Properties

Label 4205.2.a.d.1.1
Level $4205$
Weight $2$
Character 4205.1
Self dual yes
Analytic conductor $33.577$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4205,2,Mod(1,4205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4205, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4205.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4205 = 5 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4205.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: \(33.5770940499\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 145)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4205.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.414214 q^{2} +2.00000 q^{3} -1.82843 q^{4} +1.00000 q^{5} -0.828427 q^{6} -4.82843 q^{7} +1.58579 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.414214 q^{2} +2.00000 q^{3} -1.82843 q^{4} +1.00000 q^{5} -0.828427 q^{6} -4.82843 q^{7} +1.58579 q^{8} +1.00000 q^{9} -0.414214 q^{10} -0.828427 q^{11} -3.65685 q^{12} -2.00000 q^{13} +2.00000 q^{14} +2.00000 q^{15} +3.00000 q^{16} -2.82843 q^{17} -0.414214 q^{18} +4.82843 q^{19} -1.82843 q^{20} -9.65685 q^{21} +0.343146 q^{22} -3.17157 q^{23} +3.17157 q^{24} +1.00000 q^{25} +0.828427 q^{26} -4.00000 q^{27} +8.82843 q^{28} -0.828427 q^{30} -6.48528 q^{31} -4.41421 q^{32} -1.65685 q^{33} +1.17157 q^{34} -4.82843 q^{35} -1.82843 q^{36} +8.48528 q^{37} -2.00000 q^{38} -4.00000 q^{39} +1.58579 q^{40} +6.00000 q^{41} +4.00000 q^{42} +6.00000 q^{43} +1.51472 q^{44} +1.00000 q^{45} +1.31371 q^{46} +11.6569 q^{47} +6.00000 q^{48} +16.3137 q^{49} -0.414214 q^{50} -5.65685 q^{51} +3.65685 q^{52} -3.65685 q^{53} +1.65685 q^{54} -0.828427 q^{55} -7.65685 q^{56} +9.65685 q^{57} -3.65685 q^{60} +3.65685 q^{61} +2.68629 q^{62} -4.82843 q^{63} -4.17157 q^{64} -2.00000 q^{65} +0.686292 q^{66} +6.48528 q^{67} +5.17157 q^{68} -6.34315 q^{69} +2.00000 q^{70} -15.3137 q^{71} +1.58579 q^{72} -8.48528 q^{73} -3.51472 q^{74} +2.00000 q^{75} -8.82843 q^{76} +4.00000 q^{77} +1.65685 q^{78} +2.48528 q^{79} +3.00000 q^{80} -11.0000 q^{81} -2.48528 q^{82} +7.17157 q^{83} +17.6569 q^{84} -2.82843 q^{85} -2.48528 q^{86} -1.31371 q^{88} +7.65685 q^{89} -0.414214 q^{90} +9.65685 q^{91} +5.79899 q^{92} -12.9706 q^{93} -4.82843 q^{94} +4.82843 q^{95} -8.82843 q^{96} +12.4853 q^{97} -6.75736 q^{98} -0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} - 4 q^{7} + 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} - 4 q^{7} + 6 q^{8} + 2 q^{9} + 2 q^{10} + 4 q^{11} + 4 q^{12} - 4 q^{13} + 4 q^{14} + 4 q^{15} + 6 q^{16} + 2 q^{18} + 4 q^{19} + 2 q^{20} - 8 q^{21} + 12 q^{22} - 12 q^{23} + 12 q^{24} + 2 q^{25} - 4 q^{26} - 8 q^{27} + 12 q^{28} + 4 q^{30} + 4 q^{31} - 6 q^{32} + 8 q^{33} + 8 q^{34} - 4 q^{35} + 2 q^{36} - 4 q^{38} - 8 q^{39} + 6 q^{40} + 12 q^{41} + 8 q^{42} + 12 q^{43} + 20 q^{44} + 2 q^{45} - 20 q^{46} + 12 q^{47} + 12 q^{48} + 10 q^{49} + 2 q^{50} - 4 q^{52} + 4 q^{53} - 8 q^{54} + 4 q^{55} - 4 q^{56} + 8 q^{57} + 4 q^{60} - 4 q^{61} + 28 q^{62} - 4 q^{63} - 14 q^{64} - 4 q^{65} + 24 q^{66} - 4 q^{67} + 16 q^{68} - 24 q^{69} + 4 q^{70} - 8 q^{71} + 6 q^{72} - 24 q^{74} + 4 q^{75} - 12 q^{76} + 8 q^{77} - 8 q^{78} - 12 q^{79} + 6 q^{80} - 22 q^{81} + 12 q^{82} + 20 q^{83} + 24 q^{84} + 12 q^{86} + 20 q^{88} + 4 q^{89} + 2 q^{90} + 8 q^{91} - 28 q^{92} + 8 q^{93} - 4 q^{94} + 4 q^{95} - 12 q^{96} + 8 q^{97} - 22 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.414214 −0.292893 −0.146447 0.989219i \(-0.546784\pi\)
−0.146447 + 0.989219i \(0.546784\pi\)
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) −1.82843 −0.914214
\(5\) 1.00000 0.447214
\(6\) −0.828427 −0.338204
\(7\) −4.82843 −1.82497 −0.912487 0.409106i \(-0.865841\pi\)
−0.912487 + 0.409106i \(0.865841\pi\)
\(8\) 1.58579 0.560660
\(9\) 1.00000 0.333333
\(10\) −0.414214 −0.130986
\(11\) −0.828427 −0.249780 −0.124890 0.992171i \(-0.539858\pi\)
−0.124890 + 0.992171i \(0.539858\pi\)
\(12\) −3.65685 −1.05564
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 2.00000 0.534522
\(15\) 2.00000 0.516398
\(16\) 3.00000 0.750000
\(17\) −2.82843 −0.685994 −0.342997 0.939336i \(-0.611442\pi\)
−0.342997 + 0.939336i \(0.611442\pi\)
\(18\) −0.414214 −0.0976311
\(19\) 4.82843 1.10772 0.553859 0.832611i \(-0.313155\pi\)
0.553859 + 0.832611i \(0.313155\pi\)
\(20\) −1.82843 −0.408849
\(21\) −9.65685 −2.10730
\(22\) 0.343146 0.0731589
\(23\) −3.17157 −0.661319 −0.330659 0.943750i \(-0.607271\pi\)
−0.330659 + 0.943750i \(0.607271\pi\)
\(24\) 3.17157 0.647395
\(25\) 1.00000 0.200000
\(26\) 0.828427 0.162468
\(27\) −4.00000 −0.769800
\(28\) 8.82843 1.66842
\(29\) 0 0
\(30\) −0.828427 −0.151249
\(31\) −6.48528 −1.16479 −0.582395 0.812906i \(-0.697884\pi\)
−0.582395 + 0.812906i \(0.697884\pi\)
\(32\) −4.41421 −0.780330
\(33\) −1.65685 −0.288421
\(34\) 1.17157 0.200923
\(35\) −4.82843 −0.816153
\(36\) −1.82843 −0.304738
\(37\) 8.48528 1.39497 0.697486 0.716599i \(-0.254302\pi\)
0.697486 + 0.716599i \(0.254302\pi\)
\(38\) −2.00000 −0.324443
\(39\) −4.00000 −0.640513
\(40\) 1.58579 0.250735
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 4.00000 0.617213
