pi\)
\(150\) 0.546295 0.0446048
\(151\) 7.58830 0.617527 0.308764 0.951139i \(-0.400085\pi\)
0.308764 + 0.951139i \(0.400085\pi\)
\(152\) 16.2249 1.31601
\(153\) −2.70156 −0.218408
\(154\) 0 0
\(155\) −2.33897 −0.187871
\(156\) −7.77348 −0.622377
\(157\) 12.8583 1.02620 0.513102 0.858328i \(-0.328496\pi\)
0.513102 + 0.858328i \(0.328496\pi\)
\(158\) −1.40312 −0.111627
\(159\) −11.8384 −0.938849
\(160\) −5.29991 −0.418995
\(161\) 0 0
\(162\) −0.546295 −0.0429210
\(163\) 22.6525 1.77428 0.887139 0.461503i \(-0.152690\pi\)
0.887139 + 0.461503i \(0.152690\pi\)
\(164\) 18.6884 1.45932
\(165\) 1.00000 0.0778499
\(166\) −4.23781 −0.328918
\(167\) 3.21795 0.249012 0.124506 0.992219i \(-0.460265\pi\)
0.124506 + 0.992219i \(0.460265\pi\)
\(168\) 0 0
\(169\) 7.87063 0.605433
\(170\) 1.47585 0.113192
\(171\) 8.02362 0.613581
\(172\) −16.0886 −1.22674
\(173\) 18.2094 1.38443 0.692216 0.721690i \(-0.256634\pi\)
0.692216 + 0.721690i \(0.256634\pi\)
\(174\) 0.0727248 0.00551325
\(175\) 0 0
\(176\) −2.29844 −0.173251
\(177\) −0.959466 −0.0721179
\(178\) 7.30621 0.547623
\(179\) −4.75362 −0.355302 −0.177651 0.984094i \(-0.556850\pi\)
−0.177651 + 0.984094i \(0.556850\pi\)
\(180\) −1.70156 −0.126827
\(181\) −0.347316 −0.0258158 −0.0129079 0.999917i \(-0.504109\pi\)
−0.0129079 + 0.999917i \(0.504109\pi\)
\(182\) 0 0
\(183\) 13.1162 0.969578
\(184\) −7.13393 −0.525920
\(185\) 6.75362 0.496536
\(186\) −1.27777 −0.0936905
\(187\) 2.70156 0.197558
\(188\) −1.10518 −0.0806033
\(189\) 0 0
\(190\) −4.38326 −0.317995
\(191\) 11.8379 0.856558 0.428279 0.903647i \(-0.359120\pi\)
0.428279 + 0.903647i \(0.359120\pi\)
\(192\) 1.70156 0.122800
\(193\) −4.14840 −0.298608 −0.149304 0.988791i \(-0.547703\pi\)
−0.149304 + 0.988791i \(0.547703\pi\)
\(194\) −0.975577 −0.0700424
\(195\) 4.56844 0.327153
\(196\) 0 0
\(197\) 19.3663 1.37979 0.689897 0.723907i \(-0.257656\pi\)
0.689897 + 0.723907i \(0.257656\pi\)
\(198\) 0.546295 0.0388235
\(199\) −13.2622 −0.940135 −0.470067 0.882631i \(-0.655770\pi\)
−0.470067 + 0.882631i \(0.655770\pi\)
\(200\) 2.02214 0.142987
\(201\) −6.41464 −0.452454
\(202\) −6.35482 −0.447124
\(203\) 0 0
\(204\) −4.59688 −0.321846
\(205\) −10.9831 −0.767092
\(206\) −5.01986 −0.349751
\(207\) −3.52790 −0.245206
\(208\) −10.5003 −0.728063
\(209\) −8.02362 −0.555005
\(210\) 0 0
\(211\) 2.54000 0.174861 0.0874304 0.996171i \(-0.472134\pi\)
0.0874304 + 0.996171i \(0.472134\pi\)
\(212\) −20.1438 −1.38348
\(213\) 2.56844 0.175986
\(214\) 7.11268 0.486213
\(215\) 9.45518 0.644838
\(216\) −2.02214 −0.137589
\(217\) 0 0
\(218\) −0.0598226 −0.00405170
\(219\) 12.5400 0.847375
\(220\) 1.70156 0.114719
\(221\) 12.3419 0.830207
\(222\) 3.68947 0.247621
\(223\) −20.5110 −1.37352 −0.686759 0.726886i \(-0.740967\pi\)
−0.686759 + 0.726886i \(0.740967\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 2.09692 0.139485
\(227\) −19.6119 −1.30169 −0.650844 0.759211i \(-0.725585\pi\)
−0.650844 + 0.759211i \(0.725585\pi\)
\(228\) 13.6527 0.904171
\(229\) −20.4230 −1.34959 −0.674795 0.738006i \(-0.735768\pi\)
−0.674795 + 0.738006i \(0.735768\pi\)
\(230\) 1.92728 0.127081
\(231\) 0 0
\(232\) 0.269195 0.0176735
\(233\) −17.3304 −1.13535 −0.567676 0.823252i \(-0.692157\pi\)
−0.567676 + 0.823252i \(0.692157\pi\)
\(234\) 2.49571 0.163150
\(235\) 0.649507 0.0423692
\(236\) −1.63259 −0.106273
\(237\) −2.56844 −0.166838
\(238\) 0 0
\(239\) −11.6404 −0.752952 −0.376476 0.926426i \(-0.622864\pi\)
−0.376476 + 0.926426i \(0.622864\pi\)
\(240\) −2.29844 −0.148364
\(241\) 22.5429 1.45212 0.726059 0.687632i \(-0.241350\pi\)
0.726059 + 0.687632i \(0.241350\pi\)
\(242\) −0.546295 −0.0351172
\(243\) −1.00000 −0.0641500
\(244\) 22.3180 1.42877
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) −36.6554 −2.33233
\(248\) −4.72974 −0.300339
\(249\) −7.75737 −0.491603
\(250\) −0.546295 −0.0345507
\(251\) −11.4474 −0.722554 −0.361277 0.932458i \(-0.617659\pi\)
