Properties

Label 7098.2.a.bl.1.2
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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: \(1\)
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 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.561553 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.561553 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.561553 q^{10} -2.56155 q^{11} +1.00000 q^{12} -1.00000 q^{14} +0.561553 q^{15} +1.00000 q^{16} +5.68466 q^{17} -1.00000 q^{18} -7.68466 q^{19} +0.561553 q^{20} +1.00000 q^{21} +2.56155 q^{22} -1.43845 q^{23} -1.00000 q^{24} -4.68466 q^{25} +1.00000 q^{27} +1.00000 q^{28} -5.68466 q^{29} -0.561553 q^{30} +10.2462 q^{31} -1.00000 q^{32} -2.56155 q^{33} -5.68466 q^{34} +0.561553 q^{35} +1.00000 q^{36} +3.43845 q^{37} +7.68466 q^{38} -0.561553 q^{40} -7.12311 q^{41} -1.00000 q^{42} -10.5616 q^{43} -2.56155 q^{44} +0.561553 q^{45} +1.43845 q^{46} +1.00000 q^{48} +1.00000 q^{49} +4.68466 q^{50} +5.68466 q^{51} -4.24621 q^{53} -1.00000 q^{54} -1.43845 q^{55} -1.00000 q^{56} -7.68466 q^{57} +5.68466 q^{58} +14.2462 q^{59} +0.561553 q^{60} -5.68466 q^{61} -10.2462 q^{62} +1.00000 q^{63} +1.00000 q^{64} +2.56155 q^{66} -1.12311 q^{67} +5.68466 q^{68} -1.43845 q^{69} -0.561553 q^{70} -8.00000 q^{71} -1.00000 q^{72} -0.561553 q^{73} -3.43845 q^{74} -4.68466 q^{75} -7.68466 q^{76} -2.56155 q^{77} -2.87689 q^{79} +0.561553 q^{80} +1.00000 q^{81} +7.12311 q^{82} -17.1231 q^{83} +1.00000 q^{84} +3.19224 q^{85} +10.5616 q^{86} -5.68466 q^{87} +2.56155 q^{88} -10.0000 q^{89} -0.561553 q^{90} -1.43845 q^{92} +10.2462 q^{93} -4.31534 q^{95} -1.00000 q^{96} +18.4924 q^{97} -1.00000 q^{98} -2.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 3 q^{10} - q^{11} + 2 q^{12} - 2 q^{14} - 3 q^{15} + 2 q^{16} - q^{17} - 2 q^{18} - 3 q^{19} - 3 q^{20} + 2 q^{21} + q^{22} - 7 q^{23} - 2 q^{24} + 3 q^{25} + 2 q^{27} + 2 q^{28} + q^{29} + 3 q^{30} + 4 q^{31} - 2 q^{32} - q^{33} + q^{34} - 3 q^{35} + 2 q^{36} + 11 q^{37} + 3 q^{38} + 3 q^{40} - 6 q^{41} - 2 q^{42} - 17 q^{43} - q^{44} - 3 q^{45} + 7 q^{46} + 2 q^{48} + 2 q^{49} - 3 q^{50} - q^{51} + 8 q^{53} - 2 q^{54} - 7 q^{55} - 2 q^{56} - 3 q^{57} - q^{58} + 12 q^{59} - 3 q^{60} + q^{61} - 4 q^{62} + 2 q^{63} + 2 q^{64} + q^{66} + 6 q^{67} - q^{68} - 7 q^{69} + 3 q^{70} - 16 q^{71} - 2 q^{72} + 3 q^{73} - 11 q^{74} + 3 q^{75} - 3 q^{76} - q^{77} - 14 q^{79} - 3 q^{80} + 2 q^{81} + 6 q^{82} - 26 q^{83} + 2 q^{84} + 27 q^{85} + 17 q^{86} + q^{87} + q^{88} - 20 q^{89} + 3 q^{90} - 7 q^{92} + 4 q^{93} - 21 q^{95} - 2 q^{96} + 4 q^{97} - 2 q^{98} - q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0.561553 0.251134 0.125567 0.992085i \(-0.459925\pi\)
0.125567 + 0.992085i \(0.459925\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −0.561553 −0.177579
\(11\) −2.56155 −0.772337 −0.386169 0.922428i \(-0.626202\pi\)
−0.386169 + 0.922428i \(0.626202\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) 0.561553 0.144992
\(16\) 1.00000 0.250000
\(17\) 5.68466 1.37873 0.689366 0.724413i \(-0.257889\pi\)
0.689366 + 0.724413i \(0.257889\pi\)
\(18\) −1.00000 −0.235702
\(19\) −7.68466 −1.76298 −0.881491 0.472201i \(-0.843460\pi\)
−0.881491 + 0.472201i \(0.843460\pi\)
\(20\) 0.561553 0.125567
\(21\) 1.00000 0.218218
\(22\) 2.56155 0.546125
\(23\) −1.43845 −0.299937 −0.149968 0.988691i \(-0.547917\pi\)
−0.149968 + 0.988691i \(0.547917\pi\)
\(24\) −1.00000 −0.204124
\(25\) −4.68466 −0.936932
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −5.68466 −1.05561 −0.527807 0.849364i \(-0.676986\pi\)
−0.527807 + 0.849364i \(0.676986\pi\)
\(30\) −0.561553 −0.102525
\(31\) 10.2462 1.84027 0.920137 0.391597i \(-0.128077\pi\)
0.920137 + 0.391597i \(0.128077\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.56155 −0.445909
\(34\) −5.68466 −0.974911
\(35\) 0.561553 0.0949197
\(36\) 1.00000 0.166667
\(37\) 3.43845 0.565277 0.282639 0.959226i \(-0.408790\pi\)
0.282639 + 0.959226i \(0.408790\pi\)
\(38\) 7.68466 1.24662
\(39\) 0 0
\(40\) −0.561553 −0.0887893
\(41\) −7.12311 −1.11244 −0.556221 0.831034i \(-0.687749\pi\)
−0.556221 + 0.831034i \(0.687749\pi\)
\(42\) −1.00000 −0.154303
\(43\) −10.5616 −1.61062 −0.805311 0.592853i \(-0.798002\pi\)
−0.805311 + 0.592853i \(0.798002\pi\)
\(44\) −2.56155 −0.386169
\(45\) 0.561553 0.0837114
\(46\) 1.43845 0.212087
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 4.68466 0.662511
\(51\) 5.68466 0.796011
\(52\) 0 0
\(53\) −4.24621 −0.583262 −0.291631 0.956531i \(-0.594198\pi\)
−0.291631 + 0.956531i \(0.594198\pi\)
\(54\) −1.00000 −0.136083
\(55\) −1.43845 −0.193960
\(56\) −1.00000 −0.133631
\(57\) −7.68466 −1.01786
\(58\) 5.68466 0.746432
\(59\) 14.2462 1.85470 0.927349 0.374197i \(-0.122082\pi\)
0.927349 + 0.374197i \(0.122082\pi\)
\(60\) 0.561553 0.0724962
\(61\) −5.68466 −0.727846 −0.363923 0.931429i \(-0.618563\pi\)
−0.363923 + 0.931429i \(0.618563\pi\)
\(62\) −10.2462 −1.30127
