Properties

Label 637.2.a.n.1.3
Level $637$
Weight $2$
Character 637.1
Self dual yes
Analytic conductor $5.086$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.4507648.1
Defining polynomial: \( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.20475\) of defining polynomial
Character \(\chi\) \(=\) 637.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.656184 q^{2} -0.204753 q^{3} -1.56942 q^{4} -1.35996 q^{5} +0.134356 q^{6} +2.34220 q^{8} -2.95808 q^{9} +O(q^{10})\) \(q-0.656184 q^{2} -0.204753 q^{3} -1.56942 q^{4} -1.35996 q^{5} +0.134356 q^{6} +2.34220 q^{8} -2.95808 q^{9} +0.892385 q^{10} -1.90551 q^{11} +0.321345 q^{12} +1.00000 q^{13} +0.278457 q^{15} +1.60193 q^{16} +3.56633 q^{17} +1.94104 q^{18} +0.985255 q^{19} +2.13436 q^{20} +1.25037 q^{22} +1.69605 q^{23} -0.479573 q^{24} -3.15050 q^{25} -0.656184 q^{26} +1.21994 q^{27} +6.54835 q^{29} -0.182719 q^{30} +7.69550 q^{31} -5.73556 q^{32} +0.390160 q^{33} -2.34017 q^{34} +4.64247 q^{36} -2.02005 q^{37} -0.646508 q^{38} -0.204753 q^{39} -3.18530 q^{40} +9.88779 q^{41} -3.16639 q^{43} +2.99055 q^{44} +4.02287 q^{45} -1.11292 q^{46} +7.76416 q^{47} -0.328001 q^{48} +2.06731 q^{50} -0.730219 q^{51} -1.56942 q^{52} +0.354194 q^{53} -0.800502 q^{54} +2.59143 q^{55} -0.201734 q^{57} -4.29692 q^{58} +2.16385 q^{59} -0.437016 q^{60} +12.2002 q^{61} -5.04967 q^{62} +0.559715 q^{64} -1.35996 q^{65} -0.256017 q^{66} +11.3134 q^{67} -5.59709 q^{68} -0.347272 q^{69} -9.05268 q^{71} -6.92840 q^{72} -7.13619 q^{73} +1.32553 q^{74} +0.645076 q^{75} -1.54628 q^{76} +0.134356 q^{78} -5.39629 q^{79} -2.17857 q^{80} +8.62444 q^{81} -6.48821 q^{82} -2.03494 q^{83} -4.85008 q^{85} +2.07773 q^{86} -1.34080 q^{87} -4.46309 q^{88} -6.89864 q^{89} -2.63974 q^{90} -2.66182 q^{92} -1.57568 q^{93} -5.09472 q^{94} -1.33991 q^{95} +1.17437 q^{96} -14.6223 q^{97} +5.63665 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{3} + 4 q^{4} + 6 q^{5} + 4 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 8 q^{3} + 4 q^{4} + 6 q^{5} + 4 q^{6} + 6 q^{9} + 4 q^{10} + 4 q^{11} - 4 q^{12} + 6 q^{13} + 12 q^{15} + 16 q^{17} - 4 q^{18} + 2 q^{19} + 16 q^{20} - 12 q^{22} - 6 q^{23} + 12 q^{24} - 4 q^{25} + 20 q^{27} - 6 q^{29} + 6 q^{31} - 20 q^{32} + 4 q^{33} - 24 q^{36} + 8 q^{38} + 8 q^{39} + 4 q^{40} - 8 q^{41} + 2 q^{43} - 4 q^{44} + 14 q^{45} + 8 q^{46} + 30 q^{47} - 8 q^{48} + 8 q^{50} - 4 q^{51} + 4 q^{52} - 14 q^{53} - 48 q^{54} - 8 q^{55} + 4 q^{57} - 8 q^{58} + 24 q^{59} + 12 q^{60} + 28 q^{62} - 20 q^{64} + 6 q^{65} - 4 q^{66} + 16 q^{67} + 28 q^{68} - 20 q^{69} + 8 q^{71} + 28 q^{72} - 6 q^{73} - 12 q^{74} + 12 q^{75} - 16 q^{76} + 4 q^{78} - 22 q^{79} - 28 q^{80} + 46 q^{81} - 40 q^{82} + 50 q^{83} - 8 q^{85} - 16 q^{86} - 16 q^{87} - 44 q^{88} + 26 q^{89} - 40 q^{90} + 20 q^{92} + 16 q^{93} - 32 q^{94} - 6 q^{95} - 20 q^{96} - 14 q^{97} + 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.656184 −0.463992 −0.231996 0.972717i \(-0.574526\pi\)
−0.231996 + 0.972717i \(0.574526\pi\)
\(3\) −0.204753 −0.118214 −0.0591072 0.998252i \(-0.518825\pi\)
−0.0591072 + 0.998252i \(0.518825\pi\)
\(4\) −1.56942 −0.784711
\(5\) −1.35996 −0.608194 −0.304097 0.952641i \(-0.598355\pi\)
−0.304097 + 0.952641i \(0.598355\pi\)
\(6\) 0.134356 0.0548505
\(7\) 0 0
\(8\) 2.34220 0.828092
\(9\) −2.95808 −0.986025
\(10\) 0.892385 0.282197
\(11\) −1.90551 −0.574534 −0.287267 0.957851i \(-0.592747\pi\)
−0.287267 + 0.957851i \(0.592747\pi\)
\(12\) 0.321345 0.0927642
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 0.278457 0.0718972
\(16\) 1.60193 0.400483
\(17\) 3.56633 0.864963 0.432482 0.901643i \(-0.357638\pi\)
0.432482 + 0.901643i \(0.357638\pi\)
\(18\) 1.94104 0.457508
\(19\) 0.985255 0.226033 0.113016 0.993593i \(-0.463949\pi\)
0.113016 + 0.993593i \(0.463949\pi\)
\(20\) 2.13436 0.477256
\(21\) 0 0
\(22\) 1.25037 0.266579
\(23\) 1.69605 0.353651 0.176826 0.984242i \(-0.443417\pi\)
0.176826 + 0.984242i \(0.443417\pi\)
\(24\) −0.479573 −0.0978924
\(25\) −3.15050 −0.630100
\(26\) −0.656184 −0.128688
\(27\) 1.21994 0.234777
\(28\) 0 0
\(29\) 6.54835 1.21600 0.607999 0.793938i \(-0.291972\pi\)
0.607999 + 0.793938i \(0.291972\pi\)
\(30\) −0.182719 −0.0333598
\(31\) 7.69550 1.38215 0.691077 0.722781i \(-0.257137\pi\)
0.691077 + 0.722781i \(0.257137\pi\)
\(32\) −5.73556 −1.01391
\(33\) 0.390160 0.0679181
\(34\) −2.34017 −0.401336
\(35\) 0 0
\(36\) 4.64247 0.773745
\(37\) −2.02005 −0.332095 −0.166047 0.986118i \(-0.553100\pi\)
−0.166047 + 0.986118i \(0.553100\pi\)
\(38\) −0.646508 −0.104877
\(39\) −0.204753 −0.0327868
\(40\) −3.18530 −0.503640
\(41\) 9.88779 1.54421 0.772107 0.635493i \(-0.219203\pi\)
0.772107 + 0.635493i \(0.219203\pi\)
\(42\) 0 0
\(43\) −3.16639 −0.482869 −0.241435 0.970417i \(-0.577618\pi\)
−0.241435 + 0.970417i \(0.577618\pi\)
\(44\) 2.99055 0.450843
\(45\) 4.02287 0.599694
\(46\) −1.11292 −0.164091
\(47\) 7.76416 1.13252 0.566260 0.824227i \(-0.308390\pi\)
0.566260 + 0.824227i \(0.308390\pi\)
\(48\) −0.328001 −0.0473429
\(49\) 0 0
\(50\) 2.06731 0.292362
\(51\) −0.730219 −0.102251
