Properties

Label 5070.2.a.bz.1.3
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.131472.2
Defining polynomial: \(x^{4} - 2 x^{3} - 19 x^{2} + 20 x + 52\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
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 1.3
Root \(-1.32258\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.32258 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.32258 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.61335 q^{11} +1.00000 q^{12} -1.32258 q^{14} +1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -2.29078 q^{19} +1.00000 q^{20} +1.32258 q^{21} -4.61335 q^{22} +8.66799 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +1.32258 q^{28} -2.02283 q^{29} -1.00000 q^{30} +10.1321 q^{31} -1.00000 q^{32} +4.61335 q^{33} -4.00000 q^{34} +1.32258 q^{35} +1.00000 q^{36} -6.80951 q^{37} +2.29078 q^{38} -1.00000 q^{40} -4.64516 q^{41} -1.32258 q^{42} +8.60562 q^{43} +4.61335 q^{44} +1.00000 q^{45} -8.66799 q^{46} -9.10926 q^{47} +1.00000 q^{48} -5.25078 q^{49} -1.00000 q^{50} +4.00000 q^{51} +0.826674 q^{53} -1.00000 q^{54} +4.61335 q^{55} -1.32258 q^{56} -2.29078 q^{57} +2.02283 q^{58} +3.14152 q^{59} +1.00000 q^{60} +0.535898 q^{61} -10.1321 q^{62} +1.32258 q^{63} +1.00000 q^{64} -4.61335 q^{66} -3.18106 q^{67} +4.00000 q^{68} +8.66799 q^{69} -1.32258 q^{70} +11.3360 q^{71} -1.00000 q^{72} -6.28304 q^{73} +6.80951 q^{74} +1.00000 q^{75} -2.29078 q^{76} +6.10153 q^{77} +2.96774 q^{79} +1.00000 q^{80} +1.00000 q^{81} +4.64516 q^{82} -15.8719 q^{83} +1.32258 q^{84} +4.00000 q^{85} -8.60562 q^{86} -2.02283 q^{87} -4.61335 q^{88} -11.8641 q^{89} -1.00000 q^{90} +8.66799 q^{92} +10.1321 q^{93} +9.10926 q^{94} -2.29078 q^{95} -1.00000 q^{96} +8.81894 q^{97} +5.25078 q^{98} +4.61335 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + 4q^{3} + 4q^{4} + 4q^{5} - 4q^{6} - 2q^{7} - 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{2} + 4q^{3} + 4q^{4} + 4q^{5} - 4q^{6} - 2q^{7} - 4q^{8} + 4q^{9} - 4q^{10} + 2q^{11} + 4q^{12} + 2q^{14} + 4q^{15} + 4q^{16} + 16q^{17} - 4q^{18} + 4q^{20} - 2q^{21} - 2q^{22} + 4q^{23} - 4q^{24} + 4q^{25} + 4q^{27} - 2q^{28} + 8q^{29} - 4q^{30} - 4q^{31} - 4q^{32} + 2q^{33} - 16q^{34} - 2q^{35} + 4q^{36} + 10q^{37} - 4q^{40} - 4q^{41} + 2q^{42} + 14q^{43} + 2q^{44} + 4q^{45} - 4q^{46} - 8q^{47} + 4q^{48} + 14q^{49} - 4q^{50} + 16q^{51} + 8q^{53} - 4q^{54} + 2q^{55} + 2q^{56} - 8q^{58} + 6q^{59} + 4q^{60} + 16q^{61} + 4q^{62} - 2q^{63} + 4q^{64} - 2q^{66} - 12q^{67} + 16q^{68} + 4q^{69} + 2q^{70} - 16q^{71} - 4q^{72} - 12q^{73} - 10q^{74} + 4q^{75} - 8q^{77} - 10q^{79} + 4q^{80} + 4q^{81} + 4q^{82} - 16q^{83} - 2q^{84} + 16q^{85} - 14q^{86} + 8q^{87} - 2q^{88} + 4q^{89} - 4q^{90} + 4q^{92} - 4q^{93} + 8q^{94} - 4q^{96} + 36q^{97} - 14q^{98} + 2q^{99} + O(q^{100}) \)

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
\(7\) 1.32258 0.499888 0.249944 0.968260i \(-0.419588\pi\)
0.249944 + 0.968260i \(0.419588\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 4.61335 1.39098 0.695489 0.718536i \(-0.255188\pi\)
0.695489 + 0.718536i \(0.255188\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −1.32258 −0.353474
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.29078 −0.525540 −0.262770 0.964859i \(-0.584636\pi\)
−0.262770 + 0.964859i \(0.584636\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.32258 0.288611
\(22\) −4.61335 −0.983571
\(23\) 8.66799 1.80740 0.903700 0.428166i \(-0.140840\pi\)
0.903700 + 0.428166i \(0.140840\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 1.32258 0.249944
\(29\) −2.02283 −0.375629 −0.187815 0.982204i \(-0.560140\pi\)
−0.187815 + 0.982204i \(0.560140\pi\)
\(30\) −1.00000 −0.182574
\(31\) 10.1321 1.81978 0.909888 0.414853i \(-0.136167\pi\)
0.909888 + 0.414853i \(0.136167\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.61335 0.803082
\(34\) −4.00000 −0.685994
\(35\) 1.32258 0.223557
\(36\) 1.00000 0.166667
\(37\) −6.80951 −1.11948 −0.559738 0.828670i \(-0.689098\pi\)
−0.559738 + 0.828670i \(0.689098\pi\)
\(38\) 2.29078 0.371613
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −4.64516 −0.725452 −0.362726 0.931896i \(-0.618154\pi\)
−0.362726 + 0.931896i \(0.618154\pi\)
\(42\) −1.32258 −0.204078
\(43\) 8.60562 1.31235 0.656173 0.754611i \(-0.272174\pi\)
0.656173 + 0.754611i \(0.272174\pi\)
\(44\) 4.61335 0.695489
\(45\) 1.00000 0.149071
\(46\) −8.66799 −1.27802
\(47\) −9.10926 −1.32872 −0.664361 0.747412i \(-0.731296\pi\)
−0.664361 + 0.747412i \(0.731296\pi\)
\(48\) 1.00000 0.144338
\(49\) −5.25078 −0.750112
\(50\) −1.00000 −0.141421
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) 0.826674 0.113552 0.0567762 0.998387i \(-0.481918\pi\)
0.0567762 + 0.998387i \(0.481918\pi\)
\(54\) −1.00000 −0.136083
\(55\) 4.61335 0.622065
\(56\) −1.32258 −0.176737
\(57\) −2.29078 −0.303421
\(58\) 2.02283 0.265610
\(59\) 3.14152 0.408991 0.204496 0.978867i \(-0.434445\pi\)
0.204496 + 0.978867i \(0.434445\pi\)
\(60\) 1.00000 0.129099
\(61\) 0.535898 0.0686148 0.0343074 0.999411i \(-0.489077\pi\)
0.0343074 + 0.999411i \(0.489077\pi\)
\(62\) −10.1321 −1.28678
\(63\) 1.32258 0.166629
