Properties

Label 1850.2.a.bd.1.2
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 1.2
Root \(-1.53175\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.53175 q^{3} +1.00000 q^{4} +1.53175 q^{6} -4.67211 q^{7} -1.00000 q^{8} -0.653743 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.53175 q^{3} +1.00000 q^{4} +1.53175 q^{6} -4.67211 q^{7} -1.00000 q^{8} -0.653743 q^{9} +0.0451320 q^{11} -1.53175 q^{12} -5.26514 q^{13} +4.67211 q^{14} +1.00000 q^{16} -3.60861 q^{17} +0.653743 q^{18} -6.22000 q^{19} +7.15650 q^{21} -0.0451320 q^{22} +2.20164 q^{23} +1.53175 q^{24} +5.26514 q^{26} +5.59662 q^{27} -4.67211 q^{28} -4.20386 q^{29} -3.01051 q^{31} -1.00000 q^{32} -0.0691309 q^{33} +3.60861 q^{34} -0.653743 q^{36} -1.00000 q^{37} +6.22000 q^{38} +8.06487 q^{39} -7.38299 q^{41} -7.15650 q^{42} +5.54789 q^{43} +0.0451320 q^{44} -2.20164 q^{46} +4.28072 q^{47} -1.53175 q^{48} +14.8286 q^{49} +5.52749 q^{51} -5.26514 q^{52} -6.10215 q^{53} -5.59662 q^{54} +4.67211 q^{56} +9.52749 q^{57} +4.20386 q^{58} +13.5275 q^{59} -4.27299 q^{61} +3.01051 q^{62} +3.05436 q^{63} +1.00000 q^{64} +0.0691309 q^{66} -12.3751 q^{67} -3.60861 q^{68} -3.37235 q^{69} -4.49354 q^{71} +0.653743 q^{72} -7.03811 q^{73} +1.00000 q^{74} -6.22000 q^{76} -0.210861 q^{77} -8.06487 q^{78} -8.72499 q^{79} -6.61139 q^{81} +7.38299 q^{82} +0.880231 q^{83} +7.15650 q^{84} -5.54789 q^{86} +6.43926 q^{87} -0.0451320 q^{88} +9.97602 q^{89} +24.5993 q^{91} +2.20164 q^{92} +4.61135 q^{93} -4.28072 q^{94} +1.53175 q^{96} -0.240408 q^{97} -14.8286 q^{98} -0.0295047 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9} + O(q^{10}) \) \( 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{13} - q^{14} + 5 q^{16} - 9 q^{17} - 3 q^{18} + 4 q^{19} + 16 q^{21} - 3 q^{22} - 6 q^{23} - 6 q^{26} + q^{28} + 11 q^{29} + 23 q^{31} - 5 q^{32} - 20 q^{33} + 9 q^{34} + 3 q^{36} - 5 q^{37} - 4 q^{38} + 20 q^{39} - 7 q^{41} - 16 q^{42} + 17 q^{43} + 3 q^{44} + 6 q^{46} - 12 q^{47} + 30 q^{49} - 20 q^{51} + 6 q^{52} + 7 q^{53} - q^{56} - 11 q^{58} + 20 q^{59} - 9 q^{61} - 23 q^{62} + 33 q^{63} + 5 q^{64} + 20 q^{66} - 12 q^{67} - 9 q^{68} + 16 q^{69} + 6 q^{71} - 3 q^{72} + 6 q^{73} + 5 q^{74} + 4 q^{76} + q^{77} - 20 q^{78} + 20 q^{79} - 7 q^{81} + 7 q^{82} - 12 q^{83} + 16 q^{84} - 17 q^{86} + 34 q^{87} - 3 q^{88} + 12 q^{89} + 16 q^{91} - 6 q^{92} + 4 q^{93} + 12 q^{94} - 3 q^{97} - 30 q^{98} - 11 q^{99} + O(q^{100}) \)

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
\(3\) −1.53175 −0.884356 −0.442178 0.896927i \(-0.645794\pi\)
−0.442178 + 0.896927i \(0.645794\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.53175 0.625334
\(7\) −4.67211 −1.76589 −0.882946 0.469475i \(-0.844443\pi\)
−0.882946 + 0.469475i \(0.844443\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.653743 −0.217914
\(10\) 0 0
\(11\) 0.0451320 0.0136078 0.00680390 0.999977i \(-0.497834\pi\)
0.00680390 + 0.999977i \(0.497834\pi\)
\(12\) −1.53175 −0.442178
\(13\) −5.26514 −1.46029 −0.730143 0.683294i \(-0.760547\pi\)
−0.730143 + 0.683294i \(0.760547\pi\)
\(14\) 4.67211 1.24867
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.60861 −0.875217 −0.437608 0.899166i \(-0.644174\pi\)
−0.437608 + 0.899166i \(0.644174\pi\)
\(18\) 0.653743 0.154089
\(19\) −6.22000 −1.42697 −0.713483 0.700672i \(-0.752884\pi\)
−0.713483 + 0.700672i \(0.752884\pi\)
\(20\) 0 0
\(21\) 7.15650 1.56168
\(22\) −0.0451320 −0.00962217
\(23\) 2.20164 0.459073 0.229536 0.973300i \(-0.426279\pi\)
0.229536 + 0.973300i \(0.426279\pi\)
\(24\) 1.53175 0.312667
\(25\) 0 0
\(26\) 5.26514 1.03258
\(27\) 5.59662 1.07707
\(28\) −4.67211 −0.882946
\(29\) −4.20386 −0.780637 −0.390319 0.920680i \(-0.627635\pi\)
−0.390319 + 0.920680i \(0.627635\pi\)
\(30\) 0 0
\(31\) −3.01051 −0.540704 −0.270352 0.962762i \(-0.587140\pi\)
−0.270352 + 0.962762i \(0.587140\pi\)
\(32\) −1.00000 −0.176777
\(33\) −0.0691309 −0.0120341
\(34\) 3.60861 0.618872
\(35\) 0 0
\(36\) −0.653743 −0.108957
\(37\) −1.00000 −0.164399
\(38\) 6.22000 1.00902
\(39\) 8.06487 1.29141
\(40\) 0 0
\(41\) −7.38299 −1.15303 −0.576515 0.817087i \(-0.695588\pi\)
−0.576515 + 0.817087i \(0.695588\pi\)
\(42\) −7.15650 −1.10427
\(43\) 5.54789 0.846046 0.423023 0.906119i \(-0.360969\pi\)
0.423023 + 0.906119i \(0.360969\pi\)
\(44\) 0.0451320 0.00680390
\(45\) 0 0
\(46\) −2.20164 −0.324613
\(47\) 4.28072 0.624407 0.312204 0.950015i \(-0.398933\pi\)
0.312204 + 0.950015i \(0.398933\pi\)
\(48\) −1.53175 −0.221089
\(49\) 14.8286 2.11837
\(50\) 0 0
\(51\) 5.52749 0.774003
\(52\) −5.26514 −0.730143
\(53\) −6.10215 −0.838194 −0.419097 0.907941i \(-0.637653\pi\)
−0.419097 + 0.907941i \(0.637653\pi\)
\(54\) −5.59662 −0.761603
\(55\) 0 0
\(56\) 4.67211 0.624337
\(57\) 9.52749 1.26195
\(58\) 4.20386 0.551994
\(59\) 13.5275 1.76113 0.880565 0.473926i \(-0.157164\pi\)
0.880565 + 0.473926i \(0.157164\pi\)
