Properties

Label 4235.2.a.ba.1.4
Level $4235$
Weight $2$
Character 4235.1
Self dual yes
Analytic conductor $33.817$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 4235 = 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4235.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.8166452560\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.580017.1
Defining polynomial: \( x^{5} - 7x^{3} - 4x^{2} + 6x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.941569\) of defining polynomial
Character \(\chi\) \(=\) 4235.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.941569 q^{2} +1.53997 q^{3} -1.11345 q^{4} -1.00000 q^{5} +1.44999 q^{6} -1.00000 q^{7} -2.93153 q^{8} -0.628495 q^{9} +O(q^{10})\) \(q+0.941569 q^{2} +1.53997 q^{3} -1.11345 q^{4} -1.00000 q^{5} +1.44999 q^{6} -1.00000 q^{7} -2.93153 q^{8} -0.628495 q^{9} -0.941569 q^{10} -1.71467 q^{12} -1.39156 q^{13} -0.941569 q^{14} -1.53997 q^{15} -0.533340 q^{16} +2.87651 q^{17} -0.591771 q^{18} +1.39301 q^{19} +1.11345 q^{20} -1.53997 q^{21} +6.46428 q^{23} -4.51446 q^{24} +1.00000 q^{25} -1.31025 q^{26} -5.58777 q^{27} +1.11345 q^{28} +2.72955 q^{29} -1.44999 q^{30} -2.10106 q^{31} +5.36088 q^{32} +2.70843 q^{34} +1.00000 q^{35} +0.699796 q^{36} -6.54173 q^{37} +1.31162 q^{38} -2.14296 q^{39} +2.93153 q^{40} +7.54422 q^{41} -1.44999 q^{42} -5.24119 q^{43} +0.628495 q^{45} +6.08657 q^{46} +8.87972 q^{47} -0.821328 q^{48} +1.00000 q^{49} +0.941569 q^{50} +4.42974 q^{51} +1.54943 q^{52} +7.31070 q^{53} -5.26127 q^{54} +2.93153 q^{56} +2.14520 q^{57} +2.57006 q^{58} -8.87651 q^{59} +1.71467 q^{60} +1.70038 q^{61} -1.97829 q^{62} +0.628495 q^{63} +6.11432 q^{64} +1.39156 q^{65} +0.00145621 q^{67} -3.20284 q^{68} +9.95479 q^{69} +0.941569 q^{70} +6.94923 q^{71} +1.84245 q^{72} -0.0957409 q^{73} -6.15949 q^{74} +1.53997 q^{75} -1.55105 q^{76} -2.01774 q^{78} +9.24782 q^{79} +0.533340 q^{80} -6.71951 q^{81} +7.10340 q^{82} +13.2464 q^{83} +1.71467 q^{84} -2.87651 q^{85} -4.93494 q^{86} +4.20343 q^{87} +0.516504 q^{89} +0.591771 q^{90} +1.39156 q^{91} -7.19764 q^{92} -3.23557 q^{93} +8.36088 q^{94} -1.39301 q^{95} +8.25558 q^{96} +0.474323 q^{97} +0.941569 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{3} + 4 q^{4} - 5 q^{5} - 5 q^{6} - 5 q^{7} + 12 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{3} + 4 q^{4} - 5 q^{5} - 5 q^{6} - 5 q^{7} + 12 q^{8} + 11 q^{9} - 23 q^{12} + 10 q^{13} + 2 q^{15} + 10 q^{16} + 2 q^{17} + 5 q^{18} - 3 q^{19} - 4 q^{20} + 2 q^{21} + 3 q^{23} - 28 q^{24} + 5 q^{25} + 13 q^{26} - 11 q^{27} - 4 q^{28} + q^{29} + 5 q^{30} - 12 q^{31} + 29 q^{32} - 20 q^{34} + 5 q^{35} + 45 q^{36} + 9 q^{38} - 2 q^{39} - 12 q^{40} - 11 q^{41} + 5 q^{42} + 10 q^{43} - 11 q^{45} + 14 q^{46} + 16 q^{47} - 46 q^{48} + 5 q^{49} + 6 q^{51} + 30 q^{52} + 4 q^{53} - 34 q^{54} - 12 q^{56} + 34 q^{57} - 6 q^{58} - 32 q^{59} + 23 q^{60} + 40 q^{61} + q^{62} - 11 q^{63} + 38 q^{64} - 10 q^{65} + 7 q^{67} - 19 q^{68} - 24 q^{69} + 10 q^{71} + 72 q^{72} + 11 q^{73} - 37 q^{74} - 2 q^{75} + 3 q^{76} - 87 q^{78} + 13 q^{79} - 10 q^{80} + q^{81} + 4 q^{82} + 26 q^{83} + 23 q^{84} - 2 q^{85} - 17 q^{86} + 14 q^{87} + 5 q^{89} - 5 q^{90} - 10 q^{91} + 32 q^{92} + 3 q^{93} + 44 q^{94} + 3 q^{95} - 84 q^{96} - 5 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.941569 0.665790 0.332895 0.942964i \(-0.391975\pi\)
0.332895 + 0.942964i \(0.391975\pi\)
\(3\) 1.53997 0.889102 0.444551 0.895754i \(-0.353363\pi\)
0.444551 + 0.895754i \(0.353363\pi\)
\(4\) −1.11345 −0.556724
\(5\) −1.00000 −0.447214
\(6\) 1.44999 0.591955
\(7\) −1.00000 −0.377964
\(8\) −2.93153 −1.03645
\(9\) −0.628495 −0.209498
\(10\) −0.941569 −0.297750
\(11\) 0 0
\(12\) −1.71467 −0.494984
\(13\) −1.39156 −0.385948 −0.192974 0.981204i \(-0.561813\pi\)
−0.192974 + 0.981204i \(0.561813\pi\)
\(14\) −0.941569 −0.251645
\(15\) −1.53997 −0.397618
\(16\) −0.533340 −0.133335
\(17\) 2.87651 0.697656 0.348828 0.937187i \(-0.386580\pi\)
0.348828 + 0.937187i \(0.386580\pi\)
\(18\) −0.591771 −0.139482
\(19\) 1.39301 0.319579 0.159790 0.987151i \(-0.448918\pi\)
0.159790 + 0.987151i \(0.448918\pi\)
\(20\) 1.11345 0.248974
\(21\) −1.53997 −0.336049
\(22\) 0 0
\(23\) 6.46428 1.34790 0.673948 0.738779i \(-0.264597\pi\)
0.673948 + 0.738779i \(0.264597\pi\)
\(24\) −4.51446 −0.921510
\(25\) 1.00000 0.200000
\(26\) −1.31025 −0.256961
\(27\) −5.58777 −1.07537
\(28\) 1.11345 0.210422
\(29\) 2.72955 0.506865 0.253433 0.967353i \(-0.418440\pi\)
0.253433 + 0.967353i \(0.418440\pi\)
\(30\) −1.44999 −0.264730
\(31\) −2.10106 −0.377361 −0.188681 0.982038i \(-0.560421\pi\)
−0.188681 + 0.982038i \(0.560421\pi\)
\(32\) 5.36088 0.947678
\(33\) 0 0
\(34\) 2.70843 0.464492
\(35\) 1.00000 0.169031
\(36\) 0.699796 0.116633
\(37\) −6.54173 −1.07545 −0.537727 0.843119i \(-0.680717\pi\)
−0.537727 + 0.843119i \(0.680717\pi\)
\(38\) 1.31162 0.212773
\(39\) −2.14296 −0.343147
\(40\) 2.93153 0.463515
\(41\) 7.54422 1.17821 0.589104 0.808057i \(-0.299481\pi\)
0.589104 + 0.808057i \(0.299481\pi\)
\(42\) −1.44999 −0.223738
\(43\) −5.24119 −0.799273 −0.399637 0.916674i \(-0.630864\pi\)
−0.399637 + 0.916674i \(0.630864\pi\)