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 1.51472 0.228352
\(45\) 1.00000 0.149071
\(46\) 1.31371 0.193696
\(47\) 11.6569 1.70033 0.850163 0.526519i \(-0.176503\pi\)
0.850163 + 0.526519i \(0.176503\pi\)
\(48\) 6.00000 0.866025
\(49\) 16.3137 2.33053
\(50\) −0.414214 −0.0585786
\(51\) −5.65685 −0.792118
\(52\) 3.65685 0.507114
\(53\) −3.65685 −0.502308 −0.251154 0.967947i \(-0.580810\pi\)
−0.251154 + 0.967947i \(0.580810\pi\)
\(54\) 1.65685 0.225469
\(55\) −0.828427 −0.111705
\(56\) −7.65685 −1.02319
\(57\) 9.65685 1.27908
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −3.65685 −0.472098
\(61\) 3.65685 0.468212 0.234106 0.972211i \(-0.424784\pi\)
0.234106 + 0.972211i \(0.424784\pi\)
\(62\) 2.68629 0.341159
\(63\) −4.82843 −0.608325
\(64\) −4.17157 −0.521447
\(65\) −2.00000 −0.248069
\(66\) 0.686292 0.0844766
\(67\) 6.48528 0.792303 0.396152 0.918185i \(-0.370345\pi\)
0.396152 + 0.918185i \(0.370345\pi\)
\(68\) 5.17157 0.627145
\(69\) −6.34315 −0.763625
\(70\) 2.00000 0.239046
\(71\) −15.3137 −1.81740 −0.908701 0.417447i \(-0.862925\pi\)
−0.908701 + 0.417447i \(0.862925\pi\)
\(72\) 1.58579 0.186887
\(73\) −8.48528 −0.993127 −0.496564 0.868000i \(-0.665405\pi\)
−0.496564 + 0.868000i \(0.665405\pi\)
\(74\) −3.51472 −0.408578
\(75\) 2.00000 0.230940
\(76\) −8.82843 −1.01269
\(77\) 4.00000 0.455842
\(78\) 1.65685 0.187602
\(79\) 2.48528 0.279616 0.139808 0.990179i \(-0.455351\pi\)
0.139808 + 0.990179i \(0.455351\pi\)
\(80\) 3.00000 0.335410
\(81\) −11.0000 −1.22222
\(82\) −2.48528 −0.274453
\(83\) 7.17157 0.787182 0.393591 0.919286i \(-0.371233\pi\)
0.393591 + 0.919286i \(0.371233\pi\)
\(84\) 17.6569 1.92652
\(85\) −2.82843 −0.306786
\(86\) −2.48528 −0.267995
\(87\) 0 0
\(88\) −1.31371 −0.140042
\(89\) 7.65685 0.811625 0.405812 0.913956i \(-0.366989\pi\)
0.405812 + 0.913956i \(0.366989\pi\)
\(90\) −0.414214 −0.0436619
\(91\) 9.65685 1.01231
\(92\) 5.79899 0.604586
\(93\) −12.9706 −1.34498
\(94\) −4.82843 −0.498014
\(95\) 4.82843 0.495386
\(96\) −8.82843 −0.901048
\(97\) 12.4853 1.26769 0.633844 0.773461i \(-0.281476\pi\)
0.633844 + 0.773461i \(0.281476\pi\)
\(98\) −6.75736 −0.682596
\(99\) −0.828427 −0.0832601
\(100\) −1.82843 −0.182843
\(101\) −15.6569 −1.55792 −0.778958 0.627077i \(-0.784251\pi\)
−0.778958 + 0.627077i \(0.784251\pi\)
\(102\) 2.34315 0.232006
\(103\) 16.1421 1.59053 0.795266 0.606261i \(-0.207331\pi\)
0.795266 + 0.606261i \(0.207331\pi\)
\(104\) −3.17157 −0.310998
\(105\) −9.65685 −0.942412
\(106\) 1.51472 0.147122
\(107\) 20.1421 1.94721 0.973607 0.228232i \(-0.0732943\pi\)
0.973607 + 0.228232i \(0.0732943\pi\)
\(108\) 7.31371 0.703762
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0.343146 0.0327177
\(111\) 16.9706 1.61077
\(112\) −14.4853 −1.36873
\(113\) 2.82843 0.266076 0.133038 0.991111i \(-0.457527\pi\)
0.133038 + 0.991111i \(0.457527\pi\)
\(114\) −4.00000 −0.374634
\(115\) −3.17157 −0.295751
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 13.6569 1.25192
\(120\) 3.17157 0.289524
\(121\) −10.3137 −0.937610
\(122\) −1.51472 −0.137136
\(123\) 12.0000 1.08200
\(124\) 11.8579 1.06487
\(125\) 1.00000 0.0894427
\(126\) 2.00000 0.178174
\(127\) −6.00000 −0.532414 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) 10.5563 0.933058
\(129\) 12.0000 1.05654
\(130\) 0.828427 0.0726579
\(131\) 12.1421 1.06086 0.530432 0.847728i \(-0.322030\pi\)
0.530432 + 0.847728i \(0.322030\pi\)
\(132\) 3.02944 0.263679
\(133\) −23.3137 −2.02155
\(134\) −2.68629 −0.232060
\(135\) −4.00000 −0.344265
\(136\) −4.48528 −0.384610
\(137\) 5.17157 0.441837 0.220919 0.975292i \(-0.429094\pi\)
0.220919 + 0.975292i \(0.429094\pi\)
\(138\) 2.62742 0.223661
\(139\) 21.6569 1.83691 0.918455 0.395525i \(-0.129437\pi\)
0.918455 + 0.395525i \(0.129437\pi\)
\(140\) 8.82843 0.746138
\(141\) 23.3137 1.96337
\(142\) 6.34315 0.532305
\(143\) 1.65685 0.138553
\(144\) 3.00000 0.250000
\(145\) 0 0
\(146\) 3.51472 0.290880
\(147\) 32.6274 2.69106
\(148\) −15.5147 −1.27530
\(149\) 9.31371 0.763009 0.381504 0.924367i \(-0.375406\pi\)
0.381504 + 0.924367i \(0.375406\pi\)
\(150\) −0.828427 −0.0676408
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 7.65685 0.621053
\(153\) −2.82843 −0.228665
\(154\) −1.65685 −0.133513
\(155\) −6.48528 −0.520910
\(156\) 7.31371 0.585565
\(157\) −0.485281 −0.0387297 −0.0193648 0.999812i \(-0.506164\pi\)
−0.0193648 + 0.999812i \(0.506164\pi\)
\(158\) −1.02944 −0.0818976
\(159\) −7.31371 −0.580015
\(160\) −4.41421 −0.348974
\(161\) 15.3137 1.20689
\(162\) 4.55635 0.357981
\(163\) 8.34315 0.653486 0.326743 0.945113i \(-0.394049\pi\)
0.326743 + 0.945113i \(0.394049\pi\)
\(164\) −10.9706 −0.856657
\(165\) −1.65685 −0.128986
\(166\) −2.97056 −0.230560
\(167\) −2.48528 −0.192317 −0.0961584 0.995366i \(-0.530656\pi\)
−0.0961584 + 0.995366i \(0.530656\pi\)
\(168\) −15.3137 −1.18148
\(169\) −9.00000 −0.692308
\(170\) 1.17157 0.0898555
\(171\) 4.82843 0.369239
\(172\) −10.9706 −0.836498
\(173\) 17.3137 1.31634 0.658168 0.752871i \(-0.271331\pi\)
0.658168 + 0.752871i \(0.271331\pi\)
\(174\) 0 0
\(175\) −4.82843 −0.364995
\(176\) −2.48528 −0.187335
