Properties

Label 1150.2.b.g.599.3
Level $1150$
Weight $2$
Character 1150.599
Analytic conductor $9.183$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{21})\)
Defining polynomial: \(x^{4} + 11 x^{2} + 25\)
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 599.3
Root \(-2.79129i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.2.b.g.599.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -2.79129i q^{3} -1.00000 q^{4} +2.79129 q^{6} +1.79129i q^{7} -1.00000i q^{8} -4.79129 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -2.79129i q^{3} -1.00000 q^{4} +2.79129 q^{6} +1.79129i q^{7} -1.00000i q^{8} -4.79129 q^{9} -0.791288 q^{11} +2.79129i q^{12} +5.79129i q^{13} -1.79129 q^{14} +1.00000 q^{16} -0.791288i q^{17} -4.79129i q^{18} -5.79129 q^{19} +5.00000 q^{21} -0.791288i q^{22} +1.00000i q^{23} -2.79129 q^{24} -5.79129 q^{26} +5.00000i q^{27} -1.79129i q^{28} -7.58258 q^{29} -3.37386 q^{31} +1.00000i q^{32} +2.20871i q^{33} +0.791288 q^{34} +4.79129 q^{36} +4.00000i q^{37} -5.79129i q^{38} +16.1652 q^{39} -6.79129 q^{41} +5.00000i q^{42} +11.1652i q^{43} +0.791288 q^{44} -1.00000 q^{46} +4.41742i q^{47} -2.79129i q^{48} +3.79129 q^{49} -2.20871 q^{51} -5.79129i q^{52} +6.00000i q^{53} -5.00000 q^{54} +1.79129 q^{56} +16.1652i q^{57} -7.58258i q^{58} +13.5826 q^{59} +10.3739 q^{61} -3.37386i q^{62} -8.58258i q^{63} -1.00000 q^{64} -2.20871 q^{66} -11.1652i q^{67} +0.791288i q^{68} +2.79129 q^{69} +8.37386 q^{71} +4.79129i q^{72} +12.7477i q^{73} -4.00000 q^{74} +5.79129 q^{76} -1.41742i q^{77} +16.1652i q^{78} -8.00000 q^{79} -0.417424 q^{81} -6.79129i q^{82} -6.00000i q^{83} -5.00000 q^{84} -11.1652 q^{86} +21.1652i q^{87} +0.791288i q^{88} -15.1652 q^{89} -10.3739 q^{91} -1.00000i q^{92} +9.41742i q^{93} -4.41742 q^{94} +2.79129 q^{96} +7.95644i q^{97} +3.79129i q^{98} +3.79129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 2q^{6} - 10q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 2q^{6} - 10q^{9} + 6q^{11} + 2q^{14} + 4q^{16} - 14q^{19} + 20q^{21} - 2q^{24} - 14q^{26} - 12q^{29} + 14q^{31} - 6q^{34} + 10q^{36} + 28q^{39} - 18q^{41} - 6q^{44} - 4q^{46} + 6q^{49} - 18q^{51} - 20q^{54} - 2q^{56} + 36q^{59} + 14q^{61} - 4q^{64} - 18q^{66} + 2q^{69} + 6q^{71} - 16q^{74} + 14q^{76} - 32q^{79} - 20q^{81} - 20q^{84} - 8q^{86} - 24q^{89} - 14q^{91} - 36q^{94} + 2q^{96} + 6q^{99} + O(q^{100}) \)

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
\(3\) − 2.79129i − 1.61155i −0.592221 0.805775i \(-0.701749\pi\)
0.592221 0.805775i \(-0.298251\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.79129 1.13954
\(7\) 1.79129i 0.677043i 0.940959 + 0.338522i \(0.109927\pi\)
−0.940959 + 0.338522i \(0.890073\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −4.79129 −1.59710
\(10\) 0 0
\(11\) −0.791288 −0.238582 −0.119291 0.992859i \(-0.538062\pi\)
−0.119291 + 0.992859i \(0.538062\pi\)
\(12\) 2.79129i 0.805775i
\(13\) 5.79129i 1.60621i 0.595835 + 0.803107i \(0.296821\pi\)
−0.595835 + 0.803107i \(0.703179\pi\)
\(14\) −1.79129 −0.478742
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 0.791288i − 0.191915i −0.995385 0.0959577i \(-0.969409\pi\)
0.995385 0.0959577i \(-0.0305914\pi\)
\(18\) − 4.79129i − 1.12932i
\(19\) −5.79129 −1.32861 −0.664306 0.747460i \(-0.731273\pi\)
−0.664306 + 0.747460i \(0.731273\pi\)
\(20\) 0 0
\(21\) 5.00000 1.09109
\(22\) − 0.791288i − 0.168703i
\(23\) 1.00000i 0.208514i
\(24\) −2.79129 −0.569769
\(25\) 0 0
\(26\) −5.79129 −1.13576
\(27\) 5.00000i 0.962250i
\(28\) − 1.79129i − 0.338522i
\(29\) −7.58258 −1.40805 −0.704024 0.710176i \(-0.748615\pi\)
−0.704024 + 0.710176i \(0.748615\pi\)
\(30\) 0 0
\(31\) −3.37386 −0.605964 −0.302982 0.952996i \(-0.597982\pi\)
−0.302982 + 0.952996i \(0.597982\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 2.20871i 0.384487i
\(34\) 0.791288 0.135705
\(35\) 0 0
\(36\) 4.79129 0.798548
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) − 5.79129i − 0.939471i
\(39\) 16.1652 2.58850
\(40\) 0 0
\(41\) −6.79129 −1.06062 −0.530310 0.847804i \(-0.677925\pi\)
−0.530310 + 0.847804i \(0.677925\pi\)
\(42\) 5.00000i 0.771517i
\(43\) 11.1652i 1.70267i 0.524623 + 0.851335i \(0.324206\pi\)
−0.524623 + 0.851335i \(0.675794\pi\)
\(44\) 0.791288 0.119291
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 4.41742i 0.644348i 0.946681 + 0.322174i \(0.104414\pi\)
−0.946681 + 0.322174i \(0.895586\pi\)
\(48\) − 2.79129i − 0.402888i
\(49\) 3.79129 0.541613
\(50\) 0 0
\(51\) −2.20871 −0.309282
\(52\) − 5.79129i − 0.803107i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) −5.00000 −0.680414
\(55\) 0 0
\(56\) 1.79129 0.239371
\(57\) 16.1652i 2.14113i
\(58\) − 7.58258i − 0.995641i
\(59\) 13.5826 1.76830 0.884150 0.467202i \(-0.154738\pi\)
0.884150 + 0.467202i \(0.154738\pi\)
\(60\) 0 0
\(61\) 10.3739 1.32824 0.664119 0.747627i \(-0.268807\pi\)
0.664119 + 0.747627i \(0.268807\pi\)
\(62\) − 3.37386i − 0.428481i
