Properties

Label 1150.2.a.o.1.1
Level $1150$
Weight $2$
Character 1150.1
Self dual yes
Analytic conductor $9.183$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.18279623245\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Defining polynomial: \(x^{2} - x - 5\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.79129\) of defining polynomial
Character \(\chi\) \(=\) 1150.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.79129 q^{3} +1.00000 q^{4} -1.79129 q^{6} -2.79129 q^{7} +1.00000 q^{8} +0.208712 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.79129 q^{3} +1.00000 q^{4} -1.79129 q^{6} -2.79129 q^{7} +1.00000 q^{8} +0.208712 q^{9} +3.79129 q^{11} -1.79129 q^{12} -1.20871 q^{13} -2.79129 q^{14} +1.00000 q^{16} +3.79129 q^{17} +0.208712 q^{18} +1.20871 q^{19} +5.00000 q^{21} +3.79129 q^{22} -1.00000 q^{23} -1.79129 q^{24} -1.20871 q^{26} +5.00000 q^{27} -2.79129 q^{28} -1.58258 q^{29} +10.3739 q^{31} +1.00000 q^{32} -6.79129 q^{33} +3.79129 q^{34} +0.208712 q^{36} +4.00000 q^{37} +1.20871 q^{38} +2.16515 q^{39} -2.20871 q^{41} +5.00000 q^{42} +7.16515 q^{43} +3.79129 q^{44} -1.00000 q^{46} +13.5826 q^{47} -1.79129 q^{48} +0.791288 q^{49} -6.79129 q^{51} -1.20871 q^{52} -6.00000 q^{53} +5.00000 q^{54} -2.79129 q^{56} -2.16515 q^{57} -1.58258 q^{58} -4.41742 q^{59} -3.37386 q^{61} +10.3739 q^{62} -0.582576 q^{63} +1.00000 q^{64} -6.79129 q^{66} +7.16515 q^{67} +3.79129 q^{68} +1.79129 q^{69} -5.37386 q^{71} +0.208712 q^{72} +14.7477 q^{73} +4.00000 q^{74} +1.20871 q^{76} -10.5826 q^{77} +2.16515 q^{78} +8.00000 q^{79} -9.58258 q^{81} -2.20871 q^{82} +6.00000 q^{83} +5.00000 q^{84} +7.16515 q^{86} +2.83485 q^{87} +3.79129 q^{88} -3.16515 q^{89} +3.37386 q^{91} -1.00000 q^{92} -18.5826 q^{93} +13.5826 q^{94} -1.79129 q^{96} -14.9564 q^{97} +0.791288 q^{98} +0.791288 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + q^{3} + 2q^{4} + q^{6} - q^{7} + 2q^{8} + 5q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + q^{3} + 2q^{4} + q^{6} - q^{7} + 2q^{8} + 5q^{9} + 3q^{11} + q^{12} - 7q^{13} - q^{14} + 2q^{16} + 3q^{17} + 5q^{18} + 7q^{19} + 10q^{21} + 3q^{22} - 2q^{23} + q^{24} - 7q^{26} + 10q^{27} - q^{28} + 6q^{29} + 7q^{31} + 2q^{32} - 9q^{33} + 3q^{34} + 5q^{36} + 8q^{37} + 7q^{38} - 14q^{39} - 9q^{41} + 10q^{42} - 4q^{43} + 3q^{44} - 2q^{46} + 18q^{47} + q^{48} - 3q^{49} - 9q^{51} - 7q^{52} - 12q^{53} + 10q^{54} - q^{56} + 14q^{57} + 6q^{58} - 18q^{59} + 7q^{61} + 7q^{62} + 8q^{63} + 2q^{64} - 9q^{66} - 4q^{67} + 3q^{68} - q^{69} + 3q^{71} + 5q^{72} + 2q^{73} + 8q^{74} + 7q^{76} - 12q^{77} - 14q^{78} + 16q^{79} - 10q^{81} - 9q^{82} + 12q^{83} + 10q^{84} - 4q^{86} + 24q^{87} + 3q^{88} + 12q^{89} - 7q^{91} - 2q^{92} - 28q^{93} + 18q^{94} + q^{96} - 7q^{97} - 3q^{98} - 3q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.79129 −1.03420 −0.517100 0.855925i \(-0.672989\pi\)
−0.517100 + 0.855925i \(0.672989\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.79129 −0.731290
\(7\) −2.79129 −1.05501 −0.527504 0.849553i \(-0.676872\pi\)
−0.527504 + 0.849553i \(0.676872\pi\)
\(8\) 1.00000 0.353553
\(9\) 0.208712 0.0695707
\(10\) 0 0
\(11\) 3.79129 1.14312 0.571558 0.820562i \(-0.306339\pi\)
0.571558 + 0.820562i \(0.306339\pi\)
\(12\) −1.79129 −0.517100
\(13\) −1.20871 −0.335236 −0.167618 0.985852i \(-0.553608\pi\)
−0.167618 + 0.985852i \(0.553608\pi\)
\(14\) −2.79129 −0.746003
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.79129 0.919522 0.459761 0.888043i \(-0.347935\pi\)
0.459761 + 0.888043i \(0.347935\pi\)
\(18\) 0.208712 0.0491939
\(19\) 1.20871 0.277298 0.138649 0.990342i \(-0.455724\pi\)
0.138649 + 0.990342i \(0.455724\pi\)
\(20\) 0 0
\(21\) 5.00000 1.09109
\(22\) 3.79129 0.808305
\(23\) −1.00000 −0.208514
\(24\) −1.79129 −0.365645
\(25\) 0 0
\(26\) −1.20871 −0.237048
\(27\) 5.00000 0.962250
\(28\) −2.79129 −0.527504
\(29\) −1.58258 −0.293877 −0.146938 0.989146i \(-0.546942\pi\)
−0.146938 + 0.989146i \(0.546942\pi\)
\(30\) 0 0
\(31\) 10.3739 1.86320 0.931600 0.363484i \(-0.118413\pi\)
0.931600 + 0.363484i \(0.118413\pi\)
\(32\) 1.00000 0.176777
\(33\) −6.79129 −1.18221
\(34\) 3.79129 0.650201
\(35\) 0 0
\(36\) 0.208712 0.0347854
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 1.20871 0.196079
\(39\) 2.16515 0.346702
\(40\) 0 0
\(41\) −2.20871 −0.344943 −0.172471 0.985015i \(-0.555175\pi\)
−0.172471 + 0.985015i \(0.555175\pi\)
\(42\) 5.00000 0.771517
\(43\) 7.16515 1.09268 0.546338 0.837565i \(-0.316022\pi\)
0.546338 + 0.837565i \(0.316022\pi\)
\(44\) 3.79129 0.571558
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 13.5826 1.98122 0.990611 0.136710i \(-0.0436528\pi\)
0.990611 + 0.136710i \(0.0436528\pi\)
\(48\) −1.79129 −0.258550
\(49\) 0.791288 0.113041
\(50\) 0 0
\(51\) −6.79129 −0.950971
\(52\) −1.20871 −0.167618
