Properties

Label 637.2.a.m.1.4
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: \(x^{6} - 2 x^{5} - 5 x^{4} + 8 x^{3} + 7 x^{2} - 6 x - 1\)
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 1.4
Root \(1.90903\) of defining polynomial
Character \(\chi\) \(=\) 637.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.264627 q^{2} -2.90903 q^{3} -1.92997 q^{4} +1.43515 q^{5} -0.769807 q^{6} -1.03998 q^{8} +5.46247 q^{9} +O(q^{10})\) \(q+0.264627 q^{2} -2.90903 q^{3} -1.92997 q^{4} +1.43515 q^{5} -0.769807 q^{6} -1.03998 q^{8} +5.46247 q^{9} +0.379780 q^{10} +5.50474 q^{11} +5.61435 q^{12} -1.00000 q^{13} -4.17491 q^{15} +3.58474 q^{16} -4.83072 q^{17} +1.44552 q^{18} -2.82036 q^{19} -2.76981 q^{20} +1.45670 q^{22} -5.99956 q^{23} +3.02532 q^{24} -2.94033 q^{25} -0.264627 q^{26} -7.16341 q^{27} +1.04188 q^{29} -1.10479 q^{30} -9.20895 q^{31} +3.02857 q^{32} -16.0135 q^{33} -1.27834 q^{34} -10.5424 q^{36} +0.612497 q^{37} -0.746342 q^{38} +2.90903 q^{39} -1.49252 q^{40} +10.6196 q^{41} -8.43685 q^{43} -10.6240 q^{44} +7.83949 q^{45} -1.58764 q^{46} -2.40922 q^{47} -10.4281 q^{48} -0.778091 q^{50} +14.0527 q^{51} +1.92997 q^{52} -1.82959 q^{53} -1.89563 q^{54} +7.90015 q^{55} +8.20452 q^{57} +0.275709 q^{58} +0.870914 q^{59} +8.05746 q^{60} +3.33253 q^{61} -2.43693 q^{62} -6.36804 q^{64} -1.43515 q^{65} -4.23759 q^{66} -6.62741 q^{67} +9.32316 q^{68} +17.4529 q^{69} -6.85856 q^{71} -5.68083 q^{72} -3.14147 q^{73} +0.162083 q^{74} +8.55353 q^{75} +5.44322 q^{76} +0.769807 q^{78} -17.5723 q^{79} +5.14465 q^{80} +4.45118 q^{81} +2.81022 q^{82} -11.4525 q^{83} -6.93283 q^{85} -2.23261 q^{86} -3.03086 q^{87} -5.72479 q^{88} +0.995318 q^{89} +2.07454 q^{90} +11.5790 q^{92} +26.7891 q^{93} -0.637545 q^{94} -4.04765 q^{95} -8.81020 q^{96} -13.5090 q^{97} +30.0695 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{3} + 4 q^{4} - 6 q^{5} - 4 q^{6} + 6 q^{9} + O(q^{10}) \) \( 6 q - 8 q^{3} + 4 q^{4} - 6 q^{5} - 4 q^{6} + 6 q^{9} - 4 q^{10} + 4 q^{11} + 4 q^{12} - 6 q^{13} + 12 q^{15} - 16 q^{17} - 4 q^{18} - 2 q^{19} - 16 q^{20} - 12 q^{22} - 6 q^{23} - 12 q^{24} - 4 q^{25} - 20 q^{27} - 6 q^{29} - 6 q^{31} - 20 q^{32} - 4 q^{33} - 24 q^{36} - 8 q^{38} + 8 q^{39} - 4 q^{40} + 8 q^{41} + 2 q^{43} - 4 q^{44} - 14 q^{45} + 8 q^{46} - 30 q^{47} + 8 q^{48} + 8 q^{50} - 4 q^{51} - 4 q^{52} - 14 q^{53} + 48 q^{54} + 8 q^{55} + 4 q^{57} - 8 q^{58} - 24 q^{59} + 12 q^{60} - 28 q^{62} - 20 q^{64} + 6 q^{65} + 4 q^{66} + 16 q^{67} - 28 q^{68} + 20 q^{69} + 8 q^{71} + 28 q^{72} + 6 q^{73} - 12 q^{74} - 12 q^{75} + 16 q^{76} + 4 q^{78} - 22 q^{79} + 28 q^{80} + 46 q^{81} + 40 q^{82} - 50 q^{83} - 8 q^{85} - 16 q^{86} + 16 q^{87} - 44 q^{88} - 26 q^{89} + 40 q^{90} + 20 q^{92} + 16 q^{93} + 32 q^{94} - 6 q^{95} + 20 q^{96} + 14 q^{97} + 12 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.264627 0.187119 0.0935596 0.995614i \(-0.470175\pi\)
0.0935596 + 0.995614i \(0.470175\pi\)
\(3\) −2.90903 −1.67953 −0.839765 0.542949i \(-0.817308\pi\)
−0.839765 + 0.542949i \(0.817308\pi\)
\(4\) −1.92997 −0.964986
\(5\) 1.43515 0.641820 0.320910 0.947110i \(-0.396011\pi\)
0.320910 + 0.947110i \(0.396011\pi\)
\(6\) −0.769807 −0.314273
\(7\) 0 0
\(8\) −1.03998 −0.367687
\(9\) 5.46247 1.82082
\(10\) 0.379780 0.120097
\(11\) 5.50474 1.65974 0.829871 0.557955i \(-0.188414\pi\)
0.829871 + 0.557955i \(0.188414\pi\)
\(12\) 5.61435 1.62072
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −4.17491 −1.07796
\(16\) 3.58474 0.896185
\(17\) −4.83072 −1.17162 −0.585811 0.810448i \(-0.699224\pi\)
−0.585811 + 0.810448i \(0.699224\pi\)
\(18\) 1.44552 0.340711
\(19\) −2.82036 −0.647035 −0.323518 0.946222i \(-0.604865\pi\)
−0.323518 + 0.946222i \(0.604865\pi\)
\(20\) −2.76981 −0.619348
\(21\) 0 0
\(22\) 1.45670 0.310570
\(23\) −5.99956 −1.25100 −0.625498 0.780226i \(-0.715104\pi\)
−0.625498 + 0.780226i \(0.715104\pi\)
\(24\) 3.02532 0.617541
\(25\) −2.94033 −0.588067
\(26\) −0.264627 −0.0518975
\(27\) −7.16341 −1.37860
\(28\) 0 0
\(29\) 1.04188 0.193472 0.0967361 0.995310i \(-0.469160\pi\)
0.0967361 + 0.995310i \(0.469160\pi\)
\(30\) −1.10479 −0.201706
\(31\) −9.20895 −1.65398 −0.826988 0.562219i \(-0.809948\pi\)
−0.826988 + 0.562219i \(0.809948\pi\)
\(32\) 3.02857 0.535380
\(33\) −16.0135 −2.78759
\(34\) −1.27834 −0.219233
\(35\) 0 0
\(36\) −10.5424 −1.75707
\(37\) 0.612497 0.100694 0.0503470 0.998732i \(-0.483967\pi\)
0.0503470 + 0.998732i \(0.483967\pi\)
\(38\) −0.746342 −0.121073
\(39\) 2.90903 0.465818
\(40\) −1.49252 −0.235989
\(41\) 10.6196 1.65850 0.829249 0.558879i \(-0.188768\pi\)
0.829249 + 0.558879i \(0.188768\pi\)
\(42\) 0 0
\(43\) −8.43685 −1.28661 −0.643304 0.765611i \(-0.722437\pi\)
−0.643304 + 0.765611i \(0.722437\pi\)
\(44\) −10.6240 −1.60163
\(45\) 7.83949 1.16864
\(46\) −1.58764 −0.234085
\(47\) −2.40922 −0.351422 −0.175711 0.984442i \(-0.556222\pi\)
−0.175711 + 0.984442i \(0.556222\pi\)
\(48\) −10.4281 −1.50517
\(49\) 0 0
\(50\) −0.778091 −0.110039
\(51\) 14.0527 1.96778
\(52\) 1.92997 0.267639