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.61335 −0.567865
\(67\) −3.18106 −0.388628 −0.194314 0.980939i \(-0.562248\pi\)
−0.194314 + 0.980939i \(0.562248\pi\)
\(68\) 4.00000 0.485071
\(69\) 8.66799 1.04350
\(70\) −1.32258 −0.158079
\(71\) 11.3360 1.34533 0.672666 0.739946i \(-0.265149\pi\)
0.672666 + 0.739946i \(0.265149\pi\)
\(72\) −1.00000 −0.117851
\(73\) −6.28304 −0.735375 −0.367687 0.929949i \(-0.619850\pi\)
−0.367687 + 0.929949i \(0.619850\pi\)
\(74\) 6.80951 0.791589
\(75\) 1.00000 0.115470
\(76\) −2.29078 −0.262770
\(77\) 6.10153 0.695334
\(78\) 0 0
\(79\) 2.96774 0.333897 0.166948 0.985966i \(-0.446609\pi\)
0.166948 + 0.985966i \(0.446609\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 4.64516 0.512972
\(83\) −15.8719 −1.74216 −0.871082 0.491138i \(-0.836581\pi\)
−0.871082 + 0.491138i \(0.836581\pi\)
\(84\) 1.32258 0.144305
\(85\) 4.00000 0.433861
\(86\) −8.60562 −0.927968
\(87\) −2.02283 −0.216870
\(88\) −4.61335 −0.491785
\(89\) −11.8641 −1.25760 −0.628798 0.777569i \(-0.716453\pi\)
−0.628798 + 0.777569i \(0.716453\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 8.66799 0.903700
\(93\) 10.1321 1.05065
\(94\) 9.10926 0.939549
\(95\) −2.29078 −0.235029
\(96\) −1.00000 −0.102062
\(97\) 8.81894 0.895428 0.447714 0.894177i \(-0.352238\pi\)
0.447714 + 0.894177i \(0.352238\pi\)
\(98\) 5.25078 0.530409
\(99\) 4.61335 0.463660
\(100\) 1.00000 0.100000
\(101\) −7.22671 −0.719085 −0.359542 0.933129i \(-0.617067\pi\)
−0.359542 + 0.933129i \(0.617067\pi\)
\(102\) −4.00000 −0.396059
\(103\) 18.4000 1.81301 0.906505 0.422196i \(-0.138740\pi\)
0.906505 + 0.422196i \(0.138740\pi\)
\(104\) 0 0
\(105\) 1.32258 0.129071
\(106\) −0.826674 −0.0802936
\(107\) 3.07180 0.296962 0.148481 0.988915i \(-0.452562\pi\)
0.148481 + 0.988915i \(0.452562\pi\)
\(108\) 1.00000 0.0962250
\(109\) −13.2267 −1.26689 −0.633445 0.773788i \(-0.718360\pi\)
−0.633445 + 0.773788i \(0.718360\pi\)
\(110\) −4.61335 −0.439866
\(111\) −6.80951 −0.646330
\(112\) 1.32258 0.124972
\(113\) −8.08643 −0.760708 −0.380354 0.924841i \(-0.624198\pi\)
−0.380354 + 0.924841i \(0.624198\pi\)
\(114\) 2.29078 0.214551
\(115\) 8.66799 0.808294
\(116\) −2.02283 −0.187815
\(117\) 0 0
\(118\) −3.14152 −0.289201
\(119\) 5.29032 0.484963
\(120\) −1.00000 −0.0912871
\(121\) 10.2830 0.934822
\(122\) −0.535898 −0.0485180
\(123\) −4.64516 −0.418840
\(124\) 10.1321 0.909888
\(125\) 1.00000 0.0894427
\(126\) −1.32258 −0.117825
\(127\) 11.4718 1.01796 0.508980 0.860778i \(-0.330023\pi\)
0.508980 + 0.860778i \(0.330023\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.60562 0.757683
\(130\) 0 0
\(131\) −15.9906 −1.39710 −0.698551 0.715560i \(-0.746172\pi\)
−0.698551 + 0.715560i \(0.746172\pi\)
\(132\) 4.61335 0.401541
\(133\) −3.02973 −0.262711
\(134\) 3.18106 0.274802
\(135\) 1.00000 0.0860663
\(136\) −4.00000 −0.342997
\(137\) 16.0697 1.37293 0.686465 0.727163i \(-0.259162\pi\)
0.686465 + 0.727163i \(0.259162\pi\)
\(138\) −8.66799 −0.737868
\(139\) 9.32258 0.790731 0.395365 0.918524i \(-0.370618\pi\)
0.395365 + 0.918524i \(0.370618\pi\)
\(140\) 1.32258 0.111778
\(141\) −9.10926 −0.767138
\(142\) −11.3360 −0.951294
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −2.02283 −0.167987
\(146\) 6.28304 0.519988
\(147\) −5.25078 −0.433077
\(148\) −6.80951 −0.559738
\(149\) −4.08519 −0.334672 −0.167336 0.985900i \(-0.553516\pi\)
−0.167336 + 0.985900i \(0.553516\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −19.9811 −1.62604 −0.813021 0.582235i \(-0.802178\pi\)
−0.813021 + 0.582235i \(0.802178\pi\)
\(152\) 2.29078 0.185806
\(153\) 4.00000 0.323381
\(154\) −6.10153 −0.491675
\(155\) 10.1321 0.813829
\(156\) 0 0
\(157\) 7.13379 0.569338 0.284669 0.958626i \(-0.408116\pi\)
0.284669 + 0.958626i \(0.408116\pi\)
\(158\) −2.96774 −0.236101
\(159\) 0.826674 0.0655595
\(160\) −1.00000 −0.0790569
\(161\) 11.4641 0.903498
\(162\) −1.00000 −0.0785674
\(163\) −9.89470 −0.775012 −0.387506 0.921867i \(-0.626663\pi\)
−0.387506 + 0.921867i \(0.626663\pi\)
\(164\) −4.64516 −0.362726
\(165\) 4.61335 0.359149
\(166\) 15.8719 1.23190
\(167\) −24.0375 −1.86007 −0.930037 0.367465i \(-0.880226\pi\)
−0.930037 + 0.367465i \(0.880226\pi\)
\(168\) −1.32258 −0.102039
\(169\) 0 0
\(170\) −4.00000 −0.306786
\(171\) −2.29078 −0.175180
\(172\) 8.60562 0.656173
\(173\) 2.52817 0.192213 0.0961065 0.995371i \(-0.469361\pi\)
0.0961065 + 0.995371i \(0.469361\pi\)
\(174\) 2.02283 0.153350
\(175\) 1.32258 0.0999776
\(176\) 4.61335 0.347745
\(177\) 3.14152 0.236131
\(178\) 11.8641 0.889255
\(179\) −6.27568 −0.469066 −0.234533 0.972108i \(-0.575356\pi\)
−0.234533 + 0.972108i \(0.575356\pi\)
\(180\) 1.00000 0.0745356
\(181\) 12.2830 0.912991 0.456496 0.889726i \(-0.349104\pi\)
0.456496 + 0.889726i \(0.349104\pi\)
\(182\) 0 0
\(183\) 0.535898 0.0396147
\(184\) −8.66799 −0.639012
\(185\) −6.80951 −0.500645
\(186\) −10.1321 −0.742921
\(187\) 18.4534 1.34945
\(188\) −9.10926 −0.664361
\(189\) 1.32258 0.0962035
\(190\) 2.29078 0.166190
\(191\) −9.74715 −0.705279 −0.352639 0.935759i \(-0.614716\pi\)
−0.352639 + 0.935759i \(0.614716\pi\)
\(192\) 1.00000 0.0721688
