Properties

Label 1150.2.b.j.599.4
Level $1150$
Weight $2$
Character 1150.599
Analytic conductor $9.183$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.18279623245\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.77580864.1
Defining polynomial: \(x^{6} + 19 x^{4} + 105 x^{2} + 144\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.4
Root \(-2.68740i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.2.b.j.599.3

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -2.68740i q^{3} -1.00000 q^{4} +2.68740 q^{6} -4.59692i q^{7} -1.00000i q^{8} -4.22212 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -2.68740i q^{3} -1.00000 q^{4} +2.68740 q^{6} -4.59692i q^{7} -1.00000i q^{8} -4.22212 q^{9} +5.13163 q^{11} +2.68740i q^{12} +1.22212i q^{13} +4.59692 q^{14} +1.00000 q^{16} -4.68740i q^{17} -4.22212i q^{18} +4.59692 q^{19} -12.3537 q^{21} +5.13163i q^{22} +1.00000i q^{23} -2.68740 q^{24} -1.22212 q^{26} +3.28432i q^{27} +4.59692i q^{28} -3.37480 q^{29} -0.777884 q^{31} +1.00000i q^{32} -13.7907i q^{33} +4.68740 q^{34} +4.22212 q^{36} +5.81903i q^{37} +4.59692i q^{38} +3.28432 q^{39} -8.50643 q^{41} -12.3537i q^{42} -8.00000i q^{43} -5.13163 q^{44} -1.00000 q^{46} -6.44423i q^{47} -2.68740i q^{48} -14.1316 q^{49} -12.5969 q^{51} -1.22212i q^{52} +6.00000i q^{53} -3.28432 q^{54} -4.59692 q^{56} -12.3537i q^{57} -3.37480i q^{58} -9.37480 q^{59} +10.9507 q^{61} -0.777884i q^{62} +19.4087i q^{63} -1.00000 q^{64} +13.7907 q^{66} +15.6381i q^{67} +4.68740i q^{68} +2.68740 q^{69} +1.31260 q^{71} +4.22212i q^{72} +4.44423i q^{73} -5.81903 q^{74} -4.59692 q^{76} -23.5897i q^{77} +3.28432i q^{78} +4.88847 q^{79} -3.84008 q^{81} -8.50643i q^{82} +3.81903i q^{83} +12.3537 q^{84} +8.00000 q^{86} +9.06943i q^{87} -5.13163i q^{88} -8.93057 q^{89} +5.61797 q^{91} -1.00000i q^{92} +2.09048i q^{93} +6.44423 q^{94} +2.68740 q^{96} -18.0622i q^{97} -14.1316i q^{98} -21.6663 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{4} + 2q^{6} - 20q^{9} + O(q^{10}) \) \( 6q - 6q^{4} + 2q^{6} - 20q^{9} + 6q^{11} - 6q^{14} + 6q^{16} - 6q^{19} - 44q^{21} - 2q^{24} - 2q^{26} + 8q^{29} - 10q^{31} + 14q^{34} + 20q^{36} - 28q^{39} + 2q^{41} - 6q^{44} - 6q^{46} - 60q^{49} - 42q^{51} + 28q^{54} + 6q^{56} - 28q^{59} + 2q^{61} - 6q^{64} - 18q^{66} + 2q^{69} + 22q^{71} + 4q^{74} + 6q^{76} + 8q^{79} + 14q^{81} + 44q^{84} + 48q^{86} - 36q^{89} + 2q^{91} + 28q^{94} + 2q^{96} - 114q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) − 2.68740i − 1.55157i −0.630997 0.775785i \(-0.717354\pi\)
0.630997 0.775785i \(-0.282646\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.68740 1.09713
\(7\) − 4.59692i − 1.73747i −0.495277 0.868735i \(-0.664933\pi\)
0.495277 0.868735i \(-0.335067\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −4.22212 −1.40737
\(10\) 0 0
\(11\) 5.13163 1.54725 0.773623 0.633647i \(-0.218443\pi\)
0.773623 + 0.633647i \(0.218443\pi\)
\(12\) 2.68740i 0.775785i
\(13\) 1.22212i 0.338954i 0.985534 + 0.169477i \(0.0542079\pi\)
−0.985534 + 0.169477i \(0.945792\pi\)
\(14\) 4.59692 1.22858
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 4.68740i − 1.13686i −0.822731 0.568431i \(-0.807551\pi\)
0.822731 0.568431i \(-0.192449\pi\)
\(18\) − 4.22212i − 0.995162i
\(19\) 4.59692 1.05460 0.527302 0.849678i \(-0.323204\pi\)
0.527302 + 0.849678i \(0.323204\pi\)
\(20\) 0 0
\(21\) −12.3537 −2.69581
\(22\) 5.13163i 1.09407i
\(23\) 1.00000i 0.208514i
\(24\) −2.68740 −0.548563
\(25\) 0 0
\(26\) −1.22212 −0.239677
\(27\) 3.28432i 0.632067i
\(28\) 4.59692i 0.868735i
\(29\) −3.37480 −0.626684 −0.313342 0.949640i \(-0.601449\pi\)
−0.313342 + 0.949640i \(0.601449\pi\)
\(30\) 0 0
\(31\) −0.777884 −0.139712 −0.0698560 0.997557i \(-0.522254\pi\)
−0.0698560 + 0.997557i \(0.522254\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 13.7907i − 2.40066i
\(34\) 4.68740 0.803882
\(35\) 0 0
\(36\) 4.22212 0.703686
\(37\) 5.81903i 0.956643i 0.878185 + 0.478321i \(0.158755\pi\)
−0.878185 + 0.478321i \(0.841245\pi\)
\(38\) 4.59692i 0.745718i
\(39\) 3.28432 0.525911
\(40\) 0 0
\(41\) −8.50643 −1.32848 −0.664241 0.747519i \(-0.731245\pi\)
−0.664241 + 0.747519i \(0.731245\pi\)
\(42\) − 12.3537i − 1.90622i
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) −5.13163 −0.773623
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) − 6.44423i − 0.939988i −0.882670 0.469994i \(-0.844256\pi\)
0.882670 0.469994i \(-0.155744\pi\)
\(48\) − 2.68740i − 0.387893i
\(49\) −14.1316 −2.01880
\(50\) 0 0
\(51\) −12.5969 −1.76392
\(52\) − 1.22212i − 0.169477i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) −3.28432 −0.446939
\(55\) 0 0
\(56\) −4.59692 −0.614289
\(57\) − 12.3537i − 1.63629i
\(58\) − 3.37480i − 0.443133i
\(59\) −9.37480 −1.22049 −0.610247 0.792211i \(-0.708930\pi\)
−0.610247 + 0.792211i \(0.708930\pi\)
\(60\) 0 0
\(61\) 10.9507 1.40209 0.701044 0.713118i \(-0.252717\pi\)
0.701044 + 0.713118i \(0.252717\pi\)
\(62\) − 0.777884i − 0.0987913i