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.56155 0.315305
\(67\) −1.12311 −0.137209 −0.0686046 0.997644i \(-0.521855\pi\)
−0.0686046 + 0.997644i \(0.521855\pi\)
\(68\) 5.68466 0.689366
\(69\) −1.43845 −0.173169
\(70\) −0.561553 −0.0671184
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −1.00000 −0.117851
\(73\) −0.561553 −0.0657248 −0.0328624 0.999460i \(-0.510462\pi\)
−0.0328624 + 0.999460i \(0.510462\pi\)
\(74\) −3.43845 −0.399711
\(75\) −4.68466 −0.540938
\(76\) −7.68466 −0.881491
\(77\) −2.56155 −0.291916
\(78\) 0 0
\(79\) −2.87689 −0.323676 −0.161838 0.986817i \(-0.551742\pi\)
−0.161838 + 0.986817i \(0.551742\pi\)
\(80\) 0.561553 0.0627835
\(81\) 1.00000 0.111111
\(82\) 7.12311 0.786615
\(83\) −17.1231 −1.87951 −0.939753 0.341856i \(-0.888945\pi\)
−0.939753 + 0.341856i \(0.888945\pi\)
\(84\) 1.00000 0.109109
\(85\) 3.19224 0.346247
\(86\) 10.5616 1.13888
\(87\) −5.68466 −0.609459
\(88\) 2.56155 0.273062
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) −0.561553 −0.0591929
\(91\) 0 0
\(92\) −1.43845 −0.149968
\(93\) 10.2462 1.06248
\(94\) 0 0
\(95\) −4.31534 −0.442745
\(96\) −1.00000 −0.102062
\(97\) 18.4924 1.87762 0.938811 0.344434i \(-0.111929\pi\)
0.938811 + 0.344434i \(0.111929\pi\)
\(98\) −1.00000 −0.101015
\(99\) −2.56155 −0.257446
\(100\) −4.68466 −0.468466
\(101\) 3.12311 0.310761 0.155380 0.987855i \(-0.450340\pi\)
0.155380 + 0.987855i \(0.450340\pi\)
\(102\) −5.68466 −0.562865
\(103\) 11.6847 1.15132 0.575662 0.817688i \(-0.304744\pi\)
0.575662 + 0.817688i \(0.304744\pi\)
\(104\) 0 0
\(105\) 0.561553 0.0548019
\(106\) 4.24621 0.412428
\(107\) −17.1231 −1.65535 −0.827677 0.561205i \(-0.810338\pi\)
−0.827677 + 0.561205i \(0.810338\pi\)
\(108\) 1.00000 0.0962250
\(109\) −11.9309 −1.14277 −0.571385 0.820682i \(-0.693594\pi\)
−0.571385 + 0.820682i \(0.693594\pi\)
\(110\) 1.43845 0.137151
\(111\) 3.43845 0.326363
\(112\) 1.00000 0.0944911
\(113\) 12.2462 1.15203 0.576013 0.817440i \(-0.304608\pi\)
0.576013 + 0.817440i \(0.304608\pi\)
\(114\) 7.68466 0.719734
\(115\) −0.807764 −0.0753244
\(116\) −5.68466 −0.527807
\(117\) 0 0
\(118\) −14.2462 −1.31147
\(119\) 5.68466 0.521112
\(120\) −0.561553 −0.0512625
\(121\) −4.43845 −0.403495
\(122\) 5.68466 0.514665
\(123\) −7.12311 −0.642269
\(124\) 10.2462 0.920137
\(125\) −5.43845 −0.486430
\(126\) −1.00000 −0.0890871
\(127\) −13.1231 −1.16449 −0.582244 0.813014i \(-0.697825\pi\)
−0.582244 + 0.813014i \(0.697825\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −10.5616 −0.929893
\(130\) 0 0
\(131\) −5.43845 −0.475159 −0.237580 0.971368i \(-0.576354\pi\)
−0.237580 + 0.971368i \(0.576354\pi\)
\(132\) −2.56155 −0.222955
\(133\) −7.68466 −0.666344
\(134\) 1.12311 0.0970215
\(135\) 0.561553 0.0483308
\(136\) −5.68466 −0.487455
\(137\) −0.561553 −0.0479767 −0.0239883 0.999712i \(-0.507636\pi\)
−0.0239883 + 0.999712i \(0.507636\pi\)
\(138\) 1.43845 0.122449
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0.561553 0.0474599
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −3.19224 −0.265101
\(146\) 0.561553 0.0464744
\(147\) 1.00000 0.0824786
\(148\) 3.43845 0.282639
\(149\) −23.6155 −1.93466 −0.967330 0.253522i \(-0.918411\pi\)
−0.967330 + 0.253522i \(0.918411\pi\)
\(150\) 4.68466 0.382501
\(151\) 8.80776 0.716766 0.358383 0.933575i \(-0.383328\pi\)
0.358383 + 0.933575i \(0.383328\pi\)
\(152\) 7.68466 0.623308
\(153\) 5.68466 0.459577
\(154\) 2.56155 0.206416
\(155\) 5.75379 0.462155
\(156\) 0 0
\(157\) −11.4384 −0.912887 −0.456444 0.889752i \(-0.650877\pi\)
−0.456444 + 0.889752i \(0.650877\pi\)
\(158\) 2.87689 0.228873
\(159\) −4.24621 −0.336746
\(160\) −0.561553 −0.0443946
\(161\) −1.43845 −0.113366
\(162\) −1.00000 −0.0785674
\(163\) 25.1231 1.96779 0.983897 0.178738i \(-0.0572014\pi\)
0.983897 + 0.178738i \(0.0572014\pi\)
\(164\) −7.12311 −0.556221
\(165\) −1.43845 −0.111983
\(166\) 17.1231 1.32901
\(167\) 5.93087 0.458944 0.229472 0.973315i \(-0.426300\pi\)
0.229472 + 0.973315i \(0.426300\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 0 0
\(170\) −3.19224 −0.244833
\(171\) −7.68466 −0.587661
\(172\) −10.5616 −0.805311
\(173\) 3.75379 0.285395 0.142698 0.989766i \(-0.454422\pi\)
0.142698 + 0.989766i \(0.454422\pi\)
\(174\) 5.68466 0.430953
\(175\) −4.68466 −0.354127
\(176\) −2.56155 −0.193084
\(177\) 14.2462 1.07081
\(178\) 10.0000 0.749532
\(179\) −16.4924 −1.23270 −0.616351 0.787472i \(-0.711390\pi\)
−0.616351 + 0.787472i \(0.711390\pi\)
\(180\) 0.561553 0.0418557
\(181\) 16.2462 1.20757 0.603786 0.797147i \(-0.293658\pi\)
0.603786 + 0.797147i \(0.293658\pi\)
\(182\) 0 0
\(183\) −5.68466 −0.420222
\(184\) 1.43845 0.106044
\(185\) 1.93087 0.141960
\(186\) −10.2462 −0.751289
\(187\) −14.5616 −1.06485
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 4.31534 0.313068
\(191\) −13.9309 −1.00800 −0.504001 0.863703i \(-0.668139\pi\)
−0.504001 + 0.863703i \(0.668139\pi\)
\(192\) 1.00000 0.0721688
