Properties

Label 787.2.a.b.1.9
Level $787$
Weight $2$
Character 787.1
Self dual yes
Analytic conductor $6.284$
Analytic rank $0$
Dimension $37$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [787,2,Mod(1,787)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("787.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(787, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 787 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 787.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [37] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(6.28422663907\)
Analytic rank: \(0\)
Dimension: \(37\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Character \(\chi\) \(=\) 787.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.54331 q^{2} +2.10249 q^{3} +0.381797 q^{4} +1.25926 q^{5} -3.24479 q^{6} +3.14448 q^{7} +2.49738 q^{8} +1.42047 q^{9} -1.94342 q^{10} -0.917999 q^{11} +0.802724 q^{12} +2.37731 q^{13} -4.85289 q^{14} +2.64757 q^{15} -4.61782 q^{16} +4.67298 q^{17} -2.19222 q^{18} +0.491776 q^{19} +0.480780 q^{20} +6.61124 q^{21} +1.41675 q^{22} -1.93961 q^{23} +5.25073 q^{24} -3.41428 q^{25} -3.66892 q^{26} -3.32095 q^{27} +1.20055 q^{28} +5.54398 q^{29} -4.08602 q^{30} -4.99990 q^{31} +2.13195 q^{32} -1.93008 q^{33} -7.21184 q^{34} +3.95970 q^{35} +0.542331 q^{36} +0.00995484 q^{37} -0.758961 q^{38} +4.99827 q^{39} +3.14484 q^{40} +7.33823 q^{41} -10.2032 q^{42} +2.73113 q^{43} -0.350489 q^{44} +1.78874 q^{45} +2.99341 q^{46} -5.69556 q^{47} -9.70894 q^{48} +2.88774 q^{49} +5.26928 q^{50} +9.82489 q^{51} +0.907648 q^{52} -3.42166 q^{53} +5.12524 q^{54} -1.15599 q^{55} +7.85297 q^{56} +1.03395 q^{57} -8.55607 q^{58} +1.55022 q^{59} +1.01083 q^{60} +0.660781 q^{61} +7.71639 q^{62} +4.46664 q^{63} +5.94539 q^{64} +2.99364 q^{65} +2.97871 q^{66} -6.25946 q^{67} +1.78413 q^{68} -4.07801 q^{69} -6.11103 q^{70} +6.33818 q^{71} +3.54746 q^{72} -6.27766 q^{73} -0.0153634 q^{74} -7.17849 q^{75} +0.187758 q^{76} -2.88663 q^{77} -7.71386 q^{78} +8.41425 q^{79} -5.81502 q^{80} -11.2437 q^{81} -11.3251 q^{82} -2.75585 q^{83} +2.52415 q^{84} +5.88447 q^{85} -4.21498 q^{86} +11.6562 q^{87} -2.29260 q^{88} +12.8586 q^{89} -2.76057 q^{90} +7.47539 q^{91} -0.740536 q^{92} -10.5123 q^{93} +8.79000 q^{94} +0.619271 q^{95} +4.48241 q^{96} +14.0438 q^{97} -4.45666 q^{98} -1.30399 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 37 q + 12 q^{2} + 6 q^{3} + 42 q^{4} + 31 q^{5} + 4 q^{6} + 9 q^{7} + 36 q^{8} + 47 q^{9} + 4 q^{10} + 18 q^{11} + 15 q^{12} + 13 q^{13} + 8 q^{14} + 3 q^{15} + 48 q^{16} + 18 q^{17} + 17 q^{18} + 40 q^{20}+ \cdots - 35 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.54331 −1.09128 −0.545641 0.838019i \(-0.683714\pi\)
−0.545641 + 0.838019i \(0.683714\pi\)
\(3\) 2.10249 1.21387 0.606937 0.794750i \(-0.292398\pi\)
0.606937 + 0.794750i \(0.292398\pi\)
\(4\) 0.381797 0.190898
\(5\) 1.25926 0.563156 0.281578 0.959538i \(-0.409142\pi\)
0.281578 + 0.959538i \(0.409142\pi\)
\(6\) −3.24479 −1.32468
\(7\) 3.14448 1.18850 0.594250 0.804280i \(-0.297449\pi\)
0.594250 + 0.804280i \(0.297449\pi\)
\(8\) 2.49738 0.882959
\(9\) 1.42047 0.473490
\(10\) −1.94342 −0.614563
\(11\) −0.917999 −0.276787 −0.138394 0.990377i \(-0.544194\pi\)
−0.138394 + 0.990377i \(0.544194\pi\)
\(12\) 0.802724 0.231727
\(13\) 2.37731 0.659346 0.329673 0.944095i \(-0.393061\pi\)
0.329673 + 0.944095i \(0.393061\pi\)
\(14\) −4.85289 −1.29699
\(15\) 2.64757 0.683600
\(16\) −4.61782 −1.15446
\(17\) 4.67298 1.13336 0.566681 0.823937i \(-0.308227\pi\)
0.566681 + 0.823937i \(0.308227\pi\)
\(18\) −2.19222 −0.516712
\(19\) 0.491776 0.112821 0.0564105 0.998408i \(-0.482034\pi\)
0.0564105 + 0.998408i \(0.482034\pi\)
\(20\) 0.480780 0.107506
\(21\) 6.61124 1.44269
\(22\) 1.41675 0.302053
\(23\) −1.93961 −0.404436 −0.202218 0.979341i \(-0.564815\pi\)
−0.202218 + 0.979341i \(0.564815\pi\)
\(24\) 5.25073 1.07180
\(25\) −3.41428 −0.682855
\(26\) −3.66892 −0.719533
\(27\) −3.32095 −0.639117
\(28\) 1.20055 0.226883
\(29\) 5.54398 1.02949 0.514746 0.857343i \(-0.327886\pi\)
0.514746 + 0.857343i \(0.327886\pi\)
\(30\) −4.08602 −0.746002
\(31\) −4.99990 −0.898009 −0.449005 0.893529i \(-0.648221\pi\)
−0.449005 + 0.893529i \(0.648221\pi\)
\(32\) 2.13195 0.376880
\(33\) −1.93008 −0.335985
\(34\) −7.21184 −1.23682
\(35\) 3.95970 0.669311
\(36\) 0.542331 0.0903885
\(37\) 0.00995484 0.00163657 0.000818283 1.00000i \(-0.499740\pi\)
0.000818283 1.00000i \(0.499740\pi\)
\(38\) −0.758961 −0.123120
\(39\) 4.99827 0.800364
\(40\) 3.14484 0.497244
\(41\) 7.33823 1.14604 0.573020 0.819541i \(-0.305772\pi\)
0.573020 + 0.819541i \(0.305772\pi\)
\(42\) −10.2032 −1.57438
\(43\) 2.73113 0.416494 0.208247 0.978076i \(-0.433224\pi\)
0.208247 + 0.978076i \(0.433224\pi\)
\(44\) −0.350489 −0.0528382
\(45\) 1.78874 0.266649
\(46\) 2.99341 0.441354
\(47\) −5.69556 −0.830783 −0.415392 0.909643i \(-0.636355\pi\)
−0.415392 + 0.909643i \(0.636355\pi\)
\(48\) −9.70894 −1.40136
\(49\) 2.88774 0.412534
\(50\) 5.26928 0.745188
\(51\) 9.82489 1.37576
\(52\) 0.907648 0.125868
\(53\) −3.42166 −0.470002 −0.235001 0.971995i \(-0.575509\pi\)
−0.235001 + 0.971995i \(0.575509\pi\)
\(54\) 5.12524 0.697457
\(55\) −1.15599 −0.155874