\(53\) −1.82959 −0.251313 −0.125657 0.992074i \(-0.540104\pi\)
−0.125657 + 0.992074i \(0.540104\pi\)
\(54\) −1.89563 −0.257962
\(55\) 7.90015 1.06526
\(56\) 0 0
\(57\) 8.20452 1.08672
\(58\) 0.275709 0.0362024
\(59\) 0.870914 0.113383 0.0566917 0.998392i \(-0.481945\pi\)
0.0566917 + 0.998392i \(0.481945\pi\)
\(60\) 8.05746 1.04021
\(61\) 3.33253 0.426686 0.213343 0.976977i \(-0.431565\pi\)
0.213343 + 0.976977i \(0.431565\pi\)
\(62\) −2.43693 −0.309491
\(63\) 0 0
\(64\) −6.36804 −0.796005
\(65\) −1.43515 −0.178009
\(66\) −4.23759 −0.521611
\(67\) −6.62741 −0.809667 −0.404833 0.914390i \(-0.632671\pi\)
−0.404833 + 0.914390i \(0.632671\pi\)
\(68\) 9.32316 1.13060
\(69\) 17.4529 2.10108
\(70\) 0 0
\(71\) −6.85856 −0.813961 −0.406980 0.913437i \(-0.633418\pi\)
−0.406980 + 0.913437i \(0.633418\pi\)
\(72\) −5.68083 −0.669493
\(73\) −3.14147 −0.367682 −0.183841 0.982956i \(-0.558853\pi\)
−0.183841 + 0.982956i \(0.558853\pi\)
\(74\) 0.162083 0.0188418
\(75\) 8.55353 0.987676
\(76\) 5.44322 0.624380
\(77\) 0 0
\(78\) 0.769807 0.0871635
\(79\) −17.5723 −1.97704 −0.988518 0.151101i \(-0.951718\pi\)
−0.988518 + 0.151101i \(0.951718\pi\)
\(80\) 5.14465 0.575190
\(81\) 4.45118 0.494576
\(82\) 2.81022 0.310337
\(83\) −11.4525 −1.25708 −0.628538 0.777779i \(-0.716346\pi\)
−0.628538 + 0.777779i \(0.716346\pi\)
\(84\) 0 0
\(85\) −6.93283 −0.751971
\(86\) −2.23261 −0.240749
\(87\) −3.03086 −0.324943
\(88\) −5.72479 −0.610265
\(89\) 0.995318 0.105503 0.0527517 0.998608i \(-0.483201\pi\)
0.0527517 + 0.998608i \(0.483201\pi\)
\(90\) 2.07454 0.218675
\(91\) 0 0
\(92\) 11.5790 1.20719
\(93\) 26.7891 2.77790
\(94\) −0.637545 −0.0657577
\(95\) −4.04765 −0.415280
\(96\) −8.81020 −0.899188
\(97\) −13.5090 −1.37163 −0.685817 0.727774i \(-0.740555\pi\)
−0.685817 + 0.727774i \(0.740555\pi\)
\(98\) 0 0
\(99\) 30.0695 3.02210
\(100\) 5.67477 0.567477
\(101\) 1.00807 0.100306 0.0501532 0.998742i \(-0.484029\pi\)
0.0501532 + 0.998742i \(0.484029\pi\)
\(102\) 3.71873 0.368209
\(103\) 12.7754 1.25880 0.629401 0.777081i \(-0.283300\pi\)
0.629401 + 0.777081i \(0.283300\pi\)
\(104\) 1.03998 0.101978
\(105\) 0 0
\(106\) −0.484157 −0.0470255
\(107\) −0.685495 −0.0662693 −0.0331347 0.999451i \(-0.510549\pi\)
−0.0331347 + 0.999451i \(0.510549\pi\)
\(108\) 13.8252 1.33033
\(109\) −2.90344 −0.278099 −0.139050 0.990285i \(-0.544405\pi\)
−0.139050 + 0.990285i \(0.544405\pi\)
\(110\) 2.09059 0.199330
\(111\) −1.78177 −0.169119
\(112\) 0 0
\(113\) 12.0315 1.13183 0.565915 0.824464i \(-0.308523\pi\)
0.565915 + 0.824464i \(0.308523\pi\)
\(114\) 2.17113 0.203345
\(115\) −8.61029 −0.802914
\(116\) −2.01080 −0.186698
\(117\) −5.46247 −0.505006
\(118\) 0.230467 0.0212162
\(119\) 0 0
\(120\) 4.34180 0.396350
\(121\) 19.3022 1.75474
\(122\) 0.881875 0.0798412
\(123\) −30.8927 −2.78550
\(124\) 17.7730 1.59606
\(125\) −11.3956 −1.01925
\(126\) 0 0
\(127\) 15.6659 1.39012 0.695062 0.718950i \(-0.255377\pi\)
0.695062 + 0.718950i \(0.255377\pi\)
\(128\) −7.74229 −0.684328
\(129\) 24.5431 2.16090
\(130\) −0.379780 −0.0333089
\(131\) −12.1273 −1.05957 −0.529784 0.848132i \(-0.677727\pi\)
−0.529784 + 0.848132i \(0.677727\pi\)
\(132\) 30.9056 2.68998
\(133\) 0 0
\(134\) −1.75379 −0.151504
\(135\) −10.2806 −0.884813
\(136\) 5.02383 0.430790
\(137\) 15.9375 1.36163 0.680815 0.732456i \(-0.261626\pi\)
0.680815 + 0.732456i \(0.261626\pi\)
\(138\) 4.61851 0.393153
\(139\) 6.64088 0.563272 0.281636 0.959521i \(-0.409123\pi\)
0.281636 + 0.959521i \(0.409123\pi\)
\(140\) 0 0
\(141\) 7.00851 0.590223
\(142\) −1.81496 −0.152308
\(143\) −5.50474 −0.460330
\(144\) 19.5815 1.63180
\(145\) 1.49526 0.124174
\(146\) −0.831317 −0.0688003
\(147\) 0 0
\(148\) −1.18210 −0.0971683
\(149\) −19.5502 −1.60162 −0.800809 0.598920i \(-0.795597\pi\)
−0.800809 + 0.598920i \(0.795597\pi\)
\(150\) 2.26349 0.184813
\(151\) 10.6880 0.869779 0.434890 0.900484i \(-0.356787\pi\)
0.434890 + 0.900484i \(0.356787\pi\)
\(152\) 2.93311 0.237906
\(153\) −26.3877 −2.13332
\(154\) 0 0
\(155\) −13.2163 −1.06156
\(156\) −5.61435 −0.449508
\(157\) 15.0734 1.20299 0.601496 0.798876i \(-0.294572\pi\)
0.601496 + 0.798876i \(0.294572\pi\)
\(158\) −4.65009 −0.369942
\(159\) 5.32233 0.422088
\(160\) 4.34646 0.343618
\(161\) 0 0
\(162\) 1.17790 0.0925447
\(163\) −23.8135 −1.86521 −0.932607 0.360894i \(-0.882472\pi\)
−0.932607 + 0.360894i \(0.882472\pi\)
\(164\) −20.4955 −1.60043
\(165\) −22.9818 −1.78913
\(166\) −3.03064 −0.235223
\(167\) 7.12371 0.551249 0.275625 0.961265i \(-0.411115\pi\)
0.275625 + 0.961265i \(0.411115\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −1.83461 −0.140708
\(171\) −15.4061 −1.17814
\(172\) 16.2829 1.24156
\(173\) −11.2367 −0.854309 −0.427155 0.904179i \(-0.640484\pi\)
−0.427155 + 0.904179i \(0.640484\pi\)
\(174\) −0.802047 −0.0608030
\(175\) 0 0
\(176\) 19.7331 1.48744
\(177\) −2.53352 −0.190431
\(178\) 0.263387 0.0197417
\(179\) −13.1945 −0.986204 −0.493102 0.869972i \(-0.664137\pi\)
−0.493102 + 0.869972i \(0.664137\pi\)
\(180\) −15.1300 −1.12772
