Properties

Label 6045.2.a.p.1.1
Level $6045$
Weight $2$
Character 6045.1
Self dual yes
Analytic conductor $48.270$
Analytic rank $1$
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] = [6045,2,Mod(1,6045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6045.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6045.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: \(48.2695680219\)
Analytic rank: \(1\)
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: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 6045.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{2} +1.00000 q^{3} +1.00000 q^{5} -1.41421 q^{6} -0.414214 q^{7} +2.82843 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.41421 q^{2} +1.00000 q^{3} +1.00000 q^{5} -1.41421 q^{6} -0.414214 q^{7} +2.82843 q^{8} +1.00000 q^{9} -1.41421 q^{10} +0.828427 q^{11} +1.00000 q^{13} +0.585786 q^{14} +1.00000 q^{15} -4.00000 q^{16} +7.65685 q^{17} -1.41421 q^{18} -7.41421 q^{19} -0.414214 q^{21} -1.17157 q^{22} -8.82843 q^{23} +2.82843 q^{24} +1.00000 q^{25} -1.41421 q^{26} +1.00000 q^{27} +4.41421 q^{29} -1.41421 q^{30} +1.00000 q^{31} +0.828427 q^{33} -10.8284 q^{34} -0.414214 q^{35} -8.82843 q^{37} +10.4853 q^{38} +1.00000 q^{39} +2.82843 q^{40} -8.41421 q^{41} +0.585786 q^{42} -2.41421 q^{43} +1.00000 q^{45} +12.4853 q^{46} +5.07107 q^{47} -4.00000 q^{48} -6.82843 q^{49} -1.41421 q^{50} +7.65685 q^{51} -0.242641 q^{53} -1.41421 q^{54} +0.828427 q^{55} -1.17157 q^{56} -7.41421 q^{57} -6.24264 q^{58} +6.07107 q^{59} -8.24264 q^{61} -1.41421 q^{62} -0.414214 q^{63} +8.00000 q^{64} +1.00000 q^{65} -1.17157 q^{66} -8.41421 q^{67} -8.82843 q^{69} +0.585786 q^{70} -15.3137 q^{71} +2.82843 q^{72} +1.07107 q^{73} +12.4853 q^{74} +1.00000 q^{75} -0.343146 q^{77} -1.41421 q^{78} -7.07107 q^{79} -4.00000 q^{80} +1.00000 q^{81} +11.8995 q^{82} -6.17157 q^{83} +7.65685 q^{85} +3.41421 q^{86} +4.41421 q^{87} +2.34315 q^{88} -8.82843 q^{89} -1.41421 q^{90} -0.414214 q^{91} +1.00000 q^{93} -7.17157 q^{94} -7.41421 q^{95} +7.58579 q^{97} +9.65685 q^{98} +0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 4 q^{11} + 2 q^{13} + 4 q^{14} + 2 q^{15} - 8 q^{16} + 4 q^{17} - 12 q^{19} + 2 q^{21} - 8 q^{22} - 12 q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} + 2 q^{31} - 4 q^{33} - 16 q^{34} + 2 q^{35} - 12 q^{37} + 4 q^{38} + 2 q^{39} - 14 q^{41} + 4 q^{42} - 2 q^{43} + 2 q^{45} + 8 q^{46} - 4 q^{47} - 8 q^{48} - 8 q^{49} + 4 q^{51} + 8 q^{53} - 4 q^{55} - 8 q^{56} - 12 q^{57} - 4 q^{58} - 2 q^{59} - 8 q^{61} + 2 q^{63} + 16 q^{64} + 2 q^{65} - 8 q^{66} - 14 q^{67} - 12 q^{69} + 4 q^{70} - 8 q^{71} - 12 q^{73} + 8 q^{74} + 2 q^{75} - 12 q^{77} - 8 q^{80} + 2 q^{81} + 4 q^{82} - 18 q^{83} + 4 q^{85} + 4 q^{86} + 6 q^{87} + 16 q^{88} - 12 q^{89} + 2 q^{91} + 2 q^{93} - 20 q^{94} - 12 q^{95} + 18 q^{97} + 8 q^{98} - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) −1.41421 −0.577350
\(7\) −0.414214 −0.156558 −0.0782790 0.996931i \(-0.524942\pi\)
−0.0782790 + 0.996931i \(0.524942\pi\)
\(8\) 2.82843 1.00000
\(9\) 1.00000 0.333333
\(10\) −1.41421 −0.447214
\(11\) 0.828427 0.249780 0.124890 0.992171i \(-0.460142\pi\)
0.124890 + 0.992171i \(0.460142\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0.585786 0.156558
\(15\) 1.00000 0.258199
\(16\) −4.00000 −1.00000
\(17\) 7.65685 1.85706 0.928530 0.371257i \(-0.121073\pi\)
0.928530 + 0.371257i \(0.121073\pi\)
\(18\) −1.41421 −0.333333
\(19\) −7.41421 −1.70094 −0.850469 0.526026i \(-0.823682\pi\)
−0.850469 + 0.526026i \(0.823682\pi\)
\(20\) 0 0
\(21\) −0.414214 −0.0903888
\(22\) −1.17157 −0.249780
\(23\) −8.82843 −1.84085 −0.920427 0.390914i \(-0.872159\pi\)
−0.920427 + 0.390914i \(0.872159\pi\)
\(24\) 2.82843 0.577350
\(25\) 1.00000 0.200000
\(26\) −1.41421 −0.277350
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 4.41421 0.819699 0.409849 0.912153i \(-0.365581\pi\)
0.409849 + 0.912153i \(0.365581\pi\)
\(30\) −1.41421 −0.258199
\(31\) 1.00000 0.179605
\(32\) 0 0
\(33\) 0.828427 0.144211
\(34\) −10.8284 −1.85706
\(35\) −0.414214 −0.0700149
\(36\) 0 0
\(37\) −8.82843 −1.45138 −0.725692 0.688019i \(-0.758480\pi\)
−0.725692 + 0.688019i \(0.758480\pi\)
\(38\) 10.4853 1.70094
\(39\) 1.00000 0.160128
\(40\) 2.82843 0.447214
\(41\) −8.41421 −1.31408 −0.657040 0.753856i \(-0.728192\pi\)
−0.657040 + 0.753856i \(0.728192\pi\)
\(42\) 0.585786 0.0903888
\(43\) −2.41421 −0.368164 −0.184082 0.982911i \(-0.558931\pi\)
−0.184082 + 0.982911i \(0.558931\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 12.4853 1.84085
\(47\) 5.07107 0.739691 0.369846 0.929093i \(-0.379411\pi\)
0.369846 + 0.929093i \(0.379411\pi\)
\(48\) −4.00000 −0.577350
\(49\) −6.82843 −0.975490
\(50\) −1.41421 −0.200000
\(51\) 7.65685 1.07217
\(52\) 0 0
\(53\) −0.242641 −0.0333293 −0.0166646 0.999861i \(-0.505305\pi\)
−0.0166646 + 0.999861i \(0.505305\pi\)
\(54\) −1.41421 −0.192450
