Properties

Label 1850.2.a.be.1.5
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.1791440.1
Defining polynomial: \(x^{5} - 9 x^{3} + 13 x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-2.62545\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.62545 q^{3} +1.00000 q^{4} +2.62545 q^{6} -1.83227 q^{7} +1.00000 q^{8} +3.89300 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.62545 q^{3} +1.00000 q^{4} +2.62545 q^{6} -1.83227 q^{7} +1.00000 q^{8} +3.89300 q^{9} +4.19017 q^{11} +2.62545 q^{12} -0.369454 q^{13} -1.83227 q^{14} +1.00000 q^{16} -5.08317 q^{17} +3.89300 q^{18} +3.55963 q^{19} -4.81053 q^{21} +4.19017 q^{22} +5.62036 q^{23} +2.62545 q^{24} -0.369454 q^{26} +2.34453 q^{27} -1.83227 q^{28} +1.20681 q^{29} +10.1030 q^{31} +1.00000 q^{32} +11.0011 q^{33} -5.08317 q^{34} +3.89300 q^{36} +1.00000 q^{37} +3.55963 q^{38} -0.969984 q^{39} -8.01447 q^{41} -4.81053 q^{42} -2.27264 q^{43} +4.19017 q^{44} +5.62036 q^{46} +10.9154 q^{47} +2.62545 q^{48} -3.64280 q^{49} -13.3456 q^{51} -0.369454 q^{52} -9.94355 q^{53} +2.34453 q^{54} -1.83227 q^{56} +9.34563 q^{57} +1.20681 q^{58} -5.34563 q^{59} -9.79428 q^{61} +10.1030 q^{62} -7.13302 q^{63} +1.00000 q^{64} +11.0011 q^{66} +1.85073 q^{67} -5.08317 q^{68} +14.7560 q^{69} +2.86038 q^{71} +3.89300 q^{72} -8.09942 q^{73} +1.00000 q^{74} +3.55963 q^{76} -7.67751 q^{77} -0.969984 q^{78} +6.06361 q^{79} -5.52355 q^{81} -8.01447 q^{82} -8.93200 q^{83} -4.81053 q^{84} -2.27264 q^{86} +3.16843 q^{87} +4.19017 q^{88} +11.4773 q^{89} +0.676938 q^{91} +5.62036 q^{92} +26.5249 q^{93} +10.9154 q^{94} +2.62545 q^{96} +6.05864 q^{97} -3.64280 q^{98} +16.3123 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{2} + 5q^{4} - q^{7} + 5q^{8} + 3q^{9} + O(q^{10}) \) \( 5q + 5q^{2} + 5q^{4} - q^{7} + 5q^{8} + 3q^{9} + 3q^{11} - 6q^{13} - q^{14} + 5q^{16} + 9q^{17} + 3q^{18} + 4q^{19} + 16q^{21} + 3q^{22} + 6q^{23} - 6q^{26} - q^{28} + 11q^{29} + 23q^{31} + 5q^{32} + 20q^{33} + 9q^{34} + 3q^{36} + 5q^{37} + 4q^{38} + 20q^{39} - 7q^{41} + 16q^{42} - 17q^{43} + 3q^{44} + 6q^{46} + 12q^{47} + 30q^{49} - 20q^{51} - 6q^{52} - 7q^{53} - q^{56} + 11q^{58} + 20q^{59} - 9q^{61} + 23q^{62} - 33q^{63} + 5q^{64} + 20q^{66} + 12q^{67} + 9q^{68} + 16q^{69} + 6q^{71} + 3q^{72} - 6q^{73} + 5q^{74} + 4q^{76} - q^{77} + 20q^{78} + 20q^{79} - 7q^{81} - 7q^{82} + 12q^{83} + 16q^{84} - 17q^{86} - 34q^{87} + 3q^{88} + 12q^{89} + 16q^{91} + 6q^{92} - 4q^{93} + 12q^{94} + 3q^{97} + 30q^{98} - 11q^{99} + O(q^{100}) \)

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
\(3\) 2.62545 1.51581 0.757903 0.652367i \(-0.226224\pi\)
0.757903 + 0.652367i \(0.226224\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.62545 1.07184
\(7\) −1.83227 −0.692532 −0.346266 0.938136i \(-0.612551\pi\)
−0.346266 + 0.938136i \(0.612551\pi\)
\(8\) 1.00000 0.353553
\(9\) 3.89300 1.29767
\(10\) 0 0
\(11\) 4.19017 1.26338 0.631692 0.775219i \(-0.282361\pi\)
0.631692 + 0.775219i \(0.282361\pi\)
\(12\) 2.62545 0.757903
\(13\) −0.369454 −0.102468 −0.0512340 0.998687i \(-0.516315\pi\)
−0.0512340 + 0.998687i \(0.516315\pi\)
\(14\) −1.83227 −0.489694
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.08317 −1.23285 −0.616425 0.787414i \(-0.711420\pi\)
−0.616425 + 0.787414i \(0.711420\pi\)
\(18\) 3.89300 0.917589
\(19\) 3.55963 0.816634 0.408317 0.912840i \(-0.366116\pi\)
0.408317 + 0.912840i \(0.366116\pi\)
\(20\) 0 0
\(21\) −4.81053 −1.04974
\(22\) 4.19017 0.893348
\(23\) 5.62036 1.17193 0.585963 0.810338i \(-0.300716\pi\)
0.585963 + 0.810338i \(0.300716\pi\)
\(24\) 2.62545 0.535918
\(25\) 0 0
\(26\) −0.369454 −0.0724559
\(27\) 2.34453 0.451205
\(28\) −1.83227 −0.346266
\(29\) 1.20681 0.224100 0.112050 0.993703i \(-0.464258\pi\)
0.112050 + 0.993703i \(0.464258\pi\)
\(30\) 0 0
\(31\) 10.1030 1.81455 0.907276 0.420535i \(-0.138158\pi\)
0.907276 + 0.420535i \(0.138158\pi\)
\(32\) 1.00000 0.176777
\(33\) 11.0011 1.91504
\(34\) −5.08317 −0.871757
\(35\) 0 0
\(36\) 3.89300 0.648833
\(37\) 1.00000 0.164399
\(38\) 3.55963 0.577447
\(39\) −0.969984 −0.155322
\(40\) 0 0
\(41\) −8.01447 −1.25165 −0.625825 0.779964i \(-0.715238\pi\)
−0.625825 + 0.779964i \(0.715238\pi\)
\(42\) −4.81053 −0.742281
\(43\) −2.27264 −0.346575 −0.173287 0.984871i \(-0.555439\pi\)
−0.173287 + 0.984871i \(0.555439\pi\)
\(44\) 4.19017 0.631692
\(45\) 0 0
\(46\) 5.62036 0.828677
\(47\) 10.9154 1.59218 0.796090 0.605178i \(-0.206898\pi\)
0.796090 + 0.605178i \(0.206898\pi\)
\(48\) 2.62545 0.378951
\(49\) −3.64280 −0.520400
\(50\) 0 0
\(51\) −13.3456 −1.86876
\(52\) −0.369454 −0.0512340
\(53\) −9.94355 −1.36585 −0.682926 0.730488i \(-0.739293\pi\)
−0.682926 + 0.730488i \(0.739293\pi\)
\(54\) 2.34453 0.319050
\(55\) 0 0
\(56\) −1.83227 −0.244847
\(57\) 9.34563 1.23786
\(58\) 1.20681 0.158463
\(59\) −5.34563 −0.695941 −0.347971 0.937505i \(-0.613129\pi\)