\(181\) 13.7414 1.02139 0.510696 0.859761i \(-0.329388\pi\)
0.510696 + 0.859761i \(0.329388\pi\)
\(182\) 0 0
\(183\) −9.69443 −0.716633
\(184\) 6.23939 0.459974
\(185\) 0.879028 0.0646274
\(186\) 7.08912 0.519799
\(187\) −26.5919 −1.94459
\(188\) 4.64974 0.339117
\(189\) 0 0
\(190\) −1.07112 −0.0777069
\(191\) −16.3307 −1.18165 −0.590824 0.806800i \(-0.701197\pi\)
−0.590824 + 0.806800i \(0.701197\pi\)
\(192\) 18.5248 1.33692
\(193\) −14.0533 −1.01158 −0.505790 0.862656i \(-0.668799\pi\)
−0.505790 + 0.862656i \(0.668799\pi\)
\(194\) −3.57485 −0.256659
\(195\) 4.17491 0.298971
\(196\) 0 0
\(197\) −1.46898 −0.104660 −0.0523302 0.998630i \(-0.516665\pi\)
−0.0523302 + 0.998630i \(0.516665\pi\)
\(198\) 7.95719 0.565493
\(199\) 13.3772 0.948285 0.474142 0.880448i \(-0.342758\pi\)
0.474142 + 0.880448i \(0.342758\pi\)
\(200\) 3.05787 0.216224
\(201\) 19.2794 1.35986
\(202\) 0.266761 0.0187692
\(203\) 0 0
\(204\) −27.1214 −1.89888
\(205\) 15.2407 1.06446
\(206\) 3.38072 0.235546
\(207\) −32.7724 −2.27784
\(208\) −3.58474 −0.248557
\(209\) −15.5254 −1.07391
\(210\) 0 0
\(211\) 3.47044 0.238915 0.119457 0.992839i \(-0.461885\pi\)
0.119457 + 0.992839i \(0.461885\pi\)
\(212\) 3.53105 0.242514
\(213\) 19.9518 1.36707
\(214\) −0.181400 −0.0124003
\(215\) −12.1082 −0.825771
\(216\) 7.44977 0.506893
\(217\) 0 0
\(218\) −0.768328 −0.0520378
\(219\) 9.13865 0.617533
\(220\) −15.2471 −1.02796
\(221\) 4.83072 0.324950
\(222\) −0.471505 −0.0316453
\(223\) 9.91318 0.663836 0.331918 0.943308i \(-0.392304\pi\)
0.331918 + 0.943308i \(0.392304\pi\)
\(224\) 0 0
\(225\) −16.0615 −1.07077
\(226\) 3.18386 0.211787
\(227\) −12.0727 −0.801292 −0.400646 0.916233i \(-0.631214\pi\)
−0.400646 + 0.916233i \(0.631214\pi\)
\(228\) −15.8345 −1.04867
\(229\) −4.05171 −0.267745 −0.133872 0.990999i \(-0.542741\pi\)
−0.133872 + 0.990999i \(0.542741\pi\)
\(230\) −2.27851 −0.150241
\(231\) 0 0
\(232\) −1.08353 −0.0711372
\(233\) 12.5450 0.821850 0.410925 0.911669i \(-0.365206\pi\)
0.410925 + 0.911669i \(0.365206\pi\)
\(234\) −1.44552 −0.0944963
\(235\) −3.45761 −0.225549
\(236\) −1.68084 −0.109413
\(237\) 51.1184 3.32049
\(238\) 0 0
\(239\) 13.3463 0.863299 0.431649 0.902042i \(-0.357932\pi\)
0.431649 + 0.902042i \(0.357932\pi\)
\(240\) −14.9660 −0.966049
\(241\) −20.3854 −1.31314 −0.656568 0.754267i \(-0.727993\pi\)
−0.656568 + 0.754267i \(0.727993\pi\)
\(242\) 5.10787 0.328346
\(243\) 8.54160 0.547944
\(244\) −6.43169 −0.411747
\(245\) 0 0
\(246\) −8.17502 −0.521221
\(247\) 2.82036 0.179455
\(248\) 9.57708 0.608145
\(249\) 33.3157 2.11130
\(250\) −3.01558 −0.190722
\(251\) −17.1921 −1.08515 −0.542577 0.840006i \(-0.682551\pi\)
−0.542577 + 0.840006i \(0.682551\pi\)
\(252\) 0 0
\(253\) −33.0260 −2.07633
\(254\) 4.14561 0.260119
\(255\) 20.1678 1.26296
\(256\) 10.6873 0.667954
\(257\) 7.64695 0.477004 0.238502 0.971142i \(-0.423344\pi\)
0.238502 + 0.971142i \(0.423344\pi\)
\(258\) 6.49475 0.404345
\(259\) 0 0
\(260\) 2.76981 0.171776
\(261\) 5.69124 0.352279
\(262\) −3.20921 −0.198266
\(263\) −0.101037 −0.00623022 −0.00311511 0.999995i \(-0.500992\pi\)
−0.00311511 + 0.999995i \(0.500992\pi\)
\(264\) 16.6536 1.02496
\(265\) −2.62574 −0.161298
\(266\) 0 0
\(267\) −2.89541 −0.177196
\(268\) 12.7907 0.781318
\(269\) 7.56852 0.461461 0.230730 0.973018i \(-0.425889\pi\)
0.230730 + 0.973018i \(0.425889\pi\)
\(270\) −2.72052 −0.165565
\(271\) 13.8554 0.841653 0.420826 0.907141i \(-0.361740\pi\)
0.420826 + 0.907141i \(0.361740\pi\)
\(272\) −17.3169 −1.04999
\(273\) 0 0
\(274\) 4.21748 0.254787
\(275\) −16.1858 −0.976039
\(276\) −33.6837 −2.02752
\(277\) 0.552935 0.0332226 0.0166113 0.999862i \(-0.494712\pi\)
0.0166113 + 0.999862i \(0.494712\pi\)
\(278\) 1.75735 0.105399
\(279\) −50.3036 −3.01160
\(280\) 0 0
\(281\) 1.14667 0.0684043 0.0342022 0.999415i \(-0.489111\pi\)
0.0342022 + 0.999415i \(0.489111\pi\)
\(282\) 1.85464 0.110442
\(283\) 4.05396 0.240983 0.120491 0.992714i \(-0.461553\pi\)
0.120491 + 0.992714i \(0.461553\pi\)
\(284\) 13.2368 0.785461
\(285\) 11.7747 0.697476
\(286\) −1.45670 −0.0861365
\(287\) 0 0
\(288\) 16.5435 0.974833
\(289\) 6.33588 0.372699
\(290\) 0.395685 0.0232354
\(291\) 39.2982 2.30370
\(292\) 6.06296 0.354808
\(293\) −15.0649 −0.880102 −0.440051 0.897973i \(-0.645040\pi\)
−0.440051 + 0.897973i \(0.645040\pi\)
\(294\) 0 0
\(295\) 1.24990 0.0727718
\(296\) −0.636982 −0.0370238
\(297\) −39.4327 −2.28812
\(298\) −5.17351 −0.299694
\(299\) 5.99956 0.346964
\(300\) −16.5081 −0.953094
\(301\) 0 0
\(302\) 2.82834 0.162752
\(303\) −2.93250 −0.168468
\(304\) −10.1103 −0.579863
\(305\) 4.78269 0.273856
\(306\) −6.98288 −0.399185
\(307\) 19.9408 1.13808 0.569040 0.822310i \(-0.307315\pi\)
0.569040 + 0.822310i \(0.307315\pi\)
\(308\) 0 0
\(309\) −37.1642 −2.11420
\(310\) −3.49737 −0.198637
\(311\) 10.8956 0.617833 0.308916 0.951089i \(-0.400034\pi\)
0.308916 + 0.951089i \(0.400034\pi\)
\(312\) −3.02532 −0.171275
\(313\) 0.0519190 0.00293464 0.00146732 0.999999i \(-0.499533\pi\)
0.00146732 + 0.999999i \(0.499533\pi\)
