Properties

Label 2368.2.a.ba.1.2
Level $2368$
Weight $2$
Character 2368.1
Self dual yes
Analytic conductor $18.909$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2368 = 2^{6} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2368.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.9085751986\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) 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 74)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 2368.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.30278 q^{3} +2.30278 q^{5} +2.60555 q^{7} +7.90833 q^{9} +O(q^{10})\) \(q+3.30278 q^{3} +2.30278 q^{5} +2.60555 q^{7} +7.90833 q^{9} -2.30278 q^{11} -1.30278 q^{13} +7.60555 q^{15} -6.00000 q^{17} +2.00000 q^{19} +8.60555 q^{21} -3.90833 q^{23} +0.302776 q^{25} +16.2111 q^{27} +3.90833 q^{29} +0.302776 q^{31} -7.60555 q^{33} +6.00000 q^{35} -1.00000 q^{37} -4.30278 q^{39} +9.90833 q^{41} +0.605551 q^{43} +18.2111 q^{45} -4.60555 q^{47} -0.211103 q^{49} -19.8167 q^{51} +6.00000 q^{53} -5.30278 q^{55} +6.60555 q^{57} +10.6056 q^{59} -7.51388 q^{61} +20.6056 q^{63} -3.00000 q^{65} -3.51388 q^{67} -12.9083 q^{69} -6.00000 q^{71} -12.3028 q^{73} +1.00000 q^{75} -6.00000 q^{77} -9.11943 q^{79} +29.8167 q^{81} +2.78890 q^{83} -13.8167 q^{85} +12.9083 q^{87} -9.21110 q^{89} -3.39445 q^{91} +1.00000 q^{93} +4.60555 q^{95} -16.4222 q^{97} -18.2111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} + q^{5} - 2 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} + q^{5} - 2 q^{7} + 5 q^{9} - q^{11} + q^{13} + 8 q^{15} - 12 q^{17} + 4 q^{19} + 10 q^{21} + 3 q^{23} - 3 q^{25} + 18 q^{27} - 3 q^{29} - 3 q^{31} - 8 q^{33} + 12 q^{35} - 2 q^{37} - 5 q^{39} + 9 q^{41} - 6 q^{43} + 22 q^{45} - 2 q^{47} + 14 q^{49} - 18 q^{51} + 12 q^{53} - 7 q^{55} + 6 q^{57} + 14 q^{59} + 3 q^{61} + 34 q^{63} - 6 q^{65} + 11 q^{67} - 15 q^{69} - 12 q^{71} - 21 q^{73} + 2 q^{75} - 12 q^{77} + 7 q^{79} + 38 q^{81} + 20 q^{83} - 6 q^{85} + 15 q^{87} - 4 q^{89} - 14 q^{91} + 2 q^{93} + 2 q^{95} - 4 q^{97} - 22 q^{99}+O(q^{100}) \) 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
\(3\) 3.30278 1.90686 0.953429 0.301617i \(-0.0975264\pi\)
0.953429 + 0.301617i \(0.0975264\pi\)
\(4\) 0 0
\(5\) 2.30278 1.02983 0.514916 0.857240i \(-0.327823\pi\)
0.514916 + 0.857240i \(0.327823\pi\)
\(6\) 0 0
\(7\) 2.60555 0.984806 0.492403 0.870367i \(-0.336119\pi\)
0.492403 + 0.870367i \(0.336119\pi\)
\(8\) 0 0
\(9\) 7.90833 2.63611
\(10\) 0 0
\(11\) −2.30278 −0.694313 −0.347156 0.937807i \(-0.612853\pi\)
−0.347156 + 0.937807i \(0.612853\pi\)
\(12\) 0 0
\(13\) −1.30278 −0.361325 −0.180662 0.983545i \(-0.557824\pi\)
−0.180662 + 0.983545i \(0.557824\pi\)
\(14\) 0 0
\(15\) 7.60555 1.96374
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 8.60555 1.87789
\(22\) 0 0
\(23\) −3.90833 −0.814942 −0.407471 0.913218i \(-0.633589\pi\)
−0.407471 + 0.913218i \(0.633589\pi\)
\(24\) 0 0
\(25\) 0.302776 0.0605551
\(26\) 0 0
\(27\) 16.2111 3.11983
\(28\) 0 0
\(29\) 3.90833 0.725758 0.362879 0.931836i \(-0.381794\pi\)
0.362879 + 0.931836i \(0.381794\pi\)
\(30\) 0 0
\(31\) 0.302776 0.0543801 0.0271901 0.999630i \(-0.491344\pi\)
0.0271901 + 0.999630i \(0.491344\pi\)
\(32\) 0 0
\(33\) −7.60555 −1.32396
\(34\) 0 0
\(35\) 6.00000 1.01419
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) −4.30278 −0.688996
\(40\) 0 0
\(41\) 9.90833 1.54742 0.773710 0.633540i \(-0.218399\pi\)
0.773710 + 0.633540i \(0.218399\pi\)
\(42\) 0 0
\(43\) 0.605551 0.0923457 0.0461729 0.998933i \(-0.485297\pi\)
0.0461729 + 0.998933i \(0.485297\pi\)
\(44\) 0 0
\(45\) 18.2111 2.71475
\(46\) 0 0
\(47\) −4.60555 −0.671789 −0.335894 0.941900i \(-0.609039\pi\)
−0.335894 + 0.941900i \(0.609039\pi\)
\(48\) 0 0
\(49\) −0.211103 −0.0301575
\(50\) 0 0
\(51\) −19.8167 −2.77489
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −5.30278 −0.715026
\(56\) 0 0
\(57\) 6.60555 0.874927
\(58\) 0 0
\(59\) 10.6056 1.38073 0.690363 0.723464i \(-0.257451\pi\)
0.690363 + 0.723464i \(0.257451\pi\)
\(60\) 0 0
\(61\) −7.51388 −0.962054 −0.481027 0.876706i \(-0.659736\pi\)
−0.481027 + 0.876706i \(0.659736\pi\)
