Properties

Label 5610.2.a.b.1.1
Level $5610$
Weight $2$
Character 5610.1
Self dual yes
Analytic conductor $44.796$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5610.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.7960755339\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5610.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} -1.00000 q^{20} +2.00000 q^{21} -1.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{28} +2.00000 q^{29} -1.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} +1.00000 q^{40} -2.00000 q^{42} -6.00000 q^{43} +1.00000 q^{44} -1.00000 q^{45} +4.00000 q^{46} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -1.00000 q^{51} -6.00000 q^{53} +1.00000 q^{54} -1.00000 q^{55} +2.00000 q^{56} -2.00000 q^{58} +14.0000 q^{59} +1.00000 q^{60} -2.00000 q^{61} -4.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +1.00000 q^{66} +14.0000 q^{67} +1.00000 q^{68} +4.00000 q^{69} -2.00000 q^{70} -2.00000 q^{71} -1.00000 q^{72} +16.0000 q^{73} +2.00000 q^{74} -1.00000 q^{75} -2.00000 q^{77} +12.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{84} -1.00000 q^{85} +6.00000 q^{86} -2.00000 q^{87} -1.00000 q^{88} -18.0000 q^{89} +1.00000 q^{90} -4.00000 q^{92} -4.00000 q^{93} +1.00000 q^{96} +3.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 2.00000 0.534522
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) −1.00000 −0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −1.00000 −0.223607
\(21\) 2.00000 0.436436
\(22\) −1.00000 −0.213201
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −1.00000 −0.182574
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) −1.00000 −0.171499
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −2.00000 −0.308607
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 1.00000 0.150756
\(45\) −1.00000 −0.149071
\(46\) 4.00000 0.589768
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) −1.00000 −0.140028
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) −1.00000 −0.134840
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) 14.0000 1.82264 0.911322 0.411693i \(-0.135063\pi\)
0.911322 + 0.411693i \(0.135063\pi\)
\(60\) 1.00000 0.129099
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −4.00000 −0.508001
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 1.00000 0.121268
\(69\) 4.00000 0.481543
\(70\) −2.00000 −0.239046
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) −1.00000 −0.117851
\(73\) 16.0000 1.87266 0.936329 0.351123i \(-0.114200\pi\)
0.936329 + 0.351123i \(0.114200\pi\)
\(74\) 2.00000 0.232495
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 2.00000 0.218218
\(85\) −1.00000 −0.108465
\(86\) 6.00000 0.646997
\(87\) −2.00000 −0.214423
\(88\) −1.00000 −0.106600
\(89\) −18.0000 −1.90800 −0.953998 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 3.00000 0.303046
\(99\) 1.00000 0.100504
\(100\) 1.00000 0.100000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 1.00000 0.0990148
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 6.00000 0.582772
\(107\) −20.0000 −1.93347 −0.966736 0.255774i \(-0.917670\pi\)
−0.966736 + 0.255774i \(0.917670\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 1.00000 0.0953463
\(111\) 2.00000 0.189832
\(112\) −2.00000 −0.188982
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) −14.0000 −1.28880
\(119\) −2.00000 −0.183340
\(120\) −1.00000 −0.0912871
\(121\) 1.00000 0.0909091
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) −1.00000 −0.0894427
\(126\) 2.00000 0.178174
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) −16.0000 −1.39793 −0.698963 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) −14.0000 −1.20942
