Properties

Label 8470.2.a.cx.1.2
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 1.2
Root \(2.67544\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.67544 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.67544 q^{6} +1.00000 q^{7} -1.00000 q^{8} +4.15798 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.67544 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.67544 q^{6} +1.00000 q^{7} -1.00000 q^{8} +4.15798 q^{9} +1.00000 q^{10} -2.67544 q^{12} -3.11654 q^{13} -1.00000 q^{14} +2.67544 q^{15} +1.00000 q^{16} -0.564586 q^{17} -4.15798 q^{18} +7.52404 q^{19} -1.00000 q^{20} -2.67544 q^{21} +4.63400 q^{23} +2.67544 q^{24} +1.00000 q^{25} +3.11654 q^{26} -3.09812 q^{27} +1.00000 q^{28} +6.24003 q^{29} -2.67544 q^{30} +2.72257 q^{31} -1.00000 q^{32} +0.564586 q^{34} -1.00000 q^{35} +4.15798 q^{36} -0.240027 q^{37} -7.52404 q^{38} +8.33813 q^{39} +1.00000 q^{40} -2.28644 q^{41} +2.67544 q^{42} -3.78866 q^{43} -4.15798 q^{45} -4.63400 q^{46} -0.981558 q^{47} -2.67544 q^{48} +1.00000 q^{49} -1.00000 q^{50} +1.51052 q^{51} -3.11654 q^{52} -1.14198 q^{53} +3.09812 q^{54} -1.00000 q^{56} -20.1301 q^{57} -6.24003 q^{58} +8.79892 q^{59} +2.67544 q^{60} +11.9594 q^{61} -2.72257 q^{62} +4.15798 q^{63} +1.00000 q^{64} +3.11654 q^{65} -6.80218 q^{67} -0.564586 q^{68} -12.3980 q^{69} +1.00000 q^{70} +11.4260 q^{71} -4.15798 q^{72} +13.2594 q^{73} +0.240027 q^{74} -2.67544 q^{75} +7.52404 q^{76} -8.33813 q^{78} -15.3694 q^{79} -1.00000 q^{80} -4.18512 q^{81} +2.28644 q^{82} +10.5208 q^{83} -2.67544 q^{84} +0.564586 q^{85} +3.78866 q^{86} -16.6948 q^{87} -4.33244 q^{89} +4.15798 q^{90} -3.11654 q^{91} +4.63400 q^{92} -7.28408 q^{93} +0.981558 q^{94} -7.52404 q^{95} +2.67544 q^{96} -9.05134 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{2} + 6q^{4} - 6q^{5} + 6q^{7} - 6q^{8} + 18q^{9} + O(q^{10}) \) \( 6q - 6q^{2} + 6q^{4} - 6q^{5} + 6q^{7} - 6q^{8} + 18q^{9} + 6q^{10} - 6q^{14} + 6q^{16} + 6q^{17} - 18q^{18} - 6q^{20} + 6q^{25} - 12q^{27} + 6q^{28} + 12q^{29} - 6q^{32} - 6q^{34} - 6q^{35} + 18q^{36} + 24q^{37} - 24q^{39} + 6q^{40} + 12q^{41} - 18q^{43} - 18q^{45} + 24q^{47} + 6q^{49} - 6q^{50} - 12q^{51} + 36q^{53} + 12q^{54} - 6q^{56} - 12q^{57} - 12q^{58} + 30q^{59} + 36q^{61} + 18q^{63} + 6q^{64} - 12q^{67} + 6q^{68} + 6q^{70} + 6q^{71} - 18q^{72} - 6q^{73} - 24q^{74} + 24q^{78} - 24q^{79} - 6q^{80} + 54q^{81} - 12q^{82} + 24q^{83} - 6q^{85} + 18q^{86} - 24q^{87} + 36q^{89} + 18q^{90} - 24q^{94} - 6q^{98} + O(q^{100}) \)

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
\(3\) −2.67544 −1.54467 −0.772333 0.635217i \(-0.780910\pi\)
−0.772333 + 0.635217i \(0.780910\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 2.67544 1.09224
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 4.15798 1.38599
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) −2.67544 −0.772333
\(13\) −3.11654 −0.864374 −0.432187 0.901784i \(-0.642258\pi\)
−0.432187 + 0.901784i \(0.642258\pi\)
\(14\) −1.00000 −0.267261
\(15\) 2.67544 0.690796
\(16\) 1.00000 0.250000
\(17\) −0.564586 −0.136932 −0.0684661 0.997653i \(-0.521810\pi\)
−0.0684661 + 0.997653i \(0.521810\pi\)
\(18\) −4.15798 −0.980046
\(19\) 7.52404 1.72613 0.863066 0.505091i \(-0.168541\pi\)
0.863066 + 0.505091i \(0.168541\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.67544 −0.583829
\(22\) 0 0
\(23\) 4.63400 0.966256 0.483128 0.875550i \(-0.339501\pi\)
0.483128 + 0.875550i \(0.339501\pi\)
\(24\) 2.67544 0.546122
\(25\) 1.00000 0.200000
\(26\) 3.11654 0.611204
\(27\) −3.09812 −0.596233
\(28\) 1.00000 0.188982
\(29\) 6.24003 1.15874 0.579372 0.815063i \(-0.303298\pi\)
0.579372 + 0.815063i \(0.303298\pi\)
\(30\) −2.67544 −0.488466
\(31\) 2.72257 0.488988 0.244494 0.969651i \(-0.421378\pi\)
0.244494 + 0.969651i \(0.421378\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0.564586 0.0968257
\(35\) −1.00000 −0.169031
\(36\) 4.15798 0.692997
\(37\) −0.240027 −0.0394602 −0.0197301 0.999805i \(-0.506281\pi\)
−0.0197301 + 0.999805i \(0.506281\pi\)
\(38\) −7.52404 −1.22056
\(39\) 8.33813 1.33517
\(40\) 1.00000 0.158114
\(41\) −2.28644 −0.357082 −0.178541 0.983932i \(-0.557138\pi\)
−0.178541 + 0.983932i \(0.557138\pi\)
\(42\) 2.67544 0.412830
\(43\) −3.78866 −0.577765 −0.288883 0.957365i \(-0.593284\pi\)
−0.288883 + 0.957365i \(0.593284\pi\)
\(44\) 0 0
\(45\) −4.15798 −0.619836
\(46\) −4.63400 −0.683246
\(47\) −0.981558 −0.143175 −0.0715875 0.997434i \(-0.522807\pi\)
−0.0715875 + 0.997434i \(0.522807\pi\)
\(48\) −2.67544 −0.386167
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 1.51052 0.211515
\(52\) −3.11654 −0.432187
\(53\) −1.14198 −0.156862 −0.0784312 0.996920i \(-0.524991\pi\)
−0.0784312 + 0.996920i \(0.524991\pi\)
\(54\) 3.09812 0.421601
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −20.1301 −2.66630
\(58\) −6.24003 −0.819356
\(59\) 8.79892 1.14552 0.572761 0.819722i \(-0.305872\pi\)
0.572761 + 0.819722i \(0.305872\pi\)
\(60\) 2.67544 0.345398
\(61\) 11.9594 1.53125 0.765625 0.643287i \(-0.222430\pi\)
