Properties

Label 525.2.d.a.274.2
Level $525$
Weight $2$
Character 525.274
Analytic conductor $4.192$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.19214610612\)
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 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 525.274
Dual form 525.2.d.a.274.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} -1.00000 q^{14} -1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} -1.00000 q^{21} +4.00000i q^{22} -3.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} +1.00000i q^{28} +2.00000 q^{29} +5.00000i q^{32} +4.00000i q^{33} -6.00000 q^{34} -1.00000 q^{36} -6.00000i q^{37} -4.00000i q^{38} +2.00000 q^{39} +2.00000 q^{41} -1.00000i q^{42} -4.00000i q^{43} +4.00000 q^{44} -1.00000i q^{48} -1.00000 q^{49} -6.00000 q^{51} -2.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -3.00000 q^{56} -4.00000i q^{57} +2.00000i q^{58} -12.0000 q^{59} -2.00000 q^{61} -1.00000i q^{63} -7.00000 q^{64} -4.00000 q^{66} -4.00000i q^{67} +6.00000i q^{68} -3.00000i q^{72} -6.00000i q^{73} +6.00000 q^{74} -4.00000 q^{76} +4.00000i q^{77} +2.00000i q^{78} +16.0000 q^{79} +1.00000 q^{81} +2.00000i q^{82} -12.0000i q^{83} -1.00000 q^{84} +4.00000 q^{86} +2.00000i q^{87} +12.0000i q^{88} +14.0000 q^{89} +2.00000 q^{91} -5.00000 q^{96} -18.0000i q^{97} -1.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{4} - 2q^{6} - 2q^{9} + 8q^{11} - 2q^{14} - 2q^{16} - 8q^{19} - 2q^{21} - 6q^{24} + 4q^{26} + 4q^{29} - 12q^{34} - 2q^{36} + 4q^{39} + 4q^{41} + 8q^{44} - 2q^{49} - 12q^{51} + 2q^{54} - 6q^{56} - 24q^{59} - 4q^{61} - 14q^{64} - 8q^{66} + 12q^{74} - 8q^{76} + 32q^{79} + 2q^{81} - 2q^{84} + 8q^{86} + 28q^{89} + 4q^{91} - 10q^{96} - 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 0.935414 + 0.353553i \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) 1.00000i 0.577350i
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 1.00000i 0.377964i
\(8\) 3.00000i 1.06066i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) − 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
\(21\) −1.00000 −0.218218
\(22\) 4.00000i 0.852803i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) −3.00000 −0.612372
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) − 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 \(0.5\pi\)
\(32\) 5.00000i 0.883883i
\(33\) 4.00000i 0.696311i
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) − 6.00000i − 0.986394i −0.869918 0.493197i \(-0.835828\pi\)
0.869918 0.493197i \(-0.164172\pi\)
\(38\) − 4.00000i − 0.648886i
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) − 1.00000i − 0.154303i
\(43\) − 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
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) − 2.00000i − 0.277350i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −3.00000 −0.400892
\(57\) − 4.00000i − 0.529813i
\(58\) 2.00000i 0.262613i
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) − 1.00000i − 0.125988i
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 6.00000i 0.727607i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) − 3.00000i − 0.353553i
\(73\) − 6.00000i − 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 4.00000i 0.455842i
\(78\) 2.00000i 0.226455i
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000i 0.220863i
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 2.00000i 0.214423i
\(88\) 12.0000i 1.27920i
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) − 18.0000i − 1.82762i −0.406138 0.913812i \(-0.633125\pi\)
0.406138 0.913812i \(-0.366875\pi\)
\(98\) − 1.00000i − 0.101015i
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) − 6.00000i − 0.594089i
\(103\) 8.00000i 0.788263i 0.919054 + 0.394132i \(0.128955\pi\)
