Properties

Label 5290.2.a.e.1.2
Level $5290$
Weight $2$
Character 5290.1
Self dual yes
Analytic conductor $42.241$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5290 = 2 \cdot 5 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5290.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(42.2408626693\)
Analytic rank: \(1\)
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 1.2
Root \(-1.79129\) of defining polynomial
Character \(\chi\) \(=\) 5290.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.79129 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.79129 q^{6} -2.79129 q^{7} -1.00000 q^{8} +0.208712 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.79129 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.79129 q^{6} -2.79129 q^{7} -1.00000 q^{8} +0.208712 q^{9} -1.00000 q^{10} -3.79129 q^{11} +1.79129 q^{12} +1.20871 q^{13} +2.79129 q^{14} +1.79129 q^{15} +1.00000 q^{16} +3.79129 q^{17} -0.208712 q^{18} -1.20871 q^{19} +1.00000 q^{20} -5.00000 q^{21} +3.79129 q^{22} -1.79129 q^{24} +1.00000 q^{25} -1.20871 q^{26} -5.00000 q^{27} -2.79129 q^{28} -1.58258 q^{29} -1.79129 q^{30} +10.3739 q^{31} -1.00000 q^{32} -6.79129 q^{33} -3.79129 q^{34} -2.79129 q^{35} +0.208712 q^{36} +4.00000 q^{37} +1.20871 q^{38} +2.16515 q^{39} -1.00000 q^{40} -2.20871 q^{41} +5.00000 q^{42} +7.16515 q^{43} -3.79129 q^{44} +0.208712 q^{45} -13.5826 q^{47} +1.79129 q^{48} +0.791288 q^{49} -1.00000 q^{50} +6.79129 q^{51} +1.20871 q^{52} -6.00000 q^{53} +5.00000 q^{54} -3.79129 q^{55} +2.79129 q^{56} -2.16515 q^{57} +1.58258 q^{58} -4.41742 q^{59} +1.79129 q^{60} +3.37386 q^{61} -10.3739 q^{62} -0.582576 q^{63} +1.00000 q^{64} +1.20871 q^{65} +6.79129 q^{66} +7.16515 q^{67} +3.79129 q^{68} +2.79129 q^{70} -5.37386 q^{71} -0.208712 q^{72} -14.7477 q^{73} -4.00000 q^{74} +1.79129 q^{75} -1.20871 q^{76} +10.5826 q^{77} -2.16515 q^{78} -8.00000 q^{79} +1.00000 q^{80} -9.58258 q^{81} +2.20871 q^{82} +6.00000 q^{83} -5.00000 q^{84} +3.79129 q^{85} -7.16515 q^{86} -2.83485 q^{87} +3.79129 q^{88} +3.16515 q^{89} -0.208712 q^{90} -3.37386 q^{91} +18.5826 q^{93} +13.5826 q^{94} -1.20871 q^{95} -1.79129 q^{96} -14.9564 q^{97} -0.791288 q^{98} -0.791288 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - q^{3} + 2q^{4} + 2q^{5} + q^{6} - q^{7} - 2q^{8} + 5q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - q^{3} + 2q^{4} + 2q^{5} + q^{6} - q^{7} - 2q^{8} + 5q^{9} - 2q^{10} - 3q^{11} - q^{12} + 7q^{13} + q^{14} - q^{15} + 2q^{16} + 3q^{17} - 5q^{18} - 7q^{19} + 2q^{20} - 10q^{21} + 3q^{22} + q^{24} + 2q^{25} - 7q^{26} - 10q^{27} - q^{28} + 6q^{29} + q^{30} + 7q^{31} - 2q^{32} - 9q^{33} - 3q^{34} - q^{35} + 5q^{36} + 8q^{37} + 7q^{38} - 14q^{39} - 2q^{40} - 9q^{41} + 10q^{42} - 4q^{43} - 3q^{44} + 5q^{45} - 18q^{47} - q^{48} - 3q^{49} - 2q^{50} + 9q^{51} + 7q^{52} - 12q^{53} + 10q^{54} - 3q^{55} + q^{56} + 14q^{57} - 6q^{58} - 18q^{59} - q^{60} - 7q^{61} - 7q^{62} + 8q^{63} + 2q^{64} + 7q^{65} + 9q^{66} - 4q^{67} + 3q^{68} + q^{70} + 3q^{71} - 5q^{72} - 2q^{73} - 8q^{74} - q^{75} - 7q^{76} + 12q^{77} + 14q^{78} - 16q^{79} + 2q^{80} - 10q^{81} + 9q^{82} + 12q^{83} - 10q^{84} + 3q^{85} + 4q^{86} - 24q^{87} + 3q^{88} - 12q^{89} - 5q^{90} + 7q^{91} + 28q^{93} + 18q^{94} - 7q^{95} + q^{96} - 7q^{97} + 3q^{98} + 3q^{99} + O(q^{100}) \)

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 \(-0.327011\pi\)
0.517100 + 0.855925i \(0.327011\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −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
\(10\) −1.00000 −0.316228
\(11\) −3.79129 −1.14312 −0.571558 0.820562i \(-0.693661\pi\)
−0.571558 + 0.820562i \(0.693661\pi\)
\(12\) 1.79129 0.517100
\(13\) 1.20871 0.335236 0.167618 0.985852i \(-0.446392\pi\)
0.167618 + 0.985852i \(0.446392\pi\)
\(14\) 2.79129 0.746003
\(15\) 1.79129 0.462509
\(16\) 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 \(-0.544276\pi\)
−0.138649 + 0.990342i \(0.544276\pi\)
\(20\) 1.00000 0.223607
\(21\) −5.00000 −1.09109
\(22\) 3.79129 0.808305
\(23\) 0 0
\(24\) −1.79129 −0.365645
\(25\) 1.00000 0.200000
