Properties

Label 6160.2.a.bb.1.2
Level $6160$
Weight $2$
Character 6160.1
Self dual yes
Analytic conductor $49.188$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(49.1878476451\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3080)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 6160.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.41421 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{9} +1.00000 q^{11} -3.41421 q^{13} +1.41421 q^{15} +2.24264 q^{17} -5.65685 q^{19} -1.41421 q^{21} +2.00000 q^{23} +1.00000 q^{25} -5.65685 q^{27} -3.17157 q^{29} +4.58579 q^{31} +1.41421 q^{33} -1.00000 q^{35} -2.82843 q^{37} -4.82843 q^{39} +4.24264 q^{41} +2.00000 q^{43} -1.00000 q^{45} +8.24264 q^{47} +1.00000 q^{49} +3.17157 q^{51} -4.48528 q^{53} +1.00000 q^{55} -8.00000 q^{57} -4.58579 q^{59} -12.2426 q^{61} +1.00000 q^{63} -3.41421 q^{65} -9.31371 q^{67} +2.82843 q^{69} +4.48528 q^{71} -10.2426 q^{73} +1.41421 q^{75} -1.00000 q^{77} -7.17157 q^{79} -5.00000 q^{81} -17.6569 q^{83} +2.24264 q^{85} -4.48528 q^{87} -3.17157 q^{89} +3.41421 q^{91} +6.48528 q^{93} -5.65685 q^{95} -9.31371 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 2 q^{7} - 2 q^{9} + 2 q^{11} - 4 q^{13} - 4 q^{17} + 4 q^{23} + 2 q^{25} - 12 q^{29} + 12 q^{31} - 2 q^{35} - 4 q^{39} + 4 q^{43} - 2 q^{45} + 8 q^{47} + 2 q^{49} + 12 q^{51} + 8 q^{53} + 2 q^{55} - 16 q^{57} - 12 q^{59} - 16 q^{61} + 2 q^{63} - 4 q^{65} + 4 q^{67} - 8 q^{71} - 12 q^{73} - 2 q^{77} - 20 q^{79} - 10 q^{81} - 24 q^{83} - 4 q^{85} + 8 q^{87} - 12 q^{89} + 4 q^{91} - 4 q^{93} + 4 q^{97} - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.41421 0.816497 0.408248 0.912871i \(-0.366140\pi\)
0.408248 + 0.912871i \(0.366140\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −3.41421 −0.946932 −0.473466 0.880812i \(-0.656997\pi\)
−0.473466 + 0.880812i \(0.656997\pi\)
\(14\) 0 0
\(15\) 1.41421 0.365148
\(16\) 0 0
\(17\) 2.24264 0.543920 0.271960 0.962309i \(-0.412328\pi\)
0.271960 + 0.962309i \(0.412328\pi\)
\(18\) 0 0
\(19\) −5.65685 −1.29777 −0.648886 0.760886i \(-0.724765\pi\)
−0.648886 + 0.760886i \(0.724765\pi\)
\(20\) 0 0
\(21\) −1.41421 −0.308607
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) −3.17157 −0.588946 −0.294473 0.955660i \(-0.595144\pi\)
−0.294473 + 0.955660i \(0.595144\pi\)
\(30\) 0 0
\(31\) 4.58579 0.823632 0.411816 0.911267i \(-0.364895\pi\)
0.411816 + 0.911267i \(0.364895\pi\)
\(32\) 0 0
\(33\) 1.41421 0.246183
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −2.82843 −0.464991 −0.232495 0.972598i \(-0.574689\pi\)
−0.232495 + 0.972598i \(0.574689\pi\)
\(38\) 0 0
\(39\) −4.82843 −0.773167
\(40\) 0 0
\(41\) 4.24264 0.662589 0.331295 0.943527i \(-0.392515\pi\)
0.331295 + 0.943527i \(0.392515\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 8.24264 1.20231 0.601156 0.799131i \(-0.294707\pi\)
0.601156 + 0.799131i \(0.294707\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.17157 0.444109
\(52\) 0 0
\(53\) −4.48528 −0.616101 −0.308050 0.951370i \(-0.599677\pi\)
−0.308050 + 0.951370i \(0.599677\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) −8.00000 −1.05963
\(58\) 0 0
\(59\) −4.58579 −0.597019 −0.298509 0.954407i \(-0.596489\pi\)
−0.298509 + 0.954407i \(0.596489\pi\)
\(60\) 0 0
\(61\) −12.2426 −1.56751 −0.783755 0.621070i \(-0.786698\pi\)
−0.783755 + 0.621070i \(0.786698\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −3.41421 −0.423481
\(66\) 0 0
\(67\) −9.31371 −1.13785 −0.568925 0.822389i \(-0.692641\pi\)
−0.568925 + 0.822389i \(0.692641\pi\)
\(68\) 0 0
\(69\) 2.82843 0.340503
\(70\) 0 0
\(71\) 4.48528 0.532305 0.266152 0.963931i \(-0.414248\pi\)
0.266152 + 0.963931i \(0.414248\pi\)
\(72\) 0 0
\(73\) −10.2426 −1.19881 −0.599405 0.800446i \(-0.704596\pi\)
−0.599405 + 0.800446i \(0.704596\pi\)
\(74\) 0 0
\(75\) 1.41421 0.163299
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −7.17157 −0.806865 −0.403432 0.915009i \(-0.632183\pi\)
−0.403432 + 0.915009i \(0.632183\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) −17.6569 −1.93809 −0.969046 0.246881i \(-0.920594\pi\)