\(55\) 0.828427 0.111705
\(56\) −1.17157 −0.156558
\(57\) −7.41421 −0.982037
\(58\) −6.24264 −0.819699
\(59\) 6.07107 0.790386 0.395193 0.918598i \(-0.370678\pi\)
0.395193 + 0.918598i \(0.370678\pi\)
\(60\) 0 0
\(61\) −8.24264 −1.05536 −0.527681 0.849443i \(-0.676938\pi\)
−0.527681 + 0.849443i \(0.676938\pi\)
\(62\) −1.41421 −0.179605
\(63\) −0.414214 −0.0521860
\(64\) 8.00000 1.00000
\(65\) 1.00000 0.124035
\(66\) −1.17157 −0.144211
\(67\) −8.41421 −1.02796 −0.513980 0.857802i \(-0.671829\pi\)
−0.513980 + 0.857802i \(0.671829\pi\)
\(68\) 0 0
\(69\) −8.82843 −1.06282
\(70\) 0.585786 0.0700149
\(71\) −15.3137 −1.81740 −0.908701 0.417447i \(-0.862925\pi\)
−0.908701 + 0.417447i \(0.862925\pi\)
\(72\) 2.82843 0.333333
\(73\) 1.07107 0.125359 0.0626795 0.998034i \(-0.480035\pi\)
0.0626795 + 0.998034i \(0.480035\pi\)
\(74\) 12.4853 1.45138
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −0.343146 −0.0391051
\(78\) −1.41421 −0.160128
\(79\) −7.07107 −0.795557 −0.397779 0.917481i \(-0.630219\pi\)
−0.397779 + 0.917481i \(0.630219\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) 11.8995 1.31408
\(83\) −6.17157 −0.677418 −0.338709 0.940891i \(-0.609990\pi\)
−0.338709 + 0.940891i \(0.609990\pi\)
\(84\) 0 0
\(85\) 7.65685 0.830502
\(86\) 3.41421 0.368164
\(87\) 4.41421 0.473253
\(88\) 2.34315 0.249780
\(89\) −8.82843 −0.935811 −0.467906 0.883778i \(-0.654991\pi\)
−0.467906 + 0.883778i \(0.654991\pi\)
\(90\) −1.41421 −0.149071
\(91\) −0.414214 −0.0434214
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) −7.17157 −0.739691
\(95\) −7.41421 −0.760682
\(96\) 0 0
\(97\) 7.58579 0.770220 0.385110 0.922871i \(-0.374164\pi\)
0.385110 + 0.922871i \(0.374164\pi\)
\(98\) 9.65685 0.975490
\(99\) 0.828427 0.0832601
\(100\) 0 0
\(101\) 5.07107 0.504590 0.252295 0.967650i \(-0.418815\pi\)
0.252295 + 0.967650i \(0.418815\pi\)
\(102\) −10.8284 −1.07217
\(103\) 13.0711 1.28793 0.643965 0.765055i \(-0.277288\pi\)
0.643965 + 0.765055i \(0.277288\pi\)
\(104\) 2.82843 0.277350
\(105\) −0.414214 −0.0404231
\(106\) 0.343146 0.0333293
\(107\) 5.82843 0.563455 0.281728 0.959494i \(-0.409093\pi\)
0.281728 + 0.959494i \(0.409093\pi\)
\(108\) 0 0
\(109\) −3.75736 −0.359890 −0.179945 0.983677i \(-0.557592\pi\)
−0.179945 + 0.983677i \(0.557592\pi\)
\(110\) −1.17157 −0.111705
\(111\) −8.82843 −0.837957
\(112\) 1.65685 0.156558
\(113\) 5.34315 0.502641 0.251320 0.967904i \(-0.419135\pi\)
0.251320 + 0.967904i \(0.419135\pi\)
\(114\) 10.4853 0.982037
\(115\) −8.82843 −0.823255
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) −8.58579 −0.790386
\(119\) −3.17157 −0.290738
\(120\) 2.82843 0.258199
\(121\) −10.3137 −0.937610
\(122\) 11.6569 1.05536
\(123\) −8.41421 −0.758684
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0.585786 0.0521860
\(127\) 0.757359 0.0672048 0.0336024 0.999435i \(-0.489302\pi\)
0.0336024 + 0.999435i \(0.489302\pi\)
\(128\) −11.3137 −1.00000
\(129\) −2.41421 −0.212560
\(130\) −1.41421 −0.124035
\(131\) −15.4142 −1.34675 −0.673373 0.739303i \(-0.735155\pi\)
−0.673373 + 0.739303i \(0.735155\pi\)
\(132\) 0 0
\(133\) 3.07107 0.266295
\(134\) 11.8995 1.02796
\(135\) 1.00000 0.0860663
\(136\) 21.6569 1.85706
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 12.4853 1.06282
\(139\) 12.5858 1.06751 0.533756 0.845638i \(-0.320780\pi\)
0.533756 + 0.845638i \(0.320780\pi\)
\(140\) 0 0
\(141\) 5.07107 0.427061
\(142\) 21.6569 1.81740
\(143\) 0.828427 0.0692766
\(144\) −4.00000 −0.333333
\(145\) 4.41421 0.366580
\(146\) −1.51472 −0.125359
\(147\) −6.82843 −0.563199
\(148\) 0 0
\(149\) 14.0711 1.15275 0.576373 0.817186i \(-0.304467\pi\)
0.576373 + 0.817186i \(0.304467\pi\)
\(150\) −1.41421 −0.115470
\(151\) −5.82843 −0.474311 −0.237155 0.971472i \(-0.576215\pi\)
−0.237155 + 0.971472i \(0.576215\pi\)
\(152\) −20.9706 −1.70094
\(153\) 7.65685 0.619020
\(154\) 0.485281 0.0391051
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 10.0000 0.795557
\(159\) −0.242641 −0.0192427
\(160\) 0 0
\(161\) 3.65685 0.288200
\(162\) −1.41421 −0.111111
\(163\) 5.17157 0.405069 0.202534 0.979275i \(-0.435082\pi\)
0.202534 + 0.979275i \(0.435082\pi\)
\(164\) 0 0
\(165\) 0.828427 0.0644930
\(166\) 8.72792 0.677418
\(167\) −14.0000 −1.08335 −0.541676 0.840587i \(-0.682210\pi\)
−0.541676 + 0.840587i \(0.682210\pi\)
\(168\) −1.17157 −0.0903888
\(169\) 1.00000 0.0769231
\(170\) −10.8284 −0.830502
\(171\) −7.41421 −0.566979
\(172\) 0 0
\(173\) −0.171573 −0.0130444 −0.00652222 0.999979i \(-0.502076\pi\)
−0.00652222 + 0.999979i \(0.502076\pi\)
\(174\) −6.24264 −0.473253
\(175\) −0.414214 −0.0313116
\(176\) −3.31371 −0.249780
\(177\) 6.07107 0.456329
\(178\) 12.4853 0.935811
\(179\) 7.72792 0.577612 0.288806 0.957388i \(-0.406742\pi\)
0.288806 + 0.957388i \(0.406742\pi\)
\(180\) 0 0
\(181\) 7.51472 0.558565 0.279282 0.960209i \(-0.409903\pi\)
0.279282 + 0.960209i \(0.409903\pi\)
\(182\) 0.585786 0.0434214
