Properties

Label 231.2.a.d.1.2
Level $231$
Weight $2$
Character 231.1
Self dual yes
Analytic conductor $1.845$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.837.1
Defining polynomial: \(x^{3} - 6 x - 1\)
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 \(-0.167449\) of defining polynomial
Character \(\chi\) \(=\) 231.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.167449 q^{2} -1.00000 q^{3} -1.97196 q^{4} +3.80451 q^{5} +0.167449 q^{6} -1.00000 q^{7} +0.665102 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.167449 q^{2} -1.00000 q^{3} -1.97196 q^{4} +3.80451 q^{5} +0.167449 q^{6} -1.00000 q^{7} +0.665102 q^{8} +1.00000 q^{9} -0.637062 q^{10} +1.00000 q^{11} +1.97196 q^{12} +3.80451 q^{13} +0.167449 q^{14} -3.80451 q^{15} +3.83255 q^{16} +0.334898 q^{17} -0.167449 q^{18} +8.13941 q^{19} -7.50235 q^{20} +1.00000 q^{21} -0.167449 q^{22} -1.66510 q^{23} -0.665102 q^{24} +9.47431 q^{25} -0.637062 q^{26} -1.00000 q^{27} +1.97196 q^{28} +0.195488 q^{29} +0.637062 q^{30} -9.94392 q^{31} -1.97196 q^{32} -1.00000 q^{33} -0.0560785 q^{34} -3.80451 q^{35} -1.97196 q^{36} -4.47431 q^{37} -1.36294 q^{38} -3.80451 q^{39} +2.53039 q^{40} -6.27882 q^{41} -0.167449 q^{42} +2.33490 q^{43} -1.97196 q^{44} +3.80451 q^{45} +0.278820 q^{46} -12.1394 q^{47} -3.83255 q^{48} +1.00000 q^{49} -1.58647 q^{50} -0.334898 q^{51} -7.50235 q^{52} +7.94392 q^{53} +0.167449 q^{54} +3.80451 q^{55} -0.665102 q^{56} -8.13941 q^{57} -0.0327344 q^{58} +3.74843 q^{59} +7.50235 q^{60} +6.00000 q^{61} +1.66510 q^{62} -1.00000 q^{63} -7.33490 q^{64} +14.4743 q^{65} +0.167449 q^{66} -0.139410 q^{67} -0.660406 q^{68} +1.66510 q^{69} +0.637062 q^{70} +4.66980 q^{71} +0.665102 q^{72} +4.19549 q^{73} +0.749219 q^{74} -9.47431 q^{75} -16.0506 q^{76} -1.00000 q^{77} +0.637062 q^{78} +3.33020 q^{79} +14.5810 q^{80} +1.00000 q^{81} +1.05138 q^{82} -13.9439 q^{83} -1.97196 q^{84} +1.27412 q^{85} -0.390977 q^{86} -0.195488 q^{87} +0.665102 q^{88} -9.88784 q^{89} -0.637062 q^{90} -3.80451 q^{91} +3.28352 q^{92} +9.94392 q^{93} +2.03273 q^{94} +30.9665 q^{95} +1.97196 q^{96} +0.0560785 q^{97} -0.167449 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 6 q^{4} - 3 q^{7} + 3 q^{8} + 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} + 6 q^{4} - 3 q^{7} + 3 q^{8} + 3 q^{9} + 9 q^{10} + 3 q^{11} - 6 q^{12} + 12 q^{16} + 12 q^{19} - 21 q^{20} + 3 q^{21} - 6 q^{23} - 3 q^{24} + 15 q^{25} + 9 q^{26} - 3 q^{27} - 6 q^{28} + 12 q^{29} - 9 q^{30} - 6 q^{31} + 6 q^{32} - 3 q^{33} - 24 q^{34} + 6 q^{36} - 15 q^{38} + 18 q^{40} + 6 q^{41} + 6 q^{43} + 6 q^{44} - 24 q^{46} - 24 q^{47} - 12 q^{48} + 3 q^{49} - 39 q^{50} - 21 q^{52} - 3 q^{56} - 12 q^{57} - 9 q^{58} - 24 q^{59} + 21 q^{60} + 18 q^{61} + 6 q^{62} - 3 q^{63} - 21 q^{64} + 30 q^{65} + 12 q^{67} - 6 q^{68} + 6 q^{69} - 9 q^{70} + 12 q^{71} + 3 q^{72} + 24 q^{73} + 39 q^{74} - 15 q^{75} - 3 q^{76} - 3 q^{77} - 9 q^{78} + 12 q^{79} + 9 q^{80} + 3 q^{81} + 30 q^{82} - 18 q^{83} + 6 q^{84} - 18 q^{85} - 24 q^{86} - 12 q^{87} + 3 q^{88} + 18 q^{89} + 9 q^{90} - 18 q^{92} + 6 q^{93} + 15 q^{94} + 12 q^{95} - 6 q^{96} + 24 q^{97} + 3 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.167449 −0.118404 −0.0592022 0.998246i \(-0.518856\pi\)
−0.0592022 + 0.998246i \(0.518856\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.97196 −0.985980
\(5\) 3.80451 1.70143 0.850715 0.525628i \(-0.176170\pi\)
0.850715 + 0.525628i \(0.176170\pi\)
\(6\) 0.167449 0.0683608
\(7\) −1.00000 −0.377964
\(8\) 0.665102 0.235149
\(9\) 1.00000 0.333333
\(10\) −0.637062 −0.201457
\(11\) 1.00000 0.301511
\(12\) 1.97196 0.569256
\(13\) 3.80451 1.05518 0.527591 0.849499i \(-0.323095\pi\)
0.527591 + 0.849499i \(0.323095\pi\)
\(14\) 0.167449 0.0447527
\(15\) −3.80451 −0.982321
\(16\) 3.83255 0.958138
\(17\) 0.334898 0.0812248 0.0406124 0.999175i \(-0.487069\pi\)
0.0406124 + 0.999175i \(0.487069\pi\)
\(18\) −0.167449 −0.0394682
\(19\) 8.13941 1.86731 0.933654 0.358175i \(-0.116601\pi\)
0.933654 + 0.358175i \(0.116601\pi\)
\(20\) −7.50235 −1.67758
\(21\) 1.00000 0.218218
\(22\) −0.167449 −0.0357003
\(23\) −1.66510 −0.347198 −0.173599 0.984816i \(-0.555540\pi\)
−0.173599 + 0.984816i \(0.555540\pi\)
\(24\) −0.665102 −0.135763
\(25\) 9.47431 1.89486
\(26\) −0.637062 −0.124938
\(27\) −1.00000 −0.192450
\(28\) 1.97196 0.372666
\(29\) 0.195488 0.0363013 0.0181506 0.999835i \(-0.494222\pi\)
0.0181506 + 0.999835i \(0.494222\pi\)
\(30\) 0.637062 0.116311
\(31\) −9.94392 −1.78598 −0.892991 0.450075i \(-0.851397\pi\)
−0.892991 + 0.450075i \(0.851397\pi\)
\(32\) −1.97196 −0.348597
\(33\) −1.00000 −0.174078
\(34\) −0.0560785 −0.00961738
\(35\) −3.80451 −0.643080
\(36\) −1.97196 −0.328660
\(37\) −4.47431 −0.735572 −0.367786 0.929911i \(-0.619884\pi\)
−0.367786 + 0.929911i \(0.619884\pi\)
\(38\) −1.36294 −0.221098
\(39\) −3.80451 −0.609209
\(40\) 2.53039 0.400089
\(41\) −6.27882 −0.980587 −0.490293 0.871557i \(-0.663110\pi\)
−0.490293 + 0.871557i \(0.663110\pi\)
\(42\) −0.167449 −0.0258380
\(43\) 2.33490 0.356069 0.178034 0.984024i \(-0.443026\pi\)
0.178034 + 0.984024i \(0.443026\pi\)
\(44\) −1.97196 −0.297284
\(45\) 3.80451 0.567143
\(46\) 0.278820 0.0411098
\(47\) −12.1394 −1.77071 −0.885357 0.464911i \(-0.846086\pi\)
−0.885357 + 0.464911i \(0.846086\pi\)
\(48\) −3.83255 −0.553181