−0.969046 + 0.246881i \(0.920594\pi\)
\(84\) 0 0
\(85\) 2.24264 0.243249
\(86\) 0 0
\(87\) −4.48528 −0.480873
\(88\) 0 0
\(89\) −3.17157 −0.336186 −0.168093 0.985771i \(-0.553761\pi\)
−0.168093 + 0.985771i \(0.553761\pi\)
\(90\) 0 0
\(91\) 3.41421 0.357907
\(92\) 0 0
\(93\) 6.48528 0.672492
\(94\) 0 0
\(95\) −5.65685 −0.580381
\(96\) 0 0
\(97\) −9.31371 −0.945664 −0.472832 0.881153i \(-0.656768\pi\)
−0.472832 + 0.881153i \(0.656768\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 1.41421 0.140720 0.0703598 0.997522i \(-0.477585\pi\)
0.0703598 + 0.997522i \(0.477585\pi\)
\(102\) 0 0
\(103\) 13.4142 1.32174 0.660871 0.750500i \(-0.270187\pi\)
0.660871 + 0.750500i \(0.270187\pi\)
\(104\) 0 0
\(105\) −1.41421 −0.138013
\(106\) 0 0
\(107\) 11.3137 1.09374 0.546869 0.837218i \(-0.315820\pi\)
0.546869 + 0.837218i \(0.315820\pi\)
\(108\) 0 0
\(109\) −12.8284 −1.22874 −0.614370 0.789018i \(-0.710590\pi\)
−0.614370 + 0.789018i \(0.710590\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) −9.31371 −0.876160 −0.438080 0.898936i \(-0.644341\pi\)
−0.438080 + 0.898936i \(0.644341\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 0 0
\(117\) 3.41421 0.315644
\(118\) 0 0
\(119\) −2.24264 −0.205583
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 6.00000 0.541002
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −9.65685 −0.856907 −0.428454 0.903564i \(-0.640941\pi\)
−0.428454 + 0.903564i \(0.640941\pi\)
\(128\) 0 0
\(129\) 2.82843 0.249029
\(130\) 0 0
\(131\) 7.31371 0.639002 0.319501 0.947586i \(-0.396485\pi\)
0.319501 + 0.947586i \(0.396485\pi\)
\(132\) 0 0
\(133\) 5.65685 0.490511
\(134\) 0 0
\(135\) −5.65685 −0.486864
\(136\) 0 0
\(137\) −3.51472 −0.300283 −0.150141 0.988665i \(-0.547973\pi\)
−0.150141 + 0.988665i \(0.547973\pi\)
\(138\) 0 0
\(139\) −8.48528 −0.719712 −0.359856 0.933008i \(-0.617174\pi\)
−0.359856 + 0.933008i \(0.617174\pi\)
\(140\) 0 0
\(141\) 11.6569 0.981684
\(142\) 0 0
\(143\) −3.41421 −0.285511
\(144\) 0 0
\(145\) −3.17157 −0.263385
\(146\) 0 0
\(147\) 1.41421 0.116642
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −12.8284 −1.04396 −0.521981 0.852957i \(-0.674807\pi\)
−0.521981 + 0.852957i \(0.674807\pi\)
\(152\) 0 0
\(153\) −2.24264 −0.181307
\(154\) 0 0
\(155\) 4.58579 0.368339
\(156\) 0 0
\(157\) −16.1421 −1.28828 −0.644141 0.764906i \(-0.722785\pi\)
−0.644141 + 0.764906i \(0.722785\pi\)
\(158\) 0 0
\(159\) −6.34315 −0.503044
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) 1.31371 0.102898 0.0514488 0.998676i \(-0.483616\pi\)
0.0514488 + 0.998676i \(0.483616\pi\)
\(164\) 0 0
\(165\) 1.41421 0.110096
\(166\) 0 0
\(167\) 22.1421 1.71341 0.856705 0.515807i \(-0.172508\pi\)
0.856705 + 0.515807i \(0.172508\pi\)
\(168\) 0 0
\(169\) −1.34315 −0.103319
\(170\) 0 0
\(171\) 5.65685 0.432590
\(172\) 0 0
\(173\) −8.58579 −0.652765 −0.326383 0.945238i \(-0.605830\pi\)
−0.326383 + 0.945238i \(0.605830\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −6.48528 −0.487464
\(178\) 0 0
\(179\) −11.3137 −0.845626 −0.422813 0.906217i \(-0.638957\pi\)
−0.422813 + 0.906217i \(0.638957\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) −17.3137 −1.27987
\(184\) 0 0
\(185\) −2.82843 −0.207950
\(186\) 0 0
\(187\) 2.24264 0.163998
\(188\) 0 0
\(189\) 5.65685 0.411476
\(190\) 0 0
\(191\) −1.65685 −0.119886 −0.0599429 0.998202i \(-0.519092\pi\)
−0.0599429 + 0.998202i \(0.519092\pi\)
\(192\) 0 0
\(193\) −12.4853 −0.898710 −0.449355 0.893353i \(-0.648346\pi\)
−0.449355 + 0.893353i \(0.648346\pi\)
\(194\) 0 0
\(195\) −4.82843 −0.345771
\(196\) 0 0
\(197\) −18.8284 −1.34147 −0.670735 0.741697i \(-0.734021\pi\)
−0.670735 + 0.741697i \(0.734021\pi\)
\(198\) 0 0
\(199\) 21.5563 1.52809 0.764045 0.645164i \(-0.223211\pi\)
0.764045 + 0.645164i \(0.223211\pi\)
\(200\) 0 0
\(201\) −13.1716 −0.929051
\(202\) 0 0
\(203\) 3.17157 0.222601
\(204\) 0 0
\(205\) 4.24264 0.296319
\(206\) 0 0
\(207\) −2.00000 −0.139010
\(208\) 0 0
\(209\) −5.65685 −0.391293
\(210\) 0 0
