Properties

Label 1110.2.a.g.1.1
Level $1110$
Weight $2$
Character 1110.1
Self dual yes
Analytic conductor $8.863$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 1110.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -4.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -1.00000 q^{20} +4.00000 q^{21} -4.00000 q^{22} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} -6.00000 q^{29} +1.00000 q^{30} -1.00000 q^{32} +4.00000 q^{33} +2.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} +2.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} -4.00000 q^{42} +4.00000 q^{43} +4.00000 q^{44} -1.00000 q^{45} +12.0000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} -4.00000 q^{55} -4.00000 q^{56} +6.00000 q^{58} +8.00000 q^{59} -1.00000 q^{60} +10.0000 q^{61} +4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{66} +4.00000 q^{67} -2.00000 q^{68} +4.00000 q^{70} -1.00000 q^{72} -6.00000 q^{73} +1.00000 q^{74} +1.00000 q^{75} +16.0000 q^{77} -2.00000 q^{78} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -4.00000 q^{83} +4.00000 q^{84} +2.00000 q^{85} -4.00000 q^{86} -6.00000 q^{87} -4.00000 q^{88} -14.0000 q^{89} +1.00000 q^{90} +8.00000 q^{91} -12.0000 q^{94} -1.00000 q^{96} -10.0000 q^{97} -9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214
\(6\) −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
\(10\) 1.00000 0.316228
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −4.00000 −1.06904
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −1.00000 −0.223607
\(21\) 4.00000 0.872872
\(22\) −4.00000 −0.852803
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) 4.00000 0.755929
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 1.00000 0.182574
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.00000 0.696311
\(34\) 2.00000 0.342997
\(35\) −4.00000 −0.676123
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 1.00000 0.158114
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −4.00000 −0.617213
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 4.00000 0.603023
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) −2.00000 −0.280056
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) −4.00000 −0.539360
\(56\) −4.00000 −0.534522
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) −1.00000 −0.129099
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) −4.00000 −0.492366
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 1.00000 0.116248
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 16.0000 1.82337
\(78\) −2.00000 −0.226455
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −1.00000 −0.111803
\(81\) 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 \(0.570452\pi\)
\(84\) 4.00000 0.436436
\(85\) 2.00000 0.216930
\(86\) −4.00000 −0.431331
\(87\) −6.00000 −0.643268
\(88\) −4.00000 −0.426401
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 1.00000 0.105409
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −9.00000 −0.909137
\(99\) 4.00000 0.402015
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 2.00000 0.198030
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −2.00000 −0.196116
\(105\) −4.00000 −0.390360
\(106\) 6.00000 0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 1.00000 0.0962250
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 4.00000 0.381385
\(111\) −1.00000 −0.0949158
\(112\) 4.00000 0.377964
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 2.00000 0.184900
\(118\) −8.00000 −0.736460
\(119\) −8.00000 −0.733359
\(120\) 1.00000 0.0912871
\(121\) 5.00000 0.454545
\(122\) −10.0000 −0.905357
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) −4.00000 −0.356348
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.00000 0.352180
\(130\) 2.00000 0.175412
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) −1.00000 −0.0860663
\(136\) 2.00000 0.171499
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −4.00000 −0.338062