\(193\) 5.36932 0.386492 0.193246 0.981150i \(-0.438098\pi\)
0.193246 + 0.981150i \(0.438098\pi\)
\(194\) −18.4924 −1.32768
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −19.1231 −1.36246 −0.681232 0.732067i \(-0.738556\pi\)
−0.681232 + 0.732067i \(0.738556\pi\)
\(198\) 2.56155 0.182042
\(199\) −21.9309 −1.55464 −0.777319 0.629107i \(-0.783421\pi\)
−0.777319 + 0.629107i \(0.783421\pi\)
\(200\) 4.68466 0.331255
\(201\) −1.12311 −0.0792178
\(202\) −3.12311 −0.219741
\(203\) −5.68466 −0.398985
\(204\) 5.68466 0.398006
\(205\) −4.00000 −0.279372
\(206\) −11.6847 −0.814109
\(207\) −1.43845 −0.0999790
\(208\) 0 0
\(209\) 19.6847 1.36162
\(210\) −0.561553 −0.0387508
\(211\) 4.80776 0.330980 0.165490 0.986211i \(-0.447079\pi\)
0.165490 + 0.986211i \(0.447079\pi\)
\(212\) −4.24621 −0.291631
\(213\) −8.00000 −0.548151
\(214\) 17.1231 1.17051
\(215\) −5.93087 −0.404482
\(216\) −1.00000 −0.0680414
\(217\) 10.2462 0.695558
\(218\) 11.9309 0.808060
\(219\) −0.561553 −0.0379462
\(220\) −1.43845 −0.0969801
\(221\) 0 0
\(222\) −3.43845 −0.230773
\(223\) −17.6155 −1.17962 −0.589812 0.807541i \(-0.700798\pi\)
−0.589812 + 0.807541i \(0.700798\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −4.68466 −0.312311
\(226\) −12.2462 −0.814606
\(227\) −1.12311 −0.0745431 −0.0372716 0.999305i \(-0.511867\pi\)
−0.0372716 + 0.999305i \(0.511867\pi\)
\(228\) −7.68466 −0.508929
\(229\) −11.7538 −0.776712 −0.388356 0.921509i \(-0.626957\pi\)
−0.388356 + 0.921509i \(0.626957\pi\)
\(230\) 0.807764 0.0532624
\(231\) −2.56155 −0.168538
\(232\) 5.68466 0.373216
\(233\) 2.63068 0.172342 0.0861709 0.996280i \(-0.472537\pi\)
0.0861709 + 0.996280i \(0.472537\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 14.2462 0.927349
\(237\) −2.87689 −0.186874
\(238\) −5.68466 −0.368482
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0.561553 0.0362481
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 4.43845 0.285314
\(243\) 1.00000 0.0641500
\(244\) −5.68466 −0.363923
\(245\) 0.561553 0.0358763
\(246\) 7.12311 0.454153
\(247\) 0 0
\(248\) −10.2462 −0.650635
\(249\) −17.1231 −1.08513
\(250\) 5.43845 0.343958
\(251\) 12.8078 0.808419 0.404209 0.914666i \(-0.367547\pi\)
0.404209 + 0.914666i \(0.367547\pi\)
\(252\) 1.00000 0.0629941
\(253\) 3.68466 0.231652
\(254\) 13.1231 0.823417
\(255\) 3.19224 0.199906
\(256\) 1.00000 0.0625000
\(257\) 12.2462 0.763898 0.381949 0.924183i \(-0.375253\pi\)
0.381949 + 0.924183i \(0.375253\pi\)
\(258\) 10.5616 0.657534
\(259\) 3.43845 0.213655
\(260\) 0 0
\(261\) −5.68466 −0.351872
\(262\) 5.43845 0.335988
\(263\) −13.7538 −0.848095 −0.424047 0.905640i \(-0.639391\pi\)
−0.424047 + 0.905640i \(0.639391\pi\)
\(264\) 2.56155 0.157653
\(265\) −2.38447 −0.146477
\(266\) 7.68466 0.471177
\(267\) −10.0000 −0.611990
\(268\) −1.12311 −0.0686046
\(269\) 3.75379 0.228873 0.114436 0.993431i \(-0.463494\pi\)
0.114436 + 0.993431i \(0.463494\pi\)
\(270\) −0.561553 −0.0341750
\(271\) −13.1231 −0.797172 −0.398586 0.917131i \(-0.630499\pi\)
−0.398586 + 0.917131i \(0.630499\pi\)
\(272\) 5.68466 0.344683
\(273\) 0 0
\(274\) 0.561553 0.0339246
\(275\) 12.0000 0.723627
\(276\) −1.43845 −0.0865843
\(277\) 10.4924 0.630429 0.315214 0.949021i \(-0.397924\pi\)
0.315214 + 0.949021i \(0.397924\pi\)
\(278\) −12.0000 −0.719712
\(279\) 10.2462 0.613425
\(280\) −0.561553 −0.0335592
\(281\) 0.246211 0.0146877 0.00734387 0.999973i \(-0.497662\pi\)
0.00734387 + 0.999973i \(0.497662\pi\)
\(282\) 0 0
\(283\) 1.75379 0.104252 0.0521260 0.998641i \(-0.483400\pi\)
0.0521260 + 0.998641i \(0.483400\pi\)
\(284\) −8.00000 −0.474713
\(285\) −4.31534 −0.255619
\(286\) 0 0
\(287\) −7.12311 −0.420464
\(288\) −1.00000 −0.0589256
\(289\) 15.3153 0.900902
\(290\) 3.19224 0.187455
\(291\) 18.4924 1.08405
\(292\) −0.561553 −0.0328624
\(293\) 24.7386 1.44525 0.722623 0.691242i \(-0.242936\pi\)
0.722623 + 0.691242i \(0.242936\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 8.00000 0.465778
\(296\) −3.43845 −0.199856
\(297\) −2.56155 −0.148636
\(298\) 23.6155 1.36801
\(299\) 0 0
\(300\) −4.68466 −0.270469
\(301\) −10.5616 −0.608758
\(302\) −8.80776 −0.506830
\(303\) 3.12311 0.179418
\(304\) −7.68466 −0.440745
\(305\) −3.19224 −0.182787
\(306\) −5.68466 −0.324970
\(307\) −9.75379 −0.556678 −0.278339 0.960483i \(-0.589784\pi\)
−0.278339 + 0.960483i \(0.589784\pi\)
\(308\) −2.56155 −0.145958
\(309\) 11.6847 0.664717
\(310\) −5.75379 −0.326793
\(311\) −21.1231 −1.19778 −0.598891 0.800831i \(-0.704392\pi\)
−0.598891 + 0.800831i \(0.704392\pi\)
\(312\) 0 0
\(313\) 27.6155 1.56092 0.780461 0.625205i \(-0.214984\pi\)
0.780461 + 0.625205i \(0.214984\pi\)
\(314\) 11.4384 0.645509
\(315\) 0.561553 0.0316399
\(316\) −2.87689 −0.161838
\(317\) −11.1231 −0.624736 −0.312368 0.949961i \(-0.601122\pi\)
−0.312368 + 0.949961i \(0.601122\pi\)
\(318\) 4.24621 0.238116
\(319\) 14.5616 0.815290
\(320\) 0.561553 0.0313918
\(321\) −17.1231 −0.955719
\(322\) 1.43845 0.0801615
