Properties

Label 1850.2.a.v.1.2
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
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 1.2
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} -0.449490 q^{7} +1.00000 q^{8} +3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} -0.449490 q^{7} +1.00000 q^{8} +3.00000 q^{9} -4.89898 q^{11} +2.44949 q^{12} +4.00000 q^{13} -0.449490 q^{14} +1.00000 q^{16} +4.89898 q^{17} +3.00000 q^{18} +8.44949 q^{19} -1.10102 q^{21} -4.89898 q^{22} +0.898979 q^{23} +2.44949 q^{24} +4.00000 q^{26} -0.449490 q^{28} -6.44949 q^{31} +1.00000 q^{32} -12.0000 q^{33} +4.89898 q^{34} +3.00000 q^{36} +1.00000 q^{37} +8.44949 q^{38} +9.79796 q^{39} +2.00000 q^{41} -1.10102 q^{42} +4.00000 q^{43} -4.89898 q^{44} +0.898979 q^{46} -0.449490 q^{47} +2.44949 q^{48} -6.79796 q^{49} +12.0000 q^{51} +4.00000 q^{52} +7.79796 q^{53} -0.449490 q^{56} +20.6969 q^{57} -8.44949 q^{59} +12.0000 q^{61} -6.44949 q^{62} -1.34847 q^{63} +1.00000 q^{64} -12.0000 q^{66} -10.4495 q^{67} +4.89898 q^{68} +2.20204 q^{69} -4.89898 q^{71} +3.00000 q^{72} +4.00000 q^{73} +1.00000 q^{74} +8.44949 q^{76} +2.20204 q^{77} +9.79796 q^{78} +1.55051 q^{79} -9.00000 q^{81} +2.00000 q^{82} -14.4495 q^{83} -1.10102 q^{84} +4.00000 q^{86} -4.89898 q^{88} -3.79796 q^{89} -1.79796 q^{91} +0.898979 q^{92} -15.7980 q^{93} -0.449490 q^{94} +2.44949 q^{96} -2.00000 q^{97} -6.79796 q^{98} -14.6969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8} + 6 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8} + 6 q^{9} + 8 q^{13} + 4 q^{14} + 2 q^{16} + 6 q^{18} + 12 q^{19} - 12 q^{21} - 8 q^{23} + 8 q^{26} + 4 q^{28} - 8 q^{31} + 2 q^{32} - 24 q^{33} + 6 q^{36} + 2 q^{37} + 12 q^{38} + 4 q^{41} - 12 q^{42} + 8 q^{43} - 8 q^{46} + 4 q^{47} + 6 q^{49} + 24 q^{51} + 8 q^{52} - 4 q^{53} + 4 q^{56} + 12 q^{57} - 12 q^{59} + 24 q^{61} - 8 q^{62} + 12 q^{63} + 2 q^{64} - 24 q^{66} - 16 q^{67} + 24 q^{69} + 6 q^{72} + 8 q^{73} + 2 q^{74} + 12 q^{76} + 24 q^{77} + 8 q^{79} - 18 q^{81} + 4 q^{82} - 24 q^{83} - 12 q^{84} + 8 q^{86} + 12 q^{89} + 16 q^{91} - 8 q^{92} - 12 q^{93} + 4 q^{94} - 4 q^{97} + 6 q^{98} + O(q^{100}) \)

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 \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.44949 1.00000
\(7\) −0.449490 −0.169891 −0.0849456 0.996386i \(-0.527072\pi\)
−0.0849456 + 0.996386i \(0.527072\pi\)
\(8\) 1.00000 0.353553
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) −4.89898 −1.47710 −0.738549 0.674200i \(-0.764489\pi\)
−0.738549 + 0.674200i \(0.764489\pi\)
\(12\) 2.44949 0.707107
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) −0.449490 −0.120131
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.89898 1.18818 0.594089 0.804400i \(-0.297513\pi\)
0.594089 + 0.804400i \(0.297513\pi\)
\(18\) 3.00000 0.707107
\(19\) 8.44949 1.93845 0.969223 0.246185i \(-0.0791770\pi\)
0.969223 + 0.246185i \(0.0791770\pi\)
\(20\) 0 0
\(21\) −1.10102 −0.240262
\(22\) −4.89898 −1.04447
\(23\) 0.898979 0.187450 0.0937251 0.995598i \(-0.470123\pi\)
0.0937251 + 0.995598i \(0.470123\pi\)
\(24\) 2.44949 0.500000
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) −0.449490 −0.0849456
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −6.44949 −1.15836 −0.579181 0.815199i \(-0.696628\pi\)
−0.579181 + 0.815199i \(0.696628\pi\)
\(32\) 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
\(38\) 8.44949 1.37069
\(39\) 9.79796 1.56893
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) −1.10102 −0.169891
\(43\) 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
\(46\) 0.898979 0.132547
\(47\) −0.449490 −0.0655648 −0.0327824 0.999463i \(-0.510437\pi\)
−0.0327824 + 0.999463i \(0.510437\pi\)
\(48\) 2.44949 0.353553
\(49\) −6.79796 −0.971137
\(50\) 0 0
\(51\) 12.0000 1.68034
\(52\) 4.00000 0.554700
\(53\) 7.79796 1.07113 0.535566 0.844493i \(-0.320098\pi\)
0.535566 + 0.844493i \(0.320098\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.449490 −0.0600656
\(57\) 20.6969 2.74138
\(58\) 0 0
\(59\) −8.44949 −1.10003 −0.550015 0.835155i \(-0.685378\pi\)
−0.550015 + 0.835155i \(0.685378\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) −6.44949 −0.819086
\(63\) −1.34847 −0.169891
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −12.0000 −1.47710
\(67\) −10.4495 −1.27661 −0.638304 0.769784i \(-0.720364\pi\)
−0.638304 + 0.769784i \(0.720364\pi\)
\(68\) 4.89898 0.594089
\(69\) 2.20204 0.265095
