Properties

Label 2352.2.b.b.1567.2
Level $2352$
Weight $2$
Character 2352.1567
Analytic conductor $18.781$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.2
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2352.1567
Dual form 2352.2.b.b.1567.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.73205i q^{5} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.73205i q^{5} +1.00000 q^{9} +5.19615i q^{11} +6.92820i q^{13} -1.73205i q^{15} +3.46410i q^{17} +2.00000 q^{19} -6.92820i q^{23} +2.00000 q^{25} -1.00000 q^{27} -9.00000 q^{29} +1.00000 q^{31} -5.19615i q^{33} -2.00000 q^{37} -6.92820i q^{39} -3.46410i q^{41} -3.46410i q^{43} +1.73205i q^{45} -3.46410i q^{51} +9.00000 q^{53} -9.00000 q^{55} -2.00000 q^{57} -3.00000 q^{59} +6.92820i q^{61} -12.0000 q^{65} +6.92820i q^{69} -6.92820i q^{71} +6.92820i q^{73} -2.00000 q^{75} +1.73205i q^{79} +1.00000 q^{81} -15.0000 q^{83} -6.00000 q^{85} +9.00000 q^{87} +10.3923i q^{89} -1.00000 q^{93} +3.46410i q^{95} +8.66025i q^{97} +5.19615i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{9} + 4q^{19} + 4q^{25} - 2q^{27} - 18q^{29} + 2q^{31} - 4q^{37} + 18q^{53} - 18q^{55} - 4q^{57} - 6q^{59} - 24q^{65} - 4q^{75} + 2q^{81} - 30q^{83} - 12q^{85} + 18q^{87} - 2q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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
\(5\) 1.73205i 0.774597i 0.921954 + 0.387298i \(0.126592\pi\)
−0.921954 + 0.387298i \(0.873408\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.19615i 1.56670i 0.621582 + 0.783349i \(0.286490\pi\)
−0.621582 + 0.783349i \(0.713510\pi\)
\(12\) 0 0
\(13\) 6.92820i 1.92154i 0.277350 + 0.960769i \(0.410544\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) − 1.73205i − 0.447214i
\(16\) 0 0
\(17\) 3.46410i 0.840168i 0.907485 + 0.420084i \(0.137999\pi\)
−0.907485 + 0.420084i \(0.862001\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 6.92820i − 1.44463i −0.691564 0.722315i \(-0.743078\pi\)
0.691564 0.722315i \(-0.256922\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 0 0
\(33\) − 5.19615i − 0.904534i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) − 6.92820i − 1.10940i
\(40\) 0 0
\(41\) − 3.46410i − 0.541002i −0.962720 0.270501i \(-0.912811\pi\)
0.962720 0.270501i \(-0.0871893\pi\)
\(42\) 0 0
\(43\) − 3.46410i − 0.528271i −0.964486 0.264135i \(-0.914913\pi\)
0.964486 0.264135i \(-0.0850865\pi\)
\(44\) 0 0
\(45\) 1.73205i 0.258199i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) − 3.46410i − 0.485071i
\(52\) 0 0
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) −9.00000 −1.21356
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) 6.92820i 0.887066i 0.896258 + 0.443533i \(0.146275\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 6.92820i 0.834058i
\(70\) 0 0
\(71\) − 6.92820i − 0.822226i −0.911584 0.411113i \(-0.865140\pi\)
0.911584 0.411113i \(-0.134860\pi\)
\(72\) 0 0
\(73\) 6.92820i 0.810885i 0.914121 + 0.405442i \(0.132883\pi\)
−0.914121 + 0.405442i \(0.867117\pi\)
\(74\) 0 0
\(75\) −2.00000 −0.230940
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.73205i 0.194871i 0.995242 + 0.0974355i \(0.0310640\pi\)
−0.995242 + 0.0974355i \(0.968936\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −15.0000 −1.64646 −0.823232 0.567705i \(-0.807831\pi\)
−0.823232 + 0.567705i \(0.807831\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) 9.00000 0.964901
\(88\) 0 0
\(89\) 10.3923i 1.10158i 0.834643 + 0.550791i \(0.185674\pi\)
−0.834643 + 0.550791i \(0.814326\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) 3.46410i 0.355409i
\(96\) 0 0
\(97\) 8.66025i 0.879316i 0.898165 + 0.439658i \(0.144900\pi\)