\(26\) −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 \(0.546942\pi\)
\(30\) −1.79129 −0.327043
\(31\) 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
\(35\) −2.79129 −0.471814
\(36\) 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
\(40\) −1.00000 −0.158114
\(41\) −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
\(45\) 0.208712 0.0311130
\(46\) 0 0
\(47\) −13.5826 −1.98122 −0.990611 0.136710i \(-0.956347\pi\)
−0.990611 + 0.136710i \(0.956347\pi\)
\(48\) 1.79129 0.258550
\(49\) 0.791288 0.113041
\(50\) −1.00000 −0.141421
\(51\) 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
\(55\) −3.79129 −0.511217
\(56\) 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 \(0.592841\pi\)
\(60\) 1.79129 0.231254
\(61\) 3.37386 0.431979 0.215989 0.976396i \(-0.430702\pi\)
0.215989 + 0.976396i \(0.430702\pi\)
\(62\) −10.3739 −1.31748
\(63\) −0.582576 −0.0733976
\(64\) 1.00000 0.125000
\(65\) 1.20871 0.149922
\(66\) 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
\(69\) 0 0
\(70\) 2.79129 0.333623
\(71\) −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 \(-0.831446\pi\)
−0.863045 + 0.505126i \(0.831446\pi\)
\(74\) −4.00000 −0.464991
\(75\) 1.79129 0.206840
\(76\) −1.20871 −0.138649
\(77\) 10.5826 1.20600
\(78\) −2.16515 −0.245155
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.00000 0.111803
\(81\) −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
\(85\) 3.79129 0.411223
\(86\) −7.16515 −0.772638
\(87\) −2.83485 −0.303928
\(88\) 3.79129 0.404153
\(89\) 3.16515 0.335505 0.167753 0.985829i \(-0.446349\pi\)
0.167753 + 0.985829i \(0.446349\pi\)
\(90\) −0.208712 −0.0220002
\(91\) −3.37386 −0.353677
\(92\) 0 0
\(93\) 18.5826 1.92692
\(94\) 13.5826 1.40094
\(95\) −1.20871 −0.124011
\(96\) −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
\(100\) 1.00000 0.100000
\(101\) 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
\(105\) −5.00000 −0.487950
\(106\) 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 \(-0.665498\pi\)
−0.496818 + 0.867855i \(0.665498\pi\)
\(110\) 3.79129 0.361485
\(111\) 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
\(120\) −1.79129 −0.163521
\(121\) 3.37386 0.306715
\(122\) −3.37386 −0.305455
\(123\) −3.95644 −0.356740
\(124\) 10.3739 0.931600
\(125\) 1.00000 0.0894427
\(126\) 0.582576 0.0519000
\(127\) −14.7477 −1.30865 −0.654325 0.756214i \(-0.727047\pi\)
−0.654325 + 0.756214i \(0.727047\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 12.8348 1.13005
\(130\) −1.20871 −0.106011
\(131\) 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
\(135\) −5.00000 −0.430331
\(136\) −3.79129 −0.325100
\(137\) 0.791288 0.0676043 0.0338021 0.999429i \(-0.489238\pi\)
0.0338021 + 0.999429i \(0.489238\pi\)
\(138\) 0 0
\(139\) −14.7477 −1.25089 −0.625443 0.780270i \(-0.715082\pi\)
−0.625443 + 0.780270i \(0.715082\pi\)
\(140\) −2.79129 −0.235907
\(141\) −24.3303 −2.04898
\(142\) 5.37386 0.450965
\(143\) −4.58258 −0.383214
\(144\) 0.208712 0.0173927
\(145\) −1.58258 −0.131426
\(146\) 14.7477 1.22053
\(147\) 1.41742 0.116907
\(148\) 4.00000 0.328798
\(149\) −12.7913 −1.04790 −0.523952 0.851748i \(-0.675543\pi\)
−0.523952 + 0.851748i \(0.675543\pi\)
\(150\) −1.79129 −0.146258
\(151\) −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
\(155\) 10.3739 0.833249
\(156\) 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
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 9.58258 0.752878
\(163\) 22.3739 1.75246 0.876228 0.481897i \(-0.160052\pi\)
0.876228 + 0.481897i \(0.160052\pi\)
\(164\) −2.20871 −0.172471
\(165\) −6.79129 −0.528701
\(166\) −6.00000 −0.465690
\(167\) −18.3303 −1.41844 −0.709221 0.704987i \(-0.750953\pi\)
−0.709221 + 0.704987i \(0.750953\pi\)
\(168\) 5.00000 0.385758
\(169\) −11.5390 −0.887617
\(170\) −3.79129 −0.290779
\(171\) −0.252273 −0.0192918
\(172\) 7.16515 0.546338
\(173\) −14.2087 −1.08027 −0.540134 0.841579i \(-0.681627\pi\)
−0.540134 + 0.841579i \(0.681627\pi\)
\(174\) 2.83485 0.214909
\(175\) −2.79129 −0.211002
\(176\) −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 \(0.715265\pi\)
\(180\) 0.208712 0.0155565
