Properties

Label 8470.2.a.cz.1.4
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.10784448.1
Defining polynomial: \(x^{6} - 11 x^{4} - 4 x^{3} + 31 x^{2} + 22 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.930827\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.529317 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.529317 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.71982 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +0.529317 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.529317 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.71982 q^{9} -1.00000 q^{10} +0.529317 q^{12} +4.81243 q^{13} +1.00000 q^{14} +0.529317 q^{15} +1.00000 q^{16} -6.02760 q^{17} +2.71982 q^{18} -3.59370 q^{19} +1.00000 q^{20} -0.529317 q^{21} -7.06157 q^{23} -0.529317 q^{24} +1.00000 q^{25} -4.81243 q^{26} -3.02760 q^{27} -1.00000 q^{28} -0.353515 q^{29} -0.529317 q^{30} -2.24914 q^{31} -1.00000 q^{32} +6.02760 q^{34} -1.00000 q^{35} -2.71982 q^{36} +1.64649 q^{37} +3.59370 q^{38} +2.54730 q^{39} -1.00000 q^{40} -9.35973 q^{41} +0.529317 q^{42} -9.13818 q^{43} -2.71982 q^{45} +7.06157 q^{46} +10.7634 q^{47} +0.529317 q^{48} +1.00000 q^{49} -1.00000 q^{50} -3.19051 q^{51} +4.81243 q^{52} +4.64885 q^{53} +3.02760 q^{54} +1.00000 q^{56} -1.90221 q^{57} +0.353515 q^{58} +2.78483 q^{59} +0.529317 q^{60} +14.6733 q^{61} +2.24914 q^{62} +2.71982 q^{63} +1.00000 q^{64} +4.81243 q^{65} +9.43079 q^{67} -6.02760 q^{68} -3.73781 q^{69} +1.00000 q^{70} +12.2276 q^{71} +2.71982 q^{72} -5.95077 q^{73} -1.64649 q^{74} +0.529317 q^{75} -3.59370 q^{76} -2.54730 q^{78} -8.04631 q^{79} +1.00000 q^{80} +6.55691 q^{81} +9.35973 q^{82} +16.6025 q^{83} -0.529317 q^{84} -6.02760 q^{85} +9.13818 q^{86} -0.187121 q^{87} +16.0923 q^{89} +2.71982 q^{90} -4.81243 q^{91} -7.06157 q^{92} -1.19051 q^{93} -10.7634 q^{94} -3.59370 q^{95} -0.529317 q^{96} -1.23078 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{2} + 4q^{3} + 6q^{4} + 6q^{5} - 4q^{6} - 6q^{7} - 6q^{8} + 2q^{9} + O(q^{10}) \) \( 6q - 6q^{2} + 4q^{3} + 6q^{4} + 6q^{5} - 4q^{6} - 6q^{7} - 6q^{8} + 2q^{9} - 6q^{10} + 4q^{12} + 6q^{14} + 4q^{15} + 6q^{16} - 2q^{17} - 2q^{18} + 6q^{20} - 4q^{21} + 4q^{23} - 4q^{24} + 6q^{25} + 16q^{27} - 6q^{28} - 8q^{29} - 4q^{30} + 4q^{31} - 6q^{32} + 2q^{34} - 6q^{35} + 2q^{36} + 4q^{37} - 6q^{40} - 12q^{41} + 4q^{42} + 6q^{43} + 2q^{45} - 4q^{46} + 16q^{47} + 4q^{48} + 6q^{49} - 6q^{50} + 12q^{53} - 16q^{54} + 6q^{56} - 8q^{57} + 8q^{58} + 22q^{59} + 4q^{60} + 4q^{61} - 4q^{62} - 2q^{63} + 6q^{64} + 20q^{67} - 2q^{68} + 12q^{69} + 6q^{70} + 14q^{71} - 2q^{72} - 18q^{73} - 4q^{74} + 4q^{75} - 32q^{79} + 6q^{80} + 6q^{81} + 12q^{82} + 16q^{83} - 4q^{84} - 2q^{85} - 6q^{86} + 4q^{87} + 4q^{89} - 2q^{90} + 4q^{92} + 12q^{93} - 16q^{94} - 4q^{96} + 4q^{97} - 6q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0.529317 0.305601 0.152801 0.988257i \(-0.451171\pi\)
0.152801 + 0.988257i \(0.451171\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −0.529317 −0.216093
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) −2.71982 −0.906608
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 0.529317 0.152801
\(13\) 4.81243 1.33473 0.667364 0.744732i \(-0.267423\pi\)
0.667364 + 0.744732i \(0.267423\pi\)
\(14\) 1.00000 0.267261
\(15\) 0.529317 0.136669
\(16\) 1.00000 0.250000
\(17\) −6.02760 −1.46191 −0.730954 0.682427i \(-0.760924\pi\)
−0.730954 + 0.682427i \(0.760924\pi\)
\(18\) 2.71982 0.641069
\(19\) −3.59370 −0.824452 −0.412226 0.911082i \(-0.635249\pi\)
−0.412226 + 0.911082i \(0.635249\pi\)
\(20\) 1.00000 0.223607
\(21\) −0.529317 −0.115506
\(22\) 0 0
\(23\) −7.06157 −1.47244 −0.736220 0.676743i \(-0.763391\pi\)
−0.736220 + 0.676743i \(0.763391\pi\)
\(24\) −0.529317 −0.108046
\(25\) 1.00000 0.200000
\(26\) −4.81243 −0.943795
\(27\) −3.02760 −0.582661
\(28\) −1.00000 −0.188982
\(29\) −0.353515 −0.0656461 −0.0328230 0.999461i \(-0.510450\pi\)
−0.0328230 + 0.999461i \(0.510450\pi\)
\(30\) −0.529317 −0.0966395
\(31\) −2.24914 −0.403958 −0.201979 0.979390i \(-0.564737\pi\)
−0.201979 + 0.979390i \(0.564737\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.02760 1.03372
\(35\) −1.00000 −0.169031
\(36\) −2.71982 −0.453304
\(37\) 1.64649 0.270680 0.135340 0.990799i \(-0.456787\pi\)
0.135340 + 0.990799i \(0.456787\pi\)
\(38\) 3.59370 0.582976
\(39\) 2.54730 0.407894
\(40\) −1.00000 −0.158114
\(41\) −9.35973 −1.46174 −0.730872 0.682515i \(-0.760886\pi\)
−0.730872 + 0.682515i \(0.760886\pi\)
\(42\) 0.529317 0.0816753
\(43\) −9.13818 −1.39356 −0.696780 0.717285i \(-0.745385\pi\)
−0.696780 + 0.717285i \(0.745385\pi\)
\(44\) 0 0
\(45\) −2.71982 −0.405447
\(46\) 7.06157 1.04117
\(47\) 10.7634 1.57000 0.785002 0.619493i \(-0.212662\pi\)
0.785002 + 0.619493i \(0.212662\pi\)
\(48\) 0.529317 0.0764003
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −3.19051 −0.446760
\(52\) 4.81243 0.667364
\(53\) 4.64885 0.638569 0.319285 0.947659i \(-0.396557\pi\)
0.319285 + 0.947659i \(0.396557\pi\)
\(54\) 3.02760 0.412004