\(314\) 3.98883 0.225103
\(315\) 0 0
\(316\) 33.9140 1.90781
\(317\) 16.1010 0.904321 0.452161 0.891937i \(-0.350653\pi\)
0.452161 + 0.891937i \(0.350653\pi\)
\(318\) 1.40843 0.0789808
\(319\) 5.73528 0.321114
\(320\) −9.13912 −0.510892
\(321\) 1.99413 0.111301
\(322\) 0 0
\(323\) 13.6244 0.758081
\(324\) −8.59066 −0.477259
\(325\) 2.94033 0.163100
\(326\) −6.30167 −0.349017
\(327\) 8.44621 0.467077
\(328\) −11.0441 −0.609808
\(329\) 0 0
\(330\) −6.08159 −0.334781
\(331\) 30.7862 1.69216 0.846081 0.533054i \(-0.178956\pi\)
0.846081 + 0.533054i \(0.178956\pi\)
\(332\) 22.1030 1.21306
\(333\) 3.34575 0.183346
\(334\) 1.88512 0.103149
\(335\) −9.51135 −0.519661
\(336\) 0 0
\(337\) −2.41842 −0.131740 −0.0658700 0.997828i \(-0.520982\pi\)
−0.0658700 + 0.997828i \(0.520982\pi\)
\(338\) 0.264627 0.0143938
\(339\) −35.0001 −1.90094
\(340\) 13.3802 0.725642
\(341\) −50.6929 −2.74517
\(342\) −4.07687 −0.220452
\(343\) 0 0
\(344\) 8.77411 0.473069
\(345\) 25.0476 1.34852
\(346\) −2.97353 −0.159858
\(347\) −0.492527 −0.0264403 −0.0132201 0.999913i \(-0.504208\pi\)
−0.0132201 + 0.999913i \(0.504208\pi\)
\(348\) 5.84948 0.313565
\(349\) 11.9442 0.639356 0.319678 0.947526i \(-0.396425\pi\)
0.319678 + 0.947526i \(0.396425\pi\)
\(350\) 0 0
\(351\) 7.16341 0.382355
\(352\) 16.6715 0.888593
\(353\) −15.5299 −0.826575 −0.413288 0.910601i \(-0.635620\pi\)
−0.413288 + 0.910601i \(0.635620\pi\)
\(354\) −0.670436 −0.0356333
\(355\) −9.84308 −0.522417
\(356\) −1.92094 −0.101809
\(357\) 0 0
\(358\) −3.49162 −0.184538
\(359\) −8.50709 −0.448987 −0.224493 0.974476i \(-0.572073\pi\)
−0.224493 + 0.974476i \(0.572073\pi\)
\(360\) −8.15287 −0.429694
\(361\) −11.0456 −0.581345
\(362\) 3.63635 0.191122
\(363\) −56.1507 −2.94715
\(364\) 0 0
\(365\) −4.50850 −0.235985
\(366\) −2.56540 −0.134096
\(367\) −5.19084 −0.270960 −0.135480 0.990780i \(-0.543258\pi\)
−0.135480 + 0.990780i \(0.543258\pi\)
\(368\) −21.5069 −1.12112
\(369\) 58.0091 3.01983
\(370\) 0.232614 0.0120930
\(371\) 0 0
\(372\) −51.7023 −2.68064
\(373\) −10.1427 −0.525169 −0.262585 0.964909i \(-0.584575\pi\)
−0.262585 + 0.964909i \(0.584575\pi\)
\(374\) −7.03692 −0.363870
\(375\) 33.1502 1.71187
\(376\) 2.50553 0.129213
\(377\) −1.04188 −0.0536595
\(378\) 0 0
\(379\) −3.63670 −0.186805 −0.0934024 0.995628i \(-0.529774\pi\)
−0.0934024 + 0.995628i \(0.529774\pi\)
\(380\) 7.81186 0.400740
\(381\) −45.5726 −2.33476
\(382\) −4.32154 −0.221109
\(383\) −4.60281 −0.235192 −0.117596 0.993061i \(-0.537519\pi\)
−0.117596 + 0.993061i \(0.537519\pi\)
\(384\) 22.5226 1.14935
\(385\) 0 0
\(386\) −3.71888 −0.189286
\(387\) −46.0861 −2.34269
\(388\) 26.0721 1.32361
\(389\) 19.6104 0.994286 0.497143 0.867669i \(-0.334382\pi\)
0.497143 + 0.867669i \(0.334382\pi\)
\(390\) 1.10479 0.0559433
\(391\) 28.9822 1.46569
\(392\) 0 0
\(393\) 35.2788 1.77958
\(394\) −0.388731 −0.0195840
\(395\) −25.2189 −1.26890
\(396\) −58.0333 −2.91628
\(397\) 19.8635 0.996919 0.498459 0.866913i \(-0.333899\pi\)
0.498459 + 0.866913i \(0.333899\pi\)
\(398\) 3.53996 0.177442
\(399\) 0 0
\(400\) −10.5403 −0.527017
\(401\) −15.1117 −0.754644 −0.377322 0.926082i \(-0.623155\pi\)
−0.377322 + 0.926082i \(0.623155\pi\)
\(402\) 5.10183 0.254456
\(403\) 9.20895 0.458731
\(404\) −1.94554 −0.0967943
\(405\) 6.38813 0.317429
\(406\) 0 0
\(407\) 3.37164 0.167126
\(408\) −14.6145 −0.723525
\(409\) 35.2443 1.74272 0.871360 0.490644i \(-0.163238\pi\)
0.871360 + 0.490644i \(0.163238\pi\)
\(410\) 4.03310 0.199181
\(411\) −46.3626 −2.28690
\(412\) −24.6563 −1.21473
\(413\) 0 0
\(414\) −8.67246 −0.426228
\(415\) −16.4361 −0.806816
\(416\) −3.02857 −0.148488
\(417\) −19.3185 −0.946033
\(418\) −4.10842 −0.200950
\(419\) −1.50468 −0.0735084 −0.0367542 0.999324i \(-0.511702\pi\)
−0.0367542 + 0.999324i \(0.511702\pi\)
\(420\) 0 0
\(421\) −24.5079 −1.19444 −0.597221 0.802077i \(-0.703728\pi\)
−0.597221 + 0.802077i \(0.703728\pi\)
\(422\) 0.918370 0.0447056
\(423\) −13.1603 −0.639877
\(424\) 1.90272 0.0924045
\(425\) 14.2039 0.688992
\(426\) 5.27977 0.255806
\(427\) 0 0
\(428\) 1.32299 0.0639490
\(429\) 16.0135 0.773138
\(430\) −3.20414 −0.154518
\(431\) 41.0655 1.97805 0.989027 0.147732i \(-0.0471974\pi\)
0.989027 + 0.147732i \(0.0471974\pi\)
\(432\) −25.6790 −1.23548
\(433\) −6.65603 −0.319869 −0.159934 0.987128i \(-0.551128\pi\)
−0.159934 + 0.987128i \(0.551128\pi\)
\(434\) 0 0
\(435\) −4.34975 −0.208555
\(436\) 5.60357 0.268362
\(437\) 16.9209 0.809438
\(438\) 2.41833 0.115552
\(439\) −8.22990 −0.392792 −0.196396 0.980525i \(-0.562924\pi\)
−0.196396 + 0.980525i \(0.562924\pi\)
\(440\) −8.21596 −0.391681
\(441\) 0 0
\(442\) 1.27834 0.0608043
\(443\) 17.6856 0.840266 0.420133 0.907463i \(-0.361983\pi\)
0.420133 + 0.907463i \(0.361983\pi\)
\(444\) 3.43878 0.163197
\(445\) 1.42843 0.0677143
\(446\) 2.62329 0.124216
\(447\) 56.8723 2.68997
\(448\) 0 0
\(449\) −14.5250 −0.685477 −0.342738 0.939431i \(-0.611354\pi\)
−0.342738 + 0.939431i \(0.611354\pi\)
\(450\) −4.25030 −0.200361
\(451\) 58.4580 2.75268
