Properties

Label 8470.2.a.dd.1.5
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: 6.6.745749504.1
Defining polynomial: \(x^{6} - 18 x^{4} - 4 x^{3} + 81 x^{2} + 36 x - 44\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-2.23874\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.23874 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.23874 q^{6} -1.00000 q^{7} +1.00000 q^{8} +2.01195 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.23874 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.23874 q^{6} -1.00000 q^{7} +1.00000 q^{8} +2.01195 q^{9} -1.00000 q^{10} +2.23874 q^{12} +5.12829 q^{13} -1.00000 q^{14} -2.23874 q^{15} +1.00000 q^{16} -6.65568 q^{17} +2.01195 q^{18} -1.15751 q^{19} -1.00000 q^{20} -2.23874 q^{21} +3.87761 q^{23} +2.23874 q^{24} +1.00000 q^{25} +5.12829 q^{26} -2.21199 q^{27} -1.00000 q^{28} +5.89442 q^{29} -2.23874 q^{30} -6.64373 q^{31} +1.00000 q^{32} -6.65568 q^{34} +1.00000 q^{35} +2.01195 q^{36} +11.8944 q^{37} -1.15751 q^{38} +11.4809 q^{39} -1.00000 q^{40} +10.4700 q^{41} -2.23874 q^{42} +1.77464 q^{43} -2.01195 q^{45} +3.87761 q^{46} +8.71479 q^{47} +2.23874 q^{48} +1.00000 q^{49} +1.00000 q^{50} -14.9003 q^{51} +5.12829 q^{52} +8.28476 q^{53} -2.21199 q^{54} -1.00000 q^{56} -2.59135 q^{57} +5.89442 q^{58} -10.2614 q^{59} -2.23874 q^{60} -12.8132 q^{61} -6.64373 q^{62} -2.01195 q^{63} +1.00000 q^{64} -5.12829 q^{65} +14.9682 q^{67} -6.65568 q^{68} +8.68095 q^{69} +1.00000 q^{70} +7.76716 q^{71} +2.01195 q^{72} -2.54037 q^{73} +11.8944 q^{74} +2.23874 q^{75} -1.15751 q^{76} +11.4809 q^{78} +10.2605 q^{79} -1.00000 q^{80} -10.9879 q^{81} +10.4700 q^{82} -6.86423 q^{83} -2.23874 q^{84} +6.65568 q^{85} +1.77464 q^{86} +13.1961 q^{87} +15.1923 q^{89} -2.01195 q^{90} -5.12829 q^{91} +3.87761 q^{92} -14.8736 q^{93} +8.71479 q^{94} +1.15751 q^{95} +2.23874 q^{96} -0.406766 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} + 6q^{4} - 6q^{5} - 6q^{7} + 6q^{8} + 18q^{9} + O(q^{10}) \) \( 6q + 6q^{2} + 6q^{4} - 6q^{5} - 6q^{7} + 6q^{8} + 18q^{9} - 6q^{10} - 6q^{14} + 6q^{16} - 6q^{17} + 18q^{18} - 6q^{20} + 6q^{25} - 12q^{27} - 6q^{28} - 12q^{29} + 6q^{32} - 6q^{34} + 6q^{35} + 18q^{36} + 24q^{37} + 24q^{39} - 6q^{40} - 12q^{41} + 18q^{43} - 18q^{45} + 24q^{47} + 6q^{49} + 6q^{50} + 12q^{51} + 36q^{53} - 12q^{54} - 6q^{56} + 12q^{57} - 12q^{58} + 30q^{59} - 36q^{61} - 18q^{63} + 6q^{64} - 12q^{67} - 6q^{68} + 6q^{70} + 6q^{71} + 18q^{72} + 6q^{73} + 24q^{74} + 24q^{78} + 24q^{79} - 6q^{80} + 54q^{81} - 12q^{82} - 24q^{83} + 6q^{85} + 18q^{86} + 24q^{87} + 36q^{89} - 18q^{90} + 24q^{94} + 6q^{98} + O(q^{100}) \)

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
\(3\) 2.23874 1.29254 0.646268 0.763111i \(-0.276329\pi\)
0.646268 + 0.763111i \(0.276329\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 2.23874 0.913961
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 2.01195 0.670649
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 2.23874 0.646268
\(13\) 5.12829 1.42233 0.711167 0.703024i \(-0.248167\pi\)
0.711167 + 0.703024i \(0.248167\pi\)
\(14\) −1.00000 −0.267261
\(15\) −2.23874 −0.578040
\(16\) 1.00000 0.250000
\(17\) −6.65568 −1.61424 −0.807119 0.590388i \(-0.798975\pi\)
−0.807119 + 0.590388i \(0.798975\pi\)
\(18\) 2.01195 0.474221
\(19\) −1.15751 −0.265550 −0.132775 0.991146i \(-0.542389\pi\)
−0.132775 + 0.991146i \(0.542389\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.23874 −0.488533
\(22\) 0 0
\(23\) 3.87761 0.808537 0.404269 0.914640i \(-0.367526\pi\)
0.404269 + 0.914640i \(0.367526\pi\)
\(24\) 2.23874 0.456981
\(25\) 1.00000 0.200000
\(26\) 5.12829 1.00574
\(27\) −2.21199 −0.425697
\(28\) −1.00000 −0.188982
\(29\) 5.89442 1.09457 0.547283 0.836948i \(-0.315662\pi\)
0.547283 + 0.836948i \(0.315662\pi\)
\(30\) −2.23874 −0.408736
\(31\) −6.64373 −1.19325 −0.596624 0.802521i \(-0.703492\pi\)
−0.596624 + 0.802521i \(0.703492\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −6.65568 −1.14144
\(35\) 1.00000 0.169031
\(36\) 2.01195 0.335325
\(37\) 11.8944 1.95543 0.977715 0.209937i \(-0.0673259\pi\)
0.977715 + 0.209937i \(0.0673259\pi\)
\(38\) −1.15751 −0.187772
\(39\) 11.4809 1.83842
\(40\) −1.00000 −0.158114
\(41\) 10.4700 1.63514 0.817570 0.575829i \(-0.195321\pi\)
0.817570 + 0.575829i \(0.195321\pi\)
\(42\) −2.23874 −0.345445
\(43\) 1.77464 0.270630 0.135315 0.990803i \(-0.456795\pi\)
0.135315 + 0.990803i \(0.456795\pi\)
\(44\) 0 0
\(45\) −2.01195 −0.299924
\(46\) 3.87761 0.571722
\(47\) 8.71479 1.27118 0.635591 0.772026i \(-0.280756\pi\)
0.635591 + 0.772026i \(0.280756\pi\)
\(48\) 2.23874 0.323134
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −14.9003 −2.08646
\(52\) 5.12829 0.711167
\(53\) 8.28476 1.13800 0.568999 0.822338i \(-0.307331\pi\)
0.568999 + 0.822338i \(0.307331\pi\)
\(54\) −2.21199 −0.301014
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −2.59135 −0.343233
\(58\) 5.89442 0.773975
\(59\) −10.2614 −1.33593 −0.667963 0.744194i \(-0.732834\pi\)
−0.667963 + 0.744194i \(0.732834\pi\)