\(62\) 0 0
\(63\) 20.6056 2.59606
\(64\) 0 0
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) −3.51388 −0.429289 −0.214644 0.976692i \(-0.568859\pi\)
−0.214644 + 0.976692i \(0.568859\pi\)
\(68\) 0 0
\(69\) −12.9083 −1.55398
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −12.3028 −1.43993 −0.719965 0.694010i \(-0.755842\pi\)
−0.719965 + 0.694010i \(0.755842\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) −9.11943 −1.02602 −0.513008 0.858384i \(-0.671469\pi\)
−0.513008 + 0.858384i \(0.671469\pi\)
\(80\) 0 0
\(81\) 29.8167 3.31296
\(82\) 0 0
\(83\) 2.78890 0.306121 0.153061 0.988217i \(-0.451087\pi\)
0.153061 + 0.988217i \(0.451087\pi\)
\(84\) 0 0
\(85\) −13.8167 −1.49863
\(86\) 0 0
\(87\) 12.9083 1.38392
\(88\) 0 0
\(89\) −9.21110 −0.976375 −0.488187 0.872739i \(-0.662342\pi\)
−0.488187 + 0.872739i \(0.662342\pi\)
\(90\) 0 0
\(91\) −3.39445 −0.355835
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) 4.60555 0.472520
\(96\) 0 0
\(97\) −16.4222 −1.66742 −0.833711 0.552201i \(-0.813788\pi\)
−0.833711 + 0.552201i \(0.813788\pi\)
\(98\) 0 0
\(99\) −18.2111 −1.83028
\(100\) 0 0
\(101\) 12.4222 1.23606 0.618028 0.786156i \(-0.287932\pi\)
0.618028 + 0.786156i \(0.287932\pi\)
\(102\) 0 0
\(103\) 0.302776 0.0298334 0.0149167 0.999889i \(-0.495252\pi\)
0.0149167 + 0.999889i \(0.495252\pi\)
\(104\) 0 0
\(105\) 19.8167 1.93391
\(106\) 0 0
\(107\) 0.697224 0.0674032 0.0337016 0.999432i \(-0.489270\pi\)
0.0337016 + 0.999432i \(0.489270\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −3.30278 −0.313486
\(112\) 0 0
\(113\) −3.21110 −0.302075 −0.151038 0.988528i \(-0.548261\pi\)
−0.151038 + 0.988528i \(0.548261\pi\)
\(114\) 0 0
\(115\) −9.00000 −0.839254
\(116\) 0 0
\(117\) −10.3028 −0.952492
\(118\) 0 0
\(119\) −15.6333 −1.43310
\(120\) 0 0
\(121\) −5.69722 −0.517929
\(122\) 0 0
\(123\) 32.7250 2.95071
\(124\) 0 0
\(125\) −10.8167 −0.967471
\(126\) 0 0
\(127\) 19.2111 1.70471 0.852355 0.522964i \(-0.175174\pi\)
0.852355 + 0.522964i \(0.175174\pi\)
\(128\) 0 0
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) 10.6056 0.926611 0.463306 0.886199i \(-0.346663\pi\)
0.463306 + 0.886199i \(0.346663\pi\)
\(132\) 0 0
\(133\) 5.21110 0.451860
\(134\) 0 0
\(135\) 37.3305 3.21290
\(136\) 0 0
\(137\) 0.908327 0.0776036 0.0388018 0.999247i \(-0.487646\pi\)
0.0388018 + 0.999247i \(0.487646\pi\)
\(138\) 0 0
\(139\) −1.90833 −0.161862 −0.0809311 0.996720i \(-0.525789\pi\)
−0.0809311 + 0.996720i \(0.525789\pi\)
\(140\) 0 0
\(141\) −15.2111 −1.28101
\(142\) 0 0
\(143\) 3.00000 0.250873
\(144\) 0 0
\(145\) 9.00000 0.747409
\(146\) 0 0
\(147\) −0.697224 −0.0575061
\(148\) 0 0
\(149\) −19.8167 −1.62344 −0.811722 0.584044i \(-0.801469\pi\)
−0.811722 + 0.584044i \(0.801469\pi\)
\(150\) 0 0
\(151\) 20.6056 1.67686 0.838428 0.545012i \(-0.183475\pi\)
0.838428 + 0.545012i \(0.183475\pi\)
\(152\) 0 0
\(153\) −47.4500 −3.83610
\(154\) 0 0
\(155\) 0.697224 0.0560024
\(156\) 0 0
\(157\) 7.21110 0.575509 0.287754 0.957704i \(-0.407091\pi\)
0.287754 + 0.957704i \(0.407091\pi\)
\(158\) 0 0
\(159\) 19.8167 1.57156
\(160\) 0 0
\(161\) −10.1833 −0.802560
\(162\) 0 0
\(163\) 8.42221 0.659678 0.329839 0.944037i \(-0.393006\pi\)
0.329839 + 0.944037i \(0.393006\pi\)
\(164\) 0 0
\(165\) −17.5139 −1.36345
\(166\) 0 0
\(167\) 5.51388 0.426677 0.213338 0.976978i \(-0.431566\pi\)
0.213338 + 0.976978i \(0.431566\pi\)
\(168\) 0 0
\(169\) −11.3028 −0.869444
\(170\) 0 0
\(171\) 15.8167 1.20953
\(172\) 0 0
\(173\) 8.78890 0.668207 0.334104 0.942536i \(-0.391566\pi\)
0.334104 + 0.942536i \(0.391566\pi\)
\(174\) 0 0
\(175\) 0.788897 0.0596350
\(176\) 0 0
\(177\) 35.0278 2.63285
\(178\) 0 0
\(179\) −13.8167 −1.03271 −0.516353 0.856376i \(-0.672711\pi\)
−0.516353 + 0.856376i \(0.672711\pi\)
\(180\) 0 0
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) −24.8167 −1.83450
\(184\) 0 0
\(185\) −2.30278 −0.169303
\(186\) 0 0
\(187\) 13.8167 1.01037
\(188\) 0 0
\(189\) 42.2389 3.07242
\(190\) 0 0
\(191\) 5.51388 0.398970 0.199485 0.979901i \(-0.436073\pi\)
0.199485 + 0.979901i \(0.436073\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) −9.90833 −0.709550