\(49\) 1.00000 0.142857
\(50\) −1.58647 −0.224360
\(51\) −0.334898 −0.0468952
\(52\) −7.50235 −1.04039
\(53\) 7.94392 1.09118 0.545591 0.838052i \(-0.316305\pi\)
0.545591 + 0.838052i \(0.316305\pi\)
\(54\) 0.167449 0.0227869
\(55\) 3.80451 0.513000
\(56\) −0.665102 −0.0888779
\(57\) −8.13941 −1.07809
\(58\) −0.0327344 −0.00429823
\(59\) 3.74843 0.488004 0.244002 0.969775i \(-0.421540\pi\)
0.244002 + 0.969775i \(0.421540\pi\)
\(60\) 7.50235 0.968549
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 1.66510 0.211468
\(63\) −1.00000 −0.125988
\(64\) −7.33490 −0.916862
\(65\) 14.4743 1.79532
\(66\) 0.167449 0.0206116
\(67\) −0.139410 −0.0170316 −0.00851582 0.999964i \(-0.502711\pi\)
−0.00851582 + 0.999964i \(0.502711\pi\)
\(68\) −0.660406 −0.0800860
\(69\) 1.66510 0.200455
\(70\) 0.637062 0.0761435
\(71\) 4.66980 0.554203 0.277101 0.960841i \(-0.410626\pi\)
0.277101 + 0.960841i \(0.410626\pi\)
\(72\) 0.665102 0.0783830
\(73\) 4.19549 0.491045 0.245522 0.969391i \(-0.421041\pi\)
0.245522 + 0.969391i \(0.421041\pi\)
\(74\) 0.749219 0.0870950
\(75\) −9.47431 −1.09400
\(76\) −16.0506 −1.84113
\(77\) −1.00000 −0.113961
\(78\) 0.637062 0.0721331
\(79\) 3.33020 0.374677 0.187339 0.982295i \(-0.440014\pi\)
0.187339 + 0.982295i \(0.440014\pi\)
\(80\) 14.5810 1.63020
\(81\) 1.00000 0.111111
\(82\) 1.05138 0.116106
\(83\) −13.9439 −1.53054 −0.765272 0.643707i \(-0.777396\pi\)
−0.765272 + 0.643707i \(0.777396\pi\)
\(84\) −1.97196 −0.215159
\(85\) 1.27412 0.138198
\(86\) −0.390977 −0.0421601
\(87\) −0.195488 −0.0209586
\(88\) 0.665102 0.0709001
\(89\) −9.88784 −1.04811 −0.524055 0.851685i \(-0.675581\pi\)
−0.524055 + 0.851685i \(0.675581\pi\)
\(90\) −0.637062 −0.0671523
\(91\) −3.80451 −0.398821
\(92\) 3.28352 0.342330
\(93\) 9.94392 1.03114
\(94\) 2.03273 0.209661
\(95\) 30.9665 3.17709
\(96\) 1.97196 0.201262
\(97\) 0.0560785 0.00569391 0.00284695 0.999996i \(-0.499094\pi\)
0.00284695 + 0.999996i \(0.499094\pi\)
\(98\) −0.167449 −0.0169149
\(99\) 1.00000 0.100504
\(100\) −18.6830 −1.86830
\(101\) −18.8831 −1.87894 −0.939472 0.342626i \(-0.888683\pi\)
−0.939472 + 0.342626i \(0.888683\pi\)
\(102\) 0.0560785 0.00555260
\(103\) −8.27882 −0.815736 −0.407868 0.913041i \(-0.633728\pi\)
−0.407868 + 0.913041i \(0.633728\pi\)
\(104\) 2.53039 0.248125
\(105\) 3.80451 0.371282
\(106\) −1.33020 −0.129201
\(107\) −8.13941 −0.786866 −0.393433 0.919353i \(-0.628713\pi\)
−0.393433 + 0.919353i \(0.628713\pi\)
\(108\) 1.97196 0.189752
\(109\) 11.5529 1.10657 0.553286 0.832992i \(-0.313374\pi\)
0.553286 + 0.832992i \(0.313374\pi\)
\(110\) −0.637062 −0.0607415
\(111\) 4.47431 0.424683
\(112\) −3.83255 −0.362142
\(113\) −1.33020 −0.125135 −0.0625675 0.998041i \(-0.519929\pi\)
−0.0625675 + 0.998041i \(0.519929\pi\)
\(114\) 1.36294 0.127651
\(115\) −6.33490 −0.590732
\(116\) −0.385496 −0.0357924
\(117\) 3.80451 0.351727
\(118\) −0.627672 −0.0577819
\(119\) −0.334898 −0.0307001
\(120\) −2.53039 −0.230992
\(121\) 1.00000 0.0909091
\(122\) −1.00470 −0.0909608
\(123\) 6.27882 0.566142
\(124\) 19.6090 1.76094
\(125\) 17.0226 1.52254
\(126\) 0.167449 0.0149176
\(127\) 14.6137 1.29676 0.648379 0.761318i \(-0.275447\pi\)
0.648379 + 0.761318i \(0.275447\pi\)
\(128\) 5.17214 0.457157
\(129\) −2.33490 −0.205576
\(130\) −2.42371 −0.212574
\(131\) 20.5576 1.79613 0.898065 0.439863i \(-0.144973\pi\)
0.898065 + 0.439863i \(0.144973\pi\)
\(132\) 1.97196 0.171637
\(133\) −8.13941 −0.705776
\(134\) 0.0233441 0.00201662
\(135\) −3.80451 −0.327440
\(136\) 0.222741 0.0190999
\(137\) −16.2227 −1.38600 −0.693001 0.720936i \(-0.743712\pi\)
−0.693001 + 0.720936i \(0.743712\pi\)
\(138\) −0.278820 −0.0237347
\(139\) 22.5482 1.91252 0.956259 0.292522i \(-0.0944945\pi\)
0.956259 + 0.292522i \(0.0944945\pi\)
\(140\) 7.50235 0.634064
\(141\) 12.1394 1.02232
\(142\) −0.781954 −0.0656201
\(143\) 3.80451 0.318149
\(144\) 3.83255 0.319379
\(145\) 0.743738 0.0617641
\(146\) −0.702531 −0.0581419
\(147\) −1.00000 −0.0824786
\(148\) 8.82316 0.725259
\(149\) 8.19549 0.671401 0.335700 0.941969i \(-0.391027\pi\)
0.335700 + 0.941969i \(0.391027\pi\)
\(150\) 1.58647 0.129534
\(151\) −13.2741 −1.08023 −0.540116 0.841590i \(-0.681620\pi\)
−0.540116 + 0.841590i \(0.681620\pi\)
\(152\) 5.41353 0.439096
\(153\) 0.334898 0.0270749
\(154\) 0.167449 0.0134934
\(155\) −37.8318 −3.03872
\(156\) 7.50235 0.600669
\(157\) −16.9392 −1.35190 −0.675949 0.736949i \(-0.736266\pi\)
−0.675949 + 0.736949i \(0.736266\pi\)
\(158\) −0.557640 −0.0443634
\(159\) −7.94392 −0.629994
\(160\) −7.50235 −0.593113
\(161\) 1.66510 0.131228
\(162\) −0.167449 −0.0131561
\(163\) −6.79982 −0.532603 −0.266301 0.963890i \(-0.585802\pi\)
−0.266301 + 0.963890i \(0.585802\pi\)
\(164\) 12.3816 0.966839
\(165\) −3.80451 −0.296181
\(166\) 2.33490 0.181223
\(167\) −18.2227 −1.41012 −0.705059 0.709149i \(-0.749080\pi\)
−0.705059 + 0.709149i \(0.749080\pi\)
\(168\) 0.665102 0.0513137
\(169\) 1.47431 0.113408
\(170\) −0.213351 −0.0163633
\(171\) 8.13941 0.622436
\(172\) −4.60433 −0.351077
\(173\) −1.72118 −0.130859 −0.0654294 0.997857i \(-0.520842\pi\)
−0.0654294 + 0.997857i \(0.520842\pi\)
\(174\) 0.0327344 0.00248159
\(175\) −9.47431 −0.716190
\(176\) 3.83255 0.288889
\(177\) −3.74843 −0.281749
\(178\) 1.65571 0.124101
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −7.50235 −0.559192
