Properties

Label 8470.2.a.cy.1.2
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.13298000.1
Defining polynomial: \(x^{6} - x^{5} - 10 x^{4} + 3 x^{3} + 26 x^{2} + 13 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 1.2
Root \(2.79700\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.64424 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.64424 q^{6} -1.00000 q^{7} -1.00000 q^{8} +3.99201 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.64424 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.64424 q^{6} -1.00000 q^{7} -1.00000 q^{8} +3.99201 q^{9} -1.00000 q^{10} -2.64424 q^{12} -3.06037 q^{13} +1.00000 q^{14} -2.64424 q^{15} +1.00000 q^{16} +0.0683587 q^{17} -3.99201 q^{18} -2.77079 q^{19} +1.00000 q^{20} +2.64424 q^{21} +1.83120 q^{23} +2.64424 q^{24} +1.00000 q^{25} +3.06037 q^{26} -2.62310 q^{27} -1.00000 q^{28} -3.30937 q^{29} +2.64424 q^{30} +4.12571 q^{31} -1.00000 q^{32} -0.0683587 q^{34} -1.00000 q^{35} +3.99201 q^{36} -5.63448 q^{37} +2.77079 q^{38} +8.09234 q^{39} -1.00000 q^{40} +5.13257 q^{41} -2.64424 q^{42} -2.84053 q^{43} +3.99201 q^{45} -1.83120 q^{46} +11.9195 q^{47} -2.64424 q^{48} +1.00000 q^{49} -1.00000 q^{50} -0.180757 q^{51} -3.06037 q^{52} +0.123629 q^{53} +2.62310 q^{54} +1.00000 q^{56} +7.32664 q^{57} +3.30937 q^{58} -11.1929 q^{59} -2.64424 q^{60} -2.53653 q^{61} -4.12571 q^{62} -3.99201 q^{63} +1.00000 q^{64} -3.06037 q^{65} -2.20818 q^{67} +0.0683587 q^{68} -4.84213 q^{69} +1.00000 q^{70} +13.7247 q^{71} -3.99201 q^{72} -6.98905 q^{73} +5.63448 q^{74} -2.64424 q^{75} -2.77079 q^{76} -8.09234 q^{78} +12.4142 q^{79} +1.00000 q^{80} -5.03990 q^{81} -5.13257 q^{82} +3.05366 q^{83} +2.64424 q^{84} +0.0683587 q^{85} +2.84053 q^{86} +8.75076 q^{87} +13.5707 q^{89} -3.99201 q^{90} +3.06037 q^{91} +1.83120 q^{92} -10.9094 q^{93} -11.9195 q^{94} -2.77079 q^{95} +2.64424 q^{96} -2.74477 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} - 6 q^{8} + 15 q^{9} + O(q^{10}) \) \( 6 q - 6 q^{2} + q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} - 6 q^{8} + 15 q^{9} - 6 q^{10} + q^{12} - 2 q^{13} + 6 q^{14} + q^{15} + 6 q^{16} - 7 q^{17} - 15 q^{18} - 11 q^{19} + 6 q^{20} - q^{21} - 6 q^{23} - q^{24} + 6 q^{25} + 2 q^{26} + 4 q^{27} - 6 q^{28} - 2 q^{29} - q^{30} - 6 q^{32} + 7 q^{34} - 6 q^{35} + 15 q^{36} - 14 q^{37} + 11 q^{38} - 20 q^{39} - 6 q^{40} - 13 q^{41} + q^{42} - 19 q^{43} + 15 q^{45} + 6 q^{46} + 22 q^{47} + q^{48} + 6 q^{49} - 6 q^{50} + 14 q^{51} - 2 q^{52} - 10 q^{53} - 4 q^{54} + 6 q^{56} - 32 q^{57} + 2 q^{58} - 7 q^{59} + q^{60} - 22 q^{61} - 15 q^{63} + 6 q^{64} - 2 q^{65} + 5 q^{67} - 7 q^{68} - 36 q^{69} + 6 q^{70} + 8 q^{71} - 15 q^{72} - 13 q^{73} + 14 q^{74} + q^{75} - 11 q^{76} + 20 q^{78} + 16 q^{79} + 6 q^{80} + 18 q^{81} + 13 q^{82} + 5 q^{83} - q^{84} - 7 q^{85} + 19 q^{86} - 14 q^{87} + q^{89} - 15 q^{90} + 2 q^{91} - 6 q^{92} - 42 q^{93} - 22 q^{94} - 11 q^{95} - q^{96} - 3 q^{97} - 6 q^{98} + O(q^{100}) \)

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
\(3\) −2.64424 −1.52665 −0.763326 0.646013i \(-0.776435\pi\)
−0.763326 + 0.646013i \(0.776435\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 2.64424 1.07951
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 3.99201 1.33067
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −2.64424 −0.763326
\(13\) −3.06037 −0.848793 −0.424396 0.905477i \(-0.639514\pi\)
−0.424396 + 0.905477i \(0.639514\pi\)
\(14\) 1.00000 0.267261
\(15\) −2.64424 −0.682740
\(16\) 1.00000 0.250000
\(17\) 0.0683587 0.0165794 0.00828971 0.999966i \(-0.497361\pi\)
0.00828971 + 0.999966i \(0.497361\pi\)
\(18\) −3.99201 −0.940925
\(19\) −2.77079 −0.635664 −0.317832 0.948147i \(-0.602955\pi\)
−0.317832 + 0.948147i \(0.602955\pi\)
\(20\) 1.00000 0.223607
\(21\) 2.64424 0.577021
\(22\) 0 0
\(23\) 1.83120 0.381831 0.190916 0.981606i \(-0.438854\pi\)
0.190916 + 0.981606i \(0.438854\pi\)
\(24\) 2.64424 0.539753
\(25\) 1.00000 0.200000
\(26\) 3.06037 0.600187
\(27\) −2.62310 −0.504817
\(28\) −1.00000 −0.188982
\(29\) −3.30937 −0.614534 −0.307267 0.951623i \(-0.599414\pi\)
−0.307267 + 0.951623i \(0.599414\pi\)
\(30\) 2.64424 0.482770
\(31\) 4.12571 0.741000 0.370500 0.928833i \(-0.379186\pi\)
0.370500 + 0.928833i \(0.379186\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −0.0683587 −0.0117234
\(35\) −1.00000 −0.169031
\(36\) 3.99201 0.665334
\(37\) −5.63448 −0.926303 −0.463151 0.886279i \(-0.653281\pi\)
−0.463151 + 0.886279i \(0.653281\pi\)
\(38\) 2.77079 0.449482
\(39\) 8.09234 1.29581
\(40\) −1.00000 −0.158114
\(41\) 5.13257 0.801573 0.400787 0.916171i \(-0.368737\pi\)
0.400787 + 0.916171i \(0.368737\pi\)
\(42\) −2.64424 −0.408015
\(43\) −2.84053 −0.433177 −0.216589 0.976263i \(-0.569493\pi\)
−0.216589 + 0.976263i \(0.569493\pi\)
\(44\) 0 0
\(45\) 3.99201 0.595093
\(46\) −1.83120 −0.269996
\(47\) 11.9195 1.73864 0.869322 0.494247i \(-0.164556\pi\)
0.869322 + 0.494247i \(0.164556\pi\)
\(48\) −2.64424 −0.381663
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −0.180757 −0.0253110
\(52\) −3.06037 −0.424396