\(177\) 0 0
\(178\) −3.17157 −0.237719
\(179\) −23.3137 −1.74255 −0.871274 0.490797i \(-0.836706\pi\)
−0.871274 + 0.490797i \(0.836706\pi\)
\(180\) −1.82843 −0.136283
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) −4.00000 −0.296500
\(183\) 7.31371 0.540645
\(184\) −5.02944 −0.370775
\(185\) 8.48528 0.623850
\(186\) 5.37258 0.393937
\(187\) 2.34315 0.171348
\(188\) −21.3137 −1.55446
\(189\) 19.3137 1.40487
\(190\) −2.00000 −0.145095
\(191\) 20.8284 1.50709 0.753546 0.657395i \(-0.228342\pi\)
0.753546 + 0.657395i \(0.228342\pi\)
\(192\) −8.34315 −0.602115
\(193\) −4.48528 −0.322858 −0.161429 0.986884i \(-0.551610\pi\)
−0.161429 + 0.986884i \(0.551610\pi\)
\(194\) −5.17157 −0.371297
\(195\) −4.00000 −0.286446
\(196\) −29.8284 −2.13060
\(197\) −19.6569 −1.40049 −0.700246 0.713901i \(-0.746927\pi\)
−0.700246 + 0.713901i \(0.746927\pi\)
\(198\) 0.343146 0.0243863
\(199\) 12.0000 0.850657 0.425329 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) 1.58579 0.112132
\(201\) 12.9706 0.914873
\(202\) 6.48528 0.456303
\(203\) 0 0
\(204\) 10.3431 0.724165
\(205\) 6.00000 0.419058
\(206\) −6.68629 −0.465856
\(207\) −3.17157 −0.220440
\(208\) −6.00000 −0.416025
\(209\) −4.00000 −0.276686
\(210\) 4.00000 0.276026
\(211\) 0.828427 0.0570313 0.0285156 0.999593i \(-0.490922\pi\)
0.0285156 + 0.999593i \(0.490922\pi\)
\(212\) 6.68629 0.459216
\(213\) −30.6274 −2.09856
\(214\) −8.34315 −0.570326
\(215\) 6.00000 0.409197
\(216\) −6.34315 −0.431596
\(217\) 31.3137 2.12571
\(218\) −0.828427 −0.0561082
\(219\) −16.9706 −1.14676
\(220\) 1.51472 0.102122
\(221\) 5.65685 0.380521
\(222\) −7.02944 −0.471785
\(223\) −17.7990 −1.19191 −0.595954 0.803018i \(-0.703226\pi\)
−0.595954 + 0.803018i \(0.703226\pi\)
\(224\) 21.3137 1.42408
\(225\) 1.00000 0.0666667
\(226\) −1.17157 −0.0779319
\(227\) 20.1421 1.33688 0.668440 0.743766i \(-0.266962\pi\)
0.668440 + 0.743766i \(0.266962\pi\)
\(228\) −17.6569 −1.16935
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 1.31371 0.0866234
\(231\) 8.00000 0.526361
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0.828427 0.0541560
\(235\) 11.6569 0.760409
\(236\) 0 0
\(237\) 4.97056 0.322873
\(238\) −5.65685 −0.366679
\(239\) −0.686292 −0.0443925 −0.0221963 0.999754i \(-0.507066\pi\)
−0.0221963 + 0.999754i \(0.507066\pi\)
\(240\) 6.00000 0.387298
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 4.27208 0.274620
\(243\) −10.0000 −0.641500
\(244\) −6.68629 −0.428046
\(245\) 16.3137 1.04224
\(246\) −4.97056 −0.316912
\(247\) −9.65685 −0.614451
\(248\) −10.2843 −0.653052
\(249\) 14.3431 0.908960
\(250\) −0.414214 −0.0261972
\(251\) −8.82843 −0.557245 −0.278623 0.960401i \(-0.589878\pi\)
−0.278623 + 0.960401i \(0.589878\pi\)
\(252\) 8.82843 0.556139
\(253\) 2.62742 0.165184
\(254\) 2.48528 0.155940
\(255\) −5.65685 −0.354246
\(256\) 3.97056 0.248160
\(257\) 6.68629 0.417079 0.208540 0.978014i \(-0.433129\pi\)
0.208540 + 0.978014i \(0.433129\pi\)
\(258\) −4.97056 −0.309454
\(259\) −40.9706 −2.54579
\(260\) 3.65685 0.226788
\(261\) 0 0
\(262\) −5.02944 −0.310720
\(263\) 19.6569 1.21209 0.606047 0.795429i \(-0.292754\pi\)
0.606047 + 0.795429i \(0.292754\pi\)
\(264\) −2.62742 −0.161706
\(265\) −3.65685 −0.224639
\(266\) 9.65685 0.592100
\(267\) 15.3137 0.937184
\(268\) −11.8579 −0.724334
\(269\) 21.3137 1.29952 0.649760 0.760140i \(-0.274869\pi\)
0.649760 + 0.760140i \(0.274869\pi\)
\(270\) 1.65685 0.100833
\(271\) 9.79899 0.595246 0.297623 0.954683i \(-0.403806\pi\)
0.297623 + 0.954683i \(0.403806\pi\)
\(272\) −8.48528 −0.514496
\(273\) 19.3137 1.16892
\(274\) −2.14214 −0.129411
\(275\) −0.828427 −0.0499560
\(276\) 11.5980 0.698116
\(277\) −3.65685 −0.219719 −0.109860 0.993947i \(-0.535040\pi\)
−0.109860 + 0.993947i \(0.535040\pi\)
\(278\) −8.97056 −0.538019
\(279\) −6.48528 −0.388264
\(280\) −7.65685 −0.457585
\(281\) −29.3137 −1.74871 −0.874355 0.485288i \(-0.838715\pi\)
−0.874355 + 0.485288i \(0.838715\pi\)
\(282\) −9.65685 −0.575057
\(283\) 4.82843 0.287020 0.143510 0.989649i \(-0.454161\pi\)
0.143510 + 0.989649i \(0.454161\pi\)
\(284\) 28.0000 1.66149
\(285\) 9.65685 0.572023
\(286\) −0.686292 −0.0405813
\(287\) −28.9706 −1.71008
\(288\) −4.41421 −0.260110
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) 24.9706 1.46380
\(292\) 15.5147 0.907930
\(293\) −8.48528 −0.495715 −0.247858 0.968796i \(-0.579727\pi\)
−0.247858 + 0.968796i \(0.579727\pi\)
\(294\) −13.5147 −0.788194
\(295\) 0 0
\(296\) 13.4558 0.782105
\(297\) 3.31371 0.192281
\(298\) −3.85786 −0.223480
\(299\) 6.34315 0.366834
\(300\) −3.65685 −0.211129
\(301\) −28.9706 −1.66984
\(302\) 4.97056 0.286024
\(303\) −31.3137 −1.79893
\(304\) 14.4853 0.830788
\(305\) 3.65685 0.209391
\(306\) 1.17157 0.0669744
\(307\) 22.9706 1.31100 0.655500 0.755195i \(-0.272458\pi\)
0.655500 + 0.755195i \(0.272458\pi\)
\(308\) −7.31371 −0.416737
\(309\) 32.2843 1.83659
\(310\) 2.68629 0.152571
\(311\) −14.4853 −0.821385 −0.410692 0.911774i \(-0.634713\pi\)
−0.410692 + 0.911774i \(0.634713\pi\)
\(312\) −6.34315 −0.359110
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0.201010 0.0113437
