Properties

Label 6762.2.a.cg.1.4
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.138768.1
Defining polynomial: \(x^{4} - 2 x^{3} - 11 x^{2} + 12 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(3.89728\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.89728 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.89728 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -3.89728 q^{10} +4.02636 q^{11} -1.00000 q^{12} -0.619394 q^{13} -3.89728 q^{15} +1.00000 q^{16} -1.25153 q^{17} -1.00000 q^{18} -8.41395 q^{19} +3.89728 q^{20} -4.02636 q^{22} +1.00000 q^{23} +1.00000 q^{24} +10.1888 q^{25} +0.619394 q^{26} -1.00000 q^{27} -9.41395 q^{29} +3.89728 q^{30} +5.18878 q^{31} -1.00000 q^{32} -4.02636 q^{33} +1.25153 q^{34} +1.00000 q^{36} +1.38061 q^{37} +8.41395 q^{38} +0.619394 q^{39} -3.89728 q^{40} -4.41395 q^{41} +2.00000 q^{43} +4.02636 q^{44} +3.89728 q^{45} -1.00000 q^{46} -6.13607 q^{47} -1.00000 q^{48} -10.1888 q^{50} +1.25153 q^{51} -0.619394 q^{52} -1.89728 q^{53} +1.00000 q^{54} +15.6918 q^{55} +8.41395 q^{57} +9.41395 q^{58} +11.0197 q^{59} -3.89728 q^{60} -0.205443 q^{61} -5.18878 q^{62} +1.00000 q^{64} -2.41395 q^{65} +4.02636 q^{66} +12.0527 q^{67} -1.25153 q^{68} -1.00000 q^{69} +0.483327 q^{71} -1.00000 q^{72} +3.15544 q^{73} -1.38061 q^{74} -10.1888 q^{75} -8.41395 q^{76} -0.619394 q^{78} +15.4737 q^{79} +3.89728 q^{80} +1.00000 q^{81} +4.41395 q^{82} +7.13607 q^{83} -4.87755 q^{85} -2.00000 q^{86} +9.41395 q^{87} -4.02636 q^{88} +10.2085 q^{89} -3.89728 q^{90} +1.00000 q^{92} -5.18878 q^{93} +6.13607 q^{94} -32.7915 q^{95} +1.00000 q^{96} +12.0531 q^{97} +4.02636 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} - 4 q^{8} + 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} - 4 q^{8} + 4 q^{9} - 2 q^{10} + 6 q^{11} - 4 q^{12} - 2 q^{13} - 2 q^{15} + 4 q^{16} - 2 q^{17} - 4 q^{18} - 6 q^{19} + 2 q^{20} - 6 q^{22} + 4 q^{23} + 4 q^{24} + 6 q^{25} + 2 q^{26} - 4 q^{27} - 10 q^{29} + 2 q^{30} - 14 q^{31} - 4 q^{32} - 6 q^{33} + 2 q^{34} + 4 q^{36} + 6 q^{37} + 6 q^{38} + 2 q^{39} - 2 q^{40} + 10 q^{41} + 8 q^{43} + 6 q^{44} + 2 q^{45} - 4 q^{46} - 10 q^{47} - 4 q^{48} - 6 q^{50} + 2 q^{51} - 2 q^{52} + 6 q^{53} + 4 q^{54} + 22 q^{55} + 6 q^{57} + 10 q^{58} + 24 q^{59} - 2 q^{60} - 28 q^{61} + 14 q^{62} + 4 q^{64} + 18 q^{65} + 6 q^{66} + 28 q^{67} - 2 q^{68} - 4 q^{69} + 16 q^{71} - 4 q^{72} + 6 q^{73} - 6 q^{74} - 6 q^{75} - 6 q^{76} - 2 q^{78} - 4 q^{79} + 2 q^{80} + 4 q^{81} - 10 q^{82} + 14 q^{83} - 26 q^{85} - 8 q^{86} + 10 q^{87} - 6 q^{88} - 14 q^{89} - 2 q^{90} + 4 q^{92} + 14 q^{93} + 10 q^{94} - 34 q^{95} + 4 q^{96} + 6 q^{99} + O(q^{100}) \)

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
\(5\) 3.89728 1.74292 0.871458 0.490470i \(-0.163175\pi\)
0.871458 + 0.490470i \(0.163175\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.89728 −1.23243
\(11\) 4.02636 1.21399 0.606996 0.794705i \(-0.292374\pi\)
0.606996 + 0.794705i \(0.292374\pi\)
\(12\) −1.00000 −0.288675
\(13\) −0.619394 −0.171789 −0.0858945 0.996304i \(-0.527375\pi\)
−0.0858945 + 0.996304i \(0.527375\pi\)
\(14\) 0 0
\(15\) −3.89728 −1.00627
\(16\) 1.00000 0.250000
\(17\) −1.25153 −0.303540 −0.151770 0.988416i \(-0.548497\pi\)
−0.151770 + 0.988416i \(0.548497\pi\)
\(18\) −1.00000 −0.235702
\(19\) −8.41395 −1.93029 −0.965146 0.261710i \(-0.915713\pi\)
−0.965146 + 0.261710i \(0.915713\pi\)
\(20\) 3.89728 0.871458
\(21\) 0 0
\(22\) −4.02636 −0.858422
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 10.1888 2.03776
\(26\) 0.619394 0.121473
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −9.41395 −1.74813 −0.874063 0.485812i \(-0.838524\pi\)
−0.874063 + 0.485812i \(0.838524\pi\)
\(30\) 3.89728 0.711542
\(31\) 5.18878 0.931933 0.465966 0.884802i \(-0.345707\pi\)
0.465966 + 0.884802i \(0.345707\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.02636 −0.700899
\(34\) 1.25153 0.214635
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.38061 0.226970 0.113485 0.993540i \(-0.463799\pi\)
0.113485 + 0.993540i \(0.463799\pi\)
\(38\) 8.41395 1.36492
\(39\) 0.619394 0.0991824
\(40\) −3.89728 −0.616214
\(41\) −4.41395 −0.689343 −0.344672 0.938723i \(-0.612010\pi\)
−0.344672 + 0.938723i \(0.612010\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 4.02636 0.606996
\(45\) 3.89728 0.580972
\(46\) −1.00000 −0.147442
\(47\) −6.13607 −0.895037 −0.447519 0.894275i \(-0.647692\pi\)
−0.447519 + 0.894275i \(0.647692\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −10.1888 −1.44091
\(51\) 1.25153 0.175249
\(52\) −0.619394 −0.0858945
\(53\) −1.89728 −0.260611 −0.130306 0.991474i \(-0.541596\pi\)
−0.130306 + 0.991474i \(0.541596\pi\)
\(54\) 1.00000 0.136083
\(55\) 15.6918 2.11589
\(56\) 0 0
\(57\) 8.41395 1.11446
\(58\) 9.41395 1.23611
\(59\) 11.0197 1.43465 0.717323 0.696741i \(-0.245367\pi\)
0.717323 + 0.696741i \(0.245367\pi\)
\(60\) −3.89728 −0.503137