−0.361277 + 0.932458i \(0.617659\pi\)
\(252\) 0 0
\(253\) 3.52790 0.221797
\(254\) 2.98014 0.186990
\(255\) 2.70156 0.169178
\(256\) −2.89531 −0.180957
\(257\) 28.6157 1.78500 0.892498 0.451051i \(-0.148951\pi\)
0.892498 + 0.451051i \(0.148951\pi\)
\(258\) 5.16531 0.321578
\(259\) 0 0
\(260\) 7.77348 0.482091
\(261\) 0.133124 0.00824015
\(262\) −11.5314 −0.712414
\(263\) −8.67794 −0.535105 −0.267552 0.963543i \(-0.586215\pi\)
−0.267552 + 0.963543i \(0.586215\pi\)
\(264\) 2.02214 0.124454
\(265\) 11.8384 0.727230
\(266\) 0 0
\(267\) 13.3741 0.818482
\(268\) −10.9149 −0.666735
\(269\) −29.8911 −1.82249 −0.911245 0.411864i \(-0.864878\pi\)
−0.911245 + 0.411864i \(0.864878\pi\)
\(270\) 0.546295 0.0332464
\(271\) −20.9678 −1.27370 −0.636852 0.770986i \(-0.719764\pi\)
−0.636852 + 0.770986i \(0.719764\pi\)
\(272\) −6.20937 −0.376499
\(273\) 0 0
\(274\) 1.85911 0.112313
\(275\) −1.00000 −0.0603023
\(276\) −6.00295 −0.361335
\(277\) −13.7137 −0.823974 −0.411987 0.911190i \(-0.635165\pi\)
−0.411987 + 0.911190i \(0.635165\pi\)
\(278\) 11.6038 0.695952
\(279\) −2.33897 −0.140031
\(280\) 0 0
\(281\) 4.91036 0.292927 0.146464 0.989216i \(-0.453211\pi\)
0.146464 + 0.989216i \(0.453211\pi\)
\(282\) 0.354822 0.0211294
\(283\) 25.4788 1.51456 0.757279 0.653092i \(-0.226528\pi\)
0.757279 + 0.653092i \(0.226528\pi\)
\(284\) 4.37036 0.259333
\(285\) −8.02362 −0.475278
\(286\) −2.49571 −0.147575
\(287\) 0 0
\(288\) −5.29991 −0.312300
\(289\) −9.70156 −0.570680
\(290\) −0.0727248 −0.00427055
\(291\) −1.78581 −0.104686
\(292\) 21.3376 1.24869
\(293\) 2.16907 0.126718 0.0633591 0.997991i \(-0.479819\pi\)
0.0633591 + 0.997991i \(0.479819\pi\)
\(294\) 0 0
\(295\) 0.959466 0.0558622
\(296\) 13.6568 0.793784
\(297\) 1.00000 0.0580259
\(298\) 6.28634 0.364158
\(299\) 16.1170 0.932071
\(300\) 1.70156 0.0982397
\(301\) 0 0
\(302\) −4.14545 −0.238544
\(303\) −11.6326 −0.668275
\(304\) 18.4418 1.05771
\(305\) −13.1162 −0.751032
\(306\) 1.47585 0.0843687
\(307\) −12.3548 −0.705127 −0.352563 0.935788i \(-0.614690\pi\)
−0.352563 + 0.935788i \(0.614690\pi\)
\(308\) 0 0
\(309\) −9.18893 −0.522740
\(310\) 1.27777 0.0725724
\(311\) −22.6073 −1.28194 −0.640972 0.767564i \(-0.721469\pi\)
−0.640972 + 0.767564i \(0.721469\pi\)
\(312\) 9.23804 0.523001
\(313\) 34.2614 1.93657 0.968285 0.249848i \(-0.0803805\pi\)
0.968285 + 0.249848i \(0.0803805\pi\)
\(314\) −7.02442 −0.396411
\(315\) 0 0
\(316\) −4.37036 −0.245852
\(317\) 8.28929 0.465573 0.232786 0.972528i \(-0.425216\pi\)
0.232786 + 0.972528i \(0.425216\pi\)
\(318\) 6.46728 0.362667
\(319\) −0.133124 −0.00745350
\(320\) −1.70156 −0.0951202
\(321\) 13.0199 0.726698
\(322\) 0 0
\(323\) −21.6763 −1.20610
\(324\) −1.70156 −0.0945312
\(325\) −4.56844 −0.253411
\(326\) −12.3749 −0.685383
\(327\) −0.109506 −0.00605570
\(328\) −22.2094 −1.22631
\(329\) 0 0
\(330\) −0.546295 −0.0300725
\(331\) 21.4267 1.17772 0.588860 0.808235i \(-0.299577\pi\)
0.588860 + 0.808235i \(0.299577\pi\)
\(332\) −13.1996 −0.724425
\(333\) 6.75362 0.370096
\(334\) −1.75795 −0.0961906
\(335\) 6.41464 0.350469
\(336\) 0 0
\(337\) −20.2585 −1.10355 −0.551775 0.833993i \(-0.686049\pi\)
−0.551775 + 0.833993i \(0.686049\pi\)
\(338\) −4.29968 −0.233872
\(339\) 3.83844 0.208475
\(340\) 4.59688 0.249301
\(341\) 2.33897 0.126662
\(342\) −4.38326 −0.237020
\(343\) 0 0
\(344\) 19.1197 1.03087
\(345\) 3.52790 0.189936
\(346\) −9.94768 −0.534791
\(347\) −23.9431 −1.28533 −0.642667 0.766146i \(-0.722172\pi\)
−0.642667 + 0.766146i \(0.722172\pi\)
\(348\) 0.226518 0.0121427
\(349\) 25.5751 1.36901 0.684503 0.729010i \(-0.260019\pi\)
0.684503 + 0.729010i \(0.260019\pi\)
\(350\) 0 0
\(351\) 4.56844 0.243845
\(352\) 5.29991 0.282486
\(353\) −0.330628 −0.0175976 −0.00879878 0.999961i \(-0.502801\pi\)
−0.00879878 + 0.999961i \(0.502801\pi\)
\(354\) 0.524151 0.0278583
\(355\) −2.56844 −0.136319
\(356\) 22.7569 1.20611
