Properties

Label 2169.2.a.e.1.2
Level $2169$
Weight $2$
Character 2169.1
Self dual yes
Analytic conductor $17.320$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2169 = 3^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2169.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.3195521984\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: 7.7.31056073.1
Defining polynomial: \(x^{7} - 3 x^{6} - 3 x^{5} + 11 x^{4} + x^{3} - 9 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 241)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.48734\) of defining polynomial
Character \(\chi\) \(=\) 2169.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.487343 q^{2} -1.76250 q^{4} -0.961999 q^{5} -4.61392 q^{7} +1.83363 q^{8} +O(q^{10})\) \(q-0.487343 q^{2} -1.76250 q^{4} -0.961999 q^{5} -4.61392 q^{7} +1.83363 q^{8} +0.468824 q^{10} -1.93974 q^{11} -3.85571 q^{13} +2.24856 q^{14} +2.63139 q^{16} -5.40289 q^{17} -4.17145 q^{19} +1.69552 q^{20} +0.945321 q^{22} +1.42545 q^{23} -4.07456 q^{25} +1.87905 q^{26} +8.13202 q^{28} +4.85744 q^{29} -7.24699 q^{31} -4.94964 q^{32} +2.63306 q^{34} +4.43859 q^{35} +7.12597 q^{37} +2.03293 q^{38} -1.76395 q^{40} -9.18955 q^{41} +2.93624 q^{43} +3.41879 q^{44} -0.694685 q^{46} -2.48170 q^{47} +14.2883 q^{49} +1.98571 q^{50} +6.79568 q^{52} -5.64997 q^{53} +1.86603 q^{55} -8.46022 q^{56} -2.36724 q^{58} +11.9783 q^{59} -13.9214 q^{61} +3.53177 q^{62} -2.85060 q^{64} +3.70919 q^{65} -7.30682 q^{67} +9.52258 q^{68} -2.16312 q^{70} +14.8844 q^{71} +0.240264 q^{73} -3.47279 q^{74} +7.35217 q^{76} +8.94983 q^{77} -3.15128 q^{79} -2.53139 q^{80} +4.47847 q^{82} +2.46821 q^{83} +5.19758 q^{85} -1.43096 q^{86} -3.55677 q^{88} -5.55181 q^{89} +17.7900 q^{91} -2.51235 q^{92} +1.20944 q^{94} +4.01293 q^{95} +5.27964 q^{97} -6.96330 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 4q^{2} + 2q^{4} + 8q^{5} - 7q^{7} + 6q^{8} + O(q^{10}) \) \( 7q + 4q^{2} + 2q^{4} + 8q^{5} - 7q^{7} + 6q^{8} + 3q^{10} + 18q^{11} - q^{13} + 6q^{14} + 4q^{16} + 2q^{17} - 6q^{19} + 8q^{20} + 10q^{22} + 22q^{23} + 5q^{25} - 8q^{26} + 9q^{28} + 16q^{29} - 18q^{31} + 6q^{32} + 11q^{34} - 7q^{35} + 8q^{37} - 16q^{38} + 14q^{40} + 15q^{41} + 14q^{43} + 4q^{44} + 11q^{46} + 10q^{47} + 6q^{49} + 4q^{50} + 27q^{52} - 15q^{53} + 29q^{55} - 13q^{56} + 17q^{58} + 18q^{59} + 4q^{61} - 13q^{62} + 2q^{64} + 7q^{65} + 18q^{67} + 15q^{68} + 8q^{70} + 50q^{71} - 10q^{74} - 20q^{76} - 17q^{77} - 15q^{79} + 11q^{80} + 45q^{82} + 24q^{83} - 2q^{85} + 23q^{86} + 8q^{88} + 13q^{89} - 12q^{91} + 10q^{92} - 32q^{94} + 41q^{95} + q^{97} - 9q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.487343 −0.344604 −0.172302 0.985044i \(-0.555120\pi\)
−0.172302 + 0.985044i \(0.555120\pi\)
\(3\) 0 0
\(4\) −1.76250 −0.881248
\(5\) −0.961999 −0.430219 −0.215110 0.976590i \(-0.569011\pi\)
−0.215110 + 0.976590i \(0.569011\pi\)
\(6\) 0 0
\(7\) −4.61392 −1.74390 −0.871950 0.489596i \(-0.837144\pi\)
−0.871950 + 0.489596i \(0.837144\pi\)
\(8\) 1.83363 0.648285
\(9\) 0 0
\(10\) 0.468824 0.148255
\(11\) −1.93974 −0.584855 −0.292427 0.956288i \(-0.594463\pi\)
−0.292427 + 0.956288i \(0.594463\pi\)
\(12\) 0 0
\(13\) −3.85571 −1.06938 −0.534691 0.845048i \(-0.679572\pi\)
−0.534691 + 0.845048i \(0.679572\pi\)
\(14\) 2.24856 0.600954
\(15\) 0 0
\(16\) 2.63139 0.657847
\(17\) −5.40289 −1.31039 −0.655197 0.755458i \(-0.727414\pi\)
−0.655197 + 0.755458i \(0.727414\pi\)
\(18\) 0 0
\(19\) −4.17145 −0.956997 −0.478498 0.878088i \(-0.658819\pi\)
−0.478498 + 0.878088i \(0.658819\pi\)
\(20\) 1.69552 0.379130
\(21\) 0 0
\(22\) 0.945321 0.201543
\(23\) 1.42545 0.297227 0.148614 0.988895i \(-0.452519\pi\)
0.148614 + 0.988895i \(0.452519\pi\)
\(24\) 0 0
\(25\) −4.07456 −0.814911
\(26\) 1.87905 0.368513
\(27\) 0 0
\(28\) 8.13202 1.53681
\(29\) 4.85744 0.902003 0.451002 0.892523i \(-0.351067\pi\)
0.451002 + 0.892523i \(0.351067\pi\)
\(30\) 0 0
\(31\) −7.24699 −1.30160 −0.650799 0.759250i \(-0.725566\pi\)
−0.650799 + 0.759250i \(0.725566\pi\)
\(32\) −4.94964 −0.874982
\(33\) 0 0
\(34\) 2.63306 0.451567
\(35\) 4.43859 0.750259
\(36\) 0 0
\(37\) 7.12597 1.17150 0.585751 0.810491i \(-0.300800\pi\)
0.585751 + 0.810491i \(0.300800\pi\)
\(38\) 2.03293 0.329785
\(39\) 0 0
\(40\) −1.76395 −0.278905
\(41\) −9.18955 −1.43517 −0.717583 0.696473i \(-0.754752\pi\)
−0.717583 + 0.696473i \(0.754752\pi\)
\(42\) 0 0
\(43\) 2.93624 0.447772 0.223886 0.974615i \(-0.428126\pi\)
0.223886 + 0.974615i \(0.428126\pi\)
\(44\) 3.41879 0.515402
\(45\) 0 0
\(46\) −0.694685 −0.102426
\(47\) −2.48170 −0.361994 −0.180997 0.983484i \(-0.557932\pi\)
