Properties

Label 4598.2.a.w.1.2
Level $4598$
Weight $2$
Character 4598.1
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7152148494\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 4598.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.73205 q^{5} +1.00000 q^{6} +3.73205 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.73205 q^{5} +1.00000 q^{6} +3.73205 q^{7} -1.00000 q^{8} -2.00000 q^{9} -2.73205 q^{10} -1.00000 q^{12} +3.46410 q^{13} -3.73205 q^{14} -2.73205 q^{15} +1.00000 q^{16} -0.535898 q^{17} +2.00000 q^{18} +1.00000 q^{19} +2.73205 q^{20} -3.73205 q^{21} -1.00000 q^{23} +1.00000 q^{24} +2.46410 q^{25} -3.46410 q^{26} +5.00000 q^{27} +3.73205 q^{28} -1.73205 q^{29} +2.73205 q^{30} +5.46410 q^{31} -1.00000 q^{32} +0.535898 q^{34} +10.1962 q^{35} -2.00000 q^{36} +0.464102 q^{37} -1.00000 q^{38} -3.46410 q^{39} -2.73205 q^{40} -7.26795 q^{41} +3.73205 q^{42} -8.92820 q^{43} -5.46410 q^{45} +1.00000 q^{46} +5.53590 q^{47} -1.00000 q^{48} +6.92820 q^{49} -2.46410 q^{50} +0.535898 q^{51} +3.46410 q^{52} +3.92820 q^{53} -5.00000 q^{54} -3.73205 q^{56} -1.00000 q^{57} +1.73205 q^{58} +11.3923 q^{59} -2.73205 q^{60} +4.73205 q^{61} -5.46410 q^{62} -7.46410 q^{63} +1.00000 q^{64} +9.46410 q^{65} +14.0000 q^{67} -0.535898 q^{68} +1.00000 q^{69} -10.1962 q^{70} +3.80385 q^{71} +2.00000 q^{72} +2.00000 q^{73} -0.464102 q^{74} -2.46410 q^{75} +1.00000 q^{76} +3.46410 q^{78} +8.19615 q^{79} +2.73205 q^{80} +1.00000 q^{81} +7.26795 q^{82} -14.1962 q^{83} -3.73205 q^{84} -1.46410 q^{85} +8.92820 q^{86} +1.73205 q^{87} +1.80385 q^{89} +5.46410 q^{90} +12.9282 q^{91} -1.00000 q^{92} -5.46410 q^{93} -5.53590 q^{94} +2.73205 q^{95} +1.00000 q^{96} +6.53590 q^{97} -6.92820 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 4 q^{7} - 2 q^{8} - 4 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 4 q^{7} - 2 q^{8} - 4 q^{9} - 2 q^{10} - 2 q^{12} - 4 q^{14} - 2 q^{15} + 2 q^{16} - 8 q^{17} + 4 q^{18} + 2 q^{19} + 2 q^{20} - 4 q^{21} - 2 q^{23} + 2 q^{24} - 2 q^{25} + 10 q^{27} + 4 q^{28} + 2 q^{30} + 4 q^{31} - 2 q^{32} + 8 q^{34} + 10 q^{35} - 4 q^{36} - 6 q^{37} - 2 q^{38} - 2 q^{40} - 18 q^{41} + 4 q^{42} - 4 q^{43} - 4 q^{45} + 2 q^{46} + 18 q^{47} - 2 q^{48} + 2 q^{50} + 8 q^{51} - 6 q^{53} - 10 q^{54} - 4 q^{56} - 2 q^{57} + 2 q^{59} - 2 q^{60} + 6 q^{61} - 4 q^{62} - 8 q^{63} + 2 q^{64} + 12 q^{65} + 28 q^{67} - 8 q^{68} + 2 q^{69} - 10 q^{70} + 18 q^{71} + 4 q^{72} + 4 q^{73} + 6 q^{74} + 2 q^{75} + 2 q^{76} + 6 q^{79} + 2 q^{80} + 2 q^{81} + 18 q^{82} - 18 q^{83} - 4 q^{84} + 4 q^{85} + 4 q^{86} + 14 q^{89} + 4 q^{90} + 12 q^{91} - 2 q^{92} - 4 q^{93} - 18 q^{94} + 2 q^{95} + 2 q^{96} + 20 q^{97} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.73205 1.22181 0.610905 0.791704i \(-0.290806\pi\)
0.610905 + 0.791704i \(0.290806\pi\)
\(6\) 1.00000 0.408248
\(7\) 3.73205 1.41058 0.705291 0.708918i \(-0.250816\pi\)
0.705291 + 0.708918i \(0.250816\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.00000 −0.666667
\(10\) −2.73205 −0.863950
\(11\) 0 0
\(12\) −1.00000 −0.288675
\(13\) 3.46410 0.960769 0.480384 0.877058i \(-0.340497\pi\)
0.480384 + 0.877058i \(0.340497\pi\)
\(14\) −3.73205 −0.997433
\(15\) −2.73205 −0.705412
\(16\) 1.00000 0.250000
\(17\) −0.535898 −0.129974 −0.0649872 0.997886i \(-0.520701\pi\)
−0.0649872 + 0.997886i \(0.520701\pi\)
\(18\) 2.00000 0.471405
\(19\) 1.00000 0.229416
\(20\) 2.73205 0.610905
\(21\) −3.73205 −0.814400
\(22\) 0 0
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 1.00000 0.204124
\(25\) 2.46410 0.492820
\(26\) −3.46410 −0.679366
\(27\) 5.00000 0.962250
\(28\) 3.73205 0.705291
\(29\) −1.73205 −0.321634 −0.160817 0.986984i \(-0.551413\pi\)
−0.160817 + 0.986984i \(0.551413\pi\)
\(30\) 2.73205 0.498802
\(31\) 5.46410 0.981382 0.490691 0.871334i \(-0.336744\pi\)
0.490691 + 0.871334i \(0.336744\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0.535898 0.0919058
\(35\) 10.1962 1.72346
\(36\) −2.00000 −0.333333
\(37\) 0.464102 0.0762978 0.0381489 0.999272i \(-0.487854\pi\)
0.0381489 + 0.999272i \(0.487854\pi\)
\(38\) −1.00000 −0.162221
\(39\) −3.46410 −0.554700
\(40\) −2.73205 −0.431975
\(41\) −7.26795 −1.13506 −0.567531 0.823352i \(-0.692101\pi\)
−0.567531 + 0.823352i \(0.692101\pi\)
\(42\) 3.73205 0.575868
\(43\) −8.92820 −1.36154 −0.680769 0.732498i \(-0.738354\pi\)
−0.680769 + 0.732498i \(0.738354\pi\)
\(44\) 0 0
\(45\) −5.46410 −0.814540
\(46\) 1.00000 0.147442
\(47\) 5.53590 0.807494 0.403747 0.914871i \(-0.367708\pi\)
0.403747 + 0.914871i \(0.367708\pi\)
\(48\) −1.00000 −0.144338
\(49\) 6.92820 0.989743
\(50\) −2.46410 −0.348477
\(51\) 0.535898 0.0750408
\(52\) 3.46410 0.480384
\(53\) 3.92820 0.539580 0.269790 0.962919i \(-0.413046\pi\)
0.269790 + 0.962919i \(0.413046\pi\)
\(54\) −5.00000 −0.680414
\(55\) 0 0
\(56\) −3.73205 −0.498716
\(57\) −1.00000 −0.132453
\(58\) 1.73205 0.227429
\(59\) 11.3923 1.48315 0.741576 0.670869i \(-0.234079\pi\)
0.741576 + 0.670869i \(0.234079\pi\)