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 5.00000 0.680414
\(55\) 0 0
\(56\) −2.79129 −0.373002
\(57\) −2.16515 −0.286781
\(58\) −1.58258 −0.207802
\(59\) −4.41742 −0.575100 −0.287550 0.957766i \(-0.592841\pi\)
−0.287550 + 0.957766i \(0.592841\pi\)
\(60\) 0 0
\(61\) −3.37386 −0.431979 −0.215989 0.976396i \(-0.569298\pi\)
−0.215989 + 0.976396i \(0.569298\pi\)
\(62\) 10.3739 1.31748
\(63\) −0.582576 −0.0733976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −6.79129 −0.835950
\(67\) 7.16515 0.875363 0.437681 0.899130i \(-0.355800\pi\)
0.437681 + 0.899130i \(0.355800\pi\)
\(68\) 3.79129 0.459761
\(69\) 1.79129 0.215646
\(70\) 0 0
\(71\) −5.37386 −0.637760 −0.318880 0.947795i \(-0.603307\pi\)
−0.318880 + 0.947795i \(0.603307\pi\)
\(72\) 0.208712 0.0245970
\(73\) 14.7477 1.72609 0.863045 0.505126i \(-0.168554\pi\)
0.863045 + 0.505126i \(0.168554\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) 1.20871 0.138649
\(77\) −10.5826 −1.20600
\(78\) 2.16515 0.245155
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −9.58258 −1.06473
\(82\) −2.20871 −0.243911
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 5.00000 0.545545
\(85\) 0 0
\(86\) 7.16515 0.772638
\(87\) 2.83485 0.303928
\(88\) 3.79129 0.404153
\(89\) −3.16515 −0.335505 −0.167753 0.985829i \(-0.553651\pi\)
−0.167753 + 0.985829i \(0.553651\pi\)
\(90\) 0 0
\(91\) 3.37386 0.353677
\(92\) −1.00000 −0.104257
\(93\) −18.5826 −1.92692
\(94\) 13.5826 1.40094
\(95\) 0 0
\(96\) −1.79129 −0.182823
\(97\) −14.9564 −1.51860 −0.759298 0.650743i \(-0.774458\pi\)
−0.759298 + 0.650743i \(0.774458\pi\)
\(98\) 0.791288 0.0799321
\(99\) 0.791288 0.0795274
\(100\) 0 0
\(101\) 13.5826 1.35152 0.675758 0.737123i \(-0.263816\pi\)
0.675758 + 0.737123i \(0.263816\pi\)
\(102\) −6.79129 −0.672438
\(103\) −7.37386 −0.726568 −0.363284 0.931678i \(-0.618345\pi\)
−0.363284 + 0.931678i \(0.618345\pi\)
\(104\) −1.20871 −0.118524
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −13.5826 −1.31308 −0.656539 0.754292i \(-0.727980\pi\)
−0.656539 + 0.754292i \(0.727980\pi\)
\(108\) 5.00000 0.481125
\(109\) 10.3739 0.993636 0.496818 0.867855i \(-0.334502\pi\)
0.496818 + 0.867855i \(0.334502\pi\)
\(110\) 0 0
\(111\) −7.16515 −0.680086
\(112\) −2.79129 −0.263752
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −2.16515 −0.202785
\(115\) 0 0
\(116\) −1.58258 −0.146938
\(117\) −0.252273 −0.0233226
\(118\) −4.41742 −0.406657
\(119\) −10.5826 −0.970103
\(120\) 0 0
\(121\) 3.37386 0.306715
\(122\) −3.37386 −0.305455
\(123\) 3.95644 0.356740
\(124\) 10.3739 0.931600
\(125\) 0 0
\(126\) −0.582576 −0.0519000
\(127\) 14.7477 1.30865 0.654325 0.756214i \(-0.272953\pi\)
0.654325 + 0.756214i \(0.272953\pi\)
\(128\) 1.00000 0.0883883
\(129\) −12.8348 −1.13005
\(130\) 0 0
\(131\) 9.16515 0.800763 0.400381 0.916349i \(-0.368878\pi\)
0.400381 + 0.916349i \(0.368878\pi\)
\(132\) −6.79129 −0.591106
\(133\) −3.37386 −0.292551
\(134\) 7.16515 0.618975
\(135\) 0 0
\(136\) 3.79129 0.325100
\(137\) 0.791288 0.0676043 0.0338021 0.999429i \(-0.489238\pi\)
0.0338021 + 0.999429i \(0.489238\pi\)
\(138\) 1.79129 0.152485
\(139\) −14.7477 −1.25089 −0.625443 0.780270i \(-0.715082\pi\)
−0.625443 + 0.780270i \(0.715082\pi\)
\(140\) 0 0
\(141\) −24.3303 −2.04898
\(142\) −5.37386 −0.450965
\(143\) −4.58258 −0.383214
\(144\) 0.208712 0.0173927
\(145\) 0 0
\(146\) 14.7477 1.22053
\(147\) −1.41742 −0.116907
\(148\) 4.00000 0.328798
\(149\) 12.7913 1.04790 0.523952 0.851748i \(-0.324457\pi\)
0.523952 + 0.851748i \(0.324457\pi\)
\(150\) 0 0
\(151\) −6.20871 −0.505258 −0.252629 0.967563i \(-0.581295\pi\)
−0.252629 + 0.967563i \(0.581295\pi\)
\(152\) 1.20871 0.0980395
\(153\) 0.791288 0.0639718
\(154\) −10.5826 −0.852768
\(155\) 0 0
\(156\) 2.16515 0.173351
\(157\) −12.7477 −1.01738 −0.508690 0.860950i \(-0.669870\pi\)
−0.508690 + 0.860950i \(0.669870\pi\)
\(158\) 8.00000 0.636446
\(159\) 10.7477 0.852350
\(160\) 0 0
\(161\) 2.79129 0.219984
\(162\) −9.58258 −0.752878
\(163\) −22.3739 −1.75246 −0.876228 0.481897i \(-0.839948\pi\)
−0.876228 + 0.481897i \(0.839948\pi\)
\(164\) −2.20871 −0.172471
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) 18.3303 1.41844 0.709221 0.704987i \(-0.249047\pi\)
0.709221 + 0.704987i \(0.249047\pi\)
\(168\) 5.00000 0.385758
\(169\) −11.5390 −0.887617
\(170\) 0 0
\(171\) 0.252273 0.0192918
\(172\) 7.16515 0.546338
\(173\) 14.2087 1.08027 0.540134 0.841579i \(-0.318373\pi\)
0.540134 + 0.841579i \(0.318373\pi\)
\(174\) 2.83485 0.214909
\(175\) 0 0
\(176\) 3.79129 0.285779
\(177\) 7.91288 0.594768
\(178\) −3.16515 −0.237238
\(179\) −16.7477 −1.25178 −0.625892 0.779910i \(-0.715265\pi\)
−0.625892 + 0.779910i \(0.715265\pi\)
\(180\) 0 0
\(181\) 13.5390 1.00635 0.503174 0.864185i \(-0.332166\pi\)
0.503174 + 0.864185i \(0.332166\pi\)
\(182\) 3.37386 0.250087
\(183\) 6.04356 0.446753