\(452\) −23.2205 −1.09220
\(453\) −31.0918 −1.46082
\(454\) −3.19475 −0.149937
\(455\) 0 0
\(456\) −8.53250 −0.399571
\(457\) 3.78919 0.177251 0.0886255 0.996065i \(-0.471753\pi\)
0.0886255 + 0.996065i \(0.471753\pi\)
\(458\) −1.07219 −0.0501002
\(459\) 34.6045 1.61520
\(460\) 16.6176 0.774801
\(461\) −13.1107 −0.610627 −0.305314 0.952252i \(-0.598761\pi\)
−0.305314 + 0.952252i \(0.598761\pi\)
\(462\) 0 0
\(463\) 15.3027 0.711176 0.355588 0.934643i \(-0.384281\pi\)
0.355588 + 0.934643i \(0.384281\pi\)
\(464\) 3.73487 0.173387
\(465\) 38.4465 1.78292
\(466\) 3.31974 0.153784
\(467\) −30.7738 −1.42404 −0.712022 0.702158i \(-0.752220\pi\)
−0.712022 + 0.702158i \(0.752220\pi\)
\(468\) 10.5424 0.487324
\(469\) 0 0
\(470\) −0.914975 −0.0422046
\(471\) −43.8491 −2.02046
\(472\) −0.905729 −0.0416896
\(473\) −46.4427 −2.13544
\(474\) 13.5273 0.621328
\(475\) 8.29280 0.380500
\(476\) 0 0
\(477\) −9.99407 −0.457597
\(478\) 3.53178 0.161540
\(479\) 1.71740 0.0784699 0.0392350 0.999230i \(-0.487508\pi\)
0.0392350 + 0.999230i \(0.487508\pi\)
\(480\) −12.6440 −0.577117
\(481\) −0.612497 −0.0279275
\(482\) −5.39451 −0.245713
\(483\) 0 0
\(484\) −37.2527 −1.69330
\(485\) −19.3875 −0.880342
\(486\) 2.26033 0.102531
\(487\) 22.6805 1.02775 0.513877 0.857864i \(-0.328209\pi\)
0.513877 + 0.857864i \(0.328209\pi\)
\(488\) −3.46575 −0.156887
\(489\) 69.2741 3.13268
\(490\) 0 0
\(491\) −13.1366 −0.592846 −0.296423 0.955057i \(-0.595794\pi\)
−0.296423 + 0.955057i \(0.595794\pi\)
\(492\) 59.6220 2.68797
\(493\) −5.03303 −0.226676
\(494\) 0.746342 0.0335795
\(495\) 43.1543 1.93964
\(496\) −33.0117 −1.48227
\(497\) 0 0
\(498\) 8.81622 0.395064
\(499\) 14.0395 0.628495 0.314248 0.949341i \(-0.398248\pi\)
0.314248 + 0.949341i \(0.398248\pi\)
\(500\) 21.9932 0.983566
\(501\) −20.7231 −0.925840
\(502\) −4.54948 −0.203053
\(503\) −0.367865 −0.0164023 −0.00820114 0.999966i \(-0.502611\pi\)
−0.00820114 + 0.999966i \(0.502611\pi\)
\(504\) 0 0
\(505\) 1.44673 0.0643786
\(506\) −8.73957 −0.388521
\(507\) −2.90903 −0.129195
\(508\) −30.2348 −1.34145
\(509\) −41.2319 −1.82757 −0.913787 0.406194i \(-0.866856\pi\)
−0.913787 + 0.406194i \(0.866856\pi\)
\(510\) 5.33694 0.236324
\(511\) 0 0
\(512\) 18.3127 0.809315
\(513\) 20.2034 0.892002
\(514\) 2.02359 0.0892566
\(515\) 18.3347 0.807925
\(516\) −47.3675 −2.08524
\(517\) −13.2622 −0.583269
\(518\) 0 0
\(519\) 32.6879 1.43484
\(520\) 1.49252 0.0654515
\(521\) −1.04099 −0.0456065 −0.0228032 0.999740i \(-0.507259\pi\)
−0.0228032 + 0.999740i \(0.507259\pi\)
\(522\) 1.50605 0.0659182
\(523\) 20.0209 0.875451 0.437726 0.899109i \(-0.355784\pi\)
0.437726 + 0.899109i \(0.355784\pi\)
\(524\) 23.4054 1.02247
\(525\) 0 0
\(526\) −0.0267371 −0.00116579
\(527\) 44.4859 1.93784
\(528\) −57.4042 −2.49820
\(529\) 12.9947 0.564989
\(530\) −0.694840 −0.0301819
\(531\) 4.75735 0.206451
\(532\) 0 0
\(533\) −10.6196 −0.459985
\(534\) −0.766203 −0.0331568
\(535\) −0.983791 −0.0425330
\(536\) 6.89234 0.297704
\(537\) 38.3833 1.65636
\(538\) 2.00283 0.0863481
\(539\) 0 0
\(540\) 19.8413 0.853832
\(541\) −9.78749 −0.420797 −0.210399 0.977616i \(-0.567476\pi\)
−0.210399 + 0.977616i \(0.567476\pi\)
\(542\) 3.66649 0.157489
\(543\) −39.9743 −1.71546
\(544\) −14.6302 −0.627263
\(545\) −4.16689 −0.178490
\(546\) 0 0
\(547\) −2.56174 −0.109532 −0.0547660 0.998499i \(-0.517441\pi\)
−0.0547660 + 0.998499i \(0.517441\pi\)
\(548\) −30.7589 −1.31395
\(549\) 18.2038 0.776921
\(550\) −4.28319 −0.182636
\(551\) −2.93848 −0.125183
\(552\) −18.1506 −0.772541
\(553\) 0 0
\(554\) 0.146321 0.00621660
\(555\) −2.55712 −0.108544
\(556\) −12.8167 −0.543550
\(557\) −27.4442 −1.16285 −0.581424 0.813601i \(-0.697504\pi\)
−0.581424 + 0.813601i \(0.697504\pi\)
\(558\) −13.3117 −0.563528
\(559\) 8.43685 0.356841
\(560\) 0 0
\(561\) 77.3567 3.26600
\(562\) 0.303438 0.0127998
\(563\) −0.162708 −0.00685734 −0.00342867 0.999994i \(-0.501091\pi\)
−0.00342867 + 0.999994i \(0.501091\pi\)
\(564\) −13.5262 −0.569557
\(565\) 17.2671 0.726431
\(566\) 1.07278 0.0450925
\(567\) 0 0
\(568\) 7.13273 0.299283
\(569\) −12.3901 −0.519419 −0.259709 0.965687i \(-0.583627\pi\)
−0.259709 + 0.965687i \(0.583627\pi\)
\(570\) 3.11591 0.130511
\(571\) −21.8122 −0.912810 −0.456405 0.889772i \(-0.650863\pi\)
−0.456405 + 0.889772i \(0.650863\pi\)
\(572\) 10.6240 0.444212
\(573\) 47.5066 1.98462
\(574\) 0 0
\(575\) 17.6407 0.735669
\(576\) −34.7852 −1.44939
\(577\) −6.06583 −0.252524 −0.126262 0.991997i \(-0.540298\pi\)
−0.126262 + 0.991997i \(0.540298\pi\)
\(578\) 1.67664 0.0697391
\(579\) 40.8816 1.69898
\(580\) −2.88581 −0.119827
\(581\) 0 0
\(582\) 10.3993 0.431067
\(583\) −10.0714 −0.417115
\(584\) 3.26705 0.135192
\(585\) −7.83949 −0.324123
\(586\) −3.98658 −0.164684
\(587\) −20.5820 −0.849510 −0.424755 0.905308i \(-0.639640\pi\)
−0.424755 + 0.905308i \(0.639640\pi\)
\(588\) 0 0
\(589\) 25.9726 1.07018
\(590\) 0.330756 0.0136170
\(591\) 4.27331 0.175781
\(592\) 2.19564 0.0902404
\(593\) 24.0397 0.987190 0.493595 0.869692i \(-0.335682\pi\)
