Properties

Label 3822.2.a.bl.1.2
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.5188236525\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3822.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.41421 q^{11} +1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +0.414214 q^{17} -1.00000 q^{18} -3.82843 q^{19} -1.00000 q^{20} -2.41421 q^{22} -8.65685 q^{23} -1.00000 q^{24} -4.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} +2.65685 q^{29} +1.00000 q^{30} +4.24264 q^{31} -1.00000 q^{32} +2.41421 q^{33} -0.414214 q^{34} +1.00000 q^{36} -3.24264 q^{37} +3.82843 q^{38} +1.00000 q^{39} +1.00000 q^{40} -6.82843 q^{41} -7.00000 q^{43} +2.41421 q^{44} -1.00000 q^{45} +8.65685 q^{46} -7.65685 q^{47} +1.00000 q^{48} +4.00000 q^{50} +0.414214 q^{51} +1.00000 q^{52} -13.6569 q^{53} -1.00000 q^{54} -2.41421 q^{55} -3.82843 q^{57} -2.65685 q^{58} +9.89949 q^{59} -1.00000 q^{60} +5.58579 q^{61} -4.24264 q^{62} +1.00000 q^{64} -1.00000 q^{65} -2.41421 q^{66} +1.41421 q^{67} +0.414214 q^{68} -8.65685 q^{69} +5.07107 q^{71} -1.00000 q^{72} +11.8284 q^{73} +3.24264 q^{74} -4.00000 q^{75} -3.82843 q^{76} -1.00000 q^{78} +10.2426 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.82843 q^{82} -11.6569 q^{83} -0.414214 q^{85} +7.00000 q^{86} +2.65685 q^{87} -2.41421 q^{88} -0.585786 q^{89} +1.00000 q^{90} -8.65685 q^{92} +4.24264 q^{93} +7.65685 q^{94} +3.82843 q^{95} -1.00000 q^{96} -7.31371 q^{97} +2.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{11} + 2 q^{12} + 2 q^{13} - 2 q^{15} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 2 q^{19} - 2 q^{20} - 2 q^{22} - 6 q^{23} - 2 q^{24} - 8 q^{25} - 2 q^{26} + 2 q^{27} - 6 q^{29} + 2 q^{30} - 2 q^{32} + 2 q^{33} + 2 q^{34} + 2 q^{36} + 2 q^{37} + 2 q^{38} + 2 q^{39} + 2 q^{40} - 8 q^{41} - 14 q^{43} + 2 q^{44} - 2 q^{45} + 6 q^{46} - 4 q^{47} + 2 q^{48} + 8 q^{50} - 2 q^{51} + 2 q^{52} - 16 q^{53} - 2 q^{54} - 2 q^{55} - 2 q^{57} + 6 q^{58} - 2 q^{60} + 14 q^{61} + 2 q^{64} - 2 q^{65} - 2 q^{66} - 2 q^{68} - 6 q^{69} - 4 q^{71} - 2 q^{72} + 18 q^{73} - 2 q^{74} - 8 q^{75} - 2 q^{76} - 2 q^{78} + 12 q^{79} - 2 q^{80} + 2 q^{81} + 8 q^{82} - 12 q^{83} + 2 q^{85} + 14 q^{86} - 6 q^{87} - 2 q^{88} - 4 q^{89} + 2 q^{90} - 6 q^{92} + 4 q^{94} + 2 q^{95} - 2 q^{96} + 8 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 2.41421 0.727913 0.363956 0.931416i \(-0.381426\pi\)
0.363956 + 0.931416i \(0.381426\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 0.414214 0.100462 0.0502308 0.998738i \(-0.484004\pi\)
0.0502308 + 0.998738i \(0.484004\pi\)
\(18\) −1.00000 −0.235702
\(19\) −3.82843 −0.878301 −0.439151 0.898413i \(-0.644721\pi\)
−0.439151 + 0.898413i \(0.644721\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −2.41421 −0.514712
\(23\) −8.65685 −1.80508 −0.902539 0.430607i \(-0.858299\pi\)
−0.902539 + 0.430607i \(0.858299\pi\)
\(24\) −1.00000 −0.204124
\(25\) −4.00000 −0.800000
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 2.65685 0.493365 0.246683 0.969096i \(-0.420659\pi\)
0.246683 + 0.969096i \(0.420659\pi\)
\(30\) 1.00000 0.182574
\(31\) 4.24264 0.762001 0.381000 0.924575i \(-0.375580\pi\)
0.381000 + 0.924575i \(0.375580\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.41421 0.420261
\(34\) −0.414214 −0.0710370
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −3.24264 −0.533087 −0.266543 0.963823i \(-0.585882\pi\)
−0.266543 + 0.963823i \(0.585882\pi\)
\(38\) 3.82843 0.621053
\(39\) 1.00000 0.160128
\(40\) 1.00000 0.158114
\(41\) −6.82843 −1.06642 −0.533211 0.845983i \(-0.679015\pi\)
−0.533211 + 0.845983i \(0.679015\pi\)
\(42\) 0 0
\(43\) −7.00000 −1.06749 −0.533745 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) 2.41421 0.363956
\(45\) −1.00000 −0.149071
\(46\) 8.65685 1.27638
\(47\) −7.65685 −1.11687 −0.558433 0.829549i \(-0.688597\pi\)
−0.558433 + 0.829549i \(0.688597\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 4.00000 0.565685
\(51\) 0.414214 0.0580015
\(52\) 1.00000 0.138675
\(53\) −13.6569 −1.87591 −0.937957 0.346753i \(-0.887284\pi\)
−0.937957 + 0.346753i \(0.887284\pi\)
\(54\) −1.00000 −0.136083
\(55\) −2.41421 −0.325532
\(56\) 0 0
\(57\) −3.82843 −0.507088
\(58\) −2.65685 −0.348862
\(59\) 9.89949 1.28880 0.644402 0.764687i \(-0.277106\pi\)
0.644402 + 0.764687i \(0.277106\pi\)
\(60\) −1.00000 −0.129099
\(61\) 5.58579 0.715187 0.357593 0.933877i \(-0.383597\pi\)
0.357593 + 0.933877i \(0.383597\pi\)
\(62\) −4.24264 −0.538816
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) −2.41421 −0.297169
\(67\) 1.41421 0.172774 0.0863868 0.996262i \(-0.472468\pi\)
0.0863868 + 0.996262i \(0.472468\pi\)
\(68\) 0.414214 0.0502308
\(69\) −8.65685 −1.04216
\(70\) 0 0
\(71\) 5.07107 0.601825 0.300913 0.953652i \(-0.402709\pi\)
0.300913 + 0.953652i \(0.402709\pi\)
\(72\) −1.00000 −0.117851
\(73\) 11.8284 1.38441 0.692206 0.721700i \(-0.256639\pi\)
