Properties

Label 950.3.c.a.151.2
Level $950$
Weight $3$
Character 950.151
Analytic conductor $25.886$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 950.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.8856251142\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 151.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 950.151
Dual form 950.3.c.a.151.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421i q^{2} +2.82843i q^{3} -2.00000 q^{4} -4.00000 q^{6} -5.00000 q^{7} -2.82843i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.41421i q^{2} +2.82843i q^{3} -2.00000 q^{4} -4.00000 q^{6} -5.00000 q^{7} -2.82843i q^{8} +1.00000 q^{9} +5.00000 q^{11} -5.65685i q^{12} -16.9706i q^{13} -7.07107i q^{14} +4.00000 q^{16} +25.0000 q^{17} +1.41421i q^{18} +19.0000 q^{19} -14.1421i q^{21} +7.07107i q^{22} +10.0000 q^{23} +8.00000 q^{24} +24.0000 q^{26} +28.2843i q^{27} +10.0000 q^{28} -42.4264i q^{29} +42.4264i q^{31} +5.65685i q^{32} +14.1421i q^{33} +35.3553i q^{34} -2.00000 q^{36} -25.4558i q^{37} +26.8701i q^{38} +48.0000 q^{39} +42.4264i q^{41} +20.0000 q^{42} -5.00000 q^{43} -10.0000 q^{44} +14.1421i q^{46} -5.00000 q^{47} +11.3137i q^{48} -24.0000 q^{49} +70.7107i q^{51} +33.9411i q^{52} -25.4558i q^{53} -40.0000 q^{54} +14.1421i q^{56} +53.7401i q^{57} +60.0000 q^{58} -84.8528i q^{59} +95.0000 q^{61} -60.0000 q^{62} -5.00000 q^{63} -8.00000 q^{64} -20.0000 q^{66} +110.309i q^{67} -50.0000 q^{68} +28.2843i q^{69} -2.82843i q^{72} +25.0000 q^{73} +36.0000 q^{74} -38.0000 q^{76} -25.0000 q^{77} +67.8823i q^{78} -42.4264i q^{79} -71.0000 q^{81} -60.0000 q^{82} +130.000 q^{83} +28.2843i q^{84} -7.07107i q^{86} +120.000 q^{87} -14.1421i q^{88} +127.279i q^{89} +84.8528i q^{91} -20.0000 q^{92} -120.000 q^{93} -7.07107i q^{94} -16.0000 q^{96} +16.9706i q^{97} -33.9411i q^{98} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} - 8q^{6} - 10q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 4q^{4} - 8q^{6} - 10q^{7} + 2q^{9} + 10q^{11} + 8q^{16} + 50q^{17} + 38q^{19} + 20q^{23} + 16q^{24} + 48q^{26} + 20q^{28} - 4q^{36} + 96q^{39} + 40q^{42} - 10q^{43} - 20q^{44} - 10q^{47} - 48q^{49} - 80q^{54} + 120q^{58} + 190q^{61} - 120q^{62} - 10q^{63} - 16q^{64} - 40q^{66} - 100q^{68} + 50q^{73} + 72q^{74} - 76q^{76} - 50q^{77} - 142q^{81} - 120q^{82} + 260q^{83} + 240q^{87} - 40q^{92} - 240q^{93} - 32q^{96} + 10q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 0.707107i
\(3\) 2.82843i 0.942809i 0.881917 + 0.471405i \(0.156253\pi\)
−0.881917 + 0.471405i \(0.843747\pi\)
\(4\) −2.00000 −0.500000
\(5\) 0 0
\(6\) −4.00000 −0.666667
\(7\) −5.00000 −0.714286 −0.357143 0.934050i \(-0.616249\pi\)
−0.357143 + 0.934050i \(0.616249\pi\)
\(8\) − 2.82843i − 0.353553i
\(9\) 1.00000 0.111111
\(10\) 0 0
\(11\) 5.00000 0.454545 0.227273 0.973831i \(-0.427019\pi\)
0.227273 + 0.973831i \(0.427019\pi\)
\(12\) − 5.65685i − 0.471405i
\(13\) − 16.9706i − 1.30543i −0.757604 0.652714i \(-0.773630\pi\)
0.757604 0.652714i \(-0.226370\pi\)
\(14\) − 7.07107i − 0.505076i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 25.0000 1.47059 0.735294 0.677748i \(-0.237044\pi\)
0.735294 + 0.677748i \(0.237044\pi\)
\(18\) 1.41421i 0.0785674i
\(19\) 19.0000 1.00000
\(20\) 0 0
\(21\) − 14.1421i − 0.673435i
\(22\) 7.07107i 0.321412i
\(23\) 10.0000 0.434783 0.217391 0.976085i \(-0.430245\pi\)
0.217391 + 0.976085i \(0.430245\pi\)
\(24\) 8.00000 0.333333
\(25\) 0 0
\(26\) 24.0000 0.923077
\(27\) 28.2843i 1.04757i
\(28\) 10.0000 0.357143
\(29\) − 42.4264i − 1.46298i −0.681852 0.731490i \(-0.738825\pi\)
0.681852 0.731490i \(-0.261175\pi\)
\(30\) 0 0
\(31\) 42.4264i 1.36859i 0.729204 + 0.684297i \(0.239891\pi\)
−0.729204 + 0.684297i \(0.760109\pi\)
\(32\) 5.65685i 0.176777i
\(33\) 14.1421i 0.428550i
\(34\) 35.3553i 1.03986i
\(35\) 0 0
\(36\) −2.00000 −0.0555556
\(37\) − 25.4558i − 0.687996i −0.938970 0.343998i \(-0.888219\pi\)
0.938970 0.343998i \(-0.111781\pi\)
\(38\) 26.8701i 0.707107i
\(39\) 48.0000 1.23077
\(40\) 0 0
\(41\) 42.4264i 1.03479i 0.855747 + 0.517395i \(0.173098\pi\)
−0.855747 + 0.517395i \(0.826902\pi\)
\(42\) 20.0000 0.476190
\(43\) −5.00000 −0.116279 −0.0581395 0.998308i \(-0.518517\pi\)
−0.0581395 + 0.998308i \(0.518517\pi\)
\(44\) −10.0000 −0.227273
\(45\) 0 0
\(46\) 14.1421i 0.307438i
\(47\) −5.00000 −0.106383 −0.0531915 0.998584i \(-0.516939\pi\)
−0.0531915 + 0.998584i \(0.516939\pi\)
\(48\) 11.3137i 0.235702i
\(49\) −24.0000 −0.489796
\(50\) 0 0
\(51\) 70.7107i 1.38648i
\(52\) 33.9411i 0.652714i
\(53\) − 25.4558i − 0.480299i −0.970736 0.240149i \(-0.922804\pi\)
0.970736 0.240149i \(-0.0771965\pi\)
\(54\) −40.0000 −0.740741
\(55\) 0 0
\(56\) 14.1421i 0.252538i
\(57\) 53.7401i 0.942809i
\(58\) 60.0000 1.03448
\(59\) − 84.8528i − 1.43818i −0.694915 0.719092i \(-0.744558\pi\)
0.694915 0.719092i \(-0.255442\pi\)
\(60\) 0 0
\(61\) 95.0000 1.55738 0.778689 0.627411i \(-0.215885\pi\)
0.778689 + 0.627411i \(0.215885\pi\)
\(62\) −60.0000 −0.967742
\(63\) −5.00000 −0.0793651
\(64\) −8.00000 −0.125000
\(65\) 0 0
