Properties

Label 6930.2.a.bw.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: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2310)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 6930.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} -3.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.46410 q^{17} -1.00000 q^{20} -1.00000 q^{22} +1.46410 q^{23} +1.00000 q^{25} -3.46410 q^{26} -1.00000 q^{28} -8.92820 q^{29} +10.9282 q^{31} +1.00000 q^{32} +3.46410 q^{34} +1.00000 q^{35} +3.46410 q^{37} -1.00000 q^{40} -2.00000 q^{41} -1.00000 q^{44} +1.46410 q^{46} -6.92820 q^{47} +1.00000 q^{49} +1.00000 q^{50} -3.46410 q^{52} -6.00000 q^{53} +1.00000 q^{55} -1.00000 q^{56} -8.92820 q^{58} -8.00000 q^{59} -4.92820 q^{61} +10.9282 q^{62} +1.00000 q^{64} +3.46410 q^{65} -5.46410 q^{67} +3.46410 q^{68} +1.00000 q^{70} -1.07180 q^{71} +15.8564 q^{73} +3.46410 q^{74} +1.00000 q^{77} -4.00000 q^{79} -1.00000 q^{80} -2.00000 q^{82} -13.8564 q^{83} -3.46410 q^{85} -1.00000 q^{88} -8.53590 q^{89} +3.46410 q^{91} +1.46410 q^{92} -6.92820 q^{94} -10.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} - 2 q^{10} - 2 q^{11} - 2 q^{14} + 2 q^{16} - 2 q^{20} - 2 q^{22} - 4 q^{23} + 2 q^{25} - 2 q^{28} - 4 q^{29} + 8 q^{31} + 2 q^{32} + 2 q^{35} - 2 q^{40} - 4 q^{41} - 2 q^{44} - 4 q^{46} + 2 q^{49} + 2 q^{50} - 12 q^{53} + 2 q^{55} - 2 q^{56} - 4 q^{58} - 16 q^{59} + 4 q^{61} + 8 q^{62} + 2 q^{64} - 4 q^{67} + 2 q^{70} - 16 q^{71} + 4 q^{73} + 2 q^{77} - 8 q^{79} - 2 q^{80} - 4 q^{82} - 2 q^{88} - 24 q^{89} - 4 q^{92} - 20 q^{97} + 2 q^{98} + O(q^{100}) \)

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
\(13\) −3.46410 −0.960769 −0.480384 0.877058i \(-0.659503\pi\)
−0.480384 + 0.877058i \(0.659503\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 1.46410 0.305286 0.152643 0.988281i \(-0.451221\pi\)
0.152643 + 0.988281i \(0.451221\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −3.46410 −0.679366
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −8.92820 −1.65793 −0.828963 0.559304i \(-0.811069\pi\)
−0.828963 + 0.559304i \(0.811069\pi\)
\(30\) 0 0
\(31\) 10.9282 1.96276 0.981382 0.192068i \(-0.0615194\pi\)
0.981382 + 0.192068i \(0.0615194\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 3.46410 0.594089
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 3.46410 0.569495 0.284747 0.958603i \(-0.408090\pi\)
0.284747 + 0.958603i \(0.408090\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 1.46410 0.215870
\(47\) −6.92820 −1.01058 −0.505291 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −3.46410 −0.480384
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −8.92820 −1.17233
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −4.92820 −0.630992 −0.315496 0.948927i \(-0.602171\pi\)
−0.315496 + 0.948927i \(0.602171\pi\)
\(62\) 10.9282 1.38788
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.46410 0.429669
\(66\) 0 0
\(67\) −5.46410 −0.667546 −0.333773 0.942653i \(-0.608322\pi\)
−0.333773 + 0.942653i \(0.608322\pi\)
\(68\) 3.46410 0.420084
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) −1.07180 −0.127199 −0.0635994 0.997976i \(-0.520258\pi\)
−0.0635994 + 0.997976i \(0.520258\pi\)
\(72\) 0 0
\(73\) 15.8564 1.85585 0.927926 0.372764i \(-0.121590\pi\)
0.927926 + 0.372764i \(0.121590\pi\)
\(74\) 3.46410 0.402694
\(75\) 0 0
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) −13.8564 −1.52094 −0.760469 0.649374i \(-0.775031\pi\)
−0.760469 + 0.649374i \(0.775031\pi\)
\(84\) 0 0
\(85\) −3.46410 −0.375735
\(86\) 0 0
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) −8.53590 −0.904803 −0.452402 0.891814i \(-0.649433\pi\)
−0.452402 + 0.891814i \(0.649433\pi\)
\(90\) 0 0
\(91\) 3.46410 0.363137
\(92\) 1.46410 0.152643
\(93\) 0 0
\(94\) −6.92820 −0.714590
\(95\) 0 0
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 7.85641 0.781742 0.390871 0.920446i \(-0.372174\pi\)
0.390871 + 0.920446i \(0.372174\pi\)
\(102\) 0 0
\(103\) −2.92820 −0.288524 −0.144262 0.989539i \(-0.546081\pi\)
−0.144262 + 0.989539i \(0.546081\pi\)
