Properties

Label 8470.2.a.cl.1.2
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.892.1
Defining polynomial: \(x^{3} - x^{2} - 8 x + 10\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.91729\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.406728 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.406728 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.83457 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.406728 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.406728 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.83457 q^{9} -1.00000 q^{10} -0.406728 q^{12} +0.813457 q^{13} +1.00000 q^{14} +0.406728 q^{15} +1.00000 q^{16} -3.83457 q^{17} -2.83457 q^{18} -5.42784 q^{19} -1.00000 q^{20} -0.406728 q^{21} -5.42784 q^{23} -0.406728 q^{24} +1.00000 q^{25} +0.813457 q^{26} +2.37309 q^{27} +1.00000 q^{28} -6.61439 q^{29} +0.406728 q^{30} +6.00000 q^{31} +1.00000 q^{32} -3.83457 q^{34} -1.00000 q^{35} -2.83457 q^{36} +8.24130 q^{37} -5.42784 q^{38} -0.330856 q^{39} -1.00000 q^{40} +5.59327 q^{41} -0.406728 q^{42} +3.02112 q^{43} +2.83457 q^{45} -5.42784 q^{46} -5.02112 q^{47} -0.406728 q^{48} +1.00000 q^{49} +1.00000 q^{50} +1.55963 q^{51} +0.813457 q^{52} +6.61439 q^{53} +2.37309 q^{54} +1.00000 q^{56} +2.20766 q^{57} -6.61439 q^{58} +10.6480 q^{59} +0.406728 q^{60} -0.978885 q^{61} +6.00000 q^{62} -2.83457 q^{63} +1.00000 q^{64} -0.813457 q^{65} +8.00000 q^{67} -3.83457 q^{68} +2.20766 q^{69} -1.00000 q^{70} +14.8557 q^{71} -2.83457 q^{72} +6.20766 q^{73} +8.24130 q^{74} -0.406728 q^{75} -5.42784 q^{76} -0.330856 q^{78} -17.0970 q^{79} -1.00000 q^{80} +7.53851 q^{81} +5.59327 q^{82} +8.81346 q^{83} -0.406728 q^{84} +3.83457 q^{85} +3.02112 q^{86} +2.69026 q^{87} -1.18654 q^{89} +2.83457 q^{90} +0.813457 q^{91} -5.42784 q^{92} -2.44037 q^{93} -5.02112 q^{94} +5.42784 q^{95} -0.406728 q^{96} -3.42784 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{8} + 11 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{8} + 11 q^{9} - 3 q^{10} + 3 q^{14} + 3 q^{16} + 8 q^{17} + 11 q^{18} + 2 q^{19} - 3 q^{20} + 2 q^{23} + 3 q^{25} + 12 q^{27} + 3 q^{28} - 4 q^{29} + 18 q^{31} + 3 q^{32} + 8 q^{34} - 3 q^{35} + 11 q^{36} + 4 q^{37} + 2 q^{38} - 40 q^{39} - 3 q^{40} + 18 q^{41} - 8 q^{43} - 11 q^{45} + 2 q^{46} + 2 q^{47} + 3 q^{49} + 3 q^{50} + 12 q^{51} + 4 q^{53} + 12 q^{54} + 3 q^{56} - 8 q^{57} - 4 q^{58} + 10 q^{59} - 20 q^{61} + 18 q^{62} + 11 q^{63} + 3 q^{64} + 24 q^{67} + 8 q^{68} - 8 q^{69} - 3 q^{70} + 8 q^{71} + 11 q^{72} + 4 q^{73} + 4 q^{74} + 2 q^{76} - 40 q^{78} + 6 q^{79} - 3 q^{80} + 47 q^{81} + 18 q^{82} + 24 q^{83} - 8 q^{85} - 8 q^{86} - 48 q^{87} - 6 q^{89} - 11 q^{90} + 2 q^{92} + 2 q^{94} - 2 q^{95} + 8 q^{97} + 3 q^{98} + O(q^{100}) \)

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
\(3\) −0.406728 −0.234825 −0.117412 0.993083i \(-0.537460\pi\)
−0.117412 + 0.993083i \(0.537460\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −0.406728 −0.166046
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −2.83457 −0.944857
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −0.406728 −0.117412
\(13\) 0.813457 0.225612 0.112806 0.993617i \(-0.464016\pi\)
0.112806 + 0.993617i \(0.464016\pi\)
\(14\) 1.00000 0.267261
\(15\) 0.406728 0.105017
\(16\) 1.00000 0.250000
\(17\) −3.83457 −0.930020 −0.465010 0.885305i \(-0.653949\pi\)
−0.465010 + 0.885305i \(0.653949\pi\)
\(18\) −2.83457 −0.668115
\(19\) −5.42784 −1.24523 −0.622616 0.782527i \(-0.713930\pi\)
−0.622616 + 0.782527i \(0.713930\pi\)
\(20\) −1.00000 −0.223607
\(21\) −0.406728 −0.0887554
\(22\) 0 0
\(23\) −5.42784 −1.13178 −0.565892 0.824480i \(-0.691468\pi\)
−0.565892 + 0.824480i \(0.691468\pi\)
\(24\) −0.406728 −0.0830231
\(25\) 1.00000 0.200000
\(26\) 0.813457 0.159532
\(27\) 2.37309 0.456701
\(28\) 1.00000 0.188982
\(29\) −6.61439 −1.22826 −0.614130 0.789205i \(-0.710493\pi\)
−0.614130 + 0.789205i \(0.710493\pi\)
\(30\) 0.406728 0.0742581
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.83457 −0.657624
\(35\) −1.00000 −0.169031
\(36\) −2.83457 −0.472429
\(37\) 8.24130 1.35486 0.677431 0.735587i \(-0.263093\pi\)
0.677431 + 0.735587i \(0.263093\pi\)
\(38\) −5.42784 −0.880512
\(39\) −0.330856 −0.0529794
\(40\) −1.00000 −0.158114
\(41\) 5.59327 0.873522 0.436761 0.899578i \(-0.356125\pi\)
0.436761 + 0.899578i \(0.356125\pi\)
\(42\) −0.406728 −0.0627596
\(43\) 3.02112 0.460716 0.230358 0.973106i \(-0.426010\pi\)
0.230358 + 0.973106i \(0.426010\pi\)
\(44\) 0 0
\(45\) 2.83457 0.422553
\(46\) −5.42784 −0.800292
\(47\) −5.02112 −0.732405 −0.366202 0.930535i \(-0.619342\pi\)
−0.366202 + 0.930535i \(0.619342\pi\)
\(48\) −0.406728 −0.0587062
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 1.55963 0.218392
\(52\) 0.813457 0.112806
\(53\) 6.61439 0.908556 0.454278 0.890860i \(-0.349897\pi\)
0.454278 + 0.890860i \(0.349897\pi\)
\(54\) 2.37309 0.322936
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 2.20766 0.292411
\(58\) −6.61439 −0.868512
\(59\) 10.6480 1.38626 0.693128 0.720815i \(-0.256232\pi\)