\(66\) −20.0000 −0.303030
\(67\) 110.309i 1.64640i 0.567753 + 0.823199i \(0.307813\pi\)
−0.567753 + 0.823199i \(0.692187\pi\)
\(68\) −50.0000 −0.735294
\(69\) 28.2843i 0.409917i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) − 2.82843i − 0.0392837i
\(73\) 25.0000 0.342466 0.171233 0.985231i \(-0.445225\pi\)
0.171233 + 0.985231i \(0.445225\pi\)
\(74\) 36.0000 0.486486
\(75\) 0 0
\(76\) −38.0000 −0.500000
\(77\) −25.0000 −0.324675
\(78\) 67.8823i 0.870285i
\(79\) − 42.4264i − 0.537043i −0.963274 0.268522i \(-0.913465\pi\)
0.963274 0.268522i \(-0.0865351\pi\)
\(80\) 0 0
\(81\) −71.0000 −0.876543
\(82\) −60.0000 −0.731707
\(83\) 130.000 1.56627 0.783133 0.621855i \(-0.213621\pi\)
0.783133 + 0.621855i \(0.213621\pi\)
\(84\) 28.2843i 0.336718i
\(85\) 0 0
\(86\) − 7.07107i − 0.0822217i
\(87\) 120.000 1.37931
\(88\) − 14.1421i − 0.160706i
\(89\) 127.279i 1.43010i 0.699071 + 0.715052i \(0.253597\pi\)
−0.699071 + 0.715052i \(0.746403\pi\)
\(90\) 0 0
\(91\) 84.8528i 0.932449i
\(92\) −20.0000 −0.217391
\(93\) −120.000 −1.29032
\(94\) − 7.07107i − 0.0752241i
\(95\) 0 0
\(96\) −16.0000 −0.166667
\(97\) 16.9706i 0.174954i 0.996167 + 0.0874771i \(0.0278805\pi\)
−0.996167 + 0.0874771i \(0.972120\pi\)
\(98\) − 33.9411i − 0.346338i
\(99\) 5.00000 0.0505051
\(100\) 0 0
\(101\) 50.0000 0.495050 0.247525 0.968882i \(-0.420383\pi\)
0.247525 + 0.968882i \(0.420383\pi\)
\(102\) −100.000 −0.980392
\(103\) − 16.9706i − 0.164763i −0.996601 0.0823814i \(-0.973747\pi\)
0.996601 0.0823814i \(-0.0262526\pi\)
\(104\) −48.0000 −0.461538
\(105\) 0 0
\(106\) 36.0000 0.339623
\(107\) − 101.823i − 0.951620i −0.879548 0.475810i \(-0.842155\pi\)
0.879548 0.475810i \(-0.157845\pi\)
\(108\) − 56.5685i − 0.523783i
\(109\) − 127.279i − 1.16770i −0.811862 0.583850i \(-0.801546\pi\)
0.811862 0.583850i \(-0.198454\pi\)
\(110\) 0 0
\(111\) 72.0000 0.648649
\(112\) −20.0000 −0.178571
\(113\) 110.309i 0.976183i 0.872793 + 0.488091i \(0.162307\pi\)
−0.872793 + 0.488091i \(0.837693\pi\)
\(114\) −76.0000 −0.666667
\(115\) 0 0
\(116\) 84.8528i 0.731490i
\(117\) − 16.9706i − 0.145048i
\(118\) 120.000 1.01695
\(119\) −125.000 −1.05042
\(120\) 0 0
\(121\) −96.0000 −0.793388
\(122\) 134.350i 1.10123i
\(123\) −120.000 −0.975610
\(124\) − 84.8528i − 0.684297i
\(125\) 0 0
\(126\) − 7.07107i − 0.0561196i
\(127\) 229.103i 1.80396i 0.431780 + 0.901979i \(0.357886\pi\)
−0.431780 + 0.901979i \(0.642114\pi\)
\(128\) − 11.3137i − 0.0883883i
\(129\) − 14.1421i − 0.109629i
\(130\) 0 0
\(131\) −163.000 −1.24427 −0.622137 0.782908i \(-0.713735\pi\)
−0.622137 + 0.782908i \(0.713735\pi\)
\(132\) − 28.2843i − 0.214275i
\(133\) −95.0000 −0.714286
\(134\) −156.000 −1.16418
\(135\) 0 0
\(136\) − 70.7107i − 0.519931i
\(137\) −95.0000 −0.693431 −0.346715 0.937970i \(-0.612703\pi\)
−0.346715 + 0.937970i \(0.612703\pi\)
\(138\) −40.0000 −0.289855
\(139\) 125.000 0.899281 0.449640 0.893210i \(-0.351552\pi\)
0.449640 + 0.893210i \(0.351552\pi\)
\(140\) 0 0
\(141\) − 14.1421i − 0.100299i
\(142\) 0 0
\(143\) − 84.8528i − 0.593376i
\(144\) 4.00000 0.0277778
\(145\) 0 0
\(146\) 35.3553i 0.242160i
\(147\) − 67.8823i − 0.461784i
\(148\) 50.9117i 0.343998i
\(149\) 215.000 1.44295 0.721477 0.692439i \(-0.243464\pi\)
0.721477 + 0.692439i \(0.243464\pi\)
\(150\) 0 0
\(151\) 84.8528i 0.561939i 0.959717 + 0.280970i \(0.0906560\pi\)
−0.959717 + 0.280970i \(0.909344\pi\)
\(152\) − 53.7401i − 0.353553i
\(153\) 25.0000 0.163399
\(154\) − 35.3553i − 0.229580i
\(155\) 0 0
\(156\) −96.0000 −0.615385
\(157\) 190.000 1.21019 0.605096 0.796153i \(-0.293135\pi\)
0.605096 + 0.796153i \(0.293135\pi\)
\(158\) 60.0000 0.379747
\(159\) 72.0000 0.452830
\(160\) 0 0
\(161\) −50.0000 −0.310559
\(162\) − 100.409i − 0.619810i
\(163\) −110.000 −0.674847 −0.337423 0.941353i \(-0.609555\pi\)
−0.337423 + 0.941353i \(0.609555\pi\)
\(164\) − 84.8528i − 0.517395i
\(165\) 0 0
\(166\) 183.848i 1.10752i
\(167\) 59.3970i 0.355670i 0.984060 + 0.177835i \(0.0569094\pi\)
−0.984060 + 0.177835i \(0.943091\pi\)
\(168\) −40.0000 −0.238095
\(169\) −119.000 −0.704142
\(170\) 0 0
\(171\) 19.0000 0.111111
\(172\) 10.0000 0.0581395
\(173\) − 186.676i − 1.07905i −0.841969 0.539527i \(-0.818603\pi\)
0.841969 0.539527i \(-0.181397\pi\)
\(174\) 169.706i 0.975320i
\(175\) 0 0
\(176\) 20.0000 0.113636
\(177\) 240.000 1.35593
\(178\) −180.000 −1.01124
\(179\) 127.279i 0.711057i 0.934665 + 0.355529i \(0.115699\pi\)
−0.934665 + 0.355529i \(0.884301\pi\)
\(180\) 0 0
\(181\) 254.558i 1.40640i 0.710992 + 0.703200i \(0.248246\pi\)
−0.710992 + 0.703200i \(0.751754\pi\)
\(182\) −120.000 −0.659341
\(183\) 268.701i 1.46831i
\(184\) − 28.2843i − 0.153719i
\(185\) 0 0
\(186\) − 169.706i − 0.912396i
\(187\) 125.000 0.668449
\(188\) 10.0000 0.0531915
\(189\) − 141.421i − 0.748261i
\(190\) 0 0
\(191\) 293.000 1.53403 0.767016 0.641628i \(-0.221741\pi\)
0.767016 + 0.641628i \(0.221741\pi\)
\(192\) − 22.6274i − 0.117851i
\(193\) 59.3970i 0.307756i 0.988090 + 0.153878i \(0.0491763\pi\)
−0.988090 + 0.153878i \(0.950824\pi\)
\(194\) −24.0000 −0.123711
