Properties

Label 5070.2.a.o.1.1
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +6.00000 q^{19} -1.00000 q^{20} -2.00000 q^{21} +1.00000 q^{22} -3.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{28} -1.00000 q^{29} +1.00000 q^{30} -3.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +2.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -5.00000 q^{37} +6.00000 q^{38} -1.00000 q^{40} +10.0000 q^{41} -2.00000 q^{42} +5.00000 q^{43} +1.00000 q^{44} -1.00000 q^{45} -3.00000 q^{46} +3.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +14.0000 q^{53} -1.00000 q^{54} -1.00000 q^{55} +2.00000 q^{56} -6.00000 q^{57} -1.00000 q^{58} -5.00000 q^{59} +1.00000 q^{60} -10.0000 q^{61} -3.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} +2.00000 q^{68} +3.00000 q^{69} -2.00000 q^{70} +4.00000 q^{71} +1.00000 q^{72} -2.00000 q^{73} -5.00000 q^{74} -1.00000 q^{75} +6.00000 q^{76} +2.00000 q^{77} +5.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -6.00000 q^{83} -2.00000 q^{84} -2.00000 q^{85} +5.00000 q^{86} +1.00000 q^{87} +1.00000 q^{88} +10.0000 q^{89} -1.00000 q^{90} -3.00000 q^{92} +3.00000 q^{93} +3.00000 q^{94} -6.00000 q^{95} -1.00000 q^{96} -10.0000 q^{97} -3.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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 \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 2.00000 0.534522
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.00000 −0.436436
\(22\) 1.00000 0.213201
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 2.00000 0.377964
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 1.00000 0.182574
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) 2.00000 0.342997
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) −2.00000 −0.308607
\(43\) 5.00000 0.762493 0.381246 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) 1.00000 0.150756
\(45\) −1.00000 −0.149071
\(46\) −3.00000 −0.442326
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 14.0000 1.92305 0.961524 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) −1.00000 −0.136083
\(55\) −1.00000 −0.134840
\(56\) 2.00000 0.267261
\(57\) −6.00000 −0.794719
\(58\) −1.00000 −0.131306
\(59\) −5.00000 −0.650945 −0.325472 0.945552i \(-0.605523\pi\)
−0.325472 + 0.945552i \(0.605523\pi\)
\(60\) 1.00000 0.129099
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −3.00000 −0.381000
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 2.00000 0.242536
\(69\) 3.00000 0.361158
\(70\) −2.00000 −0.239046
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −5.00000 −0.581238
\(75\) −1.00000 −0.115470
\(76\) 6.00000 0.688247
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) 5.00000 0.562544 0.281272 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) −2.00000 −0.218218
\(85\) −2.00000 −0.216930
\(86\) 5.00000 0.539164
\(87\) 1.00000 0.107211
\(88\) 1.00000 0.106600
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −3.00000 −0.312772
\(93\) 3.00000 0.311086
\(94\) 3.00000 0.309426
\(95\) −6.00000 −0.615587
\(96\) −1.00000 −0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −3.00000 −0.303046
\(99\) 1.00000 0.100504
\(100\) 1.00000 0.100000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) −2.00000 −0.198030
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 14.0000 1.35980
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 5.00000 0.474579
\(112\) 2.00000 0.188982
\(113\) 17.0000 1.59923 0.799613 0.600516i \(-0.205038\pi\)
0.799613 + 0.600516i \(0.205038\pi\)
\(114\) −6.00000 −0.561951
\(115\) 3.00000 0.279751
\(116\) −1.00000 −0.0928477
\(117\) 0 0
\(118\) −5.00000 −0.460287
\(119\) 4.00000 0.366679
\(120\) 1.00000 0.0912871
\(121\) −10.0000 −0.909091
\(122\) −10.0000 −0.905357
\(123\) −10.0000 −0.901670
\(124\) −3.00000 −0.269408
\(125\) −1.00000 −0.0894427
\(126\) 2.00000 0.178174
\(127\) −14.0000 −1.24230 −0.621150 0.783692i \(-0.713334\pi\)
−0.621150 + 0.783692i \(0.713334\pi\)
\(128\) 1.00000 0.0883883
\(129\) −5.00000 −0.440225
\(130\) 0 0
\(131\) 13.0000 1.13582 0.567908 0.823092i \(-0.307753\pi\)
0.567908 + 0.823092i \(0.307753\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 12.0000 1.04053