\(44\) 0 0
\(45\) 0.628495 0.0936904
\(46\) 6.08657 0.897415
\(47\) 8.87972 1.29524 0.647620 0.761963i \(-0.275764\pi\)
0.647620 + 0.761963i \(0.275764\pi\)
\(48\) −0.821328 −0.118548
\(49\) 1.00000 0.142857
\(50\) 0.941569 0.133158
\(51\) 4.42974 0.620287
\(52\) 1.54943 0.214867
\(53\) 7.31070 1.00420 0.502101 0.864809i \(-0.332561\pi\)
0.502101 + 0.864809i \(0.332561\pi\)
\(54\) −5.26127 −0.715969
\(55\) 0 0
\(56\) 2.93153 0.391742
\(57\) 2.14520 0.284138
\(58\) 2.57006 0.337466
\(59\) −8.87651 −1.15562 −0.577812 0.816170i \(-0.696093\pi\)
−0.577812 + 0.816170i \(0.696093\pi\)
\(60\) 1.71467 0.221364
\(61\) 1.70038 0.217712 0.108856 0.994058i \(-0.465281\pi\)
0.108856 + 0.994058i \(0.465281\pi\)
\(62\) −1.97829 −0.251243
\(63\) 0.628495 0.0791829
\(64\) 6.11432 0.764290
\(65\) 1.39156 0.172601
\(66\) 0 0
\(67\) 0.00145621 0.000177904 0 8.89522e−5 1.00000i \(-0.499972\pi\)
8.89522e−5 1.00000i \(0.499972\pi\)
\(68\) −3.20284 −0.388402
\(69\) 9.95479 1.19842
\(70\) 0.941569 0.112539
\(71\) 6.94923 0.824722 0.412361 0.911020i \(-0.364704\pi\)
0.412361 + 0.911020i \(0.364704\pi\)
\(72\) 1.84245 0.217135
\(73\) −0.0957409 −0.0112056 −0.00560281 0.999984i \(-0.501783\pi\)
−0.00560281 + 0.999984i \(0.501783\pi\)
\(74\) −6.15949 −0.716026
\(75\) 1.53997 0.177820
\(76\) −1.55105 −0.177917
\(77\) 0 0
\(78\) −2.01774 −0.228464
\(79\) 9.24782 1.04046 0.520230 0.854026i \(-0.325846\pi\)
0.520230 + 0.854026i \(0.325846\pi\)
\(80\) 0.533340 0.0596293
\(81\) −6.71951 −0.746612
\(82\) 7.10340 0.784440
\(83\) 13.2464 1.45398 0.726988 0.686650i \(-0.240919\pi\)
0.726988 + 0.686650i \(0.240919\pi\)
\(84\) 1.71467 0.187086
\(85\) −2.87651 −0.312001
\(86\) −4.93494 −0.532148
\(87\) 4.20343 0.450655
\(88\) 0 0
\(89\) 0.516504 0.0547493 0.0273746 0.999625i \(-0.491285\pi\)
0.0273746 + 0.999625i \(0.491285\pi\)
\(90\) 0.591771 0.0623782
\(91\) 1.39156 0.145875
\(92\) −7.19764 −0.750405
\(93\) −3.23557 −0.335513
\(94\) 8.36088 0.862358
\(95\) −1.39301 −0.142920
\(96\) 8.25558 0.842582
\(97\) 0.474323 0.0481602 0.0240801 0.999710i \(-0.492334\pi\)
0.0240801 + 0.999710i \(0.492334\pi\)
\(98\) 0.941569 0.0951129
\(99\) 0 0
\(100\) −1.11345 −0.111345
\(101\) 9.70989 0.966170 0.483085 0.875573i \(-0.339516\pi\)
0.483085 + 0.875573i \(0.339516\pi\)
\(102\) 4.17090 0.412981
\(103\) 0.722725 0.0712122 0.0356061 0.999366i \(-0.488664\pi\)
0.0356061 + 0.999366i \(0.488664\pi\)
\(104\) 4.07939 0.400017
\(105\) 1.53997 0.150286
\(106\) 6.88353 0.668587
\(107\) 12.0970 1.16946 0.584732 0.811227i \(-0.301200\pi\)
0.584732 + 0.811227i \(0.301200\pi\)
\(108\) 6.22169 0.598682
\(109\) 2.06951 0.198223 0.0991115 0.995076i \(-0.468400\pi\)
0.0991115 + 0.995076i \(0.468400\pi\)
\(110\) 0 0
\(111\) −10.0741 −0.956187
\(112\) 0.533340 0.0503959
\(113\) 18.3350 1.72481 0.862404 0.506220i \(-0.168958\pi\)
0.862404 + 0.506220i \(0.168958\pi\)
\(114\) 2.01985 0.189176
\(115\) −6.46428 −0.602797
\(116\) −3.03921 −0.282184
\(117\) 0.874586 0.0808555
\(118\) −8.35785 −0.769402
\(119\) −2.87651 −0.263689
\(120\) 4.51446 0.412112
\(121\) 0 0
\(122\) 1.60103 0.144950
\(123\) 11.6179 1.04755
\(124\) 2.33942 0.210086
\(125\) −1.00000 −0.0894427
\(126\) 0.591771 0.0527192
\(127\) −15.6786 −1.39125 −0.695625 0.718405i \(-0.744872\pi\)
−0.695625 + 0.718405i \(0.744872\pi\)
\(128\) −4.96470 −0.438821
\(129\) −8.07127 −0.710635
\(130\) 1.31025 0.114916
\(131\) 14.8387 1.29646 0.648230 0.761445i \(-0.275510\pi\)
0.648230 + 0.761445i \(0.275510\pi\)
\(132\) 0 0
\(133\) −1.39301 −0.120790
\(134\) 0.00137112 0.000118447 0
\(135\) 5.58777 0.480919
\(136\) −8.43256 −0.723086
\(137\) 13.0351 1.11366 0.556830 0.830626i \(-0.312017\pi\)
0.556830 + 0.830626i \(0.312017\pi\)
\(138\) 9.37313 0.797894
\(139\) 18.9218 1.60492 0.802462 0.596703i \(-0.203523\pi\)
0.802462 + 0.596703i \(0.203523\pi\)
\(140\) −1.11345 −0.0941035
\(141\) 13.6745 1.15160
\(142\) 6.54318 0.549092
\(143\) 0 0
\(144\) 0.335202 0.0279335
\(145\) −2.72955 −0.226677
\(146\) −0.0901467 −0.00746060
\(147\) 1.53997 0.127015
\(148\) 7.28387 0.598730
\(149\) −5.13736 −0.420869 −0.210435 0.977608i \(-0.567488\pi\)
−0.210435 + 0.977608i \(0.567488\pi\)
\(150\) 1.44999 0.118391
\(151\) 9.51339 0.774189 0.387094 0.922040i \(-0.373479\pi\)
0.387094 + 0.922040i \(0.373479\pi\)
\(152\) −4.08365 −0.331228
\(153\) −1.80787 −0.146158
\(154\) 0 0
\(155\) 2.10106 0.168761
\(156\) 2.38607 0.191038
\(157\) −4.94602 −0.394735 −0.197368 0.980330i \(-0.563239\pi\)
−0.197368 + 0.980330i \(0.563239\pi\)
\(158\) 8.70746 0.692728
\(159\) 11.2582 0.892837
\(160\) −5.36088 −0.423814
\(161\) −6.46428 −0.509457
\(162\) −6.32688 −0.497087
\(163\) −19.1488 −1.49985 −0.749926 0.661522i \(-0.769911\pi\)
−0.749926 + 0.661522i \(0.769911\pi\)
\(164\) −8.40009 −0.655937
\(165\) 0 0
\(166\) 12.4724 0.968043
\(167\) −11.7906 −0.912382 −0.456191 0.889882i \(-0.650787\pi\)
−0.456191 + 0.889882i \(0.650787\pi\)
\(168\) 4.51446 0.348298
\(169\) −11.0636 −0.851044
\(170\) −2.70843 −0.207727
\(171\) −0.875501 −0.0669513
\(172\) 5.83579 0.444974
\(173\) 22.0859 1.67916 0.839582 0.543233i \(-0.182800\pi\)
0.839582 + 0.543233i \(0.182800\pi\)
