Properties

Label 8470.2.a.co.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.4400.1
Defining polynomial: \(x^{4} - 7 x^{2} + 11\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.14896\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -3.14896 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.14896 q^{6} -1.00000 q^{7} -1.00000 q^{8} +6.91596 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -3.14896 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.14896 q^{6} -1.00000 q^{7} -1.00000 q^{8} +6.91596 q^{9} -1.00000 q^{10} -3.14896 q^{12} +5.43886 q^{13} +1.00000 q^{14} -3.14896 q^{15} +1.00000 q^{16} -0.477092 q^{17} -6.91596 q^{18} -5.91596 q^{19} +1.00000 q^{20} +3.14896 q^{21} -2.65626 q^{23} +3.14896 q^{24} +1.00000 q^{25} -5.43886 q^{26} -12.3312 q^{27} -1.00000 q^{28} -6.56726 q^{29} +3.14896 q^{30} -9.68953 q^{31} -1.00000 q^{32} +0.477092 q^{34} -1.00000 q^{35} +6.91596 q^{36} -4.56726 q^{37} +5.91596 q^{38} -17.1268 q^{39} -1.00000 q^{40} +2.52291 q^{41} -3.14896 q^{42} +11.9003 q^{43} +6.91596 q^{45} +2.65626 q^{46} +9.42632 q^{47} -3.14896 q^{48} +1.00000 q^{49} -1.00000 q^{50} +1.50234 q^{51} +5.43886 q^{52} +6.15392 q^{53} +12.3312 q^{54} +1.00000 q^{56} +18.6291 q^{57} +6.56726 q^{58} -0.566092 q^{59} -3.14896 q^{60} +2.04888 q^{61} +9.68953 q^{62} -6.91596 q^{63} +1.00000 q^{64} +5.43886 q^{65} -1.24102 q^{67} -0.477092 q^{68} +8.36447 q^{69} +1.00000 q^{70} -3.62299 q^{71} -6.91596 q^{72} +15.6398 q^{73} +4.56726 q^{74} -3.14896 q^{75} -5.91596 q^{76} +17.1268 q^{78} -3.38007 q^{79} +1.00000 q^{80} +18.0826 q^{81} -2.52291 q^{82} +1.73868 q^{83} +3.14896 q^{84} -0.477092 q^{85} -11.9003 q^{86} +20.6801 q^{87} -9.54364 q^{89} -6.91596 q^{90} -5.43886 q^{91} -2.65626 q^{92} +30.5120 q^{93} -9.42632 q^{94} -5.91596 q^{95} +3.14896 q^{96} -3.93156 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} + 4q^{5} + 4q^{6} - 4q^{7} - 4q^{8} + 6q^{9} + O(q^{10}) \) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} + 4q^{5} + 4q^{6} - 4q^{7} - 4q^{8} + 6q^{9} - 4q^{10} - 4q^{12} + 14q^{13} + 4q^{14} - 4q^{15} + 4q^{16} + 12q^{17} - 6q^{18} - 2q^{19} + 4q^{20} + 4q^{21} + 4q^{24} + 4q^{25} - 14q^{26} - 22q^{27} - 4q^{28} + 10q^{29} + 4q^{30} - 18q^{31} - 4q^{32} - 12q^{34} - 4q^{35} + 6q^{36} + 18q^{37} + 2q^{38} - 30q^{39} - 4q^{40} + 24q^{41} - 4q^{42} + 10q^{43} + 6q^{45} - 8q^{47} - 4q^{48} + 4q^{49} - 4q^{50} + 14q^{52} + 20q^{53} + 22q^{54} + 4q^{56} + 30q^{57} - 10q^{58} - 14q^{59} - 4q^{60} + 14q^{61} + 18q^{62} - 6q^{63} + 4q^{64} + 14q^{65} + 12q^{68} - 4q^{69} + 4q^{70} - 14q^{71} - 6q^{72} + 30q^{73} - 18q^{74} - 4q^{75} - 2q^{76} + 30q^{78} + 8q^{79} + 4q^{80} + 16q^{81} - 24q^{82} + 8q^{83} + 4q^{84} + 12q^{85} - 10q^{86} + 2q^{87} - 4q^{89} - 6q^{90} - 14q^{91} - 2q^{93} + 8q^{94} - 2q^{95} + 4q^{96} - 10q^{97} - 4q^{98} + O(q^{100}) \)

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
\(3\) −3.14896 −1.81805 −0.909027 0.416738i \(-0.863173\pi\)
−0.909027 + 0.416738i \(0.863173\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 3.14896 1.28556
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 6.91596 2.30532
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −3.14896 −0.909027
\(13\) 5.43886 1.50847 0.754235 0.656605i \(-0.228008\pi\)
0.754235 + 0.656605i \(0.228008\pi\)
\(14\) 1.00000 0.267261
\(15\) −3.14896 −0.813058
\(16\) 1.00000 0.250000
\(17\) −0.477092 −0.115712 −0.0578559 0.998325i \(-0.518426\pi\)
−0.0578559 + 0.998325i \(0.518426\pi\)
\(18\) −6.91596 −1.63011
\(19\) −5.91596 −1.35721 −0.678607 0.734502i \(-0.737416\pi\)
−0.678607 + 0.734502i \(0.737416\pi\)
\(20\) 1.00000 0.223607
\(21\) 3.14896 0.687160
\(22\) 0 0
\(23\) −2.65626 −0.553869 −0.276934 0.960889i \(-0.589318\pi\)
−0.276934 + 0.960889i \(0.589318\pi\)
\(24\) 3.14896 0.642779
\(25\) 1.00000 0.200000
\(26\) −5.43886 −1.06665
\(27\) −12.3312 −2.37314
\(28\) −1.00000 −0.188982
\(29\) −6.56726 −1.21951 −0.609755 0.792590i \(-0.708732\pi\)
−0.609755 + 0.792590i \(0.708732\pi\)
\(30\) 3.14896 0.574919
\(31\) −9.68953 −1.74029 −0.870146 0.492794i \(-0.835976\pi\)
−0.870146 + 0.492794i \(0.835976\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0.477092 0.0818206
\(35\) −1.00000 −0.169031
\(36\) 6.91596 1.15266
\(37\) −4.56726 −0.750853 −0.375427 0.926852i \(-0.622504\pi\)
−0.375427 + 0.926852i \(0.622504\pi\)
\(38\) 5.91596 0.959695
\(39\) −17.1268 −2.74248
\(40\) −1.00000 −0.158114
\(41\) 2.52291 0.394012 0.197006 0.980402i \(-0.436878\pi\)
0.197006 + 0.980402i \(0.436878\pi\)
\(42\) −3.14896 −0.485895
\(43\) 11.9003 1.81479 0.907393 0.420283i \(-0.138069\pi\)
0.907393 + 0.420283i \(0.138069\pi\)
\(44\) 0 0
\(45\) 6.91596 1.03097
\(46\) 2.65626 0.391644
\(47\) 9.42632 1.37497 0.687485 0.726199i \(-0.258715\pi\)
0.687485 + 0.726199i \(0.258715\pi\)
\(48\) −3.14896 −0.454513
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 1.50234 0.210370
\(52\) 5.43886 0.754235
\(53\) 6.15392 0.845306 0.422653 0.906292i \(-0.361099\pi\)
0.422653 + 0.906292i \(0.361099\pi\)
