Properties

Label 231.2.a.c.1.1
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 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 231.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +1.00000 q^{5} -0.618034 q^{6} +1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +1.00000 q^{5} -0.618034 q^{6} +1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} -0.618034 q^{10} +1.00000 q^{11} -1.61803 q^{12} +3.47214 q^{13} -0.618034 q^{14} +1.00000 q^{15} +1.85410 q^{16} +5.23607 q^{17} -0.618034 q^{18} -6.70820 q^{19} -1.61803 q^{20} +1.00000 q^{21} -0.618034 q^{22} +5.70820 q^{23} +2.23607 q^{24} -4.00000 q^{25} -2.14590 q^{26} +1.00000 q^{27} -1.61803 q^{28} +5.00000 q^{29} -0.618034 q^{30} -5.23607 q^{31} -5.61803 q^{32} +1.00000 q^{33} -3.23607 q^{34} +1.00000 q^{35} -1.61803 q^{36} -7.00000 q^{37} +4.14590 q^{38} +3.47214 q^{39} +2.23607 q^{40} -2.47214 q^{41} -0.618034 q^{42} +5.70820 q^{43} -1.61803 q^{44} +1.00000 q^{45} -3.52786 q^{46} +0.236068 q^{47} +1.85410 q^{48} +1.00000 q^{49} +2.47214 q^{50} +5.23607 q^{51} -5.61803 q^{52} -12.1803 q^{53} -0.618034 q^{54} +1.00000 q^{55} +2.23607 q^{56} -6.70820 q^{57} -3.09017 q^{58} -11.1803 q^{59} -1.61803 q^{60} +2.00000 q^{61} +3.23607 q^{62} +1.00000 q^{63} -0.236068 q^{64} +3.47214 q^{65} -0.618034 q^{66} -9.76393 q^{67} -8.47214 q^{68} +5.70820 q^{69} -0.618034 q^{70} -2.47214 q^{71} +2.23607 q^{72} +4.52786 q^{73} +4.32624 q^{74} -4.00000 q^{75} +10.8541 q^{76} +1.00000 q^{77} -2.14590 q^{78} -14.4721 q^{79} +1.85410 q^{80} +1.00000 q^{81} +1.52786 q^{82} +6.76393 q^{83} -1.61803 q^{84} +5.23607 q^{85} -3.52786 q^{86} +5.00000 q^{87} +2.23607 q^{88} -4.47214 q^{89} -0.618034 q^{90} +3.47214 q^{91} -9.23607 q^{92} -5.23607 q^{93} -0.145898 q^{94} -6.70820 q^{95} -5.61803 q^{96} +9.70820 q^{97} -0.618034 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 2 q^{3} - q^{4} + 2 q^{5} + q^{6} + 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + q^{2} + 2 q^{3} - q^{4} + 2 q^{5} + q^{6} + 2 q^{7} + 2 q^{9} + q^{10} + 2 q^{11} - q^{12} - 2 q^{13} + q^{14} + 2 q^{15} - 3 q^{16} + 6 q^{17} + q^{18} - q^{20} + 2 q^{21} + q^{22} - 2 q^{23} - 8 q^{25} - 11 q^{26} + 2 q^{27} - q^{28} + 10 q^{29} + q^{30} - 6 q^{31} - 9 q^{32} + 2 q^{33} - 2 q^{34} + 2 q^{35} - q^{36} - 14 q^{37} + 15 q^{38} - 2 q^{39} + 4 q^{41} + q^{42} - 2 q^{43} - q^{44} + 2 q^{45} - 16 q^{46} - 4 q^{47} - 3 q^{48} + 2 q^{49} - 4 q^{50} + 6 q^{51} - 9 q^{52} - 2 q^{53} + q^{54} + 2 q^{55} + 5 q^{58} - q^{60} + 4 q^{61} + 2 q^{62} + 2 q^{63} + 4 q^{64} - 2 q^{65} + q^{66} - 24 q^{67} - 8 q^{68} - 2 q^{69} + q^{70} + 4 q^{71} + 18 q^{73} - 7 q^{74} - 8 q^{75} + 15 q^{76} + 2 q^{77} - 11 q^{78} - 20 q^{79} - 3 q^{80} + 2 q^{81} + 12 q^{82} + 18 q^{83} - q^{84} + 6 q^{85} - 16 q^{86} + 10 q^{87} + q^{90} - 2 q^{91} - 14 q^{92} - 6 q^{93} - 7 q^{94} - 9 q^{96} + 6 q^{97} + q^{98} + 2 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.618034 −0.437016 −0.218508 0.975835i \(-0.570119\pi\)
−0.218508 + 0.975835i \(0.570119\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.61803 −0.809017
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) −0.618034 −0.252311
\(7\) 1.00000 0.377964
\(8\) 2.23607 0.790569
\(9\) 1.00000 0.333333
\(10\) −0.618034 −0.195440
\(11\) 1.00000 0.301511
\(12\) −1.61803 −0.467086
\(13\) 3.47214 0.962997 0.481499 0.876447i \(-0.340093\pi\)
0.481499 + 0.876447i \(0.340093\pi\)
\(14\) −0.618034 −0.165177
\(15\) 1.00000 0.258199
\(16\) 1.85410 0.463525
\(17\) 5.23607 1.26993 0.634967 0.772540i \(-0.281014\pi\)
0.634967 + 0.772540i \(0.281014\pi\)
\(18\) −0.618034 −0.145672
\(19\) −6.70820 −1.53897 −0.769484 0.638666i \(-0.779486\pi\)
−0.769484 + 0.638666i \(0.779486\pi\)
\(20\) −1.61803 −0.361803
\(21\) 1.00000 0.218218
\(22\) −0.618034 −0.131765
\(23\) 5.70820 1.19024 0.595121 0.803636i \(-0.297104\pi\)
0.595121 + 0.803636i \(0.297104\pi\)
\(24\) 2.23607 0.456435
\(25\) −4.00000 −0.800000
\(26\) −2.14590 −0.420845
\(27\) 1.00000 0.192450
\(28\) −1.61803 −0.305780
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) −0.618034 −0.112837
\(31\) −5.23607 −0.940426 −0.470213 0.882553i \(-0.655823\pi\)
−0.470213 + 0.882553i \(0.655823\pi\)
\(32\) −5.61803 −0.993137
\(33\) 1.00000 0.174078
\(34\) −3.23607 −0.554981
\(35\) 1.00000 0.169031
\(36\) −1.61803 −0.269672
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 4.14590 0.672553
\(39\) 3.47214 0.555987
\(40\) 2.23607 0.353553
\(41\) −2.47214 −0.386083 −0.193041 0.981191i \(-0.561835\pi\)
−0.193041 + 0.981191i \(0.561835\pi\)
\(42\) −0.618034 −0.0953647
\(43\) 5.70820 0.870493 0.435246 0.900311i \(-0.356661\pi\)
0.435246 + 0.900311i \(0.356661\pi\)
\(44\) −1.61803 −0.243928
\(45\) 1.00000 0.149071
\(46\) −3.52786 −0.520155
\(47\) 0.236068 0.0344341 0.0172170 0.999852i \(-0.494519\pi\)
0.0172170 + 0.999852i \(0.494519\pi\)
\(48\) 1.85410 0.267617
\(49\) 1.00000 0.142857
\(50\) 2.47214 0.349613
\(51\) 5.23607 0.733196
\(52\) −5.61803 −0.779081
\(53\) −12.1803 −1.67310 −0.836549 0.547892i \(-0.815431\pi\)
−0.836549 + 0.547892i \(0.815431\pi\)
\(54\) −0.618034 −0.0841038
\(55\) 1.00000 0.134840
\(56\) 2.23607 0.298807
\(57\) −6.70820 −0.888523
