Properties

Label 6930.2.a.ci.1.3
Level $6930$
Weight $2$
Character 6930.1
Self dual yes
Analytic conductor $55.336$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6930.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.3363286007\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.87939\) of defining polynomial
Character \(\chi\) \(=\) 6930.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} +3.75877 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.75877 q^{17} -8.12836 q^{19} +1.00000 q^{20} -1.00000 q^{22} +7.14796 q^{23} +1.00000 q^{25} -3.75877 q^{26} +1.00000 q^{28} +9.51754 q^{29} +6.12836 q^{31} -1.00000 q^{32} +1.75877 q^{34} +1.00000 q^{35} -5.75877 q^{37} +8.12836 q^{38} -1.00000 q^{40} -4.12836 q^{41} -8.12836 q^{43} +1.00000 q^{44} -7.14796 q^{46} +6.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} +3.75877 q^{52} +0.610815 q^{53} +1.00000 q^{55} -1.00000 q^{56} -9.51754 q^{58} +10.1284 q^{59} -4.00000 q^{61} -6.12836 q^{62} +1.00000 q^{64} +3.75877 q^{65} +4.98040 q^{67} -1.75877 q^{68} -1.00000 q^{70} -13.0351 q^{71} -8.12836 q^{73} +5.75877 q^{74} -8.12836 q^{76} +1.00000 q^{77} +15.0351 q^{79} +1.00000 q^{80} +4.12836 q^{82} -8.90673 q^{83} -1.75877 q^{85} +8.12836 q^{86} -1.00000 q^{88} +16.6655 q^{89} +3.75877 q^{91} +7.14796 q^{92} -6.00000 q^{94} -8.12836 q^{95} +15.6459 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8} + O(q^{10}) \) \( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{10} + 3 q^{11} - 3 q^{14} + 3 q^{16} + 6 q^{17} - 6 q^{19} + 3 q^{20} - 3 q^{22} + 6 q^{23} + 3 q^{25} + 3 q^{28} + 6 q^{29} - 3 q^{32} - 6 q^{34} + 3 q^{35} - 6 q^{37} + 6 q^{38} - 3 q^{40} + 6 q^{41} - 6 q^{43} + 3 q^{44} - 6 q^{46} + 18 q^{47} + 3 q^{49} - 3 q^{50} + 6 q^{53} + 3 q^{55} - 3 q^{56} - 6 q^{58} + 12 q^{59} - 12 q^{61} + 3 q^{64} + 12 q^{67} + 6 q^{68} - 3 q^{70} + 6 q^{71} - 6 q^{73} + 6 q^{74} - 6 q^{76} + 3 q^{77} + 3 q^{80} - 6 q^{82} + 6 q^{85} + 6 q^{86} - 3 q^{88} + 12 q^{89} + 6 q^{92} - 18 q^{94} - 6 q^{95} + 6 q^{97} - 3 q^{98} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 3.75877 1.04250 0.521248 0.853405i \(-0.325467\pi\)
0.521248 + 0.853405i \(0.325467\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.75877 −0.426564 −0.213282 0.976991i \(-0.568415\pi\)
−0.213282 + 0.976991i \(0.568415\pi\)
\(18\) 0 0
\(19\) −8.12836 −1.86477 −0.932386 0.361463i \(-0.882277\pi\)
−0.932386 + 0.361463i \(0.882277\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 7.14796 1.49045 0.745226 0.666812i \(-0.232342\pi\)
0.745226 + 0.666812i \(0.232342\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −3.75877 −0.737156
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 9.51754 1.76736 0.883681 0.468089i \(-0.155057\pi\)
0.883681 + 0.468089i \(0.155057\pi\)
\(30\) 0 0
\(31\) 6.12836 1.10069 0.550343 0.834939i \(-0.314497\pi\)
0.550343 + 0.834939i \(0.314497\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 1.75877 0.301627
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −5.75877 −0.946736 −0.473368 0.880865i \(-0.656962\pi\)
−0.473368 + 0.880865i \(0.656962\pi\)
\(38\) 8.12836 1.31859
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −4.12836 −0.644741 −0.322370 0.946614i \(-0.604480\pi\)
−0.322370 + 0.946614i \(0.604480\pi\)
\(42\) 0 0
\(43\) −8.12836 −1.23956 −0.619781 0.784775i \(-0.712779\pi\)
−0.619781 + 0.784775i \(0.712779\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −7.14796 −1.05391
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 3.75877 0.521248
\(53\) 0.610815 0.0839018 0.0419509 0.999120i \(-0.486643\pi\)
0.0419509 + 0.999120i \(0.486643\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −9.51754 −1.24971
\(59\) 10.1284 1.31860 0.659300 0.751880i \(-0.270853\pi\)
0.659300 + 0.751880i \(0.270853\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) −6.12836 −0.778302
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.75877 0.466218
\(66\) 0 0
\(67\) 4.98040 0.608453 0.304226 0.952600i \(-0.401602\pi\)
0.304226 + 0.952600i \(0.401602\pi\)
\(68\) −1.75877 −0.213282
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) −13.0351 −1.54698 −0.773490 0.633809i \(-0.781491\pi\)
−0.773490 + 0.633809i \(0.781491\pi\)
\(72\) 0 0
\(73\) −8.12836 −0.951352 −0.475676 0.879621i \(-0.657797\pi\)
−0.475676 + 0.879621i \(0.657797\pi\)
\(74\) 5.75877 0.669443
\(75\) 0 0
\(76\) −8.12836 −0.932386
