Properties

Label 2151.2.a.k.1.14
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $0$
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: \(0\)
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 1.14
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 \(-0.357754\pi\)
0.432153 + 0.901800i \(0.357754\pi\)
\(3\) 0 0
\(4\) −0.505952 −0.252976
\(5\) 0.403117 0.180280 0.0901398 0.995929i \(-0.471269\pi\)
0.0901398 + 0.995929i \(0.471269\pi\)
\(6\) 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 \(-0.606284\pi\)
−0.327730 + 0.944772i \(0.606284\pi\)
\(12\) 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 \(-0.0806781\pi\)
0.968051 + 0.250753i \(0.0806781\pi\)
\(18\) 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 \(-0.477429\pi\)
0.0708499 + 0.997487i \(0.477429\pi\)
\(24\) 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 \(-0.668722\pi\)
−0.505582 + 0.862778i \(0.668722\pi\)
\(30\) 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 \(-0.490133\pi\)
0.0309947 + 0.999520i \(0.490133\pi\)
\(42\) 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 \(-0.654187\pi\)
−0.465672 + 0.884957i \(0.654187\pi\)
\(48\) 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 \(-0.268179\pi\)
0.665593 + 0.746315i \(0.268179\pi\)
\(54\) 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 \(-0.440143\pi\)
0.186939 + 0.982372i \(0.440143\pi\)
\(60\) 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 \(-0.261097\pi\)
0.682030 + 0.731324i \(0.261097\pi\)
\(72\) 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 \(-0.374631\pi\)
0.383755 + 0.923435i \(0.374631\pi\)
\(84\) 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 \(-0.172883\pi\)
0.856096 + 0.516818i \(0.172883\pi\)
\(90\) 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 \(-0.314364\pi\)
0.550693 + 0.834708i \(0.314364\pi\)
\(102\) 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 \(-0.264139\pi\)
0.675011 + 0.737808i \(0.264139\pi\)
\(108\) 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 \(-0.809938\pi\)
−0.826972 + 0.562243i \(0.809938\pi\)
\(114\) 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 \(-0.425449\pi\)
0.232072 + 0.972699i \(0.425449\pi\)
\(132\) 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 \(-0.430681\pi\)
0.216055 + 0.976381i \(0.430681\pi\)
\(138\) 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 \(-0.436354\pi\)
0.198620 + 0.980077i \(0.436354\pi\)
\(150\) 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 \(-0.887004\pi\)
−0.937651 + 0.347579i \(0.887004\pi\)
\(168\) 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 \(-0.136389\pi\)
0.909599 + 0.415488i \(0.136389\pi\)
\(174\) 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 \(-0.401559\pi\)
0.304354 + 0.952559i \(0.401559\pi\)
\(180\) 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 \(-0.813122\pi\)
−0.832553 + 0.553945i \(0.813122\pi\)
\(192\) 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 \(-0.422292\pi\)
0.241708 + 0.970349i \(0.422292\pi\)
\(198\) 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 \(-0.684509\pi\)
−0.547734 + 0.836653i \(0.684509\pi\)
\(228\) 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 \(-0.373370\pi\)
0.387409 + 0.921908i \(0.373370\pi\)
\(234\) 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 \(-0.580012\pi\)
−0.248725 + 0.968574i \(0.580012\pi\)
\(252\) 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 \(-0.153972\pi\)
0.885273 + 0.465072i \(0.153972\pi\)
\(258\) 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 \(-0.326099\pi\)
0.519552 + 0.854439i \(0.326099\pi\)
\(264\) 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 \(-0.576375\pi\)
−0.237643 + 0.971353i \(0.576375\pi\)
\(270\) 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 \(-0.332528\pi\)
0.502190 + 0.864758i \(0.332528\pi\)
\(282\) 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 \(-0.371171\pi\)
0.393769 + 0.919209i \(0.371171\pi\)
\(294\) 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 \(-0.602252\pi\)
−0.315739 + 0.948846i \(0.602252\pi\)
\(312\) 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 \(-0.350931\pi\)
0.451382 + 0.892331i \(0.350931\pi\)
\(318\) 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 \(-0.559687\pi\)
−0.186414 + 0.982471i \(0.559687\pi\)
\(348\) 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 \(-0.600867\pi\)
−0.311607 + 0.950211i \(0.600867\pi\)
\(354\) 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 \(-0.322683\pi\)
0.528692 + 0.848814i \(0.322683\pi\)
\(360\) 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 \(-0.430139\pi\)
0.217717 + 0.976012i \(0.430139\pi\)
\(384\) 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 \(-0.670011\pi\)
−0.509072 + 0.860724i \(0.670011\pi\)
\(390\) 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 \(-0.515876\pi\)
−0.0498568 + 0.998756i \(0.515876\pi\)
\(402\) 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 \(-0.528701\pi\)
−0.0900450 + 0.995938i \(0.528701\pi\)
\(420\) 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 \(-0.578435\pi\)
−0.243924 + 0.969794i \(0.578435\pi\)
\(432\) 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 \(-0.684962\pi\)
−0.548922 + 0.835874i \(0.684962\pi\)
