Properties

Label 1850.2.a.be.1.3
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.1791440.1
Defining polynomial: \(x^{5} - 9 x^{3} + 13 x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.332924\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.332924 q^{3} +1.00000 q^{4} -0.332924 q^{6} +3.51336 q^{7} +1.00000 q^{8} -2.88916 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.332924 q^{3} +1.00000 q^{4} -0.332924 q^{6} +3.51336 q^{7} +1.00000 q^{8} -2.88916 q^{9} -0.290044 q^{11} -0.332924 q^{12} -7.12558 q^{13} +3.51336 q^{14} +1.00000 q^{16} +6.17921 q^{17} -2.88916 q^{18} +5.83553 q^{19} -1.16968 q^{21} -0.290044 q^{22} +6.45973 q^{23} -0.332924 q^{24} -7.12558 q^{26} +1.96065 q^{27} +3.51336 q^{28} -1.18043 q^{29} +9.77838 q^{31} +1.00000 q^{32} +0.0965628 q^{33} +6.17921 q^{34} -2.88916 q^{36} +1.00000 q^{37} +5.83553 q^{38} +2.37228 q^{39} +1.64077 q^{41} -1.16968 q^{42} +5.34889 q^{43} -0.290044 q^{44} +6.45973 q^{46} -5.69256 q^{47} -0.332924 q^{48} +5.34367 q^{49} -2.05721 q^{51} -7.12558 q^{52} -9.32034 q^{53} +1.96065 q^{54} +3.51336 q^{56} -1.94279 q^{57} -1.18043 q^{58} +5.94279 q^{59} -1.27699 q^{61} +9.77838 q^{62} -10.1507 q^{63} +1.00000 q^{64} +0.0965628 q^{66} -6.04334 q^{67} +6.17921 q^{68} -2.15060 q^{69} +13.4995 q^{71} -2.88916 q^{72} -1.71348 q^{73} +1.00000 q^{74} +5.83553 q^{76} -1.01903 q^{77} +2.37228 q^{78} +8.25422 q^{79} +8.01474 q^{81} +1.64077 q^{82} -2.41807 q^{83} -1.16968 q^{84} +5.34889 q^{86} +0.392995 q^{87} -0.290044 q^{88} -10.2797 q^{89} -25.0347 q^{91} +6.45973 q^{92} -3.25546 q^{93} -5.69256 q^{94} -0.332924 q^{96} -15.1272 q^{97} +5.34367 q^{98} +0.837984 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{2} + 5 q^{4} - q^{7} + 5 q^{8} + 3 q^{9} + O(q^{10}) \) \( 5 q + 5 q^{2} + 5 q^{4} - q^{7} + 5 q^{8} + 3 q^{9} + 3 q^{11} - 6 q^{13} - q^{14} + 5 q^{16} + 9 q^{17} + 3 q^{18} + 4 q^{19} + 16 q^{21} + 3 q^{22} + 6 q^{23} - 6 q^{26} - q^{28} + 11 q^{29} + 23 q^{31} + 5 q^{32} + 20 q^{33} + 9 q^{34} + 3 q^{36} + 5 q^{37} + 4 q^{38} + 20 q^{39} - 7 q^{41} + 16 q^{42} - 17 q^{43} + 3 q^{44} + 6 q^{46} + 12 q^{47} + 30 q^{49} - 20 q^{51} - 6 q^{52} - 7 q^{53} - q^{56} + 11 q^{58} + 20 q^{59} - 9 q^{61} + 23 q^{62} - 33 q^{63} + 5 q^{64} + 20 q^{66} + 12 q^{67} + 9 q^{68} + 16 q^{69} + 6 q^{71} + 3 q^{72} - 6 q^{73} + 5 q^{74} + 4 q^{76} - q^{77} + 20 q^{78} + 20 q^{79} - 7 q^{81} - 7 q^{82} + 12 q^{83} + 16 q^{84} - 17 q^{86} - 34 q^{87} + 3 q^{88} + 12 q^{89} + 16 q^{91} + 6 q^{92} - 4 q^{93} + 12 q^{94} + 3 q^{97} + 30 q^{98} - 11 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −0.332924 −0.192214 −0.0961070 0.995371i \(-0.530639\pi\)
−0.0961070 + 0.995371i \(0.530639\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −0.332924 −0.135916
\(7\) 3.51336 1.32792 0.663962 0.747766i \(-0.268874\pi\)
0.663962 + 0.747766i \(0.268874\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.88916 −0.963054
\(10\) 0 0
\(11\) −0.290044 −0.0874516 −0.0437258 0.999044i \(-0.513923\pi\)
−0.0437258 + 0.999044i \(0.513923\pi\)
\(12\) −0.332924 −0.0961070
\(13\) −7.12558 −1.97628 −0.988140 0.153559i \(-0.950927\pi\)
−0.988140 + 0.153559i \(0.950927\pi\)
\(14\) 3.51336 0.938984
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.17921 1.49868 0.749339 0.662187i \(-0.230372\pi\)
0.749339 + 0.662187i \(0.230372\pi\)
\(18\) −2.88916 −0.680982
\(19\) 5.83553 1.33876 0.669381 0.742919i \(-0.266559\pi\)
0.669381 + 0.742919i \(0.266559\pi\)
\(20\) 0 0
\(21\) −1.16968 −0.255246
\(22\) −0.290044 −0.0618376
\(23\) 6.45973 1.34695 0.673473 0.739212i \(-0.264802\pi\)
0.673473 + 0.739212i \(0.264802\pi\)
\(24\) −0.332924 −0.0679579
\(25\) 0 0
\(26\) −7.12558 −1.39744
\(27\) 1.96065 0.377326
\(28\) 3.51336 0.663962
\(29\) −1.18043 −0.219201 −0.109600 0.993976i \(-0.534957\pi\)
−0.109600 + 0.993976i \(0.534957\pi\)
\(30\) 0 0
\(31\) 9.77838 1.75625 0.878124 0.478433i \(-0.158795\pi\)
0.878124 + 0.478433i \(0.158795\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.0965628 0.0168094
\(34\) 6.17921 1.05972
\(35\) 0 0
\(36\) −2.88916 −0.481527
\(37\) 1.00000 0.164399
\(38\) 5.83553 0.946648
\(39\) 2.37228 0.379869
\(40\) 0 0
\(41\) 1.64077 0.256245 0.128123 0.991758i \(-0.459105\pi\)
0.128123 + 0.991758i \(0.459105\pi\)
\(42\) −1.16968 −0.180486
\(43\) 5.34889 0.815698 0.407849 0.913049i \(-0.366279\pi\)
0.407849 + 0.913049i \(0.366279\pi\)
\(44\) −0.290044 −0.0437258
\(45\) 0 0
\(46\) 6.45973 0.952435
\(47\) −5.69256 −0.830346 −0.415173 0.909743i \(-0.636279\pi\)
−0.415173 + 0.909743i \(0.636279\pi\)
\(48\) −0.332924 −0.0480535
\(49\) 5.34367 0.763382
\(50\) 0 0
\(51\) −2.05721 −0.288067
\(52\) −7.12558 −0.988140
\(53\) −9.32034 −1.28025 −0.640123 0.768272i \(-0.721117\pi\)
−0.640123 + 0.768272i \(0.721117\pi\)
\(54\) 1.96065 0.266810
\(55\) 0 0
\(56\) 3.51336 0.469492
\(57\) −1.94279 −0.257329