0.493595 + 0.869692i \(0.335682\pi\)
\(594\) −10.4349 −0.428151
\(595\) 0 0
\(596\) 37.7314 1.54554
\(597\) −38.9147 −1.59267
\(598\) 1.58764 0.0649236
\(599\) 32.2523 1.31779 0.658896 0.752234i \(-0.271024\pi\)
0.658896 + 0.752234i \(0.271024\pi\)
\(600\) −8.89546 −0.363156
\(601\) −5.21454 −0.212705 −0.106353 0.994328i \(-0.533917\pi\)
−0.106353 + 0.994328i \(0.533917\pi\)
\(602\) 0 0
\(603\) −36.2020 −1.47426
\(604\) −20.6276 −0.839325
\(605\) 27.7016 1.12623
\(606\) −0.776017 −0.0315235
\(607\) −9.07048 −0.368160 −0.184080 0.982911i \(-0.558930\pi\)
−0.184080 + 0.982911i \(0.558930\pi\)
\(608\) −8.54165 −0.346410
\(609\) 0 0
\(610\) 1.26563 0.0512437
\(611\) 2.40922 0.0974668
\(612\) 50.9275 2.05862
\(613\) 20.0920 0.811507 0.405754 0.913983i \(-0.367009\pi\)
0.405754 + 0.913983i \(0.367009\pi\)
\(614\) 5.27686 0.212957
\(615\) −44.3357 −1.78779
\(616\) 0 0
\(617\) −12.9556 −0.521572 −0.260786 0.965397i \(-0.583982\pi\)
−0.260786 + 0.965397i \(0.583982\pi\)
\(618\) −9.83463 −0.395607
\(619\) 44.3644 1.78316 0.891578 0.452866i \(-0.149598\pi\)
0.891578 + 0.452866i \(0.149598\pi\)
\(620\) 25.5070 1.02439
\(621\) 42.9773 1.72462
\(622\) 2.88327 0.115608
\(623\) 0 0
\(624\) 10.4281 0.417459
\(625\) −1.65276 −0.0661105
\(626\) 0.0137392 0.000549127 0
\(627\) 45.1638 1.80367
\(628\) −29.0913 −1.16087
\(629\) −2.95880 −0.117975
\(630\) 0 0
\(631\) 6.61717 0.263426 0.131713 0.991288i \(-0.457952\pi\)
0.131713 + 0.991288i \(0.457952\pi\)
\(632\) 18.2747 0.726930
\(633\) −10.0956 −0.401265
\(634\) 4.26075 0.169216
\(635\) 22.4830 0.892210
\(636\) −10.2719 −0.407309
\(637\) 0 0
\(638\) 1.51771 0.0600866
\(639\) −37.4647 −1.48208
\(640\) −11.1114 −0.439216
\(641\) 18.9567 0.748744 0.374372 0.927279i \(-0.377858\pi\)
0.374372 + 0.927279i \(0.377858\pi\)
\(642\) 0.527699 0.0208266
\(643\) −13.4019 −0.528517 −0.264259 0.964452i \(-0.585127\pi\)
−0.264259 + 0.964452i \(0.585127\pi\)
\(644\) 0 0
\(645\) 35.2231 1.38691
\(646\) 3.60537 0.141851
\(647\) −42.7588 −1.68102 −0.840511 0.541794i \(-0.817745\pi\)
−0.840511 + 0.541794i \(0.817745\pi\)
\(648\) −4.62912 −0.181849
\(649\) 4.79416 0.188187
\(650\) 0.778091 0.0305192
\(651\) 0 0
\(652\) 45.9593 1.79991
\(653\) 10.9852 0.429884 0.214942 0.976627i \(-0.431044\pi\)
0.214942 + 0.976627i \(0.431044\pi\)
\(654\) 2.23509 0.0873990
\(655\) −17.4046 −0.680052
\(656\) 38.0684 1.48632
\(657\) −17.1602 −0.669483
\(658\) 0 0
\(659\) −17.7614 −0.691884 −0.345942 0.938256i \(-0.612441\pi\)
−0.345942 + 0.938256i \(0.612441\pi\)
\(660\) 44.3542 1.72649
\(661\) −8.18255 −0.318264 −0.159132 0.987257i \(-0.550870\pi\)
−0.159132 + 0.987257i \(0.550870\pi\)
\(662\) 8.14685 0.316636
\(663\) −14.0527 −0.545763
\(664\) 11.9103 0.462210
\(665\) 0 0
\(666\) 0.885374 0.0343076
\(667\) −6.25082 −0.242033
\(668\) −13.7486 −0.531948
\(669\) −28.8378 −1.11493
\(670\) −2.51696 −0.0972385
\(671\) 18.3447 0.708189
\(672\) 0 0
\(673\) 9.30129 0.358539 0.179269 0.983800i \(-0.442627\pi\)
0.179269 + 0.983800i \(0.442627\pi\)
\(674\) −0.639979 −0.0246511
\(675\) 21.0628 0.810708
\(676\) −1.92997 −0.0742297
\(677\) 41.1552 1.58172 0.790862 0.611994i \(-0.209633\pi\)
0.790862 + 0.611994i \(0.209633\pi\)
\(678\) −9.26195 −0.355703
\(679\) 0 0
\(680\) 7.20997 0.276490
\(681\) 35.1198 1.34579
\(682\) −13.4147 −0.513675
\(683\) 39.2842 1.50317 0.751583 0.659638i \(-0.229291\pi\)
0.751583 + 0.659638i \(0.229291\pi\)
\(684\) 29.7334 1.13689
\(685\) 22.8727 0.873921
\(686\) 0 0
\(687\) 11.7866 0.449685
\(688\) −30.2439 −1.15304
\(689\) 1.82959 0.0697017
\(690\) 6.62827 0.252334
\(691\) 3.03355 0.115402 0.0577009 0.998334i \(-0.481623\pi\)
0.0577009 + 0.998334i \(0.481623\pi\)
\(692\) 21.6865 0.824397
\(693\) 0 0
\(694\) −0.130336 −0.00494748
\(695\) 9.53069 0.361520
\(696\) 3.15202 0.119477
\(697\) −51.3002 −1.94313
\(698\) 3.16074 0.119636
\(699\) −36.4938 −1.38032
\(700\) 0 0
\(701\) −26.2320 −0.990767 −0.495384 0.868674i \(-0.664972\pi\)
−0.495384 + 0.868674i \(0.664972\pi\)
\(702\) 1.89563 0.0715459
\(703\) −1.72746 −0.0651525
\(704\) −35.0544 −1.32116
\(705\) 10.0583 0.378817
\(706\) −4.10963 −0.154668
\(707\) 0 0
\(708\) 4.88962 0.183763
\(709\) −7.87770 −0.295853 −0.147927 0.988998i \(-0.547260\pi\)
−0.147927 + 0.988998i \(0.547260\pi\)
\(710\) −2.60474 −0.0977542
\(711\) −95.9881 −3.59984
\(712\) −1.03511 −0.0387922
\(713\) 55.2497 2.06912
\(714\) 0 0
\(715\) −7.90015 −0.295449
\(716\) 25.4650 0.951673
\(717\) −38.8247 −1.44994
\(718\) −2.25120 −0.0840141
\(719\) −45.6656 −1.70304 −0.851519 0.524323i \(-0.824318\pi\)
−0.851519 + 0.524323i \(0.824318\pi\)
\(720\) 28.1025 1.04732
\(721\) 0 0
\(722\) −2.92295 −0.108781
\(723\) 59.3017 2.20545
\(724\) −26.5206 −0.985630
\(725\) −3.06348 −0.113775
\(726\) −14.8590 −0.551468
\(727\) −37.5947 −1.39431 −0.697155 0.716921i \(-0.745551\pi\)
−0.697155 + 0.716921i \(0.745551\pi\)
\(728\) 0 0
\(729\) −38.2013 −1.41486
\(730\) −1.19307 −0.0441574
\(731\) 40.7561 1.50742
\(732\) 18.7100 0.691541
\(733\) −53.1810 −1.96429 −0.982143 0.188138i \(-0.939755\pi\)
