Properties

Label 2151.2.a.j.1.7
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1758214748\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + 229 x^{12} - 14020 x^{11} + 3356 x^{10} + 23404 x^{9} - 9429 x^{8} - 21252 x^{7} + 10479 x^{6} + 9108 x^{5} - 4844 x^{4} - 1184 x^{3} + 640 x^{2} - 56 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(1.22231\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.22231 q^{2} -0.505952 q^{4} -0.403117 q^{5} -4.45690 q^{7} +3.06306 q^{8} +O(q^{10})\) \(q-1.22231 q^{2} -0.505952 q^{4} -0.403117 q^{5} -4.45690 q^{7} +3.06306 q^{8} +0.492736 q^{10} +2.17391 q^{11} +5.85396 q^{13} +5.44773 q^{14} -2.73211 q^{16} -7.98275 q^{17} +2.00460 q^{19} +0.203958 q^{20} -2.65720 q^{22} -0.679568 q^{23} -4.83750 q^{25} -7.15536 q^{26} +2.25498 q^{28} +5.44529 q^{29} -0.595692 q^{31} -2.78662 q^{32} +9.75742 q^{34} +1.79666 q^{35} +5.86330 q^{37} -2.45025 q^{38} -1.23477 q^{40} -0.396926 q^{41} +12.0393 q^{43} -1.09990 q^{44} +0.830645 q^{46} +6.38497 q^{47} +12.8640 q^{49} +5.91293 q^{50} -2.96182 q^{52} -9.69118 q^{53} -0.876342 q^{55} -13.6518 q^{56} -6.65585 q^{58} -2.87181 q^{59} -2.31211 q^{61} +0.728121 q^{62} +8.87034 q^{64} -2.35983 q^{65} +11.4712 q^{67} +4.03889 q^{68} -2.19608 q^{70} -11.4938 q^{71} +7.74678 q^{73} -7.16679 q^{74} -1.01423 q^{76} -9.68892 q^{77} -14.2836 q^{79} +1.10136 q^{80} +0.485167 q^{82} -6.99235 q^{83} +3.21799 q^{85} -14.7158 q^{86} +6.65882 q^{88} -16.1528 q^{89} -26.0905 q^{91} +0.343829 q^{92} -7.80444 q^{94} -0.808091 q^{95} -8.60226 q^{97} -15.7238 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 4q^{2} + 18q^{4} - 16q^{5} - 4q^{7} - 12q^{8} + O(q^{10}) \) \( 20q - 4q^{2} + 18q^{4} - 16q^{5} - 4q^{7} - 12q^{8} + 4q^{10} - 12q^{11} - 4q^{13} - 20q^{14} + 22q^{16} - 24q^{17} - 4q^{19} - 40q^{20} - 6q^{22} - 12q^{23} + 22q^{25} - 30q^{26} - 12q^{28} - 24q^{29} - 4q^{31} - 28q^{32} + 8q^{34} - 20q^{35} - 10q^{37} - 26q^{38} + 6q^{40} - 66q^{41} + 8q^{43} - 36q^{44} - 12q^{46} - 28q^{47} + 18q^{49} - 28q^{50} - 18q^{52} - 28q^{53} - 4q^{55} - 60q^{56} - 54q^{59} - 4q^{61} - 20q^{62} + 22q^{64} - 42q^{65} + 12q^{67} - 12q^{68} + 20q^{70} - 36q^{71} + 14q^{73} - 50q^{76} - 8q^{77} - 12q^{79} - 88q^{80} - 8q^{82} - 20q^{83} + 4q^{85} - 18q^{86} - 10q^{88} - 130q^{89} - 6q^{91} + 46q^{92} - 26q^{94} - 2q^{97} - 12q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.22231 −0.864306 −0.432153 0.901800i \(-0.642246\pi\)
−0.432153 + 0.901800i \(0.642246\pi\)
\(3\) 0 0
\(4\) −0.505952 −0.252976
\(5\) −0.403117 −0.180280 −0.0901398 0.995929i \(-0.528731\pi\)
−0.0901398 + 0.995929i \(0.528731\pi\)
\(6\) 0 0
\(7\) −4.45690 −1.68455 −0.842276 0.539047i \(-0.818785\pi\)
−0.842276 + 0.539047i \(0.818785\pi\)
\(8\) 3.06306 1.08295
\(9\) 0 0
\(10\) 0.492736 0.155817
\(11\) 2.17391 0.655460 0.327730 0.944772i \(-0.393716\pi\)
0.327730 + 0.944772i \(0.393716\pi\)
\(12\) 0 0
\(13\) 5.85396 1.62360 0.811798 0.583939i \(-0.198489\pi\)
0.811798 + 0.583939i \(0.198489\pi\)
\(14\) 5.44773 1.45597
\(15\) 0 0
\(16\) −2.73211 −0.683027
\(17\) −7.98275 −1.93610 −0.968051 0.250753i \(-0.919322\pi\)
−0.968051 + 0.250753i \(0.919322\pi\)
\(18\) 0 0
\(19\) 2.00460 0.459888 0.229944 0.973204i \(-0.426146\pi\)
0.229944 + 0.973204i \(0.426146\pi\)
\(20\) 0.203958 0.0456064
\(21\) 0 0
\(22\) −2.65720 −0.566517
\(23\) −0.679568 −0.141700 −0.0708499 0.997487i \(-0.522571\pi\)
−0.0708499 + 0.997487i \(0.522571\pi\)
\(24\) 0 0
\(25\) −4.83750 −0.967499
\(26\) −7.15536 −1.40328
\(27\) 0 0
\(28\) 2.25498 0.426151
\(29\) 5.44529 1.01116 0.505582 0.862778i \(-0.331278\pi\)
0.505582 + 0.862778i \(0.331278\pi\)
\(30\) 0 0
\(31\) −0.595692 −0.106989 −0.0534947 0.998568i \(-0.517036\pi\)
−0.0534947 + 0.998568i \(0.517036\pi\)
\(32\) −2.78662 −0.492610
\(33\) 0 0
\(34\) 9.75742 1.67338
\(35\) 1.79666 0.303690
\(36\) 0 0
\(37\) 5.86330 0.963921 0.481960 0.876193i \(-0.339925\pi\)
0.481960 + 0.876193i \(0.339925\pi\)
\(38\) −2.45025 −0.397483
\(39\) 0 0
\(40\) −1.23477 −0.195235
\(41\) −0.396926 −0.0619894 −0.0309947 0.999520i \(-0.509867\pi\)
−0.0309947 + 0.999520i \(0.509867\pi\)
\(42\) 0 0
\(43\) 12.0393 1.83598 0.917988 0.396609i \(-0.129813\pi\)
0.917988 + 0.396609i \(0.129813\pi\)
\(44\) −1.09990 −0.165815
\(45\) 0 0
\(46\) 0.830645 0.122472
\(47\) 6.38497 0.931344 0.465672 0.884957i \(-0.345813\pi\)
0.465672 + 0.884957i \(0.345813\pi\)
\(48\) 0 0
\(49\) 12.8640 1.83771
\(50\) 5.91293 0.836215
\(51\) 0 0
\(52\) −2.96182 −0.410730
\(53\) −9.69118 −1.33119 −0.665593 0.746315i \(-0.731821\pi\)