\(53\) 0.123629 0.0169818 0.00849088 0.999964i \(-0.497297\pi\)
0.00849088 + 0.999964i \(0.497297\pi\)
\(54\) 2.62310 0.356959
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 7.32664 0.970438
\(58\) 3.30937 0.434541
\(59\) −11.1929 −1.45720 −0.728598 0.684941i \(-0.759828\pi\)
−0.728598 + 0.684941i \(0.759828\pi\)
\(60\) −2.64424 −0.341370
\(61\) −2.53653 −0.324770 −0.162385 0.986728i \(-0.551919\pi\)
−0.162385 + 0.986728i \(0.551919\pi\)
\(62\) −4.12571 −0.523966
\(63\) −3.99201 −0.502946
\(64\) 1.00000 0.125000
\(65\) −3.06037 −0.379592
\(66\) 0 0
\(67\) −2.20818 −0.269773 −0.134886 0.990861i \(-0.543067\pi\)
−0.134886 + 0.990861i \(0.543067\pi\)
\(68\) 0.0683587 0.00828971
\(69\) −4.84213 −0.582924
\(70\) 1.00000 0.119523
\(71\) 13.7247 1.62883 0.814413 0.580285i \(-0.197059\pi\)
0.814413 + 0.580285i \(0.197059\pi\)
\(72\) −3.99201 −0.470462
\(73\) −6.98905 −0.818007 −0.409003 0.912533i \(-0.634124\pi\)
−0.409003 + 0.912533i \(0.634124\pi\)
\(74\) 5.63448 0.654995
\(75\) −2.64424 −0.305331
\(76\) −2.77079 −0.317832
\(77\) 0 0
\(78\) −8.09234 −0.916277
\(79\) 12.4142 1.39671 0.698354 0.715752i \(-0.253916\pi\)
0.698354 + 0.715752i \(0.253916\pi\)
\(80\) 1.00000 0.111803
\(81\) −5.03990 −0.559989
\(82\) −5.13257 −0.566798
\(83\) 3.05366 0.335183 0.167591 0.985857i \(-0.446401\pi\)
0.167591 + 0.985857i \(0.446401\pi\)
\(84\) 2.64424 0.288510
\(85\) 0.0683587 0.00741454
\(86\) 2.84053 0.306303
\(87\) 8.75076 0.938180
\(88\) 0 0
\(89\) 13.5707 1.43849 0.719246 0.694756i \(-0.244487\pi\)
0.719246 + 0.694756i \(0.244487\pi\)
\(90\) −3.99201 −0.420794
\(91\) 3.06037 0.320813
\(92\) 1.83120 0.190916
\(93\) −10.9094 −1.13125
\(94\) −11.9195 −1.22941
\(95\) −2.77079 −0.284277
\(96\) 2.64424 0.269877
\(97\) −2.74477 −0.278689 −0.139345 0.990244i \(-0.544500\pi\)
−0.139345 + 0.990244i \(0.544500\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 5.53961 0.551212 0.275606 0.961271i \(-0.411121\pi\)
0.275606 + 0.961271i \(0.411121\pi\)
\(102\) 0.180757 0.0178976
\(103\) −3.61697 −0.356390 −0.178195 0.983995i \(-0.557026\pi\)
−0.178195 + 0.983995i \(0.557026\pi\)
\(104\) 3.06037 0.300094
\(105\) 2.64424 0.258051
\(106\) −0.123629 −0.0120079
\(107\) −6.58182 −0.636288 −0.318144 0.948042i \(-0.603060\pi\)
−0.318144 + 0.948042i \(0.603060\pi\)
\(108\) −2.62310 −0.252408
\(109\) −1.14986 −0.110137 −0.0550683 0.998483i \(-0.517538\pi\)
−0.0550683 + 0.998483i \(0.517538\pi\)
\(110\) 0 0
\(111\) 14.8989 1.41414
\(112\) −1.00000 −0.0944911
\(113\) −12.1536 −1.14332 −0.571658 0.820492i \(-0.693700\pi\)
−0.571658 + 0.820492i \(0.693700\pi\)
\(114\) −7.32664 −0.686203
\(115\) 1.83120 0.170760
\(116\) −3.30937 −0.307267
\(117\) −12.2170 −1.12946
\(118\) 11.1929 1.03039
\(119\) −0.0683587 −0.00626643
\(120\) 2.64424 0.241385
\(121\) 0 0
\(122\) 2.53653 0.229647
\(123\) −13.5718 −1.22372
\(124\) 4.12571 0.370500
\(125\) 1.00000 0.0894427
\(126\) 3.99201 0.355636
\(127\) −13.7815 −1.22291 −0.611457 0.791278i \(-0.709416\pi\)
−0.611457 + 0.791278i \(0.709416\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 7.51106 0.661312
\(130\) 3.06037 0.268412
\(131\) −4.89161 −0.427382 −0.213691 0.976901i \(-0.568548\pi\)
−0.213691 + 0.976901i \(0.568548\pi\)
\(132\) 0 0
\(133\) 2.77079 0.240258
\(134\) 2.20818 0.190758
\(135\) −2.62310 −0.225761
\(136\) −0.0683587 −0.00586171
\(137\) −19.0559 −1.62805 −0.814026 0.580829i \(-0.802729\pi\)
−0.814026 + 0.580829i \(0.802729\pi\)
\(138\) 4.84213 0.412190
\(139\) −7.65318 −0.649135 −0.324567 0.945863i \(-0.605219\pi\)
−0.324567 + 0.945863i \(0.605219\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −31.5181 −2.65430
\(142\) −13.7247 −1.15175
\(143\) 0 0
\(144\) 3.99201 0.332667
\(145\) −3.30937 −0.274828
\(146\) 6.98905 0.578418
\(147\) −2.64424 −0.218093
\(148\) −5.63448 −0.463151
\(149\) 14.6774 1.20242 0.601208 0.799092i \(-0.294686\pi\)
0.601208 + 0.799092i \(0.294686\pi\)
\(150\) 2.64424 0.215901
\(151\) 11.0935 0.902774 0.451387 0.892328i \(-0.350929\pi\)
0.451387 + 0.892328i \(0.350929\pi\)
\(152\) 2.77079 0.224741
\(153\) 0.272888 0.0220617
\(154\) 0 0
\(155\) 4.12571 0.331385
\(156\) 8.09234 0.647906
\(157\) 13.1941 1.05300 0.526502 0.850174i \(-0.323503\pi\)
0.526502 + 0.850174i \(0.323503\pi\)
\(158\) −12.4142 −0.987622
\(159\) −0.326905 −0.0259253
\(160\) −1.00000 −0.0790569
\(161\) −1.83120 −0.144319
\(162\) 5.03990 0.395972
\(163\) −3.70882 −0.290497 −0.145248 0.989395i \(-0.546398\pi\)
−0.145248 + 0.989395i \(0.546398\pi\)
\(164\) 5.13257 0.400787
\(165\) 0 0
\(166\) −3.05366 −0.237010
\(167\) 8.75429 0.677428 0.338714 0.940889i \(-0.390008\pi\)
0.338714 + 0.940889i \(0.390008\pi\)
\(168\) −2.64424 −0.204008
\(169\) −3.63416 −0.279551
\(170\) −0.0683587 −0.00524287
\(171\) −11.0610 −0.845858
\(172\) −2.84053 −0.216589
\(173\) 3.03876 0.231033 0.115516 0.993306i \(-0.463148\pi\)
0.115516 + 0.993306i \(0.463148\pi\)
\(174\) −8.75076 −0.663393
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 29.5968 2.22463
\(178\) −13.5707 −1.01717
\(179\) −7.29987 −0.545618 −0.272809 0.962068i \(-0.587953\pi\)
−0.272809 + 0.962068i \(0.587953\pi\)