0.765625 + 0.643287i \(0.222430\pi\)
\(62\) −2.72257 −0.345767
\(63\) 4.15798 0.523857
\(64\) 1.00000 0.125000
\(65\) 3.11654 0.386560
\(66\) 0 0
\(67\) −6.80218 −0.831018 −0.415509 0.909589i \(-0.636397\pi\)
−0.415509 + 0.909589i \(0.636397\pi\)
\(68\) −0.564586 −0.0684661
\(69\) −12.3980 −1.49254
\(70\) 1.00000 0.119523
\(71\) 11.4260 1.35601 0.678007 0.735055i \(-0.262844\pi\)
0.678007 + 0.735055i \(0.262844\pi\)
\(72\) −4.15798 −0.490023
\(73\) 13.2594 1.55190 0.775948 0.630797i \(-0.217272\pi\)
0.775948 + 0.630797i \(0.217272\pi\)
\(74\) 0.240027 0.0279026
\(75\) −2.67544 −0.308933
\(76\) 7.52404 0.863066
\(77\) 0 0
\(78\) −8.33813 −0.944107
\(79\) −15.3694 −1.72919 −0.864595 0.502470i \(-0.832425\pi\)
−0.864595 + 0.502470i \(0.832425\pi\)
\(80\) −1.00000 −0.111803
\(81\) −4.18512 −0.465013
\(82\) 2.28644 0.252495
\(83\) 10.5208 1.15481 0.577403 0.816459i \(-0.304066\pi\)
0.577403 + 0.816459i \(0.304066\pi\)
\(84\) −2.67544 −0.291915
\(85\) 0.564586 0.0612379
\(86\) 3.78866 0.408542
\(87\) −16.6948 −1.78987
\(88\) 0 0
\(89\) −4.33244 −0.459238 −0.229619 0.973281i \(-0.573748\pi\)
−0.229619 + 0.973281i \(0.573748\pi\)
\(90\) 4.15798 0.438290
\(91\) −3.11654 −0.326703
\(92\) 4.63400 0.483128
\(93\) −7.28408 −0.755324
\(94\) 0.981558 0.101240
\(95\) −7.52404 −0.771950
\(96\) 2.67544 0.273061
\(97\) −9.05134 −0.919025 −0.459512 0.888171i \(-0.651976\pi\)
−0.459512 + 0.888171i \(0.651976\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 5.47395 0.544679 0.272339 0.962201i \(-0.412203\pi\)
0.272339 + 0.962201i \(0.412203\pi\)
\(102\) −1.51052 −0.149563
\(103\) 12.5991 1.24143 0.620715 0.784036i \(-0.286842\pi\)
0.620715 + 0.784036i \(0.286842\pi\)
\(104\) 3.11654 0.305602
\(105\) 2.67544 0.261096
\(106\) 1.14198 0.110919
\(107\) 9.86211 0.953406 0.476703 0.879064i \(-0.341832\pi\)
0.476703 + 0.879064i \(0.341832\pi\)
\(108\) −3.09812 −0.298117
\(109\) −14.9188 −1.42896 −0.714480 0.699656i \(-0.753337\pi\)
−0.714480 + 0.699656i \(0.753337\pi\)
\(110\) 0 0
\(111\) 0.642178 0.0609528
\(112\) 1.00000 0.0944911
\(113\) −1.87415 −0.176305 −0.0881527 0.996107i \(-0.528096\pi\)
−0.0881527 + 0.996107i \(0.528096\pi\)
\(114\) 20.1301 1.88536
\(115\) −4.63400 −0.432123
\(116\) 6.24003 0.579372
\(117\) −12.9585 −1.19802
\(118\) −8.79892 −0.810007
\(119\) −0.564586 −0.0517555
\(120\) −2.67544 −0.244233
\(121\) 0 0
\(122\) −11.9594 −1.08276
\(123\) 6.11724 0.551573
\(124\) 2.72257 0.244494
\(125\) −1.00000 −0.0894427
\(126\) −4.15798 −0.370423
\(127\) −13.7604 −1.22104 −0.610519 0.792002i \(-0.709039\pi\)
−0.610519 + 0.792002i \(0.709039\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 10.1363 0.892455
\(130\) −3.11654 −0.273339
\(131\) 8.75625 0.765037 0.382519 0.923948i \(-0.375057\pi\)
0.382519 + 0.923948i \(0.375057\pi\)
\(132\) 0 0
\(133\) 7.52404 0.652417
\(134\) 6.80218 0.587619
\(135\) 3.09812 0.266644
\(136\) 0.564586 0.0484128
\(137\) −3.81831 −0.326220 −0.163110 0.986608i \(-0.552153\pi\)
−0.163110 + 0.986608i \(0.552153\pi\)
\(138\) 12.3980 1.05539
\(139\) −0.0485569 −0.00411855 −0.00205927 0.999998i \(-0.500655\pi\)
−0.00205927 + 0.999998i \(0.500655\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 2.62610 0.221158
\(142\) −11.4260 −0.958847
\(143\) 0 0
\(144\) 4.15798 0.346499
\(145\) −6.24003 −0.518206
\(146\) −13.2594 −1.09736
\(147\) −2.67544 −0.220667
\(148\) −0.240027 −0.0197301
\(149\) 10.5863 0.867266 0.433633 0.901090i \(-0.357231\pi\)
0.433633 + 0.901090i \(0.357231\pi\)
\(150\) 2.67544 0.218449
\(151\) −10.5581 −0.859203 −0.429601 0.903019i \(-0.641346\pi\)
−0.429601 + 0.903019i \(0.641346\pi\)
\(152\) −7.52404 −0.610280
\(153\) −2.34754 −0.189787
\(154\) 0 0
\(155\) −2.72257 −0.218682
\(156\) 8.33813 0.667585
\(157\) 20.5831 1.64271 0.821356 0.570416i \(-0.193218\pi\)
0.821356 + 0.570416i \(0.193218\pi\)
\(158\) 15.3694 1.22272
\(159\) 3.05529 0.242300
\(160\) 1.00000 0.0790569
\(161\) 4.63400 0.365210
\(162\) 4.18512 0.328814
\(163\) −19.3407 −1.51488 −0.757439 0.652906i \(-0.773550\pi\)
−0.757439 + 0.652906i \(0.773550\pi\)
\(164\) −2.28644 −0.178541
\(165\) 0 0
\(166\) −10.5208 −0.816571
\(167\) 7.83259 0.606104 0.303052 0.952974i \(-0.401994\pi\)
0.303052 + 0.952974i \(0.401994\pi\)
\(168\) 2.67544 0.206415
\(169\) −3.28716 −0.252858
\(170\) −0.564586 −0.0433018
\(171\) 31.2848 2.39241
\(172\) −3.78866 −0.288883
\(173\) −12.1772 −0.925819 −0.462909 0.886406i \(-0.653194\pi\)
−0.462909 + 0.886406i \(0.653194\pi\)
\(174\) 16.6948 1.26563
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −23.5410 −1.76945
\(178\) 4.33244 0.324730
\(179\) 1.01844 0.0761219 0.0380610 0.999275i \(-0.487882\pi\)
0.0380610 + 0.999275i \(0.487882\pi\)
\(180\) −4.15798 −0.309918
\(181\) 1.18721 0.0882445 0.0441223 0.999026i \(-0.485951\pi\)
0.0441223 + 0.999026i \(0.485951\pi\)
\(182\) 3.11654 0.231014
\(183\) −31.9968 −2.36527
\(184\) −4.63400 −0.341623
\(185\) 0.240027 0.0176471
\(186\) 7.28408 0.534094
\(187\) 0 0
\(188\) −0.981558 −0.0715875
