Properties

Label 8470.2.a.br.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
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 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.732051 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.732051 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.732051 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.732051 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} +1.00000 q^{10} -0.732051 q^{12} -5.46410 q^{13} +1.00000 q^{14} +0.732051 q^{15} +1.00000 q^{16} -3.46410 q^{17} +2.46410 q^{18} -3.26795 q^{19} -1.00000 q^{20} +0.732051 q^{21} +2.19615 q^{23} +0.732051 q^{24} +1.00000 q^{25} +5.46410 q^{26} +4.00000 q^{27} -1.00000 q^{28} +1.26795 q^{29} -0.732051 q^{30} +2.00000 q^{31} -1.00000 q^{32} +3.46410 q^{34} +1.00000 q^{35} -2.46410 q^{36} -2.73205 q^{37} +3.26795 q^{38} +4.00000 q^{39} +1.00000 q^{40} -8.19615 q^{41} -0.732051 q^{42} -2.00000 q^{43} +2.46410 q^{45} -2.19615 q^{46} -6.92820 q^{47} -0.732051 q^{48} +1.00000 q^{49} -1.00000 q^{50} +2.53590 q^{51} -5.46410 q^{52} -10.7321 q^{53} -4.00000 q^{54} +1.00000 q^{56} +2.39230 q^{57} -1.26795 q^{58} -6.92820 q^{59} +0.732051 q^{60} -8.92820 q^{61} -2.00000 q^{62} +2.46410 q^{63} +1.00000 q^{64} +5.46410 q^{65} -4.00000 q^{67} -3.46410 q^{68} -1.60770 q^{69} -1.00000 q^{70} +2.53590 q^{71} +2.46410 q^{72} -6.39230 q^{73} +2.73205 q^{74} -0.732051 q^{75} -3.26795 q^{76} -4.00000 q^{78} +1.80385 q^{79} -1.00000 q^{80} +4.46410 q^{81} +8.19615 q^{82} -4.39230 q^{83} +0.732051 q^{84} +3.46410 q^{85} +2.00000 q^{86} -0.928203 q^{87} -3.46410 q^{89} -2.46410 q^{90} +5.46410 q^{91} +2.19615 q^{92} -1.46410 q^{93} +6.92820 q^{94} +3.26795 q^{95} +0.732051 q^{96} -16.5885 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + 2q^{10} + 2q^{12} - 4q^{13} + 2q^{14} - 2q^{15} + 2q^{16} - 2q^{18} - 10q^{19} - 2q^{20} - 2q^{21} - 6q^{23} - 2q^{24} + 2q^{25} + 4q^{26} + 8q^{27} - 2q^{28} + 6q^{29} + 2q^{30} + 4q^{31} - 2q^{32} + 2q^{35} + 2q^{36} - 2q^{37} + 10q^{38} + 8q^{39} + 2q^{40} - 6q^{41} + 2q^{42} - 4q^{43} - 2q^{45} + 6q^{46} + 2q^{48} + 2q^{49} - 2q^{50} + 12q^{51} - 4q^{52} - 18q^{53} - 8q^{54} + 2q^{56} - 16q^{57} - 6q^{58} - 2q^{60} - 4q^{61} - 4q^{62} - 2q^{63} + 2q^{64} + 4q^{65} - 8q^{67} - 24q^{69} - 2q^{70} + 12q^{71} - 2q^{72} + 8q^{73} + 2q^{74} + 2q^{75} - 10q^{76} - 8q^{78} + 14q^{79} - 2q^{80} + 2q^{81} + 6q^{82} + 12q^{83} - 2q^{84} + 4q^{86} + 12q^{87} + 2q^{90} + 4q^{91} - 6q^{92} + 4q^{93} + 10q^{95} - 2q^{96} - 2q^{97} - 2q^{98} + O(q^{100}) \)

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
\(3\) −0.732051 −0.422650 −0.211325 0.977416i \(-0.567778\pi\)
−0.211325 + 0.977416i \(0.567778\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0.732051 0.298858
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) −2.46410 −0.821367
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) −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
\(15\) 0.732051 0.189015
\(16\) 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 \(-0.622309\pi\)
−0.374859 + 0.927082i \(0.622309\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0.732051 0.159747
\(22\) 0 0
\(23\) 2.19615 0.457929 0.228965 0.973435i \(-0.426466\pi\)
0.228965 + 0.973435i \(0.426466\pi\)
\(24\) 0.732051 0.149429
\(25\) 1.00000 0.200000
\(26\) 5.46410 1.07160
\(27\) 4.00000 0.769800
\(28\) −1.00000 −0.188982
\(29\) 1.26795 0.235452 0.117726 0.993046i \(-0.462440\pi\)
0.117726 + 0.993046i \(0.462440\pi\)
\(30\) −0.732051 −0.133654
\(31\) 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 0
\(34\) 3.46410 0.594089
\(35\) 1.00000 0.169031
\(36\) −2.46410 −0.410684
\(37\) −2.73205 −0.449146 −0.224573 0.974457i \(-0.572099\pi\)
−0.224573 + 0.974457i \(0.572099\pi\)
\(38\) 3.26795 0.530131
\(39\) 4.00000 0.640513
\(40\) 1.00000 0.158114
\(41\) −8.19615 −1.28002 −0.640012 0.768365i \(-0.721071\pi\)