\(315\) −4.82843 −0.272051
\(316\) −4.54416 −0.255629
\(317\) −2.82843 −0.158860 −0.0794301 0.996840i \(-0.525310\pi\)
−0.0794301 + 0.996840i \(0.525310\pi\)
\(318\) 3.02944 0.169882
\(319\) 0 0
\(320\) −4.17157 −0.233198
\(321\) 40.2843 2.24845
\(322\) −6.34315 −0.353490
\(323\) −13.6569 −0.759888
\(324\) 20.1127 1.11737
\(325\) −2.00000 −0.110940
\(326\) −3.45584 −0.191402
\(327\) 4.00000 0.221201
\(328\) 9.51472 0.525362
\(329\) −56.2843 −3.10305
\(330\) 0.686292 0.0377791
\(331\) −21.7990 −1.19818 −0.599090 0.800681i \(-0.704471\pi\)
−0.599090 + 0.800681i \(0.704471\pi\)
\(332\) −13.1127 −0.719653
\(333\) 8.48528 0.464991
\(334\) 1.02944 0.0563283
\(335\) 6.48528 0.354329
\(336\) −28.9706 −1.58047
\(337\) 1.17157 0.0638196 0.0319098 0.999491i \(-0.489841\pi\)
0.0319098 + 0.999491i \(0.489841\pi\)
\(338\) 3.72792 0.202772
\(339\) 5.65685 0.307238
\(340\) 5.17157 0.280468
\(341\) 5.37258 0.290942
\(342\) −2.00000 −0.108148
\(343\) −44.9706 −2.42818
\(344\) 9.51472 0.512999
\(345\) −6.34315 −0.341503
\(346\) −7.17157 −0.385546
\(347\) −8.14214 −0.437093 −0.218546 0.975827i \(-0.570131\pi\)
−0.218546 + 0.975827i \(0.570131\pi\)
\(348\) 0 0
\(349\) 20.6274 1.10416 0.552080 0.833791i \(-0.313834\pi\)
0.552080 + 0.833791i \(0.313834\pi\)
\(350\) 2.00000 0.106904
\(351\) 8.00000 0.427008
\(352\) 3.65685 0.194911
\(353\) −4.34315 −0.231162 −0.115581 0.993298i \(-0.536873\pi\)
−0.115581 + 0.993298i \(0.536873\pi\)
\(354\) 0 0
\(355\) −15.3137 −0.812767
\(356\) −14.0000 −0.741999
\(357\) 27.3137 1.44559
\(358\) 9.65685 0.510381
\(359\) 3.85786 0.203610 0.101805 0.994804i \(-0.467538\pi\)
0.101805 + 0.994804i \(0.467538\pi\)
\(360\) 1.58579 0.0835783
\(361\) 4.31371 0.227037
\(362\) 2.48528 0.130623
\(363\) −20.6274 −1.08266
\(364\) −17.6569 −0.925471
\(365\) −8.48528 −0.444140
\(366\) −3.02944 −0.158351
\(367\) −18.0000 −0.939592 −0.469796 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) −9.51472 −0.495989
\(369\) 6.00000 0.312348
\(370\) −3.51472 −0.182722
\(371\) 17.6569 0.916698
\(372\) 23.7157 1.22960
\(373\) −6.97056 −0.360922 −0.180461 0.983582i \(-0.557759\pi\)
−0.180461 + 0.983582i \(0.557759\pi\)
\(374\) −0.970563 −0.0501866
\(375\) 2.00000 0.103280
\(376\) 18.4853 0.953306
\(377\) 0 0
\(378\) −8.00000 −0.411476
\(379\) −22.4853 −1.15499 −0.577496 0.816394i \(-0.695970\pi\)
−0.577496 + 0.816394i \(0.695970\pi\)
\(380\) −8.82843 −0.452889
\(381\) −12.0000 −0.614779
\(382\) −8.62742 −0.441417
\(383\) 2.48528 0.126992 0.0634960 0.997982i \(-0.479775\pi\)
0.0634960 + 0.997982i \(0.479775\pi\)
\(384\) 21.1127 1.07740
\(385\) 4.00000 0.203859
\(386\) 1.85786 0.0945628
\(387\) 6.00000 0.304997
\(388\) −22.8284 −1.15894
\(389\) 29.3137 1.48626 0.743132 0.669145i \(-0.233339\pi\)
0.743132 + 0.669145i \(0.233339\pi\)
\(390\) 1.65685 0.0838981
\(391\) 8.97056 0.453661
\(392\) 25.8701 1.30664
\(393\) 24.2843 1.22498
\(394\) 8.14214 0.410195
\(395\) 2.48528 0.125048
\(396\) 1.51472 0.0761175
\(397\) −19.6569 −0.986549 −0.493275 0.869874i \(-0.664200\pi\)
−0.493275 + 0.869874i \(0.664200\pi\)
\(398\) −4.97056 −0.249152
\(399\) −46.6274 −2.33429
\(400\) 3.00000 0.150000
\(401\) −6.68629 −0.333897 −0.166949 0.985966i \(-0.553391\pi\)
−0.166949 + 0.985966i \(0.553391\pi\)
\(402\) −5.37258 −0.267960
\(403\) 12.9706 0.646110
\(404\) 28.6274 1.42427
\(405\) −11.0000 −0.546594
\(406\) 0 0
\(407\) −7.02944 −0.348436
\(408\) −8.97056 −0.444109
\(409\) 2.97056 0.146885 0.0734424 0.997299i \(-0.476601\pi\)
0.0734424 + 0.997299i \(0.476601\pi\)
\(410\) −2.48528 −0.122739
\(411\) 10.3431 0.510190
\(412\) −29.5147 −1.45409
\(413\) 0 0
\(414\) 1.31371 0.0645653
\(415\) 7.17157 0.352039
\(416\) 8.82843 0.432849
\(417\) 43.3137 2.12108
\(418\) 1.65685 0.0810394
\(419\) 28.9706 1.41530 0.707652 0.706561i \(-0.249754\pi\)
0.707652 + 0.706561i \(0.249754\pi\)
\(420\) 17.6569 0.861566
\(421\) −18.9706 −0.924569 −0.462284 0.886732i \(-0.652970\pi\)
−0.462284 + 0.886732i \(0.652970\pi\)
\(422\) −0.343146 −0.0167041
\(423\) 11.6569 0.566776
\(424\) −5.79899 −0.281624
\(425\) −2.82843 −0.137199
\(426\) 12.6863 0.614653
\(427\) −17.6569 −0.854475
\(428\) −36.8284 −1.78017
\(429\) 3.31371 0.159987
\(430\) −2.48528 −0.119851
\(431\) 3.31371 0.159616 0.0798079 0.996810i \(-0.474569\pi\)
0.0798079 + 0.996810i \(0.474569\pi\)
\(432\) −12.0000 −0.577350
\(433\) 29.1716 1.40190 0.700948 0.713212i \(-0.252760\pi\)
0.700948 + 0.713212i \(0.252760\pi\)
\(434\) −12.9706 −0.622607
\(435\) 0 0
\(436\) −3.65685 −0.175132
\(437\) −15.3137 −0.732554
\(438\) 7.02944 0.335880
\(439\) −10.3431 −0.493651 −0.246826 0.969060i \(-0.579388\pi\)
−0.246826 + 0.969060i \(0.579388\pi\)
\(440\) −1.31371 −0.0626286
\(441\) 16.3137 0.776843
\(442\) −2.34315 −0.111452
\(443\) 7.65685 0.363788 0.181894 0.983318i \(-0.441777\pi\)
0.181894 + 0.983318i \(0.441777\pi\)
\(444\) −31.0294 −1.47259
\(445\) 7.65685 0.362970
\(446\) 7.37258 0.349102
\(447\) 18.6274 0.881047
\(448\) 20.1421 0.951626
\(449\) −11.6569 −0.550121 −0.275060 0.961427i \(-0.588698\pi\)
−0.275060 + 0.961427i \(0.588698\pi\)
\(450\) −0.414214 −0.0195262
\(451\) −4.97056 −0.234055
