Properties

Label 1850.2.a.v.1.1
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 1.1
Root \(-2.44949\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} +4.44949 q^{7} +1.00000 q^{8} +3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} +4.44949 q^{7} +1.00000 q^{8} +3.00000 q^{9} +4.89898 q^{11} -2.44949 q^{12} +4.00000 q^{13} +4.44949 q^{14} +1.00000 q^{16} -4.89898 q^{17} +3.00000 q^{18} +3.55051 q^{19} -10.8990 q^{21} +4.89898 q^{22} -8.89898 q^{23} -2.44949 q^{24} +4.00000 q^{26} +4.44949 q^{28} -1.55051 q^{31} +1.00000 q^{32} -12.0000 q^{33} -4.89898 q^{34} +3.00000 q^{36} +1.00000 q^{37} +3.55051 q^{38} -9.79796 q^{39} +2.00000 q^{41} -10.8990 q^{42} +4.00000 q^{43} +4.89898 q^{44} -8.89898 q^{46} +4.44949 q^{47} -2.44949 q^{48} +12.7980 q^{49} +12.0000 q^{51} +4.00000 q^{52} -11.7980 q^{53} +4.44949 q^{56} -8.69694 q^{57} -3.55051 q^{59} +12.0000 q^{61} -1.55051 q^{62} +13.3485 q^{63} +1.00000 q^{64} -12.0000 q^{66} -5.55051 q^{67} -4.89898 q^{68} +21.7980 q^{69} +4.89898 q^{71} +3.00000 q^{72} +4.00000 q^{73} +1.00000 q^{74} +3.55051 q^{76} +21.7980 q^{77} -9.79796 q^{78} +6.44949 q^{79} -9.00000 q^{81} +2.00000 q^{82} -9.55051 q^{83} -10.8990 q^{84} +4.00000 q^{86} +4.89898 q^{88} +15.7980 q^{89} +17.7980 q^{91} -8.89898 q^{92} +3.79796 q^{93} +4.44949 q^{94} -2.44949 q^{96} -2.00000 q^{97} +12.7980 q^{98} +14.6969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8} + 6 q^{9} + 8 q^{13} + 4 q^{14} + 2 q^{16} + 6 q^{18} + 12 q^{19} - 12 q^{21} - 8 q^{23} + 8 q^{26} + 4 q^{28} - 8 q^{31} + 2 q^{32} - 24 q^{33} + 6 q^{36} + 2 q^{37} + 12 q^{38} + 4 q^{41} - 12 q^{42} + 8 q^{43} - 8 q^{46} + 4 q^{47} + 6 q^{49} + 24 q^{51} + 8 q^{52} - 4 q^{53} + 4 q^{56} + 12 q^{57} - 12 q^{59} + 24 q^{61} - 8 q^{62} + 12 q^{63} + 2 q^{64} - 24 q^{66} - 16 q^{67} + 24 q^{69} + 6 q^{72} + 8 q^{73} + 2 q^{74} + 12 q^{76} + 24 q^{77} + 8 q^{79} - 18 q^{81} + 4 q^{82} - 24 q^{83} - 12 q^{84} + 8 q^{86} + 12 q^{89} + 16 q^{91} - 8 q^{92} - 12 q^{93} + 4 q^{94} - 4 q^{97} + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −2.44949 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −2.44949 −1.00000
\(7\) 4.44949 1.68175 0.840875 0.541230i \(-0.182041\pi\)
0.840875 + 0.541230i \(0.182041\pi\)
\(8\) 1.00000 0.353553
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 4.89898 1.47710 0.738549 0.674200i \(-0.235511\pi\)
0.738549 + 0.674200i \(0.235511\pi\)
\(12\) −2.44949 −0.707107
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 4.44949 1.18918
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.89898 −1.18818 −0.594089 0.804400i \(-0.702487\pi\)
−0.594089 + 0.804400i \(0.702487\pi\)
\(18\) 3.00000 0.707107
\(19\) 3.55051 0.814543 0.407271 0.913307i \(-0.366480\pi\)
0.407271 + 0.913307i \(0.366480\pi\)
\(20\) 0 0
\(21\) −10.8990 −2.37835
\(22\) 4.89898 1.04447
\(23\) −8.89898 −1.85557 −0.927783 0.373121i \(-0.878288\pi\)
−0.927783 + 0.373121i \(0.878288\pi\)
\(24\) −2.44949 −0.500000
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) 4.44949 0.840875
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −1.55051 −0.278480 −0.139240 0.990259i \(-0.544466\pi\)
−0.139240 + 0.990259i \(0.544466\pi\)
\(32\) 1.00000 0.176777
\(33\) −12.0000 −2.08893
\(34\) −4.89898 −0.840168
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) 1.00000 0.164399
\(38\) 3.55051 0.575969
\(39\) −9.79796 −1.56893
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) −10.8990 −1.68175
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 4.89898 0.738549
\(45\) 0 0
\(46\) −8.89898 −1.31208
\(47\) 4.44949 0.649025 0.324512 0.945881i \(-0.394800\pi\)
0.324512 + 0.945881i \(0.394800\pi\)
\(48\) −2.44949 −0.353553
\(49\) 12.7980 1.82828
\(50\) 0 0
\(51\) 12.0000 1.68034
\(52\) 4.00000 0.554700
\(53\) −11.7980 −1.62057 −0.810287 0.586033i \(-0.800689\pi\)
−0.810287 + 0.586033i \(0.800689\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.44949 0.594588
\(57\) −8.69694 −1.15194
\(58\) 0 0
\(59\) −3.55051 −0.462237 −0.231119 0.972926i \(-0.574239\pi\)
−0.231119 + 0.972926i \(0.574239\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) −1.55051 −0.196915
\(63\) 13.3485 1.68175
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −12.0000 −1.47710
\(67\) −5.55051 −0.678103 −0.339051 0.940768i \(-0.610106\pi\)
−0.339051 + 0.940768i \(0.610106\pi\)
\(68\) −4.89898 −0.594089
\(69\) 21.7980 2.62417
\(70\) 0 0
\(71\) 4.89898 0.581402 0.290701 0.956814i \(-0.406112\pi\)
0.290701 + 0.956814i \(0.406112\pi\)
