Properties

Label 2169.2.a.e.1.4
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 1.4
Root \(0.369356\) of defining polynomial
Character \(\chi\) \(=\) 2169.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.630644 q^{2} -1.60229 q^{4} +3.89634 q^{5} -3.68231 q^{7} -2.27176 q^{8} +O(q^{10})\) \(q+0.630644 q^{2} -1.60229 q^{4} +3.89634 q^{5} -3.68231 q^{7} -2.27176 q^{8} +2.45721 q^{10} +4.96431 q^{11} -1.69048 q^{13} -2.32223 q^{14} +1.77190 q^{16} -5.52260 q^{17} +4.21489 q^{19} -6.24306 q^{20} +3.13071 q^{22} +2.77495 q^{23} +10.1815 q^{25} -1.06609 q^{26} +5.90013 q^{28} +2.31253 q^{29} -0.199515 q^{31} +5.66096 q^{32} -3.48280 q^{34} -14.3476 q^{35} +1.79089 q^{37} +2.65809 q^{38} -8.85157 q^{40} +12.1960 q^{41} -5.34523 q^{43} -7.95425 q^{44} +1.75001 q^{46} +10.7345 q^{47} +6.55944 q^{49} +6.42090 q^{50} +2.70864 q^{52} -1.32229 q^{53} +19.3426 q^{55} +8.36534 q^{56} +1.45838 q^{58} -5.78578 q^{59} -0.0766435 q^{61} -0.125823 q^{62} +0.0262532 q^{64} -6.58670 q^{65} +12.6166 q^{67} +8.84880 q^{68} -9.04821 q^{70} +5.20391 q^{71} -14.0733 q^{73} +1.12941 q^{74} -6.75346 q^{76} -18.2801 q^{77} +7.82844 q^{79} +6.90393 q^{80} +7.69137 q^{82} -2.55372 q^{83} -21.5180 q^{85} -3.37094 q^{86} -11.2777 q^{88} +4.35327 q^{89} +6.22489 q^{91} -4.44627 q^{92} +6.76964 q^{94} +16.4226 q^{95} +9.02693 q^{97} +4.13668 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 4q^{2} + 2q^{4} + 8q^{5} - 7q^{7} + 6q^{8} + O(q^{10}) \) \( 7q + 4q^{2} + 2q^{4} + 8q^{5} - 7q^{7} + 6q^{8} + 3q^{10} + 18q^{11} - q^{13} + 6q^{14} + 4q^{16} + 2q^{17} - 6q^{19} + 8q^{20} + 10q^{22} + 22q^{23} + 5q^{25} - 8q^{26} + 9q^{28} + 16q^{29} - 18q^{31} + 6q^{32} + 11q^{34} - 7q^{35} + 8q^{37} - 16q^{38} + 14q^{40} + 15q^{41} + 14q^{43} + 4q^{44} + 11q^{46} + 10q^{47} + 6q^{49} + 4q^{50} + 27q^{52} - 15q^{53} + 29q^{55} - 13q^{56} + 17q^{58} + 18q^{59} + 4q^{61} - 13q^{62} + 2q^{64} + 7q^{65} + 18q^{67} + 15q^{68} + 8q^{70} + 50q^{71} - 10q^{74} - 20q^{76} - 17q^{77} - 15q^{79} + 11q^{80} + 45q^{82} + 24q^{83} - 2q^{85} + 23q^{86} + 8q^{88} + 13q^{89} - 12q^{91} + 10q^{92} - 32q^{94} + 41q^{95} + q^{97} - 9q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.630644 0.445933 0.222966 0.974826i \(-0.428426\pi\)
0.222966 + 0.974826i \(0.428426\pi\)
\(3\) 0 0
\(4\) −1.60229 −0.801144
\(5\) 3.89634 1.74250 0.871249 0.490842i \(-0.163311\pi\)
0.871249 + 0.490842i \(0.163311\pi\)
\(6\) 0 0
\(7\) −3.68231 −1.39178 −0.695892 0.718146i \(-0.744991\pi\)
−0.695892 + 0.718146i \(0.744991\pi\)
\(8\) −2.27176 −0.803189
\(9\) 0 0
\(10\) 2.45721 0.777037
\(11\) 4.96431 1.49679 0.748397 0.663250i \(-0.230824\pi\)
0.748397 + 0.663250i \(0.230824\pi\)
\(12\) 0 0
\(13\) −1.69048 −0.468855 −0.234428 0.972134i \(-0.575322\pi\)
−0.234428 + 0.972134i \(0.575322\pi\)
\(14\) −2.32223 −0.620642
\(15\) 0 0
\(16\) 1.77190 0.442975
\(17\) −5.52260 −1.33943 −0.669714 0.742619i \(-0.733583\pi\)
−0.669714 + 0.742619i \(0.733583\pi\)
\(18\) 0 0
\(19\) 4.21489 0.966961 0.483481 0.875355i \(-0.339372\pi\)
0.483481 + 0.875355i \(0.339372\pi\)
\(20\) −6.24306 −1.39599
\(21\) 0 0
\(22\) 3.13071 0.667470
\(23\) 2.77495 0.578617 0.289308 0.957236i \(-0.406575\pi\)
0.289308 + 0.957236i \(0.406575\pi\)
\(24\) 0 0
\(25\) 10.1815 2.03630
\(26\) −1.06609 −0.209078
\(27\) 0 0
\(28\) 5.90013 1.11502
\(29\) 2.31253 0.429425 0.214713 0.976677i \(-0.431119\pi\)
0.214713 + 0.976677i \(0.431119\pi\)
\(30\) 0 0
\(31\) −0.199515 −0.0358340 −0.0179170 0.999839i \(-0.505703\pi\)
−0.0179170 + 0.999839i \(0.505703\pi\)
\(32\) 5.66096 1.00073
\(33\) 0 0
\(34\) −3.48280 −0.597295
\(35\) −14.3476 −2.42518
\(36\) 0 0
\(37\) 1.79089 0.294420 0.147210 0.989105i \(-0.452971\pi\)
0.147210 + 0.989105i \(0.452971\pi\)
\(38\) 2.65809 0.431200
\(39\) 0 0
\(40\) −8.85157 −1.39956
\(41\) 12.1960 1.90470 0.952351 0.305003i \(-0.0986575\pi\)
0.952351 + 0.305003i \(0.0986575\pi\)
\(42\) 0 0
\(43\) −5.34523 −0.815139 −0.407570 0.913174i \(-0.633624\pi\)
−0.407570 + 0.913174i \(0.633624\pi\)
\(44\) −7.95425 −1.19915
\(45\) 0 0
\(46\) 1.75001 0.258024
\(47\) 10.7345 1.56579 0.782893 0.622157i \(-0.213743\pi\)
0.782893 + 0.622157i \(0.213743\pi\)
\(48\) 0 0
\(49\) 6.55944 0.937063
\(50\) 6.42090 0.908052
\(51\) 0 0
\(52\) 2.70864 0.375621
\(53\) −1.32229 −0.181631 −0.0908154 0.995868i \(-0.528947\pi\)
−0.0908154 + 0.995868i \(0.528947\pi\)
\(54\) 0 0
\(55\) 19.3426 2.60816
\(56\) 8.36534 1.11787
\(57\) 0 0
\(58\) 1.45838 0.191495
\(59\) −5.78578 −0.753244 −0.376622 0.926367i \(-0.622914\pi\)
−0.376622 + 0.926367i \(0.622914\pi\)