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 2.00000 0.171499
\(137\) −9.00000 −0.768922 −0.384461 0.923141i \(-0.625613\pi\)
−0.384461 + 0.923141i \(0.625613\pi\)
\(138\) 3.00000 0.255377
\(139\) 10.0000 0.848189 0.424094 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) −2.00000 −0.169031
\(141\) −3.00000 −0.252646
\(142\) 4.00000 0.335673
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 1.00000 0.0830455
\(146\) −2.00000 −0.165521
\(147\) 3.00000 0.247436
\(148\) −5.00000 −0.410997
\(149\) 11.0000 0.901155 0.450578 0.892737i \(-0.351218\pi\)
0.450578 + 0.892737i \(0.351218\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 24.0000 1.95309 0.976546 0.215308i \(-0.0690756\pi\)
0.976546 + 0.215308i \(0.0690756\pi\)
\(152\) 6.00000 0.486664
\(153\) 2.00000 0.161690
\(154\) 2.00000 0.161165
\(155\) 3.00000 0.240966
\(156\) 0 0
\(157\) 25.0000 1.99522 0.997609 0.0691164i \(-0.0220180\pi\)
0.997609 + 0.0691164i \(0.0220180\pi\)
\(158\) 5.00000 0.397779
\(159\) −14.0000 −1.11027
\(160\) −1.00000 −0.0790569
\(161\) −6.00000 −0.472866
\(162\) 1.00000 0.0785674
\(163\) 17.0000 1.33154 0.665771 0.746156i \(-0.268103\pi\)
0.665771 + 0.746156i \(0.268103\pi\)
\(164\) 10.0000 0.780869
\(165\) 1.00000 0.0778499
\(166\) −6.00000 −0.465690
\(167\) −7.00000 −0.541676 −0.270838 0.962625i \(-0.587301\pi\)
−0.270838 + 0.962625i \(0.587301\pi\)
\(168\) −2.00000 −0.154303
\(169\) 0 0
\(170\) −2.00000 −0.153393
\(171\) 6.00000 0.458831
\(172\) 5.00000 0.381246
\(173\) 4.00000 0.304114 0.152057 0.988372i \(-0.451410\pi\)
0.152057 + 0.988372i \(0.451410\pi\)
\(174\) 1.00000 0.0758098
\(175\) 2.00000 0.151186
\(176\) 1.00000 0.0753778
\(177\) 5.00000 0.375823
\(178\) 10.0000 0.749532
\(179\) −7.00000 −0.523205 −0.261602 0.965176i \(-0.584251\pi\)
−0.261602 + 0.965176i \(0.584251\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 16.0000 1.18927 0.594635 0.803996i \(-0.297296\pi\)
0.594635 + 0.803996i \(0.297296\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) −3.00000 −0.221163
\(185\) 5.00000 0.367607
\(186\) 3.00000 0.219971
\(187\) 2.00000 0.146254
\(188\) 3.00000 0.218797
\(189\) −2.00000 −0.145479
\(190\) −6.00000 −0.435286
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 1.00000 0.0710669
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 14.0000 0.985037
\(203\) −2.00000 −0.140372
\(204\) −2.00000 −0.140028
\(205\) −10.0000 −0.698430
\(206\) −6.00000 −0.418040
\(207\) −3.00000 −0.208514
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) 2.00000 0.138013
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 14.0000 0.961524
\(213\) −4.00000 −0.274075
\(214\) −6.00000 −0.410152
\(215\) −5.00000 −0.340997
\(216\) −1.00000 −0.0680414
\(217\) −6.00000 −0.407307
\(218\) −6.00000 −0.406371
\(219\) 2.00000 0.135147
\(220\) −1.00000 −0.0674200
\(221\) 0 0
\(222\) 5.00000 0.335578
\(223\) 10.0000 0.669650 0.334825 0.942280i \(-0.391323\pi\)
0.334825 + 0.942280i \(0.391323\pi\)
\(224\) 2.00000 0.133631
\(225\) 1.00000 0.0666667
\(226\) 17.0000 1.13082
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) −6.00000 −0.397360
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 3.00000 0.197814
\(231\) −2.00000 −0.131590
\(232\) −1.00000 −0.0656532
\(233\) 3.00000 0.196537 0.0982683 0.995160i \(-0.468670\pi\)
0.0982683 + 0.995160i \(0.468670\pi\)
\(234\) 0 0
\(235\) −3.00000 −0.195698
\(236\) −5.00000 −0.325472
\(237\) −5.00000 −0.324785
\(238\) 4.00000 0.259281
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 1.00000 0.0645497
\(241\) −7.00000 −0.450910 −0.225455 0.974254i \(-0.572387\pi\)
−0.225455 + 0.974254i \(0.572387\pi\)
\(242\) −10.0000 −0.642824
\(243\) −1.00000 −0.0641500
\(244\) −10.0000 −0.640184
\(245\) 3.00000 0.191663
\(246\) −10.0000 −0.637577
\(247\) 0 0
\(248\) −3.00000 −0.190500
\(249\) 6.00000 0.380235
\(250\) −1.00000 −0.0632456
\(251\) 3.00000 0.189358 0.0946792 0.995508i \(-0.469817\pi\)
0.0946792 + 0.995508i \(0.469817\pi\)
\(252\) 2.00000 0.125988
\(253\) −3.00000 −0.188608
\(254\) −14.0000 −0.878438
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) 21.0000 1.30994 0.654972 0.755653i \(-0.272680\pi\)
0.654972 + 0.755653i \(0.272680\pi\)
\(258\) −5.00000 −0.311286
\(259\) −10.0000 −0.621370