\(58\) −3.09017 −0.405759
\(59\) −11.1803 −1.45556 −0.727778 0.685813i \(-0.759447\pi\)
−0.727778 + 0.685813i \(0.759447\pi\)
\(60\) −1.61803 −0.208887
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 3.23607 0.410981
\(63\) 1.00000 0.125988
\(64\) −0.236068 −0.0295085
\(65\) 3.47214 0.430665
\(66\) −0.618034 −0.0760747
\(67\) −9.76393 −1.19285 −0.596427 0.802667i \(-0.703414\pi\)
−0.596427 + 0.802667i \(0.703414\pi\)
\(68\) −8.47214 −1.02740
\(69\) 5.70820 0.687187
\(70\) −0.618034 −0.0738692
\(71\) −2.47214 −0.293389 −0.146694 0.989182i \(-0.546863\pi\)
−0.146694 + 0.989182i \(0.546863\pi\)
\(72\) 2.23607 0.263523
\(73\) 4.52786 0.529946 0.264973 0.964256i \(-0.414637\pi\)
0.264973 + 0.964256i \(0.414637\pi\)
\(74\) 4.32624 0.502915
\(75\) −4.00000 −0.461880
\(76\) 10.8541 1.24505
\(77\) 1.00000 0.113961
\(78\) −2.14590 −0.242975
\(79\) −14.4721 −1.62824 −0.814121 0.580695i \(-0.802781\pi\)
−0.814121 + 0.580695i \(0.802781\pi\)
\(80\) 1.85410 0.207295
\(81\) 1.00000 0.111111
\(82\) 1.52786 0.168724
\(83\) 6.76393 0.742438 0.371219 0.928545i \(-0.378940\pi\)
0.371219 + 0.928545i \(0.378940\pi\)
\(84\) −1.61803 −0.176542
\(85\) 5.23607 0.567931
\(86\) −3.52786 −0.380419
\(87\) 5.00000 0.536056
\(88\) 2.23607 0.238366
\(89\) −4.47214 −0.474045 −0.237023 0.971504i \(-0.576172\pi\)
−0.237023 + 0.971504i \(0.576172\pi\)
\(90\) −0.618034 −0.0651465
\(91\) 3.47214 0.363979
\(92\) −9.23607 −0.962927
\(93\) −5.23607 −0.542955
\(94\) −0.145898 −0.0150482
\(95\) −6.70820 −0.688247
\(96\) −5.61803 −0.573388
\(97\) 9.70820 0.985719 0.492859 0.870109i \(-0.335952\pi\)
0.492859 + 0.870109i \(0.335952\pi\)
\(98\) −0.618034 −0.0624309
\(99\) 1.00000 0.100504
\(100\) 6.47214 0.647214
\(101\) 18.1803 1.80901 0.904506 0.426461i \(-0.140240\pi\)
0.904506 + 0.426461i \(0.140240\pi\)
\(102\) −3.23607 −0.320418
\(103\) 17.4164 1.71609 0.858045 0.513575i \(-0.171679\pi\)
0.858045 + 0.513575i \(0.171679\pi\)
\(104\) 7.76393 0.761316
\(105\) 1.00000 0.0975900
\(106\) 7.52786 0.731171
\(107\) −4.23607 −0.409516 −0.204758 0.978813i \(-0.565641\pi\)
−0.204758 + 0.978813i \(0.565641\pi\)
\(108\) −1.61803 −0.155695
\(109\) 2.76393 0.264737 0.132368 0.991201i \(-0.457742\pi\)
0.132368 + 0.991201i \(0.457742\pi\)
\(110\) −0.618034 −0.0589272
\(111\) −7.00000 −0.664411
\(112\) 1.85410 0.175196
\(113\) −0.472136 −0.0444148 −0.0222074 0.999753i \(-0.507069\pi\)
−0.0222074 + 0.999753i \(0.507069\pi\)
\(114\) 4.14590 0.388299
\(115\) 5.70820 0.532293
\(116\) −8.09017 −0.751153
\(117\) 3.47214 0.320999
\(118\) 6.90983 0.636101
\(119\) 5.23607 0.479990
\(120\) 2.23607 0.204124
\(121\) 1.00000 0.0909091
\(122\) −1.23607 −0.111908
\(123\) −2.47214 −0.222905
\(124\) 8.47214 0.760820
\(125\) −9.00000 −0.804984
\(126\) −0.618034 −0.0550588
\(127\) −12.6525 −1.12273 −0.561363 0.827570i \(-0.689723\pi\)
−0.561363 + 0.827570i \(0.689723\pi\)
\(128\) 11.3820 1.00603
\(129\) 5.70820 0.502579
\(130\) −2.14590 −0.188208
\(131\) 0.944272 0.0825014 0.0412507 0.999149i \(-0.486866\pi\)
0.0412507 + 0.999149i \(0.486866\pi\)
\(132\) −1.61803 −0.140832
\(133\) −6.70820 −0.581675
\(134\) 6.03444 0.521296
\(135\) 1.00000 0.0860663
\(136\) 11.7082 1.00397
\(137\) 19.7082 1.68379 0.841893 0.539645i \(-0.181441\pi\)
0.841893 + 0.539645i \(0.181441\pi\)
\(138\) −3.52786 −0.300312
\(139\) 14.4721 1.22751 0.613755 0.789496i \(-0.289658\pi\)
0.613755 + 0.789496i \(0.289658\pi\)
\(140\) −1.61803 −0.136749
\(141\) 0.236068 0.0198805
\(142\) 1.52786 0.128216
\(143\) 3.47214 0.290355
\(144\) 1.85410 0.154508
\(145\) 5.00000 0.415227
\(146\) −2.79837 −0.231595
\(147\) 1.00000 0.0824786
\(148\) 11.3262 0.931011
\(149\) 5.00000 0.409616 0.204808 0.978802i \(-0.434343\pi\)
0.204808 + 0.978802i \(0.434343\pi\)
\(150\) 2.47214 0.201849
\(151\) −14.1803 −1.15398 −0.576990 0.816751i \(-0.695773\pi\)
−0.576990 + 0.816751i \(0.695773\pi\)
\(152\) −15.0000 −1.21666
\(153\) 5.23607 0.423311
\(154\) −0.618034 −0.0498026
\(155\) −5.23607 −0.420571
\(156\) −5.61803 −0.449803
\(157\) −15.4164 −1.23036 −0.615182 0.788385i \(-0.710917\pi\)
−0.615182 + 0.788385i \(0.710917\pi\)
\(158\) 8.94427 0.711568
\(159\) −12.1803 −0.965964
\(160\) −5.61803 −0.444145
\(161\) 5.70820 0.449869
\(162\) −0.618034 −0.0485573
\(163\) −22.7082 −1.77864 −0.889322 0.457282i \(-0.848823\pi\)
−0.889322 + 0.457282i \(0.848823\pi\)
\(164\) 4.00000 0.312348
\(165\) 1.00000 0.0778499
\(166\) −4.18034 −0.324457
\(167\) −22.6525 −1.75290 −0.876451 0.481492i \(-0.840095\pi\)
−0.876451 + 0.481492i \(0.840095\pi\)
\(168\) 2.23607 0.172516
\(169\) −0.944272 −0.0726363
\(170\) −3.23607 −0.248195
\(171\) −6.70820 −0.512989
\(172\) −9.23607 −0.704244
\(173\) −1.52786 −0.116161 −0.0580807 0.998312i \(-0.518498\pi\)
−0.0580807 + 0.998312i \(0.518498\pi\)
\(174\) −3.09017 −0.234265
\(175\) −4.00000 −0.302372
\(176\) 1.85410 0.139758
\(177\) −11.1803 −0.840366
\(178\) 2.76393 0.207165
\(179\) 8.94427 0.668526 0.334263 0.942480i \(-0.391513\pi\)
0.334263 + 0.942480i \(0.391513\pi\)
\(180\) −1.61803 −0.120601
\(181\) −0.763932 −0.0567826 −0.0283913 0.999597i \(-0.509038\pi\)
−0.0283913 + 0.999597i \(0.509038\pi\)
\(182\) −2.14590 −0.159065
\(183\) 2.00000 0.147844
\(184\) 12.7639 0.940970