\(63\) − 8.58258i − 1.08130i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −2.20871 −0.271874
\(67\) − 11.1652i − 1.36404i −0.731333 0.682020i \(-0.761102\pi\)
0.731333 0.682020i \(-0.238898\pi\)
\(68\) 0.791288i 0.0959577i
\(69\) 2.79129 0.336032
\(70\) 0 0
\(71\) 8.37386 0.993795 0.496897 0.867809i \(-0.334473\pi\)
0.496897 + 0.867809i \(0.334473\pi\)
\(72\) 4.79129i 0.564659i
\(73\) 12.7477i 1.49201i 0.665941 + 0.746004i \(0.268030\pi\)
−0.665941 + 0.746004i \(0.731970\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) 5.79129 0.664306
\(77\) − 1.41742i − 0.161530i
\(78\) 16.1652i 1.83034i
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −0.417424 −0.0463805
\(82\) − 6.79129i − 0.749972i
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) −5.00000 −0.545545
\(85\) 0 0
\(86\) −11.1652 −1.20397
\(87\) 21.1652i 2.26914i
\(88\) 0.791288i 0.0843516i
\(89\) −15.1652 −1.60750 −0.803751 0.594965i \(-0.797166\pi\)
−0.803751 + 0.594965i \(0.797166\pi\)
\(90\) 0 0
\(91\) −10.3739 −1.08748
\(92\) − 1.00000i − 0.104257i
\(93\) 9.41742i 0.976541i
\(94\) −4.41742 −0.455623
\(95\) 0 0
\(96\) 2.79129 0.284885
\(97\) 7.95644i 0.807854i 0.914791 + 0.403927i \(0.132355\pi\)
−0.914791 + 0.403927i \(0.867645\pi\)
\(98\) 3.79129i 0.382978i
\(99\) 3.79129 0.381039
\(100\) 0 0
\(101\) 4.41742 0.439550 0.219775 0.975551i \(-0.429468\pi\)
0.219775 + 0.975551i \(0.429468\pi\)
\(102\) − 2.20871i − 0.218695i
\(103\) − 6.37386i − 0.628035i −0.949417 0.314018i \(-0.898325\pi\)
0.949417 0.314018i \(-0.101675\pi\)
\(104\) 5.79129 0.567882
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) − 4.41742i − 0.427049i −0.976938 0.213524i \(-0.931506\pi\)
0.976938 0.213524i \(-0.0684942\pi\)
\(108\) − 5.00000i − 0.481125i
\(109\) 3.37386 0.323158 0.161579 0.986860i \(-0.448341\pi\)
0.161579 + 0.986860i \(0.448341\pi\)
\(110\) 0 0
\(111\) 11.1652 1.05975
\(112\) 1.79129i 0.169261i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) −16.1652 −1.51401
\(115\) 0 0
\(116\) 7.58258 0.704024
\(117\) − 27.7477i − 2.56528i
\(118\) 13.5826i 1.25038i
\(119\) 1.41742 0.129935
\(120\) 0 0
\(121\) −10.3739 −0.943079
\(122\) 10.3739i 0.939205i
\(123\) 18.9564i 1.70924i
\(124\) 3.37386 0.302982
\(125\) 0 0
\(126\) 8.58258 0.764597
\(127\) − 12.7477i − 1.13118i −0.824687 0.565589i \(-0.808649\pi\)
0.824687 0.565589i \(-0.191351\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 31.1652 2.74394
\(130\) 0 0
\(131\) −9.16515 −0.800763 −0.400381 0.916349i \(-0.631122\pi\)
−0.400381 + 0.916349i \(0.631122\pi\)
\(132\) − 2.20871i − 0.192244i
\(133\) − 10.3739i − 0.899528i
\(134\) 11.1652 0.964522
\(135\) 0 0
\(136\) −0.791288 −0.0678524
\(137\) − 3.79129i − 0.323912i −0.986798 0.161956i \(-0.948220\pi\)
0.986798 0.161956i \(-0.0517802\pi\)
\(138\) 2.79129i 0.237610i
\(139\) −12.7477 −1.08125 −0.540624 0.841264i \(-0.681812\pi\)
−0.540624 + 0.841264i \(0.681812\pi\)
\(140\) 0 0
\(141\) 12.3303 1.03840
\(142\) 8.37386i 0.702719i
\(143\) − 4.58258i − 0.383214i
\(144\) −4.79129 −0.399274
\(145\) 0 0
\(146\) −12.7477 −1.05501
\(147\) − 10.5826i − 0.872836i
\(148\) − 4.00000i − 0.328798i
\(149\) −8.20871 −0.672484 −0.336242 0.941776i \(-0.609156\pi\)
−0.336242 + 0.941776i \(0.609156\pi\)
\(150\) 0 0
\(151\) −10.7913 −0.878183 −0.439091 0.898442i \(-0.644700\pi\)
−0.439091 + 0.898442i \(0.644700\pi\)
\(152\) 5.79129i 0.469735i
\(153\) 3.79129i 0.306507i
\(154\) 1.41742 0.114219
\(155\) 0 0
\(156\) −16.1652 −1.29425
\(157\) 14.7477i 1.17700i 0.808498 + 0.588498i \(0.200281\pi\)
−0.808498 + 0.588498i \(0.799719\pi\)
\(158\) − 8.00000i − 0.636446i
\(159\) 16.7477 1.32818
\(160\) 0 0
\(161\) −1.79129 −0.141173
\(162\) − 0.417424i − 0.0327960i
\(163\) 8.62614i 0.675651i 0.941209 + 0.337826i \(0.109691\pi\)
−0.941209 + 0.337826i \(0.890309\pi\)
\(164\) 6.79129 0.530310
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) − 18.3303i − 1.41844i −0.704987 0.709221i \(-0.749047\pi\)
0.704987 0.709221i \(-0.250953\pi\)
\(168\) − 5.00000i − 0.385758i
\(169\) −20.5390 −1.57992
\(170\) 0 0
\(171\) 27.7477 2.12192
\(172\) − 11.1652i − 0.851335i
\(173\) − 18.7913i − 1.42868i −0.699801 0.714338i \(-0.746728\pi\)
0.699801 0.714338i \(-0.253272\pi\)
\(174\) −21.1652 −1.60453
\(175\) 0 0
\(176\) −0.791288 −0.0596456
\(177\) − 37.9129i − 2.84971i
\(178\) − 15.1652i − 1.13668i
\(179\) −10.7477 −0.803323 −0.401661 0.915788i \(-0.631567\pi\)
−0.401661 + 0.915788i \(0.631567\pi\)
\(180\) 0 0
\(181\) −18.5390 −1.37799 −0.688997 0.724764i \(-0.741949\pi\)
−0.688997 + 0.724764i \(0.741949\pi\)
\(182\) − 10.3739i − 0.768962i
\(183\) − 28.9564i − 2.14052i
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) −9.41742 −0.690519
\(187\) 0.626136i 0.0457876i
\(188\) − 4.41742i − 0.322174i
\(189\) −8.95644 −0.651485
\(190\) 0 0
\(191\) 25.5826 1.85109 0.925545 0.378637i \(-0.123607\pi\)
0.925545 + 0.378637i \(0.123607\pi\)
\(192\) 2.79129i 0.201444i
\(193\) − 20.7477i − 1.49345i −0.665131 0.746727i \(-0.731624\pi\)
