Properties

Label 7098.2.a.bg.1.1
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.56155 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.56155 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.56155 q^{10} +5.56155 q^{11} -1.00000 q^{12} +1.00000 q^{14} +3.56155 q^{15} +1.00000 q^{16} +6.68466 q^{17} -1.00000 q^{18} +1.56155 q^{19} -3.56155 q^{20} +1.00000 q^{21} -5.56155 q^{22} +6.68466 q^{23} +1.00000 q^{24} +7.68466 q^{25} -1.00000 q^{27} -1.00000 q^{28} +1.56155 q^{29} -3.56155 q^{30} +6.24621 q^{31} -1.00000 q^{32} -5.56155 q^{33} -6.68466 q^{34} +3.56155 q^{35} +1.00000 q^{36} +10.6847 q^{37} -1.56155 q^{38} +3.56155 q^{40} -4.00000 q^{41} -1.00000 q^{42} +6.43845 q^{43} +5.56155 q^{44} -3.56155 q^{45} -6.68466 q^{46} -10.2462 q^{47} -1.00000 q^{48} +1.00000 q^{49} -7.68466 q^{50} -6.68466 q^{51} +4.87689 q^{53} +1.00000 q^{54} -19.8078 q^{55} +1.00000 q^{56} -1.56155 q^{57} -1.56155 q^{58} +4.24621 q^{59} +3.56155 q^{60} -1.56155 q^{61} -6.24621 q^{62} -1.00000 q^{63} +1.00000 q^{64} +5.56155 q^{66} +1.12311 q^{67} +6.68466 q^{68} -6.68466 q^{69} -3.56155 q^{70} -9.36932 q^{71} -1.00000 q^{72} +11.5616 q^{73} -10.6847 q^{74} -7.68466 q^{75} +1.56155 q^{76} -5.56155 q^{77} +16.0000 q^{79} -3.56155 q^{80} +1.00000 q^{81} +4.00000 q^{82} +2.00000 q^{83} +1.00000 q^{84} -23.8078 q^{85} -6.43845 q^{86} -1.56155 q^{87} -5.56155 q^{88} -8.00000 q^{89} +3.56155 q^{90} +6.68466 q^{92} -6.24621 q^{93} +10.2462 q^{94} -5.56155 q^{95} +1.00000 q^{96} -10.0000 q^{97} -1.00000 q^{98} +5.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} - 3q^{5} + 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} - 3q^{5} + 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + 3q^{10} + 7q^{11} - 2q^{12} + 2q^{14} + 3q^{15} + 2q^{16} + q^{17} - 2q^{18} - q^{19} - 3q^{20} + 2q^{21} - 7q^{22} + q^{23} + 2q^{24} + 3q^{25} - 2q^{27} - 2q^{28} - q^{29} - 3q^{30} - 4q^{31} - 2q^{32} - 7q^{33} - q^{34} + 3q^{35} + 2q^{36} + 9q^{37} + q^{38} + 3q^{40} - 8q^{41} - 2q^{42} + 17q^{43} + 7q^{44} - 3q^{45} - q^{46} - 4q^{47} - 2q^{48} + 2q^{49} - 3q^{50} - q^{51} + 18q^{53} + 2q^{54} - 19q^{55} + 2q^{56} + q^{57} + q^{58} - 8q^{59} + 3q^{60} + q^{61} + 4q^{62} - 2q^{63} + 2q^{64} + 7q^{66} - 6q^{67} + q^{68} - q^{69} - 3q^{70} + 6q^{71} - 2q^{72} + 19q^{73} - 9q^{74} - 3q^{75} - q^{76} - 7q^{77} + 32q^{79} - 3q^{80} + 2q^{81} + 8q^{82} + 4q^{83} + 2q^{84} - 27q^{85} - 17q^{86} + q^{87} - 7q^{88} - 16q^{89} + 3q^{90} + q^{92} + 4q^{93} + 4q^{94} - 7q^{95} + 2q^{96} - 20q^{97} - 2q^{98} + 7q^{99} + O(q^{100}) \)

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
\(5\) −3.56155 −1.59277 −0.796387 0.604787i \(-0.793258\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.56155 1.12626
\(11\) 5.56155 1.67687 0.838436 0.545001i \(-0.183471\pi\)
0.838436 + 0.545001i \(0.183471\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 1.00000 0.267261
\(15\) 3.56155 0.919589
\(16\) 1.00000 0.250000
\(17\) 6.68466 1.62127 0.810634 0.585553i \(-0.199123\pi\)
0.810634 + 0.585553i \(0.199123\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.56155 0.358245 0.179122 0.983827i \(-0.442674\pi\)
0.179122 + 0.983827i \(0.442674\pi\)
\(20\) −3.56155 −0.796387
\(21\) 1.00000 0.218218
\(22\) −5.56155 −1.18573
\(23\) 6.68466 1.39385 0.696924 0.717145i \(-0.254552\pi\)
0.696924 + 0.717145i \(0.254552\pi\)
\(24\) 1.00000 0.204124
\(25\) 7.68466 1.53693
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −1.00000 −0.188982
\(29\) 1.56155 0.289973 0.144987 0.989434i \(-0.453686\pi\)
0.144987 + 0.989434i \(0.453686\pi\)
\(30\) −3.56155 −0.650248
\(31\) 6.24621 1.12185 0.560926 0.827866i \(-0.310445\pi\)
0.560926 + 0.827866i \(0.310445\pi\)
\(32\) −1.00000 −0.176777
\(33\) −5.56155 −0.968142
\(34\) −6.68466 −1.14641
\(35\) 3.56155 0.602012
\(36\) 1.00000 0.166667
\(37\) 10.6847 1.75655 0.878274 0.478159i \(-0.158696\pi\)
0.878274 + 0.478159i \(0.158696\pi\)
\(38\) −1.56155 −0.253317
\(39\) 0 0
\(40\) 3.56155 0.563131
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) −1.00000 −0.154303
\(43\) 6.43845 0.981854 0.490927 0.871201i \(-0.336658\pi\)
0.490927 + 0.871201i \(0.336658\pi\)
\(44\) 5.56155 0.838436
\(45\) −3.56155 −0.530925
\(46\) −6.68466 −0.985599
\(47\) −10.2462 −1.49456 −0.747282 0.664507i \(-0.768641\pi\)
−0.747282 + 0.664507i \(0.768641\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) −7.68466 −1.08677
\(51\) −6.68466 −0.936039
\(52\) 0 0
\(53\) 4.87689 0.669893 0.334946 0.942237i \(-0.391282\pi\)
0.334946 + 0.942237i \(0.391282\pi\)
\(54\) 1.00000 0.136083
\(55\) −19.8078 −2.67088
\(56\) 1.00000 0.133631
\(57\) −1.56155 −0.206833
\(58\) −1.56155 −0.205042
\(59\) 4.24621 0.552810 0.276405 0.961041i \(-0.410857\pi\)
0.276405 + 0.961041i \(0.410857\pi\)
\(60\) 3.56155 0.459794
\(61\) −1.56155 −0.199936 −0.0999682 0.994991i \(-0.531874\pi\)
−0.0999682 + 0.994991i \(0.531874\pi\)
\(62\) −6.24621 −0.793270