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −26.4222 −1.87302 −0.936510 0.350640i \(-0.885964\pi\)
−0.936510 + 0.350640i \(0.885964\pi\)
\(200\) 0 0
\(201\) −11.6056 −0.818592
\(202\) 0 0
\(203\) 10.1833 0.714731
\(204\) 0 0
\(205\) 22.8167 1.59358
\(206\) 0 0
\(207\) −30.9083 −2.14828
\(208\) 0 0
\(209\) −4.60555 −0.318573
\(210\) 0 0
\(211\) 10.3028 0.709272 0.354636 0.935004i \(-0.384605\pi\)
0.354636 + 0.935004i \(0.384605\pi\)
\(212\) 0 0
\(213\) −19.8167 −1.35781
\(214\) 0 0
\(215\) 1.39445 0.0951006
\(216\) 0 0
\(217\) 0.788897 0.0535538
\(218\) 0 0
\(219\) −40.6333 −2.74574
\(220\) 0 0
\(221\) 7.81665 0.525805
\(222\) 0 0
\(223\) 5.81665 0.389512 0.194756 0.980852i \(-0.437609\pi\)
0.194756 + 0.980852i \(0.437609\pi\)
\(224\) 0 0
\(225\) 2.39445 0.159630
\(226\) 0 0
\(227\) 13.8167 0.917044 0.458522 0.888683i \(-0.348379\pi\)
0.458522 + 0.888683i \(0.348379\pi\)
\(228\) 0 0
\(229\) −24.6056 −1.62598 −0.812990 0.582277i \(-0.802162\pi\)
−0.812990 + 0.582277i \(0.802162\pi\)
\(230\) 0 0
\(231\) −19.8167 −1.30384
\(232\) 0 0
\(233\) 8.51388 0.557763 0.278881 0.960326i \(-0.410036\pi\)
0.278881 + 0.960326i \(0.410036\pi\)
\(234\) 0 0
\(235\) −10.6056 −0.691830
\(236\) 0 0
\(237\) −30.1194 −1.95647
\(238\) 0 0
\(239\) 17.5139 1.13288 0.566439 0.824103i \(-0.308321\pi\)
0.566439 + 0.824103i \(0.308321\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) 49.8444 3.19752
\(244\) 0 0
\(245\) −0.486122 −0.0310572
\(246\) 0 0
\(247\) −2.60555 −0.165787
\(248\) 0 0
\(249\) 9.21110 0.583730
\(250\) 0 0
\(251\) −21.2111 −1.33883 −0.669416 0.742887i \(-0.733456\pi\)
−0.669416 + 0.742887i \(0.733456\pi\)
\(252\) 0 0
\(253\) 9.00000 0.565825
\(254\) 0 0
\(255\) −45.6333 −2.85767
\(256\) 0 0
\(257\) 3.21110 0.200303 0.100152 0.994972i \(-0.468067\pi\)
0.100152 + 0.994972i \(0.468067\pi\)
\(258\) 0 0
\(259\) −2.60555 −0.161901
\(260\) 0 0
\(261\) 30.9083 1.91318
\(262\) 0 0
\(263\) −13.8167 −0.851971 −0.425986 0.904730i \(-0.640073\pi\)
−0.425986 + 0.904730i \(0.640073\pi\)
\(264\) 0 0
\(265\) 13.8167 0.848750
\(266\) 0 0
\(267\) −30.4222 −1.86181
\(268\) 0 0
\(269\) 21.2111 1.29326 0.646632 0.762802i \(-0.276177\pi\)
0.646632 + 0.762802i \(0.276177\pi\)
\(270\) 0 0
\(271\) 22.4222 1.36205 0.681026 0.732259i \(-0.261534\pi\)
0.681026 + 0.732259i \(0.261534\pi\)
\(272\) 0 0
\(273\) −11.2111 −0.678527
\(274\) 0 0
\(275\) −0.697224 −0.0420442
\(276\) 0 0
\(277\) −0.119429 −0.00717582 −0.00358791 0.999994i \(-0.501142\pi\)
−0.00358791 + 0.999994i \(0.501142\pi\)
\(278\) 0 0
\(279\) 2.39445 0.143352
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) 24.6056 1.46265 0.731324 0.682030i \(-0.238903\pi\)
0.731324 + 0.682030i \(0.238903\pi\)
\(284\) 0 0
\(285\) 15.2111 0.901028
\(286\) 0 0
\(287\) 25.8167 1.52391
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −54.2389 −3.17954
\(292\) 0 0
\(293\) −11.0278 −0.644248 −0.322124 0.946697i \(-0.604397\pi\)
−0.322124 + 0.946697i \(0.604397\pi\)
\(294\) 0 0
\(295\) 24.4222 1.42192
\(296\) 0 0
\(297\) −37.3305 −2.16614
\(298\) 0 0
\(299\) 5.09167 0.294459
\(300\) 0 0
\(301\) 1.57779 0.0909426
\(302\) 0 0
\(303\) 41.0278 2.35698
\(304\) 0 0
\(305\) −17.3028 −0.990754
\(306\) 0 0
\(307\) 17.9083 1.02208 0.511041 0.859556i \(-0.329260\pi\)
0.511041 + 0.859556i \(0.329260\pi\)
\(308\) 0 0
\(309\) 1.00000 0.0568880
\(310\) 0 0
\(311\) −15.9083 −0.902078 −0.451039 0.892504i \(-0.648947\pi\)
−0.451039 + 0.892504i \(0.648947\pi\)
\(312\) 0 0
\(313\) −9.02776 −0.510279 −0.255139 0.966904i \(-0.582121\pi\)
−0.255139 + 0.966904i \(0.582121\pi\)
\(314\) 0 0
\(315\) 47.4500 2.67350
\(316\) 0 0
\(317\) −9.21110 −0.517347 −0.258674 0.965965i \(-0.583285\pi\)
−0.258674 + 0.965965i \(0.583285\pi\)
\(318\) 0 0
\(319\) −9.00000 −0.503903
\(320\) 0 0
\(321\) 2.30278 0.128528
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) −0.394449 −0.0218801
\(326\) 0 0
\(327\) −6.60555 −0.365288
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −13.2111 −0.726148 −0.363074 0.931760i \(-0.618273\pi\)
−0.363074 + 0.931760i \(0.618273\pi\)
\(332\) 0 0
\(333\) −7.90833 −0.433374