\(181\) −13.5390 −1.00635 −0.503174 0.864185i \(-0.667834\pi\)
−0.503174 + 0.864185i \(0.667834\pi\)
\(182\) 3.37386 0.250087
\(183\) 6.04356 0.446753
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) −18.5826 −1.36254
\(187\) −14.3739 −1.05112
\(188\) −13.5826 −0.990611
\(189\) 13.9564 1.01518
\(190\) 1.20871 0.0876892
\(191\) −16.4174 −1.18792 −0.593962 0.804493i \(-0.702437\pi\)
−0.593962 + 0.804493i \(0.702437\pi\)
\(192\) 1.79129 0.129275
\(193\) 6.74773 0.485712 0.242856 0.970062i \(-0.421916\pi\)
0.242856 + 0.970062i \(0.421916\pi\)
\(194\) 14.9564 1.07381
\(195\) 2.16515 0.155050
\(196\) 0.791288 0.0565206
\(197\) 20.5390 1.46334 0.731672 0.681657i \(-0.238740\pi\)
0.731672 + 0.681657i \(0.238740\pi\)
\(198\) 0.791288 0.0562344
\(199\) −20.3303 −1.44118 −0.720588 0.693363i \(-0.756128\pi\)
−0.720588 + 0.693363i \(0.756128\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 12.8348 0.905300
\(202\) −13.5826 −0.955667
\(203\) 4.41742 0.310042
\(204\) 6.79129 0.475485
\(205\) −2.20871 −0.154263
\(206\) 7.37386 0.513761
\(207\) 0 0
\(208\) 1.20871 0.0838091
\(209\) 4.58258 0.316983
\(210\) 5.00000 0.345033
\(211\) −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
\(215\) 7.16515 0.488659
\(216\) 5.00000 0.340207
\(217\) −28.9564 −1.96569
\(218\) 10.3739 0.702607
\(219\) −26.4174 −1.78512
\(220\) −3.79129 −0.255609
\(221\) 4.58258 0.308257
\(222\) −7.16515 −0.480893
\(223\) 11.1652 0.747674 0.373837 0.927494i \(-0.378042\pi\)
0.373837 + 0.927494i \(0.378042\pi\)
\(224\) 2.79129 0.186501
\(225\) 0.208712 0.0139141
\(226\) 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 \(-0.318587\pi\)
0.539568 + 0.841942i \(0.318587\pi\)
\(230\) 0 0
\(231\) 18.9564 1.24724
\(232\) 1.58258 0.103901
\(233\) 7.58258 0.496751 0.248376 0.968664i \(-0.420103\pi\)
0.248376 + 0.968664i \(0.420103\pi\)
\(234\) −0.252273 −0.0164916
\(235\) −13.5826 −0.886030
\(236\) −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 \(0.467358\pi\)
\(240\) 1.79129 0.115627
\(241\) 28.0000 1.80364 0.901819 0.432113i \(-0.142232\pi\)
0.901819 + 0.432113i \(0.142232\pi\)
\(242\) −3.37386 −0.216880
\(243\) −2.16515 −0.138895
\(244\) 3.37386 0.215989
\(245\) 0.791288 0.0505535
\(246\) 3.95644 0.252253
\(247\) −1.46099 −0.0929603
\(248\) −10.3739 −0.658741
\(249\) 10.7477 0.681110
\(250\) −1.00000 −0.0632456
\(251\) −30.7913 −1.94353 −0.971764 0.235953i \(-0.924179\pi\)
−0.971764 + 0.235953i \(0.924179\pi\)
\(252\) −0.582576 −0.0366988
\(253\) 0 0
\(254\) 14.7477 0.925355
\(255\) 6.79129 0.425287
\(256\) 1.00000 0.0625000
\(257\) 22.7477 1.41896 0.709482 0.704723i \(-0.248929\pi\)
0.709482 + 0.704723i \(0.248929\pi\)
\(258\) −12.8348 −0.799063
\(259\) −11.1652 −0.693769
\(260\) 1.20871 0.0749611
\(261\) −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
\(265\) −6.00000 −0.368577
\(266\) −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 \(0.329437\pi\)
\(270\) 5.00000 0.304290
\(271\) −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
\(275\) −3.79129 −0.228623
\(276\) 0 0
\(277\) −1.16515 −0.0700072 −0.0350036 0.999387i \(-0.511144\pi\)
−0.0350036 + 0.999387i \(0.511144\pi\)
\(278\) 14.7477 0.884510
\(279\) 2.16515 0.129624
\(280\) 2.79129 0.166811
\(281\) 16.7477 0.999086 0.499543 0.866289i \(-0.333501\pi\)
0.499543 + 0.866289i \(0.333501\pi\)
\(282\) 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
\(285\) −2.16515 −0.128252
\(286\) 4.58258 0.270973
\(287\) 6.16515 0.363917
\(288\) −0.208712 −0.0122985
\(289\) −2.62614 −0.154479
\(290\) 1.58258 0.0929320
\(291\) −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
\(295\) −4.41742 −0.257192
\(296\) −4.00000 −0.232495
\(297\) 18.9564 1.09996
\(298\) 12.7913 0.740979
\(299\) 0 0
\(300\) 1.79129 0.103420
\(301\) −20.0000 −1.15278
\(302\) 6.20871 0.357271
\(303\) 24.3303 1.39774
\(304\) −1.20871 −0.0693244
\(305\) 3.37386 0.193187
\(306\) −0.791288 −0.0452349
\(307\) 16.5390 0.943931 0.471966 0.881617i \(-0.343545\pi\)
0.471966 + 0.881617i \(0.343545\pi\)
\(308\) 10.5826 0.602998
\(309\) −13.2087 −0.751417
\(310\) −10.3739 −0.589196