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 13.0000 0.803143
\(263\) 23.0000 1.41824 0.709120 0.705087i \(-0.249092\pi\)
0.709120 + 0.705087i \(0.249092\pi\)
\(264\) −1.00000 −0.0615457
\(265\) −14.0000 −0.860013
\(266\) 12.0000 0.735767
\(267\) −10.0000 −0.611990
\(268\) 0 0
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 1.00000 0.0608581
\(271\) −29.0000 −1.76162 −0.880812 0.473466i \(-0.843003\pi\)
−0.880812 + 0.473466i \(0.843003\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −9.00000 −0.543710
\(275\) 1.00000 0.0603023
\(276\) 3.00000 0.180579
\(277\) −19.0000 −1.14160 −0.570800 0.821089i \(-0.693367\pi\)
−0.570800 + 0.821089i \(0.693367\pi\)
\(278\) 10.0000 0.599760
\(279\) −3.00000 −0.179605
\(280\) −2.00000 −0.119523
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) −3.00000 −0.178647
\(283\) −13.0000 −0.772770 −0.386385 0.922338i \(-0.626276\pi\)
−0.386385 + 0.922338i \(0.626276\pi\)
\(284\) 4.00000 0.237356
\(285\) 6.00000 0.355409
\(286\) 0 0
\(287\) 20.0000 1.18056
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 1.00000 0.0587220
\(291\) 10.0000 0.586210
\(292\) −2.00000 −0.117041
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 3.00000 0.174964
\(295\) 5.00000 0.291111
\(296\) −5.00000 −0.290619
\(297\) −1.00000 −0.0580259
\(298\) 11.0000 0.637213
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 10.0000 0.576390
\(302\) 24.0000 1.38104
\(303\) −14.0000 −0.804279
\(304\) 6.00000 0.344124
\(305\) 10.0000 0.572598
\(306\) 2.00000 0.114332
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 2.00000 0.113961
\(309\) 6.00000 0.341328
\(310\) 3.00000 0.170389
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −12.0000 −0.678280 −0.339140 0.940736i \(-0.610136\pi\)
−0.339140 + 0.940736i \(0.610136\pi\)
\(314\) 25.0000 1.41083
\(315\) −2.00000 −0.112687
\(316\) 5.00000 0.281272
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) −14.0000 −0.785081
\(319\) −1.00000 −0.0559893
\(320\) −1.00000 −0.0559017
\(321\) 6.00000 0.334887
\(322\) −6.00000 −0.334367
\(323\) 12.0000 0.667698
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 17.0000 0.941543
\(327\) 6.00000 0.331801
\(328\) 10.0000 0.552158
\(329\) 6.00000 0.330791
\(330\) 1.00000 0.0550482
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −6.00000 −0.329293
\(333\) −5.00000 −0.273998
\(334\) −7.00000 −0.383023
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 0 0
\(339\) −17.0000 −0.923313
\(340\) −2.00000 −0.108465
\(341\) −3.00000 −0.162459
\(342\) 6.00000 0.324443
\(343\) −20.0000 −1.07990
\(344\) 5.00000 0.269582
\(345\) −3.00000 −0.161515
\(346\) 4.00000 0.215041
\(347\) −18.0000 −0.966291 −0.483145 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) 1.00000 0.0536056
\(349\) 8.00000 0.428230 0.214115 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 5.00000 0.265747
\(355\) −4.00000 −0.212298
\(356\) 10.0000 0.529999
\(357\) −4.00000 −0.211702
\(358\) −7.00000 −0.369961
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 17.0000 0.894737
\(362\) 16.0000 0.840941
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 10.0000 0.522708
\(367\) 36.0000 1.87918 0.939592 0.342296i \(-0.111204\pi\)
0.939592 + 0.342296i \(0.111204\pi\)
\(368\) −3.00000 −0.156386
\(369\) 10.0000 0.520579
\(370\) 5.00000 0.259938
\(371\) 28.0000 1.45369
\(372\) 3.00000 0.155543
\(373\) 37.0000 1.91579 0.957894 0.287123i \(-0.0926989\pi\)
0.957894 + 0.287123i \(0.0926989\pi\)
\(374\) 2.00000 0.103418
\(375\) 1.00000 0.0516398
\(376\) 3.00000 0.154713
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) −30.0000 −1.54100 −0.770498 0.637442i \(-0.779993\pi\)
−0.770498 + 0.637442i \(0.779993\pi\)
\(380\) −6.00000 −0.307794
\(381\) 14.0000 0.717242
\(382\) 24.0000 1.22795
\(383\) −27.0000 −1.37964 −0.689818 0.723983i \(-0.742309\pi\)
−0.689818 + 0.723983i \(0.742309\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −2.00000 −0.101929
\(386\) −4.00000 −0.203595
\(387\) 5.00000 0.254164
\(388\) −10.0000 −0.507673
\(389\) −1.00000 −0.0507020 −0.0253510 0.999679i \(-0.508070\pi\)
−0.0253510 + 0.999679i \(0.508070\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) −3.00000 −0.151523