\(135\) 1.00000 0.0860663
\(136\) −1.00000 −0.0857493
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) −4.00000 −0.340503
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 2.00000 0.167836
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) −16.0000 −1.32417
\(147\) 3.00000 0.247436
\(148\) −2.00000 −0.164399
\(149\) 16.0000 1.31077 0.655386 0.755295i \(-0.272506\pi\)
0.655386 + 0.755295i \(0.272506\pi\)
\(150\) 1.00000 0.0816497
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) 1.00000 0.0808452
\(154\) 2.00000 0.161165
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) 4.00000 0.319235 0.159617 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) −12.0000 −0.954669
\(159\) 6.00000 0.475831
\(160\) 1.00000 0.0790569
\(161\) 8.00000 0.630488
\(162\) −1.00000 −0.0785674
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 0 0
\(165\) 1.00000 0.0778499
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) −2.00000 −0.154303
\(169\) −13.0000 −1.00000
\(170\) 1.00000 0.0766965
\(171\) 0 0
\(172\) −6.00000 −0.457496
\(173\) 22.0000 1.67263 0.836315 0.548250i \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) 2.00000 0.151620
\(175\) −2.00000 −0.151186
\(176\) 1.00000 0.0753778
\(177\) −14.0000 −1.05230
\(178\) 18.0000 1.34916
\(179\) −14.0000 −1.04641 −0.523205 0.852207i \(-0.675264\pi\)
−0.523205 + 0.852207i \(0.675264\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 4.00000 0.294884
\(185\) 2.00000 0.147043
\(186\) 4.00000 0.293294
\(187\) 1.00000 0.0731272
\(188\) 0 0
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 12.0000 0.850657 0.425329 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −14.0000 −0.987484
\(202\) 0 0
\(203\) −4.00000 −0.280745
\(204\) −1.00000 −0.0700140
\(205\) 0 0
\(206\) 12.0000 0.836080
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 0 0
\(210\) 2.00000 0.138013
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −6.00000 −0.412082
\(213\) 2.00000 0.137038
\(214\) 20.0000 1.36717
\(215\) 6.00000 0.409197
\(216\) 1.00000 0.0680414
\(217\) −8.00000 −0.543075
\(218\) −18.0000 −1.21911
\(219\) −16.0000 −1.08118
\(220\) −1.00000 −0.0674200
\(221\) 0 0
\(222\) −2.00000 −0.134231
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 2.00000 0.133631
\(225\) 1.00000 0.0666667
\(226\) −10.0000 −0.665190
\(227\) −28.0000 −1.85843 −0.929213 0.369546i \(-0.879513\pi\)
−0.929213 + 0.369546i \(0.879513\pi\)
\(228\) 0 0
\(229\) −18.0000 −1.18947 −0.594737 0.803921i \(-0.702744\pi\)
−0.594737 + 0.803921i \(0.702744\pi\)
\(230\) −4.00000 −0.263752
\(231\) 2.00000 0.131590
\(232\) −2.00000 −0.131306
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 14.0000 0.911322
\(237\) −12.0000 −0.779484
\(238\) 2.00000 0.129641
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 1.00000 0.0645497
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) 0 0
\(248\) −4.00000 −0.254000
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −10.0000 −0.631194 −0.315597 0.948893i \(-0.602205\pi\)
−0.315597 + 0.948893i \(0.602205\pi\)
\(252\) −2.00000 −0.125988
\(253\) −4.00000 −0.251478
\(254\) 12.0000 0.752947
\(255\) 1.00000 0.0626224
\(256\) 1.00000 0.0625000
\(257\) −26.0000 −1.62184 −0.810918 0.585160i \(-0.801032\pi\)
−0.810918 + 0.585160i \(0.801032\pi\)
\(258\) −6.00000 −0.373544
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 16.0000 0.988483
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 1.00000 0.0615457
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 18.0000 1.10158
\(268\) 14.0000 0.855186
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) −10.0000 −0.604122
