Properties

Label 950.3.d.a.949.1
Level $950$
Weight $3$
Character 950.949
Analytic conductor $25.886$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 950.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.8856251142\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 38)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 949.1
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 950.949
Dual form 950.3.d.a.949.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421 q^{2} +2.82843 q^{3} +2.00000 q^{4} -4.00000 q^{6} -5.00000i q^{7} -2.82843 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.41421 q^{2} +2.82843 q^{3} +2.00000 q^{4} -4.00000 q^{6} -5.00000i q^{7} -2.82843 q^{8} -1.00000 q^{9} +5.00000 q^{11} +5.65685 q^{12} -16.9706 q^{13} +7.07107i q^{14} +4.00000 q^{16} +25.0000i q^{17} +1.41421 q^{18} -19.0000 q^{19} -14.1421i q^{21} -7.07107 q^{22} -10.0000i q^{23} -8.00000 q^{24} +24.0000 q^{26} -28.2843 q^{27} -10.0000i q^{28} +42.4264i q^{29} +42.4264i q^{31} -5.65685 q^{32} +14.1421 q^{33} -35.3553i q^{34} -2.00000 q^{36} +25.4558 q^{37} +26.8701 q^{38} -48.0000 q^{39} +42.4264i q^{41} +20.0000i q^{42} +5.00000i q^{43} +10.0000 q^{44} +14.1421i q^{46} -5.00000i q^{47} +11.3137 q^{48} +24.0000 q^{49} +70.7107i q^{51} -33.9411 q^{52} -25.4558 q^{53} +40.0000 q^{54} +14.1421i q^{56} -53.7401 q^{57} -60.0000i q^{58} +84.8528i q^{59} +95.0000 q^{61} -60.0000i q^{62} +5.00000i q^{63} +8.00000 q^{64} -20.0000 q^{66} -110.309 q^{67} +50.0000i q^{68} -28.2843i q^{69} +2.82843 q^{72} -25.0000i q^{73} -36.0000 q^{74} -38.0000 q^{76} -25.0000i q^{77} +67.8823 q^{78} +42.4264i q^{79} -71.0000 q^{81} -60.0000i q^{82} -130.000i q^{83} -28.2843i q^{84} -7.07107i q^{86} +120.000i q^{87} -14.1421 q^{88} -127.279i q^{89} +84.8528i q^{91} -20.0000i q^{92} +120.000i q^{93} +7.07107i q^{94} -16.0000 q^{96} -16.9706 q^{97} -33.9411 q^{98} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{4} - 16q^{6} - 4q^{9} + O(q^{10}) \) \( 4q + 8q^{4} - 16q^{6} - 4q^{9} + 20q^{11} + 16q^{16} - 76q^{19} - 32q^{24} + 96q^{26} - 8q^{36} - 192q^{39} + 40q^{44} + 96q^{49} + 160q^{54} + 380q^{61} + 32q^{64} - 80q^{66} - 144q^{74} - 152q^{76} - 284q^{81} - 64q^{96} - 20q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −0.707107
\(3\) 2.82843 0.942809 0.471405 0.881917i \(-0.343747\pi\)
0.471405 + 0.881917i \(0.343747\pi\)
\(4\) 2.00000 0.500000
\(5\) 0 0
\(6\) −4.00000 −0.666667
\(7\) − 5.00000i − 0.714286i −0.934050 0.357143i \(-0.883751\pi\)
0.934050 0.357143i \(-0.116249\pi\)
\(8\) −2.82843 −0.353553
\(9\) −1.00000 −0.111111
\(10\) 0 0
\(11\) 5.00000 0.454545 0.227273 0.973831i \(-0.427019\pi\)
0.227273 + 0.973831i \(0.427019\pi\)
\(12\) 5.65685 0.471405
\(13\) −16.9706 −1.30543 −0.652714 0.757604i \(-0.726370\pi\)
−0.652714 + 0.757604i \(0.726370\pi\)
\(14\) 7.07107i 0.505076i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 25.0000i 1.47059i 0.677748 + 0.735294i \(0.262956\pi\)
−0.677748 + 0.735294i \(0.737044\pi\)
\(18\) 1.41421 0.0785674
\(19\) −19.0000 −1.00000
\(20\) 0 0
\(21\) − 14.1421i − 0.673435i
\(22\) −7.07107 −0.321412
\(23\) − 10.0000i − 0.434783i −0.976085 0.217391i \(-0.930245\pi\)
0.976085 0.217391i \(-0.0697548\pi\)
\(24\) −8.00000 −0.333333
\(25\) 0 0
\(26\) 24.0000 0.923077
\(27\) −28.2843 −1.04757
\(28\) − 10.0000i − 0.357143i
\(29\) 42.4264i 1.46298i 0.681852 + 0.731490i \(0.261175\pi\)
−0.681852 + 0.731490i \(0.738825\pi\)
\(30\) 0 0
\(31\) 42.4264i 1.36859i 0.729204 + 0.684297i \(0.239891\pi\)
−0.729204 + 0.684297i \(0.760109\pi\)
\(32\) −5.65685 −0.176777
\(33\) 14.1421 0.428550
\(34\) − 35.3553i − 1.03986i
\(35\) 0 0
\(36\) −2.00000 −0.0555556
\(37\) 25.4558 0.687996 0.343998 0.938970i \(-0.388219\pi\)
0.343998 + 0.938970i \(0.388219\pi\)
\(38\) 26.8701 0.707107
\(39\) −48.0000 −1.23077
\(40\) 0 0
\(41\) 42.4264i 1.03479i 0.855747 + 0.517395i \(0.173098\pi\)
−0.855747 + 0.517395i \(0.826902\pi\)
\(42\) 20.0000i 0.476190i
\(43\) 5.00000i 0.116279i 0.998308 + 0.0581395i \(0.0185168\pi\)
−0.998308 + 0.0581395i \(0.981483\pi\)
\(44\) 10.0000 0.227273
\(45\) 0 0
\(46\) 14.1421i 0.307438i
\(47\) − 5.00000i − 0.106383i −0.998584 0.0531915i \(-0.983061\pi\)
0.998584 0.0531915i \(-0.0169394\pi\)
\(48\) 11.3137 0.235702
\(49\) 24.0000 0.489796
\(50\) 0 0
\(51\) 70.7107i 1.38648i
\(52\) −33.9411 −0.652714
\(53\) −25.4558 −0.480299 −0.240149 0.970736i \(-0.577196\pi\)
−0.240149 + 0.970736i \(0.577196\pi\)
\(54\) 40.0000 0.740741
\(55\) 0 0
\(56\) 14.1421i 0.252538i
\(57\) −53.7401 −0.942809
\(58\) − 60.0000i − 1.03448i
\(59\) 84.8528i 1.43818i 0.694915 + 0.719092i \(0.255442\pi\)
−0.694915 + 0.719092i \(0.744558\pi\)
\(60\) 0 0
\(61\) 95.0000 1.55738 0.778689 0.627411i \(-0.215885\pi\)
0.778689 + 0.627411i \(0.215885\pi\)
\(62\) − 60.0000i − 0.967742i
\(63\) 5.00000i 0.0793651i
\(64\) 8.00000 0.125000
\(65\) 0 0
\(66\) −20.0000 −0.303030
\(67\) −110.309 −1.64640 −0.823199 0.567753i \(-0.807813\pi\)
−0.823199 + 0.567753i \(0.807813\pi\)
\(68\) 50.0000i 0.735294i
