Properties

Label 8470.2.a.cl.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.892.1
Defining polynomial: \(x^{3} - x^{2} - 8 x + 10\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.31955\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.93923 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.93923 q^{6} +1.00000 q^{7} +1.00000 q^{8} +5.63910 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.93923 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.93923 q^{6} +1.00000 q^{7} +1.00000 q^{8} +5.63910 q^{9} -1.00000 q^{10} -2.93923 q^{12} +5.87847 q^{13} +1.00000 q^{14} +2.93923 q^{15} +1.00000 q^{16} +4.63910 q^{17} +5.63910 q^{18} +5.57834 q^{19} -1.00000 q^{20} -2.93923 q^{21} +5.57834 q^{23} -2.93923 q^{24} +1.00000 q^{25} +5.87847 q^{26} -7.75694 q^{27} +1.00000 q^{28} +9.45681 q^{29} +2.93923 q^{30} +6.00000 q^{31} +1.00000 q^{32} +4.63910 q^{34} -1.00000 q^{35} +5.63910 q^{36} +2.30013 q^{37} +5.57834 q^{38} -17.2782 q^{39} -1.00000 q^{40} +3.06077 q^{41} -2.93923 q^{42} -10.5176 q^{43} -5.63910 q^{45} +5.57834 q^{46} +8.51757 q^{47} -2.93923 q^{48} +1.00000 q^{49} +1.00000 q^{50} -13.6354 q^{51} +5.87847 q^{52} -9.45681 q^{53} -7.75694 q^{54} +1.00000 q^{56} -16.3960 q^{57} +9.45681 q^{58} +7.23937 q^{59} +2.93923 q^{60} -14.5176 q^{61} +6.00000 q^{62} +5.63910 q^{63} +1.00000 q^{64} -5.87847 q^{65} +8.00000 q^{67} +4.63910 q^{68} -16.3960 q^{69} -1.00000 q^{70} -7.15667 q^{71} +5.63910 q^{72} -12.3960 q^{73} +2.30013 q^{74} -2.93923 q^{75} +5.57834 q^{76} -17.2782 q^{78} +10.8565 q^{79} -1.00000 q^{80} +5.88216 q^{81} +3.06077 q^{82} +13.8785 q^{83} -2.93923 q^{84} -4.63910 q^{85} -10.5176 q^{86} -27.7958 q^{87} +3.87847 q^{89} -5.63910 q^{90} +5.87847 q^{91} +5.57834 q^{92} -17.6354 q^{93} +8.51757 q^{94} -5.57834 q^{95} -2.93923 q^{96} +7.57834 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{8} + 11 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{8} + 11 q^{9} - 3 q^{10} + 3 q^{14} + 3 q^{16} + 8 q^{17} + 11 q^{18} + 2 q^{19} - 3 q^{20} + 2 q^{23} + 3 q^{25} + 12 q^{27} + 3 q^{28} - 4 q^{29} + 18 q^{31} + 3 q^{32} + 8 q^{34} - 3 q^{35} + 11 q^{36} + 4 q^{37} + 2 q^{38} - 40 q^{39} - 3 q^{40} + 18 q^{41} - 8 q^{43} - 11 q^{45} + 2 q^{46} + 2 q^{47} + 3 q^{49} + 3 q^{50} + 12 q^{51} + 4 q^{53} + 12 q^{54} + 3 q^{56} - 8 q^{57} - 4 q^{58} + 10 q^{59} - 20 q^{61} + 18 q^{62} + 11 q^{63} + 3 q^{64} + 24 q^{67} + 8 q^{68} - 8 q^{69} - 3 q^{70} + 8 q^{71} + 11 q^{72} + 4 q^{73} + 4 q^{74} + 2 q^{76} - 40 q^{78} + 6 q^{79} - 3 q^{80} + 47 q^{81} + 18 q^{82} + 24 q^{83} - 8 q^{85} - 8 q^{86} - 48 q^{87} - 6 q^{89} - 11 q^{90} + 2 q^{92} + 2 q^{94} - 2 q^{95} + 8 q^{97} + 3 q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −2.93923 −1.69697 −0.848484 0.529221i \(-0.822484\pi\)
−0.848484 + 0.529221i \(0.822484\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −2.93923 −1.19994
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 5.63910 1.87970
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −2.93923 −0.848484
\(13\) 5.87847 1.63039 0.815197 0.579184i \(-0.196629\pi\)
0.815197 + 0.579184i \(0.196629\pi\)
\(14\) 1.00000 0.267261
\(15\) 2.93923 0.758907
\(16\) 1.00000 0.250000
\(17\) 4.63910 1.12515 0.562574 0.826747i \(-0.309811\pi\)
0.562574 + 0.826747i \(0.309811\pi\)
\(18\) 5.63910 1.32915
\(19\) 5.57834 1.27976 0.639879 0.768476i \(-0.278984\pi\)
0.639879 + 0.768476i \(0.278984\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.93923 −0.641394
\(22\) 0 0
\(23\) 5.57834 1.16316 0.581582 0.813488i \(-0.302434\pi\)
0.581582 + 0.813488i \(0.302434\pi\)
\(24\) −2.93923 −0.599969
\(25\) 1.00000 0.200000
\(26\) 5.87847 1.15286
\(27\) −7.75694 −1.49282
\(28\) 1.00000 0.188982
\(29\) 9.45681 1.75608 0.878042 0.478583i \(-0.158849\pi\)
0.878042 + 0.478583i \(0.158849\pi\)
\(30\) 2.93923 0.536628
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.63910 0.795599
\(35\) −1.00000 −0.169031
\(36\) 5.63910 0.939850
\(37\) 2.30013 0.378140 0.189070 0.981964i \(-0.439453\pi\)
0.189070 + 0.981964i \(0.439453\pi\)
\(38\) 5.57834 0.904926
\(39\) −17.2782 −2.76673
\(40\) −1.00000 −0.158114
\(41\) 3.06077 0.478011 0.239006 0.971018i \(-0.423179\pi\)
0.239006 + 0.971018i \(0.423179\pi\)
\(42\) −2.93923 −0.453534
\(43\) −10.5176 −1.60391 −0.801957 0.597381i \(-0.796208\pi\)
−0.801957 + 0.597381i \(0.796208\pi\)
\(44\) 0 0
\(45\) −5.63910 −0.840628
\(46\) 5.57834 0.822481
\(47\) 8.51757 1.24242 0.621208 0.783646i \(-0.286642\pi\)
0.621208 + 0.783646i \(0.286642\pi\)
\(48\) −2.93923 −0.424242
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −13.6354 −1.90934
\(52\) 5.87847 0.815197
\(53\) −9.45681 −1.29899 −0.649496 0.760365i \(-0.725020\pi\)
−0.649496 + 0.760365i \(0.725020\pi\)
\(54\) −7.75694 −1.05559
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −16.3960 −2.17171
\(58\) 9.45681 1.24174
\(59\) 7.23937 0.942485 0.471243 0.882004i \(-0.343806\pi\)
0.471243 + 0.882004i \(0.343806\pi\)
\(60\) 2.93923 0.379454
