Properties

Label 38.3.b.a.37.2
Level $38$
Weight $3$
Character 38.37
Analytic conductor $1.035$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 38.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.03542500457\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 37.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 38.37
Dual form 38.3.b.a.37.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421i q^{2} +2.82843i q^{3} -2.00000 q^{4} -1.00000 q^{5} -4.00000 q^{6} +5.00000 q^{7} -2.82843i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.41421i q^{2} +2.82843i q^{3} -2.00000 q^{4} -1.00000 q^{5} -4.00000 q^{6} +5.00000 q^{7} -2.82843i q^{8} +1.00000 q^{9} -1.41421i q^{10} +5.00000 q^{11} -5.65685i q^{12} -16.9706i q^{13} +7.07107i q^{14} -2.82843i q^{15} +4.00000 q^{16} -25.0000 q^{17} +1.41421i q^{18} +19.0000 q^{19} +2.00000 q^{20} +14.1421i q^{21} +7.07107i q^{22} -10.0000 q^{23} +8.00000 q^{24} -24.0000 q^{25} +24.0000 q^{26} +28.2843i q^{27} -10.0000 q^{28} +42.4264i q^{29} +4.00000 q^{30} -42.4264i q^{31} +5.65685i q^{32} +14.1421i q^{33} -35.3553i q^{34} -5.00000 q^{35} -2.00000 q^{36} -25.4558i q^{37} +26.8701i q^{38} +48.0000 q^{39} +2.82843i q^{40} -42.4264i q^{41} -20.0000 q^{42} +5.00000 q^{43} -10.0000 q^{44} -1.00000 q^{45} -14.1421i q^{46} +5.00000 q^{47} +11.3137i q^{48} -24.0000 q^{49} -33.9411i q^{50} -70.7107i q^{51} +33.9411i q^{52} -25.4558i q^{53} -40.0000 q^{54} -5.00000 q^{55} -14.1421i q^{56} +53.7401i q^{57} -60.0000 q^{58} +84.8528i q^{59} +5.65685i q^{60} +95.0000 q^{61} +60.0000 q^{62} +5.00000 q^{63} -8.00000 q^{64} +16.9706i q^{65} -20.0000 q^{66} +110.309i q^{67} +50.0000 q^{68} -28.2843i q^{69} -7.07107i q^{70} -2.82843i q^{72} -25.0000 q^{73} +36.0000 q^{74} -67.8823i q^{75} -38.0000 q^{76} +25.0000 q^{77} +67.8823i q^{78} +42.4264i q^{79} -4.00000 q^{80} -71.0000 q^{81} +60.0000 q^{82} -130.000 q^{83} -28.2843i q^{84} +25.0000 q^{85} +7.07107i q^{86} -120.000 q^{87} -14.1421i q^{88} -127.279i q^{89} -1.41421i q^{90} -84.8528i q^{91} +20.0000 q^{92} +120.000 q^{93} +7.07107i q^{94} -19.0000 q^{95} -16.0000 q^{96} +16.9706i q^{97} -33.9411i q^{98} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} - 2q^{5} - 8q^{6} + 10q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 4q^{4} - 2q^{5} - 8q^{6} + 10q^{7} + 2q^{9} + 10q^{11} + 8q^{16} - 50q^{17} + 38q^{19} + 4q^{20} - 20q^{23} + 16q^{24} - 48q^{25} + 48q^{26} - 20q^{28} + 8q^{30} - 10q^{35} - 4q^{36} + 96q^{39} - 40q^{42} + 10q^{43} - 20q^{44} - 2q^{45} + 10q^{47} - 48q^{49} - 80q^{54} - 10q^{55} - 120q^{58} + 190q^{61} + 120q^{62} + 10q^{63} - 16q^{64} - 40q^{66} + 100q^{68} - 50q^{73} + 72q^{74} - 76q^{76} + 50q^{77} - 8q^{80} - 142q^{81} + 120q^{82} - 260q^{83} + 50q^{85} - 240q^{87} + 40q^{92} + 240q^{93} - 38q^{95} - 32q^{96} + 10q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\) \(21\)
\(\chi(n)\) \(-1\)

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
\(5\) −1.00000 −0.200000 −0.100000 0.994987i \(-0.531884\pi\)
−0.100000 + 0.994987i \(0.531884\pi\)
\(6\) −4.00000 −0.666667
\(7\) 5.00000 0.714286 0.357143 0.934050i \(-0.383751\pi\)
0.357143 + 0.934050i \(0.383751\pi\)
\(8\) − 2.82843i − 0.353553i
\(9\) 1.00000 0.111111
\(10\) − 1.41421i − 0.141421i
\(11\) 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
\(15\) − 2.82843i − 0.188562i
\(16\) 4.00000 0.250000
\(17\) −25.0000 −1.47059 −0.735294 0.677748i \(-0.762956\pi\)
−0.735294 + 0.677748i \(0.762956\pi\)
\(18\) 1.41421i 0.0785674i
\(19\) 19.0000 1.00000
\(20\) 2.00000 0.100000
\(21\) 14.1421i 0.673435i
\(22\) 7.07107i 0.321412i
\(23\) −10.0000 −0.434783 −0.217391 0.976085i \(-0.569755\pi\)
−0.217391 + 0.976085i \(0.569755\pi\)
\(24\) 8.00000 0.333333
\(25\) −24.0000 −0.960000
\(26\) 24.0000 0.923077
\(27\) 28.2843i 1.04757i
\(28\) −10.0000 −0.357143
\(29\) 42.4264i 1.46298i 0.681852 + 0.731490i \(0.261175\pi\)
−0.681852 + 0.731490i \(0.738825\pi\)
\(30\) 4.00000 0.133333
\(31\) − 42.4264i − 1.36859i −0.729204 0.684297i \(-0.760109\pi\)
0.729204 0.684297i \(-0.239891\pi\)
\(32\) 5.65685i 0.176777i
\(33\) 14.1421i 0.428550i
\(34\) − 35.3553i − 1.03986i
\(35\) −5.00000 −0.142857
\(36\) −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
\(40\) 2.82843i 0.0707107i
\(41\) − 42.4264i − 1.03479i −0.855747 0.517395i \(-0.826902\pi\)
