Properties

Label 1850.2.a.y.1.3
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1524.1
Defining polynomial: \( x^{3} - x^{2} - 7x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.27307\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.27307 q^{3} +1.00000 q^{4} -2.27307 q^{6} +1.27307 q^{7} -1.00000 q^{8} +2.16686 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.27307 q^{3} +1.00000 q^{4} -2.27307 q^{6} +1.27307 q^{7} -1.00000 q^{8} +2.16686 q^{9} -4.43993 q^{11} +2.27307 q^{12} -5.27307 q^{13} -1.27307 q^{14} +1.00000 q^{16} -0.273073 q^{17} -2.16686 q^{18} -5.71301 q^{19} +2.89379 q^{21} +4.43993 q^{22} -6.54615 q^{23} -2.27307 q^{24} +5.27307 q^{26} -1.89379 q^{27} +1.27307 q^{28} +0.546146 q^{29} -5.98608 q^{31} -1.00000 q^{32} -10.0923 q^{33} +0.273073 q^{34} +2.16686 q^{36} -1.00000 q^{37} +5.71301 q^{38} -11.9861 q^{39} +11.8799 q^{41} -2.89379 q^{42} -10.3793 q^{43} -4.43993 q^{44} +6.54615 q^{46} +2.27307 q^{48} -5.37929 q^{49} -0.620715 q^{51} -5.27307 q^{52} +2.33372 q^{53} +1.89379 q^{54} -1.27307 q^{56} -12.9861 q^{57} -0.546146 q^{58} +2.37929 q^{59} +11.8192 q^{61} +5.98608 q^{62} +2.75857 q^{63} +1.00000 q^{64} +10.0923 q^{66} +7.15294 q^{67} -0.273073 q^{68} -14.8799 q^{69} +5.60679 q^{71} -2.16686 q^{72} -9.71301 q^{73} +1.00000 q^{74} -5.71301 q^{76} -5.65236 q^{77} +11.9861 q^{78} +11.4260 q^{79} -10.8053 q^{81} -11.8799 q^{82} +3.89379 q^{83} +2.89379 q^{84} +10.3793 q^{86} +1.24143 q^{87} +4.43993 q^{88} +13.8659 q^{89} -6.71301 q^{91} -6.54615 q^{92} -13.6068 q^{93} -2.27307 q^{96} +0.879866 q^{97} +5.37929 q^{98} -9.62071 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - q^{3} + 3 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} - q^{3} + 3 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 6 q^{9} - 5 q^{11} - q^{12} - 8 q^{13} + 4 q^{14} + 3 q^{16} + 7 q^{17} - 6 q^{18} - q^{19} + 16 q^{21} + 5 q^{22} - 4 q^{23} + q^{24} + 8 q^{26} - 13 q^{27} - 4 q^{28} - 14 q^{29} + 6 q^{31} - 3 q^{32} + q^{33} - 7 q^{34} + 6 q^{36} - 3 q^{37} + q^{38} - 12 q^{39} + 19 q^{41} - 16 q^{42} - 16 q^{43} - 5 q^{44} + 4 q^{46} - q^{48} - q^{49} - 17 q^{51} - 8 q^{52} + 6 q^{53} + 13 q^{54} + 4 q^{56} - 15 q^{57} + 14 q^{58} - 8 q^{59} + 12 q^{61} - 6 q^{62} - 22 q^{63} + 3 q^{64} - q^{66} - 3 q^{67} + 7 q^{68} - 28 q^{69} + 8 q^{71} - 6 q^{72} - 13 q^{73} + 3 q^{74} - q^{76} + 6 q^{77} + 12 q^{78} + 2 q^{79} + 15 q^{81} - 19 q^{82} + 19 q^{83} + 16 q^{84} + 16 q^{86} + 34 q^{87} + 5 q^{88} + q^{89} - 4 q^{91} - 4 q^{92} - 32 q^{93} + q^{96} - 14 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 2.27307 1.31236 0.656180 0.754605i \(-0.272171\pi\)
0.656180 + 0.754605i \(0.272171\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −2.27307 −0.927978
\(7\) 1.27307 0.481176 0.240588 0.970627i \(-0.422660\pi\)
0.240588 + 0.970627i \(0.422660\pi\)
\(8\) −1.00000 −0.353553
\(9\) 2.16686 0.722287
\(10\) 0 0
\(11\) −4.43993 −1.33869 −0.669345 0.742952i \(-0.733425\pi\)
−0.669345 + 0.742952i \(0.733425\pi\)
\(12\) 2.27307 0.656180
\(13\) −5.27307 −1.46249 −0.731244 0.682116i \(-0.761060\pi\)
−0.731244 + 0.682116i \(0.761060\pi\)
\(14\) −1.27307 −0.340243
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.273073 −0.0662299 −0.0331149 0.999452i \(-0.510543\pi\)
−0.0331149 + 0.999452i \(0.510543\pi\)
\(18\) −2.16686 −0.510734
\(19\) −5.71301 −1.31065 −0.655327 0.755346i \(-0.727469\pi\)
−0.655327 + 0.755346i \(0.727469\pi\)
\(20\) 0 0
\(21\) 2.89379 0.631476
\(22\) 4.43993 0.946597
\(23\) −6.54615 −1.36497 −0.682483 0.730902i \(-0.739100\pi\)
−0.682483 + 0.730902i \(0.739100\pi\)
\(24\) −2.27307 −0.463989
\(25\) 0 0
\(26\) 5.27307 1.03413
\(27\) −1.89379 −0.364460
\(28\) 1.27307 0.240588
\(29\) 0.546146 0.101417 0.0507084 0.998714i \(-0.483852\pi\)
0.0507084 + 0.998714i \(0.483852\pi\)
\(30\) 0 0
\(31\) −5.98608 −1.07513 −0.537566 0.843222i \(-0.680656\pi\)
−0.537566 + 0.843222i \(0.680656\pi\)
\(32\) −1.00000 −0.176777
\(33\) −10.0923 −1.75684
\(34\) 0.273073 0.0468316
\(35\) 0 0
\(36\) 2.16686 0.361143
\(37\) −1.00000 −0.164399
\(38\) 5.71301 0.926772
\(39\) −11.9861 −1.91931
\(40\) 0 0
\(41\) 11.8799 1.85532 0.927662 0.373422i \(-0.121816\pi\)
0.927662 + 0.373422i \(0.121816\pi\)
\(42\) −2.89379 −0.446521
\(43\) −10.3793 −1.58283 −0.791413 0.611282i \(-0.790654\pi\)
−0.791413 + 0.611282i \(0.790654\pi\)
\(44\) −4.43993 −0.669345
\(45\) 0 0
\(46\) 6.54615 0.965177
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 2.27307 0.328090
\(49\) −5.37929 −0.768469
\(50\) 0 0
\(51\) −0.620715 −0.0869174
\(52\) −5.27307 −0.731244
\(53\) 2.33372 0.320561 0.160281 0.987071i \(-0.448760\pi\)
0.160281 + 0.987071i \(0.448760\pi\)
\(54\) 1.89379 0.257712
\(55\) 0 0
\(56\) −1.27307 −0.170122
\(57\) −12.9861 −1.72005