\(181\) 0.725875 0.0539539 0.0269769 0.999636i \(-0.491412\pi\)
0.0269769 + 0.999636i \(0.491412\pi\)
\(182\) 0.637062 0.0472222
\(183\) −6.00000 −0.443533
\(184\) −1.10746 −0.0816432
\(185\) −17.0226 −1.25152
\(186\) −1.66510 −0.122091
\(187\) 0.334898 0.0244902
\(188\) 23.9384 1.74589
\(189\) 1.00000 0.0727393
\(190\) −5.18531 −0.376182
\(191\) 5.27412 0.381622 0.190811 0.981627i \(-0.438888\pi\)
0.190811 + 0.981627i \(0.438888\pi\)
\(192\) 7.33490 0.529351
\(193\) −19.8318 −1.42752 −0.713761 0.700390i \(-0.753010\pi\)
−0.713761 + 0.700390i \(0.753010\pi\)
\(194\) −0.00939029 −0.000674184 0
\(195\) −14.4743 −1.03653
\(196\) −1.97196 −0.140854
\(197\) 2.66980 0.190215 0.0951076 0.995467i \(-0.469680\pi\)
0.0951076 + 0.995467i \(0.469680\pi\)
\(198\) −0.167449 −0.0119001
\(199\) 13.5529 0.960743 0.480371 0.877065i \(-0.340502\pi\)
0.480371 + 0.877065i \(0.340502\pi\)
\(200\) 6.30138 0.445575
\(201\) 0.139410 0.00983322
\(202\) 3.16197 0.222475
\(203\) −0.195488 −0.0137206
\(204\) 0.660406 0.0462377
\(205\) −23.8878 −1.66840
\(206\) 1.38628 0.0965868
\(207\) −1.66510 −0.115733
\(208\) 14.5810 1.01101
\(209\) 8.13941 0.563015
\(210\) −0.637062 −0.0439615
\(211\) −4.27882 −0.294566 −0.147283 0.989094i \(-0.547053\pi\)
−0.147283 + 0.989094i \(0.547053\pi\)
\(212\) −15.6651 −1.07588
\(213\) −4.66980 −0.319969
\(214\) 1.36294 0.0931685
\(215\) 8.88315 0.605826
\(216\) −0.665102 −0.0452544
\(217\) 9.94392 0.675037
\(218\) −1.93453 −0.131023
\(219\) −4.19549 −0.283505
\(220\) −7.50235 −0.505808
\(221\) 1.27412 0.0857069
\(222\) −0.749219 −0.0502843
\(223\) −10.2694 −0.687692 −0.343846 0.939026i \(-0.611730\pi\)
−0.343846 + 0.939026i \(0.611730\pi\)
\(224\) 1.97196 0.131757
\(225\) 9.47431 0.631621
\(226\) 0.222741 0.0148165
\(227\) −0.390977 −0.0259500 −0.0129750 0.999916i \(-0.504130\pi\)
−0.0129750 + 0.999916i \(0.504130\pi\)
\(228\) 16.0506 1.06298
\(229\) 7.94392 0.524949 0.262475 0.964939i \(-0.415461\pi\)
0.262475 + 0.964939i \(0.415461\pi\)
\(230\) 1.06077 0.0699453
\(231\) 1.00000 0.0657952
\(232\) 0.130020 0.00853621
\(233\) 26.5576 1.73985 0.869924 0.493185i \(-0.164167\pi\)
0.869924 + 0.493185i \(0.164167\pi\)
\(234\) −0.637062 −0.0416461
\(235\) −46.1845 −3.01275
\(236\) −7.39176 −0.481163
\(237\) −3.33020 −0.216320
\(238\) 0.0560785 0.00363503
\(239\) 10.7998 0.698582 0.349291 0.937014i \(-0.386422\pi\)
0.349291 + 0.937014i \(0.386422\pi\)
\(240\) −14.5810 −0.941198
\(241\) −12.0833 −0.778356 −0.389178 0.921163i \(-0.627241\pi\)
−0.389178 + 0.921163i \(0.627241\pi\)
\(242\) −0.167449 −0.0107640
\(243\) −1.00000 −0.0641500
\(244\) −11.8318 −0.757451
\(245\) 3.80451 0.243061
\(246\) −1.05138 −0.0670338
\(247\) 30.9665 1.97035
\(248\) −6.61372 −0.419972
\(249\) 13.9439 0.883660
\(250\) −2.85041 −0.180276
\(251\) 4.80921 0.303554 0.151777 0.988415i \(-0.451500\pi\)
0.151777 + 0.988415i \(0.451500\pi\)
\(252\) 1.97196 0.124222
\(253\) −1.66510 −0.104684
\(254\) −2.44706 −0.153542
\(255\) −1.27412 −0.0797888
\(256\) 13.8037 0.862733
\(257\) −16.7531 −1.04503 −0.522516 0.852630i \(-0.675006\pi\)
−0.522516 + 0.852630i \(0.675006\pi\)
\(258\) 0.390977 0.0243412
\(259\) 4.47431 0.278020
\(260\) −28.5428 −1.77015
\(261\) 0.195488 0.0121004
\(262\) −3.44236 −0.212670
\(263\) −12.1394 −0.748548 −0.374274 0.927318i \(-0.622108\pi\)
−0.374274 + 0.927318i \(0.622108\pi\)
\(264\) −0.665102 −0.0409342
\(265\) 30.2227 1.85657
\(266\) 1.36294 0.0835671
\(267\) 9.88784 0.605126
\(268\) 0.274911 0.0167929
\(269\) −25.2180 −1.53757 −0.768786 0.639506i \(-0.779139\pi\)
−0.768786 + 0.639506i \(0.779139\pi\)
\(270\) 0.637062 0.0387704
\(271\) 4.13941 0.251451 0.125726 0.992065i \(-0.459874\pi\)
0.125726 + 0.992065i \(0.459874\pi\)
\(272\) 1.28352 0.0778245
\(273\) 3.80451 0.230260
\(274\) 2.71648 0.164109
\(275\) 9.47431 0.571322
\(276\) −3.28352 −0.197644
\(277\) −18.2788 −1.09827 −0.549134 0.835734i \(-0.685042\pi\)
−0.549134 + 0.835734i \(0.685042\pi\)
\(278\) −3.77569 −0.226451
\(279\) −9.94392 −0.595327
\(280\) −2.53039 −0.151220
\(281\) 6.74374 0.402298 0.201149 0.979561i \(-0.435533\pi\)
0.201149 + 0.979561i \(0.435533\pi\)
\(282\) −2.03273 −0.121048
\(283\) 23.3575 1.38846 0.694228 0.719755i \(-0.255746\pi\)
0.694228 + 0.719755i \(0.255746\pi\)
\(284\) −9.20866 −0.546433
\(285\) −30.9665 −1.83430
\(286\) −0.637062 −0.0376703
\(287\) 6.27882 0.370627
\(288\) −1.97196 −0.116199
\(289\) −16.8878 −0.993403
\(290\) −0.124538 −0.00731314
\(291\) −0.0560785 −0.00328738
\(292\) −8.27334 −0.484161
\(293\) −14.1667 −0.827625 −0.413813 0.910362i \(-0.635803\pi\)
−0.413813 + 0.910362i \(0.635803\pi\)
\(294\) 0.167449 0.00976584
\(295\) 14.2610 0.830305
\(296\) −2.97587 −0.172969
\(297\) −1.00000 −0.0580259
\(298\) −1.37233 −0.0794968
\(299\) −6.33490 −0.366357
\(300\) 18.6830 1.07866
\(301\) −2.33490 −0.134581
\(302\) 2.22274 0.127904
\(303\) 18.8831 1.08481
\(304\) 31.1947 1.78914
\(305\) 22.8271 1.30707
\(306\) −0.0560785 −0.00320579
\(307\) −12.5576 −0.716702 −0.358351 0.933587i \(-0.616661\pi\)
−0.358351 + 0.933587i \(0.616661\pi\)
\(308\) 1.97196 0.112363
\(309\) 8.27882 0.470966
\(310\) 6.33490 0.359798
\(311\) 9.33959 0.529600 0.264800 0.964303i \(-0.414694\pi\)
0.264800 + 0.964303i \(0.414694\pi\)
\(312\) −2.53039 −0.143255
\(313\) −2.99530 −0.169305 −0.0846523 0.996411i \(-0.526978\pi\)