\(174\) 3.95782 0.300042
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −13.6696 −1.02747
\(178\) 0.486324 0.0364515
\(179\) −14.5651 −1.08865 −0.544325 0.838875i \(-0.683214\pi\)
−0.544325 + 0.838875i \(0.683214\pi\)
\(180\) −0.699796 −0.0521597
\(181\) 4.14710 0.308252 0.154126 0.988051i \(-0.450744\pi\)
0.154126 + 0.988051i \(0.450744\pi\)
\(182\) 1.31025 0.0971220
\(183\) 2.61854 0.193568
\(184\) −18.9502 −1.39703
\(185\) 6.54173 0.480957
\(186\) −3.04651 −0.223381
\(187\) 0 0
\(188\) −9.88711 −0.721091
\(189\) 5.58777 0.406450
\(190\) −1.31162 −0.0951548
\(191\) −13.3933 −0.969106 −0.484553 0.874762i \(-0.661018\pi\)
−0.484553 + 0.874762i \(0.661018\pi\)
\(192\) 9.41586 0.679531
\(193\) 14.7344 1.06060 0.530302 0.847809i \(-0.322078\pi\)
0.530302 + 0.847809i \(0.322078\pi\)
\(194\) 0.446608 0.0320646
\(195\) 2.14296 0.153460
\(196\) −1.11345 −0.0795320
\(197\) 3.94112 0.280793 0.140397 0.990095i \(-0.455162\pi\)
0.140397 + 0.990095i \(0.455162\pi\)
\(198\) 0 0
\(199\) −10.2444 −0.726206 −0.363103 0.931749i \(-0.618283\pi\)
−0.363103 + 0.931749i \(0.618283\pi\)
\(200\) −2.93153 −0.207290
\(201\) 0.00224252 0.000158175 0
\(202\) 9.14253 0.643266
\(203\) −2.72955 −0.191577
\(204\) −4.93228 −0.345329
\(205\) −7.54422 −0.526911
\(206\) 0.680496 0.0474124
\(207\) −4.06277 −0.282382
\(208\) 0.742174 0.0514605
\(209\) 0 0
\(210\) 1.44999 0.100059
\(211\) 8.32284 0.572968 0.286484 0.958085i \(-0.407514\pi\)
0.286484 + 0.958085i \(0.407514\pi\)
\(212\) −8.14007 −0.559063
\(213\) 10.7016 0.733262
\(214\) 11.3902 0.778617
\(215\) 5.24119 0.357446
\(216\) 16.3807 1.11457
\(217\) 2.10106 0.142629
\(218\) 1.94859 0.131975
\(219\) −0.147438 −0.00996294
\(220\) 0 0
\(221\) −4.00283 −0.269259
\(222\) −9.48543 −0.636620
\(223\) 20.0014 1.33939 0.669695 0.742636i \(-0.266425\pi\)
0.669695 + 0.742636i \(0.266425\pi\)
\(224\) −5.36088 −0.358189
\(225\) −0.628495 −0.0418996
\(226\) 17.2636 1.14836
\(227\) −13.2057 −0.876494 −0.438247 0.898854i \(-0.644401\pi\)
−0.438247 + 0.898854i \(0.644401\pi\)
\(228\) −2.38856 −0.158187
\(229\) −25.7475 −1.70144 −0.850720 0.525618i \(-0.823834\pi\)
−0.850720 + 0.525618i \(0.823834\pi\)
\(230\) −6.08657 −0.401336
\(231\) 0 0
\(232\) −8.00176 −0.525341
\(233\) −11.9028 −0.779776 −0.389888 0.920862i \(-0.627486\pi\)
−0.389888 + 0.920862i \(0.627486\pi\)
\(234\) 0.823483 0.0538328
\(235\) −8.87972 −0.579249
\(236\) 9.88353 0.643363
\(237\) 14.2414 0.925075
\(238\) −2.70843 −0.175562
\(239\) −17.4196 −1.12678 −0.563390 0.826191i \(-0.690503\pi\)
−0.563390 + 0.826191i \(0.690503\pi\)
\(240\) 0.821328 0.0530165
\(241\) 2.30561 0.148517 0.0742587 0.997239i \(-0.476341\pi\)
0.0742587 + 0.997239i \(0.476341\pi\)
\(242\) 0 0
\(243\) 6.41547 0.411553
\(244\) −1.89329 −0.121205
\(245\) −1.00000 −0.0638877
\(246\) 10.9390 0.697447
\(247\) −1.93846 −0.123341
\(248\) 6.15931 0.391117
\(249\) 20.3990 1.29273
\(250\) −0.941569 −0.0595501
\(251\) −9.53947 −0.602126 −0.301063 0.953604i \(-0.597341\pi\)
−0.301063 + 0.953604i \(0.597341\pi\)
\(252\) −0.699796 −0.0440830
\(253\) 0 0
\(254\) −14.7625 −0.926280
\(255\) −4.42974 −0.277401
\(256\) −16.9032 −1.05645
\(257\) −29.4161 −1.83493 −0.917463 0.397821i \(-0.869767\pi\)
−0.917463 + 0.397821i \(0.869767\pi\)
\(258\) −7.59966 −0.473134
\(259\) 6.54173 0.406483
\(260\) −1.54943 −0.0960913
\(261\) −1.71551 −0.106187
\(262\) 13.9716 0.863170
\(263\) −3.05267 −0.188236 −0.0941179 0.995561i \(-0.530003\pi\)
−0.0941179 + 0.995561i \(0.530003\pi\)
\(264\) 0 0
\(265\) −7.31070 −0.449092
\(266\) −1.31162 −0.0804205
\(267\) 0.795400 0.0486777
\(268\) −0.00162141 −9.90436e−5 0
\(269\) −2.00386 −0.122178 −0.0610888 0.998132i \(-0.519457\pi\)
−0.0610888 + 0.998132i \(0.519457\pi\)
\(270\) 5.26127 0.320191
\(271\) 23.7039 1.43991 0.719955 0.694021i \(-0.244163\pi\)
0.719955 + 0.694021i \(0.244163\pi\)
\(272\) −1.53416 −0.0930221
\(273\) 2.14296 0.129698
\(274\) 12.2734 0.741464
\(275\) 0 0
\(276\) −11.0841 −0.667187
\(277\) 18.9545 1.13886 0.569432 0.822038i \(-0.307163\pi\)
0.569432 + 0.822038i \(0.307163\pi\)
\(278\) 17.8162 1.06854
\(279\) 1.32050 0.0790565
\(280\) −2.93153 −0.175192
\(281\) −1.50221 −0.0896144 −0.0448072 0.998996i \(-0.514267\pi\)
−0.0448072 + 0.998996i \(0.514267\pi\)
\(282\) 12.8755 0.766724
\(283\) −1.69585 −0.100808 −0.0504038 0.998729i \(-0.516051\pi\)
−0.0504038 + 0.998729i \(0.516051\pi\)
\(284\) −7.73760 −0.459142
\(285\) −2.14520 −0.127071
\(286\) 0 0
\(287\) −7.54422 −0.445321
\(288\) −3.36928 −0.198537
\(289\) −8.72569 −0.513276
\(290\) −2.57006 −0.150919
\(291\) 0.730443 0.0428193
\(292\) 0.106602 0.00623844
\(293\) −8.76253 −0.511912 −0.255956 0.966688i \(-0.582390\pi\)
−0.255956 + 0.966688i \(0.582390\pi\)
\(294\) 1.44999 0.0845650
\(295\) 8.87651 0.516810
\(296\) 19.1772 1.11465
\(297\) 0 0
\(298\) −4.83718 −0.280210
\(299\) −8.99541 −0.520218
\(300\) −1.71467 −0.0989968
\(301\) 5.24119 0.302097
\(302\) 8.95752 0.515447
\(303\) 14.9529 0.859023
\(304\) −0.742950 −0.0426111
\(305\) −1.70038 −0.0973636
\(306\) −1.70224 −0.0973103
\(307\) 15.6989 0.895985 0.447992 0.894037i \(-0.352139\pi\)