\(63\) 19.4087i 2.44527i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 13.7907 1.69752
\(67\) 15.6381i 1.91049i 0.295810 + 0.955247i \(0.404410\pi\)
−0.295810 + 0.955247i \(0.595590\pi\)
\(68\) 4.68740i 0.568431i
\(69\) 2.68740 0.323525
\(70\) 0 0
\(71\) 1.31260 0.155777 0.0778885 0.996962i \(-0.475182\pi\)
0.0778885 + 0.996962i \(0.475182\pi\)
\(72\) 4.22212i 0.497581i
\(73\) 4.44423i 0.520158i 0.965587 + 0.260079i \(0.0837486\pi\)
−0.965587 + 0.260079i \(0.916251\pi\)
\(74\) −5.81903 −0.676449
\(75\) 0 0
\(76\) −4.59692 −0.527302
\(77\) − 23.5897i − 2.68829i
\(78\) 3.28432i 0.371875i
\(79\) 4.88847 0.549995 0.274998 0.961445i \(-0.411323\pi\)
0.274998 + 0.961445i \(0.411323\pi\)
\(80\) 0 0
\(81\) −3.84008 −0.426676
\(82\) − 8.50643i − 0.939378i
\(83\) 3.81903i 0.419193i 0.977788 + 0.209597i \(0.0672151\pi\)
−0.977788 + 0.209597i \(0.932785\pi\)
\(84\) 12.3537 1.34790
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 9.06943i 0.972345i
\(88\) − 5.13163i − 0.547034i
\(89\) −8.93057 −0.946638 −0.473319 0.880891i \(-0.656944\pi\)
−0.473319 + 0.880891i \(0.656944\pi\)
\(90\) 0 0
\(91\) 5.61797 0.588923
\(92\) − 1.00000i − 0.104257i
\(93\) 2.09048i 0.216773i
\(94\) 6.44423 0.664672
\(95\) 0 0
\(96\) 2.68740 0.274282
\(97\) − 18.0622i − 1.83394i −0.398958 0.916969i \(-0.630628\pi\)
0.398958 0.916969i \(-0.369372\pi\)
\(98\) − 14.1316i − 1.42751i
\(99\) −21.6663 −2.17755
\(100\) 0 0
\(101\) 3.37480 0.335805 0.167903 0.985804i \(-0.446301\pi\)
0.167903 + 0.985804i \(0.446301\pi\)
\(102\) − 12.5969i − 1.24728i
\(103\) − 13.1316i − 1.29390i −0.762533 0.646949i \(-0.776045\pi\)
0.762533 0.646949i \(-0.223955\pi\)
\(104\) 1.22212 0.119838
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 12.3054i 1.18960i 0.803872 + 0.594802i \(0.202770\pi\)
−0.803872 + 0.594802i \(0.797230\pi\)
\(108\) − 3.28432i − 0.316033i
\(109\) 10.5969 1.01500 0.507500 0.861652i \(-0.330570\pi\)
0.507500 + 0.861652i \(0.330570\pi\)
\(110\) 0 0
\(111\) 15.6381 1.48430
\(112\) − 4.59692i − 0.434368i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 12.3537 1.15703
\(115\) 0 0
\(116\) 3.37480 0.313342
\(117\) − 5.15992i − 0.477035i
\(118\) − 9.37480i − 0.863020i
\(119\) −21.5476 −1.97526
\(120\) 0 0
\(121\) 15.3337 1.39397
\(122\) 10.9507i 0.991427i
\(123\) 22.8602i 2.06123i
\(124\) 0.777884 0.0698560
\(125\) 0 0
\(126\) −19.4087 −1.72907
\(127\) − 2.26326i − 0.200832i −0.994946 0.100416i \(-0.967983\pi\)
0.994946 0.100416i \(-0.0320174\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −21.4992 −1.89290
\(130\) 0 0
\(131\) 2.93057 0.256045 0.128022 0.991771i \(-0.459137\pi\)
0.128022 + 0.991771i \(0.459137\pi\)
\(132\) 13.7907i 1.20033i
\(133\) − 21.1316i − 1.83234i
\(134\) −15.6381 −1.35092
\(135\) 0 0
\(136\) −4.68740 −0.401941
\(137\) − 19.7907i − 1.69084i −0.534104 0.845419i \(-0.679351\pi\)
0.534104 0.845419i \(-0.320649\pi\)
\(138\) 2.68740i 0.228767i
\(139\) −6.26326 −0.531243 −0.265622 0.964077i \(-0.585577\pi\)
−0.265622 + 0.964077i \(0.585577\pi\)
\(140\) 0 0
\(141\) −17.3182 −1.45846
\(142\) 1.31260i 0.110151i
\(143\) 6.27145i 0.524445i
\(144\) −4.22212 −0.351843
\(145\) 0 0
\(146\) −4.44423 −0.367807
\(147\) 37.9773i 3.13232i
\(148\) − 5.81903i − 0.478321i
\(149\) 19.7907 1.62132 0.810661 0.585516i \(-0.199108\pi\)
0.810661 + 0.585516i \(0.199108\pi\)
\(150\) 0 0
\(151\) 4.06220 0.330577 0.165289 0.986245i \(-0.447144\pi\)
0.165289 + 0.986245i \(0.447144\pi\)
\(152\) − 4.59692i − 0.372859i
\(153\) 19.7907i 1.59999i
\(154\) 23.5897 1.90091
\(155\) 0 0
\(156\) −3.28432 −0.262956
\(157\) 4.62520i 0.369131i 0.982820 + 0.184566i \(0.0590878\pi\)
−0.982820 + 0.184566i \(0.940912\pi\)
\(158\) 4.88847i 0.388905i
\(159\) 16.1244 1.27875
\(160\) 0 0
\(161\) 4.59692 0.362288
\(162\) − 3.84008i − 0.301705i
\(163\) − 12.4159i − 0.972492i −0.873822 0.486246i \(-0.838366\pi\)
0.873822 0.486246i \(-0.161634\pi\)
\(164\) 8.50643 0.664241
\(165\) 0 0
\(166\) −3.81903 −0.296414
\(167\) 12.8885i 0.997339i 0.866792 + 0.498670i \(0.166178\pi\)
−0.866792 + 0.498670i \(0.833822\pi\)
\(168\) 12.3537i 0.953112i
\(169\) 11.5064 0.885110
\(170\) 0 0
\(171\) −19.4087 −1.48422
\(172\) 8.00000i 0.609994i
\(173\) 10.2432i 0.778774i 0.921074 + 0.389387i \(0.127313\pi\)
−0.921074 + 0.389387i \(0.872687\pi\)
\(174\) −9.06943 −0.687552
\(175\) 0 0
\(176\) 5.13163 0.386811
\(177\) 25.1938i 1.89368i
\(178\) − 8.93057i − 0.669374i
\(179\) 13.1938 0.986153 0.493077 0.869986i \(-0.335872\pi\)
0.493077 + 0.869986i \(0.335872\pi\)
\(180\) 0 0
\(181\) 15.7907 1.17372 0.586858 0.809690i \(-0.300364\pi\)
0.586858 + 0.809690i \(0.300364\pi\)
\(182\) 5.61797i 0.416431i
\(183\) − 29.4288i − 2.17544i
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) −2.09048 −0.153282
\(187\) − 24.0540i − 1.75900i
\(188\) 6.44423i 0.469994i
\(189\) 15.0977 1.09820
\(190\) 0 0
\(191\) 16.1244 1.16672 0.583360 0.812214i \(-0.301738\pi\)
0.583360 + 0.812214i \(0.301738\pi\)
\(192\) 2.68740i 0.193946i