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 15.0351 1.69158 0.845789 0.533517i \(-0.179130\pi\)
0.845789 + 0.533517i \(0.179130\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 4.12836 0.455901
\(83\) −8.90673 −0.977640 −0.488820 0.872385i \(-0.662573\pi\)
−0.488820 + 0.872385i \(0.662573\pi\)
\(84\) 0 0
\(85\) −1.75877 −0.190765
\(86\) 8.12836 0.876503
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 16.6655 1.76654 0.883270 0.468866i \(-0.155337\pi\)
0.883270 + 0.468866i \(0.155337\pi\)
\(90\) 0 0
\(91\) 3.75877 0.394026
\(92\) 7.14796 0.745226
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) −8.12836 −0.833952
\(96\) 0 0
\(97\) 15.6459 1.58860 0.794300 0.607526i \(-0.207838\pi\)
0.794300 + 0.607526i \(0.207838\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 10.7392 1.06859 0.534294 0.845299i \(-0.320578\pi\)
0.534294 + 0.845299i \(0.320578\pi\)
\(102\) 0 0
\(103\) 6.12836 0.603845 0.301922 0.953333i \(-0.402372\pi\)
0.301922 + 0.953333i \(0.402372\pi\)
\(104\) −3.75877 −0.368578
\(105\) 0 0
\(106\) −0.610815 −0.0593276
\(107\) 14.8675 1.43730 0.718649 0.695373i \(-0.244761\pi\)
0.718649 + 0.695373i \(0.244761\pi\)
\(108\) 0 0
\(109\) −15.2763 −1.46321 −0.731603 0.681731i \(-0.761227\pi\)
−0.731603 + 0.681731i \(0.761227\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −5.88713 −0.553814 −0.276907 0.960897i \(-0.589309\pi\)
−0.276907 + 0.960897i \(0.589309\pi\)
\(114\) 0 0
\(115\) 7.14796 0.666550
\(116\) 9.51754 0.883681
\(117\) 0 0
\(118\) −10.1284 −0.932391
\(119\) −1.75877 −0.161226
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 4.00000 0.362143
\(123\) 0 0
\(124\) 6.12836 0.550343
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −2.12836 −0.188861 −0.0944305 0.995531i \(-0.530103\pi\)
−0.0944305 + 0.995531i \(0.530103\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −3.75877 −0.329666
\(131\) −3.51754 −0.307329 −0.153664 0.988123i \(-0.549107\pi\)
−0.153664 + 0.988123i \(0.549107\pi\)
\(132\) 0 0
\(133\) −8.12836 −0.704818
\(134\) −4.98040 −0.430241
\(135\) 0 0
\(136\) 1.75877 0.150813
\(137\) 13.1480 1.12331 0.561653 0.827373i \(-0.310166\pi\)
0.561653 + 0.827373i \(0.310166\pi\)
\(138\) 0 0
\(139\) 3.64590 0.309241 0.154620 0.987974i \(-0.450585\pi\)
0.154620 + 0.987974i \(0.450585\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) 13.0351 1.09388
\(143\) 3.75877 0.314324
\(144\) 0 0
\(145\) 9.51754 0.790389
\(146\) 8.12836 0.672707
\(147\) 0 0
\(148\) −5.75877 −0.473368
\(149\) −9.51754 −0.779707 −0.389854 0.920877i \(-0.627474\pi\)
−0.389854 + 0.920877i \(0.627474\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 8.12836 0.659297
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 6.12836 0.492241
\(156\) 0 0
\(157\) −16.3851 −1.30767 −0.653835 0.756637i \(-0.726841\pi\)
−0.653835 + 0.756637i \(0.726841\pi\)
\(158\) −15.0351 −1.19613
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 7.14796 0.563338
\(162\) 0 0
\(163\) −15.2763 −1.19653 −0.598267 0.801297i \(-0.704144\pi\)
−0.598267 + 0.801297i \(0.704144\pi\)
\(164\) −4.12836 −0.322370
\(165\) 0 0
\(166\) 8.90673 0.691296
\(167\) 6.11287 0.473028 0.236514 0.971628i \(-0.423995\pi\)
0.236514 + 0.971628i \(0.423995\pi\)
\(168\) 0 0
\(169\) 1.12836 0.0867966
\(170\) 1.75877 0.134892
\(171\) 0 0
\(172\) −8.12836 −0.619781
\(173\) 25.0351 1.90338 0.951691 0.307057i \(-0.0993443\pi\)
0.951691 + 0.307057i \(0.0993443\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −16.6655 −1.24913
\(179\) −17.1634 −1.28286 −0.641428 0.767183i \(-0.721658\pi\)
−0.641428 + 0.767183i \(0.721658\pi\)
\(180\) 0 0
\(181\) −24.7939 −1.84291 −0.921456 0.388482i \(-0.873000\pi\)
−0.921456 + 0.388482i \(0.873000\pi\)
\(182\) −3.75877 −0.278619
\(183\) 0 0
\(184\) −7.14796 −0.526954
\(185\) −5.75877 −0.423393
\(186\) 0 0
\(187\) −1.75877 −0.128614
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) 8.12836 0.589693
\(191\) −9.51754 −0.688665 −0.344333 0.938848i \(-0.611895\pi\)
−0.344333 + 0.938848i \(0.611895\pi\)
\(192\) 0 0
\(193\) 12.1284 0.873018 0.436509 0.899700i \(-0.356215\pi\)
0.436509 + 0.899700i \(0.356215\pi\)
\(194\) −15.6459 −1.12331
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −14.9067 −1.06206 −0.531030 0.847353i \(-0.678195\pi\)
−0.531030 + 0.847353i \(0.678195\pi\)
\(198\) 0 0
\(199\) 21.6459 1.53444 0.767218 0.641386i \(-0.221640\pi\)