\(183\) −8.24264 −0.609314
\(184\) −24.9706 −1.84085
\(185\) −8.82843 −0.649079
\(186\) −1.41421 −0.103695
\(187\) 6.34315 0.463857
\(188\) 0 0
\(189\) −0.414214 −0.0301296
\(190\) 10.4853 0.760682
\(191\) −6.48528 −0.469258 −0.234629 0.972085i \(-0.575388\pi\)
−0.234629 + 0.972085i \(0.575388\pi\)
\(192\) 8.00000 0.577350
\(193\) 7.10051 0.511106 0.255553 0.966795i \(-0.417743\pi\)
0.255553 + 0.966795i \(0.417743\pi\)
\(194\) −10.7279 −0.770220
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) −5.82843 −0.415258 −0.207629 0.978208i \(-0.566575\pi\)
−0.207629 + 0.978208i \(0.566575\pi\)
\(198\) −1.17157 −0.0832601
\(199\) 12.4853 0.885058 0.442529 0.896754i \(-0.354081\pi\)
0.442529 + 0.896754i \(0.354081\pi\)
\(200\) 2.82843 0.200000
\(201\) −8.41421 −0.593493
\(202\) −7.17157 −0.504590
\(203\) −1.82843 −0.128330
\(204\) 0 0
\(205\) −8.41421 −0.587674
\(206\) −18.4853 −1.28793
\(207\) −8.82843 −0.613618
\(208\) −4.00000 −0.277350
\(209\) −6.14214 −0.424860
\(210\) 0.585786 0.0404231
\(211\) −17.4853 −1.20374 −0.601868 0.798595i \(-0.705577\pi\)
−0.601868 + 0.798595i \(0.705577\pi\)
\(212\) 0 0
\(213\) −15.3137 −1.04928
\(214\) −8.24264 −0.563455
\(215\) −2.41421 −0.164648
\(216\) 2.82843 0.192450
\(217\) −0.414214 −0.0281186
\(218\) 5.31371 0.359890
\(219\) 1.07107 0.0723761
\(220\) 0 0
\(221\) 7.65685 0.515056
\(222\) 12.4853 0.837957
\(223\) −19.4142 −1.30007 −0.650036 0.759903i \(-0.725246\pi\)
−0.650036 + 0.759903i \(0.725246\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) −7.55635 −0.502641
\(227\) 28.6274 1.90007 0.950034 0.312146i \(-0.101048\pi\)
0.950034 + 0.312146i \(0.101048\pi\)
\(228\) 0 0
\(229\) 25.4558 1.68217 0.841085 0.540903i \(-0.181918\pi\)
0.841085 + 0.540903i \(0.181918\pi\)
\(230\) 12.4853 0.823255
\(231\) −0.343146 −0.0225773
\(232\) 12.4853 0.819699
\(233\) −24.7990 −1.62464 −0.812318 0.583215i \(-0.801795\pi\)
−0.812318 + 0.583215i \(0.801795\pi\)
\(234\) −1.41421 −0.0924500
\(235\) 5.07107 0.330800
\(236\) 0 0
\(237\) −7.07107 −0.459315
\(238\) 4.48528 0.290738
\(239\) 1.75736 0.113674 0.0568371 0.998383i \(-0.481898\pi\)
0.0568371 + 0.998383i \(0.481898\pi\)
\(240\) −4.00000 −0.258199
\(241\) −16.7990 −1.08212 −0.541059 0.840985i \(-0.681976\pi\)
−0.541059 + 0.840985i \(0.681976\pi\)
\(242\) 14.5858 0.937610
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −6.82843 −0.436252
\(246\) 11.8995 0.758684
\(247\) −7.41421 −0.471755
\(248\) 2.82843 0.179605
\(249\) −6.17157 −0.391108
\(250\) −1.41421 −0.0894427
\(251\) −18.4142 −1.16229 −0.581147 0.813798i \(-0.697396\pi\)
−0.581147 + 0.813798i \(0.697396\pi\)
\(252\) 0 0
\(253\) −7.31371 −0.459809
\(254\) −1.07107 −0.0672048
\(255\) 7.65685 0.479491
\(256\) 0 0
\(257\) −15.3137 −0.955243 −0.477621 0.878566i \(-0.658501\pi\)
−0.477621 + 0.878566i \(0.658501\pi\)
\(258\) 3.41421 0.212560
\(259\) 3.65685 0.227226
\(260\) 0 0
\(261\) 4.41421 0.273233
\(262\) 21.7990 1.34675
\(263\) 2.92893 0.180606 0.0903028 0.995914i \(-0.471217\pi\)
0.0903028 + 0.995914i \(0.471217\pi\)
\(264\) 2.34315 0.144211
\(265\) −0.242641 −0.0149053
\(266\) −4.34315 −0.266295
\(267\) −8.82843 −0.540291
\(268\) 0 0
\(269\) 28.6274 1.74544 0.872722 0.488217i \(-0.162353\pi\)
0.872722 + 0.488217i \(0.162353\pi\)
\(270\) −1.41421 −0.0860663
\(271\) 25.4853 1.54812 0.774060 0.633112i \(-0.218223\pi\)
0.774060 + 0.633112i \(0.218223\pi\)
\(272\) −30.6274 −1.85706
\(273\) −0.414214 −0.0250693
\(274\) 0 0
\(275\) 0.828427 0.0499560
\(276\) 0 0
\(277\) −6.41421 −0.385393 −0.192696 0.981258i \(-0.561723\pi\)
−0.192696 + 0.981258i \(0.561723\pi\)
\(278\) −17.7990 −1.06751
\(279\) 1.00000 0.0598684
\(280\) −1.17157 −0.0700149
\(281\) −2.75736 −0.164490 −0.0822451 0.996612i \(-0.526209\pi\)
−0.0822451 + 0.996612i \(0.526209\pi\)
\(282\) −7.17157 −0.427061
\(283\) −9.07107 −0.539219 −0.269610 0.962970i \(-0.586895\pi\)
−0.269610 + 0.962970i \(0.586895\pi\)
\(284\) 0 0
\(285\) −7.41421 −0.439180
\(286\) −1.17157 −0.0692766
\(287\) 3.48528 0.205730
\(288\) 0 0
\(289\) 41.6274 2.44867
\(290\) −6.24264 −0.366580
\(291\) 7.58579 0.444687
\(292\) 0 0
\(293\) −0.585786 −0.0342220 −0.0171110 0.999854i \(-0.505447\pi\)
−0.0171110 + 0.999854i \(0.505447\pi\)
\(294\) 9.65685 0.563199
\(295\) 6.07107 0.353471
\(296\) −24.9706 −1.45138
\(297\) 0.828427 0.0480702
\(298\) −19.8995 −1.15275
\(299\) −8.82843 −0.510561
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) 8.24264 0.474311
\(303\) 5.07107 0.291325
\(304\) 29.6569 1.70094
\(305\) −8.24264 −0.471972
\(306\) −10.8284 −0.619020
\(307\) −25.3137 −1.44473 −0.722365 0.691512i \(-0.756945\pi\)
−0.722365 + 0.691512i \(0.756945\pi\)
\(308\) 0 0
\(309\) 13.0711 0.743587
\(310\) −1.41421 −0.0803219
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 2.82843 0.160128
\(313\) −6.41421 −0.362553 −0.181276 0.983432i \(-0.558023\pi\)
−0.181276 + 0.983432i \(0.558023\pi\)
\(314\) 14.1421 0.798087
