Properties

Label 2850.2.a.v.1.1
Level $2850$
Weight $2$
Character 2850.1
Self dual yes
Analytic conductor $22.757$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.7573645761\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 570)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} -4.00000 q^{21} -4.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +6.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} +6.00000 q^{29} -8.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -1.00000 q^{38} -6.00000 q^{39} +10.0000 q^{41} -4.00000 q^{42} +8.00000 q^{43} -4.00000 q^{44} -4.00000 q^{46} -12.0000 q^{47} -1.00000 q^{48} +9.00000 q^{49} -6.00000 q^{51} +6.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} +4.00000 q^{56} +1.00000 q^{57} +6.00000 q^{58} -4.00000 q^{59} -2.00000 q^{61} -8.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} +12.0000 q^{67} +6.00000 q^{68} +4.00000 q^{69} -16.0000 q^{71} +1.00000 q^{72} +14.0000 q^{73} -2.00000 q^{74} -1.00000 q^{76} -16.0000 q^{77} -6.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} +10.0000 q^{82} -4.00000 q^{84} +8.00000 q^{86} -6.00000 q^{87} -4.00000 q^{88} -6.00000 q^{89} +24.0000 q^{91} -4.00000 q^{92} +8.00000 q^{93} -12.0000 q^{94} -1.00000 q^{96} -14.0000 q^{97} +9.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −1.00000 −0.288675
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) −4.00000 −0.852803
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) −1.00000 −0.192450
\(28\) 4.00000 0.755929
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.00000 0.696311
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −1.00000 −0.162221
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) −4.00000 −0.617213
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 6.00000 0.832050
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) 1.00000 0.132453
\(58\) 6.00000 0.787839
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −8.00000 −1.01600
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 6.00000 0.727607
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) 1.00000 0.117851
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −16.0000 −1.82337
\(78\) −6.00000 −0.679366
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) −6.00000 −0.643268
\(88\) −4.00000 −0.426401
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 24.0000 2.51588
\(92\) −4.00000 −0.417029
\(93\) 8.00000 0.829561
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 9.00000 0.909137
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) −6.00000 −0.594089
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 4.00000 0.377964
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 6.00000 0.554700
\(118\) −4.00000 −0.368230
\(119\) 24.0000 2.20008
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) −10.0000 −0.901670
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 4.00000 0.348155
\(133\) −4.00000 −0.346844
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 4.00000 0.340503
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) −16.0000 −1.34269
\(143\) −24.0000 −2.00698
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 14.0000 1.15865
\(147\) −9.00000 −0.742307
\(148\) −2.00000 −0.164399
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 6.00000 0.485071
\(154\) −16.0000 −1.28932
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 8.00000 0.636446
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 1.00000 0.0785674
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −4.00000 −0.308607
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 8.00000 0.609994
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 4.00000 0.300658
\(178\) −6.00000 −0.449719
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 24.0000 1.77900
\(183\) 2.00000 0.147844
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) −24.0000 −1.75505
\(188\) −12.0000 −0.875190
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −4.00000 −0.284268
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) 6.00000 0.422159
\(203\) 24.0000 1.68447
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) −4.00000 −0.278019
\(208\) 6.00000 0.416025
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −2.00000 −0.137361
\(213\) 16.0000 1.09630
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) −32.0000 −2.17230
\(218\) 6.00000 0.406371
\(219\) −14.0000 −0.946032
\(220\) 0 0
\(221\) 36.0000 2.42162
\(222\) 2.00000 0.134231
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 1.00000 0.0662266
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) 16.0000 1.05272
\(232\) 6.00000 0.393919