\(193\) 17.9434i 1.29160i 0.763508 + 0.645798i \(0.223475\pi\)
−0.763508 + 0.645798i \(0.776525\pi\)
\(194\) 18.0622 1.29679
\(195\) 0 0
\(196\) 14.1316 1.00940
\(197\) 5.88123i 0.419020i 0.977806 + 0.209510i \(0.0671869\pi\)
−0.977806 + 0.209510i \(0.932813\pi\)
\(198\) − 21.6663i − 1.53976i
\(199\) −6.56863 −0.465638 −0.232819 0.972520i \(-0.574795\pi\)
−0.232819 + 0.972520i \(0.574795\pi\)
\(200\) 0 0
\(201\) 42.0257 2.96427
\(202\) 3.37480i 0.237450i
\(203\) 15.5137i 1.08885i
\(204\) 12.5969 0.881960
\(205\) 0 0
\(206\) 13.1316 0.914924
\(207\) − 4.22212i − 0.293457i
\(208\) 1.22212i 0.0847385i
\(209\) 23.5897 1.63173
\(210\) 0 0
\(211\) −15.4571 −1.06411 −0.532055 0.846710i \(-0.678580\pi\)
−0.532055 + 0.846710i \(0.678580\pi\)
\(212\) − 6.00000i − 0.412082i
\(213\) − 3.52748i − 0.241699i
\(214\) −12.3054 −0.841177
\(215\) 0 0
\(216\) 3.28432 0.223469
\(217\) 3.57587i 0.242746i
\(218\) 10.5969i 0.717714i
\(219\) 11.9434 0.807062
\(220\) 0 0
\(221\) 5.72855 0.385344
\(222\) 15.6381i 1.04956i
\(223\) 10.7496i 0.719846i 0.932982 + 0.359923i \(0.117197\pi\)
−0.932982 + 0.359923i \(0.882803\pi\)
\(224\) 4.59692 0.307144
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) − 12.3054i − 0.816736i −0.912817 0.408368i \(-0.866098\pi\)
0.912817 0.408368i \(-0.133902\pi\)
\(228\) 12.3537i 0.818147i
\(229\) −9.63806 −0.636901 −0.318451 0.947939i \(-0.603162\pi\)
−0.318451 + 0.947939i \(0.603162\pi\)
\(230\) 0 0
\(231\) −63.3949 −4.17108
\(232\) 3.37480i 0.221566i
\(233\) − 13.9434i − 0.913464i −0.889604 0.456732i \(-0.849020\pi\)
0.889604 0.456732i \(-0.150980\pi\)
\(234\) 5.15992 0.337314
\(235\) 0 0
\(236\) 9.37480 0.610247
\(237\) − 13.1373i − 0.853357i
\(238\) − 21.5476i − 1.39672i
\(239\) 26.3877 1.70688 0.853438 0.521194i \(-0.174513\pi\)
0.853438 + 0.521194i \(0.174513\pi\)
\(240\) 0 0
\(241\) −7.06943 −0.455382 −0.227691 0.973733i \(-0.573118\pi\)
−0.227691 + 0.973733i \(0.573118\pi\)
\(242\) 15.3337i 0.985684i
\(243\) 20.1728i 1.29408i
\(244\) −10.9507 −0.701044
\(245\) 0 0
\(246\) −22.8602 −1.45751
\(247\) 5.61797i 0.357463i
\(248\) 0.777884i 0.0493957i
\(249\) 10.2633 0.650408
\(250\) 0 0
\(251\) −24.9023 −1.57182 −0.785909 0.618342i \(-0.787805\pi\)
−0.785909 + 0.618342i \(0.787805\pi\)
\(252\) − 19.4087i − 1.22263i
\(253\) 5.13163i 0.322623i
\(254\) 2.26326 0.142010
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 0.444233i 0.0277105i 0.999904 + 0.0138552i \(0.00441040\pi\)
−0.999904 + 0.0138552i \(0.995590\pi\)
\(258\) − 21.4992i − 1.33848i
\(259\) 26.7496 1.66214
\(260\) 0 0
\(261\) 14.2488 0.881978
\(262\) 2.93057i 0.181051i
\(263\) 23.8812i 1.47258i 0.676666 + 0.736290i \(0.263424\pi\)
−0.676666 + 0.736290i \(0.736576\pi\)
\(264\) −13.7907 −0.848762
\(265\) 0 0
\(266\) 21.1316 1.29566
\(267\) 24.0000i 1.46878i
\(268\) − 15.6381i − 0.955247i
\(269\) −16.2633 −0.991589 −0.495794 0.868440i \(-0.665123\pi\)
−0.495794 + 0.868440i \(0.665123\pi\)
\(270\) 0 0
\(271\) 25.6098 1.55568 0.777842 0.628460i \(-0.216315\pi\)
0.777842 + 0.628460i \(0.216315\pi\)
\(272\) − 4.68740i − 0.284215i
\(273\) − 15.0977i − 0.913756i
\(274\) 19.7907 1.19560
\(275\) 0 0
\(276\) −2.68740 −0.161762
\(277\) 2.88847i 0.173551i 0.996228 + 0.0867755i \(0.0276563\pi\)
−0.996228 + 0.0867755i \(0.972344\pi\)
\(278\) − 6.26326i − 0.375646i
\(279\) 3.28432 0.196627
\(280\) 0 0
\(281\) 4.26326 0.254325 0.127163 0.991882i \(-0.459413\pi\)
0.127163 + 0.991882i \(0.459413\pi\)
\(282\) − 17.3182i − 1.03129i
\(283\) − 30.5686i − 1.81712i −0.417758 0.908558i \(-0.637184\pi\)
0.417758 0.908558i \(-0.362816\pi\)
\(284\) −1.31260 −0.0778885
\(285\) 0 0
\(286\) −6.27145 −0.370839
\(287\) 39.1033i 2.30820i
\(288\) − 4.22212i − 0.248791i
\(289\) −4.97171 −0.292454
\(290\) 0 0
\(291\) −48.5403 −2.84549
\(292\) − 4.44423i − 0.260079i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) −37.9773 −2.21488
\(295\) 0 0
\(296\) 5.81903 0.338224
\(297\) 16.8539i 0.977962i
\(298\) 19.7907i 1.14645i
\(299\) −1.22212 −0.0706768
\(300\) 0 0
\(301\) −36.7753 −2.11969
\(302\) 4.06220i 0.233753i
\(303\) − 9.06943i − 0.521025i
\(304\) 4.59692 0.263651
\(305\) 0 0
\(306\) −19.7907 −1.13136
\(307\) − 8.54853i − 0.487891i −0.969789 0.243945i \(-0.921558\pi\)
0.969789 0.243945i \(-0.0784417\pi\)
\(308\) 23.5897i 1.34415i
\(309\) −35.2899 −2.00757
\(310\) 0 0
\(311\) −7.63806 −0.433115 −0.216557 0.976270i \(-0.569483\pi\)
−0.216557 + 0.976270i \(0.569483\pi\)
\(312\) − 3.28432i − 0.185938i
\(313\) 18.2350i 1.03070i 0.856979 + 0.515351i \(0.172338\pi\)
−0.856979 + 0.515351i \(0.827662\pi\)
\(314\) −4.62520 −0.261015
\(315\) 0 0
\(316\) −4.88847 −0.274998
\(317\) − 16.8602i − 0.946962i −0.880804 0.473481i \(-0.842997\pi\)
0.880804 0.473481i \(-0.157003\pi\)
\(318\) 16.1244i 0.904211i
\(319\) −17.3182 −0.969635
\(320\) 0 0
\(321\) 33.0694 1.84576
\(322\) 4.59692i 0.256176i
\(323\) − 21.5476i − 1.19894i
\(324\) 3.84008 0.213338
\(325\) 0 0
\(326\) 12.4159 0.687656
\(327\) − 28.4781i − 1.57485i
