Properties

Label 3120.2.a.bc.1.2
Level $3120$
Weight $2$
Character 3120.1
Self dual yes
Analytic conductor $24.913$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3120.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.9133254306\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3120.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} +2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} +2.82843 q^{7} +1.00000 q^{9} -5.65685 q^{11} -1.00000 q^{13} -1.00000 q^{15} +0.828427 q^{17} -2.82843 q^{19} -2.82843 q^{21} +8.48528 q^{23} +1.00000 q^{25} -1.00000 q^{27} -8.82843 q^{29} -4.00000 q^{31} +5.65685 q^{33} +2.82843 q^{35} -11.6569 q^{37} +1.00000 q^{39} -7.65685 q^{41} -9.65685 q^{43} +1.00000 q^{45} +8.00000 q^{47} +1.00000 q^{49} -0.828427 q^{51} +13.3137 q^{53} -5.65685 q^{55} +2.82843 q^{57} +2.34315 q^{59} +6.00000 q^{61} +2.82843 q^{63} -1.00000 q^{65} -5.65685 q^{67} -8.48528 q^{69} +5.65685 q^{71} -14.4853 q^{73} -1.00000 q^{75} -16.0000 q^{77} -2.34315 q^{79} +1.00000 q^{81} -6.34315 q^{83} +0.828427 q^{85} +8.82843 q^{87} -15.6569 q^{89} -2.82843 q^{91} +4.00000 q^{93} -2.82843 q^{95} -3.17157 q^{97} -5.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{5} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{5} + 2q^{9} - 2q^{13} - 2q^{15} - 4q^{17} + 2q^{25} - 2q^{27} - 12q^{29} - 8q^{31} - 12q^{37} + 2q^{39} - 4q^{41} - 8q^{43} + 2q^{45} + 16q^{47} + 2q^{49} + 4q^{51} + 4q^{53} + 16q^{59} + 12q^{61} - 2q^{65} - 12q^{73} - 2q^{75} - 32q^{77} - 16q^{79} + 2q^{81} - 24q^{83} - 4q^{85} + 12q^{87} - 20q^{89} + 8q^{93} - 12q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.82843 1.06904 0.534522 0.845154i \(-0.320491\pi\)
0.534522 + 0.845154i \(0.320491\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.65685 −1.70561 −0.852803 0.522233i \(-0.825099\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 0.828427 0.200923 0.100462 0.994941i \(-0.467968\pi\)
0.100462 + 0.994941i \(0.467968\pi\)
\(18\) 0 0
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 0 0
\(21\) −2.82843 −0.617213
\(22\) 0 0
\(23\) 8.48528 1.76930 0.884652 0.466252i \(-0.154396\pi\)
0.884652 + 0.466252i \(0.154396\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −8.82843 −1.63940 −0.819699 0.572795i \(-0.805859\pi\)
−0.819699 + 0.572795i \(0.805859\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 5.65685 0.984732
\(34\) 0 0
\(35\) 2.82843 0.478091
\(36\) 0 0
\(37\) −11.6569 −1.91638 −0.958188 0.286141i \(-0.907627\pi\)
−0.958188 + 0.286141i \(0.907627\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −7.65685 −1.19580 −0.597900 0.801571i \(-0.703998\pi\)
−0.597900 + 0.801571i \(0.703998\pi\)
\(42\) 0 0
\(43\) −9.65685 −1.47266 −0.736328 0.676625i \(-0.763442\pi\)
−0.736328 + 0.676625i \(0.763442\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −0.828427 −0.116003
\(52\) 0 0
\(53\) 13.3137 1.82878 0.914389 0.404836i \(-0.132671\pi\)
0.914389 + 0.404836i \(0.132671\pi\)
\(54\) 0 0
\(55\) −5.65685 −0.762770
\(56\) 0 0
\(57\) 2.82843 0.374634
\(58\) 0 0
\(59\) 2.34315 0.305052 0.152526 0.988299i \(-0.451259\pi\)
0.152526 + 0.988299i \(0.451259\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 2.82843 0.356348
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −5.65685 −0.691095 −0.345547 0.938401i \(-0.612307\pi\)
−0.345547 + 0.938401i \(0.612307\pi\)
\(68\) 0 0
\(69\) −8.48528 −1.02151
\(70\) 0 0
\(71\) 5.65685 0.671345 0.335673 0.941979i \(-0.391036\pi\)
0.335673 + 0.941979i \(0.391036\pi\)
\(72\) 0 0
\(73\) −14.4853 −1.69537 −0.847687 0.530497i \(-0.822005\pi\)
−0.847687 + 0.530497i \(0.822005\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −16.0000 −1.82337
\(78\) 0 0
\(79\) −2.34315 −0.263624 −0.131812 0.991275i \(-0.542080\pi\)
−0.131812 + 0.991275i \(0.542080\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.34315 −0.696251 −0.348125 0.937448i \(-0.613182\pi\)
−0.348125 + 0.937448i \(0.613182\pi\)
\(84\) 0 0
\(85\) 0.828427 0.0898555
\(86\) 0 0