−0.919054 + 0.394132i \(0.871045\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) − 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
\(111\) 6.00000 0.569495
\(112\) − 1.00000i − 0.0944911i
\(113\) − 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
\(116\) 2.00000 0.185695
\(117\) 2.00000i 0.184900i
\(118\) − 12.0000i − 1.10469i
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) − 2.00000i − 0.181071i
\(123\) 2.00000i 0.180334i
\(124\) 0 0
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 3.00000i 0.265165i
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 4.00000i 0.348155i
\(133\) − 4.00000i − 0.346844i
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −18.0000 −1.54349
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 8.00000i − 0.668994i
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 6.00000 0.496564
\(147\) − 1.00000i − 0.0824786i
\(148\) − 6.00000i − 0.493197i
\(149\) −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 \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) − 12.0000i − 0.973329i
\(153\) − 6.00000i − 0.485071i
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) 16.0000i 1.27289i
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 8.00000i 0.619059i 0.950890 + 0.309529i \(0.100171\pi\)
−0.950890 + 0.309529i \(0.899829\pi\)
\(168\) − 3.00000i − 0.231455i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) − 4.00000i − 0.304997i
\(173\) − 10.0000i − 0.760286i −0.924928 0.380143i \(-0.875875\pi\)
0.924928 0.380143i \(-0.124125\pi\)
\(174\) −2.00000 −0.151620
\(175\) 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 \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −26.0000 −1.93256 −0.966282 0.257485i \(-0.917106\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 2.00000i 0.148250i
\(183\) − 2.00000i − 0.147844i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 24.0000i 1.75505i
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) − 7.00000i − 0.505181i
\(193\) 2.00000i 0.143963i 0.997406 + 0.0719816i \(0.0229323\pi\)
−0.997406 + 0.0719816i \(0.977068\pi\)
\(194\) 18.0000 1.29232
\(195\) 0 0
\(196\) −1.00000 −0.0714286
\(197\) − 22.0000i − 1.56744i −0.621117 0.783718i \(-0.713321\pi\)
0.621117 0.783718i \(-0.286679\pi\)
\(198\) − 4.00000i − 0.284268i
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 14.0000i 0.985037i
\(203\) 2.00000i 0.140372i
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) 2.00000i 0.138675i
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) 0 0
\(216\) 3.00000 0.204124
\(217\) 0 0
\(218\) 18.0000i 1.21911i
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 6.00000i 0.402694i
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) −5.00000 −0.334077
\(225\) 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
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 6.00000i 0.393919i
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 16.0000i 1.03931i
\(238\) − 6.00000i − 0.388922i
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 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
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) 8.00000i 0.509028i
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) − 1.00000i − 0.0629941i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) − 26.0000i − 1.62184i −0.585160 0.810918i \(-0.698968\pi\)
0.585160 0.810918i \(-0.301032\pi\)
\(258\) 4.00000i 0.249029i
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 4.00000i 0.247121i
\(263\) 16.0000i 0.986602i 0.869859 + 0.493301i \(0.164210\pi\)
−0.869859 + 0.493301i \(0.835790\pi\)
\(264\) −12.0000 −0.738549
\(265\) 0 0
\(266\) 4.00000 0.245256
\(267\) 14.0000i 0.856786i
\(268\) − 4.00000i − 0.244339i