\(211\) 0.686292 0.0472463 0.0236231 0.999721i \(-0.492480\pi\)
0.0236231 + 0.999721i \(0.492480\pi\)
\(212\) 0 0
\(213\) 6.34315 0.434625
\(214\) 0 0
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) −4.58579 −0.311303
\(218\) 0 0
\(219\) −14.4853 −0.978825
\(220\) 0 0
\(221\) −7.65685 −0.515056
\(222\) 0 0
\(223\) −4.24264 −0.284108 −0.142054 0.989859i \(-0.545371\pi\)
−0.142054 + 0.989859i \(0.545371\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) 20.9706 1.39187 0.695933 0.718107i \(-0.254991\pi\)
0.695933 + 0.718107i \(0.254991\pi\)
\(228\) 0 0
\(229\) 0.828427 0.0547440 0.0273720 0.999625i \(-0.491286\pi\)
0.0273720 + 0.999625i \(0.491286\pi\)
\(230\) 0 0
\(231\) −1.41421 −0.0930484
\(232\) 0 0
\(233\) 4.00000 0.262049 0.131024 0.991379i \(-0.458173\pi\)
0.131024 + 0.991379i \(0.458173\pi\)
\(234\) 0 0
\(235\) 8.24264 0.537691
\(236\) 0 0
\(237\) −10.1421 −0.658803
\(238\) 0 0
\(239\) 5.65685 0.365911 0.182956 0.983121i \(-0.441433\pi\)
0.182956 + 0.983121i \(0.441433\pi\)
\(240\) 0 0
\(241\) −28.7279 −1.85053 −0.925264 0.379324i \(-0.876157\pi\)
−0.925264 + 0.379324i \(0.876157\pi\)
\(242\) 0 0
\(243\) 9.89949 0.635053
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 19.3137 1.22890
\(248\) 0 0
\(249\) −24.9706 −1.58245
\(250\) 0 0
\(251\) −5.07107 −0.320083 −0.160041 0.987110i \(-0.551163\pi\)
−0.160041 + 0.987110i \(0.551163\pi\)
\(252\) 0 0
\(253\) 2.00000 0.125739
\(254\) 0 0
\(255\) 3.17157 0.198612
\(256\) 0 0
\(257\) 0.828427 0.0516759 0.0258379 0.999666i \(-0.491775\pi\)
0.0258379 + 0.999666i \(0.491775\pi\)
\(258\) 0 0
\(259\) 2.82843 0.175750
\(260\) 0 0
\(261\) 3.17157 0.196315
\(262\) 0 0
\(263\) 9.65685 0.595467 0.297734 0.954649i \(-0.403769\pi\)
0.297734 + 0.954649i \(0.403769\pi\)
\(264\) 0 0
\(265\) −4.48528 −0.275529
\(266\) 0 0
\(267\) −4.48528 −0.274495
\(268\) 0 0
\(269\) 13.3137 0.811751 0.405876 0.913928i \(-0.366967\pi\)
0.405876 + 0.913928i \(0.366967\pi\)
\(270\) 0 0
\(271\) −1.17157 −0.0711680 −0.0355840 0.999367i \(-0.511329\pi\)
−0.0355840 + 0.999367i \(0.511329\pi\)
\(272\) 0 0
\(273\) 4.82843 0.292230
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 7.79899 0.468596 0.234298 0.972165i \(-0.424721\pi\)
0.234298 + 0.972165i \(0.424721\pi\)
\(278\) 0 0
\(279\) −4.58579 −0.274544
\(280\) 0 0
\(281\) 2.48528 0.148259 0.0741297 0.997249i \(-0.476382\pi\)
0.0741297 + 0.997249i \(0.476382\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) −8.00000 −0.473879
\(286\) 0 0
\(287\) −4.24264 −0.250435
\(288\) 0 0
\(289\) −11.9706 −0.704151
\(290\) 0 0
\(291\) −13.1716 −0.772131
\(292\) 0 0
\(293\) 15.8995 0.928858 0.464429 0.885610i \(-0.346260\pi\)
0.464429 + 0.885610i \(0.346260\pi\)
\(294\) 0 0
\(295\) −4.58579 −0.266995
\(296\) 0 0
\(297\) −5.65685 −0.328244
\(298\) 0 0
\(299\) −6.82843 −0.394898
\(300\) 0 0
\(301\) −2.00000 −0.115278
\(302\) 0 0
\(303\) 2.00000 0.114897
\(304\) 0 0
\(305\) −12.2426 −0.701012
\(306\) 0 0
\(307\) 13.1716 0.751741 0.375871 0.926672i \(-0.377344\pi\)
0.375871 + 0.926672i \(0.377344\pi\)
\(308\) 0 0
\(309\) 18.9706 1.07920
\(310\) 0 0
\(311\) 31.2132 1.76994 0.884969 0.465650i \(-0.154179\pi\)
0.884969 + 0.465650i \(0.154179\pi\)
\(312\) 0 0
\(313\) −9.31371 −0.526442 −0.263221 0.964736i \(-0.584785\pi\)
−0.263221 + 0.964736i \(0.584785\pi\)
\(314\) 0 0
\(315\) 1.00000 0.0563436
\(316\) 0 0
\(317\) 14.9706 0.840831 0.420415 0.907332i \(-0.361884\pi\)
0.420415 + 0.907332i \(0.361884\pi\)
\(318\) 0 0
\(319\) −3.17157 −0.177574
\(320\) 0 0
\(321\) 16.0000 0.893033
\(322\) 0 0
\(323\) −12.6863 −0.705884
\(324\) 0 0
\(325\) −3.41421 −0.189386
\(326\) 0 0
\(327\) −18.1421 −1.00326
\(328\) 0 0
\(329\) −8.24264 −0.454431
\(330\) 0 0
\(331\) −0.686292 −0.0377220 −0.0188610 0.999822i \(-0.506004\pi\)
−0.0188610 + 0.999822i \(0.506004\pi\)
\(332\) 0 0
\(333\) 2.82843 0.154997
\(334\) 0 0
\(335\) −9.31371 −0.508862
\(336\) 0 0
\(337\) 22.6274 1.23259 0.616297 0.787514i \(-0.288632\pi\)
0.616297 + 0.787514i \(0.288632\pi\)
\(338\) 0 0
\(339\) −13.1716 −0.715382
\(340\) 0 0
