Properties

Label 1350.2.c.b.649.1
Level $1350$
Weight $2$
Character 1350.649
Analytic conductor $10.780$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1350.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.7798042729\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 54)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1350.649
Dual form 1350.2.c.b.649.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} -3.00000 q^{11} +4.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{19} +3.00000i q^{22} +6.00000i q^{23} +4.00000 q^{26} +1.00000i q^{28} -6.00000 q^{29} +5.00000 q^{31} -1.00000i q^{32} +2.00000i q^{37} +2.00000i q^{38} -6.00000 q^{41} +10.0000i q^{43} +3.00000 q^{44} +6.00000 q^{46} +6.00000i q^{47} +6.00000 q^{49} -4.00000i q^{52} -9.00000i q^{53} +1.00000 q^{56} +6.00000i q^{58} -12.0000 q^{59} +8.00000 q^{61} -5.00000i q^{62} -1.00000 q^{64} +14.0000i q^{67} +7.00000i q^{73} +2.00000 q^{74} +2.00000 q^{76} +3.00000i q^{77} -8.00000 q^{79} +6.00000i q^{82} +3.00000i q^{83} +10.0000 q^{86} -3.00000i q^{88} +18.0000 q^{89} +4.00000 q^{91} -6.00000i q^{92} +6.00000 q^{94} -1.00000i q^{97} -6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + O(q^{10}) \) \( 2 q - 2 q^{4} - 6 q^{11} - 2 q^{14} + 2 q^{16} - 4 q^{19} + 8 q^{26} - 12 q^{29} + 10 q^{31} - 12 q^{41} + 6 q^{44} + 12 q^{46} + 12 q^{49} + 2 q^{56} - 24 q^{59} + 16 q^{61} - 2 q^{64} + 4 q^{74} + 4 q^{76} - 16 q^{79} + 20 q^{86} + 36 q^{89} + 8 q^{91} + 12 q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i −0.981981 0.188982i \(-0.939481\pi\)
0.981981 0.188982i \(-0.0605189\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.00000i 0.639602i
\(23\) 6.00000i 1.25109i 0.780189 + 0.625543i \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) 1.00000i 0.188982i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 2.00000i 0.324443i
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 10.0000i 1.52499i 0.646997 + 0.762493i \(0.276025\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 6.00000i 0.875190i 0.899172 + 0.437595i \(0.144170\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) − 4.00000i − 0.554700i
\(53\) − 9.00000i − 1.23625i −0.786082 0.618123i \(-0.787894\pi\)
0.786082 0.618123i \(-0.212106\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 6.00000i 0.787839i
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) − 5.00000i − 0.635001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 14.0000i 1.71037i 0.518321 + 0.855186i \(0.326557\pi\)
−0.518321 + 0.855186i \(0.673443\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 7.00000i 0.819288i 0.912245 + 0.409644i \(0.134347\pi\)
−0.912245 + 0.409644i \(0.865653\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 3.00000i 0.341882i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 6.00000i 0.662589i
\(83\) 3.00000i 0.329293i 0.986353 + 0.164646i \(0.0526483\pi\)
−0.986353 + 0.164646i \(0.947352\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 10.0000 1.07833
\(87\) 0 0
\(88\) − 3.00000i − 0.319801i
\(89\) 18.0000 1.90800 0.953998 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) − 6.00000i − 0.625543i
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) 0 0
\(97\) − 1.00000i − 0.101535i −0.998711 0.0507673i \(-0.983833\pi\)
0.998711 0.0507673i \(-0.0161667\pi\)
\(98\) − 6.00000i − 0.606092i
\(99\) 0 0
\(100\) 0 0
\(101\) −3.00000 −0.298511 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 0 0
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) 9.00000i 0.870063i 0.900415 + 0.435031i \(0.143263\pi\)
−0.900415 + 0.435031i \(0.856737\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 1.00000i − 0.0944911i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 12.0000i 1.10469i
