Properties

Label 231.2.a.c.1.2
Level $231$
Weight $2$
Character 231.1
Self dual yes
Analytic conductor $1.845$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.84454428669\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 231.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} +1.00000 q^{5} +1.61803 q^{6} +1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} +1.00000 q^{5} +1.61803 q^{6} +1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +1.61803 q^{10} +1.00000 q^{11} +0.618034 q^{12} -5.47214 q^{13} +1.61803 q^{14} +1.00000 q^{15} -4.85410 q^{16} +0.763932 q^{17} +1.61803 q^{18} +6.70820 q^{19} +0.618034 q^{20} +1.00000 q^{21} +1.61803 q^{22} -7.70820 q^{23} -2.23607 q^{24} -4.00000 q^{25} -8.85410 q^{26} +1.00000 q^{27} +0.618034 q^{28} +5.00000 q^{29} +1.61803 q^{30} -0.763932 q^{31} -3.38197 q^{32} +1.00000 q^{33} +1.23607 q^{34} +1.00000 q^{35} +0.618034 q^{36} -7.00000 q^{37} +10.8541 q^{38} -5.47214 q^{39} -2.23607 q^{40} +6.47214 q^{41} +1.61803 q^{42} -7.70820 q^{43} +0.618034 q^{44} +1.00000 q^{45} -12.4721 q^{46} -4.23607 q^{47} -4.85410 q^{48} +1.00000 q^{49} -6.47214 q^{50} +0.763932 q^{51} -3.38197 q^{52} +10.1803 q^{53} +1.61803 q^{54} +1.00000 q^{55} -2.23607 q^{56} +6.70820 q^{57} +8.09017 q^{58} +11.1803 q^{59} +0.618034 q^{60} +2.00000 q^{61} -1.23607 q^{62} +1.00000 q^{63} +4.23607 q^{64} -5.47214 q^{65} +1.61803 q^{66} -14.2361 q^{67} +0.472136 q^{68} -7.70820 q^{69} +1.61803 q^{70} +6.47214 q^{71} -2.23607 q^{72} +13.4721 q^{73} -11.3262 q^{74} -4.00000 q^{75} +4.14590 q^{76} +1.00000 q^{77} -8.85410 q^{78} -5.52786 q^{79} -4.85410 q^{80} +1.00000 q^{81} +10.4721 q^{82} +11.2361 q^{83} +0.618034 q^{84} +0.763932 q^{85} -12.4721 q^{86} +5.00000 q^{87} -2.23607 q^{88} +4.47214 q^{89} +1.61803 q^{90} -5.47214 q^{91} -4.76393 q^{92} -0.763932 q^{93} -6.85410 q^{94} +6.70820 q^{95} -3.38197 q^{96} -3.70820 q^{97} +1.61803 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 2 q^{3} - q^{4} + 2 q^{5} + q^{6} + 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + q^{2} + 2 q^{3} - q^{4} + 2 q^{5} + q^{6} + 2 q^{7} + 2 q^{9} + q^{10} + 2 q^{11} - q^{12} - 2 q^{13} + q^{14} + 2 q^{15} - 3 q^{16} + 6 q^{17} + q^{18} - q^{20} + 2 q^{21} + q^{22} - 2 q^{23} - 8 q^{25} - 11 q^{26} + 2 q^{27} - q^{28} + 10 q^{29} + q^{30} - 6 q^{31} - 9 q^{32} + 2 q^{33} - 2 q^{34} + 2 q^{35} - q^{36} - 14 q^{37} + 15 q^{38} - 2 q^{39} + 4 q^{41} + q^{42} - 2 q^{43} - q^{44} + 2 q^{45} - 16 q^{46} - 4 q^{47} - 3 q^{48} + 2 q^{49} - 4 q^{50} + 6 q^{51} - 9 q^{52} - 2 q^{53} + q^{54} + 2 q^{55} + 5 q^{58} - q^{60} + 4 q^{61} + 2 q^{62} + 2 q^{63} + 4 q^{64} - 2 q^{65} + q^{66} - 24 q^{67} - 8 q^{68} - 2 q^{69} + q^{70} + 4 q^{71} + 18 q^{73} - 7 q^{74} - 8 q^{75} + 15 q^{76} + 2 q^{77} - 11 q^{78} - 20 q^{79} - 3 q^{80} + 2 q^{81} + 12 q^{82} + 18 q^{83} - q^{84} + 6 q^{85} - 16 q^{86} + 10 q^{87} + q^{90} - 2 q^{91} - 14 q^{92} - 6 q^{93} - 7 q^{94} - 9 q^{96} + 6 q^{97} + q^{98} + 2 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.61803 1.14412 0.572061 0.820211i \(-0.306144\pi\)
0.572061 + 0.820211i \(0.306144\pi\)
\(3\) 1.00000 0.577350
\(4\) 0.618034 0.309017
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 1.61803 0.660560
\(7\) 1.00000 0.377964
\(8\) −2.23607 −0.790569
\(9\) 1.00000 0.333333
\(10\) 1.61803 0.511667
\(11\) 1.00000 0.301511
\(12\) 0.618034 0.178411
\(13\) −5.47214 −1.51770 −0.758849 0.651267i \(-0.774238\pi\)
−0.758849 + 0.651267i \(0.774238\pi\)
\(14\) 1.61803 0.432438
\(15\) 1.00000 0.258199
\(16\) −4.85410 −1.21353
\(17\) 0.763932 0.185281 0.0926404 0.995700i \(-0.470469\pi\)
0.0926404 + 0.995700i \(0.470469\pi\)
\(18\) 1.61803 0.381374
\(19\) 6.70820 1.53897 0.769484 0.638666i \(-0.220514\pi\)
0.769484 + 0.638666i \(0.220514\pi\)
\(20\) 0.618034 0.138197
\(21\) 1.00000 0.218218
\(22\) 1.61803 0.344966
\(23\) −7.70820 −1.60727 −0.803636 0.595121i \(-0.797104\pi\)
−0.803636 + 0.595121i \(0.797104\pi\)
\(24\) −2.23607 −0.456435
\(25\) −4.00000 −0.800000
\(26\) −8.85410 −1.73643
\(27\) 1.00000 0.192450
\(28\) 0.618034 0.116797
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 1.61803 0.295411
\(31\) −0.763932 −0.137206 −0.0686031 0.997644i \(-0.521854\pi\)
−0.0686031 + 0.997644i \(0.521854\pi\)
\(32\) −3.38197 −0.597853
\(33\) 1.00000 0.174078
\(34\) 1.23607 0.211984
\(35\) 1.00000 0.169031
\(36\) 0.618034 0.103006
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 10.8541 1.76077
\(39\) −5.47214 −0.876243
\(40\) −2.23607 −0.353553
\(41\) 6.47214 1.01078 0.505389 0.862892i \(-0.331349\pi\)
0.505389 + 0.862892i \(0.331349\pi\)
\(42\) 1.61803 0.249668
\(43\) −7.70820 −1.17549 −0.587745 0.809046i \(-0.699984\pi\)
−0.587745 + 0.809046i \(0.699984\pi\)
\(44\) 0.618034 0.0931721
\(45\) 1.00000 0.149071
\(46\) −12.4721 −1.83892
\(47\) −4.23607 −0.617894 −0.308947 0.951079i \(-0.599977\pi\)
−0.308947 + 0.951079i \(0.599977\pi\)
\(48\) −4.85410 −0.700629
\(49\) 1.00000 0.142857
\(50\) −6.47214 −0.915298
\(51\) 0.763932 0.106972
\(52\) −3.38197 −0.468994
\(53\) 10.1803 1.39838 0.699189 0.714937i \(-0.253545\pi\)
0.699189 + 0.714937i \(0.253545\pi\)
\(54\) 1.61803 0.220187
\(55\) 1.00000 0.134840
