Properties

Label 1050.2.g.a.799.2
Level $1050$
Weight $2$
Character 1050.799
Analytic conductor $8.384$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.38429221223\)
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 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1050.799
Dual form 1050.2.g.a.799.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} -6.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} +1.00000 q^{21} -4.00000i q^{22} -8.00000i q^{23} +1.00000 q^{24} +6.00000 q^{26} -1.00000i q^{27} +1.00000i q^{28} +2.00000 q^{29} +1.00000i q^{32} -4.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} +4.00000i q^{38} +6.00000 q^{39} -6.00000 q^{41} +1.00000i q^{42} +4.00000i q^{43} +4.00000 q^{44} +8.00000 q^{46} +1.00000i q^{48} -1.00000 q^{49} -2.00000 q^{51} +6.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} -1.00000 q^{56} +4.00000i q^{57} +2.00000i q^{58} -4.00000 q^{59} +6.00000 q^{61} +1.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} +4.00000i q^{67} -2.00000i q^{68} +8.00000 q^{69} +8.00000 q^{71} +1.00000i q^{72} -10.0000i q^{73} +10.0000 q^{74} -4.00000 q^{76} +4.00000i q^{77} +6.00000i q^{78} +1.00000 q^{81} -6.00000i q^{82} +4.00000i q^{83} -1.00000 q^{84} -4.00000 q^{86} +2.00000i q^{87} +4.00000i q^{88} +6.00000 q^{89} -6.00000 q^{91} +8.00000i q^{92} -1.00000 q^{96} -14.0000i q^{97} -1.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} - 8q^{11} + 2q^{14} + 2q^{16} + 8q^{19} + 2q^{21} + 2q^{24} + 12q^{26} + 4q^{29} - 4q^{34} + 2q^{36} + 12q^{39} - 12q^{41} + 8q^{44} + 16q^{46} - 2q^{49} - 4q^{51} + 2q^{54} - 2q^{56} - 8q^{59} + 12q^{61} - 2q^{64} + 8q^{66} + 16q^{69} + 16q^{71} + 20q^{74} - 8q^{76} + 2q^{81} - 2q^{84} - 8q^{86} + 12q^{89} - 12q^{91} - 2q^{96} + 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(7\) − 1.00000i − 0.377964i
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 6.00000i − 1.66410i −0.554700 0.832050i \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) − 4.00000i − 0.852803i
\(23\) − 8.00000i − 1.66812i −0.551677 0.834058i \(-0.686012\pi\)
0.551677 0.834058i \(-0.313988\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) − 1.00000i − 0.192450i
\(28\) 1.00000i 0.188982i
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 4.00000i − 0.696311i
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 10.0000i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) 4.00000i 0.648886i
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 1.00000i 0.154303i
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 6.00000i 0.832050i
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 4.00000i 0.529813i
\(58\) 2.00000i 0.262613i
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) − 2.00000i − 0.242536i
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 10.0000i − 1.17041i −0.810885 0.585206i \(-0.801014\pi\)
0.810885 0.585206i \(-0.198986\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 4.00000i 0.455842i
\(78\) 6.00000i 0.679366i
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 6.00000i − 0.662589i
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 2.00000i 0.214423i
\(88\) 4.00000i 0.426401i
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 8.00000i 0.834058i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) − 14.0000i − 1.42148i −0.703452 0.710742i \(-0.748359\pi\)
0.703452 0.710742i \(-0.251641\pi\)
\(98\) − 1.00000i − 0.101015i
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) − 2.00000i − 0.198030i
\(103\) − 8.00000i − 0.788263i −0.919054 0.394132i \(-0.871045\pi\)
0.919054 0.394132i \(-0.128955\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 10.0000 0.949158
\(112\) − 1.00000i − 0.0944911i
\(113\) 14.0000i 1.31701i 0.752577 + 0.658505i \(0.228811\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) 6.00000i 0.554700i
\(118\) − 4.00000i − 0.368230i
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 6.00000i 0.543214i
