Properties

Label 637.2.a.m.1.3
Level $637$
Weight $2$
Character 637.1
Self dual yes
Analytic conductor $5.086$
Analytic rank $1$
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: \(1\)
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 \(-0.481175\pi\)
0.0591072 + 0.998252i \(0.481175\pi\)
\(4\) −1.56942 −0.784711
\(5\) 1.35996 0.608194 0.304097 0.952641i \(-0.401645\pi\)
0.304097 + 0.952641i \(0.401645\pi\)
\(6\) −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 \(-0.642362\pi\)
−0.432482 + 0.901643i \(0.642362\pi\)
\(18\) 1.94104 0.457508
\(19\) −0.985255 −0.226033 −0.113016 0.993593i \(-0.536051\pi\)
−0.113016 + 0.993593i \(0.536051\pi\)
\(20\) −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 \(-0.742863\pi\)
−0.691077 + 0.722781i \(0.742863\pi\)
\(32\) −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 \(-0.780797\pi\)
−0.772107 + 0.635493i \(0.780797\pi\)
\(42\) 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 \(-0.691610\pi\)
−0.566260 + 0.824227i \(0.691610\pi\)
\(48\) 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 \(-0.544985\pi\)
−0.140854 + 0.990030i \(0.544985\pi\)
\(60\) −0.437016 −0.0564186
\(61\) −12.2002 −1.56207 −0.781037 0.624484i \(-0.785309\pi\)
−0.781037 + 0.624484i \(0.785309\pi\)
\(62\) 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 \(-0.362866\pi\)
0.417614 + 0.908625i \(0.362866\pi\)
\(74\) 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 \(-0.464376\pi\)
0.111682 + 0.993744i \(0.464376\pi\)
\(84\) 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 \(-0.380855\pi\)
0.365627 + 0.930761i \(0.380855\pi\)
\(90\) 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 \(-0.233719\pi\)
0.742334 + 0.670030i \(0.233719\pi\)
\(98\) 0 0
\(99\) 5.63665 0.566505
\(100\) 4.94447 0.494447
\(101\) −2.81368 −0.279972 −0.139986 0.990153i \(-0.544706\pi\)
−0.139986 + 0.990153i \(0.544706\pi\)
\(102\) 0.479158 0.0474437
\(103\) −6.04988 −0.596112 −0.298056 0.954548i \(-0.596338\pi\)
−0.298056 + 0.954548i \(0.596338\pi\)
\(104\) −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 \(-0.645476\pi\)
−0.441280 + 0.897369i \(0.645476\pi\)
\(132\) 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 \(-0.300591\pi\)
0.586281 + 0.810108i \(0.300591\pi\)
\(140\) 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 \(-0.237660\pi\)
0.733982 + 0.679169i \(0.237660\pi\)
\(158\) 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 \(-0.743685\pi\)
−0.692941 + 0.720995i \(0.743685\pi\)
\(168\) 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 \(-0.579723\pi\)
−0.247846 + 0.968799i \(0.579723\pi\)
\(174\) −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 \(-0.131601\pi\)
0.915746 + 0.401758i \(0.131601\pi\)
\(182\) 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 \(-0.221269\pi\)
0.767967 + 0.640490i \(0.221269\pi\)
\(200\) −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 \(-0.767090\pi\)
−0.744035 + 0.668141i \(0.767090\pi\)
\(224\) 0 0
\(225\) 9.31943 0.621295
\(226\) 1.53984 0.102428
\(227\) −26.7374 −1.77463 −0.887313 0.461167i \(-0.847431\pi\)
−0.887313 + 0.461167i \(0.847431\pi\)
\(228\) 0.316606 0.0209678
\(229\) 16.7008 1.10362 0.551809 0.833970i \(-0.313938\pi\)
0.551809 + 0.833970i \(0.313938\pi\)
\(230\) −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 \(-0.558699\pi\)
−0.183364 + 0.983045i \(0.558699\pi\)
\(242\) 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 \(-0.691172\pi\)
−0.565126 + 0.825005i \(0.691172\pi\)
\(252\) 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 \(-0.800618\pi\)
−0.810156 + 0.586214i \(0.800618\pi\)
\(258\) 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 \(-0.557614\pi\)
−0.180013 + 0.983664i \(0.557614\pi\)
\(270\) 1.08865 0.0662533