\(56\) −2.23607 −0.298807
\(57\) 6.70820 0.888523
\(58\) 8.09017 1.06229
\(59\) 11.1803 1.45556 0.727778 0.685813i \(-0.240553\pi\)
0.727778 + 0.685813i \(0.240553\pi\)
\(60\) 0.618034 0.0797878
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −1.23607 −0.156981
\(63\) 1.00000 0.125988
\(64\) 4.23607 0.529508
\(65\) −5.47214 −0.678735
\(66\) 1.61803 0.199166
\(67\) −14.2361 −1.73921 −0.869606 0.493746i \(-0.835627\pi\)
−0.869606 + 0.493746i \(0.835627\pi\)
\(68\) 0.472136 0.0572549
\(69\) −7.70820 −0.927959
\(70\) 1.61803 0.193392
\(71\) 6.47214 0.768101 0.384051 0.923312i \(-0.374529\pi\)
0.384051 + 0.923312i \(0.374529\pi\)
\(72\) −2.23607 −0.263523
\(73\) 13.4721 1.57679 0.788397 0.615167i \(-0.210911\pi\)
0.788397 + 0.615167i \(0.210911\pi\)
\(74\) −11.3262 −1.31665
\(75\) −4.00000 −0.461880
\(76\) 4.14590 0.475567
\(77\) 1.00000 0.113961
\(78\) −8.85410 −1.00253
\(79\) −5.52786 −0.621933 −0.310967 0.950421i \(-0.600653\pi\)
−0.310967 + 0.950421i \(0.600653\pi\)
\(80\) −4.85410 −0.542705
\(81\) 1.00000 0.111111
\(82\) 10.4721 1.15645
\(83\) 11.2361 1.23332 0.616659 0.787230i \(-0.288486\pi\)
0.616659 + 0.787230i \(0.288486\pi\)
\(84\) 0.618034 0.0674330
\(85\) 0.763932 0.0828601
\(86\) −12.4721 −1.34491
\(87\) 5.00000 0.536056
\(88\) −2.23607 −0.238366
\(89\) 4.47214 0.474045 0.237023 0.971504i \(-0.423828\pi\)
0.237023 + 0.971504i \(0.423828\pi\)
\(90\) 1.61803 0.170556
\(91\) −5.47214 −0.573636
\(92\) −4.76393 −0.496674
\(93\) −0.763932 −0.0792161
\(94\) −6.85410 −0.706947
\(95\) 6.70820 0.688247
\(96\) −3.38197 −0.345170
\(97\) −3.70820 −0.376511 −0.188256 0.982120i \(-0.560283\pi\)
−0.188256 + 0.982120i \(0.560283\pi\)
\(98\) 1.61803 0.163446
\(99\) 1.00000 0.100504
\(100\) −2.47214 −0.247214
\(101\) −4.18034 −0.415959 −0.207980 0.978133i \(-0.566689\pi\)
−0.207980 + 0.978133i \(0.566689\pi\)
\(102\) 1.23607 0.122389
\(103\) −9.41641 −0.927826 −0.463913 0.885881i \(-0.653555\pi\)
−0.463913 + 0.885881i \(0.653555\pi\)
\(104\) 12.2361 1.19985
\(105\) 1.00000 0.0975900
\(106\) 16.4721 1.59992
\(107\) 0.236068 0.0228216 0.0114108 0.999935i \(-0.496368\pi\)
0.0114108 + 0.999935i \(0.496368\pi\)
\(108\) 0.618034 0.0594703
\(109\) 7.23607 0.693090 0.346545 0.938033i \(-0.387355\pi\)
0.346545 + 0.938033i \(0.387355\pi\)
\(110\) 1.61803 0.154273
\(111\) −7.00000 −0.664411
\(112\) −4.85410 −0.458670
\(113\) 8.47214 0.796992 0.398496 0.917170i \(-0.369532\pi\)
0.398496 + 0.917170i \(0.369532\pi\)
\(114\) 10.8541 1.01658
\(115\) −7.70820 −0.718794
\(116\) 3.09017 0.286915
\(117\) −5.47214 −0.505899
\(118\) 18.0902 1.66534
\(119\) 0.763932 0.0700295
\(120\) −2.23607 −0.204124
\(121\) 1.00000 0.0909091
\(122\) 3.23607 0.292980
\(123\) 6.47214 0.583573
\(124\) −0.472136 −0.0423991
\(125\) −9.00000 −0.804984
\(126\) 1.61803 0.144146
\(127\) 18.6525 1.65514 0.827570 0.561363i \(-0.189723\pi\)
0.827570 + 0.561363i \(0.189723\pi\)
\(128\) 13.6180 1.20368
\(129\) −7.70820 −0.678670
\(130\) −8.85410 −0.776556
\(131\) −16.9443 −1.48043 −0.740214 0.672371i \(-0.765276\pi\)
−0.740214 + 0.672371i \(0.765276\pi\)
\(132\) 0.618034 0.0537930
\(133\) 6.70820 0.581675
\(134\) −23.0344 −1.98987
\(135\) 1.00000 0.0860663
\(136\) −1.70820 −0.146477
\(137\) 6.29180 0.537544 0.268772 0.963204i \(-0.413382\pi\)
0.268772 + 0.963204i \(0.413382\pi\)
\(138\) −12.4721 −1.06170
\(139\) 5.52786 0.468867 0.234434 0.972132i \(-0.424676\pi\)
0.234434 + 0.972132i \(0.424676\pi\)
\(140\) 0.618034 0.0522334
\(141\) −4.23607 −0.356741
\(142\) 10.4721 0.878802
\(143\) −5.47214 −0.457603
\(144\) −4.85410 −0.404508
\(145\) 5.00000 0.415227
\(146\) 21.7984 1.80405
\(147\) 1.00000 0.0824786
\(148\) −4.32624 −0.355615
\(149\) 5.00000 0.409616 0.204808 0.978802i \(-0.434343\pi\)
0.204808 + 0.978802i \(0.434343\pi\)
\(150\) −6.47214 −0.528448
\(151\) 8.18034 0.665707 0.332853 0.942979i \(-0.391989\pi\)
0.332853 + 0.942979i \(0.391989\pi\)
\(152\) −15.0000 −1.21666
\(153\) 0.763932 0.0617602
\(154\) 1.61803 0.130385
\(155\) −0.763932 −0.0613605
\(156\) −3.38197 −0.270774
\(157\) 11.4164 0.911129 0.455564 0.890203i \(-0.349438\pi\)
0.455564 + 0.890203i \(0.349438\pi\)
\(158\) −8.94427 −0.711568
\(159\) 10.1803 0.807353
\(160\) −3.38197 −0.267368
\(161\) −7.70820 −0.607492
\(162\) 1.61803 0.127125
\(163\) −9.29180 −0.727790 −0.363895 0.931440i \(-0.618553\pi\)
−0.363895 + 0.931440i \(0.618553\pi\)
\(164\) 4.00000 0.312348
\(165\) 1.00000 0.0778499
\(166\) 18.1803 1.41107
\(167\) 8.65248 0.669549 0.334774 0.942298i \(-0.391340\pi\)
0.334774 + 0.942298i \(0.391340\pi\)
\(168\) −2.23607 −0.172516
\(169\) 16.9443 1.30341
\(170\) 1.23607 0.0948021
\(171\) 6.70820 0.512989
\(172\) −4.76393 −0.363246
\(173\) −10.4721 −0.796182 −0.398091 0.917346i \(-0.630327\pi\)
−0.398091 + 0.917346i \(0.630327\pi\)
\(174\) 8.09017 0.613314
\(175\) −4.00000 −0.302372
\(176\) −4.85410 −0.365892
\(177\) 11.1803 0.840366
\(178\) 7.23607 0.542366
\(179\) −8.94427 −0.668526 −0.334263 0.942480i \(-0.608487\pi\)
−0.334263 + 0.942480i \(0.608487\pi\)
\(180\) 0.618034 0.0460655
\(181\) −5.23607 −0.389194 −0.194597 0.980883i \(-0.562340\pi\)
−0.194597 + 0.980883i \(0.562340\pi\)
\(182\) −8.85410 −0.656310
\(183\) 2.00000 0.147844