\(328\) 8.50643i 0.469689i
\(329\) −29.6236 −1.63320
\(330\) 0 0
\(331\) 19.1517 1.05267 0.526337 0.850276i \(-0.323565\pi\)
0.526337 + 0.850276i \(0.323565\pi\)
\(332\) − 3.81903i − 0.209597i
\(333\) − 24.5686i − 1.34635i
\(334\) −12.8885 −0.705225
\(335\) 0 0
\(336\) −12.3537 −0.673952
\(337\) 1.70845i 0.0930652i 0.998917 + 0.0465326i \(0.0148171\pi\)
−0.998917 + 0.0465326i \(0.985183\pi\)
\(338\) 11.5064i 0.625867i
\(339\) 16.1244 0.875757
\(340\) 0 0
\(341\) −3.99181 −0.216169
\(342\) − 19.4087i − 1.04950i
\(343\) 32.7835i 1.77014i
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) −10.2432 −0.550676
\(347\) 10.6874i 0.573730i 0.957971 + 0.286865i \(0.0926131\pi\)
−0.957971 + 0.286865i \(0.907387\pi\)
\(348\) − 9.06943i − 0.486173i
\(349\) 12.3877 0.663096 0.331548 0.943438i \(-0.392429\pi\)
0.331548 + 0.943438i \(0.392429\pi\)
\(350\) 0 0
\(351\) −4.01382 −0.214242
\(352\) 5.13163i 0.273517i
\(353\) − 5.45710i − 0.290452i −0.989399 0.145226i \(-0.953609\pi\)
0.989399 0.145226i \(-0.0463909\pi\)
\(354\) −25.1938 −1.33904
\(355\) 0 0
\(356\) 8.93057 0.473319
\(357\) 57.9070i 3.06476i
\(358\) 13.1938i 0.697316i
\(359\) −20.1810 −1.06511 −0.532555 0.846395i \(-0.678768\pi\)
−0.532555 + 0.846395i \(0.678768\pi\)
\(360\) 0 0
\(361\) 2.13163 0.112191
\(362\) 15.7907i 0.829943i
\(363\) − 41.2076i − 2.16284i
\(364\) −5.61797 −0.294461
\(365\) 0 0
\(366\) 29.4288 1.53827
\(367\) 2.74960i 0.143528i 0.997422 + 0.0717639i \(0.0228628\pi\)
−0.997422 + 0.0717639i \(0.977137\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) 35.9151 1.86967
\(370\) 0 0
\(371\) 27.5815 1.43196
\(372\) − 2.09048i − 0.108387i
\(373\) 12.0823i 0.625598i 0.949819 + 0.312799i \(0.101267\pi\)
−0.949819 + 0.312799i \(0.898733\pi\)
\(374\) 24.0540 1.24380
\(375\) 0 0
\(376\) −6.44423 −0.332336
\(377\) − 4.12440i − 0.212417i
\(378\) 15.0977i 0.776543i
\(379\) −5.25603 −0.269984 −0.134992 0.990847i \(-0.543101\pi\)
−0.134992 + 0.990847i \(0.543101\pi\)
\(380\) 0 0
\(381\) −6.08230 −0.311605
\(382\) 16.1244i 0.824996i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) −2.68740 −0.137141
\(385\) 0 0
\(386\) −17.9434 −0.913297
\(387\) 33.7769i 1.71698i
\(388\) 18.0622i 0.916969i
\(389\) −0.325463 −0.0165017 −0.00825083 0.999966i \(-0.502626\pi\)
−0.00825083 + 0.999966i \(0.502626\pi\)
\(390\) 0 0
\(391\) 4.68740 0.237052
\(392\) 14.1316i 0.713755i
\(393\) − 7.87560i − 0.397272i
\(394\) −5.88123 −0.296292
\(395\) 0 0
\(396\) 21.6663 1.08878
\(397\) − 17.7568i − 0.891190i −0.895235 0.445595i \(-0.852992\pi\)
0.895235 0.445595i \(-0.147008\pi\)
\(398\) − 6.56863i − 0.329256i
\(399\) −56.7891 −2.84301
\(400\) 0 0
\(401\) 3.91770 0.195641 0.0978204 0.995204i \(-0.468813\pi\)
0.0978204 + 0.995204i \(0.468813\pi\)
\(402\) 42.0257i 2.09605i
\(403\) − 0.950664i − 0.0473560i
\(404\) −3.37480 −0.167903
\(405\) 0 0
\(406\) −15.5137 −0.769930
\(407\) 29.8611i 1.48016i
\(408\) 12.5969i 0.623640i
\(409\) −9.58405 −0.473901 −0.236950 0.971522i \(-0.576148\pi\)
−0.236950 + 0.971522i \(0.576148\pi\)
\(410\) 0 0
\(411\) −53.1856 −2.62345
\(412\) 13.1316i 0.646949i
\(413\) 43.0952i 2.12057i
\(414\) 4.22212 0.207506
\(415\) 0 0
\(416\) −1.22212 −0.0599192
\(417\) 16.8319i 0.824261i
\(418\) 23.5897i 1.15381i
\(419\) 20.5265 1.00279 0.501393 0.865219i \(-0.332821\pi\)
0.501393 + 0.865219i \(0.332821\pi\)
\(420\) 0 0
\(421\) 2.29155 0.111683 0.0558417 0.998440i \(-0.482216\pi\)
0.0558417 + 0.998440i \(0.482216\pi\)
\(422\) − 15.4571i − 0.752440i
\(423\) 27.2083i 1.32291i
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 3.52748 0.170907
\(427\) − 50.3393i − 2.43609i
\(428\) − 12.3054i − 0.594802i
\(429\) 16.8539 0.813714
\(430\) 0 0
\(431\) −8.83189 −0.425417 −0.212709 0.977116i \(-0.568229\pi\)
−0.212709 + 0.977116i \(0.568229\pi\)
\(432\) 3.28432i 0.158017i
\(433\) − 33.3465i − 1.60253i −0.598309 0.801266i \(-0.704160\pi\)
0.598309 0.801266i \(-0.295840\pi\)
\(434\) −3.57587 −0.171647
\(435\) 0 0
\(436\) −10.5969 −0.507500
\(437\) 4.59692i 0.219900i
\(438\) 11.9434i 0.570679i
\(439\) 6.02829 0.287714 0.143857 0.989598i \(-0.454049\pi\)
0.143857 + 0.989598i \(0.454049\pi\)
\(440\) 0 0
\(441\) 59.6654 2.84121
\(442\) 5.72855i 0.272479i
\(443\) 15.5275i 0.737733i 0.929482 + 0.368866i \(0.120254\pi\)
−0.929482 + 0.368866i \(0.879746\pi\)
\(444\) −15.6381 −0.742150
\(445\) 0 0
\(446\) −10.7496 −0.509008
\(447\) − 53.1856i − 2.51559i
\(448\) 4.59692i 0.217184i
\(449\) 18.3594 0.866433 0.433216 0.901290i \(-0.357379\pi\)
0.433216 + 0.901290i \(0.357379\pi\)
\(450\) 0 0
\(451\) −43.6519 −2.05549
\(452\) − 6.00000i − 0.282216i
\(453\) − 10.9168i − 0.512914i
\(454\) 12.3054 0.577519
\(455\) 0 0
\(456\) −12.3537 −0.578517
\(457\) − 11.4992i − 0.537910i −0.963153 0.268955i \(-0.913322\pi\)
0.963153 0.268955i \(-0.0866782\pi\)
\(458\) − 9.63806i − 0.450357i
\(459\) 15.3949 0.718572
\(460\) 0 0
\(461\) 1.33270 0.0620700 0.0310350 0.999518i \(-0.490120\pi\)
0.0310350 + 0.999518i \(0.490120\pi\)