\(60\) 0 0
\(61\) −4.27299 −0.547100 −0.273550 0.961858i \(-0.588198\pi\)
−0.273550 + 0.961858i \(0.588198\pi\)
\(62\) 3.01051 0.382335
\(63\) 3.05436 0.384813
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0.0691309 0.00850942
\(67\) −12.3751 −1.51186 −0.755932 0.654650i \(-0.772816\pi\)
−0.755932 + 0.654650i \(0.772816\pi\)
\(68\) −3.60861 −0.437608
\(69\) −3.37235 −0.405984
\(70\) 0 0
\(71\) −4.49354 −0.533285 −0.266642 0.963796i \(-0.585914\pi\)
−0.266642 + 0.963796i \(0.585914\pi\)
\(72\) 0.653743 0.0770443
\(73\) −7.03811 −0.823748 −0.411874 0.911241i \(-0.635126\pi\)
−0.411874 + 0.911241i \(0.635126\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −6.22000 −0.713483
\(77\) −0.210861 −0.0240299
\(78\) −8.06487 −0.913167
\(79\) −8.72499 −0.981638 −0.490819 0.871262i \(-0.663302\pi\)
−0.490819 + 0.871262i \(0.663302\pi\)
\(80\) 0 0
\(81\) −6.61139 −0.734599
\(82\) 7.38299 0.815315
\(83\) 0.880231 0.0966179 0.0483090 0.998832i \(-0.484617\pi\)
0.0483090 + 0.998832i \(0.484617\pi\)
\(84\) 7.15650 0.780839
\(85\) 0 0
\(86\) −5.54789 −0.598245
\(87\) 6.43926 0.690361
\(88\) −0.0451320 −0.00481108
\(89\) 9.97602 1.05746 0.528728 0.848791i \(-0.322669\pi\)
0.528728 + 0.848791i \(0.322669\pi\)
\(90\) 0 0
\(91\) 24.5993 2.57871
\(92\) 2.20164 0.229536
\(93\) 4.61135 0.478175
\(94\) −4.28072 −0.441523
\(95\) 0 0
\(96\) 1.53175 0.156334
\(97\) −0.240408 −0.0244097 −0.0122049 0.999926i \(-0.503885\pi\)
−0.0122049 + 0.999926i \(0.503885\pi\)
\(98\) −14.8286 −1.49792
\(99\) −0.0295047 −0.00296533
\(100\) 0 0
\(101\) −5.61775 −0.558987 −0.279494 0.960148i \(-0.590167\pi\)
−0.279494 + 0.960148i \(0.590167\pi\)
\(102\) −5.52749 −0.547303
\(103\) 12.5663 1.23819 0.619095 0.785316i \(-0.287499\pi\)
0.619095 + 0.785316i \(0.287499\pi\)
\(104\) 5.26514 0.516289
\(105\) 0 0
\(106\) 6.10215 0.592693
\(107\) 17.9395 1.73427 0.867137 0.498070i \(-0.165958\pi\)
0.867137 + 0.498070i \(0.165958\pi\)
\(108\) 5.59662 0.538535
\(109\) 3.09936 0.296865 0.148433 0.988923i \(-0.452577\pi\)
0.148433 + 0.988923i \(0.452577\pi\)
\(110\) 0 0
\(111\) 1.53175 0.145387
\(112\) −4.67211 −0.441473
\(113\) −4.36184 −0.410328 −0.205164 0.978728i \(-0.565773\pi\)
−0.205164 + 0.978728i \(0.565773\pi\)
\(114\) −9.52749 −0.892331
\(115\) 0 0
\(116\) −4.20386 −0.390319
\(117\) 3.44204 0.318217
\(118\) −13.5275 −1.24531
\(119\) 16.8598 1.54554
\(120\) 0 0
\(121\) −10.9980 −0.999815
\(122\) 4.27299 0.386858
\(123\) 11.3089 1.01969
\(124\) −3.01051 −0.270352
\(125\) 0 0
\(126\) −3.05436 −0.272104
\(127\) −17.6572 −1.56683 −0.783413 0.621502i \(-0.786523\pi\)
−0.783413 + 0.621502i \(0.786523\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −8.49798 −0.748206
\(130\) 0 0
\(131\) −10.3103 −0.900812 −0.450406 0.892824i \(-0.648721\pi\)
−0.450406 + 0.892824i \(0.648721\pi\)
\(132\) −0.0691309 −0.00601707
\(133\) 29.0605 2.51987
\(134\) 12.3751 1.06905
\(135\) 0 0
\(136\) 3.60861 0.309436
\(137\) −3.21863 −0.274986 −0.137493 0.990503i \(-0.543905\pi\)
−0.137493 + 0.990503i \(0.543905\pi\)
\(138\) 3.37235 0.287074
\(139\) 20.9400 1.77611 0.888055 0.459737i \(-0.152056\pi\)
0.888055 + 0.459737i \(0.152056\pi\)
\(140\) 0 0
\(141\) −6.55699 −0.552198
\(142\) 4.49354 0.377089
\(143\) −0.237626 −0.0198713
\(144\) −0.653743 −0.0544786
\(145\) 0 0
\(146\) 7.03811 0.582478
\(147\) −22.7137 −1.87340
\(148\) −1.00000 −0.0821995
\(149\) −21.9832 −1.80093 −0.900467 0.434924i \(-0.856775\pi\)
−0.900467 + 0.434924i \(0.856775\pi\)
\(150\) 0 0
\(151\) 14.1877 1.15458 0.577290 0.816539i \(-0.304110\pi\)
0.577290 + 0.816539i \(0.304110\pi\)
\(152\) 6.22000 0.504509
\(153\) 2.35910 0.190722
\(154\) 0.210861 0.0169917
\(155\) 0 0
\(156\) 8.06487 0.645706
\(157\) 19.9316 1.59072 0.795359 0.606139i \(-0.207283\pi\)
0.795359 + 0.606139i \(0.207283\pi\)
\(158\) 8.72499 0.694123
\(159\) 9.34696 0.741262
\(160\) 0 0
\(161\) −10.2863 −0.810673
\(162\) 6.61139 0.519440
\(163\) −8.65087 −0.677588 −0.338794 0.940861i \(-0.610019\pi\)
−0.338794 + 0.940861i \(0.610019\pi\)
\(164\) −7.38299 −0.576515
\(165\) 0 0
\(166\) −0.880231 −0.0683192
\(167\) 11.9640 0.925803 0.462901 0.886410i \(-0.346808\pi\)
0.462901 + 0.886410i \(0.346808\pi\)
\(168\) −7.15650 −0.552136
\(169\) 14.7216 1.13243
\(170\) 0 0
\(171\) 4.06628 0.310956
\(172\) 5.54789 0.423023
\(173\) 5.33508 0.405619 0.202809 0.979218i \(-0.434993\pi\)
0.202809 + 0.979218i \(0.434993\pi\)
\(174\) −6.43926 −0.488159
\(175\) 0 0
\(176\) 0.0451320 0.00340195
\(177\) −20.7207 −1.55747
\(178\) −9.97602 −0.747734
\(179\) −5.27905 −0.394575 −0.197288 0.980346i \(-0.563213\pi\)
−0.197288 + 0.980346i \(0.563213\pi\)
\(180\) 0 0
\(181\) 17.7448 1.31896 0.659478 0.751723i \(-0.270777\pi\)
0.659478 + 0.751723i \(0.270777\pi\)
\(182\) −24.5993 −1.82342
\(183\) 6.54515 0.483832
\(184\) −2.20164 −0.162307
\(185\) 0 0