\(311\) 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
\(315\) −0.582576 −0.0328244
\(316\) −8.00000 −0.450035
\(317\) 5.20871 0.292550 0.146275 0.989244i \(-0.453271\pi\)
0.146275 + 0.989244i \(0.453271\pi\)
\(318\) 10.7477 0.602703
\(319\) 6.00000 0.335936
\(320\) 1.00000 0.0559017
\(321\) −24.3303 −1.35799
\(322\) 0 0
\(323\) −4.58258 −0.254981
\(324\) −9.58258 −0.532365
\(325\) 1.20871 0.0670473
\(326\) −22.3739 −1.23917
\(327\) −18.5826 −1.02762
\(328\) 2.20871 0.121956
\(329\) 37.9129 2.09020
\(330\) 6.79129 0.373848
\(331\) 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
\(335\) 7.16515 0.391474
\(336\) −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
\(340\) 3.79129 0.205611
\(341\) −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 \(-0.415350\pi\)
0.262812 + 0.964847i \(0.415350\pi\)
\(348\) −2.83485 −0.151964
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 2.79129 0.149201
\(351\) −6.04356 −0.322581
\(352\) 3.79129 0.202076
\(353\) −15.1652 −0.807160 −0.403580 0.914944i \(-0.632234\pi\)
−0.403580 + 0.914944i \(0.632234\pi\)
\(354\) 7.91288 0.420565
\(355\) −5.37386 −0.285215
\(356\) 3.16515 0.167753
\(357\) −18.9564 −1.00328
\(358\) 16.7477 0.885145
\(359\) 9.16515 0.483718 0.241859 0.970311i \(-0.422243\pi\)
0.241859 + 0.970311i \(0.422243\pi\)
\(360\) −0.208712 −0.0110001
\(361\) −17.5390 −0.923106
\(362\) 13.5390 0.711595
\(363\) 6.04356 0.317205
\(364\) −3.37386 −0.176838
\(365\) −14.7477 −0.771931
\(366\) −6.04356 −0.315902
\(367\) 0.834849 0.0435787 0.0217894 0.999763i \(-0.493064\pi\)
0.0217894 + 0.999763i \(0.493064\pi\)
\(368\) 0 0
\(369\) −0.460985 −0.0239979
\(370\) −4.00000 −0.207950
\(371\) 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
\(375\) 1.79129 0.0925017
\(376\) 13.5826 0.700468
\(377\) −1.91288 −0.0985183
\(378\) −13.9564 −0.717842
\(379\) −7.37386 −0.378770 −0.189385 0.981903i \(-0.560649\pi\)
−0.189385 + 0.981903i \(0.560649\pi\)
\(380\) −1.20871 −0.0620056
\(381\) −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
\(385\) 10.5826 0.539338
\(386\) −6.74773 −0.343450
\(387\) 1.49545 0.0760182
\(388\) −14.9564 −0.759298
\(389\) −29.7042 −1.50606 −0.753031 0.657986i \(-0.771409\pi\)
−0.753031 + 0.657986i \(0.771409\pi\)
\(390\) −2.16515 −0.109637
\(391\) 0 0
\(392\) −0.791288 −0.0399661
\(393\) 16.4174 0.828150
\(394\) −20.5390 −1.03474
\(395\) −8.00000 −0.402524
\(396\) −0.791288 −0.0397637
\(397\) 16.5390 0.830069 0.415035 0.909806i \(-0.363769\pi\)
0.415035 + 0.909806i \(0.363769\pi\)
\(398\) 20.3303 1.01907
\(399\) 6.04356 0.302556
\(400\) 1.00000 0.0500000
\(401\) −22.7477 −1.13597 −0.567984 0.823040i \(-0.692276\pi\)
−0.567984 + 0.823040i \(0.692276\pi\)
\(402\) −12.8348 −0.640144
\(403\) 12.5390 0.624613
\(404\) 13.5826 0.675758
\(405\) −9.58258 −0.476162
\(406\) −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 \(0.690537\pi\)
\(410\) 2.20871 0.109081
\(411\) 1.41742 0.0699164
\(412\) −7.37386 −0.363284
\(413\) 12.3303 0.606735
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) −1.20871 −0.0592620
\(417\) −26.4174 −1.29367
\(418\) −4.58258 −0.224141
\(419\) −39.1652 −1.91334 −0.956671 0.291170i \(-0.905956\pi\)
−0.956671 + 0.291170i \(0.905956\pi\)
\(420\) −5.00000 −0.243975
\(421\) 23.1216 1.12688 0.563439 0.826158i \(-0.309478\pi\)
0.563439 + 0.826158i \(0.309478\pi\)
\(422\) 10.0000 0.486792
\(423\) −2.83485 −0.137835
\(424\) 6.00000 0.291386
\(425\) 3.79129 0.183904
\(426\) 9.62614 0.466388
\(427\) −9.41742 −0.455741
\(428\) −13.5826 −0.656539
\(429\) −8.20871 −0.396320
\(430\) −7.16515 −0.345534
\(431\) 19.9129 0.959170 0.479585 0.877496i \(-0.340787\pi\)
0.479585 + 0.877496i \(0.340787\pi\)
\(432\) −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
\(435\) −2.83485 −0.135921
\(436\) −10.3739 −0.496818
\(437\) 0 0
\(438\) 26.4174 1.26227
\(439\) 25.5390 1.21891 0.609455 0.792820i \(-0.291388\pi\)
0.609455 + 0.792820i \(0.291388\pi\)
\(440\) 3.79129 0.180743
\(441\) 0.165151 0.00786435
\(442\) −4.58258 −0.217971
\(443\) −35.2087 −1.67282 −0.836408 0.548107i \(-0.815349\pi\)
−0.836408 + 0.548107i \(0.815349\pi\)