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) − 8.00000i − 0.724286i
\(123\) 0 0
\(124\) −5.00000 −0.449013
\(125\) 0 0
\(126\) 0 0
\(127\) − 7.00000i − 0.621150i −0.950549 0.310575i \(-0.899478\pi\)
0.950549 0.310575i \(-0.100522\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −15.0000 −1.31056 −0.655278 0.755388i \(-0.727449\pi\)
−0.655278 + 0.755388i \(0.727449\pi\)
\(132\) 0 0
\(133\) 2.00000i 0.173422i
\(134\) 14.0000 1.20942
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 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
\(143\) − 12.0000i − 1.00349i
\(144\) 0 0
\(145\) 0 0
\(146\) 7.00000 0.579324
\(147\) 0 0
\(148\) − 2.00000i − 0.164399i
\(149\) −3.00000 −0.245770 −0.122885 0.992421i \(-0.539215\pi\)
−0.122885 + 0.992421i \(0.539215\pi\)
\(150\) 0 0
\(151\) 17.0000 1.38344 0.691720 0.722166i \(-0.256853\pi\)
0.691720 + 0.722166i \(0.256853\pi\)
\(152\) − 2.00000i − 0.162221i
\(153\) 0 0
\(154\) 3.00000 0.241747
\(155\) 0 0
\(156\) 0 0
\(157\) − 4.00000i − 0.319235i −0.987179 0.159617i \(-0.948974\pi\)
0.987179 0.159617i \(-0.0510260\pi\)
\(158\) 8.00000i 0.636446i
\(159\) 0 0
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) − 20.0000i − 1.56652i −0.621694 0.783260i \(-0.713555\pi\)
0.621694 0.783260i \(-0.286445\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 3.00000 0.232845
\(167\) − 6.00000i − 0.464294i −0.972681 0.232147i \(-0.925425\pi\)
0.972681 0.232147i \(-0.0745750\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) − 10.0000i − 0.762493i
\(173\) − 15.0000i − 1.14043i −0.821496 0.570214i \(-0.806860\pi\)
0.821496 0.570214i \(-0.193140\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) − 18.0000i − 1.34916i
\(179\) 9.00000 0.672692 0.336346 0.941739i \(-0.390809\pi\)
0.336346 + 0.941739i \(0.390809\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) − 4.00000i − 0.296500i
\(183\) 0 0
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) − 6.00000i − 0.437595i
\(189\) 0 0
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) − 5.00000i − 0.359908i −0.983675 0.179954i \(-0.942405\pi\)
0.983675 0.179954i \(-0.0575949\pi\)
\(194\) −1.00000 −0.0717958
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 9.00000i 0.641223i 0.947211 + 0.320612i \(0.103888\pi\)
−0.947211 + 0.320612i \(0.896112\pi\)
\(198\) 0 0
\(199\) 7.00000 0.496217 0.248108 0.968732i \(-0.420191\pi\)
0.248108 + 0.968732i \(0.420191\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 3.00000i 0.211079i
\(203\) 6.00000i 0.421117i
\(204\) 0 0
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) 4.00000i 0.277350i
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) −22.0000 −1.51454 −0.757271 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) 9.00000i 0.618123i
\(213\) 0 0
\(214\) 9.00000 0.615227
\(215\) 0 0
\(216\) 0 0
\(217\) − 5.00000i − 0.339422i
\(218\) 2.00000i 0.135457i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) − 8.00000i − 0.535720i −0.963458 0.267860i \(-0.913684\pi\)
0.963458 0.267860i \(-0.0863164\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 6.00000i − 0.393919i
\(233\) − 18.0000i − 1.17922i −0.807688 0.589610i \(-0.799282\pi\)
0.807688 0.589610i \(-0.200718\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 0 0
\(239\) −30.0000 −1.94054 −0.970269 0.242028i \(-0.922188\pi\)
−0.970269 + 0.242028i \(0.922188\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 2.00000i 0.128565i
\(243\) 0 0
\(244\) −8.00000 −0.512148
\(245\) 0 0
\(246\) 0 0
\(247\) − 8.00000i − 0.509028i
\(248\) 5.00000i 0.317500i
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) − 18.0000i − 1.13165i
\(254\) −7.00000 −0.439219
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 12.0000i 0.748539i 0.927320 + 0.374270i \(0.122107\pi\)