\(186\) −4.61135 −0.338121
\(187\) −0.162864 −0.0119098
\(188\) 4.28072 0.312204
\(189\) −26.1480 −1.90199
\(190\) 0 0
\(191\) −3.11649 −0.225501 −0.112751 0.993623i \(-0.535966\pi\)
−0.112751 + 0.993623i \(0.535966\pi\)
\(192\) −1.53175 −0.110545
\(193\) 8.78278 0.632198 0.316099 0.948726i \(-0.397627\pi\)
0.316099 + 0.948726i \(0.397627\pi\)
\(194\) 0.240408 0.0172603
\(195\) 0 0
\(196\) 14.8286 1.05919
\(197\) −7.00278 −0.498928 −0.249464 0.968384i \(-0.580254\pi\)
−0.249464 + 0.968384i \(0.580254\pi\)
\(198\) 0.0295047 0.00209681
\(199\) 17.3202 1.22780 0.613900 0.789384i \(-0.289600\pi\)
0.613900 + 0.789384i \(0.289600\pi\)
\(200\) 0 0
\(201\) 18.9556 1.33703
\(202\) 5.61775 0.395264
\(203\) 19.6409 1.37852
\(204\) 5.52749 0.387002
\(205\) 0 0
\(206\) −12.5663 −0.875533
\(207\) −1.43930 −0.100038
\(208\) −5.26514 −0.365071
\(209\) −0.280721 −0.0194179
\(210\) 0 0
\(211\) 6.77438 0.466368 0.233184 0.972433i \(-0.425086\pi\)
0.233184 + 0.972433i \(0.425086\pi\)
\(212\) −6.10215 −0.419097
\(213\) 6.88297 0.471613
\(214\) −17.9395 −1.22632
\(215\) 0 0
\(216\) −5.59662 −0.380802
\(217\) 14.0654 0.954825
\(218\) −3.09936 −0.209915
\(219\) 10.7806 0.728487
\(220\) 0 0
\(221\) 18.9998 1.27807
\(222\) −1.53175 −0.102804
\(223\) 10.2559 0.686784 0.343392 0.939192i \(-0.388424\pi\)
0.343392 + 0.939192i \(0.388424\pi\)
\(224\) 4.67211 0.312168
\(225\) 0 0
\(226\) 4.36184 0.290145
\(227\) 11.2744 0.748306 0.374153 0.927367i \(-0.377934\pi\)
0.374153 + 0.927367i \(0.377934\pi\)
\(228\) 9.52749 0.630973
\(229\) −9.40494 −0.621496 −0.310748 0.950492i \(-0.600579\pi\)
−0.310748 + 0.950492i \(0.600579\pi\)
\(230\) 0 0
\(231\) 0.322987 0.0212510
\(232\) 4.20386 0.275997
\(233\) 3.22867 0.211517 0.105759 0.994392i \(-0.466273\pi\)
0.105759 + 0.994392i \(0.466273\pi\)
\(234\) −3.44204 −0.225013
\(235\) 0 0
\(236\) 13.5275 0.880565
\(237\) 13.3645 0.868118
\(238\) −16.8598 −1.09286
\(239\) −17.8612 −1.15534 −0.577672 0.816269i \(-0.696039\pi\)
−0.577672 + 0.816269i \(0.696039\pi\)
\(240\) 0 0
\(241\) 26.6840 1.71887 0.859434 0.511248i \(-0.170816\pi\)
0.859434 + 0.511248i \(0.170816\pi\)
\(242\) 10.9980 0.706976
\(243\) −6.66286 −0.427423
\(244\) −4.27299 −0.273550
\(245\) 0 0
\(246\) −11.3089 −0.721029
\(247\) 32.7492 2.08378
\(248\) 3.01051 0.191168
\(249\) −1.34829 −0.0854447
\(250\) 0 0
\(251\) −30.1552 −1.90338 −0.951690 0.307060i \(-0.900655\pi\)
−0.951690 + 0.307060i \(0.900655\pi\)
\(252\) 3.05436 0.192406
\(253\) 0.0993641 0.00624697
\(254\) 17.6572 1.10791
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 3.15376 0.196726 0.0983632 0.995151i \(-0.468639\pi\)
0.0983632 + 0.995151i \(0.468639\pi\)
\(258\) 8.49798 0.529061
\(259\) 4.67211 0.290311
\(260\) 0 0
\(261\) 2.74824 0.170112
\(262\) 10.3103 0.636970
\(263\) −14.0332 −0.865322 −0.432661 0.901557i \(-0.642425\pi\)
−0.432661 + 0.901557i \(0.642425\pi\)
\(264\) 0.0691309 0.00425471
\(265\) 0 0
\(266\) −29.0605 −1.78182
\(267\) −15.2808 −0.935167
\(268\) −12.3751 −0.755932
\(269\) −30.1257 −1.83679 −0.918397 0.395660i \(-0.870516\pi\)
−0.918397 + 0.395660i \(0.870516\pi\)
\(270\) 0 0
\(271\) 14.8802 0.903910 0.451955 0.892041i \(-0.350727\pi\)
0.451955 + 0.892041i \(0.350727\pi\)
\(272\) −3.60861 −0.218804
\(273\) −37.6800 −2.28049
\(274\) 3.21863 0.194445
\(275\) 0 0
\(276\) −3.37235 −0.202992
\(277\) −0.147278 −0.00884908 −0.00442454 0.999990i \(-0.501408\pi\)
−0.00442454 + 0.999990i \(0.501408\pi\)
\(278\) −20.9400 −1.25590
\(279\) 1.96810 0.117827
\(280\) 0 0
\(281\) −25.5247 −1.52268 −0.761339 0.648354i \(-0.775458\pi\)
−0.761339 + 0.648354i \(0.775458\pi\)
\(282\) 6.55699 0.390463
\(283\) −0.0562691 −0.00334485 −0.00167242 0.999999i \(-0.500532\pi\)
−0.00167242 + 0.999999i \(0.500532\pi\)
\(284\) −4.49354 −0.266642
\(285\) 0 0
\(286\) 0.237626 0.0140511
\(287\) 34.4942 2.03613
\(288\) 0.653743 0.0385222
\(289\) −3.97793 −0.233996
\(290\) 0 0
\(291\) 0.368245 0.0215869
\(292\) −7.03811 −0.411874
\(293\) −13.7033 −0.800557 −0.400278 0.916394i \(-0.631087\pi\)
−0.400278 + 0.916394i \(0.631087\pi\)
\(294\) 22.7137 1.32469
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 0.252586 0.0146565
\(298\) 21.9832 1.27345
\(299\) −11.5919 −0.670377
\(300\) 0 0
\(301\) −25.9204 −1.49403
\(302\) −14.1877 −0.816411
\(303\) 8.60499 0.494344
\(304\) −6.22000 −0.356742
\(305\) 0 0
\(306\) −2.35910 −0.134861
\(307\) 17.5823 1.00348 0.501738 0.865020i \(-0.332694\pi\)
0.501738 + 0.865020i \(0.332694\pi\)
\(308\) −0.210861 −0.0120149
\(309\) −19.2484 −1.09500
\(310\) 0 0
\(311\) 0.353133 0.0200244 0.0100122 0.999950i \(-0.496813\pi\)
0.0100122 + 0.999950i \(0.496813\pi\)
\(312\) −8.06487 −0.456583
\(313\) 18.8717 1.06669 0.533346 0.845897i \(-0.320934\pi\)
0.533346 + 0.845897i \(0.320934\pi\)
\(314\) −19.9316 −1.12481
\(315\) 0 0
\(316\) −8.72499 −0.490819