−0.898165 + 0.439658i \(0.855100\pi\)
\(98\) 0 0
\(99\) 5.19615i 0.522233i
\(100\) 0 0
\(101\) − 13.8564i − 1.37876i −0.724398 0.689382i \(-0.757882\pi\)
0.724398 0.689382i \(-0.242118\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.66025i 0.837218i 0.908166 + 0.418609i \(0.137482\pi\)
−0.908166 + 0.418609i \(0.862518\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 12.0000 1.11901
\(116\) 0 0
\(117\) 6.92820i 0.640513i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −16.0000 −1.45455
\(122\) 0 0
\(123\) 3.46410i 0.312348i
\(124\) 0 0
\(125\) 12.1244i 1.08444i
\(126\) 0 0
\(127\) − 15.5885i − 1.38325i −0.722256 0.691626i \(-0.756895\pi\)
0.722256 0.691626i \(-0.243105\pi\)
\(128\) 0 0
\(129\) 3.46410i 0.304997i
\(130\) 0 0
\(131\) −3.00000 −0.262111 −0.131056 0.991375i \(-0.541837\pi\)
−0.131056 + 0.991375i \(0.541837\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) − 1.73205i − 0.149071i
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) 22.0000 1.86602 0.933008 0.359856i \(-0.117174\pi\)
0.933008 + 0.359856i \(0.117174\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −36.0000 −3.01047
\(144\) 0 0
\(145\) − 15.5885i − 1.29455i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) − 1.73205i − 0.140952i −0.997513 0.0704761i \(-0.977548\pi\)
0.997513 0.0704761i \(-0.0224519\pi\)
\(152\) 0 0
\(153\) 3.46410i 0.280056i
\(154\) 0 0
\(155\) 1.73205i 0.139122i
\(156\) 0 0
\(157\) 6.92820i 0.552931i 0.961024 + 0.276465i \(0.0891631\pi\)
−0.961024 + 0.276465i \(0.910837\pi\)
\(158\) 0 0
\(159\) −9.00000 −0.713746
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 20.7846i − 1.62798i −0.580881 0.813988i \(-0.697292\pi\)
0.580881 0.813988i \(-0.302708\pi\)
\(164\) 0 0
\(165\) 9.00000 0.700649
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −35.0000 −2.69231
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 0 0
\(173\) − 13.8564i − 1.05348i −0.850026 0.526742i \(-0.823414\pi\)
0.850026 0.526742i \(-0.176586\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 3.00000 0.225494
\(178\) 0 0
\(179\) 17.3205i 1.29460i 0.762237 + 0.647298i \(0.224101\pi\)
−0.762237 + 0.647298i \(0.775899\pi\)
\(180\) 0 0
\(181\) − 3.46410i − 0.257485i −0.991678 0.128742i \(-0.958906\pi\)
0.991678 0.128742i \(-0.0410940\pi\)
\(182\) 0 0
\(183\) − 6.92820i − 0.512148i
\(184\) 0 0
\(185\) − 3.46410i − 0.254686i
\(186\) 0 0
\(187\) −18.0000 −1.31629
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 19.0000 1.36765 0.683825 0.729646i \(-0.260315\pi\)
0.683825 + 0.729646i \(0.260315\pi\)
\(194\) 0 0
\(195\) 12.0000 0.859338
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) 0 0
\(207\) − 6.92820i − 0.481543i
\(208\) 0 0
\(209\) 10.3923i 0.718851i
\(210\) 0 0
\(211\) 13.8564i 0.953914i 0.878927 + 0.476957i \(0.158260\pi\)
−0.878927 + 0.476957i \(0.841740\pi\)
\(212\) 0 0
\(213\) 6.92820i 0.474713i
\(214\) 0 0
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) − 6.92820i − 0.468165i
\(220\) 0 0
\(221\) −24.0000 −1.61441
\(222\) 0 0
\(223\) −1.00000 −0.0669650 −0.0334825 0.999439i \(-0.510660\pi\)
−0.0334825 + 0.999439i \(0.510660\pi\)
\(224\) 0 0
\(225\) 2.00000 0.133333
\(226\) 0 0
\(227\) 9.00000 0.597351 0.298675 0.954355i \(-0.403455\pi\)
0.298675 + 0.954355i \(0.403455\pi\)
\(228\) 0 0
\(229\) 3.46410i 0.228914i 0.993428 + 0.114457i \(0.0365129\pi\)
−0.993428 + 0.114457i \(0.963487\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 1.73205i − 0.112509i
\(238\) 0 0
\(239\) 13.8564i 0.896296i 0.893959 + 0.448148i \(0.147916\pi\)
−0.893959 + 0.448148i \(0.852084\pi\)
\(240\) 0 0
\(241\) − 5.19615i − 0.334714i −0.985896 0.167357i \(-0.946477\pi\)