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 5.56155 0.684580
\(67\) 1.12311 0.137209 0.0686046 0.997644i \(-0.478145\pi\)
0.0686046 + 0.997644i \(0.478145\pi\)
\(68\) 6.68466 0.810634
\(69\) −6.68466 −0.804738
\(70\) −3.56155 −0.425687
\(71\) −9.36932 −1.11193 −0.555967 0.831205i \(-0.687652\pi\)
−0.555967 + 0.831205i \(0.687652\pi\)
\(72\) −1.00000 −0.117851
\(73\) 11.5616 1.35318 0.676589 0.736361i \(-0.263458\pi\)
0.676589 + 0.736361i \(0.263458\pi\)
\(74\) −10.6847 −1.24207
\(75\) −7.68466 −0.887348
\(76\) 1.56155 0.179122
\(77\) −5.56155 −0.633798
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) −3.56155 −0.398194
\(81\) 1.00000 0.111111
\(82\) 4.00000 0.441726
\(83\) 2.00000 0.219529 0.109764 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 1.00000 0.109109
\(85\) −23.8078 −2.58231
\(86\) −6.43845 −0.694276
\(87\) −1.56155 −0.167416
\(88\) −5.56155 −0.592864
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 3.56155 0.375421
\(91\) 0 0
\(92\) 6.68466 0.696924
\(93\) −6.24621 −0.647702
\(94\) 10.2462 1.05682
\(95\) −5.56155 −0.570603
\(96\) 1.00000 0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −1.00000 −0.101015
\(99\) 5.56155 0.558957
\(100\) 7.68466 0.768466
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 6.68466 0.661880
\(103\) −1.80776 −0.178124 −0.0890621 0.996026i \(-0.528387\pi\)
−0.0890621 + 0.996026i \(0.528387\pi\)
\(104\) 0 0
\(105\) −3.56155 −0.347572
\(106\) −4.87689 −0.473686
\(107\) −4.87689 −0.471467 −0.235734 0.971818i \(-0.575749\pi\)
−0.235734 + 0.971818i \(0.575749\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −12.9309 −1.23855 −0.619276 0.785173i \(-0.712574\pi\)
−0.619276 + 0.785173i \(0.712574\pi\)
\(110\) 19.8078 1.88860
\(111\) −10.6847 −1.01414
\(112\) −1.00000 −0.0944911
\(113\) 20.2462 1.90460 0.952302 0.305158i \(-0.0987093\pi\)
0.952302 + 0.305158i \(0.0987093\pi\)
\(114\) 1.56155 0.146253
\(115\) −23.8078 −2.22009
\(116\) 1.56155 0.144987
\(117\) 0 0
\(118\) −4.24621 −0.390895
\(119\) −6.68466 −0.612782
\(120\) −3.56155 −0.325124
\(121\) 19.9309 1.81190
\(122\) 1.56155 0.141376
\(123\) 4.00000 0.360668
\(124\) 6.24621 0.560926
\(125\) −9.56155 −0.855211
\(126\) 1.00000 0.0890871
\(127\) −10.2462 −0.909204 −0.454602 0.890695i \(-0.650219\pi\)
−0.454602 + 0.890695i \(0.650219\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −6.43845 −0.566874
\(130\) 0 0
\(131\) −1.56155 −0.136434 −0.0682168 0.997671i \(-0.521731\pi\)
−0.0682168 + 0.997671i \(0.521731\pi\)
\(132\) −5.56155 −0.484071
\(133\) −1.56155 −0.135404
\(134\) −1.12311 −0.0970215
\(135\) 3.56155 0.306530
\(136\) −6.68466 −0.573205
\(137\) 6.68466 0.571109 0.285554 0.958362i \(-0.407822\pi\)
0.285554 + 0.958362i \(0.407822\pi\)
\(138\) 6.68466 0.569036
\(139\) 10.2462 0.869072 0.434536 0.900654i \(-0.356912\pi\)
0.434536 + 0.900654i \(0.356912\pi\)
\(140\) 3.56155 0.301006
\(141\) 10.2462 0.862887
\(142\) 9.36932 0.786256
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −5.56155 −0.461862
\(146\) −11.5616 −0.956841
\(147\) −1.00000 −0.0824786
\(148\) 10.6847 0.878274
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 7.68466 0.627450
\(151\) 16.6847 1.35778 0.678889 0.734241i \(-0.262462\pi\)
0.678889 + 0.734241i \(0.262462\pi\)
\(152\) −1.56155 −0.126659
\(153\) 6.68466 0.540423
\(154\) 5.56155 0.448163
\(155\) −22.2462 −1.78686
\(156\) 0 0
\(157\) −11.8078 −0.942362 −0.471181 0.882037i \(-0.656172\pi\)
−0.471181 + 0.882037i \(0.656172\pi\)
\(158\) −16.0000 −1.27289
\(159\) −4.87689 −0.386763
\(160\) 3.56155 0.281565
\(161\) −6.68466 −0.526825
\(162\) −1.00000 −0.0785674
\(163\) 9.12311 0.714577 0.357288 0.933994i \(-0.383701\pi\)
0.357288 + 0.933994i \(0.383701\pi\)
\(164\) −4.00000 −0.312348
\(165\) 19.8078 1.54203
\(166\) −2.00000 −0.155230
\(167\) −11.8078 −0.913712 −0.456856 0.889541i \(-0.651025\pi\)
−0.456856 + 0.889541i \(0.651025\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 0 0
\(170\) 23.8078 1.82597
\(171\) 1.56155 0.119415
\(172\) 6.43845 0.490927
\(173\) 3.75379 0.285395 0.142698 0.989766i \(-0.454422\pi\)
0.142698 + 0.989766i \(0.454422\pi\)
\(174\) 1.56155 0.118381
\(175\) −7.68466 −0.580906
\(176\) 5.56155 0.419218
\(177\) −4.24621 −0.319165
\(178\) 8.00000 0.599625
\(179\) −11.1231 −0.831380 −0.415690 0.909506i \(-0.636460\pi\)
−0.415690 + 0.909506i \(0.636460\pi\)
\(180\) −3.56155 −0.265462
\(181\) 13.3693 0.993733 0.496867 0.867827i \(-0.334484\pi\)
0.496867 + 0.867827i \(0.334484\pi\)
\(182\) 0 0
\(183\) 1.56155 0.115433
\(184\) −6.68466 −0.492800
\(185\) −38.0540 −2.79778
\(186\) 6.24621 0.457994
\(187\) 37.1771 2.71866
\(188\) −10.2462 −0.747282
\(189\) 1.00000 0.0727393
\(190\) 5.56155 0.403477
\(191\) −24.0540 −1.74048 −0.870242 0.492624i \(-0.836038\pi\)
−0.870242 + 0.492624i \(0.836038\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −12.0000 −0.863779 −0.431889 0.901927i \(-0.642153\pi\)
−0.431889 + 0.901927i \(0.642153\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −5.56155 −0.395242
