Properties

Label 5225.2.a.bc.1.8
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
Dimension $30$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(41.7218350561\)
Analytic rank: \(0\)
Dimension: \(30\)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.66433 q^{2} +1.25863 q^{3} +0.769991 q^{4} -2.09478 q^{6} +4.13429 q^{7} +2.04714 q^{8} -1.41584 q^{9} +O(q^{10})\) \(q-1.66433 q^{2} +1.25863 q^{3} +0.769991 q^{4} -2.09478 q^{6} +4.13429 q^{7} +2.04714 q^{8} -1.41584 q^{9} +1.00000 q^{11} +0.969136 q^{12} +3.80507 q^{13} -6.88082 q^{14} -4.94710 q^{16} -2.64209 q^{17} +2.35643 q^{18} +1.00000 q^{19} +5.20355 q^{21} -1.66433 q^{22} +5.58456 q^{23} +2.57660 q^{24} -6.33289 q^{26} -5.55792 q^{27} +3.18337 q^{28} -6.34374 q^{29} +7.53516 q^{31} +4.13932 q^{32} +1.25863 q^{33} +4.39731 q^{34} -1.09019 q^{36} +3.00157 q^{37} -1.66433 q^{38} +4.78918 q^{39} -4.89269 q^{41} -8.66042 q^{42} +10.6638 q^{43} +0.769991 q^{44} -9.29455 q^{46} -6.54878 q^{47} -6.22657 q^{48} +10.0924 q^{49} -3.32543 q^{51} +2.92987 q^{52} +7.14902 q^{53} +9.25022 q^{54} +8.46347 q^{56} +1.25863 q^{57} +10.5581 q^{58} -7.34337 q^{59} +4.72825 q^{61} -12.5410 q^{62} -5.85351 q^{63} +3.00501 q^{64} -2.09478 q^{66} +7.94790 q^{67} -2.03439 q^{68} +7.02891 q^{69} +9.69764 q^{71} -2.89843 q^{72} -16.9975 q^{73} -4.99561 q^{74} +0.769991 q^{76} +4.13429 q^{77} -7.97078 q^{78} +3.09141 q^{79} -2.74785 q^{81} +8.14304 q^{82} -1.51906 q^{83} +4.00669 q^{84} -17.7481 q^{86} -7.98443 q^{87} +2.04714 q^{88} +8.71704 q^{89} +15.7313 q^{91} +4.30007 q^{92} +9.48400 q^{93} +10.8993 q^{94} +5.20988 q^{96} +6.68784 q^{97} -16.7970 q^{98} -1.41584 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q + 42 q^{4} + 12 q^{6} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 30 q + 42 q^{4} + 12 q^{6} + 40 q^{9} + 30 q^{11} - 4 q^{14} + 66 q^{16} + 30 q^{19} + 14 q^{21} + 22 q^{24} + 30 q^{29} + 26 q^{31} + 12 q^{34} + 78 q^{36} + 64 q^{39} + 22 q^{41} + 42 q^{44} + 28 q^{46} + 60 q^{49} + 64 q^{51} + 62 q^{54} - 32 q^{56} - 14 q^{59} + 78 q^{61} + 90 q^{64} + 12 q^{66} - 28 q^{69} + 20 q^{71} + 42 q^{74} + 42 q^{76} + 102 q^{79} + 42 q^{81} + 98 q^{84} - 52 q^{86} - 8 q^{89} + 56 q^{91} + 40 q^{94} - 74 q^{96} + 40 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.66433 −1.17686 −0.588429 0.808549i \(-0.700253\pi\)
−0.588429 + 0.808549i \(0.700253\pi\)
\(3\) 1.25863 0.726672 0.363336 0.931658i \(-0.381638\pi\)
0.363336 + 0.931658i \(0.381638\pi\)
\(4\) 0.769991 0.384996
\(5\) 0 0
\(6\) −2.09478 −0.855190
\(7\) 4.13429 1.56261 0.781307 0.624146i \(-0.214553\pi\)
0.781307 + 0.624146i \(0.214553\pi\)
\(8\) 2.04714 0.723773
\(9\) −1.41584 −0.471948
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0.969136 0.279765
\(13\) 3.80507 1.05534 0.527668 0.849451i \(-0.323066\pi\)
0.527668 + 0.849451i \(0.323066\pi\)
\(14\) −6.88082 −1.83898
\(15\) 0 0
\(16\) −4.94710 −1.23677
\(17\) −2.64209 −0.640802 −0.320401 0.947282i \(-0.603818\pi\)
−0.320401 + 0.947282i \(0.603818\pi\)
\(18\) 2.35643 0.555416
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 5.20355 1.13551
\(22\) −1.66433 −0.354836
\(23\) 5.58456 1.16446 0.582231 0.813023i \(-0.302180\pi\)
0.582231 + 0.813023i \(0.302180\pi\)
\(24\) 2.57660 0.525945
\(25\) 0 0
\(26\) −6.33289 −1.24198
\(27\) −5.55792 −1.06962
\(28\) 3.18337 0.601600
\(29\) −6.34374 −1.17800 −0.589001 0.808132i \(-0.700479\pi\)
−0.589001 + 0.808132i \(0.700479\pi\)
\(30\) 0 0
\(31\) 7.53516 1.35336 0.676678 0.736279i \(-0.263419\pi\)
0.676678 + 0.736279i \(0.263419\pi\)
\(32\) 4.13932 0.731735
\(33\) 1.25863 0.219100
\(34\) 4.39731 0.754133
\(35\) 0 0
\(36\) −1.09019 −0.181698
\(37\) 3.00157 0.493456 0.246728 0.969085i \(-0.420645\pi\)
0.246728 + 0.969085i \(0.420645\pi\)
\(38\) −1.66433 −0.269990
\(39\) 4.78918 0.766883
\(40\) 0 0
\(41\) −4.89269 −0.764110 −0.382055 0.924140i \(-0.624783\pi\)
−0.382055 + 0.924140i \(0.624783\pi\)
\(42\) −8.66042 −1.33633
\(43\) 10.6638 1.62622 0.813110 0.582110i \(-0.197773\pi\)
0.813110 + 0.582110i \(0.197773\pi\)
\(44\) 0.769991 0.116081
\(45\) 0 0
\(46\) −9.29455 −1.37041
\(47\) −6.54878 −0.955238 −0.477619 0.878567i \(-0.658500\pi\)
−0.477619 + 0.878567i \(0.658500\pi\)
\(48\) −6.22657 −0.898729
\(49\) 10.0924 1.44177
\(50\) 0 0
\(51\) −3.32543 −0.465653
\(52\) 2.92987 0.406300
\(53\) 7.14902 0.981993 0.490997 0.871161i \(-0.336633\pi\)
0.490997 + 0.871161i \(0.336633\pi\)
\(54\) 9.25022 1.25879
\(55\) 0 0
\(56\) 8.46347 1.13098
\(57\) 1.25863 0.166710
\(58\) 10.5581 1.38634
\(59\) −7.34337 −0.956025 −0.478012 0.878353i \(-0.658643\pi\)
−0.478012 + 0.878353i \(0.658643\pi\)
\(60\) 0 0
\(61\) 4.72825 0.605391 0.302695 0.953087i \(-0.402114\pi\)
0.302695 + 0.953087i \(0.402114\pi\)
\(62\) −12.5410 −1.59271