\(60\) −2.23874 −0.289020
\(61\) −12.8132 −1.64056 −0.820280 0.571962i \(-0.806182\pi\)
−0.820280 + 0.571962i \(0.806182\pi\)
\(62\) −6.64373 −0.843754
\(63\) −2.01195 −0.253482
\(64\) 1.00000 0.125000
\(65\) −5.12829 −0.636087
\(66\) 0 0
\(67\) 14.9682 1.82865 0.914327 0.404977i \(-0.132720\pi\)
0.914327 + 0.404977i \(0.132720\pi\)
\(68\) −6.65568 −0.807119
\(69\) 8.68095 1.04506
\(70\) 1.00000 0.119523
\(71\) 7.76716 0.921793 0.460896 0.887454i \(-0.347528\pi\)
0.460896 + 0.887454i \(0.347528\pi\)
\(72\) 2.01195 0.237110
\(73\) −2.54037 −0.297328 −0.148664 0.988888i \(-0.547497\pi\)
−0.148664 + 0.988888i \(0.547497\pi\)
\(74\) 11.8944 1.38270
\(75\) 2.23874 0.258507
\(76\) −1.15751 −0.132775
\(77\) 0 0
\(78\) 11.4809 1.29996
\(79\) 10.2605 1.15439 0.577197 0.816605i \(-0.304146\pi\)
0.577197 + 0.816605i \(0.304146\pi\)
\(80\) −1.00000 −0.111803
\(81\) −10.9879 −1.22088
\(82\) 10.4700 1.15622
\(83\) −6.86423 −0.753448 −0.376724 0.926326i \(-0.622949\pi\)
−0.376724 + 0.926326i \(0.622949\pi\)
\(84\) −2.23874 −0.244266
\(85\) 6.65568 0.721910
\(86\) 1.77464 0.191364
\(87\) 13.1961 1.41477
\(88\) 0 0
\(89\) 15.1923 1.61038 0.805188 0.593019i \(-0.202064\pi\)
0.805188 + 0.593019i \(0.202064\pi\)
\(90\) −2.01195 −0.212078
\(91\) −5.12829 −0.537591
\(92\) 3.87761 0.404269
\(93\) −14.8736 −1.54232
\(94\) 8.71479 0.898862
\(95\) 1.15751 0.118758
\(96\) 2.23874 0.228490
\(97\) −0.406766 −0.0413008 −0.0206504 0.999787i \(-0.506574\pi\)
−0.0206504 + 0.999787i \(0.506574\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 0.964155 0.0959370 0.0479685 0.998849i \(-0.484725\pi\)
0.0479685 + 0.998849i \(0.484725\pi\)
\(102\) −14.9003 −1.47535
\(103\) 12.4021 1.22202 0.611010 0.791623i \(-0.290764\pi\)
0.611010 + 0.791623i \(0.290764\pi\)
\(104\) 5.12829 0.502871
\(105\) 2.23874 0.218478
\(106\) 8.28476 0.804687
\(107\) 11.3466 1.09691 0.548457 0.836179i \(-0.315215\pi\)
0.548457 + 0.836179i \(0.315215\pi\)
\(108\) −2.21199 −0.212849
\(109\) −9.81672 −0.940271 −0.470135 0.882594i \(-0.655795\pi\)
−0.470135 + 0.882594i \(0.655795\pi\)
\(110\) 0 0
\(111\) 26.6285 2.52746
\(112\) −1.00000 −0.0944911
\(113\) −15.4263 −1.45118 −0.725591 0.688126i \(-0.758433\pi\)
−0.725591 + 0.688126i \(0.758433\pi\)
\(114\) −2.59135 −0.242702
\(115\) −3.87761 −0.361589
\(116\) 5.89442 0.547283
\(117\) 10.3179 0.953887
\(118\) −10.2614 −0.944643
\(119\) 6.65568 0.610125
\(120\) −2.23874 −0.204368
\(121\) 0 0
\(122\) −12.8132 −1.16005
\(123\) 23.4396 2.11348
\(124\) −6.64373 −0.596624
\(125\) −1.00000 −0.0894427
\(126\) −2.01195 −0.179239
\(127\) 15.5058 1.37592 0.687961 0.725748i \(-0.258506\pi\)
0.687961 + 0.725748i \(0.258506\pi\)
\(128\) 1.00000 0.0883883
\(129\) 3.97295 0.349798
\(130\) −5.12829 −0.449781
\(131\) −16.5157 −1.44299 −0.721493 0.692421i \(-0.756544\pi\)
−0.721493 + 0.692421i \(0.756544\pi\)
\(132\) 0 0
\(133\) 1.15751 0.100368
\(134\) 14.9682 1.29305
\(135\) 2.21199 0.190378
\(136\) −6.65568 −0.570720
\(137\) 13.8267 1.18129 0.590646 0.806931i \(-0.298873\pi\)
0.590646 + 0.806931i \(0.298873\pi\)
\(138\) 8.68095 0.738971
\(139\) 1.19890 0.101689 0.0508445 0.998707i \(-0.483809\pi\)
0.0508445 + 0.998707i \(0.483809\pi\)
\(140\) 1.00000 0.0845154
\(141\) 19.5101 1.64305
\(142\) 7.76716 0.651806
\(143\) 0 0
\(144\) 2.01195 0.167662
\(145\) −5.89442 −0.489505
\(146\) −2.54037 −0.210243
\(147\) 2.23874 0.184648
\(148\) 11.8944 0.977715
\(149\) −5.37555 −0.440382 −0.220191 0.975457i \(-0.570668\pi\)
−0.220191 + 0.975457i \(0.570668\pi\)
\(150\) 2.23874 0.182792
\(151\) 1.95930 0.159445 0.0797226 0.996817i \(-0.474597\pi\)
0.0797226 + 0.996817i \(0.474597\pi\)
\(152\) −1.15751 −0.0938861
\(153\) −13.3909 −1.08259
\(154\) 0 0
\(155\) 6.64373 0.533637
\(156\) 11.4809 0.919208
\(157\) 13.1054 1.04593 0.522964 0.852355i \(-0.324826\pi\)
0.522964 + 0.852355i \(0.324826\pi\)
\(158\) 10.2605 0.816280
\(159\) 18.5474 1.47090
\(160\) −1.00000 −0.0790569
\(161\) −3.87761 −0.305598
\(162\) −10.9879 −0.863292
\(163\) −21.5818 −1.69041 −0.845207 0.534438i \(-0.820523\pi\)
−0.845207 + 0.534438i \(0.820523\pi\)
\(164\) 10.4700 0.817570
\(165\) 0 0
\(166\) −6.86423 −0.532768
\(167\) 0.900461 0.0696797 0.0348399 0.999393i \(-0.488908\pi\)
0.0348399 + 0.999393i \(0.488908\pi\)
\(168\) −2.23874 −0.172722
\(169\) 13.2994 1.02303
\(170\) 6.65568 0.510467
\(171\) −2.32884 −0.178091
\(172\) 1.77464 0.135315
\(173\) −14.9963 −1.14015 −0.570075 0.821592i \(-0.693086\pi\)
−0.570075 + 0.821592i \(0.693086\pi\)
\(174\) 13.1961 1.00039
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −22.9727 −1.72673
\(178\) 15.1923 1.13871
\(179\) 10.7148 0.800861 0.400430 0.916327i \(-0.368861\pi\)
0.400430 + 0.916327i \(0.368861\pi\)
\(180\) −2.01195 −0.149962
\(181\) −6.20078 −0.460900 −0.230450 0.973084i \(-0.574020\pi\)
−0.230450 + 0.973084i \(0.574020\pi\)
\(182\) −5.12829 −0.380134
\(183\) −28.6854 −2.12048
\(184\) 3.87761 0.285861
\(185\) −11.8944 −0.874495
\(186\) −14.8736 −1.09058
\(187\) 0 0
\(188\) 8.71479 0.635591