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) −18.5826 −1.36254
\(187\) 14.3739 1.05112
\(188\) 13.5826 0.990611
\(189\) −13.9564 −1.01518
\(190\) 0 0
\(191\) 16.4174 1.18792 0.593962 0.804493i \(-0.297563\pi\)
0.593962 + 0.804493i \(0.297563\pi\)
\(192\) −1.79129 −0.129275
\(193\) −6.74773 −0.485712 −0.242856 0.970062i \(-0.578084\pi\)
−0.242856 + 0.970062i \(0.578084\pi\)
\(194\) −14.9564 −1.07381
\(195\) 0 0
\(196\) 0.791288 0.0565206
\(197\) −20.5390 −1.46334 −0.731672 0.681657i \(-0.761260\pi\)
−0.731672 + 0.681657i \(0.761260\pi\)
\(198\) 0.791288 0.0562344
\(199\) 20.3303 1.44118 0.720588 0.693363i \(-0.243872\pi\)
0.720588 + 0.693363i \(0.243872\pi\)
\(200\) 0 0
\(201\) −12.8348 −0.905300
\(202\) 13.5826 0.955667
\(203\) 4.41742 0.310042
\(204\) −6.79129 −0.475485
\(205\) 0 0
\(206\) −7.37386 −0.513761
\(207\) −0.208712 −0.0145065
\(208\) −1.20871 −0.0838091
\(209\) 4.58258 0.316983
\(210\) 0 0
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) −6.00000 −0.412082
\(213\) 9.62614 0.659572
\(214\) −13.5826 −0.928486
\(215\) 0 0
\(216\) 5.00000 0.340207
\(217\) −28.9564 −1.96569
\(218\) 10.3739 0.702607
\(219\) −26.4174 −1.78512
\(220\) 0 0
\(221\) −4.58258 −0.308257
\(222\) −7.16515 −0.480893
\(223\) −11.1652 −0.747674 −0.373837 0.927494i \(-0.621958\pi\)
−0.373837 + 0.927494i \(0.621958\pi\)
\(224\) −2.79129 −0.186501
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) −4.74773 −0.315118 −0.157559 0.987510i \(-0.550362\pi\)
−0.157559 + 0.987510i \(0.550362\pi\)
\(228\) −2.16515 −0.143391
\(229\) −16.3303 −1.07914 −0.539568 0.841942i \(-0.681413\pi\)
−0.539568 + 0.841942i \(0.681413\pi\)
\(230\) 0 0
\(231\) 18.9564 1.24724
\(232\) −1.58258 −0.103901
\(233\) −7.58258 −0.496751 −0.248376 0.968664i \(-0.579897\pi\)
−0.248376 + 0.968664i \(0.579897\pi\)
\(234\) −0.252273 −0.0164916
\(235\) 0 0
\(236\) −4.41742 −0.287550
\(237\) −14.3303 −0.930853
\(238\) −10.5826 −0.685966
\(239\) 3.16515 0.204737 0.102368 0.994747i \(-0.467358\pi\)
0.102368 + 0.994747i \(0.467358\pi\)
\(240\) 0 0
\(241\) −28.0000 −1.80364 −0.901819 0.432113i \(-0.857768\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) 3.37386 0.216880
\(243\) 2.16515 0.138895
\(244\) −3.37386 −0.215989
\(245\) 0 0
\(246\) 3.95644 0.252253
\(247\) −1.46099 −0.0929603
\(248\) 10.3739 0.658741
\(249\) −10.7477 −0.681110
\(250\) 0 0
\(251\) 30.7913 1.94353 0.971764 0.235953i \(-0.0758212\pi\)
0.971764 + 0.235953i \(0.0758212\pi\)
\(252\) −0.582576 −0.0366988
\(253\) −3.79129 −0.238356
\(254\) 14.7477 0.925355
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −22.7477 −1.41896 −0.709482 0.704723i \(-0.751071\pi\)
−0.709482 + 0.704723i \(0.751071\pi\)
\(258\) −12.8348 −0.799063
\(259\) −11.1652 −0.693769
\(260\) 0 0
\(261\) −0.330303 −0.0204452
\(262\) 9.16515 0.566225
\(263\) −15.7913 −0.973733 −0.486866 0.873477i \(-0.661860\pi\)
−0.486866 + 0.873477i \(0.661860\pi\)
\(264\) −6.79129 −0.417975
\(265\) 0 0
\(266\) −3.37386 −0.206865
\(267\) 5.66970 0.346980
\(268\) 7.16515 0.437681
\(269\) 16.7477 1.02113 0.510563 0.859840i \(-0.329437\pi\)
0.510563 + 0.859840i \(0.329437\pi\)
\(270\) 0 0
\(271\) −23.1216 −1.40454 −0.702268 0.711912i \(-0.747829\pi\)
−0.702268 + 0.711912i \(0.747829\pi\)
\(272\) 3.79129 0.229881
\(273\) −6.04356 −0.365773
\(274\) 0.791288 0.0478034
\(275\) 0 0
\(276\) 1.79129 0.107823
\(277\) 1.16515 0.0700072 0.0350036 0.999387i \(-0.488856\pi\)
0.0350036 + 0.999387i \(0.488856\pi\)
\(278\) −14.7477 −0.884510
\(279\) 2.16515 0.129624
\(280\) 0 0
\(281\) −16.7477 −0.999086 −0.499543 0.866289i \(-0.666499\pi\)
−0.499543 + 0.866289i \(0.666499\pi\)
\(282\) −24.3303 −1.44885
\(283\) 28.3303 1.68406 0.842031 0.539429i \(-0.181360\pi\)
0.842031 + 0.539429i \(0.181360\pi\)
\(284\) −5.37386 −0.318880
\(285\) 0 0
\(286\) −4.58258 −0.270973
\(287\) 6.16515 0.363917
\(288\) 0.208712 0.0122985
\(289\) −2.62614 −0.154479
\(290\) 0 0
\(291\) 26.7913 1.57053
\(292\) 14.7477 0.863045
\(293\) 27.4955 1.60630 0.803151 0.595776i \(-0.203155\pi\)
0.803151 + 0.595776i \(0.203155\pi\)
\(294\) −1.41742 −0.0826659
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) 18.9564 1.09996
\(298\) 12.7913 0.740979
\(299\) 1.20871 0.0699016
\(300\) 0 0
\(301\) −20.0000 −1.15278
\(302\) −6.20871 −0.357271
\(303\) −24.3303 −1.39774
\(304\) 1.20871 0.0693244
\(305\) 0 0
\(306\) 0.791288 0.0452349
\(307\) −16.5390 −0.943931 −0.471966 0.881617i \(-0.656455\pi\)
−0.471966 + 0.881617i \(0.656455\pi\)
\(308\) −10.5826 −0.602998
\(309\) 13.2087 0.751417
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 2.16515 0.122578
\(313\) 18.3739 1.03855 0.519276 0.854607i \(-0.326202\pi\)
0.519276 + 0.854607i \(0.326202\pi\)
\(314\) −12.7477 −0.719396
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) −5.20871 −0.292550 −0.146275 0.989244i \(-0.546729\pi\)