\(52\) −1.56942 −0.217640
\(53\) 0.354194 0.0486522 0.0243261 0.999704i \(-0.492256\pi\)
0.0243261 + 0.999704i \(0.492256\pi\)
\(54\) −0.800502 −0.108935
\(55\) 2.59143 0.349428
\(56\) 0 0
\(57\) −0.201734 −0.0267203
\(58\) −4.29692 −0.564214
\(59\) 2.16385 0.281709 0.140854 0.990030i \(-0.455015\pi\)
0.140854 + 0.990030i \(0.455015\pi\)
\(60\) −0.437016 −0.0564186
\(61\) 12.2002 1.56207 0.781037 0.624484i \(-0.214691\pi\)
0.781037 + 0.624484i \(0.214691\pi\)
\(62\) −5.04967 −0.641308
\(63\) 0 0
\(64\) 0.559715 0.0699644
\(65\) −1.35996 −0.168683
\(66\) −0.256017 −0.0315135
\(67\) 11.3134 1.38215 0.691076 0.722782i \(-0.257137\pi\)
0.691076 + 0.722782i \(0.257137\pi\)
\(68\) −5.59709 −0.678746
\(69\) −0.347272 −0.0418067
\(70\) 0 0
\(71\) −9.05268 −1.07436 −0.537178 0.843469i \(-0.680510\pi\)
−0.537178 + 0.843469i \(0.680510\pi\)
\(72\) −6.92840 −0.816520
\(73\) −7.13619 −0.835228 −0.417614 0.908625i \(-0.637134\pi\)
−0.417614 + 0.908625i \(0.637134\pi\)
\(74\) 1.32553 0.154089
\(75\) 0.645076 0.0744869
\(76\) −1.54628 −0.177371
\(77\) 0 0
\(78\) 0.134356 0.0152128
\(79\) −5.39629 −0.607130 −0.303565 0.952811i \(-0.598177\pi\)
−0.303565 + 0.952811i \(0.598177\pi\)
\(80\) −2.17857 −0.243571
\(81\) 8.62444 0.958271
\(82\) −6.48821 −0.716503
\(83\) −2.03494 −0.223364 −0.111682 0.993744i \(-0.535624\pi\)
−0.111682 + 0.993744i \(0.535624\pi\)
\(84\) 0 0
\(85\) −4.85008 −0.526065
\(86\) 2.07773 0.224048
\(87\) −1.34080 −0.143749
\(88\) −4.46309 −0.475767
\(89\) −6.89864 −0.731254 −0.365627 0.930761i \(-0.619145\pi\)
−0.365627 + 0.930761i \(0.619145\pi\)
\(90\) −2.63974 −0.278253
\(91\) 0 0
\(92\) −2.66182 −0.277514
\(93\) −1.57568 −0.163390
\(94\) −5.09472 −0.525480
\(95\) −1.33991 −0.137472
\(96\) 1.17437 0.119859
\(97\) −14.6223 −1.48467 −0.742334 0.670030i \(-0.766281\pi\)
−0.742334 + 0.670030i \(0.766281\pi\)
\(98\) 0 0
\(99\) 5.63665 0.566505
\(100\) 4.94447 0.494447
\(101\) 2.81368 0.279972 0.139986 0.990153i \(-0.455294\pi\)
0.139986 + 0.990153i \(0.455294\pi\)
\(102\) 0.479158 0.0474437
\(103\) 6.04988 0.596112 0.298056 0.954548i \(-0.403662\pi\)
0.298056 + 0.954548i \(0.403662\pi\)
\(104\) 2.34220 0.229671
\(105\) 0 0
\(106\) −0.232416 −0.0225743
\(107\) 17.3204 1.67442 0.837211 0.546880i \(-0.184185\pi\)
0.837211 + 0.546880i \(0.184185\pi\)
\(108\) −1.91460 −0.184232
\(109\) 10.5762 1.01302 0.506509 0.862234i \(-0.330936\pi\)
0.506509 + 0.862234i \(0.330936\pi\)
\(110\) −1.70045 −0.162132
\(111\) 0.413613 0.0392584
\(112\) 0 0
\(113\) −2.34665 −0.220755 −0.110377 0.993890i \(-0.535206\pi\)
−0.110377 + 0.993890i \(0.535206\pi\)
\(114\) 0.132375 0.0123980
\(115\) −2.30657 −0.215089
\(116\) −10.2771 −0.954208
\(117\) −2.95808 −0.273474
\(118\) −1.41988 −0.130711
\(119\) 0 0
\(120\) 0.652201 0.0595375
\(121\) −7.36902 −0.669911
\(122\) −8.00557 −0.724790
\(123\) −2.02456 −0.182548
\(124\) −12.0775 −1.08459
\(125\) 11.0844 0.991417
\(126\) 0 0
\(127\) −18.1639 −1.61179 −0.805894 0.592060i \(-0.798315\pi\)
−0.805894 + 0.592060i \(0.798315\pi\)
\(128\) 11.1038 0.981450
\(129\) 0.648328 0.0570821
\(130\) 0.892385 0.0782674
\(131\) 10.1014 0.882560 0.441280 0.897369i \(-0.354524\pi\)
0.441280 + 0.897369i \(0.354524\pi\)
\(132\) −0.612326 −0.0532961
\(133\) 0 0
\(134\) −7.42367 −0.641308
\(135\) −1.65907 −0.142790
\(136\) 8.35306 0.716269
\(137\) 6.55601 0.560118 0.280059 0.959983i \(-0.409646\pi\)
0.280059 + 0.959983i \(0.409646\pi\)
\(138\) 0.227875 0.0193980
\(139\) −13.8243 −1.17256 −0.586281 0.810108i \(-0.699409\pi\)
−0.586281 + 0.810108i \(0.699409\pi\)
\(140\) 0 0
\(141\) −1.58974 −0.133880
\(142\) 5.94023 0.498493
\(143\) −1.90551 −0.159347
\(144\) −4.73864 −0.394887
\(145\) −8.90551 −0.739563
\(146\) 4.68265 0.387539
\(147\) 0 0
\(148\) 3.17032 0.260598
\(149\) 8.09389 0.663078 0.331539 0.943442i \(-0.392432\pi\)
0.331539 + 0.943442i \(0.392432\pi\)
\(150\) −0.423288 −0.0345614
\(151\) 12.0365 0.979520 0.489760 0.871857i \(-0.337084\pi\)
0.489760 + 0.871857i \(0.337084\pi\)
\(152\) 2.30766 0.187176
\(153\) −10.5495 −0.852876
\(154\) 0 0
\(155\) −10.4656 −0.840617
\(156\) 0.321345 0.0257282
\(157\) −18.3935 −1.46796 −0.733982 0.679169i \(-0.762340\pi\)
−0.733982 + 0.679169i \(0.762340\pi\)
\(158\) 3.54096 0.281704
\(159\) −0.0725223 −0.00575139
\(160\) 7.80014 0.616655
\(161\) 0 0
\(162\) −5.65922 −0.444630
\(163\) 2.51286 0.196823 0.0984114 0.995146i \(-0.468624\pi\)
0.0984114 + 0.995146i \(0.468624\pi\)
\(164\) −15.5181 −1.21176
\(165\) −0.530603 −0.0413074
\(166\) 1.33530 0.103639
\(167\) 17.9095 1.38588 0.692941 0.720995i \(-0.256315\pi\)
0.692941 + 0.720995i \(0.256315\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 3.18254 0.244090
\(171\) −2.91446 −0.222874
\(172\) 4.96940 0.378913
\(173\) 6.51981 0.495692 0.247846 0.968799i \(-0.420277\pi\)
0.247846 + 0.968799i \(0.420277\pi\)
\(174\) 0.879809 0.0666982
\(175\) 0 0
\(176\) −3.05250 −0.230091
\(177\) −0.443055 −0.0333020
\(178\) 4.52678 0.339296
\(179\) −26.6014 −1.98828 −0.994141 0.108091i \(-0.965526\pi\)
−0.994141 + 0.108091i \(0.965526\pi\)