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) −8.00000 −0.519656
\(238\) 24.0000 1.55569
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −10.0000 −0.637577
\(247\) −6.00000 −0.381771
\(248\) −8.00000 −0.508001
\(249\) 0 0
\(250\) 0 0
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 4.00000 0.251976
\(253\) 16.0000 1.00591
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) −8.00000 −0.498058
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 12.0000 0.741362
\(263\) −20.0000 −1.23325 −0.616626 0.787256i \(-0.711501\pi\)
−0.616626 + 0.787256i \(0.711501\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) −4.00000 −0.245256
\(267\) 6.00000 0.367194
\(268\) 12.0000 0.733017
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 6.00000 0.363803
\(273\) −24.0000 −1.45255
\(274\) −2.00000 −0.120824
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 12.0000 0.719712
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 12.0000 0.714590
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) −24.0000 −1.41915
\(287\) 40.0000 2.36113
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 14.0000 0.819288
\(293\) −2.00000 −0.116841 −0.0584206 0.998292i \(-0.518606\pi\)
−0.0584206 + 0.998292i \(0.518606\pi\)
\(294\) −9.00000 −0.524891
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 4.00000 0.232104
\(298\) −10.0000 −0.579284
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) 32.0000 1.84445
\(302\) 8.00000 0.460348
\(303\) −6.00000 −0.344691
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) −16.0000 −0.911685
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) −6.00000 −0.339683
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −22.0000 −1.24153
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 14.0000 0.786318 0.393159 0.919470i \(-0.371382\pi\)
0.393159 + 0.919470i \(0.371382\pi\)
\(318\) 2.00000 0.112154
\(319\) −24.0000 −1.34374
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) −16.0000 −0.891645
\(323\) −6.00000 −0.333849
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) −6.00000 −0.331801
\(328\) 10.0000 0.552158
\(329\) −48.0000 −2.64633
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 23.0000 1.25104
\(339\) −10.0000 −0.543125
\(340\) 0 0
\(341\) 32.0000 1.73290
\(342\) −1.00000 −0.0540738
\(343\) 8.00000 0.431959
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) −32.0000 −1.71785 −0.858925 0.512101i \(-0.828867\pi\)
−0.858925 + 0.512101i \(0.828867\pi\)
\(348\) −6.00000 −0.321634
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) −4.00000 −0.213201
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) −24.0000 −1.27021
\(358\) −12.0000 −0.634220
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −2.00000 −0.105118
\(363\) −5.00000 −0.262432
\(364\) 24.0000 1.25794
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) −4.00000 −0.208514
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) −8.00000 −0.415339
\(372\) 8.00000 0.414781
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) −24.0000 −1.24101
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 36.0000 1.85409
\(378\) −4.00000 −0.205738
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) −8.00000 −0.409316
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) 8.00000 0.406663
\(388\) −14.0000 −0.710742
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) 9.00000 0.454569
\(393\) −12.0000 −0.605320
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) −16.0000 −0.802008
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) −12.0000 −0.598506
\(403\) −48.0000 −2.39105
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 24.0000 1.19110
\(407\) 8.00000 0.396545
\(408\) −6.00000 −0.297044
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) −8.00000 −0.394132
\(413\) −16.0000 −0.787309
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) 6.00000 0.294174
\(417\) −12.0000 −0.587643
\(418\) 4.00000 0.195646
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −20.0000 −0.973585
\(423\) −12.0000 −0.583460
\(424\) −2.00000 −0.0971286
\(425\) 0 0
\(426\) 16.0000 0.775203
\(427\) −8.00000 −0.387147
\(428\) 12.0000 0.580042
\(429\) 24.0000 1.15873
\(430\) 0 0
\(431\) −32.0000 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) −32.0000 −1.53605
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) 4.00000 0.191346
\(438\) −14.0000 −0.668946
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 36.0000 1.71235
\(443\) −8.00000 −0.380091 −0.190046 0.981775i \(-0.560864\pi\)
−0.190046 + 0.981775i \(0.560864\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) 10.0000 0.472984
\(448\) 4.00000 0.188982
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) −40.0000 −1.88353