0.447992 + 0.894037i \(0.352139\pi\)
\(308\) 0 0
\(309\) 1.11297 0.0633149
\(310\) 1.97829 0.112359
\(311\) 4.53471 0.257140 0.128570 0.991700i \(-0.458961\pi\)
0.128570 + 0.991700i \(0.458961\pi\)
\(312\) 6.28213 0.355656
\(313\) −2.68654 −0.151852 −0.0759261 0.997113i \(-0.524191\pi\)
−0.0759261 + 0.997113i \(0.524191\pi\)
\(314\) −4.65702 −0.262811
\(315\) −0.628495 −0.0354117
\(316\) −10.2970 −0.579249
\(317\) −14.0343 −0.788244 −0.394122 0.919058i \(-0.628951\pi\)
−0.394122 + 0.919058i \(0.628951\pi\)
\(318\) 10.6004 0.594442
\(319\) 0 0
\(320\) −6.11432 −0.341801
\(321\) 18.6290 1.03977
\(322\) −6.08657 −0.339191
\(323\) 4.00702 0.222956
\(324\) 7.48182 0.415657
\(325\) −1.39156 −0.0771897
\(326\) −18.0299 −0.998586
\(327\) 3.18698 0.176240
\(328\) −22.1161 −1.22116
\(329\) −8.87972 −0.489555
\(330\) 0 0
\(331\) 21.5349 1.18367 0.591833 0.806060i \(-0.298404\pi\)
0.591833 + 0.806060i \(0.298404\pi\)
\(332\) −14.7491 −0.809463
\(333\) 4.11144 0.225306
\(334\) −11.1016 −0.607455
\(335\) −0.00145621 −7.95613e−5 0
\(336\) 0.821328 0.0448071
\(337\) −26.0578 −1.41946 −0.709730 0.704474i \(-0.751183\pi\)
−0.709730 + 0.704474i \(0.751183\pi\)
\(338\) −10.4171 −0.566616
\(339\) 28.2353 1.53353
\(340\) 3.20284 0.173698
\(341\) 0 0
\(342\) −0.824345 −0.0445755
\(343\) −1.00000 −0.0539949
\(344\) 15.3647 0.828408
\(345\) −9.95479 −0.535948
\(346\) 20.7954 1.11797
\(347\) 29.2251 1.56888 0.784442 0.620202i \(-0.212950\pi\)
0.784442 + 0.620202i \(0.212950\pi\)
\(348\) −4.68030 −0.250890
\(349\) −7.70633 −0.412510 −0.206255 0.978498i \(-0.566128\pi\)
−0.206255 + 0.978498i \(0.566128\pi\)
\(350\) −0.941569 −0.0503290
\(351\) 7.77570 0.415036
\(352\) 0 0
\(353\) −22.2765 −1.18566 −0.592829 0.805328i \(-0.701989\pi\)
−0.592829 + 0.805328i \(0.701989\pi\)
\(354\) −12.8708 −0.684077
\(355\) −6.94923 −0.368827
\(356\) −0.575099 −0.0304802
\(357\) −4.42974 −0.234447
\(358\) −13.7141 −0.724812
\(359\) 1.69238 0.0893205 0.0446603 0.999002i \(-0.485779\pi\)
0.0446603 + 0.999002i \(0.485779\pi\)
\(360\) −1.84245 −0.0971056
\(361\) −17.0595 −0.897869
\(362\) 3.90478 0.205231
\(363\) 0 0
\(364\) −1.54943 −0.0812120
\(365\) 0.0957409 0.00501131
\(366\) 2.46553 0.128876
\(367\) −13.2925 −0.693862 −0.346931 0.937891i \(-0.612776\pi\)
−0.346931 + 0.937891i \(0.612776\pi\)
\(368\) −3.44766 −0.179722
\(369\) −4.74150 −0.246833
\(370\) 6.15949 0.320217
\(371\) −7.31070 −0.379552
\(372\) 3.60263 0.186788
\(373\) −5.17666 −0.268037 −0.134019 0.990979i \(-0.542788\pi\)
−0.134019 + 0.990979i \(0.542788\pi\)
\(374\) 0 0
\(375\) −1.53997 −0.0795237
\(376\) −26.0311 −1.34245
\(377\) −3.79833 −0.195624
\(378\) 5.26127 0.270611
\(379\) 20.8034 1.06860 0.534299 0.845295i \(-0.320576\pi\)
0.534299 + 0.845295i \(0.320576\pi\)
\(380\) 1.55105 0.0795670
\(381\) −24.1445 −1.23696
\(382\) −12.6107 −0.645221
\(383\) −5.55781 −0.283991 −0.141996 0.989867i \(-0.545352\pi\)
−0.141996 + 0.989867i \(0.545352\pi\)
\(384\) −7.64548 −0.390157
\(385\) 0 0
\(386\) 13.8734 0.706140
\(387\) 3.29406 0.167446
\(388\) −0.528134 −0.0268119
\(389\) 30.9206 1.56774 0.783869 0.620927i \(-0.213244\pi\)
0.783869 + 0.620927i \(0.213244\pi\)
\(390\) 2.01774 0.102172
\(391\) 18.5946 0.940368
\(392\) −2.93153 −0.148064
\(393\) 22.8511 1.15268
\(394\) 3.71084 0.186949
\(395\) −9.24782 −0.465308
\(396\) 0 0
\(397\) 30.4744 1.52946 0.764732 0.644348i \(-0.222871\pi\)
0.764732 + 0.644348i \(0.222871\pi\)
\(398\) −9.64581 −0.483501
\(399\) −2.14520 −0.107394
\(400\) −0.533340 −0.0266670
\(401\) −27.1836 −1.35749 −0.678743 0.734376i \(-0.737475\pi\)
−0.678743 + 0.734376i \(0.737475\pi\)
\(402\) 0.00211149 0.000105311 0
\(403\) 2.92374 0.145642
\(404\) −10.8115 −0.537890
\(405\) 6.71951 0.333895
\(406\) −2.57006 −0.127550
\(407\) 0 0
\(408\) −12.9859 −0.642897
\(409\) 12.4138 0.613822 0.306911 0.951738i \(-0.400705\pi\)
0.306911 + 0.951738i \(0.400705\pi\)
\(410\) −7.10340 −0.350812
\(411\) 20.0736 0.990158
\(412\) −0.804717 −0.0396455
\(413\) 8.87651 0.436784
\(414\) −3.82537 −0.188007
\(415\) −13.2464 −0.650238
\(416\) −7.45996 −0.365755
\(417\) 29.1390 1.42694
\(418\) 0 0
\(419\) 10.5333 0.514585 0.257292 0.966334i \(-0.417170\pi\)
0.257292 + 0.966334i \(0.417170\pi\)
\(420\) −1.71467 −0.0836676
\(421\) 19.6680 0.958560 0.479280 0.877662i \(-0.340898\pi\)
0.479280 + 0.877662i \(0.340898\pi\)
\(422\) 7.83653 0.381476
\(423\) −5.58086 −0.271351
\(424\) −21.4315 −1.04081
\(425\) 2.87651 0.139531
\(426\) 10.0763 0.488198
\(427\) −1.70038 −0.0822873
\(428\) −13.4694 −0.651068
\(429\) 0 0
\(430\) 4.93494 0.237984
\(431\) 15.7944 0.760788 0.380394 0.924824i \(-0.375788\pi\)
0.380394 + 0.924824i \(0.375788\pi\)
\(432\) 2.98018 0.143384
\(433\) 7.09301 0.340868 0.170434 0.985369i \(-0.445483\pi\)
0.170434 + 0.985369i \(0.445483\pi\)
\(434\) 1.97829 0.0949611
\(435\) −4.20343 −0.201539
\(436\) −2.30429 −0.110355
\(437\) 9.00483 0.430759
\(438\) −0.138823 −0.00663323
\(439\) −24.5227 −1.17040 −0.585202 0.810888i \(-0.698985\pi\)
−0.585202 + 0.810888i \(0.698985\pi\)
\(440\) 0 0
\(441\) −0.628495 −0.0299283
\(442\) −3.76894 −0.179270
\(443\) −10.0332 −0.476693 −0.238346 0.971180i \(-0.576605\pi\)