\(271\) 4.34932 0.264203 0.132101 0.991236i \(-0.457828\pi\)
0.132101 + 0.991236i \(0.457828\pi\)
\(272\) −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 \(-0.759607\pi\)
−0.728123 + 0.685446i \(0.759607\pi\)
\(284\) 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 \(-0.510797\pi\)
−0.0339148 + 0.999425i \(0.510797\pi\)
\(294\) 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 \(-0.757726\pi\)
−0.724060 + 0.689737i \(0.757726\pi\)
\(308\) 0 0
\(309\) −1.23873 −0.0704690
\(310\) 6.86736 0.390040
\(311\) −24.4645 −1.38726 −0.693628 0.720333i \(-0.743989\pi\)
−0.693628 + 0.720333i \(0.743989\pi\)
\(312\) −0.479573 −0.0271505
\(313\) −6.61665 −0.373995 −0.186997 0.982360i \(-0.559876\pi\)
−0.186997 + 0.982360i \(0.559876\pi\)
\(314\) −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 \(-0.651703\pi\)
−0.458750 + 0.888565i \(0.651703\pi\)
\(350\) 0 0
\(351\) 1.21994 0.0651154
\(352\) 10.9292 0.582527
\(353\) 18.9101 1.00648 0.503242 0.864146i \(-0.332140\pi\)
0.503242 + 0.864146i \(0.332140\pi\)
\(354\) 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 \(-0.502993\pi\)
−0.00940266 + 0.999956i \(0.502993\pi\)
\(368\) 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 \(-0.816532\pi\)
−0.838440 + 0.544994i \(0.816532\pi\)
\(384\) 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 \(-0.457588\pi\)
0.132848 + 0.991136i \(0.457588\pi\)
\(398\) −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 \(-0.581328\pi\)
−0.252729 + 0.967537i \(0.581328\pi\)
\(410\) 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 \(-0.769313\pi\)
−0.748682 + 0.662929i \(0.769313\pi\)
\(420\) 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 \(-0.0865837\pi\)
0.963233 + 0.268669i \(0.0865837\pi\)
\(434\) 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 \(-0.477174\pi\)
0.0716491 + 0.997430i \(0.477174\pi\)
\(440\) −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 \(-0.144493\pi\)
0.898727 + 0.438509i \(0.144493\pi\)
\(462\) 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 \(-0.296200\pi\)
0.597401 + 0.801943i \(0.296200\pi\)
\(468\) −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 \(-0.143364\pi\)
0.900277 + 0.435317i \(0.143364\pi\)
\(480\) −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 \(-0.619846\pi\)
−0.367676 + 0.929954i \(0.619846\pi\)
\(504\) 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 \(-0.360112\pi\)
0.425460 + 0.904977i \(0.360112\pi\)
\(510\) 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 \(-0.438535\pi\)
0.191899 + 0.981415i \(0.438535\pi\)
\(522\) 12.7106 0.556329
\(523\) 11.7406 0.513381 0.256690 0.966494i \(-0.417368\pi\)
0.256690 + 0.966494i \(0.417368\pi\)
\(524\) 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 \(-0.565906\pi\)
−0.205575 + 0.978641i \(0.565906\pi\)
\(564\) 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 \(-0.468021\pi\)
0.100297 + 0.994957i \(0.468021\pi\)
\(578\) 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 \(-0.772735\pi\)
−0.755765 + 0.654843i \(0.772735\pi\)
\(588\) 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 \(-0.374503\pi\)
0.384125 + 0.923281i \(0.374503\pi\)
\(594\) −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 \(-0.618114\pi\)
−0.362609 + 0.931941i \(0.618114\pi\)
\(602\) 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 \(-0.676964\pi\)
−0.527750 + 0.849400i \(0.676964\pi\)
\(608\) 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 \(-0.584591\pi\)
−0.262634 + 0.964895i \(0.584591\pi\)
\(620\) 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 \(-0.466380\pi\)
0.105423 + 0.994427i \(0.466380\pi\)
\(644\) 0 0
\(645\) −0.881702 −0.0347170
\(646\) −2.30566 −0.0907151
\(647\) 29.7346 1.16899 0.584493 0.811398i \(-0.301293\pi\)
0.584493 + 0.811398i \(0.301293\pi\)