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) − 6.00000i − 0.363803i
\(273\) 2.00000i 0.121046i
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) − 22.0000i − 1.32185i −0.750451 0.660926i \(-0.770164\pi\)
0.750451 0.660926i \(-0.229836\pi\)
\(278\) − 12.0000i − 0.719712i
\(279\) 0 0
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 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
\(286\) 8.00000 0.473050
\(287\) 2.00000i 0.118056i
\(288\) − 5.00000i − 0.294628i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 18.0000 1.05518
\(292\) − 6.00000i − 0.351123i
\(293\) 14.0000i 0.817889i 0.912559 + 0.408944i \(0.134103\pi\)
−0.912559 + 0.408944i \(0.865897\pi\)
\(294\) 1.00000 0.0583212
\(295\) 0 0
\(296\) 18.0000 1.04623
\(297\) − 4.00000i − 0.232104i
\(298\) − 6.00000i − 0.347571i
\(299\) 0 0
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 8.00000i 0.460348i
\(303\) 14.0000i 0.804279i
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) − 4.00000i − 0.228292i −0.993464 0.114146i \(-0.963587\pi\)
0.993464 0.114146i \(-0.0364132\pi\)
\(308\) 4.00000i 0.227921i
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 6.00000i 0.339683i
\(313\) 26.0000i 1.46961i 0.678280 + 0.734803i \(0.262726\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) 16.0000 0.900070
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) − 24.0000i − 1.33540i
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 18.0000i 0.995402i
\(328\) 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 \(0.535063\pi\)
\(332\) − 12.0000i − 0.658586i
\(333\) 6.00000i 0.328798i
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) 14.0000i 0.762629i 0.924445 + 0.381314i \(0.124528\pi\)
−0.924445 + 0.381314i \(0.875472\pi\)
\(338\) 9.00000i 0.489535i
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) 0 0
\(342\) 4.00000i 0.216295i
\(343\) − 1.00000i − 0.0539949i
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 10.0000 0.537603
\(347\) 28.0000i 1.50312i 0.659665 + 0.751559i \(0.270698\pi\)
−0.659665 + 0.751559i \(0.729302\pi\)
\(348\) 2.00000i 0.107211i
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 20.0000i 1.06600i
\(353\) 10.0000i 0.532246i 0.963939 + 0.266123i \(0.0857428\pi\)
−0.963939 + 0.266123i \(0.914257\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) − 6.00000i − 0.317554i
\(358\) 4.00000i 0.211407i
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) − 26.0000i − 1.36653i
\(363\) 5.00000i 0.262432i
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) − 10.0000i − 0.517780i −0.965907 0.258890i \(-0.916643\pi\)
0.965907 0.258890i \(-0.0833568\pi\)
\(374\) −24.0000 −1.24101
\(375\) 0 0
\(376\) 0 0
\(377\) − 4.00000i − 0.206010i
\(378\) 1.00000i 0.0514344i
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 8.00000i − 0.409316i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) 4.00000i 0.203331i
\(388\) − 18.0000i − 0.913812i
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) − 3.00000i − 0.151523i
\(393\) 4.00000i 0.201773i
\(394\) 22.0000 1.10834
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) 18.0000i 0.903394i 0.892171 + 0.451697i \(0.149181\pi\)
−0.892171 + 0.451697i \(0.850819\pi\)
\(398\) − 24.0000i − 1.20301i
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 4.00000i 0.199502i
\(403\) 0 0
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) − 24.0000i − 1.18964i
\(408\) − 18.0000i − 0.891133i
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) 8.00000i 0.394132i
\(413\) − 12.0000i − 0.590481i
\(414\) 0 0
\(415\) 0 0
\(416\) 10.0000 0.490290
\(417\) − 12.0000i − 0.587643i
\(418\) − 16.0000i − 0.782586i
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 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