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) 6.00000 0.498273
\(146\) 6.00000 0.496564
\(147\) 9.00000 0.742307
\(148\) −1.00000 −0.0821995
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) −16.0000 −1.28932
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −6.00000 −0.468521
\(165\) −4.00000 −0.311400
\(166\) 4.00000 0.310460
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) −4.00000 −0.308607
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 6.00000 0.454859
\(175\) 4.00000 0.302372
\(176\) 4.00000 0.301511
\(177\) 8.00000 0.601317
\(178\) 14.0000 1.04934
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) −8.00000 −0.592999
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) 12.0000 0.875190
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 1.00000 0.0721688
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 10.0000 0.717958
\(195\) −2.00000 −0.143223
\(196\) 9.00000 0.642857
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) −4.00000 −0.284268
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 4.00000 0.282138
\(202\) −6.00000 −0.422159
\(203\) −24.0000 −1.68447
\(204\) −2.00000 −0.140028
\(205\) 6.00000 0.419058
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 4.00000 0.276026
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) −4.00000 −0.272798
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 14.0000 0.948200
\(219\) −6.00000 −0.405442
\(220\) −4.00000 −0.269680
\(221\) −4.00000 −0.269069
\(222\) 1.00000 0.0671156
\(223\) −20.0000 −1.33930 −0.669650 0.742677i \(-0.733556\pi\)
−0.669650 + 0.742677i \(0.733556\pi\)
\(224\) −4.00000 −0.267261
\(225\) 1.00000 0.0666667
\(226\) −14.0000 −0.931266
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) −18.0000 −1.18947 −0.594737 0.803921i \(-0.702744\pi\)
−0.594737 + 0.803921i \(0.702744\pi\)
\(230\) 0 0
\(231\) 16.0000 1.05272
\(232\) 6.00000 0.393919
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) −2.00000 −0.130744
\(235\) −12.0000 −0.782794
\(236\) 8.00000 0.520756
\(237\) 0 0
\(238\) 8.00000 0.518563
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) −5.00000 −0.321412
\(243\) 1.00000 0.0641500
\(244\) 10.0000 0.640184
\(245\) −9.00000 −0.574989
\(246\) 6.00000 0.382546
\(247\) 0 0
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 1.00000 0.0632456
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 4.00000 0.251976
\(253\) 0 0
\(254\) −12.0000 −0.752947
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −4.00000 −0.249029
\(259\) −4.00000 −0.248548
\(260\) −2.00000 −0.124035
\(261\) −6.00000 −0.371391
\(262\) 8.00000 0.494242
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) −4.00000 −0.246183
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) −14.0000 −0.856786
\(268\) 4.00000 0.244339
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 1.00000 0.0608581
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −2.00000 −0.121268
\(273\) 8.00000 0.484182
\(274\) −10.0000 −0.604122
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 4.00000 0.239046
\(281\) −14.0000 −0.835170 −0.417585 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) −12.0000 −0.714590
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) −24.0000 −1.41668
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) −6.00000 −0.352332
\(291\) −10.0000 −0.586210
\(292\) −6.00000 −0.351123
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) −9.00000 −0.524891
\(295\) −8.00000 −0.465778
\(296\) 1.00000 0.0581238
\(297\) 4.00000 0.232104
\(298\) 10.0000 0.579284
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 16.0000 0.922225
\(302\) 0 0
\(303\) 6.00000 0.344691
\(304\) 0 0
\(305\) −10.0000 −0.572598
\(306\) 2.00000 0.114332
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 16.0000 0.911685
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −16.0000 −0.907277 −0.453638 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) −2.00000 −0.113228