\(104\) −3.46410 −0.339683
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −6.92820 −0.669775 −0.334887 0.942258i \(-0.608698\pi\)
−0.334887 + 0.942258i \(0.608698\pi\)
\(108\) 0 0
\(109\) −3.46410 −0.331801 −0.165900 0.986143i \(-0.553053\pi\)
−0.165900 + 0.986143i \(0.553053\pi\)
\(110\) 1.00000 0.0953463
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 2.39230 0.225049 0.112525 0.993649i \(-0.464106\pi\)
0.112525 + 0.993649i \(0.464106\pi\)
\(114\) 0 0
\(115\) −1.46410 −0.136528
\(116\) −8.92820 −0.828963
\(117\) 0 0
\(118\) −8.00000 −0.736460
\(119\) −3.46410 −0.317554
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −4.92820 −0.446179
\(123\) 0 0
\(124\) 10.9282 0.981382
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −10.9282 −0.969721 −0.484861 0.874591i \(-0.661130\pi\)
−0.484861 + 0.874591i \(0.661130\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 3.46410 0.303822
\(131\) −17.8564 −1.56012 −0.780061 0.625704i \(-0.784812\pi\)
−0.780061 + 0.625704i \(0.784812\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −5.46410 −0.472026
\(135\) 0 0
\(136\) 3.46410 0.297044
\(137\) −3.46410 −0.295958 −0.147979 0.988990i \(-0.547277\pi\)
−0.147979 + 0.988990i \(0.547277\pi\)
\(138\) 0 0
\(139\) −10.9282 −0.926918 −0.463459 0.886118i \(-0.653392\pi\)
−0.463459 + 0.886118i \(0.653392\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) −1.07180 −0.0899432
\(143\) 3.46410 0.289683
\(144\) 0 0
\(145\) 8.92820 0.741447
\(146\) 15.8564 1.31229
\(147\) 0 0
\(148\) 3.46410 0.284747
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) −10.9282 −0.877774
\(156\) 0 0
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) −4.00000 −0.318223
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −1.46410 −0.115387
\(162\) 0 0
\(163\) −19.3205 −1.51330 −0.756649 0.653821i \(-0.773165\pi\)
−0.756649 + 0.653821i \(0.773165\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) −13.8564 −1.07547
\(167\) 17.4641 1.35141 0.675706 0.737171i \(-0.263839\pi\)
0.675706 + 0.737171i \(0.263839\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) −3.46410 −0.265684
\(171\) 0 0
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) −8.53590 −0.639793
\(179\) 6.92820 0.517838 0.258919 0.965899i \(-0.416634\pi\)
0.258919 + 0.965899i \(0.416634\pi\)
\(180\) 0 0
\(181\) 3.46410 0.257485 0.128742 0.991678i \(-0.458906\pi\)
0.128742 + 0.991678i \(0.458906\pi\)
\(182\) 3.46410 0.256776
\(183\) 0 0
\(184\) 1.46410 0.107935
\(185\) −3.46410 −0.254686
\(186\) 0 0
\(187\) −3.46410 −0.253320
\(188\) −6.92820 −0.505291
\(189\) 0 0
\(190\) 0 0
\(191\) 9.85641 0.713185 0.356592 0.934260i \(-0.383939\pi\)
0.356592 + 0.934260i \(0.383939\pi\)
\(192\) 0 0
\(193\) 15.8564 1.14137 0.570685 0.821169i \(-0.306678\pi\)
0.570685 + 0.821169i \(0.306678\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) 18.9282 1.34178 0.670892 0.741555i \(-0.265911\pi\)
0.670892 + 0.741555i \(0.265911\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 7.85641 0.552775
\(203\) 8.92820 0.626637
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) −2.92820 −0.204018
\(207\) 0 0
\(208\) −3.46410 −0.240192
\(209\) 0 0
\(210\) 0 0
\(211\) 4.39230 0.302379 0.151189 0.988505i \(-0.451690\pi\)
0.151189 + 0.988505i \(0.451690\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −6.92820 −0.473602
\(215\) 0 0
\(216\) 0 0
\(217\) −10.9282 −0.741855
\(218\) −3.46410 −0.234619
\(219\) 0 0
\(220\) 1.00000 0.0674200
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 27.7128 1.85579 0.927894 0.372845i \(-0.121618\pi\)
0.927894 + 0.372845i \(0.121618\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 2.39230 0.159134
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 0 0
\(229\) 16.5359 1.09272 0.546361 0.837549i \(-0.316012\pi\)
0.546361 + 0.837549i \(0.316012\pi\)
\(230\) −1.46410 −0.0965400
\(231\) 0 0
\(232\) −8.92820 −0.586165
\(233\) −19.8564 −1.30084 −0.650418 0.759576i \(-0.725406\pi\)
−0.650418 + 0.759576i \(0.725406\pi\)
\(234\) 0 0
\(235\) 6.92820 0.451946