\(63\) −5.85351 −0.737473
\(64\) 3.00501 0.375626
\(65\) 0 0
\(66\) −2.09478 −0.257849
\(67\) 7.94790 0.970990 0.485495 0.874239i \(-0.338639\pi\)
0.485495 + 0.874239i \(0.338639\pi\)
\(68\) −2.03439 −0.246706
\(69\) 7.02891 0.846182
\(70\) 0 0
\(71\) 9.69764 1.15090 0.575449 0.817838i \(-0.304827\pi\)
0.575449 + 0.817838i \(0.304827\pi\)
\(72\) −2.89843 −0.341583
\(73\) −16.9975 −1.98940 −0.994702 0.102798i \(-0.967221\pi\)
−0.994702 + 0.102798i \(0.967221\pi\)
\(74\) −4.99561 −0.580727
\(75\) 0 0
\(76\) 0.769991 0.0883240
\(77\) 4.13429 0.471146
\(78\) −7.97078 −0.902513
\(79\) 3.09141 0.347811 0.173906 0.984762i \(-0.444361\pi\)
0.173906 + 0.984762i \(0.444361\pi\)
\(80\) 0 0
\(81\) −2.74785 −0.305317
\(82\) 8.14304 0.899249
\(83\) −1.51906 −0.166739 −0.0833693 0.996519i \(-0.526568\pi\)
−0.0833693 + 0.996519i \(0.526568\pi\)
\(84\) 4.00669 0.437166
\(85\) 0 0
\(86\) −17.7481 −1.91383
\(87\) −7.98443 −0.856021
\(88\) 2.04714 0.218226
\(89\) 8.71704 0.924005 0.462002 0.886879i \(-0.347131\pi\)
0.462002 + 0.886879i \(0.347131\pi\)
\(90\) 0 0
\(91\) 15.7313 1.64908
\(92\) 4.30007 0.448313
\(93\) 9.48400 0.983445
\(94\) 10.8993 1.12418
\(95\) 0 0
\(96\) 5.20988 0.531731
\(97\) 6.68784 0.679048 0.339524 0.940597i \(-0.389734\pi\)
0.339524 + 0.940597i \(0.389734\pi\)
\(98\) −16.7970 −1.69675
\(99\) −1.41584 −0.142298
\(100\) 0 0
\(101\) 17.8311 1.77426 0.887128 0.461523i \(-0.152697\pi\)
0.887128 + 0.461523i \(0.152697\pi\)
\(102\) 5.53460 0.548007
\(103\) −11.5581 −1.13886 −0.569429 0.822041i \(-0.692836\pi\)
−0.569429 + 0.822041i \(0.692836\pi\)
\(104\) 7.78951 0.763824
\(105\) 0 0
\(106\) −11.8983 −1.15567
\(107\) 8.89809 0.860210 0.430105 0.902779i \(-0.358476\pi\)
0.430105 + 0.902779i \(0.358476\pi\)
\(108\) −4.27955 −0.411800
\(109\) 5.45885 0.522863 0.261432 0.965222i \(-0.415805\pi\)
0.261432 + 0.965222i \(0.415805\pi\)
\(110\) 0 0
\(111\) 3.77788 0.358580
\(112\) −20.4527 −1.93260
\(113\) −2.45792 −0.231221 −0.115611 0.993295i \(-0.536883\pi\)
−0.115611 + 0.993295i \(0.536883\pi\)
\(114\) −2.09478 −0.196194
\(115\) 0 0
\(116\) −4.88462 −0.453526
\(117\) −5.38739 −0.498064
\(118\) 12.2218 1.12511
\(119\) −10.9232 −1.00133
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −7.86937 −0.712459
\(123\) −6.15810 −0.555257
\(124\) 5.80201 0.521036
\(125\) 0 0
\(126\) 9.74217 0.867902
\(127\) −19.9879 −1.77364 −0.886820 0.462116i \(-0.847091\pi\)
−0.886820 + 0.462116i \(0.847091\pi\)
\(128\) −13.2800 −1.17379
\(129\) 13.4218 1.18173
\(130\) 0 0
\(131\) −6.36974 −0.556527 −0.278263 0.960505i \(-0.589759\pi\)
−0.278263 + 0.960505i \(0.589759\pi\)
\(132\) 0.969136 0.0843524
\(133\) 4.13429 0.358488
\(134\) −13.2279 −1.14272
\(135\) 0 0
\(136\) −5.40874 −0.463795
\(137\) −8.57793 −0.732862 −0.366431 0.930445i \(-0.619420\pi\)
−0.366431 + 0.930445i \(0.619420\pi\)
\(138\) −11.6984 −0.995836
\(139\) 17.9590 1.52326 0.761629 0.648013i \(-0.224400\pi\)
0.761629 + 0.648013i \(0.224400\pi\)
\(140\) 0 0
\(141\) −8.24251 −0.694145
\(142\) −16.1401 −1.35444
\(143\) 3.80507 0.318196
\(144\) 7.00432 0.583693
\(145\) 0 0
\(146\) 28.2894 2.34125
\(147\) 12.7026 1.04769
\(148\) 2.31118 0.189978
\(149\) −5.57782 −0.456953 −0.228477 0.973549i \(-0.573374\pi\)
−0.228477 + 0.973549i \(0.573374\pi\)
\(150\) 0 0
\(151\) −17.4741 −1.42202 −0.711011 0.703181i \(-0.751762\pi\)
−0.711011 + 0.703181i \(0.751762\pi\)
\(152\) 2.04714 0.166045
\(153\) 3.74080 0.302425
\(154\) −6.88082 −0.554472
\(155\) 0 0
\(156\) 3.68763 0.295247
\(157\) −0.335336 −0.0267627 −0.0133814 0.999910i \(-0.504260\pi\)
−0.0133814 + 0.999910i \(0.504260\pi\)
\(158\) −5.14513 −0.409324
\(159\) 8.99799 0.713587
\(160\) 0 0
\(161\) 23.0882 1.81961
\(162\) 4.57332 0.359314
\(163\) 18.9776 1.48644 0.743219 0.669048i \(-0.233298\pi\)
0.743219 + 0.669048i \(0.233298\pi\)
\(164\) −3.76733 −0.294179
\(165\) 0 0
\(166\) 2.52822 0.196228
\(167\) −2.50778 −0.194058 −0.0970289 0.995282i \(-0.530934\pi\)
−0.0970289 + 0.995282i \(0.530934\pi\)
\(168\) 10.6524 0.821850
\(169\) 1.47856 0.113735
\(170\) 0 0
\(171\) −1.41584 −0.108272
\(172\) 8.21106 0.626087
\(173\) 8.73580 0.664170 0.332085 0.943249i \(-0.392248\pi\)
0.332085 + 0.943249i \(0.392248\pi\)
\(174\) 13.2887 1.00742
\(175\) 0 0
\(176\) −4.94710 −0.372901
\(177\) −9.24260 −0.694716
\(178\) −14.5080 −1.08742
\(179\) −7.03056 −0.525489 −0.262744 0.964865i \(-0.584628\pi\)
−0.262744 + 0.964865i \(0.584628\pi\)
\(180\) 0 0
\(181\) 4.57978 0.340412 0.170206 0.985408i \(-0.445557\pi\)
0.170206 + 0.985408i \(0.445557\pi\)
\(182\) −26.1820 −1.94074
\(183\) 5.95113 0.439920
\(184\) 11.4324 0.842806
\(185\) 0 0
\(186\) −15.7845 −1.15738
\(187\) −2.64209 −0.193209
\(188\) −5.04251 −0.367763
\(189\) −22.9781 −1.67141
\(190\) 0 0
