Properties

Label 6930.2.a.bs.1.1
Level $6930$
Weight $2$
Character 6930.1
Self dual yes
Analytic conductor $55.336$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 6930.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.12311 q^{17} +1.12311 q^{19} +1.00000 q^{20} -1.00000 q^{22} -7.12311 q^{23} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{28} +8.24621 q^{29} -7.12311 q^{31} -1.00000 q^{32} +3.12311 q^{34} -1.00000 q^{35} +7.12311 q^{37} -1.12311 q^{38} -1.00000 q^{40} -9.12311 q^{41} -5.12311 q^{43} +1.00000 q^{44} +7.12311 q^{46} +11.3693 q^{47} +1.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} -10.0000 q^{53} +1.00000 q^{55} +1.00000 q^{56} -8.24621 q^{58} -4.00000 q^{59} +3.12311 q^{61} +7.12311 q^{62} +1.00000 q^{64} +2.00000 q^{65} -9.12311 q^{67} -3.12311 q^{68} +1.00000 q^{70} +1.12311 q^{71} -0.246211 q^{73} -7.12311 q^{74} +1.12311 q^{76} -1.00000 q^{77} -3.12311 q^{79} +1.00000 q^{80} +9.12311 q^{82} -11.1231 q^{83} -3.12311 q^{85} +5.12311 q^{86} -1.00000 q^{88} -8.24621 q^{89} -2.00000 q^{91} -7.12311 q^{92} -11.3693 q^{94} +1.12311 q^{95} -2.87689 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8} - 2 q^{10} + 2 q^{11} + 4 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} - 6 q^{19} + 2 q^{20} - 2 q^{22} - 6 q^{23} + 2 q^{25} - 4 q^{26} - 2 q^{28} - 6 q^{31} - 2 q^{32} - 2 q^{34} - 2 q^{35} + 6 q^{37} + 6 q^{38} - 2 q^{40} - 10 q^{41} - 2 q^{43} + 2 q^{44} + 6 q^{46} - 2 q^{47} + 2 q^{49} - 2 q^{50} + 4 q^{52} - 20 q^{53} + 2 q^{55} + 2 q^{56} - 8 q^{59} - 2 q^{61} + 6 q^{62} + 2 q^{64} + 4 q^{65} - 10 q^{67} + 2 q^{68} + 2 q^{70} - 6 q^{71} + 16 q^{73} - 6 q^{74} - 6 q^{76} - 2 q^{77} + 2 q^{79} + 2 q^{80} + 10 q^{82} - 14 q^{83} + 2 q^{85} + 2 q^{86} - 2 q^{88} - 4 q^{91} - 6 q^{92} + 2 q^{94} - 6 q^{95} - 14 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 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
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.12311 −0.757464 −0.378732 0.925506i \(-0.623640\pi\)
−0.378732 + 0.925506i \(0.623640\pi\)
\(18\) 0 0
\(19\) 1.12311 0.257658 0.128829 0.991667i \(-0.458878\pi\)
0.128829 + 0.991667i \(0.458878\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −7.12311 −1.48527 −0.742635 0.669696i \(-0.766424\pi\)
−0.742635 + 0.669696i \(0.766424\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 8.24621 1.53128 0.765641 0.643268i \(-0.222422\pi\)
0.765641 + 0.643268i \(0.222422\pi\)
\(30\) 0 0
\(31\) −7.12311 −1.27935 −0.639674 0.768647i \(-0.720931\pi\)
−0.639674 + 0.768647i \(0.720931\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 3.12311 0.535608
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 7.12311 1.17103 0.585516 0.810661i \(-0.300892\pi\)
0.585516 + 0.810661i \(0.300892\pi\)
\(38\) −1.12311 −0.182192
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −9.12311 −1.42479 −0.712395 0.701779i \(-0.752389\pi\)
−0.712395 + 0.701779i \(0.752389\pi\)
\(42\) 0 0
\(43\) −5.12311 −0.781266 −0.390633 0.920546i \(-0.627744\pi\)
−0.390633 + 0.920546i \(0.627744\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 7.12311 1.05024
\(47\) 11.3693 1.65839 0.829193 0.558963i \(-0.188801\pi\)
0.829193 + 0.558963i \(0.188801\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −8.24621 −1.08278
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 3.12311 0.399873 0.199936 0.979809i \(-0.435926\pi\)
0.199936 + 0.979809i \(0.435926\pi\)
\(62\) 7.12311 0.904635
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −9.12311 −1.11456 −0.557282 0.830323i \(-0.688156\pi\)
−0.557282 + 0.830323i \(0.688156\pi\)
\(68\) −3.12311 −0.378732
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) 1.12311 0.133288 0.0666441 0.997777i \(-0.478771\pi\)
0.0666441 + 0.997777i \(0.478771\pi\)
\(72\) 0 0
\(73\) −0.246211 −0.0288168 −0.0144084 0.999896i \(-0.504587\pi\)
−0.0144084 + 0.999896i \(0.504587\pi\)
\(74\) −7.12311 −0.828044
\(75\) 0 0
\(76\) 1.12311 0.128829
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −3.12311 −0.351377 −0.175688 0.984446i \(-0.556215\pi\)
−0.175688 + 0.984446i \(0.556215\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 9.12311 1.00748
\(83\) −11.1231 −1.22092 −0.610460 0.792047i \(-0.709015\pi\)
−0.610460 + 0.792047i \(0.709015\pi\)
\(84\) 0 0
\(85\) −3.12311 −0.338748