\(315\) −0.414214 −0.0233383
\(316\) 0 0
\(317\) 4.58579 0.257563 0.128782 0.991673i \(-0.458893\pi\)
0.128782 + 0.991673i \(0.458893\pi\)
\(318\) 0.343146 0.0192427
\(319\) 3.65685 0.204745
\(320\) 8.00000 0.447214
\(321\) 5.82843 0.325311
\(322\) −5.17157 −0.288200
\(323\) −56.7696 −3.15874
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) −7.31371 −0.405069
\(327\) −3.75736 −0.207782
\(328\) −23.7990 −1.31408
\(329\) −2.10051 −0.115805
\(330\) −1.17157 −0.0644930
\(331\) −11.8284 −0.650149 −0.325075 0.945688i \(-0.605389\pi\)
−0.325075 + 0.945688i \(0.605389\pi\)
\(332\) 0 0
\(333\) −8.82843 −0.483795
\(334\) 19.7990 1.08335
\(335\) −8.41421 −0.459718
\(336\) 1.65685 0.0903888
\(337\) 22.4853 1.22485 0.612426 0.790528i \(-0.290194\pi\)
0.612426 + 0.790528i \(0.290194\pi\)
\(338\) −1.41421 −0.0769231
\(339\) 5.34315 0.290200
\(340\) 0 0
\(341\) 0.828427 0.0448618
\(342\) 10.4853 0.566979
\(343\) 5.72792 0.309279
\(344\) −6.82843 −0.368164
\(345\) −8.82843 −0.475307
\(346\) 0.242641 0.0130444
\(347\) −25.5563 −1.37194 −0.685968 0.727631i \(-0.740621\pi\)
−0.685968 + 0.727631i \(0.740621\pi\)
\(348\) 0 0
\(349\) 9.79899 0.524528 0.262264 0.964996i \(-0.415531\pi\)
0.262264 + 0.964996i \(0.415531\pi\)
\(350\) 0.585786 0.0313116
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 13.4853 0.717749 0.358875 0.933386i \(-0.383161\pi\)
0.358875 + 0.933386i \(0.383161\pi\)
\(354\) −8.58579 −0.456329
\(355\) −15.3137 −0.812767
\(356\) 0 0
\(357\) −3.17157 −0.167857
\(358\) −10.9289 −0.577612
\(359\) −35.2426 −1.86004 −0.930018 0.367515i \(-0.880209\pi\)
−0.930018 + 0.367515i \(0.880209\pi\)
\(360\) 2.82843 0.149071
\(361\) 35.9706 1.89319
\(362\) −10.6274 −0.558565
\(363\) −10.3137 −0.541329
\(364\) 0 0
\(365\) 1.07107 0.0560623
\(366\) 11.6569 0.609314
\(367\) 35.6569 1.86127 0.930636 0.365945i \(-0.119254\pi\)
0.930636 + 0.365945i \(0.119254\pi\)
\(368\) 35.3137 1.84085
\(369\) −8.41421 −0.438026
\(370\) 12.4853 0.649079
\(371\) 0.100505 0.00521796
\(372\) 0 0
\(373\) 1.51472 0.0784292 0.0392146 0.999231i \(-0.487514\pi\)
0.0392146 + 0.999231i \(0.487514\pi\)
\(374\) −8.97056 −0.463857
\(375\) 1.00000 0.0516398
\(376\) 14.3431 0.739691
\(377\) 4.41421 0.227344
\(378\) 0.585786 0.0301296
\(379\) −11.0711 −0.568683 −0.284341 0.958723i \(-0.591775\pi\)
−0.284341 + 0.958723i \(0.591775\pi\)
\(380\) 0 0
\(381\) 0.757359 0.0388007
\(382\) 9.17157 0.469258
\(383\) −23.4853 −1.20004 −0.600021 0.799984i \(-0.704841\pi\)
−0.600021 + 0.799984i \(0.704841\pi\)
\(384\) −11.3137 −0.577350
\(385\) −0.343146 −0.0174883
\(386\) −10.0416 −0.511106
\(387\) −2.41421 −0.122721
\(388\) 0 0
\(389\) 32.0711 1.62607 0.813034 0.582216i \(-0.197814\pi\)
0.813034 + 0.582216i \(0.197814\pi\)
\(390\) −1.41421 −0.0716115
\(391\) −67.5980 −3.41858
\(392\) −19.3137 −0.975490
\(393\) −15.4142 −0.777544
\(394\) 8.24264 0.415258
\(395\) −7.07107 −0.355784
\(396\) 0 0
\(397\) −3.65685 −0.183532 −0.0917661 0.995781i \(-0.529251\pi\)
−0.0917661 + 0.995781i \(0.529251\pi\)
\(398\) −17.6569 −0.885058
\(399\) 3.07107 0.153746
\(400\) −4.00000 −0.200000
\(401\) 11.7574 0.587135 0.293567 0.955938i \(-0.405158\pi\)
0.293567 + 0.955938i \(0.405158\pi\)
\(402\) 11.8995 0.593493
\(403\) 1.00000 0.0498135
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 2.58579 0.128330
\(407\) −7.31371 −0.362527
\(408\) 21.6569 1.07217
\(409\) 17.9706 0.888587 0.444294 0.895881i \(-0.353455\pi\)
0.444294 + 0.895881i \(0.353455\pi\)
\(410\) 11.8995 0.587674
\(411\) 0 0
\(412\) 0 0
\(413\) −2.51472 −0.123741
\(414\) 12.4853 0.613618
\(415\) −6.17157 −0.302951
\(416\) 0 0
\(417\) 12.5858 0.616329
\(418\) 8.68629 0.424860
\(419\) −18.4853 −0.903065 −0.451533 0.892255i \(-0.649123\pi\)
−0.451533 + 0.892255i \(0.649123\pi\)
\(420\) 0 0
\(421\) −0.686292 −0.0334478 −0.0167239 0.999860i \(-0.505324\pi\)
−0.0167239 + 0.999860i \(0.505324\pi\)
\(422\) 24.7279 1.20374
\(423\) 5.07107 0.246564
\(424\) −0.686292 −0.0333293
\(425\) 7.65685 0.371412
\(426\) 21.6569 1.04928
\(427\) 3.41421 0.165225
\(428\) 0 0
\(429\) 0.828427 0.0399968
\(430\) 3.41421 0.164648
\(431\) −28.2132 −1.35898 −0.679491 0.733684i \(-0.737799\pi\)
−0.679491 + 0.733684i \(0.737799\pi\)
\(432\) −4.00000 −0.192450
\(433\) 3.72792 0.179153 0.0895763 0.995980i \(-0.471449\pi\)
0.0895763 + 0.995980i \(0.471449\pi\)
\(434\) 0.585786 0.0281186
\(435\) 4.41421 0.211645
\(436\) 0 0
\(437\) 65.4558 3.13118
\(438\) −1.51472 −0.0723761
\(439\) 32.1127 1.53266 0.766328 0.642450i \(-0.222082\pi\)
0.766328 + 0.642450i \(0.222082\pi\)
\(440\) 2.34315 0.111705
\(441\) −6.82843 −0.325163
\(442\) −10.8284 −0.515056
\(443\) −39.6274 −1.88276 −0.941378 0.337354i \(-0.890468\pi\)
−0.941378 + 0.337354i \(0.890468\pi\)
\(444\) 0 0
\(445\) −8.82843 −0.418508
\(446\) 27.4558 1.30007
\(447\) 14.0711 0.665539
\(448\) −3.31371 −0.156558
\(449\) −31.6985 −1.49594 −0.747972 0.663730i \(-0.768972\pi\)