\(72\) 3.00000 0.353553
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 3.55051 0.407271
\(77\) 21.7980 2.48411
\(78\) −9.79796 −1.10940
\(79\) 6.44949 0.725624 0.362812 0.931862i \(-0.381817\pi\)
0.362812 + 0.931862i \(0.381817\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 2.00000 0.220863
\(83\) −9.55051 −1.04830 −0.524152 0.851625i \(-0.675618\pi\)
−0.524152 + 0.851625i \(0.675618\pi\)
\(84\) −10.8990 −1.18918
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 4.89898 0.522233
\(89\) 15.7980 1.67458 0.837290 0.546759i \(-0.184139\pi\)
0.837290 + 0.546759i \(0.184139\pi\)
\(90\) 0 0
\(91\) 17.7980 1.86573
\(92\) −8.89898 −0.927783
\(93\) 3.79796 0.393830
\(94\) 4.44949 0.458930
\(95\) 0 0
\(96\) −2.44949 −0.250000
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 12.7980 1.29279
\(99\) 14.6969 1.47710
\(100\) 0 0
\(101\) 10.6969 1.06439 0.532193 0.846623i \(-0.321368\pi\)
0.532193 + 0.846623i \(0.321368\pi\)
\(102\) 12.0000 1.18818
\(103\) 9.79796 0.965422 0.482711 0.875780i \(-0.339652\pi\)
0.482711 + 0.875780i \(0.339652\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −11.7980 −1.14592
\(107\) −5.55051 −0.536588 −0.268294 0.963337i \(-0.586460\pi\)
−0.268294 + 0.963337i \(0.586460\pi\)
\(108\) 0 0
\(109\) −5.79796 −0.555344 −0.277672 0.960676i \(-0.589563\pi\)
−0.277672 + 0.960676i \(0.589563\pi\)
\(110\) 0 0
\(111\) −2.44949 −0.232495
\(112\) 4.44949 0.420437
\(113\) −3.10102 −0.291719 −0.145860 0.989305i \(-0.546595\pi\)
−0.145860 + 0.989305i \(0.546595\pi\)
\(114\) −8.69694 −0.814543
\(115\) 0 0
\(116\) 0 0
\(117\) 12.0000 1.10940
\(118\) −3.55051 −0.326851
\(119\) −21.7980 −1.99822
\(120\) 0 0
\(121\) 13.0000 1.18182
\(122\) 12.0000 1.08643
\(123\) −4.89898 −0.441726
\(124\) −1.55051 −0.139240
\(125\) 0 0
\(126\) 13.3485 1.18918
\(127\) −2.65153 −0.235285 −0.117643 0.993056i \(-0.537534\pi\)
−0.117643 + 0.993056i \(0.537534\pi\)
\(128\) 1.00000 0.0883883
\(129\) −9.79796 −0.862662
\(130\) 0 0
\(131\) −10.2474 −0.895324 −0.447662 0.894203i \(-0.647743\pi\)
−0.447662 + 0.894203i \(0.647743\pi\)
\(132\) −12.0000 −1.04447
\(133\) 15.7980 1.36986
\(134\) −5.55051 −0.479491
\(135\) 0 0
\(136\) −4.89898 −0.420084
\(137\) 19.5959 1.67419 0.837096 0.547056i \(-0.184251\pi\)
0.837096 + 0.547056i \(0.184251\pi\)
\(138\) 21.7980 1.85557
\(139\) −5.79796 −0.491776 −0.245888 0.969298i \(-0.579080\pi\)
−0.245888 + 0.969298i \(0.579080\pi\)
\(140\) 0 0
\(141\) −10.8990 −0.917860
\(142\) 4.89898 0.411113
\(143\) 19.5959 1.63869
\(144\) 3.00000 0.250000
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) −31.3485 −2.58558
\(148\) 1.00000 0.0821995
\(149\) −18.6969 −1.53171 −0.765856 0.643012i \(-0.777685\pi\)
−0.765856 + 0.643012i \(0.777685\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 3.55051 0.287984
\(153\) −14.6969 −1.18818
\(154\) 21.7980 1.75653
\(155\) 0 0
\(156\) −9.79796 −0.784465
\(157\) −7.79796 −0.622345 −0.311172 0.950353i \(-0.600722\pi\)
−0.311172 + 0.950353i \(0.600722\pi\)
\(158\) 6.44949 0.513094
\(159\) 28.8990 2.29184
\(160\) 0 0
\(161\) −39.5959 −3.12060
\(162\) −9.00000 −0.707107
\(163\) −21.7980 −1.70735 −0.853674 0.520808i \(-0.825631\pi\)
−0.853674 + 0.520808i \(0.825631\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) −9.55051 −0.741263
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) −10.8990 −0.840875
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 10.6515 0.814543
\(172\) 4.00000 0.304997
\(173\) 19.7980 1.50521 0.752605 0.658472i \(-0.228797\pi\)
0.752605 + 0.658472i \(0.228797\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.89898 0.369274
\(177\) 8.69694 0.653702
\(178\) 15.7980 1.18411
\(179\) −9.34847 −0.698737 −0.349369 0.936985i \(-0.613604\pi\)
−0.349369 + 0.936985i \(0.613604\pi\)
\(180\) 0 0
\(181\) 10.6969 0.795097 0.397549 0.917581i \(-0.369861\pi\)
0.397549 + 0.917581i \(0.369861\pi\)
\(182\) 17.7980 1.31927
\(183\) −29.3939 −2.17286
\(184\) −8.89898 −0.656041
\(185\) 0 0
\(186\) 3.79796 0.278480
\(187\) −24.0000 −1.75505
\(188\) 4.44949 0.324512
\(189\) 0 0
\(190\) 0 0
\(191\) 11.3485 0.821146 0.410573 0.911828i \(-0.365329\pi\)
0.410573 + 0.911828i \(0.365329\pi\)
\(192\) −2.44949 −0.176777
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 12.7980 0.914140
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 14.6969 1.04447
\(199\) 6.44949 0.457192 0.228596 0.973521i \(-0.426586\pi\)
0.228596 + 0.973521i \(0.426586\pi\)
\(200\) 0 0
\(201\) 13.5959 0.958982
\(202\) 10.6969 0.752634