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −1.90221 −0.251954
\(58\) 0.353515 0.0464188
\(59\) 2.78483 0.362554 0.181277 0.983432i \(-0.441977\pi\)
0.181277 + 0.983432i \(0.441977\pi\)
\(60\) 0.529317 0.0683345
\(61\) 14.6733 1.87872 0.939359 0.342935i \(-0.111421\pi\)
0.939359 + 0.342935i \(0.111421\pi\)
\(62\) 2.24914 0.285641
\(63\) 2.71982 0.342666
\(64\) 1.00000 0.125000
\(65\) 4.81243 0.596908
\(66\) 0 0
\(67\) 9.43079 1.15215 0.576077 0.817395i \(-0.304583\pi\)
0.576077 + 0.817395i \(0.304583\pi\)
\(68\) −6.02760 −0.730954
\(69\) −3.73781 −0.449979
\(70\) 1.00000 0.119523
\(71\) 12.2276 1.45115 0.725575 0.688143i \(-0.241574\pi\)
0.725575 + 0.688143i \(0.241574\pi\)
\(72\) 2.71982 0.320534
\(73\) −5.95077 −0.696486 −0.348243 0.937404i \(-0.613222\pi\)
−0.348243 + 0.937404i \(0.613222\pi\)
\(74\) −1.64649 −0.191400
\(75\) 0.529317 0.0611202
\(76\) −3.59370 −0.412226
\(77\) 0 0
\(78\) −2.54730 −0.288425
\(79\) −8.04631 −0.905280 −0.452640 0.891693i \(-0.649518\pi\)
−0.452640 + 0.891693i \(0.649518\pi\)
\(80\) 1.00000 0.111803
\(81\) 6.55691 0.728546
\(82\) 9.35973 1.03361
\(83\) 16.6025 1.82236 0.911180 0.412008i \(-0.135172\pi\)
0.911180 + 0.412008i \(0.135172\pi\)
\(84\) −0.529317 −0.0577532
\(85\) −6.02760 −0.653785
\(86\) 9.13818 0.985396
\(87\) −0.187121 −0.0200615
\(88\) 0 0
\(89\) 16.0923 1.70579 0.852893 0.522086i \(-0.174846\pi\)
0.852893 + 0.522086i \(0.174846\pi\)
\(90\) 2.71982 0.286695
\(91\) −4.81243 −0.504480
\(92\) −7.06157 −0.736220
\(93\) −1.19051 −0.123450
\(94\) −10.7634 −1.11016
\(95\) −3.59370 −0.368706
\(96\) −0.529317 −0.0540231
\(97\) −1.23078 −0.124967 −0.0624835 0.998046i \(-0.519902\pi\)
−0.0624835 + 0.998046i \(0.519902\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 10.8741 1.08201 0.541007 0.841018i \(-0.318043\pi\)
0.541007 + 0.841018i \(0.318043\pi\)
\(102\) 3.19051 0.315907
\(103\) 3.42100 0.337081 0.168541 0.985695i \(-0.446095\pi\)
0.168541 + 0.985695i \(0.446095\pi\)
\(104\) −4.81243 −0.471897
\(105\) −0.529317 −0.0516560
\(106\) −4.64885 −0.451537
\(107\) 9.35136 0.904030 0.452015 0.892010i \(-0.350705\pi\)
0.452015 + 0.892010i \(0.350705\pi\)
\(108\) −3.02760 −0.291331
\(109\) −19.9100 −1.90703 −0.953514 0.301348i \(-0.902563\pi\)
−0.953514 + 0.301348i \(0.902563\pi\)
\(110\) 0 0
\(111\) 0.871512 0.0827202
\(112\) −1.00000 −0.0944911
\(113\) 4.10257 0.385938 0.192969 0.981205i \(-0.438188\pi\)
0.192969 + 0.981205i \(0.438188\pi\)
\(114\) 1.90221 0.178158
\(115\) −7.06157 −0.658495
\(116\) −0.353515 −0.0328230
\(117\) −13.0890 −1.21007
\(118\) −2.78483 −0.256364
\(119\) 6.02760 0.552549
\(120\) −0.529317 −0.0483198
\(121\) 0 0
\(122\) −14.6733 −1.32845
\(123\) −4.95426 −0.446710
\(124\) −2.24914 −0.201979
\(125\) 1.00000 0.0894427
\(126\) −2.71982 −0.242301
\(127\) −0.926321 −0.0821977 −0.0410989 0.999155i \(-0.513086\pi\)
−0.0410989 + 0.999155i \(0.513086\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.83699 −0.425873
\(130\) −4.81243 −0.422078
\(131\) 13.3755 1.16862 0.584310 0.811530i \(-0.301365\pi\)
0.584310 + 0.811530i \(0.301365\pi\)
\(132\) 0 0
\(133\) 3.59370 0.311614
\(134\) −9.43079 −0.814697
\(135\) −3.02760 −0.260574
\(136\) 6.02760 0.516862
\(137\) 19.8175 1.69312 0.846560 0.532294i \(-0.178670\pi\)
0.846560 + 0.532294i \(0.178670\pi\)
\(138\) 3.73781 0.318183
\(139\) −18.7006 −1.58616 −0.793081 0.609117i \(-0.791524\pi\)
−0.793081 + 0.609117i \(0.791524\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 5.69725 0.479795
\(142\) −12.2276 −1.02612
\(143\) 0 0
\(144\) −2.71982 −0.226652
\(145\) −0.353515 −0.0293578
\(146\) 5.95077 0.492490
\(147\) 0.529317 0.0436573
\(148\) 1.64649 0.135340
\(149\) 19.6448 1.60937 0.804684 0.593704i \(-0.202335\pi\)
0.804684 + 0.593704i \(0.202335\pi\)
\(150\) −0.529317 −0.0432185
\(151\) −1.69178 −0.137675 −0.0688374 0.997628i \(-0.521929\pi\)
−0.0688374 + 0.997628i \(0.521929\pi\)
\(152\) 3.59370 0.291488
\(153\) 16.3940 1.32538
\(154\) 0 0
\(155\) −2.24914 −0.180655
\(156\) 2.54730 0.203947
\(157\) −13.6152 −1.08661 −0.543305 0.839536i \(-0.682827\pi\)
−0.543305 + 0.839536i \(0.682827\pi\)
\(158\) 8.04631 0.640130
\(159\) 2.46072 0.195147
\(160\) −1.00000 −0.0790569
\(161\) 7.06157 0.556530
\(162\) −6.55691 −0.515160
\(163\) 14.0694 1.10200 0.551001 0.834505i \(-0.314246\pi\)
0.551001 + 0.834505i \(0.314246\pi\)
\(164\) −9.35973 −0.730872
\(165\) 0 0
\(166\) −16.6025 −1.28860
\(167\) −17.5422 −1.35745 −0.678726 0.734391i \(-0.737468\pi\)
−0.678726 + 0.734391i \(0.737468\pi\)
\(168\) 0.529317 0.0408377
\(169\) 10.1595 0.781498
\(170\) 6.02760 0.462296
\(171\) 9.77424 0.747455
\(172\) −9.13818 −0.696780
\(173\) 14.8544 1.12936 0.564681 0.825310i \(-0.308999\pi\)
0.564681 + 0.825310i \(0.308999\pi\)
\(174\) 0.187121 0.0141856
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 1.47406 0.110797
\(178\) −16.0923 −1.20617
\(179\) −13.8768 −1.03720 −0.518601 0.855016i \(-0.673547\pi\)
−0.518601 + 0.855016i \(0.673547\pi\)
\(180\) −2.71982 −0.202724
\(181\) 16.3546 1.21563 0.607815 0.794078i \(-0.292046\pi\)