\(61\) −0.205443 −0.0263042 −0.0131521 0.999914i \(-0.504187\pi\)
−0.0131521 + 0.999914i \(0.504187\pi\)
\(62\) −5.18878 −0.658976
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.41395 −0.299414
\(66\) 4.02636 0.495610
\(67\) 12.0527 1.47247 0.736237 0.676724i \(-0.236601\pi\)
0.736237 + 0.676724i \(0.236601\pi\)
\(68\) −1.25153 −0.151770
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 0.483327 0.0573604 0.0286802 0.999589i \(-0.490870\pi\)
0.0286802 + 0.999589i \(0.490870\pi\)
\(72\) −1.00000 −0.117851
\(73\) 3.15544 0.369316 0.184658 0.982803i \(-0.440882\pi\)
0.184658 + 0.982803i \(0.440882\pi\)
\(74\) −1.38061 −0.160492
\(75\) −10.1888 −1.17650
\(76\) −8.41395 −0.965146
\(77\) 0 0
\(78\) −0.619394 −0.0701326
\(79\) 15.4737 1.74092 0.870461 0.492237i \(-0.163821\pi\)
0.870461 + 0.492237i \(0.163821\pi\)
\(80\) 3.89728 0.435729
\(81\) 1.00000 0.111111
\(82\) 4.41395 0.487439
\(83\) 7.13607 0.783285 0.391643 0.920117i \(-0.371907\pi\)
0.391643 + 0.920117i \(0.371907\pi\)
\(84\) 0 0
\(85\) −4.87755 −0.529045
\(86\) −2.00000 −0.215666
\(87\) 9.41395 1.00928
\(88\) −4.02636 −0.429211
\(89\) 10.2085 1.08210 0.541050 0.840990i \(-0.318027\pi\)
0.541050 + 0.840990i \(0.318027\pi\)
\(90\) −3.89728 −0.410809
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) −5.18878 −0.538052
\(94\) 6.13607 0.632887
\(95\) −32.7915 −3.36434
\(96\) 1.00000 0.102062
\(97\) 12.0531 1.22380 0.611902 0.790934i \(-0.290405\pi\)
0.611902 + 0.790934i \(0.290405\pi\)
\(98\) 0 0
\(99\) 4.02636 0.404664
\(100\) 10.1888 1.01888
\(101\) −10.1888 −1.01382 −0.506911 0.861999i \(-0.669213\pi\)
−0.506911 + 0.861999i \(0.669213\pi\)
\(102\) −1.25153 −0.123920
\(103\) 6.95698 0.685492 0.342746 0.939428i \(-0.388643\pi\)
0.342746 + 0.939428i \(0.388643\pi\)
\(104\) 0.619394 0.0607366
\(105\) 0 0
\(106\) 1.89728 0.184280
\(107\) 11.1361 1.07656 0.538282 0.842765i \(-0.319073\pi\)
0.538282 + 0.842765i \(0.319073\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 1.36787 0.131018 0.0655089 0.997852i \(-0.479133\pi\)
0.0655089 + 0.997852i \(0.479133\pi\)
\(110\) −15.6918 −1.49616
\(111\) −1.38061 −0.131041
\(112\) 0 0
\(113\) 19.7582 1.85869 0.929346 0.369210i \(-0.120372\pi\)
0.929346 + 0.369210i \(0.120372\pi\)
\(114\) −8.41395 −0.788039
\(115\) 3.89728 0.363423
\(116\) −9.41395 −0.874063
\(117\) −0.619394 −0.0572630
\(118\) −11.0197 −1.01445
\(119\) 0 0
\(120\) 3.89728 0.355771
\(121\) 5.21155 0.473777
\(122\) 0.205443 0.0185999
\(123\) 4.41395 0.397993
\(124\) 5.18878 0.465966
\(125\) 20.2221 1.80872
\(126\) 0 0
\(127\) −15.4337 −1.36952 −0.684759 0.728770i \(-0.740092\pi\)
−0.684759 + 0.728770i \(0.740092\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.00000 −0.176090
\(130\) 2.41395 0.211718
\(131\) 20.8942 1.82554 0.912769 0.408476i \(-0.133940\pi\)
0.912769 + 0.408476i \(0.133940\pi\)
\(132\) −4.02636 −0.350449
\(133\) 0 0
\(134\) −12.0527 −1.04120
\(135\) −3.89728 −0.335424
\(136\) 1.25153 0.107318
\(137\) 6.37398 0.544566 0.272283 0.962217i \(-0.412221\pi\)
0.272283 + 0.962217i \(0.412221\pi\)
\(138\) 1.00000 0.0851257
\(139\) −2.13607 −0.181179 −0.0905894 0.995888i \(-0.528875\pi\)
−0.0905894 + 0.995888i \(0.528875\pi\)
\(140\) 0 0
\(141\) 6.13607 0.516750
\(142\) −0.483327 −0.0405599
\(143\) −2.49390 −0.208551
\(144\) 1.00000 0.0833333
\(145\) −36.6888 −3.04684
\(146\) −3.15544 −0.261146
\(147\) 0 0
\(148\) 1.38061 0.113485
\(149\) 11.8218 0.968479 0.484240 0.874935i \(-0.339096\pi\)
0.484240 + 0.874935i \(0.339096\pi\)
\(150\) 10.1888 0.831911
\(151\) 9.43368 0.767702 0.383851 0.923395i \(-0.374598\pi\)
0.383851 + 0.923395i \(0.374598\pi\)
\(152\) 8.41395 0.682462
\(153\) −1.25153 −0.101180
\(154\) 0 0
\(155\) 20.2221 1.62428
\(156\) 0.619394 0.0495912
\(157\) 3.78182 0.301822 0.150911 0.988547i \(-0.451779\pi\)
0.150911 + 0.988547i \(0.451779\pi\)
\(158\) −15.4737 −1.23102
\(159\) 1.89728 0.150464
\(160\) −3.89728 −0.308107
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 4.48333 0.351161 0.175581 0.984465i \(-0.443820\pi\)
0.175581 + 0.984465i \(0.443820\pi\)
\(164\) −4.41395 −0.344672
\(165\) −15.6918 −1.22161
\(166\) −7.13607 −0.553866
\(167\) 17.1388 1.32624 0.663119 0.748514i \(-0.269232\pi\)
0.663119 + 0.748514i \(0.269232\pi\)
\(168\) 0 0
\(169\) −12.6164 −0.970489
\(170\) 4.87755 0.374091
\(171\) −8.41395 −0.643431
\(172\) 2.00000 0.152499
\(173\) −2.84456 −0.216268 −0.108134 0.994136i \(-0.534488\pi\)
−0.108134 + 0.994136i \(0.534488\pi\)
\(174\) −9.41395 −0.713670
\(175\) 0 0
\(176\) 4.02636 0.303498
\(177\) −11.0197 −0.828293
\(178\) −10.2085 −0.765160
\(179\) 13.2782 0.992463 0.496231 0.868190i \(-0.334717\pi\)
0.496231 + 0.868190i \(0.334717\pi\)
\(180\) 3.89728 0.290486
\(181\) 14.5066 1.07827 0.539135 0.842219i \(-0.318751\pi\)
0.539135 + 0.842219i \(0.318751\pi\)
\(182\) 0 0
\(183\) 0.205443 0.0151868
\(184\) −1.00000 −0.0737210
\(185\) 5.38061 0.395590
\(186\) 5.18878 0.380460
\(187\) −5.03910 −0.368495
