Properties

Label 5070.2.a.ba.1.1
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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 1.1
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 \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0.267949 0.0807897 0.0403949 0.999184i \(-0.487138\pi\)
0.0403949 + 0.999184i \(0.487138\pi\)
\(12\) −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 \(-0.271609\pi\)
0.657511 + 0.753445i \(0.271609\pi\)
\(20\) 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 \(-0.354068\pi\)
0.442566 + 0.896736i \(0.354068\pi\)
\(32\) −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 \(-0.662016\pi\)
−0.487295 + 0.873237i \(0.662016\pi\)
\(38\) −5.73205 −0.929861
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) −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 \(-0.343733\pi\)
0.471443 + 0.881897i \(0.343733\pi\)
\(48\) −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 \(-0.231853\pi\)
0.746249 + 0.665666i \(0.231853\pi\)
\(60\) −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 \(-0.528506\pi\)
−0.0894342 + 0.995993i \(0.528506\pi\)
\(68\) −4.00000 −0.485071
\(69\) 3.46410 0.417029
\(70\) 3.00000 0.358569
\(71\) −12.9282 −1.53430 −0.767148 0.641470i \(-0.778325\pi\)
−0.767148 + 0.641470i \(0.778325\pi\)
\(72\) −1.00000 −0.117851
\(73\) 6.92820 0.810885 0.405442 0.914121i \(-0.367117\pi\)
0.405442 + 0.914121i \(0.367117\pi\)
\(74\) 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 \(-0.326152\pi\)
0.519410 + 0.854525i \(0.326152\pi\)
\(84\) 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 \(-0.769269\pi\)
−0.748589 + 0.663034i \(0.769269\pi\)
\(90\) −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 \(-0.359903\pi\)
0.426055 + 0.904697i \(0.359903\pi\)
\(98\) −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 \(-0.665822\pi\)
−0.497701 + 0.867349i \(0.665822\pi\)
\(110\) −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 \(-0.695059\pi\)
−0.575158 + 0.818042i \(0.695059\pi\)
\(138\) −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 \(-0.220532\pi\)
0.769448 + 0.638710i \(0.220532\pi\)
\(150\) 1.00000 0.0816497
\(151\) −22.7846 −1.85419 −0.927093 0.374832i \(-0.877700\pi\)
−0.927093 + 0.374832i \(0.877700\pi\)
\(152\) −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 \(-0.463417\pi\)
0.114677 + 0.993403i \(0.463417\pi\)
\(164\) 4.00000 0.312348
\(165\) −0.267949 −0.0208598
\(166\) −9.46410 −0.734557
\(167\) −0.464102 −0.0359133 −0.0179566 0.999839i \(-0.505716\pi\)
−0.0179566 + 0.999839i \(0.505716\pi\)
\(168\) −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 \(-0.615431\pi\)
−0.354740 + 0.934965i \(0.615431\pi\)
\(194\) −8.39230 −0.602532
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 11.3923 0.811668 0.405834 0.913947i \(-0.366981\pi\)
0.405834 + 0.913947i \(0.366981\pi\)
\(198\) −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 \(-0.282497\pi\)
0.631359 + 0.775491i \(0.282497\pi\)
\(224\) 3.00000 0.200446
\(225\) 1.00000 0.0666667
\(226\) 12.0000 0.798228
\(227\) 6.53590 0.433803 0.216901 0.976194i \(-0.430405\pi\)
0.216901 + 0.976194i \(0.430405\pi\)
\(228\) −5.73205 −0.379614
\(229\) −4.53590 −0.299741 −0.149870 0.988706i \(-0.547886\pi\)
−0.149870 + 0.988706i \(0.547886\pi\)
\(230\) 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 \(-0.535738\pi\)
−0.112037 + 0.993704i \(0.535738\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 25.1962 1.62303 0.811513 0.584334i \(-0.198644\pi\)
0.811513 + 0.584334i \(0.198644\pi\)
\(242\) 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 \(-0.234417\pi\)
0.740863 + 0.671656i \(0.234417\pi\)
\(272\) −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 \(-0.433744\pi\)
0.206651 + 0.978415i \(0.433744\pi\)
\(282\) 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 \(-0.826053\pi\)
−0.854364 + 0.519675i \(0.826053\pi\)
\(294\) 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 \(-0.798436\pi\)