\(189\) −3.09812 −0.225355
\(190\) 7.52404 0.545851
\(191\) 16.9155 1.22396 0.611980 0.790873i \(-0.290373\pi\)
0.611980 + 0.790873i \(0.290373\pi\)
\(192\) −2.67544 −0.193083
\(193\) 15.2274 1.09609 0.548045 0.836449i \(-0.315372\pi\)
0.548045 + 0.836449i \(0.315372\pi\)
\(194\) 9.05134 0.649849
\(195\) −8.33813 −0.597106
\(196\) 1.00000 0.0714286
\(197\) −2.83965 −0.202317 −0.101158 0.994870i \(-0.532255\pi\)
−0.101158 + 0.994870i \(0.532255\pi\)
\(198\) 0 0
\(199\) 4.77593 0.338556 0.169278 0.985568i \(-0.445856\pi\)
0.169278 + 0.985568i \(0.445856\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 18.1988 1.28365
\(202\) −5.47395 −0.385146
\(203\) 6.24003 0.437964
\(204\) 1.51052 0.105757
\(205\) 2.28644 0.159692
\(206\) −12.5991 −0.877824
\(207\) 19.2681 1.33923
\(208\) −3.11654 −0.216093
\(209\) 0 0
\(210\) −2.67544 −0.184623
\(211\) −26.0899 −1.79610 −0.898052 0.439889i \(-0.855018\pi\)
−0.898052 + 0.439889i \(0.855018\pi\)
\(212\) −1.14198 −0.0784312
\(213\) −30.5695 −2.09459
\(214\) −9.86211 −0.674160
\(215\) 3.78866 0.258385
\(216\) 3.09812 0.210800
\(217\) 2.72257 0.184820
\(218\) 14.9188 1.01043
\(219\) −35.4748 −2.39716
\(220\) 0 0
\(221\) 1.75956 0.118361
\(222\) −0.642178 −0.0431002
\(223\) 8.25283 0.552650 0.276325 0.961064i \(-0.410883\pi\)
0.276325 + 0.961064i \(0.410883\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 4.15798 0.277199
\(226\) 1.87415 0.124667
\(227\) −5.55836 −0.368921 −0.184461 0.982840i \(-0.559054\pi\)
−0.184461 + 0.982840i \(0.559054\pi\)
\(228\) −20.1301 −1.33315
\(229\) −5.96188 −0.393972 −0.196986 0.980406i \(-0.563115\pi\)
−0.196986 + 0.980406i \(0.563115\pi\)
\(230\) 4.63400 0.305557
\(231\) 0 0
\(232\) −6.24003 −0.409678
\(233\) 2.50193 0.163907 0.0819533 0.996636i \(-0.473884\pi\)
0.0819533 + 0.996636i \(0.473884\pi\)
\(234\) 12.9585 0.847126
\(235\) 0.981558 0.0640298
\(236\) 8.79892 0.572761
\(237\) 41.1199 2.67102
\(238\) 0.564586 0.0365967
\(239\) −28.8313 −1.86494 −0.932472 0.361242i \(-0.882353\pi\)
−0.932472 + 0.361242i \(0.882353\pi\)
\(240\) 2.67544 0.172699
\(241\) −1.67379 −0.107818 −0.0539092 0.998546i \(-0.517168\pi\)
−0.0539092 + 0.998546i \(0.517168\pi\)
\(242\) 0 0
\(243\) 20.4914 1.31452
\(244\) 11.9594 0.765625
\(245\) −1.00000 −0.0638877
\(246\) −6.11724 −0.390021
\(247\) −23.4490 −1.49202
\(248\) −2.72257 −0.172883
\(249\) −28.1477 −1.78379
\(250\) 1.00000 0.0632456
\(251\) −7.91255 −0.499436 −0.249718 0.968319i \(-0.580338\pi\)
−0.249718 + 0.968319i \(0.580338\pi\)
\(252\) 4.15798 0.261928
\(253\) 0 0
\(254\) 13.7604 0.863404
\(255\) −1.51052 −0.0945922
\(256\) 1.00000 0.0625000
\(257\) −9.58983 −0.598197 −0.299099 0.954222i \(-0.596686\pi\)
−0.299099 + 0.954222i \(0.596686\pi\)
\(258\) −10.1363 −0.631061
\(259\) −0.240027 −0.0149145
\(260\) 3.11654 0.193280
\(261\) 25.9459 1.60601
\(262\) −8.75625 −0.540963
\(263\) −23.6980 −1.46128 −0.730642 0.682761i \(-0.760779\pi\)
−0.730642 + 0.682761i \(0.760779\pi\)
\(264\) 0 0
\(265\) 1.14198 0.0701510
\(266\) −7.52404 −0.461328
\(267\) 11.5912 0.709369
\(268\) −6.80218 −0.415509
\(269\) 3.41365 0.208134 0.104067 0.994570i \(-0.466814\pi\)
0.104067 + 0.994570i \(0.466814\pi\)
\(270\) −3.09812 −0.188546
\(271\) −23.1613 −1.40695 −0.703474 0.710721i \(-0.748369\pi\)
−0.703474 + 0.710721i \(0.748369\pi\)
\(272\) −0.564586 −0.0342331
\(273\) 8.33813 0.504646
\(274\) 3.81831 0.230672
\(275\) 0 0
\(276\) −12.3980 −0.746271
\(277\) 4.39966 0.264350 0.132175 0.991226i \(-0.457804\pi\)
0.132175 + 0.991226i \(0.457804\pi\)
\(278\) 0.0485569 0.00291225
\(279\) 11.3204 0.677735
\(280\) 1.00000 0.0597614
\(281\) −13.5746 −0.809794 −0.404897 0.914362i \(-0.632693\pi\)
−0.404897 + 0.914362i \(0.632693\pi\)
\(282\) −2.62610 −0.156382
\(283\) −23.5943 −1.40254 −0.701268 0.712897i \(-0.747383\pi\)
−0.701268 + 0.712897i \(0.747383\pi\)
\(284\) 11.4260 0.678007
\(285\) 20.1301 1.19240
\(286\) 0 0
\(287\) −2.28644 −0.134964
\(288\) −4.15798 −0.245012
\(289\) −16.6812 −0.981250
\(290\) 6.24003 0.366427
\(291\) 24.2163 1.41959
\(292\) 13.2594 0.775948
\(293\) −30.2362 −1.76642 −0.883209 0.468979i \(-0.844622\pi\)
−0.883209 + 0.468979i \(0.844622\pi\)
\(294\) 2.67544 0.156035
\(295\) −8.79892 −0.512293
\(296\) 0.240027 0.0139513
\(297\) 0 0
\(298\) −10.5863 −0.613250
\(299\) −14.4421 −0.835206
\(300\) −2.67544 −0.154467
\(301\) −3.78866 −0.218375
\(302\) 10.5581 0.607548
\(303\) −14.6452 −0.841347
\(304\) 7.52404 0.431533
\(305\) −11.9594 −0.684796
\(306\) 2.34754 0.134200
\(307\) 5.66685 0.323424 0.161712 0.986838i \(-0.448298\pi\)
0.161712 + 0.986838i \(0.448298\pi\)
\(308\) 0 0
\(309\) −33.7082 −1.91760
\(310\) 2.72257 0.154632
\(311\) 29.7071 1.68454 0.842269 0.539058i \(-0.181219\pi\)
0.842269 + 0.539058i \(0.181219\pi\)
\(312\) −8.33813 −0.472054
\(313\) −7.84976 −0.443695 −0.221847 0.975081i \(-0.571209\pi\)
−0.221847 + 0.975081i \(0.571209\pi\)
\(314\) −20.5831 −1.16157
\(315\) −4.15798 −0.234276
\(316\) −15.3694 −0.864595
\(317\) 6.99148 0.392681 0.196340 0.980536i \(-0.437094\pi\)