\(185\) −7.00000 −0.514650
\(186\) 3.23607 0.237280
\(187\) 5.23607 0.382899
\(188\) −0.381966 −0.0278577
\(189\) 1.00000 0.0727393
\(190\) 4.14590 0.300775
\(191\) −10.7639 −0.778851 −0.389425 0.921058i \(-0.627326\pi\)
−0.389425 + 0.921058i \(0.627326\pi\)
\(192\) −0.236068 −0.0170367
\(193\) 14.6525 1.05471 0.527354 0.849646i \(-0.323184\pi\)
0.527354 + 0.849646i \(0.323184\pi\)
\(194\) −6.00000 −0.430775
\(195\) 3.47214 0.248645
\(196\) −1.61803 −0.115574
\(197\) −16.4721 −1.17359 −0.586796 0.809735i \(-0.699611\pi\)
−0.586796 + 0.809735i \(0.699611\pi\)
\(198\) −0.618034 −0.0439218
\(199\) −3.81966 −0.270769 −0.135384 0.990793i \(-0.543227\pi\)
−0.135384 + 0.990793i \(0.543227\pi\)
\(200\) −8.94427 −0.632456
\(201\) −9.76393 −0.688695
\(202\) −11.2361 −0.790567
\(203\) 5.00000 0.350931
\(204\) −8.47214 −0.593168
\(205\) −2.47214 −0.172661
\(206\) −10.7639 −0.749959
\(207\) 5.70820 0.396748
\(208\) 6.43769 0.446374
\(209\) −6.70820 −0.464016
\(210\) −0.618034 −0.0426484
\(211\) 5.41641 0.372881 0.186440 0.982466i \(-0.440305\pi\)
0.186440 + 0.982466i \(0.440305\pi\)
\(212\) 19.7082 1.35357
\(213\) −2.47214 −0.169388
\(214\) 2.61803 0.178965
\(215\) 5.70820 0.389296
\(216\) 2.23607 0.152145
\(217\) −5.23607 −0.355447
\(218\) −1.70820 −0.115694
\(219\) 4.52786 0.305965
\(220\) −1.61803 −0.109088
\(221\) 18.1803 1.22294
\(222\) 4.32624 0.290358
\(223\) −6.00000 −0.401790 −0.200895 0.979613i \(-0.564385\pi\)
−0.200895 + 0.979613i \(0.564385\pi\)
\(224\) −5.61803 −0.375371
\(225\) −4.00000 −0.266667
\(226\) 0.291796 0.0194100
\(227\) −2.00000 −0.132745 −0.0663723 0.997795i \(-0.521143\pi\)
−0.0663723 + 0.997795i \(0.521143\pi\)
\(228\) 10.8541 0.718830
\(229\) 7.23607 0.478173 0.239086 0.970998i \(-0.423152\pi\)
0.239086 + 0.970998i \(0.423152\pi\)
\(230\) −3.52786 −0.232620
\(231\) 1.00000 0.0657952
\(232\) 11.1803 0.734025
\(233\) −14.9443 −0.979032 −0.489516 0.871994i \(-0.662826\pi\)
−0.489516 + 0.871994i \(0.662826\pi\)
\(234\) −2.14590 −0.140282
\(235\) 0.236068 0.0153994
\(236\) 18.0902 1.17757
\(237\) −14.4721 −0.940066
\(238\) −3.23607 −0.209763
\(239\) 10.1246 0.654907 0.327453 0.944867i \(-0.393810\pi\)
0.327453 + 0.944867i \(0.393810\pi\)
\(240\) 1.85410 0.119682
\(241\) 25.9443 1.67122 0.835609 0.549325i \(-0.185115\pi\)
0.835609 + 0.549325i \(0.185115\pi\)
\(242\) −0.618034 −0.0397287
\(243\) 1.00000 0.0641500
\(244\) −3.23607 −0.207168
\(245\) 1.00000 0.0638877
\(246\) 1.52786 0.0974131
\(247\) −23.2918 −1.48202
\(248\) −11.7082 −0.743472
\(249\) 6.76393 0.428647
\(250\) 5.56231 0.351791
\(251\) 12.1246 0.765299 0.382649 0.923894i \(-0.375012\pi\)
0.382649 + 0.923894i \(0.375012\pi\)
\(252\) −1.61803 −0.101927
\(253\) 5.70820 0.358872
\(254\) 7.81966 0.490649
\(255\) 5.23607 0.327895
\(256\) −6.56231 −0.410144
\(257\) −7.00000 −0.436648 −0.218324 0.975876i \(-0.570059\pi\)
−0.218324 + 0.975876i \(0.570059\pi\)
\(258\) −3.52786 −0.219635
\(259\) −7.00000 −0.434959
\(260\) −5.61803 −0.348416
\(261\) 5.00000 0.309492
\(262\) −0.583592 −0.0360544
\(263\) −26.1246 −1.61091 −0.805456 0.592655i \(-0.798080\pi\)
−0.805456 + 0.592655i \(0.798080\pi\)
\(264\) 2.23607 0.137620
\(265\) −12.1803 −0.748232
\(266\) 4.14590 0.254201
\(267\) −4.47214 −0.273690
\(268\) 15.7984 0.965039
\(269\) 1.05573 0.0643689 0.0321844 0.999482i \(-0.489754\pi\)
0.0321844 + 0.999482i \(0.489754\pi\)
\(270\) −0.618034 −0.0376124
\(271\) 5.29180 0.321454 0.160727 0.986999i \(-0.448616\pi\)
0.160727 + 0.986999i \(0.448616\pi\)
\(272\) 9.70820 0.588646
\(273\) 3.47214 0.210143
\(274\) −12.1803 −0.735841
\(275\) −4.00000 −0.241209
\(276\) −9.23607 −0.555946
\(277\) −6.47214 −0.388873 −0.194436 0.980915i \(-0.562288\pi\)
−0.194436 + 0.980915i \(0.562288\pi\)
\(278\) −8.94427 −0.536442
\(279\) −5.23607 −0.313475
\(280\) 2.23607 0.133631
\(281\) 11.4721 0.684370 0.342185 0.939633i \(-0.388833\pi\)
0.342185 + 0.939633i \(0.388833\pi\)
\(282\) −0.145898 −0.00868810
\(283\) −13.7639 −0.818181 −0.409090 0.912494i \(-0.634154\pi\)
−0.409090 + 0.912494i \(0.634154\pi\)
\(284\) 4.00000 0.237356
\(285\) −6.70820 −0.397360
\(286\) −2.14590 −0.126890
\(287\) −2.47214 −0.145926
\(288\) −5.61803 −0.331046
\(289\) 10.4164 0.612730
\(290\) −3.09017 −0.181461
\(291\) 9.70820 0.569105
\(292\) −7.32624 −0.428736
\(293\) −16.0000 −0.934730 −0.467365 0.884064i \(-0.654797\pi\)
−0.467365 + 0.884064i \(0.654797\pi\)
\(294\) −0.618034 −0.0360445
\(295\) −11.1803 −0.650945
\(296\) −15.6525 −0.909782
\(297\) 1.00000 0.0580259
\(298\) −3.09017 −0.179009
\(299\) 19.8197 1.14620
\(300\) 6.47214 0.373669
\(301\) 5.70820 0.329015
\(302\) 8.76393 0.504308
\(303\) 18.1803 1.04443
\(304\) −12.4377 −0.713351
\(305\) 2.00000 0.114520
\(306\) −3.23607 −0.184994
\(307\) −20.9443 −1.19535 −0.597676 0.801737i \(-0.703909\pi\)
−0.597676 + 0.801737i \(0.703909\pi\)
\(308\) −1.61803 −0.0921960
\(309\) 17.4164 0.990785
\(310\) 3.23607 0.183796
\(311\) 9.88854 0.560728 0.280364 0.959894i \(-0.409545\pi\)
0.280364 + 0.959894i \(0.409545\pi\)
\(312\) 7.76393 0.439546
\(313\) 24.6525 1.39344 0.696720 0.717343i \(-0.254642\pi\)
0.696720 + 0.717343i \(0.254642\pi\)
\(314\) 9.52786 0.537688
\(315\) 1.00000 0.0563436