\(58\) −0.546146 −0.0717124
\(59\) 2.37929 0.309757 0.154878 0.987934i \(-0.450501\pi\)
0.154878 + 0.987934i \(0.450501\pi\)
\(60\) 0 0
\(61\) 11.8192 1.51330 0.756648 0.653823i \(-0.226836\pi\)
0.756648 + 0.653823i \(0.226836\pi\)
\(62\) 5.98608 0.760233
\(63\) 2.75857 0.347547
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 10.0923 1.24228
\(67\) 7.15294 0.873871 0.436935 0.899493i \(-0.356064\pi\)
0.436935 + 0.899493i \(0.356064\pi\)
\(68\) −0.273073 −0.0331149
\(69\) −14.8799 −1.79133
\(70\) 0 0
\(71\) 5.60679 0.665404 0.332702 0.943032i \(-0.392040\pi\)
0.332702 + 0.943032i \(0.392040\pi\)
\(72\) −2.16686 −0.255367
\(73\) −9.71301 −1.13682 −0.568411 0.822745i \(-0.692442\pi\)
−0.568411 + 0.822745i \(0.692442\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −5.71301 −0.655327
\(77\) −5.65236 −0.644146
\(78\) 11.9861 1.35716
\(79\) 11.4260 1.28553 0.642763 0.766065i \(-0.277788\pi\)
0.642763 + 0.766065i \(0.277788\pi\)
\(80\) 0 0
\(81\) −10.8053 −1.20059
\(82\) −11.8799 −1.31191
\(83\) 3.89379 0.427399 0.213699 0.976899i \(-0.431449\pi\)
0.213699 + 0.976899i \(0.431449\pi\)
\(84\) 2.89379 0.315738
\(85\) 0 0
\(86\) 10.3793 1.11923
\(87\) 1.24143 0.133095
\(88\) 4.43993 0.473298
\(89\) 13.8659 1.46979 0.734894 0.678182i \(-0.237232\pi\)
0.734894 + 0.678182i \(0.237232\pi\)
\(90\) 0 0
\(91\) −6.71301 −0.703714
\(92\) −6.54615 −0.682483
\(93\) −13.6068 −1.41096
\(94\) 0 0
\(95\) 0 0
\(96\) −2.27307 −0.231995
\(97\) 0.879866 0.0893369 0.0446684 0.999002i \(-0.485777\pi\)
0.0446684 + 0.999002i \(0.485777\pi\)
\(98\) 5.37929 0.543390
\(99\) −9.62071 −0.966918
\(100\) 0 0
\(101\) 3.66628 0.364808 0.182404 0.983224i \(-0.441612\pi\)
0.182404 + 0.983224i \(0.441612\pi\)
\(102\) 0.620715 0.0614599
\(103\) −12.3198 −1.21391 −0.606953 0.794738i \(-0.707608\pi\)
−0.606953 + 0.794738i \(0.707608\pi\)
\(104\) 5.27307 0.517067
\(105\) 0 0
\(106\) −2.33372 −0.226671
\(107\) 7.36536 0.712037 0.356018 0.934479i \(-0.384134\pi\)
0.356018 + 0.934479i \(0.384134\pi\)
\(108\) −1.89379 −0.182230
\(109\) 4.33372 0.415095 0.207548 0.978225i \(-0.433452\pi\)
0.207548 + 0.978225i \(0.433452\pi\)
\(110\) 0 0
\(111\) −2.27307 −0.215751
\(112\) 1.27307 0.120294
\(113\) 2.10621 0.198136 0.0990679 0.995081i \(-0.468414\pi\)
0.0990679 + 0.995081i \(0.468414\pi\)
\(114\) 12.9861 1.21626
\(115\) 0 0
\(116\) 0.546146 0.0507084
\(117\) −11.4260 −1.05634
\(118\) −2.37929 −0.219031
\(119\) −0.347642 −0.0318683
\(120\) 0 0
\(121\) 8.71301 0.792091
\(122\) −11.8192 −1.07006
\(123\) 27.0038 2.43485
\(124\) −5.98608 −0.537566
\(125\) 0 0
\(126\) −2.75857 −0.245753
\(127\) −13.6068 −1.20741 −0.603704 0.797209i \(-0.706309\pi\)
−0.603704 + 0.797209i \(0.706309\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −23.5929 −2.07724
\(130\) 0 0
\(131\) 4.16686 0.364060 0.182030 0.983293i \(-0.441733\pi\)
0.182030 + 0.983293i \(0.441733\pi\)
\(132\) −10.0923 −0.878421
\(133\) −7.27307 −0.630655
\(134\) −7.15294 −0.617920
\(135\) 0 0
\(136\) 0.273073 0.0234158
\(137\) 1.21243 0.103584 0.0517922 0.998658i \(-0.483507\pi\)
0.0517922 + 0.998658i \(0.483507\pi\)
\(138\) 14.8799 1.26666
\(139\) 7.53222 0.638875 0.319437 0.947607i \(-0.396506\pi\)
0.319437 + 0.947607i \(0.396506\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −5.60679 −0.470512
\(143\) 23.4121 1.95782
\(144\) 2.16686 0.180572
\(145\) 0 0
\(146\) 9.71301 0.803854
\(147\) −12.2275 −1.00851
\(148\) −1.00000 −0.0821995
\(149\) −13.9861 −1.14578 −0.572892 0.819631i \(-0.694179\pi\)
−0.572892 + 0.819631i \(0.694179\pi\)
\(150\) 0 0
\(151\) 13.7597 1.11975 0.559876 0.828577i \(-0.310849\pi\)
0.559876 + 0.828577i \(0.310849\pi\)
\(152\) 5.71301 0.463386
\(153\) −0.591711 −0.0478370
\(154\) 5.65236 0.455480
\(155\) 0 0
\(156\) −11.9861 −0.959654
\(157\) 2.56007 0.204316 0.102158 0.994768i \(-0.467425\pi\)
0.102158 + 0.994768i \(0.467425\pi\)
\(158\) −11.4260 −0.909005
\(159\) 5.30472 0.420691
\(160\) 0 0
\(161\) −8.33372 −0.656789
\(162\) 10.8053 0.848944
\(163\) −21.7130 −1.70069 −0.850347 0.526222i \(-0.823608\pi\)
−0.850347 + 0.526222i \(0.823608\pi\)
\(164\) 11.8799 0.927662
\(165\) 0 0
\(166\) −3.89379 −0.302217
\(167\) 16.8659 1.30513 0.652563 0.757734i \(-0.273694\pi\)
0.652563 + 0.757734i \(0.273694\pi\)
\(168\) −2.89379 −0.223261
\(169\) 14.8053 1.13887
\(170\) 0 0
\(171\) −12.3793 −0.946668
\(172\) −10.3793 −0.791413
\(173\) 10.7586 0.817959 0.408980 0.912544i \(-0.365885\pi\)
0.408980 + 0.912544i \(0.365885\pi\)
\(174\) −1.24143 −0.0941125
\(175\) 0 0
\(176\) −4.43993 −0.334673
\(177\) 5.40829 0.406512
\(178\) −13.8659 −1.03930
\(179\) 1.66744 0.124630 0.0623152 0.998057i \(-0.480152\pi\)
0.0623152 + 0.998057i \(0.480152\pi\)
\(180\) 0 0
\(181\) 5.97216 0.443907 0.221953 0.975057i \(-0.428757\pi\)
0.221953 + 0.975057i \(0.428757\pi\)
