Properties

Label 4830.2.a.s.1.1
Level $4830$
Weight $2$
Character 4830.1
Self dual yes
Analytic conductor $38.568$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4830.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.5677441763\)
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 \(\chi\) \(=\) 4830.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} -1.00000 q^{21} -4.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +2.00000 q^{29} +1.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +6.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} +10.0000 q^{41} -1.00000 q^{42} +4.00000 q^{43} -4.00000 q^{44} -1.00000 q^{45} -1.00000 q^{46} -12.0000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -6.00000 q^{51} -2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +4.00000 q^{55} +1.00000 q^{56} +4.00000 q^{57} +2.00000 q^{58} -4.00000 q^{59} +1.00000 q^{60} -14.0000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +4.00000 q^{66} -12.0000 q^{67} +6.00000 q^{68} +1.00000 q^{69} -1.00000 q^{70} -12.0000 q^{71} +1.00000 q^{72} +10.0000 q^{73} -6.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} -4.00000 q^{77} +2.00000 q^{78} -8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -4.00000 q^{83} -1.00000 q^{84} -6.00000 q^{85} +4.00000 q^{86} -2.00000 q^{87} -4.00000 q^{88} -6.00000 q^{89} -1.00000 q^{90} -2.00000 q^{91} -1.00000 q^{92} -4.00000 q^{93} -12.0000 q^{94} +4.00000 q^{95} -1.00000 q^{96} +2.00000 q^{97} +1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.00000 −0.218218
\(22\) −4.00000 −0.852803
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 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
\(33\) 4.00000 0.696311
\(34\) 6.00000 1.02899
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −4.00000 −0.648886
\(39\) 2.00000 0.320256
\(40\) −1.00000 −0.158114
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) −1.00000 −0.154303
\(43\) 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
\(46\) −1.00000 −0.147442
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −6.00000 −0.840168
\(52\) −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
\(56\) 1.00000 0.133631
\(57\) 4.00000 0.529813
\(58\) 2.00000 0.262613
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 1.00000 0.129099
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 4.00000 0.508001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 4.00000 0.492366
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 6.00000 0.727607
\(69\) 1.00000 0.120386
\(70\) −1.00000 −0.119523
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 1.00000 0.117851
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −6.00000 −0.697486
\(75\) −1.00000 −0.115470
\(76\) −4.00000 −0.458831
\(77\) −4.00000 −0.455842
\(78\) 2.00000 0.226455
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −1.00000 −0.109109
\(85\) −6.00000 −0.650791
\(86\) 4.00000 0.431331
\(87\) −2.00000 −0.214423
\(88\) −4.00000 −0.426401
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −1.00000 −0.105409
\(91\) −2.00000 −0.209657
\(92\) −1.00000 −0.104257
\(93\) −4.00000 −0.414781
\(94\) −12.0000 −1.23771
\(95\) 4.00000 0.410391
\(96\) −1.00000 −0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 1.00000 0.101015
\(99\) −4.00000 −0.402015
\(100\) 1.00000 0.100000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) −6.00000 −0.594089
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −2.00000 −0.196116
\(105\) 1.00000 0.0975900
\(106\) −6.00000 −0.582772
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 4.00000 0.381385
\(111\) 6.00000 0.569495
\(112\) 1.00000 0.0944911
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 4.00000 0.374634
\(115\) 1.00000 0.0932505
\(116\) 2.00000 0.185695
\(117\) −2.00000 −0.184900
\(118\) −4.00000 −0.368230
\(119\) 6.00000 0.550019