\(393\) −13.0000 −0.655763
\(394\) −12.0000 −0.604551
\(395\) −5.00000 −0.251577
\(396\) 1.00000 0.0502519
\(397\) −13.0000 −0.652451 −0.326226 0.945292i \(-0.605777\pi\)
−0.326226 + 0.945292i \(0.605777\pi\)
\(398\) 0 0
\(399\) −12.0000 −0.600751
\(400\) 1.00000 0.0500000
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 14.0000 0.696526
\(405\) −1.00000 −0.0496904
\(406\) −2.00000 −0.0992583
\(407\) −5.00000 −0.247841
\(408\) −2.00000 −0.0990148
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) −10.0000 −0.493865
\(411\) 9.00000 0.443937
\(412\) −6.00000 −0.295599
\(413\) −10.0000 −0.492068
\(414\) −3.00000 −0.147442
\(415\) 6.00000 0.294528
\(416\) 0 0
\(417\) −10.0000 −0.489702
\(418\) 6.00000 0.293470
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 2.00000 0.0975900
\(421\) 28.0000 1.36464 0.682318 0.731055i \(-0.260972\pi\)
0.682318 + 0.731055i \(0.260972\pi\)
\(422\) −8.00000 −0.389434
\(423\) 3.00000 0.145865
\(424\) 14.0000 0.679900
\(425\) 2.00000 0.0970143
\(426\) −4.00000 −0.193801
\(427\) −20.0000 −0.967868
\(428\) −6.00000 −0.290021
\(429\) 0 0
\(430\) −5.00000 −0.241121
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) −6.00000 −0.288009
\(435\) −1.00000 −0.0479463
\(436\) −6.00000 −0.287348
\(437\) −18.0000 −0.861057
\(438\) 2.00000 0.0955637
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) −1.00000 −0.0476731
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 16.0000 0.760183 0.380091 0.924949i \(-0.375893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(444\) 5.00000 0.237289
\(445\) −10.0000 −0.474045
\(446\) 10.0000 0.473514
\(447\) −11.0000 −0.520282
\(448\) 2.00000 0.0944911
\(449\) −36.0000 −1.69895 −0.849473 0.527633i \(-0.823080\pi\)
−0.849473 + 0.527633i \(0.823080\pi\)
\(450\) 1.00000 0.0471405
\(451\) 10.0000 0.470882
\(452\) 17.0000 0.799613
\(453\) −24.0000 −1.12762
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) −6.00000 −0.280976
\(457\) 14.0000 0.654892 0.327446 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(458\) 22.0000 1.02799
\(459\) −2.00000 −0.0933520
\(460\) 3.00000 0.139876
\(461\) 27.0000 1.25752 0.628758 0.777601i \(-0.283564\pi\)
0.628758 + 0.777601i \(0.283564\pi\)
\(462\) −2.00000 −0.0930484
\(463\) 10.0000 0.464739 0.232370 0.972628i \(-0.425352\pi\)
0.232370 + 0.972628i \(0.425352\pi\)
\(464\) −1.00000 −0.0464238
\(465\) −3.00000 −0.139122
\(466\) 3.00000 0.138972
\(467\) −2.00000 −0.0925490 −0.0462745 0.998929i \(-0.514735\pi\)
−0.0462745 + 0.998929i \(0.514735\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −3.00000 −0.138380
\(471\) −25.0000 −1.15194
\(472\) −5.00000 −0.230144
\(473\) 5.00000 0.229900
\(474\) −5.00000 −0.229658
\(475\) 6.00000 0.275299
\(476\) 4.00000 0.183340
\(477\) 14.0000 0.641016
\(478\) −8.00000 −0.365911
\(479\) −4.00000 −0.182765 −0.0913823 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) −7.00000 −0.318841
\(483\) 6.00000 0.273009
\(484\) −10.0000 −0.454545
\(485\) 10.0000 0.454077
\(486\) −1.00000 −0.0453609
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −10.0000 −0.452679
\(489\) −17.0000 −0.768767
\(490\) 3.00000 0.135526
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) −10.0000 −0.450835
\(493\) −2.00000 −0.0900755
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) −3.00000 −0.134704
\(497\) 8.00000 0.358849
\(498\) 6.00000 0.268866
\(499\) −40.0000 −1.79065 −0.895323 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 7.00000 0.312737
\(502\) 3.00000 0.133897
\(503\) 40.0000 1.78351 0.891756 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(504\) 2.00000 0.0890871
\(505\) −14.0000 −0.622992
\(506\) −3.00000 −0.133366
\(507\) 0 0
\(508\) −14.0000 −0.621150
\(509\) −9.00000 −0.398918 −0.199459 0.979906i \(-0.563918\pi\)
−0.199459 + 0.979906i \(0.563918\pi\)
\(510\) 2.00000 0.0885615
\(511\) −4.00000 −0.176950
\(512\) 1.00000 0.0441942
\(513\) −6.00000 −0.264906
\(514\) 21.0000 0.926270
\(515\) 6.00000 0.264392
\(516\) −5.00000 −0.220113
\(517\) 3.00000 0.131940
\(518\) −10.0000 −0.439375
\(519\) −4.00000 −0.175581
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) −1.00000 −0.0437688
\(523\) 33.0000 1.44299 0.721495 0.692420i \(-0.243455\pi\)