\(341\) 4.58579 0.248334
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 2.82843 0.152277
\(346\) 0 0
\(347\) 2.00000 0.107366 0.0536828 0.998558i \(-0.482904\pi\)
0.0536828 + 0.998558i \(0.482904\pi\)
\(348\) 0 0
\(349\) 18.3848 0.984115 0.492057 0.870563i \(-0.336245\pi\)
0.492057 + 0.870563i \(0.336245\pi\)
\(350\) 0 0
\(351\) 19.3137 1.03089
\(352\) 0 0
\(353\) −32.6274 −1.73658 −0.868291 0.496055i \(-0.834781\pi\)
−0.868291 + 0.496055i \(0.834781\pi\)
\(354\) 0 0
\(355\) 4.48528 0.238054
\(356\) 0 0
\(357\) −3.17157 −0.167857
\(358\) 0 0
\(359\) −12.1421 −0.640837 −0.320419 0.947276i \(-0.603824\pi\)
−0.320419 + 0.947276i \(0.603824\pi\)
\(360\) 0 0
\(361\) 13.0000 0.684211
\(362\) 0 0
\(363\) 1.41421 0.0742270
\(364\) 0 0
\(365\) −10.2426 −0.536124
\(366\) 0 0
\(367\) 7.27208 0.379599 0.189800 0.981823i \(-0.439216\pi\)
0.189800 + 0.981823i \(0.439216\pi\)
\(368\) 0 0
\(369\) −4.24264 −0.220863
\(370\) 0 0
\(371\) 4.48528 0.232864
\(372\) 0 0
\(373\) −22.8284 −1.18201 −0.591006 0.806667i \(-0.701269\pi\)
−0.591006 + 0.806667i \(0.701269\pi\)
\(374\) 0 0
\(375\) 1.41421 0.0730297
\(376\) 0 0
\(377\) 10.8284 0.557692
\(378\) 0 0
\(379\) −9.85786 −0.506364 −0.253182 0.967419i \(-0.581477\pi\)
−0.253182 + 0.967419i \(0.581477\pi\)
\(380\) 0 0
\(381\) −13.6569 −0.699662
\(382\) 0 0
\(383\) −4.72792 −0.241586 −0.120793 0.992678i \(-0.538544\pi\)
−0.120793 + 0.992678i \(0.538544\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 0 0
\(387\) −2.00000 −0.101666
\(388\) 0 0
\(389\) 30.2843 1.53547 0.767737 0.640765i \(-0.221383\pi\)
0.767737 + 0.640765i \(0.221383\pi\)
\(390\) 0 0
\(391\) 4.48528 0.226830
\(392\) 0 0
\(393\) 10.3431 0.521743
\(394\) 0 0
\(395\) −7.17157 −0.360841
\(396\) 0 0
\(397\) −3.85786 −0.193621 −0.0968103 0.995303i \(-0.530864\pi\)
−0.0968103 + 0.995303i \(0.530864\pi\)
\(398\) 0 0
\(399\) 8.00000 0.400501
\(400\) 0 0
\(401\) 21.3137 1.06436 0.532178 0.846633i \(-0.321374\pi\)
0.532178 + 0.846633i \(0.321374\pi\)
\(402\) 0 0
\(403\) −15.6569 −0.779923
\(404\) 0 0
\(405\) −5.00000 −0.248452
\(406\) 0 0
\(407\) −2.82843 −0.140200
\(408\) 0 0
\(409\) −11.7574 −0.581364 −0.290682 0.956820i \(-0.593882\pi\)
−0.290682 + 0.956820i \(0.593882\pi\)
\(410\) 0 0
\(411\) −4.97056 −0.245180
\(412\) 0 0
\(413\) 4.58579 0.225652
\(414\) 0 0
\(415\) −17.6569 −0.866741
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) −32.3848 −1.58210 −0.791050 0.611752i \(-0.790465\pi\)
−0.791050 + 0.611752i \(0.790465\pi\)
\(420\) 0 0
\(421\) 34.6274 1.68764 0.843819 0.536629i \(-0.180302\pi\)
0.843819 + 0.536629i \(0.180302\pi\)
\(422\) 0 0
\(423\) −8.24264 −0.400771
\(424\) 0 0
\(425\) 2.24264 0.108784
\(426\) 0 0
\(427\) 12.2426 0.592463
\(428\) 0 0
\(429\) −4.82843 −0.233119
\(430\) 0 0
\(431\) −8.14214 −0.392193 −0.196096 0.980585i \(-0.562827\pi\)
−0.196096 + 0.980585i \(0.562827\pi\)
\(432\) 0 0
\(433\) 27.4558 1.31944 0.659722 0.751510i \(-0.270674\pi\)
0.659722 + 0.751510i \(0.270674\pi\)
\(434\) 0 0
\(435\) −4.48528 −0.215053
\(436\) 0 0
\(437\) −11.3137 −0.541208
\(438\) 0 0
\(439\) −35.7990 −1.70859 −0.854296 0.519786i \(-0.826012\pi\)
−0.854296 + 0.519786i \(0.826012\pi\)
\(440\) 0 0
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) 29.1127 1.38319 0.691593 0.722287i \(-0.256909\pi\)
0.691593 + 0.722287i \(0.256909\pi\)
\(444\) 0 0
\(445\) −3.17157 −0.150347
\(446\) 0 0
\(447\) 8.48528 0.401340
\(448\) 0 0
\(449\) 18.3431 0.865667 0.432833 0.901474i \(-0.357514\pi\)
0.432833 + 0.901474i \(0.357514\pi\)
\(450\) 0 0
\(451\) 4.24264 0.199778
\(452\) 0 0
\(453\) −18.1421 −0.852392
\(454\) 0 0
\(455\) 3.41421 0.160061
\(456\) 0 0
\(457\) −3.51472 −0.164412 −0.0822058 0.996615i \(-0.526196\pi\)
−0.0822058 + 0.996615i \(0.526196\pi\)
\(458\) 0 0
\(459\) −12.6863 −0.592145
\(460\) 0 0
\(461\) 7.27208 0.338694 0.169347 0.985556i \(-0.445834\pi\)
0.169347 + 0.985556i \(0.445834\pi\)
\(462\) 0 0
\(463\) 22.0000 1.02243 0.511213 0.859454i \(-0.329196\pi\)
0.511213 + 0.859454i \(0.329196\pi\)
\(464\) 0 0
\(465\) 6.48528 0.300748