\(58\) −1.18043 −0.154998
\(59\) 5.94279 0.773686 0.386843 0.922146i \(-0.373566\pi\)
0.386843 + 0.922146i \(0.373566\pi\)
\(60\) 0 0
\(61\) −1.27699 −0.163502 −0.0817512 0.996653i \(-0.526051\pi\)
−0.0817512 + 0.996653i \(0.526051\pi\)
\(62\) 9.77838 1.24185
\(63\) −10.1507 −1.27886
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0.0965628 0.0118861
\(67\) −6.04334 −0.738312 −0.369156 0.929368i \(-0.620353\pi\)
−0.369156 + 0.929368i \(0.620353\pi\)
\(68\) 6.17921 0.749339
\(69\) −2.15060 −0.258902
\(70\) 0 0
\(71\) 13.4995 1.60210 0.801050 0.598597i \(-0.204275\pi\)
0.801050 + 0.598597i \(0.204275\pi\)
\(72\) −2.88916 −0.340491
\(73\) −1.71348 −0.200548 −0.100274 0.994960i \(-0.531972\pi\)
−0.100274 + 0.994960i \(0.531972\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 5.83553 0.669381
\(77\) −1.01903 −0.116129
\(78\) 2.37228 0.268608
\(79\) 8.25422 0.928672 0.464336 0.885659i \(-0.346293\pi\)
0.464336 + 0.885659i \(0.346293\pi\)
\(80\) 0 0
\(81\) 8.01474 0.890526
\(82\) 1.64077 0.181193
\(83\) −2.41807 −0.265418 −0.132709 0.991155i \(-0.542368\pi\)
−0.132709 + 0.991155i \(0.542368\pi\)
\(84\) −1.16968 −0.127623
\(85\) 0 0
\(86\) 5.34889 0.576785
\(87\) 0.392995 0.0421335
\(88\) −0.290044 −0.0309188
\(89\) −10.2797 −1.08965 −0.544823 0.838551i \(-0.683403\pi\)
−0.544823 + 0.838551i \(0.683403\pi\)
\(90\) 0 0
\(91\) −25.0347 −2.62435
\(92\) 6.45973 0.673473
\(93\) −3.25546 −0.337576
\(94\) −5.69256 −0.587143
\(95\) 0 0
\(96\) −0.332924 −0.0339790
\(97\) −15.1272 −1.53594 −0.767968 0.640489i \(-0.778732\pi\)
−0.767968 + 0.640489i \(0.778732\pi\)
\(98\) 5.34367 0.539793
\(99\) 0.837984 0.0842206
\(100\) 0 0
\(101\) 2.63730 0.262421 0.131210 0.991355i \(-0.458114\pi\)
0.131210 + 0.991355i \(0.458114\pi\)
\(102\) −2.05721 −0.203694
\(103\) 1.72469 0.169939 0.0849695 0.996384i \(-0.472921\pi\)
0.0849695 + 0.996384i \(0.472921\pi\)
\(104\) −7.12558 −0.698720
\(105\) 0 0
\(106\) −9.32034 −0.905271
\(107\) −10.0279 −0.969438 −0.484719 0.874670i \(-0.661078\pi\)
−0.484719 + 0.874670i \(0.661078\pi\)
\(108\) 1.96065 0.188663
\(109\) 4.87361 0.466807 0.233403 0.972380i \(-0.425014\pi\)
0.233403 + 0.972380i \(0.425014\pi\)
\(110\) 0 0
\(111\) −0.332924 −0.0315998
\(112\) 3.51336 0.331981
\(113\) 15.9290 1.49847 0.749236 0.662303i \(-0.230421\pi\)
0.749236 + 0.662303i \(0.230421\pi\)
\(114\) −1.94279 −0.181959
\(115\) 0 0
\(116\) −1.18043 −0.109600
\(117\) 20.5869 1.90326
\(118\) 5.94279 0.547078
\(119\) 21.7098 1.99013
\(120\) 0 0
\(121\) −10.9159 −0.992352
\(122\) −1.27699 −0.115614
\(123\) −0.546253 −0.0492540
\(124\) 9.77838 0.878124
\(125\) 0 0
\(126\) −10.1507 −0.904292
\(127\) −1.31265 −0.116479 −0.0582395 0.998303i \(-0.518549\pi\)
−0.0582395 + 0.998303i \(0.518549\pi\)
\(128\) 1.00000 0.0883883
\(129\) −1.78078 −0.156789
\(130\) 0 0
\(131\) 2.41562 0.211054 0.105527 0.994416i \(-0.466347\pi\)
0.105527 + 0.994416i \(0.466347\pi\)
\(132\) 0.0965628 0.00840471
\(133\) 20.5023 1.77778
\(134\) −6.04334 −0.522065
\(135\) 0 0
\(136\) 6.17921 0.529862
\(137\) −6.87366 −0.587256 −0.293628 0.955920i \(-0.594863\pi\)
−0.293628 + 0.955920i \(0.594863\pi\)
\(138\) −2.15060 −0.183071
\(139\) −9.80616 −0.831748 −0.415874 0.909422i \(-0.636524\pi\)
−0.415874 + 0.909422i \(0.636524\pi\)
\(140\) 0 0
\(141\) 1.89519 0.159604
\(142\) 13.4995 1.13286
\(143\) 2.06673 0.172829
\(144\) −2.88916 −0.240763
\(145\) 0 0
\(146\) −1.71348 −0.141809
\(147\) −1.77904 −0.146733
\(148\) 1.00000 0.0821995
\(149\) −9.07687 −0.743606 −0.371803 0.928312i \(-0.621260\pi\)
−0.371803 + 0.928312i \(0.621260\pi\)
\(150\) 0 0
\(151\) 20.1964 1.64356 0.821780 0.569805i \(-0.192981\pi\)
0.821780 + 0.569805i \(0.192981\pi\)
\(152\) 5.83553 0.473324
\(153\) −17.8527 −1.44331
\(154\) −1.01903 −0.0821157
\(155\) 0 0
\(156\) 2.37228 0.189934
\(157\) 17.2677 1.37811 0.689057 0.724707i \(-0.258025\pi\)
0.689057 + 0.724707i \(0.258025\pi\)
\(158\) 8.25422 0.656670
\(159\) 3.10297 0.246081
\(160\) 0 0
\(161\) 22.6953 1.78864
\(162\) 8.01474 0.629697
\(163\) −16.6901 −1.30727 −0.653634 0.756810i \(-0.726757\pi\)
−0.653634 + 0.756810i \(0.726757\pi\)
\(164\) 1.64077 0.128123
\(165\) 0 0
\(166\) −2.41807 −0.187679
\(167\) −1.47354 −0.114026 −0.0570130 0.998373i \(-0.518158\pi\)
−0.0570130 + 0.998373i \(0.518158\pi\)
\(168\) −1.16968 −0.0902430
\(169\) 37.7738 2.90568
\(170\) 0 0
\(171\) −16.8598 −1.28930
\(172\) 5.34889 0.407849
\(173\) −13.8432 −1.05248 −0.526240 0.850336i \(-0.676399\pi\)
−0.526240 + 0.850336i \(0.676399\pi\)
\(174\) 0.392995 0.0297929
\(175\) 0 0
\(176\) −0.290044 −0.0218629
\(177\) −1.97850 −0.148713
\(178\) −10.2797 −0.770496
\(179\) 21.7817 1.62804 0.814020 0.580836i \(-0.197274\pi\)
0.814020 + 0.580836i \(0.197274\pi\)
\(180\) 0 0
\(181\) 2.03100 0.150963 0.0754817 0.997147i \(-0.475951\pi\)
0.0754817 + 0.997147i \(0.475951\pi\)
\(182\) −25.0347 −1.85569
