Properties

Label 1950.2.e.d.1249.1
Level $1950$
Weight $2$
Character 1950.1249
Analytic conductor $15.571$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.5708283941\)
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 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1950.1249
Dual form 1950.2.e.d.1249.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -1.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{18} -2.00000 q^{19} +2.00000 q^{21} +6.00000i q^{23} +1.00000 q^{24} -1.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} +8.00000 q^{31} -1.00000i q^{32} +1.00000 q^{36} +2.00000i q^{37} +2.00000i q^{38} -1.00000 q^{39} +6.00000 q^{41} -2.00000i q^{42} +4.00000i q^{43} +6.00000 q^{46} -1.00000i q^{48} +3.00000 q^{49} +1.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -2.00000 q^{56} +2.00000i q^{57} +14.0000 q^{61} -8.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} +6.00000 q^{69} -1.00000i q^{72} +4.00000i q^{73} +2.00000 q^{74} +2.00000 q^{76} +1.00000i q^{78} +16.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} +12.0000i q^{83} -2.00000 q^{84} +4.00000 q^{86} +6.00000 q^{89} +2.00000 q^{91} -6.00000i q^{92} -8.00000i q^{93} -1.00000 q^{96} -4.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 4 q^{14} + 2 q^{16} - 4 q^{19} + 4 q^{21} + 2 q^{24} - 2 q^{26} + 16 q^{31} + 2 q^{36} - 2 q^{39} + 12 q^{41} + 12 q^{46} + 6 q^{49} + 2 q^{54} - 4 q^{56} + 28 q^{61} - 2 q^{64} + 12 q^{69} + 4 q^{74} + 4 q^{76} + 32 q^{79} + 2 q^{81} - 4 q^{84} + 8 q^{86} + 12 q^{89} + 4 q^{91} - 2 q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) − 1.00000i − 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 1.00000i − 0.277350i
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 6.00000i 1.25109i 0.780189 + 0.625543i \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 1.00000i 0.192450i
\(28\) − 2.00000i − 0.377964i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 2.00000i 0.324443i
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) − 2.00000i − 0.308607i
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 1.00000i 0.138675i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 2.00000i 0.264906i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) − 8.00000i − 1.01600i
\(63\) − 2.00000i − 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 1.00000i 0.113228i
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 6.00000i − 0.662589i
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) − 6.00000i − 0.625543i
\(93\) − 8.00000i − 0.829561i
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) − 4.00000i − 0.406138i −0.979164 0.203069i \(-0.934908\pi\)
0.979164 0.203069i \(-0.0650917\pi\)
\(98\) − 3.00000i − 0.303046i
\(99\) 0 0
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) 16.0000i 1.57653i 0.615338 + 0.788263i \(0.289020\pi\)
−0.615338 + 0.788263i \(0.710980\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 2.00000i 0.188982i
\(113\) − 12.0000i − 1.12887i −0.825479 0.564433i \(-0.809095\pi\)
0.825479 0.564433i \(-0.190905\pi\)
\(114\) 2.00000 0.187317
\(115\) 0 0
\(116\) 0 0
\(117\) 1.00000i 0.0924500i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) − 14.0000i − 1.26750i
\(123\) − 6.00000i − 0.541002i
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) − 16.0000i − 1.41977i −0.704317 0.709885i \(-0.748747\pi\)
0.704317 0.709885i \(-0.251253\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) − 4.00000i − 0.346844i
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) − 6.00000i − 0.510754i
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) − 3.00000i − 0.247436i
\(148\) − 2.00000i − 0.164399i
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) − 2.00000i − 0.162221i
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) − 16.0000i − 1.27289i
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) − 1.00000i − 0.0785674i