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −10.0000 −0.564333
\(315\) −4.00000 −0.225374
\(316\) 0 0
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 6.00000 0.336463
\(319\) −24.0000 −1.34374
\(320\) −1.00000 −0.0559017
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) −4.00000 −0.221540
\(327\) −14.0000 −0.774202
\(328\) 6.00000 0.331295
\(329\) 48.0000 2.64633
\(330\) 4.00000 0.220193
\(331\) −32.0000 −1.75888 −0.879440 0.476011i \(-0.842082\pi\)
−0.879440 + 0.476011i \(0.842082\pi\)
\(332\) −4.00000 −0.219529
\(333\) −1.00000 −0.0547997
\(334\) −8.00000 −0.437741
\(335\) −4.00000 −0.218543
\(336\) 4.00000 0.218218
\(337\) −30.0000 −1.63420 −0.817102 0.576493i \(-0.804421\pi\)
−0.817102 + 0.576493i \(0.804421\pi\)
\(338\) 9.00000 0.489535
\(339\) 14.0000 0.760376
\(340\) 2.00000 0.108465
\(341\) 0 0
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −2.00000 −0.107521
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −6.00000 −0.321634
\(349\) 30.0000 1.60586 0.802932 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) −4.00000 −0.213809
\(351\) 2.00000 0.106752
\(352\) −4.00000 −0.213201
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) −8.00000 −0.425195
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) −8.00000 −0.423405
\(358\) 16.0000 0.845626
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 1.00000 0.0527046
\(361\) −19.0000 −1.00000
\(362\) −14.0000 −0.735824
\(363\) 5.00000 0.262432
\(364\) 8.00000 0.419314
\(365\) 6.00000 0.314054
\(366\) −10.0000 −0.522708
\(367\) −20.0000 −1.04399 −0.521996 0.852948i \(-0.674812\pi\)
−0.521996 + 0.852948i \(0.674812\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) −1.00000 −0.0519875
\(371\) −24.0000 −1.24602
\(372\) 0 0
\(373\) 2.00000 0.103556 0.0517780 0.998659i \(-0.483511\pi\)
0.0517780 + 0.998659i \(0.483511\pi\)
\(374\) 8.00000 0.413670
\(375\) −1.00000 −0.0516398
\(376\) −12.0000 −0.618853
\(377\) −12.0000 −0.618031
\(378\) −4.00000 −0.205738
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 12.0000 0.614779
\(382\) 16.0000 0.818631
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −16.0000 −0.815436
\(386\) 2.00000 0.101797
\(387\) 4.00000 0.203331
\(388\) −10.0000 −0.507673
\(389\) 34.0000 1.72387 0.861934 0.507020i \(-0.169253\pi\)
0.861934 + 0.507020i \(0.169253\pi\)
\(390\) 2.00000 0.101274
\(391\) 0 0
\(392\) −9.00000 −0.454569
\(393\) −8.00000 −0.403547
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 24.0000 1.20301
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) −4.00000 −0.199502
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) −1.00000 −0.0496904
\(406\) 24.0000 1.19110
\(407\) −4.00000 −0.198273
\(408\) 2.00000 0.0990148
\(409\) −30.0000 −1.48340 −0.741702 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) −6.00000 −0.296319
\(411\) 10.0000 0.493264
\(412\) 8.00000 0.394132
\(413\) 32.0000 1.57462
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) −2.00000 −0.0980581
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) −4.00000 −0.195180
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 20.0000 0.973585
\(423\) 12.0000 0.583460
\(424\) 6.00000 0.291386
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 40.0000 1.93574
\(428\) 12.0000 0.580042
\(429\) 8.00000 0.386244
\(430\) 4.00000 0.192897
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) −30.0000 −1.44171 −0.720854 0.693087i \(-0.756250\pi\)
−0.720854 + 0.693087i \(0.756250\pi\)
\(434\) 0 0
\(435\) 6.00000 0.287678
\(436\) −14.0000 −0.670478
\(437\) 0 0
\(438\) 6.00000 0.286691
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 4.00000 0.190693
\(441\) 9.00000 0.428571
\(442\) 4.00000 0.190261
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) −1.00000 −0.0474579
\(445\) 14.0000 0.663664
\(446\) 20.0000 0.947027
\(447\) −10.0000 −0.472984