\(317\) −15.1979 −0.853601 −0.426800 0.904346i \(-0.640359\pi\)
−0.426800 + 0.904346i \(0.640359\pi\)
\(318\) −9.34696 −0.524152
\(319\) −0.189728 −0.0106228
\(320\) 0 0
\(321\) −27.4788 −1.53372
\(322\) 10.2863 0.573232
\(323\) 22.4456 1.24890
\(324\) −6.61139 −0.367300
\(325\) 0 0
\(326\) 8.65087 0.479127
\(327\) −4.74745 −0.262535
\(328\) 7.38299 0.407658
\(329\) −20.0000 −1.10264
\(330\) 0 0
\(331\) −23.0210 −1.26535 −0.632674 0.774418i \(-0.718043\pi\)
−0.632674 + 0.774418i \(0.718043\pi\)
\(332\) 0.880231 0.0483090
\(333\) 0.653743 0.0358249
\(334\) −11.9640 −0.654641
\(335\) 0 0
\(336\) 7.15650 0.390419
\(337\) −34.0776 −1.85633 −0.928163 0.372173i \(-0.878613\pi\)
−0.928163 + 0.372173i \(0.878613\pi\)
\(338\) −14.7216 −0.800752
\(339\) 6.68125 0.362876
\(340\) 0 0
\(341\) −0.135870 −0.00735779
\(342\) −4.06628 −0.219879
\(343\) −36.5761 −1.97493
\(344\) −5.54789 −0.299122
\(345\) 0 0
\(346\) −5.33508 −0.286816
\(347\) −13.8449 −0.743236 −0.371618 0.928386i \(-0.621197\pi\)
−0.371618 + 0.928386i \(0.621197\pi\)
\(348\) 6.43926 0.345181
\(349\) 11.3765 0.608970 0.304485 0.952517i \(-0.401516\pi\)
0.304485 + 0.952517i \(0.401516\pi\)
\(350\) 0 0
\(351\) −29.4670 −1.57283
\(352\) −0.0451320 −0.00240554
\(353\) −19.2631 −1.02527 −0.512636 0.858606i \(-0.671331\pi\)
−0.512636 + 0.858606i \(0.671331\pi\)
\(354\) 20.7207 1.10129
\(355\) 0 0
\(356\) 9.97602 0.528728
\(357\) −25.8250 −1.36681
\(358\) 5.27905 0.279007
\(359\) 14.6645 0.773961 0.386980 0.922088i \(-0.373518\pi\)
0.386980 + 0.922088i \(0.373518\pi\)
\(360\) 0 0
\(361\) 19.6884 1.03623
\(362\) −17.7448 −0.932643
\(363\) 16.8461 0.884192
\(364\) 24.5993 1.28935
\(365\) 0 0
\(366\) −6.54515 −0.342121
\(367\) −0.868349 −0.0453275 −0.0226637 0.999743i \(-0.507215\pi\)
−0.0226637 + 0.999743i \(0.507215\pi\)
\(368\) 2.20164 0.114768
\(369\) 4.82658 0.251262
\(370\) 0 0
\(371\) 28.5099 1.48016
\(372\) 4.61135 0.239087
\(373\) −32.4555 −1.68048 −0.840240 0.542214i \(-0.817586\pi\)
−0.840240 + 0.542214i \(0.817586\pi\)
\(374\) 0.162864 0.00842148
\(375\) 0 0
\(376\) −4.28072 −0.220761
\(377\) 22.1339 1.13995
\(378\) 26.1480 1.34491
\(379\) −30.2511 −1.55389 −0.776947 0.629566i \(-0.783233\pi\)
−0.776947 + 0.629566i \(0.783233\pi\)
\(380\) 0 0
\(381\) 27.0465 1.38563
\(382\) 3.11649 0.159453
\(383\) −14.4935 −0.740585 −0.370293 0.928915i \(-0.620743\pi\)
−0.370293 + 0.928915i \(0.620743\pi\)
\(384\) 1.53175 0.0781668
\(385\) 0 0
\(386\) −8.78278 −0.447032
\(387\) −3.62689 −0.184365
\(388\) −0.240408 −0.0122049
\(389\) 36.9033 1.87107 0.935537 0.353229i \(-0.114916\pi\)
0.935537 + 0.353229i \(0.114916\pi\)
\(390\) 0 0
\(391\) −7.94485 −0.401788
\(392\) −14.8286 −0.748958
\(393\) 15.7927 0.796639
\(394\) 7.00278 0.352795
\(395\) 0 0
\(396\) −0.0295047 −0.00148267
\(397\) −22.9365 −1.15115 −0.575575 0.817749i \(-0.695222\pi\)
−0.575575 + 0.817749i \(0.695222\pi\)
\(398\) −17.3202 −0.868185
\(399\) −44.5135 −2.22846
\(400\) 0 0
\(401\) 27.4332 1.36995 0.684973 0.728568i \(-0.259814\pi\)
0.684973 + 0.728568i \(0.259814\pi\)
\(402\) −18.9556 −0.945420
\(403\) 15.8508 0.789582
\(404\) −5.61775 −0.279494
\(405\) 0 0
\(406\) −19.6409 −0.974761
\(407\) −0.0451320 −0.00223711
\(408\) −5.52749 −0.273651
\(409\) −21.3920 −1.05777 −0.528883 0.848695i \(-0.677389\pi\)
−0.528883 + 0.848695i \(0.677389\pi\)
\(410\) 0 0
\(411\) 4.93014 0.243186
\(412\) 12.5663 0.619095
\(413\) −63.2019 −3.10996
\(414\) 1.43930 0.0707379
\(415\) 0 0
\(416\) 5.26514 0.258144
\(417\) −32.0749 −1.57071
\(418\) 0.280721 0.0137305
\(419\) 23.0891 1.12797 0.563987 0.825784i \(-0.309267\pi\)
0.563987 + 0.825784i \(0.309267\pi\)
\(420\) 0 0
\(421\) −1.83757 −0.0895576 −0.0447788 0.998997i \(-0.514258\pi\)
−0.0447788 + 0.998997i \(0.514258\pi\)
\(422\) −6.77438 −0.329772
\(423\) −2.79849 −0.136067
\(424\) 6.10215 0.296346
\(425\) 0 0
\(426\) −6.88297 −0.333481
\(427\) 19.9639 0.966120
\(428\) 17.9395 0.867137
\(429\) 0.363983 0.0175733
\(430\) 0 0
\(431\) −15.9796 −0.769710 −0.384855 0.922977i \(-0.625749\pi\)
−0.384855 + 0.922977i \(0.625749\pi\)
\(432\) 5.59662 0.269267
\(433\) 25.2525 1.21356 0.606779 0.794870i \(-0.292461\pi\)
0.606779 + 0.794870i \(0.292461\pi\)
\(434\) −14.0654 −0.675163
\(435\) 0 0
\(436\) 3.09936 0.148433
\(437\) −13.6942 −0.655082
\(438\) −10.7806 −0.515118
\(439\) −9.30792 −0.444243 −0.222121 0.975019i \(-0.571298\pi\)
−0.222121 + 0.975019i \(0.571298\pi\)
\(440\) 0 0
\(441\) −9.69410 −0.461624
\(442\) −18.9998 −0.903729
\(443\) 9.52445 0.452520 0.226260 0.974067i \(-0.427350\pi\)
0.226260 + 0.974067i \(0.427350\pi\)
\(444\) 1.53175 0.0726936
\(445\) 0 0
\(446\) −10.2559 −0.485629
\(447\) 33.6728 1.59267
\(448\) −4.67211 −0.220736
\(449\) 5.43278 0.256389 0.128194 0.991749i \(-0.459082\pi\)
0.128194 + 0.991749i \(0.459082\pi\)
\(450\) 0 0
\(451\) −0.333209 −0.0156902
\(452\) −4.36184 −0.205164