\(163\) 16.0000i 1.25322i 0.779334 + 0.626608i \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 2.00000i 0.154303i
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) − 4.00000i − 0.304997i
\(173\) − 18.0000i − 1.36851i −0.729241 0.684257i \(-0.760127\pi\)
0.729241 0.684257i \(-0.239873\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) − 6.00000i − 0.449719i
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) − 2.00000i − 0.148250i
\(183\) − 14.0000i − 1.03491i
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 0 0
\(188\) 0 0
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) − 8.00000i − 0.575853i −0.957653 0.287926i \(-0.907034\pi\)
0.957653 0.287926i \(-0.0929658\pi\)
\(194\) −4.00000 −0.287183
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 18.0000i 1.28245i 0.767354 + 0.641223i \(0.221573\pi\)
−0.767354 + 0.641223i \(0.778427\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 12.0000i 0.844317i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) − 6.00000i − 0.417029i
\(208\) − 1.00000i − 0.0693375i
\(209\) 0 0
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) − 6.00000i − 0.412082i
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 16.0000i 1.08615i
\(218\) − 4.00000i − 0.270914i
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) 0 0
\(222\) − 2.00000i − 0.134231i
\(223\) 10.0000i 0.669650i 0.942280 + 0.334825i \(0.108677\pi\)
−0.942280 + 0.334825i \(0.891323\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) −12.0000 −0.798228
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) − 2.00000i − 0.132453i
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) 0 0
\(237\) − 16.0000i − 1.03931i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 11.0000i 0.707107i
\(243\) − 1.00000i − 0.0641500i
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 2.00000i 0.127257i
\(248\) 8.00000i 0.508001i
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) −30.0000 −1.89358 −0.946792 0.321847i \(-0.895696\pi\)
−0.946792 + 0.321847i \(0.895696\pi\)
\(252\) 2.00000i 0.125988i
\(253\) 0 0
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 12.0000i − 0.748539i −0.927320 0.374270i \(-0.877893\pi\)
0.927320 0.374270i \(-0.122107\pi\)
\(258\) − 4.00000i − 0.249029i
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) − 6.00000i − 0.370681i
\(263\) − 30.0000i − 1.84988i −0.380114 0.924940i \(-0.624115\pi\)
0.380114 0.924940i \(-0.375885\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.00000 −0.245256
\(267\) − 6.00000i − 0.367194i
\(268\) 4.00000i 0.244339i
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) − 2.00000i − 0.121046i
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) −6.00000 −0.361158
\(277\) 26.0000i 1.56219i 0.624413 + 0.781094i \(0.285338\pi\)
−0.624413 + 0.781094i \(0.714662\pi\)
\(278\) 8.00000i 0.479808i
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) 28.0000i 1.66443i 0.554455 + 0.832214i \(0.312927\pi\)
−0.554455 + 0.832214i \(0.687073\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000i 0.708338i
\(288\) 1.00000i 0.0589256i
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) −4.00000 −0.234484
\(292\) − 4.00000i − 0.234082i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) 18.0000i 1.04271i
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) − 8.00000i − 0.460348i
\(303\) 12.0000i 0.689382i
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) 0 0
\(307\) − 4.00000i − 0.228292i −0.993464 0.114146i \(-0.963587\pi\)
0.993464 0.114146i \(-0.0364132\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) − 1.00000i − 0.0566139i
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) − 18.0000i − 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) 0 0
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 12.0000i 0.668734i
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) − 4.00000i − 0.221201i
\(328\) 6.00000i 0.331295i
\(329\) 0 0
\(330\) 0 0