\(195\) 0 0
\(196\) 48.0000 0.244898
\(197\) 70.0000 0.355330 0.177665 0.984091i \(-0.443146\pi\)
0.177665 + 0.984091i \(0.443146\pi\)
\(198\) 7.07107i 0.0357125i
\(199\) 173.000 0.869347 0.434673 0.900588i \(-0.356864\pi\)
0.434673 + 0.900588i \(0.356864\pi\)
\(200\) 0 0
\(201\) −312.000 −1.55224
\(202\) 70.7107i 0.350053i
\(203\) 212.132i 1.04499i
\(204\) − 141.421i − 0.693242i
\(205\) 0 0
\(206\) 24.0000 0.116505
\(207\) 10.0000 0.0483092
\(208\) − 67.8823i − 0.326357i
\(209\) 95.0000 0.454545
\(210\) 0 0
\(211\) − 84.8528i − 0.402146i −0.979576 0.201073i \(-0.935557\pi\)
0.979576 0.201073i \(-0.0644429\pi\)
\(212\) 50.9117i 0.240149i
\(213\) 0 0
\(214\) 144.000 0.672897
\(215\) 0 0
\(216\) 80.0000 0.370370
\(217\) − 212.132i − 0.977567i
\(218\) 180.000 0.825688
\(219\) 70.7107i 0.322880i
\(220\) 0 0
\(221\) − 424.264i − 1.91975i
\(222\) 101.823i 0.458664i
\(223\) − 364.867i − 1.63618i −0.575094 0.818088i \(-0.695034\pi\)
0.575094 0.818088i \(-0.304966\pi\)
\(224\) − 28.2843i − 0.126269i
\(225\) 0 0
\(226\) −156.000 −0.690265
\(227\) 67.8823i 0.299041i 0.988759 + 0.149520i \(0.0477730\pi\)
−0.988759 + 0.149520i \(0.952227\pi\)
\(228\) − 107.480i − 0.471405i
\(229\) −145.000 −0.633188 −0.316594 0.948561i \(-0.602539\pi\)
−0.316594 + 0.948561i \(0.602539\pi\)
\(230\) 0 0
\(231\) − 70.7107i − 0.306107i
\(232\) −120.000 −0.517241
\(233\) −335.000 −1.43777 −0.718884 0.695130i \(-0.755347\pi\)
−0.718884 + 0.695130i \(0.755347\pi\)
\(234\) 24.0000 0.102564
\(235\) 0 0
\(236\) 169.706i 0.719092i
\(237\) 120.000 0.506329
\(238\) − 176.777i − 0.742759i
\(239\) 197.000 0.824268 0.412134 0.911123i \(-0.364784\pi\)
0.412134 + 0.911123i \(0.364784\pi\)
\(240\) 0 0
\(241\) − 296.985i − 1.23230i −0.787628 0.616151i \(-0.788691\pi\)
0.787628 0.616151i \(-0.211309\pi\)
\(242\) − 135.765i − 0.561010i
\(243\) 53.7401i 0.221153i
\(244\) −190.000 −0.778689
\(245\) 0 0
\(246\) − 169.706i − 0.689860i
\(247\) − 322.441i − 1.30543i
\(248\) 120.000 0.483871
\(249\) 367.696i 1.47669i
\(250\) 0 0
\(251\) 173.000 0.689243 0.344622 0.938742i \(-0.388007\pi\)
0.344622 + 0.938742i \(0.388007\pi\)
\(252\) 10.0000 0.0396825
\(253\) 50.0000 0.197628
\(254\) −324.000 −1.27559
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) − 67.8823i − 0.264133i −0.991241 0.132067i \(-0.957839\pi\)
0.991241 0.132067i \(-0.0421613\pi\)
\(258\) 20.0000 0.0775194
\(259\) 127.279i 0.491426i
\(260\) 0 0
\(261\) − 42.4264i − 0.162553i
\(262\) − 230.517i − 0.879835i
\(263\) 355.000 1.34981 0.674905 0.737905i \(-0.264185\pi\)
0.674905 + 0.737905i \(0.264185\pi\)
\(264\) 40.0000 0.151515
\(265\) 0 0
\(266\) − 134.350i − 0.505076i
\(267\) −360.000 −1.34831
\(268\) − 220.617i − 0.823199i
\(269\) 381.838i 1.41947i 0.704468 + 0.709735i \(0.251186\pi\)
−0.704468 + 0.709735i \(0.748814\pi\)
\(270\) 0 0
\(271\) 110.000 0.405904 0.202952 0.979189i \(-0.434946\pi\)
0.202952 + 0.979189i \(0.434946\pi\)
\(272\) 100.000 0.367647
\(273\) −240.000 −0.879121
\(274\) − 134.350i − 0.490330i
\(275\) 0 0
\(276\) − 56.5685i − 0.204958i
\(277\) 265.000 0.956679 0.478339 0.878175i \(-0.341239\pi\)
0.478339 + 0.878175i \(0.341239\pi\)
\(278\) 176.777i 0.635887i
\(279\) 42.4264i 0.152066i
\(280\) 0 0
\(281\) − 424.264i − 1.50984i −0.655819 0.754918i \(-0.727677\pi\)
0.655819 0.754918i \(-0.272323\pi\)
\(282\) 20.0000 0.0709220
\(283\) −125.000 −0.441696 −0.220848 0.975308i \(-0.570882\pi\)
−0.220848 + 0.975308i \(0.570882\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 120.000 0.419580
\(287\) − 212.132i − 0.739136i
\(288\) 5.65685i 0.0196419i
\(289\) 336.000 1.16263
\(290\) 0 0
\(291\) −48.0000 −0.164948
\(292\) −50.0000 −0.171233
\(293\) 186.676i 0.637120i 0.947903 + 0.318560i \(0.103199\pi\)
−0.947903 + 0.318560i \(0.896801\pi\)
\(294\) 96.0000 0.326531
\(295\) 0 0
\(296\) −72.0000 −0.243243
\(297\) 141.421i 0.476166i
\(298\) 304.056i 1.02032i
\(299\) − 169.706i − 0.567577i
\(300\) 0 0
\(301\) 25.0000 0.0830565
\(302\) −120.000 −0.397351
\(303\) 141.421i 0.466737i
\(304\) 76.0000 0.250000
\(305\) 0 0
\(306\) 35.3553i 0.115540i
\(307\) 280.014i 0.912099i 0.889955 + 0.456049i \(0.150736\pi\)
−0.889955 + 0.456049i \(0.849264\pi\)
\(308\) 50.0000 0.162338
\(309\) 48.0000 0.155340
\(310\) 0 0
\(311\) −235.000 −0.755627 −0.377814 0.925882i \(-0.623324\pi\)
−0.377814 + 0.925882i \(0.623324\pi\)
\(312\) − 135.765i − 0.435143i
\(313\) 310.000 0.990415 0.495208 0.868775i \(-0.335092\pi\)
0.495208 + 0.868775i \(0.335092\pi\)
\(314\) 268.701i 0.855734i
\(315\) 0 0
\(316\) 84.8528i 0.268522i
\(317\) − 186.676i − 0.588884i −0.955669 0.294442i \(-0.904866\pi\)
0.955669 0.294442i \(-0.0951338\pi\)
\(318\) 101.823i 0.320199i
\(319\) − 212.132i − 0.664991i
\(320\) 0 0
\(321\) 288.000 0.897196
\(322\) − 70.7107i − 0.219598i
\(323\) 475.000 1.47059
\(324\) 142.000 0.438272
\(325\) 0 0
\(326\) − 155.563i − 0.477189i
\(327\) 360.000 1.10092
\(328\) 120.000 0.365854
\(329\) 25.0000 0.0759878
\(330\) 0 0
\(331\) − 296.985i − 0.897235i −0.893724 0.448618i \(-0.851917\pi\)