−0.0846523 + 0.996411i \(0.526978\pi\)
\(314\) 2.83646 0.160071
\(315\) −3.80451 −0.214360
\(316\) −6.56703 −0.369424
\(317\) 9.93453 0.557979 0.278989 0.960294i \(-0.410001\pi\)
0.278989 + 0.960294i \(0.410001\pi\)
\(318\) 1.33020 0.0745941
\(319\) 0.195488 0.0109453
\(320\) −27.9057 −1.55998
\(321\) 8.13941 0.454298
\(322\) −0.278820 −0.0155380
\(323\) 2.72588 0.151672
\(324\) −1.97196 −0.109553
\(325\) 36.0451 1.99942
\(326\) 1.13862 0.0630625
\(327\) −11.5529 −0.638879
\(328\) −4.17605 −0.230584
\(329\) 12.1394 0.669267
\(330\) 0.637062 0.0350691
\(331\) 22.5482 1.23936 0.619682 0.784853i \(-0.287262\pi\)
0.619682 + 0.784853i \(0.287262\pi\)
\(332\) 27.4969 1.50909
\(333\) −4.47431 −0.245191
\(334\) 3.05138 0.166964
\(335\) −0.530387 −0.0289781
\(336\) 3.83255 0.209083
\(337\) 13.2835 0.723599 0.361800 0.932256i \(-0.382162\pi\)
0.361800 + 0.932256i \(0.382162\pi\)
\(338\) −0.246872 −0.0134281
\(339\) 1.33020 0.0722467
\(340\) −2.51252 −0.136261
\(341\) −9.94392 −0.538494
\(342\) −1.36294 −0.0736992
\(343\) −1.00000 −0.0539949
\(344\) 1.55294 0.0837292
\(345\) 6.33490 0.341059
\(346\) 0.288210 0.0154943
\(347\) −27.7757 −1.49108 −0.745538 0.666463i \(-0.767808\pi\)
−0.745538 + 0.666463i \(0.767808\pi\)
\(348\) 0.385496 0.0206647
\(349\) −2.85589 −0.152873 −0.0764363 0.997074i \(-0.524354\pi\)
−0.0764363 + 0.997074i \(0.524354\pi\)
\(350\) 1.58647 0.0848001
\(351\) −3.80451 −0.203070
\(352\) −1.97196 −0.105106
\(353\) 24.6410 1.31151 0.655753 0.754975i \(-0.272351\pi\)
0.655753 + 0.754975i \(0.272351\pi\)
\(354\) 0.627672 0.0333604
\(355\) 17.7663 0.942937
\(356\) 19.4984 1.03342
\(357\) 0.334898 0.0177247
\(358\) 2.00939 0.106200
\(359\) −11.8878 −0.627416 −0.313708 0.949519i \(-0.601571\pi\)
−0.313708 + 0.949519i \(0.601571\pi\)
\(360\) 2.53039 0.133363
\(361\) 47.2500 2.48684
\(362\) −0.121547 −0.00638838
\(363\) −1.00000 −0.0524864
\(364\) 7.50235 0.393230
\(365\) 15.9618 0.835478
\(366\) 1.00470 0.0525163
\(367\) 3.60902 0.188389 0.0941947 0.995554i \(-0.469972\pi\)
0.0941947 + 0.995554i \(0.469972\pi\)
\(368\) −6.38159 −0.332663
\(369\) −6.27882 −0.326862
\(370\) 2.85041 0.148186
\(371\) −7.94392 −0.412428
\(372\) −19.6090 −1.01668
\(373\) 12.3349 0.638677 0.319338 0.947641i \(-0.396539\pi\)
0.319338 + 0.947641i \(0.396539\pi\)
\(374\) −0.0560785 −0.00289975
\(375\) −17.0226 −0.879041
\(376\) −8.07394 −0.416382
\(377\) 0.743738 0.0383045
\(378\) −0.167449 −0.00861266
\(379\) 23.3575 1.19979 0.599896 0.800078i \(-0.295209\pi\)
0.599896 + 0.800078i \(0.295209\pi\)
\(380\) −61.0647 −3.13255
\(381\) −14.6137 −0.748683
\(382\) −0.883148 −0.0451858
\(383\) −15.2180 −0.777606 −0.388803 0.921321i \(-0.627111\pi\)
−0.388803 + 0.921321i \(0.627111\pi\)
\(384\) −5.17214 −0.263940
\(385\) −3.80451 −0.193896
\(386\) 3.32081 0.169025
\(387\) 2.33490 0.118690
\(388\) −0.110585 −0.00561408
\(389\) 3.73057 0.189147 0.0945737 0.995518i \(-0.469851\pi\)
0.0945737 + 0.995518i \(0.469851\pi\)
\(390\) 2.42371 0.122729
\(391\) −0.557640 −0.0282011
\(392\) 0.665102 0.0335927
\(393\) −20.5576 −1.03700
\(394\) −0.447055 −0.0225223
\(395\) 12.6698 0.637487
\(396\) −1.97196 −0.0990948
\(397\) 20.8925 1.04857 0.524283 0.851544i \(-0.324333\pi\)
0.524283 + 0.851544i \(0.324333\pi\)
\(398\) −2.26943 −0.113756
\(399\) 8.13941 0.407480
\(400\) 36.3108 1.81554
\(401\) 23.2741 1.16225 0.581127 0.813813i \(-0.302612\pi\)
0.581127 + 0.813813i \(0.302612\pi\)
\(402\) −0.0233441 −0.00116430
\(403\) −37.8318 −1.88453
\(404\) 37.2368 1.85260
\(405\) 3.80451 0.189048
\(406\) 0.0327344 0.00162458
\(407\) −4.47431 −0.221783
\(408\) −0.222741 −0.0110273
\(409\) −23.8972 −1.18164 −0.590821 0.806803i \(-0.701196\pi\)
−0.590821 + 0.806803i \(0.701196\pi\)
\(410\) 4.00000 0.197546
\(411\) 16.2227 0.800209
\(412\) 16.3255 0.804300
\(413\) −3.74843 −0.184448
\(414\) 0.278820 0.0137033
\(415\) −53.0498 −2.60411
\(416\) −7.50235 −0.367833
\(417\) −22.5482 −1.10419
\(418\) −1.36294 −0.0666635
\(419\) −30.2967 −1.48009 −0.740045 0.672557i \(-0.765196\pi\)
−0.740045 + 0.672557i \(0.765196\pi\)
\(420\) −7.50235 −0.366077
\(421\) −15.5257 −0.756676 −0.378338 0.925668i \(-0.623504\pi\)
−0.378338 + 0.925668i \(0.623504\pi\)
\(422\) 0.716485 0.0348779
\(423\) −12.1394 −0.590238
\(424\) 5.28352 0.256590
\(425\) 3.17293 0.153910
\(426\) 0.781954 0.0378858
\(427\) −6.00000 −0.290360
\(428\) 16.0506 0.775835
\(429\) −3.80451 −0.183684
\(430\) −1.48748 −0.0717325
\(431\) 3.07864 0.148293 0.0741463 0.997247i \(-0.476377\pi\)
0.0741463 + 0.997247i \(0.476377\pi\)
\(432\) −3.83255 −0.184394
\(433\) 16.9392 0.814047 0.407024 0.913418i \(-0.366567\pi\)
0.407024 + 0.913418i \(0.366567\pi\)
\(434\) −1.66510 −0.0799274
\(435\) −0.743738 −0.0356595
\(436\) −22.7820 −1.09106
\(437\) −13.5529 −0.648325
\(438\) 0.702531 0.0335682
\(439\) −11.0786 −0.528754 −0.264377 0.964419i \(-0.585166\pi\)
−0.264377 + 0.964419i \(0.585166\pi\)
\(440\) 2.53039 0.120631
\(441\) 1.00000 0.0476190
\(442\) −0.213351 −0.0101481
\(443\) −14.5482 −0.691208 −0.345604 0.938380i \(-0.612326\pi\)
−0.345604 + 0.938380i \(0.612326\pi\)
\(444\) −8.82316 −0.418729
\(445\) −37.6184 −1.78328
\(446\) 1.71961 0.0814258
\(447\) −8.19549 −0.387633
\(448\) 7.33490 0.346541
\(449\) 27.4875 1.29721 0.648607 0.761123i \(-0.275352\pi\)