0.985896 0.167357i \(-0.0535232\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 13.8564i 0.881662i
\(248\) 0 0
\(249\) 15.0000 0.950586
\(250\) 0 0
\(251\) −21.0000 −1.32551 −0.662754 0.748837i \(-0.730613\pi\)
−0.662754 + 0.748837i \(0.730613\pi\)
\(252\) 0 0
\(253\) 36.0000 2.26330
\(254\) 0 0
\(255\) 6.00000 0.375735
\(256\) 0 0
\(257\) 3.46410i 0.216085i 0.994146 + 0.108042i \(0.0344582\pi\)
−0.994146 + 0.108042i \(0.965542\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −9.00000 −0.557086
\(262\) 0 0
\(263\) 17.3205i 1.06803i 0.845476 + 0.534014i \(0.179317\pi\)
−0.845476 + 0.534014i \(0.820683\pi\)
\(264\) 0 0
\(265\) 15.5885i 0.957591i
\(266\) 0 0
\(267\) − 10.3923i − 0.635999i
\(268\) 0 0
\(269\) 1.73205i 0.105605i 0.998605 + 0.0528025i \(0.0168154\pi\)
−0.998605 + 0.0528025i \(0.983185\pi\)
\(270\) 0 0
\(271\) −17.0000 −1.03268 −0.516338 0.856385i \(-0.672705\pi\)
−0.516338 + 0.856385i \(0.672705\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 10.3923i 0.626680i
\(276\) 0 0
\(277\) 28.0000 1.68236 0.841178 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) 0 0
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 0 0
\(285\) − 3.46410i − 0.205196i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 5.00000 0.294118
\(290\) 0 0
\(291\) − 8.66025i − 0.507673i
\(292\) 0 0
\(293\) − 5.19615i − 0.303562i −0.988414 0.151781i \(-0.951499\pi\)
0.988414 0.151781i \(-0.0485009\pi\)
\(294\) 0 0
\(295\) − 5.19615i − 0.302532i
\(296\) 0 0
\(297\) − 5.19615i − 0.301511i
\(298\) 0 0
\(299\) 48.0000 2.77591
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 13.8564i 0.796030i
\(304\) 0 0
\(305\) −12.0000 −0.687118
\(306\) 0 0
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 0 0
\(313\) 19.0526i 1.07691i 0.842653 + 0.538457i \(0.180993\pi\)
−0.842653 + 0.538457i \(0.819007\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −27.0000 −1.51647 −0.758236 0.651981i \(-0.773938\pi\)
−0.758236 + 0.651981i \(0.773938\pi\)
\(318\) 0 0
\(319\) − 46.7654i − 2.61836i
\(320\) 0 0
\(321\) − 8.66025i − 0.483368i
\(322\) 0 0
\(323\) 6.92820i 0.385496i
\(324\) 0 0
\(325\) 13.8564i 0.768615i
\(326\) 0 0
\(327\) 4.00000 0.221201
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 23.0000 1.25289 0.626445 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 5.19615i 0.281387i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −12.0000 −0.646058
\(346\) 0 0
\(347\) 31.1769i 1.67366i 0.547459 + 0.836832i \(0.315595\pi\)
−0.547459 + 0.836832i \(0.684405\pi\)
\(348\) 0 0
\(349\) − 20.7846i − 1.11257i −0.830990 0.556287i \(-0.812225\pi\)
0.830990 0.556287i \(-0.187775\pi\)
\(350\) 0 0
\(351\) − 6.92820i − 0.369800i
\(352\) 0 0
\(353\) − 6.92820i − 0.368751i −0.982856 0.184376i \(-0.940974\pi\)
0.982856 0.184376i \(-0.0590263\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 27.7128i 1.46263i 0.682042 + 0.731313i \(0.261092\pi\)
−0.682042 + 0.731313i \(0.738908\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 16.0000 0.839782
\(364\) 0 0
\(365\) −12.0000 −0.628109
\(366\) 0 0
\(367\) 23.0000 1.20059 0.600295 0.799779i \(-0.295050\pi\)
0.600295 + 0.799779i \(0.295050\pi\)
\(368\) 0 0
\(369\) − 3.46410i − 0.180334i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) − 12.1244i − 0.626099i
\(376\) 0 0
\(377\) − 62.3538i − 3.21139i
\(378\) 0 0
\(379\) 17.3205i 0.889695i 0.895606 + 0.444847i \(0.146742\pi\)
−0.895606 + 0.444847i \(0.853258\pi\)
\(380\) 0 0
\(381\) 15.5885i 0.798621i
\(382\) 0 0
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 3.46410i − 0.176090i
\(388\) 0 0
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 0 0