\(61\) −14.5176 −1.85878 −0.929392 0.369093i \(-0.879668\pi\)
−0.929392 + 0.369093i \(0.879668\pi\)
\(62\) 6.00000 0.762001
\(63\) 5.63910 0.710460
\(64\) 1.00000 0.125000
\(65\) −5.87847 −0.729134
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 4.63910 0.562574
\(69\) −16.3960 −1.97385
\(70\) −1.00000 −0.119523
\(71\) −7.15667 −0.849341 −0.424670 0.905348i \(-0.639610\pi\)
−0.424670 + 0.905348i \(0.639610\pi\)
\(72\) 5.63910 0.664574
\(73\) −12.3960 −1.45085 −0.725423 0.688303i \(-0.758356\pi\)
−0.725423 + 0.688303i \(0.758356\pi\)
\(74\) 2.30013 0.267385
\(75\) −2.93923 −0.339394
\(76\) 5.57834 0.639879
\(77\) 0 0
\(78\) −17.2782 −1.95637
\(79\) 10.8565 1.22146 0.610728 0.791840i \(-0.290877\pi\)
0.610728 + 0.791840i \(0.290877\pi\)
\(80\) −1.00000 −0.111803
\(81\) 5.88216 0.653574
\(82\) 3.06077 0.338005
\(83\) 13.8785 1.52336 0.761680 0.647953i \(-0.224375\pi\)
0.761680 + 0.647953i \(0.224375\pi\)
\(84\) −2.93923 −0.320697
\(85\) −4.63910 −0.503181
\(86\) −10.5176 −1.13414
\(87\) −27.7958 −2.98002
\(88\) 0 0
\(89\) 3.87847 0.411117 0.205558 0.978645i \(-0.434099\pi\)
0.205558 + 0.978645i \(0.434099\pi\)
\(90\) −5.63910 −0.594414
\(91\) 5.87847 0.616231
\(92\) 5.57834 0.581582
\(93\) −17.6354 −1.82871
\(94\) 8.51757 0.878520
\(95\) −5.57834 −0.572325
\(96\) −2.93923 −0.299984
\(97\) 7.57834 0.769463 0.384732 0.923028i \(-0.374294\pi\)
0.384732 + 0.923028i \(0.374294\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −11.7958 −1.17372 −0.586862 0.809687i \(-0.699637\pi\)
−0.586862 + 0.809687i \(0.699637\pi\)
\(102\) −13.6354 −1.35011
\(103\) 17.1178 1.68667 0.843335 0.537388i \(-0.180589\pi\)
0.843335 + 0.537388i \(0.180589\pi\)
\(104\) 5.87847 0.576431
\(105\) 2.93923 0.286840
\(106\) −9.45681 −0.918526
\(107\) −2.51757 −0.243383 −0.121691 0.992568i \(-0.538832\pi\)
−0.121691 + 0.992568i \(0.538832\pi\)
\(108\) −7.75694 −0.746412
\(109\) −10.3001 −0.986574 −0.493287 0.869867i \(-0.664205\pi\)
−0.493287 + 0.869867i \(0.664205\pi\)
\(110\) 0 0
\(111\) −6.76063 −0.641691
\(112\) 1.00000 0.0944911
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) −16.3960 −1.53563
\(115\) −5.57834 −0.520183
\(116\) 9.45681 0.878042
\(117\) 33.1493 3.06465
\(118\) 7.23937 0.666438
\(119\) 4.63910 0.425266
\(120\) 2.93923 0.268314
\(121\) 0 0
\(122\) −14.5176 −1.31436
\(123\) −8.99631 −0.811170
\(124\) 6.00000 0.538816
\(125\) −1.00000 −0.0894427
\(126\) 5.63910 0.502371
\(127\) −10.4787 −0.929837 −0.464919 0.885353i \(-0.653916\pi\)
−0.464919 + 0.885353i \(0.653916\pi\)
\(128\) 1.00000 0.0883883
\(129\) 30.9136 2.72179
\(130\) −5.87847 −0.515576
\(131\) 1.57834 0.137900 0.0689499 0.997620i \(-0.478035\pi\)
0.0689499 + 0.997620i \(0.478035\pi\)
\(132\) 0 0
\(133\) 5.57834 0.483703
\(134\) 8.00000 0.691095
\(135\) 7.75694 0.667611
\(136\) 4.63910 0.397800
\(137\) −3.27820 −0.280076 −0.140038 0.990146i \(-0.544722\pi\)
−0.140038 + 0.990146i \(0.544722\pi\)
\(138\) −16.3960 −1.39572
\(139\) 16.0571 1.36194 0.680972 0.732310i \(-0.261558\pi\)
0.680972 + 0.732310i \(0.261558\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −25.0351 −2.10834
\(142\) −7.15667 −0.600575
\(143\) 0 0
\(144\) 5.63910 0.469925
\(145\) −9.45681 −0.785345
\(146\) −12.3960 −1.02590
\(147\) −2.93923 −0.242424
\(148\) 2.30013 0.189070
\(149\) −10.7350 −0.879446 −0.439723 0.898133i \(-0.644923\pi\)
−0.439723 + 0.898133i \(0.644923\pi\)
\(150\) −2.93923 −0.239988
\(151\) 2.17860 0.177292 0.0886461 0.996063i \(-0.471746\pi\)
0.0886461 + 0.996063i \(0.471746\pi\)
\(152\) 5.57834 0.452463
\(153\) 26.1604 2.11494
\(154\) 0 0
\(155\) −6.00000 −0.481932
\(156\) −17.2782 −1.38336
\(157\) −17.1567 −1.36925 −0.684626 0.728895i \(-0.740034\pi\)
−0.684626 + 0.728895i \(0.740034\pi\)
\(158\) 10.8565 0.863700
\(159\) 27.7958 2.20435
\(160\) −1.00000 −0.0790569
\(161\) 5.57834 0.439635
\(162\) 5.88216 0.462146
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 3.06077 0.239006
\(165\) 0 0
\(166\) 13.8785 1.07718
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) −2.93923 −0.226767
\(169\) 21.5564 1.65819
\(170\) −4.63910 −0.355803
\(171\) 31.4568 2.40556
\(172\) −10.5176 −0.801957
\(173\) 5.48243 0.416821 0.208411 0.978041i \(-0.433171\pi\)
0.208411 + 0.978041i \(0.433171\pi\)
\(174\) −27.7958 −2.10719
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −21.2782 −1.59937
\(178\) 3.87847 0.290704
\(179\) 1.23937 0.0926347 0.0463174 0.998927i \(-0.485251\pi\)
0.0463174 + 0.998927i \(0.485251\pi\)
\(180\) −5.63910 −0.420314
\(181\) 8.55641 0.635993 0.317996 0.948092i \(-0.396990\pi\)
0.317996 + 0.948092i \(0.396990\pi\)
\(182\) 5.87847 0.435741
\(183\) 42.6706 3.15430
\(184\) 5.57834 0.411240
\(185\) −2.30013 −0.169109
\(186\) −17.6354 −1.29309
\(187\) 0 0
\(188\) 8.51757 0.621208
\(189\) −7.75694 −0.564234
\(190\) −5.57834 −0.404695
\(191\) −1.27820 −0.0924875 −0.0462438 0.998930i \(-0.514725\pi\)
−0.0462438 + 0.998930i \(0.514725\pi\)
\(192\) −2.93923 −0.212121