\(87\) 8.82843 0.946507
\(88\) 0 0
\(89\) −15.6569 −1.65962 −0.829812 0.558044i \(-0.811552\pi\)
−0.829812 + 0.558044i \(0.811552\pi\)
\(90\) 0 0
\(91\) −2.82843 −0.296500
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) −2.82843 −0.290191
\(96\) 0 0
\(97\) −3.17157 −0.322024 −0.161012 0.986952i \(-0.551476\pi\)
−0.161012 + 0.986952i \(0.551476\pi\)
\(98\) 0 0
\(99\) −5.65685 −0.568535
\(100\) 0 0
\(101\) 16.1421 1.60620 0.803101 0.595843i \(-0.203182\pi\)
0.803101 + 0.595843i \(0.203182\pi\)
\(102\) 0 0
\(103\) 1.65685 0.163255 0.0816274 0.996663i \(-0.473988\pi\)
0.0816274 + 0.996663i \(0.473988\pi\)
\(104\) 0 0
\(105\) −2.82843 −0.276026
\(106\) 0 0
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) 8.82843 0.845610 0.422805 0.906221i \(-0.361046\pi\)
0.422805 + 0.906221i \(0.361046\pi\)
\(110\) 0 0
\(111\) 11.6569 1.10642
\(112\) 0 0
\(113\) 6.48528 0.610084 0.305042 0.952339i \(-0.401330\pi\)
0.305042 + 0.952339i \(0.401330\pi\)
\(114\) 0 0
\(115\) 8.48528 0.791257
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 2.34315 0.214796
\(120\) 0 0
\(121\) 21.0000 1.90909
\(122\) 0 0
\(123\) 7.65685 0.690395
\(124\) 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
\(129\) 9.65685 0.850239
\(130\) 0 0
\(131\) 6.14214 0.536641 0.268320 0.963330i \(-0.413531\pi\)
0.268320 + 0.963330i \(0.413531\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −17.3137 −1.47921 −0.739605 0.673041i \(-0.764988\pi\)
−0.739605 + 0.673041i \(0.764988\pi\)
\(138\) 0 0
\(139\) −6.34315 −0.538019 −0.269009 0.963138i \(-0.586696\pi\)
−0.269009 + 0.963138i \(0.586696\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 5.65685 0.473050
\(144\) 0 0
\(145\) −8.82843 −0.733161
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 3.65685 0.299581 0.149791 0.988718i \(-0.452140\pi\)
0.149791 + 0.988718i \(0.452140\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 0.828427 0.0669744
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) 5.31371 0.424080 0.212040 0.977261i \(-0.431989\pi\)
0.212040 + 0.977261i \(0.431989\pi\)
\(158\) 0 0
\(159\) −13.3137 −1.05585
\(160\) 0 0
\(161\) 24.0000 1.89146
\(162\) 0 0
\(163\) −11.3137 −0.886158 −0.443079 0.896483i \(-0.646114\pi\)
−0.443079 + 0.896483i \(0.646114\pi\)
\(164\) 0 0
\(165\) 5.65685 0.440386
\(166\) 0 0
\(167\) −8.97056 −0.694163 −0.347081 0.937835i \(-0.612827\pi\)
−0.347081 + 0.937835i \(0.612827\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −2.82843 −0.216295
\(172\) 0 0
\(173\) −9.31371 −0.708108 −0.354054 0.935225i \(-0.615197\pi\)
−0.354054 + 0.935225i \(0.615197\pi\)
\(174\) 0 0
\(175\) 2.82843 0.213809
\(176\) 0 0
\(177\) −2.34315 −0.176122
\(178\) 0 0
\(179\) −7.51472 −0.561676 −0.280838 0.959755i \(-0.590612\pi\)
−0.280838 + 0.959755i \(0.590612\pi\)
\(180\) 0 0
\(181\) −7.65685 −0.569129 −0.284565 0.958657i \(-0.591849\pi\)
−0.284565 + 0.958657i \(0.591849\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) 0 0
\(185\) −11.6569 −0.857029
\(186\) 0 0
\(187\) −4.68629 −0.342696
\(188\) 0 0
\(189\) −2.82843 −0.205738
\(190\) 0 0
\(191\) −11.3137 −0.818631 −0.409316 0.912393i \(-0.634232\pi\)
−0.409316 + 0.912393i \(0.634232\pi\)
\(192\) 0 0
\(193\) 2.48528 0.178894 0.0894472 0.995992i \(-0.471490\pi\)
0.0894472 + 0.995992i \(0.471490\pi\)
\(194\) 0 0
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) −13.3137 −0.948562 −0.474281 0.880373i \(-0.657292\pi\)
−0.474281 + 0.880373i \(0.657292\pi\)
\(198\) 0 0
\(199\) −10.3431 −0.733206 −0.366603 0.930377i \(-0.619479\pi\)
−0.366603 + 0.930377i \(0.619479\pi\)
\(200\) 0 0
\(201\) 5.65685 0.399004
\(202\) 0 0
\(203\) −24.9706 −1.75259
\(204\) 0 0
\(205\) −7.65685 −0.534778
\(206\) 0 0
\(207\) 8.48528 0.589768
\(208\) 0 0
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −0.686292 −0.0472463 −0.0236231 0.999721i \(-0.507520\pi\)
−0.0236231 + 0.999721i \(0.507520\pi\)
\(212\) 0 0
\(213\) −5.65685 −0.387601
\(214\) 0 0
\(215\) −9.65685 −0.658592
\(216\) 0 0
