Properties

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

$q$-expansion

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

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 \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) 0 0
\(15\) 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 \(-0.651732\pi\)
−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 \(0.313988\pi\)
−0.551677 + 0.834058i \(0.686012\pi\)
\(24\) −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 \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) − 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 \(0.0986555\pi\)
−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
\(49\) 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 \(0.240986\pi\)
−0.726844 + 0.686803i \(0.759014\pi\)
\(54\) −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 \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) − 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 \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 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 \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 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 \(0.366875\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 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 \(-0.710980\pi\)
0.615338 0.788263i \(-0.289020\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) − 4.00000i − 0.386695i −0.981130 0.193347i \(-0.938066\pi\)
0.981130 0.193347i \(-0.0619344\pi\)
\(108\) − 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 \(0.228811\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) −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 \(-0.611017\pi\)
0.341743 0.939793i \(-0.388983\pi\)
\(138\) 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 \(0.188685\pi\)
−0.829396 + 0.558661i \(0.811315\pi\)
\(158\) 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 \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 24.0000i 1.85718i 0.371113 + 0.928588i \(0.378976\pi\)
−0.371113 + 0.928588i \(0.621024\pi\)
\(168\) 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 \(-0.821365\pi\)
0.846619 0.532200i \(-0.178635\pi\)
\(174\) −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 \(0.168091\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) −18.0000 −1.29232
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) − 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 \(-0.408487\pi\)
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 \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) 4.00000i 0.265489i 0.991150 + 0.132745i \(0.0423790\pi\)
−0.991150 + 0.132745i \(0.957621\pi\)
\(228\) − 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 \(-0.893773\pi\)
0.944830 0.327561i \(-0.106227\pi\)
\(234\) −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 \(0.0198685\pi\)
−0.998053 + 0.0623783i \(0.980131\pi\)
\(258\) 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 \(0.0793300\pi\)
−0.969104 + 0.246651i \(0.920670\pi\)
\(264\) 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 \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 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 \(-0.797374\pi\)
0.804141 0.594438i \(-0.202626\pi\)
\(284\) 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 \(0.274541\pi\)
−0.650545 + 0.759468i \(0.725459\pi\)
\(294\) 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 \(0.294649\pi\)
−0.601302 + 0.799022i \(0.705351\pi\)
\(308\) 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 \(-0.830123\pi\)
0.860938 0.508710i \(-0.169877\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 0 0
\(317\) 30.0000i 1.68497i 0.538721 + 0.842484i \(0.318908\pi\)
−0.538721 + 0.842484i \(0.681092\pi\)
\(318\) 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 \(0.250472\pi\)
−0.706057 + 0.708155i \(0.749528\pi\)
\(338\) − 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 \(0.270698\pi\)
−0.659665 + 0.751559i \(0.729302\pi\)
\(348\) − 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 \(0.121524\pi\)
−0.928003 + 0.372572i \(0.878476\pi\)
\(354\) 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 \(0.314644\pi\)
−0.549957 + 0.835193i \(0.685356\pi\)
\(368\) 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 \(-0.557404\pi\)
0.179364 0.983783i \(-0.442596\pi\)
\(374\) 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 \(0.304675\pi\)
−0.575841 + 0.817562i \(0.695325\pi\)
\(384\) −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 \(-0.850819\pi\)
0.892171 0.451697i \(-0.149181\pi\)
\(398\) 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 \(0.177293\pi\)
−0.848855 + 0.528626i \(0.822707\pi\)
\(434\) 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 \(-0.768363\pi\)
0.746701 0.665160i \(-0.231637\pi\)
\(444\) 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 \(0.0148953\pi\)
−0.998905 + 0.0467780i \(0.985105\pi\)
\(458\) 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 \(0.121256\pi\)
−0.928316 + 0.371792i \(0.878744\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) 36.0000i 1.66588i 0.553362 + 0.832941i \(0.313345\pi\)
−0.553362 + 0.832941i \(0.686655\pi\)
\(468\) − 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 \(0.0580167\pi\)
