Properties

Label 825.2.c.b.199.1
Level $825$
Weight $2$
Character 825.199
Analytic conductor $6.588$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.2.c.b.199.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +2.00000 q^{4} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +2.00000 q^{4} -1.00000i q^{7} -1.00000 q^{9} -1.00000 q^{11} -2.00000i q^{12} +1.00000i q^{13} +4.00000 q^{16} -6.00000i q^{17} +7.00000 q^{19} -1.00000 q^{21} -6.00000i q^{23} +1.00000i q^{27} -2.00000i q^{28} +6.00000 q^{29} -7.00000 q^{31} +1.00000i q^{33} -2.00000 q^{36} +2.00000i q^{37} +1.00000 q^{39} -6.00000 q^{41} +1.00000i q^{43} -2.00000 q^{44} -4.00000i q^{48} +6.00000 q^{49} -6.00000 q^{51} +2.00000i q^{52} +6.00000i q^{53} -7.00000i q^{57} +5.00000 q^{61} +1.00000i q^{63} +8.00000 q^{64} +5.00000i q^{67} -12.0000i q^{68} -6.00000 q^{69} -12.0000 q^{71} -14.0000i q^{73} +14.0000 q^{76} +1.00000i q^{77} +4.00000 q^{79} +1.00000 q^{81} +6.00000i q^{83} -2.00000 q^{84} -6.00000i q^{87} -6.00000 q^{89} +1.00000 q^{91} -12.0000i q^{92} +7.00000i q^{93} +17.0000i q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{4} - 2 q^{9} - 2 q^{11} + 8 q^{16} + 14 q^{19} - 2 q^{21} + 12 q^{29} - 14 q^{31} - 4 q^{36} + 2 q^{39} - 12 q^{41} - 4 q^{44} + 12 q^{49} - 12 q^{51} + 10 q^{61} + 16 q^{64} - 12 q^{69} - 24 q^{71} + 28 q^{76} + 8 q^{79} + 2 q^{81} - 4 q^{84} - 12 q^{89} + 2 q^{91} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i −0.981981 0.188982i \(-0.939481\pi\)
0.981981 0.188982i \(-0.0605189\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) − 2.00000i − 0.577350i
\(13\) 1.00000i 0.277350i 0.990338 + 0.138675i \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 0 0
\(19\) 7.00000 1.60591 0.802955 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) − 6.00000i − 1.25109i −0.780189 0.625543i \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) − 2.00000i − 0.377964i
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) 1.00000i 0.174078i
\(34\) 0 0
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 1.00000i 0.152499i 0.997089 + 0.0762493i \(0.0242945\pi\)
−0.997089 + 0.0762493i \(0.975706\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) − 4.00000i − 0.577350i
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 2.00000i 0.277350i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 7.00000i − 0.927173i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 5.00000i 0.610847i 0.952217 + 0.305424i \(0.0987981\pi\)
−0.952217 + 0.305424i \(0.901202\pi\)
\(68\) − 12.0000i − 1.45521i
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) − 14.0000i − 1.63858i −0.573382 0.819288i \(-0.694369\pi\)
0.573382 0.819288i \(-0.305631\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 14.0000 1.60591
\(77\) 1.00000i 0.113961i
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 0 0
\(87\) − 6.00000i − 0.643268i
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) − 12.0000i − 1.25109i
\(93\) 7.00000i 0.725866i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 17.0000i 1.72609i 0.505128 + 0.863044i \(0.331445\pi\)
−0.505128 + 0.863044i \(0.668555\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) − 8.00000i − 0.788263i −0.919054 0.394132i \(-0.871045\pi\)
0.919054 0.394132i \(-0.128955\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 6.00000i − 0.580042i −0.957020 0.290021i \(-0.906338\pi\)
0.957020 0.290021i \(-0.0936623\pi\)
\(108\) 2.00000i 0.192450i
\(109\) −11.0000 −1.05361 −0.526804 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) − 4.00000i − 0.377964i