\(193\) 13.7626 0.990654 0.495327 0.868707i \(-0.335048\pi\)
0.495327 + 0.868707i \(0.335048\pi\)
\(194\) −8.81894 −0.633163
\(195\) 0 0
\(196\) −5.25078 −0.375056
\(197\) −10.1415 −0.722554 −0.361277 0.932459i \(-0.617659\pi\)
−0.361277 + 0.932459i \(0.617659\pi\)
\(198\) −4.61335 −0.327857
\(199\) 9.57336 0.678638 0.339319 0.940671i \(-0.389803\pi\)
0.339319 + 0.940671i \(0.389803\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −3.18106 −0.224375
\(202\) 7.22671 0.508470
\(203\) −2.67535 −0.187773
\(204\) 4.00000 0.280056
\(205\) −4.64516 −0.324432
\(206\) −18.4000 −1.28199
\(207\) 8.66799 0.602467
\(208\) 0 0
\(209\) −10.5682 −0.731015
\(210\) −1.32258 −0.0912667
\(211\) −1.08519 −0.0747074 −0.0373537 0.999302i \(-0.511893\pi\)
−0.0373537 + 0.999302i \(0.511893\pi\)
\(212\) 0.826674 0.0567762
\(213\) 11.3360 0.776728
\(214\) −3.07180 −0.209984
\(215\) 8.60562 0.586899
\(216\) −1.00000 −0.0680414
\(217\) 13.4005 0.909685
\(218\) 13.2267 0.895826
\(219\) −6.28304 −0.424569
\(220\) 4.61335 0.311032
\(221\) 0 0
\(222\) 6.80951 0.457024
\(223\) 24.9416 1.67021 0.835106 0.550089i \(-0.185406\pi\)
0.835106 + 0.550089i \(0.185406\pi\)
\(224\) −1.32258 −0.0883686
\(225\) 1.00000 0.0666667
\(226\) 8.08643 0.537902
\(227\) 19.3205 1.28235 0.641174 0.767396i \(-0.278448\pi\)
0.641174 + 0.767396i \(0.278448\pi\)
\(228\) −2.29078 −0.151710
\(229\) 4.15491 0.274564 0.137282 0.990532i \(-0.456163\pi\)
0.137282 + 0.990532i \(0.456163\pi\)
\(230\) −8.66799 −0.571550
\(231\) 6.10153 0.401451
\(232\) 2.02283 0.132805
\(233\) −16.9665 −1.11151 −0.555756 0.831346i \(-0.687571\pi\)
−0.555756 + 0.831346i \(0.687571\pi\)
\(234\) 0 0
\(235\) −9.10926 −0.594223
\(236\) 3.14152 0.204496
\(237\) 2.96774 0.192775
\(238\) −5.29032 −0.342920
\(239\) 6.34416 0.410370 0.205185 0.978723i \(-0.434220\pi\)
0.205185 + 0.978723i \(0.434220\pi\)
\(240\) 1.00000 0.0645497
\(241\) 24.1855 1.55792 0.778962 0.627072i \(-0.215747\pi\)
0.778962 + 0.627072i \(0.215747\pi\)
\(242\) −10.2830 −0.661019
\(243\) 1.00000 0.0641500
\(244\) 0.535898 0.0343074
\(245\) −5.25078 −0.335460
\(246\) 4.64516 0.296165
\(247\) 0 0
\(248\) −10.1321 −0.643388
\(249\) −15.8719 −1.00584
\(250\) −1.00000 −0.0632456
\(251\) 8.82497 0.557027 0.278514 0.960432i \(-0.410158\pi\)
0.278514 + 0.960432i \(0.410158\pi\)
\(252\) 1.32258 0.0833147
\(253\) 39.9885 2.51406
\(254\) −11.4718 −0.719807
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) 8.38494 0.523038 0.261519 0.965198i \(-0.415777\pi\)
0.261519 + 0.965198i \(0.415777\pi\)
\(258\) −8.60562 −0.535763
\(259\) −9.00612 −0.559613
\(260\) 0 0
\(261\) −2.02283 −0.125210
\(262\) 15.9906 0.987900
\(263\) 1.92130 0.118472 0.0592361 0.998244i \(-0.481134\pi\)
0.0592361 + 0.998244i \(0.481134\pi\)
\(264\) −4.61335 −0.283932
\(265\) 0.826674 0.0507822
\(266\) 3.02973 0.185765
\(267\) −11.8641 −0.726073
\(268\) −3.18106 −0.194314
\(269\) −32.3098 −1.96996 −0.984982 0.172655i \(-0.944766\pi\)
−0.984982 + 0.172655i \(0.944766\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −23.4526 −1.42464 −0.712322 0.701853i \(-0.752356\pi\)
−0.712322 + 0.701853i \(0.752356\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −16.0697 −0.970808
\(275\) 4.61335 0.278196
\(276\) 8.66799 0.521751
\(277\) −9.00566 −0.541098 −0.270549 0.962706i \(-0.587205\pi\)
−0.270549 + 0.962706i \(0.587205\pi\)
\(278\) −9.32258 −0.559131
\(279\) 10.1321 0.606592
\(280\) −1.32258 −0.0790393
\(281\) 1.71696 0.102425 0.0512125 0.998688i \(-0.483691\pi\)
0.0512125 + 0.998688i \(0.483691\pi\)
\(282\) 9.10926 0.542449
\(283\) −1.54929 −0.0920957 −0.0460479 0.998939i \(-0.514663\pi\)
−0.0460479 + 0.998939i \(0.514663\pi\)
\(284\) 11.3360 0.672666
\(285\) −2.29078 −0.135694
\(286\) 0 0
\(287\) −6.14359 −0.362645
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 2.02283 0.118784
\(291\) 8.81894 0.516976
\(292\) −6.28304 −0.367687
\(293\) 30.4057 1.77632 0.888160 0.459535i \(-0.151984\pi\)
0.888160 + 0.459535i \(0.151984\pi\)
\(294\) 5.25078 0.306232
\(295\) 3.14152 0.182906
\(296\) 6.80951 0.395795
\(297\) 4.61335 0.267694
\(298\) 4.08519 0.236649
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 11.3816 0.656026
\(302\) 19.9811 1.14978
\(303\) −7.22671 −0.415164
\(304\) −2.29078 −0.131385
\(305\) 0.535898 0.0306855
\(306\) −4.00000 −0.228665
\(307\) 9.82622 0.560812 0.280406 0.959882i \(-0.409531\pi\)
0.280406 + 0.959882i \(0.409531\pi\)
\(308\) 6.10153 0.347667
\(309\) 18.4000 1.04674
\(310\) −10.1321 −0.575464
\(311\) 3.40049 0.192824 0.0964121 0.995342i \(-0.469263\pi\)
0.0964121 + 0.995342i \(0.469263\pi\)
\(312\) 0 0
\(313\) −16.2340 −0.917599 −0.458800 0.888540i \(-0.651720\pi\)
−0.458800 + 0.888540i \(0.651720\pi\)
\(314\) −7.13379 −0.402583
\(315\) 1.32258 0.0745189
\(316\) 2.96774 0.166948
\(317\) −23.5231 −1.32119 −0.660596 0.750742i \(-0.729696\pi\)
−0.660596 + 0.750742i \(0.729696\pi\)
\(318\) −0.826674 −0.0463576
\(319\) −9.33201 −0.522493
\(320\) 1.00000 0.0559017
\(321\) 3.07180 0.171451
\(322\) −11.4641 −0.638869
\(323\) −9.16310 −0.509849
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 9.89470 0.548016
\(327\) −13.2267 −0.731439
\(328\) 4.64516 0.256486