\(424\) −18.0000 −0.874157
\(425\) 0 0
\(426\) 0 0
\(427\) − 2.00000i − 0.0967868i
\(428\) − 4.00000i − 0.193347i
\(429\) 8.00000 0.386244
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) − 14.0000i − 0.672797i −0.941720 0.336399i \(-0.890791\pi\)
0.941720 0.336399i \(-0.109209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 18.0000 0.862044
\(437\) 0 0
\(438\) 6.00000i 0.286691i
\(439\) 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
\(443\) 36.0000i 1.71041i 0.518289 + 0.855206i \(0.326569\pi\)
−0.518289 + 0.855206i \(0.673431\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) − 6.00000i − 0.283790i
\(448\) − 7.00000i − 0.330719i
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) − 14.0000i − 0.658505i
\(453\) 8.00000i 0.375873i
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 12.0000 0.561951
\(457\) − 10.0000i − 0.467780i −0.972263 0.233890i \(-0.924854\pi\)
0.972263 0.233890i \(-0.0751456\pi\)
\(458\) 10.0000i 0.467269i
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) −10.0000 −0.465746 −0.232873 0.972507i \(-0.574813\pi\)
−0.232873 + 0.972507i \(0.574813\pi\)
\(462\) − 4.00000i − 0.186097i
\(463\) 16.0000i 0.743583i 0.928316 + 0.371792i \(0.121256\pi\)
−0.928316 + 0.371792i \(0.878744\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) − 36.0000i − 1.66588i −0.553362 0.832941i \(-0.686655\pi\)
0.553362 0.832941i \(-0.313345\pi\)
\(468\) 2.00000i 0.0924500i
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) − 36.0000i − 1.65703i
\(473\) − 16.0000i − 0.735681i
\(474\) −16.0000 −0.734904
\(475\) 0 0
\(476\) −6.00000 −0.275010
\(477\) − 6.00000i − 0.274721i
\(478\) − 24.0000i − 1.09773i
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 2.00000i 0.0910975i
\(483\) 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 \(0.941983\pi\)
\(488\) − 6.00000i − 0.271607i
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 2.00000i 0.0901670i
\(493\) 12.0000i 0.540453i
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 12.0000i 0.537733i
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) −8.00000 −0.357414
\(502\) − 20.0000i − 0.892644i
\(503\) 24.0000i 1.07011i 0.844818 + 0.535054i \(0.179709\pi\)
−0.844818 + 0.535054i \(0.820291\pi\)
\(504\) 3.00000 0.133631
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000i 0.399704i
\(508\) 0 0
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) − 11.0000i − 0.486136i
\(513\) 4.00000i 0.176604i
\(514\) 26.0000 1.14681
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 6.00000i 0.263625i
\(519\) 10.0000 0.438951
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) − 2.00000i − 0.0875376i
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 0 0
\(528\) − 4.00000i − 0.174078i
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) − 4.00000i − 0.173422i
\(533\) − 4.00000i − 0.173259i
\(534\) −14.0000 −0.605839
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) 4.00000i 0.172613i
\(538\) − 6.00000i − 0.258678i
\(539\) −4.00000 −0.172292
\(540\) 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
\(543\) − 26.0000i − 1.11577i
\(544\) −30.0000 −1.28624
\(545\) 0 0
\(546\) −2.00000 −0.0855921
\(547\) − 4.00000i − 0.171028i −0.996337 0.0855138i \(-0.972747\pi\)
0.996337 0.0855138i \(-0.0272532\pi\)
\(548\) 6.00000i 0.256307i
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) 16.0000i 0.680389i
\(554\) 22.0000 0.934690
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) 2.00000i 0.0847427i 0.999102 + 0.0423714i \(0.0134913\pi\)
−0.999102 + 0.0423714i \(0.986509\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) − 22.0000i − 0.928014i
\(563\) 4.00000i 0.168580i 0.996441 + 0.0842900i \(0.0268622\pi\)