\(86\) 5.12311 0.552439
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) −8.24621 −0.874097 −0.437048 0.899438i \(-0.643976\pi\)
−0.437048 + 0.899438i \(0.643976\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) −7.12311 −0.742635
\(93\) 0 0
\(94\) −11.3693 −1.17266
\(95\) 1.12311 0.115228
\(96\) 0 0
\(97\) −2.87689 −0.292104 −0.146052 0.989277i \(-0.546657\pi\)
−0.146052 + 0.989277i \(0.546657\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 11.3693 1.13129 0.565645 0.824649i \(-0.308627\pi\)
0.565645 + 0.824649i \(0.308627\pi\)
\(102\) 0 0
\(103\) 14.2462 1.40372 0.701860 0.712314i \(-0.252353\pi\)
0.701860 + 0.712314i \(0.252353\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) −18.2462 −1.76393 −0.881964 0.471317i \(-0.843779\pi\)
−0.881964 + 0.471317i \(0.843779\pi\)
\(108\) 0 0
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) −7.12311 −0.664233
\(116\) 8.24621 0.765641
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) 3.12311 0.286295
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −3.12311 −0.282753
\(123\) 0 0
\(124\) −7.12311 −0.639674
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −1.75379 −0.155624 −0.0778118 0.996968i \(-0.524793\pi\)
−0.0778118 + 0.996968i \(0.524793\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −2.00000 −0.175412
\(131\) 5.75379 0.502711 0.251355 0.967895i \(-0.419124\pi\)
0.251355 + 0.967895i \(0.419124\pi\)
\(132\) 0 0
\(133\) −1.12311 −0.0973856
\(134\) 9.12311 0.788116
\(135\) 0 0
\(136\) 3.12311 0.267804
\(137\) −20.2462 −1.72975 −0.864875 0.501987i \(-0.832603\pi\)
−0.864875 + 0.501987i \(0.832603\pi\)
\(138\) 0 0
\(139\) −9.12311 −0.773812 −0.386906 0.922119i \(-0.626456\pi\)
−0.386906 + 0.922119i \(0.626456\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) −1.12311 −0.0942489
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) 8.24621 0.684811
\(146\) 0.246211 0.0203766
\(147\) 0 0
\(148\) 7.12311 0.585516
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) 12.8769 1.04791 0.523953 0.851747i \(-0.324457\pi\)
0.523953 + 0.851747i \(0.324457\pi\)
\(152\) −1.12311 −0.0910959
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) −7.12311 −0.572142
\(156\) 0 0
\(157\) 5.12311 0.408868 0.204434 0.978880i \(-0.434465\pi\)
0.204434 + 0.978880i \(0.434465\pi\)
\(158\) 3.12311 0.248461
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 7.12311 0.561379
\(162\) 0 0
\(163\) 3.36932 0.263905 0.131953 0.991256i \(-0.457875\pi\)
0.131953 + 0.991256i \(0.457875\pi\)
\(164\) −9.12311 −0.712395
\(165\) 0 0
\(166\) 11.1231 0.863320
\(167\) −14.0000 −1.08335 −0.541676 0.840587i \(-0.682210\pi\)
−0.541676 + 0.840587i \(0.682210\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 3.12311 0.239531
\(171\) 0 0
\(172\) −5.12311 −0.390633
\(173\) 4.24621 0.322833 0.161417 0.986886i \(-0.448394\pi\)
0.161417 + 0.986886i \(0.448394\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 8.24621 0.618080
\(179\) −14.2462 −1.06481 −0.532406 0.846489i \(-0.678712\pi\)
−0.532406 + 0.846489i \(0.678712\pi\)
\(180\) 0 0
\(181\) −5.36932 −0.399098 −0.199549 0.979888i \(-0.563948\pi\)
−0.199549 + 0.979888i \(0.563948\pi\)
\(182\) 2.00000 0.148250
\(183\) 0 0
\(184\) 7.12311 0.525122
\(185\) 7.12311 0.523701
\(186\) 0 0
\(187\) −3.12311 −0.228384
\(188\) 11.3693 0.829193
\(189\) 0 0
\(190\) −1.12311 −0.0814786
\(191\) −6.87689 −0.497595 −0.248797 0.968556i \(-0.580035\pi\)
−0.248797 + 0.968556i \(0.580035\pi\)
\(192\) 0 0
\(193\) 2.87689 0.207083 0.103542 0.994625i \(-0.466982\pi\)
0.103542 + 0.994625i \(0.466982\pi\)
\(194\) 2.87689 0.206549
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −22.4924 −1.60252 −0.801259 0.598317i \(-0.795836\pi\)
−0.801259 + 0.598317i \(0.795836\pi\)
\(198\) 0 0
\(199\) −4.87689 −0.345714 −0.172857 0.984947i \(-0.555300\pi\)
−0.172857 + 0.984947i \(0.555300\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −11.3693 −0.799942
\(203\) −8.24621 −0.578771
\(204\) 0 0
\(205\) −9.12311 −0.637185
\(206\) −14.2462 −0.992581
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 1.12311 0.0776868
\(210\) 0 0
\(211\) 8.87689 0.611111 0.305555 0.952174i \(-0.401158\pi\)
0.305555 + 0.952174i \(0.401158\pi\)
\(212\) −10.0000 −0.686803
\(213\) 0 0
\(214\) 18.2462 1.24729
\(215\) −5.12311 −0.349393
\(216\) 0 0