\(182\) 6.71301 0.497601
\(183\) 26.8659 1.98599
\(184\) 6.54615 0.482588
\(185\) 0 0
\(186\) 13.6068 0.997698
\(187\) 1.21243 0.0886613
\(188\) 0 0
\(189\) −2.41093 −0.175369
\(190\) 0 0
\(191\) −24.8520 −1.79823 −0.899115 0.437713i \(-0.855789\pi\)
−0.899115 + 0.437713i \(0.855789\pi\)
\(192\) 2.27307 0.164045
\(193\) 7.65352 0.550912 0.275456 0.961314i \(-0.411171\pi\)
0.275456 + 0.961314i \(0.411171\pi\)
\(194\) −0.879866 −0.0631707
\(195\) 0 0
\(196\) −5.37929 −0.384235
\(197\) −14.5322 −1.03538 −0.517689 0.855569i \(-0.673208\pi\)
−0.517689 + 0.855569i \(0.673208\pi\)
\(198\) 9.62071 0.683714
\(199\) −12.8799 −0.913030 −0.456515 0.889716i \(-0.650902\pi\)
−0.456515 + 0.889716i \(0.650902\pi\)
\(200\) 0 0
\(201\) 16.2592 1.14683
\(202\) −3.66628 −0.257959
\(203\) 0.695283 0.0487993
\(204\) −0.620715 −0.0434587
\(205\) 0 0
\(206\) 12.3198 0.858361
\(207\) −14.1846 −0.985897
\(208\) −5.27307 −0.365622
\(209\) 25.3654 1.75456
\(210\) 0 0
\(211\) −19.1529 −1.31854 −0.659271 0.751905i \(-0.729135\pi\)
−0.659271 + 0.751905i \(0.729135\pi\)
\(212\) 2.33372 0.160281
\(213\) 12.7446 0.873249
\(214\) −7.36536 −0.503486
\(215\) 0 0
\(216\) 1.89379 0.128856
\(217\) −7.62071 −0.517328
\(218\) −4.33372 −0.293517
\(219\) −22.0784 −1.49192
\(220\) 0 0
\(221\) 1.43993 0.0968604
\(222\) 2.27307 0.152559
\(223\) 2.75857 0.184728 0.0923638 0.995725i \(-0.470558\pi\)
0.0923638 + 0.995725i \(0.470558\pi\)
\(224\) −1.27307 −0.0850608
\(225\) 0 0
\(226\) −2.10621 −0.140103
\(227\) −17.2592 −1.14553 −0.572765 0.819720i \(-0.694129\pi\)
−0.572765 + 0.819720i \(0.694129\pi\)
\(228\) −12.9861 −0.860024
\(229\) −22.3059 −1.47401 −0.737007 0.675885i \(-0.763762\pi\)
−0.737007 + 0.675885i \(0.763762\pi\)
\(230\) 0 0
\(231\) −12.8482 −0.845351
\(232\) −0.546146 −0.0358562
\(233\) −24.3514 −1.59532 −0.797658 0.603110i \(-0.793928\pi\)
−0.797658 + 0.603110i \(0.793928\pi\)
\(234\) 11.4260 0.746942
\(235\) 0 0
\(236\) 2.37929 0.154878
\(237\) 25.9722 1.68707
\(238\) 0.347642 0.0225343
\(239\) 1.78757 0.115629 0.0578143 0.998327i \(-0.481587\pi\)
0.0578143 + 0.998327i \(0.481587\pi\)
\(240\) 0 0
\(241\) −3.72693 −0.240072 −0.120036 0.992770i \(-0.538301\pi\)
−0.120036 + 0.992770i \(0.538301\pi\)
\(242\) −8.71301 −0.560093
\(243\) −18.8799 −1.21114
\(244\) 11.8192 0.756648
\(245\) 0 0
\(246\) −27.0038 −1.72170
\(247\) 30.1251 1.91681
\(248\) 5.98608 0.380116
\(249\) 8.85086 0.560901
\(250\) 0 0
\(251\) −9.00000 −0.568075 −0.284037 0.958813i \(-0.591674\pi\)
−0.284037 + 0.958813i \(0.591674\pi\)
\(252\) 2.75857 0.173774
\(253\) 29.0644 1.82727
\(254\) 13.6068 0.853766
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −3.42601 −0.213709 −0.106854 0.994275i \(-0.534078\pi\)
−0.106854 + 0.994275i \(0.534078\pi\)
\(258\) 23.5929 1.46883
\(259\) −1.27307 −0.0791049
\(260\) 0 0
\(261\) 1.18342 0.0732519
\(262\) −4.16686 −0.257429
\(263\) 2.57399 0.158719 0.0793595 0.996846i \(-0.474713\pi\)
0.0793595 + 0.996846i \(0.474713\pi\)
\(264\) 10.0923 0.621138
\(265\) 0 0
\(266\) 7.27307 0.445941
\(267\) 31.5183 1.92889
\(268\) 7.15294 0.436935
\(269\) −27.7319 −1.69084 −0.845422 0.534100i \(-0.820651\pi\)
−0.845422 + 0.534100i \(0.820651\pi\)
\(270\) 0 0
\(271\) 26.4588 1.60726 0.803629 0.595130i \(-0.202899\pi\)
0.803629 + 0.595130i \(0.202899\pi\)
\(272\) −0.273073 −0.0165575
\(273\) −15.2592 −0.923526
\(274\) −1.21243 −0.0732453
\(275\) 0 0
\(276\) −14.8799 −0.895663
\(277\) 0.573988 0.0344876 0.0172438 0.999851i \(-0.494511\pi\)
0.0172438 + 0.999851i \(0.494511\pi\)
\(278\) −7.53222 −0.451753
\(279\) −12.9710 −0.776553
\(280\) 0 0
\(281\) −29.2592 −1.74545 −0.872727 0.488208i \(-0.837651\pi\)
−0.872727 + 0.488208i \(0.837651\pi\)
\(282\) 0 0
\(283\) −21.0177 −1.24937 −0.624687 0.780875i \(-0.714773\pi\)
−0.624687 + 0.780875i \(0.714773\pi\)
\(284\) 5.60679 0.332702
\(285\) 0 0
\(286\) −23.4121 −1.38439
\(287\) 15.1239 0.892738
\(288\) −2.16686 −0.127683
\(289\) −16.9254 −0.995614
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) −9.71301 −0.568411
\(293\) −17.9582 −1.04913 −0.524566 0.851370i \(-0.675772\pi\)
−0.524566 + 0.851370i \(0.675772\pi\)
\(294\) 12.2275 0.713123
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 8.40829 0.487898
\(298\) 13.9861 0.810192
\(299\) 34.5183 1.99625
\(300\) 0 0
\(301\) −13.2136 −0.761618
\(302\) −13.7597 −0.791784
\(303\) 8.33372 0.478760
\(304\) −5.71301 −0.327663
\(305\) 0 0
\(306\) 0.591711 0.0338259
\(307\) −23.1985 −1.32401 −0.662004 0.749500i \(-0.730294\pi\)
−0.662004 + 0.749500i \(0.730294\pi\)
\(308\) −5.65236 −0.322073
\(309\) −28.0038 −1.59308
\(310\) 0 0
\(311\) −6.98492 −0.396078 −0.198039 0.980194i \(-0.563457\pi\)
−0.198039 + 0.980194i \(0.563457\pi\)
\(312\) 11.9861 0.678578
\(313\) −24.1669 −1.36599 −0.682996 0.730422i \(-0.739323\pi\)