\(180\) −6.31359 −0.470587
\(181\) −24.6402 −1.83149 −0.915746 0.401758i \(-0.868399\pi\)
−0.915746 + 0.401758i \(0.868399\pi\)
\(182\) 0 0
\(183\) −2.49803 −0.184660
\(184\) 3.97249 0.292856
\(185\) 2.74720 0.201978
\(186\) 1.03394 0.0758119
\(187\) −6.79569 −0.496950
\(188\) −12.1853 −0.888701
\(189\) 0 0
\(190\) 0.879227 0.0637858
\(191\) 3.02099 0.218591 0.109295 0.994009i \(-0.465141\pi\)
0.109295 + 0.994009i \(0.465141\pi\)
\(192\) −0.114604 −0.00827080
\(193\) 21.2522 1.52977 0.764884 0.644168i \(-0.222796\pi\)
0.764884 + 0.644168i \(0.222796\pi\)
\(194\) 9.59491 0.688874
\(195\) 0.278457 0.0199407
\(196\) 0 0
\(197\) −2.72191 −0.193928 −0.0969639 0.995288i \(-0.530913\pi\)
−0.0969639 + 0.995288i \(0.530913\pi\)
\(198\) −3.69868 −0.262854
\(199\) −21.6670 −1.53593 −0.767967 0.640490i \(-0.778731\pi\)
−0.767967 + 0.640490i \(0.778731\pi\)
\(200\) −7.37910 −0.521781
\(201\) −2.31646 −0.163390
\(202\) −1.84629 −0.129905
\(203\) 0 0
\(204\) 1.14602 0.0802376
\(205\) −13.4470 −0.939181
\(206\) −3.96983 −0.276591
\(207\) −5.01705 −0.348709
\(208\) 1.60193 0.111074
\(209\) −1.87741 −0.129864
\(210\) 0 0
\(211\) 18.3885 1.26592 0.632959 0.774186i \(-0.281840\pi\)
0.632959 + 0.774186i \(0.281840\pi\)
\(212\) −0.555879 −0.0381780
\(213\) 1.85357 0.127004
\(214\) −11.3653 −0.776918
\(215\) 4.30617 0.293678
\(216\) 2.85733 0.194417
\(217\) 0 0
\(218\) −6.93995 −0.470033
\(219\) 1.46116 0.0987359
\(220\) −4.06704 −0.274200
\(221\) 3.56633 0.239898
\(222\) −0.271406 −0.0182156
\(223\) 22.2216 1.48807 0.744035 0.668141i \(-0.232910\pi\)
0.744035 + 0.668141i \(0.232910\pi\)
\(224\) 0 0
\(225\) 9.31943 0.621295
\(226\) 1.53984 0.102428
\(227\) 26.7374 1.77463 0.887313 0.461167i \(-0.152569\pi\)
0.887313 + 0.461167i \(0.152569\pi\)
\(228\) 0.316606 0.0209678
\(229\) −16.7008 −1.10362 −0.551809 0.833970i \(-0.686062\pi\)
−0.551809 + 0.833970i \(0.686062\pi\)
\(230\) 1.51353 0.0997994
\(231\) 0 0
\(232\) 15.3375 1.00696
\(233\) −13.8776 −0.909149 −0.454575 0.890709i \(-0.650209\pi\)
−0.454575 + 0.890709i \(0.650209\pi\)
\(234\) 1.94104 0.126890
\(235\) −10.5590 −0.688791
\(236\) −3.39599 −0.221060
\(237\) 1.10491 0.0717715
\(238\) 0 0
\(239\) 0.465845 0.0301330 0.0150665 0.999886i \(-0.495204\pi\)
0.0150665 + 0.999886i \(0.495204\pi\)
\(240\) 0.446069 0.0287936
\(241\) 5.69314 0.366727 0.183364 0.983045i \(-0.441301\pi\)
0.183364 + 0.983045i \(0.441301\pi\)
\(242\) 4.83543 0.310833
\(243\) −5.42569 −0.348058
\(244\) −19.1473 −1.22578
\(245\) 0 0
\(246\) 1.32848 0.0847010
\(247\) 0.985255 0.0626902
\(248\) 18.0244 1.14455
\(249\) 0.416662 0.0264049
\(250\) −7.27339 −0.460010
\(251\) 17.9066 1.13025 0.565126 0.825005i \(-0.308828\pi\)
0.565126 + 0.825005i \(0.308828\pi\)
\(252\) 0 0
\(253\) −3.23185 −0.203185
\(254\) 11.9189 0.747857
\(255\) 0.993070 0.0621885
\(256\) −8.40559 −0.525350
\(257\) 25.9756 1.62031 0.810156 0.586214i \(-0.199382\pi\)
0.810156 + 0.586214i \(0.199382\pi\)
\(258\) −0.425423 −0.0264857
\(259\) 0 0
\(260\) 2.13436 0.132367
\(261\) −19.3705 −1.19901
\(262\) −6.62835 −0.409501
\(263\) 27.8402 1.71670 0.858349 0.513067i \(-0.171491\pi\)
0.858349 + 0.513067i \(0.171491\pi\)
\(264\) 0.913832 0.0562425
\(265\) −0.481690 −0.0295900
\(266\) 0 0
\(267\) 1.41252 0.0864448
\(268\) −17.7555 −1.08459
\(269\) 5.90486 0.360025 0.180013 0.983664i \(-0.442386\pi\)
0.180013 + 0.983664i \(0.442386\pi\)
\(270\) 1.08865 0.0662533
\(271\) −4.34932 −0.264203 −0.132101 0.991236i \(-0.542172\pi\)
−0.132101 + 0.991236i \(0.542172\pi\)
\(272\) 5.71303 0.346403
\(273\) 0 0
\(274\) −4.30195 −0.259890
\(275\) 6.00332 0.362014
\(276\) 0.545017 0.0328062
\(277\) 1.08051 0.0649217 0.0324609 0.999473i \(-0.489666\pi\)
0.0324609 + 0.999473i \(0.489666\pi\)
\(278\) 9.07129 0.544060
\(279\) −22.7639 −1.36284
\(280\) 0 0
\(281\) 14.4912 0.864473 0.432236 0.901760i \(-0.357725\pi\)
0.432236 + 0.901760i \(0.357725\pi\)
\(282\) 1.04316 0.0621193
\(283\) 24.4979 1.45625 0.728123 0.685446i \(-0.240393\pi\)
0.728123 + 0.685446i \(0.240393\pi\)
\(284\) 14.2075 0.843059
\(285\) 0.274351 0.0162511
\(286\) 1.25037 0.0739357
\(287\) 0 0
\(288\) 16.9662 0.999744
\(289\) −4.28126 −0.251839
\(290\) 5.84365 0.343151
\(291\) 2.99396 0.175509
\(292\) 11.1997 0.655413
\(293\) 1.16105 0.0678296 0.0339148 0.999425i \(-0.489203\pi\)
0.0339148 + 0.999425i \(0.489203\pi\)
\(294\) 0 0
\(295\) −2.94275 −0.171334
\(296\) −4.73136 −0.275005
\(297\) −2.32460 −0.134887
\(298\) −5.31108 −0.307663
\(299\) 1.69605 0.0980852
\(300\) −1.01240 −0.0584507
\(301\) 0 0
\(302\) −7.89818 −0.454489
\(303\) −0.576111 −0.0330967
\(304\) 1.57831 0.0905224
\(305\) −16.5918 −0.950044
\(306\) 6.92240 0.395728
\(307\) 25.3731 1.44812 0.724060 0.689737i \(-0.242274\pi\)
0.724060 + 0.689737i \(0.242274\pi\)
\(308\) 0 0
\(309\) −1.23873 −0.0704690
\(310\) 6.86736 0.390040
\(311\) 24.4645 1.38726 0.693628 0.720333i \(-0.256011\pi\)
0.693628 + 0.720333i \(0.256011\pi\)
\(312\) −0.479573 −0.0271505
\(313\) 6.61665 0.373995 0.186997 0.982360i \(-0.440124\pi\)
0.186997 + 0.982360i \(0.440124\pi\)