−0.640012 + 0.768365i \(0.721071\pi\)
\(42\) −0.732051 −0.112958
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 2.46410 0.367327
\(46\) −2.19615 −0.323805
\(47\) −6.92820 −1.01058 −0.505291 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) −0.732051 −0.105662
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 2.53590 0.355097
\(52\) −5.46410 −0.757735
\(53\) −10.7321 −1.47416 −0.737080 0.675805i \(-0.763796\pi\)
−0.737080 + 0.675805i \(0.763796\pi\)
\(54\) −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 \(0.648928\pi\)
\(60\) 0.732051 0.0945074
\(61\) −8.92820 −1.14314 −0.571570 0.820554i \(-0.693665\pi\)
−0.571570 + 0.820554i \(0.693665\pi\)
\(62\) −2.00000 −0.254000
\(63\) 2.46410 0.310448
\(64\) 1.00000 0.125000
\(65\) 5.46410 0.677738
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −3.46410 −0.420084
\(69\) −1.60770 −0.193544
\(70\) −1.00000 −0.119523
\(71\) 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
\(75\) −0.732051 −0.0845299
\(76\) −3.26795 −0.374859
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) 1.80385 0.202949 0.101474 0.994838i \(-0.467644\pi\)
0.101474 + 0.994838i \(0.467644\pi\)
\(80\) −1.00000 −0.111803
\(81\) 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
\(85\) 3.46410 0.375735
\(86\) 2.00000 0.215666
\(87\) −0.928203 −0.0995138
\(88\) 0 0
\(89\) −3.46410 −0.367194 −0.183597 0.983002i \(-0.558774\pi\)
−0.183597 + 0.983002i \(0.558774\pi\)
\(90\) −2.46410 −0.259739
\(91\) 5.46410 0.572793
\(92\) 2.19615 0.228965
\(93\) −1.46410 −0.151820
\(94\) 6.92820 0.714590
\(95\) 3.26795 0.335285
\(96\) 0.732051 0.0747146
\(97\) −16.5885 −1.68430 −0.842151 0.539241i \(-0.818711\pi\)
−0.842151 + 0.539241i \(0.818711\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −19.8564 −1.97579 −0.987893 0.155136i \(-0.950418\pi\)
−0.987893 + 0.155136i \(0.950418\pi\)
\(102\) −2.53590 −0.251091
\(103\) −8.39230 −0.826918 −0.413459 0.910523i \(-0.635680\pi\)
−0.413459 + 0.910523i \(0.635680\pi\)
\(104\) 5.46410 0.535799
\(105\) −0.732051 −0.0714408
\(106\) 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 \(-0.525336\pi\)
−0.0795117 + 0.996834i \(0.525336\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) −1.00000 −0.0944911
\(113\) −19.8564 −1.86793 −0.933967 0.357360i \(-0.883677\pi\)
−0.933967 + 0.357360i \(0.883677\pi\)
\(114\) −2.39230 −0.224060
\(115\) −2.19615 −0.204792
\(116\) 1.26795 0.117726
\(117\) 13.4641 1.24476
\(118\) 6.92820 0.637793
\(119\) 3.46410 0.317554
\(120\) −0.732051 −0.0668268
\(121\) 0 0
\(122\) 8.92820 0.808322
\(123\) 6.00000 0.541002
\(124\) 2.00000 0.179605
\(125\) −1.00000 −0.0894427
\(126\) −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
\(130\) −5.46410 −0.479233
\(131\) 11.6603 1.01876 0.509381 0.860541i \(-0.329875\pi\)
0.509381 + 0.860541i \(0.329875\pi\)
\(132\) 0 0
\(133\) 3.26795 0.283367
\(134\) 4.00000 0.345547
\(135\) −4.00000 −0.344265
\(136\) 3.46410 0.297044
\(137\) 12.9282 1.10453 0.552265 0.833668i \(-0.313763\pi\)
0.552265 + 0.833668i \(0.313763\pi\)
\(138\) 1.60770 0.136856
\(139\) 3.66025 0.310459 0.155229 0.987878i \(-0.450388\pi\)
0.155229 + 0.987878i \(0.450388\pi\)
\(140\) 1.00000 0.0845154
\(141\) 5.07180 0.427122
\(142\) −2.53590 −0.212808
\(143\) 0 0
\(144\) −2.46410 −0.205342
\(145\) −1.26795 −0.105297
\(146\) 6.39230 0.529031
\(147\) −0.732051 −0.0603785
\(148\) −2.73205 −0.224573
\(149\) 10.7321 0.879204 0.439602 0.898193i \(-0.355120\pi\)
0.439602 + 0.898193i \(0.355120\pi\)
\(150\) 0.732051 0.0597717
\(151\) 13.1244 1.06804 0.534022 0.845470i \(-0.320680\pi\)
0.534022 + 0.845470i \(0.320680\pi\)
\(152\) 3.26795 0.265066
\(153\) 8.53590 0.690086
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) 4.00000 0.320256
\(157\) 11.4641 0.914935 0.457467 0.889226i \(-0.348757\pi\)
0.457467 + 0.889226i \(0.348757\pi\)
\(158\) −1.80385 −0.143506
\(159\) 7.85641 0.623054
\(160\) 1.00000 0.0790569