0.607815 + 0.794078i \(0.292046\pi\)
\(182\) 4.81243 0.356721
\(183\) 7.76680 0.574138
\(184\) 7.06157 0.520586
\(185\) 1.64649 0.121052
\(186\) 1.19051 0.0872922
\(187\) 0 0
\(188\) 10.7634 0.785002
\(189\) 3.02760 0.220225
\(190\) 3.59370 0.260715
\(191\) −16.1410 −1.16792 −0.583961 0.811782i \(-0.698498\pi\)
−0.583961 + 0.811782i \(0.698498\pi\)
\(192\) 0.529317 0.0382001
\(193\) −3.72307 −0.267992 −0.133996 0.990982i \(-0.542781\pi\)
−0.133996 + 0.990982i \(0.542781\pi\)
\(194\) 1.23078 0.0883650
\(195\) 2.54730 0.182416
\(196\) 1.00000 0.0714286
\(197\) 2.01016 0.143218 0.0716091 0.997433i \(-0.477187\pi\)
0.0716091 + 0.997433i \(0.477187\pi\)
\(198\) 0 0
\(199\) −9.97363 −0.707012 −0.353506 0.935432i \(-0.615011\pi\)
−0.353506 + 0.935432i \(0.615011\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 4.99188 0.352100
\(202\) −10.8741 −0.765099
\(203\) 0.353515 0.0248119
\(204\) −3.19051 −0.223380
\(205\) −9.35973 −0.653712
\(206\) −3.42100 −0.238352
\(207\) 19.2062 1.33492
\(208\) 4.81243 0.333682
\(209\) 0 0
\(210\) 0.529317 0.0365263
\(211\) 10.8629 0.747832 0.373916 0.927463i \(-0.378015\pi\)
0.373916 + 0.927463i \(0.378015\pi\)
\(212\) 4.64885 0.319285
\(213\) 6.47228 0.443473
\(214\) −9.35136 −0.639246
\(215\) −9.13818 −0.623219
\(216\) 3.02760 0.206002
\(217\) 2.24914 0.152682
\(218\) 19.9100 1.34847
\(219\) −3.14984 −0.212847
\(220\) 0 0
\(221\) −29.0074 −1.95125
\(222\) −0.871512 −0.0584920
\(223\) −12.2446 −0.819956 −0.409978 0.912095i \(-0.634464\pi\)
−0.409978 + 0.912095i \(0.634464\pi\)
\(224\) 1.00000 0.0668153
\(225\) −2.71982 −0.181322
\(226\) −4.10257 −0.272899
\(227\) −21.0077 −1.39433 −0.697164 0.716912i \(-0.745555\pi\)
−0.697164 + 0.716912i \(0.745555\pi\)
\(228\) −1.90221 −0.125977
\(229\) −4.92949 −0.325750 −0.162875 0.986647i \(-0.552077\pi\)
−0.162875 + 0.986647i \(0.552077\pi\)
\(230\) 7.06157 0.465626
\(231\) 0 0
\(232\) 0.353515 0.0232094
\(233\) 20.1605 1.32076 0.660380 0.750932i \(-0.270395\pi\)
0.660380 + 0.750932i \(0.270395\pi\)
\(234\) 13.0890 0.855652
\(235\) 10.7634 0.702128
\(236\) 2.78483 0.181277
\(237\) −4.25904 −0.276655
\(238\) −6.02760 −0.390711
\(239\) 9.53194 0.616570 0.308285 0.951294i \(-0.400245\pi\)
0.308285 + 0.951294i \(0.400245\pi\)
\(240\) 0.529317 0.0341672
\(241\) 6.73710 0.433975 0.216987 0.976174i \(-0.430377\pi\)
0.216987 + 0.976174i \(0.430377\pi\)
\(242\) 0 0
\(243\) 12.5535 0.805306
\(244\) 14.6733 0.939359
\(245\) 1.00000 0.0638877
\(246\) 4.95426 0.315872
\(247\) −17.2944 −1.10042
\(248\) 2.24914 0.142821
\(249\) 8.78798 0.556915
\(250\) −1.00000 −0.0632456
\(251\) 8.94574 0.564650 0.282325 0.959319i \(-0.408894\pi\)
0.282325 + 0.959319i \(0.408894\pi\)
\(252\) 2.71982 0.171333
\(253\) 0 0
\(254\) 0.926321 0.0581226
\(255\) −3.19051 −0.199797
\(256\) 1.00000 0.0625000
\(257\) 16.9351 1.05639 0.528193 0.849125i \(-0.322870\pi\)
0.528193 + 0.849125i \(0.322870\pi\)
\(258\) 4.83699 0.301138
\(259\) −1.64649 −0.102308
\(260\) 4.81243 0.298454
\(261\) 0.961498 0.0595152
\(262\) −13.3755 −0.826340
\(263\) −16.0270 −0.988266 −0.494133 0.869386i \(-0.664514\pi\)
−0.494133 + 0.869386i \(0.664514\pi\)
\(264\) 0 0
\(265\) 4.64885 0.285577
\(266\) −3.59370 −0.220344
\(267\) 8.51795 0.521290
\(268\) 9.43079 0.576077
\(269\) 9.38110 0.571975 0.285988 0.958233i \(-0.407678\pi\)
0.285988 + 0.958233i \(0.407678\pi\)
\(270\) 3.02760 0.184254
\(271\) 15.8105 0.960418 0.480209 0.877154i \(-0.340561\pi\)
0.480209 + 0.877154i \(0.340561\pi\)
\(272\) −6.02760 −0.365477
\(273\) −2.54730 −0.154170
\(274\) −19.8175 −1.19722
\(275\) 0 0
\(276\) −3.73781 −0.224989
\(277\) 24.2012 1.45411 0.727054 0.686580i \(-0.240889\pi\)
0.727054 + 0.686580i \(0.240889\pi\)
\(278\) 18.7006 1.12159
\(279\) 6.11727 0.366231
\(280\) 1.00000 0.0597614
\(281\) −16.8649 −1.00608 −0.503038 0.864264i \(-0.667785\pi\)
−0.503038 + 0.864264i \(0.667785\pi\)
\(282\) −5.69725 −0.339266
\(283\) 3.49996 0.208051 0.104025 0.994575i \(-0.466828\pi\)
0.104025 + 0.994575i \(0.466828\pi\)
\(284\) 12.2276 0.725575
\(285\) −1.90221 −0.112677
\(286\) 0 0
\(287\) 9.35973 0.552487
\(288\) 2.71982 0.160267
\(289\) 19.3319 1.13717
\(290\) 0.353515 0.0207591
\(291\) −0.651473 −0.0381900
\(292\) −5.95077 −0.348243
\(293\) −10.6822 −0.624060 −0.312030 0.950072i \(-0.601009\pi\)
−0.312030 + 0.950072i \(0.601009\pi\)
\(294\) −0.529317 −0.0308704
\(295\) 2.78483 0.162139
\(296\) −1.64649 −0.0957000
\(297\) 0 0
\(298\) −19.6448 −1.13799
\(299\) −33.9833 −1.96531
\(300\) 0.529317 0.0305601
\(301\) 9.13818 0.526716
\(302\) 1.69178 0.0973508
\(303\) 5.75584 0.330664
\(304\) −3.59370 −0.206113
\(305\) 14.6733 0.840188
\(306\) −16.3940 −0.937183
\(307\) 17.4466 0.995731 0.497865 0.867254i \(-0.334117\pi\)
0.497865 + 0.867254i \(0.334117\pi\)
\(308\) 0 0
\(309\) 1.81079 0.103012
\(310\) 2.24914 0.127743
\(311\) 9.50939 0.539228 0.269614 0.962968i \(-0.413104\pi\)
0.269614 + 0.962968i \(0.413104\pi\)
\(312\) −2.54730 −0.144212
\(313\) 29.1754 1.64909 0.824545 0.565796i \(-0.191431\pi\)
