Properties

Label 3675.2.a.f.1.1
Level $3675$
Weight $2$
Character 3675.1
Self dual yes
Analytic conductor $29.345$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3675.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.3450227428\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3675.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{12} -6.00000 q^{13} -1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +8.00000 q^{19} -8.00000 q^{23} +3.00000 q^{24} +6.00000 q^{26} +1.00000 q^{27} -2.00000 q^{29} -4.00000 q^{31} -5.00000 q^{32} -2.00000 q^{34} -1.00000 q^{36} +2.00000 q^{37} -8.00000 q^{38} -6.00000 q^{39} +6.00000 q^{41} -4.00000 q^{43} +8.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} +2.00000 q^{51} +6.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} +8.00000 q^{57} +2.00000 q^{58} -4.00000 q^{59} +2.00000 q^{61} +4.00000 q^{62} +7.00000 q^{64} -4.00000 q^{67} -2.00000 q^{68} -8.00000 q^{69} -12.0000 q^{71} +3.00000 q^{72} -2.00000 q^{73} -2.00000 q^{74} -8.00000 q^{76} +6.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} -4.00000 q^{83} +4.00000 q^{86} -2.00000 q^{87} +6.00000 q^{89} +8.00000 q^{92} -4.00000 q^{93} -8.00000 q^{94} -5.00000 q^{96} -18.0000 q^{97} +O(q^{100})\)

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 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.00000 −0.235702
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −8.00000 −1.29777
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 6.00000 0.832050
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00000 1.05963
\(58\) 2.00000 0.262613
\(59\) −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 \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −2.00000 −0.242536
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 3.00000 0.353553
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) 0 0
\(78\) 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
\(82\) −6.00000 −0.662589
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) −4.00000 −0.414781
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) −18.0000 −1.82762 −0.913812 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) −2.00000 −0.198030
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −18.0000 −1.76505
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −8.00000 −0.749269
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) −6.00000 −0.554700
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) −2.00000 −0.181071
\(123\) 6.00000 0.541002
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 3.00000 0.265165
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −20.0000 −1.74741 −0.873704 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 8.00000 0.681005
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 12.0000 1.00702
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 24.0000 1.94666
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) −8.00000 −0.636446
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 8.00000 0.611775
\(172\) 4.00000 0.304997
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) −6.00000 −0.449719
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) −24.0000 −1.76930
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) 0 0
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 0 0
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 7.00000 0.505181
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 18.0000 1.29232
\(195\) 0 0
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) −10.0000 −0.703598
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) −8.00000 −0.556038
\(208\) 6.00000 0.416025
\(209\) 0 0
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 10.0000 0.686803
\(213\) −12.0000 −0.822226
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 3.00000 0.204124
\(217\) 0 0
\(218\) 18.0000 1.21911
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) −2.00000 −0.134231
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) −8.00000 −0.529813
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) −4.00000 −0.258738 −0.129369 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 11.0000 0.707107
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) −48.0000 −3.05417
\(248\) −12.0000 −0.762001
\(249\) −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
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 20.0000 1.23560
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 4.00000 0.244339
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) −10.0000 −0.604122
\(275\) 0 0
\(276\) 8.00000 0.481543
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) −8.00000 −0.476393
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −5.00000 −0.294628
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −18.0000 −1.05518
\(292\) 2.00000 0.117041
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) −14.0000 −0.810998
\(299\) 48.0000 2.77591
\(300\) 0 0
\(301\) 0 0
\(302\) −8.00000 −0.460348