\(54\) 12.3312 1.67806
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 18.6291 2.46749
\(58\) 6.56726 0.862324
\(59\) −0.566092 −0.0736989 −0.0368495 0.999321i \(-0.511732\pi\)
−0.0368495 + 0.999321i \(0.511732\pi\)
\(60\) −3.14896 −0.406529
\(61\) 2.04888 0.262332 0.131166 0.991360i \(-0.458128\pi\)
0.131166 + 0.991360i \(0.458128\pi\)
\(62\) 9.68953 1.23057
\(63\) −6.91596 −0.871329
\(64\) 1.00000 0.125000
\(65\) 5.43886 0.674608
\(66\) 0 0
\(67\) −1.24102 −0.151615 −0.0758076 0.997122i \(-0.524153\pi\)
−0.0758076 + 0.997122i \(0.524153\pi\)
\(68\) −0.477092 −0.0578559
\(69\) 8.36447 1.00696
\(70\) 1.00000 0.119523
\(71\) −3.62299 −0.429970 −0.214985 0.976617i \(-0.568970\pi\)
−0.214985 + 0.976617i \(0.568970\pi\)
\(72\) −6.91596 −0.815053
\(73\) 15.6398 1.83050 0.915248 0.402891i \(-0.131994\pi\)
0.915248 + 0.402891i \(0.131994\pi\)
\(74\) 4.56726 0.530933
\(75\) −3.14896 −0.363611
\(76\) −5.91596 −0.678607
\(77\) 0 0
\(78\) 17.1268 1.93923
\(79\) −3.38007 −0.380288 −0.190144 0.981756i \(-0.560895\pi\)
−0.190144 + 0.981756i \(0.560895\pi\)
\(80\) 1.00000 0.111803
\(81\) 18.0826 2.00918
\(82\) −2.52291 −0.278609
\(83\) 1.73868 0.190845 0.0954224 0.995437i \(-0.469580\pi\)
0.0954224 + 0.995437i \(0.469580\pi\)
\(84\) 3.14896 0.343580
\(85\) −0.477092 −0.0517479
\(86\) −11.9003 −1.28325
\(87\) 20.6801 2.21713
\(88\) 0 0
\(89\) −9.54364 −1.01162 −0.505812 0.862644i \(-0.668807\pi\)
−0.505812 + 0.862644i \(0.668807\pi\)
\(90\) −6.91596 −0.729006
\(91\) −5.43886 −0.570148
\(92\) −2.65626 −0.276934
\(93\) 30.5120 3.16394
\(94\) −9.42632 −0.972251
\(95\) −5.91596 −0.606964
\(96\) 3.14896 0.321390
\(97\) −3.93156 −0.399190 −0.199595 0.979879i \(-0.563963\pi\)
−0.199595 + 0.979879i \(0.563963\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −2.90794 −0.289351 −0.144675 0.989479i \(-0.546214\pi\)
−0.144675 + 0.989479i \(0.546214\pi\)
\(102\) −1.50234 −0.148754
\(103\) −15.9287 −1.56950 −0.784749 0.619814i \(-0.787208\pi\)
−0.784749 + 0.619814i \(0.787208\pi\)
\(104\) −5.43886 −0.533325
\(105\) 3.14896 0.307307
\(106\) −6.15392 −0.597721
\(107\) −6.17232 −0.596701 −0.298350 0.954456i \(-0.596436\pi\)
−0.298350 + 0.954456i \(0.596436\pi\)
\(108\) −12.3312 −1.18657
\(109\) −14.9240 −1.42946 −0.714729 0.699402i \(-0.753450\pi\)
−0.714729 + 0.699402i \(0.753450\pi\)
\(110\) 0 0
\(111\) 14.3821 1.36509
\(112\) −1.00000 −0.0944911
\(113\) 19.0906 1.79589 0.897946 0.440105i \(-0.145059\pi\)
0.897946 + 0.440105i \(0.145059\pi\)
\(114\) −18.6291 −1.74478
\(115\) −2.65626 −0.247698
\(116\) −6.56726 −0.609755
\(117\) 37.6149 3.47750
\(118\) 0.566092 0.0521130
\(119\) 0.477092 0.0437350
\(120\) 3.14896 0.287460
\(121\) 0 0
\(122\) −2.04888 −0.185497
\(123\) −7.94454 −0.716335
\(124\) −9.68953 −0.870146
\(125\) 1.00000 0.0894427
\(126\) 6.91596 0.616122
\(127\) 16.3815 1.45362 0.726812 0.686836i \(-0.241001\pi\)
0.726812 + 0.686836i \(0.241001\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −37.4737 −3.29938
\(130\) −5.43886 −0.477020
\(131\) 5.60812 0.489984 0.244992 0.969525i \(-0.421215\pi\)
0.244992 + 0.969525i \(0.421215\pi\)
\(132\) 0 0
\(133\) 5.91596 0.512978
\(134\) 1.24102 0.107208
\(135\) −12.3312 −1.06130
\(136\) 0.477092 0.0409103
\(137\) −2.50874 −0.214336 −0.107168 0.994241i \(-0.534178\pi\)
−0.107168 + 0.994241i \(0.534178\pi\)
\(138\) −8.36447 −0.712031
\(139\) 3.30261 0.280124 0.140062 0.990143i \(-0.455270\pi\)
0.140062 + 0.990143i \(0.455270\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −29.6831 −2.49977
\(142\) 3.62299 0.304035
\(143\) 0 0
\(144\) 6.91596 0.576330
\(145\) −6.56726 −0.545381
\(146\) −15.6398 −1.29436
\(147\) −3.14896 −0.259722
\(148\) −4.56726 −0.375427
\(149\) 6.83048 0.559574 0.279787 0.960062i \(-0.409736\pi\)
0.279787 + 0.960062i \(0.409736\pi\)
\(150\) 3.14896 0.257112
\(151\) −10.8189 −0.880433 −0.440216 0.897892i \(-0.645098\pi\)
−0.440216 + 0.897892i \(0.645098\pi\)
\(152\) 5.91596 0.479847
\(153\) −3.29955 −0.266753
\(154\) 0 0
\(155\) −9.68953 −0.778282
\(156\) −17.1268 −1.37124
\(157\) 8.01766 0.639879 0.319940 0.947438i \(-0.396337\pi\)
0.319940 + 0.947438i \(0.396337\pi\)
\(158\) 3.38007 0.268904
\(159\) −19.3784 −1.53681
\(160\) −1.00000 −0.0790569
\(161\) 2.65626 0.209343
\(162\) −18.0826 −1.42070
\(163\) −18.2991 −1.43330 −0.716648 0.697435i \(-0.754324\pi\)
−0.716648 + 0.697435i \(0.754324\pi\)
\(164\) 2.52291 0.197006
\(165\) 0 0
\(166\) −1.73868 −0.134948
\(167\) −17.9260 −1.38716 −0.693579 0.720381i \(-0.743967\pi\)
−0.693579 + 0.720381i \(0.743967\pi\)
\(168\) −3.14896 −0.242948
\(169\) 16.5812 1.27548
\(170\) 0.477092 0.0365913
\(171\) −40.9145 −3.12881
\(172\) 11.9003 0.907393
\(173\) −7.68791 −0.584501 −0.292250 0.956342i \(-0.594404\pi\)
−0.292250 + 0.956342i \(0.594404\pi\)
\(174\) −20.6801 −1.56775
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 1.78260 0.133989
\(178\) 9.54364 0.715326
\(179\) 22.7197 1.69815 0.849076 0.528271i \(-0.177160\pi\)
0.849076 + 0.528271i \(0.177160\pi\)
\(180\) 6.91596 0.515485