\(123\) − 6.00000i − 0.541002i
\(124\) 0 0
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −20.0000 −1.74741 −0.873704 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) 4.00000i 0.348155i
\(133\) − 4.00000i − 0.346844i
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) 10.0000i 0.854358i 0.904167 + 0.427179i \(0.140493\pi\)
−0.904167 + 0.427179i \(0.859507\pi\)
\(138\) 8.00000i 0.681005i
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000i 0.671345i
\(143\) 24.0000i 2.00698i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 10.0000 0.827606
\(147\) − 1.00000i − 0.0824786i
\(148\) 10.0000i 0.821995i
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) − 4.00000i − 0.324443i
\(153\) − 2.00000i − 0.161690i
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) − 10.0000i − 0.798087i −0.916932 0.399043i \(-0.869342\pi\)
0.916932 0.399043i \(-0.130658\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) 1.00000i 0.0785674i
\(163\) − 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
\(166\) −4.00000 −0.310460
\(167\) − 8.00000i − 0.619059i −0.950890 0.309529i \(-0.899829\pi\)
0.950890 0.309529i \(-0.100171\pi\)
\(168\) − 1.00000i − 0.0771517i
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) − 4.00000i − 0.304997i
\(173\) − 22.0000i − 1.67263i −0.548250 0.836315i \(-0.684706\pi\)
0.548250 0.836315i \(-0.315294\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) − 4.00000i − 0.300658i
\(178\) 6.00000i 0.449719i
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) − 6.00000i − 0.444750i
\(183\) 6.00000i 0.443533i
\(184\) −8.00000 −0.589768
\(185\) 0 0
\(186\) 0 0
\(187\) − 8.00000i − 0.585018i
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) − 2.00000i − 0.143963i −0.997406 0.0719816i \(-0.977068\pi\)
0.997406 0.0719816i \(-0.0229323\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) − 10.0000i − 0.712470i −0.934396 0.356235i \(-0.884060\pi\)
0.934396 0.356235i \(-0.115940\pi\)
\(198\) 4.00000i 0.284268i
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) − 2.00000i − 0.140720i
\(203\) − 2.00000i − 0.140372i
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 8.00000i 0.556038i
\(208\) − 6.00000i − 0.416025i
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 8.00000i 0.548151i
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 2.00000i 0.135457i
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 10.0000i 0.671156i
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) − 4.00000i − 0.264906i
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) − 2.00000i − 0.131306i
\(233\) 22.0000i 1.44127i 0.693316 + 0.720634i \(0.256149\pi\)
−0.693316 + 0.720634i \(0.743851\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) 2.00000i 0.129641i
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 5.00000i 0.321412i
\(243\) 1.00000i 0.0641500i
\(244\) −6.00000 −0.384111
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) − 24.0000i − 1.52708i
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) − 1.00000i − 0.0629941i
\(253\) 32.0000i 2.01182i
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 30.0000i − 1.87135i −0.352865 0.935674i \(-0.614792\pi\)
0.352865 0.935674i \(-0.385208\pi\)
\(258\) − 4.00000i − 0.249029i
\(259\) −10.0000 −0.621370
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) − 20.0000i − 1.23560i
\(263\) 24.0000i 1.47990i 0.672660 + 0.739952i \(0.265152\pi\)
−0.672660 + 0.739952i \(0.734848\pi\)
\(264\) −4.00000 −0.246183
\(265\) 0 0
\(266\) 4.00000 0.245256
\(267\) 6.00000i 0.367194i
\(268\) − 4.00000i − 0.244339i
\(269\) −22.0000 −1.34136 −0.670682 0.741745i \(-0.733998\pi\)
−0.670682 + 0.741745i \(0.733998\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 2.00000i 0.121268i
\(273\) − 6.00000i − 0.363137i
\(274\) −10.0000 −0.604122
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) − 10.0000i − 0.600842i −0.953807 0.300421i \(-0.902873\pi\)
0.953807 0.300421i \(-0.0971271\pi\)