−0.665593 + 0.746315i \(0.731821\pi\)
\(54\) 0 0
\(55\) −0.876342 −0.118166
\(56\) −13.6518 −1.82429
\(57\) 0 0
\(58\) −6.65585 −0.873955
\(59\) −2.87181 −0.373878 −0.186939 0.982372i \(-0.559857\pi\)
−0.186939 + 0.982372i \(0.559857\pi\)
\(60\) 0 0
\(61\) −2.31211 −0.296036 −0.148018 0.988985i \(-0.547289\pi\)
−0.148018 + 0.988985i \(0.547289\pi\)
\(62\) 0.728121 0.0924715
\(63\) 0 0
\(64\) 8.87034 1.10879
\(65\) −2.35983 −0.292701
\(66\) 0 0
\(67\) 11.4712 1.40143 0.700715 0.713442i \(-0.252865\pi\)
0.700715 + 0.713442i \(0.252865\pi\)
\(68\) 4.03889 0.489787
\(69\) 0 0
\(70\) −2.19608 −0.262481
\(71\) −11.4938 −1.36406 −0.682030 0.731324i \(-0.738903\pi\)
−0.682030 + 0.731324i \(0.738903\pi\)
\(72\) 0 0
\(73\) 7.74678 0.906692 0.453346 0.891335i \(-0.350230\pi\)
0.453346 + 0.891335i \(0.350230\pi\)
\(74\) −7.16679 −0.833122
\(75\) 0 0
\(76\) −1.01423 −0.116340
\(77\) −9.68892 −1.10416
\(78\) 0 0
\(79\) −14.2836 −1.60703 −0.803514 0.595285i \(-0.797039\pi\)
−0.803514 + 0.595285i \(0.797039\pi\)
\(80\) 1.10136 0.123136
\(81\) 0 0
\(82\) 0.485167 0.0535778
\(83\) −6.99235 −0.767510 −0.383755 0.923435i \(-0.625369\pi\)
−0.383755 + 0.923435i \(0.625369\pi\)
\(84\) 0 0
\(85\) 3.21799 0.349040
\(86\) −14.7158 −1.58684
\(87\) 0 0
\(88\) 6.65882 0.709833
\(89\) −16.1528 −1.71219 −0.856096 0.516818i \(-0.827117\pi\)
−0.856096 + 0.516818i \(0.827117\pi\)
\(90\) 0 0
\(91\) −26.0905 −2.73503
\(92\) 0.343829 0.0358466
\(93\) 0 0
\(94\) −7.80444 −0.804966
\(95\) −0.808091 −0.0829084
\(96\) 0 0
\(97\) −8.60226 −0.873427 −0.436714 0.899601i \(-0.643858\pi\)
−0.436714 + 0.899601i \(0.643858\pi\)
\(98\) −15.7238 −1.58835
\(99\) 0 0
\(100\) 2.44754 0.244754
\(101\) −11.0688 −1.10139 −0.550693 0.834708i \(-0.685636\pi\)
−0.550693 + 0.834708i \(0.685636\pi\)
\(102\) 0 0
\(103\) −5.71739 −0.563351 −0.281675 0.959510i \(-0.590890\pi\)
−0.281675 + 0.959510i \(0.590890\pi\)
\(104\) 17.9310 1.75828
\(105\) 0 0
\(106\) 11.8456 1.15055
\(107\) −13.9647 −1.35002 −0.675011 0.737808i \(-0.735861\pi\)
−0.675011 + 0.737808i \(0.735861\pi\)
\(108\) 0 0
\(109\) −7.04998 −0.675265 −0.337633 0.941278i \(-0.609626\pi\)
−0.337633 + 0.941278i \(0.609626\pi\)
\(110\) 1.07116 0.102132
\(111\) 0 0
\(112\) 12.1767 1.15059
\(113\) 17.5817 1.65394 0.826972 0.562243i \(-0.190062\pi\)
0.826972 + 0.562243i \(0.190062\pi\)
\(114\) 0 0
\(115\) 0.273946 0.0255456
\(116\) −2.75505 −0.255800
\(117\) 0 0
\(118\) 3.51025 0.323144
\(119\) 35.5784 3.26146
\(120\) 0 0
\(121\) −6.27410 −0.570373
\(122\) 2.82612 0.255865
\(123\) 0 0
\(124\) 0.301391 0.0270657
\(125\) 3.96567 0.354700
\(126\) 0 0
\(127\) 0.0188127 0.00166936 0.000834680 1.00000i \(-0.499734\pi\)
0.000834680 1.00000i \(0.499734\pi\)
\(128\) −5.26909 −0.465726
\(129\) 0 0
\(130\) 2.88445 0.252983
\(131\) −5.31238 −0.464145 −0.232072 0.972699i \(-0.574551\pi\)
−0.232072 + 0.972699i \(0.574551\pi\)
\(132\) 0 0
\(133\) −8.93433 −0.774704
\(134\) −14.0214 −1.21126
\(135\) 0 0
\(136\) −24.4516 −2.09671
\(137\) −5.05773 −0.432111 −0.216055 0.976381i \(-0.569319\pi\)
−0.216055 + 0.976381i \(0.569319\pi\)
\(138\) 0 0
\(139\) −9.89513 −0.839294 −0.419647 0.907687i \(-0.637846\pi\)
−0.419647 + 0.907687i \(0.637846\pi\)
\(140\) −0.909021 −0.0768263
\(141\) 0 0
\(142\) 14.0490 1.17897
\(143\) 12.7260 1.06420
\(144\) 0 0
\(145\) −2.19509 −0.182292
\(146\) −9.46899 −0.783659
\(147\) 0 0
\(148\) −2.96655 −0.243849
\(149\) −4.84892 −0.397239 −0.198620 0.980077i \(-0.563646\pi\)
−0.198620 + 0.980077i \(0.563646\pi\)
\(150\) 0 0
\(151\) 0.196098 0.0159583 0.00797913 0.999968i \(-0.497460\pi\)
0.00797913 + 0.999968i \(0.497460\pi\)
\(152\) 6.14021 0.498037
\(153\) 0 0
\(154\) 11.8429 0.954328
\(155\) 0.240134 0.0192880
\(156\) 0 0
\(157\) −5.28786 −0.422018 −0.211009 0.977484i \(-0.567675\pi\)
−0.211009 + 0.977484i \(0.567675\pi\)
\(158\) 17.4590 1.38896
\(159\) 0 0
\(160\) 1.12334 0.0888075
\(161\) 3.02877 0.238701
\(162\) 0 0
\(163\) −13.1799 −1.03233 −0.516165 0.856489i \(-0.672641\pi\)
−0.516165 + 0.856489i \(0.672641\pi\)
\(164\) 0.200825 0.0156818
\(165\) 0 0
\(166\) 8.54684 0.663364
\(167\) 24.2342 1.87530 0.937651 0.347579i \(-0.112996\pi\)
0.937651 + 0.347579i \(0.112996\pi\)
\(168\) 0 0
\(169\) 21.2688 1.63606
\(170\) −3.93339 −0.301677
\(171\) 0 0
\(172\) −6.09130 −0.464458
\(173\) −23.9278 −1.81920 −0.909599 0.415488i \(-0.863611\pi\)
−0.909599 + 0.415488i \(0.863611\pi\)
\(174\) 0 0
\(175\) 21.5603 1.62980
\(176\) −5.93937 −0.447697
\(177\) 0 0
\(178\) 19.7437 1.47986
\(179\) −8.14397 −0.608709 −0.304354 0.952559i \(-0.598441\pi\)