\(199\) −4.93087 −0.349540 −0.174770 0.984609i \(-0.555918\pi\)
−0.174770 + 0.984609i \(0.555918\pi\)
\(200\) −7.68466 −0.543387
\(201\) −1.12311 −0.0792178
\(202\) 6.00000 0.422159
\(203\) −1.56155 −0.109600
\(204\) −6.68466 −0.468020
\(205\) 14.2462 0.994999
\(206\) 1.80776 0.125953
\(207\) 6.68466 0.464616
\(208\) 0 0
\(209\) 8.68466 0.600730
\(210\) 3.56155 0.245770
\(211\) 7.80776 0.537509 0.268754 0.963209i \(-0.413388\pi\)
0.268754 + 0.963209i \(0.413388\pi\)
\(212\) 4.87689 0.334946
\(213\) 9.36932 0.641975
\(214\) 4.87689 0.333378
\(215\) −22.9309 −1.56387
\(216\) 1.00000 0.0680414
\(217\) −6.24621 −0.424020
\(218\) 12.9309 0.875789
\(219\) −11.5616 −0.781257
\(220\) −19.8078 −1.33544
\(221\) 0 0
\(222\) 10.6847 0.717107
\(223\) 1.75379 0.117442 0.0587212 0.998274i \(-0.481298\pi\)
0.0587212 + 0.998274i \(0.481298\pi\)
\(224\) 1.00000 0.0668153
\(225\) 7.68466 0.512311
\(226\) −20.2462 −1.34676
\(227\) 17.6155 1.16918 0.584592 0.811328i \(-0.301255\pi\)
0.584592 + 0.811328i \(0.301255\pi\)
\(228\) −1.56155 −0.103416
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 23.8078 1.56984
\(231\) 5.56155 0.365923
\(232\) −1.56155 −0.102521
\(233\) −2.00000 −0.131024 −0.0655122 0.997852i \(-0.520868\pi\)
−0.0655122 + 0.997852i \(0.520868\pi\)
\(234\) 0 0
\(235\) 36.4924 2.38050
\(236\) 4.24621 0.276405
\(237\) −16.0000 −1.03931
\(238\) 6.68466 0.433302
\(239\) −14.2462 −0.921511 −0.460755 0.887527i \(-0.652421\pi\)
−0.460755 + 0.887527i \(0.652421\pi\)
\(240\) 3.56155 0.229897
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) −19.9309 −1.28120
\(243\) −1.00000 −0.0641500
\(244\) −1.56155 −0.0999682
\(245\) −3.56155 −0.227539
\(246\) −4.00000 −0.255031
\(247\) 0 0
\(248\) −6.24621 −0.396635
\(249\) −2.00000 −0.126745
\(250\) 9.56155 0.604726
\(251\) −22.0540 −1.39203 −0.696017 0.718025i \(-0.745046\pi\)
−0.696017 + 0.718025i \(0.745046\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 37.1771 2.33730
\(254\) 10.2462 0.642904
\(255\) 23.8078 1.49090
\(256\) 1.00000 0.0625000
\(257\) 26.0000 1.62184 0.810918 0.585160i \(-0.198968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(258\) 6.43845 0.400840
\(259\) −10.6847 −0.663912
\(260\) 0 0
\(261\) 1.56155 0.0966577
\(262\) 1.56155 0.0964731
\(263\) −23.3693 −1.44101 −0.720507 0.693448i \(-0.756091\pi\)
−0.720507 + 0.693448i \(0.756091\pi\)
\(264\) 5.56155 0.342290
\(265\) −17.3693 −1.06699
\(266\) 1.56155 0.0957449
\(267\) 8.00000 0.489592
\(268\) 1.12311 0.0686046
\(269\) 31.3693 1.91262 0.956311 0.292353i \(-0.0944382\pi\)
0.956311 + 0.292353i \(0.0944382\pi\)
\(270\) −3.56155 −0.216749
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 6.68466 0.405317
\(273\) 0 0
\(274\) −6.68466 −0.403835
\(275\) 42.7386 2.57724
\(276\) −6.68466 −0.402369
\(277\) 10.8769 0.653529 0.326765 0.945106i \(-0.394042\pi\)
0.326765 + 0.945106i \(0.394042\pi\)
\(278\) −10.2462 −0.614527
\(279\) 6.24621 0.373951
\(280\) −3.56155 −0.212843
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) −10.2462 −0.610153
\(283\) 4.87689 0.289901 0.144951 0.989439i \(-0.453698\pi\)
0.144951 + 0.989439i \(0.453698\pi\)
\(284\) −9.36932 −0.555967
\(285\) 5.56155 0.329438
\(286\) 0 0
\(287\) 4.00000 0.236113
\(288\) −1.00000 −0.0589256
\(289\) 27.6847 1.62851
\(290\) 5.56155 0.326586
\(291\) 10.0000 0.586210
\(292\) 11.5616 0.676589
\(293\) −7.75379 −0.452981 −0.226491 0.974013i \(-0.572725\pi\)
−0.226491 + 0.974013i \(0.572725\pi\)
\(294\) 1.00000 0.0583212
\(295\) −15.1231 −0.880501
\(296\) −10.6847 −0.621033
\(297\) −5.56155 −0.322714
\(298\) 10.0000 0.579284
\(299\) 0 0
\(300\) −7.68466 −0.443674
\(301\) −6.43845 −0.371106
\(302\) −16.6847 −0.960094
\(303\) 6.00000 0.344691
\(304\) 1.56155 0.0895612
\(305\) 5.56155 0.318454
\(306\) −6.68466 −0.382136
\(307\) 26.2462 1.49795 0.748975 0.662598i \(-0.230546\pi\)
0.748975 + 0.662598i \(0.230546\pi\)
\(308\) −5.56155 −0.316899
\(309\) 1.80776 0.102840
\(310\) 22.2462 1.26350
\(311\) 0.492423 0.0279227 0.0139614 0.999903i \(-0.495556\pi\)
0.0139614 + 0.999903i \(0.495556\pi\)
\(312\) 0 0
\(313\) 32.2462 1.82266 0.911332 0.411673i \(-0.135055\pi\)
0.911332 + 0.411673i \(0.135055\pi\)
\(314\) 11.8078 0.666351
\(315\) 3.56155 0.200671
\(316\) 16.0000 0.900070
\(317\) 3.36932 0.189240 0.0946198 0.995513i \(-0.469836\pi\)
0.0946198 + 0.995513i \(0.469836\pi\)
\(318\) 4.87689 0.273483
\(319\) 8.68466 0.486248
\(320\) −3.56155 −0.199097
\(321\) 4.87689 0.272202
\(322\) 6.68466 0.372521
\(323\) 10.4384 0.580811
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −9.12311 −0.505282
\(327\) 12.9309 0.715079
\(328\) 4.00000 0.220863
\(329\) 10.2462 0.564892
\(330\) −19.8078 −1.09038
\(331\) 1.12311 0.0617315 0.0308657 0.999524i \(-0.490174\pi\)
0.0308657 + 0.999524i \(0.490174\pi\)
\(332\) 2.00000 0.109764
\(333\) 10.6847 0.585516
\(334\) 11.8078 0.646092
\(335\) −4.00000 −0.218543
\(336\) 1.00000 0.0545545
