Properties

Label 1150.2.b.g.599.4
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 599.4
Root \(1.79129i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.2.b.g.599.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.79129i q^{3} -1.00000 q^{4} -1.79129 q^{6} -2.79129i q^{7} -1.00000i q^{8} -0.208712 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.79129i q^{3} -1.00000 q^{4} -1.79129 q^{6} -2.79129i q^{7} -1.00000i q^{8} -0.208712 q^{9} +3.79129 q^{11} -1.79129i q^{12} +1.20871i q^{13} +2.79129 q^{14} +1.00000 q^{16} +3.79129i q^{17} -0.208712i q^{18} -1.20871 q^{19} +5.00000 q^{21} +3.79129i q^{22} +1.00000i q^{23} +1.79129 q^{24} -1.20871 q^{26} +5.00000i q^{27} +2.79129i q^{28} +1.58258 q^{29} +10.3739 q^{31} +1.00000i q^{32} +6.79129i q^{33} -3.79129 q^{34} +0.208712 q^{36} +4.00000i q^{37} -1.20871i q^{38} -2.16515 q^{39} -2.20871 q^{41} +5.00000i q^{42} -7.16515i q^{43} -3.79129 q^{44} -1.00000 q^{46} +13.5826i q^{47} +1.79129i q^{48} -0.791288 q^{49} -6.79129 q^{51} -1.20871i q^{52} +6.00000i q^{53} -5.00000 q^{54} -2.79129 q^{56} -2.16515i q^{57} +1.58258i q^{58} +4.41742 q^{59} -3.37386 q^{61} +10.3739i q^{62} +0.582576i q^{63} -1.00000 q^{64} -6.79129 q^{66} +7.16515i q^{67} -3.79129i q^{68} -1.79129 q^{69} -5.37386 q^{71} +0.208712i q^{72} -14.7477i q^{73} -4.00000 q^{74} +1.20871 q^{76} -10.5826i q^{77} -2.16515i q^{78} -8.00000 q^{79} -9.58258 q^{81} -2.20871i q^{82} -6.00000i q^{83} -5.00000 q^{84} +7.16515 q^{86} +2.83485i q^{87} -3.79129i q^{88} +3.16515 q^{89} +3.37386 q^{91} -1.00000i q^{92} +18.5826i q^{93} -13.5826 q^{94} -1.79129 q^{96} -14.9564i q^{97} -0.791288i q^{98} -0.791288 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 2q^{6} - 10q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 2q^{6} - 10q^{9} + 6q^{11} + 2q^{14} + 4q^{16} - 14q^{19} + 20q^{21} - 2q^{24} - 14q^{26} - 12q^{29} + 14q^{31} - 6q^{34} + 10q^{36} + 28q^{39} - 18q^{41} - 6q^{44} - 4q^{46} + 6q^{49} - 18q^{51} - 20q^{54} - 2q^{56} + 36q^{59} + 14q^{61} - 4q^{64} - 18q^{66} + 2q^{69} + 6q^{71} - 16q^{74} + 14q^{76} - 32q^{79} - 20q^{81} - 20q^{84} - 8q^{86} - 24q^{89} - 14q^{91} - 36q^{94} + 2q^{96} + 6q^{99} + O(q^{100}) \)

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
\(3\) 1.79129i 1.03420i 0.855925 + 0.517100i \(0.172989\pi\)
−0.855925 + 0.517100i \(0.827011\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.79129 −0.731290
\(7\) − 2.79129i − 1.05501i −0.849553 0.527504i \(-0.823128\pi\)
0.849553 0.527504i \(-0.176872\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −0.208712 −0.0695707
\(10\) 0 0
\(11\) 3.79129 1.14312 0.571558 0.820562i \(-0.306339\pi\)
0.571558 + 0.820562i \(0.306339\pi\)
\(12\) − 1.79129i − 0.517100i
\(13\) 1.20871i 0.335236i 0.985852 + 0.167618i \(0.0536076\pi\)
−0.985852 + 0.167618i \(0.946392\pi\)
\(14\) 2.79129 0.746003
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.79129i 0.919522i 0.888043 + 0.459761i \(0.152065\pi\)
−0.888043 + 0.459761i \(0.847935\pi\)
\(18\) − 0.208712i − 0.0491939i
\(19\) −1.20871 −0.277298 −0.138649 0.990342i \(-0.544276\pi\)
−0.138649 + 0.990342i \(0.544276\pi\)
\(20\) 0 0
\(21\) 5.00000 1.09109
\(22\) 3.79129i 0.808305i
\(23\) 1.00000i 0.208514i
\(24\) 1.79129 0.365645
\(25\) 0 0
\(26\) −1.20871 −0.237048
\(27\) 5.00000i 0.962250i
\(28\) 2.79129i 0.527504i
\(29\) 1.58258 0.293877 0.146938 0.989146i \(-0.453058\pi\)
0.146938 + 0.989146i \(0.453058\pi\)
\(30\) 0 0
\(31\) 10.3739 1.86320 0.931600 0.363484i \(-0.118413\pi\)
0.931600 + 0.363484i \(0.118413\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 6.79129i 1.18221i
\(34\) −3.79129 −0.650201
\(35\) 0 0
\(36\) 0.208712 0.0347854
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) − 1.20871i − 0.196079i
\(39\) −2.16515 −0.346702
\(40\) 0 0
\(41\) −2.20871 −0.344943 −0.172471 0.985015i \(-0.555175\pi\)
−0.172471 + 0.985015i \(0.555175\pi\)
\(42\) 5.00000i 0.771517i
\(43\) − 7.16515i − 1.09268i −0.837565 0.546338i \(-0.816022\pi\)
0.837565 0.546338i \(-0.183978\pi\)
\(44\) −3.79129 −0.571558
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 13.5826i 1.98122i 0.136710 + 0.990611i \(0.456347\pi\)
−0.136710 + 0.990611i \(0.543653\pi\)
\(48\) 1.79129i 0.258550i
\(49\) −0.791288 −0.113041
\(50\) 0 0
\(51\) −6.79129 −0.950971
\(52\) − 1.20871i − 0.167618i
\(53\) 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
\(56\) −2.79129 −0.373002
\(57\) − 2.16515i − 0.286781i
\(58\) 1.58258i 0.207802i
\(59\) 4.41742 0.575100 0.287550 0.957766i \(-0.407159\pi\)
0.287550 + 0.957766i \(0.407159\pi\)
\(60\) 0 0
\(61\) −3.37386 −0.431979 −0.215989 0.976396i \(-0.569298\pi\)
−0.215989 + 0.976396i \(0.569298\pi\)
\(62\) 10.3739i 1.31748i
\(63\) 0.582576i 0.0733976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −6.79129 −0.835950
\(67\) 7.16515i 0.875363i 0.899130 + 0.437681i \(0.144200\pi\)
−0.899130 + 0.437681i \(0.855800\pi\)
\(68\) − 3.79129i − 0.459761i