\(217\) −11.3137 −0.768025
\(218\) 0 0
\(219\) 14.4853 0.978825
\(220\) 0 0
\(221\) −0.828427 −0.0557260
\(222\) 0 0
\(223\) −10.8284 −0.725125 −0.362563 0.931959i \(-0.618098\pi\)
−0.362563 + 0.931959i \(0.618098\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) 4.14214 0.273720 0.136860 0.990590i \(-0.456299\pi\)
0.136860 + 0.990590i \(0.456299\pi\)
\(230\) 0 0
\(231\) 16.0000 1.05272
\(232\) 0 0
\(233\) 5.51472 0.361281 0.180641 0.983549i \(-0.442183\pi\)
0.180641 + 0.983549i \(0.442183\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) 2.34315 0.152204
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 5.31371 0.342286 0.171143 0.985246i \(-0.445254\pi\)
0.171143 + 0.985246i \(0.445254\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 2.82843 0.179969
\(248\) 0 0
\(249\) 6.34315 0.401981
\(250\) 0 0
\(251\) −10.8284 −0.683484 −0.341742 0.939794i \(-0.611017\pi\)
−0.341742 + 0.939794i \(0.611017\pi\)
\(252\) 0 0
\(253\) −48.0000 −3.01773
\(254\) 0 0
\(255\) −0.828427 −0.0518781
\(256\) 0 0
\(257\) −4.82843 −0.301189 −0.150595 0.988596i \(-0.548119\pi\)
−0.150595 + 0.988596i \(0.548119\pi\)
\(258\) 0 0
\(259\) −32.9706 −2.04869
\(260\) 0 0
\(261\) −8.82843 −0.546466
\(262\) 0 0
\(263\) −16.4853 −1.01653 −0.508263 0.861202i \(-0.669712\pi\)
−0.508263 + 0.861202i \(0.669712\pi\)
\(264\) 0 0
\(265\) 13.3137 0.817855
\(266\) 0 0
\(267\) 15.6569 0.958184
\(268\) 0 0
\(269\) −14.4853 −0.883183 −0.441592 0.897216i \(-0.645586\pi\)
−0.441592 + 0.897216i \(0.645586\pi\)
\(270\) 0 0
\(271\) −7.31371 −0.444276 −0.222138 0.975015i \(-0.571304\pi\)
−0.222138 + 0.975015i \(0.571304\pi\)
\(272\) 0 0
\(273\) 2.82843 0.171184
\(274\) 0 0
\(275\) −5.65685 −0.341121
\(276\) 0 0
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 8.34315 0.497710 0.248855 0.968541i \(-0.419946\pi\)
0.248855 + 0.968541i \(0.419946\pi\)
\(282\) 0 0
\(283\) 17.6569 1.04959 0.524796 0.851228i \(-0.324142\pi\)
0.524796 + 0.851228i \(0.324142\pi\)
\(284\) 0 0
\(285\) 2.82843 0.167542
\(286\) 0 0
\(287\) −21.6569 −1.27836
\(288\) 0 0
\(289\) −16.3137 −0.959630
\(290\) 0 0
\(291\) 3.17157 0.185921
\(292\) 0 0
\(293\) −16.6274 −0.971384 −0.485692 0.874130i \(-0.661432\pi\)
−0.485692 + 0.874130i \(0.661432\pi\)
\(294\) 0 0
\(295\) 2.34315 0.136423
\(296\) 0 0
\(297\) 5.65685 0.328244
\(298\) 0 0
\(299\) −8.48528 −0.490716
\(300\) 0 0
\(301\) −27.3137 −1.57434
\(302\) 0 0
\(303\) −16.1421 −0.927341
\(304\) 0 0
\(305\) 6.00000 0.343559
\(306\) 0 0
\(307\) 21.6569 1.23602 0.618011 0.786169i \(-0.287939\pi\)
0.618011 + 0.786169i \(0.287939\pi\)
\(308\) 0 0
\(309\) −1.65685 −0.0942551
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) −30.9706 −1.75056 −0.875280 0.483617i \(-0.839323\pi\)
−0.875280 + 0.483617i \(0.839323\pi\)
\(314\) 0 0
\(315\) 2.82843 0.159364
\(316\) 0 0
\(317\) 25.3137 1.42176 0.710880 0.703314i \(-0.248297\pi\)
0.710880 + 0.703314i \(0.248297\pi\)
\(318\) 0 0
\(319\) 49.9411 2.79617
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) −2.34315 −0.130376
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) −8.82843 −0.488213
\(328\) 0 0
\(329\) 22.6274 1.24749
\(330\) 0 0
\(331\) 8.48528 0.466393 0.233197 0.972430i \(-0.425081\pi\)
0.233197 + 0.972430i \(0.425081\pi\)
\(332\) 0 0
\(333\) −11.6569 −0.638792
\(334\) 0 0
\(335\) −5.65685 −0.309067
\(336\) 0 0
\(337\) 10.9706 0.597605 0.298802 0.954315i \(-0.403413\pi\)
0.298802 + 0.954315i \(0.403413\pi\)
\(338\) 0 0
\(339\) −6.48528 −0.352232
\(340\) 0 0
\(341\) 22.6274 1.22534
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 0 0
\(345\) −8.48528 −0.456832
\(346\) 0 0
\(347\) 9.65685 0.518407 0.259204 0.965823i \(-0.416540\pi\)
0.259204 + 0.965823i \(0.416540\pi\)
\(348\) 0 0
\(349\) 12.1421 0.649954 0.324977 0.945722i \(-0.394644\pi\)
0.324977 + 0.945722i \(0.394644\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 5.31371 0.282820 0.141410 0.989951i \(-0.454836\pi\)
0.141410 + 0.989951i \(0.454836\pi\)
\(354\) 0 0
\(355\) 5.65685 0.300235