\(337\) −20.0540 −1.09241 −0.546205 0.837652i \(-0.683928\pi\)
−0.546205 + 0.837652i \(0.683928\pi\)
\(338\) 0 0
\(339\) −20.2462 −1.09962
\(340\) −23.8078 −1.29116
\(341\) 34.7386 1.88120
\(342\) −1.56155 −0.0844391
\(343\) −1.00000 −0.0539949
\(344\) −6.43845 −0.347138
\(345\) 23.8078 1.28177
\(346\) −3.75379 −0.201805
\(347\) −24.4924 −1.31482 −0.657411 0.753532i \(-0.728348\pi\)
−0.657411 + 0.753532i \(0.728348\pi\)
\(348\) −1.56155 −0.0837080
\(349\) 3.36932 0.180355 0.0901777 0.995926i \(-0.471256\pi\)
0.0901777 + 0.995926i \(0.471256\pi\)
\(350\) 7.68466 0.410762
\(351\) 0 0
\(352\) −5.56155 −0.296432
\(353\) 18.2462 0.971148 0.485574 0.874196i \(-0.338611\pi\)
0.485574 + 0.874196i \(0.338611\pi\)
\(354\) 4.24621 0.225684
\(355\) 33.3693 1.77106
\(356\) −8.00000 −0.423999
\(357\) 6.68466 0.353790
\(358\) 11.1231 0.587874
\(359\) 23.6155 1.24638 0.623190 0.782071i \(-0.285836\pi\)
0.623190 + 0.782071i \(0.285836\pi\)
\(360\) 3.56155 0.187710
\(361\) −16.5616 −0.871661
\(362\) −13.3693 −0.702676
\(363\) −19.9309 −1.04610
\(364\) 0 0
\(365\) −41.1771 −2.15531
\(366\) −1.56155 −0.0816237
\(367\) 0.630683 0.0329214 0.0164607 0.999865i \(-0.494760\pi\)
0.0164607 + 0.999865i \(0.494760\pi\)
\(368\) 6.68466 0.348462
\(369\) −4.00000 −0.208232
\(370\) 38.0540 1.97833
\(371\) −4.87689 −0.253196
\(372\) −6.24621 −0.323851
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) −37.1771 −1.92238
\(375\) 9.56155 0.493756
\(376\) 10.2462 0.528408
\(377\) 0 0
\(378\) −1.00000 −0.0514344
\(379\) 18.8769 0.969641 0.484820 0.874614i \(-0.338885\pi\)
0.484820 + 0.874614i \(0.338885\pi\)
\(380\) −5.56155 −0.285302
\(381\) 10.2462 0.524929
\(382\) 24.0540 1.23071
\(383\) −17.5616 −0.897353 −0.448677 0.893694i \(-0.648105\pi\)
−0.448677 + 0.893694i \(0.648105\pi\)
\(384\) 1.00000 0.0510310
\(385\) 19.8078 1.00950
\(386\) 12.0000 0.610784
\(387\) 6.43845 0.327285
\(388\) −10.0000 −0.507673
\(389\) −27.1231 −1.37520 −0.687598 0.726092i \(-0.741335\pi\)
−0.687598 + 0.726092i \(0.741335\pi\)
\(390\) 0 0
\(391\) 44.6847 2.25980
\(392\) −1.00000 −0.0505076
\(393\) 1.56155 0.0787699
\(394\) −6.00000 −0.302276
\(395\) −56.9848 −2.86722
\(396\) 5.56155 0.279479
\(397\) 15.3693 0.771364 0.385682 0.922632i \(-0.373966\pi\)
0.385682 + 0.922632i \(0.373966\pi\)
\(398\) 4.93087 0.247162
\(399\) 1.56155 0.0781754
\(400\) 7.68466 0.384233
\(401\) −30.9848 −1.54731 −0.773655 0.633608i \(-0.781573\pi\)
−0.773655 + 0.633608i \(0.781573\pi\)
\(402\) 1.12311 0.0560154
\(403\) 0 0
\(404\) −6.00000 −0.298511
\(405\) −3.56155 −0.176975
\(406\) 1.56155 0.0774986
\(407\) 59.4233 2.94550
\(408\) 6.68466 0.330940
\(409\) −13.8078 −0.682750 −0.341375 0.939927i \(-0.610893\pi\)
−0.341375 + 0.939927i \(0.610893\pi\)
\(410\) −14.2462 −0.703570
\(411\) −6.68466 −0.329730
\(412\) −1.80776 −0.0890621
\(413\) −4.24621 −0.208942
\(414\) −6.68466 −0.328533
\(415\) −7.12311 −0.349660
\(416\) 0 0
\(417\) −10.2462 −0.501759
\(418\) −8.68466 −0.424781
\(419\) −20.6847 −1.01051 −0.505256 0.862970i \(-0.668602\pi\)
−0.505256 + 0.862970i \(0.668602\pi\)
\(420\) −3.56155 −0.173786
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) −7.80776 −0.380076
\(423\) −10.2462 −0.498188
\(424\) −4.87689 −0.236843
\(425\) 51.3693 2.49178
\(426\) −9.36932 −0.453945
\(427\) 1.56155 0.0755688
\(428\) −4.87689 −0.235734
\(429\) 0 0
\(430\) 22.9309 1.10582
\(431\) −29.8617 −1.43839 −0.719195 0.694809i \(-0.755489\pi\)
−0.719195 + 0.694809i \(0.755489\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −23.3693 −1.12306 −0.561529 0.827457i \(-0.689787\pi\)
−0.561529 + 0.827457i \(0.689787\pi\)
\(434\) 6.24621 0.299828
\(435\) 5.56155 0.266656
\(436\) −12.9309 −0.619276
\(437\) 10.4384 0.499339
\(438\) 11.5616 0.552432
\(439\) 8.93087 0.426247 0.213124 0.977025i \(-0.431636\pi\)
0.213124 + 0.977025i \(0.431636\pi\)
\(440\) 19.8078 0.944298
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 6.63068 0.315033 0.157517 0.987516i \(-0.449651\pi\)
0.157517 + 0.987516i \(0.449651\pi\)
\(444\) −10.6847 −0.507071
\(445\) 28.4924 1.35067
\(446\) −1.75379 −0.0830443
\(447\) 10.0000 0.472984
\(448\) −1.00000 −0.0472456
\(449\) −9.31534 −0.439618 −0.219809 0.975543i \(-0.570543\pi\)
−0.219809 + 0.975543i \(0.570543\pi\)
\(450\) −7.68466 −0.362258
\(451\) −22.2462 −1.04753
\(452\) 20.2462 0.952302
\(453\) −16.6847 −0.783914
\(454\) −17.6155 −0.826738
\(455\) 0 0
\(456\) 1.56155 0.0731264
\(457\) 16.0000 0.748448 0.374224 0.927338i \(-0.377909\pi\)
0.374224 + 0.927338i \(0.377909\pi\)
\(458\) −2.00000 −0.0934539
\(459\) −6.68466 −0.312013
\(460\) −23.8078 −1.11004
\(461\) −36.9309 −1.72004 −0.860021 0.510259i \(-0.829550\pi\)
−0.860021 + 0.510259i \(0.829550\pi\)
\(462\) −5.56155 −0.258747
\(463\) −30.5464 −1.41961 −0.709806 0.704397i \(-0.751217\pi\)
−0.709806 + 0.704397i \(0.751217\pi\)
\(464\) 1.56155 0.0724933
\(465\) 22.2462 1.03164
\(466\) 2.00000 0.0926482
\(467\) −1.56155 −0.0722600 −0.0361300 0.999347i \(-0.511503\pi\)