0.893724 0.448618i \(-0.148083\pi\)
\(332\) −260.000 −0.783133
\(333\) − 25.4558i − 0.0764440i
\(334\) −84.0000 −0.251497
\(335\) 0 0
\(336\) − 56.5685i − 0.168359i
\(337\) − 526.087i − 1.56109i −0.625099 0.780545i \(-0.714942\pi\)
0.625099 0.780545i \(-0.285058\pi\)
\(338\) − 168.291i − 0.497904i
\(339\) −312.000 −0.920354
\(340\) 0 0
\(341\) 212.132i 0.622088i
\(342\) 26.8701i 0.0785674i
\(343\) 365.000 1.06414
\(344\) 14.1421i 0.0411109i
\(345\) 0 0
\(346\) 264.000 0.763006
\(347\) −125.000 −0.360231 −0.180115 0.983646i \(-0.557647\pi\)
−0.180115 + 0.983646i \(0.557647\pi\)
\(348\) −240.000 −0.689655
\(349\) 23.0000 0.0659026 0.0329513 0.999457i \(-0.489509\pi\)
0.0329513 + 0.999457i \(0.489509\pi\)
\(350\) 0 0
\(351\) 480.000 1.36752
\(352\) 28.2843i 0.0803530i
\(353\) −410.000 −1.16147 −0.580737 0.814092i \(-0.697235\pi\)
−0.580737 + 0.814092i \(0.697235\pi\)
\(354\) 339.411i 0.958789i
\(355\) 0 0
\(356\) − 254.558i − 0.715052i
\(357\) − 353.553i − 0.990346i
\(358\) −180.000 −0.502793
\(359\) −475.000 −1.32312 −0.661560 0.749892i \(-0.730105\pi\)
−0.661560 + 0.749892i \(0.730105\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) −360.000 −0.994475
\(363\) − 271.529i − 0.748014i
\(364\) − 169.706i − 0.466224i
\(365\) 0 0
\(366\) −380.000 −1.03825
\(367\) −230.000 −0.626703 −0.313351 0.949637i \(-0.601452\pi\)
−0.313351 + 0.949637i \(0.601452\pi\)
\(368\) 40.0000 0.108696
\(369\) 42.4264i 0.114977i
\(370\) 0 0
\(371\) 127.279i 0.343071i
\(372\) 240.000 0.645161
\(373\) − 67.8823i − 0.181990i −0.995851 0.0909950i \(-0.970995\pi\)
0.995851 0.0909950i \(-0.0290047\pi\)
\(374\) 176.777i 0.472665i
\(375\) 0 0
\(376\) 14.1421i 0.0376121i
\(377\) −720.000 −1.90981
\(378\) 200.000 0.529101
\(379\) 254.558i 0.671658i 0.941923 + 0.335829i \(0.109016\pi\)
−0.941923 + 0.335829i \(0.890984\pi\)
\(380\) 0 0
\(381\) −648.000 −1.70079
\(382\) 414.365i 1.08472i
\(383\) − 144.250i − 0.376631i −0.982109 0.188316i \(-0.939697\pi\)
0.982109 0.188316i \(-0.0603028\pi\)
\(384\) 32.0000 0.0833333
\(385\) 0 0
\(386\) −84.0000 −0.217617
\(387\) −5.00000 −0.0129199
\(388\) − 33.9411i − 0.0874771i
\(389\) −553.000 −1.42159 −0.710797 0.703397i \(-0.751666\pi\)
−0.710797 + 0.703397i \(0.751666\pi\)
\(390\) 0 0
\(391\) 250.000 0.639386
\(392\) 67.8823i 0.173169i
\(393\) − 461.034i − 1.17311i
\(394\) 98.9949i 0.251256i
\(395\) 0 0
\(396\) −10.0000 −0.0252525
\(397\) −335.000 −0.843829 −0.421914 0.906636i \(-0.638642\pi\)
−0.421914 + 0.906636i \(0.638642\pi\)
\(398\) 244.659i 0.614721i
\(399\) − 268.701i − 0.673435i
\(400\) 0 0
\(401\) − 212.132i − 0.529008i −0.964385 0.264504i \(-0.914792\pi\)
0.964385 0.264504i \(-0.0852082\pi\)
\(402\) − 441.235i − 1.09760i
\(403\) 720.000 1.78660
\(404\) −100.000 −0.247525
\(405\) 0 0
\(406\) −300.000 −0.738916
\(407\) − 127.279i − 0.312725i
\(408\) 200.000 0.490196
\(409\) − 721.249i − 1.76344i −0.471769 0.881722i \(-0.656384\pi\)
0.471769 0.881722i \(-0.343616\pi\)
\(410\) 0 0
\(411\) − 268.701i − 0.653773i
\(412\) 33.9411i 0.0823814i
\(413\) 424.264i 1.02727i
\(414\) 14.1421i 0.0341597i
\(415\) 0 0
\(416\) 96.0000 0.230769
\(417\) 353.553i 0.847850i
\(418\) 134.350i 0.321412i
\(419\) 62.0000 0.147971 0.0739857 0.997259i \(-0.476428\pi\)
0.0739857 + 0.997259i \(0.476428\pi\)
\(420\) 0 0
\(421\) − 296.985i − 0.705427i −0.935731 0.352714i \(-0.885259\pi\)
0.935731 0.352714i \(-0.114741\pi\)
\(422\) 120.000 0.284360
\(423\) −5.00000 −0.0118203
\(424\) −72.0000 −0.169811
\(425\) 0 0
\(426\) 0 0
\(427\) −475.000 −1.11241
\(428\) 203.647i 0.475810i
\(429\) 240.000 0.559441
\(430\) 0 0
\(431\) 509.117i 1.18125i 0.806948 + 0.590623i \(0.201118\pi\)
−0.806948 + 0.590623i \(0.798882\pi\)
\(432\) 113.137i 0.261891i
\(433\) − 229.103i − 0.529105i −0.964371 0.264553i \(-0.914776\pi\)
0.964371 0.264553i \(-0.0852243\pi\)
\(434\) 300.000 0.691244
\(435\) 0 0
\(436\) 254.558i 0.583850i
\(437\) 190.000 0.434783
\(438\) −100.000 −0.228311
\(439\) − 806.102i − 1.83622i −0.396323 0.918111i \(-0.629714\pi\)
0.396323 0.918111i \(-0.370286\pi\)
\(440\) 0 0
\(441\) −24.0000 −0.0544218
\(442\) 600.000 1.35747
\(443\) −365.000 −0.823928 −0.411964 0.911200i \(-0.635157\pi\)
−0.411964 + 0.911200i \(0.635157\pi\)
\(444\) −144.000 −0.324324
\(445\) 0 0
\(446\) 516.000 1.15695
\(447\) 608.112i 1.36043i
\(448\) 40.0000 0.0892857
\(449\) 763.675i 1.70084i 0.526108 + 0.850418i \(0.323651\pi\)
−0.526108 + 0.850418i \(0.676349\pi\)
\(450\) 0 0
\(451\) 212.132i 0.470359i
\(452\) − 220.617i − 0.488091i
\(453\) −240.000 −0.529801
\(454\) −96.0000 −0.211454
\(455\) 0 0
\(456\) 152.000 0.333333
\(457\) 265.000 0.579869 0.289934 0.957047i \(-0.406367\pi\)
0.289934 + 0.957047i \(0.406367\pi\)
\(458\) − 205.061i − 0.447731i
\(459\) 707.107i 1.54054i
\(460\) 0 0
\(461\) −553.000 −1.19957 −0.599783 0.800163i \(-0.704746\pi\)
−0.599783 + 0.800163i \(0.704746\pi\)
\(462\) 100.000 0.216450
\(463\) −485.000 −1.04752 −0.523758 0.851867i \(-0.675470\pi\)
−0.523758 + 0.851867i \(0.675470\pi\)
\(464\) − 169.706i − 0.365745i
\(465\) 0 0