−0.180997 + 0.983484i \(0.557932\pi\)
\(48\) 0 0
\(49\) 14.2883 2.04118
\(50\) 1.98571 0.280822
\(51\) 0 0
\(52\) 6.79568 0.942391
\(53\) −5.64997 −0.776083 −0.388041 0.921642i \(-0.626848\pi\)
−0.388041 + 0.921642i \(0.626848\pi\)
\(54\) 0 0
\(55\) 1.86603 0.251616
\(56\) −8.46022 −1.13054
\(57\) 0 0
\(58\) −2.36724 −0.310834
\(59\) 11.9783 1.55944 0.779718 0.626131i \(-0.215362\pi\)
0.779718 + 0.626131i \(0.215362\pi\)
\(60\) 0 0
\(61\) −13.9214 −1.78245 −0.891223 0.453565i \(-0.850152\pi\)
−0.891223 + 0.453565i \(0.850152\pi\)
\(62\) 3.53177 0.448535
\(63\) 0 0
\(64\) −2.85060 −0.356325
\(65\) 3.70919 0.460069
\(66\) 0 0
\(67\) −7.30682 −0.892670 −0.446335 0.894866i \(-0.647271\pi\)
−0.446335 + 0.894866i \(0.647271\pi\)
\(68\) 9.52258 1.15478
\(69\) 0 0
\(70\) −2.16312 −0.258542
\(71\) 14.8844 1.76645 0.883226 0.468948i \(-0.155367\pi\)
0.883226 + 0.468948i \(0.155367\pi\)
\(72\) 0 0
\(73\) 0.240264 0.0281208 0.0140604 0.999901i \(-0.495524\pi\)
0.0140604 + 0.999901i \(0.495524\pi\)
\(74\) −3.47279 −0.403704
\(75\) 0 0
\(76\) 7.35217 0.843352
\(77\) 8.94983 1.01993
\(78\) 0 0
\(79\) −3.15128 −0.354546 −0.177273 0.984162i \(-0.556728\pi\)
−0.177273 + 0.984162i \(0.556728\pi\)
\(80\) −2.53139 −0.283018
\(81\) 0 0
\(82\) 4.47847 0.494564
\(83\) 2.46821 0.270922 0.135461 0.990783i \(-0.456749\pi\)
0.135461 + 0.990783i \(0.456749\pi\)
\(84\) 0 0
\(85\) 5.19758 0.563757
\(86\) −1.43096 −0.154304
\(87\) 0 0
\(88\) −3.55677 −0.379153
\(89\) −5.55181 −0.588491 −0.294246 0.955730i \(-0.595068\pi\)
−0.294246 + 0.955730i \(0.595068\pi\)
\(90\) 0 0
\(91\) 17.7900 1.86489
\(92\) −2.51235 −0.261931
\(93\) 0 0
\(94\) 1.20944 0.124744
\(95\) 4.01293 0.411718
\(96\) 0 0
\(97\) 5.27964 0.536067 0.268033 0.963410i \(-0.413626\pi\)
0.268033 + 0.963410i \(0.413626\pi\)
\(98\) −6.96330 −0.703400
\(99\) 0 0
\(100\) 7.18139 0.718139
\(101\) 12.4239 1.23623 0.618113 0.786089i \(-0.287897\pi\)
0.618113 + 0.786089i \(0.287897\pi\)
\(102\) 0 0
\(103\) −9.80533 −0.966148 −0.483074 0.875580i \(-0.660480\pi\)
−0.483074 + 0.875580i \(0.660480\pi\)
\(104\) −7.06994 −0.693264
\(105\) 0 0
\(106\) 2.75347 0.267441
\(107\) 9.15031 0.884593 0.442297 0.896869i \(-0.354164\pi\)
0.442297 + 0.896869i \(0.354164\pi\)
\(108\) 0 0
\(109\) 3.24534 0.310847 0.155424 0.987848i \(-0.450326\pi\)
0.155424 + 0.987848i \(0.450326\pi\)
\(110\) −0.909398 −0.0867077
\(111\) 0 0
\(112\) −12.1410 −1.14722
\(113\) 0.842874 0.0792909 0.0396455 0.999214i \(-0.487377\pi\)
0.0396455 + 0.999214i \(0.487377\pi\)
\(114\) 0 0
\(115\) −1.37128 −0.127873
\(116\) −8.56121 −0.794889
\(117\) 0 0
\(118\) −5.83752 −0.537387
\(119\) 24.9285 2.28520
\(120\) 0 0
\(121\) −7.23740 −0.657945
\(122\) 6.78448 0.614238
\(123\) 0 0
\(124\) 12.7728 1.14703
\(125\) 8.72972 0.780810
\(126\) 0 0
\(127\) 11.3416 1.00641 0.503203 0.864168i \(-0.332155\pi\)
0.503203 + 0.864168i \(0.332155\pi\)
\(128\) 11.2885 0.997773
\(129\) 0 0
\(130\) −1.80765 −0.158541
\(131\) −14.1261 −1.23421 −0.617103 0.786882i \(-0.711694\pi\)
−0.617103 + 0.786882i \(0.711694\pi\)
\(132\) 0 0
\(133\) 19.2468 1.66891
\(134\) 3.56093 0.307617
\(135\) 0 0
\(136\) −9.90689 −0.849509
\(137\) 12.9555 1.10686 0.553430 0.832896i \(-0.313319\pi\)
0.553430 + 0.832896i \(0.313319\pi\)
\(138\) 0 0
\(139\) 8.23606 0.698574 0.349287 0.937016i \(-0.386424\pi\)
0.349287 + 0.937016i \(0.386424\pi\)
\(140\) −7.82300 −0.661164
\(141\) 0 0
\(142\) −7.25380 −0.608726
\(143\) 7.47909 0.625433
\(144\) 0 0
\(145\) −4.67285 −0.388059
\(146\) −0.117091 −0.00969054
\(147\) 0 0
\(148\) −12.5595 −1.03238
\(149\) −4.48606 −0.367513 −0.183756 0.982972i \(-0.558826\pi\)
−0.183756 + 0.982972i \(0.558826\pi\)
\(150\) 0 0
\(151\) 0.0933624 0.00759773 0.00379886 0.999993i \(-0.498791\pi\)
0.00379886 + 0.999993i \(0.498791\pi\)
\(152\) −7.64889 −0.620407
\(153\) 0 0
\(154\) −4.36164 −0.351471
\(155\) 6.97160 0.559972
\(156\) 0 0
\(157\) −10.1337 −0.808760 −0.404380 0.914591i \(-0.632513\pi\)
−0.404380 + 0.914591i \(0.632513\pi\)
\(158\) 1.53575 0.122178
\(159\) 0 0
\(160\) 4.76155 0.376434
\(161\) −6.57693 −0.518334
\(162\) 0 0
\(163\) 7.49804 0.587292 0.293646 0.955914i \(-0.405131\pi\)
0.293646 + 0.955914i \(0.405131\pi\)
\(164\) 16.1965 1.26474
\(165\) 0 0
\(166\) −1.20287 −0.0933606
\(167\) −17.7922 −1.37680 −0.688399 0.725332i \(-0.741686\pi\)
−0.688399 + 0.725332i \(0.741686\pi\)
\(168\) 0 0
\(169\) 1.86650 0.143577
\(170\) −2.53301 −0.194273
\(171\) 0 0
\(172\) −5.17511 −0.394599
\(173\) −18.4927 −1.40597 −0.702987 0.711203i \(-0.748151\pi\)
−0.702987 + 0.711203i \(0.748151\pi\)
\(174\) 0 0
\(175\) 18.7997 1.42112
