Properties

Label 1850.2.a.bd.1.1
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: \( x^{5} - 9x^{3} + 13x - 4 \) Copy content Toggle raw display
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 1.1
Root \(-2.62545\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-expansion

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

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 \(-0.773776\pi\)
−0.757903 + 0.652367i \(0.773776\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.62545 1.07184
\(7\) 1.83227 0.692532 0.346266 0.938136i \(-0.387449\pi\)
0.346266 + 0.938136i \(0.387449\pi\)
\(8\) −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 \(-0.483685\pi\)
0.0512340 + 0.998687i \(0.483685\pi\)
\(14\) −1.83227 −0.489694
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.08317 1.23285 0.616425 0.787414i \(-0.288580\pi\)
0.616425 + 0.787414i \(0.288580\pi\)
\(18\) −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 \(-0.699284\pi\)
−0.585963 + 0.810338i \(0.699284\pi\)
\(24\) 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 \(-0.444561\pi\)
0.173287 + 0.984871i \(0.444561\pi\)
\(44\) 4.19017 0.631692
\(45\) 0 0
\(46\) 5.62036 0.828677
\(47\) −10.9154 −1.59218 −0.796090 0.605178i \(-0.793102\pi\)
−0.796090 + 0.605178i \(0.793102\pi\)
\(48\) −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 \(-0.260707\pi\)
0.682926 + 0.730488i \(0.260707\pi\)
\(54\) 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 \(-0.536063\pi\)
−0.113052 + 0.993589i \(0.536063\pi\)
\(68\) 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 \(-0.342816\pi\)
0.473983 + 0.880534i \(0.342816\pi\)
\(74\) 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 \(-0.336921\pi\)
0.490207 + 0.871606i \(0.336921\pi\)
\(84\) −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 \(-0.599520\pi\)
−0.307581 + 0.951522i \(0.599520\pi\)
\(98\) 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 \(-0.382337\pi\)
0.361289 + 0.932454i \(0.382337\pi\)
\(104\) −0.369454 −0.0362279
\(105\) 0 0
\(106\) −9.94355 −0.965803
\(107\) 8.21182 0.793867 0.396933 0.917847i \(-0.370074\pi\)
0.396933 + 0.917847i \(0.370074\pi\)
\(108\) −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 \(-0.490222\pi\)
0.0307138 + 0.999528i \(0.490222\pi\)
\(114\) 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 \(-0.173152\pi\)
0.855660 + 0.517539i \(0.173152\pi\)
\(128\) −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 \(-0.563808\pi\)
−0.199119 + 0.979975i \(0.563808\pi\)
\(138\) −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 \(-0.405430\pi\)
0.292750 + 0.956189i \(0.405430\pi\)
\(158\) −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 \(-0.236647\pi\)
0.736138 + 0.676831i \(0.236647\pi\)
\(164\) −8.01447 −0.625825
\(165\) 0 0
\(166\) −8.93200 −0.693257
\(167\) 5.92772 0.458700 0.229350 0.973344i \(-0.426340\pi\)
0.229350 + 0.973344i \(0.426340\pi\)
\(168\) 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 \(-0.570545\pi\)
−0.219815 + 0.975542i \(0.570545\pi\)
\(174\) 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 \(-0.109183\pi\)
0.941747 + 0.336321i \(0.109183\pi\)
\(194\) 6.05864 0.434985
\(195\) 0 0
\(196\) −3.64280 −0.260200
\(197\) −14.6067 −1.04069 −0.520343 0.853957i \(-0.674196\pi\)
−0.520343 + 0.853957i \(0.674196\pi\)
\(198\) −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 \(-0.822881\pi\)
−0.849142 + 0.528164i \(0.822881\pi\)
\(224\) −1.83227 −0.122423
\(225\) 0 0
\(226\) −0.652984 −0.0434359
\(227\) 5.57339 0.369919 0.184960 0.982746i \(-0.440785\pi\)
0.184960 + 0.982746i \(0.440785\pi\)
\(228\) −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 \(-0.626436\pi\)
−0.386848 + 0.922144i \(0.626436\pi\)
\(234\) −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 \(-0.360222\pi\)
0.425147 + 0.905124i \(0.360222\pi\)
\(258\) 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 \(-0.816939\pi\)