\(466\) 0 0
\(467\) 14.8701 0.688104 0.344052 0.938951i \(-0.388200\pi\)
0.344052 + 0.938951i \(0.388200\pi\)
\(468\) 0 0
\(469\) 9.31371 0.430067
\(470\) 0 0
\(471\) −22.8284 −1.05188
\(472\) 0 0
\(473\) 2.00000 0.0919601
\(474\) 0 0
\(475\) −5.65685 −0.259554
\(476\) 0 0
\(477\) 4.48528 0.205367
\(478\) 0 0
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) 9.65685 0.440315
\(482\) 0 0
\(483\) −2.82843 −0.128698
\(484\) 0 0
\(485\) −9.31371 −0.422914
\(486\) 0 0
\(487\) 30.2843 1.37231 0.686156 0.727455i \(-0.259297\pi\)
0.686156 + 0.727455i \(0.259297\pi\)
\(488\) 0 0
\(489\) 1.85786 0.0840155
\(490\) 0 0
\(491\) −34.4853 −1.55630 −0.778149 0.628079i \(-0.783841\pi\)
−0.778149 + 0.628079i \(0.783841\pi\)
\(492\) 0 0
\(493\) −7.11270 −0.320340
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) 0 0
\(497\) −4.48528 −0.201192
\(498\) 0 0
\(499\) −0.201010 −0.00899845 −0.00449922 0.999990i \(-0.501432\pi\)
−0.00449922 + 0.999990i \(0.501432\pi\)
\(500\) 0 0
\(501\) 31.3137 1.39899
\(502\) 0 0
\(503\) −17.6569 −0.787280 −0.393640 0.919265i \(-0.628784\pi\)
−0.393640 + 0.919265i \(0.628784\pi\)
\(504\) 0 0
\(505\) 1.41421 0.0629317
\(506\) 0 0
\(507\) −1.89949 −0.0843595
\(508\) 0 0
\(509\) −18.4853 −0.819346 −0.409673 0.912233i \(-0.634357\pi\)
−0.409673 + 0.912233i \(0.634357\pi\)
\(510\) 0 0
\(511\) 10.2426 0.453108
\(512\) 0 0
\(513\) 32.0000 1.41283
\(514\) 0 0
\(515\) 13.4142 0.591101
\(516\) 0 0
\(517\) 8.24264 0.362511
\(518\) 0 0
\(519\) −12.1421 −0.532981
\(520\) 0 0
\(521\) 37.3137 1.63474 0.817372 0.576111i \(-0.195430\pi\)
0.817372 + 0.576111i \(0.195430\pi\)
\(522\) 0 0
\(523\) −40.7696 −1.78273 −0.891364 0.453288i \(-0.850251\pi\)
−0.891364 + 0.453288i \(0.850251\pi\)
\(524\) 0 0
\(525\) −1.41421 −0.0617213
\(526\) 0 0
\(527\) 10.2843 0.447990
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) 4.58579 0.199006
\(532\) 0 0
\(533\) −14.4853 −0.627427
\(534\) 0 0
\(535\) 11.3137 0.489134
\(536\) 0 0
\(537\) −16.0000 −0.690451
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 1.31371 0.0564807 0.0282404 0.999601i \(-0.491010\pi\)
0.0282404 + 0.999601i \(0.491010\pi\)
\(542\) 0 0
\(543\) −8.48528 −0.364138
\(544\) 0 0
\(545\) −12.8284 −0.549509
\(546\) 0 0
\(547\) 15.3137 0.654767 0.327383 0.944892i \(-0.393833\pi\)
0.327383 + 0.944892i \(0.393833\pi\)
\(548\) 0 0
\(549\) 12.2426 0.522503
\(550\) 0 0
\(551\) 17.9411 0.764318
\(552\) 0 0
\(553\) 7.17157 0.304966
\(554\) 0 0
\(555\) −4.00000 −0.169791
\(556\) 0 0
\(557\) 3.79899 0.160968 0.0804842 0.996756i \(-0.474353\pi\)
0.0804842 + 0.996756i \(0.474353\pi\)
\(558\) 0 0
\(559\) −6.82843 −0.288812
\(560\) 0 0
\(561\) 3.17157 0.133904
\(562\) 0 0
\(563\) 19.5147 0.822447 0.411224 0.911534i \(-0.365102\pi\)
0.411224 + 0.911534i \(0.365102\pi\)
\(564\) 0 0
\(565\) −9.31371 −0.391831
\(566\) 0 0
\(567\) 5.00000 0.209980
\(568\) 0 0
\(569\) 44.8284 1.87931 0.939653 0.342130i \(-0.111148\pi\)
0.939653 + 0.342130i \(0.111148\pi\)
\(570\) 0 0
\(571\) 26.6274 1.11432 0.557161 0.830404i \(-0.311890\pi\)
0.557161 + 0.830404i \(0.311890\pi\)
\(572\) 0 0
\(573\) −2.34315 −0.0978863
\(574\) 0 0
\(575\) 2.00000 0.0834058
\(576\) 0 0
\(577\) 0.627417 0.0261197 0.0130599 0.999915i \(-0.495843\pi\)
0.0130599 + 0.999915i \(0.495843\pi\)
\(578\) 0 0
\(579\) −17.6569 −0.733794
\(580\) 0 0
\(581\) 17.6569 0.732530
\(582\) 0 0
\(583\) −4.48528 −0.185761
\(584\) 0 0
\(585\) 3.41421 0.141160
\(586\) 0 0
\(587\) −36.0416 −1.48760 −0.743799 0.668404i \(-0.766978\pi\)
−0.743799 + 0.668404i \(0.766978\pi\)
\(588\) 0 0
\(589\) −25.9411 −1.06889
\(590\) 0 0
\(591\) −26.6274 −1.09531
\(592\) 0 0
\(593\) 15.8995 0.652914 0.326457 0.945212i \(-0.394145\pi\)
0.326457 + 0.945212i \(0.394145\pi\)
\(594\) 0 0
\(595\) −2.24264 −0.0919393
\(596\) 0 0
\(597\) 30.4853 1.24768
\(598\) 0 0
\(599\) −27.7990 −1.13584 −0.567918 0.823085i \(-0.692251\pi\)
−0.567918 + 0.823085i \(0.692251\pi\)
\(600\) 0 0
\(601\) −40.0416 −1.63333 −0.816666 0.577110i \(-0.804180\pi\)
−0.816666 + 0.577110i \(0.804180\pi\)