\(217\) 7.12311 0.483548
\(218\) −18.0000 −1.21911
\(219\) 0 0
\(220\) 1.00000 0.0674200
\(221\) −6.24621 −0.420166
\(222\) 0 0
\(223\) 4.49242 0.300835 0.150417 0.988623i \(-0.451938\pi\)
0.150417 + 0.988623i \(0.451938\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 2.00000 0.133038
\(227\) −21.3693 −1.41833 −0.709166 0.705042i \(-0.750928\pi\)
−0.709166 + 0.705042i \(0.750928\pi\)
\(228\) 0 0
\(229\) 11.1231 0.735036 0.367518 0.930017i \(-0.380208\pi\)
0.367518 + 0.930017i \(0.380208\pi\)
\(230\) 7.12311 0.469684
\(231\) 0 0
\(232\) −8.24621 −0.541390
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 11.3693 0.741652
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) −3.12311 −0.202441
\(239\) 7.36932 0.476681 0.238341 0.971182i \(-0.423396\pi\)
0.238341 + 0.971182i \(0.423396\pi\)
\(240\) 0 0
\(241\) 22.4924 1.44886 0.724432 0.689346i \(-0.242102\pi\)
0.724432 + 0.689346i \(0.242102\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 3.12311 0.199936
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 2.24621 0.142923
\(248\) 7.12311 0.452318
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −16.4924 −1.04099 −0.520496 0.853864i \(-0.674253\pi\)
−0.520496 + 0.853864i \(0.674253\pi\)
\(252\) 0 0
\(253\) −7.12311 −0.447826
\(254\) 1.75379 0.110042
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 15.6155 0.974070 0.487035 0.873382i \(-0.338078\pi\)
0.487035 + 0.873382i \(0.338078\pi\)
\(258\) 0 0
\(259\) −7.12311 −0.442608
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) −5.75379 −0.355470
\(263\) 8.49242 0.523665 0.261833 0.965113i \(-0.415673\pi\)
0.261833 + 0.965113i \(0.415673\pi\)
\(264\) 0 0
\(265\) −10.0000 −0.614295
\(266\) 1.12311 0.0688620
\(267\) 0 0
\(268\) −9.12311 −0.557282
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) −3.12311 −0.189366
\(273\) 0 0
\(274\) 20.2462 1.22312
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −10.2462 −0.615635 −0.307818 0.951445i \(-0.599599\pi\)
−0.307818 + 0.951445i \(0.599599\pi\)
\(278\) 9.12311 0.547168
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) −13.3693 −0.797547 −0.398773 0.917050i \(-0.630564\pi\)
−0.398773 + 0.917050i \(0.630564\pi\)
\(282\) 0 0
\(283\) 14.2462 0.846849 0.423425 0.905931i \(-0.360828\pi\)
0.423425 + 0.905931i \(0.360828\pi\)
\(284\) 1.12311 0.0666441
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) 9.12311 0.538520
\(288\) 0 0
\(289\) −7.24621 −0.426248
\(290\) −8.24621 −0.484234
\(291\) 0 0
\(292\) −0.246211 −0.0144084
\(293\) 10.0000 0.584206 0.292103 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) −7.12311 −0.414022
\(297\) 0 0
\(298\) 2.00000 0.115857
\(299\) −14.2462 −0.823880
\(300\) 0 0
\(301\) 5.12311 0.295291
\(302\) −12.8769 −0.740982
\(303\) 0 0
\(304\) 1.12311 0.0644145
\(305\) 3.12311 0.178829
\(306\) 0 0
\(307\) −34.7386 −1.98264 −0.991319 0.131477i \(-0.958028\pi\)
−0.991319 + 0.131477i \(0.958028\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) 7.12311 0.404565
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) −17.6155 −0.995689 −0.497844 0.867266i \(-0.665875\pi\)
−0.497844 + 0.867266i \(0.665875\pi\)
\(314\) −5.12311 −0.289114
\(315\) 0 0
\(316\) −3.12311 −0.175688
\(317\) 8.24621 0.463153 0.231577 0.972817i \(-0.425612\pi\)
0.231577 + 0.972817i \(0.425612\pi\)
\(318\) 0 0
\(319\) 8.24621 0.461699
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −7.12311 −0.396955
\(323\) −3.50758 −0.195167
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) −3.36932 −0.186609
\(327\) 0 0
\(328\) 9.12311 0.503739
\(329\) −11.3693 −0.626811
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) −11.1231 −0.610460
\(333\) 0 0
\(334\) 14.0000 0.766046
\(335\) −9.12311 −0.498449
\(336\) 0 0
\(337\) 29.6155 1.61326 0.806630 0.591056i \(-0.201289\pi\)
0.806630 + 0.591056i \(0.201289\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) −3.12311 −0.169374
\(341\) −7.12311 −0.385738
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 5.12311 0.276219
\(345\) 0 0
\(346\) −4.24621 −0.228278
\(347\) 30.7386 1.65014 0.825068 0.565033i \(-0.191137\pi\)
0.825068 + 0.565033i \(0.191137\pi\)
\(348\) 0 0
\(349\) −5.36932 −0.287413 −0.143706 0.989620i \(-0.545902\pi\)
−0.143706 + 0.989620i \(0.545902\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) 15.6155 0.831131 0.415565 0.909563i \(-0.363584\pi\)