−0.927320 + 0.374270i \(0.877893\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) 0 0
\(262\) 15.0000i 0.926703i
\(263\) 30.0000i 1.84988i 0.380114 + 0.924940i \(0.375885\pi\)
−0.380114 + 0.924940i \(0.624115\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 2.00000 0.122628
\(267\) 0 0
\(268\) − 14.0000i − 0.855186i
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) −25.0000 −1.51864 −0.759321 0.650716i \(-0.774469\pi\)
−0.759321 + 0.650716i \(0.774469\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) 8.00000i 0.480673i 0.970690 + 0.240337i \(0.0772579\pi\)
−0.970690 + 0.240337i \(0.922742\pi\)
\(278\) − 4.00000i − 0.239904i
\(279\) 0 0
\(280\) 0 0
\(281\) 24.0000 1.43172 0.715860 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 0 0
\(283\) − 14.0000i − 0.832214i −0.909316 0.416107i \(-0.863394\pi\)
0.909316 0.416107i \(-0.136606\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) 6.00000i 0.354169i
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) − 7.00000i − 0.409644i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) 3.00000i 0.173785i
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) 10.0000 0.576390
\(302\) − 17.0000i − 0.978240i
\(303\) 0 0
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) 0 0
\(307\) − 16.0000i − 0.913168i −0.889680 0.456584i \(-0.849073\pi\)
0.889680 0.456584i \(-0.150927\pi\)
\(308\) − 3.00000i − 0.170941i
\(309\) 0 0
\(310\) 0 0
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 0 0
\(313\) 19.0000i 1.07394i 0.843600 + 0.536972i \(0.180432\pi\)
−0.843600 + 0.536972i \(0.819568\pi\)
\(314\) −4.00000 −0.225733
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) − 3.00000i − 0.168497i −0.996445 0.0842484i \(-0.973151\pi\)
0.996445 0.0842484i \(-0.0268489\pi\)
\(318\) 0 0
\(319\) 18.0000 1.00781
\(320\) 0 0
\(321\) 0 0
\(322\) − 6.00000i − 0.334367i
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) −20.0000 −1.10770
\(327\) 0 0
\(328\) − 6.00000i − 0.331295i
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) − 3.00000i − 0.164646i
\(333\) 0 0
\(334\) −6.00000 −0.328305
\(335\) 0 0
\(336\) 0 0
\(337\) − 22.0000i − 1.19842i −0.800593 0.599208i \(-0.795482\pi\)
0.800593 0.599208i \(-0.204518\pi\)
\(338\) 3.00000i 0.163178i
\(339\) 0 0
\(340\) 0 0
\(341\) −15.0000 −0.812296
\(342\) 0 0
\(343\) − 13.0000i − 0.701934i
\(344\) −10.0000 −0.539164
\(345\) 0 0
\(346\) −15.0000 −0.806405
\(347\) 3.00000i 0.161048i 0.996753 + 0.0805242i \(0.0256594\pi\)
−0.996753 + 0.0805242i \(0.974341\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 3.00000i 0.159901i
\(353\) − 6.00000i − 0.319348i −0.987170 0.159674i \(-0.948956\pi\)
0.987170 0.159674i \(-0.0510443\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −18.0000 −0.953998
\(357\) 0 0
\(358\) − 9.00000i − 0.475665i
\(359\) −18.0000 −0.950004 −0.475002 0.879985i \(-0.657553\pi\)
−0.475002 + 0.879985i \(0.657553\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 16.0000i 0.840941i
\(363\) 0 0
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) 0 0
\(367\) 17.0000i 0.887393i 0.896177 + 0.443696i \(0.146333\pi\)
−0.896177 + 0.443696i \(0.853667\pi\)
\(368\) 6.00000i 0.312772i
\(369\) 0 0
\(370\) 0 0
\(371\) −9.00000 −0.467257
\(372\) 0 0
\(373\) − 32.0000i − 1.65690i −0.560065 0.828449i \(-0.689224\pi\)
0.560065 0.828449i \(-0.310776\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) − 24.0000i − 1.23606i
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 12.0000i 0.613973i
\(383\) 24.0000i 1.22634i 0.789950 + 0.613171i \(0.210106\pi\)
−0.789950 + 0.613171i \(0.789894\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −5.00000 −0.254493
\(387\) 0 0
\(388\) 1.00000i 0.0507673i
\(389\) 21.0000 1.06474 0.532371 0.846511i \(-0.321301\pi\)