\(191\) −15.1847 −1.09873 −0.549364 0.835583i \(-0.685130\pi\)
−0.549364 + 0.835583i \(0.685130\pi\)
\(192\) 3.78220 0.272957
\(193\) −10.0701 −0.724864 −0.362432 0.932010i \(-0.618053\pi\)
−0.362432 + 0.932010i \(0.618053\pi\)
\(194\) −11.1308 −0.799143
\(195\) 0 0
\(196\) 7.77103 0.555073
\(197\) −4.77076 −0.339902 −0.169951 0.985452i \(-0.554361\pi\)
−0.169951 + 0.985452i \(0.554361\pi\)
\(198\) 2.35643 0.167464
\(199\) −1.12545 −0.0797808 −0.0398904 0.999204i \(-0.512701\pi\)
−0.0398904 + 0.999204i \(0.512701\pi\)
\(200\) 0 0
\(201\) 10.0035 0.705591
\(202\) −29.6768 −2.08805
\(203\) −26.2269 −1.84076
\(204\) −2.56055 −0.179274
\(205\) 0 0
\(206\) 19.2366 1.34027
\(207\) −7.90688 −0.549566
\(208\) −18.8240 −1.30521
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 4.95441 0.341076 0.170538 0.985351i \(-0.445450\pi\)
0.170538 + 0.985351i \(0.445450\pi\)
\(212\) 5.50468 0.378063
\(213\) 12.2058 0.836325
\(214\) −14.8093 −1.01235
\(215\) 0 0
\(216\) −11.3778 −0.774164
\(217\) 31.1526 2.11477
\(218\) −9.08533 −0.615336
\(219\) −21.3936 −1.44564
\(220\) 0 0
\(221\) −10.0534 −0.676262
\(222\) −6.28763 −0.421998
\(223\) −3.36972 −0.225653 −0.112826 0.993615i \(-0.535990\pi\)
−0.112826 + 0.993615i \(0.535990\pi\)
\(224\) 17.1131 1.14342
\(225\) 0 0
\(226\) 4.09078 0.272115
\(227\) 21.4571 1.42416 0.712078 0.702101i \(-0.247754\pi\)
0.712078 + 0.702101i \(0.247754\pi\)
\(228\) 0.969136 0.0641826
\(229\) 4.58169 0.302767 0.151383 0.988475i \(-0.451627\pi\)
0.151383 + 0.988475i \(0.451627\pi\)
\(230\) 0 0
\(231\) 5.20355 0.342369
\(232\) −12.9865 −0.852606
\(233\) 12.2022 0.799396 0.399698 0.916647i \(-0.369115\pi\)
0.399698 + 0.916647i \(0.369115\pi\)
\(234\) 8.96639 0.586151
\(235\) 0 0
\(236\) −5.65433 −0.368065
\(237\) 3.89095 0.252745
\(238\) 18.1798 1.17842
\(239\) 27.5196 1.78009 0.890047 0.455869i \(-0.150671\pi\)
0.890047 + 0.455869i \(0.150671\pi\)
\(240\) 0 0
\(241\) −8.15118 −0.525064 −0.262532 0.964923i \(-0.584557\pi\)
−0.262532 + 0.964923i \(0.584557\pi\)
\(242\) −1.66433 −0.106987
\(243\) 13.2152 0.847758
\(244\) 3.64071 0.233073
\(245\) 0 0
\(246\) 10.2491 0.653459
\(247\) 3.80507 0.242111
\(248\) 15.4255 0.979522
\(249\) −1.91194 −0.121164
\(250\) 0 0
\(251\) −15.0639 −0.950826 −0.475413 0.879763i \(-0.657701\pi\)
−0.475413 + 0.879763i \(0.657701\pi\)
\(252\) −4.50715 −0.283924
\(253\) 5.58456 0.351099
\(254\) 33.2664 2.08732
\(255\) 0 0
\(256\) 16.0922 1.00576
\(257\) −10.7664 −0.671588 −0.335794 0.941935i \(-0.609005\pi\)
−0.335794 + 0.941935i \(0.609005\pi\)
\(258\) −22.3384 −1.39073
\(259\) 12.4094 0.771081
\(260\) 0 0
\(261\) 8.98175 0.555956
\(262\) 10.6013 0.654953
\(263\) −8.75501 −0.539857 −0.269928 0.962880i \(-0.587000\pi\)
−0.269928 + 0.962880i \(0.587000\pi\)
\(264\) 2.57660 0.158579
\(265\) 0 0
\(266\) −6.88082 −0.421890
\(267\) 10.9716 0.671448
\(268\) 6.11981 0.373827
\(269\) −30.8940 −1.88364 −0.941822 0.336113i \(-0.890888\pi\)
−0.941822 + 0.336113i \(0.890888\pi\)
\(270\) 0 0
\(271\) −18.0422 −1.09598 −0.547991 0.836484i \(-0.684607\pi\)
−0.547991 + 0.836484i \(0.684607\pi\)
\(272\) 13.0707 0.792527
\(273\) 19.7999 1.19834
\(274\) 14.2765 0.862475
\(275\) 0 0
\(276\) 5.41220 0.325776
\(277\) 22.7236 1.36533 0.682666 0.730731i \(-0.260821\pi\)
0.682666 + 0.730731i \(0.260821\pi\)
\(278\) −29.8896 −1.79266
\(279\) −10.6686 −0.638714
\(280\) 0 0
\(281\) 24.8693 1.48358 0.741790 0.670632i \(-0.233977\pi\)
0.741790 + 0.670632i \(0.233977\pi\)
\(282\) 13.7183 0.816910
\(283\) −30.6371 −1.82118 −0.910592 0.413307i \(-0.864374\pi\)
−0.910592 + 0.413307i \(0.864374\pi\)
\(284\) 7.46709 0.443091
\(285\) 0 0
\(286\) −6.33289 −0.374472
\(287\) −20.2278 −1.19401
\(288\) −5.86063 −0.345341
\(289\) −10.0193 −0.589373
\(290\) 0 0
\(291\) 8.41754 0.493445
\(292\) −13.0879 −0.765912
\(293\) 6.67638 0.390038 0.195019 0.980799i \(-0.437523\pi\)
0.195019 + 0.980799i \(0.437523\pi\)
\(294\) −21.1413 −1.23298
\(295\) 0 0
\(296\) 6.14464 0.357150
\(297\) −5.55792 −0.322504
\(298\) 9.28333 0.537769
\(299\) 21.2497 1.22890
\(300\) 0 0
\(301\) 44.0874 2.54115
\(302\) 29.0826 1.67352
\(303\) 22.4427 1.28930
\(304\) −4.94710 −0.283735
\(305\) 0 0
\(306\) −6.22592 −0.355912
\(307\) 6.23523 0.355863 0.177932 0.984043i \(-0.443059\pi\)
0.177932 + 0.984043i \(0.443059\pi\)
\(308\) 3.18337 0.181389
\(309\) −14.5475 −0.827576
\(310\) 0 0
\(311\) 10.7143 0.607550 0.303775 0.952744i \(-0.401753\pi\)
0.303775 + 0.952744i \(0.401753\pi\)
\(312\) 9.80413 0.555049
\(313\) 17.2646 0.975855 0.487928 0.872884i \(-0.337753\pi\)
0.487928 + 0.872884i \(0.337753\pi\)
\(314\) 0.558109 0.0314959
\(315\) 0 0
\(316\) 2.38036 0.133906
\(317\) 25.7380 1.44559 0.722794 0.691064i \(-0.242858\pi\)
0.722794 + 0.691064i \(0.242858\pi\)
\(318\) −14.9756 −0.839791
\(319\) −6.34374 −0.355181