−0.347971 + 0.937505i \(0.613129\pi\)
\(60\) 0 0
\(61\) −9.79428 −1.25403 −0.627015 0.779008i \(-0.715723\pi\)
−0.627015 + 0.779008i \(0.715723\pi\)
\(62\) 10.1030 1.28308
\(63\) −7.13302 −0.898676
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 11.0011 1.35414
\(67\) 1.85073 0.226103 0.113052 0.993589i \(-0.463937\pi\)
0.113052 + 0.993589i \(0.463937\pi\)
\(68\) −5.08317 −0.616425
\(69\) 14.7560 1.77641
\(70\) 0 0
\(71\) 2.86038 0.339464 0.169732 0.985490i \(-0.445710\pi\)
0.169732 + 0.985490i \(0.445710\pi\)
\(72\) 3.89300 0.458795
\(73\) −8.09942 −0.947966 −0.473983 0.880534i \(-0.657184\pi\)
−0.473983 + 0.880534i \(0.657184\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 3.55963 0.408317
\(77\) −7.67751 −0.874934
\(78\) −0.969984 −0.109829
\(79\) 6.06361 0.682209 0.341105 0.940025i \(-0.389199\pi\)
0.341105 + 0.940025i \(0.389199\pi\)
\(80\) 0 0
\(81\) −5.52355 −0.613727
\(82\) −8.01447 −0.885050
\(83\) −8.93200 −0.980414 −0.490207 0.871606i \(-0.663079\pi\)
−0.490207 + 0.871606i \(0.663079\pi\)
\(84\) −4.81053 −0.524872
\(85\) 0 0
\(86\) −2.27264 −0.245065
\(87\) 3.16843 0.339692
\(88\) 4.19017 0.446674
\(89\) 11.4773 1.21659 0.608295 0.793711i \(-0.291854\pi\)
0.608295 + 0.793711i \(0.291854\pi\)
\(90\) 0 0
\(91\) 0.676938 0.0709624
\(92\) 5.62036 0.585963
\(93\) 26.5249 2.75051
\(94\) 10.9154 1.12584
\(95\) 0 0
\(96\) 2.62545 0.267959
\(97\) 6.05864 0.615162 0.307581 0.951522i \(-0.400480\pi\)
0.307581 + 0.951522i \(0.400480\pi\)
\(98\) −3.64280 −0.367978
\(99\) 16.3123 1.63945
\(100\) 0 0
\(101\) 4.96528 0.494064 0.247032 0.969007i \(-0.420545\pi\)
0.247032 + 0.969007i \(0.420545\pi\)
\(102\) −13.3456 −1.32141
\(103\) −7.33338 −0.722579 −0.361289 0.932454i \(-0.617663\pi\)
−0.361289 + 0.932454i \(0.617663\pi\)
\(104\) −0.369454 −0.0362279
\(105\) 0 0
\(106\) −9.94355 −0.965803
\(107\) −8.21182 −0.793867 −0.396933 0.917847i \(-0.629926\pi\)
−0.396933 + 0.917847i \(0.629926\pi\)
\(108\) 2.34453 0.225603
\(109\) −20.5503 −1.96836 −0.984179 0.177175i \(-0.943304\pi\)
−0.984179 + 0.177175i \(0.943304\pi\)
\(110\) 0 0
\(111\) 2.62545 0.249197
\(112\) −1.83227 −0.173133
\(113\) −0.652984 −0.0614276 −0.0307138 0.999528i \(-0.509778\pi\)
−0.0307138 + 0.999528i \(0.509778\pi\)
\(114\) 9.34563 0.875298
\(115\) 0 0
\(116\) 1.20681 0.112050
\(117\) −1.43828 −0.132969
\(118\) −5.34563 −0.492105
\(119\) 9.31373 0.853788
\(120\) 0 0
\(121\) 6.55754 0.596140
\(122\) −9.79428 −0.886733
\(123\) −21.0416 −1.89726
\(124\) 10.1030 0.907276
\(125\) 0 0
\(126\) −7.13302 −0.635460
\(127\) −19.2856 −1.71132 −0.855660 0.517539i \(-0.826848\pi\)
−0.855660 + 0.517539i \(0.826848\pi\)
\(128\) 1.00000 0.0883883
\(129\) −5.96671 −0.525340
\(130\) 0 0
\(131\) −8.82072 −0.770670 −0.385335 0.922777i \(-0.625914\pi\)
−0.385335 + 0.922777i \(0.625914\pi\)
\(132\) 11.0011 0.957522
\(133\) −6.52218 −0.565545
\(134\) 1.85073 0.159879
\(135\) 0 0
\(136\) −5.08317 −0.435878
\(137\) 4.66126 0.398239 0.199119 0.979975i \(-0.436192\pi\)
0.199119 + 0.979975i \(0.436192\pi\)
\(138\) 14.7560 1.25611
\(139\) 16.4050 1.39145 0.695727 0.718306i \(-0.255082\pi\)
0.695727 + 0.718306i \(0.255082\pi\)
\(140\) 0 0
\(141\) 28.6580 2.41344
\(142\) 2.86038 0.240037
\(143\) −1.54808 −0.129457
\(144\) 3.89300 0.324417
\(145\) 0 0
\(146\) −8.09942 −0.670313
\(147\) −9.56399 −0.788825
\(148\) 1.00000 0.0821995
\(149\) −5.86259 −0.480282 −0.240141 0.970738i \(-0.577194\pi\)
−0.240141 + 0.970738i \(0.577194\pi\)
\(150\) 0 0
\(151\) 13.1460 1.06981 0.534903 0.844913i \(-0.320348\pi\)
0.534903 + 0.844913i \(0.320348\pi\)
\(152\) 3.55963 0.288724
\(153\) −19.7888 −1.59983
\(154\) −7.67751 −0.618672
\(155\) 0 0
\(156\) −0.969984 −0.0776609
\(157\) −7.33629 −0.585500 −0.292750 0.956189i \(-0.594570\pi\)
−0.292750 + 0.956189i \(0.594570\pi\)
\(158\) 6.06361 0.482395
\(159\) −26.1063 −2.07037
\(160\) 0 0
\(161\) −10.2980 −0.811596
\(162\) −5.52355 −0.433971
\(163\) −18.7968 −1.47228 −0.736138 0.676831i \(-0.763353\pi\)
−0.736138 + 0.676831i \(0.763353\pi\)
\(164\) −8.01447 −0.625825
\(165\) 0 0
\(166\) −8.93200 −0.693257
\(167\) −5.92772 −0.458700 −0.229350 0.973344i \(-0.573660\pi\)
−0.229350 + 0.973344i \(0.573660\pi\)
\(168\) −4.81053 −0.371140
\(169\) −12.8635 −0.989500
\(170\) 0 0
\(171\) 13.8576 1.05972
\(172\) −2.27264 −0.173287
\(173\) 5.78242 0.439629 0.219815 0.975542i \(-0.429455\pi\)
0.219815 + 0.975542i \(0.429455\pi\)
\(174\) 3.16843 0.240198
\(175\) 0 0
\(176\) 4.19017 0.315846
\(177\) −14.0347 −1.05491
\(178\) 11.4773 0.860259
\(179\) 7.13581 0.533355 0.266678 0.963786i \(-0.414074\pi\)
0.266678 + 0.963786i \(0.414074\pi\)
\(180\) 0 0
\(181\) 11.5365 0.857503 0.428752 0.903422i \(-0.358954\pi\)
0.428752 + 0.903422i \(0.358954\pi\)
\(182\) 0.676938 0.0501780
\(183\) −25.7144 −1.90086
\(184\) 5.62036 0.414338