\(331\) 2.00000 0.109930 0.0549650 0.998488i \(-0.482495\pi\)
0.0549650 + 0.998488i \(0.482495\pi\)
\(332\) − 12.0000i − 0.658586i
\(333\) − 2.00000i − 0.109599i
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 14.0000i 0.762629i 0.924445 + 0.381314i \(0.124528\pi\)
−0.924445 + 0.381314i \(0.875472\pi\)
\(338\) 1.00000i 0.0543928i
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) 0 0
\(342\) − 2.00000i − 0.108148i
\(343\) 20.0000i 1.07990i
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) 24.0000i 1.28839i 0.764862 + 0.644194i \(0.222807\pi\)
−0.764862 + 0.644194i \(0.777193\pi\)
\(348\) 0 0
\(349\) −20.0000 −1.07058 −0.535288 0.844670i \(-0.679797\pi\)
−0.535288 + 0.844670i \(0.679797\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) − 6.00000i − 0.319348i −0.987170 0.159674i \(-0.948956\pi\)
0.987170 0.159674i \(-0.0510443\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) − 6.00000i − 0.317110i
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) − 2.00000i − 0.105118i
\(363\) 11.0000i 0.577350i
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) −14.0000 −0.731792
\(367\) 8.00000i 0.417597i 0.977959 + 0.208798i \(0.0669552\pi\)
−0.977959 + 0.208798i \(0.933045\pi\)
\(368\) 6.00000i 0.312772i
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 8.00000i 0.414781i
\(373\) − 26.0000i − 1.34623i −0.739538 0.673114i \(-0.764956\pi\)
0.739538 0.673114i \(-0.235044\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 2.00000i 0.102869i
\(379\) 34.0000 1.74646 0.873231 0.487306i \(-0.162020\pi\)
0.873231 + 0.487306i \(0.162020\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) − 12.0000i − 0.613171i −0.951843 0.306586i \(-0.900813\pi\)
0.951843 0.306586i \(-0.0991866\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −8.00000 −0.407189
\(387\) − 4.00000i − 0.203331i
\(388\) 4.00000i 0.203069i
\(389\) −36.0000 −1.82527 −0.912636 0.408773i \(-0.865957\pi\)
−0.912636 + 0.408773i \(0.865957\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 3.00000i 0.151523i
\(393\) − 6.00000i − 0.302660i
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) 26.0000i 1.30490i 0.757831 + 0.652451i \(0.226259\pi\)
−0.757831 + 0.652451i \(0.773741\pi\)
\(398\) − 16.0000i − 0.802008i
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 4.00000i 0.199502i
\(403\) − 8.00000i − 0.398508i
\(404\) 12.0000 0.597022
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) − 16.0000i − 0.788263i
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) 8.00000i 0.391762i
\(418\) 0 0
\(419\) −6.00000 −0.293119 −0.146560 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 0 0
\(421\) 32.0000 1.55958 0.779792 0.626038i \(-0.215325\pi\)
0.779792 + 0.626038i \(0.215325\pi\)
\(422\) 4.00000i 0.194717i
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 28.0000i 1.35501i
\(428\) 12.0000i 0.580042i
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) − 14.0000i − 0.672797i −0.941720 0.336399i \(-0.890791\pi\)
0.941720 0.336399i \(-0.109209\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) − 12.0000i − 0.574038i
\(438\) − 4.00000i − 0.191127i
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) 10.0000 0.473514
\(447\) 18.0000i 0.851371i
\(448\) − 2.00000i − 0.0944911i
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 12.0000i 0.564433i
\(453\) − 8.00000i − 0.375873i
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) − 28.0000i − 1.30978i −0.755722 0.654892i \(-0.772714\pi\)
0.755722 0.654892i \(-0.227286\pi\)
\(458\) − 4.00000i − 0.186908i
\(459\) 0 0
\(460\) 0 0
\(461\) −42.0000 −1.95614 −0.978068 0.208288i \(-0.933211\pi\)
−0.978068 + 0.208288i \(0.933211\pi\)
\(462\) 0 0
\(463\) 10.0000i 0.464739i 0.972628 + 0.232370i \(0.0746479\pi\)
−0.972628 + 0.232370i \(0.925352\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 36.0000i 1.66588i 0.553362 + 0.832941i \(0.313345\pi\)
−0.553362 + 0.832941i \(0.686655\pi\)
\(468\) − 1.00000i − 0.0462250i