\(453\) −21.7320 −1.02106
\(454\) −11.2744 −0.529132
\(455\) 0 0
\(456\) −9.52749 −0.446166
\(457\) −7.54789 −0.353076 −0.176538 0.984294i \(-0.556490\pi\)
−0.176538 + 0.984294i \(0.556490\pi\)
\(458\) 9.40494 0.439464
\(459\) −20.1960 −0.942670
\(460\) 0 0
\(461\) −35.4323 −1.65025 −0.825124 0.564952i \(-0.808895\pi\)
−0.825124 + 0.564952i \(0.808895\pi\)
\(462\) −0.322987 −0.0150267
\(463\) −20.9124 −0.971883 −0.485942 0.873991i \(-0.661523\pi\)
−0.485942 + 0.873991i \(0.661523\pi\)
\(464\) −4.20386 −0.195159
\(465\) 0 0
\(466\) −3.22867 −0.149565
\(467\) −6.62523 −0.306579 −0.153289 0.988181i \(-0.548987\pi\)
−0.153289 + 0.988181i \(0.548987\pi\)
\(468\) 3.44204 0.159109
\(469\) 57.8180 2.66979
\(470\) 0 0
\(471\) −30.5303 −1.40676
\(472\) −13.5275 −0.622653
\(473\) 0.250387 0.0115128
\(474\) −13.3645 −0.613852
\(475\) 0 0
\(476\) 16.8598 0.772769
\(477\) 3.98923 0.182654
\(478\) 17.8612 0.816952
\(479\) −29.4581 −1.34597 −0.672987 0.739655i \(-0.734989\pi\)
−0.672987 + 0.739655i \(0.734989\pi\)
\(480\) 0 0
\(481\) 5.26514 0.240070
\(482\) −26.6840 −1.21542
\(483\) 15.7560 0.716923
\(484\) −10.9980 −0.499907
\(485\) 0 0
\(486\) 6.66286 0.302233
\(487\) −8.80654 −0.399063 −0.199531 0.979891i \(-0.563942\pi\)
−0.199531 + 0.979891i \(0.563942\pi\)
\(488\) 4.27299 0.193429
\(489\) 13.2510 0.599229
\(490\) 0 0
\(491\) 5.14258 0.232082 0.116041 0.993244i \(-0.462980\pi\)
0.116041 + 0.993244i \(0.462980\pi\)
\(492\) 11.3089 0.509844
\(493\) 15.1701 0.683227
\(494\) −32.7492 −1.47345
\(495\) 0 0
\(496\) −3.01051 −0.135176
\(497\) 20.9943 0.941723
\(498\) 1.34829 0.0604185
\(499\) −12.9692 −0.580579 −0.290290 0.956939i \(-0.593752\pi\)
−0.290290 + 0.956939i \(0.593752\pi\)
\(500\) 0 0
\(501\) −18.3259 −0.818739
\(502\) 30.1552 1.34589
\(503\) −17.2091 −0.767314 −0.383657 0.923476i \(-0.625336\pi\)
−0.383657 + 0.923476i \(0.625336\pi\)
\(504\) −3.05436 −0.136052
\(505\) 0 0
\(506\) −0.0993641 −0.00441727
\(507\) −22.5499 −1.00148
\(508\) −17.6572 −0.783413
\(509\) −12.1085 −0.536698 −0.268349 0.963322i \(-0.586478\pi\)
−0.268349 + 0.963322i \(0.586478\pi\)
\(510\) 0 0
\(511\) 32.8828 1.45465
\(512\) −1.00000 −0.0441942
\(513\) −34.8110 −1.53694
\(514\) −3.15376 −0.139107
\(515\) 0 0
\(516\) −8.49798 −0.374103
\(517\) 0.193197 0.00849681
\(518\) −4.67211 −0.205281
\(519\) −8.17201 −0.358711
\(520\) 0 0
\(521\) −18.7936 −0.823362 −0.411681 0.911328i \(-0.635058\pi\)
−0.411681 + 0.911328i \(0.635058\pi\)
\(522\) −2.74824 −0.120287
\(523\) 42.9985 1.88019 0.940096 0.340909i \(-0.110735\pi\)
0.940096 + 0.340909i \(0.110735\pi\)
\(524\) −10.3103 −0.450406
\(525\) 0 0
\(526\) 14.0332 0.611875
\(527\) 10.8638 0.473233
\(528\) −0.0691309 −0.00300853
\(529\) −18.1528 −0.789252
\(530\) 0 0
\(531\) −8.84350 −0.383775
\(532\) 29.0605 1.25993
\(533\) 38.8725 1.68375
\(534\) 15.2808 0.661263
\(535\) 0 0
\(536\) 12.3751 0.534525
\(537\) 8.08619 0.348945
\(538\) 30.1257 1.29881
\(539\) 0.669244 0.0288264
\(540\) 0 0
\(541\) 30.1751 1.29733 0.648664 0.761075i \(-0.275328\pi\)
0.648664 + 0.761075i \(0.275328\pi\)
\(542\) −14.8802 −0.639161
\(543\) −27.1805 −1.16643
\(544\) 3.60861 0.154718
\(545\) 0 0
\(546\) 37.6800 1.61255
\(547\) 31.6736 1.35426 0.677132 0.735862i \(-0.263223\pi\)
0.677132 + 0.735862i \(0.263223\pi\)
\(548\) −3.21863 −0.137493
\(549\) 2.79344 0.119221
\(550\) 0 0
\(551\) 26.1480 1.11394
\(552\) 3.37235 0.143537
\(553\) 40.7641 1.73347
\(554\) 0.147278 0.00625724
\(555\) 0 0
\(556\) 20.9400 0.888055
\(557\) 0.708971 0.0300401 0.0150200 0.999887i \(-0.495219\pi\)
0.0150200 + 0.999887i \(0.495219\pi\)
\(558\) −1.96810 −0.0833163
\(559\) −29.2104 −1.23547
\(560\) 0 0
\(561\) 0.249466 0.0105325
\(562\) 25.5247 1.07670
\(563\) 29.9399 1.26182 0.630908 0.775857i \(-0.282682\pi\)
0.630908 + 0.775857i \(0.282682\pi\)
\(564\) −6.55699 −0.276099
\(565\) 0 0
\(566\) 0.0562691 0.00236517
\(567\) 30.8892 1.29722
\(568\) 4.49354 0.188545
\(569\) 21.1453 0.886456 0.443228 0.896409i \(-0.353833\pi\)
0.443228 + 0.896409i \(0.353833\pi\)
\(570\) 0 0
\(571\) −19.5326 −0.817416 −0.408708 0.912665i \(-0.634020\pi\)
−0.408708 + 0.912665i \(0.634020\pi\)
\(572\) −0.237626 −0.00993564
\(573\) 4.77368 0.199423
\(574\) −34.4942 −1.43976
\(575\) 0 0
\(576\) −0.653743 −0.0272393
\(577\) −1.08303 −0.0450873 −0.0225436 0.999746i \(-0.507176\pi\)
−0.0225436 + 0.999746i \(0.507176\pi\)
\(578\) 3.97793 0.165460
\(579\) −13.4530 −0.559088
\(580\) 0 0
\(581\) −4.11254 −0.170617
\(582\) −0.368245 −0.0152642
\(583\) −0.275402 −0.0114060
\(584\) 7.03811 0.291239
\(585\) 0 0
\(586\) 13.7033 0.566079
\(587\) −1.16839 −0.0482244 −0.0241122 0.999709i \(-0.507676\pi\)
−0.0241122 + 0.999709i \(0.507676\pi\)
\(588\) −22.7137 −0.936698
\(589\) 18.7254 0.771566
\(590\) 0 0
\(591\) 10.7265 0.441230
\(592\) −1.00000 −0.0410997
\(593\) 6.01305 0.246926 0.123463 0.992349i \(-0.460600\pi\)