\(275\) 1.00000 0.0603023
\(276\) 4.00000 0.240772
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 4.00000 0.239904
\(279\) 4.00000 0.239474
\(280\) −2.00000 −0.119523
\(281\) −26.0000 −1.55103 −0.775515 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 1.00000 0.0588235
\(290\) 2.00000 0.117444
\(291\) 0 0
\(292\) 16.0000 0.936329
\(293\) −30.0000 −1.75262 −0.876309 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) −3.00000 −0.174964
\(295\) −14.0000 −0.815112
\(296\) 2.00000 0.116248
\(297\) −1.00000 −0.0580259
\(298\) −16.0000 −0.926855
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 12.0000 0.691669
\(302\) −16.0000 −0.920697
\(303\) 0 0
\(304\) 0 0
\(305\) 2.00000 0.114520
\(306\) −1.00000 −0.0571662
\(307\) 34.0000 1.94048 0.970241 0.242140i \(-0.0778494\pi\)
0.970241 + 0.242140i \(0.0778494\pi\)
\(308\) −2.00000 −0.113961
\(309\) 12.0000 0.682656
\(310\) 4.00000 0.227185
\(311\) 26.0000 1.47432 0.737162 0.675716i \(-0.236165\pi\)
0.737162 + 0.675716i \(0.236165\pi\)
\(312\) 0 0
\(313\) 16.0000 0.904373 0.452187 0.891923i \(-0.350644\pi\)
0.452187 + 0.891923i \(0.350644\pi\)
\(314\) −4.00000 −0.225733
\(315\) 2.00000 0.112687
\(316\) 12.0000 0.675053
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −6.00000 −0.336463
\(319\) 2.00000 0.111979
\(320\) −1.00000 −0.0559017
\(321\) 20.0000 1.11629
\(322\) −8.00000 −0.445823
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 8.00000 0.443079
\(327\) −18.0000 −0.995402
\(328\) 0 0
\(329\) 0 0
\(330\) −1.00000 −0.0550482
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 0 0
\(333\) −2.00000 −0.109599
\(334\) 12.0000 0.656611
\(335\) −14.0000 −0.764902
\(336\) 2.00000 0.109109
\(337\) 16.0000 0.871576 0.435788 0.900049i \(-0.356470\pi\)
0.435788 + 0.900049i \(0.356470\pi\)
\(338\) 13.0000 0.707107
\(339\) −10.0000 −0.543125
\(340\) −1.00000 −0.0542326
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 6.00000 0.323498
\(345\) −4.00000 −0.215353
\(346\) −22.0000 −1.18273
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) −2.00000 −0.107211
\(349\) 30.0000 1.60586 0.802932 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −34.0000 −1.80964 −0.904819 0.425797i \(-0.859994\pi\)
−0.904819 + 0.425797i \(0.859994\pi\)
\(354\) 14.0000 0.744092
\(355\) 2.00000 0.106149
\(356\) −18.0000 −0.953998
\(357\) 2.00000 0.105851
\(358\) 14.0000 0.739923
\(359\) 36.0000 1.90001 0.950004 0.312239i \(-0.101079\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) 1.00000 0.0527046
\(361\) −19.0000 −1.00000
\(362\) −10.0000 −0.525588
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −16.0000 −0.837478
\(366\) −2.00000 −0.104542
\(367\) −26.0000 −1.35719 −0.678594 0.734513i \(-0.737411\pi\)
−0.678594 + 0.734513i \(0.737411\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) −2.00000 −0.103975
\(371\) 12.0000 0.623009
\(372\) −4.00000 −0.207390
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) −1.00000 −0.0517088
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 12.0000 0.614779
\(382\) 4.00000 0.204658
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 1.00000 0.0510310
\(385\) 2.00000 0.101929
\(386\) 4.00000 0.203595
\(387\) −6.00000 −0.304997
\(388\) 0 0
\(389\) −36.0000 −1.82527 −0.912636 0.408773i \(-0.865957\pi\)
−0.912636 + 0.408773i \(0.865957\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 3.00000 0.151523
\(393\) 16.0000 0.807093
\(394\) −6.00000 −0.302276
\(395\) −12.0000 −0.603786
\(396\) 1.00000 0.0502519
\(397\) −26.0000 −1.30490 −0.652451 0.757831i \(-0.726259\pi\)
−0.652451 + 0.757831i \(0.726259\pi\)
\(398\) −12.0000 −0.601506