\(56\) 7.85297 1.04940
\(57\) 1.03395 0.136951
\(58\) −8.55607 −1.12347
\(59\) 1.55022 0.201821 0.100911 0.994895i \(-0.467824\pi\)
0.100911 + 0.994895i \(0.467824\pi\)
\(60\) 1.01083 0.130498
\(61\) 0.660781 0.0846043 0.0423022 0.999105i \(-0.486531\pi\)
0.0423022 + 0.999105i \(0.486531\pi\)
\(62\) 7.71639 0.979982
\(63\) 4.46664 0.562743
\(64\) 5.94539 0.743174
\(65\) 2.99364 0.371315
\(66\) 2.97871 0.366654
\(67\) −6.25946 −0.764715 −0.382357 0.924014i \(-0.624888\pi\)
−0.382357 + 0.924014i \(0.624888\pi\)
\(68\) 1.78413 0.216357
\(69\) −4.07801 −0.490935
\(70\) −6.11103 −0.730408
\(71\) 6.33818 0.752204 0.376102 0.926578i \(-0.377264\pi\)
0.376102 + 0.926578i \(0.377264\pi\)
\(72\) 3.54746 0.418072
\(73\) −6.27766 −0.734744 −0.367372 0.930074i \(-0.619742\pi\)
−0.367372 + 0.930074i \(0.619742\pi\)
\(74\) −0.0153634 −0.00178596
\(75\) −7.17849 −0.828900
\(76\) 0.187758 0.0215374
\(77\) −2.88663 −0.328962
\(78\) −7.71386 −0.873423
\(79\) 8.41425 0.946677 0.473338 0.880881i \(-0.343049\pi\)
0.473338 + 0.880881i \(0.343049\pi\)
\(80\) −5.81502 −0.650139
\(81\) −11.2437 −1.24930
\(82\) −11.3251 −1.25065
\(83\) −2.75585 −0.302494 −0.151247 0.988496i \(-0.548329\pi\)
−0.151247 + 0.988496i \(0.548329\pi\)
\(84\) 2.52415 0.275407
\(85\) 5.88447 0.638260
\(86\) −4.21498 −0.454513
\(87\) 11.6562 1.24967
\(88\) −2.29260 −0.244392
\(89\) 12.8586 1.36301 0.681505 0.731813i \(-0.261326\pi\)
0.681505 + 0.731813i \(0.261326\pi\)
\(90\) −2.76057 −0.290989
\(91\) 7.47539 0.783634
\(92\) −0.740536 −0.0772062
\(93\) −10.5123 −1.09007
\(94\) 8.79000 0.906620
\(95\) 0.619271 0.0635358
\(96\) 4.48241 0.457484
\(97\) 14.0438 1.42593 0.712963 0.701201i \(-0.247353\pi\)
0.712963 + 0.701201i \(0.247353\pi\)
\(98\) −4.45666 −0.450191
\(99\) −1.30399 −0.131056
\(100\) −1.30356 −0.130356
\(101\) −7.55228 −0.751480 −0.375740 0.926725i \(-0.622611\pi\)
−0.375740 + 0.926725i \(0.622611\pi\)
\(102\) −15.1628 −1.50134
\(103\) −9.07007 −0.893701 −0.446850 0.894609i \(-0.647454\pi\)
−0.446850 + 0.894609i \(0.647454\pi\)
\(104\) 5.93705 0.582176
\(105\) 8.32523 0.812460
\(106\) 5.28068 0.512905
\(107\) 18.0943 1.74924 0.874619 0.484811i \(-0.161112\pi\)
0.874619 + 0.484811i \(0.161112\pi\)
\(108\) −1.26793 −0.122006
\(109\) −0.101668 −0.00973802 −0.00486901 0.999988i \(-0.501550\pi\)
−0.00486901 + 0.999988i \(0.501550\pi\)
\(110\) 1.78405 0.170103
\(111\) 0.0209300 0.00198658
\(112\) −14.5206 −1.37207
\(113\) 11.4722 1.07921 0.539606 0.841918i \(-0.318573\pi\)
0.539606 + 0.841918i \(0.318573\pi\)
\(114\) −1.59571 −0.149452
\(115\) −2.44246 −0.227761
\(116\) 2.11668 0.196528
\(117\) 3.37690 0.312194
\(118\) −2.39246 −0.220244
\(119\) 14.6941 1.34700
\(120\) 6.61201 0.603591
\(121\) −10.1573 −0.923389
\(122\) −1.01979 −0.0923273
\(123\) 15.4286 1.39115
\(124\) −1.90895 −0.171429
\(125\) −10.5957 −0.947710
\(126\) −6.89339 −0.614112
\(127\) 4.12393 0.365939 0.182970 0.983119i \(-0.441429\pi\)
0.182970 + 0.983119i \(0.441429\pi\)
\(128\) −13.4395 −1.18789
\(129\) 5.74219 0.505571
\(130\) −4.62010 −0.405210
\(131\) 13.2555 1.15814 0.579068 0.815279i \(-0.303417\pi\)
0.579068 + 0.815279i \(0.303417\pi\)
\(132\) −0.736900 −0.0641389
\(133\) 1.54638 0.134088
\(134\) 9.66027 0.834520
\(135\) −4.18192 −0.359922
\(136\) 11.6702 1.00071
\(137\) −12.3555 −1.05560 −0.527800 0.849369i \(-0.676983\pi\)
−0.527800 + 0.849369i \(0.676983\pi\)
\(138\) 6.29362 0.535749
\(139\) −20.2775 −1.71991 −0.859957 0.510366i \(-0.829510\pi\)
−0.859957 + 0.510366i \(0.829510\pi\)
\(140\) 1.51180 0.127770
\(141\) −11.9749 −1.00847
\(142\) −9.78176 −0.820867
\(143\) −2.18236 −0.182499
\(144\) −6.55948 −0.546624
\(145\) 6.98129 0.579765
\(146\) 9.68836 0.801814
\(147\) 6.07144 0.500764
\(148\) 0.00380073 0.000312418 0
\(149\) −2.91309 −0.238650 −0.119325 0.992855i \(-0.538073\pi\)
−0.119325 + 0.992855i \(0.538073\pi\)
\(150\) 11.0786 0.904565
\(151\) −5.82211 −0.473796 −0.236898 0.971534i \(-0.576131\pi\)
−0.236898 + 0.971534i \(0.576131\pi\)
\(152\) 1.22815 0.0996163
\(153\) 6.63782 0.536636
\(154\) 4.45495 0.358990
\(155\) −6.29616 −0.505719
\(156\) 1.90832 0.152788
\(157\) 6.85944 0.547443 0.273721 0.961809i \(-0.411745\pi\)
0.273721 + 0.961809i \(0.411745\pi\)
\(158\) −12.9858 −1.03309
\(159\) −7.19402 −0.570523
\(160\) 2.68467 0.212242
\(161\) −6.09905 −0.480673
\(162\) 17.3524 1.36334
\(163\) −9.63668 −0.754803 −0.377402 0.926050i \(-0.623182\pi\)
−0.377402 + 0.926050i \(0.623182\pi\)
\(164\) 2.80171 0.218777
\(165\) −2.43047 −0.189212
\(166\) 4.25312 0.330106
\(167\) −6.38436 −0.494037 −0.247018 0.969011i \(-0.579451\pi\)
−0.247018 + 0.969011i \(0.579451\pi\)
\(168\) 16.5108 1.27384
\(169\) −7.34841 −0.565262
\(170\) −9.08154 −0.696522
\(171\) 0.698553 0.0534197
\(172\) 1.04274 0.0795081
\(173\) −10.1853 −0.774376 −0.387188 0.922001i \(-0.626553\pi\)
−0.387188 + 0.922001i \(0.626553\pi\)
\(174\) −17.9891 −1.36375
\(175\) −10.7361 −0.811574
\(176\) 4.23916 0.319538
\(177\) 3.25932 0.244986
\(178\) −19.8448 −1.48743
\(179\) −6.30119 −0.470973 −0.235487 0.971878i \(-0.575668\pi\)
−0.235487 + 0.971878i \(0.575668\pi\)
\(180\) 0.682933 0.0509028
\(181\) −2.30314 −0.171191 −0.0855954 0.996330i \(-0.527279\pi\)
