Properties

Label 8470.2.a.cl.1.3
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 1.3
Root \(2.59774\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.34596 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.34596 q^{6} +1.00000 q^{7} +1.00000 q^{8} +8.19547 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +3.34596 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.34596 q^{6} +1.00000 q^{7} +1.00000 q^{8} +8.19547 q^{9} -1.00000 q^{10} +3.34596 q^{12} -6.69193 q^{13} +1.00000 q^{14} -3.34596 q^{15} +1.00000 q^{16} +7.19547 q^{17} +8.19547 q^{18} +1.84951 q^{19} -1.00000 q^{20} +3.34596 q^{21} +1.84951 q^{23} +3.34596 q^{24} +1.00000 q^{25} -6.69193 q^{26} +17.3839 q^{27} +1.00000 q^{28} -6.84242 q^{29} -3.34596 q^{30} +6.00000 q^{31} +1.00000 q^{32} +7.19547 q^{34} -1.00000 q^{35} +8.19547 q^{36} -6.54143 q^{37} +1.84951 q^{38} -22.3909 q^{39} -1.00000 q^{40} +9.34596 q^{41} +3.34596 q^{42} -0.503544 q^{43} -8.19547 q^{45} +1.84951 q^{46} -1.49646 q^{47} +3.34596 q^{48} +1.00000 q^{49} +1.00000 q^{50} +24.0758 q^{51} -6.69193 q^{52} +6.84242 q^{53} +17.3839 q^{54} +1.00000 q^{56} +6.18838 q^{57} -6.84242 q^{58} -7.88740 q^{59} -3.34596 q^{60} -4.50354 q^{61} +6.00000 q^{62} +8.19547 q^{63} +1.00000 q^{64} +6.69193 q^{65} +8.00000 q^{67} +7.19547 q^{68} +6.18838 q^{69} -1.00000 q^{70} +0.300986 q^{71} +8.19547 q^{72} +10.1884 q^{73} -6.54143 q^{74} +3.34596 q^{75} +1.84951 q^{76} -22.3909 q^{78} +12.2404 q^{79} -1.00000 q^{80} +33.5793 q^{81} +9.34596 q^{82} +1.30807 q^{83} +3.34596 q^{84} -7.19547 q^{85} -0.503544 q^{86} -22.8945 q^{87} -8.69193 q^{89} -8.19547 q^{90} -6.69193 q^{91} +1.84951 q^{92} +20.0758 q^{93} -1.49646 q^{94} -1.84951 q^{95} +3.34596 q^{96} +3.84951 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{8} + 11 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{8} + 11 q^{9} - 3 q^{10} + 3 q^{14} + 3 q^{16} + 8 q^{17} + 11 q^{18} + 2 q^{19} - 3 q^{20} + 2 q^{23} + 3 q^{25} + 12 q^{27} + 3 q^{28} - 4 q^{29} + 18 q^{31} + 3 q^{32} + 8 q^{34} - 3 q^{35} + 11 q^{36} + 4 q^{37} + 2 q^{38} - 40 q^{39} - 3 q^{40} + 18 q^{41} - 8 q^{43} - 11 q^{45} + 2 q^{46} + 2 q^{47} + 3 q^{49} + 3 q^{50} + 12 q^{51} + 4 q^{53} + 12 q^{54} + 3 q^{56} - 8 q^{57} - 4 q^{58} + 10 q^{59} - 20 q^{61} + 18 q^{62} + 11 q^{63} + 3 q^{64} + 24 q^{67} + 8 q^{68} - 8 q^{69} - 3 q^{70} + 8 q^{71} + 11 q^{72} + 4 q^{73} + 4 q^{74} + 2 q^{76} - 40 q^{78} + 6 q^{79} - 3 q^{80} + 47 q^{81} + 18 q^{82} + 24 q^{83} - 8 q^{85} - 8 q^{86} - 48 q^{87} - 6 q^{89} - 11 q^{90} + 2 q^{92} + 2 q^{94} - 2 q^{95} + 8 q^{97} + 3 q^{98} + O(q^{100}) \)

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
\(3\) 3.34596 1.93179 0.965896 0.258929i \(-0.0833695\pi\)
0.965896 + 0.258929i \(0.0833695\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 3.34596 1.36598
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 8.19547 2.73182
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 3.34596 0.965896
\(13\) −6.69193 −1.85601 −0.928003 0.372572i \(-0.878476\pi\)
−0.928003 + 0.372572i \(0.878476\pi\)
\(14\) 1.00000 0.267261
\(15\) −3.34596 −0.863924
\(16\) 1.00000 0.250000
\(17\) 7.19547 1.74516 0.872579 0.488473i \(-0.162446\pi\)
0.872579 + 0.488473i \(0.162446\pi\)
\(18\) 8.19547 1.93169
\(19\) 1.84951 0.424306 0.212153 0.977236i \(-0.431952\pi\)
0.212153 + 0.977236i \(0.431952\pi\)
\(20\) −1.00000 −0.223607
\(21\) 3.34596 0.730149
\(22\) 0 0
\(23\) 1.84951 0.385649 0.192824 0.981233i \(-0.438235\pi\)
0.192824 + 0.981233i \(0.438235\pi\)
\(24\) 3.34596 0.682992
\(25\) 1.00000 0.200000
\(26\) −6.69193 −1.31239
\(27\) 17.3839 3.34552
\(28\) 1.00000 0.188982
\(29\) −6.84242 −1.27061 −0.635303 0.772263i \(-0.719125\pi\)
−0.635303 + 0.772263i \(0.719125\pi\)
\(30\) −3.34596 −0.610887
\(31\) 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
\(34\) 7.19547 1.23401
\(35\) −1.00000 −0.169031
\(36\) 8.19547 1.36591
\(37\) −6.54143 −1.07541 −0.537703 0.843135i \(-0.680708\pi\)
−0.537703 + 0.843135i \(0.680708\pi\)
\(38\) 1.84951 0.300030
\(39\) −22.3909 −3.58542
\(40\) −1.00000 −0.158114
\(41\) 9.34596 1.45959 0.729797 0.683664i \(-0.239615\pi\)
0.729797 + 0.683664i \(0.239615\pi\)
\(42\) 3.34596 0.516293
\(43\) −0.503544 −0.0767897 −0.0383949 0.999263i \(-0.512224\pi\)
−0.0383949 + 0.999263i \(0.512224\pi\)
\(44\) 0 0
\(45\) −8.19547 −1.22171
\(46\) 1.84951 0.272695
\(47\) −1.49646 −0.218281 −0.109140 0.994026i \(-0.534810\pi\)
−0.109140 + 0.994026i \(0.534810\pi\)
\(48\) 3.34596 0.482948
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 24.0758 3.37128
\(52\) −6.69193 −0.928003
\(53\) 6.84242 0.939879 0.469939 0.882699i \(-0.344276\pi\)
0.469939 + 0.882699i \(0.344276\pi\)
\(54\) 17.3839 2.36564
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 6.18838 0.819671
\(58\) −6.84242 −0.898454
\(59\) −7.88740 −1.02685 −0.513426 0.858134i \(-0.671624\pi\)
−0.513426 + 0.858134i \(0.671624\pi\)
\(60\) −3.34596 −0.431962
\(61\) −4.50354 −0.576620 −0.288310 0.957537i \(-0.593093\pi\)