\(181\) −13.6567 −1.01509 −0.507547 0.861624i \(-0.669448\pi\)
−0.507547 + 0.861624i \(0.669448\pi\)
\(182\) 5.43886 0.403155
\(183\) −6.45184 −0.476934
\(184\) 2.65626 0.195822
\(185\) −4.56726 −0.335792
\(186\) −30.5120 −2.23725
\(187\) 0 0
\(188\) 9.42632 0.687485
\(189\) 12.3312 0.896962
\(190\) 5.91596 0.429189
\(191\) 9.42182 0.681739 0.340869 0.940111i \(-0.389279\pi\)
0.340869 + 0.940111i \(0.389279\pi\)
\(192\) −3.14896 −0.227257
\(193\) −5.09919 −0.367048 −0.183524 0.983015i \(-0.558751\pi\)
−0.183524 + 0.983015i \(0.558751\pi\)
\(194\) 3.93156 0.282270
\(195\) −17.1268 −1.22647
\(196\) 1.00000 0.0714286
\(197\) 17.1284 1.22035 0.610174 0.792268i \(-0.291100\pi\)
0.610174 + 0.792268i \(0.291100\pi\)
\(198\) 0 0
\(199\) 11.9225 0.845166 0.422583 0.906324i \(-0.361123\pi\)
0.422583 + 0.906324i \(0.361123\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 3.90794 0.275645
\(202\) 2.90794 0.204602
\(203\) 6.56726 0.460931
\(204\) 1.50234 0.105185
\(205\) 2.52291 0.176208
\(206\) 15.9287 1.10980
\(207\) −18.3706 −1.27684
\(208\) 5.43886 0.377117
\(209\) 0 0
\(210\) −3.14896 −0.217299
\(211\) 22.8331 1.57189 0.785947 0.618294i \(-0.212176\pi\)
0.785947 + 0.618294i \(0.212176\pi\)
\(212\) 6.15392 0.422653
\(213\) 11.4087 0.781708
\(214\) 6.17232 0.421931
\(215\) 11.9003 0.811597
\(216\) 12.3312 0.839031
\(217\) 9.68953 0.657768
\(218\) 14.9240 1.01078
\(219\) −49.2490 −3.32794
\(220\) 0 0
\(221\) −2.59484 −0.174548
\(222\) −14.3821 −0.965265
\(223\) −15.8782 −1.06328 −0.531640 0.846970i \(-0.678424\pi\)
−0.531640 + 0.846970i \(0.678424\pi\)
\(224\) 1.00000 0.0668153
\(225\) 6.91596 0.461064
\(226\) −19.0906 −1.26989
\(227\) −8.41830 −0.558742 −0.279371 0.960183i \(-0.590126\pi\)
−0.279371 + 0.960183i \(0.590126\pi\)
\(228\) 18.6291 1.23374
\(229\) 22.2844 1.47259 0.736296 0.676659i \(-0.236573\pi\)
0.736296 + 0.676659i \(0.236573\pi\)
\(230\) 2.65626 0.175149
\(231\) 0 0
\(232\) 6.56726 0.431162
\(233\) −12.4011 −0.812421 −0.406210 0.913780i \(-0.633150\pi\)
−0.406210 + 0.913780i \(0.633150\pi\)
\(234\) −37.6149 −2.45897
\(235\) 9.42632 0.614905
\(236\) −0.566092 −0.0368495
\(237\) 10.6437 0.691384
\(238\) −0.477092 −0.0309253
\(239\) 19.3571 1.25211 0.626055 0.779779i \(-0.284669\pi\)
0.626055 + 0.779779i \(0.284669\pi\)
\(240\) −3.14896 −0.203265
\(241\) −5.12445 −0.330095 −0.165047 0.986286i \(-0.552778\pi\)
−0.165047 + 0.986286i \(0.552778\pi\)
\(242\) 0 0
\(243\) −19.9478 −1.27965
\(244\) 2.04888 0.131166
\(245\) 1.00000 0.0638877
\(246\) 7.94454 0.506525
\(247\) −32.1761 −2.04732
\(248\) 9.68953 0.615286
\(249\) −5.47503 −0.346966
\(250\) −1.00000 −0.0632456
\(251\) −8.99152 −0.567540 −0.283770 0.958892i \(-0.591585\pi\)
−0.283770 + 0.958892i \(0.591585\pi\)
\(252\) −6.91596 −0.435664
\(253\) 0 0
\(254\) −16.3815 −1.02787
\(255\) 1.50234 0.0940805
\(256\) 1.00000 0.0625000
\(257\) −17.9476 −1.11954 −0.559770 0.828648i \(-0.689111\pi\)
−0.559770 + 0.828648i \(0.689111\pi\)
\(258\) 37.4737 2.33301
\(259\) 4.56726 0.283796
\(260\) 5.43886 0.337304
\(261\) −45.4189 −2.81136
\(262\) −5.60812 −0.346471
\(263\) 27.3312 1.68531 0.842657 0.538451i \(-0.180990\pi\)
0.842657 + 0.538451i \(0.180990\pi\)
\(264\) 0 0
\(265\) 6.15392 0.378032
\(266\) −5.91596 −0.362731
\(267\) 30.0525 1.83919
\(268\) −1.24102 −0.0758076
\(269\) −12.0775 −0.736376 −0.368188 0.929751i \(-0.620022\pi\)
−0.368188 + 0.929751i \(0.620022\pi\)
\(270\) 12.3312 0.750453
\(271\) 25.8325 1.56921 0.784607 0.619993i \(-0.212865\pi\)
0.784607 + 0.619993i \(0.212865\pi\)
\(272\) −0.477092 −0.0289280
\(273\) 17.1268 1.03656
\(274\) 2.50874 0.151558
\(275\) 0 0
\(276\) 8.36447 0.503482
\(277\) −18.2240 −1.09497 −0.547486 0.836815i \(-0.684415\pi\)
−0.547486 + 0.836815i \(0.684415\pi\)
\(278\) −3.30261 −0.198077
\(279\) −67.0124 −4.01193
\(280\) 1.00000 0.0597614
\(281\) −4.05953 −0.242171 −0.121086 0.992642i \(-0.538638\pi\)
−0.121086 + 0.992642i \(0.538638\pi\)
\(282\) 29.6831 1.76760
\(283\) −3.16574 −0.188184 −0.0940918 0.995564i \(-0.529995\pi\)
−0.0940918 + 0.995564i \(0.529995\pi\)
\(284\) −3.62299 −0.214985
\(285\) 18.6291 1.10349
\(286\) 0 0
\(287\) −2.52291 −0.148923
\(288\) −6.91596 −0.407527
\(289\) −16.7724 −0.986611
\(290\) 6.56726 0.385643
\(291\) 12.3803 0.725748
\(292\) 15.6398 0.915248
\(293\) 10.2517 0.598909 0.299455 0.954111i \(-0.403195\pi\)
0.299455 + 0.954111i \(0.403195\pi\)
\(294\) 3.14896 0.183651
\(295\) −0.566092 −0.0329592
\(296\) 4.56726 0.265467
\(297\) 0 0
\(298\) −6.83048 −0.395679
\(299\) −14.4470 −0.835494
\(300\) −3.14896 −0.181805
\(301\) −11.9003 −0.685925
\(302\) 10.8189 0.622560
\(303\) 9.15698 0.526055
\(304\) −5.91596 −0.339303
\(305\) 2.04888 0.117318
\(306\) 3.29955 0.188623
\(307\) −3.38972 −0.193461 −0.0967307 0.995311i \(-0.530839\pi\)
−0.0967307 + 0.995311i \(0.530839\pi\)
\(308\) 0 0
\(309\) 50.1587 2.85343
\(310\) 9.68953 0.550329
\(311\) −29.4523 −1.67009 −0.835043 0.550185i \(-0.814557\pi\)
−0.835043 + 0.550185i \(0.814557\pi\)
\(312\) 17.1268 0.969613
