Properties

Label 9075.2.a.bc.1.2
Level $9075$
Weight $2$
Character 9075.1
Self dual yes
Analytic conductor $72.464$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.4642398343\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 825)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 9075.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} -0.763932 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} -0.763932 q^{7} -2.23607 q^{8} +1.00000 q^{9} -1.61803 q^{12} -1.23607 q^{13} -0.472136 q^{14} +1.85410 q^{16} +2.00000 q^{17} +0.618034 q^{18} -5.00000 q^{19} -0.763932 q^{21} -2.38197 q^{23} -2.23607 q^{24} -0.763932 q^{26} +1.00000 q^{27} +1.23607 q^{28} +5.85410 q^{29} +7.00000 q^{31} +5.61803 q^{32} +1.23607 q^{34} -1.61803 q^{36} +7.47214 q^{37} -3.09017 q^{38} -1.23607 q^{39} -2.52786 q^{41} -0.472136 q^{42} -2.09017 q^{43} -1.47214 q^{46} -5.09017 q^{47} +1.85410 q^{48} -6.41641 q^{49} +2.00000 q^{51} +2.00000 q^{52} +7.61803 q^{53} +0.618034 q^{54} +1.70820 q^{56} -5.00000 q^{57} +3.61803 q^{58} -6.70820 q^{59} -7.00000 q^{61} +4.32624 q^{62} -0.763932 q^{63} -0.236068 q^{64} +9.38197 q^{67} -3.23607 q^{68} -2.38197 q^{69} -8.00000 q^{71} -2.23607 q^{72} -13.4721 q^{73} +4.61803 q^{74} +8.09017 q^{76} -0.763932 q^{78} +8.09017 q^{79} +1.00000 q^{81} -1.56231 q^{82} -5.70820 q^{83} +1.23607 q^{84} -1.29180 q^{86} +5.85410 q^{87} +10.8541 q^{89} +0.944272 q^{91} +3.85410 q^{92} +7.00000 q^{93} -3.14590 q^{94} +5.61803 q^{96} -11.4721 q^{97} -3.96556 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} - 6 q^{7} + 2 q^{9} - q^{12} + 2 q^{13} + 8 q^{14} - 3 q^{16} + 4 q^{17} - q^{18} - 10 q^{19} - 6 q^{21} - 7 q^{23} - 6 q^{26} + 2 q^{27} - 2 q^{28} + 5 q^{29} + 14 q^{31} + 9 q^{32} - 2 q^{34} - q^{36} + 6 q^{37} + 5 q^{38} + 2 q^{39} - 14 q^{41} + 8 q^{42} + 7 q^{43} + 6 q^{46} + q^{47} - 3 q^{48} + 14 q^{49} + 4 q^{51} + 4 q^{52} + 13 q^{53} - q^{54} - 10 q^{56} - 10 q^{57} + 5 q^{58} - 14 q^{61} - 7 q^{62} - 6 q^{63} + 4 q^{64} + 21 q^{67} - 2 q^{68} - 7 q^{69} - 16 q^{71} - 18 q^{73} + 7 q^{74} + 5 q^{76} - 6 q^{78} + 5 q^{79} + 2 q^{81} + 17 q^{82} + 2 q^{83} - 2 q^{84} - 16 q^{86} + 5 q^{87} + 15 q^{89} - 16 q^{91} + q^{92} + 14 q^{93} - 13 q^{94} + 9 q^{96} - 14 q^{97} - 37 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.618034 0.437016 0.218508 0.975835i \(-0.429881\pi\)
0.218508 + 0.975835i \(0.429881\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.61803 −0.809017
\(5\) 0 0
\(6\) 0.618034 0.252311
\(7\) −0.763932 −0.288739 −0.144370 0.989524i \(-0.546115\pi\)
−0.144370 + 0.989524i \(0.546115\pi\)
\(8\) −2.23607 −0.790569
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) −1.61803 −0.467086
\(13\) −1.23607 −0.342824 −0.171412 0.985199i \(-0.554833\pi\)
−0.171412 + 0.985199i \(0.554833\pi\)
\(14\) −0.472136 −0.126184
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0.618034 0.145672
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) −0.763932 −0.166704
\(22\) 0 0
\(23\) −2.38197 −0.496674 −0.248337 0.968674i \(-0.579884\pi\)
−0.248337 + 0.968674i \(0.579884\pi\)
\(24\) −2.23607 −0.456435
\(25\) 0 0
\(26\) −0.763932 −0.149819
\(27\) 1.00000 0.192450
\(28\) 1.23607 0.233595
\(29\) 5.85410 1.08708 0.543540 0.839383i \(-0.317084\pi\)
0.543540 + 0.839383i \(0.317084\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 5.61803 0.993137
\(33\) 0 0
\(34\) 1.23607 0.211984
\(35\) 0 0
\(36\) −1.61803 −0.269672
\(37\) 7.47214 1.22841 0.614206 0.789146i \(-0.289476\pi\)
0.614206 + 0.789146i \(0.289476\pi\)
\(38\) −3.09017 −0.501292
\(39\) −1.23607 −0.197929
\(40\) 0 0
\(41\) −2.52786 −0.394786 −0.197393 0.980324i \(-0.563248\pi\)
−0.197393 + 0.980324i \(0.563248\pi\)
\(42\) −0.472136 −0.0728522
\(43\) −2.09017 −0.318748 −0.159374 0.987218i \(-0.550948\pi\)
−0.159374 + 0.987218i \(0.550948\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.47214 −0.217055
\(47\) −5.09017 −0.742478 −0.371239 0.928537i \(-0.621067\pi\)
−0.371239 + 0.928537i \(0.621067\pi\)
\(48\) 1.85410 0.267617
\(49\) −6.41641 −0.916630
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 2.00000 0.277350
\(53\) 7.61803 1.04642 0.523209 0.852205i \(-0.324735\pi\)
0.523209 + 0.852205i \(0.324735\pi\)
\(54\) 0.618034 0.0841038
\(55\) 0 0
\(56\) 1.70820 0.228268
\(57\) −5.00000 −0.662266
\(58\) 3.61803 0.475071
\(59\) −6.70820 −0.873334 −0.436667 0.899623i \(-0.643841\pi\)
−0.436667 + 0.899623i \(0.643841\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 4.32624 0.549433
\(63\) −0.763932 −0.0962464
\(64\) −0.236068 −0.0295085
\(65\) 0 0
\(66\) 0 0
\(67\) 9.38197 1.14619 0.573095 0.819489i \(-0.305743\pi\)
0.573095 + 0.819489i \(0.305743\pi\)