\(180\) 3.99201 0.297547
\(181\) 10.9351 0.812800 0.406400 0.913695i \(-0.366784\pi\)
0.406400 + 0.913695i \(0.366784\pi\)
\(182\) −3.06037 −0.226849
\(183\) 6.70720 0.495811
\(184\) −1.83120 −0.134998
\(185\) −5.63448 −0.414255
\(186\) 10.9094 0.799914
\(187\) 0 0
\(188\) 11.9195 0.869322
\(189\) 2.62310 0.190803
\(190\) 2.77079 0.201014
\(191\) 22.6520 1.63904 0.819519 0.573051i \(-0.194241\pi\)
0.819519 + 0.573051i \(0.194241\pi\)
\(192\) −2.64424 −0.190832
\(193\) 16.9764 1.22199 0.610994 0.791635i \(-0.290770\pi\)
0.610994 + 0.791635i \(0.290770\pi\)
\(194\) 2.74477 0.197063
\(195\) 8.09234 0.579505
\(196\) 1.00000 0.0714286
\(197\) −26.3241 −1.87551 −0.937757 0.347293i \(-0.887101\pi\)
−0.937757 + 0.347293i \(0.887101\pi\)
\(198\) 0 0
\(199\) 16.9127 1.19891 0.599456 0.800408i \(-0.295384\pi\)
0.599456 + 0.800408i \(0.295384\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 5.83897 0.411849
\(202\) −5.53961 −0.389766
\(203\) 3.30937 0.232272
\(204\) −0.180757 −0.0126555
\(205\) 5.13257 0.358474
\(206\) 3.61697 0.252006
\(207\) 7.31016 0.508091
\(208\) −3.06037 −0.212198
\(209\) 0 0
\(210\) −2.64424 −0.182470
\(211\) 16.4483 1.13235 0.566173 0.824286i \(-0.308423\pi\)
0.566173 + 0.824286i \(0.308423\pi\)
\(212\) 0.123629 0.00849088
\(213\) −36.2915 −2.48665
\(214\) 6.58182 0.449924
\(215\) −2.84053 −0.193723
\(216\) 2.62310 0.178480
\(217\) −4.12571 −0.280072
\(218\) 1.14986 0.0778784
\(219\) 18.4807 1.24881
\(220\) 0 0
\(221\) −0.209203 −0.0140725
\(222\) −14.8989 −0.999950
\(223\) 20.4675 1.37060 0.685302 0.728259i \(-0.259670\pi\)
0.685302 + 0.728259i \(0.259670\pi\)
\(224\) 1.00000 0.0668153
\(225\) 3.99201 0.266134
\(226\) 12.1536 0.808447
\(227\) −23.7041 −1.57329 −0.786647 0.617403i \(-0.788185\pi\)
−0.786647 + 0.617403i \(0.788185\pi\)
\(228\) 7.32664 0.485219
\(229\) −26.6368 −1.76021 −0.880106 0.474778i \(-0.842528\pi\)
−0.880106 + 0.474778i \(0.842528\pi\)
\(230\) −1.83120 −0.120746
\(231\) 0 0
\(232\) 3.30937 0.217271
\(233\) −30.3368 −1.98743 −0.993714 0.111950i \(-0.964290\pi\)
−0.993714 + 0.111950i \(0.964290\pi\)
\(234\) 12.2170 0.798650
\(235\) 11.9195 0.777545
\(236\) −11.1929 −0.728598
\(237\) −32.8262 −2.13229
\(238\) 0.0683587 0.00443104
\(239\) 9.95180 0.643728 0.321864 0.946786i \(-0.395691\pi\)
0.321864 + 0.946786i \(0.395691\pi\)
\(240\) −2.64424 −0.170685
\(241\) 30.1849 1.94438 0.972191 0.234189i \(-0.0752433\pi\)
0.972191 + 0.234189i \(0.0752433\pi\)
\(242\) 0 0
\(243\) 21.1960 1.35973
\(244\) −2.53653 −0.162385
\(245\) 1.00000 0.0638877
\(246\) 13.5718 0.865303
\(247\) 8.47964 0.539547
\(248\) −4.12571 −0.261983
\(249\) −8.07461 −0.511707
\(250\) −1.00000 −0.0632456
\(251\) −1.16998 −0.0738485 −0.0369242 0.999318i \(-0.511756\pi\)
−0.0369242 + 0.999318i \(0.511756\pi\)
\(252\) −3.99201 −0.251473
\(253\) 0 0
\(254\) 13.7815 0.864730
\(255\) −0.180757 −0.0113194
\(256\) 1.00000 0.0625000
\(257\) −0.980916 −0.0611879 −0.0305939 0.999532i \(-0.509740\pi\)
−0.0305939 + 0.999532i \(0.509740\pi\)
\(258\) −7.51106 −0.467618
\(259\) 5.63448 0.350110
\(260\) −3.06037 −0.189796
\(261\) −13.2110 −0.817741
\(262\) 4.89161 0.302204
\(263\) 15.9923 0.986125 0.493063 0.869994i \(-0.335877\pi\)
0.493063 + 0.869994i \(0.335877\pi\)
\(264\) 0 0
\(265\) 0.123629 0.00759448
\(266\) −2.77079 −0.169888
\(267\) −35.8842 −2.19608
\(268\) −2.20818 −0.134886
\(269\) 15.1048 0.920953 0.460476 0.887672i \(-0.347679\pi\)
0.460476 + 0.887672i \(0.347679\pi\)
\(270\) 2.62310 0.159637
\(271\) 0.499548 0.0303454 0.0151727 0.999885i \(-0.495170\pi\)
0.0151727 + 0.999885i \(0.495170\pi\)
\(272\) 0.0683587 0.00414486
\(273\) −8.09234 −0.489771
\(274\) 19.0559 1.15121
\(275\) 0 0
\(276\) −4.84213 −0.291462
\(277\) 11.1061 0.667302 0.333651 0.942697i \(-0.391719\pi\)
0.333651 + 0.942697i \(0.391719\pi\)
\(278\) 7.65318 0.459007
\(279\) 16.4699 0.986025
\(280\) 1.00000 0.0597614
\(281\) −15.4512 −0.921742 −0.460871 0.887467i \(-0.652463\pi\)
−0.460871 + 0.887467i \(0.652463\pi\)
\(282\) 31.5181 1.87688
\(283\) −14.3139 −0.850876 −0.425438 0.904988i \(-0.639880\pi\)
−0.425438 + 0.904988i \(0.639880\pi\)
\(284\) 13.7247 0.814413
\(285\) 7.32664 0.433993
\(286\) 0 0
\(287\) −5.13257 −0.302966
\(288\) −3.99201 −0.235231
\(289\) −16.9953 −0.999725
\(290\) 3.30937 0.194333
\(291\) 7.25783 0.425462
\(292\) −6.98905 −0.409003
\(293\) 17.5749 1.02673 0.513367 0.858169i \(-0.328398\pi\)
0.513367 + 0.858169i \(0.328398\pi\)
\(294\) 2.64424 0.154215
\(295\) −11.1929 −0.651678
\(296\) 5.63448 0.327498
\(297\) 0 0
\(298\) −14.6774 −0.850237
\(299\) −5.60414 −0.324096
\(300\) −2.64424 −0.152665
\(301\) 2.84053 0.163726
\(302\) −11.0935 −0.638358
\(303\) −14.6481 −0.841509
\(304\) −2.77079 −0.158916
\(305\) −2.53653 −0.145241
\(306\) −0.272888 −0.0156000
\(307\) −5.38032 −0.307071 −0.153536 0.988143i \(-0.549066\pi\)
−0.153536 + 0.988143i \(0.549066\pi\)
\(308\) 0 0
\(309\) 9.56413 0.544084
\(310\) −4.12571 −0.234325
\(311\) 16.9028 0.958470 0.479235 0.877687i \(-0.340914\pi\)
0.479235 + 0.877687i \(0.340914\pi\)
\(312\) −8.09234 −0.458139