\(303\) 10.0000 0.574485
\(304\) −8.00000 −0.458831
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) −18.0000 −1.01905
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 10.0000 0.560772
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 16.0000 0.890264
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 12.0000 0.664619
\(327\) −18.0000 −0.995402
\(328\) 18.0000 0.993884
\(329\) 0 0
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 4.00000 0.219529
\(333\) 2.00000 0.109599
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −23.0000 −1.25104
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) −8.00000 −0.432590
\(343\) 0 0
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 2.00000 0.107211
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 24.0000 1.26844
\(359\) −36.0000 −1.90001 −0.950004 0.312239i \(-0.898921\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) −2.00000 −0.105118
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 8.00000 0.417029
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) 4.00000 0.207390
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 24.0000 1.23771
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) −4.00000 −0.204658
\(383\) −32.0000 −1.63512 −0.817562 0.575841i \(-0.804675\pi\)
−0.817562 + 0.575841i \(0.804675\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) 18.0000 0.916176
\(387\) −4.00000 −0.203331
\(388\) 18.0000 0.913812
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) −20.0000 −1.00887
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 4.00000 0.199502
\(403\) 24.0000 1.19553
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 6.00000 0.297044
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) 10.0000 0.493264
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 8.00000 0.393179
\(415\) 0 0
\(416\) 30.0000 1.47087
\(417\) 0 0
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 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
\(423\) 8.00000 0.388973
\(424\) −30.0000 −1.45693
\(425\) 0 0
\(426\) 12.0000 0.581402
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 28.0000 1.34871 0.674356 0.738406i \(-0.264421\pi\)
0.674356 + 0.738406i \(0.264421\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 18.0000 0.862044
\(437\) −64.0000 −3.06154
\(438\) 2.00000 0.0955637
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) 24.0000 1.13643
\(447\) 14.0000 0.662177
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.00000 0.282216
\(453\) 8.00000 0.375873
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) 24.0000 1.12390
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 22.0000 1.02799
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 6.00000 0.277350
\(469\) 0 0
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) 4.00000 0.182956
\(479\) −32.0000 −1.46212 −0.731059 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) −6.00000 −0.273293
\(483\) 0 0
\(484\) 11.0000 0.500000
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 6.00000 0.271607
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) −6.00000 −0.270501
\(493\) −4.00000 −0.180151
\(494\) 48.0000 2.15962
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) −12.0000 −0.535586
\(503\) −8.00000 −0.356702 −0.178351 0.983967i \(-0.557076\pi\)
−0.178351 + 0.983967i \(0.557076\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 23.0000 1.02147
\(508\) 8.00000 0.354943
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11.0000 0.486136
\(513\) 8.00000 0.353209
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 2.00000 0.0875376
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) −36.0000 −1.55933
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) −24.0000 −1.03568
\(538\) 14.0000 0.603583
\(539\) 0 0
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 12.0000 0.515444
\(543\) 2.00000 0.0858282
\(544\) −10.0000 −0.428746
\(545\) 0 0
\(546\) 0 0
\(547\) 12.0000 0.513083 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) −10.0000 −0.427179
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −16.0000 −0.681623
\(552\) −24.0000 −1.02151
\(553\) 0 0
\(554\) 14.0000 0.594803
\(555\) 0 0
\(556\) 0 0
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 4.00000 0.169334
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) −36.0000 −1.51053
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) −36.0000 −1.50655 −0.753277 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) 0 0
\(573\) 4.00000 0.167102
\(574\) 0 0
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 13.0000 0.540729
\(579\) −18.0000 −0.748054
\(580\) 0 0
\(581\) 0 0
\(582\) 18.0000 0.746124
\(583\) 0 0
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) −2.00000 −0.0821995
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −14.0000 −0.573462
\(597\) 4.00000 0.163709
\(598\) −48.0000 −1.96287
\(599\) −4.00000 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) 0 0
