Properties

Label 4029.2.a.c.1.2
Level $4029$
Weight $2$
Character 4029.1
Self dual yes
Analytic conductor $32.172$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4029,2,Mod(1,4029)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4029, 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("4029.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4029 = 3 \cdot 17 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4029.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: \(32.1717269744\)
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 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4029.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} +1.00000 q^{3} +0.414214 q^{5} +1.41421 q^{6} +1.00000 q^{7} -2.82843 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.41421 q^{2} +1.00000 q^{3} +0.414214 q^{5} +1.41421 q^{6} +1.00000 q^{7} -2.82843 q^{8} +1.00000 q^{9} +0.585786 q^{10} -5.65685 q^{11} +1.41421 q^{14} +0.414214 q^{15} -4.00000 q^{16} +1.00000 q^{17} +1.41421 q^{18} -2.17157 q^{19} +1.00000 q^{21} -8.00000 q^{22} +1.24264 q^{23} -2.82843 q^{24} -4.82843 q^{25} +1.00000 q^{27} +4.82843 q^{29} +0.585786 q^{30} -1.65685 q^{31} -5.65685 q^{33} +1.41421 q^{34} +0.414214 q^{35} +0.656854 q^{37} -3.07107 q^{38} -1.17157 q^{40} +7.65685 q^{41} +1.41421 q^{42} -6.48528 q^{43} +0.414214 q^{45} +1.75736 q^{46} -10.0711 q^{47} -4.00000 q^{48} -6.00000 q^{49} -6.82843 q^{50} +1.00000 q^{51} -12.4142 q^{53} +1.41421 q^{54} -2.34315 q^{55} -2.82843 q^{56} -2.17157 q^{57} +6.82843 q^{58} -3.58579 q^{59} -0.656854 q^{61} -2.34315 q^{62} +1.00000 q^{63} +8.00000 q^{64} -8.00000 q^{66} -10.0000 q^{67} +1.24264 q^{69} +0.585786 q^{70} -13.0711 q^{71} -2.82843 q^{72} -10.7279 q^{73} +0.928932 q^{74} -4.82843 q^{75} -5.65685 q^{77} -1.00000 q^{79} -1.65685 q^{80} +1.00000 q^{81} +10.8284 q^{82} +5.41421 q^{83} +0.414214 q^{85} -9.17157 q^{86} +4.82843 q^{87} +16.0000 q^{88} -6.34315 q^{89} +0.585786 q^{90} -1.65685 q^{93} -14.2426 q^{94} -0.899495 q^{95} +17.4142 q^{97} -8.48528 q^{98} -5.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{10} - 2 q^{15} - 8 q^{16} + 2 q^{17} - 10 q^{19} + 2 q^{21} - 16 q^{22} - 6 q^{23} - 4 q^{25} + 2 q^{27} + 4 q^{29} + 4 q^{30} + 8 q^{31} - 2 q^{35} - 10 q^{37} + 8 q^{38} - 8 q^{40} + 4 q^{41} + 4 q^{43} - 2 q^{45} + 12 q^{46} - 6 q^{47} - 8 q^{48} - 12 q^{49} - 8 q^{50} + 2 q^{51} - 22 q^{53} - 16 q^{55} - 10 q^{57} + 8 q^{58} - 10 q^{59} + 10 q^{61} - 16 q^{62} + 2 q^{63} + 16 q^{64} - 16 q^{66} - 20 q^{67} - 6 q^{69} + 4 q^{70} - 12 q^{71} + 4 q^{73} + 16 q^{74} - 4 q^{75} - 2 q^{79} + 8 q^{80} + 2 q^{81} + 16 q^{82} + 8 q^{83} - 2 q^{85} - 24 q^{86} + 4 q^{87} + 32 q^{88} - 24 q^{89} + 4 q^{90} + 8 q^{93} - 20 q^{94} + 18 q^{95} + 32 q^{97}+O(q^{100}) \) 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 \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0.414214 0.185242 0.0926210 0.995701i \(-0.470476\pi\)
0.0926210 + 0.995701i \(0.470476\pi\)
\(6\) 1.41421 0.577350
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) −2.82843 −1.00000
\(9\) 1.00000 0.333333
\(10\) 0.585786 0.185242
\(11\) −5.65685 −1.70561 −0.852803 0.522233i \(-0.825099\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 1.41421 0.377964
\(15\) 0.414214 0.106949
\(16\) −4.00000 −1.00000
\(17\) 1.00000 0.242536
\(18\) 1.41421 0.333333
\(19\) −2.17157 −0.498193 −0.249096 0.968479i \(-0.580134\pi\)
−0.249096 + 0.968479i \(0.580134\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) −8.00000 −1.70561
\(23\) 1.24264 0.259108 0.129554 0.991572i \(-0.458645\pi\)
0.129554 + 0.991572i \(0.458645\pi\)
\(24\) −2.82843 −0.577350
\(25\) −4.82843 −0.965685