0.855747 0.517395i \(-0.173098\pi\)
\(42\) −20.0000 −0.476190
\(43\) 5.00000 0.116279 0.0581395 0.998308i \(-0.481483\pi\)
0.0581395 + 0.998308i \(0.481483\pi\)
\(44\) −10.0000 −0.227273
\(45\) −1.00000 −0.0222222
\(46\) − 14.1421i − 0.307438i
\(47\) 5.00000 0.106383 0.0531915 0.998584i \(-0.483061\pi\)
0.0531915 + 0.998584i \(0.483061\pi\)
\(48\) 11.3137i 0.235702i
\(49\) −24.0000 −0.489796
\(50\) − 33.9411i − 0.678823i
\(51\) − 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
\(55\) −5.00000 −0.0909091
\(56\) − 14.1421i − 0.252538i
\(57\) 53.7401i 0.942809i
\(58\) −60.0000 −1.03448
\(59\) 84.8528i 1.43818i 0.694915 + 0.719092i \(0.255442\pi\)
−0.694915 + 0.719092i \(0.744558\pi\)
\(60\) 5.65685i 0.0942809i
\(61\) 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
\(65\) 16.9706i 0.261086i
\(66\) −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
\(70\) − 7.07107i − 0.101015i
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) − 2.82843i − 0.0392837i
\(73\) −25.0000 −0.342466 −0.171233 0.985231i \(-0.554775\pi\)
−0.171233 + 0.985231i \(0.554775\pi\)
\(74\) 36.0000 0.486486
\(75\) − 67.8823i − 0.905097i
\(76\) −38.0000 −0.500000
\(77\) 25.0000 0.324675
\(78\) 67.8823i 0.870285i
\(79\) 42.4264i 0.537043i 0.963274 + 0.268522i \(0.0865351\pi\)
−0.963274 + 0.268522i \(0.913465\pi\)
\(80\) −4.00000 −0.0500000
\(81\) −71.0000 −0.876543
\(82\) 60.0000 0.731707
\(83\) −130.000 −1.56627 −0.783133 0.621855i \(-0.786379\pi\)
−0.783133 + 0.621855i \(0.786379\pi\)
\(84\) − 28.2843i − 0.336718i
\(85\) 25.0000 0.294118
\(86\) 7.07107i 0.0822217i
\(87\) −120.000 −1.37931
\(88\) − 14.1421i − 0.160706i
\(89\) − 127.279i − 1.43010i −0.699071 0.715052i \(-0.746403\pi\)
0.699071 0.715052i \(-0.253597\pi\)
\(90\) − 1.41421i − 0.0157135i
\(91\) − 84.8528i − 0.932449i
\(92\) 20.0000 0.217391
\(93\) 120.000 1.29032
\(94\) 7.07107i 0.0752241i
\(95\) −19.0000 −0.200000
\(96\) −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
\(100\) 48.0000 0.480000
\(101\) 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
\(105\) − 14.1421i − 0.134687i
\(106\) 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 \(0.198454\pi\)
−0.811862 + 0.583850i \(0.801546\pi\)
\(110\) − 7.07107i − 0.0642824i
\(111\) 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
\(115\) 10.0000 0.0869565
\(116\) − 84.8528i − 0.731490i
\(117\) − 16.9706i − 0.145048i
\(118\) −120.000 −1.01695
\(119\) −125.000 −1.05042
\(120\) −8.00000 −0.0666667
\(121\) −96.0000 −0.793388
\(122\) 134.350i 1.10123i
\(123\) 120.000 0.975610
\(124\) 84.8528i 0.684297i
\(125\) 49.0000 0.392000
\(126\) 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
\(130\) −24.0000 −0.184615
\(131\) −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
\(135\) − 28.2843i − 0.209513i
\(136\) 70.7107i 0.519931i
\(137\) 95.0000 0.693431 0.346715 0.937970i \(-0.387297\pi\)
0.346715 + 0.937970i \(0.387297\pi\)
\(138\) 40.0000 0.289855
\(139\) 125.000 0.899281 0.449640 0.893210i \(-0.351552\pi\)
0.449640 + 0.893210i \(0.351552\pi\)
\(140\) 10.0000 0.0714286
\(141\) 14.1421i 0.100299i
\(142\) 0 0
\(143\) − 84.8528i − 0.593376i
\(144\) 4.00000 0.0277778
\(145\) − 42.4264i − 0.292596i
\(146\) − 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 \(0.243464\pi\)
\(150\) 96.0000 0.640000
\(151\) − 84.8528i − 0.561939i −0.959717 0.280970i \(-0.909344\pi\)
0.959717 0.280970i \(-0.0906560\pi\)
\(152\) − 53.7401i − 0.353553i
\(153\) −25.0000 −0.163399
\(154\) 35.3553i 0.229580i
\(155\) 42.4264i 0.273719i
\(156\) −96.0000 −0.615385
\(157\) −190.000 −1.21019 −0.605096 0.796153i \(-0.706865\pi\)
−0.605096 + 0.796153i \(0.706865\pi\)
\(158\) −60.0000 −0.379747
\(159\) 72.0000 0.452830
\(160\) − 5.65685i − 0.0353553i
\(161\) −50.0000 −0.310559
\(162\) − 100.409i − 0.619810i
\(163\) 110.000 0.674847 0.337423 0.941353i \(-0.390445\pi\)
0.337423 + 0.941353i \(0.390445\pi\)
\(164\) 84.8528i 0.517395i
\(165\) − 14.1421i − 0.0857099i
\(166\) − 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
\(170\) 35.3553i 0.207973i
\(171\) 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
\(175\) −120.000 −0.685714
\(176\) 20.0000 0.113636
\(177\) −240.000 −1.35593
\(178\) 180.000 1.01124
\(179\) − 127.279i − 0.711057i −0.934665 0.355529i \(-0.884301\pi\)
0.934665 0.355529i \(-0.115699\pi\)
\(180\) 2.00000 0.0111111