−0.288310 + 0.957537i \(0.593093\pi\)
\(62\) 6.00000 0.762001
\(63\) 8.19547 1.03253
\(64\) 1.00000 0.125000
\(65\) 6.69193 0.830031
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 7.19547 0.872579
\(69\) 6.18838 0.744994
\(70\) −1.00000 −0.119523
\(71\) 0.300986 0.0357204 0.0178602 0.999840i \(-0.494315\pi\)
0.0178602 + 0.999840i \(0.494315\pi\)
\(72\) 8.19547 0.965845
\(73\) 10.1884 1.19246 0.596230 0.802814i \(-0.296665\pi\)
0.596230 + 0.802814i \(0.296665\pi\)
\(74\) −6.54143 −0.760426
\(75\) 3.34596 0.386359
\(76\) 1.84951 0.212153
\(77\) 0 0
\(78\) −22.3909 −2.53527
\(79\) 12.2404 1.37716 0.688579 0.725161i \(-0.258235\pi\)
0.688579 + 0.725161i \(0.258235\pi\)
\(80\) −1.00000 −0.111803
\(81\) 33.5793 3.73104
\(82\) 9.34596 1.03209
\(83\) 1.30807 0.143580 0.0717899 0.997420i \(-0.477129\pi\)
0.0717899 + 0.997420i \(0.477129\pi\)
\(84\) 3.34596 0.365075
\(85\) −7.19547 −0.780458
\(86\) −0.503544 −0.0542985
\(87\) −22.8945 −2.45455
\(88\) 0 0
\(89\) −8.69193 −0.921342 −0.460671 0.887571i \(-0.652391\pi\)
−0.460671 + 0.887571i \(0.652391\pi\)
\(90\) −8.19547 −0.863878
\(91\) −6.69193 −0.701505
\(92\) 1.84951 0.192824
\(93\) 20.0758 2.08176
\(94\) −1.49646 −0.154348
\(95\) −1.84951 −0.189755
\(96\) 3.34596 0.341496
\(97\) 3.84951 0.390858 0.195429 0.980718i \(-0.437390\pi\)
0.195429 + 0.980718i \(0.437390\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −6.89448 −0.686027 −0.343013 0.939331i \(-0.611448\pi\)
−0.343013 + 0.939331i \(0.611448\pi\)
\(102\) 24.0758 2.38386
\(103\) −10.5793 −1.04241 −0.521206 0.853431i \(-0.674518\pi\)
−0.521206 + 0.853431i \(0.674518\pi\)
\(104\) −6.69193 −0.656197
\(105\) −3.34596 −0.326533
\(106\) 6.84242 0.664595
\(107\) 7.49646 0.724710 0.362355 0.932040i \(-0.381973\pi\)
0.362355 + 0.932040i \(0.381973\pi\)
\(108\) 17.3839 1.67276
\(109\) −1.45857 −0.139705 −0.0698527 0.997557i \(-0.522253\pi\)
−0.0698527 + 0.997557i \(0.522253\pi\)
\(110\) 0 0
\(111\) −21.8874 −2.07746
\(112\) 1.00000 0.0944911
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 6.18838 0.579595
\(115\) −1.84951 −0.172467
\(116\) −6.84242 −0.635303
\(117\) −54.8435 −5.07028
\(118\) −7.88740 −0.726094
\(119\) 7.19547 0.659608
\(120\) −3.34596 −0.305443
\(121\) 0 0
\(122\) −4.50354 −0.407732
\(123\) 31.2713 2.81963
\(124\) 6.00000 0.538816
\(125\) −1.00000 −0.0894427
\(126\) 8.19547 0.730111
\(127\) 19.7748 1.75473 0.877365 0.479824i \(-0.159300\pi\)
0.877365 + 0.479824i \(0.159300\pi\)
\(128\) 1.00000 0.0883883
\(129\) −1.68484 −0.148342
\(130\) 6.69193 0.586921
\(131\) −2.15049 −0.187889 −0.0939447 0.995577i \(-0.529948\pi\)
−0.0939447 + 0.995577i \(0.529948\pi\)
\(132\) 0 0
\(133\) 1.84951 0.160373
\(134\) 8.00000 0.691095
\(135\) −17.3839 −1.49616
\(136\) 7.19547 0.617006
\(137\) −8.39094 −0.716886 −0.358443 0.933552i \(-0.616692\pi\)
−0.358443 + 0.933552i \(0.616692\pi\)
\(138\) 6.18838 0.526790
\(139\) −17.9253 −1.52040 −0.760201 0.649687i \(-0.774900\pi\)
−0.760201 + 0.649687i \(0.774900\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −5.00709 −0.421673
\(142\) 0.300986 0.0252582
\(143\) 0 0
\(144\) 8.19547 0.682956
\(145\) 6.84242 0.568232
\(146\) 10.1884 0.843197
\(147\) 3.34596 0.275970
\(148\) −6.54143 −0.537703
\(149\) 0.451479 0.0369866 0.0184933 0.999829i \(-0.494113\pi\)
0.0184933 + 0.999829i \(0.494113\pi\)
\(150\) 3.34596 0.273197
\(151\) −19.2334 −1.56519 −0.782594 0.622532i \(-0.786104\pi\)
−0.782594 + 0.622532i \(0.786104\pi\)
\(152\) 1.84951 0.150015
\(153\) 58.9703 4.76746
\(154\) 0 0
\(155\) −6.00000 −0.481932
\(156\) −22.3909 −1.79271
\(157\) −9.69901 −0.774066 −0.387033 0.922066i \(-0.626500\pi\)
−0.387033 + 0.922066i \(0.626500\pi\)
\(158\) 12.2404 0.973798
\(159\) 22.8945 1.81565
\(160\) −1.00000 −0.0790569
\(161\) 1.84951 0.145762
\(162\) 33.5793 2.63824
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 9.34596 0.729797
\(165\) 0 0
\(166\) 1.30807 0.101526
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 3.34596 0.258147
\(169\) 31.7819 2.44476
\(170\) −7.19547 −0.551867
\(171\) 15.1576 1.15913
\(172\) −0.503544 −0.0383949
\(173\) 15.4965 1.17817 0.589087 0.808070i \(-0.299488\pi\)
0.589087 + 0.808070i \(0.299488\pi\)
\(174\) −22.8945 −1.73563
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −26.3909 −1.98366
\(178\) −8.69193 −0.651487
\(179\) −13.8874 −1.03799 −0.518996 0.854776i \(-0.673694\pi\)
−0.518996 + 0.854776i \(0.673694\pi\)
\(180\) −8.19547 −0.610854
\(181\) 18.7819 1.39605 0.698023 0.716075i \(-0.254063\pi\)
0.698023 + 0.716075i \(0.254063\pi\)
\(182\) −6.69193 −0.496039
\(183\) −15.0687 −1.11391
\(184\) 1.84951 0.136347
\(185\) 6.54143 0.480936
\(186\) 20.0758 1.47203
\(187\) 0 0
\(188\) −1.49646 −0.109140
\(189\) 17.3839 1.26449
\(190\) −1.84951 −0.134177
\(191\) −6.39094 −0.462432 −0.231216 0.972902i \(-0.574270\pi\)
−0.231216 + 0.972902i \(0.574270\pi\)
\(192\) 3.34596 0.241474
\(193\) 13.1955 0.949831 0.474915 0.880031i \(-0.342479\pi\)