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) 0 0
\(474\) −16.0000 −0.734904
\(475\) 0 0
\(476\) 0 0
\(477\) − 6.00000i − 0.274721i
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) − 26.0000i − 1.18427i
\(483\) 12.0000i 0.546019i
\(484\) 11.0000 0.500000
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 2.00000i 0.0906287i 0.998973 + 0.0453143i \(0.0144289\pi\)
−0.998973 + 0.0453143i \(0.985571\pi\)
\(488\) 14.0000i 0.633750i
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) 6.00000 0.270776 0.135388 0.990793i \(-0.456772\pi\)
0.135388 + 0.990793i \(0.456772\pi\)
\(492\) 6.00000i 0.270501i
\(493\) 0 0
\(494\) 2.00000 0.0899843
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) − 12.0000i − 0.537733i
\(499\) −14.0000 −0.626726 −0.313363 0.949633i \(-0.601456\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 30.0000i 1.33897i
\(503\) 18.0000i 0.802580i 0.915951 + 0.401290i \(0.131438\pi\)
−0.915951 + 0.401290i \(0.868562\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000i 0.0444116i
\(508\) 16.0000i 0.709885i
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) − 1.00000i − 0.0441942i
\(513\) − 2.00000i − 0.0883022i
\(514\) −12.0000 −0.529297
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 4.00000i 0.175750i
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 0 0
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) −30.0000 −1.30806
\(527\) 0 0
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 4.00000i 0.173422i
\(533\) − 6.00000i − 0.259889i
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) − 6.00000i − 0.258919i
\(538\) 12.0000i 0.517357i
\(539\) 0 0
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) − 8.00000i − 0.343629i
\(543\) − 2.00000i − 0.0858282i
\(544\) 0 0
\(545\) 0 0
\(546\) −2.00000 −0.0855921
\(547\) 20.0000i 0.855138i 0.903983 + 0.427569i \(0.140630\pi\)
−0.903983 + 0.427569i \(0.859370\pi\)
\(548\) − 6.00000i − 0.256307i
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) 0 0
\(552\) 6.00000i 0.255377i
\(553\) 32.0000i 1.36078i
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) 30.0000i 1.27114i 0.772043 + 0.635570i \(0.219235\pi\)
−0.772043 + 0.635570i \(0.780765\pi\)
\(558\) 8.00000i 0.338667i
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 30.0000i 1.26547i
\(563\) − 24.0000i − 1.01148i −0.862686 0.505740i \(-0.831220\pi\)
0.862686 0.505740i \(-0.168780\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 28.0000 1.17693
\(567\) 2.00000i 0.0839921i
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 16.0000i − 0.666089i −0.942911 0.333044i \(-0.891924\pi\)
0.942911 0.333044i \(-0.108076\pi\)
\(578\) − 17.0000i − 0.707107i
\(579\) −8.00000 −0.332469
\(580\) 0 0
\(581\) −24.0000 −0.995688
\(582\) 4.00000i 0.165805i
\(583\) 0 0
\(584\) −4.00000 −0.165521
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 12.0000i 0.495293i 0.968850 + 0.247647i \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) 3.00000i 0.123718i
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) 2.00000i 0.0821995i
\(593\) 6.00000i 0.246390i 0.992382 + 0.123195i \(0.0393141\pi\)
−0.992382 + 0.123195i \(0.960686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) − 16.0000i − 0.654836i
\(598\) − 6.00000i − 0.245358i
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 8.00000i 0.326056i
\(603\) 4.00000i 0.162893i
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 12.0000 0.487467
\(607\) − 40.0000i − 1.62355i −0.583970 0.811775i \(-0.698502\pi\)
0.583970 0.811775i \(-0.301498\pi\)
\(608\) 2.00000i 0.0811107i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 22.0000i 0.888572i 0.895885 + 0.444286i \(0.146543\pi\)
−0.895885 + 0.444286i \(0.853457\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) 0 0
\(617\) − 18.0000i − 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) − 16.0000i − 0.643614i
\(619\) −38.0000 −1.52735 −0.763674 0.645601i \(-0.776607\pi\)