\(181\) − 254.558i − 1.40640i −0.710992 0.703200i \(-0.751754\pi\)
0.710992 0.703200i \(-0.248246\pi\)
\(182\) 120.000 0.659341
\(183\) 268.701i 1.46831i
\(184\) 28.2843i 0.153719i
\(185\) 25.4558i 0.137599i
\(186\) 169.706i 0.912396i
\(187\) −125.000 −0.668449
\(188\) −10.0000 −0.0531915
\(189\) 141.421i 0.748261i
\(190\) − 26.8701i − 0.141421i
\(191\) 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
\(195\) −48.0000 −0.246154
\(196\) 48.0000 0.244898
\(197\) −70.0000 −0.355330 −0.177665 0.984091i \(-0.556854\pi\)
−0.177665 + 0.984091i \(0.556854\pi\)
\(198\) 7.07107i 0.0357125i
\(199\) 173.000 0.869347 0.434673 0.900588i \(-0.356864\pi\)
0.434673 + 0.900588i \(0.356864\pi\)
\(200\) 67.8823i 0.339411i
\(201\) −312.000 −1.55224
\(202\) 70.7107i 0.350053i
\(203\) 212.132i 1.04499i
\(204\) 141.421i 0.693242i
\(205\) 42.4264i 0.206958i
\(206\) 24.0000 0.116505
\(207\) −10.0000 −0.0483092
\(208\) − 67.8823i − 0.326357i
\(209\) 95.0000 0.454545
\(210\) 20.0000 0.0952381
\(211\) 84.8528i 0.402146i 0.979576 + 0.201073i \(0.0644429\pi\)
−0.979576 + 0.201073i \(0.935557\pi\)
\(212\) 50.9117i 0.240149i
\(213\) 0 0
\(214\) 144.000 0.672897
\(215\) −5.00000 −0.0232558
\(216\) 80.0000 0.370370
\(217\) − 212.132i − 0.977567i
\(218\) −180.000 −0.825688
\(219\) − 70.7107i − 0.322880i
\(220\) 10.0000 0.0454545
\(221\) 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
\(225\) −24.0000 −0.106667
\(226\) −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 \(0.602539\pi\)
\(230\) 14.1421i 0.0614875i
\(231\) 70.7107i 0.306107i
\(232\) 120.000 0.517241
\(233\) 335.000 1.43777 0.718884 0.695130i \(-0.244653\pi\)
0.718884 + 0.695130i \(0.244653\pi\)
\(234\) 24.0000 0.102564
\(235\) −5.00000 −0.0212766
\(236\) − 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 \(0.364784\pi\)
\(240\) − 11.3137i − 0.0471405i
\(241\) 296.985i 1.23230i 0.787628 + 0.616151i \(0.211309\pi\)
−0.787628 + 0.616151i \(0.788691\pi\)
\(242\) − 135.765i − 0.561010i
\(243\) 53.7401i 0.221153i
\(244\) −190.000 −0.778689
\(245\) 24.0000 0.0979592
\(246\) 169.706i 0.689860i
\(247\) − 322.441i − 1.30543i
\(248\) −120.000 −0.483871
\(249\) − 367.696i − 1.47669i
\(250\) 69.2965i 0.277186i
\(251\) 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
\(255\) 70.7107i 0.277297i
\(256\) 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
\(260\) − 33.9411i − 0.130543i
\(261\) 42.4264i 0.162553i
\(262\) − 230.517i − 0.879835i
\(263\) −355.000 −1.34981 −0.674905 0.737905i \(-0.735815\pi\)
−0.674905 + 0.737905i \(0.735815\pi\)
\(264\) 40.0000 0.151515
\(265\) 25.4558i 0.0960598i
\(266\) 134.350i 0.505076i
\(267\) 360.000 1.34831
\(268\) − 220.617i − 0.823199i
\(269\) − 381.838i − 1.41947i −0.704468 0.709735i \(-0.748814\pi\)
0.704468 0.709735i \(-0.251186\pi\)
\(270\) 40.0000 0.148148
\(271\) 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
\(275\) −120.000 −0.436364
\(276\) 56.5685i 0.204958i
\(277\) −265.000 −0.956679 −0.478339 0.878175i \(-0.658761\pi\)
−0.478339 + 0.878175i \(0.658761\pi\)
\(278\) 176.777i 0.635887i
\(279\) − 42.4264i − 0.152066i
\(280\) 14.1421i 0.0505076i
\(281\) 424.264i 1.50984i 0.655819 + 0.754918i \(0.272323\pi\)
−0.655819 + 0.754918i \(0.727677\pi\)
\(282\) −20.0000 −0.0709220
\(283\) 125.000 0.441696 0.220848 0.975308i \(-0.429118\pi\)
0.220848 + 0.975308i \(0.429118\pi\)
\(284\) 0 0
\(285\) − 53.7401i − 0.188562i
\(286\) 120.000 0.419580
\(287\) − 212.132i − 0.739136i
\(288\) 5.65685i 0.0196419i
\(289\) 336.000 1.16263
\(290\) 60.0000 0.206897
\(291\) −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
\(295\) − 84.8528i − 0.287637i
\(296\) −72.0000 −0.243243
\(297\) 141.421i 0.476166i
\(298\) 304.056i 1.02032i
\(299\) 169.706i 0.567577i
\(300\) 135.765i 0.452548i
\(301\) 25.0000 0.0830565
\(302\) 120.000 0.397351
\(303\) 141.421i 0.466737i
\(304\) 76.0000 0.250000
\(305\) −95.0000 −0.311475
\(306\) − 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
\(310\) −60.0000 −0.193548
\(311\) −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 \(-0.664908\pi\)
−0.495208 + 0.868775i \(0.664908\pi\)
\(314\) − 268.701i − 0.855734i
\(315\) −5.00000 −0.0158730
\(316\) − 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
\(320\) 8.00000 0.0250000
\(321\) 288.000 0.897196
\(322\) − 70.7107i − 0.219598i
\(323\) −475.000 −1.47059
\(324\) 142.000 0.438272