−0.0361300 + 0.999347i \(0.511503\pi\)
\(468\) 0 0
\(469\) −1.12311 −0.0518602
\(470\) −36.4924 −1.68327
\(471\) 11.8078 0.544073
\(472\) −4.24621 −0.195448
\(473\) 35.8078 1.64644
\(474\) 16.0000 0.734904
\(475\) 12.0000 0.550598
\(476\) −6.68466 −0.306391
\(477\) 4.87689 0.223298
\(478\) 14.2462 0.651607
\(479\) 12.3002 0.562010 0.281005 0.959706i \(-0.409332\pi\)
0.281005 + 0.959706i \(0.409332\pi\)
\(480\) −3.56155 −0.162562
\(481\) 0 0
\(482\) −6.00000 −0.273293
\(483\) 6.68466 0.304162
\(484\) 19.9309 0.905949
\(485\) 35.6155 1.61722
\(486\) 1.00000 0.0453609
\(487\) 13.7538 0.623244 0.311622 0.950206i \(-0.399128\pi\)
0.311622 + 0.950206i \(0.399128\pi\)
\(488\) 1.56155 0.0706882
\(489\) −9.12311 −0.412561
\(490\) 3.56155 0.160895
\(491\) 10.2462 0.462405 0.231203 0.972906i \(-0.425734\pi\)
0.231203 + 0.972906i \(0.425734\pi\)
\(492\) 4.00000 0.180334
\(493\) 10.4384 0.470124
\(494\) 0 0
\(495\) −19.8078 −0.890293
\(496\) 6.24621 0.280463
\(497\) 9.36932 0.420271
\(498\) 2.00000 0.0896221
\(499\) 1.50758 0.0674884 0.0337442 0.999431i \(-0.489257\pi\)
0.0337442 + 0.999431i \(0.489257\pi\)
\(500\) −9.56155 −0.427606
\(501\) 11.8078 0.527532
\(502\) 22.0540 0.984317
\(503\) 35.6155 1.58802 0.794009 0.607906i \(-0.207990\pi\)
0.794009 + 0.607906i \(0.207990\pi\)
\(504\) 1.00000 0.0445435
\(505\) 21.3693 0.950922
\(506\) −37.1771 −1.65272
\(507\) 0 0
\(508\) −10.2462 −0.454602
\(509\) −2.68466 −0.118995 −0.0594977 0.998228i \(-0.518950\pi\)
−0.0594977 + 0.998228i \(0.518950\pi\)
\(510\) −23.8078 −1.05423
\(511\) −11.5616 −0.511453
\(512\) −1.00000 −0.0441942
\(513\) −1.56155 −0.0689442
\(514\) −26.0000 −1.14681
\(515\) 6.43845 0.283712
\(516\) −6.43845 −0.283437
\(517\) −56.9848 −2.50619
\(518\) 10.6847 0.469457
\(519\) −3.75379 −0.164773
\(520\) 0 0
\(521\) 8.05398 0.352851 0.176426 0.984314i \(-0.443547\pi\)
0.176426 + 0.984314i \(0.443547\pi\)
\(522\) −1.56155 −0.0683473
\(523\) −8.49242 −0.371348 −0.185674 0.982611i \(-0.559447\pi\)
−0.185674 + 0.982611i \(0.559447\pi\)
\(524\) −1.56155 −0.0682168
\(525\) 7.68466 0.335386
\(526\) 23.3693 1.01895
\(527\) 41.7538 1.81882
\(528\) −5.56155 −0.242036
\(529\) 21.6847 0.942811
\(530\) 17.3693 0.754475
\(531\) 4.24621 0.184270
\(532\) −1.56155 −0.0677019
\(533\) 0 0
\(534\) −8.00000 −0.346194
\(535\) 17.3693 0.750941
\(536\) −1.12311 −0.0485108
\(537\) 11.1231 0.479997
\(538\) −31.3693 −1.35243
\(539\) 5.56155 0.239553
\(540\) 3.56155 0.153265
\(541\) −6.19224 −0.266225 −0.133113 0.991101i \(-0.542497\pi\)
−0.133113 + 0.991101i \(0.542497\pi\)
\(542\) 8.00000 0.343629
\(543\) −13.3693 −0.573732
\(544\) −6.68466 −0.286602
\(545\) 46.0540 1.97274
\(546\) 0 0
\(547\) 28.9848 1.23930 0.619651 0.784877i \(-0.287274\pi\)
0.619651 + 0.784877i \(0.287274\pi\)
\(548\) 6.68466 0.285554
\(549\) −1.56155 −0.0666455
\(550\) −42.7386 −1.82238
\(551\) 2.43845 0.103881
\(552\) 6.68466 0.284518
\(553\) −16.0000 −0.680389
\(554\) −10.8769 −0.462115
\(555\) 38.0540 1.61530
\(556\) 10.2462 0.434536
\(557\) 16.2462 0.688374 0.344187 0.938901i \(-0.388155\pi\)
0.344187 + 0.938901i \(0.388155\pi\)
\(558\) −6.24621 −0.264423
\(559\) 0 0
\(560\) 3.56155 0.150503
\(561\) −37.1771 −1.56962
\(562\) −6.00000 −0.253095
\(563\) −38.0540 −1.60378 −0.801892 0.597469i \(-0.796173\pi\)
−0.801892 + 0.597469i \(0.796173\pi\)
\(564\) 10.2462 0.431443
\(565\) −72.1080 −3.03360
\(566\) −4.87689 −0.204991
\(567\) −1.00000 −0.0419961
\(568\) 9.36932 0.393128
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) −5.56155 −0.232948
\(571\) 18.2462 0.763580 0.381790 0.924249i \(-0.375308\pi\)
0.381790 + 0.924249i \(0.375308\pi\)
\(572\) 0 0
\(573\) 24.0540 1.00487
\(574\) −4.00000 −0.166957
\(575\) 51.3693 2.14225
\(576\) 1.00000 0.0416667
\(577\) −7.75379 −0.322794 −0.161397 0.986890i \(-0.551600\pi\)
−0.161397 + 0.986890i \(0.551600\pi\)
\(578\) −27.6847 −1.15153
\(579\) 12.0000 0.498703
\(580\) −5.56155 −0.230931
\(581\) −2.00000 −0.0829740
\(582\) −10.0000 −0.414513
\(583\) 27.1231 1.12332
\(584\) −11.5616 −0.478420
\(585\) 0 0
\(586\) 7.75379 0.320306
\(587\) 37.1231 1.53223 0.766117 0.642701i \(-0.222186\pi\)
0.766117 + 0.642701i \(0.222186\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 9.75379 0.401898
\(590\) 15.1231 0.622608
\(591\) −6.00000 −0.246807
\(592\) 10.6847 0.439137
\(593\) −28.0000 −1.14982 −0.574911 0.818216i \(-0.694963\pi\)
−0.574911 + 0.818216i \(0.694963\pi\)
\(594\) 5.56155 0.228193
\(595\) 23.8078 0.976023
\(596\) −10.0000 −0.409616
\(597\) 4.93087 0.201807
\(598\) 0 0
\(599\) 30.3002 1.23803 0.619016 0.785378i \(-0.287532\pi\)
0.619016 + 0.785378i \(0.287532\pi\)
\(600\) 7.68466 0.313725
\(601\) −31.8617 −1.29967 −0.649834 0.760076i \(-0.725161\pi\)
−0.649834 + 0.760076i \(0.725161\pi\)
\(602\) 6.43845 0.262412
\(603\) 1.12311 0.0457364
\(604\) 16.6847 0.678889
\(605\) −70.9848 −2.88594
\(606\) −6.00000 −0.243733
\(607\) −28.5464 −1.15866 −0.579331 0.815092i \(-0.696686\pi\)