\(68\) −3.23607 −0.392431
\(69\) −2.38197 −0.286755
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −2.23607 −0.263523
\(73\) −13.4721 −1.57679 −0.788397 0.615167i \(-0.789089\pi\)
−0.788397 + 0.615167i \(0.789089\pi\)
\(74\) 4.61803 0.536836
\(75\) 0 0
\(76\) 8.09017 0.928006
\(77\) 0 0
\(78\) −0.763932 −0.0864983
\(79\) 8.09017 0.910215 0.455108 0.890436i \(-0.349601\pi\)
0.455108 + 0.890436i \(0.349601\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −1.56231 −0.172528
\(83\) −5.70820 −0.626557 −0.313278 0.949661i \(-0.601427\pi\)
−0.313278 + 0.949661i \(0.601427\pi\)
\(84\) 1.23607 0.134866
\(85\) 0 0
\(86\) −1.29180 −0.139298
\(87\) 5.85410 0.627626
\(88\) 0 0
\(89\) 10.8541 1.15053 0.575266 0.817966i \(-0.304898\pi\)
0.575266 + 0.817966i \(0.304898\pi\)
\(90\) 0 0
\(91\) 0.944272 0.0989866
\(92\) 3.85410 0.401818
\(93\) 7.00000 0.725866
\(94\) −3.14590 −0.324475
\(95\) 0 0
\(96\) 5.61803 0.573388
\(97\) −11.4721 −1.16482 −0.582409 0.812896i \(-0.697890\pi\)
−0.582409 + 0.812896i \(0.697890\pi\)
\(98\) −3.96556 −0.400582
\(99\) 0 0
\(100\) 0 0
\(101\) 0.562306 0.0559515 0.0279758 0.999609i \(-0.491094\pi\)
0.0279758 + 0.999609i \(0.491094\pi\)
\(102\) 1.23607 0.122389
\(103\) −2.38197 −0.234702 −0.117351 0.993090i \(-0.537440\pi\)
−0.117351 + 0.993090i \(0.537440\pi\)
\(104\) 2.76393 0.271026
\(105\) 0 0
\(106\) 4.70820 0.457301
\(107\) 2.85410 0.275916 0.137958 0.990438i \(-0.455946\pi\)
0.137958 + 0.990438i \(0.455946\pi\)
\(108\) −1.61803 −0.155695
\(109\) −4.14590 −0.397105 −0.198553 0.980090i \(-0.563624\pi\)
−0.198553 + 0.980090i \(0.563624\pi\)
\(110\) 0 0
\(111\) 7.47214 0.709224
\(112\) −1.41641 −0.133838
\(113\) −5.47214 −0.514775 −0.257388 0.966308i \(-0.582862\pi\)
−0.257388 + 0.966308i \(0.582862\pi\)
\(114\) −3.09017 −0.289421
\(115\) 0 0
\(116\) −9.47214 −0.879466
\(117\) −1.23607 −0.114275
\(118\) −4.14590 −0.381661
\(119\) −1.52786 −0.140059
\(120\) 0 0
\(121\) 0 0
\(122\) −4.32624 −0.391679
\(123\) −2.52786 −0.227930
\(124\) −11.3262 −1.01713
\(125\) 0 0
\(126\) −0.472136 −0.0420612
\(127\) 15.9443 1.41483 0.707413 0.706801i \(-0.249862\pi\)
0.707413 + 0.706801i \(0.249862\pi\)
\(128\) −11.3820 −1.00603
\(129\) −2.09017 −0.184029
\(130\) 0 0
\(131\) −13.1803 −1.15157 −0.575786 0.817601i \(-0.695304\pi\)
−0.575786 + 0.817601i \(0.695304\pi\)
\(132\) 0 0
\(133\) 3.81966 0.331207
\(134\) 5.79837 0.500903
\(135\) 0 0
\(136\) −4.47214 −0.383482
\(137\) −18.7082 −1.59835 −0.799175 0.601099i \(-0.794730\pi\)
−0.799175 + 0.601099i \(0.794730\pi\)
\(138\) −1.47214 −0.125317
\(139\) −21.1803 −1.79649 −0.898246 0.439492i \(-0.855158\pi\)
−0.898246 + 0.439492i \(0.855158\pi\)
\(140\) 0 0
\(141\) −5.09017 −0.428670
\(142\) −4.94427 −0.414914
\(143\) 0 0
\(144\) 1.85410 0.154508
\(145\) 0 0
\(146\) −8.32624 −0.689084
\(147\) −6.41641 −0.529216
\(148\) −12.0902 −0.993806
\(149\) −20.1246 −1.64867 −0.824336 0.566101i \(-0.808451\pi\)
−0.824336 + 0.566101i \(0.808451\pi\)
\(150\) 0 0
\(151\) 10.7639 0.875956 0.437978 0.898986i \(-0.355695\pi\)
0.437978 + 0.898986i \(0.355695\pi\)
\(152\) 11.1803 0.906845
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 3.32624 0.265463 0.132731 0.991152i \(-0.457625\pi\)
0.132731 + 0.991152i \(0.457625\pi\)
\(158\) 5.00000 0.397779
\(159\) 7.61803 0.604149
\(160\) 0 0
\(161\) 1.81966 0.143409
\(162\) 0.618034 0.0485573
\(163\) −13.5623 −1.06228 −0.531141 0.847284i \(-0.678237\pi\)
−0.531141 + 0.847284i \(0.678237\pi\)
\(164\) 4.09017 0.319389
\(165\) 0 0
\(166\) −3.52786 −0.273815
\(167\) −20.0344 −1.55031 −0.775156 0.631770i \(-0.782329\pi\)
−0.775156 + 0.631770i \(0.782329\pi\)
\(168\) 1.70820 0.131791
\(169\) −11.4721 −0.882472
\(170\) 0 0
\(171\) −5.00000 −0.382360
\(172\) 3.38197 0.257872
\(173\) 3.56231 0.270837 0.135419 0.990788i \(-0.456762\pi\)
0.135419 + 0.990788i \(0.456762\pi\)
\(174\) 3.61803 0.274282
\(175\) 0 0
\(176\) 0 0
\(177\) −6.70820 −0.504219
\(178\) 6.70820 0.502801
\(179\) −19.4721 −1.45542 −0.727708 0.685887i \(-0.759414\pi\)
−0.727708 + 0.685887i \(0.759414\pi\)
\(180\) 0 0
\(181\) −5.23607 −0.389194 −0.194597 0.980883i \(-0.562340\pi\)
−0.194597 + 0.980883i \(0.562340\pi\)
\(182\) 0.583592 0.0432587
\(183\) −7.00000 −0.517455
\(184\) 5.32624 0.392655
\(185\) 0 0
\(186\) 4.32624 0.317215
\(187\) 0 0
\(188\) 8.23607 0.600677
\(189\) −0.763932 −0.0555679
\(190\) 0 0
\(191\) 10.4164 0.753705 0.376852 0.926273i \(-0.377006\pi\)
0.376852 + 0.926273i \(0.377006\pi\)
\(192\) −0.236068 −0.0170367
\(193\) −21.0344 −1.51409 −0.757046 0.653361i \(-0.773358\pi\)
−0.757046 + 0.653361i \(0.773358\pi\)
\(194\) −7.09017 −0.509045
\(195\) 0 0
\(196\) 10.3820 0.741569