\(325\) 407.294i 1.25321i
\(326\) 155.563i 0.477189i
\(327\) −360.000 −1.10092
\(328\) −120.000 −0.365854
\(329\) 25.0000 0.0759878
\(330\) 20.0000 0.0606061
\(331\) 296.985i 0.897235i 0.893724 + 0.448618i \(0.148083\pi\)
−0.893724 + 0.448618i \(0.851917\pi\)
\(332\) 260.000 0.783133
\(333\) − 25.4558i − 0.0764440i
\(334\) −84.0000 −0.251497
\(335\) − 110.309i − 0.329280i
\(336\) 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
\(340\) −50.0000 −0.147059
\(341\) − 212.132i − 0.622088i
\(342\) 26.8701i 0.0785674i
\(343\) −365.000 −1.06414
\(344\) − 14.1421i − 0.0411109i
\(345\) 28.2843i 0.0819834i
\(346\) 264.000 0.763006
\(347\) 125.000 0.360231 0.180115 0.983646i \(-0.442353\pi\)
0.180115 + 0.983646i \(0.442353\pi\)
\(348\) 240.000 0.689655
\(349\) 23.0000 0.0659026 0.0329513 0.999457i \(-0.489509\pi\)
0.0329513 + 0.999457i \(0.489509\pi\)
\(350\) − 169.706i − 0.484873i
\(351\) 480.000 1.36752
\(352\) 28.2843i 0.0803530i
\(353\) 410.000 1.16147 0.580737 0.814092i \(-0.302765\pi\)
0.580737 + 0.814092i \(0.302765\pi\)
\(354\) − 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 \(0.730105\pi\)
\(360\) 2.82843i 0.00785674i
\(361\) 361.000 1.00000
\(362\) 360.000 0.994475
\(363\) − 271.529i − 0.748014i
\(364\) 169.706i 0.466224i
\(365\) 25.0000 0.0684932
\(366\) −380.000 −1.03825
\(367\) 230.000 0.626703 0.313351 0.949637i \(-0.398548\pi\)
0.313351 + 0.949637i \(0.398548\pi\)
\(368\) −40.0000 −0.108696
\(369\) − 42.4264i − 0.114977i
\(370\) −36.0000 −0.0972973
\(371\) − 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
\(375\) 138.593i 0.369581i
\(376\) − 14.1421i − 0.0376121i
\(377\) 720.000 1.90981
\(378\) −200.000 −0.529101
\(379\) − 254.558i − 0.671658i −0.941923 0.335829i \(-0.890984\pi\)
0.941923 0.335829i \(-0.109016\pi\)
\(380\) 38.0000 0.100000
\(381\) −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
\(385\) −25.0000 −0.0649351
\(386\) −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 \(0.751666\pi\)
\(390\) − 67.8823i − 0.174057i
\(391\) 250.000 0.639386
\(392\) 67.8823i 0.173169i
\(393\) − 461.034i − 1.17311i
\(394\) − 98.9949i − 0.251256i
\(395\) − 42.4264i − 0.107409i
\(396\) −10.0000 −0.0252525
\(397\) 335.000 0.843829 0.421914 0.906636i \(-0.361358\pi\)
0.421914 + 0.906636i \(0.361358\pi\)
\(398\) 244.659i 0.614721i
\(399\) 268.701i 0.673435i
\(400\) −96.0000 −0.240000
\(401\) 212.132i 0.529008i 0.964385 + 0.264504i \(0.0852082\pi\)
−0.964385 + 0.264504i \(0.914792\pi\)
\(402\) − 441.235i − 1.09760i
\(403\) −720.000 −1.78660
\(404\) −100.000 −0.247525
\(405\) 71.0000 0.175309
\(406\) −300.000 −0.738916
\(407\) − 127.279i − 0.312725i
\(408\) −200.000 −0.490196
\(409\) 721.249i 1.76344i 0.471769 + 0.881722i \(0.343616\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(410\) −60.0000 −0.146341
\(411\) 268.701i 0.653773i
\(412\) 33.9411i 0.0823814i
\(413\) 424.264i 1.02727i
\(414\) − 14.1421i − 0.0341597i
\(415\) 130.000 0.313253
\(416\) 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 \(0.476428\pi\)
\(420\) 28.2843i 0.0673435i
\(421\) 296.985i 0.705427i 0.935731 + 0.352714i \(0.114741\pi\)
−0.935731 + 0.352714i \(0.885259\pi\)
\(422\) −120.000 −0.284360
\(423\) 5.00000 0.0118203
\(424\) −72.0000 −0.169811
\(425\) 600.000 1.41176
\(426\) 0 0
\(427\) 475.000 1.11241
\(428\) 203.647i 0.475810i
\(429\) 240.000 0.559441
\(430\) − 7.07107i − 0.0164443i
\(431\) − 509.117i − 1.18125i −0.806948 0.590623i \(-0.798882\pi\)
0.806948 0.590623i \(-0.201118\pi\)
\(432\) 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
\(435\) 120.000 0.275862
\(436\) − 254.558i − 0.583850i
\(437\) −190.000 −0.434783
\(438\) 100.000 0.228311
\(439\) 806.102i 1.83622i 0.396323 + 0.918111i \(0.370286\pi\)
−0.396323 + 0.918111i \(0.629714\pi\)
\(440\) 14.1421i 0.0321412i
\(441\) −24.0000 −0.0544218
\(442\) −600.000 −1.35747
\(443\) 365.000 0.823928 0.411964 0.911200i \(-0.364843\pi\)
0.411964 + 0.911200i \(0.364843\pi\)
\(444\) −144.000 −0.324324
\(445\) 127.279i 0.286021i
\(446\) 516.000 1.15695
\(447\) 608.112i 1.36043i
\(448\) −40.0000 −0.0892857
\(449\) − 763.675i − 1.70084i −0.526108 0.850418i \(-0.676349\pi\)
0.526108 0.850418i \(-0.323651\pi\)
\(450\) − 33.9411i − 0.0754247i
\(451\) − 212.132i − 0.470359i
\(452\) − 220.617i − 0.488091i
\(453\) 240.000 0.529801
\(454\) −96.0000 −0.211454
\(455\) 84.8528i 0.186490i