\(323\) −43.6847 −2.43068
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −25.1231 −1.39144
\(327\) −11.9309 −0.659779
\(328\) 7.12311 0.393308
\(329\) 0 0
\(330\) 1.43845 0.0791839
\(331\) 11.3693 0.624914 0.312457 0.949932i \(-0.398848\pi\)
0.312457 + 0.949932i \(0.398848\pi\)
\(332\) −17.1231 −0.939753
\(333\) 3.43845 0.188426
\(334\) −5.93087 −0.324523
\(335\) −0.630683 −0.0344579
\(336\) 1.00000 0.0545545
\(337\) 8.56155 0.466377 0.233189 0.972431i \(-0.425084\pi\)
0.233189 + 0.972431i \(0.425084\pi\)
\(338\) 0 0
\(339\) 12.2462 0.665123
\(340\) 3.19224 0.173123
\(341\) −26.2462 −1.42131
\(342\) 7.68466 0.415539
\(343\) 1.00000 0.0539949
\(344\) 10.5616 0.569441
\(345\) −0.807764 −0.0434886
\(346\) −3.75379 −0.201805
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) −5.68466 −0.304730
\(349\) −24.2462 −1.29787 −0.648935 0.760844i \(-0.724785\pi\)
−0.648935 + 0.760844i \(0.724785\pi\)
\(350\) 4.68466 0.250406
\(351\) 0 0
\(352\) 2.56155 0.136531
\(353\) 2.49242 0.132658 0.0663291 0.997798i \(-0.478871\pi\)
0.0663291 + 0.997798i \(0.478871\pi\)
\(354\) −14.2462 −0.757178
\(355\) −4.49242 −0.238433
\(356\) −10.0000 −0.529999
\(357\) 5.68466 0.300864
\(358\) 16.4924 0.871652
\(359\) −22.7386 −1.20010 −0.600050 0.799963i \(-0.704852\pi\)
−0.600050 + 0.799963i \(0.704852\pi\)
\(360\) −0.561553 −0.0295964
\(361\) 40.0540 2.10810
\(362\) −16.2462 −0.853882
\(363\) −4.43845 −0.232958
\(364\) 0 0
\(365\) −0.315342 −0.0165057
\(366\) 5.68466 0.297142
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) −1.43845 −0.0749842
\(369\) −7.12311 −0.370814
\(370\) −1.93087 −0.100381
\(371\) −4.24621 −0.220452
\(372\) 10.2462 0.531241
\(373\) 23.6155 1.22277 0.611383 0.791335i \(-0.290614\pi\)
0.611383 + 0.791335i \(0.290614\pi\)
\(374\) 14.5616 0.752960
\(375\) −5.43845 −0.280840
\(376\) 0 0
\(377\) 0 0
\(378\) −1.00000 −0.0514344
\(379\) −17.7538 −0.911951 −0.455975 0.889992i \(-0.650710\pi\)
−0.455975 + 0.889992i \(0.650710\pi\)
\(380\) −4.31534 −0.221372
\(381\) −13.1231 −0.672317
\(382\) 13.9309 0.712765
\(383\) 34.4233 1.75895 0.879474 0.475947i \(-0.157895\pi\)
0.879474 + 0.475947i \(0.157895\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −1.43845 −0.0733101
\(386\) −5.36932 −0.273291
\(387\) −10.5616 −0.536874
\(388\) 18.4924 0.938811
\(389\) 20.7386 1.05149 0.525745 0.850642i \(-0.323787\pi\)
0.525745 + 0.850642i \(0.323787\pi\)
\(390\) 0 0
\(391\) −8.17708 −0.413533
\(392\) −1.00000 −0.0505076
\(393\) −5.43845 −0.274333
\(394\) 19.1231 0.963408
\(395\) −1.61553 −0.0812860
\(396\) −2.56155 −0.128723
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 21.9309 1.09930
\(399\) −7.68466 −0.384714
\(400\) −4.68466 −0.234233
\(401\) 8.24621 0.411796 0.205898 0.978573i \(-0.433988\pi\)
0.205898 + 0.978573i \(0.433988\pi\)
\(402\) 1.12311 0.0560154
\(403\) 0 0
\(404\) 3.12311 0.155380
\(405\) 0.561553 0.0279038
\(406\) 5.68466 0.282125
\(407\) −8.80776 −0.436585
\(408\) −5.68466 −0.281433
\(409\) −23.9309 −1.18331 −0.591653 0.806193i \(-0.701524\pi\)
−0.591653 + 0.806193i \(0.701524\pi\)
\(410\) 4.00000 0.197546
\(411\) −0.561553 −0.0276994
\(412\) 11.6847 0.575662
\(413\) 14.2462 0.701010
\(414\) 1.43845 0.0706958
\(415\) −9.61553 −0.472008
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) −19.6847 −0.962808
\(419\) −24.3153 −1.18788 −0.593941 0.804509i \(-0.702429\pi\)
−0.593941 + 0.804509i \(0.702429\pi\)
\(420\) 0.561553 0.0274010
\(421\) 20.2462 0.986740 0.493370 0.869820i \(-0.335765\pi\)
0.493370 + 0.869820i \(0.335765\pi\)
\(422\) −4.80776 −0.234038
\(423\) 0 0
\(424\) 4.24621 0.206214
\(425\) −26.6307 −1.29178
\(426\) 8.00000 0.387601
\(427\) −5.68466 −0.275100
\(428\) −17.1231 −0.827677
\(429\) 0 0
\(430\) 5.93087 0.286012
\(431\) 8.63068 0.415725 0.207863 0.978158i \(-0.433349\pi\)
0.207863 + 0.978158i \(0.433349\pi\)
\(432\) 1.00000 0.0481125
\(433\) −6.63068 −0.318650 −0.159325 0.987226i \(-0.550932\pi\)
−0.159325 + 0.987226i \(0.550932\pi\)
\(434\) −10.2462 −0.491834
\(435\) −3.19224 −0.153056
\(436\) −11.9309 −0.571385
\(437\) 11.0540 0.528783
\(438\) 0.561553 0.0268320
\(439\) 21.9309 1.04670 0.523352 0.852117i \(-0.324681\pi\)
0.523352 + 0.852117i \(0.324681\pi\)
\(440\) 1.43845 0.0685753
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 19.3693 0.920264 0.460132 0.887851i \(-0.347802\pi\)
0.460132 + 0.887851i \(0.347802\pi\)
\(444\) 3.43845 0.163181
\(445\) −5.61553 −0.266202
\(446\) 17.6155 0.834119
\(447\) −23.6155 −1.11698
\(448\) 1.00000 0.0472456
\(449\) 1.68466 0.0795039 0.0397520 0.999210i \(-0.487343\pi\)
0.0397520 + 0.999210i \(0.487343\pi\)
\(450\) 4.68466 0.220837
\(451\) 18.2462 0.859181
\(452\) 12.2462 0.576013
\(453\) 8.80776 0.413825
\(454\) 1.12311 0.0527100
\(455\) 0 0
\(456\) 7.68466 0.359867
\(457\) −2.63068 −0.123058 −0.0615291 0.998105i \(-0.519598\pi\)
−0.0615291 + 0.998105i \(0.519598\pi\)
\(458\) 11.7538 0.549218
\(459\) 5.68466 0.265337