\(183\) 0.425143 0.0314275
\(184\) 6.45973 0.476217
\(185\) 0 0
\(186\) −3.25546 −0.238702
\(187\) −1.79224 −0.131062
\(188\) −5.69256 −0.415173
\(189\) 6.88845 0.501061
\(190\) 0 0
\(191\) −8.44668 −0.611180 −0.305590 0.952163i \(-0.598854\pi\)
−0.305590 + 0.952163i \(0.598854\pi\)
\(192\) −0.332924 −0.0240268
\(193\) −3.64159 −0.262127 −0.131064 0.991374i \(-0.541839\pi\)
−0.131064 + 0.991374i \(0.541839\pi\)
\(194\) −15.1272 −1.08607
\(195\) 0 0
\(196\) 5.34367 0.381691
\(197\) −10.1939 −0.726288 −0.363144 0.931733i \(-0.618297\pi\)
−0.363144 + 0.931733i \(0.618297\pi\)
\(198\) 0.837984 0.0595530
\(199\) −5.25299 −0.372375 −0.186187 0.982514i \(-0.559613\pi\)
−0.186187 + 0.982514i \(0.559613\pi\)
\(200\) 0 0
\(201\) 2.01198 0.141914
\(202\) 2.63730 0.185560
\(203\) −4.14728 −0.291082
\(204\) −2.05721 −0.144033
\(205\) 0 0
\(206\) 1.72469 0.120165
\(207\) −18.6632 −1.29718
\(208\) −7.12558 −0.494070
\(209\) −1.69256 −0.117077
\(210\) 0 0
\(211\) −4.81998 −0.331821 −0.165910 0.986141i \(-0.553056\pi\)
−0.165910 + 0.986141i \(0.553056\pi\)
\(212\) −9.32034 −0.640123
\(213\) −4.49433 −0.307946
\(214\) −10.0279 −0.685496
\(215\) 0 0
\(216\) 1.96065 0.133405
\(217\) 34.3549 2.33216
\(218\) 4.87361 0.330082
\(219\) 0.570460 0.0385481
\(220\) 0 0
\(221\) −44.0304 −2.96180
\(222\) −0.332924 −0.0223444
\(223\) −3.70392 −0.248033 −0.124017 0.992280i \(-0.539578\pi\)
−0.124017 + 0.992280i \(0.539578\pi\)
\(224\) 3.51336 0.234746
\(225\) 0 0
\(226\) 15.9290 1.05958
\(227\) −6.31512 −0.419149 −0.209575 0.977793i \(-0.567208\pi\)
−0.209575 + 0.977793i \(0.567208\pi\)
\(228\) −1.94279 −0.128664
\(229\) −20.5548 −1.35830 −0.679150 0.734000i \(-0.737651\pi\)
−0.679150 + 0.734000i \(0.737651\pi\)
\(230\) 0 0
\(231\) 0.339260 0.0223216
\(232\) −1.18043 −0.0774992
\(233\) −21.4393 −1.40453 −0.702267 0.711914i \(-0.747829\pi\)
−0.702267 + 0.711914i \(0.747829\pi\)
\(234\) 20.5869 1.34581
\(235\) 0 0
\(236\) 5.94279 0.386843
\(237\) −2.74803 −0.178504
\(238\) 21.7098 1.40723
\(239\) −20.7479 −1.34207 −0.671034 0.741426i \(-0.734150\pi\)
−0.671034 + 0.741426i \(0.734150\pi\)
\(240\) 0 0
\(241\) 10.7731 0.693957 0.346978 0.937873i \(-0.387208\pi\)
0.346978 + 0.937873i \(0.387208\pi\)
\(242\) −10.9159 −0.701699
\(243\) −8.55024 −0.548498
\(244\) −1.27699 −0.0817512
\(245\) 0 0
\(246\) −0.546253 −0.0348278
\(247\) −41.5815 −2.64577
\(248\) 9.77838 0.620927
\(249\) 0.805036 0.0510171
\(250\) 0 0
\(251\) −4.46812 −0.282025 −0.141013 0.990008i \(-0.545036\pi\)
−0.141013 + 0.990008i \(0.545036\pi\)
\(252\) −10.1507 −0.639431
\(253\) −1.87361 −0.117793
\(254\) −1.31265 −0.0823631
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 1.24594 0.0777194 0.0388597 0.999245i \(-0.487627\pi\)
0.0388597 + 0.999245i \(0.487627\pi\)
\(258\) −1.78078 −0.110866
\(259\) 3.51336 0.218309
\(260\) 0 0
\(261\) 3.41046 0.211102
\(262\) 2.41562 0.149237
\(263\) −16.3230 −1.00652 −0.503259 0.864135i \(-0.667866\pi\)
−0.503259 + 0.864135i \(0.667866\pi\)
\(264\) 0.0965628 0.00594303
\(265\) 0 0
\(266\) 20.5023 1.25708
\(267\) 3.42236 0.209445
\(268\) −6.04334 −0.369156
\(269\) −18.5763 −1.13262 −0.566309 0.824193i \(-0.691629\pi\)
−0.566309 + 0.824193i \(0.691629\pi\)
\(270\) 0 0
\(271\) 16.4181 0.997327 0.498663 0.866796i \(-0.333824\pi\)
0.498663 + 0.866796i \(0.333824\pi\)
\(272\) 6.17921 0.374669
\(273\) 8.33466 0.504437
\(274\) −6.87366 −0.415253
\(275\) 0 0
\(276\) −2.15060 −0.129451
\(277\) −15.6104 −0.937937 −0.468968 0.883215i \(-0.655374\pi\)
−0.468968 + 0.883215i \(0.655374\pi\)
\(278\) −9.80616 −0.588134
\(279\) −28.2513 −1.69136
\(280\) 0 0
\(281\) −21.8665 −1.30445 −0.652224 0.758026i \(-0.726164\pi\)
−0.652224 + 0.758026i \(0.726164\pi\)
\(282\) 1.89519 0.112857
\(283\) 2.24778 0.133616 0.0668082 0.997766i \(-0.478718\pi\)
0.0668082 + 0.997766i \(0.478718\pi\)
\(284\) 13.4995 0.801050
\(285\) 0 0
\(286\) 2.06673 0.122208
\(287\) 5.76461 0.340274
\(288\) −2.88916 −0.170245
\(289\) 21.1826 1.24603
\(290\) 0 0
\(291\) 5.03622 0.295228
\(292\) −1.71348 −0.100274
\(293\) 26.8794 1.57031 0.785157 0.619297i \(-0.212582\pi\)
0.785157 + 0.619297i \(0.212582\pi\)
\(294\) −1.77904 −0.103756
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) −0.568674 −0.0329978
\(298\) −9.07687 −0.525809
\(299\) −46.0293 −2.66194
\(300\) 0 0
\(301\) 18.7926 1.08318
\(302\) 20.1964 1.16217
\(303\) −0.878021 −0.0504410
\(304\) 5.83553 0.334691
\(305\) 0 0
\(306\) −17.8527 −1.02057
\(307\) 23.9979 1.36963 0.684815 0.728717i \(-0.259883\pi\)
0.684815 + 0.728717i \(0.259883\pi\)
\(308\) −1.01903 −0.0580645
\(309\) −0.574192 −0.0326647
\(310\) 0 0
\(311\) −5.07563 −0.287812 −0.143906 0.989591i \(-0.545966\pi\)
−0.143906 + 0.989591i \(0.545966\pi\)
\(312\) 2.37228 0.134304
\(313\) −8.96950 −0.506986 −0.253493 0.967337i \(-0.581580\pi\)
−0.253493 + 0.967337i \(0.581580\pi\)
\(314\) 17.2677 0.974474