−0.983436 + 0.181257i \(0.941983\pi\)
\(488\) 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 \(0.179709\pi\)
−0.844818 + 0.535054i \(0.820291\pi\)
\(504\) 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 \(0.0278730\pi\)
−0.996169 + 0.0874539i \(0.972127\pi\)
\(524\) 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 \(0.204275\pi\)
−0.801050 + 0.598597i \(0.795725\pi\)
\(548\) 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 \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) 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 \(-0.725879\pi\)
0.651546 0.758610i \(-0.274121\pi\)
\(564\) 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 \(0.0132552\pi\)
−0.999133 + 0.0416305i \(0.986745\pi\)
\(578\) 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 \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) 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 \(-0.879500\pi\)
0.929197 0.369586i \(-0.120500\pi\)
\(594\) 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 \(0.0519090\pi\)
−0.986732 + 0.162355i \(0.948091\pi\)
\(608\) − 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 \(-0.908761\pi\)
0.959200 0.282727i \(-0.0912392\pi\)
\(614\) −28.0000 −1.12999
\(615\) 0 0
\(616\) 0 0
\(617\) − 22.0000i − 0.885687i −0.896599 0.442843i \(-0.853970\pi\)
0.896599 0.442843i \(-0.146030\pi\)
\(618\) − 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 \(-0.870966\pi\)
0.918955 0.394362i \(-0.129034\pi\)
\(644\) 0 0
\(645\) 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
\(649\) −48.0000 −1.88416
\(650\) 0 0
\(651\) 0 0
\(652\) − 4.00000i − 0.156652i
\(653\) 18.0000i 0.704394i 0.935926 + 0.352197i \(0.114565\pi\)
−0.935926 + 0.352197i \(0.885435\pi\)
\(654\) 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 \(-0.938264\pi\)
0.981251 0.192736i \(-0.0617360\pi\)
\(674\) −26.0000 −1.00148
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) − 26.0000i − 0.999261i −0.866239 0.499631i \(-0.833469\pi\)
0.866239 0.499631i \(-0.166531\pi\)
\(678\) 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 \(-0.681492\pi\)
0.539779 0.841807i \(-0.318508\pi\)
\(684\) −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 \(-0.650627\pi\)
0.455744 0.890111i \(-0.349373\pi\)
\(728\) 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 \(0.0117597\pi\)
−0.999318 + 0.0369358i \(0.988240\pi\)
\(734\) −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 \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) 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 \(0.0818843\pi\)
−0.967094 + 0.254419i \(0.918116\pi\)
\(758\) − 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 \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) 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 \(-0.931391\pi\)
0.976861 0.213877i \(-0.0686091\pi\)
\(788\) 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 \(-0.896721\pi\)
0.947823 0.318796i \(-0.103279\pi\)
\(798\) 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 \(0.245552\pi\)
−0.716919 + 0.697156i \(0.754448\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 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 \(-0.877082\pi\)
0.926363 0.376633i \(-0.122918\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) − 6.00000i − 0.204956i −0.994735 0.102478i \(-0.967323\pi\)
0.994735 0.102478i \(-0.0326771\pi\)
\(858\) 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 \(0.0877920\pi\)
−0.962206 + 0.272323i \(0.912208\pi\)
\(864\) −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 \(0.283085\pi\)
−0.629926 + 0.776655i \(0.716915\pi\)
\(878\) 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 \(0.265344\pi\)
−0.672212 + 0.740359i \(0.734656\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 28.0000 0.940678
\(887\) 24.0000i 0.805841i 0.915235 + 0.402921i \(0.132005\pi\)
−0.915235 + 0.402921i \(0.867995\pi\)
\(888\) 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 \(-0.936157\pi\)
0.979953 0.199227i \(-0.0638430\pi\)
\(908\) − 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 \(-0.786847\pi\)
0.784046 0.620703i \(-0.213153\pi\)
\(938\) 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 \(0.198875\pi\)
−0.811090 + 0.584921i \(0.801125\pi\)
\(948\) 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 \(0.0309821\pi\)
−0.995267 + 0.0971795i \(0.969018\pi\)
\(954\) 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 \(-0.958942\pi\)
0.991692 0.128631i \(-0.0410584\pi\)
\(968\) − 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 \(0.0929689\pi\)
−0.957650 + 0.287936i \(0.907031\pi\)
\(978\) 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 \(0.125020\pi\)
−0.923856 + 0.382741i \(0.874980\pi\)
\(984\) −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 \(-0.768399\pi\)
0.746775 0.665077i \(-0.231601\pi\)
\(998\) − 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.3349.2 2
5.2 odd 4 4650.2.a.h.1.1 1
5.3 odd 4 930.2.a.o.1.1 1
5.4 even 2 inner 4650.2.d.n.3349.1 2
15.8 even 4 2790.2.a.c.1.1 1
20.3 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 5.3 odd 4
2790.2.a.c.1.1 1 15.8 even 4
4650.2.a.h.1.1 1 5.2 odd 4
4650.2.d.n.3349.1 2 5.4 even 2 inner
4650.2.d.n.3349.2 2 1.1 even 1 trivial
7440.2.a.j.1.1 1 20.3 even 4