\(393\) 3.00000 0.151330
\(394\) 0 0
\(395\) −3.00000 −0.150946
\(396\) 0 0
\(397\) − 20.7846i − 1.04315i −0.853206 0.521575i \(-0.825345\pi\)
0.853206 0.521575i \(-0.174655\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 36.0000 1.79775 0.898877 0.438201i \(-0.144384\pi\)
0.898877 + 0.438201i \(0.144384\pi\)
\(402\) 0 0
\(403\) 6.92820i 0.345118i
\(404\) 0 0
\(405\) 1.73205i 0.0860663i
\(406\) 0 0
\(407\) − 10.3923i − 0.515127i
\(408\) 0 0
\(409\) − 8.66025i − 0.428222i −0.976809 0.214111i \(-0.931315\pi\)
0.976809 0.214111i \(-0.0686854\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) − 25.9808i − 1.27535i
\(416\) 0 0
\(417\) −22.0000 −1.07734
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −8.00000 −0.389896 −0.194948 0.980814i \(-0.562454\pi\)
−0.194948 + 0.980814i \(0.562454\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.92820i 0.336067i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 36.0000 1.73810
\(430\) 0 0
\(431\) − 24.2487i − 1.16802i −0.811747 0.584010i \(-0.801483\pi\)
0.811747 0.584010i \(-0.198517\pi\)
\(432\) 0 0
\(433\) − 6.92820i − 0.332948i −0.986046 0.166474i \(-0.946762\pi\)
0.986046 0.166474i \(-0.0532382\pi\)
\(434\) 0 0
\(435\) 15.5885i 0.747409i
\(436\) 0 0
\(437\) − 13.8564i − 0.662842i
\(438\) 0 0
\(439\) −25.0000 −1.19318 −0.596592 0.802544i \(-0.703479\pi\)
−0.596592 + 0.802544i \(0.703479\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 22.5167i − 1.06980i −0.844916 0.534899i \(-0.820349\pi\)
0.844916 0.534899i \(-0.179651\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) 24.0000 1.13263 0.566315 0.824189i \(-0.308369\pi\)
0.566315 + 0.824189i \(0.308369\pi\)
\(450\) 0 0
\(451\) 18.0000 0.847587
\(452\) 0 0
\(453\) 1.73205i 0.0813788i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 17.0000 0.795226 0.397613 0.917553i \(-0.369839\pi\)
0.397613 + 0.917553i \(0.369839\pi\)
\(458\) 0 0
\(459\) − 3.46410i − 0.161690i
\(460\) 0 0
\(461\) − 20.7846i − 0.968036i −0.875058 0.484018i \(-0.839177\pi\)
0.875058 0.484018i \(-0.160823\pi\)
\(462\) 0 0
\(463\) − 10.3923i − 0.482971i −0.970404 0.241486i \(-0.922365\pi\)
0.970404 0.241486i \(-0.0776347\pi\)
\(464\) 0 0
\(465\) − 1.73205i − 0.0803219i
\(466\) 0 0
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) − 6.92820i − 0.319235i
\(472\) 0 0
\(473\) 18.0000 0.827641
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 9.00000 0.412082
\(478\) 0 0
\(479\) −42.0000 −1.91903 −0.959514 0.281659i \(-0.909115\pi\)
−0.959514 + 0.281659i \(0.909115\pi\)
\(480\) 0 0
\(481\) − 13.8564i − 0.631798i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −15.0000 −0.681115
\(486\) 0 0
\(487\) − 15.5885i − 0.706380i −0.935552 0.353190i \(-0.885097\pi\)
0.935552 0.353190i \(-0.114903\pi\)
\(488\) 0 0
\(489\) 20.7846i 0.939913i
\(490\) 0 0
\(491\) 19.0526i 0.859830i 0.902869 + 0.429915i \(0.141456\pi\)
−0.902869 + 0.429915i \(0.858544\pi\)
\(492\) 0 0
\(493\) − 31.1769i − 1.40414i
\(494\) 0 0
\(495\) −9.00000 −0.404520
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 17.3205i − 0.775372i −0.921791 0.387686i \(-0.873274\pi\)
0.921791 0.387686i \(-0.126726\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) 24.0000 1.06799
\(506\) 0 0
\(507\) 35.0000 1.55440
\(508\) 0 0
\(509\) 19.0526i 0.844490i 0.906482 + 0.422245i \(0.138758\pi\)
−0.906482 + 0.422245i \(0.861242\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −2.00000 −0.0883022
\(514\) 0 0
\(515\) 6.92820i 0.305293i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 13.8564i 0.608229i
\(520\) 0 0
\(521\) − 31.1769i − 1.36589i −0.730472 0.682943i \(-0.760700\pi\)
0.730472 0.682943i \(-0.239300\pi\)