0.767218 + 0.641386i \(0.221640\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −10.7392 −0.755605
\(203\) 9.51754 0.668000
\(204\) 0 0
\(205\) −4.12836 −0.288337
\(206\) −6.12836 −0.426983
\(207\) 0 0
\(208\) 3.75877 0.260624
\(209\) −8.12836 −0.562250
\(210\) 0 0
\(211\) 22.7547 1.56649 0.783247 0.621710i \(-0.213562\pi\)
0.783247 + 0.621710i \(0.213562\pi\)
\(212\) 0.610815 0.0419509
\(213\) 0 0
\(214\) −14.8675 −1.01632
\(215\) −8.12836 −0.554349
\(216\) 0 0
\(217\) 6.12836 0.416020
\(218\) 15.2763 1.03464
\(219\) 0 0
\(220\) 1.00000 0.0674200
\(221\) −6.61081 −0.444692
\(222\) 0 0
\(223\) 3.26083 0.218361 0.109181 0.994022i \(-0.465177\pi\)
0.109181 + 0.994022i \(0.465177\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 5.88713 0.391606
\(227\) 24.4243 1.62110 0.810548 0.585673i \(-0.199170\pi\)
0.810548 + 0.585673i \(0.199170\pi\)
\(228\) 0 0
\(229\) −2.24123 −0.148105 −0.0740523 0.997254i \(-0.523593\pi\)
−0.0740523 + 0.997254i \(0.523593\pi\)
\(230\) −7.14796 −0.471322
\(231\) 0 0
\(232\) −9.51754 −0.624857
\(233\) 4.77837 0.313041 0.156521 0.987675i \(-0.449972\pi\)
0.156521 + 0.987675i \(0.449972\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) 10.1284 0.659300
\(237\) 0 0
\(238\) 1.75877 0.114004
\(239\) 28.0155 1.81217 0.906085 0.423095i \(-0.139056\pi\)
0.906085 + 0.423095i \(0.139056\pi\)
\(240\) 0 0
\(241\) −18.9067 −1.21789 −0.608945 0.793213i \(-0.708407\pi\)
−0.608945 + 0.793213i \(0.708407\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) −4.00000 −0.256074
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −30.5526 −1.94402
\(248\) −6.12836 −0.389151
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 4.16756 0.263054 0.131527 0.991313i \(-0.458012\pi\)
0.131527 + 0.991313i \(0.458012\pi\)
\(252\) 0 0
\(253\) 7.14796 0.449388
\(254\) 2.12836 0.133545
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 25.6459 1.59975 0.799874 0.600169i \(-0.204900\pi\)
0.799874 + 0.600169i \(0.204900\pi\)
\(258\) 0 0
\(259\) −5.75877 −0.357833
\(260\) 3.75877 0.233109
\(261\) 0 0
\(262\) 3.51754 0.217314
\(263\) −15.5175 −0.956853 −0.478426 0.878128i \(-0.658793\pi\)
−0.478426 + 0.878128i \(0.658793\pi\)
\(264\) 0 0
\(265\) 0.610815 0.0375220
\(266\) 8.12836 0.498381
\(267\) 0 0
\(268\) 4.98040 0.304226
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −31.2918 −1.90084 −0.950421 0.310968i \(-0.899347\pi\)
−0.950421 + 0.310968i \(0.899347\pi\)
\(272\) −1.75877 −0.106641
\(273\) 0 0
\(274\) −13.1480 −0.794297
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −6.86753 −0.412630 −0.206315 0.978486i \(-0.566147\pi\)
−0.206315 + 0.978486i \(0.566147\pi\)
\(278\) −3.64590 −0.218666
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) −29.2763 −1.74648 −0.873239 0.487292i \(-0.837985\pi\)
−0.873239 + 0.487292i \(0.837985\pi\)
\(282\) 0 0
\(283\) 7.88713 0.468841 0.234420 0.972135i \(-0.424681\pi\)
0.234420 + 0.972135i \(0.424681\pi\)
\(284\) −13.0351 −0.773490
\(285\) 0 0
\(286\) −3.75877 −0.222261
\(287\) −4.12836 −0.243689
\(288\) 0 0
\(289\) −13.9067 −0.818043
\(290\) −9.51754 −0.558889
\(291\) 0 0
\(292\) −8.12836 −0.475676
\(293\) 18.9959 1.10975 0.554876 0.831933i \(-0.312766\pi\)
0.554876 + 0.831933i \(0.312766\pi\)
\(294\) 0 0
\(295\) 10.1284 0.589696
\(296\) 5.75877 0.334722
\(297\) 0 0
\(298\) 9.51754 0.551336
\(299\) 26.8675 1.55379
\(300\) 0 0
\(301\) −8.12836 −0.468511
\(302\) −8.00000 −0.460348
\(303\) 0 0
\(304\) −8.12836 −0.466193
\(305\) −4.00000 −0.229039
\(306\) 0 0
\(307\) 28.1438 1.60625 0.803127 0.595808i \(-0.203168\pi\)
0.803127 + 0.595808i \(0.203168\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) −6.12836 −0.348067
\(311\) 16.0547 0.910378 0.455189 0.890395i \(-0.349572\pi\)
0.455189 + 0.890395i \(0.349572\pi\)
\(312\) 0 0
\(313\) 26.9959 1.52590 0.762949 0.646459i \(-0.223751\pi\)
0.762949 + 0.646459i \(0.223751\pi\)
\(314\) 16.3851 0.924663
\(315\) 0 0
\(316\) 15.0351 0.845789
\(317\) 6.65002 0.373502 0.186751 0.982407i \(-0.440204\pi\)
0.186751 + 0.982407i \(0.440204\pi\)
\(318\) 0 0
\(319\) 9.51754 0.532880
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −7.14796 −0.398340
\(323\) 14.2959 0.795446
\(324\) 0 0
\(325\) 3.75877 0.208499
\(326\) 15.2763 0.846077
\(327\) 0 0
\(328\) 4.12836 0.227950
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) −5.22163 −0.287007 −0.143503 0.989650i \(-0.545837\pi\)
−0.143503 + 0.989650i \(0.545837\pi\)