0.721495 + 0.692420i \(0.243455\pi\)
\(524\) 13.0000 0.567908
\(525\) −2.00000 −0.0872872
\(526\) 23.0000 1.00285
\(527\) −6.00000 −0.261364
\(528\) −1.00000 −0.0435194
\(529\) −14.0000 −0.608696
\(530\) −14.0000 −0.608121
\(531\) −5.00000 −0.216982
\(532\) 12.0000 0.520266
\(533\) 0 0
\(534\) −10.0000 −0.432742
\(535\) 6.00000 0.259403
\(536\) 0 0
\(537\) 7.00000 0.302072
\(538\) 18.0000 0.776035
\(539\) −3.00000 −0.129219
\(540\) 1.00000 0.0430331
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) −29.0000 −1.24566
\(543\) −16.0000 −0.686626
\(544\) 2.00000 0.0857493
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −9.00000 −0.384461
\(549\) −10.0000 −0.426790
\(550\) 1.00000 0.0426401
\(551\) −6.00000 −0.255609
\(552\) 3.00000 0.127688
\(553\) 10.0000 0.425243
\(554\) −19.0000 −0.807233
\(555\) −5.00000 −0.212238
\(556\) 10.0000 0.424094
\(557\) −34.0000 −1.44063 −0.720313 0.693649i \(-0.756002\pi\)
−0.720313 + 0.693649i \(0.756002\pi\)
\(558\) −3.00000 −0.127000
\(559\) 0 0
\(560\) −2.00000 −0.0845154
\(561\) −2.00000 −0.0844401
\(562\) −30.0000 −1.26547
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) −3.00000 −0.126323
\(565\) −17.0000 −0.715195
\(566\) −13.0000 −0.546431
\(567\) 2.00000 0.0839921
\(568\) 4.00000 0.167836
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 6.00000 0.251312
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) −24.0000 −1.00261
\(574\) 20.0000 0.834784
\(575\) −3.00000 −0.125109
\(576\) 1.00000 0.0416667
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) −13.0000 −0.540729
\(579\) 4.00000 0.166234
\(580\) 1.00000 0.0415227
\(581\) −12.0000 −0.497844
\(582\) 10.0000 0.414513
\(583\) 14.0000 0.579821
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) −18.0000 −0.742940 −0.371470 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) 3.00000 0.123718
\(589\) −18.0000 −0.741677
\(590\) 5.00000 0.205847
\(591\) 12.0000 0.493614
\(592\) −5.00000 −0.205499
\(593\) −35.0000 −1.43728 −0.718639 0.695383i \(-0.755235\pi\)
−0.718639 + 0.695383i \(0.755235\pi\)
\(594\) −1.00000 −0.0410305
\(595\) −4.00000 −0.163984
\(596\) 11.0000 0.450578
\(597\) 0 0
\(598\) 0 0
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −11.0000 −0.448699 −0.224350 0.974509i \(-0.572026\pi\)
−0.224350 + 0.974509i \(0.572026\pi\)
\(602\) 10.0000 0.407570
\(603\) 0 0
\(604\) 24.0000 0.976546
\(605\) 10.0000 0.406558
\(606\) −14.0000 −0.568711
\(607\) −10.0000 −0.405887 −0.202944 0.979190i \(-0.565051\pi\)
−0.202944 + 0.979190i \(0.565051\pi\)
\(608\) 6.00000 0.243332
\(609\) 2.00000 0.0810441
\(610\) 10.0000 0.404888
\(611\) 0 0
\(612\) 2.00000 0.0808452
\(613\) −37.0000 −1.49442 −0.747208 0.664590i \(-0.768606\pi\)
−0.747208 + 0.664590i \(0.768606\pi\)
\(614\) 0 0
\(615\) 10.0000 0.403239
\(616\) 2.00000 0.0805823
\(617\) 3.00000 0.120775 0.0603877 0.998175i \(-0.480766\pi\)
0.0603877 + 0.998175i \(0.480766\pi\)
\(618\) 6.00000 0.241355
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 3.00000 0.120483
\(621\) 3.00000 0.120386
\(622\) 0 0
\(623\) 20.0000 0.801283
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −12.0000 −0.479616
\(627\) −6.00000 −0.239617
\(628\) 25.0000 0.997609
\(629\) −10.0000 −0.398726
\(630\) −2.00000 −0.0796819
\(631\) 48.0000 1.91085 0.955425 0.295234i \(-0.0953977\pi\)
0.955425 + 0.295234i \(0.0953977\pi\)
\(632\) 5.00000 0.198889
\(633\) 8.00000 0.317971
\(634\) 12.0000 0.476581
\(635\) 14.0000 0.555573
\(636\) −14.0000 −0.555136
\(637\) 0 0
\(638\) −1.00000 −0.0395904
\(639\) 4.00000 0.158238
\(640\) −1.00000 −0.0395285
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 6.00000 0.236801
\(643\) −40.0000 −1.57745 −0.788723 0.614749i \(-0.789257\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) −6.00000 −0.236433
\(645\) 5.00000 0.196875
\(646\) 12.0000 0.472134
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 1.00000 0.0392837
\(649\) −5.00000 −0.196267
\(650\) 0 0
\(651\) 6.00000 0.235159
\(652\) 17.0000 0.665771
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 6.00000 0.234619
\(655\) −13.0000 −0.507952
\(656\) 10.0000 0.390434
\(657\) −2.00000 −0.0780274
\(658\) 6.00000 0.233904