\(334\) 0 0
\(335\) −8.09167 −0.442095
\(336\) 0 0
\(337\) 6.11943 0.333347 0.166673 0.986012i \(-0.446697\pi\)
0.166673 + 0.986012i \(0.446697\pi\)
\(338\) 0 0
\(339\) −10.6056 −0.576014
\(340\) 0 0
\(341\) −0.697224 −0.0377568
\(342\) 0 0
\(343\) −18.7889 −1.01451
\(344\) 0 0
\(345\) −29.7250 −1.60034
\(346\) 0 0
\(347\) 10.1833 0.546671 0.273335 0.961919i \(-0.411873\pi\)
0.273335 + 0.961919i \(0.411873\pi\)
\(348\) 0 0
\(349\) −28.2389 −1.51159 −0.755796 0.654807i \(-0.772750\pi\)
−0.755796 + 0.654807i \(0.772750\pi\)
\(350\) 0 0
\(351\) −21.1194 −1.12727
\(352\) 0 0
\(353\) 10.1833 0.542005 0.271002 0.962579i \(-0.412645\pi\)
0.271002 + 0.962579i \(0.412645\pi\)
\(354\) 0 0
\(355\) −13.8167 −0.733312
\(356\) 0 0
\(357\) −51.6333 −2.73272
\(358\) 0 0
\(359\) −3.21110 −0.169476 −0.0847378 0.996403i \(-0.527005\pi\)
−0.0847378 + 0.996403i \(0.527005\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) −18.8167 −0.987618
\(364\) 0 0
\(365\) −28.3305 −1.48289
\(366\) 0 0
\(367\) −3.81665 −0.199228 −0.0996139 0.995026i \(-0.531761\pi\)
−0.0996139 + 0.995026i \(0.531761\pi\)
\(368\) 0 0
\(369\) 78.3583 4.07917
\(370\) 0 0
\(371\) 15.6333 0.811641
\(372\) 0 0
\(373\) 17.8167 0.922511 0.461256 0.887267i \(-0.347399\pi\)
0.461256 + 0.887267i \(0.347399\pi\)
\(374\) 0 0
\(375\) −35.7250 −1.84483
\(376\) 0 0
\(377\) −5.09167 −0.262235
\(378\) 0 0
\(379\) 24.3305 1.24978 0.624888 0.780715i \(-0.285145\pi\)
0.624888 + 0.780715i \(0.285145\pi\)
\(380\) 0 0
\(381\) 63.4500 3.25064
\(382\) 0 0
\(383\) 36.8444 1.88266 0.941331 0.337486i \(-0.109576\pi\)
0.941331 + 0.337486i \(0.109576\pi\)
\(384\) 0 0
\(385\) −13.8167 −0.704162
\(386\) 0 0
\(387\) 4.78890 0.243433
\(388\) 0 0
\(389\) 37.1194 1.88203 0.941015 0.338365i \(-0.109874\pi\)
0.941015 + 0.338365i \(0.109874\pi\)
\(390\) 0 0
\(391\) 23.4500 1.18592
\(392\) 0 0
\(393\) 35.0278 1.76692
\(394\) 0 0
\(395\) −21.0000 −1.05662
\(396\) 0 0
\(397\) −6.18335 −0.310333 −0.155167 0.987888i \(-0.549591\pi\)
−0.155167 + 0.987888i \(0.549591\pi\)
\(398\) 0 0
\(399\) 17.2111 0.861633
\(400\) 0 0
\(401\) −7.81665 −0.390345 −0.195173 0.980769i \(-0.562527\pi\)
−0.195173 + 0.980769i \(0.562527\pi\)
\(402\) 0 0
\(403\) −0.394449 −0.0196489
\(404\) 0 0
\(405\) 68.6611 3.41180
\(406\) 0 0
\(407\) 2.30278 0.114144
\(408\) 0 0
\(409\) 31.0278 1.53422 0.767112 0.641513i \(-0.221693\pi\)
0.767112 + 0.641513i \(0.221693\pi\)
\(410\) 0 0
\(411\) 3.00000 0.147979
\(412\) 0 0
\(413\) 27.6333 1.35975
\(414\) 0 0
\(415\) 6.42221 0.315254
\(416\) 0 0
\(417\) −6.30278 −0.308648
\(418\) 0 0
\(419\) 36.1472 1.76591 0.882953 0.469462i \(-0.155552\pi\)
0.882953 + 0.469462i \(0.155552\pi\)
\(420\) 0 0
\(421\) 3.72498 0.181544 0.0907722 0.995872i \(-0.471066\pi\)
0.0907722 + 0.995872i \(0.471066\pi\)
\(422\) 0 0
\(423\) −36.4222 −1.77091
\(424\) 0 0
\(425\) −1.81665 −0.0881207
\(426\) 0 0
\(427\) −19.5778 −0.947436
\(428\) 0 0
\(429\) 9.90833 0.478379
\(430\) 0 0
\(431\) −9.21110 −0.443683 −0.221842 0.975083i \(-0.571207\pi\)
−0.221842 + 0.975083i \(0.571207\pi\)
\(432\) 0 0
\(433\) 34.9361 1.67892 0.839461 0.543421i \(-0.182871\pi\)
0.839461 + 0.543421i \(0.182871\pi\)
\(434\) 0 0
\(435\) 29.7250 1.42520
\(436\) 0 0
\(437\) −7.81665 −0.373921
\(438\) 0 0
\(439\) −30.3305 −1.44760 −0.723799 0.690011i \(-0.757606\pi\)
−0.723799 + 0.690011i \(0.757606\pi\)
\(440\) 0 0
\(441\) −1.66947 −0.0794985
\(442\) 0 0
\(443\) 32.7250 1.55481 0.777405 0.629000i \(-0.216535\pi\)
0.777405 + 0.629000i \(0.216535\pi\)
\(444\) 0 0
\(445\) −21.2111 −1.00550
\(446\) 0 0
\(447\) −65.4500 −3.09568
\(448\) 0 0
\(449\) −15.2111 −0.717856 −0.358928 0.933365i \(-0.616858\pi\)
−0.358928 + 0.933365i \(0.616858\pi\)
\(450\) 0 0
\(451\) −22.8167 −1.07439
\(452\) 0 0
\(453\) 68.0555 3.19753
\(454\) 0 0
\(455\) −7.81665 −0.366450
\(456\) 0 0
\(457\) −2.60555 −0.121883 −0.0609413 0.998141i \(-0.519410\pi\)
−0.0609413 + 0.998141i \(0.519410\pi\)
\(458\) 0 0
\(459\) −97.2666 −4.54002
\(460\) 0 0
\(461\) −12.4222 −0.578560 −0.289280 0.957245i \(-0.593416\pi\)
−0.289280 + 0.957245i \(0.593416\pi\)
\(462\) 0 0