\(332\) −8.90673 −0.488820
\(333\) 0 0
\(334\) −6.11287 −0.334482
\(335\) 4.98040 0.272108
\(336\) 0 0
\(337\) −14.7392 −0.802894 −0.401447 0.915882i \(-0.631493\pi\)
−0.401447 + 0.915882i \(0.631493\pi\)
\(338\) −1.12836 −0.0613745
\(339\) 0 0
\(340\) −1.75877 −0.0953827
\(341\) 6.12836 0.331869
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 8.12836 0.438252
\(345\) 0 0
\(346\) −25.0351 −1.34589
\(347\) 21.9026 1.17579 0.587897 0.808936i \(-0.299956\pi\)
0.587897 + 0.808936i \(0.299956\pi\)
\(348\) 0 0
\(349\) −33.8135 −1.80999 −0.904996 0.425419i \(-0.860127\pi\)
−0.904996 + 0.425419i \(0.860127\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) 14.8675 0.791319 0.395659 0.918397i \(-0.370516\pi\)
0.395659 + 0.918397i \(0.370516\pi\)
\(354\) 0 0
\(355\) −13.0351 −0.691830
\(356\) 16.6655 0.883270
\(357\) 0 0
\(358\) 17.1634 0.907116
\(359\) −17.3155 −0.913878 −0.456939 0.889498i \(-0.651054\pi\)
−0.456939 + 0.889498i \(0.651054\pi\)
\(360\) 0 0
\(361\) 47.0702 2.47738
\(362\) 24.7939 1.30314
\(363\) 0 0
\(364\) 3.75877 0.197013
\(365\) −8.12836 −0.425458
\(366\) 0 0
\(367\) −4.65002 −0.242729 −0.121364 0.992608i \(-0.538727\pi\)
−0.121364 + 0.992608i \(0.538727\pi\)
\(368\) 7.14796 0.372613
\(369\) 0 0
\(370\) 5.75877 0.299384
\(371\) 0.610815 0.0317119
\(372\) 0 0
\(373\) 4.90673 0.254061 0.127030 0.991899i \(-0.459455\pi\)
0.127030 + 0.991899i \(0.459455\pi\)
\(374\) 1.75877 0.0909439
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) 35.7743 1.84247
\(378\) 0 0
\(379\) 5.55674 0.285431 0.142715 0.989764i \(-0.454417\pi\)
0.142715 + 0.989764i \(0.454417\pi\)
\(380\) −8.12836 −0.416976
\(381\) 0 0
\(382\) 9.51754 0.486960
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) −12.1284 −0.617317
\(387\) 0 0
\(388\) 15.6459 0.794300
\(389\) −6.42427 −0.325723 −0.162862 0.986649i \(-0.552072\pi\)
−0.162862 + 0.986649i \(0.552072\pi\)
\(390\) 0 0
\(391\) −12.5716 −0.635774
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 14.9067 0.750990
\(395\) 15.0351 0.756497
\(396\) 0 0
\(397\) 12.3541 0.620035 0.310017 0.950731i \(-0.399665\pi\)
0.310017 + 0.950731i \(0.399665\pi\)
\(398\) −21.6459 −1.08501
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 23.0351 1.14746
\(404\) 10.7392 0.534294
\(405\) 0 0
\(406\) −9.51754 −0.472348
\(407\) −5.75877 −0.285452
\(408\) 0 0
\(409\) −14.7392 −0.728805 −0.364403 0.931242i \(-0.618727\pi\)
−0.364403 + 0.931242i \(0.618727\pi\)
\(410\) 4.12836 0.203885
\(411\) 0 0
\(412\) 6.12836 0.301922
\(413\) 10.1284 0.498384
\(414\) 0 0
\(415\) −8.90673 −0.437214
\(416\) −3.75877 −0.184289
\(417\) 0 0
\(418\) 8.12836 0.397571
\(419\) 4.16756 0.203598 0.101799 0.994805i \(-0.467540\pi\)
0.101799 + 0.994805i \(0.467540\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) −22.7547 −1.10768
\(423\) 0 0
\(424\) −0.610815 −0.0296638
\(425\) −1.75877 −0.0853129
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) 14.8675 0.718649
\(429\) 0 0
\(430\) 8.12836 0.391984
\(431\) 19.7588 0.951746 0.475873 0.879514i \(-0.342132\pi\)
0.475873 + 0.879514i \(0.342132\pi\)
\(432\) 0 0
\(433\) −22.4243 −1.07764 −0.538821 0.842420i \(-0.681130\pi\)
−0.538821 + 0.842420i \(0.681130\pi\)
\(434\) −6.12836 −0.294170
\(435\) 0 0
\(436\) −15.2763 −0.731603
\(437\) −58.1011 −2.77935
\(438\) 0 0
\(439\) −10.3851 −0.495652 −0.247826 0.968805i \(-0.579716\pi\)
−0.247826 + 0.968805i \(0.579716\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 0 0
\(442\) 6.61081 0.314444
\(443\) −23.1242 −1.09867 −0.549333 0.835604i \(-0.685118\pi\)
−0.549333 + 0.835604i \(0.685118\pi\)
\(444\) 0 0
\(445\) 16.6655 0.790020
\(446\) −3.26083 −0.154405
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −19.0351 −0.898321 −0.449161 0.893451i \(-0.648277\pi\)
−0.449161 + 0.893451i \(0.648277\pi\)
\(450\) 0 0
\(451\) −4.12836 −0.194397
\(452\) −5.88713 −0.276907
\(453\) 0 0
\(454\) −24.4243 −1.14629
\(455\) 3.75877 0.176214
\(456\) 0 0
\(457\) 38.1985 1.78685 0.893426 0.449211i \(-0.148295\pi\)
0.893426 + 0.449211i \(0.148295\pi\)
\(458\) 2.24123 0.104726
\(459\) 0 0
\(460\) 7.14796 0.333275
\(461\) −12.6108 −0.587344 −0.293672 0.955906i \(-0.594877\pi\)
−0.293672 + 0.955906i \(0.594877\pi\)
\(462\) 0 0
\(463\) −2.55262 −0.118630 −0.0593152 0.998239i \(-0.518892\pi\)
−0.0593152 + 0.998239i \(0.518892\pi\)
\(464\) 9.51754 0.441841
\(465\) 0 0
\(466\) −4.77837 −0.221354
\(467\) 13.8716 0.641903 0.320952 0.947096i \(-0.395997\pi\)