\(197\) 14.2361 1.01428 0.507139 0.861864i \(-0.330703\pi\)
0.507139 + 0.861864i \(0.330703\pi\)
\(198\) 0 0
\(199\) 19.7984 1.40347 0.701735 0.712438i \(-0.252409\pi\)
0.701735 + 0.712438i \(0.252409\pi\)
\(200\) 0 0
\(201\) 9.38197 0.661753
\(202\) 0.347524 0.0244517
\(203\) −4.47214 −0.313882
\(204\) −3.23607 −0.226570
\(205\) 0 0
\(206\) −1.47214 −0.102569
\(207\) −2.38197 −0.165558
\(208\) −2.29180 −0.158907
\(209\) 0 0
\(210\) 0 0
\(211\) 3.85410 0.265327 0.132664 0.991161i \(-0.457647\pi\)
0.132664 + 0.991161i \(0.457647\pi\)
\(212\) −12.3262 −0.846569
\(213\) −8.00000 −0.548151
\(214\) 1.76393 0.120580
\(215\) 0 0
\(216\) −2.23607 −0.152145
\(217\) −5.34752 −0.363014
\(218\) −2.56231 −0.173541
\(219\) −13.4721 −0.910363
\(220\) 0 0
\(221\) −2.47214 −0.166294
\(222\) 4.61803 0.309942
\(223\) −8.03444 −0.538026 −0.269013 0.963137i \(-0.586697\pi\)
−0.269013 + 0.963137i \(0.586697\pi\)
\(224\) −4.29180 −0.286758
\(225\) 0 0
\(226\) −3.38197 −0.224965
\(227\) 25.9443 1.72198 0.860991 0.508620i \(-0.169844\pi\)
0.860991 + 0.508620i \(0.169844\pi\)
\(228\) 8.09017 0.535785
\(229\) 18.0902 1.19543 0.597716 0.801708i \(-0.296075\pi\)
0.597716 + 0.801708i \(0.296075\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −13.0902 −0.859412
\(233\) −23.2705 −1.52450 −0.762251 0.647282i \(-0.775906\pi\)
−0.762251 + 0.647282i \(0.775906\pi\)
\(234\) −0.763932 −0.0499398
\(235\) 0 0
\(236\) 10.8541 0.706542
\(237\) 8.09017 0.525513
\(238\) −0.944272 −0.0612081
\(239\) 7.03444 0.455020 0.227510 0.973776i \(-0.426942\pi\)
0.227510 + 0.973776i \(0.426942\pi\)
\(240\) 0 0
\(241\) 1.61803 0.104227 0.0521134 0.998641i \(-0.483404\pi\)
0.0521134 + 0.998641i \(0.483404\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 11.3262 0.725088
\(245\) 0 0
\(246\) −1.56231 −0.0996090
\(247\) 6.18034 0.393246
\(248\) −15.6525 −0.993933
\(249\) −5.70820 −0.361743
\(250\) 0 0
\(251\) 7.52786 0.475155 0.237577 0.971369i \(-0.423647\pi\)
0.237577 + 0.971369i \(0.423647\pi\)
\(252\) 1.23607 0.0778650
\(253\) 0 0
\(254\) 9.85410 0.618301
\(255\) 0 0
\(256\) −6.56231 −0.410144
\(257\) 10.5623 0.658859 0.329429 0.944180i \(-0.393144\pi\)
0.329429 + 0.944180i \(0.393144\pi\)
\(258\) −1.29180 −0.0804237
\(259\) −5.70820 −0.354691
\(260\) 0 0
\(261\) 5.85410 0.362360
\(262\) −8.14590 −0.503255
\(263\) −27.2148 −1.67814 −0.839068 0.544027i \(-0.816899\pi\)
−0.839068 + 0.544027i \(0.816899\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 2.36068 0.144743
\(267\) 10.8541 0.664260
\(268\) −15.1803 −0.927287
\(269\) −5.85410 −0.356931 −0.178465 0.983946i \(-0.557113\pi\)
−0.178465 + 0.983946i \(0.557113\pi\)
\(270\) 0 0
\(271\) −20.6180 −1.25246 −0.626228 0.779640i \(-0.715402\pi\)
−0.626228 + 0.779640i \(0.715402\pi\)
\(272\) 3.70820 0.224843
\(273\) 0.944272 0.0571499
\(274\) −11.5623 −0.698504
\(275\) 0 0
\(276\) 3.85410 0.231990
\(277\) −8.65248 −0.519877 −0.259938 0.965625i \(-0.583702\pi\)
−0.259938 + 0.965625i \(0.583702\pi\)
\(278\) −13.0902 −0.785096
\(279\) 7.00000 0.419079
\(280\) 0 0
\(281\) −10.0902 −0.601929 −0.300965 0.953635i \(-0.597309\pi\)
−0.300965 + 0.953635i \(0.597309\pi\)
\(282\) −3.14590 −0.187336
\(283\) 13.0344 0.774817 0.387409 0.921908i \(-0.373370\pi\)
0.387409 + 0.921908i \(0.373370\pi\)
\(284\) 12.9443 0.768101
\(285\) 0 0
\(286\) 0 0
\(287\) 1.93112 0.113990
\(288\) 5.61803 0.331046
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −11.4721 −0.672509
\(292\) 21.7984 1.27565
\(293\) 16.5279 0.965568 0.482784 0.875739i \(-0.339626\pi\)
0.482784 + 0.875739i \(0.339626\pi\)
\(294\) −3.96556 −0.231276
\(295\) 0 0
\(296\) −16.7082 −0.971145
\(297\) 0 0
\(298\) −12.4377 −0.720496
\(299\) 2.94427 0.170272
\(300\) 0 0
\(301\) 1.59675 0.0920350
\(302\) 6.65248 0.382807
\(303\) 0.562306 0.0323036
\(304\) −9.27051 −0.531700
\(305\) 0 0
\(306\) 1.23607 0.0706613
\(307\) 12.1246 0.691988 0.345994 0.938237i \(-0.387542\pi\)
0.345994 + 0.938237i \(0.387542\pi\)
\(308\) 0 0
\(309\) −2.38197 −0.135505
\(310\) 0 0
\(311\) 21.4721 1.21757 0.608787 0.793334i \(-0.291657\pi\)
0.608787 + 0.793334i \(0.291657\pi\)
\(312\) 2.76393 0.156477
\(313\) −0.472136 −0.0266867 −0.0133434 0.999911i \(-0.504247\pi\)
−0.0133434 + 0.999911i \(0.504247\pi\)
\(314\) 2.05573 0.116011
\(315\) 0 0
\(316\) −13.0902 −0.736380
\(317\) −30.4164 −1.70836 −0.854178 0.519981i \(-0.825939\pi\)
−0.854178 + 0.519981i \(0.825939\pi\)
\(318\) 4.70820 0.264023
\(319\) 0 0
\(320\) 0 0
\(321\) 2.85410 0.159300
\(322\) 1.12461 0.0626722
\(323\) −10.0000 −0.556415
\(324\) −1.61803 −0.0898908
\(325\) 0 0
\(326\) −8.38197 −0.464234
\(327\) −4.14590 −0.229269
\(328\) 5.65248 0.312106
\(329\) 3.88854 0.214382
\(330\) 0 0