\(161\) −2.19615 −0.173081
\(162\) −4.46410 −0.350733
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −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
\(170\) −3.46410 −0.265684
\(171\) 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
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 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 \(0.571987\pi\)
\(180\) 2.46410 0.183663
\(181\) 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
\(185\) 2.73205 0.200864
\(186\) 1.46410 0.107353
\(187\) 0 0
\(188\) −6.92820 −0.505291
\(189\) −4.00000 −0.290957
\(190\) −3.26795 −0.237082
\(191\) −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
\(195\) −4.00000 −0.286446
\(196\) 1.00000 0.0714286
\(197\) −24.2487 −1.72765 −0.863825 0.503793i \(-0.831938\pi\)
−0.863825 + 0.503793i \(0.831938\pi\)
\(198\) 0 0
\(199\) −10.9282 −0.774680 −0.387340 0.921937i \(-0.626606\pi\)
−0.387340 + 0.921937i \(0.626606\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 2.92820 0.206540
\(202\) 19.8564 1.39709
\(203\) −1.26795 −0.0889926
\(204\) 2.53590 0.177548
\(205\) 8.19615 0.572444
\(206\) 8.39230 0.584720
\(207\) −5.41154 −0.376128
\(208\) −5.46410 −0.378867
\(209\) 0 0
\(210\) 0.732051 0.0505163
\(211\) −13.0718 −0.899900 −0.449950 0.893054i \(-0.648558\pi\)
−0.449950 + 0.893054i \(0.648558\pi\)
\(212\) −10.7321 −0.737080
\(213\) −1.85641 −0.127199
\(214\) −19.8564 −1.35736
\(215\) 2.00000 0.136399
\(216\) −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 \(-0.713122\pi\)
−0.620628 + 0.784105i \(0.713122\pi\)
\(224\) 1.00000 0.0668153
\(225\) −2.46410 −0.164273
\(226\) 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 \(0.461967\pi\)
\(230\) 2.19615 0.144810
\(231\) 0 0
\(232\) −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
\(235\) 6.92820 0.451946
\(236\) −6.92820 −0.450988
\(237\) −1.32051 −0.0857762
\(238\) −3.46410 −0.224544
\(239\) −4.73205 −0.306091 −0.153045 0.988219i \(-0.548908\pi\)
−0.153045 + 0.988219i \(0.548908\pi\)
\(240\) 0.732051 0.0472537
\(241\) −6.73205 −0.433650 −0.216825 0.976211i \(-0.569570\pi\)
−0.216825 + 0.976211i \(0.569570\pi\)
\(242\) 0 0
\(243\) −15.2679 −0.979439
\(244\) −8.92820 −0.571570
\(245\) −1.00000 −0.0638877
\(246\) −6.00000 −0.382546
\(247\) 17.8564 1.13618
\(248\) −2.00000 −0.127000
\(249\) 3.21539 0.203767
\(250\) 1.00000 0.0632456
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 2.46410 0.155224
\(253\) 0 0
\(254\) 14.9282 0.936679
\(255\) −2.53590 −0.158804
\(256\) 1.00000 0.0625000
\(257\) −6.33975 −0.395462 −0.197731 0.980256i \(-0.563357\pi\)
−0.197731 + 0.980256i \(0.563357\pi\)
\(258\) −1.46410 −0.0911510
\(259\) 2.73205 0.169761
\(260\) 5.46410 0.338869
\(261\) −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 0
\(265\) 10.7321 0.659265
\(266\) −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 \(0.425498\pi\)
\(270\) 4.00000 0.243432
\(271\) 20.3923 1.23874 0.619372 0.785098i \(-0.287387\pi\)
0.619372 + 0.785098i \(0.287387\pi\)
\(272\) −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
\(280\) −1.00000 −0.0597614
\(281\) −1.60770 −0.0959071 −0.0479535 0.998850i \(-0.515270\pi\)
−0.0479535 + 0.998850i \(0.515270\pi\)
\(282\) −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
\(285\) −2.39230 −0.141708
\(286\) 0 0
\(287\) 8.19615 0.483804
\(288\) 2.46410 0.145199
\(289\) −5.00000 −0.294118
\(290\) 1.26795 0.0744565
\(291\) 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
\(295\) 6.92820 0.403376
\(296\) 2.73205 0.158797
\(297\) 0 0
\(298\) −10.7321 −0.621691
\(299\) −12.0000 −0.693978
\(300\) −0.732051 −0.0422650
\(301\) 2.00000 0.115278
\(302\) −13.1244 −0.755222
\(303\) 14.5359 0.835066
\(304\) −3.26795 −0.187430
\(305\) 8.92820 0.511227
\(306\) −8.53590 −0.487965
\(307\) −3.60770 −0.205902 −0.102951 0.994686i \(-0.532828\pi\)
−0.102951 + 0.994686i \(0.532828\pi\)
\(308\) 0 0
\(309\) 6.14359 0.349497