−0.682996 + 0.730422i \(0.739323\pi\)
\(314\) −2.56007 −0.144473
\(315\) 0 0
\(316\) 11.4260 0.642763
\(317\) 32.2920 1.81370 0.906848 0.421457i \(-0.138481\pi\)
0.906848 + 0.421457i \(0.138481\pi\)
\(318\) −5.30472 −0.297474
\(319\) −2.42485 −0.135766
\(320\) 0 0
\(321\) 16.7420 0.934448
\(322\) 8.33372 0.464420
\(323\) 1.56007 0.0868044
\(324\) −10.8053 −0.600294
\(325\) 0 0
\(326\) 21.7130 1.20257
\(327\) 9.85086 0.544754
\(328\) −11.8799 −0.655956
\(329\) 0 0
\(330\) 0 0
\(331\) −27.9254 −1.53492 −0.767460 0.641097i \(-0.778480\pi\)
−0.767460 + 0.641097i \(0.778480\pi\)
\(332\) 3.89379 0.213699
\(333\) −2.16686 −0.118743
\(334\) −16.8659 −0.922863
\(335\) 0 0
\(336\) 2.89379 0.157869
\(337\) 8.33256 0.453903 0.226952 0.973906i \(-0.427124\pi\)
0.226952 + 0.973906i \(0.427124\pi\)
\(338\) −14.8053 −0.805302
\(339\) 4.78757 0.260025
\(340\) 0 0
\(341\) 26.5778 1.43927
\(342\) 12.3793 0.669395
\(343\) −15.7597 −0.850946
\(344\) 10.3793 0.559614
\(345\) 0 0
\(346\) −10.7586 −0.578384
\(347\) 1.99884 0.107303 0.0536516 0.998560i \(-0.482914\pi\)
0.0536516 + 0.998560i \(0.482914\pi\)
\(348\) 1.24143 0.0665476
\(349\) −21.9582 −1.17540 −0.587699 0.809080i \(-0.699966\pi\)
−0.587699 + 0.809080i \(0.699966\pi\)
\(350\) 0 0
\(351\) 9.98608 0.533017
\(352\) 4.43993 0.236649
\(353\) −11.7597 −0.625907 −0.312954 0.949768i \(-0.601318\pi\)
−0.312954 + 0.949768i \(0.601318\pi\)
\(354\) −5.40829 −0.287447
\(355\) 0 0
\(356\) 13.8659 0.734894
\(357\) −0.790215 −0.0418226
\(358\) −1.66744 −0.0881270
\(359\) 2.57399 0.135850 0.0679249 0.997690i \(-0.478362\pi\)
0.0679249 + 0.997690i \(0.478362\pi\)
\(360\) 0 0
\(361\) 13.6384 0.717812
\(362\) −5.97216 −0.313890
\(363\) 19.8053 1.03951
\(364\) −6.71301 −0.351857
\(365\) 0 0
\(366\) −26.8659 −1.40431
\(367\) 12.4867 0.651798 0.325899 0.945405i \(-0.394333\pi\)
0.325899 + 0.945405i \(0.394333\pi\)
\(368\) −6.54615 −0.341241
\(369\) 25.7420 1.34008
\(370\) 0 0
\(371\) 2.97100 0.154246
\(372\) −13.6068 −0.705479
\(373\) 14.8659 0.769729 0.384865 0.922973i \(-0.374248\pi\)
0.384865 + 0.922973i \(0.374248\pi\)
\(374\) −1.21243 −0.0626930
\(375\) 0 0
\(376\) 0 0
\(377\) −2.87987 −0.148321
\(378\) 2.41093 0.124005
\(379\) −2.39437 −0.122990 −0.0614952 0.998107i \(-0.519587\pi\)
−0.0614952 + 0.998107i \(0.519587\pi\)
\(380\) 0 0
\(381\) −30.9292 −1.58455
\(382\) 24.8520 1.27154
\(383\) 14.7725 0.754839 0.377420 0.926042i \(-0.376811\pi\)
0.377420 + 0.926042i \(0.376811\pi\)
\(384\) −2.27307 −0.115997
\(385\) 0 0
\(386\) −7.65352 −0.389554
\(387\) −22.4905 −1.14325
\(388\) 0.879866 0.0446684
\(389\) −12.6991 −0.643869 −0.321935 0.946762i \(-0.604333\pi\)
−0.321935 + 0.946762i \(0.604333\pi\)
\(390\) 0 0
\(391\) 1.78757 0.0904015
\(392\) 5.37929 0.271695
\(393\) 9.47158 0.477778
\(394\) 14.5322 0.732123
\(395\) 0 0
\(396\) −9.62071 −0.483459
\(397\) 4.98492 0.250186 0.125093 0.992145i \(-0.460077\pi\)
0.125093 + 0.992145i \(0.460077\pi\)
\(398\) 12.8799 0.645609
\(399\) −16.5322 −0.827646
\(400\) 0 0
\(401\) 27.1985 1.35823 0.679114 0.734033i \(-0.262364\pi\)
0.679114 + 0.734033i \(0.262364\pi\)
\(402\) −16.2592 −0.810933
\(403\) 31.5650 1.57237
\(404\) 3.66628 0.182404
\(405\) 0 0
\(406\) −0.695283 −0.0345063
\(407\) 4.43993 0.220079
\(408\) 0.620715 0.0307299
\(409\) 35.5499 1.75783 0.878916 0.476977i \(-0.158267\pi\)
0.878916 + 0.476977i \(0.158267\pi\)
\(410\) 0 0
\(411\) 2.75593 0.135940
\(412\) −12.3198 −0.606953
\(413\) 3.02900 0.149048
\(414\) 14.1846 0.697134
\(415\) 0 0
\(416\) 5.27307 0.258534
\(417\) 17.1213 0.838433
\(418\) −25.3654 −1.24066
\(419\) −29.5322 −1.44274 −0.721372 0.692548i \(-0.756488\pi\)
−0.721372 + 0.692548i \(0.756488\pi\)
\(420\) 0 0
\(421\) −30.4905 −1.48601 −0.743007 0.669284i \(-0.766601\pi\)
−0.743007 + 0.669284i \(0.766601\pi\)
\(422\) 19.1529 0.932350
\(423\) 0 0
\(424\) −2.33372 −0.113335
\(425\) 0 0
\(426\) −12.7446 −0.617480
\(427\) 15.0467 0.728162
\(428\) 7.36536 0.356018
\(429\) 53.2174 2.56936
\(430\) 0 0
\(431\) 21.9722 1.05836 0.529181 0.848509i \(-0.322499\pi\)
0.529181 + 0.848509i \(0.322499\pi\)
\(432\) −1.89379 −0.0911149
\(433\) −5.00000 −0.240285 −0.120142 0.992757i \(-0.538335\pi\)
−0.120142 + 0.992757i \(0.538335\pi\)
\(434\) 7.62071 0.365806
\(435\) 0 0
\(436\) 4.33372 0.207548
\(437\) 37.3982 1.78900
\(438\) 22.0784 1.05495
\(439\) 29.5171 1.40878 0.704388 0.709815i \(-0.251221\pi\)
0.704388 + 0.709815i \(0.251221\pi\)
\(440\) 0 0
\(441\) −11.6562 −0.555055
\(442\) −1.43993 −0.0684906
\(443\) 28.9861 1.37717 0.688585 0.725156i \(-0.258232\pi\)
0.688585 + 0.725156i \(0.258232\pi\)
\(444\) −2.27307 −0.107875
\(445\) 0 0
\(446\) −2.75857 −0.130622
\(447\) −31.7914 −1.50368
\(448\) 1.27307 0.0601470
\(449\) 36.6245 1.72842 0.864209 0.503133i \(-0.167819\pi\)