0.196340 + 0.980536i \(0.437094\pi\)
\(318\) −3.05529 −0.171332
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −26.3855 −1.47270
\(322\) −4.63400 −0.258243
\(323\) −4.24796 −0.236363
\(324\) −4.18512 −0.232507
\(325\) −3.11654 −0.172875
\(326\) 19.3407 1.07118
\(327\) 39.9143 2.20727
\(328\) 2.28644 0.126248
\(329\) −0.981558 −0.0541150
\(330\) 0 0
\(331\) 25.4265 1.39757 0.698784 0.715333i \(-0.253725\pi\)
0.698784 + 0.715333i \(0.253725\pi\)
\(332\) 10.5208 0.577403
\(333\) −0.998028 −0.0546916
\(334\) −7.83259 −0.428580
\(335\) 6.80218 0.371643
\(336\) −2.67544 −0.145957
\(337\) −15.5179 −0.845313 −0.422656 0.906290i \(-0.638902\pi\)
−0.422656 + 0.906290i \(0.638902\pi\)
\(338\) 3.28716 0.178798
\(339\) 5.01418 0.272333
\(340\) 0.564586 0.0306190
\(341\) 0 0
\(342\) −31.2848 −1.69169
\(343\) 1.00000 0.0539949
\(344\) 3.78866 0.204271
\(345\) 12.3980 0.667486
\(346\) 12.1772 0.654653
\(347\) 12.5627 0.674403 0.337202 0.941432i \(-0.390520\pi\)
0.337202 + 0.941432i \(0.390520\pi\)
\(348\) −16.6948 −0.894937
\(349\) 20.3193 1.08767 0.543833 0.839194i \(-0.316973\pi\)
0.543833 + 0.839194i \(0.316973\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 9.65543 0.515368
\(352\) 0 0
\(353\) 32.3745 1.72312 0.861560 0.507656i \(-0.169488\pi\)
0.861560 + 0.507656i \(0.169488\pi\)
\(354\) 23.5410 1.25119
\(355\) −11.4260 −0.606428
\(356\) −4.33244 −0.229619
\(357\) 1.51052 0.0799450
\(358\) −1.01844 −0.0538263
\(359\) 23.2783 1.22858 0.614291 0.789079i \(-0.289442\pi\)
0.614291 + 0.789079i \(0.289442\pi\)
\(360\) 4.15798 0.219145
\(361\) 37.6111 1.97953
\(362\) −1.18721 −0.0623983
\(363\) 0 0
\(364\) −3.11654 −0.163351
\(365\) −13.2594 −0.694029
\(366\) 31.9968 1.67250
\(367\) −4.14641 −0.216441 −0.108221 0.994127i \(-0.534515\pi\)
−0.108221 + 0.994127i \(0.534515\pi\)
\(368\) 4.63400 0.241564
\(369\) −9.50699 −0.494914
\(370\) −0.240027 −0.0124784
\(371\) −1.14198 −0.0592884
\(372\) −7.28408 −0.377662
\(373\) −21.9641 −1.13726 −0.568628 0.822595i \(-0.692526\pi\)
−0.568628 + 0.822595i \(0.692526\pi\)
\(374\) 0 0
\(375\) 2.67544 0.138159
\(376\) 0.981558 0.0506200
\(377\) −19.4473 −1.00159
\(378\) 3.09812 0.159350
\(379\) 20.9655 1.07692 0.538462 0.842650i \(-0.319005\pi\)
0.538462 + 0.842650i \(0.319005\pi\)
\(380\) −7.52404 −0.385975
\(381\) 36.8151 1.88610
\(382\) −16.9155 −0.865471
\(383\) −36.1904 −1.84925 −0.924623 0.380885i \(-0.875619\pi\)
−0.924623 + 0.380885i \(0.875619\pi\)
\(384\) 2.67544 0.136531
\(385\) 0 0
\(386\) −15.2274 −0.775053
\(387\) −15.7532 −0.800780
\(388\) −9.05134 −0.459512
\(389\) 27.3801 1.38823 0.694113 0.719866i \(-0.255797\pi\)
0.694113 + 0.719866i \(0.255797\pi\)
\(390\) 8.33813 0.422218
\(391\) −2.61629 −0.132312
\(392\) −1.00000 −0.0505076
\(393\) −23.4268 −1.18173
\(394\) 2.83965 0.143060
\(395\) 15.3694 0.773317
\(396\) 0 0
\(397\) −2.72549 −0.136788 −0.0683942 0.997658i \(-0.521788\pi\)
−0.0683942 + 0.997658i \(0.521788\pi\)
\(398\) −4.77593 −0.239396
\(399\) −20.1301 −1.00777
\(400\) 1.00000 0.0500000
\(401\) 26.3810 1.31741 0.658703 0.752403i \(-0.271105\pi\)
0.658703 + 0.752403i \(0.271105\pi\)
\(402\) −18.1988 −0.907675
\(403\) −8.48501 −0.422668
\(404\) 5.47395 0.272339
\(405\) 4.18512 0.207960
\(406\) −6.24003 −0.309687
\(407\) 0 0
\(408\) −1.51052 −0.0747817
\(409\) 31.2447 1.54495 0.772476 0.635044i \(-0.219018\pi\)
0.772476 + 0.635044i \(0.219018\pi\)
\(410\) −2.28644 −0.112919
\(411\) 10.2157 0.503901
\(412\) 12.5991 0.620715
\(413\) 8.79892 0.432967
\(414\) −19.2681 −0.946975
\(415\) −10.5208 −0.516445
\(416\) 3.11654 0.152801
\(417\) 0.129911 0.00636178
\(418\) 0 0
\(419\) 0.716042 0.0349809 0.0174905 0.999847i \(-0.494432\pi\)
0.0174905 + 0.999847i \(0.494432\pi\)
\(420\) 2.67544 0.130548
\(421\) 28.3733 1.38283 0.691416 0.722457i \(-0.256987\pi\)
0.691416 + 0.722457i \(0.256987\pi\)
\(422\) 26.0899 1.27004
\(423\) −4.08130 −0.198440
\(424\) 1.14198 0.0554593
\(425\) −0.564586 −0.0273864
\(426\) 30.5695 1.48110
\(427\) 11.9594 0.578758
\(428\) 9.86211 0.476703
\(429\) 0 0
\(430\) −3.78866 −0.182705
\(431\) −15.8511 −0.763519 −0.381760 0.924262i \(-0.624682\pi\)
−0.381760 + 0.924262i \(0.624682\pi\)
\(432\) −3.09812 −0.149058
\(433\) 22.2137 1.06752 0.533761 0.845635i \(-0.320778\pi\)
0.533761 + 0.845635i \(0.320778\pi\)
\(434\) −2.72257 −0.130688
\(435\) 16.6948 0.800456
\(436\) −14.9188 −0.714480
\(437\) 34.8664 1.66789
\(438\) 35.4748 1.69505
\(439\) −16.3478 −0.780238 −0.390119 0.920764i \(-0.627566\pi\)
−0.390119 + 0.920764i \(0.627566\pi\)
\(440\) 0 0
\(441\) 4.15798 0.197999
\(442\) −1.75956 −0.0836936
\(443\) 28.6492 1.36117 0.680583 0.732671i \(-0.261727\pi\)
0.680583 + 0.732671i \(0.261727\pi\)
\(444\) 0.642178 0.0304764
\(445\) 4.33244 0.205377
\(446\) −8.25283 −0.390783
\(447\) −28.3231 −1.33964
\(448\) 1.00000 0.0472456
\(449\) 23.2905 1.09914 0.549572 0.835446i \(-0.314791\pi\)
0.549572 + 0.835446i \(0.314791\pi\)
\(450\) −4.15798 −0.196009
\(451\) 0 0
\(452\) −1.87415 −0.0881527