\(448\) 4.00000 0.188982
\(449\) 42.0000 1.98210 0.991051 0.133482i \(-0.0426157\pi\)
0.991051 + 0.133482i \(0.0426157\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −24.0000 −1.13012
\(452\) 14.0000 0.658505
\(453\) 0 0
\(454\) 4.00000 0.187729
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) 18.0000 0.841085
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 26.0000 1.21094 0.605470 0.795868i \(-0.292985\pi\)
0.605470 + 0.795868i \(0.292985\pi\)
\(462\) −16.0000 −0.744387
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 2.00000 0.0924500
\(469\) 16.0000 0.738811
\(470\) 12.0000 0.553519
\(471\) 10.0000 0.460776
\(472\) −8.00000 −0.368230
\(473\) 16.0000 0.735681
\(474\) 0 0
\(475\) 0 0
\(476\) −8.00000 −0.366679
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 1.00000 0.0456435
\(481\) −2.00000 −0.0911922
\(482\) −18.0000 −0.819878
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 10.0000 0.454077
\(486\) −1.00000 −0.0453609
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) −10.0000 −0.452679
\(489\) 4.00000 0.180886
\(490\) 9.00000 0.406579
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) −6.00000 −0.270501
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) −4.00000 −0.179787
\(496\) 0 0
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) 40.0000 1.79065 0.895323 0.445418i \(-0.146945\pi\)
0.895323 + 0.445418i \(0.146945\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 8.00000 0.357414
\(502\) −24.0000 −1.07117
\(503\) 40.0000 1.78351 0.891756 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(504\) −4.00000 −0.178174
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 12.0000 0.532414
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) −2.00000 −0.0885615
\(511\) −24.0000 −1.06170
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) −8.00000 −0.352522
\(516\) 4.00000 0.176090
\(517\) 48.0000 2.11104
\(518\) 4.00000 0.175750
\(519\) 2.00000 0.0877903
\(520\) 2.00000 0.0877058
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 6.00000 0.262613
\(523\) −36.0000 −1.57417 −0.787085 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) −8.00000 −0.349482
\(525\) 4.00000 0.174574
\(526\) 12.0000 0.523225
\(527\) 0 0
\(528\) 4.00000 0.174078
\(529\) −23.0000 −1.00000
\(530\) −6.00000 −0.260623
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 14.0000 0.605839
\(535\) −12.0000 −0.518805
\(536\) −4.00000 −0.172774
\(537\) −16.0000 −0.690451
\(538\) 18.0000 0.776035
\(539\) 36.0000 1.55063
\(540\) −1.00000 −0.0430331
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) −8.00000 −0.343629
\(543\) 14.0000 0.600798
\(544\) 2.00000 0.0857493
\(545\) 14.0000 0.599694
\(546\) −8.00000 −0.342368
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 10.0000 0.427179
\(549\) 10.0000 0.426790
\(550\) −4.00000 −0.170561
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −10.0000 −0.424859
\(555\) 1.00000 0.0424476
\(556\) −4.00000 −0.169638
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) −4.00000 −0.169031
\(561\) −8.00000 −0.337760
\(562\) 14.0000 0.590554
\(563\) −44.0000 −1.85438 −0.927189 0.374593i \(-0.877783\pi\)
−0.927189 + 0.374593i \(0.877783\pi\)
\(564\) 12.0000 0.505291
\(565\) −14.0000 −0.588984
\(566\) 4.00000 0.168133
\(567\) 4.00000 0.167984
\(568\) 0 0
\(569\) 2.00000 0.0838444 0.0419222 0.999121i \(-0.486652\pi\)
0.0419222 + 0.999121i \(0.486652\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 8.00000 0.334497
\(573\) −16.0000 −0.668410
\(574\) 24.0000 1.00174
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) 13.0000 0.540729
\(579\) −2.00000 −0.0831172
\(580\) 6.00000 0.249136
\(581\) −16.0000 −0.663792
\(582\) 10.0000 0.414513
\(583\) −24.0000 −0.993978
\(584\) 6.00000 0.248282
\(585\) −2.00000 −0.0826898
\(586\) 14.0000 0.578335
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 9.00000 0.371154
\(589\) 0 0
\(590\) 8.00000 0.329355