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −8.00000 −0.399501 −0.199750 0.979847i \(-0.564013\pi\)
−0.199750 + 0.979847i \(0.564013\pi\)
\(402\) 14.0000 0.698257
\(403\) 0 0
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 4.00000 0.198517
\(407\) −2.00000 −0.0991363
\(408\) 1.00000 0.0495074
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) −10.0000 −0.493264
\(412\) −12.0000 −0.591198
\(413\) −28.0000 −1.37779
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 0 0
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) −2.00000 −0.0975900
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) 20.0000 0.973585
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 1.00000 0.0485071
\(426\) −2.00000 −0.0969003
\(427\) 4.00000 0.193574
\(428\) −20.0000 −0.966736
\(429\) 0 0
\(430\) −6.00000 −0.289346
\(431\) −26.0000 −1.25238 −0.626188 0.779672i \(-0.715386\pi\)
−0.626188 + 0.779672i \(0.715386\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 8.00000 0.384012
\(435\) 2.00000 0.0958927
\(436\) 18.0000 0.862044
\(437\) 0 0
\(438\) 16.0000 0.764510
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 1.00000 0.0476731
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 2.00000 0.0949158
\(445\) 18.0000 0.853282
\(446\) 4.00000 0.189405
\(447\) −16.0000 −0.756774
\(448\) −2.00000 −0.0944911
\(449\) 32.0000 1.51017 0.755087 0.655625i \(-0.227595\pi\)
0.755087 + 0.655625i \(0.227595\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) 10.0000 0.470360
\(453\) −16.0000 −0.751746
\(454\) 28.0000 1.31411
\(455\) 0 0
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 18.0000 0.841085
\(459\) −1.00000 −0.0466760
\(460\) 4.00000 0.186501
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) −2.00000 −0.0930484
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 2.00000 0.0928477
\(465\) 4.00000 0.185496
\(466\) −10.0000 −0.463241
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 0 0
\(469\) −28.0000 −1.29292
\(470\) 0 0
\(471\) −4.00000 −0.184310
\(472\) −14.0000 −0.644402
\(473\) −6.00000 −0.275880
\(474\) 12.0000 0.551178
\(475\) 0 0
\(476\) −2.00000 −0.0916698
\(477\) −6.00000 −0.274721
\(478\) 16.0000 0.731823
\(479\) −6.00000 −0.274147 −0.137073 0.990561i \(-0.543770\pi\)
−0.137073 + 0.990561i \(0.543770\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 10.0000 0.455488
\(483\) −8.00000 −0.364013
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −38.0000 −1.72194 −0.860972 0.508652i \(-0.830144\pi\)
−0.860972 + 0.508652i \(0.830144\pi\)
\(488\) 2.00000 0.0905357
\(489\) 8.00000 0.361773
\(490\) −3.00000 −0.135526
\(491\) −42.0000 −1.89543 −0.947717 0.319113i \(-0.896615\pi\)
−0.947717 + 0.319113i \(0.896615\pi\)
\(492\) 0 0
\(493\) 2.00000 0.0900755
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) 4.00000 0.179605
\(497\) 4.00000 0.179425
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 12.0000 0.536120
\(502\) 10.0000 0.446322
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) 4.00000 0.177822
\(507\) 13.0000 0.577350
\(508\) −12.0000 −0.532414
\(509\) 12.0000 0.531891 0.265945 0.963988i \(-0.414316\pi\)
0.265945 + 0.963988i \(0.414316\pi\)
\(510\) −1.00000 −0.0442807
\(511\) −32.0000 −1.41560
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 26.0000 1.14681
\(515\) 12.0000 0.528783
\(516\) 6.00000 0.264135
\(517\) 0 0
\(518\) −4.00000 −0.175750
\(519\) −22.0000 −0.965693
\(520\) 0 0
\(521\) −16.0000 −0.700973 −0.350486 0.936568i \(-0.613984\pi\)
−0.350486 + 0.936568i \(0.613984\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −18.0000 −0.787085 −0.393543 0.919306i \(-0.628751\pi\)
−0.393543 + 0.919306i \(0.628751\pi\)