0.824545 + 0.565796i \(0.191431\pi\)
\(314\) 13.6152 0.768349
\(315\) 2.71982 0.153245
\(316\) −8.04631 −0.452640
\(317\) −28.7999 −1.61756 −0.808782 0.588109i \(-0.799873\pi\)
−0.808782 + 0.588109i \(0.799873\pi\)
\(318\) −2.46072 −0.137990
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 4.94983 0.276273
\(322\) −7.06157 −0.393526
\(323\) 21.6614 1.20527
\(324\) 6.55691 0.364273
\(325\) 4.81243 0.266946
\(326\) −14.0694 −0.779233
\(327\) −10.5387 −0.582790
\(328\) 9.35973 0.516804
\(329\) −10.7634 −0.593406
\(330\) 0 0
\(331\) 17.9933 0.989003 0.494502 0.869177i \(-0.335351\pi\)
0.494502 + 0.869177i \(0.335351\pi\)
\(332\) 16.6025 0.911180
\(333\) −4.47815 −0.245401
\(334\) 17.5422 0.959864
\(335\) 9.43079 0.515259
\(336\) −0.529317 −0.0288766
\(337\) −23.2472 −1.26635 −0.633177 0.774007i \(-0.718249\pi\)
−0.633177 + 0.774007i \(0.718249\pi\)
\(338\) −10.1595 −0.552602
\(339\) 2.17156 0.117943
\(340\) −6.02760 −0.326892
\(341\) 0 0
\(342\) −9.77424 −0.528531
\(343\) −1.00000 −0.0539949
\(344\) 9.13818 0.492698
\(345\) −3.73781 −0.201237
\(346\) −14.8544 −0.798579
\(347\) 34.5288 1.85361 0.926803 0.375548i \(-0.122546\pi\)
0.926803 + 0.375548i \(0.122546\pi\)
\(348\) −0.187121 −0.0100308
\(349\) −12.3015 −0.658483 −0.329242 0.944246i \(-0.606793\pi\)
−0.329242 + 0.944246i \(0.606793\pi\)
\(350\) 1.00000 0.0534522
\(351\) −14.5701 −0.777694
\(352\) 0 0
\(353\) −4.73131 −0.251822 −0.125911 0.992042i \(-0.540185\pi\)
−0.125911 + 0.992042i \(0.540185\pi\)
\(354\) −1.47406 −0.0783453
\(355\) 12.2276 0.648974
\(356\) 16.0923 0.852893
\(357\) 3.19051 0.168860
\(358\) 13.8768 0.733413
\(359\) −1.31338 −0.0693178 −0.0346589 0.999399i \(-0.511034\pi\)
−0.0346589 + 0.999399i \(0.511034\pi\)
\(360\) 2.71982 0.143347
\(361\) −6.08529 −0.320278
\(362\) −16.3546 −0.859581
\(363\) 0 0
\(364\) −4.81243 −0.252240
\(365\) −5.95077 −0.311478
\(366\) −7.76680 −0.405977
\(367\) −5.39464 −0.281598 −0.140799 0.990038i \(-0.544967\pi\)
−0.140799 + 0.990038i \(0.544967\pi\)
\(368\) −7.06157 −0.368110
\(369\) 25.4568 1.32523
\(370\) −1.64649 −0.0855967
\(371\) −4.64885 −0.241356
\(372\) −1.19051 −0.0617249
\(373\) 13.7265 0.710731 0.355366 0.934727i \(-0.384356\pi\)
0.355366 + 0.934727i \(0.384356\pi\)
\(374\) 0 0
\(375\) 0.529317 0.0273338
\(376\) −10.7634 −0.555081
\(377\) −1.70127 −0.0876196
\(378\) −3.02760 −0.155723
\(379\) −12.4209 −0.638019 −0.319009 0.947752i \(-0.603350\pi\)
−0.319009 + 0.947752i \(0.603350\pi\)
\(380\) −3.59370 −0.184353
\(381\) −0.490317 −0.0251197
\(382\) 16.1410 0.825846
\(383\) −6.58507 −0.336481 −0.168241 0.985746i \(-0.553809\pi\)
−0.168241 + 0.985746i \(0.553809\pi\)
\(384\) −0.529317 −0.0270116
\(385\) 0 0
\(386\) 3.72307 0.189499
\(387\) 24.8543 1.26341
\(388\) −1.23078 −0.0624835
\(389\) −23.8620 −1.20985 −0.604925 0.796282i \(-0.706797\pi\)
−0.604925 + 0.796282i \(0.706797\pi\)
\(390\) −2.54730 −0.128987
\(391\) 42.5643 2.15257
\(392\) −1.00000 −0.0505076
\(393\) 7.07986 0.357132
\(394\) −2.01016 −0.101271
\(395\) −8.04631 −0.404854
\(396\) 0 0
\(397\) −14.1058 −0.707952 −0.353976 0.935254i \(-0.615171\pi\)
−0.353976 + 0.935254i \(0.615171\pi\)
\(398\) 9.97363 0.499933
\(399\) 1.90221 0.0952295
\(400\) 1.00000 0.0500000
\(401\) −1.50759 −0.0752855 −0.0376428 0.999291i \(-0.511985\pi\)
−0.0376428 + 0.999291i \(0.511985\pi\)
\(402\) −4.99188 −0.248972
\(403\) −10.8238 −0.539173
\(404\) 10.8741 0.541007
\(405\) 6.55691 0.325816
\(406\) −0.353515 −0.0175446
\(407\) 0 0
\(408\) 3.19051 0.157954
\(409\) 4.42801 0.218951 0.109476 0.993989i \(-0.465083\pi\)
0.109476 + 0.993989i \(0.465083\pi\)
\(410\) 9.35973 0.462244
\(411\) 10.4897 0.517419
\(412\) 3.42100 0.168541
\(413\) −2.78483 −0.137033
\(414\) −19.2062 −0.943934
\(415\) 16.6025 0.814984
\(416\) −4.81243 −0.235949
\(417\) −9.89852 −0.484733
\(418\) 0 0
\(419\) −16.3250 −0.797529 −0.398765 0.917053i \(-0.630561\pi\)
−0.398765 + 0.917053i \(0.630561\pi\)
\(420\) −0.529317 −0.0258280
\(421\) −3.12002 −0.152061 −0.0760303 0.997106i \(-0.524225\pi\)
−0.0760303 + 0.997106i \(0.524225\pi\)
\(422\) −10.8629 −0.528797
\(423\) −29.2746 −1.42338
\(424\) −4.64885 −0.225768
\(425\) −6.02760 −0.292381
\(426\) −6.47228 −0.313583
\(427\) −14.6733 −0.710089
\(428\) 9.35136 0.452015
\(429\) 0 0
\(430\) 9.13818 0.440682
\(431\) 4.93779 0.237845 0.118923 0.992904i \(-0.462056\pi\)
0.118923 + 0.992904i \(0.462056\pi\)
\(432\) −3.02760 −0.145665
\(433\) 31.6620 1.52158 0.760790 0.648998i \(-0.224812\pi\)
0.760790 + 0.648998i \(0.224812\pi\)
\(434\) −2.24914 −0.107962
\(435\) −0.187121 −0.00897178
\(436\) −19.9100 −0.953514
\(437\) 25.3772 1.21396
\(438\) 3.14984 0.150505
\(439\) −3.57840 −0.170788 −0.0853938 0.996347i \(-0.527215\pi\)
−0.0853938 + 0.996347i \(0.527215\pi\)
\(440\) 0 0
\(441\) −2.71982 −0.129515
\(442\) 29.0074 1.37974
\(443\) 11.2916 0.536482 0.268241 0.963352i \(-0.413558\pi\)
0.268241 + 0.963352i \(0.413558\pi\)
\(444\) 0.871512 0.0413601
\(445\) 16.0923 0.762850
\(446\) 12.2446 0.579797
\(447\) 10.3983 0.491824
\(448\) −1.00000 −0.0472456