\(456\) 152.000 0.333333
\(457\) −265.000 −0.579869 −0.289934 0.957047i \(-0.593633\pi\)
−0.289934 + 0.957047i \(0.593633\pi\)
\(458\) − 205.061i − 0.447731i
\(459\) − 707.107i − 1.54054i
\(460\) −20.0000 −0.0434783
\(461\) −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 \(-0.324530\pi\)
0.523758 + 0.851867i \(0.324530\pi\)
\(464\) 169.706i 0.365745i
\(465\) −120.000 −0.258065
\(466\) 473.762i 1.01666i
\(467\) −115.000 −0.246253 −0.123126 0.992391i \(-0.539292\pi\)
−0.123126 + 0.992391i \(0.539292\pi\)
\(468\) 33.9411i 0.0725238i
\(469\) 551.543i 1.17600i
\(470\) − 7.07107i − 0.0150448i
\(471\) − 537.401i − 1.14098i
\(472\) 240.000 0.508475
\(473\) 25.0000 0.0528541
\(474\) − 169.706i − 0.358029i
\(475\) −456.000 −0.960000
\(476\) 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 \(0.670903\pi\)
\(480\) 16.0000 0.0333333
\(481\) −432.000 −0.898129
\(482\) −420.000 −0.871369
\(483\) − 141.421i − 0.292798i
\(484\) 192.000 0.396694
\(485\) − 16.9706i − 0.0349909i
\(486\) −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
\(490\) 33.9411i 0.0692676i
\(491\) −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
\(495\) −5.00000 −0.0101010
\(496\) − 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 \(0.338466\pi\)
\(500\) −98.0000 −0.196000
\(501\) −168.000 −0.335329
\(502\) 244.659i 0.487368i
\(503\) −250.000 −0.497018 −0.248509 0.968630i \(-0.579941\pi\)
−0.248509 + 0.968630i \(0.579941\pi\)
\(504\) − 14.1421i − 0.0280598i
\(505\) −50.0000 −0.0990099
\(506\) − 70.7107i − 0.139744i
\(507\) − 336.583i − 0.663871i
\(508\) − 458.205i − 0.901979i
\(509\) − 169.706i − 0.333410i −0.986007 0.166705i \(-0.946687\pi\)
0.986007 0.166705i \(-0.0533127\pi\)
\(510\) −100.000 −0.196078
\(511\) −125.000 −0.244618
\(512\) 22.6274i 0.0441942i
\(513\) 537.401i 1.04757i
\(514\) 96.0000 0.186770
\(515\) 16.9706i 0.0329525i
\(516\) − 28.2843i − 0.0548145i
\(517\) 25.0000 0.0483559
\(518\) 180.000 0.347490
\(519\) 528.000 1.01734
\(520\) 48.0000 0.0923077
\(521\) − 127.279i − 0.244298i −0.992512 0.122149i \(-0.961021\pi\)
0.992512 0.122149i \(-0.0389786\pi\)
\(522\) −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
\(525\) − 339.411i − 0.646498i
\(526\) − 502.046i − 0.954460i
\(527\) 1060.66i 2.01264i
\(528\) 56.5685i 0.107137i
\(529\) −429.000 −0.810964
\(530\) −36.0000 −0.0679245
\(531\) 84.8528i 0.159798i
\(532\) −190.000 −0.357143
\(533\) −720.000 −1.35084
\(534\) 509.117i 0.953402i
\(535\) 101.823i 0.190324i
\(536\) 312.000 0.582090
\(537\) 360.000 0.670391
\(538\) 540.000 1.00372
\(539\) −120.000 −0.222635
\(540\) 56.5685i 0.104757i
\(541\) −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
\(545\) − 127.279i − 0.233540i
\(546\) 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
\(550\) − 169.706i − 0.308556i
\(551\) 806.102i 1.46298i
\(552\) −80.0000 −0.144928
\(553\) 212.132i 0.383602i
\(554\) − 374.767i − 0.676474i
\(555\) −72.0000 −0.129730
\(556\) −250.000 −0.449640
\(557\) −745.000 −1.33752 −0.668761 0.743477i \(-0.733175\pi\)
−0.668761 + 0.743477i \(0.733175\pi\)
\(558\) 60.0000 0.107527
\(559\) − 84.8528i − 0.151794i
\(560\) −20.0000 −0.0357143
\(561\) − 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
\(565\) − 110.309i − 0.195237i
\(566\) 176.777i 0.312326i
\(567\) −355.000 −0.626102
\(568\) 0 0
\(569\) 424.264i 0.745631i 0.927905 + 0.372816i \(0.121608\pi\)
−0.927905 + 0.372816i \(0.878392\pi\)
\(570\) 76.0000 0.133333
\(571\) 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
\(575\) 240.000 0.417391
\(576\) −8.00000 −0.0138889
\(577\) −25.0000 −0.0433276 −0.0216638 0.999765i \(-0.506896\pi\)
−0.0216638 + 0.999765i \(0.506896\pi\)
\(578\) 475.176i 0.822103i
\(579\) −168.000 −0.290155
\(580\) 84.8528i 0.146298i
\(581\) −650.000 −1.11876
\(582\) − 67.8823i − 0.116636i
\(583\) − 127.279i − 0.218318i
\(584\) 70.7107i 0.121080i
\(585\) 16.9706i 0.0290095i
\(586\) −264.000 −0.450512
\(587\) 725.000 1.23509 0.617547 0.786534i \(-0.288127\pi\)
0.617547 + 0.786534i \(0.288127\pi\)
\(588\) 135.765i 0.230892i
\(589\) − 806.102i − 1.36859i
\(590\) 120.000 0.203390
\(591\) − 197.990i − 0.335008i
\(592\) − 101.823i − 0.171999i
\(593\) 650.000 1.09612 0.548061 0.836439i \(-0.315366\pi\)
0.548061 + 0.836439i \(0.315366\pi\)
\(594\) −200.000 −0.336700
\(595\) 125.000 0.210084
\(596\) −430.000 −0.721477
