Properties

Label 5070.2.a.bb.1.2
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +0.561553 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +0.561553 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.12311 q^{11} +1.00000 q^{12} -0.561553 q^{14} -1.00000 q^{15} +1.00000 q^{16} -3.12311 q^{17} -1.00000 q^{18} -0.561553 q^{19} -1.00000 q^{20} +0.561553 q^{21} +4.12311 q^{22} +4.68466 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +0.561553 q^{28} +2.43845 q^{29} +1.00000 q^{30} +6.68466 q^{31} -1.00000 q^{32} -4.12311 q^{33} +3.12311 q^{34} -0.561553 q^{35} +1.00000 q^{36} +4.12311 q^{37} +0.561553 q^{38} +1.00000 q^{40} -12.2462 q^{41} -0.561553 q^{42} +0.438447 q^{43} -4.12311 q^{44} -1.00000 q^{45} -4.68466 q^{46} -7.00000 q^{47} +1.00000 q^{48} -6.68466 q^{49} -1.00000 q^{50} -3.12311 q^{51} +8.56155 q^{53} -1.00000 q^{54} +4.12311 q^{55} -0.561553 q^{56} -0.561553 q^{57} -2.43845 q^{58} -6.43845 q^{59} -1.00000 q^{60} +6.00000 q^{61} -6.68466 q^{62} +0.561553 q^{63} +1.00000 q^{64} +4.12311 q^{66} +2.24621 q^{67} -3.12311 q^{68} +4.68466 q^{69} +0.561553 q^{70} +13.1231 q^{71} -1.00000 q^{72} -9.36932 q^{73} -4.12311 q^{74} +1.00000 q^{75} -0.561553 q^{76} -2.31534 q^{77} +11.5616 q^{79} -1.00000 q^{80} +1.00000 q^{81} +12.2462 q^{82} +7.12311 q^{83} +0.561553 q^{84} +3.12311 q^{85} -0.438447 q^{86} +2.43845 q^{87} +4.12311 q^{88} +18.8078 q^{89} +1.00000 q^{90} +4.68466 q^{92} +6.68466 q^{93} +7.00000 q^{94} +0.561553 q^{95} -1.00000 q^{96} +7.12311 q^{97} +6.68466 q^{98} -4.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 3q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 3q^{7} - 2q^{8} + 2q^{9} + 2q^{10} + 2q^{12} + 3q^{14} - 2q^{15} + 2q^{16} + 2q^{17} - 2q^{18} + 3q^{19} - 2q^{20} - 3q^{21} - 3q^{23} - 2q^{24} + 2q^{25} + 2q^{27} - 3q^{28} + 9q^{29} + 2q^{30} + q^{31} - 2q^{32} - 2q^{34} + 3q^{35} + 2q^{36} - 3q^{38} + 2q^{40} - 8q^{41} + 3q^{42} + 5q^{43} - 2q^{45} + 3q^{46} - 14q^{47} + 2q^{48} - q^{49} - 2q^{50} + 2q^{51} + 13q^{53} - 2q^{54} + 3q^{56} + 3q^{57} - 9q^{58} - 17q^{59} - 2q^{60} + 12q^{61} - q^{62} - 3q^{63} + 2q^{64} - 12q^{67} + 2q^{68} - 3q^{69} - 3q^{70} + 18q^{71} - 2q^{72} + 6q^{73} + 2q^{75} + 3q^{76} - 17q^{77} + 19q^{79} - 2q^{80} + 2q^{81} + 8q^{82} + 6q^{83} - 3q^{84} - 2q^{85} - 5q^{86} + 9q^{87} + 17q^{89} + 2q^{90} - 3q^{92} + q^{93} + 14q^{94} - 3q^{95} - 2q^{96} + 6q^{97} + q^{98} + O(q^{100}) \)

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
\(7\) 0.561553 0.212247 0.106124 0.994353i \(-0.466156\pi\)
0.106124 + 0.994353i \(0.466156\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −4.12311 −1.24316 −0.621582 0.783349i \(-0.713510\pi\)
−0.621582 + 0.783349i \(0.713510\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −0.561553 −0.150081
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −3.12311 −0.757464 −0.378732 0.925506i \(-0.623640\pi\)
−0.378732 + 0.925506i \(0.623640\pi\)
\(18\) −1.00000 −0.235702
\(19\) −0.561553 −0.128829 −0.0644145 0.997923i \(-0.520518\pi\)
−0.0644145 + 0.997923i \(0.520518\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0.561553 0.122541
\(22\) 4.12311 0.879049
\(23\) 4.68466 0.976819 0.488409 0.872615i \(-0.337577\pi\)
0.488409 + 0.872615i \(0.337577\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0.561553 0.106124
\(29\) 2.43845 0.452808 0.226404 0.974033i \(-0.427303\pi\)
0.226404 + 0.974033i \(0.427303\pi\)
\(30\) 1.00000 0.182574
\(31\) 6.68466 1.20060 0.600300 0.799775i \(-0.295048\pi\)
0.600300 + 0.799775i \(0.295048\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.12311 −0.717741
\(34\) 3.12311 0.535608
\(35\) −0.561553 −0.0949197
\(36\) 1.00000 0.166667
\(37\) 4.12311 0.677834 0.338917 0.940816i \(-0.389939\pi\)
0.338917 + 0.940816i \(0.389939\pi\)
\(38\) 0.561553 0.0910959
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −12.2462 −1.91254 −0.956268 0.292490i \(-0.905516\pi\)
−0.956268 + 0.292490i \(0.905516\pi\)
\(42\) −0.561553 −0.0866495
\(43\) 0.438447 0.0668626 0.0334313 0.999441i \(-0.489357\pi\)
0.0334313 + 0.999441i \(0.489357\pi\)
\(44\) −4.12311 −0.621582
\(45\) −1.00000 −0.149071
\(46\) −4.68466 −0.690715
\(47\) −7.00000 −1.02105 −0.510527 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.68466 −0.954951
\(50\) −1.00000 −0.141421
\(51\) −3.12311 −0.437322
\(52\) 0 0
\(53\) 8.56155 1.17602 0.588010 0.808854i \(-0.299912\pi\)
0.588010 + 0.808854i \(0.299912\pi\)
\(54\) −1.00000 −0.136083
\(55\) 4.12311 0.555959
\(56\) −0.561553 −0.0750407
\(57\) −0.561553 −0.0743795
\(58\) −2.43845 −0.320184
\(59\) −6.43845 −0.838214 −0.419107 0.907937i \(-0.637657\pi\)
−0.419107 + 0.907937i \(0.637657\pi\)
\(60\) −1.00000 −0.129099
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −6.68466 −0.848952
\(63\) 0.561553 0.0707490
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.12311 0.507519
\(67\) 2.24621 0.274418 0.137209 0.990542i \(-0.456187\pi\)
0.137209 + 0.990542i \(0.456187\pi\)
\(68\) −3.12311 −0.378732
\(69\) 4.68466 0.563967
\(70\) 0.561553 0.0671184