−0.0855954 + 0.996330i \(0.527279\pi\)
\(182\) −11.5368 −0.855166
\(183\) 1.38929 0.102699
\(184\) −4.84395 −0.357100
\(185\) 0.0125357 0.000921642 0
\(186\) 16.2236 1.18958
\(187\) −4.28978 −0.313700
\(188\) −2.17455 −0.158595
\(189\) −10.4426 −0.759590
\(190\) −0.955725 −0.0693356
\(191\) 19.0312 1.37705 0.688526 0.725212i \(-0.258258\pi\)
0.688526 + 0.725212i \(0.258258\pi\)
\(192\) 12.5001 0.902120
\(193\) −18.3716 −1.32241 −0.661207 0.750204i \(-0.729955\pi\)
−0.661207 + 0.750204i \(0.729955\pi\)
\(194\) −21.6738 −1.55609
\(195\) 6.29410 0.450730
\(196\) 1.10253 0.0787520
\(197\) −14.6513 −1.04386 −0.521930 0.852989i \(-0.674788\pi\)
−0.521930 + 0.852989i \(0.674788\pi\)
\(198\) 2.01246 0.143019
\(199\) −11.3317 −0.803282 −0.401641 0.915797i \(-0.631560\pi\)
−0.401641 + 0.915797i \(0.631560\pi\)
\(200\) −8.52676 −0.602933
\(201\) −13.1605 −0.928268
\(202\) 11.6555 0.820078
\(203\) 17.4329 1.22355
\(204\) 3.75111 0.262630
\(205\) 9.24071 0.645399
\(206\) 13.9979 0.975280
\(207\) −2.75516 −0.191497
\(208\) −10.9780 −0.761187
\(209\) −0.451449 −0.0312274
\(210\) −12.8484 −0.886623
\(211\) −24.1080 −1.65967 −0.829833 0.558013i \(-0.811564\pi\)
−0.829833 + 0.558013i \(0.811564\pi\)
\(212\) −1.30638 −0.0897225
\(213\) 13.3260 0.913081
\(214\) −27.9250 −1.90891
\(215\) 3.43920 0.234551
\(216\) −8.29368 −0.564314
\(217\) −15.7221 −1.06728
\(218\) 0.156905 0.0106269
\(219\) −13.1987 −0.891887
\(220\) −0.441355 −0.0297561
\(221\) 11.1091 0.747279
\(222\) −0.0323014 −0.00216793
\(223\) −25.3974 −1.70073 −0.850367 0.526191i \(-0.823620\pi\)
−0.850367 + 0.526191i \(0.823620\pi\)
\(224\) 6.70388 0.447922
\(225\) −4.84988 −0.323325
\(226\) −17.7051 −1.17772
\(227\) 3.56104 0.236354 0.118177 0.992993i \(-0.462295\pi\)
0.118177 + 0.992993i \(0.462295\pi\)
\(228\) 0.394760 0.0261436
\(229\) 1.97251 0.130347 0.0651735 0.997874i \(-0.479240\pi\)
0.0651735 + 0.997874i \(0.479240\pi\)
\(230\) 3.76947 0.248551
\(231\) −6.06911 −0.399318
\(232\) 13.8455 0.908999
\(233\) −8.85144 −0.579877 −0.289939 0.957045i \(-0.593635\pi\)
−0.289939 + 0.957045i \(0.593635\pi\)
\(234\) −5.21159 −0.340692
\(235\) −7.17217 −0.467861
\(236\) 0.591869 0.0385274
\(237\) 17.6909 1.14915
\(238\) −22.6775 −1.46996
\(239\) −8.00959 −0.518097 −0.259049 0.965864i \(-0.583409\pi\)
−0.259049 + 0.965864i \(0.583409\pi\)
\(240\) −12.2260 −0.789187
\(241\) −10.1593 −0.654415 −0.327208 0.944952i \(-0.606108\pi\)
−0.327208 + 0.944952i \(0.606108\pi\)
\(242\) 15.6758 1.00768
\(243\) −13.6769 −0.877373
\(244\) 0.252284 0.0161508
\(245\) 3.63640 0.232321
\(246\) −23.8110 −1.51814
\(247\) 1.16910 0.0743881
\(248\) −12.4867 −0.792905
\(249\) −5.79415 −0.367189
\(250\) 16.3525 1.03422
\(251\) −9.64964 −0.609080 −0.304540 0.952500i \(-0.598503\pi\)
−0.304540 + 0.952500i \(0.598503\pi\)
\(252\) 1.70535 0.107427
\(253\) 1.78056 0.111943
\(254\) −6.36449 −0.399343
\(255\) 12.3720 0.774767
\(256\) 8.85045 0.553153
\(257\) 19.3494 1.20698 0.603491 0.797370i \(-0.293776\pi\)
0.603491 + 0.797370i \(0.293776\pi\)
\(258\) −8.86196 −0.551721
\(259\) 0.0313028 0.00194506
\(260\) 1.14296 0.0708834
\(261\) 7.87507 0.487454
\(262\) −20.4573 −1.26385
\(263\) −3.53940 −0.218249 −0.109124 0.994028i \(-0.534805\pi\)
−0.109124 + 0.994028i \(0.534805\pi\)
\(264\) −4.82016 −0.296661
\(265\) −4.30875 −0.264684
\(266\) −2.38653 −0.146328
\(267\) 27.0351 1.65452
\(268\) −2.38984 −0.145983
\(269\) 21.2672 1.29669 0.648343 0.761348i \(-0.275462\pi\)
0.648343 + 0.761348i \(0.275462\pi\)
\(270\) 6.45399 0.392777
\(271\) 10.6255 0.645453 0.322726 0.946492i \(-0.395401\pi\)
0.322726 + 0.946492i \(0.395401\pi\)
\(272\) −21.5790 −1.30842
\(273\) 15.7169 0.951233
\(274\) 19.0683 1.15196
\(275\) 3.13430 0.189005
\(276\) −1.55697 −0.0937186
\(277\) −6.19742 −0.372367 −0.186183 0.982515i \(-0.559612\pi\)
−0.186183 + 0.982515i \(0.559612\pi\)
\(278\) 31.2944 1.87691
\(279\) −7.10222 −0.425199
\(280\) 9.88889 0.590974
\(281\) −0.403247 −0.0240557 −0.0120278 0.999928i \(-0.503829\pi\)
−0.0120278 + 0.999928i \(0.503829\pi\)
\(282\) 18.4809 1.10052
\(283\) −3.24956 −0.193167 −0.0965833 0.995325i \(-0.530791\pi\)
−0.0965833 + 0.995325i \(0.530791\pi\)
\(284\) 2.41990 0.143595
\(285\) 1.30201 0.0771245
\(286\) 3.36806 0.199158
\(287\) 23.0749 1.36207
\(288\) 3.02838 0.178449
\(289\) 4.83670 0.284512
\(290\) −10.7743 −0.632687
\(291\) 29.5269 1.73090
\(292\) −2.39679 −0.140262
\(293\) 5.58232 0.326123 0.163061 0.986616i \(-0.447863\pi\)
0.163061 + 0.986616i \(0.447863\pi\)
\(294\) −9.37010 −0.546475
\(295\) 1.95212 0.113657
\(296\) 0.0248611 0.00144502
\(297\) 3.04862 0.176899
\(298\) 4.49579 0.260434
\(299\) −4.61104 −0.266664
\(300\) −2.74072 −0.158236
\(301\) 8.58799 0.495004
\(302\) 8.98530 0.517046
\(303\) −15.8786 −0.912202
\(304\) −2.27093 −0.130247
\(305\) 0.832092 0.0476454
\(306\) −10.2442 −0.585622
\(307\) 10.8518 0.619345 0.309673 0.950843i \(-0.399781\pi\)
0.309673 + 0.950843i \(0.399781\pi\)
\(308\) −1.10210 −0.0627982
\(309\) −19.0697 −1.08484
\(310\) 9.71690 0.551883
\(311\) −21.3609 −1.21127 −0.605633 0.795744i \(-0.707080\pi\)
−0.605633 + 0.795744i \(0.707080\pi\)
\(312\) 12.4826 0.706688
\(313\) 8.53183 0.482248 0.241124 0.970494i \(-0.422484\pi\)