\(236\) −8.00000 −0.520756
\(237\) 0 0
\(238\) −3.46410 −0.224544
\(239\) −21.4641 −1.38840 −0.694199 0.719783i \(-0.744241\pi\)
−0.694199 + 0.719783i \(0.744241\pi\)
\(240\) 0 0
\(241\) −18.7846 −1.21002 −0.605012 0.796217i \(-0.706832\pi\)
−0.605012 + 0.796217i \(0.706832\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) −4.92820 −0.315496
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 0 0
\(248\) 10.9282 0.693942
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 8.00000 0.504956 0.252478 0.967603i \(-0.418755\pi\)
0.252478 + 0.967603i \(0.418755\pi\)
\(252\) 0 0
\(253\) −1.46410 −0.0920473
\(254\) −10.9282 −0.685696
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −3.07180 −0.191613 −0.0958067 0.995400i \(-0.530543\pi\)
−0.0958067 + 0.995400i \(0.530543\pi\)
\(258\) 0 0
\(259\) −3.46410 −0.215249
\(260\) 3.46410 0.214834
\(261\) 0 0
\(262\) −17.8564 −1.10317
\(263\) −5.07180 −0.312740 −0.156370 0.987699i \(-0.549979\pi\)
−0.156370 + 0.987699i \(0.549979\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 0 0
\(268\) −5.46410 −0.333773
\(269\) −16.9282 −1.03213 −0.516065 0.856549i \(-0.672604\pi\)
−0.516065 + 0.856549i \(0.672604\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 3.46410 0.210042
\(273\) 0 0
\(274\) −3.46410 −0.209274
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) −10.9282 −0.655430
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) −21.3205 −1.27187 −0.635937 0.771741i \(-0.719386\pi\)
−0.635937 + 0.771741i \(0.719386\pi\)
\(282\) 0 0
\(283\) −24.3923 −1.44997 −0.724986 0.688764i \(-0.758154\pi\)
−0.724986 + 0.688764i \(0.758154\pi\)
\(284\) −1.07180 −0.0635994
\(285\) 0 0
\(286\) 3.46410 0.204837
\(287\) 2.00000 0.118056
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 8.92820 0.524282
\(291\) 0 0
\(292\) 15.8564 0.927926
\(293\) 18.7846 1.09741 0.548704 0.836016i \(-0.315121\pi\)
0.548704 + 0.836016i \(0.315121\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 3.46410 0.201347
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) −5.07180 −0.293310
\(300\) 0 0
\(301\) 0 0
\(302\) −20.0000 −1.15087
\(303\) 0 0
\(304\) 0 0
\(305\) 4.92820 0.282188
\(306\) 0 0
\(307\) −24.3923 −1.39214 −0.696071 0.717973i \(-0.745070\pi\)
−0.696071 + 0.717973i \(0.745070\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) −10.9282 −0.620680
\(311\) 21.4641 1.21712 0.608559 0.793509i \(-0.291748\pi\)
0.608559 + 0.793509i \(0.291748\pi\)
\(312\) 0 0
\(313\) 27.8564 1.57454 0.787269 0.616610i \(-0.211495\pi\)
0.787269 + 0.616610i \(0.211495\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) −11.8564 −0.665922 −0.332961 0.942941i \(-0.608048\pi\)
−0.332961 + 0.942941i \(0.608048\pi\)
\(318\) 0 0
\(319\) 8.92820 0.499883
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −1.46410 −0.0815912
\(323\) 0 0
\(324\) 0 0
\(325\) −3.46410 −0.192154
\(326\) −19.3205 −1.07006
\(327\) 0 0
\(328\) −2.00000 −0.110432
\(329\) 6.92820 0.381964
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −13.8564 −0.760469
\(333\) 0 0
\(334\) 17.4641 0.955593
\(335\) 5.46410 0.298536
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 0 0
\(340\) −3.46410 −0.187867
\(341\) −10.9282 −0.591795
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 14.9282 0.801388 0.400694 0.916212i \(-0.368769\pi\)
0.400694 + 0.916212i \(0.368769\pi\)
\(348\) 0 0
\(349\) −7.07180 −0.378545 −0.189272 0.981925i \(-0.560613\pi\)
−0.189272 + 0.981925i \(0.560613\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) 12.9282 0.688099 0.344049 0.938952i \(-0.388201\pi\)
0.344049 + 0.938952i \(0.388201\pi\)
\(354\) 0 0
\(355\) 1.07180 0.0568851
\(356\) −8.53590 −0.452402
\(357\) 0 0
\(358\) 6.92820 0.366167
\(359\) 5.46410 0.288384 0.144192 0.989550i \(-0.453942\pi\)
0.144192 + 0.989550i \(0.453942\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 3.46410 0.182069
\(363\) 0 0
\(364\) 3.46410 0.181568
\(365\) −15.8564 −0.829962
\(366\) 0 0
\(367\) 16.7846 0.876149 0.438075 0.898939i \(-0.355661\pi\)
0.438075 + 0.898939i \(0.355661\pi\)
