Properties

Label 5265.2.a.ba.1.2
Level $5265$
Weight $2$
Character 5265.1
Self dual yes
Analytic conductor $42.041$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5265.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(42.0412366642\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 3x^{7} - 8x^{6} + 31x^{5} - x^{4} - 70x^{3} + 66x^{2} - 19x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 585)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.25662\) of defining polynomial
Character \(\chi\) \(=\) 5265.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.25662 q^{2} +3.09235 q^{4} +1.00000 q^{5} +0.706571 q^{7} -2.46502 q^{8} +O(q^{10})\) \(q-2.25662 q^{2} +3.09235 q^{4} +1.00000 q^{5} +0.706571 q^{7} -2.46502 q^{8} -2.25662 q^{10} +2.33021 q^{11} -1.00000 q^{13} -1.59446 q^{14} -0.622073 q^{16} -6.04736 q^{17} -7.91884 q^{19} +3.09235 q^{20} -5.25841 q^{22} +8.95599 q^{23} +1.00000 q^{25} +2.25662 q^{26} +2.18496 q^{28} +2.40680 q^{29} -3.06432 q^{31} +6.33383 q^{32} +13.6466 q^{34} +0.706571 q^{35} -2.38079 q^{37} +17.8698 q^{38} -2.46502 q^{40} +9.45889 q^{41} +11.9359 q^{43} +7.20582 q^{44} -20.2103 q^{46} -9.72867 q^{47} -6.50076 q^{49} -2.25662 q^{50} -3.09235 q^{52} -5.90968 q^{53} +2.33021 q^{55} -1.74171 q^{56} -5.43123 q^{58} -2.97437 q^{59} -9.24156 q^{61} +6.91501 q^{62} -13.0489 q^{64} -1.00000 q^{65} -1.69448 q^{67} -18.7006 q^{68} -1.59446 q^{70} +12.4439 q^{71} +4.88899 q^{73} +5.37255 q^{74} -24.4878 q^{76} +1.64646 q^{77} -11.0254 q^{79} -0.622073 q^{80} -21.3452 q^{82} +0.521691 q^{83} -6.04736 q^{85} -26.9348 q^{86} -5.74402 q^{88} -10.7608 q^{89} -0.706571 q^{91} +27.6950 q^{92} +21.9539 q^{94} -7.91884 q^{95} -2.96306 q^{97} +14.6698 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3 q^{2} + 9 q^{4} + 8 q^{5} - 11 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 3 q^{2} + 9 q^{4} + 8 q^{5} - 11 q^{7} + 6 q^{8} - 3 q^{10} - 6 q^{11} - 8 q^{13} - 10 q^{14} + 11 q^{16} + 2 q^{17} - 10 q^{19} + 9 q^{20} + 3 q^{22} - 6 q^{23} + 8 q^{25} + 3 q^{26} - 34 q^{28} - 14 q^{29} - 31 q^{31} - q^{32} - 7 q^{34} - 11 q^{35} + q^{37} - 9 q^{38} + 6 q^{40} + 12 q^{41} + 15 q^{43} - 16 q^{44} - 32 q^{46} + 18 q^{47} + 17 q^{49} - 3 q^{50} - 9 q^{52} - 2 q^{53} - 6 q^{55} - 16 q^{56} - 42 q^{58} - 24 q^{59} - 9 q^{61} + 20 q^{62} - 30 q^{64} - 8 q^{65} - 18 q^{67} + 14 q^{68} - 10 q^{70} - 10 q^{71} + 6 q^{73} + 37 q^{74} - 53 q^{76} + 34 q^{77} - 3 q^{79} + 11 q^{80} - 34 q^{82} + 10 q^{83} + 2 q^{85} - 60 q^{86} - 14 q^{88} + 13 q^{89} + 11 q^{91} - 5 q^{92} + 17 q^{94} - 10 q^{95} - 34 q^{97} + 30 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.25662 −1.59567 −0.797837 0.602873i \(-0.794022\pi\)
−0.797837 + 0.602873i \(0.794022\pi\)
\(3\) 0 0
\(4\) 3.09235 1.54617
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.706571 0.267059 0.133529 0.991045i \(-0.457369\pi\)
0.133529 + 0.991045i \(0.457369\pi\)
\(8\) −2.46502 −0.871517
\(9\) 0 0
\(10\) −2.25662 −0.713607
\(11\) 2.33021 0.702585 0.351292 0.936266i \(-0.385742\pi\)
0.351292 + 0.936266i \(0.385742\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) −1.59446 −0.426138
\(15\) 0 0
\(16\) −0.622073 −0.155518
\(17\) −6.04736 −1.46670 −0.733351 0.679851i \(-0.762045\pi\)
−0.733351 + 0.679851i \(0.762045\pi\)
\(18\) 0 0
\(19\) −7.91884 −1.81671 −0.908353 0.418204i \(-0.862660\pi\)
−0.908353 + 0.418204i \(0.862660\pi\)
\(20\) 3.09235 0.691470
\(21\) 0 0
\(22\) −5.25841 −1.12110
\(23\) 8.95599 1.86745 0.933726 0.357988i \(-0.116537\pi\)
0.933726 + 0.357988i \(0.116537\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.25662 0.442560
\(27\) 0 0
\(28\) 2.18496 0.412919
\(29\) 2.40680 0.446931 0.223465 0.974712i \(-0.428263\pi\)
0.223465 + 0.974712i \(0.428263\pi\)
\(30\) 0 0
\(31\) −3.06432 −0.550367 −0.275184 0.961392i \(-0.588739\pi\)
−0.275184 + 0.961392i \(0.588739\pi\)
\(32\) 6.33383 1.11967
\(33\) 0 0
\(34\) 13.6466 2.34038
\(35\) 0.706571 0.119432
\(36\) 0 0
\(37\) −2.38079 −0.391400 −0.195700 0.980664i \(-0.562698\pi\)
−0.195700 + 0.980664i \(0.562698\pi\)
\(38\) 17.8698 2.89887
\(39\) 0 0
\(40\) −2.46502 −0.389754
\(41\) 9.45889 1.47723 0.738616 0.674127i \(-0.235480\pi\)
0.738616 + 0.674127i \(0.235480\pi\)
\(42\) 0 0
\(43\) 11.9359 1.82021 0.910103 0.414381i \(-0.136002\pi\)
0.910103 + 0.414381i \(0.136002\pi\)
\(44\) 7.20582 1.08632
\(45\) 0 0
\(46\) −20.2103 −2.97985
\(47\) −9.72867 −1.41907 −0.709536 0.704669i \(-0.751095\pi\)
−0.709536 + 0.704669i \(0.751095\pi\)
\(48\) 0 0
\(49\) −6.50076 −0.928680
\(50\) −2.25662 −0.319135
\(51\) 0 0
\(52\) −3.09235 −0.428832