\(60\) −2.73205 −0.352706
\(61\) 4.73205 0.605877 0.302939 0.953010i \(-0.402032\pi\)
0.302939 + 0.953010i \(0.402032\pi\)
\(62\) −5.46410 −0.693942
\(63\) −7.46410 −0.940388
\(64\) 1.00000 0.125000
\(65\) 9.46410 1.17388
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) −0.535898 −0.0649872
\(69\) 1.00000 0.120386
\(70\) −10.1962 −1.21867
\(71\) 3.80385 0.451434 0.225717 0.974193i \(-0.427528\pi\)
0.225717 + 0.974193i \(0.427528\pi\)
\(72\) 2.00000 0.235702
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −0.464102 −0.0539507
\(75\) −2.46410 −0.284530
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 3.46410 0.392232
\(79\) 8.19615 0.922139 0.461070 0.887364i \(-0.347466\pi\)
0.461070 + 0.887364i \(0.347466\pi\)
\(80\) 2.73205 0.305453
\(81\) 1.00000 0.111111
\(82\) 7.26795 0.802611
\(83\) −14.1962 −1.55823 −0.779115 0.626881i \(-0.784331\pi\)
−0.779115 + 0.626881i \(0.784331\pi\)
\(84\) −3.73205 −0.407200
\(85\) −1.46410 −0.158804
\(86\) 8.92820 0.962753
\(87\) 1.73205 0.185695
\(88\) 0 0
\(89\) 1.80385 0.191207 0.0956037 0.995419i \(-0.469522\pi\)
0.0956037 + 0.995419i \(0.469522\pi\)
\(90\) 5.46410 0.575967
\(91\) 12.9282 1.35524
\(92\) −1.00000 −0.104257
\(93\) −5.46410 −0.566601
\(94\) −5.53590 −0.570984
\(95\) 2.73205 0.280302
\(96\) 1.00000 0.102062
\(97\) 6.53590 0.663620 0.331810 0.943346i \(-0.392341\pi\)
0.331810 + 0.943346i \(0.392341\pi\)
\(98\) −6.92820 −0.699854
\(99\) 0 0
\(100\) 2.46410 0.246410
\(101\) 14.5885 1.45161 0.725803 0.687903i \(-0.241468\pi\)
0.725803 + 0.687903i \(0.241468\pi\)
\(102\) −0.535898 −0.0530618
\(103\) 0.732051 0.0721311 0.0360656 0.999349i \(-0.488517\pi\)
0.0360656 + 0.999349i \(0.488517\pi\)
\(104\) −3.46410 −0.339683
\(105\) −10.1962 −0.995043
\(106\) −3.92820 −0.381541
\(107\) −13.1962 −1.27572 −0.637860 0.770152i \(-0.720180\pi\)
−0.637860 + 0.770152i \(0.720180\pi\)
\(108\) 5.00000 0.481125
\(109\) 18.1244 1.73600 0.867999 0.496566i \(-0.165406\pi\)
0.867999 + 0.496566i \(0.165406\pi\)
\(110\) 0 0
\(111\) −0.464102 −0.0440506
\(112\) 3.73205 0.352646
\(113\) −1.26795 −0.119279 −0.0596393 0.998220i \(-0.518995\pi\)
−0.0596393 + 0.998220i \(0.518995\pi\)
\(114\) 1.00000 0.0936586
\(115\) −2.73205 −0.254765
\(116\) −1.73205 −0.160817
\(117\) −6.92820 −0.640513
\(118\) −11.3923 −1.04875
\(119\) −2.00000 −0.183340
\(120\) 2.73205 0.249401
\(121\) 0 0
\(122\) −4.73205 −0.428420
\(123\) 7.26795 0.655329
\(124\) 5.46410 0.490691
\(125\) −6.92820 −0.619677
\(126\) 7.46410 0.664955
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.92820 0.786084
\(130\) −9.46410 −0.830057
\(131\) −7.07180 −0.617866 −0.308933 0.951084i \(-0.599972\pi\)
−0.308933 + 0.951084i \(0.599972\pi\)
\(132\) 0 0
\(133\) 3.73205 0.323610
\(134\) −14.0000 −1.20942
\(135\) 13.6603 1.17569
\(136\) 0.535898 0.0459529
\(137\) 2.53590 0.216656 0.108328 0.994115i \(-0.465450\pi\)
0.108328 + 0.994115i \(0.465450\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −13.4641 −1.14201 −0.571005 0.820947i \(-0.693446\pi\)
−0.571005 + 0.820947i \(0.693446\pi\)
\(140\) 10.1962 0.861732
\(141\) −5.53590 −0.466207
\(142\) −3.80385 −0.319212
\(143\) 0 0
\(144\) −2.00000 −0.166667
\(145\) −4.73205 −0.392975
\(146\) −2.00000 −0.165521
\(147\) −6.92820 −0.571429
\(148\) 0.464102 0.0381489
\(149\) −23.3205 −1.91049 −0.955245 0.295815i \(-0.904409\pi\)
−0.955245 + 0.295815i \(0.904409\pi\)
\(150\) 2.46410 0.201193
\(151\) −17.2679 −1.40525 −0.702623 0.711562i \(-0.747988\pi\)
−0.702623 + 0.711562i \(0.747988\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 1.07180 0.0866496
\(154\) 0 0
\(155\) 14.9282 1.19906
\(156\) −3.46410 −0.277350
\(157\) −8.73205 −0.696894 −0.348447 0.937329i \(-0.613291\pi\)
−0.348447 + 0.937329i \(0.613291\pi\)
\(158\) −8.19615 −0.652051
\(159\) −3.92820 −0.311527
\(160\) −2.73205 −0.215988
\(161\) −3.73205 −0.294127
\(162\) −1.00000 −0.0785674
\(163\) −17.3205 −1.35665 −0.678323 0.734763i \(-0.737293\pi\)
−0.678323 + 0.734763i \(0.737293\pi\)
\(164\) −7.26795 −0.567531
\(165\) 0 0
\(166\) 14.1962 1.10184
\(167\) 3.46410 0.268060 0.134030 0.990977i \(-0.457208\pi\)
0.134030 + 0.990977i \(0.457208\pi\)
\(168\) 3.73205 0.287934
\(169\) −1.00000 −0.0769231
\(170\) 1.46410 0.112291
\(171\) −2.00000 −0.152944
\(172\) −8.92820 −0.680769
\(173\) 20.1244 1.53003 0.765013 0.644015i \(-0.222732\pi\)
0.765013 + 0.644015i \(0.222732\pi\)
\(174\) −1.73205 −0.131306
\(175\) 9.19615 0.695164
\(176\) 0 0
\(177\) −11.3923 −0.856298
\(178\) −1.80385 −0.135204
\(179\) −15.9282 −1.19053 −0.595265 0.803530i \(-0.702953\pi\)
−0.595265 + 0.803530i \(0.702953\pi\)
\(180\) −5.46410 −0.407270
\(181\) 3.60770 0.268158 0.134079 0.990971i \(-0.457192\pi\)
0.134079 + 0.990971i \(0.457192\pi\)
\(182\) −12.9282 −0.958302
\(183\) −4.73205 −0.349803
\(184\) 1.00000 0.0737210
\(185\) 1.26795 0.0932215
\(186\) 5.46410 0.400647
\(187\) 0 0
\(188\) 5.53590 0.403747
\(189\) 18.6603 1.35733
\(190\) −2.73205 −0.198204
\(191\) 3.39230 0.245459 0.122729 0.992440i \(-0.460835\pi\)