\(522\) 0 0
\(523\) 40.0000 1.74908 0.874539 0.484955i \(-0.161164\pi\)
0.874539 + 0.484955i \(0.161164\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.46410i 0.150899i
\(528\) 0 0
\(529\) −25.0000 −1.08696
\(530\) 0 0
\(531\) −3.00000 −0.130189
\(532\) 0 0
\(533\) 24.0000 1.03956
\(534\) 0 0
\(535\) −15.0000 −0.648507
\(536\) 0 0
\(537\) − 17.3205i − 0.747435i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 0 0
\(543\) 3.46410i 0.148659i
\(544\) 0 0
\(545\) − 6.92820i − 0.296772i
\(546\) 0 0
\(547\) 24.2487i 1.03680i 0.855138 + 0.518400i \(0.173472\pi\)
−0.855138 + 0.518400i \(0.826528\pi\)
\(548\) 0 0
\(549\) 6.92820i 0.295689i
\(550\) 0 0
\(551\) −18.0000 −0.766826
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 3.46410i 0.147043i
\(556\) 0 0
\(557\) 3.00000 0.127114 0.0635570 0.997978i \(-0.479756\pi\)
0.0635570 + 0.997978i \(0.479756\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 18.0000 0.759961
\(562\) 0 0
\(563\) 9.00000 0.379305 0.189652 0.981851i \(-0.439264\pi\)
0.189652 + 0.981851i \(0.439264\pi\)
\(564\) 0 0
\(565\) − 10.3923i − 0.437208i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) 27.7128i 1.15975i 0.814707 + 0.579873i \(0.196898\pi\)
−0.814707 + 0.579873i \(0.803102\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 13.8564i − 0.577852i
\(576\) 0 0
\(577\) 25.9808i 1.08159i 0.841153 + 0.540797i \(0.181877\pi\)
−0.841153 + 0.540797i \(0.818123\pi\)
\(578\) 0 0
\(579\) −19.0000 −0.789613
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 46.7654i 1.93682i
\(584\) 0 0
\(585\) −12.0000 −0.496139
\(586\) 0 0
\(587\) 21.0000 0.866763 0.433381 0.901211i \(-0.357320\pi\)
0.433381 + 0.901211i \(0.357320\pi\)
\(588\) 0 0
\(589\) 2.00000 0.0824086
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) 20.7846i 0.853522i 0.904365 + 0.426761i \(0.140345\pi\)
−0.904365 + 0.426761i \(0.859655\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) 27.7128i 1.13231i 0.824297 + 0.566157i \(0.191571\pi\)
−0.824297 + 0.566157i \(0.808429\pi\)
\(600\) 0 0
\(601\) − 15.5885i − 0.635866i −0.948113 0.317933i \(-0.897011\pi\)
0.948113 0.317933i \(-0.102989\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 27.7128i − 1.12669i
\(606\) 0 0
\(607\) 23.0000 0.933541 0.466771 0.884378i \(-0.345417\pi\)
0.466771 + 0.884378i \(0.345417\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) 0 0
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) 24.0000 0.966204 0.483102 0.875564i \(-0.339510\pi\)
0.483102 + 0.875564i \(0.339510\pi\)
\(618\) 0 0
\(619\) 16.0000 0.643094 0.321547 0.946894i \(-0.395797\pi\)
0.321547 + 0.946894i \(0.395797\pi\)
\(620\) 0 0
\(621\) 6.92820i 0.278019i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) − 10.3923i − 0.415029i
\(628\) 0 0
\(629\) − 6.92820i − 0.276246i
\(630\) 0 0
\(631\) 25.9808i 1.03428i 0.855901 + 0.517139i \(0.173003\pi\)
−0.855901 + 0.517139i \(0.826997\pi\)
\(632\) 0 0
\(633\) − 13.8564i − 0.550743i
\(634\) 0 0
\(635\) 27.0000 1.07146
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) − 6.92820i − 0.274075i
\(640\) 0 0
\(641\) 24.0000 0.947943 0.473972 0.880540i \(-0.342820\pi\)
0.473972 + 0.880540i \(0.342820\pi\)
\(642\) 0 0
\(643\) 32.0000 1.26196 0.630978 0.775800i \(-0.282654\pi\)
0.630978 + 0.775800i \(0.282654\pi\)
\(644\) 0 0
\(645\) −6.00000 −0.236250
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) − 15.5885i − 0.611900i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −21.0000 −0.821794 −0.410897 0.911682i \(-0.634784\pi\)
−0.410897 + 0.911682i \(0.634784\pi\)
\(654\) 0 0
\(655\) − 5.19615i − 0.203030i
\(656\) 0 0
\(657\) 6.92820i 0.270295i
\(658\) 0 0