\(313\) −1.35523 −0.0766020 −0.0383010 0.999266i \(-0.512195\pi\)
−0.0383010 + 0.999266i \(0.512195\pi\)
\(314\) −13.1941 −0.744587
\(315\) −3.99201 −0.224924
\(316\) 12.4142 0.698354
\(317\) 29.3217 1.64687 0.823434 0.567412i \(-0.192055\pi\)
0.823434 + 0.567412i \(0.192055\pi\)
\(318\) 0.326905 0.0183319
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 17.4039 0.971391
\(322\) 1.83120 0.102049
\(323\) −0.189408 −0.0105389
\(324\) −5.03990 −0.279995
\(325\) −3.06037 −0.169759
\(326\) 3.70882 0.205412
\(327\) 3.04051 0.168140
\(328\) −5.13257 −0.283399
\(329\) −11.9195 −0.657145
\(330\) 0 0
\(331\) 7.10546 0.390551 0.195276 0.980748i \(-0.437440\pi\)
0.195276 + 0.980748i \(0.437440\pi\)
\(332\) 3.05366 0.167591
\(333\) −22.4929 −1.23260
\(334\) −8.75429 −0.479014
\(335\) −2.20818 −0.120646
\(336\) 2.64424 0.144255
\(337\) −28.7650 −1.56693 −0.783463 0.621438i \(-0.786549\pi\)
−0.783463 + 0.621438i \(0.786549\pi\)
\(338\) 3.63416 0.197672
\(339\) 32.1371 1.74545
\(340\) 0.0683587 0.00370727
\(341\) 0 0
\(342\) 11.0610 0.598112
\(343\) −1.00000 −0.0539949
\(344\) 2.84053 0.153151
\(345\) −4.84213 −0.260692
\(346\) −3.03876 −0.163365
\(347\) 25.7893 1.38444 0.692222 0.721685i \(-0.256632\pi\)
0.692222 + 0.721685i \(0.256632\pi\)
\(348\) 8.75076 0.469090
\(349\) −36.6973 −1.96436 −0.982180 0.187942i \(-0.939818\pi\)
−0.982180 + 0.187942i \(0.939818\pi\)
\(350\) 1.00000 0.0534522
\(351\) 8.02766 0.428485
\(352\) 0 0
\(353\) 16.9704 0.903241 0.451620 0.892210i \(-0.350846\pi\)
0.451620 + 0.892210i \(0.350846\pi\)
\(354\) −29.5968 −1.57305
\(355\) 13.7247 0.728434
\(356\) 13.5707 0.719246
\(357\) 0.180757 0.00956667
\(358\) 7.29987 0.385810
\(359\) −21.1819 −1.11794 −0.558969 0.829189i \(-0.688803\pi\)
−0.558969 + 0.829189i \(0.688803\pi\)
\(360\) −3.99201 −0.210397
\(361\) −11.3227 −0.595932
\(362\) −10.9351 −0.574736
\(363\) 0 0
\(364\) 3.06037 0.160407
\(365\) −6.98905 −0.365824
\(366\) −6.70720 −0.350591
\(367\) 36.4966 1.90511 0.952554 0.304371i \(-0.0984461\pi\)
0.952554 + 0.304371i \(0.0984461\pi\)
\(368\) 1.83120 0.0954579
\(369\) 20.4893 1.06663
\(370\) 5.63448 0.292923
\(371\) −0.123629 −0.00641850
\(372\) −10.9094 −0.565625
\(373\) 6.23055 0.322606 0.161303 0.986905i \(-0.448430\pi\)
0.161303 + 0.986905i \(0.448430\pi\)
\(374\) 0 0
\(375\) −2.64424 −0.136548
\(376\) −11.9195 −0.614703
\(377\) 10.1279 0.521612
\(378\) −2.62310 −0.134918
\(379\) 0.463665 0.0238169 0.0119084 0.999929i \(-0.496209\pi\)
0.0119084 + 0.999929i \(0.496209\pi\)
\(380\) −2.77079 −0.142139
\(381\) 36.4417 1.86696
\(382\) −22.6520 −1.15898
\(383\) 30.4947 1.55821 0.779103 0.626896i \(-0.215675\pi\)
0.779103 + 0.626896i \(0.215675\pi\)
\(384\) 2.64424 0.134938
\(385\) 0 0
\(386\) −16.9764 −0.864076
\(387\) −11.3394 −0.576416
\(388\) −2.74477 −0.139345
\(389\) −13.9789 −0.708758 −0.354379 0.935102i \(-0.615308\pi\)
−0.354379 + 0.935102i \(0.615308\pi\)
\(390\) −8.09234 −0.409772
\(391\) 0.125178 0.00633054
\(392\) −1.00000 −0.0505076
\(393\) 12.9346 0.652463
\(394\) 26.3241 1.32619
\(395\) 12.4142 0.624627
\(396\) 0 0
\(397\) −11.9510 −0.599801 −0.299901 0.953970i \(-0.596953\pi\)
−0.299901 + 0.953970i \(0.596953\pi\)
\(398\) −16.9127 −0.847759
\(399\) −7.32664 −0.366791
\(400\) 1.00000 0.0500000
\(401\) −14.4578 −0.721990 −0.360995 0.932568i \(-0.617563\pi\)
−0.360995 + 0.932568i \(0.617563\pi\)
\(402\) −5.83897 −0.291221
\(403\) −12.6262 −0.628955
\(404\) 5.53961 0.275606
\(405\) −5.03990 −0.250435
\(406\) −3.30937 −0.164241
\(407\) 0 0
\(408\) 0.180757 0.00894880
\(409\) 3.54419 0.175249 0.0876246 0.996154i \(-0.472072\pi\)
0.0876246 + 0.996154i \(0.472072\pi\)
\(410\) −5.13257 −0.253480
\(411\) 50.3882 2.48547
\(412\) −3.61697 −0.178195
\(413\) 11.1929 0.550768
\(414\) −7.31016 −0.359275
\(415\) 3.05366 0.149898
\(416\) 3.06037 0.150047
\(417\) 20.2369 0.991003
\(418\) 0 0
\(419\) −35.5085 −1.73471 −0.867353 0.497694i \(-0.834180\pi\)
−0.867353 + 0.497694i \(0.834180\pi\)
\(420\) 2.64424 0.129026
\(421\) −26.6190 −1.29733 −0.648665 0.761074i \(-0.724672\pi\)
−0.648665 + 0.761074i \(0.724672\pi\)
\(422\) −16.4483 −0.800690
\(423\) 47.5829 2.31356
\(424\) −0.123629 −0.00600396
\(425\) 0.0683587 0.00331588
\(426\) 36.2915 1.75833
\(427\) 2.53653 0.122751
\(428\) −6.58182 −0.318144
\(429\) 0 0
\(430\) 2.84053 0.136983
\(431\) −26.0439 −1.25449 −0.627244 0.778823i \(-0.715817\pi\)
−0.627244 + 0.778823i \(0.715817\pi\)
\(432\) −2.62310 −0.126204
\(433\) 1.69218 0.0813208 0.0406604 0.999173i \(-0.487054\pi\)
0.0406604 + 0.999173i \(0.487054\pi\)
\(434\) 4.12571 0.198040
\(435\) 8.75076 0.419567
\(436\) −1.14986 −0.0550683
\(437\) −5.07387 −0.242716
\(438\) −18.4807 −0.883044
\(439\) −12.8065 −0.611223 −0.305611 0.952156i \(-0.598861\pi\)
−0.305611 + 0.952156i \(0.598861\pi\)
\(440\) 0 0
\(441\) 3.99201 0.190096
\(442\) 0.209203 0.00995075
\(443\) 0.704398 0.0334670 0.0167335 0.999860i \(-0.494673\pi\)
0.0167335 + 0.999860i \(0.494673\pi\)
\(444\) 14.8989 0.707071
\(445\) 13.5707 0.643313
\(446\) −20.4675 −0.969163
\(447\) −38.8105 −1.83567
\(448\) −1.00000 −0.0472456