\(113\) 18.0000i 1.69330i 0.532152 + 0.846649i \(0.321383\pi\)
−0.532152 + 0.846649i \(0.678617\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 12.0000 1.11417
\(117\) − 1.00000i − 0.0924500i
\(118\) 0 0
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 6.00000i 0.541002i
\(124\) −14.0000 −1.25724
\(125\) 0 0
\(126\) 0 0
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) 0 0
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 2.00000i 0.174078i
\(133\) − 7.00000i − 0.606977i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 1.00000i − 0.0836242i
\(144\) −4.00000 −0.333333
\(145\) 0 0
\(146\) 0 0
\(147\) − 6.00000i − 0.494872i
\(148\) 4.00000i 0.328798i
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −7.00000 −0.569652 −0.284826 0.958579i \(-0.591936\pi\)
−0.284826 + 0.958579i \(0.591936\pi\)
\(152\) 0 0
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 23.0000i 1.83560i 0.397043 + 0.917800i \(0.370036\pi\)
−0.397043 + 0.917800i \(0.629964\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) 0 0
\(163\) 13.0000i 1.01824i 0.860696 + 0.509119i \(0.170029\pi\)
−0.860696 + 0.509119i \(0.829971\pi\)
\(164\) −12.0000 −0.937043
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −7.00000 −0.535303
\(172\) 2.00000i 0.152499i
\(173\) 24.0000i 1.82469i 0.409426 + 0.912343i \(0.365729\pi\)
−0.409426 + 0.912343i \(0.634271\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) 0 0
\(179\) −18.0000 −1.34538 −0.672692 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 0 0
\(183\) − 5.00000i − 0.369611i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 6.00000i 0.438763i
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) − 8.00000i − 0.577350i
\(193\) 7.00000i 0.503871i 0.967744 + 0.251936i \(0.0810671\pi\)
−0.967744 + 0.251936i \(0.918933\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 12.0000 0.857143
\(197\) 24.0000i 1.70993i 0.518686 + 0.854965i \(0.326421\pi\)
−0.518686 + 0.854965i \(0.673579\pi\)
\(198\) 0 0
\(199\) −11.0000 −0.779769 −0.389885 0.920864i \(-0.627485\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) 5.00000 0.352673
\(202\) 0 0
\(203\) − 6.00000i − 0.421117i
\(204\) −12.0000 −0.840168
\(205\) 0 0
\(206\) 0 0
\(207\) 6.00000i 0.417029i
\(208\) 4.00000i 0.277350i
\(209\) −7.00000 −0.484200
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) 12.0000i 0.824163i
\(213\) 12.0000i 0.822226i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 7.00000i 0.475191i
\(218\) 0 0
\(219\) −14.0000 −0.946032
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) − 5.00000i − 0.334825i −0.985887 0.167412i \(-0.946459\pi\)
0.985887 0.167412i \(-0.0535411\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 24.0000i − 1.59294i −0.604681 0.796468i \(-0.706699\pi\)
0.604681 0.796468i \(-0.293301\pi\)
\(228\) − 14.0000i − 0.927173i
\(229\) −17.0000 −1.12339 −0.561696 0.827344i \(-0.689851\pi\)
−0.561696 + 0.827344i \(0.689851\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) 0 0
\(233\) − 12.0000i − 0.786146i −0.919507 0.393073i \(-0.871412\pi\)
0.919507 0.393073i \(-0.128588\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 4.00000i − 0.259828i
\(238\) 0 0
\(239\) 30.0000 1.94054 0.970269 0.242028i \(-0.0778125\pi\)
0.970269 + 0.242028i \(0.0778125\pi\)
\(240\) 0 0
\(241\) −13.0000 −0.837404 −0.418702 0.908124i \(-0.637515\pi\)
−0.418702 + 0.908124i \(0.637515\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) 7.00000i 0.445399i
\(248\) 0 0