\(444\) 7.16515 0.340043
\(445\) 3.16515 0.150043
\(446\) −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 \(0.298030\pi\)
\(450\) −0.208712 −0.00983879
\(451\) 8.37386 0.394310
\(452\) −6.00000 −0.282216
\(453\) −11.1216 −0.522538
\(454\) 4.74773 0.222822
\(455\) −3.37386 −0.158169
\(456\) 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 \(-0.574648\pi\)
−0.232370 + 0.972628i \(0.574648\pi\)
\(464\) −1.58258 −0.0734692
\(465\) 18.5826 0.861746
\(466\) −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
\(470\) 13.5826 0.626517
\(471\) −22.8348 −1.05217
\(472\) 4.41742 0.203328
\(473\) −27.1652 −1.24905
\(474\) 14.3303 0.658213
\(475\) −1.20871 −0.0554595
\(476\) −10.5826 −0.485052
\(477\) −1.25227 −0.0573376
\(478\) −3.16515 −0.144771
\(479\) 39.4955 1.80459 0.902297 0.431116i \(-0.141880\pi\)
0.902297 + 0.431116i \(0.141880\pi\)
\(480\) −1.79129 −0.0817607
\(481\) 4.83485 0.220450
\(482\) −28.0000 −1.27537
\(483\) 0 0
\(484\) 3.37386 0.153357
\(485\) −14.9564 −0.679137
\(486\) 2.16515 0.0982133
\(487\) 15.5826 0.706114 0.353057 0.935602i \(-0.385142\pi\)
0.353057 + 0.935602i \(0.385142\pi\)
\(488\) −3.37386 −0.152728
\(489\) 40.0780 1.81239
\(490\) −0.791288 −0.0357467
\(491\) −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
\(495\) −0.791288 −0.0355657
\(496\) 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 \(0.326488\pi\)
\(500\) 1.00000 0.0447214
\(501\) −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
\(505\) 13.5826 0.604417
\(506\) 0 0
\(507\) −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 \(0.448617\pi\)
\(510\) −6.79129 −0.300723
\(511\) 41.1652 1.82104
\(512\) −1.00000 −0.0441942
\(513\) 6.04356 0.266830
\(514\) −22.7477 −1.00336
\(515\) −7.37386 −0.324931
\(516\) 12.8348 0.565023
\(517\) 51.4955 2.26477
\(518\) 11.1652 0.490569
\(519\) −25.4519 −1.11721
\(520\) −1.20871 −0.0530055
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 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
\(525\) −5.00000 −0.218218
\(526\) 15.7913 0.688533
\(527\) 39.3303 1.71325
\(528\) −6.79129 −0.295553
\(529\) 0 0
\(530\) 6.00000 0.260623
\(531\) −0.921970 −0.0400101
\(532\) 3.37386 0.146276
\(533\) −2.66970 −0.115637
\(534\) −5.66970 −0.245352
\(535\) −13.5826 −0.587226
\(536\) −7.16515 −0.309487
\(537\) −30.0000 −1.29460
\(538\) −16.7477 −0.722046
\(539\) −3.00000 −0.129219
\(540\) −5.00000 −0.215166
\(541\) 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
\(545\) −10.3739 −0.444367
\(546\) 6.04356 0.258641
\(547\) 15.1216 0.646553 0.323276 0.946305i \(-0.395216\pi\)
0.323276 + 0.946305i \(0.395216\pi\)
\(548\) 0.791288 0.0338021
\(549\) 0.704166 0.0300531
\(550\) 3.79129 0.161661
\(551\) 1.91288 0.0814914
\(552\) 0 0
\(553\) 22.3303 0.949581
\(554\) 1.16515 0.0495025
\(555\) 7.16515 0.304144
\(556\) −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
\(560\) −2.79129 −0.117953
\(561\) −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
\(565\) −6.00000 −0.252422
\(566\) −28.3303 −1.19081
\(567\) 26.7477 1.12330
\(568\) 5.37386 0.225482
\(569\) −15.4955 −0.649603 −0.324802 0.945782i \(-0.605298\pi\)
−0.324802 + 0.945782i \(0.605298\pi\)
\(570\) 2.16515 0.0906882
\(571\) −30.1216 −1.26055 −0.630275 0.776372i \(-0.717058\pi\)
−0.630275 + 0.776372i \(0.717058\pi\)
\(572\) −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 \(-0.342335\pi\)
0.475314 + 0.879816i \(0.342335\pi\)
\(578\) 2.62614 0.109233
\(579\) 12.0871 0.502324
\(580\) −1.58258 −0.0657129
\(581\) −16.7477 −0.694813
\(582\) 26.7913 1.11053
\(583\) 22.7477 0.942115
\(584\) 14.7477 0.610265
\(585\) 0.252273 0.0104302
\(586\) −27.4955 −1.13583
\(587\) −26.2087 −1.08175 −0.540875 0.841103i \(-0.681907\pi\)
−0.540875 + 0.841103i \(0.681907\pi\)
\(588\) 1.41742 0.0584536
\(589\) −12.5390 −0.516661
\(590\) 4.41742 0.181862
\(591\) 36.7913 1.51339
\(592\) 4.00000 0.164399
\(593\) 13.9129 0.571333 0.285667 0.958329i \(-0.407785\pi\)
0.285667 + 0.958329i \(0.407785\pi\)
\(594\) −18.9564 −0.777792
\(595\) −10.5826 −0.433843
\(596\) −12.7913 −0.523952
\(597\) −36.4174 −1.49047
\(598\) 0 0