\(203\) 0 0
\(204\) 12.0000 0.840168
\(205\) 0 0
\(206\) 9.79796 0.682656
\(207\) −26.6969 −1.85557
\(208\) 4.00000 0.277350
\(209\) 17.3939 1.20316
\(210\) 0 0
\(211\) −6.69694 −0.461036 −0.230518 0.973068i \(-0.574042\pi\)
−0.230518 + 0.973068i \(0.574042\pi\)
\(212\) −11.7980 −0.810287
\(213\) −12.0000 −0.822226
\(214\) −5.55051 −0.379425
\(215\) 0 0
\(216\) 0 0
\(217\) −6.89898 −0.468333
\(218\) −5.79796 −0.392687
\(219\) −9.79796 −0.662085
\(220\) 0 0
\(221\) −19.5959 −1.31816
\(222\) −2.44949 −0.164399
\(223\) 0.449490 0.0301001 0.0150500 0.999887i \(-0.495209\pi\)
0.0150500 + 0.999887i \(0.495209\pi\)
\(224\) 4.44949 0.297294
\(225\) 0 0
\(226\) −3.10102 −0.206277
\(227\) −24.8990 −1.65260 −0.826302 0.563228i \(-0.809559\pi\)
−0.826302 + 0.563228i \(0.809559\pi\)
\(228\) −8.69694 −0.575969
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) −53.3939 −3.51306
\(232\) 0 0
\(233\) 9.79796 0.641886 0.320943 0.947099i \(-0.396000\pi\)
0.320943 + 0.947099i \(0.396000\pi\)
\(234\) 12.0000 0.784465
\(235\) 0 0
\(236\) −3.55051 −0.231119
\(237\) −15.7980 −1.02619
\(238\) −21.7980 −1.41295
\(239\) −18.0454 −1.16726 −0.583630 0.812020i \(-0.698368\pi\)
−0.583630 + 0.812020i \(0.698368\pi\)
\(240\) 0 0
\(241\) 7.79796 0.502311 0.251155 0.967947i \(-0.419189\pi\)
0.251155 + 0.967947i \(0.419189\pi\)
\(242\) 13.0000 0.835672
\(243\) 22.0454 1.41421
\(244\) 12.0000 0.768221
\(245\) 0 0
\(246\) −4.89898 −0.312348
\(247\) 14.2020 0.903654
\(248\) −1.55051 −0.0984575
\(249\) 23.3939 1.48253
\(250\) 0 0
\(251\) 21.3485 1.34750 0.673752 0.738958i \(-0.264682\pi\)
0.673752 + 0.738958i \(0.264682\pi\)
\(252\) 13.3485 0.840875
\(253\) −43.5959 −2.74085
\(254\) −2.65153 −0.166372
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −4.89898 −0.305590 −0.152795 0.988258i \(-0.548827\pi\)
−0.152795 + 0.988258i \(0.548827\pi\)
\(258\) −9.79796 −0.609994
\(259\) 4.44949 0.276478
\(260\) 0 0
\(261\) 0 0
\(262\) −10.2474 −0.633089
\(263\) 1.75255 0.108067 0.0540335 0.998539i \(-0.482792\pi\)
0.0540335 + 0.998539i \(0.482792\pi\)
\(264\) −12.0000 −0.738549
\(265\) 0 0
\(266\) 15.7980 0.968635
\(267\) −38.6969 −2.36821
\(268\) −5.55051 −0.339051
\(269\) −18.6969 −1.13997 −0.569986 0.821654i \(-0.693051\pi\)
−0.569986 + 0.821654i \(0.693051\pi\)
\(270\) 0 0
\(271\) −32.4949 −1.97392 −0.986962 0.160952i \(-0.948544\pi\)
−0.986962 + 0.160952i \(0.948544\pi\)
\(272\) −4.89898 −0.297044
\(273\) −43.5959 −2.63854
\(274\) 19.5959 1.18383
\(275\) 0 0
\(276\) 21.7980 1.31208
\(277\) 19.5959 1.17740 0.588702 0.808350i \(-0.299639\pi\)
0.588702 + 0.808350i \(0.299639\pi\)
\(278\) −5.79796 −0.347738
\(279\) −4.65153 −0.278480
\(280\) 0 0
\(281\) 27.7980 1.65829 0.829144 0.559036i \(-0.188829\pi\)
0.829144 + 0.559036i \(0.188829\pi\)
\(282\) −10.8990 −0.649025
\(283\) 15.5959 0.927081 0.463541 0.886076i \(-0.346579\pi\)
0.463541 + 0.886076i \(0.346579\pi\)
\(284\) 4.89898 0.290701
\(285\) 0 0
\(286\) 19.5959 1.15873
\(287\) 8.89898 0.525290
\(288\) 3.00000 0.176777
\(289\) 7.00000 0.411765
\(290\) 0 0
\(291\) 4.89898 0.287183
\(292\) 4.00000 0.234082
\(293\) 25.5959 1.49533 0.747665 0.664076i \(-0.231175\pi\)
0.747665 + 0.664076i \(0.231175\pi\)
\(294\) −31.3485 −1.82828
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) −18.6969 −1.08308
\(299\) −35.5959 −2.05857
\(300\) 0 0
\(301\) 17.7980 1.02586
\(302\) −8.00000 −0.460348
\(303\) −26.2020 −1.50527
\(304\) 3.55051 0.203636
\(305\) 0 0
\(306\) −14.6969 −0.840168
\(307\) 13.1464 0.750306 0.375153 0.926963i \(-0.377590\pi\)
0.375153 + 0.926963i \(0.377590\pi\)
\(308\) 21.7980 1.24205
\(309\) −24.0000 −1.36531
\(310\) 0 0
\(311\) −7.34847 −0.416693 −0.208347 0.978055i \(-0.566808\pi\)
−0.208347 + 0.978055i \(0.566808\pi\)
\(312\) −9.79796 −0.554700
\(313\) −17.5959 −0.994580 −0.497290 0.867584i \(-0.665672\pi\)
−0.497290 + 0.867584i \(0.665672\pi\)
\(314\) −7.79796 −0.440064
\(315\) 0 0
\(316\) 6.44949 0.362812
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 28.8990 1.62057
\(319\) 0 0
\(320\) 0 0
\(321\) 13.5959 0.758850
\(322\) −39.5959 −2.20659
\(323\) −17.3939 −0.967821
\(324\) −9.00000 −0.500000
\(325\) 0 0
\(326\) −21.7980 −1.20728
\(327\) 14.2020 0.785375
\(328\) 2.00000 0.110432
\(329\) 19.7980 1.09150
\(330\) 0 0
\(331\) 14.2474 0.783111 0.391555 0.920155i \(-0.371937\pi\)
0.391555 + 0.920155i \(0.371937\pi\)
\(332\) −9.55051 −0.524152
\(333\) 3.00000 0.164399
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) −10.8990 −0.594588