\(659\) 17.3205i 0.674711i 0.941377 + 0.337356i \(0.109532\pi\)
−0.941377 + 0.337356i \(0.890468\pi\)
\(660\) 0 0
\(661\) 38.1051i 1.48212i 0.671440 + 0.741059i \(0.265676\pi\)
−0.671440 + 0.741059i \(0.734324\pi\)
\(662\) 0 0
\(663\) 24.0000 0.932083
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 62.3538i 2.41435i
\(668\) 0 0
\(669\) 1.00000 0.0386622
\(670\) 0 0
\(671\) −36.0000 −1.38976
\(672\) 0 0
\(673\) −11.0000 −0.424019 −0.212009 0.977268i \(-0.568001\pi\)
−0.212009 + 0.977268i \(0.568001\pi\)
\(674\) 0 0
\(675\) −2.00000 −0.0769800
\(676\) 0 0
\(677\) − 25.9808i − 0.998522i −0.866452 0.499261i \(-0.833605\pi\)
0.866452 0.499261i \(-0.166395\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −9.00000 −0.344881
\(682\) 0 0
\(683\) − 8.66025i − 0.331375i −0.986178 0.165688i \(-0.947016\pi\)
0.986178 0.165688i \(-0.0529844\pi\)
\(684\) 0 0
\(685\) − 20.7846i − 0.794139i
\(686\) 0 0
\(687\) − 3.46410i − 0.132164i
\(688\) 0 0
\(689\) 62.3538i 2.37549i
\(690\) 0 0
\(691\) −14.0000 −0.532585 −0.266293 0.963892i \(-0.585799\pi\)
−0.266293 + 0.963892i \(0.585799\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 38.1051i 1.44541i
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 0 0
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) −27.0000 −1.01978 −0.509888 0.860241i \(-0.670313\pi\)
−0.509888 + 0.860241i \(0.670313\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) 0 0
\(711\) 1.73205i 0.0649570i
\(712\) 0 0
\(713\) − 6.92820i − 0.259463i
\(714\) 0 0
\(715\) − 62.3538i − 2.33190i
\(716\) 0 0
\(717\) − 13.8564i − 0.517477i
\(718\) 0 0
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 5.19615i 0.193247i
\(724\) 0 0
\(725\) −18.0000 −0.668503
\(726\) 0 0
\(727\) −17.0000 −0.630495 −0.315248 0.949009i \(-0.602088\pi\)
−0.315248 + 0.949009i \(0.602088\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) 0 0
\(733\) 31.1769i 1.15155i 0.817610 + 0.575773i \(0.195299\pi\)
−0.817610 + 0.575773i \(0.804701\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) − 31.1769i − 1.14686i −0.819254 0.573431i \(-0.805612\pi\)
0.819254 0.573431i \(-0.194388\pi\)
\(740\) 0 0
\(741\) − 13.8564i − 0.509028i
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) − 10.3923i − 0.380745i
\(746\) 0 0
\(747\) −15.0000 −0.548821
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) − 12.1244i − 0.442424i −0.975226 0.221212i \(-0.928999\pi\)
0.975226 0.221212i \(-0.0710013\pi\)
\(752\) 0 0
\(753\) 21.0000 0.765283
\(754\) 0 0
\(755\) 3.00000 0.109181
\(756\) 0 0
\(757\) −16.0000 −0.581530 −0.290765 0.956795i \(-0.593910\pi\)
−0.290765 + 0.956795i \(0.593910\pi\)
\(758\) 0 0
\(759\) −36.0000 −1.30672
\(760\) 0 0
\(761\) − 13.8564i − 0.502294i −0.967949 0.251147i \(-0.919192\pi\)
0.967949 0.251147i \(-0.0808078\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −6.00000 −0.216930
\(766\) 0 0
\(767\) − 20.7846i − 0.750489i
\(768\) 0 0
\(769\) 5.19615i 0.187378i 0.995602 + 0.0936890i \(0.0298659\pi\)
−0.995602 + 0.0936890i \(0.970134\pi\)
\(770\) 0 0
\(771\) − 3.46410i − 0.124757i
\(772\) 0 0
\(773\) − 27.7128i − 0.996761i −0.866959 0.498380i \(-0.833928\pi\)
0.866959 0.498380i \(-0.166072\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 6.92820i − 0.248229i
\(780\) 0 0
\(781\) 36.0000 1.28818
\(782\) 0 0
\(783\) 9.00000 0.321634
\(784\) 0 0
\(785\) −12.0000 −0.428298
\(786\) 0 0
\(787\) −32.0000 −1.14068 −0.570338 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) 0 0
\(789\) − 17.3205i − 0.616626i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −48.0000 −1.70453
\(794\) 0 0
\(795\) − 15.5885i − 0.552866i
\(796\) 0 0