\(71\) 13.1231 1.55743 0.778713 0.627380i \(-0.215873\pi\)
0.778713 + 0.627380i \(0.215873\pi\)
\(72\) −1.00000 −0.117851
\(73\) −9.36932 −1.09660 −0.548298 0.836283i \(-0.684724\pi\)
−0.548298 + 0.836283i \(0.684724\pi\)
\(74\) −4.12311 −0.479301
\(75\) 1.00000 0.115470
\(76\) −0.561553 −0.0644145
\(77\) −2.31534 −0.263858
\(78\) 0 0
\(79\) 11.5616 1.30078 0.650388 0.759602i \(-0.274606\pi\)
0.650388 + 0.759602i \(0.274606\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 12.2462 1.35237
\(83\) 7.12311 0.781862 0.390931 0.920420i \(-0.372153\pi\)
0.390931 + 0.920420i \(0.372153\pi\)
\(84\) 0.561553 0.0612704
\(85\) 3.12311 0.338748
\(86\) −0.438447 −0.0472790
\(87\) 2.43845 0.261429
\(88\) 4.12311 0.439525
\(89\) 18.8078 1.99362 0.996810 0.0798174i \(-0.0254337\pi\)
0.996810 + 0.0798174i \(0.0254337\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 4.68466 0.488409
\(93\) 6.68466 0.693167
\(94\) 7.00000 0.721995
\(95\) 0.561553 0.0576141
\(96\) −1.00000 −0.102062
\(97\) 7.12311 0.723242 0.361621 0.932325i \(-0.382223\pi\)
0.361621 + 0.932325i \(0.382223\pi\)
\(98\) 6.68466 0.675252
\(99\) −4.12311 −0.414388
\(100\) 1.00000 0.100000
\(101\) −8.87689 −0.883284 −0.441642 0.897191i \(-0.645604\pi\)
−0.441642 + 0.897191i \(0.645604\pi\)
\(102\) 3.12311 0.309234
\(103\) 4.56155 0.449463 0.224732 0.974421i \(-0.427849\pi\)
0.224732 + 0.974421i \(0.427849\pi\)
\(104\) 0 0
\(105\) −0.561553 −0.0548019
\(106\) −8.56155 −0.831572
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 1.00000 0.0962250
\(109\) 3.75379 0.359548 0.179774 0.983708i \(-0.442463\pi\)
0.179774 + 0.983708i \(0.442463\pi\)
\(110\) −4.12311 −0.393123
\(111\) 4.12311 0.391348
\(112\) 0.561553 0.0530618
\(113\) 8.68466 0.816984 0.408492 0.912762i \(-0.366055\pi\)
0.408492 + 0.912762i \(0.366055\pi\)
\(114\) 0.561553 0.0525942
\(115\) −4.68466 −0.436847
\(116\) 2.43845 0.226404
\(117\) 0 0
\(118\) 6.43845 0.592707
\(119\) −1.75379 −0.160770
\(120\) 1.00000 0.0912871
\(121\) 6.00000 0.545455
\(122\) −6.00000 −0.543214
\(123\) −12.2462 −1.10420
\(124\) 6.68466 0.600300
\(125\) −1.00000 −0.0894427
\(126\) −0.561553 −0.0500271
\(127\) −8.56155 −0.759715 −0.379857 0.925045i \(-0.624027\pi\)
−0.379857 + 0.925045i \(0.624027\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0.438447 0.0386031
\(130\) 0 0
\(131\) −2.12311 −0.185497 −0.0927483 0.995690i \(-0.529565\pi\)
−0.0927483 + 0.995690i \(0.529565\pi\)
\(132\) −4.12311 −0.358870
\(133\) −0.315342 −0.0273436
\(134\) −2.24621 −0.194043
\(135\) −1.00000 −0.0860663
\(136\) 3.12311 0.267804
\(137\) −11.8078 −1.00881 −0.504403 0.863469i \(-0.668287\pi\)
−0.504403 + 0.863469i \(0.668287\pi\)
\(138\) −4.68466 −0.398785
\(139\) 12.5616 1.06546 0.532729 0.846286i \(-0.321167\pi\)
0.532729 + 0.846286i \(0.321167\pi\)
\(140\) −0.561553 −0.0474599
\(141\) −7.00000 −0.589506
\(142\) −13.1231 −1.10127
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −2.43845 −0.202502
\(146\) 9.36932 0.775410
\(147\) −6.68466 −0.551341
\(148\) 4.12311 0.338917
\(149\) 2.19224 0.179595 0.0897975 0.995960i \(-0.471378\pi\)
0.0897975 + 0.995960i \(0.471378\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 15.3693 1.25074 0.625369 0.780329i \(-0.284949\pi\)
0.625369 + 0.780329i \(0.284949\pi\)
\(152\) 0.561553 0.0455479
\(153\) −3.12311 −0.252488
\(154\) 2.31534 0.186576
\(155\) −6.68466 −0.536925
\(156\) 0 0
\(157\) 13.8769 1.10750 0.553748 0.832684i \(-0.313197\pi\)
0.553748 + 0.832684i \(0.313197\pi\)
\(158\) −11.5616 −0.919788
\(159\) 8.56155 0.678975
\(160\) 1.00000 0.0790569
\(161\) 2.63068 0.207327
\(162\) −1.00000 −0.0785674
\(163\) 14.4384 1.13091 0.565453 0.824780i \(-0.308701\pi\)
0.565453 + 0.824780i \(0.308701\pi\)
\(164\) −12.2462 −0.956268
\(165\) 4.12311 0.320983
\(166\) −7.12311 −0.552860
\(167\) −4.36932 −0.338108 −0.169054 0.985607i \(-0.554071\pi\)
−0.169054 + 0.985607i \(0.554071\pi\)
\(168\) −0.561553 −0.0433247
\(169\) 0 0
\(170\) −3.12311 −0.239531
\(171\) −0.561553 −0.0429430
\(172\) 0.438447 0.0334313
\(173\) 20.1771 1.53404 0.767018 0.641626i \(-0.221740\pi\)
0.767018 + 0.641626i \(0.221740\pi\)
\(174\) −2.43845 −0.184858
\(175\) 0.561553 0.0424494
\(176\) −4.12311 −0.310791
\(177\) −6.43845 −0.483943
\(178\) −18.8078 −1.40970
\(179\) −12.6847 −0.948096 −0.474048 0.880499i \(-0.657208\pi\)
−0.474048 + 0.880499i \(0.657208\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 2.87689 0.213838 0.106919 0.994268i \(-0.465901\pi\)
0.106919 + 0.994268i \(0.465901\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) −4.68466 −0.345358
\(185\) −4.12311 −0.303137
\(186\) −6.68466 −0.490143
\(187\) 12.8769 0.941652
\(188\) −7.00000 −0.510527
\(189\) 0.561553 0.0408470
\(190\) −0.561553 −0.0407393
\(191\) 10.8769 0.787024 0.393512 0.919319i \(-0.371260\pi\)
0.393512 + 0.919319i \(0.371260\pi\)
\(192\) 1.00000 0.0721688
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) −7.12311 −0.511409
\(195\) 0 0
\(196\) −6.68466 −0.477476
\(197\) −12.8078 −0.912515 −0.456258 0.889848i \(-0.650810\pi\)
−0.456258 + 0.889848i \(0.650810\pi\)
\(198\) 4.12311 0.293016