\(602\) 0 0
\(603\) 9.31371 0.379284
\(604\) 0 0
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 30.1421 1.22343 0.611716 0.791078i \(-0.290480\pi\)
0.611716 + 0.791078i \(0.290480\pi\)
\(608\) 0 0
\(609\) 4.48528 0.181753
\(610\) 0 0
\(611\) −28.1421 −1.13851
\(612\) 0 0
\(613\) −3.31371 −0.133839 −0.0669197 0.997758i \(-0.521317\pi\)
−0.0669197 + 0.997758i \(0.521317\pi\)
\(614\) 0 0
\(615\) 6.00000 0.241943
\(616\) 0 0
\(617\) −3.51472 −0.141497 −0.0707486 0.997494i \(-0.522539\pi\)
−0.0707486 + 0.997494i \(0.522539\pi\)
\(618\) 0 0
\(619\) −26.9289 −1.08237 −0.541183 0.840905i \(-0.682023\pi\)
−0.541183 + 0.840905i \(0.682023\pi\)
\(620\) 0 0
\(621\) −11.3137 −0.454003
\(622\) 0 0
\(623\) 3.17157 0.127066
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −8.00000 −0.319489
\(628\) 0 0
\(629\) −6.34315 −0.252918
\(630\) 0 0
\(631\) 12.6863 0.505033 0.252517 0.967593i \(-0.418742\pi\)
0.252517 + 0.967593i \(0.418742\pi\)
\(632\) 0 0
\(633\) 0.970563 0.0385764
\(634\) 0 0
\(635\) −9.65685 −0.383221
\(636\) 0 0
\(637\) −3.41421 −0.135276
\(638\) 0 0
\(639\) −4.48528 −0.177435
\(640\) 0 0
\(641\) −8.68629 −0.343088 −0.171544 0.985176i \(-0.554876\pi\)
−0.171544 + 0.985176i \(0.554876\pi\)
\(642\) 0 0
\(643\) −37.8995 −1.49461 −0.747305 0.664481i \(-0.768653\pi\)
−0.747305 + 0.664481i \(0.768653\pi\)
\(644\) 0 0
\(645\) 2.82843 0.111369
\(646\) 0 0
\(647\) −14.8701 −0.584602 −0.292301 0.956326i \(-0.594421\pi\)
−0.292301 + 0.956326i \(0.594421\pi\)
\(648\) 0 0
\(649\) −4.58579 −0.180008
\(650\) 0 0
\(651\) −6.48528 −0.254178
\(652\) 0 0
\(653\) −5.31371 −0.207941 −0.103971 0.994580i \(-0.533155\pi\)
−0.103971 + 0.994580i \(0.533155\pi\)
\(654\) 0 0
\(655\) 7.31371 0.285770
\(656\) 0 0
\(657\) 10.2426 0.399603
\(658\) 0 0
\(659\) −26.4853 −1.03172 −0.515860 0.856673i \(-0.672528\pi\)
−0.515860 + 0.856673i \(0.672528\pi\)
\(660\) 0 0
\(661\) −9.51472 −0.370080 −0.185040 0.982731i \(-0.559241\pi\)
−0.185040 + 0.982731i \(0.559241\pi\)
\(662\) 0 0
\(663\) −10.8284 −0.420541
\(664\) 0 0
\(665\) 5.65685 0.219363
\(666\) 0 0
\(667\) −6.34315 −0.245608
\(668\) 0 0
\(669\) −6.00000 −0.231973
\(670\) 0 0
\(671\) −12.2426 −0.472622
\(672\) 0 0
\(673\) 12.6863 0.489021 0.244510 0.969647i \(-0.421373\pi\)
0.244510 + 0.969647i \(0.421373\pi\)
\(674\) 0 0
\(675\) −5.65685 −0.217732
\(676\) 0 0
\(677\) −36.3848 −1.39838 −0.699190 0.714936i \(-0.746456\pi\)
−0.699190 + 0.714936i \(0.746456\pi\)
\(678\) 0 0
\(679\) 9.31371 0.357427
\(680\) 0 0
\(681\) 29.6569 1.13645
\(682\) 0 0
\(683\) −39.4558 −1.50974 −0.754868 0.655877i \(-0.772299\pi\)
−0.754868 + 0.655877i \(0.772299\pi\)
\(684\) 0 0
\(685\) −3.51472 −0.134290
\(686\) 0 0
\(687\) 1.17157 0.0446983
\(688\) 0 0
\(689\) 15.3137 0.583406
\(690\) 0 0
\(691\) 14.2426 0.541816 0.270908 0.962605i \(-0.412676\pi\)
0.270908 + 0.962605i \(0.412676\pi\)
\(692\) 0 0
\(693\) 1.00000 0.0379869
\(694\) 0 0
\(695\) −8.48528 −0.321865
\(696\) 0 0
\(697\) 9.51472 0.360396
\(698\) 0 0
\(699\) 5.65685 0.213962
\(700\) 0 0
\(701\) 27.4558 1.03699 0.518496 0.855080i \(-0.326492\pi\)
0.518496 + 0.855080i \(0.326492\pi\)
\(702\) 0 0
\(703\) 16.0000 0.603451
\(704\) 0 0
\(705\) 11.6569 0.439023
\(706\) 0 0
\(707\) −1.41421 −0.0531870
\(708\) 0 0
\(709\) 25.3137 0.950676 0.475338 0.879803i \(-0.342326\pi\)
0.475338 + 0.879803i \(0.342326\pi\)
\(710\) 0 0
\(711\) 7.17157 0.268955
\(712\) 0 0
\(713\) 9.17157 0.343478
\(714\) 0 0
\(715\) −3.41421 −0.127684
\(716\) 0 0
\(717\) 8.00000 0.298765
\(718\) 0 0
\(719\) 0.384776 0.0143497 0.00717487 0.999974i \(-0.497716\pi\)
0.00717487 + 0.999974i \(0.497716\pi\)
\(720\) 0 0
\(721\) −13.4142 −0.499571
\(722\) 0 0
\(723\) −40.6274 −1.51095
\(724\) 0 0
\(725\) −3.17157 −0.117789
\(726\) 0 0
\(727\) 2.10051 0.0779034 0.0389517 0.999241i \(-0.487598\pi\)
0.0389517 + 0.999241i \(0.487598\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 0 0
\(731\) 4.48528 0.165894
\(732\) 0 0
\(733\) −0.786797 −0.0290610 −0.0145305 0.999894i \(-0.504625\pi\)
−0.0145305 + 0.999894i \(0.504625\pi\)