\(466\) − 473.762i − 1.01666i
\(467\) 115.000 0.246253 0.123126 0.992391i \(-0.460708\pi\)
0.123126 + 0.992391i \(0.460708\pi\)
\(468\) 33.9411i 0.0725238i
\(469\) − 551.543i − 1.17600i
\(470\) 0 0
\(471\) 537.401i 1.14098i
\(472\) −240.000 −0.508475
\(473\) −25.0000 −0.0528541
\(474\) 169.706i 0.358029i
\(475\) 0 0
\(476\) 250.000 0.525210
\(477\) − 25.4558i − 0.0533665i
\(478\) 278.600i 0.582845i
\(479\) −490.000 −1.02296 −0.511482 0.859294i \(-0.670903\pi\)
−0.511482 + 0.859294i \(0.670903\pi\)
\(480\) 0 0
\(481\) −432.000 −0.898129
\(482\) 420.000 0.871369
\(483\) − 141.421i − 0.292798i
\(484\) 192.000 0.396694
\(485\) 0 0
\(486\) −76.0000 −0.156379
\(487\) − 610.940i − 1.25450i −0.778819 0.627249i \(-0.784181\pi\)
0.778819 0.627249i \(-0.215819\pi\)
\(488\) − 268.701i − 0.550616i
\(489\) − 311.127i − 0.636252i
\(490\) 0 0
\(491\) −82.0000 −0.167006 −0.0835031 0.996508i \(-0.526611\pi\)
−0.0835031 + 0.996508i \(0.526611\pi\)
\(492\) 240.000 0.487805
\(493\) − 1060.66i − 2.15144i
\(494\) 456.000 0.923077
\(495\) 0 0
\(496\) 169.706i 0.342148i
\(497\) 0 0
\(498\) −520.000 −1.04418
\(499\) 485.000 0.971944 0.485972 0.873974i \(-0.338466\pi\)
0.485972 + 0.873974i \(0.338466\pi\)
\(500\) 0 0
\(501\) −168.000 −0.335329
\(502\) 244.659i 0.487368i
\(503\) 250.000 0.497018 0.248509 0.968630i \(-0.420059\pi\)
0.248509 + 0.968630i \(0.420059\pi\)
\(504\) 14.1421i 0.0280598i
\(505\) 0 0
\(506\) 70.7107i 0.139744i
\(507\) − 336.583i − 0.663871i
\(508\) − 458.205i − 0.901979i
\(509\) 169.706i 0.333410i 0.986007 + 0.166705i \(0.0533127\pi\)
−0.986007 + 0.166705i \(0.946687\pi\)
\(510\) 0 0
\(511\) −125.000 −0.244618
\(512\) 22.6274i 0.0441942i
\(513\) 537.401i 1.04757i
\(514\) 96.0000 0.186770
\(515\) 0 0
\(516\) 28.2843i 0.0548145i
\(517\) −25.0000 −0.0483559
\(518\) −180.000 −0.347490
\(519\) 528.000 1.01734
\(520\) 0 0
\(521\) 127.279i 0.244298i 0.992512 + 0.122149i \(0.0389786\pi\)
−0.992512 + 0.122149i \(0.961021\pi\)
\(522\) 60.0000 0.114943
\(523\) 356.382i 0.681418i 0.940169 + 0.340709i \(0.110667\pi\)
−0.940169 + 0.340709i \(0.889333\pi\)
\(524\) 326.000 0.622137
\(525\) 0 0
\(526\) 502.046i 0.954460i
\(527\) 1060.66i 2.01264i
\(528\) 56.5685i 0.107137i
\(529\) −429.000 −0.810964
\(530\) 0 0
\(531\) − 84.8528i − 0.159798i
\(532\) 190.000 0.357143
\(533\) 720.000 1.35084
\(534\) − 509.117i − 0.953402i
\(535\) 0 0
\(536\) 312.000 0.582090
\(537\) −360.000 −0.670391
\(538\) −540.000 −1.00372
\(539\) −120.000 −0.222635
\(540\) 0 0
\(541\) −25.0000 −0.0462107 −0.0231054 0.999733i \(-0.507355\pi\)
−0.0231054 + 0.999733i \(0.507355\pi\)
\(542\) 155.563i 0.287018i
\(543\) −720.000 −1.32597
\(544\) 141.421i 0.259966i
\(545\) 0 0
\(546\) − 339.411i − 0.621632i
\(547\) 16.9706i 0.0310248i 0.999880 + 0.0155124i \(0.00493795\pi\)
−0.999880 + 0.0155124i \(0.995062\pi\)
\(548\) 190.000 0.346715
\(549\) 95.0000 0.173042
\(550\) 0 0
\(551\) − 806.102i − 1.46298i
\(552\) 80.0000 0.144928
\(553\) 212.132i 0.383602i
\(554\) 374.767i 0.676474i
\(555\) 0 0
\(556\) −250.000 −0.449640
\(557\) 745.000 1.33752 0.668761 0.743477i \(-0.266825\pi\)
0.668761 + 0.743477i \(0.266825\pi\)
\(558\) −60.0000 −0.107527
\(559\) 84.8528i 0.151794i
\(560\) 0 0
\(561\) 353.553i 0.630220i
\(562\) 600.000 1.06762
\(563\) 313.955i 0.557647i 0.960342 + 0.278824i \(0.0899445\pi\)
−0.960342 + 0.278824i \(0.910056\pi\)
\(564\) 28.2843i 0.0501494i
\(565\) 0 0
\(566\) − 176.777i − 0.312326i
\(567\) 355.000 0.626102
\(568\) 0 0
\(569\) − 424.264i − 0.745631i −0.927905 0.372816i \(-0.878392\pi\)
0.927905 0.372816i \(-0.121608\pi\)
\(570\) 0 0
\(571\) 1070.00 1.87391 0.936953 0.349456i \(-0.113634\pi\)
0.936953 + 0.349456i \(0.113634\pi\)
\(572\) 169.706i 0.296688i
\(573\) 828.729i 1.44630i
\(574\) 300.000 0.522648
\(575\) 0 0
\(576\) −8.00000 −0.0138889
\(577\) 25.0000 0.0433276 0.0216638 0.999765i \(-0.493104\pi\)
0.0216638 + 0.999765i \(0.493104\pi\)
\(578\) 475.176i 0.822103i
\(579\) −168.000 −0.290155
\(580\) 0 0
\(581\) −650.000 −1.11876
\(582\) − 67.8823i − 0.116636i
\(583\) − 127.279i − 0.218318i
\(584\) − 70.7107i − 0.121080i
\(585\) 0 0
\(586\) −264.000 −0.450512
\(587\) −725.000 −1.23509 −0.617547 0.786534i \(-0.711873\pi\)
−0.617547 + 0.786534i \(0.711873\pi\)
\(588\) 135.765i 0.230892i
\(589\) 806.102i 1.36859i
\(590\) 0 0
\(591\) 197.990i 0.335008i
\(592\) − 101.823i − 0.171999i
\(593\) −650.000 −1.09612 −0.548061 0.836439i \(-0.684634\pi\)
−0.548061 + 0.836439i \(0.684634\pi\)
\(594\) −200.000 −0.336700
\(595\) 0 0
\(596\) −430.000 −0.721477
\(597\) 489.318i 0.819628i
\(598\) 240.000 0.401338
\(599\) 296.985i 0.495801i 0.968785 + 0.247901i \(0.0797406\pi\)
−0.968785 + 0.247901i \(0.920259\pi\)
\(600\) 0 0
\(601\) 848.528i 1.41186i 0.708281 + 0.705930i \(0.249471\pi\)
−0.708281 + 0.705930i \(0.750529\pi\)
\(602\) 35.3553i 0.0587298i
\(603\) 110.309i 0.182933i
\(604\) − 169.706i − 0.280970i
\(605\) 0 0
\(606\) −200.000 −0.330033
\(607\) − 271.529i − 0.447329i −0.974666 0.223665i \(-0.928198\pi\)
0.974666 0.223665i \(-0.0718021\pi\)