0.123463 + 0.992349i \(0.460600\pi\)
\(594\) −0.252586 −0.0103637
\(595\) 0 0
\(596\) −21.9832 −0.900467
\(597\) −26.5303 −1.08581
\(598\) 11.5919 0.474028
\(599\) −11.9208 −0.487072 −0.243536 0.969892i \(-0.578307\pi\)
−0.243536 + 0.969892i \(0.578307\pi\)
\(600\) 0 0
\(601\) 21.7772 0.888309 0.444155 0.895950i \(-0.353504\pi\)
0.444155 + 0.895950i \(0.353504\pi\)
\(602\) 25.9204 1.05644
\(603\) 8.09015 0.329457
\(604\) 14.1877 0.577290
\(605\) 0 0
\(606\) −8.60499 −0.349554
\(607\) −5.16486 −0.209635 −0.104818 0.994491i \(-0.533426\pi\)
−0.104818 + 0.994491i \(0.533426\pi\)
\(608\) 6.22000 0.252254
\(609\) −30.0849 −1.21910
\(610\) 0 0
\(611\) −22.5386 −0.911813
\(612\) 2.35910 0.0953611
\(613\) 39.5817 1.59869 0.799344 0.600873i \(-0.205180\pi\)
0.799344 + 0.600873i \(0.205180\pi\)
\(614\) −17.5823 −0.709565
\(615\) 0 0
\(616\) 0.210861 0.00849585
\(617\) −44.9917 −1.81130 −0.905650 0.424027i \(-0.860616\pi\)
−0.905650 + 0.424027i \(0.860616\pi\)
\(618\) 19.2484 0.774283
\(619\) 30.0465 1.20767 0.603836 0.797108i \(-0.293638\pi\)
0.603836 + 0.797108i \(0.293638\pi\)
\(620\) 0 0
\(621\) 12.3217 0.494453
\(622\) −0.353133 −0.0141594
\(623\) −46.6091 −1.86735
\(624\) 8.06487 0.322853
\(625\) 0 0
\(626\) −18.8717 −0.754265
\(627\) 0.429994 0.0171723
\(628\) 19.9316 0.795359
\(629\) 3.60861 0.143885
\(630\) 0 0
\(631\) 5.01448 0.199623 0.0998116 0.995006i \(-0.468176\pi\)
0.0998116 + 0.995006i \(0.468176\pi\)
\(632\) 8.72499 0.347061
\(633\) −10.3767 −0.412435
\(634\) 15.1979 0.603587
\(635\) 0 0
\(636\) 9.34696 0.370631
\(637\) −78.0747 −3.09343
\(638\) 0.189728 0.00751142
\(639\) 2.93762 0.116210
\(640\) 0 0
\(641\) 18.9133 0.747031 0.373515 0.927624i \(-0.378152\pi\)
0.373515 + 0.927624i \(0.378152\pi\)
\(642\) 27.4788 1.08450
\(643\) −22.9344 −0.904444 −0.452222 0.891906i \(-0.649368\pi\)
−0.452222 + 0.891906i \(0.649368\pi\)
\(644\) −10.2863 −0.405336
\(645\) 0 0
\(646\) −22.4456 −0.883109
\(647\) −10.4741 −0.411779 −0.205889 0.978575i \(-0.566009\pi\)
−0.205889 + 0.978575i \(0.566009\pi\)
\(648\) 6.61139 0.259720
\(649\) 0.610522 0.0239651
\(650\) 0 0
\(651\) −21.5447 −0.844405
\(652\) −8.65087 −0.338794
\(653\) 44.7251 1.75023 0.875115 0.483916i \(-0.160786\pi\)
0.875115 + 0.483916i \(0.160786\pi\)
\(654\) 4.74745 0.185640
\(655\) 0 0
\(656\) −7.38299 −0.288257
\(657\) 4.60111 0.179506
\(658\) 20.0000 0.779681
\(659\) 15.1056 0.588431 0.294215 0.955739i \(-0.404942\pi\)
0.294215 + 0.955739i \(0.404942\pi\)
\(660\) 0 0
\(661\) 34.9654 1.36000 0.679998 0.733214i \(-0.261981\pi\)
0.679998 + 0.733214i \(0.261981\pi\)
\(662\) 23.0210 0.894736
\(663\) −29.1030 −1.13027
\(664\) −0.880231 −0.0341596
\(665\) 0 0
\(666\) −0.653743 −0.0253320
\(667\) −9.25537 −0.358369
\(668\) 11.9640 0.462901
\(669\) −15.7094 −0.607361
\(670\) 0 0
\(671\) −0.192848 −0.00744483
\(672\) −7.15650 −0.276068
\(673\) −35.2496 −1.35877 −0.679385 0.733782i \(-0.737754\pi\)
−0.679385 + 0.733782i \(0.737754\pi\)
\(674\) 34.0776 1.31262
\(675\) 0 0
\(676\) 14.7216 0.566217
\(677\) 23.2835 0.894858 0.447429 0.894320i \(-0.352340\pi\)
0.447429 + 0.894320i \(0.352340\pi\)
\(678\) −6.68125 −0.256592
\(679\) 1.12321 0.0431049
\(680\) 0 0
\(681\) −17.2695 −0.661769
\(682\) 0.135870 0.00520274
\(683\) 11.8131 0.452017 0.226008 0.974125i \(-0.427432\pi\)
0.226008 + 0.974125i \(0.427432\pi\)
\(684\) 4.06628 0.155478
\(685\) 0 0
\(686\) 36.5761 1.39648
\(687\) 14.4060 0.549624
\(688\) 5.54789 0.211511
\(689\) 32.1286 1.22400
\(690\) 0 0
\(691\) 14.1858 0.539652 0.269826 0.962909i \(-0.413034\pi\)
0.269826 + 0.962909i \(0.413034\pi\)
\(692\) 5.33508 0.202809
\(693\) 0.137849 0.00523646
\(694\) 13.8449 0.525547
\(695\) 0 0
\(696\) −6.43926 −0.244080
\(697\) 26.6423 1.00915
\(698\) −11.3765 −0.430607
\(699\) −4.94552 −0.187057
\(700\) 0 0
\(701\) 19.2184 0.725870 0.362935 0.931815i \(-0.381775\pi\)
0.362935 + 0.931815i \(0.381775\pi\)
\(702\) 29.4670 1.11216
\(703\) 6.22000 0.234592
\(704\) 0.0451320 0.00170097
\(705\) 0 0
\(706\) 19.2631 0.724976
\(707\) 26.2468 0.987111
\(708\) −20.7207 −0.778733
\(709\) 4.52525 0.169949 0.0849746 0.996383i \(-0.472919\pi\)
0.0849746 + 0.996383i \(0.472919\pi\)
\(710\) 0 0
\(711\) 5.70390 0.213913
\(712\) −9.97602 −0.373867
\(713\) −6.62805 −0.248222
\(714\) 25.8250 0.966478
\(715\) 0 0
\(716\) −5.27905 −0.197288
\(717\) 27.3589 1.02174
\(718\) −14.6645 −0.547273
\(719\) 0.915477 0.0341415 0.0170708 0.999854i \(-0.494566\pi\)
0.0170708 + 0.999854i \(0.494566\pi\)
\(720\) 0 0
\(721\) −58.7110 −2.18651
\(722\) −19.6884 −0.732728
\(723\) −40.8732 −1.52009
\(724\) 17.7448 0.659478
\(725\) 0 0
\(726\) −16.8461 −0.625218
\(727\) 11.7429 0.435521 0.217760 0.976002i \(-0.430125\pi\)
0.217760 + 0.976002i \(0.430125\pi\)
\(728\) −24.5993 −0.911710
\(729\) 30.0400 1.11259
\(730\) 0 0
\(731\) −20.0202 −0.740473