\(120\) 1.00000 0.0912871
\(121\) 5.00000 0.454545
\(122\) −14.0000 −1.26750
\(123\) −10.0000 −0.901670
\(124\) 4.00000 0.359211
\(125\) −1.00000 −0.0894427
\(126\) 1.00000 0.0890871
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 2.00000 0.175412
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 4.00000 0.348155
\(133\) −4.00000 −0.346844
\(134\) −12.0000 −1.03664
\(135\) 1.00000 0.0860663
\(136\) 6.00000 0.514496
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 1.00000 0.0851257
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 12.0000 1.01058
\(142\) −12.0000 −1.00702
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) 10.0000 0.827606
\(147\) −1.00000 −0.0824786
\(148\) −6.00000 −0.493197
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −4.00000 −0.324443
\(153\) 6.00000 0.485071
\(154\) −4.00000 −0.322329
\(155\) −4.00000 −0.321288
\(156\) 2.00000 0.160128
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) −8.00000 −0.636446
\(159\) 6.00000 0.475831
\(160\) −1.00000 −0.0790569
\(161\) −1.00000 −0.0788110
\(162\) 1.00000 0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 10.0000 0.780869
\(165\) −4.00000 −0.311400
\(166\) −4.00000 −0.310460
\(167\) −20.0000 −1.54765 −0.773823 0.633402i \(-0.781658\pi\)
−0.773823 + 0.633402i \(0.781658\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) −6.00000 −0.460179
\(171\) −4.00000 −0.305888
\(172\) 4.00000 0.304997
\(173\) 10.0000 0.760286 0.380143 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) −2.00000 −0.151620
\(175\) 1.00000 0.0755929
\(176\) −4.00000 −0.301511
\(177\) 4.00000 0.300658
\(178\) −6.00000 −0.449719
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) −2.00000 −0.148250
\(183\) 14.0000 1.03491
\(184\) −1.00000 −0.0737210
\(185\) 6.00000 0.441129
\(186\) −4.00000 −0.293294
\(187\) −24.0000 −1.75505
\(188\) −12.0000 −0.875190
\(189\) −1.00000 −0.0727393
\(190\) 4.00000 0.290191
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 2.00000 0.143592
\(195\) −2.00000 −0.143223
\(196\) 1.00000 0.0714286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) −4.00000 −0.284268
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 1.00000 0.0707107
\(201\) 12.0000 0.846415
\(202\) 14.0000 0.985037
\(203\) 2.00000 0.140372
\(204\) −6.00000 −0.420084
\(205\) −10.0000 −0.698430
\(206\) 8.00000 0.557386
\(207\) −1.00000 −0.0695048
\(208\) −2.00000 −0.138675
\(209\) 16.0000 1.10674
\(210\) 1.00000 0.0690066
\(211\) −28.0000 −1.92760 −0.963800 0.266627i \(-0.914091\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) −6.00000 −0.412082
\(213\) 12.0000 0.822226
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) −1.00000 −0.0680414
\(217\) 4.00000 0.271538
\(218\) −6.00000 −0.406371
\(219\) −10.0000 −0.675737
\(220\) 4.00000 0.269680
\(221\) −12.0000 −0.807207
\(222\) 6.00000 0.402694
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) −14.0000 −0.931266
\(227\) −28.0000 −1.85843 −0.929213 0.369546i \(-0.879513\pi\)
−0.929213 + 0.369546i \(0.879513\pi\)
\(228\) 4.00000 0.264906
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 1.00000 0.0659380
\(231\) 4.00000 0.263181
\(232\) 2.00000 0.131306
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) −2.00000 −0.130744
\(235\) 12.0000 0.782794
\(236\) −4.00000 −0.260378
\(237\) 8.00000 0.519656
\(238\) 6.00000 0.388922
\(239\) −28.0000 −1.81117 −0.905585 0.424165i \(-0.860568\pi\)
−0.905585 + 0.424165i \(0.860568\pi\)
\(240\) 1.00000 0.0645497
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) −14.0000 −0.896258
\(245\) −1.00000 −0.0638877
\(246\) −10.0000 −0.637577
\(247\) 8.00000 0.509028
\(248\) 4.00000 0.254000
\(249\) 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 \(0.226453\pi\)
\(252\) 1.00000 0.0629941
\(253\) 4.00000 0.251478
\(254\) −12.0000 −0.752947
\(255\) 6.00000 0.375735
