Properties

Label 5070.2.a.be.1.2
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -0.267949 q^{11} -1.00000 q^{12} +3.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} -5.73205 q^{19} -1.00000 q^{20} -3.00000 q^{21} -0.267949 q^{22} -3.46410 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +3.00000 q^{28} +1.46410 q^{29} +1.00000 q^{30} -4.92820 q^{31} +1.00000 q^{32} +0.267949 q^{33} -4.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} +5.92820 q^{37} -5.73205 q^{38} -1.00000 q^{40} -4.00000 q^{41} -3.00000 q^{42} -6.00000 q^{43} -0.267949 q^{44} -1.00000 q^{45} -3.46410 q^{46} -6.46410 q^{47} -1.00000 q^{48} +2.00000 q^{49} +1.00000 q^{50} +4.00000 q^{51} -0.267949 q^{53} -1.00000 q^{54} +0.267949 q^{55} +3.00000 q^{56} +5.73205 q^{57} +1.46410 q^{58} -11.4641 q^{59} +1.00000 q^{60} -0.535898 q^{61} -4.92820 q^{62} +3.00000 q^{63} +1.00000 q^{64} +0.267949 q^{66} +1.46410 q^{67} -4.00000 q^{68} +3.46410 q^{69} -3.00000 q^{70} +12.9282 q^{71} +1.00000 q^{72} -6.92820 q^{73} +5.92820 q^{74} -1.00000 q^{75} -5.73205 q^{76} -0.803848 q^{77} +3.07180 q^{79} -1.00000 q^{80} +1.00000 q^{81} -4.00000 q^{82} -9.46410 q^{83} -3.00000 q^{84} +4.00000 q^{85} -6.00000 q^{86} -1.46410 q^{87} -0.267949 q^{88} +14.1244 q^{89} -1.00000 q^{90} -3.46410 q^{92} +4.92820 q^{93} -6.46410 q^{94} +5.73205 q^{95} -1.00000 q^{96} -8.39230 q^{97} +2.00000 q^{98} -0.267949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} + 6q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} + 6q^{7} + 2q^{8} + 2q^{9} - 2q^{10} - 4q^{11} - 2q^{12} + 6q^{14} + 2q^{15} + 2q^{16} - 8q^{17} + 2q^{18} - 8q^{19} - 2q^{20} - 6q^{21} - 4q^{22} - 2q^{24} + 2q^{25} - 2q^{27} + 6q^{28} - 4q^{29} + 2q^{30} + 4q^{31} + 2q^{32} + 4q^{33} - 8q^{34} - 6q^{35} + 2q^{36} - 2q^{37} - 8q^{38} - 2q^{40} - 8q^{41} - 6q^{42} - 12q^{43} - 4q^{44} - 2q^{45} - 6q^{47} - 2q^{48} + 4q^{49} + 2q^{50} + 8q^{51} - 4q^{53} - 2q^{54} + 4q^{55} + 6q^{56} + 8q^{57} - 4q^{58} - 16q^{59} + 2q^{60} - 8q^{61} + 4q^{62} + 6q^{63} + 2q^{64} + 4q^{66} - 4q^{67} - 8q^{68} - 6q^{70} + 12q^{71} + 2q^{72} - 2q^{74} - 2q^{75} - 8q^{76} - 12q^{77} + 20q^{79} - 2q^{80} + 2q^{81} - 8q^{82} - 12q^{83} - 6q^{84} + 8q^{85} - 12q^{86} + 4q^{87} - 4q^{88} + 4q^{89} - 2q^{90} - 4q^{93} - 6q^{94} + 8q^{95} - 2q^{96} + 4q^{97} + 4q^{98} - 4q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −0.267949 −0.0807897 −0.0403949 0.999184i \(-0.512862\pi\)
−0.0403949 + 0.999184i \(0.512862\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 3.00000 0.801784
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 1.00000 0.235702
\(19\) −5.73205 −1.31502 −0.657511 0.753445i \(-0.728391\pi\)
−0.657511 + 0.753445i \(0.728391\pi\)
\(20\) −1.00000 −0.223607
\(21\) −3.00000 −0.654654
\(22\) −0.267949 −0.0571270
\(23\) −3.46410 −0.722315 −0.361158 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 3.00000 0.566947
\(29\) 1.46410 0.271877 0.135938 0.990717i \(-0.456595\pi\)
0.135938 + 0.990717i \(0.456595\pi\)
\(30\) 1.00000 0.182574
\(31\) −4.92820 −0.885131 −0.442566 0.896736i \(-0.645932\pi\)
−0.442566 + 0.896736i \(0.645932\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.267949 0.0466440
\(34\) −4.00000 −0.685994
\(35\) −3.00000 −0.507093
\(36\) 1.00000 0.166667
\(37\) 5.92820 0.974591 0.487295 0.873237i \(-0.337984\pi\)
0.487295 + 0.873237i \(0.337984\pi\)
\(38\) −5.73205 −0.929861
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) −3.00000 −0.462910
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −0.267949 −0.0403949
\(45\) −1.00000 −0.149071
\(46\) −3.46410 −0.510754
\(47\) −6.46410 −0.942886 −0.471443 0.881897i \(-0.656267\pi\)
−0.471443 + 0.881897i \(0.656267\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.00000 0.285714
\(50\) 1.00000 0.141421
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) −0.267949 −0.0368057 −0.0184028 0.999831i \(-0.505858\pi\)
−0.0184028 + 0.999831i \(0.505858\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0.267949 0.0361303
\(56\) 3.00000 0.400892
\(57\) 5.73205 0.759229
\(58\) 1.46410 0.192246
\(59\) −11.4641 −1.49250 −0.746249 0.665666i \(-0.768147\pi\)
−0.746249 + 0.665666i \(0.768147\pi\)
\(60\) 1.00000 0.129099
\(61\) −0.535898 −0.0686148 −0.0343074 0.999411i \(-0.510923\pi\)
−0.0343074 + 0.999411i \(0.510923\pi\)
\(62\) −4.92820 −0.625882
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0.267949 0.0329823
\(67\) 1.46410 0.178868 0.0894342 0.995993i \(-0.471494\pi\)
0.0894342 + 0.995993i \(0.471494\pi\)
\(68\) −4.00000 −0.485071
\(69\) 3.46410 0.417029
\(70\) −3.00000 −0.358569
\(71\) 12.9282 1.53430 0.767148 0.641470i \(-0.221675\pi\)
0.767148 + 0.641470i \(0.221675\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.92820 −0.810885 −0.405442 0.914121i \(-0.632883\pi\)
−0.405442 + 0.914121i \(0.632883\pi\)
\(74\) 5.92820 0.689140
\(75\) −1.00000 −0.115470
\(76\) −5.73205 −0.657511
\(77\) −0.803848 −0.0916069
\(78\) 0 0
\(79\) 3.07180 0.345604 0.172802 0.984957i \(-0.444718\pi\)