0.693128 + 0.720815i \(0.256232\pi\)
\(60\) 0.406728 0.0525084
\(61\) −0.978885 −0.125333 −0.0626667 0.998035i \(-0.519961\pi\)
−0.0626667 + 0.998035i \(0.519961\pi\)
\(62\) 6.00000 0.762001
\(63\) −2.83457 −0.357123
\(64\) 1.00000 0.125000
\(65\) −0.813457 −0.100897
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −3.83457 −0.465010
\(69\) 2.20766 0.265771
\(70\) −1.00000 −0.119523
\(71\) 14.8557 1.76305 0.881523 0.472141i \(-0.156519\pi\)
0.881523 + 0.472141i \(0.156519\pi\)
\(72\) −2.83457 −0.334058
\(73\) 6.20766 0.726551 0.363276 0.931682i \(-0.381658\pi\)
0.363276 + 0.931682i \(0.381658\pi\)
\(74\) 8.24130 0.958032
\(75\) −0.406728 −0.0469650
\(76\) −5.42784 −0.622616
\(77\) 0 0
\(78\) −0.330856 −0.0374621
\(79\) −17.0970 −1.92356 −0.961781 0.273821i \(-0.911712\pi\)
−0.961781 + 0.273821i \(0.911712\pi\)
\(80\) −1.00000 −0.111803
\(81\) 7.53851 0.837613
\(82\) 5.59327 0.617674
\(83\) 8.81346 0.967403 0.483701 0.875233i \(-0.339292\pi\)
0.483701 + 0.875233i \(0.339292\pi\)
\(84\) −0.406728 −0.0443777
\(85\) 3.83457 0.415918
\(86\) 3.02112 0.325775
\(87\) 2.69026 0.288426
\(88\) 0 0
\(89\) −1.18654 −0.125773 −0.0628867 0.998021i \(-0.520031\pi\)
−0.0628867 + 0.998021i \(0.520031\pi\)
\(90\) 2.83457 0.298790
\(91\) 0.813457 0.0852734
\(92\) −5.42784 −0.565892
\(93\) −2.44037 −0.253055
\(94\) −5.02112 −0.517888
\(95\) 5.42784 0.556885
\(96\) −0.406728 −0.0415115
\(97\) −3.42784 −0.348045 −0.174022 0.984742i \(-0.555677\pi\)
−0.174022 + 0.984742i \(0.555677\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 18.6903 1.85975 0.929875 0.367875i \(-0.119915\pi\)
0.929875 + 0.367875i \(0.119915\pi\)
\(102\) 1.55963 0.154426
\(103\) 15.4615 1.52347 0.761733 0.647891i \(-0.224349\pi\)
0.761733 + 0.647891i \(0.224349\pi\)
\(104\) 0.813457 0.0797660
\(105\) 0.406728 0.0396926
\(106\) 6.61439 0.642446
\(107\) 11.0211 1.06545 0.532726 0.846288i \(-0.321168\pi\)
0.532726 + 0.846288i \(0.321168\pi\)
\(108\) 2.37309 0.228350
\(109\) −16.2413 −1.55563 −0.777817 0.628491i \(-0.783673\pi\)
−0.777817 + 0.628491i \(0.783673\pi\)
\(110\) 0 0
\(111\) −3.35197 −0.318155
\(112\) 1.00000 0.0944911
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 2.20766 0.206766
\(115\) 5.42784 0.506149
\(116\) −6.61439 −0.614130
\(117\) −2.30580 −0.213171
\(118\) 10.6480 0.980231
\(119\) −3.83457 −0.351515
\(120\) 0.406728 0.0371291
\(121\) 0 0
\(122\) −0.978885 −0.0886241
\(123\) −2.27494 −0.205125
\(124\) 6.00000 0.538816
\(125\) −1.00000 −0.0894427
\(126\) −2.83457 −0.252524
\(127\) −17.2961 −1.53478 −0.767388 0.641182i \(-0.778444\pi\)
−0.767388 + 0.641182i \(0.778444\pi\)
\(128\) 1.00000 0.0883883
\(129\) −1.22877 −0.108187
\(130\) −0.813457 −0.0713449
\(131\) −9.42784 −0.823715 −0.411857 0.911248i \(-0.635120\pi\)
−0.411857 + 0.911248i \(0.635120\pi\)
\(132\) 0 0
\(133\) −5.42784 −0.470654
\(134\) 8.00000 0.691095
\(135\) −2.37309 −0.204243
\(136\) −3.83457 −0.328812
\(137\) 13.6691 1.16783 0.583917 0.811813i \(-0.301519\pi\)
0.583917 + 0.811813i \(0.301519\pi\)
\(138\) 2.20766 0.187928
\(139\) 11.8682 1.00665 0.503324 0.864098i \(-0.332110\pi\)
0.503324 + 0.864098i \(0.332110\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 2.04223 0.171987
\(142\) 14.8557 1.24666
\(143\) 0 0
\(144\) −2.83457 −0.236214
\(145\) 6.61439 0.549295
\(146\) 6.20766 0.513749
\(147\) −0.406728 −0.0335464
\(148\) 8.24130 0.677431
\(149\) 22.2835 1.82554 0.912769 0.408476i \(-0.133940\pi\)
0.912769 + 0.408476i \(0.133940\pi\)
\(150\) −0.406728 −0.0332092
\(151\) 3.05476 0.248593 0.124296 0.992245i \(-0.460333\pi\)
0.124296 + 0.992245i \(0.460333\pi\)
\(152\) −5.42784 −0.440256
\(153\) 10.8694 0.878737
\(154\) 0 0
\(155\) −6.00000 −0.481932
\(156\) −0.330856 −0.0264897
\(157\) 4.85569 0.387526 0.193763 0.981048i \(-0.437931\pi\)
0.193763 + 0.981048i \(0.437931\pi\)
\(158\) −17.0970 −1.36016
\(159\) −2.69026 −0.213351
\(160\) −1.00000 −0.0790569
\(161\) −5.42784 −0.427774
\(162\) 7.53851 0.592282
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 5.59327 0.436761
\(165\) 0 0
\(166\) 8.81346 0.684057
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) −0.406728 −0.0313798
\(169\) −12.3383 −0.949099
\(170\) 3.83457 0.294098
\(171\) 15.3856 1.17657
\(172\) 3.02112 0.230358
\(173\) 19.0211 1.44615 0.723074 0.690770i \(-0.242728\pi\)
0.723074 + 0.690770i \(0.242728\pi\)
\(174\) 2.69026 0.203948
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −4.33086 −0.325527
\(178\) −1.18654 −0.0889352
\(179\) 4.64803 0.347410 0.173705 0.984798i \(-0.444426\pi\)
0.173705 + 0.984798i \(0.444426\pi\)
\(180\) 2.83457 0.211277
\(181\) −25.3383 −1.88338 −0.941690 0.336482i \(-0.890763\pi\)
−0.941690 + 0.336482i \(0.890763\pi\)
\(182\) 0.813457 0.0602974
\(183\) 0.398140 0.0294314
\(184\) −5.42784 −0.400146
\(185\) −8.24130 −0.605912
\(186\) −2.44037 −0.178937
\(187\) 0 0
\(188\) −5.02112 −0.366202
\(189\) 2.37309 0.172617
\(190\) 5.42784 0.393777
\(191\) 15.6691 1.13378 0.566890 0.823794i \(-0.308147\pi\)
0.566890 + 0.823794i \(0.308147\pi\)