\(591\) 2.00000 0.0822690
\(592\) −1.00000 −0.0410997
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) −4.00000 −0.164122
\(595\) 8.00000 0.327968
\(596\) −10.0000 −0.409616
\(597\) −24.0000 −0.982255
\(598\) 0 0
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) −16.0000 −0.652111
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) −5.00000 −0.203279
\(606\) −6.00000 −0.243733
\(607\) 24.0000 0.974130 0.487065 0.873366i \(-0.338067\pi\)
0.487065 + 0.873366i \(0.338067\pi\)
\(608\) 0 0
\(609\) −24.0000 −0.972529
\(610\) 10.0000 0.404888
\(611\) 24.0000 0.970936
\(612\) −2.00000 −0.0808452
\(613\) 10.0000 0.403896 0.201948 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) −12.0000 −0.484281
\(615\) 6.00000 0.241943
\(616\) −16.0000 −0.644658
\(617\) 2.00000 0.0805170 0.0402585 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) −8.00000 −0.321807
\(619\) −44.0000 −1.76851 −0.884255 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 16.0000 0.641542
\(623\) −56.0000 −2.24359
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) 10.0000 0.399043
\(629\) 2.00000 0.0797452
\(630\) 4.00000 0.159364
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) −20.0000 −0.794929
\(634\) −2.00000 −0.0794301
\(635\) −12.0000 −0.476205
\(636\) −6.00000 −0.237915
\(637\) 18.0000 0.713186
\(638\) 24.0000 0.950169
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) −12.0000 −0.473602
\(643\) 44.0000 1.73519 0.867595 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) 0 0
\(645\) −4.00000 −0.157500
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 32.0000 1.25611
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 14.0000 0.547443
\(655\) 8.00000 0.312586
\(656\) −6.00000 −0.234261
\(657\) −6.00000 −0.234082
\(658\) −48.0000 −1.87123
\(659\) −44.0000 −1.71400 −0.856998 0.515319i \(-0.827673\pi\)
−0.856998 + 0.515319i \(0.827673\pi\)
\(660\) −4.00000 −0.155700
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 32.0000 1.24372
\(663\) −4.00000 −0.155347
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) 1.00000 0.0387492
\(667\) 0 0
\(668\) 8.00000 0.309529
\(669\) −20.0000 −0.773245
\(670\) 4.00000 0.154533
\(671\) 40.0000 1.54418
\(672\) −4.00000 −0.154303
\(673\) −38.0000 −1.46479 −0.732396 0.680879i \(-0.761598\pi\)
−0.732396 + 0.680879i \(0.761598\pi\)
\(674\) 30.0000 1.15556
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) −30.0000 −1.15299 −0.576497 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) −14.0000 −0.537667
\(679\) −40.0000 −1.53506
\(680\) −2.00000 −0.0766965
\(681\) −4.00000 −0.153280
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) −10.0000 −0.382080
\(686\) −8.00000 −0.305441
\(687\) −18.0000 −0.686743
\(688\) 4.00000 0.152499
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 2.00000 0.0760286
\(693\) 16.0000 0.607790
\(694\) 12.0000 0.455514
\(695\) 4.00000 0.151729
\(696\) 6.00000 0.227429
\(697\) 12.0000 0.454532
\(698\) −30.0000 −1.13552
\(699\) 10.0000 0.378235
\(700\) 4.00000 0.151186
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 0 0
\(704\) 4.00000 0.150756
\(705\) −12.0000 −0.451946
\(706\) 26.0000 0.978523
\(707\) 24.0000 0.902613
\(708\) 8.00000 0.300658
\(709\) −14.0000 −0.525781 −0.262891 0.964826i \(-0.584676\pi\)
−0.262891 + 0.964826i \(0.584676\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 14.0000 0.524672
\(713\) 0 0
\(714\) 8.00000 0.299392
\(715\) −8.00000 −0.299183
\(716\) −16.0000 −0.597948
\(717\) 0 0
\(718\) 24.0000 0.895672
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 32.0000 1.19174
\(722\) 19.0000 0.707107
\(723\) 18.0000 0.669427
\(724\) 14.0000 0.520306
\(725\) −6.00000 −0.222834
\(726\) −5.00000 −0.185567
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) −8.00000 −0.296500
\(729\) 1.00000 0.0370370
\(730\) −6.00000 −0.222070
\(731\) −8.00000 −0.295891
\(732\) 10.0000 0.369611
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 20.0000 0.738213