\(463\) −26.6972 −1.24073 −0.620363 0.784315i \(-0.713015\pi\)
−0.620363 + 0.784315i \(0.713015\pi\)
\(464\) 0 0
\(465\) 2.30278 0.106789
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −9.15559 −0.422766
\(470\) 0 0
\(471\) 23.8167 1.09741
\(472\) 0 0
\(473\) −1.39445 −0.0641168
\(474\) 0 0
\(475\) 0.605551 0.0277846
\(476\) 0 0
\(477\) 47.4500 2.17258
\(478\) 0 0
\(479\) 13.1194 0.599442 0.299721 0.954027i \(-0.403106\pi\)
0.299721 + 0.954027i \(0.403106\pi\)
\(480\) 0 0
\(481\) 1.30278 0.0594015
\(482\) 0 0
\(483\) −33.6333 −1.53037
\(484\) 0 0
\(485\) −37.8167 −1.71717
\(486\) 0 0
\(487\) 37.2111 1.68620 0.843098 0.537760i \(-0.180729\pi\)
0.843098 + 0.537760i \(0.180729\pi\)
\(488\) 0 0
\(489\) 27.8167 1.25791
\(490\) 0 0
\(491\) 17.7250 0.799917 0.399959 0.916533i \(-0.369025\pi\)
0.399959 + 0.916533i \(0.369025\pi\)
\(492\) 0 0
\(493\) −23.4500 −1.05613
\(494\) 0 0
\(495\) −41.9361 −1.88489
\(496\) 0 0
\(497\) −15.6333 −0.701250
\(498\) 0 0
\(499\) −42.2389 −1.89087 −0.945436 0.325809i \(-0.894363\pi\)
−0.945436 + 0.325809i \(0.894363\pi\)
\(500\) 0 0
\(501\) 18.2111 0.813612
\(502\) 0 0
\(503\) 6.48612 0.289202 0.144601 0.989490i \(-0.453810\pi\)
0.144601 + 0.989490i \(0.453810\pi\)
\(504\) 0 0
\(505\) 28.6056 1.27293
\(506\) 0 0
\(507\) −37.3305 −1.65791
\(508\) 0 0
\(509\) 4.18335 0.185424 0.0927118 0.995693i \(-0.470446\pi\)
0.0927118 + 0.995693i \(0.470446\pi\)
\(510\) 0 0
\(511\) −32.0555 −1.41805
\(512\) 0 0
\(513\) 32.4222 1.43148
\(514\) 0 0
\(515\) 0.697224 0.0307234
\(516\) 0 0
\(517\) 10.6056 0.466432
\(518\) 0 0
\(519\) 29.0278 1.27418
\(520\) 0 0
\(521\) 33.6333 1.47350 0.736751 0.676164i \(-0.236359\pi\)
0.736751 + 0.676164i \(0.236359\pi\)
\(522\) 0 0
\(523\) −18.2389 −0.797530 −0.398765 0.917053i \(-0.630561\pi\)
−0.398765 + 0.917053i \(0.630561\pi\)
\(524\) 0 0
\(525\) 2.60555 0.113716
\(526\) 0 0
\(527\) −1.81665 −0.0791347
\(528\) 0 0
\(529\) −7.72498 −0.335869
\(530\) 0 0
\(531\) 83.8722 3.63974
\(532\) 0 0
\(533\) −12.9083 −0.559122
\(534\) 0 0
\(535\) 1.60555 0.0694140
\(536\) 0 0
\(537\) −45.6333 −1.96922
\(538\) 0 0
\(539\) 0.486122 0.0209387
\(540\) 0 0
\(541\) −25.9361 −1.11508 −0.557540 0.830150i \(-0.688255\pi\)
−0.557540 + 0.830150i \(0.688255\pi\)
\(542\) 0 0
\(543\) −66.0555 −2.83471
\(544\) 0 0
\(545\) −4.60555 −0.197280
\(546\) 0 0
\(547\) −20.6056 −0.881030 −0.440515 0.897745i \(-0.645204\pi\)
−0.440515 + 0.897745i \(0.645204\pi\)
\(548\) 0 0
\(549\) −59.4222 −2.53608
\(550\) 0 0
\(551\) 7.81665 0.333001
\(552\) 0 0
\(553\) −23.7611 −1.01043
\(554\) 0 0
\(555\) −7.60555 −0.322838
\(556\) 0 0
\(557\) −11.5139 −0.487859 −0.243929 0.969793i \(-0.578436\pi\)
−0.243929 + 0.969793i \(0.578436\pi\)
\(558\) 0 0
\(559\) −0.788897 −0.0333668
\(560\) 0 0
\(561\) 45.6333 1.92664
\(562\) 0 0
\(563\) −28.0555 −1.18240 −0.591199 0.806525i \(-0.701345\pi\)
−0.591199 + 0.806525i \(0.701345\pi\)
\(564\) 0 0
\(565\) −7.39445 −0.311087
\(566\) 0 0
\(567\) 77.6888 3.26262
\(568\) 0 0
\(569\) 18.4222 0.772299 0.386150 0.922436i \(-0.373805\pi\)
0.386150 + 0.922436i \(0.373805\pi\)
\(570\) 0 0
\(571\) −16.6972 −0.698757 −0.349379 0.936982i \(-0.613607\pi\)
−0.349379 + 0.936982i \(0.613607\pi\)
\(572\) 0 0
\(573\) 18.2111 0.760780
\(574\) 0 0
\(575\) −1.18335 −0.0493489
\(576\) 0 0
\(577\) 22.2389 0.925816 0.462908 0.886406i \(-0.346806\pi\)
0.462908 + 0.886406i \(0.346806\pi\)
\(578\) 0 0
\(579\) −13.2111 −0.549035
\(580\) 0 0
\(581\) 7.26662 0.301470
\(582\) 0 0
\(583\) −13.8167 −0.572227
\(584\) 0 0
\(585\) −23.7250 −0.980907
\(586\) 0 0
\(587\) −45.6333 −1.88349 −0.941744 0.336330i \(-0.890814\pi\)
−0.941744 + 0.336330i \(0.890814\pi\)
\(588\) 0 0
\(589\) 0.605551 0.0249513
\(590\) 0 0
\(591\) 19.8167 0.815148
\(592\) 0 0
\(593\) 18.4861 0.759134 0.379567 0.925164i \(-0.376073\pi\)
0.379567 + 0.925164i \(0.376073\pi\)
\(594\) 0 0
\(595\) −36.0000 −1.47586
\(596\) 0 0
\(597\) −87.2666 −3.57158
\(598\) 0 0
\(599\) 20.7889 0.849411 0.424706 0.905331i \(-0.360378\pi\)
0.424706 + 0.905331i \(0.360378\pi\)
\(600\) 0 0
\(601\) −24.3028 −0.991331 −0.495665 0.868514i \(-0.665076\pi\)
−0.495665 + 0.868514i \(0.665076\pi\)