\(331\) 21.5967 1.18706 0.593532 0.804810i \(-0.297733\pi\)
0.593532 + 0.804810i \(0.297733\pi\)
\(332\) 9.23607 0.506895
\(333\) 7.47214 0.409471
\(334\) −12.3820 −0.677511
\(335\) 0 0
\(336\) −1.41641 −0.0772714
\(337\) −22.1459 −1.20636 −0.603182 0.797604i \(-0.706101\pi\)
−0.603182 + 0.797604i \(0.706101\pi\)
\(338\) −7.09017 −0.385654
\(339\) −5.47214 −0.297206
\(340\) 0 0
\(341\) 0 0
\(342\) −3.09017 −0.167097
\(343\) 10.2492 0.553406
\(344\) 4.67376 0.251992
\(345\) 0 0
\(346\) 2.20163 0.118360
\(347\) 5.81966 0.312416 0.156208 0.987724i \(-0.450073\pi\)
0.156208 + 0.987724i \(0.450073\pi\)
\(348\) −9.47214 −0.507760
\(349\) −27.7639 −1.48617 −0.743085 0.669197i \(-0.766638\pi\)
−0.743085 + 0.669197i \(0.766638\pi\)
\(350\) 0 0
\(351\) −1.23607 −0.0659764
\(352\) 0 0
\(353\) 35.8328 1.90719 0.953594 0.301095i \(-0.0973521\pi\)
0.953594 + 0.301095i \(0.0973521\pi\)
\(354\) −4.14590 −0.220352
\(355\) 0 0
\(356\) −17.5623 −0.930800
\(357\) −1.52786 −0.0808631
\(358\) −12.0344 −0.636040
\(359\) −16.7082 −0.881825 −0.440913 0.897550i \(-0.645345\pi\)
−0.440913 + 0.897550i \(0.645345\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) −3.23607 −0.170084
\(363\) 0 0
\(364\) −1.52786 −0.0800818
\(365\) 0 0
\(366\) −4.32624 −0.226136
\(367\) 18.1246 0.946097 0.473049 0.881036i \(-0.343154\pi\)
0.473049 + 0.881036i \(0.343154\pi\)
\(368\) −4.41641 −0.230221
\(369\) −2.52786 −0.131595
\(370\) 0 0
\(371\) −5.81966 −0.302142
\(372\) −11.3262 −0.587238
\(373\) 9.74265 0.504455 0.252228 0.967668i \(-0.418837\pi\)
0.252228 + 0.967668i \(0.418837\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 11.3820 0.586980
\(377\) −7.23607 −0.372676
\(378\) −0.472136 −0.0242841
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) 15.9443 0.816850
\(382\) 6.43769 0.329381
\(383\) 31.3607 1.60246 0.801228 0.598359i \(-0.204180\pi\)
0.801228 + 0.598359i \(0.204180\pi\)
\(384\) −11.3820 −0.580834
\(385\) 0 0
\(386\) −13.0000 −0.661683
\(387\) −2.09017 −0.106249
\(388\) 18.5623 0.942358
\(389\) −38.9443 −1.97455 −0.987276 0.159013i \(-0.949169\pi\)
−0.987276 + 0.159013i \(0.949169\pi\)
\(390\) 0 0
\(391\) −4.76393 −0.240922
\(392\) 14.3475 0.724659
\(393\) −13.1803 −0.664860
\(394\) 8.79837 0.443256
\(395\) 0 0
\(396\) 0 0
\(397\) −36.2705 −1.82036 −0.910182 0.414208i \(-0.864059\pi\)
−0.910182 + 0.414208i \(0.864059\pi\)
\(398\) 12.2361 0.613339
\(399\) 3.81966 0.191222
\(400\) 0 0
\(401\) 4.88854 0.244122 0.122061 0.992523i \(-0.461050\pi\)
0.122061 + 0.992523i \(0.461050\pi\)
\(402\) 5.79837 0.289197
\(403\) −8.65248 −0.431011
\(404\) −0.909830 −0.0452657
\(405\) 0 0
\(406\) −2.76393 −0.137172
\(407\) 0 0
\(408\) −4.47214 −0.221404
\(409\) −36.3050 −1.79516 −0.897582 0.440847i \(-0.854678\pi\)
−0.897582 + 0.440847i \(0.854678\pi\)
\(410\) 0 0
\(411\) −18.7082 −0.922808
\(412\) 3.85410 0.189878
\(413\) 5.12461 0.252166
\(414\) −1.47214 −0.0723515
\(415\) 0 0
\(416\) −6.94427 −0.340471
\(417\) −21.1803 −1.03721
\(418\) 0 0
\(419\) 24.5967 1.20163 0.600815 0.799388i \(-0.294843\pi\)
0.600815 + 0.799388i \(0.294843\pi\)
\(420\) 0 0
\(421\) 2.85410 0.139100 0.0695502 0.997578i \(-0.477844\pi\)
0.0695502 + 0.997578i \(0.477844\pi\)
\(422\) 2.38197 0.115952
\(423\) −5.09017 −0.247493
\(424\) −17.0344 −0.827266
\(425\) 0 0
\(426\) −4.94427 −0.239551
\(427\) 5.34752 0.258785
\(428\) −4.61803 −0.223221
\(429\) 0 0
\(430\) 0 0
\(431\) −19.8885 −0.957997 −0.478999 0.877816i \(-0.659000\pi\)
−0.478999 + 0.877816i \(0.659000\pi\)
\(432\) 1.85410 0.0892055
\(433\) −28.0344 −1.34725 −0.673625 0.739074i \(-0.735264\pi\)
−0.673625 + 0.739074i \(0.735264\pi\)
\(434\) −3.30495 −0.158643
\(435\) 0 0
\(436\) 6.70820 0.321265
\(437\) 11.9098 0.569724
\(438\) −8.32624 −0.397843
\(439\) −9.67376 −0.461703 −0.230852 0.972989i \(-0.574151\pi\)
−0.230852 + 0.972989i \(0.574151\pi\)
\(440\) 0 0
\(441\) −6.41641 −0.305543
\(442\) −1.52786 −0.0726731
\(443\) −25.2705 −1.20064 −0.600319 0.799761i \(-0.704960\pi\)
−0.600319 + 0.799761i \(0.704960\pi\)
\(444\) −12.0902 −0.573774
\(445\) 0 0
\(446\) −4.96556 −0.235126
\(447\) −20.1246 −0.951861
\(448\) 0.180340 0.00852026
\(449\) −0.729490 −0.0344268 −0.0172134 0.999852i \(-0.505479\pi\)
−0.0172134 + 0.999852i \(0.505479\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 8.85410 0.416462
\(453\) 10.7639 0.505734
\(454\) 16.0344 0.752534
\(455\) 0 0
\(456\) 11.1803 0.523567
\(457\) 1.47214 0.0688636 0.0344318 0.999407i \(-0.489038\pi\)
0.0344318 + 0.999407i \(0.489038\pi\)
\(458\) 11.1803 0.522423
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −25.9443 −1.20835 −0.604173 0.796853i \(-0.706496\pi\)
−0.604173 + 0.796853i \(0.706496\pi\)
\(462\) 0 0