\(70\) 0 0
\(71\) −4.89898 −0.581402 −0.290701 0.956814i \(-0.593888\pi\)
−0.290701 + 0.956814i \(0.593888\pi\)
\(72\) 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
\(76\) 8.44949 0.969223
\(77\) 2.20204 0.250946
\(78\) 9.79796 1.10940
\(79\) 1.55051 0.174446 0.0872230 0.996189i \(-0.472201\pi\)
0.0872230 + 0.996189i \(0.472201\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 2.00000 0.220863
\(83\) −14.4495 −1.58604 −0.793019 0.609197i \(-0.791492\pi\)
−0.793019 + 0.609197i \(0.791492\pi\)
\(84\) −1.10102 −0.120131
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) −4.89898 −0.522233
\(89\) −3.79796 −0.402583 −0.201291 0.979531i \(-0.564514\pi\)
−0.201291 + 0.979531i \(0.564514\pi\)
\(90\) 0 0
\(91\) −1.79796 −0.188477
\(92\) 0.898979 0.0937251
\(93\) −15.7980 −1.63817
\(94\) −0.449490 −0.0463613
\(95\) 0 0
\(96\) 2.44949 0.250000
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −6.79796 −0.686698
\(99\) −14.6969 −1.47710
\(100\) 0 0
\(101\) −18.6969 −1.86041 −0.930207 0.367034i \(-0.880373\pi\)
−0.930207 + 0.367034i \(0.880373\pi\)
\(102\) 12.0000 1.18818
\(103\) −9.79796 −0.965422 −0.482711 0.875780i \(-0.660348\pi\)
−0.482711 + 0.875780i \(0.660348\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) 7.79796 0.757405
\(107\) −10.4495 −1.01019 −0.505095 0.863064i \(-0.668543\pi\)
−0.505095 + 0.863064i \(0.668543\pi\)
\(108\) 0 0
\(109\) 13.7980 1.32160 0.660802 0.750560i \(-0.270216\pi\)
0.660802 + 0.750560i \(0.270216\pi\)
\(110\) 0 0
\(111\) 2.44949 0.232495
\(112\) −0.449490 −0.0424728
\(113\) −12.8990 −1.21343 −0.606717 0.794918i \(-0.707514\pi\)
−0.606717 + 0.794918i \(0.707514\pi\)
\(114\) 20.6969 1.93845
\(115\) 0 0
\(116\) 0 0
\(117\) 12.0000 1.10940
\(118\) −8.44949 −0.777839
\(119\) −2.20204 −0.201861
\(120\) 0 0
\(121\) 13.0000 1.18182
\(122\) 12.0000 1.08643
\(123\) 4.89898 0.441726
\(124\) −6.44949 −0.579181
\(125\) 0 0
\(126\) −1.34847 −0.120131
\(127\) −17.3485 −1.53943 −0.769714 0.638389i \(-0.779601\pi\)
−0.769714 + 0.638389i \(0.779601\pi\)
\(128\) 1.00000 0.0883883
\(129\) 9.79796 0.862662
\(130\) 0 0
\(131\) 14.2474 1.24481 0.622403 0.782697i \(-0.286157\pi\)
0.622403 + 0.782697i \(0.286157\pi\)
\(132\) −12.0000 −1.04447
\(133\) −3.79796 −0.329325
\(134\) −10.4495 −0.902698
\(135\) 0 0
\(136\) 4.89898 0.420084
\(137\) −19.5959 −1.67419 −0.837096 0.547056i \(-0.815749\pi\)
−0.837096 + 0.547056i \(0.815749\pi\)
\(138\) 2.20204 0.187450
\(139\) 13.7980 1.17033 0.585164 0.810915i \(-0.301030\pi\)
0.585164 + 0.810915i \(0.301030\pi\)
\(140\) 0 0
\(141\) −1.10102 −0.0927227
\(142\) −4.89898 −0.411113
\(143\) −19.5959 −1.63869
\(144\) 3.00000 0.250000
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) −16.6515 −1.37340
\(148\) 1.00000 0.0821995
\(149\) 10.6969 0.876327 0.438164 0.898895i \(-0.355629\pi\)
0.438164 + 0.898895i \(0.355629\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 8.44949 0.685344
\(153\) 14.6969 1.18818
\(154\) 2.20204 0.177446
\(155\) 0 0
\(156\) 9.79796 0.784465
\(157\) 11.7980 0.941580 0.470790 0.882245i \(-0.343969\pi\)
0.470790 + 0.882245i \(0.343969\pi\)
\(158\) 1.55051 0.123352
\(159\) 19.1010 1.51481
\(160\) 0 0
\(161\) −0.404082 −0.0318461
\(162\) −9.00000 −0.707107
\(163\) −2.20204 −0.172477 −0.0862386 0.996275i \(-0.527485\pi\)
−0.0862386 + 0.996275i \(0.527485\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) −14.4495 −1.12150
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) −1.10102 −0.0849456
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 25.3485 1.93845
\(172\) 4.00000 0.304997
\(173\) 0.202041 0.0153609 0.00768045 0.999971i \(-0.497555\pi\)
0.00768045 + 0.999971i \(0.497555\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.89898 −0.369274
\(177\) −20.6969 −1.55568
\(178\) −3.79796 −0.284669
\(179\) 5.34847 0.399763 0.199882 0.979820i \(-0.435944\pi\)
0.199882 + 0.979820i \(0.435944\pi\)
\(180\) 0 0
\(181\) −18.6969 −1.38973 −0.694866 0.719139i \(-0.744536\pi\)
−0.694866 + 0.719139i \(0.744536\pi\)
\(182\) −1.79796 −0.133274
\(183\) 29.3939 2.17286
\(184\) 0.898979 0.0662736
\(185\) 0 0
\(186\) −15.7980 −1.15836
\(187\) −24.0000 −1.75505
\(188\) −0.449490 −0.0327824
\(189\) 0 0
\(190\) 0 0
\(191\) −3.34847 −0.242287 −0.121143 0.992635i \(-0.538656\pi\)
−0.121143 + 0.992635i \(0.538656\pi\)
\(192\) 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
\(196\) −6.79796 −0.485568
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) −14.6969 −1.04447