\(659\) −13.0000 −0.506408 −0.253204 0.967413i \(-0.581484\pi\)
−0.253204 + 0.967413i \(0.581484\pi\)
\(660\) 1.00000 0.0389249
\(661\) −12.0000 −0.466746 −0.233373 0.972387i \(-0.574976\pi\)
−0.233373 + 0.972387i \(0.574976\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) −6.00000 −0.232845
\(665\) −12.0000 −0.465340
\(666\) −5.00000 −0.193746
\(667\) 3.00000 0.116160
\(668\) −7.00000 −0.270838
\(669\) −10.0000 −0.386622
\(670\) 0 0
\(671\) −10.0000 −0.386046
\(672\) −2.00000 −0.0771517
\(673\) −16.0000 −0.616755 −0.308377 0.951264i \(-0.599786\pi\)
−0.308377 + 0.951264i \(0.599786\pi\)
\(674\) −22.0000 −0.847408
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) −17.0000 −0.652881
\(679\) −20.0000 −0.767530
\(680\) −2.00000 −0.0766965
\(681\) −20.0000 −0.766402
\(682\) −3.00000 −0.114876
\(683\) −44.0000 −1.68361 −0.841807 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) 6.00000 0.229416
\(685\) 9.00000 0.343872
\(686\) −20.0000 −0.763604
\(687\) −22.0000 −0.839352
\(688\) 5.00000 0.190623
\(689\) 0 0
\(690\) −3.00000 −0.114208
\(691\) −14.0000 −0.532585 −0.266293 0.963892i \(-0.585799\pi\)
−0.266293 + 0.963892i \(0.585799\pi\)
\(692\) 4.00000 0.152057
\(693\) 2.00000 0.0759737
\(694\) −18.0000 −0.683271
\(695\) −10.0000 −0.379322
\(696\) 1.00000 0.0379049
\(697\) 20.0000 0.757554
\(698\) 8.00000 0.302804
\(699\) −3.00000 −0.113470
\(700\) 2.00000 0.0755929
\(701\) 23.0000 0.868698 0.434349 0.900745i \(-0.356978\pi\)
0.434349 + 0.900745i \(0.356978\pi\)
\(702\) 0 0
\(703\) −30.0000 −1.13147
\(704\) 1.00000 0.0376889
\(705\) 3.00000 0.112987
\(706\) −14.0000 −0.526897
\(707\) 28.0000 1.05305
\(708\) 5.00000 0.187912
\(709\) −16.0000 −0.600893 −0.300446 0.953799i \(-0.597136\pi\)
−0.300446 + 0.953799i \(0.597136\pi\)
\(710\) −4.00000 −0.150117
\(711\) 5.00000 0.187515
\(712\) 10.0000 0.374766
\(713\) 9.00000 0.337053
\(714\) −4.00000 −0.149696
\(715\) 0 0
\(716\) −7.00000 −0.261602
\(717\) 8.00000 0.298765
\(718\) 12.0000 0.447836
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −12.0000 −0.446903
\(722\) 17.0000 0.632674
\(723\) 7.00000 0.260333
\(724\) 16.0000 0.594635
\(725\) −1.00000 −0.0371391
\(726\) 10.0000 0.371135
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 2.00000 0.0740233
\(731\) 10.0000 0.369863
\(732\) 10.0000 0.369611
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 36.0000 1.32878
\(735\) −3.00000 −0.110657
\(736\) −3.00000 −0.110581
\(737\) 0 0
\(738\) 10.0000 0.368105
\(739\) 24.0000 0.882854 0.441427 0.897297i \(-0.354472\pi\)
0.441427 + 0.897297i \(0.354472\pi\)
\(740\) 5.00000 0.183804
\(741\) 0 0
\(742\) 28.0000 1.02791
\(743\) −15.0000 −0.550297 −0.275148 0.961402i \(-0.588727\pi\)
−0.275148 + 0.961402i \(0.588727\pi\)
\(744\) 3.00000 0.109985
\(745\) −11.0000 −0.403009
\(746\) 37.0000 1.35467
\(747\) −6.00000 −0.219529
\(748\) 2.00000 0.0731272
\(749\) −12.0000 −0.438470
\(750\) 1.00000 0.0365148
\(751\) 41.0000 1.49611 0.748056 0.663636i \(-0.230988\pi\)
0.748056 + 0.663636i \(0.230988\pi\)
\(752\) 3.00000 0.109399
\(753\) −3.00000 −0.109326
\(754\) 0 0
\(755\) −24.0000 −0.873449
\(756\) −2.00000 −0.0727393
\(757\) −54.0000 −1.96266 −0.981332 0.192323i \(-0.938398\pi\)
−0.981332 + 0.192323i \(0.938398\pi\)
\(758\) −30.0000 −1.08965
\(759\) 3.00000 0.108893
\(760\) −6.00000 −0.217643
\(761\) −20.0000 −0.724999 −0.362500 0.931984i \(-0.618077\pi\)
−0.362500 + 0.931984i \(0.618077\pi\)
\(762\) 14.0000 0.507166
\(763\) −12.0000 −0.434429
\(764\) 24.0000 0.868290
\(765\) −2.00000 −0.0723102
\(766\) −27.0000 −0.975550
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −29.0000 −1.04577 −0.522883 0.852404i \(-0.675144\pi\)
−0.522883 + 0.852404i \(0.675144\pi\)
\(770\) −2.00000 −0.0720750
\(771\) −21.0000 −0.756297
\(772\) −4.00000 −0.143963
\(773\) −16.0000 −0.575480 −0.287740 0.957709i \(-0.592904\pi\)
−0.287740 + 0.957709i \(0.592904\pi\)
\(774\) 5.00000 0.179721
\(775\) −3.00000 −0.107763
\(776\) −10.0000 −0.358979
\(777\) 10.0000 0.358748
\(778\) −1.00000 −0.0358517
\(779\) 60.0000 2.14972
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) −6.00000 −0.214560
\(783\) 1.00000 0.0357371
\(784\) −3.00000 −0.107143
\(785\) −25.0000 −0.892288
\(786\) −13.0000 −0.463695