−0.806120 + 0.591753i \(0.798436\pi\)
\(308\) −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 \(-0.826761\pi\)
−0.855517 + 0.517774i \(0.826761\pi\)
\(318\) −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 \(-0.370559\pi\)
0.395536 + 0.918450i \(0.370559\pi\)
\(332\) 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 \(-0.590993\pi\)
−0.281987 + 0.959418i \(0.590993\pi\)
\(350\) 3.00000 0.160357
\(351\) 0 0
\(352\) −0.267949 −0.0142817
\(353\) −7.60770 −0.404917 −0.202458 0.979291i \(-0.564893\pi\)
−0.202458 + 0.979291i \(0.564893\pi\)
\(354\) 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 \(-0.492202\pi\)
0.0244943 + 0.999700i \(0.492202\pi\)
\(360\) −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 \(-0.419586\pi\)
0.249951 + 0.968259i \(0.419586\pi\)
\(380\) 5.73205 0.294048
\(381\) −12.6603 −0.648604
\(382\) 17.3205 0.886194
\(383\) 4.78461 0.244482 0.122241 0.992500i \(-0.460992\pi\)
0.122241 + 0.992500i \(0.460992\pi\)
\(384\) 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 \(-0.452471\pi\)
0.148764 + 0.988873i \(0.452471\pi\)
\(398\) −24.9282 −1.24954
\(399\) 17.1962 0.860884
\(400\) 1.00000 0.0500000
\(401\) −22.1244 −1.10484 −0.552419 0.833567i \(-0.686295\pi\)
−0.552419 + 0.833567i \(0.686295\pi\)
\(402\) −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 \(-0.916160\pi\)
−0.965512 + 0.260357i \(0.916160\pi\)
\(410\) −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 \(-0.678788\pi\)
−0.532608 + 0.846362i \(0.678788\pi\)
\(422\) 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 \(-0.391696\pi\)
0.333720 + 0.942672i \(0.391696\pi\)
\(432\) −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 \(-0.453838\pi\)
0.144513 + 0.989503i \(0.453838\pi\)
\(450\) −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 \(-0.444144\pi\)
0.174578 + 0.984643i \(0.444144\pi\)
\(458\) 4.53590 0.211949
\(459\) 4.00000 0.186704
\(460\) −3.46410 −0.161515
\(461\) −11.6077 −0.540624 −0.270312 0.962773i \(-0.587127\pi\)
−0.270312 + 0.962773i \(0.587127\pi\)
\(462\) −0.803848 −0.0373984
\(463\) −40.7846 −1.89542 −0.947711 0.319131i \(-0.896609\pi\)
−0.947711 + 0.319131i \(0.896609\pi\)
\(464\) 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 \(-0.669720\pi\)
−0.508285 + 0.861189i \(0.669720\pi\)
\(480\) 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 \(-0.342158\pi\)
0.475800 + 0.879553i \(0.342158\pi\)
\(488\) 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 \(-0.767940\pi\)
−0.745815 + 0.666153i \(0.767940\pi\)
\(500\) 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 \(-0.725149\pi\)
−0.649804 + 0.760102i \(0.725149\pi\)
\(510\) −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 \(-0.397049\pi\)
0.317820 + 0.948151i \(0.397049\pi\)
\(542\) −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 \(-0.658986\pi\)
−0.478959 + 0.877837i \(0.658986\pi\)
\(558\) −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 \(-0.437851\pi\)
0.194009 + 0.981000i \(0.437851\pi\)
\(578\) 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 \(-0.441013\pi\)
0.184253 + 0.982879i \(0.441013\pi\)
\(588\) −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 \(-0.871440\pi\)
−0.919542 + 0.392992i \(0.871440\pi\)
\(594\) 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 \(-0.862975\pi\)
−0.908766 + 0.417305i \(0.862975\pi\)
\(614\) 28.2487 1.14003
\(615\) −4.00000 −0.161296
\(616\) 0.803848 0.0323879
\(617\) −35.4641 −1.42773 −0.713865 0.700283i \(-0.753057\pi\)
−0.713865 + 0.700283i \(0.753057\pi\)
\(618\) 11.5885 0.466156
\(619\) 2.51666 0.101153 0.0505766 0.998720i \(-0.483894\pi\)
0.0505766 + 0.998720i \(0.483894\pi\)
\(620\) 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 \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −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 \(-0.418883\pi\)
0.252088 + 0.967704i \(0.418883\pi\)