\(608\) 107.480i 0.176777i
\(609\) −600.000 −0.985222
\(610\) 0 0
\(611\) 84.8528i 0.138875i
\(612\) −50.0000 −0.0816993
\(613\) −1055.00 −1.72104 −0.860522 0.509413i \(-0.829862\pi\)
−0.860522 + 0.509413i \(0.829862\pi\)
\(614\) −396.000 −0.644951
\(615\) 0 0
\(616\) 70.7107i 0.114790i
\(617\) 505.000 0.818476 0.409238 0.912428i \(-0.365794\pi\)
0.409238 + 0.912428i \(0.365794\pi\)
\(618\) 67.8823i 0.109842i
\(619\) −130.000 −0.210016 −0.105008 0.994471i \(-0.533487\pi\)
−0.105008 + 0.994471i \(0.533487\pi\)
\(620\) 0 0
\(621\) 282.843i 0.455463i
\(622\) − 332.340i − 0.534309i
\(623\) − 636.396i − 1.02150i
\(624\) 192.000 0.307692
\(625\) 0 0
\(626\) 438.406i 0.700329i
\(627\) 268.701i 0.428550i
\(628\) −380.000 −0.605096
\(629\) − 636.396i − 1.01176i
\(630\) 0 0
\(631\) −475.000 −0.752773 −0.376387 0.926463i \(-0.622834\pi\)
−0.376387 + 0.926463i \(0.622834\pi\)
\(632\) −120.000 −0.189873
\(633\) 240.000 0.379147
\(634\) 264.000 0.416404
\(635\) 0 0
\(636\) −144.000 −0.226415
\(637\) 407.294i 0.639393i
\(638\) 300.000 0.470219
\(639\) 0 0
\(640\) 0 0
\(641\) 848.528i 1.32376i 0.749611 + 0.661878i \(0.230240\pi\)
−0.749611 + 0.661878i \(0.769760\pi\)
\(642\) 407.294i 0.634414i
\(643\) 955.000 1.48523 0.742613 0.669721i \(-0.233586\pi\)
0.742613 + 0.669721i \(0.233586\pi\)
\(644\) 100.000 0.155280
\(645\) 0 0
\(646\) 671.751i 1.03986i
\(647\) −965.000 −1.49150 −0.745750 0.666226i \(-0.767908\pi\)
−0.745750 + 0.666226i \(0.767908\pi\)
\(648\) 200.818i 0.309905i
\(649\) − 424.264i − 0.653720i
\(650\) 0 0
\(651\) 600.000 0.921659
\(652\) 220.000 0.337423
\(653\) −935.000 −1.43185 −0.715926 0.698176i \(-0.753995\pi\)
−0.715926 + 0.698176i \(0.753995\pi\)
\(654\) 509.117i 0.778466i
\(655\) 0 0
\(656\) 169.706i 0.258698i
\(657\) 25.0000 0.0380518
\(658\) 35.3553i 0.0537315i
\(659\) 84.8528i 0.128760i 0.997925 + 0.0643800i \(0.0205070\pi\)
−0.997925 + 0.0643800i \(0.979493\pi\)
\(660\) 0 0
\(661\) 678.823i 1.02696i 0.858101 + 0.513481i \(0.171644\pi\)
−0.858101 + 0.513481i \(0.828356\pi\)
\(662\) 420.000 0.634441
\(663\) 1200.00 1.80995
\(664\) − 367.696i − 0.553758i
\(665\) 0 0
\(666\) 36.0000 0.0540541
\(667\) − 424.264i − 0.636078i
\(668\) − 118.794i − 0.177835i
\(669\) 1032.00 1.54260
\(670\) 0 0
\(671\) 475.000 0.707899
\(672\) 80.0000 0.119048
\(673\) − 186.676i − 0.277379i −0.990336 0.138690i \(-0.955711\pi\)
0.990336 0.138690i \(-0.0442890\pi\)
\(674\) 744.000 1.10386
\(675\) 0 0
\(676\) 238.000 0.352071
\(677\) 907.925i 1.34110i 0.741864 + 0.670550i \(0.233942\pi\)
−0.741864 + 0.670550i \(0.766058\pi\)
\(678\) − 441.235i − 0.650789i
\(679\) − 84.8528i − 0.124967i
\(680\) 0 0
\(681\) −192.000 −0.281938
\(682\) −300.000 −0.439883
\(683\) 1120.06i 1.63991i 0.572429 + 0.819954i \(0.306001\pi\)
−0.572429 + 0.819954i \(0.693999\pi\)
\(684\) −38.0000 −0.0555556
\(685\) 0 0
\(686\) 516.188i 0.752461i
\(687\) − 410.122i − 0.596975i
\(688\) −20.0000 −0.0290698
\(689\) −432.000 −0.626996
\(690\) 0 0
\(691\) −715.000 −1.03473 −0.517366 0.855764i \(-0.673087\pi\)
−0.517366 + 0.855764i \(0.673087\pi\)
\(692\) 373.352i 0.539527i
\(693\) −25.0000 −0.0360750
\(694\) − 176.777i − 0.254721i
\(695\) 0 0
\(696\) − 339.411i − 0.487660i
\(697\) 1060.66i 1.52175i
\(698\) 32.5269i 0.0466002i
\(699\) − 947.523i − 1.35554i
\(700\) 0 0
\(701\) −430.000 −0.613409 −0.306705 0.951805i \(-0.599226\pi\)
−0.306705 + 0.951805i \(0.599226\pi\)
\(702\) 678.823i 0.966984i
\(703\) − 483.661i − 0.687996i
\(704\) −40.0000 −0.0568182
\(705\) 0 0
\(706\) − 579.828i − 0.821285i
\(707\) −250.000 −0.353607
\(708\) −480.000 −0.677966
\(709\) −382.000 −0.538787 −0.269394 0.963030i \(-0.586823\pi\)
−0.269394 + 0.963030i \(0.586823\pi\)
\(710\) 0 0
\(711\) − 42.4264i − 0.0596715i
\(712\) 360.000 0.505618
\(713\) 424.264i 0.595041i
\(714\) 500.000 0.700280
\(715\) 0 0
\(716\) − 254.558i − 0.355529i
\(717\) 557.200i 0.777127i
\(718\) − 671.751i − 0.935587i
\(719\) −115.000 −0.159944 −0.0799722 0.996797i \(-0.525483\pi\)
−0.0799722 + 0.996797i \(0.525483\pi\)
\(720\) 0 0
\(721\) 84.8528i 0.117688i
\(722\) 510.531i 0.707107i
\(723\) 840.000 1.16183
\(724\) − 509.117i − 0.703200i
\(725\) 0 0
\(726\) 384.000 0.528926
\(727\) 1075.00 1.47868 0.739340 0.673333i \(-0.235138\pi\)
0.739340 + 0.673333i \(0.235138\pi\)
\(728\) 240.000 0.329670
\(729\) −791.000 −1.08505
\(730\) 0 0
\(731\) −125.000 −0.170999
\(732\) − 537.401i − 0.734155i
\(733\) −530.000 −0.723056 −0.361528 0.932361i \(-0.617745\pi\)
−0.361528 + 0.932361i \(0.617745\pi\)
\(734\) − 325.269i − 0.443146i
\(735\) 0 0
\(736\) 56.5685i 0.0768594i
\(737\) 551.543i 0.748363i
\(738\) −60.0000 −0.0813008
\(739\) −547.000 −0.740189 −0.370095 0.928994i \(-0.620675\pi\)
−0.370095 + 0.928994i \(0.620675\pi\)
\(740\) 0 0
\(741\) 912.000 1.23077
\(742\) −180.000 −0.242588
\(743\) − 958.837i − 1.29049i −0.763974 0.645247i \(-0.776755\pi\)
0.763974 0.645247i \(-0.223245\pi\)
\(744\) 339.411i 0.456198i
\(745\) 0 0
\(746\) 96.0000 0.128686
\(747\) 130.000 0.174029
\(748\) −250.000 −0.334225