0.474915 + 0.880031i \(0.342479\pi\)
\(194\) 3.84951 0.276379
\(195\) 22.3909 1.60345
\(196\) 1.00000 0.0714286
\(197\) 4.69193 0.334286 0.167143 0.985933i \(-0.446546\pi\)
0.167143 + 0.985933i \(0.446546\pi\)
\(198\) 0 0
\(199\) −9.49646 −0.673186 −0.336593 0.941650i \(-0.609275\pi\)
−0.336593 + 0.941650i \(0.609275\pi\)
\(200\) 1.00000 0.0707107
\(201\) 26.7677 1.88805
\(202\) −6.89448 −0.485094
\(203\) −6.84242 −0.480244
\(204\) 24.0758 1.68564
\(205\) −9.34596 −0.652750
\(206\) −10.5793 −0.737096
\(207\) 15.1576 1.05352
\(208\) −6.69193 −0.464002
\(209\) 0 0
\(210\) −3.34596 −0.230893
\(211\) 2.39094 0.164599 0.0822996 0.996608i \(-0.473774\pi\)
0.0822996 + 0.996608i \(0.473774\pi\)
\(212\) 6.84242 0.469939
\(213\) 1.00709 0.0690045
\(214\) 7.49646 0.512447
\(215\) 0.503544 0.0343414
\(216\) 17.3839 1.18282
\(217\) 6.00000 0.407307
\(218\) −1.45857 −0.0987866
\(219\) 34.0900 2.30359
\(220\) 0 0
\(221\) −48.1516 −3.23902
\(222\) −21.8874 −1.46899
\(223\) 0.188383 0.0126150 0.00630752 0.999980i \(-0.497992\pi\)
0.00630752 + 0.999980i \(0.497992\pi\)
\(224\) 1.00000 0.0668153
\(225\) 8.19547 0.546365
\(226\) −14.0000 −0.931266
\(227\) 6.99291 0.464136 0.232068 0.972700i \(-0.425451\pi\)
0.232068 + 0.972700i \(0.425451\pi\)
\(228\) 6.18838 0.409836
\(229\) −9.19547 −0.607654 −0.303827 0.952727i \(-0.598264\pi\)
−0.303827 + 0.952727i \(0.598264\pi\)
\(230\) −1.84951 −0.121953
\(231\) 0 0
\(232\) −6.84242 −0.449227
\(233\) 0.992912 0.0650479 0.0325239 0.999471i \(-0.489645\pi\)
0.0325239 + 0.999471i \(0.489645\pi\)
\(234\) −54.8435 −3.58523
\(235\) 1.49646 0.0976180
\(236\) −7.88740 −0.513426
\(237\) 40.9561 2.66038
\(238\) 7.19547 0.466413
\(239\) 1.14341 0.0739607 0.0369804 0.999316i \(-0.488226\pi\)
0.0369804 + 0.999316i \(0.488226\pi\)
\(240\) −3.34596 −0.215981
\(241\) −8.33888 −0.537154 −0.268577 0.963258i \(-0.586553\pi\)
−0.268577 + 0.963258i \(0.586553\pi\)
\(242\) 0 0
\(243\) 60.2036 3.86206
\(244\) −4.50354 −0.288310
\(245\) −1.00000 −0.0638877
\(246\) 31.2713 1.99378
\(247\) −12.3768 −0.787515
\(248\) 6.00000 0.381000
\(249\) 4.37677 0.277366
\(250\) −1.00000 −0.0632456
\(251\) 23.8874 1.50776 0.753880 0.657013i \(-0.228180\pi\)
0.753880 + 0.657013i \(0.228180\pi\)
\(252\) 8.19547 0.516266
\(253\) 0 0
\(254\) 19.7748 1.24078
\(255\) −24.0758 −1.50768
\(256\) 1.00000 0.0625000
\(257\) −31.6243 −1.97267 −0.986335 0.164753i \(-0.947317\pi\)
−0.986335 + 0.164753i \(0.947317\pi\)
\(258\) −1.68484 −0.104893
\(259\) −6.54143 −0.406465
\(260\) 6.69193 0.415016
\(261\) −56.0768 −3.47107
\(262\) −2.15049 −0.132858
\(263\) −13.3839 −0.825284 −0.412642 0.910893i \(-0.635394\pi\)
−0.412642 + 0.910893i \(0.635394\pi\)
\(264\) 0 0
\(265\) −6.84242 −0.420326
\(266\) 1.84951 0.113401
\(267\) −29.0829 −1.77984
\(268\) 8.00000 0.488678
\(269\) 5.79744 0.353476 0.176738 0.984258i \(-0.443445\pi\)
0.176738 + 0.984258i \(0.443445\pi\)
\(270\) −17.3839 −1.05795
\(271\) 20.0758 1.21952 0.609758 0.792587i \(-0.291266\pi\)
0.609758 + 0.792587i \(0.291266\pi\)
\(272\) 7.19547 0.436289
\(273\) −22.3909 −1.35516
\(274\) −8.39094 −0.506915
\(275\) 0 0
\(276\) 6.18838 0.372497
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) −17.9253 −1.07509
\(279\) 49.1728 2.94390
\(280\) −1.00000 −0.0597614
\(281\) 6.30099 0.375885 0.187943 0.982180i \(-0.439818\pi\)
0.187943 + 0.982180i \(0.439818\pi\)
\(282\) −5.00709 −0.298168
\(283\) −22.3909 −1.33100 −0.665502 0.746396i \(-0.731782\pi\)
−0.665502 + 0.746396i \(0.731782\pi\)
\(284\) 0.300986 0.0178602
\(285\) −6.18838 −0.366568
\(286\) 0 0
\(287\) 9.34596 0.551675
\(288\) 8.19547 0.482923
\(289\) 34.7748 2.04558
\(290\) 6.84242 0.401801
\(291\) 12.8803 0.755057
\(292\) 10.1884 0.596230
\(293\) 10.6919 0.624629 0.312315 0.949979i \(-0.398896\pi\)
0.312315 + 0.949979i \(0.398896\pi\)
\(294\) 3.34596 0.195141
\(295\) 7.88740 0.459222
\(296\) −6.54143 −0.380213
\(297\) 0 0
\(298\) 0.451479 0.0261535
\(299\) −12.3768 −0.715767
\(300\) 3.34596 0.193179
\(301\) −0.503544 −0.0290238
\(302\) −19.2334 −1.10676
\(303\) −23.0687 −1.32526
\(304\) 1.84951 0.106077
\(305\) 4.50354 0.257872
\(306\) 58.9703 3.37111
\(307\) 9.08287 0.518387 0.259193 0.965825i \(-0.416543\pi\)
0.259193 + 0.965825i \(0.416543\pi\)
\(308\) 0 0
\(309\) −35.3980 −2.01372
\(310\) −6.00000 −0.340777
\(311\) −11.3839 −0.645519 −0.322760 0.946481i \(-0.604611\pi\)
−0.322760 + 0.946481i \(0.604611\pi\)
\(312\) −22.3909 −1.26764
\(313\) −23.6243 −1.33532 −0.667662 0.744464i \(-0.732705\pi\)
−0.667662 + 0.744464i \(0.732705\pi\)
\(314\) −9.69901 −0.547347
\(315\) −8.19547 −0.461762
\(316\) 12.2404 0.688579
\(317\) −9.53435 −0.535502 −0.267751 0.963488i \(-0.586280\pi\)
−0.267751 + 0.963488i \(0.586280\pi\)
\(318\) 22.8945 1.28386
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 25.0829 1.39999
\(322\) 1.84951 0.103069
\(323\) 13.3081 0.740481
\(324\) 33.5793 1.86552
\(325\) −6.69193 −0.371201
\(326\) −8.00000 −0.443079
\(327\) −4.88031 −0.269882
\(328\) 9.34596 0.516044