\(192\) −0.406728 −0.0293531
\(193\) 2.16543 0.155871 0.0779355 0.996958i \(-0.475167\pi\)
0.0779355 + 0.996958i \(0.475167\pi\)
\(194\) −3.42784 −0.246105
\(195\) 0.330856 0.0236931
\(196\) 1.00000 0.0714286
\(197\) −2.81346 −0.200451 −0.100225 0.994965i \(-0.531956\pi\)
−0.100225 + 0.994965i \(0.531956\pi\)
\(198\) 0 0
\(199\) −13.0211 −0.923042 −0.461521 0.887129i \(-0.652696\pi\)
−0.461521 + 0.887129i \(0.652696\pi\)
\(200\) 1.00000 0.0707107
\(201\) −3.25383 −0.229507
\(202\) 18.6903 1.31504
\(203\) −6.61439 −0.464239
\(204\) 1.55963 0.109196
\(205\) −5.59327 −0.390651
\(206\) 15.4615 1.07725
\(207\) 15.3856 1.06937
\(208\) 0.813457 0.0564031
\(209\) 0 0
\(210\) 0.406728 0.0280669
\(211\) −19.6691 −1.35408 −0.677040 0.735946i \(-0.736738\pi\)
−0.677040 + 0.735946i \(0.736738\pi\)
\(212\) 6.61439 0.454278
\(213\) −6.04223 −0.414007
\(214\) 11.0211 0.753388
\(215\) −3.02112 −0.206038
\(216\) 2.37309 0.161468
\(217\) 6.00000 0.407307
\(218\) −16.2413 −1.10000
\(219\) −2.52483 −0.170612
\(220\) 0 0
\(221\) −3.11926 −0.209824
\(222\) −3.35197 −0.224970
\(223\) −3.79234 −0.253954 −0.126977 0.991906i \(-0.540527\pi\)
−0.126977 + 0.991906i \(0.540527\pi\)
\(224\) 1.00000 0.0668153
\(225\) −2.83457 −0.188971
\(226\) −14.0000 −0.931266
\(227\) 14.0422 0.932016 0.466008 0.884781i \(-0.345692\pi\)
0.466008 + 0.884781i \(0.345692\pi\)
\(228\) 2.20766 0.146206
\(229\) 1.83457 0.121232 0.0606160 0.998161i \(-0.480694\pi\)
0.0606160 + 0.998161i \(0.480694\pi\)
\(230\) 5.42784 0.357901
\(231\) 0 0
\(232\) −6.61439 −0.434256
\(233\) 8.04223 0.526864 0.263432 0.964678i \(-0.415146\pi\)
0.263432 + 0.964678i \(0.415146\pi\)
\(234\) −2.30580 −0.150735
\(235\) 5.02112 0.327541
\(236\) 10.6480 0.693128
\(237\) 6.95383 0.451700
\(238\) −3.83457 −0.248558
\(239\) 15.4701 1.00068 0.500338 0.865830i \(-0.333209\pi\)
0.500338 + 0.865830i \(0.333209\pi\)
\(240\) 0.406728 0.0262542
\(241\) −11.6355 −0.749509 −0.374754 0.927124i \(-0.622273\pi\)
−0.374754 + 0.927124i \(0.622273\pi\)
\(242\) 0 0
\(243\) −10.1854 −0.653393
\(244\) −0.978885 −0.0626667
\(245\) −1.00000 −0.0638877
\(246\) −2.27494 −0.145045
\(247\) −4.41532 −0.280940
\(248\) 6.00000 0.381000
\(249\) −3.58468 −0.227170
\(250\) −1.00000 −0.0632456
\(251\) 5.35197 0.337813 0.168907 0.985632i \(-0.445976\pi\)
0.168907 + 0.985632i \(0.445976\pi\)
\(252\) −2.83457 −0.178561
\(253\) 0 0
\(254\) −17.2961 −1.08525
\(255\) −1.55963 −0.0976678
\(256\) 1.00000 0.0625000
\(257\) 12.7239 0.793695 0.396848 0.917885i \(-0.370104\pi\)
0.396848 + 0.917885i \(0.370104\pi\)
\(258\) −1.22877 −0.0765001
\(259\) 8.24130 0.512089
\(260\) −0.813457 −0.0504485
\(261\) 18.7490 1.16053
\(262\) −9.42784 −0.582454
\(263\) 1.62691 0.100320 0.0501599 0.998741i \(-0.484027\pi\)
0.0501599 + 0.998741i \(0.484027\pi\)
\(264\) 0 0
\(265\) −6.61439 −0.406319
\(266\) −5.42784 −0.332802
\(267\) 0.482601 0.0295347
\(268\) 8.00000 0.488678
\(269\) 23.8768 1.45579 0.727897 0.685686i \(-0.240498\pi\)
0.727897 + 0.685686i \(0.240498\pi\)
\(270\) −2.37309 −0.144421
\(271\) −2.44037 −0.148242 −0.0741210 0.997249i \(-0.523615\pi\)
−0.0741210 + 0.997249i \(0.523615\pi\)
\(272\) −3.83457 −0.232505
\(273\) −0.330856 −0.0200243
\(274\) 13.6691 0.825783
\(275\) 0 0
\(276\) 2.20766 0.132885
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 11.8682 0.711808
\(279\) −17.0074 −1.01821
\(280\) −1.00000 −0.0597614
\(281\) 20.8557 1.24415 0.622073 0.782959i \(-0.286291\pi\)
0.622073 + 0.782959i \(0.286291\pi\)
\(282\) 2.04223 0.121613
\(283\) −0.330856 −0.0196673 −0.00983367 0.999952i \(-0.503130\pi\)
−0.00983367 + 0.999952i \(0.503130\pi\)
\(284\) 14.8557 0.881523
\(285\) −2.20766 −0.130770
\(286\) 0 0
\(287\) 5.59327 0.330160
\(288\) −2.83457 −0.167029
\(289\) −2.29606 −0.135062
\(290\) 6.61439 0.388410
\(291\) 1.39420 0.0817295
\(292\) 6.20766 0.363276
\(293\) 3.18654 0.186160 0.0930799 0.995659i \(-0.470329\pi\)
0.0930799 + 0.995659i \(0.470329\pi\)
\(294\) −0.406728 −0.0237209
\(295\) −10.6480 −0.619952
\(296\) 8.24130 0.479016
\(297\) 0 0
\(298\) 22.2835 1.29085
\(299\) −4.41532 −0.255344
\(300\) −0.406728 −0.0234825
\(301\) 3.02112 0.174134
\(302\) 3.05476 0.175782
\(303\) −7.60186 −0.436715
\(304\) −5.42784 −0.311308
\(305\) 0.978885 0.0560508
\(306\) 10.8694 0.621361
\(307\) −20.4826 −1.16900 −0.584502 0.811392i \(-0.698710\pi\)
−0.584502 + 0.811392i \(0.698710\pi\)
\(308\) 0 0
\(309\) −6.28863 −0.357747
\(310\) −6.00000 −0.340777
\(311\) 3.62691 0.205663 0.102832 0.994699i \(-0.467210\pi\)
0.102832 + 0.994699i \(0.467210\pi\)
\(312\) −0.330856 −0.0187310
\(313\) 20.7239 1.17138 0.585692 0.810534i \(-0.300823\pi\)
0.585692 + 0.810534i \(0.300823\pi\)
\(314\) 4.85569 0.274022
\(315\) 2.83457 0.159710
\(316\) −17.0970 −0.961781
\(317\) −1.80093 −0.101150 −0.0505751 0.998720i \(-0.516105\pi\)
−0.0505751 + 0.998720i \(0.516105\pi\)
\(318\) −2.69026 −0.150862
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −4.48260 −0.250194
\(322\) −5.42784 −0.302482
\(323\) 20.8135 1.15809
\(324\) 7.53851 0.418806