\(193\) 10.6391 0.765819 0.382910 0.923786i \(-0.374922\pi\)
0.382910 + 0.923786i \(0.374922\pi\)
\(194\) 7.57834 0.544093
\(195\) 17.2782 1.23732
\(196\) 1.00000 0.0714286
\(197\) −7.87847 −0.561318 −0.280659 0.959808i \(-0.590553\pi\)
−0.280659 + 0.959808i \(0.590553\pi\)
\(198\) 0 0
\(199\) 0.517571 0.0366897 0.0183448 0.999832i \(-0.494160\pi\)
0.0183448 + 0.999832i \(0.494160\pi\)
\(200\) 1.00000 0.0707107
\(201\) −23.5139 −1.65854
\(202\) −11.7958 −0.829948
\(203\) 9.45681 0.663738
\(204\) −13.6354 −0.954670
\(205\) −3.06077 −0.213773
\(206\) 17.1178 1.19266
\(207\) 31.4568 2.18640
\(208\) 5.87847 0.407599
\(209\) 0 0
\(210\) 2.93923 0.202826
\(211\) −2.72180 −0.187376 −0.0936881 0.995602i \(-0.529866\pi\)
−0.0936881 + 0.995602i \(0.529866\pi\)
\(212\) −9.45681 −0.649496
\(213\) 21.0351 1.44130
\(214\) −2.51757 −0.172098
\(215\) 10.5176 0.717292
\(216\) −7.75694 −0.527793
\(217\) 6.00000 0.407307
\(218\) −10.3001 −0.697613
\(219\) 36.4349 2.46204
\(220\) 0 0
\(221\) 27.2708 1.83443
\(222\) −6.76063 −0.453744
\(223\) −22.3960 −1.49975 −0.749875 0.661580i \(-0.769886\pi\)
−0.749875 + 0.661580i \(0.769886\pi\)
\(224\) 1.00000 0.0668153
\(225\) 5.63910 0.375940
\(226\) −14.0000 −0.931266
\(227\) −13.0351 −0.865173 −0.432586 0.901592i \(-0.642399\pi\)
−0.432586 + 0.901592i \(0.642399\pi\)
\(228\) −16.3960 −1.08585
\(229\) −6.63910 −0.438724 −0.219362 0.975644i \(-0.570398\pi\)
−0.219362 + 0.975644i \(0.570398\pi\)
\(230\) −5.57834 −0.367825
\(231\) 0 0
\(232\) 9.45681 0.620870
\(233\) −19.0351 −1.24703 −0.623517 0.781810i \(-0.714297\pi\)
−0.623517 + 0.781810i \(0.714297\pi\)
\(234\) 33.1493 2.16704
\(235\) −8.51757 −0.555625
\(236\) 7.23937 0.471243
\(237\) −31.9099 −2.07277
\(238\) 4.63910 0.300708
\(239\) −22.6135 −1.46274 −0.731372 0.681979i \(-0.761120\pi\)
−0.731372 + 0.681979i \(0.761120\pi\)
\(240\) 2.93923 0.189727
\(241\) 17.9744 1.15783 0.578916 0.815387i \(-0.303476\pi\)
0.578916 + 0.815387i \(0.303476\pi\)
\(242\) 0 0
\(243\) 5.98176 0.383730
\(244\) −14.5176 −0.929392
\(245\) −1.00000 −0.0638877
\(246\) −8.99631 −0.573584
\(247\) 32.7921 2.08651
\(248\) 6.00000 0.381000
\(249\) −40.7921 −2.58509
\(250\) −1.00000 −0.0632456
\(251\) 8.76063 0.552966 0.276483 0.961019i \(-0.410831\pi\)
0.276483 + 0.961019i \(0.410831\pi\)
\(252\) 5.63910 0.355230
\(253\) 0 0
\(254\) −10.4787 −0.657494
\(255\) 13.6354 0.853882
\(256\) 1.00000 0.0625000
\(257\) −5.09960 −0.318104 −0.159052 0.987270i \(-0.550844\pi\)
−0.159052 + 0.987270i \(0.550844\pi\)
\(258\) 30.9136 1.92460
\(259\) 2.30013 0.142923
\(260\) −5.87847 −0.364567
\(261\) 53.3279 3.30091
\(262\) 1.57834 0.0975100
\(263\) 11.7569 0.724964 0.362482 0.931991i \(-0.381929\pi\)
0.362482 + 0.931991i \(0.381929\pi\)
\(264\) 0 0
\(265\) 9.45681 0.580927
\(266\) 5.57834 0.342030
\(267\) −11.3997 −0.697652
\(268\) 8.00000 0.488678
\(269\) −11.6742 −0.711791 −0.355896 0.934526i \(-0.615824\pi\)
−0.355896 + 0.934526i \(0.615824\pi\)
\(270\) 7.75694 0.472072
\(271\) −17.6354 −1.07127 −0.535637 0.844448i \(-0.679929\pi\)
−0.535637 + 0.844448i \(0.679929\pi\)
\(272\) 4.63910 0.281287
\(273\) −17.2782 −1.04572
\(274\) −3.27820 −0.198043
\(275\) 0 0
\(276\) −16.3960 −0.986926
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 16.0571 0.963039
\(279\) 33.8346 2.02563
\(280\) −1.00000 −0.0597614
\(281\) −1.15667 −0.0690013 −0.0345007 0.999405i \(-0.510984\pi\)
−0.0345007 + 0.999405i \(0.510984\pi\)
\(282\) −25.0351 −1.49082
\(283\) −17.2782 −1.02708 −0.513541 0.858065i \(-0.671667\pi\)
−0.513541 + 0.858065i \(0.671667\pi\)
\(284\) −7.15667 −0.424670
\(285\) 16.3960 0.971218
\(286\) 0 0
\(287\) 3.06077 0.180671
\(288\) 5.63910 0.332287
\(289\) 4.52126 0.265957
\(290\) −9.45681 −0.555323
\(291\) −22.2745 −1.30575
\(292\) −12.3960 −0.725423
\(293\) −1.87847 −0.109741 −0.0548707 0.998493i \(-0.517475\pi\)
−0.0548707 + 0.998493i \(0.517475\pi\)
\(294\) −2.93923 −0.171420
\(295\) −7.23937 −0.421492
\(296\) 2.30013 0.133693
\(297\) 0 0
\(298\) −10.7350 −0.621862
\(299\) 32.7921 1.89642
\(300\) −2.93923 −0.169697
\(301\) −10.5176 −0.606223
\(302\) 2.17860 0.125365
\(303\) 34.6706 1.99177
\(304\) 5.57834 0.319940
\(305\) 14.5176 0.831274
\(306\) 26.1604 1.49549
\(307\) −8.60027 −0.490843 −0.245422 0.969416i \(-0.578926\pi\)
−0.245422 + 0.969416i \(0.578926\pi\)
\(308\) 0 0
\(309\) −50.3133 −2.86223
\(310\) −6.00000 −0.340777
\(311\) 13.7569 0.780084 0.390042 0.920797i \(-0.372460\pi\)
0.390042 + 0.920797i \(0.372460\pi\)
\(312\) −17.2782 −0.978186
\(313\) 2.90040 0.163940 0.0819701 0.996635i \(-0.473879\pi\)
0.0819701 + 0.996635i \(0.473879\pi\)
\(314\) −17.1567 −0.968207
\(315\) −5.63910 −0.317727
\(316\) 10.8565 0.610728
\(317\) 19.3353 1.08598 0.542989 0.839740i \(-0.317293\pi\)
0.542989 + 0.839740i \(0.317293\pi\)
\(318\) 27.7958 1.55871
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 7.39973 0.413013
\(322\) 5.57834 0.310869
\(323\) 25.8785 1.43992
\(324\) 5.88216 0.326787
\(325\) 5.87847 0.326079
\(326\) −8.00000 −0.443079