\(316\) 23.4164 1.31728
\(317\) 24.1803 1.35810 0.679052 0.734091i \(-0.262391\pi\)
0.679052 + 0.734091i \(0.262391\pi\)
\(318\) 7.52786 0.422142
\(319\) 5.00000 0.279946
\(320\) −0.236068 −0.0131966
\(321\) −4.23607 −0.236434
\(322\) −3.52786 −0.196600
\(323\) −35.1246 −1.95439
\(324\) −1.61803 −0.0898908
\(325\) −13.8885 −0.770398
\(326\) 14.0344 0.777296
\(327\) 2.76393 0.152846
\(328\) −5.52786 −0.305225
\(329\) 0.236068 0.0130148
\(330\) −0.618034 −0.0340217
\(331\) −11.4164 −0.627503 −0.313751 0.949505i \(-0.601586\pi\)
−0.313751 + 0.949505i \(0.601586\pi\)
\(332\) −10.9443 −0.600645
\(333\) −7.00000 −0.383598
\(334\) 14.0000 0.766046
\(335\) −9.76393 −0.533461
\(336\) 1.85410 0.101150
\(337\) −8.18034 −0.445612 −0.222806 0.974863i \(-0.571522\pi\)
−0.222806 + 0.974863i \(0.571522\pi\)
\(338\) 0.583592 0.0317432
\(339\) −0.472136 −0.0256429
\(340\) −8.47214 −0.459466
\(341\) −5.23607 −0.283549
\(342\) 4.14590 0.224184
\(343\) 1.00000 0.0539949
\(344\) 12.7639 0.688185
\(345\) 5.70820 0.307319
\(346\) 0.944272 0.0507644
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) −8.09017 −0.433679
\(349\) 1.58359 0.0847677 0.0423839 0.999101i \(-0.486505\pi\)
0.0423839 + 0.999101i \(0.486505\pi\)
\(350\) 2.47214 0.132141
\(351\) 3.47214 0.185329
\(352\) −5.61803 −0.299442
\(353\) 24.5279 1.30549 0.652743 0.757579i \(-0.273618\pi\)
0.652743 + 0.757579i \(0.273618\pi\)
\(354\) 6.90983 0.367253
\(355\) −2.47214 −0.131207
\(356\) 7.23607 0.383511
\(357\) 5.23607 0.277122
\(358\) −5.52786 −0.292157
\(359\) 23.4164 1.23587 0.617935 0.786229i \(-0.287969\pi\)
0.617935 + 0.786229i \(0.287969\pi\)
\(360\) 2.23607 0.117851
\(361\) 26.0000 1.36842
\(362\) 0.472136 0.0248149
\(363\) 1.00000 0.0524864
\(364\) −5.61803 −0.294465
\(365\) 4.52786 0.236999
\(366\) −1.23607 −0.0646103
\(367\) −19.8885 −1.03817 −0.519087 0.854722i \(-0.673728\pi\)
−0.519087 + 0.854722i \(0.673728\pi\)
\(368\) 10.5836 0.551708
\(369\) −2.47214 −0.128694
\(370\) 4.32624 0.224910
\(371\) −12.1803 −0.632372
\(372\) 8.47214 0.439260
\(373\) 4.65248 0.240896 0.120448 0.992720i \(-0.461567\pi\)
0.120448 + 0.992720i \(0.461567\pi\)
\(374\) −3.23607 −0.167333
\(375\) −9.00000 −0.464758
\(376\) 0.527864 0.0272225
\(377\) 17.3607 0.894120
\(378\) −0.618034 −0.0317882
\(379\) −31.1803 −1.60163 −0.800813 0.598914i \(-0.795599\pi\)
−0.800813 + 0.598914i \(0.795599\pi\)
\(380\) 10.8541 0.556804
\(381\) −12.6525 −0.648206
\(382\) 6.65248 0.340370
\(383\) 32.9443 1.68337 0.841687 0.539966i \(-0.181563\pi\)
0.841687 + 0.539966i \(0.181563\pi\)
\(384\) 11.3820 0.580834
\(385\) 1.00000 0.0509647
\(386\) −9.05573 −0.460924
\(387\) 5.70820 0.290164
\(388\) −15.7082 −0.797463
\(389\) 11.0557 0.560548 0.280274 0.959920i \(-0.409575\pi\)
0.280274 + 0.959920i \(0.409575\pi\)
\(390\) −2.14590 −0.108662
\(391\) 29.8885 1.51153
\(392\) 2.23607 0.112938
\(393\) 0.944272 0.0476322
\(394\) 10.1803 0.512878
\(395\) −14.4721 −0.728172
\(396\) −1.61803 −0.0813093
\(397\) 23.1246 1.16059 0.580295 0.814406i \(-0.302937\pi\)
0.580295 + 0.814406i \(0.302937\pi\)
\(398\) 2.36068 0.118330
\(399\) −6.70820 −0.335830
\(400\) −7.41641 −0.370820
\(401\) −29.7082 −1.48356 −0.741778 0.670645i \(-0.766017\pi\)
−0.741778 + 0.670645i \(0.766017\pi\)
\(402\) 6.03444 0.300971
\(403\) −18.1803 −0.905627
\(404\) −29.4164 −1.46352
\(405\) 1.00000 0.0496904
\(406\) −3.09017 −0.153363
\(407\) −7.00000 −0.346977
\(408\) 11.7082 0.579642
\(409\) 21.0557 1.04114 0.520569 0.853819i \(-0.325720\pi\)
0.520569 + 0.853819i \(0.325720\pi\)
\(410\) 1.52786 0.0754558
\(411\) 19.7082 0.972134
\(412\) −28.1803 −1.38835
\(413\) −11.1803 −0.550149
\(414\) −3.52786 −0.173385
\(415\) 6.76393 0.332028
\(416\) −19.5066 −0.956389
\(417\) 14.4721 0.708704
\(418\) 4.14590 0.202783
\(419\) −1.18034 −0.0576634 −0.0288317 0.999584i \(-0.509179\pi\)
−0.0288317 + 0.999584i \(0.509179\pi\)
\(420\) −1.61803 −0.0789520
\(421\) −13.0000 −0.633581 −0.316791 0.948495i \(-0.602605\pi\)
−0.316791 + 0.948495i \(0.602605\pi\)
\(422\) −3.34752 −0.162955
\(423\) 0.236068 0.0114780
\(424\) −27.2361 −1.32270
\(425\) −20.9443 −1.01595
\(426\) 1.52786 0.0740253
\(427\) 2.00000 0.0967868
\(428\) 6.85410 0.331306
\(429\) 3.47214 0.167636
\(430\) −3.52786 −0.170129
\(431\) 8.70820 0.419459 0.209730 0.977759i \(-0.432742\pi\)
0.209730 + 0.977759i \(0.432742\pi\)
\(432\) 1.85410 0.0892055
\(433\) −10.4721 −0.503259 −0.251629 0.967824i \(-0.580966\pi\)
−0.251629 + 0.967824i \(0.580966\pi\)
\(434\) 3.23607 0.155336
\(435\) 5.00000 0.239732
\(436\) −4.47214 −0.214176
\(437\) −38.2918 −1.83175
\(438\) −2.79837 −0.133711
\(439\) 11.1803 0.533609 0.266804 0.963751i \(-0.414032\pi\)
0.266804 + 0.963751i \(0.414032\pi\)
\(440\) 2.23607 0.106600
\(441\) 1.00000 0.0476190
\(442\) −11.2361 −0.534445
\(443\) 18.4721 0.877638 0.438819 0.898576i \(-0.355397\pi\)
0.438819 + 0.898576i \(0.355397\pi\)
\(444\) 11.3262 0.537519
\(445\) −4.47214 −0.212000
\(446\) 3.70820 0.175589
\(447\) 5.00000 0.236492
\(448\) −0.236068 −0.0111532
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 2.47214 0.116538
\(451\) −2.47214 −0.116408
\(452\) 0.763932 0.0359323
\(453\) −14.1803 −0.666250
\(454\) 1.23607 0.0580115