\(176\) −5.10421 −0.384745
\(177\) 0 0
\(178\) 2.70564 0.202796
\(179\) 17.3387 1.29595 0.647977 0.761660i \(-0.275615\pi\)
0.647977 + 0.761660i \(0.275615\pi\)
\(180\) 0 0
\(181\) 17.7124 1.31655 0.658275 0.752778i \(-0.271287\pi\)
0.658275 + 0.752778i \(0.271287\pi\)
\(182\) −8.66981 −0.642649
\(183\) 0 0
\(184\) 2.61375 0.192688
\(185\) −6.85518 −0.504003
\(186\) 0 0
\(187\) 10.4802 0.766390
\(188\) 4.37400 0.319006
\(189\) 0 0
\(190\) −1.95568 −0.141880
\(191\) −13.2440 −0.958303 −0.479151 0.877732i \(-0.659056\pi\)
−0.479151 + 0.877732i \(0.659056\pi\)
\(192\) 0 0
\(193\) −25.5030 −1.83574 −0.917872 0.396877i \(-0.870094\pi\)
−0.917872 + 0.396877i \(0.870094\pi\)
\(194\) −2.57300 −0.184731
\(195\) 0 0
\(196\) −25.1831 −1.79879
\(197\) 15.0303 1.07087 0.535433 0.844578i \(-0.320149\pi\)
0.535433 + 0.844578i \(0.320149\pi\)
\(198\) 0 0
\(199\) −19.3801 −1.37382 −0.686911 0.726742i \(-0.741034\pi\)
−0.686911 + 0.726742i \(0.741034\pi\)
\(200\) −7.47122 −0.528295
\(201\) 0 0
\(202\) −6.05472 −0.426008
\(203\) −22.4118 −1.57300
\(204\) 0 0
\(205\) 8.84034 0.617436
\(206\) 4.77856 0.332938
\(207\) 0 0
\(208\) −10.1459 −0.703489
\(209\) 8.09154 0.559704
\(210\) 0 0
\(211\) 27.9383 1.92335 0.961675 0.274192i \(-0.0884105\pi\)
0.961675 + 0.274192i \(0.0884105\pi\)
\(212\) 9.95805 0.683922
\(213\) 0 0
\(214\) −4.45934 −0.304834
\(215\) −2.82466 −0.192640
\(216\) 0 0
\(217\) 33.4370 2.26985
\(218\) −1.58160 −0.107119
\(219\) 0 0
\(220\) −3.28887 −0.221736
\(221\) 20.8320 1.40131
\(222\) 0 0
\(223\) −0.387500 −0.0259489 −0.0129745 0.999916i \(-0.504130\pi\)
−0.0129745 + 0.999916i \(0.504130\pi\)
\(224\) 22.8373 1.52588
\(225\) 0 0
\(226\) −0.410769 −0.0273240
\(227\) −10.2778 −0.682161 −0.341081 0.940034i \(-0.610793\pi\)
−0.341081 + 0.940034i \(0.610793\pi\)
\(228\) 0 0
\(229\) −10.5346 −0.696147 −0.348074 0.937467i \(-0.613164\pi\)
−0.348074 + 0.937467i \(0.613164\pi\)
\(230\) 0.668286 0.0440655
\(231\) 0 0
\(232\) 8.90673 0.584755
\(233\) −25.6725 −1.68186 −0.840932 0.541141i \(-0.817993\pi\)
−0.840932 + 0.541141i \(0.817993\pi\)
\(234\) 0 0
\(235\) 2.38740 0.155737
\(236\) −21.1116 −1.37425
\(237\) 0 0
\(238\) −12.1488 −0.787487
\(239\) 25.1312 1.62560 0.812801 0.582542i \(-0.197942\pi\)
0.812801 + 0.582542i \(0.197942\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157
\(242\) 3.52710 0.226730
\(243\) 0 0
\(244\) 24.5363 1.57078
\(245\) −13.7453 −0.878157
\(246\) 0 0
\(247\) 16.0839 1.02339
\(248\) −13.2883 −0.843806
\(249\) 0 0
\(250\) −4.25437 −0.269070
\(251\) −2.52238 −0.159211 −0.0796055 0.996826i \(-0.525366\pi\)
−0.0796055 + 0.996826i \(0.525366\pi\)
\(252\) 0 0
\(253\) −2.76501 −0.173835
\(254\) −5.52726 −0.346811
\(255\) 0 0
\(256\) 0.199817 0.0124886
\(257\) −3.92348 −0.244740 −0.122370 0.992485i \(-0.539049\pi\)
−0.122370 + 0.992485i \(0.539049\pi\)
\(258\) 0 0
\(259\) −32.8787 −2.04298
\(260\) −6.53744 −0.405435
\(261\) 0 0
\(262\) 6.88428 0.425312
\(263\) −6.24984 −0.385382 −0.192691 0.981260i \(-0.561721\pi\)
−0.192691 + 0.981260i \(0.561721\pi\)
\(264\) 0 0
\(265\) 5.43527 0.333886
\(266\) −9.37978 −0.575111
\(267\) 0 0
\(268\) 12.8782 0.786664
\(269\) −20.4943 −1.24956 −0.624780 0.780801i \(-0.714811\pi\)
−0.624780 + 0.780801i \(0.714811\pi\)
\(270\) 0 0
\(271\) 15.3179 0.930499 0.465249 0.885180i \(-0.345965\pi\)
0.465249 + 0.885180i \(0.345965\pi\)
\(272\) −14.2171 −0.862039
\(273\) 0 0
\(274\) −6.31376 −0.381428
\(275\) 7.90359 0.476605
\(276\) 0 0
\(277\) 3.26388 0.196108 0.0980539 0.995181i \(-0.468738\pi\)
0.0980539 + 0.995181i \(0.468738\pi\)
\(278\) −4.01379 −0.240731
\(279\) 0 0
\(280\) 8.13872 0.486382
\(281\) 12.4791 0.744442 0.372221 0.928144i \(-0.378596\pi\)
0.372221 + 0.928144i \(0.378596\pi\)
\(282\) 0 0
\(283\) −15.8115 −0.939899 −0.469949 0.882693i \(-0.655728\pi\)
−0.469949 + 0.882693i \(0.655728\pi\)
\(284\) −26.2337 −1.55668
\(285\) 0 0
\(286\) −3.64488 −0.215526
\(287\) 42.3999 2.50279
\(288\) 0 0
\(289\) 12.1913 0.717133
\(290\) 2.27728 0.133727
\(291\) 0 0
\(292\) −0.423465 −0.0247814
\(293\) −18.7147 −1.09333 −0.546663 0.837353i \(-0.684102\pi\)
−0.546663 + 0.837353i \(0.684102\pi\)
\(294\) 0 0
\(295\) −11.5231 −0.670899
\(296\) 13.0664 0.759467
\(297\) 0 0
\(298\) 2.18625 0.126646
\(299\) −5.49613 −0.317849
\(300\) 0 0
\(301\) −13.5476 −0.780870
\(302\) −0.0454995 −0.00261820
\(303\) 0 0
\(304\) −10.9767 −0.629557
\(305\) 13.3923 0.766843
\(306\) 0 0
\(307\) 5.23477 0.298764 0.149382 0.988780i \(-0.452272\pi\)
0.149382 + 0.988780i \(0.452272\pi\)
\(308\) −15.7740 −0.898809
\(309\) 0 0
\(310\) −3.39756 −0.192969