\(184\) 17.2361 1.27066
\(185\) −7.00000 −0.514650
\(186\) −1.23607 −0.0906329
\(187\) 0.763932 0.0558642
\(188\) −2.61803 −0.190940
\(189\) 1.00000 0.0727393
\(190\) 10.8541 0.787439
\(191\) −15.2361 −1.10244 −0.551222 0.834359i \(-0.685838\pi\)
−0.551222 + 0.834359i \(0.685838\pi\)
\(192\) 4.23607 0.305712
\(193\) −16.6525 −1.19867 −0.599336 0.800498i \(-0.704569\pi\)
−0.599336 + 0.800498i \(0.704569\pi\)
\(194\) −6.00000 −0.430775
\(195\) −5.47214 −0.391868
\(196\) 0.618034 0.0441453
\(197\) −7.52786 −0.536338 −0.268169 0.963372i \(-0.586419\pi\)
−0.268169 + 0.963372i \(0.586419\pi\)
\(198\) 1.61803 0.114989
\(199\) −26.1803 −1.85588 −0.927938 0.372736i \(-0.878420\pi\)
−0.927938 + 0.372736i \(0.878420\pi\)
\(200\) 8.94427 0.632456
\(201\) −14.2361 −1.00413
\(202\) −6.76393 −0.475909
\(203\) 5.00000 0.350931
\(204\) 0.472136 0.0330561
\(205\) 6.47214 0.452034
\(206\) −15.2361 −1.06155
\(207\) −7.70820 −0.535757
\(208\) 26.5623 1.84176
\(209\) 6.70820 0.464016
\(210\) 1.61803 0.111655
\(211\) −21.4164 −1.47437 −0.737183 0.675693i \(-0.763845\pi\)
−0.737183 + 0.675693i \(0.763845\pi\)
\(212\) 6.29180 0.432122
\(213\) 6.47214 0.443463
\(214\) 0.381966 0.0261107
\(215\) −7.70820 −0.525695
\(216\) −2.23607 −0.152145
\(217\) −0.763932 −0.0518591
\(218\) 11.7082 0.792980
\(219\) 13.4721 0.910363
\(220\) 0.618034 0.0416678
\(221\) −4.18034 −0.281200
\(222\) −11.3262 −0.760167
\(223\) −6.00000 −0.401790 −0.200895 0.979613i \(-0.564385\pi\)
−0.200895 + 0.979613i \(0.564385\pi\)
\(224\) −3.38197 −0.225967
\(225\) −4.00000 −0.266667
\(226\) 13.7082 0.911856
\(227\) −2.00000 −0.132745 −0.0663723 0.997795i \(-0.521143\pi\)
−0.0663723 + 0.997795i \(0.521143\pi\)
\(228\) 4.14590 0.274569
\(229\) 2.76393 0.182646 0.0913229 0.995821i \(-0.470890\pi\)
0.0913229 + 0.995821i \(0.470890\pi\)
\(230\) −12.4721 −0.822388
\(231\) 1.00000 0.0657952
\(232\) −11.1803 −0.734025
\(233\) 2.94427 0.192886 0.0964428 0.995339i \(-0.469254\pi\)
0.0964428 + 0.995339i \(0.469254\pi\)
\(234\) −8.85410 −0.578811
\(235\) −4.23607 −0.276331
\(236\) 6.90983 0.449792
\(237\) −5.52786 −0.359073
\(238\) 1.23607 0.0801224
\(239\) −30.1246 −1.94860 −0.974300 0.225256i \(-0.927678\pi\)
−0.974300 + 0.225256i \(0.927678\pi\)
\(240\) −4.85410 −0.313331
\(241\) 8.05573 0.518915 0.259458 0.965755i \(-0.416456\pi\)
0.259458 + 0.965755i \(0.416456\pi\)
\(242\) 1.61803 0.104011
\(243\) 1.00000 0.0641500
\(244\) 1.23607 0.0791311
\(245\) 1.00000 0.0638877
\(246\) 10.4721 0.667679
\(247\) −36.7082 −2.33569
\(248\) 1.70820 0.108471
\(249\) 11.2361 0.712057
\(250\) −14.5623 −0.921001
\(251\) −28.1246 −1.77521 −0.887605 0.460606i \(-0.847632\pi\)
−0.887605 + 0.460606i \(0.847632\pi\)
\(252\) 0.618034 0.0389325
\(253\) −7.70820 −0.484611
\(254\) 30.1803 1.89368
\(255\) 0.763932 0.0478393
\(256\) 13.5623 0.847644
\(257\) −7.00000 −0.436648 −0.218324 0.975876i \(-0.570059\pi\)
−0.218324 + 0.975876i \(0.570059\pi\)
\(258\) −12.4721 −0.776481
\(259\) −7.00000 −0.434959
\(260\) −3.38197 −0.209741
\(261\) 5.00000 0.309492
\(262\) −27.4164 −1.69379
\(263\) 14.1246 0.870961 0.435480 0.900198i \(-0.356579\pi\)
0.435480 + 0.900198i \(0.356579\pi\)
\(264\) −2.23607 −0.137620
\(265\) 10.1803 0.625373
\(266\) 10.8541 0.665508
\(267\) 4.47214 0.273690
\(268\) −8.79837 −0.537446
\(269\) 18.9443 1.15505 0.577526 0.816372i \(-0.304018\pi\)
0.577526 + 0.816372i \(0.304018\pi\)
\(270\) 1.61803 0.0984704
\(271\) 18.7082 1.13644 0.568221 0.822876i \(-0.307632\pi\)
0.568221 + 0.822876i \(0.307632\pi\)
\(272\) −3.70820 −0.224843
\(273\) −5.47214 −0.331189
\(274\) 10.1803 0.615017
\(275\) −4.00000 −0.241209
\(276\) −4.76393 −0.286755
\(277\) 2.47214 0.148536 0.0742681 0.997238i \(-0.476338\pi\)
0.0742681 + 0.997238i \(0.476338\pi\)
\(278\) 8.94427 0.536442
\(279\) −0.763932 −0.0457354
\(280\) −2.23607 −0.133631
\(281\) 2.52786 0.150800 0.0753999 0.997153i \(-0.475977\pi\)
0.0753999 + 0.997153i \(0.475977\pi\)
\(282\) −6.85410 −0.408156
\(283\) −18.2361 −1.08402 −0.542011 0.840371i \(-0.682337\pi\)
−0.542011 + 0.840371i \(0.682337\pi\)
\(284\) 4.00000 0.237356
\(285\) 6.70820 0.397360
\(286\) −8.85410 −0.523554
\(287\) 6.47214 0.382038
\(288\) −3.38197 −0.199284
\(289\) −16.4164 −0.965671
\(290\) 8.09017 0.475071
\(291\) −3.70820 −0.217379
\(292\) 8.32624 0.487256
\(293\) −16.0000 −0.934730 −0.467365 0.884064i \(-0.654797\pi\)
−0.467365 + 0.884064i \(0.654797\pi\)
\(294\) 1.61803 0.0943657
\(295\) 11.1803 0.650945
\(296\) 15.6525 0.909782
\(297\) 1.00000 0.0580259
\(298\) 8.09017 0.468651
\(299\) 42.1803 2.43935
\(300\) −2.47214 −0.142729
\(301\) −7.70820 −0.444293
\(302\) 13.2361 0.761650
\(303\) −4.18034 −0.240154
\(304\) −32.5623 −1.86758
\(305\) 2.00000 0.114520
\(306\) 1.23607 0.0706613
\(307\) −3.05573 −0.174400 −0.0871998 0.996191i \(-0.527792\pi\)
−0.0871998 + 0.996191i \(0.527792\pi\)
\(308\) 0.618034 0.0352158
\(309\) −9.41641 −0.535681
\(310\) −1.23607 −0.0702039
\(311\) −25.8885 −1.46800 −0.734002 0.679147i \(-0.762350\pi\)
−0.734002 + 0.679147i \(0.762350\pi\)
\(312\) 12.2361 0.692731
\(313\) −6.65248 −0.376020 −0.188010 0.982167i \(-0.560204\pi\)
−0.188010 + 0.982167i \(0.560204\pi\)
\(314\) 18.4721 1.04244
