Properties

Label 637.2.a.k.1.4
Level $637$
Weight $2$
Character 637.1
Self dual yes
Analytic conductor $5.086$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.746052.1
Defining polynomial: \(x^{5} - x^{4} - 7 x^{3} + 8 x + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.21332\) of defining polynomial
Character \(\chi\) \(=\) 637.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.21332 q^{2} -2.47443 q^{3} +2.89879 q^{4} +2.12280 q^{5} -5.47671 q^{6} +1.98932 q^{8} +3.12280 q^{9} +O(q^{10})\) \(q+2.21332 q^{2} -2.47443 q^{3} +2.89879 q^{4} +2.12280 q^{5} -5.47671 q^{6} +1.98932 q^{8} +3.12280 q^{9} +4.69843 q^{10} +4.78896 q^{11} -7.17286 q^{12} -1.00000 q^{13} -5.25271 q^{15} -1.39458 q^{16} +3.77828 q^{17} +6.91175 q^{18} +3.56723 q^{19} +6.15355 q^{20} +10.5995 q^{22} +4.47443 q^{23} -4.92243 q^{24} -0.493740 q^{25} -2.21332 q^{26} -0.303848 q^{27} -5.90107 q^{29} -11.6259 q^{30} +3.77116 q^{31} -7.06530 q^{32} -11.8499 q^{33} +8.36254 q^{34} +9.05234 q^{36} +5.62570 q^{37} +7.89544 q^{38} +2.47443 q^{39} +4.22292 q^{40} -10.3948 q^{41} +3.40733 q^{43} +13.8822 q^{44} +6.62906 q^{45} +9.90335 q^{46} +7.10876 q^{47} +3.45079 q^{48} -1.09280 q^{50} -9.34907 q^{51} -2.89879 q^{52} -12.3801 q^{53} -0.672514 q^{54} +10.1660 q^{55} -8.82686 q^{57} -13.0610 q^{58} -4.78896 q^{59} -15.2265 q^{60} -3.20697 q^{61} +8.34680 q^{62} -12.8486 q^{64} -2.12280 q^{65} -26.2277 q^{66} -2.89955 q^{67} +10.9524 q^{68} -11.0717 q^{69} -2.53876 q^{71} +6.21224 q^{72} -7.70071 q^{73} +12.4515 q^{74} +1.22172 q^{75} +10.3407 q^{76} +5.47671 q^{78} -5.17850 q^{79} -2.96041 q^{80} -8.61654 q^{81} -23.0071 q^{82} -3.46731 q^{83} +8.02051 q^{85} +7.54152 q^{86} +14.6018 q^{87} +9.52677 q^{88} -3.66432 q^{89} +14.6722 q^{90} +12.9704 q^{92} -9.33147 q^{93} +15.7340 q^{94} +7.57251 q^{95} +17.4826 q^{96} +5.40733 q^{97} +14.9549 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 4 q^{2} + 8 q^{4} - 2 q^{5} + 5 q^{6} + 9 q^{8} + 3 q^{9} + O(q^{10}) \) \( 5 q + 4 q^{2} + 8 q^{4} - 2 q^{5} + 5 q^{6} + 9 q^{8} + 3 q^{9} + 5 q^{10} + 11 q^{11} - 5 q^{12} - 5 q^{13} + 10 q^{16} + 5 q^{17} + 9 q^{18} - 9 q^{19} - q^{20} + 8 q^{22} + 10 q^{23} + 9 q^{25} - 4 q^{26} - 3 q^{29} - 13 q^{30} + 6 q^{31} + 22 q^{32} - 8 q^{33} + 22 q^{34} + 7 q^{36} + 4 q^{37} + 10 q^{38} - 28 q^{40} - 14 q^{41} + 2 q^{43} + 32 q^{45} + 3 q^{46} - q^{47} + 23 q^{48} + 9 q^{50} - 8 q^{51} - 8 q^{52} + 17 q^{53} - 23 q^{54} - 16 q^{57} - 27 q^{58} - 11 q^{59} - 29 q^{60} + 11 q^{61} + 23 q^{62} + 9 q^{64} + 2 q^{65} - 21 q^{66} + 13 q^{67} + 32 q^{68} - 18 q^{69} + 15 q^{71} - 19 q^{72} - 33 q^{74} + 20 q^{75} - 8 q^{76} - 5 q^{78} + 2 q^{79} - 55 q^{80} - 19 q^{81} - 34 q^{82} - 6 q^{83} - 22 q^{85} + 28 q^{86} + 8 q^{87} - 3 q^{88} + 4 q^{89} + 34 q^{90} + 21 q^{92} + 18 q^{93} - 20 q^{94} - 12 q^{95} + 37 q^{96} + 12 q^{97} + 11 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.21332 1.56505 0.782527 0.622616i \(-0.213930\pi\)
0.782527 + 0.622616i \(0.213930\pi\)
\(3\) −2.47443 −1.42861 −0.714306 0.699834i \(-0.753257\pi\)
−0.714306 + 0.699834i \(0.753257\pi\)
\(4\) 2.89879 1.44940
\(5\) 2.12280 0.949343 0.474671 0.880163i \(-0.342567\pi\)
0.474671 + 0.880163i \(0.342567\pi\)
\(6\) −5.47671 −2.23586
\(7\) 0 0
\(8\) 1.98932 0.703331
\(9\) 3.12280 1.04093
\(10\) 4.69843 1.48577
\(11\) 4.78896 1.44392 0.721962 0.691932i \(-0.243240\pi\)
0.721962 + 0.691932i \(0.243240\pi\)
\(12\) −7.17286 −2.07063
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −5.25271 −1.35624
\(16\) −1.39458 −0.348645
\(17\) 3.77828 0.916367 0.458183 0.888858i \(-0.348500\pi\)
0.458183 + 0.888858i \(0.348500\pi\)
\(18\) 6.91175 1.62912
\(19\) 3.56723 0.818379 0.409190 0.912449i \(-0.365811\pi\)
0.409190 + 0.912449i \(0.365811\pi\)
\(20\) 6.15355 1.37597
\(21\) 0 0
\(22\) 10.5995 2.25982
\(23\) 4.47443 0.932983 0.466491 0.884526i \(-0.345518\pi\)
0.466491 + 0.884526i \(0.345518\pi\)
\(24\) −4.92243 −1.00479
\(25\) −0.493740 −0.0987479
\(26\) −2.21332 −0.434068
\(27\) −0.303848 −0.0584757
\(28\) 0 0
\(29\) −5.90107 −1.09580 −0.547901 0.836543i \(-0.684573\pi\)
−0.547901 + 0.836543i \(0.684573\pi\)
\(30\) −11.6259 −2.12259
\(31\) 3.77116 0.677321 0.338660 0.940909i \(-0.390026\pi\)
0.338660 + 0.940909i \(0.390026\pi\)
\(32\) −7.06530 −1.24898
\(33\) −11.8499 −2.06281
\(34\) 8.36254 1.43416
\(35\) 0 0
\(36\) 9.05234 1.50872
\(37\) 5.62570 0.924859 0.462429 0.886656i \(-0.346978\pi\)
0.462429 + 0.886656i \(0.346978\pi\)
\(38\) 7.89544 1.28081
\(39\) 2.47443 0.396226
\(40\) 4.22292 0.667702
\(41\) −10.3948 −1.62340 −0.811698 0.584077i \(-0.801457\pi\)
−0.811698 + 0.584077i \(0.801457\pi\)
\(42\) 0 0
\(43\) 3.40733 0.519613 0.259807 0.965661i \(-0.416341\pi\)
0.259807 + 0.965661i \(0.416341\pi\)
\(44\) 13.8822 2.09282
\(45\) 6.62906 0.988201
\(46\) 9.90335 1.46017
\(47\) 7.10876 1.03692 0.518459 0.855102i \(-0.326506\pi\)
0.518459 + 0.855102i \(0.326506\pi\)
\(48\) 3.45079 0.498079
\(49\) 0 0
\(50\) −1.09280 −0.154546
\(51\) −9.34907 −1.30913