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) −4.00000 −0.249029
\(259\) −6.00000 −0.372822
\(260\) 2.00000 0.124035
\(261\) 2.00000 0.123797
\(262\) 12.0000 0.741362
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 4.00000 0.246183
\(265\) 6.00000 0.368577
\(266\) −4.00000 −0.245256
\(267\) 6.00000 0.367194
\(268\) −12.0000 −0.733017
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 1.00000 0.0608581
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 6.00000 0.363803
\(273\) 2.00000 0.121046
\(274\) 10.0000 0.604122
\(275\) −4.00000 −0.241209
\(276\) 1.00000 0.0601929
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) −4.00000 −0.239904
\(279\) 4.00000 0.239474
\(280\) −1.00000 −0.0597614
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 12.0000 0.714590
\(283\) 24.0000 1.42665 0.713326 0.700832i \(-0.247188\pi\)
0.713326 + 0.700832i \(0.247188\pi\)
\(284\) −12.0000 −0.712069
\(285\) −4.00000 −0.236940
\(286\) 8.00000 0.473050
\(287\) 10.0000 0.590281
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) −2.00000 −0.117444
\(291\) −2.00000 −0.117242
\(292\) 10.0000 0.585206
\(293\) 26.0000 1.51894 0.759468 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 4.00000 0.232889
\(296\) −6.00000 −0.348743
\(297\) 4.00000 0.232104
\(298\) 10.0000 0.579284
\(299\) 2.00000 0.115663
\(300\) −1.00000 −0.0577350
\(301\) 4.00000 0.230556
\(302\) 0 0
\(303\) −14.0000 −0.804279
\(304\) −4.00000 −0.229416
\(305\) 14.0000 0.801638
\(306\) 6.00000 0.342997
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) −4.00000 −0.227921
\(309\) −8.00000 −0.455104
\(310\) −4.00000 −0.227185
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 2.00000 0.113228
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −14.0000 −0.790066
\(315\) −1.00000 −0.0563436
\(316\) −8.00000 −0.450035
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 6.00000 0.336463
\(319\) −8.00000 −0.447914
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −1.00000 −0.0557278
\(323\) −24.0000 −1.33540
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) 12.0000 0.664619
\(327\) 6.00000 0.331801
\(328\) 10.0000 0.552158
\(329\) −12.0000 −0.661581
\(330\) −4.00000 −0.220193
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −4.00000 −0.219529
\(333\) −6.00000 −0.328798
\(334\) −20.0000 −1.09435
\(335\) 12.0000 0.655630
\(336\) −1.00000 −0.0545545
\(337\) 30.0000 1.63420 0.817102 0.576493i \(-0.195579\pi\)
0.817102 + 0.576493i \(0.195579\pi\)
\(338\) −9.00000 −0.489535
\(339\) 14.0000 0.760376
\(340\) −6.00000 −0.325396
\(341\) −16.0000 −0.866449
\(342\) −4.00000 −0.216295
\(343\) 1.00000 0.0539949
\(344\) 4.00000 0.215666
\(345\) −1.00000 −0.0538382
\(346\) 10.0000 0.537603
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) −2.00000 −0.107211
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 1.00000 0.0534522
\(351\) 2.00000 0.106752
\(352\) −4.00000 −0.213201
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 4.00000 0.212598
\(355\) 12.0000 0.636894
\(356\) −6.00000 −0.317999
\(357\) −6.00000 −0.317554
\(358\) 4.00000 0.211407
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −3.00000 −0.157895
\(362\) −22.0000 −1.15629
\(363\) −5.00000 −0.262432
\(364\) −2.00000 −0.104828
\(365\) −10.0000 −0.523424
\(366\) 14.0000 0.731792
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 10.0000 0.520579
\(370\) 6.00000 0.311925
\(371\) −6.00000 −0.311504
\(372\) −4.00000 −0.207390
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) −24.0000 −1.24101
\(375\) 1.00000 0.0516398
\(376\) −12.0000 −0.618853
\(377\) −4.00000 −0.206010
\(378\) −1.00000 −0.0514344
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 4.00000 0.205196
\(381\) 12.0000 0.614779
\(382\) 8.00000 0.409316
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 4.00000 0.203859