\(69\) − 28.2843i − 0.409917i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 2.82843 0.0392837
\(73\) − 25.0000i − 0.342466i −0.985231 0.171233i \(-0.945225\pi\)
0.985231 0.171233i \(-0.0547750\pi\)
\(74\) −36.0000 −0.486486
\(75\) 0 0
\(76\) −38.0000 −0.500000
\(77\) − 25.0000i − 0.324675i
\(78\) 67.8823 0.870285
\(79\) 42.4264i 0.537043i 0.963274 + 0.268522i \(0.0865351\pi\)
−0.963274 + 0.268522i \(0.913465\pi\)
\(80\) 0 0
\(81\) −71.0000 −0.876543
\(82\) − 60.0000i − 0.731707i
\(83\) − 130.000i − 1.56627i −0.621855 0.783133i \(-0.713621\pi\)
0.621855 0.783133i \(-0.286379\pi\)
\(84\) − 28.2843i − 0.336718i
\(85\) 0 0
\(86\) − 7.07107i − 0.0822217i
\(87\) 120.000i 1.37931i
\(88\) −14.1421 −0.160706
\(89\) − 127.279i − 1.43010i −0.699071 0.715052i \(-0.746403\pi\)
0.699071 0.715052i \(-0.253597\pi\)
\(90\) 0 0
\(91\) 84.8528i 0.932449i
\(92\) − 20.0000i − 0.217391i
\(93\) 120.000i 1.29032i
\(94\) 7.07107i 0.0752241i
\(95\) 0 0
\(96\) −16.0000 −0.166667
\(97\) −16.9706 −0.174954 −0.0874771 0.996167i \(-0.527880\pi\)
−0.0874771 + 0.996167i \(0.527880\pi\)
\(98\) −33.9411 −0.346338
\(99\) −5.00000 −0.0505051
\(100\) 0 0
\(101\) 50.0000 0.495050 0.247525 0.968882i \(-0.420383\pi\)
0.247525 + 0.968882i \(0.420383\pi\)
\(102\) − 100.000i − 0.980392i
\(103\) −16.9706 −0.164763 −0.0823814 0.996601i \(-0.526253\pi\)
−0.0823814 + 0.996601i \(0.526253\pi\)
\(104\) 48.0000 0.461538
\(105\) 0 0
\(106\) 36.0000 0.339623
\(107\) 101.823 0.951620 0.475810 0.879548i \(-0.342155\pi\)
0.475810 + 0.879548i \(0.342155\pi\)
\(108\) −56.5685 −0.523783
\(109\) 127.279i 1.16770i 0.811862 + 0.583850i \(0.198454\pi\)
−0.811862 + 0.583850i \(0.801546\pi\)
\(110\) 0 0
\(111\) 72.0000 0.648649
\(112\) − 20.0000i − 0.178571i
\(113\) 110.309 0.976183 0.488091 0.872793i \(-0.337693\pi\)
0.488091 + 0.872793i \(0.337693\pi\)
\(114\) 76.0000 0.666667
\(115\) 0 0
\(116\) 84.8528i 0.731490i
\(117\) 16.9706 0.145048
\(118\) − 120.000i − 1.01695i
\(119\) 125.000 1.05042
\(120\) 0 0
\(121\) −96.0000 −0.793388
\(122\) −134.350 −1.10123
\(123\) 120.000i 0.975610i
\(124\) 84.8528i 0.684297i
\(125\) 0 0
\(126\) − 7.07107i − 0.0561196i
\(127\) −229.103 −1.80396 −0.901979 0.431780i \(-0.857886\pi\)
−0.901979 + 0.431780i \(0.857886\pi\)
\(128\) −11.3137 −0.0883883
\(129\) 14.1421i 0.109629i
\(130\) 0 0
\(131\) −163.000 −1.24427 −0.622137 0.782908i \(-0.713735\pi\)
−0.622137 + 0.782908i \(0.713735\pi\)
\(132\) 28.2843 0.214275
\(133\) 95.0000i 0.714286i
\(134\) 156.000 1.16418
\(135\) 0 0
\(136\) − 70.7107i − 0.519931i
\(137\) − 95.0000i − 0.693431i −0.937970 0.346715i \(-0.887297\pi\)
0.937970 0.346715i \(-0.112703\pi\)
\(138\) 40.0000i 0.289855i
\(139\) −125.000 −0.899281 −0.449640 0.893210i \(-0.648448\pi\)
−0.449640 + 0.893210i \(0.648448\pi\)
\(140\) 0 0
\(141\) − 14.1421i − 0.100299i
\(142\) 0 0
\(143\) −84.8528 −0.593376
\(144\) −4.00000 −0.0277778
\(145\) 0 0
\(146\) 35.3553i 0.242160i
\(147\) 67.8823 0.461784
\(148\) 50.9117 0.343998
\(149\) −215.000 −1.44295 −0.721477 0.692439i \(-0.756536\pi\)
−0.721477 + 0.692439i \(0.756536\pi\)
\(150\) 0 0
\(151\) 84.8528i 0.561939i 0.959717 + 0.280970i \(0.0906560\pi\)
−0.959717 + 0.280970i \(0.909344\pi\)
\(152\) 53.7401 0.353553
\(153\) − 25.0000i − 0.163399i
\(154\) 35.3553i 0.229580i
\(155\) 0 0
\(156\) −96.0000 −0.615385
\(157\) 190.000i 1.21019i 0.796153 + 0.605096i \(0.206865\pi\)
−0.796153 + 0.605096i \(0.793135\pi\)
\(158\) − 60.0000i − 0.379747i
\(159\) −72.0000 −0.452830
\(160\) 0 0
\(161\) −50.0000 −0.310559
\(162\) 100.409 0.619810
\(163\) 110.000i 0.674847i 0.941353 + 0.337423i \(0.109555\pi\)
−0.941353 + 0.337423i \(0.890445\pi\)
\(164\) 84.8528i 0.517395i
\(165\) 0 0
\(166\) 183.848i 1.10752i
\(167\) −59.3970 −0.355670 −0.177835 0.984060i \(-0.556909\pi\)
−0.177835 + 0.984060i \(0.556909\pi\)
\(168\) 40.0000i 0.238095i
\(169\) 119.000 0.704142
\(170\) 0 0
\(171\) 19.0000 0.111111
\(172\) 10.0000i 0.0581395i
\(173\) −186.676 −1.07905 −0.539527 0.841969i \(-0.681397\pi\)
−0.539527 + 0.841969i \(0.681397\pi\)
\(174\) − 169.706i − 0.975320i
\(175\) 0 0
\(176\) 20.0000 0.113636
\(177\) 240.000i 1.35593i
\(178\) 180.000i 1.01124i
\(179\) − 127.279i − 0.711057i −0.934665 0.355529i \(-0.884301\pi\)
0.934665 0.355529i \(-0.115699\pi\)
\(180\) 0 0
\(181\) 254.558i 1.40640i 0.710992 + 0.703200i \(0.248246\pi\)
−0.710992 + 0.703200i \(0.751754\pi\)
\(182\) − 120.000i − 0.659341i
\(183\) 268.701 1.46831
\(184\) 28.2843i 0.153719i
\(185\) 0 0
\(186\) − 169.706i − 0.912396i
\(187\) 125.000i 0.668449i
\(188\) − 10.0000i − 0.0531915i
\(189\) 141.421i 0.748261i
\(190\) 0 0
\(191\) 293.000 1.53403 0.767016 0.641628i \(-0.221741\pi\)
0.767016 + 0.641628i \(0.221741\pi\)
\(192\) 22.6274 0.117851
\(193\) 59.3970 0.307756 0.153878 0.988090i \(-0.450824\pi\)
0.153878 + 0.988090i \(0.450824\pi\)
\(194\) 24.0000 0.123711
\(195\) 0 0
\(196\) 48.0000 0.244898
\(197\) 70.0000i 0.355330i 0.984091 + 0.177665i \(0.0568543\pi\)
−0.984091 + 0.177665i \(0.943146\pi\)