\(189\) 2.21199 0.160899
\(190\) 1.15751 0.0839743
\(191\) −0.133153 −0.00963464 −0.00481732 0.999988i \(-0.501533\pi\)
−0.00481732 + 0.999988i \(0.501533\pi\)
\(192\) 2.23874 0.161567
\(193\) −12.7468 −0.917531 −0.458766 0.888557i \(-0.651708\pi\)
−0.458766 + 0.888557i \(0.651708\pi\)
\(194\) −0.406766 −0.0292041
\(195\) −11.4809 −0.822165
\(196\) 1.00000 0.0714286
\(197\) 4.39162 0.312890 0.156445 0.987687i \(-0.449997\pi\)
0.156445 + 0.987687i \(0.449997\pi\)
\(198\) 0 0
\(199\) −0.430314 −0.0305041 −0.0152521 0.999884i \(-0.504855\pi\)
−0.0152521 + 0.999884i \(0.504855\pi\)
\(200\) 1.00000 0.0707107
\(201\) 33.5098 2.36360
\(202\) 0.964155 0.0678377
\(203\) −5.89442 −0.413707
\(204\) −14.9003 −1.04323
\(205\) −10.4700 −0.731257
\(206\) 12.4021 0.864098
\(207\) 7.80155 0.542245
\(208\) 5.12829 0.355583
\(209\) 0 0
\(210\) 2.23874 0.154488
\(211\) −8.33737 −0.573968 −0.286984 0.957935i \(-0.592653\pi\)
−0.286984 + 0.957935i \(0.592653\pi\)
\(212\) 8.28476 0.568999
\(213\) 17.3886 1.19145
\(214\) 11.3466 0.775635
\(215\) −1.77464 −0.121029
\(216\) −2.21199 −0.150507
\(217\) 6.64373 0.451006
\(218\) −9.81672 −0.664872
\(219\) −5.68723 −0.384308
\(220\) 0 0
\(221\) −34.1323 −2.29599
\(222\) 26.6285 1.78719
\(223\) 1.13218 0.0758166 0.0379083 0.999281i \(-0.487931\pi\)
0.0379083 + 0.999281i \(0.487931\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 2.01195 0.134130
\(226\) −15.4263 −1.02614
\(227\) −10.2741 −0.681915 −0.340957 0.940079i \(-0.610751\pi\)
−0.340957 + 0.940079i \(0.610751\pi\)
\(228\) −2.59135 −0.171616
\(229\) −9.23127 −0.610019 −0.305010 0.952349i \(-0.598660\pi\)
−0.305010 + 0.952349i \(0.598660\pi\)
\(230\) −3.87761 −0.255682
\(231\) 0 0
\(232\) 5.89442 0.386987
\(233\) −6.68548 −0.437980 −0.218990 0.975727i \(-0.570276\pi\)
−0.218990 + 0.975727i \(0.570276\pi\)
\(234\) 10.3179 0.674500
\(235\) −8.71479 −0.568490
\(236\) −10.2614 −0.667963
\(237\) 22.9705 1.49210
\(238\) 6.65568 0.431423
\(239\) −22.4534 −1.45239 −0.726193 0.687490i \(-0.758712\pi\)
−0.726193 + 0.687490i \(0.758712\pi\)
\(240\) −2.23874 −0.144510
\(241\) −28.6046 −1.84258 −0.921292 0.388872i \(-0.872865\pi\)
−0.921292 + 0.388872i \(0.872865\pi\)
\(242\) 0 0
\(243\) −17.9631 −1.15233
\(244\) −12.8132 −0.820280
\(245\) −1.00000 −0.0638877
\(246\) 23.4396 1.49445
\(247\) −5.93603 −0.377701
\(248\) −6.64373 −0.421877
\(249\) −15.3672 −0.973858
\(250\) −1.00000 −0.0632456
\(251\) 25.4351 1.60545 0.802725 0.596349i \(-0.203383\pi\)
0.802725 + 0.596349i \(0.203383\pi\)
\(252\) −2.01195 −0.126741
\(253\) 0 0
\(254\) 15.5058 0.972924
\(255\) 14.9003 0.933094
\(256\) 1.00000 0.0625000
\(257\) −24.9567 −1.55676 −0.778379 0.627795i \(-0.783958\pi\)
−0.778379 + 0.627795i \(0.783958\pi\)
\(258\) 3.97295 0.247345
\(259\) −11.8944 −0.739083
\(260\) −5.12829 −0.318043
\(261\) 11.8593 0.734070
\(262\) −16.5157 −1.02035
\(263\) −0.333627 −0.0205723 −0.0102862 0.999947i \(-0.503274\pi\)
−0.0102862 + 0.999947i \(0.503274\pi\)
\(264\) 0 0
\(265\) −8.28476 −0.508929
\(266\) 1.15751 0.0709712
\(267\) 34.0115 2.08147
\(268\) 14.9682 0.914327
\(269\) 1.82597 0.111331 0.0556656 0.998449i \(-0.482272\pi\)
0.0556656 + 0.998449i \(0.482272\pi\)
\(270\) 2.21199 0.134617
\(271\) 13.3284 0.809642 0.404821 0.914396i \(-0.367334\pi\)
0.404821 + 0.914396i \(0.367334\pi\)
\(272\) −6.65568 −0.403560
\(273\) −11.4809 −0.694856
\(274\) 13.8267 0.835299
\(275\) 0 0
\(276\) 8.68095 0.522532
\(277\) −15.4834 −0.930306 −0.465153 0.885230i \(-0.654001\pi\)
−0.465153 + 0.885230i \(0.654001\pi\)
\(278\) 1.19890 0.0719050
\(279\) −13.3668 −0.800252
\(280\) 1.00000 0.0597614
\(281\) 28.3105 1.68886 0.844432 0.535663i \(-0.179938\pi\)
0.844432 + 0.535663i \(0.179938\pi\)
\(282\) 19.5101 1.16181
\(283\) 13.5891 0.807791 0.403895 0.914805i \(-0.367656\pi\)
0.403895 + 0.914805i \(0.367656\pi\)
\(284\) 7.76716 0.460896
\(285\) 2.59135 0.153498
\(286\) 0 0
\(287\) −10.4700 −0.618025
\(288\) 2.01195 0.118555
\(289\) 27.2980 1.60577
\(290\) −5.89442 −0.346132
\(291\) −0.910642 −0.0533828
\(292\) −2.54037 −0.148664
\(293\) 1.83423 0.107157 0.0535785 0.998564i \(-0.482937\pi\)
0.0535785 + 0.998564i \(0.482937\pi\)
\(294\) 2.23874 0.130566
\(295\) 10.2614 0.597445
\(296\) 11.8944 0.691349
\(297\) 0 0
\(298\) −5.37555 −0.311397
\(299\) 19.8855 1.15001
\(300\) 2.23874 0.129254
\(301\) −1.77464 −0.102288
\(302\) 1.95930 0.112745
\(303\) 2.15849 0.124002
\(304\) −1.15751 −0.0663875
\(305\) 12.8132 0.733681
\(306\) −13.3909 −0.765506
\(307\) 8.45358 0.482471 0.241236 0.970467i \(-0.422447\pi\)
0.241236 + 0.970467i \(0.422447\pi\)
\(308\) 0 0
\(309\) 27.7652 1.57950
\(310\) 6.64373 0.377338
\(311\) 16.9898 0.963402 0.481701 0.876336i \(-0.340019\pi\)
0.481701 + 0.876336i \(0.340019\pi\)
\(312\) 11.4809 0.649978
\(313\) 18.0862 1.02230 0.511148 0.859493i \(-0.329221\pi\)
0.511148 + 0.859493i \(0.329221\pi\)
\(314\) 13.1054 0.739583
\(315\) 2.01195 0.113360
\(316\) 10.2605 0.577197
\(317\) −0.393194 −0.0220840 −0.0110420 0.999939i \(-0.503515\pi\)