\(185\) 0 0
\(186\) 26.5249 1.94490
\(187\) −21.2994 −1.55756
\(188\) 10.9154 0.796090
\(189\) −4.29581 −0.312474
\(190\) 0 0
\(191\) −20.6048 −1.49091 −0.745456 0.666555i \(-0.767768\pi\)
−0.745456 + 0.666555i \(0.767768\pi\)
\(192\) 2.62545 0.189476
\(193\) −26.1663 −1.88349 −0.941747 0.336321i \(-0.890817\pi\)
−0.941747 + 0.336321i \(0.890817\pi\)
\(194\) 6.05864 0.434985
\(195\) 0 0
\(196\) −3.64280 −0.260200
\(197\) 14.6067 1.04069 0.520343 0.853957i \(-0.325804\pi\)
0.520343 + 0.853957i \(0.325804\pi\)
\(198\) 16.3123 1.15927
\(199\) 5.81275 0.412055 0.206027 0.978546i \(-0.433946\pi\)
0.206027 + 0.978546i \(0.433946\pi\)
\(200\) 0 0
\(201\) 4.85901 0.342728
\(202\) 4.96528 0.349356
\(203\) −2.21121 −0.155196
\(204\) −13.3456 −0.934381
\(205\) 0 0
\(206\) −7.33338 −0.510940
\(207\) 21.8801 1.52077
\(208\) −0.369454 −0.0256170
\(209\) 14.9154 1.03172
\(210\) 0 0
\(211\) 16.0976 1.10821 0.554104 0.832448i \(-0.313061\pi\)
0.554104 + 0.832448i \(0.313061\pi\)
\(212\) −9.94355 −0.682926
\(213\) 7.50978 0.514562
\(214\) −8.21182 −0.561349
\(215\) 0 0
\(216\) 2.34453 0.159525
\(217\) −18.5114 −1.25664
\(218\) −20.5503 −1.39184
\(219\) −21.2646 −1.43693
\(220\) 0 0
\(221\) 1.87800 0.126328
\(222\) 2.62545 0.176209
\(223\) 25.3608 1.69828 0.849142 0.528164i \(-0.177119\pi\)
0.849142 + 0.528164i \(0.177119\pi\)
\(224\) −1.83227 −0.122423
\(225\) 0 0
\(226\) −0.652984 −0.0434359
\(227\) −5.57339 −0.369919 −0.184960 0.982746i \(-0.559215\pi\)
−0.184960 + 0.982746i \(0.559215\pi\)
\(228\) 9.34563 0.618929
\(229\) 9.02035 0.596081 0.298041 0.954553i \(-0.403667\pi\)
0.298041 + 0.954553i \(0.403667\pi\)
\(230\) 0 0
\(231\) −20.1569 −1.32623
\(232\) 1.20681 0.0792313
\(233\) 11.8100 0.773696 0.386848 0.922144i \(-0.373564\pi\)
0.386848 + 0.922144i \(0.373564\pi\)
\(234\) −1.43828 −0.0940236
\(235\) 0 0
\(236\) −5.34563 −0.347971
\(237\) 15.9197 1.03410
\(238\) 9.31373 0.603719
\(239\) 0.907160 0.0586793 0.0293396 0.999569i \(-0.490660\pi\)
0.0293396 + 0.999569i \(0.490660\pi\)
\(240\) 0 0
\(241\) −4.15616 −0.267722 −0.133861 0.991000i \(-0.542738\pi\)
−0.133861 + 0.991000i \(0.542738\pi\)
\(242\) 6.55754 0.421534
\(243\) −21.5354 −1.38150
\(244\) −9.79428 −0.627015
\(245\) 0 0
\(246\) −21.0416 −1.34156
\(247\) −1.31512 −0.0836789
\(248\) 10.1030 0.641541
\(249\) −23.4505 −1.48612
\(250\) 0 0
\(251\) 9.31888 0.588203 0.294101 0.955774i \(-0.404980\pi\)
0.294101 + 0.955774i \(0.404980\pi\)
\(252\) −7.13302 −0.449338
\(253\) 23.5503 1.48059
\(254\) −19.2856 −1.21009
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −13.6312 −0.850294 −0.425147 0.905124i \(-0.639778\pi\)
−0.425147 + 0.905124i \(0.639778\pi\)
\(258\) −5.96671 −0.371471
\(259\) −1.83227 −0.113852
\(260\) 0 0
\(261\) 4.69813 0.290807
\(262\) −8.82072 −0.544946
\(263\) 27.2170 1.67827 0.839137 0.543921i \(-0.183061\pi\)
0.839137 + 0.543921i \(0.183061\pi\)
\(264\) 11.0011 0.677071
\(265\) 0 0
\(266\) −6.52218 −0.399901
\(267\) 30.1331 1.84411
\(268\) 1.85073 0.113052
\(269\) 23.0550 1.40569 0.702845 0.711343i \(-0.251913\pi\)
0.702845 + 0.711343i \(0.251913\pi\)
\(270\) 0 0
\(271\) 22.9320 1.39302 0.696510 0.717547i \(-0.254735\pi\)
0.696510 + 0.717547i \(0.254735\pi\)
\(272\) −5.08317 −0.308213
\(273\) 1.77727 0.107565
\(274\) 4.66126 0.281597
\(275\) 0 0
\(276\) 14.7560 0.888206
\(277\) −11.7534 −0.706192 −0.353096 0.935587i \(-0.614871\pi\)
−0.353096 + 0.935587i \(0.614871\pi\)
\(278\) 16.4050 0.983906
\(279\) 39.3310 2.35468
\(280\) 0 0
\(281\) −29.0962 −1.73573 −0.867865 0.496799i \(-0.834508\pi\)
−0.867865 + 0.496799i \(0.834508\pi\)
\(282\) 28.6580 1.70656
\(283\) −10.1829 −0.605311 −0.302655 0.953100i \(-0.597873\pi\)
−0.302655 + 0.953100i \(0.597873\pi\)
\(284\) 2.86038 0.169732
\(285\) 0 0
\(286\) −1.54808 −0.0915396
\(287\) 14.6846 0.866807
\(288\) 3.89300 0.229397
\(289\) 8.83864 0.519920
\(290\) 0 0
\(291\) 15.9067 0.932466
\(292\) −8.09942 −0.473983
\(293\) 18.1243 1.05883 0.529415 0.848363i \(-0.322411\pi\)
0.529415 + 0.848363i \(0.322411\pi\)
\(294\) −9.56399 −0.557783
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 9.82399 0.570046
\(298\) −5.86259 −0.339611
\(299\) −2.07646 −0.120085
\(300\) 0 0
\(301\) 4.16409 0.240014
\(302\) 13.1460 0.756467
\(303\) 13.0361 0.748905
\(304\) 3.55963 0.204159
\(305\) 0 0
\(306\) −19.7888 −1.13125
\(307\) −6.15561 −0.351319 −0.175660 0.984451i \(-0.556206\pi\)
−0.175660 + 0.984451i \(0.556206\pi\)
\(308\) −7.67751 −0.437467
\(309\) −19.2534 −1.09529
\(310\) 0 0
\(311\) −29.4512 −1.67002 −0.835011 0.550234i \(-0.814539\pi\)
−0.835011 + 0.550234i \(0.814539\pi\)
\(312\) −0.969984 −0.0549145
\(313\) 13.0102 0.735378 0.367689 0.929949i \(-0.380149\pi\)
0.367689 + 0.929949i \(0.380149\pi\)
\(314\) −7.33629 −0.414011
\(315\) 0 0
\(316\) 6.06361 0.341105
\(317\) −7.39826 −0.415528 −0.207764 0.978179i \(-0.566619\pi\)