\(314\) 12.0695 0.681124
\(315\) 0 0
\(316\) 8.46906 0.476422
\(317\) 8.19803 0.460447 0.230224 0.973138i \(-0.426054\pi\)
0.230224 + 0.973138i \(0.426054\pi\)
\(318\) 0.0475880 0.00266860
\(319\) −12.4780 −0.698632
\(320\) −0.761192 −0.0425519
\(321\) −3.54640 −0.197941
\(322\) 0 0
\(323\) 3.51375 0.195510
\(324\) −13.5354 −0.751966
\(325\) −3.15050 −0.174758
\(326\) −1.64890 −0.0913242
\(327\) −2.16552 −0.119753
\(328\) 23.1592 1.27875
\(329\) 0 0
\(330\) 0.348173 0.0191663
\(331\) −0.724715 −0.0398339 −0.0199170 0.999802i \(-0.506340\pi\)
−0.0199170 + 0.999802i \(0.506340\pi\)
\(332\) 3.19369 0.175276
\(333\) 5.97547 0.327454
\(334\) −11.7519 −0.643038
\(335\) −15.3858 −0.840616
\(336\) 0 0
\(337\) −8.58299 −0.467545 −0.233773 0.972291i \(-0.575107\pi\)
−0.233773 + 0.972291i \(0.575107\pi\)
\(338\) −0.656184 −0.0356917
\(339\) 0.480485 0.0260964
\(340\) 7.61183 0.412809
\(341\) −14.6639 −0.794094
\(342\) 1.91242 0.103412
\(343\) 0 0
\(344\) −7.41630 −0.399860
\(345\) 0.472277 0.0254266
\(346\) −4.27820 −0.229997
\(347\) −25.9250 −1.39173 −0.695864 0.718173i \(-0.744979\pi\)
−0.695864 + 0.718173i \(0.744979\pi\)
\(348\) 2.10428 0.112801
\(349\) 17.1403 0.917501 0.458750 0.888565i \(-0.348297\pi\)
0.458750 + 0.888565i \(0.348297\pi\)
\(350\) 0 0
\(351\) 1.21994 0.0651154
\(352\) 10.9292 0.582527
\(353\) −18.9101 −1.00648 −0.503242 0.864146i \(-0.667860\pi\)
−0.503242 + 0.864146i \(0.667860\pi\)
\(354\) 0.290725 0.0154519
\(355\) 12.3113 0.653417
\(356\) 10.8269 0.573823
\(357\) 0 0
\(358\) 17.4554 0.922547
\(359\) 14.1049 0.744431 0.372215 0.928146i \(-0.378598\pi\)
0.372215 + 0.928146i \(0.378598\pi\)
\(360\) 9.42236 0.496602
\(361\) −18.0293 −0.948909
\(362\) 16.1685 0.849798
\(363\) 1.50883 0.0791931
\(364\) 0 0
\(365\) 9.70495 0.507980
\(366\) 1.63917 0.0856806
\(367\) 0.360258 0.0188053 0.00940266 0.999956i \(-0.497007\pi\)
0.00940266 + 0.999956i \(0.497007\pi\)
\(368\) 2.71696 0.141631
\(369\) −29.2488 −1.52263
\(370\) −1.80267 −0.0937162
\(371\) 0 0
\(372\) 2.47291 0.128214
\(373\) −25.1680 −1.30315 −0.651575 0.758584i \(-0.725892\pi\)
−0.651575 + 0.758584i \(0.725892\pi\)
\(374\) 4.45923 0.230581
\(375\) −2.26956 −0.117200
\(376\) 18.1852 0.937830
\(377\) 6.54835 0.337257
\(378\) 0 0
\(379\) 31.8947 1.63832 0.819161 0.573564i \(-0.194440\pi\)
0.819161 + 0.573564i \(0.194440\pi\)
\(380\) 2.10288 0.107876
\(381\) 3.71913 0.190537
\(382\) −1.98232 −0.101424
\(383\) 32.8172 1.67688 0.838440 0.544994i \(-0.183468\pi\)
0.838440 + 0.544994i \(0.183468\pi\)
\(384\) −2.27355 −0.116022
\(385\) 0 0
\(386\) −13.9454 −0.709800
\(387\) 9.36641 0.476122
\(388\) 22.9486 1.16504
\(389\) −25.3087 −1.28320 −0.641600 0.767040i \(-0.721729\pi\)
−0.641600 + 0.767040i \(0.721729\pi\)
\(390\) −0.182719 −0.00925233
\(391\) 6.04869 0.305895
\(392\) 0 0
\(393\) −2.06829 −0.104331
\(394\) 1.78607 0.0899810
\(395\) 7.33875 0.369253
\(396\) −8.84629 −0.444543
\(397\) −5.29395 −0.265696 −0.132848 0.991136i \(-0.542412\pi\)
−0.132848 + 0.991136i \(0.542412\pi\)
\(398\) 14.2175 0.712661
\(399\) 0 0
\(400\) −5.04689 −0.252345
\(401\) −24.6335 −1.23014 −0.615070 0.788473i \(-0.710872\pi\)
−0.615070 + 0.788473i \(0.710872\pi\)
\(402\) 1.52002 0.0758118
\(403\) 7.69550 0.383340
\(404\) −4.41586 −0.219697
\(405\) −11.7289 −0.582815
\(406\) 0 0
\(407\) 3.84924 0.190800
\(408\) −1.71032 −0.0846733
\(409\) 10.2223 0.505458 0.252729 0.967537i \(-0.418672\pi\)
0.252729 + 0.967537i \(0.418672\pi\)
\(410\) 8.82372 0.435773
\(411\) −1.34236 −0.0662140
\(412\) −9.49481 −0.467776
\(413\) 0 0
\(414\) 3.29211 0.161798
\(415\) 2.76745 0.135849
\(416\) −5.73556 −0.281209
\(417\) 2.83057 0.138614
\(418\) 1.23193 0.0602556
\(419\) 30.6503 1.49736 0.748682 0.662929i \(-0.230687\pi\)
0.748682 + 0.662929i \(0.230687\pi\)
\(420\) 0 0
\(421\) 25.9764 1.26601 0.633007 0.774146i \(-0.281820\pi\)
0.633007 + 0.774146i \(0.281820\pi\)
\(422\) −12.0662 −0.587376
\(423\) −22.9670 −1.11669
\(424\) 0.829592 0.0402885
\(425\) −11.2357 −0.545014
\(426\) −1.21628 −0.0589290
\(427\) 0 0
\(428\) −27.1830 −1.31394
\(429\) 0.390160 0.0188371
\(430\) −2.82564 −0.136264
\(431\) 14.8818 0.716829 0.358415 0.933563i \(-0.383317\pi\)
0.358415 + 0.933563i \(0.383317\pi\)
\(432\) 1.95426 0.0940242
\(433\) −40.0871 −1.92647 −0.963233 0.268669i \(-0.913416\pi\)
−0.963233 + 0.268669i \(0.913416\pi\)
\(434\) 0 0
\(435\) 1.82343 0.0874269
\(436\) −16.5986 −0.794927
\(437\) 1.67104 0.0799368
\(438\) −0.958789 −0.0458127
\(439\) −3.00243 −0.143298 −0.0716491 0.997430i \(-0.522826\pi\)
−0.0716491 + 0.997430i \(0.522826\pi\)
\(440\) 6.06963 0.289358
\(441\) 0 0
\(442\) −2.34017 −0.111311
\(443\) 1.43566 0.0682101 0.0341051 0.999418i \(-0.489142\pi\)
0.0341051 + 0.999418i \(0.489142\pi\)
\(444\) −0.649133 −0.0308065
\(445\) 9.38189 0.444744
\(446\) −14.5815 −0.690453
\(447\) −1.65725 −0.0783853
\(448\) 0 0
\(449\) 19.6313 0.926460 0.463230 0.886238i \(-0.346690\pi\)
0.463230 + 0.886238i \(0.346690\pi\)
\(450\) −6.11526 −0.288276
\(451\) −18.8413 −0.887203
\(452\) 3.68289 0.173229