\(53\) −5.90968 −0.811756 −0.405878 0.913927i \(-0.633034\pi\)
−0.405878 + 0.913927i \(0.633034\pi\)
\(54\) 0 0
\(55\) 2.33021 0.314205
\(56\) −1.74171 −0.232746
\(57\) 0 0
\(58\) −5.43123 −0.713156
\(59\) −2.97437 −0.387230 −0.193615 0.981078i \(-0.562021\pi\)
−0.193615 + 0.981078i \(0.562021\pi\)
\(60\) 0 0
\(61\) −9.24156 −1.18326 −0.591630 0.806209i \(-0.701516\pi\)
−0.591630 + 0.806209i \(0.701516\pi\)
\(62\) 6.91501 0.878207
\(63\) 0 0
\(64\) −13.0489 −1.63111
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −1.69448 −0.207014 −0.103507 0.994629i \(-0.533006\pi\)
−0.103507 + 0.994629i \(0.533006\pi\)
\(68\) −18.7006 −2.26778
\(69\) 0 0
\(70\) −1.59446 −0.190575
\(71\) 12.4439 1.47682 0.738410 0.674352i \(-0.235577\pi\)
0.738410 + 0.674352i \(0.235577\pi\)
\(72\) 0 0
\(73\) 4.88899 0.572213 0.286106 0.958198i \(-0.407639\pi\)
0.286106 + 0.958198i \(0.407639\pi\)
\(74\) 5.37255 0.624546
\(75\) 0 0
\(76\) −24.4878 −2.80895
\(77\) 1.64646 0.187631
\(78\) 0 0
\(79\) −11.0254 −1.24045 −0.620227 0.784423i \(-0.712959\pi\)
−0.620227 + 0.784423i \(0.712959\pi\)
\(80\) −0.622073 −0.0695499
\(81\) 0 0
\(82\) −21.3452 −2.35718
\(83\) 0.521691 0.0572630 0.0286315 0.999590i \(-0.490885\pi\)
0.0286315 + 0.999590i \(0.490885\pi\)
\(84\) 0 0
\(85\) −6.04736 −0.655929
\(86\) −26.9348 −2.90446
\(87\) 0 0
\(88\) −5.74402 −0.612314
\(89\) −10.7608 −1.14065 −0.570323 0.821420i \(-0.693182\pi\)
−0.570323 + 0.821420i \(0.693182\pi\)
\(90\) 0 0
\(91\) −0.706571 −0.0740687
\(92\) 27.6950 2.88741
\(93\) 0 0
\(94\) 21.9539 2.26438
\(95\) −7.91884 −0.812456
\(96\) 0 0
\(97\) −2.96306 −0.300853 −0.150427 0.988621i \(-0.548065\pi\)
−0.150427 + 0.988621i \(0.548065\pi\)
\(98\) 14.6698 1.48187
\(99\) 0 0
\(100\) 3.09235 0.309235
\(101\) −13.6715 −1.36036 −0.680181 0.733044i \(-0.738099\pi\)
−0.680181 + 0.733044i \(0.738099\pi\)
\(102\) 0 0
\(103\) −7.13071 −0.702610 −0.351305 0.936261i \(-0.614262\pi\)
−0.351305 + 0.936261i \(0.614262\pi\)
\(104\) 2.46502 0.241715
\(105\) 0 0
\(106\) 13.3359 1.29530
\(107\) 1.70776 0.165095 0.0825477 0.996587i \(-0.473694\pi\)
0.0825477 + 0.996587i \(0.473694\pi\)
\(108\) 0 0
\(109\) −3.23628 −0.309979 −0.154989 0.987916i \(-0.549534\pi\)
−0.154989 + 0.987916i \(0.549534\pi\)
\(110\) −5.25841 −0.501369
\(111\) 0 0
\(112\) −0.439539 −0.0415325
\(113\) 0.678440 0.0638223 0.0319112 0.999491i \(-0.489841\pi\)
0.0319112 + 0.999491i \(0.489841\pi\)
\(114\) 0 0
\(115\) 8.95599 0.835150
\(116\) 7.44266 0.691033
\(117\) 0 0
\(118\) 6.71204 0.617893
\(119\) −4.27289 −0.391695
\(120\) 0 0
\(121\) −5.57012 −0.506375
\(122\) 20.8547 1.88810
\(123\) 0 0
\(124\) −9.47594 −0.850964
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 14.6889 1.30343 0.651714 0.758465i \(-0.274050\pi\)
0.651714 + 0.758465i \(0.274050\pi\)
\(128\) 16.7788 1.48305
\(129\) 0 0
\(130\) 2.25662 0.197919
\(131\) 9.06388 0.791915 0.395957 0.918269i \(-0.370413\pi\)
0.395957 + 0.918269i \(0.370413\pi\)
\(132\) 0 0
\(133\) −5.59522 −0.485167
\(134\) 3.82381 0.330327
\(135\) 0 0
\(136\) 14.9069 1.27825
\(137\) 2.02984 0.173421 0.0867104 0.996234i \(-0.472365\pi\)
0.0867104 + 0.996234i \(0.472365\pi\)
\(138\) 0 0
\(139\) −2.85318 −0.242003 −0.121002 0.992652i \(-0.538611\pi\)
−0.121002 + 0.992652i \(0.538611\pi\)
\(140\) 2.18496 0.184663
\(141\) 0 0
\(142\) −28.0812 −2.35652
\(143\) −2.33021 −0.194862
\(144\) 0 0
\(145\) 2.40680 0.199874
\(146\) −11.0326 −0.913065
\(147\) 0 0
\(148\) −7.36224 −0.605173
\(149\) −8.98225 −0.735854 −0.367927 0.929855i \(-0.619932\pi\)
−0.367927 + 0.929855i \(0.619932\pi\)
\(150\) 0 0
\(151\) −2.75483 −0.224185 −0.112092 0.993698i \(-0.535755\pi\)
−0.112092 + 0.993698i \(0.535755\pi\)
\(152\) 19.5201 1.58329
\(153\) 0 0
\(154\) −3.71544 −0.299398
\(155\) −3.06432 −0.246132
\(156\) 0 0
\(157\) 24.4986 1.95520 0.977601 0.210468i \(-0.0674989\pi\)
0.977601 + 0.210468i \(0.0674989\pi\)
\(158\) 24.8802 1.97936
\(159\) 0 0
\(160\) 6.33383 0.500733
\(161\) 6.32804 0.498719
\(162\) 0 0
\(163\) −18.2489 −1.42937 −0.714684 0.699448i \(-0.753429\pi\)
−0.714684 + 0.699448i \(0.753429\pi\)
\(164\) 29.2502 2.28406
\(165\) 0 0
\(166\) −1.17726 −0.0913730
\(167\) −5.60131 −0.433442 −0.216721 0.976234i \(-0.569536\pi\)
−0.216721 + 0.976234i \(0.569536\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 13.6466 1.04665
\(171\) 0 0
\(172\) 36.9100 2.81436
\(173\) 3.60260 0.273900 0.136950 0.990578i \(-0.456270\pi\)
0.136950 + 0.990578i \(0.456270\pi\)
\(174\) 0 0
\(175\) 0.706571 0.0534117
\(176\) −1.44956 −0.109265
\(177\) 0 0
\(178\) 24.2832 1.82010
\(179\) −15.3123 −1.14449 −0.572246 0.820082i \(-0.693928\pi\)