\(69\) −1.79129 −0.215646
\(70\) 0 0
\(71\) −5.37386 −0.637760 −0.318880 0.947795i \(-0.603307\pi\)
−0.318880 + 0.947795i \(0.603307\pi\)
\(72\) 0.208712i 0.0245970i
\(73\) − 14.7477i − 1.72609i −0.505126 0.863045i \(-0.668554\pi\)
0.505126 0.863045i \(-0.331446\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) 1.20871 0.138649
\(77\) − 10.5826i − 1.20600i
\(78\) − 2.16515i − 0.245155i
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −9.58258 −1.06473
\(82\) − 2.20871i − 0.243911i
\(83\) − 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
\(86\) 7.16515 0.772638
\(87\) 2.83485i 0.303928i
\(88\) − 3.79129i − 0.404153i
\(89\) 3.16515 0.335505 0.167753 0.985829i \(-0.446349\pi\)
0.167753 + 0.985829i \(0.446349\pi\)
\(90\) 0 0
\(91\) 3.37386 0.353677
\(92\) − 1.00000i − 0.104257i
\(93\) 18.5826i 1.92692i
\(94\) −13.5826 −1.40094
\(95\) 0 0
\(96\) −1.79129 −0.182823
\(97\) − 14.9564i − 1.51860i −0.650743 0.759298i \(-0.725542\pi\)
0.650743 0.759298i \(-0.274458\pi\)
\(98\) − 0.791288i − 0.0799321i
\(99\) −0.791288 −0.0795274
\(100\) 0 0
\(101\) 13.5826 1.35152 0.675758 0.737123i \(-0.263816\pi\)
0.675758 + 0.737123i \(0.263816\pi\)
\(102\) − 6.79129i − 0.672438i
\(103\) 7.37386i 0.726568i 0.931678 + 0.363284i \(0.118345\pi\)
−0.931678 + 0.363284i \(0.881655\pi\)
\(104\) 1.20871 0.118524
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) − 13.5826i − 1.31308i −0.754292 0.656539i \(-0.772020\pi\)
0.754292 0.656539i \(-0.227980\pi\)
\(108\) − 5.00000i − 0.481125i
\(109\) −10.3739 −0.993636 −0.496818 0.867855i \(-0.665498\pi\)
−0.496818 + 0.867855i \(0.665498\pi\)
\(110\) 0 0
\(111\) −7.16515 −0.680086
\(112\) − 2.79129i − 0.263752i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 2.16515 0.202785
\(115\) 0 0
\(116\) −1.58258 −0.146938
\(117\) − 0.252273i − 0.0233226i
\(118\) 4.41742i 0.406657i
\(119\) 10.5826 0.970103
\(120\) 0 0
\(121\) 3.37386 0.306715
\(122\) − 3.37386i − 0.305455i
\(123\) − 3.95644i − 0.356740i
\(124\) −10.3739 −0.931600
\(125\) 0 0
\(126\) −0.582576 −0.0519000
\(127\) 14.7477i 1.30865i 0.756214 + 0.654325i \(0.227047\pi\)
−0.756214 + 0.654325i \(0.772953\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 12.8348 1.13005
\(130\) 0 0
\(131\) 9.16515 0.800763 0.400381 0.916349i \(-0.368878\pi\)
0.400381 + 0.916349i \(0.368878\pi\)
\(132\) − 6.79129i − 0.591106i
\(133\) 3.37386i 0.292551i
\(134\) −7.16515 −0.618975
\(135\) 0 0
\(136\) 3.79129 0.325100
\(137\) 0.791288i 0.0676043i 0.999429 + 0.0338021i \(0.0107616\pi\)
−0.999429 + 0.0338021i \(0.989238\pi\)
\(138\) − 1.79129i − 0.152485i
\(139\) 14.7477 1.25089 0.625443 0.780270i \(-0.284918\pi\)
0.625443 + 0.780270i \(0.284918\pi\)
\(140\) 0 0
\(141\) −24.3303 −2.04898
\(142\) − 5.37386i − 0.450965i
\(143\) 4.58258i 0.383214i
\(144\) −0.208712 −0.0173927
\(145\) 0 0
\(146\) 14.7477 1.22053
\(147\) − 1.41742i − 0.116907i
\(148\) − 4.00000i − 0.328798i
\(149\) −12.7913 −1.04790 −0.523952 0.851748i \(-0.675543\pi\)
−0.523952 + 0.851748i \(0.675543\pi\)
\(150\) 0 0
\(151\) −6.20871 −0.505258 −0.252629 0.967563i \(-0.581295\pi\)
−0.252629 + 0.967563i \(0.581295\pi\)
\(152\) 1.20871i 0.0980395i
\(153\) − 0.791288i − 0.0639718i
\(154\) 10.5826 0.852768
\(155\) 0 0
\(156\) 2.16515 0.173351
\(157\) − 12.7477i − 1.01738i −0.860950 0.508690i \(-0.830130\pi\)
0.860950 0.508690i \(-0.169870\pi\)
\(158\) − 8.00000i − 0.636446i
\(159\) −10.7477 −0.852350
\(160\) 0 0
\(161\) 2.79129 0.219984
\(162\) − 9.58258i − 0.752878i
\(163\) 22.3739i 1.75246i 0.481897 + 0.876228i \(0.339948\pi\)
−0.481897 + 0.876228i \(0.660052\pi\)
\(164\) 2.20871 0.172471
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) 18.3303i 1.41844i 0.704987 + 0.709221i \(0.250953\pi\)
−0.704987 + 0.709221i \(0.749047\pi\)
\(168\) − 5.00000i − 0.385758i
\(169\) 11.5390 0.887617
\(170\) 0 0
\(171\) 0.252273 0.0192918
\(172\) 7.16515i 0.546338i
\(173\) − 14.2087i − 1.08027i −0.841579 0.540134i \(-0.818373\pi\)
0.841579 0.540134i \(-0.181627\pi\)
\(174\) −2.83485 −0.214909
\(175\) 0 0
\(176\) 3.79129 0.285779
\(177\) 7.91288i 0.594768i
\(178\) 3.16515i 0.237238i
\(179\) 16.7477 1.25178 0.625892 0.779910i \(-0.284735\pi\)
0.625892 + 0.779910i \(0.284735\pi\)
\(180\) 0 0
\(181\) 13.5390 1.00635 0.503174 0.864185i \(-0.332166\pi\)
0.503174 + 0.864185i \(0.332166\pi\)
\(182\) 3.37386i 0.250087i
\(183\) − 6.04356i − 0.446753i
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) −18.5826 −1.36254
\(187\) 14.3739i 1.05112i
\(188\) − 13.5826i − 0.990611i
\(189\) 13.9564 1.01518
\(190\) 0 0
\(191\) 16.4174 1.18792 0.593962 0.804493i \(-0.297563\pi\)
0.593962 + 0.804493i \(0.297563\pi\)
\(192\) − 1.79129i − 0.129275i
\(193\) 6.74773i 0.485712i 0.970062 + 0.242856i \(0.0780843\pi\)
−0.970062 + 0.242856i \(0.921916\pi\)
\(194\) 14.9564 1.07381
\(195\) 0 0
\(196\) 0.791288 0.0565206
\(197\) − 20.5390i − 1.46334i −0.681657 0.731672i \(-0.738740\pi\)