0.172802 + 0.984957i \(0.444718\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −4.00000 −0.441726
\(83\) −9.46410 −1.03882 −0.519410 0.854525i \(-0.673848\pi\)
−0.519410 + 0.854525i \(0.673848\pi\)
\(84\) −3.00000 −0.327327
\(85\) 4.00000 0.433861
\(86\) −6.00000 −0.646997
\(87\) −1.46410 −0.156968
\(88\) −0.267949 −0.0285635
\(89\) 14.1244 1.49718 0.748589 0.663034i \(-0.230731\pi\)
0.748589 + 0.663034i \(0.230731\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −3.46410 −0.361158
\(93\) 4.92820 0.511031
\(94\) −6.46410 −0.666721
\(95\) 5.73205 0.588096
\(96\) −1.00000 −0.102062
\(97\) −8.39230 −0.852109 −0.426055 0.904697i \(-0.640097\pi\)
−0.426055 + 0.904697i \(0.640097\pi\)
\(98\) 2.00000 0.202031
\(99\) −0.267949 −0.0269299
\(100\) 1.00000 0.100000
\(101\) −12.3923 −1.23308 −0.616540 0.787323i \(-0.711466\pi\)
−0.616540 + 0.787323i \(0.711466\pi\)
\(102\) 4.00000 0.396059
\(103\) 11.5885 1.14184 0.570922 0.821004i \(-0.306586\pi\)
0.570922 + 0.821004i \(0.306586\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) −0.267949 −0.0260255
\(107\) −12.9282 −1.24982 −0.624908 0.780698i \(-0.714864\pi\)
−0.624908 + 0.780698i \(0.714864\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 10.3923 0.995402 0.497701 0.867349i \(-0.334178\pi\)
0.497701 + 0.867349i \(0.334178\pi\)
\(110\) 0.267949 0.0255480
\(111\) −5.92820 −0.562680
\(112\) 3.00000 0.283473
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 5.73205 0.536856
\(115\) 3.46410 0.323029
\(116\) 1.46410 0.135938
\(117\) 0 0
\(118\) −11.4641 −1.05536
\(119\) −12.0000 −1.10004
\(120\) 1.00000 0.0912871
\(121\) −10.9282 −0.993473
\(122\) −0.535898 −0.0485180
\(123\) 4.00000 0.360668
\(124\) −4.92820 −0.442566
\(125\) −1.00000 −0.0894427
\(126\) 3.00000 0.267261
\(127\) 12.6603 1.12342 0.561708 0.827336i \(-0.310144\pi\)
0.561708 + 0.827336i \(0.310144\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) 14.3205 1.25119 0.625594 0.780149i \(-0.284857\pi\)
0.625594 + 0.780149i \(0.284857\pi\)
\(132\) 0.267949 0.0233220
\(133\) −17.1962 −1.49110
\(134\) 1.46410 0.126479
\(135\) 1.00000 0.0860663
\(136\) −4.00000 −0.342997
\(137\) 13.4641 1.15032 0.575158 0.818042i \(-0.304941\pi\)
0.575158 + 0.818042i \(0.304941\pi\)
\(138\) 3.46410 0.294884
\(139\) −19.7846 −1.67811 −0.839054 0.544048i \(-0.816891\pi\)
−0.839054 + 0.544048i \(0.816891\pi\)
\(140\) −3.00000 −0.253546
\(141\) 6.46410 0.544376
\(142\) 12.9282 1.08491
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −1.46410 −0.121587
\(146\) −6.92820 −0.573382
\(147\) −2.00000 −0.164957
\(148\) 5.92820 0.487295
\(149\) −18.7846 −1.53890 −0.769448 0.638710i \(-0.779468\pi\)
−0.769448 + 0.638710i \(0.779468\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 22.7846 1.85419 0.927093 0.374832i \(-0.122300\pi\)
0.927093 + 0.374832i \(0.122300\pi\)
\(152\) −5.73205 −0.464931
\(153\) −4.00000 −0.323381
\(154\) −0.803848 −0.0647759
\(155\) 4.92820 0.395843
\(156\) 0 0
\(157\) 21.1962 1.69164 0.845819 0.533471i \(-0.179113\pi\)
0.845819 + 0.533471i \(0.179113\pi\)
\(158\) 3.07180 0.244379
\(159\) 0.267949 0.0212498
\(160\) −1.00000 −0.0790569
\(161\) −10.3923 −0.819028
\(162\) 1.00000 0.0785674
\(163\) −2.92820 −0.229355 −0.114677 0.993403i \(-0.536583\pi\)
−0.114677 + 0.993403i \(0.536583\pi\)
\(164\) −4.00000 −0.312348
\(165\) −0.267949 −0.0208598
\(166\) −9.46410 −0.734557
\(167\) 0.464102 0.0359133 0.0179566 0.999839i \(-0.494284\pi\)
0.0179566 + 0.999839i \(0.494284\pi\)
\(168\) −3.00000 −0.231455
\(169\) 0 0
\(170\) 4.00000 0.306786
\(171\) −5.73205 −0.438341
\(172\) −6.00000 −0.457496
\(173\) 2.12436 0.161512 0.0807559 0.996734i \(-0.474267\pi\)
0.0807559 + 0.996734i \(0.474267\pi\)
\(174\) −1.46410 −0.110993
\(175\) 3.00000 0.226779
\(176\) −0.267949 −0.0201974
\(177\) 11.4641 0.861695
\(178\) 14.1244 1.05867
\(179\) −9.07180 −0.678058 −0.339029 0.940776i \(-0.610098\pi\)
−0.339029 + 0.940776i \(0.610098\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −16.9282 −1.25826 −0.629132 0.777299i \(-0.716589\pi\)
−0.629132 + 0.777299i \(0.716589\pi\)
\(182\) 0 0
\(183\) 0.535898 0.0396147
\(184\) −3.46410 −0.255377
\(185\) −5.92820 −0.435850
\(186\) 4.92820 0.361353
\(187\) 1.07180 0.0783775
\(188\) −6.46410 −0.471443
\(189\) −3.00000 −0.218218
\(190\) 5.73205 0.415847
\(191\) −17.3205 −1.25327 −0.626634 0.779314i \(-0.715568\pi\)
−0.626634 + 0.779314i \(0.715568\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 9.85641 0.709480 0.354740 0.934965i \(-0.384569\pi\)
0.354740 + 0.934965i \(0.384569\pi\)
\(194\) −8.39230 −0.602532
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −11.3923 −0.811668 −0.405834 0.913947i \(-0.633019\pi\)
−0.405834 + 0.913947i \(0.633019\pi\)
\(198\) −0.267949 −0.0190423
\(199\) 24.9282 1.76711 0.883557 0.468324i \(-0.155142\pi\)
0.883557 + 0.468324i \(0.155142\pi\)
\(200\) 1.00000 0.0707107
\(201\) −1.46410 −0.103270
\(202\) −12.3923 −0.871920
\(203\) 4.39230 0.308279
\(204\) 4.00000 0.280056
\(205\) 4.00000 0.279372
\(206\) 11.5885 0.807406
\(207\) −3.46410 −0.240772