\(463\) 33.2705 1.54621 0.773106 0.634277i \(-0.218702\pi\)
0.773106 + 0.634277i \(0.218702\pi\)
\(464\) 10.8541 0.503889
\(465\) 0 0
\(466\) −14.3820 −0.666232
\(467\) −4.88854 −0.226215 −0.113107 0.993583i \(-0.536080\pi\)
−0.113107 + 0.993583i \(0.536080\pi\)
\(468\) 2.00000 0.0924500
\(469\) −7.16718 −0.330950
\(470\) 0 0
\(471\) 3.32624 0.153265
\(472\) 15.0000 0.690431
\(473\) 0 0
\(474\) 5.00000 0.229658
\(475\) 0 0
\(476\) 2.47214 0.113310
\(477\) 7.61803 0.348806
\(478\) 4.34752 0.198851
\(479\) −33.4164 −1.52683 −0.763417 0.645906i \(-0.776480\pi\)
−0.763417 + 0.645906i \(0.776480\pi\)
\(480\) 0 0
\(481\) −9.23607 −0.421128
\(482\) 1.00000 0.0455488
\(483\) 1.81966 0.0827974
\(484\) 0 0
\(485\) 0 0
\(486\) 0.618034 0.0280346
\(487\) 14.1803 0.642573 0.321286 0.946982i \(-0.395885\pi\)
0.321286 + 0.946982i \(0.395885\pi\)
\(488\) 15.6525 0.708554
\(489\) −13.5623 −0.613309
\(490\) 0 0
\(491\) −31.7984 −1.43504 −0.717520 0.696538i \(-0.754723\pi\)
−0.717520 + 0.696538i \(0.754723\pi\)
\(492\) 4.09017 0.184399
\(493\) 11.7082 0.527311
\(494\) 3.81966 0.171855
\(495\) 0 0
\(496\) 12.9787 0.582761
\(497\) 6.11146 0.274136
\(498\) −3.52786 −0.158087
\(499\) −11.1803 −0.500501 −0.250250 0.968181i \(-0.580513\pi\)
−0.250250 + 0.968181i \(0.580513\pi\)
\(500\) 0 0
\(501\) −20.0344 −0.895073
\(502\) 4.65248 0.207650
\(503\) −21.8885 −0.975962 −0.487981 0.872854i \(-0.662266\pi\)
−0.487981 + 0.872854i \(0.662266\pi\)
\(504\) 1.70820 0.0760895
\(505\) 0 0
\(506\) 0 0
\(507\) −11.4721 −0.509495
\(508\) −25.7984 −1.14462
\(509\) 0.527864 0.0233972 0.0116986 0.999932i \(-0.496276\pi\)
0.0116986 + 0.999932i \(0.496276\pi\)
\(510\) 0 0
\(511\) 10.2918 0.455282
\(512\) 18.7082 0.826794
\(513\) −5.00000 −0.220755
\(514\) 6.52786 0.287932
\(515\) 0 0
\(516\) 3.38197 0.148883
\(517\) 0 0
\(518\) −3.52786 −0.155005
\(519\) 3.56231 0.156368
\(520\) 0 0
\(521\) −15.3607 −0.672964 −0.336482 0.941690i \(-0.609237\pi\)
−0.336482 + 0.941690i \(0.609237\pi\)
\(522\) 3.61803 0.158357
\(523\) 8.23607 0.360138 0.180069 0.983654i \(-0.442368\pi\)
0.180069 + 0.983654i \(0.442368\pi\)
\(524\) 21.3262 0.931641
\(525\) 0 0
\(526\) −16.8197 −0.733372
\(527\) 14.0000 0.609850
\(528\) 0 0
\(529\) −17.3262 −0.753315
\(530\) 0 0
\(531\) −6.70820 −0.291111
\(532\) −6.18034 −0.267952
\(533\) 3.12461 0.135342
\(534\) 6.70820 0.290292
\(535\) 0 0
\(536\) −20.9787 −0.906142
\(537\) −19.4721 −0.840285
\(538\) −3.61803 −0.155985
\(539\) 0 0
\(540\) 0 0
\(541\) −0.618034 −0.0265714 −0.0132857 0.999912i \(-0.504229\pi\)
−0.0132857 + 0.999912i \(0.504229\pi\)
\(542\) −12.7426 −0.547344
\(543\) −5.23607 −0.224701
\(544\) 11.2361 0.481742
\(545\) 0 0
\(546\) 0.583592 0.0249754
\(547\) −25.3607 −1.08434 −0.542172 0.840267i \(-0.682398\pi\)
−0.542172 + 0.840267i \(0.682398\pi\)
\(548\) 30.2705 1.29309
\(549\) −7.00000 −0.298753
\(550\) 0 0
\(551\) −29.2705 −1.24697
\(552\) 5.32624 0.226700
\(553\) −6.18034 −0.262815
\(554\) −5.34752 −0.227195
\(555\) 0 0
\(556\) 34.2705 1.45339
\(557\) 31.3951 1.33025 0.665127 0.746730i \(-0.268377\pi\)
0.665127 + 0.746730i \(0.268377\pi\)
\(558\) 4.32624 0.183144
\(559\) 2.58359 0.109274
\(560\) 0 0
\(561\) 0 0
\(562\) −6.23607 −0.263053
\(563\) 36.3262 1.53097 0.765484 0.643455i \(-0.222500\pi\)
0.765484 + 0.643455i \(0.222500\pi\)
\(564\) 8.23607 0.346801
\(565\) 0 0
\(566\) 8.05573 0.338608
\(567\) −0.763932 −0.0320821
\(568\) 17.8885 0.750587
\(569\) 2.43769 0.102193 0.0510967 0.998694i \(-0.483728\pi\)
0.0510967 + 0.998694i \(0.483728\pi\)
\(570\) 0 0
\(571\) −0.0901699 −0.00377349 −0.00188675 0.999998i \(-0.500601\pi\)
−0.00188675 + 0.999998i \(0.500601\pi\)
\(572\) 0 0
\(573\) 10.4164 0.435152
\(574\) 1.19350 0.0498155
\(575\) 0 0
\(576\) −0.236068 −0.00983617
\(577\) 46.0132 1.91555 0.957776 0.287514i \(-0.0928289\pi\)
0.957776 + 0.287514i \(0.0928289\pi\)
\(578\) −8.03444 −0.334189
\(579\) −21.0344 −0.874162
\(580\) 0 0
\(581\) 4.36068 0.180911
\(582\) −7.09017 −0.293897
\(583\) 0 0
\(584\) 30.1246 1.24657
\(585\) 0 0
\(586\) 10.2148 0.421969
\(587\) 19.3820 0.799979 0.399990 0.916520i \(-0.369014\pi\)
0.399990 + 0.916520i \(0.369014\pi\)
\(588\) 10.3820 0.428145
\(589\) −35.0000 −1.44215
\(590\) 0 0
\(591\) 14.2361 0.585594
\(592\) 13.8541 0.569400
\(593\) 38.8885 1.59696 0.798481 0.602021i \(-0.205638\pi\)
0.798481 + 0.602021i \(0.205638\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 32.5623 1.33380
\(597\) 19.7984 0.810294
\(598\) 1.81966 0.0744114
\(599\) −22.2361 −0.908541 −0.454271 0.890864i \(-0.650100\pi\)
−0.454271 + 0.890864i \(0.650100\pi\)
\(600\) 0 0
\(601\) −34.3607 −1.40160 −0.700801 0.713357i \(-0.747174\pi\)
−0.700801 + 0.713357i \(0.747174\pi\)