\(337\) −3.59592 −0.195882 −0.0979411 0.995192i \(-0.531226\pi\)
−0.0979411 + 0.995192i \(0.531226\pi\)
\(338\) 3.00000 0.163178
\(339\) 7.59592 0.412554
\(340\) 0 0
\(341\) −7.59592 −0.411342
\(342\) 10.6515 0.575969
\(343\) 25.7980 1.39296
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 19.7980 1.06434
\(347\) 2.20204 0.118212 0.0591059 0.998252i \(-0.481175\pi\)
0.0591059 + 0.998252i \(0.481175\pi\)
\(348\) 0 0
\(349\) 7.10102 0.380109 0.190054 0.981774i \(-0.439134\pi\)
0.190054 + 0.981774i \(0.439134\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4.89898 0.261116
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 8.69694 0.462237
\(355\) 0 0
\(356\) 15.7980 0.837290
\(357\) 53.3939 2.82590
\(358\) −9.34847 −0.494082
\(359\) −12.8990 −0.680782 −0.340391 0.940284i \(-0.610559\pi\)
−0.340391 + 0.940284i \(0.610559\pi\)
\(360\) 0 0
\(361\) −6.39388 −0.336520
\(362\) 10.6969 0.562219
\(363\) −31.8434 −1.67134
\(364\) 17.7980 0.932867
\(365\) 0 0
\(366\) −29.3939 −1.53644
\(367\) −2.65153 −0.138409 −0.0692044 0.997603i \(-0.522046\pi\)
−0.0692044 + 0.997603i \(0.522046\pi\)
\(368\) −8.89898 −0.463891
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −52.4949 −2.72540
\(372\) 3.79796 0.196915
\(373\) −11.7980 −0.610875 −0.305438 0.952212i \(-0.598803\pi\)
−0.305438 + 0.952212i \(0.598803\pi\)
\(374\) −24.0000 −1.24101
\(375\) 0 0
\(376\) 4.44949 0.229465
\(377\) 0 0
\(378\) 0 0
\(379\) −31.5959 −1.62297 −0.811487 0.584371i \(-0.801341\pi\)
−0.811487 + 0.584371i \(0.801341\pi\)
\(380\) 0 0
\(381\) 6.49490 0.332744
\(382\) 11.3485 0.580638
\(383\) −34.6969 −1.77293 −0.886465 0.462795i \(-0.846847\pi\)
−0.886465 + 0.462795i \(0.846847\pi\)
\(384\) −2.44949 −0.125000
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 12.0000 0.609994
\(388\) −2.00000 −0.101535
\(389\) −25.7980 −1.30801 −0.654004 0.756491i \(-0.726912\pi\)
−0.654004 + 0.756491i \(0.726912\pi\)
\(390\) 0 0
\(391\) 43.5959 2.20474
\(392\) 12.7980 0.646395
\(393\) 25.1010 1.26618
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) 14.6969 0.738549
\(397\) −16.2020 −0.813157 −0.406579 0.913616i \(-0.633278\pi\)
−0.406579 + 0.913616i \(0.633278\pi\)
\(398\) 6.44949 0.323284
\(399\) −38.6969 −1.93727
\(400\) 0 0
\(401\) −23.7980 −1.18841 −0.594207 0.804312i \(-0.702534\pi\)
−0.594207 + 0.804312i \(0.702534\pi\)
\(402\) 13.5959 0.678103
\(403\) −6.20204 −0.308946
\(404\) 10.6969 0.532193
\(405\) 0 0
\(406\) 0 0
\(407\) 4.89898 0.242833
\(408\) 12.0000 0.594089
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) −48.0000 −2.36767
\(412\) 9.79796 0.482711
\(413\) −15.7980 −0.777367
\(414\) −26.6969 −1.31208
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) 14.2020 0.695477
\(418\) 17.3939 0.850762
\(419\) 5.79796 0.283249 0.141624 0.989920i \(-0.454767\pi\)
0.141624 + 0.989920i \(0.454767\pi\)
\(420\) 0 0
\(421\) −19.5959 −0.955047 −0.477523 0.878619i \(-0.658465\pi\)
−0.477523 + 0.878619i \(0.658465\pi\)
\(422\) −6.69694 −0.326002
\(423\) 13.3485 0.649025
\(424\) −11.7980 −0.572960
\(425\) 0 0
\(426\) −12.0000 −0.581402
\(427\) 53.3939 2.58391
\(428\) −5.55051 −0.268294
\(429\) −48.0000 −2.31746
\(430\) 0 0
\(431\) −15.7526 −0.758774 −0.379387 0.925238i \(-0.623865\pi\)
−0.379387 + 0.925238i \(0.623865\pi\)
\(432\) 0 0
\(433\) 21.3939 1.02812 0.514062 0.857753i \(-0.328140\pi\)
0.514062 + 0.857753i \(0.328140\pi\)
\(434\) −6.89898 −0.331162
\(435\) 0 0
\(436\) −5.79796 −0.277672
\(437\) −31.5959 −1.51144
\(438\) −9.79796 −0.468165
\(439\) 20.6515 0.985644 0.492822 0.870130i \(-0.335965\pi\)
0.492822 + 0.870130i \(0.335965\pi\)
\(440\) 0 0
\(441\) 38.3939 1.82828
\(442\) −19.5959 −0.932083
\(443\) 24.6515 1.17123 0.585615 0.810589i \(-0.300853\pi\)
0.585615 + 0.810589i \(0.300853\pi\)
\(444\) −2.44949 −0.116248
\(445\) 0 0
\(446\) 0.449490 0.0212840
\(447\) 45.7980 2.16617
\(448\) 4.44949 0.210219
\(449\) −15.7980 −0.745552 −0.372776 0.927921i \(-0.621594\pi\)
−0.372776 + 0.927921i \(0.621594\pi\)
\(450\) 0 0
\(451\) 9.79796 0.461368
\(452\) −3.10102 −0.145860
\(453\) 19.5959 0.920697
\(454\) −24.8990 −1.16857
\(455\) 0 0
\(456\) −8.69694 −0.407271
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) 0 0
\(461\) −33.7980 −1.57413 −0.787064 0.616871i \(-0.788400\pi\)
−0.787064 + 0.616871i \(0.788400\pi\)
\(462\) −53.3939 −2.48411
\(463\) −20.4949 −0.952479 −0.476239 0.879316i \(-0.658000\pi\)
−0.476239 + 0.879316i \(0.658000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 9.79796 0.453882