\(208\) 0 0
\(209\) 1.53590 0.106240
\(210\) 3.00000 0.207020
\(211\) −8.07180 −0.555685 −0.277843 0.960627i \(-0.589619\pi\)
−0.277843 + 0.960627i \(0.589619\pi\)
\(212\) −0.267949 −0.0184028
\(213\) −12.9282 −0.885826
\(214\) −12.9282 −0.883754
\(215\) 6.00000 0.409197
\(216\) −1.00000 −0.0680414
\(217\) −14.7846 −1.00364
\(218\) 10.3923 0.703856
\(219\) 6.92820 0.468165
\(220\) 0.267949 0.0180651
\(221\) 0 0
\(222\) −5.92820 −0.397875
\(223\) −18.8564 −1.26272 −0.631359 0.775491i \(-0.717503\pi\)
−0.631359 + 0.775491i \(0.717503\pi\)
\(224\) 3.00000 0.200446
\(225\) 1.00000 0.0666667
\(226\) −12.0000 −0.798228
\(227\) −6.53590 −0.433803 −0.216901 0.976194i \(-0.569595\pi\)
−0.216901 + 0.976194i \(0.569595\pi\)
\(228\) 5.73205 0.379614
\(229\) 4.53590 0.299741 0.149870 0.988706i \(-0.452114\pi\)
0.149870 + 0.988706i \(0.452114\pi\)
\(230\) 3.46410 0.228416
\(231\) 0.803848 0.0528893
\(232\) 1.46410 0.0961230
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 6.46410 0.421671
\(236\) −11.4641 −0.746249
\(237\) −3.07180 −0.199535
\(238\) −12.0000 −0.777844
\(239\) 3.46410 0.224074 0.112037 0.993704i \(-0.464262\pi\)
0.112037 + 0.993704i \(0.464262\pi\)
\(240\) 1.00000 0.0645497
\(241\) −25.1962 −1.62303 −0.811513 0.584334i \(-0.801356\pi\)
−0.811513 + 0.584334i \(0.801356\pi\)
\(242\) −10.9282 −0.702492
\(243\) −1.00000 −0.0641500
\(244\) −0.535898 −0.0343074
\(245\) −2.00000 −0.127775
\(246\) 4.00000 0.255031
\(247\) 0 0
\(248\) −4.92820 −0.312941
\(249\) 9.46410 0.599763
\(250\) −1.00000 −0.0632456
\(251\) −19.5359 −1.23309 −0.616547 0.787318i \(-0.711469\pi\)
−0.616547 + 0.787318i \(0.711469\pi\)
\(252\) 3.00000 0.188982
\(253\) 0.928203 0.0583556
\(254\) 12.6603 0.794375
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) −14.5359 −0.906724 −0.453362 0.891326i \(-0.649776\pi\)
−0.453362 + 0.891326i \(0.649776\pi\)
\(258\) 6.00000 0.373544
\(259\) 17.7846 1.10508
\(260\) 0 0
\(261\) 1.46410 0.0906256
\(262\) 14.3205 0.884724
\(263\) −24.5167 −1.51176 −0.755881 0.654709i \(-0.772791\pi\)
−0.755881 + 0.654709i \(0.772791\pi\)
\(264\) 0.267949 0.0164911
\(265\) 0.267949 0.0164600
\(266\) −17.1962 −1.05436
\(267\) −14.1244 −0.864397
\(268\) 1.46410 0.0894342
\(269\) −17.0718 −1.04089 −0.520443 0.853896i \(-0.674233\pi\)
−0.520443 + 0.853896i \(0.674233\pi\)
\(270\) 1.00000 0.0608581
\(271\) −24.3923 −1.48173 −0.740863 0.671656i \(-0.765583\pi\)
−0.740863 + 0.671656i \(0.765583\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 13.4641 0.813396
\(275\) −0.267949 −0.0161579
\(276\) 3.46410 0.208514
\(277\) −11.5885 −0.696283 −0.348141 0.937442i \(-0.613187\pi\)
−0.348141 + 0.937442i \(0.613187\pi\)
\(278\) −19.7846 −1.18660
\(279\) −4.92820 −0.295044
\(280\) −3.00000 −0.179284
\(281\) −6.92820 −0.413302 −0.206651 0.978415i \(-0.566256\pi\)
−0.206651 + 0.978415i \(0.566256\pi\)
\(282\) 6.46410 0.384932
\(283\) 6.39230 0.379983 0.189992 0.981786i \(-0.439154\pi\)
0.189992 + 0.981786i \(0.439154\pi\)
\(284\) 12.9282 0.767148
\(285\) −5.73205 −0.339537
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) −1.46410 −0.0859750
\(291\) 8.39230 0.491966
\(292\) −6.92820 −0.405442
\(293\) 29.2487 1.70873 0.854364 0.519675i \(-0.173947\pi\)
0.854364 + 0.519675i \(0.173947\pi\)
\(294\) −2.00000 −0.116642
\(295\) 11.4641 0.667466
\(296\) 5.92820 0.344570
\(297\) 0.267949 0.0155480
\(298\) −18.7846 −1.08816
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) −18.0000 −1.03750
\(302\) 22.7846 1.31111
\(303\) 12.3923 0.711919
\(304\) −5.73205 −0.328756
\(305\) 0.535898 0.0306855
\(306\) −4.00000 −0.228665
\(307\) 28.2487 1.61224 0.806120 0.591753i \(-0.201564\pi\)
0.806120 + 0.591753i \(0.201564\pi\)
\(308\) −0.803848 −0.0458035
\(309\) −11.5885 −0.659244
\(310\) 4.92820 0.279903
\(311\) −18.9282 −1.07332 −0.536660 0.843799i \(-0.680314\pi\)
−0.536660 + 0.843799i \(0.680314\pi\)
\(312\) 0 0
\(313\) −33.3205 −1.88339 −0.941693 0.336473i \(-0.890766\pi\)
−0.941693 + 0.336473i \(0.890766\pi\)
\(314\) 21.1962 1.19617
\(315\) −3.00000 −0.169031
\(316\) 3.07180 0.172802
\(317\) 30.4641 1.71103 0.855517 0.517774i \(-0.173239\pi\)
0.855517 + 0.517774i \(0.173239\pi\)
\(318\) 0.267949 0.0150258
\(319\) −0.392305 −0.0219649
\(320\) −1.00000 −0.0559017
\(321\) 12.9282 0.721582
\(322\) −10.3923 −0.579141
\(323\) 22.9282 1.27576
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −2.92820 −0.162178
\(327\) −10.3923 −0.574696
\(328\) −4.00000 −0.220863
\(329\) −19.3923 −1.06913
\(330\) −0.267949 −0.0147501
\(331\) −14.3923 −0.791073 −0.395536 0.918450i \(-0.629441\pi\)
−0.395536 + 0.918450i \(0.629441\pi\)
\(332\) −9.46410 −0.519410
\(333\) 5.92820 0.324864
\(334\) 0.464102 0.0253945
\(335\) −1.46410 −0.0799924
\(336\) −3.00000 −0.163663
\(337\) 26.3923 1.43768 0.718840 0.695175i \(-0.244673\pi\)
0.718840 + 0.695175i \(0.244673\pi\)
\(338\) 0 0
\(339\) 12.0000 0.651751
\(340\) 4.00000 0.216930
\(341\) 1.32051 0.0715095
\(342\) −5.73205 −0.309954
\(343\) −15.0000 −0.809924
\(344\) −6.00000 −0.323498