0.692206 + 0.721700i \(0.256639\pi\)
\(74\) 3.24264 0.376949
\(75\) −4.00000 −0.461880
\(76\) −3.82843 −0.439151
\(77\) 0 0
\(78\) −1.00000 −0.113228
\(79\) 10.2426 1.15239 0.576194 0.817313i \(-0.304537\pi\)
0.576194 + 0.817313i \(0.304537\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 6.82843 0.754074
\(83\) −11.6569 −1.27951 −0.639753 0.768581i \(-0.720963\pi\)
−0.639753 + 0.768581i \(0.720963\pi\)
\(84\) 0 0
\(85\) −0.414214 −0.0449278
\(86\) 7.00000 0.754829
\(87\) 2.65685 0.284845
\(88\) −2.41421 −0.257356
\(89\) −0.585786 −0.0620932 −0.0310466 0.999518i \(-0.509884\pi\)
−0.0310466 + 0.999518i \(0.509884\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −8.65685 −0.902539
\(93\) 4.24264 0.439941
\(94\) 7.65685 0.789744
\(95\) 3.82843 0.392788
\(96\) −1.00000 −0.102062
\(97\) −7.31371 −0.742595 −0.371297 0.928514i \(-0.621087\pi\)
−0.371297 + 0.928514i \(0.621087\pi\)
\(98\) 0 0
\(99\) 2.41421 0.242638
\(100\) −4.00000 −0.400000
\(101\) 1.07107 0.106575 0.0532876 0.998579i \(-0.483030\pi\)
0.0532876 + 0.998579i \(0.483030\pi\)
\(102\) −0.414214 −0.0410133
\(103\) 8.07107 0.795266 0.397633 0.917545i \(-0.369832\pi\)
0.397633 + 0.917545i \(0.369832\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 13.6569 1.32647
\(107\) 4.58579 0.443325 0.221662 0.975123i \(-0.428852\pi\)
0.221662 + 0.975123i \(0.428852\pi\)
\(108\) 1.00000 0.0962250
\(109\) 0.414214 0.0396745 0.0198372 0.999803i \(-0.493685\pi\)
0.0198372 + 0.999803i \(0.493685\pi\)
\(110\) 2.41421 0.230186
\(111\) −3.24264 −0.307778
\(112\) 0 0
\(113\) −18.2426 −1.71612 −0.858062 0.513547i \(-0.828331\pi\)
−0.858062 + 0.513547i \(0.828331\pi\)
\(114\) 3.82843 0.358565
\(115\) 8.65685 0.807256
\(116\) 2.65685 0.246683
\(117\) 1.00000 0.0924500
\(118\) −9.89949 −0.911322
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) −5.17157 −0.470143
\(122\) −5.58579 −0.505713
\(123\) −6.82843 −0.615699
\(124\) 4.24264 0.381000
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −2.92893 −0.259901 −0.129950 0.991521i \(-0.541482\pi\)
−0.129950 + 0.991521i \(0.541482\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −7.00000 −0.616316
\(130\) 1.00000 0.0877058
\(131\) −17.2426 −1.50650 −0.753248 0.657736i \(-0.771514\pi\)
−0.753248 + 0.657736i \(0.771514\pi\)
\(132\) 2.41421 0.210130
\(133\) 0 0
\(134\) −1.41421 −0.122169
\(135\) −1.00000 −0.0860663
\(136\) −0.414214 −0.0355185
\(137\) −7.24264 −0.618781 −0.309390 0.950935i \(-0.600125\pi\)
−0.309390 + 0.950935i \(0.600125\pi\)
\(138\) 8.65685 0.736920
\(139\) 6.82843 0.579180 0.289590 0.957151i \(-0.406481\pi\)
0.289590 + 0.957151i \(0.406481\pi\)
\(140\) 0 0
\(141\) −7.65685 −0.644823
\(142\) −5.07107 −0.425555
\(143\) 2.41421 0.201887
\(144\) 1.00000 0.0833333
\(145\) −2.65685 −0.220640
\(146\) −11.8284 −0.978928
\(147\) 0 0
\(148\) −3.24264 −0.266543
\(149\) −2.82843 −0.231714 −0.115857 0.993266i \(-0.536961\pi\)
−0.115857 + 0.993266i \(0.536961\pi\)
\(150\) 4.00000 0.326599
\(151\) −13.2426 −1.07767 −0.538835 0.842411i \(-0.681136\pi\)
−0.538835 + 0.842411i \(0.681136\pi\)
\(152\) 3.82843 0.310526
\(153\) 0.414214 0.0334872
\(154\) 0 0
\(155\) −4.24264 −0.340777
\(156\) 1.00000 0.0800641
\(157\) 2.89949 0.231405 0.115702 0.993284i \(-0.463088\pi\)
0.115702 + 0.993284i \(0.463088\pi\)
\(158\) −10.2426 −0.814861
\(159\) −13.6569 −1.08306
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −11.6569 −0.913035 −0.456518 0.889714i \(-0.650903\pi\)
−0.456518 + 0.889714i \(0.650903\pi\)
\(164\) −6.82843 −0.533211
\(165\) −2.41421 −0.187946
\(166\) 11.6569 0.904747
\(167\) −13.8284 −1.07008 −0.535038 0.844828i \(-0.679703\pi\)
−0.535038 + 0.844828i \(0.679703\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0.414214 0.0317687
\(171\) −3.82843 −0.292767
\(172\) −7.00000 −0.533745
\(173\) −16.2426 −1.23491 −0.617453 0.786608i \(-0.711835\pi\)
−0.617453 + 0.786608i \(0.711835\pi\)
\(174\) −2.65685 −0.201416
\(175\) 0 0
\(176\) 2.41421 0.181978
\(177\) 9.89949 0.744092
\(178\) 0.585786 0.0439065
\(179\) −19.5563 −1.46171 −0.730855 0.682533i \(-0.760878\pi\)
−0.730855 + 0.682533i \(0.760878\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −18.1421 −1.34849 −0.674247 0.738506i \(-0.735532\pi\)
−0.674247 + 0.738506i \(0.735532\pi\)
\(182\) 0 0
\(183\) 5.58579 0.412913
\(184\) 8.65685 0.638192
\(185\) 3.24264 0.238404
\(186\) −4.24264 −0.311086
\(187\) 1.00000 0.0731272
\(188\) −7.65685 −0.558433
\(189\) 0 0
\(190\) −3.82843 −0.277743
\(191\) 5.48528 0.396901 0.198451 0.980111i \(-0.436409\pi\)
0.198451 + 0.980111i \(0.436409\pi\)
\(192\) 1.00000 0.0721688
\(193\) 12.3848 0.891476 0.445738 0.895164i \(-0.352941\pi\)
0.445738 + 0.895164i \(0.352941\pi\)
\(194\) 7.31371 0.525094
\(195\) −1.00000 −0.0716115
\(196\) 0 0
\(197\) 5.07107 0.361299 0.180649 0.983548i \(-0.442180\pi\)