0.681657 0.731672i \(-0.261260\pi\)
\(198\) − 0.791288i − 0.0562344i
\(199\) −20.3303 −1.44118 −0.720588 0.693363i \(-0.756128\pi\)
−0.720588 + 0.693363i \(0.756128\pi\)
\(200\) 0 0
\(201\) −12.8348 −0.905300
\(202\) 13.5826i 0.955667i
\(203\) − 4.41742i − 0.310042i
\(204\) 6.79129 0.475485
\(205\) 0 0
\(206\) −7.37386 −0.513761
\(207\) − 0.208712i − 0.0145065i
\(208\) 1.20871i 0.0838091i
\(209\) −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
\(213\) − 9.62614i − 0.659572i
\(214\) 13.5826 0.928486
\(215\) 0 0
\(216\) 5.00000 0.340207
\(217\) − 28.9564i − 1.96569i
\(218\) − 10.3739i − 0.702607i
\(219\) 26.4174 1.78512
\(220\) 0 0
\(221\) −4.58258 −0.308257
\(222\) − 7.16515i − 0.480893i
\(223\) 11.1652i 0.747674i 0.927494 + 0.373837i \(0.121958\pi\)
−0.927494 + 0.373837i \(0.878042\pi\)
\(224\) 2.79129 0.186501
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) − 4.74773i − 0.315118i −0.987510 0.157559i \(-0.949638\pi\)
0.987510 0.157559i \(-0.0503624\pi\)
\(228\) 2.16515i 0.143391i
\(229\) 16.3303 1.07914 0.539568 0.841942i \(-0.318587\pi\)
0.539568 + 0.841942i \(0.318587\pi\)
\(230\) 0 0
\(231\) 18.9564 1.24724
\(232\) − 1.58258i − 0.103901i
\(233\) 7.58258i 0.496751i 0.968664 + 0.248376i \(0.0798967\pi\)
−0.968664 + 0.248376i \(0.920103\pi\)
\(234\) 0.252273 0.0164916
\(235\) 0 0
\(236\) −4.41742 −0.287550
\(237\) − 14.3303i − 0.930853i
\(238\) 10.5826i 0.685966i
\(239\) −3.16515 −0.204737 −0.102368 0.994747i \(-0.532642\pi\)
−0.102368 + 0.994747i \(0.532642\pi\)
\(240\) 0 0
\(241\) −28.0000 −1.80364 −0.901819 0.432113i \(-0.857768\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) 3.37386i 0.216880i
\(243\) − 2.16515i − 0.138895i
\(244\) 3.37386 0.215989
\(245\) 0 0
\(246\) 3.95644 0.252253
\(247\) − 1.46099i − 0.0929603i
\(248\) − 10.3739i − 0.658741i
\(249\) 10.7477 0.681110
\(250\) 0 0
\(251\) 30.7913 1.94353 0.971764 0.235953i \(-0.0758212\pi\)
0.971764 + 0.235953i \(0.0758212\pi\)
\(252\) − 0.582576i − 0.0366988i
\(253\) 3.79129i 0.238356i
\(254\) −14.7477 −0.925355
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 22.7477i − 1.41896i −0.704723 0.709482i \(-0.748929\pi\)
0.704723 0.709482i \(-0.251071\pi\)
\(258\) 12.8348i 0.799063i
\(259\) 11.1652 0.693769
\(260\) 0 0
\(261\) −0.330303 −0.0204452
\(262\) 9.16515i 0.566225i
\(263\) 15.7913i 0.973733i 0.873477 + 0.486866i \(0.161860\pi\)
−0.873477 + 0.486866i \(0.838140\pi\)
\(264\) 6.79129 0.417975
\(265\) 0 0
\(266\) −3.37386 −0.206865
\(267\) 5.66970i 0.346980i
\(268\) − 7.16515i − 0.437681i
\(269\) −16.7477 −1.02113 −0.510563 0.859840i \(-0.670563\pi\)
−0.510563 + 0.859840i \(0.670563\pi\)
\(270\) 0 0
\(271\) −23.1216 −1.40454 −0.702268 0.711912i \(-0.747829\pi\)
−0.702268 + 0.711912i \(0.747829\pi\)
\(272\) 3.79129i 0.229881i
\(273\) 6.04356i 0.365773i
\(274\) −0.791288 −0.0478034
\(275\) 0 0
\(276\) 1.79129 0.107823
\(277\) 1.16515i 0.0700072i 0.999387 + 0.0350036i \(0.0111443\pi\)
−0.999387 + 0.0350036i \(0.988856\pi\)
\(278\) 14.7477i 0.884510i
\(279\) −2.16515 −0.129624
\(280\) 0 0
\(281\) −16.7477 −0.999086 −0.499543 0.866289i \(-0.666499\pi\)
−0.499543 + 0.866289i \(0.666499\pi\)
\(282\) − 24.3303i − 1.44885i
\(283\) − 28.3303i − 1.68406i −0.539429 0.842031i \(-0.681360\pi\)
0.539429 0.842031i \(-0.318640\pi\)
\(284\) 5.37386 0.318880
\(285\) 0 0
\(286\) −4.58258 −0.270973
\(287\) 6.16515i 0.363917i
\(288\) − 0.208712i − 0.0122985i
\(289\) 2.62614 0.154479
\(290\) 0 0
\(291\) 26.7913 1.57053
\(292\) 14.7477i 0.863045i
\(293\) − 27.4955i − 1.60630i −0.595776 0.803151i \(-0.703155\pi\)
0.595776 0.803151i \(-0.296845\pi\)
\(294\) 1.41742 0.0826659
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) 18.9564i 1.09996i
\(298\) − 12.7913i − 0.740979i
\(299\) −1.20871 −0.0699016
\(300\) 0 0
\(301\) −20.0000 −1.15278
\(302\) − 6.20871i − 0.357271i
\(303\) 24.3303i 1.39774i
\(304\) −1.20871 −0.0693244
\(305\) 0 0
\(306\) 0.791288 0.0452349
\(307\) − 16.5390i − 0.943931i −0.881617 0.471966i \(-0.843545\pi\)
0.881617 0.471966i \(-0.156455\pi\)
\(308\) 10.5826i 0.602998i
\(309\) −13.2087 −0.751417
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 2.16515i 0.122578i
\(313\) − 18.3739i − 1.03855i −0.854607 0.519276i \(-0.826202\pi\)
0.854607 0.519276i \(-0.173798\pi\)
\(314\) 12.7477 0.719396
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) − 5.20871i − 0.292550i −0.989244 0.146275i \(-0.953271\pi\)
0.989244 0.146275i \(-0.0467285\pi\)
\(318\) − 10.7477i − 0.602703i
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) 24.3303 1.35799
\(322\) 2.79129i 0.155552i
\(323\) − 4.58258i − 0.254981i
\(324\) 9.58258 0.532365
\(325\) 0 0
\(326\) −22.3739 −1.23917
\(327\) − 18.5826i − 1.02762i
\(328\) 2.20871i 0.121956i
\(329\) 37.9129 2.09020
\(330\) 0 0
\(331\) 6.74773 0.370889 0.185444 0.982655i \(-0.440628\pi\)
0.185444 + 0.982655i \(0.440628\pi\)
\(332\) 6.00000i 0.329293i
\(333\) − 0.834849i − 0.0457494i