\(327\) 30.2745 1.67418
\(328\) 3.06077 0.169002
\(329\) 8.51757 0.469589
\(330\) 0 0
\(331\) 10.5176 0.578098 0.289049 0.957314i \(-0.406661\pi\)
0.289049 + 0.957314i \(0.406661\pi\)
\(332\) 13.8785 0.761680
\(333\) 12.9707 0.710789
\(334\) −8.00000 −0.437741
\(335\) −8.00000 −0.437087
\(336\) −2.93923 −0.160348
\(337\) −7.27820 −0.396469 −0.198234 0.980155i \(-0.563521\pi\)
−0.198234 + 0.980155i \(0.563521\pi\)
\(338\) 21.5564 1.17251
\(339\) 41.1493 2.23492
\(340\) −4.63910 −0.251591
\(341\) 0 0
\(342\) 31.4568 1.70099
\(343\) 1.00000 0.0539949
\(344\) −10.5176 −0.567069
\(345\) 16.3960 0.882733
\(346\) 5.48243 0.294737
\(347\) 14.2745 0.766296 0.383148 0.923687i \(-0.374840\pi\)
0.383148 + 0.923687i \(0.374840\pi\)
\(348\) −27.7958 −1.49001
\(349\) −0.396041 −0.0211996 −0.0105998 0.999944i \(-0.503374\pi\)
−0.0105998 + 0.999944i \(0.503374\pi\)
\(350\) 1.00000 0.0534522
\(351\) −45.5989 −2.43389
\(352\) 0 0
\(353\) −5.09960 −0.271424 −0.135712 0.990748i \(-0.543332\pi\)
−0.135712 + 0.990748i \(0.543332\pi\)
\(354\) −21.2782 −1.13092
\(355\) 7.15667 0.379837
\(356\) 3.87847 0.205558
\(357\) −13.6354 −0.721662
\(358\) 1.23937 0.0655026
\(359\) 18.3704 0.969554 0.484777 0.874638i \(-0.338901\pi\)
0.484777 + 0.874638i \(0.338901\pi\)
\(360\) −5.63910 −0.297207
\(361\) 12.1178 0.637781
\(362\) 8.55641 0.449715
\(363\) 0 0
\(364\) 5.87847 0.308116
\(365\) 12.3960 0.648838
\(366\) 42.6706 2.23043
\(367\) 17.1178 0.893544 0.446772 0.894648i \(-0.352574\pi\)
0.446772 + 0.894648i \(0.352574\pi\)
\(368\) 5.57834 0.290791
\(369\) 17.2600 0.898518
\(370\) −2.30013 −0.119578
\(371\) −9.45681 −0.490973
\(372\) −17.6354 −0.914353
\(373\) 6.35721 0.329164 0.164582 0.986363i \(-0.447373\pi\)
0.164582 + 0.986363i \(0.447373\pi\)
\(374\) 0 0
\(375\) 2.93923 0.151781
\(376\) 8.51757 0.439260
\(377\) 55.5915 2.86311
\(378\) −7.75694 −0.398974
\(379\) −5.03514 −0.258638 −0.129319 0.991603i \(-0.541279\pi\)
−0.129319 + 0.991603i \(0.541279\pi\)
\(380\) −5.57834 −0.286163
\(381\) 30.7995 1.57790
\(382\) −1.27820 −0.0653986
\(383\) −7.43118 −0.379716 −0.189858 0.981812i \(-0.560803\pi\)
−0.189858 + 0.981812i \(0.560803\pi\)
\(384\) −2.93923 −0.149992
\(385\) 0 0
\(386\) 10.6391 0.541516
\(387\) −59.3097 −3.01488
\(388\) 7.57834 0.384732
\(389\) −5.15667 −0.261454 −0.130727 0.991418i \(-0.541731\pi\)
−0.130727 + 0.991418i \(0.541731\pi\)
\(390\) 17.2782 0.874916
\(391\) 25.8785 1.30873
\(392\) 1.00000 0.0505076
\(393\) −4.63910 −0.234012
\(394\) −7.87847 −0.396912
\(395\) −10.8565 −0.546252
\(396\) 0 0
\(397\) −19.6354 −0.985473 −0.492736 0.870179i \(-0.664003\pi\)
−0.492736 + 0.870179i \(0.664003\pi\)
\(398\) 0.517571 0.0259435
\(399\) −16.3960 −0.820829
\(400\) 1.00000 0.0500000
\(401\) −5.11784 −0.255573 −0.127786 0.991802i \(-0.540787\pi\)
−0.127786 + 0.991802i \(0.540787\pi\)
\(402\) −23.5139 −1.17277
\(403\) 35.2708 1.75696
\(404\) −11.7958 −0.586862
\(405\) −5.88216 −0.292287
\(406\) 9.45681 0.469333
\(407\) 0 0
\(408\) −13.6354 −0.675053
\(409\) −2.21744 −0.109645 −0.0548226 0.998496i \(-0.517459\pi\)
−0.0548226 + 0.998496i \(0.517459\pi\)
\(410\) −3.06077 −0.151160
\(411\) 9.63541 0.475280
\(412\) 17.1178 0.843335
\(413\) 7.23937 0.356226
\(414\) 31.4568 1.54602
\(415\) −13.8785 −0.681267
\(416\) 5.87847 0.288216
\(417\) −47.1955 −2.31117
\(418\) 0 0
\(419\) −4.51757 −0.220698 −0.110349 0.993893i \(-0.535197\pi\)
−0.110349 + 0.993893i \(0.535197\pi\)
\(420\) 2.93923 0.143420
\(421\) −12.3133 −0.600116 −0.300058 0.953921i \(-0.597006\pi\)
−0.300058 + 0.953921i \(0.597006\pi\)
\(422\) −2.72180 −0.132495
\(423\) 48.0315 2.33537
\(424\) −9.45681 −0.459263
\(425\) 4.63910 0.225029
\(426\) 21.0351 1.01916
\(427\) −14.5176 −0.702555
\(428\) −2.51757 −0.121691
\(429\) 0 0
\(430\) 10.5176 0.507202
\(431\) 2.17860 0.104940 0.0524698 0.998623i \(-0.483291\pi\)
0.0524698 + 0.998623i \(0.483291\pi\)
\(432\) −7.75694 −0.373206
\(433\) 5.02193 0.241339 0.120669 0.992693i \(-0.461496\pi\)
0.120669 + 0.992693i \(0.461496\pi\)
\(434\) 6.00000 0.288009
\(435\) 27.7958 1.33271
\(436\) −10.3001 −0.493287
\(437\) 31.1178 1.48857
\(438\) 36.4349 1.74093
\(439\) −7.15667 −0.341569 −0.170785 0.985308i \(-0.554630\pi\)
−0.170785 + 0.985308i \(0.554630\pi\)
\(440\) 0 0
\(441\) 5.63910 0.268529
\(442\) 27.2708 1.29714
\(443\) −19.5139 −0.927132 −0.463566 0.886062i \(-0.653430\pi\)
−0.463566 + 0.886062i \(0.653430\pi\)
\(444\) −6.76063 −0.320845
\(445\) −3.87847 −0.183857
\(446\) −22.3960 −1.06048
\(447\) 31.5527 1.49239
\(448\) 1.00000 0.0472456
\(449\) 1.36090 0.0642248 0.0321124 0.999484i \(-0.489777\pi\)
0.0321124 + 0.999484i \(0.489777\pi\)
\(450\) 5.63910 0.265830
\(451\) 0 0
\(452\) −14.0000 −0.658505
\(453\) −6.40343 −0.300859
\(454\) −13.0351 −0.611770
\(455\) −5.87847 −0.275587
\(456\) −16.3960 −0.767815
\(457\) −4.47874 −0.209506 −0.104753 0.994498i \(-0.533405\pi\)
−0.104753 + 0.994498i \(0.533405\pi\)
\(458\) −6.63910 −0.310225
\(459\) −35.9852 −1.67965
\(460\) −5.57834 −0.260091