\(356\) 0 0
\(357\) −2.34315 −0.124012
\(358\) 0 0
\(359\) 28.2843 1.49279 0.746393 0.665505i \(-0.231784\pi\)
0.746393 + 0.665505i \(0.231784\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) −21.0000 −1.10221
\(364\) 0 0
\(365\) −14.4853 −0.758194
\(366\) 0 0
\(367\) −25.6569 −1.33928 −0.669638 0.742687i \(-0.733551\pi\)
−0.669638 + 0.742687i \(0.733551\pi\)
\(368\) 0 0
\(369\) −7.65685 −0.398600
\(370\) 0 0
\(371\) 37.6569 1.95505
\(372\) 0 0
\(373\) −2.68629 −0.139091 −0.0695455 0.997579i \(-0.522155\pi\)
−0.0695455 + 0.997579i \(0.522155\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 8.82843 0.454687
\(378\) 0 0
\(379\) 7.51472 0.386005 0.193003 0.981198i \(-0.438177\pi\)
0.193003 + 0.981198i \(0.438177\pi\)
\(380\) 0 0
\(381\) 9.65685 0.494736
\(382\) 0 0
\(383\) 29.6569 1.51539 0.757697 0.652606i \(-0.226324\pi\)
0.757697 + 0.652606i \(0.226324\pi\)
\(384\) 0 0
\(385\) −16.0000 −0.815436
\(386\) 0 0
\(387\) −9.65685 −0.490885
\(388\) 0 0
\(389\) −6.48528 −0.328817 −0.164408 0.986392i \(-0.552571\pi\)
−0.164408 + 0.986392i \(0.552571\pi\)
\(390\) 0 0
\(391\) 7.02944 0.355494
\(392\) 0 0
\(393\) −6.14214 −0.309830
\(394\) 0 0
\(395\) −2.34315 −0.117896
\(396\) 0 0
\(397\) 30.2843 1.51992 0.759962 0.649968i \(-0.225218\pi\)
0.759962 + 0.649968i \(0.225218\pi\)
\(398\) 0 0
\(399\) 8.00000 0.400501
\(400\) 0 0
\(401\) −26.9706 −1.34685 −0.673423 0.739258i \(-0.735177\pi\)
−0.673423 + 0.739258i \(0.735177\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 65.9411 3.26858
\(408\) 0 0
\(409\) −3.65685 −0.180820 −0.0904099 0.995905i \(-0.528818\pi\)
−0.0904099 + 0.995905i \(0.528818\pi\)
\(410\) 0 0
\(411\) 17.3137 0.854022
\(412\) 0 0
\(413\) 6.62742 0.326114
\(414\) 0 0
\(415\) −6.34315 −0.311373
\(416\) 0 0
\(417\) 6.34315 0.310625
\(418\) 0 0
\(419\) 10.8284 0.529003 0.264502 0.964385i \(-0.414793\pi\)
0.264502 + 0.964385i \(0.414793\pi\)
\(420\) 0 0
\(421\) −24.1421 −1.17662 −0.588308 0.808637i \(-0.700206\pi\)
−0.588308 + 0.808637i \(0.700206\pi\)
\(422\) 0 0
\(423\) 8.00000 0.388973
\(424\) 0 0
\(425\) 0.828427 0.0401846
\(426\) 0 0
\(427\) 16.9706 0.821263
\(428\) 0 0
\(429\) −5.65685 −0.273115
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) −22.9706 −1.10389 −0.551947 0.833879i \(-0.686115\pi\)
−0.551947 + 0.833879i \(0.686115\pi\)
\(434\) 0 0
\(435\) 8.82843 0.423291
\(436\) 0 0
\(437\) −24.0000 −1.14808
\(438\) 0 0
\(439\) −22.6274 −1.07995 −0.539974 0.841682i \(-0.681566\pi\)
−0.539974 + 0.841682i \(0.681566\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −30.3431 −1.44165 −0.720823 0.693119i \(-0.756236\pi\)
−0.720823 + 0.693119i \(0.756236\pi\)
\(444\) 0 0
\(445\) −15.6569 −0.742206
\(446\) 0 0
\(447\) −3.65685 −0.172963
\(448\) 0 0
\(449\) 26.2843 1.24043 0.620216 0.784431i \(-0.287045\pi\)
0.620216 + 0.784431i \(0.287045\pi\)
\(450\) 0 0
\(451\) 43.3137 2.03956
\(452\) 0 0
\(453\) 12.0000 0.563809
\(454\) 0 0
\(455\) −2.82843 −0.132599
\(456\) 0 0
\(457\) 20.8284 0.974313 0.487156 0.873315i \(-0.338034\pi\)
0.487156 + 0.873315i \(0.338034\pi\)
\(458\) 0 0
\(459\) −0.828427 −0.0386677
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) 3.79899 0.176554 0.0882770 0.996096i \(-0.471864\pi\)
0.0882770 + 0.996096i \(0.471864\pi\)
\(464\) 0 0
\(465\) 4.00000 0.185496
\(466\) 0 0
\(467\) −7.31371 −0.338438 −0.169219 0.985578i \(-0.554125\pi\)
−0.169219 + 0.985578i \(0.554125\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) −5.31371 −0.244843
\(472\) 0 0
\(473\) 54.6274 2.51177
\(474\) 0 0
\(475\) −2.82843 −0.129777
\(476\) 0 0
\(477\) 13.3137 0.609593
\(478\) 0 0
\(479\) 11.3137 0.516937 0.258468 0.966020i \(-0.416782\pi\)
0.258468 + 0.966020i \(0.416782\pi\)
\(480\) 0 0
\(481\) 11.6569 0.531507
\(482\) 0 0
\(483\) −24.0000 −1.09204
\(484\) 0 0
\(485\) −3.17157 −0.144014
\(486\) 0 0
\(487\) −16.4853 −0.747019 −0.373510 0.927626i \(-0.621846\pi\)
−0.373510 + 0.927626i \(0.621846\pi\)
\(488\) 0 0
\(489\) 11.3137 0.511624
\(490\) 0 0