\(52\) −2.89879 −0.401990
\(53\) −12.3801 −1.70053 −0.850266 0.526354i \(-0.823559\pi\)
−0.850266 + 0.526354i \(0.823559\pi\)
\(54\) −0.672514 −0.0915176
\(55\) 10.1660 1.37078
\(56\) 0 0
\(57\) −8.82686 −1.16915
\(58\) −13.0610 −1.71499
\(59\) −4.78896 −0.623469 −0.311734 0.950169i \(-0.600910\pi\)
−0.311734 + 0.950169i \(0.600910\pi\)
\(60\) −15.2265 −1.96573
\(61\) −3.20697 −0.410610 −0.205305 0.978698i \(-0.565819\pi\)
−0.205305 + 0.978698i \(0.565819\pi\)
\(62\) 8.34680 1.06004
\(63\) 0 0
\(64\) −12.8486 −1.60608
\(65\) −2.12280 −0.263300
\(66\) −26.2277 −3.22841
\(67\) −2.89955 −0.354237 −0.177118 0.984190i \(-0.556678\pi\)
−0.177118 + 0.984190i \(0.556678\pi\)
\(68\) 10.9524 1.32818
\(69\) −11.0717 −1.33287
\(70\) 0 0
\(71\) −2.53876 −0.301295 −0.150648 0.988588i \(-0.548136\pi\)
−0.150648 + 0.988588i \(0.548136\pi\)
\(72\) 6.21224 0.732120
\(73\) −7.70071 −0.901300 −0.450650 0.892701i \(-0.648808\pi\)
−0.450650 + 0.892701i \(0.648808\pi\)
\(74\) 12.4515 1.44745
\(75\) 1.22172 0.141072
\(76\) 10.3407 1.18616
\(77\) 0 0
\(78\) 5.47671 0.620115
\(79\) −5.17850 −0.582626 −0.291313 0.956628i \(-0.594092\pi\)
−0.291313 + 0.956628i \(0.594092\pi\)
\(80\) −2.96041 −0.330984
\(81\) −8.61654 −0.957393
\(82\) −23.0071 −2.54071
\(83\) −3.46731 −0.380587 −0.190294 0.981727i \(-0.560944\pi\)
−0.190294 + 0.981727i \(0.560944\pi\)
\(84\) 0 0
\(85\) 8.02051 0.869946
\(86\) 7.54152 0.813223
\(87\) 14.6018 1.56548
\(88\) 9.52677 1.01556
\(89\) −3.66432 −0.388417 −0.194209 0.980960i \(-0.562214\pi\)
−0.194209 + 0.980960i \(0.562214\pi\)
\(90\) 14.6722 1.54659
\(91\) 0 0
\(92\) 12.9704 1.35226
\(93\) −9.33147 −0.967629
\(94\) 15.7340 1.62283
\(95\) 7.57251 0.776923
\(96\) 17.4826 1.78431
\(97\) 5.40733 0.549031 0.274516 0.961583i \(-0.411482\pi\)
0.274516 + 0.961583i \(0.411482\pi\)
\(98\) 0 0
\(99\) 14.9549 1.50303
\(100\) −1.43125 −0.143125
\(101\) −9.31724 −0.927100 −0.463550 0.886071i \(-0.653425\pi\)
−0.463550 + 0.886071i \(0.653425\pi\)
\(102\) −20.6925 −2.04886
\(103\) −7.30636 −0.719917 −0.359958 0.932968i \(-0.617209\pi\)
−0.359958 + 0.932968i \(0.617209\pi\)
\(104\) −1.98932 −0.195069
\(105\) 0 0
\(106\) −27.4011 −2.66143
\(107\) 6.74729 0.652286 0.326143 0.945321i \(-0.394251\pi\)
0.326143 + 0.945321i \(0.394251\pi\)
\(108\) −0.880794 −0.0847545
\(109\) 4.17645 0.400031 0.200016 0.979793i \(-0.435901\pi\)
0.200016 + 0.979793i \(0.435901\pi\)
\(110\) 22.5006 2.14535
\(111\) −13.9204 −1.32126
\(112\) 0 0
\(113\) 5.90107 0.555126 0.277563 0.960707i \(-0.410473\pi\)
0.277563 + 0.960707i \(0.410473\pi\)
\(114\) −19.5367 −1.82978
\(115\) 9.49830 0.885721
\(116\) −17.1060 −1.58825
\(117\) −3.12280 −0.288703
\(118\) −10.5995 −0.975763
\(119\) 0 0
\(120\) −10.4493 −0.953888
\(121\) 11.9341 1.08492
\(122\) −7.09805 −0.642628
\(123\) 25.7212 2.31920
\(124\) 10.9318 0.981707
\(125\) −11.6621 −1.04309
\(126\) 0 0
\(127\) −10.5268 −0.934100 −0.467050 0.884231i \(-0.654683\pi\)
−0.467050 + 0.884231i \(0.654683\pi\)
\(128\) −14.3075 −1.26462
\(129\) −8.43120 −0.742326
\(130\) −4.69843 −0.412080
\(131\) −5.42409 −0.473905 −0.236952 0.971521i \(-0.576149\pi\)
−0.236952 + 0.971521i \(0.576149\pi\)
\(132\) −34.3505 −2.98983
\(133\) 0 0
\(134\) −6.41765 −0.554400
\(135\) −0.645008 −0.0555135
\(136\) 7.51620 0.644509
\(137\) 22.2447 1.90050 0.950248 0.311494i \(-0.100829\pi\)
0.950248 + 0.311494i \(0.100829\pi\)
\(138\) −24.5051 −2.08602
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −17.5901 −1.48135
\(142\) −5.61909 −0.471544
\(143\) −4.78896 −0.400473
\(144\) −4.35499 −0.362916
\(145\) −12.5268 −1.04029
\(146\) −17.0441 −1.41058
\(147\) 0 0
\(148\) 16.3077 1.34049
\(149\) 2.95472 0.242060 0.121030 0.992649i \(-0.461380\pi\)
0.121030 + 0.992649i \(0.461380\pi\)
\(150\) 2.70407 0.220786
\(151\) −18.5547 −1.50996 −0.754981 0.655747i \(-0.772354\pi\)
−0.754981 + 0.655747i \(0.772354\pi\)
\(152\) 7.09637 0.575592
\(153\) 11.7988 0.953875
\(154\) 0 0
\(155\) 8.00541 0.643010
\(156\) 7.17286 0.574288
\(157\) 9.79964 0.782096 0.391048 0.920370i \(-0.372113\pi\)
0.391048 + 0.920370i \(0.372113\pi\)
\(158\) −11.4617 −0.911842
\(159\) 30.6336 2.42940
\(160\) −14.9982 −1.18571
\(161\) 0 0
\(162\) −19.0712 −1.49837
\(163\) 13.8342 1.08358 0.541788 0.840515i \(-0.317747\pi\)
0.541788 + 0.840515i \(0.317747\pi\)
\(164\) −30.1324 −2.35295
\(165\) −25.1550 −1.95831
\(166\) −7.67428 −0.595640
\(167\) −17.3534 −1.34285 −0.671424 0.741073i \(-0.734317\pi\)
−0.671424 + 0.741073i \(0.734317\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 17.7520 1.36151
\(171\) 11.1397 0.851877
\(172\) 9.87716 0.753126
\(173\) 2.96138 0.225149 0.112575 0.993643i \(-0.464090\pi\)
0.112575 + 0.993643i \(0.464090\pi\)
\(174\) 32.3184 2.45005
\(175\) 0 0
\(176\) −6.67859 −0.503418
\(177\) 11.8499 0.890695
\(178\) −8.11032 −0.607894
\(179\) −5.66888 −0.423712 −0.211856 0.977301i \(-0.567951\pi\)
−0.211856 + 0.977301i \(0.567951\pi\)
\(180\) 19.2163 1.43230
\(181\) −7.17645 −0.533421 −0.266711 0.963777i \(-0.585937\pi\)