−0.572246 + 0.820082i \(0.693928\pi\)
\(180\) 0 0
\(181\) −7.23734 −0.537947 −0.268974 0.963148i \(-0.586684\pi\)
−0.268974 + 0.963148i \(0.586684\pi\)
\(182\) 1.59446 0.118190
\(183\) 0 0
\(184\) −22.0767 −1.62752
\(185\) −2.38079 −0.175039
\(186\) 0 0
\(187\) −14.0916 −1.03048
\(188\) −30.0844 −2.19413
\(189\) 0 0
\(190\) 17.8698 1.29641
\(191\) 19.3679 1.40141 0.700705 0.713452i \(-0.252869\pi\)
0.700705 + 0.713452i \(0.252869\pi\)
\(192\) 0 0
\(193\) 5.05546 0.363900 0.181950 0.983308i \(-0.441759\pi\)
0.181950 + 0.983308i \(0.441759\pi\)
\(194\) 6.68651 0.480064
\(195\) 0 0
\(196\) −20.1026 −1.43590
\(197\) −6.35430 −0.452725 −0.226363 0.974043i \(-0.572683\pi\)
−0.226363 + 0.974043i \(0.572683\pi\)
\(198\) 0 0
\(199\) −16.7816 −1.18961 −0.594807 0.803868i \(-0.702772\pi\)
−0.594807 + 0.803868i \(0.702772\pi\)
\(200\) −2.46502 −0.174303
\(201\) 0 0
\(202\) 30.8514 2.17069
\(203\) 1.70057 0.119357
\(204\) 0 0
\(205\) 9.45889 0.660638
\(206\) 16.0913 1.12114
\(207\) 0 0
\(208\) 0.622073 0.0431330
\(209\) −18.4526 −1.27639
\(210\) 0 0
\(211\) −11.3631 −0.782268 −0.391134 0.920334i \(-0.627917\pi\)
−0.391134 + 0.920334i \(0.627917\pi\)
\(212\) −18.2748 −1.25512
\(213\) 0 0
\(214\) −3.85377 −0.263439
\(215\) 11.9359 0.814021
\(216\) 0 0
\(217\) −2.16516 −0.146980
\(218\) 7.30305 0.494625
\(219\) 0 0
\(220\) 7.20582 0.485817
\(221\) 6.04736 0.406790
\(222\) 0 0
\(223\) −20.0857 −1.34504 −0.672518 0.740080i \(-0.734787\pi\)
−0.672518 + 0.740080i \(0.734787\pi\)
\(224\) 4.47530 0.299018
\(225\) 0 0
\(226\) −1.53098 −0.101840
\(227\) 11.3482 0.753210 0.376605 0.926374i \(-0.377091\pi\)
0.376605 + 0.926374i \(0.377091\pi\)
\(228\) 0 0
\(229\) −3.68135 −0.243271 −0.121635 0.992575i \(-0.538814\pi\)
−0.121635 + 0.992575i \(0.538814\pi\)
\(230\) −20.2103 −1.33263
\(231\) 0 0
\(232\) −5.93281 −0.389508
\(233\) −12.5286 −0.820778 −0.410389 0.911911i \(-0.634607\pi\)
−0.410389 + 0.911911i \(0.634607\pi\)
\(234\) 0 0
\(235\) −9.72867 −0.634628
\(236\) −9.19779 −0.598725
\(237\) 0 0
\(238\) 9.64230 0.625018
\(239\) −3.32887 −0.215327 −0.107663 0.994187i \(-0.534337\pi\)
−0.107663 + 0.994187i \(0.534337\pi\)
\(240\) 0 0
\(241\) −26.7838 −1.72530 −0.862649 0.505804i \(-0.831196\pi\)
−0.862649 + 0.505804i \(0.831196\pi\)
\(242\) 12.5697 0.808009
\(243\) 0 0
\(244\) −28.5781 −1.82953
\(245\) −6.50076 −0.415318
\(246\) 0 0
\(247\) 7.91884 0.503864
\(248\) 7.55361 0.479654
\(249\) 0 0
\(250\) −2.25662 −0.142721
\(251\) 6.05718 0.382326 0.191163 0.981558i \(-0.438774\pi\)
0.191163 + 0.981558i \(0.438774\pi\)
\(252\) 0 0
\(253\) 20.8693 1.31204
\(254\) −33.1473 −2.07985
\(255\) 0 0
\(256\) −11.7657 −0.735356
\(257\) 3.95270 0.246563 0.123281 0.992372i \(-0.460658\pi\)
0.123281 + 0.992372i \(0.460658\pi\)
\(258\) 0 0
\(259\) −1.68220 −0.104527
\(260\) −3.09235 −0.191779
\(261\) 0 0
\(262\) −20.4538 −1.26364
\(263\) −11.3011 −0.696859 −0.348429 0.937335i \(-0.613285\pi\)
−0.348429 + 0.937335i \(0.613285\pi\)
\(264\) 0 0
\(265\) −5.90968 −0.363029
\(266\) 12.6263 0.774169
\(267\) 0 0
\(268\) −5.23993 −0.320080
\(269\) 20.7773 1.26681 0.633407 0.773819i \(-0.281656\pi\)
0.633407 + 0.773819i \(0.281656\pi\)
\(270\) 0 0
\(271\) −1.72567 −0.104827 −0.0524136 0.998625i \(-0.516691\pi\)
−0.0524136 + 0.998625i \(0.516691\pi\)
\(272\) 3.76190 0.228099
\(273\) 0 0
\(274\) −4.58058 −0.276723
\(275\) 2.33021 0.140517
\(276\) 0 0
\(277\) −3.46722 −0.208325 −0.104163 0.994560i \(-0.533216\pi\)
−0.104163 + 0.994560i \(0.533216\pi\)
\(278\) 6.43855 0.386158
\(279\) 0 0
\(280\) −1.74171 −0.104087
\(281\) −1.17263 −0.0699534 −0.0349767 0.999388i \(-0.511136\pi\)
−0.0349767 + 0.999388i \(0.511136\pi\)
\(282\) 0 0
\(283\) 21.8122 1.29660 0.648301 0.761384i \(-0.275480\pi\)
0.648301 + 0.761384i \(0.275480\pi\)
\(284\) 38.4809 2.28342
\(285\) 0 0
\(286\) 5.25841 0.310936
\(287\) 6.68338 0.394507
\(288\) 0 0
\(289\) 19.5706 1.15121
\(290\) −5.43123 −0.318933
\(291\) 0 0
\(292\) 15.1185 0.884741
\(293\) 27.8919 1.62946 0.814732 0.579838i \(-0.196884\pi\)
0.814732 + 0.579838i \(0.196884\pi\)
\(294\) 0 0
\(295\) −2.97437 −0.173175
\(296\) 5.86870 0.341112
\(297\) 0 0
\(298\) 20.2695 1.17418
\(299\) −8.95599 −0.517938
\(300\) 0 0
\(301\) 8.43355 0.486102
\(302\) 6.21661 0.357726
\(303\) 0 0
\(304\) 4.92610 0.282531
\(305\) −9.24156 −0.529170
\(306\) 0 0
\(307\) −0.574330 −0.0327787 −0.0163894 0.999866i \(-0.505217\pi\)
−0.0163894 + 0.999866i \(0.505217\pi\)
\(308\) 5.09142 0.290111
\(309\) 0 0
\(310\) 6.91501 0.392746
\(311\) −23.1071 −1.31029 −0.655143 0.755505i \(-0.727392\pi\)
−0.655143 + 0.755505i \(0.727392\pi\)
