Properties

Label 3850.2.a.bd.1.2
Level $3850$
Weight $2$
Character 3850.1
Self dual yes
Analytic conductor $30.742$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.7424047782\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) 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 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 3850.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.732051 q^{3} +1.00000 q^{4} -0.732051 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +0.732051 q^{3} +1.00000 q^{4} -0.732051 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} +1.00000 q^{11} +0.732051 q^{12} -5.46410 q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.46410 q^{17} +2.46410 q^{18} +3.26795 q^{19} -0.732051 q^{21} -1.00000 q^{22} -2.19615 q^{23} -0.732051 q^{24} +5.46410 q^{26} -4.00000 q^{27} -1.00000 q^{28} -1.26795 q^{29} +2.00000 q^{31} -1.00000 q^{32} +0.732051 q^{33} +3.46410 q^{34} -2.46410 q^{36} +2.73205 q^{37} -3.26795 q^{38} -4.00000 q^{39} +8.19615 q^{41} +0.732051 q^{42} -2.00000 q^{43} +1.00000 q^{44} +2.19615 q^{46} +6.92820 q^{47} +0.732051 q^{48} +1.00000 q^{49} -2.53590 q^{51} -5.46410 q^{52} +10.7321 q^{53} +4.00000 q^{54} +1.00000 q^{56} +2.39230 q^{57} +1.26795 q^{58} -6.92820 q^{59} +8.92820 q^{61} -2.00000 q^{62} +2.46410 q^{63} +1.00000 q^{64} -0.732051 q^{66} +4.00000 q^{67} -3.46410 q^{68} -1.60770 q^{69} +2.53590 q^{71} +2.46410 q^{72} -6.39230 q^{73} -2.73205 q^{74} +3.26795 q^{76} -1.00000 q^{77} +4.00000 q^{78} -1.80385 q^{79} +4.46410 q^{81} -8.19615 q^{82} -4.39230 q^{83} -0.732051 q^{84} +2.00000 q^{86} -0.928203 q^{87} -1.00000 q^{88} -3.46410 q^{89} +5.46410 q^{91} -2.19615 q^{92} +1.46410 q^{93} -6.92820 q^{94} -0.732051 q^{96} +16.5885 q^{97} -1.00000 q^{98} -2.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{18} + 10 q^{19} + 2 q^{21} - 2 q^{22} + 6 q^{23} + 2 q^{24} + 4 q^{26} - 8 q^{27} - 2 q^{28} - 6 q^{29} + 4 q^{31} - 2 q^{32} - 2 q^{33} + 2 q^{36} + 2 q^{37} - 10 q^{38} - 8 q^{39} + 6 q^{41} - 2 q^{42} - 4 q^{43} + 2 q^{44} - 6 q^{46} - 2 q^{48} + 2 q^{49} - 12 q^{51} - 4 q^{52} + 18 q^{53} + 8 q^{54} + 2 q^{56} - 16 q^{57} + 6 q^{58} + 4 q^{61} - 4 q^{62} - 2 q^{63} + 2 q^{64} + 2 q^{66} + 8 q^{67} - 24 q^{69} + 12 q^{71} - 2 q^{72} + 8 q^{73} - 2 q^{74} + 10 q^{76} - 2 q^{77} + 8 q^{78} - 14 q^{79} + 2 q^{81} - 6 q^{82} + 12 q^{83} + 2 q^{84} + 4 q^{86} + 12 q^{87} - 2 q^{88} + 4 q^{91} + 6 q^{92} - 4 q^{93} + 2 q^{96} + 2 q^{97} - 2 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0.732051 0.422650 0.211325 0.977416i \(-0.432222\pi\)
0.211325 + 0.977416i \(0.432222\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −0.732051 −0.298858
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) −2.46410 −0.821367
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0.732051 0.211325
\(13\) −5.46410 −1.51547 −0.757735 0.652563i \(-0.773694\pi\)
−0.757735 + 0.652563i \(0.773694\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.46410 −0.840168 −0.420084 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 2.46410 0.580794
\(19\) 3.26795 0.749719 0.374859 0.927082i \(-0.377691\pi\)
0.374859 + 0.927082i \(0.377691\pi\)
\(20\) 0 0
\(21\) −0.732051 −0.159747
\(22\) −1.00000 −0.213201
\(23\) −2.19615 −0.457929 −0.228965 0.973435i \(-0.573534\pi\)
−0.228965 + 0.973435i \(0.573534\pi\)
\(24\) −0.732051 −0.149429
\(25\) 0 0
\(26\) 5.46410 1.07160
\(27\) −4.00000 −0.769800
\(28\) −1.00000 −0.188982
\(29\) −1.26795 −0.235452 −0.117726 0.993046i \(-0.537560\pi\)
−0.117726 + 0.993046i \(0.537560\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.732051 0.127434
\(34\) 3.46410 0.594089
\(35\) 0 0
\(36\) −2.46410 −0.410684
\(37\) 2.73205 0.449146 0.224573 0.974457i \(-0.427901\pi\)
0.224573 + 0.974457i \(0.427901\pi\)
\(38\) −3.26795 −0.530131
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 8.19615 1.28002 0.640012 0.768365i \(-0.278929\pi\)
0.640012 + 0.768365i \(0.278929\pi\)
\(42\) 0.732051 0.112958
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 2.19615 0.323805
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0.732051 0.105662
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.53590 −0.355097
\(52\) −5.46410 −0.757735
\(53\) 10.7321 1.47416 0.737080 0.675805i \(-0.236204\pi\)
0.737080 + 0.675805i \(0.236204\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 2.39230 0.316869
\(58\) 1.26795 0.166490
\(59\) −6.92820 −0.901975 −0.450988 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) 0 0
\(61\) 8.92820 1.14314 0.571570 0.820554i \(-0.306335\pi\)
0.571570 + 0.820554i \(0.306335\pi\)
\(62\) −2.00000 −0.254000
\(63\) 2.46410 0.310448
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −0.732051 −0.0901092