−0.266711 + 0.963777i \(0.585937\pi\)
\(182\) 0 0
\(183\) 7.93541 0.586603
\(184\) 8.90107 0.656196
\(185\) 11.9422 0.878008
\(186\) −20.6536 −1.51439
\(187\) 18.0940 1.32316
\(188\) 20.6068 1.50291
\(189\) 0 0
\(190\) 16.7604 1.21593
\(191\) 11.8818 0.859734 0.429867 0.902892i \(-0.358560\pi\)
0.429867 + 0.902892i \(0.358560\pi\)
\(192\) 31.7930 2.29446
\(193\) 22.9702 1.65343 0.826714 0.562622i \(-0.190207\pi\)
0.826714 + 0.562622i \(0.190207\pi\)
\(194\) 11.9682 0.859264
\(195\) 5.25271 0.376154
\(196\) 0 0
\(197\) 16.9216 1.20561 0.602806 0.797888i \(-0.294049\pi\)
0.602806 + 0.797888i \(0.294049\pi\)
\(198\) 33.1001 2.35232
\(199\) −10.0591 −0.713068 −0.356534 0.934282i \(-0.616042\pi\)
−0.356534 + 0.934282i \(0.616042\pi\)
\(200\) −0.982206 −0.0694525
\(201\) 7.17474 0.506067
\(202\) −20.6221 −1.45096
\(203\) 0 0
\(204\) −27.1010 −1.89745
\(205\) −22.0661 −1.54116
\(206\) −16.1713 −1.12671
\(207\) 13.9727 0.971171
\(208\) 1.39458 0.0966968
\(209\) 17.0833 1.18168
\(210\) 0 0
\(211\) −24.4609 −1.68396 −0.841978 0.539512i \(-0.818609\pi\)
−0.841978 + 0.539512i \(0.818609\pi\)
\(212\) −35.8872 −2.46475
\(213\) 6.28198 0.430434
\(214\) 14.9339 1.02086
\(215\) 7.23307 0.493291
\(216\) −0.604452 −0.0411277
\(217\) 0 0
\(218\) 9.24382 0.626071
\(219\) 19.0548 1.28761
\(220\) 29.4691 1.98680
\(221\) −3.77828 −0.254154
\(222\) −30.8103 −2.06785
\(223\) 29.2625 1.95956 0.979780 0.200076i \(-0.0641188\pi\)
0.979780 + 0.200076i \(0.0641188\pi\)
\(224\) 0 0
\(225\) −1.54185 −0.102790
\(226\) 13.0610 0.868803
\(227\) 10.0737 0.668615 0.334307 0.942464i \(-0.391498\pi\)
0.334307 + 0.942464i \(0.391498\pi\)
\(228\) −25.5873 −1.69456
\(229\) 11.1399 0.736148 0.368074 0.929796i \(-0.380017\pi\)
0.368074 + 0.929796i \(0.380017\pi\)
\(230\) 21.0228 1.38620
\(231\) 0 0
\(232\) −11.7391 −0.770711
\(233\) −17.0833 −1.11917 −0.559583 0.828774i \(-0.689039\pi\)
−0.559583 + 0.828774i \(0.689039\pi\)
\(234\) −6.91175 −0.451835
\(235\) 15.0904 0.984392
\(236\) −13.8822 −0.903654
\(237\) 12.8138 0.832347
\(238\) 0 0
\(239\) 6.92142 0.447710 0.223855 0.974622i \(-0.428136\pi\)
0.223855 + 0.974622i \(0.428136\pi\)
\(240\) 7.32532 0.472848
\(241\) −6.49625 −0.418460 −0.209230 0.977866i \(-0.567096\pi\)
−0.209230 + 0.977866i \(0.567096\pi\)
\(242\) 26.4140 1.69796
\(243\) 22.2325 1.42622
\(244\) −9.29634 −0.595137
\(245\) 0 0
\(246\) 56.9293 3.62968
\(247\) −3.56723 −0.226978
\(248\) 7.50205 0.476381
\(249\) 8.57962 0.543711
\(250\) −25.8119 −1.63249
\(251\) 9.86804 0.622865 0.311433 0.950268i \(-0.399191\pi\)
0.311433 + 0.950268i \(0.399191\pi\)
\(252\) 0 0
\(253\) 21.4278 1.34716
\(254\) −23.2991 −1.46192
\(255\) −19.8462 −1.24282
\(256\) −5.96994 −0.373121
\(257\) −6.86468 −0.428207 −0.214104 0.976811i \(-0.568683\pi\)
−0.214104 + 0.976811i \(0.568683\pi\)
\(258\) −18.6610 −1.16178
\(259\) 0 0
\(260\) −6.15355 −0.381627
\(261\) −18.4278 −1.14065
\(262\) −12.0052 −0.741687
\(263\) −0.126551 −0.00780345 −0.00390172 0.999992i \(-0.501242\pi\)
−0.00390172 + 0.999992i \(0.501242\pi\)
\(264\) −23.5733 −1.45084
\(265\) −26.2803 −1.61439
\(266\) 0 0
\(267\) 9.06710 0.554897
\(268\) −8.40521 −0.513430
\(269\) 4.24308 0.258705 0.129353 0.991599i \(-0.458710\pi\)
0.129353 + 0.991599i \(0.458710\pi\)
\(270\) −1.42761 −0.0868816
\(271\) −1.56723 −0.0952026 −0.0476013 0.998866i \(-0.515158\pi\)
−0.0476013 + 0.998866i \(0.515158\pi\)
\(272\) −5.26911 −0.319487
\(273\) 0 0
\(274\) 49.2348 2.97438
\(275\) −2.36450 −0.142585
\(276\) −32.0944 −1.93186
\(277\) −12.7452 −0.765785 −0.382892 0.923793i \(-0.625072\pi\)
−0.382892 + 0.923793i \(0.625072\pi\)
\(278\) 8.85329 0.530985
\(279\) 11.7766 0.705045
\(280\) 0 0
\(281\) 4.62986 0.276194 0.138097 0.990419i \(-0.455901\pi\)
0.138097 + 0.990419i \(0.455901\pi\)
\(282\) −38.9326 −2.31840
\(283\) −3.64832 −0.216870 −0.108435 0.994104i \(-0.534584\pi\)
−0.108435 + 0.994104i \(0.534584\pi\)
\(284\) −7.35934 −0.436697
\(285\) −18.7376 −1.10992
\(286\) −10.5995 −0.626762
\(287\) 0 0
\(288\) −22.0635 −1.30010
\(289\) −2.72462 −0.160272
\(290\) −27.7258 −1.62811
\(291\) −13.3801 −0.784353
\(292\) −22.3228 −1.30634
\(293\) 21.0415 1.22926 0.614630 0.788816i \(-0.289305\pi\)
0.614630 + 0.788816i \(0.289305\pi\)
\(294\) 0 0
\(295\) −10.1660 −0.591886
\(296\) 11.1913 0.650482
\(297\) −1.45512 −0.0844345
\(298\) 6.53976 0.378838
\(299\) −4.47443 −0.258763
\(300\) 3.54152 0.204470
\(301\) 0 0
\(302\) −41.0676 −2.36317
\(303\) 23.0548 1.32447
\(304\) −4.97480 −0.285324
\(305\) −6.80774 −0.389810
\(306\) 26.1145 1.49287
\(307\) 4.95861 0.283003 0.141502 0.989938i \(-0.454807\pi\)
0.141502 + 0.989938i \(0.454807\pi\)
\(308\) 0 0
\(309\) 18.0791 1.02848
\(310\) 17.7185 1.00635
\(311\) 2.42158 0.137315 0.0686575 0.997640i \(-0.478128\pi\)
0.0686575 + 0.997640i \(0.478128\pi\)
\(312\) 4.92243 0.278678
\(313\) −13.9605 −0.789096 −0.394548 0.918875i \(-0.629099\pi\)
−0.394548 + 0.918875i \(0.629099\pi\)
\(314\) 21.6897 1.22402
\(315\) 0 0
\(316\) −15.0114 −0.844457