\(315\) 1.00000 0.0563436
\(316\) −3.41641 −0.192188
\(317\) 1.81966 0.102202 0.0511011 0.998693i \(-0.483727\pi\)
0.0511011 + 0.998693i \(0.483727\pi\)
\(318\) 16.4721 0.923712
\(319\) 5.00000 0.279946
\(320\) 4.23607 0.236803
\(321\) 0.236068 0.0131760
\(322\) −12.4721 −0.695045
\(323\) 5.12461 0.285141
\(324\) 0.618034 0.0343352
\(325\) 21.8885 1.21416
\(326\) −15.0344 −0.832681
\(327\) 7.23607 0.400155
\(328\) −14.4721 −0.799090
\(329\) −4.23607 −0.233542
\(330\) 1.61803 0.0890698
\(331\) 15.4164 0.847362 0.423681 0.905811i \(-0.360738\pi\)
0.423681 + 0.905811i \(0.360738\pi\)
\(332\) 6.94427 0.381116
\(333\) −7.00000 −0.383598
\(334\) 14.0000 0.766046
\(335\) −14.2361 −0.777799
\(336\) −4.85410 −0.264813
\(337\) 14.1803 0.772452 0.386226 0.922404i \(-0.373778\pi\)
0.386226 + 0.922404i \(0.373778\pi\)
\(338\) 27.4164 1.49126
\(339\) 8.47214 0.460143
\(340\) 0.472136 0.0256052
\(341\) −0.763932 −0.0413692
\(342\) 10.8541 0.586923
\(343\) 1.00000 0.0539949
\(344\) 17.2361 0.929307
\(345\) −7.70820 −0.414996
\(346\) −16.9443 −0.910930
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 3.09017 0.165650
\(349\) 28.4164 1.52110 0.760548 0.649282i \(-0.224931\pi\)
0.760548 + 0.649282i \(0.224931\pi\)
\(350\) −6.47214 −0.345950
\(351\) −5.47214 −0.292081
\(352\) −3.38197 −0.180259
\(353\) 33.4721 1.78154 0.890771 0.454452i \(-0.150165\pi\)
0.890771 + 0.454452i \(0.150165\pi\)
\(354\) 18.0902 0.961482
\(355\) 6.47214 0.343505
\(356\) 2.76393 0.146488
\(357\) 0.763932 0.0404316
\(358\) −14.4721 −0.764876
\(359\) −3.41641 −0.180311 −0.0901556 0.995928i \(-0.528736\pi\)
−0.0901556 + 0.995928i \(0.528736\pi\)
\(360\) −2.23607 −0.117851
\(361\) 26.0000 1.36842
\(362\) −8.47214 −0.445286
\(363\) 1.00000 0.0524864
\(364\) −3.38197 −0.177263
\(365\) 13.4721 0.705164
\(366\) 3.23607 0.169152
\(367\) 15.8885 0.829375 0.414688 0.909964i \(-0.363891\pi\)
0.414688 + 0.909964i \(0.363891\pi\)
\(368\) 37.4164 1.95047
\(369\) 6.47214 0.336926
\(370\) −11.3262 −0.588823
\(371\) 10.1803 0.528537
\(372\) −0.472136 −0.0244791
\(373\) −26.6525 −1.38001 −0.690006 0.723803i \(-0.742392\pi\)
−0.690006 + 0.723803i \(0.742392\pi\)
\(374\) 1.23607 0.0639156
\(375\) −9.00000 −0.464758
\(376\) 9.47214 0.488488
\(377\) −27.3607 −1.40915
\(378\) 1.61803 0.0832227
\(379\) −8.81966 −0.453036 −0.226518 0.974007i \(-0.572734\pi\)
−0.226518 + 0.974007i \(0.572734\pi\)
\(380\) 4.14590 0.212680
\(381\) 18.6525 0.955595
\(382\) −24.6525 −1.26133
\(383\) 15.0557 0.769312 0.384656 0.923060i \(-0.374320\pi\)
0.384656 + 0.923060i \(0.374320\pi\)
\(384\) 13.6180 0.694942
\(385\) 1.00000 0.0509647
\(386\) −26.9443 −1.37143
\(387\) −7.70820 −0.391830
\(388\) −2.29180 −0.116348
\(389\) 28.9443 1.46753 0.733766 0.679402i \(-0.237761\pi\)
0.733766 + 0.679402i \(0.237761\pi\)
\(390\) −8.85410 −0.448345
\(391\) −5.88854 −0.297796
\(392\) −2.23607 −0.112938
\(393\) −16.9443 −0.854725
\(394\) −12.1803 −0.613637
\(395\) −5.52786 −0.278137
\(396\) 0.618034 0.0310574
\(397\) −17.1246 −0.859460 −0.429730 0.902958i \(-0.641391\pi\)
−0.429730 + 0.902958i \(0.641391\pi\)
\(398\) −42.3607 −2.12335
\(399\) 6.70820 0.335830
\(400\) 19.4164 0.970820
\(401\) −16.2918 −0.813573 −0.406787 0.913523i \(-0.633351\pi\)
−0.406787 + 0.913523i \(0.633351\pi\)
\(402\) −23.0344 −1.14885
\(403\) 4.18034 0.208238
\(404\) −2.58359 −0.128539
\(405\) 1.00000 0.0496904
\(406\) 8.09017 0.401508
\(407\) −7.00000 −0.346977
\(408\) −1.70820 −0.0845687
\(409\) 38.9443 1.92567 0.962835 0.270090i \(-0.0870534\pi\)
0.962835 + 0.270090i \(0.0870534\pi\)
\(410\) 10.4721 0.517182
\(411\) 6.29180 0.310351
\(412\) −5.81966 −0.286714
\(413\) 11.1803 0.550149
\(414\) −12.4721 −0.612972
\(415\) 11.2361 0.551557
\(416\) 18.5066 0.907360
\(417\) 5.52786 0.270701
\(418\) 10.8541 0.530891
\(419\) 21.1803 1.03473 0.517364 0.855766i \(-0.326913\pi\)
0.517364 + 0.855766i \(0.326913\pi\)
\(420\) 0.618034 0.0301570
\(421\) −13.0000 −0.633581 −0.316791 0.948495i \(-0.602605\pi\)
−0.316791 + 0.948495i \(0.602605\pi\)
\(422\) −34.6525 −1.68686
\(423\) −4.23607 −0.205965
\(424\) −22.7639 −1.10551
\(425\) −3.05573 −0.148225
\(426\) 10.4721 0.507377
\(427\) 2.00000 0.0967868
\(428\) 0.145898 0.00705225
\(429\) −5.47214 −0.264197
\(430\) −12.4721 −0.601460
\(431\) −4.70820 −0.226786 −0.113393 0.993550i \(-0.536172\pi\)
−0.113393 + 0.993550i \(0.536172\pi\)
\(432\) −4.85410 −0.233543
\(433\) −1.52786 −0.0734245 −0.0367122 0.999326i \(-0.511688\pi\)
−0.0367122 + 0.999326i \(0.511688\pi\)
\(434\) −1.23607 −0.0593332
\(435\) 5.00000 0.239732
\(436\) 4.47214 0.214176
\(437\) −51.7082 −2.47354
\(438\) 21.7984 1.04157
\(439\) −11.1803 −0.533609 −0.266804 0.963751i \(-0.585968\pi\)
−0.266804 + 0.963751i \(0.585968\pi\)
\(440\) −2.23607 −0.106600
\(441\) 1.00000 0.0476190
\(442\) −6.76393 −0.321727
\(443\) 9.52786 0.452682 0.226341 0.974048i \(-0.427324\pi\)
0.226341 + 0.974048i \(0.427324\pi\)
\(444\) −4.32624 −0.205314
\(445\) 4.47214 0.212000
\(446\) −9.70820 −0.459697
\(447\) 5.00000 0.236492
\(448\) 4.23607 0.200135
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) −6.47214 −0.305099
\(451\) 6.47214 0.304761
\(452\) 5.23607 0.246284
\(453\) 8.18034 0.384346