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −3.46410 −0.420084
\(69\) −1.60770 −0.193544
\(70\) 0 0
\(71\) 2.53590 0.300956 0.150478 0.988613i \(-0.451919\pi\)
0.150478 + 0.988613i \(0.451919\pi\)
\(72\) 2.46410 0.290397
\(73\) −6.39230 −0.748163 −0.374081 0.927396i \(-0.622042\pi\)
−0.374081 + 0.927396i \(0.622042\pi\)
\(74\) −2.73205 −0.317594
\(75\) 0 0
\(76\) 3.26795 0.374859
\(77\) −1.00000 −0.113961
\(78\) 4.00000 0.452911
\(79\) −1.80385 −0.202949 −0.101474 0.994838i \(-0.532356\pi\)
−0.101474 + 0.994838i \(0.532356\pi\)
\(80\) 0 0
\(81\) 4.46410 0.496011
\(82\) −8.19615 −0.905114
\(83\) −4.39230 −0.482118 −0.241059 0.970510i \(-0.577495\pi\)
−0.241059 + 0.970510i \(0.577495\pi\)
\(84\) −0.732051 −0.0798733
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) −0.928203 −0.0995138
\(88\) −1.00000 −0.106600
\(89\) −3.46410 −0.367194 −0.183597 0.983002i \(-0.558774\pi\)
−0.183597 + 0.983002i \(0.558774\pi\)
\(90\) 0 0
\(91\) 5.46410 0.572793
\(92\) −2.19615 −0.228965
\(93\) 1.46410 0.151820
\(94\) −6.92820 −0.714590
\(95\) 0 0
\(96\) −0.732051 −0.0747146
\(97\) 16.5885 1.68430 0.842151 0.539241i \(-0.181289\pi\)
0.842151 + 0.539241i \(0.181289\pi\)
\(98\) −1.00000 −0.101015
\(99\) −2.46410 −0.247652
\(100\) 0 0
\(101\) 19.8564 1.97579 0.987893 0.155136i \(-0.0495815\pi\)
0.987893 + 0.155136i \(0.0495815\pi\)
\(102\) 2.53590 0.251091
\(103\) 8.39230 0.826918 0.413459 0.910523i \(-0.364320\pi\)
0.413459 + 0.910523i \(0.364320\pi\)
\(104\) 5.46410 0.535799
\(105\) 0 0
\(106\) −10.7321 −1.04239
\(107\) 19.8564 1.91959 0.959796 0.280700i \(-0.0905665\pi\)
0.959796 + 0.280700i \(0.0905665\pi\)
\(108\) −4.00000 −0.384900
\(109\) 1.66025 0.159023 0.0795117 0.996834i \(-0.474664\pi\)
0.0795117 + 0.996834i \(0.474664\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) −1.00000 −0.0944911
\(113\) 19.8564 1.86793 0.933967 0.357360i \(-0.116323\pi\)
0.933967 + 0.357360i \(0.116323\pi\)
\(114\) −2.39230 −0.224060
\(115\) 0 0
\(116\) −1.26795 −0.117726
\(117\) 13.4641 1.24476
\(118\) 6.92820 0.637793
\(119\) 3.46410 0.317554
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −8.92820 −0.808322
\(123\) 6.00000 0.541002
\(124\) 2.00000 0.179605
\(125\) 0 0
\(126\) −2.46410 −0.219520
\(127\) −14.9282 −1.32466 −0.662332 0.749211i \(-0.730433\pi\)
−0.662332 + 0.749211i \(0.730433\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.46410 −0.128907
\(130\) 0 0
\(131\) −11.6603 −1.01876 −0.509381 0.860541i \(-0.670125\pi\)
−0.509381 + 0.860541i \(0.670125\pi\)
\(132\) 0.732051 0.0637168
\(133\) −3.26795 −0.283367
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 3.46410 0.297044
\(137\) −12.9282 −1.10453 −0.552265 0.833668i \(-0.686237\pi\)
−0.552265 + 0.833668i \(0.686237\pi\)
\(138\) 1.60770 0.136856
\(139\) −3.66025 −0.310459 −0.155229 0.987878i \(-0.549612\pi\)
−0.155229 + 0.987878i \(0.549612\pi\)
\(140\) 0 0
\(141\) 5.07180 0.427122
\(142\) −2.53590 −0.212808
\(143\) −5.46410 −0.456931
\(144\) −2.46410 −0.205342
\(145\) 0 0
\(146\) 6.39230 0.529031
\(147\) 0.732051 0.0603785
\(148\) 2.73205 0.224573
\(149\) −10.7321 −0.879204 −0.439602 0.898193i \(-0.644880\pi\)
−0.439602 + 0.898193i \(0.644880\pi\)
\(150\) 0 0
\(151\) −13.1244 −1.06804 −0.534022 0.845470i \(-0.679320\pi\)
−0.534022 + 0.845470i \(0.679320\pi\)
\(152\) −3.26795 −0.265066
\(153\) 8.53590 0.690086
\(154\) 1.00000 0.0805823
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) −11.4641 −0.914935 −0.457467 0.889226i \(-0.651243\pi\)
−0.457467 + 0.889226i \(0.651243\pi\)
\(158\) 1.80385 0.143506
\(159\) 7.85641 0.623054
\(160\) 0 0
\(161\) 2.19615 0.173081
\(162\) −4.46410 −0.350733
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 8.19615 0.640012
\(165\) 0 0
\(166\) 4.39230 0.340909
\(167\) 13.8564 1.07224 0.536120 0.844141i \(-0.319889\pi\)
0.536120 + 0.844141i \(0.319889\pi\)
\(168\) 0.732051 0.0564789
\(169\) 16.8564 1.29665
\(170\) 0 0
\(171\) −8.05256 −0.615795
\(172\) −2.00000 −0.152499
\(173\) 12.9282 0.982913 0.491457 0.870902i \(-0.336465\pi\)
0.491457 + 0.870902i \(0.336465\pi\)
\(174\) 0.928203 0.0703669
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) −5.07180 −0.381220
\(178\) 3.46410 0.259645
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) −5.46410 −0.405026
\(183\) 6.53590 0.483148
\(184\) 2.19615 0.161903
\(185\) 0 0
\(186\) −1.46410 −0.107353
\(187\) −3.46410 −0.253320
\(188\) 6.92820 0.505291
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0.732051 0.0528312
\(193\) 8.39230 0.604091 0.302046 0.953293i \(-0.402330\pi\)
0.302046 + 0.953293i \(0.402330\pi\)
\(194\) −16.5885 −1.19098
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −24.2487 −1.72765 −0.863825 0.503793i \(-0.831938\pi\)