\(278\) − 4.00000i − 0.239904i
\(279\) 0 0
\(280\) 0 0
\(281\) 26.0000 1.55103 0.775515 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) 0 0
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −24.0000 −1.41915
\(287\) 6.00000i 0.354169i
\(288\) − 1.00000i − 0.0589256i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 10.0000i 0.585206i
\(293\) − 30.0000i − 1.75262i −0.481749 0.876309i \(-0.659998\pi\)
0.481749 0.876309i \(-0.340002\pi\)
\(294\) 1.00000 0.0583212
\(295\) 0 0
\(296\) −10.0000 −0.581238
\(297\) 4.00000i 0.232104i
\(298\) − 6.00000i − 0.347571i
\(299\) −48.0000 −2.77591
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) − 8.00000i − 0.460348i
\(303\) − 2.00000i − 0.114897i
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) 28.0000i 1.59804i 0.601302 + 0.799022i \(0.294649\pi\)
−0.601302 + 0.799022i \(0.705351\pi\)
\(308\) − 4.00000i − 0.227921i
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) − 6.00000i − 0.339683i
\(313\) − 10.0000i − 0.565233i −0.959233 0.282617i \(-0.908798\pi\)
0.959233 0.282617i \(-0.0912024\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) 0 0
\(317\) − 18.0000i − 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) 6.00000i 0.336463i
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) − 8.00000i − 0.445823i
\(323\) 8.00000i 0.445132i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 20.0000 1.10770
\(327\) 2.00000i 0.110600i
\(328\) 6.00000i 0.331295i
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) − 4.00000i − 0.219529i
\(333\) 10.0000i 0.547997i
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) 18.0000i 0.980522i 0.871576 + 0.490261i \(0.163099\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) − 23.0000i − 1.25104i
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) 0 0
\(342\) − 4.00000i − 0.216295i
\(343\) 1.00000i 0.0539949i
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 22.0000 1.18273
\(347\) 12.0000i 0.644194i 0.946707 + 0.322097i \(0.104388\pi\)
−0.946707 + 0.322097i \(0.895612\pi\)
\(348\) − 2.00000i − 0.107211i
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) − 4.00000i − 0.213201i
\(353\) 30.0000i 1.59674i 0.602168 + 0.798369i \(0.294304\pi\)
−0.602168 + 0.798369i \(0.705696\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 2.00000i 0.105851i
\(358\) 12.0000i 0.634220i
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) − 18.0000i − 0.946059i
\(363\) 5.00000i 0.262432i
\(364\) 6.00000 0.314485
\(365\) 0 0
\(366\) −6.00000 −0.313625
\(367\) 32.0000i 1.67039i 0.549957 + 0.835193i \(0.314644\pi\)
−0.549957 + 0.835193i \(0.685356\pi\)
\(368\) − 8.00000i − 0.417029i
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) − 22.0000i − 1.13912i −0.821951 0.569558i \(-0.807114\pi\)
0.821951 0.569558i \(-0.192886\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) 0 0
\(377\) − 12.0000i − 0.618031i
\(378\) − 1.00000i − 0.0514344i
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 16.0000i 0.817562i 0.912633 + 0.408781i \(0.134046\pi\)
−0.912633 + 0.408781i \(0.865954\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) − 4.00000i − 0.203331i
\(388\) 14.0000i 0.710742i
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 1.00000i 0.0505076i
\(393\) − 20.0000i − 1.00887i
\(394\) 10.0000 0.503793
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) 6.00000i 0.301131i 0.988600 + 0.150566i \(0.0481095\pi\)
−0.988600 + 0.150566i \(0.951890\pi\)
\(398\) − 8.00000i − 0.401004i
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) − 4.00000i − 0.199502i
\(403\) 0 0
\(404\) 2.00000 0.0995037
\(405\) 0 0
\(406\) 2.00000 0.0992583
\(407\) 40.0000i 1.98273i
\(408\) 2.00000i 0.0990148i
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) −10.0000 −0.493264
\(412\) 8.00000i 0.394132i
\(413\) 4.00000i 0.196827i
\(414\) −8.00000 −0.393179
\(415\) 0 0
\(416\) 6.00000 0.294174
\(417\) − 4.00000i − 0.195881i
\(418\) − 16.0000i − 0.782586i