\(368\) 1.46410 0.0763216
\(369\) 0 0
\(370\) −3.46410 −0.180090
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) 34.7846 1.80108 0.900539 0.434774i \(-0.143172\pi\)
0.900539 + 0.434774i \(0.143172\pi\)
\(374\) −3.46410 −0.179124
\(375\) 0 0
\(376\) −6.92820 −0.357295
\(377\) 30.9282 1.59288
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 9.85641 0.504298
\(383\) 1.07180 0.0547663 0.0273831 0.999625i \(-0.491283\pi\)
0.0273831 + 0.999625i \(0.491283\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 15.8564 0.807070
\(387\) 0 0
\(388\) −10.0000 −0.507673
\(389\) −28.9282 −1.46672 −0.733359 0.679842i \(-0.762049\pi\)
−0.733359 + 0.679842i \(0.762049\pi\)
\(390\) 0 0
\(391\) 5.07180 0.256492
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −2.00000 −0.100759
\(395\) 4.00000 0.201262
\(396\) 0 0
\(397\) 34.7846 1.74579 0.872895 0.487909i \(-0.162240\pi\)
0.872895 + 0.487909i \(0.162240\pi\)
\(398\) 18.9282 0.948785
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 19.8564 0.991582 0.495791 0.868442i \(-0.334878\pi\)
0.495791 + 0.868442i \(0.334878\pi\)
\(402\) 0 0
\(403\) −37.8564 −1.88576
\(404\) 7.85641 0.390871
\(405\) 0 0
\(406\) 8.92820 0.443099
\(407\) −3.46410 −0.171709
\(408\) 0 0
\(409\) −7.07180 −0.349678 −0.174839 0.984597i \(-0.555940\pi\)
−0.174839 + 0.984597i \(0.555940\pi\)
\(410\) 2.00000 0.0987730
\(411\) 0 0
\(412\) −2.92820 −0.144262
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) 13.8564 0.680184
\(416\) −3.46410 −0.169842
\(417\) 0 0
\(418\) 0 0
\(419\) 2.92820 0.143052 0.0715260 0.997439i \(-0.477213\pi\)
0.0715260 + 0.997439i \(0.477213\pi\)
\(420\) 0 0
\(421\) −31.8564 −1.55259 −0.776293 0.630372i \(-0.782902\pi\)
−0.776293 + 0.630372i \(0.782902\pi\)
\(422\) 4.39230 0.213814
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 3.46410 0.168034
\(426\) 0 0
\(427\) 4.92820 0.238492
\(428\) −6.92820 −0.334887
\(429\) 0 0
\(430\) 0 0
\(431\) 32.3923 1.56028 0.780141 0.625603i \(-0.215147\pi\)
0.780141 + 0.625603i \(0.215147\pi\)
\(432\) 0 0
\(433\) −23.8564 −1.14647 −0.573233 0.819393i \(-0.694311\pi\)
−0.573233 + 0.819393i \(0.694311\pi\)
\(434\) −10.9282 −0.524571
\(435\) 0 0
\(436\) −3.46410 −0.165900
\(437\) 0 0
\(438\) 0 0
\(439\) 34.9282 1.66703 0.833516 0.552495i \(-0.186324\pi\)
0.833516 + 0.552495i \(0.186324\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) 6.14359 0.291891 0.145945 0.989293i \(-0.453378\pi\)
0.145945 + 0.989293i \(0.453378\pi\)
\(444\) 0 0
\(445\) 8.53590 0.404640
\(446\) 27.7128 1.31224
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 11.8564 0.559538 0.279769 0.960067i \(-0.409742\pi\)
0.279769 + 0.960067i \(0.409742\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) 2.39230 0.112525
\(453\) 0 0
\(454\) 8.00000 0.375459
\(455\) −3.46410 −0.162400
\(456\) 0 0
\(457\) −11.8564 −0.554619 −0.277310 0.960781i \(-0.589443\pi\)
−0.277310 + 0.960781i \(0.589443\pi\)
\(458\) 16.5359 0.772672
\(459\) 0 0
\(460\) −1.46410 −0.0682641
\(461\) −25.7128 −1.19757 −0.598783 0.800912i \(-0.704349\pi\)
−0.598783 + 0.800912i \(0.704349\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) −8.92820 −0.414481
\(465\) 0 0
\(466\) −19.8564 −0.919830
\(467\) 20.7846 0.961797 0.480899 0.876776i \(-0.340311\pi\)
0.480899 + 0.876776i \(0.340311\pi\)
\(468\) 0 0
\(469\) 5.46410 0.252309
\(470\) 6.92820 0.319574
\(471\) 0 0
\(472\) −8.00000 −0.368230
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −3.46410 −0.158777
\(477\) 0 0
\(478\) −21.4641 −0.981745
\(479\) −21.8564 −0.998645 −0.499322 0.866416i \(-0.666418\pi\)
−0.499322 + 0.866416i \(0.666418\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) −18.7846 −0.855616
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 10.0000 0.454077
\(486\) 0 0
\(487\) −41.8564 −1.89669 −0.948347 0.317234i \(-0.897246\pi\)
−0.948347 + 0.317234i \(0.897246\pi\)
\(488\) −4.92820 −0.223089
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) −1.85641 −0.0837785 −0.0418892 0.999122i \(-0.513338\pi\)
−0.0418892 + 0.999122i \(0.513338\pi\)