−0.304354 + 0.952559i \(0.598441\pi\)
\(180\) 0 0
\(181\) −22.0482 −1.63883 −0.819414 0.573203i \(-0.805701\pi\)
−0.819414 + 0.573203i \(0.805701\pi\)
\(182\) 31.8908 2.36390
\(183\) 0 0
\(184\) −2.08156 −0.153454
\(185\) −2.36360 −0.173775
\(186\) 0 0
\(187\) −17.3538 −1.26904
\(188\) −3.23049 −0.235608
\(189\) 0 0
\(190\) 0.987739 0.0716582
\(191\) 23.0122 1.66511 0.832553 0.553945i \(-0.186878\pi\)
0.832553 + 0.553945i \(0.186878\pi\)
\(192\) 0 0
\(193\) 7.80810 0.562039 0.281020 0.959702i \(-0.409327\pi\)
0.281020 + 0.959702i \(0.409327\pi\)
\(194\) 10.5147 0.754908
\(195\) 0 0
\(196\) −6.50856 −0.464897
\(197\) −6.78508 −0.483417 −0.241708 0.970349i \(-0.577708\pi\)
−0.241708 + 0.970349i \(0.577708\pi\)
\(198\) 0 0
\(199\) −9.79051 −0.694031 −0.347015 0.937859i \(-0.612805\pi\)
−0.347015 + 0.937859i \(0.612805\pi\)
\(200\) −14.8175 −1.04776
\(201\) 0 0
\(202\) 13.5295 0.951934
\(203\) −24.2691 −1.70336
\(204\) 0 0
\(205\) 0.160008 0.0111754
\(206\) 6.98843 0.486907
\(207\) 0 0
\(208\) −15.9936 −1.10896
\(209\) 4.35783 0.301438
\(210\) 0 0
\(211\) −11.8919 −0.818673 −0.409337 0.912383i \(-0.634240\pi\)
−0.409337 + 0.912383i \(0.634240\pi\)
\(212\) 4.90327 0.336758
\(213\) 0 0
\(214\) 17.0693 1.16683
\(215\) −4.85325 −0.330989
\(216\) 0 0
\(217\) 2.65494 0.180229
\(218\) 8.61727 0.583635
\(219\) 0 0
\(220\) 0.443387 0.0298931
\(221\) −46.7307 −3.14345
\(222\) 0 0
\(223\) 21.0443 1.40923 0.704614 0.709591i \(-0.251120\pi\)
0.704614 + 0.709591i \(0.251120\pi\)
\(224\) 12.4197 0.829827
\(225\) 0 0
\(226\) −21.4903 −1.42951
\(227\) 16.5049 1.09547 0.547734 0.836653i \(-0.315491\pi\)
0.547734 + 0.836653i \(0.315491\pi\)
\(228\) 0 0
\(229\) −5.18227 −0.342454 −0.171227 0.985232i \(-0.554773\pi\)
−0.171227 + 0.985232i \(0.554773\pi\)
\(230\) −0.334848 −0.0220792
\(231\) 0 0
\(232\) 16.6792 1.09505
\(233\) −11.8271 −0.774817 −0.387409 0.921908i \(-0.626630\pi\)
−0.387409 + 0.921908i \(0.626630\pi\)
\(234\) 0 0
\(235\) −2.57389 −0.167902
\(236\) 1.45300 0.0945820
\(237\) 0 0
\(238\) −43.4879 −2.81890
\(239\) −1.00000 −0.0646846
\(240\) 0 0
\(241\) −15.1921 −0.978607 −0.489304 0.872113i \(-0.662749\pi\)
−0.489304 + 0.872113i \(0.662749\pi\)
\(242\) 7.66891 0.492976
\(243\) 0 0
\(244\) 1.16982 0.0748899
\(245\) −5.18570 −0.331302
\(246\) 0 0
\(247\) 11.7349 0.746671
\(248\) −1.82464 −0.115865
\(249\) 0 0
\(250\) −4.84728 −0.306569
\(251\) 7.88110 0.497451 0.248725 0.968574i \(-0.419988\pi\)
0.248725 + 0.968574i \(0.419988\pi\)
\(252\) 0 0
\(253\) −1.47732 −0.0928785
\(254\) −0.0229950 −0.00144284
\(255\) 0 0
\(256\) −11.3002 −0.706263
\(257\) −28.3840 −1.77055 −0.885273 0.465072i \(-0.846028\pi\)
−0.885273 + 0.465072i \(0.846028\pi\)
\(258\) 0 0
\(259\) −26.1322 −1.62377
\(260\) 1.19396 0.0740463
\(261\) 0 0
\(262\) 6.49339 0.401163
\(263\) −16.8514 −1.03910 −0.519552 0.854439i \(-0.673901\pi\)
−0.519552 + 0.854439i \(0.673901\pi\)
\(264\) 0 0
\(265\) 3.90668 0.239986
\(266\) 10.9205 0.669581
\(267\) 0 0
\(268\) −5.80387 −0.354528
\(269\) 7.79527 0.475286 0.237643 0.971353i \(-0.423625\pi\)
0.237643 + 0.971353i \(0.423625\pi\)
\(270\) 0 0
\(271\) 1.75917 0.106862 0.0534309 0.998572i \(-0.482984\pi\)
0.0534309 + 0.998572i \(0.482984\pi\)
\(272\) 21.8098 1.32241
\(273\) 0 0
\(274\) 6.18212 0.373476
\(275\) −10.5163 −0.634157
\(276\) 0 0
\(277\) 28.0737 1.68679 0.843394 0.537296i \(-0.180554\pi\)
0.843394 + 0.537296i \(0.180554\pi\)
\(278\) 12.0949 0.725407
\(279\) 0 0
\(280\) 5.50326 0.328883
\(281\) −16.8365 −1.00438 −0.502190 0.864758i \(-0.667472\pi\)
−0.502190 + 0.864758i \(0.667472\pi\)
\(282\) 0 0
\(283\) 14.3487 0.852941 0.426470 0.904502i \(-0.359757\pi\)
0.426470 + 0.904502i \(0.359757\pi\)
\(284\) 5.81530 0.345075
\(285\) 0 0
\(286\) −15.5551 −0.919795
\(287\) 1.76906 0.104424
\(288\) 0 0
\(289\) 46.7244 2.74849
\(290\) 2.68309 0.157556
\(291\) 0 0
\(292\) −3.91950 −0.229371
\(293\) −13.4805 −0.787538 −0.393769 0.919209i \(-0.628829\pi\)
−0.393769 + 0.919209i \(0.628829\pi\)
\(294\) 0 0
\(295\) 1.15768 0.0674025
\(296\) 17.9596 1.04388
\(297\) 0 0
\(298\) 5.92690 0.343336
\(299\) −3.97816 −0.230063
\(300\) 0 0
\(301\) −53.6580 −3.09280
\(302\) −0.239693 −0.0137928
\(303\) 0 0
\(304\) −5.47680 −0.314116
\(305\) 0.932052 0.0533692
\(306\) 0 0
\(307\) −14.9529 −0.853408 −0.426704 0.904391i \(-0.640325\pi\)
−0.426704 + 0.904391i \(0.640325\pi\)
\(308\) 4.90213 0.279325
\(309\) 0 0
\(310\) −0.293518 −0.0166707
\(311\) 11.1362 0.631478 0.315739 0.948846i \(-0.397748\pi\)
0.315739 + 0.948846i \(0.397748\pi\)
\(312\) 0 0