\(453\) −2.46452 −0.115793
\(454\) −17.5447 −0.823413
\(455\) 0 0
\(456\) −0.472501 −0.0221269
\(457\) 10.6421 0.497815 0.248907 0.968527i \(-0.419929\pi\)
0.248907 + 0.968527i \(0.419929\pi\)
\(458\) 10.9588 0.512070
\(459\) 4.35070 0.203073
\(460\) 3.61998 0.168782
\(461\) −38.5930 −1.79745 −0.898727 0.438509i \(-0.855507\pi\)
−0.898727 + 0.438509i \(0.855507\pi\)
\(462\) 0 0
\(463\) −27.9993 −1.30124 −0.650618 0.759405i \(-0.725490\pi\)
−0.650618 + 0.759405i \(0.725490\pi\)
\(464\) 10.4900 0.486987
\(465\) 2.14287 0.0993730
\(466\) 9.10623 0.421838
\(467\) −25.8199 −1.19480 −0.597401 0.801943i \(-0.703800\pi\)
−0.597401 + 0.801943i \(0.703800\pi\)
\(468\) 4.64247 0.214598
\(469\) 0 0
\(470\) 6.92862 0.319594
\(471\) 3.76614 0.173535
\(472\) 5.06816 0.233281
\(473\) 6.03359 0.277425
\(474\) −0.725023 −0.0333014
\(475\) −3.10405 −0.142423
\(476\) 0 0
\(477\) −1.04773 −0.0479723
\(478\) −0.305680 −0.0139815
\(479\) −39.4071 −1.80055 −0.900277 0.435317i \(-0.856636\pi\)
−0.900277 + 0.435317i \(0.856636\pi\)
\(480\) −1.59711 −0.0728976
\(481\) −2.02005 −0.0921065
\(482\) −3.73575 −0.170159
\(483\) 0 0
\(484\) 11.5651 0.525687
\(485\) 19.8858 0.902966
\(486\) 3.56025 0.161496
\(487\) −34.0959 −1.54503 −0.772517 0.634994i \(-0.781003\pi\)
−0.772517 + 0.634994i \(0.781003\pi\)
\(488\) 28.5753 1.29354
\(489\) −0.514517 −0.0232673
\(490\) 0 0
\(491\) −9.70414 −0.437942 −0.218971 0.975731i \(-0.570270\pi\)
−0.218971 + 0.975731i \(0.570270\pi\)
\(492\) 3.17739 0.143248
\(493\) 23.3536 1.05179
\(494\) −0.646508 −0.0290878
\(495\) −7.66563 −0.344545
\(496\) 12.3277 0.553529
\(497\) 0 0
\(498\) −0.273407 −0.0122516
\(499\) 18.0463 0.807862 0.403931 0.914789i \(-0.367644\pi\)
0.403931 + 0.914789i \(0.367644\pi\)
\(500\) −17.3961 −0.777976
\(501\) −3.66704 −0.163831
\(502\) −11.7500 −0.524428
\(503\) 16.4922 0.735352 0.367676 0.929954i \(-0.380154\pi\)
0.367676 + 0.929954i \(0.380154\pi\)
\(504\) 0 0
\(505\) −3.82650 −0.170277
\(506\) 2.12069 0.0942761
\(507\) −0.204753 −0.00909341
\(508\) 28.5069 1.26479
\(509\) −19.1977 −0.850921 −0.425460 0.904977i \(-0.639888\pi\)
−0.425460 + 0.904977i \(0.639888\pi\)
\(510\) −0.651637 −0.0288550
\(511\) 0 0
\(512\) −16.6921 −0.737692
\(513\) 1.20195 0.0530673
\(514\) −17.0448 −0.751812
\(515\) −8.22760 −0.362552
\(516\) −1.01750 −0.0447930
\(517\) −14.7947 −0.650670
\(518\) 0 0
\(519\) −1.33495 −0.0585979
\(520\) −3.18530 −0.139685
\(521\) −8.76034 −0.383797 −0.191899 0.981415i \(-0.561465\pi\)
−0.191899 + 0.981415i \(0.561465\pi\)
\(522\) 12.7106 0.556329
\(523\) −11.7406 −0.513381 −0.256690 0.966494i \(-0.582632\pi\)
−0.256690 + 0.966494i \(0.582632\pi\)
\(524\) −15.8533 −0.692555
\(525\) 0 0
\(526\) −18.2683 −0.796534
\(527\) 27.4447 1.19551
\(528\) 0.625010 0.0272001
\(529\) −20.1234 −0.874931
\(530\) 0.316077 0.0137295
\(531\) −6.40082 −0.277772
\(532\) 0 0
\(533\) 9.88779 0.428288
\(534\) −0.926872 −0.0401097
\(535\) −23.5550 −1.01837
\(536\) 26.4982 1.14455
\(537\) 5.44673 0.235044
\(538\) −3.87467 −0.167049
\(539\) 0 0
\(540\) 2.60378 0.112049
\(541\) 16.4587 0.707614 0.353807 0.935318i \(-0.384887\pi\)
0.353807 + 0.935318i \(0.384887\pi\)
\(542\) 2.85396 0.122588
\(543\) 5.04516 0.216509
\(544\) −20.4549 −0.876997
\(545\) −14.3833 −0.616112
\(546\) 0 0
\(547\) −8.19375 −0.350339 −0.175170 0.984538i \(-0.556047\pi\)
−0.175170 + 0.984538i \(0.556047\pi\)
\(548\) −10.2891 −0.439531
\(549\) −36.0891 −1.54025
\(550\) −3.93928 −0.167972
\(551\) 6.45179 0.274856
\(552\) −0.813381 −0.0346198
\(553\) 0 0
\(554\) −0.709015 −0.0301232
\(555\) −0.562498 −0.0238767
\(556\) 21.6962 0.920123
\(557\) −13.8144 −0.585335 −0.292667 0.956214i \(-0.594543\pi\)
−0.292667 + 0.956214i \(0.594543\pi\)
\(558\) 14.9373 0.632346
\(559\) −3.16639 −0.133924
\(560\) 0 0
\(561\) 1.39144 0.0587467
\(562\) −9.50889 −0.401108
\(563\) 9.75559 0.411149 0.205575 0.978641i \(-0.434094\pi\)
0.205575 + 0.978641i \(0.434094\pi\)
\(564\) 2.49497 0.105057
\(565\) 3.19136 0.134262
\(566\) −16.0751 −0.675687
\(567\) 0 0
\(568\) −21.2032 −0.889666
\(569\) −1.43264 −0.0600593 −0.0300296 0.999549i \(-0.509560\pi\)
−0.0300296 + 0.999549i \(0.509560\pi\)
\(570\) −0.180025 −0.00754040
\(571\) 16.2872 0.681598 0.340799 0.940136i \(-0.389303\pi\)
0.340799 + 0.940136i \(0.389303\pi\)
\(572\) 2.99055 0.125041
\(573\) −0.618557 −0.0258406
\(574\) 0 0
\(575\) −5.34342 −0.222836
\(576\) −1.65568 −0.0689867
\(577\) −4.81846 −0.200595 −0.100297 0.994957i \(-0.531979\pi\)
−0.100297 + 0.994957i \(0.531979\pi\)
\(578\) 2.80929 0.116851
\(579\) −4.35146 −0.180841
\(580\) 13.9765 0.580343
\(581\) 0 0
\(582\) −1.96459 −0.0814349
\(583\) −0.674920 −0.0279523
\(584\) −16.7144 −0.691645
\(585\) 4.02287 0.166325
\(586\) −0.761866 −0.0314724
\(587\) 36.6215 1.51153 0.755765 0.654843i \(-0.227265\pi\)
0.755765 + 0.654843i \(0.227265\pi\)
\(588\) 0 0
\(589\) 7.58203 0.312412
\(590\) 1.93099 0.0794974
\(591\) 0.557319 0.0229251
\(592\) −3.23599 −0.132998
\(593\) −18.7081 −0.768251 −0.384125 0.923281i \(-0.625497\pi\)
−0.384125 + 0.923281i \(0.625497\pi\)