−0.146275 + 0.989244i \(0.546729\pi\)
\(318\) 10.7477 0.602703
\(319\) −6.00000 −0.335936
\(320\) 0 0
\(321\) 24.3303 1.35799
\(322\) 2.79129 0.155552
\(323\) 4.58258 0.254981
\(324\) −9.58258 −0.532365
\(325\) 0 0
\(326\) −22.3739 −1.23917
\(327\) −18.5826 −1.02762
\(328\) −2.20871 −0.121956
\(329\) −37.9129 −2.09020
\(330\) 0 0
\(331\) 6.74773 0.370889 0.185444 0.982655i \(-0.440628\pi\)
0.185444 + 0.982655i \(0.440628\pi\)
\(332\) 6.00000 0.329293
\(333\) 0.834849 0.0457494
\(334\) 18.3303 1.00299
\(335\) 0 0
\(336\) 5.00000 0.272772
\(337\) 16.7913 0.914680 0.457340 0.889292i \(-0.348802\pi\)
0.457340 + 0.889292i \(0.348802\pi\)
\(338\) −11.5390 −0.627640
\(339\) 10.7477 0.583736
\(340\) 0 0
\(341\) 39.3303 2.12986
\(342\) 0.252273 0.0136414
\(343\) 17.3303 0.935748
\(344\) 7.16515 0.386319
\(345\) 0 0
\(346\) 14.2087 0.763865
\(347\) −9.79129 −0.525624 −0.262812 0.964847i \(-0.584650\pi\)
−0.262812 + 0.964847i \(0.584650\pi\)
\(348\) 2.83485 0.151964
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) −6.04356 −0.322581
\(352\) 3.79129 0.202076
\(353\) 15.1652 0.807160 0.403580 0.914944i \(-0.367766\pi\)
0.403580 + 0.914944i \(0.367766\pi\)
\(354\) 7.91288 0.420565
\(355\) 0 0
\(356\) −3.16515 −0.167753
\(357\) 18.9564 1.00328
\(358\) −16.7477 −0.885145
\(359\) −9.16515 −0.483718 −0.241859 0.970311i \(-0.577757\pi\)
−0.241859 + 0.970311i \(0.577757\pi\)
\(360\) 0 0
\(361\) −17.5390 −0.923106
\(362\) 13.5390 0.711595
\(363\) −6.04356 −0.317205
\(364\) 3.37386 0.176838
\(365\) 0 0
\(366\) 6.04356 0.315902
\(367\) 0.834849 0.0435787 0.0217894 0.999763i \(-0.493064\pi\)
0.0217894 + 0.999763i \(0.493064\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −0.460985 −0.0239979
\(370\) 0 0
\(371\) 16.7477 0.869499
\(372\) −18.5826 −0.963462
\(373\) 14.7477 0.763608 0.381804 0.924243i \(-0.375303\pi\)
0.381804 + 0.924243i \(0.375303\pi\)
\(374\) 14.3739 0.743255
\(375\) 0 0
\(376\) 13.5826 0.700468
\(377\) 1.91288 0.0985183
\(378\) −13.9564 −0.717842
\(379\) 7.37386 0.378770 0.189385 0.981903i \(-0.439351\pi\)
0.189385 + 0.981903i \(0.439351\pi\)
\(380\) 0 0
\(381\) −26.4174 −1.35341
\(382\) 16.4174 0.839989
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) −1.79129 −0.0914113
\(385\) 0 0
\(386\) −6.74773 −0.343450
\(387\) 1.49545 0.0760182
\(388\) −14.9564 −0.759298
\(389\) 29.7042 1.50606 0.753031 0.657986i \(-0.228591\pi\)
0.753031 + 0.657986i \(0.228591\pi\)
\(390\) 0 0
\(391\) −3.79129 −0.191734
\(392\) 0.791288 0.0399661
\(393\) −16.4174 −0.828150
\(394\) −20.5390 −1.03474
\(395\) 0 0
\(396\) 0.791288 0.0397637
\(397\) −16.5390 −0.830069 −0.415035 0.909806i \(-0.636231\pi\)
−0.415035 + 0.909806i \(0.636231\pi\)
\(398\) 20.3303 1.01907
\(399\) 6.04356 0.302556
\(400\) 0 0
\(401\) 22.7477 1.13597 0.567984 0.823040i \(-0.307724\pi\)
0.567984 + 0.823040i \(0.307724\pi\)
\(402\) −12.8348 −0.640144
\(403\) −12.5390 −0.624613
\(404\) 13.5826 0.675758
\(405\) 0 0
\(406\) 4.41742 0.219233
\(407\) 15.1652 0.751709
\(408\) −6.79129 −0.336219
\(409\) −22.7913 −1.12696 −0.563478 0.826131i \(-0.690537\pi\)
−0.563478 + 0.826131i \(0.690537\pi\)
\(410\) 0 0
\(411\) −1.41742 −0.0699164
\(412\) −7.37386 −0.363284
\(413\) 12.3303 0.606735
\(414\) −0.208712 −0.0102576
\(415\) 0 0
\(416\) −1.20871 −0.0592620
\(417\) 26.4174 1.29367
\(418\) 4.58258 0.224141
\(419\) 39.1652 1.91334 0.956671 0.291170i \(-0.0940444\pi\)
0.956671 + 0.291170i \(0.0940444\pi\)
\(420\) 0 0
\(421\) −23.1216 −1.12688 −0.563439 0.826158i \(-0.690522\pi\)
−0.563439 + 0.826158i \(0.690522\pi\)
\(422\) −10.0000 −0.486792
\(423\) 2.83485 0.137835
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 9.62614 0.466388
\(427\) 9.41742 0.455741
\(428\) −13.5826 −0.656539
\(429\) 8.20871 0.396320
\(430\) 0 0
\(431\) −19.9129 −0.959170 −0.479585 0.877496i \(-0.659213\pi\)
−0.479585 + 0.877496i \(0.659213\pi\)
\(432\) 5.00000 0.240563
\(433\) −1.53901 −0.0739603 −0.0369802 0.999316i \(-0.511774\pi\)
−0.0369802 + 0.999316i \(0.511774\pi\)
\(434\) −28.9564 −1.38995
\(435\) 0 0
\(436\) 10.3739 0.496818
\(437\) −1.20871 −0.0578205
\(438\) −26.4174 −1.26227
\(439\) 25.5390 1.21891 0.609455 0.792820i \(-0.291388\pi\)
0.609455 + 0.792820i \(0.291388\pi\)
\(440\) 0 0
\(441\) 0.165151 0.00786435
\(442\) −4.58258 −0.217971
\(443\) 35.2087 1.67282 0.836408 0.548107i \(-0.184651\pi\)
0.836408 + 0.548107i \(0.184651\pi\)
\(444\) −7.16515 −0.340043
\(445\) 0 0
\(446\) −11.1652 −0.528685
\(447\) −22.9129 −1.08374
\(448\) −2.79129 −0.131876
\(449\) 25.1216 1.18556 0.592781 0.805364i \(-0.298030\pi\)
0.592781 + 0.805364i \(0.298030\pi\)
\(450\) 0 0
\(451\) −8.37386 −0.394310
\(452\) −6.00000 −0.282216
\(453\) 11.1216 0.522538
\(454\) −4.74773 −0.222822
\(455\) 0 0
\(456\) −2.16515 −0.101393
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −16.3303 −0.763065