0.180649 + 0.983548i \(0.442180\pi\)
\(198\) −2.41421 −0.171571
\(199\) −10.0711 −0.713919 −0.356960 0.934120i \(-0.616187\pi\)
−0.356960 + 0.934120i \(0.616187\pi\)
\(200\) 4.00000 0.282843
\(201\) 1.41421 0.0997509
\(202\) −1.07107 −0.0753601
\(203\) 0 0
\(204\) 0.414214 0.0290008
\(205\) 6.82843 0.476918
\(206\) −8.07107 −0.562338
\(207\) −8.65685 −0.601693
\(208\) 1.00000 0.0693375
\(209\) −9.24264 −0.639327
\(210\) 0 0
\(211\) −16.3137 −1.12308 −0.561541 0.827449i \(-0.689791\pi\)
−0.561541 + 0.827449i \(0.689791\pi\)
\(212\) −13.6569 −0.937957
\(213\) 5.07107 0.347464
\(214\) −4.58579 −0.313478
\(215\) 7.00000 0.477396
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −0.414214 −0.0280541
\(219\) 11.8284 0.799291
\(220\) −2.41421 −0.162766
\(221\) 0.414214 0.0278630
\(222\) 3.24264 0.217632
\(223\) 15.6569 1.04846 0.524230 0.851577i \(-0.324353\pi\)
0.524230 + 0.851577i \(0.324353\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 18.2426 1.21348
\(227\) 5.07107 0.336579 0.168289 0.985738i \(-0.446176\pi\)
0.168289 + 0.985738i \(0.446176\pi\)
\(228\) −3.82843 −0.253544
\(229\) 23.8995 1.57932 0.789662 0.613543i \(-0.210256\pi\)
0.789662 + 0.613543i \(0.210256\pi\)
\(230\) −8.65685 −0.570816
\(231\) 0 0
\(232\) −2.65685 −0.174431
\(233\) −10.8284 −0.709394 −0.354697 0.934981i \(-0.615416\pi\)
−0.354697 + 0.934981i \(0.615416\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 7.65685 0.499478
\(236\) 9.89949 0.644402
\(237\) 10.2426 0.665331
\(238\) 0 0
\(239\) 14.7279 0.952670 0.476335 0.879264i \(-0.341965\pi\)
0.476335 + 0.879264i \(0.341965\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 12.3431 0.795092 0.397546 0.917582i \(-0.369862\pi\)
0.397546 + 0.917582i \(0.369862\pi\)
\(242\) 5.17157 0.332441
\(243\) 1.00000 0.0641500
\(244\) 5.58579 0.357593
\(245\) 0 0
\(246\) 6.82843 0.435365
\(247\) −3.82843 −0.243597
\(248\) −4.24264 −0.269408
\(249\) −11.6569 −0.738723
\(250\) −9.00000 −0.569210
\(251\) −3.58579 −0.226333 −0.113166 0.993576i \(-0.536099\pi\)
−0.113166 + 0.993576i \(0.536099\pi\)
\(252\) 0 0
\(253\) −20.8995 −1.31394
\(254\) 2.92893 0.183778
\(255\) −0.414214 −0.0259391
\(256\) 1.00000 0.0625000
\(257\) −5.17157 −0.322594 −0.161297 0.986906i \(-0.551568\pi\)
−0.161297 + 0.986906i \(0.551568\pi\)
\(258\) 7.00000 0.435801
\(259\) 0 0
\(260\) −1.00000 −0.0620174
\(261\) 2.65685 0.164455
\(262\) 17.2426 1.06525
\(263\) 9.65685 0.595467 0.297734 0.954649i \(-0.403769\pi\)
0.297734 + 0.954649i \(0.403769\pi\)
\(264\) −2.41421 −0.148585
\(265\) 13.6569 0.838934
\(266\) 0 0
\(267\) −0.585786 −0.0358495
\(268\) 1.41421 0.0863868
\(269\) 16.7279 1.01992 0.509960 0.860198i \(-0.329660\pi\)
0.509960 + 0.860198i \(0.329660\pi\)
\(270\) 1.00000 0.0608581
\(271\) 14.2426 0.865179 0.432589 0.901591i \(-0.357600\pi\)
0.432589 + 0.901591i \(0.357600\pi\)
\(272\) 0.414214 0.0251154
\(273\) 0 0
\(274\) 7.24264 0.437544
\(275\) −9.65685 −0.582330
\(276\) −8.65685 −0.521081
\(277\) −16.5858 −0.996543 −0.498272 0.867021i \(-0.666032\pi\)
−0.498272 + 0.867021i \(0.666032\pi\)
\(278\) −6.82843 −0.409542
\(279\) 4.24264 0.254000
\(280\) 0 0
\(281\) 1.31371 0.0783693 0.0391846 0.999232i \(-0.487524\pi\)
0.0391846 + 0.999232i \(0.487524\pi\)
\(282\) 7.65685 0.455959
\(283\) 10.8284 0.643683 0.321842 0.946794i \(-0.395698\pi\)
0.321842 + 0.946794i \(0.395698\pi\)
\(284\) 5.07107 0.300913
\(285\) 3.82843 0.226776
\(286\) −2.41421 −0.142755
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −16.8284 −0.989907
\(290\) 2.65685 0.156016
\(291\) −7.31371 −0.428737
\(292\) 11.8284 0.692206
\(293\) −6.14214 −0.358827 −0.179414 0.983774i \(-0.557420\pi\)
−0.179414 + 0.983774i \(0.557420\pi\)
\(294\) 0 0
\(295\) −9.89949 −0.576371
\(296\) 3.24264 0.188475
\(297\) 2.41421 0.140087
\(298\) 2.82843 0.163846
\(299\) −8.65685 −0.500639
\(300\) −4.00000 −0.230940
\(301\) 0 0
\(302\) 13.2426 0.762028
\(303\) 1.07107 0.0615312
\(304\) −3.82843 −0.219575
\(305\) −5.58579 −0.319841
\(306\) −0.414214 −0.0236790
\(307\) 11.6569 0.665292 0.332646 0.943052i \(-0.392059\pi\)
0.332646 + 0.943052i \(0.392059\pi\)
\(308\) 0 0
\(309\) 8.07107 0.459147
\(310\) 4.24264 0.240966
\(311\) −21.7990 −1.23611 −0.618054 0.786136i \(-0.712079\pi\)
−0.618054 + 0.786136i \(0.712079\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 21.0711 1.19101 0.595504 0.803353i \(-0.296953\pi\)
0.595504 + 0.803353i \(0.296953\pi\)
\(314\) −2.89949 −0.163628
\(315\) 0 0
\(316\) 10.2426 0.576194
\(317\) 18.0416 1.01332 0.506659 0.862146i \(-0.330880\pi\)
0.506659 + 0.862146i \(0.330880\pi\)
\(318\) 13.6569 0.765838
\(319\) 6.41421 0.359127
\(320\) −1.00000 −0.0559017
\(321\) 4.58579 0.255954
\(322\) 0 0
\(323\) −1.58579 −0.0882355
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) 11.6569 0.645613
\(327\) 0.414214 0.0229061
\(328\) 6.82843 0.377037