\(334\) −18.3303 −1.00299
\(335\) 0 0
\(336\) 5.00000 0.272772
\(337\) 16.7913i 0.914680i 0.889292 + 0.457340i \(0.151198\pi\)
−0.889292 + 0.457340i \(0.848802\pi\)
\(338\) 11.5390i 0.627640i
\(339\) −10.7477 −0.583736
\(340\) 0 0
\(341\) 39.3303 2.12986
\(342\) 0.252273i 0.0136414i
\(343\) − 17.3303i − 0.935748i
\(344\) −7.16515 −0.386319
\(345\) 0 0
\(346\) 14.2087 0.763865
\(347\) − 9.79129i − 0.525624i −0.964847 0.262812i \(-0.915350\pi\)
0.964847 0.262812i \(-0.0846499\pi\)
\(348\) − 2.83485i − 0.151964i
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) −6.04356 −0.322581
\(352\) 3.79129i 0.202076i
\(353\) − 15.1652i − 0.807160i −0.914944 0.403580i \(-0.867766\pi\)
0.914944 0.403580i \(-0.132234\pi\)
\(354\) −7.91288 −0.420565
\(355\) 0 0
\(356\) −3.16515 −0.167753
\(357\) 18.9564i 1.00328i
\(358\) 16.7477i 0.885145i
\(359\) 9.16515 0.483718 0.241859 0.970311i \(-0.422243\pi\)
0.241859 + 0.970311i \(0.422243\pi\)
\(360\) 0 0
\(361\) −17.5390 −0.923106
\(362\) 13.5390i 0.711595i
\(363\) 6.04356i 0.317205i
\(364\) −3.37386 −0.176838
\(365\) 0 0
\(366\) 6.04356 0.315902
\(367\) 0.834849i 0.0435787i 0.999763 + 0.0217894i \(0.00693632\pi\)
−0.999763 + 0.0217894i \(0.993064\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) 0.460985 0.0239979
\(370\) 0 0
\(371\) 16.7477 0.869499
\(372\) − 18.5826i − 0.963462i
\(373\) − 14.7477i − 0.763608i −0.924243 0.381804i \(-0.875303\pi\)
0.924243 0.381804i \(-0.124697\pi\)
\(374\) −14.3739 −0.743255
\(375\) 0 0
\(376\) 13.5826 0.700468
\(377\) 1.91288i 0.0985183i
\(378\) 13.9564i 0.717842i
\(379\) −7.37386 −0.378770 −0.189385 0.981903i \(-0.560649\pi\)
−0.189385 + 0.981903i \(0.560649\pi\)
\(380\) 0 0
\(381\) −26.4174 −1.35341
\(382\) 16.4174i 0.839989i
\(383\) − 24.0000i − 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) 1.79129 0.0914113
\(385\) 0 0
\(386\) −6.74773 −0.343450
\(387\) 1.49545i 0.0760182i
\(388\) 14.9564i 0.759298i
\(389\) −29.7042 −1.50606 −0.753031 0.657986i \(-0.771409\pi\)
−0.753031 + 0.657986i \(0.771409\pi\)
\(390\) 0 0
\(391\) −3.79129 −0.191734
\(392\) 0.791288i 0.0399661i
\(393\) 16.4174i 0.828150i
\(394\) 20.5390 1.03474
\(395\) 0 0
\(396\) 0.791288 0.0397637
\(397\) − 16.5390i − 0.830069i −0.909806 0.415035i \(-0.863769\pi\)
0.909806 0.415035i \(-0.136231\pi\)
\(398\) − 20.3303i − 1.01907i
\(399\) −6.04356 −0.302556
\(400\) 0 0
\(401\) 22.7477 1.13597 0.567984 0.823040i \(-0.307724\pi\)
0.567984 + 0.823040i \(0.307724\pi\)
\(402\) − 12.8348i − 0.640144i
\(403\) 12.5390i 0.624613i
\(404\) −13.5826 −0.675758
\(405\) 0 0
\(406\) 4.41742 0.219233
\(407\) 15.1652i 0.751709i
\(408\) 6.79129i 0.336219i
\(409\) 22.7913 1.12696 0.563478 0.826131i \(-0.309463\pi\)
0.563478 + 0.826131i \(0.309463\pi\)
\(410\) 0 0
\(411\) −1.41742 −0.0699164
\(412\) − 7.37386i − 0.363284i
\(413\) − 12.3303i − 0.606735i
\(414\) 0.208712 0.0102576
\(415\) 0 0
\(416\) −1.20871 −0.0592620
\(417\) 26.4174i 1.29367i
\(418\) − 4.58258i − 0.224141i
\(419\) −39.1652 −1.91334 −0.956671 0.291170i \(-0.905956\pi\)
−0.956671 + 0.291170i \(0.905956\pi\)
\(420\) 0 0
\(421\) −23.1216 −1.12688 −0.563439 0.826158i \(-0.690522\pi\)
−0.563439 + 0.826158i \(0.690522\pi\)
\(422\) − 10.0000i − 0.486792i
\(423\) − 2.83485i − 0.137835i
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 9.62614 0.466388
\(427\) 9.41742i 0.455741i
\(428\) 13.5826i 0.656539i
\(429\) −8.20871 −0.396320
\(430\) 0 0
\(431\) −19.9129 −0.959170 −0.479585 0.877496i \(-0.659213\pi\)
−0.479585 + 0.877496i \(0.659213\pi\)
\(432\) 5.00000i 0.240563i
\(433\) 1.53901i 0.0739603i 0.999316 + 0.0369802i \(0.0117738\pi\)
−0.999316 + 0.0369802i \(0.988226\pi\)
\(434\) 28.9564 1.38995
\(435\) 0 0
\(436\) 10.3739 0.496818
\(437\) − 1.20871i − 0.0578205i
\(438\) 26.4174i 1.26227i
\(439\) −25.5390 −1.21891 −0.609455 0.792820i \(-0.708612\pi\)
−0.609455 + 0.792820i \(0.708612\pi\)
\(440\) 0 0
\(441\) 0.165151 0.00786435
\(442\) − 4.58258i − 0.217971i
\(443\) − 35.2087i − 1.67282i −0.548107 0.836408i \(-0.684651\pi\)
0.548107 0.836408i \(-0.315349\pi\)
\(444\) 7.16515 0.340043
\(445\) 0 0
\(446\) −11.1652 −0.528685
\(447\) − 22.9129i − 1.08374i
\(448\) 2.79129i 0.131876i
\(449\) −25.1216 −1.18556 −0.592781 0.805364i \(-0.701970\pi\)
−0.592781 + 0.805364i \(0.701970\pi\)
\(450\) 0 0
\(451\) −8.37386 −0.394310
\(452\) − 6.00000i − 0.282216i
\(453\) − 11.1216i − 0.522538i
\(454\) 4.74773 0.222822
\(455\) 0 0
\(456\) −2.16515 −0.101393
\(457\) 10.0000i 0.467780i 0.972263 + 0.233890i \(0.0751456\pi\)
−0.972263 + 0.233890i \(0.924854\pi\)
\(458\) 16.3303i 0.763065i
\(459\) −18.9564 −0.884811
\(460\) 0 0
\(461\) −1.25227 −0.0583242 −0.0291621 0.999575i \(-0.509284\pi\)
−0.0291621 + 0.999575i \(0.509284\pi\)
\(462\) 18.9564i 0.881933i
\(463\) − 10.0000i − 0.464739i −0.972628 0.232370i \(-0.925352\pi\)
0.972628 0.232370i \(-0.0746479\pi\)
\(464\) 1.58258 0.0734692
\(465\) 0 0
\(466\) −7.58258 −0.351256