\(453\) 28.2475 1.32718
\(454\) 5.55836 0.260867
\(455\) 3.11654 0.146106
\(456\) 20.1301 0.942679
\(457\) 10.3649 0.484849 0.242424 0.970170i \(-0.422057\pi\)
0.242424 + 0.970170i \(0.422057\pi\)
\(458\) 5.96188 0.278580
\(459\) 1.74915 0.0816436
\(460\) −4.63400 −0.216061
\(461\) −0.698098 −0.0325137 −0.0162568 0.999868i \(-0.505175\pi\)
−0.0162568 + 0.999868i \(0.505175\pi\)
\(462\) 0 0
\(463\) 10.4790 0.487002 0.243501 0.969901i \(-0.421704\pi\)
0.243501 + 0.969901i \(0.421704\pi\)
\(464\) 6.24003 0.289686
\(465\) 7.28408 0.337791
\(466\) −2.50193 −0.115900
\(467\) −39.3583 −1.82129 −0.910644 0.413193i \(-0.864414\pi\)
−0.910644 + 0.413193i \(0.864414\pi\)
\(468\) −12.9585 −0.599009
\(469\) −6.80218 −0.314095
\(470\) −0.981558 −0.0452759
\(471\) −55.0689 −2.53744
\(472\) −8.79892 −0.405003
\(473\) 0 0
\(474\) −41.1199 −1.88870
\(475\) 7.52404 0.345226
\(476\) −0.564586 −0.0258778
\(477\) −4.74832 −0.217411
\(478\) 28.8313 1.31871
\(479\) −27.4511 −1.25427 −0.627136 0.778910i \(-0.715773\pi\)
−0.627136 + 0.778910i \(0.715773\pi\)
\(480\) −2.67544 −0.122117
\(481\) 0.748054 0.0341083
\(482\) 1.67379 0.0762391
\(483\) −12.3980 −0.564128
\(484\) 0 0
\(485\) 9.05134 0.411000
\(486\) −20.4914 −0.929508
\(487\) −36.5980 −1.65841 −0.829206 0.558943i \(-0.811207\pi\)
−0.829206 + 0.558943i \(0.811207\pi\)
\(488\) −11.9594 −0.541379
\(489\) 51.7448 2.33998
\(490\) 1.00000 0.0451754
\(491\) 3.39001 0.152989 0.0764944 0.997070i \(-0.475627\pi\)
0.0764944 + 0.997070i \(0.475627\pi\)
\(492\) 6.11724 0.275786
\(493\) −3.52303 −0.158669
\(494\) 23.4490 1.05502
\(495\) 0 0
\(496\) 2.72257 0.122247
\(497\) 11.4260 0.512525
\(498\) 28.1477 1.26133
\(499\) 22.9786 1.02867 0.514333 0.857591i \(-0.328040\pi\)
0.514333 + 0.857591i \(0.328040\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −20.9556 −0.936228
\(502\) 7.91255 0.353154
\(503\) 31.6613 1.41171 0.705854 0.708357i \(-0.250563\pi\)
0.705854 + 0.708357i \(0.250563\pi\)
\(504\) −4.15798 −0.185211
\(505\) −5.47395 −0.243588
\(506\) 0 0
\(507\) 8.79459 0.390582
\(508\) −13.7604 −0.610519
\(509\) 0.742359 0.0329045 0.0164522 0.999865i \(-0.494763\pi\)
0.0164522 + 0.999865i \(0.494763\pi\)
\(510\) 1.51052 0.0668868
\(511\) 13.2594 0.586562
\(512\) −1.00000 −0.0441942
\(513\) −23.3104 −1.02918
\(514\) 9.58983 0.422989
\(515\) −12.5991 −0.555184
\(516\) 10.1363 0.446227
\(517\) 0 0
\(518\) 0.240027 0.0105462
\(519\) 32.5795 1.43008
\(520\) −3.11654 −0.136669
\(521\) 20.2911 0.888969 0.444485 0.895786i \(-0.353387\pi\)
0.444485 + 0.895786i \(0.353387\pi\)
\(522\) −25.9459 −1.13562
\(523\) −31.4418 −1.37485 −0.687426 0.726255i \(-0.741259\pi\)
−0.687426 + 0.726255i \(0.741259\pi\)
\(524\) 8.75625 0.382519
\(525\) −2.67544 −0.116766
\(526\) 23.6980 1.03328
\(527\) −1.53713 −0.0669582
\(528\) 0 0
\(529\) −1.52605 −0.0663498
\(530\) −1.14198 −0.0496043
\(531\) 36.5858 1.58769
\(532\) 7.52404 0.326208
\(533\) 7.12580 0.308652
\(534\) −11.5912 −0.501600
\(535\) −9.86211 −0.426376
\(536\) 6.80218 0.293809
\(537\) −2.72478 −0.117583
\(538\) −3.41365 −0.147173
\(539\) 0 0
\(540\) 3.09812 0.133322
\(541\) −3.03964 −0.130684 −0.0653422 0.997863i \(-0.520814\pi\)
−0.0653422 + 0.997863i \(0.520814\pi\)
\(542\) 23.1613 0.994863
\(543\) −3.17631 −0.136308
\(544\) 0.564586 0.0242064
\(545\) 14.9188 0.639050
\(546\) −8.33813 −0.356839
\(547\) 18.1983 0.778102 0.389051 0.921216i \(-0.372803\pi\)
0.389051 + 0.921216i \(0.372803\pi\)
\(548\) −3.81831 −0.163110
\(549\) 49.7272 2.12231
\(550\) 0 0
\(551\) 46.9502 2.00015
\(552\) 12.3980 0.527694
\(553\) −15.3694 −0.653572
\(554\) −4.39966 −0.186924
\(555\) −0.642178 −0.0272589
\(556\) −0.0485569 −0.00205927
\(557\) 12.1608 0.515268 0.257634 0.966243i \(-0.417057\pi\)
0.257634 + 0.966243i \(0.417057\pi\)
\(558\) −11.3204 −0.479231
\(559\) 11.8075 0.499405
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 13.5746 0.572611
\(563\) 38.4298 1.61962 0.809812 0.586689i \(-0.199569\pi\)
0.809812 + 0.586689i \(0.199569\pi\)
\(564\) 2.62610 0.110579
\(565\) 1.87415 0.0788462
\(566\) 23.5943 0.991743
\(567\) −4.18512 −0.175758
\(568\) −11.4260 −0.479424
\(569\) −19.7245 −0.826893 −0.413446 0.910528i \(-0.635675\pi\)
−0.413446 + 0.910528i \(0.635675\pi\)
\(570\) −20.1301 −0.843158
\(571\) 25.4847 1.06650 0.533251 0.845957i \(-0.320970\pi\)
0.533251 + 0.845957i \(0.320970\pi\)
\(572\) 0 0
\(573\) −45.2563 −1.89061
\(574\) 2.28644 0.0954342
\(575\) 4.63400 0.193251
\(576\) 4.15798 0.173249
\(577\) −27.2044 −1.13254 −0.566268 0.824221i \(-0.691613\pi\)
−0.566268 + 0.824221i \(0.691613\pi\)
\(578\) 16.6812 0.693848
\(579\) −40.7400 −1.69309
\(580\) −6.24003 −0.259103
\(581\) 10.5208 0.436476
\(582\) −24.2163 −1.00380
\(583\) 0 0
\(584\) −13.2594 −0.548678
\(585\) 12.9585 0.535770
\(586\) 30.2362 1.24905
\(587\) 28.3154 1.16870 0.584350 0.811502i \(-0.301350\pi\)
0.584350 + 0.811502i \(0.301350\pi\)
\(588\) −2.67544 −0.110333
\(589\) 20.4847 0.844058
\(590\) 8.79892 0.362246
\(591\) 7.59732 0.312512