\(315\) 0 0
\(316\) 8.25422 0.464336
\(317\) −22.0181 −1.23666 −0.618330 0.785918i \(-0.712191\pi\)
−0.618330 + 0.785918i \(0.712191\pi\)
\(318\) 3.10297 0.174006
\(319\) 0.342377 0.0191695
\(320\) 0 0
\(321\) 3.33855 0.186339
\(322\) 22.6953 1.26476
\(323\) 36.0589 2.00637
\(324\) 8.01474 0.445263
\(325\) 0 0
\(326\) −16.6901 −0.924379
\(327\) −1.62254 −0.0897268
\(328\) 1.64077 0.0905964
\(329\) −20.0000 −1.10264
\(330\) 0 0
\(331\) −10.7134 −0.588864 −0.294432 0.955672i \(-0.595130\pi\)
−0.294432 + 0.955672i \(0.595130\pi\)
\(332\) −2.41807 −0.132709
\(333\) −2.88916 −0.158325
\(334\) −1.47354 −0.0806286
\(335\) 0 0
\(336\) −1.16968 −0.0638114
\(337\) 1.34097 0.0730471 0.0365235 0.999333i \(-0.488372\pi\)
0.0365235 + 0.999333i \(0.488372\pi\)
\(338\) 37.7738 2.05463
\(339\) −5.30315 −0.288027
\(340\) 0 0
\(341\) −2.83616 −0.153587
\(342\) −16.8598 −0.911673
\(343\) −5.81926 −0.314211
\(344\) 5.34889 0.288393
\(345\) 0 0
\(346\) −13.8432 −0.744216
\(347\) 0.883744 0.0474419 0.0237209 0.999719i \(-0.492449\pi\)
0.0237209 + 0.999719i \(0.492449\pi\)
\(348\) 0.392995 0.0210667
\(349\) −9.00521 −0.482038 −0.241019 0.970520i \(-0.577482\pi\)
−0.241019 + 0.970520i \(0.577482\pi\)
\(350\) 0 0
\(351\) −13.9707 −0.745702
\(352\) −0.290044 −0.0154594
\(353\) 6.79030 0.361411 0.180706 0.983537i \(-0.442162\pi\)
0.180706 + 0.983537i \(0.442162\pi\)
\(354\) −1.97850 −0.105156
\(355\) 0 0
\(356\) −10.2797 −0.544823
\(357\) −7.22771 −0.382531
\(358\) 21.7817 1.15120
\(359\) −10.2263 −0.539722 −0.269861 0.962899i \(-0.586978\pi\)
−0.269861 + 0.962899i \(0.586978\pi\)
\(360\) 0 0
\(361\) 15.0534 0.792286
\(362\) 2.03100 0.106747
\(363\) 3.63416 0.190744
\(364\) −25.0347 −1.31217
\(365\) 0 0
\(366\) 0.425143 0.0222226
\(367\) 17.1583 0.895657 0.447829 0.894119i \(-0.352198\pi\)
0.447829 + 0.894119i \(0.352198\pi\)
\(368\) 6.45973 0.336737
\(369\) −4.74045 −0.246778
\(370\) 0 0
\(371\) −32.7457 −1.70007
\(372\) −3.25546 −0.168788
\(373\) 17.1601 0.888515 0.444257 0.895899i \(-0.353468\pi\)
0.444257 + 0.895899i \(0.353468\pi\)
\(374\) −1.79224 −0.0926747
\(375\) 0 0
\(376\) −5.69256 −0.293571
\(377\) 8.41126 0.433202
\(378\) 6.88845 0.354304
\(379\) −27.5435 −1.41482 −0.707408 0.706805i \(-0.750136\pi\)
−0.707408 + 0.706805i \(0.750136\pi\)
\(380\) 0 0
\(381\) 0.437014 0.0223889
\(382\) −8.44668 −0.432170
\(383\) −3.49954 −0.178818 −0.0894091 0.995995i \(-0.528498\pi\)
−0.0894091 + 0.995995i \(0.528498\pi\)
\(384\) −0.332924 −0.0169895
\(385\) 0 0
\(386\) −3.64159 −0.185352
\(387\) −15.4538 −0.785561
\(388\) −15.1272 −0.767968
\(389\) −13.0015 −0.659203 −0.329602 0.944120i \(-0.606914\pi\)
−0.329602 + 0.944120i \(0.606914\pi\)
\(390\) 0 0
\(391\) 39.9160 2.01864
\(392\) 5.34367 0.269896
\(393\) −0.804219 −0.0405675
\(394\) −10.1939 −0.513563
\(395\) 0 0
\(396\) 0.837984 0.0421103
\(397\) 26.6658 1.33832 0.669160 0.743118i \(-0.266654\pi\)
0.669160 + 0.743118i \(0.266654\pi\)
\(398\) −5.25299 −0.263309
\(399\) −6.82572 −0.341713
\(400\) 0 0
\(401\) 20.3546 1.01646 0.508231 0.861221i \(-0.330300\pi\)
0.508231 + 0.861221i \(0.330300\pi\)
\(402\) 2.01198 0.100348
\(403\) −69.6766 −3.47084
\(404\) 2.63730 0.131210
\(405\) 0 0
\(406\) −4.14728 −0.205826
\(407\) −0.290044 −0.0143770
\(408\) −2.05721 −0.101847
\(409\) 29.9603 1.48144 0.740720 0.671814i \(-0.234485\pi\)
0.740720 + 0.671814i \(0.234485\pi\)
\(410\) 0 0
\(411\) 2.28841 0.112879
\(412\) 1.72469 0.0849695
\(413\) 20.8791 1.02740
\(414\) −18.6632 −0.917246
\(415\) 0 0
\(416\) −7.12558 −0.349360
\(417\) 3.26471 0.159874
\(418\) −1.69256 −0.0827859
\(419\) 23.3149 1.13901 0.569504 0.821988i \(-0.307135\pi\)
0.569504 + 0.821988i \(0.307135\pi\)
\(420\) 0 0
\(421\) 24.0289 1.17110 0.585548 0.810638i \(-0.300880\pi\)
0.585548 + 0.810638i \(0.300880\pi\)
\(422\) −4.81998 −0.234633
\(423\) 16.4467 0.799667
\(424\) −9.32034 −0.452636
\(425\) 0 0
\(426\) −4.49433 −0.217751
\(427\) −4.48654 −0.217119
\(428\) −10.0279 −0.484719
\(429\) −0.688066 −0.0332201
\(430\) 0 0
\(431\) −19.2917 −0.929247 −0.464624 0.885508i \(-0.653810\pi\)
−0.464624 + 0.885508i \(0.653810\pi\)
\(432\) 1.96065 0.0943316
\(433\) 12.5816 0.604634 0.302317 0.953207i \(-0.402240\pi\)
0.302317 + 0.953207i \(0.402240\pi\)
\(434\) 34.3549 1.64909
\(435\) 0 0
\(436\) 4.87361 0.233403
\(437\) 37.6959 1.80324
\(438\) 0.570460 0.0272576
\(439\) 14.0428 0.670225 0.335113 0.942178i \(-0.391226\pi\)
0.335113 + 0.942178i \(0.391226\pi\)
\(440\) 0 0
\(441\) −15.4387 −0.735178
\(442\) −44.0304 −2.09431
\(443\) 24.5696 1.16734 0.583668 0.811992i \(-0.301617\pi\)
0.583668 + 0.811992i \(0.301617\pi\)
\(444\) −0.332924 −0.0157999
\(445\) 0 0
\(446\) −3.70392 −0.175386
\(447\) 3.02191 0.142932
\(448\) 3.51336 0.165990
\(449\) −12.7574 −0.602061 −0.301030 0.953615i \(-0.597331\pi\)
−0.301030 + 0.953615i \(0.597331\pi\)
\(450\) 0 0