0.241124 + 0.970494i \(0.422484\pi\)
\(314\) −10.5862 −0.597415
\(315\) 5.62464 0.316912
\(316\) 3.21253 0.180719
\(317\) −7.28886 −0.409383 −0.204692 0.978827i \(-0.565619\pi\)
−0.204692 + 0.978827i \(0.565619\pi\)
\(318\) 11.1026 0.622602
\(319\) −5.08937 −0.284950
\(320\) 7.48677 0.418523
\(321\) 38.0430 2.12335
\(322\) 9.41271 0.524550
\(323\) 2.29805 0.127867
\(324\) −4.29280 −0.238489
\(325\) −8.11678 −0.450238
\(326\) 14.8724 0.823704
\(327\) −0.213756 −0.0118207
\(328\) 18.3264 1.01191
\(329\) −17.9096 −0.987387
\(330\) 3.75096 0.206484
\(331\) 3.46556 0.190484 0.0952422 0.995454i \(-0.469637\pi\)
0.0952422 + 0.995454i \(0.469637\pi\)
\(332\) −1.05217 −0.0577456
\(333\) 0.0141406 0.000774898 0
\(334\) 9.85303 0.539134
\(335\) −7.88226 −0.430654
\(336\) −30.5295 −1.66552
\(337\) 12.2351 0.666489 0.333244 0.942840i \(-0.391857\pi\)
0.333244 + 0.942840i \(0.391857\pi\)
\(338\) 11.3409 0.616861
\(339\) 24.1201 1.31003
\(340\) 2.24667 0.121843
\(341\) 4.58991 0.248557
\(342\) −1.07808 −0.0582960
\(343\) −12.9309 −0.698204
\(344\) 6.82069 0.367747
\(345\) −5.13525 −0.276473
\(346\) 15.7191 0.845063
\(347\) 15.5937 0.837111 0.418556 0.908191i \(-0.362536\pi\)
0.418556 + 0.908191i \(0.362536\pi\)
\(348\) 4.45029 0.238561
\(349\) 9.55242 0.511329 0.255665 0.966766i \(-0.417706\pi\)
0.255665 + 0.966766i \(0.417706\pi\)
\(350\) 16.5691 0.885657
\(351\) −7.89491 −0.421399
\(352\) −1.95713 −0.104315
\(353\) −9.09226 −0.483932 −0.241966 0.970285i \(-0.577792\pi\)
−0.241966 + 0.970285i \(0.577792\pi\)
\(354\) −5.03014 −0.267349
\(355\) 7.98139 0.423608
\(356\) 4.90938 0.260196
\(357\) 30.8941 1.63509
\(358\) 9.72468 0.513965
\(359\) 17.2094 0.908277 0.454139 0.890931i \(-0.349947\pi\)
0.454139 + 0.890931i \(0.349947\pi\)
\(360\) 4.46716 0.235440
\(361\) −18.7582 −0.987271
\(362\) 3.55445 0.186818
\(363\) −21.3556 −1.12088
\(364\) 2.85408 0.149594
\(365\) −7.90518 −0.413776
\(366\) −2.14410 −0.112074
\(367\) 8.17947 0.426965 0.213483 0.976947i \(-0.431519\pi\)
0.213483 + 0.976947i \(0.431519\pi\)
\(368\) 8.95677 0.466904
\(369\) 10.4237 0.542639
\(370\) −0.0193464 −0.00100577
\(371\) −10.7593 −0.558597
\(372\) −4.01355 −0.208093
\(373\) 3.65734 0.189370 0.0946850 0.995507i \(-0.469816\pi\)
0.0946850 + 0.995507i \(0.469816\pi\)
\(374\) 6.62046 0.342336
\(375\) −22.2774 −1.15040
\(376\) −14.2240 −0.733547
\(377\) 13.1798 0.678792
\(378\) 16.1162 0.828928
\(379\) −13.6390 −0.700589 −0.350294 0.936640i \(-0.613918\pi\)
−0.350294 + 0.936640i \(0.613918\pi\)
\(380\) 0.236436 0.0121289
\(381\) 8.67052 0.444204
\(382\) −29.3710 −1.50275
\(383\) −32.5444 −1.66294 −0.831470 0.555570i \(-0.812500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(384\) −28.2564 −1.44195
\(385\) −3.63500 −0.185257
\(386\) 28.3530 1.44313
\(387\) 3.87950 0.197206
\(388\) 5.36186 0.272207
\(389\) 5.60065 0.283964 0.141982 0.989869i \(-0.454652\pi\)
0.141982 + 0.989869i \(0.454652\pi\)
\(390\) −9.71372 −0.491873
\(391\) −9.06374 −0.458373
\(392\) 7.21179 0.364250
\(393\) 27.8695 1.40583
\(394\) 22.6114 1.13915
\(395\) 10.5957 0.533127
\(396\) −0.497859 −0.0250184
\(397\) 12.8170 0.643264 0.321632 0.946865i \(-0.395769\pi\)
0.321632 + 0.946865i \(0.395769\pi\)
\(398\) 17.4883 0.876608
\(399\) 3.25124 0.162766
\(400\) 15.7665 0.788327
\(401\) 39.6940 1.98222 0.991111 0.133037i \(-0.0424728\pi\)
0.991111 + 0.133037i \(0.0424728\pi\)
\(402\) 20.3106 1.01300
\(403\) −11.8863 −0.592099
\(404\) −2.88344 −0.143456
\(405\) −14.1587 −0.703549
\(406\) −26.9044 −1.33524
\(407\) −0.00913853 −0.000452980 0
\(408\) 24.5365 1.21474
\(409\) −9.10948 −0.450435 −0.225217 0.974309i \(-0.572309\pi\)
−0.225217 + 0.974309i \(0.572309\pi\)
\(410\) −14.2613 −0.704313
\(411\) −25.9773 −1.28137
\(412\) −3.46292 −0.170606
\(413\) 4.87463 0.239865
\(414\) 4.25205 0.208977
\(415\) −3.47032 −0.170351
\(416\) 5.06831 0.248494
\(417\) −42.6333 −2.08776
\(418\) 0.696725 0.0340779
\(419\) −1.62496 −0.0793847 −0.0396924 0.999212i \(-0.512638\pi\)
−0.0396924 + 0.999212i \(0.512638\pi\)
\(420\) 3.17855 0.155097
\(421\) 14.4927 0.706330 0.353165 0.935561i \(-0.385105\pi\)
0.353165 + 0.935561i \(0.385105\pi\)
\(422\) 37.2061 1.81116
\(423\) −8.09038 −0.393368
\(424\) −8.54521 −0.414992
\(425\) −15.9548 −0.773923
\(426\) −20.5661 −0.996429
\(427\) 2.07781 0.100552
\(428\) 6.90833 0.333927
\(429\) −4.58840 −0.221530
\(430\) −5.30773 −0.255962
\(431\) −14.0946 −0.678913 −0.339457 0.940622i \(-0.610243\pi\)
−0.339457 + 0.940622i \(0.610243\pi\)
\(432\) 15.3356 0.737832
\(433\) −15.4666 −0.743280 −0.371640 0.928377i \(-0.621204\pi\)
−0.371640 + 0.928377i \(0.621204\pi\)
\(434\) 24.2640 1.16471
\(435\) 14.6781 0.703761
\(436\) −0.0388165 −0.00185897
\(437\) −0.953852 −0.0456289
\(438\) 20.3697 0.973301
\(439\) 20.3349 0.970530 0.485265 0.874367i \(-0.338723\pi\)
0.485265 + 0.874367i \(0.338723\pi\)
\(440\) −2.88696 −0.137631
\(441\) 4.10194 0.195331
\(442\) −17.1447 −0.815493
\(443\) −36.0512 −1.71285 −0.856423 0.516275i \(-0.827318\pi\)
−0.856423 + 0.516275i \(0.827318\pi\)
\(444\) 0.00799099 0.000379236 0
\(445\) 16.1923 0.767587
\(446\) 39.1959 1.85598
\(447\) −6.12475 −0.289691
\(448\) 18.6951 0.883263
\(449\) −22.8583 −1.07875 −0.539376 0.842065i \(-0.681340\pi\)