\(601\) −18.0000 −0.734235 −0.367118 0.930175i \(-0.619655\pi\)
−0.367118 + 0.930175i \(0.619655\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) −10.0000 −0.406222
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) −40.0000 −1.62221
\(609\) 0 0
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) −2.00000 −0.0808452
\(613\) 18.0000 0.727013 0.363507 0.931592i \(-0.381579\pi\)
0.363507 + 0.931592i \(0.381579\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) −8.00000 −0.321807
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 24.0000 0.962312
\(623\) 0 0
\(624\) 6.00000 0.240192
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 24.0000 0.954669
\(633\) −20.0000 −0.794929
\(634\) 2.00000 0.0794301
\(635\) 0 0
\(636\) 10.0000 0.396526
\(637\) 0 0
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) −12.0000 −0.473602
\(643\) 28.0000 1.10421 0.552106 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −16.0000 −0.629512
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 3.00000 0.117851
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) 22.0000 0.860927 0.430463 0.902608i \(-0.358350\pi\)
0.430463 + 0.902608i \(0.358350\pi\)
\(654\) 18.0000 0.703856
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) 12.0000 0.466393
\(663\) −12.0000 −0.466041
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 16.0000 0.619522
\(668\) −8.00000 −0.309529
\(669\) −24.0000 −0.927894
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) −23.0000 −0.884615
\(677\) 46.0000 1.76792 0.883962 0.467559i \(-0.154866\pi\)
0.883962 + 0.467559i \(0.154866\pi\)
\(678\) 6.00000 0.230429
\(679\) 0 0
\(680\) 0 0
\(681\) −4.00000 −0.153280
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) −8.00000 −0.305888
\(685\) 0 0
\(686\) 0 0
\(687\) −22.0000 −0.839352
\(688\) 4.00000 0.152499
\(689\) 60.0000 2.28582
\(690\) 0 0
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −20.0000 −0.759190
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) 12.0000 0.454532
\(698\) 14.0000 0.529908
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 6.00000 0.226455
\(703\) 16.0000 0.603451
\(704\) 0 0
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 0 0
\(708\) 4.00000 0.150329
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 18.0000 0.674579
\(713\) 32.0000 1.19841
\(714\) 0 0
\(715\) 0 0
\(716\) 24.0000 0.896922
\(717\) −4.00000 −0.149383
\(718\) 36.0000 1.34351
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −45.0000 −1.67473
\(723\) 6.00000 0.223142
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 11.0000 0.408248
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) −2.00000 −0.0739221
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 40.0000 1.47442
\(737\) 0 0
\(738\) −6.00000 −0.220863
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) −48.0000 −1.76332
\(742\) 0 0
\(743\) 48.0000 1.76095 0.880475 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) −12.0000 −0.439941
\(745\) 0 0
\(746\) −10.0000 −0.366126
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) −8.00000 −0.291730
\(753\) 12.0000 0.437304
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) 0 0
\(761\) 54.0000 1.95750 0.978749 0.205061i \(-0.0657392\pi\)
0.978749 + 0.205061i \(0.0657392\pi\)
\(762\) 8.00000 0.289809
\(763\) 0 0
\(764\) −4.00000 −0.144715
\(765\) 0 0
\(766\) 32.0000 1.15621
\(767\) 24.0000 0.866590
\(768\) −17.0000 −0.613435
\(769\) −26.0000 −0.937584 −0.468792 0.883309i \(-0.655311\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 18.0000 0.647834
\(773\) 30.0000 1.07903 0.539513 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) −54.0000 −1.93849
\(777\) 0 0
\(778\) −30.0000 −1.07555
\(779\) 48.0000 1.71978
\(780\) 0 0
\(781\) 0 0
\(782\) 16.0000 0.572159
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) 0 0
\(786\) 20.0000 0.713376
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 18.0000 0.641223
\(789\) −16.0000 −0.569615
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −12.0000 −0.426132
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) 54.0000 1.91278 0.956389 0.292096i \(-0.0943526\pi\)
0.956389 + 0.292096i \(0.0943526\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) −18.0000 −0.635602
\(803\) 0 0
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) −24.0000 −0.845364
\(807\) −14.0000 −0.492823
\(808\) 30.0000 1.05540
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) −12.0000 −0.420858
\(814\) 0 0
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) −32.0000 −1.11954
\(818\) −22.0000 −0.769212
\(819\) 0 0
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) −10.0000 −0.348790
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 24.0000 0.836080
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 8.00000 0.278019
\(829\) 42.0000 1.45872 0.729360 0.684130i \(-0.239818\pi\)
0.729360 + 0.684130i \(0.239818\pi\)
\(830\) 0 0
\(831\) −14.0000 −0.485655
\(832\) −42.0000 −1.45609