\(467\) − 25.9129i − 1.19911i −0.800335 0.599553i \(-0.795345\pi\)
0.800335 0.599553i \(-0.204655\pi\)
\(468\) 0.252273i 0.0116613i
\(469\) 20.0000 0.923514
\(470\) 0 0
\(471\) 22.8348 1.05217
\(472\) − 4.41742i − 0.203328i
\(473\) − 27.1652i − 1.24905i
\(474\) 14.3303 0.658213
\(475\) 0 0
\(476\) −10.5826 −0.485052
\(477\) − 1.25227i − 0.0573376i
\(478\) − 3.16515i − 0.144771i
\(479\) 39.4955 1.80459 0.902297 0.431116i \(-0.141880\pi\)
0.902297 + 0.431116i \(0.141880\pi\)
\(480\) 0 0
\(481\) −4.83485 −0.220450
\(482\) − 28.0000i − 1.27537i
\(483\) 5.00000i 0.227508i
\(484\) −3.37386 −0.153357
\(485\) 0 0
\(486\) 2.16515 0.0982133
\(487\) − 15.5826i − 0.706114i −0.935602 0.353057i \(-0.885142\pi\)
0.935602 0.353057i \(-0.114858\pi\)
\(488\) 3.37386i 0.152728i
\(489\) −40.0780 −1.81239
\(490\) 0 0
\(491\) −16.7477 −0.755814 −0.377907 0.925843i \(-0.623356\pi\)
−0.377907 + 0.925843i \(0.623356\pi\)
\(492\) 3.95644i 0.178370i
\(493\) 6.00000i 0.270226i
\(494\) 1.46099 0.0657328
\(495\) 0 0
\(496\) 10.3739 0.465800
\(497\) 15.0000i 0.672842i
\(498\) 10.7477i 0.481617i
\(499\) −23.1652 −1.03701 −0.518507 0.855073i \(-0.673512\pi\)
−0.518507 + 0.855073i \(0.673512\pi\)
\(500\) 0 0
\(501\) −32.8348 −1.46695
\(502\) 30.7913i 1.37428i
\(503\) 18.7913i 0.837862i 0.908018 + 0.418931i \(0.137595\pi\)
−0.908018 + 0.418931i \(0.862405\pi\)
\(504\) 0.582576 0.0259500
\(505\) 0 0
\(506\) −3.79129 −0.168543
\(507\) 20.6697i 0.917973i
\(508\) − 14.7477i − 0.654325i
\(509\) −7.25227 −0.321451 −0.160726 0.986999i \(-0.551383\pi\)
−0.160726 + 0.986999i \(0.551383\pi\)
\(510\) 0 0
\(511\) −41.1652 −1.82104
\(512\) 1.00000i 0.0441942i
\(513\) − 6.04356i − 0.266830i
\(514\) 22.7477 1.00336
\(515\) 0 0
\(516\) −12.8348 −0.565023
\(517\) 51.4955i 2.26477i
\(518\) 11.1652i 0.490569i
\(519\) 25.4519 1.11721
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) − 0.330303i − 0.0144570i
\(523\) − 1.16515i − 0.0509485i −0.999675 0.0254743i \(-0.991890\pi\)
0.999675 0.0254743i \(-0.00810958\pi\)
\(524\) −9.16515 −0.400381
\(525\) 0 0
\(526\) −15.7913 −0.688533
\(527\) 39.3303i 1.71325i
\(528\) 6.79129i 0.295553i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −0.921970 −0.0400101
\(532\) − 3.37386i − 0.146276i
\(533\) − 2.66970i − 0.115637i
\(534\) −5.66970 −0.245352
\(535\) 0 0
\(536\) 7.16515 0.309487
\(537\) 30.0000i 1.29460i
\(538\) − 16.7477i − 0.722046i
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 38.3303 1.64795 0.823974 0.566627i \(-0.191752\pi\)
0.823974 + 0.566627i \(0.191752\pi\)
\(542\) − 23.1216i − 0.993157i
\(543\) 24.2523i 1.04076i
\(544\) −3.79129 −0.162550
\(545\) 0 0
\(546\) −6.04356 −0.258641
\(547\) − 15.1216i − 0.646553i −0.946305 0.323276i \(-0.895216\pi\)
0.946305 0.323276i \(-0.104784\pi\)
\(548\) − 0.791288i − 0.0338021i
\(549\) 0.704166 0.0300531
\(550\) 0 0
\(551\) −1.91288 −0.0814914
\(552\) 1.79129i 0.0762423i
\(553\) 22.3303i 0.949581i
\(554\) −1.16515 −0.0495025
\(555\) 0 0
\(556\) −14.7477 −0.625443
\(557\) − 30.3303i − 1.28514i −0.766229 0.642568i \(-0.777869\pi\)
0.766229 0.642568i \(-0.222131\pi\)
\(558\) − 2.16515i − 0.0916582i
\(559\) 8.66061 0.366305
\(560\) 0 0
\(561\) −25.7477 −1.08707
\(562\) − 16.7477i − 0.706460i
\(563\) 3.16515i 0.133395i 0.997773 + 0.0666976i \(0.0212463\pi\)
−0.997773 + 0.0666976i \(0.978754\pi\)
\(564\) 24.3303 1.02449
\(565\) 0 0
\(566\) 28.3303 1.19081
\(567\) 26.7477i 1.12330i
\(568\) 5.37386i 0.225482i
\(569\) −15.4955 −0.649603 −0.324802 0.945782i \(-0.605298\pi\)
−0.324802 + 0.945782i \(0.605298\pi\)
\(570\) 0 0
\(571\) 30.1216 1.26055 0.630275 0.776372i \(-0.282942\pi\)
0.630275 + 0.776372i \(0.282942\pi\)
\(572\) − 4.58258i − 0.191607i
\(573\) 29.4083i 1.22855i
\(574\) −6.16515 −0.257328
\(575\) 0 0
\(576\) 0.208712 0.00869634
\(577\) − 22.8348i − 0.950627i −0.879816 0.475314i \(-0.842335\pi\)
0.879816 0.475314i \(-0.157665\pi\)
\(578\) 2.62614i 0.109233i
\(579\) −12.0871 −0.502324
\(580\) 0 0
\(581\) −16.7477 −0.694813
\(582\) 26.7913i 1.11053i
\(583\) 22.7477i 0.942115i
\(584\) −14.7477 −0.610265
\(585\) 0 0
\(586\) 27.4955 1.13583
\(587\) 26.2087i 1.08175i 0.841103 + 0.540875i \(0.181907\pi\)
−0.841103 + 0.540875i \(0.818093\pi\)
\(588\) 1.41742i 0.0584536i
\(589\) −12.5390 −0.516661
\(590\) 0 0
\(591\) 36.7913 1.51339
\(592\) 4.00000i 0.164399i
\(593\) 13.9129i 0.571333i 0.958329 + 0.285667i \(0.0922150\pi\)
−0.958329 + 0.285667i \(0.907785\pi\)
\(594\) −18.9564 −0.777792
\(595\) 0 0
\(596\) 12.7913 0.523952
\(597\) − 36.4174i − 1.49047i
\(598\) − 1.20871i − 0.0494279i
\(599\) 40.1216 1.63932 0.819662 0.572848i \(-0.194161\pi\)
0.819662 + 0.572848i \(0.194161\pi\)
\(600\) 0 0
\(601\) −22.7913 −0.929676 −0.464838 0.885396i \(-0.653887\pi\)
−0.464838 + 0.885396i \(0.653887\pi\)
\(602\) − 20.0000i − 0.815139i
\(603\) − 1.49545i − 0.0608996i
\(604\) 6.20871 0.252629
\(605\) 0 0
\(606\) −24.3303 −0.988351
\(607\) 28.0000i 1.13648i 0.822861 + 0.568242i \(0.192376\pi\)