\(313\) 6.43716 0.363850 0.181925 0.983312i \(-0.441767\pi\)
0.181925 + 0.983312i \(0.441767\pi\)
\(314\) 6.46342 0.364752
\(315\) 0 0
\(316\) 7.22681 0.406540
\(317\) −16.0732 −0.902763 −0.451382 0.892331i \(-0.649069\pi\)
−0.451382 + 0.892331i \(0.649069\pi\)
\(318\) 0 0
\(319\) 11.8376 0.662778
\(320\) −3.57579 −0.199893
\(321\) 0 0
\(322\) −3.70211 −0.206310
\(323\) −16.0023 −0.890390
\(324\) 0 0
\(325\) −28.3185 −1.57083
\(326\) 16.1100 0.892248
\(327\) 0 0
\(328\) −1.21581 −0.0671317
\(329\) −28.4572 −1.56890
\(330\) 0 0
\(331\) 11.3812 0.625567 0.312784 0.949824i \(-0.398739\pi\)
0.312784 + 0.949824i \(0.398739\pi\)
\(332\) 3.53779 0.194162
\(333\) 0 0
\(334\) −29.6218 −1.62083
\(335\) −4.62424 −0.252649
\(336\) 0 0
\(337\) 6.74602 0.367479 0.183740 0.982975i \(-0.441180\pi\)
0.183740 + 0.982975i \(0.441180\pi\)
\(338\) −25.9971 −1.41406
\(339\) 0 0
\(340\) −1.62815 −0.0882986
\(341\) −1.29498 −0.0701272
\(342\) 0 0
\(343\) −26.1353 −1.41117
\(344\) 36.8770 1.98828
\(345\) 0 0
\(346\) 29.2473 1.57234
\(347\) 6.94503 0.372829 0.186414 0.982471i \(-0.440313\pi\)
0.186414 + 0.982471i \(0.440313\pi\)
\(348\) 0 0
\(349\) 15.7152 0.841217 0.420609 0.907242i \(-0.361817\pi\)
0.420609 + 0.907242i \(0.361817\pi\)
\(350\) −26.3534 −1.40865
\(351\) 0 0
\(352\) −6.05787 −0.322886
\(353\) 11.7091 0.623215 0.311607 0.950211i \(-0.399133\pi\)
0.311607 + 0.950211i \(0.399133\pi\)
\(354\) 0 0
\(355\) 4.63334 0.245912
\(356\) 8.17253 0.433143
\(357\) 0 0
\(358\) 9.95448 0.526110
\(359\) −20.0346 −1.05738 −0.528692 0.848814i \(-0.677317\pi\)
−0.528692 + 0.848814i \(0.677317\pi\)
\(360\) 0 0
\(361\) −14.9816 −0.788503
\(362\) 26.9497 1.41645
\(363\) 0 0
\(364\) 13.2005 0.691897
\(365\) −3.12286 −0.163458
\(366\) 0 0
\(367\) −20.9033 −1.09114 −0.545572 0.838064i \(-0.683688\pi\)
−0.545572 + 0.838064i \(0.683688\pi\)
\(368\) 1.85666 0.0967848
\(369\) 0 0
\(370\) 2.88906 0.150195
\(371\) 43.1927 2.24245
\(372\) 0 0
\(373\) 2.89285 0.149786 0.0748930 0.997192i \(-0.476138\pi\)
0.0748930 + 0.997192i \(0.476138\pi\)
\(374\) 21.2118 1.09684
\(375\) 0 0
\(376\) 19.5575 1.00860
\(377\) 31.8765 1.64172
\(378\) 0 0
\(379\) 23.5553 1.20995 0.604977 0.796243i \(-0.293182\pi\)
0.604977 + 0.796243i \(0.293182\pi\)
\(380\) 0.408855 0.0209738
\(381\) 0 0
\(382\) −28.1281 −1.43916
\(383\) −8.52161 −0.435434 −0.217717 0.976012i \(-0.569861\pi\)
−0.217717 + 0.976012i \(0.569861\pi\)
\(384\) 0 0
\(385\) 3.90577 0.199057
\(386\) −9.54394 −0.485774
\(387\) 0 0
\(388\) 4.35233 0.220956
\(389\) 20.0809 1.01814 0.509072 0.860724i \(-0.329989\pi\)
0.509072 + 0.860724i \(0.329989\pi\)
\(390\) 0 0
\(391\) 5.42483 0.274345
\(392\) 39.4032 1.99016
\(393\) 0 0
\(394\) 8.29349 0.417820
\(395\) 5.75796 0.289714
\(396\) 0 0
\(397\) 25.3899 1.27428 0.637141 0.770748i \(-0.280117\pi\)
0.637141 + 0.770748i \(0.280117\pi\)
\(398\) 11.9671 0.599855
\(399\) 0 0
\(400\) 13.2166 0.660828
\(401\) 1.99676 0.0997136 0.0498568 0.998756i \(-0.484124\pi\)
0.0498568 + 0.998756i \(0.484124\pi\)
\(402\) 0 0
\(403\) −3.48715 −0.173707
\(404\) 5.60027 0.278624
\(405\) 0 0
\(406\) 29.6645 1.47222
\(407\) 12.7463 0.631811
\(408\) 0 0
\(409\) −23.0181 −1.13817 −0.569085 0.822279i \(-0.692702\pi\)
−0.569085 + 0.822279i \(0.692702\pi\)
\(410\) −0.195579 −0.00965898
\(411\) 0 0
\(412\) 2.89272 0.142514
\(413\) 12.7994 0.629816
\(414\) 0 0
\(415\) 2.81874 0.138366
\(416\) −16.3128 −0.799799
\(417\) 0 0
\(418\) −5.32664 −0.260534
\(419\) 3.68635 0.180090 0.0900450 0.995938i \(-0.471299\pi\)
0.0900450 + 0.995938i \(0.471299\pi\)
\(420\) 0 0
\(421\) 19.3486 0.942992 0.471496 0.881868i \(-0.343714\pi\)
0.471496 + 0.881868i \(0.343714\pi\)
\(422\) 14.5356 0.707584
\(423\) 0 0
\(424\) −29.6846 −1.44161
\(425\) 38.6165 1.87318
\(426\) 0 0
\(427\) 10.3049 0.498687
\(428\) 7.06549 0.341523
\(429\) 0 0
\(430\) 5.93219 0.286076
\(431\) 10.1280 0.487848 0.243924 0.969794i \(-0.421565\pi\)
0.243924 + 0.969794i \(0.421565\pi\)
\(432\) 0 0
\(433\) −34.6573 −1.66552 −0.832761 0.553632i \(-0.813241\pi\)
−0.832761 + 0.553632i \(0.813241\pi\)
\(434\) −3.24517 −0.155773
\(435\) 0 0
\(436\) 3.56695 0.170826
\(437\) −1.36227 −0.0651660
\(438\) 0 0
\(439\) 3.02301 0.144280 0.0721402 0.997395i \(-0.477017\pi\)
0.0721402 + 0.997395i \(0.477017\pi\)
\(440\) −2.68429 −0.127968
\(441\) 0 0
\(442\) 57.1195 2.71690
\(443\) 23.1069 1.09784 0.548922 0.835874i \(-0.315038\pi\)
0.548922 + 0.835874i \(0.315038\pi\)
\(444\) 0 0
\(445\) 6.51147 0.308673
\(446\) −25.7227 −1.21800
\(447\) 0 0
\(448\) −39.5343 −1.86782