\(449\) −18.4066 −0.868660 −0.434330 0.900754i \(-0.643015\pi\)
−0.434330 + 0.900754i \(0.643015\pi\)
\(450\) −3.99201 −0.188185
\(451\) 0 0
\(452\) −12.1536 −0.571658
\(453\) −29.3338 −1.37822
\(454\) 23.7041 1.11249
\(455\) 3.06037 0.143472
\(456\) −7.32664 −0.343101
\(457\) −16.9955 −0.795016 −0.397508 0.917599i \(-0.630125\pi\)
−0.397508 + 0.917599i \(0.630125\pi\)
\(458\) 26.6368 1.24466
\(459\) −0.179312 −0.00836957
\(460\) 1.83120 0.0853801
\(461\) −13.3276 −0.620729 −0.310364 0.950618i \(-0.600451\pi\)
−0.310364 + 0.950618i \(0.600451\pi\)
\(462\) 0 0
\(463\) −34.9814 −1.62572 −0.812861 0.582457i \(-0.802091\pi\)
−0.812861 + 0.582457i \(0.802091\pi\)
\(464\) −3.30937 −0.153634
\(465\) −10.9094 −0.505910
\(466\) 30.3368 1.40532
\(467\) −3.57861 −0.165599 −0.0827993 0.996566i \(-0.526386\pi\)
−0.0827993 + 0.996566i \(0.526386\pi\)
\(468\) −12.2170 −0.564731
\(469\) 2.20818 0.101964
\(470\) −11.9195 −0.549807
\(471\) −34.8884 −1.60757
\(472\) 11.1929 0.515197
\(473\) 0 0
\(474\) 32.8262 1.50776
\(475\) −2.77079 −0.127133
\(476\) −0.0683587 −0.00313322
\(477\) 0.493528 0.0225971
\(478\) −9.95180 −0.455185
\(479\) −18.3300 −0.837518 −0.418759 0.908097i \(-0.637535\pi\)
−0.418759 + 0.908097i \(0.637535\pi\)
\(480\) 2.64424 0.120693
\(481\) 17.2436 0.786239
\(482\) −30.1849 −1.37489
\(483\) 4.84213 0.220325
\(484\) 0 0
\(485\) −2.74477 −0.124634
\(486\) −21.1960 −0.961471
\(487\) 0.487041 0.0220699 0.0110350 0.999939i \(-0.496487\pi\)
0.0110350 + 0.999939i \(0.496487\pi\)
\(488\) 2.53653 0.114823
\(489\) 9.80700 0.443488
\(490\) −1.00000 −0.0451754
\(491\) −6.09167 −0.274913 −0.137457 0.990508i \(-0.543893\pi\)
−0.137457 + 0.990508i \(0.543893\pi\)
\(492\) −13.5718 −0.611862
\(493\) −0.226224 −0.0101886
\(494\) −8.47964 −0.381517
\(495\) 0 0
\(496\) 4.12571 0.185250
\(497\) −13.7247 −0.615639
\(498\) 8.07461 0.361832
\(499\) 42.3141 1.89424 0.947120 0.320879i \(-0.103978\pi\)
0.947120 + 0.320879i \(0.103978\pi\)
\(500\) 1.00000 0.0447214
\(501\) −23.1485 −1.03420
\(502\) 1.16998 0.0522188
\(503\) −32.3677 −1.44320 −0.721602 0.692308i \(-0.756594\pi\)
−0.721602 + 0.692308i \(0.756594\pi\)
\(504\) 3.99201 0.177818
\(505\) 5.53961 0.246510
\(506\) 0 0
\(507\) 9.60960 0.426777
\(508\) −13.7815 −0.611457
\(509\) 23.7319 1.05190 0.525950 0.850516i \(-0.323710\pi\)
0.525950 + 0.850516i \(0.323710\pi\)
\(510\) 0.180757 0.00800405
\(511\) 6.98905 0.309178
\(512\) −1.00000 −0.0441942
\(513\) 7.26808 0.320893
\(514\) 0.980916 0.0432664
\(515\) −3.61697 −0.159383
\(516\) 7.51106 0.330656
\(517\) 0 0
\(518\) −5.63448 −0.247565
\(519\) −8.03522 −0.352707
\(520\) 3.06037 0.134206
\(521\) −30.8219 −1.35033 −0.675167 0.737665i \(-0.735928\pi\)
−0.675167 + 0.737665i \(0.735928\pi\)
\(522\) 13.2110 0.578230
\(523\) −37.6079 −1.64448 −0.822240 0.569141i \(-0.807276\pi\)
−0.822240 + 0.569141i \(0.807276\pi\)
\(524\) −4.89161 −0.213691
\(525\) 2.64424 0.115404
\(526\) −15.9923 −0.697296
\(527\) 0.282028 0.0122853
\(528\) 0 0
\(529\) −19.6467 −0.854205
\(530\) −0.123629 −0.00537011
\(531\) −44.6823 −1.93905
\(532\) 2.77079 0.120129
\(533\) −15.7075 −0.680369
\(534\) 35.8842 1.55286
\(535\) −6.58182 −0.284557
\(536\) 2.20818 0.0953790
\(537\) 19.3026 0.832969
\(538\) −15.1048 −0.651212
\(539\) 0 0
\(540\) −2.62310 −0.112880
\(541\) 7.18833 0.309050 0.154525 0.987989i \(-0.450615\pi\)
0.154525 + 0.987989i \(0.450615\pi\)
\(542\) −0.499548 −0.0214574
\(543\) −28.9150 −1.24086
\(544\) −0.0683587 −0.00293086
\(545\) −1.14986 −0.0492546
\(546\) 8.09234 0.346320
\(547\) −17.0757 −0.730103 −0.365052 0.930987i \(-0.618949\pi\)
−0.365052 + 0.930987i \(0.618949\pi\)
\(548\) −19.0559 −0.814026
\(549\) −10.1259 −0.432161
\(550\) 0 0
\(551\) 9.16957 0.390637
\(552\) 4.84213 0.206095
\(553\) −12.4142 −0.527906
\(554\) −11.1061 −0.471854
\(555\) 14.8989 0.632424
\(556\) −7.65318 −0.324567
\(557\) −35.7390 −1.51431 −0.757156 0.653234i \(-0.773412\pi\)
−0.757156 + 0.653234i \(0.773412\pi\)
\(558\) −16.4699 −0.697225
\(559\) 8.69307 0.367678
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 15.4512 0.651770
\(563\) 18.6595 0.786404 0.393202 0.919452i \(-0.371367\pi\)
0.393202 + 0.919452i \(0.371367\pi\)
\(564\) −31.5181 −1.32715
\(565\) −12.1536 −0.511307
\(566\) 14.3139 0.601660
\(567\) 5.03990 0.211656
\(568\) −13.7247 −0.575877
\(569\) −39.7394 −1.66596 −0.832981 0.553301i \(-0.813368\pi\)
−0.832981 + 0.553301i \(0.813368\pi\)
\(570\) −7.32664 −0.306879
\(571\) −13.3763 −0.559779 −0.279890 0.960032i \(-0.590298\pi\)
−0.279890 + 0.960032i \(0.590298\pi\)
\(572\) 0 0
\(573\) −59.8972 −2.50224
\(574\) 5.13257 0.214229
\(575\) 1.83120 0.0763663
\(576\) 3.99201 0.166334
\(577\) 19.8754 0.827422 0.413711 0.910408i \(-0.364232\pi\)
0.413711 + 0.910408i \(0.364232\pi\)
\(578\) 16.9953 0.706912
\(579\) −44.8897 −1.86555
\(580\) −3.30937 −0.137414
\(581\) −3.05366 −0.126687
\(582\) −7.25783 −0.300847
\(583\) 0 0
\(584\) 6.98905 0.289209
\(585\) −12.2170 −0.505111
\(586\) −17.5749 −0.726010
\(587\) 5.68609 0.234690 0.117345 0.993091i \(-0.462562\pi\)
0.117345 + 0.993091i \(0.462562\pi\)
\(588\) −2.64424 −0.109047