\(454\) −3.23607 −0.151876
\(455\) −5.47214 −0.256538
\(456\) −15.0000 −0.702439
\(457\) 10.7639 0.503516 0.251758 0.967790i \(-0.418991\pi\)
0.251758 + 0.967790i \(0.418991\pi\)
\(458\) 4.47214 0.208969
\(459\) 0.763932 0.0356573
\(460\) −4.76393 −0.222119
\(461\) −28.0000 −1.30409 −0.652045 0.758180i \(-0.726089\pi\)
−0.652045 + 0.758180i \(0.726089\pi\)
\(462\) 1.61803 0.0752778
\(463\) −27.1803 −1.26318 −0.631589 0.775304i \(-0.717597\pi\)
−0.631589 + 0.775304i \(0.717597\pi\)
\(464\) −24.2705 −1.12673
\(465\) −0.763932 −0.0354265
\(466\) 4.76393 0.220685
\(467\) 6.81966 0.315576 0.157788 0.987473i \(-0.449564\pi\)
0.157788 + 0.987473i \(0.449564\pi\)
\(468\) −3.38197 −0.156331
\(469\) −14.2361 −0.657361
\(470\) −6.85410 −0.316156
\(471\) 11.4164 0.526040
\(472\) −25.0000 −1.15072
\(473\) −7.70820 −0.354424
\(474\) −8.94427 −0.410824
\(475\) −26.8328 −1.23117
\(476\) 0.472136 0.0216403
\(477\) 10.1803 0.466126
\(478\) −48.7426 −2.22944
\(479\) −1.70820 −0.0780498 −0.0390249 0.999238i \(-0.512425\pi\)
−0.0390249 + 0.999238i \(0.512425\pi\)
\(480\) −3.38197 −0.154365
\(481\) 38.3050 1.74656
\(482\) 13.0344 0.593703
\(483\) −7.70820 −0.350735
\(484\) 0.618034 0.0280925
\(485\) −3.70820 −0.168381
\(486\) 1.61803 0.0733955
\(487\) −0.944272 −0.0427890 −0.0213945 0.999771i \(-0.506811\pi\)
−0.0213945 + 0.999771i \(0.506811\pi\)
\(488\) −4.47214 −0.202444
\(489\) −9.29180 −0.420190
\(490\) 1.61803 0.0730953
\(491\) 22.1246 0.998470 0.499235 0.866467i \(-0.333614\pi\)
0.499235 + 0.866467i \(0.333614\pi\)
\(492\) 4.00000 0.180334
\(493\) 3.81966 0.172029
\(494\) −59.3951 −2.67231
\(495\) 1.00000 0.0449467
\(496\) 3.70820 0.166503
\(497\) 6.47214 0.290315
\(498\) 18.1803 0.814681
\(499\) 2.23607 0.100100 0.0500501 0.998747i \(-0.484062\pi\)
0.0500501 + 0.998747i \(0.484062\pi\)
\(500\) −5.56231 −0.248754
\(501\) 8.65248 0.386564
\(502\) −45.5066 −2.03106
\(503\) 15.7082 0.700394 0.350197 0.936676i \(-0.386115\pi\)
0.350197 + 0.936676i \(0.386115\pi\)
\(504\) −2.23607 −0.0996024
\(505\) −4.18034 −0.186023
\(506\) −12.4721 −0.554454
\(507\) 16.9443 0.752522
\(508\) 11.5279 0.511466
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 1.23607 0.0547340
\(511\) 13.4721 0.595972
\(512\) −5.29180 −0.233867
\(513\) 6.70820 0.296174
\(514\) −11.3262 −0.499579
\(515\) −9.41641 −0.414937
\(516\) −4.76393 −0.209720
\(517\) −4.23607 −0.186302
\(518\) −11.3262 −0.497646
\(519\) −10.4721 −0.459676
\(520\) 12.2361 0.536587
\(521\) −24.3050 −1.06482 −0.532410 0.846487i \(-0.678713\pi\)
−0.532410 + 0.846487i \(0.678713\pi\)
\(522\) 8.09017 0.354097
\(523\) 9.65248 0.422073 0.211037 0.977478i \(-0.432316\pi\)
0.211037 + 0.977478i \(0.432316\pi\)
\(524\) −10.4721 −0.457477
\(525\) −4.00000 −0.174574
\(526\) 22.8541 0.996486
\(527\) −0.583592 −0.0254217
\(528\) −4.85410 −0.211248
\(529\) 36.4164 1.58332
\(530\) 16.4721 0.715504
\(531\) 11.1803 0.485185
\(532\) 4.14590 0.179747
\(533\) −35.4164 −1.53405
\(534\) 7.23607 0.313135
\(535\) 0.236068 0.0102061
\(536\) 31.8328 1.37497
\(537\) −8.94427 −0.385974
\(538\) 30.6525 1.32152
\(539\) 1.00000 0.0430730
\(540\) 0.618034 0.0265959
\(541\) −19.0557 −0.819270 −0.409635 0.912250i \(-0.634344\pi\)
−0.409635 + 0.912250i \(0.634344\pi\)
\(542\) 30.2705 1.30023
\(543\) −5.23607 −0.224701
\(544\) −2.58359 −0.110771
\(545\) 7.23607 0.309959
\(546\) −8.85410 −0.378921
\(547\) −38.8328 −1.66037 −0.830186 0.557487i \(-0.811766\pi\)
−0.830186 + 0.557487i \(0.811766\pi\)
\(548\) 3.88854 0.166110
\(549\) 2.00000 0.0853579
\(550\) −6.47214 −0.275973
\(551\) 33.5410 1.42890
\(552\) 17.2361 0.733616
\(553\) −5.52786 −0.235069
\(554\) 4.00000 0.169944
\(555\) −7.00000 −0.297133
\(556\) 3.41641 0.144888
\(557\) −21.4721 −0.909804 −0.454902 0.890542i \(-0.650326\pi\)
−0.454902 + 0.890542i \(0.650326\pi\)
\(558\) −1.23607 −0.0523269
\(559\) 42.1803 1.78404
\(560\) −4.85410 −0.205123
\(561\) 0.763932 0.0322532
\(562\) 4.09017 0.172533
\(563\) 3.34752 0.141081 0.0705407 0.997509i \(-0.477528\pi\)
0.0705407 + 0.997509i \(0.477528\pi\)
\(564\) −2.61803 −0.110239
\(565\) 8.47214 0.356425
\(566\) −29.5066 −1.24025
\(567\) 1.00000 0.0419961
\(568\) −14.4721 −0.607237
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 10.8541 0.454628
\(571\) 26.4721 1.10782 0.553912 0.832575i \(-0.313134\pi\)
0.553912 + 0.832575i \(0.313134\pi\)
\(572\) −3.38197 −0.141407
\(573\) −15.2361 −0.636496
\(574\) 10.4721 0.437099
\(575\) 30.8328 1.28582
\(576\) 4.23607 0.176503
\(577\) 38.6525 1.60912 0.804562 0.593869i \(-0.202400\pi\)
0.804562 + 0.593869i \(0.202400\pi\)
\(578\) −26.5623 −1.10485
\(579\) −16.6525 −0.692053
\(580\) 3.09017 0.128312
\(581\) 11.2361 0.466151
\(582\) −6.00000 −0.248708
\(583\) 10.1803 0.421627
\(584\) −30.1246 −1.24657
\(585\) −5.47214 −0.226245
\(586\) −25.8885 −1.06945
\(587\) −30.0132 −1.23878 −0.619388 0.785085i \(-0.712619\pi\)
−0.619388 + 0.785085i \(0.712619\pi\)
\(588\) 0.618034 0.0254873
\(589\) −5.12461 −0.211156
\(590\) 18.0902 0.744761
\(591\) −7.52786 −0.309655
\(592\) 33.9787 1.39652
\(593\) 30.8328 1.26615 0.633076 0.774090i \(-0.281792\pi\)
0.633076 + 0.774090i \(0.281792\pi\)
\(594\) 1.61803 0.0663887
\(595\) 0.763932 0.0313182