\(329\) 0 0
\(330\) 2.41421 0.132898
\(331\) −29.8995 −1.64342 −0.821712 0.569902i \(-0.806981\pi\)
−0.821712 + 0.569902i \(0.806981\pi\)
\(332\) −11.6569 −0.639753
\(333\) −3.24264 −0.177696
\(334\) 13.8284 0.756658
\(335\) −1.41421 −0.0772667
\(336\) 0 0
\(337\) −17.4853 −0.952484 −0.476242 0.879314i \(-0.658001\pi\)
−0.476242 + 0.879314i \(0.658001\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −18.2426 −0.990804
\(340\) −0.414214 −0.0224639
\(341\) 10.2426 0.554670
\(342\) 3.82843 0.207018
\(343\) 0 0
\(344\) 7.00000 0.377415
\(345\) 8.65685 0.466069
\(346\) 16.2426 0.873210
\(347\) −22.5858 −1.21247 −0.606234 0.795286i \(-0.707321\pi\)
−0.606234 + 0.795286i \(0.707321\pi\)
\(348\) 2.65685 0.142422
\(349\) −14.7279 −0.788368 −0.394184 0.919032i \(-0.628973\pi\)
−0.394184 + 0.919032i \(0.628973\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) −2.41421 −0.128678
\(353\) −19.4558 −1.03553 −0.517765 0.855523i \(-0.673236\pi\)
−0.517765 + 0.855523i \(0.673236\pi\)
\(354\) −9.89949 −0.526152
\(355\) −5.07107 −0.269144
\(356\) −0.585786 −0.0310466
\(357\) 0 0
\(358\) 19.5563 1.03359
\(359\) −29.4558 −1.55462 −0.777310 0.629118i \(-0.783416\pi\)
−0.777310 + 0.629118i \(0.783416\pi\)
\(360\) 1.00000 0.0527046
\(361\) −4.34315 −0.228587
\(362\) 18.1421 0.953529
\(363\) −5.17157 −0.271437
\(364\) 0 0
\(365\) −11.8284 −0.619128
\(366\) −5.58579 −0.291974
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) −8.65685 −0.451270
\(369\) −6.82843 −0.355474
\(370\) −3.24264 −0.168577
\(371\) 0 0
\(372\) 4.24264 0.219971
\(373\) 4.24264 0.219676 0.109838 0.993950i \(-0.464967\pi\)
0.109838 + 0.993950i \(0.464967\pi\)
\(374\) −1.00000 −0.0517088
\(375\) 9.00000 0.464758
\(376\) 7.65685 0.394872
\(377\) 2.65685 0.136835
\(378\) 0 0
\(379\) −24.3848 −1.25256 −0.626281 0.779597i \(-0.715424\pi\)
−0.626281 + 0.779597i \(0.715424\pi\)
\(380\) 3.82843 0.196394
\(381\) −2.92893 −0.150054
\(382\) −5.48528 −0.280651
\(383\) 11.3431 0.579608 0.289804 0.957086i \(-0.406410\pi\)
0.289804 + 0.957086i \(0.406410\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −12.3848 −0.630369
\(387\) −7.00000 −0.355830
\(388\) −7.31371 −0.371297
\(389\) −8.00000 −0.405616 −0.202808 0.979219i \(-0.565007\pi\)
−0.202808 + 0.979219i \(0.565007\pi\)
\(390\) 1.00000 0.0506370
\(391\) −3.58579 −0.181341
\(392\) 0 0
\(393\) −17.2426 −0.869776
\(394\) −5.07107 −0.255477
\(395\) −10.2426 −0.515363
\(396\) 2.41421 0.121319
\(397\) 6.68629 0.335575 0.167788 0.985823i \(-0.446338\pi\)
0.167788 + 0.985823i \(0.446338\pi\)
\(398\) 10.0711 0.504817
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −21.4558 −1.07145 −0.535727 0.844391i \(-0.679962\pi\)
−0.535727 + 0.844391i \(0.679962\pi\)
\(402\) −1.41421 −0.0705346
\(403\) 4.24264 0.211341
\(404\) 1.07107 0.0532876
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −7.82843 −0.388041
\(408\) −0.414214 −0.0205066
\(409\) 33.9706 1.67974 0.839868 0.542791i \(-0.182632\pi\)
0.839868 + 0.542791i \(0.182632\pi\)
\(410\) −6.82843 −0.337232
\(411\) −7.24264 −0.357253
\(412\) 8.07107 0.397633
\(413\) 0 0
\(414\) 8.65685 0.425461
\(415\) 11.6569 0.572212
\(416\) −1.00000 −0.0490290
\(417\) 6.82843 0.334390
\(418\) 9.24264 0.452072
\(419\) 10.5563 0.515711 0.257856 0.966183i \(-0.416984\pi\)
0.257856 + 0.966183i \(0.416984\pi\)
\(420\) 0 0
\(421\) −33.1127 −1.61381 −0.806907 0.590678i \(-0.798860\pi\)
−0.806907 + 0.590678i \(0.798860\pi\)
\(422\) 16.3137 0.794139
\(423\) −7.65685 −0.372289
\(424\) 13.6569 0.663235
\(425\) −1.65685 −0.0803692
\(426\) −5.07107 −0.245694
\(427\) 0 0
\(428\) 4.58579 0.221662
\(429\) 2.41421 0.116559
\(430\) −7.00000 −0.337570
\(431\) 15.1716 0.730789 0.365394 0.930853i \(-0.380934\pi\)
0.365394 + 0.930853i \(0.380934\pi\)
\(432\) 1.00000 0.0481125
\(433\) 26.6274 1.27963 0.639816 0.768528i \(-0.279011\pi\)
0.639816 + 0.768528i \(0.279011\pi\)
\(434\) 0 0
\(435\) −2.65685 −0.127386
\(436\) 0.414214 0.0198372
\(437\) 33.1421 1.58540
\(438\) −11.8284 −0.565184
\(439\) −38.2132 −1.82382 −0.911908 0.410394i \(-0.865391\pi\)
−0.911908 + 0.410394i \(0.865391\pi\)
\(440\) 2.41421 0.115093
\(441\) 0 0
\(442\) −0.414214 −0.0197021
\(443\) 0.828427 0.0393598 0.0196799 0.999806i \(-0.493735\pi\)
0.0196799 + 0.999806i \(0.493735\pi\)
\(444\) −3.24264 −0.153889
\(445\) 0.585786 0.0277689
\(446\) −15.6569 −0.741374
\(447\) −2.82843 −0.133780
\(448\) 0 0
\(449\) 18.5563 0.875728 0.437864 0.899041i \(-0.355735\pi\)
0.437864 + 0.899041i \(0.355735\pi\)
\(450\) 4.00000 0.188562
\(451\) −16.4853 −0.776262
\(452\) −18.2426 −0.858062
\(453\) −13.2426 −0.622194
\(454\) −5.07107 −0.237997
\(455\) 0 0
\(456\) 3.82843 0.179283
\(457\) −22.2426 −1.04047 −0.520233 0.854024i \(-0.674155\pi\)
−0.520233 + 0.854024i \(0.674155\pi\)
\(458\) −23.8995 −1.11675
\(459\) 0.414214 0.0193338
\(460\) 8.65685 0.403628