\(199\) 2.87689 0.203938 0.101969 0.994788i \(-0.467486\pi\)
0.101969 + 0.994788i \(0.467486\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 2.24621 0.158436
\(202\) 8.87689 0.624576
\(203\) 1.36932 0.0961072
\(204\) −3.12311 −0.218661
\(205\) 12.2462 0.855312
\(206\) −4.56155 −0.317818
\(207\) 4.68466 0.325606
\(208\) 0 0
\(209\) 2.31534 0.160156
\(210\) 0.561553 0.0387508
\(211\) −21.9309 −1.50978 −0.754892 0.655850i \(-0.772311\pi\)
−0.754892 + 0.655850i \(0.772311\pi\)
\(212\) 8.56155 0.588010
\(213\) 13.1231 0.899180
\(214\) −2.00000 −0.136717
\(215\) −0.438447 −0.0299018
\(216\) −1.00000 −0.0680414
\(217\) 3.75379 0.254824
\(218\) −3.75379 −0.254239
\(219\) −9.36932 −0.633120
\(220\) 4.12311 0.277980
\(221\) 0 0
\(222\) −4.12311 −0.276725
\(223\) −27.3002 −1.82816 −0.914078 0.405539i \(-0.867084\pi\)
−0.914078 + 0.405539i \(0.867084\pi\)
\(224\) −0.561553 −0.0375203
\(225\) 1.00000 0.0666667
\(226\) −8.68466 −0.577695
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) −0.561553 −0.0371897
\(229\) 24.2462 1.60223 0.801117 0.598507i \(-0.204239\pi\)
0.801117 + 0.598507i \(0.204239\pi\)
\(230\) 4.68466 0.308897
\(231\) −2.31534 −0.152338
\(232\) −2.43845 −0.160092
\(233\) −5.31534 −0.348220 −0.174110 0.984726i \(-0.555705\pi\)
−0.174110 + 0.984726i \(0.555705\pi\)
\(234\) 0 0
\(235\) 7.00000 0.456630
\(236\) −6.43845 −0.419107
\(237\) 11.5616 0.751004
\(238\) 1.75379 0.113681
\(239\) 11.3693 0.735420 0.367710 0.929941i \(-0.380142\pi\)
0.367710 + 0.929941i \(0.380142\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 28.1231 1.81157 0.905784 0.423739i \(-0.139283\pi\)
0.905784 + 0.423739i \(0.139283\pi\)
\(242\) −6.00000 −0.385695
\(243\) 1.00000 0.0641500
\(244\) 6.00000 0.384111
\(245\) 6.68466 0.427067
\(246\) 12.2462 0.780790
\(247\) 0 0
\(248\) −6.68466 −0.424476
\(249\) 7.12311 0.451408
\(250\) 1.00000 0.0632456
\(251\) −8.12311 −0.512726 −0.256363 0.966581i \(-0.582524\pi\)
−0.256363 + 0.966581i \(0.582524\pi\)
\(252\) 0.561553 0.0353745
\(253\) −19.3153 −1.21435
\(254\) 8.56155 0.537200
\(255\) 3.12311 0.195576
\(256\) 1.00000 0.0625000
\(257\) −5.56155 −0.346920 −0.173460 0.984841i \(-0.555495\pi\)
−0.173460 + 0.984841i \(0.555495\pi\)
\(258\) −0.438447 −0.0272965
\(259\) 2.31534 0.143868
\(260\) 0 0
\(261\) 2.43845 0.150936
\(262\) 2.12311 0.131166
\(263\) −13.0000 −0.801614 −0.400807 0.916162i \(-0.631270\pi\)
−0.400807 + 0.916162i \(0.631270\pi\)
\(264\) 4.12311 0.253760
\(265\) −8.56155 −0.525932
\(266\) 0.315342 0.0193348
\(267\) 18.8078 1.15102
\(268\) 2.24621 0.137209
\(269\) 21.3693 1.30291 0.651455 0.758687i \(-0.274159\pi\)
0.651455 + 0.758687i \(0.274159\pi\)
\(270\) 1.00000 0.0608581
\(271\) 13.1771 0.800451 0.400225 0.916417i \(-0.368932\pi\)
0.400225 + 0.916417i \(0.368932\pi\)
\(272\) −3.12311 −0.189366
\(273\) 0 0
\(274\) 11.8078 0.713333
\(275\) −4.12311 −0.248633
\(276\) 4.68466 0.281983
\(277\) −1.00000 −0.0600842 −0.0300421 0.999549i \(-0.509564\pi\)
−0.0300421 + 0.999549i \(0.509564\pi\)
\(278\) −12.5616 −0.753392
\(279\) 6.68466 0.400200
\(280\) 0.561553 0.0335592
\(281\) −16.2462 −0.969168 −0.484584 0.874745i \(-0.661029\pi\)
−0.484584 + 0.874745i \(0.661029\pi\)
\(282\) 7.00000 0.416844
\(283\) 15.5616 0.925038 0.462519 0.886609i \(-0.346946\pi\)
0.462519 + 0.886609i \(0.346946\pi\)
\(284\) 13.1231 0.778713
\(285\) 0.561553 0.0332635
\(286\) 0 0
\(287\) −6.87689 −0.405930
\(288\) −1.00000 −0.0589256
\(289\) −7.24621 −0.426248
\(290\) 2.43845 0.143191
\(291\) 7.12311 0.417564
\(292\) −9.36932 −0.548298
\(293\) −3.93087 −0.229644 −0.114822 0.993386i \(-0.536630\pi\)
−0.114822 + 0.993386i \(0.536630\pi\)
\(294\) 6.68466 0.389857
\(295\) 6.43845 0.374861
\(296\) −4.12311 −0.239651
\(297\) −4.12311 −0.239247
\(298\) −2.19224 −0.126993
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 0.246211 0.0141914
\(302\) −15.3693 −0.884405
\(303\) −8.87689 −0.509964
\(304\) −0.561553 −0.0322073
\(305\) −6.00000 −0.343559
\(306\) 3.12311 0.178536
\(307\) −25.6155 −1.46196 −0.730978 0.682401i \(-0.760936\pi\)
−0.730978 + 0.682401i \(0.760936\pi\)
\(308\) −2.31534 −0.131929
\(309\) 4.56155 0.259498
\(310\) 6.68466 0.379663
\(311\) 30.7386 1.74303 0.871514 0.490371i \(-0.163139\pi\)
0.871514 + 0.490371i \(0.163139\pi\)
\(312\) 0 0
\(313\) 31.3693 1.77310 0.886549 0.462634i \(-0.153096\pi\)
0.886549 + 0.462634i \(0.153096\pi\)
\(314\) −13.8769 −0.783118
\(315\) −0.561553 −0.0316399
\(316\) 11.5616 0.650388
\(317\) 24.8078 1.39334 0.696671 0.717390i \(-0.254664\pi\)
0.696671 + 0.717390i \(0.254664\pi\)
\(318\) −8.56155 −0.480108
\(319\) −10.0540 −0.562915
\(320\) −1.00000 −0.0559017
\(321\) 2.00000 0.111629
\(322\) −2.63068 −0.146602
\(323\) 1.75379 0.0975834
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −14.4384 −0.799672
\(327\) 3.75379 0.207585
\(328\) 12.2462 0.676184
\(329\) −3.93087 −0.216716
\(330\) −4.12311 −0.226969
\(331\) −30.7386 −1.68955 −0.844774 0.535123i \(-0.820265\pi\)
−0.844774 + 0.535123i \(0.820265\pi\)
\(332\) 7.12311 0.390931
\(333\) 4.12311 0.225945
\(334\) 4.36932 0.239078