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −4.00000 −0.138260
\(838\) −12.0000 −0.414533
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 26.0000 0.896019
\(843\) 18.0000 0.619953
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) 0 0
\(848\) 10.0000 0.343401
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) −16.0000 −0.548473
\(852\) 12.0000 0.411113
\(853\) −30.0000 −1.02718 −0.513590 0.858036i \(-0.671685\pi\)
−0.513590 + 0.858036i \(0.671685\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 36.0000 1.23045
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −28.0000 −0.953684
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) 2.00000 0.0679628
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) −54.0000 −1.82867
\(873\) −18.0000 −0.609208
\(874\) 64.0000 2.16483
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) 58.0000 1.95852 0.979260 0.202606i \(-0.0649409\pi\)
0.979260 + 0.202606i \(0.0649409\pi\)
\(878\) 28.0000 0.944954
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 16.0000 0.537227 0.268614 0.963248i \(-0.413434\pi\)
0.268614 + 0.963248i \(0.413434\pi\)
\(888\) 6.00000 0.201347
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 24.0000 0.803579
\(893\) 64.0000 2.14168
\(894\) −14.0000 −0.468230
\(895\) 0 0
\(896\) 0 0
\(897\) 48.0000 1.60267
\(898\) 30.0000 1.00111
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) −20.0000 −0.666297
\(902\) 0 0
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 4.00000 0.132745
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) −8.00000 −0.264906
\(913\) 0 0
\(914\) 18.0000 0.595387
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) −2.00000 −0.0658665
\(923\) 72.0000 2.36991
\(924\) 0 0
\(925\) 0 0
\(926\) −24.0000 −0.788689
\(927\) 8.00000 0.262754
\(928\) 10.0000 0.328266
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −18.0000 −0.589610
\(933\) −24.0000 −0.785725
\(934\) −28.0000 −0.916188
\(935\) 0 0
\(936\) −18.0000 −0.588348
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 14.0000 0.456145
\(943\) −48.0000 −1.56310
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 0 0
\(947\) −20.0000 −0.649913 −0.324956 0.945729i \(-0.605350\pi\)
−0.324956 + 0.945729i \(0.605350\pi\)
\(948\) −8.00000 −0.259828
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) −2.00000 −0.0648544
\(952\) 0 0
\(953\) 2.00000 0.0647864 0.0323932 0.999475i \(-0.489687\pi\)
0.0323932 + 0.999475i \(0.489687\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) 4.00000 0.129369
\(957\) 0 0
\(958\) 32.0000 1.03387
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 12.0000 0.386896
\(963\) 12.0000 0.386695
\(964\) −6.00000 −0.193247
\(965\) 0 0
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) −33.0000 −1.06066
\(969\) 16.0000 0.513994
\(970\) 0 0
\(971\) −20.0000 −0.641831 −0.320915 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 10.0000 0.319928 0.159964 0.987123i \(-0.448862\pi\)
0.159964 + 0.987123i \(0.448862\pi\)
\(978\) 12.0000 0.383718
\(979\) 0 0
\(980\) 0 0
\(981\) −18.0000 −0.574696
\(982\) 0 0
\(983\) −16.0000 −0.510321 −0.255160 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(984\) 18.0000 0.573819
\(985\) 0 0
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) 48.0000 1.52708
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) −24.0000 −0.762385 −0.381193 0.924496i \(-0.624487\pi\)
−0.381193 + 0.924496i \(0.624487\pi\)
\(992\) 20.0000 0.635001
\(993\) −12.0000 −0.380808
\(994\) 0 0
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) 2.00000 0.0633406 0.0316703 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) −20.0000 −0.633089
\(999\) 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 3675.2.a.f.1.1 1
5.4 even 2 735.2.a.f.1.1 1
7.6 odd 2 525.2.a.a.1.1 1
15.14 odd 2 2205.2.a.b.1.1 1
21.20 even 2 1575.2.a.h.1.1 1
28.27 even 2 8400.2.a.co.1.1 1
35.4 even 6 735.2.i.b.226.1 2
35.9 even 6 735.2.i.b.361.1 2
35.13 even 4 525.2.d.b.274.2 2
35.19 odd 6 735.2.i.a.361.1 2
35.24 odd 6 735.2.i.a.226.1 2
35.27 even 4 525.2.d.b.274.1 2
35.34 odd 2 105.2.a.a.1.1 1
105.62 odd 4 1575.2.d.b.1324.2 2
105.83 odd 4 1575.2.d.b.1324.1 2
105.104 even 2 315.2.a.a.1.1 1
140.139 even 2 1680.2.a.f.1.1 1
280.69 odd 2 6720.2.a.p.1.1 1
280.139 even 2 6720.2.a.bk.1.1 1
420.419 odd 2 5040.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.a.a.1.1 1 35.34 odd 2
315.2.a.a.1.1 1 105.104 even 2
525.2.a.a.1.1 1 7.6 odd 2
525.2.d.b.274.1 2 35.27 even 4
525.2.d.b.274.2 2 35.13 even 4
735.2.a.f.1.1 1 5.4 even 2
735.2.i.a.226.1 2 35.24 odd 6
735.2.i.a.361.1 2 35.19 odd 6
735.2.i.b.226.1 2 35.4 even 6
735.2.i.b.361.1 2 35.9 even 6
1575.2.a.h.1.1 1 21.20 even 2
1575.2.d.b.1324.1 2 105.83 odd 4
1575.2.d.b.1324.2 2 105.62 odd 4
1680.2.a.f.1.1 1 140.139 even 2
2205.2.a.b.1.1 1 15.14 odd 2
3675.2.a.f.1.1 1 1.1 even 1 trivial
5040.2.a.d.1.1 1 420.419 odd 2
6720.2.a.p.1.1 1 280.69 odd 2
6720.2.a.bk.1.1 1 280.139 even 2
8400.2.a.co.1.1 1 28.27 even 2