\(188\) −6.13607 −0.447519
\(189\) 0 0
\(190\) 32.7915 2.37895
\(191\) −15.2419 −1.10286 −0.551431 0.834221i \(-0.685918\pi\)
−0.551431 + 0.834221i \(0.685918\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 11.9500 0.860179 0.430090 0.902786i \(-0.358482\pi\)
0.430090 + 0.902786i \(0.358482\pi\)
\(194\) −12.0531 −0.865360
\(195\) 2.41395 0.172867
\(196\) 0 0
\(197\) −18.5391 −1.32086 −0.660428 0.750889i \(-0.729625\pi\)
−0.660428 + 0.750889i \(0.729625\pi\)
\(198\) −4.02636 −0.286141
\(199\) −8.91753 −0.632147 −0.316073 0.948735i \(-0.602365\pi\)
−0.316073 + 0.948735i \(0.602365\pi\)
\(200\) −10.1888 −0.720456
\(201\) −12.0527 −0.850133
\(202\) 10.1888 0.716880
\(203\) 0 0
\(204\) 1.25153 0.0876244
\(205\) −17.2024 −1.20147
\(206\) −6.95698 −0.484716
\(207\) 1.00000 0.0695048
\(208\) −0.619394 −0.0429473
\(209\) −33.8776 −2.34336
\(210\) 0 0
\(211\) 17.1694 1.18199 0.590996 0.806675i \(-0.298735\pi\)
0.590996 + 0.806675i \(0.298735\pi\)
\(212\) −1.89728 −0.130306
\(213\) −0.483327 −0.0331170
\(214\) −11.1361 −0.761246
\(215\) 7.79456 0.531584
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −1.36787 −0.0926436
\(219\) −3.15544 −0.213225
\(220\) 15.6918 1.05794
\(221\) 0.775189 0.0521448
\(222\) 1.38061 0.0926602
\(223\) −9.39422 −0.629084 −0.314542 0.949244i \(-0.601851\pi\)
−0.314542 + 0.949244i \(0.601851\pi\)
\(224\) 0 0
\(225\) 10.1888 0.679252
\(226\) −19.7582 −1.31429
\(227\) 25.3179 1.68041 0.840203 0.542272i \(-0.182436\pi\)
0.840203 + 0.542272i \(0.182436\pi\)
\(228\) 8.41395 0.557228
\(229\) −11.9600 −0.790341 −0.395170 0.918608i \(-0.629314\pi\)
−0.395170 + 0.918608i \(0.629314\pi\)
\(230\) −3.89728 −0.256979
\(231\) 0 0
\(232\) 9.41395 0.618056
\(233\) −8.86429 −0.580719 −0.290360 0.956918i \(-0.593775\pi\)
−0.290360 + 0.956918i \(0.593775\pi\)
\(234\) 0.619394 0.0404911
\(235\) −23.9140 −1.55997
\(236\) 11.0197 0.717323
\(237\) −15.4737 −1.00512
\(238\) 0 0
\(239\) 9.93062 0.642359 0.321179 0.947018i \(-0.395921\pi\)
0.321179 + 0.947018i \(0.395921\pi\)
\(240\) −3.89728 −0.251568
\(241\) −12.2840 −0.791282 −0.395641 0.918405i \(-0.629477\pi\)
−0.395641 + 0.918405i \(0.629477\pi\)
\(242\) −5.21155 −0.335011
\(243\) −1.00000 −0.0641500
\(244\) −0.205443 −0.0131521
\(245\) 0 0
\(246\) −4.41395 −0.281423
\(247\) 5.21155 0.331603
\(248\) −5.18878 −0.329488
\(249\) −7.13607 −0.452230
\(250\) −20.2221 −1.27896
\(251\) 6.81481 0.430147 0.215073 0.976598i \(-0.431001\pi\)
0.215073 + 0.976598i \(0.431001\pi\)
\(252\) 0 0
\(253\) 4.02636 0.253135
\(254\) 15.4337 0.968395
\(255\) 4.87755 0.305444
\(256\) 1.00000 0.0625000
\(257\) −23.5000 −1.46589 −0.732945 0.680288i \(-0.761855\pi\)
−0.732945 + 0.680288i \(0.761855\pi\)
\(258\) 2.00000 0.124515
\(259\) 0 0
\(260\) −2.41395 −0.149707
\(261\) −9.41395 −0.582709
\(262\) −20.8942 −1.29085
\(263\) 18.8534 1.16255 0.581275 0.813707i \(-0.302554\pi\)
0.581275 + 0.813707i \(0.302554\pi\)
\(264\) 4.02636 0.247805
\(265\) −7.39422 −0.454224
\(266\) 0 0
\(267\) −10.2085 −0.624751
\(268\) 12.0527 0.736237
\(269\) −26.7976 −1.63388 −0.816940 0.576723i \(-0.804331\pi\)
−0.816940 + 0.576723i \(0.804331\pi\)
\(270\) 3.89728 0.237181
\(271\) −28.4803 −1.73005 −0.865027 0.501725i \(-0.832699\pi\)
−0.865027 + 0.501725i \(0.832699\pi\)
\(272\) −1.25153 −0.0758850
\(273\) 0 0
\(274\) −6.37398 −0.385066
\(275\) 41.0237 2.47382
\(276\) −1.00000 −0.0601929
\(277\) 19.3969 1.16545 0.582724 0.812670i \(-0.301987\pi\)
0.582724 + 0.812670i \(0.301987\pi\)
\(278\) 2.13607 0.128113
\(279\) 5.18878 0.310644
\(280\) 0 0
\(281\) 22.3903 1.33569 0.667847 0.744299i \(-0.267216\pi\)
0.667847 + 0.744299i \(0.267216\pi\)
\(282\) −6.13607 −0.365397
\(283\) −10.0395 −0.596784 −0.298392 0.954443i \(-0.596450\pi\)
−0.298392 + 0.954443i \(0.596450\pi\)
\(284\) 0.483327 0.0286802
\(285\) 32.7915 1.94240
\(286\) 2.49390 0.147468
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −15.4337 −0.907863
\(290\) 36.6888 2.15444
\(291\) −12.0531 −0.706564
\(292\) 3.15544 0.184658
\(293\) −25.0228 −1.46185 −0.730924 0.682459i \(-0.760910\pi\)
−0.730924 + 0.682459i \(0.760910\pi\)
\(294\) 0 0
\(295\) 42.9469 2.50047
\(296\) −1.38061 −0.0802461
\(297\) −4.02636 −0.233633
\(298\) −11.8218 −0.684818
\(299\) −0.619394 −0.0358205
\(300\) −10.1888 −0.588250
\(301\) 0 0
\(302\) −9.43368 −0.542847
\(303\) 10.1888 0.585330
\(304\) −8.41395 −0.482573
\(305\) −0.800667 −0.0458461
\(306\) 1.25153 0.0715451
\(307\) −27.6221 −1.57648 −0.788238 0.615370i \(-0.789007\pi\)
−0.788238 + 0.615370i \(0.789007\pi\)
\(308\) 0 0
\(309\) −6.95698 −0.395769
\(310\) −20.2221 −1.14854
\(311\) 12.8337 0.727730 0.363865 0.931452i \(-0.381457\pi\)
0.363865 + 0.931452i \(0.381457\pi\)
\(312\) −0.619394 −0.0350663
\(313\) −11.3306 −0.640443 −0.320222 0.947343i \(-0.603757\pi\)
−0.320222 + 0.947343i \(0.603757\pi\)
\(314\) −3.78182 −0.213420
\(315\) 0 0
\(316\) 15.4737 0.870461
\(317\) 29.0197 1.62991 0.814955 0.579524i \(-0.196762\pi\)