\(596\) 3.09017 0.126578
\(597\) −26.1803 −1.07149
\(598\) 68.2492 2.79092
\(599\) −34.4721 −1.40849 −0.704247 0.709955i \(-0.748715\pi\)
−0.704247 + 0.709955i \(0.748715\pi\)
\(600\) 8.94427 0.365148
\(601\) 27.0000 1.10135 0.550676 0.834719i \(-0.314370\pi\)
0.550676 + 0.834719i \(0.314370\pi\)
\(602\) −12.4721 −0.508326
\(603\) −14.2361 −0.579738
\(604\) 5.05573 0.205715
\(605\) 1.00000 0.0406558
\(606\) −6.76393 −0.274766
\(607\) 29.1803 1.18439 0.592197 0.805793i \(-0.298261\pi\)
0.592197 + 0.805793i \(0.298261\pi\)
\(608\) −22.6869 −0.920076
\(609\) 5.00000 0.202610
\(610\) 3.23607 0.131025
\(611\) 23.1803 0.937776
\(612\) 0.472136 0.0190850
\(613\) −35.5967 −1.43774 −0.718870 0.695145i \(-0.755340\pi\)
−0.718870 + 0.695145i \(0.755340\pi\)
\(614\) −4.94427 −0.199535
\(615\) 6.47214 0.260982
\(616\) −2.23607 −0.0900937
\(617\) 23.5279 0.947196 0.473598 0.880741i \(-0.342955\pi\)
0.473598 + 0.880741i \(0.342955\pi\)
\(618\) −15.2361 −0.612885
\(619\) 14.0689 0.565476 0.282738 0.959197i \(-0.408757\pi\)
0.282738 + 0.959197i \(0.408757\pi\)
\(620\) −0.472136 −0.0189614
\(621\) −7.70820 −0.309320
\(622\) −41.8885 −1.67958
\(623\) 4.47214 0.179172
\(624\) 26.5623 1.06334
\(625\) 11.0000 0.440000
\(626\) −10.7639 −0.430213
\(627\) 6.70820 0.267900
\(628\) 7.05573 0.281554
\(629\) −5.34752 −0.213220
\(630\) 1.61803 0.0644640
\(631\) −0.360680 −0.0143584 −0.00717922 0.999974i \(-0.502285\pi\)
−0.00717922 + 0.999974i \(0.502285\pi\)
\(632\) 12.3607 0.491681
\(633\) −21.4164 −0.851226
\(634\) 2.94427 0.116932
\(635\) 18.6525 0.740201
\(636\) 6.29180 0.249486
\(637\) −5.47214 −0.216814
\(638\) 8.09017 0.320293
\(639\) 6.47214 0.256034
\(640\) 13.6180 0.538300
\(641\) 20.5410 0.811321 0.405661 0.914024i \(-0.367041\pi\)
0.405661 + 0.914024i \(0.367041\pi\)
\(642\) 0.381966 0.0150750
\(643\) 45.9574 1.81238 0.906192 0.422866i \(-0.138976\pi\)
0.906192 + 0.422866i \(0.138976\pi\)
\(644\) −4.76393 −0.187725
\(645\) −7.70820 −0.303510
\(646\) 8.29180 0.326236
\(647\) 43.6525 1.71616 0.858078 0.513519i \(-0.171659\pi\)
0.858078 + 0.513519i \(0.171659\pi\)
\(648\) −2.23607 −0.0878410
\(649\) 11.1803 0.438867
\(650\) 35.4164 1.38915
\(651\) −0.763932 −0.0299409
\(652\) −5.74265 −0.224899
\(653\) −27.0557 −1.05877 −0.529386 0.848381i \(-0.677578\pi\)
−0.529386 + 0.848381i \(0.677578\pi\)
\(654\) 11.7082 0.457827
\(655\) −16.9443 −0.662067
\(656\) −31.4164 −1.22660
\(657\) 13.4721 0.525598
\(658\) −6.85410 −0.267201
\(659\) −43.5410 −1.69612 −0.848059 0.529902i \(-0.822229\pi\)
−0.848059 + 0.529902i \(0.822229\pi\)
\(660\) 0.618034 0.0240569
\(661\) −26.5410 −1.03233 −0.516163 0.856490i \(-0.672640\pi\)
−0.516163 + 0.856490i \(0.672640\pi\)
\(662\) 24.9443 0.969487
\(663\) −4.18034 −0.162351
\(664\) −25.1246 −0.975024
\(665\) 6.70820 0.260133
\(666\) −11.3262 −0.438883
\(667\) −38.5410 −1.49231
\(668\) 5.34752 0.206902
\(669\) −6.00000 −0.231973
\(670\) −23.0344 −0.889898
\(671\) 2.00000 0.0772091
\(672\) −3.38197 −0.130462
\(673\) −45.5967 −1.75763 −0.878813 0.477167i \(-0.841664\pi\)
−0.878813 + 0.477167i \(0.841664\pi\)
\(674\) 22.9443 0.883780
\(675\) −4.00000 −0.153960
\(676\) 10.4721 0.402774
\(677\) −43.3050 −1.66434 −0.832172 0.554517i \(-0.812903\pi\)
−0.832172 + 0.554517i \(0.812903\pi\)
\(678\) 13.7082 0.526460
\(679\) −3.70820 −0.142308
\(680\) −1.70820 −0.0655066
\(681\) −2.00000 −0.0766402
\(682\) −1.23607 −0.0473315
\(683\) −39.0132 −1.49280 −0.746398 0.665499i \(-0.768219\pi\)
−0.746398 + 0.665499i \(0.768219\pi\)
\(684\) 4.14590 0.158522
\(685\) 6.29180 0.240397
\(686\) 1.61803 0.0617768
\(687\) 2.76393 0.105451
\(688\) 37.4164 1.42649
\(689\) −55.7082 −2.12231
\(690\) −12.4721 −0.474806
\(691\) 2.00000 0.0760836 0.0380418 0.999276i \(-0.487888\pi\)
0.0380418 + 0.999276i \(0.487888\pi\)
\(692\) −6.47214 −0.246034
\(693\) 1.00000 0.0379869
\(694\) 45.3050 1.71975
\(695\) 5.52786 0.209684
\(696\) −11.1803 −0.423790
\(697\) 4.94427 0.187278
\(698\) 45.9787 1.74032
\(699\) 2.94427 0.111363
\(700\) −2.47214 −0.0934380
\(701\) 39.8885 1.50657 0.753285 0.657695i \(-0.228468\pi\)
0.753285 + 0.657695i \(0.228468\pi\)
\(702\) −8.85410 −0.334177
\(703\) −46.9574 −1.77103
\(704\) 4.23607 0.159653
\(705\) −4.23607 −0.159540
\(706\) 54.1591 2.03830
\(707\) −4.18034 −0.157218
\(708\) 6.90983 0.259687
\(709\) 39.7214 1.49177 0.745883 0.666076i \(-0.232028\pi\)
0.745883 + 0.666076i \(0.232028\pi\)
\(710\) 10.4721 0.393012
\(711\) −5.52786 −0.207311
\(712\) −10.0000 −0.374766
\(713\) 5.88854 0.220528
\(714\) 1.23607 0.0462587
\(715\) −5.47214 −0.204646
\(716\) −5.52786 −0.206586
\(717\) −30.1246 −1.12502
\(718\) −5.52786 −0.206298
\(719\) 12.2361 0.456328 0.228164 0.973623i \(-0.426728\pi\)
0.228164 + 0.973623i \(0.426728\pi\)
\(720\) −4.85410 −0.180902
\(721\) −9.41641 −0.350685
\(722\) 42.0689 1.56564
\(723\) 8.05573 0.299596
\(724\) −3.23607 −0.120268
\(725\) −20.0000 −0.742781
\(726\) 1.61803 0.0600509
\(727\) 1.81966 0.0674875 0.0337437 0.999431i \(-0.489257\pi\)
0.0337437 + 0.999431i \(0.489257\pi\)
\(728\) 12.2361 0.453499
\(729\) 1.00000 0.0370370
\(730\) 21.7984 0.806794
\(731\) −5.88854 −0.217796
\(732\) 1.23607 0.0456864