\(461\) 36.1530 1.68381 0.841906 0.539624i \(-0.181434\pi\)
0.841906 + 0.539624i \(0.181434\pi\)
\(462\) 0 0
\(463\) 21.0922 0.980238 0.490119 0.871655i \(-0.336953\pi\)
0.490119 + 0.871655i \(0.336953\pi\)
\(464\) 9.45681 0.439021
\(465\) 17.6354 0.817823
\(466\) −19.0351 −0.881786
\(467\) 19.1311 0.885279 0.442640 0.896700i \(-0.354042\pi\)
0.442640 + 0.896700i \(0.354042\pi\)
\(468\) 33.1493 1.53233
\(469\) 8.00000 0.369406
\(470\) −8.51757 −0.392886
\(471\) 50.4275 2.32358
\(472\) 7.23937 0.333219
\(473\) 0 0
\(474\) −31.9099 −1.46567
\(475\) 5.57834 0.255952
\(476\) 4.63910 0.212633
\(477\) −53.3279 −2.44172
\(478\) −22.6135 −1.03432
\(479\) −5.27820 −0.241167 −0.120584 0.992703i \(-0.538477\pi\)
−0.120584 + 0.992703i \(0.538477\pi\)
\(480\) 2.93923 0.134157
\(481\) 13.5213 0.616517
\(482\) 17.9744 0.818710
\(483\) −16.3960 −0.746046
\(484\) 0 0
\(485\) −7.57834 −0.344115
\(486\) 5.98176 0.271338
\(487\) −19.8140 −0.897859 −0.448929 0.893567i \(-0.648194\pi\)
−0.448929 + 0.893567i \(0.648194\pi\)
\(488\) −14.5176 −0.657180
\(489\) 23.5139 1.06333
\(490\) −1.00000 −0.0451754
\(491\) 6.47874 0.292381 0.146191 0.989256i \(-0.453299\pi\)
0.146191 + 0.989256i \(0.453299\pi\)
\(492\) −8.99631 −0.405585
\(493\) 43.8711 1.97585
\(494\) 32.7921 1.47539
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) −7.15667 −0.321021
\(498\) −40.7921 −1.82794
\(499\) −21.2394 −0.950805 −0.475402 0.879768i \(-0.657698\pi\)
−0.475402 + 0.879768i \(0.657698\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 23.5139 1.05052
\(502\) 8.76063 0.391006
\(503\) −42.2357 −1.88320 −0.941598 0.336739i \(-0.890676\pi\)
−0.941598 + 0.336739i \(0.890676\pi\)
\(504\) 5.63910 0.251186
\(505\) 11.7958 0.524905
\(506\) 0 0
\(507\) −63.3593 −2.81389
\(508\) −10.4787 −0.464919
\(509\) 38.8698 1.72287 0.861436 0.507867i \(-0.169566\pi\)
0.861436 + 0.507867i \(0.169566\pi\)
\(510\) 13.6354 0.603786
\(511\) −12.3960 −0.548369
\(512\) 1.00000 0.0441942
\(513\) −43.2708 −1.91045
\(514\) −5.09960 −0.224934
\(515\) −17.1178 −0.754302
\(516\) 30.9136 1.36090
\(517\) 0 0
\(518\) 2.30013 0.101062
\(519\) −16.1141 −0.707332
\(520\) −5.87847 −0.257788
\(521\) 25.9488 1.13684 0.568418 0.822740i \(-0.307556\pi\)
0.568418 + 0.822740i \(0.307556\pi\)
\(522\) 53.3279 2.33410
\(523\) −42.7482 −1.86925 −0.934625 0.355636i \(-0.884264\pi\)
−0.934625 + 0.355636i \(0.884264\pi\)
\(524\) 1.57834 0.0689499
\(525\) −2.93923 −0.128279
\(526\) 11.7569 0.512627
\(527\) 27.8346 1.21249
\(528\) 0 0
\(529\) 8.11784 0.352949
\(530\) 9.45681 0.410777
\(531\) 40.8235 1.77159
\(532\) 5.57834 0.241852
\(533\) 17.9926 0.779347
\(534\) −11.3997 −0.493315
\(535\) 2.51757 0.108844
\(536\) 8.00000 0.345547
\(537\) −3.64279 −0.157198
\(538\) −11.6742 −0.503312
\(539\) 0 0
\(540\) 7.75694 0.333806
\(541\) 6.73501 0.289561 0.144780 0.989464i \(-0.453752\pi\)
0.144780 + 0.989464i \(0.453752\pi\)
\(542\) −17.6354 −0.757506
\(543\) −25.1493 −1.07926
\(544\) 4.63910 0.198900
\(545\) 10.3001 0.441209
\(546\) −17.2782 −0.739439
\(547\) −23.7569 −1.01577 −0.507887 0.861424i \(-0.669573\pi\)
−0.507887 + 0.861424i \(0.669573\pi\)
\(548\) −3.27820 −0.140038
\(549\) −81.8661 −3.49396
\(550\) 0 0
\(551\) 52.7532 2.24736
\(552\) −16.3960 −0.697862
\(553\) 10.8565 0.461667
\(554\) −22.0000 −0.934690
\(555\) 6.76063 0.286973
\(556\) 16.0571 0.680972
\(557\) 34.1918 1.44875 0.724377 0.689404i \(-0.242128\pi\)
0.724377 + 0.689404i \(0.242128\pi\)
\(558\) 33.8346 1.43233
\(559\) −61.8272 −2.61501
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −1.15667 −0.0487913
\(563\) 31.3485 1.32118 0.660591 0.750746i \(-0.270306\pi\)
0.660591 + 0.750746i \(0.270306\pi\)
\(564\) −25.0351 −1.05417
\(565\) 14.0000 0.588984
\(566\) −17.2782 −0.726257
\(567\) 5.88216 0.247028
\(568\) −7.15667 −0.300287
\(569\) 38.5490 1.61606 0.808030 0.589142i \(-0.200534\pi\)
0.808030 + 0.589142i \(0.200534\pi\)
\(570\) 16.3960 0.686755
\(571\) −26.9136 −1.12630 −0.563150 0.826355i \(-0.690411\pi\)
−0.563150 + 0.826355i \(0.690411\pi\)
\(572\) 0 0
\(573\) 3.75694 0.156948
\(574\) 3.06077 0.127754
\(575\) 5.57834 0.232633
\(576\) 5.63910 0.234963
\(577\) −15.3353 −0.638416 −0.319208 0.947685i \(-0.603417\pi\)
−0.319208 + 0.947685i \(0.603417\pi\)
\(578\) 4.52126 0.188060
\(579\) −31.2708 −1.29957
\(580\) −9.45681 −0.392673
\(581\) 13.8785 0.575776
\(582\) −22.2745 −0.923308
\(583\) 0 0
\(584\) −12.3960 −0.512952
\(585\) −33.1493 −1.37055
\(586\) −1.87847 −0.0775989
\(587\) −21.2526 −0.877188 −0.438594 0.898685i \(-0.644523\pi\)
−0.438594 + 0.898685i \(0.644523\pi\)
\(588\) −2.93923 −0.121212
\(589\) 33.4700 1.37911
\(590\) −7.23937 −0.298040
\(591\) 23.1567 0.952538
\(592\) 2.30013 0.0945349
\(593\) −19.1955 −0.788265 −0.394133 0.919054i \(-0.628955\pi\)
−0.394133 + 0.919054i \(0.628955\pi\)
\(594\) 0 0
\(595\) −4.63910 −0.190185
\(596\) −10.7350 −0.439723
\(597\) −1.52126 −0.0622612
\(598\) 32.7921 1.34097
\(599\) 3.39973 0.138909 0.0694547 0.997585i \(-0.477874\pi\)