0.122729 + 0.992440i \(0.460835\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 19.4641 1.40106 0.700528 0.713625i \(-0.252948\pi\)
0.700528 + 0.713625i \(0.252948\pi\)
\(194\) −6.53590 −0.469250
\(195\) −9.46410 −0.677738
\(196\) 6.92820 0.494872
\(197\) 18.9282 1.34858 0.674289 0.738467i \(-0.264450\pi\)
0.674289 + 0.738467i \(0.264450\pi\)
\(198\) 0 0
\(199\) 3.07180 0.217754 0.108877 0.994055i \(-0.465275\pi\)
0.108877 + 0.994055i \(0.465275\pi\)
\(200\) −2.46410 −0.174238
\(201\) −14.0000 −0.987484
\(202\) −14.5885 −1.02644
\(203\) −6.46410 −0.453691
\(204\) 0.535898 0.0375204
\(205\) −19.8564 −1.38683
\(206\) −0.732051 −0.0510044
\(207\) 2.00000 0.139010
\(208\) 3.46410 0.240192
\(209\) 0 0
\(210\) 10.1962 0.703601
\(211\) 20.1244 1.38542 0.692709 0.721217i \(-0.256417\pi\)
0.692709 + 0.721217i \(0.256417\pi\)
\(212\) 3.92820 0.269790
\(213\) −3.80385 −0.260635
\(214\) 13.1962 0.902070
\(215\) −24.3923 −1.66354
\(216\) −5.00000 −0.340207
\(217\) 20.3923 1.38432
\(218\) −18.1244 −1.22754
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) −1.85641 −0.124875
\(222\) 0.464102 0.0311485
\(223\) −11.1244 −0.744942 −0.372471 0.928044i \(-0.621489\pi\)
−0.372471 + 0.928044i \(0.621489\pi\)
\(224\) −3.73205 −0.249358
\(225\) −4.92820 −0.328547
\(226\) 1.26795 0.0843427
\(227\) −12.3923 −0.822506 −0.411253 0.911521i \(-0.634909\pi\)
−0.411253 + 0.911521i \(0.634909\pi\)
\(228\) −1.00000 −0.0662266
\(229\) 8.39230 0.554579 0.277290 0.960786i \(-0.410564\pi\)
0.277290 + 0.960786i \(0.410564\pi\)
\(230\) 2.73205 0.180146
\(231\) 0 0
\(232\) 1.73205 0.113715
\(233\) −11.7321 −0.768592 −0.384296 0.923210i \(-0.625556\pi\)
−0.384296 + 0.923210i \(0.625556\pi\)
\(234\) 6.92820 0.452911
\(235\) 15.1244 0.986604
\(236\) 11.3923 0.741576
\(237\) −8.19615 −0.532397
\(238\) 2.00000 0.129641
\(239\) 10.1244 0.654890 0.327445 0.944870i \(-0.393812\pi\)
0.327445 + 0.944870i \(0.393812\pi\)
\(240\) −2.73205 −0.176353
\(241\) 14.7321 0.948975 0.474487 0.880262i \(-0.342633\pi\)
0.474487 + 0.880262i \(0.342633\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 4.73205 0.302939
\(245\) 18.9282 1.20928
\(246\) −7.26795 −0.463388
\(247\) 3.46410 0.220416
\(248\) −5.46410 −0.346971
\(249\) 14.1962 0.899645
\(250\) 6.92820 0.438178
\(251\) 15.4641 0.976085 0.488043 0.872820i \(-0.337711\pi\)
0.488043 + 0.872820i \(0.337711\pi\)
\(252\) −7.46410 −0.470194
\(253\) 0 0
\(254\) −4.00000 −0.250982
\(255\) 1.46410 0.0916856
\(256\) 1.00000 0.0625000
\(257\) 31.3205 1.95372 0.976860 0.213881i \(-0.0686104\pi\)
0.976860 + 0.213881i \(0.0686104\pi\)
\(258\) −8.92820 −0.555846
\(259\) 1.73205 0.107624
\(260\) 9.46410 0.586939
\(261\) 3.46410 0.214423
\(262\) 7.07180 0.436897
\(263\) −16.7846 −1.03498 −0.517492 0.855688i \(-0.673134\pi\)
−0.517492 + 0.855688i \(0.673134\pi\)
\(264\) 0 0
\(265\) 10.7321 0.659265
\(266\) −3.73205 −0.228827
\(267\) −1.80385 −0.110394
\(268\) 14.0000 0.855186
\(269\) 11.7846 0.718520 0.359260 0.933237i \(-0.383029\pi\)
0.359260 + 0.933237i \(0.383029\pi\)
\(270\) −13.6603 −0.831337
\(271\) 27.3205 1.65960 0.829801 0.558059i \(-0.188454\pi\)
0.829801 + 0.558059i \(0.188454\pi\)
\(272\) −0.535898 −0.0324936
\(273\) −12.9282 −0.782450
\(274\) −2.53590 −0.153199
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) −9.12436 −0.548229 −0.274115 0.961697i \(-0.588385\pi\)
−0.274115 + 0.961697i \(0.588385\pi\)
\(278\) 13.4641 0.807523
\(279\) −10.9282 −0.654254
\(280\) −10.1962 −0.609337
\(281\) −22.0526 −1.31555 −0.657773 0.753216i \(-0.728501\pi\)
−0.657773 + 0.753216i \(0.728501\pi\)
\(282\) 5.53590 0.329658
\(283\) 14.5359 0.864069 0.432035 0.901857i \(-0.357796\pi\)
0.432035 + 0.901857i \(0.357796\pi\)
\(284\) 3.80385 0.225717
\(285\) −2.73205 −0.161833
\(286\) 0 0
\(287\) −27.1244 −1.60110
\(288\) 2.00000 0.117851
\(289\) −16.7128 −0.983107
\(290\) 4.73205 0.277876
\(291\) −6.53590 −0.383141
\(292\) 2.00000 0.117041
\(293\) −12.1244 −0.708312 −0.354156 0.935186i \(-0.615232\pi\)
−0.354156 + 0.935186i \(0.615232\pi\)
\(294\) 6.92820 0.404061
\(295\) 31.1244 1.81213
\(296\) −0.464102 −0.0269754
\(297\) 0 0
\(298\) 23.3205 1.35092
\(299\) −3.46410 −0.200334
\(300\) −2.46410 −0.142265
\(301\) −33.3205 −1.92056
\(302\) 17.2679 0.993659
\(303\) −14.5885 −0.838085
\(304\) 1.00000 0.0573539
\(305\) 12.9282 0.740267
\(306\) −1.07180 −0.0612705
\(307\) −15.5885 −0.889680 −0.444840 0.895610i \(-0.646740\pi\)
−0.444840 + 0.895610i \(0.646740\pi\)
\(308\) 0 0
\(309\) −0.732051 −0.0416449
\(310\) −14.9282 −0.847865
\(311\) −22.3923 −1.26975 −0.634876 0.772614i \(-0.718949\pi\)
−0.634876 + 0.772614i \(0.718949\pi\)
\(312\) 3.46410 0.196116
\(313\) 20.4641 1.15670 0.578350 0.815789i \(-0.303697\pi\)
0.578350 + 0.815789i \(0.303697\pi\)
\(314\) 8.73205 0.492778
\(315\) −20.3923 −1.14898
\(316\) 8.19615 0.461070
\(317\) 20.7846 1.16738 0.583690 0.811977i \(-0.301608\pi\)
0.583690 + 0.811977i \(0.301608\pi\)
\(318\) 3.92820 0.220283