\(357\) 0 0
\(358\) 2.59688 0.137249
\(359\) −23.6248 −1.24687 −0.623435 0.781875i \(-0.714263\pi\)
−0.623435 + 0.781875i \(0.714263\pi\)
\(360\) 2.02214 0.106576
\(361\) 45.3784 2.38834
\(362\) 0.189737 0.00997235
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −12.5400 −0.656374
\(366\) −7.16531 −0.374537
\(367\) 22.3730 1.16786 0.583932 0.811803i \(-0.301514\pi\)
0.583932 + 0.811803i \(0.301514\pi\)
\(368\) −8.10867 −0.422694
\(369\) −10.9831 −0.571756
\(370\) −3.68947 −0.191806
\(371\) 0 0
\(372\) −3.97991 −0.206349
\(373\) −11.9922 −0.620934 −0.310467 0.950584i \(-0.600485\pi\)
−0.310467 + 0.950584i \(0.600485\pi\)
\(374\) −1.47585 −0.0763143
\(375\) −1.00000 −0.0516398
\(376\) 1.31340 0.0677333
\(377\) −0.608168 −0.0313222
\(378\) 0 0
\(379\) 18.2088 0.935323 0.467662 0.883908i \(-0.345097\pi\)
0.467662 + 0.883908i \(0.345097\pi\)
\(380\) −13.6527 −0.700368
\(381\) 5.45518 0.279477
\(382\) −6.46696 −0.330879
\(383\) −22.3494 −1.14200 −0.571001 0.820949i \(-0.693445\pi\)
−0.571001 + 0.820949i \(0.693445\pi\)
\(384\) −11.5294 −0.588356
\(385\) 0 0
\(386\) 2.26625 0.115349
\(387\) 9.45518 0.480634
\(388\) −3.03866 −0.154265
\(389\) 25.4863 1.29221 0.646103 0.763250i \(-0.276397\pi\)
0.646103 + 0.763250i \(0.276397\pi\)
\(390\) −2.49571 −0.126375
\(391\) 9.53085 0.481996
\(392\) 0 0
\(393\) −21.1084 −1.06478
\(394\) −10.5797 −0.532999
\(395\) 2.56844 0.129232
\(396\) 1.70156 0.0855067
\(397\) −10.3548 −0.519694 −0.259847 0.965650i \(-0.583672\pi\)
−0.259847 + 0.965650i \(0.583672\pi\)
\(398\) 7.24509 0.363163
\(399\) 0 0
\(400\) 2.29844 0.114922
\(401\) 0.0939709 0.00469268 0.00234634 0.999997i \(-0.499253\pi\)
0.00234634 + 0.999997i \(0.499253\pi\)
\(402\) 3.50429 0.174778
\(403\) 10.6855 0.532280
\(404\) −19.7936 −0.984767
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −6.75362 −0.334764
\(408\) 5.46295 0.270456
\(409\) −2.90741 −0.143762 −0.0718811 0.997413i \(-0.522900\pi\)
−0.0718811 + 0.997413i \(0.522900\pi\)
\(410\) 6.00000 0.296319
\(411\) 3.40312 0.167864
\(412\) −15.6355 −0.770308
\(413\) 0 0
\(414\) 1.92728 0.0947204
\(415\) 7.75737 0.380794
\(416\) 24.2123 1.18711
\(417\) 21.2410 1.04018
\(418\) 4.38326 0.214392
\(419\) −16.2660 −0.794645 −0.397323 0.917679i \(-0.630061\pi\)
−0.397323 + 0.917679i \(0.630061\pi\)
\(420\) 0 0
\(421\) −16.1047 −0.784894 −0.392447 0.919775i \(-0.628371\pi\)
−0.392447 + 0.919775i \(0.628371\pi\)
\(422\) −1.38759 −0.0675468
\(423\) 0.649507 0.0315801
\(424\) 23.9390 1.16258
\(425\) −2.70156 −0.131045
\(426\) −1.40312 −0.0679816
\(427\) 0 0
\(428\) 22.1541 1.07086
\(429\) −4.56844 −0.220566
\(430\) −5.16531 −0.249094
\(431\) −19.3221 −0.930711 −0.465355 0.885124i \(-0.654073\pi\)
−0.465355 + 0.885124i \(0.654073\pi\)
\(432\) −2.29844 −0.110584
\(433\) −37.0462 −1.78033 −0.890163 0.455643i \(-0.849409\pi\)
−0.890163 + 0.455643i \(0.849409\pi\)
\(434\) 0 0
\(435\) −0.133124 −0.00638279
\(436\) −0.186331 −0.00892366
\(437\) −28.3066 −1.35409
\(438\) −6.85054 −0.327331
\(439\) −14.4750 −0.690856 −0.345428 0.938445i \(-0.612266\pi\)
−0.345428 + 0.938445i \(0.612266\pi\)
\(440\) −2.02214 −0.0964019
\(441\) 0 0
\(442\) −6.74233 −0.320700
\(443\) −26.4146 −1.25500 −0.627499 0.778618i \(-0.715921\pi\)
−0.627499 + 0.778618i \(0.715921\pi\)
\(444\) 11.4917 0.545372
\(445\) −13.3741 −0.633994
\(446\) 11.2050 0.530574
\(447\) 11.5072 0.544274
\(448\) 0 0
\(449\) −13.7526 −0.649023 −0.324511 0.945882i \(-0.605200\pi\)
−0.324511 + 0.945882i \(0.605200\pi\)
\(450\) −0.546295 −0.0257526
\(451\) 10.9831 0.517173
\(452\) 6.53134 0.307208
\(453\) −7.58830 −0.356530
\(454\) 10.7139 0.502828
\(455\) 0 0
\(456\) −16.2249 −0.759801
\(457\) −9.98473 −0.467066 −0.233533 0.972349i \(-0.575029\pi\)
−0.233533 + 0.972349i \(0.575029\pi\)
\(458\) 11.1570 0.521331
\(459\) 2.70156 0.126098
\(460\) 6.00295 0.279889
\(461\) −3.31911 −0.154586 −0.0772931 0.997008i \(-0.524628\pi\)