\(455\) 3.47214 0.162776
\(456\) −15.0000 −0.702439
\(457\) 15.2361 0.712713 0.356357 0.934350i \(-0.384019\pi\)
0.356357 + 0.934350i \(0.384019\pi\)
\(458\) −4.47214 −0.208969
\(459\) 5.23607 0.244399
\(460\) −9.23607 −0.430634
\(461\) −28.0000 −1.30409 −0.652045 0.758180i \(-0.726089\pi\)
−0.652045 + 0.758180i \(0.726089\pi\)
\(462\) −0.618034 −0.0287535
\(463\) −4.81966 −0.223989 −0.111994 0.993709i \(-0.535724\pi\)
−0.111994 + 0.993709i \(0.535724\pi\)
\(464\) 9.27051 0.430373
\(465\) −5.23607 −0.242817
\(466\) 9.23607 0.427853
\(467\) 29.1803 1.35031 0.675153 0.737678i \(-0.264078\pi\)
0.675153 + 0.737678i \(0.264078\pi\)
\(468\) −5.61803 −0.259694
\(469\) −9.76393 −0.450856
\(470\) −0.145898 −0.00672977
\(471\) −15.4164 −0.710351
\(472\) −25.0000 −1.15072
\(473\) 5.70820 0.262463
\(474\) 8.94427 0.410824
\(475\) 26.8328 1.23117
\(476\) −8.47214 −0.388320
\(477\) −12.1803 −0.557699
\(478\) −6.25735 −0.286205
\(479\) 11.7082 0.534961 0.267481 0.963563i \(-0.413809\pi\)
0.267481 + 0.963563i \(0.413809\pi\)
\(480\) −5.61803 −0.256427
\(481\) −24.3050 −1.10821
\(482\) −16.0344 −0.730349
\(483\) 5.70820 0.259732
\(484\) −1.61803 −0.0735470
\(485\) 9.70820 0.440827
\(486\) −0.618034 −0.0280346
\(487\) 16.9443 0.767818 0.383909 0.923371i \(-0.374578\pi\)
0.383909 + 0.923371i \(0.374578\pi\)
\(488\) 4.47214 0.202444
\(489\) −22.7082 −1.02690
\(490\) −0.618034 −0.0279199
\(491\) −18.1246 −0.817952 −0.408976 0.912545i \(-0.634114\pi\)
−0.408976 + 0.912545i \(0.634114\pi\)
\(492\) 4.00000 0.180334
\(493\) 26.1803 1.17910
\(494\) 14.3951 0.647667
\(495\) 1.00000 0.0449467
\(496\) −9.70820 −0.435911
\(497\) −2.47214 −0.110890
\(498\) −4.18034 −0.187326
\(499\) −2.23607 −0.100100 −0.0500501 0.998747i \(-0.515938\pi\)
−0.0500501 + 0.998747i \(0.515938\pi\)
\(500\) 14.5623 0.651246
\(501\) −22.6525 −1.01204
\(502\) −7.49342 −0.334448
\(503\) 2.29180 0.102186 0.0510931 0.998694i \(-0.483729\pi\)
0.0510931 + 0.998694i \(0.483729\pi\)
\(504\) 2.23607 0.0996024
\(505\) 18.1803 0.809015
\(506\) −3.52786 −0.156833
\(507\) −0.944272 −0.0419366
\(508\) 20.4721 0.908304
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) −3.23607 −0.143295
\(511\) 4.52786 0.200301
\(512\) −18.7082 −0.826794
\(513\) −6.70820 −0.296174
\(514\) 4.32624 0.190822
\(515\) 17.4164 0.767459
\(516\) −9.23607 −0.406595
\(517\) 0.236068 0.0103823
\(518\) 4.32624 0.190084
\(519\) −1.52786 −0.0670658
\(520\) 7.76393 0.340471
\(521\) 38.3050 1.67817 0.839085 0.544000i \(-0.183091\pi\)
0.839085 + 0.544000i \(0.183091\pi\)
\(522\) −3.09017 −0.135253
\(523\) −21.6525 −0.946797 −0.473398 0.880848i \(-0.656973\pi\)
−0.473398 + 0.880848i \(0.656973\pi\)
\(524\) −1.52786 −0.0667451
\(525\) −4.00000 −0.174574
\(526\) 16.1459 0.703995
\(527\) −27.4164 −1.19428
\(528\) 1.85410 0.0806894
\(529\) 9.58359 0.416678
\(530\) 7.52786 0.326990
\(531\) −11.1803 −0.485185
\(532\) 10.8541 0.470585
\(533\) −8.58359 −0.371797
\(534\) 2.76393 0.119607
\(535\) −4.23607 −0.183141
\(536\) −21.8328 −0.943034
\(537\) 8.94427 0.385974
\(538\) −0.652476 −0.0281302
\(539\) 1.00000 0.0430730
\(540\) −1.61803 −0.0696291
\(541\) −36.9443 −1.58836 −0.794179 0.607684i \(-0.792099\pi\)
−0.794179 + 0.607684i \(0.792099\pi\)
\(542\) −3.27051 −0.140480
\(543\) −0.763932 −0.0327835
\(544\) −29.4164 −1.26122
\(545\) 2.76393 0.118394
\(546\) −2.14590 −0.0918360
\(547\) 14.8328 0.634205 0.317103 0.948391i \(-0.397290\pi\)
0.317103 + 0.948391i \(0.397290\pi\)
\(548\) −31.8885 −1.36221
\(549\) 2.00000 0.0853579
\(550\) 2.47214 0.105412
\(551\) −33.5410 −1.42890
\(552\) 12.7639 0.543269
\(553\) −14.4721 −0.615418
\(554\) 4.00000 0.169944
\(555\) −7.00000 −0.297133
\(556\) −23.4164 −0.993077
\(557\) −12.5279 −0.530823 −0.265411 0.964135i \(-0.585508\pi\)
−0.265411 + 0.964135i \(0.585508\pi\)
\(558\) 3.23607 0.136994
\(559\) 19.8197 0.838282
\(560\) 1.85410 0.0783501
\(561\) 5.23607 0.221067
\(562\) −7.09017 −0.299081
\(563\) 34.6525 1.46043 0.730214 0.683219i \(-0.239420\pi\)
0.730214 + 0.683219i \(0.239420\pi\)
\(564\) −0.381966 −0.0160837
\(565\) −0.472136 −0.0198629
\(566\) 8.50658 0.357558
\(567\) 1.00000 0.0419961
\(568\) −5.52786 −0.231944
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 4.14590 0.173653
\(571\) 17.5279 0.733518 0.366759 0.930316i \(-0.380467\pi\)
0.366759 + 0.930316i \(0.380467\pi\)
\(572\) −5.61803 −0.234902
\(573\) −10.7639 −0.449670
\(574\) 1.52786 0.0637718
\(575\) −22.8328 −0.952194
\(576\) −0.236068 −0.00983617
\(577\) 7.34752 0.305881 0.152941 0.988235i \(-0.451126\pi\)
0.152941 + 0.988235i \(0.451126\pi\)
\(578\) −6.43769 −0.267773
\(579\) 14.6525 0.608936
\(580\) −8.09017 −0.335926
\(581\) 6.76393 0.280615
\(582\) −6.00000 −0.248708
\(583\) −12.1803 −0.504458
\(584\) 10.1246 0.418959
\(585\) 3.47214 0.143555
\(586\) 9.88854 0.408492
\(587\) 46.0132 1.89917 0.949583 0.313515i \(-0.101507\pi\)
0.949583 + 0.313515i \(0.101507\pi\)
\(588\) −1.61803 −0.0667266
\(589\) 35.1246 1.44728
\(590\) 6.90983 0.284473
\(591\) −16.4721 −0.677573
\(592\) −12.9787 −0.533422
\(593\) −22.8328 −0.937631 −0.468816 0.883296i \(-0.655319\pi\)
−0.468816 + 0.883296i \(0.655319\pi\)
\(594\) −0.618034 −0.0253582
\(595\) 5.23607 0.214658