\(319\) 0 0
\(320\) 2.73205 0.152726
\(321\) 13.1962 0.736537
\(322\) 3.73205 0.207979
\(323\) −0.535898 −0.0298182
\(324\) 1.00000 0.0555556
\(325\) 8.53590 0.473486
\(326\) 17.3205 0.959294
\(327\) −18.1244 −1.00228
\(328\) 7.26795 0.401305
\(329\) 20.6603 1.13904
\(330\) 0 0
\(331\) −19.7846 −1.08746 −0.543730 0.839260i \(-0.682989\pi\)
−0.543730 + 0.839260i \(0.682989\pi\)
\(332\) −14.1962 −0.779115
\(333\) −0.928203 −0.0508652
\(334\) −3.46410 −0.189547
\(335\) 38.2487 2.08975
\(336\) −3.73205 −0.203600
\(337\) −13.8038 −0.751943 −0.375972 0.926631i \(-0.622691\pi\)
−0.375972 + 0.926631i \(0.622691\pi\)
\(338\) 1.00000 0.0543928
\(339\) 1.26795 0.0688655
\(340\) −1.46410 −0.0794021
\(341\) 0 0
\(342\) 2.00000 0.108148
\(343\) −0.267949 −0.0144679
\(344\) 8.92820 0.481376
\(345\) 2.73205 0.147089
\(346\) −20.1244 −1.08189
\(347\) −28.0526 −1.50594 −0.752970 0.658055i \(-0.771380\pi\)
−0.752970 + 0.658055i \(0.771380\pi\)
\(348\) 1.73205 0.0928477
\(349\) 26.7846 1.43375 0.716874 0.697203i \(-0.245572\pi\)
0.716874 + 0.697203i \(0.245572\pi\)
\(350\) −9.19615 −0.491555
\(351\) 17.3205 0.924500
\(352\) 0 0
\(353\) 0.464102 0.0247016 0.0123508 0.999924i \(-0.496069\pi\)
0.0123508 + 0.999924i \(0.496069\pi\)
\(354\) 11.3923 0.605494
\(355\) 10.3923 0.551566
\(356\) 1.80385 0.0956037
\(357\) 2.00000 0.105851
\(358\) 15.9282 0.841832
\(359\) 7.60770 0.401519 0.200759 0.979641i \(-0.435659\pi\)
0.200759 + 0.979641i \(0.435659\pi\)
\(360\) 5.46410 0.287983
\(361\) 1.00000 0.0526316
\(362\) −3.60770 −0.189616
\(363\) 0 0
\(364\) 12.9282 0.677622
\(365\) 5.46410 0.286004
\(366\) 4.73205 0.247348
\(367\) 4.85641 0.253502 0.126751 0.991935i \(-0.459545\pi\)
0.126751 + 0.991935i \(0.459545\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 14.5359 0.756709
\(370\) −1.26795 −0.0659175
\(371\) 14.6603 0.761122
\(372\) −5.46410 −0.283300
\(373\) −27.1962 −1.40816 −0.704082 0.710119i \(-0.748641\pi\)
−0.704082 + 0.710119i \(0.748641\pi\)
\(374\) 0 0
\(375\) 6.92820 0.357771
\(376\) −5.53590 −0.285492
\(377\) −6.00000 −0.309016
\(378\) −18.6603 −0.959780
\(379\) 10.0718 0.517353 0.258677 0.965964i \(-0.416714\pi\)
0.258677 + 0.965964i \(0.416714\pi\)
\(380\) 2.73205 0.140151
\(381\) −4.00000 −0.204926
\(382\) −3.39230 −0.173565
\(383\) 26.5359 1.35592 0.677961 0.735098i \(-0.262864\pi\)
0.677961 + 0.735098i \(0.262864\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −19.4641 −0.990697
\(387\) 17.8564 0.907692
\(388\) 6.53590 0.331810
\(389\) 30.5885 1.55090 0.775448 0.631411i \(-0.217524\pi\)
0.775448 + 0.631411i \(0.217524\pi\)
\(390\) 9.46410 0.479233
\(391\) 0.535898 0.0271015
\(392\) −6.92820 −0.349927
\(393\) 7.07180 0.356725
\(394\) −18.9282 −0.953589
\(395\) 22.3923 1.12668
\(396\) 0 0
\(397\) 26.3923 1.32459 0.662296 0.749242i \(-0.269582\pi\)
0.662296 + 0.749242i \(0.269582\pi\)
\(398\) −3.07180 −0.153975
\(399\) −3.73205 −0.186836
\(400\) 2.46410 0.123205
\(401\) 22.0526 1.10125 0.550626 0.834752i \(-0.314389\pi\)
0.550626 + 0.834752i \(0.314389\pi\)
\(402\) 14.0000 0.698257
\(403\) 18.9282 0.942881
\(404\) 14.5885 0.725803
\(405\) 2.73205 0.135757
\(406\) 6.46410 0.320808
\(407\) 0 0
\(408\) −0.535898 −0.0265309
\(409\) 6.53590 0.323179 0.161590 0.986858i \(-0.448338\pi\)
0.161590 + 0.986858i \(0.448338\pi\)
\(410\) 19.8564 0.980638
\(411\) −2.53590 −0.125087
\(412\) 0.732051 0.0360656
\(413\) 42.5167 2.09211
\(414\) −2.00000 −0.0982946
\(415\) −38.7846 −1.90386
\(416\) −3.46410 −0.169842
\(417\) 13.4641 0.659340
\(418\) 0 0
\(419\) 26.4449 1.29192 0.645958 0.763373i \(-0.276458\pi\)
0.645958 + 0.763373i \(0.276458\pi\)
\(420\) −10.1962 −0.497521
\(421\) −36.3205 −1.77015 −0.885077 0.465445i \(-0.845894\pi\)
−0.885077 + 0.465445i \(0.845894\pi\)
\(422\) −20.1244 −0.979638
\(423\) −11.0718 −0.538329
\(424\) −3.92820 −0.190770
\(425\) −1.32051 −0.0640541
\(426\) 3.80385 0.184297
\(427\) 17.6603 0.854640
\(428\) −13.1962 −0.637860
\(429\) 0 0
\(430\) 24.3923 1.17630
\(431\) 6.33975 0.305375 0.152687 0.988275i \(-0.451207\pi\)
0.152687 + 0.988275i \(0.451207\pi\)
\(432\) 5.00000 0.240563
\(433\) −22.7846 −1.09496 −0.547479 0.836819i \(-0.684412\pi\)
−0.547479 + 0.836819i \(0.684412\pi\)
\(434\) −20.3923 −0.978862
\(435\) 4.73205 0.226884
\(436\) 18.1244 0.867999
\(437\) −1.00000 −0.0478365
\(438\) 2.00000 0.0955637
\(439\) 10.5359 0.502851 0.251425 0.967877i \(-0.419101\pi\)
0.251425 + 0.967877i \(0.419101\pi\)
\(440\) 0 0
\(441\) −13.8564 −0.659829
\(442\) 1.85641 0.0883003
\(443\) 18.0000 0.855206 0.427603 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(444\) −0.464102 −0.0220253
\(445\) 4.92820 0.233619
\(446\) 11.1244 0.526754
\(447\) 23.3205 1.10302
\(448\) 3.73205 0.176323
\(449\) 0.143594 0.00677660 0.00338830 0.999994i \(-0.498921\pi\)
0.00338830 + 0.999994i \(0.498921\pi\)
\(450\) 4.92820 0.232318
\(451\) 0 0
\(452\) −1.26795 −0.0596393
\(453\) 17.2679 0.811319
\(454\) 12.3923 0.581600
\(455\) 35.3205 1.65585
\(456\) 1.00000 0.0468293