−0.207764 + 0.978179i \(0.566619\pi\)
\(318\) −26.1063 −1.46397
\(319\) 5.05676 0.283124
\(320\) 0 0
\(321\) −21.5598 −1.20335
\(322\) −10.2980 −0.573885
\(323\) −18.0942 −1.00679
\(324\) −5.52355 −0.306864
\(325\) 0 0
\(326\) −18.7968 −1.04106
\(327\) −53.9537 −2.98365
\(328\) −8.01447 −0.442525
\(329\) −20.0000 −1.10264
\(330\) 0 0
\(331\) 33.2545 1.82783 0.913916 0.405903i \(-0.133043\pi\)
0.913916 + 0.405903i \(0.133043\pi\)
\(332\) −8.93200 −0.490207
\(333\) 3.89300 0.213335
\(334\) −5.92772 −0.324350
\(335\) 0 0
\(336\) −4.81053 −0.262436
\(337\) 22.6288 1.23267 0.616334 0.787485i \(-0.288617\pi\)
0.616334 + 0.787485i \(0.288617\pi\)
\(338\) −12.8635 −0.699682
\(339\) −1.71438 −0.0931123
\(340\) 0 0
\(341\) 42.3333 2.29248
\(342\) 13.8576 0.749334
\(343\) 19.5004 1.05293
\(344\) −2.27264 −0.122533
\(345\) 0 0
\(346\) 5.78242 0.310865
\(347\) −24.1396 −1.29588 −0.647941 0.761691i \(-0.724370\pi\)
−0.647941 + 0.761691i \(0.724370\pi\)
\(348\) 3.16843 0.169846
\(349\) −10.3702 −0.555102 −0.277551 0.960711i \(-0.589523\pi\)
−0.277551 + 0.960711i \(0.589523\pi\)
\(350\) 0 0
\(351\) −0.866196 −0.0462341
\(352\) 4.19017 0.223337
\(353\) −3.92699 −0.209013 −0.104506 0.994524i \(-0.533326\pi\)
−0.104506 + 0.994524i \(0.533326\pi\)
\(354\) −14.0347 −0.745935
\(355\) 0 0
\(356\) 11.4773 0.608295
\(357\) 24.4528 1.29418
\(358\) 7.13581 0.377139
\(359\) −9.85179 −0.519957 −0.259979 0.965614i \(-0.583716\pi\)
−0.259979 + 0.965614i \(0.583716\pi\)
\(360\) 0 0
\(361\) −6.32907 −0.333109
\(362\) 11.5365 0.606346
\(363\) 17.2165 0.903632
\(364\) 0.676938 0.0354812
\(365\) 0 0
\(366\) −25.7144 −1.34411
\(367\) 33.2559 1.73594 0.867972 0.496614i \(-0.165423\pi\)
0.867972 + 0.496614i \(0.165423\pi\)
\(368\) 5.62036 0.292981
\(369\) −31.2003 −1.62422
\(370\) 0 0
\(371\) 18.2192 0.945896
\(372\) 26.5249 1.37525
\(373\) −28.5387 −1.47768 −0.738839 0.673882i \(-0.764625\pi\)
−0.738839 + 0.673882i \(0.764625\pi\)
\(374\) −21.2994 −1.10136
\(375\) 0 0
\(376\) 10.9154 0.562921
\(377\) −0.445862 −0.0229631
\(378\) −4.29581 −0.220953
\(379\) 0.476554 0.0244789 0.0122395 0.999925i \(-0.496104\pi\)
0.0122395 + 0.999925i \(0.496104\pi\)
\(380\) 0 0
\(381\) −50.6334 −2.59403
\(382\) −20.6048 −1.05423
\(383\) 7.13962 0.364818 0.182409 0.983223i \(-0.441611\pi\)
0.182409 + 0.983223i \(0.441611\pi\)
\(384\) 2.62545 0.133980
\(385\) 0 0
\(386\) −26.1663 −1.33183
\(387\) −8.84740 −0.449738
\(388\) 6.05864 0.307581
\(389\) −24.2060 −1.22729 −0.613646 0.789581i \(-0.710298\pi\)
−0.613646 + 0.789581i \(0.710298\pi\)
\(390\) 0 0
\(391\) −28.5693 −1.44481
\(392\) −3.64280 −0.183989
\(393\) −23.1584 −1.16819
\(394\) 14.6067 0.735876
\(395\) 0 0
\(396\) 16.3123 0.819726
\(397\) 20.7491 1.04137 0.520684 0.853750i \(-0.325677\pi\)
0.520684 + 0.853750i \(0.325677\pi\)
\(398\) 5.81275 0.291367
\(399\) −17.1237 −0.857256
\(400\) 0 0
\(401\) −34.8410 −1.73988 −0.869939 0.493159i \(-0.835842\pi\)
−0.869939 + 0.493159i \(0.835842\pi\)
\(402\) 4.85901 0.242346
\(403\) −3.73259 −0.185934
\(404\) 4.96528 0.247032
\(405\) 0 0
\(406\) −2.21121 −0.109740
\(407\) 4.19017 0.207699
\(408\) −13.3456 −0.660707
\(409\) 25.2132 1.24671 0.623356 0.781938i \(-0.285769\pi\)
0.623356 + 0.781938i \(0.285769\pi\)
\(410\) 0 0
\(411\) 12.2379 0.603652
\(412\) −7.33338 −0.361289
\(413\) 9.79462 0.481962
\(414\) 21.8801 1.07535
\(415\) 0 0
\(416\) −0.369454 −0.0181140
\(417\) 43.0705 2.10917
\(418\) 14.9154 0.729538
\(419\) 5.69695 0.278314 0.139157 0.990270i \(-0.455561\pi\)
0.139157 + 0.990270i \(0.455561\pi\)
\(420\) 0 0
\(421\) 3.38092 0.164776 0.0823880 0.996600i \(-0.473745\pi\)
0.0823880 + 0.996600i \(0.473745\pi\)
\(422\) 16.0976 0.783621
\(423\) 42.4938 2.06612
\(424\) −9.94355 −0.482901
\(425\) 0 0
\(426\) 7.50978 0.363850
\(427\) 17.9457 0.868455
\(428\) −8.21182 −0.396933
\(429\) −4.06440 −0.196231
\(430\) 0 0
\(431\) −0.381731 −0.0183873 −0.00919367 0.999958i \(-0.502926\pi\)
−0.00919367 + 0.999958i \(0.502926\pi\)
\(432\) 2.34453 0.112801
\(433\) −21.6393 −1.03992 −0.519958 0.854192i \(-0.674052\pi\)
−0.519958 + 0.854192i \(0.674052\pi\)
\(434\) −18.5114 −0.888575
\(435\) 0 0
\(436\) −20.5503 −0.984179
\(437\) 20.0064 0.957035
\(438\) −21.2646 −1.01606
\(439\) 26.4663 1.26317 0.631583 0.775308i \(-0.282405\pi\)
0.631583 + 0.775308i \(0.282405\pi\)
\(440\) 0 0
\(441\) −14.1814 −0.675305
\(442\) 1.87800 0.0893273
\(443\) −4.65489 −0.221161 −0.110580 0.993867i \(-0.535271\pi\)
−0.110580 + 0.993867i \(0.535271\pi\)
\(444\) 2.62545 0.124598
\(445\) 0 0
\(446\) 25.3608 1.20087
\(447\) −15.3920 −0.728015
\(448\) −1.83227 −0.0865665
\(449\) −26.5531 −1.25312 −0.626558 0.779375i \(-0.715537\pi\)
−0.626558 + 0.779375i \(0.715537\pi\)
\(450\) 0 0
\(451\) −33.5820 −1.58131
\(452\) −0.652984 −0.0307138
\(453\) 34.5142 1.62162