\(491\) 38.1421 1.72133 0.860665 0.509171i \(-0.170048\pi\)
0.860665 + 0.509171i \(0.170048\pi\)
\(492\) 0 0
\(493\) −7.31371 −0.329393
\(494\) 0 0
\(495\) −5.65685 −0.254257
\(496\) 0 0
\(497\) 16.0000 0.717698
\(498\) 0 0
\(499\) −0.485281 −0.0217242 −0.0108621 0.999941i \(-0.503458\pi\)
−0.0108621 + 0.999941i \(0.503458\pi\)
\(500\) 0 0
\(501\) 8.97056 0.400775
\(502\) 0 0
\(503\) 23.5147 1.04847 0.524235 0.851574i \(-0.324351\pi\)
0.524235 + 0.851574i \(0.324351\pi\)
\(504\) 0 0
\(505\) 16.1421 0.718316
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) −37.3137 −1.65390 −0.826951 0.562275i \(-0.809926\pi\)
−0.826951 + 0.562275i \(0.809926\pi\)
\(510\) 0 0
\(511\) −40.9706 −1.81243
\(512\) 0 0
\(513\) 2.82843 0.124878
\(514\) 0 0
\(515\) 1.65685 0.0730097
\(516\) 0 0
\(517\) −45.2548 −1.99031
\(518\) 0 0
\(519\) 9.31371 0.408826
\(520\) 0 0
\(521\) 26.9706 1.18160 0.590801 0.806817i \(-0.298812\pi\)
0.590801 + 0.806817i \(0.298812\pi\)
\(522\) 0 0
\(523\) 10.6274 0.464704 0.232352 0.972632i \(-0.425358\pi\)
0.232352 + 0.972632i \(0.425358\pi\)
\(524\) 0 0
\(525\) −2.82843 −0.123443
\(526\) 0 0
\(527\) −3.31371 −0.144347
\(528\) 0 0
\(529\) 49.0000 2.13043
\(530\) 0 0
\(531\) 2.34315 0.101684
\(532\) 0 0
\(533\) 7.65685 0.331655
\(534\) 0 0
\(535\) 4.00000 0.172935
\(536\) 0 0
\(537\) 7.51472 0.324284
\(538\) 0 0
\(539\) −5.65685 −0.243658
\(540\) 0 0
\(541\) 14.4853 0.622771 0.311385 0.950284i \(-0.399207\pi\)
0.311385 + 0.950284i \(0.399207\pi\)
\(542\) 0 0
\(543\) 7.65685 0.328587
\(544\) 0 0
\(545\) 8.82843 0.378168
\(546\) 0 0
\(547\) −0.686292 −0.0293437 −0.0146719 0.999892i \(-0.504670\pi\)
−0.0146719 + 0.999892i \(0.504670\pi\)
\(548\) 0 0
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) 24.9706 1.06378
\(552\) 0 0
\(553\) −6.62742 −0.281826
\(554\) 0 0
\(555\) 11.6569 0.494806
\(556\) 0 0
\(557\) 10.6863 0.452793 0.226396 0.974035i \(-0.427306\pi\)
0.226396 + 0.974035i \(0.427306\pi\)
\(558\) 0 0
\(559\) 9.65685 0.408441
\(560\) 0 0
\(561\) 4.68629 0.197855
\(562\) 0 0
\(563\) 30.3431 1.27881 0.639406 0.768870i \(-0.279181\pi\)
0.639406 + 0.768870i \(0.279181\pi\)
\(564\) 0 0
\(565\) 6.48528 0.272838
\(566\) 0 0
\(567\) 2.82843 0.118783
\(568\) 0 0
\(569\) 31.6569 1.32712 0.663562 0.748121i \(-0.269044\pi\)
0.663562 + 0.748121i \(0.269044\pi\)
\(570\) 0 0
\(571\) −20.9706 −0.877591 −0.438795 0.898587i \(-0.644595\pi\)
−0.438795 + 0.898587i \(0.644595\pi\)
\(572\) 0 0
\(573\) 11.3137 0.472637
\(574\) 0 0
\(575\) 8.48528 0.353861
\(576\) 0 0
\(577\) −23.4558 −0.976480 −0.488240 0.872710i \(-0.662361\pi\)
−0.488240 + 0.872710i \(0.662361\pi\)
\(578\) 0 0
\(579\) −2.48528 −0.103285
\(580\) 0 0
\(581\) −17.9411 −0.744323
\(582\) 0 0
\(583\) −75.3137 −3.11918
\(584\) 0 0
\(585\) −1.00000 −0.0413449
\(586\) 0 0
\(587\) −2.62742 −0.108445 −0.0542226 0.998529i \(-0.517268\pi\)
−0.0542226 + 0.998529i \(0.517268\pi\)
\(588\) 0 0
\(589\) 11.3137 0.466173
\(590\) 0 0
\(591\) 13.3137 0.547653
\(592\) 0 0
\(593\) −0.343146 −0.0140913 −0.00704565 0.999975i \(-0.502243\pi\)
−0.00704565 + 0.999975i \(0.502243\pi\)
\(594\) 0 0
\(595\) 2.34315 0.0960596
\(596\) 0 0
\(597\) 10.3431 0.423317
\(598\) 0 0
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 0 0
\(601\) 29.3137 1.19573 0.597866 0.801596i \(-0.296016\pi\)
0.597866 + 0.801596i \(0.296016\pi\)
\(602\) 0 0
\(603\) −5.65685 −0.230365
\(604\) 0 0
\(605\) 21.0000 0.853771
\(606\) 0 0
\(607\) −28.9706 −1.17588 −0.587939 0.808905i \(-0.700061\pi\)
−0.587939 + 0.808905i \(0.700061\pi\)
\(608\) 0 0
\(609\) 24.9706 1.01186
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) 22.2843 0.900053 0.450027 0.893015i \(-0.351414\pi\)
0.450027 + 0.893015i \(0.351414\pi\)
\(614\) 0 0
\(615\) 7.65685 0.308754
\(616\) 0 0
\(617\) 2.00000 0.0805170 0.0402585 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) 0 0
\(619\) 34.8284 1.39987 0.699936 0.714205i \(-0.253212\pi\)
0.699936 + 0.714205i \(0.253212\pi\)
\(620\) 0 0
\(621\) −8.48528 −0.340503