\(452\) 10.0000 0.470360
\(453\) −8.00000 −0.375873
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 6.00000 0.280362
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 16.0000 0.744387
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 6.00000 0.277350
\(469\) 48.0000 2.21643
\(470\) 0 0
\(471\) 22.0000 1.01371
\(472\) −4.00000 −0.184115
\(473\) −32.0000 −1.47136
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 24.0000 1.10004
\(477\) −2.00000 −0.0915737
\(478\) 16.0000 0.731823
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) −22.0000 −1.00207
\(483\) 16.0000 0.728025
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −10.0000 −0.450835
\(493\) 36.0000 1.62136
\(494\) −6.00000 −0.269953
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) −64.0000 −2.87079
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 4.00000 0.178529
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 4.00000 0.178174
\(505\) 0 0
\(506\) 16.0000 0.711287
\(507\) −23.0000 −1.02147
\(508\) 16.0000 0.709885
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 56.0000 2.47729
\(512\) 1.00000 0.0441942
\(513\) 1.00000 0.0441511
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) −8.00000 −0.352180
\(517\) 48.0000 2.11104
\(518\) −8.00000 −0.351500
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 6.00000 0.262613
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −20.0000 −0.872041
\(527\) −48.0000 −2.09091
\(528\) 4.00000 0.174078
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) −4.00000 −0.173422
\(533\) 60.0000 2.59889
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) 12.0000 0.517838
\(538\) 14.0000 0.603583
\(539\) −36.0000 −1.55063
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) −8.00000 −0.343629
\(543\) 2.00000 0.0858282
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) −24.0000 −1.02711
\(547\) 36.0000 1.53925 0.769624 0.638497i \(-0.220443\pi\)
0.769624 + 0.638497i \(0.220443\pi\)
\(548\) −2.00000 −0.0854358
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 4.00000 0.170251
\(553\) 32.0000 1.36078
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) 26.0000 1.10166 0.550828 0.834619i \(-0.314312\pi\)
0.550828 + 0.834619i \(0.314312\pi\)
\(558\) −8.00000 −0.338667
\(559\) 48.0000 2.03018
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) −22.0000 −0.928014
\(563\) 20.0000 0.842900 0.421450 0.906852i \(-0.361521\pi\)
0.421450 + 0.906852i \(0.361521\pi\)
\(564\) 12.0000 0.505291
\(565\) 0 0
\(566\) 16.0000 0.672530
\(567\) 4.00000 0.167984
\(568\) −16.0000 −0.671345
\(569\) 34.0000 1.42535 0.712677 0.701492i \(-0.247483\pi\)
0.712677 + 0.701492i \(0.247483\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) −24.0000 −1.00349
\(573\) 8.00000 0.334205
\(574\) 40.0000 1.66957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 19.0000 0.790296
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) 0 0
\(582\) 14.0000 0.580319
\(583\) 8.00000 0.331326
\(584\) 14.0000 0.579324
\(585\) 0 0
\(586\) −2.00000 −0.0826192
\(587\) −32.0000 −1.32078 −0.660391 0.750922i \(-0.729609\pi\)
−0.660391 + 0.750922i \(0.729609\pi\)
\(588\) −9.00000 −0.371154
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) −2.00000 −0.0821995
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 16.0000 0.654836
\(598\) −24.0000 −0.981433
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 32.0000 1.30422
\(603\) 12.0000 0.488678
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) −24.0000 −0.974130 −0.487065 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) −72.0000 −2.91281
\(612\) 6.00000 0.242536
\(613\) −46.0000 −1.85792 −0.928961 0.370177i \(-0.879297\pi\)
−0.928961 + 0.370177i \(0.879297\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) −16.0000 −0.644658
\(617\) 14.0000 0.563619 0.281809 0.959470i \(-0.409065\pi\)
0.281809 + 0.959470i \(0.409065\pi\)
\(618\) 8.00000 0.321807
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) −8.00000 −0.320771
\(623\) −24.0000 −0.961540
\(624\) −6.00000 −0.240192
\(625\) 0 0
\(626\) 6.00000 0.239808
\(627\) −4.00000 −0.159745
\(628\) −22.0000 −0.877896
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 8.00000 0.318223
\(633\) 20.0000 0.794929
\(634\) 14.0000 0.556011
\(635\) 0 0
\(636\) 2.00000 0.0793052
\(637\) 54.0000 2.13956
\(638\) −24.0000 −0.950169
\(639\) −16.0000 −0.632950
\(640\) 0 0
\(641\) −14.0000 −0.552967 −0.276483 0.961019i \(-0.589169\pi\)
−0.276483 + 0.961019i \(0.589169\pi\)
\(642\) −12.0000 −0.473602
\(643\) 40.0000 1.57745 0.788723 0.614749i \(-0.210743\pi\)
0.788723 + 0.614749i \(0.210743\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 1.00000 0.0392837