\(602\) 0 0
\(603\) −27.7889 −1.13165
\(604\) 0 0
\(605\) −13.1194 −0.533381
\(606\) 0 0
\(607\) 13.4861 0.547385 0.273692 0.961817i \(-0.411755\pi\)
0.273692 + 0.961817i \(0.411755\pi\)
\(608\) 0 0
\(609\) 33.6333 1.36289
\(610\) 0 0
\(611\) 6.00000 0.242734
\(612\) 0 0
\(613\) 29.8167 1.20428 0.602142 0.798389i \(-0.294314\pi\)
0.602142 + 0.798389i \(0.294314\pi\)
\(614\) 0 0
\(615\) 75.3583 3.03874
\(616\) 0 0
\(617\) −42.5694 −1.71378 −0.856890 0.515500i \(-0.827606\pi\)
−0.856890 + 0.515500i \(0.827606\pi\)
\(618\) 0 0
\(619\) −6.30278 −0.253330 −0.126665 0.991946i \(-0.540427\pi\)
−0.126665 + 0.991946i \(0.540427\pi\)
\(620\) 0 0
\(621\) −63.3583 −2.54248
\(622\) 0 0
\(623\) −24.0000 −0.961540
\(624\) 0 0
\(625\) −26.4222 −1.05689
\(626\) 0 0
\(627\) −15.2111 −0.607473
\(628\) 0 0
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) −14.6972 −0.585087 −0.292544 0.956252i \(-0.594502\pi\)
−0.292544 + 0.956252i \(0.594502\pi\)
\(632\) 0 0
\(633\) 34.0278 1.35248
\(634\) 0 0
\(635\) 44.2389 1.75557
\(636\) 0 0
\(637\) 0.275019 0.0108967
\(638\) 0 0
\(639\) −47.4500 −1.87709
\(640\) 0 0
\(641\) −20.5139 −0.810249 −0.405125 0.914261i \(-0.632772\pi\)
−0.405125 + 0.914261i \(0.632772\pi\)
\(642\) 0 0
\(643\) −8.18335 −0.322720 −0.161360 0.986896i \(-0.551588\pi\)
−0.161360 + 0.986896i \(0.551588\pi\)
\(644\) 0 0
\(645\) 4.60555 0.181343
\(646\) 0 0
\(647\) −20.9361 −0.823082 −0.411541 0.911391i \(-0.635009\pi\)
−0.411541 + 0.911391i \(0.635009\pi\)
\(648\) 0 0
\(649\) −24.4222 −0.958655
\(650\) 0 0
\(651\) 2.60555 0.102120
\(652\) 0 0
\(653\) 3.90833 0.152945 0.0764723 0.997072i \(-0.475634\pi\)
0.0764723 + 0.997072i \(0.475634\pi\)
\(654\) 0 0
\(655\) 24.4222 0.954255
\(656\) 0 0
\(657\) −97.2944 −3.79581
\(658\) 0 0
\(659\) −16.8806 −0.657574 −0.328787 0.944404i \(-0.606640\pi\)
−0.328787 + 0.944404i \(0.606640\pi\)
\(660\) 0 0
\(661\) 30.5139 1.18685 0.593426 0.804888i \(-0.297775\pi\)
0.593426 + 0.804888i \(0.297775\pi\)
\(662\) 0 0
\(663\) 25.8167 1.00264
\(664\) 0 0
\(665\) 12.0000 0.465340
\(666\) 0 0
\(667\) −15.2750 −0.591451
\(668\) 0 0
\(669\) 19.2111 0.742744
\(670\) 0 0
\(671\) 17.3028 0.667966
\(672\) 0 0
\(673\) 20.6972 0.797819 0.398910 0.916990i \(-0.369389\pi\)
0.398910 + 0.916990i \(0.369389\pi\)
\(674\) 0 0
\(675\) 4.90833 0.188922
\(676\) 0 0
\(677\) −14.2389 −0.547244 −0.273622 0.961837i \(-0.588222\pi\)
−0.273622 + 0.961837i \(0.588222\pi\)
\(678\) 0 0
\(679\) −42.7889 −1.64209
\(680\) 0 0
\(681\) 45.6333 1.74867
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) 2.09167 0.0799187
\(686\) 0 0
\(687\) −81.2666 −3.10051
\(688\) 0 0
\(689\) −7.81665 −0.297791
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 0 0
\(693\) −47.4500 −1.80247
\(694\) 0 0
\(695\) −4.39445 −0.166691
\(696\) 0 0
\(697\) −59.4500 −2.25183
\(698\) 0 0
\(699\) 28.1194 1.06357
\(700\) 0 0
\(701\) −40.1194 −1.51529 −0.757645 0.652667i \(-0.773650\pi\)
−0.757645 + 0.652667i \(0.773650\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) 0 0
\(705\) −35.0278 −1.31922
\(706\) 0 0
\(707\) 32.3667 1.21727
\(708\) 0 0
\(709\) 41.3305 1.55220 0.776100 0.630609i \(-0.217195\pi\)
0.776100 + 0.630609i \(0.217195\pi\)
\(710\) 0 0
\(711\) −72.1194 −2.70469
\(712\) 0 0
\(713\) −1.18335 −0.0443167
\(714\) 0 0
\(715\) 6.90833 0.258357
\(716\) 0 0
\(717\) 57.8444 2.16024
\(718\) 0 0
\(719\) 51.6333 1.92560 0.962799 0.270220i \(-0.0870963\pi\)
0.962799 + 0.270220i \(0.0870963\pi\)
\(720\) 0 0
\(721\) 0.788897 0.0293801
\(722\) 0 0
\(723\) 26.4222 0.982652
\(724\) 0 0
\(725\) 1.18335 0.0439484
\(726\) 0 0
\(727\) −19.0917 −0.708071 −0.354035 0.935232i \(-0.615191\pi\)
−0.354035 + 0.935232i \(0.615191\pi\)
\(728\) 0 0
\(729\) 75.1749 2.78426
\(730\) 0 0
\(731\) −3.63331 −0.134383
\(732\) 0 0
\(733\) 13.6333 0.503558 0.251779 0.967785i \(-0.418984\pi\)
0.251779 + 0.967785i \(0.418984\pi\)
\(734\) 0 0
\(735\) −1.60555 −0.0592217
\(736\) 0 0
\(737\) 8.09167 0.298061
\(738\) 0 0
\(739\) −2.66947 −0.0981980 −0.0490990 0.998794i \(-0.515635\pi\)
−0.0490990 + 0.998794i \(0.515635\pi\)
\(740\) 0 0
\(741\) −8.60555 −0.316133