0.665131 0.746727i \(-0.268376\pi\)
\(194\) −7.95644 −0.571239
\(195\) 0 0
\(196\) −3.79129 −0.270806
\(197\) 11.5390i 0.822121i 0.911608 + 0.411060i \(0.134841\pi\)
−0.911608 + 0.411060i \(0.865159\pi\)
\(198\) 3.79129i 0.269435i
\(199\) 16.3303 1.15762 0.578812 0.815461i \(-0.303516\pi\)
0.578812 + 0.815461i \(0.303516\pi\)
\(200\) 0 0
\(201\) −31.1652 −2.19822
\(202\) 4.41742i 0.310809i
\(203\) − 13.5826i − 0.953310i
\(204\) 2.20871 0.154641
\(205\) 0 0
\(206\) 6.37386 0.444088
\(207\) − 4.79129i − 0.333018i
\(208\) 5.79129i 0.401554i
\(209\) 4.58258 0.316983
\(210\) 0 0
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) − 6.00000i − 0.412082i
\(213\) − 23.3739i − 1.60155i
\(214\) 4.41742 0.301969
\(215\) 0 0
\(216\) 5.00000 0.340207
\(217\) − 6.04356i − 0.410264i
\(218\) 3.37386i 0.228507i
\(219\) 35.5826 2.40445
\(220\) 0 0
\(221\) 4.58258 0.308257
\(222\) 11.1652i 0.749356i
\(223\) − 7.16515i − 0.479814i −0.970796 0.239907i \(-0.922883\pi\)
0.970796 0.239907i \(-0.0771170\pi\)
\(224\) −1.79129 −0.119685
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 22.7477i 1.50982i 0.655829 + 0.754910i \(0.272319\pi\)
−0.655829 + 0.754910i \(0.727681\pi\)
\(228\) − 16.1652i − 1.07056i
\(229\) −20.3303 −1.34346 −0.671732 0.740794i \(-0.734449\pi\)
−0.671732 + 0.740794i \(0.734449\pi\)
\(230\) 0 0
\(231\) −3.95644 −0.260315
\(232\) 7.58258i 0.497820i
\(233\) − 1.58258i − 0.103678i −0.998655 0.0518390i \(-0.983492\pi\)
0.998655 0.0518390i \(-0.0165083\pi\)
\(234\) 27.7477 1.81393
\(235\) 0 0
\(236\) −13.5826 −0.884150
\(237\) 22.3303i 1.45051i
\(238\) 1.41742i 0.0918780i
\(239\) 15.1652 0.980952 0.490476 0.871455i \(-0.336823\pi\)
0.490476 + 0.871455i \(0.336823\pi\)
\(240\) 0 0
\(241\) −28.0000 −1.80364 −0.901819 0.432113i \(-0.857768\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) − 10.3739i − 0.666857i
\(243\) 16.1652i 1.03699i
\(244\) −10.3739 −0.664119
\(245\) 0 0
\(246\) −18.9564 −1.20862
\(247\) − 33.5390i − 2.13404i
\(248\) 3.37386i 0.214241i
\(249\) −16.7477 −1.06134
\(250\) 0 0
\(251\) 26.2087 1.65428 0.827140 0.561996i \(-0.189967\pi\)
0.827140 + 0.561996i \(0.189967\pi\)
\(252\) 8.58258i 0.540651i
\(253\) − 0.791288i − 0.0497478i
\(254\) 12.7477 0.799864
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 4.74773i 0.296155i 0.988976 + 0.148078i \(0.0473085\pi\)
−0.988976 + 0.148078i \(0.952691\pi\)
\(258\) 31.1652i 1.94026i
\(259\) −7.16515 −0.445221
\(260\) 0 0
\(261\) 36.3303 2.24879
\(262\) − 9.16515i − 0.566225i
\(263\) 11.2087i 0.691159i 0.938390 + 0.345579i \(0.112318\pi\)
−0.938390 + 0.345579i \(0.887682\pi\)
\(264\) 2.20871 0.135937
\(265\) 0 0
\(266\) 10.3739 0.636062
\(267\) 42.3303i 2.59057i
\(268\) 11.1652i 0.682020i
\(269\) 10.7477 0.655300 0.327650 0.944799i \(-0.393743\pi\)
0.327650 + 0.944799i \(0.393743\pi\)
\(270\) 0 0
\(271\) 18.1216 1.10081 0.550404 0.834898i \(-0.314474\pi\)
0.550404 + 0.834898i \(0.314474\pi\)
\(272\) − 0.791288i − 0.0479789i
\(273\) 28.9564i 1.75252i
\(274\) 3.79129 0.229040
\(275\) 0 0
\(276\) −2.79129 −0.168016
\(277\) − 17.1652i − 1.03135i −0.856783 0.515677i \(-0.827540\pi\)
0.856783 0.515677i \(-0.172460\pi\)
\(278\) − 12.7477i − 0.764558i
\(279\) 16.1652 0.967782
\(280\) 0 0
\(281\) 10.7477 0.641156 0.320578 0.947222i \(-0.396123\pi\)
0.320578 + 0.947222i \(0.396123\pi\)
\(282\) 12.3303i 0.734259i
\(283\) 8.33030i 0.495185i 0.968864 + 0.247593i \(0.0796394\pi\)
−0.968864 + 0.247593i \(0.920361\pi\)
\(284\) −8.37386 −0.496897
\(285\) 0 0
\(286\) 4.58258 0.270973
\(287\) − 12.1652i − 0.718086i
\(288\) − 4.79129i − 0.282329i
\(289\) 16.3739 0.963168
\(290\) 0 0
\(291\) 22.2087 1.30190
\(292\) − 12.7477i − 0.746004i
\(293\) 27.4955i 1.60630i 0.595776 + 0.803151i \(0.296845\pi\)
−0.595776 + 0.803151i \(0.703155\pi\)
\(294\) 10.5826 0.617188
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) − 3.95644i − 0.229576i
\(298\) − 8.20871i − 0.475518i
\(299\) −5.79129 −0.334919
\(300\) 0 0
\(301\) −20.0000 −1.15278
\(302\) − 10.7913i − 0.620969i
\(303\) − 12.3303i − 0.708357i
\(304\) −5.79129 −0.332153
\(305\) 0 0
\(306\) −3.79129 −0.216734
\(307\) 15.5390i 0.886858i 0.896309 + 0.443429i \(0.146238\pi\)
−0.896309 + 0.443429i \(0.853762\pi\)
\(308\) 1.41742i 0.0807652i
\(309\) −17.7913 −1.01211
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) − 16.1652i − 0.915171i
\(313\) − 4.62614i − 0.261485i −0.991416 0.130742i \(-0.958264\pi\)
0.991416 0.130742i \(-0.0417361\pi\)
\(314\) −14.7477 −0.832262
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) − 9.79129i − 0.549934i −0.961454 0.274967i \(-0.911333\pi\)
0.961454 0.274967i \(-0.0886669\pi\)
\(318\) 16.7477i 0.939166i
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) −12.3303 −0.688210
\(322\) − 1.79129i − 0.0998246i
\(323\) 4.58258i 0.254981i
\(324\) 0.417424 0.0231902
\(325\) 0 0
\(326\) −8.62614 −0.477758
\(327\) − 9.41742i − 0.520785i