−0.238346 + 0.971180i \(0.576605\pi\)
\(444\) 11.2169 0.532332
\(445\) −0.516504 −0.0244846
\(446\) 18.8327 0.891753
\(447\) −7.91138 −0.374195
\(448\) −6.11432 −0.288874
\(449\) 28.3911 1.33986 0.669929 0.742425i \(-0.266325\pi\)
0.669929 + 0.742425i \(0.266325\pi\)
\(450\) −0.591771 −0.0278964
\(451\) 0 0
\(452\) −20.4150 −0.960242
\(453\) 14.6503 0.688333
\(454\) −12.4341 −0.583561
\(455\) −1.39156 −0.0652372
\(456\) −6.28870 −0.294496
\(457\) −17.9107 −0.837825 −0.418913 0.908027i \(-0.637589\pi\)
−0.418913 + 0.908027i \(0.637589\pi\)
\(458\) −24.2430 −1.13280
\(459\) −16.0733 −0.750236
\(460\) 7.19764 0.335591
\(461\) −22.3455 −1.04073 −0.520366 0.853943i \(-0.674204\pi\)
−0.520366 + 0.853943i \(0.674204\pi\)
\(462\) 0 0
\(463\) −31.7804 −1.47696 −0.738480 0.674275i \(-0.764456\pi\)
−0.738480 + 0.674275i \(0.764456\pi\)
\(464\) −1.45578 −0.0675830
\(465\) 3.23557 0.150046
\(466\) −11.2073 −0.519167
\(467\) 13.4270 0.621328 0.310664 0.950520i \(-0.399449\pi\)
0.310664 + 0.950520i \(0.399449\pi\)
\(468\) −0.973806 −0.0450142
\(469\) −0.00145621 −6.72415e−5 0
\(470\) −8.36088 −0.385658
\(471\) −7.61672 −0.350960
\(472\) 26.0217 1.19775
\(473\) 0 0
\(474\) 13.4092 0.615906
\(475\) 1.39301 0.0639158
\(476\) 3.20284 0.146802
\(477\) −4.59473 −0.210378
\(478\) −16.4018 −0.750199
\(479\) 9.65688 0.441234 0.220617 0.975361i \(-0.429193\pi\)
0.220617 + 0.975361i \(0.429193\pi\)
\(480\) −8.25558 −0.376814
\(481\) 9.10319 0.415070
\(482\) 2.17089 0.0988814
\(483\) −9.95479 −0.452959
\(484\) 0 0
\(485\) −0.474323 −0.0215379
\(486\) 6.04061 0.274008
\(487\) −34.1522 −1.54759 −0.773793 0.633439i \(-0.781643\pi\)
−0.773793 + 0.633439i \(0.781643\pi\)
\(488\) −4.98472 −0.225647
\(489\) −29.4886 −1.33352
\(490\) −0.941569 −0.0425358
\(491\) 1.54632 0.0697845 0.0348923 0.999391i \(-0.488891\pi\)
0.0348923 + 0.999391i \(0.488891\pi\)
\(492\) −12.9359 −0.583195
\(493\) 7.85159 0.353618
\(494\) −1.82519 −0.0821193
\(495\) 0 0
\(496\) 1.12058 0.0503155
\(497\) −6.94923 −0.311716
\(498\) 19.2071 0.860689
\(499\) −14.9721 −0.670241 −0.335121 0.942175i \(-0.608777\pi\)
−0.335121 + 0.942175i \(0.608777\pi\)
\(500\) 1.11345 0.0497949
\(501\) −18.1571 −0.811201
\(502\) −8.98207 −0.400889
\(503\) 3.62122 0.161462 0.0807311 0.996736i \(-0.474275\pi\)
0.0807311 + 0.996736i \(0.474275\pi\)
\(504\) −1.84245 −0.0820692
\(505\) −9.70989 −0.432084
\(506\) 0 0
\(507\) −17.0376 −0.756664
\(508\) 17.4573 0.774542
\(509\) −2.39707 −0.106248 −0.0531240 0.998588i \(-0.516918\pi\)
−0.0531240 + 0.998588i \(0.516918\pi\)
\(510\) −4.17090 −0.184691
\(511\) 0.0957409 0.00423533
\(512\) −5.98618 −0.264554
\(513\) −7.78384 −0.343665
\(514\) −27.6973 −1.22168
\(515\) −0.722725 −0.0318471
\(516\) 8.98693 0.395628
\(517\) 0 0
\(518\) 6.15949 0.270632
\(519\) 34.0117 1.49295
\(520\) −4.07939 −0.178893
\(521\) 41.8248 1.83238 0.916189 0.400747i \(-0.131250\pi\)
0.916189 + 0.400747i \(0.131250\pi\)
\(522\) −1.61527 −0.0706985
\(523\) −28.7142 −1.25559 −0.627793 0.778380i \(-0.716042\pi\)
−0.627793 + 0.778380i \(0.716042\pi\)
\(524\) −16.5221 −0.721770
\(525\) −1.53997 −0.0672098
\(526\) −2.87430 −0.125326
\(527\) −6.04372 −0.263268
\(528\) 0 0
\(529\) 18.7869 0.816822
\(530\) −6.88353 −0.299001
\(531\) 5.57884 0.242101
\(532\) 1.55105 0.0672464
\(533\) −10.4982 −0.454728
\(534\) 0.748924 0.0324091
\(535\) −12.0970 −0.523000
\(536\) −0.00426892 −0.000184389 0
\(537\) −22.4299 −0.967920
\(538\) −1.88677 −0.0813446
\(539\) 0 0
\(540\) −6.22169 −0.267739
\(541\) 41.4908 1.78383 0.891914 0.452205i \(-0.149362\pi\)
0.891914 + 0.452205i \(0.149362\pi\)
\(542\) 22.3189 0.958677
\(543\) 6.38641 0.274067
\(544\) 15.4206 0.661153
\(545\) −2.06951 −0.0886480
\(546\) 2.01774 0.0863513
\(547\) 12.1246 0.518412 0.259206 0.965822i \(-0.416539\pi\)
0.259206 + 0.965822i \(0.416539\pi\)
\(548\) −14.5139 −0.620001
\(549\) −1.06868 −0.0456102
\(550\) 0 0
\(551\) 3.80230 0.161984
\(552\) −29.1827 −1.24210
\(553\) −9.24782 −0.393257
\(554\) 17.8470 0.758245
\(555\) 10.0741 0.427620
\(556\) −21.0684 −0.893500
\(557\) 2.35811 0.0999163 0.0499581 0.998751i \(-0.484091\pi\)
0.0499581 + 0.998751i \(0.484091\pi\)
\(558\) 1.24335 0.0526350
\(559\) 7.29341 0.308478
\(560\) −0.533340 −0.0225377
\(561\) 0 0
\(562\) −1.41444 −0.0596644
\(563\) 42.3956 1.78676 0.893380 0.449302i \(-0.148327\pi\)
0.893380 + 0.449302i \(0.148327\pi\)
\(564\) −15.2258 −0.641123
\(565\) −18.3350 −0.771358
\(566\) −1.59676 −0.0671167
\(567\) 6.71951 0.282193
\(568\) −20.3719 −0.854784
\(569\) −2.97701 −0.124803 −0.0624013 0.998051i \(-0.519876\pi\)
−0.0624013 + 0.998051i \(0.519876\pi\)
\(570\) −2.01985 −0.0846023
\(571\) 46.8050 1.95873 0.979364 0.202107i \(-0.0647788\pi\)
0.979364 + 0.202107i \(0.0647788\pi\)
\(572\) 0 0
\(573\) −20.6253 −0.861634
\(574\) −7.10340 −0.296490
\(575\) 6.46428 0.269579
\(576\) −3.84282 −0.160117
\(577\) −26.4035 −1.09919 −0.549596 0.835431i \(-0.685218\pi\)
−0.549596 + 0.835431i \(0.685218\pi\)
\(578\) −8.21584 −0.341734
\(579\) 22.6905 0.942985
\(580\) 3.03921 0.126197
\(581\) −13.2464 −0.549552
\(582\) 0.687762 0.0285087
\(583\) 0 0