\(310\) 2.00000 0.113592
\(311\) 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 \(-0.391738\pi\)
0.333596 + 0.942716i \(0.391738\pi\)
\(314\) −11.4641 −0.646957
\(315\) −2.46410 −0.138836
\(316\) 1.80385 0.101474
\(317\) −0.588457 −0.0330511 −0.0165255 0.999863i \(-0.505260\pi\)
−0.0165255 + 0.999863i \(0.505260\pi\)
\(318\) −7.85641 −0.440565
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −14.5359 −0.811315
\(322\) 2.19615 0.122387
\(323\) 11.3205 0.629890
\(324\) 4.46410 0.248006
\(325\) −5.46410 −0.303094
\(326\) 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
\(335\) 4.00000 0.218543
\(336\) 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
\(340\) 3.46410 0.187867
\(341\) 0 0
\(342\) −8.05256 −0.435433
\(343\) −1.00000 −0.0539949
\(344\) 2.00000 0.107833
\(345\) 1.60770 0.0865554
\(346\) −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 \(-0.250330\pi\)
0.706374 + 0.707839i \(0.250330\pi\)
\(350\) 1.00000 0.0534522
\(351\) −21.8564 −1.16661
\(352\) 0 0
\(353\) 3.80385 0.202458 0.101229 0.994863i \(-0.467722\pi\)
0.101229 + 0.994863i \(0.467722\pi\)
\(354\) −5.07180 −0.269563
\(355\) −2.53590 −0.134592
\(356\) −3.46410 −0.183597
\(357\) −2.53590 −0.134214
\(358\) 6.00000 0.317110
\(359\) −4.05256 −0.213886 −0.106943 0.994265i \(-0.534106\pi\)
−0.106943 + 0.994265i \(0.534106\pi\)
\(360\) −2.46410 −0.129870
\(361\) −8.32051 −0.437921
\(362\) −14.0000 −0.735824
\(363\) 0 0
\(364\) 5.46410 0.286397
\(365\) 6.39230 0.334589
\(366\) −6.53590 −0.341637
\(367\) −8.39230 −0.438075 −0.219037 0.975716i \(-0.570292\pi\)
−0.219037 + 0.975716i \(0.570292\pi\)
\(368\) 2.19615 0.114482
\(369\) 20.1962 1.05137
\(370\) −2.73205 −0.142033
\(371\) 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 \(0.480273\pi\)
\(374\) 0 0
\(375\) 0.732051 0.0378029
\(376\) 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 \(0.164413\pi\)
\(380\) 3.26795 0.167642
\(381\) 10.9282 0.559869
\(382\) 12.0000 0.613973
\(383\) −26.5359 −1.35592 −0.677961 0.735098i \(-0.737136\pi\)
−0.677961 + 0.735098i \(0.737136\pi\)
\(384\) 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 \(0.692154\pi\)
\(390\) 4.00000 0.202548
\(391\) −7.60770 −0.384738
\(392\) −1.00000 −0.0505076
\(393\) −8.53590 −0.430579
\(394\) 24.2487 1.22163
\(395\) −1.80385 −0.0907614
\(396\) 0 0
\(397\) 9.60770 0.482196 0.241098 0.970501i \(-0.422492\pi\)
0.241098 + 0.970501i \(0.422492\pi\)
\(398\) 10.9282 0.547781
\(399\) −2.39230 −0.119765
\(400\) 1.00000 0.0500000
\(401\) 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
\(405\) −4.46410 −0.221823
\(406\) 1.26795 0.0629273
\(407\) 0 0
\(408\) −2.53590 −0.125546
\(409\) −35.1244 −1.73679 −0.868394 0.495875i \(-0.834847\pi\)
−0.868394 + 0.495875i \(0.834847\pi\)
\(410\) −8.19615 −0.404779
\(411\) −9.46410 −0.466830
\(412\) −8.39230 −0.413459
\(413\) 6.92820 0.340915
\(414\) 5.41154 0.265963
\(415\) 4.39230 0.215610
\(416\) 5.46410 0.267900
\(417\) −2.67949 −0.131215
\(418\) 0 0
\(419\) 17.0718 0.834012 0.417006 0.908904i \(-0.363079\pi\)
0.417006 + 0.908904i \(0.363079\pi\)
\(420\) −0.732051 −0.0357204
\(421\) −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
\(425\) −3.46410 −0.168034
\(426\) 1.85641 0.0899432
\(427\) 8.92820 0.432066
\(428\) 19.8564 0.959796
\(429\) 0 0
\(430\) −2.00000 −0.0964486
\(431\) 21.1244 1.01752 0.508762 0.860907i \(-0.330103\pi\)
0.508762 + 0.860907i \(0.330103\pi\)
\(432\) 4.00000 0.192450
\(433\) −7.80385 −0.375029 −0.187514 0.982262i \(-0.560043\pi\)
−0.187514 + 0.982262i \(0.560043\pi\)
\(434\) 2.00000 0.0960031
\(435\) 0.928203 0.0445039
\(436\) −1.66025 −0.0795117
\(437\) −7.17691 −0.343318
\(438\) −4.67949 −0.223595
\(439\) 22.2487 1.06187 0.530937 0.847412i \(-0.321840\pi\)
0.530937 + 0.847412i \(0.321840\pi\)
\(440\) 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
\(445\) 3.46410 0.164214
\(446\) 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 \(0.346891\pi\)