\(449\) −13.0483 −0.615787 −0.307894 0.951421i \(-0.599624\pi\)
−0.307894 + 0.951421i \(0.599624\pi\)
\(450\) 0 0
\(451\) −0.862882 −0.0406315
\(452\) −8.89548 −0.418408
\(453\) 0 0
\(454\) −20.1741 −0.946819
\(455\) 10.5175 0.493070
\(456\) 0 0
\(457\) −22.9529 −1.07369 −0.536846 0.843680i \(-0.680384\pi\)
−0.536846 + 0.843680i \(0.680384\pi\)
\(458\) 6.33436 0.295985
\(459\) 0 0
\(460\) −0.138603 −0.00646242
\(461\) −29.2496 −1.36229 −0.681145 0.732148i \(-0.738518\pi\)
−0.681145 + 0.732148i \(0.738518\pi\)
\(462\) 0 0
\(463\) −0.209955 −0.00975742 −0.00487871 0.999988i \(-0.501553\pi\)
−0.00487871 + 0.999988i \(0.501553\pi\)
\(464\) −14.8771 −0.690653
\(465\) 0 0
\(466\) 14.4564 0.669679
\(467\) −25.6201 −1.18556 −0.592778 0.805366i \(-0.701969\pi\)
−0.592778 + 0.805366i \(0.701969\pi\)
\(468\) 0 0
\(469\) −51.1260 −2.36078
\(470\) 3.14610 0.145119
\(471\) 0 0
\(472\) −8.79651 −0.404892
\(473\) 26.1724 1.20341
\(474\) 0 0
\(475\) −9.69726 −0.444941
\(476\) −18.0009 −0.825072
\(477\) 0 0
\(478\) 1.22231 0.0559073
\(479\) 31.5384 1.44103 0.720513 0.693441i \(-0.243906\pi\)
0.720513 + 0.693441i \(0.243906\pi\)
\(480\) 0 0
\(481\) 34.3235 1.56502
\(482\) 18.5695 0.845816
\(483\) 0 0
\(484\) 3.17439 0.144291
\(485\) 3.46772 0.157461
\(486\) 0 0
\(487\) −13.2957 −0.602486 −0.301243 0.953547i \(-0.597402\pi\)
−0.301243 + 0.953547i \(0.597402\pi\)
\(488\) −7.08213 −0.320593
\(489\) 0 0
\(490\) 6.33855 0.286346
\(491\) −13.8843 −0.626588 −0.313294 0.949656i \(-0.601432\pi\)
−0.313294 + 0.949656i \(0.601432\pi\)
\(492\) 0 0
\(493\) −43.4684 −1.95772
\(494\) −14.3437 −0.645352
\(495\) 0 0
\(496\) 1.62749 0.0730767
\(497\) 51.2267 2.29783
\(498\) 0 0
\(499\) 7.25368 0.324719 0.162360 0.986732i \(-0.448090\pi\)
0.162360 + 0.986732i \(0.448090\pi\)
\(500\) −2.00644 −0.0897305
\(501\) 0 0
\(502\) −9.63317 −0.429949
\(503\) −6.18887 −0.275948 −0.137974 0.990436i \(-0.544059\pi\)
−0.137974 + 0.990436i \(0.544059\pi\)
\(504\) 0 0
\(505\) 4.46202 0.198557
\(506\) 1.80575 0.0802754
\(507\) 0 0
\(508\) −0.00951833 −0.000422308 0
\(509\) −9.55970 −0.423726 −0.211863 0.977299i \(-0.567953\pi\)
−0.211863 + 0.977299i \(0.567953\pi\)
\(510\) 0 0
\(511\) −34.5267 −1.52737
\(512\) 24.3506 1.07615
\(513\) 0 0
\(514\) 34.6941 1.53029
\(515\) 2.30478 0.101561
\(516\) 0 0
\(517\) 13.8804 0.610458
\(518\) 31.9417 1.40344
\(519\) 0 0
\(520\) −7.22830 −0.316982
\(521\) −38.5824 −1.69033 −0.845163 0.534509i \(-0.820497\pi\)
−0.845163 + 0.534509i \(0.820497\pi\)
\(522\) 0 0
\(523\) 24.4407 1.06872 0.534359 0.845258i \(-0.320553\pi\)
0.534359 + 0.845258i \(0.320553\pi\)
\(524\) 2.68781 0.117417
\(525\) 0 0
\(526\) 20.5977 0.898104
\(527\) 4.75526 0.207142
\(528\) 0 0
\(529\) −22.5382 −0.979921
\(530\) −4.77519 −0.207421
\(531\) 0 0
\(532\) 4.52034 0.195982
\(533\) −2.32359 −0.100646
\(534\) 0 0
\(535\) 5.62943 0.243381
\(536\) 35.1369 1.51768
\(537\) 0 0
\(538\) −9.52825 −0.410792
\(539\) 27.9652 1.20455
\(540\) 0 0
\(541\) 7.78761 0.334815 0.167408 0.985888i \(-0.446460\pi\)
0.167408 + 0.985888i \(0.446460\pi\)
\(542\) −2.15025 −0.0923612
\(543\) 0 0
\(544\) 22.2449 0.953743
\(545\) 2.84197 0.121737
\(546\) 0 0
\(547\) 0.826426 0.0353354 0.0176677 0.999844i \(-0.494376\pi\)
0.0176677 + 0.999844i \(0.494376\pi\)
\(548\) 2.55897 0.109314
\(549\) 0 0
\(550\) 12.8542 0.548105
\(551\) 10.9156 0.465022
\(552\) 0 0
\(553\) 63.6606 2.70712
\(554\) −34.3149 −1.45790
\(555\) 0 0
\(556\) 5.00646 0.212321
\(557\) −27.4842 −1.16454 −0.582272 0.812994i \(-0.697836\pi\)
−0.582272 + 0.812994i \(0.697836\pi\)
\(558\) 0 0
\(559\) 70.4775 2.98088
\(560\) −4.90866 −0.207429
\(561\) 0 0
\(562\) 20.5794 0.868090
\(563\) −9.93084 −0.418535 −0.209267 0.977858i \(-0.567108\pi\)
−0.209267 + 0.977858i \(0.567108\pi\)
\(564\) 0 0
\(565\) −7.08748 −0.298172
\(566\) −17.5386 −0.737201
\(567\) 0 0
\(568\) −35.2061 −1.47722
\(569\) −35.7909 −1.50043 −0.750216 0.661193i \(-0.770050\pi\)
−0.750216 + 0.661193i \(0.770050\pi\)
\(570\) 0 0
\(571\) 26.5947 1.11295 0.556476 0.830864i \(-0.312153\pi\)
0.556476 + 0.830864i \(0.312153\pi\)
\(572\) −6.43874 −0.269217
\(573\) 0 0
\(574\) −2.16234 −0.0902545
\(575\) 3.28741 0.137094
\(576\) 0 0
\(577\) −18.2595 −0.760154 −0.380077 0.924955i \(-0.624102\pi\)
−0.380077 + 0.924955i \(0.624102\pi\)
\(578\) −57.1118 −2.37554
\(579\) 0 0
\(580\) 1.11061 0.0461156
\(581\) 31.1643 1.29291
\(582\) 0 0
\(583\) −21.0678 −0.872538
\(584\) 23.7288 0.981906
\(585\) 0 0
\(586\) 16.4774 0.680673
\(587\) −33.6391 −1.38843 −0.694217 0.719766i \(-0.744249\pi\)
−0.694217 + 0.719766i \(0.744249\pi\)