−0.579331 + 0.815092i \(0.696686\pi\)
\(608\) −1.56155 −0.0633293
\(609\) 1.56155 0.0632773
\(610\) −5.56155 −0.225181
\(611\) 0 0
\(612\) 6.68466 0.270211
\(613\) 1.80776 0.0730149 0.0365075 0.999333i \(-0.488377\pi\)
0.0365075 + 0.999333i \(0.488377\pi\)
\(614\) −26.2462 −1.05921
\(615\) −14.2462 −0.574463
\(616\) 5.56155 0.224081
\(617\) −6.19224 −0.249290 −0.124645 0.992201i \(-0.539779\pi\)
−0.124645 + 0.992201i \(0.539779\pi\)
\(618\) −1.80776 −0.0727189
\(619\) −30.9309 −1.24322 −0.621608 0.783328i \(-0.713520\pi\)
−0.621608 + 0.783328i \(0.713520\pi\)
\(620\) −22.2462 −0.893429
\(621\) −6.68466 −0.268246
\(622\) −0.492423 −0.0197443
\(623\) 8.00000 0.320513
\(624\) 0 0
\(625\) −4.36932 −0.174773
\(626\) −32.2462 −1.28882
\(627\) −8.68466 −0.346832
\(628\) −11.8078 −0.471181
\(629\) 71.4233 2.84783
\(630\) −3.56155 −0.141896
\(631\) 12.6847 0.504968 0.252484 0.967601i \(-0.418752\pi\)
0.252484 + 0.967601i \(0.418752\pi\)
\(632\) −16.0000 −0.636446
\(633\) −7.80776 −0.310331
\(634\) −3.36932 −0.133813
\(635\) 36.4924 1.44816
\(636\) −4.87689 −0.193381
\(637\) 0 0
\(638\) −8.68466 −0.343829
\(639\) −9.36932 −0.370644
\(640\) 3.56155 0.140783
\(641\) −30.1080 −1.18919 −0.594596 0.804024i \(-0.702688\pi\)
−0.594596 + 0.804024i \(0.702688\pi\)
\(642\) −4.87689 −0.192476
\(643\) 0.192236 0.00758105 0.00379052 0.999993i \(-0.498793\pi\)
0.00379052 + 0.999993i \(0.498793\pi\)
\(644\) −6.68466 −0.263412
\(645\) 22.9309 0.902902
\(646\) −10.4384 −0.410695
\(647\) −10.6307 −0.417935 −0.208968 0.977923i \(-0.567010\pi\)
−0.208968 + 0.977923i \(0.567010\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 23.6155 0.926991
\(650\) 0 0
\(651\) 6.24621 0.244808
\(652\) 9.12311 0.357288
\(653\) −29.5616 −1.15683 −0.578416 0.815742i \(-0.696329\pi\)
−0.578416 + 0.815742i \(0.696329\pi\)
\(654\) −12.9309 −0.505637
\(655\) 5.56155 0.217308
\(656\) −4.00000 −0.156174
\(657\) 11.5616 0.451059
\(658\) −10.2462 −0.399439
\(659\) 12.8769 0.501613 0.250806 0.968037i \(-0.419304\pi\)
0.250806 + 0.968037i \(0.419304\pi\)
\(660\) 19.8078 0.771016
\(661\) −44.7386 −1.74013 −0.870066 0.492936i \(-0.835924\pi\)
−0.870066 + 0.492936i \(0.835924\pi\)
\(662\) −1.12311 −0.0436507
\(663\) 0 0
\(664\) −2.00000 −0.0776151
\(665\) 5.56155 0.215668
\(666\) −10.6847 −0.414022
\(667\) 10.4384 0.404178
\(668\) −11.8078 −0.456856
\(669\) −1.75379 −0.0678054
\(670\) 4.00000 0.154533
\(671\) −8.68466 −0.335268
\(672\) −1.00000 −0.0385758
\(673\) −37.8078 −1.45738 −0.728691 0.684843i \(-0.759871\pi\)
−0.728691 + 0.684843i \(0.759871\pi\)
\(674\) 20.0540 0.772450
\(675\) −7.68466 −0.295783
\(676\) 0 0
\(677\) 31.3693 1.20562 0.602810 0.797884i \(-0.294048\pi\)
0.602810 + 0.797884i \(0.294048\pi\)
\(678\) 20.2462 0.777551
\(679\) 10.0000 0.383765
\(680\) 23.8078 0.912986
\(681\) −17.6155 −0.675029
\(682\) −34.7386 −1.33021
\(683\) 0.684658 0.0261977 0.0130989 0.999914i \(-0.495830\pi\)
0.0130989 + 0.999914i \(0.495830\pi\)
\(684\) 1.56155 0.0597075
\(685\) −23.8078 −0.909648
\(686\) 1.00000 0.0381802
\(687\) −2.00000 −0.0763048
\(688\) 6.43845 0.245463
\(689\) 0 0
\(690\) −23.8078 −0.906346
\(691\) −36.4924 −1.38824 −0.694119 0.719861i \(-0.744206\pi\)
−0.694119 + 0.719861i \(0.744206\pi\)
\(692\) 3.75379 0.142698
\(693\) −5.56155 −0.211266
\(694\) 24.4924 0.929720
\(695\) −36.4924 −1.38424
\(696\) 1.56155 0.0591905
\(697\) −26.7386 −1.01280
\(698\) −3.36932 −0.127531
\(699\) 2.00000 0.0756469
\(700\) −7.68466 −0.290453
\(701\) 3.61553 0.136557 0.0682783 0.997666i \(-0.478249\pi\)
0.0682783 + 0.997666i \(0.478249\pi\)
\(702\) 0 0
\(703\) 16.6847 0.629274
\(704\) 5.56155 0.209609
\(705\) −36.4924 −1.37438
\(706\) −18.2462 −0.686705
\(707\) 6.00000 0.225653
\(708\) −4.24621 −0.159582
\(709\) 31.7538 1.19254 0.596269 0.802784i \(-0.296649\pi\)
0.596269 + 0.802784i \(0.296649\pi\)
\(710\) −33.3693 −1.25233
\(711\) 16.0000 0.600047
\(712\) 8.00000 0.299813
\(713\) 41.7538 1.56369
\(714\) −6.68466 −0.250167
\(715\) 0 0
\(716\) −11.1231 −0.415690
\(717\) 14.2462 0.532035
\(718\) −23.6155 −0.881324
\(719\) −13.7538 −0.512930 −0.256465 0.966554i \(-0.582558\pi\)
−0.256465 + 0.966554i \(0.582558\pi\)
\(720\) −3.56155 −0.132731
\(721\) 1.80776 0.0673247
\(722\) 16.5616 0.616357
\(723\) −6.00000 −0.223142
\(724\) 13.3693 0.496867
\(725\) 12.0000 0.445669
\(726\) 19.9309 0.739704
\(727\) 48.9309 1.81475 0.907373 0.420327i \(-0.138085\pi\)
0.907373 + 0.420327i \(0.138085\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 41.1771 1.52403
\(731\) 43.0388 1.59185
\(732\) 1.56155 0.0577167
\(733\) 22.8769 0.844977 0.422489 0.906368i \(-0.361157\pi\)
0.422489 + 0.906368i \(0.361157\pi\)
\(734\) −0.630683 −0.0232789
\(735\) 3.56155 0.131370
\(736\) −6.68466 −0.246400
\(737\) 6.24621 0.230082
\(738\) 4.00000 0.147242
\(739\) −17.6155 −0.647998 −0.323999 0.946057i \(-0.605027\pi\)
−0.323999 + 0.946057i \(0.605027\pi\)
\(740\) −38.0540 −1.39889
\(741\) 0 0
\(742\) 4.87689 0.179036