0.814955 + 0.579524i \(0.196762\pi\)
\(318\) −1.89728 −0.106394
\(319\) −37.9039 −2.12221
\(320\) 3.89728 0.217865
\(321\) −11.1361 −0.621555
\(322\) 0 0
\(323\) 10.5303 0.585921
\(324\) 1.00000 0.0555556
\(325\) −6.31087 −0.350064
\(326\) −4.48333 −0.248309
\(327\) −1.36787 −0.0756432
\(328\) 4.41395 0.243720
\(329\) 0 0
\(330\) 15.6918 0.863807
\(331\) 29.1024 1.59961 0.799806 0.600259i \(-0.204936\pi\)
0.799806 + 0.600259i \(0.204936\pi\)
\(332\) 7.13607 0.391643
\(333\) 1.38061 0.0756567
\(334\) −17.1388 −0.937792
\(335\) 46.9728 2.56640
\(336\) 0 0
\(337\) 2.54215 0.138480 0.0692399 0.997600i \(-0.477943\pi\)
0.0692399 + 0.997600i \(0.477943\pi\)
\(338\) 12.6164 0.686239
\(339\) −19.7582 −1.07312
\(340\) −4.87755 −0.264522
\(341\) 20.8919 1.13136
\(342\) 8.41395 0.454974
\(343\) 0 0
\(344\) −2.00000 −0.107833
\(345\) −3.89728 −0.209822
\(346\) 2.84456 0.152925
\(347\) 20.3776 1.09392 0.546962 0.837157i \(-0.315784\pi\)
0.546962 + 0.837157i \(0.315784\pi\)
\(348\) 9.41395 0.504641
\(349\) 5.63876 0.301836 0.150918 0.988546i \(-0.451777\pi\)
0.150918 + 0.988546i \(0.451777\pi\)
\(350\) 0 0
\(351\) 0.619394 0.0330608
\(352\) −4.02636 −0.214606
\(353\) −4.69759 −0.250027 −0.125014 0.992155i \(-0.539897\pi\)
−0.125014 + 0.992155i \(0.539897\pi\)
\(354\) 11.0197 0.585692
\(355\) 1.88366 0.0999743
\(356\) 10.2085 0.541050
\(357\) 0 0
\(358\) −13.2782 −0.701777
\(359\) −28.4303 −1.50049 −0.750246 0.661158i \(-0.770065\pi\)
−0.750246 + 0.661158i \(0.770065\pi\)
\(360\) −3.89728 −0.205405
\(361\) 51.7946 2.72603
\(362\) −14.5066 −0.762452
\(363\) −5.21155 −0.273536
\(364\) 0 0
\(365\) 12.2976 0.643686
\(366\) −0.205443 −0.0107387
\(367\) −6.27177 −0.327384 −0.163692 0.986511i \(-0.552340\pi\)
−0.163692 + 0.986511i \(0.552340\pi\)
\(368\) 1.00000 0.0521286
\(369\) −4.41395 −0.229781
\(370\) −5.38061 −0.279724
\(371\) 0 0
\(372\) −5.18878 −0.269026
\(373\) −6.27396 −0.324853 −0.162427 0.986721i \(-0.551932\pi\)
−0.162427 + 0.986721i \(0.551932\pi\)
\(374\) 5.03910 0.260565
\(375\) −20.2221 −1.04427
\(376\) 6.13607 0.316443
\(377\) 5.83095 0.300309
\(378\) 0 0
\(379\) −28.2248 −1.44981 −0.724906 0.688848i \(-0.758117\pi\)
−0.724906 + 0.688848i \(0.758117\pi\)
\(380\) −32.7915 −1.68217
\(381\) 15.4337 0.790691
\(382\) 15.2419 0.779841
\(383\) 9.44658 0.482698 0.241349 0.970438i \(-0.422410\pi\)
0.241349 + 0.970438i \(0.422410\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −11.9500 −0.608239
\(387\) 2.00000 0.101666
\(388\) 12.0531 0.611902
\(389\) 5.64183 0.286052 0.143026 0.989719i \(-0.454317\pi\)
0.143026 + 0.989719i \(0.454317\pi\)
\(390\) −2.41395 −0.122235
\(391\) −1.25153 −0.0632925
\(392\) 0 0
\(393\) −20.8942 −1.05397
\(394\) 18.5391 0.933987
\(395\) 60.3051 3.03428
\(396\) 4.02636 0.202332
\(397\) −32.7388 −1.64311 −0.821557 0.570127i \(-0.806894\pi\)
−0.821557 + 0.570127i \(0.806894\pi\)
\(398\) 8.91753 0.446995
\(399\) 0 0
\(400\) 10.1888 0.509439
\(401\) −9.88490 −0.493628 −0.246814 0.969063i \(-0.579384\pi\)
−0.246814 + 0.969063i \(0.579384\pi\)
\(402\) 12.0527 0.601135
\(403\) −3.21390 −0.160096
\(404\) −10.1888 −0.506911
\(405\) 3.89728 0.193657
\(406\) 0 0
\(407\) 5.55881 0.275540
\(408\) −1.25153 −0.0619598
\(409\) −25.1061 −1.24142 −0.620710 0.784041i \(-0.713155\pi\)
−0.620710 + 0.784041i \(0.713155\pi\)
\(410\) 17.2024 0.849566
\(411\) −6.37398 −0.314405
\(412\) 6.95698 0.342746
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) 27.8112 1.36520
\(416\) 0.619394 0.0303683
\(417\) 2.13607 0.104604
\(418\) 33.8776 1.65701
\(419\) −1.66548 −0.0813640 −0.0406820 0.999172i \(-0.512953\pi\)
−0.0406820 + 0.999172i \(0.512953\pi\)
\(420\) 0 0
\(421\) 4.61887 0.225110 0.112555 0.993645i \(-0.464097\pi\)
0.112555 + 0.993645i \(0.464097\pi\)
\(422\) −17.1694 −0.835794
\(423\) −6.13607 −0.298346
\(424\) 1.89728 0.0921400
\(425\) −12.7515 −0.618540
\(426\) 0.483327 0.0234173
\(427\) 0 0
\(428\) 11.1361 0.538282
\(429\) 2.49390 0.120407
\(430\) −7.79456 −0.375887
\(431\) −32.8674 −1.58316 −0.791582 0.611062i \(-0.790742\pi\)
−0.791582 + 0.611062i \(0.790742\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 21.0079 1.00957 0.504787 0.863244i \(-0.331571\pi\)
0.504787 + 0.863244i \(0.331571\pi\)
\(434\) 0 0
\(435\) 36.6888 1.75909
\(436\) 1.36787 0.0655089
\(437\) −8.41395 −0.402494
\(438\) 3.15544 0.150773
\(439\) −7.87004 −0.375617 −0.187808 0.982206i \(-0.560138\pi\)
−0.187808 + 0.982206i \(0.560138\pi\)
\(440\) −15.6918 −0.748079
\(441\) 0 0
\(442\) −0.775189 −0.0368720
\(443\) 16.5561 0.786605 0.393303 0.919409i \(-0.371332\pi\)
0.393303 + 0.919409i \(0.371332\pi\)
\(444\) −1.38061 −0.0655207
\(445\) 39.7854 1.88601
\(446\) 9.39422 0.444829
\(447\) −11.8218 −0.559152
\(448\) 0 0
\(449\) −10.4272 −0.492090 −0.246045 0.969258i \(-0.579131\pi\)
−0.246045 + 0.969258i \(0.579131\pi\)
\(450\) −10.1888 −0.480304
\(451\) −17.7721 −0.836858
\(452\) 19.7582 0.929346