\(449\) −9.27050 −0.437502 −0.218751 0.975781i \(-0.570198\pi\)
−0.218751 + 0.975781i \(0.570198\pi\)
\(450\) 2.71982 0.128214
\(451\) 0 0
\(452\) 4.10257 0.192969
\(453\) −0.895485 −0.0420736
\(454\) 21.0077 0.985938
\(455\) −4.81243 −0.225610
\(456\) 1.90221 0.0890790
\(457\) 41.7715 1.95399 0.976993 0.213271i \(-0.0684117\pi\)
0.976993 + 0.213271i \(0.0684117\pi\)
\(458\) 4.92949 0.230340
\(459\) 18.2491 0.851797
\(460\) −7.06157 −0.329247
\(461\) −14.6501 −0.682325 −0.341162 0.940004i \(-0.610821\pi\)
−0.341162 + 0.940004i \(0.610821\pi\)
\(462\) 0 0
\(463\) −28.0291 −1.30262 −0.651312 0.758810i \(-0.725781\pi\)
−0.651312 + 0.758810i \(0.725781\pi\)
\(464\) −0.353515 −0.0164115
\(465\) −1.19051 −0.0552085
\(466\) −20.1605 −0.933918
\(467\) 13.2230 0.611887 0.305943 0.952050i \(-0.401028\pi\)
0.305943 + 0.952050i \(0.401028\pi\)
\(468\) −13.0890 −0.605037
\(469\) −9.43079 −0.435474
\(470\) −10.7634 −0.496479
\(471\) −7.20674 −0.332069
\(472\) −2.78483 −0.128182
\(473\) 0 0
\(474\) 4.25904 0.195624
\(475\) −3.59370 −0.164890
\(476\) 6.02760 0.276274
\(477\) −12.6441 −0.578932
\(478\) −9.53194 −0.435981
\(479\) 18.6878 0.853866 0.426933 0.904283i \(-0.359594\pi\)
0.426933 + 0.904283i \(0.359594\pi\)
\(480\) −0.529317 −0.0241599
\(481\) 7.92359 0.361285
\(482\) −6.73710 −0.306866
\(483\) 3.73781 0.170076
\(484\) 0 0
\(485\) −1.23078 −0.0558869
\(486\) −12.5535 −0.569437
\(487\) −11.4532 −0.518992 −0.259496 0.965744i \(-0.583556\pi\)
−0.259496 + 0.965744i \(0.583556\pi\)
\(488\) −14.6733 −0.664227
\(489\) 7.44717 0.336773
\(490\) −1.00000 −0.0451754
\(491\) −32.7871 −1.47966 −0.739831 0.672793i \(-0.765095\pi\)
−0.739831 + 0.672793i \(0.765095\pi\)
\(492\) −4.95426 −0.223355
\(493\) 2.13085 0.0959684
\(494\) 17.2944 0.778114
\(495\) 0 0
\(496\) −2.24914 −0.100989
\(497\) −12.2276 −0.548483
\(498\) −8.78798 −0.393799
\(499\) −31.0534 −1.39014 −0.695071 0.718941i \(-0.744627\pi\)
−0.695071 + 0.718941i \(0.744627\pi\)
\(500\) 1.00000 0.0447214
\(501\) −9.28535 −0.414839
\(502\) −8.94574 −0.399268
\(503\) 25.5004 1.13701 0.568503 0.822681i \(-0.307523\pi\)
0.568503 + 0.822681i \(0.307523\pi\)
\(504\) −2.71982 −0.121151
\(505\) 10.8741 0.483891
\(506\) 0 0
\(507\) 5.37758 0.238827
\(508\) −0.926321 −0.0410989
\(509\) −27.3509 −1.21231 −0.606153 0.795348i \(-0.707288\pi\)
−0.606153 + 0.795348i \(0.707288\pi\)
\(510\) 3.19051 0.141278
\(511\) 5.95077 0.263247
\(512\) −1.00000 −0.0441942
\(513\) 10.8803 0.480377
\(514\) −16.9351 −0.746977
\(515\) 3.42100 0.150747
\(516\) −4.83699 −0.212937
\(517\) 0 0
\(518\) 1.64649 0.0723424
\(519\) 7.86269 0.345134
\(520\) −4.81243 −0.211039
\(521\) 23.0349 1.00918 0.504589 0.863359i \(-0.331644\pi\)
0.504589 + 0.863359i \(0.331644\pi\)
\(522\) −0.961498 −0.0420836
\(523\) −43.6113 −1.90699 −0.953494 0.301413i \(-0.902542\pi\)
−0.953494 + 0.301413i \(0.902542\pi\)
\(524\) 13.3755 0.584310
\(525\) −0.529317 −0.0231013
\(526\) 16.0270 0.698810
\(527\) 13.5569 0.590548
\(528\) 0 0
\(529\) 26.8658 1.16808
\(530\) −4.64885 −0.201933
\(531\) −7.57425 −0.328694
\(532\) 3.59370 0.155807
\(533\) −45.0430 −1.95103
\(534\) −8.51795 −0.368608
\(535\) 9.35136 0.404295
\(536\) −9.43079 −0.407348
\(537\) −7.34523 −0.316970
\(538\) −9.38110 −0.404448
\(539\) 0 0
\(540\) −3.02760 −0.130287
\(541\) −28.6593 −1.23216 −0.616079 0.787684i \(-0.711280\pi\)
−0.616079 + 0.787684i \(0.711280\pi\)
\(542\) −15.8105 −0.679118
\(543\) 8.65678 0.371498
\(544\) 6.02760 0.258431
\(545\) −19.9100 −0.852849
\(546\) 2.54730 0.109014
\(547\) 0.317491 0.0135749 0.00678747 0.999977i \(-0.497839\pi\)
0.00678747 + 0.999977i \(0.497839\pi\)
\(548\) 19.8175 0.846560
\(549\) −39.9087 −1.70326
\(550\) 0 0
\(551\) 1.27043 0.0541221
\(552\) 3.73781 0.159092
\(553\) 8.04631 0.342164
\(554\) −24.2012 −1.02821
\(555\) 0.871512 0.0369936
\(556\) −18.7006 −0.793081
\(557\) −2.72599 −0.115504 −0.0577519 0.998331i \(-0.518393\pi\)
−0.0577519 + 0.998331i \(0.518393\pi\)
\(558\) −6.11727 −0.258965
\(559\) −43.9769 −1.86002
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 16.8649 0.711404
\(563\) 7.36860 0.310550 0.155275 0.987871i \(-0.450374\pi\)
0.155275 + 0.987871i \(0.450374\pi\)
\(564\) 5.69725 0.239898
\(565\) 4.10257 0.172597
\(566\) −3.49996 −0.147114
\(567\) −6.55691 −0.275365
\(568\) −12.2276 −0.513059
\(569\) −12.7784 −0.535700 −0.267850 0.963461i \(-0.586313\pi\)
−0.267850 + 0.963461i \(0.586313\pi\)
\(570\) 1.90221 0.0796747
\(571\) 13.5141 0.565546 0.282773 0.959187i \(-0.408746\pi\)
0.282773 + 0.959187i \(0.408746\pi\)
\(572\) 0 0
\(573\) −8.54370 −0.356918
\(574\) −9.35973 −0.390667
\(575\) −7.06157 −0.294488
\(576\) −2.71982 −0.113326
\(577\) −4.80939 −0.200218 −0.100109 0.994976i \(-0.531919\pi\)
−0.100109 + 0.994976i \(0.531919\pi\)
\(578\) −19.3319 −0.804102
\(579\) −1.97068 −0.0818987
\(580\) −0.353515 −0.0146789
\(581\) −16.6025 −0.688788
\(582\) 0.651473 0.0270044
\(583\) 0 0
\(584\) 5.95077 0.246245
\(585\) −13.0890 −0.541162
\(586\) 10.6822 0.441277
\(587\) 16.1586 0.666936 0.333468 0.942761i \(-0.391781\pi\)