\(317\) 3.06862 0.172351 0.0861753 0.996280i \(-0.472535\pi\)
0.0861753 + 0.996280i \(0.472535\pi\)
\(318\) 67.8019 3.80214
\(319\) −28.2600 −1.58225
\(320\) −27.2750 −1.52472
\(321\) −16.6957 −0.931863
\(322\) 0 0
\(323\) 13.4780 0.749936
\(324\) −24.9776 −1.38764
\(325\) 0.493740 0.0273877
\(326\) 30.6195 1.69586
\(327\) −10.3343 −0.571489
\(328\) −20.6786 −1.14179
\(329\) 0 0
\(330\) −55.6761 −3.06487
\(331\) 13.6052 0.747810 0.373905 0.927467i \(-0.378019\pi\)
0.373905 + 0.927467i \(0.378019\pi\)
\(332\) −10.0510 −0.551622
\(333\) 17.5679 0.962715
\(334\) −38.4087 −2.10163
\(335\) −6.15516 −0.336292
\(336\) 0 0
\(337\) −35.1646 −1.91554 −0.957769 0.287538i \(-0.907163\pi\)
−0.957769 + 0.287538i \(0.907163\pi\)
\(338\) 2.21332 0.120389
\(339\) −14.6018 −0.793060
\(340\) 23.2498 1.26090
\(341\) 18.0599 0.978000
\(342\) 24.6558 1.33323
\(343\) 0 0
\(344\) 6.77828 0.365460
\(345\) −23.5029 −1.26535
\(346\) 6.55448 0.352371
\(347\) −5.47102 −0.293700 −0.146850 0.989159i \(-0.546913\pi\)
−0.146850 + 0.989159i \(0.546913\pi\)
\(348\) 42.3276 2.26899
\(349\) −4.34196 −0.232420 −0.116210 0.993225i \(-0.537075\pi\)
−0.116210 + 0.993225i \(0.537075\pi\)
\(350\) 0 0
\(351\) 0.303848 0.0162182
\(352\) −33.8354 −1.80343
\(353\) −27.5992 −1.46896 −0.734479 0.678631i \(-0.762573\pi\)
−0.734479 + 0.678631i \(0.762573\pi\)
\(354\) 26.2277 1.39399
\(355\) −5.38927 −0.286033
\(356\) −10.6221 −0.562971
\(357\) 0 0
\(358\) −12.5470 −0.663132
\(359\) 6.62855 0.349841 0.174921 0.984583i \(-0.444033\pi\)
0.174921 + 0.984583i \(0.444033\pi\)
\(360\) 13.1873 0.695033
\(361\) −6.27485 −0.330255
\(362\) −15.8838 −0.834833
\(363\) −29.5301 −1.54993
\(364\) 0 0
\(365\) −16.3470 −0.855643
\(366\) 17.5636 0.918065
\(367\) −31.2074 −1.62901 −0.814506 0.580156i \(-0.802992\pi\)
−0.814506 + 0.580156i \(0.802992\pi\)
\(368\) −6.23995 −0.325280
\(369\) −32.4609 −1.68985
\(370\) 26.4319 1.37413
\(371\) 0 0
\(372\) −27.0500 −1.40248
\(373\) −15.7746 −0.816778 −0.408389 0.912808i \(-0.633909\pi\)
−0.408389 + 0.912808i \(0.633909\pi\)
\(374\) 40.0479 2.07083
\(375\) 28.8570 1.49017
\(376\) 14.1416 0.729297
\(377\) 5.90107 0.303921
\(378\) 0 0
\(379\) 31.6512 1.62581 0.812907 0.582393i \(-0.197884\pi\)
0.812907 + 0.582393i \(0.197884\pi\)
\(380\) 21.9511 1.12607
\(381\) 26.0477 1.33447
\(382\) 26.2982 1.34553
\(383\) −12.3935 −0.633278 −0.316639 0.948546i \(-0.602554\pi\)
−0.316639 + 0.948546i \(0.602554\pi\)
\(384\) 35.4030 1.80665
\(385\) 0 0
\(386\) 50.8404 2.58771
\(387\) 10.6404 0.540882
\(388\) 15.6747 0.795765
\(389\) −14.0741 −0.713585 −0.356792 0.934184i \(-0.616130\pi\)
−0.356792 + 0.934184i \(0.616130\pi\)
\(390\) 11.6259 0.588702
\(391\) 16.9056 0.854954
\(392\) 0 0
\(393\) 13.4215 0.677026
\(394\) 37.4529 1.88685
\(395\) −10.9929 −0.553112
\(396\) 43.3513 2.17848
\(397\) 6.97305 0.349967 0.174984 0.984571i \(-0.444013\pi\)
0.174984 + 0.984571i \(0.444013\pi\)
\(398\) −22.2639 −1.11599
\(399\) 0 0
\(400\) 0.688560 0.0344280
\(401\) 2.73682 0.136670 0.0683352 0.997662i \(-0.478231\pi\)
0.0683352 + 0.997662i \(0.478231\pi\)
\(402\) 15.8800 0.792023
\(403\) −3.77116 −0.187855
\(404\) −27.0088 −1.34374
\(405\) −18.2911 −0.908894
\(406\) 0 0
\(407\) 26.9412 1.33543
\(408\) −18.5983 −0.920753
\(409\) 24.5154 1.21221 0.606104 0.795386i \(-0.292732\pi\)
0.606104 + 0.795386i \(0.292732\pi\)
\(410\) −48.8393 −2.41200
\(411\) −55.0430 −2.71507
\(412\) −21.1796 −1.04345
\(413\) 0 0
\(414\) 30.9261 1.51994
\(415\) −7.36040 −0.361308
\(416\) 7.06530 0.346405
\(417\) −9.89771 −0.484693
\(418\) 37.8109 1.84939
\(419\) 3.01252 0.147171 0.0735856 0.997289i \(-0.476556\pi\)
0.0735856 + 0.997289i \(0.476556\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −54.1398 −2.63548
\(423\) 22.1992 1.07936
\(424\) −24.6279 −1.19604
\(425\) −1.86548 −0.0904893
\(426\) 13.9040 0.673653
\(427\) 0 0
\(428\) 19.5590 0.945421
\(429\) 11.8499 0.572120
\(430\) 16.0091 0.772028
\(431\) −18.7942 −0.905286 −0.452643 0.891692i \(-0.649519\pi\)
−0.452643 + 0.891692i \(0.649519\pi\)
\(432\) 0.423741 0.0203873
\(433\) 7.76911 0.373360 0.186680 0.982421i \(-0.440227\pi\)
0.186680 + 0.982421i \(0.440227\pi\)
\(434\) 0 0
\(435\) 30.9966 1.48617
\(436\) 12.1067 0.579804
\(437\) 15.9613 0.763534
\(438\) 42.1745 2.01518
\(439\) 37.9681 1.81212 0.906060 0.423150i \(-0.139076\pi\)
0.906060 + 0.423150i \(0.139076\pi\)
\(440\) 20.2234 0.964112
\(441\) 0 0
\(442\) −8.36254 −0.397766
\(443\) 35.6270 1.69269 0.846344 0.532637i \(-0.178799\pi\)
0.846344 + 0.532637i \(0.178799\pi\)
\(444\) −40.3523 −1.91504
\(445\) −7.77860 −0.368741
\(446\) 64.7673 3.06682
\(447\) −7.31125 −0.345810
\(448\) 0 0
\(449\) −8.05285 −0.380038 −0.190019 0.981780i \(-0.560855\pi\)
−0.190019 + 0.981780i \(0.560855\pi\)
\(450\) −3.41261 −0.160872
\(451\) −49.7803 −2.34406
\(452\) 17.1060 0.804598
\(453\) 45.9123 2.15715
\(454\) 22.2963 1.04642
\(455\) 0 0
\(456\) −17.5595 −0.822297
\(457\) 15.5976 0.729626 0.364813 0.931081i \(-0.381133\pi\)
0.364813 + 0.931081i \(0.381133\pi\)