−0.539376 + 0.842065i \(0.681340\pi\)
\(450\) 7.48485 0.352839
\(451\) −6.73649 −0.317209
\(452\) 4.38004 0.206020
\(453\) −12.2409 −0.575129
\(454\) −5.49578 −0.257930
\(455\) 9.41342 0.441308
\(456\) 2.58218 0.120922
\(457\) 27.9339 1.30669 0.653346 0.757059i \(-0.273365\pi\)
0.653346 + 0.757059i \(0.273365\pi\)
\(458\) −3.04419 −0.142246
\(459\) −15.5187 −0.724351
\(460\) −0.932524 −0.0434791
\(461\) 21.1066 0.983033 0.491516 0.870868i \(-0.336443\pi\)
0.491516 + 0.870868i \(0.336443\pi\)
\(462\) 9.36649 0.435769
\(463\) 28.0736 1.30469 0.652345 0.757922i \(-0.273785\pi\)
0.652345 + 0.757922i \(0.273785\pi\)
\(464\) −25.6011 −1.18850
\(465\) −13.2376 −0.613880
\(466\) 13.6605 0.632810
\(467\) −4.91492 −0.227435 −0.113718 0.993513i \(-0.536276\pi\)
−0.113718 + 0.993513i \(0.536276\pi\)
\(468\) 1.28929 0.0595973
\(469\) −19.6827 −0.908864
\(470\) 11.0689 0.510568
\(471\) 14.4219 0.664527
\(472\) 3.87149 0.178200
\(473\) −2.50718 −0.115280
\(474\) −27.3025 −1.25404
\(475\) −1.67906 −0.0770404
\(476\) 5.61015 0.257141
\(477\) −4.86037 −0.222541
\(478\) 12.3613 0.565391
\(479\) 31.5688 1.44241 0.721207 0.692720i \(-0.243588\pi\)
0.721207 + 0.692720i \(0.243588\pi\)
\(480\) 5.64450 0.257635
\(481\) 0.0236657 0.00107906
\(482\) 15.6789 0.714152
\(483\) −12.8232 −0.583476
\(484\) −3.87802 −0.176273
\(485\) 17.6847 0.803019
\(486\) 21.1076 0.957462
\(487\) 26.3909 1.19588 0.597942 0.801540i \(-0.295985\pi\)
0.597942 + 0.801540i \(0.295985\pi\)
\(488\) 1.65022 0.0747021
\(489\) −20.2610 −0.916236
\(490\) −5.61208 −0.253528
\(491\) 33.9322 1.53134 0.765669 0.643235i \(-0.222408\pi\)
0.765669 + 0.643235i \(0.222408\pi\)
\(492\) 5.89058 0.265568
\(493\) 25.9069 1.16679
\(494\) −1.80428 −0.0811785
\(495\) −1.64206 −0.0738049
\(496\) 23.0887 1.03671
\(497\) 19.9303 0.893995
\(498\) 8.94215 0.400707
\(499\) 4.42373 0.198033 0.0990167 0.995086i \(-0.468430\pi\)
0.0990167 + 0.995086i \(0.468430\pi\)
\(500\) −4.04541 −0.180916
\(501\) −13.4231 −0.599698
\(502\) 14.8924 0.664678
\(503\) 22.7962 1.01643 0.508216 0.861230i \(-0.330305\pi\)
0.508216 + 0.861230i \(0.330305\pi\)
\(504\) 11.1549 0.496879
\(505\) −9.51025 −0.423201
\(506\) −2.74795 −0.122161
\(507\) −15.4500 −0.686157
\(508\) 1.57450 0.0698572
\(509\) −14.4925 −0.642369 −0.321185 0.947017i \(-0.604081\pi\)
−0.321185 + 0.947017i \(0.604081\pi\)
\(510\) −19.0939 −0.845490
\(511\) −19.7400 −0.873244
\(512\) 13.2200 0.584246
\(513\) −1.63316 −0.0721058
\(514\) −29.8621 −1.31716
\(515\) −11.4215 −0.503293
\(516\) 2.19235 0.0965128
\(517\) 5.22852 0.229950
\(518\) −0.0483098 −0.00212261
\(519\) −21.4145 −0.939994
\(520\) 7.47626 0.327856
\(521\) −16.3460 −0.716131 −0.358065 0.933697i \(-0.616564\pi\)
−0.358065 + 0.933697i \(0.616564\pi\)
\(522\) −12.1536 −0.531951
\(523\) 11.3842 0.497795 0.248897 0.968530i \(-0.419932\pi\)
0.248897 + 0.968530i \(0.419932\pi\)
\(524\) 5.06090 0.221086
\(525\) −22.5726 −0.985149
\(526\) 5.46238 0.238171
\(527\) −23.3644 −1.01777
\(528\) 8.91279 0.387879
\(529\) −19.2379 −0.836431
\(530\) 6.64972 0.288845
\(531\) 2.20204 0.0955605
\(532\) 0.590402 0.0255972
\(533\) 17.4452 0.755637
\(534\) −41.7235 −1.80555
\(535\) 22.7853 0.985094
\(536\) −15.6323 −0.675212
\(537\) −13.2482 −0.571702
\(538\) −32.8219 −1.41505
\(539\) −2.65094 −0.114184
\(540\) −1.59664 −0.0687086
\(541\) 14.1027 0.606322 0.303161 0.952939i \(-0.401958\pi\)
0.303161 + 0.952939i \(0.401958\pi\)
\(542\) −16.3984 −0.704371
\(543\) −4.84233 −0.207804
\(544\) 9.96256 0.427141
\(545\) −0.128026 −0.00548403
\(546\) −24.2561 −1.03806
\(547\) 15.9165 0.680541 0.340270 0.940328i \(-0.389481\pi\)
0.340270 + 0.940328i \(0.389481\pi\)
\(548\) −4.71728 −0.201512
\(549\) 0.938620 0.0400593
\(550\) −4.83719 −0.206258
\(551\) 2.72640 0.116148
\(552\) −10.1844 −0.433475
\(553\) 26.4584 1.12513
\(554\) 9.56452 0.406358
\(555\) 0.0263562 0.00111876
\(556\) −7.74188 −0.328329
\(557\) 19.8147 0.839574 0.419787 0.907623i \(-0.362105\pi\)
0.419787 + 0.907623i \(0.362105\pi\)
\(558\) 10.9609 0.464012
\(559\) 6.49275 0.274614
\(560\) −18.2852 −0.772691
\(561\) −9.01924 −0.380792
\(562\) 0.622334 0.0262516
\(563\) −36.3627 −1.53250 −0.766252 0.642540i \(-0.777881\pi\)
−0.766252 + 0.642540i \(0.777881\pi\)
\(564\) −4.57197 −0.192515
\(565\) 14.4464 0.607764
\(566\) 5.01507 0.210799
\(567\) −35.3555 −1.48479
\(568\) 15.8289 0.664165
\(569\) −23.0447 −0.966085 −0.483043 0.875597i \(-0.660468\pi\)
−0.483043 + 0.875597i \(0.660468\pi\)
\(570\) −2.00940 −0.0841647
\(571\) −32.5529 −1.36229 −0.681147 0.732146i \(-0.738519\pi\)
−0.681147 + 0.732146i \(0.738519\pi\)
\(572\) −0.833220 −0.0348387
\(573\) 40.0130 1.67157
\(574\) −35.6117 −1.48640
\(575\) 6.62236 0.276171
\(576\) 8.44525 0.351886
\(577\) 23.8906 0.994579 0.497289 0.867585i \(-0.334329\pi\)
0.497289 + 0.867585i \(0.334329\pi\)
\(578\) −7.46451 −0.310483
\(579\) −38.6261 −1.60524
\(580\) 2.66543 0.110676
\(581\) −8.66570 −0.359514
\(582\) −45.5690 −1.88890
\(583\) 3.14108 0.130090
\(584\) −15.6777 −0.648749
\(585\) 4.25237 0.175814
\(586\) −8.61524 −0.355892
\(587\) −13.1090 −0.541066 −0.270533 0.962711i \(-0.587200\pi\)
−0.270533 + 0.962711i \(0.587200\pi\)
\(588\) 2.31806 0.0955950