\(602\) 0.986844 0.0402208
\(603\) 9.38197 0.382063
\(604\) −17.4164 −0.708664
\(605\) 0 0
\(606\) 0.347524 0.0141172
\(607\) −33.0000 −1.33943 −0.669714 0.742619i \(-0.733583\pi\)
−0.669714 + 0.742619i \(0.733583\pi\)
\(608\) −28.0902 −1.13921
\(609\) −4.47214 −0.181220
\(610\) 0 0
\(611\) 6.29180 0.254539
\(612\) −3.23607 −0.130810
\(613\) 8.23607 0.332652 0.166326 0.986071i \(-0.446810\pi\)
0.166326 + 0.986071i \(0.446810\pi\)
\(614\) 7.49342 0.302410
\(615\) 0 0
\(616\) 0 0
\(617\) 38.2492 1.53986 0.769928 0.638131i \(-0.220292\pi\)
0.769928 + 0.638131i \(0.220292\pi\)
\(618\) −1.47214 −0.0592180
\(619\) 7.11146 0.285834 0.142917 0.989735i \(-0.454352\pi\)
0.142917 + 0.989735i \(0.454352\pi\)
\(620\) 0 0
\(621\) −2.38197 −0.0955850
\(622\) 13.2705 0.532099
\(623\) −8.29180 −0.332204
\(624\) −2.29180 −0.0917453
\(625\) 0 0
\(626\) −0.291796 −0.0116625
\(627\) 0 0
\(628\) −5.38197 −0.214764
\(629\) 14.9443 0.595867
\(630\) 0 0
\(631\) 38.6312 1.53788 0.768942 0.639319i \(-0.220784\pi\)
0.768942 + 0.639319i \(0.220784\pi\)
\(632\) −18.0902 −0.719588
\(633\) 3.85410 0.153187
\(634\) −18.7984 −0.746579
\(635\) 0 0
\(636\) −12.3262 −0.488767
\(637\) 7.93112 0.314242
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) −0.236068 −0.00932412 −0.00466206 0.999989i \(-0.501484\pi\)
−0.00466206 + 0.999989i \(0.501484\pi\)
\(642\) 1.76393 0.0696168
\(643\) 29.3262 1.15651 0.578257 0.815855i \(-0.303733\pi\)
0.578257 + 0.815855i \(0.303733\pi\)
\(644\) −2.94427 −0.116021
\(645\) 0 0
\(646\) −6.18034 −0.243162
\(647\) 17.5967 0.691800 0.345900 0.938271i \(-0.387574\pi\)
0.345900 + 0.938271i \(0.387574\pi\)
\(648\) −2.23607 −0.0878410
\(649\) 0 0
\(650\) 0 0
\(651\) −5.34752 −0.209586
\(652\) 21.9443 0.859404
\(653\) −23.4377 −0.917188 −0.458594 0.888646i \(-0.651647\pi\)
−0.458594 + 0.888646i \(0.651647\pi\)
\(654\) −2.56231 −0.100194
\(655\) 0 0
\(656\) −4.68692 −0.182993
\(657\) −13.4721 −0.525598
\(658\) 2.40325 0.0936885
\(659\) −12.9656 −0.505066 −0.252533 0.967588i \(-0.581264\pi\)
−0.252533 + 0.967588i \(0.581264\pi\)
\(660\) 0 0
\(661\) 3.90983 0.152075 0.0760374 0.997105i \(-0.475773\pi\)
0.0760374 + 0.997105i \(0.475773\pi\)
\(662\) 13.3475 0.518766
\(663\) −2.47214 −0.0960098
\(664\) 12.7639 0.495337
\(665\) 0 0
\(666\) 4.61803 0.178945
\(667\) −13.9443 −0.539924
\(668\) 32.4164 1.25423
\(669\) −8.03444 −0.310629
\(670\) 0 0
\(671\) 0 0
\(672\) −4.29180 −0.165560
\(673\) 27.3050 1.05253 0.526264 0.850321i \(-0.323592\pi\)
0.526264 + 0.850321i \(0.323592\pi\)
\(674\) −13.6869 −0.527200
\(675\) 0 0
\(676\) 18.5623 0.713935
\(677\) −11.9443 −0.459056 −0.229528 0.973302i \(-0.573718\pi\)
−0.229528 + 0.973302i \(0.573718\pi\)
\(678\) −3.38197 −0.129884
\(679\) 8.76393 0.336329
\(680\) 0 0
\(681\) 25.9443 0.994187
\(682\) 0 0
\(683\) 2.29180 0.0876931 0.0438466 0.999038i \(-0.486039\pi\)
0.0438466 + 0.999038i \(0.486039\pi\)
\(684\) 8.09017 0.309335
\(685\) 0 0
\(686\) 6.33437 0.241847
\(687\) 18.0902 0.690183
\(688\) −3.87539 −0.147748
\(689\) −9.41641 −0.358737
\(690\) 0 0
\(691\) 33.3050 1.26698 0.633490 0.773751i \(-0.281622\pi\)
0.633490 + 0.773751i \(0.281622\pi\)
\(692\) −5.76393 −0.219112
\(693\) 0 0
\(694\) 3.59675 0.136531
\(695\) 0 0
\(696\) −13.0902 −0.496182
\(697\) −5.05573 −0.191499
\(698\) −17.1591 −0.649480
\(699\) −23.2705 −0.880172
\(700\) 0 0
\(701\) −13.5836 −0.513045 −0.256523 0.966538i \(-0.582577\pi\)
−0.256523 + 0.966538i \(0.582577\pi\)
\(702\) −0.763932 −0.0288328
\(703\) −37.3607 −1.40908
\(704\) 0 0
\(705\) 0 0
\(706\) 22.1459 0.833472
\(707\) −0.429563 −0.0161554
\(708\) 10.8541 0.407922
\(709\) 41.1803 1.54656 0.773280 0.634065i \(-0.218615\pi\)
0.773280 + 0.634065i \(0.218615\pi\)
\(710\) 0 0
\(711\) 8.09017 0.303405
\(712\) −24.2705 −0.909576
\(713\) −16.6738 −0.624437
\(714\) −0.944272 −0.0353385
\(715\) 0 0
\(716\) 31.5066 1.17746
\(717\) 7.03444 0.262706
\(718\) −10.3262 −0.385372
\(719\) 38.0902 1.42052 0.710262 0.703938i \(-0.248577\pi\)
0.710262 + 0.703938i \(0.248577\pi\)
\(720\) 0 0
\(721\) 1.81966 0.0677677
\(722\) 3.70820 0.138005
\(723\) 1.61803 0.0601753
\(724\) 8.47214 0.314864
\(725\) 0 0
\(726\) 0 0
\(727\) −37.3262 −1.38435 −0.692177 0.721728i \(-0.743348\pi\)
−0.692177 + 0.721728i \(0.743348\pi\)
\(728\) −2.11146 −0.0782558
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −4.18034 −0.154615
\(732\) 11.3262 0.418630
\(733\) 1.85410 0.0684828 0.0342414 0.999414i \(-0.489098\pi\)
0.0342414 + 0.999414i \(0.489098\pi\)
\(734\) 11.2016 0.413460
\(735\) 0 0
\(736\) −13.3820 −0.493266
\(737\) 0 0
\(738\) −1.56231 −0.0575093
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 6.18034 0.227040
\(742\) −3.59675 −0.132041