\(749\) 509.117i 0.679729i
\(750\) 0 0
\(751\) 169.706i 0.225973i 0.993597 + 0.112986i \(0.0360417\pi\)
−0.993597 + 0.112986i \(0.963958\pi\)
\(752\) −20.0000 −0.0265957
\(753\) 489.318i 0.649825i
\(754\) − 1018.23i − 1.35044i
\(755\) 0 0
\(756\) 282.843i 0.374131i
\(757\) −1055.00 −1.39366 −0.696830 0.717237i \(-0.745407\pi\)
−0.696830 + 0.717237i \(0.745407\pi\)
\(758\) −360.000 −0.474934
\(759\) 141.421i 0.186326i
\(760\) 0 0
\(761\) 215.000 0.282523 0.141261 0.989972i \(-0.454884\pi\)
0.141261 + 0.989972i \(0.454884\pi\)
\(762\) − 916.410i − 1.20264i
\(763\) 636.396i 0.834071i
\(764\) −586.000 −0.767016
\(765\) 0 0
\(766\) 204.000 0.266319
\(767\) −1440.00 −1.87744
\(768\) 45.2548i 0.0589256i
\(769\) −145.000 −0.188557 −0.0942783 0.995546i \(-0.530054\pi\)
−0.0942783 + 0.995546i \(0.530054\pi\)
\(770\) 0 0
\(771\) 192.000 0.249027
\(772\) − 118.794i − 0.153878i
\(773\) 407.294i 0.526900i 0.964673 + 0.263450i \(0.0848604\pi\)
−0.964673 + 0.263450i \(0.915140\pi\)
\(774\) − 7.07107i − 0.00913575i
\(775\) 0 0
\(776\) 48.0000 0.0618557
\(777\) −360.000 −0.463320
\(778\) − 782.060i − 1.00522i
\(779\) 806.102i 1.03479i
\(780\) 0 0
\(781\) 0 0
\(782\) 353.553i 0.452114i
\(783\) 1200.00 1.53257
\(784\) −96.0000 −0.122449
\(785\) 0 0
\(786\) 652.000 0.829517
\(787\) 186.676i 0.237200i 0.992942 + 0.118600i \(0.0378406\pi\)
−0.992942 + 0.118600i \(0.962159\pi\)
\(788\) −140.000 −0.177665
\(789\) 1004.09i 1.27261i
\(790\) 0 0
\(791\) − 551.543i − 0.697273i
\(792\) − 14.1421i − 0.0178562i
\(793\) − 1612.20i − 2.03304i
\(794\) − 473.762i − 0.596677i
\(795\) 0 0
\(796\) −346.000 −0.434673
\(797\) − 704.278i − 0.883662i −0.897098 0.441831i \(-0.854329\pi\)
0.897098 0.441831i \(-0.145671\pi\)
\(798\) 380.000 0.476190
\(799\) −125.000 −0.156446
\(800\) 0 0
\(801\) 127.279i 0.158900i
\(802\) 300.000 0.374065
\(803\) 125.000 0.155666
\(804\) 624.000 0.776119
\(805\) 0 0
\(806\) 1018.23i 1.26332i
\(807\) −1080.00 −1.33829
\(808\) − 141.421i − 0.175026i
\(809\) −457.000 −0.564895 −0.282447 0.959283i \(-0.591146\pi\)
−0.282447 + 0.959283i \(0.591146\pi\)
\(810\) 0 0
\(811\) − 509.117i − 0.627764i −0.949462 0.313882i \(-0.898370\pi\)
0.949462 0.313882i \(-0.101630\pi\)
\(812\) − 424.264i − 0.522493i
\(813\) 311.127i 0.382690i
\(814\) 180.000 0.221130
\(815\) 0 0
\(816\) 282.843i 0.346621i
\(817\) −95.0000 −0.116279
\(818\) 1020.00 1.24694
\(819\) 84.8528i 0.103605i
\(820\) 0 0
\(821\) 167.000 0.203410 0.101705 0.994815i \(-0.467570\pi\)
0.101705 + 0.994815i \(0.467570\pi\)
\(822\) 380.000 0.462287
\(823\) 1315.00 1.59781 0.798906 0.601455i \(-0.205412\pi\)
0.798906 + 0.601455i \(0.205412\pi\)
\(824\) −48.0000 −0.0582524
\(825\) 0 0
\(826\) −600.000 −0.726392
\(827\) − 534.573i − 0.646400i −0.946331 0.323200i \(-0.895241\pi\)
0.946331 0.323200i \(-0.104759\pi\)
\(828\) −20.0000 −0.0241546
\(829\) − 763.675i − 0.921201i −0.887608 0.460600i \(-0.847634\pi\)
0.887608 0.460600i \(-0.152366\pi\)
\(830\) 0 0
\(831\) 749.533i 0.901965i
\(832\) 135.765i 0.163178i
\(833\) −600.000 −0.720288
\(834\) −500.000 −0.599520
\(835\) 0 0
\(836\) −190.000 −0.227273
\(837\) −1200.00 −1.43369
\(838\) 87.6812i 0.104632i
\(839\) 339.411i 0.404543i 0.979330 + 0.202271i \(0.0648323\pi\)
−0.979330 + 0.202271i \(0.935168\pi\)
\(840\) 0 0
\(841\) −959.000 −1.14031
\(842\) 420.000 0.498812
\(843\) 1200.00 1.42349
\(844\) 169.706i 0.201073i
\(845\) 0 0
\(846\) − 7.07107i − 0.00835824i
\(847\) 480.000 0.566706
\(848\) − 101.823i − 0.120075i
\(849\) − 353.553i − 0.416435i
\(850\) 0 0
\(851\) − 254.558i − 0.299129i
\(852\) 0 0
\(853\) −770.000 −0.902696 −0.451348 0.892348i \(-0.649057\pi\)
−0.451348 + 0.892348i \(0.649057\pi\)
\(854\) − 671.751i − 0.786594i
\(855\) 0 0
\(856\) −288.000 −0.336449
\(857\) − 1255.82i − 1.46537i −0.680568 0.732685i \(-0.738267\pi\)
0.680568 0.732685i \(-0.261733\pi\)
\(858\) 339.411i 0.395584i
\(859\) 557.000 0.648428 0.324214 0.945984i \(-0.394900\pi\)
0.324214 + 0.945984i \(0.394900\pi\)
\(860\) 0 0
\(861\) 600.000 0.696864
\(862\) −720.000 −0.835267
\(863\) 992.778i 1.15038i 0.818020 + 0.575190i \(0.195072\pi\)
−0.818020 + 0.575190i \(0.804928\pi\)
\(864\) −160.000 −0.185185
\(865\) 0 0
\(866\) 324.000 0.374134
\(867\) 950.352i 1.09614i
\(868\) 424.264i 0.488783i
\(869\) − 212.132i − 0.244111i
\(870\) 0 0
\(871\) 1872.00 2.14925
\(872\) −360.000 −0.412844
\(873\) 16.9706i 0.0194394i
\(874\) 268.701i 0.307438i
\(875\) 0 0
\(876\) − 141.421i − 0.161440i
\(877\) 186.676i 0.212858i 0.994320 + 0.106429i \(0.0339416\pi\)
−0.994320 + 0.106429i \(0.966058\pi\)
\(878\) 1140.00 1.29841
\(879\) −528.000 −0.600683
\(880\) 0 0
\(881\) −25.0000 −0.0283768 −0.0141884 0.999899i \(-0.504516\pi\)
−0.0141884 + 0.999899i \(0.504516\pi\)
\(882\) − 33.9411i − 0.0384820i
\(883\) −965.000 −1.09287 −0.546433 0.837503i \(-0.684015\pi\)
−0.546433 + 0.837503i \(0.684015\pi\)
\(884\) 848.528i 0.959873i
\(885\) 0 0
\(886\) − 516.188i − 0.582605i
\(887\) − 780.646i − 0.880097i −0.897974 0.440048i \(-0.854961\pi\)
0.897974 0.440048i \(-0.145039\pi\)