\(597\) 489.318i 0.819628i
\(598\) −240.000 −0.401338
\(599\) − 296.985i − 0.495801i −0.968785 0.247901i \(-0.920259\pi\)
0.968785 0.247901i \(-0.0797406\pi\)
\(600\) −192.000 −0.320000
\(601\) − 848.528i − 1.41186i −0.708281 0.705930i \(-0.750529\pi\)
0.708281 0.705930i \(-0.249471\pi\)
\(602\) 35.3553i 0.0587298i
\(603\) 110.309i 0.182933i
\(604\) 169.706i 0.280970i
\(605\) 96.0000 0.158678
\(606\) −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
\(610\) − 134.350i − 0.220246i
\(611\) − 84.8528i − 0.138875i
\(612\) 50.0000 0.0816993
\(613\) 1055.00 1.72104 0.860522 0.509413i \(-0.170138\pi\)
0.860522 + 0.509413i \(0.170138\pi\)
\(614\) −396.000 −0.644951
\(615\) −120.000 −0.195122
\(616\) − 70.7107i − 0.114790i
\(617\) −505.000 −0.818476 −0.409238 0.912428i \(-0.634206\pi\)
−0.409238 + 0.912428i \(0.634206\pi\)
\(618\) 67.8823i 0.109842i
\(619\) −130.000 −0.210016 −0.105008 0.994471i \(-0.533487\pi\)
−0.105008 + 0.994471i \(0.533487\pi\)
\(620\) − 84.8528i − 0.136859i
\(621\) − 282.843i − 0.455463i
\(622\) − 332.340i − 0.534309i
\(623\) − 636.396i − 1.02150i
\(624\) 192.000 0.307692
\(625\) 551.000 0.881600
\(626\) − 438.406i − 0.700329i
\(627\) 268.701i 0.428550i
\(628\) 380.000 0.605096
\(629\) 636.396i 1.01176i
\(630\) − 7.07107i − 0.0112239i
\(631\) −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
\(635\) − 229.103i − 0.360791i
\(636\) −144.000 −0.226415
\(637\) 407.294i 0.639393i
\(638\) −300.000 −0.470219
\(639\) 0 0
\(640\) 11.3137i 0.0176777i
\(641\) − 848.528i − 1.32376i −0.749611 0.661878i \(-0.769760\pi\)
0.749611 0.661878i \(-0.230240\pi\)
\(642\) 407.294i 0.634414i
\(643\) −955.000 −1.48523 −0.742613 0.669721i \(-0.766414\pi\)
−0.742613 + 0.669721i \(0.766414\pi\)
\(644\) 100.000 0.155280
\(645\) − 14.1421i − 0.0219258i
\(646\) − 671.751i − 1.03986i
\(647\) 965.000 1.49150 0.745750 0.666226i \(-0.232092\pi\)
0.745750 + 0.666226i \(0.232092\pi\)
\(648\) 200.818i 0.309905i
\(649\) 424.264i 0.653720i
\(650\) −576.000 −0.886154
\(651\) 600.000 0.921659
\(652\) −220.000 −0.337423
\(653\) 935.000 1.43185 0.715926 0.698176i \(-0.246005\pi\)
0.715926 + 0.698176i \(0.246005\pi\)
\(654\) − 509.117i − 0.778466i
\(655\) 163.000 0.248855
\(656\) − 169.706i − 0.258698i
\(657\) −25.0000 −0.0380518
\(658\) 35.3553i 0.0537315i
\(659\) − 84.8528i − 0.128760i −0.997925 0.0643800i \(-0.979493\pi\)
0.997925 0.0643800i \(-0.0205070\pi\)
\(660\) 28.2843i 0.0428550i
\(661\) − 678.823i − 1.02696i −0.858101 0.513481i \(-0.828356\pi\)
0.858101 0.513481i \(-0.171644\pi\)
\(662\) −420.000 −0.634441
\(663\) −1200.00 −1.80995
\(664\) 367.696i 0.553758i
\(665\) −95.0000 −0.142857
\(666\) 36.0000 0.0540541
\(667\) − 424.264i − 0.636078i
\(668\) − 118.794i − 0.177835i
\(669\) 1032.00 1.54260
\(670\) 156.000 0.232836
\(671\) 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
\(675\) − 678.823i − 1.00566i
\(676\) 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
\(680\) − 70.7107i − 0.103986i
\(681\) −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
\(685\) −95.0000 −0.138686
\(686\) − 516.188i − 0.752461i
\(687\) − 410.122i − 0.596975i
\(688\) 20.0000 0.0290698
\(689\) −432.000 −0.626996
\(690\) −40.0000 −0.0579710
\(691\) −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
\(695\) −125.000 −0.179856
\(696\) 339.411i 0.487660i
\(697\) 1060.66i 1.52175i
\(698\) 32.5269i 0.0466002i
\(699\) 947.523i 1.35554i
\(700\) 240.000 0.342857
\(701\) −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
\(705\) − 14.1421i − 0.0200598i
\(706\) 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
\(715\) 84.8528i 0.118675i
\(716\) 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 \(0.525483\pi\)
\(720\) −4.00000 −0.00555556
\(721\) − 84.8528i − 0.117688i
\(722\) 510.531i 0.707107i
\(723\) −840.000 −1.16183
\(724\) 509.117i 0.703200i
\(725\) − 1018.23i − 1.40446i
\(726\) 384.000 0.528926
\(727\) −1075.00 −1.47868 −0.739340 0.673333i \(-0.764862\pi\)
−0.739340 + 0.673333i \(0.764862\pi\)
\(728\) −240.000 −0.329670
\(729\) −791.000 −1.08505
\(730\) 35.3553i 0.0484320i
\(731\) −125.000 −0.170999
\(732\) − 537.401i − 0.734155i
\(733\) 530.000 0.723056 0.361528 0.932361i \(-0.382255\pi\)
0.361528 + 0.932361i \(0.382255\pi\)
\(734\) 325.269i 0.443146i
\(735\) 67.8823i 0.0923568i
\(736\) − 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 \(0.620675\pi\)