−0.747972 + 0.663730i \(0.768972\pi\)
\(450\) −1.41421 −0.0666667
\(451\) −6.97056 −0.328231
\(452\) 0 0
\(453\) −5.82843 −0.273843
\(454\) −40.4853 −1.90007
\(455\) −0.414214 −0.0194186
\(456\) −20.9706 −0.982037
\(457\) −36.0416 −1.68596 −0.842978 0.537948i \(-0.819200\pi\)
−0.842978 + 0.537948i \(0.819200\pi\)
\(458\) −36.0000 −1.68217
\(459\) 7.65685 0.357391
\(460\) 0 0
\(461\) −10.6274 −0.494968 −0.247484 0.968892i \(-0.579604\pi\)
−0.247484 + 0.968892i \(0.579604\pi\)
\(462\) 0.485281 0.0225773
\(463\) −9.17157 −0.426239 −0.213120 0.977026i \(-0.568362\pi\)
−0.213120 + 0.977026i \(0.568362\pi\)
\(464\) −17.6569 −0.819699
\(465\) 1.00000 0.0463739
\(466\) 35.0711 1.62464
\(467\) −31.1421 −1.44109 −0.720543 0.693410i \(-0.756107\pi\)
−0.720543 + 0.693410i \(0.756107\pi\)
\(468\) 0 0
\(469\) 3.48528 0.160935
\(470\) −7.17157 −0.330800
\(471\) −10.0000 −0.460776
\(472\) 17.1716 0.790386
\(473\) −2.00000 −0.0919601
\(474\) 10.0000 0.459315
\(475\) −7.41421 −0.340187
\(476\) 0 0
\(477\) −0.242641 −0.0111098
\(478\) −2.48528 −0.113674
\(479\) 15.5858 0.712133 0.356066 0.934461i \(-0.384118\pi\)
0.356066 + 0.934461i \(0.384118\pi\)
\(480\) 0 0
\(481\) −8.82843 −0.402542
\(482\) 23.7574 1.08212
\(483\) 3.65685 0.166393
\(484\) 0 0
\(485\) 7.58579 0.344453
\(486\) −1.41421 −0.0641500
\(487\) 8.24264 0.373510 0.186755 0.982407i \(-0.440203\pi\)
0.186755 + 0.982407i \(0.440203\pi\)
\(488\) −23.3137 −1.05536
\(489\) 5.17157 0.233867
\(490\) 9.65685 0.436252
\(491\) 32.8995 1.48473 0.742367 0.669994i \(-0.233703\pi\)
0.742367 + 0.669994i \(0.233703\pi\)
\(492\) 0 0
\(493\) 33.7990 1.52223
\(494\) 10.4853 0.471755
\(495\) 0.828427 0.0372350
\(496\) −4.00000 −0.179605
\(497\) 6.34315 0.284529
\(498\) 8.72792 0.391108
\(499\) 20.5147 0.918365 0.459182 0.888342i \(-0.348142\pi\)
0.459182 + 0.888342i \(0.348142\pi\)
\(500\) 0 0
\(501\) −14.0000 −0.625474
\(502\) 26.0416 1.16229
\(503\) 32.4558 1.44713 0.723567 0.690254i \(-0.242501\pi\)
0.723567 + 0.690254i \(0.242501\pi\)
\(504\) −1.17157 −0.0521860
\(505\) 5.07107 0.225660
\(506\) 10.3431 0.459809
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) −17.0711 −0.756662 −0.378331 0.925670i \(-0.623502\pi\)
−0.378331 + 0.925670i \(0.623502\pi\)
\(510\) −10.8284 −0.479491
\(511\) −0.443651 −0.0196260
\(512\) 22.6274 1.00000
\(513\) −7.41421 −0.327346
\(514\) 21.6569 0.955243
\(515\) 13.0711 0.575980
\(516\) 0 0
\(517\) 4.20101 0.184760
\(518\) −5.17157 −0.227226
\(519\) −0.171573 −0.00753121
\(520\) 2.82843 0.124035
\(521\) 17.0711 0.747897 0.373949 0.927449i \(-0.378004\pi\)
0.373949 + 0.927449i \(0.378004\pi\)
\(522\) −6.24264 −0.273233
\(523\) −21.1005 −0.922661 −0.461330 0.887228i \(-0.652628\pi\)
−0.461330 + 0.887228i \(0.652628\pi\)
\(524\) 0 0
\(525\) −0.414214 −0.0180778
\(526\) −4.14214 −0.180606
\(527\) 7.65685 0.333538
\(528\) −3.31371 −0.144211
\(529\) 54.9411 2.38874
\(530\) 0.343146 0.0149053
\(531\) 6.07107 0.263462
\(532\) 0 0
\(533\) −8.41421 −0.364460
\(534\) 12.4853 0.540291
\(535\) 5.82843 0.251985
\(536\) −23.7990 −1.02796
\(537\) 7.72792 0.333484
\(538\) −40.4853 −1.74544
\(539\) −5.65685 −0.243658
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) −36.0416 −1.54812
\(543\) 7.51472 0.322487
\(544\) 0 0
\(545\) −3.75736 −0.160948
\(546\) 0.585786 0.0250693
\(547\) 14.7279 0.629720 0.314860 0.949138i \(-0.398042\pi\)
0.314860 + 0.949138i \(0.398042\pi\)
\(548\) 0 0
\(549\) −8.24264 −0.351787
\(550\) −1.17157 −0.0499560
\(551\) −32.7279 −1.39426
\(552\) −24.9706 −1.06282
\(553\) 2.92893 0.124551
\(554\) 9.07107 0.385393
\(555\) −8.82843 −0.374746
\(556\) 0 0
\(557\) −40.9706 −1.73598 −0.867989 0.496583i \(-0.834588\pi\)
−0.867989 + 0.496583i \(0.834588\pi\)
\(558\) −1.41421 −0.0598684
\(559\) −2.41421 −0.102110
\(560\) 1.65685 0.0700149
\(561\) 6.34315 0.267808
\(562\) 3.89949 0.164490
\(563\) 35.6569 1.50276 0.751379 0.659871i \(-0.229389\pi\)
0.751379 + 0.659871i \(0.229389\pi\)
\(564\) 0 0
\(565\) 5.34315 0.224788
\(566\) 12.8284 0.539219
\(567\) −0.414214 −0.0173953
\(568\) −43.3137 −1.81740
\(569\) −7.51472 −0.315033 −0.157517 0.987516i \(-0.550349\pi\)
−0.157517 + 0.987516i \(0.550349\pi\)
\(570\) 10.4853 0.439180
\(571\) −29.4558 −1.23269 −0.616344 0.787477i \(-0.711387\pi\)
−0.616344 + 0.787477i \(0.711387\pi\)
\(572\) 0 0
\(573\) −6.48528 −0.270927
\(574\) −4.92893 −0.205730
\(575\) −8.82843 −0.368171
\(576\) 8.00000 0.333333
\(577\) −7.65685 −0.318759 −0.159380 0.987217i \(-0.550949\pi\)
−0.159380 + 0.987217i \(0.550949\pi\)
\(578\) −58.8701 −2.44867
\(579\) 7.10051 0.295087
\(580\) 0 0
\(581\) 2.55635 0.106055
\(582\) −10.7279 −0.444687
\(583\) −0.201010 −0.00832499
\(584\) 3.02944 0.125359
\(585\) 1.00000 0.0413449
\(586\) 0.828427 0.0342220
\(587\) −9.14214 −0.377336 −0.188668 0.982041i \(-0.560417\pi\)
−0.188668 + 0.982041i \(0.560417\pi\)
\(588\) 0 0
\(589\) −7.41421 −0.305497