\(312\) 0 0
\(313\) 25.4854 1.44052 0.720260 0.693704i \(-0.244022\pi\)
0.720260 + 0.693704i \(0.244022\pi\)
\(314\) −55.2841 −3.11986
\(315\) 0 0
\(316\) −34.0944 −1.91796
\(317\) −32.3557 −1.81728 −0.908638 0.417585i \(-0.862877\pi\)
−0.908638 + 0.417585i \(0.862877\pi\)
\(318\) 0 0
\(319\) 5.60834 0.314007
\(320\) −13.0489 −0.729457
\(321\) 0 0
\(322\) −14.2800 −0.795793
\(323\) 47.8881 2.66457
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 41.1810 2.28080
\(327\) 0 0
\(328\) −23.3164 −1.28743
\(329\) −6.87399 −0.378976
\(330\) 0 0
\(331\) 30.1696 1.65827 0.829135 0.559048i \(-0.188833\pi\)
0.829135 + 0.559048i \(0.188833\pi\)
\(332\) 1.61325 0.0885386
\(333\) 0 0
\(334\) 12.6400 0.691633
\(335\) −1.69448 −0.0925795
\(336\) 0 0
\(337\) 6.57838 0.358347 0.179174 0.983817i \(-0.442658\pi\)
0.179174 + 0.983817i \(0.442658\pi\)
\(338\) −2.25662 −0.122744
\(339\) 0 0
\(340\) −18.7006 −1.01418
\(341\) −7.14050 −0.386680
\(342\) 0 0
\(343\) −9.53924 −0.515071
\(344\) −29.4222 −1.58634
\(345\) 0 0
\(346\) −8.12970 −0.437055
\(347\) 2.01063 0.107936 0.0539682 0.998543i \(-0.482813\pi\)
0.0539682 + 0.998543i \(0.482813\pi\)
\(348\) 0 0
\(349\) 14.0039 0.749610 0.374805 0.927104i \(-0.377710\pi\)
0.374805 + 0.927104i \(0.377710\pi\)
\(350\) −1.59446 −0.0852277
\(351\) 0 0
\(352\) 14.7591 0.786665
\(353\) −34.5998 −1.84156 −0.920780 0.390081i \(-0.872447\pi\)
−0.920780 + 0.390081i \(0.872447\pi\)
\(354\) 0 0
\(355\) 12.4439 0.660454
\(356\) −33.2763 −1.76364
\(357\) 0 0
\(358\) 34.5540 1.82624
\(359\) −22.7679 −1.20164 −0.600822 0.799382i \(-0.705160\pi\)
−0.600822 + 0.799382i \(0.705160\pi\)
\(360\) 0 0
\(361\) 43.7080 2.30042
\(362\) 16.3319 0.858388
\(363\) 0 0
\(364\) −2.18496 −0.114523
\(365\) 4.88899 0.255901
\(366\) 0 0
\(367\) −11.3939 −0.594758 −0.297379 0.954760i \(-0.596112\pi\)
−0.297379 + 0.954760i \(0.596112\pi\)
\(368\) −5.57128 −0.290423
\(369\) 0 0
\(370\) 5.37255 0.279306
\(371\) −4.17560 −0.216787
\(372\) 0 0
\(373\) 2.59519 0.134374 0.0671870 0.997740i \(-0.478598\pi\)
0.0671870 + 0.997740i \(0.478598\pi\)
\(374\) 31.7995 1.64431
\(375\) 0 0
\(376\) 23.9814 1.23675
\(377\) −2.40680 −0.123956
\(378\) 0 0
\(379\) 2.38710 0.122617 0.0613084 0.998119i \(-0.480473\pi\)
0.0613084 + 0.998119i \(0.480473\pi\)
\(380\) −24.4878 −1.25620
\(381\) 0 0
\(382\) −43.7060 −2.23619
\(383\) 32.2726 1.64905 0.824525 0.565825i \(-0.191442\pi\)
0.824525 + 0.565825i \(0.191442\pi\)
\(384\) 0 0
\(385\) 1.64646 0.0839113
\(386\) −11.4083 −0.580666
\(387\) 0 0
\(388\) −9.16282 −0.465172
\(389\) −7.50105 −0.380318 −0.190159 0.981753i \(-0.560900\pi\)
−0.190159 + 0.981753i \(0.560900\pi\)
\(390\) 0 0
\(391\) −54.1601 −2.73899
\(392\) 16.0245 0.809360
\(393\) 0 0
\(394\) 14.3393 0.722402
\(395\) −11.0254 −0.554748
\(396\) 0 0
\(397\) −22.5214 −1.13032 −0.565158 0.824983i \(-0.691185\pi\)
−0.565158 + 0.824983i \(0.691185\pi\)
\(398\) 37.8697 1.89824
\(399\) 0 0
\(400\) −0.622073 −0.0311036
\(401\) −2.53261 −0.126472 −0.0632362 0.997999i \(-0.520142\pi\)
−0.0632362 + 0.997999i \(0.520142\pi\)
\(402\) 0 0
\(403\) 3.06432 0.152644
\(404\) −42.2770 −2.10336
\(405\) 0 0
\(406\) −3.83755 −0.190454
\(407\) −5.54775 −0.274992
\(408\) 0 0
\(409\) −6.37084 −0.315018 −0.157509 0.987518i \(-0.550346\pi\)
−0.157509 + 0.987518i \(0.550346\pi\)
\(410\) −21.3452 −1.05416
\(411\) 0 0
\(412\) −22.0506 −1.08636
\(413\) −2.10160 −0.103413
\(414\) 0 0
\(415\) 0.521691 0.0256088
\(416\) −6.33383 −0.310541
\(417\) 0 0
\(418\) 41.6405 2.03670
\(419\) −30.9358 −1.51131 −0.755657 0.654968i \(-0.772682\pi\)
−0.755657 + 0.654968i \(0.772682\pi\)
\(420\) 0 0
\(421\) −13.9644 −0.680584 −0.340292 0.940320i \(-0.610526\pi\)
−0.340292 + 0.940320i \(0.610526\pi\)
\(422\) 25.6422 1.24824
\(423\) 0 0
\(424\) 14.5675 0.707459
\(425\) −6.04736 −0.293340
\(426\) 0 0
\(427\) −6.52982 −0.316000
\(428\) 5.28099 0.255266
\(429\) 0 0
\(430\) −26.9348 −1.29891
\(431\) −20.2027 −0.973131 −0.486566 0.873644i \(-0.661751\pi\)
−0.486566 + 0.873644i \(0.661751\pi\)
\(432\) 0 0
\(433\) −13.7657 −0.661537 −0.330768 0.943712i \(-0.607308\pi\)
−0.330768 + 0.943712i \(0.607308\pi\)
\(434\) 4.88594 0.234533
\(435\) 0 0
\(436\) −10.0077 −0.479282
\(437\) −70.9210 −3.39261
\(438\) 0 0
\(439\) 7.76756 0.370725 0.185363 0.982670i \(-0.440654\pi\)
0.185363 + 0.982670i \(0.440654\pi\)
\(440\) −5.74402 −0.273835
\(441\) 0 0
\(442\) −13.6466 −0.649104
\(443\) −16.4315 −0.780682 −0.390341 0.920670i \(-0.627643\pi\)
−0.390341 + 0.920670i \(0.627643\pi\)
\(444\) 0 0
\(445\) −10.7608 −0.510113
\(446\) 45.3258 2.14624
\(447\) 0 0
\(448\) −9.21998 −0.435603