\(450\) 2.46410 0.116159
\(451\) 0 0
\(452\) −19.8564 −0.933967
\(453\) −9.60770 −0.451409
\(454\) −6.92820 −0.325157
\(455\) −5.46410 −0.256161
\(456\) −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
\(460\) −2.19615 −0.102396
\(461\) −1.60770 −0.0748778 −0.0374389 0.999299i \(-0.511920\pi\)
−0.0374389 + 0.999299i \(0.511920\pi\)
\(462\) 0 0
\(463\) −17.5167 −0.814068 −0.407034 0.913413i \(-0.633437\pi\)
−0.407034 + 0.913413i \(0.633437\pi\)
\(464\) 1.26795 0.0588631
\(465\) 1.46410 0.0678961
\(466\) 19.8564 0.919830
\(467\) 4.05256 0.187530 0.0937650 0.995594i \(-0.470110\pi\)
0.0937650 + 0.995594i \(0.470110\pi\)
\(468\) 13.4641 0.622378
\(469\) 4.00000 0.184703
\(470\) −6.92820 −0.319574
\(471\) −8.39230 −0.386697
\(472\) 6.92820 0.318896
\(473\) 0 0
\(474\) 1.32051 0.0606529
\(475\) −3.26795 −0.149944
\(476\) 3.46410 0.158777
\(477\) 26.4449 1.21083
\(478\) 4.73205 0.216439
\(479\) −8.78461 −0.401379 −0.200690 0.979655i \(-0.564318\pi\)
−0.200690 + 0.979655i \(0.564318\pi\)
\(480\) −0.732051 −0.0334134
\(481\) 14.9282 0.680667
\(482\) 6.73205 0.306637
\(483\) 1.60770 0.0731527
\(484\) 0 0
\(485\) 16.5885 0.753243
\(486\) 15.2679 0.692568
\(487\) −6.19615 −0.280774 −0.140387 0.990097i \(-0.544835\pi\)
−0.140387 + 0.990097i \(0.544835\pi\)
\(488\) 8.92820 0.404161
\(489\) 2.92820 0.132418
\(490\) 1.00000 0.0451754
\(491\) −27.7128 −1.25066 −0.625331 0.780360i \(-0.715036\pi\)
−0.625331 + 0.780360i \(0.715036\pi\)
\(492\) 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 \(0.149225\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −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
\(505\) 19.8564 0.883598
\(506\) 0 0
\(507\) −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 \(0.313689\pi\)
\(510\) 2.53590 0.112291
\(511\) 6.39230 0.282779
\(512\) −1.00000 −0.0441942
\(513\) −13.0718 −0.577134
\(514\) 6.33975 0.279634
\(515\) 8.39230 0.369809
\(516\) 1.46410 0.0644535
\(517\) 0 0
\(518\) −2.73205 −0.120039
\(519\) −9.46410 −0.415428
\(520\) −5.46410 −0.239617
\(521\) −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
\(525\) 0.732051 0.0319493
\(526\) −24.0000 −1.04645
\(527\) −6.92820 −0.301797
\(528\) 0 0
\(529\) −18.1769 −0.790301
\(530\) −10.7321 −0.466170
\(531\) 17.0718 0.740853
\(532\) 3.26795 0.141684
\(533\) 44.7846 1.93984
\(534\) −2.53590 −0.109739
\(535\) −19.8564 −0.858467
\(536\) 4.00000 0.172774
\(537\) 4.39230 0.189542
\(538\) −7.60770 −0.327991
\(539\) 0 0
\(540\) −4.00000 −0.172133
\(541\) 17.2679 0.742407 0.371204 0.928552i \(-0.378945\pi\)
0.371204 + 0.928552i \(0.378945\pi\)
\(542\) −20.3923 −0.875924
\(543\) −10.2487 −0.439814
\(544\) 3.46410 0.148522
\(545\) 1.66025 0.0711175
\(546\) 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
\(555\) −2.00000 −0.0848953
\(556\) 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
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 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
\(565\) 19.8564 0.835365
\(566\) 23.7128 0.996724
\(567\) −4.46410 −0.187475
\(568\) −2.53590 −0.106404
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 2.39230 0.100203
\(571\) −24.3923 −1.02079 −0.510393 0.859941i \(-0.670500\pi\)
−0.510393 + 0.859941i \(0.670500\pi\)
\(572\) 0 0
\(573\) 8.78461 0.366982
\(574\) −8.19615 −0.342101
\(575\) 2.19615 0.0915859
\(576\) −2.46410 −0.102671
\(577\) −3.41154 −0.142024 −0.0710122 0.997475i \(-0.522623\pi\)
−0.0710122 + 0.997475i \(0.522623\pi\)
\(578\) 5.00000 0.207973
\(579\) −6.14359 −0.255319
\(580\) −1.26795 −0.0526487
\(581\) 4.39230 0.182224
\(582\) −12.1436 −0.503368
\(583\) 0 0
\(584\) 6.39230 0.264515
\(585\) −13.4641 −0.556672
\(586\) 14.5359 0.600472
\(587\) −42.5885 −1.75781 −0.878907 0.476993i \(-0.841727\pi\)
−0.878907 + 0.476993i \(0.841727\pi\)
\(588\) −0.732051 −0.0301893
\(589\) −6.53590 −0.269307
\(590\) −6.92820 −0.285230