−0.996441 + 0.0842900i \(0.973138\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 20.0000 0.840663
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) − 8.00000i − 0.334497i
\(573\) − 8.00000i − 0.334205i
\(574\) −2.00000 −0.0834784
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) − 34.0000i − 1.41544i −0.706494 0.707719i \(-0.749724\pi\)
0.706494 0.707719i \(-0.250276\pi\)
\(578\) − 19.0000i − 0.790296i
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 18.0000i 0.746124i
\(583\) 24.0000i 0.993978i
\(584\) 18.0000 0.744845
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) − 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
\(591\) 22.0000 0.904959
\(592\) 6.00000i 0.246598i
\(593\) − 6.00000i − 0.246390i −0.992382 0.123195i \(-0.960686\pi\)
0.992382 0.123195i \(-0.0393141\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) − 24.0000i − 0.982255i
\(598\) 0 0
\(599\) −48.0000 −1.96123 −0.980613 0.195952i \(-0.937220\pi\)
−0.980613 + 0.195952i \(0.937220\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 4.00000i 0.163028i
\(603\) 4.00000i 0.162893i
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) −14.0000 −0.568711
\(607\) 16.0000i 0.649420i 0.945814 + 0.324710i \(0.105267\pi\)
−0.945814 + 0.324710i \(0.894733\pi\)
\(608\) − 20.0000i − 0.811107i
\(609\) −2.00000 −0.0810441
\(610\) 0 0
\(611\) 0 0
\(612\) − 6.00000i − 0.242536i
\(613\) − 26.0000i − 1.05013i −0.851062 0.525065i \(-0.824041\pi\)
0.851062 0.525065i \(-0.175959\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) −12.0000 −0.483494
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) − 8.00000i − 0.321807i
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 24.0000i − 0.962312i
\(623\) 14.0000i 0.560898i
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) −26.0000 −1.03917
\(627\) − 16.0000i − 0.638978i
\(628\) 2.00000i 0.0798087i
\(629\) 36.0000 1.43541
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 48.0000i 1.90934i
\(633\) 4.00000i 0.158986i
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 2.00000i 0.0792429i
\(638\) 8.00000i 0.316723i
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 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
\(646\) 24.0000 0.944267
\(647\) 40.0000i 1.57256i 0.617869 + 0.786281i \(0.287996\pi\)
−0.617869 + 0.786281i \(0.712004\pi\)
\(648\) 3.00000i 0.117851i
\(649\) −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
\(656\) −2.00000 −0.0780869
\(657\) 6.00000i 0.234082i
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) − 4.00000i − 0.155464i
\(663\) 12.0000i 0.466041i
\(664\) 36.0000 1.39707
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 0 0
\(668\) 8.00000i 0.309529i
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) − 5.00000i − 0.192879i
\(673\) 34.0000i 1.31060i 0.755367 + 0.655302i \(0.227459\pi\)
−0.755367 + 0.655302i \(0.772541\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) 18.0000i 0.691796i 0.938272 + 0.345898i \(0.112426\pi\)
−0.938272 + 0.345898i \(0.887574\pi\)
\(678\) 14.0000i 0.537667i
\(679\) 18.0000 0.690777
\(680\) 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
\(687\) 10.0000i 0.381524i
\(688\) 4.00000i 0.152499i
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) − 10.0000i − 0.380143i
\(693\) − 4.00000i − 0.151947i
\(694\) −28.0000 −1.06287
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) 12.0000i 0.454532i
\(698\) 2.00000i 0.0757011i
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) − 2.00000i − 0.0754851i
\(703\) 24.0000i 0.905177i
\(704\) −28.0000 −1.05529
\(705\) 0 0
\(706\) −10.0000 −0.376355
\(707\) 14.0000i 0.526524i
\(708\) − 12.0000i − 0.450988i