\(743\) 0.347524 0.0127494 0.00637471 0.999980i \(-0.497971\pi\)
0.00637471 + 0.999980i \(0.497971\pi\)
\(744\) −15.6525 −0.573848
\(745\) 0 0
\(746\) 6.02129 0.220455
\(747\) −5.70820 −0.208852
\(748\) 0 0
\(749\) −2.18034 −0.0796679
\(750\) 0 0
\(751\) −45.8885 −1.67450 −0.837248 0.546823i \(-0.815837\pi\)
−0.837248 + 0.546823i \(0.815837\pi\)
\(752\) −9.43769 −0.344157
\(753\) 7.52786 0.274331
\(754\) −4.47214 −0.162866
\(755\) 0 0
\(756\) 1.23607 0.0449554
\(757\) 24.1803 0.878849 0.439425 0.898279i \(-0.355182\pi\)
0.439425 + 0.898279i \(0.355182\pi\)
\(758\) −3.09017 −0.112240
\(759\) 0 0
\(760\) 0 0
\(761\) −39.0344 −1.41500 −0.707499 0.706715i \(-0.750176\pi\)
−0.707499 + 0.706715i \(0.750176\pi\)
\(762\) 9.85410 0.356976
\(763\) 3.16718 0.114660
\(764\) −16.8541 −0.609760
\(765\) 0 0
\(766\) 19.3820 0.700299
\(767\) 8.29180 0.299399
\(768\) −6.56231 −0.236797
\(769\) −12.7639 −0.460279 −0.230140 0.973158i \(-0.573918\pi\)
−0.230140 + 0.973158i \(0.573918\pi\)
\(770\) 0 0
\(771\) 10.5623 0.380392
\(772\) 34.0344 1.22493
\(773\) 10.5066 0.377895 0.188948 0.981987i \(-0.439492\pi\)
0.188948 + 0.981987i \(0.439492\pi\)
\(774\) −1.29180 −0.0464327
\(775\) 0 0
\(776\) 25.6525 0.920870
\(777\) −5.70820 −0.204781
\(778\) −24.0689 −0.862911
\(779\) 12.6393 0.452851
\(780\) 0 0
\(781\) 0 0
\(782\) −2.94427 −0.105287
\(783\) 5.85410 0.209209
\(784\) −11.8967 −0.424881
\(785\) 0 0
\(786\) −8.14590 −0.290555
\(787\) −8.20163 −0.292356 −0.146178 0.989258i \(-0.546697\pi\)
−0.146178 + 0.989258i \(0.546697\pi\)
\(788\) −23.0344 −0.820568
\(789\) −27.2148 −0.968872
\(790\) 0 0
\(791\) 4.18034 0.148636
\(792\) 0 0
\(793\) 8.65248 0.307258
\(794\) −22.4164 −0.795529
\(795\) 0 0
\(796\) −32.0344 −1.13543
\(797\) −28.7082 −1.01690 −0.508448 0.861092i \(-0.669781\pi\)
−0.508448 + 0.861092i \(0.669781\pi\)
\(798\) 2.36068 0.0835672
\(799\) −10.1803 −0.360155
\(800\) 0 0
\(801\) 10.8541 0.383511
\(802\) 3.02129 0.106685
\(803\) 0 0
\(804\) −15.1803 −0.535369
\(805\) 0 0
\(806\) −5.34752 −0.188359
\(807\) −5.85410 −0.206074
\(808\) −1.25735 −0.0442336
\(809\) −45.3262 −1.59359 −0.796793 0.604253i \(-0.793472\pi\)
−0.796793 + 0.604253i \(0.793472\pi\)
\(810\) 0 0
\(811\) 34.6312 1.21607 0.608033 0.793912i \(-0.291959\pi\)
0.608033 + 0.793912i \(0.291959\pi\)
\(812\) 7.23607 0.253936
\(813\) −20.6180 −0.723106
\(814\) 0 0
\(815\) 0 0
\(816\) 3.70820 0.129813
\(817\) 10.4508 0.365629
\(818\) −22.4377 −0.784516
\(819\) 0.944272 0.0329955
\(820\) 0 0
\(821\) −43.1803 −1.50700 −0.753502 0.657445i \(-0.771637\pi\)
−0.753502 + 0.657445i \(0.771637\pi\)
\(822\) −11.5623 −0.403282
\(823\) −25.4721 −0.887903 −0.443951 0.896051i \(-0.646424\pi\)
−0.443951 + 0.896051i \(0.646424\pi\)
\(824\) 5.32624 0.185548
\(825\) 0 0
\(826\) 3.16718 0.110200
\(827\) −24.5836 −0.854855 −0.427428 0.904050i \(-0.640580\pi\)
−0.427428 + 0.904050i \(0.640580\pi\)
\(828\) 3.85410 0.133939
\(829\) 2.88854 0.100323 0.0501616 0.998741i \(-0.484026\pi\)
0.0501616 + 0.998741i \(0.484026\pi\)
\(830\) 0 0
\(831\) −8.65248 −0.300151
\(832\) 0.291796 0.0101162
\(833\) −12.8328 −0.444631
\(834\) −13.0902 −0.453276
\(835\) 0 0
\(836\) 0 0
\(837\) 7.00000 0.241955
\(838\) 15.2016 0.525131
\(839\) −26.3050 −0.908148 −0.454074 0.890964i \(-0.650030\pi\)
−0.454074 + 0.890964i \(0.650030\pi\)
\(840\) 0 0
\(841\) 5.27051 0.181742
\(842\) 1.76393 0.0607891
\(843\) −10.0902 −0.347524
\(844\) −6.23607 −0.214654
\(845\) 0 0
\(846\) −3.14590 −0.108158
\(847\) 0 0
\(848\) 14.1246 0.485041
\(849\) 13.0344 0.447341
\(850\) 0 0
\(851\) −17.7984 −0.610120
\(852\) 12.9443 0.443463
\(853\) −21.3607 −0.731376 −0.365688 0.930738i \(-0.619166\pi\)
−0.365688 + 0.930738i \(0.619166\pi\)
\(854\) 3.30495 0.113093
\(855\) 0 0
\(856\) −6.38197 −0.218131
\(857\) 35.0132 1.19603 0.598013 0.801486i \(-0.295957\pi\)
0.598013 + 0.801486i \(0.295957\pi\)
\(858\) 0 0
\(859\) −5.72949 −0.195488 −0.0977438 0.995212i \(-0.531163\pi\)
−0.0977438 + 0.995212i \(0.531163\pi\)
\(860\) 0 0
\(861\) 1.93112 0.0658123
\(862\) −12.2918 −0.418660
\(863\) 6.03444 0.205415 0.102707 0.994712i \(-0.467249\pi\)
0.102707 + 0.994712i \(0.467249\pi\)
\(864\) 5.61803 0.191129
\(865\) 0 0
\(866\) −17.3262 −0.588770
\(867\) −13.0000 −0.441503
\(868\) 8.65248 0.293684
\(869\) 0 0
\(870\) 0 0
\(871\) −11.5967 −0.392941
\(872\) 9.27051 0.313939
\(873\) −11.4721 −0.388273
\(874\) 7.36068 0.248979
\(875\) 0 0
\(876\) 21.7984 0.736499
\(877\) 16.4721 0.556225 0.278112 0.960549i \(-0.410291\pi\)
0.278112 + 0.960549i \(0.410291\pi\)
\(878\) −5.97871 −0.201772
\(879\) 16.5279 0.557471
\(880\) 0 0
\(881\) −23.8541 −0.803665 −0.401833 0.915713i \(-0.631627\pi\)