−0.763674 + 0.645601i \(0.776607\pi\)
\(620\) 0 0
\(621\) −6.00000 −0.240772
\(622\) − 24.0000i − 0.962312i
\(623\) 12.0000i 0.480770i
\(624\) −1.00000 −0.0400320
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) − 2.00000i − 0.0798087i
\(629\) 0 0
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 16.0000i 0.636446i
\(633\) 4.00000i 0.158986i
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) − 3.00000i − 0.118864i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 12.0000i 0.473602i
\(643\) 4.00000i 0.157745i 0.996885 + 0.0788723i \(0.0251319\pi\)
−0.996885 + 0.0788723i \(0.974868\pi\)
\(644\) 12.0000 0.472866
\(645\) 0 0
\(646\) 0 0
\(647\) 6.00000i 0.235884i 0.993020 + 0.117942i \(0.0376297\pi\)
−0.993020 + 0.117942i \(0.962370\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) − 16.0000i − 0.626608i
\(653\) − 30.0000i − 1.17399i −0.809590 0.586995i \(-0.800311\pi\)
0.809590 0.586995i \(-0.199689\pi\)
\(654\) −4.00000 −0.156412
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) − 4.00000i − 0.156055i
\(658\) 0 0
\(659\) 18.0000 0.701180 0.350590 0.936529i \(-0.385981\pi\)
0.350590 + 0.936529i \(0.385981\pi\)
\(660\) 0 0
\(661\) −16.0000 −0.622328 −0.311164 0.950356i \(-0.600719\pi\)
−0.311164 + 0.950356i \(0.600719\pi\)
\(662\) − 2.00000i − 0.0777322i
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 0 0
\(668\) − 12.0000i − 0.464294i
\(669\) 10.0000 0.386622
\(670\) 0 0
\(671\) 0 0
\(672\) − 2.00000i − 0.0771517i
\(673\) − 2.00000i − 0.0770943i −0.999257 0.0385472i \(-0.987727\pi\)
0.999257 0.0385472i \(-0.0122730\pi\)
\(674\) 14.0000 0.539260
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) − 42.0000i − 1.61419i −0.590421 0.807096i \(-0.701038\pi\)
0.590421 0.807096i \(-0.298962\pi\)
\(678\) 12.0000i 0.460857i
\(679\) 8.00000 0.307012
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 36.0000i 1.37750i 0.724998 + 0.688751i \(0.241841\pi\)
−0.724998 + 0.688751i \(0.758159\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) − 4.00000i − 0.152610i
\(688\) 4.00000i 0.152499i
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) 26.0000 0.989087 0.494543 0.869153i \(-0.335335\pi\)
0.494543 + 0.869153i \(0.335335\pi\)
\(692\) 18.0000i 0.684257i
\(693\) 0 0
\(694\) 24.0000 0.911028
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 20.0000i 0.757011i
\(699\) 0 0
\(700\) 0 0
\(701\) 36.0000 1.35970 0.679851 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(702\) − 1.00000i − 0.0377426i
\(703\) − 4.00000i − 0.150863i
\(704\) 0 0
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) − 24.0000i − 0.902613i
\(708\) 0 0
\(709\) −20.0000 −0.751116 −0.375558 0.926799i \(-0.622549\pi\)
−0.375558 + 0.926799i \(0.622549\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) 6.00000i 0.224860i
\(713\) 48.0000i 1.79761i
\(714\) 0 0
\(715\) 0 0
\(716\) −6.00000 −0.224231
\(717\) 0 0
\(718\) − 24.0000i − 0.895672i
\(719\) −12.0000 −0.447524 −0.223762 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) 0 0
\(721\) −32.0000 −1.19174
\(722\) 15.0000i 0.558242i
\(723\) − 26.0000i − 0.966950i
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 11.0000 0.408248
\(727\) − 4.00000i − 0.148352i −0.997245 0.0741759i \(-0.976367\pi\)
0.997245 0.0741759i \(-0.0236326\pi\)
\(728\) 2.00000i 0.0741249i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 14.0000i 0.517455i
\(733\) − 50.0000i − 1.84679i −0.383849 0.923396i \(-0.625402\pi\)
0.383849 0.923396i \(-0.374598\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) 0 0
\(738\) 6.00000i 0.220863i
\(739\) −38.0000 −1.39785 −0.698926 0.715194i \(-0.746338\pi\)
−0.698926 + 0.715194i \(0.746338\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) 12.0000i 0.440534i
\(743\) − 36.0000i − 1.32071i −0.750953 0.660356i \(-0.770405\pi\)
0.750953 0.660356i \(-0.229595\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) −26.0000 −0.951928