\(328\) 6.79129i 0.374986i
\(329\) −7.91288 −0.436251
\(330\) 0 0
\(331\) −20.7477 −1.14040 −0.570199 0.821507i \(-0.693134\pi\)
−0.570199 + 0.821507i \(0.693134\pi\)
\(332\) 6.00000i 0.329293i
\(333\) − 19.1652i − 1.05024i
\(334\) 18.3303 1.00299
\(335\) 0 0
\(336\) 5.00000 0.272772
\(337\) 12.2087i 0.665051i 0.943094 + 0.332525i \(0.107901\pi\)
−0.943094 + 0.332525i \(0.892099\pi\)
\(338\) − 20.5390i − 1.11718i
\(339\) 16.7477 0.909612
\(340\) 0 0
\(341\) 2.66970 0.144572
\(342\) 27.7477i 1.50043i
\(343\) 19.3303i 1.04374i
\(344\) 11.1652 0.601985
\(345\) 0 0
\(346\) 18.7913 1.01023
\(347\) − 5.20871i − 0.279618i −0.990178 0.139809i \(-0.955351\pi\)
0.990178 0.139809i \(-0.0446489\pi\)
\(348\) − 21.1652i − 1.13457i
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) −28.9564 −1.54558
\(352\) − 0.791288i − 0.0421758i
\(353\) 3.16515i 0.168464i 0.996446 + 0.0842320i \(0.0268437\pi\)
−0.996446 + 0.0842320i \(0.973156\pi\)
\(354\) 37.9129 2.01505
\(355\) 0 0
\(356\) 15.1652 0.803751
\(357\) − 3.95644i − 0.209397i
\(358\) − 10.7477i − 0.568035i
\(359\) −9.16515 −0.483718 −0.241859 0.970311i \(-0.577757\pi\)
−0.241859 + 0.970311i \(0.577757\pi\)
\(360\) 0 0
\(361\) 14.5390 0.765211
\(362\) − 18.5390i − 0.974389i
\(363\) 28.9564i 1.51982i
\(364\) 10.3739 0.543738
\(365\) 0 0
\(366\) 28.9564 1.51358
\(367\) 19.1652i 1.00041i 0.865906 + 0.500206i \(0.166743\pi\)
−0.865906 + 0.500206i \(0.833257\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) 32.5390 1.69391
\(370\) 0 0
\(371\) −10.7477 −0.557994
\(372\) − 9.41742i − 0.488271i
\(373\) 12.7477i 0.660052i 0.943972 + 0.330026i \(0.107058\pi\)
−0.943972 + 0.330026i \(0.892942\pi\)
\(374\) −0.626136 −0.0323767
\(375\) 0 0
\(376\) 4.41742 0.227811
\(377\) − 43.9129i − 2.26163i
\(378\) − 8.95644i − 0.460670i
\(379\) 6.37386 0.327403 0.163702 0.986510i \(-0.447657\pi\)
0.163702 + 0.986510i \(0.447657\pi\)
\(380\) 0 0
\(381\) −35.5826 −1.82295
\(382\) 25.5826i 1.30892i
\(383\) − 24.0000i − 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) −2.79129 −0.142442
\(385\) 0 0
\(386\) 20.7477 1.05603
\(387\) − 53.4955i − 2.71933i
\(388\) − 7.95644i − 0.403927i
\(389\) 20.7042 1.04974 0.524871 0.851182i \(-0.324113\pi\)
0.524871 + 0.851182i \(0.324113\pi\)
\(390\) 0 0
\(391\) 0.791288 0.0400171
\(392\) − 3.79129i − 0.191489i
\(393\) 25.5826i 1.29047i
\(394\) −11.5390 −0.581327
\(395\) 0 0
\(396\) −3.79129 −0.190519
\(397\) 15.5390i 0.779881i 0.920840 + 0.389940i \(0.127504\pi\)
−0.920840 + 0.389940i \(0.872496\pi\)
\(398\) 16.3303i 0.818564i
\(399\) −28.9564 −1.44964
\(400\) 0 0
\(401\) −4.74773 −0.237090 −0.118545 0.992949i \(-0.537823\pi\)
−0.118545 + 0.992949i \(0.537823\pi\)
\(402\) − 31.1652i − 1.55438i
\(403\) − 19.5390i − 0.973308i
\(404\) −4.41742 −0.219775
\(405\) 0 0
\(406\) 13.5826 0.674092
\(407\) − 3.16515i − 0.156891i
\(408\) 2.20871i 0.109348i
\(409\) 18.2087 0.900363 0.450181 0.892937i \(-0.351359\pi\)
0.450181 + 0.892937i \(0.351359\pi\)
\(410\) 0 0
\(411\) −10.5826 −0.522000
\(412\) 6.37386i 0.314018i
\(413\) 24.3303i 1.19722i
\(414\) 4.79129 0.235479
\(415\) 0 0
\(416\) −5.79129 −0.283941
\(417\) 35.5826i 1.74249i
\(418\) 4.58258i 0.224141i
\(419\) −20.8348 −1.01785 −0.508924 0.860811i \(-0.669957\pi\)
−0.508924 + 0.860811i \(0.669957\pi\)
\(420\) 0 0
\(421\) 18.1216 0.883192 0.441596 0.897214i \(-0.354412\pi\)
0.441596 + 0.897214i \(0.354412\pi\)
\(422\) − 10.0000i − 0.486792i
\(423\) − 21.1652i − 1.02908i
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 23.3739 1.13247
\(427\) 18.5826i 0.899274i
\(428\) 4.41742i 0.213524i
\(429\) −12.7913 −0.617569
\(430\) 0 0
\(431\) 25.9129 1.24818 0.624090 0.781353i \(-0.285470\pi\)
0.624090 + 0.781353i \(0.285470\pi\)
\(432\) 5.00000i 0.240563i
\(433\) − 30.5390i − 1.46761i −0.679359 0.733806i \(-0.737742\pi\)
0.679359 0.733806i \(-0.262258\pi\)
\(434\) 6.04356 0.290100
\(435\) 0 0
\(436\) −3.37386 −0.161579
\(437\) − 5.79129i − 0.277035i
\(438\) 35.5826i 1.70020i
\(439\) 6.53901 0.312090 0.156045 0.987750i \(-0.450125\pi\)
0.156045 + 0.987750i \(0.450125\pi\)
\(440\) 0 0
\(441\) −18.1652 −0.865007
\(442\) 4.58258i 0.217971i
\(443\) − 39.7913i − 1.89054i −0.326288 0.945271i \(-0.605798\pi\)
0.326288 0.945271i \(-0.394202\pi\)
\(444\) −11.1652 −0.529875
\(445\) 0 0
\(446\) 7.16515 0.339280
\(447\) 22.9129i 1.08374i
\(448\) − 1.79129i − 0.0846304i
\(449\) 16.1216 0.760825 0.380412 0.924817i \(-0.375782\pi\)
0.380412 + 0.924817i \(0.375782\pi\)
\(450\) 0 0
\(451\) 5.37386 0.253045
\(452\) − 6.00000i − 0.282216i
\(453\) 30.1216i 1.41524i
\(454\) −22.7477 −1.06760
\(455\) 0 0
\(456\) 16.1652 0.757003
\(457\) 10.0000i 0.467780i 0.972263 + 0.233890i \(0.0751456\pi\)
−0.972263 + 0.233890i \(0.924854\pi\)
\(458\) − 20.3303i − 0.949973i
\(459\) 3.95644 0.184671
\(460\) 0 0
\(461\) −28.7477 −1.33892 −0.669458 0.742850i \(-0.733473\pi\)
−0.669458 + 0.742850i \(0.733473\pi\)
\(462\) − 3.95644i − 0.184070i
