Properties

Label 6930.2.a.cj.1.3
Level $6930$
Weight $2$
Character 6930.1
Self dual yes
Analytic conductor $55.336$
Analytic rank $1$
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: \(1\)
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 \(-0.431585\pi\)
0.213282 + 0.976991i \(0.431585\pi\)
\(18\) 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 \(-0.767658\pi\)
−0.745226 + 0.666812i \(0.767658\pi\)
\(24\) 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 \(-0.844943\pi\)
−0.883681 + 0.468089i \(0.844943\pi\)
\(30\) 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 \(-0.395520\pi\)
0.322370 + 0.946614i \(0.395520\pi\)
\(42\) 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 \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 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 \(-0.513357\pi\)
−0.0419509 + 0.999120i \(0.513357\pi\)
\(54\) 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 \(-0.729147\pi\)
−0.659300 + 0.751880i \(0.729147\pi\)
\(60\) 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 \(-0.218509\pi\)
0.773490 + 0.633809i \(0.218509\pi\)
\(72\) 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 \(-0.337427\pi\)
0.488820 + 0.872385i \(0.337427\pi\)
\(84\) 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 \(-0.844663\pi\)
−0.883270 + 0.468866i \(0.844663\pi\)
\(90\) 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 \(-0.679422\pi\)
−0.534294 + 0.845299i \(0.679422\pi\)
\(102\) 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 \(-0.755239\pi\)
−0.718649 + 0.695373i \(0.755239\pi\)
\(108\) 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 \(-0.410691\pi\)
0.276907 + 0.960897i \(0.410691\pi\)
\(114\) 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 \(-0.450893\pi\)
0.153664 + 0.988123i \(0.450893\pi\)
\(132\) 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 \(-0.689834\pi\)
−0.561653 + 0.827373i \(0.689834\pi\)
\(138\) 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 \(-0.372526\pi\)
0.389854 + 0.920877i \(0.372526\pi\)
\(150\) 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 \(-0.576005\pi\)
−0.236514 + 0.971628i \(0.576005\pi\)
\(168\) 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 \(-0.900656\pi\)
−0.951691 + 0.307057i \(0.900656\pi\)
\(174\) 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 \(-0.278342\pi\)
0.641428 + 0.767183i \(0.278342\pi\)
\(180\) 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 \(-0.388105\pi\)
0.344333 + 0.938848i \(0.388105\pi\)
\(192\) 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 \(-0.321805\pi\)
0.531030 + 0.847353i \(0.321805\pi\)
\(198\) 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 \(-0.800830\pi\)
−0.810548 + 0.585673i \(0.800830\pi\)
\(228\) 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 \(-0.550028\pi\)
−0.156521 + 0.987675i \(0.550028\pi\)
\(234\) 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 \(-0.860944\pi\)
−0.906085 + 0.423095i \(0.860944\pi\)
\(240\) 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 \(-0.541988\pi\)
−0.131527 + 0.991313i \(0.541988\pi\)
\(252\) 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 \(-0.795100\pi\)
−0.799874 + 0.600169i \(0.795100\pi\)
\(258\) 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 \(-0.341207\pi\)
0.478426 + 0.878128i \(0.341207\pi\)
\(264\) 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 \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 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 \(-0.162015\pi\)
0.873239 + 0.487292i \(0.162015\pi\)
\(282\) 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 \(-0.687234\pi\)
−0.554876 + 0.831933i \(0.687234\pi\)
\(294\) 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 \(-0.650428\pi\)
−0.455189 + 0.890395i \(0.650428\pi\)
\(312\) 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 \(-0.559796\pi\)
−0.186751 + 0.982407i \(0.559796\pi\)
\(318\) 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 \(-0.700044\pi\)
−0.587897 + 0.808936i \(0.700044\pi\)
\(348\) 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 \(-0.629484\pi\)
−0.395659 + 0.918397i \(0.629484\pi\)
\(354\) 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 \(-0.348946\pi\)
0.456939 + 0.889498i \(0.348946\pi\)
\(360\) 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 \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) 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 \(-0.447928\pi\)
0.162862 + 0.986649i \(0.447928\pi\)
\(390\) 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 \(-0.532460\pi\)
−0.101799 + 0.994805i \(0.532460\pi\)
\(420\) 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 \(-0.657868\pi\)
−0.475873 + 0.879514i \(0.657868\pi\)
\(432\) 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 \(-0.314882\pi\)
0.549333 + 0.835604i \(0.314882\pi\)
\(444\) 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 \(-0.351723\pi\)
0.449161 + 0.893451i \(0.351723\pi\)
\(450\) 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 \(-0.405123\pi\)