−0.863825 + 0.503793i \(0.831938\pi\)
\(198\) 2.46410 0.175116
\(199\) −10.9282 −0.774680 −0.387340 0.921937i \(-0.626606\pi\)
−0.387340 + 0.921937i \(0.626606\pi\)
\(200\) 0 0
\(201\) 2.92820 0.206540
\(202\) −19.8564 −1.39709
\(203\) 1.26795 0.0889926
\(204\) −2.53590 −0.177548
\(205\) 0 0
\(206\) −8.39230 −0.584720
\(207\) 5.41154 0.376128
\(208\) −5.46410 −0.378867
\(209\) 3.26795 0.226049
\(210\) 0 0
\(211\) 13.0718 0.899900 0.449950 0.893054i \(-0.351442\pi\)
0.449950 + 0.893054i \(0.351442\pi\)
\(212\) 10.7321 0.737080
\(213\) 1.85641 0.127199
\(214\) −19.8564 −1.35736
\(215\) 0 0
\(216\) 4.00000 0.272166
\(217\) −2.00000 −0.135769
\(218\) −1.66025 −0.112447
\(219\) −4.67949 −0.316211
\(220\) 0 0
\(221\) 18.9282 1.27325
\(222\) −2.00000 −0.134231
\(223\) 18.5359 1.24126 0.620628 0.784105i \(-0.286878\pi\)
0.620628 + 0.784105i \(0.286878\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −19.8564 −1.32083
\(227\) 6.92820 0.459841 0.229920 0.973209i \(-0.426153\pi\)
0.229920 + 0.973209i \(0.426153\pi\)
\(228\) 2.39230 0.158434
\(229\) 3.60770 0.238403 0.119202 0.992870i \(-0.461967\pi\)
0.119202 + 0.992870i \(0.461967\pi\)
\(230\) 0 0
\(231\) −0.732051 −0.0481654
\(232\) 1.26795 0.0832449
\(233\) −19.8564 −1.30084 −0.650418 0.759576i \(-0.725406\pi\)
−0.650418 + 0.759576i \(0.725406\pi\)
\(234\) −13.4641 −0.880176
\(235\) 0 0
\(236\) −6.92820 −0.450988
\(237\) −1.32051 −0.0857762
\(238\) −3.46410 −0.224544
\(239\) 4.73205 0.306091 0.153045 0.988219i \(-0.451092\pi\)
0.153045 + 0.988219i \(0.451092\pi\)
\(240\) 0 0
\(241\) 6.73205 0.433650 0.216825 0.976211i \(-0.430430\pi\)
0.216825 + 0.976211i \(0.430430\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 15.2679 0.979439
\(244\) 8.92820 0.571570
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) −17.8564 −1.13618
\(248\) −2.00000 −0.127000
\(249\) −3.21539 −0.203767
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 2.46410 0.155224
\(253\) −2.19615 −0.138071
\(254\) 14.9282 0.936679
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.33975 0.395462 0.197731 0.980256i \(-0.436643\pi\)
0.197731 + 0.980256i \(0.436643\pi\)
\(258\) 1.46410 0.0911510
\(259\) −2.73205 −0.169761
\(260\) 0 0
\(261\) 3.12436 0.193393
\(262\) 11.6603 0.720373
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) −0.732051 −0.0450546
\(265\) 0 0
\(266\) 3.26795 0.200371
\(267\) −2.53590 −0.155194
\(268\) 4.00000 0.244339
\(269\) 7.60770 0.463849 0.231925 0.972734i \(-0.425498\pi\)
0.231925 + 0.972734i \(0.425498\pi\)
\(270\) 0 0
\(271\) −20.3923 −1.23874 −0.619372 0.785098i \(-0.712613\pi\)
−0.619372 + 0.785098i \(0.712613\pi\)
\(272\) −3.46410 −0.210042
\(273\) 4.00000 0.242091
\(274\) 12.9282 0.781021
\(275\) 0 0
\(276\) −1.60770 −0.0967719
\(277\) −20.9282 −1.25745 −0.628727 0.777626i \(-0.716424\pi\)
−0.628727 + 0.777626i \(0.716424\pi\)
\(278\) 3.66025 0.219527
\(279\) −4.92820 −0.295044
\(280\) 0 0
\(281\) 1.60770 0.0959071 0.0479535 0.998850i \(-0.484730\pi\)
0.0479535 + 0.998850i \(0.484730\pi\)
\(282\) −5.07180 −0.302021
\(283\) −23.7128 −1.40958 −0.704790 0.709416i \(-0.748959\pi\)
−0.704790 + 0.709416i \(0.748959\pi\)
\(284\) 2.53590 0.150478
\(285\) 0 0
\(286\) 5.46410 0.323099
\(287\) −8.19615 −0.483804
\(288\) 2.46410 0.145199
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) 12.1436 0.711870
\(292\) −6.39230 −0.374081
\(293\) −14.5359 −0.849196 −0.424598 0.905382i \(-0.639585\pi\)
−0.424598 + 0.905382i \(0.639585\pi\)
\(294\) −0.732051 −0.0426941
\(295\) 0 0
\(296\) −2.73205 −0.158797
\(297\) −4.00000 −0.232104
\(298\) 10.7321 0.621691
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) 2.00000 0.115278
\(302\) 13.1244 0.755222
\(303\) 14.5359 0.835066
\(304\) 3.26795 0.187430
\(305\) 0 0
\(306\) −8.53590 −0.487965
\(307\) −3.60770 −0.205902 −0.102951 0.994686i \(-0.532828\pi\)
−0.102951 + 0.994686i \(0.532828\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 6.14359 0.349497
\(310\) 0 0
\(311\) 11.0718 0.627824 0.313912 0.949452i \(-0.398360\pi\)
0.313912 + 0.949452i \(0.398360\pi\)
\(312\) 4.00000 0.226455
\(313\) −11.8038 −0.667193 −0.333596 0.942716i \(-0.608262\pi\)
−0.333596 + 0.942716i \(0.608262\pi\)
\(314\) 11.4641 0.646957
\(315\) 0 0
\(316\) −1.80385 −0.101474
\(317\) 0.588457 0.0330511 0.0165255 0.999863i \(-0.494740\pi\)
0.0165255 + 0.999863i \(0.494740\pi\)
\(318\) −7.85641 −0.440565
\(319\) −1.26795 −0.0709915
\(320\) 0 0
\(321\) 14.5359 0.811315
\(322\) −2.19615 −0.122387
\(323\) −11.3205 −0.629890
\(324\) 4.46410 0.248006
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 1.21539 0.0672112
\(328\) −8.19615 −0.452557
\(329\) −6.92820 −0.381964
\(330\) 0 0
\(331\) 22.7846 1.25236 0.626178 0.779680i \(-0.284618\pi\)