\(325\) 0.813457 0.0451225
\(326\) −8.00000 −0.443079
\(327\) 6.60580 0.365301
\(328\) 5.59327 0.308837
\(329\) −5.02112 −0.276823
\(330\) 0 0
\(331\) −3.02112 −0.166056 −0.0830278 0.996547i \(-0.526459\pi\)
−0.0830278 + 0.996547i \(0.526459\pi\)
\(332\) 8.81346 0.483701
\(333\) −23.3606 −1.28015
\(334\) −8.00000 −0.437741
\(335\) −8.00000 −0.437087
\(336\) −0.406728 −0.0221889
\(337\) 9.66914 0.526712 0.263356 0.964699i \(-0.415171\pi\)
0.263356 + 0.964699i \(0.415171\pi\)
\(338\) −12.3383 −0.671114
\(339\) 5.69420 0.309266
\(340\) 3.83457 0.207959
\(341\) 0 0
\(342\) 15.3856 0.831959
\(343\) 1.00000 0.0539949
\(344\) 3.02112 0.162888
\(345\) −2.20766 −0.118856
\(346\) 19.0211 1.02258
\(347\) −9.39420 −0.504307 −0.252154 0.967687i \(-0.581139\pi\)
−0.252154 + 0.967687i \(0.581139\pi\)
\(348\) 2.69026 0.144213
\(349\) 18.2077 0.974634 0.487317 0.873225i \(-0.337976\pi\)
0.487317 + 0.873225i \(0.337976\pi\)
\(350\) 1.00000 0.0534522
\(351\) 1.93040 0.103037
\(352\) 0 0
\(353\) 12.7239 0.677225 0.338612 0.940926i \(-0.390042\pi\)
0.338612 + 0.940926i \(0.390042\pi\)
\(354\) −4.33086 −0.230182
\(355\) −14.8557 −0.788458
\(356\) −1.18654 −0.0628867
\(357\) 1.55963 0.0825443
\(358\) 4.64803 0.245656
\(359\) −29.8432 −1.57506 −0.787531 0.616275i \(-0.788641\pi\)
−0.787531 + 0.616275i \(0.788641\pi\)
\(360\) 2.83457 0.149395
\(361\) 10.4615 0.550605
\(362\) −25.3383 −1.33175
\(363\) 0 0
\(364\) 0.813457 0.0426367
\(365\) −6.20766 −0.324924
\(366\) 0.398140 0.0208111
\(367\) 15.4615 0.807083 0.403541 0.914961i \(-0.367779\pi\)
0.403541 + 0.914961i \(0.367779\pi\)
\(368\) −5.42784 −0.282946
\(369\) −15.8545 −0.825354
\(370\) −8.24130 −0.428445
\(371\) 6.61439 0.343402
\(372\) −2.44037 −0.126527
\(373\) 8.10951 0.419895 0.209947 0.977713i \(-0.432671\pi\)
0.209947 + 0.977713i \(0.432671\pi\)
\(374\) 0 0
\(375\) 0.406728 0.0210034
\(376\) −5.02112 −0.258944
\(377\) −5.38052 −0.277111
\(378\) 2.37309 0.122058
\(379\) 22.0422 1.13223 0.566117 0.824325i \(-0.308445\pi\)
0.566117 + 0.824325i \(0.308445\pi\)
\(380\) 5.42784 0.278442
\(381\) 7.03480 0.360404
\(382\) 15.6691 0.801703
\(383\) 38.2499 1.95448 0.977239 0.212141i \(-0.0680437\pi\)
0.977239 + 0.212141i \(0.0680437\pi\)
\(384\) −0.406728 −0.0207558
\(385\) 0 0
\(386\) 2.16543 0.110217
\(387\) −8.56357 −0.435311
\(388\) −3.42784 −0.174022
\(389\) 16.8557 0.854617 0.427309 0.904106i \(-0.359462\pi\)
0.427309 + 0.904106i \(0.359462\pi\)
\(390\) 0.330856 0.0167535
\(391\) 20.8135 1.05258
\(392\) 1.00000 0.0505076
\(393\) 3.83457 0.193429
\(394\) −2.81346 −0.141740
\(395\) 17.0970 0.860243
\(396\) 0 0
\(397\) −4.44037 −0.222856 −0.111428 0.993773i \(-0.535542\pi\)
−0.111428 + 0.993773i \(0.535542\pi\)
\(398\) −13.0211 −0.652690
\(399\) 2.20766 0.110521
\(400\) 1.00000 0.0500000
\(401\) −3.46149 −0.172858 −0.0864292 0.996258i \(-0.527546\pi\)
−0.0864292 + 0.996258i \(0.527546\pi\)
\(402\) −3.25383 −0.162286
\(403\) 4.88074 0.243127
\(404\) 18.6903 0.929875
\(405\) −7.53851 −0.374592
\(406\) −6.61439 −0.328267
\(407\) 0 0
\(408\) 1.55963 0.0772132
\(409\) 17.2624 0.853572 0.426786 0.904353i \(-0.359646\pi\)
0.426786 + 0.904353i \(0.359646\pi\)
\(410\) −5.59327 −0.276232
\(411\) −5.55963 −0.274236
\(412\) 15.4615 0.761733
\(413\) 10.6480 0.523955
\(414\) 15.3856 0.756162
\(415\) −8.81346 −0.432636
\(416\) 0.813457 0.0398830
\(417\) −4.82714 −0.236386
\(418\) 0 0
\(419\) 9.02112 0.440710 0.220355 0.975420i \(-0.429278\pi\)
0.220355 + 0.975420i \(0.429278\pi\)
\(420\) 0.406728 0.0198463
\(421\) 31.7114 1.54552 0.772759 0.634700i \(-0.218876\pi\)
0.772759 + 0.634700i \(0.218876\pi\)
\(422\) −19.6691 −0.957479
\(423\) 14.2327 0.692018
\(424\) 6.61439 0.321223
\(425\) −3.83457 −0.186004
\(426\) −6.04223 −0.292747
\(427\) −0.978885 −0.0473716
\(428\) 11.0211 0.532726
\(429\) 0 0
\(430\) −3.02112 −0.145691
\(431\) 3.05476 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(432\) 2.37309 0.114175
\(433\) 27.9104 1.34129 0.670645 0.741778i \(-0.266017\pi\)
0.670645 + 0.741778i \(0.266017\pi\)
\(434\) 6.00000 0.288009
\(435\) −2.69026 −0.128988
\(436\) −16.2413 −0.777817
\(437\) 29.4615 1.40933
\(438\) −2.52483 −0.120641
\(439\) 14.8557 0.709023 0.354512 0.935052i \(-0.384647\pi\)
0.354512 + 0.935052i \(0.384647\pi\)
\(440\) 0 0
\(441\) −2.83457 −0.134980
\(442\) −3.11926 −0.148368
\(443\) 0.746173 0.0354517 0.0177259 0.999843i \(-0.494357\pi\)
0.0177259 + 0.999843i \(0.494357\pi\)
\(444\) −3.35197 −0.159078
\(445\) 1.18654 0.0562475
\(446\) −3.79234 −0.179573
\(447\) −9.06335 −0.428682
\(448\) 1.00000 0.0472456
\(449\) 9.83457 0.464122 0.232061 0.972701i \(-0.425453\pi\)
0.232061 + 0.972701i \(0.425453\pi\)
\(450\) −2.83457 −0.133623
\(451\) 0 0
\(452\) −14.0000 −0.658505
\(453\) −1.24246 −0.0583757
\(454\) 14.0422 0.659035
\(455\) −0.813457 −0.0381354
\(456\) 2.20766 0.103383
\(457\) −11.2961 −0.528407 −0.264204 0.964467i \(-0.585109\pi\)
−0.264204 + 0.964467i \(0.585109\pi\)
\(458\) 1.83457 0.0857239
\(459\) −9.09977 −0.424741
\(460\) 5.42784 0.253075