\(199\) 1.55051 0.109913 0.0549564 0.998489i \(-0.482498\pi\)
0.0549564 + 0.998489i \(0.482498\pi\)
\(200\) 0 0
\(201\) −25.5959 −1.80540
\(202\) −18.6969 −1.31551
\(203\) 0 0
\(204\) 12.0000 0.840168
\(205\) 0 0
\(206\) −9.79796 −0.682656
\(207\) 2.69694 0.187450
\(208\) 4.00000 0.277350
\(209\) −41.3939 −2.86327
\(210\) 0 0
\(211\) 22.6969 1.56252 0.781261 0.624205i \(-0.214577\pi\)
0.781261 + 0.624205i \(0.214577\pi\)
\(212\) 7.79796 0.535566
\(213\) −12.0000 −0.822226
\(214\) −10.4495 −0.714312
\(215\) 0 0
\(216\) 0 0
\(217\) 2.89898 0.196796
\(218\) 13.7980 0.934516
\(219\) 9.79796 0.662085
\(220\) 0 0
\(221\) 19.5959 1.31816
\(222\) 2.44949 0.164399
\(223\) −4.44949 −0.297960 −0.148980 0.988840i \(-0.547599\pi\)
−0.148980 + 0.988840i \(0.547599\pi\)
\(224\) −0.449490 −0.0300328
\(225\) 0 0
\(226\) −12.8990 −0.858027
\(227\) −15.1010 −1.00229 −0.501145 0.865363i \(-0.667088\pi\)
−0.501145 + 0.865363i \(0.667088\pi\)
\(228\) 20.6969 1.37069
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 5.39388 0.354891
\(232\) 0 0
\(233\) −9.79796 −0.641886 −0.320943 0.947099i \(-0.604000\pi\)
−0.320943 + 0.947099i \(0.604000\pi\)
\(234\) 12.0000 0.784465
\(235\) 0 0
\(236\) −8.44949 −0.550015
\(237\) 3.79796 0.246704
\(238\) −2.20204 −0.142737
\(239\) 26.0454 1.68474 0.842369 0.538902i \(-0.181161\pi\)
0.842369 + 0.538902i \(0.181161\pi\)
\(240\) 0 0
\(241\) −11.7980 −0.759973 −0.379987 0.924992i \(-0.624071\pi\)
−0.379987 + 0.924992i \(0.624071\pi\)
\(242\) 13.0000 0.835672
\(243\) −22.0454 −1.41421
\(244\) 12.0000 0.768221
\(245\) 0 0
\(246\) 4.89898 0.312348
\(247\) 33.7980 2.15051
\(248\) −6.44949 −0.409543
\(249\) −35.3939 −2.24300
\(250\) 0 0
\(251\) 6.65153 0.419841 0.209920 0.977718i \(-0.432679\pi\)
0.209920 + 0.977718i \(0.432679\pi\)
\(252\) −1.34847 −0.0849456
\(253\) −4.40408 −0.276882
\(254\) −17.3485 −1.08854
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 4.89898 0.305590 0.152795 0.988258i \(-0.451173\pi\)
0.152795 + 0.988258i \(0.451173\pi\)
\(258\) 9.79796 0.609994
\(259\) −0.449490 −0.0279299
\(260\) 0 0
\(261\) 0 0
\(262\) 14.2474 0.880210
\(263\) 26.2474 1.61849 0.809244 0.587473i \(-0.199877\pi\)
0.809244 + 0.587473i \(0.199877\pi\)
\(264\) −12.0000 −0.738549
\(265\) 0 0
\(266\) −3.79796 −0.232868
\(267\) −9.30306 −0.569338
\(268\) −10.4495 −0.638304
\(269\) 10.6969 0.652204 0.326102 0.945335i \(-0.394265\pi\)
0.326102 + 0.945335i \(0.394265\pi\)
\(270\) 0 0
\(271\) 16.4949 1.00199 0.500997 0.865449i \(-0.332967\pi\)
0.500997 + 0.865449i \(0.332967\pi\)
\(272\) 4.89898 0.297044
\(273\) −4.40408 −0.266547
\(274\) −19.5959 −1.18383
\(275\) 0 0
\(276\) 2.20204 0.132547
\(277\) −19.5959 −1.17740 −0.588702 0.808350i \(-0.700361\pi\)
−0.588702 + 0.808350i \(0.700361\pi\)
\(278\) 13.7980 0.827547
\(279\) −19.3485 −1.15836
\(280\) 0 0
\(281\) 8.20204 0.489293 0.244646 0.969612i \(-0.421328\pi\)
0.244646 + 0.969612i \(0.421328\pi\)
\(282\) −1.10102 −0.0655648
\(283\) −23.5959 −1.40263 −0.701316 0.712851i \(-0.747404\pi\)
−0.701316 + 0.712851i \(0.747404\pi\)
\(284\) −4.89898 −0.290701
\(285\) 0 0
\(286\) −19.5959 −1.15873
\(287\) −0.898979 −0.0530651
\(288\) 3.00000 0.176777
\(289\) 7.00000 0.411765
\(290\) 0 0
\(291\) −4.89898 −0.287183
\(292\) 4.00000 0.234082
\(293\) −13.5959 −0.794282 −0.397141 0.917758i \(-0.629998\pi\)
−0.397141 + 0.917758i \(0.629998\pi\)
\(294\) −16.6515 −0.971137
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) 10.6969 0.619657
\(299\) 3.59592 0.207957
\(300\) 0 0
\(301\) −1.79796 −0.103633
\(302\) −8.00000 −0.460348
\(303\) −45.7980 −2.63102
\(304\) 8.44949 0.484611
\(305\) 0 0
\(306\) 14.6969 0.840168
\(307\) −21.1464 −1.20689 −0.603445 0.797404i \(-0.706206\pi\)
−0.603445 + 0.797404i \(0.706206\pi\)
\(308\) 2.20204 0.125473
\(309\) −24.0000 −1.36531
\(310\) 0 0
\(311\) 7.34847 0.416693 0.208347 0.978055i \(-0.433192\pi\)
0.208347 + 0.978055i \(0.433192\pi\)
\(312\) 9.79796 0.554700
\(313\) 21.5959 1.22067 0.610337 0.792142i \(-0.291034\pi\)
0.610337 + 0.792142i \(0.291034\pi\)
\(314\) 11.7980 0.665797
\(315\) 0 0
\(316\) 1.55051 0.0872230
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 19.1010 1.07113
\(319\) 0 0
\(320\) 0 0
\(321\) −25.5959 −1.42862
\(322\) −0.404082 −0.0225186
\(323\) 41.3939 2.30322
\(324\) −9.00000 −0.500000
\(325\) 0 0
\(326\) −2.20204 −0.121960
\(327\) 33.7980 1.86903
\(328\) 2.00000 0.110432
\(329\) 0.202041 0.0111389