\(345\) −3.46410 −0.186501
\(346\) 2.12436 0.114206
\(347\) 4.39230 0.235791 0.117896 0.993026i \(-0.462385\pi\)
0.117896 + 0.993026i \(0.462385\pi\)
\(348\) −1.46410 −0.0784841
\(349\) 10.5359 0.563974 0.281987 0.959418i \(-0.409007\pi\)
0.281987 + 0.959418i \(0.409007\pi\)
\(350\) 3.00000 0.160357
\(351\) 0 0
\(352\) −0.267949 −0.0142817
\(353\) 7.60770 0.404917 0.202458 0.979291i \(-0.435107\pi\)
0.202458 + 0.979291i \(0.435107\pi\)
\(354\) 11.4641 0.609310
\(355\) −12.9282 −0.686158
\(356\) 14.1244 0.748589
\(357\) 12.0000 0.635107
\(358\) −9.07180 −0.479459
\(359\) −0.928203 −0.0489887 −0.0244943 0.999700i \(-0.507798\pi\)
−0.0244943 + 0.999700i \(0.507798\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 13.8564 0.729285
\(362\) −16.9282 −0.889727
\(363\) 10.9282 0.573582
\(364\) 0 0
\(365\) 6.92820 0.362639
\(366\) 0.535898 0.0280119
\(367\) 21.3205 1.11292 0.556461 0.830874i \(-0.312159\pi\)
0.556461 + 0.830874i \(0.312159\pi\)
\(368\) −3.46410 −0.180579
\(369\) −4.00000 −0.208232
\(370\) −5.92820 −0.308193
\(371\) −0.803848 −0.0417337
\(372\) 4.92820 0.255515
\(373\) 9.07180 0.469720 0.234860 0.972029i \(-0.424537\pi\)
0.234860 + 0.972029i \(0.424537\pi\)
\(374\) 1.07180 0.0554213
\(375\) 1.00000 0.0516398
\(376\) −6.46410 −0.333361
\(377\) 0 0
\(378\) −3.00000 −0.154303
\(379\) −9.73205 −0.499902 −0.249951 0.968259i \(-0.580414\pi\)
−0.249951 + 0.968259i \(0.580414\pi\)
\(380\) 5.73205 0.294048
\(381\) −12.6603 −0.648604
\(382\) −17.3205 −0.886194
\(383\) −4.78461 −0.244482 −0.122241 0.992500i \(-0.539008\pi\)
−0.122241 + 0.992500i \(0.539008\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0.803848 0.0409679
\(386\) 9.85641 0.501678
\(387\) −6.00000 −0.304997
\(388\) −8.39230 −0.426055
\(389\) −17.8564 −0.905356 −0.452678 0.891674i \(-0.649531\pi\)
−0.452678 + 0.891674i \(0.649531\pi\)
\(390\) 0 0
\(391\) 13.8564 0.700749
\(392\) 2.00000 0.101015
\(393\) −14.3205 −0.722374
\(394\) −11.3923 −0.573936
\(395\) −3.07180 −0.154559
\(396\) −0.267949 −0.0134650
\(397\) −5.92820 −0.297528 −0.148764 0.988873i \(-0.547529\pi\)
−0.148764 + 0.988873i \(0.547529\pi\)
\(398\) 24.9282 1.24954
\(399\) 17.1962 0.860884
\(400\) 1.00000 0.0500000
\(401\) 22.1244 1.10484 0.552419 0.833567i \(-0.313705\pi\)
0.552419 + 0.833567i \(0.313705\pi\)
\(402\) −1.46410 −0.0730228
\(403\) 0 0
\(404\) −12.3923 −0.616540
\(405\) −1.00000 −0.0496904
\(406\) 4.39230 0.217986
\(407\) −1.58846 −0.0787369
\(408\) 4.00000 0.198030
\(409\) 39.0526 1.93102 0.965512 0.260357i \(-0.0838403\pi\)
0.965512 + 0.260357i \(0.0838403\pi\)
\(410\) 4.00000 0.197546
\(411\) −13.4641 −0.664135
\(412\) 11.5885 0.570922
\(413\) −34.3923 −1.69233
\(414\) −3.46410 −0.170251
\(415\) 9.46410 0.464574
\(416\) 0 0
\(417\) 19.7846 0.968857
\(418\) 1.53590 0.0751232
\(419\) 9.85641 0.481517 0.240758 0.970585i \(-0.422604\pi\)
0.240758 + 0.970585i \(0.422604\pi\)
\(420\) 3.00000 0.146385
\(421\) 21.8564 1.06522 0.532608 0.846362i \(-0.321212\pi\)
0.532608 + 0.846362i \(0.321212\pi\)
\(422\) −8.07180 −0.392929
\(423\) −6.46410 −0.314295
\(424\) −0.267949 −0.0130128
\(425\) −4.00000 −0.194029
\(426\) −12.9282 −0.626373
\(427\) −1.60770 −0.0778018
\(428\) −12.9282 −0.624908
\(429\) 0 0
\(430\) 6.00000 0.289346
\(431\) −13.8564 −0.667440 −0.333720 0.942672i \(-0.608304\pi\)
−0.333720 + 0.942672i \(0.608304\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 8.78461 0.422161 0.211081 0.977469i \(-0.432302\pi\)
0.211081 + 0.977469i \(0.432302\pi\)
\(434\) −14.7846 −0.709684
\(435\) 1.46410 0.0701983
\(436\) 10.3923 0.497701
\(437\) 19.8564 0.949861
\(438\) 6.92820 0.331042
\(439\) 13.3205 0.635753 0.317877 0.948132i \(-0.397030\pi\)
0.317877 + 0.948132i \(0.397030\pi\)
\(440\) 0.267949 0.0127740
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) −19.8564 −0.943406 −0.471703 0.881757i \(-0.656361\pi\)
−0.471703 + 0.881757i \(0.656361\pi\)
\(444\) −5.92820 −0.281340
\(445\) −14.1244 −0.669559
\(446\) −18.8564 −0.892877
\(447\) 18.7846 0.888482
\(448\) 3.00000 0.141737
\(449\) −6.12436 −0.289026 −0.144513 0.989503i \(-0.546162\pi\)
−0.144513 + 0.989503i \(0.546162\pi\)
\(450\) 1.00000 0.0471405
\(451\) 1.07180 0.0504689
\(452\) −12.0000 −0.564433
\(453\) −22.7846 −1.07051
\(454\) −6.53590 −0.306745
\(455\) 0 0
\(456\) 5.73205 0.268428
\(457\) −7.46410 −0.349156 −0.174578 0.984643i \(-0.555856\pi\)
−0.174578 + 0.984643i \(0.555856\pi\)
\(458\) 4.53590 0.211949
\(459\) 4.00000 0.186704
\(460\) 3.46410 0.161515
\(461\) 11.6077 0.540624 0.270312 0.962773i \(-0.412873\pi\)
0.270312 + 0.962773i \(0.412873\pi\)
\(462\) 0.803848 0.0373984
\(463\) 40.7846 1.89542 0.947711 0.319131i \(-0.103391\pi\)
0.947711 + 0.319131i \(0.103391\pi\)
\(464\) 1.46410 0.0679692
\(465\) −4.92820 −0.228540
\(466\) −18.0000 −0.833834
\(467\) −24.3923 −1.12874 −0.564371 0.825522i \(-0.690881\pi\)
−0.564371 + 0.825522i \(0.690881\pi\)
\(468\) 0 0
\(469\) 4.39230 0.202818
\(470\) 6.46410 0.298167
\(471\) −21.1962 −0.976667
\(472\) −11.4641 −0.527678
\(473\) 1.60770 0.0739219