0.333468 + 0.942761i \(0.391781\pi\)
\(588\) 0.529317 0.0218286
\(589\) 8.08275 0.333044
\(590\) −2.78483 −0.114650
\(591\) 1.06401 0.0437676
\(592\) 1.64649 0.0676701
\(593\) 30.4646 1.25103 0.625516 0.780211i \(-0.284889\pi\)
0.625516 + 0.780211i \(0.284889\pi\)
\(594\) 0 0
\(595\) 6.02760 0.247107
\(596\) 19.6448 0.804684
\(597\) −5.27921 −0.216064
\(598\) 33.9833 1.38968
\(599\) 30.4537 1.24430 0.622152 0.782896i \(-0.286258\pi\)
0.622152 + 0.782896i \(0.286258\pi\)
\(600\) −0.529317 −0.0216093
\(601\) −17.0612 −0.695940 −0.347970 0.937506i \(-0.613129\pi\)
−0.347970 + 0.937506i \(0.613129\pi\)
\(602\) −9.13818 −0.372445
\(603\) −25.6501 −1.04455
\(604\) −1.69178 −0.0688374
\(605\) 0 0
\(606\) −5.75584 −0.233815
\(607\) 15.7032 0.637374 0.318687 0.947860i \(-0.396758\pi\)
0.318687 + 0.947860i \(0.396758\pi\)
\(608\) 3.59370 0.145744
\(609\) 0.187121 0.00758254
\(610\) −14.6733 −0.594103
\(611\) 51.7982 2.09553
\(612\) 16.3940 0.662688
\(613\) −17.2732 −0.697657 −0.348829 0.937186i \(-0.613421\pi\)
−0.348829 + 0.937186i \(0.613421\pi\)
\(614\) −17.4466 −0.704088
\(615\) −4.95426 −0.199775
\(616\) 0 0
\(617\) −11.2271 −0.451986 −0.225993 0.974129i \(-0.572563\pi\)
−0.225993 + 0.974129i \(0.572563\pi\)
\(618\) −1.81079 −0.0728408
\(619\) 18.3288 0.736696 0.368348 0.929688i \(-0.379924\pi\)
0.368348 + 0.929688i \(0.379924\pi\)
\(620\) −2.24914 −0.0903277
\(621\) 21.3796 0.857933
\(622\) −9.50939 −0.381292
\(623\) −16.0923 −0.644726
\(624\) 2.54730 0.101974
\(625\) 1.00000 0.0400000
\(626\) −29.1754 −1.16608
\(627\) 0 0
\(628\) −13.6152 −0.543305
\(629\) −9.92435 −0.395710
\(630\) −2.71982 −0.108360
\(631\) −35.0446 −1.39510 −0.697552 0.716534i \(-0.745727\pi\)
−0.697552 + 0.716534i \(0.745727\pi\)
\(632\) 8.04631 0.320065
\(633\) 5.74991 0.228538
\(634\) 28.7999 1.14379
\(635\) −0.926321 −0.0367599
\(636\) 2.46072 0.0975737
\(637\) 4.81243 0.190675
\(638\) 0 0
\(639\) −33.2569 −1.31562
\(640\) −1.00000 −0.0395285
\(641\) 20.3744 0.804739 0.402370 0.915477i \(-0.368187\pi\)
0.402370 + 0.915477i \(0.368187\pi\)
\(642\) −4.94983 −0.195354
\(643\) −1.34168 −0.0529108 −0.0264554 0.999650i \(-0.508422\pi\)
−0.0264554 + 0.999650i \(0.508422\pi\)
\(644\) 7.06157 0.278265
\(645\) −4.83699 −0.190456
\(646\) −21.6614 −0.852257
\(647\) 21.1028 0.829637 0.414818 0.909904i \(-0.363845\pi\)
0.414818 + 0.909904i \(0.363845\pi\)
\(648\) −6.55691 −0.257580
\(649\) 0 0
\(650\) −4.81243 −0.188759
\(651\) 1.19051 0.0466597
\(652\) 14.0694 0.551001
\(653\) 48.3564 1.89233 0.946166 0.323681i \(-0.104921\pi\)
0.946166 + 0.323681i \(0.104921\pi\)
\(654\) 10.5387 0.412095
\(655\) 13.3755 0.522623
\(656\) −9.35973 −0.365436
\(657\) 16.1851 0.631439
\(658\) 10.7634 0.419601
\(659\) −15.9811 −0.622536 −0.311268 0.950322i \(-0.600754\pi\)
−0.311268 + 0.950322i \(0.600754\pi\)
\(660\) 0 0
\(661\) 31.1212 1.21048 0.605238 0.796045i \(-0.293078\pi\)
0.605238 + 0.796045i \(0.293078\pi\)
\(662\) −17.9933 −0.699331
\(663\) −15.3541 −0.596303
\(664\) −16.6025 −0.644302
\(665\) 3.59370 0.139358
\(666\) 4.47815 0.173525
\(667\) 2.49637 0.0966598
\(668\) −17.5422 −0.678726
\(669\) −6.48125 −0.250580
\(670\) −9.43079 −0.364343
\(671\) 0 0
\(672\) 0.529317 0.0204188
\(673\) −38.4561 −1.48237 −0.741186 0.671300i \(-0.765736\pi\)
−0.741186 + 0.671300i \(0.765736\pi\)
\(674\) 23.2472 0.895448
\(675\) −3.02760 −0.116532
\(676\) 10.1595 0.390749
\(677\) 3.05095 0.117257 0.0586287 0.998280i \(-0.481327\pi\)
0.0586287 + 0.998280i \(0.481327\pi\)
\(678\) −2.17156 −0.0833983
\(679\) 1.23078 0.0472331
\(680\) 6.02760 0.231148
\(681\) −11.1197 −0.426108
\(682\) 0 0
\(683\) −29.6414 −1.13420 −0.567099 0.823650i \(-0.691934\pi\)
−0.567099 + 0.823650i \(0.691934\pi\)
\(684\) 9.77424 0.373728
\(685\) 19.8175 0.757186
\(686\) 1.00000 0.0381802
\(687\) −2.60926 −0.0995495
\(688\) −9.13818 −0.348390
\(689\) 22.3723 0.852316
\(690\) 3.73781 0.142296
\(691\) −29.4999 −1.12223 −0.561114 0.827738i \(-0.689627\pi\)
−0.561114 + 0.827738i \(0.689627\pi\)
\(692\) 14.8544 0.564681
\(693\) 0 0
\(694\) −34.5288 −1.31070
\(695\) −18.7006 −0.709353
\(696\) 0.187121 0.00709281
\(697\) 56.4167 2.13693
\(698\) 12.3015 0.465618
\(699\) 10.6713 0.403625
\(700\) −1.00000 −0.0377964
\(701\) 1.49369 0.0564158 0.0282079 0.999602i \(-0.491020\pi\)
0.0282079 + 0.999602i \(0.491020\pi\)
\(702\) 14.5701 0.549913
\(703\) −5.91698 −0.223163
\(704\) 0 0
\(705\) 5.69725 0.214571
\(706\) 4.73131 0.178065
\(707\) −10.8741 −0.408962
\(708\) 1.47406 0.0553985
\(709\) −36.0120 −1.35246 −0.676229 0.736691i \(-0.736387\pi\)
−0.676229 + 0.736691i \(0.736387\pi\)
\(710\) −12.2276 −0.458894
\(711\) 21.8845 0.820734
\(712\) −16.0923 −0.603086
\(713\) 15.8825 0.594803
\(714\) −3.19051 −0.119402
\(715\) 0 0
\(716\) −13.8768 −0.518601
\(717\) 5.04541 0.188424
\(718\) 1.31338 0.0490151
\(719\) 14.9481 0.557472 0.278736 0.960368i \(-0.410085\pi\)
0.278736 + 0.960368i \(0.410085\pi\)
\(720\) −2.71982 −0.101362
\(721\) −3.42100 −0.127405
\(722\) 6.08529 0.226471
\(723\) 3.56606 0.132623
