Properties

Label 4650.2.d.n.3349.1
Level $4650$
Weight $2$
Character 4650.3349
Analytic conductor $37.130$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(37.1304369399\)
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 930)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3349.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4650.3349
Dual form 4650.2.d.n.3349.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} +6.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} +4.00000i q^{22} -8.00000i q^{23} -1.00000 q^{24} +6.00000 q^{26} -1.00000i q^{27} -6.00000 q^{29} -1.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} -6.00000 q^{39} +10.0000 q^{41} -4.00000i q^{43} +4.00000 q^{44} -8.00000 q^{46} +1.00000i q^{48} +7.00000 q^{49} +2.00000 q^{51} -6.00000i q^{52} -10.0000i q^{53} -1.00000 q^{54} -4.00000i q^{57} +6.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} +1.00000i q^{62} -1.00000 q^{64} -4.00000 q^{66} +4.00000i q^{67} +2.00000i q^{68} +8.00000 q^{69} -1.00000i q^{72} +2.00000i q^{73} +2.00000 q^{74} +4.00000 q^{76} +6.00000i q^{78} +1.00000 q^{81} -10.0000i q^{82} +4.00000i q^{83} -4.00000 q^{86} -6.00000i q^{87} -4.00000i q^{88} +14.0000 q^{89} +8.00000i q^{92} -1.00000i q^{93} +1.00000 q^{96} -18.0000i q^{97} -7.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} - 8q^{11} + 2q^{16} - 8q^{19} - 2q^{24} + 12q^{26} - 12q^{29} - 2q^{31} - 4q^{34} + 2q^{36} - 12q^{39} + 20q^{41} + 8q^{44} - 16q^{46} + 14q^{49} + 4q^{51} - 2q^{54} + 24q^{59} - 4q^{61} - 2q^{64} - 8q^{66} + 16q^{69} + 4q^{74} + 8q^{76} + 2q^{81} - 8q^{86} + 28q^{89} + 2q^{96} + 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times\).

\(n\) \(1801\) \(2977\) \(3101\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 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 \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 2.00000i − 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 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
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) − 1.00000i − 0.176777i
\(33\) − 4.00000i − 0.696311i
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 4.00000i 0.648886i
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 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
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) − 6.00000i − 0.832050i
\(53\) − 10.0000i − 1.37361i −0.726844 0.686803i \(-0.759014\pi\)
0.726844 0.686803i \(-0.240986\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) − 4.00000i − 0.529813i
\(58\) 6.00000i 0.787839i
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 1.00000i 0.127000i
\(63\) 0 0
\(64\) −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
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 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
\(82\) − 10.0000i − 1.10432i
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) − 6.00000i − 0.643268i
\(88\) − 4.00000i − 0.426401i
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 8.00000i 0.834058i
\(93\) − 1.00000i − 0.103695i
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) − 18.0000i − 1.82762i −0.406138 0.913812i \(-0.633125\pi\)
0.406138 0.913812i \(-0.366875\pi\)
\(98\) − 7.00000i − 0.707107i
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) − 2.00000i − 0.198030i
\(103\) 16.0000i 1.57653i 0.615338 + 0.788263i \(0.289020\pi\)
−0.615338 + 0.788263i \(0.710980\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) 4.00000i 0.386695i 0.981130 + 0.193347i \(0.0619344\pi\)
−0.981130 + 0.193347i \(0.938066\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) − 14.0000i − 1.31701i −0.752577 0.658505i \(-0.771189\pi\)
0.752577 0.658505i \(-0.228811\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) − 6.00000i − 0.554700i
\(118\) − 12.0000i − 1.10469i
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 2.00000i 0.181071i
\(123\) 10.0000i 0.901670i
\(124\) 1.00000 0.0898027