\(444\) 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 \(-0.400376\pi\)
0.307894 + 0.951421i \(0.400376\pi\)
\(450\) 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 \(-0.261482\pi\)
0.681145 + 0.732148i \(0.261482\pi\)
\(462\) 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 \(-0.298031\pi\)
0.592778 + 0.805366i \(0.298031\pi\)
\(468\) 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 \(-0.756094\pi\)
−0.720513 + 0.693441i \(0.756094\pi\)
\(480\) 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 \(-0.398568\pi\)
0.313294 + 0.949656i \(0.398568\pi\)
\(492\) 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 \(-0.455941\pi\)
0.137974 + 0.990436i \(0.455941\pi\)
\(504\) 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 \(-0.432047\pi\)
0.211863 + 0.977299i \(0.432047\pi\)
\(510\) 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 \(-0.179503\pi\)
0.845163 + 0.534509i \(0.179503\pi\)
\(522\) 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 \(-0.302164\pi\)
0.582272 + 0.812994i \(0.302164\pi\)
\(558\) 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 \(-0.432892\pi\)
0.209267 + 0.977858i \(0.432892\pi\)
\(564\) 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 \(-0.229950\pi\)
0.750216 + 0.661193i \(0.229950\pi\)
\(570\) 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 \(-0.255751\pi\)
0.694217 + 0.719766i \(0.255751\pi\)
\(588\) 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 \(-0.840605\pi\)
−0.877221 + 0.480088i \(0.840605\pi\)
\(594\) 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 \(-0.199641\pi\)
0.809679 + 0.586873i \(0.199641\pi\)
\(600\) 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 \(-0.544062\pi\)
−0.137983 + 0.990435i \(0.544062\pi\)
\(618\) 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 \(-0.523231\pi\)
−0.0729174 + 0.997338i \(0.523231\pi\)
\(642\) 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 \(-0.461080\pi\)
0.121967 + 0.992534i \(0.461080\pi\)
\(648\) 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 \(-0.511704\pi\)
−0.0367615 + 0.999324i \(0.511704\pi\)
\(654\) 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 \(-0.249521\pi\)
0.708170 + 0.706042i \(0.249521\pi\)
\(660\) 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 \(-0.618668\pi\)
−0.364232 + 0.931308i \(0.618668\pi\)
\(678\) 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 \(-0.623279\pi\)
−0.377681 + 0.925936i \(0.623279\pi\)
\(684\) 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 \(-0.744937\pi\)
−0.695770 + 0.718265i \(0.744937\pi\)
\(702\) 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 \(-0.389644\pi\)
0.339791 + 0.940501i \(0.389644\pi\)
\(720\) 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 \(-0.801714\pi\)
−0.812171 + 0.583420i \(0.801714\pi\)
\(744\) 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 \(-0.846911\pi\)
−0.886559 + 0.462616i \(0.846911\pi\)
\(762\) 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 \(-0.489549\pi\)
0.0328257 + 0.999461i \(0.489549\pi\)
\(774\) 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 \(-0.659302\pi\)
−0.479831 + 0.877361i \(0.659302\pi\)
\(798\) 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 \(-0.647602\pi\)
−0.447266 + 0.894401i \(0.647602\pi\)
\(810\) 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 \(-0.353539\pi\)
0.444056 + 0.895999i \(0.353539\pi\)
\(822\) 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 \(-0.365116\pi\)
0.411183 + 0.911553i \(0.365116\pi\)
\(828\) 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 \(-0.370059\pi\)
0.396978 + 0.917828i \(0.370059\pi\)
\(840\) 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 \(-0.780853\pi\)
−0.772219 + 0.635356i \(0.780853\pi\)
\(858\) 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 \(-0.503235\pi\)
−0.0101633 + 0.999948i \(0.503235\pi\)
\(864\) 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 \(-0.622128\pi\)
−0.374331 + 0.927295i \(0.622128\pi\)
\(882\) 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 \(-0.568179\pi\)
−0.212556 + 0.977149i \(0.568179\pi\)
\(888\) 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 \(-0.594106\pi\)
−0.291354 + 0.956615i \(0.594106\pi\)
\(912\) 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 \(-0.271943\pi\)
0.656722 + 0.754133i \(0.271943\pi\)
\(930\) 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 \(-0.431179\pi\)
0.214526 + 0.976718i \(0.431179\pi\)
\(942\) 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 \(-0.598250\pi\)
−0.303783 + 0.952741i \(0.598250\pi\)
\(948\) 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 \(-0.476402\pi\)
0.0740684 + 0.997253i \(0.476402\pi\)
\(954\) 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 \(-0.657269\pi\)
−0.474219 + 0.880407i \(0.657269\pi\)
\(972\) 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 \(-0.631029\pi\)
−0.400112 + 0.916466i \(0.631029\pi\)
\(978\) 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 \(-0.533547\pi\)
−0.105195 + 0.994452i \(0.533547\pi\)
\(984\) 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.k.1.14 yes 20
3.2 odd 2 2151.2.a.j.1.7 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.7 20 3.2 odd 2
2151.2.a.k.1.14 yes 20 1.1 even 1 trivial