\(453\) −9.43368 −0.443233
\(454\) −25.3179 −1.18823
\(455\) 0 0
\(456\) −8.41395 −0.394019
\(457\) 25.8082 1.20726 0.603628 0.797266i \(-0.293721\pi\)
0.603628 + 0.797266i \(0.293721\pi\)
\(458\) 11.9600 0.558855
\(459\) 1.25153 0.0584163
\(460\) 3.89728 0.181712
\(461\) −19.1660 −0.892650 −0.446325 0.894871i \(-0.647267\pi\)
−0.446325 + 0.894871i \(0.647267\pi\)
\(462\) 0 0
\(463\) 11.6891 0.543240 0.271620 0.962405i \(-0.412441\pi\)
0.271620 + 0.962405i \(0.412441\pi\)
\(464\) −9.41395 −0.437032
\(465\) −20.2221 −0.937779
\(466\) 8.86429 0.410630
\(467\) 0.837576 0.0387584 0.0193792 0.999812i \(-0.493831\pi\)
0.0193792 + 0.999812i \(0.493831\pi\)
\(468\) −0.619394 −0.0286315
\(469\) 0 0
\(470\) 23.9140 1.10307
\(471\) −3.78182 −0.174257
\(472\) −11.0197 −0.507224
\(473\) 8.05271 0.370264
\(474\) 15.4737 0.710728
\(475\) −85.7279 −3.93347
\(476\) 0 0
\(477\) −1.89728 −0.0868704
\(478\) −9.93062 −0.454216
\(479\) 3.58605 0.163851 0.0819254 0.996638i \(-0.473893\pi\)
0.0819254 + 0.996638i \(0.473893\pi\)
\(480\) 3.89728 0.177886
\(481\) −0.855139 −0.0389910
\(482\) 12.2840 0.559521
\(483\) 0 0
\(484\) 5.21155 0.236889
\(485\) 46.9742 2.13299
\(486\) 1.00000 0.0453609
\(487\) −15.0361 −0.681349 −0.340674 0.940181i \(-0.610655\pi\)
−0.340674 + 0.940181i \(0.610655\pi\)
\(488\) 0.205443 0.00929995
\(489\) −4.48333 −0.202743
\(490\) 0 0
\(491\) −6.13302 −0.276779 −0.138390 0.990378i \(-0.544193\pi\)
−0.138390 + 0.990378i \(0.544193\pi\)
\(492\) 4.41395 0.198996
\(493\) 11.7818 0.530626
\(494\) −5.21155 −0.234479
\(495\) 15.6918 0.705296
\(496\) 5.18878 0.232983
\(497\) 0 0
\(498\) 7.13607 0.319775
\(499\) −21.4925 −0.962137 −0.481068 0.876683i \(-0.659751\pi\)
−0.481068 + 0.876683i \(0.659751\pi\)
\(500\) 20.2221 0.904361
\(501\) −17.1388 −0.765704
\(502\) −6.81481 −0.304160
\(503\) −14.0922 −0.628339 −0.314169 0.949367i \(-0.601726\pi\)
−0.314169 + 0.949367i \(0.601726\pi\)
\(504\) 0 0
\(505\) −39.7085 −1.76701
\(506\) −4.02636 −0.178993
\(507\) 12.6164 0.560312
\(508\) −15.4337 −0.684759
\(509\) 11.9575 0.530007 0.265003 0.964247i \(-0.414627\pi\)
0.265003 + 0.964247i \(0.414627\pi\)
\(510\) −4.87755 −0.215982
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 8.41395 0.371485
\(514\) 23.5000 1.03654
\(515\) 27.1133 1.19475
\(516\) −2.00000 −0.0880451
\(517\) −24.7060 −1.08657
\(518\) 0 0
\(519\) 2.84456 0.124863
\(520\) 2.41395 0.105859
\(521\) 22.3776 0.980379 0.490189 0.871616i \(-0.336928\pi\)
0.490189 + 0.871616i \(0.336928\pi\)
\(522\) 9.41395 0.412037
\(523\) 15.2660 0.667537 0.333768 0.942655i \(-0.391680\pi\)
0.333768 + 0.942655i \(0.391680\pi\)
\(524\) 20.8942 0.912769
\(525\) 0 0
\(526\) −18.8534 −0.822046
\(527\) −6.49390 −0.282879
\(528\) −4.02636 −0.175225
\(529\) 1.00000 0.0434783
\(530\) 7.39422 0.321185
\(531\) 11.0197 0.478215
\(532\) 0 0
\(533\) 2.73398 0.118422
\(534\) 10.2085 0.441765
\(535\) 43.4004 1.87636
\(536\) −12.0527 −0.520598
\(537\) −13.2782 −0.572999
\(538\) 26.7976 1.15533
\(539\) 0 0
\(540\) −3.89728 −0.167712
\(541\) 4.79151 0.206003 0.103002 0.994681i \(-0.467155\pi\)
0.103002 + 0.994681i \(0.467155\pi\)
\(542\) 28.4803 1.22333
\(543\) −14.5066 −0.622540
\(544\) 1.25153 0.0536588
\(545\) 5.33096 0.228353
\(546\) 0 0
\(547\) 20.5888 0.880312 0.440156 0.897921i \(-0.354923\pi\)
0.440156 + 0.897921i \(0.354923\pi\)
\(548\) 6.37398 0.272283
\(549\) −0.205443 −0.00876808
\(550\) −41.0237 −1.74926
\(551\) 79.2085 3.37440
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) −19.3969 −0.824097
\(555\) −5.38061 −0.228394
\(556\) −2.13607 −0.0905894
\(557\) 26.6979 1.13123 0.565614 0.824670i \(-0.308639\pi\)
0.565614 + 0.824670i \(0.308639\pi\)
\(558\) −5.18878 −0.219659
\(559\) −1.23879 −0.0523952
\(560\) 0 0
\(561\) 5.03910 0.212751
\(562\) −22.3903 −0.944478
\(563\) 21.0070 0.885339 0.442670 0.896685i \(-0.354031\pi\)
0.442670 + 0.896685i \(0.354031\pi\)
\(564\) 6.13607 0.258375
\(565\) 77.0031 3.23954
\(566\) 10.0395 0.421990
\(567\) 0 0
\(568\) −0.483327 −0.0202800
\(569\) −16.3381 −0.684929 −0.342465 0.939531i \(-0.611262\pi\)
−0.342465 + 0.939531i \(0.611262\pi\)
\(570\) −32.7915 −1.37349
\(571\) −3.04967 −0.127625 −0.0638124 0.997962i \(-0.520326\pi\)
−0.0638124 + 0.997962i \(0.520326\pi\)
\(572\) −2.49390 −0.104275
\(573\) 15.2419 0.636738
\(574\) 0 0
\(575\) 10.1888 0.424902
\(576\) 1.00000 0.0416667
\(577\) 3.89388 0.162104 0.0810521 0.996710i \(-0.474172\pi\)
0.0810521 + 0.996710i \(0.474172\pi\)
\(578\) 15.4337 0.641956
\(579\) −11.9500 −0.496625
\(580\) −36.6888 −1.52342
\(581\) 0 0
\(582\) 12.0531 0.499616
\(583\) −7.63912 −0.316380
\(584\) −3.15544 −0.130573
\(585\) −2.41395 −0.0998046
\(586\) 25.0228 1.03368
\(587\) −36.4701 −1.50528 −0.752641 0.658431i \(-0.771220\pi\)
−0.752641 + 0.658431i \(0.771220\pi\)
\(588\) 0 0
\(589\) −43.6582 −1.79890
\(590\) −42.9469 −1.76810
\(591\) 18.5391 0.762597
\(592\) 1.38061 0.0567426