\(459\) 18.9564 0.884811
\(460\) 0 0
\(461\) −1.25227 −0.0583242 −0.0291621 0.999575i \(-0.509284\pi\)
−0.0291621 + 0.999575i \(0.509284\pi\)
\(462\) 18.9564 0.881933
\(463\) 10.0000 0.464739 0.232370 0.972628i \(-0.425352\pi\)
0.232370 + 0.972628i \(0.425352\pi\)
\(464\) −1.58258 −0.0734692
\(465\) 0 0
\(466\) −7.58258 −0.351256
\(467\) −25.9129 −1.19911 −0.599553 0.800335i \(-0.704655\pi\)
−0.599553 + 0.800335i \(0.704655\pi\)
\(468\) −0.252273 −0.0116613
\(469\) −20.0000 −0.923514
\(470\) 0 0
\(471\) 22.8348 1.05217
\(472\) −4.41742 −0.203328
\(473\) 27.1652 1.24905
\(474\) −14.3303 −0.658213
\(475\) 0 0
\(476\) −10.5826 −0.485052
\(477\) −1.25227 −0.0573376
\(478\) 3.16515 0.144771
\(479\) −39.4955 −1.80459 −0.902297 0.431116i \(-0.858120\pi\)
−0.902297 + 0.431116i \(0.858120\pi\)
\(480\) 0 0
\(481\) −4.83485 −0.220450
\(482\) −28.0000 −1.27537
\(483\) −5.00000 −0.227508
\(484\) 3.37386 0.153357
\(485\) 0 0
\(486\) 2.16515 0.0982133
\(487\) −15.5826 −0.706114 −0.353057 0.935602i \(-0.614858\pi\)
−0.353057 + 0.935602i \(0.614858\pi\)
\(488\) −3.37386 −0.152728
\(489\) 40.0780 1.81239
\(490\) 0 0
\(491\) −16.7477 −0.755814 −0.377907 0.925843i \(-0.623356\pi\)
−0.377907 + 0.925843i \(0.623356\pi\)
\(492\) 3.95644 0.178370
\(493\) −6.00000 −0.270226
\(494\) −1.46099 −0.0657328
\(495\) 0 0
\(496\) 10.3739 0.465800
\(497\) 15.0000 0.672842
\(498\) −10.7477 −0.481617
\(499\) 23.1652 1.03701 0.518507 0.855073i \(-0.326488\pi\)
0.518507 + 0.855073i \(0.326488\pi\)
\(500\) 0 0
\(501\) −32.8348 −1.46695
\(502\) 30.7913 1.37428
\(503\) −18.7913 −0.837862 −0.418931 0.908018i \(-0.637595\pi\)
−0.418931 + 0.908018i \(0.637595\pi\)
\(504\) −0.582576 −0.0259500
\(505\) 0 0
\(506\) −3.79129 −0.168543
\(507\) 20.6697 0.917973
\(508\) 14.7477 0.654325
\(509\) 7.25227 0.321451 0.160726 0.986999i \(-0.448617\pi\)
0.160726 + 0.986999i \(0.448617\pi\)
\(510\) 0 0
\(511\) −41.1652 −1.82104
\(512\) 1.00000 0.0441942
\(513\) 6.04356 0.266830
\(514\) −22.7477 −1.00336
\(515\) 0 0
\(516\) −12.8348 −0.565023
\(517\) 51.4955 2.26477
\(518\) −11.1652 −0.490569
\(519\) −25.4519 −1.11721
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) −0.330303 −0.0144570
\(523\) 1.16515 0.0509485 0.0254743 0.999675i \(-0.491890\pi\)
0.0254743 + 0.999675i \(0.491890\pi\)
\(524\) 9.16515 0.400381
\(525\) 0 0
\(526\) −15.7913 −0.688533
\(527\) 39.3303 1.71325
\(528\) −6.79129 −0.295553
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −0.921970 −0.0400101
\(532\) −3.37386 −0.146276
\(533\) 2.66970 0.115637
\(534\) 5.66970 0.245352
\(535\) 0 0
\(536\) 7.16515 0.309487
\(537\) 30.0000 1.29460
\(538\) 16.7477 0.722046
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) 38.3303 1.64795 0.823974 0.566627i \(-0.191752\pi\)
0.823974 + 0.566627i \(0.191752\pi\)
\(542\) −23.1216 −0.993157
\(543\) −24.2523 −1.04076
\(544\) 3.79129 0.162550
\(545\) 0 0
\(546\) −6.04356 −0.258641
\(547\) −15.1216 −0.646553 −0.323276 0.946305i \(-0.604784\pi\)
−0.323276 + 0.946305i \(0.604784\pi\)
\(548\) 0.791288 0.0338021
\(549\) −0.704166 −0.0300531
\(550\) 0 0
\(551\) −1.91288 −0.0814914
\(552\) 1.79129 0.0762423
\(553\) −22.3303 −0.949581
\(554\) 1.16515 0.0495025
\(555\) 0 0
\(556\) −14.7477 −0.625443
\(557\) −30.3303 −1.28514 −0.642568 0.766229i \(-0.722131\pi\)
−0.642568 + 0.766229i \(0.722131\pi\)
\(558\) 2.16515 0.0916582
\(559\) −8.66061 −0.366305
\(560\) 0 0
\(561\) −25.7477 −1.08707
\(562\) −16.7477 −0.706460
\(563\) −3.16515 −0.133395 −0.0666976 0.997773i \(-0.521246\pi\)
−0.0666976 + 0.997773i \(0.521246\pi\)
\(564\) −24.3303 −1.02449
\(565\) 0 0
\(566\) 28.3303 1.19081
\(567\) 26.7477 1.12330
\(568\) −5.37386 −0.225482
\(569\) 15.4955 0.649603 0.324802 0.945782i \(-0.394702\pi\)
0.324802 + 0.945782i \(0.394702\pi\)
\(570\) 0 0
\(571\) 30.1216 1.26055 0.630275 0.776372i \(-0.282942\pi\)
0.630275 + 0.776372i \(0.282942\pi\)
\(572\) −4.58258 −0.191607
\(573\) −29.4083 −1.22855
\(574\) 6.16515 0.257328
\(575\) 0 0
\(576\) 0.208712 0.00869634
\(577\) −22.8348 −0.950627 −0.475314 0.879816i \(-0.657665\pi\)
−0.475314 + 0.879816i \(0.657665\pi\)
\(578\) −2.62614 −0.109233
\(579\) 12.0871 0.502324
\(580\) 0 0
\(581\) −16.7477 −0.694813
\(582\) 26.7913 1.11053
\(583\) −22.7477 −0.942115
\(584\) 14.7477 0.610265
\(585\) 0 0
\(586\) 27.4955 1.13583
\(587\) 26.2087 1.08175 0.540875 0.841103i \(-0.318093\pi\)
0.540875 + 0.841103i \(0.318093\pi\)
\(588\) −1.41742 −0.0584536
\(589\) 12.5390 0.516661
\(590\) 0 0
\(591\) 36.7913 1.51339
\(592\) 4.00000 0.164399
\(593\) −13.9129 −0.571333 −0.285667 0.958329i \(-0.592215\pi\)
−0.285667 + 0.958329i \(0.592215\pi\)
\(594\) 18.9564 0.777792
\(595\) 0 0
\(596\) 12.7913 0.523952
\(597\) −36.4174 −1.49047
\(598\) 1.20871 0.0494279
\(599\) −40.1216 −1.63932 −0.819662 0.572848i \(-0.805839\pi\)
−0.819662 + 0.572848i \(0.805839\pi\)