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 32.0000 1.25418
\(652\) 16.0000 0.626608
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) −6.00000 −0.234619
\(655\) 0 0
\(656\) 10.0000 0.390434
\(657\) 14.0000 0.546192
\(658\) −48.0000 −1.87123
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 4.00000 0.155464
\(663\) −36.0000 −1.39812
\(664\) 0 0
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) −24.0000 −0.929284
\(668\) 0 0
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) −4.00000 −0.154303
\(673\) −22.0000 −0.848038 −0.424019 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 30.0000 1.15299 0.576497 0.817099i \(-0.304419\pi\)
0.576497 + 0.817099i \(0.304419\pi\)
\(678\) −10.0000 −0.384048
\(679\) −56.0000 −2.14908
\(680\) 0 0
\(681\) −4.00000 −0.153280
\(682\) 32.0000 1.22534
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) −6.00000 −0.228914
\(688\) 8.00000 0.304997
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 6.00000 0.228086
\(693\) −16.0000 −0.607790
\(694\) −32.0000 −1.21470
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) 60.0000 2.27266
\(698\) −2.00000 −0.0757011
\(699\) 10.0000 0.378235
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) −6.00000 −0.226455
\(703\) 2.00000 0.0754314
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −26.0000 −0.978523
\(707\) 24.0000 0.902613
\(708\) 4.00000 0.150329
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) −6.00000 −0.224860
\(713\) 32.0000 1.19841
\(714\) −24.0000 −0.898177
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) −16.0000 −0.597531
\(718\) −16.0000 −0.597115
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) −32.0000 −1.19174
\(722\) 1.00000 0.0372161
\(723\) 22.0000 0.818189
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 24.0000 0.889499
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 48.0000 1.77534
\(732\) 2.00000 0.0739221
\(733\) 50.0000 1.84679 0.923396 0.383849i \(-0.125402\pi\)
0.923396 + 0.383849i \(0.125402\pi\)
\(734\) −4.00000 −0.147643
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) −48.0000 −1.76810
\(738\) 10.0000 0.368105
\(739\) 44.0000 1.61857 0.809283 0.587419i \(-0.199856\pi\)
0.809283 + 0.587419i \(0.199856\pi\)
\(740\) 0 0
\(741\) 6.00000 0.220416
\(742\) −8.00000 −0.293689
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) −10.0000 −0.366126
\(747\) 0 0
\(748\) −24.0000 −0.877527
\(749\) 48.0000 1.75388
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) −12.0000 −0.437595
\(753\) −4.00000 −0.145768
\(754\) 36.0000 1.31104
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 4.00000 0.145287
\(759\) −16.0000 −0.580763
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) −16.0000 −0.579619
\(763\) 24.0000 0.868858
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) −24.0000 −0.866590
\(768\) −1.00000 −0.0360844
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 2.00000 0.0719816
\(773\) −10.0000 −0.359675 −0.179838 0.983696i \(-0.557557\pi\)
−0.179838 + 0.983696i \(0.557557\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) −14.0000 −0.502571
\(777\) 8.00000 0.286998
\(778\) −26.0000 −0.932145
\(779\) −10.0000 −0.358287
\(780\) 0 0
\(781\) 64.0000 2.29010
\(782\) −24.0000 −0.858238
\(783\) −6.00000 −0.214423
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) −12.0000 −0.428026
\(787\) 52.0000 1.85360 0.926800 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) −6.00000 −0.213741
\(789\) 20.0000 0.712019
\(790\) 0 0
\(791\) 40.0000 1.42224
\(792\) −4.00000 −0.142134
\(793\) −12.0000 −0.426132
\(794\) −22.0000 −0.780751
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 38.0000 1.34603 0.673015 0.739629i \(-0.264999\pi\)
0.673015 + 0.739629i \(0.264999\pi\)
\(798\) 4.00000 0.141598
\(799\) −72.0000 −2.54718
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) −22.0000 −0.776847
\(803\) −56.0000 −1.97620
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) −48.0000 −1.69073
\(807\) −14.0000 −0.492823
\(808\) 6.00000 0.211079
\(809\) 42.0000 1.47664 0.738321 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 24.0000 0.842235
\(813\) 8.00000 0.280572
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) −8.00000 −0.279885
\(818\) −6.00000 −0.209785
\(819\) 24.0000 0.838628
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 2.00000 0.0697580
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) −16.0000 −0.556711
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) −4.00000 −0.139010
\(829\) −26.0000 −0.903017 −0.451509 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 6.00000 0.208013
\(833\) 54.0000 1.87099