−0.822861 + 0.568242i \(0.807624\pi\)
\(608\) − 1.20871i − 0.0490198i
\(609\) 7.91288 0.320646
\(610\) 0 0
\(611\) −16.4174 −0.664178
\(612\) 0.791288i 0.0319859i
\(613\) 14.0000i 0.565455i 0.959200 + 0.282727i \(0.0912392\pi\)
−0.959200 + 0.282727i \(0.908761\pi\)
\(614\) 16.5390 0.667460
\(615\) 0 0
\(616\) −10.5826 −0.426384
\(617\) − 44.8693i − 1.80637i −0.429251 0.903185i \(-0.641222\pi\)
0.429251 0.903185i \(-0.358778\pi\)
\(618\) − 13.2087i − 0.531332i
\(619\) −2.79129 −0.112191 −0.0560957 0.998425i \(-0.517865\pi\)
−0.0560957 + 0.998425i \(0.517865\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) 12.0000i 0.481156i
\(623\) − 8.83485i − 0.353961i
\(624\) −2.16515 −0.0866754
\(625\) 0 0
\(626\) 18.3739 0.734367
\(627\) − 8.20871i − 0.327824i
\(628\) 12.7477i 0.508690i
\(629\) −15.1652 −0.604674
\(630\) 0 0
\(631\) −17.9129 −0.713100 −0.356550 0.934276i \(-0.616047\pi\)
−0.356550 + 0.934276i \(0.616047\pi\)
\(632\) 8.00000i 0.318223i
\(633\) − 17.9129i − 0.711973i
\(634\) 5.20871 0.206864
\(635\) 0 0
\(636\) 10.7477 0.426175
\(637\) − 0.956439i − 0.0378955i
\(638\) 6.00000i 0.237542i
\(639\) 1.12159 0.0443694
\(640\) 0 0
\(641\) −3.16515 −0.125016 −0.0625080 0.998044i \(-0.519910\pi\)
−0.0625080 + 0.998044i \(0.519910\pi\)
\(642\) 24.3303i 0.960240i
\(643\) − 20.7477i − 0.818210i −0.912487 0.409105i \(-0.865841\pi\)
0.912487 0.409105i \(-0.134159\pi\)
\(644\) −2.79129 −0.109992
\(645\) 0 0
\(646\) 4.58258 0.180299
\(647\) − 2.83485i − 0.111449i −0.998446 0.0557247i \(-0.982253\pi\)
0.998446 0.0557247i \(-0.0177469\pi\)
\(648\) 9.58258i 0.376439i
\(649\) 16.7477 0.657406
\(650\) 0 0
\(651\) 51.8693 2.03292
\(652\) − 22.3739i − 0.876228i
\(653\) 35.5390i 1.39075i 0.718647 + 0.695375i \(0.244762\pi\)
−0.718647 + 0.695375i \(0.755238\pi\)
\(654\) 18.5826 0.726636
\(655\) 0 0
\(656\) −2.20871 −0.0862357
\(657\) 3.07803i 0.120085i
\(658\) 37.9129i 1.47800i
\(659\) 27.1652 1.05820 0.529102 0.848558i \(-0.322529\pi\)
0.529102 + 0.848558i \(0.322529\pi\)
\(660\) 0 0
\(661\) −39.3739 −1.53147 −0.765733 0.643159i \(-0.777624\pi\)
−0.765733 + 0.643159i \(0.777624\pi\)
\(662\) 6.74773i 0.262258i
\(663\) − 8.20871i − 0.318800i
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) 0.834849 0.0323497
\(667\) 1.58258i 0.0612776i
\(668\) − 18.3303i − 0.709221i
\(669\) −20.0000 −0.773245
\(670\) 0 0
\(671\) −12.7913 −0.493802
\(672\) 5.00000i 0.192879i
\(673\) 38.0000i 1.46479i 0.680879 + 0.732396i \(0.261598\pi\)
−0.680879 + 0.732396i \(0.738402\pi\)
\(674\) −16.7913 −0.646776
\(675\) 0 0
\(676\) −11.5390 −0.443808
\(677\) 30.6606i 1.17838i 0.807993 + 0.589191i \(0.200554\pi\)
−0.807993 + 0.589191i \(0.799446\pi\)
\(678\) − 10.7477i − 0.412764i
\(679\) −41.7477 −1.60213
\(680\) 0 0
\(681\) 8.50455 0.325895
\(682\) 39.3303i 1.50604i
\(683\) 2.37386i 0.0908334i 0.998968 + 0.0454167i \(0.0144616\pi\)
−0.998968 + 0.0454167i \(0.985538\pi\)
\(684\) −0.252273 −0.00964590
\(685\) 0 0
\(686\) 17.3303 0.661674
\(687\) 29.2523i 1.11604i
\(688\) − 7.16515i − 0.273169i
\(689\) −7.25227 −0.276290
\(690\) 0 0
\(691\) 15.2523 0.580224 0.290112 0.956993i \(-0.406307\pi\)
0.290112 + 0.956993i \(0.406307\pi\)
\(692\) 14.2087i 0.540134i
\(693\) 2.20871i 0.0839020i
\(694\) 9.79129 0.371672
\(695\) 0 0
\(696\) 2.83485 0.107455
\(697\) − 8.37386i − 0.317183i
\(698\) − 26.0000i − 0.984115i
\(699\) −13.5826 −0.513740
\(700\) 0 0
\(701\) 9.62614 0.363574 0.181787 0.983338i \(-0.441812\pi\)
0.181787 + 0.983338i \(0.441812\pi\)
\(702\) − 6.04356i − 0.228100i
\(703\) − 4.83485i − 0.182350i
\(704\) −3.79129 −0.142890
\(705\) 0 0
\(706\) 15.1652 0.570748
\(707\) − 37.9129i − 1.42586i
\(708\) − 7.91288i − 0.297384i
\(709\) −34.5390 −1.29714 −0.648570 0.761155i \(-0.724633\pi\)
−0.648570 + 0.761155i \(0.724633\pi\)
\(710\) 0 0
\(711\) 1.66970 0.0626185
\(712\) − 3.16515i − 0.118619i
\(713\) 10.3739i 0.388504i
\(714\) −18.9564 −0.709427
\(715\) 0 0
\(716\) −16.7477 −0.625892
\(717\) − 5.66970i − 0.211739i
\(718\) 9.16515i 0.342040i
\(719\) 29.5390 1.10162 0.550810 0.834631i \(-0.314319\pi\)
0.550810 + 0.834631i \(0.314319\pi\)
\(720\) 0 0
\(721\) 20.5826 0.766535
\(722\) − 17.5390i − 0.652735i
\(723\) − 50.1561i − 1.86532i
\(724\) −13.5390 −0.503174
\(725\) 0 0
\(726\) −6.04356 −0.224298
\(727\) 2.12159i 0.0786854i 0.999226 + 0.0393427i \(0.0125264\pi\)
−0.999226 + 0.0393427i \(0.987474\pi\)
\(728\) − 3.37386i − 0.125044i
\(729\) −24.8693 −0.921086
\(730\) 0 0
\(731\) 27.1652 1.00474
\(732\) 6.04356i 0.223376i
\(733\) 26.0000i 0.960332i 0.877178 + 0.480166i \(0.159424\pi\)
−0.877178 + 0.480166i \(0.840576\pi\)
\(734\) −0.834849 −0.0308148
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 27.1652i 1.00064i
\(738\) 0.460985i 0.0169691i
\(739\) −8.00000 −0.294285 −0.147142 0.989115i \(-0.547008\pi\)
−0.147142 + 0.989115i \(0.547008\pi\)
\(740\) 0 0
\(741\) 2.61704 0.0961395
\(742\) 16.7477i 0.614828i