\(329\) −1.49646 −0.0825023
\(330\) 0 0
\(331\) 0.503544 0.0276773 0.0138386 0.999904i \(-0.495595\pi\)
0.0138386 + 0.999904i \(0.495595\pi\)
\(332\) 1.30807 0.0717899
\(333\) −53.6101 −2.93782
\(334\) −8.00000 −0.437741
\(335\) −8.00000 −0.437087
\(336\) 3.34596 0.182537
\(337\) −12.3909 −0.674978 −0.337489 0.941330i \(-0.609578\pi\)
−0.337489 + 0.941330i \(0.609578\pi\)
\(338\) 31.7819 1.72871
\(339\) −46.8435 −2.54419
\(340\) −7.19547 −0.390229
\(341\) 0 0
\(342\) 15.1576 0.819628
\(343\) 1.00000 0.0539949
\(344\) −0.503544 −0.0271493
\(345\) −6.18838 −0.333171
\(346\) 15.4965 0.833095
\(347\) −20.8803 −1.12091 −0.560457 0.828184i \(-0.689374\pi\)
−0.560457 + 0.828184i \(0.689374\pi\)
\(348\) −22.8945 −1.22727
\(349\) 22.1884 1.18772 0.593858 0.804570i \(-0.297604\pi\)
0.593858 + 0.804570i \(0.297604\pi\)
\(350\) 1.00000 0.0534522
\(351\) −116.331 −6.20931
\(352\) 0 0
\(353\) −31.6243 −1.68319 −0.841596 0.540108i \(-0.818383\pi\)
−0.841596 + 0.540108i \(0.818383\pi\)
\(354\) −26.3909 −1.40266
\(355\) −0.300986 −0.0159747
\(356\) −8.69193 −0.460671
\(357\) 24.0758 1.27423
\(358\) −13.8874 −0.733972
\(359\) −30.5273 −1.61117 −0.805584 0.592482i \(-0.798148\pi\)
−0.805584 + 0.592482i \(0.798148\pi\)
\(360\) −8.19547 −0.431939
\(361\) −15.5793 −0.819964
\(362\) 18.7819 0.987154
\(363\) 0 0
\(364\) −6.69193 −0.350752
\(365\) −10.1884 −0.533284
\(366\) −15.0687 −0.787653
\(367\) −10.5793 −0.552236 −0.276118 0.961124i \(-0.589048\pi\)
−0.276118 + 0.961124i \(0.589048\pi\)
\(368\) 1.84951 0.0964122
\(369\) 76.5946 3.98735
\(370\) 6.54143 0.340073
\(371\) 6.84242 0.355241
\(372\) 20.0758 1.04088
\(373\) −36.4667 −1.88818 −0.944088 0.329695i \(-0.893054\pi\)
−0.944088 + 0.329695i \(0.893054\pi\)
\(374\) 0 0
\(375\) −3.34596 −0.172785
\(376\) −1.49646 −0.0771738
\(377\) 45.7890 2.35825
\(378\) 17.3839 0.894129
\(379\) 14.9929 0.770134 0.385067 0.922889i \(-0.374178\pi\)
0.385067 + 0.922889i \(0.374178\pi\)
\(380\) −1.84951 −0.0948777
\(381\) 66.1657 3.38977
\(382\) −6.39094 −0.326989
\(383\) 35.1813 1.79768 0.898840 0.438277i \(-0.144411\pi\)
0.898840 + 0.438277i \(0.144411\pi\)
\(384\) 3.34596 0.170748
\(385\) 0 0
\(386\) 13.1955 0.671632
\(387\) −4.12678 −0.209776
\(388\) 3.84951 0.195429
\(389\) 2.30099 0.116665 0.0583323 0.998297i \(-0.481422\pi\)
0.0583323 + 0.998297i \(0.481422\pi\)
\(390\) 22.3909 1.13381
\(391\) 13.3081 0.673018
\(392\) 1.00000 0.0505076
\(393\) −7.19547 −0.362963
\(394\) 4.69193 0.236376
\(395\) −12.2404 −0.615884
\(396\) 0 0
\(397\) 18.0758 0.907197 0.453599 0.891206i \(-0.350140\pi\)
0.453599 + 0.891206i \(0.350140\pi\)
\(398\) −9.49646 −0.476014
\(399\) 6.18838 0.309807
\(400\) 1.00000 0.0500000
\(401\) 22.5793 1.12756 0.563779 0.825926i \(-0.309347\pi\)
0.563779 + 0.825926i \(0.309347\pi\)
\(402\) 26.7677 1.33505
\(403\) −40.1516 −2.00009
\(404\) −6.89448 −0.343013
\(405\) −33.5793 −1.66857
\(406\) −6.84242 −0.339584
\(407\) 0 0
\(408\) 24.0758 1.19193
\(409\) −1.04498 −0.0516708 −0.0258354 0.999666i \(-0.508225\pi\)
−0.0258354 + 0.999666i \(0.508225\pi\)
\(410\) −9.34596 −0.461564
\(411\) −28.0758 −1.38488
\(412\) −10.5793 −0.521206
\(413\) −7.88740 −0.388113
\(414\) 15.1576 0.744954
\(415\) −1.30807 −0.0642108
\(416\) −6.69193 −0.328099
\(417\) −59.9774 −2.93710
\(418\) 0 0
\(419\) 5.49646 0.268519 0.134260 0.990946i \(-0.457134\pi\)
0.134260 + 0.990946i \(0.457134\pi\)
\(420\) −3.34596 −0.163266
\(421\) 2.60197 0.126812 0.0634062 0.997988i \(-0.479804\pi\)
0.0634062 + 0.997988i \(0.479804\pi\)
\(422\) 2.39094 0.116389
\(423\) −12.2642 −0.596304
\(424\) 6.84242 0.332297
\(425\) 7.19547 0.349032
\(426\) 1.00709 0.0487936
\(427\) −4.50354 −0.217942
\(428\) 7.49646 0.362355
\(429\) 0 0
\(430\) 0.503544 0.0242830
\(431\) −19.2334 −0.926438 −0.463219 0.886244i \(-0.653306\pi\)
−0.463219 + 0.886244i \(0.653306\pi\)
\(432\) 17.3839 0.836381
\(433\) −8.93237 −0.429263 −0.214631 0.976695i \(-0.568855\pi\)
−0.214631 + 0.976695i \(0.568855\pi\)
\(434\) 6.00000 0.288009
\(435\) 22.8945 1.09771
\(436\) −1.45857 −0.0698527
\(437\) 3.42068 0.163633
\(438\) 34.0900 1.62888
\(439\) 0.300986 0.0143653 0.00718264 0.999974i \(-0.497714\pi\)
0.00718264 + 0.999974i \(0.497714\pi\)
\(440\) 0 0
\(441\) 8.19547 0.390260
\(442\) −48.1516 −2.29034
\(443\) 30.7677 1.46182 0.730909 0.682475i \(-0.239096\pi\)
0.730909 + 0.682475i \(0.239096\pi\)
\(444\) −21.8874 −1.03873
\(445\) 8.69193 0.412037
\(446\) 0.188383 0.00892018
\(447\) 1.51063 0.0714504
\(448\) 1.00000 0.0472456
\(449\) −1.19547 −0.0564177 −0.0282089 0.999602i \(-0.508980\pi\)
−0.0282089 + 0.999602i \(0.508980\pi\)
\(450\) 8.19547 0.386338
\(451\) 0 0
\(452\) −14.0000 −0.658505
\(453\) −64.3541 −3.02362
\(454\) 6.99291 0.328194
\(455\) 6.69193 0.313722
\(456\) 6.18838 0.289798
\(457\) 25.7748 1.20569 0.602847 0.797857i \(-0.294033\pi\)
0.602847 + 0.797857i \(0.294033\pi\)
\(458\) −9.19547 −0.429676
\(459\) 125.085 5.83847
\(460\) −1.84951 −0.0862337
\(461\) −11.5722 −0.538973 −0.269486 0.963004i \(-0.586854\pi\)