\(596\) −8.09017 −0.331386
\(597\) −3.81966 −0.156328
\(598\) −12.2492 −0.500908
\(599\) −25.5279 −1.04304 −0.521520 0.853239i \(-0.674635\pi\)
−0.521520 + 0.853239i \(0.674635\pi\)
\(600\) −8.94427 −0.365148
\(601\) 27.0000 1.10135 0.550676 0.834719i \(-0.314370\pi\)
0.550676 + 0.834719i \(0.314370\pi\)
\(602\) −3.52786 −0.143785
\(603\) −9.76393 −0.397618
\(604\) 22.9443 0.933589
\(605\) 1.00000 0.0406558
\(606\) −11.2361 −0.456434
\(607\) 6.81966 0.276801 0.138401 0.990376i \(-0.455804\pi\)
0.138401 + 0.990376i \(0.455804\pi\)
\(608\) 37.6869 1.52841
\(609\) 5.00000 0.202610
\(610\) −1.23607 −0.0500469
\(611\) 0.819660 0.0331599
\(612\) −8.47214 −0.342466
\(613\) 13.5967 0.549167 0.274584 0.961563i \(-0.411460\pi\)
0.274584 + 0.961563i \(0.411460\pi\)
\(614\) 12.9443 0.522388
\(615\) −2.47214 −0.0996861
\(616\) 2.23607 0.0900937
\(617\) 32.4721 1.30728 0.653639 0.756806i \(-0.273241\pi\)
0.653639 + 0.756806i \(0.273241\pi\)
\(618\) −10.7639 −0.432989
\(619\) −44.0689 −1.77128 −0.885639 0.464374i \(-0.846279\pi\)
−0.885639 + 0.464374i \(0.846279\pi\)
\(620\) 8.47214 0.340249
\(621\) 5.70820 0.229062
\(622\) −6.11146 −0.245047
\(623\) −4.47214 −0.179172
\(624\) 6.43769 0.257714
\(625\) 11.0000 0.440000
\(626\) −15.2361 −0.608956
\(627\) −6.70820 −0.267900
\(628\) 24.9443 0.995385
\(629\) −36.6525 −1.46143
\(630\) −0.618034 −0.0246231
\(631\) 44.3607 1.76597 0.882985 0.469400i \(-0.155530\pi\)
0.882985 + 0.469400i \(0.155530\pi\)
\(632\) −32.3607 −1.28724
\(633\) 5.41641 0.215283
\(634\) −14.9443 −0.593513
\(635\) −12.6525 −0.502098
\(636\) 19.7082 0.781481
\(637\) 3.47214 0.137571
\(638\) −3.09017 −0.122341
\(639\) −2.47214 −0.0977962
\(640\) 11.3820 0.449912
\(641\) −46.5410 −1.83826 −0.919130 0.393955i \(-0.871107\pi\)
−0.919130 + 0.393955i \(0.871107\pi\)
\(642\) 2.61803 0.103326
\(643\) −47.9574 −1.89126 −0.945628 0.325250i \(-0.894552\pi\)
−0.945628 + 0.325250i \(0.894552\pi\)
\(644\) −9.23607 −0.363952
\(645\) 5.70820 0.224760
\(646\) 21.7082 0.854098
\(647\) 12.3475 0.485431 0.242716 0.970097i \(-0.421962\pi\)
0.242716 + 0.970097i \(0.421962\pi\)
\(648\) 2.23607 0.0878410
\(649\) −11.1803 −0.438867
\(650\) 8.58359 0.336676
\(651\) −5.23607 −0.205218
\(652\) 36.7426 1.43895
\(653\) −44.9443 −1.75881 −0.879403 0.476079i \(-0.842058\pi\)
−0.879403 + 0.476079i \(0.842058\pi\)
\(654\) −1.70820 −0.0667961
\(655\) 0.944272 0.0368958
\(656\) −4.58359 −0.178959
\(657\) 4.52786 0.176649
\(658\) −0.145898 −0.00568770
\(659\) 23.5410 0.917028 0.458514 0.888687i \(-0.348382\pi\)
0.458514 + 0.888687i \(0.348382\pi\)
\(660\) −1.61803 −0.0629819
\(661\) 40.5410 1.57686 0.788431 0.615123i \(-0.210894\pi\)
0.788431 + 0.615123i \(0.210894\pi\)
\(662\) 7.05573 0.274229
\(663\) 18.1803 0.706066
\(664\) 15.1246 0.586949
\(665\) −6.70820 −0.260133
\(666\) 4.32624 0.167638
\(667\) 28.5410 1.10511
\(668\) 36.6525 1.41813
\(669\) −6.00000 −0.231973
\(670\) 6.03444 0.233131
\(671\) 2.00000 0.0772091
\(672\) −5.61803 −0.216720
\(673\) 3.59675 0.138644 0.0693222 0.997594i \(-0.477916\pi\)
0.0693222 + 0.997594i \(0.477916\pi\)
\(674\) 5.05573 0.194739
\(675\) −4.00000 −0.153960
\(676\) 1.52786 0.0587640
\(677\) 19.3050 0.741950 0.370975 0.928643i \(-0.379024\pi\)
0.370975 + 0.928643i \(0.379024\pi\)
\(678\) 0.291796 0.0112064
\(679\) 9.70820 0.372567
\(680\) 11.7082 0.448989
\(681\) −2.00000 −0.0766402
\(682\) 3.23607 0.123915
\(683\) 37.0132 1.41627 0.708135 0.706078i \(-0.249537\pi\)
0.708135 + 0.706078i \(0.249537\pi\)
\(684\) 10.8541 0.415017
\(685\) 19.7082 0.753012
\(686\) −0.618034 −0.0235966
\(687\) 7.23607 0.276073
\(688\) 10.5836 0.403496
\(689\) −42.2918 −1.61119
\(690\) −3.52786 −0.134303
\(691\) 2.00000 0.0760836 0.0380418 0.999276i \(-0.487888\pi\)
0.0380418 + 0.999276i \(0.487888\pi\)
\(692\) 2.47214 0.0939765
\(693\) 1.00000 0.0379869
\(694\) −17.3050 −0.656887
\(695\) 14.4721 0.548959
\(696\) 11.1803 0.423790
\(697\) −12.9443 −0.490299
\(698\) −0.978714 −0.0370449
\(699\) −14.9443 −0.565244
\(700\) 6.47214 0.244624
\(701\) 4.11146 0.155288 0.0776438 0.996981i \(-0.475260\pi\)
0.0776438 + 0.996981i \(0.475260\pi\)
\(702\) −2.14590 −0.0809917
\(703\) 46.9574 1.77103
\(704\) −0.236068 −0.00889715
\(705\) 0.236068 0.00889083
\(706\) −15.1591 −0.570519
\(707\) 18.1803 0.683742
\(708\) 18.0902 0.679870
\(709\) −49.7214 −1.86732 −0.933662 0.358154i \(-0.883406\pi\)
−0.933662 + 0.358154i \(0.883406\pi\)
\(710\) 1.52786 0.0573397
\(711\) −14.4721 −0.542748
\(712\) −10.0000 −0.374766
\(713\) −29.8885 −1.11933
\(714\) −3.23607 −0.121107
\(715\) 3.47214 0.129851
\(716\) −14.4721 −0.540849
\(717\) 10.1246 0.378111
\(718\) −14.4721 −0.540095
\(719\) 7.76393 0.289546 0.144773 0.989465i \(-0.453755\pi\)
0.144773 + 0.989465i \(0.453755\pi\)
\(720\) 1.85410 0.0690983
\(721\) 17.4164 0.648621
\(722\) −16.0689 −0.598022
\(723\) 25.9443 0.964878
\(724\) 1.23607 0.0459381
\(725\) −20.0000 −0.742781
\(726\) −0.618034 −0.0229374
\(727\) 24.1803 0.896799 0.448400 0.893833i \(-0.351994\pi\)
0.448400 + 0.893833i \(0.351994\pi\)
\(728\) 7.76393 0.287750
\(729\) 1.00000 0.0370370
\(730\) −2.79837 −0.103572
\(731\) 29.8885 1.10547
\(732\) −3.23607 −0.119609