\(797\) 8.66025i 0.306762i 0.988167 + 0.153381i \(0.0490162\pi\)
−0.988167 + 0.153381i \(0.950984\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 10.3923i 0.367194i
\(802\) 0 0
\(803\) −36.0000 −1.27041
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 1.73205i − 0.0609711i
\(808\) 0 0
\(809\) −24.0000 −0.843795 −0.421898 0.906644i \(-0.638636\pi\)
−0.421898 + 0.906644i \(0.638636\pi\)
\(810\) 0 0
\(811\) −8.00000 −0.280918 −0.140459 0.990086i \(-0.544858\pi\)
−0.140459 + 0.990086i \(0.544858\pi\)
\(812\) 0 0
\(813\) 17.0000 0.596216
\(814\) 0 0
\(815\) 36.0000 1.26102
\(816\) 0 0
\(817\) − 6.92820i − 0.242387i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 45.0000 1.57051 0.785255 0.619172i \(-0.212532\pi\)
0.785255 + 0.619172i \(0.212532\pi\)
\(822\) 0 0
\(823\) 3.46410i 0.120751i 0.998176 + 0.0603755i \(0.0192298\pi\)
−0.998176 + 0.0603755i \(0.980770\pi\)
\(824\) 0 0
\(825\) − 10.3923i − 0.361814i
\(826\) 0 0
\(827\) 36.3731i 1.26482i 0.774636 + 0.632408i \(0.217933\pi\)
−0.774636 + 0.632408i \(0.782067\pi\)
\(828\) 0 0
\(829\) − 10.3923i − 0.360940i −0.983581 0.180470i \(-0.942238\pi\)
0.983581 0.180470i \(-0.0577618\pi\)
\(830\) 0 0
\(831\) −28.0000 −0.971309
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) − 20.7846i − 0.719281i
\(836\) 0 0
\(837\) −1.00000 −0.0345651
\(838\) 0 0
\(839\) −18.0000 −0.621429 −0.310715 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 60.6218i − 2.08545i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −14.0000 −0.480479
\(850\) 0 0
\(851\) 13.8564i 0.474991i
\(852\) 0 0
\(853\) 6.92820i 0.237217i 0.992941 + 0.118609i \(0.0378434\pi\)
−0.992941 + 0.118609i \(0.962157\pi\)
\(854\) 0 0
\(855\) 3.46410i 0.118470i
\(856\) 0 0
\(857\) 48.4974i 1.65664i 0.560255 + 0.828320i \(0.310703\pi\)
−0.560255 + 0.828320i \(0.689297\pi\)
\(858\) 0 0
\(859\) −10.0000 −0.341196 −0.170598 0.985341i \(-0.554570\pi\)
−0.170598 + 0.985341i \(0.554570\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 31.1769i − 1.06127i −0.847599 0.530637i \(-0.821953\pi\)
0.847599 0.530637i \(-0.178047\pi\)
\(864\) 0 0
\(865\) 24.0000 0.816024
\(866\) 0 0
\(867\) −5.00000 −0.169809
\(868\) 0 0
\(869\) −9.00000 −0.305304
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 8.66025i 0.293105i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 8.00000 0.270141 0.135070 0.990836i \(-0.456874\pi\)
0.135070 + 0.990836i \(0.456874\pi\)
\(878\) 0 0
\(879\) 5.19615i 0.175262i
\(880\) 0 0
\(881\) 55.4256i 1.86734i 0.358139 + 0.933668i \(0.383411\pi\)
−0.358139 + 0.933668i \(0.616589\pi\)
\(882\) 0 0
\(883\) 31.1769i 1.04919i 0.851353 + 0.524593i \(0.175783\pi\)
−0.851353 + 0.524593i \(0.824217\pi\)
\(884\) 0 0
\(885\) 5.19615i 0.174667i
\(886\) 0 0
\(887\) 12.0000 0.402921 0.201460 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 5.19615i 0.174078i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −30.0000 −1.00279
\(896\) 0 0
\(897\) −48.0000 −1.60267
\(898\) 0 0
\(899\) −9.00000 −0.300167
\(900\) 0 0
\(901\) 31.1769i 1.03865i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 6.00000 0.199447
\(906\) 0 0
\(907\) 55.4256i 1.84038i 0.391475 + 0.920189i \(0.371965\pi\)
−0.391475 + 0.920189i \(0.628035\pi\)
\(908\) 0 0
\(909\) − 13.8564i − 0.459588i
\(910\) 0 0
\(911\) 20.7846i 0.688625i 0.938855 + 0.344312i \(0.111888\pi\)
−0.938855 + 0.344312i \(0.888112\pi\)
\(912\) 0 0
\(913\) − 77.9423i − 2.57951i
\(914\) 0 0
\(915\) 12.0000 0.396708
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) − 3.46410i − 0.114270i −0.998366 0.0571351i \(-0.981803\pi\)
0.998366 0.0571351i \(-0.0181966\pi\)
\(920\) 0 0
\(921\) 16.0000 0.527218
\(922\) 0 0