−0.269486 + 0.963004i \(0.586854\pi\)
\(462\) 0 0
\(463\) −32.9182 −1.52984 −0.764919 0.644126i \(-0.777221\pi\)
−0.764919 + 0.644126i \(0.777221\pi\)
\(464\) −6.84242 −0.317651
\(465\) −20.0758 −0.930992
\(466\) 0.992912 0.0459958
\(467\) −14.6399 −0.677452 −0.338726 0.940885i \(-0.609996\pi\)
−0.338726 + 0.940885i \(0.609996\pi\)
\(468\) −54.8435 −2.53514
\(469\) 8.00000 0.369406
\(470\) 1.49646 0.0690264
\(471\) −32.4525 −1.49533
\(472\) −7.88740 −0.363047
\(473\) 0 0
\(474\) 40.9561 1.88118
\(475\) 1.84951 0.0848612
\(476\) 7.19547 0.329804
\(477\) 56.0768 2.56758
\(478\) 1.14341 0.0522981
\(479\) −10.3909 −0.474774 −0.237387 0.971415i \(-0.576291\pi\)
−0.237387 + 0.971415i \(0.576291\pi\)
\(480\) −3.34596 −0.152722
\(481\) 43.7748 1.99596
\(482\) −8.33888 −0.379825
\(483\) 6.18838 0.281581
\(484\) 0 0
\(485\) −3.84951 −0.174797
\(486\) 60.2036 2.73089
\(487\) 39.3091 1.78127 0.890634 0.454722i \(-0.150261\pi\)
0.890634 + 0.454722i \(0.150261\pi\)
\(488\) −4.50354 −0.203866
\(489\) −26.7677 −1.21048
\(490\) −1.00000 −0.0451754
\(491\) −23.7748 −1.07294 −0.536471 0.843919i \(-0.680243\pi\)
−0.536471 + 0.843919i \(0.680243\pi\)
\(492\) 31.2713 1.40982
\(493\) −49.2344 −2.21741
\(494\) −12.3768 −0.556857
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) 0.300986 0.0135011
\(498\) 4.37677 0.196128
\(499\) −6.11260 −0.273638 −0.136819 0.990596i \(-0.543688\pi\)
−0.136819 + 0.990596i \(0.543688\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −26.7677 −1.19589
\(502\) 23.8874 1.06615
\(503\) 13.1586 0.586715 0.293358 0.956003i \(-0.405227\pi\)
0.293358 + 0.956003i \(0.405227\pi\)
\(504\) 8.19547 0.365055
\(505\) 6.89448 0.306801
\(506\) 0 0
\(507\) 106.341 4.72277
\(508\) 19.7748 0.877365
\(509\) 34.1799 1.51500 0.757499 0.652836i \(-0.226421\pi\)
0.757499 + 0.652836i \(0.226421\pi\)
\(510\) −24.0758 −1.06609
\(511\) 10.1884 0.450708
\(512\) 1.00000 0.0441942
\(513\) 32.1516 1.41953
\(514\) −31.6243 −1.39489
\(515\) 10.5793 0.466181
\(516\) −1.68484 −0.0741709
\(517\) 0 0
\(518\) −6.54143 −0.287414
\(519\) 51.8506 2.27599
\(520\) 6.69193 0.293460
\(521\) −26.6778 −1.16877 −0.584387 0.811475i \(-0.698665\pi\)
−0.584387 + 0.811475i \(0.698665\pi\)
\(522\) −56.0768 −2.45442
\(523\) −25.4880 −1.11451 −0.557256 0.830341i \(-0.688146\pi\)
−0.557256 + 0.830341i \(0.688146\pi\)
\(524\) −2.15049 −0.0939447
\(525\) 3.34596 0.146030
\(526\) −13.3839 −0.583564
\(527\) 43.1728 1.88064
\(528\) 0 0
\(529\) −19.5793 −0.851275
\(530\) −6.84242 −0.297216
\(531\) −64.6409 −2.80518
\(532\) 1.84951 0.0801863
\(533\) −62.5425 −2.70902
\(534\) −29.0829 −1.25854
\(535\) −7.49646 −0.324100
\(536\) 8.00000 0.345547
\(537\) −46.4667 −2.00519
\(538\) 5.79744 0.249945
\(539\) 0 0
\(540\) −17.3839 −0.748082
\(541\) −4.45148 −0.191384 −0.0956920 0.995411i \(-0.530506\pi\)
−0.0956920 + 0.995411i \(0.530506\pi\)
\(542\) 20.0758 0.862329
\(543\) 62.8435 2.69687
\(544\) 7.19547 0.308503
\(545\) 1.45857 0.0624781
\(546\) −22.3909 −0.958244
\(547\) 1.38385 0.0591693 0.0295846 0.999562i \(-0.490582\pi\)
0.0295846 + 0.999562i \(0.490582\pi\)
\(548\) −8.39094 −0.358443
\(549\) −36.9087 −1.57522
\(550\) 0 0
\(551\) −12.6551 −0.539126
\(552\) 6.18838 0.263395
\(553\) 12.2404 0.520517
\(554\) −22.0000 −0.934690
\(555\) 21.8874 0.929068
\(556\) −17.9253 −0.760201
\(557\) 6.70610 0.284147 0.142073 0.989856i \(-0.454623\pi\)
0.142073 + 0.989856i \(0.454623\pi\)
\(558\) 49.1728 2.08165
\(559\) 3.36968 0.142522
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 6.30099 0.265791
\(563\) −3.59488 −0.151506 −0.0757532 0.997127i \(-0.524136\pi\)
−0.0757532 + 0.997127i \(0.524136\pi\)
\(564\) −5.00709 −0.210836
\(565\) 14.0000 0.588984
\(566\) −22.3909 −0.941161
\(567\) 33.5793 1.41020
\(568\) 0.300986 0.0126291
\(569\) −31.7606 −1.33147 −0.665737 0.746186i \(-0.731883\pi\)
−0.665737 + 0.746186i \(0.731883\pi\)
\(570\) −6.18838 −0.259203
\(571\) 5.68484 0.237903 0.118952 0.992900i \(-0.462047\pi\)
0.118952 + 0.992900i \(0.462047\pi\)
\(572\) 0 0
\(573\) −21.3839 −0.893323
\(574\) 9.34596 0.390093
\(575\) 1.84951 0.0771298
\(576\) 8.19547 0.341478
\(577\) 13.5343 0.563442 0.281721 0.959496i \(-0.409095\pi\)
0.281721 + 0.959496i \(0.409095\pi\)
\(578\) 34.7748 1.44644
\(579\) 44.1516 1.83488
\(580\) 6.84242 0.284116
\(581\) 1.30807 0.0542680
\(582\) 12.8803 0.533906
\(583\) 0 0
\(584\) 10.1884 0.421598
\(585\) 54.8435 2.26750
\(586\) 10.6919 0.441679
\(587\) −0.0520650 −0.00214895 −0.00107448 0.999999i \(-0.500342\pi\)
−0.00107448 + 0.999999i \(0.500342\pi\)
\(588\) 3.34596 0.137985
\(589\) 11.0970 0.457246
\(590\) 7.88740 0.324719
\(591\) 15.6990 0.645771
\(592\) −6.54143 −0.268851
\(593\) −31.9774 −1.31315 −0.656576 0.754260i \(-0.727996\pi\)
−0.656576 + 0.754260i \(0.727996\pi\)
\(594\) 0 0
\(595\) −7.19547 −0.294986
\(596\) 0.451479 0.0184933
\(597\) −31.7748 −1.30046
\(598\) −12.3768 −0.506124
\(599\) 21.0829 0.861423 0.430711 0.902490i \(-0.358263\pi\)