\(461\) 23.4853 1.09382 0.546909 0.837192i \(-0.315804\pi\)
0.546909 + 0.837192i \(0.315804\pi\)
\(462\) 0 0
\(463\) 31.8701 1.48113 0.740564 0.671986i \(-0.234559\pi\)
0.740564 + 0.671986i \(0.234559\pi\)
\(464\) 2.65685 0.123341
\(465\) −4.24264 −0.196748
\(466\) 10.8284 0.501617
\(467\) −9.10051 −0.421121 −0.210561 0.977581i \(-0.567529\pi\)
−0.210561 + 0.977581i \(0.567529\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) −7.65685 −0.353184
\(471\) 2.89949 0.133602
\(472\) −9.89949 −0.455661
\(473\) −16.8995 −0.777040
\(474\) −10.2426 −0.470460
\(475\) 15.3137 0.702641
\(476\) 0 0
\(477\) −13.6569 −0.625304
\(478\) −14.7279 −0.673639
\(479\) 9.48528 0.433394 0.216697 0.976239i \(-0.430472\pi\)
0.216697 + 0.976239i \(0.430472\pi\)
\(480\) 1.00000 0.0456435
\(481\) −3.24264 −0.147852
\(482\) −12.3431 −0.562215
\(483\) 0 0
\(484\) −5.17157 −0.235071
\(485\) 7.31371 0.332098
\(486\) −1.00000 −0.0453609
\(487\) 37.5980 1.70373 0.851864 0.523764i \(-0.175473\pi\)
0.851864 + 0.523764i \(0.175473\pi\)
\(488\) −5.58579 −0.252857
\(489\) −11.6569 −0.527141
\(490\) 0 0
\(491\) 34.7696 1.56913 0.784564 0.620048i \(-0.212887\pi\)
0.784564 + 0.620048i \(0.212887\pi\)
\(492\) −6.82843 −0.307849
\(493\) 1.10051 0.0495643
\(494\) 3.82843 0.172249
\(495\) −2.41421 −0.108511
\(496\) 4.24264 0.190500
\(497\) 0 0
\(498\) 11.6569 0.522356
\(499\) 22.7279 1.01744 0.508721 0.860932i \(-0.330119\pi\)
0.508721 + 0.860932i \(0.330119\pi\)
\(500\) 9.00000 0.402492
\(501\) −13.8284 −0.617809
\(502\) 3.58579 0.160041
\(503\) 11.0711 0.493635 0.246817 0.969062i \(-0.420615\pi\)
0.246817 + 0.969062i \(0.420615\pi\)
\(504\) 0 0
\(505\) −1.07107 −0.0476619
\(506\) 20.8995 0.929096
\(507\) 1.00000 0.0444116
\(508\) −2.92893 −0.129950
\(509\) 5.14214 0.227921 0.113961 0.993485i \(-0.463646\pi\)
0.113961 + 0.993485i \(0.463646\pi\)
\(510\) 0.414214 0.0183417
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −3.82843 −0.169029
\(514\) 5.17157 0.228108
\(515\) −8.07107 −0.355654
\(516\) −7.00000 −0.308158
\(517\) −18.4853 −0.812982
\(518\) 0 0
\(519\) −16.2426 −0.712973
\(520\) 1.00000 0.0438529
\(521\) 10.5563 0.462482 0.231241 0.972896i \(-0.425721\pi\)
0.231241 + 0.972896i \(0.425721\pi\)
\(522\) −2.65685 −0.116287
\(523\) 5.27208 0.230532 0.115266 0.993335i \(-0.463228\pi\)
0.115266 + 0.993335i \(0.463228\pi\)
\(524\) −17.2426 −0.753248
\(525\) 0 0
\(526\) −9.65685 −0.421059
\(527\) 1.75736 0.0765518
\(528\) 2.41421 0.105065
\(529\) 51.9411 2.25831
\(530\) −13.6569 −0.593216
\(531\) 9.89949 0.429601
\(532\) 0 0
\(533\) −6.82843 −0.295772
\(534\) 0.585786 0.0253495
\(535\) −4.58579 −0.198261
\(536\) −1.41421 −0.0610847
\(537\) −19.5563 −0.843919
\(538\) −16.7279 −0.721192
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) −25.3848 −1.09138 −0.545688 0.837988i \(-0.683732\pi\)
−0.545688 + 0.837988i \(0.683732\pi\)
\(542\) −14.2426 −0.611774
\(543\) −18.1421 −0.778554
\(544\) −0.414214 −0.0177593
\(545\) −0.414214 −0.0177430
\(546\) 0 0
\(547\) −25.7990 −1.10309 −0.551543 0.834147i \(-0.685961\pi\)
−0.551543 + 0.834147i \(0.685961\pi\)
\(548\) −7.24264 −0.309390
\(549\) 5.58579 0.238396
\(550\) 9.65685 0.411770
\(551\) −10.1716 −0.433324
\(552\) 8.65685 0.368460
\(553\) 0 0
\(554\) 16.5858 0.704663
\(555\) 3.24264 0.137642
\(556\) 6.82843 0.289590
\(557\) −8.48528 −0.359533 −0.179766 0.983709i \(-0.557534\pi\)
−0.179766 + 0.983709i \(0.557534\pi\)
\(558\) −4.24264 −0.179605
\(559\) −7.00000 −0.296068
\(560\) 0 0
\(561\) 1.00000 0.0422200
\(562\) −1.31371 −0.0554154
\(563\) −8.41421 −0.354617 −0.177308 0.984155i \(-0.556739\pi\)
−0.177308 + 0.984155i \(0.556739\pi\)
\(564\) −7.65685 −0.322412
\(565\) 18.2426 0.767474
\(566\) −10.8284 −0.455153
\(567\) 0 0
\(568\) −5.07107 −0.212777
\(569\) −18.4853 −0.774943 −0.387472 0.921882i \(-0.626651\pi\)
−0.387472 + 0.921882i \(0.626651\pi\)
\(570\) −3.82843 −0.160355
\(571\) 30.2843 1.26736 0.633679 0.773596i \(-0.281544\pi\)
0.633679 + 0.773596i \(0.281544\pi\)
\(572\) 2.41421 0.100943
\(573\) 5.48528 0.229151
\(574\) 0 0
\(575\) 34.6274 1.44406
\(576\) 1.00000 0.0416667
\(577\) −21.1716 −0.881384 −0.440692 0.897658i \(-0.645267\pi\)
−0.440692 + 0.897658i \(0.645267\pi\)
\(578\) 16.8284 0.699970
\(579\) 12.3848 0.514694
\(580\) −2.65685 −0.110320
\(581\) 0 0
\(582\) 7.31371 0.303163
\(583\) −32.9706 −1.36550
\(584\) −11.8284 −0.489464
\(585\) −1.00000 −0.0413449
\(586\) 6.14214 0.253729
\(587\) 37.1127 1.53180 0.765902 0.642957i \(-0.222293\pi\)
0.765902 + 0.642957i \(0.222293\pi\)
\(588\) 0 0
\(589\) −16.2426 −0.669266
\(590\) 9.89949 0.407556
\(591\) 5.07107 0.208596
\(592\) −3.24264 −0.133272
\(593\) −7.41421 −0.304465 −0.152233 0.988345i \(-0.548646\pi\)
−0.152233 + 0.988345i \(0.548646\pi\)
\(594\) −2.41421 −0.0990564
\(595\) 0 0
\(596\) −2.82843 −0.115857
\(597\) −10.0711 −0.412181