−0.0772931 + 0.997008i \(0.524628\pi\)
\(462\) 0 0
\(463\) −20.1726 −0.937500 −0.468750 0.883331i \(-0.655295\pi\)
−0.468750 + 0.883331i \(0.655295\pi\)
\(464\) 0.305977 0.0142046
\(465\) 2.33897 0.108467
\(466\) 9.46751 0.438574
\(467\) −3.69241 −0.170864 −0.0854322 0.996344i \(-0.527227\pi\)
−0.0854322 + 0.996344i \(0.527227\pi\)
\(468\) 7.77348 0.359329
\(469\) 0 0
\(470\) −0.354822 −0.0163667
\(471\) −12.8583 −0.592479
\(472\) 1.94018 0.0893039
\(473\) −9.45518 −0.434750
\(474\) 1.40312 0.0644476
\(475\) 8.02362 0.368149
\(476\) 0 0
\(477\) 11.8384 0.542045
\(478\) 6.35907 0.290857
\(479\) −41.0516 −1.87569 −0.937847 0.347049i \(-0.887184\pi\)
−0.937847 + 0.347049i \(0.887184\pi\)
\(480\) 5.29991 0.241907
\(481\) −30.8535 −1.40680
\(482\) −12.3151 −0.560937
\(483\) 0 0
\(484\) −1.70156 −0.0773437
\(485\) 1.78581 0.0810893
\(486\) 0.546295 0.0247804
\(487\) −8.08402 −0.366322 −0.183161 0.983083i \(-0.558633\pi\)
−0.183161 + 0.983083i \(0.558633\pi\)
\(488\) −26.5229 −1.20063
\(489\) −22.6525 −1.02438
\(490\) 0 0
\(491\) −3.47348 −0.156756 −0.0783779 0.996924i \(-0.524974\pi\)
−0.0783779 + 0.996924i \(0.524974\pi\)
\(492\) −18.6884 −0.842538
\(493\) −0.359642 −0.0161975
\(494\) 20.0247 0.900952
\(495\) −1.00000 −0.0449467
\(496\) −5.37598 −0.241389
\(497\) 0 0
\(498\) 4.23781 0.189901
\(499\) −29.0719 −1.30144 −0.650719 0.759319i \(-0.725532\pi\)
−0.650719 + 0.759319i \(0.725532\pi\)
\(500\) −1.70156 −0.0760962
\(501\) −3.21795 −0.143767
\(502\) 6.25366 0.279115
\(503\) −30.6109 −1.36487 −0.682435 0.730946i \(-0.739079\pi\)
−0.682435 + 0.730946i \(0.739079\pi\)
\(504\) 0 0
\(505\) 11.6326 0.517643
\(506\) −1.92728 −0.0856778
\(507\) −7.87063 −0.349547
\(508\) 9.28233 0.411837
\(509\) 0.696166 0.0308570 0.0154285 0.999881i \(-0.495089\pi\)
0.0154285 + 0.999881i \(0.495089\pi\)
\(510\) −1.47585 −0.0653517
\(511\) 0 0
\(512\) −21.4771 −0.949161
\(513\) −8.02362 −0.354251
\(514\) −15.6326 −0.689524
\(515\) 9.18893 0.404913
\(516\) 16.0886 0.708260
\(517\) −0.649507 −0.0285653
\(518\) 0 0
\(519\) −18.2094 −0.799303
\(520\) −9.23804 −0.405115
\(521\) −14.2142 −0.622735 −0.311368 0.950290i \(-0.600787\pi\)
−0.311368 + 0.950290i \(0.600787\pi\)
\(522\) −0.0727248 −0.00318308
\(523\) −5.04830 −0.220747 −0.110373 0.993890i \(-0.535205\pi\)
−0.110373 + 0.993890i \(0.535205\pi\)
\(524\) −35.9173 −1.56906
\(525\) 0 0
\(526\) 4.74071 0.206705
\(527\) 6.31888 0.275255
\(528\) 2.29844 0.100027
\(529\) −10.5539 −0.458865
\(530\) −6.46728 −0.280921
\(531\) 0.959466 0.0416373
\(532\) 0 0
\(533\) 50.1755 2.17334
\(534\) −7.30621 −0.316170
\(535\) −13.0199 −0.562898
\(536\) 12.9713 0.560276
\(537\) 4.75362 0.205134
\(538\) 16.3293 0.704008
\(539\) 0 0
\(540\) 1.70156 0.0732236
\(541\) 36.1154 1.55272 0.776361 0.630288i \(-0.217063\pi\)
0.776361 + 0.630288i \(0.217063\pi\)
\(542\) 11.4546 0.492017
\(543\) 0.347316 0.0149048
\(544\) 14.3180 0.613881
\(545\) 0.109506 0.00469073
\(546\) 0 0
\(547\) 11.4213 0.488341 0.244171 0.969732i \(-0.421484\pi\)
0.244171 + 0.969732i \(0.421484\pi\)
\(548\) 5.79063 0.247363
\(549\) −13.1162 −0.559786
\(550\) 0.546295 0.0232941
\(551\) 1.06813 0.0455040
\(552\) 7.13393 0.303640
\(553\) 0 0
\(554\) 7.49170 0.318292
\(555\) −6.75362 −0.286675
\(556\) 36.1429 1.53280
\(557\) 3.52733 0.149458 0.0747288 0.997204i \(-0.476191\pi\)
0.0747288 + 0.997204i \(0.476191\pi\)
\(558\) 1.27777 0.0540922
\(559\) −43.1954 −1.82697
\(560\) 0 0
\(561\) −2.70156 −0.114060
\(562\) −2.68250 −0.113155
\(563\) 10.5400 0.444208 0.222104 0.975023i \(-0.428708\pi\)
0.222104 + 0.975023i \(0.428708\pi\)
\(564\) 1.10518 0.0465363
\(565\) −3.83844 −0.161484
\(566\) −13.9189 −0.585056
\(567\) 0 0
\(568\) −5.19375 −0.217925
\(569\) −15.6173 −0.654712 −0.327356 0.944901i \(-0.606158\pi\)
−0.327356 + 0.944901i \(0.606158\pi\)
\(570\) 4.38326 0.183595
\(571\) −9.94737 −0.416284 −0.208142 0.978099i \(-0.566742\pi\)