\(311\) 26.6878 1.51332 0.756662 0.653806i \(-0.226829\pi\)
0.756662 + 0.653806i \(0.226829\pi\)
\(312\) 0 0
\(313\) 6.24141 0.352786 0.176393 0.984320i \(-0.443557\pi\)
0.176393 + 0.984320i \(0.443557\pi\)
\(314\) 4.93861 0.278702
\(315\) 0 0
\(316\) 5.55412 0.312443
\(317\) −6.89526 −0.387276 −0.193638 0.981073i \(-0.562029\pi\)
−0.193638 + 0.981073i \(0.562029\pi\)
\(318\) 0 0
\(319\) −9.42218 −0.527541
\(320\) 2.74227 0.153298
\(321\) 0 0
\(322\) 3.20522 0.178620
\(323\) 22.5379 1.25404
\(324\) 0 0
\(325\) 15.7103 0.871451
\(326\) −3.65412 −0.202383
\(327\) 0 0
\(328\) −16.8502 −0.930397
\(329\) 11.4504 0.631281
\(330\) 0 0
\(331\) −13.5697 −0.745860 −0.372930 0.927859i \(-0.621647\pi\)
−0.372930 + 0.927859i \(0.621647\pi\)
\(332\) −4.35022 −0.238749
\(333\) 0 0
\(334\) 8.67089 0.474450
\(335\) 7.02916 0.384044
\(336\) 0 0
\(337\) −20.2288 −1.10193 −0.550966 0.834528i \(-0.685740\pi\)
−0.550966 + 0.834528i \(0.685740\pi\)
\(338\) −0.909627 −0.0494772
\(339\) 0 0
\(340\) −9.16072 −0.496810
\(341\) 14.0573 0.761245
\(342\) 0 0
\(343\) −33.6276 −1.81572
\(344\) 5.38397 0.290284
\(345\) 0 0
\(346\) 9.01229 0.484504
\(347\) −1.52458 −0.0818437 −0.0409219 0.999162i \(-0.513029\pi\)
−0.0409219 + 0.999162i \(0.513029\pi\)
\(348\) 0 0
\(349\) 16.5585 0.886358 0.443179 0.896433i \(-0.353851\pi\)
0.443179 + 0.896433i \(0.353851\pi\)
\(350\) −9.16191 −0.489724
\(351\) 0 0
\(352\) 9.60104 0.511737
\(353\) −34.6802 −1.84584 −0.922920 0.384992i \(-0.874204\pi\)
−0.922920 + 0.384992i \(0.874204\pi\)
\(354\) 0 0
\(355\) −14.3188 −0.759961
\(356\) 9.78505 0.518607
\(357\) 0 0
\(358\) −8.44989 −0.446590
\(359\) −18.1341 −0.957084 −0.478542 0.878065i \(-0.658835\pi\)
−0.478542 + 0.878065i \(0.658835\pi\)
\(360\) 0 0
\(361\) −1.59899 −0.0841573
\(362\) −8.63200 −0.453688
\(363\) 0 0
\(364\) −31.3547 −1.64343
\(365\) −0.231134 −0.0120981
\(366\) 0 0
\(367\) −10.9879 −0.573566 −0.286783 0.957996i \(-0.592586\pi\)
−0.286783 + 0.957996i \(0.592586\pi\)
\(368\) 3.75092 0.195530
\(369\) 0 0
\(370\) 3.34082 0.173681
\(371\) 26.0685 1.35341
\(372\) 0 0
\(373\) 5.23646 0.271134 0.135567 0.990768i \(-0.456714\pi\)
0.135567 + 0.990768i \(0.456714\pi\)
\(374\) −5.10747 −0.264101
\(375\) 0 0
\(376\) −4.55052 −0.234675
\(377\) −18.7289 −0.964586
\(378\) 0 0
\(379\) 8.07101 0.414580 0.207290 0.978280i \(-0.433536\pi\)
0.207290 + 0.978280i \(0.433536\pi\)
\(380\) −7.07278 −0.362826
\(381\) 0 0
\(382\) 6.45438 0.330235
\(383\) −2.46503 −0.125957 −0.0629785 0.998015i \(-0.520060\pi\)
−0.0629785 + 0.998015i \(0.520060\pi\)
\(384\) 0 0
\(385\) −8.60973 −0.438792
\(386\) 12.4287 0.632604
\(387\) 0 0
\(388\) −9.30535 −0.472408
\(389\) 28.7159 1.45595 0.727977 0.685601i \(-0.240461\pi\)
0.727977 + 0.685601i \(0.240461\pi\)
\(390\) 0 0
\(391\) −7.70157 −0.389485
\(392\) 26.1994 1.32327
\(393\) 0 0
\(394\) −7.32492 −0.369024
\(395\) 3.03153 0.152533
\(396\) 0 0
\(397\) 12.0605 0.605301 0.302651 0.953102i \(-0.402129\pi\)
0.302651 + 0.953102i \(0.402129\pi\)
\(398\) 9.44478 0.473424
\(399\) 0 0
\(400\) −10.7217 −0.536087
\(401\) 31.7082 1.58343 0.791716 0.610890i \(-0.209188\pi\)
0.791716 + 0.610890i \(0.209188\pi\)
\(402\) 0 0
\(403\) 27.9423 1.39190
\(404\) −21.8971 −1.08942
\(405\) 0 0
\(406\) 10.9223 0.542063
\(407\) −13.8225 −0.685158
\(408\) 0 0
\(409\) −14.1893 −0.701617 −0.350809 0.936447i \(-0.614093\pi\)
−0.350809 + 0.936447i \(0.614093\pi\)
\(410\) −4.30828 −0.212771
\(411\) 0 0
\(412\) 17.2819 0.851416
\(413\) −55.2668 −2.71950
\(414\) 0 0
\(415\) −2.37442 −0.116556
\(416\) 19.0844 0.935689
\(417\) 0 0
\(418\) −3.94336 −0.192876
\(419\) −31.1014 −1.51941 −0.759703 0.650271i \(-0.774655\pi\)
−0.759703 + 0.650271i \(0.774655\pi\)
\(420\) 0 0
\(421\) −27.4415 −1.33742 −0.668709 0.743524i \(-0.733153\pi\)
−0.668709 + 0.743524i \(0.733153\pi\)
\(422\) −13.6155 −0.662794
\(423\) 0 0
\(424\) −10.3599 −0.503123
\(425\) 22.0144 1.06786
\(426\) 0 0
\(427\) 64.2321 3.10841
\(428\) −16.1274 −0.779546
\(429\) 0 0
\(430\) 1.37658 0.0663846
\(431\) 14.1005 0.679199 0.339599 0.940570i \(-0.389709\pi\)
0.339599 + 0.940570i \(0.389709\pi\)
\(432\) 0 0
\(433\) −17.7403 −0.852546 −0.426273 0.904595i \(-0.640174\pi\)
−0.426273 + 0.904595i \(0.640174\pi\)
\(434\) −16.2953 −0.782200
\(435\) 0 0
\(436\) −5.71991 −0.273934
\(437\) −5.94621 −0.284446
\(438\) 0 0
\(439\) −2.52447 −0.120486 −0.0602431 0.998184i \(-0.519188\pi\)
−0.0602431 + 0.998184i \(0.519188\pi\)
\(440\) 3.42161 0.163119
\(441\) 0 0
\(442\) −10.1523 −0.482897
\(443\) 16.8382 0.800008 0.400004 0.916513i \(-0.369009\pi\)
0.400004 + 0.916513i \(0.369009\pi\)