\(598\) 8.65685 0.354005
\(599\) 20.5147 0.838209 0.419104 0.907938i \(-0.362344\pi\)
0.419104 + 0.907938i \(0.362344\pi\)
\(600\) 4.00000 0.163299
\(601\) 29.4558 1.20153 0.600764 0.799426i \(-0.294863\pi\)
0.600764 + 0.799426i \(0.294863\pi\)
\(602\) 0 0
\(603\) 1.41421 0.0575912
\(604\) −13.2426 −0.538835
\(605\) 5.17157 0.210254
\(606\) −1.07107 −0.0435092
\(607\) −35.1838 −1.42807 −0.714033 0.700113i \(-0.753133\pi\)
−0.714033 + 0.700113i \(0.753133\pi\)
\(608\) 3.82843 0.155263
\(609\) 0 0
\(610\) 5.58579 0.226162
\(611\) −7.65685 −0.309763
\(612\) 0.414214 0.0167436
\(613\) 6.07107 0.245208 0.122604 0.992456i \(-0.460875\pi\)
0.122604 + 0.992456i \(0.460875\pi\)
\(614\) −11.6569 −0.470432
\(615\) 6.82843 0.275349
\(616\) 0 0
\(617\) −25.5858 −1.03004 −0.515022 0.857177i \(-0.672216\pi\)
−0.515022 + 0.857177i \(0.672216\pi\)
\(618\) −8.07107 −0.324666
\(619\) −21.2843 −0.855487 −0.427744 0.903900i \(-0.640691\pi\)
−0.427744 + 0.903900i \(0.640691\pi\)
\(620\) −4.24264 −0.170389
\(621\) −8.65685 −0.347388
\(622\) 21.7990 0.874060
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) 11.0000 0.440000
\(626\) −21.0711 −0.842169
\(627\) −9.24264 −0.369116
\(628\) 2.89949 0.115702
\(629\) −1.34315 −0.0535547
\(630\) 0 0
\(631\) 0.615224 0.0244917 0.0122458 0.999925i \(-0.496102\pi\)
0.0122458 + 0.999925i \(0.496102\pi\)
\(632\) −10.2426 −0.407430
\(633\) −16.3137 −0.648412
\(634\) −18.0416 −0.716525
\(635\) 2.92893 0.116231
\(636\) −13.6569 −0.541529
\(637\) 0 0
\(638\) −6.41421 −0.253941
\(639\) 5.07107 0.200608
\(640\) 1.00000 0.0395285
\(641\) −42.5269 −1.67971 −0.839856 0.542809i \(-0.817361\pi\)
−0.839856 + 0.542809i \(0.817361\pi\)
\(642\) −4.58579 −0.180987
\(643\) −15.6863 −0.618607 −0.309303 0.950963i \(-0.600096\pi\)
−0.309303 + 0.950963i \(0.600096\pi\)
\(644\) 0 0
\(645\) 7.00000 0.275625
\(646\) 1.58579 0.0623919
\(647\) 11.0294 0.433612 0.216806 0.976215i \(-0.430436\pi\)
0.216806 + 0.976215i \(0.430436\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 23.8995 0.938137
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) −11.6569 −0.456518
\(653\) −21.4853 −0.840784 −0.420392 0.907343i \(-0.638107\pi\)
−0.420392 + 0.907343i \(0.638107\pi\)
\(654\) −0.414214 −0.0161970
\(655\) 17.2426 0.673726
\(656\) −6.82843 −0.266605
\(657\) 11.8284 0.461471
\(658\) 0 0
\(659\) −11.4558 −0.446256 −0.223128 0.974789i \(-0.571627\pi\)
−0.223128 + 0.974789i \(0.571627\pi\)
\(660\) −2.41421 −0.0939731
\(661\) −34.9706 −1.36020 −0.680099 0.733121i \(-0.738063\pi\)
−0.680099 + 0.733121i \(0.738063\pi\)
\(662\) 29.8995 1.16208
\(663\) 0.414214 0.0160867
\(664\) 11.6569 0.452374
\(665\) 0 0
\(666\) 3.24264 0.125650
\(667\) −23.0000 −0.890564
\(668\) −13.8284 −0.535038
\(669\) 15.6569 0.605329
\(670\) 1.41421 0.0546358
\(671\) 13.4853 0.520594
\(672\) 0 0
\(673\) 38.1127 1.46914 0.734568 0.678535i \(-0.237385\pi\)
0.734568 + 0.678535i \(0.237385\pi\)
\(674\) 17.4853 0.673508
\(675\) −4.00000 −0.153960
\(676\) 1.00000 0.0384615
\(677\) −38.7696 −1.49003 −0.745017 0.667045i \(-0.767559\pi\)
−0.745017 + 0.667045i \(0.767559\pi\)
\(678\) 18.2426 0.700604
\(679\) 0 0
\(680\) 0.414214 0.0158844
\(681\) 5.07107 0.194324
\(682\) −10.2426 −0.392211
\(683\) 23.5858 0.902485 0.451243 0.892401i \(-0.350981\pi\)
0.451243 + 0.892401i \(0.350981\pi\)
\(684\) −3.82843 −0.146384
\(685\) 7.24264 0.276727
\(686\) 0 0
\(687\) 23.8995 0.911823
\(688\) −7.00000 −0.266872
\(689\) −13.6569 −0.520285
\(690\) −8.65685 −0.329561
\(691\) 23.1716 0.881488 0.440744 0.897633i \(-0.354715\pi\)
0.440744 + 0.897633i \(0.354715\pi\)
\(692\) −16.2426 −0.617453
\(693\) 0 0
\(694\) 22.5858 0.857345
\(695\) −6.82843 −0.259017
\(696\) −2.65685 −0.100708
\(697\) −2.82843 −0.107134
\(698\) 14.7279 0.557460
\(699\) −10.8284 −0.409569
\(700\) 0 0
\(701\) 34.8284 1.31545 0.657726 0.753257i \(-0.271519\pi\)
0.657726 + 0.753257i \(0.271519\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 12.4142 0.468211
\(704\) 2.41421 0.0909891
\(705\) 7.65685 0.288374
\(706\) 19.4558 0.732230
\(707\) 0 0
\(708\) 9.89949 0.372046
\(709\) 40.8284 1.53334 0.766672 0.642039i \(-0.221911\pi\)
0.766672 + 0.642039i \(0.221911\pi\)
\(710\) 5.07107 0.190314
\(711\) 10.2426 0.384129
\(712\) 0.585786 0.0219533
\(713\) −36.7279 −1.37547
\(714\) 0 0
\(715\) −2.41421 −0.0902865
\(716\) −19.5563 −0.730855
\(717\) 14.7279 0.550024
\(718\) 29.4558 1.09928
\(719\) −10.5858 −0.394783 −0.197392 0.980325i \(-0.563247\pi\)
−0.197392 + 0.980325i \(0.563247\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) 4.34315 0.161635
\(723\) 12.3431 0.459047
\(724\) −18.1421 −0.674247
\(725\) −10.6274 −0.394692
\(726\) 5.17157 0.191935
\(727\) −7.78680 −0.288796 −0.144398 0.989520i \(-0.546125\pi\)
−0.144398 + 0.989520i \(0.546125\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 11.8284 0.437790