\(449\) −9.97715 −0.470851 −0.235425 0.971892i \(-0.575648\pi\)
−0.235425 + 0.971892i \(0.575648\pi\)
\(450\) 0 0
\(451\) 22.0412 1.03788
\(452\) 2.09797 0.0986804
\(453\) 0 0
\(454\) −25.6087 −1.20188
\(455\) −0.706571 −0.0331245
\(456\) 0 0
\(457\) −17.3560 −0.811879 −0.405939 0.913900i \(-0.633056\pi\)
−0.405939 + 0.913900i \(0.633056\pi\)
\(458\) 8.30743 0.388181
\(459\) 0 0
\(460\) 27.6950 1.29129
\(461\) 10.5929 0.493361 0.246681 0.969097i \(-0.420660\pi\)
0.246681 + 0.969097i \(0.420660\pi\)
\(462\) 0 0
\(463\) −3.60744 −0.167652 −0.0838260 0.996480i \(-0.526714\pi\)
−0.0838260 + 0.996480i \(0.526714\pi\)
\(464\) −1.49720 −0.0695059
\(465\) 0 0
\(466\) 28.2724 1.30969
\(467\) −6.96611 −0.322353 −0.161177 0.986926i \(-0.551529\pi\)
−0.161177 + 0.986926i \(0.551529\pi\)
\(468\) 0 0
\(469\) −1.19727 −0.0552849
\(470\) 21.9539 1.01266
\(471\) 0 0
\(472\) 7.33189 0.337478
\(473\) 27.8131 1.27885
\(474\) 0 0
\(475\) −7.91884 −0.363341
\(476\) −13.2133 −0.605629
\(477\) 0 0
\(478\) 7.51200 0.343591
\(479\) 22.0606 1.00797 0.503987 0.863711i \(-0.331866\pi\)
0.503987 + 0.863711i \(0.331866\pi\)
\(480\) 0 0
\(481\) 2.38079 0.108555
\(482\) 60.4410 2.75301
\(483\) 0 0
\(484\) −17.2248 −0.782944
\(485\) −2.96306 −0.134546
\(486\) 0 0
\(487\) −5.19759 −0.235525 −0.117763 0.993042i \(-0.537572\pi\)
−0.117763 + 0.993042i \(0.537572\pi\)
\(488\) 22.7807 1.03123
\(489\) 0 0
\(490\) 14.6698 0.662712
\(491\) 2.23657 0.100935 0.0504675 0.998726i \(-0.483929\pi\)
0.0504675 + 0.998726i \(0.483929\pi\)
\(492\) 0 0
\(493\) −14.5548 −0.655514
\(494\) −17.8698 −0.804002
\(495\) 0 0
\(496\) 1.90623 0.0855922
\(497\) 8.79250 0.394398
\(498\) 0 0
\(499\) −21.2893 −0.953038 −0.476519 0.879164i \(-0.658102\pi\)
−0.476519 + 0.879164i \(0.658102\pi\)
\(500\) 3.09235 0.138294
\(501\) 0 0
\(502\) −13.6688 −0.610068
\(503\) 33.5371 1.49534 0.747672 0.664068i \(-0.231171\pi\)
0.747672 + 0.664068i \(0.231171\pi\)
\(504\) 0 0
\(505\) −13.6715 −0.608372
\(506\) −47.0942 −2.09359
\(507\) 0 0
\(508\) 45.4232 2.01533
\(509\) 3.07482 0.136289 0.0681444 0.997675i \(-0.478292\pi\)
0.0681444 + 0.997675i \(0.478292\pi\)
\(510\) 0 0
\(511\) 3.45442 0.152814
\(512\) −7.00695 −0.309666
\(513\) 0 0
\(514\) −8.91975 −0.393433
\(515\) −7.13071 −0.314217
\(516\) 0 0
\(517\) −22.6698 −0.997018
\(518\) 3.79609 0.166791
\(519\) 0 0
\(520\) 2.46502 0.108098
\(521\) −32.8855 −1.44074 −0.720371 0.693589i \(-0.756028\pi\)
−0.720371 + 0.693589i \(0.756028\pi\)
\(522\) 0 0
\(523\) 19.0090 0.831206 0.415603 0.909546i \(-0.363571\pi\)
0.415603 + 0.909546i \(0.363571\pi\)
\(524\) 28.0287 1.22444
\(525\) 0 0
\(526\) 25.5024 1.11196
\(527\) 18.5310 0.807224
\(528\) 0 0
\(529\) 57.2097 2.48738
\(530\) 13.3359 0.579275
\(531\) 0 0
\(532\) −17.3024 −0.750153
\(533\) −9.45889 −0.409710
\(534\) 0 0
\(535\) 1.70776 0.0738330
\(536\) 4.17694 0.180416
\(537\) 0 0
\(538\) −46.8865 −2.02142
\(539\) −15.1481 −0.652476
\(540\) 0 0
\(541\) −13.4399 −0.577825 −0.288912 0.957356i \(-0.593294\pi\)
−0.288912 + 0.957356i \(0.593294\pi\)
\(542\) 3.89420 0.167270
\(543\) 0 0
\(544\) −38.3030 −1.64223
\(545\) −3.23628 −0.138627
\(546\) 0 0
\(547\) 11.3254 0.484239 0.242119 0.970246i \(-0.422157\pi\)
0.242119 + 0.970246i \(0.422157\pi\)
\(548\) 6.27697 0.268139
\(549\) 0 0
\(550\) −5.25841 −0.224219
\(551\) −19.0590 −0.811943
\(552\) 0 0
\(553\) −7.79022 −0.331274
\(554\) 7.82421 0.332419
\(555\) 0 0
\(556\) −8.82302 −0.374179
\(557\) −4.71355 −0.199720 −0.0998598 0.995002i \(-0.531839\pi\)
−0.0998598 + 0.995002i \(0.531839\pi\)
\(558\) 0 0
\(559\) −11.9359 −0.504835
\(560\) −0.439539 −0.0185739
\(561\) 0 0
\(562\) 2.64619 0.111623
\(563\) −35.3159 −1.48839 −0.744193 0.667964i \(-0.767166\pi\)
−0.744193 + 0.667964i \(0.767166\pi\)
\(564\) 0 0
\(565\) 0.678440 0.0285422
\(566\) −49.2220 −2.06895
\(567\) 0 0
\(568\) −30.6745 −1.28707
\(569\) 26.5359 1.11244 0.556222 0.831034i \(-0.312251\pi\)
0.556222 + 0.831034i \(0.312251\pi\)
\(570\) 0 0
\(571\) −33.7542 −1.41257 −0.706284 0.707928i \(-0.749630\pi\)
−0.706284 + 0.707928i \(0.749630\pi\)
\(572\) −7.20582 −0.301291
\(573\) 0 0
\(574\) −15.0819 −0.629505
\(575\) 8.95599 0.373491
\(576\) 0 0
\(577\) −30.7531 −1.28027 −0.640133 0.768264i \(-0.721121\pi\)
−0.640133 + 0.768264i \(0.721121\pi\)
\(578\) −44.1635 −1.83696
\(579\) 0 0
\(580\) 7.44266 0.309040
\(581\) 0.368611 0.0152926
\(582\) 0 0
\(583\) −13.7708 −0.570328
\(584\) −12.0515 −0.498693
\(585\) 0 0
\(586\) −62.9416 −2.60009
\(587\) −5.23724 −0.216164 −0.108082 0.994142i \(-0.534471\pi\)
−0.108082 + 0.994142i \(0.534471\pi\)
\(588\) 0 0