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 4.00000i 0.348155i
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) 22.0000i 1.87959i 0.341743 + 0.939793i \(0.388983\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) − 8.00000i − 0.681005i
\(139\) 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
\(143\) − 24.0000i − 2.00698i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 7.00000i 0.577350i
\(148\) − 2.00000i − 0.164399i
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) − 4.00000i − 0.324443i
\(153\) 2.00000i 0.161690i
\(154\) 0 0
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) − 14.0000i − 1.11732i −0.829396 0.558661i \(-0.811315\pi\)
0.829396 0.558661i \(-0.188685\pi\)
\(158\) 0 0
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) 0 0
\(162\) − 1.00000i − 0.0785674i
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) − 24.0000i − 1.85718i −0.371113 0.928588i \(-0.621024\pi\)
0.371113 0.928588i \(-0.378976\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 4.00000i 0.304997i
\(173\) 14.0000i 1.06440i 0.846619 + 0.532200i \(0.178635\pi\)
−0.846619 + 0.532200i \(0.821365\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 12.0000i 0.901975i
\(178\) − 14.0000i − 1.04934i
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) − 2.00000i − 0.147844i
\(184\) 8.00000 0.589768
\(185\) 0 0
\(186\) −1.00000 −0.0733236
\(187\) 8.00000i 0.585018i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) − 14.0000i − 1.00774i −0.863779 0.503871i \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) −18.0000 −1.29232
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) 10.0000i 0.712470i 0.934396 + 0.356235i \(0.115940\pi\)
−0.934396 + 0.356235i \(0.884060\pi\)
\(198\) − 4.00000i − 0.284268i
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) − 6.00000i − 0.422159i
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) 8.00000i 0.556038i
\(208\) 6.00000i 0.416025i
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 10.0000i 0.686803i
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) − 18.0000i − 1.21911i
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 2.00000i 0.134231i
\(223\) − 16.0000i − 1.07144i −0.844396 0.535720i \(-0.820040\pi\)
0.844396 0.535720i \(-0.179960\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) − 4.00000i − 0.265489i −0.991150 0.132745i \(-0.957621\pi\)
0.991150 0.132745i \(-0.0423790\pi\)
\(228\) 4.00000i 0.264906i
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 6.00000i − 0.393919i
\(233\) 10.0000i 0.655122i 0.944830 + 0.327561i \(0.106227\pi\)
−0.944830 + 0.327561i \(0.893773\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) 1.00000i 0.0641500i
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 10.0000 0.637577
\(247\) − 24.0000i − 1.52708i
\(248\) − 1.00000i − 0.0635001i
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 32.0000i 2.01182i
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 2.00000i − 0.124757i −0.998053 0.0623783i \(-0.980131\pi\)
0.998053 0.0623783i \(-0.0198685\pi\)
\(258\) − 4.00000i − 0.249029i
\(259\) 0 0
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 4.00000i 0.247121i
\(263\) − 8.00000i − 0.493301i −0.969104 0.246651i \(-0.920670\pi\)
0.969104 0.246651i \(-0.0793300\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) 0 0
\(267\) 14.0000i 0.856786i
\(268\) − 4.00000i − 0.244339i
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) − 2.00000i − 0.121268i
\(273\) 0 0
\(274\) 22.0000 1.32907
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) − 4.00000i − 0.239904i
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 20.0000i 1.18888i 0.804141 + 0.594438i \(0.202626\pi\)
−0.804141 + 0.594438i \(0.797374\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −24.0000 −1.41915
\(287\) 0 0
\(288\) 1.00000i 0.0589256i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 18.0000 1.05518
\(292\) − 2.00000i − 0.117041i
\(293\) − 26.0000i − 1.51894i −0.650545 0.759468i \(-0.725459\pi\)