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) 42.0000i 1.57402i
\(713\) 0 0
\(714\) 6.00000 0.224544
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) − 24.0000i − 0.896296i
\(718\) − 32.0000i − 1.19423i
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) − 3.00000i − 0.111648i
\(723\) 2.00000i 0.0743808i
\(724\) −26.0000 −0.966282
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) 40.0000i 1.48352i 0.670667 + 0.741759i \(0.266008\pi\)
−0.670667 + 0.741759i \(0.733992\pi\)
\(728\) 6.00000i 0.222375i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) − 2.00000i − 0.0739221i
\(733\) − 18.0000i − 0.664845i −0.943131 0.332423i \(-0.892134\pi\)
0.943131 0.332423i \(-0.107866\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 16.0000i − 0.589368i
\(738\) − 2.00000i − 0.0736210i
\(739\) −36.0000 −1.32428 −0.662141 0.749380i \(-0.730352\pi\)
−0.662141 + 0.749380i \(0.730352\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) − 6.00000i − 0.220267i
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 10.0000 0.366126
\(747\) 12.0000i 0.439057i
\(748\) 24.0000i 0.877527i
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) − 20.0000i − 0.728841i
\(754\) 4.00000 0.145671
\(755\) 0 0
\(756\) 1.00000 0.0363696
\(757\) 10.0000i 0.363456i 0.983349 + 0.181728i \(0.0581691\pi\)
−0.983349 + 0.181728i \(0.941831\pi\)
\(758\) − 12.0000i − 0.435860i
\(759\) 0 0
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) 18.0000i 0.651644i
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 0 0
\(767\) 24.0000i 0.866590i
\(768\) − 17.0000i − 0.613435i
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) 26.0000 0.936367
\(772\) 2.00000i 0.0719816i
\(773\) 14.0000i 0.503545i 0.967786 + 0.251773i \(0.0810135\pi\)
−0.967786 + 0.251773i \(0.918987\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 54.0000 1.93849
\(777\) 6.00000i 0.215249i
\(778\) − 6.00000i − 0.215110i
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) − 2.00000i − 0.0714742i
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) −4.00000 −0.142675
\(787\) 44.0000i 1.56843i 0.620489 + 0.784215i \(0.286934\pi\)
−0.620489 + 0.784215i \(0.713066\pi\)
\(788\) − 22.0000i − 0.783718i
\(789\) −16.0000 −0.569615
\(790\) 0 0
\(791\) 14.0000 0.497783
\(792\) − 12.0000i − 0.426401i
\(793\) 4.00000i 0.142044i
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) −24.0000 −0.850657
\(797\) 26.0000i 0.920967i 0.887668 + 0.460484i \(0.152324\pi\)
−0.887668 + 0.460484i \(0.847676\pi\)
\(798\) 4.00000i 0.141598i
\(799\) 0 0
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) − 30.0000i − 1.05934i
\(803\) − 24.0000i − 0.846942i
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) − 6.00000i − 0.211210i
\(808\) 42.0000i 1.47755i
\(809\) −42.0000 −1.47664 −0.738321 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) 0 0
\(811\) 44.0000 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) 2.00000i 0.0701862i
\(813\) 16.0000i 0.561144i
\(814\) 24.0000 0.841200
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 16.0000i 0.559769i
\(818\) 22.0000i 0.769212i
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) 38.0000 1.32621 0.663105 0.748527i \(-0.269238\pi\)
0.663105 + 0.748527i \(0.269238\pi\)
\(822\) − 6.00000i − 0.209274i
\(823\) 24.0000i 0.836587i 0.908312 + 0.418294i \(0.137372\pi\)
−0.908312 + 0.418294i \(0.862628\pi\)
\(824\) −24.0000 −0.836080
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 0 0
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) 22.0000 0.763172
\(832\) 14.0000i 0.485363i
\(833\) − 6.00000i − 0.207888i
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) −16.0000 −0.553372
\(837\) 0 0
\(838\) 12.0000i 0.414533i
\(839\) 8.00000 0.276191 0.138095 0.990419i \(-0.455902\pi\)