\(742\) 0 0
\(743\) 29.4500 1.08041 0.540207 0.841532i \(-0.318346\pi\)
0.540207 + 0.841532i \(0.318346\pi\)
\(744\) 0 0
\(745\) −45.6333 −1.67188
\(746\) 0 0
\(747\) 22.0555 0.806969
\(748\) 0 0
\(749\) 1.81665 0.0663791
\(750\) 0 0
\(751\) −14.0000 −0.510867 −0.255434 0.966827i \(-0.582218\pi\)
−0.255434 + 0.966827i \(0.582218\pi\)
\(752\) 0 0
\(753\) −70.0555 −2.55296
\(754\) 0 0
\(755\) 47.4500 1.72688
\(756\) 0 0
\(757\) −5.69722 −0.207069 −0.103535 0.994626i \(-0.533015\pi\)
−0.103535 + 0.994626i \(0.533015\pi\)
\(758\) 0 0
\(759\) 29.7250 1.07895
\(760\) 0 0
\(761\) 16.8806 0.611920 0.305960 0.952044i \(-0.401023\pi\)
0.305960 + 0.952044i \(0.401023\pi\)
\(762\) 0 0
\(763\) −5.21110 −0.188655
\(764\) 0 0
\(765\) −109.267 −3.95054
\(766\) 0 0
\(767\) −13.8167 −0.498890
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) 10.6056 0.381950
\(772\) 0 0
\(773\) −22.0555 −0.793282 −0.396641 0.917974i \(-0.629824\pi\)
−0.396641 + 0.917974i \(0.629824\pi\)
\(774\) 0 0
\(775\) 0.0916731 0.00329299
\(776\) 0 0
\(777\) −8.60555 −0.308722
\(778\) 0 0
\(779\) 19.8167 0.710005
\(780\) 0 0
\(781\) 13.8167 0.494399
\(782\) 0 0
\(783\) 63.3583 2.26424
\(784\) 0 0
\(785\) 16.6056 0.592678
\(786\) 0 0
\(787\) 10.7889 0.384583 0.192291 0.981338i \(-0.438408\pi\)
0.192291 + 0.981338i \(0.438408\pi\)
\(788\) 0 0
\(789\) −45.6333 −1.62459
\(790\) 0 0
\(791\) −8.36669 −0.297485
\(792\) 0 0
\(793\) 9.78890 0.347614
\(794\) 0 0
\(795\) 45.6333 1.61845
\(796\) 0 0
\(797\) −22.3305 −0.790988 −0.395494 0.918469i \(-0.629427\pi\)
−0.395494 + 0.918469i \(0.629427\pi\)
\(798\) 0 0
\(799\) 27.6333 0.977596
\(800\) 0 0
\(801\) −72.8444 −2.57383
\(802\) 0 0
\(803\) 28.3305 0.999763
\(804\) 0 0
\(805\) −23.4500 −0.826503
\(806\) 0 0
\(807\) 70.0555 2.46607
\(808\) 0 0
\(809\) −35.4500 −1.24635 −0.623177 0.782081i \(-0.714158\pi\)
−0.623177 + 0.782081i \(0.714158\pi\)
\(810\) 0 0
\(811\) −7.14719 −0.250972 −0.125486 0.992095i \(-0.540049\pi\)
−0.125486 + 0.992095i \(0.540049\pi\)
\(812\) 0 0
\(813\) 74.0555 2.59724
\(814\) 0 0
\(815\) 19.3944 0.679358
\(816\) 0 0
\(817\) 1.21110 0.0423711
\(818\) 0 0
\(819\) −26.8444 −0.938020
\(820\) 0 0
\(821\) 3.21110 0.112068 0.0560341 0.998429i \(-0.482154\pi\)
0.0560341 + 0.998429i \(0.482154\pi\)
\(822\) 0 0
\(823\) −44.8444 −1.56318 −0.781589 0.623794i \(-0.785590\pi\)
−0.781589 + 0.623794i \(0.785590\pi\)
\(824\) 0 0
\(825\) −2.30278 −0.0801724
\(826\) 0 0
\(827\) 34.6056 1.20335 0.601676 0.798740i \(-0.294500\pi\)
0.601676 + 0.798740i \(0.294500\pi\)
\(828\) 0 0
\(829\) 27.7250 0.962928 0.481464 0.876466i \(-0.340105\pi\)
0.481464 + 0.876466i \(0.340105\pi\)
\(830\) 0 0
\(831\) −0.394449 −0.0136833
\(832\) 0 0
\(833\) 1.26662 0.0438856
\(834\) 0 0
\(835\) 12.6972 0.439406
\(836\) 0 0
\(837\) 4.90833 0.169657
\(838\) 0 0
\(839\) 12.9722 0.447852 0.223926 0.974606i \(-0.428113\pi\)
0.223926 + 0.974606i \(0.428113\pi\)
\(840\) 0 0
\(841\) −13.7250 −0.473275
\(842\) 0 0
\(843\) −39.6333 −1.36504
\(844\) 0 0
\(845\) −26.0278 −0.895382
\(846\) 0 0
\(847\) −14.8444 −0.510060
\(848\) 0 0
\(849\) 81.2666 2.78906
\(850\) 0 0
\(851\) 3.90833 0.133976
\(852\) 0 0
\(853\) −42.5416 −1.45660 −0.728299 0.685260i \(-0.759689\pi\)
−0.728299 + 0.685260i \(0.759689\pi\)
\(854\) 0 0
\(855\) 36.4222 1.24561
\(856\) 0 0
\(857\) 42.8444 1.46354 0.731769 0.681553i \(-0.238695\pi\)
0.731769 + 0.681553i \(0.238695\pi\)
\(858\) 0 0
\(859\) 48.0555 1.63963 0.819816 0.572626i \(-0.194075\pi\)
0.819816 + 0.572626i \(0.194075\pi\)
\(860\) 0 0
\(861\) 85.2666 2.90588
\(862\) 0 0
\(863\) −12.0000 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) 0 0
\(865\) 20.2389 0.688142
\(866\) 0 0
\(867\) 62.7527 2.13119
\(868\) 0 0
\(869\) 21.0000 0.712376
\(870\) 0 0
\(871\) 4.57779 0.155113
\(872\) 0 0
\(873\) −129.872 −4.39551
\(874\) 0 0
\(875\) −28.1833 −0.952771
\(876\) 0 0
\(877\) 7.21110 0.243502 0.121751 0.992561i \(-0.461149\pi\)
0.121751 + 0.992561i \(0.461149\pi\)
\(878\) 0 0
\(879\) −36.4222 −1.22849
\(880\) 0 0
\(881\) −28.5416 −0.961592 −0.480796 0.876832i \(-0.659652\pi\)
−0.480796 + 0.876832i \(0.659652\pi\)
\(882\) 0 0