\(452\) −5.17157 −0.243250
\(453\) −24.0000 −1.12762
\(454\) −8.34315 −0.391563
\(455\) 9.65685 0.452720
\(456\) 15.3137 0.717130
\(457\) 19.6569 0.919509 0.459754 0.888046i \(-0.347937\pi\)
0.459754 + 0.888046i \(0.347937\pi\)
\(458\) −0.828427 −0.0387099
\(459\) 11.3137 0.528079
\(460\) 5.79899 0.270379
\(461\) 35.6569 1.66071 0.830353 0.557238i \(-0.188139\pi\)
0.830353 + 0.557238i \(0.188139\pi\)
\(462\) −3.31371 −0.154168
\(463\) 21.7990 1.01308 0.506542 0.862215i \(-0.330923\pi\)
0.506542 + 0.862215i \(0.330923\pi\)
\(464\) 0 0
\(465\) −12.9706 −0.601495
\(466\) −7.45584 −0.345385
\(467\) −10.9706 −0.507657 −0.253829 0.967249i \(-0.581690\pi\)
−0.253829 + 0.967249i \(0.581690\pi\)
\(468\) 3.65685 0.169038
\(469\) −31.3137 −1.44593
\(470\) −4.82843 −0.222719
\(471\) −0.970563 −0.0447212
\(472\) 0 0
\(473\) −4.97056 −0.228547
\(474\) −2.05887 −0.0945672
\(475\) 4.82843 0.221543
\(476\) −24.9706 −1.14452
\(477\) −3.65685 −0.167436
\(478\) 0.284271 0.0130023
\(479\) −7.17157 −0.327678 −0.163839 0.986487i \(-0.552388\pi\)
−0.163839 + 0.986487i \(0.552388\pi\)
\(480\) −8.82843 −0.402961
\(481\) −16.9706 −0.773791
\(482\) −4.14214 −0.188669
\(483\) 30.6274 1.39360
\(484\) 18.8579 0.857176
\(485\) 12.4853 0.566927
\(486\) 4.14214 0.187891
\(487\) 9.79899 0.444035 0.222017 0.975043i \(-0.428736\pi\)
0.222017 + 0.975043i \(0.428736\pi\)
\(488\) 5.79899 0.262508
\(489\) 16.6863 0.754580
\(490\) −6.75736 −0.305266
\(491\) 7.45584 0.336478 0.168239 0.985746i \(-0.446192\pi\)
0.168239 + 0.985746i \(0.446192\pi\)
\(492\) −21.9411 −0.989182
\(493\) 0 0
\(494\) 4.00000 0.179969
\(495\) −0.828427 −0.0372350
\(496\) −19.4558 −0.873593
\(497\) 73.9411 3.31671
\(498\) −5.94113 −0.266228
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) −1.82843 −0.0817697
\(501\) −4.97056 −0.222068
\(502\) 3.65685 0.163213
\(503\) 30.0000 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) −7.65685 −0.341063
\(505\) −15.6569 −0.696721
\(506\) −1.08831 −0.0483814
\(507\) −18.0000 −0.799408
\(508\) 10.9706 0.486740
\(509\) 0.627417 0.0278098 0.0139049 0.999903i \(-0.495574\pi\)
0.0139049 + 0.999903i \(0.495574\pi\)
\(510\) 2.34315 0.103756
\(511\) 40.9706 1.81243
\(512\) −22.7574 −1.00574
\(513\) −19.3137 −0.852721
\(514\) −2.76955 −0.122160
\(515\) 16.1421 0.711307
\(516\) −21.9411 −0.965904
\(517\) −9.65685 −0.424708
\(518\) 16.9706 0.745644
\(519\) 34.6274 1.51997
\(520\) −3.17157 −0.139083
\(521\) −21.3137 −0.933771 −0.466885 0.884318i \(-0.654624\pi\)
−0.466885 + 0.884318i \(0.654624\pi\)
\(522\) 0 0
\(523\) 2.48528 0.108674 0.0543369 0.998523i \(-0.482696\pi\)
0.0543369 + 0.998523i \(0.482696\pi\)
\(524\) −22.2010 −0.969856
\(525\) −9.65685 −0.421460
\(526\) −8.14214 −0.355014
\(527\) 18.3431 0.799040
\(528\) −4.97056 −0.216316
\(529\) −12.9411 −0.562658
\(530\) 1.51472 0.0657952
\(531\) 0 0
\(532\) 42.6274 1.84813
\(533\) −12.0000 −0.519778
\(534\) −6.34315 −0.274495
\(535\) 20.1421 0.870820
\(536\) 10.2843 0.444213
\(537\) −46.6274 −2.01212
\(538\) −8.82843 −0.380621
\(539\) −13.5147 −0.582120
\(540\) 7.31371 0.314732
\(541\) 5.02944 0.216232 0.108116 0.994138i \(-0.465518\pi\)
0.108116 + 0.994138i \(0.465518\pi\)
\(542\) −4.05887 −0.174344
\(543\) −12.0000 −0.514969
\(544\) 12.4853 0.535302
\(545\) 2.00000 0.0856706
\(546\) −8.00000 −0.342368
\(547\) −2.48528 −0.106263 −0.0531315 0.998588i \(-0.516920\pi\)
−0.0531315 + 0.998588i \(0.516920\pi\)
\(548\) −9.45584 −0.403934
\(549\) 3.65685 0.156071
\(550\) 0.343146 0.0146318
\(551\) 0 0
\(552\) −10.0589 −0.428134
\(553\) −12.0000 −0.510292
\(554\) 1.51472 0.0643542
\(555\) 16.9706 0.720360
\(556\) −39.5980 −1.67933
\(557\) −27.9411 −1.18390 −0.591952 0.805973i \(-0.701642\pi\)
−0.591952 + 0.805973i \(0.701642\pi\)
\(558\) 2.68629 0.113720
\(559\) −12.0000 −0.507546
\(560\) −14.4853 −0.612115
\(561\) 4.68629 0.197855
\(562\) 12.1421 0.512185
\(563\) −7.65685 −0.322698 −0.161349 0.986897i \(-0.551584\pi\)
−0.161349 + 0.986897i \(0.551584\pi\)
\(564\) −42.6274 −1.79494
\(565\) 2.82843 0.118993
\(566\) −2.00000 −0.0840663
\(567\) 53.1127 2.23052
\(568\) −24.2843 −1.01895
\(569\) −27.6569 −1.15944 −0.579718 0.814817i \(-0.696837\pi\)
−0.579718 + 0.814817i \(0.696837\pi\)
\(570\) −4.00000 −0.167542
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) −3.02944 −0.126667
\(573\) 41.6569 1.74024
\(574\) 12.0000 0.500870
\(575\) −3.17157 −0.132264
\(576\) −4.17157 −0.173816
\(577\) −23.7990 −0.990765 −0.495382 0.868675i \(-0.664972\pi\)
−0.495382 + 0.868675i \(0.664972\pi\)
\(578\) 3.72792 0.155061
\(579\) −8.97056 −0.372804
\(580\) 0 0
\(581\) −34.6274 −1.43659
\(582\) −10.3431 −0.428737
\(583\) 3.02944 0.125466
\(584\) −13.4558 −0.556807
\(585\) −2.00000 −0.0826898
\(586\) 3.51472 0.145192
\(587\) −29.7990 −1.22994 −0.614968 0.788552i \(-0.710831\pi\)
−0.614968 + 0.788552i \(0.710831\pi\)
\(588\) −59.6569 −2.46021
\(589\) −31.3137 −1.29026
\(590\) 0 0
\(591\) −39.3137 −1.61715
\(592\) 25.4558 1.04623
\(593\) −7.65685 −0.314429 −0.157215 0.987564i \(-0.550251\pi\)
−0.157215 + 0.987564i \(0.550251\pi\)
\(594\) −1.37258 −0.0563178
\(595\) 13.6569 0.559876