\(335\) −2.24621 −0.122724
\(336\) 0.561553 0.0306352
\(337\) −6.00000 −0.326841 −0.163420 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) 0 0
\(339\) 8.68466 0.471686
\(340\) 3.12311 0.169374
\(341\) −27.5616 −1.49254
\(342\) 0.561553 0.0303653
\(343\) −7.68466 −0.414933
\(344\) −0.438447 −0.0236395
\(345\) −4.68466 −0.252214
\(346\) −20.1771 −1.08473
\(347\) 0.876894 0.0470742 0.0235371 0.999723i \(-0.492507\pi\)
0.0235371 + 0.999723i \(0.492507\pi\)
\(348\) 2.43845 0.130714
\(349\) −8.49242 −0.454589 −0.227294 0.973826i \(-0.572988\pi\)
−0.227294 + 0.973826i \(0.572988\pi\)
\(350\) −0.561553 −0.0300163
\(351\) 0 0
\(352\) 4.12311 0.219762
\(353\) 25.8617 1.37648 0.688241 0.725482i \(-0.258383\pi\)
0.688241 + 0.725482i \(0.258383\pi\)
\(354\) 6.43845 0.342200
\(355\) −13.1231 −0.696502
\(356\) 18.8078 0.996810
\(357\) −1.75379 −0.0928203
\(358\) 12.6847 0.670405
\(359\) −13.1231 −0.692611 −0.346306 0.938122i \(-0.612564\pi\)
−0.346306 + 0.938122i \(0.612564\pi\)
\(360\) 1.00000 0.0527046
\(361\) −18.6847 −0.983403
\(362\) −2.87689 −0.151206
\(363\) 6.00000 0.314918
\(364\) 0 0
\(365\) 9.36932 0.490412
\(366\) −6.00000 −0.313625
\(367\) 12.4924 0.652099 0.326050 0.945353i \(-0.394282\pi\)
0.326050 + 0.945353i \(0.394282\pi\)
\(368\) 4.68466 0.244205
\(369\) −12.2462 −0.637512
\(370\) 4.12311 0.214350
\(371\) 4.80776 0.249607
\(372\) 6.68466 0.346583
\(373\) −25.8078 −1.33628 −0.668138 0.744038i \(-0.732908\pi\)
−0.668138 + 0.744038i \(0.732908\pi\)
\(374\) −12.8769 −0.665848
\(375\) −1.00000 −0.0516398
\(376\) 7.00000 0.360997
\(377\) 0 0
\(378\) −0.561553 −0.0288832
\(379\) 17.6847 0.908400 0.454200 0.890900i \(-0.349925\pi\)
0.454200 + 0.890900i \(0.349925\pi\)
\(380\) 0.561553 0.0288071
\(381\) −8.56155 −0.438622
\(382\) −10.8769 −0.556510
\(383\) −0.192236 −0.00982280 −0.00491140 0.999988i \(-0.501563\pi\)
−0.00491140 + 0.999988i \(0.501563\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 2.31534 0.118001
\(386\) 0 0
\(387\) 0.438447 0.0222875
\(388\) 7.12311 0.361621
\(389\) −18.0540 −0.915373 −0.457686 0.889114i \(-0.651322\pi\)
−0.457686 + 0.889114i \(0.651322\pi\)
\(390\) 0 0
\(391\) −14.6307 −0.739905
\(392\) 6.68466 0.337626
\(393\) −2.12311 −0.107097
\(394\) 12.8078 0.645246
\(395\) −11.5616 −0.581725
\(396\) −4.12311 −0.207194
\(397\) 20.1231 1.00995 0.504975 0.863134i \(-0.331502\pi\)
0.504975 + 0.863134i \(0.331502\pi\)
\(398\) −2.87689 −0.144206
\(399\) −0.315342 −0.0157868
\(400\) 1.00000 0.0500000
\(401\) −0.315342 −0.0157474 −0.00787370 0.999969i \(-0.502506\pi\)
−0.00787370 + 0.999969i \(0.502506\pi\)
\(402\) −2.24621 −0.112031
\(403\) 0 0
\(404\) −8.87689 −0.441642
\(405\) −1.00000 −0.0496904
\(406\) −1.36932 −0.0679581
\(407\) −17.0000 −0.842659
\(408\) 3.12311 0.154617
\(409\) 18.8078 0.929984 0.464992 0.885315i \(-0.346057\pi\)
0.464992 + 0.885315i \(0.346057\pi\)
\(410\) −12.2462 −0.604797
\(411\) −11.8078 −0.582434
\(412\) 4.56155 0.224732
\(413\) −3.61553 −0.177909
\(414\) −4.68466 −0.230238
\(415\) −7.12311 −0.349660
\(416\) 0 0
\(417\) 12.5616 0.615142
\(418\) −2.31534 −0.113247
\(419\) 32.4924 1.58736 0.793679 0.608336i \(-0.208163\pi\)
0.793679 + 0.608336i \(0.208163\pi\)
\(420\) −0.561553 −0.0274010
\(421\) 32.4924 1.58358 0.791792 0.610791i \(-0.209148\pi\)
0.791792 + 0.610791i \(0.209148\pi\)
\(422\) 21.9309 1.06758
\(423\) −7.00000 −0.340352
\(424\) −8.56155 −0.415786
\(425\) −3.12311 −0.151493
\(426\) −13.1231 −0.635817
\(427\) 3.36932 0.163053
\(428\) 2.00000 0.0966736
\(429\) 0 0
\(430\) 0.438447 0.0211438
\(431\) 22.7386 1.09528 0.547641 0.836714i \(-0.315526\pi\)
0.547641 + 0.836714i \(0.315526\pi\)
\(432\) 1.00000 0.0481125
\(433\) −18.7386 −0.900521 −0.450261 0.892897i \(-0.648669\pi\)
−0.450261 + 0.892897i \(0.648669\pi\)
\(434\) −3.75379 −0.180188
\(435\) −2.43845 −0.116915
\(436\) 3.75379 0.179774
\(437\) −2.63068 −0.125843
\(438\) 9.36932 0.447683
\(439\) 25.1231 1.19906 0.599530 0.800352i \(-0.295354\pi\)
0.599530 + 0.800352i \(0.295354\pi\)
\(440\) −4.12311 −0.196561
\(441\) −6.68466 −0.318317
\(442\) 0 0
\(443\) −13.1231 −0.623498 −0.311749 0.950165i \(-0.600915\pi\)
−0.311749 + 0.950165i \(0.600915\pi\)
\(444\) 4.12311 0.195674
\(445\) −18.8078 −0.891574
\(446\) 27.3002 1.29270
\(447\) 2.19224 0.103689
\(448\) 0.561553 0.0265309
\(449\) 20.1771 0.952215 0.476108 0.879387i \(-0.342047\pi\)
0.476108 + 0.879387i \(0.342047\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 50.4924 2.37760
\(452\) 8.68466 0.408492
\(453\) 15.3693 0.722113
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) 0.561553 0.0262971
\(457\) −20.2462 −0.947078 −0.473539 0.880773i \(-0.657024\pi\)
−0.473539 + 0.880773i \(0.657024\pi\)
\(458\) −24.2462 −1.13295
\(459\) −3.12311 −0.145774
\(460\) −4.68466 −0.218423
\(461\) 30.0540 1.39975 0.699877 0.714264i \(-0.253238\pi\)
0.699877 + 0.714264i \(0.253238\pi\)
\(462\) 2.31534 0.107719
\(463\) 7.61553 0.353924 0.176962 0.984218i \(-0.443373\pi\)
0.176962 + 0.984218i \(0.443373\pi\)
\(464\) 2.43845 0.113202
\(465\) −6.68466 −0.309994