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 2.00000i 0.125988i
\(253\) 6.00000i 0.377217i
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 24.0000i 1.49708i 0.663090 + 0.748539i \(0.269245\pi\)
−0.663090 + 0.748539i \(0.730755\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 18.0000i 1.10993i 0.831875 + 0.554964i \(0.187268\pi\)
−0.831875 + 0.554964i \(0.812732\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6.00000i 0.367194i
\(268\) 10.0000i 0.610847i
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) − 24.0000i − 1.45521i
\(273\) − 1.00000i − 0.0605228i
\(274\) 0 0
\(275\) 0 0
\(276\) −12.0000 −0.722315
\(277\) − 13.0000i − 0.781094i −0.920583 0.390547i \(-0.872286\pi\)
0.920583 0.390547i \(-0.127714\pi\)
\(278\) 0 0
\(279\) 7.00000 0.419079
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 0 0
\(283\) − 29.0000i − 1.72387i −0.507018 0.861936i \(-0.669252\pi\)
0.507018 0.861936i \(-0.330748\pi\)
\(284\) −24.0000 −1.42414
\(285\) 0 0
\(286\) 0 0
\(287\) 6.00000i 0.354169i
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 17.0000 0.996558
\(292\) − 28.0000i − 1.63858i
\(293\) − 18.0000i − 1.05157i −0.850617 0.525786i \(-0.823771\pi\)
0.850617 0.525786i \(-0.176229\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 1.00000i − 0.0580259i
\(298\) 0 0
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) 0 0
\(303\) 0 0
\(304\) 28.0000 1.60591
\(305\) 0 0
\(306\) 0 0
\(307\) 5.00000i 0.285365i 0.989769 + 0.142683i \(0.0455728\pi\)
−0.989769 + 0.142683i \(0.954427\pi\)
\(308\) 2.00000i 0.113961i
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) 13.0000i 0.734803i 0.930062 + 0.367402i \(0.119753\pi\)
−0.930062 + 0.367402i \(0.880247\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) − 30.0000i − 1.68497i −0.538721 0.842484i \(-0.681092\pi\)
0.538721 0.842484i \(-0.318908\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) − 42.0000i − 2.33694i
\(324\) 2.00000 0.111111
\(325\) 0 0
\(326\) 0 0
\(327\) 11.0000i 0.608301i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 12.0000i 0.658586i
\(333\) − 2.00000i − 0.109599i
\(334\) 0 0
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) − 13.0000i − 0.708155i −0.935216 0.354078i \(-0.884795\pi\)
0.935216 0.354078i \(-0.115205\pi\)
\(338\) 0 0
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) 7.00000 0.379071
\(342\) 0 0
\(343\) − 13.0000i − 0.701934i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 24.0000i 1.28839i 0.764862 + 0.644194i \(0.222807\pi\)
−0.764862 + 0.644194i \(0.777193\pi\)
\(348\) − 12.0000i − 0.643268i
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −12.0000 −0.635999
\(357\) 6.00000i 0.317554i
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) − 1.00000i − 0.0524864i
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) − 37.0000i − 1.93138i −0.259690 0.965692i \(-0.583620\pi\)
0.259690 0.965692i \(-0.416380\pi\)
\(368\) − 24.0000i − 1.25109i
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) 14.0000i 0.725866i
\(373\) 31.0000i 1.60512i 0.596572 + 0.802560i \(0.296529\pi\)
−0.596572 + 0.802560i \(0.703471\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.00000i 0.309016i
\(378\) 0 0
\(379\) 19.0000 0.975964 0.487982 0.872854i \(-0.337733\pi\)
0.487982 + 0.872854i \(0.337733\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 0 0
\(383\) 6.00000i 0.306586i 0.988181 + 0.153293i \(0.0489878\pi\)
−0.988181 + 0.153293i \(0.951012\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 1.00000i − 0.0508329i
\(388\) 34.0000i 1.72609i