\(596\) −17.0294 −0.697553
\(597\) 24.0000 0.982255
\(598\) −2.62742 −0.107443
\(599\) 37.7990 1.54442 0.772212 0.635364i \(-0.219150\pi\)
0.772212 + 0.635364i \(0.219150\pi\)
\(600\) 3.17157 0.129479
\(601\) −2.00000 −0.0815817 −0.0407909 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) 12.0000 0.489083
\(603\) 6.48528 0.264101
\(604\) 21.9411 0.892772
\(605\) −10.3137 −0.419312
\(606\) 12.9706 0.526893
\(607\) −9.02944 −0.366494 −0.183247 0.983067i \(-0.558661\pi\)
−0.183247 + 0.983067i \(0.558661\pi\)
\(608\) −21.3137 −0.864385
\(609\) 0 0
\(610\) −1.51472 −0.0613292
\(611\) −23.3137 −0.943172
\(612\) 5.17157 0.209048
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −9.51472 −0.383983
\(615\) 12.0000 0.483887
\(616\) 6.34315 0.255573
\(617\) 9.17157 0.369234 0.184617 0.982811i \(-0.440896\pi\)
0.184617 + 0.982811i \(0.440896\pi\)
\(618\) −13.3726 −0.537924
\(619\) 9.79899 0.393855 0.196927 0.980418i \(-0.436904\pi\)
0.196927 + 0.980418i \(0.436904\pi\)
\(620\) 11.8579 0.476223
\(621\) 12.6863 0.509083
\(622\) 6.00000 0.240578
\(623\) −36.9706 −1.48119
\(624\) −12.0000 −0.480384
\(625\) 1.00000 0.0400000
\(626\) 2.48528 0.0993318
\(627\) −8.00000 −0.319489
\(628\) 0.887302 0.0354072
\(629\) −24.0000 −0.956943
\(630\) 2.00000 0.0796819
\(631\) 36.9706 1.47177 0.735887 0.677104i \(-0.236765\pi\)
0.735887 + 0.677104i \(0.236765\pi\)
\(632\) 3.94113 0.156770
\(633\) 1.65685 0.0658540
\(634\) 1.17157 0.0465291
\(635\) −6.00000 −0.238103
\(636\) 13.3726 0.530257
\(637\) −32.6274 −1.29275
\(638\) 0 0
\(639\) −15.3137 −0.605801
\(640\) 10.5563 0.417276
\(641\) −0.627417 −0.0247815 −0.0123907 0.999923i \(-0.503944\pi\)
−0.0123907 + 0.999923i \(0.503944\pi\)
\(642\) −16.6863 −0.658555
\(643\) 19.4558 0.767264 0.383632 0.923486i \(-0.374673\pi\)
0.383632 + 0.923486i \(0.374673\pi\)
\(644\) −28.0000 −1.10335
\(645\) 12.0000 0.472500
\(646\) 5.65685 0.222566
\(647\) −41.1127 −1.61631 −0.808153 0.588972i \(-0.799533\pi\)
−0.808153 + 0.588972i \(0.799533\pi\)
\(648\) −17.4437 −0.685251
\(649\) 0 0
\(650\) 0.828427 0.0324936
\(651\) 62.6274 2.45456
\(652\) −15.2548 −0.597425
\(653\) 17.1716 0.671976 0.335988 0.941866i \(-0.390930\pi\)
0.335988 + 0.941866i \(0.390930\pi\)
\(654\) −1.65685 −0.0647881
\(655\) 12.1421 0.474432
\(656\) 18.0000 0.702782
\(657\) −8.48528 −0.331042
\(658\) 23.3137 0.908863
\(659\) −1.79899 −0.0700787 −0.0350393 0.999386i \(-0.511156\pi\)
−0.0350393 + 0.999386i \(0.511156\pi\)
\(660\) 3.02944 0.117921
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) 9.02944 0.350939
\(663\) 11.3137 0.439388
\(664\) 11.3726 0.441342
\(665\) −23.3137 −0.904067
\(666\) −3.51472 −0.136193
\(667\) 0 0
\(668\) 4.54416 0.175819
\(669\) −35.5980 −1.37630
\(670\) −2.68629 −0.103780
\(671\) −3.02944 −0.116950
\(672\) 42.6274 1.64439
\(673\) 22.9706 0.885450 0.442725 0.896657i \(-0.354012\pi\)
0.442725 + 0.896657i \(0.354012\pi\)
\(674\) −0.485281 −0.0186923
\(675\) −4.00000 −0.153960
\(676\) 16.4558 0.632917
\(677\) 36.7696 1.41317 0.706584 0.707629i \(-0.250235\pi\)
0.706584 + 0.707629i \(0.250235\pi\)
\(678\) −2.34315 −0.0899880
\(679\) −60.2843 −2.31350
\(680\) −4.48528 −0.172003
\(681\) 40.2843 1.54370
\(682\) −2.22540 −0.0852148
\(683\) 11.8579 0.453729 0.226864 0.973926i \(-0.427153\pi\)
0.226864 + 0.973926i \(0.427153\pi\)
\(684\) −8.82843 −0.337563
\(685\) 5.17157 0.197596
\(686\) 18.6274 0.711198
\(687\) 4.00000 0.152610
\(688\) 18.0000 0.686244
\(689\) 7.31371 0.278630
\(690\) 2.62742 0.100024
\(691\) −44.9706 −1.71076 −0.855380 0.518000i \(-0.826677\pi\)
−0.855380 + 0.518000i \(0.826677\pi\)
\(692\) −31.6569 −1.20341
\(693\) 4.00000 0.151947
\(694\) 3.37258 0.128022
\(695\) 21.6569 0.821491
\(696\) 0 0
\(697\) −16.9706 −0.642806
\(698\) −8.54416 −0.323401
\(699\) 36.0000 1.36165
\(700\) 8.82843 0.333683
\(701\) 6.68629 0.252538 0.126269 0.991996i \(-0.459700\pi\)
0.126269 + 0.991996i \(0.459700\pi\)
\(702\) −3.31371 −0.125068
\(703\) 40.9706 1.54523
\(704\) 3.45584 0.130247
\(705\) 23.3137 0.878045
\(706\) 1.79899 0.0677059
\(707\) 75.5980 2.84315
\(708\) 0 0
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 6.34315 0.238054
\(711\) 2.48528 0.0932053
\(712\) 12.1421 0.455046
\(713\) 20.5685 0.770298
\(714\) −11.3137 −0.423405
\(715\) 1.65685 0.0619628
\(716\) 42.6274 1.59306
\(717\) −1.37258 −0.0512601
\(718\) −1.59798 −0.0596361
\(719\) −34.6274 −1.29138 −0.645692 0.763598i \(-0.723431\pi\)
−0.645692 + 0.763598i \(0.723431\pi\)
\(720\) 3.00000 0.111803
\(721\) −77.9411 −2.90268
\(722\) −1.78680 −0.0664977
\(723\) 20.0000 0.743808
\(724\) 10.9706 0.407718
\(725\) 0 0
\(726\) 8.54416 0.317103
\(727\) 23.9411 0.887927 0.443964 0.896045i \(-0.353572\pi\)
0.443964 + 0.896045i \(0.353572\pi\)
\(728\) 15.3137 0.567564
\(729\) 13.0000 0.481481
\(730\) 3.51472 0.130086
\(731\) −16.9706 −0.627679
\(732\) −13.3726 −0.494265
\(733\) 22.8284 0.843187 0.421594 0.906785i \(-0.361471\pi\)
0.421594 + 0.906785i \(0.361471\pi\)
\(734\) 7.45584 0.275200
\(735\) 32.6274 1.20348
\(736\) 14.0000 0.516047
\(737\) −5.37258 −0.197902
\(738\) −2.48528 −0.0914845