\(589\) −2.45883 −0.101314
\(590\) −3.01272 −0.124032
\(591\) −30.8042 −1.26711
\(592\) −0.0459697 −0.00188934
\(593\) −0.00581924 −0.000238967 0 −0.000119484 1.00000i \(-0.500038\pi\)
−0.000119484 1.00000i \(0.500038\pi\)
\(594\) −4.70496 −0.193047
\(595\) 18.5036 0.758573
\(596\) −1.11221 −0.0455578
\(597\) −23.8248 −0.975083
\(598\) 7.11626 0.291005
\(599\) −36.4163 −1.48793 −0.743965 0.668219i \(-0.767057\pi\)
−0.743965 + 0.668219i \(0.767057\pi\)
\(600\) −17.9274 −0.731885
\(601\) −15.5649 −0.634907 −0.317454 0.948274i \(-0.602828\pi\)
−0.317454 + 0.948274i \(0.602828\pi\)
\(602\) −13.2539 −0.540189
\(603\) −8.89138 −0.362085
\(604\) −2.22286 −0.0904469
\(605\) −12.7906 −0.520012
\(606\) 24.5056 0.995471
\(607\) −33.2538 −1.34973 −0.674865 0.737941i \(-0.735798\pi\)
−0.674865 + 0.737941i \(0.735798\pi\)
\(608\) 1.04844 0.0425199
\(609\) 36.6526 1.48524
\(610\) −1.28417 −0.0519946
\(611\) −13.5401 −0.547774
\(612\) 2.53430 0.102443
\(613\) 14.6747 0.592705 0.296353 0.955079i \(-0.404230\pi\)
0.296353 + 0.955079i \(0.404230\pi\)
\(614\) −16.7477 −0.675881
\(615\) 19.4285 0.783433
\(616\) −7.20901 −0.290459
\(617\) −2.79841 −0.112660 −0.0563298 0.998412i \(-0.517940\pi\)
−0.0563298 + 0.998412i \(0.517940\pi\)
\(618\) 29.4305 1.18387
\(619\) −25.0599 −1.00724 −0.503622 0.863924i \(-0.668000\pi\)
−0.503622 + 0.863924i \(0.668000\pi\)
\(620\) −2.40385 −0.0965410
\(621\) 6.44134 0.258482
\(622\) 32.9665 1.32183
\(623\) 40.4336 1.61994
\(624\) −23.0811 −0.923985
\(625\) 3.72867 0.149147
\(626\) −13.1672 −0.526269
\(627\) −0.949168 −0.0379061
\(628\) 2.61891 0.104506
\(629\) 0.0465187 0.00185482
\(630\) −8.68054 −0.345841
\(631\) 5.16456 0.205598 0.102799 0.994702i \(-0.467220\pi\)
0.102799 + 0.994702i \(0.467220\pi\)
\(632\) 21.0136 0.835876
\(633\) −50.6869 −2.01462
\(634\) 11.2489 0.446753
\(635\) 5.19308 0.206081
\(636\) −2.74665 −0.108912
\(637\) 6.86504 0.272003
\(638\) 7.85446 0.310961
\(639\) 9.00320 0.356161
\(640\) −16.9237 −0.668969
\(641\) 7.13556 0.281838 0.140919 0.990021i \(-0.454994\pi\)
0.140919 + 0.990021i \(0.454994\pi\)
\(642\) −58.7121 −2.31718
\(643\) −33.7559 −1.33120 −0.665601 0.746307i \(-0.731825\pi\)
−0.665601 + 0.746307i \(0.731825\pi\)
\(644\) −2.32860 −0.0917596
\(645\) 7.23088 0.284716
\(646\) −3.54660 −0.139539
\(647\) 5.75214 0.226140 0.113070 0.993587i \(-0.463932\pi\)
0.113070 + 0.993587i \(0.463932\pi\)
\(648\) −28.0798 −1.10308
\(649\) −1.42310 −0.0558615
\(650\) 12.5267 0.491337
\(651\) −33.0556 −1.29555
\(652\) −3.67925 −0.144091
\(653\) −48.0198 −1.87916 −0.939581 0.342327i \(-0.888785\pi\)
−0.939581 + 0.342327i \(0.888785\pi\)
\(654\) 0.329891 0.0128998
\(655\) 16.6920 0.652211
\(656\) −33.8867 −1.32305
\(657\) −8.91723 −0.347894
\(658\) 27.6400 1.07752
\(659\) −24.1192 −0.939552 −0.469776 0.882786i \(-0.655665\pi\)
−0.469776 + 0.882786i \(0.655665\pi\)
\(660\) −0.927945 −0.0361202
\(661\) 45.0095 1.75067 0.875333 0.483520i \(-0.160642\pi\)
0.875333 + 0.483520i \(0.160642\pi\)
\(662\) −5.34842 −0.207872
\(663\) 23.3568 0.907102
\(664\) −6.88241 −0.267089
\(665\) 1.94728 0.0755124
\(666\) −0.0218232 −0.000845633 0
\(667\) −10.7532 −0.416364
\(668\) −2.43753 −0.0943108
\(669\) −53.3977 −2.06448
\(670\) 12.1647 0.469965
\(671\) −0.606596 −0.0234174
\(672\) 14.0948 0.543720
\(673\) 50.0165 1.92799 0.963997 0.265915i \(-0.0856740\pi\)
0.963997 + 0.265915i \(0.0856740\pi\)
\(674\) −18.8825 −0.727328
\(675\) 11.3386 0.436424
\(676\) −2.80560 −0.107908
\(677\) 11.4711 0.440871 0.220436 0.975402i \(-0.429252\pi\)
0.220436 + 0.975402i \(0.429252\pi\)
\(678\) −37.2248 −1.42961
\(679\) 44.1603 1.69472
\(680\) 14.6958 0.563557
\(681\) 7.48705 0.286905
\(682\) −7.08363 −0.271246
\(683\) 11.1992 0.428525 0.214262 0.976776i \(-0.431265\pi\)
0.214262 + 0.976776i \(0.431265\pi\)
\(684\) 0.266705 0.0101977
\(685\) −15.5587 −0.594468
\(686\) 19.9564 0.761938
\(687\) 4.14718 0.158225
\(688\) −12.6119 −0.480824
\(689\) −8.13435 −0.309894
\(690\) 7.92527 0.301710
\(691\) 8.42528 0.320513 0.160256 0.987075i \(-0.448768\pi\)
0.160256 + 0.987075i \(0.448768\pi\)
\(692\) −3.88872 −0.147827
\(693\) −4.10037 −0.155760
\(694\) −24.0658 −0.913525
\(695\) −25.5345 −0.968580
\(696\) 29.1100 1.10341
\(697\) 34.2914 1.29888
\(698\) −14.7423 −0.558005
\(699\) −18.6101 −0.703898
\(700\) −4.09901 −0.154928
\(701\) 29.4021 1.11050 0.555250 0.831683i \(-0.312622\pi\)
0.555250 + 0.831683i \(0.312622\pi\)
\(702\) 12.1843 0.459866
\(703\) 0.00489555 0.000184639 0
\(704\) −5.45786 −0.205701
\(705\) −15.0794 −0.567924
\(706\) 14.0322 0.528107
\(707\) −23.7480 −0.893135
\(708\) 1.24440 0.0467674
\(709\) −26.8200 −1.00725 −0.503624 0.863923i \(-0.668000\pi\)
−0.503624 + 0.863923i \(0.668000\pi\)
\(710\) −12.3177 −0.462276
\(711\) 11.9522 0.448242
\(712\) 32.1129 1.20348
\(713\) 9.69785 0.363188
\(714\) −47.6792 −1.78435
\(715\) −2.74815 −0.102775
\(716\) −2.40578 −0.0899080
\(717\) −16.8401 −0.628905
\(718\) −26.5594 −0.991188
\(719\) 35.9642 1.34124 0.670620 0.741801i \(-0.266028\pi\)
0.670620 + 0.741801i \(0.266028\pi\)
\(720\) −8.26006 −0.307834
\(721\) −28.5206 −1.06216
\(722\) 28.9496 1.07739
\(723\) −21.3598 −0.794378
\(724\) −0.879330 −0.0326801