\(584\) 0.280667 0.0116141
\(585\) −0.874586 −0.0361597
\(586\) −8.25053 −0.340826
\(587\) −33.7101 −1.39136 −0.695682 0.718349i \(-0.744898\pi\)
−0.695682 + 0.718349i \(0.744898\pi\)
\(588\) −1.71467 −0.0707120
\(589\) −2.92680 −0.120597
\(590\) 8.35785 0.344087
\(591\) 6.06921 0.249654
\(592\) 3.48897 0.143396
\(593\) −34.4146 −1.41324 −0.706620 0.707594i \(-0.749781\pi\)
−0.706620 + 0.707594i \(0.749781\pi\)
\(594\) 0 0
\(595\) 2.87651 0.117925
\(596\) 5.72018 0.234308
\(597\) −15.7761 −0.645671
\(598\) −8.46981 −0.346356
\(599\) 40.4876 1.65428 0.827140 0.561996i \(-0.189967\pi\)
0.827140 + 0.561996i \(0.189967\pi\)
\(600\) −4.51446 −0.184302
\(601\) 22.7587 0.928346 0.464173 0.885744i \(-0.346352\pi\)
0.464173 + 0.885744i \(0.346352\pi\)
\(602\) 4.93494 0.201133
\(603\) −0.000915220 0 −3.72706e−5 0
\(604\) −10.5927 −0.431009
\(605\) 0 0
\(606\) 14.0792 0.571929
\(607\) −3.61857 −0.146873 −0.0734365 0.997300i \(-0.523397\pi\)
−0.0734365 + 0.997300i \(0.523397\pi\)
\(608\) 7.46777 0.302858
\(609\) −4.20343 −0.170332
\(610\) −1.60103 −0.0648237
\(611\) −12.3566 −0.499896
\(612\) 2.01297 0.0813694
\(613\) 33.2875 1.34447 0.672235 0.740337i \(-0.265334\pi\)
0.672235 + 0.740337i \(0.265334\pi\)
\(614\) 14.7816 0.596538
\(615\) −11.6179 −0.468478
\(616\) 0 0
\(617\) 14.2743 0.574662 0.287331 0.957831i \(-0.407232\pi\)
0.287331 + 0.957831i \(0.407232\pi\)
\(618\) 1.04794 0.0421544
\(619\) −40.8425 −1.64160 −0.820800 0.571216i \(-0.806472\pi\)
−0.820800 + 0.571216i \(0.806472\pi\)
\(620\) −2.33942 −0.0939533
\(621\) −36.1209 −1.44948
\(622\) 4.26975 0.171201
\(623\) −0.516504 −0.0206933
\(624\) 1.14292 0.0457536
\(625\) 1.00000 0.0400000
\(626\) −2.52956 −0.101102
\(627\) 0 0
\(628\) 5.50713 0.219758
\(629\) −18.8173 −0.750297
\(630\) −0.591771 −0.0235767
\(631\) 19.6251 0.781264 0.390632 0.920547i \(-0.372256\pi\)
0.390632 + 0.920547i \(0.372256\pi\)
\(632\) −27.1102 −1.07839
\(633\) 12.8169 0.509426
\(634\) −13.2142 −0.524805
\(635\) 15.6786 0.622186
\(636\) −12.5355 −0.497063
\(637\) −1.39156 −0.0551355
\(638\) 0 0
\(639\) −4.36756 −0.172778
\(640\) 4.96470 0.196247
\(641\) −1.26648 −0.0500228 −0.0250114 0.999687i \(-0.507962\pi\)
−0.0250114 + 0.999687i \(0.507962\pi\)
\(642\) 17.5405 0.692270
\(643\) −1.93346 −0.0762481 −0.0381241 0.999273i \(-0.512138\pi\)
−0.0381241 + 0.999273i \(0.512138\pi\)
\(644\) 7.19764 0.283627
\(645\) 8.07127 0.317806
\(646\) 3.77288 0.148442
\(647\) 7.99759 0.314418 0.157209 0.987565i \(-0.449750\pi\)
0.157209 + 0.987565i \(0.449750\pi\)
\(648\) 19.6984 0.773827
\(649\) 0 0
\(650\) −1.31025 −0.0513921
\(651\) 3.23557 0.126812
\(652\) 21.3212 0.835003
\(653\) 46.2529 1.81002 0.905009 0.425393i \(-0.139864\pi\)
0.905009 + 0.425393i \(0.139864\pi\)
\(654\) 3.00076 0.117339
\(655\) −14.8387 −0.579795
\(656\) −4.02364 −0.157097
\(657\) 0.0601727 0.00234756
\(658\) −8.36088 −0.325941
\(659\) −15.8183 −0.616193 −0.308097 0.951355i \(-0.599692\pi\)
−0.308097 + 0.951355i \(0.599692\pi\)
\(660\) 0 0
\(661\) 15.2510 0.593195 0.296597 0.955003i \(-0.404148\pi\)
0.296597 + 0.955003i \(0.404148\pi\)
\(662\) 20.2766 0.788073
\(663\) −6.16423 −0.239399
\(664\) −38.8321 −1.50698
\(665\) 1.39301 0.0540187
\(666\) 3.87121 0.150006
\(667\) 17.6446 0.683202
\(668\) 13.1282 0.507945
\(669\) 30.8015 1.19085
\(670\) −0.00137112 −5.29711e−5 0
\(671\) 0 0
\(672\) −8.25558 −0.318466
\(673\) 13.4813 0.519666 0.259833 0.965654i \(-0.416333\pi\)
0.259833 + 0.965654i \(0.416333\pi\)
\(674\) −24.5352 −0.945062
\(675\) −5.58777 −0.215073
\(676\) 12.3187 0.473796
\(677\) −2.21002 −0.0849381 −0.0424690 0.999098i \(-0.513522\pi\)
−0.0424690 + 0.999098i \(0.513522\pi\)
\(678\) 26.5855 1.02101
\(679\) −0.474323 −0.0182028
\(680\) 8.43256 0.323374
\(681\) −20.3364 −0.779293
\(682\) 0 0
\(683\) 44.4249 1.69987 0.849935 0.526888i \(-0.176641\pi\)
0.849935 + 0.526888i \(0.176641\pi\)
\(684\) 0.974825 0.0372733
\(685\) −13.0351 −0.498044
\(686\) −0.941569 −0.0359493
\(687\) −39.6503 −1.51275
\(688\) 2.79534 0.106571
\(689\) −10.1732 −0.387570
\(690\) −9.37313 −0.356829
\(691\) 29.0538 1.10526 0.552629 0.833427i \(-0.313625\pi\)
0.552629 + 0.833427i \(0.313625\pi\)
\(692\) −24.5915 −0.934830
\(693\) 0 0
\(694\) 27.5174 1.04455
\(695\) −18.9218 −0.717744
\(696\) −12.3225 −0.467082
\(697\) 21.7010 0.821985
\(698\) −7.25604 −0.274645
\(699\) −18.3299 −0.693300
\(700\) 1.11345 0.0420844
\(701\) 26.7632 1.01083 0.505416 0.862876i \(-0.331339\pi\)
0.505416 + 0.862876i \(0.331339\pi\)
\(702\) 7.32136 0.276327
\(703\) −9.11271 −0.343692
\(704\) 0 0
\(705\) −13.6745 −0.515012
\(706\) −20.9749 −0.789399
\(707\) −9.70989 −0.365178
\(708\) 15.2203 0.572015
\(709\) 9.52139 0.357583 0.178792 0.983887i \(-0.442781\pi\)
0.178792 + 0.983887i \(0.442781\pi\)
\(710\) −6.54318 −0.245561
\(711\) −5.81220 −0.217975
\(712\) −1.51414 −0.0567449
\(713\) −13.5818 −0.508644
\(714\) −4.17090 −0.156092
\(715\) 0 0
\(716\) 16.2175 0.606077
\(717\) −26.8257 −1.00182
\(718\) 1.59350 0.0594687
\(719\) −28.4864 −1.06236 −0.531182 0.847258i \(-0.678252\pi\)
−0.531182 + 0.847258i \(0.678252\pi\)
\(720\) −0.335202 −0.0124922
\(721\) −0.722725 −0.0269157