0.138095 + 0.990419i \(0.455902\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 38.0000i 1.30957i
\(843\) − 22.0000i − 0.757720i
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) 5.00000i 0.171802i
\(848\) − 6.00000i − 0.206041i
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 10.0000i − 0.342393i −0.985237 0.171197i \(-0.945237\pi\)
0.985237 0.171197i \(-0.0547634\pi\)
\(854\) 2.00000 0.0684386
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 14.0000i 0.478231i 0.970991 + 0.239115i \(0.0768574\pi\)
−0.970991 + 0.239115i \(0.923143\pi\)
\(858\) 8.00000i 0.273115i
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 0 0
\(861\) −2.00000 −0.0681598
\(862\) − 24.0000i − 0.817443i
\(863\) − 24.0000i − 0.816970i −0.912765 0.408485i \(-0.866057\pi\)
0.912765 0.408485i \(-0.133943\pi\)
\(864\) 5.00000 0.170103
\(865\) 0 0
\(866\) 14.0000 0.475739
\(867\) − 19.0000i − 0.645274i
\(868\) 0 0
\(869\) 64.0000 2.17105
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 54.0000i 1.82867i
\(873\) 18.0000i 0.609208i
\(874\) 0 0
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) − 46.0000i − 1.55331i −0.629926 0.776655i \(-0.716915\pi\)
0.629926 0.776655i \(-0.283085\pi\)
\(878\) 24.0000i 0.809961i
\(879\) −14.0000 −0.472208
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 1.00000i 0.0336718i
\(883\) − 28.0000i − 0.942275i −0.882060 0.471138i \(-0.843844\pi\)
0.882060 0.471138i \(-0.156156\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) − 8.00000i − 0.268614i −0.990940 0.134307i \(-0.957119\pi\)
0.990940 0.134307i \(-0.0428808\pi\)
\(888\) 18.0000i 0.604040i
\(889\) 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
\(896\) −3.00000 −0.100223
\(897\) 0 0
\(898\) 30.0000i 1.00111i
\(899\) 0 0
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 8.00000i 0.266371i
\(903\) 4.00000i 0.133112i
\(904\) 42.0000 1.39690
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) 4.00000i 0.132818i 0.997792 + 0.0664089i \(0.0211542\pi\)
−0.997792 + 0.0664089i \(0.978846\pi\)
\(908\) 12.0000i 0.398234i
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 4.00000i 0.132453i
\(913\) − 48.0000i − 1.58857i
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) 4.00000i 0.132092i
\(918\) 6.00000i 0.198030i
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) − 10.0000i − 0.329332i
\(923\) 0 0
\(924\) −4.00000 −0.131590
\(925\) 0 0
\(926\) −16.0000 −0.525793
\(927\) − 8.00000i − 0.262754i
\(928\) 10.0000i 0.328266i
\(929\) −26.0000 −0.853032 −0.426516 0.904480i \(-0.640259\pi\)
−0.426516 + 0.904480i \(0.640259\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) − 6.00000i − 0.196537i
\(933\) − 24.0000i − 0.785725i
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) − 42.0000i − 1.37208i −0.727564 0.686040i \(-0.759347\pi\)
0.727564 0.686040i \(-0.240653\pi\)
\(938\) 4.00000i 0.130605i
\(939\) −26.0000 −0.848478
\(940\) 0 0
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) − 2.00000i − 0.0651635i
\(943\) 0 0
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) − 44.0000i − 1.42981i −0.699223 0.714904i \(-0.746470\pi\)
0.699223 0.714904i \(-0.253530\pi\)
\(948\) 16.0000i 0.519656i
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) − 18.0000i − 0.583383i
\(953\) 26.0000i 0.842223i 0.907009 + 0.421111i \(0.138360\pi\)
−0.907009 + 0.421111i \(0.861640\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) 8.00000i 0.258603i
\(958\) 16.0000i 0.516937i
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) − 12.0000i − 0.386896i
\(963\) 4.00000i 0.128898i
\(964\) 2.00000 0.0644157
\(965\) 0 0
\(966\) 0 0
\(967\) − 40.0000i − 1.28631i −0.765735 0.643157i \(-0.777624\pi\)