\(313\) −22.0645 −1.24716 −0.623579 0.781761i \(-0.714322\pi\)
−0.623579 + 0.781761i \(0.714322\pi\)
\(314\) −8.01766 −0.452463
\(315\) −6.91596 −0.389670
\(316\) −3.38007 −0.190144
\(317\) 21.1159 1.18598 0.592992 0.805208i \(-0.297946\pi\)
0.592992 + 0.805208i \(0.297946\pi\)
\(318\) 19.3784 1.08669
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 19.4364 1.08483
\(322\) −2.65626 −0.148028
\(323\) 2.82246 0.157046
\(324\) 18.0826 1.00459
\(325\) 5.43886 0.301694
\(326\) 18.2991 1.01349
\(327\) 46.9950 2.59883
\(328\) −2.52291 −0.139304
\(329\) −9.42632 −0.519690
\(330\) 0 0
\(331\) −31.1409 −1.71166 −0.855830 0.517258i \(-0.826953\pi\)
−0.855830 + 0.517258i \(0.826953\pi\)
\(332\) 1.73868 0.0954224
\(333\) −31.5870 −1.73096
\(334\) 17.9260 0.980869
\(335\) −1.24102 −0.0678044
\(336\) 3.14896 0.171790
\(337\) −1.60099 −0.0872115 −0.0436057 0.999049i \(-0.513885\pi\)
−0.0436057 + 0.999049i \(0.513885\pi\)
\(338\) −16.5812 −0.901901
\(339\) −60.1156 −3.26503
\(340\) −0.477092 −0.0258740
\(341\) 0 0
\(342\) 40.9145 2.21240
\(343\) −1.00000 −0.0539949
\(344\) −11.9003 −0.641624
\(345\) 8.36447 0.450328
\(346\) 7.68791 0.413304
\(347\) 19.7193 1.05859 0.529293 0.848439i \(-0.322457\pi\)
0.529293 + 0.848439i \(0.322457\pi\)
\(348\) 20.6801 1.10857
\(349\) 15.3791 0.823223 0.411611 0.911359i \(-0.364966\pi\)
0.411611 + 0.911359i \(0.364966\pi\)
\(350\) 1.00000 0.0534522
\(351\) −67.0677 −3.57981
\(352\) 0 0
\(353\) −10.3793 −0.552436 −0.276218 0.961095i \(-0.589081\pi\)
−0.276218 + 0.961095i \(0.589081\pi\)
\(354\) −1.78260 −0.0947442
\(355\) −3.62299 −0.192288
\(356\) −9.54364 −0.505812
\(357\) −1.50234 −0.0795125
\(358\) −22.7197 −1.20077
\(359\) 23.8319 1.25780 0.628900 0.777486i \(-0.283505\pi\)
0.628900 + 0.777486i \(0.283505\pi\)
\(360\) −6.91596 −0.364503
\(361\) 15.9985 0.842028
\(362\) 13.6567 0.717780
\(363\) 0 0
\(364\) −5.43886 −0.285074
\(365\) 15.6398 0.818623
\(366\) 6.45184 0.337243
\(367\) −19.3945 −1.01238 −0.506192 0.862421i \(-0.668947\pi\)
−0.506192 + 0.862421i \(0.668947\pi\)
\(368\) −2.65626 −0.138467
\(369\) 17.4483 0.908323
\(370\) 4.56726 0.237441
\(371\) −6.15392 −0.319495
\(372\) 30.5120 1.58197
\(373\) 10.9015 0.564459 0.282230 0.959347i \(-0.408926\pi\)
0.282230 + 0.959347i \(0.408926\pi\)
\(374\) 0 0
\(375\) −3.14896 −0.162612
\(376\) −9.42632 −0.486125
\(377\) −35.7184 −1.83959
\(378\) −12.3312 −0.634248
\(379\) 6.80767 0.349686 0.174843 0.984596i \(-0.444058\pi\)
0.174843 + 0.984596i \(0.444058\pi\)
\(380\) −5.91596 −0.303482
\(381\) −51.5847 −2.64277
\(382\) −9.42182 −0.482062
\(383\) −24.2311 −1.23815 −0.619075 0.785332i \(-0.712492\pi\)
−0.619075 + 0.785332i \(0.712492\pi\)
\(384\) 3.14896 0.160695
\(385\) 0 0
\(386\) 5.09919 0.259542
\(387\) 82.3023 4.18366
\(388\) −3.93156 −0.199595
\(389\) −18.2552 −0.925574 −0.462787 0.886470i \(-0.653150\pi\)
−0.462787 + 0.886470i \(0.653150\pi\)
\(390\) 17.1268 0.867248
\(391\) 1.26728 0.0640892
\(392\) −1.00000 −0.0505076
\(393\) −17.6598 −0.890817
\(394\) −17.1284 −0.862916
\(395\) −3.38007 −0.170070
\(396\) 0 0
\(397\) 17.6742 0.887043 0.443522 0.896264i \(-0.353729\pi\)
0.443522 + 0.896264i \(0.353729\pi\)
\(398\) −11.9225 −0.597623
\(399\) −18.6291 −0.932622
\(400\) 1.00000 0.0500000
\(401\) 30.4187 1.51904 0.759520 0.650485i \(-0.225434\pi\)
0.759520 + 0.650485i \(0.225434\pi\)
\(402\) −3.90794 −0.194910
\(403\) −52.7001 −2.62518
\(404\) −2.90794 −0.144675
\(405\) 18.0826 0.898531
\(406\) −6.56726 −0.325928
\(407\) 0 0
\(408\) −1.50234 −0.0743771
\(409\) 15.4065 0.761802 0.380901 0.924616i \(-0.375614\pi\)
0.380901 + 0.924616i \(0.375614\pi\)
\(410\) −2.52291 −0.124598
\(411\) 7.89992 0.389674
\(412\) −15.9287 −0.784749
\(413\) 0.566092 0.0278556
\(414\) 18.3706 0.902865
\(415\) 1.73868 0.0853484
\(416\) −5.43886 −0.266662
\(417\) −10.3998 −0.509280
\(418\) 0 0
\(419\) 3.21914 0.157265 0.0786327 0.996904i \(-0.474945\pi\)
0.0786327 + 0.996904i \(0.474945\pi\)
\(420\) 3.14896 0.153654
\(421\) 13.1605 0.641402 0.320701 0.947180i \(-0.396081\pi\)
0.320701 + 0.947180i \(0.396081\pi\)
\(422\) −22.8331 −1.11150
\(423\) 65.1920 3.16974
\(424\) −6.15392 −0.298861
\(425\) −0.477092 −0.0231424
\(426\) −11.4087 −0.552751
\(427\) −2.04888 −0.0991522
\(428\) −6.17232 −0.298350
\(429\) 0 0
\(430\) −11.9003 −0.573886
\(431\) 2.38429 0.114847 0.0574236 0.998350i \(-0.481711\pi\)
0.0574236 + 0.998350i \(0.481711\pi\)
\(432\) −12.3312 −0.593285
\(433\) 9.02568 0.433747 0.216873 0.976200i \(-0.430414\pi\)
0.216873 + 0.976200i \(0.430414\pi\)
\(434\) −9.68953 −0.465113
\(435\) 20.6801 0.991533
\(436\) −14.9240 −0.714729
\(437\) 15.7143 0.751718
\(438\) 49.2490 2.35321
\(439\) 23.1241 1.10366 0.551828 0.833958i \(-0.313931\pi\)
0.551828 + 0.833958i \(0.313931\pi\)
\(440\) 0 0
\(441\) 6.91596 0.329331
\(442\) 2.59484 0.123424
\(443\) −6.78799 −0.322507 −0.161254 0.986913i \(-0.551554\pi\)
−0.161254 + 0.986913i \(0.551554\pi\)
\(444\) 14.3821 0.682546
\(445\) −9.54364 −0.452412
\(446\) 15.8782 0.751853
\(447\) −21.5089 −1.01734
\(448\) −1.00000 −0.0472456