0.532371 + 0.846511i \(0.321301\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 6.00000i 0.303046i
\(393\) 0 0
\(394\) 9.00000 0.453413
\(395\) 0 0
\(396\) 0 0
\(397\) 20.0000i 1.00377i 0.864934 + 0.501886i \(0.167360\pi\)
−0.864934 + 0.501886i \(0.832640\pi\)
\(398\) − 7.00000i − 0.350878i
\(399\) 0 0
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 0 0
\(403\) 20.0000i 0.996271i
\(404\) 3.00000 0.149256
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) − 6.00000i − 0.297409i
\(408\) 0 0
\(409\) −23.0000 −1.13728 −0.568638 0.822588i \(-0.692530\pi\)
−0.568638 + 0.822588i \(0.692530\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 4.00000i − 0.197066i
\(413\) 12.0000i 0.590481i
\(414\) 0 0
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) 0 0
\(418\) − 6.00000i − 0.293470i
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 22.0000i 1.07094i
\(423\) 0 0
\(424\) 9.00000 0.437079
\(425\) 0 0
\(426\) 0 0
\(427\) − 8.00000i − 0.387147i
\(428\) − 9.00000i − 0.435031i
\(429\) 0 0
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) − 29.0000i − 1.39365i −0.717241 0.696826i \(-0.754595\pi\)
0.717241 0.696826i \(-0.245405\pi\)
\(434\) −5.00000 −0.240008
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) − 12.0000i − 0.574038i
\(438\) 0 0
\(439\) 19.0000 0.906821 0.453410 0.891302i \(-0.350207\pi\)
0.453410 + 0.891302i \(0.350207\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.0000i 0.570137i 0.958507 + 0.285069i \(0.0920164\pi\)
−0.958507 + 0.285069i \(0.907984\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) 0 0
\(448\) 1.00000i 0.0472456i
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 18.0000 0.847587
\(452\) − 6.00000i − 0.282216i
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) − 1.00000i − 0.0467780i −0.999726 0.0233890i \(-0.992554\pi\)
0.999726 0.0233890i \(-0.00744563\pi\)
\(458\) 14.0000i 0.654177i
\(459\) 0 0
\(460\) 0 0
\(461\) −21.0000 −0.978068 −0.489034 0.872265i \(-0.662651\pi\)
−0.489034 + 0.872265i \(0.662651\pi\)
\(462\) 0 0
\(463\) 13.0000i 0.604161i 0.953282 + 0.302081i \(0.0976812\pi\)
−0.953282 + 0.302081i \(0.902319\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) − 27.0000i − 1.24941i −0.780860 0.624705i \(-0.785219\pi\)
0.780860 0.624705i \(-0.214781\pi\)
\(468\) 0 0
\(469\) 14.0000 0.646460
\(470\) 0 0
\(471\) 0 0
\(472\) − 12.0000i − 0.552345i
\(473\) − 30.0000i − 1.37940i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 30.0000i 1.37217i
\(479\) −6.00000 −0.274147 −0.137073 0.990561i \(-0.543770\pi\)
−0.137073 + 0.990561i \(0.543770\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 10.0000i 0.455488i
\(483\) 0 0
\(484\) 2.00000 0.0909091
\(485\) 0 0
\(486\) 0 0
\(487\) − 16.0000i − 0.725029i −0.931978 0.362515i \(-0.881918\pi\)
0.931978 0.362515i \(-0.118082\pi\)
\(488\) 8.00000i 0.362143i
\(489\) 0 0
\(490\) 0 0
\(491\) 39.0000 1.76005 0.880023 0.474932i \(-0.157527\pi\)
0.880023 + 0.474932i \(0.157527\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 5.00000 0.224507
\(497\) 0 0
\(498\) 0 0
\(499\) −14.0000 −0.626726 −0.313363 0.949633i \(-0.601456\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 18.0000i 0.802580i 0.915951 + 0.401290i \(0.131438\pi\)
−0.915951 + 0.401290i \(0.868562\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −18.0000 −0.800198
\(507\) 0 0
\(508\) 7.00000i 0.310575i
\(509\) 15.0000 0.664863 0.332432 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) 7.00000 0.309662
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) 12.0000 0.529297
\(515\) 0 0
\(516\) 0 0
\(517\) − 18.0000i − 0.791639i
\(518\) − 2.00000i − 0.0878750i
\(519\) 0 0
\(520\) 0 0
\(521\) 36.0000 1.57719 0.788594 0.614914i \(-0.210809\pi\)
0.788594 + 0.614914i \(0.210809\pi\)
\(522\) 0 0
\(523\) 16.0000i 0.699631i 0.936819 + 0.349816i \(0.113756\pi\)