\(461\) 7.41926 0.345549 0.172775 0.984961i \(-0.444727\pi\)
0.172775 + 0.984961i \(0.444727\pi\)
\(462\) 0 0
\(463\) −10.1740 −0.472827 −0.236413 0.971653i \(-0.575972\pi\)
−0.236413 + 0.971653i \(0.575972\pi\)
\(464\) −6.61439 −0.307065
\(465\) 2.44037 0.113169
\(466\) 8.04223 0.372549
\(467\) −32.4912 −1.50351 −0.751756 0.659441i \(-0.770793\pi\)
−0.751756 + 0.659441i \(0.770793\pi\)
\(468\) −2.30580 −0.106586
\(469\) 8.00000 0.369406
\(470\) 5.02112 0.231607
\(471\) −1.97495 −0.0910007
\(472\) 10.6480 0.490115
\(473\) 0 0
\(474\) 6.95383 0.319400
\(475\) −5.42784 −0.249047
\(476\) −3.83457 −0.175757
\(477\) −18.7490 −0.858456
\(478\) 15.4701 0.707585
\(479\) 11.6691 0.533177 0.266588 0.963810i \(-0.414104\pi\)
0.266588 + 0.963810i \(0.414104\pi\)
\(480\) 0.406728 0.0185645
\(481\) 6.70394 0.305673
\(482\) −11.6355 −0.529983
\(483\) 2.20766 0.100452
\(484\) 0 0
\(485\) 3.42784 0.155650
\(486\) −10.1854 −0.462019
\(487\) −5.49513 −0.249008 −0.124504 0.992219i \(-0.539734\pi\)
−0.124504 + 0.992219i \(0.539734\pi\)
\(488\) −0.978885 −0.0443120
\(489\) 3.25383 0.147143
\(490\) −1.00000 −0.0451754
\(491\) 13.2961 0.600043 0.300021 0.953932i \(-0.403006\pi\)
0.300021 + 0.953932i \(0.403006\pi\)
\(492\) −2.27494 −0.102562
\(493\) 25.3633 1.14231
\(494\) −4.41532 −0.198654
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) 14.8557 0.666369
\(498\) −3.58468 −0.160634
\(499\) −24.6480 −1.10340 −0.551699 0.834044i \(-0.686020\pi\)
−0.551699 + 0.834044i \(0.686020\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 3.25383 0.145370
\(502\) 5.35197 0.238870
\(503\) −38.9230 −1.73549 −0.867745 0.497010i \(-0.834431\pi\)
−0.867745 + 0.497010i \(0.834431\pi\)
\(504\) −2.83457 −0.126262
\(505\) −18.6903 −0.831706
\(506\) 0 0
\(507\) 5.01833 0.222872
\(508\) −17.2961 −0.767388
\(509\) −39.0497 −1.73085 −0.865423 0.501042i \(-0.832950\pi\)
−0.865423 + 0.501042i \(0.832950\pi\)
\(510\) −1.55963 −0.0690616
\(511\) 6.20766 0.274611
\(512\) 1.00000 0.0441942
\(513\) −12.8807 −0.568699
\(514\) 12.7239 0.561227
\(515\) −15.4615 −0.681314
\(516\) −1.22877 −0.0540937
\(517\) 0 0
\(518\) 8.24130 0.362102
\(519\) −7.73643 −0.339592
\(520\) −0.813457 −0.0356724
\(521\) −33.2710 −1.45763 −0.728815 0.684711i \(-0.759928\pi\)
−0.728815 + 0.684711i \(0.759928\pi\)
\(522\) 18.7490 0.820619
\(523\) 40.2362 1.75941 0.879703 0.475523i \(-0.157741\pi\)
0.879703 + 0.475523i \(0.157741\pi\)
\(524\) −9.42784 −0.411857
\(525\) −0.406728 −0.0177511
\(526\) 1.62691 0.0709368
\(527\) −23.0074 −1.00222
\(528\) 0 0
\(529\) 6.46149 0.280934
\(530\) −6.61439 −0.287311
\(531\) −30.1826 −1.30981
\(532\) −5.42784 −0.235327
\(533\) 4.54989 0.197077
\(534\) 0.482601 0.0208842
\(535\) −11.0211 −0.476484
\(536\) 8.00000 0.345547
\(537\) −1.89049 −0.0815805
\(538\) 23.8768 1.02940
\(539\) 0 0
\(540\) −2.37309 −0.102121
\(541\) −26.2835 −1.13002 −0.565009 0.825085i \(-0.691127\pi\)
−0.565009 + 0.825085i \(0.691127\pi\)
\(542\) −2.44037 −0.104823
\(543\) 10.3058 0.442264
\(544\) −3.83457 −0.164406
\(545\) 16.2413 0.695701
\(546\) −0.330856 −0.0141593
\(547\) −13.6269 −0.582645 −0.291322 0.956625i \(-0.594095\pi\)
−0.291322 + 0.956625i \(0.594095\pi\)
\(548\) 13.6691 0.583917
\(549\) 2.77472 0.118422
\(550\) 0 0
\(551\) 35.9019 1.52947
\(552\) 2.20766 0.0939642
\(553\) −17.0970 −0.727038
\(554\) −22.0000 −0.934690
\(555\) 3.35197 0.142283
\(556\) 11.8682 0.503324
\(557\) −14.8979 −0.631245 −0.315623 0.948885i \(-0.602213\pi\)
−0.315623 + 0.948885i \(0.602213\pi\)
\(558\) −17.0074 −0.719982
\(559\) 2.45755 0.103943
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 20.8557 0.879744
\(563\) −39.7536 −1.67541 −0.837707 0.546119i \(-0.816104\pi\)
−0.837707 + 0.546119i \(0.816104\pi\)
\(564\) 2.04223 0.0859934
\(565\) 14.0000 0.588984
\(566\) −0.330856 −0.0139069
\(567\) 7.53851 0.316588
\(568\) 14.8557 0.623331
\(569\) −8.78840 −0.368429 −0.184215 0.982886i \(-0.558974\pi\)
−0.184215 + 0.982886i \(0.558974\pi\)
\(570\) −2.20766 −0.0924686
\(571\) 5.22877 0.218817 0.109409 0.993997i \(-0.465104\pi\)
0.109409 + 0.993997i \(0.465104\pi\)
\(572\) 0 0
\(573\) −6.37309 −0.266239
\(574\) 5.59327 0.233459
\(575\) −5.42784 −0.226357
\(576\) −2.83457 −0.118107
\(577\) 5.80093 0.241496 0.120748 0.992683i \(-0.461471\pi\)
0.120748 + 0.992683i \(0.461471\pi\)
\(578\) −2.29606 −0.0955034
\(579\) −0.880741 −0.0366024
\(580\) 6.61439 0.274647
\(581\) 8.81346 0.365644
\(582\) 1.39420 0.0577915
\(583\) 0 0
\(584\) 6.20766 0.256875
\(585\) 2.30580 0.0953332
\(586\) 3.18654 0.131635
\(587\) 25.3046 1.04443 0.522217 0.852812i \(-0.325105\pi\)
0.522217 + 0.852812i \(0.325105\pi\)
\(588\) −0.406728 −0.0167732
\(589\) −32.5671 −1.34190
\(590\) −10.6480 −0.438372
\(591\) 1.14431 0.0470707
\(592\) 8.24130 0.338715
\(593\) 23.1729 0.951595 0.475798 0.879555i \(-0.342159\pi\)
0.475798 + 0.879555i \(0.342159\pi\)
\(594\) 0 0
\(595\) 3.83457 0.157202
\(596\) 22.2835 0.912769
\(597\) 5.29606 0.216753
\(598\) −4.41532 −0.180556
\(599\) −8.48260 −0.346590 −0.173295 0.984870i \(-0.555441\pi\)