0.648607 + 0.761123i \(0.275352\pi\)
\(450\) −1.58647 −0.0747867
\(451\) −6.27882 −0.295658
\(452\) 2.62311 0.123381
\(453\) 13.2741 0.623673
\(454\) 0.0654688 0.00307260
\(455\) −14.4743 −0.678566
\(456\) −5.41353 −0.253512
\(457\) −7.94392 −0.371601 −0.185800 0.982587i \(-0.559488\pi\)
−0.185800 + 0.982587i \(0.559488\pi\)
\(458\) −1.33020 −0.0621563
\(459\) −0.334898 −0.0156317
\(460\) 12.4922 0.582450
\(461\) 8.26943 0.385146 0.192573 0.981283i \(-0.438317\pi\)
0.192573 + 0.981283i \(0.438317\pi\)
\(462\) −0.167449 −0.00779044
\(463\) 9.73904 0.452612 0.226306 0.974056i \(-0.427335\pi\)
0.226306 + 0.974056i \(0.427335\pi\)
\(464\) 0.749219 0.0347816
\(465\) 37.8318 1.75441
\(466\) −4.44706 −0.206006
\(467\) −40.6970 −1.88323 −0.941617 0.336685i \(-0.890694\pi\)
−0.941617 + 0.336685i \(0.890694\pi\)
\(468\) −7.50235 −0.346796
\(469\) 0.139410 0.00643735
\(470\) 7.73356 0.356723
\(471\) 16.9392 0.780518
\(472\) 2.49309 0.114754
\(473\) 2.33490 0.107359
\(474\) 0.557640 0.0256132
\(475\) 77.1153 3.53829
\(476\) 0.660406 0.0302697
\(477\) 7.94392 0.363727
\(478\) −1.80842 −0.0827152
\(479\) 37.8318 1.72858 0.864289 0.502996i \(-0.167769\pi\)
0.864289 + 0.502996i \(0.167769\pi\)
\(480\) 7.50235 0.342434
\(481\) −17.0226 −0.776162
\(482\) 2.02334 0.0921608
\(483\) −1.66510 −0.0757647
\(484\) −1.97196 −0.0896346
\(485\) 0.213351 0.00968778
\(486\) 0.167449 0.00759565
\(487\) 40.5576 1.83784 0.918921 0.394442i \(-0.129062\pi\)
0.918921 + 0.394442i \(0.129062\pi\)
\(488\) 3.99061 0.180646
\(489\) 6.79982 0.307498
\(490\) −0.637062 −0.0287795
\(491\) 31.0786 1.40256 0.701280 0.712886i \(-0.252612\pi\)
0.701280 + 0.712886i \(0.252612\pi\)
\(492\) −12.3816 −0.558205
\(493\) 0.0654688 0.00294856
\(494\) −5.18531 −0.233298
\(495\) 3.80451 0.171000
\(496\) −38.1106 −1.71122
\(497\) −4.66980 −0.209469
\(498\) −2.33490 −0.104629
\(499\) 15.3575 0.687494 0.343747 0.939062i \(-0.388304\pi\)
0.343747 + 0.939062i \(0.388304\pi\)
\(500\) −33.5678 −1.50120
\(501\) 18.2227 0.814132
\(502\) −0.805298 −0.0359422
\(503\) 14.8925 0.664025 0.332013 0.943275i \(-0.392272\pi\)
0.332013 + 0.943275i \(0.392272\pi\)
\(504\) −0.665102 −0.0296260
\(505\) −71.8412 −3.19689
\(506\) 0.278820 0.0123951
\(507\) −1.47431 −0.0654763
\(508\) −28.8177 −1.27858
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0.213351 0.00944735
\(511\) −4.19549 −0.185597
\(512\) −12.6557 −0.559309
\(513\) −8.13941 −0.359364
\(514\) 2.80530 0.123736
\(515\) −31.4969 −1.38792
\(516\) 4.60433 0.202694
\(517\) −12.1394 −0.533891
\(518\) −0.749219 −0.0329188
\(519\) 1.72118 0.0755514
\(520\) 9.62689 0.422167
\(521\) 27.6924 1.21322 0.606612 0.794998i \(-0.292528\pi\)
0.606612 + 0.794998i \(0.292528\pi\)
\(522\) −0.0327344 −0.00143274
\(523\) 18.4088 0.804962 0.402481 0.915428i \(-0.368148\pi\)
0.402481 + 0.915428i \(0.368148\pi\)
\(524\) −40.5389 −1.77095
\(525\) 9.47431 0.413493
\(526\) 2.03273 0.0886314
\(527\) −3.33020 −0.145066
\(528\) −3.83255 −0.166790
\(529\) −20.2274 −0.879454
\(530\) −5.06077 −0.219826
\(531\) 3.74843 0.162668
\(532\) 16.0506 0.695882
\(533\) −23.8878 −1.03470
\(534\) −1.65571 −0.0716496
\(535\) −30.9665 −1.33880
\(536\) −0.0927218 −0.00400497
\(537\) 12.0000 0.517838
\(538\) 4.22274 0.182055
\(539\) 1.00000 0.0430730
\(540\) 7.50235 0.322850
\(541\) −4.26943 −0.183557 −0.0917786 0.995779i \(-0.529255\pi\)
−0.0917786 + 0.995779i \(0.529255\pi\)
\(542\) −0.693141 −0.0297729
\(543\) −0.725875 −0.0311503
\(544\) −0.660406 −0.0283147
\(545\) 43.9533 1.88275
\(546\) −0.637062 −0.0272638
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 31.9906 1.36657
\(549\) 6.00000 0.256074
\(550\) −1.58647 −0.0676471
\(551\) 1.59116 0.0677857
\(552\) 1.10746 0.0471367
\(553\) −3.33020 −0.141615
\(554\) 3.06077 0.130040
\(555\) 17.0226 0.722567
\(556\) −44.4643 −1.88570
\(557\) −12.4743 −0.528553 −0.264277 0.964447i \(-0.585133\pi\)
−0.264277 + 0.964447i \(0.585133\pi\)
\(558\) 1.66510 0.0704894
\(559\) 8.88315 0.375717
\(560\) −14.5810 −0.616159
\(561\) −0.334898 −0.0141394
\(562\) −1.12923 −0.0476338
\(563\) −32.7710 −1.38113 −0.690566 0.723269i \(-0.742639\pi\)
−0.690566 + 0.723269i \(0.742639\pi\)
\(564\) −23.9384 −1.00799
\(565\) −5.06077 −0.212908
\(566\) −3.91119 −0.164399
\(567\) −1.00000 −0.0419961
\(568\) 3.10589 0.130320
\(569\) −29.2180 −1.22488 −0.612442 0.790515i \(-0.709813\pi\)
−0.612442 + 0.790515i \(0.709813\pi\)
\(570\) 5.18531 0.217189
\(571\) 32.4455 1.35780 0.678901 0.734230i \(-0.262457\pi\)
0.678901 + 0.734230i \(0.262457\pi\)
\(572\) −7.50235 −0.313689
\(573\) −5.27412 −0.220330
\(574\) −1.05138 −0.0438839
\(575\) −15.7757 −0.657892
\(576\) −7.33490 −0.305621
\(577\) −14.8831 −0.619594 −0.309797 0.950803i \(-0.600261\pi\)
−0.309797 + 0.950803i \(0.600261\pi\)
\(578\) 2.82786 0.117623
\(579\) 19.8318 0.824180
\(580\) −1.46662 −0.0608982
\(581\) 13.9439 0.578491
\(582\) 0.00939029 0.000389240 0
\(583\) 7.94392 0.329004
\(584\) 2.79043 0.115469
\(585\) 14.4743 0.598439
\(586\) 2.37220 0.0979945
\(587\) −4.53039 −0.186989 −0.0934945 0.995620i \(-0.529804\pi\)
−0.0934945 + 0.995620i \(0.529804\pi\)
\(588\) 1.97196 0.0813223
\(589\) −80.9377 −3.33498
\(590\) −2.38799 −0.0983118
\(591\) −2.66980 −0.109821
\(592\) −17.1480 −0.704779
\(593\) 8.28821 0.340356 0.170178 0.985413i \(-0.445566\pi\)