0.320952 + 0.947096i \(0.395997\pi\)
\(468\) 0 0
\(469\) 4.98040 0.229973
\(470\) −6.00000 −0.276759
\(471\) 0 0
\(472\) −10.1284 −0.466195
\(473\) −8.12836 −0.373742
\(474\) 0 0
\(475\) −8.12836 −0.372955
\(476\) −1.75877 −0.0806131
\(477\) 0 0
\(478\) −28.0155 −1.28140
\(479\) −28.5134 −1.30281 −0.651406 0.758730i \(-0.725820\pi\)
−0.651406 + 0.758730i \(0.725820\pi\)
\(480\) 0 0
\(481\) −21.6459 −0.986968
\(482\) 18.9067 0.861178
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 15.6459 0.710444
\(486\) 0 0
\(487\) 40.2567 1.82421 0.912103 0.409961i \(-0.134458\pi\)
0.912103 + 0.409961i \(0.134458\pi\)
\(488\) 4.00000 0.181071
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) −0.225748 −0.0101879 −0.00509393 0.999987i \(-0.501621\pi\)
−0.00509393 + 0.999987i \(0.501621\pi\)
\(492\) 0 0
\(493\) −16.7392 −0.753894
\(494\) 30.5526 1.37463
\(495\) 0 0
\(496\) 6.12836 0.275171
\(497\) −13.0351 −0.584703
\(498\) 0 0
\(499\) 9.07428 0.406221 0.203110 0.979156i \(-0.434895\pi\)
0.203110 + 0.979156i \(0.434895\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −4.16756 −0.186007
\(503\) −31.9573 −1.42491 −0.712453 0.701720i \(-0.752416\pi\)
−0.712453 + 0.701720i \(0.752416\pi\)
\(504\) 0 0
\(505\) 10.7392 0.477887
\(506\) −7.14796 −0.317765
\(507\) 0 0
\(508\) −2.12836 −0.0944305
\(509\) −4.55262 −0.201791 −0.100896 0.994897i \(-0.532171\pi\)
−0.100896 + 0.994897i \(0.532171\pi\)
\(510\) 0 0
\(511\) −8.12836 −0.359577
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −25.6459 −1.13119
\(515\) 6.12836 0.270048
\(516\) 0 0
\(517\) 6.00000 0.263880
\(518\) 5.75877 0.253026
\(519\) 0 0
\(520\) −3.75877 −0.164833
\(521\) −4.89124 −0.214289 −0.107145 0.994243i \(-0.534171\pi\)
−0.107145 + 0.994243i \(0.534171\pi\)
\(522\) 0 0
\(523\) 23.4047 1.02341 0.511707 0.859160i \(-0.329013\pi\)
0.511707 + 0.859160i \(0.329013\pi\)
\(524\) −3.51754 −0.153664
\(525\) 0 0
\(526\) 15.5175 0.676597
\(527\) −10.7784 −0.469513
\(528\) 0 0
\(529\) 28.0933 1.22145
\(530\) −0.610815 −0.0265321
\(531\) 0 0
\(532\) −8.12836 −0.352409
\(533\) −15.5175 −0.672139
\(534\) 0 0
\(535\) 14.8675 0.642779
\(536\) −4.98040 −0.215120
\(537\) 0 0
\(538\) 6.00000 0.258678
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −12.9804 −0.558071 −0.279035 0.960281i \(-0.590015\pi\)
−0.279035 + 0.960281i \(0.590015\pi\)
\(542\) 31.2918 1.34410
\(543\) 0 0
\(544\) 1.75877 0.0754067
\(545\) −15.2763 −0.654365
\(546\) 0 0
\(547\) −21.1242 −0.903207 −0.451604 0.892219i \(-0.649148\pi\)
−0.451604 + 0.892219i \(0.649148\pi\)
\(548\) 13.1480 0.561653
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) −77.3620 −3.29573
\(552\) 0 0
\(553\) 15.0351 0.639357
\(554\) 6.86753 0.291773
\(555\) 0 0
\(556\) 3.64590 0.154620
\(557\) −23.1634 −0.981466 −0.490733 0.871310i \(-0.663271\pi\)
−0.490733 + 0.871310i \(0.663271\pi\)
\(558\) 0 0
\(559\) −30.5526 −1.29224
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 29.2763 1.23495
\(563\) −37.6459 −1.58659 −0.793293 0.608840i \(-0.791635\pi\)
−0.793293 + 0.608840i \(0.791635\pi\)
\(564\) 0 0
\(565\) −5.88713 −0.247673
\(566\) −7.88713 −0.331521
\(567\) 0 0
\(568\) 13.0351 0.546940
\(569\) 19.7980 0.829974 0.414987 0.909827i \(-0.363786\pi\)
0.414987 + 0.909827i \(0.363786\pi\)
\(570\) 0 0
\(571\) 8.53714 0.357268 0.178634 0.983916i \(-0.442832\pi\)
0.178634 + 0.983916i \(0.442832\pi\)
\(572\) 3.75877 0.157162
\(573\) 0 0
\(574\) 4.12836 0.172314
\(575\) 7.14796 0.298090
\(576\) 0 0
\(577\) 10.4825 0.436390 0.218195 0.975905i \(-0.429983\pi\)
0.218195 + 0.975905i \(0.429983\pi\)
\(578\) 13.9067 0.578444
\(579\) 0 0
\(580\) 9.51754 0.395194
\(581\) −8.90673 −0.369513
\(582\) 0 0
\(583\) 0.610815 0.0252974
\(584\) 8.12836 0.336354
\(585\) 0 0
\(586\) −18.9959 −0.784713
\(587\) 32.9067 1.35821 0.679103 0.734043i \(-0.262369\pi\)
0.679103 + 0.734043i \(0.262369\pi\)
\(588\) 0 0
\(589\) −49.8135 −2.05253
\(590\) −10.1284 −0.416978
\(591\) 0 0
\(592\) −5.75877 −0.236684
\(593\) 43.5722 1.78930 0.894648 0.446771i \(-0.147426\pi\)
0.894648 + 0.446771i \(0.147426\pi\)
\(594\) 0 0
\(595\) −1.75877 −0.0721026
\(596\) −9.51754 −0.389854
\(597\) 0 0
\(598\) −26.8675 −1.09869
\(599\) 48.8093 1.99430 0.997148 0.0754754i \(-0.0240474\pi\)
0.997148 + 0.0754754i \(0.0240474\pi\)
\(600\) 0 0
\(601\) 38.9959 1.59068 0.795338 0.606167i \(-0.207294\pi\)
0.795338 + 0.606167i \(0.207294\pi\)