0.864209 + 0.503133i \(0.167819\pi\)
\(450\) 0 0
\(451\) −52.7458 −2.48370
\(452\) 2.10621 0.0990679
\(453\) 31.2769 1.46952
\(454\) 17.2592 0.810012
\(455\) 0 0
\(456\) 12.9861 0.608129
\(457\) 19.7446 0.923616 0.461808 0.886980i \(-0.347201\pi\)
0.461808 + 0.886980i \(0.347201\pi\)
\(458\) 22.3059 1.04229
\(459\) 0.517142 0.0241381
\(460\) 0 0
\(461\) 34.9733 1.62887 0.814435 0.580255i \(-0.197047\pi\)
0.814435 + 0.580255i \(0.197047\pi\)
\(462\) 12.8482 0.597753
\(463\) −24.1707 −1.12331 −0.561653 0.827373i \(-0.689834\pi\)
−0.561653 + 0.827373i \(0.689834\pi\)
\(464\) 0.546146 0.0253542
\(465\) 0 0
\(466\) 24.3514 1.12806
\(467\) 7.63844 0.353465 0.176732 0.984259i \(-0.443447\pi\)
0.176732 + 0.984259i \(0.443447\pi\)
\(468\) −11.4260 −0.528168
\(469\) 9.10621 0.420486
\(470\) 0 0
\(471\) 5.81922 0.268135
\(472\) −2.37929 −0.109515
\(473\) 46.0833 2.11891
\(474\) −25.9722 −1.19294
\(475\) 0 0
\(476\) −0.347642 −0.0159341
\(477\) 5.05685 0.231537
\(478\) −1.78757 −0.0817618
\(479\) −4.01392 −0.183401 −0.0917004 0.995787i \(-0.529230\pi\)
−0.0917004 + 0.995787i \(0.529230\pi\)
\(480\) 0 0
\(481\) 5.27307 0.240431
\(482\) 3.72693 0.169757
\(483\) −18.9432 −0.861943
\(484\) 8.71301 0.396046
\(485\) 0 0
\(486\) 18.8799 0.856408
\(487\) 43.3842 1.96593 0.982964 0.183798i \(-0.0588393\pi\)
0.982964 + 0.183798i \(0.0588393\pi\)
\(488\) −11.8192 −0.535031
\(489\) −49.3552 −2.23192
\(490\) 0 0
\(491\) −20.7307 −0.935565 −0.467782 0.883844i \(-0.654947\pi\)
−0.467782 + 0.883844i \(0.654947\pi\)
\(492\) 27.0038 1.21743
\(493\) −0.149138 −0.00671682
\(494\) −30.1251 −1.35539
\(495\) 0 0
\(496\) −5.98608 −0.268783
\(497\) 7.13786 0.320177
\(498\) −8.85086 −0.396617
\(499\) 5.92659 0.265311 0.132655 0.991162i \(-0.457650\pi\)
0.132655 + 0.991162i \(0.457650\pi\)
\(500\) 0 0
\(501\) 38.3375 1.71279
\(502\) 9.00000 0.401690
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) −2.75857 −0.122877
\(505\) 0 0
\(506\) −29.0644 −1.29207
\(507\) 33.6535 1.49461
\(508\) −13.6068 −0.603704
\(509\) −9.07837 −0.402392 −0.201196 0.979551i \(-0.564483\pi\)
−0.201196 + 0.979551i \(0.564483\pi\)
\(510\) 0 0
\(511\) −12.3654 −0.547012
\(512\) −1.00000 −0.0441942
\(513\) 10.8192 0.477680
\(514\) 3.42601 0.151115
\(515\) 0 0
\(516\) −23.5929 −1.03862
\(517\) 0 0
\(518\) 1.27307 0.0559356
\(519\) 24.4550 1.07346
\(520\) 0 0
\(521\) −7.16686 −0.313986 −0.156993 0.987600i \(-0.550180\pi\)
−0.156993 + 0.987600i \(0.550180\pi\)
\(522\) −1.18342 −0.0517970
\(523\) −29.3970 −1.28544 −0.642721 0.766101i \(-0.722194\pi\)
−0.642721 + 0.766101i \(0.722194\pi\)
\(524\) 4.16686 0.182030
\(525\) 0 0
\(526\) −2.57399 −0.112231
\(527\) 1.63464 0.0712058
\(528\) −10.0923 −0.439211
\(529\) 19.8520 0.863131
\(530\) 0 0
\(531\) 5.15558 0.223733
\(532\) −7.27307 −0.315328
\(533\) −62.6434 −2.71339
\(534\) −31.5183 −1.36393
\(535\) 0 0
\(536\) −7.15294 −0.308960
\(537\) 3.79021 0.163560
\(538\) 27.7319 1.19561
\(539\) 23.8837 1.02874
\(540\) 0 0
\(541\) 26.0038 1.11799 0.558995 0.829171i \(-0.311187\pi\)
0.558995 + 0.829171i \(0.311187\pi\)
\(542\) −26.4588 −1.13650
\(543\) 13.5751 0.582565
\(544\) 0.273073 0.0117079
\(545\) 0 0
\(546\) 15.2592 0.653031
\(547\) −8.77746 −0.375297 −0.187648 0.982236i \(-0.560087\pi\)
−0.187648 + 0.982236i \(0.560087\pi\)
\(548\) 1.21243 0.0517922
\(549\) 25.6106 1.09303
\(550\) 0 0
\(551\) −3.12013 −0.132922
\(552\) 14.8799 0.633329
\(553\) 14.5461 0.618565
\(554\) −0.573988 −0.0243864
\(555\) 0 0
\(556\) 7.53222 0.319437
\(557\) −21.6663 −0.918030 −0.459015 0.888429i \(-0.651798\pi\)
−0.459015 + 0.888429i \(0.651798\pi\)
\(558\) 12.9710 0.549106
\(559\) 54.7307 2.31486
\(560\) 0 0
\(561\) 2.75593 0.116355
\(562\) 29.2592 1.23422
\(563\) 19.4437 0.819456 0.409728 0.912208i \(-0.365624\pi\)
0.409728 + 0.912208i \(0.365624\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 21.0177 0.883441
\(567\) −13.7559 −0.577695
\(568\) −5.60679 −0.235256
\(569\) 21.7446 0.911583 0.455792 0.890087i \(-0.349356\pi\)
0.455792 + 0.890087i \(0.349356\pi\)
\(570\) 0 0
\(571\) 1.54498 0.0646556 0.0323278 0.999477i \(-0.489708\pi\)
0.0323278 + 0.999477i \(0.489708\pi\)
\(572\) 23.4121 0.978909
\(573\) −56.4905 −2.35992
\(574\) −15.1239 −0.631261
\(575\) 0 0
\(576\) 2.16686 0.0902858
\(577\) −37.8837 −1.57712 −0.788559 0.614959i \(-0.789172\pi\)
−0.788559 + 0.614959i \(0.789172\pi\)
\(578\) 16.9254 0.704005
\(579\) 17.3970 0.722995
\(580\) 0 0
\(581\) 4.95708 0.205654
\(582\) −2.00000 −0.0829027
\(583\) −10.3616 −0.429132
\(584\) 9.71301 0.401927
\(585\) 0 0
\(586\) 17.9582 0.741848
\(587\) −38.8053 −1.60167 −0.800833 0.598888i \(-0.795610\pi\)
−0.800833 + 0.598888i \(0.795610\pi\)
\(588\) −12.2275 −0.504254
\(589\) 34.1985 1.40912
\(590\) 0 0
\(591\) −33.0328 −1.35879
\(592\) −1.00000 −0.0410997