\(329\) −12.0477 −0.664213
\(330\) −4.61335 −0.253957
\(331\) −18.0904 −0.994338 −0.497169 0.867654i \(-0.665627\pi\)
−0.497169 + 0.867654i \(0.665627\pi\)
\(332\) −15.8719 −0.871082
\(333\) −6.80951 −0.373159
\(334\) 24.0375 1.31527
\(335\) −3.18106 −0.173800
\(336\) 1.32258 0.0721526
\(337\) 5.69900 0.310444 0.155222 0.987880i \(-0.450391\pi\)
0.155222 + 0.987880i \(0.450391\pi\)
\(338\) 0 0
\(339\) −8.08643 −0.439195
\(340\) 4.00000 0.216930
\(341\) 46.7429 2.53127
\(342\) 2.29078 0.123871
\(343\) −16.2026 −0.874860
\(344\) −8.60562 −0.463984
\(345\) 8.66799 0.466669
\(346\) −2.52817 −0.135915
\(347\) −15.7471 −0.845351 −0.422676 0.906281i \(-0.638909\pi\)
−0.422676 + 0.906281i \(0.638909\pi\)
\(348\) −2.02283 −0.108435
\(349\) −15.3205 −0.820088 −0.410044 0.912066i \(-0.634487\pi\)
−0.410044 + 0.912066i \(0.634487\pi\)
\(350\) −1.32258 −0.0706949
\(351\) 0 0
\(352\) −4.61335 −0.245893
\(353\) 1.19993 0.0638657 0.0319328 0.999490i \(-0.489834\pi\)
0.0319328 + 0.999490i \(0.489834\pi\)
\(354\) −3.14152 −0.166970
\(355\) 11.3360 0.601651
\(356\) −11.8641 −0.628798
\(357\) 5.29032 0.279993
\(358\) 6.27568 0.331680
\(359\) 16.2830 0.859386 0.429693 0.902975i \(-0.358622\pi\)
0.429693 + 0.902975i \(0.358622\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −13.7523 −0.723808
\(362\) −12.2830 −0.645582
\(363\) 10.2830 0.539720
\(364\) 0 0
\(365\) −6.28304 −0.328870
\(366\) −0.535898 −0.0280119
\(367\) −19.6345 −1.02491 −0.512456 0.858714i \(-0.671264\pi\)
−0.512456 + 0.858714i \(0.671264\pi\)
\(368\) 8.66799 0.451850
\(369\) −4.64516 −0.241817
\(370\) 6.80951 0.354009
\(371\) 1.09334 0.0567635
\(372\) 10.1321 0.525324
\(373\) −2.55748 −0.132421 −0.0662106 0.997806i \(-0.521091\pi\)
−0.0662106 + 0.997806i \(0.521091\pi\)
\(374\) −18.4534 −0.954204
\(375\) 1.00000 0.0516398
\(376\) 9.10926 0.469774
\(377\) 0 0
\(378\) −1.32258 −0.0680262
\(379\) −15.1088 −0.776087 −0.388044 0.921641i \(-0.626849\pi\)
−0.388044 + 0.921641i \(0.626849\pi\)
\(380\) −2.29078 −0.117514
\(381\) 11.4718 0.587720
\(382\) 9.74715 0.498707
\(383\) 18.7303 0.957076 0.478538 0.878067i \(-0.341167\pi\)
0.478538 + 0.878067i \(0.341167\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 6.10153 0.310963
\(386\) −13.7626 −0.700498
\(387\) 8.60562 0.437448
\(388\) 8.81894 0.447714
\(389\) −6.96899 −0.353342 −0.176671 0.984270i \(-0.556533\pi\)
−0.176671 + 0.984270i \(0.556533\pi\)
\(390\) 0 0
\(391\) 34.6719 1.75344
\(392\) 5.25078 0.265205
\(393\) −15.9906 −0.806617
\(394\) 10.1415 0.510922
\(395\) 2.96774 0.149323
\(396\) 4.61335 0.231830
\(397\) −18.5721 −0.932108 −0.466054 0.884756i \(-0.654325\pi\)
−0.466054 + 0.884756i \(0.654325\pi\)
\(398\) −9.57336 −0.479869
\(399\) −3.02973 −0.151676
\(400\) 1.00000 0.0500000
\(401\) 29.6904 1.48267 0.741333 0.671138i \(-0.234194\pi\)
0.741333 + 0.671138i \(0.234194\pi\)
\(402\) 3.18106 0.158657
\(403\) 0 0
\(404\) −7.22671 −0.359542
\(405\) 1.00000 0.0496904
\(406\) 2.67535 0.132775
\(407\) −31.4147 −1.55717
\(408\) −4.00000 −0.198030
\(409\) 24.3355 1.20331 0.601657 0.798755i \(-0.294507\pi\)
0.601657 + 0.798755i \(0.294507\pi\)
\(410\) 4.64516 0.229408
\(411\) 16.0697 0.792661
\(412\) 18.4000 0.906505
\(413\) 4.15491 0.204450
\(414\) −8.66799 −0.426008
\(415\) −15.8719 −0.779119
\(416\) 0 0
\(417\) 9.32258 0.456529
\(418\) 10.5682 0.516906
\(419\) −2.88255 −0.140822 −0.0704109 0.997518i \(-0.522431\pi\)
−0.0704109 + 0.997518i \(0.522431\pi\)
\(420\) 1.32258 0.0645353
\(421\) 4.94707 0.241106 0.120553 0.992707i \(-0.461533\pi\)
0.120553 + 0.992707i \(0.461533\pi\)
\(422\) 1.08519 0.0528261
\(423\) −9.10926 −0.442907
\(424\) −0.826674 −0.0401468
\(425\) 4.00000 0.194029
\(426\) −11.3360 −0.549230
\(427\) 0.708768 0.0342997
\(428\) 3.07180 0.148481
\(429\) 0 0
\(430\) −8.60562 −0.415000
\(431\) −17.7626 −0.855595 −0.427797 0.903875i \(-0.640710\pi\)
−0.427797 + 0.903875i \(0.640710\pi\)
\(432\) 1.00000 0.0481125
\(433\) 35.8375 1.72224 0.861121 0.508400i \(-0.169763\pi\)
0.861121 + 0.508400i \(0.169763\pi\)
\(434\) −13.4005 −0.643244
\(435\) −2.02283 −0.0969871
\(436\) −13.2267 −0.633445
\(437\) −19.8564 −0.949861
\(438\) 6.28304 0.300215
\(439\) 10.0904 0.481588 0.240794 0.970576i \(-0.422592\pi\)
0.240794 + 0.970576i \(0.422592\pi\)
\(440\) −4.61335 −0.219933
\(441\) −5.25078 −0.250037
\(442\) 0 0
\(443\) 13.6981 0.650816 0.325408 0.945574i \(-0.394498\pi\)
0.325408 + 0.945574i \(0.394498\pi\)
\(444\) −6.80951 −0.323165
\(445\) −11.8641 −0.562414
\(446\) −24.9416 −1.18102
\(447\) −4.08519 −0.193223
\(448\) 1.32258 0.0624860
\(449\) 15.3128 0.722655 0.361327 0.932439i \(-0.382324\pi\)
0.361327 + 0.932439i \(0.382324\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −21.4298 −1.00909
\(452\) −8.08643 −0.380354
\(453\) −19.9811 −0.938795
\(454\) −19.3205 −0.906756
\(455\) 0 0
\(456\) 2.29078 0.107275
\(457\) −15.5554 −0.727651 −0.363826 0.931467i \(-0.618530\pi\)
−0.363826 + 0.931467i \(0.618530\pi\)
\(458\) −4.15491 −0.194146
\(459\) 4.00000 0.186704
\(460\) 8.66799 0.404147
\(461\) −15.6431 −0.728571 −0.364286 0.931287i \(-0.618687\pi\)
−0.364286 + 0.931287i \(0.618687\pi\)