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) −36.0000 −1.82060
\(392\) 0 0
\(393\) − 18.0000i − 0.907980i
\(394\) 0 0
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) − 25.0000i − 1.25471i −0.778732 0.627357i \(-0.784137\pi\)
0.778732 0.627357i \(-0.215863\pi\)
\(398\) 0 0
\(399\) −7.00000 −0.350438
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) − 7.00000i − 0.348695i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 2.00000i − 0.0991363i
\(408\) 0 0
\(409\) 13.0000 0.642809 0.321404 0.946942i \(-0.395845\pi\)
0.321404 + 0.946942i \(0.395845\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 16.0000i − 0.788263i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 4.00000i − 0.195881i
\(418\) 0 0
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 5.00000i − 0.241967i
\(428\) − 12.0000i − 0.580042i
\(429\) −1.00000 −0.0482805
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 4.00000i 0.192450i
\(433\) − 11.0000i − 0.528626i −0.964437 0.264313i \(-0.914855\pi\)
0.964437 0.264313i \(-0.0851452\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −22.0000 −1.05361
\(437\) − 42.0000i − 2.00913i
\(438\) 0 0
\(439\) −35.0000 −1.67046 −0.835229 0.549902i \(-0.814665\pi\)
−0.835229 + 0.549902i \(0.814665\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) 30.0000i 1.42534i 0.701498 + 0.712672i \(0.252515\pi\)
−0.701498 + 0.712672i \(0.747485\pi\)
\(444\) 4.00000 0.189832
\(445\) 0 0
\(446\) 0 0
\(447\) − 6.00000i − 0.283790i
\(448\) − 8.00000i − 0.377964i
\(449\) −24.0000 −1.13263 −0.566315 0.824189i \(-0.691631\pi\)
−0.566315 + 0.824189i \(0.691631\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) 36.0000i 1.69330i
\(453\) 7.00000i 0.328889i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 26.0000i 1.21623i 0.793849 + 0.608114i \(0.208074\pi\)
−0.793849 + 0.608114i \(0.791926\pi\)
\(458\) 0 0
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) 24.0000 1.11417
\(465\) 0 0
\(466\) 0 0
\(467\) − 18.0000i − 0.832941i −0.909149 0.416470i \(-0.863267\pi\)
0.909149 0.416470i \(-0.136733\pi\)
\(468\) − 2.00000i − 0.0924500i
\(469\) 5.00000 0.230879
\(470\) 0 0
\(471\) 23.0000 1.05978
\(472\) 0 0
\(473\) − 1.00000i − 0.0459800i
\(474\) 0 0
\(475\) 0 0
\(476\) −12.0000 −0.550019
\(477\) − 6.00000i − 0.274721i
\(478\) 0 0
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) 6.00000i 0.273009i
\(484\) 2.00000 0.0909091
\(485\) 0 0
\(486\) 0 0
\(487\) − 13.0000i − 0.589086i −0.955638 0.294543i \(-0.904833\pi\)
0.955638 0.294543i \(-0.0951675\pi\)
\(488\) 0 0
\(489\) 13.0000 0.587880
\(490\) 0 0
\(491\) 18.0000 0.812329 0.406164 0.913800i \(-0.366866\pi\)
0.406164 + 0.913800i \(0.366866\pi\)
\(492\) 12.0000i 0.541002i
\(493\) − 36.0000i − 1.62136i
\(494\) 0 0
\(495\) 0 0
\(496\) −28.0000 −1.25724
\(497\) 12.0000i 0.538274i
\(498\) 0 0
\(499\) 13.0000 0.581960 0.290980 0.956729i \(-0.406019\pi\)
0.290980 + 0.956729i \(0.406019\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 6.00000i − 0.267527i −0.991013 0.133763i \(-0.957294\pi\)
0.991013 0.133763i \(-0.0427062\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 12.0000i − 0.532939i
\(508\) 16.0000i 0.709885i
\(509\) −36.0000 −1.59567 −0.797836 0.602875i \(-0.794022\pi\)
−0.797836 + 0.602875i \(0.794022\pi\)
\(510\) 0 0
\(511\) −14.0000 −0.619324
\(512\) 0 0
\(513\) 7.00000i 0.309058i
\(514\) 0 0
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) 0 0
\(518\) 0 0
\(519\) 24.0000 1.05348