\(462\) − 63.3949i − 2.94940i
\(463\) − 35.8190i − 1.66465i −0.554287 0.832326i \(-0.687009\pi\)
0.554287 0.832326i \(-0.312991\pi\)
\(464\) −3.37480 −0.156671
\(465\) 0 0
\(466\) 13.9434 0.645917
\(467\) − 23.7625i − 1.09960i −0.835298 0.549798i \(-0.814705\pi\)
0.835298 0.549798i \(-0.185295\pi\)
\(468\) 5.15992i 0.238517i
\(469\) 71.8869 3.31943
\(470\) 0 0
\(471\) 12.4298 0.572733
\(472\) 9.37480i 0.431510i
\(473\) − 41.0531i − 1.88762i
\(474\) 13.1373 0.603414
\(475\) 0 0
\(476\) 21.5476 0.987632
\(477\) − 25.3327i − 1.15990i
\(478\) 26.3877i 1.20694i
\(479\) 2.04210 0.0933060 0.0466530 0.998911i \(-0.485145\pi\)
0.0466530 + 0.998911i \(0.485145\pi\)
\(480\) 0 0
\(481\) −7.11153 −0.324258
\(482\) − 7.06943i − 0.322004i
\(483\) − 12.3537i − 0.562115i
\(484\) −15.3337 −0.696984
\(485\) 0 0
\(486\) −20.1728 −0.915056
\(487\) − 18.6252i − 0.843988i −0.906599 0.421994i \(-0.861330\pi\)
0.906599 0.421994i \(-0.138670\pi\)
\(488\) − 10.9507i − 0.495713i
\(489\) −33.3666 −1.50889
\(490\) 0 0
\(491\) 33.7204 1.52178 0.760889 0.648882i \(-0.224763\pi\)
0.760889 + 0.648882i \(0.224763\pi\)
\(492\) − 22.8602i − 1.03062i
\(493\) 15.8190i 0.712453i
\(494\) −5.61797 −0.252764
\(495\) 0 0
\(496\) −0.777884 −0.0349280
\(497\) − 6.03391i − 0.270658i
\(498\) 10.2633i 0.459908i
\(499\) 16.8885 0.756032 0.378016 0.925799i \(-0.376607\pi\)
0.378016 + 0.925799i \(0.376607\pi\)
\(500\) 0 0
\(501\) 34.6365 1.54744
\(502\) − 24.9023i − 1.11144i
\(503\) − 11.4031i − 0.508438i −0.967147 0.254219i \(-0.918182\pi\)
0.967147 0.254219i \(-0.0818185\pi\)
\(504\) 19.4087 0.864533
\(505\) 0 0
\(506\) −5.13163 −0.228129
\(507\) − 30.9224i − 1.37331i
\(508\) 2.26326i 0.100416i
\(509\) 1.87560 0.0831346 0.0415673 0.999136i \(-0.486765\pi\)
0.0415673 + 0.999136i \(0.486765\pi\)
\(510\) 0 0
\(511\) 20.4298 0.903759
\(512\) 1.00000i 0.0441942i
\(513\) 15.0977i 0.666581i
\(514\) −0.444233 −0.0195943
\(515\) 0 0
\(516\) 21.4992 0.946449
\(517\) − 33.0694i − 1.45439i
\(518\) 26.7496i 1.17531i
\(519\) 27.5275 1.20832
\(520\) 0 0
\(521\) 5.11153 0.223940 0.111970 0.993712i \(-0.464284\pi\)
0.111970 + 0.993712i \(0.464284\pi\)
\(522\) 14.2488i 0.623653i
\(523\) − 19.4571i − 0.850799i −0.905006 0.425400i \(-0.860134\pi\)
0.905006 0.425400i \(-0.139866\pi\)
\(524\) −2.93057 −0.128022
\(525\) 0 0
\(526\) −23.8812 −1.04127
\(527\) 3.64625i 0.158833i
\(528\) − 13.7907i − 0.600165i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 39.5815 1.71769
\(532\) 21.1316i 0.916172i
\(533\) − 10.3958i − 0.450294i
\(534\) −24.0000 −1.03858
\(535\) 0 0
\(536\) 15.6381 0.675461
\(537\) − 35.4571i − 1.53009i
\(538\) − 16.2633i − 0.701159i
\(539\) −72.5183 −3.12359
\(540\) 0 0
\(541\) −2.70750 −0.116404 −0.0582022 0.998305i \(-0.518537\pi\)
−0.0582022 + 0.998305i \(0.518537\pi\)
\(542\) 25.6098i 1.10003i
\(543\) − 42.4360i − 1.82111i
\(544\) 4.68740 0.200971
\(545\) 0 0
\(546\) 15.0977 0.646123
\(547\) − 1.66635i − 0.0712479i −0.999365 0.0356240i \(-0.988658\pi\)
0.999365 0.0356240i \(-0.0113419\pi\)
\(548\) 19.7907i 0.845419i
\(549\) −46.2350 −1.97326
\(550\) 0 0
\(551\) −15.5137 −0.660904
\(552\) − 2.68740i − 0.114383i
\(553\) − 22.4719i − 0.955601i
\(554\) −2.88847 −0.122719
\(555\) 0 0
\(556\) 6.26326 0.265622
\(557\) − 20.9306i − 0.886857i −0.896310 0.443428i \(-0.853762\pi\)
0.896310 0.443428i \(-0.146238\pi\)
\(558\) 3.28432i 0.139036i
\(559\) 9.77693 0.413520
\(560\) 0 0
\(561\) −64.6427 −2.72922
\(562\) 4.26326i 0.179835i
\(563\) 15.2761i 0.643812i 0.946772 + 0.321906i \(0.104324\pi\)
−0.946772 + 0.321906i \(0.895676\pi\)
\(564\) 17.3182 0.729229
\(565\) 0 0
\(566\) 30.5686 1.28490
\(567\) 17.6525i 0.741337i
\(568\) − 1.31260i − 0.0550755i
\(569\) −20.3877 −0.854695 −0.427348 0.904087i \(-0.640552\pi\)
−0.427348 + 0.904087i \(0.640552\pi\)
\(570\) 0 0
\(571\) 5.49357 0.229899 0.114949 0.993371i \(-0.463329\pi\)
0.114949 + 0.993371i \(0.463329\pi\)
\(572\) − 6.27145i − 0.262223i
\(573\) − 43.3327i − 1.81025i
\(574\) −39.1033 −1.63214
\(575\) 0 0
\(576\) 4.22212 0.175922
\(577\) 17.2761i 0.719215i 0.933104 + 0.359607i \(0.117089\pi\)
−0.933104 + 0.359607i \(0.882911\pi\)
\(578\) − 4.97171i − 0.206796i
\(579\) 48.2212 2.00400
\(580\) 0 0
\(581\) 17.5558 0.728336
\(582\) − 48.5403i − 2.01206i
\(583\) 30.7898i 1.27518i
\(584\) 4.44423 0.183904
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) 31.8247i 1.31354i 0.754089 + 0.656772i \(0.228079\pi\)
−0.754089 + 0.656772i \(0.771921\pi\)
\(588\) − 37.9773i − 1.56616i
\(589\) −3.57587 −0.147341
\(590\) 0 0
\(591\) 15.8052 0.650140
\(592\) 5.81903i 0.239161i
\(593\) 12.4442i 0.511023i 0.966806 + 0.255512i \(0.0822439\pi\)
−0.966806 + 0.255512i \(0.917756\pi\)
\(594\) −16.8539 −0.691524
\(595\) 0 0
\(596\) −19.7907 −0.810661
\(597\) 17.6525i 0.722470i
\(598\) − 1.22212i − 0.0499761i
\(599\) −17.4370 −0.712456 −0.356228 0.934399i \(-0.615937\pi\)
−0.356228 + 0.934399i \(0.615937\pi\)
\(600\) 0 0