−0.0110420 + 0.999939i \(0.503515\pi\)
\(318\) 18.5474 1.04009
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 25.4020 1.41780
\(322\) −3.87761 −0.216091
\(323\) 7.70398 0.428661
\(324\) −10.9879 −0.610439
\(325\) 5.12829 0.284467
\(326\) −21.5818 −1.19530
\(327\) −21.9771 −1.21533
\(328\) 10.4700 0.578109
\(329\) −8.71479 −0.480462
\(330\) 0 0
\(331\) −34.5395 −1.89846 −0.949232 0.314577i \(-0.898137\pi\)
−0.949232 + 0.314577i \(0.898137\pi\)
\(332\) −6.86423 −0.376724
\(333\) 23.9309 1.31141
\(334\) 0.900461 0.0492710
\(335\) −14.9682 −0.817799
\(336\) −2.23874 −0.122133
\(337\) 2.36075 0.128598 0.0642991 0.997931i \(-0.479519\pi\)
0.0642991 + 0.997931i \(0.479519\pi\)
\(338\) 13.2994 0.723392
\(339\) −34.5354 −1.87570
\(340\) 6.65568 0.360955
\(341\) 0 0
\(342\) −2.32884 −0.125929
\(343\) −1.00000 −0.0539949
\(344\) 1.77464 0.0956820
\(345\) −8.68095 −0.467367
\(346\) −14.9963 −0.806208
\(347\) −6.85646 −0.368074 −0.184037 0.982919i \(-0.558917\pi\)
−0.184037 + 0.982919i \(0.558917\pi\)
\(348\) 13.1961 0.707383
\(349\) −8.34033 −0.446448 −0.223224 0.974767i \(-0.571658\pi\)
−0.223224 + 0.974767i \(0.571658\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −11.3437 −0.605484
\(352\) 0 0
\(353\) 19.7691 1.05220 0.526101 0.850422i \(-0.323653\pi\)
0.526101 + 0.850422i \(0.323653\pi\)
\(354\) −22.9727 −1.22098
\(355\) −7.76716 −0.412238
\(356\) 15.1923 0.805188
\(357\) 14.9003 0.788608
\(358\) 10.7148 0.566294
\(359\) 0.257703 0.0136010 0.00680051 0.999977i \(-0.497835\pi\)
0.00680051 + 0.999977i \(0.497835\pi\)
\(360\) −2.01195 −0.106039
\(361\) −17.6602 −0.929483
\(362\) −6.20078 −0.325906
\(363\) 0 0
\(364\) −5.12829 −0.268796
\(365\) 2.54037 0.132969
\(366\) −28.6854 −1.49941
\(367\) 25.6755 1.34025 0.670125 0.742248i \(-0.266241\pi\)
0.670125 + 0.742248i \(0.266241\pi\)
\(368\) 3.87761 0.202134
\(369\) 21.0651 1.09661
\(370\) −11.8944 −0.618361
\(371\) −8.28476 −0.430123
\(372\) −14.8736 −0.771159
\(373\) −5.70957 −0.295631 −0.147815 0.989015i \(-0.547224\pi\)
−0.147815 + 0.989015i \(0.547224\pi\)
\(374\) 0 0
\(375\) −2.23874 −0.115608
\(376\) 8.71479 0.449431
\(377\) 30.2283 1.55684
\(378\) 2.21199 0.113772
\(379\) 2.16686 0.111304 0.0556520 0.998450i \(-0.482276\pi\)
0.0556520 + 0.998450i \(0.482276\pi\)
\(380\) 1.15751 0.0593788
\(381\) 34.7135 1.77843
\(382\) −0.133153 −0.00681272
\(383\) 10.3043 0.526523 0.263262 0.964725i \(-0.415202\pi\)
0.263262 + 0.964725i \(0.415202\pi\)
\(384\) 2.23874 0.114245
\(385\) 0 0
\(386\) −12.7468 −0.648793
\(387\) 3.57048 0.181498
\(388\) −0.406766 −0.0206504
\(389\) 5.86030 0.297129 0.148564 0.988903i \(-0.452535\pi\)
0.148564 + 0.988903i \(0.452535\pi\)
\(390\) −11.4809 −0.581358
\(391\) −25.8081 −1.30517
\(392\) 1.00000 0.0505076
\(393\) −36.9744 −1.86511
\(394\) 4.39162 0.221246
\(395\) −10.2605 −0.516261
\(396\) 0 0
\(397\) −9.65824 −0.484733 −0.242367 0.970185i \(-0.577924\pi\)
−0.242367 + 0.970185i \(0.577924\pi\)
\(398\) −0.430314 −0.0215697
\(399\) 2.59135 0.129730
\(400\) 1.00000 0.0500000
\(401\) −3.76869 −0.188199 −0.0940996 0.995563i \(-0.529997\pi\)
−0.0940996 + 0.995563i \(0.529997\pi\)
\(402\) 33.5098 1.67132
\(403\) −34.0710 −1.69720
\(404\) 0.964155 0.0479685
\(405\) 10.9879 0.545994
\(406\) −5.89442 −0.292535
\(407\) 0 0
\(408\) −14.9003 −0.737676
\(409\) 7.59552 0.375575 0.187787 0.982210i \(-0.439868\pi\)
0.187787 + 0.982210i \(0.439868\pi\)
\(410\) −10.4700 −0.517077
\(411\) 30.9543 1.52686
\(412\) 12.4021 0.611010
\(413\) 10.2614 0.504933
\(414\) 7.80155 0.383425
\(415\) 6.86423 0.336952
\(416\) 5.12829 0.251435
\(417\) 2.68401 0.131437
\(418\) 0 0
\(419\) −10.0288 −0.489937 −0.244968 0.969531i \(-0.578778\pi\)
−0.244968 + 0.969531i \(0.578778\pi\)
\(420\) 2.23874 0.109239
\(421\) −4.21566 −0.205459 −0.102729 0.994709i \(-0.532758\pi\)
−0.102729 + 0.994709i \(0.532758\pi\)
\(422\) −8.33737 −0.405857
\(423\) 17.5337 0.852518
\(424\) 8.28476 0.402343
\(425\) −6.65568 −0.322848
\(426\) 17.3886 0.842483
\(427\) 12.8132 0.620073
\(428\) 11.3466 0.548457
\(429\) 0 0
\(430\) −1.77464 −0.0855806
\(431\) −23.7780 −1.14534 −0.572672 0.819784i \(-0.694093\pi\)
−0.572672 + 0.819784i \(0.694093\pi\)
\(432\) −2.21199 −0.106424
\(433\) −33.5935 −1.61440 −0.807199 0.590279i \(-0.799018\pi\)
−0.807199 + 0.590279i \(0.799018\pi\)
\(434\) 6.64373 0.318909
\(435\) −13.1961 −0.632702
\(436\) −9.81672 −0.470135
\(437\) −4.48835 −0.214707
\(438\) −5.68723 −0.271747
\(439\) −27.7198 −1.32299 −0.661496 0.749949i \(-0.730078\pi\)
−0.661496 + 0.749949i \(0.730078\pi\)
\(440\) 0 0
\(441\) 2.01195 0.0958071
\(442\) −34.1323 −1.62351
\(443\) 28.5238 1.35521 0.677603 0.735427i \(-0.263019\pi\)
0.677603 + 0.735427i \(0.263019\pi\)
\(444\) 26.6285 1.26373
\(445\) −15.1923 −0.720182
\(446\) 1.13218 0.0536104
\(447\) −12.0344 −0.569210
\(448\) −1.00000 −0.0472456
\(449\) −0.982968 −0.0463891 −0.0231946 0.999731i \(-0.507384\pi\)
−0.0231946 + 0.999731i \(0.507384\pi\)
\(450\) 2.01195 0.0948442
\(451\) 0 0
\(452\) −15.4263 −0.725591