\(457\) 33.4449 1.56448 0.782242 0.622974i \(-0.214076\pi\)
0.782242 + 0.622974i \(0.214076\pi\)
\(458\) −8.39230 −0.392147
\(459\) −2.67949 −0.125068
\(460\) −2.73205 −0.127383
\(461\) −17.8038 −0.829208 −0.414604 0.910002i \(-0.636080\pi\)
−0.414604 + 0.910002i \(0.636080\pi\)
\(462\) 0 0
\(463\) 26.0000 1.20832 0.604161 0.796862i \(-0.293508\pi\)
0.604161 + 0.796862i \(0.293508\pi\)
\(464\) −1.73205 −0.0804084
\(465\) −14.9282 −0.692279
\(466\) 11.7321 0.543477
\(467\) −21.7128 −1.00475 −0.502375 0.864650i \(-0.667540\pi\)
−0.502375 + 0.864650i \(0.667540\pi\)
\(468\) −6.92820 −0.320256
\(469\) 52.2487 2.41262
\(470\) −15.1244 −0.697635
\(471\) 8.73205 0.402352
\(472\) −11.3923 −0.524373
\(473\) 0 0
\(474\) 8.19615 0.376462
\(475\) 2.46410 0.113061
\(476\) −2.00000 −0.0916698
\(477\) −7.85641 −0.359720
\(478\) −10.1244 −0.463077
\(479\) −28.2679 −1.29160 −0.645798 0.763508i \(-0.723475\pi\)
−0.645798 + 0.763508i \(0.723475\pi\)
\(480\) 2.73205 0.124700
\(481\) 1.60770 0.0733046
\(482\) −14.7321 −0.671027
\(483\) 3.73205 0.169814
\(484\) 0 0
\(485\) 17.8564 0.810818
\(486\) 16.0000 0.725775
\(487\) −35.1769 −1.59402 −0.797009 0.603967i \(-0.793586\pi\)
−0.797009 + 0.603967i \(0.793586\pi\)
\(488\) −4.73205 −0.214210
\(489\) 17.3205 0.783260
\(490\) −18.9282 −0.855089
\(491\) 24.3923 1.10081 0.550405 0.834898i \(-0.314473\pi\)
0.550405 + 0.834898i \(0.314473\pi\)
\(492\) 7.26795 0.327664
\(493\) 0.928203 0.0418042
\(494\) −3.46410 −0.155857
\(495\) 0 0
\(496\) 5.46410 0.245345
\(497\) 14.1962 0.636784
\(498\) −14.1962 −0.636145
\(499\) 24.7321 1.10716 0.553579 0.832796i \(-0.313262\pi\)
0.553579 + 0.832796i \(0.313262\pi\)
\(500\) −6.92820 −0.309839
\(501\) −3.46410 −0.154765
\(502\) −15.4641 −0.690197
\(503\) 16.6603 0.742844 0.371422 0.928464i \(-0.378870\pi\)
0.371422 + 0.928464i \(0.378870\pi\)
\(504\) 7.46410 0.332478
\(505\) 39.8564 1.77359
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 4.00000 0.177471
\(509\) 20.0718 0.889667 0.444833 0.895613i \(-0.353263\pi\)
0.444833 + 0.895613i \(0.353263\pi\)
\(510\) −1.46410 −0.0648315
\(511\) 7.46410 0.330192
\(512\) −1.00000 −0.0441942
\(513\) 5.00000 0.220755
\(514\) −31.3205 −1.38149
\(515\) 2.00000 0.0881305
\(516\) 8.92820 0.393042
\(517\) 0 0
\(518\) −1.73205 −0.0761019
\(519\) −20.1244 −0.883361
\(520\) −9.46410 −0.415028
\(521\) 17.3205 0.758825 0.379413 0.925228i \(-0.376126\pi\)
0.379413 + 0.925228i \(0.376126\pi\)
\(522\) −3.46410 −0.151620
\(523\) 26.6603 1.16577 0.582886 0.812554i \(-0.301924\pi\)
0.582886 + 0.812554i \(0.301924\pi\)
\(524\) −7.07180 −0.308933
\(525\) −9.19615 −0.401353
\(526\) 16.7846 0.731844
\(527\) −2.92820 −0.127555
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) −10.7321 −0.466170
\(531\) −22.7846 −0.988768
\(532\) 3.73205 0.161805
\(533\) −25.1769 −1.09053
\(534\) 1.80385 0.0780601
\(535\) −36.0526 −1.55869
\(536\) −14.0000 −0.604708
\(537\) 15.9282 0.687353
\(538\) −11.7846 −0.508071
\(539\) 0 0
\(540\) 13.6603 0.587844
\(541\) −35.1769 −1.51237 −0.756187 0.654356i \(-0.772940\pi\)
−0.756187 + 0.654356i \(0.772940\pi\)
\(542\) −27.3205 −1.17352
\(543\) −3.60770 −0.154821
\(544\) 0.535898 0.0229765
\(545\) 49.5167 2.12106
\(546\) 12.9282 0.553276
\(547\) −15.1962 −0.649741 −0.324870 0.945759i \(-0.605321\pi\)
−0.324870 + 0.945759i \(0.605321\pi\)
\(548\) 2.53590 0.108328
\(549\) −9.46410 −0.403918
\(550\) 0 0
\(551\) −1.73205 −0.0737878
\(552\) −1.00000 −0.0425628
\(553\) 30.5885 1.30075
\(554\) 9.12436 0.387657
\(555\) −1.26795 −0.0538214
\(556\) −13.4641 −0.571005
\(557\) −39.1769 −1.65998 −0.829990 0.557779i \(-0.811654\pi\)
−0.829990 + 0.557779i \(0.811654\pi\)
\(558\) 10.9282 0.462628
\(559\) −30.9282 −1.30812
\(560\) 10.1962 0.430866
\(561\) 0 0
\(562\) 22.0526 0.930231
\(563\) −7.33975 −0.309333 −0.154667 0.987967i \(-0.549430\pi\)
−0.154667 + 0.987967i \(0.549430\pi\)
\(564\) −5.53590 −0.233103
\(565\) −3.46410 −0.145736
\(566\) −14.5359 −0.610989
\(567\) 3.73205 0.156731
\(568\) −3.80385 −0.159606
\(569\) −15.7128 −0.658715 −0.329358 0.944205i \(-0.606832\pi\)
−0.329358 + 0.944205i \(0.606832\pi\)
\(570\) 2.73205 0.114433
\(571\) −28.9808 −1.21281 −0.606404 0.795157i \(-0.707388\pi\)
−0.606404 + 0.795157i \(0.707388\pi\)
\(572\) 0 0
\(573\) −3.39230 −0.141716
\(574\) 27.1244 1.13215
\(575\) −2.46410 −0.102760
\(576\) −2.00000 −0.0833333
\(577\) 29.1769 1.21465 0.607325 0.794453i \(-0.292242\pi\)
0.607325 + 0.794453i \(0.292242\pi\)
\(578\) 16.7128 0.695161
\(579\) −19.4641 −0.808900
\(580\) −4.73205 −0.196488
\(581\) −52.9808 −2.19801
\(582\) 6.53590 0.270922
\(583\) 0 0
\(584\) −2.00000 −0.0827606
\(585\) −18.9282 −0.782585
\(586\) 12.1244 0.500853
\(587\) 41.3731 1.70765 0.853825 0.520561i \(-0.174277\pi\)
0.853825 + 0.520561i \(0.174277\pi\)
\(588\) −6.92820 −0.285714
\(589\) 5.46410 0.225144
\(590\) −31.1244 −1.28137
\(591\) −18.9282 −0.778602
\(592\) 0.464102 0.0190745
\(593\) −26.1244 −1.07280 −0.536399 0.843964i \(-0.680216\pi\)