\(458\) 24.6563 1.15211
\(459\) −1.14802 −0.0535852
\(460\) 27.5336 1.28376
\(461\) 25.6991 1.19692 0.598462 0.801151i \(-0.295779\pi\)
0.598462 + 0.801151i \(0.295779\pi\)
\(462\) 0 0
\(463\) −20.5209 −0.953685 −0.476842 0.878989i \(-0.658219\pi\)
−0.476842 + 0.878989i \(0.658219\pi\)
\(464\) 8.22952 0.382046
\(465\) −19.8088 −0.918611
\(466\) −37.8109 −1.75156
\(467\) −11.8248 −0.547187 −0.273594 0.961845i \(-0.588212\pi\)
−0.273594 + 0.961845i \(0.588212\pi\)
\(468\) −9.05234 −0.418445
\(469\) 0 0
\(470\) 33.4000 1.54063
\(471\) −24.2485 −1.11731
\(472\) −9.52677 −0.438505
\(473\) 16.3176 0.750283
\(474\) 28.3611 1.30267
\(475\) −1.76128 −0.0808133
\(476\) 0 0
\(477\) −38.6604 −1.77014
\(478\) 15.3193 0.700690
\(479\) −22.6552 −1.03514 −0.517571 0.855640i \(-0.673164\pi\)
−0.517571 + 0.855640i \(0.673164\pi\)
\(480\) 37.1119 1.69392
\(481\) −5.62570 −0.256510
\(482\) −14.3783 −0.654913
\(483\) 0 0
\(484\) 34.5945 1.57248
\(485\) 11.4787 0.521219
\(486\) 49.2078 2.23211
\(487\) −32.7167 −1.48254 −0.741268 0.671209i \(-0.765775\pi\)
−0.741268 + 0.671209i \(0.765775\pi\)
\(488\) −6.37969 −0.288795
\(489\) −34.2317 −1.54801
\(490\) 0 0
\(491\) 6.17281 0.278575 0.139288 0.990252i \(-0.455519\pi\)
0.139288 + 0.990252i \(0.455519\pi\)
\(492\) 74.5605 3.36145
\(493\) −22.2959 −1.00416
\(494\) −7.89544 −0.355232
\(495\) 31.7463 1.42689
\(496\) −5.25919 −0.236145
\(497\) 0 0
\(498\) 18.9895 0.850938
\(499\) −14.6387 −0.655317 −0.327659 0.944796i \(-0.606260\pi\)
−0.327659 + 0.944796i \(0.606260\pi\)
\(500\) −33.8060 −1.51185
\(501\) 42.9398 1.91841
\(502\) 21.8412 0.974818
\(503\) −12.7787 −0.569774 −0.284887 0.958561i \(-0.591956\pi\)
−0.284887 + 0.958561i \(0.591956\pi\)
\(504\) 0 0
\(505\) −19.7786 −0.880136
\(506\) 47.4267 2.10837
\(507\) −2.47443 −0.109893
\(508\) −30.5149 −1.35388
\(509\) −11.6853 −0.517940 −0.258970 0.965885i \(-0.583383\pi\)
−0.258970 + 0.965885i \(0.583383\pi\)
\(510\) −43.9260 −1.94507
\(511\) 0 0
\(512\) 15.4017 0.680664
\(513\) −1.08390 −0.0478553
\(514\) −15.1938 −0.670168
\(515\) −15.5099 −0.683448
\(516\) −24.4403 −1.07592
\(517\) 34.0435 1.49723
\(518\) 0 0
\(519\) −7.32772 −0.321651
\(520\) −4.22292 −0.185187
\(521\) −8.47675 −0.371373 −0.185687 0.982609i \(-0.559451\pi\)
−0.185687 + 0.982609i \(0.559451\pi\)
\(522\) −40.7867 −1.78519
\(523\) −32.7108 −1.43034 −0.715172 0.698949i \(-0.753652\pi\)
−0.715172 + 0.698949i \(0.753652\pi\)
\(524\) −15.7233 −0.686876
\(525\) 0 0
\(526\) −0.280097 −0.0122128
\(527\) 14.2485 0.620674
\(528\) 16.5257 0.719188
\(529\) −2.97949 −0.129543
\(530\) −58.1668 −2.52661
\(531\) −14.9549 −0.648989
\(532\) 0 0
\(533\) 10.3948 0.450249
\(534\) 20.0684 0.868445
\(535\) 14.3231 0.619243
\(536\) −5.76814 −0.249146
\(537\) 14.0272 0.605320
\(538\) 9.39131 0.404888
\(539\) 0 0
\(540\) −1.86975 −0.0804610
\(541\) −28.1705 −1.21115 −0.605573 0.795790i \(-0.707056\pi\)
−0.605573 + 0.795790i \(0.707056\pi\)
\(542\) −3.46879 −0.148997
\(543\) 17.7576 0.762052
\(544\) −26.6947 −1.14452
\(545\) 8.86574 0.379767
\(546\) 0 0
\(547\) −18.5377 −0.792615 −0.396307 0.918118i \(-0.629709\pi\)
−0.396307 + 0.918118i \(0.629709\pi\)
\(548\) 64.4829 2.75457
\(549\) −10.0147 −0.427417
\(550\) −5.23339 −0.223153
\(551\) −21.0505 −0.896781
\(552\) −22.0251 −0.937449
\(553\) 0 0
\(554\) −28.2092 −1.19850
\(555\) −29.5501 −1.25433
\(556\) 11.5952 0.491745
\(557\) 4.00283 0.169605 0.0848027 0.996398i \(-0.472974\pi\)
0.0848027 + 0.996398i \(0.472974\pi\)
\(558\) 26.0653 1.10343
\(559\) −3.40733 −0.144115
\(560\) 0 0
\(561\) −44.7723 −1.89029
\(562\) 10.2474 0.432259
\(563\) 17.8620 0.752794 0.376397 0.926459i \(-0.377163\pi\)
0.376397 + 0.926459i \(0.377163\pi\)
\(564\) −50.9901 −2.14707
\(565\) 12.5268 0.527005
\(566\) −8.07490 −0.339413
\(567\) 0 0
\(568\) −5.05041 −0.211910
\(569\) −37.4672 −1.57071 −0.785353 0.619048i \(-0.787519\pi\)
−0.785353 + 0.619048i \(0.787519\pi\)
\(570\) −41.4724 −1.73709
\(571\) 17.5703 0.735293 0.367646 0.929966i \(-0.380164\pi\)
0.367646 + 0.929966i \(0.380164\pi\)
\(572\) −13.8822 −0.580444
\(573\) −29.4006 −1.22823
\(574\) 0 0
\(575\) −2.20920 −0.0921301
\(576\) −40.1236 −1.67182
\(577\) 34.2494 1.42582 0.712910 0.701256i \(-0.247377\pi\)
0.712910 + 0.701256i \(0.247377\pi\)
\(578\) −6.03047 −0.250835
\(579\) −56.8380 −2.36211
\(580\) −36.3125 −1.50780
\(581\) 0 0
\(582\) −29.6144 −1.22756
\(583\) −59.2876 −2.45544
\(584\) −15.3192 −0.633912
\(585\) −6.62906 −0.274078
\(586\) 46.5717 1.92386
\(587\) 29.4494 1.21551 0.607754 0.794126i \(-0.292071\pi\)
0.607754 + 0.794126i \(0.292071\pi\)
\(588\) 0 0
\(589\) 13.4526 0.554305
\(590\) −22.5006 −0.926334
\(591\) −41.8712 −1.72235
\(592\) −7.84549 −0.322448
\(593\) 34.0001 1.39622 0.698109 0.715992i \(-0.254025\pi\)
0.698109 + 0.715992i \(0.254025\pi\)
\(594\) −3.22064 −0.132145
\(595\) 0 0
\(596\) 8.56514 0.350842
\(597\) 24.8904 1.01870
\(598\) −9.90335 −0.404978
\(599\) 21.4418 0.876087 0.438043 0.898954i \(-0.355672\pi\)
0.438043 + 0.898954i \(0.355672\pi\)