\(599\) −40.1216 −1.63932 −0.819662 0.572848i \(-0.805839\pi\)
−0.819662 + 0.572848i \(0.805839\pi\)
\(600\) −1.79129 −0.0731290
\(601\) −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
\(605\) 3.37386 0.137167
\(606\) −24.3303 −0.988351
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) 1.20871 0.0490198
\(609\) 7.91288 0.320646
\(610\) −3.37386 −0.136604
\(611\) −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
\(615\) −3.95644 −0.159539
\(616\) −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 \(-0.517865\pi\)
−0.0560957 + 0.998425i \(0.517865\pi\)
\(620\) 10.3739 0.416624
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) −8.83485 −0.353961
\(624\) 2.16515 0.0866754
\(625\) 1.00000 0.0400000
\(626\) −18.3739 −0.734367
\(627\) 8.20871 0.327824
\(628\) −12.7477 −0.508690
\(629\) 15.1652 0.604674
\(630\) 0.582576 0.0232104
\(631\) 17.9129 0.713100 0.356550 0.934276i \(-0.383953\pi\)
0.356550 + 0.934276i \(0.383953\pi\)
\(632\) 8.00000 0.318223
\(633\) −17.9129 −0.711973
\(634\) −5.20871 −0.206864
\(635\) −14.7477 −0.585246
\(636\) −10.7477 −0.426175
\(637\) 0.956439 0.0378955
\(638\) −6.00000 −0.237542
\(639\) −1.12159 −0.0443694
\(640\) −1.00000 −0.0395285
\(641\) 3.16515 0.125016 0.0625080 0.998044i \(-0.480090\pi\)
0.0625080 + 0.998044i \(0.480090\pi\)
\(642\) 24.3303 0.960240
\(643\) 20.7477 0.818210 0.409105 0.912487i \(-0.365841\pi\)
0.409105 + 0.912487i \(0.365841\pi\)
\(644\) 0 0
\(645\) 12.8348 0.505372
\(646\) 4.58258 0.180299
\(647\) 2.83485 0.111449 0.0557247 0.998446i \(-0.482253\pi\)
0.0557247 + 0.998446i \(0.482253\pi\)
\(648\) 9.58258 0.376439
\(649\) 16.7477 0.657406
\(650\) −1.20871 −0.0474096
\(651\) −51.8693 −2.03292
\(652\) 22.3739 0.876228
\(653\) 35.5390 1.39075 0.695375 0.718647i \(-0.255238\pi\)
0.695375 + 0.718647i \(0.255238\pi\)
\(654\) 18.5826 0.726636
\(655\) 9.16515 0.358112
\(656\) −2.20871 −0.0862357
\(657\) −3.07803 −0.120085
\(658\) −37.9129 −1.47800
\(659\) 27.1652 1.05820 0.529102 0.848558i \(-0.322529\pi\)
0.529102 + 0.848558i \(0.322529\pi\)
\(660\) −6.79129 −0.264351
\(661\) 39.3739 1.53147 0.765733 0.643159i \(-0.222376\pi\)
0.765733 + 0.643159i \(0.222376\pi\)
\(662\) −6.74773 −0.262258
\(663\) 8.20871 0.318800
\(664\) −6.00000 −0.232845
\(665\) 3.37386 0.130833
\(666\) −0.834849 −0.0323497
\(667\) 0 0
\(668\) −18.3303 −0.709221
\(669\) 20.0000 0.773245
\(670\) −7.16515 −0.276814
\(671\) −12.7913 −0.493802
\(672\) 5.00000 0.192879
\(673\) 38.0000 1.46479 0.732396 0.680879i \(-0.238402\pi\)
0.732396 + 0.680879i \(0.238402\pi\)
\(674\) −16.7913 −0.646776
\(675\) −5.00000 −0.192450
\(676\) −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
\(680\) −3.79129 −0.145389
\(681\) −8.50455 −0.325895
\(682\) 39.3303 1.50604
\(683\) 2.37386 0.0908334 0.0454167 0.998968i \(-0.485538\pi\)
0.0454167 + 0.998968i \(0.485538\pi\)
\(684\) −0.252273 −0.00964590
\(685\) 0.791288 0.0302336
\(686\) −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
\(695\) −14.7477 −0.559413
\(696\) 2.83485 0.107455
\(697\) −8.37386 −0.317183
\(698\) −26.0000 −0.984115
\(699\) 13.5826 0.513740
\(700\) −2.79129 −0.105501
\(701\) −9.62614 −0.363574 −0.181787 0.983338i \(-0.558188\pi\)
−0.181787 + 0.983338i \(0.558188\pi\)
\(702\) 6.04356 0.228100
\(703\) −4.83485 −0.182350
\(704\) −3.79129 −0.142890
\(705\) −24.3303 −0.916332
\(706\) 15.1652 0.570748
\(707\) −37.9129 −1.42586
\(708\) −7.91288 −0.297384
\(709\) −34.5390 −1.29714 −0.648570 0.761155i \(-0.724633\pi\)
−0.648570 + 0.761155i \(0.724633\pi\)
\(710\) 5.37386 0.201678
\(711\) −1.66970 −0.0626185
\(712\) −3.16515 −0.118619
\(713\) 0 0
\(714\) 18.9564 0.709427
\(715\) −4.58258 −0.171379
\(716\) −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 \(0.685681\pi\)
\(720\) 0.208712 0.00777824
\(721\) 20.5826 0.766535
\(722\) 17.5390 0.652735
\(723\) 50.1561 1.86532
\(724\) −13.5390 −0.503174
\(725\) −1.58258 −0.0587754
\(726\) −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
\(730\) 14.7477 0.545838
\(731\) 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
\(735\) 1.41742 0.0522825
\(736\) 0 0