−0.982143 + 0.188138i \(0.939755\pi\)
\(734\) −1.37363 −0.0507018
\(735\) 0 0
\(736\) −18.1701 −0.669758
\(737\) −36.4822 −1.34384
\(738\) 15.3508 0.565069
\(739\) 41.9633 1.54364 0.771822 0.635839i \(-0.219346\pi\)
0.771822 + 0.635839i \(0.219346\pi\)
\(740\) −1.69650 −0.0623646
\(741\) −8.20452 −0.301401
\(742\) 0 0
\(743\) −38.5424 −1.41398 −0.706991 0.707222i \(-0.749948\pi\)
−0.706991 + 0.707222i \(0.749948\pi\)
\(744\) −27.8600 −1.02140
\(745\) −28.0576 −1.02795
\(746\) −2.68403 −0.0982693
\(747\) −62.5590 −2.28891
\(748\) 51.3216 1.87650
\(749\) 0 0
\(750\) 8.77242 0.320323
\(751\) 36.2434 1.32254 0.661270 0.750148i \(-0.270018\pi\)
0.661270 + 0.750148i \(0.270018\pi\)
\(752\) −8.63644 −0.314939
\(753\) 50.0123 1.82255
\(754\) −0.275709 −0.0100407
\(755\) 15.3390 0.558242
\(756\) 0 0
\(757\) 19.4752 0.707837 0.353919 0.935276i \(-0.384849\pi\)
0.353919 + 0.935276i \(0.384849\pi\)
\(758\) −0.962368 −0.0349548
\(759\) 96.0738 3.48726
\(760\) 4.20946 0.152693
\(761\) 51.9059 1.88159 0.940793 0.338981i \(-0.110082\pi\)
0.940793 + 0.338981i \(0.110082\pi\)
\(762\) −12.0597 −0.436878
\(763\) 0 0
\(764\) 31.5178 1.14028
\(765\) −37.8704 −1.36921
\(766\) −1.21803 −0.0440090
\(767\) −0.870914 −0.0314469
\(768\) −31.0896 −1.12185
\(769\) 7.31376 0.263741 0.131870 0.991267i \(-0.457902\pi\)
0.131870 + 0.991267i \(0.457902\pi\)
\(770\) 0 0
\(771\) −22.2452 −0.801142
\(772\) 27.1225 0.976161
\(773\) 14.1844 0.510178 0.255089 0.966918i \(-0.417895\pi\)
0.255089 + 0.966918i \(0.417895\pi\)
\(774\) −12.1956 −0.438362
\(775\) 27.0774 0.972649
\(776\) 14.0491 0.504332
\(777\) 0 0
\(778\) 5.18943 0.186050
\(779\) −29.9510 −1.07311
\(780\) −8.05746 −0.288503
\(781\) −37.7546 −1.35097
\(782\) 7.66946 0.274259
\(783\) −7.46341 −0.266721
\(784\) 0 0
\(785\) 21.6327 0.772104
\(786\) 9.33569 0.332993
\(787\) −31.2777 −1.11493 −0.557465 0.830201i \(-0.688226\pi\)
−0.557465 + 0.830201i \(0.688226\pi\)
\(788\) 2.83509 0.100996
\(789\) 0.293921 0.0104639
\(790\) −6.67360 −0.237436
\(791\) 0 0
\(792\) −31.2715 −1.11119
\(793\) −3.33253 −0.118342
\(794\) 5.25640 0.186543
\(795\) 7.63836 0.270905
\(796\) −25.8176 −0.915082
\(797\) −20.2422 −0.717017 −0.358509 0.933526i \(-0.616715\pi\)
−0.358509 + 0.933526i \(0.616715\pi\)
\(798\) 0 0
\(799\) 11.6383 0.411733
\(800\) −8.90500 −0.314839
\(801\) 5.43689 0.192103
\(802\) −3.99897 −0.141208
\(803\) −17.2930 −0.610257
\(804\) −37.2086 −1.31225
\(805\) 0 0
\(806\) 2.43693 0.0858373
\(807\) −22.0171 −0.775037
\(808\) −1.04836 −0.0368813
\(809\) 7.88265 0.277139 0.138570 0.990353i \(-0.455750\pi\)
0.138570 + 0.990353i \(0.455750\pi\)
\(810\) 1.69047 0.0593970
\(811\) 5.99962 0.210675 0.105338 0.994437i \(-0.466408\pi\)
0.105338 + 0.994437i \(0.466408\pi\)
\(812\) 0 0
\(813\) −40.3057 −1.41358
\(814\) 0.892226 0.0312725
\(815\) −34.1760 −1.19713
\(816\) 50.3754 1.76349
\(817\) 23.7950 0.832480
\(818\) 9.32659 0.326096
\(819\) 0 0
\(820\) −29.4142 −1.02719
\(821\) −19.1692 −0.669011 −0.334505 0.942394i \(-0.608569\pi\)
−0.334505 + 0.942394i \(0.608569\pi\)
\(822\) −12.2688 −0.427923
\(823\) −30.3735 −1.05875 −0.529376 0.848387i \(-0.677574\pi\)
−0.529376 + 0.848387i \(0.677574\pi\)
\(824\) −13.2861 −0.462845
\(825\) 47.0850 1.63929
\(826\) 0 0
\(827\) −14.6870 −0.510717 −0.255359 0.966846i \(-0.582193\pi\)
−0.255359 + 0.966846i \(0.582193\pi\)
\(828\) 63.2499 2.19809
\(829\) 34.9985 1.21555 0.607774 0.794110i \(-0.292062\pi\)
0.607774 + 0.794110i \(0.292062\pi\)
\(830\) −4.34943 −0.150971
\(831\) −1.60851 −0.0557985
\(832\) 6.36804 0.220772
\(833\) 0 0
\(834\) −5.11220 −0.177021
\(835\) 10.2236 0.353803
\(836\) 29.9635 1.03631
\(837\) 65.9675 2.28017
\(838\) −0.398178 −0.0137548
\(839\) −27.6333 −0.954008 −0.477004 0.878901i \(-0.658277\pi\)
−0.477004 + 0.878901i \(0.658277\pi\)
\(840\) 0 0
\(841\) −27.9145 −0.962568
\(842\) −6.48544 −0.223503
\(843\) −3.33569 −0.114887
\(844\) −6.69785 −0.230550
\(845\) 1.43515 0.0493708
\(846\) −3.48257 −0.119733
\(847\) 0 0
\(848\) −6.55859 −0.225223
\(849\) −11.7931 −0.404738
\(850\) 3.75874 0.128924
\(851\) −3.67472 −0.125968
\(852\) −38.5064 −1.31921
\(853\) −32.6336 −1.11735 −0.558676 0.829386i \(-0.688690\pi\)
−0.558676 + 0.829386i \(0.688690\pi\)
\(854\) 0 0
\(855\) −22.1102 −0.756152
\(856\) 0.712898 0.0243664
\(857\) 18.8742 0.644730 0.322365 0.946616i \(-0.395522\pi\)
0.322365 + 0.946616i \(0.395522\pi\)
\(858\) 4.23759 0.144669
\(859\) −15.8242 −0.539915 −0.269957 0.962872i \(-0.587010\pi\)
−0.269957 + 0.962872i \(0.587010\pi\)
\(860\) 23.3684 0.796857
\(861\) 0 0
\(862\) 10.8670 0.370132
\(863\) 52.3212 1.78104 0.890518 0.454948i \(-0.150342\pi\)
0.890518 + 0.454948i \(0.150342\pi\)
\(864\) −21.6949 −0.738075
\(865\) −16.1264 −0.548313
\(866\) −1.76136 −0.0598536
\(867\) −18.4313 −0.625959
\(868\) 0 0
\(869\) −96.7309 −3.28137
\(870\) −1.15106 −0.0390246
\(871\) 6.62741 0.224561
\(872\) 3.01951 0.102253
\(873\) −73.7927 −2.49750
\(874\) 4.47773 0.151461
\(875\) 0 0
\(876\) −17.6373 −0.595911