\(330\) 0 0
\(331\) −10.2474 −0.563251 −0.281625 0.959524i \(-0.590874\pi\)
−0.281625 + 0.959524i \(0.590874\pi\)
\(332\) −14.4495 −0.793019
\(333\) 3.00000 0.164399
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) −1.10102 −0.0600656
\(337\) 35.5959 1.93903 0.969517 0.245026i \(-0.0787964\pi\)
0.969517 + 0.245026i \(0.0787964\pi\)
\(338\) 3.00000 0.163178
\(339\) −31.5959 −1.71605
\(340\) 0 0
\(341\) 31.5959 1.71101
\(342\) 25.3485 1.37069
\(343\) 6.20204 0.334879
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 0.202041 0.0108618
\(347\) 21.7980 1.17018 0.585088 0.810970i \(-0.301060\pi\)
0.585088 + 0.810970i \(0.301060\pi\)
\(348\) 0 0
\(349\) 16.8990 0.904582 0.452291 0.891871i \(-0.350607\pi\)
0.452291 + 0.891871i \(0.350607\pi\)
\(350\) 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 \(0.551044\pi\)
\(354\) −20.6969 −1.10003
\(355\) 0 0
\(356\) −3.79796 −0.201291
\(357\) −5.39388 −0.285474
\(358\) 5.34847 0.282675
\(359\) −3.10102 −0.163666 −0.0818328 0.996646i \(-0.526077\pi\)
−0.0818328 + 0.996646i \(0.526077\pi\)
\(360\) 0 0
\(361\) 52.3939 2.75757
\(362\) −18.6969 −0.982689
\(363\) 31.8434 1.67134
\(364\) −1.79796 −0.0942387
\(365\) 0 0
\(366\) 29.3939 1.53644
\(367\) −17.3485 −0.905583 −0.452791 0.891617i \(-0.649572\pi\)
−0.452791 + 0.891617i \(0.649572\pi\)
\(368\) 0.898979 0.0468625
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −3.50510 −0.181976
\(372\) −15.7980 −0.819086
\(373\) 7.79796 0.403763 0.201882 0.979410i \(-0.435294\pi\)
0.201882 + 0.979410i \(0.435294\pi\)
\(374\) −24.0000 −1.24101
\(375\) 0 0
\(376\) −0.449490 −0.0231807
\(377\) 0 0
\(378\) 0 0
\(379\) 7.59592 0.390176 0.195088 0.980786i \(-0.437501\pi\)
0.195088 + 0.980786i \(0.437501\pi\)
\(380\) 0 0
\(381\) −42.4949 −2.17708
\(382\) −3.34847 −0.171323
\(383\) −5.30306 −0.270974 −0.135487 0.990779i \(-0.543260\pi\)
−0.135487 + 0.990779i \(0.543260\pi\)
\(384\) 2.44949 0.125000
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 12.0000 0.609994
\(388\) −2.00000 −0.101535
\(389\) −6.20204 −0.314456 −0.157228 0.987562i \(-0.550256\pi\)
−0.157228 + 0.987562i \(0.550256\pi\)
\(390\) 0 0
\(391\) 4.40408 0.222724
\(392\) −6.79796 −0.343349
\(393\) 34.8990 1.76042
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) −14.6969 −0.738549
\(397\) −35.7980 −1.79665 −0.898324 0.439334i \(-0.855215\pi\)
−0.898324 + 0.439334i \(0.855215\pi\)
\(398\) 1.55051 0.0777201
\(399\) −9.30306 −0.465736
\(400\) 0 0
\(401\) −4.20204 −0.209840 −0.104920 0.994481i \(-0.533459\pi\)
−0.104920 + 0.994481i \(0.533459\pi\)
\(402\) −25.5959 −1.27661
\(403\) −25.7980 −1.28509
\(404\) −18.6969 −0.930207
\(405\) 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
\(413\) 3.79796 0.186885
\(414\) 2.69694 0.132547
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) 33.7980 1.65509
\(418\) −41.3939 −2.02464
\(419\) −13.7980 −0.674074 −0.337037 0.941491i \(-0.609425\pi\)
−0.337037 + 0.941491i \(0.609425\pi\)
\(420\) 0 0
\(421\) 19.5959 0.955047 0.477523 0.878619i \(-0.341535\pi\)
0.477523 + 0.878619i \(0.341535\pi\)
\(422\) 22.6969 1.10487
\(423\) −1.34847 −0.0655648
\(424\) 7.79796 0.378702
\(425\) 0 0
\(426\) −12.0000 −0.581402
\(427\) −5.39388 −0.261028
\(428\) −10.4495 −0.505095
\(429\) −48.0000 −2.31746
\(430\) 0 0
\(431\) −40.2474 −1.93865 −0.969326 0.245780i \(-0.920956\pi\)
−0.969326 + 0.245780i \(0.920956\pi\)
\(432\) 0 0
\(433\) −37.3939 −1.79704 −0.898518 0.438938i \(-0.855355\pi\)
−0.898518 + 0.438938i \(0.855355\pi\)
\(434\) 2.89898 0.139155
\(435\) 0 0
\(436\) 13.7980 0.660802
\(437\) 7.59592 0.363362
\(438\) 9.79796 0.468165
\(439\) 35.3485 1.68709 0.843545 0.537058i \(-0.180464\pi\)
0.843545 + 0.537058i \(0.180464\pi\)
\(440\) 0 0
\(441\) −20.3939 −0.971137
\(442\) 19.5959 0.932083
\(443\) 39.3485 1.86950 0.934751 0.355303i \(-0.115622\pi\)
0.934751 + 0.355303i \(0.115622\pi\)
\(444\) 2.44949 0.116248
\(445\) 0 0
\(446\) −4.44949 −0.210689
\(447\) 26.2020 1.23931
\(448\) −0.449490 −0.0212364
\(449\) 3.79796 0.179237 0.0896184 0.995976i \(-0.471435\pi\)
0.0896184 + 0.995976i \(0.471435\pi\)
\(450\) 0 0
\(451\) −9.79796 −0.461368
\(452\) −12.8990 −0.606717
\(453\) −19.5959 −0.920697
\(454\) −15.1010 −0.708726
\(455\) 0 0
\(456\) 20.6969 0.969223
\(457\) −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
\(461\) −14.2020 −0.661455 −0.330727 0.943726i \(-0.607294\pi\)
−0.330727 + 0.943726i \(0.607294\pi\)
\(462\) 5.39388 0.250946