\(462\) −6.10153 −0.283869
\(463\) 13.7170 0.637481 0.318741 0.947842i \(-0.396740\pi\)
0.318741 + 0.947842i \(0.396740\pi\)
\(464\) −2.02283 −0.0939073
\(465\) 10.1321 0.469864
\(466\) 16.9665 0.785958
\(467\) 0.943666 0.0436677 0.0218338 0.999762i \(-0.493050\pi\)
0.0218338 + 0.999762i \(0.493050\pi\)
\(468\) 0 0
\(469\) −4.20720 −0.194271
\(470\) 9.10926 0.420179
\(471\) 7.13379 0.328708
\(472\) −3.14152 −0.144600
\(473\) 39.7008 1.82544
\(474\) −2.96774 −0.136313
\(475\) −2.29078 −0.105108
\(476\) 5.29032 0.242481
\(477\) 0.826674 0.0378508
\(478\) −6.34416 −0.290175
\(479\) 11.1020 0.507263 0.253631 0.967301i \(-0.418375\pi\)
0.253631 + 0.967301i \(0.418375\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) −24.1855 −1.10162
\(483\) 11.4641 0.521635
\(484\) 10.2830 0.467411
\(485\) 8.81894 0.400448
\(486\) −1.00000 −0.0453609
\(487\) 32.9416 1.49273 0.746363 0.665539i \(-0.231798\pi\)
0.746363 + 0.665539i \(0.231798\pi\)
\(488\) −0.535898 −0.0242590
\(489\) −9.89470 −0.447454
\(490\) 5.25078 0.237206
\(491\) 28.4513 1.28399 0.641996 0.766708i \(-0.278107\pi\)
0.641996 + 0.766708i \(0.278107\pi\)
\(492\) −4.64516 −0.209420
\(493\) −8.09130 −0.364414
\(494\) 0 0
\(495\) 4.61335 0.207355
\(496\) 10.1321 0.454944
\(497\) 14.9927 0.672516
\(498\) 15.8719 0.711235
\(499\) 4.10926 0.183956 0.0919779 0.995761i \(-0.470681\pi\)
0.0919779 + 0.995761i \(0.470681\pi\)
\(500\) 1.00000 0.0447214
\(501\) −24.0375 −1.07391
\(502\) −8.82497 −0.393878
\(503\) 5.73601 0.255756 0.127878 0.991790i \(-0.459183\pi\)
0.127878 + 0.991790i \(0.459183\pi\)
\(504\) −1.32258 −0.0589124
\(505\) −7.22671 −0.321584
\(506\) −39.9885 −1.77771
\(507\) 0 0
\(508\) 11.4718 0.508980
\(509\) −15.1415 −0.671136 −0.335568 0.942016i \(-0.608928\pi\)
−0.335568 + 0.942016i \(0.608928\pi\)
\(510\) −4.00000 −0.177123
\(511\) −8.30983 −0.367605
\(512\) −1.00000 −0.0441942
\(513\) −2.29078 −0.101140
\(514\) −8.38494 −0.369844
\(515\) 18.4000 0.810802
\(516\) 8.60562 0.378841
\(517\) −42.0243 −1.84822
\(518\) 9.00612 0.395706
\(519\) 2.52817 0.110974
\(520\) 0 0
\(521\) 8.93027 0.391242 0.195621 0.980680i \(-0.437328\pi\)
0.195621 + 0.980680i \(0.437328\pi\)
\(522\) 2.02283 0.0885367
\(523\) −9.81687 −0.429262 −0.214631 0.976695i \(-0.568855\pi\)
−0.214631 + 0.976695i \(0.568855\pi\)
\(524\) −15.9906 −0.698551
\(525\) 1.32258 0.0577221
\(526\) −1.92130 −0.0837725
\(527\) 40.5283 1.76544
\(528\) 4.61335 0.200771
\(529\) 52.1340 2.26669
\(530\) −0.826674 −0.0359084
\(531\) 3.14152 0.136330
\(532\) −3.02973 −0.131356
\(533\) 0 0
\(534\) 11.8641 0.513411
\(535\) 3.07180 0.132805
\(536\) 3.18106 0.137401
\(537\) −6.27568 −0.270816
\(538\) 32.3098 1.39298
\(539\) −24.2237 −1.04339
\(540\) 1.00000 0.0430331
\(541\) 3.80826 0.163730 0.0818650 0.996643i \(-0.473912\pi\)
0.0818650 + 0.996643i \(0.473912\pi\)
\(542\) 23.4526 1.00738
\(543\) 12.2830 0.527116
\(544\) −4.00000 −0.171499
\(545\) −13.2267 −0.566570
\(546\) 0 0
\(547\) 45.5847 1.94906 0.974530 0.224257i \(-0.0719954\pi\)
0.974530 + 0.224257i \(0.0719954\pi\)
\(548\) 16.0697 0.686465
\(549\) 0.535898 0.0228716
\(550\) −4.61335 −0.196714
\(551\) 4.63384 0.197408
\(552\) −8.66799 −0.368934
\(553\) 3.92507 0.166911
\(554\) 9.00566 0.382614
\(555\) −6.80951 −0.289047
\(556\) 9.32258 0.395365
\(557\) 36.8591 1.56177 0.780885 0.624674i \(-0.214768\pi\)
0.780885 + 0.624674i \(0.214768\pi\)
\(558\) −10.1321 −0.428925
\(559\) 0 0
\(560\) 1.32258 0.0558892
\(561\) 18.4534 0.779104
\(562\) −1.71696 −0.0724254
\(563\) 33.7739 1.42340 0.711701 0.702483i \(-0.247925\pi\)
0.711701 + 0.702483i \(0.247925\pi\)
\(564\) −9.10926 −0.383569
\(565\) −8.08643 −0.340199
\(566\) 1.54929 0.0651215
\(567\) 1.32258 0.0555431
\(568\) −11.3360 −0.475647
\(569\) −25.4318 −1.06616 −0.533079 0.846065i \(-0.678965\pi\)
−0.533079 + 0.846065i \(0.678965\pi\)
\(570\) 2.29078 0.0959500
\(571\) −13.3682 −0.559443 −0.279722 0.960081i \(-0.590242\pi\)
−0.279722 + 0.960081i \(0.590242\pi\)
\(572\) 0 0
\(573\) −9.74715 −0.407193
\(574\) 6.14359 0.256429
\(575\) 8.66799 0.361480
\(576\) 1.00000 0.0416667
\(577\) 9.57428 0.398582 0.199291 0.979940i \(-0.436136\pi\)
0.199291 + 0.979940i \(0.436136\pi\)
\(578\) 1.00000 0.0415945
\(579\) 13.7626 0.571954
\(580\) −2.02283 −0.0839933
\(581\) −20.9918 −0.870887
\(582\) −8.81894 −0.365557
\(583\) 3.81374 0.157949
\(584\) 6.28304 0.259994
\(585\) 0 0
\(586\) −30.4057 −1.25605
\(587\) 1.40049 0.0578045 0.0289023 0.999582i \(-0.490799\pi\)
0.0289023 + 0.999582i \(0.490799\pi\)
\(588\) −5.25078 −0.216539
\(589\) −23.2103 −0.956365
\(590\) −3.14152 −0.129334
\(591\) −10.1415 −0.417166
\(592\) −6.80951 −0.279869
\(593\) −10.7303 −0.440643 −0.220321 0.975427i \(-0.570711\pi\)
−0.220321 + 0.975427i \(0.570711\pi\)
\(594\) −4.61335 −0.189288
\(595\) 5.29032 0.216882
\(596\) −4.08519 −0.167336
\(597\) 9.57336 0.391812
\(598\) 0 0
\(599\) −40.7967 −1.66691 −0.833453 0.552590i \(-0.813640\pi\)
−0.833453 + 0.552590i \(0.813640\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 42.8832 1.74924 0.874621 0.484808i \(-0.161110\pi\)
0.874621 + 0.484808i \(0.161110\pi\)
\(602\) −11.3816 −0.463880