\(724\) 16.3546 0.607815
\(725\) −0.353515 −0.0131292
\(726\) 0 0
\(727\) 21.3542 0.791984 0.395992 0.918254i \(-0.370401\pi\)
0.395992 + 0.918254i \(0.370401\pi\)
\(728\) 4.81243 0.178360
\(729\) −13.0260 −0.482444
\(730\) 5.95077 0.220248
\(731\) 55.0813 2.03726
\(732\) 7.76680 0.287069
\(733\) 26.1929 0.967455 0.483728 0.875219i \(-0.339283\pi\)
0.483728 + 0.875219i \(0.339283\pi\)
\(734\) 5.39464 0.199120
\(735\) 0.529317 0.0195241
\(736\) 7.06157 0.260293
\(737\) 0 0
\(738\) −25.4568 −0.937078
\(739\) −0.973255 −0.0358018 −0.0179009 0.999840i \(-0.505698\pi\)
−0.0179009 + 0.999840i \(0.505698\pi\)
\(740\) 1.64649 0.0605260
\(741\) −9.15424 −0.336289
\(742\) 4.64885 0.170665
\(743\) 47.9841 1.76037 0.880183 0.474635i \(-0.157420\pi\)
0.880183 + 0.474635i \(0.157420\pi\)
\(744\) 1.19051 0.0436461
\(745\) 19.6448 0.719731
\(746\) −13.7265 −0.502563
\(747\) −45.1559 −1.65217
\(748\) 0 0
\(749\) −9.35136 −0.341691
\(750\) −0.529317 −0.0193279
\(751\) 12.1660 0.443944 0.221972 0.975053i \(-0.428751\pi\)
0.221972 + 0.975053i \(0.428751\pi\)
\(752\) 10.7634 0.392501
\(753\) 4.73513 0.172558
\(754\) 1.70127 0.0619564
\(755\) −1.69178 −0.0615700
\(756\) 3.02760 0.110113
\(757\) 33.8459 1.23015 0.615076 0.788468i \(-0.289125\pi\)
0.615076 + 0.788468i \(0.289125\pi\)
\(758\) 12.4209 0.451147
\(759\) 0 0
\(760\) 3.59370 0.130357
\(761\) 14.6460 0.530917 0.265459 0.964122i \(-0.414477\pi\)
0.265459 + 0.964122i \(0.414477\pi\)
\(762\) 0.490317 0.0177623
\(763\) 19.9100 0.720789
\(764\) −16.1410 −0.583961
\(765\) 16.3940 0.592726
\(766\) 6.58507 0.237928
\(767\) 13.4018 0.483911
\(768\) 0.529317 0.0191001
\(769\) 15.5100 0.559304 0.279652 0.960101i \(-0.409781\pi\)
0.279652 + 0.960101i \(0.409781\pi\)
\(770\) 0 0
\(771\) 8.96405 0.322832
\(772\) −3.72307 −0.133996
\(773\) 40.5521 1.45856 0.729278 0.684217i \(-0.239856\pi\)
0.729278 + 0.684217i \(0.239856\pi\)
\(774\) −24.8543 −0.893368
\(775\) −2.24914 −0.0807915
\(776\) 1.23078 0.0441825
\(777\) −0.871512 −0.0312653
\(778\) 23.8620 0.855493
\(779\) 33.6361 1.20514
\(780\) 2.54730 0.0912079
\(781\) 0 0
\(782\) −42.5643 −1.52210
\(783\) 1.07030 0.0382494
\(784\) 1.00000 0.0357143
\(785\) −13.6152 −0.485947
\(786\) −7.07986 −0.252530
\(787\) −28.5706 −1.01843 −0.509216 0.860639i \(-0.670065\pi\)
−0.509216 + 0.860639i \(0.670065\pi\)
\(788\) 2.01016 0.0716091
\(789\) −8.48335 −0.302015
\(790\) 8.04631 0.286275
\(791\) −4.10257 −0.145871
\(792\) 0 0
\(793\) 70.6140 2.50758
\(794\) 14.1058 0.500598
\(795\) 2.46072 0.0872726
\(796\) −9.97363 −0.353506
\(797\) −16.0691 −0.569196 −0.284598 0.958647i \(-0.591860\pi\)
−0.284598 + 0.958647i \(0.591860\pi\)
\(798\) −1.90221 −0.0673374
\(799\) −64.8775 −2.29520
\(800\) −1.00000 −0.0353553
\(801\) −43.7683 −1.54648
\(802\) 1.50759 0.0532349
\(803\) 0 0
\(804\) 4.99188 0.176050
\(805\) 7.06157 0.248888
\(806\) 10.8238 0.381253
\(807\) 4.96557 0.174796
\(808\) −10.8741 −0.382549
\(809\) −30.7932 −1.08263 −0.541315 0.840820i \(-0.682073\pi\)
−0.541315 + 0.840820i \(0.682073\pi\)
\(810\) −6.55691 −0.230386
\(811\) −21.9746 −0.771632 −0.385816 0.922576i \(-0.626080\pi\)
−0.385816 + 0.922576i \(0.626080\pi\)
\(812\) 0.353515 0.0124059
\(813\) 8.36875 0.293505
\(814\) 0 0
\(815\) 14.0694 0.492830
\(816\) −3.19051 −0.111690
\(817\) 32.8399 1.14892
\(818\) −4.42801 −0.154822
\(819\) 13.0890 0.457365
\(820\) −9.35973 −0.326856
\(821\) 45.6665 1.59377 0.796885 0.604130i \(-0.206479\pi\)
0.796885 + 0.604130i \(0.206479\pi\)
\(822\) −10.4897 −0.365871
\(823\) 2.44948 0.0853834 0.0426917 0.999088i \(-0.486407\pi\)
0.0426917 + 0.999088i \(0.486407\pi\)
\(824\) −3.42100 −0.119176
\(825\) 0 0
\(826\) 2.78483 0.0968967
\(827\) 31.9237 1.11009 0.555047 0.831819i \(-0.312700\pi\)
0.555047 + 0.831819i \(0.312700\pi\)
\(828\) 19.2062 0.667462
\(829\) 17.0171 0.591029 0.295515 0.955338i \(-0.404509\pi\)
0.295515 + 0.955338i \(0.404509\pi\)
\(830\) −16.6025 −0.576281
\(831\) 12.8101 0.444377
\(832\) 4.81243 0.166841
\(833\) −6.02760 −0.208844
\(834\) 9.89852 0.342758
\(835\) −17.5422 −0.607071
\(836\) 0 0
\(837\) 6.80949 0.235370
\(838\) 16.3250 0.563938
\(839\) 46.6872 1.61182 0.805911 0.592037i \(-0.201676\pi\)
0.805911 + 0.592037i \(0.201676\pi\)
\(840\) 0.529317 0.0182632
\(841\) −28.8750 −0.995691
\(842\) 3.12002 0.107523
\(843\) −8.92688 −0.307458
\(844\) 10.8629 0.373916
\(845\) 10.1595 0.349496
\(846\) 29.2746 1.00648
\(847\) 0 0
\(848\) 4.64885 0.159642
\(849\) 1.85258 0.0635805
\(850\) 6.02760 0.206745
\(851\) −11.6268 −0.398561
\(852\) 6.47228 0.221737
\(853\) 13.8541 0.474357 0.237178 0.971466i \(-0.423777\pi\)
0.237178 + 0.971466i \(0.423777\pi\)
\(854\) 14.6733 0.502108
\(855\) 9.77424 0.334272
\(856\) −9.35136 −0.319623
\(857\) 30.0551 1.02666 0.513331 0.858191i \(-0.328411\pi\)
0.513331 + 0.858191i \(0.328411\pi\)
\(858\) 0 0
\(859\) 44.4219 1.51566 0.757828 0.652454i \(-0.226260\pi\)
0.757828 + 0.652454i \(0.226260\pi\)
\(860\) −9.13818 −0.311610
\(861\) 4.95426 0.168841
\(862\) −4.93779 −0.168182