\(622\) 0 0
\(623\) −44.2843 −1.77421
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −16.0000 −0.638978
\(628\) 0 0
\(629\) −9.65685 −0.385044
\(630\) 0 0
\(631\) 33.6569 1.33986 0.669929 0.742425i \(-0.266324\pi\)
0.669929 + 0.742425i \(0.266324\pi\)
\(632\) 0 0
\(633\) 0.686292 0.0272776
\(634\) 0 0
\(635\) −9.65685 −0.383221
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 5.65685 0.223782
\(640\) 0 0
\(641\) −4.62742 −0.182772 −0.0913860 0.995816i \(-0.529130\pi\)
−0.0913860 + 0.995816i \(0.529130\pi\)
\(642\) 0 0
\(643\) −39.5980 −1.56159 −0.780796 0.624786i \(-0.785186\pi\)
−0.780796 + 0.624786i \(0.785186\pi\)
\(644\) 0 0
\(645\) 9.65685 0.380238
\(646\) 0 0
\(647\) −8.48528 −0.333591 −0.166795 0.985992i \(-0.553342\pi\)
−0.166795 + 0.985992i \(0.553342\pi\)
\(648\) 0 0
\(649\) −13.2548 −0.520298
\(650\) 0 0
\(651\) 11.3137 0.443419
\(652\) 0 0
\(653\) −42.2843 −1.65471 −0.827356 0.561678i \(-0.810156\pi\)
−0.827356 + 0.561678i \(0.810156\pi\)
\(654\) 0 0
\(655\) 6.14214 0.239993
\(656\) 0 0
\(657\) −14.4853 −0.565125
\(658\) 0 0
\(659\) 7.51472 0.292732 0.146366 0.989231i \(-0.453242\pi\)
0.146366 + 0.989231i \(0.453242\pi\)
\(660\) 0 0
\(661\) −8.14214 −0.316692 −0.158346 0.987384i \(-0.550616\pi\)
−0.158346 + 0.987384i \(0.550616\pi\)
\(662\) 0 0
\(663\) 0.828427 0.0321734
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) −74.9117 −2.90059
\(668\) 0 0
\(669\) 10.8284 0.418651
\(670\) 0 0
\(671\) −33.9411 −1.31028
\(672\) 0 0
\(673\) 32.6274 1.25769 0.628847 0.777529i \(-0.283527\pi\)
0.628847 + 0.777529i \(0.283527\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 12.3431 0.474386 0.237193 0.971463i \(-0.423773\pi\)
0.237193 + 0.971463i \(0.423773\pi\)
\(678\) 0 0
\(679\) −8.97056 −0.344259
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 0 0
\(683\) −33.6569 −1.28784 −0.643922 0.765091i \(-0.722694\pi\)
−0.643922 + 0.765091i \(0.722694\pi\)
\(684\) 0 0
\(685\) −17.3137 −0.661523
\(686\) 0 0
\(687\) −4.14214 −0.158032
\(688\) 0 0
\(689\) −13.3137 −0.507212
\(690\) 0 0
\(691\) 27.7990 1.05752 0.528762 0.848770i \(-0.322657\pi\)
0.528762 + 0.848770i \(0.322657\pi\)
\(692\) 0 0
\(693\) −16.0000 −0.607790
\(694\) 0 0
\(695\) −6.34315 −0.240609
\(696\) 0 0
\(697\) −6.34315 −0.240264
\(698\) 0 0
\(699\) −5.51472 −0.208586
\(700\) 0 0
\(701\) 0.142136 0.00536839 0.00268419 0.999996i \(-0.499146\pi\)
0.00268419 + 0.999996i \(0.499146\pi\)
\(702\) 0 0
\(703\) 32.9706 1.24351
\(704\) 0 0
\(705\) −8.00000 −0.301297
\(706\) 0 0
\(707\) 45.6569 1.71710
\(708\) 0 0
\(709\) −7.17157 −0.269334 −0.134667 0.990891i \(-0.542996\pi\)
−0.134667 + 0.990891i \(0.542996\pi\)
\(710\) 0 0
\(711\) −2.34315 −0.0878748
\(712\) 0 0
\(713\) −33.9411 −1.27111
\(714\) 0 0
\(715\) 5.65685 0.211554
\(716\) 0 0
\(717\) 16.0000 0.597531
\(718\) 0 0
\(719\) 29.6569 1.10601 0.553007 0.833177i \(-0.313480\pi\)
0.553007 + 0.833177i \(0.313480\pi\)
\(720\) 0 0
\(721\) 4.68629 0.174527
\(722\) 0 0
\(723\) −5.31371 −0.197619
\(724\) 0 0
\(725\) −8.82843 −0.327880
\(726\) 0 0
\(727\) 45.9411 1.70386 0.851931 0.523654i \(-0.175432\pi\)
0.851931 + 0.523654i \(0.175432\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) −0.343146 −0.0126744 −0.00633719 0.999980i \(-0.502017\pi\)
−0.00633719 + 0.999980i \(0.502017\pi\)
\(734\) 0 0
\(735\) −1.00000 −0.0368856
\(736\) 0 0
\(737\) 32.0000 1.17874
\(738\) 0 0
\(739\) 14.1421 0.520227 0.260113 0.965578i \(-0.416240\pi\)
0.260113 + 0.965578i \(0.416240\pi\)
\(740\) 0 0
\(741\) −2.82843 −0.103905
\(742\) 0 0
\(743\) 36.2843 1.33114 0.665570 0.746335i \(-0.268188\pi\)
0.665570 + 0.746335i \(0.268188\pi\)
\(744\) 0 0
\(745\) 3.65685 0.133977
\(746\) 0 0
\(747\) −6.34315 −0.232084
\(748\) 0 0
\(749\) 11.3137 0.413394
\(750\) 0 0
\(751\) −11.3137 −0.412843 −0.206422 0.978463i \(-0.566182\pi\)
−0.206422 + 0.978463i \(0.566182\pi\)
\(752\) 0 0
\(753\) 10.8284 0.394610
\(754\) 0 0
\(755\) −12.0000 −0.436725
\(756\) 0 0