\(735\) −9.00000 −0.331970
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 6.00000 0.220863
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 1.00000 0.0367607
\(741\) 0 0
\(742\) 24.0000 0.881068
\(743\) 28.0000 1.02722 0.513610 0.858024i \(-0.328308\pi\)
0.513610 + 0.858024i \(0.328308\pi\)
\(744\) 0 0
\(745\) 10.0000 0.366372
\(746\) −2.00000 −0.0732252
\(747\) −4.00000 −0.146352
\(748\) −8.00000 −0.292509
\(749\) 48.0000 1.75388
\(750\) 1.00000 0.0365148
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 12.0000 0.437595
\(753\) 24.0000 0.874609
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 0 0
\(761\) −54.0000 −1.95750 −0.978749 0.205061i \(-0.934261\pi\)
−0.978749 + 0.205061i \(0.934261\pi\)
\(762\) −12.0000 −0.434714
\(763\) −56.0000 −2.02734
\(764\) −16.0000 −0.578860
\(765\) 2.00000 0.0723102
\(766\) −24.0000 −0.867155
\(767\) 16.0000 0.577727
\(768\) 1.00000 0.0360844
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 16.0000 0.576600
\(771\) −18.0000 −0.648254
\(772\) −2.00000 −0.0719816
\(773\) 42.0000 1.51064 0.755318 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) −4.00000 −0.143499
\(778\) −34.0000 −1.21896
\(779\) 0 0
\(780\) −2.00000 −0.0716115
\(781\) 0 0
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) 9.00000 0.321429
\(785\) −10.0000 −0.356915
\(786\) 8.00000 0.285351
\(787\) −12.0000 −0.427754 −0.213877 0.976861i \(-0.568609\pi\)
−0.213877 + 0.976861i \(0.568609\pi\)
\(788\) 2.00000 0.0712470
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) 56.0000 1.99113
\(792\) −4.00000 −0.142134
\(793\) 20.0000 0.710221
\(794\) −18.0000 −0.638796
\(795\) 6.00000 0.212798
\(796\) −24.0000 −0.850657
\(797\) 34.0000 1.20434 0.602171 0.798367i \(-0.294303\pi\)
0.602171 + 0.798367i \(0.294303\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) −1.00000 −0.0353553
\(801\) −14.0000 −0.494666
\(802\) 6.00000 0.211867
\(803\) −24.0000 −0.846942
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) −18.0000 −0.633630
\(808\) −6.00000 −0.211079
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 1.00000 0.0351364
\(811\) 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
\(814\) 4.00000 0.140200
\(815\) −4.00000 −0.140114
\(816\) −2.00000 −0.0700140
\(817\) 0 0
\(818\) 30.0000 1.04893
\(819\) 8.00000 0.279543
\(820\) 6.00000 0.209529
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) −10.0000 −0.348790
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) −8.00000 −0.278693
\(825\) 4.00000 0.139262
\(826\) −32.0000 −1.11342
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) −4.00000 −0.138842
\(831\) 10.0000 0.346896
\(832\) 2.00000 0.0693375
\(833\) −18.0000 −0.623663
\(834\) 4.00000 0.138509
\(835\) −8.00000 −0.276851
\(836\) 0 0
\(837\) 0 0
\(838\) −20.0000 −0.690889
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 4.00000 0.138013
\(841\) 7.00000 0.241379
\(842\) 6.00000 0.206774
\(843\) −14.0000 −0.482186
\(844\) −20.0000 −0.688428
\(845\) 9.00000 0.309609
\(846\) −12.0000 −0.412568
\(847\) 20.0000 0.687208
\(848\) −6.00000 −0.206041
\(849\) −4.00000 −0.137280
\(850\) 2.00000 0.0685994
\(851\) 0 0
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) −40.0000 −1.36877
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) −8.00000 −0.273115
\(859\) −8.00000 −0.272956 −0.136478 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) −4.00000 −0.136399
\(861\) −24.0000 −0.817918
\(862\) 0 0
\(863\) −12.0000 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −2.00000 −0.0680020
\(866\) 30.0000 1.01944
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) −6.00000 −0.203419
\(871\) 8.00000 0.271070
\(872\) 14.0000 0.474100
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) −4.00000 −0.135225
\(876\) −6.00000 −0.202721
\(877\) 26.0000 0.877958 0.438979 0.898497i \(-0.355340\pi\)
0.438979 + 0.898497i \(0.355340\pi\)