0.170178 + 0.985413i \(0.445566\pi\)
\(594\) 0.167449 0.00687052
\(595\) −1.27412 −0.0522340
\(596\) −16.1612 −0.661988
\(597\) −13.5529 −0.554685
\(598\) 1.06077 0.0433783
\(599\) 1.99061 0.0813341 0.0406671 0.999173i \(-0.487052\pi\)
0.0406671 + 0.999173i \(0.487052\pi\)
\(600\) −6.30138 −0.257253
\(601\) −8.47431 −0.345674 −0.172837 0.984950i \(-0.555293\pi\)
−0.172837 + 0.984950i \(0.555293\pi\)
\(602\) 0.390977 0.0159350
\(603\) −0.139410 −0.00567721
\(604\) 26.1761 1.06509
\(605\) 3.80451 0.154675
\(606\) −3.16197 −0.128446
\(607\) −22.9665 −0.932181 −0.466090 0.884737i \(-0.654338\pi\)
−0.466090 + 0.884737i \(0.654338\pi\)
\(608\) −16.0506 −0.650938
\(609\) 0.195488 0.00792159
\(610\) −3.82237 −0.154763
\(611\) −46.1845 −1.86843
\(612\) −0.660406 −0.0266953
\(613\) 3.55294 0.143502 0.0717510 0.997423i \(-0.477141\pi\)
0.0717510 + 0.997423i \(0.477141\pi\)
\(614\) 2.10277 0.0848608
\(615\) 23.8878 0.963251
\(616\) −0.665102 −0.0267977
\(617\) 22.6698 0.912652 0.456326 0.889813i \(-0.349165\pi\)
0.456326 + 0.889813i \(0.349165\pi\)
\(618\) −1.38628 −0.0557644
\(619\) 8.71648 0.350345 0.175173 0.984538i \(-0.443952\pi\)
0.175173 + 0.984538i \(0.443952\pi\)
\(620\) 74.6028 2.99612
\(621\) 1.66510 0.0668182
\(622\) −1.56391 −0.0627070
\(623\) 9.88784 0.396148
\(624\) −14.5810 −0.583707
\(625\) 17.3910 0.695639
\(626\) 0.501561 0.0200464
\(627\) −8.13941 −0.325057
\(628\) 33.4035 1.33294
\(629\) −1.49844 −0.0597467
\(630\) 0.637062 0.0253812
\(631\) −11.8878 −0.473248 −0.236624 0.971601i \(-0.576041\pi\)
−0.236624 + 0.971601i \(0.576041\pi\)
\(632\) 2.21492 0.0881049
\(633\) 4.27882 0.170068
\(634\) −1.66353 −0.0660672
\(635\) 55.5981 2.20634
\(636\) 15.6651 0.621162
\(637\) 3.80451 0.150740
\(638\) −0.0327344 −0.00129597
\(639\) 4.66980 0.184734
\(640\) 19.6775 0.777821
\(641\) −4.89254 −0.193244 −0.0966218 0.995321i \(-0.530804\pi\)
−0.0966218 + 0.995321i \(0.530804\pi\)
\(642\) −1.36294 −0.0537909
\(643\) 22.7804 0.898371 0.449185 0.893439i \(-0.351714\pi\)
0.449185 + 0.893439i \(0.351714\pi\)
\(644\) −3.28352 −0.129389
\(645\) −8.88315 −0.349774
\(646\) −0.456446 −0.0179586
\(647\) 31.2453 1.22838 0.614190 0.789158i \(-0.289483\pi\)
0.614190 + 0.789158i \(0.289483\pi\)
\(648\) 0.665102 0.0261277
\(649\) 3.74843 0.147139
\(650\) −6.03573 −0.236741
\(651\) −9.94392 −0.389733
\(652\) 13.4090 0.525136
\(653\) 28.2882 1.10700 0.553502 0.832848i \(-0.313291\pi\)
0.553502 + 0.832848i \(0.313291\pi\)
\(654\) 1.93453 0.0756462
\(655\) 78.2118 3.05599
\(656\) −24.0639 −0.939537
\(657\) 4.19549 0.163682
\(658\) −2.03273 −0.0792442
\(659\) 30.2967 1.18019 0.590096 0.807333i \(-0.299090\pi\)
0.590096 + 0.807333i \(0.299090\pi\)
\(660\) 7.50235 0.292028
\(661\) −35.7196 −1.38933 −0.694666 0.719333i \(-0.744448\pi\)
−0.694666 + 0.719333i \(0.744448\pi\)
\(662\) −3.77569 −0.146746
\(663\) −1.27412 −0.0494829
\(664\) −9.27412 −0.359906
\(665\) −30.9665 −1.20083
\(666\) 0.749219 0.0290317
\(667\) −0.325508 −0.0126037
\(668\) 35.9345 1.39035
\(669\) 10.2694 0.397039
\(670\) 0.0888128 0.00343114
\(671\) 6.00000 0.231627
\(672\) −1.97196 −0.0760700
\(673\) 34.9377 1.34675 0.673374 0.739302i \(-0.264844\pi\)
0.673374 + 0.739302i \(0.264844\pi\)
\(674\) −2.22431 −0.0856774
\(675\) −9.47431 −0.364666
\(676\) −2.90728 −0.111818
\(677\) 20.0094 0.769023 0.384512 0.923120i \(-0.374370\pi\)
0.384512 + 0.923120i \(0.374370\pi\)
\(678\) −0.222741 −0.00855433
\(679\) −0.0560785 −0.00215209
\(680\) 0.847422 0.0324972
\(681\) 0.390977 0.0149823
\(682\) 1.66510 0.0637600
\(683\) 25.0498 0.958504 0.479252 0.877677i \(-0.340908\pi\)
0.479252 + 0.877677i \(0.340908\pi\)
\(684\) −16.0506 −0.613710
\(685\) −61.7196 −2.35818
\(686\) 0.167449 0.00639324
\(687\) −7.94392 −0.303080
\(688\) 8.94862 0.341163
\(689\) 30.2227 1.15139
\(690\) −1.06077 −0.0403830
\(691\) −38.8271 −1.47705 −0.738526 0.674225i \(-0.764478\pi\)
−0.738526 + 0.674225i \(0.764478\pi\)
\(692\) 3.39410 0.129024
\(693\) −1.00000 −0.0379869
\(694\) 4.65102 0.176550
\(695\) 85.7851 3.25401
\(696\) −0.130020 −0.00492838
\(697\) −2.10277 −0.0796480
\(698\) 0.478217 0.0181008
\(699\) −26.5576 −1.00450
\(700\) 18.6830 0.706150
\(701\) −41.7757 −1.57785 −0.788923 0.614492i \(-0.789361\pi\)
−0.788923 + 0.614492i \(0.789361\pi\)
\(702\) 0.637062 0.0240444
\(703\) −36.4182 −1.37354
\(704\) −7.33490 −0.276444
\(705\) 46.1845 1.73941
\(706\) −4.12611 −0.155288
\(707\) 18.8831 0.710174
\(708\) 7.39176 0.277799
\(709\) −13.4041 −0.503403 −0.251702 0.967805i \(-0.580990\pi\)
−0.251702 + 0.967805i \(0.580990\pi\)
\(710\) −2.97495 −0.111648
\(711\) 3.33020 0.124892
\(712\) −6.57642 −0.246462
\(713\) 16.5576 0.620088
\(714\) −0.0560785 −0.00209868
\(715\) 14.4743 0.541308
\(716\) 23.6635 0.884348
\(717\) −10.7998 −0.403327
\(718\) 1.99061 0.0742889
\(719\) 5.47900 0.204332 0.102166 0.994767i \(-0.467423\pi\)
0.102166 + 0.994767i \(0.467423\pi\)
\(720\) 14.5810 0.543401
\(721\) 8.27882 0.308319
\(722\) −7.91197 −0.294453
\(723\) 12.0833 0.449384
\(724\) −1.43140 −0.0531975
\(725\) 1.85212 0.0687859
\(726\) 0.167449 0.00621462
\(727\) −32.8831 −1.21957 −0.609784 0.792567i \(-0.708744\pi\)
−0.609784 + 0.792567i \(0.708744\pi\)
\(728\) −2.53039 −0.0937824
\(729\) 1.00000 0.0370370
\(730\) −2.67279 −0.0989243
\(731\) 0.781954 0.0289216