−0.173295 + 0.984870i \(0.555441\pi\)
\(600\) −0.406728 −0.0166046
\(601\) 32.4490 1.32362 0.661810 0.749671i \(-0.269788\pi\)
0.661810 + 0.749671i \(0.269788\pi\)
\(602\) 3.02112 0.123131
\(603\) −22.6766 −0.923462
\(604\) 3.05476 0.124296
\(605\) 0 0
\(606\) −7.60186 −0.308804
\(607\) 33.9441 1.37775 0.688874 0.724881i \(-0.258105\pi\)
0.688874 + 0.724881i \(0.258105\pi\)
\(608\) −5.42784 −0.220128
\(609\) 2.69026 0.109015
\(610\) 0.978885 0.0396339
\(611\) −4.08446 −0.165240
\(612\) 10.8694 0.439368
\(613\) −8.77123 −0.354267 −0.177133 0.984187i \(-0.556682\pi\)
−0.177133 + 0.984187i \(0.556682\pi\)
\(614\) −20.4826 −0.826610
\(615\) 2.27494 0.0917345
\(616\) 0 0
\(617\) −32.1095 −1.29268 −0.646340 0.763049i \(-0.723701\pi\)
−0.646340 + 0.763049i \(0.723701\pi\)
\(618\) −6.28863 −0.252966
\(619\) −28.2749 −1.13647 −0.568233 0.822868i \(-0.692373\pi\)
−0.568233 + 0.822868i \(0.692373\pi\)
\(620\) −6.00000 −0.240966
\(621\) −12.8807 −0.516886
\(622\) 3.62691 0.145426
\(623\) −1.18654 −0.0475378
\(624\) −0.330856 −0.0132448
\(625\) 1.00000 0.0400000
\(626\) 20.7239 0.828294
\(627\) 0 0
\(628\) 4.85569 0.193763
\(629\) −31.6019 −1.26005
\(630\) 2.83457 0.112932
\(631\) 34.9230 1.39026 0.695131 0.718883i \(-0.255346\pi\)
0.695131 + 0.718883i \(0.255346\pi\)
\(632\) −17.0970 −0.680082
\(633\) 8.00000 0.317971
\(634\) −1.80093 −0.0715241
\(635\) 17.2961 0.686373
\(636\) −2.69026 −0.106676
\(637\) 0.813457 0.0322303
\(638\) 0 0
\(639\) −42.1095 −1.66583
\(640\) −1.00000 −0.0395285
\(641\) −5.25383 −0.207514 −0.103757 0.994603i \(-0.533086\pi\)
−0.103757 + 0.994603i \(0.533086\pi\)
\(642\) −4.48260 −0.176914
\(643\) −3.17796 −0.125326 −0.0626632 0.998035i \(-0.519959\pi\)
−0.0626632 + 0.998035i \(0.519959\pi\)
\(644\) −5.42784 −0.213887
\(645\) 1.22877 0.0483829
\(646\) 20.8135 0.818895
\(647\) 37.9190 1.49075 0.745375 0.666645i \(-0.232270\pi\)
0.745375 + 0.666645i \(0.232270\pi\)
\(648\) 7.53851 0.296141
\(649\) 0 0
\(650\) 0.813457 0.0319064
\(651\) −2.44037 −0.0956457
\(652\) −8.00000 −0.313304
\(653\) 29.5374 1.15589 0.577943 0.816077i \(-0.303856\pi\)
0.577943 + 0.816077i \(0.303856\pi\)
\(654\) 6.60580 0.258307
\(655\) 9.42784 0.368376
\(656\) 5.59327 0.218381
\(657\) −17.5961 −0.686487
\(658\) −5.02112 −0.195743
\(659\) −15.1865 −0.591584 −0.295792 0.955252i \(-0.595583\pi\)
−0.295792 + 0.955252i \(0.595583\pi\)
\(660\) 0 0
\(661\) −44.6766 −1.73772 −0.868859 0.495060i \(-0.835146\pi\)
−0.868859 + 0.495060i \(0.835146\pi\)
\(662\) −3.02112 −0.117419
\(663\) 1.26869 0.0492719
\(664\) 8.81346 0.342028
\(665\) 5.42784 0.210483
\(666\) −23.3606 −0.905203
\(667\) 35.9019 1.39013
\(668\) −8.00000 −0.309529
\(669\) 1.54245 0.0596347
\(670\) −8.00000 −0.309067
\(671\) 0 0
\(672\) −0.406728 −0.0156899
\(673\) −38.0000 −1.46479 −0.732396 0.680879i \(-0.761598\pi\)
−0.732396 + 0.680879i \(0.761598\pi\)
\(674\) 9.66914 0.372442
\(675\) 2.37309 0.0913401
\(676\) −12.3383 −0.474550
\(677\) 48.7325 1.87294 0.936471 0.350745i \(-0.114072\pi\)
0.936471 + 0.350745i \(0.114072\pi\)
\(678\) 5.69420 0.218684
\(679\) −3.42784 −0.131549
\(680\) 3.83457 0.147049
\(681\) −5.71137 −0.218860
\(682\) 0 0
\(683\) −0.482601 −0.0184662 −0.00923310 0.999957i \(-0.502939\pi\)
−0.00923310 + 0.999957i \(0.502939\pi\)
\(684\) 15.3856 0.588284
\(685\) −13.6691 −0.522271
\(686\) 1.00000 0.0381802
\(687\) −0.746173 −0.0284683
\(688\) 3.02112 0.115179
\(689\) 5.38052 0.204981
\(690\) −2.20766 −0.0840441
\(691\) −25.5037 −0.970207 −0.485104 0.874457i \(-0.661218\pi\)
−0.485104 + 0.874457i \(0.661218\pi\)
\(692\) 19.0211 0.723074
\(693\) 0 0
\(694\) −9.39420 −0.356599
\(695\) −11.8682 −0.450187
\(696\) 2.69026 0.101974
\(697\) −21.4478 −0.812393
\(698\) 18.2077 0.689170
\(699\) −3.27100 −0.123721
\(700\) 1.00000 0.0377964
\(701\) −33.8682 −1.27918 −0.639592 0.768714i \(-0.720897\pi\)
−0.639592 + 0.768714i \(0.720897\pi\)
\(702\) 1.93040 0.0728584
\(703\) −44.7325 −1.68712
\(704\) 0 0
\(705\) −2.04223 −0.0769148
\(706\) 12.7239 0.478870
\(707\) 18.6903 0.702920
\(708\) −4.33086 −0.162764
\(709\) −15.9749 −0.599952 −0.299976 0.953947i \(-0.596979\pi\)
−0.299976 + 0.953947i \(0.596979\pi\)
\(710\) −14.8557 −0.557524
\(711\) 48.4626 1.81749
\(712\) −1.18654 −0.0444676
\(713\) −32.5671 −1.21965
\(714\) 1.55963 0.0583677
\(715\) 0 0
\(716\) 4.64803 0.173705
\(717\) −6.29212 −0.234983
\(718\) −29.8432 −1.11374
\(719\) −13.4364 −0.501094 −0.250547 0.968104i \(-0.580611\pi\)
−0.250547 + 0.968104i \(0.580611\pi\)
\(720\) 2.83457 0.105638
\(721\) 15.4615 0.575816
\(722\) 10.4615 0.389336
\(723\) 4.73249 0.176003
\(724\) −25.3383 −0.941690
\(725\) −6.61439 −0.245652
\(726\) 0 0
\(727\) 6.64803 0.246562 0.123281 0.992372i \(-0.460658\pi\)
0.123281 + 0.992372i \(0.460658\pi\)
\(728\) 0.813457 0.0301487
\(729\) −18.4729 −0.684180
\(730\) −6.20766 −0.229756
\(731\) −11.5847 −0.428475
\(732\) 0.398140 0.0147157
\(733\) −38.0285 −1.40462 −0.702308 0.711873i \(-0.747847\pi\)
−0.702308 + 0.711873i \(0.747847\pi\)
\(734\) 15.4615 0.570694