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 20.0000i 0.973585i
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) − 6.00000i − 0.290360i
\(428\) − 12.0000i − 0.580042i
\(429\) −24.0000 −1.15873
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) − 2.00000i − 0.0961139i −0.998845 0.0480569i \(-0.984697\pi\)
0.998845 0.0480569i \(-0.0153029\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) − 32.0000i − 1.53077i
\(438\) 10.0000i 0.477818i
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 12.0000i 0.570782i
\(443\) 4.00000i 0.190046i 0.995475 + 0.0950229i \(0.0302924\pi\)
−0.995475 + 0.0950229i \(0.969708\pi\)
\(444\) −10.0000 −0.474579
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) − 6.00000i − 0.283790i
\(448\) 1.00000i 0.0472456i
\(449\) −34.0000 −1.60456 −0.802280 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) − 14.0000i − 0.658505i
\(453\) − 8.00000i − 0.375873i
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 10.0000i 0.467780i 0.972263 + 0.233890i \(0.0751456\pi\)
−0.972263 + 0.233890i \(0.924854\pi\)
\(458\) 2.00000i 0.0934539i
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) 22.0000 1.02464 0.512321 0.858794i \(-0.328786\pi\)
0.512321 + 0.858794i \(0.328786\pi\)
\(462\) − 4.00000i − 0.186097i
\(463\) 32.0000i 1.48717i 0.668644 + 0.743583i \(0.266875\pi\)
−0.668644 + 0.743583i \(0.733125\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −22.0000 −1.01913
\(467\) 28.0000i 1.29569i 0.761774 + 0.647843i \(0.224329\pi\)
−0.761774 + 0.647843i \(0.775671\pi\)
\(468\) − 6.00000i − 0.277350i
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) 4.00000i 0.184115i
\(473\) − 16.0000i − 0.735681i
\(474\) 0 0
\(475\) 0 0
\(476\) −2.00000 −0.0916698
\(477\) 6.00000i 0.274721i
\(478\) 0 0
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) −60.0000 −2.73576
\(482\) 2.00000i 0.0910975i
\(483\) − 8.00000i − 0.364013i
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 8.00000i 0.362515i 0.983436 + 0.181257i \(0.0580167\pi\)
−0.983436 + 0.181257i \(0.941983\pi\)
\(488\) − 6.00000i − 0.271607i
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 6.00000i 0.270501i
\(493\) 4.00000i 0.180151i
\(494\) 24.0000 1.07981
\(495\) 0 0
\(496\) 0 0
\(497\) − 8.00000i − 0.358849i
\(498\) − 4.00000i − 0.179244i
\(499\) 44.0000 1.96971 0.984855 0.173379i \(-0.0554684\pi\)
0.984855 + 0.173379i \(0.0554684\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) − 12.0000i − 0.535586i
\(503\) − 24.0000i − 1.07011i −0.844818 0.535054i \(-0.820291\pi\)
0.844818 0.535054i \(-0.179709\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) −32.0000 −1.42257
\(507\) − 23.0000i − 1.02147i
\(508\) 0 0
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) 1.00000i 0.0441942i
\(513\) − 4.00000i − 0.176604i
\(514\) 30.0000 1.32324
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) − 10.0000i − 0.439375i
\(519\) 22.0000 0.965693
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) − 2.00000i − 0.0875376i
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 0 0
\(528\) − 4.00000i − 0.174078i
\(529\) −41.0000 −1.78261
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 4.00000i 0.173422i
\(533\) 36.0000i 1.55933i
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) 12.0000i 0.517838i
\(538\) − 22.0000i − 0.948487i
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 0 0
\(543\) − 18.0000i − 0.772454i
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 6.00000 0.256776
\(547\) − 12.0000i − 0.513083i −0.966533 0.256541i \(-0.917417\pi\)
0.966533 0.256541i \(-0.0825830\pi\)
\(548\) − 10.0000i − 0.427179i
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) − 8.00000i − 0.340503i
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) − 2.00000i − 0.0847427i −0.999102 0.0423714i \(-0.986509\pi\)
0.999102 0.0423714i \(-0.0134913\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 26.0000i 1.09674i