\(60\) 0 0
\(61\) −0.0766435 −0.00981320 −0.00490660 0.999988i \(-0.501562\pi\)
−0.00490660 + 0.999988i \(0.501562\pi\)
\(62\) −0.125823 −0.0159795
\(63\) 0 0
\(64\) 0.0262532 0.00328165
\(65\) −6.58670 −0.816979
\(66\) 0 0
\(67\) 12.6166 1.54136 0.770681 0.637221i \(-0.219916\pi\)
0.770681 + 0.637221i \(0.219916\pi\)
\(68\) 8.84880 1.07307
\(69\) 0 0
\(70\) −9.04821 −1.08147
\(71\) 5.20391 0.617591 0.308796 0.951128i \(-0.400074\pi\)
0.308796 + 0.951128i \(0.400074\pi\)
\(72\) 0 0
\(73\) −14.0733 −1.64715 −0.823577 0.567204i \(-0.808025\pi\)
−0.823577 + 0.567204i \(0.808025\pi\)
\(74\) 1.12941 0.131291
\(75\) 0 0
\(76\) −6.75346 −0.774675
\(77\) −18.2801 −2.08322
\(78\) 0 0
\(79\) 7.82844 0.880769 0.440384 0.897809i \(-0.354842\pi\)
0.440384 + 0.897809i \(0.354842\pi\)
\(80\) 6.90393 0.771883
\(81\) 0 0
\(82\) 7.69137 0.849370
\(83\) −2.55372 −0.280307 −0.140153 0.990130i \(-0.544760\pi\)
−0.140153 + 0.990130i \(0.544760\pi\)
\(84\) 0 0
\(85\) −21.5180 −2.33395
\(86\) −3.37094 −0.363497
\(87\) 0 0
\(88\) −11.2777 −1.20221
\(89\) 4.35327 0.461446 0.230723 0.973019i \(-0.425891\pi\)
0.230723 + 0.973019i \(0.425891\pi\)
\(90\) 0 0
\(91\) 6.22489 0.652546
\(92\) −4.44627 −0.463555
\(93\) 0 0
\(94\) 6.76964 0.698235
\(95\) 16.4226 1.68493
\(96\) 0 0
\(97\) 9.02693 0.916546 0.458273 0.888811i \(-0.348468\pi\)
0.458273 + 0.888811i \(0.348468\pi\)
\(98\) 4.13668 0.417867
\(99\) 0 0
\(100\) −16.3137 −1.63137
\(101\) −6.94323 −0.690877 −0.345439 0.938441i \(-0.612270\pi\)
−0.345439 + 0.938441i \(0.612270\pi\)
\(102\) 0 0
\(103\) −0.105009 −0.0103469 −0.00517343 0.999987i \(-0.501647\pi\)
−0.00517343 + 0.999987i \(0.501647\pi\)
\(104\) 3.84037 0.376580
\(105\) 0 0
\(106\) −0.833897 −0.0809952
\(107\) 18.1842 1.75793 0.878966 0.476885i \(-0.158234\pi\)
0.878966 + 0.476885i \(0.158234\pi\)
\(108\) 0 0
\(109\) 14.6576 1.40394 0.701972 0.712205i \(-0.252303\pi\)
0.701972 + 0.712205i \(0.252303\pi\)
\(110\) 12.1983 1.16307
\(111\) 0 0
\(112\) −6.52470 −0.616526
\(113\) −1.08836 −0.102384 −0.0511922 0.998689i \(-0.516302\pi\)
−0.0511922 + 0.998689i \(0.516302\pi\)
\(114\) 0 0
\(115\) 10.8122 1.00824
\(116\) −3.70533 −0.344031
\(117\) 0 0
\(118\) −3.64877 −0.335896
\(119\) 20.3360 1.86420
\(120\) 0 0
\(121\) 13.6443 1.24040
\(122\) −0.0483348 −0.00437603
\(123\) 0 0
\(124\) 0.319681 0.0287082
\(125\) 20.1889 1.80575
\(126\) 0 0
\(127\) 0.364538 0.0323475 0.0161737 0.999869i \(-0.494852\pi\)
0.0161737 + 0.999869i \(0.494852\pi\)
\(128\) −11.3054 −0.999263
\(129\) 0 0
\(130\) −4.15386 −0.364318
\(131\) −5.48865 −0.479546 −0.239773 0.970829i \(-0.577073\pi\)
−0.239773 + 0.970829i \(0.577073\pi\)
\(132\) 0 0
\(133\) −15.5205 −1.34580
\(134\) 7.95658 0.687344
\(135\) 0 0
\(136\) 12.5460 1.07581
\(137\) 17.1533 1.46551 0.732754 0.680493i \(-0.238234\pi\)
0.732754 + 0.680493i \(0.238234\pi\)
\(138\) 0 0
\(139\) −19.6074 −1.66308 −0.831539 0.555467i \(-0.812540\pi\)
−0.831539 + 0.555467i \(0.812540\pi\)
\(140\) 22.9889 1.94292
\(141\) 0 0
\(142\) 3.28182 0.275404
\(143\) −8.39207 −0.701780
\(144\) 0 0
\(145\) 9.01039 0.748272
\(146\) −8.87525 −0.734521
\(147\) 0 0
\(148\) −2.86951 −0.235873
\(149\) −19.0508 −1.56070 −0.780350 0.625343i \(-0.784959\pi\)
−0.780350 + 0.625343i \(0.784959\pi\)
\(150\) 0 0
\(151\) −4.76802 −0.388016 −0.194008 0.981000i \(-0.562149\pi\)
−0.194008 + 0.981000i \(0.562149\pi\)
\(152\) −9.57522 −0.776653
\(153\) 0 0
\(154\) −11.5283 −0.928974
\(155\) −0.777379 −0.0624406
\(156\) 0 0
\(157\) 6.12724 0.489007 0.244503 0.969648i \(-0.421375\pi\)
0.244503 + 0.969648i \(0.421375\pi\)
\(158\) 4.93696 0.392764
\(159\) 0 0
\(160\) 22.0571 1.74376
\(161\) −10.2182 −0.805310
\(162\) 0 0
\(163\) −1.20414 −0.0943159 −0.0471579 0.998887i \(-0.515016\pi\)
−0.0471579 + 0.998887i \(0.515016\pi\)
\(164\) −19.5416 −1.52594
\(165\) 0 0
\(166\) −1.61049 −0.124998
\(167\) 0.900391 0.0696744 0.0348372 0.999393i \(-0.488909\pi\)
0.0348372 + 0.999393i \(0.488909\pi\)
\(168\) 0 0
\(169\) −10.1423 −0.780175
\(170\) −13.5702 −1.04079
\(171\) 0 0
\(172\) 8.56459 0.653044
\(173\) −12.3564 −0.939436 −0.469718 0.882817i \(-0.655644\pi\)
−0.469718 + 0.882817i \(0.655644\pi\)
\(174\) 0 0
\(175\) −37.4915 −2.83409
\(176\) 8.79626 0.663043
\(177\) 0 0
\(178\) 2.74537 0.205774
\(179\) 10.0258 0.749367 0.374683 0.927153i \(-0.377751\pi\)
0.374683 + 0.927153i \(0.377751\pi\)
\(180\) 0 0
\(181\) −17.0491 −1.26725 −0.633623 0.773642i \(-0.718433\pi\)
−0.633623 + 0.773642i \(0.718433\pi\)
\(182\) 3.92569 0.290992
\(183\) 0 0