\(588\) 0 0
\(589\) −1.19413 −0.0492031
\(590\) −1.41504 −0.0582564
\(591\) 0 0
\(592\) −16.0192 −0.658384
\(593\) 42.7234 1.75444 0.877221 0.480088i \(-0.159395\pi\)
0.877221 + 0.480088i \(0.159395\pi\)
\(594\) 0 0
\(595\) −14.3423 −0.587975
\(596\) 2.45332 0.100492
\(597\) 0 0
\(598\) 4.86256 0.198845
\(599\) −39.6329 −1.61936 −0.809679 0.586873i \(-0.800359\pi\)
−0.809679 + 0.586873i \(0.800359\pi\)
\(600\) 0 0
\(601\) −0.109685 −0.00447414 −0.00223707 0.999997i \(-0.500712\pi\)
−0.00223707 + 0.999997i \(0.500712\pi\)
\(602\) 65.5869 2.67312
\(603\) 0 0
\(604\) −0.0992163 −0.00403705
\(605\) 2.52920 0.102827
\(606\) 0 0
\(607\) 24.6085 0.998827 0.499414 0.866364i \(-0.333549\pi\)
0.499414 + 0.866364i \(0.333549\pi\)
\(608\) −5.58607 −0.226545
\(609\) 0 0
\(610\) −1.13926 −0.0461273
\(611\) 37.3774 1.51213
\(612\) 0 0
\(613\) −18.4684 −0.745931 −0.372965 0.927845i \(-0.621659\pi\)
−0.372965 + 0.927845i \(0.621659\pi\)
\(614\) 18.2771 0.737605
\(615\) 0 0
\(616\) −29.6777 −1.19575
\(617\) 6.85488 0.275967 0.137983 0.990435i \(-0.455938\pi\)
0.137983 + 0.990435i \(0.455938\pi\)
\(618\) 0 0
\(619\) 23.9096 0.961007 0.480503 0.876993i \(-0.340454\pi\)
0.480503 + 0.876993i \(0.340454\pi\)
\(620\) −0.121496 −0.00487940
\(621\) 0 0
\(622\) −13.6120 −0.545790
\(623\) 71.9914 2.88427
\(624\) 0 0
\(625\) 22.5889 0.903554
\(626\) −7.86823 −0.314478
\(627\) 0 0
\(628\) 2.67540 0.106760
\(629\) −46.8053 −1.86625
\(630\) 0 0
\(631\) 30.1531 1.20037 0.600187 0.799859i \(-0.295093\pi\)
0.600187 + 0.799859i \(0.295093\pi\)
\(632\) −43.7514 −1.74034
\(633\) 0 0
\(634\) 19.6465 0.780263
\(635\) −0.00758374 −0.000300951 0
\(636\) 0 0
\(637\) 75.3053 2.98370
\(638\) −14.4692 −0.572842
\(639\) 0 0
\(640\) 2.12406 0.0839609
\(641\) 3.69224 0.145835 0.0729174 0.997338i \(-0.476769\pi\)
0.0729174 + 0.997338i \(0.476769\pi\)
\(642\) 0 0
\(643\) 41.0966 1.62069 0.810346 0.585952i \(-0.199279\pi\)
0.810346 + 0.585952i \(0.199279\pi\)
\(644\) −1.53241 −0.0603855
\(645\) 0 0
\(646\) 19.5598 0.769569
\(647\) −6.20476 −0.243934 −0.121967 0.992534i \(-0.538920\pi\)
−0.121967 + 0.992534i \(0.538920\pi\)
\(648\) 0 0
\(649\) −6.24306 −0.245062
\(650\) 34.6140 1.35767
\(651\) 0 0
\(652\) 6.66839 0.261154
\(653\) 1.87880 0.0735231 0.0367615 0.999324i \(-0.488296\pi\)
0.0367615 + 0.999324i \(0.488296\pi\)
\(654\) 0 0
\(655\) 2.14151 0.0836758
\(656\) 1.08444 0.0423404
\(657\) 0 0
\(658\) 34.7836 1.35601
\(659\) −36.3588 −1.41634 −0.708170 0.706042i \(-0.750479\pi\)
−0.708170 + 0.706042i \(0.750479\pi\)
\(660\) 0 0
\(661\) −48.1205 −1.87167 −0.935836 0.352436i \(-0.885353\pi\)
−0.935836 + 0.352436i \(0.885353\pi\)
\(662\) −13.9114 −0.540681
\(663\) 0 0
\(664\) −21.4180 −0.831179
\(665\) 3.60158 0.139663
\(666\) 0 0
\(667\) −3.70045 −0.143282
\(668\) −12.2614 −0.474406
\(669\) 0 0
\(670\) 5.65226 0.218366
\(671\) −5.02633 −0.194039
\(672\) 0 0
\(673\) 17.7655 0.684811 0.342405 0.939552i \(-0.388758\pi\)
0.342405 + 0.939552i \(0.388758\pi\)
\(674\) −8.24575 −0.317614
\(675\) 0 0
\(676\) −10.7610 −0.413884
\(677\) 18.9540 0.728463 0.364232 0.931308i \(-0.381332\pi\)
0.364232 + 0.931308i \(0.381332\pi\)
\(678\) 0 0
\(679\) 38.3395 1.47133
\(680\) 9.85688 0.377994
\(681\) 0 0
\(682\) 1.58287 0.0606113
\(683\) 19.7409 0.755363 0.377681 0.925936i \(-0.376721\pi\)
0.377681 + 0.925936i \(0.376721\pi\)
\(684\) 0 0
\(685\) 2.03886 0.0779007
\(686\) 31.9455 1.21968
\(687\) 0 0
\(688\) −32.8927 −1.25402
\(689\) −56.7317 −2.16131
\(690\) 0 0
\(691\) −13.4678 −0.512338 −0.256169 0.966632i \(-0.582460\pi\)
−0.256169 + 0.966632i \(0.582460\pi\)
\(692\) 12.1063 0.460213
\(693\) 0 0
\(694\) −8.48900 −0.322238
\(695\) 3.98890 0.151308
\(696\) 0 0
\(697\) 3.16856 0.120018
\(698\) −19.2089 −0.727069
\(699\) 0 0
\(700\) −10.9085 −0.412301
\(701\) 36.8430 1.39154 0.695770 0.718265i \(-0.255063\pi\)
0.695770 + 0.718265i \(0.255063\pi\)
\(702\) 0 0
\(703\) 11.7536 0.443295
\(704\) 19.2834 0.726769
\(705\) 0 0
\(706\) −14.3122 −0.538648
\(707\) 49.3325 1.85534
\(708\) 0 0
\(709\) −2.68748 −0.100931 −0.0504653 0.998726i \(-0.516070\pi\)
−0.0504653 + 0.998726i \(0.516070\pi\)
\(710\) −5.66339 −0.212543
\(711\) 0 0
\(712\) −49.4769 −1.85422
\(713\) 0.404813 0.0151604
\(714\) 0 0
\(715\) −5.13007 −0.191854
\(716\) 4.12046 0.153989
\(717\) 0 0
\(718\) 24.4885 0.913903
\(719\) −18.2224 −0.679582 −0.339791 0.940501i \(-0.610356\pi\)
−0.339791 + 0.940501i \(0.610356\pi\)
\(720\) 0 0
\(721\) 25.4818 0.948993
\(722\) 18.3122 0.681508
\(723\) 0 0
\(724\) 11.1553 0.414584
\(725\) −26.3416 −0.978301
\(726\) 0 0