\(451\) −0.475896 −0.0224091
\(452\) 15.9290 0.749236
\(453\) −6.72387 −0.315915
\(454\) −6.31512 −0.296383
\(455\) 0 0
\(456\) −1.94279 −0.0909795
\(457\) −3.34889 −0.156654 −0.0783272 0.996928i \(-0.524958\pi\)
−0.0783272 + 0.996928i \(0.524958\pi\)
\(458\) −20.5548 −0.960463
\(459\) 12.1152 0.565491
\(460\) 0 0
\(461\) −0.859501 −0.0400310 −0.0200155 0.999800i \(-0.506372\pi\)
−0.0200155 + 0.999800i \(0.506372\pi\)
\(462\) 0.339260 0.0157838
\(463\) −13.8711 −0.644643 −0.322321 0.946630i \(-0.604463\pi\)
−0.322321 + 0.946630i \(0.604463\pi\)
\(464\) −1.18043 −0.0548002
\(465\) 0 0
\(466\) −21.4393 −0.993155
\(467\) −40.9281 −1.89392 −0.946962 0.321344i \(-0.895865\pi\)
−0.946962 + 0.321344i \(0.895865\pi\)
\(468\) 20.5869 0.951632
\(469\) −21.2324 −0.980422
\(470\) 0 0
\(471\) −5.74885 −0.264893
\(472\) 5.94279 0.273539
\(473\) −1.55141 −0.0713341
\(474\) −2.74803 −0.126221
\(475\) 0 0
\(476\) 21.7098 0.995065
\(477\) 26.9280 1.23295
\(478\) −20.7479 −0.948986
\(479\) 8.58990 0.392483 0.196241 0.980556i \(-0.437126\pi\)
0.196241 + 0.980556i \(0.437126\pi\)
\(480\) 0 0
\(481\) −7.12558 −0.324898
\(482\) 10.7731 0.490702
\(483\) −7.55583 −0.343802
\(484\) −10.9159 −0.496176
\(485\) 0 0
\(486\) −8.55024 −0.387847
\(487\) −25.8389 −1.17087 −0.585436 0.810718i \(-0.699077\pi\)
−0.585436 + 0.810718i \(0.699077\pi\)
\(488\) −1.27699 −0.0578068
\(489\) 5.55654 0.251275
\(490\) 0 0
\(491\) 11.4864 0.518376 0.259188 0.965827i \(-0.416545\pi\)
0.259188 + 0.965827i \(0.416545\pi\)
\(492\) −0.546253 −0.0246270
\(493\) −7.29413 −0.328511
\(494\) −41.5815 −1.87084
\(495\) 0 0
\(496\) 9.77838 0.439062
\(497\) 47.4287 2.12747
\(498\) 0.805036 0.0360745
\(499\) −9.74599 −0.436290 −0.218145 0.975916i \(-0.570001\pi\)
−0.218145 + 0.975916i \(0.570001\pi\)
\(500\) 0 0
\(501\) 0.490578 0.0219174
\(502\) −4.46812 −0.199422
\(503\) −17.4803 −0.779408 −0.389704 0.920940i \(-0.627423\pi\)
−0.389704 + 0.920940i \(0.627423\pi\)
\(504\) −10.1507 −0.452146
\(505\) 0 0
\(506\) −1.87361 −0.0832920
\(507\) −12.5758 −0.558512
\(508\) −1.31265 −0.0582395
\(509\) −16.3273 −0.723695 −0.361847 0.932237i \(-0.617854\pi\)
−0.361847 + 0.932237i \(0.617854\pi\)
\(510\) 0 0
\(511\) −6.02007 −0.266312
\(512\) 1.00000 0.0441942
\(513\) 11.4414 0.505151
\(514\) 1.24594 0.0549559
\(515\) 0 0
\(516\) −1.78078 −0.0783943
\(517\) 1.65109 0.0726151
\(518\) 3.51336 0.154368
\(519\) 4.60875 0.202301
\(520\) 0 0
\(521\) −4.78335 −0.209562 −0.104781 0.994495i \(-0.533414\pi\)
−0.104781 + 0.994495i \(0.533414\pi\)
\(522\) 3.41046 0.149272
\(523\) 24.1224 1.05480 0.527399 0.849618i \(-0.323167\pi\)
0.527399 + 0.849618i \(0.323167\pi\)
\(524\) 2.41562 0.105527
\(525\) 0 0
\(526\) −16.3230 −0.711716
\(527\) 60.4226 2.63205
\(528\) 0.0965628 0.00420236
\(529\) 18.7281 0.814264
\(530\) 0 0
\(531\) −17.1697 −0.745101
\(532\) 20.5023 0.888888
\(533\) −11.6914 −0.506412
\(534\) 3.42236 0.148100
\(535\) 0 0
\(536\) −6.04334 −0.261033
\(537\) −7.25166 −0.312932
\(538\) −18.5763 −0.800881
\(539\) −1.54990 −0.0667590
\(540\) 0 0
\(541\) 4.11745 0.177023 0.0885115 0.996075i \(-0.471789\pi\)
0.0885115 + 0.996075i \(0.471789\pi\)
\(542\) 16.4181 0.705217
\(543\) −0.676171 −0.0290173
\(544\) 6.17921 0.264931
\(545\) 0 0
\(546\) 8.33466 0.356691
\(547\) −9.22742 −0.394536 −0.197268 0.980350i \(-0.563207\pi\)
−0.197268 + 0.980350i \(0.563207\pi\)
\(548\) −6.87366 −0.293628
\(549\) 3.68944 0.157462
\(550\) 0 0
\(551\) −6.88845 −0.293458
\(552\) −2.15060 −0.0915357
\(553\) 29.0000 1.23321
\(554\) −15.6104 −0.663222
\(555\) 0 0
\(556\) −9.80616 −0.415874
\(557\) 2.25125 0.0953885 0.0476943 0.998862i \(-0.484813\pi\)
0.0476943 + 0.998862i \(0.484813\pi\)
\(558\) −28.2513 −1.19597
\(559\) −38.1139 −1.61205
\(560\) 0 0
\(561\) 0.596681 0.0251919
\(562\) −21.8665 −0.922384
\(563\) 5.79316 0.244153 0.122076 0.992521i \(-0.461045\pi\)
0.122076 + 0.992521i \(0.461045\pi\)
\(564\) 1.89519 0.0798020
\(565\) 0 0
\(566\) 2.24778 0.0944811
\(567\) 28.1586 1.18255
\(568\) 13.4995 0.566428
\(569\) −34.4807 −1.44551 −0.722753 0.691106i \(-0.757124\pi\)
−0.722753 + 0.691106i \(0.757124\pi\)
\(570\) 0 0
\(571\) −19.2152 −0.804133 −0.402066 0.915611i \(-0.631708\pi\)
−0.402066 + 0.915611i \(0.631708\pi\)
\(572\) 2.06673 0.0864144
\(573\) 2.81211 0.117477
\(574\) 5.76461 0.240610
\(575\) 0 0
\(576\) −2.88916 −0.120382
\(577\) 6.33354 0.263669 0.131834 0.991272i \(-0.457913\pi\)
0.131834 + 0.991272i \(0.457913\pi\)
\(578\) 21.1826 0.881079
\(579\) 1.21237 0.0503845
\(580\) 0 0
\(581\) −8.49555 −0.352455
\(582\) 5.03622 0.208758
\(583\) 2.70331 0.111960
\(584\) −1.71348 −0.0709044
\(585\) 0 0
\(586\) 26.8794 1.11038
\(587\) −21.9099 −0.904320 −0.452160 0.891937i \(-0.649346\pi\)
−0.452160 + 0.891937i \(0.649346\pi\)
\(588\) −1.77904 −0.0733664
\(589\) 57.0620 2.35120
\(590\) 0 0
\(591\) 3.39381 0.139603
\(592\) 1.00000 0.0410997