\(592\) −0.240027 −0.00986505
\(593\) −0.594543 −0.0244150 −0.0122075 0.999925i \(-0.503886\pi\)
−0.0122075 + 0.999925i \(0.503886\pi\)
\(594\) 0 0
\(595\) 0.564586 0.0231458
\(596\) 10.5863 0.433633
\(597\) −12.7777 −0.522957
\(598\) 14.4421 0.590580
\(599\) 5.55684 0.227046 0.113523 0.993535i \(-0.463786\pi\)
0.113523 + 0.993535i \(0.463786\pi\)
\(600\) 2.67544 0.109224
\(601\) 3.76921 0.153749 0.0768747 0.997041i \(-0.475506\pi\)
0.0768747 + 0.997041i \(0.475506\pi\)
\(602\) 3.78866 0.154414
\(603\) −28.2834 −1.15179
\(604\) −10.5581 −0.429601
\(605\) 0 0
\(606\) 14.6452 0.594922
\(607\) 20.3035 0.824092 0.412046 0.911163i \(-0.364814\pi\)
0.412046 + 0.911163i \(0.364814\pi\)
\(608\) −7.52404 −0.305140
\(609\) −16.6948 −0.676508
\(610\) 11.9594 0.484224
\(611\) 3.05907 0.123757
\(612\) −2.34754 −0.0948937
\(613\) 36.3948 1.46997 0.734986 0.678082i \(-0.237189\pi\)
0.734986 + 0.678082i \(0.237189\pi\)
\(614\) −5.66685 −0.228696
\(615\) −6.11724 −0.246671
\(616\) 0 0
\(617\) 25.2056 1.01474 0.507371 0.861728i \(-0.330618\pi\)
0.507371 + 0.861728i \(0.330618\pi\)
\(618\) 33.7082 1.35594
\(619\) 42.8248 1.72128 0.860638 0.509218i \(-0.170065\pi\)
0.860638 + 0.509218i \(0.170065\pi\)
\(620\) −2.72257 −0.109341
\(621\) −14.3567 −0.576114
\(622\) −29.7071 −1.19115
\(623\) −4.33244 −0.173576
\(624\) 8.33813 0.333792
\(625\) 1.00000 0.0400000
\(626\) 7.84976 0.313739
\(627\) 0 0
\(628\) 20.5831 0.821356
\(629\) 0.135516 0.00540337
\(630\) 4.15798 0.165658
\(631\) −10.9354 −0.435331 −0.217666 0.976023i \(-0.569844\pi\)
−0.217666 + 0.976023i \(0.569844\pi\)
\(632\) 15.3694 0.611361
\(633\) 69.8021 2.77438
\(634\) −6.99148 −0.277667
\(635\) 13.7604 0.546065
\(636\) 3.05529 0.121150
\(637\) −3.11654 −0.123482
\(638\) 0 0
\(639\) 47.5091 1.87943
\(640\) 1.00000 0.0395285
\(641\) 21.8420 0.862708 0.431354 0.902183i \(-0.358036\pi\)
0.431354 + 0.902183i \(0.358036\pi\)
\(642\) 26.3855 1.04135
\(643\) 0.551560 0.0217514 0.0108757 0.999941i \(-0.496538\pi\)
0.0108757 + 0.999941i \(0.496538\pi\)
\(644\) 4.63400 0.182605
\(645\) −10.1363 −0.399118
\(646\) 4.24796 0.167134
\(647\) 1.00700 0.0395894 0.0197947 0.999804i \(-0.493699\pi\)
0.0197947 + 0.999804i \(0.493699\pi\)
\(648\) 4.18512 0.164407
\(649\) 0 0
\(650\) 3.11654 0.122241
\(651\) −7.28408 −0.285485
\(652\) −19.3407 −0.757439
\(653\) −19.2293 −0.752500 −0.376250 0.926518i \(-0.622787\pi\)
−0.376250 + 0.926518i \(0.622787\pi\)
\(654\) −39.9143 −1.56077
\(655\) −8.75625 −0.342135
\(656\) −2.28644 −0.0892706
\(657\) 55.1324 2.15092
\(658\) 0.981558 0.0382651
\(659\) −2.36031 −0.0919447 −0.0459723 0.998943i \(-0.514639\pi\)
−0.0459723 + 0.998943i \(0.514639\pi\)
\(660\) 0 0
\(661\) 24.9127 0.968991 0.484495 0.874794i \(-0.339003\pi\)
0.484495 + 0.874794i \(0.339003\pi\)
\(662\) −25.4265 −0.988229
\(663\) −4.70759 −0.182828
\(664\) −10.5208 −0.408285
\(665\) −7.52404 −0.291770
\(666\) 0.998028 0.0386728
\(667\) 28.9163 1.11964
\(668\) 7.83259 0.303052
\(669\) −22.0800 −0.853661
\(670\) −6.80218 −0.262791
\(671\) 0 0
\(672\) 2.67544 0.103207
\(673\) −4.02852 −0.155288 −0.0776440 0.996981i \(-0.524740\pi\)
−0.0776440 + 0.996981i \(0.524740\pi\)
\(674\) 15.5179 0.597726
\(675\) −3.09812 −0.119247
\(676\) −3.28716 −0.126429
\(677\) 15.2440 0.585874 0.292937 0.956132i \(-0.405367\pi\)
0.292937 + 0.956132i \(0.405367\pi\)
\(678\) −5.01418 −0.192569
\(679\) −9.05134 −0.347359
\(680\) −0.564586 −0.0216509
\(681\) 14.8711 0.569861
\(682\) 0 0
\(683\) 33.8181 1.29401 0.647006 0.762485i \(-0.276021\pi\)
0.647006 + 0.762485i \(0.276021\pi\)
\(684\) 31.2848 1.19621
\(685\) 3.81831 0.145890
\(686\) −1.00000 −0.0381802
\(687\) 15.9507 0.608556
\(688\) −3.78866 −0.144441
\(689\) 3.55902 0.135588
\(690\) −12.3980 −0.471984
\(691\) 1.75430 0.0667368 0.0333684 0.999443i \(-0.489377\pi\)
0.0333684 + 0.999443i \(0.489377\pi\)
\(692\) −12.1772 −0.462909
\(693\) 0 0
\(694\) −12.5627 −0.476875
\(695\) 0.0485569 0.00184187
\(696\) 16.6948 0.632816
\(697\) 1.29089 0.0488961
\(698\) −20.3193 −0.769096
\(699\) −6.69376 −0.253181
\(700\) 1.00000 0.0377964
\(701\) 13.1026 0.494877 0.247438 0.968904i \(-0.420411\pi\)
0.247438 + 0.968904i \(0.420411\pi\)
\(702\) −9.65543 −0.364421
\(703\) −1.80597 −0.0681135
\(704\) 0 0
\(705\) −2.62610 −0.0989047
\(706\) −32.3745 −1.21843
\(707\) 5.47395 0.205869
\(708\) −23.5410 −0.884725
\(709\) −5.43312 −0.204045 −0.102023 0.994782i \(-0.532531\pi\)
−0.102023 + 0.994782i \(0.532531\pi\)
\(710\) 11.4260 0.428810
\(711\) −63.9056 −2.39665
\(712\) 4.33244 0.162365
\(713\) 12.6164 0.472488
\(714\) −1.51052 −0.0565297
\(715\) 0 0
\(716\) 1.01844 0.0380610
\(717\) 77.1366 2.88072
\(718\) −23.2783 −0.868739
\(719\) −48.9750 −1.82646 −0.913230 0.407445i \(-0.866420\pi\)
−0.913230 + 0.407445i \(0.866420\pi\)
\(720\) −4.15798 −0.154959
\(721\) 12.5991 0.469216
\(722\) −37.6111 −1.39974
\(723\) 4.47813 0.166543
\(724\) 1.18721 0.0441223
\(725\) 6.24003 0.231749
\(726\) 0 0
\(727\) 17.7501 0.658316 0.329158 0.944275i \(-0.393235\pi\)