\(589\) −11.4315 −0.471026
\(590\) 11.1929 0.460806
\(591\) 69.6072 2.86326
\(592\) −5.63448 −0.231576
\(593\) −28.1824 −1.15731 −0.578656 0.815571i \(-0.696423\pi\)
−0.578656 + 0.815571i \(0.696423\pi\)
\(594\) 0 0
\(595\) −0.0683587 −0.00280243
\(596\) 14.6774 0.601208
\(597\) −44.7214 −1.83032
\(598\) 5.60414 0.229170
\(599\) 36.2189 1.47987 0.739933 0.672681i \(-0.234857\pi\)
0.739933 + 0.672681i \(0.234857\pi\)
\(600\) 2.64424 0.107951
\(601\) 14.3423 0.585034 0.292517 0.956260i \(-0.405507\pi\)
0.292517 + 0.956260i \(0.405507\pi\)
\(602\) −2.84053 −0.115772
\(603\) −8.81509 −0.358978
\(604\) 11.0935 0.451387
\(605\) 0 0
\(606\) 14.6481 0.595037
\(607\) 3.54637 0.143943 0.0719714 0.997407i \(-0.477071\pi\)
0.0719714 + 0.997407i \(0.477071\pi\)
\(608\) 2.77079 0.112370
\(609\) −8.75076 −0.354599
\(610\) 2.53653 0.102701
\(611\) −36.4781 −1.47575
\(612\) 0.272888 0.0110309
\(613\) 20.4324 0.825259 0.412629 0.910899i \(-0.364610\pi\)
0.412629 + 0.910899i \(0.364610\pi\)
\(614\) 5.38032 0.217132
\(615\) −13.5718 −0.547266
\(616\) 0 0
\(617\) 19.3755 0.780028 0.390014 0.920809i \(-0.372470\pi\)
0.390014 + 0.920809i \(0.372470\pi\)
\(618\) −9.56413 −0.384726
\(619\) −6.00523 −0.241371 −0.120685 0.992691i \(-0.538509\pi\)
−0.120685 + 0.992691i \(0.538509\pi\)
\(620\) 4.12571 0.165693
\(621\) −4.80343 −0.192755
\(622\) −16.9028 −0.677741
\(623\) −13.5707 −0.543699
\(624\) 8.09234 0.323953
\(625\) 1.00000 0.0400000
\(626\) 1.35523 0.0541658
\(627\) 0 0
\(628\) 13.1941 0.526502
\(629\) −0.385166 −0.0153576
\(630\) 3.99201 0.159045
\(631\) 21.7929 0.867562 0.433781 0.901018i \(-0.357179\pi\)
0.433781 + 0.901018i \(0.357179\pi\)
\(632\) −12.4142 −0.493811
\(633\) −43.4932 −1.72870
\(634\) −29.3217 −1.16451
\(635\) −13.7815 −0.546903
\(636\) −0.326905 −0.0129626
\(637\) −3.06037 −0.121256
\(638\) 0 0
\(639\) 54.7892 2.16743
\(640\) −1.00000 −0.0395285
\(641\) −41.0344 −1.62076 −0.810380 0.585904i \(-0.800739\pi\)
−0.810380 + 0.585904i \(0.800739\pi\)
\(642\) −17.4039 −0.686877
\(643\) −24.7624 −0.976534 −0.488267 0.872694i \(-0.662371\pi\)
−0.488267 + 0.872694i \(0.662371\pi\)
\(644\) −1.83120 −0.0721594
\(645\) 7.51106 0.295748
\(646\) 0.189408 0.00745215
\(647\) 44.1102 1.73415 0.867076 0.498175i \(-0.165996\pi\)
0.867076 + 0.498175i \(0.165996\pi\)
\(648\) 5.03990 0.197986
\(649\) 0 0
\(650\) 3.06037 0.120037
\(651\) 10.9094 0.427572
\(652\) −3.70882 −0.145248
\(653\) −22.0286 −0.862046 −0.431023 0.902341i \(-0.641847\pi\)
−0.431023 + 0.902341i \(0.641847\pi\)
\(654\) −3.04051 −0.118893
\(655\) −4.89161 −0.191131
\(656\) 5.13257 0.200393
\(657\) −27.9003 −1.08850
\(658\) 11.9195 0.464672
\(659\) −19.5066 −0.759868 −0.379934 0.925014i \(-0.624053\pi\)
−0.379934 + 0.925014i \(0.624053\pi\)
\(660\) 0 0
\(661\) 8.75637 0.340583 0.170292 0.985394i \(-0.445529\pi\)
0.170292 + 0.985394i \(0.445529\pi\)
\(662\) −7.10546 −0.276161
\(663\) 0.553182 0.0214838
\(664\) −3.05366 −0.118505
\(665\) 2.77079 0.107447
\(666\) 22.4929 0.871582
\(667\) −6.06011 −0.234648
\(668\) 8.75429 0.338714
\(669\) −54.1209 −2.09244
\(670\) 2.20818 0.0853096
\(671\) 0 0
\(672\) −2.64424 −0.102004
\(673\) 2.41083 0.0929306 0.0464653 0.998920i \(-0.485204\pi\)
0.0464653 + 0.998920i \(0.485204\pi\)
\(674\) 28.7650 1.10798
\(675\) −2.62310 −0.100963
\(676\) −3.63416 −0.139776
\(677\) −2.32293 −0.0892776 −0.0446388 0.999003i \(-0.514214\pi\)
−0.0446388 + 0.999003i \(0.514214\pi\)
\(678\) −32.1371 −1.23422
\(679\) 2.74477 0.105335
\(680\) −0.0683587 −0.00262144
\(681\) 62.6792 2.40187
\(682\) 0 0
\(683\) −4.35856 −0.166776 −0.0833879 0.996517i \(-0.526574\pi\)
−0.0833879 + 0.996517i \(0.526574\pi\)
\(684\) −11.0610 −0.422929
\(685\) −19.0559 −0.728087
\(686\) 1.00000 0.0381802
\(687\) 70.4342 2.68723
\(688\) −2.84053 −0.108294
\(689\) −0.378350 −0.0144140
\(690\) 4.84213 0.184337
\(691\) −20.4837 −0.779236 −0.389618 0.920977i \(-0.627393\pi\)
−0.389618 + 0.920977i \(0.627393\pi\)
\(692\) 3.03876 0.115516
\(693\) 0 0
\(694\) −25.7893 −0.978950
\(695\) −7.65318 −0.290302
\(696\) −8.75076 −0.331697
\(697\) 0.350856 0.0132896
\(698\) 36.6973 1.38901
\(699\) 80.2177 3.03411
\(700\) −1.00000 −0.0377964
\(701\) 25.5850 0.966333 0.483166 0.875529i \(-0.339487\pi\)
0.483166 + 0.875529i \(0.339487\pi\)
\(702\) −8.02766 −0.302984
\(703\) 15.6120 0.588817
\(704\) 0 0
\(705\) −31.5181 −1.18704
\(706\) −16.9704 −0.638688
\(707\) −5.53961 −0.208339
\(708\) 29.5968 1.11232
\(709\) −20.1461 −0.756604 −0.378302 0.925682i \(-0.623492\pi\)
−0.378302 + 0.925682i \(0.623492\pi\)
\(710\) −13.7247 −0.515080
\(711\) 49.5576 1.85856
\(712\) −13.5707 −0.508584
\(713\) 7.55500 0.282937
\(714\) −0.180757 −0.00676465
\(715\) 0 0
\(716\) −7.29987 −0.272809
\(717\) −26.3150 −0.982750
\(718\) 21.1819 0.790501
\(719\) −41.3909 −1.54362 −0.771811 0.635852i \(-0.780649\pi\)
−0.771811 + 0.635852i \(0.780649\pi\)
\(720\) 3.99201 0.148773
\(721\) 3.61697 0.134703
\(722\) 11.3227 0.421387
\(723\) −79.8162 −2.96840
\(724\) 10.9351 0.406400
\(725\) −3.30937 −0.122907
\(726\) 0 0
\(727\) −18.3813 −0.681723 −0.340862 0.940113i \(-0.610719\pi\)