\(648\) 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 \(-0.424538\pi\)
0.234857 + 0.972030i \(0.424538\pi\)
\(662\) 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 \(-0.538361\pi\)
−0.120224 + 0.992747i \(0.538361\pi\)
\(678\) 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 \(-0.476926\pi\)
0.0724260 + 0.997374i \(0.476926\pi\)
\(692\) 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 \(-0.291299\pi\)
0.609678 + 0.792649i \(0.291299\pi\)
\(720\) −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 \(-0.405526\pi\)
0.292460 + 0.956278i \(0.405526\pi\)
\(728\) 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 \(-0.642958\pi\)
−0.434170 + 0.900831i \(0.642958\pi\)
\(734\) 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 \(-0.249365\pi\)
0.708515 + 0.705696i \(0.249365\pi\)
\(762\) 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 \(-0.371796\pi\)
0.391963 + 0.919981i \(0.371796\pi\)
\(770\) 0 0
\(771\) −5.31859 −0.191544
\(772\) −33.3537 −1.20043
\(773\) −4.12806 −0.148476 −0.0742381 0.997241i \(-0.523652\pi\)
−0.0742381 + 0.997241i \(0.523652\pi\)
\(774\) −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 \(-0.742958\pi\)
−0.691291 + 0.722576i \(0.742958\pi\)
\(788\) 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 \(-0.615043\pi\)
−0.353601 + 0.935397i \(0.615043\pi\)
\(798\) 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 \(-0.634826\pi\)
−0.411017 + 0.911628i \(0.634826\pi\)
\(812\) 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 \(-0.434000\pi\)
0.205861 + 0.978581i \(0.434000\pi\)
\(830\) −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 \(-0.485193\pi\)
0.0465013 + 0.998918i \(0.485193\pi\)
\(840\) 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 \(-0.486429\pi\)
0.0426220 + 0.999091i \(0.486429\pi\)
\(854\) 0 0
\(855\) 3.96355 0.135551
\(856\) 40.5677 1.38658
\(857\) −46.2110 −1.57854 −0.789270 0.614047i \(-0.789541\pi\)
−0.789270 + 0.614047i \(0.789541\pi\)
\(858\) −0.256017 −0.00874027
\(859\) 10.9170 0.372482 0.186241 0.982504i \(-0.440369\pi\)
0.186241 + 0.982504i \(0.440369\pi\)
\(860\) 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 \(-0.589573\pi\)
−0.277703 + 0.960667i \(0.589573\pi\)
\(882\) 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 \(-0.456508\pi\)
0.136209 + 0.990680i \(0.456508\pi\)
\(888\) −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 \(-0.361867\pi\)
0.420465 + 0.907309i \(0.361867\pi\)
\(930\) 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 \(-0.628021\pi\)
−0.391433 + 0.920206i \(0.628021\pi\)
\(938\) 0 0
\(939\) −1.35478 −0.0442116
\(940\) 16.5715 0.540502
\(941\) 42.2934 1.37872 0.689362 0.724417i \(-0.257891\pi\)
0.689362 + 0.724417i \(0.257891\pi\)
\(942\) −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 \(-0.863907\pi\)
−0.909984 + 0.414642i \(0.863907\pi\)
\(972\) −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 \(-0.728648\pi\)
−0.658120 + 0.752913i \(0.728648\pi\)
\(984\) −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 \(-0.543586\pi\)
−0.136501 + 0.990640i \(0.543586\pi\)
\(998\) −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.m.1.3 6
3.2 odd 2 5733.2.a.bu.1.4 6
7.2 even 3 637.2.e.o.508.4 12
7.3 odd 6 637.2.e.n.79.4 12
7.4 even 3 637.2.e.o.79.4 12
7.5 odd 6 637.2.e.n.508.4 12
7.6 odd 2 637.2.a.n.1.3 yes 6
13.12 even 2 8281.2.a.cc.1.4 6
21.20 even 2 5733.2.a.br.1.4 6
91.90 odd 2 8281.2.a.cd.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.m.1.3 6 1.1 even 1 trivial
637.2.a.n.1.3 yes 6 7.6 odd 2
637.2.e.n.79.4 12 7.3 odd 6
637.2.e.n.508.4 12 7.5 odd 6
637.2.e.o.79.4 12 7.4 even 3
637.2.e.o.508.4 12 7.2 even 3
5733.2.a.br.1.4 6 21.20 even 2
5733.2.a.bu.1.4 6 3.2 odd 2
8281.2.a.cc.1.4 6 13.12 even 2
8281.2.a.cd.1.4 6 91.90 odd 2