0.329158 + 0.944275i \(0.393235\pi\)
\(728\) 3.11654 0.115507
\(729\) −42.2682 −1.56549
\(730\) 13.2594 0.490753
\(731\) 2.13902 0.0791147
\(732\) −31.9968 −1.18264
\(733\) −28.0948 −1.03770 −0.518852 0.854864i \(-0.673641\pi\)
−0.518852 + 0.854864i \(0.673641\pi\)
\(734\) 4.14641 0.153047
\(735\) 2.67544 0.0986851
\(736\) −4.63400 −0.170811
\(737\) 0 0
\(738\) 9.50699 0.349957
\(739\) 10.5129 0.386723 0.193361 0.981128i \(-0.438061\pi\)
0.193361 + 0.981128i \(0.438061\pi\)
\(740\) 0.240027 0.00882357
\(741\) 62.7364 2.30468
\(742\) 1.14198 0.0419233
\(743\) −33.7047 −1.23650 −0.618252 0.785980i \(-0.712159\pi\)
−0.618252 + 0.785980i \(0.712159\pi\)
\(744\) 7.28408 0.267047
\(745\) −10.5863 −0.387853
\(746\) 21.9641 0.804161
\(747\) 43.7452 1.60055
\(748\) 0 0
\(749\) 9.86211 0.360354
\(750\) −2.67544 −0.0976933
\(751\) −21.5277 −0.785557 −0.392778 0.919633i \(-0.628486\pi\)
−0.392778 + 0.919633i \(0.628486\pi\)
\(752\) −0.981558 −0.0357937
\(753\) 21.1696 0.771462
\(754\) 19.4473 0.708229
\(755\) 10.5581 0.384247
\(756\) −3.09812 −0.112678
\(757\) 30.8831 1.12247 0.561233 0.827658i \(-0.310327\pi\)
0.561233 + 0.827658i \(0.310327\pi\)
\(758\) −20.9655 −0.761501
\(759\) 0 0
\(760\) 7.52404 0.272925
\(761\) −7.53569 −0.273169 −0.136584 0.990628i \(-0.543612\pi\)
−0.136584 + 0.990628i \(0.543612\pi\)
\(762\) −36.8151 −1.33367
\(763\) −14.9188 −0.540096
\(764\) 16.9155 0.611980
\(765\) 2.34754 0.0848755
\(766\) 36.1904 1.30761
\(767\) −27.4222 −0.990159
\(768\) −2.67544 −0.0965417
\(769\) −52.9068 −1.90787 −0.953934 0.300017i \(-0.903008\pi\)
−0.953934 + 0.300017i \(0.903008\pi\)
\(770\) 0 0
\(771\) 25.6570 0.924015
\(772\) 15.2274 0.548045
\(773\) 40.8245 1.46836 0.734178 0.678957i \(-0.237568\pi\)
0.734178 + 0.678957i \(0.237568\pi\)
\(774\) 15.7532 0.566237
\(775\) 2.72257 0.0977976
\(776\) 9.05134 0.324924
\(777\) 0.642178 0.0230380
\(778\) −27.3801 −0.981623
\(779\) −17.2033 −0.616371
\(780\) −8.33813 −0.298553
\(781\) 0 0
\(782\) 2.61629 0.0935584
\(783\) −19.3324 −0.690882
\(784\) 1.00000 0.0357143
\(785\) −20.5831 −0.734643
\(786\) 23.4268 0.835607
\(787\) 30.7705 1.09685 0.548425 0.836200i \(-0.315228\pi\)
0.548425 + 0.836200i \(0.315228\pi\)
\(788\) −2.83965 −0.101158
\(789\) 63.4027 2.25720
\(790\) −15.3694 −0.546818
\(791\) −1.87415 −0.0666372
\(792\) 0 0
\(793\) −37.2721 −1.32357
\(794\) 2.72549 0.0967240
\(795\) −3.05529 −0.108360
\(796\) 4.77593 0.169278
\(797\) 48.5308 1.71905 0.859525 0.511093i \(-0.170759\pi\)
0.859525 + 0.511093i \(0.170759\pi\)
\(798\) 20.1301 0.712598
\(799\) 0.554174 0.0196053
\(800\) −1.00000 −0.0353553
\(801\) −18.0142 −0.636501
\(802\) −26.3810 −0.931547
\(803\) 0 0
\(804\) 18.1988 0.641823
\(805\) −4.63400 −0.163327
\(806\) 8.48501 0.298872
\(807\) −9.13301 −0.321497
\(808\) −5.47395 −0.192573
\(809\) 4.68981 0.164885 0.0824424 0.996596i \(-0.473728\pi\)
0.0824424 + 0.996596i \(0.473728\pi\)
\(810\) −4.18512 −0.147050
\(811\) −30.7640 −1.08027 −0.540135 0.841578i \(-0.681627\pi\)
−0.540135 + 0.841578i \(0.681627\pi\)
\(812\) 6.24003 0.218982
\(813\) 61.9667 2.17327
\(814\) 0 0
\(815\) 19.3407 0.677474
\(816\) 1.51052 0.0528786
\(817\) −28.5060 −0.997299
\(818\) −31.2447 −1.09245
\(819\) −12.9585 −0.452808
\(820\) 2.28644 0.0798460
\(821\) 55.6318 1.94156 0.970782 0.239961i \(-0.0771347\pi\)
0.970782 + 0.239961i \(0.0771347\pi\)
\(822\) −10.2157 −0.356312
\(823\) −31.3173 −1.09165 −0.545826 0.837898i \(-0.683784\pi\)
−0.545826 + 0.837898i \(0.683784\pi\)
\(824\) −12.5991 −0.438912
\(825\) 0 0
\(826\) −8.79892 −0.306154
\(827\) −37.9025 −1.31800 −0.659000 0.752143i \(-0.729020\pi\)
−0.659000 + 0.752143i \(0.729020\pi\)
\(828\) 19.2681 0.669613
\(829\) 52.1297 1.81054 0.905269 0.424838i \(-0.139669\pi\)
0.905269 + 0.424838i \(0.139669\pi\)
\(830\) 10.5208 0.365182
\(831\) −11.7710 −0.408333
\(832\) −3.11654 −0.108047
\(833\) −0.564586 −0.0195617
\(834\) −0.129911 −0.00449846
\(835\) −7.83259 −0.271058
\(836\) 0 0
\(837\) −8.43485 −0.291551
\(838\) −0.716042 −0.0247353
\(839\) −22.4013 −0.773380 −0.386690 0.922210i \(-0.626382\pi\)
−0.386690 + 0.922210i \(0.626382\pi\)
\(840\) −2.67544 −0.0923115
\(841\) 9.93794 0.342687
\(842\) −28.3733 −0.977810
\(843\) 36.3181 1.25086
\(844\) −26.0899 −0.898052
\(845\) 3.28716 0.113082
\(846\) 4.08130 0.140318
\(847\) 0 0
\(848\) −1.14198 −0.0392156
\(849\) 63.1252 2.16645
\(850\) 0.564586 0.0193651
\(851\) −1.11228 −0.0381286
\(852\) −30.5695 −1.04730
\(853\) −20.6326 −0.706447 −0.353223 0.935539i \(-0.614914\pi\)
−0.353223 + 0.935539i \(0.614914\pi\)
\(854\) −11.9594 −0.409244
\(855\) −31.2848 −1.06992
\(856\) −9.86211 −0.337080
\(857\) −18.5793 −0.634656 −0.317328 0.948316i \(-0.602786\pi\)
−0.317328 + 0.948316i \(0.602786\pi\)
\(858\) 0 0
\(859\) −52.2925 −1.78420 −0.892099 0.451840i \(-0.850768\pi\)
−0.892099 + 0.451840i \(0.850768\pi\)
\(860\) 3.78866 0.129192
\(861\) 6.11724 0.208475
\(862\) 15.8511 0.539890
\(863\) 8.28381 0.281984 0.140992 0.990011i \(-0.454971\pi\)