\(463\) − 10.0000i − 0.464739i −0.972628 0.232370i \(-0.925352\pi\)
0.972628 0.232370i \(-0.0746479\pi\)
\(464\) −7.58258 −0.352012
\(465\) 0 0
\(466\) 1.58258 0.0733114
\(467\) 19.9129i 0.921458i 0.887541 + 0.460729i \(0.152412\pi\)
−0.887541 + 0.460729i \(0.847588\pi\)
\(468\) 27.7477i 1.28264i
\(469\) 20.0000 0.923514
\(470\) 0 0
\(471\) 41.1652 1.89679
\(472\) − 13.5826i − 0.625189i
\(473\) − 8.83485i − 0.406227i
\(474\) −22.3303 −1.02566
\(475\) 0 0
\(476\) −1.41742 −0.0649675
\(477\) − 28.7477i − 1.31627i
\(478\) 15.1652i 0.693638i
\(479\) −15.4955 −0.708005 −0.354003 0.935244i \(-0.615180\pi\)
−0.354003 + 0.935244i \(0.615180\pi\)
\(480\) 0 0
\(481\) −23.1652 −1.05624
\(482\) − 28.0000i − 1.27537i
\(483\) 5.00000i 0.227508i
\(484\) 10.3739 0.471539
\(485\) 0 0
\(486\) −16.1652 −0.733266
\(487\) − 6.41742i − 0.290801i −0.989373 0.145401i \(-0.953553\pi\)
0.989373 0.145401i \(-0.0464471\pi\)
\(488\) − 10.3739i − 0.469603i
\(489\) 24.0780 1.08885
\(490\) 0 0
\(491\) 10.7477 0.485038 0.242519 0.970147i \(-0.422026\pi\)
0.242519 + 0.970147i \(0.422026\pi\)
\(492\) − 18.9564i − 0.854622i
\(493\) 6.00000i 0.270226i
\(494\) 33.5390 1.50899
\(495\) 0 0
\(496\) −3.37386 −0.151491
\(497\) 15.0000i 0.672842i
\(498\) − 16.7477i − 0.750484i
\(499\) −4.83485 −0.216438 −0.108219 0.994127i \(-0.534515\pi\)
−0.108219 + 0.994127i \(0.534515\pi\)
\(500\) 0 0
\(501\) −51.1652 −2.28589
\(502\) 26.2087i 1.16975i
\(503\) 14.2087i 0.633535i 0.948503 + 0.316768i \(0.102598\pi\)
−0.948503 + 0.316768i \(0.897402\pi\)
\(504\) −8.58258 −0.382298
\(505\) 0 0
\(506\) 0.791288 0.0351770
\(507\) 57.3303i 2.54613i
\(508\) 12.7477i 0.565589i
\(509\) −34.7477 −1.54017 −0.770083 0.637944i \(-0.779785\pi\)
−0.770083 + 0.637944i \(0.779785\pi\)
\(510\) 0 0
\(511\) −22.8348 −1.01015
\(512\) 1.00000i 0.0441942i
\(513\) − 28.9564i − 1.27846i
\(514\) −4.74773 −0.209413
\(515\) 0 0
\(516\) −31.1652 −1.37197
\(517\) − 3.49545i − 0.153730i
\(518\) − 7.16515i − 0.314819i
\(519\) −52.4519 −2.30238
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 36.3303i 1.59013i
\(523\) 17.1652i 0.750580i 0.926908 + 0.375290i \(0.122457\pi\)
−0.926908 + 0.375290i \(0.877543\pi\)
\(524\) 9.16515 0.400381
\(525\) 0 0
\(526\) −11.2087 −0.488723
\(527\) 2.66970i 0.116294i
\(528\) 2.20871i 0.0961219i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −65.0780 −2.82415
\(532\) 10.3739i 0.449764i
\(533\) − 39.3303i − 1.70358i
\(534\) −42.3303 −1.83181
\(535\) 0 0
\(536\) −11.1652 −0.482261
\(537\) 30.0000i 1.29460i
\(538\) 10.7477i 0.463367i
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 1.66970 0.0717859 0.0358929 0.999356i \(-0.488572\pi\)
0.0358929 + 0.999356i \(0.488572\pi\)
\(542\) 18.1216i 0.778389i
\(543\) 51.7477i 2.22071i
\(544\) 0.791288 0.0339262
\(545\) 0 0
\(546\) −28.9564 −1.23922
\(547\) 26.1216i 1.11688i 0.829545 + 0.558439i \(0.188600\pi\)
−0.829545 + 0.558439i \(0.811400\pi\)
\(548\) 3.79129i 0.161956i
\(549\) −49.7042 −2.12132
\(550\) 0 0
\(551\) 43.9129 1.87075
\(552\) − 2.79129i − 0.118805i
\(553\) − 14.3303i − 0.609386i
\(554\) 17.1652 0.729277
\(555\) 0 0
\(556\) 12.7477 0.540624
\(557\) 6.33030i 0.268224i 0.990966 + 0.134112i \(0.0428181\pi\)
−0.990966 + 0.134112i \(0.957182\pi\)
\(558\) 16.1652i 0.684325i
\(559\) −64.6606 −2.73485
\(560\) 0 0
\(561\) 1.74773 0.0737891
\(562\) 10.7477i 0.453366i
\(563\) − 15.1652i − 0.639135i −0.947564 0.319567i \(-0.896462\pi\)
0.947564 0.319567i \(-0.103538\pi\)
\(564\) −12.3303 −0.519199
\(565\) 0 0
\(566\) −8.33030 −0.350149
\(567\) − 0.747727i − 0.0314016i
\(568\) − 8.37386i − 0.351360i
\(569\) 39.4955 1.65574 0.827868 0.560923i \(-0.189554\pi\)
0.827868 + 0.560923i \(0.189554\pi\)
\(570\) 0 0
\(571\) −11.1216 −0.465424 −0.232712 0.972546i \(-0.574760\pi\)
−0.232712 + 0.972546i \(0.574760\pi\)
\(572\) 4.58258i 0.191607i
\(573\) − 71.4083i − 2.98313i
\(574\) 12.1652 0.507764
\(575\) 0 0
\(576\) 4.79129 0.199637
\(577\) − 41.1652i − 1.71373i −0.515543 0.856864i \(-0.672410\pi\)
0.515543 0.856864i \(-0.327590\pi\)
\(578\) 16.3739i 0.681063i
\(579\) −57.9129 −2.40678
\(580\) 0 0
\(581\) 10.7477 0.445891
\(582\) 22.2087i 0.920581i
\(583\) − 4.74773i − 0.196631i
\(584\) 12.7477 0.527505
\(585\) 0 0
\(586\) −27.4955 −1.13583
\(587\) 30.7913i 1.27089i 0.772145 + 0.635446i \(0.219184\pi\)
−0.772145 + 0.635446i \(0.780816\pi\)
\(588\) 10.5826i 0.436418i
\(589\) 19.5390 0.805091
\(590\) 0 0
\(591\) 32.2087 1.32489
\(592\) 4.00000i 0.164399i
\(593\) − 31.9129i − 1.31050i −0.755410 0.655252i \(-0.772562\pi\)
0.755410 0.655252i \(-0.227438\pi\)
\(594\) 3.95644 0.162335
\(595\) 0 0
\(596\) 8.20871 0.336242
\(597\) − 45.5826i − 1.86557i
\(598\) − 5.79129i − 0.236823i
\(599\) −1.12159 −0.0458270 −0.0229135 0.999737i \(-0.507294\pi\)
−0.0229135 + 0.999737i \(0.507294\pi\)
\(600\) 0 0
\(601\) −18.2087 −0.742749 −0.371374 0.928483i \(-0.621113\pi\)
−0.371374 + 0.928483i \(0.621113\pi\)
\(602\) − 20.0000i − 0.815139i