0.650545 0.759468i \(-0.274541\pi\)
\(294\) 7.00000 0.408248
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 4.00000i 0.232104i
\(298\) − 10.0000i − 0.579284i
\(299\) 48.0000 2.77591
\(300\) 0 0
\(301\) 0 0
\(302\) − 8.00000i − 0.460348i
\(303\) 6.00000i 0.344691i
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) − 28.0000i − 1.59804i −0.601302 0.799022i \(-0.705351\pi\)
0.601302 0.799022i \(-0.294649\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) − 6.00000i − 0.339683i
\(313\) 18.0000i 1.01742i 0.860938 + 0.508710i \(0.169877\pi\)
−0.860938 + 0.508710i \(0.830123\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 0 0
\(317\) − 30.0000i − 1.68497i −0.538721 0.842484i \(-0.681092\pi\)
0.538721 0.842484i \(-0.318908\pi\)
\(318\) − 10.0000i − 0.560772i
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 8.00000i 0.445132i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 18.0000i 0.995402i
\(328\) 10.0000i 0.552158i
\(329\) 0 0
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) − 4.00000i − 0.219529i
\(333\) − 2.00000i − 0.109599i
\(334\) −24.0000 −1.31322
\(335\) 0 0
\(336\) 0 0
\(337\) − 26.0000i − 1.41631i −0.706057 0.708155i \(-0.749528\pi\)
0.706057 0.708155i \(-0.250472\pi\)
\(338\) 23.0000i 1.25104i
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) − 4.00000i − 0.216295i
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) − 28.0000i − 1.50312i −0.659665 0.751559i \(-0.729302\pi\)
0.659665 0.751559i \(-0.270698\pi\)
\(348\) 6.00000i 0.321634i
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 4.00000i 0.213201i
\(353\) − 14.0000i − 0.745145i −0.928003 0.372572i \(-0.878476\pi\)
0.928003 0.372572i \(-0.121524\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) 4.00000i 0.211407i
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 10.0000i 0.525588i
\(363\) 5.00000i 0.262432i
\(364\) 0 0
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) − 32.0000i − 1.67039i −0.549957 0.835193i \(-0.685356\pi\)
0.549957 0.835193i \(-0.314644\pi\)
\(368\) − 8.00000i − 0.417029i
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 0 0
\(372\) 1.00000i 0.0518476i
\(373\) 38.0000i 1.96757i 0.179364 + 0.983783i \(0.442596\pi\)
−0.179364 + 0.983783i \(0.557404\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) 0 0
\(377\) − 36.0000i − 1.85409i
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 24.0000i 1.22795i
\(383\) − 32.0000i − 1.63512i −0.575841 0.817562i \(-0.695325\pi\)
0.575841 0.817562i \(-0.304675\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 4.00000i 0.203331i
\(388\) 18.0000i 0.913812i
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 7.00000i 0.353553i
\(393\) − 4.00000i − 0.201773i
\(394\) 10.0000 0.503793
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) 18.0000i 0.903394i 0.892171 + 0.451697i \(0.149181\pi\)
−0.892171 + 0.451697i \(0.850819\pi\)
\(398\) − 8.00000i − 0.401004i
\(399\) 0 0
\(400\) 0 0
\(401\) 26.0000 1.29838 0.649189 0.760627i \(-0.275108\pi\)
0.649189 + 0.760627i \(0.275108\pi\)
\(402\) 4.00000i 0.199502i
\(403\) − 6.00000i − 0.298881i
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) − 8.00000i − 0.396545i
\(408\) 2.00000i 0.0990148i
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) −22.0000 −1.08518
\(412\) − 16.0000i − 0.788263i
\(413\) 0 0
\(414\) 8.00000 0.393179
\(415\) 0 0
\(416\) 6.00000 0.294174
\(417\) 4.00000i 0.195881i
\(418\) − 16.0000i − 0.782586i
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) 38.0000 1.85201 0.926003 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) − 4.00000i − 0.194717i
\(423\) 0 0
\(424\) 10.0000 0.485643
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) − 4.00000i − 0.193347i
\(429\) 24.0000 1.15873
\(430\) 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
\(433\) − 22.0000i − 1.05725i −0.848855 0.528626i \(-0.822707\pi\)
0.848855 0.528626i \(-0.177293\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −18.0000 −0.862044
\(437\) 32.0000i 1.53077i