0.0694547 + 0.997585i \(0.477874\pi\)
\(600\) −2.93923 −0.119994
\(601\) 7.90409 0.322415 0.161207 0.986921i \(-0.448461\pi\)
0.161207 + 0.986921i \(0.448461\pi\)
\(602\) −10.5176 −0.428664
\(603\) 45.1128 1.83714
\(604\) 2.17860 0.0886461
\(605\) 0 0
\(606\) 34.6706 1.40839
\(607\) 23.7181 0.962688 0.481344 0.876532i \(-0.340149\pi\)
0.481344 + 0.876532i \(0.340149\pi\)
\(608\) 5.57834 0.226231
\(609\) −27.7958 −1.12634
\(610\) 14.5176 0.587799
\(611\) 50.0703 2.02563
\(612\) 26.1604 1.05747
\(613\) −40.9136 −1.65249 −0.826243 0.563314i \(-0.809526\pi\)
−0.826243 + 0.563314i \(0.809526\pi\)
\(614\) −8.60027 −0.347079
\(615\) 8.99631 0.362766
\(616\) 0 0
\(617\) −30.3572 −1.22214 −0.611068 0.791578i \(-0.709260\pi\)
−0.611068 + 0.791578i \(0.709260\pi\)
\(618\) −50.3133 −2.02390
\(619\) −34.9963 −1.40662 −0.703310 0.710883i \(-0.748295\pi\)
−0.703310 + 0.710883i \(0.748295\pi\)
\(620\) −6.00000 −0.240966
\(621\) −43.2708 −1.73640
\(622\) 13.7569 0.551603
\(623\) 3.87847 0.155388
\(624\) −17.2782 −0.691682
\(625\) 1.00000 0.0400000
\(626\) 2.90040 0.115923
\(627\) 0 0
\(628\) −17.1567 −0.684626
\(629\) 10.6706 0.425463
\(630\) −5.63910 −0.224667
\(631\) 38.2357 1.52214 0.761069 0.648671i \(-0.224675\pi\)
0.761069 + 0.648671i \(0.224675\pi\)
\(632\) 10.8565 0.431850
\(633\) 8.00000 0.317971
\(634\) 19.3353 0.767902
\(635\) 10.4787 0.415836
\(636\) 27.7958 1.10217
\(637\) 5.87847 0.232913
\(638\) 0 0
\(639\) −40.3572 −1.59651
\(640\) −1.00000 −0.0395285
\(641\) −25.5139 −1.00774 −0.503869 0.863780i \(-0.668091\pi\)
−0.503869 + 0.863780i \(0.668091\pi\)
\(642\) 7.39973 0.292044
\(643\) −37.8528 −1.49277 −0.746385 0.665514i \(-0.768212\pi\)
−0.746385 + 0.665514i \(0.768212\pi\)
\(644\) 5.57834 0.219817
\(645\) −30.9136 −1.21722
\(646\) 25.8785 1.01817
\(647\) −24.7094 −0.971426 −0.485713 0.874118i \(-0.661440\pi\)
−0.485713 + 0.874118i \(0.661440\pi\)
\(648\) 5.88216 0.231073
\(649\) 0 0
\(650\) 5.87847 0.230573
\(651\) −17.6354 −0.691186
\(652\) −8.00000 −0.313304
\(653\) 16.7789 0.656608 0.328304 0.944572i \(-0.393523\pi\)
0.328304 + 0.944572i \(0.393523\pi\)
\(654\) 30.2745 1.18383
\(655\) −1.57834 −0.0616707
\(656\) 3.06077 0.119503
\(657\) −69.9025 −2.72716
\(658\) 8.51757 0.332050
\(659\) −10.1215 −0.394279 −0.197139 0.980375i \(-0.563165\pi\)
−0.197139 + 0.980375i \(0.563165\pi\)
\(660\) 0 0
\(661\) 23.1128 0.898984 0.449492 0.893284i \(-0.351605\pi\)
0.449492 + 0.893284i \(0.351605\pi\)
\(662\) 10.5176 0.408777
\(663\) −80.1553 −3.11298
\(664\) 13.8785 0.538589
\(665\) −5.57834 −0.216319
\(666\) 12.9707 0.502604
\(667\) 52.7532 2.04261
\(668\) −8.00000 −0.309529
\(669\) 65.8272 2.54503
\(670\) −8.00000 −0.309067
\(671\) 0 0
\(672\) −2.93923 −0.113383
\(673\) −38.0000 −1.46479 −0.732396 0.680879i \(-0.761598\pi\)
−0.732396 + 0.680879i \(0.761598\pi\)
\(674\) −7.27820 −0.280346
\(675\) −7.75694 −0.298565
\(676\) 21.5564 0.829093
\(677\) −8.83092 −0.339400 −0.169700 0.985496i \(-0.554280\pi\)
−0.169700 + 0.985496i \(0.554280\pi\)
\(678\) 41.1493 1.58033
\(679\) 7.57834 0.290830
\(680\) −4.63910 −0.177901
\(681\) 38.3133 1.46817
\(682\) 0 0
\(683\) 11.3997 0.436199 0.218099 0.975927i \(-0.430014\pi\)
0.218099 + 0.975927i \(0.430014\pi\)
\(684\) 31.4568 1.20278
\(685\) 3.27820 0.125254
\(686\) 1.00000 0.0381802
\(687\) 19.5139 0.744501
\(688\) −10.5176 −0.400979
\(689\) −55.5915 −2.11787
\(690\) 16.3960 0.624187
\(691\) −0.0826952 −0.00314587 −0.00157294 0.999999i \(-0.500501\pi\)
−0.00157294 + 0.999999i \(0.500501\pi\)
\(692\) 5.48243 0.208411
\(693\) 0 0
\(694\) 14.2745 0.541853
\(695\) −16.0571 −0.609079
\(696\) −27.7958 −1.05360
\(697\) 14.1992 0.537833
\(698\) −0.396041 −0.0149904
\(699\) 55.9488 2.11618
\(700\) 1.00000 0.0377964
\(701\) −38.0571 −1.43740 −0.718698 0.695322i \(-0.755262\pi\)
−0.718698 + 0.695322i \(0.755262\pi\)
\(702\) −45.5989 −1.72102
\(703\) 12.8309 0.483927
\(704\) 0 0
\(705\) 25.0351 0.942878
\(706\) −5.09960 −0.191926
\(707\) −11.7958 −0.443626
\(708\) −21.2782 −0.799684
\(709\) 36.4275 1.36806 0.684032 0.729452i \(-0.260225\pi\)
0.684032 + 0.729452i \(0.260225\pi\)
\(710\) 7.15667 0.268585
\(711\) 61.2211 2.29597
\(712\) 3.87847 0.145352
\(713\) 33.4700 1.25346
\(714\) −13.6354 −0.510292
\(715\) 0 0
\(716\) 1.23937 0.0463174
\(717\) 66.4663 2.48223
\(718\) 18.3704 0.685578
\(719\) 37.3097 1.39142 0.695708 0.718325i \(-0.255091\pi\)
0.695708 + 0.718325i \(0.255091\pi\)
\(720\) −5.63910 −0.210157
\(721\) 17.1178 0.637502
\(722\) 12.1178 0.450979
\(723\) −52.8309 −1.96480
\(724\) 8.55641 0.317996
\(725\) 9.45681 0.351217
\(726\) 0 0
\(727\) 3.23937 0.120142 0.0600708 0.998194i \(-0.480867\pi\)
0.0600708 + 0.998194i \(0.480867\pi\)
\(728\) 5.87847 0.217871
\(729\) −35.2283 −1.30475
\(730\) 12.3960 0.458798
\(731\) −48.7921 −1.80464
\(732\) 42.6706 1.57715
\(733\) 26.3522 0.973340 0.486670 0.873586i \(-0.338211\pi\)
0.486670 + 0.873586i \(0.338211\pi\)
\(734\) 17.1178 0.631831
\(735\) 2.93923 0.108415