\(524\) −16.0000 −0.698963
\(525\) 2.00000 0.0872872
\(526\) 16.0000 0.697633
\(527\) 4.00000 0.174243
\(528\) −1.00000 −0.0435194
\(529\) −7.00000 −0.304348
\(530\) −6.00000 −0.260623
\(531\) 14.0000 0.607548
\(532\) 0 0
\(533\) 0 0
\(534\) −18.0000 −0.778936
\(535\) 20.0000 0.864675
\(536\) −14.0000 −0.604708
\(537\) 14.0000 0.604145
\(538\) 6.00000 0.258678
\(539\) −3.00000 −0.129219
\(540\) 1.00000 0.0430331
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) −8.00000 −0.343629
\(543\) −10.0000 −0.429141
\(544\) −1.00000 −0.0428746
\(545\) −18.0000 −0.771035
\(546\) 0 0
\(547\) 44.0000 1.88130 0.940652 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) 10.0000 0.427179
\(549\) −2.00000 −0.0853579
\(550\) −1.00000 −0.0426401
\(551\) 0 0
\(552\) −4.00000 −0.170251
\(553\) −24.0000 −1.02058
\(554\) −10.0000 −0.424859
\(555\) −2.00000 −0.0848953
\(556\) −4.00000 −0.169638
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) −4.00000 −0.169334
\(559\) 0 0
\(560\) 2.00000 0.0845154
\(561\) −1.00000 −0.0422200
\(562\) 26.0000 1.09674
\(563\) −8.00000 −0.337160 −0.168580 0.985688i \(-0.553918\pi\)
−0.168580 + 0.985688i \(0.553918\pi\)
\(564\) 0 0
\(565\) −10.0000 −0.420703
\(566\) 20.0000 0.840663
\(567\) −2.00000 −0.0839921
\(568\) 2.00000 0.0839181
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 4.00000 0.167102
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 4.00000 0.166234
\(580\) −2.00000 −0.0830455
\(581\) 0 0
\(582\) 0 0
\(583\) −6.00000 −0.248495
\(584\) −16.0000 −0.662085
\(585\) 0 0
\(586\) 30.0000 1.23929
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) 3.00000 0.123718
\(589\) 0 0
\(590\) 14.0000 0.576371
\(591\) −6.00000 −0.246807
\(592\) −2.00000 −0.0821995
\(593\) −2.00000 −0.0821302 −0.0410651 0.999156i \(-0.513075\pi\)
−0.0410651 + 0.999156i \(0.513075\pi\)
\(594\) 1.00000 0.0410305
\(595\) 2.00000 0.0819920
\(596\) 16.0000 0.655386
\(597\) −12.0000 −0.491127
\(598\) 0 0
\(599\) 4.00000 0.163436 0.0817178 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(600\) 1.00000 0.0408248
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) −12.0000 −0.489083
\(603\) 14.0000 0.570124
\(604\) 16.0000 0.651031
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) −18.0000 −0.730597 −0.365299 0.930890i \(-0.619033\pi\)
−0.365299 + 0.930890i \(0.619033\pi\)
\(608\) 0 0
\(609\) 4.00000 0.162088
\(610\) −2.00000 −0.0809776
\(611\) 0 0
\(612\) 1.00000 0.0404226
\(613\) 44.0000 1.77714 0.888572 0.458738i \(-0.151698\pi\)
0.888572 + 0.458738i \(0.151698\pi\)
\(614\) −34.0000 −1.37213
\(615\) 0 0
\(616\) 2.00000 0.0805823
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) −12.0000 −0.482711
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) −4.00000 −0.160644
\(621\) 4.00000 0.160514
\(622\) −26.0000 −1.04251
\(623\) 36.0000 1.44231
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −16.0000 −0.639489
\(627\) 0 0
\(628\) 4.00000 0.159617
\(629\) −2.00000 −0.0797452
\(630\) −2.00000 −0.0796819
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −12.0000 −0.477334
\(633\) 20.0000 0.794929
\(634\) 18.0000 0.714871
\(635\) 12.0000 0.476205
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) −2.00000 −0.0791808
\(639\) −2.00000 −0.0791188
\(640\) 1.00000 0.0395285
\(641\) 48.0000 1.89589 0.947943 0.318440i \(-0.103159\pi\)
0.947943 + 0.318440i \(0.103159\pi\)
\(642\) −20.0000 −0.789337
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 8.00000 0.315244
\(645\) −6.00000 −0.236250
\(646\) 0 0
\(647\) −48.0000 −1.88707 −0.943537 0.331266i \(-0.892524\pi\)
−0.943537 + 0.331266i \(0.892524\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 14.0000 0.549548