0.626178 + 0.779680i \(0.284618\pi\)
\(332\) −4.39230 −0.241059
\(333\) −6.73205 −0.368914
\(334\) −13.8564 −0.758189
\(335\) 0 0
\(336\) −0.732051 −0.0399366
\(337\) 18.7846 1.02326 0.511631 0.859205i \(-0.329041\pi\)
0.511631 + 0.859205i \(0.329041\pi\)
\(338\) −16.8564 −0.916868
\(339\) 14.5359 0.789482
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 8.05256 0.435433
\(343\) −1.00000 −0.0539949
\(344\) 2.00000 0.107833
\(345\) 0 0
\(346\) −12.9282 −0.695025
\(347\) 36.9282 1.98241 0.991205 0.132336i \(-0.0422478\pi\)
0.991205 + 0.132336i \(0.0422478\pi\)
\(348\) −0.928203 −0.0497569
\(349\) −26.3923 −1.41275 −0.706374 0.707839i \(-0.749670\pi\)
−0.706374 + 0.707839i \(0.749670\pi\)
\(350\) 0 0
\(351\) 21.8564 1.16661
\(352\) −1.00000 −0.0533002
\(353\) −3.80385 −0.202458 −0.101229 0.994863i \(-0.532278\pi\)
−0.101229 + 0.994863i \(0.532278\pi\)
\(354\) 5.07180 0.269563
\(355\) 0 0
\(356\) −3.46410 −0.183597
\(357\) 2.53590 0.134214
\(358\) 6.00000 0.317110
\(359\) 4.05256 0.213886 0.106943 0.994265i \(-0.465894\pi\)
0.106943 + 0.994265i \(0.465894\pi\)
\(360\) 0 0
\(361\) −8.32051 −0.437921
\(362\) −14.0000 −0.735824
\(363\) 0.732051 0.0384227
\(364\) 5.46410 0.286397
\(365\) 0 0
\(366\) −6.53590 −0.341637
\(367\) 8.39230 0.438075 0.219037 0.975716i \(-0.429708\pi\)
0.219037 + 0.975716i \(0.429708\pi\)
\(368\) −2.19615 −0.114482
\(369\) −20.1962 −1.05137
\(370\) 0 0
\(371\) −10.7321 −0.557180
\(372\) 1.46410 0.0759101
\(373\) 2.39230 0.123869 0.0619344 0.998080i \(-0.480273\pi\)
0.0619344 + 0.998080i \(0.480273\pi\)
\(374\) 3.46410 0.179124
\(375\) 0 0
\(376\) −6.92820 −0.357295
\(377\) 6.92820 0.356821
\(378\) −4.00000 −0.205738
\(379\) 33.8564 1.73909 0.869543 0.493857i \(-0.164413\pi\)
0.869543 + 0.493857i \(0.164413\pi\)
\(380\) 0 0
\(381\) −10.9282 −0.559869
\(382\) 12.0000 0.613973
\(383\) 26.5359 1.35592 0.677961 0.735098i \(-0.262864\pi\)
0.677961 + 0.735098i \(0.262864\pi\)
\(384\) −0.732051 −0.0373573
\(385\) 0 0
\(386\) −8.39230 −0.427157
\(387\) 4.92820 0.250515
\(388\) 16.5885 0.842151
\(389\) −22.3923 −1.13533 −0.567667 0.823258i \(-0.692154\pi\)
−0.567667 + 0.823258i \(0.692154\pi\)
\(390\) 0 0
\(391\) 7.60770 0.384738
\(392\) −1.00000 −0.0505076
\(393\) −8.53590 −0.430579
\(394\) 24.2487 1.22163
\(395\) 0 0
\(396\) −2.46410 −0.123826
\(397\) −9.60770 −0.482196 −0.241098 0.970501i \(-0.577508\pi\)
−0.241098 + 0.970501i \(0.577508\pi\)
\(398\) 10.9282 0.547781
\(399\) −2.39230 −0.119765
\(400\) 0 0
\(401\) 9.46410 0.472615 0.236307 0.971678i \(-0.424063\pi\)
0.236307 + 0.971678i \(0.424063\pi\)
\(402\) −2.92820 −0.146046
\(403\) −10.9282 −0.544373
\(404\) 19.8564 0.987893
\(405\) 0 0
\(406\) −1.26795 −0.0629273
\(407\) 2.73205 0.135423
\(408\) 2.53590 0.125546
\(409\) 35.1244 1.73679 0.868394 0.495875i \(-0.165153\pi\)
0.868394 + 0.495875i \(0.165153\pi\)
\(410\) 0 0
\(411\) −9.46410 −0.466830
\(412\) 8.39230 0.413459
\(413\) 6.92820 0.340915
\(414\) −5.41154 −0.265963
\(415\) 0 0
\(416\) 5.46410 0.267900
\(417\) −2.67949 −0.131215
\(418\) −3.26795 −0.159841
\(419\) 17.0718 0.834012 0.417006 0.908904i \(-0.363079\pi\)
0.417006 + 0.908904i \(0.363079\pi\)
\(420\) 0 0
\(421\) −8.14359 −0.396894 −0.198447 0.980112i \(-0.563590\pi\)
−0.198447 + 0.980112i \(0.563590\pi\)
\(422\) −13.0718 −0.636325
\(423\) −17.0718 −0.830059
\(424\) −10.7321 −0.521194
\(425\) 0 0
\(426\) −1.85641 −0.0899432
\(427\) −8.92820 −0.432066
\(428\) 19.8564 0.959796
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) −21.1244 −1.01752 −0.508762 0.860907i \(-0.669897\pi\)
−0.508762 + 0.860907i \(0.669897\pi\)
\(432\) −4.00000 −0.192450
\(433\) 7.80385 0.375029 0.187514 0.982262i \(-0.439957\pi\)
0.187514 + 0.982262i \(0.439957\pi\)
\(434\) 2.00000 0.0960031
\(435\) 0 0
\(436\) 1.66025 0.0795117
\(437\) −7.17691 −0.343318
\(438\) 4.67949 0.223595
\(439\) −22.2487 −1.06187 −0.530937 0.847412i \(-0.678160\pi\)
−0.530937 + 0.847412i \(0.678160\pi\)
\(440\) 0 0
\(441\) −2.46410 −0.117338
\(442\) −18.9282 −0.900323
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) −18.5359 −0.877700
\(447\) −7.85641 −0.371595
\(448\) −1.00000 −0.0472456
\(449\) 19.6077 0.925344 0.462672 0.886529i \(-0.346891\pi\)
0.462672 + 0.886529i \(0.346891\pi\)
\(450\) 0 0
\(451\) 8.19615 0.385942
\(452\) 19.8564 0.933967
\(453\) −9.60770 −0.451409
\(454\) −6.92820 −0.325157
\(455\) 0 0
\(456\) −2.39230 −0.112030
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) −3.60770 −0.168577
\(459\) 13.8564 0.646762
\(460\) 0 0
\(461\) 1.60770 0.0748778 0.0374389 0.999299i \(-0.488080\pi\)
0.0374389 + 0.999299i \(0.488080\pi\)
\(462\) 0.732051 0.0340581
\(463\) 17.5167 0.814068 0.407034 0.913413i \(-0.366563\pi\)
0.407034 + 0.913413i \(0.366563\pi\)