−0.536399 + 0.843964i \(0.680216\pi\)
\(594\) 0 0
\(595\) −5.46410 −0.224006
\(596\) −23.3205 −0.955245
\(597\) −3.07180 −0.125720
\(598\) 3.46410 0.141658
\(599\) −40.3923 −1.65038 −0.825192 0.564852i \(-0.808933\pi\)
−0.825192 + 0.564852i \(0.808933\pi\)
\(600\) 2.46410 0.100597
\(601\) 22.7846 0.929404 0.464702 0.885467i \(-0.346162\pi\)
0.464702 + 0.885467i \(0.346162\pi\)
\(602\) 33.3205 1.35804
\(603\) −28.0000 −1.14025
\(604\) −17.2679 −0.702623
\(605\) 0 0
\(606\) 14.5885 0.592616
\(607\) 6.73205 0.273246 0.136623 0.990623i \(-0.456375\pi\)
0.136623 + 0.990623i \(0.456375\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 6.46410 0.261939
\(610\) −12.9282 −0.523448
\(611\) 19.1769 0.775815
\(612\) 1.07180 0.0433248
\(613\) 11.8038 0.476753 0.238376 0.971173i \(-0.423385\pi\)
0.238376 + 0.971173i \(0.423385\pi\)
\(614\) 15.5885 0.629099
\(615\) 19.8564 0.800688
\(616\) 0 0
\(617\) 11.9282 0.480211 0.240106 0.970747i \(-0.422818\pi\)
0.240106 + 0.970747i \(0.422818\pi\)
\(618\) 0.732051 0.0294474
\(619\) 13.6077 0.546939 0.273470 0.961881i \(-0.411829\pi\)
0.273470 + 0.961881i \(0.411829\pi\)
\(620\) 14.9282 0.599531
\(621\) −5.00000 −0.200643
\(622\) 22.3923 0.897850
\(623\) 6.73205 0.269714
\(624\) −3.46410 −0.138675
\(625\) −31.2487 −1.24995
\(626\) −20.4641 −0.817910
\(627\) 0 0
\(628\) −8.73205 −0.348447
\(629\) −0.248711 −0.00991677
\(630\) 20.3923 0.812449
\(631\) 27.2487 1.08475 0.542377 0.840135i \(-0.317524\pi\)
0.542377 + 0.840135i \(0.317524\pi\)
\(632\) −8.19615 −0.326025
\(633\) −20.1244 −0.799871
\(634\) −20.7846 −0.825462
\(635\) 10.9282 0.433673
\(636\) −3.92820 −0.155763
\(637\) 24.0000 0.950915
\(638\) 0 0
\(639\) −7.60770 −0.300956
\(640\) −2.73205 −0.107994
\(641\) −44.1962 −1.74564 −0.872822 0.488040i \(-0.837712\pi\)
−0.872822 + 0.488040i \(0.837712\pi\)
\(642\) −13.1962 −0.520811
\(643\) 37.7128 1.48725 0.743624 0.668598i \(-0.233105\pi\)
0.743624 + 0.668598i \(0.233105\pi\)
\(644\) −3.73205 −0.147063
\(645\) 24.3923 0.960446
\(646\) 0.535898 0.0210846
\(647\) −30.8564 −1.21309 −0.606545 0.795049i \(-0.707445\pi\)
−0.606545 + 0.795049i \(0.707445\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) −8.53590 −0.334805
\(651\) −20.3923 −0.799237
\(652\) −17.3205 −0.678323
\(653\) −37.3205 −1.46046 −0.730232 0.683199i \(-0.760588\pi\)
−0.730232 + 0.683199i \(0.760588\pi\)
\(654\) 18.1244 0.708718
\(655\) −19.3205 −0.754915
\(656\) −7.26795 −0.283766
\(657\) −4.00000 −0.156055
\(658\) −20.6603 −0.805421
\(659\) −17.3205 −0.674711 −0.337356 0.941377i \(-0.609532\pi\)
−0.337356 + 0.941377i \(0.609532\pi\)
\(660\) 0 0
\(661\) −29.2487 −1.13764 −0.568822 0.822461i \(-0.692600\pi\)
−0.568822 + 0.822461i \(0.692600\pi\)
\(662\) 19.7846 0.768951
\(663\) 1.85641 0.0720969
\(664\) 14.1962 0.550918
\(665\) 10.1962 0.395390
\(666\) 0.928203 0.0359671
\(667\) 1.73205 0.0670653
\(668\) 3.46410 0.134030
\(669\) 11.1244 0.430092
\(670\) −38.2487 −1.47768
\(671\) 0 0
\(672\) 3.73205 0.143967
\(673\) 16.5885 0.639438 0.319719 0.947512i \(-0.396411\pi\)
0.319719 + 0.947512i \(0.396411\pi\)
\(674\) 13.8038 0.531704
\(675\) 12.3205 0.474217
\(676\) −1.00000 −0.0384615
\(677\) 19.3397 0.743287 0.371643 0.928376i \(-0.378794\pi\)
0.371643 + 0.928376i \(0.378794\pi\)
\(678\) −1.26795 −0.0486953
\(679\) 24.3923 0.936091
\(680\) 1.46410 0.0561457
\(681\) 12.3923 0.474874
\(682\) 0 0
\(683\) 18.7846 0.718773 0.359387 0.933189i \(-0.382986\pi\)
0.359387 + 0.933189i \(0.382986\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 6.92820 0.264713
\(686\) 0.267949 0.0102303
\(687\) −8.39230 −0.320186
\(688\) −8.92820 −0.340385
\(689\) 13.6077 0.518412
\(690\) −2.73205 −0.104007
\(691\) −10.0526 −0.382417 −0.191209 0.981549i \(-0.561241\pi\)
−0.191209 + 0.981549i \(0.561241\pi\)
\(692\) 20.1244 0.765013
\(693\) 0 0
\(694\) 28.0526 1.06486
\(695\) −36.7846 −1.39532
\(696\) −1.73205 −0.0656532
\(697\) 3.89488 0.147529
\(698\) −26.7846 −1.01381
\(699\) 11.7321 0.443747
\(700\) 9.19615 0.347582
\(701\) −2.53590 −0.0957796 −0.0478898 0.998853i \(-0.515250\pi\)
−0.0478898 + 0.998853i \(0.515250\pi\)
\(702\) −17.3205 −0.653720
\(703\) 0.464102 0.0175039
\(704\) 0 0
\(705\) −15.1244 −0.569616
\(706\) −0.464102 −0.0174667
\(707\) 54.4449 2.04761
\(708\) −11.3923 −0.428149
\(709\) −17.8564 −0.670611 −0.335306 0.942109i \(-0.608840\pi\)
−0.335306 + 0.942109i \(0.608840\pi\)
\(710\) −10.3923 −0.390016
\(711\) −16.3923 −0.614759
\(712\) −1.80385 −0.0676020
\(713\) −5.46410 −0.204632
\(714\) −2.00000 −0.0748481
\(715\) 0 0
\(716\) −15.9282 −0.595265
\(717\) −10.1244 −0.378101
\(718\) −7.60770 −0.283917
\(719\) −16.0000 −0.596699 −0.298350 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) −5.46410 −0.203635
\(721\) 2.73205 0.101747
\(722\) −1.00000 −0.0372161
\(723\) −14.7321 −0.547891
\(724\) 3.60770 0.134079
\(725\) −4.26795 −0.158508
\(726\) 0 0
\(727\) 6.32051 0.234415 0.117207 0.993107i \(-0.462606\pi\)
0.117207 + 0.993107i \(0.462606\pi\)