\(725\) −18.9287 −0.702994
\(726\) 32.9582 1.22319
\(727\) −1.77216 −0.0657256 −0.0328628 0.999460i \(-0.510462\pi\)
−0.0328628 + 0.999460i \(0.510462\pi\)
\(728\) 18.6689 0.691916
\(729\) 4.97548 0.184277
\(730\) 12.2001 0.451546
\(731\) 12.7625 0.472039
\(732\) 0.530425 0.0196051
\(733\) 42.9239 1.58543 0.792715 0.609592i \(-0.208667\pi\)
0.792715 + 0.609592i \(0.208667\pi\)
\(734\) −12.6234 −0.465940
\(735\) 7.64549 0.282008
\(736\) −4.13515 −0.152424
\(737\) 5.74618 0.211663
\(738\) −16.0870 −0.592172
\(739\) −40.1728 −1.47778 −0.738891 0.673825i \(-0.764650\pi\)
−0.738891 + 0.673825i \(0.764650\pi\)
\(740\) 0.00478608 0.000175940 0
\(741\) 2.45803 0.0902978
\(742\) 16.6050 0.609588
\(743\) −37.6445 −1.38104 −0.690521 0.723313i \(-0.742619\pi\)
−0.690521 + 0.723313i \(0.742619\pi\)
\(744\) −26.2531 −0.962487
\(745\) −3.66832 −0.134397
\(746\) −5.64440 −0.206656
\(747\) −3.91460 −0.143228
\(748\) −1.63783 −0.0598848
\(749\) 56.8970 2.07897
\(750\) 34.3809 1.25541
\(751\) 10.5774 0.385973 0.192987 0.981201i \(-0.438183\pi\)
0.192987 + 0.981201i \(0.438183\pi\)
\(752\) 26.3011 0.959103
\(753\) −20.2883 −0.739346
\(754\) −20.3404 −0.740754
\(755\) −7.33152 −0.266821
\(756\) −3.98697 −0.145005
\(757\) 6.11713 0.222331 0.111165 0.993802i \(-0.464542\pi\)
0.111165 + 0.993802i \(0.464542\pi\)
\(758\) 21.0492 0.764541
\(759\) 3.74361 0.135884
\(760\) 1.54656 0.0560995
\(761\) −42.3024 −1.53346 −0.766730 0.641969i \(-0.778118\pi\)
−0.766730 + 0.641969i \(0.778118\pi\)
\(762\) −13.3813 −0.484753
\(763\) −0.319692 −0.0115736
\(764\) 7.26606 0.262877
\(765\) 8.35871 0.302210
\(766\) 50.2259 1.81474
\(767\) 3.68535 0.133070
\(768\) 18.6080 0.671458
\(769\) 9.78310 0.352787 0.176394 0.984320i \(-0.443557\pi\)
0.176394 + 0.984320i \(0.443557\pi\)
\(770\) 5.60992 0.202167
\(771\) 40.6819 1.46512
\(772\) −7.01420 −0.252447
\(773\) 50.6466 1.82163 0.910815 0.412815i \(-0.135454\pi\)
0.910815 + 0.412815i \(0.135454\pi\)
\(774\) −5.98725 −0.215207
\(775\) 17.0711 0.613210
\(776\) 35.0726 1.25903
\(777\) 0.0658138 0.00236106
\(778\) −8.64353 −0.309886
\(779\) 3.60876 0.129297
\(780\) 2.40307 0.0860435
\(781\) −5.81844 −0.208200
\(782\) 13.9881 0.500215
\(783\) −18.4113 −0.657965
\(784\) −13.3351 −0.476252
\(785\) 8.63778 0.308296
\(786\) −43.0112 −1.53416
\(787\) 1.00000 0.0356462
\(788\) −5.59380 −0.199271
\(789\) −7.44156 −0.264927
\(790\) −16.3524 −0.581792
\(791\) 36.0740 1.28264
\(792\) −3.25656 −0.115717
\(793\) 1.57088 0.0557836
\(794\) −19.7805 −0.701983
\(795\) −9.05910 −0.321293
\(796\) −4.32640 −0.153345
\(797\) 18.1740 0.643755 0.321877 0.946781i \(-0.395686\pi\)
0.321877 + 0.946781i \(0.395686\pi\)
\(798\) −5.01767 −0.177624
\(799\) −26.6152 −0.941579
\(800\) −7.27908 −0.257354
\(801\) 18.2653 0.645372
\(802\) −61.2600 −2.16317
\(803\) 5.76288 0.203368
\(804\) −5.02462 −0.177205
\(805\) −7.68026 −0.270694
\(806\) 18.3442 0.646148
\(807\) 44.7142 1.57401
\(808\) −18.8610 −0.663526
\(809\) 19.5962 0.688966 0.344483 0.938793i \(-0.388054\pi\)
0.344483 + 0.938793i \(0.388054\pi\)
\(810\) 21.8512 0.767771
\(811\) 47.5922 1.67119 0.835594 0.549348i \(-0.185124\pi\)
0.835594 + 0.549348i \(0.185124\pi\)
\(812\) 6.65584 0.233574
\(813\) 22.3400 0.783498
\(814\) 0.0141036 0.000494330 0
\(815\) −12.1350 −0.425072
\(816\) −45.3696 −1.58825
\(817\) 1.34311 0.0469893
\(818\) 14.0587 0.491552
\(819\) 10.6186 0.371043
\(820\) 3.52807 0.123206
\(821\) −53.1186 −1.85385 −0.926927 0.375243i \(-0.877559\pi\)
−0.926927 + 0.375243i \(0.877559\pi\)
\(822\) 40.0909 1.39833
\(823\) −25.6754 −0.894989 −0.447494 0.894287i \(-0.647684\pi\)
−0.447494 + 0.894287i \(0.647684\pi\)
\(824\) −22.6515 −0.789101
\(825\) 6.58984 0.229429
\(826\) −7.52305 −0.261760
\(827\) −11.0337 −0.383680 −0.191840 0.981426i \(-0.561446\pi\)
−0.191840 + 0.981426i \(0.561446\pi\)
\(828\) −1.05191 −0.0365564
\(829\) 33.3089 1.15687 0.578433 0.815730i \(-0.303664\pi\)
0.578433 + 0.815730i \(0.303664\pi\)
\(830\) 5.35576 0.185901
\(831\) −13.0300 −0.452006
\(832\) 14.1340 0.490009
\(833\) 13.4943 0.467550
\(834\) 65.7962 2.27834
\(835\) −8.03954 −0.278220
\(836\) −0.172362 −0.00596126
\(837\) 16.6044 0.573933
\(838\) 2.50782 0.0866312
\(839\) 30.7148 1.06039 0.530197 0.847875i \(-0.322118\pi\)
0.530197 + 0.847875i \(0.322118\pi\)
\(840\) 20.7913 0.717368
\(841\) 1.73576 0.0598539
\(842\) −22.3667 −0.770806
\(843\) −0.847823 −0.0292006
\(844\) −9.20437 −0.316827
\(845\) −9.25352 −0.318331
\(846\) 12.4859 0.429276
\(847\) −31.9393 −1.09745
\(848\) 15.8006 0.542596
\(849\) −6.83218 −0.234480
\(850\) 24.6232 0.844569
\(851\) −0.0193085 −0.000661886 0
\(852\) 5.08781 0.174306
\(853\) 32.9507 1.12821 0.564104 0.825703i \(-0.309222\pi\)
0.564104 + 0.825703i \(0.309222\pi\)
\(854\) −3.20670 −0.109731
\(855\) 0.879656 0.0300836
\(856\) 45.1883 1.54451
\(857\) −27.1765 −0.928332 −0.464166 0.885748i \(-0.653646\pi\)
−0.464166 + 0.885748i \(0.653646\pi\)
\(858\) 7.08132 0.241752
\(859\) 14.7938 0.504757 0.252379 0.967629i \(-0.418787\pi\)
0.252379 + 0.967629i \(0.418787\pi\)
\(860\) 1.31307 0.0447754
\(861\) 48.5148 1.65338
\(862\) 21.7523 0.740886
\(863\) −37.2987 −1.26966 −0.634832 0.772650i \(-0.718931\pi\)
−0.634832 + 0.772650i \(0.718931\pi\)