\(454\) −5.57339 −0.261572
\(455\) 0 0
\(456\) 9.34563 0.437649
\(457\) 4.27264 0.199866 0.0999329 0.994994i \(-0.468137\pi\)
0.0999329 + 0.994994i \(0.468137\pi\)
\(458\) 9.02035 0.421493
\(459\) −11.9177 −0.556269
\(460\) 0 0
\(461\) 32.1873 1.49911 0.749555 0.661942i \(-0.230267\pi\)
0.749555 + 0.661942i \(0.230267\pi\)
\(462\) −20.1569 −0.937786
\(463\) 18.4014 0.855186 0.427593 0.903971i \(-0.359362\pi\)
0.427593 + 0.903971i \(0.359362\pi\)
\(464\) 1.20681 0.0560250
\(465\) 0 0
\(466\) 11.8100 0.547086
\(467\) −3.06777 −0.141959 −0.0709797 0.997478i \(-0.522613\pi\)
−0.0709797 + 0.997478i \(0.522613\pi\)
\(468\) −1.43828 −0.0664847
\(469\) −3.39104 −0.156584
\(470\) 0 0
\(471\) −19.2611 −0.887504
\(472\) −5.34563 −0.246052
\(473\) −9.52276 −0.437857
\(474\) 15.9197 0.731217
\(475\) 0 0
\(476\) 9.31373 0.426894
\(477\) −38.7102 −1.77242
\(478\) 0.907160 0.0414925
\(479\) 8.78798 0.401533 0.200767 0.979639i \(-0.435657\pi\)
0.200767 + 0.979639i \(0.435657\pi\)
\(480\) 0 0
\(481\) −0.369454 −0.0168457
\(482\) −4.15616 −0.189308
\(483\) −27.0369 −1.23022
\(484\) 6.55754 0.298070
\(485\) 0 0
\(486\) −21.5354 −0.976866
\(487\) −22.4814 −1.01873 −0.509366 0.860550i \(-0.670120\pi\)
−0.509366 + 0.860550i \(0.670120\pi\)
\(488\) −9.79428 −0.443366
\(489\) −49.3500 −2.23168
\(490\) 0 0
\(491\) −0.0441752 −0.00199360 −0.000996799 1.00000i \(-0.500317\pi\)
−0.000996799 1.00000i \(0.500317\pi\)
\(492\) −21.0416 −0.948629
\(493\) −6.13445 −0.276282
\(494\) −1.31512 −0.0591699
\(495\) 0 0
\(496\) 10.1030 0.453638
\(497\) −5.24097 −0.235090
\(498\) −23.4505 −1.05084
\(499\) 28.2445 1.26440 0.632199 0.774806i \(-0.282153\pi\)
0.632199 + 0.774806i \(0.282153\pi\)
\(500\) 0 0
\(501\) −15.5629 −0.695301
\(502\) 9.31888 0.415922
\(503\) −19.7713 −0.881560 −0.440780 0.897615i \(-0.645298\pi\)
−0.440780 + 0.897615i \(0.645298\pi\)
\(504\) −7.13302 −0.317730
\(505\) 0 0
\(506\) 23.5503 1.04694
\(507\) −33.7725 −1.49989
\(508\) −19.2856 −0.855660
\(509\) 43.4809 1.92726 0.963628 0.267247i \(-0.0861141\pi\)
0.963628 + 0.267247i \(0.0861141\pi\)
\(510\) 0 0
\(511\) 14.8403 0.656496
\(512\) 1.00000 0.0441942
\(513\) 8.34565 0.368470
\(514\) −13.6312 −0.601249
\(515\) 0 0
\(516\) −5.96671 −0.262670
\(517\) 45.7376 2.01154
\(518\) −1.83227 −0.0805052
\(519\) 15.1815 0.666393
\(520\) 0 0
\(521\) −18.0325 −0.790019 −0.395009 0.918677i \(-0.629259\pi\)
−0.395009 + 0.918677i \(0.629259\pi\)
\(522\) 4.69813 0.205632
\(523\) 31.6788 1.38522 0.692610 0.721313i \(-0.256461\pi\)
0.692610 + 0.721313i \(0.256461\pi\)
\(524\) −8.82072 −0.385335
\(525\) 0 0
\(526\) 27.2170 1.18672
\(527\) −51.3553 −2.23707
\(528\) 11.0011 0.478761
\(529\) 8.58844 0.373410
\(530\) 0 0
\(531\) −20.8105 −0.903100
\(532\) −6.52218 −0.282773
\(533\) 2.96098 0.128254
\(534\) 30.1331 1.30398
\(535\) 0 0
\(536\) 1.85073 0.0799395
\(537\) 18.7347 0.808463
\(538\) 23.0550 0.993973
\(539\) −15.2639 −0.657465
\(540\) 0 0
\(541\) 9.39442 0.403898 0.201949 0.979396i \(-0.435273\pi\)
0.201949 + 0.979396i \(0.435273\pi\)
\(542\) 22.9320 0.985014
\(543\) 30.2886 1.29981
\(544\) −5.08317 −0.217939
\(545\) 0 0
\(546\) 1.77727 0.0760601
\(547\) 24.7824 1.05962 0.529809 0.848117i \(-0.322263\pi\)
0.529809 + 0.848117i \(0.322263\pi\)
\(548\) 4.66126 0.199119
\(549\) −38.1291 −1.62731
\(550\) 0 0
\(551\) 4.29581 0.183008
\(552\) 14.7560 0.628056
\(553\) −11.1102 −0.472452
\(554\) −11.7534 −0.499353
\(555\) 0 0
\(556\) 16.4050 0.695727
\(557\) −22.1627 −0.939062 −0.469531 0.882916i \(-0.655577\pi\)
−0.469531 + 0.882916i \(0.655577\pi\)
\(558\) 39.3310 1.66501
\(559\) 0.839637 0.0355128
\(560\) 0 0
\(561\) −55.9205 −2.36096
\(562\) −29.0962 −1.22735
\(563\) 6.46846 0.272613 0.136306 0.990667i \(-0.456477\pi\)
0.136306 + 0.990667i \(0.456477\pi\)
\(564\) 28.6580 1.20672
\(565\) 0 0
\(566\) −10.1829 −0.428019
\(567\) 10.1206 0.425026
\(568\) 2.86038 0.120019
\(569\) 8.26550 0.346508 0.173254 0.984877i \(-0.444572\pi\)
0.173254 + 0.984877i \(0.444572\pi\)
\(570\) 0 0
\(571\) 34.5179 1.44453 0.722264 0.691618i \(-0.243102\pi\)
0.722264 + 0.691618i \(0.243102\pi\)
\(572\) −1.54808 −0.0647283
\(573\) −54.0970 −2.25993
\(574\) 14.6846 0.612925
\(575\) 0 0
\(576\) 3.89300 0.162208
\(577\) −3.05347 −0.127117 −0.0635587 0.997978i \(-0.520245\pi\)
−0.0635587 + 0.997978i \(0.520245\pi\)
\(578\) 8.83864 0.367639
\(579\) −68.6985 −2.85501
\(580\) 0 0
\(581\) 16.3658 0.678968
\(582\) 15.9067 0.659353
\(583\) −41.6652 −1.72559
\(584\) −8.09942 −0.335156
\(585\) 0 0
\(586\) 18.1243 0.748706
\(587\) −35.1344 −1.45015 −0.725076 0.688668i \(-0.758196\pi\)
−0.725076 + 0.688668i \(0.758196\pi\)
\(588\) −9.56399 −0.394412
\(589\) 35.9629 1.48183
\(590\) 0 0
\(591\) 38.3492 1.57748
\(592\) 1.00000 0.0410997
\(593\) −12.0400 −0.494424 −0.247212 0.968961i \(-0.579514\pi\)
−0.247212 + 0.968961i \(0.579514\pi\)