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) − 17.0000i − 0.743358i −0.928361 0.371679i \(-0.878782\pi\)
0.928361 0.371679i \(-0.121218\pi\)
\(524\) 36.0000 1.57267
\(525\) 0 0
\(526\) 0 0
\(527\) 42.0000i 1.82955i
\(528\) 4.00000i 0.174078i
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) − 14.0000i − 0.606977i
\(533\) − 6.00000i − 0.259889i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 18.0000i 0.776757i
\(538\) 0 0
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) −37.0000 −1.59075 −0.795377 0.606115i \(-0.792727\pi\)
−0.795377 + 0.606115i \(0.792727\pi\)
\(542\) 0 0
\(543\) − 5.00000i − 0.214571i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 28.0000i − 1.19719i −0.801050 0.598597i \(-0.795725\pi\)
0.801050 0.598597i \(-0.204275\pi\)
\(548\) 0 0
\(549\) −5.00000 −0.213395
\(550\) 0 0
\(551\) 42.0000 1.78926
\(552\) 0 0
\(553\) − 4.00000i − 0.170097i
\(554\) 0 0
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) − 12.0000i − 0.508456i −0.967144 0.254228i \(-0.918179\pi\)
0.967144 0.254228i \(-0.0818214\pi\)
\(558\) 0 0
\(559\) −1.00000 −0.0422955
\(560\) 0 0
\(561\) 6.00000 0.253320
\(562\) 0 0
\(563\) − 30.0000i − 1.26435i −0.774826 0.632175i \(-0.782163\pi\)
0.774826 0.632175i \(-0.217837\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 1.00000i − 0.0419961i
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) −31.0000 −1.29731 −0.648655 0.761083i \(-0.724668\pi\)
−0.648655 + 0.761083i \(0.724668\pi\)
\(572\) − 2.00000i − 0.0836242i
\(573\) 12.0000i 0.501307i
\(574\) 0 0
\(575\) 0 0
\(576\) −8.00000 −0.333333
\(577\) − 1.00000i − 0.0416305i −0.999783 0.0208153i \(-0.993374\pi\)
0.999783 0.0208153i \(-0.00662619\pi\)
\(578\) 0 0
\(579\) 7.00000 0.290910
\(580\) 0 0
\(581\) 6.00000 0.248922
\(582\) 0 0
\(583\) − 6.00000i − 0.248495i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 18.0000i − 0.742940i −0.928445 0.371470i \(-0.878854\pi\)
0.928445 0.371470i \(-0.121146\pi\)
\(588\) − 12.0000i − 0.494872i
\(589\) −49.0000 −2.01901
\(590\) 0 0
\(591\) 24.0000 0.987228
\(592\) 8.00000i 0.328798i
\(593\) − 6.00000i − 0.246390i −0.992382 0.123195i \(-0.960686\pi\)
0.992382 0.123195i \(-0.0393141\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 12.0000 0.491539
\(597\) 11.0000i 0.450200i
\(598\) 0 0
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) 17.0000 0.693444 0.346722 0.937968i \(-0.387295\pi\)
0.346722 + 0.937968i \(0.387295\pi\)
\(602\) 0 0
\(603\) − 5.00000i − 0.203616i
\(604\) −14.0000 −0.569652
\(605\) 0 0
\(606\) 0 0
\(607\) − 40.0000i − 1.62355i −0.583970 0.811775i \(-0.698502\pi\)
0.583970 0.811775i \(-0.301498\pi\)
\(608\) 0 0
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) 0 0
\(612\) 12.0000i 0.485071i
\(613\) − 14.0000i − 0.565455i −0.959200 0.282727i \(-0.908761\pi\)
0.959200 0.282727i \(-0.0912392\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12.0000i 0.483102i 0.970388 + 0.241551i \(0.0776561\pi\)
−0.970388 + 0.241551i \(0.922344\pi\)
\(618\) 0 0
\(619\) −47.0000 −1.88909 −0.944545 0.328383i \(-0.893496\pi\)
−0.944545 + 0.328383i \(0.893496\pi\)
\(620\) 0 0
\(621\) 6.00000 0.240772
\(622\) 0 0
\(623\) 6.00000i 0.240385i
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) 0 0
\(627\) 7.00000i 0.279553i
\(628\) 46.0000i 1.83560i
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −19.0000 −0.756378 −0.378189 0.925728i \(-0.623453\pi\)
−0.378189 + 0.925728i \(0.623453\pi\)
\(632\) 0 0
\(633\) − 5.00000i − 0.198732i
\(634\) 0 0
\(635\) 0 0
\(636\) 12.0000 0.475831
\(637\) 6.00000i 0.237729i