\(593\) −26.9466 −1.10656 −0.553282 0.832994i \(-0.686625\pi\)
−0.553282 + 0.832994i \(0.686625\pi\)
\(594\) 4.02636 0.165203
\(595\) 0 0
\(596\) 11.8218 0.484240
\(597\) 8.91753 0.364970
\(598\) 0.619394 0.0253289
\(599\) −39.2507 −1.60374 −0.801869 0.597499i \(-0.796161\pi\)
−0.801869 + 0.597499i \(0.796161\pi\)
\(600\) 10.1888 0.415955
\(601\) −7.94999 −0.324287 −0.162143 0.986767i \(-0.551841\pi\)
−0.162143 + 0.986767i \(0.551841\pi\)
\(602\) 0 0
\(603\) 12.0527 0.490824
\(604\) 9.43368 0.383851
\(605\) 20.3109 0.825754
\(606\) −10.1888 −0.413891
\(607\) 18.1221 0.735554 0.367777 0.929914i \(-0.380119\pi\)
0.367777 + 0.929914i \(0.380119\pi\)
\(608\) 8.41395 0.341231
\(609\) 0 0
\(610\) 0.800667 0.0324181
\(611\) 3.80064 0.153758
\(612\) −1.25153 −0.0505900
\(613\) −38.6655 −1.56168 −0.780842 0.624728i \(-0.785210\pi\)
−0.780842 + 0.624728i \(0.785210\pi\)
\(614\) 27.6221 1.11474
\(615\) 17.2024 0.693668
\(616\) 0 0
\(617\) 11.9509 0.481124 0.240562 0.970634i \(-0.422668\pi\)
0.240562 + 0.970634i \(0.422668\pi\)
\(618\) 6.95698 0.279851
\(619\) −14.9303 −0.600098 −0.300049 0.953924i \(-0.597003\pi\)
−0.300049 + 0.953924i \(0.597003\pi\)
\(620\) 20.2221 0.812140
\(621\) −1.00000 −0.0401286
\(622\) −12.8337 −0.514583
\(623\) 0 0
\(624\) 0.619394 0.0247956
\(625\) 27.8674 1.11469
\(626\) 11.3306 0.452862
\(627\) 33.8776 1.35294
\(628\) 3.78182 0.150911
\(629\) −1.72787 −0.0688945
\(630\) 0 0
\(631\) 27.8917 1.11035 0.555176 0.831733i \(-0.312651\pi\)
0.555176 + 0.831733i \(0.312651\pi\)
\(632\) −15.4737 −0.615509
\(633\) −17.1694 −0.682423
\(634\) −29.0197 −1.15252
\(635\) −60.1493 −2.38695
\(636\) 1.89728 0.0752320
\(637\) 0 0
\(638\) 37.9039 1.50063
\(639\) 0.483327 0.0191201
\(640\) −3.89728 −0.154053
\(641\) 25.9539 1.02512 0.512559 0.858652i \(-0.328697\pi\)
0.512559 + 0.858652i \(0.328697\pi\)
\(642\) 11.1361 0.439506
\(643\) 49.0759 1.93536 0.967682 0.252175i \(-0.0811459\pi\)
0.967682 + 0.252175i \(0.0811459\pi\)
\(644\) 0 0
\(645\) −7.79456 −0.306910
\(646\) −10.5303 −0.414309
\(647\) 18.3143 0.720008 0.360004 0.932951i \(-0.382775\pi\)
0.360004 + 0.932951i \(0.382775\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 44.3694 1.74165
\(650\) 6.31087 0.247533
\(651\) 0 0
\(652\) 4.48333 0.175581
\(653\) −1.16905 −0.0457486 −0.0228743 0.999738i \(-0.507282\pi\)
−0.0228743 + 0.999738i \(0.507282\pi\)
\(654\) 1.36787 0.0534878
\(655\) 81.4307 3.18176
\(656\) −4.41395 −0.172336
\(657\) 3.15544 0.123105
\(658\) 0 0
\(659\) −4.40121 −0.171447 −0.0857234 0.996319i \(-0.527320\pi\)
−0.0857234 + 0.996319i \(0.527320\pi\)
\(660\) −15.6918 −0.610804
\(661\) 11.2818 0.438812 0.219406 0.975634i \(-0.429588\pi\)
0.219406 + 0.975634i \(0.429588\pi\)
\(662\) −29.1024 −1.13110
\(663\) −0.775189 −0.0301058
\(664\) −7.13607 −0.276933
\(665\) 0 0
\(666\) −1.38061 −0.0534974
\(667\) −9.41395 −0.364510
\(668\) 17.1388 0.663119
\(669\) 9.39422 0.363202
\(670\) −46.9728 −1.81472
\(671\) −0.827185 −0.0319331
\(672\) 0 0
\(673\) 6.54966 0.252471 0.126235 0.992000i \(-0.459711\pi\)
0.126235 + 0.992000i \(0.459711\pi\)
\(674\) −2.54215 −0.0979200
\(675\) −10.1888 −0.392166
\(676\) −12.6164 −0.485244
\(677\) 4.75066 0.182583 0.0912913 0.995824i \(-0.470901\pi\)
0.0912913 + 0.995824i \(0.470901\pi\)
\(678\) 19.7582 0.758808
\(679\) 0 0
\(680\) 4.87755 0.187046
\(681\) −25.3179 −0.970182
\(682\) −20.8919 −0.799992
\(683\) 18.3894 0.703652 0.351826 0.936065i \(-0.385561\pi\)
0.351826 + 0.936065i \(0.385561\pi\)
\(684\) −8.41395 −0.321715
\(685\) 24.8412 0.949132
\(686\) 0 0
\(687\) 11.9600 0.456303
\(688\) 2.00000 0.0762493
\(689\) 1.17516 0.0447702
\(690\) 3.89728 0.148367
\(691\) −32.4500 −1.23446 −0.617228 0.786784i \(-0.711744\pi\)
−0.617228 + 0.786784i \(0.711744\pi\)
\(692\) −2.84456 −0.108134
\(693\) 0 0
\(694\) −20.3776 −0.773522
\(695\) −8.32485 −0.315779
\(696\) −9.41395 −0.356835
\(697\) 5.52418 0.209243
\(698\) −5.63876 −0.213430
\(699\) 8.86429 0.335278
\(700\) 0 0
\(701\) −33.9894 −1.28376 −0.641882 0.766804i \(-0.721846\pi\)
−0.641882 + 0.766804i \(0.721846\pi\)
\(702\) −0.619394 −0.0233775
\(703\) −11.6164 −0.438119
\(704\) 4.02636 0.151749
\(705\) 23.9140 0.900652
\(706\) 4.69759 0.176796
\(707\) 0 0
\(708\) −11.0197 −0.414147
\(709\) −16.6624 −0.625771 −0.312885 0.949791i \(-0.601296\pi\)
−0.312885 + 0.949791i \(0.601296\pi\)
\(710\) −1.88366 −0.0706925
\(711\) 15.4737 0.580307
\(712\) −10.2085 −0.382580
\(713\) 5.18878 0.194321
\(714\) 0 0
\(715\) −9.71943 −0.363486
\(716\) 13.2782 0.496231
\(717\) −9.93062 −0.370866
\(718\) 28.4303 1.06101
\(719\) −33.1932 −1.23790 −0.618950 0.785431i \(-0.712441\pi\)
−0.618950 + 0.785431i \(0.712441\pi\)
\(720\) 3.89728 0.145243
\(721\) 0 0
\(722\) −51.7946 −1.92759
\(723\) 12.2840 0.456847
\(724\) 14.5066 0.539135
\(725\) −95.9167 −3.56226
\(726\) 5.21155 0.193419
\(727\) 27.3991 1.01618 0.508089 0.861305i \(-0.330352\pi\)
0.508089 + 0.861305i \(0.330352\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −12.2976 −0.455155