\(463\) 28.4949 1.32427 0.662135 0.749384i \(-0.269650\pi\)
0.662135 + 0.749384i \(0.269650\pi\)
\(464\) 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
\(469\) 4.69694 0.216884
\(470\) 0 0
\(471\) 28.8990 1.33159
\(472\) −8.44949 −0.388919
\(473\) −19.5959 −0.901021
\(474\) 3.79796 0.174446
\(475\) 0 0
\(476\) −2.20204 −0.100930
\(477\) 23.3939 1.07113
\(478\) 26.0454 1.19129
\(479\) −29.1464 −1.33173 −0.665867 0.746070i \(-0.731938\pi\)
−0.665867 + 0.746070i \(0.731938\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) −11.7980 −0.537382
\(483\) −0.989795 −0.0450372
\(484\) 13.0000 0.590909
\(485\) 0 0
\(486\) −22.0454 −1.00000
\(487\) 29.3939 1.33196 0.665982 0.745968i \(-0.268013\pi\)
0.665982 + 0.745968i \(0.268013\pi\)
\(488\) 12.0000 0.543214
\(489\) −5.39388 −0.243920
\(490\) 0 0
\(491\) −38.6969 −1.74637 −0.873184 0.487390i \(-0.837949\pi\)
−0.873184 + 0.487390i \(0.837949\pi\)
\(492\) 4.89898 0.220863
\(493\) 0 0
\(494\) 33.7980 1.52064
\(495\) 0 0
\(496\) −6.44949 −0.289591
\(497\) 2.20204 0.0987750
\(498\) −35.3939 −1.58604
\(499\) −25.3485 −1.13475 −0.567377 0.823458i \(-0.692042\pi\)
−0.567377 + 0.823458i \(0.692042\pi\)
\(500\) 0 0
\(501\) 19.5959 0.875481
\(502\) 6.65153 0.296872
\(503\) 17.7980 0.793572 0.396786 0.917911i \(-0.370126\pi\)
0.396786 + 0.917911i \(0.370126\pi\)
\(504\) −1.34847 −0.0600656
\(505\) 0 0
\(506\) −4.40408 −0.195785
\(507\) 7.34847 0.326357
\(508\) −17.3485 −0.769714
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) −1.79796 −0.0795370
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 4.89898 0.216085
\(515\) 0 0
\(516\) 9.79796 0.431331
\(517\) 2.20204 0.0968457
\(518\) −0.449490 −0.0197494
\(519\) 0.494897 0.0217236
\(520\) 0 0
\(521\) −1.79796 −0.0787700 −0.0393850 0.999224i \(-0.512540\pi\)
−0.0393850 + 0.999224i \(0.512540\pi\)
\(522\) 0 0
\(523\) 0.898979 0.0393096 0.0196548 0.999807i \(-0.493743\pi\)
0.0196548 + 0.999807i \(0.493743\pi\)
\(524\) 14.2474 0.622403
\(525\) 0 0
\(526\) 26.2474 1.14444
\(527\) −31.5959 −1.37634
\(528\) −12.0000 −0.522233
\(529\) −22.1918 −0.964862
\(530\) 0 0
\(531\) −25.3485 −1.10003
\(532\) −3.79796 −0.164662
\(533\) 8.00000 0.346518
\(534\) −9.30306 −0.402583
\(535\) 0 0
\(536\) −10.4495 −0.451349
\(537\) 13.1010 0.565351
\(538\) 10.6969 0.461178
\(539\) 33.3031 1.43446
\(540\) 0 0
\(541\) 39.5959 1.70236 0.851181 0.524873i \(-0.175887\pi\)
0.851181 + 0.524873i \(0.175887\pi\)
\(542\) 16.4949 0.708517
\(543\) −45.7980 −1.96538
\(544\) 4.89898 0.210042
\(545\) 0 0
\(546\) −4.40408 −0.188477
\(547\) 18.6969 0.799423 0.399712 0.916641i \(-0.369110\pi\)
0.399712 + 0.916641i \(0.369110\pi\)
\(548\) −19.5959 −0.837096
\(549\) 36.0000 1.53644
\(550\) 0 0
\(551\) 0 0
\(552\) 2.20204 0.0937251
\(553\) −0.696938 −0.0296368
\(554\) −19.5959 −0.832551
\(555\) 0 0
\(556\) 13.7980 0.585164
\(557\) −12.0000 −0.508456 −0.254228 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) −19.3485 −0.819086
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) −58.7878 −2.48202
\(562\) 8.20204 0.345982
\(563\) −5.30306 −0.223497 −0.111749 0.993736i \(-0.535645\pi\)
−0.111749 + 0.993736i \(0.535645\pi\)
\(564\) −1.10102 −0.0463613
\(565\) 0 0
\(566\) −23.5959 −0.991810
\(567\) 4.04541 0.169891
\(568\) −4.89898 −0.205557
\(569\) −17.5959 −0.737659 −0.368830 0.929497i \(-0.620241\pi\)
−0.368830 + 0.929497i \(0.620241\pi\)
\(570\) 0 0
\(571\) 33.3939 1.39749 0.698745 0.715371i \(-0.253742\pi\)
0.698745 + 0.715371i \(0.253742\pi\)
\(572\) −19.5959 −0.819346
\(573\) −8.20204 −0.342645
\(574\) −0.898979 −0.0375227
\(575\) 0 0
\(576\) 3.00000 0.125000
\(577\) −1.30306 −0.0542472 −0.0271236 0.999632i \(-0.508635\pi\)
−0.0271236 + 0.999632i \(0.508635\pi\)
\(578\) 7.00000 0.291162
\(579\) 34.2929 1.42516
\(580\) 0 0
\(581\) 6.49490 0.269454
\(582\) −4.89898 −0.203069
\(583\) −38.2020 −1.58217
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) −13.5959 −0.561642
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) −16.6515 −0.686698
\(589\) −54.4949 −2.24542
\(590\) 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
\(596\) 10.6969 0.438164
\(597\) 3.79796 0.155440
\(598\) 3.59592 0.147048
\(599\) −27.5959 −1.12754 −0.563769 0.825932i \(-0.690649\pi\)
−0.563769 + 0.825932i \(0.690649\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) −1.79796 −0.0732793
\(603\) −31.3485 −1.27661
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) −45.7980 −1.86041