\(460\) −0.807764 −0.0376622
\(461\) −9.05398 −0.421686 −0.210843 0.977520i \(-0.567621\pi\)
−0.210843 + 0.977520i \(0.567621\pi\)
\(462\) 2.56155 0.119174
\(463\) −3.68466 −0.171241 −0.0856203 0.996328i \(-0.527287\pi\)
−0.0856203 + 0.996328i \(0.527287\pi\)
\(464\) −5.68466 −0.263904
\(465\) 5.75379 0.266826
\(466\) −2.63068 −0.121864
\(467\) −15.6847 −0.725799 −0.362900 0.931828i \(-0.618213\pi\)
−0.362900 + 0.931828i \(0.618213\pi\)
\(468\) 0 0
\(469\) −1.12311 −0.0518602
\(470\) 0 0
\(471\) −11.4384 −0.527056
\(472\) −14.2462 −0.655735
\(473\) 27.0540 1.24394
\(474\) 2.87689 0.132140
\(475\) 36.0000 1.65179
\(476\) 5.68466 0.260556
\(477\) −4.24621 −0.194421
\(478\) 16.0000 0.731823
\(479\) 21.3002 0.973230 0.486615 0.873616i \(-0.338231\pi\)
0.486615 + 0.873616i \(0.338231\pi\)
\(480\) −0.561553 −0.0256313
\(481\) 0 0
\(482\) −14.0000 −0.637683
\(483\) −1.43845 −0.0654516
\(484\) −4.43845 −0.201748
\(485\) 10.3845 0.471535
\(486\) −1.00000 −0.0453609
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 5.68466 0.257332
\(489\) 25.1231 1.13611
\(490\) −0.561553 −0.0253684
\(491\) −17.1231 −0.772755 −0.386377 0.922341i \(-0.626274\pi\)
−0.386377 + 0.922341i \(0.626274\pi\)
\(492\) −7.12311 −0.321134
\(493\) −32.3153 −1.45541
\(494\) 0 0
\(495\) −1.43845 −0.0646534
\(496\) 10.2462 0.460068
\(497\) −8.00000 −0.358849
\(498\) 17.1231 0.767305
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) −5.43845 −0.243215
\(501\) 5.93087 0.264972
\(502\) −12.8078 −0.571638
\(503\) 5.12311 0.228428 0.114214 0.993456i \(-0.463565\pi\)
0.114214 + 0.993456i \(0.463565\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 1.75379 0.0780426
\(506\) −3.68466 −0.163803
\(507\) 0 0
\(508\) −13.1231 −0.582244
\(509\) −25.0540 −1.11050 −0.555249 0.831684i \(-0.687377\pi\)
−0.555249 + 0.831684i \(0.687377\pi\)
\(510\) −3.19224 −0.141355
\(511\) −0.561553 −0.0248416
\(512\) −1.00000 −0.0441942
\(513\) −7.68466 −0.339286
\(514\) −12.2462 −0.540157
\(515\) 6.56155 0.289137
\(516\) −10.5616 −0.464946
\(517\) 0 0
\(518\) −3.43845 −0.151077
\(519\) 3.75379 0.164773
\(520\) 0 0
\(521\) −9.68466 −0.424293 −0.212146 0.977238i \(-0.568045\pi\)
−0.212146 + 0.977238i \(0.568045\pi\)
\(522\) 5.68466 0.248811
\(523\) 38.2462 1.67239 0.836195 0.548432i \(-0.184775\pi\)
0.836195 + 0.548432i \(0.184775\pi\)
\(524\) −5.43845 −0.237580
\(525\) −4.68466 −0.204455
\(526\) 13.7538 0.599694
\(527\) 58.2462 2.53724
\(528\) −2.56155 −0.111477
\(529\) −20.9309 −0.910038
\(530\) 2.38447 0.103575
\(531\) 14.2462 0.618233
\(532\) −7.68466 −0.333172
\(533\) 0 0
\(534\) 10.0000 0.432742
\(535\) −9.61553 −0.415716
\(536\) 1.12311 0.0485108
\(537\) −16.4924 −0.711701
\(538\) −3.75379 −0.161837
\(539\) −2.56155 −0.110334
\(540\) 0.561553 0.0241654
\(541\) 4.06913 0.174946 0.0874728 0.996167i \(-0.472121\pi\)
0.0874728 + 0.996167i \(0.472121\pi\)
\(542\) 13.1231 0.563686
\(543\) 16.2462 0.697192
\(544\) −5.68466 −0.243728
\(545\) −6.69981 −0.286988
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −0.561553 −0.0239883
\(549\) −5.68466 −0.242615
\(550\) −12.0000 −0.511682
\(551\) 43.6847 1.86103
\(552\) 1.43845 0.0612244
\(553\) −2.87689 −0.122338
\(554\) −10.4924 −0.445780
\(555\) 1.93087 0.0819609
\(556\) 12.0000 0.508913
\(557\) −21.3693 −0.905447 −0.452724 0.891651i \(-0.649548\pi\)
−0.452724 + 0.891651i \(0.649548\pi\)
\(558\) −10.2462 −0.433757
\(559\) 0 0
\(560\) 0.561553 0.0237299
\(561\) −14.5616 −0.614789
\(562\) −0.246211 −0.0103858
\(563\) 31.0540 1.30877 0.654385 0.756162i \(-0.272928\pi\)
0.654385 + 0.756162i \(0.272928\pi\)
\(564\) 0 0
\(565\) 6.87689 0.289313
\(566\) −1.75379 −0.0737172
\(567\) 1.00000 0.0419961
\(568\) 8.00000 0.335673
\(569\) −41.2311 −1.72850 −0.864248 0.503066i \(-0.832205\pi\)
−0.864248 + 0.503066i \(0.832205\pi\)
\(570\) 4.31534 0.180750
\(571\) 1.75379 0.0733938 0.0366969 0.999326i \(-0.488316\pi\)
0.0366969 + 0.999326i \(0.488316\pi\)
\(572\) 0 0
\(573\) −13.9309 −0.581970
\(574\) 7.12311 0.297313
\(575\) 6.73863 0.281020
\(576\) 1.00000 0.0416667
\(577\) 30.0000 1.24892 0.624458 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) −15.3153 −0.637034
\(579\) 5.36932 0.223141
\(580\) −3.19224 −0.132550
\(581\) −17.1231 −0.710386
\(582\) −18.4924 −0.766536
\(583\) 10.8769 0.450475
\(584\) 0.561553 0.0232372
\(585\) 0 0
\(586\) −24.7386 −1.02194
\(587\) 11.3693 0.469262 0.234631 0.972085i \(-0.424612\pi\)
0.234631 + 0.972085i \(0.424612\pi\)
\(588\) 1.00000 0.0412393
\(589\) −78.7386 −3.24437
\(590\) −8.00000 −0.329355
\(591\) −19.1231 −0.786619
\(592\) 3.43845 0.141319
\(593\) 3.75379 0.154150 0.0770748 0.997025i \(-0.475442\pi\)
0.0770748 + 0.997025i \(0.475442\pi\)
\(594\) 2.56155 0.105102
\(595\) 3.19224 0.130869
\(596\) −23.6155 −0.967330
\(597\) −21.9309 −0.897571
\(598\) 0 0
\(599\) 17.4384 0.712516 0.356258 0.934388i \(-0.384052\pi\)
0.356258 + 0.934388i \(0.384052\pi\)
\(600\) 4.68466 0.191250