\(877\) −54.0162 −1.82400 −0.911999 0.410193i \(-0.865461\pi\)
−0.911999 + 0.410193i \(0.865461\pi\)
\(878\) −2.17785 −0.0734989
\(879\) 43.8243 1.47816
\(880\) 28.3200 0.954667
\(881\) 42.0823 1.41779 0.708895 0.705314i \(-0.249194\pi\)
0.708895 + 0.705314i \(0.249194\pi\)
\(882\) 0 0
\(883\) 36.5314 1.22938 0.614689 0.788769i \(-0.289281\pi\)
0.614689 + 0.788769i \(0.289281\pi\)
\(884\) −9.32316 −0.313572
\(885\) −3.63599 −0.122222
\(886\) 4.68007 0.157230
\(887\) 11.4648 0.384950 0.192475 0.981302i \(-0.438349\pi\)
0.192475 + 0.981302i \(0.438349\pi\)
\(888\) 1.85300 0.0621827
\(889\) 0 0
\(890\) 0.378001 0.0126706
\(891\) 24.5026 0.820869
\(892\) −19.1322 −0.640592
\(893\) 6.79488 0.227382
\(894\) 15.0499 0.503345
\(895\) −18.9361 −0.632965
\(896\) 0 0
\(897\) −17.4529 −0.582736
\(898\) −3.84370 −0.128266
\(899\) −9.59462 −0.319999
\(900\) 30.9982 1.03327
\(901\) 8.83823 0.294444
\(902\) 15.4695 0.515079
\(903\) 0 0
\(904\) −12.5125 −0.416159
\(905\) 19.7211 0.655550
\(906\) −8.22772 −0.273348
\(907\) −5.04665 −0.167571 −0.0837856 0.996484i \(-0.526701\pi\)
−0.0837856 + 0.996484i \(0.526701\pi\)
\(908\) 23.2999 0.773236
\(909\) 5.50653 0.182640
\(910\) 0 0
\(911\) 47.5236 1.57453 0.787263 0.616618i \(-0.211497\pi\)
0.787263 + 0.616618i \(0.211497\pi\)
\(912\) 29.4111 0.973898
\(913\) −63.0431 −2.08642
\(914\) 1.00272 0.0331671
\(915\) −13.9130 −0.459950
\(916\) 7.81969 0.258370
\(917\) 0 0
\(918\) 9.15726 0.302235
\(919\) −45.6698 −1.50651 −0.753254 0.657730i \(-0.771517\pi\)
−0.753254 + 0.657730i \(0.771517\pi\)
\(920\) 8.95449 0.295221
\(921\) −58.0084 −1.91144
\(922\) −3.46945 −0.114260
\(923\) 6.85856 0.225752
\(924\) 0 0
\(925\) −1.80095 −0.0592148
\(926\) 4.04950 0.133075
\(927\) 69.7855 2.29206
\(928\) 3.15540 0.103581
\(929\) 44.4449 1.45819 0.729095 0.684412i \(-0.239941\pi\)
0.729095 + 0.684412i \(0.239941\pi\)
\(930\) 10.1740 0.333618
\(931\) 0 0
\(932\) −24.2115 −0.793074
\(933\) −31.6957 −1.03767
\(934\) −8.14357 −0.266466
\(935\) −38.1634 −1.24808
\(936\) 5.68083 0.185684
\(937\) −46.9796 −1.53476 −0.767379 0.641194i \(-0.778439\pi\)
−0.767379 + 0.641194i \(0.778439\pi\)
\(938\) 0 0
\(939\) −0.151034 −0.00492881
\(940\) 6.67309 0.217652
\(941\) −35.5654 −1.15940 −0.579699 0.814830i \(-0.696830\pi\)
−0.579699 + 0.814830i \(0.696830\pi\)
\(942\) −11.6036 −0.378067
\(943\) −63.7128 −2.07477
\(944\) 3.12200 0.101613
\(945\) 0 0
\(946\) −12.2900 −0.399581
\(947\) −22.3592 −0.726576 −0.363288 0.931677i \(-0.618346\pi\)
−0.363288 + 0.931677i \(0.618346\pi\)
\(948\) −98.6570 −3.20423
\(949\) 3.14147 0.101977
\(950\) 2.19450 0.0711989
\(951\) −46.8383 −1.51884
\(952\) 0 0
\(953\) −46.7684 −1.51498 −0.757488 0.652849i \(-0.773574\pi\)
−0.757488 + 0.652849i \(0.773574\pi\)
\(954\) −2.64470 −0.0856252
\(955\) −23.4371 −0.758406
\(956\) −25.7579 −0.833071
\(957\) −16.6841 −0.539321
\(958\) 0.454469 0.0146832
\(959\) 0 0
\(960\) 26.5860 0.858059
\(961\) 53.8048 1.73564
\(962\) −0.162083 −0.00522577
\(963\) −3.74450 −0.120665
\(964\) 39.3432 1.26716
\(965\) −20.1687 −0.649253
\(966\) 0 0
\(967\) 8.22976 0.264651 0.132326 0.991206i \(-0.457756\pi\)
0.132326 + 0.991206i \(0.457756\pi\)
\(968\) −20.0738 −0.645196
\(969\) −39.6338 −1.27322
\(970\) −5.13046 −0.164729
\(971\) 23.6326 0.758407 0.379204 0.925313i \(-0.376198\pi\)
0.379204 + 0.925313i \(0.376198\pi\)
\(972\) −16.4850 −0.528758
\(973\) 0 0
\(974\) 6.00187 0.192312
\(975\) −8.55353 −0.273932
\(976\) 11.9462 0.382390
\(977\) −26.6428 −0.852379 −0.426190 0.904634i \(-0.640144\pi\)
−0.426190 + 0.904634i \(0.640144\pi\)
\(978\) 18.3318 0.586185
\(979\) 5.47897 0.175109
\(980\) 0 0
\(981\) −15.8600 −0.506370
\(982\) −3.47629 −0.110933
\(983\) −43.6302 −1.39159 −0.695793 0.718242i \(-0.744947\pi\)
−0.695793 + 0.718242i \(0.744947\pi\)
\(984\) 32.1276 1.02419
\(985\) −2.10821 −0.0671732
\(986\) −1.33187 −0.0424155
\(987\) 0 0
\(988\) −5.44322 −0.173172
\(989\) 50.6174 1.60954
\(990\) 11.4198 0.362945
\(991\) −7.60816 −0.241681 −0.120841 0.992672i \(-0.538559\pi\)
−0.120841 + 0.992672i \(0.538559\pi\)
\(992\) −27.8899 −0.885506
\(993\) −89.5581 −2.84204
\(994\) 0 0
\(995\) 19.1983 0.608628
\(996\) −64.2984 −2.03737
\(997\) −19.4356 −0.615532 −0.307766 0.951462i \(-0.599581\pi\)
−0.307766 + 0.951462i \(0.599581\pi\)
\(998\) 3.71523 0.117604
\(999\) −4.38757 −0.138817
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.1.4 6
3.2 odd 2 5733.2.a.bu.1.3 6
7.2 even 3 637.2.e.o.508.3 12
7.3 odd 6 637.2.e.n.79.3 12
7.4 even 3 637.2.e.o.79.3 12
7.5 odd 6 637.2.e.n.508.3 12
7.6 odd 2 637.2.a.n.1.4 yes 6
13.12 even 2 8281.2.a.cc.1.3 6
21.20 even 2 5733.2.a.br.1.3 6
91.90 odd 2 8281.2.a.cd.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.m.1.4 6 1.1 even 1 trivial
637.2.a.n.1.4 yes 6 7.6 odd 2
637.2.e.n.79.3 12 7.3 odd 6
637.2.e.n.508.3 12 7.5 odd 6
637.2.e.o.79.3 12 7.4 even 3
637.2.e.o.508.3 12 7.2 even 3
5733.2.a.br.1.3 6 21.20 even 2
5733.2.a.bu.1.3 6 3.2 odd 2
8281.2.a.cc.1.3 6 13.12 even 2
8281.2.a.cd.1.3 6 91.90 odd 2