\(740\) − 50.9117i − 0.0687996i
\(741\) 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
\(745\) −215.000 −0.288591
\(746\) 96.0000 0.128686
\(747\) −130.000 −0.174029
\(748\) 250.000 0.334225
\(749\) − 509.117i − 0.679729i
\(750\) −196.000 −0.261333
\(751\) − 169.706i − 0.225973i −0.993597 0.112986i \(-0.963958\pi\)
0.993597 0.112986i \(-0.0360417\pi\)
\(752\) 20.0000 0.0265957
\(753\) 489.318i 0.649825i
\(754\) 1018.23i 1.35044i
\(755\) 84.8528i 0.112388i
\(756\) − 282.843i − 0.374131i
\(757\) 1055.00 1.39366 0.696830 0.717237i \(-0.254593\pi\)
0.696830 + 0.717237i \(0.254593\pi\)
\(758\) 360.000 0.474934
\(759\) − 141.421i − 0.186326i
\(760\) 53.7401i 0.0707107i
\(761\) 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
\(765\) 25.0000 0.0326797
\(766\) 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 \(0.530054\pi\)
\(770\) − 35.3553i − 0.0459160i
\(771\) 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
\(775\) 1018.23i 1.31385i
\(776\) 48.0000 0.0618557
\(777\) 360.000 0.463320
\(778\) − 782.060i − 1.00522i
\(779\) − 806.102i − 1.03479i
\(780\) 96.0000 0.123077
\(781\) 0 0
\(782\) 353.553i 0.452114i
\(783\) −1200.00 −1.53257
\(784\) −96.0000 −0.122449
\(785\) 190.000 0.242038
\(786\) 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
\(790\) 60.0000 0.0759494
\(791\) 551.543i 0.697273i
\(792\) − 14.1421i − 0.0178562i
\(793\) − 1612.20i − 2.03304i
\(794\) 473.762i 0.596677i
\(795\) −72.0000 −0.0905660
\(796\) −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
\(800\) − 135.765i − 0.169706i
\(801\) − 127.279i − 0.158900i
\(802\) −300.000 −0.374065
\(803\) −125.000 −0.155666
\(804\) 624.000 0.776119
\(805\) 50.0000 0.0621118
\(806\) − 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 \(0.591146\pi\)
\(810\) 100.409i 0.123962i
\(811\) 509.117i 0.627764i 0.949462 + 0.313882i \(0.101630\pi\)
−0.949462 + 0.313882i \(0.898370\pi\)
\(812\) − 424.264i − 0.522493i
\(813\) 311.127i 0.382690i
\(814\) 180.000 0.221130
\(815\) −110.000 −0.134969
\(816\) − 282.843i − 0.346621i
\(817\) 95.0000 0.116279
\(818\) −1020.00 −1.24694
\(819\) − 84.8528i − 0.103605i
\(820\) − 84.8528i − 0.103479i
\(821\) 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 \(-0.794588\pi\)
−0.798906 + 0.601455i \(0.794588\pi\)
\(824\) −48.0000 −0.0582524
\(825\) − 339.411i − 0.411408i
\(826\) −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 \(0.152366\pi\)
−0.887608 + 0.460600i \(0.847634\pi\)
\(830\) 183.848i 0.221503i
\(831\) − 749.533i − 0.901965i
\(832\) 135.765i 0.163178i
\(833\) 600.000 0.720288
\(834\) −500.000 −0.599520
\(835\) − 59.3970i − 0.0711341i
\(836\) −190.000 −0.227273
\(837\) 1200.00 1.43369
\(838\) 87.6812i 0.104632i
\(839\) − 339.411i − 0.404543i −0.979330 0.202271i \(-0.935168\pi\)
0.979330 0.202271i \(-0.0648323\pi\)
\(840\) −40.0000 −0.0476190
\(841\) −959.000 −1.14031
\(842\) −420.000 −0.498812
\(843\) −1200.00 −1.42349
\(844\) − 169.706i − 0.201073i
\(845\) 119.000 0.140828
\(846\) 7.07107i 0.00835824i
\(847\) −480.000 −0.566706
\(848\) − 101.823i − 0.120075i
\(849\) 353.553i 0.416435i
\(850\) 848.528i 0.998268i
\(851\) 254.558i 0.299129i
\(852\) 0 0
\(853\) 770.000 0.902696 0.451348 0.892348i \(-0.350943\pi\)
0.451348 + 0.892348i \(0.350943\pi\)
\(854\) 671.751i 0.786594i
\(855\) −19.0000 −0.0222222
\(856\) −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 \(0.394900\pi\)
\(860\) 10.0000 0.0116279
\(861\) 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
\(865\) 186.676i 0.215811i
\(866\) 324.000 0.374134
\(867\) 950.352i 1.09614i
\(868\) 424.264i 0.488783i
\(869\) 212.132i 0.244111i
\(870\) 169.706i 0.195064i
\(871\) 1872.00 2.14925
\(872\) 360.000 0.412844
\(873\) 16.9706i 0.0194394i
\(874\) − 268.701i − 0.307438i
\(875\) 245.000 0.280000
\(876\) 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
\(880\) −20.0000 −0.0227273
\(881\) −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 \(-0.315985\pi\)
0.546433 + 0.837503i \(0.315985\pi\)
\(884\) − 848.528i − 0.959873i
\(885\) 240.000 0.271186
\(886\) 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
\(890\) −180.000 −0.202247
\(891\) −355.000 −0.398429
\(892\) 729.734i 0.818088i
\(893\) 95.0000 0.106383
\(894\) −860.000 −0.961969
\(895\) 127.279i 0.142211i
\(896\) − 56.5685i − 0.0631345i