\(438\) 2.00000i 0.0955637i
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) − 12.0000i − 0.570782i
\(443\) 28.0000i 1.33032i 0.746701 + 0.665160i \(0.231637\pi\)
−0.746701 + 0.665160i \(0.768363\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) 10.0000i 0.472984i
\(448\) 0 0
\(449\) −42.0000 −1.98210 −0.991051 0.133482i \(-0.957384\pi\)
−0.991051 + 0.133482i \(0.957384\pi\)
\(450\) 0 0
\(451\) −40.0000 −1.88353
\(452\) 14.0000i 0.658505i
\(453\) 8.00000i 0.375873i
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) − 2.00000i − 0.0935561i −0.998905 0.0467780i \(-0.985105\pi\)
0.998905 0.0467780i \(-0.0148953\pi\)
\(458\) − 10.0000i − 0.467269i
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) − 16.0000i − 0.743583i −0.928316 0.371792i \(-0.878744\pi\)
0.928316 0.371792i \(-0.121256\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) − 36.0000i − 1.66588i −0.553362 0.832941i \(-0.686655\pi\)
0.553362 0.832941i \(-0.313345\pi\)
\(468\) 6.00000i 0.277350i
\(469\) 0 0
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 12.0000i 0.552345i
\(473\) 16.0000i 0.735681i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 10.0000i 0.457869i
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) − 18.0000i − 0.819878i
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 8.00000i − 0.362515i −0.983436 0.181257i \(-0.941983\pi\)
0.983436 0.181257i \(-0.0580167\pi\)
\(488\) − 2.00000i − 0.0905357i
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) − 10.0000i − 0.450835i
\(493\) 12.0000i 0.540453i
\(494\) −24.0000 −1.07981
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) 0 0
\(498\) 4.00000i 0.179244i
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) − 12.0000i − 0.535586i
\(503\) − 24.0000i − 1.07011i −0.844818 0.535054i \(-0.820291\pi\)
0.844818 0.535054i \(-0.179709\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 32.0000 1.42257
\(507\) − 23.0000i − 1.02147i
\(508\) 0 0
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) 4.00000i 0.176604i
\(514\) −2.00000 −0.0882162
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) − 6.00000i − 0.262613i
\(523\) − 4.00000i − 0.174908i −0.996169 0.0874539i \(-0.972127\pi\)
0.996169 0.0874539i \(-0.0278730\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 2.00000i 0.0871214i
\(528\) − 4.00000i − 0.174078i
\(529\) −41.0000 −1.78261
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 60.0000i 2.59889i
\(534\) 14.0000 0.605839
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) − 4.00000i − 0.172613i
\(538\) − 10.0000i − 0.431131i
\(539\) −28.0000 −1.20605
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 16.0000i 0.687259i
\(543\) − 10.0000i − 0.429141i
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) − 28.0000i − 1.19719i −0.801050 0.598597i \(-0.795725\pi\)
0.801050 0.598597i \(-0.204275\pi\)
\(548\) − 22.0000i − 0.939793i
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 8.00000i 0.340503i
\(553\) 0 0
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) − 1.00000i − 0.0423334i
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) − 10.0000i − 0.421825i
\(563\) 36.0000i 1.51722i 0.651546 + 0.758610i \(0.274121\pi\)
−0.651546 + 0.758610i \(0.725879\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 20.0000 0.840663
\(567\) 0 0
\(568\) 0 0
\(569\) −34.0000 −1.42535 −0.712677 0.701492i \(-0.752517\pi\)
−0.712677 + 0.701492i \(0.752517\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 24.0000i 1.00349i
\(573\) − 24.0000i − 1.00261i
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) − 13.0000i − 0.540729i
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) 0 0
\(582\) − 18.0000i − 0.746124i
\(583\) 40.0000i 1.65663i
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) − 7.00000i − 0.288675i
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) −10.0000 −0.411345
\(592\) 2.00000i 0.0821995i
\(593\) 18.0000i 0.739171i 0.929197 + 0.369586i \(0.120500\pi\)