\(589\) 24.2658 0.999856
\(590\) 6.71204 0.276330
\(591\) 0 0
\(592\) 1.48103 0.0608698
\(593\) −15.8490 −0.650839 −0.325420 0.945570i \(-0.605506\pi\)
−0.325420 + 0.945570i \(0.605506\pi\)
\(594\) 0 0
\(595\) −4.27289 −0.175171
\(596\) −27.7762 −1.13776
\(597\) 0 0
\(598\) 20.2103 0.826460
\(599\) −26.3235 −1.07555 −0.537775 0.843088i \(-0.680735\pi\)
−0.537775 + 0.843088i \(0.680735\pi\)
\(600\) 0 0
\(601\) 1.80372 0.0735754 0.0367877 0.999323i \(-0.488287\pi\)
0.0367877 + 0.999323i \(0.488287\pi\)
\(602\) −19.0314 −0.775660
\(603\) 0 0
\(604\) −8.51889 −0.346629
\(605\) −5.57012 −0.226458
\(606\) 0 0
\(607\) 12.5387 0.508929 0.254464 0.967082i \(-0.418101\pi\)
0.254464 + 0.967082i \(0.418101\pi\)
\(608\) −50.1566 −2.03412
\(609\) 0 0
\(610\) 20.8547 0.844383
\(611\) 9.72867 0.393580
\(612\) 0 0
\(613\) 9.01785 0.364228 0.182114 0.983277i \(-0.441706\pi\)
0.182114 + 0.983277i \(0.441706\pi\)
\(614\) 1.29605 0.0523042
\(615\) 0 0
\(616\) −4.05856 −0.163524
\(617\) −14.6819 −0.591071 −0.295536 0.955332i \(-0.595498\pi\)
−0.295536 + 0.955332i \(0.595498\pi\)
\(618\) 0 0
\(619\) −41.6709 −1.67490 −0.837448 0.546517i \(-0.815953\pi\)
−0.837448 + 0.546517i \(0.815953\pi\)
\(620\) −9.47594 −0.380563
\(621\) 0 0
\(622\) 52.1441 2.09079
\(623\) −7.60330 −0.304620
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −57.5110 −2.29860
\(627\) 0 0
\(628\) 75.7582 3.02308
\(629\) 14.3975 0.574067
\(630\) 0 0
\(631\) −19.8681 −0.790936 −0.395468 0.918480i \(-0.629418\pi\)
−0.395468 + 0.918480i \(0.629418\pi\)
\(632\) 27.1778 1.08108
\(633\) 0 0
\(634\) 73.0146 2.89978
\(635\) 14.6889 0.582911
\(636\) 0 0
\(637\) 6.50076 0.257569
\(638\) −12.6559 −0.501053
\(639\) 0 0
\(640\) 16.7788 0.663242
\(641\) −22.3153 −0.881401 −0.440700 0.897654i \(-0.645270\pi\)
−0.440700 + 0.897654i \(0.645270\pi\)
\(642\) 0 0
\(643\) 15.4444 0.609070 0.304535 0.952501i \(-0.401499\pi\)
0.304535 + 0.952501i \(0.401499\pi\)
\(644\) 19.5685 0.771107
\(645\) 0 0
\(646\) −108.065 −4.25178
\(647\) 25.0635 0.985349 0.492675 0.870214i \(-0.336019\pi\)
0.492675 + 0.870214i \(0.336019\pi\)
\(648\) 0 0
\(649\) −6.93091 −0.272062
\(650\) 2.25662 0.0885121
\(651\) 0 0
\(652\) −56.4321 −2.21005
\(653\) 2.98488 0.116807 0.0584036 0.998293i \(-0.481399\pi\)
0.0584036 + 0.998293i \(0.481399\pi\)
\(654\) 0 0
\(655\) 9.06388 0.354155
\(656\) −5.88412 −0.229736
\(657\) 0 0
\(658\) 15.5120 0.604721
\(659\) 37.4330 1.45818 0.729092 0.684416i \(-0.239943\pi\)
0.729092 + 0.684416i \(0.239943\pi\)
\(660\) 0 0
\(661\) 15.3726 0.597925 0.298963 0.954265i \(-0.403359\pi\)
0.298963 + 0.954265i \(0.403359\pi\)
\(662\) −68.0814 −2.64606
\(663\) 0 0
\(664\) −1.28598 −0.0499057
\(665\) −5.59522 −0.216973
\(666\) 0 0
\(667\) 21.5552 0.834622
\(668\) −17.3212 −0.670178
\(669\) 0 0
\(670\) 3.82381 0.147727
\(671\) −21.5348 −0.831341
\(672\) 0 0
\(673\) −0.543557 −0.0209526 −0.0104763 0.999945i \(-0.503335\pi\)
−0.0104763 + 0.999945i \(0.503335\pi\)
\(674\) −14.8449 −0.571805
\(675\) 0 0
\(676\) 3.09235 0.118937
\(677\) 20.0587 0.770917 0.385458 0.922725i \(-0.374043\pi\)
0.385458 + 0.922725i \(0.374043\pi\)
\(678\) 0 0
\(679\) −2.09361 −0.0803454
\(680\) 14.9069 0.571653
\(681\) 0 0
\(682\) 16.1134 0.617015
\(683\) −35.5213 −1.35919 −0.679593 0.733589i \(-0.737844\pi\)
−0.679593 + 0.733589i \(0.737844\pi\)
\(684\) 0 0
\(685\) 2.02984 0.0775561
\(686\) 21.5265 0.821885
\(687\) 0 0
\(688\) −7.42500 −0.283075
\(689\) 5.90968 0.225141
\(690\) 0 0
\(691\) −51.3512 −1.95349 −0.976746 0.214401i \(-0.931220\pi\)
−0.976746 + 0.214401i \(0.931220\pi\)
\(692\) 11.1405 0.423498
\(693\) 0 0
\(694\) −4.53724 −0.172231
\(695\) −2.85318 −0.108227
\(696\) 0 0
\(697\) −57.2014 −2.16666
\(698\) −31.6015 −1.19613
\(699\) 0 0
\(700\) 2.18496 0.0825839
\(701\) −26.2661 −0.992057 −0.496029 0.868306i \(-0.665209\pi\)
−0.496029 + 0.868306i \(0.665209\pi\)
\(702\) 0 0
\(703\) 18.8531 0.711059
\(704\) −30.4067 −1.14600
\(705\) 0 0
\(706\) 78.0787 2.93853
\(707\) −9.65986 −0.363296
\(708\) 0 0
\(709\) −25.7362 −0.966545 −0.483273 0.875470i \(-0.660552\pi\)
−0.483273 + 0.875470i \(0.660552\pi\)
\(710\) −28.0812 −1.05387
\(711\) 0 0
\(712\) 26.5257 0.994093
\(713\) −27.4440 −1.02778
\(714\) 0 0
\(715\) −2.33021 −0.0871449
\(716\) −47.3509 −1.76959
\(717\) 0 0
\(718\) 51.3786 1.91743
\(719\) 21.6829 0.808634 0.404317 0.914619i \(-0.367509\pi\)
0.404317 + 0.914619i \(0.367509\pi\)
\(720\) 0 0
\(721\) −5.03835 −0.187638
\(722\) −98.6326 −3.67073
\(723\) 0 0
\(724\) −22.3804 −0.831760
\(725\) 2.40680 0.0893862
\(726\) 0 0
\(727\) −49.3288 −1.82950 −0.914752 0.404016i \(-0.867614\pi\)