\(607\) −2.69694 −0.109465 −0.0547327 0.998501i \(-0.517431\pi\)
−0.0547327 + 0.998501i \(0.517431\pi\)
\(608\) 8.44949 0.342672
\(609\) 0 0
\(610\) 0 0
\(611\) −1.79796 −0.0727376
\(612\) 14.6969 0.594089
\(613\) 20.2020 0.815953 0.407976 0.912992i \(-0.366235\pi\)
0.407976 + 0.912992i \(0.366235\pi\)
\(614\) −21.1464 −0.853400
\(615\) 0 0
\(616\) 2.20204 0.0887228
\(617\) −19.5959 −0.788902 −0.394451 0.918917i \(-0.629065\pi\)
−0.394451 + 0.918917i \(0.629065\pi\)
\(618\) −24.0000 −0.965422
\(619\) −3.10102 −0.124641 −0.0623203 0.998056i \(-0.519850\pi\)
−0.0623203 + 0.998056i \(0.519850\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 7.34847 0.294647
\(623\) 1.70714 0.0683953
\(624\) 9.79796 0.392232
\(625\) 0 0
\(626\) 21.5959 0.863146
\(627\) −101.394 −4.04928
\(628\) 11.7980 0.470790
\(629\) 4.89898 0.195335
\(630\) 0 0
\(631\) 4.24745 0.169088 0.0845441 0.996420i \(-0.473057\pi\)
0.0845441 + 0.996420i \(0.473057\pi\)
\(632\) 1.55051 0.0616760
\(633\) 55.5959 2.20974
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) 19.1010 0.757405
\(637\) −27.1918 −1.07738
\(638\) 0 0
\(639\) −14.6969 −0.581402
\(640\) 0 0
\(641\) −1.79796 −0.0710151 −0.0355076 0.999369i \(-0.511305\pi\)
−0.0355076 + 0.999369i \(0.511305\pi\)
\(642\) −25.5959 −1.01019
\(643\) −32.8990 −1.29741 −0.648705 0.761040i \(-0.724689\pi\)
−0.648705 + 0.761040i \(0.724689\pi\)
\(644\) −0.404082 −0.0159231
\(645\) 0 0
\(646\) 41.3939 1.62862
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) −9.00000 −0.353553
\(649\) 41.3939 1.62485
\(650\) 0 0
\(651\) 7.10102 0.278311
\(652\) −2.20204 −0.0862386
\(653\) 34.0000 1.33052 0.665261 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) 33.7980 1.32160
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) 12.0000 0.468165
\(658\) 0.202041 0.00787638
\(659\) −7.59592 −0.295895 −0.147947 0.988995i \(-0.547267\pi\)
−0.147947 + 0.988995i \(0.547267\pi\)
\(660\) 0 0
\(661\) 47.1918 1.83555 0.917775 0.397101i \(-0.129984\pi\)
0.917775 + 0.397101i \(0.129984\pi\)
\(662\) −10.2474 −0.398278
\(663\) 48.0000 1.86417
\(664\) −14.4495 −0.560749
\(665\) 0 0
\(666\) 3.00000 0.116248
\(667\) 0 0
\(668\) 8.00000 0.309529
\(669\) −10.8990 −0.421379
\(670\) 0 0
\(671\) −58.7878 −2.26948
\(672\) −1.10102 −0.0424728
\(673\) 24.0000 0.925132 0.462566 0.886585i \(-0.346929\pi\)
0.462566 + 0.886585i \(0.346929\pi\)
\(674\) 35.5959 1.37110
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) −31.5959 −1.21343
\(679\) 0.898979 0.0344997
\(680\) 0 0
\(681\) −36.9898 −1.41745
\(682\) 31.5959 1.20987
\(683\) −23.5959 −0.902873 −0.451436 0.892303i \(-0.649088\pi\)
−0.451436 + 0.892303i \(0.649088\pi\)
\(684\) 25.3485 0.969223
\(685\) 0 0
\(686\) 6.20204 0.236795
\(687\) −24.4949 −0.934539
\(688\) 4.00000 0.152499
\(689\) 31.1918 1.18831
\(690\) 0 0
\(691\) 18.2020 0.692438 0.346219 0.938154i \(-0.387465\pi\)
0.346219 + 0.938154i \(0.387465\pi\)
\(692\) 0.202041 0.00768045
\(693\) 6.60612 0.250946
\(694\) 21.7980 0.827439
\(695\) 0 0
\(696\) 0 0
\(697\) 9.79796 0.371124
\(698\) 16.8990 0.639636
\(699\) −24.0000 −0.907763
\(700\) 0 0
\(701\) 5.79796 0.218986 0.109493 0.993988i \(-0.465077\pi\)
0.109493 + 0.993988i \(0.465077\pi\)
\(702\) 0 0
\(703\) 8.44949 0.318679
\(704\) −4.89898 −0.184637
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 8.40408 0.316068
\(708\) −20.6969 −0.777839
\(709\) −13.7980 −0.518193 −0.259097 0.965851i \(-0.583425\pi\)
−0.259097 + 0.965851i \(0.583425\pi\)
\(710\) 0 0
\(711\) 4.65153 0.174446
\(712\) −3.79796 −0.142335
\(713\) −5.79796 −0.217135
\(714\) −5.39388 −0.201861
\(715\) 0 0
\(716\) 5.34847 0.199882
\(717\) 63.7980 2.38258
\(718\) −3.10102 −0.115729
\(719\) 12.4041 0.462594 0.231297 0.972883i \(-0.425703\pi\)
0.231297 + 0.972883i \(0.425703\pi\)
\(720\) 0 0
\(721\) 4.40408 0.164017
\(722\) 52.3939 1.94990
\(723\) −28.8990 −1.07476
\(724\) −18.6969 −0.694866
\(725\) 0 0
\(726\) 31.8434 1.18182
\(727\) −8.89898 −0.330045 −0.165022 0.986290i \(-0.552770\pi\)
−0.165022 + 0.986290i \(0.552770\pi\)
\(728\) −1.79796 −0.0666368
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 19.5959 0.724781
\(732\) 29.3939 1.08643
\(733\) 21.5959 0.797663 0.398832 0.917024i \(-0.369416\pi\)
0.398832 + 0.917024i \(0.369416\pi\)
\(734\) −17.3485 −0.640344
\(735\) 0 0
\(736\) 0.898979 0.0331368
\(737\) 51.1918 1.88568
\(738\) 6.00000 0.220863
\(739\) −26.2020 −0.963858 −0.481929 0.876210i \(-0.660064\pi\)