\(474\) −3.07180 −0.141092
\(475\) −5.73205 −0.263005
\(476\) −12.0000 −0.550019
\(477\) −0.267949 −0.0122686
\(478\) 3.46410 0.158444
\(479\) 22.2487 1.01657 0.508285 0.861189i \(-0.330280\pi\)
0.508285 + 0.861189i \(0.330280\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) −25.1962 −1.14765
\(483\) 10.3923 0.472866
\(484\) −10.9282 −0.496737
\(485\) 8.39230 0.381075
\(486\) −1.00000 −0.0453609
\(487\) −21.0000 −0.951601 −0.475800 0.879553i \(-0.657842\pi\)
−0.475800 + 0.879553i \(0.657842\pi\)
\(488\) −0.535898 −0.0242590
\(489\) 2.92820 0.132418
\(490\) −2.00000 −0.0903508
\(491\) −5.39230 −0.243351 −0.121676 0.992570i \(-0.538827\pi\)
−0.121676 + 0.992570i \(0.538827\pi\)
\(492\) 4.00000 0.180334
\(493\) −5.85641 −0.263759
\(494\) 0 0
\(495\) 0.267949 0.0120434
\(496\) −4.92820 −0.221283
\(497\) 38.7846 1.73973
\(498\) 9.46410 0.424097
\(499\) 33.3205 1.49163 0.745815 0.666153i \(-0.232060\pi\)
0.745815 + 0.666153i \(0.232060\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −0.464102 −0.0207345
\(502\) −19.5359 −0.871930
\(503\) 3.73205 0.166404 0.0832020 0.996533i \(-0.473485\pi\)
0.0832020 + 0.996533i \(0.473485\pi\)
\(504\) 3.00000 0.133631
\(505\) 12.3923 0.551450
\(506\) 0.928203 0.0412637
\(507\) 0 0
\(508\) 12.6603 0.561708
\(509\) 29.3205 1.29961 0.649804 0.760102i \(-0.274851\pi\)
0.649804 + 0.760102i \(0.274851\pi\)
\(510\) −4.00000 −0.177123
\(511\) −20.7846 −0.919457
\(512\) 1.00000 0.0441942
\(513\) 5.73205 0.253076
\(514\) −14.5359 −0.641151
\(515\) −11.5885 −0.510648
\(516\) 6.00000 0.264135
\(517\) 1.73205 0.0761755
\(518\) 17.7846 0.781411
\(519\) −2.12436 −0.0932489
\(520\) 0 0
\(521\) −6.60770 −0.289488 −0.144744 0.989469i \(-0.546236\pi\)
−0.144744 + 0.989469i \(0.546236\pi\)
\(522\) 1.46410 0.0640820
\(523\) −41.7128 −1.82397 −0.911987 0.410219i \(-0.865452\pi\)
−0.911987 + 0.410219i \(0.865452\pi\)
\(524\) 14.3205 0.625594
\(525\) −3.00000 −0.130931
\(526\) −24.5167 −1.06898
\(527\) 19.7128 0.858704
\(528\) 0.267949 0.0116610
\(529\) −11.0000 −0.478261
\(530\) 0.267949 0.0116390
\(531\) −11.4641 −0.497500
\(532\) −17.1962 −0.745548
\(533\) 0 0
\(534\) −14.1244 −0.611221
\(535\) 12.9282 0.558935
\(536\) 1.46410 0.0632396
\(537\) 9.07180 0.391477
\(538\) −17.0718 −0.736017
\(539\) −0.535898 −0.0230828
\(540\) 1.00000 0.0430331
\(541\) −14.7846 −0.635640 −0.317820 0.948151i \(-0.602951\pi\)
−0.317820 + 0.948151i \(0.602951\pi\)
\(542\) −24.3923 −1.04774
\(543\) 16.9282 0.726459
\(544\) −4.00000 −0.171499
\(545\) −10.3923 −0.445157
\(546\) 0 0
\(547\) −5.32051 −0.227488 −0.113744 0.993510i \(-0.536284\pi\)
−0.113744 + 0.993510i \(0.536284\pi\)
\(548\) 13.4641 0.575158
\(549\) −0.535898 −0.0228716
\(550\) −0.267949 −0.0114254
\(551\) −8.39230 −0.357524
\(552\) 3.46410 0.147442
\(553\) 9.21539 0.391878
\(554\) −11.5885 −0.492346
\(555\) 5.92820 0.251638
\(556\) −19.7846 −0.839054
\(557\) 22.6077 0.957919 0.478959 0.877837i \(-0.341014\pi\)
0.478959 + 0.877837i \(0.341014\pi\)
\(558\) −4.92820 −0.208627
\(559\) 0 0
\(560\) −3.00000 −0.126773
\(561\) −1.07180 −0.0452513
\(562\) −6.92820 −0.292249
\(563\) 15.3205 0.645682 0.322841 0.946453i \(-0.395362\pi\)
0.322841 + 0.946453i \(0.395362\pi\)
\(564\) 6.46410 0.272188
\(565\) 12.0000 0.504844
\(566\) 6.39230 0.268689
\(567\) 3.00000 0.125988
\(568\) 12.9282 0.542455
\(569\) 16.3205 0.684191 0.342096 0.939665i \(-0.388863\pi\)
0.342096 + 0.939665i \(0.388863\pi\)
\(570\) −5.73205 −0.240089
\(571\) −10.8564 −0.454326 −0.227163 0.973857i \(-0.572945\pi\)
−0.227163 + 0.973857i \(0.572945\pi\)
\(572\) 0 0
\(573\) 17.3205 0.723575
\(574\) −12.0000 −0.500870
\(575\) −3.46410 −0.144463
\(576\) 1.00000 0.0416667
\(577\) −9.32051 −0.388018 −0.194009 0.981000i \(-0.562149\pi\)
−0.194009 + 0.981000i \(0.562149\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −9.85641 −0.409618
\(580\) −1.46410 −0.0607935
\(581\) −28.3923 −1.17791
\(582\) 8.39230 0.347872
\(583\) 0.0717968 0.00297352
\(584\) −6.92820 −0.286691
\(585\) 0 0
\(586\) 29.2487 1.20825
\(587\) −8.92820 −0.368506 −0.184253 0.982879i \(-0.558987\pi\)
−0.184253 + 0.982879i \(0.558987\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 28.2487 1.16397
\(590\) 11.4641 0.471970
\(591\) 11.3923 0.468617
\(592\) 5.92820 0.243648
\(593\) 44.7846 1.83908 0.919542 0.392992i \(-0.128560\pi\)
0.919542 + 0.392992i \(0.128560\pi\)
\(594\) 0.267949 0.0109941
\(595\) 12.0000 0.491952
\(596\) −18.7846 −0.769448
\(597\) −24.9282 −1.02024
\(598\) 0 0
\(599\) 28.9282 1.18197 0.590987 0.806681i \(-0.298738\pi\)
0.590987 + 0.806681i \(0.298738\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 30.7128 1.25280 0.626401 0.779501i \(-0.284527\pi\)
0.626401 + 0.779501i \(0.284527\pi\)
\(602\) −18.0000 −0.733625
\(603\) 1.46410 0.0596228
\(604\) 22.7846 0.927093
\(605\) 10.9282 0.444295
\(606\) 12.3923 0.503403
\(607\) −21.4449 −0.870420 −0.435210 0.900329i \(-0.643326\pi\)
−0.435210 + 0.900329i \(0.643326\pi\)
\(608\) −5.73205 −0.232465
\(609\) −4.39230 −0.177985
\(610\) 0.535898 0.0216979