\(733\) −43.8885 −1.62106 −0.810530 0.585697i \(-0.800821\pi\)
−0.810530 + 0.585697i \(0.800821\pi\)
\(734\) 25.7082 0.948907
\(735\) 1.00000 0.0368856
\(736\) 26.0689 0.960912
\(737\) −14.2361 −0.524392
\(738\) 10.4721 0.385485
\(739\) −24.0689 −0.885388 −0.442694 0.896673i \(-0.645977\pi\)
−0.442694 + 0.896673i \(0.645977\pi\)
\(740\) −4.32624 −0.159036
\(741\) −36.7082 −1.34851
\(742\) 16.4721 0.604711
\(743\) 25.1803 0.923777 0.461889 0.886938i \(-0.347172\pi\)
0.461889 + 0.886938i \(0.347172\pi\)
\(744\) 1.70820 0.0626258
\(745\) 5.00000 0.183186
\(746\) −43.1246 −1.57890
\(747\) 11.2361 0.411106
\(748\) 0.472136 0.0172630
\(749\) 0.236068 0.00862574
\(750\) −14.5623 −0.531740
\(751\) 44.2361 1.61420 0.807099 0.590417i \(-0.201037\pi\)
0.807099 + 0.590417i \(0.201037\pi\)
\(752\) 20.5623 0.749830
\(753\) −28.1246 −1.02492
\(754\) −44.2705 −1.61224
\(755\) 8.18034 0.297713
\(756\) 0.618034 0.0224777
\(757\) 37.7214 1.37101 0.685503 0.728070i \(-0.259582\pi\)
0.685503 + 0.728070i \(0.259582\pi\)
\(758\) −14.2705 −0.518328
\(759\) −7.70820 −0.279790
\(760\) −15.0000 −0.544107
\(761\) −43.7771 −1.58692 −0.793459 0.608624i \(-0.791722\pi\)
−0.793459 + 0.608624i \(0.791722\pi\)
\(762\) 30.1803 1.09332
\(763\) 7.23607 0.261963
\(764\) −9.41641 −0.340674
\(765\) 0.763932 0.0276200
\(766\) 24.3607 0.880187
\(767\) −61.1803 −2.20909
\(768\) 13.5623 0.489388
\(769\) −3.94427 −0.142234 −0.0711170 0.997468i \(-0.522656\pi\)
−0.0711170 + 0.997468i \(0.522656\pi\)
\(770\) 1.61803 0.0583099
\(771\) −7.00000 −0.252099
\(772\) −10.2918 −0.370410
\(773\) 3.47214 0.124884 0.0624420 0.998049i \(-0.480111\pi\)
0.0624420 + 0.998049i \(0.480111\pi\)
\(774\) −12.4721 −0.448302
\(775\) 3.05573 0.109765
\(776\) 8.29180 0.297658
\(777\) −7.00000 −0.251124
\(778\) 46.8328 1.67904
\(779\) 43.4164 1.55555
\(780\) −3.38197 −0.121094
\(781\) 6.47214 0.231591
\(782\) −9.52786 −0.340716
\(783\) 5.00000 0.178685
\(784\) −4.85410 −0.173361
\(785\) 11.4164 0.407469
\(786\) −27.4164 −0.977911
\(787\) −27.6525 −0.985704 −0.492852 0.870113i \(-0.664046\pi\)
−0.492852 + 0.870113i \(0.664046\pi\)
\(788\) −4.65248 −0.165738
\(789\) 14.1246 0.502849
\(790\) −8.94427 −0.318223
\(791\) 8.47214 0.301234
\(792\) −2.23607 −0.0794552
\(793\) −10.9443 −0.388642
\(794\) −27.7082 −0.983327
\(795\) 10.1803 0.361059
\(796\) −16.1803 −0.573497
\(797\) −11.4721 −0.406364 −0.203182 0.979141i \(-0.565128\pi\)
−0.203182 + 0.979141i \(0.565128\pi\)
\(798\) 10.8541 0.384231
\(799\) −3.23607 −0.114484
\(800\) 13.5279 0.478282
\(801\) 4.47214 0.158015
\(802\) −26.3607 −0.930828
\(803\) 13.4721 0.475421
\(804\) −8.79837 −0.310295
\(805\) −7.70820 −0.271678
\(806\) 6.76393 0.238249
\(807\) 18.9443 0.666870
\(808\) 9.34752 0.328845
\(809\) 6.30495 0.221670 0.110835 0.993839i \(-0.464647\pi\)
0.110835 + 0.993839i \(0.464647\pi\)
\(810\) 1.61803 0.0568519
\(811\) 28.7082 1.00808 0.504041 0.863680i \(-0.331846\pi\)
0.504041 + 0.863680i \(0.331846\pi\)
\(812\) 3.09017 0.108444
\(813\) 18.7082 0.656125
\(814\) −11.3262 −0.396984
\(815\) −9.29180 −0.325477
\(816\) −3.70820 −0.129813
\(817\) −51.7082 −1.80904
\(818\) 63.0132 2.20320
\(819\) −5.47214 −0.191212
\(820\) 4.00000 0.139686
\(821\) 1.47214 0.0513779 0.0256889 0.999670i \(-0.491822\pi\)
0.0256889 + 0.999670i \(0.491822\pi\)
\(822\) 10.1803 0.355080
\(823\) 5.18034 0.180575 0.0902876 0.995916i \(-0.471221\pi\)
0.0902876 + 0.995916i \(0.471221\pi\)
\(824\) 21.0557 0.733511
\(825\) −4.00000 −0.139262
\(826\) 18.0902 0.629438
\(827\) 43.6525 1.51795 0.758973 0.651122i \(-0.225702\pi\)
0.758973 + 0.651122i \(0.225702\pi\)
\(828\) −4.76393 −0.165558
\(829\) −35.7771 −1.24259 −0.621295 0.783577i \(-0.713393\pi\)
−0.621295 + 0.783577i \(0.713393\pi\)
\(830\) 18.1803 0.631049
\(831\) 2.47214 0.0857574
\(832\) −23.1803 −0.803634
\(833\) 0.763932 0.0264687
\(834\) 8.94427 0.309715
\(835\) 8.65248 0.299431
\(836\) 4.14590 0.143389
\(837\) −0.763932 −0.0264054
\(838\) 34.2705 1.18386
\(839\) 43.5410 1.50320 0.751601 0.659617i \(-0.229282\pi\)
0.751601 + 0.659617i \(0.229282\pi\)
\(840\) −2.23607 −0.0771517
\(841\) −4.00000 −0.137931
\(842\) −21.0344 −0.724895
\(843\) 2.52786 0.0870643
\(844\) −13.2361 −0.455604
\(845\) 16.9443 0.582901
\(846\) −6.85410 −0.235649
\(847\) 1.00000 0.0343604
\(848\) −49.4164 −1.69697
\(849\) −18.2361 −0.625860
\(850\) −4.94427 −0.169587
\(851\) 53.9574 1.84964
\(852\) 4.00000 0.137038
\(853\) −2.58359 −0.0884605 −0.0442303 0.999021i \(-0.514084\pi\)
−0.0442303 + 0.999021i \(0.514084\pi\)
\(854\) 3.23607 0.110736
\(855\) 6.70820 0.229416
\(856\) −0.527864 −0.0180420
\(857\) 35.8885 1.22593 0.612965 0.790110i \(-0.289977\pi\)
0.612965 + 0.790110i \(0.289977\pi\)
\(858\) −8.85410 −0.302274
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) −4.76393 −0.162449
\(861\) 6.47214 0.220570
\(862\) −7.61803 −0.259471
\(863\) −38.7639 −1.31954 −0.659770 0.751468i \(-0.729346\pi\)
−0.659770 + 0.751468i \(0.729346\pi\)
\(864\) −3.38197 −0.115057
\(865\) −10.4721 −0.356063
\(866\) −2.47214 −0.0840066
\(867\) −16.4164 −0.557530
\(868\) −0.472136 −0.0160253
\(869\) −5.52786 −0.187520
\(870\) 8.09017 0.274282
\(871\) 77.9017 2.63960
\(872\) −16.1803 −0.547935