\(184\) −6.30402 −0.464739
\(185\) 6.97790 0.513026
\(186\) 0 0
\(187\) −27.4159 −2.00485
\(188\) −17.1997 −1.25442
\(189\) 0 0
\(190\) 10.3568 0.751365
\(191\) 23.7744 1.72026 0.860129 0.510077i \(-0.170383\pi\)
0.860129 + 0.510077i \(0.170383\pi\)
\(192\) 0 0
\(193\) 7.49373 0.539411 0.269705 0.962943i \(-0.413074\pi\)
0.269705 + 0.962943i \(0.413074\pi\)
\(194\) 5.69278 0.408718
\(195\) 0 0
\(196\) −10.5101 −0.750723
\(197\) −1.85647 −0.132268 −0.0661339 0.997811i \(-0.521066\pi\)
−0.0661339 + 0.997811i \(0.521066\pi\)
\(198\) 0 0
\(199\) 3.46667 0.245746 0.122873 0.992422i \(-0.460789\pi\)
0.122873 + 0.992422i \(0.460789\pi\)
\(200\) −23.1299 −1.63553
\(201\) 0 0
\(202\) −4.37871 −0.308085
\(203\) −8.51545 −0.597667
\(204\) 0 0
\(205\) 47.5200 3.31894
\(206\) −0.0662234 −0.00461400
\(207\) 0 0
\(208\) −2.99537 −0.207691
\(209\) 20.9240 1.44734
\(210\) 0 0
\(211\) −2.33201 −0.160542 −0.0802711 0.996773i \(-0.525579\pi\)
−0.0802711 + 0.996773i \(0.525579\pi\)
\(212\) 2.11869 0.145512
\(213\) 0 0
\(214\) 11.4678 0.783919
\(215\) −20.8268 −1.42038
\(216\) 0 0
\(217\) 0.734678 0.0498732
\(218\) 9.24373 0.626065
\(219\) 0 0
\(220\) −30.9925 −2.08951
\(221\) 9.33587 0.627998
\(222\) 0 0
\(223\) −1.75535 −0.117547 −0.0587734 0.998271i \(-0.518719\pi\)
−0.0587734 + 0.998271i \(0.518719\pi\)
\(224\) −20.8455 −1.39280
\(225\) 0 0
\(226\) −0.686369 −0.0456566
\(227\) −4.35631 −0.289139 −0.144569 0.989495i \(-0.546180\pi\)
−0.144569 + 0.989495i \(0.546180\pi\)
\(228\) 0 0
\(229\) 6.24160 0.412457 0.206228 0.978504i \(-0.433881\pi\)
0.206228 + 0.978504i \(0.433881\pi\)
\(230\) 6.81862 0.449607
\(231\) 0 0
\(232\) −5.25351 −0.344910
\(233\) −13.8315 −0.906135 −0.453067 0.891476i \(-0.649670\pi\)
−0.453067 + 0.891476i \(0.649670\pi\)
\(234\) 0 0
\(235\) 41.8252 2.72838
\(236\) 9.27048 0.603457
\(237\) 0 0
\(238\) 12.8248 0.831306
\(239\) 19.3924 1.25439 0.627194 0.778863i \(-0.284203\pi\)
0.627194 + 0.778863i \(0.284203\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157
\(242\) 8.60473 0.553133
\(243\) 0 0
\(244\) 0.122805 0.00786178
\(245\) 25.5578 1.63283
\(246\) 0 0
\(247\) −7.12519 −0.453365
\(248\) 0.453251 0.0287815
\(249\) 0 0
\(250\) 12.7320 0.805242
\(251\) 14.4441 0.911705 0.455852 0.890055i \(-0.349334\pi\)
0.455852 + 0.890055i \(0.349334\pi\)
\(252\) 0 0
\(253\) 13.7757 0.866071
\(254\) 0.229894 0.0144248
\(255\) 0 0
\(256\) −7.18218 −0.448886
\(257\) −28.9004 −1.80276 −0.901379 0.433031i \(-0.857444\pi\)
−0.901379 + 0.433031i \(0.857444\pi\)
\(258\) 0 0
\(259\) −6.59460 −0.409769
\(260\) 10.5538 0.654518
\(261\) 0 0
\(262\) −3.46139 −0.213845
\(263\) −26.3078 −1.62221 −0.811103 0.584903i \(-0.801133\pi\)
−0.811103 + 0.584903i \(0.801133\pi\)
\(264\) 0 0
\(265\) −5.15211 −0.316491
\(266\) −9.78794 −0.600137
\(267\) 0 0
\(268\) −20.2154 −1.23485
\(269\) 9.42450 0.574622 0.287311 0.957837i \(-0.407239\pi\)
0.287311 + 0.957837i \(0.407239\pi\)
\(270\) 0 0
\(271\) −5.86519 −0.356285 −0.178142 0.984005i \(-0.557009\pi\)
−0.178142 + 0.984005i \(0.557009\pi\)
\(272\) −9.78551 −0.593334
\(273\) 0 0
\(274\) 10.8177 0.653519
\(275\) 50.5440 3.04792
\(276\) 0 0
\(277\) 5.60395 0.336708 0.168354 0.985727i \(-0.446155\pi\)
0.168354 + 0.985727i \(0.446155\pi\)
\(278\) −12.3653 −0.741621
\(279\) 0 0
\(280\) 32.5943 1.94788
\(281\) 4.71669 0.281374 0.140687 0.990054i \(-0.455069\pi\)
0.140687 + 0.990054i \(0.455069\pi\)
\(282\) 0 0
\(283\) 19.2068 1.14173 0.570864 0.821044i \(-0.306608\pi\)
0.570864 + 0.821044i \(0.306608\pi\)
\(284\) −8.33817 −0.494779
\(285\) 0 0
\(286\) −5.29241 −0.312947
\(287\) −44.9097 −2.65094
\(288\) 0 0
\(289\) 13.4992 0.794069
\(290\) 5.68235 0.333679
\(291\) 0 0
\(292\) 22.5495 1.31961
\(293\) −13.0140 −0.760289 −0.380144 0.924927i \(-0.624126\pi\)
−0.380144 + 0.924927i \(0.624126\pi\)
\(294\) 0 0
\(295\) −22.5434 −1.31253
\(296\) −4.06847 −0.236475
\(297\) 0 0
\(298\) −12.0143 −0.695968
\(299\) −4.69100 −0.271288
\(300\) 0 0
\(301\) 19.6828 1.13450
\(302\) −3.00692 −0.173029
\(303\) 0 0
\(304\) 7.46836 0.428340
\(305\) −0.298630 −0.0170995
\(306\) 0 0
\(307\) −11.5372 −0.658463 −0.329232 0.944249i \(-0.606790\pi\)
−0.329232 + 0.944249i \(0.606790\pi\)
\(308\) 29.2900 1.66896
\(309\) 0 0
\(310\) −0.490250 −0.0278443
\(311\) 23.1542 1.31296 0.656478 0.754345i \(-0.272045\pi\)
0.656478 + 0.754345i \(0.272045\pi\)
\(312\) 0 0
\(313\) −7.88902 −0.445914 −0.222957 0.974828i \(-0.571571\pi\)
−0.222957 + 0.974828i \(0.571571\pi\)
\(314\) 3.86411 0.218064
\(315\) 0 0
\(316\) −12.5434 −0.705623