\(386\) −14.0000 −0.712581
\(387\) 4.00000 0.203331
\(388\) 2.00000 0.101535
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) −2.00000 −0.101274
\(391\) −6.00000 −0.303433
\(392\) 1.00000 0.0505076
\(393\) −12.0000 −0.605320
\(394\) −18.0000 −0.906827
\(395\) 8.00000 0.402524
\(396\) −4.00000 −0.201008
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 24.0000 1.20301
\(399\) 4.00000 0.200250
\(400\) 1.00000 0.0500000
\(401\) −34.0000 −1.69788 −0.848939 0.528490i \(-0.822758\pi\)
−0.848939 + 0.528490i \(0.822758\pi\)
\(402\) 12.0000 0.598506
\(403\) −8.00000 −0.398508
\(404\) 14.0000 0.696526
\(405\) −1.00000 −0.0496904
\(406\) 2.00000 0.0992583
\(407\) 24.0000 1.18964
\(408\) −6.00000 −0.297044
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) −10.0000 −0.493865
\(411\) −10.0000 −0.493264
\(412\) 8.00000 0.394132
\(413\) −4.00000 −0.196827
\(414\) −1.00000 −0.0491473
\(415\) 4.00000 0.196352
\(416\) −2.00000 −0.0980581
\(417\) 4.00000 0.195881
\(418\) 16.0000 0.782586
\(419\) −40.0000 −1.95413 −0.977064 0.212946i \(-0.931694\pi\)
−0.977064 + 0.212946i \(0.931694\pi\)
\(420\) 1.00000 0.0487950
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) −28.0000 −1.36302
\(423\) −12.0000 −0.583460
\(424\) −6.00000 −0.291386
\(425\) 6.00000 0.291043
\(426\) 12.0000 0.581402
\(427\) −14.0000 −0.677507
\(428\) 0 0
\(429\) −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 \(0.756250\pi\)
\(434\) 4.00000 0.192006
\(435\) 2.00000 0.0958927
\(436\) −6.00000 −0.287348
\(437\) 4.00000 0.191346
\(438\) −10.0000 −0.477818
\(439\) 36.0000 1.71819 0.859093 0.511819i \(-0.171028\pi\)
0.859093 + 0.511819i \(0.171028\pi\)
\(440\) 4.00000 0.190693
\(441\) 1.00000 0.0476190
\(442\) −12.0000 −0.570782
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) 6.00000 0.284747
\(445\) 6.00000 0.284427
\(446\) −16.0000 −0.757622
\(447\) −10.0000 −0.472984
\(448\) 1.00000 0.0472456
\(449\) −22.0000 −1.03824 −0.519122 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 1.00000 0.0471405
\(451\) −40.0000 −1.88353
\(452\) −14.0000 −0.658505
\(453\) 0 0
\(454\) −28.0000 −1.31411
\(455\) 2.00000 0.0937614
\(456\) 4.00000 0.187317
\(457\) 14.0000 0.654892 0.327446 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(458\) −22.0000 −1.02799
\(459\) −6.00000 −0.280056
\(460\) 1.00000 0.0466252
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 4.00000 0.186097
\(463\) 36.0000 1.67306 0.836531 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) 2.00000 0.0928477
\(465\) 4.00000 0.185496
\(466\) 26.0000 1.20443
\(467\) 4.00000 0.185098 0.0925490 0.995708i \(-0.470499\pi\)
0.0925490 + 0.995708i \(0.470499\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −12.0000 −0.554109
\(470\) 12.0000 0.553519
\(471\) 14.0000 0.645086
\(472\) −4.00000 −0.184115
\(473\) −16.0000 −0.735681
\(474\) 8.00000 0.367452
\(475\) −4.00000 −0.183533
\(476\) 6.00000 0.275010
\(477\) −6.00000 −0.274721
\(478\) −28.0000 −1.28069
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 1.00000 0.0456435
\(481\) 12.0000 0.547153
\(482\) −18.0000 −0.819878
\(483\) 1.00000 0.0455016
\(484\) 5.00000 0.227273
\(485\) −2.00000 −0.0908153
\(486\) −1.00000 −0.0453609
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) −14.0000 −0.633750
\(489\) −12.0000 −0.542659
\(490\) −1.00000 −0.0451754
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) −10.0000 −0.450835
\(493\) 12.0000 0.540453
\(494\) 8.00000 0.359937
\(495\) 4.00000 0.179787
\(496\) 4.00000 0.179605
\(497\) −12.0000 −0.538274
\(498\) 4.00000 0.179244
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 20.0000 0.893534
\(502\) 24.0000 1.07117
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 1.00000 0.0445435
\(505\) −14.0000 −0.622992
\(506\) 4.00000 0.177822
\(507\) 9.00000 0.399704
\(508\) −12.0000 −0.532414