\(732\) 11.8318 0.437315
\(733\) −35.8972 −1.32589 −0.662947 0.748666i \(-0.730695\pi\)
−0.662947 + 0.748666i \(0.730695\pi\)
\(734\) −0.604328 −0.0223062
\(735\) −3.80451 −0.140332
\(736\) 3.28352 0.121032
\(737\) −0.139410 −0.00513523
\(738\) 1.05138 0.0387020
\(739\) 29.6651 1.09125 0.545624 0.838030i \(-0.316293\pi\)
0.545624 + 0.838030i \(0.316293\pi\)
\(740\) 33.5678 1.23398
\(741\) −30.9665 −1.13758
\(742\) 1.33020 0.0488333
\(743\) −18.7998 −0.689698 −0.344849 0.938658i \(-0.612070\pi\)
−0.344849 + 0.938658i \(0.612070\pi\)
\(744\) 6.61372 0.242471
\(745\) 31.1798 1.14234
\(746\) −2.06547 −0.0756222
\(747\) −13.9439 −0.510181
\(748\) −0.660406 −0.0241469
\(749\) 8.13941 0.297408
\(750\) 2.85041 0.104082
\(751\) −9.59116 −0.349986 −0.174993 0.984570i \(-0.555990\pi\)
−0.174993 + 0.984570i \(0.555990\pi\)
\(752\) −46.5249 −1.69659
\(753\) −4.80921 −0.175257
\(754\) −0.124538 −0.00453542
\(755\) −50.5016 −1.83794
\(756\) −1.97196 −0.0717195
\(757\) 2.74374 0.0997229 0.0498614 0.998756i \(-0.484122\pi\)
0.0498614 + 0.998756i \(0.484122\pi\)
\(758\) −3.91119 −0.142061
\(759\) 1.66510 0.0604394
\(760\) 20.5959 0.747090
\(761\) 6.39098 0.231673 0.115836 0.993268i \(-0.463045\pi\)
0.115836 + 0.993268i \(0.463045\pi\)
\(762\) 2.44706 0.0886475
\(763\) −11.5529 −0.418245
\(764\) −10.4004 −0.376272
\(765\) 1.27412 0.0460661
\(766\) 2.54825 0.0920720
\(767\) 14.2610 0.514933
\(768\) −13.8037 −0.498099
\(769\) 17.7951 0.641708 0.320854 0.947129i \(-0.396030\pi\)
0.320854 + 0.947129i \(0.396030\pi\)
\(770\) 0.637062 0.0229581
\(771\) 16.7531 0.603349
\(772\) 39.1075 1.40751
\(773\) −31.5257 −1.13390 −0.566950 0.823752i \(-0.691877\pi\)
−0.566950 + 0.823752i \(0.691877\pi\)
\(774\) −0.390977 −0.0140534
\(775\) −94.2118 −3.38419
\(776\) 0.0372979 0.00133892
\(777\) −4.47431 −0.160515
\(778\) −0.624681 −0.0223959
\(779\) −51.1059 −1.83106
\(780\) 28.5428 1.02200
\(781\) 4.66980 0.167098
\(782\) 0.0933763 0.00333913
\(783\) −0.195488 −0.00698619
\(784\) 3.83255 0.136877
\(785\) −64.4455 −2.30016
\(786\) 3.44236 0.122785
\(787\) 31.8606 1.13571 0.567854 0.823130i \(-0.307774\pi\)
0.567854 + 0.823130i \(0.307774\pi\)
\(788\) −5.26473 −0.187548
\(789\) 12.1394 0.432174
\(790\) −2.12155 −0.0754813
\(791\) 1.33020 0.0472966
\(792\) 0.665102 0.0236334
\(793\) 22.8271 0.810613
\(794\) −3.49844 −0.124155
\(795\) −30.2227 −1.07189
\(796\) −26.7259 −0.947274
\(797\) 43.8590 1.55357 0.776783 0.629768i \(-0.216850\pi\)
0.776783 + 0.629768i \(0.216850\pi\)
\(798\) −1.36294 −0.0482475
\(799\) −4.06547 −0.143826
\(800\) −18.6830 −0.660543
\(801\) −9.88784 −0.349370
\(802\) −3.89723 −0.137616
\(803\) 4.19549 0.148056
\(804\) −0.274911 −0.00969536
\(805\) 6.33490 0.223276
\(806\) 6.33490 0.223137
\(807\) 25.2180 0.887717
\(808\) −12.5592 −0.441832
\(809\) 6.74374 0.237097 0.118549 0.992948i \(-0.462176\pi\)
0.118549 + 0.992948i \(0.462176\pi\)
\(810\) −0.637062 −0.0223841
\(811\) 3.97275 0.139502 0.0697510 0.997564i \(-0.477780\pi\)
0.0697510 + 0.997564i \(0.477780\pi\)
\(812\) 0.385496 0.0135282
\(813\) −4.13941 −0.145175
\(814\) 0.749219 0.0262601
\(815\) −25.8700 −0.906186
\(816\) −1.28352 −0.0449320
\(817\) 19.0047 0.664890
\(818\) 4.00157 0.139912
\(819\) −3.80451 −0.132940
\(820\) 47.1059 1.64501
\(821\) 49.5896 1.73069 0.865344 0.501178i \(-0.167100\pi\)
0.865344 + 0.501178i \(0.167100\pi\)
\(822\) −2.71648 −0.0947483
\(823\) 23.1908 0.808380 0.404190 0.914675i \(-0.367553\pi\)
0.404190 + 0.914675i \(0.367553\pi\)
\(824\) −5.50626 −0.191820
\(825\) −9.47431 −0.329853
\(826\) 0.627672 0.0218395
\(827\) 27.7484 0.964908 0.482454 0.875921i \(-0.339746\pi\)
0.482454 + 0.875921i \(0.339746\pi\)
\(828\) 3.28352 0.114110
\(829\) 0.269430 0.00935768 0.00467884 0.999989i \(-0.498511\pi\)
0.00467884 + 0.999989i \(0.498511\pi\)
\(830\) 8.88315 0.308339
\(831\) 18.2788 0.634085
\(832\) −27.9057 −0.967456
\(833\) 0.334898 0.0116035
\(834\) 3.77569 0.130741
\(835\) −69.3286 −2.39922
\(836\) −16.0506 −0.555122
\(837\) 9.94392 0.343712
\(838\) 5.07316 0.175249
\(839\) −27.3575 −0.944484 −0.472242 0.881469i \(-0.656555\pi\)
−0.472242 + 0.881469i \(0.656555\pi\)
\(840\) 2.53039 0.0873066
\(841\) −28.9618 −0.998682
\(842\) 2.59976 0.0895938
\(843\) −6.74374 −0.232267
\(844\) 8.43767 0.290436
\(845\) 5.60902 0.192956
\(846\) 2.03273 0.0698868
\(847\) −1.00000 −0.0343604
\(848\) 30.4455 1.04550
\(849\) −23.3575 −0.801626
\(850\) −0.531305 −0.0182236
\(851\) 7.45018 0.255389
\(852\) 9.20866 0.315483
\(853\) −28.5482 −0.977473 −0.488737 0.872431i \(-0.662542\pi\)
−0.488737 + 0.872431i \(0.662542\pi\)
\(854\) 1.00470 0.0343800
\(855\) 30.9665 1.05903
\(856\) −5.41353 −0.185031
\(857\) −39.8972 −1.36286 −0.681432 0.731882i \(-0.738642\pi\)
−0.681432 + 0.731882i \(0.738642\pi\)
\(858\) 0.637062 0.0217490
\(859\) 20.5576 0.701418 0.350709 0.936485i \(-0.385941\pi\)
0.350709 + 0.936485i \(0.385941\pi\)
\(860\) −17.5172 −0.597332
\(861\) −6.27882 −0.213982
\(862\) −0.515515 −0.0175585
\(863\) −9.66510 −0.329004 −0.164502 0.986377i \(-0.552602\pi\)
−0.164502 + 0.986377i \(0.552602\pi\)
\(864\) 1.97196 0.0670875
\(865\) −6.54825 −0.222647
\(866\) −2.83646 −0.0963868
\(867\) 16.8878 0.573541
\(868\) −19.6090 −0.665574
\(869\) 3.33020 0.112969
\(870\) 0.124538 0.00422224
\(871\) −0.530387 −0.0179715
\(872\) 7.68388 0.260209