\(317\) 2.38325 0.133857 0.0669283 0.997758i \(-0.478680\pi\)
0.0669283 + 0.997758i \(0.478680\pi\)
\(318\) 0 0
\(319\) 11.4801 0.642761
\(320\) 0.102292 0.00571827
\(321\) 0 0
\(322\) −6.44407 −0.359114
\(323\) −23.2772 −1.29518
\(324\) 0 0
\(325\) −17.2116 −0.954729
\(326\) −0.759387 −0.0420585
\(327\) 0 0
\(328\) −27.7065 −1.52984
\(329\) −39.5278 −2.17924
\(330\) 0 0
\(331\) −3.05598 −0.167972 −0.0839860 0.996467i \(-0.526765\pi\)
−0.0839860 + 0.996467i \(0.526765\pi\)
\(332\) 4.09179 0.224566
\(333\) 0 0
\(334\) 0.567827 0.0310701
\(335\) 49.1586 2.68582
\(336\) 0 0
\(337\) −16.1755 −0.881134 −0.440567 0.897720i \(-0.645223\pi\)
−0.440567 + 0.897720i \(0.645223\pi\)
\(338\) −6.39617 −0.347906
\(339\) 0 0
\(340\) 34.4780 1.86983
\(341\) −0.990454 −0.0536361
\(342\) 0 0
\(343\) 1.62227 0.0875943
\(344\) 12.1431 0.654711
\(345\) 0 0
\(346\) −7.79246 −0.418925
\(347\) −24.4716 −1.31370 −0.656851 0.754020i \(-0.728112\pi\)
−0.656851 + 0.754020i \(0.728112\pi\)
\(348\) 0 0
\(349\) −13.0468 −0.698380 −0.349190 0.937052i \(-0.613543\pi\)
−0.349190 + 0.937052i \(0.613543\pi\)
\(350\) −23.6438 −1.26381
\(351\) 0 0
\(352\) 28.1028 1.49788
\(353\) −23.6788 −1.26029 −0.630147 0.776476i \(-0.717005\pi\)
−0.630147 + 0.776476i \(0.717005\pi\)
\(354\) 0 0
\(355\) 20.2762 1.07615
\(356\) −6.97519 −0.369685
\(357\) 0 0
\(358\) 6.32274 0.334167
\(359\) 25.5499 1.34847 0.674236 0.738516i \(-0.264473\pi\)
0.674236 + 0.738516i \(0.264473\pi\)
\(360\) 0 0
\(361\) −1.23473 −0.0649857
\(362\) −10.7519 −0.565107
\(363\) 0 0
\(364\) −9.97406 −0.522783
\(365\) −54.8344 −2.87016
\(366\) 0 0
\(367\) 8.35766 0.436266 0.218133 0.975919i \(-0.430003\pi\)
0.218133 + 0.975919i \(0.430003\pi\)
\(368\) 4.91693 0.256313
\(369\) 0 0
\(370\) 4.40058 0.228775
\(371\) 4.86910 0.252791
\(372\) 0 0
\(373\) −24.6934 −1.27858 −0.639288 0.768968i \(-0.720771\pi\)
−0.639288 + 0.768968i \(0.720771\pi\)
\(374\) −17.2897 −0.894028
\(375\) 0 0
\(376\) −24.3862 −1.25762
\(377\) −3.90928 −0.201338
\(378\) 0 0
\(379\) 18.6593 0.958465 0.479232 0.877688i \(-0.340915\pi\)
0.479232 + 0.877688i \(0.340915\pi\)
\(380\) −26.3138 −1.34987
\(381\) 0 0
\(382\) 14.9932 0.767119
\(383\) −11.9780 −0.612049 −0.306025 0.952024i \(-0.598999\pi\)
−0.306025 + 0.952024i \(0.598999\pi\)
\(384\) 0 0
\(385\) −71.2257 −3.63000
\(386\) 4.72588 0.240541
\(387\) 0 0
\(388\) −14.4637 −0.734285
\(389\) 7.62670 0.386689 0.193344 0.981131i \(-0.438067\pi\)
0.193344 + 0.981131i \(0.438067\pi\)
\(390\) 0 0
\(391\) −15.3249 −0.775016
\(392\) −14.9015 −0.752639
\(393\) 0 0
\(394\) −1.17077 −0.0589826
\(395\) 30.5023 1.53474
\(396\) 0 0
\(397\) −33.4013 −1.67636 −0.838181 0.545392i \(-0.816381\pi\)
−0.838181 + 0.545392i \(0.816381\pi\)
\(398\) 2.18624 0.109586
\(399\) 0 0
\(400\) 18.0406 0.902030
\(401\) 1.73530 0.0866566 0.0433283 0.999061i \(-0.486204\pi\)
0.0433283 + 0.999061i \(0.486204\pi\)
\(402\) 0 0
\(403\) 0.337277 0.0168010
\(404\) 11.1251 0.553492
\(405\) 0 0
\(406\) −5.37022 −0.266519
\(407\) 8.89050 0.440686
\(408\) 0 0
\(409\) −3.81908 −0.188841 −0.0944207 0.995532i \(-0.530100\pi\)
−0.0944207 + 0.995532i \(0.530100\pi\)
\(410\) 29.9682 1.48002
\(411\) 0 0
\(412\) 0.168255 0.00828932
\(413\) 21.3050 1.04835
\(414\) 0 0
\(415\) −9.95015 −0.488434
\(416\) −9.56976 −0.469196
\(417\) 0 0
\(418\) 13.1956 0.645418
\(419\) −0.640088 −0.0312703 −0.0156352 0.999878i \(-0.504977\pi\)
−0.0156352 + 0.999878i \(0.504977\pi\)
\(420\) 0 0
\(421\) 13.2765 0.647057 0.323528 0.946218i \(-0.395131\pi\)
0.323528 + 0.946218i \(0.395131\pi\)
\(422\) −1.47067 −0.0715910
\(423\) 0 0
\(424\) 3.00394 0.145884
\(425\) −56.2283 −2.72748
\(426\) 0 0
\(427\) 0.282226 0.0136579
\(428\) −29.1363 −1.40836
\(429\) 0 0
\(430\) −13.1343 −0.633393
\(431\) 26.8180 1.29178 0.645888 0.763432i \(-0.276487\pi\)
0.645888 + 0.763432i \(0.276487\pi\)
\(432\) 0 0
\(433\) −20.7392 −0.996664 −0.498332 0.866986i \(-0.666054\pi\)
−0.498332 + 0.866986i \(0.666054\pi\)
\(434\) 0.463320 0.0222401
\(435\) 0 0
\(436\) −23.4857 −1.12476
\(437\) 11.6961 0.559500
\(438\) 0 0
\(439\) −12.7382 −0.607960 −0.303980 0.952678i \(-0.598316\pi\)
−0.303980 + 0.952678i \(0.598316\pi\)
\(440\) −43.9419 −2.09485
\(441\) 0 0
\(442\) 5.88761 0.280045
\(443\) −23.4058 −1.11204 −0.556022 0.831167i \(-0.687673\pi\)
−0.556022 + 0.831167i \(0.687673\pi\)
\(444\) 0 0
\(445\) 16.9618 0.804068
\(446\) −1.10700 −0.0524180
\(447\) 0 0
\(448\) −0.0966726 −0.00456735
\(449\) −35.8887 −1.69369 −0.846846 0.531838i \(-0.821502\pi\)
−0.846846 + 0.531838i \(0.821502\pi\)
\(450\) 0 0