\(198\) 7.07107 0.0357125
\(199\) −173.000 −0.869347 −0.434673 0.900588i \(-0.643136\pi\)
−0.434673 + 0.900588i \(0.643136\pi\)
\(200\) 0 0
\(201\) −312.000 −1.55224
\(202\) −70.7107 −0.350053
\(203\) 212.132 1.04499
\(204\) 141.421i 0.693242i
\(205\) 0 0
\(206\) 24.0000 0.116505
\(207\) 10.0000i 0.0483092i
\(208\) −67.8823 −0.326357
\(209\) −95.0000 −0.454545
\(210\) 0 0
\(211\) − 84.8528i − 0.402146i −0.979576 0.201073i \(-0.935557\pi\)
0.979576 0.201073i \(-0.0644429\pi\)
\(212\) −50.9117 −0.240149
\(213\) 0 0
\(214\) −144.000 −0.672897
\(215\) 0 0
\(216\) 80.0000 0.370370
\(217\) 212.132 0.977567
\(218\) − 180.000i − 0.825688i
\(219\) − 70.7107i − 0.322880i
\(220\) 0 0
\(221\) − 424.264i − 1.91975i
\(222\) −101.823 −0.458664
\(223\) −364.867 −1.63618 −0.818088 0.575094i \(-0.804966\pi\)
−0.818088 + 0.575094i \(0.804966\pi\)
\(224\) 28.2843i 0.126269i
\(225\) 0 0
\(226\) −156.000 −0.690265
\(227\) −67.8823 −0.299041 −0.149520 0.988759i \(-0.547773\pi\)
−0.149520 + 0.988759i \(0.547773\pi\)
\(228\) −107.480 −0.471405
\(229\) 145.000 0.633188 0.316594 0.948561i \(-0.397461\pi\)
0.316594 + 0.948561i \(0.397461\pi\)
\(230\) 0 0
\(231\) − 70.7107i − 0.306107i
\(232\) − 120.000i − 0.517241i
\(233\) 335.000i 1.43777i 0.695130 + 0.718884i \(0.255347\pi\)
−0.695130 + 0.718884i \(0.744653\pi\)
\(234\) −24.0000 −0.102564
\(235\) 0 0
\(236\) 169.706i 0.719092i
\(237\) 120.000i 0.506329i
\(238\) −176.777 −0.742759
\(239\) −197.000 −0.824268 −0.412134 0.911123i \(-0.635216\pi\)
−0.412134 + 0.911123i \(0.635216\pi\)
\(240\) 0 0
\(241\) − 296.985i − 1.23230i −0.787628 0.616151i \(-0.788691\pi\)
0.787628 0.616151i \(-0.211309\pi\)
\(242\) 135.765 0.561010
\(243\) 53.7401 0.221153
\(244\) 190.000 0.778689
\(245\) 0 0
\(246\) − 169.706i − 0.689860i
\(247\) 322.441 1.30543
\(248\) − 120.000i − 0.483871i
\(249\) − 367.696i − 1.47669i
\(250\) 0 0
\(251\) 173.000 0.689243 0.344622 0.938742i \(-0.388007\pi\)
0.344622 + 0.938742i \(0.388007\pi\)
\(252\) 10.0000i 0.0396825i
\(253\) − 50.0000i − 0.197628i
\(254\) 324.000 1.27559
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) 67.8823 0.264133 0.132067 0.991241i \(-0.457839\pi\)
0.132067 + 0.991241i \(0.457839\pi\)
\(258\) − 20.0000i − 0.0775194i
\(259\) − 127.279i − 0.491426i
\(260\) 0 0
\(261\) − 42.4264i − 0.162553i
\(262\) 230.517 0.879835
\(263\) − 355.000i − 1.34981i −0.737905 0.674905i \(-0.764185\pi\)
0.737905 0.674905i \(-0.235815\pi\)
\(264\) −40.0000 −0.151515
\(265\) 0 0
\(266\) − 134.350i − 0.505076i
\(267\) − 360.000i − 1.34831i
\(268\) −220.617 −0.823199
\(269\) − 381.838i − 1.41947i −0.704468 0.709735i \(-0.748814\pi\)
0.704468 0.709735i \(-0.251186\pi\)
\(270\) 0 0
\(271\) 110.000 0.405904 0.202952 0.979189i \(-0.434946\pi\)
0.202952 + 0.979189i \(0.434946\pi\)
\(272\) 100.000i 0.367647i
\(273\) 240.000i 0.879121i
\(274\) 134.350i 0.490330i
\(275\) 0 0
\(276\) − 56.5685i − 0.204958i
\(277\) 265.000i 0.956679i 0.878175 + 0.478339i \(0.158761\pi\)
−0.878175 + 0.478339i \(0.841239\pi\)
\(278\) 176.777 0.635887
\(279\) − 42.4264i − 0.152066i
\(280\) 0 0
\(281\) − 424.264i − 1.50984i −0.655819 0.754918i \(-0.727677\pi\)
0.655819 0.754918i \(-0.272323\pi\)
\(282\) 20.0000i 0.0709220i
\(283\) 125.000i 0.441696i 0.975308 + 0.220848i \(0.0708825\pi\)
−0.975308 + 0.220848i \(0.929118\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 120.000 0.419580
\(287\) 212.132 0.739136
\(288\) 5.65685 0.0196419
\(289\) −336.000 −1.16263
\(290\) 0 0
\(291\) −48.0000 −0.164948
\(292\) − 50.0000i − 0.171233i
\(293\) 186.676 0.637120 0.318560 0.947903i \(-0.396801\pi\)
0.318560 + 0.947903i \(0.396801\pi\)
\(294\) −96.0000 −0.326531
\(295\) 0 0
\(296\) −72.0000 −0.243243
\(297\) −141.421 −0.476166
\(298\) 304.056 1.02032
\(299\) 169.706i 0.567577i
\(300\) 0 0
\(301\) 25.0000 0.0830565
\(302\) − 120.000i − 0.397351i
\(303\) 141.421 0.466737
\(304\) −76.0000 −0.250000
\(305\) 0 0
\(306\) 35.3553i 0.115540i
\(307\) −280.014 −0.912099 −0.456049 0.889955i \(-0.650736\pi\)
−0.456049 + 0.889955i \(0.650736\pi\)
\(308\) − 50.0000i − 0.162338i
\(309\) −48.0000 −0.155340
\(310\) 0 0
\(311\) −235.000 −0.755627 −0.377814 0.925882i \(-0.623324\pi\)
−0.377814 + 0.925882i \(0.623324\pi\)
\(312\) 135.765 0.435143
\(313\) − 310.000i − 0.990415i −0.868775 0.495208i \(-0.835092\pi\)
0.868775 0.495208i \(-0.164908\pi\)
\(314\) − 268.701i − 0.855734i
\(315\) 0 0
\(316\) 84.8528i 0.268522i
\(317\) 186.676 0.588884 0.294442 0.955669i \(-0.404866\pi\)
0.294442 + 0.955669i \(0.404866\pi\)
\(318\) 101.823 0.320199
\(319\) 212.132i 0.664991i
\(320\) 0 0
\(321\) 288.000 0.897196
\(322\) 70.7107 0.219598
\(323\) − 475.000i − 1.47059i
\(324\) −142.000 −0.438272
\(325\) 0 0
\(326\) − 155.563i − 0.477189i
\(327\) 360.000i 1.10092i
\(328\) − 120.000i − 0.365854i
\(329\) −25.0000 −0.0759878
\(330\) 0 0
\(331\) − 296.985i − 0.897235i −0.893724 0.448618i \(-0.851917\pi\)
0.893724 0.448618i \(-0.148083\pi\)
\(332\) − 260.000i − 0.783133i
\(333\) −25.4558 −0.0764440
\(334\) 84.0000 0.251497