\(878\) 24.0000 0.809961
\(879\) −14.0000 −0.472208
\(880\) −4.00000 −0.134840
\(881\) −46.0000 −1.54978 −0.774890 0.632096i \(-0.782195\pi\)
−0.774890 + 0.632096i \(0.782195\pi\)
\(882\) −9.00000 −0.303046
\(883\) −36.0000 −1.21150 −0.605748 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) −4.00000 −0.134535
\(885\) −8.00000 −0.268917
\(886\) 4.00000 0.134383
\(887\) −44.0000 −1.47738 −0.738688 0.674048i \(-0.764554\pi\)
−0.738688 + 0.674048i \(0.764554\pi\)
\(888\) 1.00000 0.0335578
\(889\) 48.0000 1.60987
\(890\) −14.0000 −0.469281
\(891\) 4.00000 0.134005
\(892\) −20.0000 −0.669650
\(893\) 0 0
\(894\) 10.0000 0.334450
\(895\) 16.0000 0.534821
\(896\) −4.00000 −0.133631
\(897\) 0 0
\(898\) −42.0000 −1.40156
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 12.0000 0.399778
\(902\) 24.0000 0.799113
\(903\) 16.0000 0.532447
\(904\) −14.0000 −0.465633
\(905\) −14.0000 −0.465376
\(906\) 0 0
\(907\) 4.00000 0.132818 0.0664089 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) −4.00000 −0.132745
\(909\) 6.00000 0.199007
\(910\) 8.00000 0.265197
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) 2.00000 0.0661541
\(915\) −10.0000 −0.330590
\(916\) −18.0000 −0.594737
\(917\) −32.0000 −1.05673
\(918\) 2.00000 0.0660098
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) −26.0000 −0.856264
\(923\) 0 0
\(924\) 16.0000 0.526361
\(925\) −1.00000 −0.0328798
\(926\) −8.00000 −0.262896
\(927\) 8.00000 0.262754
\(928\) 6.00000 0.196960
\(929\) 50.0000 1.64045 0.820223 0.572043i \(-0.193849\pi\)
0.820223 + 0.572043i \(0.193849\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 10.0000 0.327561
\(933\) −16.0000 −0.523816
\(934\) −36.0000 −1.17796
\(935\) 8.00000 0.261628
\(936\) −2.00000 −0.0653720
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) −16.0000 −0.522419
\(939\) 6.00000 0.195803
\(940\) −12.0000 −0.391397
\(941\) −10.0000 −0.325991 −0.162995 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) −10.0000 −0.325818
\(943\) 0 0
\(944\) 8.00000 0.260378
\(945\) −4.00000 −0.130120
\(946\) −16.0000 −0.520205
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 2.00000 0.0648544
\(952\) 8.00000 0.259281
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) 6.00000 0.194257
\(955\) 16.0000 0.517748
\(956\) 0 0
\(957\) −24.0000 −0.775810
\(958\) 0 0
\(959\) 40.0000 1.29167
\(960\) −1.00000 −0.0322749
\(961\) −31.0000 −1.00000
\(962\) 2.00000 0.0644826
\(963\) 12.0000 0.386695
\(964\) 18.0000 0.579741
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) 48.0000 1.54358 0.771788 0.635880i \(-0.219363\pi\)
0.771788 + 0.635880i \(0.219363\pi\)
\(968\) −5.00000 −0.160706
\(969\) 0 0
\(970\) −10.0000 −0.321081
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 1.00000 0.0320750
\(973\) −16.0000 −0.512936
\(974\) −8.00000 −0.256337
\(975\) 2.00000 0.0640513
\(976\) 10.0000 0.320092
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) −4.00000 −0.127906
\(979\) −56.0000 −1.78977
\(980\) −9.00000 −0.287494
\(981\) −14.0000 −0.446986
\(982\) −36.0000 −1.14881
\(983\) −4.00000 −0.127580 −0.0637901 0.997963i \(-0.520319\pi\)
−0.0637901 + 0.997963i \(0.520319\pi\)
\(984\) 6.00000 0.191273
\(985\) −2.00000 −0.0637253
\(986\) −12.0000 −0.382158
\(987\) 48.0000 1.52786
\(988\) 0 0
\(989\) 0 0
\(990\) 4.00000 0.127128
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 0 0
\(993\) −32.0000 −1.01549
\(994\) 0 0
\(995\) 24.0000 0.760851
\(996\) −4.00000 −0.126745
\(997\) −54.0000 −1.71020 −0.855099 0.518465i \(-0.826503\pi\)
−0.855099 + 0.518465i \(0.826503\pi\)
\(998\) −40.0000 −1.26618
\(999\) −1.00000 −0.0316386
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 1110.2.a.g.1.1 1
3.2 odd 2 3330.2.a.ba.1.1 1
4.3 odd 2 8880.2.a.a.1.1 1
5.4 even 2 5550.2.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.g.1.1 1 1.1 even 1 trivial
3330.2.a.ba.1.1 1 3.2 odd 2
5550.2.a.w.1.1 1 5.4 even 2
8880.2.a.a.1.1 1 4.3 odd 2