\(591\) 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 \(0.0435189\pi\)
\(594\) 0 0
\(595\) −3.46410 −0.142014
\(596\) 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 \(0.671967\pi\)
\(600\) 0.732051 0.0298858
\(601\) −39.5167 −1.61192 −0.805959 0.591971i \(-0.798350\pi\)
−0.805959 + 0.591971i \(0.798350\pi\)
\(602\) −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
\(610\) −8.92820 −0.361492
\(611\) 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
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) −17.3205 −0.697297 −0.348649 0.937253i \(-0.613359\pi\)
−0.348649 + 0.937253i \(0.613359\pi\)
\(618\) −6.14359 −0.247132
\(619\) −12.7846 −0.513857 −0.256928 0.966430i \(-0.582710\pi\)
−0.256928 + 0.966430i \(0.582710\pi\)
\(620\) −2.00000 −0.0803219
\(621\) 8.78461 0.352514
\(622\) −11.0718 −0.443939
\(623\) 3.46410 0.138786
\(624\) 4.00000 0.160128
\(625\) 1.00000 0.0400000
\(626\) −11.8038 −0.471777
\(627\) 0 0
\(628\) 11.4641 0.457467
\(629\) 9.46410 0.377358
\(630\) 2.46410 0.0981722
\(631\) −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
\(635\) 14.9282 0.592408
\(636\) 7.85641 0.311527
\(637\) −5.46410 −0.216496
\(638\) 0 0
\(639\) −6.24871 −0.247195
\(640\) 1.00000 0.0395285
\(641\) 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 \(-0.242170\pi\)
0.724285 + 0.689500i \(0.242170\pi\)
\(644\) −2.19615 −0.0865405
\(645\) −1.46410 −0.0576489
\(646\) −11.3205 −0.445399
\(647\) −33.4641 −1.31561 −0.657805 0.753188i \(-0.728515\pi\)
−0.657805 + 0.753188i \(0.728515\pi\)
\(648\) −4.46410 −0.175366
\(649\) 0 0
\(650\) 5.46410 0.214320
\(651\) 1.46410 0.0573827
\(652\) −4.00000 −0.156652
\(653\) −5.66025 −0.221503 −0.110751 0.993848i \(-0.535326\pi\)
−0.110751 + 0.993848i \(0.535326\pi\)
\(654\) −1.21539 −0.0475255
\(655\) −11.6603 −0.455604
\(656\) −8.19615 −0.320006
\(657\) 15.7513 0.614516
\(658\) −6.92820 −0.270089
\(659\) 18.2487 0.710869 0.355434 0.934701i \(-0.384333\pi\)
0.355434 + 0.934701i \(0.384333\pi\)
\(660\) 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
\(665\) −3.26795 −0.126726
\(666\) −6.73205 −0.260862
\(667\) 2.78461 0.107821
\(668\) 13.8564 0.536120
\(669\) 13.5692 0.524616
\(670\) −4.00000 −0.154533
\(671\) 0 0
\(672\) −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
\(675\) 4.00000 0.153960
\(676\) 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
\(680\) −3.46410 −0.132842
\(681\) −5.07180 −0.194352
\(682\) 0 0
\(683\) −42.2487 −1.61660 −0.808301 0.588769i \(-0.799613\pi\)
−0.808301 + 0.588769i \(0.799613\pi\)
\(684\) 8.05256 0.307897
\(685\) −12.9282 −0.493961
\(686\) 1.00000 0.0381802
\(687\) −2.64102 −0.100761
\(688\) −2.00000 −0.0762493
\(689\) 58.6410 2.23404
\(690\) −1.60770 −0.0612039
\(691\) −6.53590 −0.248637 −0.124319 0.992242i \(-0.539675\pi\)
−0.124319 + 0.992242i \(0.539675\pi\)
\(692\) 12.9282 0.491457
\(693\) 0 0
\(694\) −36.9282 −1.40178
\(695\) −3.66025 −0.138841
\(696\) 0.928203 0.0351835
\(697\) 28.3923 1.07544
\(698\) −26.3923 −0.998963
\(699\) 14.5359 0.549798
\(700\) −1.00000 −0.0377964
\(701\) 23.9090 0.903029 0.451515 0.892264i \(-0.350884\pi\)
0.451515 + 0.892264i \(0.350884\pi\)
\(702\) 21.8564 0.824917
\(703\) 8.92820 0.336734
\(704\) 0 0
\(705\) −5.07180 −0.191015
\(706\) −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 \(0.555999\pi\)
\(710\) 2.53590 0.0951706
\(711\) −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 \(0.327139\pi\)
\(720\) 2.46410 0.0918316
\(721\) 8.39230 0.312546
\(722\) 8.32051 0.309657
\(723\) 4.92820 0.183282
\(724\) 14.0000 0.520306
\(725\) 1.26795 0.0470905
\(726\) 0 0
\(727\) −4.00000 −0.148352 −0.0741759 0.997245i \(-0.523633\pi\)
−0.0741759 + 0.997245i \(0.523633\pi\)
\(728\) −5.46410 −0.202513
\(729\) −2.21539 −0.0820515
\(730\) −6.39230 −0.236590
\(731\) 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
\(735\) 0.732051 0.0270021
\(736\) −2.19615 −0.0809513
\(737\) 0 0