\(736\) 5.57834 0.205620
\(737\) 0 0
\(738\) 17.2600 0.635348
\(739\) 11.9223 0.438570 0.219285 0.975661i \(-0.429628\pi\)
0.219285 + 0.975661i \(0.429628\pi\)
\(740\) −2.30013 −0.0845546
\(741\) −96.3836 −3.54074
\(742\) −9.45681 −0.347170
\(743\) 26.0703 0.956426 0.478213 0.878244i \(-0.341285\pi\)
0.478213 + 0.878244i \(0.341285\pi\)
\(744\) −17.6354 −0.646545
\(745\) 10.7350 0.393300
\(746\) 6.35721 0.232754
\(747\) 78.2621 2.86346
\(748\) 0 0
\(749\) −2.51757 −0.0919901
\(750\) 2.93923 0.107326
\(751\) 36.6003 1.33556 0.667781 0.744357i \(-0.267244\pi\)
0.667781 + 0.744357i \(0.267244\pi\)
\(752\) 8.51757 0.310604
\(753\) −25.7496 −0.938366
\(754\) 55.5915 2.02452
\(755\) −2.17860 −0.0792875
\(756\) −7.75694 −0.282117
\(757\) −43.0922 −1.56621 −0.783107 0.621887i \(-0.786366\pi\)
−0.783107 + 0.621887i \(0.786366\pi\)
\(758\) −5.03514 −0.182885
\(759\) 0 0
\(760\) −5.57834 −0.202348
\(761\) 44.0959 1.59848 0.799238 0.601015i \(-0.205237\pi\)
0.799238 + 0.601015i \(0.205237\pi\)
\(762\) 30.7995 1.11575
\(763\) −10.3001 −0.372890
\(764\) −1.27820 −0.0462438
\(765\) −26.1604 −0.945830
\(766\) −7.43118 −0.268500
\(767\) 42.5564 1.53662
\(768\) −2.93923 −0.106061
\(769\) 8.33897 0.300711 0.150355 0.988632i \(-0.451958\pi\)
0.150355 + 0.988632i \(0.451958\pi\)
\(770\) 0 0
\(771\) 14.9889 0.539813
\(772\) 10.6391 0.382910
\(773\) −19.2782 −0.693389 −0.346694 0.937978i \(-0.612696\pi\)
−0.346694 + 0.937978i \(0.612696\pi\)
\(774\) −59.3097 −2.13184
\(775\) 6.00000 0.215526
\(776\) 7.57834 0.272046
\(777\) −6.76063 −0.242536
\(778\) −5.15667 −0.184876
\(779\) 17.0740 0.611739
\(780\) 17.2782 0.618659
\(781\) 0 0
\(782\) 25.8785 0.925412
\(783\) −73.3559 −2.62153
\(784\) 1.00000 0.0357143
\(785\) 17.1567 0.612348
\(786\) −4.63910 −0.165471
\(787\) −43.5915 −1.55387 −0.776935 0.629580i \(-0.783227\pi\)
−0.776935 + 0.629580i \(0.783227\pi\)
\(788\) −7.87847 −0.280659
\(789\) −34.5564 −1.23024
\(790\) −10.8565 −0.386258
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) −85.3411 −3.03055
\(794\) −19.6354 −0.696835
\(795\) −27.7958 −0.985815
\(796\) 0.517571 0.0183448
\(797\) −51.1493 −1.81180 −0.905900 0.423491i \(-0.860805\pi\)
−0.905900 + 0.423491i \(0.860805\pi\)
\(798\) −16.3960 −0.580414
\(799\) 39.5139 1.39790
\(800\) 1.00000 0.0353553
\(801\) 21.8711 0.772777
\(802\) −5.11784 −0.180717
\(803\) 0 0
\(804\) −23.5139 −0.829271
\(805\) −5.57834 −0.196611
\(806\) 35.2708 1.24236
\(807\) 34.3133 1.20789
\(808\) −11.7958 −0.414974
\(809\) −45.1567 −1.58762 −0.793812 0.608163i \(-0.791907\pi\)
−0.793812 + 0.608163i \(0.791907\pi\)
\(810\) −5.88216 −0.206678
\(811\) −39.2914 −1.37971 −0.689854 0.723948i \(-0.742325\pi\)
−0.689854 + 0.723948i \(0.742325\pi\)
\(812\) 9.45681 0.331869
\(813\) 51.8346 1.81792
\(814\) 0 0
\(815\) 8.00000 0.280228
\(816\) −13.6354 −0.477335
\(817\) −58.6706 −2.05262
\(818\) −2.21744 −0.0775309
\(819\) 33.1493 1.15833
\(820\) −3.06077 −0.106887
\(821\) −25.6486 −0.895143 −0.447572 0.894248i \(-0.647711\pi\)
−0.447572 + 0.894248i \(0.647711\pi\)
\(822\) 9.63541 0.336073
\(823\) −51.6486 −1.80036 −0.900179 0.435520i \(-0.856564\pi\)
−0.900179 + 0.435520i \(0.856564\pi\)
\(824\) 17.1178 0.596328
\(825\) 0 0
\(826\) 7.23937 0.251890
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 31.4568 1.09320
\(829\) 5.36090 0.186192 0.0930958 0.995657i \(-0.470324\pi\)
0.0930958 + 0.995657i \(0.470324\pi\)
\(830\) −13.8785 −0.481729
\(831\) 64.6632 2.24314
\(832\) 5.87847 0.203799
\(833\) 4.63910 0.160735
\(834\) −47.1955 −1.63425
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) −46.5416 −1.60871
\(838\) −4.51757 −0.156057
\(839\) 16.9649 0.585692 0.292846 0.956160i \(-0.405398\pi\)
0.292846 + 0.956160i \(0.405398\pi\)
\(840\) 2.93923 0.101413
\(841\) 60.4312 2.08383
\(842\) −12.3133 −0.424346
\(843\) 3.39973 0.117093
\(844\) −2.72180 −0.0936881
\(845\) −21.5564 −0.741563
\(846\) 48.0315 1.65136
\(847\) 0 0
\(848\) −9.45681 −0.324748
\(849\) 50.7847 1.74293
\(850\) 4.63910 0.159120
\(851\) 12.8309 0.439838
\(852\) 21.0351 0.720652
\(853\) −31.0666 −1.06370 −0.531850 0.846839i \(-0.678503\pi\)
−0.531850 + 0.846839i \(0.678503\pi\)
\(854\) −14.5176 −0.496781
\(855\) −31.4568 −1.07580
\(856\) −2.51757 −0.0860488
\(857\) −37.5966 −1.28427 −0.642137 0.766590i \(-0.721952\pi\)
−0.642137 + 0.766590i \(0.721952\pi\)
\(858\) 0 0
\(859\) −14.0388 −0.478999 −0.239499 0.970897i \(-0.576983\pi\)
−0.239499 + 0.970897i \(0.576983\pi\)
\(860\) 10.5176 0.358646
\(861\) −8.99631 −0.306593
\(862\) 2.17860 0.0742035
\(863\) 38.8053 1.32095 0.660474 0.750849i \(-0.270355\pi\)
0.660474 + 0.750849i \(0.270355\pi\)
\(864\) −7.75694 −0.263896
\(865\) −5.48243 −0.186408
\(866\) 5.02193 0.170652
\(867\) −13.2891 −0.451320
\(868\) 6.00000 0.203653
\(869\) 0 0
\(870\) 27.7958 0.942365
\(871\) 47.0278 1.59347
\(872\) −10.3001 −0.348807
\(873\) 42.7350 1.44636
\(874\) 31.1178 1.05258
\(875\) −1.00000 −0.0338062
\(876\) 36.4349 1.23102