−0.929197 + 0.369586i \(0.879500\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 8.00000i 0.327418i
\(598\) − 48.0000i − 1.96287i
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) − 4.00000i − 0.162893i
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) − 8.00000i − 0.324710i −0.986732 0.162355i \(-0.948091\pi\)
0.986732 0.162355i \(-0.0519090\pi\)
\(608\) 4.00000i 0.162221i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) − 2.00000i − 0.0808452i
\(613\) 14.0000i 0.565455i 0.959200 + 0.282727i \(0.0912392\pi\)
−0.959200 + 0.282727i \(0.908761\pi\)
\(614\) −28.0000 −1.12999
\(615\) 0 0
\(616\) 0 0
\(617\) 22.0000i 0.885687i 0.896599 + 0.442843i \(0.146030\pi\)
−0.896599 + 0.442843i \(0.853970\pi\)
\(618\) 16.0000i 0.643614i
\(619\) −12.0000 −0.482321 −0.241160 0.970485i \(-0.577528\pi\)
−0.241160 + 0.970485i \(0.577528\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 0 0
\(623\) 0 0
\(624\) −6.00000 −0.240192
\(625\) 0 0
\(626\) 18.0000 0.719425
\(627\) 16.0000i 0.638978i
\(628\) 14.0000i 0.558661i
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) 4.00000i 0.158986i
\(634\) −30.0000 −1.19145
\(635\) 0 0
\(636\) −10.0000 −0.396526
\(637\) 42.0000i 1.66410i
\(638\) − 24.0000i − 0.950169i
\(639\) 0 0
\(640\) 0 0
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 4.00000i 0.157867i
\(643\) 20.0000i 0.788723i 0.918955 + 0.394362i \(0.129034\pi\)
−0.918955 + 0.394362i \(0.870966\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) − 24.0000i − 0.943537i −0.881722 0.471769i \(-0.843616\pi\)
0.881722 0.471769i \(-0.156384\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −48.0000 −1.88416
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000i 0.156652i
\(653\) − 18.0000i − 0.704394i −0.935926 0.352197i \(-0.885435\pi\)
0.935926 0.352197i \(-0.114565\pi\)
\(654\) 18.0000 0.703856
\(655\) 0 0
\(656\) 10.0000 0.390434
\(657\) − 2.00000i − 0.0780274i
\(658\) 0 0
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 20.0000i 0.777322i
\(663\) 12.0000i 0.466041i
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 48.0000i 1.85857i
\(668\) 24.0000i 0.928588i
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) 10.0000i 0.385472i 0.981251 + 0.192736i \(0.0617360\pi\)
−0.981251 + 0.192736i \(0.938264\pi\)
\(674\) −26.0000 −1.00148
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 26.0000i 0.999261i 0.866239 + 0.499631i \(0.166531\pi\)
−0.866239 + 0.499631i \(0.833469\pi\)
\(678\) − 14.0000i − 0.537667i
\(679\) 0 0
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) − 4.00000i − 0.153168i
\(683\) 44.0000i 1.68361i 0.539779 + 0.841807i \(0.318508\pi\)
−0.539779 + 0.841807i \(0.681492\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) 0 0
\(687\) 10.0000i 0.381524i
\(688\) − 4.00000i − 0.152499i
\(689\) 60.0000 2.28582
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) − 14.0000i − 0.532200i
\(693\) 0 0
\(694\) −28.0000 −1.06287
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) − 20.0000i − 0.757554i
\(698\) 30.0000i 1.13552i
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) −50.0000 −1.88847 −0.944237 0.329267i \(-0.893198\pi\)
−0.944237 + 0.329267i \(0.893198\pi\)
\(702\) − 6.00000i − 0.226455i
\(703\) − 8.00000i − 0.301726i
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) 0 0
\(708\) − 12.0000i − 0.450988i
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 14.0000i 0.524672i
\(713\) 8.00000i 0.299602i
\(714\) 0 0
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) − 16.0000i − 0.597115i
\(719\) −16.0000 −0.596699 −0.298350 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 3.00000i 0.111648i
\(723\) 18.0000i 0.669427i
\(724\) 10.0000 0.371647
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) 48.0000i 1.78022i 0.455744 + 0.890111i \(0.349373\pi\)
−0.455744 + 0.890111i \(0.650627\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 2.00000i 0.0739221i
\(733\) − 2.00000i − 0.0738717i −0.999318 0.0369358i \(-0.988240\pi\)