\(335\) 0 0
\(336\) − 56.5685i − 0.168359i
\(337\) 526.087 1.56109 0.780545 0.625099i \(-0.214942\pi\)
0.780545 + 0.625099i \(0.214942\pi\)
\(338\) −168.291 −0.497904
\(339\) 312.000 0.920354
\(340\) 0 0
\(341\) 212.132i 0.622088i
\(342\) −26.8701 −0.0785674
\(343\) − 365.000i − 1.06414i
\(344\) − 14.1421i − 0.0411109i
\(345\) 0 0
\(346\) 264.000 0.763006
\(347\) − 125.000i − 0.360231i −0.983646 0.180115i \(-0.942353\pi\)
0.983646 0.180115i \(-0.0576471\pi\)
\(348\) 240.000i 0.689655i
\(349\) −23.0000 −0.0659026 −0.0329513 0.999457i \(-0.510491\pi\)
−0.0329513 + 0.999457i \(0.510491\pi\)
\(350\) 0 0
\(351\) 480.000 1.36752
\(352\) −28.2843 −0.0803530
\(353\) 410.000i 1.16147i 0.814092 + 0.580737i \(0.197235\pi\)
−0.814092 + 0.580737i \(0.802765\pi\)
\(354\) − 339.411i − 0.958789i
\(355\) 0 0
\(356\) − 254.558i − 0.715052i
\(357\) 353.553 0.990346
\(358\) 180.000i 0.502793i
\(359\) 475.000 1.32312 0.661560 0.749892i \(-0.269895\pi\)
0.661560 + 0.749892i \(0.269895\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) − 360.000i − 0.994475i
\(363\) −271.529 −0.748014
\(364\) 169.706i 0.466224i
\(365\) 0 0
\(366\) −380.000 −1.03825
\(367\) − 230.000i − 0.626703i −0.949637 0.313351i \(-0.898548\pi\)
0.949637 0.313351i \(-0.101452\pi\)
\(368\) − 40.0000i − 0.108696i
\(369\) − 42.4264i − 0.114977i
\(370\) 0 0
\(371\) 127.279i 0.343071i
\(372\) 240.000i 0.645161i
\(373\) −67.8823 −0.181990 −0.0909950 0.995851i \(-0.529005\pi\)
−0.0909950 + 0.995851i \(0.529005\pi\)
\(374\) − 176.777i − 0.472665i
\(375\) 0 0
\(376\) 14.1421i 0.0376121i
\(377\) − 720.000i − 1.90981i
\(378\) − 200.000i − 0.529101i
\(379\) − 254.558i − 0.671658i −0.941923 0.335829i \(-0.890984\pi\)
0.941923 0.335829i \(-0.109016\pi\)
\(380\) 0 0
\(381\) −648.000 −1.70079
\(382\) −414.365 −1.08472
\(383\) −144.250 −0.376631 −0.188316 0.982109i \(-0.560303\pi\)
−0.188316 + 0.982109i \(0.560303\pi\)
\(384\) −32.0000 −0.0833333
\(385\) 0 0
\(386\) −84.0000 −0.217617
\(387\) − 5.00000i − 0.0129199i
\(388\) −33.9411 −0.0874771
\(389\) 553.000 1.42159 0.710797 0.703397i \(-0.248334\pi\)
0.710797 + 0.703397i \(0.248334\pi\)
\(390\) 0 0
\(391\) 250.000 0.639386
\(392\) −67.8823 −0.173169
\(393\) −461.034 −1.17311
\(394\) − 98.9949i − 0.251256i
\(395\) 0 0
\(396\) −10.0000 −0.0252525
\(397\) − 335.000i − 0.843829i −0.906636 0.421914i \(-0.861358\pi\)
0.906636 0.421914i \(-0.138642\pi\)
\(398\) 244.659 0.614721
\(399\) 268.701i 0.673435i
\(400\) 0 0
\(401\) − 212.132i − 0.529008i −0.964385 0.264504i \(-0.914792\pi\)
0.964385 0.264504i \(-0.0852082\pi\)
\(402\) 441.235 1.09760
\(403\) − 720.000i − 1.78660i
\(404\) 100.000 0.247525
\(405\) 0 0
\(406\) −300.000 −0.738916
\(407\) 127.279 0.312725
\(408\) − 200.000i − 0.490196i
\(409\) 721.249i 1.76344i 0.471769 + 0.881722i \(0.343616\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(410\) 0 0
\(411\) − 268.701i − 0.653773i
\(412\) −33.9411 −0.0823814
\(413\) 424.264 1.02727
\(414\) − 14.1421i − 0.0341597i
\(415\) 0 0
\(416\) 96.0000 0.230769
\(417\) −353.553 −0.847850
\(418\) 134.350 0.321412
\(419\) −62.0000 −0.147971 −0.0739857 0.997259i \(-0.523572\pi\)
−0.0739857 + 0.997259i \(0.523572\pi\)
\(420\) 0 0
\(421\) − 296.985i − 0.705427i −0.935731 0.352714i \(-0.885259\pi\)
0.935731 0.352714i \(-0.114741\pi\)
\(422\) 120.000i 0.284360i
\(423\) 5.00000i 0.0118203i
\(424\) 72.0000 0.169811
\(425\) 0 0
\(426\) 0 0
\(427\) − 475.000i − 1.11241i
\(428\) 203.647 0.475810
\(429\) −240.000 −0.559441
\(430\) 0 0
\(431\) 509.117i 1.18125i 0.806948 + 0.590623i \(0.201118\pi\)
−0.806948 + 0.590623i \(0.798882\pi\)
\(432\) −113.137 −0.261891
\(433\) −229.103 −0.529105 −0.264553 0.964371i \(-0.585224\pi\)
−0.264553 + 0.964371i \(0.585224\pi\)
\(434\) −300.000 −0.691244
\(435\) 0 0
\(436\) 254.558i 0.583850i
\(437\) 190.000i 0.434783i
\(438\) 100.000i 0.228311i
\(439\) 806.102i 1.83622i 0.396323 + 0.918111i \(0.370286\pi\)
−0.396323 + 0.918111i \(0.629714\pi\)
\(440\) 0 0
\(441\) −24.0000 −0.0544218
\(442\) 600.000i 1.35747i
\(443\) 365.000i 0.823928i 0.911200 + 0.411964i \(0.135157\pi\)
−0.911200 + 0.411964i \(0.864843\pi\)
\(444\) 144.000 0.324324
\(445\) 0 0
\(446\) 516.000 1.15695
\(447\) −608.112 −1.36043
\(448\) − 40.0000i − 0.0892857i
\(449\) − 763.675i − 1.70084i −0.526108 0.850418i \(-0.676349\pi\)
0.526108 0.850418i \(-0.323651\pi\)
\(450\) 0 0
\(451\) 212.132i 0.470359i
\(452\) 220.617 0.488091
\(453\) 240.000i 0.529801i
\(454\) 96.0000 0.211454
\(455\) 0 0
\(456\) 152.000 0.333333
\(457\) 265.000i 0.579869i 0.957047 + 0.289934i \(0.0936335\pi\)
−0.957047 + 0.289934i \(0.906367\pi\)
\(458\) −205.061 −0.447731
\(459\) − 707.107i − 1.54054i
\(460\) 0 0
\(461\) −553.000 −1.19957 −0.599783 0.800163i \(-0.704746\pi\)
−0.599783 + 0.800163i \(0.704746\pi\)
\(462\) 100.000i 0.216450i
\(463\) 485.000i 1.04752i 0.851867 + 0.523758i \(0.175470\pi\)
−0.851867 + 0.523758i \(0.824530\pi\)
\(464\) 169.706i 0.365745i
\(465\) 0 0
\(466\) − 473.762i − 1.01666i