\(451\) 60.5449 2.85095
\(452\) 1.74387 0.0820247
\(453\) 0 0
\(454\) −2.74728 −0.128936
\(455\) 24.2543 1.13706
\(456\) 0 0
\(457\) −23.4014 −1.09467 −0.547336 0.836913i \(-0.684358\pi\)
−0.547336 + 0.836913i \(0.684358\pi\)
\(458\) 3.93623 0.183928
\(459\) 0 0
\(460\) −17.3242 −0.807744
\(461\) −17.9146 −0.834366 −0.417183 0.908822i \(-0.636983\pi\)
−0.417183 + 0.908822i \(0.636983\pi\)
\(462\) 0 0
\(463\) −32.0159 −1.48790 −0.743952 0.668233i \(-0.767051\pi\)
−0.743952 + 0.668233i \(0.767051\pi\)
\(464\) 4.09757 0.190225
\(465\) 0 0
\(466\) −8.72279 −0.404075
\(467\) 12.2949 0.568940 0.284470 0.958685i \(-0.408182\pi\)
0.284470 + 0.958685i \(0.408182\pi\)
\(468\) 0 0
\(469\) −46.4583 −2.14524
\(470\) 26.3768 1.21667
\(471\) 0 0
\(472\) 13.1439 0.604997
\(473\) −26.5353 −1.22010
\(474\) 0 0
\(475\) 42.9138 1.96902
\(476\) −32.5841 −1.49349
\(477\) 0 0
\(478\) 12.2297 0.559373
\(479\) −15.1555 −0.692472 −0.346236 0.938148i \(-0.612540\pi\)
−0.346236 + 0.938148i \(0.612540\pi\)
\(480\) 0 0
\(481\) −3.02746 −0.138040
\(482\) −0.630644 −0.0287251
\(483\) 0 0
\(484\) −21.8622 −0.993735
\(485\) 35.1720 1.59708
\(486\) 0 0
\(487\) −15.5008 −0.702407 −0.351204 0.936299i \(-0.614227\pi\)
−0.351204 + 0.936299i \(0.614227\pi\)
\(488\) 0.174116 0.00788186
\(489\) 0 0
\(490\) 16.1179 0.728133
\(491\) −12.5838 −0.567901 −0.283950 0.958839i \(-0.591645\pi\)
−0.283950 + 0.958839i \(0.591645\pi\)
\(492\) 0 0
\(493\) −12.7712 −0.575184
\(494\) −4.49346 −0.202170
\(495\) 0 0
\(496\) −0.353521 −0.0158736
\(497\) −19.1625 −0.859553
\(498\) 0 0
\(499\) −41.9638 −1.87856 −0.939280 0.343152i \(-0.888505\pi\)
−0.939280 + 0.343152i \(0.888505\pi\)
\(500\) −32.3484 −1.44666
\(501\) 0 0
\(502\) 9.10910 0.406559
\(503\) −7.23458 −0.322574 −0.161287 0.986908i \(-0.551564\pi\)
−0.161287 + 0.986908i \(0.551564\pi\)
\(504\) 0 0
\(505\) −27.0532 −1.20385
\(506\) 8.68756 0.386209
\(507\) 0 0
\(508\) −0.584094 −0.0259150
\(509\) −43.2945 −1.91899 −0.959497 0.281718i \(-0.909096\pi\)
−0.959497 + 0.281718i \(0.909096\pi\)
\(510\) 0 0
\(511\) 51.8223 2.29248
\(512\) 18.0813 0.799090
\(513\) 0 0
\(514\) −18.2259 −0.803909
\(515\) −0.409152 −0.0180294
\(516\) 0 0
\(517\) 53.2893 2.34366
\(518\) −4.15885 −0.182729
\(519\) 0 0
\(520\) 14.9634 0.656189
\(521\) 22.7560 0.996961 0.498480 0.866901i \(-0.333892\pi\)
0.498480 + 0.866901i \(0.333892\pi\)
\(522\) 0 0
\(523\) 36.7529 1.60709 0.803547 0.595241i \(-0.202944\pi\)
0.803547 + 0.595241i \(0.202944\pi\)
\(524\) 8.79440 0.384185
\(525\) 0 0
\(526\) −16.5908 −0.723395
\(527\) 1.10184 0.0479970
\(528\) 0 0
\(529\) −15.2997 −0.665203
\(530\) −3.24915 −0.141134
\(531\) 0 0
\(532\) 24.8684 1.07818
\(533\) −20.6172 −0.893030
\(534\) 0 0
\(535\) 70.8518 3.06319
\(536\) −28.6619 −1.23801
\(537\) 0 0
\(538\) 5.94351 0.256243
\(539\) 32.5631 1.40259
\(540\) 0 0
\(541\) −19.7726 −0.850089 −0.425045 0.905172i \(-0.639742\pi\)
−0.425045 + 0.905172i \(0.639742\pi\)
\(542\) −3.69885 −0.158879
\(543\) 0 0
\(544\) −31.2633 −1.34040
\(545\) 57.1110 2.44637
\(546\) 0 0
\(547\) −9.94319 −0.425140 −0.212570 0.977146i \(-0.568183\pi\)
−0.212570 + 0.977146i \(0.568183\pi\)
\(548\) −27.4846 −1.17408
\(549\) 0 0
\(550\) 31.8753 1.35917
\(551\) 9.74703 0.415238
\(552\) 0 0
\(553\) −28.8268 −1.22584
\(554\) 3.53410 0.150149
\(555\) 0 0
\(556\) 31.4167 1.33236
\(557\) 7.84163 0.332260 0.166130 0.986104i \(-0.446873\pi\)
0.166130 + 0.986104i \(0.446873\pi\)
\(558\) 0 0
\(559\) 9.03601 0.382182
\(560\) −25.4225 −1.07430
\(561\) 0 0
\(562\) 2.97455 0.125474
\(563\) 19.2105 0.809625 0.404813 0.914400i \(-0.367337\pi\)
0.404813 + 0.914400i \(0.367337\pi\)
\(564\) 0 0
\(565\) −4.24063 −0.178405
\(566\) 12.1127 0.509134
\(567\) 0 0
\(568\) −11.8221 −0.496043
\(569\) −25.1338 −1.05366 −0.526831 0.849970i \(-0.676620\pi\)
−0.526831 + 0.849970i \(0.676620\pi\)
\(570\) 0 0
\(571\) −18.1912 −0.761277 −0.380638 0.924724i \(-0.624296\pi\)
−0.380638 + 0.924724i \(0.624296\pi\)
\(572\) 13.4465 0.562227
\(573\) 0 0
\(574\) −28.3220 −1.18214
\(575\) 28.2531 1.17824
\(576\) 0 0
\(577\) 28.9417 1.20486 0.602429 0.798172i \(-0.294200\pi\)
0.602429 + 0.798172i \(0.294200\pi\)
\(578\) 8.51317 0.354101
\(579\) 0 0
\(580\) −14.4372 −0.599474
\(581\) 9.40358 0.390126
\(582\) 0 0
\(583\) −6.56427 −0.271864
\(584\) 31.9712 1.32298
\(585\) 0 0
\(586\) −8.20724 −0.339038
\(587\) 21.9772 0.907098 0.453549 0.891231i \(-0.350158\pi\)
0.453549 + 0.891231i \(0.350158\pi\)
\(588\) 0 0
\(589\) −0.840934 −0.0346501
\(590\) −14.2168 −0.585298
\(591\) 0 0
\(592\) 3.17327 0.130421