0.293672 + 0.955906i \(0.405123\pi\)
\(462\) 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 \(-0.604003\pi\)
−0.320952 + 0.947096i \(0.604003\pi\)
\(468\) 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 \(-0.274180\pi\)
0.651406 + 0.758730i \(0.274180\pi\)
\(480\) 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 \(-0.498379\pi\)
0.00509393 + 0.999987i \(0.498379\pi\)
\(492\) 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 \(-0.247584\pi\)
0.712453 + 0.701720i \(0.247584\pi\)
\(504\) 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 \(-0.467829\pi\)
0.100896 + 0.994897i \(0.467829\pi\)
\(510\) 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 \(-0.465829\pi\)
0.107145 + 0.994243i \(0.465829\pi\)
\(522\) 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 \(-0.336729\pi\)
0.490733 + 0.871310i \(0.336729\pi\)
\(558\) 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 \(-0.208365\pi\)
0.793293 + 0.608840i \(0.208365\pi\)
\(564\) 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 \(-0.636214\pi\)
−0.414987 + 0.909827i \(0.636214\pi\)
\(570\) 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 \(-0.737631\pi\)
−0.679103 + 0.734043i \(0.737631\pi\)
\(588\) 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 \(-0.852574\pi\)
−0.894648 + 0.446771i \(0.852574\pi\)
\(594\) 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 \(-0.975953\pi\)
−0.997148 + 0.0754754i \(0.975953\pi\)
\(600\) 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 \(-0.523030\pi\)
−0.0722884 + 0.997384i \(0.523030\pi\)
\(618\) 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 \(-0.747728\pi\)
−0.702041 + 0.712137i \(0.747728\pi\)
\(642\) 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 \(-0.425402\pi\)
0.232217 + 0.972664i \(0.425402\pi\)
\(648\) 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 \(-0.870524\pi\)
−0.918407 + 0.395637i \(0.870524\pi\)
\(654\) 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 \(-0.492425\pi\)
0.0237940 + 0.999717i \(0.492425\pi\)
\(660\) 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 \(-0.426178\pi\)
0.229845 + 0.973227i \(0.426178\pi\)
\(678\) 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 \(-0.385079\pi\)
0.353242 + 0.935532i \(0.385079\pi\)
\(684\) 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 \(-0.663583\pi\)
−0.491588 + 0.870828i \(0.663583\pi\)
\(702\) 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 \(-0.613526\pi\)
−0.349138 + 0.937071i \(0.613526\pi\)
\(720\) 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 \(-0.541192\pi\)
−0.129046 + 0.991639i \(0.541192\pi\)
\(744\) 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 \(-0.461540\pi\)
0.120531 + 0.992710i \(0.461540\pi\)
\(762\) 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 \(-0.385057\pi\)
0.353307 + 0.935507i \(0.385057\pi\)
\(774\) 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 \(-0.416273\pi\)
0.260013 + 0.965605i \(0.416273\pi\)
\(798\) 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 \(-0.656359\pi\)
−0.471700 + 0.881759i \(0.656359\pi\)
\(810\) 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 \(-0.231152\pi\)
0.747712 + 0.664023i \(0.231152\pi\)
\(822\) 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 \(-0.609863\pi\)
−0.338334 + 0.941026i \(0.609863\pi\)
\(828\) 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 \(-0.779410\pi\)
−0.769331 + 0.638850i \(0.779410\pi\)
\(840\) 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 \(-0.404656\pi\)
0.295074 + 0.955474i \(0.404656\pi\)
\(858\) 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 \(-0.500187\pi\)
−0.000586729 1.00000i \(0.500187\pi\)
\(864\) 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 \(-0.525043\pi\)
−0.0785923 + 0.996907i \(0.525043\pi\)
\(882\) 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 \(-0.518414\pi\)
−0.0578169 + 0.998327i \(0.518414\pi\)
\(888\) 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 \(-0.186886\pi\)
0.832540 + 0.553966i \(0.186886\pi\)
\(912\) 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 \(-0.418891\pi\)
0.252062 + 0.967711i \(0.418891\pi\)
\(930\) 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 \(-0.523642\pi\)
−0.0742056 + 0.997243i \(0.523642\pi\)
\(942\) 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 \(-0.395566\pi\)
0.322234 + 0.946660i \(0.395566\pi\)
\(948\) 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 \(-0.163498\pi\)
0.870959 + 0.491355i \(0.163498\pi\)
\(954\) 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 \(-0.774687\pi\)
−0.759767 + 0.650196i \(0.774687\pi\)
\(972\) 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 \(-0.559931\pi\)
−0.187167 + 0.982328i \(0.559931\pi\)
\(978\) 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 \(-0.530504\pi\)
−0.0956851 + 0.995412i \(0.530504\pi\)
\(984\) 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.cj.1.3 yes 3
3.2 odd 2 6930.2.a.ci.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6930.2.a.ci.1.3 3 3.2 odd 2
6930.2.a.cj.1.3 yes 3 1.1 even 1 trivial