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 6.00000 0.265684
\(511\) 10.0000 0.442374
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) 18.0000 0.793946
\(515\) −8.00000 −0.352522
\(516\) −4.00000 −0.176090
\(517\) 48.0000 2.11104
\(518\) −6.00000 −0.263625
\(519\) −10.0000 −0.438951
\(520\) 2.00000 0.0877058
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 2.00000 0.0875376
\(523\) −32.0000 −1.39926 −0.699631 0.714504i \(-0.746652\pi\)
−0.699631 + 0.714504i \(0.746652\pi\)
\(524\) 12.0000 0.524222
\(525\) −1.00000 −0.0436436
\(526\) −16.0000 −0.697633
\(527\) 24.0000 1.04546
\(528\) 4.00000 0.174078
\(529\) 1.00000 0.0434783
\(530\) 6.00000 0.260623
\(531\) −4.00000 −0.173585
\(532\) −4.00000 −0.173422
\(533\) −20.0000 −0.866296
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) −4.00000 −0.172613
\(538\) −10.0000 −0.431131
\(539\) −4.00000 −0.172292
\(540\) 1.00000 0.0430331
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) −20.0000 −0.859074
\(543\) 22.0000 0.944110
\(544\) 6.00000 0.257248
\(545\) 6.00000 0.257012
\(546\) 2.00000 0.0855921
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 10.0000 0.427179
\(549\) −14.0000 −0.597505
\(550\) −4.00000 −0.170561
\(551\) −8.00000 −0.340811
\(552\) 1.00000 0.0425628
\(553\) −8.00000 −0.340195
\(554\) −14.0000 −0.594803
\(555\) −6.00000 −0.254686
\(556\) −4.00000 −0.169638
\(557\) 42.0000 1.77960 0.889799 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) 4.00000 0.169334
\(559\) −8.00000 −0.338364
\(560\) −1.00000 −0.0422577
\(561\) 24.0000 1.01328
\(562\) 6.00000 0.253095
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 12.0000 0.505291
\(565\) 14.0000 0.588984
\(566\) 24.0000 1.00880
\(567\) 1.00000 0.0419961
\(568\) −12.0000 −0.503509
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) −4.00000 −0.167542
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 8.00000 0.334497
\(573\) −8.00000 −0.334205
\(574\) 10.0000 0.417392
\(575\) −1.00000 −0.0417029
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 19.0000 0.790296
\(579\) 14.0000 0.581820
\(580\) −2.00000 −0.0830455
\(581\) −4.00000 −0.165948
\(582\) −2.00000 −0.0829027
\(583\) 24.0000 0.993978
\(584\) 10.0000 0.413803
\(585\) 2.00000 0.0826898
\(586\) 26.0000 1.07405
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −16.0000 −0.659269
\(590\) 4.00000 0.164677
\(591\) 18.0000 0.740421
\(592\) −6.00000 −0.246598
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 4.00000 0.164122
\(595\) −6.00000 −0.245976
\(596\) 10.0000 0.409616
\(597\) −24.0000 −0.982255
\(598\) 2.00000 0.0817861
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 4.00000 0.163028
\(603\) −12.0000 −0.488678
\(604\) 0 0
\(605\) −5.00000 −0.203279
\(606\) −14.0000 −0.568711
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) −4.00000 −0.162221
\(609\) −2.00000 −0.0810441
\(610\) 14.0000 0.566843
\(611\) 24.0000 0.970936
\(612\) 6.00000 0.242536
\(613\) 42.0000 1.69636 0.848182 0.529705i \(-0.177697\pi\)
0.848182 + 0.529705i \(0.177697\pi\)
\(614\) 20.0000 0.807134
\(615\) 10.0000 0.403239
\(616\) −4.00000 −0.161165
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) −8.00000 −0.321807
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) −4.00000 −0.160644
\(621\) 1.00000 0.0401286
\(622\) 24.0000 0.962312
\(623\) −6.00000 −0.240385
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) −16.0000 −0.638978
\(628\) −14.0000 −0.558661
\(629\) −36.0000 −1.43541
\(630\) −1.00000 −0.0398410
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −8.00000 −0.318223
\(633\) 28.0000 1.11290
\(634\) −18.0000 −0.714871
\(635\) 12.0000 0.476205
\(636\) 6.00000 0.237915
\(637\) −2.00000 −0.0792429
\(638\) −8.00000 −0.316723
\(639\) −12.0000 −0.474713
\(640\) −1.00000 −0.0395285
\(641\) −26.0000 −1.02694 −0.513469 0.858108i \(-0.671640\pi\)