0.999318 0.0369358i \(-0.0117597\pi\)
\(734\) −32.0000 −1.18114
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) − 16.0000i − 0.589368i
\(738\) 10.0000i 0.368105i
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) 24.0000 0.881662
\(742\) 0 0
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) 1.00000 0.0366618
\(745\) 0 0
\(746\) 38.0000 1.39128
\(747\) − 4.00000i − 0.146352i
\(748\) − 8.00000i − 0.292509i
\(749\) 0 0
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 0 0
\(753\) 12.0000i 0.437304i
\(754\) −36.0000 −1.31104
\(755\) 0 0
\(756\) 0 0
\(757\) − 14.0000i − 0.508839i −0.967094 0.254419i \(-0.918116\pi\)
0.967094 0.254419i \(-0.0818843\pi\)
\(758\) 12.0000i 0.435860i
\(759\) −32.0000 −1.16153
\(760\) 0 0
\(761\) 2.00000 0.0724999 0.0362500 0.999343i \(-0.488459\pi\)
0.0362500 + 0.999343i \(0.488459\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 24.0000 0.868290
\(765\) 0 0
\(766\) −32.0000 −1.15621
\(767\) 72.0000i 2.59977i
\(768\) 1.00000i 0.0360844i
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) 2.00000 0.0720282
\(772\) 14.0000i 0.503871i
\(773\) 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
\(776\) 18.0000 0.646162
\(777\) 0 0
\(778\) − 18.0000i − 0.645331i
\(779\) −40.0000 −1.43315
\(780\) 0 0
\(781\) 0 0
\(782\) 16.0000i 0.572159i
\(783\) 6.00000i 0.214423i
\(784\) 7.00000 0.250000
\(785\) 0 0
\(786\) −4.00000 −0.142675
\(787\) 12.0000i 0.427754i 0.976861 + 0.213877i \(0.0686091\pi\)
−0.976861 + 0.213877i \(0.931391\pi\)
\(788\) − 10.0000i − 0.356235i
\(789\) 8.00000 0.284808
\(790\) 0 0
\(791\) 0 0
\(792\) 4.00000i 0.142134i
\(793\) − 12.0000i − 0.426132i
\(794\) 18.0000 0.638796
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) 18.0000i 0.637593i 0.947823 + 0.318796i \(0.103279\pi\)
−0.947823 + 0.318796i \(0.896721\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) − 26.0000i − 0.918092i
\(803\) − 8.00000i − 0.282314i
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) −6.00000 −0.211341
\(807\) 10.0000i 0.352017i
\(808\) 6.00000i 0.211079i
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 0 0
\(813\) − 16.0000i − 0.561144i
\(814\) −8.00000 −0.280400
\(815\) 0 0
\(816\) 2.00000 0.0700140
\(817\) 16.0000i 0.559769i
\(818\) 10.0000i 0.349642i
\(819\) 0 0
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 22.0000i 0.767338i
\(823\) − 40.0000i − 1.39431i −0.716919 0.697156i \(-0.754448\pi\)
0.716919 0.697156i \(-0.245552\pi\)
\(824\) −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 \(-0.0669038\pi\)
\(828\) − 8.00000i − 0.278019i
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) − 6.00000i − 0.208013i
\(833\) − 14.0000i − 0.485071i
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) −16.0000 −0.553372
\(837\) 1.00000i 0.0345651i
\(838\) − 20.0000i − 0.690889i
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) − 38.0000i − 1.30957i
\(843\) 10.0000i 0.344418i
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) − 10.0000i − 0.343401i
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 16.0000 0.548473
\(852\) 0 0
\(853\) 22.0000i 0.753266i 0.926363 + 0.376633i \(0.122918\pi\)
−0.926363 + 0.376633i \(0.877082\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) 6.00000i 0.204956i 0.994735 + 0.102478i \(0.0326771\pi\)
−0.994735 + 0.102478i \(0.967323\pi\)
\(858\) − 24.0000i − 0.819346i
\(859\) 52.0000 1.77422 0.887109 0.461561i \(-0.152710\pi\)
0.887109 + 0.461561i \(0.152710\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 24.0000i − 0.817443i
\(863\) − 16.0000i − 0.544646i −0.962206 0.272323i \(-0.912208\pi\)
0.962206 0.272323i \(-0.0877920\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −22.0000 −0.747590
\(867\) 13.0000i 0.441503i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 18.0000i 0.609557i
\(873\) 18.0000i 0.609208i
\(874\) 32.0000 1.08242
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) − 46.0000i − 1.55331i −0.629926 0.776655i \(-0.716915\pi\)
0.629926 0.776655i \(-0.283085\pi\)