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) −8.00000 −0.313304
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) 18.0000 0.703856
\(655\) 16.0000 0.625172
\(656\) 0 0
\(657\) 16.0000 0.624219
\(658\) 0 0
\(659\) −42.0000 −1.63609 −0.818044 0.575156i \(-0.804941\pi\)
−0.818044 + 0.575156i \(0.804941\pi\)
\(660\) 1.00000 0.0389249
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) 8.00000 0.310929
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) −8.00000 −0.309761
\(668\) −12.0000 −0.464294
\(669\) 4.00000 0.154649
\(670\) 14.0000 0.540867
\(671\) −2.00000 −0.0772091
\(672\) −2.00000 −0.0771517
\(673\) −20.0000 −0.770943 −0.385472 0.922720i \(-0.625961\pi\)
−0.385472 + 0.922720i \(0.625961\pi\)
\(674\) −16.0000 −0.616297
\(675\) −1.00000 −0.0384900
\(676\) −13.0000 −0.500000
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 10.0000 0.384048
\(679\) 0 0
\(680\) 1.00000 0.0383482
\(681\) 28.0000 1.07296
\(682\) −4.00000 −0.153168
\(683\) −44.0000 −1.68361 −0.841807 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) 0 0
\(685\) −10.0000 −0.382080
\(686\) −20.0000 −0.763604
\(687\) 18.0000 0.686743
\(688\) −6.00000 −0.228748
\(689\) 0 0
\(690\) 4.00000 0.152277
\(691\) 12.0000 0.456502 0.228251 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) 22.0000 0.836315
\(693\) −2.00000 −0.0759737
\(694\) 4.00000 0.151838
\(695\) 4.00000 0.151729
\(696\) 2.00000 0.0758098
\(697\) 0 0
\(698\) −30.0000 −1.13552
\(699\) −10.0000 −0.378235
\(700\) −2.00000 −0.0755929
\(701\) −12.0000 −0.453234 −0.226617 0.973984i \(-0.572767\pi\)
−0.226617 + 0.973984i \(0.572767\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 34.0000 1.27961
\(707\) 0 0
\(708\) −14.0000 −0.526152
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) −2.00000 −0.0750587
\(711\) 12.0000 0.450035
\(712\) 18.0000 0.674579
\(713\) −16.0000 −0.599205
\(714\) −2.00000 −0.0748481
\(715\) 0 0
\(716\) −14.0000 −0.523205
\(717\) 16.0000 0.597531
\(718\) −36.0000 −1.34351
\(719\) −34.0000 −1.26799 −0.633993 0.773339i \(-0.718585\pi\)
−0.633993 + 0.773339i \(0.718585\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 24.0000 0.893807
\(722\) 19.0000 0.707107
\(723\) 10.0000 0.371904
\(724\) 10.0000 0.371647
\(725\) 2.00000 0.0742781
\(726\) 1.00000 0.0371135
\(727\) 12.0000 0.445055 0.222528 0.974926i \(-0.428569\pi\)
0.222528 + 0.974926i \(0.428569\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 16.0000 0.592187
\(731\) −6.00000 −0.221918
\(732\) 2.00000 0.0739221
\(733\) −36.0000 −1.32969 −0.664845 0.746981i \(-0.731502\pi\)
−0.664845 + 0.746981i \(0.731502\pi\)
\(734\) 26.0000 0.959678
\(735\) −3.00000 −0.110657
\(736\) 4.00000 0.147442
\(737\) 14.0000 0.515697
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 2.00000 0.0735215
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 4.00000 0.146647
\(745\) −16.0000 −0.586195
\(746\) 4.00000 0.146450
\(747\) 0 0
\(748\) 1.00000 0.0365636
\(749\) 40.0000 1.46157
\(750\) −1.00000 −0.0365148
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 10.0000 0.364420
\(754\) 0 0
\(755\) −16.0000 −0.582300
\(756\) 2.00000 0.0727393
\(757\) 4.00000 0.145382 0.0726912 0.997354i \(-0.476841\pi\)
0.0726912 + 0.997354i \(0.476841\pi\)
\(758\) 4.00000 0.145287
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) −12.0000 −0.434714
\(763\) −36.0000 −1.30329
\(764\) −4.00000 −0.144715
\(765\) −1.00000 −0.0361551
\(766\) 0 0
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) −2.00000 −0.0720750
\(771\) 26.0000 0.936367
\(772\) −4.00000 −0.143963
\(773\) 42.0000 1.51064 0.755318 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 6.00000 0.215666