\(732\) 6.54515 0.241916
\(733\) −27.6776 −1.02230 −0.511148 0.859493i \(-0.670780\pi\)
−0.511148 + 0.859493i \(0.670780\pi\)
\(734\) 0.868349 0.0320514
\(735\) 0 0
\(736\) −2.20164 −0.0811534
\(737\) −0.558514 −0.0205731
\(738\) −4.82658 −0.177669
\(739\) −6.52187 −0.239911 −0.119956 0.992779i \(-0.538275\pi\)
−0.119956 + 0.992779i \(0.538275\pi\)
\(740\) 0 0
\(741\) −50.1635 −1.84280
\(742\) −28.5099 −1.04663
\(743\) −43.0091 −1.57785 −0.788925 0.614489i \(-0.789362\pi\)
−0.788925 + 0.614489i \(0.789362\pi\)
\(744\) −4.61135 −0.169060
\(745\) 0 0
\(746\) 32.4555 1.18828
\(747\) −0.575445 −0.0210544
\(748\) −0.162864 −0.00595489
\(749\) −83.8152 −3.06254
\(750\) 0 0
\(751\) −29.7068 −1.08402 −0.542008 0.840373i \(-0.682336\pi\)
−0.542008 + 0.840373i \(0.682336\pi\)
\(752\) 4.28072 0.156102
\(753\) 46.1902 1.68327
\(754\) −22.1339 −0.806069
\(755\) 0 0
\(756\) −26.1480 −0.950994
\(757\) −32.7918 −1.19184 −0.595920 0.803044i \(-0.703212\pi\)
−0.595920 + 0.803044i \(0.703212\pi\)
\(758\) 30.2511 1.09877
\(759\) −0.152201 −0.00552455
\(760\) 0 0
\(761\) 22.9411 0.831616 0.415808 0.909452i \(-0.363499\pi\)
0.415808 + 0.909452i \(0.363499\pi\)
\(762\) −27.0465 −0.979790
\(763\) −14.4806 −0.524232
\(764\) −3.11649 −0.112751
\(765\) 0 0
\(766\) 14.4935 0.523673
\(767\) −71.2241 −2.57175
\(768\) −1.53175 −0.0552723
\(769\) 10.8123 0.389902 0.194951 0.980813i \(-0.437545\pi\)
0.194951 + 0.980813i \(0.437545\pi\)
\(770\) 0 0
\(771\) −4.83078 −0.173976
\(772\) 8.78278 0.316099
\(773\) 37.9989 1.36673 0.683363 0.730079i \(-0.260517\pi\)
0.683363 + 0.730079i \(0.260517\pi\)
\(774\) 3.62689 0.130366
\(775\) 0 0
\(776\) 0.240408 0.00863014
\(777\) −7.15650 −0.256738
\(778\) −36.9033 −1.32305
\(779\) 45.9222 1.64533
\(780\) 0 0
\(781\) −0.202802 −0.00725683
\(782\) 7.94485 0.284107
\(783\) −23.5274 −0.840801
\(784\) 14.8286 0.529593
\(785\) 0 0
\(786\) −15.7927 −0.563309
\(787\) −31.7730 −1.13258 −0.566292 0.824205i \(-0.691623\pi\)
−0.566292 + 0.824205i \(0.691623\pi\)
\(788\) −7.00278 −0.249464
\(789\) 21.4953 0.765252
\(790\) 0 0
\(791\) 20.3790 0.724594
\(792\) 0.0295047 0.00104840
\(793\) 22.4979 0.798923
\(794\) 22.9365 0.813986
\(795\) 0 0
\(796\) 17.3202 0.613900
\(797\) −9.02551 −0.319700 −0.159850 0.987141i \(-0.551101\pi\)
−0.159850 + 0.987141i \(0.551101\pi\)
\(798\) 44.5135 1.57576
\(799\) −15.4475 −0.546492
\(800\) 0 0
\(801\) −6.52175 −0.230435
\(802\) −27.4332 −0.968698
\(803\) −0.317643 −0.0112094
\(804\) 18.9556 0.668513
\(805\) 0 0
\(806\) −15.8508 −0.558319
\(807\) 46.1450 1.62438
\(808\) 5.61775 0.197632
\(809\) 29.7376 1.04552 0.522759 0.852481i \(-0.324903\pi\)
0.522759 + 0.852481i \(0.324903\pi\)
\(810\) 0 0
\(811\) 15.7650 0.553585 0.276793 0.960930i \(-0.410729\pi\)
0.276793 + 0.960930i \(0.410729\pi\)
\(812\) 19.6409 0.689260
\(813\) −22.7928 −0.799378
\(814\) 0.0451320 0.00158187
\(815\) 0 0
\(816\) 5.52749 0.193501
\(817\) −34.5079 −1.20728
\(818\) 21.3920 0.747954
\(819\) −16.0816 −0.561937
\(820\) 0 0
\(821\) −4.27798 −0.149303 −0.0746513 0.997210i \(-0.523784\pi\)
−0.0746513 + 0.997210i \(0.523784\pi\)
\(822\) −4.93014 −0.171958
\(823\) 13.8985 0.484471 0.242236 0.970217i \(-0.422119\pi\)
0.242236 + 0.970217i \(0.422119\pi\)
\(824\) −12.5663 −0.437766
\(825\) 0 0
\(826\) 63.2019 2.19908
\(827\) 30.4944 1.06039 0.530197 0.847875i \(-0.322118\pi\)
0.530197 + 0.847875i \(0.322118\pi\)
\(828\) −1.43930 −0.0500192
\(829\) 0.493020 0.0171233 0.00856165 0.999963i \(-0.497275\pi\)
0.00856165 + 0.999963i \(0.497275\pi\)
\(830\) 0 0
\(831\) 0.225593 0.00782574
\(832\) −5.26514 −0.182536
\(833\) −53.5107 −1.85404
\(834\) 32.0749 1.11066
\(835\) 0 0
\(836\) −0.280721 −0.00970894
\(837\) −16.8487 −0.582376
\(838\) −23.0891 −0.797598
\(839\) −27.9308 −0.964278 −0.482139 0.876095i \(-0.660140\pi\)
−0.482139 + 0.876095i \(0.660140\pi\)
\(840\) 0 0
\(841\) −11.3276 −0.390606
\(842\) 1.83757 0.0633268
\(843\) 39.0975 1.34659
\(844\) 6.77438 0.233184
\(845\) 0 0
\(846\) 2.79849 0.0962141
\(847\) 51.3837 1.76556
\(848\) −6.10215 −0.209549
\(849\) 0.0861901 0.00295804
\(850\) 0 0
\(851\) −2.20164 −0.0754711
\(852\) 6.88297 0.235807
\(853\) −31.4029 −1.07521 −0.537607 0.843195i \(-0.680672\pi\)
−0.537607 + 0.843195i \(0.680672\pi\)
\(854\) −19.9639 −0.683150
\(855\) 0 0
\(856\) −17.9395 −0.613158
\(857\) −39.7806 −1.35888 −0.679441 0.733731i \(-0.737777\pi\)
−0.679441 + 0.733731i \(0.737777\pi\)
\(858\) −0.363983 −0.0124262
\(859\) 41.8673 1.42849 0.714247 0.699894i \(-0.246769\pi\)
0.714247 + 0.699894i \(0.246769\pi\)
\(860\) 0 0
\(861\) −52.8364 −1.80066
\(862\) 15.9796 0.544267
\(863\) 8.52387 0.290156 0.145078 0.989420i \(-0.453657\pi\)
0.145078 + 0.989420i \(0.453657\pi\)
\(864\) −5.59662 −0.190401
\(865\) 0 0
\(866\) −25.2525 −0.858116
\(867\) 6.09319 0.206936
\(868\) 14.0654 0.477412
\(869\) −0.393776 −0.0133579