\(464\) −1.26795 −0.0588631
\(465\) 0 0
\(466\) 19.8564 0.919830
\(467\) −4.05256 −0.187530 −0.0937650 0.995594i \(-0.529890\pi\)
−0.0937650 + 0.995594i \(0.529890\pi\)
\(468\) 13.4641 0.622378
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) −8.39230 −0.386697
\(472\) 6.92820 0.318896
\(473\) −2.00000 −0.0919601
\(474\) 1.32051 0.0606529
\(475\) 0 0
\(476\) 3.46410 0.158777
\(477\) −26.4449 −1.21083
\(478\) −4.73205 −0.216439
\(479\) 8.78461 0.401379 0.200690 0.979655i \(-0.435682\pi\)
0.200690 + 0.979655i \(0.435682\pi\)
\(480\) 0 0
\(481\) −14.9282 −0.680667
\(482\) −6.73205 −0.306637
\(483\) 1.60770 0.0731527
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) −15.2679 −0.692568
\(487\) 6.19615 0.280774 0.140387 0.990097i \(-0.455165\pi\)
0.140387 + 0.990097i \(0.455165\pi\)
\(488\) −8.92820 −0.404161
\(489\) 2.92820 0.132418
\(490\) 0 0
\(491\) 27.7128 1.25066 0.625331 0.780360i \(-0.284964\pi\)
0.625331 + 0.780360i \(0.284964\pi\)
\(492\) 6.00000 0.270501
\(493\) 4.39230 0.197819
\(494\) 17.8564 0.803398
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) −2.53590 −0.113751
\(498\) 3.21539 0.144085
\(499\) 39.8564 1.78422 0.892109 0.451820i \(-0.149225\pi\)
0.892109 + 0.451820i \(0.149225\pi\)
\(500\) 0 0
\(501\) 10.1436 0.453182
\(502\) −12.0000 −0.535586
\(503\) 32.7846 1.46179 0.730897 0.682488i \(-0.239102\pi\)
0.730897 + 0.682488i \(0.239102\pi\)
\(504\) −2.46410 −0.109760
\(505\) 0 0
\(506\) 2.19615 0.0976309
\(507\) 12.3397 0.548027
\(508\) −14.9282 −0.662332
\(509\) 24.9282 1.10492 0.552462 0.833538i \(-0.313689\pi\)
0.552462 + 0.833538i \(0.313689\pi\)
\(510\) 0 0
\(511\) 6.39230 0.282779
\(512\) −1.00000 −0.0441942
\(513\) −13.0718 −0.577134
\(514\) −6.33975 −0.279634
\(515\) 0 0
\(516\) −1.46410 −0.0644535
\(517\) 6.92820 0.304702
\(518\) 2.73205 0.120039
\(519\) 9.46410 0.415428
\(520\) 0 0
\(521\) −10.3923 −0.455295 −0.227648 0.973744i \(-0.573103\pi\)
−0.227648 + 0.973744i \(0.573103\pi\)
\(522\) −3.12436 −0.136749
\(523\) −33.1769 −1.45073 −0.725363 0.688367i \(-0.758328\pi\)
−0.725363 + 0.688367i \(0.758328\pi\)
\(524\) −11.6603 −0.509381
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) −6.92820 −0.301797
\(528\) 0.732051 0.0318584
\(529\) −18.1769 −0.790301
\(530\) 0 0
\(531\) 17.0718 0.740853
\(532\) −3.26795 −0.141684
\(533\) −44.7846 −1.93984
\(534\) 2.53590 0.109739
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) −4.39230 −0.189542
\(538\) −7.60770 −0.327991
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −17.2679 −0.742407 −0.371204 0.928552i \(-0.621055\pi\)
−0.371204 + 0.928552i \(0.621055\pi\)
\(542\) 20.3923 0.875924
\(543\) 10.2487 0.439814
\(544\) 3.46410 0.148522
\(545\) 0 0
\(546\) −4.00000 −0.171184
\(547\) 12.7846 0.546630 0.273315 0.961925i \(-0.411880\pi\)
0.273315 + 0.961925i \(0.411880\pi\)
\(548\) −12.9282 −0.552265
\(549\) −22.0000 −0.938937
\(550\) 0 0
\(551\) −4.14359 −0.176523
\(552\) 1.60770 0.0684280
\(553\) 1.80385 0.0767074
\(554\) 20.9282 0.889154
\(555\) 0 0
\(556\) −3.66025 −0.155229
\(557\) 46.3923 1.96571 0.982853 0.184393i \(-0.0590320\pi\)
0.982853 + 0.184393i \(0.0590320\pi\)
\(558\) 4.92820 0.208627
\(559\) 10.9282 0.462214
\(560\) 0 0
\(561\) −2.53590 −0.107066
\(562\) −1.60770 −0.0678165
\(563\) −18.9282 −0.797729 −0.398864 0.917010i \(-0.630596\pi\)
−0.398864 + 0.917010i \(0.630596\pi\)
\(564\) 5.07180 0.213561
\(565\) 0 0
\(566\) 23.7128 0.996724
\(567\) −4.46410 −0.187475
\(568\) −2.53590 −0.106404
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 24.3923 1.02079 0.510393 0.859941i \(-0.329500\pi\)
0.510393 + 0.859941i \(0.329500\pi\)
\(572\) −5.46410 −0.228466
\(573\) −8.78461 −0.366982
\(574\) 8.19615 0.342101
\(575\) 0 0
\(576\) −2.46410 −0.102671
\(577\) 3.41154 0.142024 0.0710122 0.997475i \(-0.477377\pi\)
0.0710122 + 0.997475i \(0.477377\pi\)
\(578\) 5.00000 0.207973
\(579\) 6.14359 0.255319
\(580\) 0 0
\(581\) 4.39230 0.182224
\(582\) −12.1436 −0.503368
\(583\) 10.7321 0.444476
\(584\) 6.39230 0.264515
\(585\) 0 0
\(586\) 14.5359 0.600472
\(587\) 42.5885 1.75781 0.878907 0.476993i \(-0.158273\pi\)
0.878907 + 0.476993i \(0.158273\pi\)
\(588\) 0.732051 0.0301893
\(589\) 6.53590 0.269307
\(590\) 0 0
\(591\) −17.7513 −0.730190
\(592\) 2.73205 0.112287
\(593\) 48.2487 1.98134 0.990669 0.136293i \(-0.0435189\pi\)
0.990669 + 0.136293i \(0.0435189\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −10.7321 −0.439602
\(597\) −8.00000 −0.327418
\(598\) −12.0000 −0.490716
\(599\) −25.1769 −1.02870 −0.514350 0.857580i \(-0.671967\pi\)
−0.514350 + 0.857580i \(0.671967\pi\)
\(600\) 0 0
\(601\) 39.5167 1.61192 0.805959 0.591971i \(-0.201650\pi\)
0.805959 + 0.591971i \(0.201650\pi\)
\(602\) −2.00000 −0.0815139