\(602\) 8.12836 0.331287
\(603\) 0 0
\(604\) 8.00000 0.325515
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 42.5526 1.72716 0.863579 0.504214i \(-0.168218\pi\)
0.863579 + 0.504214i \(0.168218\pi\)
\(608\) 8.12836 0.329648
\(609\) 0 0
\(610\) 4.00000 0.161955
\(611\) 22.5526 0.912381
\(612\) 0 0
\(613\) 48.9377 1.97657 0.988287 0.152605i \(-0.0487661\pi\)
0.988287 + 0.152605i \(0.0487661\pi\)
\(614\) −28.1438 −1.13579
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) 3.59121 0.144577 0.0722884 0.997384i \(-0.476970\pi\)
0.0722884 + 0.997384i \(0.476970\pi\)
\(618\) 0 0
\(619\) 28.1046 1.12962 0.564810 0.825221i \(-0.308949\pi\)
0.564810 + 0.825221i \(0.308949\pi\)
\(620\) 6.12836 0.246121
\(621\) 0 0
\(622\) −16.0547 −0.643734
\(623\) 16.6655 0.667689
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −26.9959 −1.07897
\(627\) 0 0
\(628\) −16.3851 −0.653835
\(629\) 10.1284 0.403844
\(630\) 0 0
\(631\) −23.4593 −0.933902 −0.466951 0.884283i \(-0.654648\pi\)
−0.466951 + 0.884283i \(0.654648\pi\)
\(632\) −15.0351 −0.598063
\(633\) 0 0
\(634\) −6.65002 −0.264106
\(635\) −2.12836 −0.0844612
\(636\) 0 0
\(637\) 3.75877 0.148928
\(638\) −9.51754 −0.376803
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 35.5485 1.40408 0.702041 0.712137i \(-0.252272\pi\)
0.702041 + 0.712137i \(0.252272\pi\)
\(642\) 0 0
\(643\) −13.5567 −0.534626 −0.267313 0.963610i \(-0.586136\pi\)
−0.267313 + 0.963610i \(0.586136\pi\)
\(644\) 7.14796 0.281669
\(645\) 0 0
\(646\) −14.2959 −0.562465
\(647\) −11.8135 −0.464435 −0.232217 0.972664i \(-0.574598\pi\)
−0.232217 + 0.972664i \(0.574598\pi\)
\(648\) 0 0
\(649\) 10.1284 0.397573
\(650\) −3.75877 −0.147431
\(651\) 0 0
\(652\) −15.2763 −0.598267
\(653\) 46.9377 1.83681 0.918407 0.395637i \(-0.129476\pi\)
0.918407 + 0.395637i \(0.129476\pi\)
\(654\) 0 0
\(655\) −3.51754 −0.137442
\(656\) −4.12836 −0.161185
\(657\) 0 0
\(658\) −6.00000 −0.233904
\(659\) −1.22163 −0.0475879 −0.0237940 0.999717i \(-0.507575\pi\)
−0.0237940 + 0.999717i \(0.507575\pi\)
\(660\) 0 0
\(661\) −8.28043 −0.322071 −0.161036 0.986949i \(-0.551483\pi\)
−0.161036 + 0.986949i \(0.551483\pi\)
\(662\) 5.22163 0.202944
\(663\) 0 0
\(664\) 8.90673 0.345648
\(665\) −8.12836 −0.315204
\(666\) 0 0
\(667\) 68.0310 2.63417
\(668\) 6.11287 0.236514
\(669\) 0 0
\(670\) −4.98040 −0.192410
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) −30.1094 −1.16063 −0.580315 0.814392i \(-0.697071\pi\)
−0.580315 + 0.814392i \(0.697071\pi\)
\(674\) 14.7392 0.567732
\(675\) 0 0
\(676\) 1.12836 0.0433983
\(677\) −11.9608 −0.459691 −0.229845 0.973227i \(-0.573822\pi\)
−0.229845 + 0.973227i \(0.573822\pi\)
\(678\) 0 0
\(679\) 15.6459 0.600434
\(680\) 1.75877 0.0674458
\(681\) 0 0
\(682\) −6.12836 −0.234667
\(683\) −18.4635 −0.706485 −0.353242 0.935532i \(-0.614921\pi\)
−0.353242 + 0.935532i \(0.614921\pi\)
\(684\) 0 0
\(685\) 13.1480 0.502358
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −8.12836 −0.309891
\(689\) 2.29591 0.0874673
\(690\) 0 0
\(691\) −15.7006 −0.597278 −0.298639 0.954366i \(-0.596533\pi\)
−0.298639 + 0.954366i \(0.596533\pi\)
\(692\) 25.0351 0.951691
\(693\) 0 0
\(694\) −21.9026 −0.831412
\(695\) 3.64590 0.138297
\(696\) 0 0
\(697\) 7.26083 0.275024
\(698\) 33.8135 1.27986
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) 26.0310 0.983176 0.491588 0.870828i \(-0.336417\pi\)
0.491588 + 0.870828i \(0.336417\pi\)
\(702\) 0 0
\(703\) 46.8093 1.76545
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −14.8675 −0.559547
\(707\) 10.7392 0.403888
\(708\) 0 0
\(709\) −25.2918 −0.949853 −0.474927 0.880025i \(-0.657525\pi\)
−0.474927 + 0.880025i \(0.657525\pi\)
\(710\) 13.0351 0.489198
\(711\) 0 0
\(712\) −16.6655 −0.624566
\(713\) 43.8052 1.64052
\(714\) 0 0
\(715\) 3.75877 0.140570
\(716\) −17.1634 −0.641428
\(717\) 0 0
\(718\) 17.3155 0.646209
\(719\) 18.7237 0.698276 0.349138 0.937071i \(-0.386474\pi\)
0.349138 + 0.937071i \(0.386474\pi\)
\(720\) 0 0
\(721\) 6.12836 0.228232
\(722\) −47.0702 −1.75177
\(723\) 0 0
\(724\) −24.7939 −0.921456
\(725\) 9.51754 0.353473
\(726\) 0 0
\(727\) 34.2959 1.27196 0.635982 0.771703i \(-0.280595\pi\)
0.635982 + 0.771703i \(0.280595\pi\)
\(728\) −3.75877 −0.139309
\(729\) 0 0
\(730\) 8.12836 0.300844
\(731\) 14.2959 0.528753
\(732\) 0 0
\(733\) 32.3506 1.19490 0.597448 0.801907i \(-0.296181\pi\)
0.597448 + 0.801907i \(0.296181\pi\)
\(734\) 4.65002 0.171635