\(601\) −0.916751 −0.0373951 −0.0186975 0.999825i \(-0.505952\pi\)
−0.0186975 + 0.999825i \(0.505952\pi\)
\(602\) − 36.7753i − 1.49885i
\(603\) − 66.0257i − 2.68878i
\(604\) −4.06220 −0.165289
\(605\) 0 0
\(606\) 9.06943 0.368421
\(607\) 36.7753i 1.49266i 0.665574 + 0.746332i \(0.268187\pi\)
−0.665574 + 0.746332i \(0.731813\pi\)
\(608\) 4.59692i 0.186430i
\(609\) 41.6914 1.68942
\(610\) 0 0
\(611\) 7.87560 0.318613
\(612\) − 19.7907i − 0.799994i
\(613\) − 4.38766i − 0.177216i −0.996067 0.0886080i \(-0.971758\pi\)
0.996067 0.0886080i \(-0.0282418\pi\)
\(614\) 8.54853 0.344991
\(615\) 0 0
\(616\) −23.5897 −0.950455
\(617\) 40.4499i 1.62845i 0.580549 + 0.814225i \(0.302838\pi\)
−0.580549 + 0.814225i \(0.697162\pi\)
\(618\) − 35.2899i − 1.41957i
\(619\) 39.8165 1.60036 0.800180 0.599760i \(-0.204737\pi\)
0.800180 + 0.599760i \(0.204737\pi\)
\(620\) 0 0
\(621\) −3.28432 −0.131795
\(622\) − 7.63806i − 0.306258i
\(623\) 41.0531i 1.64476i
\(624\) 3.28432 0.131478
\(625\) 0 0
\(626\) −18.2350 −0.728816
\(627\) − 63.3949i − 2.53175i
\(628\) − 4.62520i − 0.184566i
\(629\) 27.2761 1.08757
\(630\) 0 0
\(631\) 25.9013 1.03112 0.515558 0.856855i \(-0.327585\pi\)
0.515558 + 0.856855i \(0.327585\pi\)
\(632\) − 4.88847i − 0.194453i
\(633\) 41.5394i 1.65104i
\(634\) 16.8602 0.669603
\(635\) 0 0
\(636\) −16.1244 −0.639374
\(637\) − 17.2705i − 0.684282i
\(638\) − 17.3182i − 0.685635i
\(639\) −5.54195 −0.219236
\(640\) 0 0
\(641\) −43.8448 −1.73176 −0.865882 0.500248i \(-0.833242\pi\)
−0.865882 + 0.500248i \(0.833242\pi\)
\(642\) 33.0694i 1.30515i
\(643\) − 3.94343i − 0.155514i −0.996972 0.0777568i \(-0.975224\pi\)
0.996972 0.0777568i \(-0.0247758\pi\)
\(644\) −4.59692 −0.181144
\(645\) 0 0
\(646\) 21.5476 0.847778
\(647\) 24.3456i 0.957123i 0.878054 + 0.478561i \(0.158842\pi\)
−0.878054 + 0.478561i \(0.841158\pi\)
\(648\) 3.84008i 0.150853i
\(649\) −48.1080 −1.88841
\(650\) 0 0
\(651\) 9.60978 0.376637
\(652\) 12.4159i 0.486246i
\(653\) − 37.6921i − 1.47500i −0.675344 0.737502i \(-0.736005\pi\)
0.675344 0.737502i \(-0.263995\pi\)
\(654\) 28.4781 1.11358
\(655\) 0 0
\(656\) −8.50643 −0.332120
\(657\) − 18.7641i − 0.732056i
\(658\) − 29.6236i − 1.15485i
\(659\) −24.6107 −0.958698 −0.479349 0.877624i \(-0.659127\pi\)
−0.479349 + 0.877624i \(0.659127\pi\)
\(660\) 0 0
\(661\) 27.3126 1.06234 0.531169 0.847266i \(-0.321753\pi\)
0.531169 + 0.847266i \(0.321753\pi\)
\(662\) 19.1517i 0.744353i
\(663\) − 15.3949i − 0.597888i
\(664\) 3.81903 0.148207
\(665\) 0 0
\(666\) 24.5686 0.952015
\(667\) − 3.37480i − 0.130673i
\(668\) − 12.8885i − 0.498670i
\(669\) 28.8885 1.11689
\(670\) 0 0
\(671\) 56.1948 2.16938
\(672\) − 12.3537i − 0.476556i
\(673\) − 22.5265i − 0.868334i −0.900832 0.434167i \(-0.857043\pi\)
0.900832 0.434167i \(-0.142957\pi\)
\(674\) −1.70845 −0.0658070
\(675\) 0 0
\(676\) −11.5064 −0.442555
\(677\) − 6.00000i − 0.230599i −0.993331 0.115299i \(-0.963217\pi\)
0.993331 0.115299i \(-0.0367827\pi\)
\(678\) 16.1244i 0.619254i
\(679\) −83.0304 −3.18641
\(680\) 0 0
\(681\) −33.0694 −1.26722
\(682\) − 3.99181i − 0.152854i
\(683\) 31.9974i 1.22435i 0.790723 + 0.612174i \(0.209705\pi\)
−0.790723 + 0.612174i \(0.790295\pi\)
\(684\) 19.4087 0.742111
\(685\) 0 0
\(686\) −32.7835 −1.25168
\(687\) 25.9013i 0.988197i
\(688\) − 8.00000i − 0.304997i
\(689\) −7.33270 −0.279354
\(690\) 0 0
\(691\) −2.80617 −0.106752 −0.0533758 0.998574i \(-0.516998\pi\)
−0.0533758 + 0.998574i \(0.516998\pi\)
\(692\) − 10.2432i − 0.389387i
\(693\) 99.5984i 3.78343i
\(694\) −10.6874 −0.405688
\(695\) 0 0
\(696\) 9.06943 0.343776
\(697\) 39.8730i 1.51030i
\(698\) 12.3877i 0.468880i
\(699\) −37.4716 −1.41730
\(700\) 0 0
\(701\) 8.37385 0.316276 0.158138 0.987417i \(-0.449451\pi\)
0.158138 + 0.987417i \(0.449451\pi\)
\(702\) − 4.01382i − 0.151492i
\(703\) 26.7496i 1.00888i
\(704\) −5.13163 −0.193406
\(705\) 0 0
\(706\) 5.45710 0.205381
\(707\) − 15.5137i − 0.583451i
\(708\) − 25.1938i − 0.946842i
\(709\) −21.2139 −0.796706 −0.398353 0.917232i \(-0.630418\pi\)
−0.398353 + 0.917232i \(0.630418\pi\)
\(710\) 0 0
\(711\) −20.6397 −0.774048
\(712\) 8.93057i 0.334687i
\(713\) − 0.777884i − 0.0291320i
\(714\) −57.9070 −2.16711
\(715\) 0 0
\(716\) −13.1938 −0.493077
\(717\) − 70.9142i − 2.64834i
\(718\) − 20.1810i − 0.753147i
\(719\) 28.1106 1.04835 0.524174 0.851611i \(-0.324374\pi\)
0.524174 + 0.851611i \(0.324374\pi\)
\(720\) 0 0
\(721\) −60.3650 −2.24811
\(722\) 2.13163i 0.0793311i
\(723\) 18.9984i 0.706558i
\(724\) −15.7907 −0.586858
\(725\) 0 0
\(726\) 41.2076 1.52936
\(727\) − 39.0330i − 1.44765i −0.689982 0.723826i \(-0.742382\pi\)
0.689982 0.723826i \(-0.257618\pi\)
\(728\) − 5.61797i − 0.208216i
\(729\) 42.6921 1.58119
\(730\) 0 0
\(731\) −37.4992 −1.38696
\(732\) 29.4288i 1.08772i
\(733\) − 47.1373i − 1.74105i −0.492120 0.870527i \(-0.663778\pi\)
0.492120 0.870527i \(-0.336222\pi\)
\(734\) −2.74960 −0.101490
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 80.2488i 2.95600i
\(738\) 35.9151i 1.32205i