\(738\) −20.1962 −0.743431
\(739\) 43.7128 1.60800 0.804001 0.594628i \(-0.202701\pi\)
0.804001 + 0.594628i \(0.202701\pi\)
\(740\) 2.73205 0.100432
\(741\) −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
\(745\) −10.7321 −0.393192
\(746\) −2.39230 −0.0875885
\(747\) 10.8231 0.395996
\(748\) 0 0
\(749\) −19.8564 −0.725537
\(750\) −0.732051 −0.0267307
\(751\) −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
\(755\) −13.1244 −0.477644
\(756\) −4.00000 −0.145479
\(757\) 29.3731 1.06758 0.533791 0.845616i \(-0.320767\pi\)
0.533791 + 0.845616i \(0.320767\pi\)
\(758\) −33.8564 −1.22972
\(759\) 0 0
\(760\) −3.26795 −0.118541
\(761\) 29.6603 1.07518 0.537592 0.843205i \(-0.319334\pi\)
0.537592 + 0.843205i \(0.319334\pi\)
\(762\) −10.9282 −0.395887
\(763\) 1.66025 0.0601052
\(764\) −12.0000 −0.434145
\(765\) −8.53590 −0.308616
\(766\) 26.5359 0.958781
\(767\) 37.8564 1.36692
\(768\) −0.732051 −0.0264156
\(769\) 7.80385 0.281414 0.140707 0.990051i \(-0.455062\pi\)
0.140707 + 0.990051i \(0.455062\pi\)
\(770\) 0 0
\(771\) 4.64102 0.167142
\(772\) 8.39230 0.302046
\(773\) 12.9282 0.464995 0.232498 0.972597i \(-0.425310\pi\)
0.232498 + 0.972597i \(0.425310\pi\)
\(774\) −4.92820 −0.177141
\(775\) 2.00000 0.0718421
\(776\) 16.5885 0.595491
\(777\) −2.00000 −0.0717496
\(778\) 22.3923 0.802803
\(779\) 26.7846 0.959658
\(780\) −4.00000 −0.143223
\(781\) 0 0
\(782\) 7.60770 0.272051
\(783\) 5.07180 0.181251
\(784\) 1.00000 0.0357143
\(785\) −11.4641 −0.409171
\(786\) 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
\(790\) 1.80385 0.0641780
\(791\) 19.8564 0.706013
\(792\) 0 0
\(793\) 48.7846 1.73239
\(794\) −9.60770 −0.340964
\(795\) −7.85641 −0.278638
\(796\) −10.9282 −0.387340
\(797\) −46.3923 −1.64330 −0.821650 0.569993i \(-0.806946\pi\)
−0.821650 + 0.569993i \(0.806946\pi\)
\(798\) 2.39230 0.0846867
\(799\) 24.0000 0.849059
\(800\) −1.00000 −0.0353553
\(801\) 8.53590 0.301601
\(802\) −9.46410 −0.334189
\(803\) 0 0
\(804\) 2.92820 0.103270
\(805\) 2.19615 0.0774042
\(806\) 10.9282 0.384930
\(807\) −5.56922 −0.196046
\(808\) 19.8564 0.698546
\(809\) −31.1769 −1.09612 −0.548061 0.836438i \(-0.684634\pi\)
−0.548061 + 0.836438i \(0.684634\pi\)
\(810\) 4.46410 0.156853
\(811\) 18.8756 0.662814 0.331407 0.943488i \(-0.392477\pi\)
0.331407 + 0.943488i \(0.392477\pi\)
\(812\) −1.26795 −0.0444963
\(813\) −14.9282 −0.523555
\(814\) 0 0
\(815\) 4.00000 0.140114
\(816\) 2.53590 0.0887742
\(817\) 6.53590 0.228662
\(818\) 35.1244 1.22809
\(819\) −13.4641 −0.470474
\(820\) 8.19615 0.286222
\(821\) 47.9090 1.67203 0.836017 0.548703i \(-0.184878\pi\)
0.836017 + 0.548703i \(0.184878\pi\)
\(822\) 9.46410 0.330098
\(823\) −25.1244 −0.875780 −0.437890 0.899029i \(-0.644274\pi\)
−0.437890 + 0.899029i \(0.644274\pi\)
\(824\) 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 \(0.796840\pi\)
\(830\) −4.39230 −0.152459
\(831\) 15.3205 0.531463
\(832\) −5.46410 −0.189434
\(833\) −3.46410 −0.120024
\(834\) 2.67949 0.0927832
\(835\) −13.8564 −0.479521
\(836\) 0 0
\(837\) 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 \(0.522787\pi\)
\(840\) 0.732051 0.0252582
\(841\) −27.3923 −0.944562
\(842\) 8.14359 0.280647
\(843\) 1.17691 0.0405351
\(844\) −13.0718 −0.449950
\(845\) −16.8564 −0.579878
\(846\) −17.0718 −0.586940
\(847\) 0 0
\(848\) −10.7321 −0.368540
\(849\) 17.3590 0.595759
\(850\) 3.46410 0.118818
\(851\) −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
\(855\) −8.05256 −0.275392
\(856\) −19.8564 −0.678678
\(857\) 20.5359 0.701493 0.350746 0.936470i \(-0.385928\pi\)
0.350746 + 0.936470i \(0.385928\pi\)
\(858\) 0 0
\(859\) 28.7846 0.982118 0.491059 0.871126i \(-0.336610\pi\)
0.491059 + 0.871126i \(0.336610\pi\)
\(860\) 2.00000 0.0681994
\(861\) −6.00000 −0.204479
\(862\) −21.1244 −0.719498
\(863\) 37.5167 1.27708 0.638541 0.769588i \(-0.279538\pi\)
0.638541 + 0.769588i \(0.279538\pi\)
\(864\) −4.00000 −0.136083