\(449\) 13.1114 0.618767 0.309383 0.950937i \(-0.399877\pi\)
0.309383 + 0.950937i \(0.399877\pi\)
\(450\) −6.91596 −0.326021
\(451\) 0 0
\(452\) 19.0906 0.897946
\(453\) 34.0684 1.60067
\(454\) 8.41830 0.395090
\(455\) −5.43886 −0.254978
\(456\) −18.6291 −0.872388
\(457\) −19.0200 −0.889718 −0.444859 0.895601i \(-0.646746\pi\)
−0.444859 + 0.895601i \(0.646746\pi\)
\(458\) −22.2844 −1.04128
\(459\) 5.88312 0.274600
\(460\) −2.65626 −0.123849
\(461\) 18.8767 0.879177 0.439588 0.898199i \(-0.355124\pi\)
0.439588 + 0.898199i \(0.355124\pi\)
\(462\) 0 0
\(463\) 13.6194 0.632945 0.316473 0.948602i \(-0.397501\pi\)
0.316473 + 0.948602i \(0.397501\pi\)
\(464\) −6.56726 −0.304877
\(465\) 30.5120 1.41496
\(466\) 12.4011 0.574468
\(467\) −18.9801 −0.878296 −0.439148 0.898415i \(-0.644720\pi\)
−0.439148 + 0.898415i \(0.644720\pi\)
\(468\) 37.6149 1.73875
\(469\) 1.24102 0.0573052
\(470\) −9.42632 −0.434804
\(471\) −25.2473 −1.16333
\(472\) 0.566092 0.0260565
\(473\) 0 0
\(474\) −10.6437 −0.488882
\(475\) −5.91596 −0.271443
\(476\) 0.477092 0.0218675
\(477\) 42.5602 1.94870
\(478\) −19.3571 −0.885375
\(479\) −5.51532 −0.252001 −0.126001 0.992030i \(-0.540214\pi\)
−0.126001 + 0.992030i \(0.540214\pi\)
\(480\) 3.14896 0.143730
\(481\) −24.8407 −1.13264
\(482\) 5.12445 0.233412
\(483\) −8.36447 −0.380596
\(484\) 0 0
\(485\) −3.93156 −0.178523
\(486\) 19.9478 0.904849
\(487\) 28.0086 1.26919 0.634596 0.772844i \(-0.281167\pi\)
0.634596 + 0.772844i \(0.281167\pi\)
\(488\) −2.04888 −0.0927484
\(489\) 57.6231 2.60581
\(490\) −1.00000 −0.0451754
\(491\) −12.9545 −0.584626 −0.292313 0.956323i \(-0.594425\pi\)
−0.292313 + 0.956323i \(0.594425\pi\)
\(492\) −7.94454 −0.358167
\(493\) 3.13319 0.141112
\(494\) 32.1761 1.44767
\(495\) 0 0
\(496\) −9.68953 −0.435073
\(497\) 3.62299 0.162513
\(498\) 5.47503 0.245342
\(499\) 3.93985 0.176372 0.0881859 0.996104i \(-0.471893\pi\)
0.0881859 + 0.996104i \(0.471893\pi\)
\(500\) 1.00000 0.0447214
\(501\) 56.4484 2.52193
\(502\) 8.99152 0.401311
\(503\) 28.7158 1.28037 0.640186 0.768220i \(-0.278857\pi\)
0.640186 + 0.768220i \(0.278857\pi\)
\(504\) 6.91596 0.308061
\(505\) −2.90794 −0.129401
\(506\) 0 0
\(507\) −52.2137 −2.31889
\(508\) 16.3815 0.726812
\(509\) 6.73822 0.298666 0.149333 0.988787i \(-0.452287\pi\)
0.149333 + 0.988787i \(0.452287\pi\)
\(510\) −1.50234 −0.0665249
\(511\) −15.6398 −0.691863
\(512\) −1.00000 −0.0441942
\(513\) 72.9508 3.22086
\(514\) 17.9476 0.791635
\(515\) −15.9287 −0.701901
\(516\) −37.4737 −1.64969
\(517\) 0 0
\(518\) −4.56726 −0.200674
\(519\) 24.2089 1.06265
\(520\) −5.43886 −0.238510
\(521\) 32.2003 1.41072 0.705361 0.708848i \(-0.250785\pi\)
0.705361 + 0.708848i \(0.250785\pi\)
\(522\) 45.4189 1.98793
\(523\) −28.4244 −1.24291 −0.621457 0.783449i \(-0.713459\pi\)
−0.621457 + 0.783449i \(0.713459\pi\)
\(524\) 5.60812 0.244992
\(525\) 3.14896 0.137432
\(526\) −27.3312 −1.19170
\(527\) 4.62280 0.201372
\(528\) 0 0
\(529\) −15.9443 −0.693229
\(530\) −6.15392 −0.267309
\(531\) −3.91507 −0.169899
\(532\) 5.91596 0.256489
\(533\) 13.7218 0.594355
\(534\) −30.0525 −1.30050
\(535\) −6.17232 −0.266853
\(536\) 1.24102 0.0536041
\(537\) −71.5435 −3.08733
\(538\) 12.0775 0.520696
\(539\) 0 0
\(540\) −12.3312 −0.530650
\(541\) 21.6078 0.928993 0.464496 0.885575i \(-0.346235\pi\)
0.464496 + 0.885575i \(0.346235\pi\)
\(542\) −25.8325 −1.10960
\(543\) 43.0044 1.84550
\(544\) 0.477092 0.0204552
\(545\) −14.9240 −0.639273
\(546\) −17.1268 −0.732958
\(547\) 31.9393 1.36562 0.682812 0.730594i \(-0.260757\pi\)
0.682812 + 0.730594i \(0.260757\pi\)
\(548\) −2.50874 −0.107168
\(549\) 14.1700 0.604759
\(550\) 0 0
\(551\) 38.8516 1.65514
\(552\) −8.36447 −0.356015
\(553\) 3.38007 0.143735
\(554\) 18.2240 0.774262
\(555\) 14.3821 0.610487
\(556\) 3.30261 0.140062
\(557\) 18.6958 0.792169 0.396084 0.918214i \(-0.370369\pi\)
0.396084 + 0.918214i \(0.370369\pi\)
\(558\) 67.0124 2.83686
\(559\) 64.7244 2.73755
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 4.05953 0.171241
\(563\) 41.8680 1.76453 0.882263 0.470758i \(-0.156019\pi\)
0.882263 + 0.470758i \(0.156019\pi\)
\(564\) −29.6831 −1.24988
\(565\) 19.0906 0.803148
\(566\) 3.16574 0.133066
\(567\) −18.0826 −0.759397
\(568\) 3.62299 0.152017
\(569\) 14.7563 0.618618 0.309309 0.950962i \(-0.399902\pi\)
0.309309 + 0.950962i \(0.399902\pi\)
\(570\) −18.6291 −0.780288
\(571\) −43.4272 −1.81737 −0.908686 0.417481i \(-0.862913\pi\)
−0.908686 + 0.417481i \(0.862913\pi\)
\(572\) 0 0
\(573\) −29.6689 −1.23944
\(574\) 2.52291 0.105304
\(575\) −2.65626 −0.110774
\(576\) 6.91596 0.288165
\(577\) −19.1867 −0.798754 −0.399377 0.916787i \(-0.630774\pi\)
−0.399377 + 0.916787i \(0.630774\pi\)
\(578\) 16.7724 0.697639
\(579\) 16.0572 0.667313
\(580\) −6.56726 −0.272691
\(581\) −1.73868 −0.0721326
\(582\) −12.3803 −0.513182
\(583\) 0 0
\(584\) −15.6398 −0.647178
\(585\) 37.6149 1.55519
\(586\) −10.2517 −0.423493
\(587\) 3.76583 0.155432 0.0777161 0.996976i \(-0.475237\pi\)
0.0777161 + 0.996976i \(0.475237\pi\)
\(588\) −3.14896 −0.129861