\(593\) 22.0923 0.907222 0.453611 0.891200i \(-0.350136\pi\)
0.453611 + 0.891200i \(0.350136\pi\)
\(594\) −8.40829 −0.344996
\(595\) 0 0
\(596\) −13.9861 −0.572892
\(597\) −29.2769 −1.19822
\(598\) −34.5183 −1.41156
\(599\) −17.8471 −0.729211 −0.364606 0.931162i \(-0.618796\pi\)
−0.364606 + 0.931162i \(0.618796\pi\)
\(600\) 0 0
\(601\) 46.1100 1.88087 0.940433 0.339978i \(-0.110420\pi\)
0.940433 + 0.339978i \(0.110420\pi\)
\(602\) 13.2136 0.538546
\(603\) 15.4994 0.631185
\(604\) 13.7597 0.559876
\(605\) 0 0
\(606\) −8.33372 −0.338534
\(607\) −47.3842 −1.92327 −0.961634 0.274337i \(-0.911542\pi\)
−0.961634 + 0.274337i \(0.911542\pi\)
\(608\) 5.71301 0.231693
\(609\) 1.58043 0.0640422
\(610\) 0 0
\(611\) 0 0
\(612\) −0.591711 −0.0239185
\(613\) −15.7876 −0.637654 −0.318827 0.947813i \(-0.603289\pi\)
−0.318827 + 0.947813i \(0.603289\pi\)
\(614\) 23.1985 0.936215
\(615\) 0 0
\(616\) 5.65236 0.227740
\(617\) −26.1390 −1.05232 −0.526159 0.850386i \(-0.676368\pi\)
−0.526159 + 0.850386i \(0.676368\pi\)
\(618\) 28.0038 1.12648
\(619\) 31.3047 1.25824 0.629121 0.777307i \(-0.283415\pi\)
0.629121 + 0.777307i \(0.283415\pi\)
\(620\) 0 0
\(621\) 12.3970 0.497475
\(622\) 6.98492 0.280070
\(623\) 17.6524 0.707227
\(624\) −11.9861 −0.479827
\(625\) 0 0
\(626\) 24.1669 0.965902
\(627\) 57.6573 2.30261
\(628\) 2.56007 0.102158
\(629\) 0.273073 0.0108881
\(630\) 0 0
\(631\) −15.3047 −0.609271 −0.304636 0.952469i \(-0.598535\pi\)
−0.304636 + 0.952469i \(0.598535\pi\)
\(632\) −11.4260 −0.454502
\(633\) −43.5360 −1.73040
\(634\) −32.2920 −1.28248
\(635\) 0 0
\(636\) 5.30472 0.210346
\(637\) 28.3654 1.12388
\(638\) 2.42485 0.0960008
\(639\) 12.1491 0.480612
\(640\) 0 0
\(641\) −12.8520 −0.507624 −0.253812 0.967254i \(-0.581685\pi\)
−0.253812 + 0.967254i \(0.581685\pi\)
\(642\) −16.7420 −0.660754
\(643\) 0.139018 0.00548233 0.00274117 0.999996i \(-0.499127\pi\)
0.00274117 + 0.999996i \(0.499127\pi\)
\(644\) −8.33372 −0.328395
\(645\) 0 0
\(646\) −1.56007 −0.0613800
\(647\) −46.8659 −1.84249 −0.921245 0.388982i \(-0.872827\pi\)
−0.921245 + 0.388982i \(0.872827\pi\)
\(648\) 10.8053 0.424472
\(649\) −10.5639 −0.414668
\(650\) 0 0
\(651\) −17.3224 −0.678920
\(652\) −21.7130 −0.850347
\(653\) 7.48550 0.292930 0.146465 0.989216i \(-0.453210\pi\)
0.146465 + 0.989216i \(0.453210\pi\)
\(654\) −9.85086 −0.385199
\(655\) 0 0
\(656\) 11.8799 0.463831
\(657\) −21.0467 −0.821111
\(658\) 0 0
\(659\) −22.7914 −0.887826 −0.443913 0.896070i \(-0.646410\pi\)
−0.443913 + 0.896070i \(0.646410\pi\)
\(660\) 0 0
\(661\) 16.4249 0.638853 0.319426 0.947611i \(-0.396510\pi\)
0.319426 + 0.947611i \(0.396510\pi\)
\(662\) 27.9254 1.08535
\(663\) 3.27307 0.127116
\(664\) −3.89379 −0.151108
\(665\) 0 0
\(666\) 2.16686 0.0839641
\(667\) −3.57515 −0.138430
\(668\) 16.8659 0.652563
\(669\) 6.27043 0.242429
\(670\) 0 0
\(671\) −52.4765 −2.02583
\(672\) −2.89379 −0.111630
\(673\) −51.2035 −1.97375 −0.986874 0.161490i \(-0.948370\pi\)
−0.986874 + 0.161490i \(0.948370\pi\)
\(674\) −8.33256 −0.320958
\(675\) 0 0
\(676\) 14.8053 0.569435
\(677\) 1.00116 0.0384778 0.0192389 0.999815i \(-0.493876\pi\)
0.0192389 + 0.999815i \(0.493876\pi\)
\(678\) −4.78757 −0.183866
\(679\) 1.12013 0.0429868
\(680\) 0 0
\(681\) −39.2313 −1.50335
\(682\) −26.5778 −1.01772
\(683\) −28.3526 −1.08488 −0.542441 0.840094i \(-0.682500\pi\)
−0.542441 + 0.840094i \(0.682500\pi\)
\(684\) −12.3793 −0.473334
\(685\) 0 0
\(686\) 15.7597 0.601709
\(687\) −50.7029 −1.93444
\(688\) −10.3793 −0.395707
\(689\) −12.3059 −0.468817
\(690\) 0 0
\(691\) 26.3842 1.00370 0.501852 0.864953i \(-0.332652\pi\)
0.501852 + 0.864953i \(0.332652\pi\)
\(692\) 10.7586 0.408980
\(693\) −12.2479 −0.465258
\(694\) −1.99884 −0.0758749
\(695\) 0 0
\(696\) −1.24143 −0.0470562
\(697\) −3.24407 −0.122878
\(698\) 21.9582 0.831132
\(699\) −55.3526 −2.09363
\(700\) 0 0
\(701\) 40.5801 1.53269 0.766345 0.642429i \(-0.222073\pi\)
0.766345 + 0.642429i \(0.222073\pi\)
\(702\) −9.98608 −0.376900
\(703\) 5.71301 0.215470
\(704\) −4.43993 −0.167336
\(705\) 0 0
\(706\) 11.7597 0.442583
\(707\) 4.66744 0.175537
\(708\) 5.40829 0.203256
\(709\) 14.5461 0.546292 0.273146 0.961973i \(-0.411936\pi\)
0.273146 + 0.961973i \(0.411936\pi\)
\(710\) 0 0
\(711\) 24.7586 0.928519
\(712\) −13.8659 −0.519648
\(713\) 39.1857 1.46752
\(714\) 0.790215 0.0295730
\(715\) 0 0
\(716\) 1.66744 0.0623152
\(717\) 4.06329 0.151746
\(718\) −2.57399 −0.0960604
\(719\) 21.6701 0.808158 0.404079 0.914724i \(-0.367592\pi\)
0.404079 + 0.914724i \(0.367592\pi\)
\(720\) 0 0
\(721\) −15.6840 −0.584103
\(722\) −13.6384 −0.507570
\(723\) −8.47158 −0.315061
\(724\) 5.97216 0.221953
\(725\) 0 0
\(726\) −19.8053 −0.735044
\(727\) −28.8799 −1.07109 −0.535547 0.844505i \(-0.679895\pi\)
−0.535547 + 0.844505i \(0.679895\pi\)