\(444\) 0 0
\(445\) 5.34084 0.253180
\(446\) 0.188846 0.00894211
\(447\) 0 0
\(448\) 13.1524 0.621394
\(449\) 30.5688 1.44263 0.721314 0.692608i \(-0.243538\pi\)
0.721314 + 0.692608i \(0.243538\pi\)
\(450\) 0 0
\(451\) 17.8254 0.839364
\(452\) −1.48556 −0.0698750
\(453\) 0 0
\(454\) 5.00882 0.235075
\(455\) −17.1139 −0.802313
\(456\) 0 0
\(457\) −32.7043 −1.52984 −0.764922 0.644123i \(-0.777223\pi\)
−0.764922 + 0.644123i \(0.777223\pi\)
\(458\) 5.13398 0.239895
\(459\) 0 0
\(460\) 2.41688 0.112688
\(461\) −12.3932 −0.577210 −0.288605 0.957448i \(-0.593191\pi\)
−0.288605 + 0.957448i \(0.593191\pi\)
\(462\) 0 0
\(463\) 12.1379 0.564094 0.282047 0.959401i \(-0.408987\pi\)
0.282047 + 0.959401i \(0.408987\pi\)
\(464\) 12.7818 0.593380
\(465\) 0 0
\(466\) 12.5113 0.579577
\(467\) −5.38203 −0.249051 −0.124525 0.992216i \(-0.539741\pi\)
−0.124525 + 0.992216i \(0.539741\pi\)
\(468\) 0 0
\(469\) 33.7131 1.55673
\(470\) −1.16348 −0.0536675
\(471\) 0 0
\(472\) 21.9637 1.01096
\(473\) −5.69555 −0.261882
\(474\) 0 0
\(475\) 16.9968 0.779868
\(476\) −43.9365 −2.01382
\(477\) 0 0
\(478\) −12.2475 −0.560188
\(479\) 25.3752 1.15942 0.579711 0.814822i \(-0.303166\pi\)
0.579711 + 0.814822i \(0.303166\pi\)
\(480\) 0 0
\(481\) −27.4757 −1.25278
\(482\) 0.487343 0.0221979
\(483\) 0 0
\(484\) 12.7559 0.579813
\(485\) −5.07901 −0.230626
\(486\) 0 0
\(487\) −10.4243 −0.472369 −0.236185 0.971708i \(-0.575897\pi\)
−0.236185 + 0.971708i \(0.575897\pi\)
\(488\) −25.5266 −1.15553
\(489\) 0 0
\(490\) 6.69869 0.302616
\(491\) −34.2858 −1.54730 −0.773649 0.633614i \(-0.781571\pi\)
−0.773649 + 0.633614i \(0.781571\pi\)
\(492\) 0 0
\(493\) −26.2442 −1.18198
\(494\) −7.83839 −0.352666
\(495\) 0 0
\(496\) −19.0696 −0.856252
\(497\) −68.6754 −3.08051
\(498\) 0 0
\(499\) 15.9672 0.714792 0.357396 0.933953i \(-0.383665\pi\)
0.357396 + 0.933953i \(0.383665\pi\)
\(500\) −15.3861 −0.688087
\(501\) 0 0
\(502\) 1.22926 0.0548647
\(503\) 4.53174 0.202060 0.101030 0.994883i \(-0.467786\pi\)
0.101030 + 0.994883i \(0.467786\pi\)
\(504\) 0 0
\(505\) −11.9518 −0.531849
\(506\) 1.34751 0.0599041
\(507\) 0 0
\(508\) −19.9896 −0.886893
\(509\) 4.08798 0.181197 0.0905983 0.995888i \(-0.471122\pi\)
0.0905983 + 0.995888i \(0.471122\pi\)
\(510\) 0 0
\(511\) −1.10856 −0.0490398
\(512\) −22.6744 −1.00208
\(513\) 0 0
\(514\) 1.91208 0.0843384
\(515\) 9.43272 0.415655
\(516\) 0 0
\(517\) 4.81387 0.211714
\(518\) 16.0232 0.704019
\(519\) 0 0
\(520\) 6.80127 0.298256
\(521\) −34.2884 −1.50220 −0.751102 0.660186i \(-0.770477\pi\)
−0.751102 + 0.660186i \(0.770477\pi\)
\(522\) 0 0
\(523\) 33.7724 1.47676 0.738382 0.674382i \(-0.235590\pi\)
0.738382 + 0.674382i \(0.235590\pi\)
\(524\) 24.8973 1.08764
\(525\) 0 0
\(526\) 3.04582 0.132804
\(527\) 39.1547 1.70561
\(528\) 0 0
\(529\) −20.9681 −0.911656
\(530\) −2.64884 −0.115058
\(531\) 0 0
\(532\) −33.9223 −1.47072
\(533\) 35.4322 1.53474
\(534\) 0 0
\(535\) −8.80259 −0.380569
\(536\) −13.3980 −0.578705
\(537\) 0 0
\(538\) 9.98777 0.430603
\(539\) −27.7156 −1.19380
\(540\) 0 0
\(541\) −31.0330 −1.33421 −0.667106 0.744963i \(-0.732467\pi\)
−0.667106 + 0.744963i \(0.732467\pi\)
\(542\) −7.46510 −0.320653
\(543\) 0 0
\(544\) 26.7424 1.14657
\(545\) −3.12202 −0.133733
\(546\) 0 0
\(547\) −33.0104 −1.41142 −0.705712 0.708499i \(-0.749373\pi\)
−0.705712 + 0.708499i \(0.749373\pi\)
\(548\) −22.8340 −0.975418
\(549\) 0 0
\(550\) −3.85176 −0.164240
\(551\) −20.2626 −0.863214
\(552\) 0 0
\(553\) 14.5398 0.618293
\(554\) −1.59063 −0.0675795
\(555\) 0 0
\(556\) −14.5160 −0.615617
\(557\) −15.4826 −0.656019 −0.328010 0.944674i \(-0.606378\pi\)
−0.328010 + 0.944674i \(0.606378\pi\)
\(558\) 0 0
\(559\) −11.3213 −0.478840
\(560\) 11.6797 0.493555
\(561\) 0 0
\(562\) −6.08161 −0.256537
\(563\) −6.21072 −0.261751 −0.130875 0.991399i \(-0.541779\pi\)
−0.130875 + 0.991399i \(0.541779\pi\)
\(564\) 0 0
\(565\) −0.810845 −0.0341125
\(566\) 7.70565 0.323893
\(567\) 0 0
\(568\) 27.2924 1.14516
\(569\) 20.3014 0.851078 0.425539 0.904940i \(-0.360084\pi\)
0.425539 + 0.904940i \(0.360084\pi\)
\(570\) 0 0
\(571\) −38.5209 −1.61205 −0.806024 0.591882i \(-0.798385\pi\)
−0.806024 + 0.591882i \(0.798385\pi\)
\(572\) −13.1819 −0.551161
\(573\) 0 0
\(574\) −20.6633 −0.862469
\(575\) −5.80809 −0.242214
\(576\) 0 0
\(577\) 19.3481 0.805472 0.402736 0.915316i \(-0.368059\pi\)
0.402736 + 0.915316i \(0.368059\pi\)
\(578\) −5.94133 −0.247127
\(579\) 0 0
\(580\) 8.23588 0.341976
\(581\) −11.3881 −0.472460
\(582\) 0 0
\(583\) 10.9595 0.453896
\(584\) 0.440555 0.0182303
\(585\) 0 0
\(586\) 9.12050 0.376764