−0.839137 + 0.543921i \(0.816939\pi\)
\(264\) 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 \(-0.385129\pi\)
0.353096 + 0.935587i \(0.385129\pi\)
\(278\) −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 \(-0.402127\pi\)
0.302655 + 0.953100i \(0.402127\pi\)
\(284\) 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 \(-0.677589\pi\)
−0.529415 + 0.848363i \(0.677589\pi\)
\(294\) −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 \(-0.443794\pi\)
0.175660 + 0.984451i \(0.443794\pi\)
\(308\) 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 \(-0.619851\pi\)
−0.367689 + 0.929949i \(0.619851\pi\)
\(314\) −7.33629 −0.414011
\(315\) 0 0
\(316\) 6.06361 0.341105
\(317\) 7.39826 0.415528 0.207764 0.978179i \(-0.433381\pi\)
0.207764 + 0.978179i \(0.433381\pi\)
\(318\) 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 \(-0.711383\pi\)
−0.616334 + 0.787485i \(0.711383\pi\)
\(338\) 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 \(-0.275630\pi\)
0.647941 + 0.761691i \(0.275630\pi\)
\(348\) −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 \(-0.466674\pi\)
0.104506 + 0.994524i \(0.466674\pi\)
\(354\) −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 \(-0.834577\pi\)
−0.867972 + 0.496614i \(0.834577\pi\)
\(368\) −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 \(-0.235375\pi\)
0.738839 + 0.673882i \(0.235375\pi\)
\(374\) −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 \(-0.558389\pi\)
−0.182409 + 0.983223i \(0.558389\pi\)
\(384\) 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 \(-0.674323\pi\)
−0.520684 + 0.853750i \(0.674323\pi\)
\(398\) −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 \(-0.325948\pi\)
0.519958 + 0.854192i \(0.325948\pi\)
\(434\) −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 \(-0.464729\pi\)
0.110580 + 0.993867i \(0.464729\pi\)
\(444\) 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 \(-0.531863\pi\)
−0.0999329 + 0.994994i \(0.531863\pi\)
\(458\) −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 \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(464\) 1.20681 0.0560250
\(465\) 0 0
\(466\) 11.8100 0.547086
\(467\) 3.06777 0.141959 0.0709797 0.997478i \(-0.477387\pi\)
0.0709797 + 0.997478i \(0.477387\pi\)
\(468\) 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 \(-0.329880\pi\)
0.509366 + 0.860550i \(0.329880\pi\)
\(488\) 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 \(-0.354702\pi\)
0.440780 + 0.897615i \(0.354702\pi\)
\(504\) −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 \(-0.743539\pi\)
−0.692610 + 0.721313i \(0.743539\pi\)
\(524\) −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 \(-0.677737\pi\)
−0.529809 + 0.848117i \(0.677737\pi\)
\(548\) −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 \(-0.344423\pi\)
0.469531 + 0.882916i \(0.344423\pi\)
\(558\) −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 \(-0.543523\pi\)
−0.136306 + 0.990667i \(0.543523\pi\)
\(564\) 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 \(-0.479755\pi\)
0.0635587 + 0.997978i \(0.479755\pi\)
\(578\) −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 \(-0.241804\pi\)
0.725076 + 0.688668i \(0.241804\pi\)
\(588\) 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 \(-0.420486\pi\)
0.247212 + 0.968961i \(0.420486\pi\)
\(594\) 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 \(-0.301376\pi\)
0.584282 + 0.811551i \(0.301376\pi\)
\(608\) −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 \(-0.450526\pi\)
0.154801 + 0.987946i \(0.450526\pi\)
\(614\) −6.15561 −0.248420
\(615\) 0 0
\(616\) −7.67751 −0.309336
\(617\) −36.5388 −1.47100 −0.735498 0.677527i \(-0.763052\pi\)
−0.735498 + 0.677527i \(0.763052\pi\)
\(618\) 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 \(-0.508578\pi\)
−0.0269468 + 0.999637i \(0.508578\pi\)
\(644\) −10.2980 −0.405798
\(645\) 0 0
\(646\) −18.0942 −0.711906