\(787\) 25.0000 0.891154 0.445577 0.895244i \(-0.352999\pi\)
0.445577 + 0.895244i \(0.352999\pi\)
\(788\) −12.0000 −0.427482
\(789\) −23.0000 −0.818822
\(790\) −5.00000 −0.177892
\(791\) 34.0000 1.20890
\(792\) 1.00000 0.0355335
\(793\) 0 0
\(794\) −13.0000 −0.461353
\(795\) 14.0000 0.496529
\(796\) 0 0
\(797\) −52.0000 −1.84193 −0.920967 0.389640i \(-0.872599\pi\)
−0.920967 + 0.389640i \(0.872599\pi\)
\(798\) −12.0000 −0.424795
\(799\) 6.00000 0.212265
\(800\) 1.00000 0.0353553
\(801\) 10.0000 0.353333
\(802\) −12.0000 −0.423735
\(803\) −2.00000 −0.0705785
\(804\) 0 0
\(805\) 6.00000 0.211472
\(806\) 0 0
\(807\) −18.0000 −0.633630
\(808\) 14.0000 0.492518
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 52.0000 1.82597 0.912983 0.407997i \(-0.133772\pi\)
0.912983 + 0.407997i \(0.133772\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 29.0000 1.01707
\(814\) −5.00000 −0.175250
\(815\) −17.0000 −0.595484
\(816\) −2.00000 −0.0700140
\(817\) 30.0000 1.04957
\(818\) −26.0000 −0.909069
\(819\) 0 0
\(820\) −10.0000 −0.349215
\(821\) 23.0000 0.802706 0.401353 0.915924i \(-0.368540\pi\)
0.401353 + 0.915924i \(0.368540\pi\)
\(822\) 9.00000 0.313911
\(823\) −22.0000 −0.766872 −0.383436 0.923567i \(-0.625259\pi\)
−0.383436 + 0.923567i \(0.625259\pi\)
\(824\) −6.00000 −0.209020
\(825\) −1.00000 −0.0348155
\(826\) −10.0000 −0.347945
\(827\) −18.0000 −0.625921 −0.312961 0.949766i \(-0.601321\pi\)
−0.312961 + 0.949766i \(0.601321\pi\)
\(828\) −3.00000 −0.104257
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) 6.00000 0.208263
\(831\) 19.0000 0.659103
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) −10.0000 −0.346272
\(835\) 7.00000 0.242245
\(836\) 6.00000 0.207514
\(837\) 3.00000 0.103695
\(838\) 4.00000 0.138178
\(839\) 38.0000 1.31191 0.655953 0.754802i \(-0.272267\pi\)
0.655953 + 0.754802i \(0.272267\pi\)
\(840\) 2.00000 0.0690066
\(841\) −28.0000 −0.965517
\(842\) 28.0000 0.964944
\(843\) 30.0000 1.03325
\(844\) −8.00000 −0.275371
\(845\) 0 0
\(846\) 3.00000 0.103142
\(847\) −20.0000 −0.687208
\(848\) 14.0000 0.480762
\(849\) 13.0000 0.446159
\(850\) 2.00000 0.0685994
\(851\) 15.0000 0.514193
\(852\) −4.00000 −0.137038
\(853\) 7.00000 0.239675 0.119838 0.992793i \(-0.461763\pi\)
0.119838 + 0.992793i \(0.461763\pi\)
\(854\) −20.0000 −0.684386
\(855\) −6.00000 −0.205196
\(856\) −6.00000 −0.205076
\(857\) −35.0000 −1.19558 −0.597789 0.801654i \(-0.703954\pi\)
−0.597789 + 0.801654i \(0.703954\pi\)
\(858\) 0 0
\(859\) 18.0000 0.614152 0.307076 0.951685i \(-0.400649\pi\)
0.307076 + 0.951685i \(0.400649\pi\)
\(860\) −5.00000 −0.170499
\(861\) −20.0000 −0.681598
\(862\) 12.0000 0.408722
\(863\) −51.0000 −1.73606 −0.868030 0.496512i \(-0.834614\pi\)
−0.868030 + 0.496512i \(0.834614\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −4.00000 −0.136004
\(866\) 16.0000 0.543702
\(867\) 13.0000 0.441503
\(868\) −6.00000 −0.203653
\(869\) 5.00000 0.169613
\(870\) −1.00000 −0.0339032
\(871\) 0 0
\(872\) −6.00000 −0.203186
\(873\) −10.0000 −0.338449
\(874\) −18.0000 −0.608859
\(875\) −2.00000 −0.0676123
\(876\) 2.00000 0.0675737
\(877\) 13.0000 0.438979 0.219489 0.975615i \(-0.429561\pi\)
0.219489 + 0.975615i \(0.429561\pi\)
\(878\) 28.0000 0.944954
\(879\) 14.0000 0.472208
\(880\) −1.00000 −0.0337100
\(881\) 54.0000 1.81931 0.909653 0.415369i \(-0.136347\pi\)
0.909653 + 0.415369i \(0.136347\pi\)
\(882\) −3.00000 −0.101015
\(883\) −1.00000 −0.0336527 −0.0168263 0.999858i \(-0.505356\pi\)
−0.0168263 + 0.999858i \(0.505356\pi\)
\(884\) 0 0
\(885\) −5.00000 −0.168073
\(886\) 16.0000 0.537531
\(887\) 1.00000 0.0335767 0.0167884 0.999859i \(-0.494656\pi\)
0.0167884 + 0.999859i \(0.494656\pi\)
\(888\) 5.00000 0.167789
\(889\) −28.0000 −0.939090
\(890\) −10.0000 −0.335201
\(891\) 1.00000 0.0335013
\(892\) 10.0000 0.334825
\(893\) 18.0000 0.602347
\(894\) −11.0000 −0.367895
\(895\) 7.00000 0.233984
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) −36.0000 −1.20134
\(899\) 3.00000 0.100056
\(900\) 1.00000 0.0333333
\(901\) 28.0000 0.932815
\(902\) 10.0000 0.332964
\(903\) −10.0000 −0.332779
\(904\) 17.0000 0.565412
\(905\) −16.0000 −0.531858
\(906\) −24.0000 −0.797347
\(907\) −53.0000 −1.75984 −0.879918 0.475125i \(-0.842403\pi\)