−0.340862 + 0.940113i \(0.610719\pi\)
\(728\) −3.06037 −0.113425
\(729\) −40.9277 −1.51584
\(730\) 6.98905 0.258676
\(731\) −0.194175 −0.00718183
\(732\) 6.70720 0.247905
\(733\) −47.3817 −1.75008 −0.875042 0.484048i \(-0.839166\pi\)
−0.875042 + 0.484048i \(0.839166\pi\)
\(734\) −36.4966 −1.34711
\(735\) −2.64424 −0.0975343
\(736\) −1.83120 −0.0674989
\(737\) 0 0
\(738\) −20.4893 −0.754220
\(739\) 36.4944 1.34247 0.671234 0.741246i \(-0.265765\pi\)
0.671234 + 0.741246i \(0.265765\pi\)
\(740\) −5.63448 −0.207128
\(741\) −22.4222 −0.823700
\(742\) 0.123629 0.00453857
\(743\) −19.4544 −0.713714 −0.356857 0.934159i \(-0.616152\pi\)
−0.356857 + 0.934159i \(0.616152\pi\)
\(744\) 10.9094 0.399957
\(745\) 14.6774 0.537737
\(746\) −6.23055 −0.228117
\(747\) 12.1902 0.446017
\(748\) 0 0
\(749\) 6.58182 0.240494
\(750\) 2.64424 0.0965540
\(751\) −34.0903 −1.24397 −0.621986 0.783029i \(-0.713674\pi\)
−0.621986 + 0.783029i \(0.713674\pi\)
\(752\) 11.9195 0.434661
\(753\) 3.09371 0.112741
\(754\) −10.1279 −0.368835
\(755\) 11.0935 0.403733
\(756\) 2.62310 0.0954014
\(757\) −8.47015 −0.307853 −0.153926 0.988082i \(-0.549192\pi\)
−0.153926 + 0.988082i \(0.549192\pi\)
\(758\) −0.463665 −0.0168411
\(759\) 0 0
\(760\) 2.77079 0.100507
\(761\) −14.8964 −0.539993 −0.269997 0.962861i \(-0.587023\pi\)
−0.269997 + 0.962861i \(0.587023\pi\)
\(762\) −36.4417 −1.32014
\(763\) 1.14986 0.0416277
\(764\) 22.6520 0.819519
\(765\) 0.272888 0.00986630
\(766\) −30.4947 −1.10182
\(767\) 34.2545 1.23686
\(768\) −2.64424 −0.0954158
\(769\) −26.5771 −0.958396 −0.479198 0.877707i \(-0.659072\pi\)
−0.479198 + 0.877707i \(0.659072\pi\)
\(770\) 0 0
\(771\) 2.59378 0.0934126
\(772\) 16.9764 0.610994
\(773\) 23.5243 0.846112 0.423056 0.906104i \(-0.360957\pi\)
0.423056 + 0.906104i \(0.360957\pi\)
\(774\) 11.3394 0.407587
\(775\) 4.12571 0.148200
\(776\) 2.74477 0.0985315
\(777\) −14.8989 −0.534496
\(778\) 13.9789 0.501167
\(779\) −14.2213 −0.509531
\(780\) 8.09234 0.289752
\(781\) 0 0
\(782\) −0.125178 −0.00447637
\(783\) 8.68081 0.310227
\(784\) 1.00000 0.0357143
\(785\) 13.1941 0.470918
\(786\) −12.9346 −0.461361
\(787\) −9.85816 −0.351405 −0.175703 0.984443i \(-0.556220\pi\)
−0.175703 + 0.984443i \(0.556220\pi\)
\(788\) −26.3241 −0.937757
\(789\) −42.2874 −1.50547
\(790\) −12.4142 −0.441678
\(791\) 12.1536 0.432133
\(792\) 0 0
\(793\) 7.76271 0.275662
\(794\) 11.9510 0.424123
\(795\) −0.326905 −0.0115941
\(796\) 16.9127 0.599456
\(797\) −38.6360 −1.36856 −0.684279 0.729220i \(-0.739883\pi\)
−0.684279 + 0.729220i \(0.739883\pi\)
\(798\) 7.32664 0.259360
\(799\) 0.814804 0.0288257
\(800\) −1.00000 −0.0353553
\(801\) 54.1743 1.91416
\(802\) 14.4578 0.510524
\(803\) 0 0
\(804\) 5.83897 0.205925
\(805\) −1.83120 −0.0645413
\(806\) 12.6262 0.444738
\(807\) −39.9406 −1.40598
\(808\) −5.53961 −0.194883
\(809\) −30.9594 −1.08847 −0.544237 0.838932i \(-0.683181\pi\)
−0.544237 + 0.838932i \(0.683181\pi\)
\(810\) 5.03990 0.177084
\(811\) 49.4329 1.73582 0.867912 0.496717i \(-0.165461\pi\)
0.867912 + 0.496717i \(0.165461\pi\)
\(812\) 3.30937 0.116136
\(813\) −1.32092 −0.0463268
\(814\) 0 0
\(815\) −3.70882 −0.129914
\(816\) −0.180757 −0.00632776
\(817\) 7.87053 0.275355
\(818\) −3.54419 −0.123920
\(819\) 12.2170 0.426896
\(820\) 5.13257 0.179237
\(821\) −23.1278 −0.807167 −0.403584 0.914943i \(-0.632236\pi\)
−0.403584 + 0.914943i \(0.632236\pi\)
\(822\) −50.3882 −1.75749
\(823\) −36.0554 −1.25681 −0.628406 0.777885i \(-0.716292\pi\)
−0.628406 + 0.777885i \(0.716292\pi\)
\(824\) 3.61697 0.126003
\(825\) 0 0
\(826\) −11.1929 −0.389452
\(827\) 10.3899 0.361293 0.180647 0.983548i \(-0.442181\pi\)
0.180647 + 0.983548i \(0.442181\pi\)
\(828\) 7.31016 0.254046
\(829\) 37.4379 1.30027 0.650136 0.759818i \(-0.274712\pi\)
0.650136 + 0.759818i \(0.274712\pi\)
\(830\) −3.05366 −0.105994
\(831\) −29.3672 −1.01874
\(832\) −3.06037 −0.106099
\(833\) 0.0683587 0.00236849
\(834\) −20.2369 −0.700745
\(835\) 8.75429 0.302955
\(836\) 0 0
\(837\) −10.8222 −0.374069
\(838\) 35.5085 1.22662
\(839\) −32.6640 −1.12769 −0.563843 0.825882i \(-0.690678\pi\)
−0.563843 + 0.825882i \(0.690678\pi\)
\(840\) −2.64424 −0.0912350
\(841\) −18.0481 −0.622348
\(842\) 26.6190 0.917351
\(843\) 40.8567 1.40718
\(844\) 16.4483 0.566173
\(845\) −3.63416 −0.125019
\(846\) −47.5829 −1.63593
\(847\) 0 0
\(848\) 0.123629 0.00424544
\(849\) 37.8495 1.29899
\(850\) −0.0683587 −0.00234468
\(851\) −10.3179 −0.353692
\(852\) −36.2915 −1.24333
\(853\) 21.4274 0.733659 0.366830 0.930288i \(-0.380443\pi\)
0.366830 + 0.930288i \(0.380443\pi\)
\(854\) −2.53653 −0.0867983
\(855\) −11.0610 −0.378279
\(856\) 6.58182 0.224962
\(857\) 46.6805 1.59457 0.797287 0.603600i \(-0.206268\pi\)
0.797287 + 0.603600i \(0.206268\pi\)
\(858\) 0 0
\(859\) 34.0149 1.16057 0.580287 0.814412i \(-0.302940\pi\)
0.580287 + 0.814412i \(0.302940\pi\)
\(860\) −2.84053 −0.0968614
\(861\) 13.5718 0.462524
\(862\) 26.0439 0.887057
\(863\) −5.55865 −0.189219 −0.0946094 0.995514i \(-0.530160\pi\)
−0.0946094 + 0.995514i \(0.530160\pi\)
\(864\) 2.62310 0.0892398