\(865\) −12.9282 −0.439572
\(866\) 7.80385 0.265186
\(867\) 3.66025 0.124309
\(868\) −2.00000 −0.0678844
\(869\) 0 0
\(870\) −0.928203 −0.0314690
\(871\) 21.8564 0.740576
\(872\) 1.66025 0.0562233
\(873\) 40.8756 1.38343
\(874\) 7.17691 0.242763
\(875\) 1.00000 0.0338062
\(876\) 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 \(-0.897778\pi\)
−0.948876 + 0.315649i \(0.897778\pi\)
\(884\) 18.9282 0.636624
\(885\) −5.07180 −0.170487
\(886\) 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
\(890\) −3.46410 −0.116117
\(891\) 0 0
\(892\) −18.5359 −0.620628
\(893\) 22.6410 0.757653
\(894\) 7.85641 0.262758
\(895\) 6.00000 0.200558
\(896\) 1.00000 0.0334077
\(897\) 8.78461 0.293310
\(898\) −19.6077 −0.654317
\(899\) 2.53590 0.0845769
\(900\) −2.46410 −0.0821367
\(901\) 37.1769 1.23854
\(902\) 0 0
\(903\) −1.46410 −0.0487223
\(904\) 19.8564 0.660414
\(905\) −14.0000 −0.465376
\(906\) 9.60770 0.319194
\(907\) 59.0333 1.96017 0.980085 0.198580i \(-0.0636331\pi\)
0.980085 + 0.198580i \(0.0636331\pi\)
\(908\) 6.92820 0.229920
\(909\) 48.9282 1.62285
\(910\) 5.46410 0.181133
\(911\) 2.53590 0.0840181 0.0420090 0.999117i \(-0.486624\pi\)
0.0420090 + 0.999117i \(0.486624\pi\)
\(912\) 2.39230 0.0792171
\(913\) 0 0
\(914\) 2.00000 0.0661541
\(915\) −6.53590 −0.216070
\(916\) 3.60770 0.119202
\(917\) −11.6603 −0.385056
\(918\) 13.8564 0.457330
\(919\) −47.3731 −1.56269 −0.781347 0.624097i \(-0.785467\pi\)
−0.781347 + 0.624097i \(0.785467\pi\)
\(920\) 2.19615 0.0724050
\(921\) 2.64102 0.0870244
\(922\) 1.60770 0.0529466
\(923\) −13.8564 −0.456089
\(924\) 0 0
\(925\) −2.73205 −0.0898293
\(926\) 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 \(0.775311\pi\)
\(930\) −1.46410 −0.0480098
\(931\) −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
\(940\) 6.92820 0.225973
\(941\) −26.7846 −0.873153 −0.436577 0.899667i \(-0.643809\pi\)
−0.436577 + 0.899667i \(0.643809\pi\)
\(942\) 8.39230 0.273436
\(943\) −18.0000 −0.586161
\(944\) −6.92820 −0.225494
\(945\) 4.00000 0.130120
\(946\) 0 0
\(947\) 12.6795 0.412028 0.206014 0.978549i \(-0.433951\pi\)
0.206014 + 0.978549i \(0.433951\pi\)
\(948\) −1.32051 −0.0428881
\(949\) 34.9282 1.13382
\(950\) 3.26795 0.106026
\(951\) 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
\(955\) 12.0000 0.388311
\(956\) −4.73205 −0.153045
\(957\) 0 0
\(958\) 8.78461 0.283818
\(959\) −12.9282 −0.417473
\(960\) 0.732051 0.0236268
\(961\) −27.0000 −0.870968
\(962\) −14.9282 −0.481305
\(963\) −48.9282 −1.57669
\(964\) −6.73205 −0.216825
\(965\) −8.39230 −0.270158
\(966\) −1.60770 −0.0517267
\(967\) −13.0718 −0.420361 −0.210180 0.977663i \(-0.567405\pi\)
−0.210180 + 0.977663i \(0.567405\pi\)
\(968\) 0 0
\(969\) −8.28719 −0.266223
\(970\) −16.5885 −0.532623
\(971\) 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
\(975\) 4.00000 0.128103
\(976\) −8.92820 −0.285785
\(977\) 21.7128 0.694654 0.347327 0.937744i \(-0.387089\pi\)
0.347327 + 0.937744i \(0.387089\pi\)
\(978\) −2.92820 −0.0936336
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 4.09103 0.130617
\(982\) 27.7128 0.884351
\(983\) 25.1769 0.803019 0.401509 0.915855i \(-0.368486\pi\)
0.401509 + 0.915855i \(0.368486\pi\)
\(984\) −6.00000 −0.191273
\(985\) 24.2487 0.772628
\(986\) 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
\(995\) 10.9282 0.346447
\(996\) 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 8470.2.a.br.1.1 2
11.10 odd 2 770.2.a.j.1.1 2
33.32 even 2 6930.2.a.bv.1.2 2
44.43 even 2 6160.2.a.t.1.2 2
55.32 even 4 3850.2.c.x.1849.4 4
55.43 even 4 3850.2.c.x.1849.1 4
55.54 odd 2 3850.2.a.bd.1.2 2
77.76 even 2 5390.2.a.bs.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.j.1.1 2 11.10 odd 2
3850.2.a.bd.1.2 2 55.54 odd 2
3850.2.c.x.1849.1 4 55.43 even 4
3850.2.c.x.1849.4 4 55.32 even 4
5390.2.a.bs.1.2 2 77.76 even 2
6160.2.a.t.1.2 2 44.43 even 2
6930.2.a.bv.1.2 2 33.32 even 2
8470.2.a.br.1.1 2 1.1 even 1 trivial