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 12.0000 0.554700
\(469\) −24.6969 −1.14040
\(470\) 0 0
\(471\) 19.1010 0.880129
\(472\) −3.55051 −0.163425
\(473\) 19.5959 0.901021
\(474\) −15.7980 −0.725624
\(475\) 0 0
\(476\) −21.7980 −0.999108
\(477\) −35.3939 −1.62057
\(478\) −18.0454 −0.825378
\(479\) 5.14643 0.235146 0.117573 0.993064i \(-0.462489\pi\)
0.117573 + 0.993064i \(0.462489\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 7.79796 0.355187
\(483\) 96.9898 4.41319
\(484\) 13.0000 0.590909
\(485\) 0 0
\(486\) 22.0454 1.00000
\(487\) −29.3939 −1.33196 −0.665982 0.745968i \(-0.731987\pi\)
−0.665982 + 0.745968i \(0.731987\pi\)
\(488\) 12.0000 0.543214
\(489\) 53.3939 2.41455
\(490\) 0 0
\(491\) −9.30306 −0.419841 −0.209921 0.977718i \(-0.567321\pi\)
−0.209921 + 0.977718i \(0.567321\pi\)
\(492\) −4.89898 −0.220863
\(493\) 0 0
\(494\) 14.2020 0.638980
\(495\) 0 0
\(496\) −1.55051 −0.0696200
\(497\) 21.7980 0.977772
\(498\) 23.3939 1.04830
\(499\) −10.6515 −0.476828 −0.238414 0.971164i \(-0.576628\pi\)
−0.238414 + 0.971164i \(0.576628\pi\)
\(500\) 0 0
\(501\) −19.5959 −0.875481
\(502\) 21.3485 0.952829
\(503\) −1.79796 −0.0801670 −0.0400835 0.999196i \(-0.512762\pi\)
−0.0400835 + 0.999196i \(0.512762\pi\)
\(504\) 13.3485 0.594588
\(505\) 0 0
\(506\) −43.5959 −1.93807
\(507\) −7.34847 −0.326357
\(508\) −2.65153 −0.117643
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) 17.7980 0.787335
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −4.89898 −0.216085
\(515\) 0 0
\(516\) −9.79796 −0.431331
\(517\) 21.7980 0.958673
\(518\) 4.44949 0.195499
\(519\) −48.4949 −2.12869
\(520\) 0 0
\(521\) 17.7980 0.779743 0.389871 0.920869i \(-0.372519\pi\)
0.389871 + 0.920869i \(0.372519\pi\)
\(522\) 0 0
\(523\) −8.89898 −0.389125 −0.194563 0.980890i \(-0.562329\pi\)
−0.194563 + 0.980890i \(0.562329\pi\)
\(524\) −10.2474 −0.447662
\(525\) 0 0
\(526\) 1.75255 0.0764149
\(527\) 7.59592 0.330883
\(528\) −12.0000 −0.522233
\(529\) 56.1918 2.44312
\(530\) 0 0
\(531\) −10.6515 −0.462237
\(532\) 15.7980 0.684928
\(533\) 8.00000 0.346518
\(534\) −38.6969 −1.67458
\(535\) 0 0
\(536\) −5.55051 −0.239746
\(537\) 22.8990 0.988164
\(538\) −18.6969 −0.806082
\(539\) 62.6969 2.70055
\(540\) 0 0
\(541\) 0.404082 0.0173728 0.00868642 0.999962i \(-0.497235\pi\)
0.00868642 + 0.999962i \(0.497235\pi\)
\(542\) −32.4949 −1.39578
\(543\) −26.2020 −1.12444
\(544\) −4.89898 −0.210042
\(545\) 0 0
\(546\) −43.5959 −1.86573
\(547\) −10.6969 −0.457368 −0.228684 0.973501i \(-0.573442\pi\)
−0.228684 + 0.973501i \(0.573442\pi\)
\(548\) 19.5959 0.837096
\(549\) 36.0000 1.53644
\(550\) 0 0
\(551\) 0 0
\(552\) 21.7980 0.927783
\(553\) 28.6969 1.22032
\(554\) 19.5959 0.832551
\(555\) 0 0
\(556\) −5.79796 −0.245888
\(557\) −12.0000 −0.508456 −0.254228 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) −4.65153 −0.196915
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 58.7878 2.48202
\(562\) 27.7980 1.17259
\(563\) −34.6969 −1.46230 −0.731151 0.682216i \(-0.761016\pi\)
−0.731151 + 0.682216i \(0.761016\pi\)
\(564\) −10.8990 −0.458930
\(565\) 0 0
\(566\) 15.5959 0.655545
\(567\) −40.0454 −1.68175
\(568\) 4.89898 0.205557
\(569\) 21.5959 0.905348 0.452674 0.891676i \(-0.350470\pi\)
0.452674 + 0.891676i \(0.350470\pi\)
\(570\) 0 0
\(571\) −25.3939 −1.06270 −0.531350 0.847152i \(-0.678315\pi\)
−0.531350 + 0.847152i \(0.678315\pi\)
\(572\) 19.5959 0.819346
\(573\) −27.7980 −1.16128
\(574\) 8.89898 0.371436
\(575\) 0 0
\(576\) 3.00000 0.125000
\(577\) −30.6969 −1.27793 −0.638965 0.769236i \(-0.720637\pi\)
−0.638965 + 0.769236i \(0.720637\pi\)
\(578\) 7.00000 0.291162
\(579\) −34.2929 −1.42516
\(580\) 0 0
\(581\) −42.4949 −1.76299
\(582\) 4.89898 0.203069
\(583\) −57.7980 −2.39375
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) 25.5959 1.05736
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) −31.3485 −1.29279
\(589\) −5.50510 −0.226834
\(590\) 0 0
\(591\) 4.89898 0.201517
\(592\) 1.00000 0.0410997
\(593\) 24.0000 0.985562 0.492781 0.870153i \(-0.335980\pi\)
0.492781 + 0.870153i \(0.335980\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −18.6969 −0.765856
\(597\) −15.7980 −0.646567
\(598\) −35.5959 −1.45563
\(599\) 11.5959 0.473796 0.236898 0.971534i \(-0.423869\pi\)
0.236898 + 0.971534i \(0.423869\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) 17.7980 0.725391
\(603\) −16.6515 −0.678103
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) −26.2020 −1.06439
\(607\) 26.6969 1.08360 0.541798 0.840509i \(-0.317744\pi\)