\(603\) −9.85641 −0.401384
\(604\) −13.1244 −0.534022
\(605\) 0 0
\(606\) −14.5359 −0.590481
\(607\) −7.07180 −0.287035 −0.143518 0.989648i \(-0.545841\pi\)
−0.143518 + 0.989648i \(0.545841\pi\)
\(608\) −3.26795 −0.132533
\(609\) 0.928203 0.0376127
\(610\) 0 0
\(611\) −37.8564 −1.53151
\(612\) 8.53590 0.345043
\(613\) −27.1769 −1.09767 −0.548833 0.835932i \(-0.684928\pi\)
−0.548833 + 0.835932i \(0.684928\pi\)
\(614\) 3.60770 0.145595
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) 17.3205 0.697297 0.348649 0.937253i \(-0.386641\pi\)
0.348649 + 0.937253i \(0.386641\pi\)
\(618\) −6.14359 −0.247132
\(619\) −12.7846 −0.513857 −0.256928 0.966430i \(-0.582710\pi\)
−0.256928 + 0.966430i \(0.582710\pi\)
\(620\) 0 0
\(621\) 8.78461 0.352514
\(622\) −11.0718 −0.443939
\(623\) 3.46410 0.138786
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) 11.8038 0.471777
\(627\) 2.39230 0.0955395
\(628\) −11.4641 −0.457467
\(629\) −9.46410 −0.377358
\(630\) 0 0
\(631\) −33.0718 −1.31657 −0.658284 0.752770i \(-0.728717\pi\)
−0.658284 + 0.752770i \(0.728717\pi\)
\(632\) 1.80385 0.0717532
\(633\) 9.56922 0.380342
\(634\) −0.588457 −0.0233706
\(635\) 0 0
\(636\) 7.85641 0.311527
\(637\) −5.46410 −0.216496
\(638\) 1.26795 0.0501986
\(639\) −6.24871 −0.247195
\(640\) 0 0
\(641\) 47.5692 1.87887 0.939436 0.342725i \(-0.111350\pi\)
0.939436 + 0.342725i \(0.111350\pi\)
\(642\) −14.5359 −0.573686
\(643\) −36.7321 −1.44857 −0.724285 0.689500i \(-0.757830\pi\)
−0.724285 + 0.689500i \(0.757830\pi\)
\(644\) 2.19615 0.0865405
\(645\) 0 0
\(646\) 11.3205 0.445399
\(647\) 33.4641 1.31561 0.657805 0.753188i \(-0.271485\pi\)
0.657805 + 0.753188i \(0.271485\pi\)
\(648\) −4.46410 −0.175366
\(649\) −6.92820 −0.271956
\(650\) 0 0
\(651\) −1.46410 −0.0573827
\(652\) 4.00000 0.156652
\(653\) 5.66025 0.221503 0.110751 0.993848i \(-0.464674\pi\)
0.110751 + 0.993848i \(0.464674\pi\)
\(654\) −1.21539 −0.0475255
\(655\) 0 0
\(656\) 8.19615 0.320006
\(657\) 15.7513 0.614516
\(658\) 6.92820 0.270089
\(659\) −18.2487 −0.710869 −0.355434 0.934701i \(-0.615667\pi\)
−0.355434 + 0.934701i \(0.615667\pi\)
\(660\) 0 0
\(661\) −46.0000 −1.78919 −0.894596 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) −22.7846 −0.885549
\(663\) 13.8564 0.538138
\(664\) 4.39230 0.170454
\(665\) 0 0
\(666\) 6.73205 0.260862
\(667\) 2.78461 0.107821
\(668\) 13.8564 0.536120
\(669\) 13.5692 0.524616
\(670\) 0 0
\(671\) 8.92820 0.344669
\(672\) 0.732051 0.0282395
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) −18.7846 −0.723556
\(675\) 0 0
\(676\) 16.8564 0.648323
\(677\) 31.8564 1.22434 0.612171 0.790726i \(-0.290297\pi\)
0.612171 + 0.790726i \(0.290297\pi\)
\(678\) −14.5359 −0.558248
\(679\) −16.5885 −0.636607
\(680\) 0 0
\(681\) 5.07180 0.194352
\(682\) −2.00000 −0.0765840
\(683\) 42.2487 1.61660 0.808301 0.588769i \(-0.200387\pi\)
0.808301 + 0.588769i \(0.200387\pi\)
\(684\) −8.05256 −0.307897
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 2.64102 0.100761
\(688\) −2.00000 −0.0762493
\(689\) −58.6410 −2.23404
\(690\) 0 0
\(691\) −6.53590 −0.248637 −0.124319 0.992242i \(-0.539675\pi\)
−0.124319 + 0.992242i \(0.539675\pi\)
\(692\) 12.9282 0.491457
\(693\) 2.46410 0.0936035
\(694\) −36.9282 −1.40178
\(695\) 0 0
\(696\) 0.928203 0.0351835
\(697\) −28.3923 −1.07544
\(698\) 26.3923 0.998963
\(699\) −14.5359 −0.549798
\(700\) 0 0
\(701\) −23.9090 −0.903029 −0.451515 0.892264i \(-0.649116\pi\)
−0.451515 + 0.892264i \(0.649116\pi\)
\(702\) −21.8564 −0.824917
\(703\) 8.92820 0.336734
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 3.80385 0.143160
\(707\) −19.8564 −0.746777
\(708\) −5.07180 −0.190610
\(709\) −9.32051 −0.350039 −0.175020 0.984565i \(-0.555999\pi\)
−0.175020 + 0.984565i \(0.555999\pi\)
\(710\) 0 0
\(711\) 4.44486 0.166695
\(712\) 3.46410 0.129823
\(713\) −4.39230 −0.164493
\(714\) −2.53590 −0.0949036
\(715\) 0 0
\(716\) −6.00000 −0.224231
\(717\) 3.46410 0.129369
\(718\) −4.05256 −0.151240
\(719\) 27.7128 1.03351 0.516757 0.856132i \(-0.327139\pi\)
0.516757 + 0.856132i \(0.327139\pi\)
\(720\) 0 0
\(721\) −8.39230 −0.312546
\(722\) 8.32051 0.309657
\(723\) 4.92820 0.183282
\(724\) 14.0000 0.520306
\(725\) 0 0
\(726\) −0.732051 −0.0271690
\(727\) 4.00000 0.148352 0.0741759 0.997245i \(-0.476367\pi\)
0.0741759 + 0.997245i \(0.476367\pi\)
\(728\) −5.46410 −0.202513
\(729\) −2.21539 −0.0820515
\(730\) 0 0
\(731\) 6.92820 0.256249
\(732\) 6.53590 0.241574
\(733\) −26.0000 −0.960332 −0.480166 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) −8.39230 −0.309766
\(735\) 0 0
\(736\) 2.19615 0.0809513
\(737\) 4.00000 0.147342
\(738\) 20.1962 0.743431
\(739\) −43.7128 −1.60800 −0.804001 0.594628i \(-0.797299\pi\)