\(467\) 115.000i 0.246253i 0.992391 + 0.123126i \(0.0392920\pi\)
−0.992391 + 0.123126i \(0.960708\pi\)
\(468\) 33.9411 0.0725238
\(469\) 551.543i 1.17600i
\(470\) 0 0
\(471\) 537.401i 1.14098i
\(472\) − 240.000i − 0.508475i
\(473\) 25.0000i 0.0528541i
\(474\) − 169.706i − 0.358029i
\(475\) 0 0
\(476\) 250.000 0.525210
\(477\) 25.4558 0.0533665
\(478\) 278.600 0.582845
\(479\) 490.000 1.02296 0.511482 0.859294i \(-0.329097\pi\)
0.511482 + 0.859294i \(0.329097\pi\)
\(480\) 0 0
\(481\) −432.000 −0.898129
\(482\) 420.000i 0.871369i
\(483\) −141.421 −0.292798
\(484\) −192.000 −0.396694
\(485\) 0 0
\(486\) −76.0000 −0.156379
\(487\) 610.940 1.25450 0.627249 0.778819i \(-0.284181\pi\)
0.627249 + 0.778819i \(0.284181\pi\)
\(488\) −268.701 −0.550616
\(489\) 311.127i 0.636252i
\(490\) 0 0
\(491\) −82.0000 −0.167006 −0.0835031 0.996508i \(-0.526611\pi\)
−0.0835031 + 0.996508i \(0.526611\pi\)
\(492\) 240.000i 0.487805i
\(493\) −1060.66 −2.15144
\(494\) −456.000 −0.923077
\(495\) 0 0
\(496\) 169.706i 0.342148i
\(497\) 0 0
\(498\) 520.000i 1.04418i
\(499\) −485.000 −0.971944 −0.485972 0.873974i \(-0.661534\pi\)
−0.485972 + 0.873974i \(0.661534\pi\)
\(500\) 0 0
\(501\) −168.000 −0.335329
\(502\) −244.659 −0.487368
\(503\) − 250.000i − 0.497018i −0.968630 0.248509i \(-0.920059\pi\)
0.968630 0.248509i \(-0.0799405\pi\)
\(504\) − 14.1421i − 0.0280598i
\(505\) 0 0
\(506\) 70.7107i 0.139744i
\(507\) 336.583 0.663871
\(508\) −458.205 −0.901979
\(509\) − 169.706i − 0.333410i −0.986007 0.166705i \(-0.946687\pi\)
0.986007 0.166705i \(-0.0533127\pi\)
\(510\) 0 0
\(511\) −125.000 −0.244618
\(512\) −22.6274 −0.0441942
\(513\) 537.401 1.04757
\(514\) −96.0000 −0.186770
\(515\) 0 0
\(516\) 28.2843i 0.0548145i
\(517\) − 25.0000i − 0.0483559i
\(518\) 180.000i 0.347490i
\(519\) −528.000 −1.01734
\(520\) 0 0
\(521\) 127.279i 0.244298i 0.992512 + 0.122149i \(0.0389786\pi\)
−0.992512 + 0.122149i \(0.961021\pi\)
\(522\) 60.0000i 0.114943i
\(523\) 356.382 0.681418 0.340709 0.940169i \(-0.389333\pi\)
0.340709 + 0.940169i \(0.389333\pi\)
\(524\) −326.000 −0.622137
\(525\) 0 0
\(526\) 502.046i 0.954460i
\(527\) −1060.66 −2.01264
\(528\) 56.5685 0.107137
\(529\) 429.000 0.810964
\(530\) 0 0
\(531\) − 84.8528i − 0.159798i
\(532\) 190.000i 0.357143i
\(533\) − 720.000i − 1.35084i
\(534\) 509.117i 0.953402i
\(535\) 0 0
\(536\) 312.000 0.582090
\(537\) − 360.000i − 0.670391i
\(538\) 540.000i 1.00372i
\(539\) 120.000 0.222635
\(540\) 0 0
\(541\) −25.0000 −0.0462107 −0.0231054 0.999733i \(-0.507355\pi\)
−0.0231054 + 0.999733i \(0.507355\pi\)
\(542\) −155.563 −0.287018
\(543\) 720.000i 1.32597i
\(544\) − 141.421i − 0.259966i
\(545\) 0 0
\(546\) − 339.411i − 0.621632i
\(547\) −16.9706 −0.0310248 −0.0155124 0.999880i \(-0.504938\pi\)
−0.0155124 + 0.999880i \(0.504938\pi\)
\(548\) − 190.000i − 0.346715i
\(549\) −95.0000 −0.173042
\(550\) 0 0
\(551\) − 806.102i − 1.46298i
\(552\) 80.0000i 0.144928i
\(553\) 212.132 0.383602
\(554\) − 374.767i − 0.676474i
\(555\) 0 0
\(556\) −250.000 −0.449640
\(557\) 745.000i 1.33752i 0.743477 + 0.668761i \(0.233175\pi\)
−0.743477 + 0.668761i \(0.766825\pi\)
\(558\) 60.0000i 0.107527i
\(559\) − 84.8528i − 0.151794i
\(560\) 0 0
\(561\) 353.553i 0.630220i
\(562\) 600.000i 1.06762i
\(563\) 313.955 0.557647 0.278824 0.960342i \(-0.410056\pi\)
0.278824 + 0.960342i \(0.410056\pi\)
\(564\) − 28.2843i − 0.0501494i
\(565\) 0 0
\(566\) − 176.777i − 0.312326i
\(567\) 355.000i 0.626102i
\(568\) 0 0
\(569\) 424.264i 0.745631i 0.927905 + 0.372816i \(0.121608\pi\)
−0.927905 + 0.372816i \(0.878392\pi\)
\(570\) 0 0
\(571\) 1070.00 1.87391 0.936953 0.349456i \(-0.113634\pi\)
0.936953 + 0.349456i \(0.113634\pi\)
\(572\) −169.706 −0.296688
\(573\) 828.729 1.44630
\(574\) −300.000 −0.522648
\(575\) 0 0
\(576\) −8.00000 −0.0138889
\(577\) 25.0000i 0.0433276i 0.999765 + 0.0216638i \(0.00689633\pi\)
−0.999765 + 0.0216638i \(0.993104\pi\)
\(578\) 475.176 0.822103
\(579\) 168.000 0.290155
\(580\) 0 0
\(581\) −650.000 −1.11876
\(582\) 67.8823 0.116636
\(583\) −127.279 −0.218318
\(584\) 70.7107i 0.121080i
\(585\) 0 0
\(586\) −264.000 −0.450512
\(587\) − 725.000i − 1.23509i −0.786534 0.617547i \(-0.788127\pi\)
0.786534 0.617547i \(-0.211873\pi\)
\(588\) 135.765 0.230892
\(589\) − 806.102i − 1.36859i
\(590\) 0 0
\(591\) 197.990i 0.335008i
\(592\) 101.823 0.171999
\(593\) 650.000i 1.09612i 0.836439 + 0.548061i \(0.184634\pi\)
−0.836439 + 0.548061i \(0.815366\pi\)
\(594\) 200.000 0.336700
\(595\) 0 0
\(596\) −430.000 −0.721477
\(597\) −489.318 −0.819628
\(598\) − 240.000i − 0.401338i
\(599\) − 296.985i − 0.495801i −0.968785 0.247901i \(-0.920259\pi\)
0.968785 0.247901i \(-0.0797406\pi\)
\(600\) 0 0
\(601\) 848.528i 1.41186i 0.708281 + 0.705930i \(0.249471\pi\)
−0.708281 + 0.705930i \(0.750529\pi\)
\(602\) −35.3553 −0.0587298
\(603\) 110.309 0.182933
\(604\) 169.706i 0.280970i
\(605\) 0 0
\(606\) −200.000 −0.330033
\(607\) 271.529 0.447329 0.223665 0.974666i \(-0.428198\pi\)
0.223665 + 0.974666i \(0.428198\pi\)
\(608\) 107.480 0.176777