\(563\) − 44.0000i − 1.85438i −0.374593 0.927189i \(-0.622217\pi\)
0.374593 0.927189i \(-0.377783\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 4.00000 0.168133
\(567\) − 1.00000i − 0.0419961i
\(568\) − 8.00000i − 0.335673i
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) − 24.0000i − 1.00349i
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 34.0000i 1.41544i 0.706494 + 0.707719i \(0.250276\pi\)
−0.706494 + 0.707719i \(0.749724\pi\)
\(578\) 13.0000i 0.540729i
\(579\) 2.00000 0.0831172
\(580\) 0 0
\(581\) 4.00000 0.165948
\(582\) 14.0000i 0.580319i
\(583\) 24.0000i 0.993978i
\(584\) −10.0000 −0.413803
\(585\) 0 0
\(586\) 30.0000 1.23929
\(587\) − 28.0000i − 1.15568i −0.816149 0.577842i \(-0.803895\pi\)
0.816149 0.577842i \(-0.196105\pi\)
\(588\) 1.00000i 0.0412393i
\(589\) 0 0
\(590\) 0 0
\(591\) 10.0000 0.411345
\(592\) − 10.0000i − 0.410997i
\(593\) − 18.0000i − 0.739171i −0.929197 0.369586i \(-0.879500\pi\)
0.929197 0.369586i \(-0.120500\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) − 8.00000i − 0.327418i
\(598\) − 48.0000i − 1.96287i
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 4.00000i 0.163028i
\(603\) − 4.00000i − 0.162893i
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 2.00000 0.0812444
\(607\) 48.0000i 1.94826i 0.225989 + 0.974130i \(0.427439\pi\)
−0.225989 + 0.974130i \(0.572561\pi\)
\(608\) 4.00000i 0.162221i
\(609\) 2.00000 0.0810441
\(610\) 0 0
\(611\) 0 0
\(612\) 2.00000i 0.0808452i
\(613\) 42.0000i 1.69636i 0.529705 + 0.848182i \(0.322303\pi\)
−0.529705 + 0.848182i \(0.677697\pi\)
\(614\) −28.0000 −1.12999
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) − 22.0000i − 0.885687i −0.896599 0.442843i \(-0.853970\pi\)
0.896599 0.442843i \(-0.146030\pi\)
\(618\) 8.00000i 0.321807i
\(619\) 44.0000 1.76851 0.884255 0.467005i \(-0.154667\pi\)
0.884255 + 0.467005i \(0.154667\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) − 8.00000i − 0.320771i
\(623\) − 6.00000i − 0.240385i
\(624\) 6.00000 0.240192
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) − 16.0000i − 0.638978i
\(628\) 10.0000i 0.399043i
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) 20.0000i 0.794929i
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 6.00000i 0.237729i
\(638\) − 8.00000i − 0.316723i
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) − 12.0000i − 0.473602i
\(643\) 4.00000i 0.157745i 0.996885 + 0.0788723i \(0.0251319\pi\)
−0.996885 + 0.0788723i \(0.974868\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) 24.0000i 0.943537i 0.881722 + 0.471769i \(0.156384\pi\)
−0.881722 + 0.471769i \(0.843616\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 20.0000i 0.783260i
\(653\) 18.0000i 0.704394i 0.935926 + 0.352197i \(0.114565\pi\)
−0.935926 + 0.352197i \(0.885435\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 10.0000i 0.390137i
\(658\) 0 0
\(659\) 28.0000 1.09073 0.545363 0.838200i \(-0.316392\pi\)
0.545363 + 0.838200i \(0.316392\pi\)
\(660\) 0 0
\(661\) −2.00000 −0.0777910 −0.0388955 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) − 4.00000i − 0.155464i
\(663\) 12.0000i 0.466041i
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) − 16.0000i − 0.619522i
\(668\) 8.00000i 0.309529i
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) −24.0000 −0.926510
\(672\) 1.00000i 0.0385758i
\(673\) − 2.00000i − 0.0770943i −0.999257 0.0385472i \(-0.987727\pi\)
0.999257 0.0385472i \(-0.0122730\pi\)
\(674\) −18.0000 −0.693334
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) − 18.0000i − 0.691796i −0.938272 0.345898i \(-0.887574\pi\)
0.938272 0.345898i \(-0.112426\pi\)
\(678\) − 14.0000i − 0.537667i
\(679\) −14.0000 −0.537271
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) − 12.0000i − 0.459167i −0.973289 0.229584i \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) 2.00000i 0.0763048i
\(688\) 4.00000i 0.152499i
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) 22.0000i 0.836315i