\(611\) 0 0
\(612\) −4.00000 −0.161690
\(613\) 45.0000 1.81753 0.908766 0.417305i \(-0.137025\pi\)
0.908766 + 0.417305i \(0.137025\pi\)
\(614\) 28.2487 1.14003
\(615\) −4.00000 −0.161296
\(616\) −0.803848 −0.0323879
\(617\) 35.4641 1.42773 0.713865 0.700283i \(-0.246943\pi\)
0.713865 + 0.700283i \(0.246943\pi\)
\(618\) −11.5885 −0.466156
\(619\) −2.51666 −0.101153 −0.0505766 0.998720i \(-0.516106\pi\)
−0.0505766 + 0.998720i \(0.516106\pi\)
\(620\) 4.92820 0.197921
\(621\) 3.46410 0.139010
\(622\) −18.9282 −0.758952
\(623\) 42.3731 1.69764
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −33.3205 −1.33176
\(627\) −1.53590 −0.0613379
\(628\) 21.1962 0.845819
\(629\) −23.7128 −0.945492
\(630\) −3.00000 −0.119523
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 3.07180 0.122190
\(633\) 8.07180 0.320825
\(634\) 30.4641 1.20988
\(635\) −12.6603 −0.502407
\(636\) 0.267949 0.0106249
\(637\) 0 0
\(638\) −0.392305 −0.0155315
\(639\) 12.9282 0.511432
\(640\) −1.00000 −0.0395285
\(641\) −14.4641 −0.571298 −0.285649 0.958334i \(-0.592209\pi\)
−0.285649 + 0.958334i \(0.592209\pi\)
\(642\) 12.9282 0.510235
\(643\) −12.7846 −0.504176 −0.252088 0.967704i \(-0.581117\pi\)
−0.252088 + 0.967704i \(0.581117\pi\)
\(644\) −10.3923 −0.409514
\(645\) −6.00000 −0.236250
\(646\) 22.9282 0.902098
\(647\) 35.7321 1.40477 0.702386 0.711796i \(-0.252118\pi\)
0.702386 + 0.711796i \(0.252118\pi\)
\(648\) 1.00000 0.0392837
\(649\) 3.07180 0.120579
\(650\) 0 0
\(651\) 14.7846 0.579455
\(652\) −2.92820 −0.114677
\(653\) −1.58846 −0.0621611 −0.0310806 0.999517i \(-0.509895\pi\)
−0.0310806 + 0.999517i \(0.509895\pi\)
\(654\) −10.3923 −0.406371
\(655\) −14.3205 −0.559549
\(656\) −4.00000 −0.156174
\(657\) −6.92820 −0.270295
\(658\) −19.3923 −0.755991
\(659\) −39.7128 −1.54699 −0.773496 0.633801i \(-0.781494\pi\)
−0.773496 + 0.633801i \(0.781494\pi\)
\(660\) −0.267949 −0.0104299
\(661\) 21.6077 0.840442 0.420221 0.907422i \(-0.361953\pi\)
0.420221 + 0.907422i \(0.361953\pi\)
\(662\) −14.3923 −0.559373
\(663\) 0 0
\(664\) −9.46410 −0.367278
\(665\) 17.1962 0.666838
\(666\) 5.92820 0.229713
\(667\) −5.07180 −0.196381
\(668\) 0.464102 0.0179566
\(669\) 18.8564 0.729031
\(670\) −1.46410 −0.0565632
\(671\) 0.143594 0.00554337
\(672\) −3.00000 −0.115728
\(673\) 16.7846 0.646999 0.323500 0.946228i \(-0.395141\pi\)
0.323500 + 0.946228i \(0.395141\pi\)
\(674\) 26.3923 1.01659
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 10.9282 0.420005 0.210002 0.977701i \(-0.432653\pi\)
0.210002 + 0.977701i \(0.432653\pi\)
\(678\) 12.0000 0.460857
\(679\) −25.1769 −0.966201
\(680\) 4.00000 0.153393
\(681\) 6.53590 0.250456
\(682\) 1.32051 0.0505649
\(683\) 41.3205 1.58109 0.790543 0.612407i \(-0.209799\pi\)
0.790543 + 0.612407i \(0.209799\pi\)
\(684\) −5.73205 −0.219170
\(685\) −13.4641 −0.514437
\(686\) −15.0000 −0.572703
\(687\) −4.53590 −0.173055
\(688\) −6.00000 −0.228748
\(689\) 0 0
\(690\) −3.46410 −0.131876
\(691\) 15.0526 0.572626 0.286313 0.958136i \(-0.407570\pi\)
0.286313 + 0.958136i \(0.407570\pi\)
\(692\) 2.12436 0.0807559
\(693\) −0.803848 −0.0305356
\(694\) 4.39230 0.166730
\(695\) 19.7846 0.750473
\(696\) −1.46410 −0.0554966
\(697\) 16.0000 0.606043
\(698\) 10.5359 0.398790
\(699\) 18.0000 0.680823
\(700\) 3.00000 0.113389
\(701\) 4.39230 0.165895 0.0829475 0.996554i \(-0.473567\pi\)
0.0829475 + 0.996554i \(0.473567\pi\)
\(702\) 0 0
\(703\) −33.9808 −1.28161
\(704\) −0.267949 −0.0100987
\(705\) −6.46410 −0.243452
\(706\) 7.60770 0.286319
\(707\) −37.1769 −1.39818
\(708\) 11.4641 0.430847
\(709\) −27.3205 −1.02604 −0.513022 0.858376i \(-0.671474\pi\)
−0.513022 + 0.858376i \(0.671474\pi\)
\(710\) −12.9282 −0.485187
\(711\) 3.07180 0.115201
\(712\) 14.1244 0.529333
\(713\) 17.0718 0.639344
\(714\) 12.0000 0.449089
\(715\) 0 0
\(716\) −9.07180 −0.339029
\(717\) −3.46410 −0.129369
\(718\) −0.928203 −0.0346402
\(719\) −16.0000 −0.596699 −0.298350 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 34.7654 1.29473
\(722\) 13.8564 0.515682
\(723\) 25.1962 0.937055
\(724\) −16.9282 −0.629132
\(725\) 1.46410 0.0543754
\(726\) 10.9282 0.405584
\(727\) 4.66025 0.172839 0.0864196 0.996259i \(-0.472457\pi\)
0.0864196 + 0.996259i \(0.472457\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 6.92820 0.256424
\(731\) 24.0000 0.887672
\(732\) 0.535898 0.0198074
\(733\) 20.8564 0.770349 0.385174 0.922844i \(-0.374141\pi\)
0.385174 + 0.922844i \(0.374141\pi\)
\(734\) 21.3205 0.786954
\(735\) 2.00000 0.0737711
\(736\) −3.46410 −0.127688
\(737\) −0.392305 −0.0144507
\(738\) −4.00000 −0.147242
\(739\) −35.8372 −1.31829 −0.659146 0.752015i \(-0.729082\pi\)
−0.659146 + 0.752015i \(0.729082\pi\)
\(740\) −5.92820 −0.217925
\(741\) 0 0
\(742\) −0.803848 −0.0295102
\(743\) −10.9282 −0.400917 −0.200458 0.979702i \(-0.564243\pi\)
−0.200458 + 0.979702i \(0.564243\pi\)
\(744\) 4.92820 0.180677
\(745\) 18.7846 0.688215
\(746\) 9.07180 0.332142
\(747\) −9.46410 −0.346273
\(748\) 1.07180 0.0391888
\(749\) −38.7846 −1.41716
\(750\) 1.00000 0.0365148