\(731\) −2.50305 −0.0925788
\(732\) 0.205443 0.00759338
\(733\) −1.09880 −0.0405851 −0.0202925 0.999794i \(-0.506460\pi\)
−0.0202925 + 0.999794i \(0.506460\pi\)
\(734\) 6.27177 0.231495
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 48.5285 1.78757
\(738\) 4.41395 0.162480
\(739\) −49.5137 −1.82139 −0.910695 0.413080i \(-0.864453\pi\)
−0.910695 + 0.413080i \(0.864453\pi\)
\(740\) 5.38061 0.197795
\(741\) −5.21155 −0.191451
\(742\) 0 0
\(743\) 10.6054 0.389075 0.194538 0.980895i \(-0.437679\pi\)
0.194538 + 0.980895i \(0.437679\pi\)
\(744\) 5.18878 0.190230
\(745\) 46.0728 1.68798
\(746\) 6.27396 0.229706
\(747\) 7.13607 0.261095
\(748\) −5.03910 −0.184248
\(749\) 0 0
\(750\) 20.2221 0.738408
\(751\) −4.17857 −0.152478 −0.0762390 0.997090i \(-0.524291\pi\)
−0.0762390 + 0.997090i \(0.524291\pi\)
\(752\) −6.13607 −0.223759
\(753\) −6.81481 −0.248345
\(754\) −5.83095 −0.212351
\(755\) 36.7657 1.33804
\(756\) 0 0
\(757\) −18.0758 −0.656978 −0.328489 0.944508i \(-0.606539\pi\)
−0.328489 + 0.944508i \(0.606539\pi\)
\(758\) 28.2248 1.02517
\(759\) −4.02636 −0.146148
\(760\) 32.7915 1.18947
\(761\) −3.81769 −0.138391 −0.0691955 0.997603i \(-0.522043\pi\)
−0.0691955 + 0.997603i \(0.522043\pi\)
\(762\) −15.4337 −0.559103
\(763\) 0 0
\(764\) −15.2419 −0.551431
\(765\) −4.87755 −0.176348
\(766\) −9.44658 −0.341319
\(767\) −6.82555 −0.246456
\(768\) −1.00000 −0.0360844
\(769\) 11.4105 0.411475 0.205737 0.978607i \(-0.434041\pi\)
0.205737 + 0.978607i \(0.434041\pi\)
\(770\) 0 0
\(771\) 23.5000 0.846332
\(772\) 11.9500 0.430090
\(773\) 25.6558 0.922775 0.461388 0.887199i \(-0.347352\pi\)
0.461388 + 0.887199i \(0.347352\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 52.8674 1.89905
\(776\) −12.0531 −0.432680
\(777\) 0 0
\(778\) −5.64183 −0.202269
\(779\) 37.1388 1.33063
\(780\) 2.41395 0.0864333
\(781\) 1.94605 0.0696351
\(782\) 1.25153 0.0447545
\(783\) 9.41395 0.336427
\(784\) 0 0
\(785\) 14.7388 0.526050
\(786\) 20.8942 0.745273
\(787\) 35.6303 1.27008 0.635042 0.772478i \(-0.280983\pi\)
0.635042 + 0.772478i \(0.280983\pi\)
\(788\) −18.5391 −0.660428
\(789\) −18.8534 −0.671198
\(790\) −60.3051 −2.14556
\(791\) 0 0
\(792\) −4.02636 −0.143070
\(793\) 0.127250 0.00451878
\(794\) 32.7388 1.16186
\(795\) 7.39422 0.262246
\(796\) −8.91753 −0.316073
\(797\) 37.7840 1.33838 0.669189 0.743092i \(-0.266642\pi\)
0.669189 + 0.743092i \(0.266642\pi\)
\(798\) 0 0
\(799\) 7.67946 0.271680
\(800\) −10.1888 −0.360228
\(801\) 10.2085 0.360700
\(802\) 9.88490 0.349048
\(803\) 12.7049 0.448347
\(804\) −12.0527 −0.425066
\(805\) 0 0
\(806\) 3.21390 0.113205
\(807\) 26.7976 0.943321
\(808\) 10.1888 0.358440
\(809\) 13.5800 0.477446 0.238723 0.971088i \(-0.423271\pi\)
0.238723 + 0.971088i \(0.423271\pi\)
\(810\) −3.89728 −0.136936
\(811\) 46.7245 1.64072 0.820359 0.571848i \(-0.193773\pi\)
0.820359 + 0.571848i \(0.193773\pi\)
\(812\) 0 0
\(813\) 28.4803 0.998847
\(814\) −5.55881 −0.194836
\(815\) 17.4728 0.612045
\(816\) 1.25153 0.0438122
\(817\) −16.8279 −0.588734
\(818\) 25.1061 0.877816
\(819\) 0 0
\(820\) −17.2024 −0.600734
\(821\) 34.5922 1.20728 0.603638 0.797259i \(-0.293717\pi\)
0.603638 + 0.797259i \(0.293717\pi\)
\(822\) 6.37398 0.222318
\(823\) 41.4242 1.44396 0.721978 0.691916i \(-0.243233\pi\)
0.721978 + 0.691916i \(0.243233\pi\)
\(824\) −6.95698 −0.242358
\(825\) −41.0237 −1.42826
\(826\) 0 0
\(827\) 50.4046 1.75274 0.876370 0.481638i \(-0.159958\pi\)
0.876370 + 0.481638i \(0.159958\pi\)
\(828\) 1.00000 0.0347524
\(829\) −44.7157 −1.55304 −0.776520 0.630093i \(-0.783017\pi\)
−0.776520 + 0.630093i \(0.783017\pi\)
\(830\) −27.8112 −0.965342
\(831\) −19.3969 −0.672872
\(832\) −0.619394 −0.0214736
\(833\) 0 0
\(834\) −2.13607 −0.0739659
\(835\) 66.7946 2.31152
\(836\) −33.8776 −1.17168
\(837\) −5.18878 −0.179351
\(838\) 1.66548 0.0575330
\(839\) 24.1691 0.834408 0.417204 0.908813i \(-0.363010\pi\)
0.417204 + 0.908813i \(0.363010\pi\)
\(840\) 0 0
\(841\) 59.6225 2.05595
\(842\) −4.61887 −0.159177
\(843\) −22.3903 −0.771163
\(844\) 17.1694 0.590996
\(845\) −49.1694 −1.69148
\(846\) 6.13607 0.210962
\(847\) 0 0
\(848\) −1.89728 −0.0651528
\(849\) 10.0395 0.344553
\(850\) 12.7515 0.437374
\(851\) 1.38061 0.0473266
\(852\) −0.483327 −0.0165585
\(853\) −17.3969 −0.595660 −0.297830 0.954619i \(-0.596263\pi\)
−0.297830 + 0.954619i \(0.596263\pi\)
\(854\) 0 0
\(855\) −32.7915 −1.12145
\(856\) −11.1361 −0.380623
\(857\) −34.9837 −1.19502 −0.597510 0.801861i \(-0.703843\pi\)
−0.597510 + 0.801861i \(0.703843\pi\)
\(858\) −2.49390 −0.0851404
\(859\) 26.9269 0.918733 0.459366 0.888247i \(-0.348077\pi\)
0.459366 + 0.888247i \(0.348077\pi\)
\(860\) 7.79456 0.265792
\(861\) 0 0
\(862\) 32.8674 1.11947
\(863\) 55.7160 1.89660 0.948298 0.317382i \(-0.102804\pi\)
0.948298 + 0.317382i \(0.102804\pi\)
\(864\) 1.00000 0.0340207
\(865\) −11.0861 −0.376937
\(866\) −21.0079 −0.713876