\(320\) 0 0
\(321\) 11.1994 0.625091
\(322\) −38.4264 −2.14142
\(323\) −2.64209 −0.147010
\(324\) −2.11582 −0.117546
\(325\) 0 0
\(326\) −31.5849 −1.74933
\(327\) 6.87069 0.379950
\(328\) −10.0160 −0.553042
\(329\) −27.0746 −1.49267
\(330\) 0 0
\(331\) 19.3860 1.06555 0.532776 0.846256i \(-0.321149\pi\)
0.532776 + 0.846256i \(0.321149\pi\)
\(332\) −1.16966 −0.0641936
\(333\) −4.24976 −0.232886
\(334\) 4.17377 0.228379
\(335\) 0 0
\(336\) −25.7425 −1.40437
\(337\) 23.5387 1.28223 0.641117 0.767443i \(-0.278471\pi\)
0.641117 + 0.767443i \(0.278471\pi\)
\(338\) −2.46081 −0.133850
\(339\) −3.09361 −0.168022
\(340\) 0 0
\(341\) 7.53516 0.408052
\(342\) 2.35643 0.127421
\(343\) 12.7847 0.690309
\(344\) 21.8304 1.17701
\(345\) 0 0
\(346\) −14.5392 −0.781634
\(347\) −25.5573 −1.37199 −0.685993 0.727608i \(-0.740632\pi\)
−0.685993 + 0.727608i \(0.740632\pi\)
\(348\) −6.14794 −0.329564
\(349\) 6.75120 0.361383 0.180692 0.983540i \(-0.442166\pi\)
0.180692 + 0.983540i \(0.442166\pi\)
\(350\) 0 0
\(351\) −21.1483 −1.12881
\(352\) 4.13932 0.220626
\(353\) −18.1264 −0.964771 −0.482386 0.875959i \(-0.660230\pi\)
−0.482386 + 0.875959i \(0.660230\pi\)
\(354\) 15.3827 0.817582
\(355\) 0 0
\(356\) 6.71205 0.355738
\(357\) −13.7483 −0.727636
\(358\) 11.7012 0.618426
\(359\) 30.0381 1.58535 0.792675 0.609644i \(-0.208688\pi\)
0.792675 + 0.609644i \(0.208688\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −7.62226 −0.400617
\(363\) 1.25863 0.0660611
\(364\) 12.1129 0.634890
\(365\) 0 0
\(366\) −9.90464 −0.517724
\(367\) 3.71003 0.193662 0.0968309 0.995301i \(-0.469129\pi\)
0.0968309 + 0.995301i \(0.469129\pi\)
\(368\) −27.6274 −1.44018
\(369\) 6.92729 0.360620
\(370\) 0 0
\(371\) 29.5561 1.53448
\(372\) 7.30259 0.378622
\(373\) 22.8775 1.18455 0.592277 0.805735i \(-0.298229\pi\)
0.592277 + 0.805735i \(0.298229\pi\)
\(374\) 4.39731 0.227380
\(375\) 0 0
\(376\) −13.4063 −0.691376
\(377\) −24.1384 −1.24319
\(378\) 38.2431 1.96701
\(379\) 1.48762 0.0764140 0.0382070 0.999270i \(-0.487835\pi\)
0.0382070 + 0.999270i \(0.487835\pi\)
\(380\) 0 0
\(381\) −25.1574 −1.28885
\(382\) 25.2724 1.29305
\(383\) −29.7901 −1.52220 −0.761100 0.648634i \(-0.775341\pi\)
−0.761100 + 0.648634i \(0.775341\pi\)
\(384\) −16.7146 −0.852962
\(385\) 0 0
\(386\) 16.7600 0.853062
\(387\) −15.0983 −0.767491
\(388\) 5.14958 0.261430
\(389\) −10.6431 −0.539628 −0.269814 0.962913i \(-0.586962\pi\)
−0.269814 + 0.962913i \(0.586962\pi\)
\(390\) 0 0
\(391\) −14.7549 −0.746190
\(392\) 20.6605 1.04351
\(393\) −8.01716 −0.404412
\(394\) 7.94011 0.400017
\(395\) 0 0
\(396\) −1.09019 −0.0547840
\(397\) 10.1616 0.509997 0.254999 0.966941i \(-0.417925\pi\)
0.254999 + 0.966941i \(0.417925\pi\)
\(398\) 1.87311 0.0938907
\(399\) 5.20355 0.260503
\(400\) 0 0
\(401\) 25.4722 1.27202 0.636011 0.771680i \(-0.280583\pi\)
0.636011 + 0.771680i \(0.280583\pi\)
\(402\) −16.6491 −0.830381
\(403\) 28.6718 1.42825
\(404\) 13.7298 0.683081
\(405\) 0 0
\(406\) 43.6501 2.16632
\(407\) 3.00157 0.148782
\(408\) −6.80761 −0.337027
\(409\) 4.41274 0.218196 0.109098 0.994031i \(-0.465204\pi\)
0.109098 + 0.994031i \(0.465204\pi\)
\(410\) 0 0
\(411\) −10.7965 −0.532550
\(412\) −8.89967 −0.438455
\(413\) −30.3596 −1.49390
\(414\) 13.1596 0.646761
\(415\) 0 0
\(416\) 15.7504 0.772226
\(417\) 22.6037 1.10691
\(418\) −1.66433 −0.0814050
\(419\) −12.1463 −0.593384 −0.296692 0.954973i \(-0.595883\pi\)
−0.296692 + 0.954973i \(0.595883\pi\)
\(420\) 0 0
\(421\) 15.9860 0.779108 0.389554 0.921004i \(-0.372629\pi\)
0.389554 + 0.921004i \(0.372629\pi\)
\(422\) −8.24577 −0.401398
\(423\) 9.27206 0.450823
\(424\) 14.6350 0.710740
\(425\) 0 0
\(426\) −20.3144 −0.984236
\(427\) 19.5480 0.945992
\(428\) 6.85145 0.331177
\(429\) 4.78918 0.231224
\(430\) 0 0
\(431\) 2.80822 0.135267 0.0676336 0.997710i \(-0.478455\pi\)
0.0676336 + 0.997710i \(0.478455\pi\)
\(432\) 27.4956 1.32288
\(433\) −17.0633 −0.820011 −0.410006 0.912083i \(-0.634473\pi\)
−0.410006 + 0.912083i \(0.634473\pi\)
\(434\) −51.8481 −2.48879
\(435\) 0 0
\(436\) 4.20327 0.201300
\(437\) 5.58456 0.267146
\(438\) 35.6060 1.70132
\(439\) 4.59307 0.219215 0.109608 0.993975i \(-0.465041\pi\)
0.109608 + 0.993975i \(0.465041\pi\)
\(440\) 0 0
\(441\) −14.2892 −0.680439
\(442\) 16.7321 0.795864
\(443\) 13.2967 0.631747 0.315873 0.948801i \(-0.397703\pi\)
0.315873 + 0.948801i \(0.397703\pi\)
\(444\) 2.90893 0.138052
\(445\) 0 0
\(446\) 5.60832 0.265561
\(447\) −7.02043 −0.332055
\(448\) 12.4236 0.586959
\(449\) −4.22753 −0.199510 −0.0997548 0.995012i \(-0.531806\pi\)
−0.0997548 + 0.995012i \(0.531806\pi\)
\(450\) 0 0
\(451\) −4.89269 −0.230388
\(452\) −1.89257 −0.0890192
\(453\) −21.9935 −1.03334
\(454\) −35.7116 −1.67603
\(455\) 0 0
\(456\) 2.57660 0.120660