−0.513469 + 0.858108i \(0.671640\pi\)
\(642\) 0 0
\(643\) 24.0000 0.946468 0.473234 0.880937i \(-0.343087\pi\)
0.473234 + 0.880937i \(0.343087\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 4.00000 0.157500
\(646\) −24.0000 −0.944267
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 1.00000 0.0392837
\(649\) 16.0000 0.628055
\(650\) −2.00000 −0.0784465
\(651\) −4.00000 −0.156772
\(652\) 12.0000 0.469956
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 6.00000 0.234619
\(655\) −12.0000 −0.468879
\(656\) 10.0000 0.390434
\(657\) 10.0000 0.390137
\(658\) −12.0000 −0.467809
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) −4.00000 −0.155700
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) −28.0000 −1.08825
\(663\) 12.0000 0.466041
\(664\) −4.00000 −0.155230
\(665\) 4.00000 0.155113
\(666\) −6.00000 −0.232495
\(667\) −2.00000 −0.0774403
\(668\) −20.0000 −0.773823
\(669\) 16.0000 0.618596
\(670\) 12.0000 0.463600
\(671\) 56.0000 2.16186
\(672\) −1.00000 −0.0385758
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 30.0000 1.15556
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) 26.0000 0.999261 0.499631 0.866239i \(-0.333469\pi\)
0.499631 + 0.866239i \(0.333469\pi\)
\(678\) 14.0000 0.537667
\(679\) 2.00000 0.0767530
\(680\) −6.00000 −0.230089
\(681\) 28.0000 1.07296
\(682\) −16.0000 −0.612672
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) −4.00000 −0.152944
\(685\) −10.0000 −0.382080
\(686\) 1.00000 0.0381802
\(687\) 22.0000 0.839352
\(688\) 4.00000 0.152499
\(689\) 12.0000 0.457164
\(690\) −1.00000 −0.0380693
\(691\) 44.0000 1.67384 0.836919 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) 10.0000 0.380143
\(693\) −4.00000 −0.151947
\(694\) 28.0000 1.06287
\(695\) 4.00000 0.151729
\(696\) −2.00000 −0.0758098
\(697\) 60.0000 2.27266
\(698\) −30.0000 −1.13552
\(699\) −26.0000 −0.983410
\(700\) 1.00000 0.0377964
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 2.00000 0.0754851
\(703\) 24.0000 0.905177
\(704\) −4.00000 −0.150756
\(705\) −12.0000 −0.451946
\(706\) 26.0000 0.978523
\(707\) 14.0000 0.526524
\(708\) 4.00000 0.150329
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 12.0000 0.450352
\(711\) −8.00000 −0.300023
\(712\) −6.00000 −0.224860
\(713\) −4.00000 −0.149801
\(714\) −6.00000 −0.224544
\(715\) −8.00000 −0.299183
\(716\) 4.00000 0.149487
\(717\) 28.0000 1.04568
\(718\) −32.0000 −1.19423
\(719\) −16.0000 −0.596699 −0.298350 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 8.00000 0.297936
\(722\) −3.00000 −0.111648
\(723\) 18.0000 0.669427
\(724\) −22.0000 −0.817624
\(725\) 2.00000 0.0742781
\(726\) −5.00000 −0.185567
\(727\) −40.0000 −1.48352 −0.741759 0.670667i \(-0.766008\pi\)
−0.741759 + 0.670667i \(0.766008\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) −10.0000 −0.370117
\(731\) 24.0000 0.887672
\(732\) 14.0000 0.517455
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 8.00000 0.295285
\(735\) 1.00000 0.0368856
\(736\) −1.00000 −0.0368605
\(737\) 48.0000 1.76810
\(738\) 10.0000 0.368105
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 6.00000 0.220564
\(741\) −8.00000 −0.293887
\(742\) −6.00000 −0.220267
\(743\) −8.00000 −0.293492 −0.146746 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) −4.00000 −0.146647
\(745\) −10.0000 −0.366372
\(746\) −22.0000 −0.805477
\(747\) −4.00000 −0.146352
\(748\) −24.0000 −0.877527
\(749\) 0 0
\(750\) 1.00000 0.0365148
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) −12.0000 −0.437595
\(753\) −24.0000 −0.874609
\(754\) −4.00000 −0.145671
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) −8.00000 −0.290573
\(759\) −4.00000 −0.145191
\(760\) 4.00000 0.145095
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 12.0000 0.434714
\(763\) −6.00000 −0.217215
\(764\) 8.00000 0.289430
\(765\) −6.00000 −0.216930