\(751\) 21.1769 0.772757 0.386378 0.922340i \(-0.373726\pi\)
0.386378 + 0.922340i \(0.373726\pi\)
\(752\) −6.46410 −0.235722
\(753\) 19.5359 0.711928
\(754\) 0 0
\(755\) −22.7846 −0.829217
\(756\) −3.00000 −0.109109
\(757\) 21.7321 0.789865 0.394932 0.918710i \(-0.370768\pi\)
0.394932 + 0.918710i \(0.370768\pi\)
\(758\) −9.73205 −0.353484
\(759\) −0.928203 −0.0336916
\(760\) 5.73205 0.207923
\(761\) 7.98076 0.289302 0.144651 0.989483i \(-0.453794\pi\)
0.144651 + 0.989483i \(0.453794\pi\)
\(762\) −12.6603 −0.458633
\(763\) 31.1769 1.12868
\(764\) −17.3205 −0.626634
\(765\) 4.00000 0.144620
\(766\) −4.78461 −0.172875
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 15.7128 0.566619 0.283309 0.959029i \(-0.408568\pi\)
0.283309 + 0.959029i \(0.408568\pi\)
\(770\) 0.803848 0.0289687
\(771\) 14.5359 0.523498
\(772\) 9.85641 0.354740
\(773\) 32.6077 1.17282 0.586409 0.810015i \(-0.300541\pi\)
0.586409 + 0.810015i \(0.300541\pi\)
\(774\) −6.00000 −0.215666
\(775\) −4.92820 −0.177026
\(776\) −8.39230 −0.301266
\(777\) −17.7846 −0.638019
\(778\) −17.8564 −0.640183
\(779\) 22.9282 0.821488
\(780\) 0 0
\(781\) −3.46410 −0.123955
\(782\) 13.8564 0.495504
\(783\) −1.46410 −0.0523227
\(784\) 2.00000 0.0714286
\(785\) −21.1962 −0.756523
\(786\) −14.3205 −0.510796
\(787\) 1.46410 0.0521896 0.0260948 0.999659i \(-0.491693\pi\)
0.0260948 + 0.999659i \(0.491693\pi\)
\(788\) −11.3923 −0.405834
\(789\) 24.5167 0.872816
\(790\) −3.07180 −0.109290
\(791\) −36.0000 −1.28001
\(792\) −0.267949 −0.00952116
\(793\) 0 0
\(794\) −5.92820 −0.210384
\(795\) −0.267949 −0.00950318
\(796\) 24.9282 0.883557
\(797\) 10.1436 0.359305 0.179652 0.983730i \(-0.442503\pi\)
0.179652 + 0.983730i \(0.442503\pi\)
\(798\) 17.1962 0.608737
\(799\) 25.8564 0.914734
\(800\) 1.00000 0.0353553
\(801\) 14.1244 0.499060
\(802\) 22.1244 0.781238
\(803\) 1.85641 0.0655112
\(804\) −1.46410 −0.0516349
\(805\) 10.3923 0.366281
\(806\) 0 0
\(807\) 17.0718 0.600956
\(808\) −12.3923 −0.435960
\(809\) −6.78461 −0.238534 −0.119267 0.992862i \(-0.538054\pi\)
−0.119267 + 0.992862i \(0.538054\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 23.5885 0.828303 0.414151 0.910208i \(-0.364078\pi\)
0.414151 + 0.910208i \(0.364078\pi\)
\(812\) 4.39230 0.154140
\(813\) 24.3923 0.855475
\(814\) −1.58846 −0.0556754
\(815\) 2.92820 0.102570
\(816\) 4.00000 0.140028
\(817\) 34.3923 1.20323
\(818\) 39.0526 1.36544
\(819\) 0 0
\(820\) 4.00000 0.139686
\(821\) −14.6795 −0.512318 −0.256159 0.966635i \(-0.582457\pi\)
−0.256159 + 0.966635i \(0.582457\pi\)
\(822\) −13.4641 −0.469614
\(823\) 8.41154 0.293208 0.146604 0.989195i \(-0.453166\pi\)
0.146604 + 0.989195i \(0.453166\pi\)
\(824\) 11.5885 0.403703
\(825\) 0.267949 0.00932879
\(826\) −34.3923 −1.19666
\(827\) −46.4974 −1.61687 −0.808437 0.588583i \(-0.799686\pi\)
−0.808437 + 0.588583i \(0.799686\pi\)
\(828\) −3.46410 −0.120386
\(829\) 4.67949 0.162525 0.0812627 0.996693i \(-0.474105\pi\)
0.0812627 + 0.996693i \(0.474105\pi\)
\(830\) 9.46410 0.328504
\(831\) 11.5885 0.401999
\(832\) 0 0
\(833\) −8.00000 −0.277184
\(834\) 19.7846 0.685085
\(835\) −0.464102 −0.0160609
\(836\) 1.53590 0.0531202
\(837\) 4.92820 0.170344
\(838\) 9.85641 0.340484
\(839\) 19.1769 0.662061 0.331030 0.943620i \(-0.392604\pi\)
0.331030 + 0.943620i \(0.392604\pi\)
\(840\) 3.00000 0.103510
\(841\) −26.8564 −0.926083
\(842\) 21.8564 0.753222
\(843\) 6.92820 0.238620
\(844\) −8.07180 −0.277843
\(845\) 0 0
\(846\) −6.46410 −0.222240
\(847\) −32.7846 −1.12649
\(848\) −0.267949 −0.00920141
\(849\) −6.39230 −0.219383
\(850\) −4.00000 −0.137199
\(851\) −20.5359 −0.703962
\(852\) −12.9282 −0.442913
\(853\) 24.6410 0.843692 0.421846 0.906667i \(-0.361382\pi\)
0.421846 + 0.906667i \(0.361382\pi\)
\(854\) −1.60770 −0.0550142
\(855\) 5.73205 0.196032
\(856\) −12.9282 −0.441877
\(857\) 28.9282 0.988169 0.494084 0.869414i \(-0.335503\pi\)
0.494084 + 0.869414i \(0.335503\pi\)
\(858\) 0 0
\(859\) −6.07180 −0.207167 −0.103584 0.994621i \(-0.533031\pi\)
−0.103584 + 0.994621i \(0.533031\pi\)
\(860\) 6.00000 0.204598
\(861\) 12.0000 0.408959
\(862\) −13.8564 −0.471951
\(863\) 2.92820 0.0996772 0.0498386 0.998757i \(-0.484129\pi\)
0.0498386 + 0.998757i \(0.484129\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −2.12436 −0.0722303
\(866\) 8.78461 0.298513
\(867\) 1.00000 0.0339618
\(868\) −14.7846 −0.501822
\(869\) −0.823085 −0.0279213
\(870\) 1.46410 0.0496377
\(871\) 0 0
\(872\) 10.3923 0.351928
\(873\) −8.39230 −0.284036
\(874\) 19.8564 0.671653
\(875\) −3.00000 −0.101419
\(876\) 6.92820 0.234082
\(877\) −21.7128 −0.733190 −0.366595 0.930381i \(-0.619476\pi\)
−0.366595 + 0.930381i \(0.619476\pi\)
\(878\) 13.3205 0.449545
\(879\) −29.2487 −0.986535
\(880\) 0.267949 0.00903257
\(881\) −39.1051 −1.31748 −0.658742 0.752369i \(-0.728911\pi\)
−0.658742 + 0.752369i \(0.728911\pi\)
\(882\) 2.00000 0.0673435
\(883\) 13.3205 0.448271 0.224135 0.974558i \(-0.428044\pi\)
0.224135 + 0.974558i \(0.428044\pi\)
\(884\) 0 0
\(885\) −11.4641 −0.385362
\(886\) −19.8564 −0.667089