\(457\) 39.3140 1.83903 0.919517 0.393051i \(-0.128580\pi\)
0.919517 + 0.393051i \(0.128580\pi\)
\(458\) −7.62545 −0.356314
\(459\) 14.6846 0.685417
\(460\) 0 0
\(461\) −32.6330 −1.51987 −0.759936 0.649998i \(-0.774770\pi\)
−0.759936 + 0.649998i \(0.774770\pi\)
\(462\) −8.66042 −0.402919
\(463\) 9.25218 0.429985 0.214993 0.976616i \(-0.431027\pi\)
0.214993 + 0.976616i \(0.431027\pi\)
\(464\) 31.3831 1.45692
\(465\) 0 0
\(466\) −20.3085 −0.940775
\(467\) 30.3491 1.40439 0.702194 0.711986i \(-0.252204\pi\)
0.702194 + 0.711986i \(0.252204\pi\)
\(468\) −4.14824 −0.191753
\(469\) 32.8589 1.51728
\(470\) 0 0
\(471\) −0.422065 −0.0194477
\(472\) −15.0329 −0.691945
\(473\) 10.6638 0.490324
\(474\) −6.47582 −0.297444
\(475\) 0 0
\(476\) −8.41076 −0.385506
\(477\) −10.1219 −0.463450
\(478\) −45.8016 −2.09492
\(479\) 22.0685 1.00834 0.504168 0.863606i \(-0.331799\pi\)
0.504168 + 0.863606i \(0.331799\pi\)
\(480\) 0 0
\(481\) 11.4212 0.520762
\(482\) 13.5662 0.617926
\(483\) 29.0596 1.32226
\(484\) 0.769991 0.0349996
\(485\) 0 0
\(486\) −21.9945 −0.997691
\(487\) −27.3741 −1.24044 −0.620220 0.784428i \(-0.712957\pi\)
−0.620220 + 0.784428i \(0.712957\pi\)
\(488\) 9.67939 0.438165
\(489\) 23.8858 1.08015
\(490\) 0 0
\(491\) 24.6060 1.11045 0.555227 0.831699i \(-0.312631\pi\)
0.555227 + 0.831699i \(0.312631\pi\)
\(492\) −4.74168 −0.213771
\(493\) 16.7608 0.754866
\(494\) −6.33289 −0.284930
\(495\) 0 0
\(496\) −37.2772 −1.67379
\(497\) 40.0928 1.79841
\(498\) 3.18210 0.142593
\(499\) −22.4290 −1.00406 −0.502031 0.864850i \(-0.667414\pi\)
−0.502031 + 0.864850i \(0.667414\pi\)
\(500\) 0 0
\(501\) −3.15637 −0.141016
\(502\) 25.0713 1.11899
\(503\) −32.2644 −1.43860 −0.719298 0.694701i \(-0.755537\pi\)
−0.719298 + 0.694701i \(0.755537\pi\)
\(504\) −11.9830 −0.533763
\(505\) 0 0
\(506\) −9.29455 −0.413193
\(507\) 1.86096 0.0826482
\(508\) −15.3905 −0.682843
\(509\) −41.7294 −1.84962 −0.924811 0.380426i \(-0.875777\pi\)
−0.924811 + 0.380426i \(0.875777\pi\)
\(510\) 0 0
\(511\) −70.2725 −3.10867
\(512\) −0.222805 −0.00984669
\(513\) −5.55792 −0.245388
\(514\) 17.9188 0.790364
\(515\) 0 0
\(516\) 10.3347 0.454960
\(517\) −6.54878 −0.288015
\(518\) −20.6533 −0.907453
\(519\) 10.9952 0.482634
\(520\) 0 0
\(521\) −24.2787 −1.06367 −0.531836 0.846848i \(-0.678498\pi\)
−0.531836 + 0.846848i \(0.678498\pi\)
\(522\) −14.9486 −0.654282
\(523\) −3.57111 −0.156154 −0.0780769 0.996947i \(-0.524878\pi\)
−0.0780769 + 0.996947i \(0.524878\pi\)
\(524\) −4.90464 −0.214260
\(525\) 0 0
\(526\) 14.5712 0.635335
\(527\) −19.9086 −0.867233
\(528\) −6.22657 −0.270977
\(529\) 8.18736 0.355972
\(530\) 0 0
\(531\) 10.3971 0.451194
\(532\) 3.18337 0.138016
\(533\) −18.6170 −0.806393
\(534\) −18.2603 −0.790199
\(535\) 0 0
\(536\) 16.2705 0.702777
\(537\) −8.84889 −0.381858
\(538\) 51.4179 2.21678
\(539\) 10.0924 0.434709
\(540\) 0 0
\(541\) −29.0917 −1.25075 −0.625376 0.780324i \(-0.715054\pi\)
−0.625376 + 0.780324i \(0.715054\pi\)
\(542\) 30.0281 1.28982
\(543\) 5.76426 0.247368
\(544\) −10.9365 −0.468897
\(545\) 0 0
\(546\) −32.9535 −1.41028
\(547\) 7.61063 0.325407 0.162704 0.986675i \(-0.447979\pi\)
0.162704 + 0.986675i \(0.447979\pi\)
\(548\) −6.60493 −0.282149
\(549\) −6.69447 −0.285713
\(550\) 0 0
\(551\) −6.34374 −0.270252
\(552\) 14.3892 0.612444
\(553\) 12.7808 0.543495
\(554\) −37.8196 −1.60680
\(555\) 0 0
\(556\) 13.8282 0.586448
\(557\) −34.8324 −1.47590 −0.737948 0.674858i \(-0.764205\pi\)
−0.737948 + 0.674858i \(0.764205\pi\)
\(558\) 17.7561 0.751675
\(559\) 40.5766 1.71621
\(560\) 0 0
\(561\) −3.32543 −0.140400
\(562\) −41.3908 −1.74596
\(563\) −40.2832 −1.69773 −0.848867 0.528607i \(-0.822715\pi\)
−0.848867 + 0.528607i \(0.822715\pi\)
\(564\) −6.34666 −0.267243
\(565\) 0 0
\(566\) 50.9901 2.14328
\(567\) −11.3604 −0.477092
\(568\) 19.8524 0.832989
\(569\) 22.4278 0.940221 0.470111 0.882608i \(-0.344214\pi\)
0.470111 + 0.882608i \(0.344214\pi\)
\(570\) 0 0
\(571\) 28.5385 1.19430 0.597149 0.802131i \(-0.296300\pi\)
0.597149 + 0.802131i \(0.296300\pi\)
\(572\) 2.92987 0.122504
\(573\) −19.1120 −0.798415
\(574\) 33.6657 1.40518
\(575\) 0 0
\(576\) −4.25462 −0.177276
\(577\) 35.9350 1.49600 0.747998 0.663701i \(-0.231015\pi\)
0.747998 + 0.663701i \(0.231015\pi\)
\(578\) 16.6755 0.693608
\(579\) −12.6746 −0.526738
\(580\) 0 0
\(581\) −6.28024 −0.260548
\(582\) −14.0096 −0.580715
\(583\) 7.14902 0.296082
\(584\) −34.7962 −1.43988
\(585\) 0 0
\(586\) −11.1117 −0.459020
\(587\) −17.4907 −0.721920 −0.360960 0.932581i \(-0.617551\pi\)
−0.360960 + 0.932581i \(0.617551\pi\)
\(588\) 9.78086 0.403356
\(589\) 7.53516 0.310481
\(590\) 0 0
\(591\) −6.00463 −0.246997
\(592\) −14.8491 −0.610293
\(593\) 32.0403 1.31574 0.657869 0.753132i \(-0.271458\pi\)
0.657869 + 0.753132i \(0.271458\pi\)
\(594\) 9.25022 0.379541