\(878\) − 8.00000i − 0.269987i
\(879\) 26.0000 0.876958
\(880\) 0 0
\(881\) −22.0000 −0.741199 −0.370599 0.928793i \(-0.620848\pi\)
−0.370599 + 0.928793i \(0.620848\pi\)
\(882\) 7.00000i 0.235702i
\(883\) − 44.0000i − 1.48072i −0.672212 0.740359i \(-0.734656\pi\)
0.672212 0.740359i \(-0.265344\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 28.0000 0.940678
\(887\) − 24.0000i − 0.805841i −0.915235 0.402921i \(-0.867995\pi\)
0.915235 0.402921i \(-0.132005\pi\)
\(888\) − 2.00000i − 0.0671156i
\(889\) 0 0
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 16.0000i 0.535720i
\(893\) 0 0
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) 0 0
\(897\) 48.0000i 1.60267i
\(898\) 42.0000i 1.40156i
\(899\) 6.00000 0.200111
\(900\) 0 0
\(901\) −20.0000 −0.666297
\(902\) 40.0000i 1.33185i
\(903\) 0 0
\(904\) 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 \(0.936157\pi\)
\(908\) 4.00000i 0.132745i
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −16.0000 −0.530104 −0.265052 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) − 16.0000i − 0.529523i
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) 2.00000i 0.0660098i
\(919\) 56.0000 1.84727 0.923635 0.383274i \(-0.125203\pi\)
0.923635 + 0.383274i \(0.125203\pi\)
\(920\) 0 0
\(921\) 28.0000 0.922631
\(922\) − 6.00000i − 0.197599i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) −16.0000 −0.525793
\(927\) − 16.0000i − 0.525509i
\(928\) 6.00000i 0.196960i
\(929\) 38.0000 1.24674 0.623370 0.781927i \(-0.285763\pi\)
0.623370 + 0.781927i \(0.285763\pi\)
\(930\) 0 0
\(931\) −28.0000 −0.917663
\(932\) − 10.0000i − 0.327561i
\(933\) 0 0
\(934\) −36.0000 −1.17796
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) 38.0000i 1.24141i 0.784046 + 0.620703i \(0.213153\pi\)
−0.784046 + 0.620703i \(0.786847\pi\)
\(938\) 0 0
\(939\) −18.0000 −0.587408
\(940\) 0 0
\(941\) 22.0000 0.717180 0.358590 0.933495i \(-0.383258\pi\)
0.358590 + 0.933495i \(0.383258\pi\)
\(942\) − 14.0000i − 0.456145i
\(943\) − 80.0000i − 2.60516i
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) − 36.0000i − 1.16984i −0.811090 0.584921i \(-0.801125\pi\)
0.811090 0.584921i \(-0.198875\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 30.0000 0.972817
\(952\) 0 0
\(953\) − 6.00000i − 0.194359i −0.995267 0.0971795i \(-0.969018\pi\)
0.995267 0.0971795i \(-0.0309821\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) 0 0
\(957\) 24.0000i 0.775810i
\(958\) 24.0000i 0.775405i
\(959\) 0 0
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 12.0000i 0.386896i
\(963\) − 4.00000i − 0.128898i
\(964\) −18.0000 −0.579741
\(965\) 0 0
\(966\) 0 0
\(967\) 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
\(973\) 0 0
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) − 18.0000i − 0.575871i −0.957650 0.287936i \(-0.907031\pi\)
0.957650 0.287936i \(-0.0929689\pi\)
\(978\) − 4.00000i − 0.127906i
\(979\) −56.0000 −1.78977
\(980\) 0 0
\(981\) −18.0000 −0.574696
\(982\) 20.0000i 0.638226i
\(983\) − 24.0000i − 0.765481i −0.923856 0.382741i \(-0.874980\pi\)
0.923856 0.382741i \(-0.125020\pi\)
\(984\) −10.0000 −0.318788
\(985\) 0 0
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 24.0000i 0.763542i
\(989\) −32.0000 −1.01754
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 1.00000i 0.0317500i
\(993\) − 20.0000i − 0.634681i
\(994\) 0 0
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) 42.0000i 1.33015i 0.746775 + 0.665077i \(0.231601\pi\)
−0.746775 + 0.665077i \(0.768399\pi\)
\(998\) 4.00000i 0.126618i
\(999\) 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 4650.2.d.n.3349.1 2
5.2 odd 4 930.2.a.o.1.1 1
5.3 odd 4 4650.2.a.h.1.1 1
5.4 even 2 inner 4650.2.d.n.3349.2 2
15.2 even 4 2790.2.a.c.1.1 1
20.7 even 4 7440.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.o.1.1 1 5.2 odd 4
2790.2.a.c.1.1 1 15.2 even 4
4650.2.a.h.1.1 1 5.3 odd 4
4650.2.d.n.3349.1 2 1.1 even 1 trivial
4650.2.d.n.3349.2 2 5.4 even 2 inner
7440.2.a.j.1.1 1 20.7 even 4