0.541798 + 0.840509i \(0.317744\pi\)
\(608\) 3.55051 0.143992
\(609\) 0 0
\(610\) 0 0
\(611\) 17.7980 0.720028
\(612\) −14.6969 −0.594089
\(613\) 39.7980 1.60742 0.803712 0.595018i \(-0.202855\pi\)
0.803712 + 0.595018i \(0.202855\pi\)
\(614\) 13.1464 0.530547
\(615\) 0 0
\(616\) 21.7980 0.878265
\(617\) 19.5959 0.788902 0.394451 0.918917i \(-0.370935\pi\)
0.394451 + 0.918917i \(0.370935\pi\)
\(618\) −24.0000 −0.965422
\(619\) −12.8990 −0.518454 −0.259227 0.965816i \(-0.583468\pi\)
−0.259227 + 0.965816i \(0.583468\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −7.34847 −0.294647
\(623\) 70.2929 2.81622
\(624\) −9.79796 −0.392232
\(625\) 0 0
\(626\) −17.5959 −0.703274
\(627\) −42.6061 −1.70152
\(628\) −7.79796 −0.311172
\(629\) −4.89898 −0.195335
\(630\) 0 0
\(631\) −20.2474 −0.806038 −0.403019 0.915192i \(-0.632039\pi\)
−0.403019 + 0.915192i \(0.632039\pi\)
\(632\) 6.44949 0.256547
\(633\) 16.4041 0.652004
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) 28.8990 1.14592
\(637\) 51.1918 2.02829
\(638\) 0 0
\(639\) 14.6969 0.581402
\(640\) 0 0
\(641\) 17.7980 0.702977 0.351489 0.936192i \(-0.385676\pi\)
0.351489 + 0.936192i \(0.385676\pi\)
\(642\) 13.5959 0.536588
\(643\) −23.1010 −0.911015 −0.455508 0.890232i \(-0.650542\pi\)
−0.455508 + 0.890232i \(0.650542\pi\)
\(644\) −39.5959 −1.56030
\(645\) 0 0
\(646\) −17.3939 −0.684353
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) −9.00000 −0.353553
\(649\) −17.3939 −0.682769
\(650\) 0 0
\(651\) 16.8990 0.662323
\(652\) −21.7980 −0.853674
\(653\) 34.0000 1.33052 0.665261 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) 14.2020 0.555344
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) 12.0000 0.468165
\(658\) 19.7980 0.771805
\(659\) 31.5959 1.23080 0.615401 0.788214i \(-0.288994\pi\)
0.615401 + 0.788214i \(0.288994\pi\)
\(660\) 0 0
\(661\) −31.1918 −1.21322 −0.606611 0.794999i \(-0.707471\pi\)
−0.606611 + 0.794999i \(0.707471\pi\)
\(662\) 14.2474 0.553743
\(663\) 48.0000 1.86417
\(664\) −9.55051 −0.370632
\(665\) 0 0
\(666\) 3.00000 0.116248
\(667\) 0 0
\(668\) 8.00000 0.309529
\(669\) −1.10102 −0.0425679
\(670\) 0 0
\(671\) 58.7878 2.26948
\(672\) −10.8990 −0.420437
\(673\) 24.0000 0.925132 0.462566 0.886585i \(-0.346929\pi\)
0.462566 + 0.886585i \(0.346929\pi\)
\(674\) −3.59592 −0.138510
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 7.59592 0.291719
\(679\) −8.89898 −0.341511
\(680\) 0 0
\(681\) 60.9898 2.33713
\(682\) −7.59592 −0.290863
\(683\) 15.5959 0.596761 0.298381 0.954447i \(-0.403554\pi\)
0.298381 + 0.954447i \(0.403554\pi\)
\(684\) 10.6515 0.407271
\(685\) 0 0
\(686\) 25.7980 0.984971
\(687\) 24.4949 0.934539
\(688\) 4.00000 0.152499
\(689\) −47.1918 −1.79787
\(690\) 0 0
\(691\) 37.7980 1.43790 0.718951 0.695061i \(-0.244623\pi\)
0.718951 + 0.695061i \(0.244623\pi\)
\(692\) 19.7980 0.752605
\(693\) 65.3939 2.48411
\(694\) 2.20204 0.0835883
\(695\) 0 0
\(696\) 0 0
\(697\) −9.79796 −0.371124
\(698\) 7.10102 0.268778
\(699\) −24.0000 −0.907763
\(700\) 0 0
\(701\) −13.7980 −0.521142 −0.260571 0.965455i \(-0.583911\pi\)
−0.260571 + 0.965455i \(0.583911\pi\)
\(702\) 0 0
\(703\) 3.55051 0.133910
\(704\) 4.89898 0.184637
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 47.5959 1.79003
\(708\) 8.69694 0.326851
\(709\) 5.79796 0.217747 0.108873 0.994056i \(-0.465276\pi\)
0.108873 + 0.994056i \(0.465276\pi\)
\(710\) 0 0
\(711\) 19.3485 0.725624
\(712\) 15.7980 0.592054
\(713\) 13.7980 0.516738
\(714\) 53.3939 1.99822
\(715\) 0 0
\(716\) −9.34847 −0.349369
\(717\) 44.2020 1.65076
\(718\) −12.8990 −0.481386
\(719\) 51.5959 1.92420 0.962102 0.272692i \(-0.0879138\pi\)
0.962102 + 0.272692i \(0.0879138\pi\)
\(720\) 0 0
\(721\) 43.5959 1.62360
\(722\) −6.39388 −0.237955
\(723\) −19.1010 −0.710375
\(724\) 10.6969 0.397549
\(725\) 0 0
\(726\) −31.8434 −1.18182
\(727\) 0.898979 0.0333413 0.0166707 0.999861i \(-0.494693\pi\)
0.0166707 + 0.999861i \(0.494693\pi\)
\(728\) 17.7980 0.659636
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −19.5959 −0.724781
\(732\) −29.3939 −1.08643
\(733\) −17.5959 −0.649920 −0.324960 0.945728i \(-0.605351\pi\)
−0.324960 + 0.945728i \(0.605351\pi\)
\(734\) −2.65153 −0.0978698
\(735\) 0 0
\(736\) −8.89898 −0.328021
\(737\) −27.1918 −1.00162
\(738\) 6.00000 0.220863
\(739\) −45.7980 −1.68471 −0.842353 0.538927i \(-0.818830\pi\)
−0.842353 + 0.538927i \(0.818830\pi\)
\(740\) 0 0
\(741\) −34.7878 −1.27796
\(742\) −52.4949 −1.92715