0.430711 + 0.902490i \(0.358263\pi\)
\(600\) 3.34596 0.136598
\(601\) 21.6469 0.882997 0.441499 0.897262i \(-0.354447\pi\)
0.441499 + 0.897262i \(0.354447\pi\)
\(602\) −0.503544 −0.0205229
\(603\) 65.5638 2.66996
\(604\) −19.2334 −0.782594
\(605\) 0 0
\(606\) −23.0687 −0.937102
\(607\) −21.6622 −0.879241 −0.439621 0.898184i \(-0.644887\pi\)
−0.439621 + 0.898184i \(0.644887\pi\)
\(608\) 1.84951 0.0750074
\(609\) −22.8945 −0.927731
\(610\) 4.50354 0.182343
\(611\) 10.0142 0.405130
\(612\) 58.9703 2.38373
\(613\) −8.31516 −0.335846 −0.167923 0.985800i \(-0.553706\pi\)
−0.167923 + 0.985800i \(0.553706\pi\)
\(614\) 9.08287 0.366555
\(615\) −31.2713 −1.26098
\(616\) 0 0
\(617\) 12.4667 0.501891 0.250946 0.968001i \(-0.419258\pi\)
0.250946 + 0.968001i \(0.419258\pi\)
\(618\) −35.3980 −1.42392
\(619\) 5.27125 0.211869 0.105935 0.994373i \(-0.466217\pi\)
0.105935 + 0.994373i \(0.466217\pi\)
\(620\) −6.00000 −0.240966
\(621\) 32.1516 1.29020
\(622\) −11.3839 −0.456451
\(623\) −8.69193 −0.348235
\(624\) −22.3909 −0.896355
\(625\) 1.00000 0.0400000
\(626\) −23.6243 −0.944217
\(627\) 0 0
\(628\) −9.69901 −0.387033
\(629\) −47.0687 −1.87675
\(630\) −8.19547 −0.326515
\(631\) −17.1586 −0.683075 −0.341537 0.939868i \(-0.610948\pi\)
−0.341537 + 0.939868i \(0.610948\pi\)
\(632\) 12.2404 0.486899
\(633\) 8.00000 0.317971
\(634\) −9.53435 −0.378657
\(635\) −19.7748 −0.784739
\(636\) 22.8945 0.907825
\(637\) −6.69193 −0.265144
\(638\) 0 0
\(639\) 2.46672 0.0975820
\(640\) −1.00000 −0.0395285
\(641\) 24.7677 0.978266 0.489133 0.872209i \(-0.337313\pi\)
0.489133 + 0.872209i \(0.337313\pi\)
\(642\) 25.0829 0.989942
\(643\) 1.03080 0.0406509 0.0203254 0.999793i \(-0.493530\pi\)
0.0203254 + 0.999793i \(0.493530\pi\)
\(644\) 1.84951 0.0728808
\(645\) 1.68484 0.0663405
\(646\) 13.3081 0.523599
\(647\) 12.7904 0.502841 0.251420 0.967878i \(-0.419102\pi\)
0.251420 + 0.967878i \(0.419102\pi\)
\(648\) 33.5793 1.31912
\(649\) 0 0
\(650\) −6.69193 −0.262479
\(651\) 20.0758 0.786832
\(652\) −8.00000 −0.313304
\(653\) −22.3162 −0.873301 −0.436651 0.899631i \(-0.643835\pi\)
−0.436651 + 0.899631i \(0.643835\pi\)
\(654\) −4.88031 −0.190835
\(655\) 2.15049 0.0840267
\(656\) 9.34596 0.364899
\(657\) 83.4986 3.25759
\(658\) −1.49646 −0.0583379
\(659\) −22.6919 −0.883952 −0.441976 0.897027i \(-0.645722\pi\)
−0.441976 + 0.897027i \(0.645722\pi\)
\(660\) 0 0
\(661\) 43.5638 1.69443 0.847217 0.531247i \(-0.178276\pi\)
0.847217 + 0.531247i \(0.178276\pi\)
\(662\) 0.503544 0.0195708
\(663\) −161.113 −6.25712
\(664\) 1.30807 0.0507631
\(665\) −1.84951 −0.0717208
\(666\) −53.6101 −2.07735
\(667\) −12.6551 −0.490008
\(668\) −8.00000 −0.309529
\(669\) 0.630322 0.0243697
\(670\) −8.00000 −0.309067
\(671\) 0 0
\(672\) 3.34596 0.129073
\(673\) −38.0000 −1.46479 −0.732396 0.680879i \(-0.761598\pi\)
−0.732396 + 0.680879i \(0.761598\pi\)
\(674\) −12.3909 −0.477281
\(675\) 17.3839 0.669105
\(676\) 31.7819 1.22238
\(677\) 16.0984 0.618713 0.309356 0.950946i \(-0.399886\pi\)
0.309356 + 0.950946i \(0.399886\pi\)
\(678\) −46.8435 −1.79901
\(679\) 3.84951 0.147731
\(680\) −7.19547 −0.275934
\(681\) 23.3980 0.896614
\(682\) 0 0
\(683\) 29.0829 1.11282 0.556412 0.830906i \(-0.312177\pi\)
0.556412 + 0.830906i \(0.312177\pi\)
\(684\) 15.1576 0.579565
\(685\) 8.39094 0.320601
\(686\) 1.00000 0.0381802
\(687\) −30.7677 −1.17386
\(688\) −0.503544 −0.0191974
\(689\) −45.7890 −1.74442
\(690\) −6.18838 −0.235588
\(691\) 7.58641 0.288601 0.144300 0.989534i \(-0.453907\pi\)
0.144300 + 0.989534i \(0.453907\pi\)
\(692\) 15.4965 0.589087
\(693\) 0 0
\(694\) −20.8803 −0.792606
\(695\) 17.9253 0.679945
\(696\) −22.8945 −0.867813
\(697\) 67.2486 2.54722
\(698\) 22.1884 0.839843
\(699\) 3.32225 0.125659
\(700\) 1.00000 0.0377964
\(701\) −4.07471 −0.153900 −0.0769499 0.997035i \(-0.524518\pi\)
−0.0769499 + 0.997035i \(0.524518\pi\)
\(702\) −116.331 −4.39065
\(703\) −12.0984 −0.456301
\(704\) 0 0
\(705\) 5.00709 0.188578
\(706\) −31.6243 −1.19020
\(707\) −6.89448 −0.259294
\(708\) −26.3909 −0.991832
\(709\) −46.4525 −1.74456 −0.872281 0.489005i \(-0.837360\pi\)
−0.872281 + 0.489005i \(0.837360\pi\)
\(710\) −0.300986 −0.0112958
\(711\) 100.316 3.76215
\(712\) −8.69193 −0.325744
\(713\) 11.0970 0.415588
\(714\) 24.0758 0.901013
\(715\) 0 0
\(716\) −13.8874 −0.518996
\(717\) 3.82579 0.142877
\(718\) −30.5273 −1.13927
\(719\) −17.8732 −0.666559 −0.333279 0.942828i \(-0.608155\pi\)
−0.333279 + 0.942828i \(0.608155\pi\)
\(720\) −8.19547 −0.305427
\(721\) −10.5793 −0.393995
\(722\) −15.5793 −0.579802
\(723\) −27.9016 −1.03767
\(724\) 18.7819 0.698023
\(725\) −6.84242 −0.254121
\(726\) 0 0
\(727\) −11.8874 −0.440879 −0.220440 0.975401i \(-0.570749\pi\)
−0.220440 + 0.975401i \(0.570749\pi\)
\(728\) −6.69193 −0.248019
\(729\) 100.701 3.72967
\(730\) −10.1884 −0.377089
\(731\) −3.62323 −0.134010
\(732\) −15.0687 −0.556955
\(733\) 31.6764 1.16999 0.584997 0.811036i \(-0.301096\pi\)
0.584997 + 0.811036i \(0.301096\pi\)
\(734\) −10.5793 −0.390490