\(887\) 55.7321 1.87130 0.935650 0.352930i \(-0.114815\pi\)
0.935650 + 0.352930i \(0.114815\pi\)
\(888\) −5.92820 −0.198937
\(889\) 37.9808 1.27383
\(890\) −14.1244 −0.473449
\(891\) −0.267949 −0.00897664
\(892\) −18.8564 −0.631359
\(893\) 37.0526 1.23992
\(894\) 18.7846 0.628251
\(895\) 9.07180 0.303237
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) −6.12436 −0.204372
\(899\) −7.21539 −0.240647
\(900\) 1.00000 0.0333333
\(901\) 1.07180 0.0357067
\(902\) 1.07180 0.0356869
\(903\) 18.0000 0.599002
\(904\) −12.0000 −0.399114
\(905\) 16.9282 0.562713
\(906\) −22.7846 −0.756968
\(907\) −20.9282 −0.694910 −0.347455 0.937697i \(-0.612954\pi\)
−0.347455 + 0.937697i \(0.612954\pi\)
\(908\) −6.53590 −0.216901
\(909\) −12.3923 −0.411027
\(910\) 0 0
\(911\) 12.7846 0.423573 0.211787 0.977316i \(-0.432072\pi\)
0.211787 + 0.977316i \(0.432072\pi\)
\(912\) 5.73205 0.189807
\(913\) 2.53590 0.0839260
\(914\) −7.46410 −0.246891
\(915\) −0.535898 −0.0177163
\(916\) 4.53590 0.149870
\(917\) 42.9615 1.41871
\(918\) 4.00000 0.132020
\(919\) −47.9615 −1.58210 −0.791052 0.611748i \(-0.790466\pi\)
−0.791052 + 0.611748i \(0.790466\pi\)
\(920\) 3.46410 0.114208
\(921\) −28.2487 −0.930827
\(922\) 11.6077 0.382279
\(923\) 0 0
\(924\) 0.803848 0.0264446
\(925\) 5.92820 0.194918
\(926\) 40.7846 1.34027
\(927\) 11.5885 0.380615
\(928\) 1.46410 0.0480615
\(929\) −57.8564 −1.89821 −0.949104 0.314964i \(-0.898007\pi\)
−0.949104 + 0.314964i \(0.898007\pi\)
\(930\) −4.92820 −0.161602
\(931\) −11.4641 −0.375721
\(932\) −18.0000 −0.589610
\(933\) 18.9282 0.619682
\(934\) −24.3923 −0.798141
\(935\) −1.07180 −0.0350515
\(936\) 0 0
\(937\) −21.7128 −0.709327 −0.354663 0.934994i \(-0.615405\pi\)
−0.354663 + 0.934994i \(0.615405\pi\)
\(938\) 4.39230 0.143414
\(939\) 33.3205 1.08737
\(940\) 6.46410 0.210836
\(941\) −60.4974 −1.97216 −0.986080 0.166273i \(-0.946827\pi\)
−0.986080 + 0.166273i \(0.946827\pi\)
\(942\) −21.1962 −0.690608
\(943\) 13.8564 0.451227
\(944\) −11.4641 −0.373125
\(945\) 3.00000 0.0975900
\(946\) 1.60770 0.0522707
\(947\) −34.3923 −1.11760 −0.558800 0.829303i \(-0.688738\pi\)
−0.558800 + 0.829303i \(0.688738\pi\)
\(948\) −3.07180 −0.0997673
\(949\) 0 0
\(950\) −5.73205 −0.185972
\(951\) −30.4641 −0.987866
\(952\) −12.0000 −0.388922
\(953\) 53.3205 1.72722 0.863610 0.504160i \(-0.168198\pi\)
0.863610 + 0.504160i \(0.168198\pi\)
\(954\) −0.267949 −0.00867518
\(955\) 17.3205 0.560478
\(956\) 3.46410 0.112037
\(957\) 0.392305 0.0126814
\(958\) 22.2487 0.718823
\(959\) 40.3923 1.30434
\(960\) 1.00000 0.0322749
\(961\) −6.71281 −0.216542
\(962\) 0 0
\(963\) −12.9282 −0.416606
\(964\) −25.1962 −0.811513
\(965\) −9.85641 −0.317289
\(966\) 10.3923 0.334367
\(967\) −1.14359 −0.0367755 −0.0183877 0.999831i \(-0.505853\pi\)
−0.0183877 + 0.999831i \(0.505853\pi\)
\(968\) −10.9282 −0.351246
\(969\) −22.9282 −0.736560
\(970\) 8.39230 0.269461
\(971\) −41.3923 −1.32834 −0.664171 0.747581i \(-0.731215\pi\)
−0.664171 + 0.747581i \(0.731215\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −59.3538 −1.90280
\(974\) −21.0000 −0.672883
\(975\) 0 0
\(976\) −0.535898 −0.0171537
\(977\) −5.46410 −0.174812 −0.0874060 0.996173i \(-0.527858\pi\)
−0.0874060 + 0.996173i \(0.527858\pi\)
\(978\) 2.92820 0.0936336
\(979\) −3.78461 −0.120957
\(980\) −2.00000 −0.0638877
\(981\) 10.3923 0.331801
\(982\) −5.39230 −0.172075
\(983\) −46.1769 −1.47281 −0.736407 0.676538i \(-0.763479\pi\)
−0.736407 + 0.676538i \(0.763479\pi\)
\(984\) 4.00000 0.127515
\(985\) 11.3923 0.362989
\(986\) −5.85641 −0.186506
\(987\) 19.3923 0.617264
\(988\) 0 0
\(989\) 20.7846 0.660912
\(990\) 0.267949 0.00851598
\(991\) 39.1769 1.24450 0.622248 0.782820i \(-0.286220\pi\)
0.622248 + 0.782820i \(0.286220\pi\)
\(992\) −4.92820 −0.156471
\(993\) 14.3923 0.456726
\(994\) 38.7846 1.23017
\(995\) −24.9282 −0.790277
\(996\) 9.46410 0.299882
\(997\) 61.9808 1.96295 0.981475 0.191589i \(-0.0613641\pi\)
0.981475 + 0.191589i \(0.0613641\pi\)
\(998\) 33.3205 1.05474
\(999\) −5.92820 −0.187560
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 5070.2.a.be.1.2 2
13.2 odd 12 390.2.bb.a.121.2 4
13.5 odd 4 5070.2.b.p.1351.2 4
13.7 odd 12 390.2.bb.a.361.2 yes 4
13.8 odd 4 5070.2.b.p.1351.3 4
13.12 even 2 5070.2.a.ba.1.1 2
39.2 even 12 1170.2.bs.d.901.1 4
39.20 even 12 1170.2.bs.d.361.1 4
65.2 even 12 1950.2.y.d.199.1 4
65.7 even 12 1950.2.y.e.49.2 4
65.28 even 12 1950.2.y.e.199.2 4
65.33 even 12 1950.2.y.d.49.1 4
65.54 odd 12 1950.2.bc.a.901.1 4
65.59 odd 12 1950.2.bc.a.751.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.bb.a.121.2 4 13.2 odd 12
390.2.bb.a.361.2 yes 4 13.7 odd 12
1170.2.bs.d.361.1 4 39.20 even 12
1170.2.bs.d.901.1 4 39.2 even 12
1950.2.y.d.49.1 4 65.33 even 12
1950.2.y.d.199.1 4 65.2 even 12
1950.2.y.e.49.2 4 65.7 even 12
1950.2.y.e.199.2 4 65.28 even 12
1950.2.bc.a.751.1 4 65.59 odd 12
1950.2.bc.a.901.1 4 65.54 odd 12
5070.2.a.ba.1.1 2 13.12 even 2
5070.2.a.be.1.2 2 1.1 even 1 trivial
5070.2.b.p.1351.2 4 13.5 odd 4
5070.2.b.p.1351.3 4 13.8 odd 4