\(603\) −3.18106 −0.129543
\(604\) −19.9811 −0.813021
\(605\) 10.2830 0.418065
\(606\) 7.22671 0.293565
\(607\) −6.87233 −0.278939 −0.139470 0.990226i \(-0.544540\pi\)
−0.139470 + 0.990226i \(0.544540\pi\)
\(608\) 2.29078 0.0929032
\(609\) −2.67535 −0.108411
\(610\) −0.535898 −0.0216979
\(611\) 0 0
\(612\) 4.00000 0.161690
\(613\) −0.761361 −0.0307511 −0.0153755 0.999882i \(-0.504894\pi\)
−0.0153755 + 0.999882i \(0.504894\pi\)
\(614\) −9.82622 −0.396554
\(615\) −4.64516 −0.187311
\(616\) −6.10153 −0.245838
\(617\) −24.6022 −0.990448 −0.495224 0.868765i \(-0.664914\pi\)
−0.495224 + 0.868765i \(0.664914\pi\)
\(618\) −18.4000 −0.740158
\(619\) 17.2035 0.691468 0.345734 0.938333i \(-0.387630\pi\)
0.345734 + 0.938333i \(0.387630\pi\)
\(620\) 10.1321 0.406914
\(621\) 8.66799 0.347834
\(622\) −3.40049 −0.136347
\(623\) −15.6913 −0.628657
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 16.2340 0.648841
\(627\) −10.5682 −0.422052
\(628\) 7.13379 0.284669
\(629\) −27.2380 −1.08605
\(630\) −1.32258 −0.0526928
\(631\) 2.40777 0.0958517 0.0479259 0.998851i \(-0.484739\pi\)
0.0479259 + 0.998851i \(0.484739\pi\)
\(632\) −2.96774 −0.118050
\(633\) −1.08519 −0.0431323
\(634\) 23.5231 0.934223
\(635\) 11.4718 0.455246
\(636\) 0.826674 0.0327797
\(637\) 0 0
\(638\) 9.33201 0.369458
\(639\) 11.3360 0.448444
\(640\) −1.00000 −0.0395285
\(641\) −19.4775 −0.769315 −0.384657 0.923059i \(-0.625680\pi\)
−0.384657 + 0.923059i \(0.625680\pi\)
\(642\) −3.07180 −0.121234
\(643\) 23.9543 0.944667 0.472334 0.881420i \(-0.343412\pi\)
0.472334 + 0.881420i \(0.343412\pi\)
\(644\) 11.4641 0.451749
\(645\) 8.60562 0.338846
\(646\) 9.16310 0.360517
\(647\) 18.2108 0.715940 0.357970 0.933733i \(-0.383469\pi\)
0.357970 + 0.933733i \(0.383469\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 14.4930 0.568898
\(650\) 0 0
\(651\) 13.4005 0.525207
\(652\) −9.89470 −0.387506
\(653\) −48.1585 −1.88459 −0.942294 0.334787i \(-0.891336\pi\)
−0.942294 + 0.334787i \(0.891336\pi\)
\(654\) 13.2267 0.517205
\(655\) −15.9906 −0.624803
\(656\) −4.64516 −0.181363
\(657\) −6.28304 −0.245125
\(658\) 12.0477 0.469669
\(659\) −36.8254 −1.43451 −0.717257 0.696809i \(-0.754603\pi\)
−0.717257 + 0.696809i \(0.754603\pi\)
\(660\) 4.61335 0.179575
\(661\) 22.5318 0.876384 0.438192 0.898881i \(-0.355619\pi\)
0.438192 + 0.898881i \(0.355619\pi\)
\(662\) 18.0904 0.703103
\(663\) 0 0
\(664\) 15.8719 0.615948
\(665\) −3.02973 −0.117488
\(666\) 6.80951 0.263863
\(667\) −17.5338 −0.678912
\(668\) −24.0375 −0.930037
\(669\) 24.9416 0.964298
\(670\) 3.18106 0.122895
\(671\) 2.47229 0.0954417
\(672\) −1.32258 −0.0510196
\(673\) 23.7437 0.915254 0.457627 0.889144i \(-0.348700\pi\)
0.457627 + 0.889144i \(0.348700\pi\)
\(674\) −5.69900 −0.219517
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 1.16559 0.0447975 0.0223987 0.999749i \(-0.492870\pi\)
0.0223987 + 0.999749i \(0.492870\pi\)
\(678\) 8.08643 0.310558
\(679\) 11.6638 0.447614
\(680\) −4.00000 −0.153393
\(681\) 19.3205 0.740363
\(682\) −46.7429 −1.78988
\(683\) 13.8719 0.530792 0.265396 0.964139i \(-0.414497\pi\)
0.265396 + 0.964139i \(0.414497\pi\)
\(684\) −2.29078 −0.0875900
\(685\) 16.0697 0.613993
\(686\) 16.2026 0.618620
\(687\) 4.15491 0.158520
\(688\) 8.60562 0.328086
\(689\) 0 0
\(690\) −8.66799 −0.329985
\(691\) −41.2190 −1.56804 −0.784022 0.620733i \(-0.786835\pi\)
−0.784022 + 0.620733i \(0.786835\pi\)
\(692\) 2.52817 0.0961065
\(693\) 6.10153 0.231778
\(694\) 15.7471 0.597753
\(695\) 9.32258 0.353626
\(696\) 2.02283 0.0766750
\(697\) −18.5806 −0.703792
\(698\) 15.3205 0.579890
\(699\) −16.9665 −0.641732
\(700\) 1.32258 0.0499888
\(701\) 23.0112 0.869122 0.434561 0.900642i \(-0.356904\pi\)
0.434561 + 0.900642i \(0.356904\pi\)
\(702\) 0 0
\(703\) 15.5991 0.588329
\(704\) 4.61335 0.173872
\(705\) −9.10926 −0.343075
\(706\) −1.19993 −0.0451599
\(707\) −9.55790 −0.359462
\(708\) 3.14152 0.118066
\(709\) −17.0831 −0.641570 −0.320785 0.947152i \(-0.603947\pi\)
−0.320785 + 0.947152i \(0.603947\pi\)
\(710\) −11.3360 −0.425431
\(711\) 2.96774 0.111299
\(712\) 11.8641 0.444627
\(713\) 87.8248 3.28906
\(714\) −5.29032 −0.197985
\(715\) 0 0
\(716\) −6.27568 −0.234533
\(717\) 6.34416 0.236927
\(718\) −16.2830 −0.607678
\(719\) −43.7128 −1.63021 −0.815106 0.579311i \(-0.803322\pi\)
−0.815106 + 0.579311i \(0.803322\pi\)
\(720\) 1.00000 0.0372678
\(721\) 24.3355 0.906302
\(722\) 13.7523 0.511809
\(723\) 24.1855 0.899467
\(724\) 12.2830 0.456496
\(725\) −2.02283 −0.0751259
\(726\) −10.2830 −0.381640
\(727\) −31.8453 −1.18108 −0.590538 0.807010i \(-0.701084\pi\)
−0.590538 + 0.807010i \(0.701084\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 6.28304 0.232546
\(731\) 34.4225 1.27316
\(732\) 0.535898 0.0198074
\(733\) 5.96774 0.220423 0.110212 0.993908i \(-0.464847\pi\)
0.110212 + 0.993908i \(0.464847\pi\)
\(734\) 19.6345 0.724722
\(735\) −5.25078 −0.193678
\(736\) −8.66799 −0.319506
\(737\) −14.6753 −0.540573
\(738\) 4.64516 0.170991
\(739\) −10.8292 −0.398357 −0.199179 0.979963i \(-0.563827\pi\)
−0.199179 + 0.979963i \(0.563827\pi\)
\(740\) −6.80951 −0.250322
\(741\) 0 0
\(742\) −1.09334 −0.0401378