\(593\) −9.40482 −0.386210 −0.193105 0.981178i \(-0.561856\pi\)
−0.193105 + 0.981178i \(0.561856\pi\)
\(594\) −0.568674 −0.0233330
\(595\) 0 0
\(596\) −9.07687 −0.371803
\(597\) 1.74885 0.0715756
\(598\) −46.0293 −1.88228
\(599\) 16.4095 0.670474 0.335237 0.942134i \(-0.391184\pi\)
0.335237 + 0.942134i \(0.391184\pi\)
\(600\) 0 0
\(601\) −7.01392 −0.286104 −0.143052 0.989715i \(-0.545692\pi\)
−0.143052 + 0.989715i \(0.545692\pi\)
\(602\) 18.7926 0.765927
\(603\) 17.4602 0.711034
\(604\) 20.1964 0.821780
\(605\) 0 0
\(606\) −0.878021 −0.0356672
\(607\) 16.5621 0.672234 0.336117 0.941820i \(-0.390886\pi\)
0.336117 + 0.941820i \(0.390886\pi\)
\(608\) 5.83553 0.236662
\(609\) 1.38073 0.0559500
\(610\) 0 0
\(611\) 40.5628 1.64099
\(612\) −17.8527 −0.721653
\(613\) 23.9370 0.966804 0.483402 0.875398i \(-0.339401\pi\)
0.483402 + 0.875398i \(0.339401\pi\)
\(614\) 23.9979 0.968475
\(615\) 0 0
\(616\) −1.01903 −0.0410578
\(617\) −13.9489 −0.561563 −0.280781 0.959772i \(-0.590594\pi\)
−0.280781 + 0.959772i \(0.590594\pi\)
\(618\) −0.574192 −0.0230974
\(619\) 21.6942 0.871964 0.435982 0.899955i \(-0.356401\pi\)
0.435982 + 0.899955i \(0.356401\pi\)
\(620\) 0 0
\(621\) 12.6652 0.508238
\(622\) −5.07563 −0.203514
\(623\) −36.1163 −1.44697
\(624\) 2.37228 0.0949671
\(625\) 0 0
\(626\) −8.96950 −0.358493
\(627\) 0.563495 0.0225038
\(628\) 17.2677 0.689057
\(629\) 6.17921 0.246381
\(630\) 0 0
\(631\) −32.4902 −1.29342 −0.646708 0.762738i \(-0.723855\pi\)
−0.646708 + 0.762738i \(0.723855\pi\)
\(632\) 8.25422 0.328335
\(633\) 1.60469 0.0637806
\(634\) −22.0181 −0.874451
\(635\) 0 0
\(636\) 3.10297 0.123041
\(637\) −38.0768 −1.50866
\(638\) 0.342377 0.0135549
\(639\) −39.0024 −1.54291
\(640\) 0 0
\(641\) −28.1621 −1.11234 −0.556168 0.831070i \(-0.687729\pi\)
−0.556168 + 0.831070i \(0.687729\pi\)
\(642\) 3.33855 0.131762
\(643\) 8.32452 0.328287 0.164144 0.986436i \(-0.447514\pi\)
0.164144 + 0.986436i \(0.447514\pi\)
\(644\) 22.6953 0.894321
\(645\) 0 0
\(646\) 36.0589 1.41872
\(647\) 1.75827 0.0691249 0.0345624 0.999403i \(-0.488996\pi\)
0.0345624 + 0.999403i \(0.488996\pi\)
\(648\) 8.01474 0.314849
\(649\) −1.72367 −0.0676600
\(650\) 0 0
\(651\) −11.4376 −0.448275
\(652\) −16.6901 −0.653634
\(653\) 24.7416 0.968213 0.484106 0.875009i \(-0.339145\pi\)
0.484106 + 0.875009i \(0.339145\pi\)
\(654\) −1.62254 −0.0634464
\(655\) 0 0
\(656\) 1.64077 0.0640613
\(657\) 4.95053 0.193138
\(658\) −20.0000 −0.779681
\(659\) 18.2640 0.711466 0.355733 0.934588i \(-0.384231\pi\)
0.355733 + 0.934588i \(0.384231\pi\)
\(660\) 0 0
\(661\) 2.61857 0.101851 0.0509253 0.998702i \(-0.483783\pi\)
0.0509253 + 0.998702i \(0.483783\pi\)
\(662\) −10.7134 −0.416390
\(663\) 14.6588 0.569300
\(664\) −2.41807 −0.0938394
\(665\) 0 0
\(666\) −2.88916 −0.111953
\(667\) −7.62527 −0.295252
\(668\) −1.47354 −0.0570130
\(669\) 1.23313 0.0476754
\(670\) 0 0
\(671\) 0.370385 0.0142986
\(672\) −1.16968 −0.0451215
\(673\) −48.4181 −1.86638 −0.933190 0.359384i \(-0.882987\pi\)
−0.933190 + 0.359384i \(0.882987\pi\)
\(674\) 1.34097 0.0516521
\(675\) 0 0
\(676\) 37.7738 1.45284
\(677\) −7.49862 −0.288195 −0.144098 0.989563i \(-0.546028\pi\)
−0.144098 + 0.989563i \(0.546028\pi\)
\(678\) −5.30315 −0.203666
\(679\) −53.1473 −2.03961
\(680\) 0 0
\(681\) 2.10246 0.0805664
\(682\) −2.83616 −0.108602
\(683\) −33.2987 −1.27414 −0.637070 0.770806i \(-0.719854\pi\)
−0.637070 + 0.770806i \(0.719854\pi\)
\(684\) −16.8598 −0.644650
\(685\) 0 0
\(686\) −5.81926 −0.222181
\(687\) 6.84320 0.261084
\(688\) 5.34889 0.203924
\(689\) 66.4128 2.53012
\(690\) 0 0
\(691\) 38.3695 1.45964 0.729822 0.683638i \(-0.239603\pi\)
0.729822 + 0.683638i \(0.239603\pi\)
\(692\) −13.8432 −0.526240
\(693\) 2.94414 0.111839
\(694\) 0.883744 0.0335465
\(695\) 0 0
\(696\) 0.392995 0.0148964
\(697\) 10.1387 0.384029
\(698\) −9.00521 −0.340852
\(699\) 7.13766 0.269971
\(700\) 0 0
\(701\) −40.6339 −1.53472 −0.767360 0.641217i \(-0.778430\pi\)
−0.767360 + 0.641217i \(0.778430\pi\)
\(702\) −13.9707 −0.527291
\(703\) 5.83553 0.220091
\(704\) −0.290044 −0.0109315
\(705\) 0 0
\(706\) 6.79030 0.255556
\(707\) 9.26577 0.348475
\(708\) −1.97850 −0.0743566
\(709\) 11.1957 0.420464 0.210232 0.977651i \(-0.432578\pi\)
0.210232 + 0.977651i \(0.432578\pi\)
\(710\) 0 0
\(711\) −23.8478 −0.894361
\(712\) −10.2797 −0.385248
\(713\) 63.1656 2.36557
\(714\) −7.22771 −0.270490
\(715\) 0 0
\(716\) 21.7817 0.814020
\(717\) 6.90748 0.257964
\(718\) −10.2263 −0.381641
\(719\) 30.2226 1.12711 0.563556 0.826078i \(-0.309433\pi\)
0.563556 + 0.826078i \(0.309433\pi\)
\(720\) 0 0
\(721\) 6.05946 0.225666
\(722\) 15.0534 0.560231
\(723\) −3.58663 −0.133388
\(724\) 2.03100 0.0754817
\(725\) 0 0
\(726\) 3.63416 0.134876
\(727\) −19.1911 −0.711758 −0.355879 0.934532i \(-0.615818\pi\)
−0.355879 + 0.934532i \(0.615818\pi\)
\(728\) −25.0347 −0.927847
\(729\) −21.1976 −0.785097