\(870\) 0 0
\(871\) 65.1568 2.20775
\(872\) −3.09936 −0.104958
\(873\) 0.157165 0.00531922
\(874\) 13.6942 0.463213
\(875\) 0 0
\(876\) 10.7806 0.364243
\(877\) 37.1163 1.25333 0.626664 0.779289i \(-0.284420\pi\)
0.626664 + 0.779289i \(0.284420\pi\)
\(878\) 9.30792 0.314127
\(879\) 20.9901 0.707977
\(880\) 0 0
\(881\) 29.8633 1.00612 0.503060 0.864252i \(-0.332208\pi\)
0.503060 + 0.864252i \(0.332208\pi\)
\(882\) 9.69410 0.326417
\(883\) 9.98952 0.336174 0.168087 0.985772i \(-0.446241\pi\)
0.168087 + 0.985772i \(0.446241\pi\)
\(884\) 18.9998 0.639033
\(885\) 0 0
\(886\) −9.52445 −0.319980
\(887\) −23.2703 −0.781340 −0.390670 0.920531i \(-0.627757\pi\)
−0.390670 + 0.920531i \(0.627757\pi\)
\(888\) −1.53175 −0.0514022
\(889\) 82.4965 2.76684
\(890\) 0 0
\(891\) −0.298385 −0.00999628
\(892\) 10.2559 0.343392
\(893\) −26.6261 −0.891008
\(894\) −33.6728 −1.12619
\(895\) 0 0
\(896\) 4.67211 0.156084
\(897\) 17.7559 0.592852
\(898\) −5.43278 −0.181294
\(899\) 12.6558 0.422094
\(900\) 0 0
\(901\) 22.0203 0.733602
\(902\) 0.333209 0.0110946
\(903\) 39.7035 1.32125
\(904\) 4.36184 0.145073
\(905\) 0 0
\(906\) 21.7320 0.721998
\(907\) 34.0053 1.12913 0.564564 0.825389i \(-0.309044\pi\)
0.564564 + 0.825389i \(0.309044\pi\)
\(908\) 11.2744 0.374153
\(909\) 3.67256 0.121811
\(910\) 0 0
\(911\) −29.2297 −0.968424 −0.484212 0.874951i \(-0.660894\pi\)
−0.484212 + 0.874951i \(0.660894\pi\)
\(912\) 9.52749 0.315487
\(913\) 0.0397266 0.00131476
\(914\) 7.54789 0.249662
\(915\) 0 0
\(916\) −9.40494 −0.310748
\(917\) 48.1707 1.59074
\(918\) 20.1960 0.666568
\(919\) 43.0238 1.41923 0.709613 0.704592i \(-0.248870\pi\)
0.709613 + 0.704592i \(0.248870\pi\)
\(920\) 0 0
\(921\) −26.9317 −0.887430
\(922\) 35.4323 1.16690
\(923\) 23.6591 0.778748
\(924\) 0.322987 0.0106255
\(925\) 0 0
\(926\) 20.9124 0.687225
\(927\) −8.21510 −0.269819
\(928\) 4.20386 0.137998
\(929\) 10.7786 0.353635 0.176817 0.984244i \(-0.443420\pi\)
0.176817 + 0.984244i \(0.443420\pi\)
\(930\) 0 0
\(931\) −92.2340 −3.02285
\(932\) 3.22867 0.105759
\(933\) −0.540912 −0.0177087
\(934\) 6.62523 0.216784
\(935\) 0 0
\(936\) −3.44204 −0.112507
\(937\) −36.5755 −1.19487 −0.597435 0.801918i \(-0.703813\pi\)
−0.597435 + 0.801918i \(0.703813\pi\)
\(938\) −57.8180 −1.88782
\(939\) −28.9067 −0.943336
\(940\) 0 0
\(941\) 58.9579 1.92197 0.960986 0.276596i \(-0.0892064\pi\)
0.960986 + 0.276596i \(0.0892064\pi\)
\(942\) 30.5303 0.994730
\(943\) −16.2547 −0.529325
\(944\) 13.5275 0.440282
\(945\) 0 0
\(946\) −0.250387 −0.00814079
\(947\) −9.63990 −0.313255 −0.156627 0.987658i \(-0.550062\pi\)
−0.156627 + 0.987658i \(0.550062\pi\)
\(948\) 13.3645 0.434059
\(949\) 37.0566 1.20291
\(950\) 0 0
\(951\) 23.2794 0.754887
\(952\) −16.8598 −0.546430
\(953\) −31.4809 −1.01977 −0.509884 0.860243i \(-0.670312\pi\)
−0.509884 + 0.860243i \(0.670312\pi\)
\(954\) −3.98923 −0.129156
\(955\) 0 0
\(956\) −17.8612 −0.577672
\(957\) 0.290616 0.00939430
\(958\) 29.4581 0.951747
\(959\) 15.0378 0.485596
\(960\) 0 0
\(961\) −21.9368 −0.707639
\(962\) −5.26514 −0.169755
\(963\) −11.7278 −0.377923
\(964\) 26.6840 0.859434
\(965\) 0 0
\(966\) −15.7560 −0.506941
\(967\) −25.5439 −0.821438 −0.410719 0.911762i \(-0.634722\pi\)
−0.410719 + 0.911762i \(0.634722\pi\)
\(968\) 10.9980 0.353488
\(969\) −34.3810 −1.10448
\(970\) 0 0
\(971\) 43.7224 1.40312 0.701559 0.712611i \(-0.252488\pi\)
0.701559 + 0.712611i \(0.252488\pi\)
\(972\) −6.66286 −0.213711
\(973\) −97.8341 −3.13642
\(974\) 8.80654 0.282180
\(975\) 0 0
\(976\) −4.27299 −0.136775
\(977\) 44.6598 1.42879 0.714396 0.699741i \(-0.246701\pi\)
0.714396 + 0.699741i \(0.246701\pi\)
\(978\) −13.2510 −0.423719
\(979\) 0.450237 0.0143896
\(980\) 0 0
\(981\) −2.02619 −0.0646912
\(982\) −5.14258 −0.164106
\(983\) −9.17840 −0.292745 −0.146373 0.989230i \(-0.546760\pi\)
−0.146373 + 0.989230i \(0.546760\pi\)
\(984\) −11.3089 −0.360514
\(985\) 0 0
\(986\) −15.1701 −0.483114
\(987\) 30.6350 0.975123
\(988\) 32.7492 1.04189
\(989\) 12.2144 0.388397
\(990\) 0 0
\(991\) 16.1246 0.512216 0.256108 0.966648i \(-0.417560\pi\)
0.256108 + 0.966648i \(0.417560\pi\)
\(992\) 3.01051 0.0955839
\(993\) 35.2624 1.11902
\(994\) −20.9943 −0.665899
\(995\) 0 0
\(996\) −1.34829 −0.0427223
\(997\) −8.99979 −0.285026 −0.142513 0.989793i \(-0.545518\pi\)
−0.142513 + 0.989793i \(0.545518\pi\)
\(998\) 12.9692 0.410532
\(999\) −5.59662 −0.177069
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.bd.1.2 5
5.2 odd 4 370.2.b.d.149.4 10
5.3 odd 4 370.2.b.d.149.7 yes 10
5.4 even 2 1850.2.a.be.1.4 5
15.2 even 4 3330.2.d.p.1999.6 10
15.8 even 4 3330.2.d.p.1999.1 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.d.149.4 10 5.2 odd 4
370.2.b.d.149.7 yes 10 5.3 odd 4
1850.2.a.bd.1.2 5 1.1 even 1 trivial
1850.2.a.be.1.4 5 5.4 even 2
3330.2.d.p.1999.1 10 15.8 even 4
3330.2.d.p.1999.6 10 15.2 even 4