\(743\) −13.7526 −0.504532 −0.252266 0.967658i \(-0.581176\pi\)
−0.252266 + 0.967658i \(0.581176\pi\)
\(744\) 3.79796 0.139240
\(745\) 0 0
\(746\) −11.7980 −0.431954
\(747\) −28.6515 −1.04830
\(748\) −24.0000 −0.877527
\(749\) −24.6969 −0.902406
\(750\) 0 0
\(751\) 30.6969 1.12015 0.560074 0.828443i \(-0.310773\pi\)
0.560074 + 0.828443i \(0.310773\pi\)
\(752\) 4.44949 0.162256
\(753\) −52.2929 −1.90566
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −33.5959 −1.22106 −0.610532 0.791991i \(-0.709044\pi\)
−0.610532 + 0.791991i \(0.709044\pi\)
\(758\) −31.5959 −1.14762
\(759\) 106.788 3.87615
\(760\) 0 0
\(761\) 25.1918 0.913203 0.456602 0.889671i \(-0.349066\pi\)
0.456602 + 0.889671i \(0.349066\pi\)
\(762\) 6.49490 0.235285
\(763\) −25.7980 −0.933949
\(764\) 11.3485 0.410573
\(765\) 0 0
\(766\) −34.6969 −1.25365
\(767\) −14.2020 −0.512806
\(768\) −2.44949 −0.0883883
\(769\) −35.7980 −1.29091 −0.645454 0.763799i \(-0.723332\pi\)
−0.645454 + 0.763799i \(0.723332\pi\)
\(770\) 0 0
\(771\) 12.0000 0.432169
\(772\) 14.0000 0.503871
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 12.0000 0.431331
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) −10.8990 −0.390999
\(778\) −25.7980 −0.924902
\(779\) 7.10102 0.254420
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 43.5959 1.55899
\(783\) 0 0
\(784\) 12.7980 0.457070
\(785\) 0 0
\(786\) 25.1010 0.895324
\(787\) −28.7423 −1.02455 −0.512277 0.858820i \(-0.671198\pi\)
−0.512277 + 0.858820i \(0.671198\pi\)
\(788\) −2.00000 −0.0712470
\(789\) −4.29286 −0.152830
\(790\) 0 0
\(791\) −13.7980 −0.490599
\(792\) 14.6969 0.522233
\(793\) 48.0000 1.70453
\(794\) −16.2020 −0.574989
\(795\) 0 0
\(796\) 6.44949 0.228596
\(797\) −43.5959 −1.54425 −0.772123 0.635473i \(-0.780805\pi\)
−0.772123 + 0.635473i \(0.780805\pi\)
\(798\) −38.6969 −1.36986
\(799\) −21.7980 −0.771156
\(800\) 0 0
\(801\) 47.3939 1.67458
\(802\) −23.7980 −0.840335
\(803\) 19.5959 0.691525
\(804\) 13.5959 0.479491
\(805\) 0 0
\(806\) −6.20204 −0.218458
\(807\) 45.7980 1.61216
\(808\) 10.6969 0.376317
\(809\) −33.1918 −1.16696 −0.583481 0.812127i \(-0.698310\pi\)
−0.583481 + 0.812127i \(0.698310\pi\)
\(810\) 0 0
\(811\) 26.2020 0.920078 0.460039 0.887899i \(-0.347835\pi\)
0.460039 + 0.887899i \(0.347835\pi\)
\(812\) 0 0
\(813\) 79.5959 2.79155
\(814\) 4.89898 0.171709
\(815\) 0 0
\(816\) 12.0000 0.420084
\(817\) 14.2020 0.496867
\(818\) −10.0000 −0.349642
\(819\) 53.3939 1.86573
\(820\) 0 0
\(821\) −6.40408 −0.223504 −0.111752 0.993736i \(-0.535646\pi\)
−0.111752 + 0.993736i \(0.535646\pi\)
\(822\) −48.0000 −1.67419
\(823\) 33.3485 1.16245 0.581227 0.813741i \(-0.302573\pi\)
0.581227 + 0.813741i \(0.302573\pi\)
\(824\) 9.79796 0.341328
\(825\) 0 0
\(826\) −15.7980 −0.549681
\(827\) −36.4949 −1.26905 −0.634526 0.772902i \(-0.718805\pi\)
−0.634526 + 0.772902i \(0.718805\pi\)
\(828\) −26.6969 −0.927783
\(829\) 8.40408 0.291886 0.145943 0.989293i \(-0.453378\pi\)
0.145943 + 0.989293i \(0.453378\pi\)
\(830\) 0 0
\(831\) −48.0000 −1.66510
\(832\) 4.00000 0.138675
\(833\) −62.6969 −2.17232
\(834\) 14.2020 0.491776
\(835\) 0 0
\(836\) 17.3939 0.601580
\(837\) 0 0
\(838\) 5.79796 0.200287
\(839\) 14.2020 0.490309 0.245154 0.969484i \(-0.421161\pi\)
0.245154 + 0.969484i \(0.421161\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −19.5959 −0.675320
\(843\) −68.0908 −2.34517
\(844\) −6.69694 −0.230518
\(845\) 0 0
\(846\) 13.3485 0.458930
\(847\) 57.8434 1.98752
\(848\) −11.7980 −0.405144
\(849\) −38.2020 −1.31109
\(850\) 0 0
\(851\) −8.89898 −0.305053
\(852\) −12.0000 −0.411113
\(853\) −37.5959 −1.28726 −0.643630 0.765337i \(-0.722572\pi\)
−0.643630 + 0.765337i \(0.722572\pi\)
\(854\) 53.3939 1.82710
\(855\) 0 0
\(856\) −5.55051 −0.189713
\(857\) 9.59592 0.327790 0.163895 0.986478i \(-0.447594\pi\)
0.163895 + 0.986478i \(0.447594\pi\)
\(858\) −48.0000 −1.63869
\(859\) −55.1464 −1.88157 −0.940786 0.339001i \(-0.889911\pi\)
−0.940786 + 0.339001i \(0.889911\pi\)
\(860\) 0 0
\(861\) −21.7980 −0.742872
\(862\) −15.7526 −0.536534
\(863\) −42.7423 −1.45497 −0.727483 0.686126i \(-0.759310\pi\)
−0.727483 + 0.686126i \(0.759310\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 21.3939 0.726994
\(867\) −17.1464 −0.582323
\(868\) −6.89898 −0.234167
\(869\) 31.5959 1.07182
\(870\) 0 0
\(871\) −22.2020 −0.752287
\(872\) −5.79796 −0.196344
\(873\) −6.00000 −0.203069
\(874\) −31.5959 −1.06875
\(875\) 0 0
\(876\) −9.79796 −0.331042
\(877\) 15.3939 0.519814 0.259907 0.965634i \(-0.416308\pi\)