\(739\) −14.4853 −0.532850 −0.266425 0.963856i \(-0.585842\pi\)
−0.266425 + 0.963856i \(0.585842\pi\)
\(740\) −15.5147 −0.570332
\(741\) −19.3137 −0.709507
\(742\) −7.31371 −0.268495
\(743\) 52.6274 1.93071 0.965356 0.260935i \(-0.0840309\pi\)
0.965356 + 0.260935i \(0.0840309\pi\)
\(744\) −20.5685 −0.754079
\(745\) 9.31371 0.341228
\(746\) 2.88730 0.105712
\(747\) 7.17157 0.262394
\(748\) −4.28427 −0.156648
\(749\) −97.2548 −3.55361
\(750\) −0.828427 −0.0302499
\(751\) −16.1421 −0.589035 −0.294517 0.955646i \(-0.595159\pi\)
−0.294517 + 0.955646i \(0.595159\pi\)
\(752\) 34.9706 1.27525
\(753\) −17.6569 −0.643452
\(754\) 0 0
\(755\) −12.0000 −0.436725
\(756\) −35.3137 −1.28435
\(757\) −19.5147 −0.709275 −0.354637 0.935004i \(-0.615396\pi\)
−0.354637 + 0.935004i \(0.615396\pi\)
\(758\) 9.31371 0.338289
\(759\) 5.25483 0.190738
\(760\) 7.65685 0.277743
\(761\) 8.62742 0.312744 0.156372 0.987698i \(-0.450020\pi\)
0.156372 + 0.987698i \(0.450020\pi\)
\(762\) 4.97056 0.180064
\(763\) −9.65685 −0.349602
\(764\) −38.0833 −1.37780
\(765\) −2.82843 −0.102262
\(766\) −1.02944 −0.0371951
\(767\) 0 0
\(768\) 7.94113 0.286551
\(769\) 15.6569 0.564601 0.282300 0.959326i \(-0.408903\pi\)
0.282300 + 0.959326i \(0.408903\pi\)
\(770\) −1.65685 −0.0597089
\(771\) 13.3726 0.481602
\(772\) 8.20101 0.295161
\(773\) 8.48528 0.305194 0.152597 0.988288i \(-0.451236\pi\)
0.152597 + 0.988288i \(0.451236\pi\)
\(774\) −2.48528 −0.0893316
\(775\) −6.48528 −0.232958
\(776\) 19.7990 0.710742
\(777\) −81.9411 −2.93962
\(778\) −12.1421 −0.435317
\(779\) 28.9706 1.03798
\(780\) 7.31371 0.261873
\(781\) 12.6863 0.453951
\(782\) −3.71573 −0.132874
\(783\) 0 0
\(784\) 48.9411 1.74790
\(785\) −0.485281 −0.0173204
\(786\) −10.0589 −0.358788
\(787\) 17.7990 0.634465 0.317233 0.948348i \(-0.397246\pi\)
0.317233 + 0.948348i \(0.397246\pi\)
\(788\) 35.9411 1.28035
\(789\) 39.3137 1.39961
\(790\) −1.02944 −0.0366257
\(791\) −13.6569 −0.485582
\(792\) −1.31371 −0.0466806
\(793\) −7.31371 −0.259717
\(794\) 8.14214 0.288954
\(795\) −7.31371 −0.259391
\(796\) −21.9411 −0.777683
\(797\) 5.85786 0.207496 0.103748 0.994604i \(-0.466916\pi\)
0.103748 + 0.994604i \(0.466916\pi\)
\(798\) 19.3137 0.683698
\(799\) −32.9706 −1.16641
\(800\) −4.41421 −0.156066
\(801\) 7.65685 0.270542
\(802\) 2.76955 0.0977963
\(803\) 7.02944 0.248063
\(804\) −23.7157 −0.836389
\(805\) 15.3137 0.539737
\(806\) −5.37258 −0.189241
\(807\) 42.6274 1.50056
\(808\) −24.8284 −0.873461
\(809\) −42.2843 −1.48664 −0.743318 0.668938i \(-0.766749\pi\)
−0.743318 + 0.668938i \(0.766749\pi\)
\(810\) 4.55635 0.160094
\(811\) 37.6569 1.32231 0.661155 0.750249i \(-0.270066\pi\)
0.661155 + 0.750249i \(0.270066\pi\)
\(812\) 0 0
\(813\) 19.5980 0.687331
\(814\) 2.91169 0.102055
\(815\) 8.34315 0.292248
\(816\) −16.9706 −0.594089
\(817\) 28.9706 1.01355
\(818\) −1.23045 −0.0430216
\(819\) 9.65685 0.337438
\(820\) −10.9706 −0.383109
\(821\) −22.6863 −0.791757 −0.395879 0.918303i \(-0.629560\pi\)
−0.395879 + 0.918303i \(0.629560\pi\)
\(822\) −4.28427 −0.149431
\(823\) 30.9706 1.07957 0.539783 0.841804i \(-0.318506\pi\)
0.539783 + 0.841804i \(0.318506\pi\)
\(824\) 25.5980 0.891748
\(825\) −1.65685 −0.0576843
\(826\) 0 0
\(827\) −17.3137 −0.602057 −0.301028 0.953615i \(-0.597330\pi\)
−0.301028 + 0.953615i \(0.597330\pi\)
\(828\) 5.79899 0.201529
\(829\) −20.6274 −0.716420 −0.358210 0.933641i \(-0.616613\pi\)
−0.358210 + 0.933641i \(0.616613\pi\)
\(830\) −2.97056 −0.103110
\(831\) −7.31371 −0.253710
\(832\) 8.34315 0.289247
\(833\) −46.1421 −1.59873
\(834\) −17.9411 −0.621250
\(835\) −2.48528 −0.0860067
\(836\) 7.31371 0.252950
\(837\) 25.9411 0.896656
\(838\) −12.0000 −0.414533
\(839\) −2.48528 −0.0858014 −0.0429007 0.999079i \(-0.513660\pi\)
−0.0429007 + 0.999079i \(0.513660\pi\)
\(840\) −15.3137 −0.528373
\(841\) 0 0
\(842\) 7.85786 0.270800
\(843\) −58.6274 −2.01924
\(844\) −1.51472 −0.0521388
\(845\) −9.00000 −0.309609
\(846\) −4.82843 −0.166005
\(847\) 49.7990 1.71111
\(848\) −10.9706 −0.376731
\(849\) 9.65685 0.331422
\(850\) 1.17157 0.0401846
\(851\) −26.9117 −0.922521
\(852\) 56.0000 1.91853
\(853\) −51.1127 −1.75007 −0.875033 0.484064i \(-0.839160\pi\)
−0.875033 + 0.484064i \(0.839160\pi\)
\(854\) 7.31371 0.250270
\(855\) 4.82843 0.165129
\(856\) 31.9411 1.09173
\(857\) 3.37258 0.115205 0.0576026 0.998340i \(-0.481654\pi\)
0.0576026 + 0.998340i \(0.481654\pi\)
\(858\) −1.37258 −0.0468592
\(859\) 56.4264 1.92524 0.962622 0.270848i \(-0.0873041\pi\)
0.962622 + 0.270848i \(0.0873041\pi\)
\(860\) −10.9706 −0.374093
\(861\) −57.9411 −1.97463
\(862\) −1.37258 −0.0467504
\(863\) −36.1421 −1.23029 −0.615146 0.788413i \(-0.710903\pi\)
−0.615146 + 0.788413i \(0.710903\pi\)
\(864\) 17.6569 0.600698
\(865\) 17.3137 0.588684
\(866\) −12.0833 −0.410606
\(867\) −18.0000 −0.611312
\(868\) −57.2548 −1.94336
\(869\) −2.05887 −0.0698425
\(870\) 0 0
\(871\) −12.9706 −0.439491
\(872\) 3.17157 0.107403
\(873\) 12.4853 0.422563
\(874\) 6.34315 0.214560
\(875\) −4.82843 −0.163231
\(876\) 31.0294 1.04839
\(877\) 38.2843 1.29277 0.646384 0.763012i \(-0.276280\pi\)
0.646384 + 0.763012i \(0.276280\pi\)
\(878\) 4.28427 0.144587