−0.936819 + 0.349816i \(0.886244\pi\)
\(524\) 15.0000 0.655278
\(525\) 0 0
\(526\) 30.0000 1.30806
\(527\) 0 0
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) − 2.00000i − 0.0867110i
\(533\) − 24.0000i − 1.03956i
\(534\) 0 0
\(535\) 0 0
\(536\) −14.0000 −0.604708
\(537\) 0 0
\(538\) − 18.0000i − 0.776035i
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) 25.0000i 1.07384i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 8.00000i 0.342055i 0.985266 + 0.171028i \(0.0547087\pi\)
−0.985266 + 0.171028i \(0.945291\pi\)
\(548\) − 6.00000i − 0.256307i
\(549\) 0 0
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 8.00000i 0.340195i
\(554\) 8.00000 0.339887
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 27.0000i 1.14403i 0.820244 + 0.572013i \(0.193837\pi\)
−0.820244 + 0.572013i \(0.806163\pi\)
\(558\) 0 0
\(559\) −40.0000 −1.69182
\(560\) 0 0
\(561\) 0 0
\(562\) − 24.0000i − 1.01238i
\(563\) − 3.00000i − 0.126435i −0.998000 0.0632175i \(-0.979864\pi\)
0.998000 0.0632175i \(-0.0201362\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −14.0000 −0.588464
\(567\) 0 0
\(568\) 0 0
\(569\) 12.0000 0.503066 0.251533 0.967849i \(-0.419065\pi\)
0.251533 + 0.967849i \(0.419065\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 12.0000i 0.501745i
\(573\) 0 0
\(574\) 6.00000 0.250435
\(575\) 0 0
\(576\) 0 0
\(577\) 38.0000i 1.58196i 0.611842 + 0.790980i \(0.290429\pi\)
−0.611842 + 0.790980i \(0.709571\pi\)
\(578\) − 17.0000i − 0.707107i
\(579\) 0 0
\(580\) 0 0
\(581\) 3.00000 0.124461
\(582\) 0 0
\(583\) 27.0000i 1.11823i
\(584\) −7.00000 −0.289662
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) − 3.00000i − 0.123823i −0.998082 0.0619116i \(-0.980280\pi\)
0.998082 0.0619116i \(-0.0197197\pi\)
\(588\) 0 0
\(589\) −10.0000 −0.412043
\(590\) 0 0
\(591\) 0 0
\(592\) 2.00000i 0.0821995i
\(593\) − 18.0000i − 0.739171i −0.929197 0.369586i \(-0.879500\pi\)
0.929197 0.369586i \(-0.120500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 3.00000 0.122885
\(597\) 0 0
\(598\) 24.0000i 0.981433i
\(599\) 42.0000 1.71607 0.858037 0.513588i \(-0.171684\pi\)
0.858037 + 0.513588i \(0.171684\pi\)
\(600\) 0 0
\(601\) 35.0000 1.42768 0.713840 0.700309i \(-0.246954\pi\)
0.713840 + 0.700309i \(0.246954\pi\)
\(602\) − 10.0000i − 0.407570i
\(603\) 0 0
\(604\) −17.0000 −0.691720
\(605\) 0 0
\(606\) 0 0
\(607\) 32.0000i 1.29884i 0.760430 + 0.649420i \(0.224988\pi\)
−0.760430 + 0.649420i \(0.775012\pi\)
\(608\) 2.00000i 0.0811107i
\(609\) 0 0
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) 34.0000i 1.37325i 0.727013 + 0.686624i \(0.240908\pi\)
−0.727013 + 0.686624i \(0.759092\pi\)
\(614\) −16.0000 −0.645707
\(615\) 0 0
\(616\) −3.00000 −0.120873
\(617\) − 42.0000i − 1.69086i −0.534089 0.845428i \(-0.679345\pi\)
0.534089 0.845428i \(-0.320655\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 6.00000i 0.240578i
\(623\) − 18.0000i − 0.721155i
\(624\) 0 0
\(625\) 0 0
\(626\) 19.0000 0.759393
\(627\) 0 0
\(628\) 4.00000i 0.159617i
\(629\) 0 0
\(630\) 0 0
\(631\) −25.0000 −0.995234 −0.497617 0.867397i \(-0.665792\pi\)
−0.497617 + 0.867397i \(0.665792\pi\)
\(632\) − 8.00000i − 0.318223i
\(633\) 0 0
\(634\) −3.00000 −0.119145
\(635\) 0 0
\(636\) 0 0
\(637\) 24.0000i 0.950915i
\(638\) − 18.0000i − 0.712627i
\(639\) 0 0
\(640\) 0 0
\(641\) 42.0000 1.65890 0.829450 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) 0 0
\(643\) 4.00000i 0.157745i 0.996885 + 0.0788723i \(0.0251319\pi\)
−0.996885 + 0.0788723i \(0.974868\pi\)
\(644\) −6.00000 −0.236433
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) 0 0
\(652\) 20.0000i 0.783260i
\(653\) − 39.0000i − 1.52619i −0.646288 0.763094i \(-0.723679\pi\)
0.646288 0.763094i \(-0.276321\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) − 6.00000i − 0.233904i