\(693\) − 4.00000i − 0.151947i
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) 2.00000 0.0758098
\(697\) − 12.0000i − 0.454532i
\(698\) − 22.0000i − 0.832712i
\(699\) −22.0000 −0.832116
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) − 6.00000i − 0.226455i
\(703\) − 40.0000i − 1.50863i
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) 2.00000i 0.0752177i
\(708\) 4.00000i 0.150329i
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 6.00000i − 0.224860i
\(713\) 0 0
\(714\) −2.00000 −0.0748481
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 8.00000i 0.298557i
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) − 3.00000i − 0.111648i
\(723\) 2.00000i 0.0743808i
\(724\) 18.0000 0.668965
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) − 8.00000i − 0.296704i −0.988935 0.148352i \(-0.952603\pi\)
0.988935 0.148352i \(-0.0473968\pi\)
\(728\) 6.00000i 0.222375i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) − 6.00000i − 0.221766i
\(733\) − 6.00000i − 0.221615i −0.993842 0.110808i \(-0.964656\pi\)
0.993842 0.110808i \(-0.0353437\pi\)
\(734\) −32.0000 −1.18114
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) − 16.0000i − 0.589368i
\(738\) 6.00000i 0.220863i
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) 24.0000 0.881662
\(742\) − 6.00000i − 0.220267i
\(743\) − 24.0000i − 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 22.0000 0.805477
\(747\) − 4.00000i − 0.146352i
\(748\) 8.00000i 0.292509i
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) 48.0000 1.75154 0.875772 0.482724i \(-0.160353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) 0 0
\(753\) − 12.0000i − 0.437304i
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 1.00000 0.0363696
\(757\) 6.00000i 0.218074i 0.994038 + 0.109037i \(0.0347767\pi\)
−0.994038 + 0.109037i \(0.965223\pi\)
\(758\) 20.0000i 0.726433i
\(759\) −32.0000 −1.16153
\(760\) 0 0
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 0 0
\(763\) − 2.00000i − 0.0724049i
\(764\) 0 0
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) 24.0000i 0.866590i
\(768\) 1.00000i 0.0360844i
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) 2.00000i 0.0719816i
\(773\) 2.00000i 0.0719350i 0.999353 + 0.0359675i \(0.0114513\pi\)
−0.999353 + 0.0359675i \(0.988549\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) −14.0000 −0.502571
\(777\) − 10.0000i − 0.358748i
\(778\) 26.0000i 0.932145i
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 16.0000i 0.572159i
\(783\) − 2.00000i − 0.0714742i
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 20.0000 0.713376
\(787\) − 36.0000i − 1.28326i −0.767014 0.641631i \(-0.778258\pi\)
0.767014 0.641631i \(-0.221742\pi\)
\(788\) 10.0000i 0.356235i
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) 14.0000 0.497783
\(792\) − 4.00000i − 0.142134i
\(793\) − 36.0000i − 1.27840i
\(794\) −6.00000 −0.212932
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) 6.00000i 0.212531i 0.994338 + 0.106265i \(0.0338893\pi\)
−0.994338 + 0.106265i \(0.966111\pi\)
\(798\) 4.00000i 0.141598i
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 18.0000i 0.635602i
\(803\) 40.0000i 1.41157i
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) − 22.0000i − 0.774437i
\(808\) 2.00000i 0.0703598i
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 2.00000i 0.0701862i
\(813\) 0 0
\(814\) −40.0000 −1.40200
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) 16.0000i 0.559769i
\(818\) 22.0000i 0.769212i
\(819\) 6.00000 0.209657
\(820\) 0 0
\(821\) 38.0000 1.32621 0.663105 0.748527i \(-0.269238\pi\)
0.663105 + 0.748527i \(0.269238\pi\)
\(822\) − 10.0000i − 0.348790i
\(823\) 56.0000i 1.95204i 0.217687 + 0.976019i \(0.430149\pi\)
−0.217687 + 0.976019i \(0.569851\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) −4.00000 −0.139178
\(827\) − 36.0000i − 1.25184i −0.779886 0.625921i \(-0.784723\pi\)
0.779886 0.625921i \(-0.215277\pi\)