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 6.24621 0.228997
\(745\) 35.6155 1.30485
\(746\) 6.00000 0.219676
\(747\) 2.00000 0.0731762
\(748\) 37.1771 1.35933
\(749\) 4.87689 0.178198
\(750\) −9.56155 −0.349139
\(751\) −2.24621 −0.0819654 −0.0409827 0.999160i \(-0.513049\pi\)
−0.0409827 + 0.999160i \(0.513049\pi\)
\(752\) −10.2462 −0.373641
\(753\) 22.0540 0.803692
\(754\) 0 0
\(755\) −59.4233 −2.16264
\(756\) 1.00000 0.0363696
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −18.8769 −0.685640
\(759\) −37.1771 −1.34944
\(760\) 5.56155 0.201739
\(761\) −39.1231 −1.41821 −0.709106 0.705102i \(-0.750901\pi\)
−0.709106 + 0.705102i \(0.750901\pi\)
\(762\) −10.2462 −0.371181
\(763\) 12.9309 0.468129
\(764\) −24.0540 −0.870242
\(765\) −23.8078 −0.860772
\(766\) 17.5616 0.634525
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 8.43845 0.304298 0.152149 0.988358i \(-0.451381\pi\)
0.152149 + 0.988358i \(0.451381\pi\)
\(770\) −19.8078 −0.713822
\(771\) −26.0000 −0.936367
\(772\) −12.0000 −0.431889
\(773\) −12.9309 −0.465091 −0.232546 0.972586i \(-0.574705\pi\)
−0.232546 + 0.972586i \(0.574705\pi\)
\(774\) −6.43845 −0.231425
\(775\) 48.0000 1.72421
\(776\) 10.0000 0.358979
\(777\) 10.6847 0.383310
\(778\) 27.1231 0.972410
\(779\) −6.24621 −0.223794
\(780\) 0 0
\(781\) −52.1080 −1.86457
\(782\) −44.6847 −1.59792
\(783\) −1.56155 −0.0558053
\(784\) 1.00000 0.0357143
\(785\) 42.0540 1.50097
\(786\) −1.56155 −0.0556987
\(787\) 17.0691 0.608449 0.304224 0.952600i \(-0.401603\pi\)
0.304224 + 0.952600i \(0.401603\pi\)
\(788\) 6.00000 0.213741
\(789\) 23.3693 0.831970
\(790\) 56.9848 2.02743
\(791\) −20.2462 −0.719872
\(792\) −5.56155 −0.197621
\(793\) 0 0
\(794\) −15.3693 −0.545437
\(795\) 17.3693 0.616026
\(796\) −4.93087 −0.174770
\(797\) −3.75379 −0.132966 −0.0664830 0.997788i \(-0.521178\pi\)
−0.0664830 + 0.997788i \(0.521178\pi\)
\(798\) −1.56155 −0.0552784
\(799\) −68.4924 −2.42309
\(800\) −7.68466 −0.271694
\(801\) −8.00000 −0.282666
\(802\) 30.9848 1.09411
\(803\) 64.3002 2.26910
\(804\) −1.12311 −0.0396089
\(805\) 23.8078 0.839113
\(806\) 0 0
\(807\) −31.3693 −1.10425
\(808\) 6.00000 0.211079
\(809\) −55.3693 −1.94668 −0.973341 0.229364i \(-0.926335\pi\)
−0.973341 + 0.229364i \(0.926335\pi\)
\(810\) 3.56155 0.125140
\(811\) 2.05398 0.0721248 0.0360624 0.999350i \(-0.488518\pi\)
0.0360624 + 0.999350i \(0.488518\pi\)
\(812\) −1.56155 −0.0547998
\(813\) 8.00000 0.280572
\(814\) −59.4233 −2.08279
\(815\) −32.4924 −1.13816
\(816\) −6.68466 −0.234010
\(817\) 10.0540 0.351744
\(818\) 13.8078 0.482777
\(819\) 0 0
\(820\) 14.2462 0.497499
\(821\) 30.8769 1.07761 0.538806 0.842430i \(-0.318876\pi\)
0.538806 + 0.842430i \(0.318876\pi\)
\(822\) 6.68466 0.233154
\(823\) 30.7386 1.07148 0.535741 0.844383i \(-0.320032\pi\)
0.535741 + 0.844383i \(0.320032\pi\)
\(824\) 1.80776 0.0629764
\(825\) −42.7386 −1.48797
\(826\) 4.24621 0.147745
\(827\) 22.0540 0.766892 0.383446 0.923563i \(-0.374737\pi\)
0.383446 + 0.923563i \(0.374737\pi\)
\(828\) 6.68466 0.232308
\(829\) −42.0540 −1.46059 −0.730297 0.683129i \(-0.760619\pi\)
−0.730297 + 0.683129i \(0.760619\pi\)
\(830\) 7.12311 0.247247
\(831\) −10.8769 −0.377315
\(832\) 0 0
\(833\) 6.68466 0.231610
\(834\) 10.2462 0.354797
\(835\) 42.0540 1.45534
\(836\) 8.68466 0.300365
\(837\) −6.24621 −0.215901
\(838\) 20.6847 0.714540
\(839\) 25.7538 0.889120 0.444560 0.895749i \(-0.353360\pi\)
0.444560 + 0.895749i \(0.353360\pi\)
\(840\) 3.56155 0.122885
\(841\) −26.5616 −0.915916
\(842\) −10.0000 −0.344623
\(843\) −6.00000 −0.206651
\(844\) 7.80776 0.268754
\(845\) 0 0
\(846\) 10.2462 0.352272
\(847\) −19.9309 −0.684833
\(848\) 4.87689 0.167473
\(849\) −4.87689 −0.167375
\(850\) −51.3693 −1.76195
\(851\) 71.4233 2.44836
\(852\) 9.36932 0.320988
\(853\) 10.8769 0.372418 0.186209 0.982510i \(-0.440380\pi\)
0.186209 + 0.982510i \(0.440380\pi\)
\(854\) −1.56155 −0.0534352
\(855\) −5.56155 −0.190201
\(856\) 4.87689 0.166689
\(857\) 38.0000 1.29806 0.649028 0.760765i \(-0.275176\pi\)
0.649028 + 0.760765i \(0.275176\pi\)
\(858\) 0 0
\(859\) 0.384472 0.0131180 0.00655901 0.999978i \(-0.497912\pi\)
0.00655901 + 0.999978i \(0.497912\pi\)
\(860\) −22.9309 −0.781936
\(861\) −4.00000 −0.136320
\(862\) 29.8617 1.01709
\(863\) 42.7386 1.45484 0.727420 0.686192i \(-0.240719\pi\)
0.727420 + 0.686192i \(0.240719\pi\)
\(864\) 1.00000 0.0340207
\(865\) −13.3693 −0.454570
\(866\) 23.3693 0.794122
\(867\) −27.6847 −0.940220
\(868\) −6.24621 −0.212010
\(869\) 88.9848 3.01860
\(870\) −5.56155 −0.188554
\(871\) 0 0
\(872\) 12.9309 0.437895
\(873\) −10.0000 −0.338449
\(874\) −10.4384 −0.353086
\(875\) 9.56155 0.323239
\(876\) −11.5616 −0.390629
\(877\) 22.9848 0.776143 0.388072 0.921629i \(-0.373141\pi\)
0.388072 + 0.921629i \(0.373141\pi\)
\(878\) −8.93087 −0.301402
\(879\) 7.75379 0.261529
\(880\) −19.8078 −0.667720
\(881\) −14.3002 −0.481786 −0.240893 0.970552i \(-0.577440\pi\)