\(864\) −7.08010 −0.240870
\(865\) −12.8259 −0.436094
\(866\) 23.8698 0.811128
\(867\) 10.1691 0.345361
\(868\) −6.00264 −0.203743
\(869\) −7.72427 −0.262028
\(870\) −22.6528 −0.768003
\(871\) −14.8807 −0.504212
\(872\) −0.253904 −0.00859827
\(873\) 19.9487 0.675162
\(874\) 1.47209 0.0497941
\(875\) −33.3180 −1.12635
\(876\) −5.03923 −0.170260
\(877\) −39.6680 −1.33949 −0.669747 0.742589i \(-0.733597\pi\)
−0.669747 + 0.742589i \(0.733597\pi\)
\(878\) −31.3830 −1.05912
\(879\) 11.7368 0.395872
\(880\) 5.33818 0.179950
\(881\) 41.7817 1.40766 0.703830 0.710369i \(-0.251472\pi\)
0.703830 + 0.710369i \(0.251472\pi\)
\(882\) −6.33056 −0.213161
\(883\) −37.7142 −1.26918 −0.634592 0.772847i \(-0.718832\pi\)
−0.634592 + 0.772847i \(0.718832\pi\)
\(884\) 4.24142 0.142654
\(885\) 4.10432 0.137965
\(886\) 55.6381 1.86920
\(887\) 25.7369 0.864162 0.432081 0.901835i \(-0.357779\pi\)
0.432081 + 0.901835i \(0.357779\pi\)
\(888\) 0.0522702 0.00175407
\(889\) 12.9676 0.434919
\(890\) −24.9897 −0.837655
\(891\) 10.3217 0.345789
\(892\) −9.69663 −0.324667
\(893\) −2.80094 −0.0937298
\(894\) 9.45236 0.316134
\(895\) −7.93481 −0.265231
\(896\) −42.2601 −1.41181
\(897\) −9.69468 −0.323696
\(898\) 35.2774 1.17722
\(899\) −27.7194 −0.924494
\(900\) −1.85167 −0.0617223
\(901\) −15.9893 −0.532682
\(902\) 10.3965 0.346165
\(903\) 18.0562 0.600872
\(904\) 28.6504 0.952899
\(905\) −2.90024 −0.0964072
\(906\) 18.8915 0.627628
\(907\) −52.6909 −1.74957 −0.874787 0.484508i \(-0.838999\pi\)
−0.874787 + 0.484508i \(0.838999\pi\)
\(908\) 1.35959 0.0451197
\(909\) −10.7278 −0.355819
\(910\) −14.5278 −0.481592
\(911\) 0.00578822 0.000191772 0 9.58861e−5 1.00000i \(-0.499969\pi\)
9.58861e−5 1.00000i \(0.499969\pi\)
\(912\) −4.77462 −0.158103
\(913\) 2.52987 0.0837263
\(914\) −43.1106 −1.42597
\(915\) 1.74947 0.0578356
\(916\) 0.753098 0.0248830
\(917\) 41.6815 1.37645
\(918\) 23.9501 0.790472
\(919\) 51.0704 1.68466 0.842328 0.538965i \(-0.181184\pi\)
0.842328 + 0.538965i \(0.181184\pi\)
\(920\) −6.09976 −0.201103
\(921\) 22.8158 0.751807
\(922\) −32.5740 −1.07277
\(923\) 15.0678 0.495963
\(924\) −2.31717 −0.0762291
\(925\) −0.0339886 −0.00111754
\(926\) −43.3262 −1.42379
\(927\) −12.8838 −0.423158
\(928\) 11.8195 0.387994
\(929\) −1.83007 −0.0600426 −0.0300213 0.999549i \(-0.509558\pi\)
−0.0300213 + 0.999549i \(0.509558\pi\)
\(930\) 20.4297 0.669916
\(931\) 1.42012 0.0465425
\(932\) −3.37945 −0.110698
\(933\) −44.9112 −1.47033
\(934\) 7.58523 0.248196
\(935\) −5.40193 −0.176662
\(936\) 8.43341 0.275655
\(937\) −0.504327 −0.0164757 −0.00823783 0.999966i \(-0.502622\pi\)
−0.00823783 + 0.999966i \(0.502622\pi\)
\(938\) 30.3765 0.991828
\(939\) 17.9381 0.585388
\(940\) −2.73831 −0.0893138
\(941\) 34.3603 1.12011 0.560056 0.828455i \(-0.310780\pi\)
0.560056 + 0.828455i \(0.310780\pi\)
\(942\) −22.2574 −0.725187
\(943\) −14.2333 −0.463500
\(944\) −7.15864 −0.232994
\(945\) −13.1500 −0.427768
\(946\) 3.86935 0.125803
\(947\) 24.7169 0.803192 0.401596 0.915817i \(-0.368456\pi\)
0.401596 + 0.915817i \(0.368456\pi\)
\(948\) 6.75432 0.219370
\(949\) −14.9239 −0.484451
\(950\) 2.59130 0.0840729
\(951\) −15.3248 −0.496939
\(952\) 36.6967 1.18935
\(953\) 13.2820 0.430245 0.215122 0.976587i \(-0.430985\pi\)
0.215122 + 0.976587i \(0.430985\pi\)
\(954\) 7.50105 0.242855
\(955\) 23.9652 0.775495
\(956\) −3.05804 −0.0989039
\(957\) −10.7004 −0.345893
\(958\) −48.7203 −1.57408
\(959\) −38.8515 −1.25458
\(960\) 15.7409 0.508034
\(961\) −6.00095 −0.193579
\(962\) −0.0365235 −0.00117756
\(963\) 25.7024 0.828247
\(964\) −3.87877 −0.124927
\(965\) −23.1345 −0.744725
\(966\) 19.7901 0.636737
\(967\) −1.03353 −0.0332361 −0.0166181 0.999862i \(-0.505290\pi\)
−0.0166181 + 0.999862i \(0.505290\pi\)
\(968\) −25.3666 −0.815314
\(969\) 4.83164 0.155215
\(970\) −27.2929 −0.876321
\(971\) −61.0345 −1.95869 −0.979344 0.202199i \(-0.935191\pi\)
−0.979344 + 0.202199i \(0.935191\pi\)
\(972\) −5.22179 −0.167489
\(973\) −63.7621 −2.04412
\(974\) −40.7292 −1.30505
\(975\) −17.0655 −0.546532
\(976\) −3.05137 −0.0976720
\(977\) 4.88191 0.156186 0.0780931 0.996946i \(-0.475117\pi\)
0.0780931 + 0.996946i \(0.475117\pi\)
\(978\) 31.2690 0.999873
\(979\) −11.8042 −0.377264
\(980\) 1.38836 0.0443497
\(981\) −0.144416 −0.00461086
\(982\) −52.3678 −1.67112
\(983\) −18.3456 −0.585133 −0.292567 0.956245i \(-0.594509\pi\)
−0.292567 + 0.956245i \(0.594509\pi\)
\(984\) 38.5311 1.22833
\(985\) −18.4497 −0.587856
\(986\) −39.9823 −1.27330
\(987\) −37.6547 −1.19856
\(988\) 0.446359 0.0142006
\(989\) −5.29733 −0.168445
\(990\) 2.53420 0.0805421
\(991\) −34.0250 −1.08084 −0.540420 0.841396i \(-0.681735\pi\)
−0.540420 + 0.841396i \(0.681735\pi\)
\(992\) −10.6596 −0.338441
\(993\) 7.28631 0.231224
\(994\) −30.7585 −0.975601
\(995\) −14.2695 −0.452373
\(996\) −2.21219 −0.0700958
\(997\) 2.02210 0.0640406 0.0320203 0.999487i \(-0.489806\pi\)
0.0320203 + 0.999487i \(0.489806\pi\)
\(998\) −6.82718 −0.216111
\(999\) −0.0330595 −0.00104596
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 787.2.a.b.1.9 37
3.2 odd 2 7083.2.a.g.1.29 37
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
787.2.a.b.1.9 37 1.1 even 1 trivial
7083.2.a.g.1.29 37 3.2 odd 2