\(647\) −11.2690 −0.443028 −0.221514 0.975157i \(-0.571100\pi\)
−0.221514 + 0.975157i \(0.571100\pi\)
\(648\) 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 \(-0.303806\pi\)
0.578070 + 0.815987i \(0.303806\pi\)
\(654\) −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 \(-0.542522\pi\)
−0.133191 + 0.991090i \(0.542522\pi\)
\(674\) 22.6288 0.871627
\(675\) 0 0
\(676\) −12.8635 −0.494750
\(677\) 15.6913 0.603065 0.301532 0.953456i \(-0.402502\pi\)
0.301532 + 0.953456i \(0.402502\pi\)
\(678\) 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 \(-0.386791\pi\)
0.348206 + 0.937418i \(0.386791\pi\)
\(684\) 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 \(-0.368320\pi\)
0.401985 + 0.915646i \(0.368320\pi\)
\(728\) −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 \(-0.537312\pi\)
−0.116951 + 0.993138i \(0.537312\pi\)
\(734\) 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 \(-0.565085\pi\)
−0.203048 + 0.979169i \(0.565085\pi\)
\(744\) 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 \(-0.704622\pi\)
−0.599470 + 0.800397i \(0.704622\pi\)
\(758\) −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 \(-0.688881\pi\)
−0.559171 + 0.829052i \(0.688881\pi\)
\(774\) −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 \(-0.394339\pi\)
0.325880 + 0.945411i \(0.394339\pi\)
\(788\) −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 \(-0.244551\pi\)
0.719106 + 0.694900i \(0.244551\pi\)
\(798\) 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 \(-0.473920\pi\)
0.0818407 + 0.996645i \(0.473920\pi\)
\(824\) −7.33338 −0.255470
\(825\) 0 0
\(826\) 9.79462 0.340798
\(827\) 28.4859 0.990550 0.495275 0.868736i \(-0.335067\pi\)
0.495275 + 0.868736i \(0.335067\pi\)
\(828\) −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 \(-0.400002\pi\)
0.309010 + 0.951059i \(0.400002\pi\)
\(854\) 17.9457 0.614091
\(855\) 0 0
\(856\) −8.21182 −0.280674
\(857\) −24.3118 −0.830474 −0.415237 0.909713i \(-0.636301\pi\)
−0.415237 + 0.909713i \(0.636301\pi\)
\(858\) 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 \(-0.629791\pi\)
−0.396546 + 0.918015i \(0.629791\pi\)
\(864\) 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 \(-0.799585\pi\)
−0.808250 + 0.588839i \(0.799585\pi\)
\(878\) −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 \(-0.779016\pi\)
−0.768539 + 0.639803i \(0.779016\pi\)
\(884\) 1.87800 0.0631639
\(885\) 0 0
\(886\) −4.65489 −0.156384
\(887\) 17.5417 0.588992 0.294496 0.955653i \(-0.404848\pi\)
0.294496 + 0.955653i \(0.404848\pi\)
\(888\) −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 \(-0.489202\pi\)
0.0339173 + 0.999425i \(0.489202\pi\)
\(908\) 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 \(-0.494854\pi\)
0.0161672 + 0.999869i \(0.494854\pi\)
\(938\) 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 \(-0.899397\pi\)
−0.950469 + 0.310819i \(0.899397\pi\)
\(948\) −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 \(-0.688337\pi\)
−0.557755 + 0.830006i \(0.688337\pi\)
\(954\) −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 \(-0.598990\pi\)
−0.305998 + 0.952032i \(0.598990\pi\)
\(968\) −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 \(-0.358647\pi\)
0.429622 + 0.903009i \(0.358647\pi\)
\(978\) 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 \(-0.463912\pi\)
0.113130 + 0.993580i \(0.463912\pi\)
\(984\) −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 \(-0.372094\pi\)
0.391103 + 0.920347i \(0.372094\pi\)
\(998\) −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.bd.1.1 5
5.2 odd 4 370.2.b.d.149.5 10
5.3 odd 4 370.2.b.d.149.6 yes 10
5.4 even 2 1850.2.a.be.1.5 5
15.2 even 4 3330.2.d.p.1999.8 10
15.8 even 4 3330.2.d.p.1999.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.d.149.5 10 5.2 odd 4
370.2.b.d.149.6 yes 10 5.3 odd 4
1850.2.a.bd.1.1 5 1.1 even 1 trivial
1850.2.a.be.1.5 5 5.4 even 2
3330.2.d.p.1999.3 10 15.8 even 4
3330.2.d.p.1999.8 10 15.2 even 4