\(923\) 48.0000 1.57994
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 0 0
\(927\) 4.00000 0.131377
\(928\) 0 0
\(929\) 58.8897i 1.93211i 0.258338 + 0.966055i \(0.416825\pi\)
−0.258338 + 0.966055i \(0.583175\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 6.00000 0.196431
\(934\) 0 0
\(935\) − 31.1769i − 1.01959i
\(936\) 0 0
\(937\) 50.2295i 1.64093i 0.571700 + 0.820463i \(0.306284\pi\)
−0.571700 + 0.820463i \(0.693716\pi\)
\(938\) 0 0
\(939\) − 19.0526i − 0.621757i
\(940\) 0 0
\(941\) − 15.5885i − 0.508169i −0.967182 0.254085i \(-0.918226\pi\)
0.967182 0.254085i \(-0.0817742\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 24.2487i − 0.787977i −0.919115 0.393989i \(-0.871095\pi\)
0.919115 0.393989i \(-0.128905\pi\)
\(948\) 0 0
\(949\) −48.0000 −1.55815
\(950\) 0 0
\(951\) 27.0000 0.875535
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 46.7654i 1.51171i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) 8.66025i 0.279073i
\(964\) 0 0
\(965\) 32.9090i 1.05938i
\(966\) 0 0
\(967\) − 46.7654i − 1.50387i −0.659236 0.751936i \(-0.729120\pi\)
0.659236 0.751936i \(-0.270880\pi\)
\(968\) 0 0
\(969\) − 6.92820i − 0.222566i
\(970\) 0 0
\(971\) 21.0000 0.673922 0.336961 0.941519i \(-0.390601\pi\)
0.336961 + 0.941519i \(0.390601\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) − 13.8564i − 0.443760i
\(976\) 0 0
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) −54.0000 −1.72585
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 0 0
\(983\) 54.0000 1.72233 0.861166 0.508323i \(-0.169735\pi\)
0.861166 + 0.508323i \(0.169735\pi\)
\(984\) 0 0
\(985\) 10.3923i 0.331126i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) 46.7654i 1.48555i 0.669541 + 0.742775i \(0.266491\pi\)
−0.669541 + 0.742775i \(0.733509\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 27.7128i − 0.878555i
\(996\) 0 0
\(997\) 27.7128i 0.877674i 0.898567 + 0.438837i \(0.144609\pi\)
−0.898567 + 0.438837i \(0.855391\pi\)
\(998\) 0 0
\(999\) 2.00000 0.0632772
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 2352.2.b.b.1567.2 2
3.2 odd 2 7056.2.b.j.1567.1 2
4.3 odd 2 2352.2.b.f.1567.2 2
7.2 even 3 2352.2.bl.k.31.1 2
7.3 odd 6 2352.2.bl.e.607.1 2
7.4 even 3 336.2.bl.f.271.1 yes 2
7.5 odd 6 336.2.bl.b.31.1 2
7.6 odd 2 2352.2.b.f.1567.1 2
12.11 even 2 7056.2.b.f.1567.1 2
21.5 even 6 1008.2.cs.k.703.1 2
21.11 odd 6 1008.2.cs.l.271.1 2
21.20 even 2 7056.2.b.f.1567.2 2
28.3 even 6 2352.2.bl.k.607.1 2
28.11 odd 6 336.2.bl.b.271.1 yes 2
28.19 even 6 336.2.bl.f.31.1 yes 2
28.23 odd 6 2352.2.bl.e.31.1 2
28.27 even 2 inner 2352.2.b.b.1567.1 2
56.5 odd 6 1344.2.bl.g.703.1 2
56.11 odd 6 1344.2.bl.g.1279.1 2
56.19 even 6 1344.2.bl.c.703.1 2
56.53 even 6 1344.2.bl.c.1279.1 2
84.11 even 6 1008.2.cs.k.271.1 2
84.47 odd 6 1008.2.cs.l.703.1 2
84.83 odd 2 7056.2.b.j.1567.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.2.bl.b.31.1 2 7.5 odd 6
336.2.bl.b.271.1 yes 2 28.11 odd 6
336.2.bl.f.31.1 yes 2 28.19 even 6
336.2.bl.f.271.1 yes 2 7.4 even 3
1008.2.cs.k.271.1 2 84.11 even 6
1008.2.cs.k.703.1 2 21.5 even 6
1008.2.cs.l.271.1 2 21.11 odd 6
1008.2.cs.l.703.1 2 84.47 odd 6
1344.2.bl.c.703.1 2 56.19 even 6
1344.2.bl.c.1279.1 2 56.53 even 6
1344.2.bl.g.703.1 2 56.5 odd 6
1344.2.bl.g.1279.1 2 56.11 odd 6
2352.2.b.b.1567.1 2 28.27 even 2 inner
2352.2.b.b.1567.2 2 1.1 even 1 trivial
2352.2.b.f.1567.1 2 7.6 odd 2
2352.2.b.f.1567.2 2 4.3 odd 2
2352.2.bl.e.31.1 2 28.23 odd 6
2352.2.bl.e.607.1 2 7.3 odd 6
2352.2.bl.k.31.1 2 7.2 even 3
2352.2.bl.k.607.1 2 28.3 even 6
7056.2.b.f.1567.1 2 12.11 even 2
7056.2.b.f.1567.2 2 21.20 even 2
7056.2.b.j.1567.1 2 3.2 odd 2
7056.2.b.j.1567.2 2 84.83 odd 2