\(453\) 4.38635 0.206089
\(454\) −10.2741 −0.482187
\(455\) 5.12829 0.240418
\(456\) −2.59135 −0.121351
\(457\) 4.98742 0.233302 0.116651 0.993173i \(-0.462784\pi\)
0.116651 + 0.993173i \(0.462784\pi\)
\(458\) −9.23127 −0.431349
\(459\) 14.7223 0.687177
\(460\) −3.87761 −0.180794
\(461\) 1.28781 0.0599791 0.0299896 0.999550i \(-0.490453\pi\)
0.0299896 + 0.999550i \(0.490453\pi\)
\(462\) 0 0
\(463\) −17.4015 −0.808717 −0.404358 0.914601i \(-0.632505\pi\)
−0.404358 + 0.914601i \(0.632505\pi\)
\(464\) 5.89442 0.273641
\(465\) 14.8736 0.689745
\(466\) −6.68548 −0.309699
\(467\) 6.97072 0.322566 0.161283 0.986908i \(-0.448437\pi\)
0.161283 + 0.986908i \(0.448437\pi\)
\(468\) 10.3179 0.476943
\(469\) −14.9682 −0.691166
\(470\) −8.71479 −0.401983
\(471\) 29.3397 1.35190
\(472\) −10.2614 −0.472321
\(473\) 0 0
\(474\) 22.9705 1.05507
\(475\) −1.15751 −0.0531100
\(476\) 6.65568 0.305062
\(477\) 16.6685 0.763198
\(478\) −22.4534 −1.02699
\(479\) −26.7035 −1.22011 −0.610056 0.792358i \(-0.708853\pi\)
−0.610056 + 0.792358i \(0.708853\pi\)
\(480\) −2.23874 −0.102184
\(481\) 60.9981 2.78127
\(482\) −28.6046 −1.30290
\(483\) −8.68095 −0.394997
\(484\) 0 0
\(485\) 0.406766 0.0184703
\(486\) −17.9631 −0.814822
\(487\) 4.32910 0.196170 0.0980851 0.995178i \(-0.468728\pi\)
0.0980851 + 0.995178i \(0.468728\pi\)
\(488\) −12.8132 −0.580026
\(489\) −48.3159 −2.18492
\(490\) −1.00000 −0.0451754
\(491\) 12.8945 0.581923 0.290961 0.956735i \(-0.406025\pi\)
0.290961 + 0.956735i \(0.406025\pi\)
\(492\) 23.4396 1.05674
\(493\) −39.2313 −1.76689
\(494\) −5.93603 −0.267075
\(495\) 0 0
\(496\) −6.64373 −0.298312
\(497\) −7.76716 −0.348405
\(498\) −15.3672 −0.688622
\(499\) −3.01676 −0.135049 −0.0675243 0.997718i \(-0.521510\pi\)
−0.0675243 + 0.997718i \(0.521510\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 2.01590 0.0900636
\(502\) 25.4351 1.13523
\(503\) −6.17711 −0.275424 −0.137712 0.990472i \(-0.543975\pi\)
−0.137712 + 0.990472i \(0.543975\pi\)
\(504\) −2.01195 −0.0896193
\(505\) −0.964155 −0.0429043
\(506\) 0 0
\(507\) 29.7739 1.32230
\(508\) 15.5058 0.687961
\(509\) 33.6703 1.49241 0.746205 0.665717i \(-0.231874\pi\)
0.746205 + 0.665717i \(0.231874\pi\)
\(510\) 14.9003 0.659797
\(511\) 2.54037 0.112380
\(512\) 1.00000 0.0441942
\(513\) 2.56039 0.113044
\(514\) −24.9567 −1.10079
\(515\) −12.4021 −0.546504
\(516\) 3.97295 0.174899
\(517\) 0 0
\(518\) −11.8944 −0.522611
\(519\) −33.5729 −1.47369
\(520\) −5.12829 −0.224891
\(521\) 24.5687 1.07637 0.538187 0.842826i \(-0.319110\pi\)
0.538187 + 0.842826i \(0.319110\pi\)
\(522\) 11.8593 0.519066
\(523\) 44.4679 1.94444 0.972222 0.234059i \(-0.0752010\pi\)
0.972222 + 0.234059i \(0.0752010\pi\)
\(524\) −16.5157 −0.721493
\(525\) −2.23874 −0.0977065
\(526\) −0.333627 −0.0145468
\(527\) 44.2185 1.92619
\(528\) 0 0
\(529\) −7.96415 −0.346268
\(530\) −8.28476 −0.359867
\(531\) −20.6455 −0.895939
\(532\) 1.15751 0.0501842
\(533\) 53.6933 2.32571
\(534\) 34.0115 1.47182
\(535\) −11.3466 −0.490555
\(536\) 14.9682 0.646527
\(537\) 23.9876 1.03514
\(538\) 1.82597 0.0787231
\(539\) 0 0
\(540\) 2.21199 0.0951888
\(541\) −1.30850 −0.0562569 −0.0281284 0.999604i \(-0.508955\pi\)
−0.0281284 + 0.999604i \(0.508955\pi\)
\(542\) 13.3284 0.572503
\(543\) −13.8819 −0.595730
\(544\) −6.65568 −0.285360
\(545\) 9.81672 0.420502
\(546\) −11.4809 −0.491338
\(547\) −17.4553 −0.746335 −0.373168 0.927764i \(-0.621728\pi\)
−0.373168 + 0.927764i \(0.621728\pi\)
\(548\) 13.8267 0.590646
\(549\) −25.7795 −1.10024
\(550\) 0 0
\(551\) −6.82282 −0.290662
\(552\) 8.68095 0.369486
\(553\) −10.2605 −0.436320
\(554\) −15.4834 −0.657826
\(555\) −26.6285 −1.13032
\(556\) 1.19890 0.0508445
\(557\) −0.219818 −0.00931400 −0.00465700 0.999989i \(-0.501482\pi\)
−0.00465700 + 0.999989i \(0.501482\pi\)
\(558\) −13.3668 −0.565863
\(559\) 9.10086 0.384925
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 28.3105 1.19421
\(563\) 22.4246 0.945084 0.472542 0.881308i \(-0.343336\pi\)
0.472542 + 0.881308i \(0.343336\pi\)
\(564\) 19.5101 0.821525
\(565\) 15.4263 0.648988
\(566\) 13.5891 0.571194
\(567\) 10.9879 0.461449
\(568\) 7.76716 0.325903
\(569\) 3.86730 0.162126 0.0810629 0.996709i \(-0.474169\pi\)
0.0810629 + 0.996709i \(0.474169\pi\)
\(570\) 2.59135 0.108540
\(571\) 25.4899 1.06672 0.533360 0.845888i \(-0.320929\pi\)
0.533360 + 0.845888i \(0.320929\pi\)
\(572\) 0 0
\(573\) −0.298095 −0.0124531
\(574\) −10.4700 −0.437010
\(575\) 3.87761 0.161707
\(576\) 2.01195 0.0838312
\(577\) −33.6011 −1.39883 −0.699415 0.714716i \(-0.746556\pi\)
−0.699415 + 0.714716i \(0.746556\pi\)
\(578\) 27.2980 1.13545
\(579\) −28.5366 −1.18594
\(580\) −5.89442 −0.244752
\(581\) 6.86423 0.284776
\(582\) −0.910642 −0.0377473
\(583\) 0 0
\(584\) −2.54037 −0.105121
\(585\) −10.3179 −0.426591
\(586\) 1.83423 0.0757715
\(587\) 38.8396 1.60308 0.801542 0.597939i \(-0.204013\pi\)
0.801542 + 0.597939i \(0.204013\pi\)
\(588\) 2.23874 0.0923240
\(589\) 7.69015 0.316867
\(590\) 10.2614 0.422457
\(591\) 9.83168 0.404421