\(730\) 0 0
\(731\) 33.0519 1.22247
\(732\) 0.425143 0.0157137
\(733\) 5.39567 0.199294 0.0996468 0.995023i \(-0.468229\pi\)
0.0996468 + 0.995023i \(0.468229\pi\)
\(734\) 17.1583 0.633325
\(735\) 0 0
\(736\) 6.45973 0.238109
\(737\) 1.75284 0.0645665
\(738\) −4.74045 −0.174498
\(739\) 24.7127 0.909072 0.454536 0.890728i \(-0.349805\pi\)
0.454536 + 0.890728i \(0.349805\pi\)
\(740\) 0 0
\(741\) 13.8435 0.508554
\(742\) −32.7457 −1.20213
\(743\) 45.4537 1.66753 0.833767 0.552116i \(-0.186180\pi\)
0.833767 + 0.552116i \(0.186180\pi\)
\(744\) −3.25546 −0.119351
\(745\) 0 0
\(746\) 17.1601 0.628275
\(747\) 6.98620 0.255612
\(748\) −1.79224 −0.0655309
\(749\) −35.2317 −1.28734
\(750\) 0 0
\(751\) 49.6360 1.81124 0.905621 0.424088i \(-0.139405\pi\)
0.905621 + 0.424088i \(0.139405\pi\)
\(752\) −5.69256 −0.207586
\(753\) 1.48755 0.0542093
\(754\) 8.41126 0.306320
\(755\) 0 0
\(756\) 6.88845 0.250530
\(757\) −27.9481 −1.01579 −0.507896 0.861419i \(-0.669576\pi\)
−0.507896 + 0.861419i \(0.669576\pi\)
\(758\) −27.5435 −1.00043
\(759\) 0.623769 0.0226414
\(760\) 0 0
\(761\) −53.4744 −1.93844 −0.969222 0.246188i \(-0.920822\pi\)
−0.969222 + 0.246188i \(0.920822\pi\)
\(762\) 0.437014 0.0158313
\(763\) 17.1227 0.619884
\(764\) −8.44668 −0.305590
\(765\) 0 0
\(766\) −3.49954 −0.126444
\(767\) −42.3458 −1.52902
\(768\) −0.332924 −0.0120134
\(769\) −8.46659 −0.305313 −0.152657 0.988279i \(-0.548783\pi\)
−0.152657 + 0.988279i \(0.548783\pi\)
\(770\) 0 0
\(771\) −0.414803 −0.0149388
\(772\) −3.64159 −0.131064
\(773\) −0.530866 −0.0190939 −0.00954697 0.999954i \(-0.503039\pi\)
−0.00954697 + 0.999954i \(0.503039\pi\)
\(774\) −15.4538 −0.555475
\(775\) 0 0
\(776\) −15.1272 −0.543035
\(777\) −1.16968 −0.0419621
\(778\) −13.0015 −0.466127
\(779\) 9.57477 0.343052
\(780\) 0 0
\(781\) −3.91546 −0.140106
\(782\) 39.9160 1.42739
\(783\) −2.31441 −0.0827102
\(784\) 5.34367 0.190846
\(785\) 0 0
\(786\) −0.804219 −0.0286855
\(787\) −2.16917 −0.0773227 −0.0386613 0.999252i \(-0.512309\pi\)
−0.0386613 + 0.999252i \(0.512309\pi\)
\(788\) −10.1939 −0.363144
\(789\) 5.43432 0.193467
\(790\) 0 0
\(791\) 55.9642 1.98986
\(792\) 0.837984 0.0297765
\(793\) 9.09932 0.323126
\(794\) 26.6658 0.946336
\(795\) 0 0
\(796\) −5.25299 −0.186187
\(797\) 26.2510 0.929857 0.464928 0.885348i \(-0.346080\pi\)
0.464928 + 0.885348i \(0.346080\pi\)
\(798\) −6.82572 −0.241628
\(799\) −35.1755 −1.24442
\(800\) 0 0
\(801\) 29.6997 1.04939
\(802\) 20.3546 0.718747
\(803\) 0.496986 0.0175382
\(804\) 2.01198 0.0709569
\(805\) 0 0
\(806\) −69.6766 −2.45425
\(807\) 6.18451 0.217705
\(808\) 2.63730 0.0927798
\(809\) 6.67443 0.234661 0.117330 0.993093i \(-0.462566\pi\)
0.117330 + 0.993093i \(0.462566\pi\)
\(810\) 0 0
\(811\) −8.24768 −0.289615 −0.144808 0.989460i \(-0.546256\pi\)
−0.144808 + 0.989460i \(0.546256\pi\)
\(812\) −4.14728 −0.145541
\(813\) −5.46598 −0.191700
\(814\) −0.290044 −0.0101660
\(815\) 0 0
\(816\) −2.05721 −0.0720167
\(817\) 31.2136 1.09203
\(818\) 29.9603 1.04754
\(819\) 72.3292 2.52739
\(820\) 0 0
\(821\) −9.61631 −0.335611 −0.167806 0.985820i \(-0.553668\pi\)
−0.167806 + 0.985820i \(0.553668\pi\)
\(822\) 2.28841 0.0798174
\(823\) 32.7309 1.14093 0.570464 0.821322i \(-0.306763\pi\)
0.570464 + 0.821322i \(0.306763\pi\)
\(824\) 1.72469 0.0600825
\(825\) 0 0
\(826\) 20.8791 0.726478
\(827\) −39.9956 −1.39078 −0.695391 0.718631i \(-0.744769\pi\)
−0.695391 + 0.718631i \(0.744769\pi\)
\(828\) −18.6632 −0.648591
\(829\) 36.8620 1.28027 0.640134 0.768263i \(-0.278879\pi\)
0.640134 + 0.768263i \(0.278879\pi\)
\(830\) 0 0
\(831\) 5.19708 0.180285
\(832\) −7.12558 −0.247035
\(833\) 33.0197 1.14406
\(834\) 3.26471 0.113048
\(835\) 0 0
\(836\) −1.69256 −0.0585385
\(837\) 19.1719 0.662679
\(838\) 23.3149 0.805400
\(839\) 36.7629 1.26919 0.634597 0.772843i \(-0.281166\pi\)
0.634597 + 0.772843i \(0.281166\pi\)
\(840\) 0 0
\(841\) −27.6066 −0.951951
\(842\) 24.0289 0.828089
\(843\) 7.27991 0.250733
\(844\) −4.81998 −0.165910
\(845\) 0 0
\(846\) 16.4467 0.565450
\(847\) −38.3514 −1.31777
\(848\) −9.32034 −0.320062
\(849\) −0.748340 −0.0256830
\(850\) 0 0
\(851\) 6.45973 0.221437
\(852\) −4.49433 −0.153973
\(853\) 15.9547 0.546278 0.273139 0.961975i \(-0.411938\pi\)
0.273139 + 0.961975i \(0.411938\pi\)
\(854\) −4.48654 −0.153526
\(855\) 0 0
\(856\) −10.0279 −0.342748
\(857\) −10.2157 −0.348963 −0.174481 0.984660i \(-0.555825\pi\)
−0.174481 + 0.984660i \(0.555825\pi\)
\(858\) −0.688066 −0.0234902
\(859\) 7.41899 0.253133 0.126566 0.991958i \(-0.459604\pi\)
0.126566 + 0.991958i \(0.459604\pi\)
\(860\) 0 0
\(861\) −1.91918 −0.0654055
\(862\) −19.2917 −0.657077
\(863\) 9.35839 0.318563 0.159282 0.987233i \(-0.449082\pi\)
0.159282 + 0.987233i \(0.449082\pi\)
\(864\) 1.96065 0.0667025
\(865\) 0 0
\(866\) 12.5816 0.427541
\(867\) −7.05220 −0.239505
\(868\) 34.3549 1.16608
\(869\) −2.39409 −0.0812139