\(589\) 57.3229 2.36195
\(590\) 0.566092 0.0233056
\(591\) −53.9367 −2.21866
\(592\) −4.56726 −0.187713
\(593\) 37.8401 1.55391 0.776954 0.629558i \(-0.216764\pi\)
0.776954 + 0.629558i \(0.216764\pi\)
\(594\) 0 0
\(595\) 0.477092 0.0195589
\(596\) 6.83048 0.279787
\(597\) −37.5436 −1.53656
\(598\) 14.4470 0.590784
\(599\) −32.2167 −1.31634 −0.658169 0.752870i \(-0.728669\pi\)
−0.658169 + 0.752870i \(0.728669\pi\)
\(600\) 3.14896 0.128556
\(601\) 3.26376 0.133132 0.0665658 0.997782i \(-0.478796\pi\)
0.0665658 + 0.997782i \(0.478796\pi\)
\(602\) 11.9003 0.485022
\(603\) −8.58287 −0.349521
\(604\) −10.8189 −0.440216
\(605\) 0 0
\(606\) −9.15698 −0.371977
\(607\) 19.3916 0.787081 0.393541 0.919307i \(-0.371250\pi\)
0.393541 + 0.919307i \(0.371250\pi\)
\(608\) 5.91596 0.239924
\(609\) −20.6801 −0.837998
\(610\) −2.04888 −0.0829567
\(611\) 51.2685 2.07410
\(612\) −3.29955 −0.133376
\(613\) 29.8111 1.20406 0.602029 0.798474i \(-0.294359\pi\)
0.602029 + 0.798474i \(0.294359\pi\)
\(614\) 3.38972 0.136798
\(615\) −7.94454 −0.320355
\(616\) 0 0
\(617\) −3.60999 −0.145333 −0.0726664 0.997356i \(-0.523151\pi\)
−0.0726664 + 0.997356i \(0.523151\pi\)
\(618\) −50.1587 −2.01768
\(619\) 26.7648 1.07577 0.537884 0.843019i \(-0.319224\pi\)
0.537884 + 0.843019i \(0.319224\pi\)
\(620\) −9.68953 −0.389141
\(621\) 32.7549 1.31441
\(622\) 29.4523 1.18093
\(623\) 9.54364 0.382358
\(624\) −17.1268 −0.685620
\(625\) 1.00000 0.0400000
\(626\) 22.0645 0.881873
\(627\) 0 0
\(628\) 8.01766 0.319940
\(629\) 2.17900 0.0868826
\(630\) 6.91596 0.275538
\(631\) −42.4651 −1.69051 −0.845254 0.534364i \(-0.820551\pi\)
−0.845254 + 0.534364i \(0.820551\pi\)
\(632\) 3.38007 0.134452
\(633\) −71.9005 −2.85779
\(634\) −21.1159 −0.838618
\(635\) 16.3815 0.650080
\(636\) −19.3784 −0.768405
\(637\) 5.43886 0.215496
\(638\) 0 0
\(639\) −25.0564 −0.991218
\(640\) −1.00000 −0.0395285
\(641\) −12.7639 −0.504144 −0.252072 0.967708i \(-0.581112\pi\)
−0.252072 + 0.967708i \(0.581112\pi\)
\(642\) −19.4364 −0.767093
\(643\) 1.15422 0.0455181 0.0227591 0.999741i \(-0.492755\pi\)
0.0227591 + 0.999741i \(0.492755\pi\)
\(644\) 2.65626 0.104671
\(645\) −37.4737 −1.47553
\(646\) −2.82246 −0.111048
\(647\) −11.1492 −0.438321 −0.219161 0.975689i \(-0.570332\pi\)
−0.219161 + 0.975689i \(0.570332\pi\)
\(648\) −18.0826 −0.710351
\(649\) 0 0
\(650\) −5.43886 −0.213330
\(651\) −30.5120 −1.19586
\(652\) −18.2991 −0.716648
\(653\) 16.0342 0.627468 0.313734 0.949511i \(-0.398420\pi\)
0.313734 + 0.949511i \(0.398420\pi\)
\(654\) −46.9950 −1.83765
\(655\) 5.60812 0.219127
\(656\) 2.52291 0.0985030
\(657\) 108.164 4.21988
\(658\) 9.42632 0.367476
\(659\) 29.5362 1.15057 0.575283 0.817955i \(-0.304892\pi\)
0.575283 + 0.817955i \(0.304892\pi\)
\(660\) 0 0
\(661\) 9.49650 0.369371 0.184686 0.982798i \(-0.440873\pi\)
0.184686 + 0.982798i \(0.440873\pi\)
\(662\) 31.1409 1.21033
\(663\) 8.17105 0.317337
\(664\) −1.73868 −0.0674739
\(665\) 5.91596 0.229411
\(666\) 31.5870 1.22397
\(667\) 17.4444 0.675449
\(668\) −17.9260 −0.693579
\(669\) 49.9997 1.93310
\(670\) 1.24102 0.0479449
\(671\) 0 0
\(672\) −3.14896 −0.121474
\(673\) 11.5388 0.444788 0.222394 0.974957i \(-0.428613\pi\)
0.222394 + 0.974957i \(0.428613\pi\)
\(674\) 1.60099 0.0616678
\(675\) −12.3312 −0.474628
\(676\) 16.5812 0.637740
\(677\) −48.8523 −1.87755 −0.938774 0.344533i \(-0.888037\pi\)
−0.938774 + 0.344533i \(0.888037\pi\)
\(678\) 60.1156 2.30872
\(679\) 3.93156 0.150880
\(680\) 0.477092 0.0182956
\(681\) 26.5089 1.01582
\(682\) 0 0
\(683\) 35.4736 1.35736 0.678679 0.734435i \(-0.262553\pi\)
0.678679 + 0.734435i \(0.262553\pi\)
\(684\) −40.9145 −1.56440
\(685\) −2.50874 −0.0958539
\(686\) 1.00000 0.0381802
\(687\) −70.1726 −2.67725
\(688\) 11.9003 0.453697
\(689\) 33.4703 1.27512
\(690\) −8.36447 −0.318430
\(691\) −30.1855 −1.14831 −0.574156 0.818746i \(-0.694670\pi\)
−0.574156 + 0.818746i \(0.694670\pi\)
\(692\) −7.68791 −0.292250
\(693\) 0 0
\(694\) −19.7193 −0.748534
\(695\) 3.30261 0.125275
\(696\) −20.6801 −0.783875
\(697\) −1.20366 −0.0455919
\(698\) −15.3791 −0.582106
\(699\) 39.0505 1.47702
\(700\) −1.00000 −0.0377964
\(701\) −11.2689 −0.425621 −0.212810 0.977094i \(-0.568262\pi\)
−0.212810 + 0.977094i \(0.568262\pi\)
\(702\) 67.0677 2.53131
\(703\) 27.0197 1.01907
\(704\) 0 0
\(705\) −29.6831 −1.11793
\(706\) 10.3793 0.390631
\(707\) 2.90794 0.109364
\(708\) 1.78260 0.0669943
\(709\) −43.0830 −1.61802 −0.809008 0.587797i \(-0.799995\pi\)
−0.809008 + 0.587797i \(0.799995\pi\)
\(710\) 3.62299 0.135968
\(711\) −23.3764 −0.876685
\(712\) 9.54364 0.357663
\(713\) 25.7379 0.963893
\(714\) 1.50234 0.0562238
\(715\) 0 0
\(716\) 22.7197 0.849076
\(717\) −60.9549 −2.27640
\(718\) −23.8319 −0.889399
\(719\) 27.1078 1.01095 0.505475 0.862841i \(-0.331317\pi\)
0.505475 + 0.862841i \(0.331317\pi\)
\(720\) 6.91596 0.257742
\(721\) 15.9287 0.593214
\(722\) −15.9985 −0.595404
\(723\) 16.1367 0.600130
\(724\) −13.6567 −0.507547
\(725\) −6.56726 −0.243902
\(726\) 0 0
\(727\) 45.5273 1.68851 0.844257 0.535939i \(-0.180042\pi\)