\(601\) −19.1231 −0.780048 −0.390024 0.920805i \(-0.627533\pi\)
−0.390024 + 0.920805i \(0.627533\pi\)
\(602\) 10.5616 0.430457
\(603\) −1.12311 −0.0457364
\(604\) 8.80776 0.358383
\(605\) −2.49242 −0.101331
\(606\) −3.12311 −0.126867
\(607\) −22.5616 −0.915745 −0.457873 0.889018i \(-0.651388\pi\)
−0.457873 + 0.889018i \(0.651388\pi\)
\(608\) 7.68466 0.311654
\(609\) −5.68466 −0.230354
\(610\) 3.19224 0.129250
\(611\) 0 0
\(612\) 5.68466 0.229789
\(613\) 25.5464 1.03181 0.515905 0.856646i \(-0.327456\pi\)
0.515905 + 0.856646i \(0.327456\pi\)
\(614\) 9.75379 0.393631
\(615\) −4.00000 −0.161296
\(616\) 2.56155 0.103208
\(617\) −28.4233 −1.14428 −0.572139 0.820156i \(-0.693886\pi\)
−0.572139 + 0.820156i \(0.693886\pi\)
\(618\) −11.6847 −0.470026
\(619\) −31.6847 −1.27351 −0.636757 0.771065i \(-0.719725\pi\)
−0.636757 + 0.771065i \(0.719725\pi\)
\(620\) 5.75379 0.231078
\(621\) −1.43845 −0.0577229
\(622\) 21.1231 0.846959
\(623\) −10.0000 −0.400642
\(624\) 0 0
\(625\) 20.3693 0.814773
\(626\) −27.6155 −1.10374
\(627\) 19.6847 0.786130
\(628\) −11.4384 −0.456444
\(629\) 19.5464 0.779366
\(630\) −0.561553 −0.0223728
\(631\) −5.93087 −0.236104 −0.118052 0.993007i \(-0.537665\pi\)
−0.118052 + 0.993007i \(0.537665\pi\)
\(632\) 2.87689 0.114437
\(633\) 4.80776 0.191091
\(634\) 11.1231 0.441755
\(635\) −7.36932 −0.292442
\(636\) −4.24621 −0.168373
\(637\) 0 0
\(638\) −14.5616 −0.576497
\(639\) −8.00000 −0.316475
\(640\) −0.561553 −0.0221973
\(641\) −3.75379 −0.148266 −0.0741329 0.997248i \(-0.523619\pi\)
−0.0741329 + 0.997248i \(0.523619\pi\)
\(642\) 17.1231 0.675795
\(643\) −36.8078 −1.45156 −0.725778 0.687929i \(-0.758520\pi\)
−0.725778 + 0.687929i \(0.758520\pi\)
\(644\) −1.43845 −0.0566828
\(645\) −5.93087 −0.233528
\(646\) 43.6847 1.71875
\(647\) 2.24621 0.0883077 0.0441538 0.999025i \(-0.485941\pi\)
0.0441538 + 0.999025i \(0.485941\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −36.4924 −1.43245
\(650\) 0 0
\(651\) 10.2462 0.401581
\(652\) 25.1231 0.983897
\(653\) 11.9309 0.466891 0.233446 0.972370i \(-0.425000\pi\)
0.233446 + 0.972370i \(0.425000\pi\)
\(654\) 11.9309 0.466534
\(655\) −3.05398 −0.119329
\(656\) −7.12311 −0.278111
\(657\) −0.561553 −0.0219083
\(658\) 0 0
\(659\) −3.36932 −0.131250 −0.0656250 0.997844i \(-0.520904\pi\)
−0.0656250 + 0.997844i \(0.520904\pi\)
\(660\) −1.43845 −0.0559915
\(661\) −16.2462 −0.631904 −0.315952 0.948775i \(-0.602324\pi\)
−0.315952 + 0.948775i \(0.602324\pi\)
\(662\) −11.3693 −0.441881
\(663\) 0 0
\(664\) 17.1231 0.664505
\(665\) −4.31534 −0.167342
\(666\) −3.43845 −0.133237
\(667\) 8.17708 0.316618
\(668\) 5.93087 0.229472
\(669\) −17.6155 −0.681056
\(670\) 0.630683 0.0243654
\(671\) 14.5616 0.562143
\(672\) −1.00000 −0.0385758
\(673\) −13.1922 −0.508523 −0.254262 0.967135i \(-0.581832\pi\)
−0.254262 + 0.967135i \(0.581832\pi\)
\(674\) −8.56155 −0.329779
\(675\) −4.68466 −0.180313
\(676\) 0 0
\(677\) −30.4924 −1.17192 −0.585959 0.810340i \(-0.699282\pi\)
−0.585959 + 0.810340i \(0.699282\pi\)
\(678\) −12.2462 −0.470313
\(679\) 18.4924 0.709674
\(680\) −3.19224 −0.122417
\(681\) −1.12311 −0.0430375
\(682\) 26.2462 1.00502
\(683\) −47.6847 −1.82460 −0.912301 0.409519i \(-0.865696\pi\)
−0.912301 + 0.409519i \(0.865696\pi\)
\(684\) −7.68466 −0.293830
\(685\) −0.315342 −0.0120486
\(686\) −1.00000 −0.0381802
\(687\) −11.7538 −0.448435
\(688\) −10.5616 −0.402655
\(689\) 0 0
\(690\) 0.807764 0.0307511
\(691\) 16.4924 0.627401 0.313701 0.949522i \(-0.398431\pi\)
0.313701 + 0.949522i \(0.398431\pi\)
\(692\) 3.75379 0.142698
\(693\) −2.56155 −0.0973053
\(694\) 20.0000 0.759190
\(695\) 6.73863 0.255611
\(696\) 5.68466 0.215476
\(697\) −40.4924 −1.53376
\(698\) 24.2462 0.917733
\(699\) 2.63068 0.0995016
\(700\) −4.68466 −0.177063
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) −26.4233 −0.996573
\(704\) −2.56155 −0.0965422
\(705\) 0 0
\(706\) −2.49242 −0.0938036
\(707\) 3.12311 0.117456
\(708\) 14.2462 0.535405
\(709\) −0.246211 −0.00924666 −0.00462333 0.999989i \(-0.501472\pi\)
−0.00462333 + 0.999989i \(0.501472\pi\)
\(710\) 4.49242 0.168598
\(711\) −2.87689 −0.107892
\(712\) 10.0000 0.374766
\(713\) −14.7386 −0.551966
\(714\) −5.68466 −0.212743
\(715\) 0 0
\(716\) −16.4924 −0.616351
\(717\) −16.0000 −0.597531
\(718\) 22.7386 0.848598
\(719\) 34.8769 1.30069 0.650344 0.759640i \(-0.274625\pi\)
0.650344 + 0.759640i \(0.274625\pi\)
\(720\) 0.561553 0.0209278
\(721\) 11.6847 0.435159
\(722\) −40.0540 −1.49065
\(723\) 14.0000 0.520666
\(724\) 16.2462 0.603786
\(725\) 26.6307 0.989039
\(726\) 4.43845 0.164726
\(727\) 21.9309 0.813371 0.406685 0.913568i \(-0.366684\pi\)
0.406685 + 0.913568i \(0.366684\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0.315342 0.0116713
\(731\) −60.0388 −2.22062
\(732\) −5.68466 −0.210111
\(733\) 0.384472 0.0142008 0.00710040 0.999975i \(-0.497740\pi\)
0.00710040 + 0.999975i \(0.497740\pi\)
\(734\) 16.0000 0.590571
\(735\) 0.561553 0.0207132