\(603\) 53.4955i 2.17850i
\(604\) 10.7913 0.439091
\(605\) 0 0
\(606\) 12.3303 0.500884
\(607\) 28.0000i 1.13648i 0.822861 + 0.568242i \(0.192376\pi\)
−0.822861 + 0.568242i \(0.807624\pi\)
\(608\) − 5.79129i − 0.234868i
\(609\) −37.9129 −1.53631
\(610\) 0 0
\(611\) −25.5826 −1.03496
\(612\) − 3.79129i − 0.153254i
\(613\) 14.0000i 0.565455i 0.959200 + 0.282727i \(0.0912392\pi\)
−0.959200 + 0.282727i \(0.908761\pi\)
\(614\) −15.5390 −0.627104
\(615\) 0 0
\(616\) −1.41742 −0.0571097
\(617\) 23.8693i 0.960943i 0.877010 + 0.480471i \(0.159534\pi\)
−0.877010 + 0.480471i \(0.840466\pi\)
\(618\) − 17.7913i − 0.715671i
\(619\) 1.79129 0.0719979 0.0359990 0.999352i \(-0.488539\pi\)
0.0359990 + 0.999352i \(0.488539\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) 12.0000i 0.481156i
\(623\) − 27.1652i − 1.08835i
\(624\) 16.1652 0.647124
\(625\) 0 0
\(626\) 4.62614 0.184898
\(627\) − 12.7913i − 0.510835i
\(628\) − 14.7477i − 0.588498i
\(629\) 3.16515 0.126203
\(630\) 0 0
\(631\) 27.9129 1.11119 0.555597 0.831452i \(-0.312490\pi\)
0.555597 + 0.831452i \(0.312490\pi\)
\(632\) 8.00000i 0.318223i
\(633\) 27.9129i 1.10944i
\(634\) 9.79129 0.388862
\(635\) 0 0
\(636\) −16.7477 −0.664091
\(637\) 21.9564i 0.869946i
\(638\) 6.00000i 0.237542i
\(639\) −40.1216 −1.58719
\(640\) 0 0
\(641\) 15.1652 0.598987 0.299494 0.954098i \(-0.403182\pi\)
0.299494 + 0.954098i \(0.403182\pi\)
\(642\) − 12.3303i − 0.486638i
\(643\) 6.74773i 0.266104i 0.991109 + 0.133052i \(0.0424778\pi\)
−0.991109 + 0.133052i \(0.957522\pi\)
\(644\) 1.79129 0.0705866
\(645\) 0 0
\(646\) −4.58258 −0.180299
\(647\) − 21.1652i − 0.832088i −0.909344 0.416044i \(-0.863416\pi\)
0.909344 0.416044i \(-0.136584\pi\)
\(648\) 0.417424i 0.0163980i
\(649\) −10.7477 −0.421885
\(650\) 0 0
\(651\) −16.8693 −0.661161
\(652\) − 8.62614i − 0.337826i
\(653\) 3.46099i 0.135439i 0.997704 + 0.0677194i \(0.0215723\pi\)
−0.997704 + 0.0677194i \(0.978428\pi\)
\(654\) 9.41742 0.368250
\(655\) 0 0
\(656\) −6.79129 −0.265155
\(657\) − 61.0780i − 2.38288i
\(658\) − 7.91288i − 0.308476i
\(659\) 8.83485 0.344157 0.172078 0.985083i \(-0.444952\pi\)
0.172078 + 0.985083i \(0.444952\pi\)
\(660\) 0 0
\(661\) −25.6261 −0.996741 −0.498371 0.866964i \(-0.666068\pi\)
−0.498371 + 0.866964i \(0.666068\pi\)
\(662\) − 20.7477i − 0.806383i
\(663\) − 12.7913i − 0.496772i
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) 19.1652 0.742635
\(667\) − 7.58258i − 0.293599i
\(668\) 18.3303i 0.709221i
\(669\) −20.0000 −0.773245
\(670\) 0 0
\(671\) −8.20871 −0.316894
\(672\) 5.00000i 0.192879i
\(673\) 38.0000i 1.46479i 0.680879 + 0.732396i \(0.261598\pi\)
−0.680879 + 0.732396i \(0.738402\pi\)
\(674\) −12.2087 −0.470262
\(675\) 0 0
\(676\) 20.5390 0.789962
\(677\) − 42.6606i − 1.63958i −0.572664 0.819790i \(-0.694090\pi\)
0.572664 0.819790i \(-0.305910\pi\)
\(678\) 16.7477i 0.643193i
\(679\) −14.2523 −0.546952
\(680\) 0 0
\(681\) 63.4955 2.43315
\(682\) 2.66970i 0.102228i
\(683\) − 11.3739i − 0.435209i −0.976037 0.217604i \(-0.930176\pi\)
0.976037 0.217604i \(-0.0698243\pi\)
\(684\) −27.7477 −1.06096
\(685\) 0 0
\(686\) −19.3303 −0.738034
\(687\) 56.7477i 2.16506i
\(688\) 11.1652i 0.425667i
\(689\) −34.7477 −1.32378
\(690\) 0 0
\(691\) 42.7477 1.62620 0.813100 0.582124i \(-0.197778\pi\)
0.813100 + 0.582124i \(0.197778\pi\)
\(692\) 18.7913i 0.714338i
\(693\) 6.79129i 0.257980i
\(694\) 5.20871 0.197720
\(695\) 0 0
\(696\) 21.1652 0.802263
\(697\) 5.37386i 0.203550i
\(698\) − 26.0000i − 0.984115i
\(699\) −4.41742 −0.167082
\(700\) 0 0
\(701\) 23.3739 0.882819 0.441409 0.897306i \(-0.354479\pi\)
0.441409 + 0.897306i \(0.354479\pi\)
\(702\) − 28.9564i − 1.09289i
\(703\) − 23.1652i − 0.873690i
\(704\) 0.791288 0.0298228
\(705\) 0 0
\(706\) −3.16515 −0.119122
\(707\) 7.91288i 0.297594i
\(708\) 37.9129i 1.42485i
\(709\) −2.46099 −0.0924242 −0.0462121 0.998932i \(-0.514715\pi\)
−0.0462121 + 0.998932i \(0.514715\pi\)
\(710\) 0 0
\(711\) 38.3303 1.43750
\(712\) 15.1652i 0.568338i
\(713\) − 3.37386i − 0.126352i
\(714\) 3.95644 0.148066
\(715\) 0 0
\(716\) 10.7477 0.401661
\(717\) − 42.3303i − 1.58085i
\(718\) − 9.16515i − 0.342040i
\(719\) −2.53901 −0.0946893 −0.0473446 0.998879i \(-0.515076\pi\)
−0.0473446 + 0.998879i \(0.515076\pi\)
\(720\) 0 0
\(721\) 11.4174 0.425207
\(722\) 14.5390i 0.541086i
\(723\) 78.1561i 2.90666i
\(724\) 18.5390 0.688997
\(725\) 0 0
\(726\) −28.9564 −1.07467
\(727\) − 39.1216i − 1.45094i −0.688254 0.725470i \(-0.741623\pi\)
0.688254 0.725470i \(-0.258377\pi\)
\(728\) 10.3739i 0.384481i
\(729\) 43.8693 1.62479
\(730\) 0 0
\(731\) 8.83485 0.326769
\(732\) 28.9564i 1.07026i
\(733\) 26.0000i 0.960332i 0.877178 + 0.480166i \(0.159424\pi\)
−0.877178 + 0.480166i \(0.840576\pi\)
\(734\) −19.1652 −0.707399
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 8.83485i 0.325436i
\(738\) 32.5390i 1.19778i
\(739\) −8.00000 −0.294285 −0.147142 0.989115i \(-0.547008\pi\)
−0.147142 + 0.989115i \(0.547008\pi\)