\(727\) −7.91643 −0.293604 −0.146802 0.989166i \(-0.546898\pi\)
−0.146802 + 0.989166i \(0.546898\pi\)
\(728\) −79.9167 −2.96191
\(729\) 0 0
\(730\) 3.81711 0.141278
\(731\) −96.1068 −3.55464
\(732\) 0 0
\(733\) 1.87435 0.0692307 0.0346154 0.999401i \(-0.488979\pi\)
0.0346154 + 0.999401i \(0.488979\pi\)
\(734\) 25.5504 0.943082
\(735\) 0 0
\(736\) 1.89370 0.0698027
\(737\) 24.9374 0.918580
\(738\) 0 0
\(739\) −27.7482 −1.02074 −0.510368 0.859956i \(-0.670491\pi\)
−0.510368 + 0.859956i \(0.670491\pi\)
\(740\) 1.19587 0.0439610
\(741\) 0 0
\(742\) −52.7949 −1.93816
\(743\) 44.2764 1.62434 0.812171 0.583420i \(-0.198286\pi\)
0.812171 + 0.583420i \(0.198286\pi\)
\(744\) 0 0
\(745\) 1.95469 0.0716141
\(746\) −3.53597 −0.129461
\(747\) 0 0
\(748\) 8.78020 0.321036
\(749\) 62.2395 2.27418
\(750\) 0 0
\(751\) −38.6448 −1.41017 −0.705084 0.709124i \(-0.749091\pi\)
−0.705084 + 0.709124i \(0.749091\pi\)
\(752\) −17.4444 −0.636134
\(753\) 0 0
\(754\) −38.9630 −1.41895
\(755\) −0.0790506 −0.00287695
\(756\) 0 0
\(757\) −24.7185 −0.898408 −0.449204 0.893429i \(-0.648292\pi\)
−0.449204 + 0.893429i \(0.648292\pi\)
\(758\) −28.7919 −1.04577
\(759\) 0 0
\(760\) −2.47523 −0.0897859
\(761\) 48.9136 1.77312 0.886559 0.462616i \(-0.153089\pi\)
0.886559 + 0.462616i \(0.153089\pi\)
\(762\) 0 0
\(763\) 31.4211 1.13752
\(764\) −11.6431 −0.421232
\(765\) 0 0
\(766\) 10.4161 0.376348
\(767\) −16.8114 −0.607026
\(768\) 0 0
\(769\) −44.2367 −1.59522 −0.797609 0.603175i \(-0.793902\pi\)
−0.797609 + 0.603175i \(0.793902\pi\)
\(770\) −4.77408 −0.172046
\(771\) 0 0
\(772\) −3.95052 −0.142182
\(773\) −1.82530 −0.0656513 −0.0328257 0.999461i \(-0.510451\pi\)
−0.0328257 + 0.999461i \(0.510451\pi\)
\(774\) 0 0
\(775\) 2.88166 0.103512
\(776\) −26.3492 −0.945882
\(777\) 0 0
\(778\) −24.5452 −0.879988
\(779\) −0.795679 −0.0285082
\(780\) 0 0
\(781\) −24.9865 −0.894087
\(782\) −6.63084 −0.237118
\(783\) 0 0
\(784\) −35.1458 −1.25521
\(785\) 2.13163 0.0760811
\(786\) 0 0
\(787\) −4.69705 −0.167432 −0.0837159 0.996490i \(-0.526679\pi\)
−0.0837159 + 0.996490i \(0.526679\pi\)
\(788\) 3.43292 0.122293
\(789\) 0 0
\(790\) −7.03803 −0.250402
\(791\) −78.3598 −2.78615
\(792\) 0 0
\(793\) −13.5350 −0.480642
\(794\) −31.0344 −1.10137
\(795\) 0 0
\(796\) 4.95353 0.175573
\(797\) 27.0924 0.959663 0.479831 0.877361i \(-0.340698\pi\)
0.479831 + 0.877361i \(0.340698\pi\)
\(798\) 0 0
\(799\) −50.9697 −1.80318
\(800\) 13.4803 0.476600
\(801\) 0 0
\(802\) −2.44067 −0.0861830
\(803\) 16.8408 0.594300
\(804\) 0 0
\(805\) −1.22095 −0.0430329
\(806\) 4.26239 0.150136
\(807\) 0 0
\(808\) −33.9043 −1.19275
\(809\) 25.4431 0.894533 0.447266 0.894401i \(-0.352398\pi\)
0.447266 + 0.894401i \(0.352398\pi\)
\(810\) 0 0
\(811\) −1.52467 −0.0535386 −0.0267693 0.999642i \(-0.508522\pi\)
−0.0267693 + 0.999642i \(0.508522\pi\)
\(812\) 12.2790 0.430909
\(813\) 0 0
\(814\) −15.5800 −0.546078
\(815\) 5.31305 0.186108
\(816\) 0 0
\(817\) 24.1340 0.844343
\(818\) 28.1353 0.983727
\(819\) 0 0
\(820\) −0.0809562 −0.00282711
\(821\) −25.4471 −0.888111 −0.444056 0.895999i \(-0.646461\pi\)
−0.444056 + 0.895999i \(0.646461\pi\)
\(822\) 0 0
\(823\) 49.2594 1.71708 0.858538 0.512750i \(-0.171373\pi\)
0.858538 + 0.512750i \(0.171373\pi\)
\(824\) −17.5127 −0.610083
\(825\) 0 0
\(826\) −15.6448 −0.544354
\(827\) −23.6493 −0.822367 −0.411183 0.911553i \(-0.634884\pi\)
−0.411183 + 0.911553i \(0.634884\pi\)
\(828\) 0 0
\(829\) 23.2134 0.806234 0.403117 0.915149i \(-0.367927\pi\)
0.403117 + 0.915149i \(0.367927\pi\)
\(830\) −3.44538 −0.119591
\(831\) 0 0
\(832\) 51.9266 1.80023
\(833\) −102.690 −3.55800
\(834\) 0 0
\(835\) −9.76924 −0.338079
\(836\) −2.20485 −0.0762565
\(837\) 0 0
\(838\) −4.50587 −0.155653
\(839\) −22.9974 −0.793957 −0.396978 0.917828i \(-0.629941\pi\)
−0.396978 + 0.917828i \(0.629941\pi\)
\(840\) 0 0
\(841\) 0.651179 0.0224545
\(842\) −23.6500 −0.815033
\(843\) 0 0
\(844\) 6.01674 0.207105
\(845\) −8.57382 −0.294948
\(846\) 0 0
\(847\) 27.9631 0.960822
\(848\) 26.4774 0.909236
\(849\) 0 0
\(850\) −47.2015 −1.61900
\(851\) −3.98452 −0.136587
\(852\) 0 0
\(853\) 5.94006 0.203384 0.101692 0.994816i \(-0.467574\pi\)
0.101692 + 0.994816i \(0.467574\pi\)
\(854\) −12.5958 −0.431018
\(855\) 0 0
\(856\) −42.7748 −1.46201
\(857\) 45.2128 1.54444 0.772219 0.635356i \(-0.219147\pi\)
0.772219 + 0.635356i \(0.219147\pi\)
\(858\) 0 0
\(859\) 37.9258 1.29401 0.647006 0.762485i \(-0.276021\pi\)
0.647006 + 0.762485i \(0.276021\pi\)
\(860\) 2.45551 0.0837322
\(861\) 0 0
\(862\) −12.3796 −0.421650