\(734\) 0 0
\(735\) 1.41421 0.0521641
\(736\) 0 0
\(737\) −9.31371 −0.343075
\(738\) 0 0
\(739\) −9.65685 −0.355233 −0.177617 0.984100i \(-0.556839\pi\)
−0.177617 + 0.984100i \(0.556839\pi\)
\(740\) 0 0
\(741\) 27.3137 1.00339
\(742\) 0 0
\(743\) 12.9706 0.475844 0.237922 0.971284i \(-0.423534\pi\)
0.237922 + 0.971284i \(0.423534\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) 17.6569 0.646031
\(748\) 0 0
\(749\) −11.3137 −0.413394
\(750\) 0 0
\(751\) 9.45584 0.345049 0.172524 0.985005i \(-0.444808\pi\)
0.172524 + 0.985005i \(0.444808\pi\)
\(752\) 0 0
\(753\) −7.17157 −0.261347
\(754\) 0 0
\(755\) −12.8284 −0.466874
\(756\) 0 0
\(757\) 36.3431 1.32091 0.660457 0.750864i \(-0.270363\pi\)
0.660457 + 0.750864i \(0.270363\pi\)
\(758\) 0 0
\(759\) 2.82843 0.102665
\(760\) 0 0
\(761\) −10.3848 −0.376448 −0.188224 0.982126i \(-0.560273\pi\)
−0.188224 + 0.982126i \(0.560273\pi\)
\(762\) 0 0
\(763\) 12.8284 0.464420
\(764\) 0 0
\(765\) −2.24264 −0.0810828
\(766\) 0 0
\(767\) 15.6569 0.565336
\(768\) 0 0
\(769\) −24.2426 −0.874212 −0.437106 0.899410i \(-0.643997\pi\)
−0.437106 + 0.899410i \(0.643997\pi\)
\(770\) 0 0
\(771\) 1.17157 0.0421932
\(772\) 0 0
\(773\) 22.9706 0.826194 0.413097 0.910687i \(-0.364447\pi\)
0.413097 + 0.910687i \(0.364447\pi\)
\(774\) 0 0
\(775\) 4.58579 0.164726
\(776\) 0 0
\(777\) 4.00000 0.143499
\(778\) 0 0
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 4.48528 0.160496
\(782\) 0 0
\(783\) 17.9411 0.641164
\(784\) 0 0
\(785\) −16.1421 −0.576138
\(786\) 0 0
\(787\) 42.8284 1.52667 0.763334 0.646004i \(-0.223561\pi\)
0.763334 + 0.646004i \(0.223561\pi\)
\(788\) 0 0
\(789\) 13.6569 0.486197
\(790\) 0 0
\(791\) 9.31371 0.331157
\(792\) 0 0
\(793\) 41.7990 1.48433
\(794\) 0 0
\(795\) −6.34315 −0.224968
\(796\) 0 0
\(797\) −3.45584 −0.122412 −0.0612061 0.998125i \(-0.519495\pi\)
−0.0612061 + 0.998125i \(0.519495\pi\)
\(798\) 0 0
\(799\) 18.4853 0.653962
\(800\) 0 0
\(801\) 3.17157 0.112062
\(802\) 0 0
\(803\) −10.2426 −0.361455
\(804\) 0 0
\(805\) −2.00000 −0.0704907
\(806\) 0 0
\(807\) 18.8284 0.662792
\(808\) 0 0
\(809\) −48.6274 −1.70965 −0.854824 0.518917i \(-0.826335\pi\)
−0.854824 + 0.518917i \(0.826335\pi\)
\(810\) 0 0
\(811\) 22.1421 0.777516 0.388758 0.921340i \(-0.372904\pi\)
0.388758 + 0.921340i \(0.372904\pi\)
\(812\) 0 0
\(813\) −1.65685 −0.0581084
\(814\) 0 0
\(815\) 1.31371 0.0460172
\(816\) 0 0
\(817\) −11.3137 −0.395817
\(818\) 0 0
\(819\) −3.41421 −0.119302
\(820\) 0 0
\(821\) −45.5980 −1.59138 −0.795690 0.605704i \(-0.792892\pi\)
−0.795690 + 0.605704i \(0.792892\pi\)
\(822\) 0 0
\(823\) −24.8284 −0.865465 −0.432732 0.901522i \(-0.642450\pi\)
−0.432732 + 0.901522i \(0.642450\pi\)
\(824\) 0 0
\(825\) 1.41421 0.0492366
\(826\) 0 0
\(827\) −42.9706 −1.49423 −0.747116 0.664693i \(-0.768562\pi\)
−0.747116 + 0.664693i \(0.768562\pi\)
\(828\) 0 0
\(829\) 0.343146 0.0119179 0.00595897 0.999982i \(-0.498103\pi\)
0.00595897 + 0.999982i \(0.498103\pi\)
\(830\) 0 0
\(831\) 11.0294 0.382607
\(832\) 0 0
\(833\) 2.24264 0.0777029
\(834\) 0 0
\(835\) 22.1421 0.766260
\(836\) 0 0
\(837\) −25.9411 −0.896656
\(838\) 0 0
\(839\) 51.0122 1.76114 0.880568 0.473919i \(-0.157161\pi\)
0.880568 + 0.473919i \(0.157161\pi\)
\(840\) 0 0
\(841\) −18.9411 −0.653142
\(842\) 0 0
\(843\) 3.51472 0.121053
\(844\) 0 0
\(845\) −1.34315 −0.0462056
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 0 0
\(849\) 5.65685 0.194143
\(850\) 0 0
\(851\) −5.65685 −0.193914
\(852\) 0 0
\(853\) 34.9289 1.19594 0.597972 0.801517i \(-0.295973\pi\)
0.597972 + 0.801517i \(0.295973\pi\)
\(854\) 0 0
\(855\) 5.65685 0.193460
\(856\) 0 0
\(857\) −44.5858 −1.52302 −0.761511 0.648152i \(-0.775542\pi\)
−0.761511 + 0.648152i \(0.775542\pi\)
\(858\) 0 0
\(859\) 49.5563 1.69084 0.845420 0.534101i \(-0.179350\pi\)
0.845420 + 0.534101i \(0.179350\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) −47.4558 −1.61542 −0.807708 0.589583i \(-0.799292\pi\)
−0.807708 + 0.589583i \(0.799292\pi\)
\(864\) 0 0
\(865\) −8.58579 −0.291925
\(866\) 0 0