\(740\) 0 0
\(741\) −93.6170 −3.43911
\(742\) − 10.7477i − 0.394561i
\(743\) − 12.9564i − 0.475326i −0.971348 0.237663i \(-0.923619\pi\)
0.971348 0.237663i \(-0.0763813\pi\)
\(744\) 9.41742 0.345260
\(745\) 0 0
\(746\) −12.7477 −0.466727
\(747\) 28.7477i 1.05182i
\(748\) − 0.626136i − 0.0228938i
\(749\) 7.91288 0.289130
\(750\) 0 0
\(751\) −8.74773 −0.319209 −0.159605 0.987181i \(-0.551022\pi\)
−0.159605 + 0.987181i \(0.551022\pi\)
\(752\) 4.41742i 0.161087i
\(753\) − 73.1561i − 2.66596i
\(754\) 43.9129 1.59921
\(755\) 0 0
\(756\) 8.95644 0.325743
\(757\) 10.3303i 0.375461i 0.982221 + 0.187731i \(0.0601132\pi\)
−0.982221 + 0.187731i \(0.939887\pi\)
\(758\) 6.37386i 0.231509i
\(759\) −2.20871 −0.0801712
\(760\) 0 0
\(761\) −11.0436 −0.400329 −0.200164 0.979762i \(-0.564148\pi\)
−0.200164 + 0.979762i \(0.564148\pi\)
\(762\) − 35.5826i − 1.28902i
\(763\) 6.04356i 0.218792i
\(764\) −25.5826 −0.925545
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 78.6606i 2.84027i
\(768\) − 2.79129i − 0.100722i
\(769\) 40.3303 1.45435 0.727174 0.686453i \(-0.240833\pi\)
0.727174 + 0.686453i \(0.240833\pi\)
\(770\) 0 0
\(771\) 13.2523 0.477269
\(772\) 20.7477i 0.746727i
\(773\) 33.4955i 1.20475i 0.798214 + 0.602374i \(0.205778\pi\)
−0.798214 + 0.602374i \(0.794222\pi\)
\(774\) 53.4955 1.92285
\(775\) 0 0
\(776\) 7.95644 0.285620
\(777\) 20.0000i 0.717496i
\(778\) 20.7042i 0.742280i
\(779\) 39.3303 1.40915
\(780\) 0 0
\(781\) −6.62614 −0.237102
\(782\) 0.791288i 0.0282964i
\(783\) − 37.9129i − 1.35490i
\(784\) 3.79129 0.135403
\(785\) 0 0
\(786\) −25.5826 −0.912500
\(787\) 17.5826i 0.626751i 0.949629 + 0.313376i \(0.101460\pi\)
−0.949629 + 0.313376i \(0.898540\pi\)
\(788\) − 11.5390i − 0.411060i
\(789\) 31.2867 1.11384
\(790\) 0 0
\(791\) −10.7477 −0.382145
\(792\) − 3.79129i − 0.134718i
\(793\) 60.0780i 2.13343i
\(794\) −15.5390 −0.551459
\(795\) 0 0
\(796\) −16.3303 −0.578812
\(797\) 4.08712i 0.144773i 0.997377 + 0.0723866i \(0.0230616\pi\)
−0.997377 + 0.0723866i \(0.976938\pi\)
\(798\) − 28.9564i − 1.02505i
\(799\) 3.49545 0.123660
\(800\) 0 0
\(801\) 72.6606 2.56734
\(802\) − 4.74773i − 0.167648i
\(803\) − 10.0871i − 0.355967i
\(804\) 31.1652 1.09911
\(805\) 0 0
\(806\) 19.5390 0.688232
\(807\) − 30.0000i − 1.05605i
\(808\) − 4.41742i − 0.155404i
\(809\) −33.9564 −1.19384 −0.596922 0.802299i \(-0.703610\pi\)
−0.596922 + 0.802299i \(0.703610\pi\)
\(810\) 0 0
\(811\) −2.08712 −0.0732887 −0.0366444 0.999328i \(-0.511667\pi\)
−0.0366444 + 0.999328i \(0.511667\pi\)
\(812\) 13.5826i 0.476655i
\(813\) − 50.5826i − 1.77401i
\(814\) 3.16515 0.110938
\(815\) 0 0
\(816\) −2.20871 −0.0773204
\(817\) − 64.6606i − 2.26219i
\(818\) 18.2087i 0.636653i
\(819\) 49.7042 1.73680
\(820\) 0 0
\(821\) 21.1652 0.738669 0.369334 0.929297i \(-0.379586\pi\)
0.369334 + 0.929297i \(0.379586\pi\)
\(822\) − 10.5826i − 0.369110i
\(823\) 22.8348i 0.795973i 0.917391 + 0.397986i \(0.130291\pi\)
−0.917391 + 0.397986i \(0.869709\pi\)
\(824\) −6.37386 −0.222044
\(825\) 0 0
\(826\) −24.3303 −0.846560
\(827\) 23.0780i 0.802502i 0.915968 + 0.401251i \(0.131424\pi\)
−0.915968 + 0.401251i \(0.868576\pi\)
\(828\) 4.79129i 0.166509i
\(829\) −23.4955 −0.816031 −0.408015 0.912975i \(-0.633779\pi\)
−0.408015 + 0.912975i \(0.633779\pi\)
\(830\) 0 0
\(831\) −47.9129 −1.66208
\(832\) − 5.79129i − 0.200777i
\(833\) − 3.00000i − 0.103944i
\(834\) −35.5826 −1.23212
\(835\) 0 0
\(836\) −4.58258 −0.158492
\(837\) − 16.8693i − 0.583089i
\(838\) − 20.8348i − 0.719728i
\(839\) 31.5826 1.09035 0.545176 0.838322i \(-0.316463\pi\)
0.545176 + 0.838322i \(0.316463\pi\)
\(840\) 0 0
\(841\) 28.4955 0.982602
\(842\) 18.1216i 0.624511i
\(843\) − 30.0000i − 1.03325i
\(844\) 10.0000 0.344214
\(845\) 0 0
\(846\) 21.1652 0.727673
\(847\) − 18.5826i − 0.638505i
\(848\) 6.00000i 0.206041i
\(849\) 23.2523 0.798016
\(850\) 0 0
\(851\) −4.00000 −0.137118
\(852\) 23.3739i 0.800775i
\(853\) 40.5390i 1.38803i 0.719961 + 0.694015i \(0.244160\pi\)
−0.719961 + 0.694015i \(0.755840\pi\)
\(854\) −18.5826 −0.635883
\(855\) 0 0
\(856\) −4.41742 −0.150984
\(857\) 9.16515i 0.313076i 0.987672 + 0.156538i \(0.0500333\pi\)
−0.987672 + 0.156538i \(0.949967\pi\)
\(858\) − 12.7913i − 0.436687i
\(859\) 26.7477 0.912621 0.456310 0.889821i \(-0.349171\pi\)
0.456310 + 0.889821i \(0.349171\pi\)
\(860\) 0 0
\(861\) −33.9564 −1.15723
\(862\) 25.9129i 0.882596i
\(863\) − 22.4174i − 0.763098i −0.924349 0.381549i \(-0.875391\pi\)
0.924349 0.381549i \(-0.124609\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) 30.5390 1.03776
\(867\) − 45.7042i − 1.55219i
\(868\) 6.04356i 0.205132i
\(869\) 6.33030 0.214741
\(870\) 0 0
\(871\) 64.6606 2.19094
\(872\) − 3.37386i − 0.114253i
\(873\) − 38.1216i − 1.29022i
\(874\) 5.79129 0.195893
\(875\) 0 0
\(876\) −35.5826 −1.20222
\(877\) 42.7042i 1.44202i 0.692926 + 0.721009i \(0.256321\pi\)
−0.692926 + 0.721009i \(0.743679\pi\)
\(878\) 6.53901i 0.220681i