\(609\) 600.000 0.985222
\(610\) 0 0
\(611\) 84.8528i 0.138875i
\(612\) − 50.0000i − 0.0816993i
\(613\) 1055.00i 1.72104i 0.509413 + 0.860522i \(0.329862\pi\)
−0.509413 + 0.860522i \(0.670138\pi\)
\(614\) 396.000 0.644951
\(615\) 0 0
\(616\) 70.7107i 0.114790i
\(617\) 505.000i 0.818476i 0.912428 + 0.409238i \(0.134206\pi\)
−0.912428 + 0.409238i \(0.865794\pi\)
\(618\) 67.8823 0.109842
\(619\) 130.000 0.210016 0.105008 0.994471i \(-0.466513\pi\)
0.105008 + 0.994471i \(0.466513\pi\)
\(620\) 0 0
\(621\) 282.843i 0.455463i
\(622\) 332.340 0.534309
\(623\) −636.396 −1.02150
\(624\) −192.000 −0.307692
\(625\) 0 0
\(626\) 438.406i 0.700329i
\(627\) −268.701 −0.428550
\(628\) 380.000i 0.605096i
\(629\) 636.396i 1.01176i
\(630\) 0 0
\(631\) −475.000 −0.752773 −0.376387 0.926463i \(-0.622834\pi\)
−0.376387 + 0.926463i \(0.622834\pi\)
\(632\) − 120.000i − 0.189873i
\(633\) − 240.000i − 0.379147i
\(634\) −264.000 −0.416404
\(635\) 0 0
\(636\) −144.000 −0.226415
\(637\) −407.294 −0.639393
\(638\) − 300.000i − 0.470219i
\(639\) 0 0
\(640\) 0 0
\(641\) 848.528i 1.32376i 0.749611 + 0.661878i \(0.230240\pi\)
−0.749611 + 0.661878i \(0.769760\pi\)
\(642\) −407.294 −0.634414
\(643\) − 955.000i − 1.48523i −0.669721 0.742613i \(-0.733586\pi\)
0.669721 0.742613i \(-0.266414\pi\)
\(644\) −100.000 −0.155280
\(645\) 0 0
\(646\) 671.751i 1.03986i
\(647\) − 965.000i − 1.49150i −0.666226 0.745750i \(-0.732092\pi\)
0.666226 0.745750i \(-0.267908\pi\)
\(648\) 200.818 0.309905
\(649\) 424.264i 0.653720i
\(650\) 0 0
\(651\) 600.000 0.921659
\(652\) 220.000i 0.337423i
\(653\) 935.000i 1.43185i 0.698176 + 0.715926i \(0.253995\pi\)
−0.698176 + 0.715926i \(0.746005\pi\)
\(654\) − 509.117i − 0.778466i
\(655\) 0 0
\(656\) 169.706i 0.258698i
\(657\) 25.0000i 0.0380518i
\(658\) 35.3553 0.0537315
\(659\) − 84.8528i − 0.128760i −0.997925 0.0643800i \(-0.979493\pi\)
0.997925 0.0643800i \(-0.0205070\pi\)
\(660\) 0 0
\(661\) 678.823i 1.02696i 0.858101 + 0.513481i \(0.171644\pi\)
−0.858101 + 0.513481i \(0.828356\pi\)
\(662\) 420.000i 0.634441i
\(663\) − 1200.00i − 1.80995i
\(664\) 367.696i 0.553758i
\(665\) 0 0
\(666\) 36.0000 0.0540541
\(667\) 424.264 0.636078
\(668\) −118.794 −0.177835
\(669\) −1032.00 −1.54260
\(670\) 0 0
\(671\) 475.000 0.707899
\(672\) 80.0000i 0.119048i
\(673\) −186.676 −0.277379 −0.138690 0.990336i \(-0.544289\pi\)
−0.138690 + 0.990336i \(0.544289\pi\)
\(674\) −744.000 −1.10386
\(675\) 0 0
\(676\) 238.000 0.352071
\(677\) −907.925 −1.34110 −0.670550 0.741864i \(-0.733942\pi\)
−0.670550 + 0.741864i \(0.733942\pi\)
\(678\) −441.235 −0.650789
\(679\) 84.8528i 0.124967i
\(680\) 0 0
\(681\) −192.000 −0.281938
\(682\) − 300.000i − 0.439883i
\(683\) 1120.06 1.63991 0.819954 0.572429i \(-0.193999\pi\)
0.819954 + 0.572429i \(0.193999\pi\)
\(684\) 38.0000 0.0555556
\(685\) 0 0
\(686\) 516.188i 0.752461i
\(687\) 410.122 0.596975
\(688\) 20.0000i 0.0290698i
\(689\) 432.000 0.626996
\(690\) 0 0
\(691\) −715.000 −1.03473 −0.517366 0.855764i \(-0.673087\pi\)
−0.517366 + 0.855764i \(0.673087\pi\)
\(692\) −373.352 −0.539527
\(693\) 25.0000i 0.0360750i
\(694\) 176.777i 0.254721i
\(695\) 0 0
\(696\) − 339.411i − 0.487660i
\(697\) −1060.66 −1.52175
\(698\) 32.5269 0.0466002
\(699\) 947.523i 1.35554i
\(700\) 0 0
\(701\) −430.000 −0.613409 −0.306705 0.951805i \(-0.599226\pi\)
−0.306705 + 0.951805i \(0.599226\pi\)
\(702\) −678.823 −0.966984
\(703\) −483.661 −0.687996
\(704\) 40.0000 0.0568182
\(705\) 0 0
\(706\) − 579.828i − 0.821285i
\(707\) − 250.000i − 0.353607i
\(708\) 480.000i 0.677966i
\(709\) 382.000 0.538787 0.269394 0.963030i \(-0.413177\pi\)
0.269394 + 0.963030i \(0.413177\pi\)
\(710\) 0 0
\(711\) − 42.4264i − 0.0596715i
\(712\) 360.000i 0.505618i
\(713\) 424.264 0.595041
\(714\) −500.000 −0.700280
\(715\) 0 0
\(716\) − 254.558i − 0.355529i
\(717\) −557.200 −0.777127
\(718\) −671.751 −0.935587
\(719\) 115.000 0.159944 0.0799722 0.996797i \(-0.474517\pi\)
0.0799722 + 0.996797i \(0.474517\pi\)
\(720\) 0 0
\(721\) 84.8528i 0.117688i
\(722\) −510.531 −0.707107
\(723\) − 840.000i − 1.16183i
\(724\) 509.117i 0.703200i
\(725\) 0 0
\(726\) 384.000 0.528926
\(727\) 1075.00i 1.47868i 0.673333 + 0.739340i \(0.264862\pi\)
−0.673333 + 0.739340i \(0.735138\pi\)
\(728\) − 240.000i − 0.329670i
\(729\) 791.000 1.08505
\(730\) 0 0
\(731\) −125.000 −0.170999
\(732\) 537.401 0.734155
\(733\) 530.000i 0.723056i 0.932361 + 0.361528i \(0.117745\pi\)
−0.932361 + 0.361528i \(0.882255\pi\)
\(734\) 325.269i 0.443146i
\(735\) 0 0
\(736\) 56.5685i 0.0768594i
\(737\) −551.543 −0.748363
\(738\) 60.0000i 0.0813008i
\(739\) 547.000 0.740189 0.370095 0.928994i \(-0.379325\pi\)
0.370095 + 0.928994i \(0.379325\pi\)
\(740\) 0 0
\(741\) 912.000 1.23077
\(742\) − 180.000i − 0.242588i
\(743\) −958.837 −1.29049 −0.645247 0.763974i \(-0.723245\pi\)
−0.645247 + 0.763974i \(0.723245\pi\)
\(744\) − 339.411i − 0.456198i
\(745\) 0 0
\(746\) 96.0000 0.128686
\(747\) 130.000i 0.174029i
\(748\) 250.000i 0.334225i
\(749\) − 509.117i − 0.679729i
\(750\) 0 0