0.844257 + 0.535939i \(0.180042\pi\)
\(728\) 5.43886 0.201578
\(729\) 8.56700 0.317296
\(730\) −15.6398 −0.578854
\(731\) −5.67756 −0.209992
\(732\) −6.45184 −0.238467
\(733\) −23.4004 −0.864312 −0.432156 0.901799i \(-0.642247\pi\)
−0.432156 + 0.901799i \(0.642247\pi\)
\(734\) 19.3945 0.715864
\(735\) −3.14896 −0.116151
\(736\) 2.65626 0.0979111
\(737\) 0 0
\(738\) −17.4483 −0.642282
\(739\) 41.0880 1.51145 0.755723 0.654892i \(-0.227286\pi\)
0.755723 + 0.654892i \(0.227286\pi\)
\(740\) −4.56726 −0.167896
\(741\) 101.321 3.72213
\(742\) 6.15392 0.225917
\(743\) 16.0312 0.588128 0.294064 0.955786i \(-0.404992\pi\)
0.294064 + 0.955786i \(0.404992\pi\)
\(744\) −30.5120 −1.11862
\(745\) 6.83048 0.250249
\(746\) −10.9015 −0.399133
\(747\) 12.0246 0.439958
\(748\) 0 0
\(749\) 6.17232 0.225532
\(750\) 3.14896 0.114984
\(751\) 5.53050 0.201811 0.100905 0.994896i \(-0.467826\pi\)
0.100905 + 0.994896i \(0.467826\pi\)
\(752\) 9.42632 0.343743
\(753\) 28.3140 1.03182
\(754\) 35.7184 1.30079
\(755\) −10.8189 −0.393741
\(756\) 12.3312 0.448481
\(757\) 25.0946 0.912077 0.456039 0.889960i \(-0.349268\pi\)
0.456039 + 0.889960i \(0.349268\pi\)
\(758\) −6.80767 −0.247266
\(759\) 0 0
\(760\) 5.91596 0.214594
\(761\) 3.21785 0.116647 0.0583236 0.998298i \(-0.481424\pi\)
0.0583236 + 0.998298i \(0.481424\pi\)
\(762\) 51.5847 1.86872
\(763\) 14.9240 0.540284
\(764\) 9.42182 0.340869
\(765\) −3.29955 −0.119295
\(766\) 24.2311 0.875505
\(767\) −3.07890 −0.111173
\(768\) −3.14896 −0.113628
\(769\) 9.89520 0.356830 0.178415 0.983955i \(-0.442903\pi\)
0.178415 + 0.983955i \(0.442903\pi\)
\(770\) 0 0
\(771\) 56.5163 2.03538
\(772\) −5.09919 −0.183524
\(773\) 18.3508 0.660033 0.330017 0.943975i \(-0.392946\pi\)
0.330017 + 0.943975i \(0.392946\pi\)
\(774\) −82.3023 −2.95829
\(775\) −9.68953 −0.348058
\(776\) 3.93156 0.141135
\(777\) −14.3821 −0.515956
\(778\) 18.2552 0.654480
\(779\) −14.9254 −0.534758
\(780\) −17.1268 −0.613237
\(781\) 0 0
\(782\) −1.26728 −0.0453179
\(783\) 80.9822 2.89407
\(784\) 1.00000 0.0357143
\(785\) 8.01766 0.286163
\(786\) 17.6598 0.629903
\(787\) 6.04479 0.215473 0.107737 0.994179i \(-0.465640\pi\)
0.107737 + 0.994179i \(0.465640\pi\)
\(788\) 17.1284 0.610174
\(789\) −86.0649 −3.06399
\(790\) 3.38007 0.120258
\(791\) −19.0906 −0.678784
\(792\) 0 0
\(793\) 11.1436 0.395720
\(794\) −17.6742 −0.627234
\(795\) −19.3784 −0.687283
\(796\) 11.9225 0.422583
\(797\) −6.98324 −0.247359 −0.123679 0.992322i \(-0.539469\pi\)
−0.123679 + 0.992322i \(0.539469\pi\)
\(798\) 18.6291 0.659464
\(799\) −4.49722 −0.159100
\(800\) −1.00000 −0.0353553
\(801\) −66.0034 −2.33211
\(802\) −30.4187 −1.07412
\(803\) 0 0
\(804\) 3.90794 0.137822
\(805\) 2.65626 0.0936209
\(806\) 52.7001 1.85628
\(807\) 38.0315 1.33877
\(808\) 2.90794 0.102301
\(809\) −12.1699 −0.427870 −0.213935 0.976848i \(-0.568628\pi\)
−0.213935 + 0.976848i \(0.568628\pi\)
\(810\) −18.0826 −0.635357
\(811\) −45.0840 −1.58311 −0.791557 0.611096i \(-0.790729\pi\)
−0.791557 + 0.611096i \(0.790729\pi\)
\(812\) 6.56726 0.230466
\(813\) −81.3456 −2.85292
\(814\) 0 0
\(815\) −18.2991 −0.640989
\(816\) 1.50234 0.0525926
\(817\) −70.4019 −2.46305
\(818\) −15.4065 −0.538675
\(819\) −37.6149 −1.31437
\(820\) 2.52291 0.0881038
\(821\) 3.79471 0.132436 0.0662182 0.997805i \(-0.478907\pi\)
0.0662182 + 0.997805i \(0.478907\pi\)
\(822\) −7.89992 −0.275541
\(823\) −39.5469 −1.37852 −0.689259 0.724515i \(-0.742064\pi\)
−0.689259 + 0.724515i \(0.742064\pi\)
\(824\) 15.9287 0.554901
\(825\) 0 0
\(826\) −0.566092 −0.0196969
\(827\) 21.2066 0.737424 0.368712 0.929544i \(-0.379799\pi\)
0.368712 + 0.929544i \(0.379799\pi\)
\(828\) −18.3706 −0.638422
\(829\) −16.6338 −0.577716 −0.288858 0.957372i \(-0.593276\pi\)
−0.288858 + 0.957372i \(0.593276\pi\)
\(830\) −1.73868 −0.0603504
\(831\) 57.3865 1.99072
\(832\) 5.43886 0.188559
\(833\) −0.477092 −0.0165303
\(834\) 10.3998 0.360115
\(835\) −17.9260 −0.620356
\(836\) 0 0
\(837\) 119.484 4.12995
\(838\) −3.21914 −0.111203
\(839\) 44.0630 1.52122 0.760612 0.649207i \(-0.224899\pi\)
0.760612 + 0.649207i \(0.224899\pi\)
\(840\) −3.14896 −0.108649
\(841\) 14.1289 0.487205
\(842\) −13.1605 −0.453540
\(843\) 12.7833 0.440280
\(844\) 22.8331 0.785947
\(845\) 16.5812 0.570412
\(846\) −65.1920 −2.24135
\(847\) 0 0
\(848\) 6.15392 0.211326
\(849\) 9.96879 0.342128
\(850\) 0.477092 0.0163641
\(851\) 12.1318 0.415874
\(852\) 11.4087 0.390854
\(853\) 8.75092 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(854\) 2.04888 0.0701112
\(855\) −40.9145 −1.39925
\(856\) 6.17232 0.210966
\(857\) 0.827106 0.0282534 0.0141267 0.999900i \(-0.495503\pi\)
0.0141267 + 0.999900i \(0.495503\pi\)
\(858\) 0 0
\(859\) 11.5033 0.392488 0.196244 0.980555i \(-0.437125\pi\)
0.196244 + 0.980555i \(0.437125\pi\)
\(860\) 11.9003 0.405799
\(861\) 7.94454 0.270749
\(862\) −2.38429 −0.0812093
\(863\) 13.9927 0.476318 0.238159 0.971226i \(-0.423456\pi\)
0.238159 + 0.971226i \(0.423456\pi\)
\(864\) 12.3312 0.419516
\(865\) −7.68791 −0.261397