\(600\) 0 0
\(601\) −22.7913 −0.929676 −0.464838 0.885396i \(-0.653887\pi\)
−0.464838 + 0.885396i \(0.653887\pi\)
\(602\) −20.0000 −0.815139
\(603\) 1.49545 0.0608996
\(604\) −6.20871 −0.252629
\(605\) 0 0
\(606\) −24.3303 −0.988351
\(607\) 28.0000 1.13648 0.568242 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) 1.20871 0.0490198
\(609\) −7.91288 −0.320646
\(610\) 0 0
\(611\) −16.4174 −0.664178
\(612\) 0.791288 0.0319859
\(613\) −14.0000 −0.565455 −0.282727 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) −16.5390 −0.667460
\(615\) 0 0
\(616\) −10.5826 −0.426384
\(617\) −44.8693 −1.80637 −0.903185 0.429251i \(-0.858778\pi\)
−0.903185 + 0.429251i \(0.858778\pi\)
\(618\) 13.2087 0.531332
\(619\) 2.79129 0.112191 0.0560957 0.998425i \(-0.482135\pi\)
0.0560957 + 0.998425i \(0.482135\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) 12.0000 0.481156
\(623\) 8.83485 0.353961
\(624\) 2.16515 0.0866754
\(625\) 0 0
\(626\) 18.3739 0.734367
\(627\) −8.20871 −0.327824
\(628\) −12.7477 −0.508690
\(629\) 15.1652 0.604674
\(630\) 0 0
\(631\) −17.9129 −0.713100 −0.356550 0.934276i \(-0.616047\pi\)
−0.356550 + 0.934276i \(0.616047\pi\)
\(632\) 8.00000 0.318223
\(633\) 17.9129 0.711973
\(634\) −5.20871 −0.206864
\(635\) 0 0
\(636\) 10.7477 0.426175
\(637\) −0.956439 −0.0378955
\(638\) −6.00000 −0.237542
\(639\) −1.12159 −0.0443694
\(640\) 0 0
\(641\) −3.16515 −0.125016 −0.0625080 0.998044i \(-0.519910\pi\)
−0.0625080 + 0.998044i \(0.519910\pi\)
\(642\) 24.3303 0.960240
\(643\) 20.7477 0.818210 0.409105 0.912487i \(-0.365841\pi\)
0.409105 + 0.912487i \(0.365841\pi\)
\(644\) 2.79129 0.109992
\(645\) 0 0
\(646\) 4.58258 0.180299
\(647\) −2.83485 −0.111449 −0.0557247 0.998446i \(-0.517747\pi\)
−0.0557247 + 0.998446i \(0.517747\pi\)
\(648\) −9.58258 −0.376439
\(649\) −16.7477 −0.657406
\(650\) 0 0
\(651\) 51.8693 2.03292
\(652\) −22.3739 −0.876228
\(653\) −35.5390 −1.39075 −0.695375 0.718647i \(-0.744762\pi\)
−0.695375 + 0.718647i \(0.744762\pi\)
\(654\) −18.5826 −0.726636
\(655\) 0 0
\(656\) −2.20871 −0.0862357
\(657\) 3.07803 0.120085
\(658\) −37.9129 −1.47800
\(659\) −27.1652 −1.05820 −0.529102 0.848558i \(-0.677471\pi\)
−0.529102 + 0.848558i \(0.677471\pi\)
\(660\) 0 0
\(661\) −39.3739 −1.53147 −0.765733 0.643159i \(-0.777624\pi\)
−0.765733 + 0.643159i \(0.777624\pi\)
\(662\) 6.74773 0.262258
\(663\) 8.20871 0.318800
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) 0.834849 0.0323497
\(667\) 1.58258 0.0612776
\(668\) 18.3303 0.709221
\(669\) 20.0000 0.773245
\(670\) 0 0
\(671\) −12.7913 −0.493802
\(672\) 5.00000 0.192879
\(673\) −38.0000 −1.46479 −0.732396 0.680879i \(-0.761598\pi\)
−0.732396 + 0.680879i \(0.761598\pi\)
\(674\) 16.7913 0.646776
\(675\) 0 0
\(676\) −11.5390 −0.443808
\(677\) 30.6606 1.17838 0.589191 0.807993i \(-0.299446\pi\)
0.589191 + 0.807993i \(0.299446\pi\)
\(678\) 10.7477 0.412764
\(679\) 41.7477 1.60213
\(680\) 0 0
\(681\) 8.50455 0.325895
\(682\) 39.3303 1.50604
\(683\) −2.37386 −0.0908334 −0.0454167 0.998968i \(-0.514462\pi\)
−0.0454167 + 0.998968i \(0.514462\pi\)
\(684\) 0.252273 0.00964590
\(685\) 0 0
\(686\) 17.3303 0.661674
\(687\) 29.2523 1.11604
\(688\) 7.16515 0.273169
\(689\) 7.25227 0.276290
\(690\) 0 0
\(691\) 15.2523 0.580224 0.290112 0.956993i \(-0.406307\pi\)
0.290112 + 0.956993i \(0.406307\pi\)
\(692\) 14.2087 0.540134
\(693\) −2.20871 −0.0839020
\(694\) −9.79129 −0.371672
\(695\) 0 0
\(696\) 2.83485 0.107455
\(697\) −8.37386 −0.317183
\(698\) 26.0000 0.984115
\(699\) 13.5826 0.513740
\(700\) 0 0
\(701\) 9.62614 0.363574 0.181787 0.983338i \(-0.441812\pi\)
0.181787 + 0.983338i \(0.441812\pi\)
\(702\) −6.04356 −0.228100
\(703\) 4.83485 0.182350
\(704\) 3.79129 0.142890
\(705\) 0 0
\(706\) 15.1652 0.570748
\(707\) −37.9129 −1.42586
\(708\) 7.91288 0.297384
\(709\) 34.5390 1.29714 0.648570 0.761155i \(-0.275367\pi\)
0.648570 + 0.761155i \(0.275367\pi\)
\(710\) 0 0
\(711\) 1.66970 0.0626185
\(712\) −3.16515 −0.118619
\(713\) −10.3739 −0.388504
\(714\) 18.9564 0.709427
\(715\) 0 0
\(716\) −16.7477 −0.625892
\(717\) −5.66970 −0.211739
\(718\) −9.16515 −0.342040
\(719\) −29.5390 −1.10162 −0.550810 0.834631i \(-0.685681\pi\)
−0.550810 + 0.834631i \(0.685681\pi\)
\(720\) 0 0
\(721\) 20.5826 0.766535
\(722\) −17.5390 −0.652735
\(723\) 50.1561 1.86532
\(724\) 13.5390 0.503174
\(725\) 0 0
\(726\) −6.04356 −0.224298
\(727\) 2.12159 0.0786854 0.0393427 0.999226i \(-0.487474\pi\)
0.0393427 + 0.999226i \(0.487474\pi\)
\(728\) 3.37386 0.125044
\(729\) 24.8693 0.921086
\(730\) 0 0
\(731\) 27.1652 1.00474
\(732\) 6.04356 0.223376
\(733\) −26.0000 −0.960332 −0.480166 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) 0.834849 0.0308148
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 27.1652 1.00064
\(738\) −0.460985 −0.0169691