\(722\) −16.0627 −0.597792
\(723\) 3.55057 0.132047
\(724\) −4.61758 −0.171611
\(725\) 2.72955 0.101373
\(726\) 0 0
\(727\) 48.7611 1.80845 0.904225 0.427057i \(-0.140449\pi\)
0.904225 + 0.427057i \(0.140449\pi\)
\(728\) −4.07939 −0.151192
\(729\) 30.0382 1.11252
\(730\) 0.0901467 0.00333648
\(731\) −15.0763 −0.557618
\(732\) −2.91560 −0.107764
\(733\) 29.8881 1.10394 0.551972 0.833863i \(-0.313876\pi\)
0.551972 + 0.833863i \(0.313876\pi\)
\(734\) −12.5158 −0.461967
\(735\) −1.53997 −0.0568026
\(736\) 34.6542 1.27737
\(737\) 0 0
\(738\) −4.46445 −0.164339
\(739\) 15.1162 0.556058 0.278029 0.960573i \(-0.410319\pi\)
0.278029 + 0.960573i \(0.410319\pi\)
\(740\) −7.28387 −0.267760
\(741\) −2.98516 −0.109663
\(742\) −6.88353 −0.252702
\(743\) −42.9453 −1.57551 −0.787756 0.615988i \(-0.788757\pi\)
−0.787756 + 0.615988i \(0.788757\pi\)
\(744\) 9.48515 0.347742
\(745\) 5.13736 0.188218
\(746\) −4.87418 −0.178457
\(747\) −8.32527 −0.304606
\(748\) 0 0
\(749\) −12.0970 −0.442016
\(750\) −1.44999 −0.0529461
\(751\) −17.3139 −0.631793 −0.315896 0.948794i \(-0.602305\pi\)
−0.315896 + 0.948794i \(0.602305\pi\)
\(752\) −4.73592 −0.172701
\(753\) −14.6905 −0.535351
\(754\) −3.57639 −0.130244
\(755\) −9.51339 −0.346228
\(756\) −6.22169 −0.226281
\(757\) 10.0094 0.363798 0.181899 0.983317i \(-0.441776\pi\)
0.181899 + 0.983317i \(0.441776\pi\)
\(758\) 19.5878 0.711462
\(759\) 0 0
\(760\) 4.08365 0.148130
\(761\) −47.8699 −1.73528 −0.867642 0.497190i \(-0.834365\pi\)
−0.867642 + 0.497190i \(0.834365\pi\)
\(762\) −22.7338 −0.823557
\(763\) −2.06951 −0.0749212
\(764\) 14.9128 0.539524
\(765\) 1.80787 0.0653637
\(766\) −5.23307 −0.189078
\(767\) 12.3522 0.446011
\(768\) −26.0305 −0.939294
\(769\) −54.1437 −1.95247 −0.976236 0.216708i \(-0.930468\pi\)
−0.976236 + 0.216708i \(0.930468\pi\)
\(770\) 0 0
\(771\) −45.2999 −1.63144
\(772\) −16.4060 −0.590464
\(773\) 44.3052 1.59355 0.796774 0.604278i \(-0.206538\pi\)
0.796774 + 0.604278i \(0.206538\pi\)
\(774\) 3.10158 0.111484
\(775\) −2.10106 −0.0754723
\(776\) −1.39049 −0.0499157
\(777\) 10.0741 0.361405
\(778\) 29.1139 1.04378
\(779\) 10.5092 0.376531
\(780\) −2.38607 −0.0854349
\(781\) 0 0
\(782\) 17.5081 0.626087
\(783\) −15.2521 −0.545066
\(784\) −0.533340 −0.0190479
\(785\) 4.94602 0.176531
\(786\) 21.5159 0.767446
\(787\) −30.0712 −1.07192 −0.535961 0.844243i \(-0.680051\pi\)
−0.535961 + 0.844243i \(0.680051\pi\)
\(788\) −4.38823 −0.156324
\(789\) −4.70102 −0.167361
\(790\) −8.70746 −0.309797
\(791\) −18.3350 −0.651916
\(792\) 0 0
\(793\) −2.36618 −0.0840255
\(794\) 28.6937 1.01830
\(795\) −11.2582 −0.399289
\(796\) 11.4066 0.404296
\(797\) −38.1915 −1.35281 −0.676406 0.736529i \(-0.736463\pi\)
−0.676406 + 0.736529i \(0.736463\pi\)
\(798\) −2.01985 −0.0715020
\(799\) 25.5426 0.903633
\(800\) 5.36088 0.189536
\(801\) −0.324620 −0.0114699
\(802\) −25.5953 −0.903801
\(803\) 0 0
\(804\) −0.00249693 −8.80598e−5 0
\(805\) 6.46428 0.227836
\(806\) 2.75291 0.0969670
\(807\) −3.08589 −0.108628
\(808\) −28.4648 −1.00139
\(809\) 1.13775 0.0400011 0.0200005 0.999800i \(-0.493633\pi\)
0.0200005 + 0.999800i \(0.493633\pi\)
\(810\) 6.32688 0.222304
\(811\) −48.3009 −1.69607 −0.848037 0.529937i \(-0.822216\pi\)
−0.848037 + 0.529937i \(0.822216\pi\)
\(812\) 3.03921 0.106656
\(813\) 36.5033 1.28023
\(814\) 0 0
\(815\) 19.1488 0.670754
\(816\) −2.36256 −0.0827061
\(817\) −7.30104 −0.255431
\(818\) 11.6884 0.408677
\(819\) −0.874586 −0.0305605
\(820\) 8.40009 0.293344
\(821\) −42.5334 −1.48443 −0.742213 0.670164i \(-0.766224\pi\)
−0.742213 + 0.670164i \(0.766224\pi\)
\(822\) 18.9007 0.659237
\(823\) −16.3459 −0.569781 −0.284891 0.958560i \(-0.591957\pi\)
−0.284891 + 0.958560i \(0.591957\pi\)
\(824\) −2.11869 −0.0738080
\(825\) 0 0
\(826\) 8.35785 0.290807
\(827\) 39.8338 1.38516 0.692578 0.721343i \(-0.256475\pi\)
0.692578 + 0.721343i \(0.256475\pi\)
\(828\) 4.52367 0.157209
\(829\) 17.9400 0.623081 0.311540 0.950233i \(-0.399155\pi\)
0.311540 + 0.950233i \(0.399155\pi\)
\(830\) −12.4724 −0.432922
\(831\) 29.1893 1.01257
\(832\) −8.50842 −0.294976
\(833\) 2.87651 0.0996652
\(834\) 27.4364 0.950043
\(835\) 11.7906 0.408030
\(836\) 0 0
\(837\) 11.7402 0.405802
\(838\) 9.91782 0.342605
\(839\) −51.5966 −1.78131 −0.890656 0.454677i \(-0.849755\pi\)
−0.890656 + 0.454677i \(0.849755\pi\)
\(840\) −4.51446 −0.155764
\(841\) −21.5495 −0.743087
\(842\) 18.5188 0.638200
\(843\) −2.31336 −0.0796763
\(844\) −9.26704 −0.318985
\(845\) 11.0636 0.380598
\(846\) −5.25477 −0.180663
\(847\) 0 0
\(848\) −3.89909 −0.133895
\(849\) −2.61155 −0.0896282
\(850\) 2.70843 0.0928985
\(851\) −42.2876 −1.44960
\(852\) −11.9157 −0.408224
\(853\) 28.2200 0.966235 0.483117 0.875556i \(-0.339504\pi\)
0.483117 + 0.875556i \(0.339504\pi\)
\(854\) −1.60103 −0.0547860
\(855\) 0.875501 0.0299415
\(856\) −35.4627 −1.21209
\(857\) −6.27743 −0.214433 −0.107216 0.994236i \(-0.534194\pi\)
−0.107216 + 0.994236i \(0.534194\pi\)
\(858\) 0 0
\(859\) 25.9580 0.885675 0.442837 0.896602i \(-0.353972\pi\)
0.442837 + 0.896602i \(0.353972\pi\)
\(860\) −5.83579 −0.198999
\(861\) −11.6179 −0.395936
\(862\) 14.8715 0.506525