\(866\) −9.02568 −0.306705
\(867\) 52.8156 1.79371
\(868\) 9.68953 0.328884
\(869\) 0 0
\(870\) −20.6801 −0.701119
\(871\) −6.74976 −0.228707
\(872\) 14.9240 0.505390
\(873\) −27.1905 −0.920260
\(874\) −15.7143 −0.531545
\(875\) −1.00000 −0.0338062
\(876\) −49.2490 −1.66397
\(877\) −6.87711 −0.232223 −0.116112 0.993236i \(-0.537043\pi\)
−0.116112 + 0.993236i \(0.537043\pi\)
\(878\) −23.1241 −0.780402
\(879\) −32.2821 −1.08885
\(880\) 0 0
\(881\) −40.9746 −1.38047 −0.690235 0.723585i \(-0.742493\pi\)
−0.690235 + 0.723585i \(0.742493\pi\)
\(882\) −6.91596 −0.232872
\(883\) 22.6100 0.760887 0.380443 0.924804i \(-0.375771\pi\)
0.380443 + 0.924804i \(0.375771\pi\)
\(884\) −2.59484 −0.0872739
\(885\) 1.78260 0.0599215
\(886\) 6.78799 0.228047
\(887\) 3.14606 0.105634 0.0528172 0.998604i \(-0.483180\pi\)
0.0528172 + 0.998604i \(0.483180\pi\)
\(888\) −14.3821 −0.482633
\(889\) −16.3815 −0.549418
\(890\) 9.54364 0.319903
\(891\) 0 0
\(892\) −15.8782 −0.531640
\(893\) −55.7657 −1.86613
\(894\) 21.5089 0.719365
\(895\) 22.7197 0.759436
\(896\) 1.00000 0.0334077
\(897\) 45.4932 1.51897
\(898\) −13.1114 −0.437534
\(899\) 63.6337 2.12230
\(900\) 6.91596 0.230532
\(901\) −2.93599 −0.0978119
\(902\) 0 0
\(903\) 37.4737 1.24705
\(904\) −19.0906 −0.634944
\(905\) −13.6567 −0.453964
\(906\) −34.0684 −1.13185
\(907\) 6.57950 0.218469 0.109234 0.994016i \(-0.465160\pi\)
0.109234 + 0.994016i \(0.465160\pi\)
\(908\) −8.41830 −0.279371
\(909\) −20.1112 −0.667045
\(910\) 5.43886 0.180297
\(911\) −10.0045 −0.331464 −0.165732 0.986171i \(-0.552999\pi\)
−0.165732 + 0.986171i \(0.552999\pi\)
\(912\) 18.6291 0.616872
\(913\) 0 0
\(914\) 19.0200 0.629125
\(915\) −6.45184 −0.213291
\(916\) 22.2844 0.736296
\(917\) −5.60812 −0.185196
\(918\) −5.88312 −0.194172
\(919\) 5.27944 0.174153 0.0870764 0.996202i \(-0.472248\pi\)
0.0870764 + 0.996202i \(0.472248\pi\)
\(920\) 2.65626 0.0875744
\(921\) 10.6741 0.351723
\(922\) −18.8767 −0.621672
\(923\) −19.7050 −0.648596
\(924\) 0 0
\(925\) −4.56726 −0.150171
\(926\) −13.6194 −0.447560
\(927\) −110.162 −3.61819
\(928\) 6.56726 0.215581
\(929\) −19.7726 −0.648717 −0.324359 0.945934i \(-0.605148\pi\)
−0.324359 + 0.945934i \(0.605148\pi\)
\(930\) −30.5120 −1.00053
\(931\) −5.91596 −0.193888
\(932\) −12.4011 −0.406210
\(933\) 92.7441 3.03630
\(934\) 18.9801 0.621049
\(935\) 0 0
\(936\) −37.6149 −1.22948
\(937\) −46.0026 −1.50284 −0.751419 0.659825i \(-0.770630\pi\)
−0.751419 + 0.659825i \(0.770630\pi\)
\(938\) −1.24102 −0.0405209
\(939\) 69.4801 2.26740
\(940\) 9.42632 0.307453
\(941\) −37.3066 −1.21616 −0.608081 0.793875i \(-0.708060\pi\)
−0.608081 + 0.793875i \(0.708060\pi\)
\(942\) 25.2473 0.822602
\(943\) −6.70150 −0.218231
\(944\) −0.566092 −0.0184247
\(945\) 12.3312 0.401134
\(946\) 0 0
\(947\) 49.7593 1.61696 0.808479 0.588524i \(-0.200291\pi\)
0.808479 + 0.588524i \(0.200291\pi\)
\(948\) 10.6437 0.345692
\(949\) 85.0626 2.76125
\(950\) 5.91596 0.191939
\(951\) −66.4930 −2.15618
\(952\) −0.477092 −0.0154626
\(953\) 20.5931 0.667076 0.333538 0.942737i \(-0.391757\pi\)
0.333538 + 0.942737i \(0.391757\pi\)
\(954\) −42.5602 −1.37794
\(955\) 9.42182 0.304883
\(956\) 19.3571 0.626055
\(957\) 0 0
\(958\) 5.51532 0.178192
\(959\) 2.50874 0.0810114
\(960\) −3.14896 −0.101632
\(961\) 62.8871 2.02862
\(962\) 24.8407 0.800897
\(963\) −42.6875 −1.37559
\(964\) −5.12445 −0.165047
\(965\) −5.09919 −0.164149
\(966\) 8.36447 0.269122
\(967\) −36.0618 −1.15967 −0.579834 0.814735i \(-0.696883\pi\)
−0.579834 + 0.814735i \(0.696883\pi\)
\(968\) 0 0
\(969\) −8.88781 −0.285517
\(970\) 3.93156 0.126235
\(971\) 4.29180 0.137730 0.0688651 0.997626i \(-0.478062\pi\)
0.0688651 + 0.997626i \(0.478062\pi\)
\(972\) −19.9478 −0.639825
\(973\) −3.30261 −0.105877
\(974\) −28.0086 −0.897454
\(975\) −17.1268 −0.548496
\(976\) 2.04888 0.0655830
\(977\) 36.2757 1.16056 0.580281 0.814416i \(-0.302943\pi\)
0.580281 + 0.814416i \(0.302943\pi\)
\(978\) −57.6231 −1.84258
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −103.214 −3.29536
\(982\) 12.9545 0.413393
\(983\) 18.9549 0.604567 0.302284 0.953218i \(-0.402251\pi\)
0.302284 + 0.953218i \(0.402251\pi\)
\(984\) 7.94454 0.253263
\(985\) 17.1284 0.545756
\(986\) −3.13319 −0.0997811
\(987\) 29.6831 0.944824
\(988\) −32.1761 −1.02366
\(989\) −31.6104 −1.00515
\(990\) 0 0
\(991\) −7.43374 −0.236141 −0.118070 0.993005i \(-0.537671\pi\)
−0.118070 + 0.993005i \(0.537671\pi\)
\(992\) 9.68953 0.307643
\(993\) 98.0615 3.11189
\(994\) −3.62299 −0.114914
\(995\) 11.9225 0.377970
\(996\) −5.47503 −0.173483
\(997\) 21.9062 0.693777 0.346888 0.937906i \(-0.387238\pi\)
0.346888 + 0.937906i \(0.387238\pi\)
\(998\) −3.93985 −0.124714
\(999\) 56.3198 1.78188
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.co.1.1 4
11.3 even 5 770.2.n.f.141.2 yes 8
11.4 even 5 770.2.n.f.71.2 8
11.10 odd 2 8470.2.a.cs.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.f.71.2 8 11.4 even 5
770.2.n.f.141.2 yes 8 11.3 even 5
8470.2.a.co.1.1 4 1.1 even 1 trivial
8470.2.a.cs.1.1 4 11.10 odd 2