\(466\) 5.31534 0.246228
\(467\) −17.8617 −0.826543 −0.413271 0.910608i \(-0.635614\pi\)
−0.413271 + 0.910608i \(0.635614\pi\)
\(468\) 0 0
\(469\) 1.26137 0.0582445
\(470\) −7.00000 −0.322886
\(471\) 13.8769 0.639414
\(472\) 6.43845 0.296354
\(473\) −1.80776 −0.0831211
\(474\) −11.5616 −0.531040
\(475\) −0.561553 −0.0257658
\(476\) −1.75379 −0.0803848
\(477\) 8.56155 0.392007
\(478\) −11.3693 −0.520020
\(479\) 38.2462 1.74751 0.873757 0.486363i \(-0.161677\pi\)
0.873757 + 0.486363i \(0.161677\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) −28.1231 −1.28097
\(483\) 2.63068 0.119700
\(484\) 6.00000 0.272727
\(485\) −7.12311 −0.323444
\(486\) −1.00000 −0.0453609
\(487\) −13.0540 −0.591532 −0.295766 0.955260i \(-0.595575\pi\)
−0.295766 + 0.955260i \(0.595575\pi\)
\(488\) −6.00000 −0.271607
\(489\) 14.4384 0.652929
\(490\) −6.68466 −0.301982
\(491\) −17.4384 −0.786986 −0.393493 0.919328i \(-0.628733\pi\)
−0.393493 + 0.919328i \(0.628733\pi\)
\(492\) −12.2462 −0.552102
\(493\) −7.61553 −0.342986
\(494\) 0 0
\(495\) 4.12311 0.185320
\(496\) 6.68466 0.300150
\(497\) 7.36932 0.330559
\(498\) −7.12311 −0.319194
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −4.36932 −0.195207
\(502\) 8.12311 0.362552
\(503\) −43.0540 −1.91968 −0.959841 0.280545i \(-0.909485\pi\)
−0.959841 + 0.280545i \(0.909485\pi\)
\(504\) −0.561553 −0.0250136
\(505\) 8.87689 0.395017
\(506\) 19.3153 0.858672
\(507\) 0 0
\(508\) −8.56155 −0.379857
\(509\) −40.5464 −1.79719 −0.898594 0.438782i \(-0.855410\pi\)
−0.898594 + 0.438782i \(0.855410\pi\)
\(510\) −3.12311 −0.138293
\(511\) −5.26137 −0.232749
\(512\) −1.00000 −0.0441942
\(513\) −0.561553 −0.0247932
\(514\) 5.56155 0.245310
\(515\) −4.56155 −0.201006
\(516\) 0.438447 0.0193016
\(517\) 28.8617 1.26934
\(518\) −2.31534 −0.101730
\(519\) 20.1771 0.885676
\(520\) 0 0
\(521\) −6.31534 −0.276680 −0.138340 0.990385i \(-0.544177\pi\)
−0.138340 + 0.990385i \(0.544177\pi\)
\(522\) −2.43845 −0.106728
\(523\) −29.8078 −1.30340 −0.651701 0.758476i \(-0.725944\pi\)
−0.651701 + 0.758476i \(0.725944\pi\)
\(524\) −2.12311 −0.0927483
\(525\) 0.561553 0.0245082
\(526\) 13.0000 0.566827
\(527\) −20.8769 −0.909412
\(528\) −4.12311 −0.179435
\(529\) −1.05398 −0.0458250
\(530\) 8.56155 0.371890
\(531\) −6.43845 −0.279405
\(532\) −0.315342 −0.0136718
\(533\) 0 0
\(534\) −18.8078 −0.813892
\(535\) −2.00000 −0.0864675
\(536\) −2.24621 −0.0970215
\(537\) −12.6847 −0.547383
\(538\) −21.3693 −0.921297
\(539\) 27.5616 1.18716
\(540\) −1.00000 −0.0430331
\(541\) 27.3693 1.17670 0.588349 0.808607i \(-0.299778\pi\)
0.588349 + 0.808607i \(0.299778\pi\)
\(542\) −13.1771 −0.566004
\(543\) 2.87689 0.123459
\(544\) 3.12311 0.133902
\(545\) −3.75379 −0.160795
\(546\) 0 0
\(547\) −5.61553 −0.240103 −0.120051 0.992768i \(-0.538306\pi\)
−0.120051 + 0.992768i \(0.538306\pi\)
\(548\) −11.8078 −0.504403
\(549\) 6.00000 0.256074
\(550\) 4.12311 0.175810
\(551\) −1.36932 −0.0583349
\(552\) −4.68466 −0.199392
\(553\) 6.49242 0.276086
\(554\) 1.00000 0.0424859
\(555\) −4.12311 −0.175016
\(556\) 12.5616 0.532729
\(557\) 8.56155 0.362765 0.181382 0.983413i \(-0.441943\pi\)
0.181382 + 0.983413i \(0.441943\pi\)
\(558\) −6.68466 −0.282984
\(559\) 0 0
\(560\) −0.561553 −0.0237299
\(561\) 12.8769 0.543663
\(562\) 16.2462 0.685305
\(563\) −32.9848 −1.39015 −0.695073 0.718939i \(-0.744628\pi\)
−0.695073 + 0.718939i \(0.744628\pi\)
\(564\) −7.00000 −0.294753
\(565\) −8.68466 −0.365366
\(566\) −15.5616 −0.654101
\(567\) 0.561553 0.0235830
\(568\) −13.1231 −0.550633
\(569\) 26.1771 1.09740 0.548700 0.836019i \(-0.315123\pi\)
0.548700 + 0.836019i \(0.315123\pi\)
\(570\) −0.561553 −0.0235209
\(571\) 23.6847 0.991172 0.495586 0.868559i \(-0.334953\pi\)
0.495586 + 0.868559i \(0.334953\pi\)
\(572\) 0 0
\(573\) 10.8769 0.454389
\(574\) 6.87689 0.287036
\(575\) 4.68466 0.195364
\(576\) 1.00000 0.0416667
\(577\) 40.7386 1.69597 0.847986 0.530019i \(-0.177815\pi\)
0.847986 + 0.530019i \(0.177815\pi\)
\(578\) 7.24621 0.301403
\(579\) 0 0
\(580\) −2.43845 −0.101251
\(581\) 4.00000 0.165948
\(582\) −7.12311 −0.295262
\(583\) −35.3002 −1.46198
\(584\) 9.36932 0.387705
\(585\) 0 0
\(586\) 3.93087 0.162383
\(587\) 40.2462 1.66114 0.830569 0.556915i \(-0.188015\pi\)
0.830569 + 0.556915i \(0.188015\pi\)
\(588\) −6.68466 −0.275671
\(589\) −3.75379 −0.154672
\(590\) −6.43845 −0.265067
\(591\) −12.8078 −0.526841
\(592\) 4.12311 0.169459
\(593\) −33.1771 −1.36242 −0.681210 0.732088i \(-0.738546\pi\)
−0.681210 + 0.732088i \(0.738546\pi\)
\(594\) 4.12311 0.169173
\(595\) 1.75379 0.0718983
\(596\) 2.19224 0.0897975
\(597\) 2.87689 0.117743
\(598\) 0 0
\(599\) −14.0000 −0.572024 −0.286012 0.958226i \(-0.592330\pi\)
−0.286012 + 0.958226i \(0.592330\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 29.9848 1.22311 0.611554 0.791203i \(-0.290545\pi\)
0.611554 + 0.791203i \(0.290545\pi\)
\(602\) −0.246211 −0.0100348
\(603\) 2.24621 0.0914728
\(604\) 15.3693 0.625369
\(605\) −6.00000 −0.243935
\(606\) 8.87689 0.360599
\(607\) 32.4233 1.31602 0.658010 0.753009i \(-0.271398\pi\)
0.658010 + 0.753009i \(0.271398\pi\)