\(863\) 0.597129 0.0203265 0.0101633 0.999948i \(-0.496765\pi\)
0.0101633 + 0.999948i \(0.496765\pi\)
\(864\) 0 0
\(865\) 9.64571 0.327964
\(866\) 42.3620 1.43952
\(867\) 0 0
\(868\) −1.34327 −0.0455936
\(869\) −31.0513 −1.05334
\(870\) 0 0
\(871\) 67.1518 2.27535
\(872\) −21.5945 −0.731281
\(873\) 0 0
\(874\) 1.66511 0.0563233
\(875\) −17.6746 −0.597510
\(876\) 0 0
\(877\) 10.9382 0.369357 0.184679 0.982799i \(-0.440876\pi\)
0.184679 + 0.982799i \(0.440876\pi\)
\(878\) −3.69506 −0.124702
\(879\) 0 0
\(880\) 2.39426 0.0807106
\(881\) 22.2215 0.748663 0.374331 0.927295i \(-0.377872\pi\)
0.374331 + 0.927295i \(0.377872\pi\)
\(882\) 0 0
\(883\) −56.4126 −1.89843 −0.949217 0.314623i \(-0.898122\pi\)
−0.949217 + 0.314623i \(0.898122\pi\)
\(884\) 23.6435 0.795216
\(885\) 0 0
\(886\) −28.2439 −0.948873
\(887\) 12.6609 0.425112 0.212556 0.977149i \(-0.431821\pi\)
0.212556 + 0.977149i \(0.431821\pi\)
\(888\) 0 0
\(889\) −0.0838465 −0.00281212
\(890\) −7.95905 −0.266788
\(891\) 0 0
\(892\) −10.6474 −0.356501
\(893\) 12.7993 0.428314
\(894\) 0 0
\(895\) 3.28298 0.109738
\(896\) 23.4838 0.784539
\(897\) 0 0
\(898\) 15.9491 0.532228
\(899\) −3.24371 −0.108184
\(900\) 0 0
\(901\) 77.3623 2.57731
\(902\) 1.05471 0.0351181
\(903\) 0 0
\(904\) 53.8536 1.79115
\(905\) 8.88800 0.295447
\(906\) 0 0
\(907\) −40.0352 −1.32935 −0.664673 0.747134i \(-0.731429\pi\)
−0.664673 + 0.747134i \(0.731429\pi\)
\(908\) −8.35068 −0.277127
\(909\) 0 0
\(910\) −12.8557 −0.426163
\(911\) 17.5878 0.582708 0.291354 0.956615i \(-0.405894\pi\)
0.291354 + 0.956615i \(0.405894\pi\)
\(912\) 0 0
\(913\) −15.2008 −0.503072
\(914\) 28.0556 0.927998
\(915\) 0 0
\(916\) 2.62198 0.0866327
\(917\) 23.6768 0.781876
\(918\) 0 0
\(919\) −15.7906 −0.520883 −0.260442 0.965490i \(-0.583868\pi\)
−0.260442 + 0.965490i \(0.583868\pi\)
\(920\) 0.839112 0.0276647
\(921\) 0 0
\(922\) 35.7522 1.17744
\(923\) −67.2841 −2.21468
\(924\) 0 0
\(925\) −28.3637 −0.932593
\(926\) 0.256630 0.00843339
\(927\) 0 0
\(928\) −15.1740 −0.498110
\(929\) −40.0331 −1.31344 −0.656722 0.754133i \(-0.728057\pi\)
−0.656722 + 0.754133i \(0.728057\pi\)
\(930\) 0 0
\(931\) 25.7872 0.845142
\(932\) 5.98393 0.196010
\(933\) 0 0
\(934\) 31.3158 1.02468
\(935\) 6.99563 0.228781
\(936\) 0 0
\(937\) −39.9832 −1.30620 −0.653098 0.757274i \(-0.726531\pi\)
−0.653098 + 0.757274i \(0.726531\pi\)
\(938\) 62.4920 2.04044
\(939\) 0 0
\(940\) 1.30227 0.0424753
\(941\) −13.1615 −0.429052 −0.214526 0.976718i \(-0.568821\pi\)
−0.214526 + 0.976718i \(0.568821\pi\)
\(942\) 0 0
\(943\) 0.269738 0.00878389
\(944\) 7.84609 0.255369
\(945\) 0 0
\(946\) −31.9908 −1.04011
\(947\) 18.6968 0.607565 0.303783 0.952741i \(-0.401750\pi\)
0.303783 + 0.952741i \(0.401750\pi\)
\(948\) 0 0
\(949\) 45.3493 1.47210
\(950\) 11.8531 0.384565
\(951\) 0 0
\(952\) 108.979 3.53202
\(953\) −4.57308 −0.148137 −0.0740684 0.997253i \(-0.523598\pi\)
−0.0740684 + 0.997253i \(0.523598\pi\)
\(954\) 0 0
\(955\) −9.27663 −0.300185
\(956\) 0.505952 0.0163637
\(957\) 0 0
\(958\) −38.5498 −1.24549
\(959\) 22.5418 0.727913
\(960\) 0 0
\(961\) −30.6452 −0.988553
\(962\) −41.9541 −1.35265
\(963\) 0 0
\(964\) 7.68646 0.247564
\(965\) −3.14758 −0.101324
\(966\) 0 0
\(967\) −30.1643 −0.970020 −0.485010 0.874509i \(-0.661184\pi\)
−0.485010 + 0.874509i \(0.661184\pi\)
\(968\) −19.2179 −0.617688
\(969\) 0 0
\(970\) −4.23864 −0.136095
\(971\) 29.5541 0.948437 0.474219 0.880407i \(-0.342731\pi\)
0.474219 + 0.880407i \(0.342731\pi\)
\(972\) 0 0
\(973\) 44.1017 1.41383
\(974\) 16.2515 0.520732
\(975\) 0 0
\(976\) 6.31694 0.202200
\(977\) 25.0126 0.800223 0.400112 0.916466i \(-0.368971\pi\)
0.400112 + 0.916466i \(0.368971\pi\)
\(978\) 0 0
\(979\) −35.1147 −1.12227
\(980\) 2.62372 0.0838115
\(981\) 0 0
\(982\) 16.9709 0.541563
\(983\) 6.59634 0.210391 0.105195 0.994452i \(-0.466453\pi\)
0.105195 + 0.994452i \(0.466453\pi\)
\(984\) 0 0
\(985\) 2.73518 0.0871502
\(986\) 53.1320 1.69207
\(987\) 0 0
\(988\) −5.93727 −0.188890
\(989\) −8.18153 −0.260157
\(990\) 0 0
\(991\) −5.17310 −0.164329 −0.0821645 0.996619i \(-0.526183\pi\)
−0.0821645 + 0.996619i \(0.526183\pi\)
\(992\) 1.65997 0.0527040
\(993\) 0 0
\(994\) −62.6150 −1.98603
\(995\) 3.94673 0.125120
\(996\) 0 0
\(997\) −32.1810 −1.01918 −0.509591 0.860417i \(-0.670203\pi\)
−0.509591 + 0.860417i \(0.670203\pi\)
\(998\) −8.86627 −0.280657
\(999\) 0 0
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 2151.2.a.j.1.7 20
3.2 odd 2 2151.2.a.k.1.14 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.7 20 1.1 even 1 trivial
2151.2.a.k.1.14 yes 20 3.2 odd 2