−0.804001 + 0.594628i \(0.797299\pi\)
\(740\) 0 0
\(741\) −13.0718 −0.480204
\(742\) 10.7321 0.393986
\(743\) −8.78461 −0.322276 −0.161138 0.986932i \(-0.551516\pi\)
−0.161138 + 0.986932i \(0.551516\pi\)
\(744\) −1.46410 −0.0536766
\(745\) 0 0
\(746\) −2.39230 −0.0875885
\(747\) 10.8231 0.395996
\(748\) −3.46410 −0.126660
\(749\) −19.8564 −0.725537
\(750\) 0 0
\(751\) −32.3923 −1.18201 −0.591006 0.806667i \(-0.701269\pi\)
−0.591006 + 0.806667i \(0.701269\pi\)
\(752\) 6.92820 0.252646
\(753\) 8.78461 0.320129
\(754\) −6.92820 −0.252310
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) −29.3731 −1.06758 −0.533791 0.845616i \(-0.679233\pi\)
−0.533791 + 0.845616i \(0.679233\pi\)
\(758\) −33.8564 −1.22972
\(759\) −1.60770 −0.0583556
\(760\) 0 0
\(761\) −29.6603 −1.07518 −0.537592 0.843205i \(-0.680666\pi\)
−0.537592 + 0.843205i \(0.680666\pi\)
\(762\) 10.9282 0.395887
\(763\) −1.66025 −0.0601052
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) −26.5359 −0.958781
\(767\) 37.8564 1.36692
\(768\) 0.732051 0.0264156
\(769\) −7.80385 −0.281414 −0.140707 0.990051i \(-0.544938\pi\)
−0.140707 + 0.990051i \(0.544938\pi\)
\(770\) 0 0
\(771\) 4.64102 0.167142
\(772\) 8.39230 0.302046
\(773\) −12.9282 −0.464995 −0.232498 0.972597i \(-0.574690\pi\)
−0.232498 + 0.972597i \(0.574690\pi\)
\(774\) −4.92820 −0.177141
\(775\) 0 0
\(776\) −16.5885 −0.595491
\(777\) −2.00000 −0.0717496
\(778\) 22.3923 0.802803
\(779\) 26.7846 0.959658
\(780\) 0 0
\(781\) 2.53590 0.0907416
\(782\) −7.60770 −0.272051
\(783\) 5.07180 0.181251
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 8.53590 0.304465
\(787\) −45.8564 −1.63460 −0.817302 0.576209i \(-0.804531\pi\)
−0.817302 + 0.576209i \(0.804531\pi\)
\(788\) −24.2487 −0.863825
\(789\) 17.5692 0.625481
\(790\) 0 0
\(791\) −19.8564 −0.706013
\(792\) 2.46410 0.0875580
\(793\) −48.7846 −1.73239
\(794\) 9.60770 0.340964
\(795\) 0 0
\(796\) −10.9282 −0.387340
\(797\) 46.3923 1.64330 0.821650 0.569993i \(-0.193054\pi\)
0.821650 + 0.569993i \(0.193054\pi\)
\(798\) 2.39230 0.0846867
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 8.53590 0.301601
\(802\) −9.46410 −0.334189
\(803\) −6.39230 −0.225580
\(804\) 2.92820 0.103270
\(805\) 0 0
\(806\) 10.9282 0.384930
\(807\) 5.56922 0.196046
\(808\) −19.8564 −0.698546
\(809\) 31.1769 1.09612 0.548061 0.836438i \(-0.315366\pi\)
0.548061 + 0.836438i \(0.315366\pi\)
\(810\) 0 0
\(811\) −18.8756 −0.662814 −0.331407 0.943488i \(-0.607523\pi\)
−0.331407 + 0.943488i \(0.607523\pi\)
\(812\) 1.26795 0.0444963
\(813\) −14.9282 −0.523555
\(814\) −2.73205 −0.0957583
\(815\) 0 0
\(816\) −2.53590 −0.0887742
\(817\) −6.53590 −0.228662
\(818\) −35.1244 −1.22809
\(819\) −13.4641 −0.470474
\(820\) 0 0
\(821\) −47.9090 −1.67203 −0.836017 0.548703i \(-0.815122\pi\)
−0.836017 + 0.548703i \(0.815122\pi\)
\(822\) 9.46410 0.330098
\(823\) 25.1244 0.875780 0.437890 0.899029i \(-0.355726\pi\)
0.437890 + 0.899029i \(0.355726\pi\)
\(824\) −8.39230 −0.292360
\(825\) 0 0
\(826\) −6.92820 −0.241063
\(827\) −1.85641 −0.0645536 −0.0322768 0.999479i \(-0.510276\pi\)
−0.0322768 + 0.999479i \(0.510276\pi\)
\(828\) 5.41154 0.188064
\(829\) −46.2487 −1.60628 −0.803142 0.595788i \(-0.796840\pi\)
−0.803142 + 0.595788i \(0.796840\pi\)
\(830\) 0 0
\(831\) −15.3205 −0.531463
\(832\) −5.46410 −0.189434
\(833\) −3.46410 −0.120024
\(834\) 2.67949 0.0927832
\(835\) 0 0
\(836\) 3.26795 0.113024
\(837\) −8.00000 −0.276520
\(838\) −17.0718 −0.589735
\(839\) −4.14359 −0.143053 −0.0715264 0.997439i \(-0.522787\pi\)
−0.0715264 + 0.997439i \(0.522787\pi\)
\(840\) 0 0
\(841\) −27.3923 −0.944562
\(842\) 8.14359 0.280647
\(843\) 1.17691 0.0405351
\(844\) 13.0718 0.449950
\(845\) 0 0
\(846\) 17.0718 0.586940
\(847\) −1.00000 −0.0343604
\(848\) 10.7321 0.368540
\(849\) −17.3590 −0.595759
\(850\) 0 0
\(851\) −6.00000 −0.205677
\(852\) 1.85641 0.0635994
\(853\) 13.2154 0.452486 0.226243 0.974071i \(-0.427356\pi\)
0.226243 + 0.974071i \(0.427356\pi\)
\(854\) 8.92820 0.305517
\(855\) 0 0
\(856\) −19.8564 −0.678678
\(857\) 20.5359 0.701493 0.350746 0.936470i \(-0.385928\pi\)
0.350746 + 0.936470i \(0.385928\pi\)
\(858\) 4.00000 0.136558
\(859\) 28.7846 0.982118 0.491059 0.871126i \(-0.336610\pi\)
0.491059 + 0.871126i \(0.336610\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 21.1244 0.719498
\(863\) −37.5167 −1.27708 −0.638541 0.769588i \(-0.720462\pi\)
−0.638541 + 0.769588i \(0.720462\pi\)
\(864\) 4.00000 0.136083
\(865\) 0 0
\(866\) −7.80385 −0.265186
\(867\) −3.66025 −0.124309
\(868\) −2.00000 −0.0678844
\(869\) −1.80385 −0.0611913
\(870\) 0 0
\(871\) −21.8564 −0.740576
\(872\) −1.66025 −0.0562233
\(873\) −40.8756 −1.38343
\(874\) 7.17691 0.242763
\(875\) 0 0
\(876\) −4.67949 −0.158105