\(735\) 0.406728 0.0150024
\(736\) −5.42784 −0.200073
\(737\) 0 0
\(738\) −15.8545 −0.583613
\(739\) 52.6343 1.93619 0.968093 0.250592i \(-0.0806252\pi\)
0.968093 + 0.250592i \(0.0806252\pi\)
\(740\) −8.24130 −0.302956
\(741\) 1.79583 0.0659716
\(742\) 6.61439 0.242822
\(743\) −28.0845 −1.03032 −0.515159 0.857094i \(-0.672267\pi\)
−0.515159 + 0.857094i \(0.672267\pi\)
\(744\) −2.44037 −0.0894683
\(745\) −22.2835 −0.816405
\(746\) 8.10951 0.296910
\(747\) −24.9824 −0.914057
\(748\) 0 0
\(749\) 11.0211 0.402703
\(750\) 0.406728 0.0148516
\(751\) 48.4826 1.76916 0.884578 0.466393i \(-0.154447\pi\)
0.884578 + 0.466393i \(0.154447\pi\)
\(752\) −5.02112 −0.183101
\(753\) −2.17680 −0.0793270
\(754\) −5.38052 −0.195947
\(755\) −3.05476 −0.111174
\(756\) 2.37309 0.0863083
\(757\) −11.8260 −0.429823 −0.214911 0.976634i \(-0.568946\pi\)
−0.214911 + 0.976634i \(0.568946\pi\)
\(758\) 22.0422 0.800610
\(759\) 0 0
\(760\) 5.42784 0.196889
\(761\) 19.5510 0.708725 0.354362 0.935108i \(-0.384698\pi\)
0.354362 + 0.935108i \(0.384698\pi\)
\(762\) 7.03480 0.254844
\(763\) −16.2413 −0.587975
\(764\) 15.6691 0.566890
\(765\) −10.8694 −0.392983
\(766\) 38.2499 1.38202
\(767\) 8.66171 0.312756
\(768\) −0.406728 −0.0146765
\(769\) −6.07587 −0.219102 −0.109551 0.993981i \(-0.534941\pi\)
−0.109551 + 0.993981i \(0.534941\pi\)
\(770\) 0 0
\(771\) −5.17517 −0.186379
\(772\) 2.16543 0.0779355
\(773\) −2.33086 −0.0838351 −0.0419175 0.999121i \(-0.513347\pi\)
−0.0419175 + 0.999121i \(0.513347\pi\)
\(774\) −8.56357 −0.307811
\(775\) 6.00000 0.215526
\(776\) −3.42784 −0.123052
\(777\) −3.35197 −0.120251
\(778\) 16.8557 0.604306
\(779\) −30.3594 −1.08774
\(780\) 0.330856 0.0118465
\(781\) 0 0
\(782\) 20.8135 0.744288
\(783\) −15.6965 −0.560948
\(784\) 1.00000 0.0357143
\(785\) −4.85569 −0.173307
\(786\) 3.83457 0.136775
\(787\) 17.3805 0.619549 0.309774 0.950810i \(-0.399747\pi\)
0.309774 + 0.950810i \(0.399747\pi\)
\(788\) −2.81346 −0.100225
\(789\) −0.661712 −0.0235576
\(790\) 17.0970 0.608284
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) −0.796281 −0.0282768
\(794\) −4.44037 −0.157583
\(795\) 2.69026 0.0954137
\(796\) −13.0211 −0.461521
\(797\) −15.6942 −0.555917 −0.277959 0.960593i \(-0.589658\pi\)
−0.277959 + 0.960593i \(0.589658\pi\)
\(798\) 2.20766 0.0781503
\(799\) 19.2538 0.681151
\(800\) 1.00000 0.0353553
\(801\) 3.36334 0.118838
\(802\) −3.46149 −0.122229
\(803\) 0 0
\(804\) −3.25383 −0.114754
\(805\) 5.42784 0.191306
\(806\) 4.88074 0.171917
\(807\) −9.71137 −0.341857
\(808\) 18.6903 0.657521
\(809\) −23.1443 −0.813711 −0.406855 0.913493i \(-0.633375\pi\)
−0.406855 + 0.913493i \(0.633375\pi\)
\(810\) −7.53851 −0.264876
\(811\) 27.6218 0.969933 0.484967 0.874533i \(-0.338832\pi\)
0.484967 + 0.874533i \(0.338832\pi\)
\(812\) −6.61439 −0.232119
\(813\) 0.992568 0.0348109
\(814\) 0 0
\(815\) 8.00000 0.280228
\(816\) 1.55963 0.0545980
\(817\) −16.3981 −0.573698
\(818\) 17.2624 0.603566
\(819\) −2.30580 −0.0805712
\(820\) −5.59327 −0.195326
\(821\) 39.5123 1.37899 0.689494 0.724291i \(-0.257833\pi\)
0.689494 + 0.724291i \(0.257833\pi\)
\(822\) −5.55963 −0.193914
\(823\) 13.5123 0.471009 0.235505 0.971873i \(-0.424326\pi\)
0.235505 + 0.971873i \(0.424326\pi\)
\(824\) 15.4615 0.538626
\(825\) 0 0
\(826\) 10.6480 0.370492
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 15.3856 0.534687
\(829\) 13.8346 0.480495 0.240247 0.970712i \(-0.422771\pi\)
0.240247 + 0.970712i \(0.422771\pi\)
\(830\) −8.81346 −0.305920
\(831\) 8.94803 0.310404
\(832\) 0.813457 0.0282015
\(833\) −3.83457 −0.132860
\(834\) −4.82714 −0.167150
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) 14.2385 0.492155
\(838\) 9.02112 0.311629
\(839\) 44.0422 1.52051 0.760253 0.649627i \(-0.225075\pi\)
0.760253 + 0.649627i \(0.225075\pi\)
\(840\) 0.406728 0.0140335
\(841\) 14.7501 0.508625
\(842\) 31.7114 1.09285
\(843\) −8.48260 −0.292156
\(844\) −19.6691 −0.677040
\(845\) 12.3383 0.424450
\(846\) 14.2327 0.489331
\(847\) 0 0
\(848\) 6.61439 0.227139
\(849\) 0.134569 0.00461838
\(850\) −3.83457 −0.131525
\(851\) −44.7325 −1.53341
\(852\) −6.04223 −0.207003
\(853\) 29.8095 1.02066 0.510329 0.859979i \(-0.329524\pi\)
0.510329 + 0.859979i \(0.329524\pi\)
\(854\) −0.978885 −0.0334968
\(855\) −15.3856 −0.526177
\(856\) 11.0211 0.376694
\(857\) −42.7575 −1.46057 −0.730285 0.683143i \(-0.760613\pi\)
−0.730285 + 0.683143i \(0.760613\pi\)
\(858\) 0 0
\(859\) 6.31717 0.215539 0.107770 0.994176i \(-0.465629\pi\)
0.107770 + 0.994176i \(0.465629\pi\)
\(860\) −3.02112 −0.103019
\(861\) −2.27494 −0.0775298
\(862\) 3.05476 0.104045
\(863\) −48.3680 −1.64647 −0.823233 0.567704i \(-0.807832\pi\)
−0.823233 + 0.567704i \(0.807832\pi\)
\(864\) 2.37309 0.0807340
\(865\) −19.0211 −0.646737
\(866\) 27.9104 0.948436
\(867\) 0.933872 0.0317160
\(868\) 6.00000 0.203653
\(869\) 0 0
\(870\) −2.69026 −0.0912083
\(871\) 6.50765 0.220503
\(872\) −16.2413 −0.550000
\(873\) 9.71647 0.328853
\(874\) 29.4615 0.996550
\(875\) −1.00000 −0.0338062
\(876\) −2.52483 −0.0853061