\(492\) 0 0
\(493\) −30.9282 −1.39294
\(494\) 0 0
\(495\) 0 0
\(496\) 10.9282 0.490691
\(497\) 1.07180 0.0480767
\(498\) 0 0
\(499\) −12.7846 −0.572318 −0.286159 0.958182i \(-0.592379\pi\)
−0.286159 + 0.958182i \(0.592379\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 8.00000 0.357057
\(503\) 19.6077 0.874264 0.437132 0.899397i \(-0.355994\pi\)
0.437132 + 0.899397i \(0.355994\pi\)
\(504\) 0 0
\(505\) −7.85641 −0.349605
\(506\) −1.46410 −0.0650873
\(507\) 0 0
\(508\) −10.9282 −0.484861
\(509\) −0.928203 −0.0411419 −0.0205709 0.999788i \(-0.506548\pi\)
−0.0205709 + 0.999788i \(0.506548\pi\)
\(510\) 0 0
\(511\) −15.8564 −0.701446
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −3.07180 −0.135491
\(515\) 2.92820 0.129032
\(516\) 0 0
\(517\) 6.92820 0.304702
\(518\) −3.46410 −0.152204
\(519\) 0 0
\(520\) 3.46410 0.151911
\(521\) −34.1051 −1.49417 −0.747086 0.664727i \(-0.768548\pi\)
−0.747086 + 0.664727i \(0.768548\pi\)
\(522\) 0 0
\(523\) −19.3205 −0.844827 −0.422413 0.906403i \(-0.638817\pi\)
−0.422413 + 0.906403i \(0.638817\pi\)
\(524\) −17.8564 −0.780061
\(525\) 0 0
\(526\) −5.07180 −0.221141
\(527\) 37.8564 1.64905
\(528\) 0 0
\(529\) −20.8564 −0.906800
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) 0 0
\(533\) 6.92820 0.300094
\(534\) 0 0
\(535\) 6.92820 0.299532
\(536\) −5.46410 −0.236013
\(537\) 0 0
\(538\) −16.9282 −0.729827
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −0.535898 −0.0230401 −0.0115200 0.999934i \(-0.503667\pi\)
−0.0115200 + 0.999934i \(0.503667\pi\)
\(542\) 8.00000 0.343629
\(543\) 0 0
\(544\) 3.46410 0.148522
\(545\) 3.46410 0.148386
\(546\) 0 0
\(547\) 38.6410 1.65217 0.826085 0.563545i \(-0.190563\pi\)
0.826085 + 0.563545i \(0.190563\pi\)
\(548\) −3.46410 −0.147979
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) 0 0
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) −14.0000 −0.594803
\(555\) 0 0
\(556\) −10.9282 −0.463459
\(557\) 16.1436 0.684026 0.342013 0.939695i \(-0.388891\pi\)
0.342013 + 0.939695i \(0.388891\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −21.3205 −0.899351
\(563\) 21.0718 0.888070 0.444035 0.896009i \(-0.353547\pi\)
0.444035 + 0.896009i \(0.353547\pi\)
\(564\) 0 0
\(565\) −2.39230 −0.100645
\(566\) −24.3923 −1.02529
\(567\) 0 0
\(568\) −1.07180 −0.0449716
\(569\) −6.67949 −0.280019 −0.140009 0.990150i \(-0.544713\pi\)
−0.140009 + 0.990150i \(0.544713\pi\)
\(570\) 0 0
\(571\) 21.1769 0.886226 0.443113 0.896466i \(-0.353874\pi\)
0.443113 + 0.896466i \(0.353874\pi\)
\(572\) 3.46410 0.144841
\(573\) 0 0
\(574\) 2.00000 0.0834784
\(575\) 1.46410 0.0610573
\(576\) 0 0
\(577\) −15.8564 −0.660111 −0.330055 0.943962i \(-0.607067\pi\)
−0.330055 + 0.943962i \(0.607067\pi\)
\(578\) −5.00000 −0.207973
\(579\) 0 0
\(580\) 8.92820 0.370723
\(581\) 13.8564 0.574861
\(582\) 0 0
\(583\) 6.00000 0.248495
\(584\) 15.8564 0.656143
\(585\) 0 0
\(586\) 18.7846 0.775985
\(587\) 22.9282 0.946348 0.473174 0.880969i \(-0.343108\pi\)
0.473174 + 0.880969i \(0.343108\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 8.00000 0.329355
\(591\) 0 0
\(592\) 3.46410 0.142374
\(593\) 12.2487 0.502994 0.251497 0.967858i \(-0.419077\pi\)
0.251497 + 0.967858i \(0.419077\pi\)
\(594\) 0 0
\(595\) 3.46410 0.142014
\(596\) −6.00000 −0.245770
\(597\) 0 0
\(598\) −5.07180 −0.207401
\(599\) −23.7128 −0.968879 −0.484440 0.874825i \(-0.660976\pi\)
−0.484440 + 0.874825i \(0.660976\pi\)
\(600\) 0 0
\(601\) 3.07180 0.125301 0.0626506 0.998036i \(-0.480045\pi\)
0.0626506 + 0.998036i \(0.480045\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −20.0000 −0.813788
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 4.92820 0.199537
\(611\) 24.0000 0.970936
\(612\) 0 0
\(613\) −43.8564 −1.77134 −0.885672 0.464312i \(-0.846302\pi\)
−0.885672 + 0.464312i \(0.846302\pi\)
\(614\) −24.3923 −0.984393
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) −3.46410 −0.139459 −0.0697297 0.997566i \(-0.522214\pi\)
−0.0697297 + 0.997566i \(0.522214\pi\)
\(618\) 0 0