\(873\) −3.70820 −0.125504
\(874\) −83.6656 −2.83003
\(875\) −9.00000 −0.304256
\(876\) 8.32624 0.281318
\(877\) 31.4164 1.06086 0.530428 0.847730i \(-0.322031\pi\)
0.530428 + 0.847730i \(0.322031\pi\)
\(878\) −18.0902 −0.610514
\(879\) −16.0000 −0.539667
\(880\) −4.85410 −0.163632
\(881\) 19.1115 0.643881 0.321941 0.946760i \(-0.395665\pi\)
0.321941 + 0.946760i \(0.395665\pi\)
\(882\) 1.61803 0.0544820
\(883\) −37.1803 −1.25122 −0.625609 0.780137i \(-0.715149\pi\)
−0.625609 + 0.780137i \(0.715149\pi\)
\(884\) −2.58359 −0.0868956
\(885\) 11.1803 0.375823
\(886\) 15.4164 0.517924
\(887\) 15.2361 0.511577 0.255789 0.966733i \(-0.417665\pi\)
0.255789 + 0.966733i \(0.417665\pi\)
\(888\) 15.6525 0.525263
\(889\) 18.6525 0.625584
\(890\) 7.23607 0.242554
\(891\) 1.00000 0.0335013
\(892\) −3.70820 −0.124160
\(893\) −28.4164 −0.950919
\(894\) 8.09017 0.270576
\(895\) −8.94427 −0.298974
\(896\) 13.6180 0.454947
\(897\) 42.1803 1.40836
\(898\) 32.3607 1.07989
\(899\) −3.81966 −0.127393
\(900\) −2.47214 −0.0824045
\(901\) 7.77709 0.259092
\(902\) 10.4721 0.348684
\(903\) −7.70820 −0.256513
\(904\) −18.9443 −0.630077
\(905\) −5.23607 −0.174053
\(906\) 13.2361 0.439739
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) −1.23607 −0.0410204
\(909\) −4.18034 −0.138653
\(910\) −8.85410 −0.293511
\(911\) −41.4164 −1.37219 −0.686093 0.727513i \(-0.740676\pi\)
−0.686093 + 0.727513i \(0.740676\pi\)
\(912\) −32.5623 −1.07825
\(913\) 11.2361 0.371860
\(914\) 17.4164 0.576084
\(915\) 2.00000 0.0661180
\(916\) 1.70820 0.0564406
\(917\) −16.9443 −0.559549
\(918\) 1.23607 0.0407963
\(919\) −9.59675 −0.316567 −0.158284 0.987394i \(-0.550596\pi\)
−0.158284 + 0.987394i \(0.550596\pi\)
\(920\) 17.2361 0.568256
\(921\) −3.05573 −0.100690
\(922\) −45.3050 −1.49204
\(923\) −35.4164 −1.16575
\(924\) 0.618034 0.0203318
\(925\) 28.0000 0.920634
\(926\) −43.9787 −1.44523
\(927\) −9.41641 −0.309275
\(928\) −16.9098 −0.555092
\(929\) −22.8885 −0.750949 −0.375474 0.926833i \(-0.622520\pi\)
−0.375474 + 0.926833i \(0.622520\pi\)
\(930\) −1.23607 −0.0405323
\(931\) 6.70820 0.219853
\(932\) 1.81966 0.0596049
\(933\) −25.8885 −0.847553
\(934\) 11.0344 0.361058
\(935\) 0.763932 0.0249832
\(936\) 12.2361 0.399948
\(937\) −4.11146 −0.134315 −0.0671577 0.997742i \(-0.521393\pi\)
−0.0671577 + 0.997742i \(0.521393\pi\)
\(938\) −23.0344 −0.752101
\(939\) −6.65248 −0.217095
\(940\) −2.61803 −0.0853909
\(941\) −15.6393 −0.509827 −0.254914 0.966964i \(-0.582047\pi\)
−0.254914 + 0.966964i \(0.582047\pi\)
\(942\) 18.4721 0.601855
\(943\) −49.8885 −1.62459
\(944\) −54.2705 −1.76635
\(945\) 1.00000 0.0325300
\(946\) −12.4721 −0.405504
\(947\) 21.4164 0.695940 0.347970 0.937506i \(-0.386871\pi\)
0.347970 + 0.937506i \(0.386871\pi\)
\(948\) −3.41641 −0.110960
\(949\) −73.7214 −2.39310
\(950\) −43.4164 −1.40861
\(951\) 1.81966 0.0590065
\(952\) −1.70820 −0.0553632
\(953\) −26.7771 −0.867395 −0.433697 0.901059i \(-0.642791\pi\)
−0.433697 + 0.901059i \(0.642791\pi\)
\(954\) 16.4721 0.533305
\(955\) −15.2361 −0.493028
\(956\) −18.6180 −0.602150
\(957\) 5.00000 0.161627
\(958\) −2.76393 −0.0892986
\(959\) 6.29180 0.203173
\(960\) 4.23607 0.136719
\(961\) −30.4164 −0.981174
\(962\) 61.9787 1.99827
\(963\) 0.236068 0.00760718
\(964\) 4.97871 0.160354
\(965\) −16.6525 −0.536062
\(966\) −12.4721 −0.401284
\(967\) 37.1935 1.19606 0.598031 0.801473i \(-0.295950\pi\)
0.598031 + 0.801473i \(0.295950\pi\)
\(968\) −2.23607 −0.0718699
\(969\) 5.12461 0.164626
\(970\) −6.00000 −0.192648
\(971\) −8.12461 −0.260731 −0.130366 0.991466i \(-0.541615\pi\)
−0.130366 + 0.991466i \(0.541615\pi\)
\(972\) 0.618034 0.0198234
\(973\) 5.52786 0.177215
\(974\) −1.52786 −0.0489559
\(975\) 21.8885 0.700994
\(976\) −9.70820 −0.310752
\(977\) −18.1803 −0.581641 −0.290820 0.956778i \(-0.593928\pi\)
−0.290820 + 0.956778i \(0.593928\pi\)
\(978\) −15.0344 −0.480748
\(979\) 4.47214 0.142930
\(980\) 0.618034 0.0197424
\(981\) 7.23607 0.231030
\(982\) 35.7984 1.14237
\(983\) 0.583592 0.0186137 0.00930685 0.999957i \(-0.497037\pi\)
0.00930685 + 0.999957i \(0.497037\pi\)
\(984\) −14.4721 −0.461355
\(985\) −7.52786 −0.239858
\(986\) 6.18034 0.196822
\(987\) −4.23607 −0.134836
\(988\) −22.6869 −0.721767
\(989\) 59.4164 1.88933
\(990\) 1.61803 0.0514245
\(991\) −6.81966 −0.216634 −0.108317 0.994116i \(-0.534546\pi\)
−0.108317 + 0.994116i \(0.534546\pi\)
\(992\) 2.58359 0.0820291
\(993\) 15.4164 0.489225
\(994\) 10.4721 0.332156
\(995\) −26.1803 −0.829973
\(996\) 6.94427 0.220038
\(997\) 9.05573 0.286798 0.143399 0.989665i \(-0.454197\pi\)
0.143399 + 0.989665i \(0.454197\pi\)
\(998\) 3.61803 0.114527
\(999\) −7.00000 −0.221470
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 231.2.a.c.1.2 2
3.2 odd 2 693.2.a.f.1.1 2
4.3 odd 2 3696.2.a.be.1.1 2
5.4 even 2 5775.2.a.be.1.1 2
7.6 odd 2 1617.2.a.p.1.2 2
11.10 odd 2 2541.2.a.t.1.1 2
21.20 even 2 4851.2.a.w.1.1 2
33.32 even 2 7623.2.a.bm.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.c.1.2 2 1.1 even 1 trivial
693.2.a.f.1.1 2 3.2 odd 2
1617.2.a.p.1.2 2 7.6 odd 2
2541.2.a.t.1.1 2 11.10 odd 2
3696.2.a.be.1.1 2 4.3 odd 2
4851.2.a.w.1.1 2 21.20 even 2
5775.2.a.be.1.1 2 5.4 even 2
7623.2.a.bm.1.2 2 33.32 even 2