0.259907 + 0.965634i \(0.416308\pi\)
\(878\) 20.6515 0.696955
\(879\) −62.6969 −2.11472
\(880\) 0 0
\(881\) −5.39388 −0.181724 −0.0908622 0.995863i \(-0.528962\pi\)
−0.0908622 + 0.995863i \(0.528962\pi\)
\(882\) 38.3939 1.29279
\(883\) 42.6969 1.43687 0.718433 0.695596i \(-0.244860\pi\)
0.718433 + 0.695596i \(0.244860\pi\)
\(884\) −19.5959 −0.659082
\(885\) 0 0
\(886\) 24.6515 0.828184
\(887\) −20.0454 −0.673059 −0.336529 0.941673i \(-0.609253\pi\)
−0.336529 + 0.941673i \(0.609253\pi\)
\(888\) −2.44949 −0.0821995
\(889\) −11.7980 −0.395691
\(890\) 0 0
\(891\) −44.0908 −1.47710
\(892\) 0.449490 0.0150500
\(893\) 15.7980 0.528659
\(894\) 45.7980 1.53171
\(895\) 0 0
\(896\) 4.44949 0.148647
\(897\) 87.1918 2.91125
\(898\) −15.7980 −0.527185
\(899\) 0 0
\(900\) 0 0
\(901\) 57.7980 1.92553
\(902\) 9.79796 0.326236
\(903\) −43.5959 −1.45078
\(904\) −3.10102 −0.103138
\(905\) 0 0
\(906\) 19.5959 0.651031
\(907\) 0.898979 0.0298501 0.0149251 0.999889i \(-0.495249\pi\)
0.0149251 + 0.999889i \(0.495249\pi\)
\(908\) −24.8990 −0.826302
\(909\) 32.0908 1.06439
\(910\) 0 0
\(911\) −31.8434 −1.05502 −0.527509 0.849550i \(-0.676874\pi\)
−0.527509 + 0.849550i \(0.676874\pi\)
\(912\) −8.69694 −0.287984
\(913\) −46.7878 −1.54845
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) −45.5959 −1.50571
\(918\) 0 0
\(919\) 25.1464 0.829504 0.414752 0.909934i \(-0.363868\pi\)
0.414752 + 0.909934i \(0.363868\pi\)
\(920\) 0 0
\(921\) −32.2020 −1.06109
\(922\) −33.7980 −1.11308
\(923\) 19.5959 0.645007
\(924\) −53.3939 −1.75653
\(925\) 0 0
\(926\) −20.4949 −0.673504
\(927\) 29.3939 0.965422
\(928\) 0 0
\(929\) −1.59592 −0.0523604 −0.0261802 0.999657i \(-0.508334\pi\)
−0.0261802 + 0.999657i \(0.508334\pi\)
\(930\) 0 0
\(931\) 45.4393 1.48921
\(932\) 9.79796 0.320943
\(933\) 18.0000 0.589294
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) 12.0000 0.392232
\(937\) 8.00000 0.261349 0.130674 0.991425i \(-0.458286\pi\)
0.130674 + 0.991425i \(0.458286\pi\)
\(938\) −24.6969 −0.806384
\(939\) 43.1010 1.40655
\(940\) 0 0
\(941\) −38.2929 −1.24831 −0.624156 0.781300i \(-0.714557\pi\)
−0.624156 + 0.781300i \(0.714557\pi\)
\(942\) 19.1010 0.622345
\(943\) −17.7980 −0.579581
\(944\) −3.55051 −0.115559
\(945\) 0 0
\(946\) 19.5959 0.637118
\(947\) 25.3939 0.825190 0.412595 0.910915i \(-0.364622\pi\)
0.412595 + 0.910915i \(0.364622\pi\)
\(948\) −15.7980 −0.513094
\(949\) 16.0000 0.519382
\(950\) 0 0
\(951\) −44.0908 −1.42974
\(952\) −21.7980 −0.706476
\(953\) −21.7980 −0.706105 −0.353053 0.935603i \(-0.614856\pi\)
−0.353053 + 0.935603i \(0.614856\pi\)
\(954\) −35.3939 −1.14592
\(955\) 0 0
\(956\) −18.0454 −0.583630
\(957\) 0 0
\(958\) 5.14643 0.166274
\(959\) 87.1918 2.81557
\(960\) 0 0
\(961\) −28.5959 −0.922449
\(962\) 4.00000 0.128965
\(963\) −16.6515 −0.536588
\(964\) 7.79796 0.251155
\(965\) 0 0
\(966\) 96.9898 3.12060
\(967\) 3.50510 0.112716 0.0563582 0.998411i \(-0.482051\pi\)
0.0563582 + 0.998411i \(0.482051\pi\)
\(968\) 13.0000 0.417836
\(969\) 42.6061 1.36871
\(970\) 0 0
\(971\) −35.1010 −1.12645 −0.563223 0.826305i \(-0.690439\pi\)
−0.563223 + 0.826305i \(0.690439\pi\)
\(972\) 22.0454 0.707107
\(973\) −25.7980 −0.827045
\(974\) −29.3939 −0.941841
\(975\) 0 0
\(976\) 12.0000 0.384111
\(977\) −10.4041 −0.332856 −0.166428 0.986054i \(-0.553223\pi\)
−0.166428 + 0.986054i \(0.553223\pi\)
\(978\) 53.3939 1.70735
\(979\) 77.3939 2.47352
\(980\) 0 0
\(981\) −17.3939 −0.555344
\(982\) −9.30306 −0.296873
\(983\) 4.94439 0.157701 0.0788507 0.996886i \(-0.474875\pi\)
0.0788507 + 0.996886i \(0.474875\pi\)
\(984\) −4.89898 −0.156174
\(985\) 0 0
\(986\) 0 0
\(987\) −48.4949 −1.54361
\(988\) 14.2020 0.451827
\(989\) −35.5959 −1.13188
\(990\) 0 0
\(991\) −31.8434 −1.01154 −0.505769 0.862669i \(-0.668791\pi\)
−0.505769 + 0.862669i \(0.668791\pi\)
\(992\) −1.55051 −0.0492287
\(993\) −34.8990 −1.10749
\(994\) 21.7980 0.691389
\(995\) 0 0
\(996\) 23.3939 0.741263
\(997\) −22.0000 −0.696747 −0.348373 0.937356i \(-0.613266\pi\)
−0.348373 + 0.937356i \(0.613266\pi\)
\(998\) −10.6515 −0.337168
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1850.2.a.v.1.1 2
5.2 odd 4 370.2.b.c.149.4 yes 4
5.3 odd 4 370.2.b.c.149.1 4
5.4 even 2 1850.2.a.s.1.2 2
15.2 even 4 3330.2.d.m.1999.2 4
15.8 even 4 3330.2.d.m.1999.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.c.149.1 4 5.3 odd 4
370.2.b.c.149.4 yes 4 5.2 odd 4
1850.2.a.s.1.2 2 5.4 even 2
1850.2.a.v.1.1 2 1.1 even 1 trivial
3330.2.d.m.1999.2 4 15.2 even 4
3330.2.d.m.1999.3 4 15.8 even 4