\(659\) 21.0000 0.818044 0.409022 0.912525i \(-0.365870\pi\)
0.409022 + 0.912525i \(0.365870\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 10.0000i 0.388661i
\(663\) 0 0
\(664\) −3.00000 −0.116423
\(665\) 0 0
\(666\) 0 0
\(667\) − 36.0000i − 1.39393i
\(668\) 6.00000i 0.232147i
\(669\) 0 0
\(670\) 0 0
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) 19.0000i 0.732396i 0.930537 + 0.366198i \(0.119341\pi\)
−0.930537 + 0.366198i \(0.880659\pi\)
\(674\) −22.0000 −0.847408
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 42.0000i 1.61419i 0.590421 + 0.807096i \(0.298962\pi\)
−0.590421 + 0.807096i \(0.701038\pi\)
\(678\) 0 0
\(679\) −1.00000 −0.0383765
\(680\) 0 0
\(681\) 0 0
\(682\) 15.0000i 0.574380i
\(683\) 36.0000i 1.37750i 0.724998 + 0.688751i \(0.241841\pi\)
−0.724998 + 0.688751i \(0.758159\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −13.0000 −0.496342
\(687\) 0 0
\(688\) 10.0000i 0.381246i
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) 44.0000 1.67384 0.836919 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) 15.0000i 0.570214i
\(693\) 0 0
\(694\) 3.00000 0.113878
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) − 10.0000i − 0.378506i
\(699\) 0 0
\(700\) 0 0
\(701\) 9.00000 0.339925 0.169963 0.985451i \(-0.445635\pi\)
0.169963 + 0.985451i \(0.445635\pi\)
\(702\) 0 0
\(703\) − 4.00000i − 0.150863i
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 3.00000i 0.112827i
\(708\) 0 0
\(709\) −44.0000 −1.65245 −0.826227 0.563337i \(-0.809517\pi\)
−0.826227 + 0.563337i \(0.809517\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 18.0000i 0.674579i
\(713\) 30.0000i 1.12351i
\(714\) 0 0
\(715\) 0 0
\(716\) −9.00000 −0.336346
\(717\) 0 0
\(718\) 18.0000i 0.671754i
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 15.0000i 0.558242i
\(723\) 0 0
\(724\) 16.0000 0.594635
\(725\) 0 0
\(726\) 0 0
\(727\) − 1.00000i − 0.0370879i −0.999828 0.0185440i \(-0.994097\pi\)
0.999828 0.0185440i \(-0.00590307\pi\)
\(728\) 4.00000i 0.148250i
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 22.0000i 0.812589i 0.913742 + 0.406294i \(0.133179\pi\)
−0.913742 + 0.406294i \(0.866821\pi\)
\(734\) 17.0000 0.627481
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) − 42.0000i − 1.54709i
\(738\) 0 0
\(739\) 16.0000 0.588570 0.294285 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 9.00000i 0.330400i
\(743\) − 12.0000i − 0.440237i −0.975473 0.220119i \(-0.929356\pi\)
0.975473 0.220119i \(-0.0706445\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −32.0000 −1.17160
\(747\) 0 0
\(748\) 0 0
\(749\) 9.00000 0.328853
\(750\) 0 0
\(751\) 41.0000 1.49611 0.748056 0.663636i \(-0.230988\pi\)
0.748056 + 0.663636i \(0.230988\pi\)
\(752\) 6.00000i 0.218797i
\(753\) 0 0
\(754\) −24.0000 −0.874028
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) 20.0000i 0.726433i
\(759\) 0 0
\(760\) 0 0
\(761\) 48.0000 1.74000 0.869999 0.493053i \(-0.164119\pi\)
0.869999 + 0.493053i \(0.164119\pi\)
\(762\) 0 0
\(763\) 2.00000i 0.0724049i
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) − 48.0000i − 1.73318i
\(768\) 0 0
\(769\) 31.0000 1.11789 0.558944 0.829205i \(-0.311207\pi\)
0.558944 + 0.829205i \(0.311207\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 5.00000i 0.179954i
\(773\) 18.0000i 0.647415i 0.946157 + 0.323708i \(0.104929\pi\)
−0.946157 + 0.323708i \(0.895071\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 1.00000 0.0358979
\(777\) 0 0
\(778\) − 21.0000i − 0.752886i
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 6.00000 0.214286
\(785\) 0 0
\(786\) 0 0
\(787\) 32.0000i 1.14068i 0.821410 + 0.570338i \(0.193188\pi\)
−0.821410 + 0.570338i \(0.806812\pi\)
\(788\) − 9.00000i − 0.320612i
\(789\) 0 0