−0.914752 + 0.404016i \(0.867614\pi\)
\(728\) 1.74171 0.0645522
\(729\) 0 0
\(730\) −11.0326 −0.408335
\(731\) −72.1807 −2.66970
\(732\) 0 0
\(733\) 26.1165 0.964636 0.482318 0.875996i \(-0.339795\pi\)
0.482318 + 0.875996i \(0.339795\pi\)
\(734\) 25.7118 0.949039
\(735\) 0 0
\(736\) 56.7257 2.09094
\(737\) −3.94850 −0.145445
\(738\) 0 0
\(739\) −20.3143 −0.747275 −0.373637 0.927575i \(-0.621890\pi\)
−0.373637 + 0.927575i \(0.621890\pi\)
\(740\) −7.36224 −0.270641
\(741\) 0 0
\(742\) 9.42277 0.345921
\(743\) −9.47288 −0.347526 −0.173763 0.984787i \(-0.555593\pi\)
−0.173763 + 0.984787i \(0.555593\pi\)
\(744\) 0 0
\(745\) −8.98225 −0.329084
\(746\) −5.85637 −0.214417
\(747\) 0 0
\(748\) −43.5762 −1.59330
\(749\) 1.20665 0.0440902
\(750\) 0 0
\(751\) −6.05844 −0.221076 −0.110538 0.993872i \(-0.535257\pi\)
−0.110538 + 0.993872i \(0.535257\pi\)
\(752\) 6.05194 0.220692
\(753\) 0 0
\(754\) 5.43123 0.197794
\(755\) −2.75483 −0.100258
\(756\) 0 0
\(757\) 19.3009 0.701503 0.350752 0.936469i \(-0.385926\pi\)
0.350752 + 0.936469i \(0.385926\pi\)
\(758\) −5.38678 −0.195657
\(759\) 0 0
\(760\) 19.5201 0.708069
\(761\) 31.5087 1.14219 0.571095 0.820884i \(-0.306519\pi\)
0.571095 + 0.820884i \(0.306519\pi\)
\(762\) 0 0
\(763\) −2.28666 −0.0827826
\(764\) 59.8922 2.16682
\(765\) 0 0
\(766\) −72.8270 −2.63135
\(767\) 2.97437 0.107398
\(768\) 0 0
\(769\) 9.31924 0.336060 0.168030 0.985782i \(-0.446259\pi\)
0.168030 + 0.985782i \(0.446259\pi\)
\(770\) −3.71544 −0.133895
\(771\) 0 0
\(772\) 15.6332 0.562653
\(773\) −24.3144 −0.874529 −0.437265 0.899333i \(-0.644053\pi\)
−0.437265 + 0.899333i \(0.644053\pi\)
\(774\) 0 0
\(775\) −3.06432 −0.110073
\(776\) 7.30401 0.262199
\(777\) 0 0
\(778\) 16.9270 0.606864
\(779\) −74.9035 −2.68370
\(780\) 0 0
\(781\) 28.9969 1.03759
\(782\) 122.219 4.37054
\(783\) 0 0
\(784\) 4.04395 0.144427
\(785\) 24.4986 0.874393
\(786\) 0 0
\(787\) −42.8300 −1.52672 −0.763362 0.645970i \(-0.776453\pi\)
−0.763362 + 0.645970i \(0.776453\pi\)
\(788\) −19.6497 −0.699992
\(789\) 0 0
\(790\) 24.8802 0.885196
\(791\) 0.479366 0.0170443
\(792\) 0 0
\(793\) 9.24156 0.328177
\(794\) 50.8223 1.80362
\(795\) 0 0
\(796\) −51.8945 −1.83935
\(797\) 30.8397 1.09240 0.546199 0.837656i \(-0.316074\pi\)
0.546199 + 0.837656i \(0.316074\pi\)
\(798\) 0 0
\(799\) 58.8328 2.08135
\(800\) 6.33383 0.223935
\(801\) 0 0
\(802\) 5.71514 0.201809
\(803\) 11.3924 0.402028
\(804\) 0 0
\(805\) 6.32804 0.223034
\(806\) −6.91501 −0.243571
\(807\) 0 0
\(808\) 33.7005 1.18558
\(809\) 14.4671 0.508636 0.254318 0.967121i \(-0.418149\pi\)
0.254318 + 0.967121i \(0.418149\pi\)
\(810\) 0 0
\(811\) 40.9902 1.43936 0.719681 0.694305i \(-0.244288\pi\)
0.719681 + 0.694305i \(0.244288\pi\)
\(812\) 5.25876 0.184546
\(813\) 0 0
\(814\) 12.5192 0.438797
\(815\) −18.2489 −0.639233
\(816\) 0 0
\(817\) −94.5184 −3.30678
\(818\) 14.3766 0.502666
\(819\) 0 0
\(820\) 29.2502 1.02146
\(821\) 47.2815 1.65014 0.825068 0.565033i \(-0.191137\pi\)
0.825068 + 0.565033i \(0.191137\pi\)
\(822\) 0 0
\(823\) 29.8208 1.03949 0.519744 0.854322i \(-0.326027\pi\)
0.519744 + 0.854322i \(0.326027\pi\)
\(824\) 17.5774 0.612336
\(825\) 0 0
\(826\) 4.74253 0.165014
\(827\) −3.90807 −0.135897 −0.0679484 0.997689i \(-0.521645\pi\)
−0.0679484 + 0.997689i \(0.521645\pi\)
\(828\) 0 0
\(829\) −7.76600 −0.269724 −0.134862 0.990864i \(-0.543059\pi\)
−0.134862 + 0.990864i \(0.543059\pi\)
\(830\) −1.17726 −0.0408633
\(831\) 0 0
\(832\) 13.0489 0.452390
\(833\) 39.3124 1.36210
\(834\) 0 0
\(835\) −5.60131 −0.193841
\(836\) −57.0618 −1.97352
\(837\) 0 0
\(838\) 69.8105 2.41156
\(839\) −15.6099 −0.538914 −0.269457 0.963012i \(-0.586844\pi\)
−0.269457 + 0.963012i \(0.586844\pi\)
\(840\) 0 0
\(841\) −23.2073 −0.800253
\(842\) 31.5124 1.08599
\(843\) 0 0
\(844\) −35.1387 −1.20952
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) −3.93569 −0.135232
\(848\) 3.67625 0.126243
\(849\) 0 0
\(850\) 13.6466 0.468075
\(851\) −21.3223 −0.730921
\(852\) 0 0
\(853\) −12.3460 −0.422719 −0.211360 0.977408i \(-0.567789\pi\)
−0.211360 + 0.977408i \(0.567789\pi\)
\(854\) 14.7353 0.504233
\(855\) 0 0
\(856\) −4.20967 −0.143884
\(857\) 36.9006 1.26050 0.630251 0.776392i \(-0.282952\pi\)
0.630251 + 0.776392i \(0.282952\pi\)
\(858\) 0 0
\(859\) −28.3978 −0.968920 −0.484460 0.874814i \(-0.660984\pi\)
−0.484460 + 0.874814i \(0.660984\pi\)
\(860\) 36.9100 1.25862
\(861\) 0 0
\(862\) 45.5900 1.55280
\(863\) 26.6238 0.906283 0.453142 0.891438i \(-0.350303\pi\)
0.453142 + 0.891438i \(0.350303\pi\)
\(864\) 0 0
\(865\) 3.60260 0.122492
\(866\) 31.0640 1.05560
\(867\) 0 0
\(868\) −6.69542 −0.227257