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) 42.0000 1.65890 0.829450 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) 0 0
\(643\) − 32.0000i − 1.26196i −0.775800 0.630978i \(-0.782654\pi\)
0.775800 0.630978i \(-0.217346\pi\)
\(644\) −12.0000 −0.472866
\(645\) 0 0
\(646\) 0 0
\(647\) − 24.0000i − 0.943537i −0.881722 0.471769i \(-0.843616\pi\)
0.881722 0.471769i \(-0.156384\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 7.00000 0.274352
\(652\) 26.0000i 1.01824i
\(653\) 24.0000i 0.939193i 0.882881 + 0.469596i \(0.155601\pi\)
−0.882881 + 0.469596i \(0.844399\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −24.0000 −0.937043
\(657\) 14.0000i 0.546192i
\(658\) 0 0
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 0 0
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) 0 0
\(663\) − 6.00000i − 0.233021i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 36.0000i − 1.39393i
\(668\) 0 0
\(669\) −5.00000 −0.193311
\(670\) 0 0
\(671\) −5.00000 −0.193023
\(672\) 0 0
\(673\) 22.0000i 0.848038i 0.905653 + 0.424019i \(0.139381\pi\)
−0.905653 + 0.424019i \(0.860619\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 24.0000 0.923077
\(677\) 30.0000i 1.15299i 0.817099 + 0.576497i \(0.195581\pi\)
−0.817099 + 0.576497i \(0.804419\pi\)
\(678\) 0 0
\(679\) 17.0000 0.652400
\(680\) 0 0
\(681\) −24.0000 −0.919682
\(682\) 0 0
\(683\) − 30.0000i − 1.14792i −0.818884 0.573959i \(-0.805407\pi\)
0.818884 0.573959i \(-0.194593\pi\)
\(684\) −14.0000 −0.535303
\(685\) 0 0
\(686\) 0 0
\(687\) 17.0000i 0.648590i
\(688\) 4.00000i 0.152499i
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) 48.0000i 1.82469i
\(693\) − 1.00000i − 0.0379869i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 36.0000i 1.36360i
\(698\) 0 0
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) −36.0000 −1.35970 −0.679851 0.733351i \(-0.737955\pi\)
−0.679851 + 0.733351i \(0.737955\pi\)
\(702\) 0 0
\(703\) 14.0000i 0.528020i
\(704\) −8.00000 −0.301511
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −17.0000 −0.638448 −0.319224 0.947679i \(-0.603422\pi\)
−0.319224 + 0.947679i \(0.603422\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 42.0000i 1.57291i
\(714\) 0 0
\(715\) 0 0
\(716\) −36.0000 −1.34538
\(717\) − 30.0000i − 1.12037i
\(718\) 0 0
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 13.0000i 0.483475i
\(724\) 10.0000 0.371647
\(725\) 0 0
\(726\) 0 0
\(727\) 5.00000i 0.185440i 0.995692 + 0.0927199i \(0.0295561\pi\)
−0.995692 + 0.0927199i \(0.970444\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 6.00000 0.221918
\(732\) − 10.0000i − 0.369611i
\(733\) − 14.0000i − 0.517102i −0.965998 0.258551i \(-0.916755\pi\)
0.965998 0.258551i \(-0.0832450\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 5.00000i − 0.184177i
\(738\) 0 0
\(739\) −32.0000 −1.17714 −0.588570 0.808447i \(-0.700309\pi\)
−0.588570 + 0.808447i \(0.700309\pi\)
\(740\) 0 0
\(741\) 7.00000 0.257151
\(742\) 0 0
\(743\) 6.00000i 0.220119i 0.993925 + 0.110059i \(0.0351041\pi\)
−0.993925 + 0.110059i \(0.964896\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 6.00000i − 0.219529i
\(748\) 12.0000i 0.438763i
\(749\) −6.00000 −0.219235
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) 18.0000i 0.655956i
\(754\) 0 0
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 29.0000i 1.05402i 0.849858 + 0.527011i \(0.176688\pi\)
−0.849858 + 0.527011i \(0.823312\pi\)
\(758\) 0 0
\(759\) 6.00000 0.217786
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 11.0000i 0.398227i
\(764\) −24.0000 −0.868290