−0.208142 + 0.978099i \(0.566742\pi\)
\(572\) −7.77348 −0.325026
\(573\) −11.8379 −0.494534
\(574\) 0 0
\(575\) −3.52790 −0.147124
\(576\) −1.70156 −0.0708984
\(577\) −29.1030 −1.21158 −0.605788 0.795626i \(-0.707142\pi\)
−0.605788 + 0.795626i \(0.707142\pi\)
\(578\) 5.29991 0.220447
\(579\) 4.14840 0.172402
\(580\) −0.226518 −0.00940566
\(581\) 0 0
\(582\) 0.975577 0.0404390
\(583\) −11.8384 −0.490298
\(584\) −25.3577 −1.04931
\(585\) −4.56844 −0.188882
\(586\) −1.18495 −0.0489498
\(587\) −12.7536 −0.526398 −0.263199 0.964742i \(-0.584778\pi\)
−0.263199 + 0.964742i \(0.584778\pi\)
\(588\) 0 0
\(589\) −18.7670 −0.773282
\(590\) −0.524151 −0.0215790
\(591\) −19.3663 −0.796625
\(592\) 15.5228 0.637982
\(593\) 3.91893 0.160931 0.0804656 0.996757i \(-0.474359\pi\)
0.0804656 + 0.996757i \(0.474359\pi\)
\(594\) −0.546295 −0.0224147
\(595\) 0 0
\(596\) 19.5803 0.802039
\(597\) 13.2622 0.542787
\(598\) −8.80464 −0.360048
\(599\) −11.9420 −0.487936 −0.243968 0.969783i \(-0.578449\pi\)
−0.243968 + 0.969783i \(0.578449\pi\)
\(600\) −2.02214 −0.0825537
\(601\) −16.6878 −0.680710 −0.340355 0.940297i \(-0.610547\pi\)
−0.340355 + 0.940297i \(0.610547\pi\)
\(602\) 0 0
\(603\) 6.41464 0.261225
\(604\) −12.9120 −0.525381
\(605\) 1.00000 0.0406558
\(606\) 6.35482 0.258147
\(607\) −4.72518 −0.191789 −0.0958946 0.995391i \(-0.530571\pi\)
−0.0958946 + 0.995391i \(0.530571\pi\)
\(608\) −42.5245 −1.72459
\(609\) 0 0
\(610\) 7.16531 0.290115
\(611\) −2.96723 −0.120041
\(612\) 4.59688 0.185818
\(613\) −6.84938 −0.276644 −0.138322 0.990387i \(-0.544171\pi\)
−0.138322 + 0.990387i \(0.544171\pi\)
\(614\) 6.74937 0.272383
\(615\) 10.9831 0.442881
\(616\) 0 0
\(617\) 21.0725 0.848347 0.424173 0.905581i \(-0.360565\pi\)
0.424173 + 0.905581i \(0.360565\pi\)
\(618\) 5.01986 0.201929
\(619\) 29.2209 1.17449 0.587243 0.809410i \(-0.300213\pi\)
0.587243 + 0.809410i \(0.300213\pi\)
\(620\) 3.97991 0.159837
\(621\) 3.52790 0.141570
\(622\) 12.3503 0.495200
\(623\) 0 0
\(624\) 10.5003 0.420347
\(625\) 1.00000 0.0400000
\(626\) −18.7168 −0.748075
\(627\) 8.02362 0.320432
\(628\) −21.8792 −0.873075
\(629\) −18.2453 −0.727488
\(630\) 0 0
\(631\) −23.8201 −0.948265 −0.474132 0.880454i \(-0.657238\pi\)
−0.474132 + 0.880454i \(0.657238\pi\)
\(632\) 5.19375 0.206596
\(633\) −2.54000 −0.100956
\(634\) −4.52839 −0.179846
\(635\) −5.45518 −0.216482
\(636\) 20.1438 0.798755
\(637\) 0 0
\(638\) 0.0727248 0.00287920
\(639\) −2.56844 −0.101606
\(640\) 11.5294 0.455739
\(641\) −30.9179 −1.22118 −0.610591 0.791946i \(-0.709068\pi\)
−0.610591 + 0.791946i \(0.709068\pi\)
\(642\) −7.11268 −0.280715
\(643\) −12.4224 −0.489892 −0.244946 0.969537i \(-0.578770\pi\)
−0.244946 + 0.969537i \(0.578770\pi\)
\(644\) 0 0
\(645\) −9.45518 −0.372297
\(646\) 11.8416 0.465903
\(647\) −31.6924 −1.24596 −0.622979 0.782239i \(-0.714078\pi\)
−0.622979 + 0.782239i \(0.714078\pi\)
\(648\) 2.02214 0.0794373
\(649\) −0.959466 −0.0376623
\(650\) 2.49571 0.0978899
\(651\) 0 0
\(652\) −38.5446 −1.50952
\(653\) 25.4037 0.994124 0.497062 0.867715i \(-0.334412\pi\)
0.497062 + 0.867715i \(0.334412\pi\)
\(654\) 0.0598226 0.00233925
\(655\) 21.1084 0.824775
\(656\) −25.2439 −0.985610
\(657\) −12.5400 −0.489232
\(658\) 0 0
\(659\) −17.1004 −0.666135 −0.333068 0.942903i \(-0.608084\pi\)
−0.333068 + 0.942903i \(0.608084\pi\)
\(660\) −1.70156 −0.0662332
\(661\) −17.8675 −0.694963 −0.347482 0.937687i \(-0.612963\pi\)
−0.347482 + 0.937687i \(0.612963\pi\)
\(662\) −11.7053 −0.454940
\(663\) −12.3419 −0.479320
\(664\) 15.6865 0.608755
\(665\) 0 0
\(666\) −3.68947 −0.142964
\(667\) −0.469648 −0.0181848
\(668\) −5.47553 −0.211855
\(669\) 20.5110 0.793001
\(670\) −3.50429 −0.135382
\(671\) 13.1162 0.506346
\(672\) 0 0
\(673\) −30.7973 −1.18715 −0.593575 0.804779i \(-0.702284\pi\)
−0.593575 + 0.804779i \(0.702284\pi\)
\(674\) 11.0671 0.426289
\(675\) −1.00000 −0.0384900