\(775\) 4.00000 0.143684
\(776\) 0 0
\(777\) −4.00000 −0.143499
\(778\) 36.0000 1.29066
\(779\) 0 0
\(780\) 0 0
\(781\) −2.00000 −0.0715656
\(782\) 4.00000 0.143040
\(783\) −2.00000 −0.0714742
\(784\) −3.00000 −0.107143
\(785\) −4.00000 −0.142766
\(786\) −16.0000 −0.570701
\(787\) −16.0000 −0.570338 −0.285169 0.958477i \(-0.592050\pi\)
−0.285169 + 0.958477i \(0.592050\pi\)
\(788\) 6.00000 0.213741
\(789\) 16.0000 0.569615
\(790\) 12.0000 0.426941
\(791\) −20.0000 −0.711118
\(792\) −1.00000 −0.0355335
\(793\) 0 0
\(794\) 26.0000 0.922705
\(795\) −6.00000 −0.212798
\(796\) 12.0000 0.425329
\(797\) −50.0000 −1.77109 −0.885545 0.464553i \(-0.846215\pi\)
−0.885545 + 0.464553i \(0.846215\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −18.0000 −0.635999
\(802\) 8.00000 0.282490
\(803\) 16.0000 0.564628
\(804\) −14.0000 −0.493742
\(805\) −8.00000 −0.281963
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) 0 0
\(809\) 32.0000 1.12506 0.562530 0.826777i \(-0.309828\pi\)
0.562530 + 0.826777i \(0.309828\pi\)
\(810\) 1.00000 0.0351364
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) −4.00000 −0.140372
\(813\) −8.00000 −0.280572
\(814\) 2.00000 0.0701000
\(815\) 8.00000 0.280228
\(816\) −1.00000 −0.0350070
\(817\) 0 0
\(818\) −6.00000 −0.209785
\(819\) 0 0
\(820\) 0 0
\(821\) −10.0000 −0.349002 −0.174501 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 10.0000 0.348790
\(823\) −10.0000 −0.348578 −0.174289 0.984695i \(-0.555763\pi\)
−0.174289 + 0.984695i \(0.555763\pi\)
\(824\) 12.0000 0.418040
\(825\) −1.00000 −0.0348155
\(826\) 28.0000 0.974245
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) −4.00000 −0.139010
\(829\) 18.0000 0.625166 0.312583 0.949890i \(-0.398806\pi\)
0.312583 + 0.949890i \(0.398806\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) −3.00000 −0.103944
\(834\) −4.00000 −0.138509
\(835\) 12.0000 0.415277
\(836\) 0 0
\(837\) −4.00000 −0.138260
\(838\) −24.0000 −0.829066
\(839\) 34.0000 1.17381 0.586905 0.809656i \(-0.300346\pi\)
0.586905 + 0.809656i \(0.300346\pi\)
\(840\) 2.00000 0.0690066
\(841\) −25.0000 −0.862069
\(842\) −14.0000 −0.482472
\(843\) 26.0000 0.895488
\(844\) −20.0000 −0.688428
\(845\) 13.0000 0.447214
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) −6.00000 −0.206041
\(849\) 20.0000 0.686398
\(850\) −1.00000 −0.0342997
\(851\) 8.00000 0.274236
\(852\) 2.00000 0.0685189
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) 20.0000 0.683586
\(857\) −54.0000 −1.84460 −0.922302 0.386469i \(-0.873695\pi\)
−0.922302 + 0.386469i \(0.873695\pi\)
\(858\) 0 0
\(859\) −32.0000 −1.09183 −0.545913 0.837842i \(-0.683817\pi\)
−0.545913 + 0.837842i \(0.683817\pi\)
\(860\) 6.00000 0.204598
\(861\) 0 0
\(862\) 26.0000 0.885564
\(863\) 16.0000 0.544646 0.272323 0.962206i \(-0.412208\pi\)
0.272323 + 0.962206i \(0.412208\pi\)
\(864\) 1.00000 0.0340207
\(865\) −22.0000 −0.748022
\(866\) 2.00000 0.0679628
\(867\) −1.00000 −0.0339618
\(868\) −8.00000 −0.271538
\(869\) 12.0000 0.407072
\(870\) −2.00000 −0.0678064
\(871\) 0 0
\(872\) −18.0000 −0.609557
\(873\) 0 0
\(874\) 0 0
\(875\) 2.00000 0.0676123
\(876\) −16.0000 −0.540590
\(877\) 50.0000 1.68838 0.844190 0.536044i \(-0.180082\pi\)
0.844190 + 0.536044i \(0.180082\pi\)
\(878\) −20.0000 −0.674967
\(879\) 30.0000 1.01187
\(880\) −1.00000 −0.0337100
\(881\) 32.0000 1.07811 0.539054 0.842271i \(-0.318782\pi\)
0.539054 + 0.842271i \(0.318782\pi\)
\(882\) 3.00000 0.101015
\(883\) 26.0000 0.874970 0.437485 0.899226i \(-0.355869\pi\)
0.437485 + 0.899226i \(0.355869\pi\)
\(884\) 0 0
\(885\) 14.0000 0.470605
\(886\) 4.00000 0.134383