\(608\) 0.561553 0.0227740
\(609\) 1.36932 0.0554875
\(610\) 6.00000 0.242933
\(611\) 0 0
\(612\) −3.12311 −0.126244
\(613\) −19.8769 −0.802820 −0.401410 0.915898i \(-0.631480\pi\)
−0.401410 + 0.915898i \(0.631480\pi\)
\(614\) 25.6155 1.03376
\(615\) 12.2462 0.493815
\(616\) 2.31534 0.0932878
\(617\) 29.3153 1.18019 0.590096 0.807333i \(-0.299090\pi\)
0.590096 + 0.807333i \(0.299090\pi\)
\(618\) −4.56155 −0.183493
\(619\) 20.5616 0.826439 0.413219 0.910632i \(-0.364404\pi\)
0.413219 + 0.910632i \(0.364404\pi\)
\(620\) −6.68466 −0.268462
\(621\) 4.68466 0.187989
\(622\) −30.7386 −1.23251
\(623\) 10.5616 0.423140
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −31.3693 −1.25377
\(627\) 2.31534 0.0924658
\(628\) 13.8769 0.553748
\(629\) −12.8769 −0.513435
\(630\) 0.561553 0.0223728
\(631\) 13.7538 0.547530 0.273765 0.961797i \(-0.411731\pi\)
0.273765 + 0.961797i \(0.411731\pi\)
\(632\) −11.5616 −0.459894
\(633\) −21.9309 −0.871674
\(634\) −24.8078 −0.985242
\(635\) 8.56155 0.339755
\(636\) 8.56155 0.339488
\(637\) 0 0
\(638\) 10.0540 0.398041
\(639\) 13.1231 0.519142
\(640\) 1.00000 0.0395285
\(641\) −11.4384 −0.451792 −0.225896 0.974151i \(-0.572531\pi\)
−0.225896 + 0.974151i \(0.572531\pi\)
\(642\) −2.00000 −0.0789337
\(643\) −21.7538 −0.857886 −0.428943 0.903332i \(-0.641114\pi\)
−0.428943 + 0.903332i \(0.641114\pi\)
\(644\) 2.63068 0.103663
\(645\) −0.438447 −0.0172638
\(646\) −1.75379 −0.0690019
\(647\) 35.0540 1.37811 0.689057 0.724707i \(-0.258025\pi\)
0.689057 + 0.724707i \(0.258025\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 26.5464 1.04204
\(650\) 0 0
\(651\) 3.75379 0.147123
\(652\) 14.4384 0.565453
\(653\) −44.5616 −1.74383 −0.871914 0.489659i \(-0.837121\pi\)
−0.871914 + 0.489659i \(0.837121\pi\)
\(654\) −3.75379 −0.146785
\(655\) 2.12311 0.0829566
\(656\) −12.2462 −0.478134
\(657\) −9.36932 −0.365532
\(658\) 3.93087 0.153241
\(659\) −6.05398 −0.235829 −0.117915 0.993024i \(-0.537621\pi\)
−0.117915 + 0.993024i \(0.537621\pi\)
\(660\) 4.12311 0.160492
\(661\) −29.6155 −1.15191 −0.575955 0.817481i \(-0.695370\pi\)
−0.575955 + 0.817481i \(0.695370\pi\)
\(662\) 30.7386 1.19469
\(663\) 0 0
\(664\) −7.12311 −0.276430
\(665\) 0.315342 0.0122284
\(666\) −4.12311 −0.159767
\(667\) 11.4233 0.442312
\(668\) −4.36932 −0.169054
\(669\) −27.3002 −1.05549
\(670\) 2.24621 0.0867787
\(671\) −24.7386 −0.955024
\(672\) −0.561553 −0.0216624
\(673\) 25.7538 0.992736 0.496368 0.868112i \(-0.334667\pi\)
0.496368 + 0.868112i \(0.334667\pi\)
\(674\) 6.00000 0.231111
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −37.1231 −1.42676 −0.713378 0.700779i \(-0.752836\pi\)
−0.713378 + 0.700779i \(0.752836\pi\)
\(678\) −8.68466 −0.333532
\(679\) 4.00000 0.153506
\(680\) −3.12311 −0.119766
\(681\) 20.0000 0.766402
\(682\) 27.5616 1.05539
\(683\) −17.1231 −0.655197 −0.327599 0.944817i \(-0.606239\pi\)
−0.327599 + 0.944817i \(0.606239\pi\)
\(684\) −0.561553 −0.0214715
\(685\) 11.8078 0.451151
\(686\) 7.68466 0.293402
\(687\) 24.2462 0.925051
\(688\) 0.438447 0.0167156
\(689\) 0 0
\(690\) 4.68466 0.178342
\(691\) 20.5616 0.782198 0.391099 0.920349i \(-0.372095\pi\)
0.391099 + 0.920349i \(0.372095\pi\)
\(692\) 20.1771 0.767018
\(693\) −2.31534 −0.0879526
\(694\) −0.876894 −0.0332865
\(695\) −12.5616 −0.476487
\(696\) −2.43845 −0.0924291
\(697\) 38.2462 1.44868
\(698\) 8.49242 0.321443
\(699\) −5.31534 −0.201045
\(700\) 0.561553 0.0212247
\(701\) −36.3002 −1.37104 −0.685520 0.728054i \(-0.740425\pi\)
−0.685520 + 0.728054i \(0.740425\pi\)
\(702\) 0 0
\(703\) −2.31534 −0.0873248
\(704\) −4.12311 −0.155395
\(705\) 7.00000 0.263635
\(706\) −25.8617 −0.973319
\(707\) −4.98485 −0.187474
\(708\) −6.43845 −0.241972
\(709\) −34.2462 −1.28614 −0.643072 0.765806i \(-0.722340\pi\)
−0.643072 + 0.765806i \(0.722340\pi\)
\(710\) 13.1231 0.492501
\(711\) 11.5616 0.433592
\(712\) −18.8078 −0.704851
\(713\) 31.3153 1.17277
\(714\) 1.75379 0.0656339
\(715\) 0 0
\(716\) −12.6847 −0.474048
\(717\) 11.3693 0.424595
\(718\) 13.1231 0.489750
\(719\) 44.9848 1.67765 0.838826 0.544400i \(-0.183243\pi\)
0.838826 + 0.544400i \(0.183243\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 2.56155 0.0953972
\(722\) 18.6847 0.695371
\(723\) 28.1231 1.04591
\(724\) 2.87689 0.106919
\(725\) 2.43845 0.0905617
\(726\) −6.00000 −0.222681
\(727\) 38.4233 1.42504 0.712521 0.701651i \(-0.247554\pi\)
0.712521 + 0.701651i \(0.247554\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −9.36932 −0.346774
\(731\) −1.36932 −0.0506460
\(732\) 6.00000 0.221766
\(733\) −20.0691 −0.741270 −0.370635 0.928779i \(-0.620860\pi\)
−0.370635 + 0.928779i \(0.620860\pi\)
\(734\) −12.4924 −0.461104
\(735\) 6.68466 0.246567
\(736\) −4.68466 −0.172679
\(737\) −9.26137 −0.341147
\(738\) 12.2462 0.450789
\(739\) 27.5464 1.01331 0.506655 0.862149i \(-0.330882\pi\)
0.506655 + 0.862149i \(0.330882\pi\)
\(740\) −4.12311 −0.151568
\(741\) 0 0
\(742\) −4.80776 −0.176499
\(743\) −13.1771 −0.483420 −0.241710 0.970349i \(-0.577708\pi\)
−0.241710 + 0.970349i \(0.577708\pi\)
\(744\) −6.68466 −0.245071