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) −1.00000 −0.0360844
\(769\) −26.0000 −0.937584 −0.468792 0.883309i \(-0.655311\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) 4.00000 0.144150
\(771\) −18.0000 −0.648254
\(772\) −14.0000 −0.503871
\(773\) −54.0000 −1.94225 −0.971123 0.238581i \(-0.923318\pi\)
−0.971123 + 0.238581i \(0.923318\pi\)
\(774\) 4.00000 0.143777
\(775\) 4.00000 0.143684
\(776\) 2.00000 0.0717958
\(777\) 6.00000 0.215249
\(778\) 26.0000 0.932145
\(779\) −40.0000 −1.43315
\(780\) −2.00000 −0.0716115
\(781\) 48.0000 1.71758
\(782\) −6.00000 −0.214560
\(783\) −2.00000 −0.0714742
\(784\) 1.00000 0.0357143
\(785\) 14.0000 0.499681
\(786\) −12.0000 −0.428026
\(787\) 8.00000 0.285169 0.142585 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) −18.0000 −0.641223
\(789\) 16.0000 0.569615
\(790\) 8.00000 0.284627
\(791\) −14.0000 −0.497783
\(792\) −4.00000 −0.142134
\(793\) 28.0000 0.994309
\(794\) −2.00000 −0.0709773
\(795\) −6.00000 −0.212798
\(796\) 24.0000 0.850657
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 4.00000 0.141598
\(799\) −72.0000 −2.54718
\(800\) 1.00000 0.0353553
\(801\) −6.00000 −0.212000
\(802\) −34.0000 −1.20058
\(803\) −40.0000 −1.41157
\(804\) 12.0000 0.423207
\(805\) 1.00000 0.0352454
\(806\) −8.00000 −0.281788
\(807\) 10.0000 0.352017
\(808\) 14.0000 0.492518
\(809\) −22.0000 −0.773479 −0.386739 0.922189i \(-0.626399\pi\)
−0.386739 + 0.922189i \(0.626399\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 2.00000 0.0701862
\(813\) 20.0000 0.701431
\(814\) 24.0000 0.841200
\(815\) −12.0000 −0.420342
\(816\) −6.00000 −0.210042
\(817\) −16.0000 −0.559769
\(818\) 10.0000 0.349642
\(819\) −2.00000 −0.0698857
\(820\) −10.0000 −0.349215
\(821\) 2.00000 0.0698005 0.0349002 0.999391i \(-0.488889\pi\)
0.0349002 + 0.999391i \(0.488889\pi\)
\(822\) −10.0000 −0.348790
\(823\) −44.0000 −1.53374 −0.766872 0.641800i \(-0.778188\pi\)
−0.766872 + 0.641800i \(0.778188\pi\)
\(824\) 8.00000 0.278693
\(825\) 4.00000 0.139262
\(826\) −4.00000 −0.139178
\(827\) 16.0000 0.556375 0.278187 0.960527i \(-0.410266\pi\)
0.278187 + 0.960527i \(0.410266\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 4.00000 0.138842
\(831\) 14.0000 0.485655
\(832\) −2.00000 −0.0693375
\(833\) 6.00000 0.207888
\(834\) 4.00000 0.138509
\(835\) 20.0000 0.692129
\(836\) 16.0000 0.553372
\(837\) −4.00000 −0.138260
\(838\) −40.0000 −1.38178
\(839\) 32.0000 1.10476 0.552381 0.833592i \(-0.313719\pi\)
0.552381 + 0.833592i \(0.313719\pi\)
\(840\) 1.00000 0.0345033
\(841\) −25.0000 −0.862069
\(842\) 26.0000 0.896019
\(843\) −6.00000 −0.206651
\(844\) −28.0000 −0.963800
\(845\) 9.00000 0.309609
\(846\) −12.0000 −0.412568
\(847\) 5.00000 0.171802
\(848\) −6.00000 −0.206041
\(849\) −24.0000 −0.823678
\(850\) 6.00000 0.205798
\(851\) 6.00000 0.205677
\(852\) 12.0000 0.411113
\(853\) 22.0000 0.753266 0.376633 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) −14.0000 −0.479070
\(855\) 4.00000 0.136797
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) −8.00000 −0.273115
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) −4.00000 −0.136399
\(861\) −10.0000 −0.340799
\(862\) 0 0
\(863\) −16.0000 −0.544646 −0.272323 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −10.0000 −0.340010
\(866\) −30.0000 −1.01944
\(867\) −19.0000 −0.645274
\(868\) 4.00000 0.135769
\(869\) 32.0000 1.08553
\(870\) 2.00000 0.0678064
\(871\) 24.0000 0.813209
\(872\) −6.00000 −0.203186
\(873\) 2.00000 0.0676897
\(874\) 4.00000 0.135302
\(875\) −1.00000 −0.0338062
\(876\) −10.0000 −0.337869
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) 36.0000 1.21494
\(879\) −26.0000 −0.876958
\(880\) 4.00000 0.134840
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 1.00000 0.0336718
\(883\) −28.0000 −0.942275 −0.471138 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) −12.0000 −0.403604