\(743\) 9.95644i 0.365266i 0.983181 + 0.182633i \(0.0584621\pi\)
−0.983181 + 0.182633i \(0.941538\pi\)
\(744\) 18.5826 0.681270
\(745\) 0 0
\(746\) 14.7477 0.539953
\(747\) 1.25227i 0.0458183i
\(748\) − 14.3739i − 0.525561i
\(749\) −37.9129 −1.38531
\(750\) 0 0
\(751\) 18.7477 0.684114 0.342057 0.939679i \(-0.388876\pi\)
0.342057 + 0.939679i \(0.388876\pi\)
\(752\) 13.5826i 0.495306i
\(753\) 55.1561i 2.01000i
\(754\) −1.91288 −0.0696629
\(755\) 0 0
\(756\) −13.9564 −0.507591
\(757\) − 26.3303i − 0.956991i −0.878090 0.478496i \(-0.841182\pi\)
0.878090 0.478496i \(-0.158818\pi\)
\(758\) − 7.37386i − 0.267831i
\(759\) −6.79129 −0.246508
\(760\) 0 0
\(761\) −33.9564 −1.23092 −0.615460 0.788168i \(-0.711030\pi\)
−0.615460 + 0.788168i \(0.711030\pi\)
\(762\) − 26.4174i − 0.957002i
\(763\) 28.9564i 1.04829i
\(764\) −16.4174 −0.593962
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 5.33939i 0.192794i
\(768\) 1.79129i 0.0646375i
\(769\) 3.66970 0.132333 0.0661663 0.997809i \(-0.478923\pi\)
0.0661663 + 0.997809i \(0.478923\pi\)
\(770\) 0 0
\(771\) 40.7477 1.46749
\(772\) − 6.74773i − 0.242856i
\(773\) − 21.4955i − 0.773138i −0.922261 0.386569i \(-0.873660\pi\)
0.922261 0.386569i \(-0.126340\pi\)
\(774\) −1.49545 −0.0537530
\(775\) 0 0
\(776\) −14.9564 −0.536905
\(777\) 20.0000i 0.717496i
\(778\) − 29.7042i − 1.06495i
\(779\) 2.66970 0.0956518
\(780\) 0 0
\(781\) −20.3739 −0.729034
\(782\) − 3.79129i − 0.135576i
\(783\) 7.91288i 0.282783i
\(784\) −0.791288 −0.0282603
\(785\) 0 0
\(786\) −16.4174 −0.585590
\(787\) 8.41742i 0.300049i 0.988682 + 0.150024i \(0.0479352\pi\)
−0.988682 + 0.150024i \(0.952065\pi\)
\(788\) 20.5390i 0.731672i
\(789\) −28.2867 −1.00703
\(790\) 0 0
\(791\) 16.7477 0.595481
\(792\) 0.791288i 0.0281172i
\(793\) − 4.07803i − 0.144815i
\(794\) 16.5390 0.586948
\(795\) 0 0
\(796\) 20.3303 0.720588
\(797\) 49.9129i 1.76800i 0.467482 + 0.884002i \(0.345161\pi\)
−0.467482 + 0.884002i \(0.654839\pi\)
\(798\) − 6.04356i − 0.213940i
\(799\) −51.4955 −1.82178
\(800\) 0 0
\(801\) −0.660606 −0.0233413
\(802\) 22.7477i 0.803250i
\(803\) − 55.9129i − 1.97312i
\(804\) 12.8348 0.452650
\(805\) 0 0
\(806\) −12.5390 −0.441668
\(807\) − 30.0000i − 1.05605i
\(808\) − 13.5826i − 0.477833i
\(809\) −11.0436 −0.388271 −0.194135 0.980975i \(-0.562190\pi\)
−0.194135 + 0.980975i \(0.562190\pi\)
\(810\) 0 0
\(811\) −47.9129 −1.68245 −0.841224 0.540686i \(-0.818165\pi\)
−0.841224 + 0.540686i \(0.818165\pi\)
\(812\) 4.41742i 0.155021i
\(813\) − 41.4174i − 1.45257i
\(814\) −15.1652 −0.531538
\(815\) 0 0
\(816\) −6.79129 −0.237743
\(817\) 8.66061i 0.302996i
\(818\) 22.7913i 0.796879i
\(819\) −0.704166 −0.0246056
\(820\) 0 0
\(821\) 2.83485 0.0989369 0.0494684 0.998776i \(-0.484247\pi\)
0.0494684 + 0.998776i \(0.484247\pi\)
\(822\) − 1.41742i − 0.0494383i
\(823\) 41.1652i 1.43493i 0.696596 + 0.717463i \(0.254697\pi\)
−0.696596 + 0.717463i \(0.745303\pi\)
\(824\) 7.37386 0.256881
\(825\) 0 0
\(826\) 12.3303 0.429026
\(827\) − 41.0780i − 1.42842i −0.699930 0.714212i \(-0.746785\pi\)
0.699930 0.714212i \(-0.253215\pi\)
\(828\) 0.208712i 0.00725325i
\(829\) 31.4955 1.09388 0.546941 0.837171i \(-0.315792\pi\)
0.546941 + 0.837171i \(0.315792\pi\)
\(830\) 0 0
\(831\) −2.08712 −0.0724014
\(832\) − 1.20871i − 0.0419046i
\(833\) − 3.00000i − 0.103944i
\(834\) −26.4174 −0.914761
\(835\) 0 0
\(836\) 4.58258 0.158492
\(837\) 51.8693i 1.79287i
\(838\) − 39.1652i − 1.35294i
\(839\) 22.4174 0.773935 0.386968 0.922093i \(-0.373522\pi\)
0.386968 + 0.922093i \(0.373522\pi\)
\(840\) 0 0
\(841\) −26.4955 −0.913636
\(842\) − 23.1216i − 0.796823i
\(843\) − 30.0000i − 1.03325i
\(844\) 10.0000 0.344214
\(845\) 0 0
\(846\) 2.83485 0.0974641
\(847\) − 9.41742i − 0.323587i
\(848\) 6.00000i 0.206041i
\(849\) 50.7477 1.74166
\(850\) 0 0
\(851\) −4.00000 −0.137118
\(852\) 9.62614i 0.329786i
\(853\) 8.46099i 0.289699i 0.989454 + 0.144849i \(0.0462697\pi\)
−0.989454 + 0.144849i \(0.953730\pi\)
\(854\) −9.41742 −0.322258
\(855\) 0 0
\(856\) −13.5826 −0.464243
\(857\) − 9.16515i − 0.313076i −0.987672 0.156538i \(-0.949967\pi\)
0.987672 0.156538i \(-0.0500333\pi\)
\(858\) − 8.20871i − 0.280241i
\(859\) −0.747727 −0.0255121 −0.0127561 0.999919i \(-0.504060\pi\)
−0.0127561 + 0.999919i \(0.504060\pi\)
\(860\) 0 0
\(861\) −11.0436 −0.376364
\(862\) − 19.9129i − 0.678235i
\(863\) − 31.5826i − 1.07508i −0.843237 0.537542i \(-0.819353\pi\)
0.843237 0.537542i \(-0.180647\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) −1.53901 −0.0522979
\(867\) 4.70417i 0.159762i
\(868\) 28.9564i 0.982846i
\(869\) −30.3303 −1.02889
\(870\) 0 0
\(871\) −8.66061 −0.293453
\(872\) 10.3739i 0.351303i
\(873\) 3.12159i 0.105650i
\(874\) 1.20871 0.0408853
\(875\) 0 0
\(876\) −26.4174 −0.892562
\(877\) − 7.70417i − 0.260151i −0.991504 0.130076i \(-0.958478\pi\)
0.991504 0.130076i \(-0.0415220\pi\)
\(878\) − 25.5390i − 0.861900i
\(879\) 49.2523 1.66124
\(880\) 0 0