\(739\) 24.5265 0.902223 0.451111 0.892468i \(-0.351028\pi\)
0.451111 + 0.892468i \(0.351028\pi\)
\(740\) 0 0
\(741\) 15.0977 0.554629
\(742\) 27.5815i 1.01255i
\(743\) 7.34651i 0.269517i 0.990878 + 0.134759i \(0.0430259\pi\)
−0.990878 + 0.134759i \(0.956974\pi\)
\(744\) 2.09048 0.0766409
\(745\) 0 0
\(746\) −12.0823 −0.442364
\(747\) − 16.1244i − 0.589961i
\(748\) 24.0540i 0.879502i
\(749\) 56.5667 2.06690
\(750\) 0 0
\(751\) 11.2359 0.410005 0.205002 0.978761i \(-0.434280\pi\)
0.205002 + 0.978761i \(0.434280\pi\)
\(752\) − 6.44423i − 0.234997i
\(753\) 66.9224i 2.43879i
\(754\) 4.12440 0.150202
\(755\) 0 0
\(756\) −15.0977 −0.549099
\(757\) 49.1794i 1.78745i 0.448611 + 0.893727i \(0.351919\pi\)
−0.448611 + 0.893727i \(0.648081\pi\)
\(758\) − 5.25603i − 0.190908i
\(759\) 13.7907 0.500572
\(760\) 0 0
\(761\) 43.2478 1.56773 0.783867 0.620929i \(-0.213245\pi\)
0.783867 + 0.620929i \(0.213245\pi\)
\(762\) − 6.08230i − 0.220338i
\(763\) − 48.7131i − 1.76353i
\(764\) −16.1244 −0.583360
\(765\) 0 0
\(766\) 0 0
\(767\) − 11.4571i − 0.413692i
\(768\) − 2.68740i − 0.0969732i
\(769\) 39.6638 1.43031 0.715156 0.698964i \(-0.246355\pi\)
0.715156 + 0.698964i \(0.246355\pi\)
\(770\) 0 0
\(771\) 1.19383 0.0429948
\(772\) − 17.9434i − 0.645798i
\(773\) 2.18097i 0.0784440i 0.999231 + 0.0392220i \(0.0124879\pi\)
−0.999231 + 0.0392220i \(0.987512\pi\)
\(774\) −33.7769 −1.21409
\(775\) 0 0
\(776\) −18.0622 −0.648395
\(777\) − 71.8869i − 2.57893i
\(778\) − 0.325463i − 0.0116684i
\(779\) −39.1033 −1.40102
\(780\) 0 0
\(781\) 6.73578 0.241025
\(782\) 4.68740i 0.167621i
\(783\) − 11.0839i − 0.396106i
\(784\) −14.1316 −0.504701
\(785\) 0 0
\(786\) 7.87560 0.280913
\(787\) 4.76407i 0.169821i 0.996389 + 0.0849103i \(0.0270604\pi\)
−0.996389 + 0.0849103i \(0.972940\pi\)
\(788\) − 5.88123i − 0.209510i
\(789\) 64.1784 2.28481
\(790\) 0 0
\(791\) 27.5815 0.980685
\(792\) 21.6663i 0.769880i
\(793\) 13.3830i 0.475244i
\(794\) 17.7568 0.630166
\(795\) 0 0
\(796\) 6.56863 0.232819
\(797\) 39.7204i 1.40697i 0.710711 + 0.703484i \(0.248373\pi\)
−0.710711 + 0.703484i \(0.751627\pi\)
\(798\) − 56.7891i − 2.01031i
\(799\) −30.2067 −1.06864
\(800\) 0 0
\(801\) 37.7059 1.33227
\(802\) 3.91770i 0.138339i
\(803\) 22.8062i 0.804812i
\(804\) −42.0257 −1.48213
\(805\) 0 0
\(806\) 0.950664 0.0334857
\(807\) 43.7059i 1.53852i
\(808\) − 3.37480i − 0.118725i
\(809\) −46.8941 −1.64871 −0.824354 0.566074i \(-0.808462\pi\)
−0.824354 + 0.566074i \(0.808462\pi\)
\(810\) 0 0
\(811\) −36.8319 −1.29334 −0.646671 0.762769i \(-0.723839\pi\)
−0.646671 + 0.762769i \(0.723839\pi\)
\(812\) − 15.5137i − 0.544423i
\(813\) − 68.8237i − 2.41375i
\(814\) −29.8611 −1.04663
\(815\) 0 0
\(816\) −12.5969 −0.440980
\(817\) − 36.7753i − 1.28661i
\(818\) − 9.58405i − 0.335099i
\(819\) −23.7197 −0.828834
\(820\) 0 0
\(821\) −18.6107 −0.649519 −0.324759 0.945797i \(-0.605283\pi\)
−0.324759 + 0.945797i \(0.605283\pi\)
\(822\) − 53.1856i − 1.85506i
\(823\) 9.31823i 0.324813i 0.986724 + 0.162407i \(0.0519256\pi\)
−0.986724 + 0.162407i \(0.948074\pi\)
\(824\) −13.1316 −0.457462
\(825\) 0 0
\(826\) −43.0952 −1.49947
\(827\) 53.0129i 1.84344i 0.387858 + 0.921719i \(0.373215\pi\)
−0.387858 + 0.921719i \(0.626785\pi\)
\(828\) 4.22212i 0.146729i
\(829\) 25.2761 0.877876 0.438938 0.898517i \(-0.355355\pi\)
0.438938 + 0.898517i \(0.355355\pi\)
\(830\) 0 0
\(831\) 7.76246 0.269277
\(832\) − 1.22212i − 0.0423693i
\(833\) 66.2406i 2.29510i
\(834\) −16.8319 −0.582841
\(835\) 0 0
\(836\) −23.5897 −0.815866
\(837\) − 2.55481i − 0.0883073i
\(838\) 20.5265i 0.709077i
\(839\) −11.6946 −0.403744 −0.201872 0.979412i \(-0.564702\pi\)
−0.201872 + 0.979412i \(0.564702\pi\)
\(840\) 0 0
\(841\) −17.6107 −0.607267
\(842\) 2.29155i 0.0789720i
\(843\) − 11.4571i − 0.394603i
\(844\) 15.4571 0.532055
\(845\) 0 0
\(846\) −27.2083 −0.935441
\(847\) − 70.4875i − 2.42198i
\(848\) 6.00000i 0.206041i
\(849\) −82.1501 −2.81938
\(850\) 0 0
\(851\) −5.81903 −0.199474
\(852\) 3.52748i 0.120850i
\(853\) − 38.6371i − 1.32291i −0.749985 0.661455i \(-0.769939\pi\)
0.749985 0.661455i \(-0.230061\pi\)
\(854\) 50.3393 1.72257
\(855\) 0 0
\(856\) 12.3054 0.420589
\(857\) 26.2488i 0.896642i 0.893873 + 0.448321i \(0.147978\pi\)
−0.893873 + 0.448321i \(0.852022\pi\)
\(858\) 16.8539i 0.575383i
\(859\) −37.5558 −1.28139 −0.640693 0.767797i \(-0.721353\pi\)
−0.640693 + 0.767797i \(0.721353\pi\)
\(860\) 0 0
\(861\) 105.086 3.58133
\(862\) − 8.83189i − 0.300816i
\(863\) 21.9855i 0.748396i 0.927349 + 0.374198i \(0.122082\pi\)
−0.927349 + 0.374198i \(0.877918\pi\)
\(864\) −3.28432 −0.111735
\(865\) 0 0
\(866\) 33.3465 1.13316
\(867\) 13.3610i 0.453763i
\(868\) − 3.57587i − 0.121373i
\(869\) 25.0858 0.850978
\(870\) 0 0
\(871\) −19.1115 −0.647570
\(872\) − 10.5969i − 0.358857i
\(873\) 76.2607i 2.58103i
\(874\) −4.59692 −0.155493
\(875\) 0 0
\(876\) −11.9434 −0.403531
\(877\) − 36.5064i − 1.23273i −0.787459 0.616367i \(-0.788604\pi\)
0.787459 0.616367i \(-0.211396\pi\)