\(885\) −4.00000 −0.134459
\(886\) −20.0000 −0.671913
\(887\) 4.00000 0.134307 0.0671534 0.997743i \(-0.478608\pi\)
0.0671534 + 0.997743i \(0.478608\pi\)
\(888\) 6.00000 0.201347
\(889\) −12.0000 −0.402467
\(890\) 6.00000 0.201120
\(891\) −4.00000 −0.134005
\(892\) −16.0000 −0.535720
\(893\) 48.0000 1.60626
\(894\) −10.0000 −0.334450
\(895\) −4.00000 −0.133705
\(896\) 1.00000 0.0334077
\(897\) −2.00000 −0.0667781
\(898\) −22.0000 −0.734150
\(899\) 8.00000 0.266815
\(900\) 1.00000 0.0333333
\(901\) −36.0000 −1.19933
\(902\) −40.0000 −1.33185
\(903\) −4.00000 −0.133112
\(904\) −14.0000 −0.465633
\(905\) 22.0000 0.731305
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) −28.0000 −0.929213
\(909\) 14.0000 0.464351
\(910\) 2.00000 0.0662994
\(911\) −16.0000 −0.530104 −0.265052 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 4.00000 0.132453
\(913\) 16.0000 0.529523
\(914\) 14.0000 0.463079
\(915\) −14.0000 −0.462826
\(916\) −22.0000 −0.726900
\(917\) 12.0000 0.396275
\(918\) −6.00000 −0.198030
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 1.00000 0.0329690
\(921\) −20.0000 −0.659022
\(922\) 30.0000 0.987997
\(923\) 24.0000 0.789970
\(924\) 4.00000 0.131590
\(925\) −6.00000 −0.197279
\(926\) 36.0000 1.18303
\(927\) 8.00000 0.262754
\(928\) 2.00000 0.0656532
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 4.00000 0.131165
\(931\) −4.00000 −0.131095
\(932\) 26.0000 0.851658
\(933\) −24.0000 −0.785725
\(934\) 4.00000 0.130884
\(935\) 24.0000 0.784884
\(936\) −2.00000 −0.0653720
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) −12.0000 −0.391814
\(939\) −10.0000 −0.326338
\(940\) 12.0000 0.391397
\(941\) 50.0000 1.62995 0.814977 0.579494i \(-0.196750\pi\)
0.814977 + 0.579494i \(0.196750\pi\)
\(942\) 14.0000 0.456145
\(943\) −10.0000 −0.325645
\(944\) −4.00000 −0.130189
\(945\) 1.00000 0.0325300
\(946\) −16.0000 −0.520205
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 8.00000 0.259828
\(949\) −20.0000 −0.649227
\(950\) −4.00000 −0.129777
\(951\) 18.0000 0.583690
\(952\) 6.00000 0.194461
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) −6.00000 −0.194257
\(955\) −8.00000 −0.258874
\(956\) −28.0000 −0.905585
\(957\) 8.00000 0.258603
\(958\) 24.0000 0.775405
\(959\) 10.0000 0.322917
\(960\) 1.00000 0.0322749
\(961\) −15.0000 −0.483871
\(962\) 12.0000 0.386896
\(963\) 0 0
\(964\) −18.0000 −0.579741
\(965\) 14.0000 0.450676
\(966\) 1.00000 0.0321745
\(967\) −44.0000 −1.41494 −0.707472 0.706741i \(-0.750165\pi\)
−0.707472 + 0.706741i \(0.750165\pi\)
\(968\) 5.00000 0.160706
\(969\) 24.0000 0.770991
\(970\) −2.00000 −0.0642161
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −4.00000 −0.128234
\(974\) −20.0000 −0.640841
\(975\) 2.00000 0.0640513
\(976\) −14.0000 −0.448129
\(977\) −38.0000 −1.21573 −0.607864 0.794041i \(-0.707973\pi\)
−0.607864 + 0.794041i \(0.707973\pi\)
\(978\) −12.0000 −0.383718
\(979\) 24.0000 0.767043
\(980\) −1.00000 −0.0319438
\(981\) −6.00000 −0.191565
\(982\) 28.0000 0.893516
\(983\) −16.0000 −0.510321 −0.255160 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(984\) −10.0000 −0.318788
\(985\) 18.0000 0.573528
\(986\) 12.0000 0.382158
\(987\) 12.0000 0.381964
\(988\) 8.00000 0.254514
\(989\) −4.00000 −0.127193
\(990\) 4.00000 0.127128
\(991\) 56.0000 1.77890 0.889449 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) 4.00000 0.127000
\(993\) 28.0000 0.888553
\(994\) −12.0000 −0.380617
\(995\) −24.0000 −0.760851
\(996\) 4.00000 0.126745
\(997\) −18.0000 −0.570066 −0.285033 0.958518i \(-0.592005\pi\)
−0.285033 + 0.958518i \(0.592005\pi\)
\(998\) −20.0000 −0.633089
\(999\) 6.00000 0.189832
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4830.2.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4830.2.a.s.1.1 1 1.1 even 1 trivial