\(879\) 76.7477 2.58864
\(880\) 0 0
\(881\) 30.3303 1.02185 0.510927 0.859624i \(-0.329302\pi\)
0.510927 + 0.859624i \(0.329302\pi\)
\(882\) − 18.1652i − 0.611652i
\(883\) − 34.9564i − 1.17638i −0.808724 0.588189i \(-0.799841\pi\)
0.808724 0.588189i \(-0.200159\pi\)
\(884\) −4.58258 −0.154129
\(885\) 0 0
\(886\) 39.7913 1.33681
\(887\) − 15.1652i − 0.509196i −0.967047 0.254598i \(-0.918057\pi\)
0.967047 0.254598i \(-0.0819431\pi\)
\(888\) − 11.1652i − 0.374678i
\(889\) 22.8348 0.765856
\(890\) 0 0
\(891\) 0.330303 0.0110656
\(892\) 7.16515i 0.239907i
\(893\) − 25.5826i − 0.856088i
\(894\) −22.9129 −0.766321
\(895\) 0 0
\(896\) 1.79129 0.0598427
\(897\) 16.1652i 0.539739i
\(898\) 16.1216i 0.537984i
\(899\) 25.5826 0.853227
\(900\) 0 0
\(901\) 4.74773 0.158170
\(902\) 5.37386i 0.178930i
\(903\) 55.8258i 1.85776i
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) −30.1216 −1.00072
\(907\) − 6.74773i − 0.224055i −0.993705 0.112027i \(-0.964266\pi\)
0.993705 0.112027i \(-0.0357344\pi\)
\(908\) − 22.7477i − 0.754910i
\(909\) −21.1652 −0.702004
\(910\) 0 0
\(911\) 4.41742 0.146356 0.0731779 0.997319i \(-0.476686\pi\)
0.0731779 + 0.997319i \(0.476686\pi\)
\(912\) 16.1652i 0.535282i
\(913\) 4.74773i 0.157127i
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 20.3303 0.671732
\(917\) − 16.4174i − 0.542151i
\(918\) 3.95644i 0.130582i
\(919\) 55.1652 1.81973 0.909865 0.414904i \(-0.136185\pi\)
0.909865 + 0.414904i \(0.136185\pi\)
\(920\) 0 0
\(921\) 43.3739 1.42922
\(922\) − 28.7477i − 0.946756i
\(923\) 48.4955i 1.59625i
\(924\) 3.95644 0.130157
\(925\) 0 0
\(926\) 10.0000 0.328620
\(927\) 30.5390i 1.00303i
\(928\) − 7.58258i − 0.248910i
\(929\) −15.4955 −0.508389 −0.254195 0.967153i \(-0.581810\pi\)
−0.254195 + 0.967153i \(0.581810\pi\)
\(930\) 0 0
\(931\) −21.9564 −0.719593
\(932\) 1.58258i 0.0518390i
\(933\) − 33.4955i − 1.09659i
\(934\) −19.9129 −0.651569
\(935\) 0 0
\(936\) −27.7477 −0.906963
\(937\) − 44.6261i − 1.45787i −0.684582 0.728936i \(-0.740015\pi\)
0.684582 0.728936i \(-0.259985\pi\)
\(938\) 20.0000i 0.653023i
\(939\) −12.9129 −0.421396
\(940\) 0 0
\(941\) 32.0436 1.04459 0.522295 0.852765i \(-0.325076\pi\)
0.522295 + 0.852765i \(0.325076\pi\)
\(942\) 41.1652i 1.34123i
\(943\) − 6.79129i − 0.221155i
\(944\) 13.5826 0.442075
\(945\) 0 0
\(946\) 8.83485 0.287246
\(947\) − 2.53901i − 0.0825069i −0.999149 0.0412534i \(-0.986865\pi\)
0.999149 0.0412534i \(-0.0131351\pi\)
\(948\) − 22.3303i − 0.725255i
\(949\) −73.8258 −2.39649
\(950\) 0 0
\(951\) −27.3303 −0.886246
\(952\) − 1.41742i − 0.0459390i
\(953\) 5.53901i 0.179426i 0.995968 + 0.0897131i \(0.0285950\pi\)
−0.995968 + 0.0897131i \(0.971405\pi\)
\(954\) 28.7477 0.930742
\(955\) 0 0
\(956\) −15.1652 −0.490476
\(957\) − 16.7477i − 0.541377i
\(958\) − 15.4955i − 0.500635i
\(959\) 6.79129 0.219302
\(960\) 0 0
\(961\) −19.6170 −0.632808
\(962\) − 23.1652i − 0.746874i
\(963\) 21.1652i 0.682037i
\(964\) 28.0000 0.901819
\(965\) 0 0
\(966\) −5.00000 −0.160872
\(967\) 32.7477i 1.05310i 0.850145 + 0.526548i \(0.176514\pi\)
−0.850145 + 0.526548i \(0.823486\pi\)
\(968\) 10.3739i 0.333429i
\(969\) 12.7913 0.410915
\(970\) 0 0
\(971\) 15.9564 0.512067 0.256033 0.966668i \(-0.417584\pi\)
0.256033 + 0.966668i \(0.417584\pi\)
\(972\) − 16.1652i − 0.518497i
\(973\) − 22.8348i − 0.732052i
\(974\) 6.41742 0.205628
\(975\) 0 0
\(976\) 10.3739 0.332059
\(977\) 34.1216i 1.09165i 0.837900 + 0.545823i \(0.183783\pi\)
−0.837900 + 0.545823i \(0.816217\pi\)
\(978\) 24.0780i 0.769930i
\(979\) 12.0000 0.383522
\(980\) 0 0
\(981\) −16.1652 −0.516114
\(982\) 10.7477i 0.342974i
\(983\) − 14.3739i − 0.458455i −0.973373 0.229228i \(-0.926380\pi\)
0.973373 0.229228i \(-0.0736200\pi\)
\(984\) 18.9564 0.604309
\(985\) 0 0
\(986\) −6.00000 −0.191079
\(987\) 22.0871i 0.703041i
\(988\) 33.5390i 1.06702i
\(989\) −11.1652 −0.355031
\(990\) 0 0
\(991\) −33.2087 −1.05491 −0.527455 0.849583i \(-0.676854\pi\)
−0.527455 + 0.849583i \(0.676854\pi\)
\(992\) − 3.37386i − 0.107120i
\(993\) 57.9129i 1.83781i
\(994\) −15.0000 −0.475771
\(995\) 0 0
\(996\) 16.7477 0.530672
\(997\) 43.4955i 1.37751i 0.724992 + 0.688757i \(0.241843\pi\)
−0.724992 + 0.688757i \(0.758157\pi\)
\(998\) − 4.83485i − 0.153044i
\(999\) −20.0000 −0.632772
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.g.599.3 4
5.2 odd 4 230.2.a.a.1.1 2
5.3 odd 4 1150.2.a.o.1.2 2
5.4 even 2 inner 1150.2.b.g.599.2 4
15.2 even 4 2070.2.a.x.1.1 2
20.3 even 4 9200.2.a.bs.1.1 2
20.7 even 4 1840.2.a.n.1.2 2
40.27 even 4 7360.2.a.bk.1.1 2
40.37 odd 4 7360.2.a.bq.1.2 2
115.22 even 4 5290.2.a.e.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.a.1.1 2 5.2 odd 4
1150.2.a.o.1.2 2 5.3 odd 4
1150.2.b.g.599.2 4 5.4 even 2 inner
1150.2.b.g.599.3 4 1.1 even 1 trivial
1840.2.a.n.1.2 2 20.7 even 4
2070.2.a.x.1.1 2 15.2 even 4
5290.2.a.e.1.1 2 115.22 even 4
7360.2.a.bk.1.1 2 40.27 even 4
7360.2.a.bq.1.2 2 40.37 odd 4
9200.2.a.bs.1.1 2 20.3 even 4