\(737\) −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 \(0.452992\pi\)
\(740\) 4.00000 0.147043
\(741\) −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
\(745\) −12.7913 −0.468637
\(746\) −14.7477 −0.539953
\(747\) 1.25227 0.0458183
\(748\) −14.3739 −0.525561
\(749\) 37.9129 1.38531
\(750\) −1.79129 −0.0654086
\(751\) −18.7477 −0.684114 −0.342057 0.939679i \(-0.611124\pi\)
−0.342057 + 0.939679i \(0.611124\pi\)
\(752\) −13.5826 −0.495306
\(753\) −55.1561 −2.01000
\(754\) 1.91288 0.0696629
\(755\) −6.20871 −0.225958
\(756\) 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
\(759\) 0 0
\(760\) 1.20871 0.0438446
\(761\) −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
\(765\) 0.791288 0.0286091
\(766\) −24.0000 −0.867155
\(767\) −5.33939 −0.192794
\(768\) 1.79129 0.0646375
\(769\) 3.66970 0.132333 0.0661663 0.997809i \(-0.478923\pi\)
0.0661663 + 0.997809i \(0.478923\pi\)
\(770\) −10.5826 −0.381370
\(771\) 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
\(775\) 10.3739 0.372640
\(776\) 14.9564 0.536905
\(777\) −20.0000 −0.717496
\(778\) 29.7042 1.06495
\(779\) 2.66970 0.0956518
\(780\) 2.16515 0.0775249
\(781\) 20.3739 0.729034
\(782\) 0 0
\(783\) 7.91288 0.282783
\(784\) 0.791288 0.0282603
\(785\) −12.7477 −0.454986
\(786\) −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
\(790\) 8.00000 0.284627
\(791\) 16.7477 0.595481
\(792\) 0.791288 0.0281172
\(793\) 4.07803 0.144815
\(794\) −16.5390 −0.586948
\(795\) −10.7477 −0.381183
\(796\) −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
\(800\) −1.00000 −0.0353553
\(801\) 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 \(0.437810\pi\)
\(810\) 9.58258 0.336697
\(811\) −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
\(815\) 22.3739 0.783722
\(816\) 6.79129 0.237743
\(817\) −8.66061 −0.302996
\(818\) 22.7913 0.796879
\(819\) −0.704166 −0.0246056
\(820\) −2.20871 −0.0771316
\(821\) 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 \(-0.245303\pi\)
0.717463 + 0.696596i \(0.245303\pi\)
\(824\) 7.37386 0.256881
\(825\) −6.79129 −0.236442
\(826\) −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 0
\(829\) −31.4955 −1.09388 −0.546941 0.837171i \(-0.684208\pi\)
−0.546941 + 0.837171i \(0.684208\pi\)
\(830\) −6.00000 −0.208263
\(831\) −2.08712 −0.0724014
\(832\) 1.20871 0.0419046
\(833\) 3.00000 0.103944
\(834\) 26.4174 0.914761
\(835\) −18.3303 −0.634346
\(836\) 4.58258 0.158492
\(837\) −51.8693 −1.79287
\(838\) 39.1652 1.35294
\(839\) 22.4174 0.773935 0.386968 0.922093i \(-0.373522\pi\)
0.386968 + 0.922093i \(0.373522\pi\)
\(840\) 5.00000 0.172516
\(841\) −26.4955 −0.913636
\(842\) −23.1216 −0.796823
\(843\) 30.0000 1.03325
\(844\) −10.0000 −0.344214
\(845\) −11.5390 −0.396954
\(846\) 2.83485 0.0974641
\(847\) −9.41742 −0.323587
\(848\) −6.00000 −0.206041
\(849\) 50.7477 1.74166
\(850\) −3.79129 −0.130040
\(851\) 0 0
\(852\) −9.62614 −0.329786
\(853\) 8.46099 0.289699 0.144849 0.989454i \(-0.453730\pi\)
0.144849 + 0.989454i \(0.453730\pi\)
\(854\) 9.41742 0.322258
\(855\) −0.252273 −0.00862755
\(856\) 13.5826 0.464243
\(857\) 9.16515 0.313076 0.156538 0.987672i \(-0.449967\pi\)
0.156538 + 0.987672i \(0.449967\pi\)
\(858\) 8.20871 0.280241
\(859\) 0.747727 0.0255121 0.0127561 0.999919i \(-0.495940\pi\)
0.0127561 + 0.999919i \(0.495940\pi\)
\(860\) 7.16515 0.244330
\(861\) 11.0436 0.376364
\(862\) −19.9129 −0.678235
\(863\) −31.5826 −1.07508 −0.537542 0.843237i \(-0.680647\pi\)
−0.537542 + 0.843237i \(0.680647\pi\)
\(864\) 5.00000 0.170103
\(865\) −14.2087 −0.483111
\(866\) 1.53901 0.0522979
\(867\) −4.70417 −0.159762
\(868\) −28.9564 −0.982846
\(869\) 30.3303 1.02889
\(870\) 2.83485 0.0961104
\(871\) 8.66061 0.293453
\(872\) 10.3739 0.351303
\(873\) −3.12159 −0.105650
\(874\) 0 0
\(875\) −2.79129 −0.0943628
\(876\) −26.4174 −0.892562
\(877\) 7.70417 0.260151 0.130076 0.991504i \(-0.458478\pi\)
0.130076 + 0.991504i \(0.458478\pi\)
\(878\) −25.5390 −0.861900
\(879\) 49.2523 1.66124
\(880\) −3.79129 −0.127804
\(881\) 6.33030 0.213273 0.106637 0.994298i \(-0.465992\pi\)
0.106637 + 0.994298i \(0.465992\pi\)