\(590\) −8.58579 −0.353471
\(591\) −5.82843 −0.239749
\(592\) 35.3137 1.45138
\(593\) 24.8701 1.02129 0.510645 0.859791i \(-0.329406\pi\)
0.510645 + 0.859791i \(0.329406\pi\)
\(594\) −1.17157 −0.0480702
\(595\) −3.17157 −0.130022
\(596\) 0 0
\(597\) 12.4853 0.510989
\(598\) 12.4853 0.510561
\(599\) −11.5147 −0.470479 −0.235239 0.971937i \(-0.575587\pi\)
−0.235239 + 0.971937i \(0.575587\pi\)
\(600\) 2.82843 0.115470
\(601\) −32.1838 −1.31280 −0.656402 0.754412i \(-0.727922\pi\)
−0.656402 + 0.754412i \(0.727922\pi\)
\(602\) −1.41421 −0.0576390
\(603\) −8.41421 −0.342653
\(604\) 0 0
\(605\) −10.3137 −0.419312
\(606\) −7.17157 −0.291325
\(607\) −45.5980 −1.85076 −0.925382 0.379035i \(-0.876256\pi\)
−0.925382 + 0.379035i \(0.876256\pi\)
\(608\) 0 0
\(609\) −1.82843 −0.0740916
\(610\) 11.6569 0.471972
\(611\) 5.07107 0.205153
\(612\) 0 0
\(613\) −28.6274 −1.15625 −0.578125 0.815948i \(-0.696215\pi\)
−0.578125 + 0.815948i \(0.696215\pi\)
\(614\) 35.7990 1.44473
\(615\) −8.41421 −0.339294
\(616\) −0.970563 −0.0391051
\(617\) −14.8284 −0.596970 −0.298485 0.954414i \(-0.596481\pi\)
−0.298485 + 0.954414i \(0.596481\pi\)
\(618\) −18.4853 −0.743587
\(619\) 0.798990 0.0321141 0.0160571 0.999871i \(-0.494889\pi\)
0.0160571 + 0.999871i \(0.494889\pi\)
\(620\) 0 0
\(621\) −8.82843 −0.354273
\(622\) 33.9411 1.36092
\(623\) 3.65685 0.146509
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) 9.07107 0.362553
\(627\) −6.14214 −0.245293
\(628\) 0 0
\(629\) −67.5980 −2.69531
\(630\) 0.585786 0.0233383
\(631\) −3.85786 −0.153579 −0.0767896 0.997047i \(-0.524467\pi\)
−0.0767896 + 0.997047i \(0.524467\pi\)
\(632\) −20.0000 −0.795557
\(633\) −17.4853 −0.694978
\(634\) −6.48528 −0.257563
\(635\) 0.757359 0.0300549
\(636\) 0 0
\(637\) −6.82843 −0.270552
\(638\) −5.17157 −0.204745
\(639\) −15.3137 −0.605801
\(640\) −11.3137 −0.447214
\(641\) −4.41421 −0.174351 −0.0871755 0.996193i \(-0.527784\pi\)
−0.0871755 + 0.996193i \(0.527784\pi\)
\(642\) −8.24264 −0.325311
\(643\) 32.8701 1.29627 0.648134 0.761526i \(-0.275550\pi\)
0.648134 + 0.761526i \(0.275550\pi\)
\(644\) 0 0
\(645\) −2.41421 −0.0950596
\(646\) 80.2843 3.15874
\(647\) −22.2426 −0.874448 −0.437224 0.899353i \(-0.644038\pi\)
−0.437224 + 0.899353i \(0.644038\pi\)
\(648\) 2.82843 0.111111
\(649\) 5.02944 0.197423
\(650\) −1.41421 −0.0554700
\(651\) −0.414214 −0.0162343
\(652\) 0 0
\(653\) −50.1127 −1.96106 −0.980531 0.196366i \(-0.937086\pi\)
−0.980531 + 0.196366i \(0.937086\pi\)
\(654\) 5.31371 0.207782
\(655\) −15.4142 −0.602283
\(656\) 33.6569 1.31408
\(657\) 1.07107 0.0417863
\(658\) 2.97056 0.115805
\(659\) 33.6985 1.31271 0.656353 0.754454i \(-0.272098\pi\)
0.656353 + 0.754454i \(0.272098\pi\)
\(660\) 0 0
\(661\) 16.1421 0.627856 0.313928 0.949447i \(-0.398355\pi\)
0.313928 + 0.949447i \(0.398355\pi\)
\(662\) 16.7279 0.650149
\(663\) 7.65685 0.297368
\(664\) −17.4558 −0.677418
\(665\) 3.07107 0.119091
\(666\) 12.4853 0.483795
\(667\) −38.9706 −1.50895
\(668\) 0 0
\(669\) −19.4142 −0.750597
\(670\) 11.8995 0.459718
\(671\) −6.82843 −0.263609
\(672\) 0 0
\(673\) 8.75736 0.337571 0.168786 0.985653i \(-0.446015\pi\)
0.168786 + 0.985653i \(0.446015\pi\)
\(674\) −31.7990 −1.22485
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −23.6985 −0.910807 −0.455403 0.890285i \(-0.650505\pi\)
−0.455403 + 0.890285i \(0.650505\pi\)
\(678\) −7.55635 −0.290200
\(679\) −3.14214 −0.120584
\(680\) 21.6569 0.830502
\(681\) 28.6274 1.09701
\(682\) −1.17157 −0.0448618
\(683\) 43.4558 1.66279 0.831396 0.555681i \(-0.187542\pi\)
0.831396 + 0.555681i \(0.187542\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −8.10051 −0.309279
\(687\) 25.4558 0.971201
\(688\) 9.65685 0.368164
\(689\) −0.242641 −0.00924387
\(690\) 12.4853 0.475307
\(691\) 11.8579 0.451094 0.225547 0.974232i \(-0.427583\pi\)
0.225547 + 0.974232i \(0.427583\pi\)
\(692\) 0 0
\(693\) −0.343146 −0.0130350
\(694\) 36.1421 1.37194
\(695\) 12.5858 0.477406
\(696\) 12.4853 0.473253
\(697\) −64.4264 −2.44032
\(698\) −13.8579 −0.524528
\(699\) −24.7990 −0.937984
\(700\) 0 0
\(701\) 10.9706 0.414352 0.207176 0.978304i \(-0.433573\pi\)
0.207176 + 0.978304i \(0.433573\pi\)
\(702\) −1.41421 −0.0533761
\(703\) 65.4558 2.46871
\(704\) 6.62742 0.249780
\(705\) 5.07107 0.190987
\(706\) −19.0711 −0.717749
\(707\) −2.10051 −0.0789976
\(708\) 0 0
\(709\) 12.2843 0.461345 0.230673 0.973031i \(-0.425907\pi\)
0.230673 + 0.973031i \(0.425907\pi\)
\(710\) 21.6569 0.812767
\(711\) −7.07107 −0.265186
\(712\) −24.9706 −0.935811
\(713\) −8.82843 −0.330627
\(714\) 4.48528 0.167857
\(715\) 0.828427 0.0309814
\(716\) 0 0
\(717\) 1.75736 0.0656298
\(718\) 49.8406 1.86004
\(719\) −2.27208 −0.0847342 −0.0423671 0.999102i \(-0.513490\pi\)
−0.0423671 + 0.999102i \(0.513490\pi\)
\(720\) −4.00000 −0.149071
\(721\) −5.41421 −0.201636
\(722\) −50.8701 −1.89319
\(723\) −16.7990 −0.624761
\(724\) 0 0
\(725\) 4.41421 0.163940