\(834\) −12.0000 −0.415526
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) 8.00000 0.276520
\(838\) 20.0000 0.690889
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −26.0000 −0.896019
\(843\) 22.0000 0.757720
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) 20.0000 0.687208
\(848\) −2.00000 −0.0686803
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 16.0000 0.548151
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 24.0000 0.819346
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 0 0
\(861\) −40.0000 −1.36320
\(862\) −32.0000 −1.08992
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −14.0000 −0.475739
\(867\) −19.0000 −0.645274
\(868\) −32.0000 −1.08615
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) 72.0000 2.43963
\(872\) 6.00000 0.203186
\(873\) −14.0000 −0.473828
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) −14.0000 −0.473016
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 32.0000 1.07995
\(879\) 2.00000 0.0674583
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 9.00000 0.303046
\(883\) 16.0000 0.538443 0.269221 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 36.0000 1.21081
\(885\) 0 0
\(886\) −8.00000 −0.268765
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 2.00000 0.0671156
\(889\) 64.0000 2.14649
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 16.0000 0.535720
\(893\) 12.0000 0.401565
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) 4.00000 0.133631
\(897\) 24.0000 0.801337
\(898\) 10.0000 0.333704
\(899\) −48.0000 −1.60089
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) −40.0000 −1.33185
\(903\) −32.0000 −1.06489
\(904\) 10.0000 0.332595
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) 4.00000 0.132745
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 1.00000 0.0331133
\(913\) 0 0
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) 48.0000 1.58510
\(918\) −6.00000 −0.198030
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) −18.0000 −0.592798
\(923\) −96.0000 −3.15988
\(924\) 16.0000 0.526361
\(925\) 0 0
\(926\) −20.0000 −0.657241
\(927\) −8.00000 −0.262754
\(928\) 6.00000 0.196960
\(929\) −14.0000 −0.459325 −0.229663 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 0 0
\(931\) −9.00000 −0.294963
\(932\) −10.0000 −0.327561
\(933\) 8.00000 0.261908
\(934\) 8.00000 0.261768
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) 30.0000 0.980057 0.490029 0.871706i \(-0.336986\pi\)
0.490029 + 0.871706i \(0.336986\pi\)
\(938\) 48.0000 1.56726
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) 46.0000 1.49956 0.749779 0.661689i \(-0.230160\pi\)
0.749779 + 0.661689i \(0.230160\pi\)
\(942\) 22.0000 0.716799
\(943\) −40.0000 −1.30258
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −32.0000 −1.04041
\(947\) 24.0000 0.779895 0.389948 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) −8.00000 −0.259828
\(949\) 84.0000 2.72676
\(950\) 0 0
\(951\) −14.0000 −0.453981
\(952\) 24.0000 0.777844
\(953\) 42.0000 1.36051 0.680257 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) 24.0000 0.775810
\(958\) 24.0000 0.775405
\(959\) −8.00000 −0.258333
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −12.0000 −0.386896
\(963\) 12.0000 0.386695
\(964\) −22.0000 −0.708572
\(965\) 0 0
\(966\) 16.0000 0.514792
\(967\) −52.0000 −1.67221 −0.836104 0.548572i \(-0.815172\pi\)
−0.836104 + 0.548572i \(0.815172\pi\)
\(968\) 5.00000 0.160706
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) 28.0000 0.898563 0.449281 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 48.0000 1.53881
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) −16.0000 −0.511624
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) −12.0000 −0.382935
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) −10.0000 −0.318788
\(985\) 0 0
\(986\) 36.0000 1.14647
\(987\) 48.0000 1.52786
\(988\) −6.00000 −0.190885
\(989\) −32.0000 −1.01754
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) −8.00000 −0.254000
\(993\) −4.00000 −0.126936
\(994\) −64.0000 −2.02996
\(995\) 0 0
\(996\) 0 0
\(997\) −46.0000 −1.45683 −0.728417 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) −28.0000 −0.886325
\(999\) 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 2850.2.a.v.1.1 1
3.2 odd 2 8550.2.a.r.1.1 1
5.2 odd 4 2850.2.d.a.799.2 2
5.3 odd 4 2850.2.d.a.799.1 2
5.4 even 2 570.2.a.e.1.1 1
15.14 odd 2 1710.2.a.l.1.1 1
20.19 odd 2 4560.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.e.1.1 1 5.4 even 2
1710.2.a.l.1.1 1 15.14 odd 2
2850.2.a.v.1.1 1 1.1 even 1 trivial
2850.2.d.a.799.1 2 5.3 odd 4
2850.2.d.a.799.2 2 5.2 odd 4
4560.2.a.q.1.1 1 20.19 odd 2
8550.2.a.r.1.1 1 3.2 odd 2