\(587\) 20.0183 0.826243 0.413122 0.910676i \(-0.364438\pi\)
0.413122 + 0.910676i \(0.364438\pi\)
\(588\) 0 0
\(589\) 30.2305 1.24562
\(590\) 5.61569 0.231194
\(591\) 0 0
\(592\) 18.7512 0.770668
\(593\) −12.5904 −0.517027 −0.258513 0.966008i \(-0.583233\pi\)
−0.258513 + 0.966008i \(0.583233\pi\)
\(594\) 0 0
\(595\) −23.9812 −0.983135
\(596\) 7.90667 0.323870
\(597\) 0 0
\(598\) 2.67850 0.109532
\(599\) −24.4282 −0.998107 −0.499054 0.866571i \(-0.666319\pi\)
−0.499054 + 0.866571i \(0.666319\pi\)
\(600\) 0 0
\(601\) 9.02256 0.368038 0.184019 0.982923i \(-0.441089\pi\)
0.184019 + 0.982923i \(0.441089\pi\)
\(602\) 6.60233 0.269091
\(603\) 0 0
\(604\) −0.164551 −0.00669548
\(605\) 6.96237 0.283061
\(606\) 0 0
\(607\) −19.0044 −0.771364 −0.385682 0.922632i \(-0.626034\pi\)
−0.385682 + 0.922632i \(0.626034\pi\)
\(608\) 20.6472 0.837355
\(609\) 0 0
\(610\) −6.52666 −0.264257
\(611\) 9.56873 0.387110
\(612\) 0 0
\(613\) 3.38617 0.136766 0.0683831 0.997659i \(-0.478216\pi\)
0.0683831 + 0.997659i \(0.478216\pi\)
\(614\) −2.55113 −0.102955
\(615\) 0 0
\(616\) 16.4106 0.661204
\(617\) 33.9920 1.36847 0.684234 0.729263i \(-0.260137\pi\)
0.684234 + 0.729263i \(0.260137\pi\)
\(618\) 0 0
\(619\) 29.6477 1.19164 0.595822 0.803117i \(-0.296826\pi\)
0.595822 + 0.803117i \(0.296826\pi\)
\(620\) −12.2874 −0.493475
\(621\) 0 0
\(622\) −13.0061 −0.521497
\(623\) 25.6156 1.02627
\(624\) 0 0
\(625\) 11.9748 0.478992
\(626\) −3.04171 −0.121571
\(627\) 0 0
\(628\) 17.8607 0.712718
\(629\) −38.5008 −1.53513
\(630\) 0 0
\(631\) 14.6241 0.582176 0.291088 0.956696i \(-0.405983\pi\)
0.291088 + 0.956696i \(0.405983\pi\)
\(632\) −5.77827 −0.229847
\(633\) 0 0
\(634\) 3.36036 0.133457
\(635\) −10.9106 −0.432975
\(636\) 0 0
\(637\) −55.0915 −2.18280
\(638\) 4.59184 0.181793
\(639\) 0 0
\(640\) −10.8595 −0.429261
\(641\) −4.17547 −0.164921 −0.0824606 0.996594i \(-0.526278\pi\)
−0.0824606 + 0.996594i \(0.526278\pi\)
\(642\) 0 0
\(643\) 23.1928 0.914635 0.457317 0.889304i \(-0.348810\pi\)
0.457317 + 0.889304i \(0.348810\pi\)
\(644\) 11.5918 0.456781
\(645\) 0 0
\(646\) −10.9837 −0.432148
\(647\) 31.0975 1.22257 0.611284 0.791411i \(-0.290653\pi\)
0.611284 + 0.791411i \(0.290653\pi\)
\(648\) 0 0
\(649\) −23.2347 −0.912043
\(650\) −7.65632 −0.300305
\(651\) 0 0
\(652\) −13.2153 −0.517550
\(653\) 31.5439 1.23441 0.617204 0.786803i \(-0.288265\pi\)
0.617204 + 0.786803i \(0.288265\pi\)
\(654\) 0 0
\(655\) 13.5893 0.530979
\(656\) −24.1813 −0.944120
\(657\) 0 0
\(658\) −5.58027 −0.217542
\(659\) −25.2754 −0.984590 −0.492295 0.870428i \(-0.663842\pi\)
−0.492295 + 0.870428i \(0.663842\pi\)
\(660\) 0 0
\(661\) −21.7628 −0.846476 −0.423238 0.906018i \(-0.639107\pi\)
−0.423238 + 0.906018i \(0.639107\pi\)
\(662\) 6.61312 0.257026
\(663\) 0 0
\(664\) 4.52578 0.175634
\(665\) −18.5154 −0.717995
\(666\) 0 0
\(667\) 6.92404 0.268100
\(668\) 31.3586 1.21330
\(669\) 0 0
\(670\) −3.42561 −0.132343
\(671\) 27.0038 1.04247
\(672\) 0 0
\(673\) −37.2710 −1.43669 −0.718346 0.695686i \(-0.755101\pi\)
−0.718346 + 0.695686i \(0.755101\pi\)
\(674\) 9.85835 0.379730
\(675\) 0 0
\(676\) −3.28970 −0.126527
\(677\) 33.5515 1.28949 0.644745 0.764398i \(-0.276964\pi\)
0.644745 + 0.764398i \(0.276964\pi\)
\(678\) 0 0
\(679\) −24.3599 −0.934846
\(680\) 9.53043 0.365475
\(681\) 0 0
\(682\) −6.85073 −0.262328
\(683\) −8.94737 −0.342362 −0.171181 0.985240i \(-0.554758\pi\)
−0.171181 + 0.985240i \(0.554758\pi\)
\(684\) 0 0
\(685\) −12.4631 −0.476192
\(686\) 16.3882 0.625704
\(687\) 0 0
\(688\) 7.72638 0.294566
\(689\) 21.7846 0.829929
\(690\) 0 0
\(691\) 7.04969 0.268183 0.134092 0.990969i \(-0.457188\pi\)
0.134092 + 0.990969i \(0.457188\pi\)
\(692\) 32.5933 1.23901
\(693\) 0 0
\(694\) 0.742994 0.0282036
\(695\) −7.92309 −0.300540
\(696\) 0 0
\(697\) 49.6502 1.88063
\(698\) −8.06969 −0.305442
\(699\) 0 0
\(700\) −33.1344 −1.25236
\(701\) 5.19179 0.196091 0.0980456 0.995182i \(-0.468741\pi\)
0.0980456 + 0.995182i \(0.468741\pi\)
\(702\) 0 0
\(703\) −29.7256 −1.12112
\(704\) 5.52943 0.208398
\(705\) 0 0
\(706\) 16.9012 0.636083
\(707\) −57.3230 −2.15586
\(708\) 0 0
\(709\) −16.3067 −0.612410 −0.306205 0.951966i \(-0.599059\pi\)
−0.306205 + 0.951966i \(0.599059\pi\)
\(710\) 6.97816 0.261886
\(711\) 0 0
\(712\) −10.1800 −0.381510
\(713\) −10.3302 −0.386870
\(714\) 0 0
\(715\) −7.19488 −0.269073
\(716\) −30.5594 −1.14206
\(717\) 0 0
\(718\) 8.83756 0.329815
\(719\) 27.4968 1.02546 0.512729 0.858551i \(-0.328635\pi\)
0.512729 + 0.858551i \(0.328635\pi\)
\(720\) 0 0
\(721\) 45.2410 1.68486
\(722\) 0.779256 0.0290009
\(723\) 0 0
\(724\) −31.2180 −1.16021