\(644\) 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 \(-0.638047\pi\)
−0.420221 + 0.907422i \(0.638047\pi\)
\(662\) −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 \(-0.790201\pi\)
−0.790543 + 0.612407i \(0.790201\pi\)
\(684\) 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 \(-0.592430\pi\)
−0.286313 + 0.958136i \(0.592430\pi\)
\(692\) 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 \(-0.328526\pi\)
0.513022 + 0.858376i \(0.328526\pi\)
\(710\) 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 \(-0.625859\pi\)
−0.385174 + 0.922844i \(0.625859\pi\)
\(734\) −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 \(-0.270918\pi\)
0.659146 + 0.752015i \(0.270918\pi\)
\(740\) −5.92820 −0.217925
\(741\) 0 0
\(742\) −0.803848 −0.0295102
\(743\) 10.9282 0.400917 0.200458 0.979702i \(-0.435757\pi\)
0.200458 + 0.979702i \(0.435757\pi\)
\(744\) 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 \(-0.546206\pi\)
−0.144651 + 0.989483i \(0.546206\pi\)
\(762\) 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 \(-0.591432\pi\)
−0.283309 + 0.959029i \(0.591432\pi\)
\(770\) 0.803848 0.0289687
\(771\) 14.5359 0.523498
\(772\) −9.85641 −0.354740
\(773\) −32.6077 −1.17282 −0.586409 0.810015i \(-0.699459\pi\)
−0.586409 + 0.810015i \(0.699459\pi\)
\(774\) 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 \(-0.508307\pi\)
−0.0260948 + 0.999659i \(0.508307\pi\)
\(788\) 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 \(-0.635922\pi\)
−0.414151 + 0.910208i \(0.635922\pi\)
\(812\) −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 \(-0.417543\pi\)
0.256159 + 0.966635i \(0.417543\pi\)
\(822\) −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 \(-0.200314\pi\)
0.808437 + 0.588583i \(0.200314\pi\)
\(828\) −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 \(-0.607396\pi\)
−0.331030 + 0.943620i \(0.607396\pi\)
\(840\) −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 \(-0.638618\pi\)
−0.421846 + 0.906667i \(0.638618\pi\)
\(854\) −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 \(-0.515871\pi\)
−0.0498386 + 0.998757i \(0.515871\pi\)
\(864\) 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 \(-0.380524\pi\)
0.366595 + 0.930381i \(0.380524\pi\)
\(878\) −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 \(-0.101993\pi\)
0.949104 + 0.314964i \(0.101993\pi\)
\(930\) 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 \(-0.0531732\pi\)
0.986080 + 0.166273i \(0.0531732\pi\)
\(942\) 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 \(-0.311262\pi\)
0.558800 + 0.829303i \(0.311262\pi\)
\(948\) −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 \(-0.494147\pi\)
0.0183877 + 0.999831i \(0.494147\pi\)
\(968\) 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 \(-0.472142\pi\)
0.0874060 + 0.996173i \(0.472142\pi\)
\(978\) 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 \(-0.236521\pi\)
0.736407 + 0.676538i \(0.236521\pi\)
\(984\) 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.ba.1.1 2
13.5 odd 4 5070.2.b.p.1351.3 4
13.6 odd 12 390.2.bb.a.361.2 yes 4
13.8 odd 4 5070.2.b.p.1351.2 4
13.11 odd 12 390.2.bb.a.121.2 4
13.12 even 2 5070.2.a.be.1.2 2
39.11 even 12 1170.2.bs.d.901.1 4
39.32 even 12 1170.2.bs.d.361.1 4
65.19 odd 12 1950.2.bc.a.751.1 4
65.24 odd 12 1950.2.bc.a.901.1 4
65.32 even 12 1950.2.y.e.49.2 4
65.37 even 12 1950.2.y.d.199.1 4
65.58 even 12 1950.2.y.d.49.1 4
65.63 even 12 1950.2.y.e.199.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.bb.a.121.2 4 13.11 odd 12
390.2.bb.a.361.2 yes 4 13.6 odd 12
1170.2.bs.d.361.1 4 39.32 even 12
1170.2.bs.d.901.1 4 39.11 even 12
1950.2.y.d.49.1 4 65.58 even 12
1950.2.y.d.199.1 4 65.37 even 12
1950.2.y.e.49.2 4 65.32 even 12
1950.2.y.e.199.2 4 65.63 even 12
1950.2.bc.a.751.1 4 65.19 odd 12
1950.2.bc.a.901.1 4 65.24 odd 12
5070.2.a.ba.1.1 2 1.1 even 1 trivial
5070.2.a.be.1.2 2 13.12 even 2
5070.2.b.p.1351.2 4 13.8 odd 4
5070.2.b.p.1351.3 4 13.5 odd 4