\(745\) −2.19224 −0.0803173
\(746\) 25.8078 0.944889
\(747\) 7.12311 0.260621
\(748\) 12.8769 0.470826
\(749\) 1.12311 0.0410374
\(750\) 1.00000 0.0365148
\(751\) −23.8078 −0.868758 −0.434379 0.900730i \(-0.643032\pi\)
−0.434379 + 0.900730i \(0.643032\pi\)
\(752\) −7.00000 −0.255264
\(753\) −8.12311 −0.296022
\(754\) 0 0
\(755\) −15.3693 −0.559347
\(756\) 0.561553 0.0204235
\(757\) −28.4233 −1.03306 −0.516531 0.856268i \(-0.672777\pi\)
−0.516531 + 0.856268i \(0.672777\pi\)
\(758\) −17.6847 −0.642336
\(759\) −19.3153 −0.701102
\(760\) −0.561553 −0.0203697
\(761\) 17.9309 0.649994 0.324997 0.945715i \(-0.394637\pi\)
0.324997 + 0.945715i \(0.394637\pi\)
\(762\) 8.56155 0.310152
\(763\) 2.10795 0.0763129
\(764\) 10.8769 0.393512
\(765\) 3.12311 0.112916
\(766\) 0.192236 0.00694577
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 33.3153 1.20138 0.600691 0.799481i \(-0.294892\pi\)
0.600691 + 0.799481i \(0.294892\pi\)
\(770\) −2.31534 −0.0834391
\(771\) −5.56155 −0.200294
\(772\) 0 0
\(773\) −24.1771 −0.869589 −0.434795 0.900530i \(-0.643179\pi\)
−0.434795 + 0.900530i \(0.643179\pi\)
\(774\) −0.438447 −0.0157597
\(775\) 6.68466 0.240120
\(776\) −7.12311 −0.255705
\(777\) 2.31534 0.0830624
\(778\) 18.0540 0.647266
\(779\) 6.87689 0.246390
\(780\) 0 0
\(781\) −54.1080 −1.93613
\(782\) 14.6307 0.523192
\(783\) 2.43845 0.0871430
\(784\) −6.68466 −0.238738
\(785\) −13.8769 −0.495288
\(786\) 2.12311 0.0757287
\(787\) −31.3153 −1.11627 −0.558136 0.829750i \(-0.688483\pi\)
−0.558136 + 0.829750i \(0.688483\pi\)
\(788\) −12.8078 −0.456258
\(789\) −13.0000 −0.462812
\(790\) 11.5616 0.411342
\(791\) 4.87689 0.173402
\(792\) 4.12311 0.146508
\(793\) 0 0
\(794\) −20.1231 −0.714142
\(795\) −8.56155 −0.303647
\(796\) 2.87689 0.101969
\(797\) 13.1231 0.464844 0.232422 0.972615i \(-0.425335\pi\)
0.232422 + 0.972615i \(0.425335\pi\)
\(798\) 0.315342 0.0111630
\(799\) 21.8617 0.773413
\(800\) −1.00000 −0.0353553
\(801\) 18.8078 0.664540
\(802\) 0.315342 0.0111351
\(803\) 38.6307 1.36325
\(804\) 2.24621 0.0792178
\(805\) −2.63068 −0.0927194
\(806\) 0 0
\(807\) 21.3693 0.752236
\(808\) 8.87689 0.312288
\(809\) −14.4924 −0.509526 −0.254763 0.967003i \(-0.581998\pi\)
−0.254763 + 0.967003i \(0.581998\pi\)
\(810\) 1.00000 0.0351364
\(811\) −47.5464 −1.66958 −0.834790 0.550569i \(-0.814411\pi\)
−0.834790 + 0.550569i \(0.814411\pi\)
\(812\) 1.36932 0.0480536
\(813\) 13.1771 0.462140
\(814\) 17.0000 0.595850
\(815\) −14.4384 −0.505757
\(816\) −3.12311 −0.109331
\(817\) −0.246211 −0.00861384
\(818\) −18.8078 −0.657598
\(819\) 0 0
\(820\) 12.2462 0.427656
\(821\) 8.43845 0.294504 0.147252 0.989099i \(-0.452957\pi\)
0.147252 + 0.989099i \(0.452957\pi\)
\(822\) 11.8078 0.411843
\(823\) 5.05398 0.176171 0.0880853 0.996113i \(-0.471925\pi\)
0.0880853 + 0.996113i \(0.471925\pi\)
\(824\) −4.56155 −0.158909
\(825\) −4.12311 −0.143548
\(826\) 3.61553 0.125800
\(827\) 28.2462 0.982217 0.491109 0.871098i \(-0.336592\pi\)
0.491109 + 0.871098i \(0.336592\pi\)
\(828\) 4.68466 0.162803
\(829\) 12.3845 0.430130 0.215065 0.976600i \(-0.431004\pi\)
0.215065 + 0.976600i \(0.431004\pi\)
\(830\) 7.12311 0.247247
\(831\) −1.00000 −0.0346896
\(832\) 0 0
\(833\) 20.8769 0.723342
\(834\) −12.5616 −0.434971
\(835\) 4.36932 0.151206
\(836\) 2.31534 0.0800778
\(837\) 6.68466 0.231056
\(838\) −32.4924 −1.12243
\(839\) −36.7386 −1.26836 −0.634179 0.773186i \(-0.718662\pi\)
−0.634179 + 0.773186i \(0.718662\pi\)
\(840\) 0.561553 0.0193754
\(841\) −23.0540 −0.794965
\(842\) −32.4924 −1.11976
\(843\) −16.2462 −0.559549
\(844\) −21.9309 −0.754892
\(845\) 0 0
\(846\) 7.00000 0.240665
\(847\) 3.36932 0.115771
\(848\) 8.56155 0.294005
\(849\) 15.5616 0.534071
\(850\) 3.12311 0.107122
\(851\) 19.3153 0.662121
\(852\) 13.1231 0.449590
\(853\) −2.30019 −0.0787569 −0.0393784 0.999224i \(-0.512538\pi\)
−0.0393784 + 0.999224i \(0.512538\pi\)
\(854\) −3.36932 −0.115296
\(855\) 0.561553 0.0192047
\(856\) −2.00000 −0.0683586
\(857\) −6.19224 −0.211523 −0.105761 0.994392i \(-0.533728\pi\)
−0.105761 + 0.994392i \(0.533728\pi\)
\(858\) 0 0
\(859\) 39.7926 1.35771 0.678853 0.734274i \(-0.262477\pi\)
0.678853 + 0.734274i \(0.262477\pi\)
\(860\) −0.438447 −0.0149509
\(861\) −6.87689 −0.234364
\(862\) −22.7386 −0.774481
\(863\) −2.43845 −0.0830057 −0.0415029 0.999138i \(-0.513215\pi\)
−0.0415029 + 0.999138i \(0.513215\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −20.1771 −0.686041
\(866\) 18.7386 0.636765
\(867\) −7.24621 −0.246094
\(868\) 3.75379 0.127412
\(869\) −47.6695 −1.61708
\(870\) 2.43845 0.0826711
\(871\) 0 0
\(872\) −3.75379 −0.127119
\(873\) 7.12311 0.241081
\(874\) 2.63068 0.0889842
\(875\) −0.561553 −0.0189839
\(876\) −9.36932 −0.316560
\(877\) −8.43845 −0.284946 −0.142473 0.989799i \(-0.545505\pi\)
−0.142473 + 0.989799i \(0.545505\pi\)
\(878\) −25.1231 −0.847864
\(879\) −3.93087 −0.132585
\(880\) 4.12311 0.138990
\(881\) −24.5616 −0.827500 −0.413750 0.910391i \(-0.635781\pi\)
−0.413750 + 0.910391i \(0.635781\pi\)
\(882\) 6.68466 0.225084
\(883\) −19.1771 −0.645360 −0.322680 0.946508i \(-0.604584\pi\)