\(594\) 1.52537 0.0625866
\(595\) 0 0
\(596\) −12.7027 −0.520325
\(597\) 4.43639 0.181569
\(598\) −1.11292 −0.0455108
\(599\) −25.4194 −1.03861 −0.519305 0.854589i \(-0.673809\pi\)
−0.519305 + 0.854589i \(0.673809\pi\)
\(600\) 1.51090 0.0616820
\(601\) 17.7790 0.725219 0.362609 0.931941i \(-0.381886\pi\)
0.362609 + 0.931941i \(0.381886\pi\)
\(602\) 0 0
\(603\) −33.4659 −1.36284
\(604\) −18.8904 −0.768640
\(605\) 10.0216 0.407436
\(606\) 0.378035 0.0153566
\(607\) 26.0047 1.05550 0.527750 0.849400i \(-0.323036\pi\)
0.527750 + 0.849400i \(0.323036\pi\)
\(608\) −5.65098 −0.229178
\(609\) 0 0
\(610\) 10.8873 0.440813
\(611\) 7.76416 0.314104
\(612\) 16.5566 0.669261
\(613\) 26.8230 1.08337 0.541685 0.840582i \(-0.317787\pi\)
0.541685 + 0.840582i \(0.317787\pi\)
\(614\) −16.6494 −0.671916
\(615\) 2.75332 0.111025
\(616\) 0 0
\(617\) −18.6491 −0.750784 −0.375392 0.926866i \(-0.622492\pi\)
−0.375392 + 0.926866i \(0.622492\pi\)
\(618\) 0.812836 0.0326971
\(619\) 13.0685 0.525269 0.262634 0.964895i \(-0.415409\pi\)
0.262634 + 0.964895i \(0.415409\pi\)
\(620\) 16.4249 0.659642
\(621\) 2.06908 0.0830291
\(622\) −16.0532 −0.643676
\(623\) 0 0
\(624\) −0.328001 −0.0131306
\(625\) 0.678175 0.0271270
\(626\) −4.34174 −0.173531
\(627\) 0.384407 0.0153517
\(628\) 28.8672 1.15193
\(629\) −7.20418 −0.287250
\(630\) 0 0
\(631\) −49.1745 −1.95761 −0.978804 0.204801i \(-0.934345\pi\)
−0.978804 + 0.204801i \(0.934345\pi\)
\(632\) −12.6392 −0.502760
\(633\) −3.76511 −0.149650
\(634\) −5.37941 −0.213644
\(635\) 24.7023 0.980279
\(636\) 0.113818 0.00451318
\(637\) 0 0
\(638\) 8.18784 0.324160
\(639\) 26.7785 1.05934
\(640\) −15.1008 −0.596912
\(641\) 4.37503 0.172803 0.0864016 0.996260i \(-0.472463\pi\)
0.0864016 + 0.996260i \(0.472463\pi\)
\(642\) 2.32709 0.0918429
\(643\) −5.34651 −0.210846 −0.105423 0.994427i \(-0.533620\pi\)
−0.105423 + 0.994427i \(0.533620\pi\)
\(644\) 0 0
\(645\) −0.881702 −0.0347170
\(646\) −2.30566 −0.0907151
\(647\) −29.7346 −1.16899 −0.584493 0.811398i \(-0.698707\pi\)
−0.584493 + 0.811398i \(0.698707\pi\)
\(648\) 20.2001 0.793537
\(649\) −4.12324 −0.161851
\(650\) 2.06731 0.0810865
\(651\) 0 0
\(652\) −3.94375 −0.154449
\(653\) −2.54262 −0.0995005 −0.0497503 0.998762i \(-0.515843\pi\)
−0.0497503 + 0.998762i \(0.515843\pi\)
\(654\) 1.42098 0.0555646
\(655\) −13.7375 −0.536768
\(656\) 15.8396 0.618432
\(657\) 21.1094 0.823556
\(658\) 0 0
\(659\) 1.87019 0.0728523 0.0364261 0.999336i \(-0.488403\pi\)
0.0364261 + 0.999336i \(0.488403\pi\)
\(660\) 0.832740 0.0324144
\(661\) −12.0763 −0.469714 −0.234857 0.972030i \(-0.575462\pi\)
−0.234857 + 0.972030i \(0.575462\pi\)
\(662\) 0.475546 0.0184826
\(663\) −0.730219 −0.0283593
\(664\) −4.76624 −0.184966
\(665\) 0 0
\(666\) −3.92101 −0.151936
\(667\) 11.1063 0.430040
\(668\) −28.1076 −1.08752
\(669\) −4.54995 −0.175911
\(670\) 10.0959 0.390039
\(671\) −23.2476 −0.897464
\(672\) 0 0
\(673\) −7.81691 −0.301320 −0.150660 0.988586i \(-0.548140\pi\)
−0.150660 + 0.988586i \(0.548140\pi\)
\(674\) 5.63202 0.216937
\(675\) −3.84341 −0.147933
\(676\) −1.56942 −0.0603624
\(677\) 6.25628 0.240448 0.120224 0.992747i \(-0.461639\pi\)
0.120224 + 0.992747i \(0.461639\pi\)
\(678\) −0.315287 −0.0121085
\(679\) 0 0
\(680\) −11.3598 −0.435630
\(681\) −5.47458 −0.209786
\(682\) 9.62220 0.368453
\(683\) 22.3656 0.855795 0.427898 0.903827i \(-0.359254\pi\)
0.427898 + 0.903827i \(0.359254\pi\)
\(684\) 4.57402 0.174892
\(685\) −8.91593 −0.340660
\(686\) 0 0
\(687\) 3.41954 0.130464
\(688\) −5.07234 −0.193381
\(689\) 0.354194 0.0134937
\(690\) −0.309901 −0.0117977
\(691\) −3.80771 −0.144852 −0.0724260 0.997374i \(-0.523074\pi\)
−0.0724260 + 0.997374i \(0.523074\pi\)
\(692\) −10.2323 −0.388975
\(693\) 0 0
\(694\) 17.0116 0.645751
\(695\) 18.8005 0.713145
\(696\) −3.14041 −0.119037
\(697\) 35.2632 1.33569
\(698\) −11.2472 −0.425713
\(699\) 2.84148 0.107475
\(700\) 0 0
\(701\) −8.14218 −0.307526 −0.153763 0.988108i \(-0.549139\pi\)
−0.153763 + 0.988108i \(0.549139\pi\)
\(702\) −0.800502 −0.0302130
\(703\) −1.99027 −0.0750643
\(704\) −1.06654 −0.0401969
\(705\) 2.16198 0.0814250
\(706\) 12.4085 0.467001
\(707\) 0 0
\(708\) 0.695340 0.0261325
\(709\) −24.0319 −0.902536 −0.451268 0.892388i \(-0.649028\pi\)
−0.451268 + 0.892388i \(0.649028\pi\)
\(710\) −8.07848 −0.303180
\(711\) 15.9626 0.598646
\(712\) −16.1580 −0.605546
\(713\) 13.0520 0.488800
\(714\) 0 0
\(715\) 2.59143 0.0969138
\(716\) 41.7488 1.56023
\(717\) −0.0953833 −0.00356215
\(718\) −9.25544 −0.345410
\(719\) −32.6960 −1.21936 −0.609678 0.792649i \(-0.708701\pi\)
−0.609678 + 0.792649i \(0.708701\pi\)
\(720\) 6.44437 0.240168
\(721\) 0 0
\(722\) 11.8305 0.440286
\(723\) −1.16569 −0.0433525
\(724\) 38.6709 1.43719
\(725\) −20.6306 −0.766201
\(726\) −0.990071 −0.0367450
\(727\) −15.7712 −0.584921 −0.292460 0.956278i \(-0.594474\pi\)
−0.292460 + 0.956278i \(0.594474\pi\)
\(728\) 0 0
\(729\) −24.7624 −0.917126
\(730\) −6.36823 −0.235699
\(731\) −11.2924 −0.417664
\(732\) 3.92046 0.144905
\(733\) 23.5094 0.868340 0.434170 0.900831i \(-0.357042\pi\)