\(869\) −25.6915 −0.871524
\(870\) 0 0
\(871\) 1.69448 0.0574153
\(872\) 7.97749 0.270152
\(873\) 0 0
\(874\) 160.042 5.41350
\(875\) 0.706571 0.0238865
\(876\) 0 0
\(877\) −51.2039 −1.72903 −0.864516 0.502605i \(-0.832375\pi\)
−0.864516 + 0.502605i \(0.832375\pi\)
\(878\) −17.5285 −0.591557
\(879\) 0 0
\(880\) −1.44956 −0.0488647
\(881\) 29.0213 0.977751 0.488876 0.872354i \(-0.337407\pi\)
0.488876 + 0.872354i \(0.337407\pi\)
\(882\) 0 0
\(883\) 21.5185 0.724154 0.362077 0.932148i \(-0.382068\pi\)
0.362077 + 0.932148i \(0.382068\pi\)
\(884\) 18.7006 0.628968
\(885\) 0 0
\(886\) 37.0796 1.24571
\(887\) −19.2022 −0.644745 −0.322373 0.946613i \(-0.604480\pi\)
−0.322373 + 0.946613i \(0.604480\pi\)
\(888\) 0 0
\(889\) 10.3787 0.348092
\(890\) 24.2832 0.813974
\(891\) 0 0
\(892\) −62.1120 −2.07966
\(893\) 77.0398 2.57804
\(894\) 0 0
\(895\) −15.3123 −0.511833
\(896\) 11.8554 0.396062
\(897\) 0 0
\(898\) 22.5147 0.751324
\(899\) −7.37519 −0.245976
\(900\) 0 0
\(901\) 35.7380 1.19060
\(902\) −49.7387 −1.65612
\(903\) 0 0
\(904\) −1.67237 −0.0556222
\(905\) −7.23734 −0.240577
\(906\) 0 0
\(907\) 47.4831 1.57665 0.788326 0.615258i \(-0.210948\pi\)
0.788326 + 0.615258i \(0.210948\pi\)
\(908\) 35.0927 1.16459
\(909\) 0 0
\(910\) 1.59446 0.0528560
\(911\) 14.0104 0.464186 0.232093 0.972694i \(-0.425443\pi\)
0.232093 + 0.972694i \(0.425443\pi\)
\(912\) 0 0
\(913\) 1.21565 0.0402321
\(914\) 39.1659 1.29549
\(915\) 0 0
\(916\) −11.3840 −0.376139
\(917\) 6.40427 0.211488
\(918\) 0 0
\(919\) 47.7699 1.57579 0.787893 0.615813i \(-0.211172\pi\)
0.787893 + 0.615813i \(0.211172\pi\)
\(920\) −22.0767 −0.727847
\(921\) 0 0
\(922\) −23.9042 −0.787243
\(923\) −12.4439 −0.409596
\(924\) 0 0
\(925\) −2.38079 −0.0782800
\(926\) 8.14064 0.267518
\(927\) 0 0
\(928\) 15.2442 0.500417
\(929\) −36.2956 −1.19082 −0.595409 0.803423i \(-0.703010\pi\)
−0.595409 + 0.803423i \(0.703010\pi\)
\(930\) 0 0
\(931\) 51.4785 1.68714
\(932\) −38.7429 −1.26907
\(933\) 0 0
\(934\) 15.7199 0.514371
\(935\) −14.0916 −0.460845
\(936\) 0 0
\(937\) −16.6768 −0.544808 −0.272404 0.962183i \(-0.587819\pi\)
−0.272404 + 0.962183i \(0.587819\pi\)
\(938\) 2.70179 0.0882166
\(939\) 0 0
\(940\) −30.0844 −0.981246
\(941\) −48.8290 −1.59178 −0.795890 0.605441i \(-0.792997\pi\)
−0.795890 + 0.605441i \(0.792997\pi\)
\(942\) 0 0
\(943\) 84.7137 2.75866
\(944\) 1.85028 0.0602213
\(945\) 0 0
\(946\) −62.7638 −2.04063
\(947\) 10.0714 0.327277 0.163639 0.986520i \(-0.447677\pi\)
0.163639 + 0.986520i \(0.447677\pi\)
\(948\) 0 0
\(949\) −4.88899 −0.158703
\(950\) 17.8698 0.579774
\(951\) 0 0
\(952\) 10.5328 0.341369
\(953\) −19.2233 −0.622703 −0.311351 0.950295i \(-0.600782\pi\)
−0.311351 + 0.950295i \(0.600782\pi\)
\(954\) 0 0
\(955\) 19.3679 0.626729
\(956\) −10.2940 −0.332933
\(957\) 0 0
\(958\) −49.7825 −1.60840
\(959\) 1.43422 0.0463135
\(960\) 0 0
\(961\) −21.6100 −0.697096
\(962\) −5.37255 −0.173218
\(963\) 0 0
\(964\) −82.8249 −2.66761
\(965\) 5.05546 0.162741
\(966\) 0 0
\(967\) −23.9667 −0.770717 −0.385358 0.922767i \(-0.625922\pi\)
−0.385358 + 0.922767i \(0.625922\pi\)
\(968\) 13.7305 0.441314
\(969\) 0 0
\(970\) 6.68651 0.214691
\(971\) −22.5951 −0.725111 −0.362556 0.931962i \(-0.618096\pi\)
−0.362556 + 0.931962i \(0.618096\pi\)
\(972\) 0 0
\(973\) −2.01597 −0.0646291
\(974\) 11.7290 0.375821
\(975\) 0 0
\(976\) 5.74893 0.184019
\(977\) 9.54650 0.305420 0.152710 0.988271i \(-0.451200\pi\)
0.152710 + 0.988271i \(0.451200\pi\)
\(978\) 0 0
\(979\) −25.0750 −0.801401
\(980\) −20.1026 −0.642155
\(981\) 0 0
\(982\) −5.04709 −0.161059
\(983\) −29.8467 −0.951963 −0.475981 0.879455i \(-0.657907\pi\)
−0.475981 + 0.879455i \(0.657907\pi\)
\(984\) 0 0
\(985\) −6.35430 −0.202465
\(986\) 32.8447 1.04599
\(987\) 0 0
\(988\) 24.4878 0.779061
\(989\) 106.898 3.39915
\(990\) 0 0
\(991\) −24.1499 −0.767146 −0.383573 0.923511i \(-0.625307\pi\)
−0.383573 + 0.923511i \(0.625307\pi\)
\(992\) −19.4088 −0.616232
\(993\) 0 0
\(994\) −19.8414 −0.629330
\(995\) −16.7816 −0.532012
\(996\) 0 0
\(997\) 31.9786 1.01277 0.506387 0.862306i \(-0.330981\pi\)
0.506387 + 0.862306i \(0.330981\pi\)
\(998\) 48.0419 1.52074
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5265.2.a.ba.1.2 8
3.2 odd 2 5265.2.a.bf.1.7 8
9.2 odd 6 585.2.i.e.391.2 yes 16
9.4 even 3 1755.2.i.f.586.7 16
9.5 odd 6 585.2.i.e.196.2 16
9.7 even 3 1755.2.i.f.1171.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.i.e.196.2 16 9.5 odd 6
585.2.i.e.391.2 yes 16 9.2 odd 6
1755.2.i.f.586.7 16 9.4 even 3
1755.2.i.f.1171.7 16 9.7 even 3
5265.2.a.ba.1.2 8 1.1 even 1 trivial
5265.2.a.bf.1.7 8 3.2 odd 2