\(726\) 14.5858 0.541329
\(727\) −24.0416 −0.891655 −0.445827 0.895119i \(-0.647090\pi\)
−0.445827 + 0.895119i \(0.647090\pi\)
\(728\) −1.17157 −0.0434214
\(729\) 1.00000 0.0370370
\(730\) −1.51472 −0.0560623
\(731\) −18.4853 −0.683703
\(732\) 0 0
\(733\) −11.2426 −0.415256 −0.207628 0.978208i \(-0.566574\pi\)
−0.207628 + 0.978208i \(0.566574\pi\)
\(734\) −50.4264 −1.86127
\(735\) −6.82843 −0.251870
\(736\) 0 0
\(737\) −6.97056 −0.256764
\(738\) 11.8995 0.438026
\(739\) −23.4558 −0.862837 −0.431419 0.902152i \(-0.641987\pi\)
−0.431419 + 0.902152i \(0.641987\pi\)
\(740\) 0 0
\(741\) −7.41421 −0.272368
\(742\) −0.142136 −0.00521796
\(743\) −25.4853 −0.934964 −0.467482 0.884003i \(-0.654839\pi\)
−0.467482 + 0.884003i \(0.654839\pi\)
\(744\) 2.82843 0.103695
\(745\) 14.0711 0.515524
\(746\) −2.14214 −0.0784292
\(747\) −6.17157 −0.225806
\(748\) 0 0
\(749\) −2.41421 −0.0882134
\(750\) −1.41421 −0.0516398
\(751\) 16.4558 0.600482 0.300241 0.953863i \(-0.402933\pi\)
0.300241 + 0.953863i \(0.402933\pi\)
\(752\) −20.2843 −0.739691
\(753\) −18.4142 −0.671051
\(754\) −6.24264 −0.227344
\(755\) −5.82843 −0.212118
\(756\) 0 0
\(757\) −34.0122 −1.23619 −0.618097 0.786102i \(-0.712096\pi\)
−0.618097 + 0.786102i \(0.712096\pi\)
\(758\) 15.6569 0.568683
\(759\) −7.31371 −0.265471
\(760\) −20.9706 −0.760682
\(761\) 49.0711 1.77882 0.889412 0.457106i \(-0.151114\pi\)
0.889412 + 0.457106i \(0.151114\pi\)
\(762\) −1.07107 −0.0388007
\(763\) 1.55635 0.0563436
\(764\) 0 0
\(765\) 7.65685 0.276834
\(766\) 33.2132 1.20004
\(767\) 6.07107 0.219214
\(768\) 0 0
\(769\) −13.2132 −0.476480 −0.238240 0.971206i \(-0.576571\pi\)
−0.238240 + 0.971206i \(0.576571\pi\)
\(770\) 0.485281 0.0174883
\(771\) −15.3137 −0.551510
\(772\) 0 0
\(773\) −18.6569 −0.671040 −0.335520 0.942033i \(-0.608912\pi\)
−0.335520 + 0.942033i \(0.608912\pi\)
\(774\) 3.41421 0.122721
\(775\) 1.00000 0.0359211
\(776\) 21.4558 0.770220
\(777\) 3.65685 0.131189
\(778\) −45.3553 −1.62607
\(779\) 62.3848 2.23517
\(780\) 0 0
\(781\) −12.6863 −0.453951
\(782\) 95.5980 3.41858
\(783\) 4.41421 0.157751
\(784\) 27.3137 0.975490
\(785\) −10.0000 −0.356915
\(786\) 21.7990 0.777544
\(787\) −45.3553 −1.61674 −0.808372 0.588673i \(-0.799651\pi\)
−0.808372 + 0.588673i \(0.799651\pi\)
\(788\) 0 0
\(789\) 2.92893 0.104273
\(790\) 10.0000 0.355784
\(791\) −2.21320 −0.0786925
\(792\) 2.34315 0.0832601
\(793\) −8.24264 −0.292705
\(794\) 5.17157 0.183532
\(795\) −0.242641 −0.00860558
\(796\) 0 0
\(797\) 54.8701 1.94360 0.971799 0.235812i \(-0.0757751\pi\)
0.971799 + 0.235812i \(0.0757751\pi\)
\(798\) −4.34315 −0.153746
\(799\) 38.8284 1.37365
\(800\) 0 0
\(801\) −8.82843 −0.311937
\(802\) −16.6274 −0.587135
\(803\) 0.887302 0.0313122
\(804\) 0 0
\(805\) 3.65685 0.128887
\(806\) −1.41421 −0.0498135
\(807\) 28.6274 1.00773
\(808\) 14.3431 0.504590
\(809\) −34.5563 −1.21494 −0.607468 0.794344i \(-0.707815\pi\)
−0.607468 + 0.794344i \(0.707815\pi\)
\(810\) −1.41421 −0.0496904
\(811\) −50.5269 −1.77424 −0.887120 0.461539i \(-0.847297\pi\)
−0.887120 + 0.461539i \(0.847297\pi\)
\(812\) 0 0
\(813\) 25.4853 0.893808
\(814\) 10.3431 0.362527
\(815\) 5.17157 0.181152
\(816\) −30.6274 −1.07217
\(817\) 17.8995 0.626224
\(818\) −25.4142 −0.888587
\(819\) −0.414214 −0.0144738
\(820\) 0 0
\(821\) 48.1421 1.68017 0.840086 0.542453i \(-0.182504\pi\)
0.840086 + 0.542453i \(0.182504\pi\)
\(822\) 0 0
\(823\) 13.5858 0.473571 0.236785 0.971562i \(-0.423906\pi\)
0.236785 + 0.971562i \(0.423906\pi\)
\(824\) 36.9706 1.28793
\(825\) 0.828427 0.0288421
\(826\) 3.55635 0.123741
\(827\) −46.1127 −1.60350 −0.801748 0.597662i \(-0.796096\pi\)
−0.801748 + 0.597662i \(0.796096\pi\)
\(828\) 0 0
\(829\) −19.7990 −0.687647 −0.343824 0.939034i \(-0.611722\pi\)
−0.343824 + 0.939034i \(0.611722\pi\)
\(830\) 8.72792 0.302951
\(831\) −6.41421 −0.222507
\(832\) 8.00000 0.277350
\(833\) −52.2843 −1.81154
\(834\) −17.7990 −0.616329
\(835\) −14.0000 −0.484490
\(836\) 0 0
\(837\) 1.00000 0.0345651
\(838\) 26.1421 0.903065
\(839\) 20.8995 0.721531 0.360765 0.932657i \(-0.382516\pi\)
0.360765 + 0.932657i \(0.382516\pi\)
\(840\) −1.17157 −0.0404231
\(841\) −9.51472 −0.328094
\(842\) 0.970563 0.0334478
\(843\) −2.75736 −0.0949685
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) −7.17157 −0.246564
\(847\) 4.27208 0.146790
\(848\) 0.970563 0.0333293
\(849\) −9.07107 −0.311318
\(850\) −10.8284 −0.371412
\(851\) 77.9411 2.67179
\(852\) 0 0
\(853\) 25.2426 0.864292 0.432146 0.901804i \(-0.357757\pi\)
0.432146 + 0.901804i \(0.357757\pi\)
\(854\) −4.82843 −0.165225
\(855\) −7.41421 −0.253561
\(856\) 16.4853 0.563455
\(857\) 56.7696 1.93921 0.969605 0.244674i \(-0.0786808\pi\)
0.969605 + 0.244674i \(0.0786808\pi\)
\(858\) −1.17157 −0.0399968
\(859\) 12.0000 0.409435 0.204717 0.978821i \(-0.434372\pi\)
0.204717 + 0.978821i \(0.434372\pi\)
\(860\) 0 0
\(861\) 3.48528 0.118778