\(747\) − 12.0000i − 0.439057i
\(748\) 0 0
\(749\) 24.0000 0.876941
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) 30.0000i 1.09326i
\(754\) 0 0
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) − 34.0000i − 1.23575i −0.786276 0.617876i \(-0.787994\pi\)
0.786276 0.617876i \(-0.212006\pi\)
\(758\) − 34.0000i − 1.23494i
\(759\) 0 0
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 16.0000i 0.579619i
\(763\) 8.00000i 0.289619i
\(764\) 0 0
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) 0 0
\(768\) − 1.00000i − 0.0360844i
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) 8.00000i 0.287926i
\(773\) 6.00000i 0.215805i 0.994161 + 0.107903i \(0.0344134\pi\)
−0.994161 + 0.107903i \(0.965587\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 4.00000 0.143592
\(777\) 4.00000i 0.143499i
\(778\) 36.0000i 1.29066i
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) −6.00000 −0.214013
\(787\) − 16.0000i − 0.570338i −0.958477 0.285169i \(-0.907950\pi\)
0.958477 0.285169i \(-0.0920498\pi\)
\(788\) − 18.0000i − 0.641223i
\(789\) −30.0000 −1.06803
\(790\) 0 0
\(791\) 24.0000 0.853342
\(792\) 0 0
\(793\) − 14.0000i − 0.497155i
\(794\) 26.0000 0.922705
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) − 30.0000i − 1.06265i −0.847167 0.531327i \(-0.821693\pi\)
0.847167 0.531327i \(-0.178307\pi\)
\(798\) 4.00000i 0.141598i
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 18.0000i 0.635602i
\(803\) 0 0
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) 12.0000i 0.422420i
\(808\) − 12.0000i − 0.422159i
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 26.0000 0.912983 0.456492 0.889728i \(-0.349106\pi\)
0.456492 + 0.889728i \(0.349106\pi\)
\(812\) 0 0
\(813\) − 8.00000i − 0.280572i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 8.00000i − 0.279885i
\(818\) 14.0000i 0.489499i
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) − 6.00000i − 0.209274i
\(823\) − 32.0000i − 1.11545i −0.830026 0.557725i \(-0.811674\pi\)
0.830026 0.557725i \(-0.188326\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) 0 0
\(827\) − 12.0000i − 0.417281i −0.977992 0.208640i \(-0.933096\pi\)
0.977992 0.208640i \(-0.0669038\pi\)
\(828\) 6.00000i 0.208514i
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) 26.0000 0.901930
\(832\) 1.00000i 0.0346688i
\(833\) 0 0
\(834\) 8.00000 0.277017
\(835\) 0 0
\(836\) 0 0
\(837\) 8.00000i 0.276520i
\(838\) 6.00000i 0.207267i
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) − 32.0000i − 1.10279i
\(843\) 30.0000i 1.03325i
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) − 22.0000i − 0.755929i
\(848\) 6.00000i 0.206041i
\(849\) 28.0000 0.960958
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) − 26.0000i − 0.890223i −0.895475 0.445112i \(-0.853164\pi\)
0.895475 0.445112i \(-0.146836\pi\)
\(854\) 28.0000 0.958140
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) − 12.0000i − 0.409912i −0.978771 0.204956i \(-0.934295\pi\)
0.978771 0.204956i \(-0.0657052\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) 12.0000 0.408959
\(862\) − 24.0000i − 0.817443i
\(863\) 12.0000i 0.408485i 0.978920 + 0.204242i \(0.0654731\pi\)
−0.978920 + 0.204242i \(0.934527\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −14.0000 −0.475739
\(867\) − 17.0000i − 0.577350i
\(868\) − 16.0000i − 0.543075i
\(869\) 0 0
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 4.00000i 0.135457i
\(873\) 4.00000i 0.135379i
\(874\) −12.0000 −0.405906
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) − 58.0000i − 1.95852i −0.202606 0.979260i \(-0.564941\pi\)
0.202606 0.979260i \(-0.435059\pi\)
\(878\) 32.0000i 1.07995i
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 3.00000i 0.101015i
\(883\) 28.0000i 0.942275i 0.882060 + 0.471138i \(0.156156\pi\)
−0.882060 + 0.471138i \(0.843844\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 30.0000i − 1.00730i −0.863907 0.503651i \(-0.831990\pi\)