\(728\) 6.71301 0.248801
\(729\) −10.4994 −0.388867
\(730\) 0 0
\(731\) 2.83430 0.104830
\(732\) 26.8659 0.992994
\(733\) 21.9165 0.809503 0.404752 0.914427i \(-0.367358\pi\)
0.404752 + 0.914427i \(0.367358\pi\)
\(734\) −12.4867 −0.460891
\(735\) 0 0
\(736\) 6.54615 0.241294
\(737\) −31.7586 −1.16984
\(738\) −25.7420 −0.947576
\(739\) −29.7040 −1.09268 −0.546341 0.837563i \(-0.683980\pi\)
−0.546341 + 0.837563i \(0.683980\pi\)
\(740\) 0 0
\(741\) 68.4765 2.51555
\(742\) −2.97100 −0.109069
\(743\) 39.5171 1.44974 0.724872 0.688884i \(-0.241899\pi\)
0.724872 + 0.688884i \(0.241899\pi\)
\(744\) 13.6068 0.498849
\(745\) 0 0
\(746\) −14.8659 −0.544281
\(747\) 8.43729 0.308704
\(748\) 1.21243 0.0443307
\(749\) 9.37665 0.342615
\(750\) 0 0
\(751\) −11.6384 −0.424693 −0.212346 0.977194i \(-0.568110\pi\)
−0.212346 + 0.977194i \(0.568110\pi\)
\(752\) 0 0
\(753\) −20.4577 −0.745518
\(754\) 2.87987 0.104879
\(755\) 0 0
\(756\) −2.41093 −0.0876847
\(757\) −21.0923 −0.766612 −0.383306 0.923621i \(-0.625215\pi\)
−0.383306 + 0.923621i \(0.625215\pi\)
\(758\) 2.39437 0.0869674
\(759\) 66.0656 2.39803
\(760\) 0 0
\(761\) −16.8988 −0.612579 −0.306290 0.951938i \(-0.599088\pi\)
−0.306290 + 0.951938i \(0.599088\pi\)
\(762\) 30.9292 1.12045
\(763\) 5.51714 0.199734
\(764\) −24.8520 −0.899115
\(765\) 0 0
\(766\) −14.7725 −0.533752
\(767\) −12.5461 −0.453015
\(768\) 2.27307 0.0820225
\(769\) 28.6524 1.03323 0.516615 0.856218i \(-0.327192\pi\)
0.516615 + 0.856218i \(0.327192\pi\)
\(770\) 0 0
\(771\) −7.78757 −0.280463
\(772\) 7.65352 0.275456
\(773\) 10.3198 0.371177 0.185589 0.982628i \(-0.440581\pi\)
0.185589 + 0.982628i \(0.440581\pi\)
\(774\) 22.4905 0.808403
\(775\) 0 0
\(776\) −0.879866 −0.0315854
\(777\) −2.89379 −0.103814
\(778\) 12.6991 0.455284
\(779\) −67.8697 −2.43169
\(780\) 0 0
\(781\) −24.8938 −0.890770
\(782\) −1.78757 −0.0639235
\(783\) −1.03428 −0.0369623
\(784\) −5.37929 −0.192117
\(785\) 0 0
\(786\) −9.47158 −0.337840
\(787\) −16.3337 −0.582234 −0.291117 0.956687i \(-0.594027\pi\)
−0.291117 + 0.956687i \(0.594027\pi\)
\(788\) −14.5322 −0.517689
\(789\) 5.85086 0.208296
\(790\) 0 0
\(791\) 2.68136 0.0953383
\(792\) 9.62071 0.341857
\(793\) −62.3236 −2.21318
\(794\) −4.98492 −0.176908
\(795\) 0 0
\(796\) −12.8799 −0.456515
\(797\) −2.96719 −0.105103 −0.0525517 0.998618i \(-0.516735\pi\)
−0.0525517 + 0.998618i \(0.516735\pi\)
\(798\) 16.5322 0.585234
\(799\) 0 0
\(800\) 0 0
\(801\) 30.0456 1.06161
\(802\) −27.1985 −0.960413
\(803\) 43.1251 1.52185
\(804\) 16.2592 0.573416
\(805\) 0 0
\(806\) −31.5650 −1.11183
\(807\) −63.0366 −2.21899
\(808\) −3.66628 −0.128979
\(809\) −7.87870 −0.277001 −0.138500 0.990362i \(-0.544228\pi\)
−0.138500 + 0.990362i \(0.544228\pi\)
\(810\) 0 0
\(811\) −48.6396 −1.70797 −0.853984 0.520300i \(-0.825820\pi\)
−0.853984 + 0.520300i \(0.825820\pi\)
\(812\) 0.695283 0.0243997
\(813\) 60.1428 2.10930
\(814\) −4.43993 −0.155620
\(815\) 0 0
\(816\) −0.620715 −0.0217294
\(817\) 59.2969 2.07454
\(818\) −35.5499 −1.24297
\(819\) −14.5461 −0.508283
\(820\) 0 0
\(821\) −7.68020 −0.268041 −0.134020 0.990979i \(-0.542789\pi\)
−0.134020 + 0.990979i \(0.542789\pi\)
\(822\) −2.75593 −0.0961241
\(823\) 26.3654 0.919039 0.459519 0.888168i \(-0.348022\pi\)
0.459519 + 0.888168i \(0.348022\pi\)
\(824\) 12.3198 0.429181
\(825\) 0 0
\(826\) −3.02900 −0.105393
\(827\) 17.8343 0.620159 0.310080 0.950711i \(-0.399644\pi\)
0.310080 + 0.950711i \(0.399644\pi\)
\(828\) −14.1846 −0.492948
\(829\) −5.17962 −0.179896 −0.0899478 0.995946i \(-0.528670\pi\)
−0.0899478 + 0.995946i \(0.528670\pi\)
\(830\) 0 0
\(831\) 1.30472 0.0452601
\(832\) −5.27307 −0.182811
\(833\) 1.46894 0.0508956
\(834\) −17.1213 −0.592862
\(835\) 0 0
\(836\) 25.3654 0.877279
\(837\) 11.3364 0.391842
\(838\) 29.5322 1.02017
\(839\) −6.15294 −0.212423 −0.106212 0.994344i \(-0.533872\pi\)
−0.106212 + 0.994344i \(0.533872\pi\)
\(840\) 0 0
\(841\) −28.7017 −0.989715
\(842\) 30.4905 1.05077
\(843\) −66.5082 −2.29066
\(844\) −19.1529 −0.659271
\(845\) 0 0
\(846\) 0 0
\(847\) 11.0923 0.381136
\(848\) 2.33372 0.0801403
\(849\) −47.7748 −1.63963
\(850\) 0 0
\(851\) 6.54615 0.224399
\(852\) 12.7446 0.436625
\(853\) 30.8837 1.05744 0.528718 0.848797i \(-0.322673\pi\)
0.528718 + 0.848797i \(0.322673\pi\)
\(854\) −15.0467 −0.514888
\(855\) 0 0
\(856\) −7.36536 −0.251743
\(857\) −15.1529 −0.517615 −0.258807 0.965929i \(-0.583329\pi\)
−0.258807 + 0.965929i \(0.583329\pi\)
\(858\) −53.2174 −1.81681
\(859\) −7.28583 −0.248589 −0.124295 0.992245i \(-0.539667\pi\)
−0.124295 + 0.992245i \(0.539667\pi\)
\(860\) 0 0
\(861\) 34.3778 1.17159
\(862\) −21.9722 −0.748375
\(863\) 51.7357 1.76110 0.880552 0.473950i \(-0.157172\pi\)
0.880552 + 0.473950i \(0.157172\pi\)
\(864\) 1.89379 0.0644280
\(865\) 0 0
\(866\) 5.00000 0.169907