\(757\) 19.9411 0.724773 0.362386 0.932028i \(-0.381962\pi\)
0.362386 + 0.932028i \(0.381962\pi\)
\(758\) 0 0
\(759\) 48.0000 1.74229
\(760\) 0 0
\(761\) 27.6569 1.00256 0.501280 0.865285i \(-0.332863\pi\)
0.501280 + 0.865285i \(0.332863\pi\)
\(762\) 0 0
\(763\) 24.9706 0.903995
\(764\) 0 0
\(765\) 0.828427 0.0299518
\(766\) 0 0
\(767\) −2.34315 −0.0846061
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 4.82843 0.173892
\(772\) 0 0
\(773\) −53.3137 −1.91756 −0.958780 0.284148i \(-0.908289\pi\)
−0.958780 + 0.284148i \(0.908289\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 0 0
\(777\) 32.9706 1.18281
\(778\) 0 0
\(779\) 21.6569 0.775937
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 0 0
\(783\) 8.82843 0.315502
\(784\) 0 0
\(785\) 5.31371 0.189654
\(786\) 0 0
\(787\) 24.0000 0.855508 0.427754 0.903895i \(-0.359305\pi\)
0.427754 + 0.903895i \(0.359305\pi\)
\(788\) 0 0
\(789\) 16.4853 0.586892
\(790\) 0 0
\(791\) 18.3431 0.652207
\(792\) 0 0
\(793\) −6.00000 −0.213066
\(794\) 0 0
\(795\) −13.3137 −0.472189
\(796\) 0 0
\(797\) 16.6274 0.588973 0.294487 0.955656i \(-0.404851\pi\)
0.294487 + 0.955656i \(0.404851\pi\)
\(798\) 0 0
\(799\) 6.62742 0.234461
\(800\) 0 0
\(801\) −15.6569 −0.553208
\(802\) 0 0
\(803\) 81.9411 2.89164
\(804\) 0 0
\(805\) 24.0000 0.845889
\(806\) 0 0
\(807\) 14.4853 0.509906
\(808\) 0 0
\(809\) 13.3137 0.468085 0.234043 0.972226i \(-0.424804\pi\)
0.234043 + 0.972226i \(0.424804\pi\)
\(810\) 0 0
\(811\) 1.85786 0.0652384 0.0326192 0.999468i \(-0.489615\pi\)
0.0326192 + 0.999468i \(0.489615\pi\)
\(812\) 0 0
\(813\) 7.31371 0.256503
\(814\) 0 0
\(815\) −11.3137 −0.396302
\(816\) 0 0
\(817\) 27.3137 0.955586
\(818\) 0 0
\(819\) −2.82843 −0.0988332
\(820\) 0 0
\(821\) 34.2843 1.19653 0.598265 0.801299i \(-0.295857\pi\)
0.598265 + 0.801299i \(0.295857\pi\)
\(822\) 0 0
\(823\) 52.9706 1.84644 0.923219 0.384275i \(-0.125548\pi\)
0.923219 + 0.384275i \(0.125548\pi\)
\(824\) 0 0
\(825\) 5.65685 0.196946
\(826\) 0 0
\(827\) −9.65685 −0.335802 −0.167901 0.985804i \(-0.553699\pi\)
−0.167901 + 0.985804i \(0.553699\pi\)
\(828\) 0 0
\(829\) −53.3137 −1.85166 −0.925831 0.377938i \(-0.876633\pi\)
−0.925831 + 0.377938i \(0.876633\pi\)
\(830\) 0 0
\(831\) −26.0000 −0.901930
\(832\) 0 0
\(833\) 0.828427 0.0287033
\(834\) 0 0
\(835\) −8.97056 −0.310439
\(836\) 0 0
\(837\) 4.00000 0.138260
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 48.9411 1.68763
\(842\) 0 0
\(843\) −8.34315 −0.287353
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 59.3970 2.04090
\(848\) 0 0
\(849\) −17.6569 −0.605982
\(850\) 0 0
\(851\) −98.9117 −3.39065
\(852\) 0 0
\(853\) 38.2843 1.31083 0.655414 0.755270i \(-0.272494\pi\)
0.655414 + 0.755270i \(0.272494\pi\)
\(854\) 0 0
\(855\) −2.82843 −0.0967302
\(856\) 0 0
\(857\) −20.8284 −0.711486 −0.355743 0.934584i \(-0.615772\pi\)
−0.355743 + 0.934584i \(0.615772\pi\)
\(858\) 0 0
\(859\) −37.9411 −1.29453 −0.647267 0.762263i \(-0.724088\pi\)
−0.647267 + 0.762263i \(0.724088\pi\)
\(860\) 0 0
\(861\) 21.6569 0.738064
\(862\) 0 0
\(863\) 28.2843 0.962808 0.481404 0.876499i \(-0.340127\pi\)
0.481404 + 0.876499i \(0.340127\pi\)
\(864\) 0 0
\(865\) −9.31371 −0.316676
\(866\) 0 0
\(867\) 16.3137 0.554043
\(868\) 0 0
\(869\) 13.2548 0.449639
\(870\) 0 0
\(871\) 5.65685 0.191675
\(872\) 0 0
\(873\) −3.17157 −0.107341
\(874\) 0 0
\(875\) 2.82843 0.0956183
\(876\) 0 0
\(877\) −51.2548 −1.73075 −0.865376 0.501122i \(-0.832921\pi\)
−0.865376 + 0.501122i \(0.832921\pi\)
\(878\) 0 0
\(879\) 16.6274 0.560829
\(880\) 0 0
\(881\) −10.2843 −0.346486 −0.173243 0.984879i \(-0.555425\pi\)
−0.173243 + 0.984879i \(0.555425\pi\)
\(882\) 0 0
\(883\) −31.3137 −1.05379 −0.526895 0.849930i \(-0.676644\pi\)
−0.526895 + 0.849930i \(0.676644\pi\)
\(884\) 0 0
\(885\) −2.34315 −0.0787640
\(886\) 0 0
\(887\) 40.4853 1.35936 0.679681 0.733508i \(-0.262118\pi\)
0.679681 + 0.733508i \(0.262118\pi\)
\(888\) 0 0
\(889\) −27.3137 −0.916072
\(890\) 0 0
\(891\) −5.65685 −0.189512
\(892\) 0 0
\(893\) −22.6274 −0.757198