−0.481929 + 0.876210i \(0.660064\pi\)
\(740\) 0 0
\(741\) 82.7878 3.04128
\(742\) −3.50510 −0.128676
\(743\) −38.2474 −1.40316 −0.701581 0.712589i \(-0.747522\pi\)
−0.701581 + 0.712589i \(0.747522\pi\)
\(744\) −15.7980 −0.579181
\(745\) 0 0
\(746\) 7.79796 0.285504
\(747\) −43.3485 −1.58604
\(748\) −24.0000 −0.877527
\(749\) 4.69694 0.171622
\(750\) 0 0
\(751\) 1.30306 0.0475494 0.0237747 0.999717i \(-0.492432\pi\)
0.0237747 + 0.999717i \(0.492432\pi\)
\(752\) −0.449490 −0.0163912
\(753\) 16.2929 0.593745
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 5.59592 0.203387 0.101694 0.994816i \(-0.467574\pi\)
0.101694 + 0.994816i \(0.467574\pi\)
\(758\) 7.59592 0.275896
\(759\) −10.7878 −0.391571
\(760\) 0 0
\(761\) −53.1918 −1.92820 −0.964101 0.265535i \(-0.914451\pi\)
−0.964101 + 0.265535i \(0.914451\pi\)
\(762\) −42.4949 −1.53943
\(763\) −6.20204 −0.224529
\(764\) −3.34847 −0.121143
\(765\) 0 0
\(766\) −5.30306 −0.191607
\(767\) −33.7980 −1.22037
\(768\) 2.44949 0.0883883
\(769\) −16.2020 −0.584261 −0.292130 0.956379i \(-0.594364\pi\)
−0.292130 + 0.956379i \(0.594364\pi\)
\(770\) 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
\(777\) −1.10102 −0.0394989
\(778\) −6.20204 −0.222354
\(779\) 16.8990 0.605469
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 4.40408 0.157490
\(783\) 0 0
\(784\) −6.79796 −0.242784
\(785\) 0 0
\(786\) 34.8990 1.24481
\(787\) 44.7423 1.59489 0.797446 0.603390i \(-0.206184\pi\)
0.797446 + 0.603390i \(0.206184\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 64.2929 2.28889
\(790\) 0 0
\(791\) 5.79796 0.206152
\(792\) −14.6969 −0.522233
\(793\) 48.0000 1.70453
\(794\) −35.7980 −1.27042
\(795\) 0 0
\(796\) 1.55051 0.0549564
\(797\) −4.40408 −0.156001 −0.0780003 0.996953i \(-0.524854\pi\)
−0.0780003 + 0.996953i \(0.524854\pi\)
\(798\) −9.30306 −0.329325
\(799\) −2.20204 −0.0779026
\(800\) 0 0
\(801\) −11.3939 −0.402583
\(802\) −4.20204 −0.148379
\(803\) −19.5959 −0.691525
\(804\) −25.5959 −0.902698
\(805\) 0 0
\(806\) −25.7980 −0.908694
\(807\) 26.2020 0.922356
\(808\) −18.6969 −0.657756
\(809\) 45.1918 1.58886 0.794430 0.607356i \(-0.207770\pi\)
0.794430 + 0.607356i \(0.207770\pi\)
\(810\) 0 0
\(811\) 45.7980 1.60818 0.804092 0.594505i \(-0.202652\pi\)
0.804092 + 0.594505i \(0.202652\pi\)
\(812\) 0 0
\(813\) 40.4041 1.41703
\(814\) −4.89898 −0.171709
\(815\) 0 0
\(816\) 12.0000 0.420084
\(817\) 33.7980 1.18244
\(818\) −10.0000 −0.349642
\(819\) −5.39388 −0.188477
\(820\) 0 0
\(821\) −45.5959 −1.59131 −0.795654 0.605751i \(-0.792873\pi\)
−0.795654 + 0.605751i \(0.792873\pi\)
\(822\) −48.0000 −1.67419
\(823\) 18.6515 0.650151 0.325076 0.945688i \(-0.394610\pi\)
0.325076 + 0.945688i \(0.394610\pi\)
\(824\) −9.79796 −0.341328
\(825\) 0 0
\(826\) 3.79796 0.132148
\(827\) 12.4949 0.434490 0.217245 0.976117i \(-0.430293\pi\)
0.217245 + 0.976117i \(0.430293\pi\)
\(828\) 2.69694 0.0937251
\(829\) 47.5959 1.65307 0.826537 0.562882i \(-0.190307\pi\)
0.826537 + 0.562882i \(0.190307\pi\)
\(830\) 0 0
\(831\) −48.0000 −1.66510
\(832\) 4.00000 0.138675
\(833\) −33.3031 −1.15388
\(834\) 33.7980 1.17033
\(835\) 0 0
\(836\) −41.3939 −1.43164
\(837\) 0 0
\(838\) −13.7980 −0.476643
\(839\) 33.7980 1.16684 0.583418 0.812172i \(-0.301715\pi\)
0.583418 + 0.812172i \(0.301715\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 19.5959 0.675320
\(843\) 20.0908 0.691964
\(844\) 22.6969 0.781261
\(845\) 0 0
\(846\) −1.34847 −0.0463613
\(847\) −5.84337 −0.200780
\(848\) 7.79796 0.267783
\(849\) −57.7980 −1.98362
\(850\) 0 0
\(851\) 0.898979 0.0308166
\(852\) −12.0000 −0.411113
\(853\) 1.59592 0.0546432 0.0273216 0.999627i \(-0.491302\pi\)
0.0273216 + 0.999627i \(0.491302\pi\)
\(854\) −5.39388 −0.184575
\(855\) 0 0
\(856\) −10.4495 −0.357156
\(857\) −29.5959 −1.01098 −0.505489 0.862833i \(-0.668688\pi\)
−0.505489 + 0.862833i \(0.668688\pi\)
\(858\) −48.0000 −1.63869
\(859\) −20.8536 −0.711515 −0.355757 0.934578i \(-0.615777\pi\)
−0.355757 + 0.934578i \(0.615777\pi\)
\(860\) 0 0
\(861\) −2.20204 −0.0750454
\(862\) −40.2474 −1.37083
\(863\) 30.7423 1.04648 0.523241 0.852185i \(-0.324723\pi\)
0.523241 + 0.852185i \(0.324723\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −37.3939 −1.27070
\(867\) 17.1464 0.582323
\(868\) 2.89898 0.0983978
\(869\) −7.59592 −0.257674
\(870\) 0 0
\(871\) −41.7980 −1.41627
\(872\) 13.7980 0.467258
\(873\) −6.00000 −0.203069
\(874\) 7.59592 0.256936
\(875\) 0 0