\(887\) 32.0000 1.07445 0.537227 0.843437i \(-0.319472\pi\)
0.537227 + 0.843437i \(0.319472\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 24.0000 0.804934
\(890\) −18.0000 −0.603361
\(891\) 1.00000 0.0335013
\(892\) −4.00000 −0.133930
\(893\) 0 0
\(894\) 16.0000 0.535120
\(895\) 14.0000 0.467968
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) −32.0000 −1.06785
\(899\) 8.00000 0.266815
\(900\) 1.00000 0.0333333
\(901\) −6.00000 −0.199889
\(902\) 0 0
\(903\) −12.0000 −0.399335
\(904\) −10.0000 −0.332595
\(905\) −10.0000 −0.332411
\(906\) 16.0000 0.531564
\(907\) 48.0000 1.59381 0.796907 0.604102i \(-0.206468\pi\)
0.796907 + 0.604102i \(0.206468\pi\)
\(908\) −28.0000 −0.929213
\(909\) 0 0
\(910\) 0 0
\(911\) −18.0000 −0.596367 −0.298183 0.954509i \(-0.596381\pi\)
−0.298183 + 0.954509i \(0.596381\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 10.0000 0.330771
\(915\) −2.00000 −0.0661180
\(916\) −18.0000 −0.594737
\(917\) 32.0000 1.05673
\(918\) 1.00000 0.0330049
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) −4.00000 −0.131876
\(921\) −34.0000 −1.12034
\(922\) 12.0000 0.395199
\(923\) 0 0
\(924\) 2.00000 0.0657952
\(925\) −2.00000 −0.0657596
\(926\) −4.00000 −0.131448
\(927\) −12.0000 −0.394132
\(928\) −2.00000 −0.0656532
\(929\) 28.0000 0.918650 0.459325 0.888268i \(-0.348091\pi\)
0.459325 + 0.888268i \(0.348091\pi\)
\(930\) −4.00000 −0.131165
\(931\) 0 0
\(932\) 10.0000 0.327561
\(933\) −26.0000 −0.851202
\(934\) 8.00000 0.261768
\(935\) −1.00000 −0.0327035
\(936\) 0 0
\(937\) 46.0000 1.50275 0.751377 0.659873i \(-0.229390\pi\)
0.751377 + 0.659873i \(0.229390\pi\)
\(938\) 28.0000 0.914232
\(939\) −16.0000 −0.522140
\(940\) 0 0
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) 4.00000 0.130327
\(943\) 0 0
\(944\) 14.0000 0.455661
\(945\) −2.00000 −0.0650600
\(946\) 6.00000 0.195077
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) −12.0000 −0.389742
\(949\) 0 0
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 2.00000 0.0648204
\(953\) 10.0000 0.323932 0.161966 0.986796i \(-0.448217\pi\)
0.161966 + 0.986796i \(0.448217\pi\)
\(954\) 6.00000 0.194257
\(955\) 4.00000 0.129437
\(956\) −16.0000 −0.517477
\(957\) −2.00000 −0.0646508
\(958\) 6.00000 0.193851
\(959\) −20.0000 −0.645834
\(960\) 1.00000 0.0322749
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −20.0000 −0.644491
\(964\) −10.0000 −0.322078
\(965\) 4.00000 0.128765
\(966\) 8.00000 0.257396
\(967\) −56.0000 −1.80084 −0.900419 0.435023i \(-0.856740\pi\)
−0.900419 + 0.435023i \(0.856740\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 0 0
\(971\) −30.0000 −0.962746 −0.481373 0.876516i \(-0.659862\pi\)
−0.481373 + 0.876516i \(0.659862\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 8.00000 0.256468
\(974\) 38.0000 1.21760
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 46.0000 1.47167 0.735835 0.677161i \(-0.236790\pi\)
0.735835 + 0.677161i \(0.236790\pi\)
\(978\) −8.00000 −0.255812
\(979\) −18.0000 −0.575282
\(980\) 3.00000 0.0958315
\(981\) 18.0000 0.574696
\(982\) 42.0000 1.34027
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) −2.00000 −0.0636930
\(987\) 0 0
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 1.00000 0.0317821
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) −4.00000 −0.127000
\(993\) 8.00000 0.253872
\(994\) −4.00000 −0.126872
\(995\) −12.0000 −0.380426
\(996\) 0 0
\(997\) 46.0000 1.45683 0.728417 0.685134i \(-0.240256\pi\)
0.728417 + 0.685134i \(0.240256\pi\)
\(998\) 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 5610.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5610.2.a.b.1.1 1 1.1 even 1 trivial