\(619\) −30.5359 −1.22734 −0.613671 0.789562i \(-0.710308\pi\)
−0.613671 + 0.789562i \(0.710308\pi\)
\(620\) −10.9282 −0.438887
\(621\) 0 0
\(622\) 21.4641 0.860632
\(623\) 8.53590 0.341984
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 27.8564 1.11337
\(627\) 0 0
\(628\) −6.00000 −0.239426
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −2.92820 −0.116570 −0.0582850 0.998300i \(-0.518563\pi\)
−0.0582850 + 0.998300i \(0.518563\pi\)
\(632\) −4.00000 −0.159111
\(633\) 0 0
\(634\) −11.8564 −0.470878
\(635\) 10.9282 0.433673
\(636\) 0 0
\(637\) −3.46410 −0.137253
\(638\) 8.92820 0.353471
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 38.0000 1.50091 0.750455 0.660922i \(-0.229834\pi\)
0.750455 + 0.660922i \(0.229834\pi\)
\(642\) 0 0
\(643\) −33.8564 −1.33517 −0.667583 0.744535i \(-0.732671\pi\)
−0.667583 + 0.744535i \(0.732671\pi\)
\(644\) −1.46410 −0.0576937
\(645\) 0 0
\(646\) 0 0
\(647\) 30.9282 1.21591 0.607957 0.793970i \(-0.291989\pi\)
0.607957 + 0.793970i \(0.291989\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) −3.46410 −0.135873
\(651\) 0 0
\(652\) −19.3205 −0.756649
\(653\) −16.9282 −0.662452 −0.331226 0.943551i \(-0.607462\pi\)
−0.331226 + 0.943551i \(0.607462\pi\)
\(654\) 0 0
\(655\) 17.8564 0.697708
\(656\) −2.00000 −0.0780869
\(657\) 0 0
\(658\) 6.92820 0.270089
\(659\) 9.85641 0.383951 0.191976 0.981400i \(-0.438511\pi\)
0.191976 + 0.981400i \(0.438511\pi\)
\(660\) 0 0
\(661\) 43.4641 1.69056 0.845279 0.534325i \(-0.179434\pi\)
0.845279 + 0.534325i \(0.179434\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) −13.8564 −0.537733
\(665\) 0 0
\(666\) 0 0
\(667\) −13.0718 −0.506142
\(668\) 17.4641 0.675706
\(669\) 0 0
\(670\) 5.46410 0.211097
\(671\) 4.92820 0.190251
\(672\) 0 0
\(673\) −35.8564 −1.38216 −0.691081 0.722777i \(-0.742865\pi\)
−0.691081 + 0.722777i \(0.742865\pi\)
\(674\) 2.00000 0.0770371
\(675\) 0 0
\(676\) −1.00000 −0.0384615
\(677\) −19.8564 −0.763144 −0.381572 0.924339i \(-0.624617\pi\)
−0.381572 + 0.924339i \(0.624617\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) −3.46410 −0.132842
\(681\) 0 0
\(682\) −10.9282 −0.418463
\(683\) −37.5692 −1.43755 −0.718773 0.695245i \(-0.755296\pi\)
−0.718773 + 0.695245i \(0.755296\pi\)
\(684\) 0 0
\(685\) 3.46410 0.132357
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 0 0
\(689\) 20.7846 0.791831
\(690\) 0 0
\(691\) −14.5359 −0.552972 −0.276486 0.961018i \(-0.589170\pi\)
−0.276486 + 0.961018i \(0.589170\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 14.9282 0.566667
\(695\) 10.9282 0.414530
\(696\) 0 0
\(697\) −6.92820 −0.262424
\(698\) −7.07180 −0.267671
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) 10.7846 0.407329 0.203665 0.979041i \(-0.434715\pi\)
0.203665 + 0.979041i \(0.434715\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 12.9282 0.486559
\(707\) −7.85641 −0.295471
\(708\) 0 0
\(709\) 9.71281 0.364772 0.182386 0.983227i \(-0.441618\pi\)
0.182386 + 0.983227i \(0.441618\pi\)
\(710\) 1.07180 0.0402238
\(711\) 0 0
\(712\) −8.53590 −0.319896
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) −3.46410 −0.129550
\(716\) 6.92820 0.258919
\(717\) 0 0
\(718\) 5.46410 0.203918
\(719\) −29.4641 −1.09883 −0.549413 0.835551i \(-0.685149\pi\)
−0.549413 + 0.835551i \(0.685149\pi\)
\(720\) 0 0
\(721\) 2.92820 0.109052
\(722\) −19.0000 −0.707107
\(723\) 0 0
\(724\) 3.46410 0.128742
\(725\) −8.92820 −0.331585
\(726\) 0 0
\(727\) 18.1436 0.672909 0.336454 0.941700i \(-0.390772\pi\)
0.336454 + 0.941700i \(0.390772\pi\)
\(728\) 3.46410 0.128388
\(729\) 0 0
\(730\) −15.8564 −0.586872
\(731\) 0 0
\(732\) 0 0
\(733\) 36.5359 1.34948 0.674742 0.738054i \(-0.264255\pi\)
0.674742 + 0.738054i \(0.264255\pi\)
\(734\) 16.7846 0.619531
\(735\) 0 0
\(736\) 1.46410 0.0539675
\(737\) 5.46410 0.201273
\(738\) 0 0
\(739\) −14.5359 −0.534712 −0.267356 0.963598i \(-0.586150\pi\)
−0.267356 + 0.963598i \(0.586150\pi\)
\(740\) −3.46410 −0.127343
\(741\) 0 0
\(742\) 6.00000 0.220267
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 34.7846 1.27356
\(747\) 0 0