\(735\) −3.34596 −0.123418
\(736\) 1.84951 0.0681737
\(737\) 0 0
\(738\) 76.5946 2.81948
\(739\) −28.5567 −1.05047 −0.525237 0.850956i \(-0.676023\pi\)
−0.525237 + 0.850956i \(0.676023\pi\)
\(740\) 6.54143 0.240468
\(741\) −41.4122 −1.52132
\(742\) 6.84242 0.251193
\(743\) −13.9858 −0.513090 −0.256545 0.966532i \(-0.582584\pi\)
−0.256545 + 0.966532i \(0.582584\pi\)
\(744\) 20.0758 0.736014
\(745\) −0.451479 −0.0165409
\(746\) −36.4667 −1.33514
\(747\) 10.7203 0.392234
\(748\) 0 0
\(749\) 7.49646 0.273915
\(750\) −3.34596 −0.122177
\(751\) 18.9171 0.690296 0.345148 0.938548i \(-0.387829\pi\)
0.345148 + 0.938548i \(0.387829\pi\)
\(752\) −1.49646 −0.0545701
\(753\) 79.9264 2.91268
\(754\) 45.7890 1.66754
\(755\) 19.2334 0.699974
\(756\) 17.3839 0.632245
\(757\) 10.9182 0.396829 0.198414 0.980118i \(-0.436421\pi\)
0.198414 + 0.980118i \(0.436421\pi\)
\(758\) 14.9929 0.544567
\(759\) 0 0
\(760\) −1.84951 −0.0670887
\(761\) 30.3531 1.10030 0.550149 0.835067i \(-0.314571\pi\)
0.550149 + 0.835067i \(0.314571\pi\)
\(762\) 66.1657 2.39693
\(763\) −1.45857 −0.0528036
\(764\) −6.39094 −0.231216
\(765\) −58.9703 −2.13207
\(766\) 35.1813 1.27115
\(767\) 52.7819 1.90584
\(768\) 3.34596 0.120737
\(769\) 19.7369 0.711731 0.355865 0.934537i \(-0.384186\pi\)
0.355865 + 0.934537i \(0.384186\pi\)
\(770\) 0 0
\(771\) −105.814 −3.81079
\(772\) 13.1955 0.474915
\(773\) −24.3909 −0.877281 −0.438641 0.898663i \(-0.644540\pi\)
−0.438641 + 0.898663i \(0.644540\pi\)
\(774\) −4.12678 −0.148334
\(775\) 6.00000 0.215526
\(776\) 3.84951 0.138189
\(777\) −21.8874 −0.785206
\(778\) 2.30099 0.0824943
\(779\) 17.2854 0.619315
\(780\) 22.3909 0.801724
\(781\) 0 0
\(782\) 13.3081 0.475896
\(783\) −118.948 −4.25084
\(784\) 1.00000 0.0357143
\(785\) 9.69901 0.346173
\(786\) −7.19547 −0.256654
\(787\) −33.7890 −1.20445 −0.602223 0.798328i \(-0.705718\pi\)
−0.602223 + 0.798328i \(0.705718\pi\)
\(788\) 4.69193 0.167143
\(789\) −44.7819 −1.59428
\(790\) −12.2404 −0.435496
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) 30.1374 1.07021
\(794\) 18.0758 0.641485
\(795\) −22.8945 −0.811984
\(796\) −9.49646 −0.336593
\(797\) 36.8435 1.30506 0.652532 0.757761i \(-0.273707\pi\)
0.652532 + 0.757761i \(0.273707\pi\)
\(798\) 6.18838 0.219066
\(799\) −10.7677 −0.380934
\(800\) 1.00000 0.0353553
\(801\) −71.2344 −2.51694
\(802\) 22.5793 0.797304
\(803\) 0 0
\(804\) 26.7677 0.944024
\(805\) −1.84951 −0.0651866
\(806\) −40.1516 −1.41428
\(807\) 19.3980 0.682843
\(808\) −6.89448 −0.242547
\(809\) −37.6990 −1.32543 −0.662713 0.748873i \(-0.730595\pi\)
−0.662713 + 0.748873i \(0.730595\pi\)
\(810\) −33.5793 −1.17986
\(811\) −38.3304 −1.34596 −0.672981 0.739660i \(-0.734987\pi\)
−0.672981 + 0.739660i \(0.734987\pi\)
\(812\) −6.84242 −0.240122
\(813\) 67.1728 2.35585
\(814\) 0 0
\(815\) 8.00000 0.280228
\(816\) 24.0758 0.842821
\(817\) −0.931308 −0.0325823
\(818\) −1.04498 −0.0365368
\(819\) −54.8435 −1.91639
\(820\) −9.34596 −0.326375
\(821\) 18.1363 0.632962 0.316481 0.948599i \(-0.397499\pi\)
0.316481 + 0.948599i \(0.397499\pi\)
\(822\) −28.0758 −0.979255
\(823\) −7.86368 −0.274111 −0.137055 0.990563i \(-0.543764\pi\)
−0.137055 + 0.990563i \(0.543764\pi\)
\(824\) −10.5793 −0.368548
\(825\) 0 0
\(826\) −7.88740 −0.274438
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 15.1576 0.526762
\(829\) 2.80453 0.0974053 0.0487027 0.998813i \(-0.484491\pi\)
0.0487027 + 0.998813i \(0.484491\pi\)
\(830\) −1.30807 −0.0454039
\(831\) −73.6112 −2.55354
\(832\) −6.69193 −0.232001
\(833\) 7.19547 0.249308
\(834\) −59.9774 −2.07685
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) 104.303 3.60524
\(838\) 5.49646 0.189872
\(839\) 36.9929 1.27714 0.638569 0.769565i \(-0.279527\pi\)
0.638569 + 0.769565i \(0.279527\pi\)
\(840\) −3.34596 −0.115447
\(841\) 17.8187 0.614438
\(842\) 2.60197 0.0896699
\(843\) 21.0829 0.726133
\(844\) 2.39094 0.0822996
\(845\) −31.7819 −1.09333
\(846\) −12.2642 −0.421651
\(847\) 0 0
\(848\) 6.84242 0.234970
\(849\) −74.9193 −2.57122
\(850\) 7.19547 0.246803
\(851\) −12.0984 −0.414729
\(852\) 1.00709 0.0345023
\(853\) 49.2571 1.68653 0.843265 0.537498i \(-0.180630\pi\)
0.843265 + 0.537498i \(0.180630\pi\)
\(854\) −4.50354 −0.154108
\(855\) −15.1576 −0.518378
\(856\) 7.49646 0.256224
\(857\) 20.3541 0.695283 0.347642 0.937627i \(-0.386983\pi\)
0.347642 + 0.937627i \(0.386983\pi\)
\(858\) 0 0
\(859\) −34.2783 −1.16956 −0.584781 0.811191i \(-0.698820\pi\)
−0.584781 + 0.811191i \(0.698820\pi\)
\(860\) 0.503544 0.0171707
\(861\) 31.2713 1.06572
\(862\) −19.2334 −0.655091
\(863\) −12.4373 −0.423371 −0.211685 0.977338i \(-0.567895\pi\)
−0.211685 + 0.977338i \(0.567895\pi\)
\(864\) 17.3839 0.591411
\(865\) −15.4965 −0.526895
\(866\) −8.93237 −0.303534
\(867\) 116.355 3.95163
\(868\) 6.00000 0.203653
\(869\) 0 0
\(870\) 22.8945 0.776196
\(871\) −53.5354 −1.81398
\(872\) −1.45857 −0.0493933
\(873\) 31.5485 1.06776
\(874\) 3.42068 0.115706
\(875\) −1.00000 −0.0338062
\(876\) 34.0900 1.15179