\(593\) −15.0138 −0.616544 −0.308272 0.951298i \(-0.599751\pi\)
−0.308272 + 0.951298i \(0.599751\pi\)
\(594\) 0 0
\(595\) 79.2359 3.24836
\(596\) 30.5248 1.25035
\(597\) 0 0
\(598\) −2.95835 −0.120976
\(599\) 31.7086 1.29558 0.647789 0.761820i \(-0.275694\pi\)
0.647789 + 0.761820i \(0.275694\pi\)
\(600\) 0 0
\(601\) 5.63682 0.229931 0.114965 0.993369i \(-0.463324\pi\)
0.114965 + 0.993369i \(0.463324\pi\)
\(602\) 12.4129 0.505910
\(603\) 0 0
\(604\) 7.63973 0.310856
\(605\) 53.1631 2.16139
\(606\) 0 0
\(607\) 39.7534 1.61354 0.806770 0.590866i \(-0.201214\pi\)
0.806770 + 0.590866i \(0.201214\pi\)
\(608\) 23.8603 0.967664
\(609\) 0 0
\(610\) −0.188329 −0.00762522
\(611\) −18.1465 −0.734127
\(612\) 0 0
\(613\) 30.3066 1.22407 0.612035 0.790831i \(-0.290351\pi\)
0.612035 + 0.790831i \(0.290351\pi\)
\(614\) −7.27588 −0.293631
\(615\) 0 0
\(616\) 41.5281 1.67322
\(617\) −18.2240 −0.733670 −0.366835 0.930286i \(-0.619559\pi\)
−0.366835 + 0.930286i \(0.619559\pi\)
\(618\) 0 0
\(619\) −26.3431 −1.05882 −0.529409 0.848366i \(-0.677586\pi\)
−0.529409 + 0.848366i \(0.677586\pi\)
\(620\) 1.24559 0.0500239
\(621\) 0 0
\(622\) 14.6021 0.585491
\(623\) −16.0301 −0.642233
\(624\) 0 0
\(625\) 27.7553 1.11021
\(626\) −4.97516 −0.198848
\(627\) 0 0
\(628\) −9.81759 −0.391765
\(629\) −9.89035 −0.394354
\(630\) 0 0
\(631\) 18.4187 0.733236 0.366618 0.930372i \(-0.380516\pi\)
0.366618 + 0.930372i \(0.380516\pi\)
\(632\) −17.7844 −0.707424
\(633\) 0 0
\(634\) 1.50298 0.0596910
\(635\) 1.42036 0.0563654
\(636\) 0 0
\(637\) −11.0886 −0.439347
\(638\) 7.23985 0.286629
\(639\) 0 0
\(640\) −44.0496 −1.74121
\(641\) 12.6956 0.501446 0.250723 0.968059i \(-0.419332\pi\)
0.250723 + 0.968059i \(0.419332\pi\)
\(642\) 0 0
\(643\) −35.1591 −1.38654 −0.693270 0.720678i \(-0.743831\pi\)
−0.693270 + 0.720678i \(0.743831\pi\)
\(644\) 16.3725 0.645169
\(645\) 0 0
\(646\) −14.6796 −0.577561
\(647\) −19.7535 −0.776588 −0.388294 0.921535i \(-0.626936\pi\)
−0.388294 + 0.921535i \(0.626936\pi\)
\(648\) 0 0
\(649\) −28.7224 −1.12745
\(650\) −10.8544 −0.425745
\(651\) 0 0
\(652\) 1.92939 0.0755606
\(653\) 44.0129 1.72236 0.861178 0.508303i \(-0.169727\pi\)
0.861178 + 0.508303i \(0.169727\pi\)
\(654\) 0 0
\(655\) −21.3857 −0.835608
\(656\) 21.6102 0.843736
\(657\) 0 0
\(658\) −24.9280 −0.971793
\(659\) 32.2032 1.25446 0.627229 0.778835i \(-0.284189\pi\)
0.627229 + 0.778835i \(0.284189\pi\)
\(660\) 0 0
\(661\) 29.5485 1.14930 0.574651 0.818398i \(-0.305138\pi\)
0.574651 + 0.818398i \(0.305138\pi\)
\(662\) −1.92724 −0.0749043
\(663\) 0 0
\(664\) 5.80143 0.225139
\(665\) −60.4734 −2.34506
\(666\) 0 0
\(667\) 6.41714 0.248473
\(668\) −1.44269 −0.0558192
\(669\) 0 0
\(670\) 31.0016 1.19770
\(671\) −0.380482 −0.0146883
\(672\) 0 0
\(673\) 28.4424 1.09637 0.548186 0.836356i \(-0.315319\pi\)
0.548186 + 0.836356i \(0.315319\pi\)
\(674\) −10.2010 −0.392927
\(675\) 0 0
\(676\) 16.2508 0.625032
\(677\) −16.9259 −0.650514 −0.325257 0.945626i \(-0.605451\pi\)
−0.325257 + 0.945626i \(0.605451\pi\)
\(678\) 0 0
\(679\) −33.2400 −1.27563
\(680\) 48.8837 1.87460
\(681\) 0 0
\(682\) −0.624625 −0.0239181
\(683\) 29.0324 1.11090 0.555448 0.831551i \(-0.312547\pi\)
0.555448 + 0.831551i \(0.312547\pi\)
\(684\) 0 0
\(685\) 66.8353 2.55365
\(686\) 1.02307 0.0390612
\(687\) 0 0
\(688\) −9.47121 −0.361087
\(689\) 2.23531 0.0851586
\(690\) 0 0
\(691\) −21.5798 −0.820935 −0.410467 0.911875i \(-0.634634\pi\)
−0.410467 + 0.911875i \(0.634634\pi\)
\(692\) 19.7984 0.752623
\(693\) 0 0
\(694\) −15.4329 −0.585823
\(695\) −76.3972 −2.89791
\(696\) 0 0
\(697\) −67.3540 −2.55121
\(698\) −8.22790 −0.311431
\(699\) 0 0
\(700\) 60.0721 2.27051
\(701\) 13.2247 0.499489 0.249744 0.968312i \(-0.419653\pi\)
0.249744 + 0.968312i \(0.419653\pi\)
\(702\) 0 0
\(703\) 7.54838 0.284692
\(704\) 0.130329 0.00491196
\(705\) 0 0
\(706\) −14.9329 −0.562006
\(707\) 25.5672 0.961552
\(708\) 0 0
\(709\) 18.8314 0.707228 0.353614 0.935391i \(-0.384953\pi\)
0.353614 + 0.935391i \(0.384953\pi\)
\(710\) 12.7871 0.479891
\(711\) 0 0
\(712\) −9.88960 −0.370628
\(713\) −0.553644 −0.0207341
\(714\) 0 0
\(715\) −32.6984 −1.22285
\(716\) −16.0643 −0.600351
\(717\) 0 0
\(718\) 16.1129 0.601328
\(719\) 37.7936 1.40946 0.704731 0.709475i \(-0.251068\pi\)
0.704731 + 0.709475i \(0.251068\pi\)
\(720\) 0 0
\(721\) 0.386677 0.0144006
\(722\) −0.778675 −0.0289793
\(723\) 0 0
\(724\) 27.3175 1.01525
\(725\) 23.5450 0.874438
\(726\) 0 0
\(727\) −9.96692 −0.369653 −0.184826 0.982771i \(-0.559172\pi\)
−0.184826 + 0.982771i \(0.559172\pi\)