\(897\) −480.000 −0.535117
\(898\) 1080.00 1.20267
\(899\) 1800.00 2.00222
\(900\) 48.0000 0.0533333
\(901\) 636.396i 0.706322i
\(902\) 300.000 0.332594
\(903\) 70.7107i 0.0783064i
\(904\) 312.000 0.345133
\(905\) 254.558i 0.281280i
\(906\) 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
\(910\) −120.000 −0.131868
\(911\) 933.381i 1.02457i 0.858816 + 0.512284i \(0.171200\pi\)
−0.858816 + 0.512284i \(0.828800\pi\)
\(912\) 214.960i 0.235702i
\(913\) −650.000 −0.711939
\(914\) − 374.767i − 0.410029i
\(915\) − 268.701i − 0.293662i
\(916\) 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 \(0.594557\pi\)
\(920\) − 28.2843i − 0.0307438i
\(921\) −792.000 −0.859935
\(922\) − 782.060i − 0.848221i
\(923\) 0 0
\(924\) − 141.421i − 0.153053i
\(925\) 610.940i 0.660476i
\(926\) 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 \(0.630766\pi\)
\(930\) − 169.706i − 0.182479i
\(931\) −456.000 −0.489796
\(932\) −670.000 −0.718884
\(933\) − 664.680i − 0.712412i
\(934\) − 162.635i − 0.174127i
\(935\) 125.000 0.133690
\(936\) −48.0000 −0.0512821
\(937\) 335.000 0.357524 0.178762 0.983892i \(-0.442791\pi\)
0.178762 + 0.983892i \(0.442791\pi\)
\(938\) −780.000 −0.831557
\(939\) − 876.812i − 0.933773i
\(940\) 10.0000 0.0106383
\(941\) − 424.264i − 0.450865i −0.974259 0.225433i \(-0.927620\pi\)
0.974259 0.225433i \(-0.0723795\pi\)
\(942\) 760.000 0.806794
\(943\) 424.264i 0.449909i
\(944\) 339.411i 0.359546i
\(945\) − 141.421i − 0.149652i
\(946\) 35.3553i 0.0373735i
\(947\) −1210.00 −1.27772 −0.638860 0.769323i \(-0.720594\pi\)
−0.638860 + 0.769323i \(0.720594\pi\)
\(948\) 240.000 0.253165
\(949\) 424.264i 0.447064i
\(950\) − 644.881i − 0.678823i
\(951\) 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
\(955\) −293.000 −0.306806
\(956\) −394.000 −0.412134
\(957\) −600.000 −0.626959
\(958\) − 692.965i − 0.723345i
\(959\) 475.000 0.495308
\(960\) 22.6274i 0.0235702i
\(961\) −839.000 −0.873049
\(962\) − 610.940i − 0.635073i
\(963\) − 101.823i − 0.105736i
\(964\) − 593.970i − 0.616151i
\(965\) − 59.3970i − 0.0615513i
\(966\) 200.000 0.207039
\(967\) 350.000 0.361944 0.180972 0.983488i \(-0.442076\pi\)
0.180972 + 0.983488i \(0.442076\pi\)
\(968\) 271.529i 0.280505i
\(969\) − 1343.50i − 1.38648i
\(970\) 24.0000 0.0247423
\(971\) 254.558i 0.262161i 0.991372 + 0.131081i \(0.0418447\pi\)
−0.991372 + 0.131081i \(0.958155\pi\)
\(972\) − 107.480i − 0.110576i
\(973\) 625.000 0.642343
\(974\) 864.000 0.887064
\(975\) −1152.00 −1.18154
\(976\) 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
\(980\) −48.0000 −0.0489796
\(981\) 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
\(985\) 70.0000 0.0710660
\(986\) 1500.00 1.52130
\(987\) 70.7107i 0.0716420i
\(988\) 644.881i 0.652714i
\(989\) −50.0000 −0.0505561
\(990\) − 7.07107i − 0.00714249i
\(991\) 381.838i 0.385305i 0.981267 + 0.192653i \(0.0617091\pi\)
−0.981267 + 0.192653i \(0.938291\pi\)
\(992\) 240.000 0.241935
\(993\) −840.000 −0.845921
\(994\) 0 0
\(995\) −173.000 −0.173869
\(996\) 735.391i 0.738344i
\(997\) −265.000 −0.265797 −0.132899 0.991130i \(-0.542428\pi\)
−0.132899 + 0.991130i \(0.542428\pi\)
\(998\) 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 38.3.b.a.37.2 yes 2
3.2 odd 2 342.3.d.a.37.1 2
4.3 odd 2 304.3.e.c.113.1 2
5.2 odd 4 950.3.d.a.949.2 4
5.3 odd 4 950.3.d.a.949.3 4
5.4 even 2 950.3.c.a.151.1 2
8.3 odd 2 1216.3.e.i.1025.2 2
8.5 even 2 1216.3.e.j.1025.1 2
12.11 even 2 2736.3.o.h.721.1 2
19.18 odd 2 inner 38.3.b.a.37.1 2
57.56 even 2 342.3.d.a.37.2 2
76.75 even 2 304.3.e.c.113.2 2
95.18 even 4 950.3.d.a.949.1 4
95.37 even 4 950.3.d.a.949.4 4
95.94 odd 2 950.3.c.a.151.2 2
152.37 odd 2 1216.3.e.j.1025.2 2
152.75 even 2 1216.3.e.i.1025.1 2
228.227 odd 2 2736.3.o.h.721.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.3.b.a.37.1 2 19.18 odd 2 inner
38.3.b.a.37.2 yes 2 1.1 even 1 trivial
304.3.e.c.113.1 2 4.3 odd 2
304.3.e.c.113.2 2 76.75 even 2
342.3.d.a.37.1 2 3.2 odd 2
342.3.d.a.37.2 2 57.56 even 2
950.3.c.a.151.1 2 5.4 even 2
950.3.c.a.151.2 2 95.94 odd 2
950.3.d.a.949.1 4 95.18 even 4
950.3.d.a.949.2 4 5.2 odd 4
950.3.d.a.949.3 4 5.3 odd 4
950.3.d.a.949.4 4 95.37 even 4
1216.3.e.i.1025.1 2 152.75 even 2
1216.3.e.i.1025.2 2 8.3 odd 2
1216.3.e.j.1025.1 2 8.5 even 2
1216.3.e.j.1025.2 2 152.37 odd 2
2736.3.o.h.721.1 2 12.11 even 2
2736.3.o.h.721.2 2 228.227 odd 2