\(725\) −19.7919 −0.735053
\(726\) 0 0
\(727\) −15.2319 −0.564919 −0.282460 0.959279i \(-0.591150\pi\)
−0.282460 + 0.959279i \(0.591150\pi\)
\(728\) 32.6201 1.20898
\(729\) 0 0
\(730\) 0.112642 0.00416905
\(731\) −15.8642 −0.586758
\(732\) 0 0
\(733\) 6.76475 0.249862 0.124931 0.992165i \(-0.460129\pi\)
0.124931 + 0.992165i \(0.460129\pi\)
\(734\) 5.35490 0.197653
\(735\) 0 0
\(736\) −7.05548 −0.260068
\(737\) 14.1734 0.522082
\(738\) 0 0
\(739\) 17.9519 0.660372 0.330186 0.943916i \(-0.392889\pi\)
0.330186 + 0.943916i \(0.392889\pi\)
\(740\) 12.0822 0.444151
\(741\) 0 0
\(742\) −12.7043 −0.466390
\(743\) 17.2096 0.631357 0.315679 0.948866i \(-0.397768\pi\)
0.315679 + 0.948866i \(0.397768\pi\)
\(744\) 0 0
\(745\) 4.31559 0.158111
\(746\) −2.55195 −0.0934336
\(747\) 0 0
\(748\) −18.4714 −0.675380
\(749\) −42.2188 −1.54264
\(750\) 0 0
\(751\) 4.98853 0.182034 0.0910170 0.995849i \(-0.470988\pi\)
0.0910170 + 0.995849i \(0.470988\pi\)
\(752\) −6.53032 −0.238136
\(753\) 0 0
\(754\) 9.12739 0.332400
\(755\) −0.0898146 −0.00326869
\(756\) 0 0
\(757\) −19.5585 −0.710864 −0.355432 0.934702i \(-0.615666\pi\)
−0.355432 + 0.934702i \(0.615666\pi\)
\(758\) −3.93335 −0.142866
\(759\) 0 0
\(760\) 7.35823 0.266911
\(761\) 41.0235 1.48710 0.743550 0.668681i \(-0.233141\pi\)
0.743550 + 0.668681i \(0.233141\pi\)
\(762\) 0 0
\(763\) −14.9738 −0.542087
\(764\) 23.3425 0.844503
\(765\) 0 0
\(766\) 1.20131 0.0434053
\(767\) −46.1847 −1.66763
\(768\) 0 0
\(769\) 11.7068 0.422159 0.211079 0.977469i \(-0.432302\pi\)
0.211079 + 0.977469i \(0.432302\pi\)
\(770\) 4.19589 0.151209
\(771\) 0 0
\(772\) 44.9489 1.61775
\(773\) 6.66278 0.239643 0.119822 0.992795i \(-0.461768\pi\)
0.119822 + 0.992795i \(0.461768\pi\)
\(774\) 0 0
\(775\) 29.5283 1.06069
\(776\) 9.68090 0.347524
\(777\) 0 0
\(778\) −13.9945 −0.501727
\(779\) 38.3338 1.37345
\(780\) 0 0
\(781\) −28.8719 −1.03312
\(782\) 3.75331 0.134218
\(783\) 0 0
\(784\) 37.5980 1.34279
\(785\) 9.74865 0.347944
\(786\) 0 0
\(787\) 2.88890 0.102978 0.0514891 0.998674i \(-0.483603\pi\)
0.0514891 + 0.998674i \(0.483603\pi\)
\(788\) −26.4909 −0.943698
\(789\) 0 0
\(790\) −1.47739 −0.0525633
\(791\) −3.88896 −0.138275
\(792\) 0 0
\(793\) 53.6767 1.90612
\(794\) −5.87762 −0.208589
\(795\) 0 0
\(796\) 34.1574 1.21068
\(797\) −18.0912 −0.640824 −0.320412 0.947278i \(-0.603821\pi\)
−0.320412 + 0.947278i \(0.603821\pi\)
\(798\) 0 0
\(799\) 13.4084 0.474355
\(800\) 20.1676 0.713033
\(801\) 0 0
\(802\) −15.4528 −0.545656
\(803\) −0.466051 −0.0164466
\(804\) 0 0
\(805\) 6.32700 0.222997
\(806\) −13.6175 −0.479655
\(807\) 0 0
\(808\) 22.7809 0.801428
\(809\) −8.04121 −0.282714 −0.141357 0.989959i \(-0.545147\pi\)
−0.141357 + 0.989959i \(0.545147\pi\)
\(810\) 0 0
\(811\) 24.5111 0.860701 0.430350 0.902662i \(-0.358390\pi\)
0.430350 + 0.902662i \(0.358390\pi\)
\(812\) 39.5008 1.38621
\(813\) 0 0
\(814\) 6.73632 0.236108
\(815\) −7.21311 −0.252664
\(816\) 0 0
\(817\) −12.2484 −0.428517
\(818\) 6.91507 0.241780
\(819\) 0 0
\(820\) −15.5811 −0.544115
\(821\) 10.6539 0.371823 0.185912 0.982566i \(-0.440476\pi\)
0.185912 + 0.982566i \(0.440476\pi\)
\(822\) 0 0
\(823\) 25.6473 0.894009 0.447004 0.894532i \(-0.352491\pi\)
0.447004 + 0.894532i \(0.352491\pi\)
\(824\) −17.9793 −0.626339
\(825\) 0 0
\(826\) 26.9339 0.937150
\(827\) −18.0150 −0.626442 −0.313221 0.949680i \(-0.601408\pi\)
−0.313221 + 0.949680i \(0.601408\pi\)
\(828\) 0 0
\(829\) 10.1189 0.351444 0.175722 0.984440i \(-0.443774\pi\)
0.175722 + 0.984440i \(0.443774\pi\)
\(830\) 1.15716 0.0401655
\(831\) 0 0
\(832\) 10.9911 0.381047
\(833\) −77.1981 −2.67476
\(834\) 0 0
\(835\) 17.1160 0.592325
\(836\) −14.2613 −0.493238
\(837\) 0 0
\(838\) 15.1571 0.523593
\(839\) 46.0365 1.58936 0.794679 0.607030i \(-0.207639\pi\)
0.794679 + 0.607030i \(0.207639\pi\)
\(840\) 0 0
\(841\) −5.40531 −0.186390
\(842\) 13.3734 0.460879
\(843\) 0 0
\(844\) −49.2411 −1.69495
\(845\) −1.79557 −0.0617696
\(846\) 0 0
\(847\) 33.3928 1.14739
\(848\) −14.8673 −0.510544
\(849\) 0 0
\(850\) −10.7286 −0.367987
\(851\) 10.1577 0.348202
\(852\) 0 0
\(853\) −55.0001 −1.88317 −0.941584 0.336780i \(-0.890662\pi\)
−0.941584 + 0.336780i \(0.890662\pi\)
\(854\) −31.3031 −1.07117
\(855\) 0 0
\(856\) 16.7783 0.573469
\(857\) 31.5202 1.07671 0.538354 0.842719i \(-0.319046\pi\)
0.538354 + 0.842719i \(0.319046\pi\)
\(858\) 0 0
\(859\) −37.2208 −1.26996 −0.634979 0.772529i \(-0.718991\pi\)
−0.634979 + 0.772529i \(0.718991\pi\)
\(860\) 4.97846 0.169764
\(861\) 0 0
\(862\) −6.87180 −0.234054
\(863\) −30.3224 −1.03218 −0.516092 0.856533i \(-0.672614\pi\)
−0.516092 + 0.856533i \(0.672614\pi\)