0.415565 + 0.909563i \(0.363584\pi\)
\(354\) 0 0
\(355\) 1.12311 0.0596083
\(356\) −8.24621 −0.437048
\(357\) 0 0
\(358\) 14.2462 0.752936
\(359\) 7.36932 0.388938 0.194469 0.980909i \(-0.437702\pi\)
0.194469 + 0.980909i \(0.437702\pi\)
\(360\) 0 0
\(361\) −17.7386 −0.933612
\(362\) 5.36932 0.282205
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) −0.246211 −0.0128873
\(366\) 0 0
\(367\) −28.4924 −1.48729 −0.743646 0.668573i \(-0.766905\pi\)
−0.743646 + 0.668573i \(0.766905\pi\)
\(368\) −7.12311 −0.371318
\(369\) 0 0
\(370\) −7.12311 −0.370313
\(371\) 10.0000 0.519174
\(372\) 0 0
\(373\) −22.2462 −1.15187 −0.575933 0.817497i \(-0.695361\pi\)
−0.575933 + 0.817497i \(0.695361\pi\)
\(374\) 3.12311 0.161492
\(375\) 0 0
\(376\) −11.3693 −0.586328
\(377\) 16.4924 0.849403
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 1.12311 0.0576141
\(381\) 0 0
\(382\) 6.87689 0.351853
\(383\) −5.61553 −0.286940 −0.143470 0.989655i \(-0.545826\pi\)
−0.143470 + 0.989655i \(0.545826\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) −2.87689 −0.146430
\(387\) 0 0
\(388\) −2.87689 −0.146052
\(389\) −11.3693 −0.576447 −0.288224 0.957563i \(-0.593065\pi\)
−0.288224 + 0.957563i \(0.593065\pi\)
\(390\) 0 0
\(391\) 22.2462 1.12504
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 22.4924 1.13315
\(395\) −3.12311 −0.157140
\(396\) 0 0
\(397\) −27.3693 −1.37363 −0.686813 0.726834i \(-0.740991\pi\)
−0.686813 + 0.726834i \(0.740991\pi\)
\(398\) 4.87689 0.244457
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 32.0000 1.59800 0.799002 0.601329i \(-0.205362\pi\)
0.799002 + 0.601329i \(0.205362\pi\)
\(402\) 0 0
\(403\) −14.2462 −0.709654
\(404\) 11.3693 0.565645
\(405\) 0 0
\(406\) 8.24621 0.409253
\(407\) 7.12311 0.353079
\(408\) 0 0
\(409\) 16.2462 0.803323 0.401662 0.915788i \(-0.368433\pi\)
0.401662 + 0.915788i \(0.368433\pi\)
\(410\) 9.12311 0.450558
\(411\) 0 0
\(412\) 14.2462 0.701860
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) −11.1231 −0.546012
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) −1.12311 −0.0549329
\(419\) 0.492423 0.0240564 0.0120282 0.999928i \(-0.496171\pi\)
0.0120282 + 0.999928i \(0.496171\pi\)
\(420\) 0 0
\(421\) −8.24621 −0.401896 −0.200948 0.979602i \(-0.564402\pi\)
−0.200948 + 0.979602i \(0.564402\pi\)
\(422\) −8.87689 −0.432120
\(423\) 0 0
\(424\) 10.0000 0.485643
\(425\) −3.12311 −0.151493
\(426\) 0 0
\(427\) −3.12311 −0.151138
\(428\) −18.2462 −0.881964
\(429\) 0 0
\(430\) 5.12311 0.247058
\(431\) 17.6155 0.848510 0.424255 0.905543i \(-0.360536\pi\)
0.424255 + 0.905543i \(0.360536\pi\)
\(432\) 0 0
\(433\) −1.12311 −0.0539730 −0.0269865 0.999636i \(-0.508591\pi\)
−0.0269865 + 0.999636i \(0.508591\pi\)
\(434\) −7.12311 −0.341920
\(435\) 0 0
\(436\) 18.0000 0.862044
\(437\) −8.00000 −0.382692
\(438\) 0 0
\(439\) −22.7386 −1.08526 −0.542628 0.839973i \(-0.682571\pi\)
−0.542628 + 0.839973i \(0.682571\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 0 0
\(442\) 6.24621 0.297102
\(443\) −27.6155 −1.31205 −0.656027 0.754738i \(-0.727764\pi\)
−0.656027 + 0.754738i \(0.727764\pi\)
\(444\) 0 0
\(445\) −8.24621 −0.390908
\(446\) −4.49242 −0.212722
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −30.7386 −1.45065 −0.725323 0.688409i \(-0.758310\pi\)
−0.725323 + 0.688409i \(0.758310\pi\)
\(450\) 0 0
\(451\) −9.12311 −0.429590
\(452\) −2.00000 −0.0940721
\(453\) 0 0
\(454\) 21.3693 1.00291
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) 11.3693 0.531834 0.265917 0.963996i \(-0.414325\pi\)
0.265917 + 0.963996i \(0.414325\pi\)
\(458\) −11.1231 −0.519749
\(459\) 0 0
\(460\) −7.12311 −0.332117
\(461\) 13.1231 0.611204 0.305602 0.952159i \(-0.401142\pi\)
0.305602 + 0.952159i \(0.401142\pi\)
\(462\) 0 0
\(463\) 22.7386 1.05675 0.528377 0.849010i \(-0.322801\pi\)
0.528377 + 0.849010i \(0.322801\pi\)
\(464\) 8.24621 0.382821
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 12.4924 0.578080 0.289040 0.957317i \(-0.406664\pi\)
0.289040 + 0.957317i \(0.406664\pi\)
\(468\) 0 0
\(469\) 9.12311 0.421266
\(470\) −11.3693 −0.524427
\(471\) 0 0
\(472\) 4.00000 0.184115
\(473\) −5.12311 −0.235561
\(474\) 0 0
\(475\) 1.12311 0.0515316
\(476\) 3.12311 0.143147
\(477\) 0 0
\(478\) −7.36932 −0.337065