\(736\) 1.43845 0.0530219
\(737\) 2.87689 0.105972
\(738\) 7.12311 0.262205
\(739\) −42.1080 −1.54897 −0.774483 0.632595i \(-0.781990\pi\)
−0.774483 + 0.632595i \(0.781990\pi\)
\(740\) 1.93087 0.0709802
\(741\) 0 0
\(742\) 4.24621 0.155883
\(743\) 46.1080 1.69154 0.845768 0.533550i \(-0.179143\pi\)
0.845768 + 0.533550i \(0.179143\pi\)
\(744\) −10.2462 −0.375644
\(745\) −13.2614 −0.485859
\(746\) −23.6155 −0.864626
\(747\) −17.1231 −0.626502
\(748\) −14.5616 −0.532423
\(749\) −17.1231 −0.625665
\(750\) 5.43845 0.198584
\(751\) 10.2462 0.373890 0.186945 0.982370i \(-0.440141\pi\)
0.186945 + 0.982370i \(0.440141\pi\)
\(752\) 0 0
\(753\) 12.8078 0.466741
\(754\) 0 0
\(755\) 4.94602 0.180004
\(756\) 1.00000 0.0363696
\(757\) 1.50758 0.0547938 0.0273969 0.999625i \(-0.491278\pi\)
0.0273969 + 0.999625i \(0.491278\pi\)
\(758\) 17.7538 0.644847
\(759\) 3.68466 0.133745
\(760\) 4.31534 0.156534
\(761\) 3.12311 0.113212 0.0566062 0.998397i \(-0.481972\pi\)
0.0566062 + 0.998397i \(0.481972\pi\)
\(762\) 13.1231 0.475400
\(763\) −11.9309 −0.431926
\(764\) −13.9309 −0.504001
\(765\) 3.19224 0.115416
\(766\) −34.4233 −1.24376
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 20.5616 0.741469 0.370734 0.928739i \(-0.379106\pi\)
0.370734 + 0.928739i \(0.379106\pi\)
\(770\) 1.43845 0.0518380
\(771\) 12.2462 0.441037
\(772\) 5.36932 0.193246
\(773\) −24.0691 −0.865706 −0.432853 0.901464i \(-0.642493\pi\)
−0.432853 + 0.901464i \(0.642493\pi\)
\(774\) 10.5616 0.379627
\(775\) −48.0000 −1.72421
\(776\) −18.4924 −0.663839
\(777\) 3.43845 0.123354
\(778\) −20.7386 −0.743516
\(779\) 54.7386 1.96122
\(780\) 0 0
\(781\) 20.4924 0.733277
\(782\) 8.17708 0.292412
\(783\) −5.68466 −0.203153
\(784\) 1.00000 0.0357143
\(785\) −6.42329 −0.229257
\(786\) 5.43845 0.193983
\(787\) 33.3002 1.18702 0.593512 0.804825i \(-0.297741\pi\)
0.593512 + 0.804825i \(0.297741\pi\)
\(788\) −19.1231 −0.681232
\(789\) −13.7538 −0.489648
\(790\) 1.61553 0.0574779
\(791\) 12.2462 0.435425
\(792\) 2.56155 0.0910208
\(793\) 0 0
\(794\) −18.0000 −0.638796
\(795\) −2.38447 −0.0845685
\(796\) −21.9309 −0.777319
\(797\) 6.63068 0.234871 0.117435 0.993081i \(-0.462533\pi\)
0.117435 + 0.993081i \(0.462533\pi\)
\(798\) 7.68466 0.272034
\(799\) 0 0
\(800\) 4.68466 0.165628
\(801\) −10.0000 −0.353333
\(802\) −8.24621 −0.291184
\(803\) 1.43845 0.0507617
\(804\) −1.12311 −0.0396089
\(805\) −0.807764 −0.0284699
\(806\) 0 0
\(807\) 3.75379 0.132140
\(808\) −3.12311 −0.109870
\(809\) 30.4924 1.07206 0.536028 0.844200i \(-0.319924\pi\)
0.536028 + 0.844200i \(0.319924\pi\)
\(810\) −0.561553 −0.0197310
\(811\) −37.4384 −1.31464 −0.657321 0.753611i \(-0.728310\pi\)
−0.657321 + 0.753611i \(0.728310\pi\)
\(812\) −5.68466 −0.199492
\(813\) −13.1231 −0.460247
\(814\) 8.80776 0.308712
\(815\) 14.1080 0.494180
\(816\) 5.68466 0.199003
\(817\) 81.1619 2.83950
\(818\) 23.9309 0.836723
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) −55.6155 −1.94100 −0.970498 0.241111i \(-0.922488\pi\)
−0.970498 + 0.241111i \(0.922488\pi\)
\(822\) 0.561553 0.0195864
\(823\) −28.4924 −0.993183 −0.496592 0.867984i \(-0.665415\pi\)
−0.496592 + 0.867984i \(0.665415\pi\)
\(824\) −11.6847 −0.407054
\(825\) 12.0000 0.417786
\(826\) −14.2462 −0.495689
\(827\) 1.93087 0.0671429 0.0335715 0.999436i \(-0.489312\pi\)
0.0335715 + 0.999436i \(0.489312\pi\)
\(828\) −1.43845 −0.0499895
\(829\) 26.3153 0.913970 0.456985 0.889475i \(-0.348929\pi\)
0.456985 + 0.889475i \(0.348929\pi\)
\(830\) 9.61553 0.333760
\(831\) 10.4924 0.363978
\(832\) 0 0
\(833\) 5.68466 0.196962
\(834\) −12.0000 −0.415526
\(835\) 3.33050 0.115257
\(836\) 19.6847 0.680808
\(837\) 10.2462 0.354161
\(838\) 24.3153 0.839960
\(839\) −38.7386 −1.33741 −0.668703 0.743530i \(-0.733150\pi\)
−0.668703 + 0.743530i \(0.733150\pi\)
\(840\) −0.561553 −0.0193754
\(841\) 3.31534 0.114322
\(842\) −20.2462 −0.697731
\(843\) 0.246211 0.00847997
\(844\) 4.80776 0.165490
\(845\) 0 0
\(846\) 0 0
\(847\) −4.43845 −0.152507
\(848\) −4.24621 −0.145815
\(849\) 1.75379 0.0601899
\(850\) 26.6307 0.913425
\(851\) −4.94602 −0.169548
\(852\) −8.00000 −0.274075
\(853\) 2.63068 0.0900729 0.0450364 0.998985i \(-0.485660\pi\)
0.0450364 + 0.998985i \(0.485660\pi\)
\(854\) 5.68466 0.194525
\(855\) −4.31534 −0.147582
\(856\) 17.1231 0.585256
\(857\) −1.50758 −0.0514979 −0.0257489 0.999668i \(-0.508197\pi\)
−0.0257489 + 0.999668i \(0.508197\pi\)
\(858\) 0 0
\(859\) 22.2462 0.759031 0.379515 0.925185i \(-0.376091\pi\)
0.379515 + 0.925185i \(0.376091\pi\)
\(860\) −5.93087 −0.202241
\(861\) −7.12311 −0.242755
\(862\) −8.63068 −0.293962
\(863\) −7.36932 −0.250854 −0.125427 0.992103i \(-0.540030\pi\)
−0.125427 + 0.992103i \(0.540030\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 2.10795 0.0716725
\(866\) 6.63068 0.225320
\(867\) 15.3153 0.520136
\(868\) 10.2462 0.347779
\(869\) 7.36932 0.249987
\(870\) 3.19224 0.108227
\(871\) 0 0
\(872\) 11.9309 0.404030