\(888\) − 203.647i − 0.229332i
\(889\) − 1145.51i − 1.28854i
\(890\) 0 0
\(891\) −355.000 −0.398429
\(892\) 729.734i 0.818088i
\(893\) −95.0000 −0.106383
\(894\) −860.000 −0.961969
\(895\) 0 0
\(896\) 56.5685i 0.0631345i
\(897\) 480.000 0.535117
\(898\) −1080.00 −1.20267
\(899\) 1800.00 2.00222
\(900\) 0 0
\(901\) − 636.396i − 0.706322i
\(902\) −300.000 −0.332594
\(903\) 70.7107i 0.0783064i
\(904\) 312.000 0.345133
\(905\) 0 0
\(906\) − 339.411i − 0.374626i
\(907\) − 313.955i − 0.346147i −0.984909 0.173074i \(-0.944630\pi\)
0.984909 0.173074i \(-0.0553698\pi\)
\(908\) − 135.765i − 0.149520i
\(909\) 50.0000 0.0550055
\(910\) 0 0
\(911\) − 933.381i − 1.02457i −0.858816 0.512284i \(-0.828800\pi\)
0.858816 0.512284i \(-0.171200\pi\)
\(912\) 214.960i 0.235702i
\(913\) 650.000 0.711939
\(914\) 374.767i 0.410029i
\(915\) 0 0
\(916\) 290.000 0.316594
\(917\) 815.000 0.888768
\(918\) −1000.00 −1.08932
\(919\) −538.000 −0.585419 −0.292709 0.956201i \(-0.594557\pi\)
−0.292709 + 0.956201i \(0.594557\pi\)
\(920\) 0 0
\(921\) −792.000 −0.859935
\(922\) − 782.060i − 0.848221i
\(923\) 0 0
\(924\) 141.421i 0.153053i
\(925\) 0 0
\(926\) − 685.894i − 0.740706i
\(927\) − 16.9706i − 0.0183070i
\(928\) 240.000 0.258621
\(929\) −742.000 −0.798708 −0.399354 0.916797i \(-0.630766\pi\)
−0.399354 + 0.916797i \(0.630766\pi\)
\(930\) 0 0
\(931\) −456.000 −0.489796
\(932\) 670.000 0.718884
\(933\) − 664.680i − 0.712412i
\(934\) 162.635i 0.174127i
\(935\) 0 0
\(936\) −48.0000 −0.0512821
\(937\) −335.000 −0.357524 −0.178762 0.983892i \(-0.557209\pi\)
−0.178762 + 0.983892i \(0.557209\pi\)
\(938\) 780.000 0.831557
\(939\) 876.812i 0.933773i
\(940\) 0 0
\(941\) 424.264i 0.450865i 0.974259 + 0.225433i \(0.0723795\pi\)
−0.974259 + 0.225433i \(0.927620\pi\)
\(942\) −760.000 −0.806794
\(943\) 424.264i 0.449909i
\(944\) − 339.411i − 0.359546i
\(945\) 0 0
\(946\) − 35.3553i − 0.0373735i
\(947\) 1210.00 1.27772 0.638860 0.769323i \(-0.279406\pi\)
0.638860 + 0.769323i \(0.279406\pi\)
\(948\) −240.000 −0.253165
\(949\) − 424.264i − 0.447064i
\(950\) 0 0
\(951\) 528.000 0.555205
\(952\) 353.553i 0.371380i
\(953\) 992.778i 1.04174i 0.853636 + 0.520870i \(0.174392\pi\)
−0.853636 + 0.520870i \(0.825608\pi\)
\(954\) 36.0000 0.0377358
\(955\) 0 0
\(956\) −394.000 −0.412134
\(957\) 600.000 0.626959
\(958\) − 692.965i − 0.723345i
\(959\) 475.000 0.495308
\(960\) 0 0
\(961\) −839.000 −0.873049
\(962\) − 610.940i − 0.635073i
\(963\) − 101.823i − 0.105736i
\(964\) 593.970i 0.616151i
\(965\) 0 0
\(966\) 200.000 0.207039
\(967\) −350.000 −0.361944 −0.180972 0.983488i \(-0.557924\pi\)
−0.180972 + 0.983488i \(0.557924\pi\)
\(968\) 271.529i 0.280505i
\(969\) 1343.50i 1.38648i
\(970\) 0 0
\(971\) − 254.558i − 0.262161i −0.991372 0.131081i \(-0.958155\pi\)
0.991372 0.131081i \(-0.0418447\pi\)
\(972\) − 107.480i − 0.110576i
\(973\) −625.000 −0.642343
\(974\) 864.000 0.887064
\(975\) 0 0
\(976\) 380.000 0.389344
\(977\) − 398.808i − 0.408197i −0.978950 0.204098i \(-0.934574\pi\)
0.978950 0.204098i \(-0.0654262\pi\)
\(978\) 440.000 0.449898
\(979\) 636.396i 0.650047i
\(980\) 0 0
\(981\) − 127.279i − 0.129744i
\(982\) − 115.966i − 0.118091i
\(983\) − 695.793i − 0.707826i −0.935278 0.353913i \(-0.884851\pi\)
0.935278 0.353913i \(-0.115149\pi\)
\(984\) 339.411i 0.344930i
\(985\) 0 0
\(986\) 1500.00 1.52130
\(987\) 70.7107i 0.0716420i
\(988\) 644.881i 0.652714i
\(989\) −50.0000 −0.0505561
\(990\) 0 0
\(991\) − 381.838i − 0.385305i −0.981267 0.192653i \(-0.938291\pi\)
0.981267 0.192653i \(-0.0617091\pi\)
\(992\) −240.000 −0.241935
\(993\) 840.000 0.845921
\(994\) 0 0
\(995\) 0 0
\(996\) − 735.391i − 0.738344i
\(997\) 265.000 0.265797 0.132899 0.991130i \(-0.457572\pi\)
0.132899 + 0.991130i \(0.457572\pi\)
\(998\) 685.894i 0.687268i
\(999\) 720.000 0.720721
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 950.3.c.a.151.2 2
5.2 odd 4 950.3.d.a.949.1 4
5.3 odd 4 950.3.d.a.949.4 4
5.4 even 2 38.3.b.a.37.1 2
15.14 odd 2 342.3.d.a.37.2 2
19.18 odd 2 inner 950.3.c.a.151.1 2
20.19 odd 2 304.3.e.c.113.2 2
40.19 odd 2 1216.3.e.i.1025.1 2
40.29 even 2 1216.3.e.j.1025.2 2
60.59 even 2 2736.3.o.h.721.2 2
95.18 even 4 950.3.d.a.949.2 4
95.37 even 4 950.3.d.a.949.3 4
95.94 odd 2 38.3.b.a.37.2 yes 2
285.284 even 2 342.3.d.a.37.1 2
380.379 even 2 304.3.e.c.113.1 2
760.189 odd 2 1216.3.e.j.1025.1 2
760.379 even 2 1216.3.e.i.1025.2 2
1140.1139 odd 2 2736.3.o.h.721.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.3.b.a.37.1 2 5.4 even 2
38.3.b.a.37.2 yes 2 95.94 odd 2
304.3.e.c.113.1 2 380.379 even 2
304.3.e.c.113.2 2 20.19 odd 2
342.3.d.a.37.1 2 285.284 even 2
342.3.d.a.37.2 2 15.14 odd 2
950.3.c.a.151.1 2 19.18 odd 2 inner
950.3.c.a.151.2 2 1.1 even 1 trivial
950.3.d.a.949.1 4 5.2 odd 4
950.3.d.a.949.2 4 95.18 even 4
950.3.d.a.949.3 4 95.37 even 4
950.3.d.a.949.4 4 5.3 odd 4
1216.3.e.i.1025.1 2 40.19 odd 2
1216.3.e.i.1025.2 2 760.379 even 2
1216.3.e.j.1025.1 2 760.189 odd 2
1216.3.e.j.1025.2 2 40.29 even 2
2736.3.o.h.721.1 2 1140.1139 odd 2
2736.3.o.h.721.2 2 60.59 even 2