\(676\) −13.3924 −0.515091
\(677\) −1.54800 −0.0594944 −0.0297472 0.999557i \(-0.509470\pi\)
−0.0297472 + 0.999557i \(0.509470\pi\)
\(678\) −2.09692 −0.0805317
\(679\) 0 0
\(680\) −5.46295 −0.209494
\(681\) 19.6119 0.751530
\(682\) −1.27777 −0.0489283
\(683\) −28.2767 −1.08198 −0.540989 0.841030i \(-0.681950\pi\)
−0.540989 + 0.841030i \(0.681950\pi\)
\(684\) −13.6527 −0.522023
\(685\) −3.40312 −0.130027
\(686\) 0 0
\(687\) 20.4230 0.779186
\(688\) 21.7321 0.828530
\(689\) −54.0832 −2.06041
\(690\) −1.92728 −0.0733701
\(691\) 11.4061 0.433907 0.216954 0.976182i \(-0.430388\pi\)
0.216954 + 0.976182i \(0.430388\pi\)
\(692\) −30.9844 −1.17785
\(693\) 0 0
\(694\) 13.0800 0.496510
\(695\) −21.2410 −0.805717
\(696\) −0.269195 −0.0102038
\(697\) 29.6715 1.12389
\(698\) −13.9716 −0.528831
\(699\) 17.3304 0.655496
\(700\) 0 0
\(701\) −4.13312 −0.156106 −0.0780530 0.996949i \(-0.524870\pi\)
−0.0780530 + 0.996949i \(0.524870\pi\)
\(702\) −2.49571 −0.0941946
\(703\) 54.1884 2.04376
\(704\) 1.70156 0.0641300
\(705\) −0.649507 −0.0244619
\(706\) 0.180620 0.00679774
\(707\) 0 0
\(708\) 1.63259 0.0613565
\(709\) 38.5867 1.44915 0.724576 0.689195i \(-0.242036\pi\)
0.724576 + 0.689195i \(0.242036\pi\)
\(710\) 1.40312 0.0526583
\(711\) 2.56844 0.0963240
\(712\) −27.0444 −1.01353
\(713\) 8.25167 0.309027
\(714\) 0 0
\(715\) 4.56844 0.170850
\(716\) 8.08857 0.302284
\(717\) 11.6404 0.434717
\(718\) 12.9061 0.481652
\(719\) 13.4920 0.503165 0.251583 0.967836i \(-0.419049\pi\)
0.251583 + 0.967836i \(0.419049\pi\)
\(720\) 2.29844 0.0856577
\(721\) 0 0
\(722\) −24.7900 −0.922588
\(723\) −22.5429 −0.838381
\(724\) 0.590980 0.0219636
\(725\) 0.133124 0.00494409
\(726\) 0.546295 0.0202749
\(727\) 15.0923 0.559743 0.279872 0.960037i \(-0.409708\pi\)
0.279872 + 0.960037i \(0.409708\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 6.85054 0.253550
\(731\) −25.5438 −0.944770
\(732\) −22.3180 −0.824899
\(733\) −5.07134 −0.187314 −0.0936572 0.995605i \(-0.529856\pi\)
−0.0936572 + 0.995605i \(0.529856\pi\)
\(734\) −12.2223 −0.451132
\(735\) 0 0
\(736\) 18.6976 0.689202
\(737\) −6.41464 −0.236286
\(738\) 6.00000 0.220863
\(739\) −22.7021 −0.835112 −0.417556 0.908651i \(-0.637113\pi\)
−0.417556 + 0.908651i \(0.637113\pi\)
\(740\) −11.4917 −0.422443
\(741\) 36.6554 1.34657
\(742\) 0 0
\(743\) 29.2727 1.07391 0.536955 0.843611i \(-0.319574\pi\)
0.536955 + 0.843611i \(0.319574\pi\)
\(744\) 4.72974 0.173401
\(745\) −11.5072 −0.421592
\(746\) 6.55129 0.239860
\(747\) 7.75737 0.283827
\(748\) −4.59688 −0.168078
\(749\) 0 0
\(750\) 0.546295 0.0199479
\(751\) 7.65211 0.279229 0.139615 0.990206i \(-0.455414\pi\)
0.139615 + 0.990206i \(0.455414\pi\)
\(752\) 1.49285 0.0544387
\(753\) 11.4474 0.417167
\(754\) 0.332239 0.0120994
\(755\) 7.58830 0.276167
\(756\) 0 0
\(757\) −20.9576 −0.761717 −0.380858 0.924633i \(-0.624371\pi\)
−0.380858 + 0.924633i \(0.624371\pi\)
\(758\) −9.94737 −0.361305
\(759\) −3.52790 −0.128055
\(760\) 16.2249 0.588539
\(761\) 8.16509 0.295984 0.147992 0.988989i \(-0.452719\pi\)
0.147992 + 0.988989i \(0.452719\pi\)
\(762\) −2.98014 −0.107959
\(763\) 0 0
\(764\) −20.1429 −0.728743
\(765\) −2.70156 −0.0976752
\(766\) 12.2094 0.441143
\(767\) −4.38326 −0.158270
\(768\) 2.89531 0.104476
\(769\) 46.3583 1.67172 0.835862 0.548939i \(-0.184968\pi\)
0.835862 + 0.548939i \(0.184968\pi\)
\(770\) 0 0
\(771\) −28.6157 −1.03057
\(772\) 7.05876 0.254050
\(773\) 24.2169 0.871021 0.435510 0.900184i \(-0.356568\pi\)
0.435510 + 0.900184i \(0.356568\pi\)
\(774\) −5.16531 −0.185663
\(775\) −2.33897 −0.0840184
\(776\) 3.61116 0.129633
\(777\) 0 0
\(778\) −13.9230 −0.499165
\(779\) −88.1241 −3.15737
\(780\) −7.77348 −0.278335
\(781\) 2.56844 0.0919060
\(782\) −5.20665 −0.186190
\(783\) −0.133124 −0.00475745
\(784\) 0 0
\(785\) 12.8583 0.458933
\(786\) 11.5314 0.411312
\(787\) −47.1916 −1.68220 −0.841100 0.540880i \(-0.818091\pi\)