\(594\) 9.82399 0.403083
\(595\) 0 0
\(596\) −5.86259 −0.240141
\(597\) 15.2611 0.624595
\(598\) −2.07646 −0.0849129
\(599\) −17.4701 −0.713810 −0.356905 0.934141i \(-0.616168\pi\)
−0.356905 + 0.934141i \(0.616168\pi\)
\(600\) 0 0
\(601\) 38.7044 1.57878 0.789392 0.613890i \(-0.210396\pi\)
0.789392 + 0.613890i \(0.210396\pi\)
\(602\) 4.16409 0.169716
\(603\) 7.20491 0.293406
\(604\) 13.1460 0.534903
\(605\) 0 0
\(606\) 13.0361 0.529556
\(607\) −28.7903 −1.16856 −0.584282 0.811551i \(-0.698624\pi\)
−0.584282 + 0.811551i \(0.698624\pi\)
\(608\) 3.55963 0.144362
\(609\) −5.80542 −0.235247
\(610\) 0 0
\(611\) −4.03275 −0.163148
\(612\) −19.7888 −0.799915
\(613\) −7.66539 −0.309602 −0.154801 0.987946i \(-0.549474\pi\)
−0.154801 + 0.987946i \(0.549474\pi\)
\(614\) −6.15561 −0.248420
\(615\) 0 0
\(616\) −7.67751 −0.309336
\(617\) 36.5388 1.47100 0.735498 0.677527i \(-0.236948\pi\)
0.735498 + 0.677527i \(0.236948\pi\)
\(618\) −19.2534 −0.774487
\(619\) −45.8085 −1.84120 −0.920599 0.390510i \(-0.872299\pi\)
−0.920599 + 0.390510i \(0.872299\pi\)
\(620\) 0 0
\(621\) 13.1771 0.528779
\(622\) −29.4512 −1.18088
\(623\) −21.0294 −0.842527
\(624\) −0.969984 −0.0388304
\(625\) 0 0
\(626\) 13.0102 0.519991
\(627\) 39.1598 1.56389
\(628\) −7.33629 −0.292750
\(629\) −5.08317 −0.202679
\(630\) 0 0
\(631\) 3.42682 0.136420 0.0682098 0.997671i \(-0.478271\pi\)
0.0682098 + 0.997671i \(0.478271\pi\)
\(632\) 6.06361 0.241197
\(633\) 42.2636 1.67983
\(634\) −7.39826 −0.293823
\(635\) 0 0
\(636\) −26.1063 −1.03518
\(637\) 1.34585 0.0533244
\(638\) 5.05676 0.200199
\(639\) 11.1354 0.440511
\(640\) 0 0
\(641\) 38.3241 1.51371 0.756855 0.653583i \(-0.226735\pi\)
0.756855 + 0.653583i \(0.226735\pi\)
\(642\) −21.5598 −0.850896
\(643\) 1.36660 0.0538935 0.0269468 0.999637i \(-0.491422\pi\)
0.0269468 + 0.999637i \(0.491422\pi\)
\(644\) −10.2980 −0.405798
\(645\) 0 0
\(646\) −18.0942 −0.711906
\(647\) 11.2690 0.443028 0.221514 0.975157i \(-0.428900\pi\)
0.221514 + 0.975157i \(0.428900\pi\)
\(648\) −5.52355 −0.216985
\(649\) −22.3991 −0.879241
\(650\) 0 0
\(651\) −48.6008 −1.90482
\(652\) −18.7968 −0.736138
\(653\) −29.5439 −1.15614 −0.578070 0.815987i \(-0.696194\pi\)
−0.578070 + 0.815987i \(0.696194\pi\)
\(654\) −53.9537 −2.10976
\(655\) 0 0
\(656\) −8.01447 −0.312912
\(657\) −31.5311 −1.23014
\(658\) −20.0000 −0.779681
\(659\) −49.9126 −1.94432 −0.972159 0.234324i \(-0.924712\pi\)
−0.972159 + 0.234324i \(0.924712\pi\)
\(660\) 0 0
\(661\) 25.9950 1.01109 0.505545 0.862800i \(-0.331292\pi\)
0.505545 + 0.862800i \(0.331292\pi\)
\(662\) 33.2545 1.29247
\(663\) 4.93059 0.191488
\(664\) −8.93200 −0.346629
\(665\) 0 0
\(666\) 3.89300 0.150851
\(667\) 6.78273 0.262628
\(668\) −5.92772 −0.229350
\(669\) 66.5836 2.57427
\(670\) 0 0
\(671\) −41.0397 −1.58432
\(672\) −4.81053 −0.185570
\(673\) 6.91054 0.266382 0.133191 0.991090i \(-0.457478\pi\)
0.133191 + 0.991090i \(0.457478\pi\)
\(674\) 22.6288 0.871627
\(675\) 0 0
\(676\) −12.8635 −0.494750
\(677\) −15.6913 −0.603065 −0.301532 0.953456i \(-0.597498\pi\)
−0.301532 + 0.953456i \(0.597498\pi\)
\(678\) −1.71438 −0.0658403
\(679\) −11.1011 −0.426019
\(680\) 0 0
\(681\) −14.6327 −0.560725
\(682\) 42.3333 1.62103
\(683\) −18.2002 −0.696412 −0.348206 0.937418i \(-0.613209\pi\)
−0.348206 + 0.937418i \(0.613209\pi\)
\(684\) 13.8576 0.529859
\(685\) 0 0
\(686\) 19.5004 0.744531
\(687\) 23.6825 0.903544
\(688\) −2.27264 −0.0866437
\(689\) 3.67368 0.139956
\(690\) 0 0
\(691\) −2.58658 −0.0983982 −0.0491991 0.998789i \(-0.515667\pi\)
−0.0491991 + 0.998789i \(0.515667\pi\)
\(692\) 5.78242 0.219815
\(693\) −29.8886 −1.13537
\(694\) −24.1396 −0.916327
\(695\) 0 0
\(696\) 3.16843 0.120099
\(697\) 40.7389 1.54310
\(698\) −10.3702 −0.392516
\(699\) 31.0065 1.17277
\(700\) 0 0
\(701\) −26.5092 −1.00124 −0.500620 0.865667i \(-0.666895\pi\)
−0.500620 + 0.865667i \(0.666895\pi\)
\(702\) −0.866196 −0.0326925
\(703\) 3.55963 0.134254
\(704\) 4.19017 0.157923
\(705\) 0 0
\(706\) −3.92699 −0.147794
\(707\) −9.09773 −0.342155
\(708\) −14.0347 −0.527456
\(709\) −25.0474 −0.940674 −0.470337 0.882487i \(-0.655868\pi\)
−0.470337 + 0.882487i \(0.655868\pi\)
\(710\) 0 0
\(711\) 23.6056 0.885281
\(712\) 11.4773 0.430129
\(713\) 56.7825 2.12652
\(714\) 24.4528 0.915121
\(715\) 0 0
\(716\) 7.13581 0.266678
\(717\) 2.38171 0.0889464
\(718\) −9.85179 −0.367665
\(719\) 24.9551 0.930668 0.465334 0.885135i \(-0.345934\pi\)
0.465334 + 0.885135i \(0.345934\pi\)
\(720\) 0 0
\(721\) 13.4367 0.500409
\(722\) −6.32907 −0.235544
\(723\) −10.9118 −0.405814
\(724\) 11.5365 0.428752
\(725\) 0 0
\(726\) 17.2165 0.638964
\(727\) −21.6774 −0.803970 −0.401985 0.915646i \(-0.631680\pi\)
−0.401985 + 0.915646i \(0.631680\pi\)
\(728\) 0.676938 0.0250890
\(729\) −39.9695 −1.48035
\(730\) 0 0
\(731\) 11.5522 0.427275