\(739\) 8.00000 0.294285 0.147142 0.989115i \(-0.452992\pi\)
0.147142 + 0.989115i \(0.452992\pi\)
\(740\) 0 0
\(741\) 2.61704 0.0961395
\(742\) 16.7477 0.614828
\(743\) −9.95644 −0.365266 −0.182633 0.983181i \(-0.558462\pi\)
−0.182633 + 0.983181i \(0.558462\pi\)
\(744\) −18.5826 −0.681270
\(745\) 0 0
\(746\) 14.7477 0.539953
\(747\) 1.25227 0.0458183
\(748\) 14.3739 0.525561
\(749\) 37.9129 1.38531
\(750\) 0 0
\(751\) 18.7477 0.684114 0.342057 0.939679i \(-0.388876\pi\)
0.342057 + 0.939679i \(0.388876\pi\)
\(752\) 13.5826 0.495306
\(753\) −55.1561 −2.01000
\(754\) 1.91288 0.0696629
\(755\) 0 0
\(756\) −13.9564 −0.507591
\(757\) −26.3303 −0.956991 −0.478496 0.878090i \(-0.658818\pi\)
−0.478496 + 0.878090i \(0.658818\pi\)
\(758\) 7.37386 0.267831
\(759\) 6.79129 0.246508
\(760\) 0 0
\(761\) −33.9564 −1.23092 −0.615460 0.788168i \(-0.711030\pi\)
−0.615460 + 0.788168i \(0.711030\pi\)
\(762\) −26.4174 −0.957002
\(763\) −28.9564 −1.04829
\(764\) 16.4174 0.593962
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 5.33939 0.192794
\(768\) −1.79129 −0.0646375
\(769\) −3.66970 −0.132333 −0.0661663 0.997809i \(-0.521077\pi\)
−0.0661663 + 0.997809i \(0.521077\pi\)
\(770\) 0 0
\(771\) 40.7477 1.46749
\(772\) −6.74773 −0.242856
\(773\) 21.4955 0.773138 0.386569 0.922261i \(-0.373660\pi\)
0.386569 + 0.922261i \(0.373660\pi\)
\(774\) 1.49545 0.0537530
\(775\) 0 0
\(776\) −14.9564 −0.536905
\(777\) 20.0000 0.717496
\(778\) 29.7042 1.06495
\(779\) −2.66970 −0.0956518
\(780\) 0 0
\(781\) −20.3739 −0.729034
\(782\) −3.79129 −0.135576
\(783\) −7.91288 −0.282783
\(784\) 0.791288 0.0282603
\(785\) 0 0
\(786\) −16.4174 −0.585590
\(787\) 8.41742 0.300049 0.150024 0.988682i \(-0.452065\pi\)
0.150024 + 0.988682i \(0.452065\pi\)
\(788\) −20.5390 −0.731672
\(789\) 28.2867 1.00703
\(790\) 0 0
\(791\) 16.7477 0.595481
\(792\) 0.791288 0.0281172
\(793\) 4.07803 0.144815
\(794\) −16.5390 −0.586948
\(795\) 0 0
\(796\) 20.3303 0.720588
\(797\) 49.9129 1.76800 0.884002 0.467482i \(-0.154839\pi\)
0.884002 + 0.467482i \(0.154839\pi\)
\(798\) 6.04356 0.213940
\(799\) 51.4955 1.82178
\(800\) 0 0
\(801\) −0.660606 −0.0233413
\(802\) 22.7477 0.803250
\(803\) 55.9129 1.97312
\(804\) −12.8348 −0.452650
\(805\) 0 0
\(806\) −12.5390 −0.441668
\(807\) −30.0000 −1.05605
\(808\) 13.5826 0.477833
\(809\) 11.0436 0.388271 0.194135 0.980975i \(-0.437810\pi\)
0.194135 + 0.980975i \(0.437810\pi\)
\(810\) 0 0
\(811\) −47.9129 −1.68245 −0.841224 0.540686i \(-0.818165\pi\)
−0.841224 + 0.540686i \(0.818165\pi\)
\(812\) 4.41742 0.155021
\(813\) 41.4174 1.45257
\(814\) 15.1652 0.531538
\(815\) 0 0
\(816\) −6.79129 −0.237743
\(817\) 8.66061 0.302996
\(818\) −22.7913 −0.796879
\(819\) 0.704166 0.0246056
\(820\) 0 0
\(821\) 2.83485 0.0989369 0.0494684 0.998776i \(-0.484247\pi\)
0.0494684 + 0.998776i \(0.484247\pi\)
\(822\) −1.41742 −0.0494383
\(823\) −41.1652 −1.43493 −0.717463 0.696596i \(-0.754697\pi\)
−0.717463 + 0.696596i \(0.754697\pi\)
\(824\) −7.37386 −0.256881
\(825\) 0 0
\(826\) 12.3303 0.429026
\(827\) −41.0780 −1.42842 −0.714212 0.699930i \(-0.753215\pi\)
−0.714212 + 0.699930i \(0.753215\pi\)
\(828\) −0.208712 −0.00725325
\(829\) −31.4955 −1.09388 −0.546941 0.837171i \(-0.684208\pi\)
−0.546941 + 0.837171i \(0.684208\pi\)
\(830\) 0 0
\(831\) −2.08712 −0.0724014
\(832\) −1.20871 −0.0419046
\(833\) 3.00000 0.103944
\(834\) 26.4174 0.914761
\(835\) 0 0
\(836\) 4.58258 0.158492
\(837\) 51.8693 1.79287
\(838\) 39.1652 1.35294
\(839\) −22.4174 −0.773935 −0.386968 0.922093i \(-0.626478\pi\)
−0.386968 + 0.922093i \(0.626478\pi\)
\(840\) 0 0
\(841\) −26.4955 −0.913636
\(842\) −23.1216 −0.796823
\(843\) 30.0000 1.03325
\(844\) −10.0000 −0.344214
\(845\) 0 0
\(846\) 2.83485 0.0974641
\(847\) −9.41742 −0.323587
\(848\) −6.00000 −0.206041
\(849\) −50.7477 −1.74166
\(850\) 0 0
\(851\) −4.00000 −0.137118
\(852\) 9.62614 0.329786
\(853\) −8.46099 −0.289699 −0.144849 0.989454i \(-0.546270\pi\)
−0.144849 + 0.989454i \(0.546270\pi\)
\(854\) 9.41742 0.322258
\(855\) 0 0
\(856\) −13.5826 −0.464243
\(857\) −9.16515 −0.313076 −0.156538 0.987672i \(-0.550033\pi\)
−0.156538 + 0.987672i \(0.550033\pi\)
\(858\) 8.20871 0.280241
\(859\) 0.747727 0.0255121 0.0127561 0.999919i \(-0.495940\pi\)
0.0127561 + 0.999919i \(0.495940\pi\)
\(860\) 0 0
\(861\) −11.0436 −0.376364
\(862\) −19.9129 −0.678235
\(863\) 31.5826 1.07508 0.537542 0.843237i \(-0.319353\pi\)
0.537542 + 0.843237i \(0.319353\pi\)
\(864\) 5.00000 0.170103
\(865\) 0 0
\(866\) −1.53901 −0.0522979
\(867\) 4.70417 0.159762
\(868\) −28.9564 −0.982846
\(869\) 30.3303 1.02889
\(870\) 0 0
\(871\) −8.66061 −0.293453
\(872\) 10.3739 0.351303
\(873\) −3.12159 −0.105650
\(874\) −1.20871 −0.0408853
\(875\) 0 0
\(876\) −26.4174 −0.892562
\(877\) −7.70417 −0.260151 −0.130076 0.991504i \(-0.541522\pi\)
−0.130076 + 0.991504i \(0.541522\pi\)