\(751\) 169.706i 0.225973i 0.993597 + 0.112986i \(0.0360417\pi\)
−0.993597 + 0.112986i \(0.963958\pi\)
\(752\) − 20.0000i − 0.0265957i
\(753\) 489.318 0.649825
\(754\) 1018.23i 1.35044i
\(755\) 0 0
\(756\) 282.843i 0.374131i
\(757\) − 1055.00i − 1.39366i −0.717237 0.696830i \(-0.754593\pi\)
0.717237 0.696830i \(-0.245407\pi\)
\(758\) 360.000i 0.474934i
\(759\) − 141.421i − 0.186326i
\(760\) 0 0
\(761\) 215.000 0.282523 0.141261 0.989972i \(-0.454884\pi\)
0.141261 + 0.989972i \(0.454884\pi\)
\(762\) 916.410 1.20264
\(763\) 636.396 0.834071
\(764\) 586.000 0.767016
\(765\) 0 0
\(766\) 204.000 0.266319
\(767\) − 1440.00i − 1.87744i
\(768\) 45.2548 0.0589256
\(769\) 145.000 0.188557 0.0942783 0.995546i \(-0.469946\pi\)
0.0942783 + 0.995546i \(0.469946\pi\)
\(770\) 0 0
\(771\) 192.000 0.249027
\(772\) 118.794 0.153878
\(773\) 407.294 0.526900 0.263450 0.964673i \(-0.415140\pi\)
0.263450 + 0.964673i \(0.415140\pi\)
\(774\) 7.07107i 0.00913575i
\(775\) 0 0
\(776\) 48.0000 0.0618557
\(777\) − 360.000i − 0.463320i
\(778\) −782.060 −1.00522
\(779\) − 806.102i − 1.03479i
\(780\) 0 0
\(781\) 0 0
\(782\) −353.553 −0.452114
\(783\) − 1200.00i − 1.53257i
\(784\) 96.0000 0.122449
\(785\) 0 0
\(786\) 652.000 0.829517
\(787\) −186.676 −0.237200 −0.118600 0.992942i \(-0.537841\pi\)
−0.118600 + 0.992942i \(0.537841\pi\)
\(788\) 140.000i 0.177665i
\(789\) − 1004.09i − 1.27261i
\(790\) 0 0
\(791\) − 551.543i − 0.697273i
\(792\) 14.1421 0.0178562
\(793\) −1612.20 −2.03304
\(794\) 473.762i 0.596677i
\(795\) 0 0
\(796\) −346.000 −0.434673
\(797\) 704.278 0.883662 0.441831 0.897098i \(-0.354329\pi\)
0.441831 + 0.897098i \(0.354329\pi\)
\(798\) − 380.000i − 0.476190i
\(799\) 125.000 0.156446
\(800\) 0 0
\(801\) 127.279i 0.158900i
\(802\) 300.000i 0.374065i
\(803\) − 125.000i − 0.155666i
\(804\) −624.000 −0.776119
\(805\) 0 0
\(806\) 1018.23i 1.26332i
\(807\) − 1080.00i − 1.33829i
\(808\) −141.421 −0.175026
\(809\) 457.000 0.564895 0.282447 0.959283i \(-0.408854\pi\)
0.282447 + 0.959283i \(0.408854\pi\)
\(810\) 0 0
\(811\) − 509.117i − 0.627764i −0.949462 0.313882i \(-0.898370\pi\)
0.949462 0.313882i \(-0.101630\pi\)
\(812\) 424.264 0.522493
\(813\) 311.127 0.382690
\(814\) −180.000 −0.221130
\(815\) 0 0
\(816\) 282.843i 0.346621i
\(817\) − 95.0000i − 0.116279i
\(818\) − 1020.00i − 1.24694i
\(819\) − 84.8528i − 0.103605i
\(820\) 0 0
\(821\) 167.000 0.203410 0.101705 0.994815i \(-0.467570\pi\)
0.101705 + 0.994815i \(0.467570\pi\)
\(822\) 380.000i 0.462287i
\(823\) − 1315.00i − 1.59781i −0.601455 0.798906i \(-0.705412\pi\)
0.601455 0.798906i \(-0.294588\pi\)
\(824\) 48.0000 0.0582524
\(825\) 0 0
\(826\) −600.000 −0.726392
\(827\) 534.573 0.646400 0.323200 0.946331i \(-0.395241\pi\)
0.323200 + 0.946331i \(0.395241\pi\)
\(828\) 20.0000i 0.0241546i
\(829\) 763.675i 0.921201i 0.887608 + 0.460600i \(0.152366\pi\)
−0.887608 + 0.460600i \(0.847634\pi\)
\(830\) 0 0
\(831\) 749.533i 0.901965i
\(832\) −135.765 −0.163178
\(833\) 600.000i 0.720288i
\(834\) 500.000 0.599520
\(835\) 0 0
\(836\) −190.000 −0.227273
\(837\) − 1200.00i − 1.43369i
\(838\) 87.6812 0.104632
\(839\) − 339.411i − 0.404543i −0.979330 0.202271i \(-0.935168\pi\)
0.979330 0.202271i \(-0.0648323\pi\)
\(840\) 0 0
\(841\) −959.000 −1.14031
\(842\) 420.000i 0.498812i
\(843\) − 1200.00i − 1.42349i
\(844\) − 169.706i − 0.201073i
\(845\) 0 0
\(846\) − 7.07107i − 0.00835824i
\(847\) 480.000i 0.566706i
\(848\) −101.823 −0.120075
\(849\) 353.553i 0.416435i
\(850\) 0 0
\(851\) − 254.558i − 0.299129i
\(852\) 0 0
\(853\) 770.000i 0.902696i 0.892348 + 0.451348i \(0.149057\pi\)
−0.892348 + 0.451348i \(0.850943\pi\)
\(854\) 671.751i 0.786594i
\(855\) 0 0
\(856\) −288.000 −0.336449
\(857\) 1255.82 1.46537 0.732685 0.680568i \(-0.238267\pi\)
0.732685 + 0.680568i \(0.238267\pi\)
\(858\) 339.411 0.395584
\(859\) −557.000 −0.648428 −0.324214 0.945984i \(-0.605100\pi\)
−0.324214 + 0.945984i \(0.605100\pi\)
\(860\) 0 0
\(861\) 600.000 0.696864
\(862\) − 720.000i − 0.835267i
\(863\) 992.778 1.15038 0.575190 0.818020i \(-0.304928\pi\)
0.575190 + 0.818020i \(0.304928\pi\)
\(864\) 160.000 0.185185
\(865\) 0 0
\(866\) 324.000 0.374134
\(867\) −950.352 −1.09614
\(868\) 424.264 0.488783
\(869\) 212.132i 0.244111i
\(870\) 0 0
\(871\) 1872.00 2.14925
\(872\) − 360.000i − 0.412844i
\(873\) 16.9706 0.0194394
\(874\) − 268.701i − 0.307438i
\(875\) 0 0
\(876\) − 141.421i − 0.161440i
\(877\) −186.676 −0.212858 −0.106429 0.994320i \(-0.533942\pi\)
−0.106429 + 0.994320i \(0.533942\pi\)
\(878\) − 1140.00i − 1.29841i
\(879\) 528.000 0.600683
\(880\) 0 0
\(881\) −25.0000 −0.0283768 −0.0141884 0.999899i \(-0.504516\pi\)
−0.0141884 + 0.999899i \(0.504516\pi\)
\(882\) 33.9411 0.0384820
\(883\) 965.000i 1.09287i 0.837503 + 0.546433i \(0.184015\pi\)
−0.837503 + 0.546433i \(0.815985\pi\)
\(884\) − 848.528i − 0.959873i
\(885\) 0 0
\(886\) − 516.188i − 0.582605i
\(887\) 780.646 0.880097 0.440048 0.897974i \(-0.354961\pi\)
0.440048 + 0.897974i \(0.354961\pi\)
\(888\) −203.647 −0.229332