\(600\) 2.43040 0.0992206
\(601\) −40.4039 −1.64811 −0.824054 0.566511i \(-0.808293\pi\)
−0.824054 + 0.566511i \(0.808293\pi\)
\(602\) 0 0
\(603\) −9.05472 −0.368737
\(604\) −53.7863 −2.18853
\(605\) 25.3337 1.02996
\(606\) 51.0278 2.07286
\(607\) 43.8913 1.78149 0.890746 0.454501i \(-0.150182\pi\)
0.890746 + 0.454501i \(0.150182\pi\)
\(608\) −25.2036 −1.02214
\(609\) 0 0
\(610\) −15.0677 −0.610074
\(611\) −7.10876 −0.287590
\(612\) 34.2022 1.38254
\(613\) −14.3155 −0.578199 −0.289100 0.957299i \(-0.593356\pi\)
−0.289100 + 0.957299i \(0.593356\pi\)
\(614\) 10.9750 0.442915
\(615\) 54.6009 2.20172
\(616\) 0 0
\(617\) −36.9097 −1.48593 −0.742965 0.669330i \(-0.766581\pi\)
−0.742965 + 0.669330i \(0.766581\pi\)
\(618\) 40.0148 1.60963
\(619\) −14.2929 −0.574481 −0.287240 0.957859i \(-0.592738\pi\)
−0.287240 + 0.957859i \(0.592738\pi\)
\(620\) 23.2060 0.931976
\(621\) −1.35955 −0.0545568
\(622\) 5.35973 0.214906
\(623\) 0 0
\(624\) −3.45079 −0.138142
\(625\) −22.2875 −0.891501
\(626\) −30.8991 −1.23498
\(627\) −42.2715 −1.68816
\(628\) 28.4071 1.13357
\(629\) 21.2554 0.847510
\(630\) 0 0
\(631\) −0.0431064 −0.00171604 −0.000858019 1.00000i \(-0.500273\pi\)
−0.000858019 1.00000i \(0.500273\pi\)
\(632\) −10.3017 −0.409779
\(633\) 60.5267 2.40572
\(634\) 6.79184 0.269738
\(635\) −22.3462 −0.886781
\(636\) 88.8004 3.52116
\(637\) 0 0
\(638\) −62.5484 −2.47632
\(639\) −7.92803 −0.313628
\(640\) −30.3720 −1.20056
\(641\) 42.6655 1.68519 0.842594 0.538550i \(-0.181028\pi\)
0.842594 + 0.538550i \(0.181028\pi\)
\(642\) −36.9530 −1.45842
\(643\) 5.49737 0.216795 0.108398 0.994108i \(-0.465428\pi\)
0.108398 + 0.994108i \(0.465428\pi\)
\(644\) 0 0
\(645\) −17.8977 −0.704722
\(646\) 29.8311 1.17369
\(647\) 38.1867 1.50127 0.750637 0.660715i \(-0.229747\pi\)
0.750637 + 0.660715i \(0.229747\pi\)
\(648\) −17.1411 −0.673364
\(649\) −22.9341 −0.900242
\(650\) 1.09280 0.0428633
\(651\) 0 0
\(652\) 40.1024 1.57053
\(653\) 38.5019 1.50670 0.753349 0.657621i \(-0.228437\pi\)
0.753349 + 0.657621i \(0.228437\pi\)
\(654\) −22.8732 −0.894412
\(655\) −11.5142 −0.449898
\(656\) 14.4964 0.565990
\(657\) −24.0477 −0.938192
\(658\) 0 0
\(659\) 19.4843 0.759002 0.379501 0.925191i \(-0.376096\pi\)
0.379501 + 0.925191i \(0.376096\pi\)
\(660\) −72.9191 −2.83837
\(661\) 41.6667 1.62065 0.810324 0.585983i \(-0.199291\pi\)
0.810324 + 0.585983i \(0.199291\pi\)
\(662\) 30.1127 1.17036
\(663\) 9.34907 0.363088
\(664\) −6.89760 −0.267679
\(665\) 0 0
\(666\) 38.8834 1.50670
\(667\) −26.4039 −1.02236
\(668\) −50.3040 −1.94632
\(669\) −72.4079 −2.79945
\(670\) −13.6234 −0.526316
\(671\) −15.3580 −0.592890
\(672\) 0 0
\(673\) −14.3157 −0.551830 −0.275915 0.961182i \(-0.588981\pi\)
−0.275915 + 0.961182i \(0.588981\pi\)
\(674\) −77.8306 −2.99792
\(675\) 0.150022 0.00577435
\(676\) 2.89879 0.111492
\(677\) 29.5281 1.13486 0.567429 0.823423i \(-0.307938\pi\)
0.567429 + 0.823423i \(0.307938\pi\)
\(678\) −32.3184 −1.24118
\(679\) 0 0
\(680\) 15.9554 0.611860
\(681\) −24.9266 −0.955191
\(682\) 39.9724 1.53062
\(683\) 47.0699 1.80108 0.900539 0.434774i \(-0.143172\pi\)
0.900539 + 0.434774i \(0.143172\pi\)
\(684\) 32.2918 1.23471
\(685\) 47.2210 1.80422
\(686\) 0 0
\(687\) −27.5650 −1.05167
\(688\) −4.75180 −0.181161
\(689\) 12.3801 0.471643
\(690\) −52.0194 −1.98034
\(691\) 30.8668 1.17423 0.587113 0.809505i \(-0.300264\pi\)
0.587113 + 0.809505i \(0.300264\pi\)
\(692\) 8.58442 0.326331
\(693\) 0 0
\(694\) −12.1091 −0.459656
\(695\) 8.49118 0.322089
\(696\) 29.0476 1.10105
\(697\) −39.2745 −1.48763
\(698\) −9.61016 −0.363750
\(699\) 42.2715 1.59885
\(700\) 0 0
\(701\) 6.48958 0.245108 0.122554 0.992462i \(-0.460892\pi\)
0.122554 + 0.992462i \(0.460892\pi\)
\(702\) 0.672514 0.0253824
\(703\) 20.0682 0.756885
\(704\) −61.5315 −2.31905
\(705\) −37.3402 −1.40631
\(706\) −61.0860 −2.29900
\(707\) 0 0
\(708\) 34.3505 1.29097
\(709\) −13.3738 −0.502263 −0.251131 0.967953i \(-0.580803\pi\)
−0.251131 + 0.967953i \(0.580803\pi\)
\(710\) −11.9282 −0.447657
\(711\) −16.1714 −0.606474
\(712\) −7.28951 −0.273186
\(713\) 16.8738 0.631929
\(714\) 0 0
\(715\) −10.1660 −0.380186
\(716\) −16.4329 −0.614126
\(717\) −17.1266 −0.639603
\(718\) 14.6711 0.547521
\(719\) 16.7410 0.624333 0.312166 0.950027i \(-0.398945\pi\)
0.312166 + 0.950027i \(0.398945\pi\)
\(720\) −9.24476 −0.344532
\(721\) 0 0
\(722\) −13.8883 −0.516868
\(723\) 16.0745 0.597817
\(724\) −20.8030 −0.773139
\(725\) 2.91359 0.108208
\(726\) −65.3596 −2.42572
\(727\) −38.8138 −1.43952 −0.719761 0.694221i \(-0.755749\pi\)
−0.719761 + 0.694221i \(0.755749\pi\)
\(728\) 0 0
\(729\) −29.1632 −1.08012
\(730\) −36.1812 −1.33913
\(731\) 12.8738 0.476156
\(732\) 23.0031 0.850220
\(733\) −37.7279 −1.39351 −0.696756 0.717309i \(-0.745374\pi\)
−0.696756 + 0.717309i \(0.745374\pi\)
\(734\) −69.0719 −2.54949
\(735\) 0 0
\(736\) −31.6132 −1.16528
\(737\) −13.8858 −0.511491
\(738\) −71.8464 −2.64470
\(739\) −9.22952 −0.339514 −0.169757 0.985486i \(-0.554298\pi\)
−0.169757 + 0.985486i \(0.554298\pi\)