\(873\) 18.4924 0.625874
\(874\) −11.0540 −0.373906
\(875\) −5.43845 −0.183853
\(876\) −0.561553 −0.0189731
\(877\) 23.7538 0.802108 0.401054 0.916054i \(-0.368644\pi\)
0.401054 + 0.916054i \(0.368644\pi\)
\(878\) −21.9309 −0.740131
\(879\) 24.7386 0.834413
\(880\) −1.43845 −0.0484900
\(881\) 26.1771 0.881928 0.440964 0.897525i \(-0.354637\pi\)
0.440964 + 0.897525i \(0.354637\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −2.56155 −0.0862031 −0.0431016 0.999071i \(-0.513724\pi\)
−0.0431016 + 0.999071i \(0.513724\pi\)
\(884\) 0 0
\(885\) 8.00000 0.268917
\(886\) −19.3693 −0.650725
\(887\) −19.5076 −0.655000 −0.327500 0.944851i \(-0.606206\pi\)
−0.327500 + 0.944851i \(0.606206\pi\)
\(888\) −3.43845 −0.115387
\(889\) −13.1231 −0.440135
\(890\) 5.61553 0.188233
\(891\) −2.56155 −0.0858152
\(892\) −17.6155 −0.589812
\(893\) 0 0
\(894\) 23.6155 0.789821
\(895\) −9.26137 −0.309573
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −1.68466 −0.0562178
\(899\) −58.2462 −1.94262
\(900\) −4.68466 −0.156155
\(901\) −24.1383 −0.804162
\(902\) −18.2462 −0.607532
\(903\) −10.5616 −0.351466
\(904\) −12.2462 −0.407303
\(905\) 9.12311 0.303262
\(906\) −8.80776 −0.292618
\(907\) −40.4924 −1.34453 −0.672264 0.740311i \(-0.734678\pi\)
−0.672264 + 0.740311i \(0.734678\pi\)
\(908\) −1.12311 −0.0372716
\(909\) 3.12311 0.103587
\(910\) 0 0
\(911\) 50.4233 1.67060 0.835299 0.549796i \(-0.185294\pi\)
0.835299 + 0.549796i \(0.185294\pi\)
\(912\) −7.68466 −0.254464
\(913\) 43.8617 1.45161
\(914\) 2.63068 0.0870153
\(915\) −3.19224 −0.105532
\(916\) −11.7538 −0.388356
\(917\) −5.43845 −0.179593
\(918\) −5.68466 −0.187622
\(919\) 10.8769 0.358796 0.179398 0.983777i \(-0.442585\pi\)
0.179398 + 0.983777i \(0.442585\pi\)
\(920\) 0.807764 0.0266312
\(921\) −9.75379 −0.321398
\(922\) 9.05398 0.298177
\(923\) 0 0
\(924\) −2.56155 −0.0842689
\(925\) −16.1080 −0.529626
\(926\) 3.68466 0.121085
\(927\) 11.6847 0.383775
\(928\) 5.68466 0.186608
\(929\) 15.6155 0.512329 0.256164 0.966633i \(-0.417541\pi\)
0.256164 + 0.966633i \(0.417541\pi\)
\(930\) −5.75379 −0.188674
\(931\) −7.68466 −0.251855
\(932\) 2.63068 0.0861709
\(933\) −21.1231 −0.691539
\(934\) 15.6847 0.513218
\(935\) −8.17708 −0.267419
\(936\) 0 0
\(937\) 18.9848 0.620208 0.310104 0.950703i \(-0.399636\pi\)
0.310104 + 0.950703i \(0.399636\pi\)
\(938\) 1.12311 0.0366707
\(939\) 27.6155 0.901199
\(940\) 0 0
\(941\) 34.0000 1.10837 0.554184 0.832394i \(-0.313030\pi\)
0.554184 + 0.832394i \(0.313030\pi\)
\(942\) 11.4384 0.372685
\(943\) 10.2462 0.333663
\(944\) 14.2462 0.463675
\(945\) 0.561553 0.0182673
\(946\) −27.0540 −0.879601
\(947\) −45.7926 −1.48806 −0.744030 0.668146i \(-0.767088\pi\)
−0.744030 + 0.668146i \(0.767088\pi\)
\(948\) −2.87689 −0.0934372
\(949\) 0 0
\(950\) −36.0000 −1.16799
\(951\) −11.1231 −0.360691
\(952\) −5.68466 −0.184241
\(953\) −13.3693 −0.433075 −0.216537 0.976274i \(-0.569476\pi\)
−0.216537 + 0.976274i \(0.569476\pi\)
\(954\) 4.24621 0.137476
\(955\) −7.82292 −0.253144
\(956\) −16.0000 −0.517477
\(957\) 14.5616 0.470708
\(958\) −21.3002 −0.688178
\(959\) −0.561553 −0.0181335
\(960\) 0.561553 0.0181240
\(961\) 73.9848 2.38661
\(962\) 0 0
\(963\) −17.1231 −0.551784
\(964\) 14.0000 0.450910
\(965\) 3.01515 0.0970613
\(966\) 1.43845 0.0462813
\(967\) 17.4384 0.560783 0.280391 0.959886i \(-0.409536\pi\)
0.280391 + 0.959886i \(0.409536\pi\)
\(968\) 4.43845 0.142657
\(969\) −43.6847 −1.40335
\(970\) −10.3845 −0.333425
\(971\) −14.2462 −0.457183 −0.228591 0.973522i \(-0.573412\pi\)
−0.228591 + 0.973522i \(0.573412\pi\)
\(972\) 1.00000 0.0320750
\(973\) 12.0000 0.384702
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) −5.68466 −0.181961
\(977\) 10.3153 0.330017 0.165009 0.986292i \(-0.447235\pi\)
0.165009 + 0.986292i \(0.447235\pi\)
\(978\) −25.1231 −0.803348
\(979\) 25.6155 0.818676
\(980\) 0.561553 0.0179381
\(981\) −11.9309 −0.380923
\(982\) 17.1231 0.546420
\(983\) −32.1771 −1.02629 −0.513145 0.858302i \(-0.671520\pi\)
−0.513145 + 0.858302i \(0.671520\pi\)
\(984\) 7.12311 0.227076
\(985\) −10.7386 −0.342161
\(986\) 32.3153 1.02913
\(987\) 0 0
\(988\) 0 0
\(989\) 15.1922 0.483085
\(990\) 1.43845 0.0457169
\(991\) 33.6155 1.06783 0.533916 0.845537i \(-0.320720\pi\)
0.533916 + 0.845537i \(0.320720\pi\)
\(992\) −10.2462 −0.325318
\(993\) 11.3693 0.360794
\(994\) 8.00000 0.253745
\(995\) −12.3153 −0.390423
\(996\) −17.1231 −0.542566
\(997\) −8.73863 −0.276755 −0.138378 0.990380i \(-0.544189\pi\)
−0.138378 + 0.990380i \(0.544189\pi\)
\(998\) 36.0000 1.13956
\(999\) 3.43845 0.108788
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.bl.1.2 2
13.12 even 2 546.2.a.j.1.1 2
39.38 odd 2 1638.2.a.u.1.2 2
52.51 odd 2 4368.2.a.be.1.1 2
91.90 odd 2 3822.2.a.bo.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.j.1.1 2 13.12 even 2
1638.2.a.u.1.2 2 39.38 odd 2
3822.2.a.bo.1.2 2 91.90 odd 2
4368.2.a.be.1.1 2 52.51 odd 2
7098.2.a.bl.1.2 2 1.1 even 1 trivial