\(877\) 24.1940 0.816972 0.408486 0.912764i \(-0.366057\pi\)
0.408486 + 0.912764i \(0.366057\pi\)
\(878\) 14.8557 0.501355
\(879\) −1.29606 −0.0437149
\(880\) 0 0
\(881\) 12.3731 0.416860 0.208430 0.978037i \(-0.433165\pi\)
0.208430 + 0.978037i \(0.433165\pi\)
\(882\) −2.83457 −0.0954450
\(883\) −14.5248 −0.488799 −0.244400 0.969675i \(-0.578591\pi\)
−0.244400 + 0.969675i \(0.578591\pi\)
\(884\) −3.11926 −0.104912
\(885\) 4.33086 0.145580
\(886\) 0.746173 0.0250682
\(887\) 25.3805 0.852194 0.426097 0.904677i \(-0.359888\pi\)
0.426097 + 0.904677i \(0.359888\pi\)
\(888\) −3.35197 −0.112485
\(889\) −17.2961 −0.580091
\(890\) 1.18654 0.0397730
\(891\) 0 0
\(892\) −3.79234 −0.126977
\(893\) 27.2538 0.912015
\(894\) −9.06335 −0.303124
\(895\) −4.64803 −0.155366
\(896\) 1.00000 0.0334077
\(897\) 1.79583 0.0599612
\(898\) 9.83457 0.328184
\(899\) −39.6863 −1.32361
\(900\) −2.83457 −0.0944857
\(901\) −25.3633 −0.844975
\(902\) 0 0
\(903\) −1.22877 −0.0408910
\(904\) −14.0000 −0.465633
\(905\) 25.3383 0.842273
\(906\) −1.24246 −0.0412779
\(907\) −7.93272 −0.263402 −0.131701 0.991290i \(-0.542044\pi\)
−0.131701 + 0.991290i \(0.542044\pi\)
\(908\) 14.0422 0.466008
\(909\) −52.9789 −1.75720
\(910\) −0.813457 −0.0269658
\(911\) 43.7364 1.44905 0.724526 0.689247i \(-0.242059\pi\)
0.724526 + 0.689247i \(0.242059\pi\)
\(912\) 2.20766 0.0731029
\(913\) 0 0
\(914\) −11.2961 −0.373640
\(915\) −0.398140 −0.0131621
\(916\) 1.83457 0.0606160
\(917\) −9.42784 −0.311335
\(918\) −9.09977 −0.300337
\(919\) 32.7661 1.08085 0.540427 0.841391i \(-0.318263\pi\)
0.540427 + 0.841391i \(0.318263\pi\)
\(920\) 5.42784 0.178951
\(921\) 8.33086 0.274511
\(922\) 7.41926 0.244340
\(923\) 12.0845 0.397765
\(924\) 0 0
\(925\) 8.24130 0.270972
\(926\) −10.1740 −0.334339
\(927\) −43.8267 −1.43946
\(928\) −6.61439 −0.217128
\(929\) 59.4478 1.95042 0.975210 0.221283i \(-0.0710246\pi\)
0.975210 + 0.221283i \(0.0710246\pi\)
\(930\) 2.44037 0.0800229
\(931\) −5.42784 −0.177890
\(932\) 8.04223 0.263432
\(933\) −1.47517 −0.0482949
\(934\) −32.4912 −1.06314
\(935\) 0 0
\(936\) −2.30580 −0.0753675
\(937\) −53.2824 −1.74066 −0.870330 0.492470i \(-0.836094\pi\)
−0.870330 + 0.492470i \(0.836094\pi\)
\(938\) 8.00000 0.261209
\(939\) −8.42900 −0.275070
\(940\) 5.02112 0.163771
\(941\) 5.72506 0.186632 0.0933158 0.995637i \(-0.470253\pi\)
0.0933158 + 0.995637i \(0.470253\pi\)
\(942\) −1.97495 −0.0643472
\(943\) −30.3594 −0.988638
\(944\) 10.6480 0.346564
\(945\) −2.37309 −0.0771965
\(946\) 0 0
\(947\) −27.4056 −0.890561 −0.445281 0.895391i \(-0.646896\pi\)
−0.445281 + 0.895391i \(0.646896\pi\)
\(948\) 6.95383 0.225850
\(949\) 5.04966 0.163919
\(950\) −5.42784 −0.176102
\(951\) 0.732489 0.0237526
\(952\) −3.83457 −0.124279
\(953\) 10.7998 0.349839 0.174919 0.984583i \(-0.444033\pi\)
0.174919 + 0.984583i \(0.444033\pi\)
\(954\) −18.7490 −0.607020
\(955\) −15.6691 −0.507042
\(956\) 15.4701 0.500338
\(957\) 0 0
\(958\) 11.6691 0.377013
\(959\) 13.6691 0.441400
\(960\) 0.406728 0.0131271
\(961\) 5.00000 0.161290
\(962\) 6.70394 0.216144
\(963\) −31.2401 −1.00670
\(964\) −11.6355 −0.374754
\(965\) −2.16543 −0.0697076
\(966\) 2.20766 0.0710302
\(967\) −11.2538 −0.361899 −0.180949 0.983492i \(-0.557917\pi\)
−0.180949 + 0.983492i \(0.557917\pi\)
\(968\) 0 0
\(969\) −8.46542 −0.271949
\(970\) 3.42784 0.110061
\(971\) 33.2401 1.06673 0.533363 0.845886i \(-0.320928\pi\)
0.533363 + 0.845886i \(0.320928\pi\)
\(972\) −10.1854 −0.326696
\(973\) 11.8682 0.380477
\(974\) −5.49513 −0.176075
\(975\) −0.330856 −0.0105959
\(976\) −0.978885 −0.0313333
\(977\) −2.74617 −0.0878578 −0.0439289 0.999035i \(-0.513988\pi\)
−0.0439289 + 0.999035i \(0.513988\pi\)
\(978\) 3.25383 0.104046
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 46.0371 1.46985
\(982\) 13.2961 0.424294
\(983\) 39.4615 1.25863 0.629313 0.777152i \(-0.283336\pi\)
0.629313 + 0.777152i \(0.283336\pi\)
\(984\) −2.27494 −0.0725225
\(985\) 2.81346 0.0896442
\(986\) 25.3633 0.807733
\(987\) 2.04223 0.0650049
\(988\) −4.41532 −0.140470
\(989\) −16.3981 −0.521431
\(990\) 0 0
\(991\) 20.9652 0.665982 0.332991 0.942930i \(-0.391942\pi\)
0.332991 + 0.942930i \(0.391942\pi\)
\(992\) 6.00000 0.190500
\(993\) 1.22877 0.0389939
\(994\) 14.8557 0.471194
\(995\) 13.0211 0.412797
\(996\) −3.58468 −0.113585
\(997\) −21.9441 −0.694976 −0.347488 0.937684i \(-0.612965\pi\)
−0.347488 + 0.937684i \(0.612965\pi\)
\(998\) −24.6480 −0.780220
\(999\) 19.5573 0.618766
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 8470.2.a.cl.1.2 3
11.10 odd 2 770.2.a.l.1.2 3
33.32 even 2 6930.2.a.cl.1.2 3
44.43 even 2 6160.2.a.bi.1.2 3
55.32 even 4 3850.2.c.z.1849.2 6
55.43 even 4 3850.2.c.z.1849.5 6
55.54 odd 2 3850.2.a.bu.1.2 3
77.76 even 2 5390.2.a.bz.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.l.1.2 3 11.10 odd 2
3850.2.a.bu.1.2 3 55.54 odd 2
3850.2.c.z.1849.2 6 55.32 even 4
3850.2.c.z.1849.5 6 55.43 even 4
5390.2.a.bz.1.2 3 77.76 even 2
6160.2.a.bi.1.2 3 44.43 even 2
6930.2.a.cl.1.2 3 33.32 even 2
8470.2.a.cl.1.2 3 1.1 even 1 trivial