\(867\) −16.9289 −0.574937
\(868\) 0 0
\(869\) −7.17157 −0.243279
\(870\) 0 0
\(871\) 31.7990 1.07747
\(872\) 0 0
\(873\) 9.31371 0.315221
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −4.68629 −0.158245 −0.0791224 0.996865i \(-0.525212\pi\)
−0.0791224 + 0.996865i \(0.525212\pi\)
\(878\) 0 0
\(879\) 22.4853 0.758410
\(880\) 0 0
\(881\) 29.3137 0.987604 0.493802 0.869574i \(-0.335607\pi\)
0.493802 + 0.869574i \(0.335607\pi\)
\(882\) 0 0
\(883\) 39.4558 1.32779 0.663897 0.747824i \(-0.268901\pi\)
0.663897 + 0.747824i \(0.268901\pi\)
\(884\) 0 0
\(885\) −6.48528 −0.218000
\(886\) 0 0
\(887\) −7.79899 −0.261864 −0.130932 0.991391i \(-0.541797\pi\)
−0.130932 + 0.991391i \(0.541797\pi\)
\(888\) 0 0
\(889\) 9.65685 0.323880
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) 0 0
\(893\) −46.6274 −1.56033
\(894\) 0 0
\(895\) −11.3137 −0.378176
\(896\) 0 0
\(897\) −9.65685 −0.322433
\(898\) 0 0
\(899\) −14.5442 −0.485075
\(900\) 0 0
\(901\) −10.0589 −0.335110
\(902\) 0 0
\(903\) −2.82843 −0.0941242
\(904\) 0 0
\(905\) −6.00000 −0.199447
\(906\) 0 0
\(907\) −27.9411 −0.927770 −0.463885 0.885895i \(-0.653545\pi\)
−0.463885 + 0.885895i \(0.653545\pi\)
\(908\) 0 0
\(909\) −1.41421 −0.0469065
\(910\) 0 0
\(911\) −21.8579 −0.724183 −0.362092 0.932142i \(-0.617937\pi\)
−0.362092 + 0.932142i \(0.617937\pi\)
\(912\) 0 0
\(913\) −17.6569 −0.584357
\(914\) 0 0
\(915\) −17.3137 −0.572374
\(916\) 0 0
\(917\) −7.31371 −0.241520
\(918\) 0 0
\(919\) 0.970563 0.0320159 0.0160080 0.999872i \(-0.494904\pi\)
0.0160080 + 0.999872i \(0.494904\pi\)
\(920\) 0 0
\(921\) 18.6274 0.613794
\(922\) 0 0
\(923\) −15.3137 −0.504057
\(924\) 0 0
\(925\) −2.82843 −0.0929981
\(926\) 0 0
\(927\) −13.4142 −0.440581
\(928\) 0 0
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) −5.65685 −0.185396
\(932\) 0 0
\(933\) 44.1421 1.44515
\(934\) 0 0
\(935\) 2.24264 0.0733422
\(936\) 0 0
\(937\) 9.55635 0.312192 0.156096 0.987742i \(-0.450109\pi\)
0.156096 + 0.987742i \(0.450109\pi\)
\(938\) 0 0
\(939\) −13.1716 −0.429838
\(940\) 0 0
\(941\) 45.0122 1.46736 0.733678 0.679498i \(-0.237802\pi\)
0.733678 + 0.679498i \(0.237802\pi\)
\(942\) 0 0
\(943\) 8.48528 0.276319
\(944\) 0 0
\(945\) 5.65685 0.184017
\(946\) 0 0
\(947\) −21.5147 −0.699134 −0.349567 0.936911i \(-0.613671\pi\)
−0.349567 + 0.936911i \(0.613671\pi\)
\(948\) 0 0
\(949\) 34.9706 1.13519
\(950\) 0 0
\(951\) 21.1716 0.686535
\(952\) 0 0
\(953\) 23.5147 0.761716 0.380858 0.924633i \(-0.375629\pi\)
0.380858 + 0.924633i \(0.375629\pi\)
\(954\) 0 0
\(955\) −1.65685 −0.0536145
\(956\) 0 0
\(957\) −4.48528 −0.144989
\(958\) 0 0
\(959\) 3.51472 0.113496
\(960\) 0 0
\(961\) −9.97056 −0.321631
\(962\) 0 0
\(963\) −11.3137 −0.364579
\(964\) 0 0
\(965\) −12.4853 −0.401915
\(966\) 0 0
\(967\) 22.0000 0.707472 0.353736 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) 0 0
\(969\) −17.9411 −0.576352
\(970\) 0 0
\(971\) 10.0416 0.322251 0.161126 0.986934i \(-0.448488\pi\)
0.161126 + 0.986934i \(0.448488\pi\)
\(972\) 0 0
\(973\) 8.48528 0.272026
\(974\) 0 0
\(975\) −4.82843 −0.154633
\(976\) 0 0
\(977\) −26.2843 −0.840908 −0.420454 0.907314i \(-0.638129\pi\)
−0.420454 + 0.907314i \(0.638129\pi\)
\(978\) 0 0
\(979\) −3.17157 −0.101364
\(980\) 0 0
\(981\) 12.8284 0.409580
\(982\) 0 0
\(983\) 48.2426 1.53870 0.769351 0.638827i \(-0.220580\pi\)
0.769351 + 0.638827i \(0.220580\pi\)
\(984\) 0 0
\(985\) −18.8284 −0.599924
\(986\) 0 0
\(987\) −11.6569 −0.371042
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) −32.7696 −1.04096 −0.520480 0.853874i \(-0.674247\pi\)
−0.520480 + 0.853874i \(0.674247\pi\)
\(992\) 0 0
\(993\) −0.970563 −0.0307999
\(994\) 0 0
\(995\) 21.5563 0.683382
\(996\) 0 0
\(997\) −37.8406 −1.19842 −0.599212 0.800590i \(-0.704519\pi\)
−0.599212 + 0.800590i \(0.704519\pi\)
\(998\) 0 0
\(999\) 16.0000 0.506218
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 6160.2.a.bb.1.2 2
4.3 odd 2 3080.2.a.i.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3080.2.a.i.1.1 2 4.3 odd 2
6160.2.a.bb.1.2 2 1.1 even 1 trivial