\(735\) 0 0
\(736\) −7.14796 −0.263477
\(737\) 4.98040 0.183455
\(738\) 0 0
\(739\) −27.0114 −0.993629 −0.496815 0.867857i \(-0.665497\pi\)
−0.496815 + 0.867857i \(0.665497\pi\)
\(740\) −5.75877 −0.211697
\(741\) 0 0
\(742\) −0.610815 −0.0224237
\(743\) 7.03508 0.258092 0.129046 0.991639i \(-0.458808\pi\)
0.129046 + 0.991639i \(0.458808\pi\)
\(744\) 0 0
\(745\) −9.51754 −0.348696
\(746\) −4.90673 −0.179648
\(747\) 0 0
\(748\) −1.75877 −0.0643070
\(749\) 14.8675 0.543248
\(750\) 0 0
\(751\) 41.9026 1.52905 0.764524 0.644595i \(-0.222974\pi\)
0.764524 + 0.644595i \(0.222974\pi\)
\(752\) 6.00000 0.218797
\(753\) 0 0
\(754\) −35.7743 −1.30282
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) 43.0898 1.56612 0.783062 0.621944i \(-0.213657\pi\)
0.783062 + 0.621944i \(0.213657\pi\)
\(758\) −5.55674 −0.201830
\(759\) 0 0
\(760\) 8.12836 0.294846
\(761\) −6.65002 −0.241063 −0.120531 0.992710i \(-0.538460\pi\)
−0.120531 + 0.992710i \(0.538460\pi\)
\(762\) 0 0
\(763\) −15.2763 −0.553040
\(764\) −9.51754 −0.344333
\(765\) 0 0
\(766\) −6.00000 −0.216789
\(767\) 38.0702 1.37463
\(768\) 0 0
\(769\) 31.5877 1.13908 0.569541 0.821963i \(-0.307121\pi\)
0.569541 + 0.821963i \(0.307121\pi\)
\(770\) −1.00000 −0.0360375
\(771\) 0 0
\(772\) 12.1284 0.436509
\(773\) −19.6459 −0.706614 −0.353307 0.935507i \(-0.614943\pi\)
−0.353307 + 0.935507i \(0.614943\pi\)
\(774\) 0 0
\(775\) 6.12836 0.220137
\(776\) −15.6459 −0.561655
\(777\) 0 0
\(778\) 6.42427 0.230321
\(779\) 33.5567 1.20230
\(780\) 0 0
\(781\) −13.0351 −0.466432
\(782\) 12.5716 0.449560
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −16.3851 −0.584808
\(786\) 0 0
\(787\) −33.4748 −1.19325 −0.596624 0.802521i \(-0.703492\pi\)
−0.596624 + 0.802521i \(0.703492\pi\)
\(788\) −14.9067 −0.531030
\(789\) 0 0
\(790\) −15.0351 −0.534924
\(791\) −5.88713 −0.209322
\(792\) 0 0
\(793\) −15.0351 −0.533911
\(794\) −12.3541 −0.438431
\(795\) 0 0
\(796\) 21.6459 0.767218
\(797\) −14.6810 −0.520027 −0.260013 0.965605i \(-0.583727\pi\)
−0.260013 + 0.965605i \(0.583727\pi\)
\(798\) 0 0
\(799\) −10.5526 −0.373325
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 0 0
\(803\) −8.12836 −0.286843
\(804\) 0 0
\(805\) 7.14796 0.251932
\(806\) −23.0351 −0.811376
\(807\) 0 0
\(808\) −10.7392 −0.377803
\(809\) 26.8331 0.943400 0.471700 0.881759i \(-0.343641\pi\)
0.471700 + 0.881759i \(0.343641\pi\)
\(810\) 0 0
\(811\) 28.7202 1.00850 0.504251 0.863557i \(-0.331769\pi\)
0.504251 + 0.863557i \(0.331769\pi\)
\(812\) 9.51754 0.334000
\(813\) 0 0
\(814\) 5.75877 0.201845
\(815\) −15.2763 −0.535106
\(816\) 0 0
\(817\) 66.0702 2.31150
\(818\) 14.7392 0.515343
\(819\) 0 0
\(820\) −4.12836 −0.144168
\(821\) −42.8485 −1.49542 −0.747712 0.664023i \(-0.768848\pi\)
−0.747712 + 0.664023i \(0.768848\pi\)
\(822\) 0 0
\(823\) −53.8444 −1.87690 −0.938449 0.345417i \(-0.887737\pi\)
−0.938449 + 0.345417i \(0.887737\pi\)
\(824\) −6.12836 −0.213491
\(825\) 0 0
\(826\) −10.1284 −0.352411
\(827\) 19.4593 0.676668 0.338334 0.941026i \(-0.390137\pi\)
0.338334 + 0.941026i \(0.390137\pi\)
\(828\) 0 0
\(829\) −28.4587 −0.988413 −0.494206 0.869345i \(-0.664541\pi\)
−0.494206 + 0.869345i \(0.664541\pi\)
\(830\) 8.90673 0.309157
\(831\) 0 0
\(832\) 3.75877 0.130312
\(833\) −1.75877 −0.0609378
\(834\) 0 0
\(835\) 6.11287 0.211545
\(836\) −8.12836 −0.281125
\(837\) 0 0
\(838\) −4.16756 −0.143966
\(839\) 44.5681 1.53866 0.769331 0.638850i \(-0.220590\pi\)
0.769331 + 0.638850i \(0.220590\pi\)
\(840\) 0 0
\(841\) 61.5836 2.12357
\(842\) −14.0000 −0.482472
\(843\) 0 0
\(844\) 22.7547 0.783247
\(845\) 1.12836 0.0388166
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0.610815 0.0209755
\(849\) 0 0
\(850\) 1.75877 0.0603253
\(851\) −41.1634 −1.41106
\(852\) 0 0
\(853\) −5.71957 −0.195834 −0.0979172 0.995195i \(-0.531218\pi\)
−0.0979172 + 0.995195i \(0.531218\pi\)
\(854\) 4.00000 0.136877
\(855\) 0 0
\(856\) −14.8675 −0.508162
\(857\) −17.2763 −0.590148 −0.295074 0.955474i \(-0.595344\pi\)
−0.295074 + 0.955474i \(0.595344\pi\)
\(858\) 0 0
\(859\) 15.1088 0.515504 0.257752 0.966211i \(-0.417018\pi\)
0.257752 + 0.966211i \(0.417018\pi\)
\(860\) −8.12836 −0.277175
\(861\) 0 0
\(862\) −19.7588 −0.672986
\(863\) 0.0344725 0.00117346 0.000586729 1.00000i \(-0.499813\pi\)
0.000586729 1.00000i \(0.499813\pi\)
\(864\) 0 0
\(865\) 25.0351 0.851218
\(866\) 22.4243 0.762008
\(867\) 0 0
\(868\) 6.12836 0.208010