\(732\) −25.7144 −0.950432
\(733\) 6.33267 0.233903 0.116951 0.993138i \(-0.462688\pi\)
0.116951 + 0.993138i \(0.462688\pi\)
\(734\) 33.2559 1.22750
\(735\) 0 0
\(736\) 5.62036 0.207169
\(737\) 7.75489 0.285655
\(738\) −31.2003 −1.14850
\(739\) 12.8076 0.471136 0.235568 0.971858i \(-0.424305\pi\)
0.235568 + 0.971858i \(0.424305\pi\)
\(740\) 0 0
\(741\) −3.45278 −0.126841
\(742\) 18.2192 0.668849
\(743\) 11.0694 0.406097 0.203048 0.979169i \(-0.434915\pi\)
0.203048 + 0.979169i \(0.434915\pi\)
\(744\) 26.5249 0.972452
\(745\) 0 0
\(746\) −28.5387 −1.04488
\(747\) −34.7723 −1.27225
\(748\) −21.2994 −0.778782
\(749\) 15.0463 0.549778
\(750\) 0 0
\(751\) −46.5316 −1.69796 −0.848981 0.528423i \(-0.822784\pi\)
−0.848981 + 0.528423i \(0.822784\pi\)
\(752\) 10.9154 0.398045
\(753\) 24.4663 0.891601
\(754\) −0.445862 −0.0162374
\(755\) 0 0
\(756\) −4.29581 −0.156237
\(757\) 32.9872 1.19894 0.599470 0.800397i \(-0.295378\pi\)
0.599470 + 0.800397i \(0.295378\pi\)
\(758\) 0.476554 0.0173092
\(759\) 61.8301 2.24429
\(760\) 0 0
\(761\) 28.7913 1.04369 0.521843 0.853042i \(-0.325245\pi\)
0.521843 + 0.853042i \(0.325245\pi\)
\(762\) −50.6334 −1.83426
\(763\) 37.6536 1.36315
\(764\) −20.6048 −0.745456
\(765\) 0 0
\(766\) 7.13962 0.257965
\(767\) 1.97496 0.0713118
\(768\) 2.62545 0.0947379
\(769\) 41.9025 1.51104 0.755521 0.655124i \(-0.227384\pi\)
0.755521 + 0.655124i \(0.227384\pi\)
\(770\) 0 0
\(771\) −35.7882 −1.28888
\(772\) −26.1663 −0.941747
\(773\) 31.0931 1.11834 0.559171 0.829052i \(-0.311119\pi\)
0.559171 + 0.829052i \(0.311119\pi\)
\(774\) −8.84740 −0.318013
\(775\) 0 0
\(776\) 6.05864 0.217493
\(777\) −4.81053 −0.172577
\(778\) −24.2060 −0.867827
\(779\) −28.5285 −1.02214
\(780\) 0 0
\(781\) 11.9855 0.428874
\(782\) −28.5693 −1.02163
\(783\) 2.82941 0.101115
\(784\) −3.64280 −0.130100
\(785\) 0 0
\(786\) −23.1584 −0.826032
\(787\) −18.2842 −0.651760 −0.325880 0.945411i \(-0.605661\pi\)
−0.325880 + 0.945411i \(0.605661\pi\)
\(788\) 14.6067 0.520343
\(789\) 71.4570 2.54394
\(790\) 0 0
\(791\) 1.19644 0.0425406
\(792\) 16.3123 0.579634
\(793\) 3.61854 0.128498
\(794\) 20.7491 0.736358
\(795\) 0 0
\(796\) 5.81275 0.206027
\(797\) −40.6025 −1.43821 −0.719106 0.694900i \(-0.755449\pi\)
−0.719106 + 0.694900i \(0.755449\pi\)
\(798\) −17.1237 −0.606172
\(799\) −55.4851 −1.96292
\(800\) 0 0
\(801\) 44.6811 1.57873
\(802\) −34.8410 −1.23028
\(803\) −33.9380 −1.19764
\(804\) 4.85901 0.171364
\(805\) 0 0
\(806\) −3.73259 −0.131475
\(807\) 60.5299 2.13075
\(808\) 4.96528 0.174678
\(809\) 41.1512 1.44680 0.723400 0.690429i \(-0.242578\pi\)
0.723400 + 0.690429i \(0.242578\pi\)
\(810\) 0 0
\(811\) −6.71866 −0.235924 −0.117962 0.993018i \(-0.537636\pi\)
−0.117962 + 0.993018i \(0.537636\pi\)
\(812\) −2.21121 −0.0775981
\(813\) 60.2069 2.11155
\(814\) 4.19017 0.146865
\(815\) 0 0
\(816\) −13.3456 −0.467190
\(817\) −8.08975 −0.283025
\(818\) 25.2132 0.881558
\(819\) 2.63532 0.0920856
\(820\) 0 0
\(821\) −11.5263 −0.402272 −0.201136 0.979563i \(-0.564463\pi\)
−0.201136 + 0.979563i \(0.564463\pi\)
\(822\) 12.2379 0.426847
\(823\) −4.69569 −0.163681 −0.0818407 0.996645i \(-0.526080\pi\)
−0.0818407 + 0.996645i \(0.526080\pi\)
\(824\) −7.33338 −0.255470
\(825\) 0 0
\(826\) 9.79462 0.340798
\(827\) −28.4859 −0.990550 −0.495275 0.868736i \(-0.664933\pi\)
−0.495275 + 0.868736i \(0.664933\pi\)
\(828\) 21.8801 0.760385
\(829\) −40.2771 −1.39888 −0.699441 0.714691i \(-0.746567\pi\)
−0.699441 + 0.714691i \(0.746567\pi\)
\(830\) 0 0
\(831\) −30.8579 −1.07045
\(832\) −0.369454 −0.0128085
\(833\) 18.5170 0.641575
\(834\) 43.0705 1.49141
\(835\) 0 0
\(836\) 14.9154 0.515861
\(837\) 23.6868 0.818736
\(838\) 5.69695 0.196798
\(839\) −9.99007 −0.344895 −0.172448 0.985019i \(-0.555168\pi\)
−0.172448 + 0.985019i \(0.555168\pi\)
\(840\) 0 0
\(841\) −27.5436 −0.949779
\(842\) 3.38092 0.116514
\(843\) −76.3906 −2.63103
\(844\) 16.0976 0.554104
\(845\) 0 0
\(846\) 42.4938 1.46097
\(847\) −12.0152 −0.412846
\(848\) −9.94355 −0.341463
\(849\) −26.7347 −0.917533
\(850\) 0 0
\(851\) 5.62036 0.192663
\(852\) 7.50978 0.257281
\(853\) −18.0500 −0.618019 −0.309010 0.951059i \(-0.599998\pi\)
−0.309010 + 0.951059i \(0.599998\pi\)
\(854\) 17.9457 0.614091
\(855\) 0 0
\(856\) −8.21182 −0.280674
\(857\) 24.3118 0.830474 0.415237 0.909713i \(-0.363699\pi\)
0.415237 + 0.909713i \(0.363699\pi\)
\(858\) −4.06440 −0.138756
\(859\) 35.2112 1.20139 0.600696 0.799477i \(-0.294890\pi\)
0.600696 + 0.799477i \(0.294890\pi\)
\(860\) 0 0
\(861\) 38.5538 1.31391
\(862\) −0.381731 −0.0130018
\(863\) 23.2986 0.793093 0.396546 0.918015i \(-0.370209\pi\)
0.396546 + 0.918015i \(0.370209\pi\)
\(864\) 2.34453 0.0797626
\(865\) 0 0
\(866\) −21.6393 −0.735332
\(867\) 23.2054 0.788098
\(868\) −18.5114 −0.628318
\(869\) 25.4076 0.861893
\(870\) 0 0
\(871\) −0.683761 −0.0231684
\(872\) −20.5503 −0.695920