\(878\) 25.5390 0.861900
\(879\) −49.2523 −1.66124
\(880\) 0 0
\(881\) −6.33030 −0.213273 −0.106637 0.994298i \(-0.534008\pi\)
−0.106637 + 0.994298i \(0.534008\pi\)
\(882\) 0.165151 0.00556094
\(883\) 12.0436 0.405298 0.202649 0.979251i \(-0.435045\pi\)
0.202649 + 0.979251i \(0.435045\pi\)
\(884\) −4.58258 −0.154129
\(885\) 0 0
\(886\) 35.2087 1.18286
\(887\) 3.16515 0.106275 0.0531377 0.998587i \(-0.483078\pi\)
0.0531377 + 0.998587i \(0.483078\pi\)
\(888\) −7.16515 −0.240447
\(889\) −41.1652 −1.38063
\(890\) 0 0
\(891\) −36.3303 −1.21711
\(892\) −11.1652 −0.373837
\(893\) 16.4174 0.549388
\(894\) −22.9129 −0.766321
\(895\) 0 0
\(896\) −2.79129 −0.0932504
\(897\) −2.16515 −0.0722923
\(898\) 25.1216 0.838318
\(899\) −16.4174 −0.547552
\(900\) 0 0
\(901\) −22.7477 −0.757837
\(902\) −8.37386 −0.278819
\(903\) 35.8258 1.19221
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 11.1216 0.369490
\(907\) 20.7477 0.688917 0.344458 0.938802i \(-0.388063\pi\)
0.344458 + 0.938802i \(0.388063\pi\)
\(908\) −4.74773 −0.157559
\(909\) 2.83485 0.0940260
\(910\) 0 0
\(911\) 13.5826 0.450011 0.225005 0.974358i \(-0.427760\pi\)
0.225005 + 0.974358i \(0.427760\pi\)
\(912\) −2.16515 −0.0716953
\(913\) 22.7477 0.752840
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) −16.3303 −0.539568
\(917\) −25.5826 −0.844811
\(918\) 18.9564 0.625656
\(919\) −36.8348 −1.21507 −0.607535 0.794293i \(-0.707841\pi\)
−0.607535 + 0.794293i \(0.707841\pi\)
\(920\) 0 0
\(921\) 29.6261 0.976214
\(922\) −1.25227 −0.0412414
\(923\) 6.49545 0.213800
\(924\) 18.9564 0.623621
\(925\) 0 0
\(926\) 10.0000 0.328620
\(927\) −1.53901 −0.0505479
\(928\) −1.58258 −0.0519506
\(929\) −39.4955 −1.29580 −0.647902 0.761724i \(-0.724353\pi\)
−0.647902 + 0.761724i \(0.724353\pi\)
\(930\) 0 0
\(931\) 0.956439 0.0313460
\(932\) −7.58258 −0.248376
\(933\) −21.4955 −0.703730
\(934\) −25.9129 −0.847895
\(935\) 0 0
\(936\) −0.252273 −0.00824580
\(937\) −58.3739 −1.90699 −0.953495 0.301407i \(-0.902544\pi\)
−0.953495 + 0.301407i \(0.902544\pi\)
\(938\) −20.0000 −0.653023
\(939\) −32.9129 −1.07407
\(940\) 0 0
\(941\) 54.9564 1.79153 0.895764 0.444529i \(-0.146629\pi\)
0.895764 + 0.444529i \(0.146629\pi\)
\(942\) 22.8348 0.744000
\(943\) 2.20871 0.0719256
\(944\) −4.41742 −0.143775
\(945\) 0 0
\(946\) 27.1652 0.883215
\(947\) 29.5390 0.959889 0.479945 0.877299i \(-0.340657\pi\)
0.479945 + 0.877299i \(0.340657\pi\)
\(948\) −14.3303 −0.465427
\(949\) −17.8258 −0.578649
\(950\) 0 0
\(951\) 9.33030 0.302556
\(952\) −10.5826 −0.342983
\(953\) 26.5390 0.859683 0.429842 0.902904i \(-0.358569\pi\)
0.429842 + 0.902904i \(0.358569\pi\)
\(954\) −1.25227 −0.0405438
\(955\) 0 0
\(956\) 3.16515 0.102368
\(957\) 10.7477 0.347425
\(958\) −39.4955 −1.27604
\(959\) −2.20871 −0.0713230
\(960\) 0 0
\(961\) 76.6170 2.47152
\(962\) −4.83485 −0.155882
\(963\) −2.83485 −0.0913517
\(964\) −28.0000 −0.901819
\(965\) 0 0
\(966\) −5.00000 −0.160872
\(967\) 5.25227 0.168902 0.0844509 0.996428i \(-0.473086\pi\)
0.0844509 + 0.996428i \(0.473086\pi\)
\(968\) 3.37386 0.108440
\(969\) −8.20871 −0.263702
\(970\) 0 0
\(971\) −6.95644 −0.223243 −0.111621 0.993751i \(-0.535604\pi\)
−0.111621 + 0.993751i \(0.535604\pi\)
\(972\) 2.16515 0.0694473
\(973\) 41.1652 1.31969
\(974\) −15.5826 −0.499298
\(975\) 0 0
\(976\) −3.37386 −0.107995
\(977\) −7.12159 −0.227840 −0.113920 0.993490i \(-0.536341\pi\)
−0.113920 + 0.993490i \(0.536341\pi\)
\(978\) 40.0780 1.28155
\(979\) −12.0000 −0.383522
\(980\) 0 0
\(981\) 2.16515 0.0691280
\(982\) −16.7477 −0.534441
\(983\) 0.626136 0.0199707 0.00998533 0.999950i \(-0.496822\pi\)
0.00998533 + 0.999950i \(0.496822\pi\)
\(984\) 3.95644 0.126127
\(985\) 0 0
\(986\) −6.00000 −0.191079
\(987\) 67.9129 2.16169
\(988\) −1.46099 −0.0464801
\(989\) −7.16515 −0.227839
\(990\) 0 0
\(991\) −37.7913 −1.20048 −0.600240 0.799820i \(-0.704928\pi\)
−0.600240 + 0.799820i \(0.704928\pi\)
\(992\) 10.3739 0.329370
\(993\) −12.0871 −0.383573
\(994\) 15.0000 0.475771
\(995\) 0 0
\(996\) −10.7477 −0.340555
\(997\) −11.4955 −0.364065 −0.182032 0.983293i \(-0.558268\pi\)
−0.182032 + 0.983293i \(0.558268\pi\)
\(998\) 23.1652 0.733280
\(999\) 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.a.o.1.1 2
4.3 odd 2 9200.2.a.bs.1.2 2
5.2 odd 4 1150.2.b.g.599.4 4
5.3 odd 4 1150.2.b.g.599.1 4
5.4 even 2 230.2.a.a.1.2 2
15.14 odd 2 2070.2.a.x.1.2 2
20.19 odd 2 1840.2.a.n.1.1 2
40.19 odd 2 7360.2.a.bk.1.2 2
40.29 even 2 7360.2.a.bq.1.1 2
115.114 odd 2 5290.2.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.a.1.2 2 5.4 even 2
1150.2.a.o.1.1 2 1.1 even 1 trivial
1150.2.b.g.599.1 4 5.3 odd 4
1150.2.b.g.599.4 4 5.2 odd 4
1840.2.a.n.1.1 2 20.19 odd 2
2070.2.a.x.1.2 2 15.14 odd 2
5290.2.a.e.1.2 2 115.114 odd 2
7360.2.a.bk.1.2 2 40.19 odd 2
7360.2.a.bq.1.1 2 40.29 even 2
9200.2.a.bs.1.2 2 4.3 odd 2