\(740\) 34.6180 1.27258
\(741\) 8.82686 0.324263
\(742\) 0 0
\(743\) −3.56327 −0.130724 −0.0653619 0.997862i \(-0.520820\pi\)
−0.0653619 + 0.997862i \(0.520820\pi\)
\(744\) −18.5633 −0.680563
\(745\) 6.27228 0.229798
\(746\) −34.9143 −1.27830
\(747\) −10.8277 −0.396165
\(748\) 52.4508 1.91779
\(749\) 0 0
\(750\) 63.8698 2.33220
\(751\) 51.2106 1.86870 0.934350 0.356357i \(-0.115981\pi\)
0.934350 + 0.356357i \(0.115981\pi\)
\(752\) −9.91374 −0.361517
\(753\) −24.4178 −0.889833
\(754\) 13.0610 0.475653
\(755\) −39.3879 −1.43347
\(756\) 0 0
\(757\) 25.2305 0.917019 0.458509 0.888690i \(-0.348384\pi\)
0.458509 + 0.888690i \(0.348384\pi\)
\(758\) 70.0543 2.54449
\(759\) −53.0217 −1.92456
\(760\) 15.0641 0.546434
\(761\) −3.64744 −0.132220 −0.0661099 0.997812i \(-0.521059\pi\)
−0.0661099 + 0.997812i \(0.521059\pi\)
\(762\) 57.6520 2.08851
\(763\) 0 0
\(764\) 34.4428 1.24610
\(765\) 25.0464 0.905555
\(766\) −27.4308 −0.991115
\(767\) 4.78896 0.172919
\(768\) 14.7722 0.533045
\(769\) −21.9882 −0.792914 −0.396457 0.918053i \(-0.629760\pi\)
−0.396457 + 0.918053i \(0.629760\pi\)
\(770\) 0 0
\(771\) 16.9862 0.611742
\(772\) 66.5858 2.39647
\(773\) −21.8590 −0.786215 −0.393108 0.919493i \(-0.628600\pi\)
−0.393108 + 0.919493i \(0.628600\pi\)
\(774\) 23.5506 0.846510
\(775\) −1.86197 −0.0668840
\(776\) 10.7569 0.386151
\(777\) 0 0
\(778\) −31.1505 −1.11680
\(779\) −37.0807 −1.32855
\(780\) 15.2265 0.545197
\(781\) −12.1580 −0.435048
\(782\) 37.4176 1.33805
\(783\) 1.79303 0.0640777
\(784\) 0 0
\(785\) 20.8026 0.742477
\(786\) 29.7061 1.05958
\(787\) −39.8673 −1.42111 −0.710557 0.703639i \(-0.751557\pi\)
−0.710557 + 0.703639i \(0.751557\pi\)
\(788\) 49.0522 1.74741
\(789\) 0.313141 0.0111481
\(790\) −24.3308 −0.865651
\(791\) 0 0
\(792\) 29.7502 1.05713
\(793\) 3.20697 0.113883
\(794\) 15.4336 0.547718
\(795\) 65.0288 2.30633
\(796\) −29.1591 −1.03352
\(797\) 40.1971 1.42385 0.711927 0.702253i \(-0.247822\pi\)
0.711927 + 0.702253i \(0.247822\pi\)
\(798\) 0 0
\(799\) 26.8589 0.950198
\(800\) 3.48842 0.123334
\(801\) −11.4429 −0.404316
\(802\) 6.05747 0.213897
\(803\) −36.8784 −1.30141
\(804\) 20.7981 0.733492
\(805\) 0 0
\(806\) −8.34680 −0.294003
\(807\) −10.4992 −0.369589
\(808\) −18.5350 −0.652058
\(809\) 2.53849 0.0892485 0.0446243 0.999004i \(-0.485791\pi\)
0.0446243 + 0.999004i \(0.485791\pi\)
\(810\) −40.4842 −1.42247
\(811\) 41.7062 1.46450 0.732251 0.681035i \(-0.238470\pi\)
0.732251 + 0.681035i \(0.238470\pi\)
\(812\) 0 0
\(813\) 3.87801 0.136008
\(814\) 59.6296 2.09002
\(815\) 29.3671 1.02869
\(816\) 13.0380 0.456423
\(817\) 12.1547 0.425241
\(818\) 54.2604 1.89717
\(819\) 0 0
\(820\) −63.9650 −2.23375
\(821\) 31.9304 1.11438 0.557189 0.830386i \(-0.311880\pi\)
0.557189 + 0.830386i \(0.311880\pi\)
\(822\) −121.828 −4.24924
\(823\) −34.2531 −1.19399 −0.596995 0.802245i \(-0.703639\pi\)
−0.596995 + 0.802245i \(0.703639\pi\)
\(824\) −14.5347 −0.506340
\(825\) 5.85078 0.203698
\(826\) 0 0
\(827\) 36.9755 1.28576 0.642882 0.765965i \(-0.277739\pi\)
0.642882 + 0.765965i \(0.277739\pi\)
\(828\) 40.5040 1.40761
\(829\) 19.9895 0.694263 0.347131 0.937817i \(-0.387156\pi\)
0.347131 + 0.937817i \(0.387156\pi\)
\(830\) −16.2909 −0.565466
\(831\) 31.5371 1.09401
\(832\) 12.8486 0.445446
\(833\) 0 0
\(834\) −21.9068 −0.758571
\(835\) −36.8378 −1.27482
\(836\) 49.5210 1.71272
\(837\) −1.14586 −0.0396068
\(838\) 6.66768 0.230331
\(839\) −12.8147 −0.442411 −0.221206 0.975227i \(-0.570999\pi\)
−0.221206 + 0.975227i \(0.570999\pi\)
\(840\) 0 0
\(841\) 5.82265 0.200781
\(842\) −22.1332 −0.762761
\(843\) −11.4562 −0.394574
\(844\) −70.9070 −2.44072
\(845\) 2.12280 0.0730264
\(846\) 49.1340 1.68926
\(847\) 0 0
\(848\) 17.2650 0.592882
\(849\) 9.02750 0.309823
\(850\) −4.12892 −0.141621
\(851\) 25.1718 0.862877
\(852\) 18.2102 0.623870
\(853\) 30.1839 1.03348 0.516739 0.856143i \(-0.327146\pi\)
0.516739 + 0.856143i \(0.327146\pi\)
\(854\) 0 0
\(855\) 23.6474 0.808724
\(856\) 13.4225 0.458773
\(857\) 53.2327 1.81839 0.909197 0.416366i \(-0.136696\pi\)
0.909197 + 0.416366i \(0.136696\pi\)
\(858\) 26.2277 0.895399
\(859\) 12.2719 0.418713 0.209357 0.977839i \(-0.432863\pi\)
0.209357 + 0.977839i \(0.432863\pi\)
\(860\) 20.9672 0.714975
\(861\) 0 0
\(862\) −41.5977 −1.41682
\(863\) 24.4453 0.832127 0.416064 0.909335i \(-0.363409\pi\)
0.416064 + 0.909335i \(0.363409\pi\)
\(864\) 2.14678 0.0730349
\(865\) 6.28640 0.213744
\(866\) 17.1956 0.584329
\(867\) 6.74189 0.228967
\(868\) 0 0
\(869\) −24.7996 −0.841268
\(870\) 68.6054 2.32594
\(871\) 2.89955 0.0982477
\(872\) 8.30829 0.281354
\(873\) 16.8860 0.571504
\(874\) 35.3276 1.19497
\(875\) 0 0
\(876\) 55.2361 1.86625
\(877\) −52.8753 −1.78547 −0.892736 0.450580i \(-0.851217\pi\)
−0.892736 + 0.450580i \(0.851217\pi\)
\(878\) 84.0357 2.83607
\(879\) −52.0658 −1.75614
\(880\) −14.1773 −0.477916
\(881\) −55.0118 −1.85339 −0.926697 0.375809i \(-0.877365\pi\)
−0.926697 + 0.375809i \(0.877365\pi\)
\(882\) 0 0
\(883\) 44.1730 1.48654 0.743269 0.668992i \(-0.233274\pi\)