\(873\) 23.5863 0.798275
\(874\) 20.0064 0.676726
\(875\) 0 0
\(876\) −21.2646 −0.718466
\(877\) 47.8713 1.61650 0.808250 0.588839i \(-0.200415\pi\)
0.808250 + 0.588839i \(0.200415\pi\)
\(878\) 26.4663 0.893194
\(879\) 47.5844 1.60498
\(880\) 0 0
\(881\) −40.1274 −1.35193 −0.675963 0.736936i \(-0.736272\pi\)
−0.675963 + 0.736936i \(0.736272\pi\)
\(882\) −14.1814 −0.477513
\(883\) 45.6747 1.53708 0.768539 0.639803i \(-0.220984\pi\)
0.768539 + 0.639803i \(0.220984\pi\)
\(884\) 1.87800 0.0631639
\(885\) 0 0
\(886\) −4.65489 −0.156384
\(887\) −17.5417 −0.588992 −0.294496 0.955653i \(-0.595152\pi\)
−0.294496 + 0.955653i \(0.595152\pi\)
\(888\) 2.62545 0.0881044
\(889\) 35.3364 1.18514
\(890\) 0 0
\(891\) −23.1446 −0.775374
\(892\) 25.3608 0.849142
\(893\) 38.8549 1.30023
\(894\) −15.3920 −0.514784
\(895\) 0 0
\(896\) −1.83227 −0.0612117
\(897\) −5.45166 −0.182026
\(898\) −26.5531 −0.886087
\(899\) 12.1924 0.406641
\(900\) 0 0
\(901\) 50.5448 1.68389
\(902\) −33.5820 −1.11816
\(903\) 10.9326 0.363815
\(904\) −0.652984 −0.0217179
\(905\) 0 0
\(906\) 34.5142 1.14666
\(907\) −2.04294 −0.0678347 −0.0339173 0.999425i \(-0.510798\pi\)
−0.0339173 + 0.999425i \(0.510798\pi\)
\(908\) −5.57339 −0.184960
\(909\) 19.3299 0.641131
\(910\) 0 0
\(911\) 54.3145 1.79952 0.899760 0.436385i \(-0.143742\pi\)
0.899760 + 0.436385i \(0.143742\pi\)
\(912\) 9.34563 0.309465
\(913\) −37.4266 −1.23864
\(914\) 4.27264 0.141326
\(915\) 0 0
\(916\) 9.02035 0.298041
\(917\) 16.1619 0.533713
\(918\) −11.9177 −0.393341
\(919\) 40.9771 1.35171 0.675855 0.737034i \(-0.263774\pi\)
0.675855 + 0.737034i \(0.263774\pi\)
\(920\) 0 0
\(921\) −16.1613 −0.532531
\(922\) 32.1873 1.06003
\(923\) −1.05678 −0.0347842
\(924\) −20.1569 −0.663115
\(925\) 0 0
\(926\) 18.4014 0.604708
\(927\) −28.5488 −0.937667
\(928\) 1.20681 0.0396156
\(929\) −22.2458 −0.729860 −0.364930 0.931035i \(-0.618907\pi\)
−0.364930 + 0.931035i \(0.618907\pi\)
\(930\) 0 0
\(931\) −12.9670 −0.424976
\(932\) 11.8100 0.386848
\(933\) −77.3226 −2.53143
\(934\) −3.06777 −0.100380
\(935\) 0 0
\(936\) −1.43828 −0.0470118
\(937\) −0.989771 −0.0323344 −0.0161672 0.999869i \(-0.505146\pi\)
−0.0161672 + 0.999869i \(0.505146\pi\)
\(938\) −3.39104 −0.110721
\(939\) 34.1576 1.11469
\(940\) 0 0
\(941\) 0.255111 0.00831639 0.00415820 0.999991i \(-0.498676\pi\)
0.00415820 + 0.999991i \(0.498676\pi\)
\(942\) −19.2611 −0.627560
\(943\) −45.0442 −1.46684
\(944\) −5.34563 −0.173985
\(945\) 0 0
\(946\) −9.52276 −0.309612
\(947\) 58.4983 1.90094 0.950469 0.310819i \(-0.100603\pi\)
0.950469 + 0.310819i \(0.100603\pi\)
\(948\) 15.9197 0.517048
\(949\) 2.99236 0.0971362
\(950\) 0 0
\(951\) −19.4238 −0.629860
\(952\) 9.31373 0.301860
\(953\) 34.4365 1.11551 0.557755 0.830006i \(-0.311663\pi\)
0.557755 + 0.830006i \(0.311663\pi\)
\(954\) −38.7102 −1.25329
\(955\) 0 0
\(956\) 0.907160 0.0293396
\(957\) 13.2763 0.429161
\(958\) 8.78798 0.283927
\(959\) −8.54068 −0.275793
\(960\) 0 0
\(961\) 71.0706 2.29260
\(962\) −0.369454 −0.0119117
\(963\) −31.9686 −1.03017
\(964\) −4.15616 −0.133861
\(965\) 0 0
\(966\) −27.0369 −0.869898
\(967\) 19.0310 0.611997 0.305998 0.952032i \(-0.401010\pi\)
0.305998 + 0.952032i \(0.401010\pi\)
\(968\) 6.55754 0.210767
\(969\) −47.5054 −1.52609
\(970\) 0 0
\(971\) 50.9856 1.63621 0.818103 0.575071i \(-0.195026\pi\)
0.818103 + 0.575071i \(0.195026\pi\)
\(972\) −21.5354 −0.690748
\(973\) −30.0583 −0.963626
\(974\) −22.4814 −0.720352
\(975\) 0 0
\(976\) −9.79428 −0.313507
\(977\) −26.8574 −0.859245 −0.429622 0.903009i \(-0.641353\pi\)
−0.429622 + 0.903009i \(0.641353\pi\)
\(978\) −49.3500 −1.57804
\(979\) 48.0918 1.53702
\(980\) 0 0
\(981\) −80.0022 −2.55427
\(982\) −0.0441752 −0.00140969
\(983\) −7.09389 −0.226260 −0.113130 0.993580i \(-0.536088\pi\)
−0.113130 + 0.993580i \(0.536088\pi\)
\(984\) −21.0416 −0.670782
\(985\) 0 0
\(986\) −6.13445 −0.195361
\(987\) −52.5091 −1.67138
\(988\) −1.31512 −0.0418395
\(989\) −12.7731 −0.406160
\(990\) 0 0
\(991\) −27.2573 −0.865856 −0.432928 0.901429i \(-0.642520\pi\)
−0.432928 + 0.901429i \(0.642520\pi\)
\(992\) 10.1030 0.320771
\(993\) 87.3081 2.77064
\(994\) −5.24097 −0.166234
\(995\) 0 0
\(996\) −23.4505 −0.743059
\(997\) −24.6984 −0.782207 −0.391103 0.920347i \(-0.627906\pi\)
−0.391103 + 0.920347i \(0.627906\pi\)
\(998\) 28.2445 0.894064
\(999\) 2.34453 0.0741777
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 1850.2.a.be.1.5 5
5.2 odd 4 370.2.b.d.149.6 yes 10
5.3 odd 4 370.2.b.d.149.5 10
5.4 even 2 1850.2.a.bd.1.1 5
15.2 even 4 3330.2.d.p.1999.3 10
15.8 even 4 3330.2.d.p.1999.8 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.d.149.5 10 5.3 odd 4
370.2.b.d.149.6 yes 10 5.2 odd 4
1850.2.a.bd.1.1 5 5.4 even 2
1850.2.a.be.1.5 5 1.1 even 1 trivial
3330.2.d.p.1999.3 10 15.2 even 4
3330.2.d.p.1999.8 10 15.8 even 4