\(748\) −3.46410 −0.126660
\(749\) 6.92820 0.253151
\(750\) 0 0
\(751\) −26.9282 −0.982624 −0.491312 0.870984i \(-0.663483\pi\)
−0.491312 + 0.870984i \(0.663483\pi\)
\(752\) −6.92820 −0.252646
\(753\) 0 0
\(754\) 30.9282 1.12634
\(755\) 20.0000 0.727875
\(756\) 0 0
\(757\) −4.53590 −0.164860 −0.0824300 0.996597i \(-0.526268\pi\)
−0.0824300 + 0.996597i \(0.526268\pi\)
\(758\) −4.00000 −0.145287
\(759\) 0 0
\(760\) 0 0
\(761\) −21.7128 −0.787089 −0.393544 0.919306i \(-0.628751\pi\)
−0.393544 + 0.919306i \(0.628751\pi\)
\(762\) 0 0
\(763\) 3.46410 0.125409
\(764\) 9.85641 0.356592
\(765\) 0 0
\(766\) 1.07180 0.0387256
\(767\) 27.7128 1.00065
\(768\) 0 0
\(769\) 28.6410 1.03282 0.516411 0.856341i \(-0.327268\pi\)
0.516411 + 0.856341i \(0.327268\pi\)
\(770\) −1.00000 −0.0360375
\(771\) 0 0
\(772\) 15.8564 0.570685
\(773\) 42.0000 1.51064 0.755318 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 0 0
\(775\) 10.9282 0.392553
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) −28.9282 −1.03713
\(779\) 0 0
\(780\) 0 0
\(781\) 1.07180 0.0383519
\(782\) 5.07180 0.181367
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 6.00000 0.214149
\(786\) 0 0
\(787\) 21.4641 0.765113 0.382556 0.923932i \(-0.375044\pi\)
0.382556 + 0.923932i \(0.375044\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 0 0
\(790\) 4.00000 0.142314
\(791\) −2.39230 −0.0850606
\(792\) 0 0
\(793\) 17.0718 0.606237
\(794\) 34.7846 1.23446
\(795\) 0 0
\(796\) 18.9282 0.670892
\(797\) −33.7128 −1.19417 −0.597085 0.802178i \(-0.703674\pi\)
−0.597085 + 0.802178i \(0.703674\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 19.8564 0.701154
\(803\) −15.8564 −0.559560
\(804\) 0 0
\(805\) 1.46410 0.0516028
\(806\) −37.8564 −1.33344
\(807\) 0 0
\(808\) 7.85641 0.276387
\(809\) 7.17691 0.252327 0.126163 0.992009i \(-0.459734\pi\)
0.126163 + 0.992009i \(0.459734\pi\)
\(810\) 0 0
\(811\) −35.7128 −1.25405 −0.627023 0.779001i \(-0.715727\pi\)
−0.627023 + 0.779001i \(0.715727\pi\)
\(812\) 8.92820 0.313319
\(813\) 0 0
\(814\) −3.46410 −0.121417
\(815\) 19.3205 0.676768
\(816\) 0 0
\(817\) 0 0
\(818\) −7.07180 −0.247260
\(819\) 0 0
\(820\) 2.00000 0.0698430
\(821\) 42.7846 1.49319 0.746597 0.665277i \(-0.231687\pi\)
0.746597 + 0.665277i \(0.231687\pi\)
\(822\) 0 0
\(823\) 28.7846 1.00337 0.501684 0.865051i \(-0.332714\pi\)
0.501684 + 0.865051i \(0.332714\pi\)
\(824\) −2.92820 −0.102009
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) 30.9282 1.07548 0.537740 0.843111i \(-0.319278\pi\)
0.537740 + 0.843111i \(0.319278\pi\)
\(828\) 0 0
\(829\) −16.2487 −0.564341 −0.282171 0.959364i \(-0.591054\pi\)
−0.282171 + 0.959364i \(0.591054\pi\)
\(830\) 13.8564 0.480963
\(831\) 0 0
\(832\) −3.46410 −0.120096
\(833\) 3.46410 0.120024
\(834\) 0 0
\(835\) −17.4641 −0.604370
\(836\) 0 0
\(837\) 0 0
\(838\) 2.92820 0.101153
\(839\) 27.3205 0.943209 0.471604 0.881810i \(-0.343675\pi\)
0.471604 + 0.881810i \(0.343675\pi\)
\(840\) 0 0
\(841\) 50.7128 1.74872
\(842\) −31.8564 −1.09784
\(843\) 0 0
\(844\) 4.39230 0.151189
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 3.46410 0.118818
\(851\) 5.07180 0.173859
\(852\) 0 0
\(853\) 26.3923 0.903655 0.451828 0.892105i \(-0.350772\pi\)
0.451828 + 0.892105i \(0.350772\pi\)
\(854\) 4.92820 0.168640
\(855\) 0 0
\(856\) −6.92820 −0.236801
\(857\) 8.53590 0.291581 0.145790 0.989316i \(-0.453427\pi\)
0.145790 + 0.989316i \(0.453427\pi\)
\(858\) 0 0
\(859\) 36.3923 1.24169 0.620845 0.783934i \(-0.286790\pi\)
0.620845 + 0.783934i \(0.286790\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 32.3923 1.10329
\(863\) −24.1051 −0.820548 −0.410274 0.911962i \(-0.634567\pi\)
−0.410274 + 0.911962i \(0.634567\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) −23.8564 −0.810674
\(867\) 0 0
\(868\) −10.9282 −0.370927
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) 18.9282 0.641358
\(872\) −3.46410 −0.117309
\(873\) 0 0
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 20.1436 0.680201 0.340100 0.940389i \(-0.389539\pi\)
0.340100 + 0.940389i \(0.389539\pi\)