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) − 16.0000i − 0.577350i
\(769\) −29.0000 −1.04577 −0.522883 0.852404i \(-0.675144\pi\)
−0.522883 + 0.852404i \(0.675144\pi\)
\(770\) 0 0
\(771\) 24.0000 0.864339
\(772\) 14.0000i 0.503871i
\(773\) 18.0000i 0.647415i 0.946157 + 0.323708i \(0.104929\pi\)
−0.946157 + 0.323708i \(0.895071\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 2.00000i − 0.0717496i
\(778\) 0 0
\(779\) −42.0000 −1.50481
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 0 0
\(783\) 6.00000i 0.214423i
\(784\) 24.0000 0.857143
\(785\) 0 0
\(786\) 0 0
\(787\) − 55.0000i − 1.96054i −0.197667 0.980269i \(-0.563337\pi\)
0.197667 0.980269i \(-0.436663\pi\)
\(788\) 48.0000i 1.70993i
\(789\) 18.0000 0.640817
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) 5.00000i 0.177555i
\(794\) 0 0
\(795\) 0 0
\(796\) −22.0000 −0.779769
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) 14.0000i 0.494049i
\(804\) 10.0000 0.352673
\(805\) 0 0
\(806\) 0 0
\(807\) 18.0000i 0.633630i
\(808\) 0 0
\(809\) −12.0000 −0.421898 −0.210949 0.977497i \(-0.567655\pi\)
−0.210949 + 0.977497i \(0.567655\pi\)
\(810\) 0 0
\(811\) −7.00000 −0.245803 −0.122902 0.992419i \(-0.539220\pi\)
−0.122902 + 0.992419i \(0.539220\pi\)
\(812\) − 12.0000i − 0.421117i
\(813\) 16.0000i 0.561144i
\(814\) 0 0
\(815\) 0 0
\(816\) −24.0000 −0.840168
\(817\) 7.00000i 0.244899i
\(818\) 0 0
\(819\) −1.00000 −0.0349428
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 31.0000i 1.08059i 0.841475 + 0.540296i \(0.181688\pi\)
−0.841475 + 0.540296i \(0.818312\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 42.0000i 1.46048i 0.683189 + 0.730242i \(0.260592\pi\)
−0.683189 + 0.730242i \(0.739408\pi\)
\(828\) 12.0000i 0.417029i
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) −13.0000 −0.450965
\(832\) 8.00000i 0.277350i
\(833\) − 36.0000i − 1.24733i
\(834\) 0 0
\(835\) 0 0
\(836\) −14.0000 −0.484200
\(837\) − 7.00000i − 0.241955i
\(838\) 0 0
\(839\) 30.0000 1.03572 0.517858 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) − 12.0000i − 0.413302i
\(844\) 10.0000 0.344214
\(845\) 0 0
\(846\) 0 0
\(847\) − 1.00000i − 0.0343604i
\(848\) 24.0000i 0.824163i
\(849\) −29.0000 −0.995277
\(850\) 0 0
\(851\) 12.0000 0.411355
\(852\) 24.0000i 0.822226i
\(853\) 19.0000i 0.650548i 0.945620 + 0.325274i \(0.105456\pi\)
−0.945620 + 0.325274i \(0.894544\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 30.0000i 1.02478i 0.858753 + 0.512390i \(0.171240\pi\)
−0.858753 + 0.512390i \(0.828760\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) 6.00000 0.204479
\(862\) 0 0
\(863\) 12.0000i 0.408485i 0.978920 + 0.204242i \(0.0654731\pi\)
−0.978920 + 0.204242i \(0.934527\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 19.0000i 0.645274i
\(868\) 14.0000i 0.475191i
\(869\) −4.00000 −0.135691
\(870\) 0 0
\(871\) −5.00000 −0.169419
\(872\) 0 0
\(873\) − 17.0000i − 0.575363i
\(874\) 0 0
\(875\) 0 0
\(876\) −28.0000 −0.946032
\(877\) 53.0000i 1.78968i 0.446384 + 0.894841i \(0.352711\pi\)
−0.446384 + 0.894841i \(0.647289\pi\)
\(878\) 0 0
\(879\) −18.0000 −0.607125
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) 7.00000i 0.235569i 0.993039 + 0.117784i \(0.0375792\pi\)
−0.993039 + 0.117784i \(0.962421\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) 0 0
\(887\) − 6.00000i − 0.201460i −0.994914 0.100730i \(-0.967882\pi\)
0.994914 0.100730i \(-0.0321179\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) − 10.0000i − 0.334825i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 6.00000i − 0.200334i