\(595\) 0 0
\(596\) −4.29487 −0.175925
\(597\) −1.41652 −0.0579744
\(598\) −35.3664 −1.44624
\(599\) −2.86659 −0.117126 −0.0585629 0.998284i \(-0.518652\pi\)
−0.0585629 + 0.998284i \(0.518652\pi\)
\(600\) 0 0
\(601\) 0.391875 0.0159849 0.00799245 0.999968i \(-0.497456\pi\)
0.00799245 + 0.999968i \(0.497456\pi\)
\(602\) −73.3759 −2.99058
\(603\) −11.2530 −0.458257
\(604\) −13.4549 −0.547472
\(605\) 0 0
\(606\) −37.3521 −1.51733
\(607\) 23.3599 0.948151 0.474075 0.880484i \(-0.342782\pi\)
0.474075 + 0.880484i \(0.342782\pi\)
\(608\) 4.13932 0.167871
\(609\) −33.0100 −1.33763
\(610\) 0 0
\(611\) −24.9186 −1.00810
\(612\) 2.88038 0.116432
\(613\) 21.9771 0.887648 0.443824 0.896114i \(-0.353621\pi\)
0.443824 + 0.896114i \(0.353621\pi\)
\(614\) −10.3775 −0.418800
\(615\) 0 0
\(616\) 8.46347 0.341003
\(617\) 20.9896 0.845010 0.422505 0.906361i \(-0.361151\pi\)
0.422505 + 0.906361i \(0.361151\pi\)
\(618\) 24.2118 0.973940
\(619\) 4.96934 0.199735 0.0998673 0.995001i \(-0.468158\pi\)
0.0998673 + 0.995001i \(0.468158\pi\)
\(620\) 0 0
\(621\) −31.0386 −1.24554
\(622\) −17.8320 −0.715000
\(623\) 36.0388 1.44386
\(624\) −23.6926 −0.948461
\(625\) 0 0
\(626\) −28.7340 −1.14844
\(627\) 1.25863 0.0502649
\(628\) −0.258206 −0.0103035
\(629\) −7.93044 −0.316207
\(630\) 0 0
\(631\) 12.8910 0.513181 0.256591 0.966520i \(-0.417401\pi\)
0.256591 + 0.966520i \(0.417401\pi\)
\(632\) 6.32855 0.251736
\(633\) 6.23578 0.247850
\(634\) −42.8364 −1.70125
\(635\) 0 0
\(636\) 6.92837 0.274728
\(637\) 38.4021 1.52155
\(638\) 10.5581 0.417998
\(639\) −13.7303 −0.543164
\(640\) 0 0
\(641\) 6.24599 0.246702 0.123351 0.992363i \(-0.460636\pi\)
0.123351 + 0.992363i \(0.460636\pi\)
\(642\) −18.6395 −0.735643
\(643\) 4.14158 0.163328 0.0816640 0.996660i \(-0.473977\pi\)
0.0816640 + 0.996660i \(0.473977\pi\)
\(644\) 17.7777 0.700540
\(645\) 0 0
\(646\) 4.39731 0.173010
\(647\) 0.463373 0.0182171 0.00910853 0.999959i \(-0.497101\pi\)
0.00910853 + 0.999959i \(0.497101\pi\)
\(648\) −5.62523 −0.220980
\(649\) −7.34337 −0.288252
\(650\) 0 0
\(651\) 39.2096 1.53675
\(652\) 14.6126 0.572272
\(653\) −5.98508 −0.234214 −0.117107 0.993119i \(-0.537362\pi\)
−0.117107 + 0.993119i \(0.537362\pi\)
\(654\) −11.4351 −0.447147
\(655\) 0 0
\(656\) 24.2046 0.945031
\(657\) 24.0658 0.938896
\(658\) 45.0610 1.75666
\(659\) 27.1167 1.05632 0.528158 0.849146i \(-0.322883\pi\)
0.528158 + 0.849146i \(0.322883\pi\)
\(660\) 0 0
\(661\) 33.8878 1.31808 0.659041 0.752107i \(-0.270962\pi\)
0.659041 + 0.752107i \(0.270962\pi\)
\(662\) −32.2647 −1.25400
\(663\) −12.6535 −0.491420
\(664\) −3.10973 −0.120681
\(665\) 0 0
\(666\) 7.07300 0.274073
\(667\) −35.4270 −1.37174
\(668\) −1.93097 −0.0747114
\(669\) −4.24123 −0.163976
\(670\) 0 0
\(671\) 4.72825 0.182532
\(672\) 21.5391 0.830891
\(673\) −0.103566 −0.00399219 −0.00199610 0.999998i \(-0.500635\pi\)
−0.00199610 + 0.999998i \(0.500635\pi\)
\(674\) −39.1761 −1.50901
\(675\) 0 0
\(676\) 1.13848 0.0437876
\(677\) −36.4846 −1.40222 −0.701109 0.713054i \(-0.747311\pi\)
−0.701109 + 0.713054i \(0.747311\pi\)
\(678\) 5.14879 0.197738
\(679\) 27.6495 1.06109
\(680\) 0 0
\(681\) 27.0066 1.03489
\(682\) −12.5410 −0.480219
\(683\) 11.0961 0.424579 0.212290 0.977207i \(-0.431908\pi\)
0.212290 + 0.977207i \(0.431908\pi\)
\(684\) −1.09019 −0.0416844
\(685\) 0 0
\(686\) −21.2780 −0.812396
\(687\) 5.76667 0.220012
\(688\) −52.7550 −2.01127
\(689\) 27.2025 1.03633
\(690\) 0 0
\(691\) 11.8894 0.452293 0.226146 0.974093i \(-0.427387\pi\)
0.226146 + 0.974093i \(0.427387\pi\)
\(692\) 6.72649 0.255703
\(693\) −5.85351 −0.222357
\(694\) 42.5357 1.61463
\(695\) 0 0
\(696\) −16.3452 −0.619565
\(697\) 12.9269 0.489643
\(698\) −11.2362 −0.425297
\(699\) 15.3581 0.580898
\(700\) 0 0
\(701\) 46.3021 1.74881 0.874403 0.485201i \(-0.161254\pi\)
0.874403 + 0.485201i \(0.161254\pi\)
\(702\) 35.1977 1.32845
\(703\) 3.00157 0.113206
\(704\) 3.00501 0.113255
\(705\) 0 0
\(706\) 30.1683 1.13540
\(707\) 73.7188 2.77248
\(708\) −7.11672 −0.267463
\(709\) 1.98659 0.0746080 0.0373040 0.999304i \(-0.488123\pi\)
0.0373040 + 0.999304i \(0.488123\pi\)
\(710\) 0 0
\(711\) −4.37696 −0.164149
\(712\) 17.8450 0.668770
\(713\) 42.0806 1.57593
\(714\) 22.8817 0.856324
\(715\) 0 0
\(716\) −5.41347 −0.202311
\(717\) 34.6370 1.29354
\(718\) −49.9933 −1.86573
\(719\) 25.3582 0.945700 0.472850 0.881143i \(-0.343225\pi\)
0.472850 + 0.881143i \(0.343225\pi\)
\(720\) 0 0
\(721\) −47.7847 −1.77960
\(722\) −1.66433 −0.0619399
\(723\) −10.2593 −0.381549
\(724\) 3.52639 0.131057
\(725\) 0 0
\(726\) −2.09478 −0.0777445
\(727\) −30.4622 −1.12978 −0.564891 0.825166i \(-0.691082\pi\)
−0.564891 + 0.825166i \(0.691082\pi\)
\(728\) 32.2041 1.19356
\(729\) 24.8767 0.921358
\(730\) 0 0
\(731\) −28.1749 −1.04208