\(883\) 26.4222 0.889178 0.444589 0.895735i \(-0.353350\pi\)
0.444589 + 0.895735i \(0.353350\pi\)
\(884\) 0 0
\(885\) 80.6611 2.71139
\(886\) 0 0
\(887\) 0.422205 0.0141763 0.00708813 0.999975i \(-0.497744\pi\)
0.00708813 + 0.999975i \(0.497744\pi\)
\(888\) 0 0
\(889\) 50.0555 1.67881
\(890\) 0 0
\(891\) −68.6611 −2.30023
\(892\) 0 0
\(893\) −9.21110 −0.308238
\(894\) 0 0
\(895\) −31.8167 −1.06351
\(896\) 0 0
\(897\) 16.8167 0.561492
\(898\) 0 0
\(899\) 1.18335 0.0394668
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 5.21110 0.173415
\(904\) 0 0
\(905\) −46.0555 −1.53094
\(906\) 0 0
\(907\) 26.0000 0.863316 0.431658 0.902037i \(-0.357929\pi\)
0.431658 + 0.902037i \(0.357929\pi\)
\(908\) 0 0
\(909\) 98.2389 3.25838
\(910\) 0 0
\(911\) 17.5778 0.582378 0.291189 0.956665i \(-0.405949\pi\)
0.291189 + 0.956665i \(0.405949\pi\)
\(912\) 0 0
\(913\) −6.42221 −0.212544
\(914\) 0 0
\(915\) −57.1472 −1.88923
\(916\) 0 0
\(917\) 27.6333 0.912532
\(918\) 0 0
\(919\) 9.57779 0.315942 0.157971 0.987444i \(-0.449505\pi\)
0.157971 + 0.987444i \(0.449505\pi\)
\(920\) 0 0
\(921\) 59.1472 1.94897
\(922\) 0 0
\(923\) 7.81665 0.257288
\(924\) 0 0
\(925\) −0.302776 −0.00995520
\(926\) 0 0
\(927\) 2.39445 0.0786440
\(928\) 0 0
\(929\) −18.4861 −0.606510 −0.303255 0.952909i \(-0.598073\pi\)
−0.303255 + 0.952909i \(0.598073\pi\)
\(930\) 0 0
\(931\) −0.422205 −0.0138372
\(932\) 0 0
\(933\) −52.5416 −1.72014
\(934\) 0 0
\(935\) 31.8167 1.04052
\(936\) 0 0
\(937\) −18.0917 −0.591029 −0.295515 0.955338i \(-0.595491\pi\)
−0.295515 + 0.955338i \(0.595491\pi\)
\(938\) 0 0
\(939\) −29.8167 −0.973030
\(940\) 0 0
\(941\) −13.8167 −0.450410 −0.225205 0.974311i \(-0.572305\pi\)
−0.225205 + 0.974311i \(0.572305\pi\)
\(942\) 0 0
\(943\) −38.7250 −1.26106
\(944\) 0 0
\(945\) 97.2666 3.16408
\(946\) 0 0
\(947\) −3.63331 −0.118067 −0.0590333 0.998256i \(-0.518802\pi\)
−0.0590333 + 0.998256i \(0.518802\pi\)
\(948\) 0 0
\(949\) 16.0278 0.520283
\(950\) 0 0
\(951\) −30.4222 −0.986508
\(952\) 0 0
\(953\) 49.7527 1.61165 0.805825 0.592154i \(-0.201722\pi\)
0.805825 + 0.592154i \(0.201722\pi\)
\(954\) 0 0
\(955\) 12.6972 0.410873
\(956\) 0 0
\(957\) −29.7250 −0.960872
\(958\) 0 0
\(959\) 2.36669 0.0764245
\(960\) 0 0
\(961\) −30.9083 −0.997043
\(962\) 0 0
\(963\) 5.51388 0.177682
\(964\) 0 0
\(965\) −9.21110 −0.296516
\(966\) 0 0
\(967\) 6.72498 0.216261 0.108130 0.994137i \(-0.465514\pi\)
0.108130 + 0.994137i \(0.465514\pi\)
\(968\) 0 0
\(969\) −39.6333 −1.27321
\(970\) 0 0
\(971\) −22.5416 −0.723395 −0.361698 0.932295i \(-0.617803\pi\)
−0.361698 + 0.932295i \(0.617803\pi\)
\(972\) 0 0
\(973\) −4.97224 −0.159403
\(974\) 0 0
\(975\) −1.30278 −0.0417222
\(976\) 0 0
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) 21.2111 0.677910
\(980\) 0 0
\(981\) −15.8167 −0.504987
\(982\) 0 0
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 0 0
\(985\) 13.8167 0.440235
\(986\) 0 0
\(987\) −39.6333 −1.26154
\(988\) 0 0
\(989\) −2.36669 −0.0752564
\(990\) 0 0
\(991\) −50.6972 −1.61045 −0.805225 0.592969i \(-0.797956\pi\)
−0.805225 + 0.592969i \(0.797956\pi\)
\(992\) 0 0
\(993\) −43.6333 −1.38466
\(994\) 0 0
\(995\) −60.8444 −1.92890
\(996\) 0 0
\(997\) 52.4222 1.66023 0.830114 0.557594i \(-0.188275\pi\)
0.830114 + 0.557594i \(0.188275\pi\)
\(998\) 0 0
\(999\) −16.2111 −0.512897
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 2368.2.a.ba.1.2 2
4.3 odd 2 2368.2.a.s.1.1 2
8.3 odd 2 74.2.a.a.1.2 2
8.5 even 2 592.2.a.f.1.1 2
24.5 odd 2 5328.2.a.bf.1.2 2
24.11 even 2 666.2.a.j.1.2 2
40.3 even 4 1850.2.b.i.149.4 4
40.19 odd 2 1850.2.a.u.1.1 2
40.27 even 4 1850.2.b.i.149.1 4
56.27 even 2 3626.2.a.a.1.1 2
88.43 even 2 8954.2.a.p.1.2 2
296.147 odd 2 2738.2.a.l.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.2 2 8.3 odd 2
592.2.a.f.1.1 2 8.5 even 2
666.2.a.j.1.2 2 24.11 even 2
1850.2.a.u.1.1 2 40.19 odd 2
1850.2.b.i.149.1 4 40.27 even 4
1850.2.b.i.149.4 4 40.3 even 4
2368.2.a.s.1.1 2 4.3 odd 2
2368.2.a.ba.1.2 2 1.1 even 1 trivial
2738.2.a.l.1.2 2 296.147 odd 2
3626.2.a.a.1.1 2 56.27 even 2
5328.2.a.bf.1.2 2 24.5 odd 2
8954.2.a.p.1.2 2 88.43 even 2