0.743269 + 0.668992i \(0.233274\pi\)
\(884\) −10.9524 −0.368371
\(885\) 25.1550 0.845575
\(886\) 78.8539 2.64915
\(887\) 5.08659 0.170791 0.0853955 0.996347i \(-0.472785\pi\)
0.0853955 + 0.996347i \(0.472785\pi\)
\(888\) −27.6921 −0.929286
\(889\) 0 0
\(890\) −17.2165 −0.577100
\(891\) −41.2642 −1.38240
\(892\) 84.8259 2.84018
\(893\) 25.3586 0.848593
\(894\) −16.1822 −0.541212
\(895\) −12.0339 −0.402248
\(896\) 0 0
\(897\) 11.0717 0.369672
\(898\) −17.8236 −0.594780
\(899\) −22.2539 −0.742209
\(900\) −4.46950 −0.148983
\(901\) −46.7753 −1.55831
\(902\) −110.180 −3.66859
\(903\) 0 0
\(904\) 11.7391 0.390437
\(905\) −15.2341 −0.506400
\(906\) 101.619 3.37606
\(907\) −18.1253 −0.601840 −0.300920 0.953649i \(-0.597294\pi\)
−0.300920 + 0.953649i \(0.597294\pi\)
\(908\) 29.2016 0.969088
\(909\) −29.0958 −0.965048
\(910\) 0 0
\(911\) −9.65804 −0.319985 −0.159993 0.987118i \(-0.551147\pi\)
−0.159993 + 0.987118i \(0.551147\pi\)
\(912\) 12.3098 0.407617
\(913\) −16.6048 −0.549539
\(914\) 34.5226 1.14190
\(915\) 16.8453 0.556887
\(916\) 32.2924 1.06697
\(917\) 0 0
\(918\) −2.54095 −0.0838637
\(919\) 47.7603 1.57547 0.787733 0.616017i \(-0.211255\pi\)
0.787733 + 0.616017i \(0.211255\pi\)
\(920\) 18.8952 0.622955
\(921\) −12.2697 −0.404301
\(922\) 56.8803 1.87325
\(923\) 2.53876 0.0835643
\(924\) 0 0
\(925\) −2.77763 −0.0913279
\(926\) −45.4193 −1.49257
\(927\) −22.8163 −0.749384
\(928\) 41.6928 1.36863
\(929\) −33.9811 −1.11488 −0.557442 0.830216i \(-0.688217\pi\)
−0.557442 + 0.830216i \(0.688217\pi\)
\(930\) −43.8433 −1.43768
\(931\) 0 0
\(932\) −49.5210 −1.62212
\(933\) −5.99202 −0.196170
\(934\) −26.1721 −0.856378
\(935\) 38.4099 1.25614
\(936\) −6.21224 −0.203053
\(937\) 24.7948 0.810012 0.405006 0.914314i \(-0.367269\pi\)
0.405006 + 0.914314i \(0.367269\pi\)
\(938\) 0 0
\(939\) 34.5443 1.12731
\(940\) 43.7441 1.42677
\(941\) 8.24196 0.268680 0.134340 0.990935i \(-0.457109\pi\)
0.134340 + 0.990935i \(0.457109\pi\)
\(942\) −53.6697 −1.74865
\(943\) −46.5108 −1.51460
\(944\) 6.67859 0.217370
\(945\) 0 0
\(946\) 36.1160 1.17423
\(947\) 19.9729 0.649031 0.324515 0.945880i \(-0.394799\pi\)
0.324515 + 0.945880i \(0.394799\pi\)
\(948\) 37.1446 1.20640
\(949\) 7.70071 0.249976
\(950\) −3.89829 −0.126477
\(951\) −7.59307 −0.246222
\(952\) 0 0
\(953\) −21.5341 −0.697557 −0.348778 0.937205i \(-0.613403\pi\)
−0.348778 + 0.937205i \(0.613403\pi\)
\(954\) −85.5679 −2.77036
\(955\) 25.2225 0.816182
\(956\) 20.0638 0.648909
\(957\) 69.9273 2.26043
\(958\) −50.1432 −1.62005
\(959\) 0 0
\(960\) 67.4900 2.17823
\(961\) −16.7783 −0.541237
\(962\) −12.4515 −0.401452
\(963\) 21.0704 0.678985
\(964\) −18.8313 −0.606515
\(965\) 48.7610 1.56967
\(966\) 0 0
\(967\) 43.2887 1.39207 0.696036 0.718007i \(-0.254945\pi\)
0.696036 + 0.718007i \(0.254945\pi\)
\(968\) 23.7408 0.763057
\(969\) −33.3503 −1.07137
\(970\) 25.4060 0.815737
\(971\) 52.6713 1.69030 0.845151 0.534528i \(-0.179510\pi\)
0.845151 + 0.534528i \(0.179510\pi\)
\(972\) 64.4476 2.06716
\(973\) 0 0
\(974\) −72.4127 −2.32025
\(975\) −1.22172 −0.0391265
\(976\) 4.47238 0.143157
\(977\) 15.4061 0.492885 0.246442 0.969157i \(-0.420738\pi\)
0.246442 + 0.969157i \(0.420738\pi\)
\(978\) −75.7658 −2.42272
\(979\) −17.5483 −0.560845
\(980\) 0 0
\(981\) 13.0422 0.416405
\(982\) 13.6624 0.435985
\(983\) −7.58146 −0.241811 −0.120906 0.992664i \(-0.538580\pi\)
−0.120906 + 0.992664i \(0.538580\pi\)
\(984\) 51.1677 1.63117
\(985\) 35.9211 1.14454
\(986\) −49.3480 −1.57156
\(987\) 0 0
\(988\) −10.3407 −0.328981
\(989\) 15.2459 0.484790
\(990\) 70.2647 2.23316
\(991\) −19.0185 −0.604141 −0.302071 0.953286i \(-0.597678\pi\)
−0.302071 + 0.953286i \(0.597678\pi\)
\(992\) −26.6444 −0.845960
\(993\) −33.6651 −1.06833
\(994\) 0 0
\(995\) −21.3533 −0.676946
\(996\) 24.8706 0.788054
\(997\) −46.0998 −1.46000 −0.729998 0.683449i \(-0.760479\pi\)
−0.729998 + 0.683449i \(0.760479\pi\)
\(998\) −32.4001 −1.02561
\(999\) −1.70936 −0.0540817
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 637.2.a.k.1.4 5
3.2 odd 2 5733.2.a.bm.1.2 5
7.2 even 3 637.2.e.m.508.2 10
7.3 odd 6 91.2.e.c.79.2 yes 10
7.4 even 3 637.2.e.m.79.2 10
7.5 odd 6 91.2.e.c.53.2 10
7.6 odd 2 637.2.a.l.1.4 5
13.12 even 2 8281.2.a.bx.1.2 5
21.5 even 6 819.2.j.h.235.4 10
21.17 even 6 819.2.j.h.352.4 10
21.20 even 2 5733.2.a.bl.1.2 5
28.3 even 6 1456.2.r.p.625.5 10
28.19 even 6 1456.2.r.p.417.5 10
91.12 odd 6 1183.2.e.f.508.4 10
91.38 odd 6 1183.2.e.f.170.4 10
91.90 odd 2 8281.2.a.bw.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.c.53.2 10 7.5 odd 6
91.2.e.c.79.2 yes 10 7.3 odd 6
637.2.a.k.1.4 5 1.1 even 1 trivial
637.2.a.l.1.4 5 7.6 odd 2
637.2.e.m.79.2 10 7.4 even 3
637.2.e.m.508.2 10 7.2 even 3
819.2.j.h.235.4 10 21.5 even 6
819.2.j.h.352.4 10 21.17 even 6
1183.2.e.f.170.4 10 91.38 odd 6
1183.2.e.f.508.4 10 91.12 odd 6
1456.2.r.p.417.5 10 28.19 even 6
1456.2.r.p.625.5 10 28.3 even 6
5733.2.a.bl.1.2 5 21.20 even 2
5733.2.a.bm.1.2 5 3.2 odd 2
8281.2.a.bw.1.2 5 91.90 odd 2
8281.2.a.bx.1.2 5 13.12 even 2