\(728\) −12.9282 −0.479151
\(729\) 13.0000 0.481481
\(730\) −5.46410 −0.202235
\(731\) 4.78461 0.176965
\(732\) −4.73205 −0.174902
\(733\) 18.9282 0.699129 0.349565 0.936912i \(-0.386330\pi\)
0.349565 + 0.936912i \(0.386330\pi\)
\(734\) −4.85641 −0.179253
\(735\) −18.9282 −0.698177
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) −14.5359 −0.535074
\(739\) −14.3397 −0.527496 −0.263748 0.964592i \(-0.584959\pi\)
−0.263748 + 0.964592i \(0.584959\pi\)
\(740\) 1.26795 0.0466107
\(741\) −3.46410 −0.127257
\(742\) −14.6603 −0.538195
\(743\) −19.4641 −0.714069 −0.357034 0.934091i \(-0.616212\pi\)
−0.357034 + 0.934091i \(0.616212\pi\)
\(744\) 5.46410 0.200324
\(745\) −63.7128 −2.33426
\(746\) 27.1962 0.995722
\(747\) 28.3923 1.03882
\(748\) 0 0
\(749\) −49.2487 −1.79951
\(750\) −6.92820 −0.252982
\(751\) −12.0000 −0.437886 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) 5.53590 0.201873
\(753\) −15.4641 −0.563543
\(754\) 6.00000 0.218507
\(755\) −47.1769 −1.71694
\(756\) 18.6603 0.678667
\(757\) 51.3731 1.86719 0.933593 0.358335i \(-0.116656\pi\)
0.933593 + 0.358335i \(0.116656\pi\)
\(758\) −10.0718 −0.365824
\(759\) 0 0
\(760\) −2.73205 −0.0991019
\(761\) 44.7654 1.62274 0.811372 0.584531i \(-0.198721\pi\)
0.811372 + 0.584531i \(0.198721\pi\)
\(762\) 4.00000 0.144905
\(763\) 67.6410 2.44877
\(764\) 3.39230 0.122729
\(765\) 2.92820 0.105869
\(766\) −26.5359 −0.958781
\(767\) 39.4641 1.42497
\(768\) −1.00000 −0.0360844
\(769\) −3.58846 −0.129403 −0.0647015 0.997905i \(-0.520610\pi\)
−0.0647015 + 0.997905i \(0.520610\pi\)
\(770\) 0 0
\(771\) −31.3205 −1.12798
\(772\) 19.4641 0.700528
\(773\) 27.3923 0.985233 0.492616 0.870247i \(-0.336041\pi\)
0.492616 + 0.870247i \(0.336041\pi\)
\(774\) −17.8564 −0.641835
\(775\) 13.4641 0.483645
\(776\) −6.53590 −0.234625
\(777\) −1.73205 −0.0621370
\(778\) −30.5885 −1.09665
\(779\) −7.26795 −0.260401
\(780\) −9.46410 −0.338869
\(781\) 0 0
\(782\) −0.535898 −0.0191637
\(783\) −8.66025 −0.309492
\(784\) 6.92820 0.247436
\(785\) −23.8564 −0.851472
\(786\) −7.07180 −0.252243
\(787\) 25.0526 0.893027 0.446514 0.894777i \(-0.352665\pi\)
0.446514 + 0.894777i \(0.352665\pi\)
\(788\) 18.9282 0.674289
\(789\) 16.7846 0.597548
\(790\) −22.3923 −0.796682
\(791\) −4.73205 −0.168252
\(792\) 0 0
\(793\) 16.3923 0.582108
\(794\) −26.3923 −0.936628
\(795\) −10.7321 −0.380627
\(796\) 3.07180 0.108877
\(797\) 32.3923 1.14739 0.573697 0.819068i \(-0.305509\pi\)
0.573697 + 0.819068i \(0.305509\pi\)
\(798\) 3.73205 0.132113
\(799\) −2.96668 −0.104954
\(800\) −2.46410 −0.0871191
\(801\) −3.60770 −0.127472
\(802\) −22.0526 −0.778703
\(803\) 0 0
\(804\) −14.0000 −0.493742
\(805\) −10.1962 −0.359367
\(806\) −18.9282 −0.666718
\(807\) −11.7846 −0.414838
\(808\) −14.5885 −0.513220
\(809\) −8.80385 −0.309527 −0.154763 0.987952i \(-0.549462\pi\)
−0.154763 + 0.987952i \(0.549462\pi\)
\(810\) −2.73205 −0.0959945
\(811\) −46.7654 −1.64215 −0.821077 0.570817i \(-0.806626\pi\)
−0.821077 + 0.570817i \(0.806626\pi\)
\(812\) −6.46410 −0.226845
\(813\) −27.3205 −0.958172
\(814\) 0 0
\(815\) −47.3205 −1.65757
\(816\) 0.535898 0.0187602
\(817\) −8.92820 −0.312358
\(818\) −6.53590 −0.228522
\(819\) −25.8564 −0.903496
\(820\) −19.8564 −0.693416
\(821\) −3.94744 −0.137767 −0.0688833 0.997625i \(-0.521944\pi\)
−0.0688833 + 0.997625i \(0.521944\pi\)
\(822\) 2.53590 0.0884496
\(823\) −28.4641 −0.992196 −0.496098 0.868266i \(-0.665234\pi\)
−0.496098 + 0.868266i \(0.665234\pi\)
\(824\) −0.732051 −0.0255022
\(825\) 0 0
\(826\) −42.5167 −1.47934
\(827\) −13.3205 −0.463199 −0.231600 0.972811i \(-0.574396\pi\)
−0.231600 + 0.972811i \(0.574396\pi\)
\(828\) 2.00000 0.0695048
\(829\) −50.6410 −1.75884 −0.879418 0.476051i \(-0.842068\pi\)
−0.879418 + 0.476051i \(0.842068\pi\)
\(830\) 38.7846 1.34623
\(831\) 9.12436 0.316520
\(832\) 3.46410 0.120096
\(833\) −3.71281 −0.128641
\(834\) −13.4641 −0.466224
\(835\) 9.46410 0.327519
\(836\) 0 0
\(837\) 27.3205 0.944335
\(838\) −26.4449 −0.913523
\(839\) −36.9808 −1.27672 −0.638359 0.769739i \(-0.720386\pi\)
−0.638359 + 0.769739i \(0.720386\pi\)
\(840\) 10.1962 0.351801
\(841\) −26.0000 −0.896552
\(842\) 36.3205 1.25169
\(843\) 22.0526 0.759530
\(844\) 20.1244 0.692709
\(845\) −2.73205 −0.0939854
\(846\) 11.0718 0.380656
\(847\) 0 0
\(848\) 3.92820 0.134895
\(849\) −14.5359 −0.498871
\(850\) 1.32051 0.0452931
\(851\) −0.464102 −0.0159092
\(852\) −3.80385 −0.130318
\(853\) 22.1051 0.756865 0.378432 0.925629i \(-0.376463\pi\)
0.378432 + 0.925629i \(0.376463\pi\)
\(854\) −17.6603 −0.604321
\(855\) −5.46410 −0.186868
\(856\) 13.1962 0.451035
\(857\) −23.4641 −0.801518 −0.400759 0.916183i \(-0.631254\pi\)
−0.400759 + 0.916183i \(0.631254\pi\)
\(858\) 0 0
\(859\) −58.0526 −1.98073 −0.990364 0.138490i \(-0.955775\pi\)
−0.990364 + 0.138490i \(0.955775\pi\)
\(860\) −24.3923 −0.831771
\(861\) 27.1244 0.924396
\(862\) −6.33975 −0.215933
\(863\) −29.0718 −0.989615 −0.494808 0.869002i \(-0.664762\pi\)
−0.494808 + 0.869002i \(0.664762\pi\)