\(862\) 39.8995 1.35898
\(863\) −30.7990 −1.04841 −0.524205 0.851592i \(-0.675637\pi\)
−0.524205 + 0.851592i \(0.675637\pi\)
\(864\) 0 0
\(865\) −0.171573 −0.00583365
\(866\) −5.27208 −0.179153
\(867\) 41.6274 1.41374
\(868\) 0 0
\(869\) −5.85786 −0.198714
\(870\) −6.24264 −0.211645
\(871\) −8.41421 −0.285105
\(872\) −10.6274 −0.359890
\(873\) 7.58579 0.256740
\(874\) −92.5685 −3.13118
\(875\) −0.414214 −0.0140030
\(876\) 0 0
\(877\) 27.1716 0.917519 0.458759 0.888561i \(-0.348294\pi\)
0.458759 + 0.888561i \(0.348294\pi\)
\(878\) −45.4142 −1.53266
\(879\) −0.585786 −0.0197581
\(880\) −3.31371 −0.111705
\(881\) 23.9289 0.806186 0.403093 0.915159i \(-0.367935\pi\)
0.403093 + 0.915159i \(0.367935\pi\)
\(882\) 9.65685 0.325163
\(883\) −56.6985 −1.90806 −0.954028 0.299718i \(-0.903108\pi\)
−0.954028 + 0.299718i \(0.903108\pi\)
\(884\) 0 0
\(885\) 6.07107 0.204077
\(886\) 56.0416 1.88276
\(887\) 10.0294 0.336756 0.168378 0.985723i \(-0.446147\pi\)
0.168378 + 0.985723i \(0.446147\pi\)
\(888\) −24.9706 −0.837957
\(889\) −0.313708 −0.0105214
\(890\) 12.4853 0.418508
\(891\) 0.828427 0.0277534
\(892\) 0 0
\(893\) −37.5980 −1.25817
\(894\) −19.8995 −0.665539
\(895\) 7.72792 0.258316
\(896\) 4.68629 0.156558
\(897\) −8.82843 −0.294773
\(898\) 44.8284 1.49594
\(899\) 4.41421 0.147222
\(900\) 0 0
\(901\) −1.85786 −0.0618944
\(902\) 9.85786 0.328231
\(903\) 1.00000 0.0332779
\(904\) 15.1127 0.502641
\(905\) 7.51472 0.249798
\(906\) 8.24264 0.273843
\(907\) 25.1716 0.835808 0.417904 0.908491i \(-0.362765\pi\)
0.417904 + 0.908491i \(0.362765\pi\)
\(908\) 0 0
\(909\) 5.07107 0.168197
\(910\) 0.585786 0.0194186
\(911\) 18.2132 0.603430 0.301715 0.953398i \(-0.402441\pi\)
0.301715 + 0.953398i \(0.402441\pi\)
\(912\) 29.6569 0.982037
\(913\) −5.11270 −0.169206
\(914\) 50.9706 1.68596
\(915\) −8.24264 −0.272493
\(916\) 0 0
\(917\) 6.38478 0.210844
\(918\) −10.8284 −0.357391
\(919\) −21.3431 −0.704045 −0.352023 0.935991i \(-0.614506\pi\)
−0.352023 + 0.935991i \(0.614506\pi\)
\(920\) −24.9706 −0.823255
\(921\) −25.3137 −0.834115
\(922\) 15.0294 0.494968
\(923\) −15.3137 −0.504057
\(924\) 0 0
\(925\) −8.82843 −0.290277
\(926\) 12.9706 0.426239
\(927\) 13.0711 0.429310
\(928\) 0 0
\(929\) −2.78680 −0.0914318 −0.0457159 0.998954i \(-0.514557\pi\)
−0.0457159 + 0.998954i \(0.514557\pi\)
\(930\) −1.41421 −0.0463739
\(931\) 50.6274 1.65925
\(932\) 0 0
\(933\) −24.0000 −0.785725
\(934\) 44.0416 1.44109
\(935\) 6.34315 0.207443
\(936\) 2.82843 0.0924500
\(937\) 23.9411 0.782122 0.391061 0.920365i \(-0.372108\pi\)
0.391061 + 0.920365i \(0.372108\pi\)
\(938\) −4.92893 −0.160935
\(939\) −6.41421 −0.209320
\(940\) 0 0
\(941\) −35.8995 −1.17029 −0.585145 0.810929i \(-0.698962\pi\)
−0.585145 + 0.810929i \(0.698962\pi\)
\(942\) 14.1421 0.460776
\(943\) 74.2843 2.41903
\(944\) −24.2843 −0.790386
\(945\) −0.414214 −0.0134744
\(946\) 2.82843 0.0919601
\(947\) −47.6569 −1.54864 −0.774320 0.632794i \(-0.781908\pi\)
−0.774320 + 0.632794i \(0.781908\pi\)
\(948\) 0 0
\(949\) 1.07107 0.0347683
\(950\) 10.4853 0.340187
\(951\) 4.58579 0.148704
\(952\) −8.97056 −0.290738
\(953\) −30.2426 −0.979655 −0.489828 0.871819i \(-0.662940\pi\)
−0.489828 + 0.871819i \(0.662940\pi\)
\(954\) 0.343146 0.0111098
\(955\) −6.48528 −0.209859
\(956\) 0 0
\(957\) 3.65685 0.118209
\(958\) −22.0416 −0.712133
\(959\) 0 0
\(960\) 8.00000 0.258199
\(961\) 1.00000 0.0322581
\(962\) 12.4853 0.402542
\(963\) 5.82843 0.187818
\(964\) 0 0
\(965\) 7.10051 0.228573
\(966\) −5.17157 −0.166393
\(967\) 33.1716 1.06673 0.533363 0.845887i \(-0.320928\pi\)
0.533363 + 0.845887i \(0.320928\pi\)
\(968\) −29.1716 −0.937610
\(969\) −56.7696 −1.82370
\(970\) −10.7279 −0.344453
\(971\) −38.9706 −1.25062 −0.625312 0.780374i \(-0.715028\pi\)
−0.625312 + 0.780374i \(0.715028\pi\)
\(972\) 0 0
\(973\) −5.21320 −0.167128
\(974\) −11.6569 −0.373510
\(975\) 1.00000 0.0320256
\(976\) 32.9706 1.05536
\(977\) −37.4558 −1.19832 −0.599159 0.800630i \(-0.704498\pi\)
−0.599159 + 0.800630i \(0.704498\pi\)
\(978\) −7.31371 −0.233867
\(979\) −7.31371 −0.233747
\(980\) 0 0
\(981\) −3.75736 −0.119963
\(982\) −46.5269 −1.48473
\(983\) −14.8579 −0.473892 −0.236946 0.971523i \(-0.576146\pi\)
−0.236946 + 0.971523i \(0.576146\pi\)
\(984\) −23.7990 −0.758684
\(985\) −5.82843 −0.185709
\(986\) −47.7990 −1.52223
\(987\) −2.10051 −0.0668598
\(988\) 0 0
\(989\) 21.3137 0.677737
\(990\) −1.17157 −0.0372350
\(991\) 45.4142 1.44263 0.721315 0.692607i \(-0.243538\pi\)
0.721315 + 0.692607i \(0.243538\pi\)
\(992\) 0 0
\(993\) −11.8284 −0.375364
\(994\) −8.97056 −0.284529
\(995\) 12.4853 0.395810
\(996\) 0 0
\(997\) 16.2426 0.514410 0.257205 0.966357i \(-0.417199\pi\)
0.257205 + 0.966357i \(0.417199\pi\)
\(998\) −29.0122 −0.918365
\(999\) −8.82843 −0.279319
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 6045.2.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6045.2.a.p.1.1 2 1.1 even 1 trivial