\(479\) 16.4924 0.753558 0.376779 0.926303i \(-0.377032\pi\)
0.376779 + 0.926303i \(0.377032\pi\)
\(480\) 0 0
\(481\) 14.2462 0.649571
\(482\) −22.4924 −1.02450
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −2.87689 −0.130633
\(486\) 0 0
\(487\) 4.49242 0.203571 0.101786 0.994806i \(-0.467544\pi\)
0.101786 + 0.994806i \(0.467544\pi\)
\(488\) −3.12311 −0.141376
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) −6.24621 −0.281888 −0.140944 0.990018i \(-0.545014\pi\)
−0.140944 + 0.990018i \(0.545014\pi\)
\(492\) 0 0
\(493\) −25.7538 −1.15989
\(494\) −2.24621 −0.101062
\(495\) 0 0
\(496\) −7.12311 −0.319837
\(497\) −1.12311 −0.0503782
\(498\) 0 0
\(499\) −18.7386 −0.838856 −0.419428 0.907789i \(-0.637769\pi\)
−0.419428 + 0.907789i \(0.637769\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 16.4924 0.736093
\(503\) 0.246211 0.0109780 0.00548901 0.999985i \(-0.498253\pi\)
0.00548901 + 0.999985i \(0.498253\pi\)
\(504\) 0 0
\(505\) 11.3693 0.505928
\(506\) 7.12311 0.316661
\(507\) 0 0
\(508\) −1.75379 −0.0778118
\(509\) −40.2462 −1.78388 −0.891941 0.452152i \(-0.850657\pi\)
−0.891941 + 0.452152i \(0.850657\pi\)
\(510\) 0 0
\(511\) 0.246211 0.0108917
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −15.6155 −0.688771
\(515\) 14.2462 0.627763
\(516\) 0 0
\(517\) 11.3693 0.500022
\(518\) 7.12311 0.312971
\(519\) 0 0
\(520\) −2.00000 −0.0877058
\(521\) 12.2462 0.536516 0.268258 0.963347i \(-0.413552\pi\)
0.268258 + 0.963347i \(0.413552\pi\)
\(522\) 0 0
\(523\) 0.492423 0.0215321 0.0107661 0.999942i \(-0.496573\pi\)
0.0107661 + 0.999942i \(0.496573\pi\)
\(524\) 5.75379 0.251355
\(525\) 0 0
\(526\) −8.49242 −0.370287
\(527\) 22.2462 0.969060
\(528\) 0 0
\(529\) 27.7386 1.20603
\(530\) 10.0000 0.434372
\(531\) 0 0
\(532\) −1.12311 −0.0486928
\(533\) −18.2462 −0.790331
\(534\) 0 0
\(535\) −18.2462 −0.788853
\(536\) 9.12311 0.394058
\(537\) 0 0
\(538\) 14.0000 0.603583
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −11.7538 −0.505335 −0.252667 0.967553i \(-0.581308\pi\)
−0.252667 + 0.967553i \(0.581308\pi\)
\(542\) 12.0000 0.515444
\(543\) 0 0
\(544\) 3.12311 0.133902
\(545\) 18.0000 0.771035
\(546\) 0 0
\(547\) 5.61553 0.240103 0.120051 0.992768i \(-0.461694\pi\)
0.120051 + 0.992768i \(0.461694\pi\)
\(548\) −20.2462 −0.864875
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) 9.26137 0.394547
\(552\) 0 0
\(553\) 3.12311 0.132808
\(554\) 10.2462 0.435320
\(555\) 0 0
\(556\) −9.12311 −0.386906
\(557\) −32.7386 −1.38718 −0.693590 0.720370i \(-0.743972\pi\)
−0.693590 + 0.720370i \(0.743972\pi\)
\(558\) 0 0
\(559\) −10.2462 −0.433369
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 13.3693 0.563951
\(563\) 31.1231 1.31168 0.655841 0.754899i \(-0.272314\pi\)
0.655841 + 0.754899i \(0.272314\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) −14.2462 −0.598813
\(567\) 0 0
\(568\) −1.12311 −0.0471245
\(569\) −31.1231 −1.30475 −0.652374 0.757897i \(-0.726227\pi\)
−0.652374 + 0.757897i \(0.726227\pi\)
\(570\) 0 0
\(571\) 15.6155 0.653490 0.326745 0.945113i \(-0.394048\pi\)
0.326745 + 0.945113i \(0.394048\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) −9.12311 −0.380791
\(575\) −7.12311 −0.297054
\(576\) 0 0
\(577\) −22.8769 −0.952378 −0.476189 0.879343i \(-0.657982\pi\)
−0.476189 + 0.879343i \(0.657982\pi\)
\(578\) 7.24621 0.301403
\(579\) 0 0
\(580\) 8.24621 0.342405
\(581\) 11.1231 0.461464
\(582\) 0 0
\(583\) −10.0000 −0.414158
\(584\) 0.246211 0.0101883
\(585\) 0 0
\(586\) −10.0000 −0.413096
\(587\) −20.9848 −0.866137 −0.433069 0.901361i \(-0.642569\pi\)
−0.433069 + 0.901361i \(0.642569\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 4.00000 0.164677
\(591\) 0 0
\(592\) 7.12311 0.292758
\(593\) 35.1231 1.44233 0.721167 0.692762i \(-0.243606\pi\)
0.721167 + 0.692762i \(0.243606\pi\)
\(594\) 0 0
\(595\) 3.12311 0.128035
\(596\) −2.00000 −0.0819232
\(597\) 0 0
\(598\) 14.2462 0.582571
\(599\) −5.12311 −0.209324 −0.104662 0.994508i \(-0.533376\pi\)
−0.104662 + 0.994508i \(0.533376\pi\)
\(600\) 0 0
\(601\) 18.9848 0.774408 0.387204 0.921994i \(-0.373441\pi\)
0.387204 + 0.921994i \(0.373441\pi\)
\(602\) −5.12311 −0.208802
\(603\) 0 0
\(604\) 12.8769 0.523953
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) −32.9848 −1.33881 −0.669407 0.742896i \(-0.733452\pi\)
−0.669407 + 0.742896i \(0.733452\pi\)