\(732\) 4.58232 0.169367
\(733\) −17.8574 −0.659577 −0.329789 0.944055i \(-0.606977\pi\)
−0.329789 + 0.944055i \(0.606977\pi\)
\(734\) −6.17471 −0.227913
\(735\) 0 0
\(736\) 23.1163 0.852077
\(737\) 7.94790 0.292765
\(738\) −11.5293 −0.424399
\(739\) 29.8916 1.09958 0.549789 0.835303i \(-0.314708\pi\)
0.549789 + 0.835303i \(0.314708\pi\)
\(740\) 0 0
\(741\) 4.78918 0.175935
\(742\) −49.1911 −1.80586
\(743\) −48.7230 −1.78747 −0.893737 0.448591i \(-0.851926\pi\)
−0.893737 + 0.448591i \(0.851926\pi\)
\(744\) 19.4151 0.711791
\(745\) 0 0
\(746\) −38.0757 −1.39405
\(747\) 2.15075 0.0786920
\(748\) −2.03439 −0.0743846
\(749\) 36.7873 1.34418
\(750\) 0 0
\(751\) 49.7261 1.81453 0.907265 0.420559i \(-0.138166\pi\)
0.907265 + 0.420559i \(0.138166\pi\)
\(752\) 32.3975 1.18141
\(753\) −18.9599 −0.690939
\(754\) 40.1742 1.46306
\(755\) 0 0
\(756\) −17.6929 −0.643485
\(757\) 26.1383 0.950014 0.475007 0.879982i \(-0.342445\pi\)
0.475007 + 0.879982i \(0.342445\pi\)
\(758\) −2.47589 −0.0899284
\(759\) 7.02891 0.255133
\(760\) 0 0
\(761\) 23.4157 0.848819 0.424410 0.905470i \(-0.360482\pi\)
0.424410 + 0.905470i \(0.360482\pi\)
\(762\) 41.8702 1.51680
\(763\) 22.5685 0.817034
\(764\) −11.6921 −0.423005
\(765\) 0 0
\(766\) 49.5804 1.79141
\(767\) −27.9420 −1.00893
\(768\) 20.2542 0.730859
\(769\) 44.2884 1.59708 0.798541 0.601940i \(-0.205606\pi\)
0.798541 + 0.601940i \(0.205606\pi\)
\(770\) 0 0
\(771\) −13.5509 −0.488024
\(772\) −7.75391 −0.279069
\(773\) 24.3892 0.877220 0.438610 0.898678i \(-0.355471\pi\)
0.438610 + 0.898678i \(0.355471\pi\)
\(774\) 25.1286 0.903229
\(775\) 0 0
\(776\) 13.6909 0.491476
\(777\) 15.6188 0.560323
\(778\) 17.7137 0.635065
\(779\) −4.89269 −0.175299
\(780\) 0 0
\(781\) 9.69764 0.347009
\(782\) 24.5571 0.878160
\(783\) 35.2580 1.26002
\(784\) −49.9279 −1.78314
\(785\) 0 0
\(786\) 13.3432 0.475936
\(787\) −52.1888 −1.86033 −0.930165 0.367141i \(-0.880337\pi\)
−0.930165 + 0.367141i \(0.880337\pi\)
\(788\) −3.67344 −0.130861
\(789\) −11.0193 −0.392299
\(790\) 0 0
\(791\) −10.1617 −0.361310
\(792\) −2.89843 −0.102991
\(793\) 17.9913 0.638891
\(794\) −16.9123 −0.600194
\(795\) 0 0
\(796\) −0.866584 −0.0307152
\(797\) −7.66970 −0.271675 −0.135837 0.990731i \(-0.543372\pi\)
−0.135837 + 0.990731i \(0.543372\pi\)
\(798\) −8.66042 −0.306576
\(799\) 17.3025 0.612119
\(800\) 0 0
\(801\) −12.3420 −0.436083
\(802\) −42.3942 −1.49699
\(803\) −16.9975 −0.599828
\(804\) 7.70259 0.271649
\(805\) 0 0
\(806\) −47.7193 −1.68084
\(807\) −38.8842 −1.36879
\(808\) 36.5027 1.28416
\(809\) −1.23293 −0.0433476 −0.0216738 0.999765i \(-0.506900\pi\)
−0.0216738 + 0.999765i \(0.506900\pi\)
\(810\) 0 0
\(811\) 12.2439 0.429940 0.214970 0.976621i \(-0.431035\pi\)
0.214970 + 0.976621i \(0.431035\pi\)
\(812\) −20.1944 −0.708686
\(813\) −22.7084 −0.796420
\(814\) −4.99561 −0.175096
\(815\) 0 0
\(816\) 16.4512 0.575907
\(817\) 10.6638 0.373080
\(818\) −7.34425 −0.256786
\(819\) −22.2730 −0.778283
\(820\) 0 0
\(821\) 21.2934 0.743145 0.371572 0.928404i \(-0.378819\pi\)
0.371572 + 0.928404i \(0.378819\pi\)
\(822\) 17.9689 0.626736
\(823\) 30.8939 1.07690 0.538448 0.842659i \(-0.319011\pi\)
0.538448 + 0.842659i \(0.319011\pi\)
\(824\) −23.6611 −0.824275
\(825\) 0 0
\(826\) 50.5284 1.75811
\(827\) −6.54249 −0.227505 −0.113752 0.993509i \(-0.536287\pi\)
−0.113752 + 0.993509i \(0.536287\pi\)
\(828\) −6.08823 −0.211580
\(829\) −46.6368 −1.61976 −0.809881 0.586594i \(-0.800468\pi\)
−0.809881 + 0.586594i \(0.800468\pi\)
\(830\) 0 0
\(831\) 28.6007 0.992148
\(832\) 11.4343 0.396412
\(833\) −26.6650 −0.923886
\(834\) −37.6200 −1.30267
\(835\) 0 0
\(836\) 0.769991 0.0266307
\(837\) −41.8799 −1.44758
\(838\) 20.2154 0.698329
\(839\) 20.6612 0.713305 0.356653 0.934237i \(-0.383918\pi\)
0.356653 + 0.934237i \(0.383918\pi\)
\(840\) 0 0
\(841\) 11.2430 0.387690
\(842\) −26.6059 −0.916899
\(843\) 31.3014 1.07808
\(844\) 3.81485 0.131313
\(845\) 0 0
\(846\) −15.4318 −0.530555
\(847\) 4.13429 0.142056
\(848\) −35.3669 −1.21450
\(849\) −38.5608 −1.32340
\(850\) 0 0
\(851\) 16.7625 0.574610
\(852\) 9.39833 0.321981
\(853\) −21.9069 −0.750077 −0.375038 0.927009i \(-0.622370\pi\)
−0.375038 + 0.927009i \(0.622370\pi\)
\(854\) −32.5343 −1.11330
\(855\) 0 0
\(856\) 18.2156 0.622597
\(857\) 27.5927 0.942547 0.471274 0.881987i \(-0.343794\pi\)
0.471274 + 0.881987i \(0.343794\pi\)
\(858\) −7.97078 −0.272118
\(859\) 23.5111 0.802187 0.401094 0.916037i \(-0.368630\pi\)
0.401094 + 0.916037i \(0.368630\pi\)
\(860\) 0 0
\(861\) −25.4594 −0.867653
\(862\) −4.67381 −0.159190
\(863\) 17.3336 0.590042 0.295021 0.955491i \(-0.404673\pi\)
0.295021 + 0.955491i \(0.404673\pi\)
\(864\) −23.0060 −0.782680
\(865\) 0 0
\(866\) 28.3990 0.965037
\(867\) −12.6107 −0.428280
\(868\) 23.9872 0.814178