0.863907 0.503651i \(-0.168010\pi\)
\(888\) 2.00000i 0.0671156i
\(889\) 32.0000 1.07325
\(890\) 0 0
\(891\) 0 0
\(892\) − 10.0000i − 0.334825i
\(893\) 0 0
\(894\) 18.0000 0.602010
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) − 6.00000i − 0.200334i
\(898\) − 30.0000i − 1.00111i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 8.00000i 0.266223i
\(904\) 12.0000 0.399114
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) − 28.0000i − 0.929725i −0.885383 0.464862i \(-0.846104\pi\)
0.885383 0.464862i \(-0.153896\pi\)
\(908\) − 12.0000i − 0.398234i
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 2.00000i 0.0662266i
\(913\) 0 0
\(914\) −28.0000 −0.926158
\(915\) 0 0
\(916\) −4.00000 −0.132164
\(917\) 12.0000i 0.396275i
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 42.0000i 1.38320i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 10.0000 0.328620
\(927\) − 16.0000i − 0.525509i
\(928\) 0 0
\(929\) 54.0000 1.77168 0.885841 0.463988i \(-0.153582\pi\)
0.885841 + 0.463988i \(0.153582\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 0 0
\(933\) − 24.0000i − 0.785725i
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) −1.00000 −0.0326860
\(937\) 2.00000i 0.0653372i 0.999466 + 0.0326686i \(0.0104006\pi\)
−0.999466 + 0.0326686i \(0.989599\pi\)
\(938\) − 8.00000i − 0.261209i
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) − 2.00000i − 0.0651635i
\(943\) 36.0000i 1.17232i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 36.0000i 1.16984i 0.811090 + 0.584921i \(0.198875\pi\)
−0.811090 + 0.584921i \(0.801125\pi\)
\(948\) 16.0000i 0.519656i
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) − 24.0000i − 0.777436i −0.921357 0.388718i \(-0.872918\pi\)
0.921357 0.388718i \(-0.127082\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) − 2.00000i − 0.0644826i
\(963\) 12.0000i 0.386695i
\(964\) −26.0000 −0.837404
\(965\) 0 0
\(966\) 12.0000 0.386094
\(967\) 50.0000i 1.60789i 0.594703 + 0.803946i \(0.297270\pi\)
−0.594703 + 0.803946i \(0.702730\pi\)
\(968\) − 11.0000i − 0.353553i
\(969\) 0 0
\(970\) 0 0
\(971\) −30.0000 −0.962746 −0.481373 0.876516i \(-0.659862\pi\)
−0.481373 + 0.876516i \(0.659862\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) − 16.0000i − 0.512936i
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) 14.0000 0.448129
\(977\) 42.0000i 1.34370i 0.740688 + 0.671850i \(0.234500\pi\)
−0.740688 + 0.671850i \(0.765500\pi\)
\(978\) − 16.0000i − 0.511624i
\(979\) 0 0
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) − 6.00000i − 0.191468i
\(983\) − 36.0000i − 1.14822i −0.818778 0.574111i \(-0.805348\pi\)
0.818778 0.574111i \(-0.194652\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) − 2.00000i − 0.0636285i
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) − 8.00000i − 0.254000i
\(993\) − 2.00000i − 0.0634681i
\(994\) 0 0
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) − 46.0000i − 1.45683i −0.685134 0.728417i \(-0.740256\pi\)
0.685134 0.728417i \(-0.259744\pi\)
\(998\) 14.0000i 0.443162i
\(999\) −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 1950.2.e.d.1249.1 2
3.2 odd 2 5850.2.e.o.5149.2 2
5.2 odd 4 1950.2.a.o.1.1 1
5.3 odd 4 390.2.a.d.1.1 1
5.4 even 2 inner 1950.2.e.d.1249.2 2
15.2 even 4 5850.2.a.g.1.1 1
15.8 even 4 1170.2.a.k.1.1 1
15.14 odd 2 5850.2.e.o.5149.1 2
20.3 even 4 3120.2.a.j.1.1 1
60.23 odd 4 9360.2.a.g.1.1 1
65.8 even 4 5070.2.b.m.1351.1 2
65.18 even 4 5070.2.b.m.1351.2 2
65.38 odd 4 5070.2.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.d.1.1 1 5.3 odd 4
1170.2.a.k.1.1 1 15.8 even 4
1950.2.a.o.1.1 1 5.2 odd 4
1950.2.e.d.1249.1 2 1.1 even 1 trivial
1950.2.e.d.1249.2 2 5.4 even 2 inner
3120.2.a.j.1.1 1 20.3 even 4
5070.2.a.t.1.1 1 65.38 odd 4
5070.2.b.m.1351.1 2 65.8 even 4
5070.2.b.m.1351.2 2 65.18 even 4
5850.2.a.g.1.1 1 15.2 even 4
5850.2.e.o.5149.1 2 15.14 odd 2
5850.2.e.o.5149.2 2 3.2 odd 2
9360.2.a.g.1.1 1 60.23 odd 4