\(731\) −2.89949 −0.107242
\(732\) 5.58579 0.206457
\(733\) −15.2721 −0.564087 −0.282044 0.959402i \(-0.591012\pi\)
−0.282044 + 0.959402i \(0.591012\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) 8.65685 0.319096
\(737\) 3.41421 0.125764
\(738\) 6.82843 0.251358
\(739\) 34.2843 1.26117 0.630584 0.776121i \(-0.282816\pi\)
0.630584 + 0.776121i \(0.282816\pi\)
\(740\) 3.24264 0.119202
\(741\) −3.82843 −0.140641
\(742\) 0 0
\(743\) 52.6274 1.93071 0.965356 0.260935i \(-0.0840309\pi\)
0.965356 + 0.260935i \(0.0840309\pi\)
\(744\) −4.24264 −0.155543
\(745\) 2.82843 0.103626
\(746\) −4.24264 −0.155334
\(747\) −11.6569 −0.426502
\(748\) 1.00000 0.0365636
\(749\) 0 0
\(750\) −9.00000 −0.328634
\(751\) 16.4437 0.600037 0.300019 0.953933i \(-0.403007\pi\)
0.300019 + 0.953933i \(0.403007\pi\)
\(752\) −7.65685 −0.279217
\(753\) −3.58579 −0.130673
\(754\) −2.65685 −0.0967569
\(755\) 13.2426 0.481949
\(756\) 0 0
\(757\) −5.07107 −0.184311 −0.0921555 0.995745i \(-0.529376\pi\)
−0.0921555 + 0.995745i \(0.529376\pi\)
\(758\) 24.3848 0.885695
\(759\) −20.8995 −0.758604
\(760\) −3.82843 −0.138872
\(761\) −6.82843 −0.247530 −0.123765 0.992312i \(-0.539497\pi\)
−0.123765 + 0.992312i \(0.539497\pi\)
\(762\) 2.92893 0.106104
\(763\) 0 0
\(764\) 5.48528 0.198451
\(765\) −0.414214 −0.0149759
\(766\) −11.3431 −0.409845
\(767\) 9.89949 0.357450
\(768\) 1.00000 0.0360844
\(769\) 0.857864 0.0309354 0.0154677 0.999880i \(-0.495076\pi\)
0.0154677 + 0.999880i \(0.495076\pi\)
\(770\) 0 0
\(771\) −5.17157 −0.186250
\(772\) 12.3848 0.445738
\(773\) 8.02944 0.288799 0.144399 0.989519i \(-0.453875\pi\)
0.144399 + 0.989519i \(0.453875\pi\)
\(774\) 7.00000 0.251610
\(775\) −16.9706 −0.609601
\(776\) 7.31371 0.262547
\(777\) 0 0
\(778\) 8.00000 0.286814
\(779\) 26.1421 0.936639
\(780\) −1.00000 −0.0358057
\(781\) 12.2426 0.438076
\(782\) 3.58579 0.128227
\(783\) 2.65685 0.0949482
\(784\) 0 0
\(785\) −2.89949 −0.103487
\(786\) 17.2426 0.615025
\(787\) 10.1716 0.362577 0.181289 0.983430i \(-0.441973\pi\)
0.181289 + 0.983430i \(0.441973\pi\)
\(788\) 5.07107 0.180649
\(789\) 9.65685 0.343793
\(790\) 10.2426 0.364417
\(791\) 0 0
\(792\) −2.41421 −0.0857853
\(793\) 5.58579 0.198357
\(794\) −6.68629 −0.237288
\(795\) 13.6569 0.484359
\(796\) −10.0711 −0.356960
\(797\) −6.72792 −0.238315 −0.119158 0.992875i \(-0.538019\pi\)
−0.119158 + 0.992875i \(0.538019\pi\)
\(798\) 0 0
\(799\) −3.17157 −0.112202
\(800\) 4.00000 0.141421
\(801\) −0.585786 −0.0206977
\(802\) 21.4558 0.757632
\(803\) 28.5563 1.00773
\(804\) 1.41421 0.0498755
\(805\) 0 0
\(806\) −4.24264 −0.149441
\(807\) 16.7279 0.588851
\(808\) −1.07107 −0.0376800
\(809\) −13.5563 −0.476616 −0.238308 0.971190i \(-0.576593\pi\)
−0.238308 + 0.971190i \(0.576593\pi\)
\(810\) 1.00000 0.0351364
\(811\) −21.3431 −0.749459 −0.374730 0.927134i \(-0.622264\pi\)
−0.374730 + 0.927134i \(0.622264\pi\)
\(812\) 0 0
\(813\) 14.2426 0.499511
\(814\) 7.82843 0.274386
\(815\) 11.6569 0.408322
\(816\) 0.414214 0.0145004
\(817\) 26.7990 0.937578
\(818\) −33.9706 −1.18775
\(819\) 0 0
\(820\) 6.82843 0.238459
\(821\) 28.4853 0.994143 0.497072 0.867710i \(-0.334409\pi\)
0.497072 + 0.867710i \(0.334409\pi\)
\(822\) 7.24264 0.252616
\(823\) −20.5269 −0.715523 −0.357762 0.933813i \(-0.616460\pi\)
−0.357762 + 0.933813i \(0.616460\pi\)
\(824\) −8.07107 −0.281169
\(825\) −9.65685 −0.336209
\(826\) 0 0
\(827\) 27.5269 0.957205 0.478602 0.878032i \(-0.341144\pi\)
0.478602 + 0.878032i \(0.341144\pi\)
\(828\) −8.65685 −0.300846
\(829\) 51.5269 1.78960 0.894802 0.446464i \(-0.147317\pi\)
0.894802 + 0.446464i \(0.147317\pi\)
\(830\) −11.6569 −0.404615
\(831\) −16.5858 −0.575355
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) −6.82843 −0.236449
\(835\) 13.8284 0.478552
\(836\) −9.24264 −0.319663
\(837\) 4.24264 0.146647
\(838\) −10.5563 −0.364663
\(839\) 56.6274 1.95500 0.977498 0.210946i \(-0.0676543\pi\)
0.977498 + 0.210946i \(0.0676543\pi\)
\(840\) 0 0
\(841\) −21.9411 −0.756591
\(842\) 33.1127 1.14114
\(843\) 1.31371 0.0452465
\(844\) −16.3137 −0.561541
\(845\) −1.00000 −0.0344010
\(846\) 7.65685 0.263248
\(847\) 0 0
\(848\) −13.6569 −0.468978
\(849\) 10.8284 0.371631
\(850\) 1.65685 0.0568296
\(851\) 28.0711 0.962264
\(852\) 5.07107 0.173732
\(853\) 33.6569 1.15239 0.576194 0.817313i \(-0.304537\pi\)
0.576194 + 0.817313i \(0.304537\pi\)
\(854\) 0 0
\(855\) 3.82843 0.130929
\(856\) −4.58579 −0.156739
\(857\) −44.8284 −1.53131 −0.765655 0.643252i \(-0.777585\pi\)
−0.765655 + 0.643252i \(0.777585\pi\)
\(858\) −2.41421 −0.0824199
\(859\) −48.9706 −1.67085 −0.835427 0.549601i \(-0.814780\pi\)
−0.835427 + 0.549601i \(0.814780\pi\)
\(860\) 7.00000 0.238698
\(861\) 0 0
\(862\) −15.1716 −0.516746
\(863\) −10.9706 −0.373442 −0.186721 0.982413i \(-0.559786\pi\)
−0.186721 + 0.982413i \(0.559786\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 16.2426 0.552266