\(867\) 15.4337 0.524155
\(868\) 0 0
\(869\) 62.3025 2.11347
\(870\) −36.6888 −1.24387
\(871\) −7.46538 −0.252955
\(872\) −1.36787 −0.0463218
\(873\) 12.0531 0.407935
\(874\) 8.41395 0.284606
\(875\) 0 0
\(876\) −3.15544 −0.106612
\(877\) −22.6976 −0.766443 −0.383222 0.923656i \(-0.625185\pi\)
−0.383222 + 0.923656i \(0.625185\pi\)
\(878\) 7.87004 0.265601
\(879\) 25.0228 0.843998
\(880\) 15.6918 0.528972
\(881\) 1.80423 0.0607861 0.0303930 0.999538i \(-0.490324\pi\)
0.0303930 + 0.999538i \(0.490324\pi\)
\(882\) 0 0
\(883\) −36.5861 −1.23122 −0.615610 0.788051i \(-0.711090\pi\)
−0.615610 + 0.788051i \(0.711090\pi\)
\(884\) 0.775189 0.0260724
\(885\) −42.9469 −1.44365
\(886\) −16.5561 −0.556214
\(887\) −19.3408 −0.649401 −0.324701 0.945817i \(-0.605264\pi\)
−0.324701 + 0.945817i \(0.605264\pi\)
\(888\) 1.38061 0.0463301
\(889\) 0 0
\(890\) −39.7854 −1.33361
\(891\) 4.02636 0.134888
\(892\) −9.39422 −0.314542
\(893\) 51.6286 1.72768
\(894\) 11.8218 0.395380
\(895\) 51.7490 1.72978
\(896\) 0 0
\(897\) 0.619394 0.0206810
\(898\) 10.4272 0.347961
\(899\) −48.8469 −1.62914
\(900\) 10.1888 0.339626
\(901\) 2.37450 0.0791059
\(902\) 17.7721 0.591748
\(903\) 0 0
\(904\) −19.7582 −0.657147
\(905\) 56.5364 1.87933
\(906\) 9.43368 0.313413
\(907\) −5.58677 −0.185506 −0.0927528 0.995689i \(-0.529567\pi\)
−0.0927528 + 0.995689i \(0.529567\pi\)
\(908\) 25.3179 0.840203
\(909\) −10.1888 −0.337941
\(910\) 0 0
\(911\) −29.3170 −0.971315 −0.485657 0.874149i \(-0.661420\pi\)
−0.485657 + 0.874149i \(0.661420\pi\)
\(912\) 8.41395 0.278614
\(913\) 28.7324 0.950902
\(914\) −25.8082 −0.853659
\(915\) 0.800667 0.0264692
\(916\) −11.9600 −0.395170
\(917\) 0 0
\(918\) −1.25153 −0.0413066
\(919\) −25.5135 −0.841612 −0.420806 0.907151i \(-0.638253\pi\)
−0.420806 + 0.907151i \(0.638253\pi\)
\(920\) −3.89728 −0.128489
\(921\) 27.6221 0.910179
\(922\) 19.1660 0.631199
\(923\) −0.299370 −0.00985388
\(924\) 0 0
\(925\) 14.0667 0.462510
\(926\) −11.6891 −0.384129
\(927\) 6.95698 0.228497
\(928\) 9.41395 0.309028
\(929\) −5.12173 −0.168039 −0.0840193 0.996464i \(-0.526776\pi\)
−0.0840193 + 0.996464i \(0.526776\pi\)
\(930\) 20.2221 0.663110
\(931\) 0 0
\(932\) −8.86429 −0.290360
\(933\) −12.8337 −0.420155
\(934\) −0.837576 −0.0274063
\(935\) −19.6388 −0.642256
\(936\) 0.619394 0.0202455
\(937\) −39.6816 −1.29634 −0.648171 0.761494i \(-0.724466\pi\)
−0.648171 + 0.761494i \(0.724466\pi\)
\(938\) 0 0
\(939\) 11.3306 0.369760
\(940\) −23.9140 −0.779987
\(941\) −10.3003 −0.335781 −0.167890 0.985806i \(-0.553695\pi\)
−0.167890 + 0.985806i \(0.553695\pi\)
\(942\) 3.78182 0.123218
\(943\) −4.41395 −0.143738
\(944\) 11.0197 0.358662
\(945\) 0 0
\(946\) −8.05271 −0.261816
\(947\) 20.2309 0.657417 0.328708 0.944431i \(-0.393387\pi\)
0.328708 + 0.944431i \(0.393387\pi\)
\(948\) −15.4737 −0.502561
\(949\) −1.95446 −0.0634444
\(950\) 85.7279 2.78138
\(951\) −29.0197 −0.941029
\(952\) 0 0
\(953\) −29.4213 −0.953049 −0.476525 0.879161i \(-0.658104\pi\)
−0.476525 + 0.879161i \(0.658104\pi\)
\(954\) 1.89728 0.0614267
\(955\) −59.4018 −1.92220
\(956\) 9.93062 0.321179
\(957\) 37.9039 1.22526
\(958\) −3.58605 −0.115860
\(959\) 0 0
\(960\) −3.89728 −0.125784
\(961\) −4.07655 −0.131502
\(962\) 0.855139 0.0275708
\(963\) 11.1361 0.358855
\(964\) −12.2840 −0.395641
\(965\) 46.5725 1.49922
\(966\) 0 0
\(967\) −8.05542 −0.259045 −0.129522 0.991576i \(-0.541344\pi\)
−0.129522 + 0.991576i \(0.541344\pi\)
\(968\) −5.21155 −0.167506
\(969\) −10.5303 −0.338282
\(970\) −46.9742 −1.50825
\(971\) −45.1658 −1.44944 −0.724721 0.689043i \(-0.758031\pi\)
−0.724721 + 0.689043i \(0.758031\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 15.0361 0.481786
\(975\) 6.31087 0.202110
\(976\) −0.205443 −0.00657606
\(977\) 20.3898 0.652327 0.326163 0.945313i \(-0.394244\pi\)
0.326163 + 0.945313i \(0.394244\pi\)
\(978\) 4.48333 0.143361
\(979\) 41.1031 1.31366
\(980\) 0 0
\(981\) 1.36787 0.0436726
\(982\) 6.13302 0.195713
\(983\) −49.4534 −1.57732 −0.788660 0.614830i \(-0.789225\pi\)
−0.788660 + 0.614830i \(0.789225\pi\)
\(984\) −4.41395 −0.140712
\(985\) −72.2521 −2.30214
\(986\) −11.7818 −0.375210
\(987\) 0 0
\(988\) 5.21155 0.165802
\(989\) 2.00000 0.0635963
\(990\) −15.6918 −0.498719
\(991\) −58.8518 −1.86949 −0.934744 0.355322i \(-0.884371\pi\)
−0.934744 + 0.355322i \(0.884371\pi\)
\(992\) −5.18878 −0.164744
\(993\) −29.1024 −0.923536
\(994\) 0 0
\(995\) −34.7541 −1.10178
\(996\) −7.13607 −0.226115
\(997\) −50.3419 −1.59434 −0.797172 0.603752i \(-0.793672\pi\)
−0.797172 + 0.603752i \(0.793672\pi\)
\(998\) 21.4925 0.680333
\(999\) −1.38061 −0.0436804
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 6762.2.a.cg.1.4 4
7.2 even 3 966.2.i.n.277.1 8
7.4 even 3 966.2.i.n.415.1 yes 8
7.6 odd 2 6762.2.a.ch.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.i.n.277.1 8 7.2 even 3
966.2.i.n.415.1 yes 8 7.4 even 3
6762.2.a.cg.1.4 4 1.1 even 1 trivial
6762.2.a.ch.1.1 4 7.6 odd 2