\(733\) −8.11146 −0.299603 −0.149802 0.988716i \(-0.547864\pi\)
−0.149802 + 0.988716i \(0.547864\pi\)
\(734\) 12.2918 0.453698
\(735\) 1.00000 0.0368856
\(736\) −32.0689 −1.18207
\(737\) −9.76393 −0.359659
\(738\) 1.52786 0.0562415
\(739\) 34.0689 1.25324 0.626622 0.779323i \(-0.284437\pi\)
0.626622 + 0.779323i \(0.284437\pi\)
\(740\) 11.3262 0.416361
\(741\) −23.2918 −0.855646
\(742\) 7.52786 0.276357
\(743\) 2.81966 0.103443 0.0517216 0.998662i \(-0.483529\pi\)
0.0517216 + 0.998662i \(0.483529\pi\)
\(744\) −11.7082 −0.429244
\(745\) 5.00000 0.183186
\(746\) −2.87539 −0.105275
\(747\) 6.76393 0.247479
\(748\) −8.47214 −0.309772
\(749\) −4.23607 −0.154783
\(750\) 5.56231 0.203107
\(751\) 39.7639 1.45101 0.725503 0.688219i \(-0.241607\pi\)
0.725503 + 0.688219i \(0.241607\pi\)
\(752\) 0.437694 0.0159611
\(753\) 12.1246 0.441845
\(754\) −10.7295 −0.390745
\(755\) −14.1803 −0.516075
\(756\) −1.61803 −0.0588473
\(757\) −51.7214 −1.87984 −0.939922 0.341388i \(-0.889103\pi\)
−0.939922 + 0.341388i \(0.889103\pi\)
\(758\) 19.2705 0.699936
\(759\) 5.70820 0.207195
\(760\) −15.0000 −0.544107
\(761\) 27.7771 1.00692 0.503459 0.864019i \(-0.332060\pi\)
0.503459 + 0.864019i \(0.332060\pi\)
\(762\) 7.81966 0.283276
\(763\) 2.76393 0.100061
\(764\) 17.4164 0.630104
\(765\) 5.23607 0.189310
\(766\) −20.3607 −0.735661
\(767\) −38.8197 −1.40170
\(768\) −6.56231 −0.236797
\(769\) 13.9443 0.502843 0.251422 0.967878i \(-0.419102\pi\)
0.251422 + 0.967878i \(0.419102\pi\)
\(770\) −0.618034 −0.0222724
\(771\) −7.00000 −0.252099
\(772\) −23.7082 −0.853277
\(773\) −5.47214 −0.196819 −0.0984095 0.995146i \(-0.531376\pi\)
−0.0984095 + 0.995146i \(0.531376\pi\)
\(774\) −3.52786 −0.126806
\(775\) 20.9443 0.752340
\(776\) 21.7082 0.779279
\(777\) −7.00000 −0.251124
\(778\) −6.83282 −0.244968
\(779\) 16.5836 0.594169
\(780\) −5.61803 −0.201158
\(781\) −2.47214 −0.0884600
\(782\) −18.4721 −0.660562
\(783\) 5.00000 0.178685
\(784\) 1.85410 0.0662179
\(785\) −15.4164 −0.550235
\(786\) −0.583592 −0.0208160
\(787\) 3.65248 0.130197 0.0650984 0.997879i \(-0.479264\pi\)
0.0650984 + 0.997879i \(0.479264\pi\)
\(788\) 26.6525 0.949455
\(789\) −26.1246 −0.930061
\(790\) 8.94427 0.318223
\(791\) −0.472136 −0.0167872
\(792\) 2.23607 0.0794552
\(793\) 6.94427 0.246598
\(794\) −14.2918 −0.507197
\(795\) −12.1803 −0.431992
\(796\) 6.18034 0.219056
\(797\) −2.52786 −0.0895415 −0.0447708 0.998997i \(-0.514256\pi\)
−0.0447708 + 0.998997i \(0.514256\pi\)
\(798\) 4.14590 0.146763
\(799\) 1.23607 0.0437289
\(800\) 22.4721 0.794510
\(801\) −4.47214 −0.158015
\(802\) 18.3607 0.648338
\(803\) 4.52786 0.159785
\(804\) 15.7984 0.557166
\(805\) 5.70820 0.201188
\(806\) 11.2361 0.395774
\(807\) 1.05573 0.0371634
\(808\) 40.6525 1.43015
\(809\) −56.3050 −1.97958 −0.989788 0.142545i \(-0.954471\pi\)
−0.989788 + 0.142545i \(0.954471\pi\)
\(810\) −0.618034 −0.0217155
\(811\) 15.2918 0.536968 0.268484 0.963284i \(-0.413477\pi\)
0.268484 + 0.963284i \(0.413477\pi\)
\(812\) −8.09017 −0.283909
\(813\) 5.29180 0.185591
\(814\) 4.32624 0.151635
\(815\) −22.7082 −0.795434
\(816\) 9.70820 0.339855
\(817\) −38.2918 −1.33966
\(818\) −13.0132 −0.454994
\(819\) 3.47214 0.121326
\(820\) 4.00000 0.139686
\(821\) −7.47214 −0.260779 −0.130390 0.991463i \(-0.541623\pi\)
−0.130390 + 0.991463i \(0.541623\pi\)
\(822\) −12.1803 −0.424838
\(823\) −17.1803 −0.598869 −0.299435 0.954117i \(-0.596798\pi\)
−0.299435 + 0.954117i \(0.596798\pi\)
\(824\) 38.9443 1.35669
\(825\) −4.00000 −0.139262
\(826\) 6.90983 0.240424
\(827\) 12.3475 0.429365 0.214683 0.976684i \(-0.431128\pi\)
0.214683 + 0.976684i \(0.431128\pi\)
\(828\) −9.23607 −0.320976
\(829\) 35.7771 1.24259 0.621295 0.783577i \(-0.286607\pi\)
0.621295 + 0.783577i \(0.286607\pi\)
\(830\) −4.18034 −0.145102
\(831\) −6.47214 −0.224516
\(832\) −0.819660 −0.0284166
\(833\) 5.23607 0.181419
\(834\) −8.94427 −0.309715
\(835\) −22.6525 −0.783921
\(836\) 10.8541 0.375397
\(837\) −5.23607 −0.180985
\(838\) 0.729490 0.0251998
\(839\) −23.5410 −0.812726 −0.406363 0.913712i \(-0.633203\pi\)
−0.406363 + 0.913712i \(0.633203\pi\)
\(840\) 2.23607 0.0771517
\(841\) −4.00000 −0.137931
\(842\) 8.03444 0.276885
\(843\) 11.4721 0.395121
\(844\) −8.76393 −0.301667
\(845\) −0.944272 −0.0324839
\(846\) −0.145898 −0.00501608
\(847\) 1.00000 0.0343604
\(848\) −22.5836 −0.775524
\(849\) −13.7639 −0.472377
\(850\) 12.9443 0.443985
\(851\) −39.9574 −1.36972
\(852\) 4.00000 0.137038
\(853\) −29.4164 −1.00720 −0.503599 0.863937i \(-0.667991\pi\)
−0.503599 + 0.863937i \(0.667991\pi\)
\(854\) −1.23607 −0.0422974
\(855\) −6.70820 −0.229416
\(856\) −9.47214 −0.323751
\(857\) 0.111456 0.00380727 0.00190364 0.999998i \(-0.499394\pi\)
0.00190364 + 0.999998i \(0.499394\pi\)
\(858\) −2.14590 −0.0732598
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) −9.23607 −0.314947
\(861\) −2.47214 −0.0842502
\(862\) −5.38197 −0.183310
\(863\) −43.2361 −1.47177 −0.735886 0.677105i \(-0.763234\pi\)
−0.735886 + 0.677105i \(0.763234\pi\)
\(864\) −5.61803 −0.191129
\(865\) −1.52786 −0.0519489
\(866\) 6.47214 0.219932
\(867\) 10.4164 0.353760
\(868\) 8.47214 0.287563
\(869\) −14.4721 −0.490934
\(870\) −3.09017 −0.104767
\(871\) −33.9017 −1.14872