\(608\) −1.12311 −0.0455479
\(609\) 0 0
\(610\) −3.12311 −0.126451
\(611\) 22.7386 0.919907
\(612\) 0 0
\(613\) 24.0000 0.969351 0.484675 0.874694i \(-0.338938\pi\)
0.484675 + 0.874694i \(0.338938\pi\)
\(614\) 34.7386 1.40194
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) −19.7538 −0.795258 −0.397629 0.917546i \(-0.630167\pi\)
−0.397629 + 0.917546i \(0.630167\pi\)
\(618\) 0 0
\(619\) −14.0000 −0.562708 −0.281354 0.959604i \(-0.590783\pi\)
−0.281354 + 0.959604i \(0.590783\pi\)
\(620\) −7.12311 −0.286071
\(621\) 0 0
\(622\) 8.00000 0.320771
\(623\) 8.24621 0.330377
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 17.6155 0.704058
\(627\) 0 0
\(628\) 5.12311 0.204434
\(629\) −22.2462 −0.887015
\(630\) 0 0
\(631\) −28.4924 −1.13427 −0.567133 0.823626i \(-0.691947\pi\)
−0.567133 + 0.823626i \(0.691947\pi\)
\(632\) 3.12311 0.124230
\(633\) 0 0
\(634\) −8.24621 −0.327499
\(635\) −1.75379 −0.0695970
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) −8.24621 −0.326471
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −30.7386 −1.21410 −0.607052 0.794662i \(-0.707648\pi\)
−0.607052 + 0.794662i \(0.707648\pi\)
\(642\) 0 0
\(643\) −38.2462 −1.50828 −0.754142 0.656712i \(-0.771947\pi\)
−0.754142 + 0.656712i \(0.771947\pi\)
\(644\) 7.12311 0.280690
\(645\) 0 0
\(646\) 3.50758 0.138004
\(647\) −18.8769 −0.742127 −0.371064 0.928607i \(-0.621007\pi\)
−0.371064 + 0.928607i \(0.621007\pi\)
\(648\) 0 0
\(649\) −4.00000 −0.157014
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) 3.36932 0.131953
\(653\) −48.7386 −1.90729 −0.953645 0.300934i \(-0.902702\pi\)
−0.953645 + 0.300934i \(0.902702\pi\)
\(654\) 0 0
\(655\) 5.75379 0.224819
\(656\) −9.12311 −0.356197
\(657\) 0 0
\(658\) 11.3693 0.443222
\(659\) −15.5076 −0.604089 −0.302045 0.953294i \(-0.597669\pi\)
−0.302045 + 0.953294i \(0.597669\pi\)
\(660\) 0 0
\(661\) −21.3693 −0.831170 −0.415585 0.909554i \(-0.636423\pi\)
−0.415585 + 0.909554i \(0.636423\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 11.1231 0.431660
\(665\) −1.12311 −0.0435522
\(666\) 0 0
\(667\) −58.7386 −2.27437
\(668\) −14.0000 −0.541676
\(669\) 0 0
\(670\) 9.12311 0.352456
\(671\) 3.12311 0.120566
\(672\) 0 0
\(673\) −39.8617 −1.53656 −0.768279 0.640116i \(-0.778886\pi\)
−0.768279 + 0.640116i \(0.778886\pi\)
\(674\) −29.6155 −1.14075
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 14.0000 0.538064 0.269032 0.963131i \(-0.413296\pi\)
0.269032 + 0.963131i \(0.413296\pi\)
\(678\) 0 0
\(679\) 2.87689 0.110405
\(680\) 3.12311 0.119766
\(681\) 0 0
\(682\) 7.12311 0.272758
\(683\) −27.6155 −1.05668 −0.528339 0.849033i \(-0.677185\pi\)
−0.528339 + 0.849033i \(0.677185\pi\)
\(684\) 0 0
\(685\) −20.2462 −0.773568
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −5.12311 −0.195317
\(689\) −20.0000 −0.761939
\(690\) 0 0
\(691\) 2.00000 0.0760836 0.0380418 0.999276i \(-0.487888\pi\)
0.0380418 + 0.999276i \(0.487888\pi\)
\(692\) 4.24621 0.161417
\(693\) 0 0
\(694\) −30.7386 −1.16682
\(695\) −9.12311 −0.346059
\(696\) 0 0
\(697\) 28.4924 1.07923
\(698\) 5.36932 0.203232
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) −9.50758 −0.359096 −0.179548 0.983749i \(-0.557464\pi\)
−0.179548 + 0.983749i \(0.557464\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −15.6155 −0.587698
\(707\) −11.3693 −0.427587
\(708\) 0 0
\(709\) 42.9848 1.61433 0.807165 0.590326i \(-0.201001\pi\)
0.807165 + 0.590326i \(0.201001\pi\)
\(710\) −1.12311 −0.0421494
\(711\) 0 0
\(712\) 8.24621 0.309040
\(713\) 50.7386 1.90018
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) −14.2462 −0.532406
\(717\) 0 0
\(718\) −7.36932 −0.275020
\(719\) 32.0000 1.19340 0.596699 0.802465i \(-0.296479\pi\)
0.596699 + 0.802465i \(0.296479\pi\)
\(720\) 0 0
\(721\) −14.2462 −0.530557
\(722\) 17.7386 0.660164
\(723\) 0 0
\(724\) −5.36932 −0.199549
\(725\) 8.24621 0.306257
\(726\) 0 0
\(727\) −46.2462 −1.71518 −0.857589 0.514336i \(-0.828038\pi\)
−0.857589 + 0.514336i \(0.828038\pi\)
\(728\) 2.00000 0.0741249
\(729\) 0 0
\(730\) 0.246211 0.00911269
\(731\) 16.0000 0.591781
\(732\) 0 0
\(733\) −16.7386 −0.618256 −0.309128 0.951021i \(-0.600037\pi\)
−0.309128 + 0.951021i \(0.600037\pi\)
\(734\) 28.4924 1.05167
\(735\) 0 0
\(736\) 7.12311 0.262561
\(737\) −9.12311 −0.336054
\(738\) 0 0
\(739\) 24.8769 0.915111 0.457556 0.889181i \(-0.348725\pi\)