−0.879918 + 0.475125i \(0.842403\pi\)
\(908\) 20.0000 0.663723
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) 34.0000 1.12647 0.563235 0.826297i \(-0.309557\pi\)
0.563235 + 0.826297i \(0.309557\pi\)
\(912\) −6.00000 −0.198680
\(913\) −6.00000 −0.198571
\(914\) 14.0000 0.463079
\(915\) −10.0000 −0.330590
\(916\) 22.0000 0.726900
\(917\) 26.0000 0.858596
\(918\) −2.00000 −0.0660098
\(919\) −56.0000 −1.84727 −0.923635 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) 3.00000 0.0989071
\(921\) 0 0
\(922\) 27.0000 0.889198
\(923\) 0 0
\(924\) −2.00000 −0.0657952
\(925\) −5.00000 −0.164399
\(926\) 10.0000 0.328620
\(927\) −6.00000 −0.197066
\(928\) −1.00000 −0.0328266
\(929\) 24.0000 0.787414 0.393707 0.919236i \(-0.371192\pi\)
0.393707 + 0.919236i \(0.371192\pi\)
\(930\) −3.00000 −0.0983739
\(931\) −18.0000 −0.589926
\(932\) 3.00000 0.0982683
\(933\) 0 0
\(934\) −2.00000 −0.0654420
\(935\) −2.00000 −0.0654070
\(936\) 0 0
\(937\) −18.0000 −0.588034 −0.294017 0.955800i \(-0.594992\pi\)
−0.294017 + 0.955800i \(0.594992\pi\)
\(938\) 0 0
\(939\) 12.0000 0.391605
\(940\) −3.00000 −0.0978492
\(941\) 26.0000 0.847576 0.423788 0.905761i \(-0.360700\pi\)
0.423788 + 0.905761i \(0.360700\pi\)
\(942\) −25.0000 −0.814544
\(943\) −30.0000 −0.976934
\(944\) −5.00000 −0.162736
\(945\) 2.00000 0.0650600
\(946\) 5.00000 0.162564
\(947\) 52.0000 1.68977 0.844886 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) −5.00000 −0.162392
\(949\) 0 0
\(950\) 6.00000 0.194666
\(951\) −12.0000 −0.389127
\(952\) 4.00000 0.129641
\(953\) 15.0000 0.485898 0.242949 0.970039i \(-0.421885\pi\)
0.242949 + 0.970039i \(0.421885\pi\)
\(954\) 14.0000 0.453267
\(955\) −24.0000 −0.776622
\(956\) −8.00000 −0.258738
\(957\) 1.00000 0.0323254
\(958\) −4.00000 −0.129234
\(959\) −18.0000 −0.581250
\(960\) 1.00000 0.0322749
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) −7.00000 −0.225455
\(965\) 4.00000 0.128765
\(966\) 6.00000 0.193047
\(967\) 16.0000 0.514525 0.257263 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) −10.0000 −0.321412
\(969\) −12.0000 −0.385496
\(970\) 10.0000 0.321081
\(971\) −40.0000 −1.28366 −0.641831 0.766846i \(-0.721825\pi\)
−0.641831 + 0.766846i \(0.721825\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 20.0000 0.641171
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −39.0000 −1.24772 −0.623860 0.781536i \(-0.714437\pi\)
−0.623860 + 0.781536i \(0.714437\pi\)
\(978\) −17.0000 −0.543600
\(979\) 10.0000 0.319601
\(980\) 3.00000 0.0958315
\(981\) −6.00000 −0.191565
\(982\) 24.0000 0.765871
\(983\) −25.0000 −0.797376 −0.398688 0.917087i \(-0.630534\pi\)
−0.398688 + 0.917087i \(0.630534\pi\)
\(984\) −10.0000 −0.318788
\(985\) 12.0000 0.382352
\(986\) −2.00000 −0.0636930
\(987\) −6.00000 −0.190982
\(988\) 0 0
\(989\) −15.0000 −0.476972
\(990\) −1.00000 −0.0317821
\(991\) 39.0000 1.23888 0.619438 0.785046i \(-0.287361\pi\)
0.619438 + 0.785046i \(0.287361\pi\)
\(992\) −3.00000 −0.0952501
\(993\) −4.00000 −0.126936
\(994\) 8.00000 0.253745
\(995\) 0 0
\(996\) 6.00000 0.190117
\(997\) −2.00000 −0.0633406 −0.0316703 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) −40.0000 −1.26618
\(999\) 5.00000 0.158193
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 5070.2.a.o.1.1 1
13.3 even 3 390.2.i.a.61.1 2
13.5 odd 4 5070.2.b.g.1351.1 2
13.8 odd 4 5070.2.b.g.1351.2 2
13.9 even 3 390.2.i.a.211.1 yes 2
13.12 even 2 5070.2.a.f.1.1 1
39.29 odd 6 1170.2.i.k.451.1 2
39.35 odd 6 1170.2.i.k.991.1 2
65.3 odd 12 1950.2.z.h.1699.2 4
65.9 even 6 1950.2.i.s.601.1 2
65.22 odd 12 1950.2.z.h.1849.2 4
65.29 even 6 1950.2.i.s.451.1 2
65.42 odd 12 1950.2.z.h.1699.1 4
65.48 odd 12 1950.2.z.h.1849.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.a.61.1 2 13.3 even 3
390.2.i.a.211.1 yes 2 13.9 even 3
1170.2.i.k.451.1 2 39.29 odd 6
1170.2.i.k.991.1 2 39.35 odd 6
1950.2.i.s.451.1 2 65.29 even 6
1950.2.i.s.601.1 2 65.9 even 6
1950.2.z.h.1699.1 4 65.42 odd 12
1950.2.z.h.1699.2 4 65.3 odd 12
1950.2.z.h.1849.1 4 65.48 odd 12
1950.2.z.h.1849.2 4 65.22 odd 12
5070.2.a.f.1.1 1 13.12 even 2
5070.2.a.o.1.1 1 1.1 even 1 trivial
5070.2.b.g.1351.1 2 13.5 odd 4
5070.2.b.g.1351.2 2 13.8 odd 4