\(898\) 0 0
\(899\) −42.0000 −1.40078
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) − 1.00000i − 0.0332779i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 20.0000i 0.664089i 0.943264 + 0.332045i \(0.107738\pi\)
−0.943264 + 0.332045i \(0.892262\pi\)
\(908\) − 48.0000i − 1.59294i
\(909\) 0 0
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) − 28.0000i − 0.927173i
\(913\) − 6.00000i − 0.198571i
\(914\) 0 0
\(915\) 0 0
\(916\) −34.0000 −1.12339
\(917\) − 18.0000i − 0.594412i
\(918\) 0 0
\(919\) 43.0000 1.41844 0.709220 0.704988i \(-0.249047\pi\)
0.709220 + 0.704988i \(0.249047\pi\)
\(920\) 0 0
\(921\) 5.00000 0.164756
\(922\) 0 0
\(923\) − 12.0000i − 0.394985i
\(924\) 2.00000 0.0657952
\(925\) 0 0
\(926\) 0 0
\(927\) 8.00000i 0.262754i
\(928\) 0 0
\(929\) 24.0000 0.787414 0.393707 0.919236i \(-0.371192\pi\)
0.393707 + 0.919236i \(0.371192\pi\)
\(930\) 0 0
\(931\) 42.0000 1.37649
\(932\) − 24.0000i − 0.786146i
\(933\) − 18.0000i − 0.589294i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 7.00000i − 0.228680i −0.993442 0.114340i \(-0.963525\pi\)
0.993442 0.114340i \(-0.0364753\pi\)
\(938\) 0 0
\(939\) 13.0000 0.424239
\(940\) 0 0
\(941\) −24.0000 −0.782378 −0.391189 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) 0 0
\(943\) 36.0000i 1.17232i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 6.00000i − 0.194974i −0.995237 0.0974869i \(-0.968920\pi\)
0.995237 0.0974869i \(-0.0310804\pi\)
\(948\) − 8.00000i − 0.259828i
\(949\) 14.0000 0.454459
\(950\) 0 0
\(951\) −30.0000 −0.972817
\(952\) 0 0
\(953\) − 42.0000i − 1.36051i −0.732974 0.680257i \(-0.761868\pi\)
0.732974 0.680257i \(-0.238132\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 60.0000 1.94054
\(957\) 6.00000i 0.193952i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) 6.00000i 0.193347i
\(964\) −26.0000 −0.837404
\(965\) 0 0
\(966\) 0 0
\(967\) − 52.0000i − 1.67221i −0.548572 0.836104i \(-0.684828\pi\)
0.548572 0.836104i \(-0.315172\pi\)
\(968\) 0 0
\(969\) −42.0000 −1.34923
\(970\) 0 0
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) − 2.00000i − 0.0641500i
\(973\) − 4.00000i − 0.128234i
\(974\) 0 0
\(975\) 0 0
\(976\) 20.0000 0.640184
\(977\) − 42.0000i − 1.34370i −0.740688 0.671850i \(-0.765500\pi\)
0.740688 0.671850i \(-0.234500\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) 11.0000 0.351203
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 14.0000i 0.445399i
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) 5.00000 0.158830 0.0794151 0.996842i \(-0.474695\pi\)
0.0794151 + 0.996842i \(0.474695\pi\)
\(992\) 0 0
\(993\) 4.00000i 0.126936i
\(994\) 0 0
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) − 10.0000i − 0.316703i −0.987383 0.158352i \(-0.949382\pi\)
0.987383 0.158352i \(-0.0506179\pi\)
\(998\) 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 825.2.c.b.199.1 2
3.2 odd 2 2475.2.c.h.199.1 2
5.2 odd 4 825.2.a.b.1.1 1
5.3 odd 4 825.2.a.c.1.1 yes 1
5.4 even 2 inner 825.2.c.b.199.2 2
15.2 even 4 2475.2.a.f.1.1 1
15.8 even 4 2475.2.a.e.1.1 1
15.14 odd 2 2475.2.c.h.199.2 2
55.32 even 4 9075.2.a.i.1.1 1
55.43 even 4 9075.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.b.1.1 1 5.2 odd 4
825.2.a.c.1.1 yes 1 5.3 odd 4
825.2.c.b.199.1 2 1.1 even 1 trivial
825.2.c.b.199.2 2 5.4 even 2 inner
2475.2.a.e.1.1 1 15.8 even 4
2475.2.a.f.1.1 1 15.2 even 4
2475.2.c.h.199.1 2 3.2 odd 2
2475.2.c.h.199.2 2 15.14 odd 2
9075.2.a.i.1.1 1 55.32 even 4
9075.2.a.l.1.1 1 55.43 even 4