0.765735 0.643157i \(-0.222376\pi\)
\(968\) 15.0000i 0.482118i
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) − 12.0000i − 0.384702i
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 30.0000i 0.959785i 0.877327 + 0.479893i \(0.159324\pi\)
−0.877327 + 0.479893i \(0.840676\pi\)
\(978\) − 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 \(0.874980\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) 8.00000i 0.254514i
\(989\) 0 0
\(990\) 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
\(994\) 0 0
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) 26.0000i 0.823428i 0.911313 + 0.411714i \(0.135070\pi\)
−0.911313 + 0.411714i \(0.864930\pi\)
\(998\) − 4.00000i − 0.126618i
\(999\) −6.00000 −0.189832
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 525.2.d.a.274.2 2
3.2 odd 2 1575.2.d.a.1324.1 2
5.2 odd 4 21.2.a.a.1.1 1
5.3 odd 4 525.2.a.d.1.1 1
5.4 even 2 inner 525.2.d.a.274.1 2
15.2 even 4 63.2.a.a.1.1 1
15.8 even 4 1575.2.a.c.1.1 1
15.14 odd 2 1575.2.d.a.1324.2 2
20.3 even 4 8400.2.a.bn.1.1 1
20.7 even 4 336.2.a.a.1.1 1
35.2 odd 12 147.2.e.b.67.1 2
35.12 even 12 147.2.e.c.67.1 2
35.13 even 4 3675.2.a.n.1.1 1
35.17 even 12 147.2.e.c.79.1 2
35.27 even 4 147.2.a.a.1.1 1
35.32 odd 12 147.2.e.b.79.1 2
40.27 even 4 1344.2.a.s.1.1 1
40.37 odd 4 1344.2.a.g.1.1 1
45.2 even 12 567.2.f.b.190.1 2
45.7 odd 12 567.2.f.g.190.1 2
45.22 odd 12 567.2.f.g.379.1 2
45.32 even 12 567.2.f.b.379.1 2
55.32 even 4 2541.2.a.j.1.1 1
60.47 odd 4 1008.2.a.l.1.1 1
65.12 odd 4 3549.2.a.c.1.1 1
80.27 even 4 5376.2.c.l.2689.2 2
80.37 odd 4 5376.2.c.r.2689.1 2
80.67 even 4 5376.2.c.l.2689.1 2
80.77 odd 4 5376.2.c.r.2689.2 2
85.67 odd 4 6069.2.a.b.1.1 1
95.37 even 4 7581.2.a.d.1.1 1
105.2 even 12 441.2.e.a.361.1 2
105.17 odd 12 441.2.e.b.226.1 2
105.32 even 12 441.2.e.a.226.1 2
105.47 odd 12 441.2.e.b.361.1 2
105.62 odd 4 441.2.a.f.1.1 1
120.77 even 4 4032.2.a.h.1.1 1
120.107 odd 4 4032.2.a.k.1.1 1
140.27 odd 4 2352.2.a.v.1.1 1
140.47 odd 12 2352.2.q.e.1537.1 2
140.67 even 12 2352.2.q.x.961.1 2
140.87 odd 12 2352.2.q.e.961.1 2
140.107 even 12 2352.2.q.x.1537.1 2
165.32 odd 4 7623.2.a.g.1.1 1
280.27 odd 4 9408.2.a.m.1.1 1
280.237 even 4 9408.2.a.bv.1.1 1
420.167 even 4 7056.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.2.a.a.1.1 1 5.2 odd 4
63.2.a.a.1.1 1 15.2 even 4
147.2.a.a.1.1 1 35.27 even 4
147.2.e.b.67.1 2 35.2 odd 12
147.2.e.b.79.1 2 35.32 odd 12
147.2.e.c.67.1 2 35.12 even 12
147.2.e.c.79.1 2 35.17 even 12
336.2.a.a.1.1 1 20.7 even 4
441.2.a.f.1.1 1 105.62 odd 4
441.2.e.a.226.1 2 105.32 even 12
441.2.e.a.361.1 2 105.2 even 12
441.2.e.b.226.1 2 105.17 odd 12
441.2.e.b.361.1 2 105.47 odd 12
525.2.a.d.1.1 1 5.3 odd 4
525.2.d.a.274.1 2 5.4 even 2 inner
525.2.d.a.274.2 2 1.1 even 1 trivial
567.2.f.b.190.1 2 45.2 even 12
567.2.f.b.379.1 2 45.32 even 12
567.2.f.g.190.1 2 45.7 odd 12
567.2.f.g.379.1 2 45.22 odd 12
1008.2.a.l.1.1 1 60.47 odd 4
1344.2.a.g.1.1 1 40.37 odd 4
1344.2.a.s.1.1 1 40.27 even 4
1575.2.a.c.1.1 1 15.8 even 4
1575.2.d.a.1324.1 2 3.2 odd 2
1575.2.d.a.1324.2 2 15.14 odd 2
2352.2.a.v.1.1 1 140.27 odd 4
2352.2.q.e.961.1 2 140.87 odd 12
2352.2.q.e.1537.1 2 140.47 odd 12
2352.2.q.x.961.1 2 140.67 even 12
2352.2.q.x.1537.1 2 140.107 even 12
2541.2.a.j.1.1 1 55.32 even 4
3549.2.a.c.1.1 1 65.12 odd 4
3675.2.a.n.1.1 1 35.13 even 4
4032.2.a.h.1.1 1 120.77 even 4
4032.2.a.k.1.1 1 120.107 odd 4
5376.2.c.l.2689.1 2 80.67 even 4
5376.2.c.l.2689.2 2 80.27 even 4
5376.2.c.r.2689.1 2 80.37 odd 4
5376.2.c.r.2689.2 2 80.77 odd 4
6069.2.a.b.1.1 1 85.67 odd 4
7056.2.a.p.1.1 1 420.167 even 4
7581.2.a.d.1.1 1 95.37 even 4
7623.2.a.g.1.1 1 165.32 odd 4
8400.2.a.bn.1.1 1 20.3 even 4
9408.2.a.m.1.1 1 280.27 odd 4
9408.2.a.bv.1.1 1 280.237 even 4