\(878\) 34.9282 1.17877
\(879\) 0 0
\(880\) 1.00000 0.0337100
\(881\) −0.535898 −0.0180549 −0.00902744 0.999959i \(-0.502874\pi\)
−0.00902744 + 0.999959i \(0.502874\pi\)
\(882\) 0 0
\(883\) 8.39230 0.282424 0.141212 0.989979i \(-0.454900\pi\)
0.141212 + 0.989979i \(0.454900\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 6.14359 0.206398
\(887\) 6.53590 0.219454 0.109727 0.993962i \(-0.465002\pi\)
0.109727 + 0.993962i \(0.465002\pi\)
\(888\) 0 0
\(889\) 10.9282 0.366520
\(890\) 8.53590 0.286124
\(891\) 0 0
\(892\) 27.7128 0.927894
\(893\) 0 0
\(894\) 0 0
\(895\) −6.92820 −0.231584
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 11.8564 0.395653
\(899\) −97.5692 −3.25412
\(900\) 0 0
\(901\) −20.7846 −0.692436
\(902\) 2.00000 0.0665927
\(903\) 0 0
\(904\) 2.39230 0.0795669
\(905\) −3.46410 −0.115151
\(906\) 0 0
\(907\) −26.5359 −0.881110 −0.440555 0.897726i \(-0.645218\pi\)
−0.440555 + 0.897726i \(0.645218\pi\)
\(908\) 8.00000 0.265489
\(909\) 0 0
\(910\) −3.46410 −0.114834
\(911\) 42.6410 1.41276 0.706380 0.707833i \(-0.250327\pi\)
0.706380 + 0.707833i \(0.250327\pi\)
\(912\) 0 0
\(913\) 13.8564 0.458580
\(914\) −11.8564 −0.392175
\(915\) 0 0
\(916\) 16.5359 0.546361
\(917\) 17.8564 0.589670
\(918\) 0 0
\(919\) 12.7846 0.421725 0.210863 0.977516i \(-0.432373\pi\)
0.210863 + 0.977516i \(0.432373\pi\)
\(920\) −1.46410 −0.0482700
\(921\) 0 0
\(922\) −25.7128 −0.846806
\(923\) 3.71281 0.122209
\(924\) 0 0
\(925\) 3.46410 0.113899
\(926\) 4.00000 0.131448
\(927\) 0 0
\(928\) −8.92820 −0.293083
\(929\) −27.4641 −0.901068 −0.450534 0.892759i \(-0.648766\pi\)
−0.450534 + 0.892759i \(0.648766\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −19.8564 −0.650418
\(933\) 0 0
\(934\) 20.7846 0.680093
\(935\) 3.46410 0.113288
\(936\) 0 0
\(937\) −14.7846 −0.482992 −0.241496 0.970402i \(-0.577638\pi\)
−0.241496 + 0.970402i \(0.577638\pi\)
\(938\) 5.46410 0.178409
\(939\) 0 0
\(940\) 6.92820 0.225973
\(941\) 21.7128 0.707817 0.353909 0.935280i \(-0.384852\pi\)
0.353909 + 0.935280i \(0.384852\pi\)
\(942\) 0 0
\(943\) −2.92820 −0.0953554
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) 0 0
\(947\) 44.7846 1.45530 0.727652 0.685946i \(-0.240612\pi\)
0.727652 + 0.685946i \(0.240612\pi\)
\(948\) 0 0
\(949\) −54.9282 −1.78304
\(950\) 0 0
\(951\) 0 0
\(952\) −3.46410 −0.112272
\(953\) 23.0718 0.747369 0.373684 0.927556i \(-0.378094\pi\)
0.373684 + 0.927556i \(0.378094\pi\)
\(954\) 0 0
\(955\) −9.85641 −0.318946
\(956\) −21.4641 −0.694199
\(957\) 0 0
\(958\) −21.8564 −0.706148
\(959\) 3.46410 0.111862
\(960\) 0 0
\(961\) 88.4256 2.85244
\(962\) −12.0000 −0.386896
\(963\) 0 0
\(964\) −18.7846 −0.605012
\(965\) −15.8564 −0.510436
\(966\) 0 0
\(967\) −8.78461 −0.282494 −0.141247 0.989974i \(-0.545111\pi\)
−0.141247 + 0.989974i \(0.545111\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 10.0000 0.321081
\(971\) 58.9282 1.89110 0.945548 0.325483i \(-0.105527\pi\)
0.945548 + 0.325483i \(0.105527\pi\)
\(972\) 0 0
\(973\) 10.9282 0.350342
\(974\) −41.8564 −1.34117
\(975\) 0 0
\(976\) −4.92820 −0.157748
\(977\) 1.60770 0.0514347 0.0257174 0.999669i \(-0.491813\pi\)
0.0257174 + 0.999669i \(0.491813\pi\)
\(978\) 0 0
\(979\) 8.53590 0.272808
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) −1.85641 −0.0592403
\(983\) 26.6410 0.849716 0.424858 0.905260i \(-0.360324\pi\)
0.424858 + 0.905260i \(0.360324\pi\)
\(984\) 0 0
\(985\) 2.00000 0.0637253
\(986\) −30.9282 −0.984955
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −10.9282 −0.347146 −0.173573 0.984821i \(-0.555531\pi\)
−0.173573 + 0.984821i \(0.555531\pi\)
\(992\) 10.9282 0.346971
\(993\) 0 0
\(994\) 1.07180 0.0339953
\(995\) −18.9282 −0.600064
\(996\) 0 0
\(997\) 1.60770 0.0509162 0.0254581 0.999676i \(-0.491896\pi\)
0.0254581 + 0.999676i \(0.491896\pi\)
\(998\) −12.7846 −0.404690
\(999\) 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.bw.1.1 2
3.2 odd 2 2310.2.a.z.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.z.1.1 2 3.2 odd 2
6930.2.a.bw.1.1 2 1.1 even 1 trivial