\(882\) −0.165151 −0.00556094
\(883\) −12.0436 −0.405298 −0.202649 0.979251i \(-0.564955\pi\)
−0.202649 + 0.979251i \(0.564955\pi\)
\(884\) 4.58258 0.154129
\(885\) −7.91288 −0.265989
\(886\) 35.2087 1.18286
\(887\) −3.16515 −0.106275 −0.0531377 0.998587i \(-0.516922\pi\)
−0.0531377 + 0.998587i \(0.516922\pi\)
\(888\) −7.16515 −0.240447
\(889\) 41.1652 1.38063
\(890\) −3.16515 −0.106096
\(891\) 36.3303 1.21711
\(892\) 11.1652 0.373837
\(893\) 16.4174 0.549388
\(894\) 22.9129 0.766321
\(895\) −16.7477 −0.559815
\(896\) 2.79129 0.0932504
\(897\) 0 0
\(898\) −25.1216 −0.838318
\(899\) −16.4174 −0.547552
\(900\) 0.208712 0.00695707
\(901\) −22.7477 −0.757837
\(902\) −8.37386 −0.278819
\(903\) −35.8258 −1.19221
\(904\) 6.00000 0.199557
\(905\) −13.5390 −0.450052
\(906\) 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
\(910\) 3.37386 0.111842
\(911\) −13.5826 −0.450011 −0.225005 0.974358i \(-0.572240\pi\)
−0.225005 + 0.974358i \(0.572240\pi\)
\(912\) −2.16515 −0.0716953
\(913\) −22.7477 −0.752840
\(914\) −10.0000 −0.330771
\(915\) 6.04356 0.199794
\(916\) 16.3303 0.539568
\(917\) −25.5826 −0.844811
\(918\) 18.9564 0.625656
\(919\) 36.8348 1.21507 0.607535 0.794293i \(-0.292159\pi\)
0.607535 + 0.794293i \(0.292159\pi\)
\(920\) 0 0
\(921\) 29.6261 0.976214
\(922\) 1.25227 0.0412414
\(923\) −6.49545 −0.213800
\(924\) 18.9564 0.623621
\(925\) 4.00000 0.131519
\(926\) 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 \(0.724353\pi\)
\(930\) −18.5826 −0.609347
\(931\) −0.956439 −0.0313460
\(932\) 7.58258 0.248376
\(933\) 21.4955 0.703730
\(934\) 25.9129 0.847895
\(935\) −14.3739 −0.470076
\(936\) −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
\(940\) −13.5826 −0.443015
\(941\) −54.9564 −1.79153 −0.895764 0.444529i \(-0.853371\pi\)
−0.895764 + 0.444529i \(0.853371\pi\)
\(942\) 22.8348 0.744000
\(943\) 0 0
\(944\) −4.41742 −0.143775
\(945\) 13.9564 0.454003
\(946\) 27.1652 0.883215
\(947\) −29.5390 −0.959889 −0.479945 0.877299i \(-0.659343\pi\)
−0.479945 + 0.877299i \(0.659343\pi\)
\(948\) −14.3303 −0.465427
\(949\) −17.8258 −0.578649
\(950\) 1.20871 0.0392158
\(951\) 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
\(955\) −16.4174 −0.531255
\(956\) 3.16515 0.102368
\(957\) 10.7477 0.347425
\(958\) −39.4955 −1.27604
\(959\) −2.20871 −0.0713230
\(960\) 1.79129 0.0578136
\(961\) 76.6170 2.47152
\(962\) −4.83485 −0.155882
\(963\) −2.83485 −0.0913517
\(964\) 28.0000 0.901819
\(965\) 6.74773 0.217217
\(966\) 0 0
\(967\) −5.25227 −0.168902 −0.0844509 0.996428i \(-0.526914\pi\)
−0.0844509 + 0.996428i \(0.526914\pi\)
\(968\) −3.37386 −0.108440
\(969\) −8.20871 −0.263702
\(970\) 14.9564 0.480222
\(971\) 6.95644 0.223243 0.111621 0.993751i \(-0.464396\pi\)
0.111621 + 0.993751i \(0.464396\pi\)
\(972\) −2.16515 −0.0694473
\(973\) 41.1652 1.31969
\(974\) −15.5826 −0.499298
\(975\) 2.16515 0.0693403
\(976\) 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
\(980\) 0.791288 0.0252768
\(981\) −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
\(985\) 20.5390 0.654427
\(986\) 6.00000 0.191079
\(987\) 67.9129 2.16169
\(988\) −1.46099 −0.0464801
\(989\) 0 0
\(990\) 0.791288 0.0251488
\(991\) −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
\(995\) −20.3303 −0.644514
\(996\) 10.7477 0.340555
\(997\) 11.4955 0.364065 0.182032 0.983293i \(-0.441732\pi\)
0.182032 + 0.983293i \(0.441732\pi\)
\(998\) −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 5290.2.a.e.1.2 2
23.22 odd 2 230.2.a.a.1.2 2
69.68 even 2 2070.2.a.x.1.2 2
92.91 even 2 1840.2.a.n.1.1 2
115.22 even 4 1150.2.b.g.599.1 4
115.68 even 4 1150.2.b.g.599.4 4
115.114 odd 2 1150.2.a.o.1.1 2
184.45 odd 2 7360.2.a.bq.1.1 2
184.91 even 2 7360.2.a.bk.1.2 2
460.459 even 2 9200.2.a.bs.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.a.1.2 2 23.22 odd 2
1150.2.a.o.1.1 2 115.114 odd 2
1150.2.b.g.599.1 4 115.22 even 4
1150.2.b.g.599.4 4 115.68 even 4
1840.2.a.n.1.1 2 92.91 even 2
2070.2.a.x.1.2 2 69.68 even 2
5290.2.a.e.1.2 2 1.1 even 1 trivial
7360.2.a.bk.1.2 2 184.91 even 2
7360.2.a.bq.1.1 2 184.45 odd 2
9200.2.a.bs.1.2 2 460.459 even 2