\(879\) −16.9706 −0.572403
\(880\) −2.48528 −0.0837788
\(881\) −29.3137 −0.987604 −0.493802 0.869574i \(-0.664393\pi\)
−0.493802 + 0.869574i \(0.664393\pi\)
\(882\) −6.75736 −0.227532
\(883\) −14.4853 −0.487469 −0.243734 0.969842i \(-0.578372\pi\)
−0.243734 + 0.969842i \(0.578372\pi\)
\(884\) −10.3431 −0.347878
\(885\) 0 0
\(886\) −3.17157 −0.106551
\(887\) 6.68629 0.224504 0.112252 0.993680i \(-0.464194\pi\)
0.112252 + 0.993680i \(0.464194\pi\)
\(888\) 26.9117 0.903097
\(889\) 28.9706 0.971641
\(890\) −3.17157 −0.106311
\(891\) 9.11270 0.305287
\(892\) 32.5442 1.08966
\(893\) 56.2843 1.88348
\(894\) −7.71573 −0.258053
\(895\) −23.3137 −0.779291
\(896\) −50.9706 −1.70281
\(897\) 12.6863 0.423583
\(898\) 4.82843 0.161127
\(899\) 0 0
\(900\) −1.82843 −0.0609476
\(901\) 10.3431 0.344580
\(902\) 2.05887 0.0685530
\(903\) −57.9411 −1.92816
\(904\) 4.48528 0.149178
\(905\) −6.00000 −0.199447
\(906\) 9.94113 0.330272
\(907\) 10.0000 0.332045 0.166022 0.986122i \(-0.446908\pi\)
0.166022 + 0.986122i \(0.446908\pi\)
\(908\) −36.8284 −1.22219
\(909\) −15.6569 −0.519305
\(910\) −4.00000 −0.132599
\(911\) −32.1421 −1.06492 −0.532458 0.846456i \(-0.678732\pi\)
−0.532458 + 0.846456i \(0.678732\pi\)
\(912\) 28.9706 0.959311
\(913\) −5.94113 −0.196623
\(914\) −8.14214 −0.269318
\(915\) 7.31371 0.241784
\(916\) −3.65685 −0.120826
\(917\) −58.6274 −1.93605
\(918\) −4.68629 −0.154671
\(919\) −36.0000 −1.18753 −0.593765 0.804638i \(-0.702359\pi\)
−0.593765 + 0.804638i \(0.702359\pi\)
\(920\) −5.02944 −0.165816
\(921\) 45.9411 1.51381
\(922\) −14.7696 −0.486409
\(923\) 30.6274 1.00811
\(924\) −14.6274 −0.481207
\(925\) 8.48528 0.278994
\(926\) −9.02944 −0.296726
\(927\) 16.1421 0.530177
\(928\) 0 0
\(929\) −4.62742 −0.151821 −0.0759103 0.997115i \(-0.524186\pi\)
−0.0759103 + 0.997115i \(0.524186\pi\)
\(930\) 5.37258 0.176174
\(931\) 78.7696 2.58157
\(932\) −32.9117 −1.07806
\(933\) −28.9706 −0.948454
\(934\) 4.54416 0.148689
\(935\) 2.34315 0.0766291
\(936\) −3.17157 −0.103666
\(937\) 19.6569 0.642161 0.321081 0.947052i \(-0.395954\pi\)
0.321081 + 0.947052i \(0.395954\pi\)
\(938\) 12.9706 0.423504
\(939\) −12.0000 −0.391605
\(940\) −21.3137 −0.695177
\(941\) −27.9411 −0.910855 −0.455427 0.890273i \(-0.650514\pi\)
−0.455427 + 0.890273i \(0.650514\pi\)
\(942\) 0.402020 0.0130985
\(943\) −19.0294 −0.619684
\(944\) 0 0
\(945\) 19.3137 0.628275
\(946\) 2.05887 0.0669398
\(947\) −44.9117 −1.45943 −0.729717 0.683749i \(-0.760348\pi\)
−0.729717 + 0.683749i \(0.760348\pi\)
\(948\) −9.08831 −0.295175
\(949\) 16.9706 0.550888
\(950\) −2.00000 −0.0648886
\(951\) −5.65685 −0.183436
\(952\) 21.6569 0.701903
\(953\) 29.3137 0.949564 0.474782 0.880103i \(-0.342527\pi\)
0.474782 + 0.880103i \(0.342527\pi\)
\(954\) 1.51472 0.0490408
\(955\) 20.8284 0.673992
\(956\) 1.25483 0.0405842
\(957\) 0 0
\(958\) 2.97056 0.0959745
\(959\) −24.9706 −0.806342
\(960\) −8.34315 −0.269274
\(961\) 11.0589 0.356738
\(962\) 7.02944 0.226638
\(963\) 20.1421 0.649071
\(964\) −18.2843 −0.588897
\(965\) −4.48528 −0.144386
\(966\) −12.6863 −0.408175
\(967\) −14.9706 −0.481421 −0.240710 0.970597i \(-0.577380\pi\)
−0.240710 + 0.970597i \(0.577380\pi\)
\(968\) −16.3553 −0.525681
\(969\) −27.3137 −0.877443
\(970\) −5.17157 −0.166049
\(971\) −28.1421 −0.903124 −0.451562 0.892240i \(-0.649133\pi\)
−0.451562 + 0.892240i \(0.649133\pi\)
\(972\) 18.2843 0.586468
\(973\) −104.569 −3.35231
\(974\) −4.05887 −0.130055
\(975\) −4.00000 −0.128103
\(976\) 10.9706 0.351159
\(977\) −2.68629 −0.0859421 −0.0429710 0.999076i \(-0.513682\pi\)
−0.0429710 + 0.999076i \(0.513682\pi\)
\(978\) −6.91169 −0.221011
\(979\) −6.34315 −0.202728
\(980\) −29.8284 −0.952834
\(981\) 2.00000 0.0638551
\(982\) −3.08831 −0.0985520
\(983\) 9.31371 0.297061 0.148531 0.988908i \(-0.452546\pi\)
0.148531 + 0.988908i \(0.452546\pi\)
\(984\) 19.0294 0.606636
\(985\) −19.6569 −0.626319
\(986\) 0 0
\(987\) −112.569 −3.58310
\(988\) 17.6569 0.561739
\(989\) −19.0294 −0.605101
\(990\) 0.343146 0.0109059
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) 28.6274 0.908921
\(993\) −43.5980 −1.38354
\(994\) −30.6274 −0.971443
\(995\) 12.0000 0.380426
\(996\) −26.2254 −0.830983
\(997\) −6.82843 −0.216258 −0.108129 0.994137i \(-0.534486\pi\)
−0.108129 + 0.994137i \(0.534486\pi\)
\(998\) −14.9117 −0.472021
\(999\) −33.9411 −1.07385
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 4205.2.a.d.1.1 2
29.28 even 2 145.2.a.b.1.2 2
87.86 odd 2 1305.2.a.n.1.1 2
116.115 odd 2 2320.2.a.k.1.2 2
145.28 odd 4 725.2.b.c.349.2 4
145.57 odd 4 725.2.b.c.349.3 4
145.144 even 2 725.2.a.c.1.1 2
203.202 odd 2 7105.2.a.e.1.2 2
232.115 odd 2 9280.2.a.w.1.2 2
232.173 even 2 9280.2.a.be.1.1 2
435.434 odd 2 6525.2.a.p.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.b.1.2 2 29.28 even 2
725.2.a.c.1.1 2 145.144 even 2
725.2.b.c.349.2 4 145.28 odd 4
725.2.b.c.349.3 4 145.57 odd 4
1305.2.a.n.1.1 2 87.86 odd 2
2320.2.a.k.1.2 2 116.115 odd 2
4205.2.a.d.1.1 2 1.1 even 1 trivial
6525.2.a.p.1.2 2 435.434 odd 2
7105.2.a.e.1.2 2 203.202 odd 2
9280.2.a.w.1.2 2 232.115 odd 2
9280.2.a.be.1.1 2 232.173 even 2