\(864\) 0 0
\(865\) 17.7900 0.604877
\(866\) 8.64563 0.293791
\(867\) 0 0
\(868\) −58.9327 −2.00031
\(869\) 6.11267 0.207358
\(870\) 0 0
\(871\) 28.1730 0.954605
\(872\) 5.95075 0.201518
\(873\) 0 0
\(874\) 2.89784 0.0980210
\(875\) −40.2783 −1.36165
\(876\) 0 0
\(877\) 27.0037 0.911849 0.455925 0.890018i \(-0.349309\pi\)
0.455925 + 0.890018i \(0.349309\pi\)
\(878\) 1.23028 0.0415200
\(879\) 0 0
\(880\) 4.91025 0.165525
\(881\) 0.565506 0.0190524 0.00952619 0.999955i \(-0.496968\pi\)
0.00952619 + 0.999955i \(0.496968\pi\)
\(882\) 0 0
\(883\) 23.8838 0.803752 0.401876 0.915694i \(-0.368358\pi\)
0.401876 + 0.915694i \(0.368358\pi\)
\(884\) −36.7163 −1.23490
\(885\) 0 0
\(886\) −8.20600 −0.275686
\(887\) 10.1717 0.341533 0.170767 0.985311i \(-0.445376\pi\)
0.170767 + 0.985311i \(0.445376\pi\)
\(888\) 0 0
\(889\) −52.3294 −1.75507
\(890\) −2.60282 −0.0872468
\(891\) 0 0
\(892\) 0.682968 0.0228675
\(893\) 10.3523 0.346427
\(894\) 0 0
\(895\) −16.6798 −0.557544
\(896\) −52.0843 −1.74001
\(897\) 0 0
\(898\) −14.8975 −0.497135
\(899\) −35.2018 −1.17405
\(900\) 0 0
\(901\) 30.5262 1.01697
\(902\) −8.68707 −0.289248
\(903\) 0 0
\(904\) 1.54552 0.0514031
\(905\) −17.0393 −0.566405
\(906\) 0 0
\(907\) −44.1661 −1.46651 −0.733255 0.679954i \(-0.762000\pi\)
−0.733255 + 0.679954i \(0.762000\pi\)
\(908\) 18.1146 0.601154
\(909\) 0 0
\(910\) 8.34036 0.276480
\(911\) 7.07316 0.234344 0.117172 0.993112i \(-0.462617\pi\)
0.117172 + 0.993112i \(0.462617\pi\)
\(912\) 0 0
\(913\) −4.78770 −0.158450
\(914\) 15.9382 0.527190
\(915\) 0 0
\(916\) 18.5672 0.613479
\(917\) 65.1769 2.15233
\(918\) 0 0
\(919\) 23.6016 0.778544 0.389272 0.921123i \(-0.372727\pi\)
0.389272 + 0.921123i \(0.372727\pi\)
\(920\) −2.51442 −0.0828981
\(921\) 0 0
\(922\) 6.03976 0.198909
\(923\) −57.3899 −1.88901
\(924\) 0 0
\(925\) −29.0352 −0.954670
\(926\) −5.91530 −0.194389
\(927\) 0 0
\(928\) −24.0426 −0.789236
\(929\) −31.8496 −1.04495 −0.522475 0.852654i \(-0.674991\pi\)
−0.522475 + 0.852654i \(0.674991\pi\)
\(930\) 0 0
\(931\) −59.6029 −1.95341
\(932\) 45.2478 1.48214
\(933\) 0 0
\(934\) 2.62290 0.0858238
\(935\) −10.0820 −0.329716
\(936\) 0 0
\(937\) 44.1729 1.44307 0.721533 0.692380i \(-0.243438\pi\)
0.721533 + 0.692380i \(0.243438\pi\)
\(938\) −16.4299 −0.536454
\(939\) 0 0
\(940\) −4.20778 −0.137243
\(941\) 34.5303 1.12566 0.562828 0.826574i \(-0.309713\pi\)
0.562828 + 0.826574i \(0.309713\pi\)
\(942\) 0 0
\(943\) −13.0993 −0.426571
\(944\) 31.5194 1.02587
\(945\) 0 0
\(946\) 2.77569 0.0902454
\(947\) 3.46191 0.112497 0.0562485 0.998417i \(-0.482086\pi\)
0.0562485 + 0.998417i \(0.482086\pi\)
\(948\) 0 0
\(949\) −0.926389 −0.0300719
\(950\) −8.28329 −0.268745
\(951\) 0 0
\(952\) 45.7097 1.48146
\(953\) −60.2596 −1.95200 −0.976000 0.217771i \(-0.930122\pi\)
−0.976000 + 0.217771i \(0.930122\pi\)
\(954\) 0 0
\(955\) 12.7407 0.412280
\(956\) −44.2936 −1.43256
\(957\) 0 0
\(958\) −12.3664 −0.399541
\(959\) −59.7755 −1.93025
\(960\) 0 0
\(961\) 21.5188 0.694156
\(962\) 13.3901 0.431713
\(963\) 0 0
\(964\) 1.76250 0.0567662
\(965\) 24.5338 0.789772
\(966\) 0 0
\(967\) −39.9168 −1.28364 −0.641819 0.766856i \(-0.721820\pi\)
−0.641819 + 0.766856i \(0.721820\pi\)
\(968\) −13.2707 −0.426536
\(969\) 0 0
\(970\) 2.47522 0.0794746
\(971\) −43.7104 −1.40273 −0.701366 0.712801i \(-0.747426\pi\)
−0.701366 + 0.712801i \(0.747426\pi\)
\(972\) 0 0
\(973\) −38.0006 −1.21824
\(974\) 5.08020 0.162780
\(975\) 0 0
\(976\) −36.6325 −1.17258
\(977\) 0.955388 0.0305656 0.0152828 0.999883i \(-0.495135\pi\)
0.0152828 + 0.999883i \(0.495135\pi\)
\(978\) 0 0
\(979\) 10.7691 0.344182
\(980\) 24.2261 0.773874
\(981\) 0 0
\(982\) 16.7090 0.533205
\(983\) 48.7488 1.55485 0.777423 0.628978i \(-0.216526\pi\)
0.777423 + 0.628978i \(0.216526\pi\)
\(984\) 0 0
\(985\) −14.4591 −0.460707
\(986\) 12.7899 0.407315
\(987\) 0 0
\(988\) −28.3478 −0.901865
\(989\) 4.18547 0.133090
\(990\) 0 0
\(991\) 0.488986 0.0155332 0.00776658 0.999970i \(-0.497528\pi\)
0.00776658 + 0.999970i \(0.497528\pi\)
\(992\) 35.8700 1.13887
\(993\) 0 0
\(994\) 33.4685 1.06156
\(995\) 18.6437 0.591044
\(996\) 0 0
\(997\) 39.5339 1.25205 0.626026 0.779802i \(-0.284680\pi\)
0.626026 + 0.779802i \(0.284680\pi\)
\(998\) −7.78153 −0.246320
\(999\) 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 2169.2.a.e.1.2 7
3.2 odd 2 241.2.a.a.1.6 7
12.11 even 2 3856.2.a.j.1.5 7
15.14 odd 2 6025.2.a.f.1.2 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
241.2.a.a.1.6 7 3.2 odd 2
2169.2.a.e.1.2 7 1.1 even 1 trivial
3856.2.a.j.1.5 7 12.11 even 2
6025.2.a.f.1.2 7 15.14 odd 2