−0.841100 + 0.540880i \(0.818091\pi\)
\(788\) −32.9530 −1.17390
\(789\) 8.67794 0.308943
\(790\) −1.40312 −0.0499209
\(791\) 0 0
\(792\) −2.02214 −0.0718537
\(793\) 59.9206 2.12784
\(794\) 5.65678 0.200752
\(795\) −11.8384 −0.419866
\(796\) 22.5665 0.799849
\(797\) 6.97474 0.247058 0.123529 0.992341i \(-0.460579\pi\)
0.123529 + 0.992341i \(0.460579\pi\)
\(798\) 0 0
\(799\) −1.75468 −0.0620763
\(800\) −5.29991 −0.187380
\(801\) −13.3741 −0.472551
\(802\) −0.0513358 −0.00181273
\(803\) 12.5400 0.442527
\(804\) 10.9149 0.384939
\(805\) 0 0
\(806\) −5.83740 −0.205614
\(807\) 29.8911 1.05222
\(808\) 23.5228 0.827528
\(809\) −39.1428 −1.37619 −0.688093 0.725622i \(-0.741552\pi\)
−0.688093 + 0.725622i \(0.741552\pi\)
\(810\) −0.546295 −0.0191948
\(811\) 22.1116 0.776444 0.388222 0.921566i \(-0.373089\pi\)
0.388222 + 0.921566i \(0.373089\pi\)
\(812\) 0 0
\(813\) 20.9678 0.735373
\(814\) 3.68947 0.129316
\(815\) 22.6525 0.793481
\(816\) 6.20937 0.217372
\(817\) 75.8647 2.65417
\(818\) 1.58830 0.0555337
\(819\) 0 0
\(820\) 18.6884 0.652627
\(821\) −49.1868 −1.71663 −0.858316 0.513122i \(-0.828489\pi\)
−0.858316 + 0.513122i \(0.828489\pi\)
\(822\) −1.85911 −0.0648439
\(823\) −4.04853 −0.141123 −0.0705615 0.997507i \(-0.522479\pi\)
−0.0705615 + 0.997507i \(0.522479\pi\)
\(824\) 18.5813 0.647312
\(825\) 1.00000 0.0348155
\(826\) 0 0
\(827\) 20.5217 0.713610 0.356805 0.934179i \(-0.383866\pi\)
0.356805 + 0.934179i \(0.383866\pi\)
\(828\) 6.00295 0.208617
\(829\) −22.7665 −0.790714 −0.395357 0.918528i \(-0.629379\pi\)
−0.395357 + 0.918528i \(0.629379\pi\)
\(830\) −4.23781 −0.147097
\(831\) 13.7137 0.475722
\(832\) 7.77348 0.269497
\(833\) 0 0
\(834\) −11.6038 −0.401808
\(835\) 3.21795 0.111362
\(836\) 13.6527 0.472188
\(837\) 2.33897 0.0808467
\(838\) 8.88602 0.306963
\(839\) −9.06473 −0.312949 −0.156475 0.987682i \(-0.550013\pi\)
−0.156475 + 0.987682i \(0.550013\pi\)
\(840\) 0 0
\(841\) −28.9823 −0.999389
\(842\) 8.79790 0.303196
\(843\) −4.91036 −0.169122
\(844\) −4.32197 −0.148768
\(845\) 7.87063 0.270758
\(846\) −0.354822 −0.0121990
\(847\) 0 0
\(848\) 27.2099 0.934392
\(849\) −25.4788 −0.874430
\(850\) 1.47585 0.0506212
\(851\) −23.8261 −0.816749
\(852\) −4.37036 −0.149726
\(853\) 13.0854 0.448035 0.224018 0.974585i \(-0.428083\pi\)
0.224018 + 0.974585i \(0.428083\pi\)
\(854\) 0 0
\(855\) 8.02362 0.274402
\(856\) −26.3280 −0.899874
\(857\) −38.4589 −1.31373 −0.656866 0.754007i \(-0.728118\pi\)
−0.656866 + 0.754007i \(0.728118\pi\)
\(858\) 2.49571 0.0852023
\(859\) 0.435576 0.0148617 0.00743083 0.999972i \(-0.497635\pi\)
0.00743083 + 0.999972i \(0.497635\pi\)
\(860\) −16.0886 −0.548616
\(861\) 0 0
\(862\) 10.5555 0.359523
\(863\) 38.7362 1.31860 0.659298 0.751882i \(-0.270854\pi\)
0.659298 + 0.751882i \(0.270854\pi\)
\(864\) 5.29991 0.180307
\(865\) 18.2094 0.619137
\(866\) 20.2381 0.687719
\(867\) 9.70156 0.329482
\(868\) 0 0
\(869\) −2.56844 −0.0871283
\(870\) 0.0727248 0.00246560
\(871\) −29.3049 −0.992959
\(872\) 0.221437 0.00749881
\(873\) 1.78581 0.0604404
\(874\) 15.4637 0.523068
\(875\) 0 0
\(876\) −21.3376 −0.720930
\(877\) −34.8044 −1.17526 −0.587630 0.809130i \(-0.699939\pi\)
−0.587630 + 0.809130i \(0.699939\pi\)
\(878\) 7.90764 0.266870
\(879\) −2.16907 −0.0731608
\(880\) −2.29844 −0.0774803
\(881\) −28.8583 −0.972261 −0.486130 0.873886i \(-0.661592\pi\)
−0.486130 + 0.873886i \(0.661592\pi\)
\(882\) 0 0
\(883\) −27.5988 −0.928772 −0.464386 0.885633i \(-0.653725\pi\)
−0.464386 + 0.885633i \(0.653725\pi\)
\(884\) −21.0005 −0.706325
\(885\) −0.959466 −0.0322521
\(886\) 14.4302 0.484791
\(887\) −34.0939 −1.14476 −0.572380 0.819988i \(-0.693980\pi\)
−0.572380 + 0.819988i \(0.693980\pi\)
\(888\) −13.6568 −0.458292
\(889\) 0 0
\(890\) 7.30621 0.244905
\(891\) −1.00000 −0.0335013
\(892\) 34.9007 1.16856
\(893\) 5.21140 0.174393
\(894\) −6.28634 −0.210247
\(895\) −4.75362 −0.158896