\(869\) 15.0351 0.510030
\(870\) 0 0
\(871\) 18.7202 0.634309
\(872\) 15.2763 0.517321
\(873\) 0 0
\(874\) 58.1011 1.96530
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 36.9377 1.24730 0.623649 0.781705i \(-0.285650\pi\)
0.623649 + 0.781705i \(0.285650\pi\)
\(878\) 10.3851 0.350479
\(879\) 0 0
\(880\) 1.00000 0.0337100
\(881\) 4.66550 0.157185 0.0785923 0.996907i \(-0.474957\pi\)
0.0785923 + 0.996907i \(0.474957\pi\)
\(882\) 0 0
\(883\) −0.754652 −0.0253961 −0.0126980 0.999919i \(-0.504042\pi\)
−0.0126980 + 0.999919i \(0.504042\pi\)
\(884\) −6.61081 −0.222346
\(885\) 0 0
\(886\) 23.1242 0.776874
\(887\) 3.44387 0.115634 0.0578169 0.998327i \(-0.481586\pi\)
0.0578169 + 0.998327i \(0.481586\pi\)
\(888\) 0 0
\(889\) −2.12836 −0.0713828
\(890\) −16.6655 −0.558629
\(891\) 0 0
\(892\) 3.26083 0.109181
\(893\) −48.7701 −1.63203
\(894\) 0 0
\(895\) −17.1634 −0.573710
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 19.0351 0.635209
\(899\) 58.3269 1.94531
\(900\) 0 0
\(901\) −1.07428 −0.0357895
\(902\) 4.12836 0.137459
\(903\) 0 0
\(904\) 5.88713 0.195803
\(905\) −24.7939 −0.824176
\(906\) 0 0
\(907\) 12.2412 0.406463 0.203232 0.979131i \(-0.434856\pi\)
0.203232 + 0.979131i \(0.434856\pi\)
\(908\) 24.4243 0.810548
\(909\) 0 0
\(910\) −3.75877 −0.124602
\(911\) −50.2567 −1.66508 −0.832540 0.553966i \(-0.813114\pi\)
−0.832540 + 0.553966i \(0.813114\pi\)
\(912\) 0 0
\(913\) −8.90673 −0.294770
\(914\) −38.1985 −1.26349
\(915\) 0 0
\(916\) −2.24123 −0.0740523
\(917\) −3.51754 −0.116159
\(918\) 0 0
\(919\) 26.9567 0.889219 0.444609 0.895725i \(-0.353342\pi\)
0.444609 + 0.895725i \(0.353342\pi\)
\(920\) −7.14796 −0.235661
\(921\) 0 0
\(922\) 12.6108 0.415315
\(923\) −48.9959 −1.61272
\(924\) 0 0
\(925\) −5.75877 −0.189347
\(926\) 2.55262 0.0838844
\(927\) 0 0
\(928\) −9.51754 −0.312429
\(929\) −15.3655 −0.504125 −0.252062 0.967711i \(-0.581109\pi\)
−0.252062 + 0.967711i \(0.581109\pi\)
\(930\) 0 0
\(931\) −8.12836 −0.266396
\(932\) 4.77837 0.156521
\(933\) 0 0
\(934\) −13.8716 −0.453894
\(935\) −1.75877 −0.0575179
\(936\) 0 0
\(937\) −34.1985 −1.11722 −0.558608 0.829431i \(-0.688665\pi\)
−0.558608 + 0.829431i \(0.688665\pi\)
\(938\) −4.98040 −0.162616
\(939\) 0 0
\(940\) 6.00000 0.195698
\(941\) 4.55262 0.148411 0.0742056 0.997243i \(-0.476358\pi\)
0.0742056 + 0.997243i \(0.476358\pi\)
\(942\) 0 0
\(943\) −29.5093 −0.960955
\(944\) 10.1284 0.329650
\(945\) 0 0
\(946\) 8.12836 0.264276
\(947\) −19.8324 −0.644468 −0.322234 0.946660i \(-0.604434\pi\)
−0.322234 + 0.946660i \(0.604434\pi\)
\(948\) 0 0
\(949\) −30.5526 −0.991780
\(950\) 8.12836 0.263719
\(951\) 0 0
\(952\) 1.75877 0.0570021
\(953\) −53.7743 −1.74192 −0.870959 0.491355i \(-0.836502\pi\)
−0.870959 + 0.491355i \(0.836502\pi\)
\(954\) 0 0
\(955\) −9.51754 −0.307980
\(956\) 28.0155 0.906085
\(957\) 0 0
\(958\) 28.5134 0.921227
\(959\) 13.1480 0.424570
\(960\) 0 0
\(961\) 6.55674 0.211508
\(962\) 21.6459 0.697892
\(963\) 0 0
\(964\) −18.9067 −0.608945
\(965\) 12.1284 0.390426
\(966\) 0 0
\(967\) −2.12836 −0.0684433 −0.0342217 0.999414i \(-0.510895\pi\)
−0.0342217 + 0.999414i \(0.510895\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) −15.6459 −0.502359
\(971\) 47.3500 1.51953 0.759767 0.650196i \(-0.225313\pi\)
0.759767 + 0.650196i \(0.225313\pi\)
\(972\) 0 0
\(973\) 3.64590 0.116882
\(974\) −40.2567 −1.28991
\(975\) 0 0
\(976\) −4.00000 −0.128037
\(977\) 11.7006 0.374335 0.187167 0.982328i \(-0.440069\pi\)
0.187167 + 0.982328i \(0.440069\pi\)
\(978\) 0 0
\(979\) 16.6655 0.532632
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) 0.225748 0.00720391
\(983\) 6.00000 0.191370 0.0956851 0.995412i \(-0.469496\pi\)
0.0956851 + 0.995412i \(0.469496\pi\)
\(984\) 0 0
\(985\) −14.9067 −0.474968
\(986\) 16.7392 0.533084
\(987\) 0 0
\(988\) −30.5526 −0.972008
\(989\) −58.1011 −1.84751
\(990\) 0 0
\(991\) 56.4243 1.79238 0.896188 0.443675i \(-0.146325\pi\)
0.896188 + 0.443675i \(0.146325\pi\)
\(992\) −6.12836 −0.194575
\(993\) 0 0
\(994\) 13.0351 0.413448
\(995\) 21.6459 0.686221
\(996\) 0 0
\(997\) 20.4979 0.649176 0.324588 0.945855i \(-0.394774\pi\)
0.324588 + 0.945855i \(0.394774\pi\)
\(998\) −9.07428 −0.287241
\(999\) 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 6930.2.a.ci.1.3 3
3.2 odd 2 6930.2.a.cj.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6930.2.a.ci.1.3 3 1.1 even 1 trivial
6930.2.a.cj.1.3 yes 3 3.2 odd 2