\(592\) 11.8944 0.488857
\(593\) −29.8393 −1.22535 −0.612677 0.790333i \(-0.709907\pi\)
−0.612677 + 0.790333i \(0.709907\pi\)
\(594\) 0 0
\(595\) −6.65568 −0.272856
\(596\) −5.37555 −0.220191
\(597\) −0.963359 −0.0394277
\(598\) 19.8855 0.813179
\(599\) 26.4186 1.07944 0.539718 0.841846i \(-0.318531\pi\)
0.539718 + 0.841846i \(0.318531\pi\)
\(600\) 2.23874 0.0913961
\(601\) −31.2101 −1.27309 −0.636543 0.771241i \(-0.719636\pi\)
−0.636543 + 0.771241i \(0.719636\pi\)
\(602\) −1.77464 −0.0723288
\(603\) 30.1152 1.22639
\(604\) 1.95930 0.0797226
\(605\) 0 0
\(606\) 2.15849 0.0876827
\(607\) −29.3084 −1.18959 −0.594795 0.803878i \(-0.702767\pi\)
−0.594795 + 0.803878i \(0.702767\pi\)
\(608\) −1.15751 −0.0469431
\(609\) −13.1961 −0.534731
\(610\) 12.8132 0.518791
\(611\) 44.6920 1.80804
\(612\) −13.3909 −0.541294
\(613\) −20.1964 −0.815725 −0.407863 0.913043i \(-0.633726\pi\)
−0.407863 + 0.913043i \(0.633726\pi\)
\(614\) 8.45358 0.341159
\(615\) −23.4396 −0.945176
\(616\) 0 0
\(617\) −2.08426 −0.0839091 −0.0419546 0.999120i \(-0.513358\pi\)
−0.0419546 + 0.999120i \(0.513358\pi\)
\(618\) 27.7652 1.11688
\(619\) −3.03990 −0.122184 −0.0610919 0.998132i \(-0.519458\pi\)
−0.0610919 + 0.998132i \(0.519458\pi\)
\(620\) 6.64373 0.266819
\(621\) −8.57723 −0.344192
\(622\) 16.9898 0.681228
\(623\) −15.1923 −0.608665
\(624\) 11.4809 0.459604
\(625\) 1.00000 0.0400000
\(626\) 18.0862 0.722872
\(627\) 0 0
\(628\) 13.1054 0.522964
\(629\) −79.1654 −3.15653
\(630\) 2.01195 0.0801579
\(631\) −11.3562 −0.452081 −0.226041 0.974118i \(-0.572578\pi\)
−0.226041 + 0.974118i \(0.572578\pi\)
\(632\) 10.2605 0.408140
\(633\) −18.6652 −0.741874
\(634\) −0.393194 −0.0156157
\(635\) −15.5058 −0.615331
\(636\) 18.5474 0.735452
\(637\) 5.12829 0.203190
\(638\) 0 0
\(639\) 15.6271 0.618200
\(640\) −1.00000 −0.0395285
\(641\) −34.4272 −1.35979 −0.679896 0.733308i \(-0.737975\pi\)
−0.679896 + 0.733308i \(0.737975\pi\)
\(642\) 25.4020 1.00254
\(643\) −31.4442 −1.24004 −0.620020 0.784586i \(-0.712875\pi\)
−0.620020 + 0.784586i \(0.712875\pi\)
\(644\) −3.87761 −0.152799
\(645\) −3.97295 −0.156435
\(646\) 7.70398 0.303109
\(647\) −48.9448 −1.92422 −0.962110 0.272662i \(-0.912096\pi\)
−0.962110 + 0.272662i \(0.912096\pi\)
\(648\) −10.9879 −0.431646
\(649\) 0 0
\(650\) 5.12829 0.201148
\(651\) 14.8736 0.582941
\(652\) −21.5818 −0.845207
\(653\) −17.0563 −0.667463 −0.333732 0.942668i \(-0.608308\pi\)
−0.333732 + 0.942668i \(0.608308\pi\)
\(654\) −21.9771 −0.859371
\(655\) 16.5157 0.645323
\(656\) 10.4700 0.408785
\(657\) −5.11110 −0.199403
\(658\) −8.71479 −0.339738
\(659\) 39.3755 1.53385 0.766926 0.641736i \(-0.221786\pi\)
0.766926 + 0.641736i \(0.221786\pi\)
\(660\) 0 0
\(661\) −47.5548 −1.84967 −0.924833 0.380374i \(-0.875795\pi\)
−0.924833 + 0.380374i \(0.875795\pi\)
\(662\) −34.5395 −1.34242
\(663\) −76.4132 −2.96764
\(664\) −6.86423 −0.266384
\(665\) −1.15751 −0.0448861
\(666\) 23.9309 0.927305
\(667\) 22.8562 0.884997
\(668\) 0.900461 0.0348399
\(669\) 2.53466 0.0979956
\(670\) −14.9682 −0.578271
\(671\) 0 0
\(672\) −2.23874 −0.0863612
\(673\) 30.7041 1.18356 0.591779 0.806100i \(-0.298426\pi\)
0.591779 + 0.806100i \(0.298426\pi\)
\(674\) 2.36075 0.0909326
\(675\) −2.21199 −0.0851395
\(676\) 13.2994 0.511516
\(677\) −30.7722 −1.18267 −0.591335 0.806426i \(-0.701399\pi\)
−0.591335 + 0.806426i \(0.701399\pi\)
\(678\) −34.5354 −1.32632
\(679\) 0.406766 0.0156102
\(680\) 6.65568 0.255234
\(681\) −23.0010 −0.881400
\(682\) 0 0
\(683\) −10.3871 −0.397451 −0.198726 0.980055i \(-0.563680\pi\)
−0.198726 + 0.980055i \(0.563680\pi\)
\(684\) −2.32884 −0.0890455
\(685\) −13.8267 −0.528289
\(686\) −1.00000 −0.0381802
\(687\) −20.6664 −0.788472
\(688\) 1.77464 0.0676574
\(689\) 42.4867 1.61861
\(690\) −8.68095 −0.330478
\(691\) 20.9815 0.798175 0.399088 0.916913i \(-0.369327\pi\)
0.399088 + 0.916913i \(0.369327\pi\)
\(692\) −14.9963 −0.570075
\(693\) 0 0
\(694\) −6.85646 −0.260267
\(695\) −1.19890 −0.0454767
\(696\) 13.1961 0.500195
\(697\) −69.6850 −2.63951
\(698\) −8.34033 −0.315686
\(699\) −14.9670 −0.566105
\(700\) −1.00000 −0.0377964
\(701\) 30.6295 1.15686 0.578431 0.815731i \(-0.303665\pi\)
0.578431 + 0.815731i \(0.303665\pi\)
\(702\) −11.3437 −0.428142
\(703\) −13.7679 −0.519264
\(704\) 0 0
\(705\) −19.5101 −0.734794
\(706\) 19.7691 0.744019
\(707\) −0.964155 −0.0362608
\(708\) −22.9727 −0.863367
\(709\) 41.2618 1.54962 0.774810 0.632194i \(-0.217845\pi\)
0.774810 + 0.632194i \(0.217845\pi\)
\(710\) −7.76716 −0.291496
\(711\) 20.6436 0.774194
\(712\) 15.1923 0.569354
\(713\) −25.7618 −0.964786
\(714\) 14.9003 0.557630
\(715\) 0 0
\(716\) 10.7148 0.400430
\(717\) −50.2672 −1.87726
\(718\) 0.257703 0.00961738
\(719\) 26.9646 1.00561 0.502804 0.864400i \(-0.332302\pi\)
0.502804 + 0.864400i \(0.332302\pi\)
\(720\) −2.01195 −0.0749809
\(721\) −12.4021 −0.461880
\(722\) −17.6602 −0.657244
\(723\) −64.0382 −2.38161
\(724\) −6.20078 −0.230450
\(725\) 5.89442 0.218913
\(726\) 0 0
\(727\) −29.0208 −1.07632 −0.538161 0.842842i \(-0.680881\pi\)