0.434170 + 0.900831i \(0.357042\pi\)
\(734\) −0.236396 −0.00872552
\(735\) 0 0
\(736\) −9.72781 −0.358572
\(737\) −21.5578 −0.794093
\(738\) 19.1926 0.706490
\(739\) −22.5267 −0.828658 −0.414329 0.910127i \(-0.635984\pi\)
−0.414329 + 0.910127i \(0.635984\pi\)
\(740\) −4.31151 −0.158494
\(741\) −0.201734 −0.00741089
\(742\) 0 0
\(743\) −13.9795 −0.512857 −0.256429 0.966563i \(-0.582546\pi\)
−0.256429 + 0.966563i \(0.582546\pi\)
\(744\) −3.69055 −0.135302
\(745\) −11.0074 −0.403280
\(746\) 16.5149 0.604652
\(747\) 6.01952 0.220243
\(748\) 10.6653 0.389963
\(749\) 0 0
\(750\) 1.48925 0.0543798
\(751\) 2.54591 0.0929017 0.0464509 0.998921i \(-0.485209\pi\)
0.0464509 + 0.998921i \(0.485209\pi\)
\(752\) 12.4377 0.453555
\(753\) −3.66643 −0.133612
\(754\) −4.29692 −0.156485
\(755\) −16.3692 −0.595738
\(756\) 0 0
\(757\) 19.6921 0.715721 0.357861 0.933775i \(-0.383506\pi\)
0.357861 + 0.933775i \(0.383506\pi\)
\(758\) −20.9288 −0.760168
\(759\) 0.661732 0.0240193
\(760\) −3.13833 −0.113839
\(761\) −39.0905 −1.41703 −0.708515 0.705696i \(-0.750635\pi\)
−0.708515 + 0.705696i \(0.750635\pi\)
\(762\) −2.44043 −0.0884075
\(763\) 0 0
\(764\) −4.74120 −0.171531
\(765\) 14.3469 0.518714
\(766\) −21.5341 −0.778059
\(767\) 2.16385 0.0781320
\(768\) 1.72107 0.0621039
\(769\) −21.7389 −0.783926 −0.391963 0.919981i \(-0.628204\pi\)
−0.391963 + 0.919981i \(0.628204\pi\)
\(770\) 0 0
\(771\) −5.31859 −0.191544
\(772\) −33.3537 −1.20043
\(773\) 4.12806 0.148476 0.0742381 0.997241i \(-0.476348\pi\)
0.0742381 + 0.997241i \(0.476348\pi\)
\(774\) −6.14609 −0.220917
\(775\) −24.2447 −0.870895
\(776\) −34.2483 −1.22944
\(777\) 0 0
\(778\) 16.6071 0.595395
\(779\) 9.74199 0.349043
\(780\) −0.437016 −0.0156477
\(781\) 17.2500 0.617254
\(782\) −3.96905 −0.141933
\(783\) 7.98857 0.285488
\(784\) 0 0
\(785\) 25.0145 0.892807
\(786\) 1.35718 0.0484089
\(787\) 38.7863 1.38258 0.691291 0.722576i \(-0.257042\pi\)
0.691291 + 0.722576i \(0.257042\pi\)
\(788\) 4.27182 0.152177
\(789\) −5.70037 −0.202938
\(790\) −4.81557 −0.171330
\(791\) 0 0
\(792\) 13.2022 0.469118
\(793\) 12.2002 0.433242
\(794\) 3.47381 0.123281
\(795\) 0.0986276 0.00349796
\(796\) 34.0047 1.20526
\(797\) 19.9651 0.707201 0.353601 0.935397i \(-0.384957\pi\)
0.353601 + 0.935397i \(0.384957\pi\)
\(798\) 0 0
\(799\) 27.6896 0.979587
\(800\) 18.0699 0.638867
\(801\) 20.4067 0.721035
\(802\) 16.1641 0.570775
\(803\) 13.5981 0.479866
\(804\) 3.63550 0.128214
\(805\) 0 0
\(806\) −5.04967 −0.177867
\(807\) −1.20904 −0.0425602
\(808\) 6.59020 0.231842
\(809\) −30.2441 −1.06332 −0.531662 0.846956i \(-0.678432\pi\)
−0.531662 + 0.846956i \(0.678432\pi\)
\(810\) 7.69633 0.270421
\(811\) 23.4099 0.822034 0.411017 0.911628i \(-0.365174\pi\)
0.411017 + 0.911628i \(0.365174\pi\)
\(812\) 0 0
\(813\) 0.890538 0.0312325
\(814\) −2.52581 −0.0885295
\(815\) −3.41740 −0.119706
\(816\) −1.16976 −0.0409498
\(817\) −3.11970 −0.109144
\(818\) −6.70768 −0.234528
\(819\) 0 0
\(820\) 21.1041 0.736986
\(821\) −46.5536 −1.62473 −0.812366 0.583147i \(-0.801821\pi\)
−0.812366 + 0.583147i \(0.801821\pi\)
\(822\) 0.880838 0.0307228
\(823\) −5.68031 −0.198003 −0.0990017 0.995087i \(-0.531565\pi\)
−0.0990017 + 0.995087i \(0.531565\pi\)
\(824\) 14.1700 0.493636
\(825\) −1.22920 −0.0427953
\(826\) 0 0
\(827\) 48.3198 1.68024 0.840122 0.542398i \(-0.182483\pi\)
0.840122 + 0.542398i \(0.182483\pi\)
\(828\) 7.87388 0.273636
\(829\) −11.8545 −0.411722 −0.205861 0.978581i \(-0.566000\pi\)
−0.205861 + 0.978581i \(0.566000\pi\)
\(830\) −1.81595 −0.0630327
\(831\) −0.221239 −0.00767468
\(832\) 0.559715 0.0194046
\(833\) 0 0
\(834\) −1.85738 −0.0643157
\(835\) −24.3563 −0.842884
\(836\) 2.94646 0.101905
\(837\) 9.38802 0.324497
\(838\) −20.1122 −0.694765
\(839\) −2.69386 −0.0930025 −0.0465013 0.998918i \(-0.514807\pi\)
−0.0465013 + 0.998918i \(0.514807\pi\)
\(840\) 0 0
\(841\) 13.8809 0.478652
\(842\) −17.0453 −0.587420
\(843\) −2.96712 −0.102193
\(844\) −28.8593 −0.993380
\(845\) −1.35996 −0.0467841
\(846\) 15.0706 0.518137
\(847\) 0 0
\(848\) 0.567394 0.0194844
\(849\) −5.01602 −0.172149
\(850\) 7.37271 0.252882
\(851\) −3.42612 −0.117446
\(852\) −2.90903 −0.0996617
\(853\) −2.48965 −0.0852440 −0.0426220 0.999091i \(-0.513571\pi\)
−0.0426220 + 0.999091i \(0.513571\pi\)
\(854\) 0 0
\(855\) 3.96355 0.135551
\(856\) 40.5677 1.38658
\(857\) 46.2110 1.57854 0.789270 0.614047i \(-0.210459\pi\)
0.789270 + 0.614047i \(0.210459\pi\)
\(858\) −0.256017 −0.00874027
\(859\) −10.9170 −0.372482 −0.186241 0.982504i \(-0.559631\pi\)
−0.186241 + 0.982504i \(0.559631\pi\)
\(860\) −6.75820 −0.230453
\(861\) 0 0
\(862\) −9.76517 −0.332603
\(863\) −29.2330 −0.995104 −0.497552 0.867434i \(-0.665768\pi\)
−0.497552 + 0.867434i \(0.665768\pi\)
\(864\) −6.99701 −0.238043
\(865\) −8.86670 −0.301477
\(866\) 26.3045 0.893865
\(867\) 0.876603 0.0297710
\(868\) 0 0
\(869\) 10.2827 0.348817
\(870\) −1.19651 −0.0405654
\(871\) 11.3134 0.383340
\(872\) 24.7716 0.838873
\(873\) 43.2538 1.46392
\(874\) −1.09651 −0.0370901
\(875\) 0 0
\(876\) −2.29318 −0.0774792