\(878\) 6.02829i 0.203445i
\(879\) 16.1244 0.543862
\(880\) 0 0
\(881\) −17.4571 −0.588145 −0.294072 0.955783i \(-0.595011\pi\)
−0.294072 + 0.955783i \(0.595011\pi\)
\(882\) 59.6654i 2.00904i
\(883\) − 22.9444i − 0.772140i −0.922470 0.386070i \(-0.873832\pi\)
0.922470 0.386070i \(-0.126168\pi\)
\(884\) −5.72855 −0.192672
\(885\) 0 0
\(886\) −15.5275 −0.521656
\(887\) − 11.7223i − 0.393595i −0.980444 0.196798i \(-0.936946\pi\)
0.980444 0.196798i \(-0.0630542\pi\)
\(888\) − 15.6381i − 0.524779i
\(889\) −10.4040 −0.348940
\(890\) 0 0
\(891\) −19.7059 −0.660172
\(892\) − 10.7496i − 0.359923i
\(893\) − 29.6236i − 0.991316i
\(894\) 53.1856 1.77879
\(895\) 0 0
\(896\) −4.59692 −0.153572
\(897\) 3.28432i 0.109660i
\(898\) 18.3594i 0.612660i
\(899\) 2.62520 0.0875554
\(900\) 0 0
\(901\) 28.1244 0.936960
\(902\) − 43.6519i − 1.45345i
\(903\) 98.8300i 3.28886i
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 10.9168 0.362685
\(907\) − 1.91770i − 0.0636763i −0.999493 0.0318381i \(-0.989864\pi\)
0.999493 0.0318381i \(-0.0101361\pi\)
\(908\) 12.3054i 0.408368i
\(909\) −14.2488 −0.472603
\(910\) 0 0
\(911\) −56.8319 −1.88292 −0.941462 0.337118i \(-0.890548\pi\)
−0.941462 + 0.337118i \(0.890548\pi\)
\(912\) − 12.3537i − 0.409073i
\(913\) 19.5979i 0.648595i
\(914\) 11.4992 0.380360
\(915\) 0 0
\(916\) 9.63806 0.318451
\(917\) − 13.4716i − 0.444870i
\(918\) 15.3949i 0.508107i
\(919\) 26.0257 0.858509 0.429255 0.903183i \(-0.358776\pi\)
0.429255 + 0.903183i \(0.358776\pi\)
\(920\) 0 0
\(921\) −22.9733 −0.756997
\(922\) 1.33270i 0.0438901i
\(923\) 1.60415i 0.0528013i
\(924\) 63.3949 2.08554
\(925\) 0 0
\(926\) 35.8190 1.17709
\(927\) 55.4433i 1.82100i
\(928\) − 3.37480i − 0.110783i
\(929\) 17.1115 0.561411 0.280706 0.959794i \(-0.409432\pi\)
0.280706 + 0.959794i \(0.409432\pi\)
\(930\) 0 0
\(931\) −64.9619 −2.12904
\(932\) 13.9434i 0.456732i
\(933\) 20.5265i 0.672008i
\(934\) 23.7625 0.777531
\(935\) 0 0
\(936\) −5.15992 −0.168657
\(937\) 16.3738i 0.534910i 0.963570 + 0.267455i \(0.0861827\pi\)
−0.963570 + 0.267455i \(0.913817\pi\)
\(938\) 71.8869i 2.34719i
\(939\) 49.0047 1.59921
\(940\) 0 0
\(941\) −16.4097 −0.534940 −0.267470 0.963566i \(-0.586188\pi\)
−0.267470 + 0.963566i \(0.586188\pi\)
\(942\) 12.4298i 0.404984i
\(943\) − 8.50643i − 0.277008i
\(944\) −9.37480 −0.305124
\(945\) 0 0
\(946\) 41.0531 1.33475
\(947\) − 21.3886i − 0.695037i −0.937673 0.347518i \(-0.887024\pi\)
0.937673 0.347518i \(-0.112976\pi\)
\(948\) 13.1373i 0.426678i
\(949\) −5.43137 −0.176310
\(950\) 0 0
\(951\) −45.3100 −1.46928
\(952\) 21.5476i 0.698361i
\(953\) 16.7276i 0.541860i 0.962599 + 0.270930i \(0.0873312\pi\)
−0.962599 + 0.270930i \(0.912669\pi\)
\(954\) 25.3327 0.820176
\(955\) 0 0
\(956\) −26.3877 −0.853438
\(957\) 46.5410i 1.50446i
\(958\) 2.04210i 0.0659773i
\(959\) −90.9764 −2.93778
\(960\) 0 0
\(961\) −30.3949 −0.980481
\(962\) − 7.11153i − 0.229285i
\(963\) − 51.9547i − 1.67422i
\(964\) 7.06943 0.227691
\(965\) 0 0
\(966\) 12.3537 0.397475
\(967\) 45.2617i 1.45552i 0.685834 + 0.727758i \(0.259438\pi\)
−0.685834 + 0.727758i \(0.740562\pi\)
\(968\) − 15.3337i − 0.492842i
\(969\) −57.9070 −1.86024
\(970\) 0 0
\(971\) 45.2560 1.45234 0.726168 0.687518i \(-0.241300\pi\)
0.726168 + 0.687518i \(0.241300\pi\)
\(972\) − 20.1728i − 0.647042i
\(973\) 28.7917i 0.923019i
\(974\) 18.6252 0.596790
\(975\) 0 0
\(976\) 10.9507 0.350522
\(977\) − 13.3465i − 0.426993i −0.976944 0.213496i \(-0.931515\pi\)
0.976944 0.213496i \(-0.0684852\pi\)
\(978\) − 33.3666i − 1.06695i
\(979\) −45.8284 −1.46468
\(980\) 0 0
\(981\) −44.7414 −1.42848
\(982\) 33.7204i 1.07606i
\(983\) − 5.13163i − 0.163674i −0.996646 0.0818368i \(-0.973921\pi\)
0.996646 0.0818368i \(-0.0260786\pi\)
\(984\) 22.8602 0.728756
\(985\) 0 0
\(986\) −15.8190 −0.503781
\(987\) 79.6104i 2.53403i
\(988\) − 5.61797i − 0.178731i
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) 17.7989 0.565402 0.282701 0.959208i \(-0.408770\pi\)
0.282701 + 0.959208i \(0.408770\pi\)
\(992\) − 0.777884i − 0.0246978i
\(993\) − 51.4684i − 1.63330i
\(994\) 6.03391 0.191384
\(995\) 0 0
\(996\) −10.2633 −0.325204
\(997\) 40.3877i 1.27909i 0.768754 + 0.639545i \(0.220877\pi\)
−0.768754 + 0.639545i \(0.779123\pi\)
\(998\) 16.8885i 0.534595i
\(999\) −19.1115 −0.604662
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 1150.2.b.j.599.4 6
5.2 odd 4 1150.2.a.q.1.1 3
5.3 odd 4 230.2.a.d.1.3 3
5.4 even 2 inner 1150.2.b.j.599.3 6
15.8 even 4 2070.2.a.z.1.1 3
20.3 even 4 1840.2.a.r.1.1 3
20.7 even 4 9200.2.a.cf.1.3 3
40.3 even 4 7360.2.a.ce.1.3 3
40.13 odd 4 7360.2.a.bz.1.1 3
115.68 even 4 5290.2.a.r.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.3 3 5.3 odd 4
1150.2.a.q.1.1 3 5.2 odd 4
1150.2.b.j.599.3 6 5.4 even 2 inner
1150.2.b.j.599.4 6 1.1 even 1 trivial
1840.2.a.r.1.1 3 20.3 even 4
2070.2.a.z.1.1 3 15.8 even 4
5290.2.a.r.1.3 3 115.68 even 4
7360.2.a.bz.1.1 3 40.13 odd 4
7360.2.a.ce.1.3 3 40.3 even 4
9200.2.a.cf.1.3 3 20.7 even 4