−0.538161 + 0.842842i \(0.680881\pi\)
\(728\) −5.12829 −0.190067
\(729\) −7.25091 −0.268552
\(730\) 2.54037 0.0940235
\(731\) −11.8114 −0.436861
\(732\) −28.6854 −1.06024
\(733\) 28.4461 1.05068 0.525339 0.850893i \(-0.323938\pi\)
0.525339 + 0.850893i \(0.323938\pi\)
\(734\) 25.6755 0.947700
\(735\) −2.23874 −0.0825771
\(736\) 3.87761 0.142931
\(737\) 0 0
\(738\) 21.0651 0.775417
\(739\) −24.4968 −0.901128 −0.450564 0.892744i \(-0.648777\pi\)
−0.450564 + 0.892744i \(0.648777\pi\)
\(740\) −11.8944 −0.437247
\(741\) −13.2892 −0.488192
\(742\) −8.28476 −0.304143
\(743\) −6.04973 −0.221943 −0.110972 0.993824i \(-0.535396\pi\)
−0.110972 + 0.993824i \(0.535396\pi\)
\(744\) −14.8736 −0.545291
\(745\) 5.37555 0.196945
\(746\) −5.70957 −0.209042
\(747\) −13.8105 −0.505299
\(748\) 0 0
\(749\) −11.3466 −0.414595
\(750\) −2.23874 −0.0817472
\(751\) 5.08525 0.185564 0.0927818 0.995686i \(-0.470424\pi\)
0.0927818 + 0.995686i \(0.470424\pi\)
\(752\) 8.71479 0.317796
\(753\) 56.9426 2.07510
\(754\) 30.2283 1.10085
\(755\) −1.95930 −0.0713061
\(756\) 2.21199 0.0804493
\(757\) 25.4296 0.924254 0.462127 0.886814i \(-0.347086\pi\)
0.462127 + 0.886814i \(0.347086\pi\)
\(758\) 2.16686 0.0787039
\(759\) 0 0
\(760\) 1.15751 0.0419871
\(761\) 41.6911 1.51130 0.755650 0.654976i \(-0.227321\pi\)
0.755650 + 0.654976i \(0.227321\pi\)
\(762\) 34.7135 1.25754
\(763\) 9.81672 0.355389
\(764\) −0.133153 −0.00481732
\(765\) 13.3909 0.484148
\(766\) 10.3043 0.372308
\(767\) −52.6237 −1.90013
\(768\) 2.23874 0.0807835
\(769\) −14.8389 −0.535106 −0.267553 0.963543i \(-0.586215\pi\)
−0.267553 + 0.963543i \(0.586215\pi\)
\(770\) 0 0
\(771\) −55.8716 −2.01216
\(772\) −12.7468 −0.458766
\(773\) 25.3422 0.911497 0.455749 0.890109i \(-0.349372\pi\)
0.455749 + 0.890109i \(0.349372\pi\)
\(774\) 3.57048 0.128338
\(775\) −6.64373 −0.238650
\(776\) −0.406766 −0.0146020
\(777\) −26.6285 −0.955291
\(778\) 5.86030 0.210102
\(779\) −12.1191 −0.434211
\(780\) −11.4809 −0.411082
\(781\) 0 0
\(782\) −25.8081 −0.922896
\(783\) −13.0384 −0.465954
\(784\) 1.00000 0.0357143
\(785\) −13.1054 −0.467753
\(786\) −36.9744 −1.31883
\(787\) 17.7718 0.633496 0.316748 0.948510i \(-0.397409\pi\)
0.316748 + 0.948510i \(0.397409\pi\)
\(788\) 4.39162 0.156445
\(789\) −0.746903 −0.0265905
\(790\) −10.2605 −0.365051
\(791\) 15.4263 0.548495
\(792\) 0 0
\(793\) −65.7098 −2.33342
\(794\) −9.65824 −0.342758
\(795\) −18.5474 −0.657808
\(796\) −0.430314 −0.0152521
\(797\) −3.36014 −0.119022 −0.0595112 0.998228i \(-0.518954\pi\)
−0.0595112 + 0.998228i \(0.518954\pi\)
\(798\) 2.59135 0.0917329
\(799\) −58.0028 −2.05199
\(800\) 1.00000 0.0353553
\(801\) 30.5661 1.08000
\(802\) −3.76869 −0.133077
\(803\) 0 0
\(804\) 33.5098 1.18180
\(805\) 3.87761 0.136668
\(806\) −34.0710 −1.20010
\(807\) 4.08787 0.143900
\(808\) 0.964155 0.0339189
\(809\) −17.3076 −0.608502 −0.304251 0.952592i \(-0.598406\pi\)
−0.304251 + 0.952592i \(0.598406\pi\)
\(810\) 10.9879 0.386076
\(811\) −12.3583 −0.433960 −0.216980 0.976176i \(-0.569621\pi\)
−0.216980 + 0.976176i \(0.569621\pi\)
\(812\) −5.89442 −0.206853
\(813\) 29.8388 1.04649
\(814\) 0 0
\(815\) 21.5818 0.755977
\(816\) −14.9003 −0.521615
\(817\) −2.05415 −0.0718657
\(818\) 7.59552 0.265571
\(819\) −10.3179 −0.360535
\(820\) −10.4700 −0.365628
\(821\) −5.55685 −0.193935 −0.0969677 0.995288i \(-0.530914\pi\)
−0.0969677 + 0.995288i \(0.530914\pi\)
\(822\) 30.9543 1.07965
\(823\) −52.8246 −1.84135 −0.920676 0.390329i \(-0.872361\pi\)
−0.920676 + 0.390329i \(0.872361\pi\)
\(824\) 12.4021 0.432049
\(825\) 0 0
\(826\) 10.2614 0.357041
\(827\) −49.4908 −1.72097 −0.860483 0.509480i \(-0.829838\pi\)
−0.860483 + 0.509480i \(0.829838\pi\)
\(828\) 7.80155 0.271123
\(829\) −48.2163 −1.67462 −0.837311 0.546727i \(-0.815873\pi\)
−0.837311 + 0.546727i \(0.815873\pi\)
\(830\) 6.86423 0.238261
\(831\) −34.6632 −1.20245
\(832\) 5.12829 0.177792
\(833\) −6.65568 −0.230606
\(834\) 2.68401 0.0929398
\(835\) −0.900461 −0.0311617
\(836\) 0 0
\(837\) 14.6959 0.507963
\(838\) −10.0288 −0.346438
\(839\) 22.3658 0.772154 0.386077 0.922467i \(-0.373830\pi\)
0.386077 + 0.922467i \(0.373830\pi\)
\(840\) 2.23874 0.0772438
\(841\) 5.74413 0.198073
\(842\) −4.21566 −0.145281
\(843\) 63.3798 2.18292
\(844\) −8.33737 −0.286984
\(845\) −13.2994 −0.457513
\(846\) 17.5337 0.602821
\(847\) 0 0
\(848\) 8.28476 0.284500
\(849\) 30.4225 1.04410
\(850\) −6.65568 −0.228288
\(851\) 46.1219 1.58104
\(852\) 17.3886 0.595725
\(853\) 24.9471 0.854173 0.427087 0.904211i \(-0.359540\pi\)
0.427087 + 0.904211i \(0.359540\pi\)
\(854\) 12.8132 0.438458
\(855\) 2.32884 0.0796447
\(856\) 11.3466 0.387818
\(857\) −19.7163 −0.673496 −0.336748 0.941595i \(-0.609327\pi\)
−0.336748 + 0.941595i \(0.609327\pi\)
\(858\) 0 0
\(859\) 15.4209 0.526156 0.263078 0.964775i \(-0.415262\pi\)
0.263078 + 0.964775i \(0.415262\pi\)
\(860\) −1.77464 −0.0605146
\(861\) −23.4396 −0.798819
\(862\) −23.7780 −0.809881
\(863\) −14.2128 −0.483810 −0.241905 0.970300i \(-0.577772\pi\)