\(828\) − 8.00000i − 0.278019i
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 6.00000i 0.208013i
\(833\) − 2.00000i − 0.0692959i
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) 0 0
\(838\) 36.0000i 1.24360i
\(839\) −56.0000 −1.93333 −0.966667 0.256036i \(-0.917584\pi\)
−0.966667 + 0.256036i \(0.917584\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 6.00000i 0.206774i
\(843\) 26.0000i 0.895488i
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 0 0
\(847\) − 5.00000i − 0.171802i
\(848\) − 6.00000i − 0.206041i
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) −80.0000 −2.74236
\(852\) − 8.00000i − 0.274075i
\(853\) − 14.0000i − 0.479351i −0.970853 0.239675i \(-0.922959\pi\)
0.970853 0.239675i \(-0.0770410\pi\)
\(854\) 6.00000 0.205316
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 42.0000i 1.43469i 0.696717 + 0.717346i \(0.254643\pi\)
−0.696717 + 0.717346i \(0.745357\pi\)
\(858\) − 24.0000i − 0.819346i
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) 32.0000i 1.08929i 0.838666 + 0.544646i \(0.183336\pi\)
−0.838666 + 0.544646i \(0.816664\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 2.00000 0.0679628
\(867\) 13.0000i 0.441503i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) − 2.00000i − 0.0677285i
\(873\) 14.0000i 0.473828i
\(874\) 32.0000 1.08242
\(875\) 0 0
\(876\) −10.0000 −0.337869
\(877\) − 2.00000i − 0.0675352i −0.999430 0.0337676i \(-0.989249\pi\)
0.999430 0.0337676i \(-0.0107506\pi\)
\(878\) 24.0000i 0.809961i
\(879\) 30.0000 1.01187
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 1.00000i 0.0336718i
\(883\) − 20.0000i − 0.673054i −0.941674 0.336527i \(-0.890748\pi\)
0.941674 0.336527i \(-0.109252\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) − 24.0000i − 0.805841i −0.915235 0.402921i \(-0.867995\pi\)
0.915235 0.402921i \(-0.132005\pi\)
\(888\) − 10.0000i − 0.335578i
\(889\) 0 0
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) − 16.0000i − 0.535720i
\(893\) 0 0
\(894\) 6.00000 0.200670
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) − 48.0000i − 1.60267i
\(898\) − 34.0000i − 1.13459i
\(899\) 0 0
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 24.0000i 0.799113i
\(903\) 4.00000i 0.133112i
\(904\) 14.0000 0.465633
\(905\) 0 0
\(906\) 8.00000 0.265782
\(907\) 12.0000i 0.398453i 0.979953 + 0.199227i \(0.0638430\pi\)
−0.979953 + 0.199227i \(0.936157\pi\)
\(908\) − 12.0000i − 0.398234i
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 4.00000i 0.132453i
\(913\) − 16.0000i − 0.529523i
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) −2.00000 −0.0660819
\(917\) 20.0000i 0.660458i
\(918\) 2.00000i 0.0660098i
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) −28.0000 −0.922631
\(922\) 22.0000i 0.724531i
\(923\) − 48.0000i − 1.57994i
\(924\) 4.00000 0.131590
\(925\) 0 0
\(926\) −32.0000 −1.05159
\(927\) 8.00000i 0.262754i
\(928\) 2.00000i 0.0656532i
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) − 22.0000i − 0.720634i
\(933\) − 8.00000i − 0.261908i
\(934\) −28.0000 −0.916188
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) − 22.0000i − 0.718709i −0.933201 0.359354i \(-0.882997\pi\)
0.933201 0.359354i \(-0.117003\pi\)
\(938\) 4.00000i 0.130605i
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) −26.0000 −0.847576 −0.423788 0.905761i \(-0.639300\pi\)
−0.423788 + 0.905761i \(0.639300\pi\)
\(942\) 10.0000i 0.325818i
\(943\) 48.0000i 1.56310i
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) 4.00000i 0.129983i 0.997886 + 0.0649913i \(0.0207020\pi\)
−0.997886 + 0.0649913i \(0.979298\pi\)
\(948\) 0 0
\(949\) −60.0000 −1.94768
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) − 2.00000i − 0.0648204i
\(953\) − 26.0000i − 0.842223i −0.907009 0.421111i \(-0.861640\pi\)
0.907009 0.421111i \(-0.138360\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 0 0
\(957\) − 8.00000i − 0.258603i
\(958\) 16.0000i 0.516937i
\(959\) 10.0000 0.322917