\(863\) 7.62600 0.259592 0.129796 0.991541i \(-0.458568\pi\)
0.129796 + 0.991541i \(0.458568\pi\)
\(864\) −29.9553 −1.01910
\(865\) −22.0859 −0.750945
\(866\) 6.67856 0.226947
\(867\) −13.4373 −0.456355
\(868\) −2.33942 −0.0794050
\(869\) 0 0
\(870\) −3.95782 −0.134183
\(871\) −0.00202640 −6.86619e−5 0
\(872\) −6.06682 −0.205448
\(873\) −0.298109 −0.0100895
\(874\) 8.47867 0.286795
\(875\) 1.00000 0.0338062
\(876\) 0.164165 0.00554661
\(877\) 25.0934 0.847344 0.423672 0.905816i \(-0.360741\pi\)
0.423672 + 0.905816i \(0.360741\pi\)
\(878\) −23.0898 −0.779243
\(879\) −13.4940 −0.455142
\(880\) 0 0
\(881\) 27.4989 0.926463 0.463231 0.886237i \(-0.346690\pi\)
0.463231 + 0.886237i \(0.346690\pi\)
\(882\) −0.591771 −0.0199260
\(883\) −26.9277 −0.906190 −0.453095 0.891462i \(-0.649680\pi\)
−0.453095 + 0.891462i \(0.649680\pi\)
\(884\) 4.45694 0.149903
\(885\) 13.6696 0.459497
\(886\) −9.44697 −0.317377
\(887\) 34.6125 1.16217 0.581087 0.813841i \(-0.302628\pi\)
0.581087 + 0.813841i \(0.302628\pi\)
\(888\) 29.5324 0.991041
\(889\) 15.6786 0.525843
\(890\) −0.486324 −0.0163016
\(891\) 0 0
\(892\) −22.2705 −0.745670
\(893\) 12.3696 0.413932
\(894\) −7.44911 −0.249136
\(895\) 14.5651 0.486859
\(896\) 4.96470 0.165859
\(897\) −13.8527 −0.462527
\(898\) 26.7322 0.892064
\(899\) −5.73495 −0.191271
\(900\) 0.699796 0.0233265
\(901\) 21.0293 0.700587
\(902\) 0 0
\(903\) 8.07127 0.268595
\(904\) −53.7494 −1.78768
\(905\) −4.14710 −0.137854
\(906\) 13.7943 0.458285
\(907\) 19.9016 0.660823 0.330411 0.943837i \(-0.392813\pi\)
0.330411 + 0.943837i \(0.392813\pi\)
\(908\) 14.7039 0.487965
\(909\) −6.10261 −0.202411
\(910\) −1.31025 −0.0434343
\(911\) 50.4666 1.67203 0.836016 0.548704i \(-0.184879\pi\)
0.836016 + 0.548704i \(0.184879\pi\)
\(912\) −1.14412 −0.0378856
\(913\) 0 0
\(914\) −16.8641 −0.557816
\(915\) −2.61854 −0.0865662
\(916\) 28.6685 0.947232
\(917\) −14.8387 −0.490016
\(918\) −15.1341 −0.499500
\(919\) 6.02847 0.198861 0.0994304 0.995045i \(-0.468298\pi\)
0.0994304 + 0.995045i \(0.468298\pi\)
\(920\) 18.9502 0.624770
\(921\) 24.1759 0.796622
\(922\) −21.0398 −0.692909
\(923\) −9.67025 −0.318300
\(924\) 0 0
\(925\) −6.54173 −0.215091
\(926\) −29.9234 −0.983345
\(927\) −0.454229 −0.0149188
\(928\) 14.6328 0.480345
\(929\) 30.9289 1.01474 0.507371 0.861727i \(-0.330617\pi\)
0.507371 + 0.861727i \(0.330617\pi\)
\(930\) 3.04651 0.0998990
\(931\) 1.39301 0.0456542
\(932\) 13.2531 0.434120
\(933\) 6.98332 0.228624
\(934\) 12.6425 0.413674
\(935\) 0 0
\(936\) −2.56387 −0.0838028
\(937\) 17.9725 0.587135 0.293568 0.955938i \(-0.405157\pi\)
0.293568 + 0.955938i \(0.405157\pi\)
\(938\) −0.00137112 −4.47687e−5 0
\(939\) −4.13719 −0.135012
\(940\) 9.88711 0.322482
\(941\) −11.1858 −0.364646 −0.182323 0.983239i \(-0.558362\pi\)
−0.182323 + 0.983239i \(0.558362\pi\)
\(942\) −7.17167 −0.233665
\(943\) 48.7679 1.58810
\(944\) 4.73420 0.154085
\(945\) −5.58777 −0.181770
\(946\) 0 0
\(947\) 36.2130 1.17677 0.588383 0.808583i \(-0.299765\pi\)
0.588383 + 0.808583i \(0.299765\pi\)
\(948\) −15.8570 −0.515011
\(949\) 0.133229 0.00432480
\(950\) 1.31162 0.0425545
\(951\) −21.6124 −0.700829
\(952\) 8.43256 0.273301
\(953\) −28.0283 −0.907926 −0.453963 0.891020i \(-0.649990\pi\)
−0.453963 + 0.891020i \(0.649990\pi\)
\(954\) −4.32626 −0.140068
\(955\) 13.3933 0.433398
\(956\) 19.3958 0.627306
\(957\) 0 0
\(958\) 9.09262 0.293769
\(959\) −13.0351 −0.420924
\(960\) −9.41586 −0.303896
\(961\) −26.5856 −0.857598
\(962\) 8.57128 0.276349
\(963\) −7.60291 −0.245000
\(964\) −2.56718 −0.0826832
\(965\) −14.7344 −0.474317
\(966\) −9.37313 −0.301575
\(967\) 60.9621 1.96041 0.980205 0.197985i \(-0.0634397\pi\)
0.980205 + 0.197985i \(0.0634397\pi\)
\(968\) 0 0
\(969\) 6.17068 0.198231
\(970\) −0.446608 −0.0143397
\(971\) −26.9575 −0.865107 −0.432554 0.901608i \(-0.642387\pi\)
−0.432554 + 0.901608i \(0.642387\pi\)
\(972\) −7.14329 −0.229121
\(973\) −18.9218 −0.606605
\(974\) −32.1567 −1.03037
\(975\) −2.14296 −0.0686295
\(976\) −0.906883 −0.0290286
\(977\) −3.67207 −0.117480 −0.0587400 0.998273i \(-0.518708\pi\)
−0.0587400 + 0.998273i \(0.518708\pi\)
\(978\) −27.7656 −0.887845
\(979\) 0 0
\(980\) 1.11345 0.0355678
\(981\) −1.30067 −0.0415273
\(982\) 1.45597 0.0464618
\(983\) 25.0044 0.797515 0.398758 0.917056i \(-0.369442\pi\)
0.398758 + 0.917056i \(0.369442\pi\)
\(984\) −34.0581 −1.08573
\(985\) −3.94112 −0.125575
\(986\) 7.39281 0.235435
\(987\) −13.6745 −0.435264
\(988\) 2.15837 0.0686669
\(989\) −33.8805 −1.07734
\(990\) 0 0
\(991\) −5.78499 −0.183766 −0.0918831 0.995770i \(-0.529289\pi\)
−0.0918831 + 0.995770i \(0.529289\pi\)
\(992\) −11.2635 −0.357617
\(993\) 33.1631 1.05240
\(994\) −6.54318 −0.207537
\(995\) 10.2444 0.324769
\(996\) −22.7132 −0.719695
\(997\) 2.01189 0.0637171 0.0318585 0.999492i \(-0.489857\pi\)
0.0318585 + 0.999492i \(0.489857\pi\)
\(998\) −14.0972 −0.446240
\(999\) 36.5537 1.15651
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 4235.2.a.ba.1.4 5
11.10 odd 2 4235.2.a.bb.1.2 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4235.2.a.ba.1.4 5 1.1 even 1 trivial
4235.2.a.bb.1.2 yes 5 11.10 odd 2