\(863\) 16.6118 0.565472 0.282736 0.959198i \(-0.408758\pi\)
0.282736 + 0.959198i \(0.408758\pi\)
\(864\) 3.02760 0.103001
\(865\) 14.8544 0.505066
\(866\) −31.6620 −1.07592
\(867\) 10.2327 0.347521
\(868\) 2.24914 0.0763408
\(869\) 0 0
\(870\) 0.187121 0.00634401
\(871\) 45.3850 1.53781
\(872\) 19.9100 0.674236
\(873\) 3.34751 0.113296
\(874\) −25.3772 −0.858396
\(875\) −1.00000 −0.0338062
\(876\) −3.14984 −0.106423
\(877\) −30.4224 −1.02729 −0.513646 0.858002i \(-0.671705\pi\)
−0.513646 + 0.858002i \(0.671705\pi\)
\(878\) 3.57840 0.120765
\(879\) −5.65426 −0.190714
\(880\) 0 0
\(881\) −28.2470 −0.951666 −0.475833 0.879536i \(-0.657853\pi\)
−0.475833 + 0.879536i \(0.657853\pi\)
\(882\) 2.71982 0.0915812
\(883\) 4.84763 0.163136 0.0815679 0.996668i \(-0.474007\pi\)
0.0815679 + 0.996668i \(0.474007\pi\)
\(884\) −29.0074 −0.975624
\(885\) 1.47406 0.0495499
\(886\) −11.2916 −0.379350
\(887\) 25.4879 0.855800 0.427900 0.903826i \(-0.359254\pi\)
0.427900 + 0.903826i \(0.359254\pi\)
\(888\) −0.871512 −0.0292460
\(889\) 0.926321 0.0310678
\(890\) −16.0923 −0.539417
\(891\) 0 0
\(892\) −12.2446 −0.409978
\(893\) −38.6805 −1.29439
\(894\) −10.3983 −0.347772
\(895\) −13.8768 −0.463851
\(896\) 1.00000 0.0334077
\(897\) −17.9879 −0.600599
\(898\) 9.27050 0.309360
\(899\) 0.795105 0.0265182
\(900\) −2.71982 −0.0906608
\(901\) −28.0214 −0.933529
\(902\) 0 0
\(903\) 4.83699 0.160965
\(904\) −4.10257 −0.136450
\(905\) 16.3546 0.543647
\(906\) 0.895485 0.0297505
\(907\) 16.2939 0.541030 0.270515 0.962716i \(-0.412806\pi\)
0.270515 + 0.962716i \(0.412806\pi\)
\(908\) −21.0077 −0.697164
\(909\) −29.5756 −0.980962
\(910\) 4.81243 0.159530
\(911\) −15.5268 −0.514427 −0.257213 0.966355i \(-0.582804\pi\)
−0.257213 + 0.966355i \(0.582804\pi\)
\(912\) −1.90221 −0.0629884
\(913\) 0 0
\(914\) −41.7715 −1.38168
\(915\) 7.76680 0.256762
\(916\) −4.92949 −0.162875
\(917\) −13.3755 −0.441697
\(918\) −18.2491 −0.602311
\(919\) −14.5051 −0.478478 −0.239239 0.970961i \(-0.576898\pi\)
−0.239239 + 0.970961i \(0.576898\pi\)
\(920\) 7.06157 0.232813
\(921\) 9.23478 0.304296
\(922\) 14.6501 0.482476
\(923\) 58.8445 1.93689
\(924\) 0 0
\(925\) 1.64649 0.0541361
\(926\) 28.0291 0.921095
\(927\) −9.30452 −0.305601
\(928\) 0.353515 0.0116047
\(929\) −25.0735 −0.822636 −0.411318 0.911492i \(-0.634931\pi\)
−0.411318 + 0.911492i \(0.634931\pi\)
\(930\) 1.19051 0.0390383
\(931\) −3.59370 −0.117779
\(932\) 20.1605 0.660380
\(933\) 5.03348 0.164789
\(934\) −13.2230 −0.432669
\(935\) 0 0
\(936\) 13.0890 0.427826
\(937\) 36.1117 1.17972 0.589859 0.807506i \(-0.299183\pi\)
0.589859 + 0.807506i \(0.299183\pi\)
\(938\) 9.43079 0.307926
\(939\) 15.4430 0.503964
\(940\) 10.7634 0.351064
\(941\) −25.8134 −0.841494 −0.420747 0.907178i \(-0.638232\pi\)
−0.420747 + 0.907178i \(0.638232\pi\)
\(942\) 7.20674 0.234808
\(943\) 66.0944 2.15233
\(944\) 2.78483 0.0906385
\(945\) 3.02760 0.0984878
\(946\) 0 0
\(947\) 21.6527 0.703617 0.351809 0.936072i \(-0.385567\pi\)
0.351809 + 0.936072i \(0.385567\pi\)
\(948\) −4.25904 −0.138327
\(949\) −28.6377 −0.929618
\(950\) 3.59370 0.116595
\(951\) −15.2443 −0.494329
\(952\) −6.02760 −0.195356
\(953\) 50.6680 1.64130 0.820649 0.571432i \(-0.193612\pi\)
0.820649 + 0.571432i \(0.193612\pi\)
\(954\) 12.6441 0.409367
\(955\) −16.1410 −0.522311
\(956\) 9.53194 0.308285
\(957\) 0 0
\(958\) −18.6878 −0.603774
\(959\) −19.8175 −0.639939
\(960\) 0.529317 0.0170836
\(961\) −25.9414 −0.836818
\(962\) −7.92359 −0.255467
\(963\) −25.4341 −0.819601
\(964\) 6.73710 0.216987
\(965\) −3.72307 −0.119850
\(966\) −3.73781 −0.120262
\(967\) −38.1065 −1.22542 −0.612711 0.790307i \(-0.709921\pi\)
−0.612711 + 0.790307i \(0.709921\pi\)
\(968\) 0 0
\(969\) 11.4657 0.368333
\(970\) 1.23078 0.0395180
\(971\) 14.7852 0.474481 0.237241 0.971451i \(-0.423757\pi\)
0.237241 + 0.971451i \(0.423757\pi\)
\(972\) 12.5535 0.402653
\(973\) 18.7006 0.599513
\(974\) 11.4532 0.366983
\(975\) 2.54730 0.0815788
\(976\) 14.6733 0.469679
\(977\) −44.2971 −1.41719 −0.708595 0.705616i \(-0.750671\pi\)
−0.708595 + 0.705616i \(0.750671\pi\)
\(978\) −7.44717 −0.238134
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 54.1516 1.72893
\(982\) 32.7871 1.04628
\(983\) 50.9868 1.62623 0.813113 0.582106i \(-0.197771\pi\)
0.813113 + 0.582106i \(0.197771\pi\)
\(984\) 4.95426 0.157936
\(985\) 2.01016 0.0640491
\(986\) −2.13085 −0.0678599
\(987\) −5.69725 −0.181346
\(988\) −17.2944 −0.550210
\(989\) 64.5299 2.05193
\(990\) 0 0
\(991\) 32.2027 1.02295 0.511477 0.859297i \(-0.329099\pi\)
0.511477 + 0.859297i \(0.329099\pi\)
\(992\) 2.24914 0.0714103
\(993\) 9.52417 0.302240
\(994\) 12.2276 0.387836
\(995\) −9.97363 −0.316185
\(996\) 8.78798 0.278458
\(997\) 14.4477 0.457562 0.228781 0.973478i \(-0.426526\pi\)
0.228781 + 0.973478i \(0.426526\pi\)
\(998\) 31.0534 0.982979
\(999\) −4.98489 −0.157715
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.cz.1.4 6
11.10 odd 2 8470.2.a.df.1.3 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cz.1.4 6 1.1 even 1 trivial
8470.2.a.df.1.3 yes 6 11.10 odd 2