\(894\) 0 0
\(895\) −7.51472 −0.251189
\(896\) 0 0
\(897\) 8.48528 0.283315
\(898\) 0 0
\(899\) 35.3137 1.17778
\(900\) 0 0
\(901\) 11.0294 0.367444
\(902\) 0 0
\(903\) 27.3137 0.908943
\(904\) 0 0
\(905\) −7.65685 −0.254522
\(906\) 0 0
\(907\) −8.28427 −0.275075 −0.137537 0.990497i \(-0.543919\pi\)
−0.137537 + 0.990497i \(0.543919\pi\)
\(908\) 0 0
\(909\) 16.1421 0.535401
\(910\) 0 0
\(911\) −24.9706 −0.827312 −0.413656 0.910433i \(-0.635748\pi\)
−0.413656 + 0.910433i \(0.635748\pi\)
\(912\) 0 0
\(913\) 35.8823 1.18753
\(914\) 0 0
\(915\) −6.00000 −0.198354
\(916\) 0 0
\(917\) 17.3726 0.573693
\(918\) 0 0
\(919\) −41.9411 −1.38351 −0.691755 0.722132i \(-0.743162\pi\)
−0.691755 + 0.722132i \(0.743162\pi\)
\(920\) 0 0
\(921\) −21.6569 −0.713618
\(922\) 0 0
\(923\) −5.65685 −0.186198
\(924\) 0 0
\(925\) −11.6569 −0.383275
\(926\) 0 0
\(927\) 1.65685 0.0544182
\(928\) 0 0
\(929\) −33.5980 −1.10231 −0.551157 0.834402i \(-0.685813\pi\)
−0.551157 + 0.834402i \(0.685813\pi\)
\(930\) 0 0
\(931\) −2.82843 −0.0926980
\(932\) 0 0
\(933\) −24.0000 −0.785725
\(934\) 0 0
\(935\) −4.68629 −0.153258
\(936\) 0 0
\(937\) 16.6274 0.543194 0.271597 0.962411i \(-0.412448\pi\)
0.271597 + 0.962411i \(0.412448\pi\)
\(938\) 0 0
\(939\) 30.9706 1.01069
\(940\) 0 0
\(941\) 54.9706 1.79199 0.895995 0.444065i \(-0.146464\pi\)
0.895995 + 0.444065i \(0.146464\pi\)
\(942\) 0 0
\(943\) −64.9706 −2.11573
\(944\) 0 0
\(945\) −2.82843 −0.0920087
\(946\) 0 0
\(947\) −30.3431 −0.986020 −0.493010 0.870024i \(-0.664103\pi\)
−0.493010 + 0.870024i \(0.664103\pi\)
\(948\) 0 0
\(949\) 14.4853 0.470212
\(950\) 0 0
\(951\) −25.3137 −0.820853
\(952\) 0 0
\(953\) −27.8579 −0.902405 −0.451202 0.892422i \(-0.649005\pi\)
−0.451202 + 0.892422i \(0.649005\pi\)
\(954\) 0 0
\(955\) −11.3137 −0.366103
\(956\) 0 0
\(957\) −49.9411 −1.61437
\(958\) 0 0
\(959\) −48.9706 −1.58134
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 4.00000 0.128898
\(964\) 0 0
\(965\) 2.48528 0.0800040
\(966\) 0 0
\(967\) 7.51472 0.241657 0.120829 0.992673i \(-0.461445\pi\)
0.120829 + 0.992673i \(0.461445\pi\)
\(968\) 0 0
\(969\) 2.34315 0.0752727
\(970\) 0 0
\(971\) −15.5147 −0.497891 −0.248946 0.968517i \(-0.580084\pi\)
−0.248946 + 0.968517i \(0.580084\pi\)
\(972\) 0 0
\(973\) −17.9411 −0.575166
\(974\) 0 0
\(975\) 1.00000 0.0320256
\(976\) 0 0
\(977\) −8.34315 −0.266921 −0.133460 0.991054i \(-0.542609\pi\)
−0.133460 + 0.991054i \(0.542609\pi\)
\(978\) 0 0
\(979\) 88.5685 2.83066
\(980\) 0 0
\(981\) 8.82843 0.281870
\(982\) 0 0
\(983\) 2.34315 0.0747347 0.0373674 0.999302i \(-0.488103\pi\)
0.0373674 + 0.999302i \(0.488103\pi\)
\(984\) 0 0
\(985\) −13.3137 −0.424210
\(986\) 0 0
\(987\) −22.6274 −0.720239
\(988\) 0 0
\(989\) −81.9411 −2.60558
\(990\) 0 0
\(991\) 42.9117 1.36313 0.681567 0.731755i \(-0.261299\pi\)
0.681567 + 0.731755i \(0.261299\pi\)
\(992\) 0 0
\(993\) −8.48528 −0.269272
\(994\) 0 0
\(995\) −10.3431 −0.327900
\(996\) 0 0
\(997\) 61.3137 1.94182 0.970912 0.239435i \(-0.0769623\pi\)
0.970912 + 0.239435i \(0.0769623\pi\)
\(998\) 0 0
\(999\) 11.6569 0.368807
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 3120.2.a.bc.1.2 2
3.2 odd 2 9360.2.a.ch.1.2 2
4.3 odd 2 390.2.a.h.1.1 2
12.11 even 2 1170.2.a.o.1.1 2
20.3 even 4 1950.2.e.o.1249.2 4
20.7 even 4 1950.2.e.o.1249.3 4
20.19 odd 2 1950.2.a.bd.1.2 2
52.31 even 4 5070.2.b.q.1351.1 4
52.47 even 4 5070.2.b.q.1351.4 4
52.51 odd 2 5070.2.a.bc.1.2 2
60.23 odd 4 5850.2.e.bk.5149.4 4
60.47 odd 4 5850.2.e.bk.5149.1 4
60.59 even 2 5850.2.a.cl.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.h.1.1 2 4.3 odd 2
1170.2.a.o.1.1 2 12.11 even 2
1950.2.a.bd.1.2 2 20.19 odd 2
1950.2.e.o.1249.2 4 20.3 even 4
1950.2.e.o.1249.3 4 20.7 even 4
3120.2.a.bc.1.2 2 1.1 even 1 trivial
5070.2.a.bc.1.2 2 52.51 odd 2
5070.2.b.q.1351.1 4 52.31 even 4
5070.2.b.q.1351.4 4 52.47 even 4
5850.2.a.cl.1.2 2 60.59 even 2
5850.2.e.bk.5149.1 4 60.47 odd 4
5850.2.e.bk.5149.4 4 60.23 odd 4
9360.2.a.ch.1.2 2 3.2 odd 2