0.457556 + 0.889181i \(0.348725\pi\)
\(740\) 7.12311 0.261851
\(741\) 0 0
\(742\) −10.0000 −0.367112
\(743\) −2.73863 −0.100471 −0.0502354 0.998737i \(-0.515997\pi\)
−0.0502354 + 0.998737i \(0.515997\pi\)
\(744\) 0 0
\(745\) −2.00000 −0.0732743
\(746\) 22.2462 0.814492
\(747\) 0 0
\(748\) −3.12311 −0.114192
\(749\) 18.2462 0.666702
\(750\) 0 0
\(751\) −38.2462 −1.39562 −0.697812 0.716281i \(-0.745843\pi\)
−0.697812 + 0.716281i \(0.745843\pi\)
\(752\) 11.3693 0.414596
\(753\) 0 0
\(754\) −16.4924 −0.600619
\(755\) 12.8769 0.468638
\(756\) 0 0
\(757\) −49.8617 −1.81226 −0.906128 0.423004i \(-0.860976\pi\)
−0.906128 + 0.423004i \(0.860976\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) −1.12311 −0.0407393
\(761\) 34.1080 1.23641 0.618206 0.786016i \(-0.287860\pi\)
0.618206 + 0.786016i \(0.287860\pi\)
\(762\) 0 0
\(763\) −18.0000 −0.651644
\(764\) −6.87689 −0.248797
\(765\) 0 0
\(766\) 5.61553 0.202897
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) −18.0000 −0.649097 −0.324548 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) 2.87689 0.103542
\(773\) −52.2462 −1.87917 −0.939583 0.342322i \(-0.888787\pi\)
−0.939583 + 0.342322i \(0.888787\pi\)
\(774\) 0 0
\(775\) −7.12311 −0.255870
\(776\) 2.87689 0.103274
\(777\) 0 0
\(778\) 11.3693 0.407610
\(779\) −10.2462 −0.367109
\(780\) 0 0
\(781\) 1.12311 0.0401879
\(782\) −22.2462 −0.795523
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 5.12311 0.182851
\(786\) 0 0
\(787\) −6.24621 −0.222653 −0.111327 0.993784i \(-0.535510\pi\)
−0.111327 + 0.993784i \(0.535510\pi\)
\(788\) −22.4924 −0.801259
\(789\) 0 0
\(790\) 3.12311 0.111115
\(791\) 2.00000 0.0711118
\(792\) 0 0
\(793\) 6.24621 0.221809
\(794\) 27.3693 0.971301
\(795\) 0 0
\(796\) −4.87689 −0.172857
\(797\) 7.75379 0.274653 0.137327 0.990526i \(-0.456149\pi\)
0.137327 + 0.990526i \(0.456149\pi\)
\(798\) 0 0
\(799\) −35.5076 −1.25617
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −32.0000 −1.12996
\(803\) −0.246211 −0.00868861
\(804\) 0 0
\(805\) 7.12311 0.251056
\(806\) 14.2462 0.501801
\(807\) 0 0
\(808\) −11.3693 −0.399971
\(809\) 7.12311 0.250435 0.125218 0.992129i \(-0.460037\pi\)
0.125218 + 0.992129i \(0.460037\pi\)
\(810\) 0 0
\(811\) 39.8617 1.39973 0.699867 0.714273i \(-0.253242\pi\)
0.699867 + 0.714273i \(0.253242\pi\)
\(812\) −8.24621 −0.289385
\(813\) 0 0
\(814\) −7.12311 −0.249665
\(815\) 3.36932 0.118022
\(816\) 0 0
\(817\) −5.75379 −0.201300
\(818\) −16.2462 −0.568035
\(819\) 0 0
\(820\) −9.12311 −0.318593
\(821\) 42.9848 1.50018 0.750091 0.661335i \(-0.230010\pi\)
0.750091 + 0.661335i \(0.230010\pi\)
\(822\) 0 0
\(823\) 21.7538 0.758289 0.379145 0.925337i \(-0.376218\pi\)
0.379145 + 0.925337i \(0.376218\pi\)
\(824\) −14.2462 −0.496290
\(825\) 0 0
\(826\) −4.00000 −0.139178
\(827\) −29.7538 −1.03464 −0.517320 0.855792i \(-0.673070\pi\)
−0.517320 + 0.855792i \(0.673070\pi\)
\(828\) 0 0
\(829\) 42.3542 1.47102 0.735510 0.677513i \(-0.236942\pi\)
0.735510 + 0.677513i \(0.236942\pi\)
\(830\) 11.1231 0.386089
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) −3.12311 −0.108209
\(834\) 0 0
\(835\) −14.0000 −0.484490
\(836\) 1.12311 0.0388434
\(837\) 0 0
\(838\) −0.492423 −0.0170105
\(839\) −5.75379 −0.198643 −0.0993214 0.995055i \(-0.531667\pi\)
−0.0993214 + 0.995055i \(0.531667\pi\)
\(840\) 0 0
\(841\) 39.0000 1.34483
\(842\) 8.24621 0.284183
\(843\) 0 0
\(844\) 8.87689 0.305555
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) −10.0000 −0.343401
\(849\) 0 0
\(850\) 3.12311 0.107122
\(851\) −50.7386 −1.73930
\(852\) 0 0
\(853\) −4.24621 −0.145388 −0.0726938 0.997354i \(-0.523160\pi\)
−0.0726938 + 0.997354i \(0.523160\pi\)
\(854\) 3.12311 0.106870
\(855\) 0 0
\(856\) 18.2462 0.623643
\(857\) 10.6307 0.363137 0.181569 0.983378i \(-0.441883\pi\)
0.181569 + 0.983378i \(0.441883\pi\)
\(858\) 0 0
\(859\) 14.4924 0.494475 0.247238 0.968955i \(-0.420477\pi\)
0.247238 + 0.968955i \(0.420477\pi\)
\(860\) −5.12311 −0.174696
\(861\) 0 0
\(862\) −17.6155 −0.599987
\(863\) 42.3542 1.44175 0.720876 0.693064i \(-0.243740\pi\)
0.720876 + 0.693064i \(0.243740\pi\)
\(864\) 0 0
\(865\) 4.24621 0.144376
\(866\) 1.12311 0.0381647
\(867\) 0 0
\(868\) 7.12311 0.241774
\(869\) −3.12311 −0.105944
\(870\) 0 0
\(871\) −18.2462 −0.618249
\(872\) −18.0000 −0.609557