\(889\) 1145.51i 1.28854i
\(890\) 0 0
\(891\) −355.000 −0.398429
\(892\) −729.734 −0.818088
\(893\) 95.0000i 0.106383i
\(894\) 860.000 0.961969
\(895\) 0 0
\(896\) 56.5685i 0.0631345i
\(897\) 480.000i 0.535117i
\(898\) 1080.00i 1.20267i
\(899\) −1800.00 −2.00222
\(900\) 0 0
\(901\) − 636.396i − 0.706322i
\(902\) − 300.000i − 0.332594i
\(903\) 70.7107 0.0783064
\(904\) −312.000 −0.345133
\(905\) 0 0
\(906\) − 339.411i − 0.374626i
\(907\) 313.955 0.346147 0.173074 0.984909i \(-0.444630\pi\)
0.173074 + 0.984909i \(0.444630\pi\)
\(908\) −135.765 −0.149520
\(909\) −50.0000 −0.0550055
\(910\) 0 0
\(911\) − 933.381i − 1.02457i −0.858816 0.512284i \(-0.828800\pi\)
0.858816 0.512284i \(-0.171200\pi\)
\(912\) −214.960 −0.235702
\(913\) − 650.000i − 0.711939i
\(914\) − 374.767i − 0.410029i
\(915\) 0 0
\(916\) 290.000 0.316594
\(917\) 815.000i 0.888768i
\(918\) 1000.00i 1.08932i
\(919\) 538.000 0.585419 0.292709 0.956201i \(-0.405443\pi\)
0.292709 + 0.956201i \(0.405443\pi\)
\(920\) 0 0
\(921\) −792.000 −0.859935
\(922\) 782.060 0.848221
\(923\) 0 0
\(924\) − 141.421i − 0.153053i
\(925\) 0 0
\(926\) − 685.894i − 0.740706i
\(927\) 16.9706 0.0183070
\(928\) − 240.000i − 0.258621i
\(929\) 742.000 0.798708 0.399354 0.916797i \(-0.369234\pi\)
0.399354 + 0.916797i \(0.369234\pi\)
\(930\) 0 0
\(931\) −456.000 −0.489796
\(932\) 670.000i 0.718884i
\(933\) −664.680 −0.712412
\(934\) − 162.635i − 0.174127i
\(935\) 0 0
\(936\) −48.0000 −0.0512821
\(937\) − 335.000i − 0.357524i −0.983892 0.178762i \(-0.942791\pi\)
0.983892 0.178762i \(-0.0572092\pi\)
\(938\) − 780.000i − 0.831557i
\(939\) − 876.812i − 0.933773i
\(940\) 0 0
\(941\) 424.264i 0.450865i 0.974259 + 0.225433i \(0.0723795\pi\)
−0.974259 + 0.225433i \(0.927620\pi\)
\(942\) − 760.000i − 0.806794i
\(943\) 424.264 0.449909
\(944\) 339.411i 0.359546i
\(945\) 0 0
\(946\) − 35.3553i − 0.0373735i
\(947\) 1210.00i 1.27772i 0.769323 + 0.638860i \(0.220594\pi\)
−0.769323 + 0.638860i \(0.779406\pi\)
\(948\) 240.000i 0.253165i
\(949\) 424.264i 0.447064i
\(950\) 0 0
\(951\) 528.000 0.555205
\(952\) −353.553 −0.371380
\(953\) 992.778 1.04174 0.520870 0.853636i \(-0.325608\pi\)
0.520870 + 0.853636i \(0.325608\pi\)
\(954\) −36.0000 −0.0377358
\(955\) 0 0
\(956\) −394.000 −0.412134
\(957\) 600.000i 0.626959i
\(958\) −692.965 −0.723345
\(959\) −475.000 −0.495308
\(960\) 0 0
\(961\) −839.000 −0.873049
\(962\) 610.940 0.635073
\(963\) −101.823 −0.105736
\(964\) − 593.970i − 0.616151i
\(965\) 0 0
\(966\) 200.000 0.207039
\(967\) − 350.000i − 0.361944i −0.983488 0.180972i \(-0.942076\pi\)
0.983488 0.180972i \(-0.0579244\pi\)
\(968\) 271.529 0.280505
\(969\) − 1343.50i − 1.38648i
\(970\) 0 0
\(971\) − 254.558i − 0.262161i −0.991372 0.131081i \(-0.958155\pi\)
0.991372 0.131081i \(-0.0418447\pi\)
\(972\) 107.480 0.110576
\(973\) 625.000i 0.642343i
\(974\) −864.000 −0.887064
\(975\) 0 0
\(976\) 380.000 0.389344
\(977\) 398.808 0.408197 0.204098 0.978950i \(-0.434574\pi\)
0.204098 + 0.978950i \(0.434574\pi\)
\(978\) − 440.000i − 0.449898i
\(979\) − 636.396i − 0.650047i
\(980\) 0 0
\(981\) − 127.279i − 0.129744i
\(982\) 115.966 0.118091
\(983\) −695.793 −0.707826 −0.353913 0.935278i \(-0.615149\pi\)
−0.353913 + 0.935278i \(0.615149\pi\)
\(984\) − 339.411i − 0.344930i
\(985\) 0 0
\(986\) 1500.00 1.52130
\(987\) −70.7107 −0.0716420
\(988\) 644.881 0.652714
\(989\) 50.0000 0.0505561
\(990\) 0 0
\(991\) − 381.838i − 0.385305i −0.981267 0.192653i \(-0.938291\pi\)
0.981267 0.192653i \(-0.0617091\pi\)
\(992\) − 240.000i − 0.241935i
\(993\) − 840.000i − 0.845921i
\(994\) 0 0
\(995\) 0 0
\(996\) − 735.391i − 0.738344i
\(997\) 265.000i 0.265797i 0.991130 + 0.132899i \(0.0424285\pi\)
−0.991130 + 0.132899i \(0.957572\pi\)
\(998\) 685.894 0.687268
\(999\) −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.d.a.949.1 4
5.2 odd 4 38.3.b.a.37.1 2
5.3 odd 4 950.3.c.a.151.2 2
5.4 even 2 inner 950.3.d.a.949.4 4
15.2 even 4 342.3.d.a.37.2 2
19.18 odd 2 inner 950.3.d.a.949.3 4
20.7 even 4 304.3.e.c.113.2 2
40.27 even 4 1216.3.e.i.1025.1 2
40.37 odd 4 1216.3.e.j.1025.2 2
60.47 odd 4 2736.3.o.h.721.2 2
95.18 even 4 950.3.c.a.151.1 2
95.37 even 4 38.3.b.a.37.2 yes 2
95.94 odd 2 inner 950.3.d.a.949.2 4
285.227 odd 4 342.3.d.a.37.1 2
380.227 odd 4 304.3.e.c.113.1 2
760.37 even 4 1216.3.e.j.1025.1 2
760.227 odd 4 1216.3.e.i.1025.2 2
1140.227 even 4 2736.3.o.h.721.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.3.b.a.37.1 2 5.2 odd 4
38.3.b.a.37.2 yes 2 95.37 even 4
304.3.e.c.113.1 2 380.227 odd 4
304.3.e.c.113.2 2 20.7 even 4
342.3.d.a.37.1 2 285.227 odd 4
342.3.d.a.37.2 2 15.2 even 4
950.3.c.a.151.1 2 95.18 even 4
950.3.c.a.151.2 2 5.3 odd 4
950.3.d.a.949.1 4 1.1 even 1 trivial
950.3.d.a.949.2 4 95.94 odd 2 inner
950.3.d.a.949.3 4 19.18 odd 2 inner
950.3.d.a.949.4 4 5.4 even 2 inner
1216.3.e.i.1025.1 2 40.27 even 4
1216.3.e.i.1025.2 2 760.227 odd 4
1216.3.e.j.1025.1 2 760.37 even 4
1216.3.e.j.1025.2 2 40.37 odd 4
2736.3.o.h.721.1 2 1140.227 even 4
2736.3.o.h.721.2 2 60.47 odd 4