\(896\) 0 0
\(897\) −16.1170 −0.538131
\(898\) 7.51295 0.250710
\(899\) −0.311373 −0.0103849
\(900\) −1.70156 −0.0567187
\(901\) −31.9823 −1.06548
\(902\) −6.00000 −0.199778
\(903\) 0 0
\(904\) −7.76188 −0.258156
\(905\) −0.347316 −0.0115452
\(906\) 4.14545 0.137723
\(907\) −26.7979 −0.889810 −0.444905 0.895578i \(-0.646763\pi\)
−0.444905 + 0.895578i \(0.646763\pi\)
\(908\) 33.3709 1.10745
\(909\) 11.6326 0.385829
\(910\) 0 0
\(911\) −27.2083 −0.901451 −0.450726 0.892663i \(-0.648835\pi\)
−0.450726 + 0.892663i \(0.648835\pi\)
\(912\) −18.4418 −0.610669
\(913\) −7.75737 −0.256731
\(914\) 5.45460 0.180422
\(915\) 13.1162 0.433608
\(916\) 34.7510 1.14820
\(917\) 0 0
\(918\) −1.47585 −0.0487103
\(919\) 1.61134 0.0531533 0.0265767 0.999647i \(-0.491539\pi\)
0.0265767 + 0.999647i \(0.491539\pi\)
\(920\) −7.13393 −0.235199
\(921\) 12.3548 0.407105
\(922\) 1.81321 0.0597149
\(923\) 11.7338 0.386221
\(924\) 0 0
\(925\) 6.75362 0.222058
\(926\) 11.0202 0.362146
\(927\) 9.18893 0.301804
\(928\) −0.705544 −0.0231606
\(929\) −53.7396 −1.76314 −0.881570 0.472053i \(-0.843513\pi\)
−0.881570 + 0.472053i \(0.843513\pi\)
\(930\) −1.27777 −0.0418997
\(931\) 0 0
\(932\) 29.4888 0.965936
\(933\) 22.6073 0.740131
\(934\) 2.01715 0.0660030
\(935\) 2.70156 0.0883505
\(936\) −9.23804 −0.301955
\(937\) 26.0601 0.851348 0.425674 0.904877i \(-0.360037\pi\)
0.425674 + 0.904877i \(0.360037\pi\)
\(938\) 0 0
\(939\) −34.2614 −1.11808
\(940\) −1.10518 −0.0360469
\(941\) 57.6176 1.87828 0.939139 0.343537i \(-0.111625\pi\)
0.939139 + 0.343537i \(0.111625\pi\)
\(942\) 7.02442 0.228868
\(943\) 38.7473 1.26178
\(944\) 2.20527 0.0717755
\(945\) 0 0
\(946\) 5.16531 0.167939
\(947\) 58.3245 1.89529 0.947646 0.319323i \(-0.103455\pi\)
0.947646 + 0.319323i \(0.103455\pi\)
\(948\) 4.37036 0.141943
\(949\) 57.2882 1.85965
\(950\) −4.38326 −0.142212
\(951\) −8.28929 −0.268799
\(952\) 0 0
\(953\) 2.27670 0.0737496 0.0368748 0.999320i \(-0.488260\pi\)
0.0368748 + 0.999320i \(0.488260\pi\)
\(954\) −6.46728 −0.209386
\(955\) 11.8379 0.383064
\(956\) 19.8068 0.640597
\(957\) 0.133124 0.00430328
\(958\) 22.4263 0.724559
\(959\) 0 0
\(960\) 1.70156 0.0549177
\(961\) −25.5292 −0.823523
\(962\) 16.8551 0.543430
\(963\) −13.0199 −0.419559
\(964\) −38.3582 −1.23544
\(965\) −4.14840 −0.133542
\(966\) 0 0
\(967\) 51.7348 1.66368 0.831840 0.555016i \(-0.187288\pi\)
0.831840 + 0.555016i \(0.187288\pi\)
\(968\) 2.02214 0.0649942
\(969\) 21.6763 0.696343
\(970\) −0.975577 −0.0313239
\(971\) 21.6846 0.695893 0.347947 0.937514i \(-0.386879\pi\)
0.347947 + 0.937514i \(0.386879\pi\)
\(972\) 1.70156 0.0545776
\(973\) 0 0
\(974\) 4.41626 0.141506
\(975\) 4.56844 0.146307
\(976\) −30.1468 −0.964975
\(977\) −15.7235 −0.503041 −0.251520 0.967852i \(-0.580931\pi\)
−0.251520 + 0.967852i \(0.580931\pi\)
\(978\) 12.3749 0.395706
\(979\) 13.3741 0.427438
\(980\) 0 0
\(981\) 0.109506 0.00349626
\(982\) 1.89754 0.0605530
\(983\) −26.0134 −0.829699 −0.414849 0.909890i \(-0.636166\pi\)
−0.414849 + 0.909890i \(0.636166\pi\)
\(984\) 22.2094 0.708009
\(985\) 19.3663 0.617063
\(986\) 0.196471 0.00625690
\(987\) 0 0
\(988\) 62.3714 1.98430
\(989\) −33.3570 −1.06069
\(990\) 0.546295 0.0173624
\(991\) 39.8932 1.26725 0.633624 0.773641i \(-0.281567\pi\)
0.633624 + 0.773641i \(0.281567\pi\)
\(992\) 12.3963 0.393584
\(993\) −21.4267 −0.679957
\(994\) 0 0
\(995\) −13.2622 −0.420441
\(996\) 13.1996 0.418247
\(997\) −10.9914 −0.348102 −0.174051 0.984737i \(-0.555686\pi\)
−0.174051 + 0.984737i \(0.555686\pi\)
\(998\) 15.8818 0.502731
\(999\) −6.75362 −0.213675
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 8085.2.a.bn.1.2 4
7.6 odd 2 1155.2.a.u.1.2 4
21.20 even 2 3465.2.a.bl.1.3 4
35.34 odd 2 5775.2.a.bz.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.u.1.2 4 7.6 odd 2
3465.2.a.bl.1.3 4 21.20 even 2
5775.2.a.bz.1.3 4 35.34 odd 2
8085.2.a.bn.1.2 4 1.1 even 1 trivial