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 32.0000i 1.13635i
\(794\) 20.0000 0.709773
\(795\) 0 0
\(796\) −7.00000 −0.248108
\(797\) 39.0000i 1.38145i 0.723117 + 0.690725i \(0.242709\pi\)
−0.723117 + 0.690725i \(0.757291\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) − 12.0000i − 0.423735i
\(803\) − 21.0000i − 0.741074i
\(804\) 0 0
\(805\) 0 0
\(806\) 20.0000 0.704470
\(807\) 0 0
\(808\) − 3.00000i − 0.105540i
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) − 6.00000i − 0.210559i
\(813\) 0 0
\(814\) −6.00000 −0.210300
\(815\) 0 0
\(816\) 0 0
\(817\) − 20.0000i − 0.699711i
\(818\) 23.0000i 0.804176i
\(819\) 0 0
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0 0
\(823\) 31.0000i 1.08059i 0.841475 + 0.540296i \(0.181688\pi\)
−0.841475 + 0.540296i \(0.818312\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 4.00000i − 0.138675i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) −6.00000 −0.207514
\(837\) 0 0
\(838\) 12.0000i 0.414533i
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) − 8.00000i − 0.275698i
\(843\) 0 0
\(844\) 22.0000 0.757271
\(845\) 0 0
\(846\) 0 0
\(847\) 2.00000i 0.0687208i
\(848\) − 9.00000i − 0.309061i
\(849\) 0 0
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) 10.0000i 0.342393i 0.985237 + 0.171197i \(0.0547634\pi\)
−0.985237 + 0.171197i \(0.945237\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) −9.00000 −0.307614
\(857\) − 48.0000i − 1.63965i −0.572615 0.819824i \(-0.694071\pi\)
0.572615 0.819824i \(-0.305929\pi\)
\(858\) 0 0
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 18.0000i − 0.613082i
\(863\) 36.0000i 1.22545i 0.790295 + 0.612727i \(0.209928\pi\)
−0.790295 + 0.612727i \(0.790072\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −29.0000 −0.985460
\(867\) 0 0
\(868\) 5.00000i 0.169711i
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) −56.0000 −1.89749
\(872\) − 2.00000i − 0.0677285i
\(873\) 0 0
\(874\) −12.0000 −0.405906
\(875\) 0 0
\(876\) 0 0
\(877\) − 22.0000i − 0.742887i −0.928456 0.371444i \(-0.878863\pi\)
0.928456 0.371444i \(-0.121137\pi\)
\(878\) − 19.0000i − 0.641219i
\(879\) 0 0
\(880\) 0 0
\(881\) −36.0000 −1.21287 −0.606435 0.795133i \(-0.707401\pi\)
−0.606435 + 0.795133i \(0.707401\pi\)
\(882\) 0 0
\(883\) − 2.00000i − 0.0673054i −0.999434 0.0336527i \(-0.989286\pi\)
0.999434 0.0336527i \(-0.0107140\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 12.0000i 0.402921i 0.979497 + 0.201460i \(0.0645687\pi\)
−0.979497 + 0.201460i \(0.935431\pi\)
\(888\) 0 0
\(889\) −7.00000 −0.234772
\(890\) 0 0
\(891\) 0 0
\(892\) 8.00000i 0.267860i
\(893\) − 12.0000i − 0.401565i
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) − 18.0000i − 0.600668i
\(899\) −30.0000 −1.00056
\(900\) 0 0
\(901\) 0 0
\(902\) − 18.0000i − 0.599334i
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 0 0
\(907\) 26.0000i 0.863316i 0.902037 + 0.431658i \(0.142071\pi\)
−0.902037 + 0.431658i \(0.857929\pi\)
\(908\) 12.0000i 0.398234i
\(909\) 0 0
\(910\) 0 0
\(911\) 6.00000 0.198789 0.0993944 0.995048i \(-0.468309\pi\)
0.0993944 + 0.995048i \(0.468309\pi\)
\(912\) 0 0
\(913\) − 9.00000i − 0.297857i
\(914\) −1.00000 −0.0330771
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 15.0000i 0.495344i
\(918\) 0 0
\(919\) −29.0000 −0.956622 −0.478311 0.878191i \(-0.658751\pi\)
−0.478311 + 0.878191i \(0.658751\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 21.0000i 0.691598i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 13.0000 0.427207
\(927\) 0 0
\(928\) 6.00000i 0.196960i
\(929\) 12.0000 0.393707 0.196854 0.980433i \(-0.436928\pi\)
0.196854 + 0.980433i \(0.436928\pi\)
\(930\) 0 0
\(931\) −12.0000 −0.393284
\(932\) 18.0000i 0.589610i
\(933\) 0 0
\(934\) −27.0000 −0.883467
\(935\) 0 0
\(936\) 0 0