−0.240893 + 0.970552i \(0.577440\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 14.4384 0.485892 0.242946 0.970040i \(-0.421886\pi\)
0.242946 + 0.970040i \(0.421886\pi\)
\(884\) 0 0
\(885\) 15.1231 0.508358
\(886\) −6.63068 −0.222762
\(887\) −37.8617 −1.27127 −0.635636 0.771989i \(-0.719262\pi\)
−0.635636 + 0.771989i \(0.719262\pi\)
\(888\) 10.6847 0.358554
\(889\) 10.2462 0.343647
\(890\) −28.4924 −0.955068
\(891\) 5.56155 0.186319
\(892\) 1.75379 0.0587212
\(893\) −16.0000 −0.535420
\(894\) −10.0000 −0.334450
\(895\) 39.6155 1.32420
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 9.31534 0.310857
\(899\) 9.75379 0.325307
\(900\) 7.68466 0.256155
\(901\) 32.6004 1.08608
\(902\) 22.2462 0.740718
\(903\) 6.43845 0.214258
\(904\) −20.2462 −0.673379
\(905\) −47.6155 −1.58279
\(906\) 16.6847 0.554311
\(907\) 24.4924 0.813258 0.406629 0.913593i \(-0.366704\pi\)
0.406629 + 0.913593i \(0.366704\pi\)
\(908\) 17.6155 0.584592
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −4.43845 −0.147052 −0.0735262 0.997293i \(-0.523425\pi\)
−0.0735262 + 0.997293i \(0.523425\pi\)
\(912\) −1.56155 −0.0517082
\(913\) 11.1231 0.368121
\(914\) −16.0000 −0.529233
\(915\) −5.56155 −0.183859
\(916\) 2.00000 0.0660819
\(917\) 1.56155 0.0515670
\(918\) 6.68466 0.220627
\(919\) −17.8617 −0.589204 −0.294602 0.955620i \(-0.595187\pi\)
−0.294602 + 0.955620i \(0.595187\pi\)
\(920\) 23.8078 0.784919
\(921\) −26.2462 −0.864842
\(922\) 36.9309 1.21625
\(923\) 0 0
\(924\) 5.56155 0.182962
\(925\) 82.1080 2.69969
\(926\) 30.5464 1.00382
\(927\) −1.80776 −0.0593748
\(928\) −1.56155 −0.0512605
\(929\) 40.8769 1.34113 0.670564 0.741852i \(-0.266052\pi\)
0.670564 + 0.741852i \(0.266052\pi\)
\(930\) −22.2462 −0.729482
\(931\) 1.56155 0.0511778
\(932\) −2.00000 −0.0655122
\(933\) −0.492423 −0.0161212
\(934\) 1.56155 0.0510956
\(935\) −132.408 −4.33021
\(936\) 0 0
\(937\) −19.8617 −0.648855 −0.324427 0.945911i \(-0.605172\pi\)
−0.324427 + 0.945911i \(0.605172\pi\)
\(938\) 1.12311 0.0366707
\(939\) −32.2462 −1.05232
\(940\) 36.4924 1.19025
\(941\) 26.4924 0.863628 0.431814 0.901963i \(-0.357874\pi\)
0.431814 + 0.901963i \(0.357874\pi\)
\(942\) −11.8078 −0.384718
\(943\) −26.7386 −0.870730
\(944\) 4.24621 0.138202
\(945\) −3.56155 −0.115857
\(946\) −35.8078 −1.16421
\(947\) 27.8078 0.903631 0.451815 0.892111i \(-0.350777\pi\)
0.451815 + 0.892111i \(0.350777\pi\)
\(948\) −16.0000 −0.519656
\(949\) 0 0
\(950\) −12.0000 −0.389331
\(951\) −3.36932 −0.109258
\(952\) 6.68466 0.216651
\(953\) −0.738634 −0.0239267 −0.0119633 0.999928i \(-0.503808\pi\)
−0.0119633 + 0.999928i \(0.503808\pi\)
\(954\) −4.87689 −0.157895
\(955\) 85.6695 2.77220
\(956\) −14.2462 −0.460755
\(957\) −8.68466 −0.280735
\(958\) −12.3002 −0.397401
\(959\) −6.68466 −0.215859
\(960\) 3.56155 0.114949
\(961\) 8.01515 0.258553
\(962\) 0 0
\(963\) −4.87689 −0.157156
\(964\) 6.00000 0.193247
\(965\) 42.7386 1.37581
\(966\) −6.68466 −0.215075
\(967\) −2.93087 −0.0942504 −0.0471252 0.998889i \(-0.515006\pi\)
−0.0471252 + 0.998889i \(0.515006\pi\)
\(968\) −19.9309 −0.640602
\(969\) −10.4384 −0.335331
\(970\) −35.6155 −1.14355
\(971\) 4.00000 0.128366 0.0641831 0.997938i \(-0.479556\pi\)
0.0641831 + 0.997938i \(0.479556\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −10.2462 −0.328478
\(974\) −13.7538 −0.440700
\(975\) 0 0
\(976\) −1.56155 −0.0499841
\(977\) 46.7926 1.49703 0.748514 0.663119i \(-0.230768\pi\)
0.748514 + 0.663119i \(0.230768\pi\)
\(978\) 9.12311 0.291725
\(979\) −44.4924 −1.42198
\(980\) −3.56155 −0.113770
\(981\) −12.9309 −0.412851
\(982\) −10.2462 −0.326970
\(983\) −2.43845 −0.0777744 −0.0388872 0.999244i \(-0.512381\pi\)
−0.0388872 + 0.999244i \(0.512381\pi\)
\(984\) −4.00000 −0.127515
\(985\) −21.3693 −0.680883
\(986\) −10.4384 −0.332428
\(987\) −10.2462 −0.326140
\(988\) 0 0
\(989\) 43.0388 1.36855
\(990\) 19.8078 0.629532
\(991\) −30.2462 −0.960803 −0.480401 0.877049i \(-0.659509\pi\)
−0.480401 + 0.877049i \(0.659509\pi\)
\(992\) −6.24621 −0.198317
\(993\) −1.12311 −0.0356407
\(994\) −9.36932 −0.297177
\(995\) 17.5616 0.556739
\(996\) −2.00000 −0.0633724
\(997\) −34.6307 −1.09676 −0.548382 0.836228i \(-0.684756\pi\)
−0.548382 + 0.836228i \(0.684756\pi\)
\(998\) −1.50758 −0.0477215
\(999\) −10.6847 −0.338048
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 7098.2.a.bg.1.1 2
13.5 odd 4 546.2.c.e.337.4 yes 4
13.8 odd 4 546.2.c.e.337.1 4
13.12 even 2 7098.2.a.bv.1.2 2
39.5 even 4 1638.2.c.h.883.1 4
39.8 even 4 1638.2.c.h.883.4 4
52.31 even 4 4368.2.h.n.337.4 4
52.47 even 4 4368.2.h.n.337.1 4
91.34 even 4 3822.2.c.h.883.2 4
91.83 even 4 3822.2.c.h.883.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.c.e.337.1 4 13.8 odd 4
546.2.c.e.337.4 yes 4 13.5 odd 4
1638.2.c.h.883.1 4 39.5 even 4
1638.2.c.h.883.4 4 39.8 even 4
3822.2.c.h.883.2 4 91.34 even 4
3822.2.c.h.883.3 4 91.83 even 4
4368.2.h.n.337.1 4 52.47 even 4
4368.2.h.n.337.4 4 52.31 even 4
7098.2.a.bg.1.1 2 1.1 even 1 trivial
7098.2.a.bv.1.2 2 13.12 even 2