\(873\) 0 0
\(874\) 8.00000 0.270604
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −22.7386 −0.767829 −0.383915 0.923369i \(-0.625424\pi\)
−0.383915 + 0.923369i \(0.625424\pi\)
\(878\) 22.7386 0.767392
\(879\) 0 0
\(880\) 1.00000 0.0337100
\(881\) 30.4924 1.02732 0.513658 0.857995i \(-0.328290\pi\)
0.513658 + 0.857995i \(0.328290\pi\)
\(882\) 0 0
\(883\) −53.6155 −1.80431 −0.902153 0.431416i \(-0.858014\pi\)
−0.902153 + 0.431416i \(0.858014\pi\)
\(884\) −6.24621 −0.210083
\(885\) 0 0
\(886\) 27.6155 0.927762
\(887\) −38.9848 −1.30898 −0.654491 0.756069i \(-0.727117\pi\)
−0.654491 + 0.756069i \(0.727117\pi\)
\(888\) 0 0
\(889\) 1.75379 0.0588202
\(890\) 8.24621 0.276414
\(891\) 0 0
\(892\) 4.49242 0.150417
\(893\) 12.7689 0.427296
\(894\) 0 0
\(895\) −14.2462 −0.476198
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 30.7386 1.02576
\(899\) −58.7386 −1.95904
\(900\) 0 0
\(901\) 31.2311 1.04046
\(902\) 9.12311 0.303766
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) −5.36932 −0.178482
\(906\) 0 0
\(907\) 7.86174 0.261045 0.130522 0.991445i \(-0.458335\pi\)
0.130522 + 0.991445i \(0.458335\pi\)
\(908\) −21.3693 −0.709166
\(909\) 0 0
\(910\) 2.00000 0.0662994
\(911\) −30.8769 −1.02300 −0.511499 0.859284i \(-0.670909\pi\)
−0.511499 + 0.859284i \(0.670909\pi\)
\(912\) 0 0
\(913\) −11.1231 −0.368121
\(914\) −11.3693 −0.376064
\(915\) 0 0
\(916\) 11.1231 0.367518
\(917\) −5.75379 −0.190007
\(918\) 0 0
\(919\) 55.6155 1.83459 0.917293 0.398212i \(-0.130369\pi\)
0.917293 + 0.398212i \(0.130369\pi\)
\(920\) 7.12311 0.234842
\(921\) 0 0
\(922\) −13.1231 −0.432186
\(923\) 2.24621 0.0739349
\(924\) 0 0
\(925\) 7.12311 0.234206
\(926\) −22.7386 −0.747238
\(927\) 0 0
\(928\) −8.24621 −0.270695
\(929\) 52.7386 1.73030 0.865149 0.501515i \(-0.167224\pi\)
0.865149 + 0.501515i \(0.167224\pi\)
\(930\) 0 0
\(931\) 1.12311 0.0368083
\(932\) −6.00000 −0.196537
\(933\) 0 0
\(934\) −12.4924 −0.408765
\(935\) −3.12311 −0.102136
\(936\) 0 0
\(937\) 25.2311 0.824263 0.412131 0.911124i \(-0.364784\pi\)
0.412131 + 0.911124i \(0.364784\pi\)
\(938\) −9.12311 −0.297880
\(939\) 0 0
\(940\) 11.3693 0.370826
\(941\) −32.3542 −1.05472 −0.527358 0.849643i \(-0.676817\pi\)
−0.527358 + 0.849643i \(0.676817\pi\)
\(942\) 0 0
\(943\) 64.9848 2.11620
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 5.12311 0.166567
\(947\) −9.86174 −0.320463 −0.160232 0.987079i \(-0.551224\pi\)
−0.160232 + 0.987079i \(0.551224\pi\)
\(948\) 0 0
\(949\) −0.492423 −0.0159847
\(950\) −1.12311 −0.0364384
\(951\) 0 0
\(952\) −3.12311 −0.101220
\(953\) 18.9848 0.614979 0.307490 0.951551i \(-0.400511\pi\)
0.307490 + 0.951551i \(0.400511\pi\)
\(954\) 0 0
\(955\) −6.87689 −0.222531
\(956\) 7.36932 0.238341
\(957\) 0 0
\(958\) −16.4924 −0.532846
\(959\) 20.2462 0.653784
\(960\) 0 0
\(961\) 19.7386 0.636730
\(962\) −14.2462 −0.459316
\(963\) 0 0
\(964\) 22.4924 0.724432
\(965\) 2.87689 0.0926105
\(966\) 0 0
\(967\) −52.4924 −1.68804 −0.844021 0.536310i \(-0.819818\pi\)
−0.844021 + 0.536310i \(0.819818\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 2.87689 0.0923715
\(971\) −4.00000 −0.128366 −0.0641831 0.997938i \(-0.520444\pi\)
−0.0641831 + 0.997938i \(0.520444\pi\)
\(972\) 0 0
\(973\) 9.12311 0.292473
\(974\) −4.49242 −0.143947
\(975\) 0 0
\(976\) 3.12311 0.0999682
\(977\) 45.2311 1.44707 0.723535 0.690288i \(-0.242516\pi\)
0.723535 + 0.690288i \(0.242516\pi\)
\(978\) 0 0
\(979\) −8.24621 −0.263550
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) 6.24621 0.199325
\(983\) −19.8617 −0.633491 −0.316746 0.948511i \(-0.602590\pi\)
−0.316746 + 0.948511i \(0.602590\pi\)
\(984\) 0 0
\(985\) −22.4924 −0.716668
\(986\) 25.7538 0.820168
\(987\) 0 0
\(988\) 2.24621 0.0714615
\(989\) 36.4924 1.16039
\(990\) 0 0
\(991\) −53.4773 −1.69876 −0.849381 0.527781i \(-0.823024\pi\)
−0.849381 + 0.527781i \(0.823024\pi\)
\(992\) 7.12311 0.226159
\(993\) 0 0
\(994\) 1.12311 0.0356227
\(995\) −4.87689 −0.154608
\(996\) 0 0
\(997\) −48.7386 −1.54357 −0.771784 0.635885i \(-0.780635\pi\)
−0.771784 + 0.635885i \(0.780635\pi\)
\(998\) 18.7386 0.593161
\(999\) 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.bs.1.1 2
3.2 odd 2 6930.2.a.bx.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6930.2.a.bs.1.1 2 1.1 even 1 trivial
6930.2.a.bx.1.2 yes 2 3.2 odd 2