0.140992 + 0.990011i \(0.454971\pi\)
\(864\) 3.09812 0.105400
\(865\) 12.1772 0.414039
\(866\) −22.2137 −0.754853
\(867\) 44.6297 1.51570
\(868\) 2.72257 0.0924101
\(869\) 0 0
\(870\) −16.6948 −0.566008
\(871\) 21.1993 0.718310
\(872\) 14.9188 0.505213
\(873\) −37.6354 −1.27376
\(874\) −34.8664 −1.17937
\(875\) −1.00000 −0.0338062
\(876\) −35.4748 −1.19858
\(877\) 35.7102 1.20585 0.602923 0.797799i \(-0.294003\pi\)
0.602923 + 0.797799i \(0.294003\pi\)
\(878\) 16.3478 0.551712
\(879\) 80.8952 2.72853
\(880\) 0 0
\(881\) −31.7041 −1.06814 −0.534069 0.845441i \(-0.679338\pi\)
−0.534069 + 0.845441i \(0.679338\pi\)
\(882\) −4.15798 −0.140007
\(883\) −48.1427 −1.62013 −0.810065 0.586340i \(-0.800568\pi\)
−0.810065 + 0.586340i \(0.800568\pi\)
\(884\) 1.75956 0.0591803
\(885\) 23.5410 0.791322
\(886\) −28.6492 −0.962489
\(887\) 39.9088 1.34001 0.670004 0.742358i \(-0.266293\pi\)
0.670004 + 0.742358i \(0.266293\pi\)
\(888\) −0.642178 −0.0215501
\(889\) −13.7604 −0.461509
\(890\) −4.33244 −0.145224
\(891\) 0 0
\(892\) 8.25283 0.276325
\(893\) −7.38528 −0.247139
\(894\) 28.3231 0.947267
\(895\) −1.01844 −0.0340428
\(896\) −1.00000 −0.0334077
\(897\) 38.6389 1.29011
\(898\) −23.2905 −0.777213
\(899\) 16.9889 0.566612
\(900\) 4.15798 0.138599
\(901\) 0.644744 0.0214795
\(902\) 0 0
\(903\) 10.1363 0.337316
\(904\) 1.87415 0.0623334
\(905\) −1.18721 −0.0394642
\(906\) −28.2475 −0.938459
\(907\) −43.4697 −1.44339 −0.721693 0.692213i \(-0.756636\pi\)
−0.721693 + 0.692213i \(0.756636\pi\)
\(908\) −5.55836 −0.184461
\(909\) 22.7606 0.754922
\(910\) −3.11654 −0.103312
\(911\) −19.2575 −0.638028 −0.319014 0.947750i \(-0.603352\pi\)
−0.319014 + 0.947750i \(0.603352\pi\)
\(912\) −20.1301 −0.666575
\(913\) 0 0
\(914\) −10.3649 −0.342840
\(915\) 31.9968 1.05778
\(916\) −5.96188 −0.196986
\(917\) 8.75625 0.289157
\(918\) −1.74915 −0.0577307
\(919\) −8.40644 −0.277303 −0.138651 0.990341i \(-0.544277\pi\)
−0.138651 + 0.990341i \(0.544277\pi\)
\(920\) 4.63400 0.152778
\(921\) −15.1613 −0.499583
\(922\) 0.698098 0.0229906
\(923\) −35.6096 −1.17210
\(924\) 0 0
\(925\) −0.240027 −0.00789204
\(926\) −10.4790 −0.344362
\(927\) 52.3870 1.72062
\(928\) −6.24003 −0.204839
\(929\) 35.8044 1.17470 0.587351 0.809332i \(-0.300171\pi\)
0.587351 + 0.809332i \(0.300171\pi\)
\(930\) −7.28408 −0.238854
\(931\) 7.52404 0.246590
\(932\) 2.50193 0.0819533
\(933\) −79.4797 −2.60205
\(934\) 39.3583 1.28784
\(935\) 0 0
\(936\) 12.9585 0.423563
\(937\) 51.9019 1.69556 0.847780 0.530348i \(-0.177939\pi\)
0.847780 + 0.530348i \(0.177939\pi\)
\(938\) 6.80218 0.222099
\(939\) 21.0016 0.685360
\(940\) 0.981558 0.0320149
\(941\) −56.8484 −1.85321 −0.926603 0.376042i \(-0.877285\pi\)
−0.926603 + 0.376042i \(0.877285\pi\)
\(942\) 55.0689 1.79424
\(943\) −10.5954 −0.345033
\(944\) 8.79892 0.286381
\(945\) 3.09812 0.100782
\(946\) 0 0
\(947\) −20.4586 −0.664815 −0.332408 0.943136i \(-0.607861\pi\)
−0.332408 + 0.943136i \(0.607861\pi\)
\(948\) 41.1199 1.33551
\(949\) −41.3235 −1.34142
\(950\) −7.52404 −0.244112
\(951\) −18.7053 −0.606561
\(952\) 0.564586 0.0182983
\(953\) 55.2868 1.79091 0.895457 0.445148i \(-0.146849\pi\)
0.895457 + 0.445148i \(0.146849\pi\)
\(954\) 4.74832 0.153732
\(955\) −16.9155 −0.547372
\(956\) −28.8313 −0.932472
\(957\) 0 0
\(958\) 27.4511 0.886904
\(959\) −3.81831 −0.123300
\(960\) 2.67544 0.0863495
\(961\) −23.5876 −0.760891
\(962\) −0.748054 −0.0241182
\(963\) 41.0065 1.32142
\(964\) −1.67379 −0.0539092
\(965\) −15.2274 −0.490187
\(966\) 12.3980 0.398899
\(967\) −56.2874 −1.81008 −0.905040 0.425327i \(-0.860159\pi\)
−0.905040 + 0.425327i \(0.860159\pi\)
\(968\) 0 0
\(969\) 11.3652 0.365102
\(970\) −9.05134 −0.290621
\(971\) −5.39008 −0.172976 −0.0864880 0.996253i \(-0.527564\pi\)
−0.0864880 + 0.996253i \(0.527564\pi\)
\(972\) 20.4914 0.657262
\(973\) −0.0485569 −0.00155666
\(974\) 36.5980 1.17267
\(975\) 8.33813 0.267034
\(976\) 11.9594 0.382813
\(977\) 10.2278 0.327217 0.163609 0.986525i \(-0.447687\pi\)
0.163609 + 0.986525i \(0.447687\pi\)
\(978\) −51.7448 −1.65462
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −62.0320 −1.98053
\(982\) −3.39001 −0.108179
\(983\) 29.2357 0.932474 0.466237 0.884660i \(-0.345609\pi\)
0.466237 + 0.884660i \(0.345609\pi\)
\(984\) −6.11724 −0.195010
\(985\) 2.83965 0.0904788
\(986\) 3.52303 0.112196
\(987\) 2.62610 0.0835897
\(988\) −23.4490 −0.746012
\(989\) −17.5567 −0.558269
\(990\) 0 0
\(991\) 51.3625 1.63158 0.815792 0.578346i \(-0.196302\pi\)
0.815792 + 0.578346i \(0.196302\pi\)
\(992\) −2.72257 −0.0864417
\(993\) −68.0271 −2.15878
\(994\) −11.4260 −0.362410
\(995\) −4.77593 −0.151407
\(996\) −28.1477 −0.891895
\(997\) 29.1018 0.921664 0.460832 0.887487i \(-0.347551\pi\)
0.460832 + 0.887487i \(0.347551\pi\)
\(998\) −22.9786 −0.727376
\(999\) 0.743632 0.0235275
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.cx.1.2 6
11.10 odd 2 8470.2.a.dd.1.2 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cx.1.2 6 1.1 even 1 trivial
8470.2.a.dd.1.2 yes 6 11.10 odd 2