\(866\) −26.6274 −0.904836
\(867\) −16.8284 −0.571523
\(868\) 0 0
\(869\) 24.7279 0.838837
\(870\) 2.65685 0.0900758
\(871\) 1.41421 0.0479188
\(872\) −0.414214 −0.0140270
\(873\) −7.31371 −0.247532
\(874\) −33.1421 −1.12105
\(875\) 0 0
\(876\) 11.8284 0.399646
\(877\) −8.48528 −0.286528 −0.143264 0.989685i \(-0.545760\pi\)
−0.143264 + 0.989685i \(0.545760\pi\)
\(878\) 38.2132 1.28963
\(879\) −6.14214 −0.207169
\(880\) −2.41421 −0.0813831
\(881\) 41.3848 1.39429 0.697144 0.716931i \(-0.254454\pi\)
0.697144 + 0.716931i \(0.254454\pi\)
\(882\) 0 0
\(883\) 22.0294 0.741350 0.370675 0.928763i \(-0.379126\pi\)
0.370675 + 0.928763i \(0.379126\pi\)
\(884\) 0.414214 0.0139315
\(885\) −9.89949 −0.332768
\(886\) −0.828427 −0.0278316
\(887\) −59.1127 −1.98481 −0.992405 0.123013i \(-0.960744\pi\)
−0.992405 + 0.123013i \(0.960744\pi\)
\(888\) 3.24264 0.108816
\(889\) 0 0
\(890\) −0.585786 −0.0196356
\(891\) 2.41421 0.0808792
\(892\) 15.6569 0.524230
\(893\) 29.3137 0.980946
\(894\) 2.82843 0.0945968
\(895\) 19.5563 0.653697
\(896\) 0 0
\(897\) −8.65685 −0.289044
\(898\) −18.5563 −0.619233
\(899\) 11.2721 0.375945
\(900\) −4.00000 −0.133333
\(901\) −5.65685 −0.188457
\(902\) 16.4853 0.548900
\(903\) 0 0
\(904\) 18.2426 0.606741
\(905\) 18.1421 0.603065
\(906\) 13.2426 0.439957
\(907\) 51.1716 1.69912 0.849562 0.527489i \(-0.176866\pi\)
0.849562 + 0.527489i \(0.176866\pi\)
\(908\) 5.07107 0.168289
\(909\) 1.07107 0.0355251
\(910\) 0 0
\(911\) −9.97056 −0.330339 −0.165170 0.986265i \(-0.552817\pi\)
−0.165170 + 0.986265i \(0.552817\pi\)
\(912\) −3.82843 −0.126772
\(913\) −28.1421 −0.931369
\(914\) 22.2426 0.735721
\(915\) −5.58579 −0.184660
\(916\) 23.8995 0.789662
\(917\) 0 0
\(918\) −0.414214 −0.0136711
\(919\) −12.1421 −0.400532 −0.200266 0.979742i \(-0.564181\pi\)
−0.200266 + 0.979742i \(0.564181\pi\)
\(920\) −8.65685 −0.285408
\(921\) 11.6569 0.384106
\(922\) −23.4853 −0.773447
\(923\) 5.07107 0.166916
\(924\) 0 0
\(925\) 12.9706 0.426469
\(926\) −31.8701 −1.04732
\(927\) 8.07107 0.265089
\(928\) −2.65685 −0.0872155
\(929\) 1.85786 0.0609546 0.0304773 0.999535i \(-0.490297\pi\)
0.0304773 + 0.999535i \(0.490297\pi\)
\(930\) 4.24264 0.139122
\(931\) 0 0
\(932\) −10.8284 −0.354697
\(933\) −21.7990 −0.713667
\(934\) 9.10051 0.297778
\(935\) −1.00000 −0.0327035
\(936\) −1.00000 −0.0326860
\(937\) −41.6985 −1.36223 −0.681115 0.732176i \(-0.738505\pi\)
−0.681115 + 0.732176i \(0.738505\pi\)
\(938\) 0 0
\(939\) 21.0711 0.687628
\(940\) 7.65685 0.249739
\(941\) −38.0000 −1.23876 −0.619382 0.785090i \(-0.712617\pi\)
−0.619382 + 0.785090i \(0.712617\pi\)
\(942\) −2.89949 −0.0944706
\(943\) 59.1127 1.92497
\(944\) 9.89949 0.322201
\(945\) 0 0
\(946\) 16.8995 0.549450
\(947\) 21.2426 0.690293 0.345147 0.938549i \(-0.387829\pi\)
0.345147 + 0.938549i \(0.387829\pi\)
\(948\) 10.2426 0.332666
\(949\) 11.8284 0.383967
\(950\) −15.3137 −0.496842
\(951\) 18.0416 0.585040
\(952\) 0 0
\(953\) 36.2843 1.17536 0.587681 0.809092i \(-0.300041\pi\)
0.587681 + 0.809092i \(0.300041\pi\)
\(954\) 13.6569 0.442157
\(955\) −5.48528 −0.177500
\(956\) 14.7279 0.476335
\(957\) 6.41421 0.207342
\(958\) −9.48528 −0.306456
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) −13.0000 −0.419355
\(962\) 3.24264 0.104547
\(963\) 4.58579 0.147775
\(964\) 12.3431 0.397546
\(965\) −12.3848 −0.398680
\(966\) 0 0
\(967\) −1.58579 −0.0509955 −0.0254977 0.999675i \(-0.508117\pi\)
−0.0254977 + 0.999675i \(0.508117\pi\)
\(968\) 5.17157 0.166221
\(969\) −1.58579 −0.0509428
\(970\) −7.31371 −0.234829
\(971\) 31.5147 1.01136 0.505678 0.862722i \(-0.331242\pi\)
0.505678 + 0.862722i \(0.331242\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −37.5980 −1.20472
\(975\) −4.00000 −0.128103
\(976\) 5.58579 0.178797
\(977\) −47.8701 −1.53150 −0.765749 0.643139i \(-0.777632\pi\)
−0.765749 + 0.643139i \(0.777632\pi\)
\(978\) 11.6569 0.372745
\(979\) −1.41421 −0.0451985
\(980\) 0 0
\(981\) 0.414214 0.0132248
\(982\) −34.7696 −1.10954
\(983\) 13.6863 0.436525 0.218262 0.975890i \(-0.429961\pi\)
0.218262 + 0.975890i \(0.429961\pi\)
\(984\) 6.82843 0.217682
\(985\) −5.07107 −0.161578
\(986\) −1.10051 −0.0350472
\(987\) 0 0
\(988\) −3.82843 −0.121798
\(989\) 60.5980 1.92690
\(990\) 2.41421 0.0767287
\(991\) 37.3137 1.18531 0.592655 0.805457i \(-0.298080\pi\)
0.592655 + 0.805457i \(0.298080\pi\)
\(992\) −4.24264 −0.134704
\(993\) −29.8995 −0.948832
\(994\) 0 0
\(995\) 10.0711 0.319274
\(996\) −11.6569 −0.369362
\(997\) 58.2843 1.84588 0.922941 0.384942i \(-0.125779\pi\)
0.922941 + 0.384942i \(0.125779\pi\)
\(998\) −22.7279 −0.719440
\(999\) −3.24264 −0.102593
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 3822.2.a.bl.1.2 yes 2
7.6 odd 2 3822.2.a.bj.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3822.2.a.bj.1.2 2 7.6 odd 2
3822.2.a.bl.1.2 yes 2 1.1 even 1 trivial