\(867\) −38.4727 −1.30660
\(868\) −7.62071 −0.258664
\(869\) −50.7307 −1.72092
\(870\) 0 0
\(871\) −37.7180 −1.27802
\(872\) −4.33372 −0.146758
\(873\) 1.90655 0.0645268
\(874\) −37.3982 −1.26501
\(875\) 0 0
\(876\) −22.0784 −0.745959
\(877\) 50.4109 1.70226 0.851128 0.524958i \(-0.175919\pi\)
0.851128 + 0.524958i \(0.175919\pi\)
\(878\) −29.5171 −0.996155
\(879\) −40.8204 −1.37684
\(880\) 0 0
\(881\) −45.0822 −1.51886 −0.759428 0.650591i \(-0.774521\pi\)
−0.759428 + 0.650591i \(0.774521\pi\)
\(882\) 11.6562 0.392483
\(883\) 17.9988 0.605709 0.302855 0.953037i \(-0.402060\pi\)
0.302855 + 0.953037i \(0.402060\pi\)
\(884\) 1.43993 0.0484302
\(885\) 0 0
\(886\) −28.9861 −0.973806
\(887\) 38.2502 1.28432 0.642158 0.766572i \(-0.278039\pi\)
0.642158 + 0.766572i \(0.278039\pi\)
\(888\) 2.27307 0.0762793
\(889\) −17.3224 −0.580976
\(890\) 0 0
\(891\) 47.9748 1.60722
\(892\) 2.75857 0.0923638
\(893\) 0 0
\(894\) 31.7914 1.06326
\(895\) 0 0
\(896\) −1.27307 −0.0425304
\(897\) 78.4626 2.61979
\(898\) −36.6245 −1.22218
\(899\) −3.26927 −0.109036
\(900\) 0 0
\(901\) −0.637276 −0.0212307
\(902\) 52.7458 1.75624
\(903\) −30.0354 −0.999517
\(904\) −2.10621 −0.0700516
\(905\) 0 0
\(906\) −31.2769 −1.03910
\(907\) −10.0734 −0.334482 −0.167241 0.985916i \(-0.553486\pi\)
−0.167241 + 0.985916i \(0.553486\pi\)
\(908\) −17.2592 −0.572765
\(909\) 7.94432 0.263496
\(910\) 0 0
\(911\) −5.56123 −0.184252 −0.0921259 0.995747i \(-0.529366\pi\)
−0.0921259 + 0.995747i \(0.529366\pi\)
\(912\) −12.9861 −0.430012
\(913\) −17.2882 −0.572154
\(914\) −19.7446 −0.653095
\(915\) 0 0
\(916\) −22.3059 −0.737007
\(917\) 5.30472 0.175177
\(918\) −0.517142 −0.0170682
\(919\) −3.75973 −0.124022 −0.0620111 0.998075i \(-0.519751\pi\)
−0.0620111 + 0.998075i \(0.519751\pi\)
\(920\) 0 0
\(921\) −52.7319 −1.73757
\(922\) −34.9733 −1.15178
\(923\) −29.5650 −0.973145
\(924\) −12.8482 −0.422675
\(925\) 0 0
\(926\) 24.1707 0.794297
\(927\) −26.6953 −0.876788
\(928\) −0.546146 −0.0179281
\(929\) 3.40597 0.111746 0.0558731 0.998438i \(-0.482206\pi\)
0.0558731 + 0.998438i \(0.482206\pi\)
\(930\) 0 0
\(931\) 30.7319 1.00720
\(932\) −24.3514 −0.797658
\(933\) −15.8772 −0.519797
\(934\) −7.63844 −0.249937
\(935\) 0 0
\(936\) 11.4260 0.373471
\(937\) −34.8053 −1.13704 −0.568520 0.822670i \(-0.692484\pi\)
−0.568520 + 0.822670i \(0.692484\pi\)
\(938\) −9.10621 −0.297328
\(939\) −54.9330 −1.79267
\(940\) 0 0
\(941\) −1.63844 −0.0534115 −0.0267058 0.999643i \(-0.508502\pi\)
−0.0267058 + 0.999643i \(0.508502\pi\)
\(942\) −5.81922 −0.189600
\(943\) −77.7673 −2.53245
\(944\) 2.37929 0.0774391
\(945\) 0 0
\(946\) −46.0833 −1.49830
\(947\) −30.8520 −1.00256 −0.501278 0.865286i \(-0.667137\pi\)
−0.501278 + 0.865286i \(0.667137\pi\)
\(948\) 25.9722 0.843536
\(949\) 51.2174 1.66259
\(950\) 0 0
\(951\) 73.4020 2.38022
\(952\) 0.347642 0.0112671
\(953\) 34.0467 1.10288 0.551441 0.834214i \(-0.314078\pi\)
0.551441 + 0.834214i \(0.314078\pi\)
\(954\) −5.05685 −0.163721
\(955\) 0 0
\(956\) 1.78757 0.0578143
\(957\) −5.51186 −0.178173
\(958\) 4.01392 0.129684
\(959\) 1.54351 0.0498424
\(960\) 0 0
\(961\) 4.83314 0.155908
\(962\) −5.27307 −0.170011
\(963\) 15.9597 0.514295
\(964\) −3.72693 −0.120036
\(965\) 0 0
\(966\) 18.9432 0.609486
\(967\) −58.2757 −1.87402 −0.937010 0.349302i \(-0.886419\pi\)
−0.937010 + 0.349302i \(0.886419\pi\)
\(968\) −8.71301 −0.280047
\(969\) 3.54615 0.113919
\(970\) 0 0
\(971\) 57.6611 1.85043 0.925217 0.379439i \(-0.123883\pi\)
0.925217 + 0.379439i \(0.123883\pi\)
\(972\) −18.8799 −0.605572
\(973\) 9.58907 0.307411
\(974\) −43.3842 −1.39012
\(975\) 0 0
\(976\) 11.8192 0.378324
\(977\) 3.78261 0.121016 0.0605082 0.998168i \(-0.480728\pi\)
0.0605082 + 0.998168i \(0.480728\pi\)
\(978\) 49.3552 1.57821
\(979\) −61.5639 −1.96759
\(980\) 0 0
\(981\) 9.39057 0.299818
\(982\) 20.7307 0.661544
\(983\) −41.3047 −1.31742 −0.658708 0.752399i \(-0.728897\pi\)
−0.658708 + 0.752399i \(0.728897\pi\)
\(984\) −27.0038 −0.860850
\(985\) 0 0
\(986\) 0.149138 0.00474951
\(987\) 0 0
\(988\) 30.1251 0.958407
\(989\) 67.9443 2.16050
\(990\) 0 0
\(991\) −9.89263 −0.314250 −0.157125 0.987579i \(-0.550222\pi\)
−0.157125 + 0.987579i \(0.550222\pi\)
\(992\) 5.98608 0.190058
\(993\) −63.4765 −2.01437
\(994\) −7.13786 −0.226399
\(995\) 0 0
\(996\) 8.85086 0.280450
\(997\) 44.5221 1.41003 0.705015 0.709193i \(-0.250940\pi\)
0.705015 + 0.709193i \(0.250940\pi\)
\(998\) −5.92659 −0.187603
\(999\) 1.89379 0.0599168
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.y.1.3 3
5.2 odd 4 1850.2.b.p.149.1 6
5.3 odd 4 1850.2.b.p.149.6 6
5.4 even 2 1850.2.a.bc.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.y.1.3 3 1.1 even 1 trivial
1850.2.a.bc.1.1 yes 3 5.4 even 2
1850.2.b.p.149.1 6 5.2 odd 4
1850.2.b.p.149.6 6 5.3 odd 4