\(743\) 19.8503 0.728237 0.364118 0.931353i \(-0.381370\pi\)
0.364118 + 0.931353i \(0.381370\pi\)
\(744\) −10.1321 −0.371460
\(745\) −4.08519 −0.149670
\(746\) 2.55748 0.0936359
\(747\) −15.8719 −0.580721
\(748\) 18.4534 0.674724
\(749\) 4.06270 0.148448
\(750\) −1.00000 −0.0365148
\(751\) −15.3308 −0.559428 −0.279714 0.960083i \(-0.590240\pi\)
−0.279714 + 0.960083i \(0.590240\pi\)
\(752\) −9.10926 −0.332181
\(753\) 8.82497 0.321600
\(754\) 0 0
\(755\) −19.9811 −0.727188
\(756\) 1.32258 0.0481018
\(757\) 45.1633 1.64149 0.820744 0.571297i \(-0.193559\pi\)
0.820744 + 0.571297i \(0.193559\pi\)
\(758\) 15.1088 0.548776
\(759\) 39.9885 1.45149
\(760\) 2.29078 0.0830952
\(761\) −21.7058 −0.786835 −0.393418 0.919360i \(-0.628707\pi\)
−0.393418 + 0.919360i \(0.628707\pi\)
\(762\) −11.4718 −0.415581
\(763\) −17.4934 −0.633303
\(764\) −9.74715 −0.352639
\(765\) 4.00000 0.144620
\(766\) −18.7303 −0.676755
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 42.1132 1.51864 0.759321 0.650717i \(-0.225531\pi\)
0.759321 + 0.650717i \(0.225531\pi\)
\(770\) −6.10153 −0.219884
\(771\) 8.38494 0.301976
\(772\) 13.7626 0.495327
\(773\) −29.7605 −1.07041 −0.535206 0.844722i \(-0.679766\pi\)
−0.535206 + 0.844722i \(0.679766\pi\)
\(774\) −8.60562 −0.309323
\(775\) 10.1321 0.363955
\(776\) −8.81894 −0.316582
\(777\) −9.00612 −0.323093
\(778\) 6.96899 0.249850
\(779\) 10.6410 0.381254
\(780\) 0 0
\(781\) 52.2969 1.87133
\(782\) −34.6719 −1.23987
\(783\) −2.02283 −0.0722899
\(784\) −5.25078 −0.187528
\(785\) 7.13379 0.254616
\(786\) 15.9906 0.570365
\(787\) −41.0448 −1.46309 −0.731545 0.681793i \(-0.761200\pi\)
−0.731545 + 0.681793i \(0.761200\pi\)
\(788\) −10.1415 −0.361277
\(789\) 1.92130 0.0684000
\(790\) −2.96774 −0.105587
\(791\) −10.6950 −0.380269
\(792\) −4.61335 −0.163928
\(793\) 0 0
\(794\) 18.5721 0.659100
\(795\) 0.826674 0.0293191
\(796\) 9.57336 0.339319
\(797\) 45.2526 1.60293 0.801464 0.598043i \(-0.204055\pi\)
0.801464 + 0.598043i \(0.204055\pi\)
\(798\) 3.02973 0.107251
\(799\) −36.4370 −1.28905
\(800\) −1.00000 −0.0353553
\(801\) −11.8641 −0.419199
\(802\) −29.6904 −1.04840
\(803\) −28.9859 −1.02289
\(804\) −3.18106 −0.112187
\(805\) 11.4641 0.404056
\(806\) 0 0
\(807\) −32.3098 −1.13736
\(808\) 7.22671 0.254235
\(809\) −28.4534 −1.00037 −0.500184 0.865919i \(-0.666734\pi\)
−0.500184 + 0.865919i \(0.666734\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 44.4114 1.55949 0.779747 0.626095i \(-0.215348\pi\)
0.779747 + 0.626095i \(0.215348\pi\)
\(812\) −2.67535 −0.0938863
\(813\) −23.4526 −0.822518
\(814\) 31.4147 1.10108
\(815\) −9.89470 −0.346596
\(816\) 4.00000 0.140028
\(817\) −19.7135 −0.689690
\(818\) −24.3355 −0.850871
\(819\) 0 0
\(820\) −4.64516 −0.162216
\(821\) −45.1683 −1.57638 −0.788192 0.615429i \(-0.788983\pi\)
−0.788192 + 0.615429i \(0.788983\pi\)
\(822\) −16.0697 −0.560496
\(823\) 2.41799 0.0842859 0.0421430 0.999112i \(-0.486582\pi\)
0.0421430 + 0.999112i \(0.486582\pi\)
\(824\) −18.4000 −0.640996
\(825\) 4.61335 0.160616
\(826\) −4.15491 −0.144568
\(827\) 4.26832 0.148424 0.0742120 0.997242i \(-0.476356\pi\)
0.0742120 + 0.997242i \(0.476356\pi\)
\(828\) 8.66799 0.301233
\(829\) −42.9852 −1.49294 −0.746468 0.665421i \(-0.768252\pi\)
−0.746468 + 0.665421i \(0.768252\pi\)
\(830\) 15.8719 0.550921
\(831\) −9.00566 −0.312403
\(832\) 0 0
\(833\) −21.0031 −0.727715
\(834\) −9.32258 −0.322815
\(835\) −24.0375 −0.831851
\(836\) −10.5682 −0.365507
\(837\) 10.1321 0.350216
\(838\) 2.88255 0.0995761
\(839\) −23.3016 −0.804462 −0.402231 0.915538i \(-0.631765\pi\)
−0.402231 + 0.915538i \(0.631765\pi\)
\(840\) −1.32258 −0.0456333
\(841\) −24.9082 −0.858903
\(842\) −4.94707 −0.170487
\(843\) 1.71696 0.0591351
\(844\) −1.08519 −0.0373537
\(845\) 0 0
\(846\) 9.10926 0.313183
\(847\) 13.6001 0.467307
\(848\) 0.826674 0.0283881
\(849\) −1.54929 −0.0531715
\(850\) −4.00000 −0.137199
\(851\) −59.0247 −2.02334
\(852\) 11.3360 0.388364
\(853\) 20.8418 0.713609 0.356804 0.934179i \(-0.383866\pi\)
0.356804 + 0.934179i \(0.383866\pi\)
\(854\) −0.708768 −0.0242536
\(855\) −2.29078 −0.0783429
\(856\) −3.07180 −0.104992
\(857\) −36.7250 −1.25450 −0.627250 0.778818i \(-0.715820\pi\)
−0.627250 + 0.778818i \(0.715820\pi\)
\(858\) 0 0
\(859\) 0.914812 0.0312130 0.0156065 0.999878i \(-0.495032\pi\)
0.0156065 + 0.999878i \(0.495032\pi\)
\(860\) 8.60562 0.293449
\(861\) −6.14359 −0.209373
\(862\) 17.7626 0.604997
\(863\) −33.6104 −1.14411 −0.572056 0.820215i \(-0.693854\pi\)
−0.572056 + 0.820215i \(0.693854\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 2.52817 0.0859603
\(866\) −35.8375 −1.21781
\(867\) −1.00000 −0.0339618
\(868\) 13.4005 0.454842
\(869\) 13.6912 0.464443
\(870\) 2.02283 0.0685802
\(871\) 0 0
\(872\) 13.2267 0.447913
\(873\) 8.81894 0.298476
\(874\) 19.8564 0.671653
\(875\) 1.32258 0.0447114
\(876\) −6.28304 −0.212284
\(877\) 24.3213 0.821273 0.410637 0.911799i \(-0.365307\pi\)
0.410637 + 0.911799i \(0.365307\pi\)
\(878\) −10.0904 −0.340534
\(879\) 30.4057 1.02556
\(880\) 4.61335 0.155516
\(881\) −9.90413 −0.333679 −0.166839 0.985984i \(-0.553356\pi\)
−0.166839 + 0.985984i \(0.553356\pi\)
\(882\) 5.25078 0.176803