\(876\) 9.79796 0.331042
\(877\) −43.3939 −1.46531 −0.732654 0.680602i \(-0.761718\pi\)
−0.732654 + 0.680602i \(0.761718\pi\)
\(878\) 35.3485 1.19295
\(879\) −33.3031 −1.12328
\(880\) 0 0
\(881\) 53.3939 1.79889 0.899443 0.437039i \(-0.143973\pi\)
0.899443 + 0.437039i \(0.143973\pi\)
\(882\) −20.3939 −0.686698
\(883\) 13.3031 0.447684 0.223842 0.974625i \(-0.428140\pi\)
0.223842 + 0.974625i \(0.428140\pi\)
\(884\) 19.5959 0.659082
\(885\) 0 0
\(886\) 39.3485 1.32194
\(887\) 24.0454 0.807366 0.403683 0.914899i \(-0.367730\pi\)
0.403683 + 0.914899i \(0.367730\pi\)
\(888\) 2.44949 0.0821995
\(889\) 7.79796 0.261535
\(890\) 0 0
\(891\) 44.0908 1.47710
\(892\) −4.44949 −0.148980
\(893\) −3.79796 −0.127094
\(894\) 26.2020 0.876327
\(895\) 0 0
\(896\) −0.449490 −0.0150164
\(897\) 8.80816 0.294096
\(898\) 3.79796 0.126740
\(899\) 0 0
\(900\) 0 0
\(901\) 38.2020 1.27269
\(902\) −9.79796 −0.326236
\(903\) −4.40408 −0.146559
\(904\) −12.8990 −0.429014
\(905\) 0 0
\(906\) −19.5959 −0.651031
\(907\) −8.89898 −0.295486 −0.147743 0.989026i \(-0.547201\pi\)
−0.147743 + 0.989026i \(0.547201\pi\)
\(908\) −15.1010 −0.501145
\(909\) −56.0908 −1.86041
\(910\) 0 0
\(911\) 31.8434 1.05502 0.527509 0.849550i \(-0.323126\pi\)
0.527509 + 0.849550i \(0.323126\pi\)
\(912\) 20.6969 0.685344
\(913\) 70.7878 2.34273
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) −6.40408 −0.211481
\(918\) 0 0
\(919\) −9.14643 −0.301713 −0.150856 0.988556i \(-0.548203\pi\)
−0.150856 + 0.988556i \(0.548203\pi\)
\(920\) 0 0
\(921\) −51.7980 −1.70680
\(922\) −14.2020 −0.467719
\(923\) −19.5959 −0.645007
\(924\) 5.39388 0.177446
\(925\) 0 0
\(926\) 28.4949 0.936400
\(927\) −29.3939 −0.965422
\(928\) 0 0
\(929\) 37.5959 1.23348 0.616741 0.787166i \(-0.288453\pi\)
0.616741 + 0.787166i \(0.288453\pi\)
\(930\) 0 0
\(931\) −57.4393 −1.88250
\(932\) −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 \(0.458286\pi\)
\(938\) 4.69694 0.153360
\(939\) 52.8990 1.72629
\(940\) 0 0
\(941\) 30.2929 0.987519 0.493759 0.869599i \(-0.335622\pi\)
0.493759 + 0.869599i \(0.335622\pi\)
\(942\) 28.8990 0.941580
\(943\) 1.79796 0.0585496
\(944\) −8.44949 −0.275007
\(945\) 0 0
\(946\) −19.5959 −0.637118
\(947\) −33.3939 −1.08516 −0.542578 0.840006i \(-0.682552\pi\)
−0.542578 + 0.840006i \(0.682552\pi\)
\(948\) 3.79796 0.123352
\(949\) 16.0000 0.519382
\(950\) 0 0
\(951\) 44.0908 1.42974
\(952\) −2.20204 −0.0713686
\(953\) −2.20204 −0.0713311 −0.0356656 0.999364i \(-0.511355\pi\)
−0.0356656 + 0.999364i \(0.511355\pi\)
\(954\) 23.3939 0.757405
\(955\) 0 0
\(956\) 26.0454 0.842369
\(957\) 0 0
\(958\) −29.1464 −0.941678
\(959\) 8.80816 0.284430
\(960\) 0 0
\(961\) 10.5959 0.341804
\(962\) 4.00000 0.128965
\(963\) −31.3485 −1.01019
\(964\) −11.7980 −0.379987
\(965\) 0 0
\(966\) −0.989795 −0.0318461
\(967\) 52.4949 1.68812 0.844061 0.536247i \(-0.180158\pi\)
0.844061 + 0.536247i \(0.180158\pi\)
\(968\) 13.0000 0.417836
\(969\) 101.394 3.25724
\(970\) 0 0
\(971\) −44.8990 −1.44088 −0.720438 0.693519i \(-0.756059\pi\)
−0.720438 + 0.693519i \(0.756059\pi\)
\(972\) −22.0454 −0.707107
\(973\) −6.20204 −0.198828
\(974\) 29.3939 0.941841
\(975\) 0 0
\(976\) 12.0000 0.384111
\(977\) −49.5959 −1.58671 −0.793357 0.608757i \(-0.791669\pi\)
−0.793357 + 0.608757i \(0.791669\pi\)
\(978\) −5.39388 −0.172477
\(979\) 18.6061 0.594654
\(980\) 0 0
\(981\) 41.3939 1.32160
\(982\) −38.6969 −1.23487
\(983\) −48.9444 −1.56108 −0.780542 0.625104i \(-0.785057\pi\)
−0.780542 + 0.625104i \(0.785057\pi\)
\(984\) 4.89898 0.156174
\(985\) 0 0
\(986\) 0 0
\(987\) 0.494897 0.0157528
\(988\) 33.7980 1.07526
\(989\) 3.59592 0.114344
\(990\) 0 0
\(991\) 31.8434 1.01154 0.505769 0.862669i \(-0.331209\pi\)
0.505769 + 0.862669i \(0.331209\pi\)
\(992\) −6.44949 −0.204772
\(993\) −25.1010 −0.796557
\(994\) 2.20204 0.0698445
\(995\) 0 0
\(996\) −35.3939 −1.12150
\(997\) −22.0000 −0.696747 −0.348373 0.937356i \(-0.613266\pi\)
−0.348373 + 0.937356i \(0.613266\pi\)
\(998\) −25.3485 −0.802392
\(999\) 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.1.2 2
5.2 odd 4 370.2.b.c.149.3 yes 4
5.3 odd 4 370.2.b.c.149.2 4
5.4 even 2 1850.2.a.s.1.1 2
15.2 even 4 3330.2.d.m.1999.1 4
15.8 even 4 3330.2.d.m.1999.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.c.149.2 4 5.3 odd 4
370.2.b.c.149.3 yes 4 5.2 odd 4
1850.2.a.s.1.1 2 5.4 even 2
1850.2.a.v.1.2 2 1.1 even 1 trivial
3330.2.d.m.1999.1 4 15.2 even 4
3330.2.d.m.1999.4 4 15.8 even 4