\(877\) −34.4809 −1.16434 −0.582169 0.813068i \(-0.697796\pi\)
−0.582169 + 0.813068i \(0.697796\pi\)
\(878\) 0.300986 0.0101578
\(879\) 35.7748 1.20665
\(880\) 0 0
\(881\) 27.3839 0.922585 0.461293 0.887248i \(-0.347386\pi\)
0.461293 + 0.887248i \(0.347386\pi\)
\(882\) 8.19547 0.275956
\(883\) 22.0900 0.743386 0.371693 0.928356i \(-0.378777\pi\)
0.371693 + 0.928356i \(0.378777\pi\)
\(884\) −48.1516 −1.61951
\(885\) 26.3909 0.887122
\(886\) 30.7677 1.03366
\(887\) −25.7890 −0.865909 −0.432954 0.901416i \(-0.642529\pi\)
−0.432954 + 0.901416i \(0.642529\pi\)
\(888\) −21.8874 −0.734493
\(889\) 19.7748 0.663225
\(890\) 8.69193 0.291354
\(891\) 0 0
\(892\) 0.188383 0.00630752
\(893\) −2.76771 −0.0926178
\(894\) 1.51063 0.0505231
\(895\) 13.8874 0.464204
\(896\) 1.00000 0.0334077
\(897\) −41.4122 −1.38271
\(898\) −1.19547 −0.0398934
\(899\) −41.0545 −1.36924
\(900\) 8.19547 0.273182
\(901\) 49.2344 1.64024
\(902\) 0 0
\(903\) −1.68484 −0.0560679
\(904\) −14.0000 −0.465633
\(905\) −18.7819 −0.624331
\(906\) −64.3541 −2.13802
\(907\) −45.4596 −1.50946 −0.754731 0.656034i \(-0.772233\pi\)
−0.754731 + 0.656034i \(0.772233\pi\)
\(908\) 6.99291 0.232068
\(909\) −56.5035 −1.87410
\(910\) 6.69193 0.221835
\(911\) −15.8506 −0.525153 −0.262576 0.964911i \(-0.584572\pi\)
−0.262576 + 0.964911i \(0.584572\pi\)
\(912\) 6.18838 0.204918
\(913\) 0 0
\(914\) 25.7748 0.852554
\(915\) 15.0687 0.498156
\(916\) −9.19547 −0.303827
\(917\) −2.15049 −0.0710155
\(918\) 125.085 4.12842
\(919\) −18.6314 −0.614593 −0.307296 0.951614i \(-0.599424\pi\)
−0.307296 + 0.951614i \(0.599424\pi\)
\(920\) −1.84951 −0.0609764
\(921\) 30.3909 1.00142
\(922\) −11.5722 −0.381111
\(923\) −2.01418 −0.0662974
\(924\) 0 0
\(925\) −6.54143 −0.215081
\(926\) −32.9182 −1.08176
\(927\) −86.7025 −2.84768
\(928\) −6.84242 −0.224613
\(929\) −29.2486 −0.959616 −0.479808 0.877374i \(-0.659294\pi\)
−0.479808 + 0.877374i \(0.659294\pi\)
\(930\) −20.0758 −0.658311
\(931\) 1.84951 0.0606151
\(932\) 0.992912 0.0325239
\(933\) −38.0900 −1.24701
\(934\) −14.6399 −0.479031
\(935\) 0 0
\(936\) −54.8435 −1.79262
\(937\) 46.4441 1.51726 0.758631 0.651521i \(-0.225869\pi\)
0.758631 + 0.651521i \(0.225869\pi\)
\(938\) 8.00000 0.261209
\(939\) −79.0460 −2.57957
\(940\) 1.49646 0.0488090
\(941\) 39.2713 1.28021 0.640103 0.768289i \(-0.278892\pi\)
0.640103 + 0.768289i \(0.278892\pi\)
\(942\) −32.4525 −1.05736
\(943\) 17.2854 0.562891
\(944\) −7.88740 −0.256713
\(945\) −17.3839 −0.565497
\(946\) 0 0
\(947\) 54.2415 1.76261 0.881306 0.472546i \(-0.156665\pi\)
0.881306 + 0.472546i \(0.156665\pi\)
\(948\) 40.9561 1.33019
\(949\) −68.1799 −2.21321
\(950\) 1.84951 0.0600059
\(951\) −31.9016 −1.03448
\(952\) 7.19547 0.233207
\(953\) −59.3612 −1.92290 −0.961449 0.274983i \(-0.911328\pi\)
−0.961449 + 0.274983i \(0.911328\pi\)
\(954\) 56.0768 1.81555
\(955\) 6.39094 0.206806
\(956\) 1.14341 0.0369804
\(957\) 0 0
\(958\) −10.3909 −0.335716
\(959\) −8.39094 −0.270958
\(960\) −3.34596 −0.107991
\(961\) 5.00000 0.161290
\(962\) 43.7748 1.41136
\(963\) 61.4370 1.97978
\(964\) −8.33888 −0.268577
\(965\) −13.1955 −0.424777
\(966\) 6.18838 0.199108
\(967\) 18.7677 0.603529 0.301764 0.953383i \(-0.402424\pi\)
0.301764 + 0.953383i \(0.402424\pi\)
\(968\) 0 0
\(969\) 44.5283 1.43046
\(970\) −3.84951 −0.123600
\(971\) −59.4370 −1.90742 −0.953712 0.300722i \(-0.902772\pi\)
−0.953712 + 0.300722i \(0.902772\pi\)
\(972\) 60.2036 1.93103
\(973\) −17.9253 −0.574658
\(974\) 39.3091 1.25955
\(975\) −22.3909 −0.717084
\(976\) −4.50354 −0.144155
\(977\) −32.7677 −1.04833 −0.524166 0.851616i \(-0.675623\pi\)
−0.524166 + 0.851616i \(0.675623\pi\)
\(978\) −26.7677 −0.855937
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −11.9536 −0.381650
\(982\) −23.7748 −0.758684
\(983\) 13.4207 0.428053 0.214027 0.976828i \(-0.431342\pi\)
0.214027 + 0.976828i \(0.431342\pi\)
\(984\) 31.2713 0.996891
\(985\) −4.69193 −0.149497
\(986\) −49.2344 −1.56794
\(987\) −5.00709 −0.159377
\(988\) −12.3768 −0.393757
\(989\) −0.931308 −0.0296139
\(990\) 0 0
\(991\) −38.1657 −1.21237 −0.606187 0.795322i \(-0.707302\pi\)
−0.606187 + 0.795322i \(0.707302\pi\)
\(992\) 6.00000 0.190500
\(993\) 1.68484 0.0534668
\(994\) 0.300986 0.00954669
\(995\) 9.49646 0.301058
\(996\) 4.37677 0.138683
\(997\) 33.6622 1.06609 0.533046 0.846086i \(-0.321047\pi\)
0.533046 + 0.846086i \(0.321047\pi\)
\(998\) −6.11260 −0.193491
\(999\) −113.715 −3.59779
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.1.3 3
11.10 odd 2 770.2.a.l.1.3 3
33.32 even 2 6930.2.a.cl.1.3 3
44.43 even 2 6160.2.a.bi.1.1 3
55.32 even 4 3850.2.c.z.1849.1 6
55.43 even 4 3850.2.c.z.1849.6 6
55.54 odd 2 3850.2.a.bu.1.1 3
77.76 even 2 5390.2.a.bz.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.l.1.3 3 11.10 odd 2
3850.2.a.bu.1.1 3 55.54 odd 2
3850.2.c.z.1849.1 6 55.32 even 4
3850.2.c.z.1849.6 6 55.43 even 4
5390.2.a.bz.1.1 3 77.76 even 2
6160.2.a.bi.1.1 3 44.43 even 2
6930.2.a.cl.1.3 3 33.32 even 2
8470.2.a.cl.1.3 3 1.1 even 1 trivial