\(883\) 43.9718 1.47977 0.739884 0.672734i \(-0.234880\pi\)
0.739884 + 0.672734i \(0.234880\pi\)
\(884\) 0 0
\(885\) 3.14152 0.105601
\(886\) −13.6981 −0.460196
\(887\) 50.5614 1.69769 0.848843 0.528645i \(-0.177300\pi\)
0.848843 + 0.528645i \(0.177300\pi\)
\(888\) 6.80951 0.228512
\(889\) 15.1724 0.508866
\(890\) 11.8641 0.397687
\(891\) 4.61335 0.154553
\(892\) 24.9416 0.835106
\(893\) 20.8673 0.698297
\(894\) 4.08519 0.136629
\(895\) −6.27568 −0.209773
\(896\) −1.32258 −0.0441843
\(897\) 0 0
\(898\) −15.3128 −0.510994
\(899\) −20.4954 −0.683562
\(900\) 1.00000 0.0333333
\(901\) 3.30669 0.110162
\(902\) 21.4298 0.713533
\(903\) 11.3816 0.378757
\(904\) 8.08643 0.268951
\(905\) 12.2830 0.408302
\(906\) 19.9811 0.663829
\(907\) 14.2435 0.472948 0.236474 0.971638i \(-0.424008\pi\)
0.236474 + 0.971638i \(0.424008\pi\)
\(908\) 19.3205 0.641174
\(909\) −7.22671 −0.239695
\(910\) 0 0
\(911\) 21.8108 0.722623 0.361311 0.932445i \(-0.382329\pi\)
0.361311 + 0.932445i \(0.382329\pi\)
\(912\) −2.29078 −0.0758551
\(913\) −73.2226 −2.42331
\(914\) 15.5554 0.514527
\(915\) 0.535898 0.0177163
\(916\) 4.15491 0.137282
\(917\) −21.1488 −0.698395
\(918\) −4.00000 −0.132020
\(919\) −5.30578 −0.175022 −0.0875108 0.996164i \(-0.527891\pi\)
−0.0875108 + 0.996164i \(0.527891\pi\)
\(920\) −8.66799 −0.285775
\(921\) 9.82622 0.323785
\(922\) 15.6431 0.515178
\(923\) 0 0
\(924\) 6.10153 0.200726
\(925\) −6.80951 −0.223895
\(926\) −13.7170 −0.450767
\(927\) 18.4000 0.604336
\(928\) 2.02283 0.0664025
\(929\) 44.3408 1.45477 0.727386 0.686228i \(-0.240735\pi\)
0.727386 + 0.686228i \(0.240735\pi\)
\(930\) −10.1321 −0.332244
\(931\) 12.0284 0.394214
\(932\) −16.9665 −0.555756
\(933\) 3.40049 0.111327
\(934\) −0.943666 −0.0308777
\(935\) 18.4534 0.603491
\(936\) 0 0
\(937\) 38.5283 1.25867 0.629333 0.777136i \(-0.283328\pi\)
0.629333 + 0.777136i \(0.283328\pi\)
\(938\) 4.20720 0.137370
\(939\) −16.2340 −0.529776
\(940\) −9.10926 −0.297111
\(941\) −28.8637 −0.940929 −0.470465 0.882419i \(-0.655914\pi\)
−0.470465 + 0.882419i \(0.655914\pi\)
\(942\) −7.13379 −0.232431
\(943\) −40.2642 −1.31118
\(944\) 3.14152 0.102248
\(945\) 1.32258 0.0430235
\(946\) −39.7008 −1.29078
\(947\) −49.2346 −1.59991 −0.799955 0.600060i \(-0.795143\pi\)
−0.799955 + 0.600060i \(0.795143\pi\)
\(948\) 2.96774 0.0963877
\(949\) 0 0
\(950\) 2.29078 0.0743226
\(951\) −23.5231 −0.762790
\(952\) −5.29032 −0.171460
\(953\) −26.5097 −0.858732 −0.429366 0.903131i \(-0.641263\pi\)
−0.429366 + 0.903131i \(0.641263\pi\)
\(954\) −0.826674 −0.0267645
\(955\) −9.74715 −0.315410
\(956\) 6.34416 0.205185
\(957\) −9.33201 −0.301661
\(958\) −11.1020 −0.358689
\(959\) 21.2535 0.686311
\(960\) 1.00000 0.0322749
\(961\) 71.6592 2.31159
\(962\) 0 0
\(963\) 3.07180 0.0989873
\(964\) 24.1855 0.778962
\(965\) 13.7626 0.443034
\(966\) −11.4641 −0.368851
\(967\) −52.3877 −1.68468 −0.842338 0.538950i \(-0.818821\pi\)
−0.842338 + 0.538950i \(0.818821\pi\)
\(968\) −10.2830 −0.330510
\(969\) −9.16310 −0.294361
\(970\) −8.81894 −0.283159
\(971\) −28.6431 −0.919200 −0.459600 0.888126i \(-0.652007\pi\)
−0.459600 + 0.888126i \(0.652007\pi\)
\(972\) 1.00000 0.0320750
\(973\) 12.3299 0.395277
\(974\) −32.9416 −1.05552
\(975\) 0 0
\(976\) 0.535898 0.0171537
\(977\) 40.5499 1.29731 0.648654 0.761084i \(-0.275332\pi\)
0.648654 + 0.761084i \(0.275332\pi\)
\(978\) 9.89470 0.316397
\(979\) −54.7335 −1.74929
\(980\) −5.25078 −0.167730
\(981\) −13.2267 −0.422296
\(982\) −28.4513 −0.907919
\(983\) 22.4160 0.714958 0.357479 0.933921i \(-0.383636\pi\)
0.357479 + 0.933921i \(0.383636\pi\)
\(984\) 4.64516 0.148082
\(985\) −10.1415 −0.323136
\(986\) 8.09130 0.257680
\(987\) −12.0477 −0.383483
\(988\) 0 0
\(989\) 74.5934 2.37193
\(990\) −4.61335 −0.146622
\(991\) −28.5688 −0.907518 −0.453759 0.891125i \(-0.649917\pi\)
−0.453759 + 0.891125i \(0.649917\pi\)
\(992\) −10.1321 −0.321694
\(993\) −18.0904 −0.574081
\(994\) −14.9927 −0.475540
\(995\) 9.57336 0.303496
\(996\) −15.8719 −0.502919
\(997\) −34.4000 −1.08946 −0.544730 0.838611i \(-0.683368\pi\)
−0.544730 + 0.838611i \(0.683368\pi\)
\(998\) −4.10926 −0.130076
\(999\) −6.80951 −0.215443
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.bz.1.3 4
13.2 odd 12 390.2.bb.c.121.2 8
13.5 odd 4 5070.2.b.ba.1351.7 8
13.7 odd 12 390.2.bb.c.361.2 yes 8
13.8 odd 4 5070.2.b.ba.1351.2 8
13.12 even 2 5070.2.a.ca.1.2 4
39.2 even 12 1170.2.bs.f.901.4 8
39.20 even 12 1170.2.bs.f.361.4 8
65.2 even 12 1950.2.y.k.199.4 8
65.7 even 12 1950.2.y.j.49.1 8
65.28 even 12 1950.2.y.j.199.1 8
65.33 even 12 1950.2.y.k.49.4 8
65.54 odd 12 1950.2.bc.g.901.3 8
65.59 odd 12 1950.2.bc.g.751.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.bb.c.121.2 8 13.2 odd 12
390.2.bb.c.361.2 yes 8 13.7 odd 12
1170.2.bs.f.361.4 8 39.20 even 12
1170.2.bs.f.901.4 8 39.2 even 12
1950.2.y.j.49.1 8 65.7 even 12
1950.2.y.j.199.1 8 65.28 even 12
1950.2.y.k.49.4 8 65.33 even 12
1950.2.y.k.199.4 8 65.2 even 12
1950.2.bc.g.751.3 8 65.59 odd 12
1950.2.bc.g.901.3 8 65.54 odd 12
5070.2.a.bz.1.3 4 1.1 even 1 trivial
5070.2.a.ca.1.2 4 13.12 even 2
5070.2.b.ba.1351.2 8 13.8 odd 4
5070.2.b.ba.1351.7 8 13.5 odd 4