\(877\) −31.7131 −1.07087 −0.535437 0.844575i \(-0.679853\pi\)
−0.535437 + 0.844575i \(0.679853\pi\)
\(878\) −7.15667 −0.241526
\(879\) 5.52126 0.186228
\(880\) 0 0
\(881\) 2.24306 0.0755706 0.0377853 0.999286i \(-0.487970\pi\)
0.0377853 + 0.999286i \(0.487970\pi\)
\(882\) 5.63910 0.189878
\(883\) 24.4349 0.822299 0.411150 0.911568i \(-0.365127\pi\)
0.411150 + 0.911568i \(0.365127\pi\)
\(884\) 27.2708 0.917217
\(885\) 21.2782 0.715259
\(886\) −19.5139 −0.655582
\(887\) −35.5915 −1.19505 −0.597524 0.801851i \(-0.703849\pi\)
−0.597524 + 0.801851i \(0.703849\pi\)
\(888\) −6.76063 −0.226872
\(889\) −10.4787 −0.351446
\(890\) −3.87847 −0.130007
\(891\) 0 0
\(892\) −22.3960 −0.749875
\(893\) 47.5139 1.58999
\(894\) 31.5527 1.05528
\(895\) −1.23937 −0.0414275
\(896\) 1.00000 0.0334077
\(897\) −96.3836 −3.21816
\(898\) 1.36090 0.0454138
\(899\) 56.7408 1.89241
\(900\) 5.63910 0.187970
\(901\) −43.8711 −1.46156
\(902\) 0 0
\(903\) 30.9136 1.02874
\(904\) −14.0000 −0.465633
\(905\) −8.55641 −0.284425
\(906\) −6.40343 −0.212740
\(907\) 17.3923 0.577503 0.288752 0.957404i \(-0.406760\pi\)
0.288752 + 0.957404i \(0.406760\pi\)
\(908\) −13.0351 −0.432586
\(909\) −66.5176 −2.20625
\(910\) −5.87847 −0.194869
\(911\) 52.1141 1.72662 0.863309 0.504675i \(-0.168388\pi\)
0.863309 + 0.504675i \(0.168388\pi\)
\(912\) −16.3960 −0.542927
\(913\) 0 0
\(914\) −4.47874 −0.148143
\(915\) −42.6706 −1.41064
\(916\) −6.63910 −0.219362
\(917\) 1.57834 0.0521213
\(918\) −35.9852 −1.18769
\(919\) −12.1347 −0.400288 −0.200144 0.979766i \(-0.564141\pi\)
−0.200144 + 0.979766i \(0.564141\pi\)
\(920\) −5.57834 −0.183912
\(921\) 25.2782 0.832945
\(922\) 36.1530 1.19064
\(923\) −42.0703 −1.38476
\(924\) 0 0
\(925\) 2.30013 0.0756279
\(926\) 21.0922 0.693133
\(927\) 96.5292 3.17044
\(928\) 9.45681 0.310435
\(929\) 23.8008 0.780879 0.390439 0.920629i \(-0.372323\pi\)
0.390439 + 0.920629i \(0.372323\pi\)
\(930\) 17.6354 0.578288
\(931\) 5.57834 0.182823
\(932\) −19.0351 −0.623517
\(933\) −40.4349 −1.32378
\(934\) 19.1311 0.625987
\(935\) 0 0
\(936\) 33.1493 1.08352
\(937\) −9.16170 −0.299300 −0.149650 0.988739i \(-0.547815\pi\)
−0.149650 + 0.988739i \(0.547815\pi\)
\(938\) 8.00000 0.261209
\(939\) −8.52496 −0.278201
\(940\) −8.51757 −0.277813
\(941\) −0.996308 −0.0324787 −0.0162393 0.999868i \(-0.505169\pi\)
−0.0162393 + 0.999868i \(0.505169\pi\)
\(942\) 50.4275 1.64302
\(943\) 17.0740 0.556005
\(944\) 7.23937 0.235621
\(945\) 7.75694 0.252333
\(946\) 0 0
\(947\) −18.8359 −0.612086 −0.306043 0.952018i \(-0.599005\pi\)
−0.306043 + 0.952018i \(0.599005\pi\)
\(948\) −31.9099 −1.03639
\(949\) −72.8698 −2.36545
\(950\) 5.57834 0.180985
\(951\) −56.8309 −1.84287
\(952\) 4.63910 0.150354
\(953\) −21.4386 −0.694463 −0.347232 0.937779i \(-0.612878\pi\)
−0.347232 + 0.937779i \(0.612878\pi\)
\(954\) −53.3279 −1.72655
\(955\) 1.27820 0.0413617
\(956\) −22.6135 −0.731372
\(957\) 0 0
\(958\) −5.27820 −0.170531
\(959\) −3.27820 −0.105859
\(960\) 2.93923 0.0948634
\(961\) 5.00000 0.161290
\(962\) 13.5213 0.435943
\(963\) −14.1968 −0.457487
\(964\) 17.9744 0.578916
\(965\) −10.6391 −0.342485
\(966\) −16.3960 −0.527534
\(967\) −31.5139 −1.01342 −0.506709 0.862117i \(-0.669138\pi\)
−0.506709 + 0.862117i \(0.669138\pi\)
\(968\) 0 0
\(969\) −76.0629 −2.44349
\(970\) −7.57834 −0.243326
\(971\) 16.1968 0.519781 0.259891 0.965638i \(-0.416313\pi\)
0.259891 + 0.965638i \(0.416313\pi\)
\(972\) 5.98176 0.191865
\(973\) 16.0571 0.514766
\(974\) −19.8140 −0.634882
\(975\) −17.2782 −0.553345
\(976\) −14.5176 −0.464696
\(977\) 17.5139 0.560319 0.280159 0.959954i \(-0.409613\pi\)
0.280159 + 0.959954i \(0.409613\pi\)
\(978\) 23.5139 0.751891
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −58.0835 −1.85446
\(982\) 6.47874 0.206745
\(983\) 41.1178 1.31146 0.655728 0.754997i \(-0.272362\pi\)
0.655728 + 0.754997i \(0.272362\pi\)
\(984\) −8.99631 −0.286792
\(985\) 7.87847 0.251029
\(986\) 43.8711 1.39714
\(987\) −25.0351 −0.796877
\(988\) 32.7921 1.04326
\(989\) −58.6706 −1.86562
\(990\) 0 0
\(991\) −2.79947 −0.0889280 −0.0444640 0.999011i \(-0.514158\pi\)
−0.0444640 + 0.999011i \(0.514158\pi\)
\(992\) 6.00000 0.190500
\(993\) −30.9136 −0.981014
\(994\) −7.15667 −0.226996
\(995\) −0.517571 −0.0164081
\(996\) −40.7921 −1.29255
\(997\) −11.7181 −0.371116 −0.185558 0.982633i \(-0.559409\pi\)
−0.185558 + 0.982633i \(0.559409\pi\)
\(998\) −21.2394 −0.672320
\(999\) −17.8420 −0.564496
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8470.2.a.cl.1.1 3
11.10 odd 2 770.2.a.l.1.1 3
33.32 even 2 6930.2.a.cl.1.1 3
44.43 even 2 6160.2.a.bi.1.3 3
55.32 even 4 3850.2.c.z.1849.3 6
55.43 even 4 3850.2.c.z.1849.4 6
55.54 odd 2 3850.2.a.bu.1.3 3
77.76 even 2 5390.2.a.bz.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.l.1.1 3 11.10 odd 2
3850.2.a.bu.1.3 3 55.54 odd 2
3850.2.c.z.1849.3 6 55.32 even 4
3850.2.c.z.1849.4 6 55.43 even 4
5390.2.a.bz.1.3 3 77.76 even 2
6160.2.a.bi.1.3 3 44.43 even 2
6930.2.a.cl.1.1 3 33.32 even 2
8470.2.a.cl.1.1 3 1.1 even 1 trivial