\(937\) 11.0000i 0.359354i 0.983726 + 0.179677i \(0.0575053\pi\)
−0.983726 + 0.179677i \(0.942495\pi\)
\(938\) − 14.0000i − 0.457116i
\(939\) 0 0
\(940\) 0 0
\(941\) −33.0000 −1.07577 −0.537885 0.843018i \(-0.680776\pi\)
−0.537885 + 0.843018i \(0.680776\pi\)
\(942\) 0 0
\(943\) − 36.0000i − 1.17232i
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −30.0000 −0.975384
\(947\) 51.0000i 1.65728i 0.559784 + 0.828639i \(0.310884\pi\)
−0.559784 + 0.828639i \(0.689116\pi\)
\(948\) 0 0
\(949\) −28.0000 −0.908918
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 36.0000i 1.16615i 0.812417 + 0.583077i \(0.198151\pi\)
−0.812417 + 0.583077i \(0.801849\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 30.0000 0.970269
\(957\) 0 0
\(958\) 6.00000i 0.193851i
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 8.00000i 0.257930i
\(963\) 0 0
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) 23.0000i 0.739630i 0.929105 + 0.369815i \(0.120579\pi\)
−0.929105 + 0.369815i \(0.879421\pi\)
\(968\) − 2.00000i − 0.0642824i
\(969\) 0 0
\(970\) 0 0
\(971\) −27.0000 −0.866471 −0.433236 0.901281i \(-0.642628\pi\)
−0.433236 + 0.901281i \(0.642628\pi\)
\(972\) 0 0
\(973\) − 4.00000i − 0.128234i
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) − 42.0000i − 1.34370i −0.740688 0.671850i \(-0.765500\pi\)
0.740688 0.671850i \(-0.234500\pi\)
\(978\) 0 0
\(979\) −54.0000 −1.72585
\(980\) 0 0
\(981\) 0 0
\(982\) − 39.0000i − 1.24454i
\(983\) − 6.00000i − 0.191370i −0.995412 0.0956851i \(-0.969496\pi\)
0.995412 0.0956851i \(-0.0305042\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 8.00000i 0.254514i
\(989\) −60.0000 −1.90789
\(990\) 0 0
\(991\) 47.0000 1.49300 0.746502 0.665383i \(-0.231732\pi\)
0.746502 + 0.665383i \(0.231732\pi\)
\(992\) − 5.00000i − 0.158750i
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 28.0000i − 0.886769i −0.896332 0.443384i \(-0.853778\pi\)
0.896332 0.443384i \(-0.146222\pi\)
\(998\) 14.0000i 0.443162i
\(999\) 0 0
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 1350.2.c.b.649.1 2
3.2 odd 2 1350.2.c.k.649.2 2
5.2 odd 4 1350.2.a.r.1.1 1
5.3 odd 4 54.2.a.a.1.1 1
5.4 even 2 inner 1350.2.c.b.649.2 2
15.2 even 4 1350.2.a.h.1.1 1
15.8 even 4 54.2.a.b.1.1 yes 1
15.14 odd 2 1350.2.c.k.649.1 2
20.3 even 4 432.2.a.g.1.1 1
35.13 even 4 2646.2.a.a.1.1 1
40.3 even 4 1728.2.a.d.1.1 1
40.13 odd 4 1728.2.a.c.1.1 1
45.13 odd 12 162.2.c.c.55.1 2
45.23 even 12 162.2.c.b.55.1 2
45.38 even 12 162.2.c.b.109.1 2
45.43 odd 12 162.2.c.c.109.1 2
55.43 even 4 6534.2.a.bc.1.1 1
60.23 odd 4 432.2.a.b.1.1 1
65.38 odd 4 9126.2.a.u.1.1 1
105.83 odd 4 2646.2.a.bd.1.1 1
120.53 even 4 1728.2.a.y.1.1 1
120.83 odd 4 1728.2.a.z.1.1 1
165.98 odd 4 6534.2.a.b.1.1 1
180.23 odd 12 1296.2.i.o.865.1 2
180.43 even 12 1296.2.i.c.433.1 2
180.83 odd 12 1296.2.i.o.433.1 2
180.103 even 12 1296.2.i.c.865.1 2
195.38 even 4 9126.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
54.2.a.a.1.1 1 5.3 odd 4
54.2.a.b.1.1 yes 1 15.8 even 4
162.2.c.b.55.1 2 45.23 even 12
162.2.c.b.109.1 2 45.38 even 12
162.2.c.c.55.1 2 45.13 odd 12
162.2.c.c.109.1 2 45.43 odd 12
432.2.a.b.1.1 1 60.23 odd 4
432.2.a.g.1.1 1 20.3 even 4
1296.2.i.c.433.1 2 180.43 even 12
1296.2.i.c.865.1 2 180.103 even 12
1296.2.i.o.433.1 2 180.83 odd 12
1296.2.i.o.865.1 2 180.23 odd 12
1350.2.a.h.1.1 1 15.2 even 4
1350.2.a.r.1.1 1 5.2 odd 4
1350.2.c.b.649.1 2 1.1 even 1 trivial
1350.2.c.b.649.2 2 5.4 even 2 inner
1350.2.c.k.649.1 2 15.14 odd 2
1350.2.c.k.649.2 2 3.2 odd 2
1728.2.a.c.1.1 1 40.13 odd 4
1728.2.a.d.1.1 1 40.3 even 4
1728.2.a.y.1.1 1 120.53 even 4
1728.2.a.z.1.1 1 120.83 odd 4
2646.2.a.a.1.1 1 35.13 even 4
2646.2.a.bd.1.1 1 105.83 odd 4
6534.2.a.b.1.1 1 165.98 odd 4
6534.2.a.bc.1.1 1 55.43 even 4
9126.2.a.r.1.1 1 195.38 even 4
9126.2.a.u.1.1 1 65.38 odd 4