\(877\) −7.38110 −0.249242 −0.124621 0.992204i \(-0.539772\pi\)
−0.124621 + 0.992204i \(0.539772\pi\)
\(878\) 1.97015 0.0664892
\(879\) −0.237730 −0.00801843
\(880\) 4.15129 0.139940
\(881\) 16.4854 0.555407 0.277703 0.960667i \(-0.410427\pi\)
0.277703 + 0.960667i \(0.410427\pi\)
\(882\) 0 0
\(883\) 12.4427 0.418732 0.209366 0.977837i \(-0.432860\pi\)
0.209366 + 0.977837i \(0.432860\pi\)
\(884\) −5.59709 −0.188250
\(885\) 0.602538 0.0202541
\(886\) −0.942055 −0.0316490
\(887\) −8.11331 −0.272418 −0.136209 0.990680i \(-0.543492\pi\)
−0.136209 + 0.990680i \(0.543492\pi\)
\(888\) 0.968763 0.0325095
\(889\) 0 0
\(890\) −6.15624 −0.206358
\(891\) −16.4340 −0.550559
\(892\) −34.8751 −1.16771
\(893\) 7.64968 0.255987
\(894\) 1.08746 0.0363702
\(895\) 36.1769 1.20926
\(896\) 0 0
\(897\) −0.347272 −0.0115951
\(898\) −12.8818 −0.429870
\(899\) 50.3929 1.68070
\(900\) −14.6261 −0.487537
\(901\) 1.26317 0.0420824
\(902\) 12.3634 0.411655
\(903\) 0 0
\(904\) −5.49633 −0.182805
\(905\) 33.5098 1.11390
\(906\) 1.61718 0.0537272
\(907\) −18.4796 −0.613605 −0.306803 0.951773i \(-0.599259\pi\)
−0.306803 + 0.951773i \(0.599259\pi\)
\(908\) −41.9623 −1.39257
\(909\) −8.32308 −0.276059
\(910\) 0 0
\(911\) 15.8660 0.525663 0.262831 0.964842i \(-0.415344\pi\)
0.262831 + 0.964842i \(0.415344\pi\)
\(912\) −0.323165 −0.0107010
\(913\) 3.87761 0.128330
\(914\) −6.98315 −0.230982
\(915\) 3.39723 0.112309
\(916\) 26.2106 0.866022
\(917\) 0 0
\(918\) −2.85486 −0.0942244
\(919\) 43.4568 1.43351 0.716753 0.697327i \(-0.245627\pi\)
0.716753 + 0.697327i \(0.245627\pi\)
\(920\) −5.40244 −0.178113
\(921\) −5.19523 −0.171189
\(922\) 25.3241 0.834004
\(923\) −9.05268 −0.297973
\(924\) 0 0
\(925\) 6.36418 0.209253
\(926\) 18.3727 0.603763
\(927\) −17.8960 −0.587782
\(928\) −37.5585 −1.23292
\(929\) −25.6312 −0.840931 −0.420465 0.907309i \(-0.638133\pi\)
−0.420465 + 0.907309i \(0.638133\pi\)
\(930\) −1.40611 −0.0461083
\(931\) 0 0
\(932\) 21.7798 0.713420
\(933\) −5.00920 −0.163994
\(934\) 16.9426 0.554379
\(935\) 9.24189 0.302242
\(936\) −6.92840 −0.226462
\(937\) 23.9639 0.782867 0.391433 0.920206i \(-0.371979\pi\)
0.391433 + 0.920206i \(0.371979\pi\)
\(938\) 0 0
\(939\) −1.35478 −0.0442116
\(940\) 16.5715 0.540502
\(941\) −42.2934 −1.37872 −0.689362 0.724417i \(-0.742109\pi\)
−0.689362 + 0.724417i \(0.742109\pi\)
\(942\) −2.47128 −0.0805186
\(943\) 16.7702 0.546113
\(944\) 3.46634 0.112820
\(945\) 0 0
\(946\) −3.95914 −0.128723
\(947\) −49.5957 −1.61165 −0.805823 0.592157i \(-0.798277\pi\)
−0.805823 + 0.592157i \(0.798277\pi\)
\(948\) −1.73407 −0.0563199
\(949\) −7.13619 −0.231650
\(950\) 2.03683 0.0660833
\(951\) −1.67857 −0.0544315
\(952\) 0 0
\(953\) 18.7156 0.606259 0.303129 0.952949i \(-0.401969\pi\)
0.303129 + 0.952949i \(0.401969\pi\)
\(954\) 0.687505 0.0222588
\(955\) −4.10843 −0.132946
\(956\) −0.731107 −0.0236457
\(957\) 2.55491 0.0825884
\(958\) 25.8583 0.835443
\(959\) 0 0
\(960\) 0.155857 0.00503025
\(961\) 28.2208 0.910348
\(962\) 1.32553 0.0427367
\(963\) −51.2349 −1.65102
\(964\) −8.93494 −0.287775
\(965\) −28.9022 −0.930395
\(966\) 0 0
\(967\) −34.3284 −1.10393 −0.551964 0.833868i \(-0.686121\pi\)
−0.551964 + 0.833868i \(0.686121\pi\)
\(968\) −17.2597 −0.554748
\(969\) −0.719451 −0.0231121
\(970\) −13.0487 −0.418969
\(971\) 56.7118 1.81997 0.909984 0.414642i \(-0.136093\pi\)
0.909984 + 0.414642i \(0.136093\pi\)
\(972\) 8.51520 0.273125
\(973\) 0 0
\(974\) 22.3732 0.716884
\(975\) 0.645076 0.0206590
\(976\) 19.5439 0.625585
\(977\) −39.8809 −1.27590 −0.637951 0.770077i \(-0.720218\pi\)
−0.637951 + 0.770077i \(0.720218\pi\)
\(978\) 0.337618 0.0107958
\(979\) 13.1454 0.420130
\(980\) 0 0
\(981\) −31.2853 −0.998862
\(982\) 6.36770 0.203201
\(983\) 41.2678 1.31624 0.658120 0.752913i \(-0.271352\pi\)
0.658120 + 0.752913i \(0.271352\pi\)
\(984\) −4.74192 −0.151167
\(985\) 3.70169 0.117946
\(986\) −15.3243 −0.488024
\(987\) 0 0
\(988\) −1.54628 −0.0491937
\(989\) −5.37036 −0.170767
\(990\) 5.03007 0.159866
\(991\) −26.1784 −0.831583 −0.415792 0.909460i \(-0.636495\pi\)
−0.415792 + 0.909460i \(0.636495\pi\)
\(992\) −44.1380 −1.40138
\(993\) 0.148388 0.00470894
\(994\) 0 0
\(995\) 29.4663 0.934145
\(996\) −0.653918 −0.0207202
\(997\) 8.62009 0.273001 0.136501 0.990640i \(-0.456414\pi\)
0.136501 + 0.990640i \(0.456414\pi\)
\(998\) −11.8417 −0.374842
\(999\) −2.46434 −0.0779681
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 637.2.a.n.1.3 yes 6
3.2 odd 2 5733.2.a.br.1.4 6
7.2 even 3 637.2.e.n.508.4 12
7.3 odd 6 637.2.e.o.79.4 12
7.4 even 3 637.2.e.n.79.4 12
7.5 odd 6 637.2.e.o.508.4 12
7.6 odd 2 637.2.a.m.1.3 6
13.12 even 2 8281.2.a.cd.1.4 6
21.20 even 2 5733.2.a.bu.1.4 6
91.90 odd 2 8281.2.a.cc.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.m.1.3 6 7.6 odd 2
637.2.a.n.1.3 yes 6 1.1 even 1 trivial
637.2.e.n.79.4 12 7.4 even 3
637.2.e.n.508.4 12 7.2 even 3
637.2.e.o.79.4 12 7.3 odd 6
637.2.e.o.508.4 12 7.5 odd 6
5733.2.a.br.1.4 6 3.2 odd 2
5733.2.a.bu.1.4 6 21.20 even 2
8281.2.a.cc.1.4 6 91.90 odd 2
8281.2.a.cd.1.4 6 13.12 even 2