−0.401833 + 0.915713i \(0.631627\pi\)
\(882\) −3.96556 −0.133527
\(883\) −18.7639 −0.631457 −0.315728 0.948850i \(-0.602249\pi\)
−0.315728 + 0.948850i \(0.602249\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) −15.6180 −0.524698
\(887\) 53.8328 1.80753 0.903765 0.428030i \(-0.140792\pi\)
0.903765 + 0.428030i \(0.140792\pi\)
\(888\) −16.7082 −0.560691
\(889\) −12.1803 −0.408515
\(890\) 0 0
\(891\) 0 0
\(892\) 13.0000 0.435272
\(893\) 25.4508 0.851680
\(894\) −12.4377 −0.415979
\(895\) 0 0
\(896\) 8.69505 0.290481
\(897\) 2.94427 0.0983064
\(898\) −0.450850 −0.0150450
\(899\) 40.9787 1.36672
\(900\) 0 0
\(901\) 15.2361 0.507587
\(902\) 0 0
\(903\) 1.59675 0.0531364
\(904\) 12.2361 0.406966
\(905\) 0 0
\(906\) 6.65248 0.221014
\(907\) −9.11146 −0.302541 −0.151270 0.988492i \(-0.548336\pi\)
−0.151270 + 0.988492i \(0.548336\pi\)
\(908\) −41.9787 −1.39311
\(909\) 0.562306 0.0186505
\(910\) 0 0
\(911\) −58.9017 −1.95150 −0.975750 0.218887i \(-0.929757\pi\)
−0.975750 + 0.218887i \(0.929757\pi\)
\(912\) −9.27051 −0.306977
\(913\) 0 0
\(914\) 0.909830 0.0300945
\(915\) 0 0
\(916\) −29.2705 −0.967125
\(917\) 10.0689 0.332504
\(918\) 1.23607 0.0407963
\(919\) 17.7639 0.585978 0.292989 0.956116i \(-0.405350\pi\)
0.292989 + 0.956116i \(0.405350\pi\)
\(920\) 0 0
\(921\) 12.1246 0.399520
\(922\) −16.0344 −0.528066
\(923\) 9.88854 0.325485
\(924\) 0 0
\(925\) 0 0
\(926\) 20.5623 0.675719
\(927\) −2.38197 −0.0782340
\(928\) 32.8885 1.07962
\(929\) −4.14590 −0.136023 −0.0680113 0.997685i \(-0.521665\pi\)
−0.0680113 + 0.997685i \(0.521665\pi\)
\(930\) 0 0
\(931\) 32.0820 1.05145
\(932\) 37.6525 1.23335
\(933\) 21.4721 0.702966
\(934\) −3.02129 −0.0988595
\(935\) 0 0
\(936\) 2.76393 0.0903419
\(937\) 41.7984 1.36549 0.682747 0.730655i \(-0.260785\pi\)
0.682747 + 0.730655i \(0.260785\pi\)
\(938\) −4.42956 −0.144630
\(939\) −0.472136 −0.0154076
\(940\) 0 0
\(941\) −3.70820 −0.120884 −0.0604420 0.998172i \(-0.519251\pi\)
−0.0604420 + 0.998172i \(0.519251\pi\)
\(942\) 2.05573 0.0669792
\(943\) 6.02129 0.196080
\(944\) −12.4377 −0.404812
\(945\) 0 0
\(946\) 0 0
\(947\) 15.5623 0.505707 0.252853 0.967505i \(-0.418631\pi\)
0.252853 + 0.967505i \(0.418631\pi\)
\(948\) −13.0902 −0.425149
\(949\) 16.6525 0.540562
\(950\) 0 0
\(951\) −30.4164 −0.986320
\(952\) 3.41641 0.110726
\(953\) 37.9098 1.22802 0.614010 0.789298i \(-0.289555\pi\)
0.614010 + 0.789298i \(0.289555\pi\)
\(954\) 4.70820 0.152434
\(955\) 0 0
\(956\) −11.3820 −0.368119
\(957\) 0 0
\(958\) −20.6525 −0.667251
\(959\) 14.2918 0.461506
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) −5.70820 −0.184040
\(963\) 2.85410 0.0919722
\(964\) −2.61803 −0.0843212
\(965\) 0 0
\(966\) 1.12461 0.0361838
\(967\) −48.3262 −1.55407 −0.777034 0.629459i \(-0.783276\pi\)
−0.777034 + 0.629459i \(0.783276\pi\)
\(968\) 0 0
\(969\) −10.0000 −0.321246
\(970\) 0 0
\(971\) 35.5410 1.14057 0.570283 0.821448i \(-0.306834\pi\)
0.570283 + 0.821448i \(0.306834\pi\)
\(972\) −1.61803 −0.0518985
\(973\) 16.1803 0.518718
\(974\) 8.76393 0.280814
\(975\) 0 0
\(976\) −12.9787 −0.415439
\(977\) 34.9098 1.11686 0.558432 0.829550i \(-0.311403\pi\)
0.558432 + 0.829550i \(0.311403\pi\)
\(978\) −8.38197 −0.268026
\(979\) 0 0
\(980\) 0 0
\(981\) −4.14590 −0.132368
\(982\) −19.6525 −0.627136
\(983\) 19.0000 0.606006 0.303003 0.952990i \(-0.402011\pi\)
0.303003 + 0.952990i \(0.402011\pi\)
\(984\) 5.65248 0.180194
\(985\) 0 0
\(986\) 7.23607 0.230443
\(987\) 3.88854 0.123774
\(988\) −10.0000 −0.318142
\(989\) 4.97871 0.158314
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 39.3262 1.24861
\(993\) 21.5967 0.685352
\(994\) 3.77709 0.119802
\(995\) 0 0
\(996\) 9.23607 0.292656
\(997\) −24.7082 −0.782517 −0.391258 0.920281i \(-0.627960\pi\)
−0.391258 + 0.920281i \(0.627960\pi\)
\(998\) −6.90983 −0.218727
\(999\) 7.47214 0.236408
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 9075.2.a.bc.1.2 2
5.4 even 2 9075.2.a.bt.1.1 2
11.2 odd 10 825.2.n.d.301.1 yes 4
11.6 odd 10 825.2.n.d.751.1 yes 4
11.10 odd 2 9075.2.a.by.1.1 2
55.2 even 20 825.2.bx.c.499.1 8
55.13 even 20 825.2.bx.c.499.2 8
55.17 even 20 825.2.bx.c.124.2 8
55.24 odd 10 825.2.n.b.301.1 4
55.28 even 20 825.2.bx.c.124.1 8
55.39 odd 10 825.2.n.b.751.1 yes 4
55.54 odd 2 9075.2.a.z.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.n.b.301.1 4 55.24 odd 10
825.2.n.b.751.1 yes 4 55.39 odd 10
825.2.n.d.301.1 yes 4 11.2 odd 10
825.2.n.d.751.1 yes 4 11.6 odd 10
825.2.bx.c.124.1 8 55.28 even 20
825.2.bx.c.124.2 8 55.17 even 20
825.2.bx.c.499.1 8 55.2 even 20
825.2.bx.c.499.2 8 55.13 even 20
9075.2.a.z.1.2 2 55.54 odd 2
9075.2.a.bc.1.2 2 1.1 even 1 trivial
9075.2.a.bt.1.1 2 5.4 even 2
9075.2.a.by.1.1 2 11.10 odd 2