\(881\) −6.33030 −0.213273 −0.106637 0.994298i \(-0.534008\pi\)
−0.106637 + 0.994298i \(0.534008\pi\)
\(882\) 0.165151i 0.00556094i
\(883\) − 12.0436i − 0.405298i −0.979251 0.202649i \(-0.935045\pi\)
0.979251 0.202649i \(-0.0649551\pi\)
\(884\) 4.58258 0.154129
\(885\) 0 0
\(886\) 35.2087 1.18286
\(887\) 3.16515i 0.106275i 0.998587 + 0.0531377i \(0.0169222\pi\)
−0.998587 + 0.0531377i \(0.983078\pi\)
\(888\) 7.16515i 0.240447i
\(889\) 41.1652 1.38063
\(890\) 0 0
\(891\) −36.3303 −1.21711
\(892\) − 11.1652i − 0.373837i
\(893\) − 16.4174i − 0.549388i
\(894\) 22.9129 0.766321
\(895\) 0 0
\(896\) −2.79129 −0.0932504
\(897\) − 2.16515i − 0.0722923i
\(898\) − 25.1216i − 0.838318i
\(899\) 16.4174 0.547552
\(900\) 0 0
\(901\) −22.7477 −0.757837
\(902\) − 8.37386i − 0.278819i
\(903\) − 35.8258i − 1.19221i
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 11.1216 0.369490
\(907\) 20.7477i 0.688917i 0.938802 + 0.344458i \(0.111937\pi\)
−0.938802 + 0.344458i \(0.888063\pi\)
\(908\) 4.74773i 0.157559i
\(909\) −2.83485 −0.0940260
\(910\) 0 0
\(911\) 13.5826 0.450011 0.225005 0.974358i \(-0.427760\pi\)
0.225005 + 0.974358i \(0.427760\pi\)
\(912\) − 2.16515i − 0.0716953i
\(913\) − 22.7477i − 0.752840i
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) −16.3303 −0.539568
\(917\) − 25.5826i − 0.844811i
\(918\) − 18.9564i − 0.625656i
\(919\) 36.8348 1.21507 0.607535 0.794293i \(-0.292159\pi\)
0.607535 + 0.794293i \(0.292159\pi\)
\(920\) 0 0
\(921\) 29.6261 0.976214
\(922\) − 1.25227i − 0.0412414i
\(923\) − 6.49545i − 0.213800i
\(924\) −18.9564 −0.623621
\(925\) 0 0
\(926\) 10.0000 0.328620
\(927\) − 1.53901i − 0.0505479i
\(928\) 1.58258i 0.0519506i
\(929\) 39.4955 1.29580 0.647902 0.761724i \(-0.275647\pi\)
0.647902 + 0.761724i \(0.275647\pi\)
\(930\) 0 0
\(931\) 0.956439 0.0313460
\(932\) − 7.58258i − 0.248376i
\(933\) 21.4955i 0.703730i
\(934\) 25.9129 0.847895
\(935\) 0 0
\(936\) −0.252273 −0.00824580
\(937\) − 58.3739i − 1.90699i −0.301407 0.953495i \(-0.597456\pi\)
0.301407 0.953495i \(-0.402544\pi\)
\(938\) 20.0000i 0.653023i
\(939\) 32.9129 1.07407
\(940\) 0 0
\(941\) 54.9564 1.79153 0.895764 0.444529i \(-0.146629\pi\)
0.895764 + 0.444529i \(0.146629\pi\)
\(942\) 22.8348i 0.744000i
\(943\) − 2.20871i − 0.0719256i
\(944\) 4.41742 0.143775
\(945\) 0 0
\(946\) 27.1652 0.883215
\(947\) 29.5390i 0.959889i 0.877299 + 0.479945i \(0.159343\pi\)
−0.877299 + 0.479945i \(0.840657\pi\)
\(948\) 14.3303i 0.465427i
\(949\) 17.8258 0.578649
\(950\) 0 0
\(951\) 9.33030 0.302556
\(952\) − 10.5826i − 0.342983i
\(953\) − 26.5390i − 0.859683i −0.902904 0.429842i \(-0.858569\pi\)
0.902904 0.429842i \(-0.141431\pi\)
\(954\) 1.25227 0.0405438
\(955\) 0 0
\(956\) 3.16515 0.102368
\(957\) 10.7477i 0.347425i
\(958\) 39.4955i 1.27604i
\(959\) 2.20871 0.0713230
\(960\) 0 0
\(961\) 76.6170 2.47152
\(962\) − 4.83485i − 0.155882i
\(963\) 2.83485i 0.0913517i
\(964\) 28.0000 0.901819
\(965\) 0 0
\(966\) −5.00000 −0.160872
\(967\) 5.25227i 0.168902i 0.996428 + 0.0844509i \(0.0269136\pi\)
−0.996428 + 0.0844509i \(0.973086\pi\)
\(968\) − 3.37386i − 0.108440i
\(969\) 8.20871 0.263702
\(970\) 0 0
\(971\) −6.95644 −0.223243 −0.111621 0.993751i \(-0.535604\pi\)
−0.111621 + 0.993751i \(0.535604\pi\)
\(972\) 2.16515i 0.0694473i
\(973\) − 41.1652i − 1.31969i
\(974\) 15.5826 0.499298
\(975\) 0 0
\(976\) −3.37386 −0.107995
\(977\) − 7.12159i − 0.227840i −0.993490 0.113920i \(-0.963659\pi\)
0.993490 0.113920i \(-0.0363407\pi\)
\(978\) − 40.0780i − 1.28155i
\(979\) 12.0000 0.383522
\(980\) 0 0
\(981\) 2.16515 0.0691280
\(982\) − 16.7477i − 0.534441i
\(983\) − 0.626136i − 0.0199707i −0.999950 0.00998533i \(-0.996822\pi\)
0.999950 0.00998533i \(-0.00317848\pi\)
\(984\) −3.95644 −0.126127
\(985\) 0 0
\(986\) −6.00000 −0.191079
\(987\) 67.9129i 2.16169i
\(988\) 1.46099i 0.0464801i
\(989\) 7.16515 0.227839
\(990\) 0 0
\(991\) −37.7913 −1.20048 −0.600240 0.799820i \(-0.704928\pi\)
−0.600240 + 0.799820i \(0.704928\pi\)
\(992\) 10.3739i 0.329370i
\(993\) 12.0871i 0.383573i
\(994\) −15.0000 −0.475771
\(995\) 0 0
\(996\) −10.7477 −0.340555
\(997\) − 11.4955i − 0.364065i −0.983293 0.182032i \(-0.941732\pi\)
0.983293 0.182032i \(-0.0582676\pi\)
\(998\) − 23.1652i − 0.733280i
\(999\) −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.599.4 4
5.2 odd 4 230.2.a.a.1.2 2
5.3 odd 4 1150.2.a.o.1.1 2
5.4 even 2 inner 1150.2.b.g.599.1 4
15.2 even 4 2070.2.a.x.1.2 2
20.3 even 4 9200.2.a.bs.1.2 2
20.7 even 4 1840.2.a.n.1.1 2
40.27 even 4 7360.2.a.bk.1.2 2
40.37 odd 4 7360.2.a.bq.1.1 2
115.22 even 4 5290.2.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.a.1.2 2 5.2 odd 4
1150.2.a.o.1.1 2 5.3 odd 4
1150.2.b.g.599.1 4 5.4 even 2 inner
1150.2.b.g.599.4 4 1.1 even 1 trivial
1840.2.a.n.1.1 2 20.7 even 4
2070.2.a.x.1.2 2 15.2 even 4
5290.2.a.e.1.2 2 115.22 even 4
7360.2.a.bk.1.2 2 40.27 even 4
7360.2.a.bq.1.1 2 40.37 odd 4
9200.2.a.bs.1.2 2 20.3 even 4