\(728\) −14.1415 −0.524118
\(729\) 0 0
\(730\) −34.5810 −1.27990
\(731\) 29.5196 1.09182
\(732\) 0 0
\(733\) 46.8282 1.72964 0.864819 0.502083i \(-0.167433\pi\)
0.864819 + 0.502083i \(0.167433\pi\)
\(734\) 5.27071 0.194546
\(735\) 0 0
\(736\) 15.7089 0.579037
\(737\) 62.6326 2.30710
\(738\) 0 0
\(739\) 31.1213 1.14482 0.572408 0.819969i \(-0.306009\pi\)
0.572408 + 0.819969i \(0.306009\pi\)
\(740\) −11.1806 −0.411007
\(741\) 0 0
\(742\) 3.07067 0.112728
\(743\) 22.3465 0.819815 0.409907 0.912127i \(-0.365561\pi\)
0.409907 + 0.912127i \(0.365561\pi\)
\(744\) 0 0
\(745\) −74.2283 −2.71952
\(746\) −15.5728 −0.570159
\(747\) 0 0
\(748\) 43.9282 1.60617
\(749\) −66.9599 −2.44666
\(750\) 0 0
\(751\) −41.3430 −1.50863 −0.754313 0.656515i \(-0.772030\pi\)
−0.754313 + 0.656515i \(0.772030\pi\)
\(752\) 19.0204 0.693604
\(753\) 0 0
\(754\) −2.46537 −0.0897834
\(755\) −18.5778 −0.676116
\(756\) 0 0
\(757\) −5.70532 −0.207364 −0.103682 0.994611i \(-0.533062\pi\)
−0.103682 + 0.994611i \(0.533062\pi\)
\(758\) 11.7674 0.427411
\(759\) 0 0
\(760\) −37.3083 −1.35332
\(761\) −15.1656 −0.549754 −0.274877 0.961479i \(-0.588637\pi\)
−0.274877 + 0.961479i \(0.588637\pi\)
\(762\) 0 0
\(763\) −53.9739 −1.95399
\(764\) −38.0935 −1.37817
\(765\) 0 0
\(766\) −7.55388 −0.272933
\(767\) 9.78075 0.353162
\(768\) 0 0
\(769\) 5.23538 0.188793 0.0943964 0.995535i \(-0.469908\pi\)
0.0943964 + 0.995535i \(0.469908\pi\)
\(770\) −44.9181 −1.61874
\(771\) 0 0
\(772\) −12.0071 −0.432145
\(773\) 27.0422 0.972641 0.486321 0.873780i \(-0.338339\pi\)
0.486321 + 0.873780i \(0.338339\pi\)
\(774\) 0 0
\(775\) −2.03136 −0.0729687
\(776\) −20.5070 −0.736160
\(777\) 0 0
\(778\) 4.80973 0.172437
\(779\) 51.4050 1.84177
\(780\) 0 0
\(781\) 25.8338 0.924407
\(782\) −9.66459 −0.345605
\(783\) 0 0
\(784\) 11.6227 0.415096
\(785\) 23.8738 0.852093
\(786\) 0 0
\(787\) −25.6221 −0.913329 −0.456665 0.889639i \(-0.650956\pi\)
−0.456665 + 0.889639i \(0.650956\pi\)
\(788\) 2.97459 0.105966
\(789\) 0 0
\(790\) 19.2361 0.684390
\(791\) 4.00769 0.142497
\(792\) 0 0
\(793\) 0.129565 0.00460097
\(794\) −21.0643 −0.747545
\(795\) 0 0
\(796\) −5.55460 −0.196878
\(797\) −28.8151 −1.02068 −0.510341 0.859972i \(-0.670481\pi\)
−0.510341 + 0.859972i \(0.670481\pi\)
\(798\) 0 0
\(799\) −59.2823 −2.09726
\(800\) 57.6371 2.03778
\(801\) 0 0
\(802\) 1.09436 0.0386430
\(803\) −69.8642 −2.46545
\(804\) 0 0
\(805\) −39.8137 −1.40325
\(806\) 0.212702 0.00749210
\(807\) 0 0
\(808\) 15.7734 0.554905
\(809\) −20.6265 −0.725188 −0.362594 0.931947i \(-0.618109\pi\)
−0.362594 + 0.931947i \(0.618109\pi\)
\(810\) 0 0
\(811\) 45.2088 1.58749 0.793747 0.608248i \(-0.208127\pi\)
0.793747 + 0.608248i \(0.208127\pi\)
\(812\) 13.6442 0.478817
\(813\) 0 0
\(814\) 5.60675 0.196516
\(815\) −4.69176 −0.164345
\(816\) 0 0
\(817\) −22.5295 −0.788208
\(818\) −2.40848 −0.0842106
\(819\) 0 0
\(820\) −76.1407 −2.65895
\(821\) −23.2674 −0.812039 −0.406019 0.913864i \(-0.633083\pi\)
−0.406019 + 0.913864i \(0.633083\pi\)
\(822\) 0 0
\(823\) 2.12031 0.0739095 0.0369547 0.999317i \(-0.488234\pi\)
0.0369547 + 0.999317i \(0.488234\pi\)
\(824\) 0.238556 0.00831049
\(825\) 0 0
\(826\) 13.4359 0.467495
\(827\) 16.5417 0.575213 0.287606 0.957749i \(-0.407141\pi\)
0.287606 + 0.957749i \(0.407141\pi\)
\(828\) 0 0
\(829\) 2.61675 0.0908835 0.0454418 0.998967i \(-0.485530\pi\)
0.0454418 + 0.998967i \(0.485530\pi\)
\(830\) −6.27501 −0.217809
\(831\) 0 0
\(832\) −0.0443806 −0.00153862
\(833\) −36.2252 −1.25513
\(834\) 0 0
\(835\) 3.50823 0.121407
\(836\) −33.5263 −1.15953
\(837\) 0 0
\(838\) −0.403668 −0.0139445
\(839\) −40.9480 −1.41368 −0.706841 0.707373i \(-0.749880\pi\)
−0.706841 + 0.707373i \(0.749880\pi\)
\(840\) 0 0
\(841\) −23.6522 −0.815594
\(842\) 8.37275 0.288544
\(843\) 0 0
\(844\) 3.73655 0.128617
\(845\) −39.5178 −1.35945
\(846\) 0 0
\(847\) −50.2428 −1.72636
\(848\) −2.34297 −0.0804580
\(849\) 0 0
\(850\) −35.4601 −1.21627
\(851\) 4.96961 0.170356
\(852\) 0 0
\(853\) 17.3933 0.595537 0.297768 0.954638i \(-0.403758\pi\)
0.297768 + 0.954638i \(0.403758\pi\)
\(854\) 0.177984 0.00609049
\(855\) 0 0
\(856\) −41.3101 −1.41195
\(857\) −16.7101 −0.570805 −0.285402 0.958408i \(-0.592127\pi\)
−0.285402 + 0.958408i \(0.592127\pi\)
\(858\) 0 0
\(859\) 13.9877 0.477255 0.238628 0.971111i \(-0.423302\pi\)
0.238628 + 0.971111i \(0.423302\pi\)
\(860\) 33.3706 1.13793
\(861\) 0 0
\(862\) 16.9126 0.576045
\(863\) 23.1575 0.788290 0.394145 0.919048i \(-0.371041\pi\)
0.394145 + 0.919048i \(0.371041\pi\)
\(864\) 0 0
\(865\) −48.1446 −1.63696