\(865\) 3.03876 0.103321
\(866\) −1.69218 −0.0575025
\(867\) 44.9397 1.52623
\(868\) −4.12571 −0.140036
\(869\) 0 0
\(870\) −8.75076 −0.296679
\(871\) 6.75785 0.228981
\(872\) 1.14986 0.0389392
\(873\) −10.9571 −0.370843
\(874\) 5.07387 0.171626
\(875\) −1.00000 −0.0338062
\(876\) 18.4807 0.624406
\(877\) 18.3183 0.618564 0.309282 0.950970i \(-0.399911\pi\)
0.309282 + 0.950970i \(0.399911\pi\)
\(878\) 12.8065 0.432200
\(879\) −46.4721 −1.56747
\(880\) 0 0
\(881\) −0.0269589 −0.000908270 0 −0.000454135 1.00000i \(-0.500145\pi\)
−0.000454135 1.00000i \(0.500145\pi\)
\(882\) −3.99201 −0.134418
\(883\) −5.28336 −0.177799 −0.0888997 0.996041i \(-0.528335\pi\)
−0.0888997 + 0.996041i \(0.528335\pi\)
\(884\) −0.209203 −0.00703625
\(885\) 29.5968 0.994886
\(886\) −0.704398 −0.0236647
\(887\) −39.7555 −1.33486 −0.667430 0.744672i \(-0.732606\pi\)
−0.667430 + 0.744672i \(0.732606\pi\)
\(888\) −14.8989 −0.499975
\(889\) 13.7815 0.462218
\(890\) −13.5707 −0.454891
\(891\) 0 0
\(892\) 20.4675 0.685302
\(893\) −33.0266 −1.10519
\(894\) 38.8105 1.29802
\(895\) −7.29987 −0.244008
\(896\) 1.00000 0.0334077
\(897\) 14.8187 0.494782
\(898\) 18.4066 0.614235
\(899\) −13.6535 −0.455369
\(900\) 3.99201 0.133067
\(901\) 0.00845113 0.000281548 0
\(902\) 0 0
\(903\) −7.51106 −0.249952
\(904\) 12.1536 0.404223
\(905\) 10.9351 0.363495
\(906\) 29.3338 0.974551
\(907\) −5.61591 −0.186473 −0.0932367 0.995644i \(-0.529721\pi\)
−0.0932367 + 0.995644i \(0.529721\pi\)
\(908\) −23.7041 −0.786647
\(909\) 22.1142 0.733481
\(910\) −3.06037 −0.101450
\(911\) −28.1487 −0.932609 −0.466304 0.884624i \(-0.654415\pi\)
−0.466304 + 0.884624i \(0.654415\pi\)
\(912\) 7.32664 0.242609
\(913\) 0 0
\(914\) 16.9955 0.562161
\(915\) 6.70720 0.221733
\(916\) −26.6368 −0.880106
\(917\) 4.89161 0.161535
\(918\) 0.179312 0.00591818
\(919\) −47.6030 −1.57028 −0.785140 0.619319i \(-0.787409\pi\)
−0.785140 + 0.619319i \(0.787409\pi\)
\(920\) −1.83120 −0.0603728
\(921\) 14.2269 0.468791
\(922\) 13.3276 0.438921
\(923\) −42.0027 −1.38254
\(924\) 0 0
\(925\) −5.63448 −0.185261
\(926\) 34.9814 1.14956
\(927\) −14.4390 −0.474238
\(928\) 3.30937 0.108635
\(929\) −38.1848 −1.25280 −0.626401 0.779501i \(-0.715473\pi\)
−0.626401 + 0.779501i \(0.715473\pi\)
\(930\) 10.9094 0.357732
\(931\) −2.77079 −0.0908091
\(932\) −30.3368 −0.993714
\(933\) −44.6951 −1.46325
\(934\) 3.57861 0.117096
\(935\) 0 0
\(936\) 12.2170 0.399325
\(937\) 0.397595 0.0129889 0.00649443 0.999979i \(-0.497933\pi\)
0.00649443 + 0.999979i \(0.497933\pi\)
\(938\) −2.20818 −0.0720998
\(939\) 3.58355 0.116945
\(940\) 11.9195 0.388772
\(941\) 6.77789 0.220953 0.110477 0.993879i \(-0.464762\pi\)
0.110477 + 0.993879i \(0.464762\pi\)
\(942\) 34.8884 1.13673
\(943\) 9.39876 0.306066
\(944\) −11.1929 −0.364299
\(945\) 2.62310 0.0853296
\(946\) 0 0
\(947\) −10.5378 −0.342432 −0.171216 0.985234i \(-0.554770\pi\)
−0.171216 + 0.985234i \(0.554770\pi\)
\(948\) −32.8262 −1.06614
\(949\) 21.3891 0.694318
\(950\) 2.77079 0.0898964
\(951\) −77.5335 −2.51420
\(952\) 0.0683587 0.00221552
\(953\) 9.44810 0.306054 0.153027 0.988222i \(-0.451098\pi\)
0.153027 + 0.988222i \(0.451098\pi\)
\(954\) −0.493528 −0.0159786
\(955\) 22.6520 0.733000
\(956\) 9.95180 0.321864
\(957\) 0 0
\(958\) 18.3300 0.592215
\(959\) 19.0559 0.615346
\(960\) −2.64424 −0.0853425
\(961\) −13.9785 −0.450920
\(962\) −17.2436 −0.555955
\(963\) −26.2747 −0.846689
\(964\) 30.1849 0.972191
\(965\) 16.9764 0.546490
\(966\) −4.84213 −0.155793
\(967\) −47.0752 −1.51384 −0.756918 0.653509i \(-0.773296\pi\)
−0.756918 + 0.653509i \(0.773296\pi\)
\(968\) 0 0
\(969\) 0.500840 0.0160893
\(970\) 2.74477 0.0881293
\(971\) 17.2012 0.552012 0.276006 0.961156i \(-0.410989\pi\)
0.276006 + 0.961156i \(0.410989\pi\)
\(972\) 21.1960 0.679863
\(973\) 7.65318 0.245350
\(974\) −0.487041 −0.0156058
\(975\) 8.09234 0.259162
\(976\) −2.53653 −0.0811924
\(977\) −6.75798 −0.216207 −0.108103 0.994140i \(-0.534478\pi\)
−0.108103 + 0.994140i \(0.534478\pi\)
\(978\) −9.80700 −0.313593
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −4.59025 −0.146555
\(982\) 6.09167 0.194393
\(983\) −45.4819 −1.45065 −0.725323 0.688408i \(-0.758310\pi\)
−0.725323 + 0.688408i \(0.758310\pi\)
\(984\) 13.5718 0.432652
\(985\) −26.3241 −0.838755
\(986\) 0.226224 0.00720444
\(987\) 31.5181 1.00323
\(988\) 8.47964 0.269773
\(989\) −5.20158 −0.165401
\(990\) 0 0
\(991\) −11.7391 −0.372906 −0.186453 0.982464i \(-0.559699\pi\)
−0.186453 + 0.982464i \(0.559699\pi\)
\(992\) −4.12571 −0.130991
\(993\) −18.7885 −0.596236
\(994\) 13.7247 0.435322
\(995\) 16.9127 0.536170
\(996\) −8.07461 −0.255854
\(997\) 17.1623 0.543536 0.271768 0.962363i \(-0.412392\pi\)
0.271768 + 0.962363i \(0.412392\pi\)
\(998\) −42.3141 −1.33943
\(999\) 14.7798 0.467613
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.cy.1.2 6
11.7 odd 10 770.2.n.i.71.3 12
11.8 odd 10 770.2.n.i.141.3 yes 12
11.10 odd 2 8470.2.a.de.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.i.71.3 12 11.7 odd 10
770.2.n.i.141.3 yes 12 11.8 odd 10
8470.2.a.cy.1.2 6 1.1 even 1 trivial
8470.2.a.de.1.2 6 11.10 odd 2