−0.241905 + 0.970300i \(0.577772\pi\)
\(864\) −2.21199 −0.0752534
\(865\) 14.9963 0.509891
\(866\) −33.5935 −1.14155
\(867\) 61.1132 2.07551
\(868\) 6.64373 0.225503
\(869\) 0 0
\(870\) −13.1961 −0.447388
\(871\) 76.7612 2.60095
\(872\) −9.81672 −0.332436
\(873\) −0.818392 −0.0276984
\(874\) −4.48835 −0.151821
\(875\) 1.00000 0.0338062
\(876\) −5.68723 −0.192154
\(877\) 6.27739 0.211972 0.105986 0.994368i \(-0.466200\pi\)
0.105986 + 0.994368i \(0.466200\pi\)
\(878\) −27.7198 −0.935496
\(879\) 4.10637 0.138504
\(880\) 0 0
\(881\) 0.955481 0.0321910 0.0160955 0.999870i \(-0.494876\pi\)
0.0160955 + 0.999870i \(0.494876\pi\)
\(882\) 2.01195 0.0677458
\(883\) 8.50497 0.286215 0.143107 0.989707i \(-0.454291\pi\)
0.143107 + 0.989707i \(0.454291\pi\)
\(884\) −34.1323 −1.14799
\(885\) 22.9727 0.772219
\(886\) 28.5238 0.958276
\(887\) −2.18620 −0.0734056 −0.0367028 0.999326i \(-0.511685\pi\)
−0.0367028 + 0.999326i \(0.511685\pi\)
\(888\) 26.6285 0.893593
\(889\) −15.5058 −0.520049
\(890\) −15.1923 −0.509246
\(891\) 0 0
\(892\) 1.13218 0.0379083
\(893\) −10.0874 −0.337563
\(894\) −12.0344 −0.402492
\(895\) −10.7148 −0.358156
\(896\) −1.00000 −0.0334077
\(897\) 44.5185 1.48643
\(898\) −0.982968 −0.0328021
\(899\) −39.1609 −1.30609
\(900\) 2.01195 0.0670649
\(901\) −55.1407 −1.83700
\(902\) 0 0
\(903\) −3.97295 −0.132211
\(904\) −15.4263 −0.513070
\(905\) 6.20078 0.206121
\(906\) 4.38635 0.145727
\(907\) 0.266857 0.00886082 0.00443041 0.999990i \(-0.498590\pi\)
0.00443041 + 0.999990i \(0.498590\pi\)
\(908\) −10.2741 −0.340957
\(909\) 1.93983 0.0643401
\(910\) 5.12829 0.170001
\(911\) −20.3552 −0.674397 −0.337198 0.941434i \(-0.609479\pi\)
−0.337198 + 0.941434i \(0.609479\pi\)
\(912\) −2.59135 −0.0858082
\(913\) 0 0
\(914\) 4.98742 0.164969
\(915\) 28.6854 0.948309
\(916\) −9.23127 −0.305010
\(917\) 16.5157 0.545398
\(918\) 14.7223 0.485908
\(919\) 8.19185 0.270224 0.135112 0.990830i \(-0.456861\pi\)
0.135112 + 0.990830i \(0.456861\pi\)
\(920\) −3.87761 −0.127841
\(921\) 18.9254 0.623612
\(922\) 1.28781 0.0424116
\(923\) 39.8323 1.31110
\(924\) 0 0
\(925\) 11.8944 0.391086
\(926\) −17.4015 −0.571849
\(927\) 24.9525 0.819547
\(928\) 5.89442 0.193494
\(929\) −36.0873 −1.18398 −0.591992 0.805943i \(-0.701658\pi\)
−0.591992 + 0.805943i \(0.701658\pi\)
\(930\) 14.8736 0.487724
\(931\) −1.15751 −0.0379357
\(932\) −6.68548 −0.218990
\(933\) 38.0357 1.24523
\(934\) 6.97072 0.228089
\(935\) 0 0
\(936\) 10.3179 0.337250
\(937\) 42.0240 1.37286 0.686432 0.727194i \(-0.259176\pi\)
0.686432 + 0.727194i \(0.259176\pi\)
\(938\) −14.9682 −0.488728
\(939\) 40.4904 1.32135
\(940\) −8.71479 −0.284245
\(941\) −57.0071 −1.85838 −0.929189 0.369605i \(-0.879493\pi\)
−0.929189 + 0.369605i \(0.879493\pi\)
\(942\) 29.3397 0.955938
\(943\) 40.5986 1.32207
\(944\) −10.2614 −0.333982
\(945\) −2.21199 −0.0719560
\(946\) 0 0
\(947\) 42.6517 1.38599 0.692997 0.720940i \(-0.256290\pi\)
0.692997 + 0.720940i \(0.256290\pi\)
\(948\) 22.9705 0.746048
\(949\) −13.0278 −0.422900
\(950\) −1.15751 −0.0375544
\(951\) −0.880257 −0.0285443
\(952\) 6.65568 0.215712
\(953\) −11.5905 −0.375453 −0.187726 0.982221i \(-0.560112\pi\)
−0.187726 + 0.982221i \(0.560112\pi\)
\(954\) 16.6685 0.539663
\(955\) 0.133153 0.00430874
\(956\) −22.4534 −0.726193
\(957\) 0 0
\(958\) −26.7035 −0.862750
\(959\) −13.8267 −0.446486
\(960\) −2.23874 −0.0722550
\(961\) 13.1391 0.423843
\(962\) 60.9981 1.96666
\(963\) 22.8287 0.735645
\(964\) −28.6046 −0.921292
\(965\) 12.7468 0.410332
\(966\) −8.68095 −0.279305
\(967\) −38.6220 −1.24200 −0.621000 0.783811i \(-0.713273\pi\)
−0.621000 + 0.783811i \(0.713273\pi\)
\(968\) 0 0
\(969\) 17.2472 0.554060
\(970\) 0.406766 0.0130605
\(971\) −53.2173 −1.70782 −0.853912 0.520418i \(-0.825776\pi\)
−0.853912 + 0.520418i \(0.825776\pi\)
\(972\) −17.9631 −0.576166
\(973\) −1.19890 −0.0384348
\(974\) 4.32910 0.138713
\(975\) 11.4809 0.367683
\(976\) −12.8132 −0.410140
\(977\) 3.80768 0.121819 0.0609093 0.998143i \(-0.480600\pi\)
0.0609093 + 0.998143i \(0.480600\pi\)
\(978\) −48.3159 −1.54497
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −19.7507 −0.630592
\(982\) 12.8945 0.411481
\(983\) 27.5427 0.878474 0.439237 0.898371i \(-0.355249\pi\)
0.439237 + 0.898371i \(0.355249\pi\)
\(984\) 23.4396 0.747227
\(985\) −4.39162 −0.139929
\(986\) −39.2313 −1.24938
\(987\) −19.5101 −0.621014
\(988\) −5.93603 −0.188850
\(989\) 6.88134 0.218814
\(990\) 0 0
\(991\) −59.3959 −1.88677 −0.943386 0.331697i \(-0.892379\pi\)
−0.943386 + 0.331697i \(0.892379\pi\)
\(992\) −6.64373 −0.210939
\(993\) −77.3249 −2.45383
\(994\) −7.76716 −0.246360
\(995\) 0.430314 0.0136419
\(996\) −15.3672 −0.486929
\(997\) 32.6231 1.03318 0.516592 0.856232i \(-0.327201\pi\)
0.516592 + 0.856232i \(0.327201\pi\)
\(998\) −3.01676 −0.0954937
\(999\) −26.3103 −0.832422
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.dd.1.5 yes 6
11.10 odd 2 8470.2.a.cx.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cx.1.5 6 11.10 odd 2
8470.2.a.dd.1.5 yes 6 1.1 even 1 trivial