−0.322680 + 0.946508i \(0.604584\pi\)
\(884\) 0 0
\(885\) 6.43845 0.216426
\(886\) 13.1231 0.440879
\(887\) −33.1080 −1.11166 −0.555828 0.831297i \(-0.687599\pi\)
−0.555828 + 0.831297i \(0.687599\pi\)
\(888\) −4.12311 −0.138362
\(889\) −4.80776 −0.161247
\(890\) 18.8078 0.630438
\(891\) −4.12311 −0.138129
\(892\) −27.3002 −0.914078
\(893\) 3.93087 0.131542
\(894\) −2.19224 −0.0733193
\(895\) 12.6847 0.424001
\(896\) −0.561553 −0.0187602
\(897\) 0 0
\(898\) −20.1771 −0.673318
\(899\) 16.3002 0.543642
\(900\) 1.00000 0.0333333
\(901\) −26.7386 −0.890793
\(902\) −50.4924 −1.68121
\(903\) 0.246211 0.00819340
\(904\) −8.68466 −0.288847
\(905\) −2.87689 −0.0956312
\(906\) −15.3693 −0.510611
\(907\) −53.9157 −1.79024 −0.895121 0.445823i \(-0.852911\pi\)
−0.895121 + 0.445823i \(0.852911\pi\)
\(908\) 20.0000 0.663723
\(909\) −8.87689 −0.294428
\(910\) 0 0
\(911\) −18.6307 −0.617262 −0.308631 0.951182i \(-0.599871\pi\)
−0.308631 + 0.951182i \(0.599871\pi\)
\(912\) −0.561553 −0.0185949
\(913\) −29.3693 −0.971983
\(914\) 20.2462 0.669685
\(915\) −6.00000 −0.198354
\(916\) 24.2462 0.801117
\(917\) −1.19224 −0.0393711
\(918\) 3.12311 0.103078
\(919\) 16.6307 0.548596 0.274298 0.961645i \(-0.411555\pi\)
0.274298 + 0.961645i \(0.411555\pi\)
\(920\) 4.68466 0.154449
\(921\) −25.6155 −0.844060
\(922\) −30.0540 −0.989775
\(923\) 0 0
\(924\) −2.31534 −0.0761691
\(925\) 4.12311 0.135567
\(926\) −7.61553 −0.250262
\(927\) 4.56155 0.149821
\(928\) −2.43845 −0.0800460
\(929\) 25.1231 0.824262 0.412131 0.911125i \(-0.364785\pi\)
0.412131 + 0.911125i \(0.364785\pi\)
\(930\) 6.68466 0.219199
\(931\) 3.75379 0.123025
\(932\) −5.31534 −0.174110
\(933\) 30.7386 1.00634
\(934\) 17.8617 0.584454
\(935\) −12.8769 −0.421119
\(936\) 0 0
\(937\) 53.2311 1.73898 0.869491 0.493948i \(-0.164447\pi\)
0.869491 + 0.493948i \(0.164447\pi\)
\(938\) −1.26137 −0.0411851
\(939\) 31.3693 1.02370
\(940\) 7.00000 0.228315
\(941\) 25.3693 0.827016 0.413508 0.910500i \(-0.364303\pi\)
0.413508 + 0.910500i \(0.364303\pi\)
\(942\) −13.8769 −0.452134
\(943\) −57.3693 −1.86820
\(944\) −6.43845 −0.209554
\(945\) −0.561553 −0.0182673
\(946\) 1.80776 0.0587755
\(947\) 43.8617 1.42532 0.712658 0.701512i \(-0.247491\pi\)
0.712658 + 0.701512i \(0.247491\pi\)
\(948\) 11.5616 0.375502
\(949\) 0 0
\(950\) 0.561553 0.0182192
\(951\) 24.8078 0.804447
\(952\) 1.75379 0.0568406
\(953\) 51.1771 1.65779 0.828894 0.559406i \(-0.188971\pi\)
0.828894 + 0.559406i \(0.188971\pi\)
\(954\) −8.56155 −0.277191
\(955\) −10.8769 −0.351968
\(956\) 11.3693 0.367710
\(957\) −10.0540 −0.324999
\(958\) −38.2462 −1.23568
\(959\) −6.63068 −0.214116
\(960\) −1.00000 −0.0322749
\(961\) 13.6847 0.441441
\(962\) 0 0
\(963\) 2.00000 0.0644491
\(964\) 28.1231 0.905784
\(965\) 0 0
\(966\) −2.63068 −0.0846408
\(967\) −15.6847 −0.504385 −0.252192 0.967677i \(-0.581152\pi\)
−0.252192 + 0.967677i \(0.581152\pi\)
\(968\) −6.00000 −0.192847
\(969\) 1.75379 0.0563398
\(970\) 7.12311 0.228709
\(971\) −4.31534 −0.138486 −0.0692430 0.997600i \(-0.522058\pi\)
−0.0692430 + 0.997600i \(0.522058\pi\)
\(972\) 1.00000 0.0320750
\(973\) 7.05398 0.226140
\(974\) 13.0540 0.418276
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) 48.3002 1.54526 0.772630 0.634857i \(-0.218941\pi\)
0.772630 + 0.634857i \(0.218941\pi\)
\(978\) −14.4384 −0.461691
\(979\) −77.5464 −2.47839
\(980\) 6.68466 0.213534
\(981\) 3.75379 0.119849
\(982\) 17.4384 0.556483
\(983\) −32.1231 −1.02457 −0.512284 0.858816i \(-0.671200\pi\)
−0.512284 + 0.858816i \(0.671200\pi\)
\(984\) 12.2462 0.390395
\(985\) 12.8078 0.408089
\(986\) 7.61553 0.242528
\(987\) −3.93087 −0.125121
\(988\) 0 0
\(989\) 2.05398 0.0653126
\(990\) −4.12311 −0.131041
\(991\) 49.4233 1.56998 0.784991 0.619507i \(-0.212667\pi\)
0.784991 + 0.619507i \(0.212667\pi\)
\(992\) −6.68466 −0.212238
\(993\) −30.7386 −0.975461
\(994\) −7.36932 −0.233741
\(995\) −2.87689 −0.0912037
\(996\) 7.12311 0.225704
\(997\) −37.5464 −1.18911 −0.594553 0.804056i \(-0.702671\pi\)
−0.594553 + 0.804056i \(0.702671\pi\)
\(998\) 4.00000 0.126618
\(999\) 4.12311 0.130449
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.bb.1.2 2
13.4 even 6 390.2.i.g.211.2 yes 4
13.5 odd 4 5070.2.b.r.1351.4 4
13.8 odd 4 5070.2.b.r.1351.1 4
13.10 even 6 390.2.i.g.61.2 4
13.12 even 2 5070.2.a.bi.1.1 2
39.17 odd 6 1170.2.i.o.991.2 4
39.23 odd 6 1170.2.i.o.451.2 4
65.4 even 6 1950.2.i.bi.601.1 4
65.17 odd 12 1950.2.z.n.1849.4 8
65.23 odd 12 1950.2.z.n.1699.4 8
65.43 odd 12 1950.2.z.n.1849.1 8
65.49 even 6 1950.2.i.bi.451.1 4
65.62 odd 12 1950.2.z.n.1699.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.g.61.2 4 13.10 even 6
390.2.i.g.211.2 yes 4 13.4 even 6
1170.2.i.o.451.2 4 39.23 odd 6
1170.2.i.o.991.2 4 39.17 odd 6
1950.2.i.bi.451.1 4 65.49 even 6
1950.2.i.bi.601.1 4 65.4 even 6
1950.2.z.n.1699.1 8 65.62 odd 12
1950.2.z.n.1699.4 8 65.23 odd 12
1950.2.z.n.1849.1 8 65.43 odd 12
1950.2.z.n.1849.4 8 65.17 odd 12
5070.2.a.bb.1.2 2 1.1 even 1 trivial
5070.2.a.bi.1.1 2 13.12 even 2
5070.2.b.r.1351.1 4 13.8 odd 4
5070.2.b.r.1351.4 4 13.5 odd 4