\(873\) 0.0560785 0.00189797
\(874\) 2.26943 0.0767646
\(875\) −17.0226 −0.575467
\(876\) 8.27334 0.279530
\(877\) −25.7212 −0.868543 −0.434271 0.900782i \(-0.642994\pi\)
−0.434271 + 0.900782i \(0.642994\pi\)
\(878\) 1.85511 0.0626069
\(879\) 14.1667 0.477830
\(880\) 14.5810 0.491525
\(881\) −14.6316 −0.492950 −0.246475 0.969149i \(-0.579272\pi\)
−0.246475 + 0.969149i \(0.579272\pi\)
\(882\) −0.167449 −0.00563831
\(883\) 18.6877 0.628890 0.314445 0.949276i \(-0.398182\pi\)
0.314445 + 0.949276i \(0.398182\pi\)
\(884\) −2.51252 −0.0845053
\(885\) −14.2610 −0.479377
\(886\) 2.43609 0.0818421
\(887\) 38.6137 1.29652 0.648261 0.761418i \(-0.275497\pi\)
0.648261 + 0.761418i \(0.275497\pi\)
\(888\) 2.97587 0.0998636
\(889\) −14.6137 −0.490128
\(890\) 6.29917 0.211149
\(891\) 1.00000 0.0335013
\(892\) 20.2509 0.678051
\(893\) −98.8076 −3.30647
\(894\) 1.37233 0.0458975
\(895\) −45.6541 −1.52605
\(896\) −5.17214 −0.172789
\(897\) 6.33490 0.211516
\(898\) −4.60276 −0.153596
\(899\) −1.94392 −0.0648334
\(900\) −18.6830 −0.622765
\(901\) 2.66041 0.0886310
\(902\) 1.05138 0.0350072
\(903\) 2.33490 0.0777006
\(904\) −0.884720 −0.0294254
\(905\) 2.76160 0.0917987
\(906\) −2.22274 −0.0738456
\(907\) 49.0965 1.63022 0.815111 0.579304i \(-0.196676\pi\)
0.815111 + 0.579304i \(0.196676\pi\)
\(908\) 0.770991 0.0255862
\(909\) −18.8831 −0.626314
\(910\) 2.42371 0.0803452
\(911\) 29.3753 0.973248 0.486624 0.873612i \(-0.338228\pi\)
0.486624 + 0.873612i \(0.338228\pi\)
\(912\) −31.1947 −1.03296
\(913\) −13.9439 −0.461476
\(914\) 1.33020 0.0439992
\(915\) −22.8271 −0.754640
\(916\) −15.6651 −0.517590
\(917\) −20.5576 −0.678873
\(918\) 0.0560785 0.00185087
\(919\) −27.0047 −0.890803 −0.445401 0.895331i \(-0.646939\pi\)
−0.445401 + 0.895331i \(0.646939\pi\)
\(920\) −4.21335 −0.138910
\(921\) 12.5576 0.413788
\(922\) −1.38471 −0.0456030
\(923\) 17.7663 0.584785
\(924\) −1.97196 −0.0648727
\(925\) −42.3910 −1.39381
\(926\) −1.63079 −0.0535912
\(927\) −8.27882 −0.271912
\(928\) −0.385496 −0.0126545
\(929\) 57.8496 1.89798 0.948992 0.315299i \(-0.102105\pi\)
0.948992 + 0.315299i \(0.102105\pi\)
\(930\) −6.33490 −0.207730
\(931\) 8.13941 0.266758
\(932\) −52.3706 −1.71546
\(933\) −9.33959 −0.305765
\(934\) 6.81469 0.222983
\(935\) 1.27412 0.0416683
\(936\) 2.53039 0.0827083
\(937\) 25.2180 0.823838 0.411919 0.911221i \(-0.364859\pi\)
0.411919 + 0.911221i \(0.364859\pi\)
\(938\) −0.0233441 −0.000762211 0
\(939\) 2.99530 0.0977481
\(940\) 91.0741 2.97051
\(941\) −34.2788 −1.11746 −0.558729 0.829350i \(-0.688711\pi\)
−0.558729 + 0.829350i \(0.688711\pi\)
\(942\) −2.83646 −0.0924169
\(943\) 10.4549 0.340458
\(944\) 14.3661 0.467575
\(945\) 3.80451 0.123761
\(946\) −0.390977 −0.0127118
\(947\) −18.1573 −0.590032 −0.295016 0.955492i \(-0.595325\pi\)
−0.295016 + 0.955492i \(0.595325\pi\)
\(948\) 6.56703 0.213287
\(949\) 15.9618 0.518141
\(950\) −12.9129 −0.418950
\(951\) −9.93453 −0.322149
\(952\) −0.222741 −0.00721909
\(953\) −19.8045 −0.641531 −0.320766 0.947159i \(-0.603940\pi\)
−0.320766 + 0.947159i \(0.603940\pi\)
\(954\) −1.33020 −0.0430669
\(955\) 20.0655 0.649303
\(956\) −21.2968 −0.688788
\(957\) −0.195488 −0.00631924
\(958\) −6.33490 −0.204671
\(959\) 16.2227 0.523860
\(960\) 27.9057 0.900653
\(961\) 67.8816 2.18973
\(962\) 2.85041 0.0919010
\(963\) −8.13941 −0.262289
\(964\) 23.8279 0.767444
\(965\) −75.4502 −2.42883
\(966\) 0.278820 0.00897088
\(967\) −0.278820 −0.00896624 −0.00448312 0.999990i \(-0.501427\pi\)
−0.00448312 + 0.999990i \(0.501427\pi\)
\(968\) 0.665102 0.0213772
\(969\) −2.72588 −0.0875677
\(970\) −0.0357255 −0.00114708
\(971\) 25.3481 0.813458 0.406729 0.913549i \(-0.366669\pi\)
0.406729 + 0.913549i \(0.366669\pi\)
\(972\) 1.97196 0.0632507
\(973\) −22.5482 −0.722864
\(974\) −6.79134 −0.217609
\(975\) −36.0451 −1.15437
\(976\) 22.9953 0.736062
\(977\) 6.71648 0.214879 0.107440 0.994212i \(-0.465735\pi\)
0.107440 + 0.994212i \(0.465735\pi\)
\(978\) −1.13862 −0.0364092
\(979\) −9.88784 −0.316017
\(980\) −7.50235 −0.239654
\(981\) 11.5529 0.368857
\(982\) −5.20409 −0.166069
\(983\) 9.99061 0.318651 0.159325 0.987226i \(-0.449068\pi\)
0.159325 + 0.987226i \(0.449068\pi\)
\(984\) 4.17605 0.133128
\(985\) 10.1573 0.323638
\(986\) −0.0109627 −0.000349123 0
\(987\) −12.1394 −0.386402
\(988\) −61.0647 −1.94273
\(989\) −3.88784 −0.123626
\(990\) −0.637062 −0.0202472
\(991\) −26.7064 −0.848358 −0.424179 0.905578i \(-0.639437\pi\)
−0.424179 + 0.905578i \(0.639437\pi\)
\(992\) 19.6090 0.622587
\(993\) −22.5482 −0.715547
\(994\) 0.781954 0.0248021
\(995\) 51.5623 1.63464
\(996\) −27.4969 −0.871272
\(997\) 54.3333 1.72075 0.860377 0.509658i \(-0.170228\pi\)
0.860377 + 0.509658i \(0.170228\pi\)
\(998\) −2.57159 −0.0814024
\(999\) 4.47431 0.141561
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.d.1.2 3
3.2 odd 2 693.2.a.m.1.2 3
4.3 odd 2 3696.2.a.bp.1.3 3
5.4 even 2 5775.2.a.bw.1.2 3
7.6 odd 2 1617.2.a.s.1.2 3
11.10 odd 2 2541.2.a.bi.1.2 3
21.20 even 2 4851.2.a.bp.1.2 3
33.32 even 2 7623.2.a.cb.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.d.1.2 3 1.1 even 1 trivial
693.2.a.m.1.2 3 3.2 odd 2
1617.2.a.s.1.2 3 7.6 odd 2
2541.2.a.bi.1.2 3 11.10 odd 2
3696.2.a.bp.1.3 3 4.3 odd 2
4851.2.a.bp.1.2 3 21.20 even 2
5775.2.a.bw.1.2 3 5.4 even 2
7623.2.a.cb.1.2 3 33.32 even 2