\(877\) −13.3205 −0.449802 −0.224901 0.974382i \(-0.572206\pi\)
−0.224901 + 0.974382i \(0.572206\pi\)
\(878\) 22.2487 0.750858
\(879\) −10.6410 −0.358913
\(880\) 0 0
\(881\) 24.9282 0.839853 0.419926 0.907558i \(-0.362056\pi\)
0.419926 + 0.907558i \(0.362056\pi\)
\(882\) 2.46410 0.0829706
\(883\) 56.3923 1.89775 0.948876 0.315649i \(-0.102222\pi\)
0.948876 + 0.315649i \(0.102222\pi\)
\(884\) 18.9282 0.636624
\(885\) 0 0
\(886\) 0 0
\(887\) 13.8564 0.465253 0.232626 0.972566i \(-0.425268\pi\)
0.232626 + 0.972566i \(0.425268\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 14.9282 0.500676
\(890\) 0 0
\(891\) 4.46410 0.149553
\(892\) 18.5359 0.620628
\(893\) 22.6410 0.757653
\(894\) 7.85641 0.262758
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 8.78461 0.293310
\(898\) −19.6077 −0.654317
\(899\) −2.53590 −0.0845769
\(900\) 0 0
\(901\) −37.1769 −1.23854
\(902\) −8.19615 −0.272902
\(903\) 1.46410 0.0487223
\(904\) −19.8564 −0.660414
\(905\) 0 0
\(906\) 9.60770 0.319194
\(907\) −59.0333 −1.96017 −0.980085 0.198580i \(-0.936367\pi\)
−0.980085 + 0.198580i \(0.936367\pi\)
\(908\) 6.92820 0.229920
\(909\) −48.9282 −1.62285
\(910\) 0 0
\(911\) 2.53590 0.0840181 0.0420090 0.999117i \(-0.486624\pi\)
0.0420090 + 0.999117i \(0.486624\pi\)
\(912\) 2.39230 0.0792171
\(913\) −4.39230 −0.145364
\(914\) 2.00000 0.0661541
\(915\) 0 0
\(916\) 3.60770 0.119202
\(917\) 11.6603 0.385056
\(918\) −13.8564 −0.457330
\(919\) 47.3731 1.56269 0.781347 0.624097i \(-0.214533\pi\)
0.781347 + 0.624097i \(0.214533\pi\)
\(920\) 0 0
\(921\) −2.64102 −0.0870244
\(922\) −1.60770 −0.0529466
\(923\) −13.8564 −0.456089
\(924\) −0.732051 −0.0240827
\(925\) 0 0
\(926\) −17.5167 −0.575633
\(927\) −20.6795 −0.679204
\(928\) 1.26795 0.0416225
\(929\) −46.3923 −1.52208 −0.761041 0.648704i \(-0.775311\pi\)
−0.761041 + 0.648704i \(0.775311\pi\)
\(930\) 0 0
\(931\) 3.26795 0.107103
\(932\) −19.8564 −0.650418
\(933\) 8.10512 0.265350
\(934\) 4.05256 0.132604
\(935\) 0 0
\(936\) −13.4641 −0.440088
\(937\) 42.7846 1.39771 0.698856 0.715262i \(-0.253693\pi\)
0.698856 + 0.715262i \(0.253693\pi\)
\(938\) 4.00000 0.130605
\(939\) −8.64102 −0.281989
\(940\) 0 0
\(941\) 26.7846 0.873153 0.436577 0.899667i \(-0.356191\pi\)
0.436577 + 0.899667i \(0.356191\pi\)
\(942\) 8.39230 0.273436
\(943\) −18.0000 −0.586161
\(944\) −6.92820 −0.225494
\(945\) 0 0
\(946\) 2.00000 0.0650256
\(947\) −12.6795 −0.412028 −0.206014 0.978549i \(-0.566049\pi\)
−0.206014 + 0.978549i \(0.566049\pi\)
\(948\) −1.32051 −0.0428881
\(949\) 34.9282 1.13382
\(950\) 0 0
\(951\) 0.430781 0.0139690
\(952\) −3.46410 −0.112272
\(953\) −33.4641 −1.08401 −0.542004 0.840376i \(-0.682334\pi\)
−0.542004 + 0.840376i \(0.682334\pi\)
\(954\) 26.4449 0.856184
\(955\) 0 0
\(956\) 4.73205 0.153045
\(957\) −0.928203 −0.0300045
\(958\) −8.78461 −0.283818
\(959\) 12.9282 0.417473
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 14.9282 0.481305
\(963\) −48.9282 −1.57669
\(964\) 6.73205 0.216825
\(965\) 0 0
\(966\) −1.60770 −0.0517267
\(967\) −13.0718 −0.420361 −0.210180 0.977663i \(-0.567405\pi\)
−0.210180 + 0.977663i \(0.567405\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −8.28719 −0.266223
\(970\) 0 0
\(971\) 29.0718 0.932958 0.466479 0.884532i \(-0.345522\pi\)
0.466479 + 0.884532i \(0.345522\pi\)
\(972\) 15.2679 0.489720
\(973\) 3.66025 0.117342
\(974\) −6.19615 −0.198538
\(975\) 0 0
\(976\) 8.92820 0.285785
\(977\) −21.7128 −0.694654 −0.347327 0.937744i \(-0.612911\pi\)
−0.347327 + 0.937744i \(0.612911\pi\)
\(978\) −2.92820 −0.0936336
\(979\) −3.46410 −0.110713
\(980\) 0 0
\(981\) −4.09103 −0.130617
\(982\) −27.7128 −0.884351
\(983\) −25.1769 −0.803019 −0.401509 0.915855i \(-0.631514\pi\)
−0.401509 + 0.915855i \(0.631514\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) −4.39230 −0.139879
\(987\) −5.07180 −0.161437
\(988\) −17.8564 −0.568088
\(989\) 4.39230 0.139667
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 16.6795 0.529308
\(994\) 2.53590 0.0804338
\(995\) 0 0
\(996\) −3.21539 −0.101884
\(997\) 37.7128 1.19438 0.597188 0.802101i \(-0.296284\pi\)
0.597188 + 0.802101i \(0.296284\pi\)
\(998\) −39.8564 −1.26163
\(999\) −10.9282 −0.345753
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 3850.2.a.bd.1.2 2
5.2 odd 4 3850.2.c.x.1849.1 4
5.3 odd 4 3850.2.c.x.1849.4 4
5.4 even 2 770.2.a.j.1.1 2
15.14 odd 2 6930.2.a.bv.1.2 2
20.19 odd 2 6160.2.a.t.1.2 2
35.34 odd 2 5390.2.a.bs.1.2 2
55.54 odd 2 8470.2.a.br.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.j.1.1 2 5.4 even 2
3850.2.a.bd.1.2 2 1.1 even 1 trivial
3850.2.c.x.1849.1 4 5.2 odd 4
3850.2.c.x.1849.4 4 5.3 odd 4
5390.2.a.bs.1.2 2 35.34 odd 2
6160.2.a.t.1.2 2 20.19 odd 2
6930.2.a.bv.1.2 2 15.14 odd 2
8470.2.a.br.1.1 2 55.54 odd 2