\(866\) −13.0791 −0.444445
\(867\) 0 0
\(868\) −1.17716 −0.0399556
\(869\) 38.8628 1.31833
\(870\) 0 0
\(871\) −21.3281 −0.722676
\(872\) −33.2986 −1.12763
\(873\) 0 0
\(874\) 7.37608 0.249499
\(875\) −74.3418 −2.51321
\(876\) 0 0
\(877\) 40.3104 1.36119 0.680594 0.732661i \(-0.261722\pi\)
0.680594 + 0.732661i \(0.261722\pi\)
\(878\) −8.03326 −0.271110
\(879\) 0 0
\(880\) 34.2733 1.15535
\(881\) 23.2156 0.782152 0.391076 0.920358i \(-0.372103\pi\)
0.391076 + 0.920358i \(0.372103\pi\)
\(882\) 0 0
\(883\) −0.920055 −0.0309623 −0.0154812 0.999880i \(-0.504928\pi\)
−0.0154812 + 0.999880i \(0.504928\pi\)
\(884\) −14.9587 −0.503117
\(885\) 0 0
\(886\) −14.7608 −0.495897
\(887\) 28.9082 0.970643 0.485321 0.874336i \(-0.338703\pi\)
0.485321 + 0.874336i \(0.338703\pi\)
\(888\) 0 0
\(889\) −1.34234 −0.0450207
\(890\) 10.6969 0.358561
\(891\) 0 0
\(892\) 2.81257 0.0941720
\(893\) 45.2446 1.51405
\(894\) 0 0
\(895\) 39.0641 1.30577
\(896\) 41.6299 1.39076
\(897\) 0 0
\(898\) −22.6330 −0.755273
\(899\) −0.461384 −0.0153880
\(900\) 0 0
\(901\) 7.30250 0.243282
\(902\) 38.1823 1.27133
\(903\) 0 0
\(904\) 2.47250 0.0822341
\(905\) −66.4290 −2.20817
\(906\) 0 0
\(907\) 36.3884 1.20826 0.604128 0.796887i \(-0.293522\pi\)
0.604128 + 0.796887i \(0.293522\pi\)
\(908\) 6.98007 0.231642
\(909\) 0 0
\(910\) 15.2958 0.507052
\(911\) 10.8289 0.358776 0.179388 0.983778i \(-0.442588\pi\)
0.179388 + 0.983778i \(0.442588\pi\)
\(912\) 0 0
\(913\) −12.6774 −0.419562
\(914\) −14.7580 −0.488150
\(915\) 0 0
\(916\) −10.0008 −0.330437
\(917\) 20.2110 0.667424
\(918\) 0 0
\(919\) 28.3998 0.936823 0.468412 0.883510i \(-0.344826\pi\)
0.468412 + 0.883510i \(0.344826\pi\)
\(920\) −24.5626 −0.809806
\(921\) 0 0
\(922\) −11.2977 −0.372071
\(923\) −8.79712 −0.289561
\(924\) 0 0
\(925\) 18.2339 0.599526
\(926\) −20.1906 −0.663505
\(927\) 0 0
\(928\) 13.0911 0.429737
\(929\) 32.8431 1.07755 0.538774 0.842451i \(-0.318888\pi\)
0.538774 + 0.842451i \(0.318888\pi\)
\(930\) 0 0
\(931\) 27.6473 0.906104
\(932\) 22.1621 0.725944
\(933\) 0 0
\(934\) 7.75371 0.253709
\(935\) −106.822 −3.49345
\(936\) 0 0
\(937\) 4.11911 0.134566 0.0672828 0.997734i \(-0.478567\pi\)
0.0672828 + 0.997734i \(0.478567\pi\)
\(938\) −29.2986 −0.956635
\(939\) 0 0
\(940\) −67.0161 −2.18582
\(941\) 8.26409 0.269402 0.134701 0.990886i \(-0.456993\pi\)
0.134701 + 0.990886i \(0.456993\pi\)
\(942\) 0 0
\(943\) 33.8434 1.10209
\(944\) −10.2518 −0.333668
\(945\) 0 0
\(946\) −16.7344 −0.544081
\(947\) −43.3860 −1.40986 −0.704928 0.709279i \(-0.749021\pi\)
−0.704928 + 0.709279i \(0.749021\pi\)
\(948\) 0 0
\(949\) 23.7907 0.772277
\(950\) 27.0634 0.878051
\(951\) 0 0
\(952\) −46.1985 −1.49730
\(953\) 26.7790 0.867457 0.433728 0.901044i \(-0.357198\pi\)
0.433728 + 0.901044i \(0.357198\pi\)
\(954\) 0 0
\(955\) 92.6333 2.99754
\(956\) −31.0722 −1.00495
\(957\) 0 0
\(958\) −9.55772 −0.308796
\(959\) −63.1640 −2.03967
\(960\) 0 0
\(961\) −30.9602 −0.998716
\(962\) −1.90925 −0.0615567
\(963\) 0 0
\(964\) 1.60229 0.0516062
\(965\) 29.1981 0.939922
\(966\) 0 0
\(967\) 7.46107 0.239932 0.119966 0.992778i \(-0.461721\pi\)
0.119966 + 0.992778i \(0.461721\pi\)
\(968\) −30.9967 −0.996272
\(969\) 0 0
\(970\) 22.1810 0.712190
\(971\) 21.4365 0.687931 0.343966 0.938982i \(-0.388230\pi\)
0.343966 + 0.938982i \(0.388230\pi\)
\(972\) 0 0
\(973\) 72.2006 2.31465
\(974\) −9.77548 −0.313226
\(975\) 0 0
\(976\) −0.135805 −0.00434700
\(977\) 30.9584 0.990446 0.495223 0.868766i \(-0.335086\pi\)
0.495223 + 0.868766i \(0.335086\pi\)
\(978\) 0 0
\(979\) 21.6110 0.690690
\(980\) −40.9510 −1.30813
\(981\) 0 0
\(982\) −7.93592 −0.253246
\(983\) −13.6397 −0.435040 −0.217520 0.976056i \(-0.569797\pi\)
−0.217520 + 0.976056i \(0.569797\pi\)
\(984\) 0 0
\(985\) −7.23343 −0.230476
\(986\) −8.05406 −0.256494
\(987\) 0 0
\(988\) 11.4166 0.363211
\(989\) −14.8327 −0.471653
\(990\) 0 0
\(991\) −46.0458 −1.46269 −0.731347 0.682006i \(-0.761108\pi\)
−0.731347 + 0.682006i \(0.761108\pi\)
\(992\) −1.12945 −0.0358600
\(993\) 0 0
\(994\) −12.0847 −0.383303
\(995\) 13.5073 0.428211
\(996\) 0 0
\(997\) 57.1775 1.81083 0.905415 0.424527i \(-0.139560\pi\)
0.905415 + 0.424527i \(0.139560\pi\)
\(998\) −26.4643 −0.837711
\(999\) 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.1.4 7
3.2 odd 2 241.2.a.a.1.4 7
12.11 even 2 3856.2.a.j.1.1 7
15.14 odd 2 6025.2.a.f.1.4 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
241.2.a.a.1.4 7 3.2 odd 2
2169.2.a.e.1.4 7 1.1 even 1 trivial
3856.2.a.j.1.1 7 12.11 even 2
6025.2.a.f.1.4 7 15.14 odd 2