\(869\) 3.09141 0.104869
\(870\) 0 0
\(871\) 30.2423 1.02472
\(872\) 11.1750 0.378435
\(873\) −9.46895 −0.320475
\(874\) −9.29455 −0.314393
\(875\) 0 0
\(876\) −16.4729 −0.556567
\(877\) −18.4351 −0.622509 −0.311254 0.950327i \(-0.600749\pi\)
−0.311254 + 0.950327i \(0.600749\pi\)
\(878\) −7.64437 −0.257985
\(879\) 8.40310 0.283430
\(880\) 0 0
\(881\) −25.3130 −0.852818 −0.426409 0.904530i \(-0.640222\pi\)
−0.426409 + 0.904530i \(0.640222\pi\)
\(882\) 23.7820 0.800780
\(883\) −50.7804 −1.70890 −0.854449 0.519535i \(-0.826105\pi\)
−0.854449 + 0.519535i \(0.826105\pi\)
\(884\) −7.74099 −0.260358
\(885\) 0 0
\(886\) −22.1301 −0.743477
\(887\) −27.0222 −0.907315 −0.453658 0.891176i \(-0.649881\pi\)
−0.453658 + 0.891176i \(0.649881\pi\)
\(888\) 7.73384 0.259531
\(889\) −82.6358 −2.77152
\(890\) 0 0
\(891\) −2.74785 −0.0920564
\(892\) −2.59465 −0.0868753
\(893\) −6.54878 −0.219147
\(894\) 11.6843 0.390782
\(895\) 0 0
\(896\) −54.9032 −1.83419
\(897\) 26.7455 0.893007
\(898\) 7.03600 0.234794
\(899\) −47.8011 −1.59426
\(900\) 0 0
\(901\) −18.8884 −0.629263
\(902\) 8.14304 0.271134
\(903\) 55.4898 1.84659
\(904\) −5.03170 −0.167352
\(905\) 0 0
\(906\) 36.6043 1.21610
\(907\) −12.2103 −0.405436 −0.202718 0.979237i \(-0.564977\pi\)
−0.202718 + 0.979237i \(0.564977\pi\)
\(908\) 16.5218 0.548294
\(909\) −25.2460 −0.837357
\(910\) 0 0
\(911\) −40.0238 −1.32605 −0.663024 0.748598i \(-0.730727\pi\)
−0.663024 + 0.748598i \(0.730727\pi\)
\(912\) −6.22657 −0.206182
\(913\) −1.51906 −0.0502736
\(914\) −65.4315 −2.16428
\(915\) 0 0
\(916\) 3.52786 0.116564
\(917\) −26.3344 −0.869637
\(918\) −24.4399 −0.806638
\(919\) −12.6684 −0.417891 −0.208945 0.977927i \(-0.567003\pi\)
−0.208945 + 0.977927i \(0.567003\pi\)
\(920\) 0 0
\(921\) 7.84786 0.258596
\(922\) 54.3121 1.78867
\(923\) 36.9002 1.21458
\(924\) 4.00669 0.131810
\(925\) 0 0
\(926\) −15.3987 −0.506032
\(927\) 16.3645 0.537482
\(928\) −26.2587 −0.861985
\(929\) 26.1480 0.857888 0.428944 0.903331i \(-0.358886\pi\)
0.428944 + 0.903331i \(0.358886\pi\)
\(930\) 0 0
\(931\) 10.0924 0.330764
\(932\) 9.39562 0.307764
\(933\) 13.4853 0.441489
\(934\) −50.5108 −1.65277
\(935\) 0 0
\(936\) −11.0287 −0.360486
\(937\) 7.59737 0.248195 0.124098 0.992270i \(-0.460396\pi\)
0.124098 + 0.992270i \(0.460396\pi\)
\(938\) −54.6880 −1.78563
\(939\) 21.7298 0.709126
\(940\) 0 0
\(941\) 11.6224 0.378880 0.189440 0.981892i \(-0.439333\pi\)
0.189440 + 0.981892i \(0.439333\pi\)
\(942\) 0.702454 0.0228872
\(943\) −27.3235 −0.889777
\(944\) 36.3283 1.18239
\(945\) 0 0
\(946\) −17.7481 −0.577041
\(947\) −30.7195 −0.998251 −0.499126 0.866530i \(-0.666345\pi\)
−0.499126 + 0.866530i \(0.666345\pi\)
\(948\) 2.99600 0.0973055
\(949\) −64.6766 −2.09949
\(950\) 0 0
\(951\) 32.3946 1.05047
\(952\) −22.3613 −0.724733
\(953\) 43.2509 1.40103 0.700516 0.713636i \(-0.252953\pi\)
0.700516 + 0.713636i \(0.252953\pi\)
\(954\) 16.8462 0.545415
\(955\) 0 0
\(956\) 21.1898 0.685328
\(957\) −7.98443 −0.258100
\(958\) −36.7292 −1.18667
\(959\) −35.4637 −1.14518
\(960\) 0 0
\(961\) 25.7787 0.831570
\(962\) −19.0086 −0.612863
\(963\) −12.5983 −0.405975
\(964\) −6.27634 −0.202147
\(965\) 0 0
\(966\) −48.3647 −1.55611
\(967\) 9.17709 0.295115 0.147558 0.989053i \(-0.452859\pi\)
0.147558 + 0.989053i \(0.452859\pi\)
\(968\) 2.04714 0.0657976
\(969\) −3.32543 −0.106828
\(970\) 0 0
\(971\) −45.3774 −1.45623 −0.728115 0.685455i \(-0.759603\pi\)
−0.728115 + 0.685455i \(0.759603\pi\)
\(972\) 10.1756 0.326383
\(973\) 74.2475 2.38027
\(974\) 45.5595 1.45982
\(975\) 0 0
\(976\) −23.3911 −0.748731
\(977\) 13.2101 0.422630 0.211315 0.977418i \(-0.432225\pi\)
0.211315 + 0.977418i \(0.432225\pi\)
\(978\) −39.7538 −1.27119
\(979\) 8.71704 0.278598
\(980\) 0 0
\(981\) −7.72889 −0.246765
\(982\) −40.9525 −1.30685
\(983\) −21.0957 −0.672848 −0.336424 0.941711i \(-0.609218\pi\)
−0.336424 + 0.941711i \(0.609218\pi\)
\(984\) −12.6065 −0.401880
\(985\) 0 0
\(986\) −27.8954 −0.888371
\(987\) −34.0769 −1.08468
\(988\) 2.92987 0.0932116
\(989\) 59.5529 1.89367
\(990\) 0 0
\(991\) 1.29349 0.0410892 0.0205446 0.999789i \(-0.493460\pi\)
0.0205446 + 0.999789i \(0.493460\pi\)
\(992\) 31.1904 0.990297
\(993\) 24.3999 0.774307
\(994\) −66.7277 −2.11647
\(995\) 0 0
\(996\) −1.47218 −0.0466477
\(997\) 0.429360 0.0135980 0.00679898 0.999977i \(-0.497836\pi\)
0.00679898 + 0.999977i \(0.497836\pi\)
\(998\) 37.3293 1.18164
\(999\) −16.6825 −0.527812
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 5225.2.a.bc.1.8 30
5.2 odd 4 1045.2.b.e.419.8 30
5.3 odd 4 1045.2.b.e.419.23 yes 30
5.4 even 2 inner 5225.2.a.bc.1.23 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.b.e.419.8 30 5.2 odd 4
1045.2.b.e.419.23 yes 30 5.3 odd 4
5225.2.a.bc.1.8 30 1.1 even 1 trivial
5225.2.a.bc.1.23 30 5.4 even 2 inner