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) − 60.0000i − 1.93448i
\(963\) − 12.0000i − 0.386695i
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) 8.00000 0.257396
\(967\) 8.00000i 0.257263i 0.991692 + 0.128631i \(0.0410584\pi\)
−0.991692 + 0.128631i \(0.958942\pi\)
\(968\) − 5.00000i − 0.160706i
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) 4.00000i 0.128234i
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) 18.0000i 0.575871i 0.957650 + 0.287936i \(0.0929689\pi\)
−0.957650 + 0.287936i \(0.907031\pi\)
\(978\) 20.0000i 0.639529i
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 12.0000i 0.382935i
\(983\) 24.0000i 0.765481i 0.923856 + 0.382741i \(0.125020\pi\)
−0.923856 + 0.382741i \(0.874980\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) 24.0000i 0.763542i
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) − 4.00000i − 0.126936i
\(994\) 8.00000 0.253745
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) 14.0000i 0.443384i 0.975117 + 0.221692i \(0.0711580\pi\)
−0.975117 + 0.221692i \(0.928842\pi\)
\(998\) 44.0000i 1.39280i
\(999\) −10.0000 −0.316386
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 1050.2.g.a.799.2 2
3.2 odd 2 3150.2.g.r.2899.1 2
5.2 odd 4 1050.2.a.i.1.1 1
5.3 odd 4 42.2.a.a.1.1 1
5.4 even 2 inner 1050.2.g.a.799.1 2
15.2 even 4 3150.2.a.bo.1.1 1
15.8 even 4 126.2.a.a.1.1 1
15.14 odd 2 3150.2.g.r.2899.2 2
20.3 even 4 336.2.a.d.1.1 1
20.7 even 4 8400.2.a.k.1.1 1
35.3 even 12 294.2.e.a.79.1 2
35.13 even 4 294.2.a.g.1.1 1
35.18 odd 12 294.2.e.c.79.1 2
35.23 odd 12 294.2.e.c.67.1 2
35.27 even 4 7350.2.a.f.1.1 1
35.33 even 12 294.2.e.a.67.1 2
40.3 even 4 1344.2.a.i.1.1 1
40.13 odd 4 1344.2.a.q.1.1 1
45.13 odd 12 1134.2.f.g.379.1 2
45.23 even 12 1134.2.f.j.379.1 2
45.38 even 12 1134.2.f.j.757.1 2
45.43 odd 12 1134.2.f.g.757.1 2
55.43 even 4 5082.2.a.d.1.1 1
60.23 odd 4 1008.2.a.j.1.1 1
65.38 odd 4 7098.2.a.f.1.1 1
80.3 even 4 5376.2.c.e.2689.2 2
80.13 odd 4 5376.2.c.bc.2689.1 2
80.43 even 4 5376.2.c.e.2689.1 2
80.53 odd 4 5376.2.c.bc.2689.2 2
105.23 even 12 882.2.g.h.361.1 2
105.38 odd 12 882.2.g.j.667.1 2
105.53 even 12 882.2.g.h.667.1 2
105.68 odd 12 882.2.g.j.361.1 2
105.83 odd 4 882.2.a.b.1.1 1
120.53 even 4 4032.2.a.e.1.1 1
120.83 odd 4 4032.2.a.m.1.1 1
140.3 odd 12 2352.2.q.n.961.1 2
140.23 even 12 2352.2.q.i.1537.1 2
140.83 odd 4 2352.2.a.l.1.1 1
140.103 odd 12 2352.2.q.n.1537.1 2
140.123 even 12 2352.2.q.i.961.1 2
280.13 even 4 9408.2.a.n.1.1 1
280.83 odd 4 9408.2.a.bw.1.1 1
420.83 even 4 7056.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.a.a.1.1 1 5.3 odd 4
126.2.a.a.1.1 1 15.8 even 4
294.2.a.g.1.1 1 35.13 even 4
294.2.e.a.67.1 2 35.33 even 12
294.2.e.a.79.1 2 35.3 even 12
294.2.e.c.67.1 2 35.23 odd 12
294.2.e.c.79.1 2 35.18 odd 12
336.2.a.d.1.1 1 20.3 even 4
882.2.a.b.1.1 1 105.83 odd 4
882.2.g.h.361.1 2 105.23 even 12
882.2.g.h.667.1 2 105.53 even 12
882.2.g.j.361.1 2 105.68 odd 12
882.2.g.j.667.1 2 105.38 odd 12
1008.2.a.j.1.1 1 60.23 odd 4
1050.2.a.i.1.1 1 5.2 odd 4
1050.2.g.a.799.1 2 5.4 even 2 inner
1050.2.g.a.799.2 2 1.1 even 1 trivial
1134.2.f.g.379.1 2 45.13 odd 12
1134.2.f.g.757.1 2 45.43 odd 12
1134.2.f.j.379.1 2 45.23 even 12
1134.2.f.j.757.1 2 45.38 even 12
1344.2.a.i.1.1 1 40.3 even 4
1344.2.a.q.1.1 1 40.13 odd 4
2352.2.a.l.1.1 1 140.83 odd 4
2352.2.q.i.961.1 2 140.123 even 12
2352.2.q.i.1537.1 2 140.23 even 12
2352.2.q.n.961.1 2 140.3 odd 12
2352.2.q.n.1537.1 2 140.103 odd 12
3150.2.a.bo.1.1 1 15.2 even 4
3150.2.g.r.2899.1 2 3.2 odd 2
3150.2.g.r.2899.2 2 15.14 odd 2
4032.2.a.e.1.1 1 120.53 even 4
4032.2.a.m.1.1 1 120.83 odd 4
5082.2.a.d.1.1 1 55.43 even 4
5376.2.c.e.2689.1 2 80.43 even 4
5376.2.c.e.2689.2 2 80.3 even 4
5376.2.c.bc.2689.1 2 80.13 odd 4
5376.2.c.bc.2689.2 2 80.53 odd 4
7056.2.a.k.1.1 1 420.83 even 4
7098.2.a.f.1.1 1 65.38 odd 4
7350.2.a.f.1.1 1 35.27 even 4
8400.2.a.k.1.1 1 20.7 even 4
9408.2.a.n.1.1 1 280.13 even 4
9408.2.a.bw.1.1 1 280.83 odd 4