\(870\) 0 0
\(871\) 43.0623 1.45911
\(872\) 4.87361 0.165041
\(873\) 43.7050 1.47919
\(874\) 37.6959 1.27508
\(875\) 0 0
\(876\) 0.570460 0.0192741
\(877\) −13.1486 −0.443997 −0.221998 0.975047i \(-0.571258\pi\)
−0.221998 + 0.975047i \(0.571258\pi\)
\(878\) 14.0428 0.473921
\(879\) −8.94882 −0.301836
\(880\) 0 0
\(881\) 26.7781 0.902178 0.451089 0.892479i \(-0.351036\pi\)
0.451089 + 0.892479i \(0.351036\pi\)
\(882\) −15.4387 −0.519849
\(883\) −16.7245 −0.562824 −0.281412 0.959587i \(-0.590803\pi\)
−0.281412 + 0.959587i \(0.590803\pi\)
\(884\) −44.0304 −1.48090
\(885\) 0 0
\(886\) 24.5696 0.825432
\(887\) 18.1469 0.609312 0.304656 0.952462i \(-0.401458\pi\)
0.304656 + 0.952462i \(0.401458\pi\)
\(888\) −0.332924 −0.0111722
\(889\) −4.61181 −0.154675
\(890\) 0 0
\(891\) −2.32463 −0.0778780
\(892\) −3.70392 −0.124017
\(893\) −33.2191 −1.11164
\(894\) 3.02191 0.101068
\(895\) 0 0
\(896\) 3.51336 0.117373
\(897\) 15.3243 0.511663
\(898\) −12.7574 −0.425721
\(899\) −11.5427 −0.384971
\(900\) 0 0
\(901\) −57.5923 −1.91868
\(902\) −0.475896 −0.0158456
\(903\) −6.25650 −0.208203
\(904\) 15.9290 0.529790
\(905\) 0 0
\(906\) −6.72387 −0.223386
\(907\) 50.1481 1.66514 0.832570 0.553920i \(-0.186869\pi\)
0.832570 + 0.553920i \(0.186869\pi\)
\(908\) −6.31512 −0.209575
\(909\) −7.61958 −0.252725
\(910\) 0 0
\(911\) 2.64691 0.0876960 0.0438480 0.999038i \(-0.486038\pi\)
0.0438480 + 0.999038i \(0.486038\pi\)
\(912\) −1.94279 −0.0643322
\(913\) 0.701348 0.0232112
\(914\) −3.34889 −0.110771
\(915\) 0 0
\(916\) −20.5548 −0.679150
\(917\) 8.48693 0.280263
\(918\) 12.1152 0.399862
\(919\) −39.5369 −1.30420 −0.652101 0.758132i \(-0.726112\pi\)
−0.652101 + 0.758132i \(0.726112\pi\)
\(920\) 0 0
\(921\) −7.98947 −0.263262
\(922\) −0.859501 −0.0283062
\(923\) −96.1920 −3.16620
\(924\) 0.339260 0.0111608
\(925\) 0 0
\(926\) −13.8711 −0.455831
\(927\) −4.98292 −0.163660
\(928\) −1.18043 −0.0387496
\(929\) −39.2999 −1.28939 −0.644693 0.764441i \(-0.723015\pi\)
−0.644693 + 0.764441i \(0.723015\pi\)
\(930\) 0 0
\(931\) 31.1832 1.02199
\(932\) −21.4393 −0.702267
\(933\) 1.68980 0.0553216
\(934\) −40.9281 −1.33921
\(935\) 0 0
\(936\) 20.5869 0.672905
\(937\) −27.7584 −0.906826 −0.453413 0.891301i \(-0.649794\pi\)
−0.453413 + 0.891301i \(0.649794\pi\)
\(938\) −21.2324 −0.693263
\(939\) 2.98617 0.0974499
\(940\) 0 0
\(941\) 48.2212 1.57197 0.785983 0.618249i \(-0.212158\pi\)
0.785983 + 0.618249i \(0.212158\pi\)
\(942\) −5.74885 −0.187308
\(943\) 10.5989 0.345149
\(944\) 5.94279 0.193421
\(945\) 0 0
\(946\) −1.55141 −0.0504408
\(947\) −26.5111 −0.861495 −0.430748 0.902472i \(-0.641750\pi\)
−0.430748 + 0.902472i \(0.641750\pi\)
\(948\) −2.74803 −0.0892519
\(949\) 12.2095 0.396339
\(950\) 0 0
\(951\) 7.33037 0.237703
\(952\) 21.7098 0.703617
\(953\) −31.8487 −1.03168 −0.515840 0.856685i \(-0.672520\pi\)
−0.515840 + 0.856685i \(0.672520\pi\)
\(954\) 26.9280 0.871825
\(955\) 0 0
\(956\) −20.7479 −0.671034
\(957\) −0.113986 −0.00368464
\(958\) 8.58990 0.277527
\(959\) −24.1496 −0.779832
\(960\) 0 0
\(961\) 64.6166 2.08441
\(962\) −7.12558 −0.229738
\(963\) 28.9723 0.933620
\(964\) 10.7731 0.346978
\(965\) 0 0
\(966\) −7.55583 −0.243105
\(967\) 19.4699 0.626109 0.313054 0.949735i \(-0.398648\pi\)
0.313054 + 0.949735i \(0.398648\pi\)
\(968\) −10.9159 −0.350849
\(969\) −12.0049 −0.385653
\(970\) 0 0
\(971\) −21.8178 −0.700168 −0.350084 0.936718i \(-0.613847\pi\)
−0.350084 + 0.936718i \(0.613847\pi\)
\(972\) −8.55024 −0.274249
\(973\) −34.4525 −1.10450
\(974\) −25.8389 −0.827932
\(975\) 0 0
\(976\) −1.27699 −0.0408756
\(977\) −41.8705 −1.33956 −0.669779 0.742561i \(-0.733611\pi\)
−0.669779 + 0.742561i \(0.733611\pi\)
\(978\) 5.55654 0.177679
\(979\) 2.98157 0.0952913
\(980\) 0 0
\(981\) −14.0806 −0.449560
\(982\) 11.4864 0.366547
\(983\) −37.0175 −1.18067 −0.590337 0.807157i \(-0.701005\pi\)
−0.590337 + 0.807157i \(0.701005\pi\)
\(984\) −0.546253 −0.0174139
\(985\) 0 0
\(986\) −7.29413 −0.232292
\(987\) 6.65849 0.211942
\(988\) −41.5815 −1.32288
\(989\) 34.5524 1.09870
\(990\) 0 0
\(991\) −28.4749 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(992\) 9.77838 0.310464
\(993\) 3.56677 0.113188
\(994\) 47.4287 1.50435
\(995\) 0 0
\(996\) 0.805036 0.0255085
\(997\) −14.2442 −0.451118 −0.225559 0.974229i \(-0.572421\pi\)
−0.225559 + 0.974229i \(0.572421\pi\)
\(998\) −9.74599 −0.308504
\(999\) 1.96065 0.0620321
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 1850.2.a.be.1.3 5
5.2 odd 4 370.2.b.d.149.8 yes 10
5.3 odd 4 370.2.b.d.149.3 10
5.4 even 2 1850.2.a.bd.1.3 5
15.2 even 4 3330.2.d.p.1999.4 10
15.8 even 4 3330.2.d.p.1999.9 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.d.149.3 10 5.3 odd 4
370.2.b.d.149.8 yes 10 5.2 odd 4
1850.2.a.bd.1.3 5 5.4 even 2
1850.2.a.be.1.3 5 1.1 even 1 trivial
3330.2.d.p.1999.4 10 15.2 even 4
3330.2.d.p.1999.9 10 15.8 even 4