\(864\) −5.00000 −0.170103
\(865\) 54.9808 1.86940
\(866\) 22.7846 0.774253
\(867\) 16.7128 0.567597
\(868\) 20.3923 0.692160
\(869\) 0 0
\(870\) −4.73205 −0.160432
\(871\) 48.4974 1.64327
\(872\) −18.1244 −0.613768
\(873\) −13.0718 −0.442413
\(874\) 1.00000 0.0338255
\(875\) −25.8564 −0.874106
\(876\) −2.00000 −0.0675737
\(877\) 10.1436 0.342525 0.171262 0.985225i \(-0.445215\pi\)
0.171262 + 0.985225i \(0.445215\pi\)
\(878\) −10.5359 −0.355569
\(879\) 12.1244 0.408944
\(880\) 0 0
\(881\) −20.4641 −0.689453 −0.344727 0.938703i \(-0.612028\pi\)
−0.344727 + 0.938703i \(0.612028\pi\)
\(882\) 13.8564 0.466569
\(883\) 6.58846 0.221719 0.110860 0.993836i \(-0.464640\pi\)
0.110860 + 0.993836i \(0.464640\pi\)
\(884\) −1.85641 −0.0624377
\(885\) −31.1244 −1.04623
\(886\) −18.0000 −0.604722
\(887\) 12.3397 0.414328 0.207164 0.978306i \(-0.433577\pi\)
0.207164 + 0.978306i \(0.433577\pi\)
\(888\) 0.464102 0.0155742
\(889\) 14.9282 0.500676
\(890\) −4.92820 −0.165194
\(891\) 0 0
\(892\) −11.1244 −0.372471
\(893\) 5.53590 0.185252
\(894\) −23.3205 −0.779954
\(895\) −43.5167 −1.45460
\(896\) −3.73205 −0.124679
\(897\) 3.46410 0.115663
\(898\) −0.143594 −0.00479178
\(899\) −9.46410 −0.315645
\(900\) −4.92820 −0.164273
\(901\) −2.10512 −0.0701316
\(902\) 0 0
\(903\) 33.3205 1.10884
\(904\) 1.26795 0.0421714
\(905\) 9.85641 0.327638
\(906\) −17.2679 −0.573689
\(907\) −19.2487 −0.639143 −0.319571 0.947562i \(-0.603539\pi\)
−0.319571 + 0.947562i \(0.603539\pi\)
\(908\) −12.3923 −0.411253
\(909\) −29.1769 −0.967737
\(910\) −35.3205 −1.17086
\(911\) −26.5885 −0.880915 −0.440457 0.897773i \(-0.645184\pi\)
−0.440457 + 0.897773i \(0.645184\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 0 0
\(914\) −33.4449 −1.10626
\(915\) −12.9282 −0.427393
\(916\) 8.39230 0.277290
\(917\) −26.3923 −0.871551
\(918\) 2.67949 0.0884364
\(919\) 22.1051 0.729181 0.364590 0.931168i \(-0.381209\pi\)
0.364590 + 0.931168i \(0.381209\pi\)
\(920\) 2.73205 0.0900730
\(921\) 15.5885 0.513657
\(922\) 17.8038 0.586338
\(923\) 13.1769 0.433723
\(924\) 0 0
\(925\) 1.14359 0.0376011
\(926\) −26.0000 −0.854413
\(927\) −1.46410 −0.0480874
\(928\) 1.73205 0.0568574
\(929\) 32.0333 1.05098 0.525490 0.850800i \(-0.323882\pi\)
0.525490 + 0.850800i \(0.323882\pi\)
\(930\) 14.9282 0.489515
\(931\) 6.92820 0.227063
\(932\) −11.7321 −0.384296
\(933\) 22.3923 0.733091
\(934\) 21.7128 0.710465
\(935\) 0 0
\(936\) 6.92820 0.226455
\(937\) 54.1244 1.76817 0.884083 0.467330i \(-0.154784\pi\)
0.884083 + 0.467330i \(0.154784\pi\)
\(938\) −52.2487 −1.70598
\(939\) −20.4641 −0.667821
\(940\) 15.1244 0.493302
\(941\) −24.2487 −0.790485 −0.395243 0.918577i \(-0.629340\pi\)
−0.395243 + 0.918577i \(0.629340\pi\)
\(942\) −8.73205 −0.284506
\(943\) 7.26795 0.236677
\(944\) 11.3923 0.370788
\(945\) 50.9808 1.65840
\(946\) 0 0
\(947\) −31.7128 −1.03053 −0.515264 0.857032i \(-0.672306\pi\)
−0.515264 + 0.857032i \(0.672306\pi\)
\(948\) −8.19615 −0.266199
\(949\) 6.92820 0.224899
\(950\) −2.46410 −0.0799460
\(951\) −20.7846 −0.673987
\(952\) 2.00000 0.0648204
\(953\) 36.9808 1.19792 0.598962 0.800777i \(-0.295580\pi\)
0.598962 + 0.800777i \(0.295580\pi\)
\(954\) 7.85641 0.254361
\(955\) 9.26795 0.299904
\(956\) 10.1244 0.327445
\(957\) 0 0
\(958\) 28.2679 0.913296
\(959\) 9.46410 0.305612
\(960\) −2.73205 −0.0881766
\(961\) −1.14359 −0.0368901
\(962\) −1.60770 −0.0518342
\(963\) 26.3923 0.850480
\(964\) 14.7321 0.474487
\(965\) 53.1769 1.71183
\(966\) −3.73205 −0.120077
\(967\) −36.5167 −1.17430 −0.587148 0.809479i \(-0.699749\pi\)
−0.587148 + 0.809479i \(0.699749\pi\)
\(968\) 0 0
\(969\) 0.535898 0.0172155
\(970\) −17.8564 −0.573335
\(971\) 13.5359 0.434388 0.217194 0.976128i \(-0.430310\pi\)
0.217194 + 0.976128i \(0.430310\pi\)
\(972\) −16.0000 −0.513200
\(973\) −50.2487 −1.61090
\(974\) 35.1769 1.12714
\(975\) −8.53590 −0.273368
\(976\) 4.73205 0.151469
\(977\) −27.4641 −0.878654 −0.439327 0.898327i \(-0.644783\pi\)
−0.439327 + 0.898327i \(0.644783\pi\)
\(978\) −17.3205 −0.553849
\(979\) 0 0
\(980\) 18.9282 0.604639
\(981\) −36.2487 −1.15733
\(982\) −24.3923 −0.778390
\(983\) −40.9808 −1.30708 −0.653542 0.756891i \(-0.726718\pi\)
−0.653542 + 0.756891i \(0.726718\pi\)
\(984\) −7.26795 −0.231694
\(985\) 51.7128 1.64771
\(986\) −0.928203 −0.0295600
\(987\) −20.6603 −0.657623
\(988\) 3.46410 0.110208
\(989\) 8.92820 0.283900
\(990\) 0 0
\(991\) 22.0526 0.700523 0.350261 0.936652i \(-0.386093\pi\)
0.350261 + 0.936652i \(0.386093\pi\)
\(992\) −5.46410 −0.173485
\(993\) 19.7846 0.627846
\(994\) −14.1962 −0.450275
\(995\) 8.39230 0.266054
\(996\) 14.1962 0.449822
\(997\) 27.6603 0.876009 0.438005 0.898973i \(-0.355685\pi\)
0.438005 + 0.898973i \(0.355685\pi\)
\(998\) −24.7321 −0.782879
\(999\) 2.32051 0.0734176
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 4598.2.a.w.1.2 2
11.10 odd 2 4598.2.a.bf.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4598.2.a.w.1.2 2 1.1 even 1 trivial
4598.2.a.bf.1.2 yes 2 11.10 odd 2