\(872\) 6.18034 0.209293
\(873\) 9.70820 0.328573
\(874\) 23.6656 0.800502
\(875\) −9.00000 −0.304256
\(876\) −7.32624 −0.247531
\(877\) 4.58359 0.154777 0.0773885 0.997001i \(-0.475342\pi\)
0.0773885 + 0.997001i \(0.475342\pi\)
\(878\) −6.90983 −0.233195
\(879\) −16.0000 −0.539667
\(880\) 1.85410 0.0625018
\(881\) 54.8885 1.84924 0.924621 0.380888i \(-0.124382\pi\)
0.924621 + 0.380888i \(0.124382\pi\)
\(882\) −0.618034 −0.0208103
\(883\) −14.8197 −0.498721 −0.249361 0.968411i \(-0.580220\pi\)
−0.249361 + 0.968411i \(0.580220\pi\)
\(884\) −29.4164 −0.989381
\(885\) −11.1803 −0.375823
\(886\) −11.4164 −0.383542
\(887\) 10.7639 0.361417 0.180709 0.983537i \(-0.442161\pi\)
0.180709 + 0.983537i \(0.442161\pi\)
\(888\) −15.6525 −0.525263
\(889\) −12.6525 −0.424350
\(890\) 2.76393 0.0926472
\(891\) 1.00000 0.0335013
\(892\) 9.70820 0.325055
\(893\) −1.58359 −0.0529929
\(894\) −3.09017 −0.103351
\(895\) 8.94427 0.298974
\(896\) 11.3820 0.380245
\(897\) 19.8197 0.661759
\(898\) −12.3607 −0.412481
\(899\) −26.1803 −0.873163
\(900\) 6.47214 0.215738
\(901\) −63.7771 −2.12472
\(902\) 1.52786 0.0508723
\(903\) 5.70820 0.189957
\(904\) −1.05573 −0.0351130
\(905\) −0.763932 −0.0253940
\(906\) 8.76393 0.291162
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 3.23607 0.107393
\(909\) 18.1803 0.603004
\(910\) −2.14590 −0.0711358
\(911\) −14.5836 −0.483176 −0.241588 0.970379i \(-0.577668\pi\)
−0.241588 + 0.970379i \(0.577668\pi\)
\(912\) −12.4377 −0.411853
\(913\) 6.76393 0.223853
\(914\) −9.41641 −0.311467
\(915\) 2.00000 0.0661180
\(916\) −11.7082 −0.386850
\(917\) 0.944272 0.0311826
\(918\) −3.23607 −0.106806
\(919\) 39.5967 1.30618 0.653088 0.757282i \(-0.273473\pi\)
0.653088 + 0.757282i \(0.273473\pi\)
\(920\) 12.7639 0.420814
\(921\) −20.9443 −0.690137
\(922\) 17.3050 0.569908
\(923\) −8.58359 −0.282532
\(924\) −1.61803 −0.0532294
\(925\) 28.0000 0.920634
\(926\) 2.97871 0.0978866
\(927\) 17.4164 0.572030
\(928\) −28.0902 −0.922105
\(929\) 12.8885 0.422859 0.211430 0.977393i \(-0.432188\pi\)
0.211430 + 0.977393i \(0.432188\pi\)
\(930\) 3.23607 0.106115
\(931\) −6.70820 −0.219853
\(932\) 24.1803 0.792053
\(933\) 9.88854 0.323736
\(934\) −18.0344 −0.590105
\(935\) 5.23607 0.171238
\(936\) 7.76393 0.253772
\(937\) −39.8885 −1.30310 −0.651551 0.758605i \(-0.725881\pi\)
−0.651551 + 0.758605i \(0.725881\pi\)
\(938\) 6.03444 0.197032
\(939\) 24.6525 0.804503
\(940\) −0.381966 −0.0124584
\(941\) −60.3607 −1.96770 −0.983851 0.178990i \(-0.942717\pi\)
−0.983851 + 0.178990i \(0.942717\pi\)
\(942\) 9.52786 0.310435
\(943\) −14.1115 −0.459532
\(944\) −20.7295 −0.674687
\(945\) 1.00000 0.0325300
\(946\) −3.52786 −0.114701
\(947\) −5.41641 −0.176010 −0.0880048 0.996120i \(-0.528049\pi\)
−0.0880048 + 0.996120i \(0.528049\pi\)
\(948\) 23.4164 0.760530
\(949\) 15.7214 0.510337
\(950\) −16.5836 −0.538043
\(951\) 24.1803 0.784101
\(952\) 11.7082 0.379465
\(953\) 44.7771 1.45047 0.725236 0.688500i \(-0.241731\pi\)
0.725236 + 0.688500i \(0.241731\pi\)
\(954\) 7.52786 0.243724
\(955\) −10.7639 −0.348313
\(956\) −16.3820 −0.529831
\(957\) 5.00000 0.161627
\(958\) −7.23607 −0.233787
\(959\) 19.7082 0.636411
\(960\) −0.236068 −0.00761906
\(961\) −3.58359 −0.115600
\(962\) 15.0213 0.484306
\(963\) −4.23607 −0.136505
\(964\) −41.9787 −1.35204
\(965\) 14.6525 0.471680
\(966\) −3.52786 −0.113507
\(967\) −61.1935 −1.96785 −0.983925 0.178582i \(-0.942849\pi\)
−0.983925 + 0.178582i \(0.942849\pi\)
\(968\) 2.23607 0.0718699
\(969\) −35.1246 −1.12837
\(970\) −6.00000 −0.192648
\(971\) 32.1246 1.03093 0.515464 0.856911i \(-0.327620\pi\)
0.515464 + 0.856911i \(0.327620\pi\)
\(972\) −1.61803 −0.0518985
\(973\) 14.4721 0.463955
\(974\) −10.4721 −0.335549
\(975\) −13.8885 −0.444789
\(976\) 3.70820 0.118697
\(977\) 4.18034 0.133741 0.0668705 0.997762i \(-0.478699\pi\)
0.0668705 + 0.997762i \(0.478699\pi\)
\(978\) 14.0344 0.448772
\(979\) −4.47214 −0.142930
\(980\) −1.61803 −0.0516862
\(981\) 2.76393 0.0882456
\(982\) 11.2016 0.357458
\(983\) 27.4164 0.874448 0.437224 0.899353i \(-0.355962\pi\)
0.437224 + 0.899353i \(0.355962\pi\)
\(984\) −5.52786 −0.176222
\(985\) −16.4721 −0.524846
\(986\) −16.1803 −0.515287
\(987\) 0.236068 0.00751413
\(988\) 37.6869 1.19898
\(989\) 32.5836 1.03610
\(990\) −0.618034 −0.0196424
\(991\) −29.1803 −0.926944 −0.463472 0.886112i \(-0.653397\pi\)
−0.463472 + 0.886112i \(0.653397\pi\)
\(992\) 29.4164 0.933972
\(993\) −11.4164 −0.362289
\(994\) 1.52786 0.0484609
\(995\) −3.81966 −0.121091
\(996\) −10.9443 −0.346783
\(997\) 26.9443 0.853334 0.426667 0.904409i \(-0.359688\pi\)
0.426667 + 0.904409i \(0.359688\pi\)
\(998\) 1.38197 0.0437454
\(999\) −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.1.1 2
3.2 odd 2 693.2.a.f.1.2 2
4.3 odd 2 3696.2.a.be.1.2 2
5.4 even 2 5775.2.a.be.1.2 2
7.6 odd 2 1617.2.a.p.1.1 2
11.10 odd 2 2541.2.a.t.1.2 2
21.20 even 2 4851.2.a.w.1.2 2
33.32 even 2 7623.2.a.bm.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.c.1.1 2 1.1 even 1 trivial
693.2.a.f.1.2 2 3.2 odd 2
1617.2.a.p.1.1 2 7.6 odd 2
2541.2.a.t.1.2 2 11.10 odd 2
3696.2.a.be.1.2 2 4.3 odd 2
4851.2.a.w.1.2 2 21.20 even 2
5775.2.a.be.1.2 2 5.4 even 2
7623.2.a.bm.1.1 2 33.32 even 2