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 4.82843 0.896616 0.448308 0.893879i \(-0.352027\pi\)
0.448308 + 0.893879i \(0.352027\pi\)
\(30\) 0.585786 0.106949
\(31\) −1.65685 −0.297580 −0.148790 0.988869i \(-0.547538\pi\)
−0.148790 + 0.988869i \(0.547538\pi\)
\(32\) 0 0
\(33\) −5.65685 −0.984732
\(34\) 1.41421 0.242536
\(35\) 0.414214 0.0700149
\(36\) 0 0
\(37\) 0.656854 0.107986 0.0539931 0.998541i \(-0.482805\pi\)
0.0539931 + 0.998541i \(0.482805\pi\)
\(38\) −3.07107 −0.498193
\(39\) 0 0
\(40\) −1.17157 −0.185242
\(41\) 7.65685 1.19580 0.597900 0.801571i \(-0.296002\pi\)
0.597900 + 0.801571i \(0.296002\pi\)
\(42\) 1.41421 0.218218
\(43\) −6.48528 −0.988996 −0.494498 0.869179i \(-0.664648\pi\)
−0.494498 + 0.869179i \(0.664648\pi\)
\(44\) 0 0
\(45\) 0.414214 0.0617473
\(46\) 1.75736 0.259108
\(47\) −10.0711 −1.46902 −0.734508 0.678600i \(-0.762587\pi\)
−0.734508 + 0.678600i \(0.762587\pi\)
\(48\) −4.00000 −0.577350
\(49\) −6.00000 −0.857143
\(50\) −6.82843 −0.965685
\(51\) 1.00000 0.140028
\(52\) 0 0
\(53\) −12.4142 −1.70522 −0.852612 0.522545i \(-0.824983\pi\)
−0.852612 + 0.522545i \(0.824983\pi\)
\(54\) 1.41421 0.192450
\(55\) −2.34315 −0.315950
\(56\) −2.82843 −0.377964
\(57\) −2.17157 −0.287632
\(58\) 6.82843 0.896616
\(59\) −3.58579 −0.466830 −0.233415 0.972377i \(-0.574990\pi\)
−0.233415 + 0.972377i \(0.574990\pi\)
\(60\) 0 0
\(61\) −0.656854 −0.0841016 −0.0420508 0.999115i \(-0.513389\pi\)
−0.0420508 + 0.999115i \(0.513389\pi\)
\(62\) −2.34315 −0.297580
\(63\) 1.00000 0.125988
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) −8.00000 −0.984732
\(67\) −10.0000 −1.22169 −0.610847 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) 0 0
\(69\) 1.24264 0.149596
\(70\) 0.585786 0.0700149
\(71\) −13.0711 −1.55125 −0.775625 0.631194i \(-0.782565\pi\)
−0.775625 + 0.631194i \(0.782565\pi\)
\(72\) −2.82843 −0.333333
\(73\) −10.7279 −1.25561 −0.627804 0.778371i \(-0.716046\pi\)
−0.627804 + 0.778371i \(0.716046\pi\)
\(74\) 0.928932 0.107986
\(75\) −4.82843 −0.557539
\(76\) 0 0
\(77\) −5.65685 −0.644658
\(78\) 0 0
\(79\) −1.00000 −0.112509
\(80\) −1.65685 −0.185242
\(81\) 1.00000 0.111111
\(82\) 10.8284 1.19580
\(83\) 5.41421 0.594287 0.297144 0.954833i \(-0.403966\pi\)
0.297144 + 0.954833i \(0.403966\pi\)
\(84\) 0 0
\(85\) 0.414214 0.0449278
\(86\) −9.17157 −0.988996
\(87\) 4.82843 0.517662
\(88\) 16.0000 1.70561
\(89\) −6.34315 −0.672372 −0.336186 0.941796i \(-0.609137\pi\)
−0.336186 + 0.941796i \(0.609137\pi\)
\(90\) 0.585786 0.0617473
\(91\) 0 0
\(92\) 0 0
\(93\) −1.65685 −0.171808
\(94\) −14.2426 −1.46902
\(95\) −0.899495 −0.0922862
\(96\) 0 0
\(97\) 17.4142 1.76815 0.884073 0.467349i \(-0.154791\pi\)
0.884073 + 0.467349i \(0.154791\pi\)
\(98\) −8.48528 −0.857143
\(99\) −5.65685 −0.568535
\(100\) 0 0
\(101\) 4.24264 0.422159 0.211079 0.977469i \(-0.432302\pi\)
0.211079 + 0.977469i \(0.432302\pi\)
\(102\) 1.41421 0.140028
\(103\) −2.24264 −0.220974 −0.110487 0.993878i \(-0.535241\pi\)
−0.110487 + 0.993878i \(0.535241\pi\)
\(104\) 0 0
\(105\) 0.414214 0.0404231
\(106\) −17.5563 −1.70522
\(107\) −3.41421 −0.330064 −0.165032 0.986288i \(-0.552773\pi\)
−0.165032 + 0.986288i \(0.552773\pi\)
\(108\) 0 0
\(109\) 8.48528 0.812743 0.406371 0.913708i \(-0.366794\pi\)
0.406371 + 0.913708i \(0.366794\pi\)
\(110\) −3.31371 −0.315950
\(111\) 0.656854 0.0623458
\(112\) −4.00000 −0.377964
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) −3.07107 −0.287632
\(115\) 0.514719 0.0479978
\(116\) 0 0
\(117\) 0 0
\(118\) −5.07107 −0.466830
\(119\) 1.00000 0.0916698
\(120\) −1.17157 −0.106949
\(121\) 21.0000 1.90909
\(122\) −0.928932 −0.0841016
\(123\) 7.65685 0.690395
\(124\) 0 0
\(125\) −4.07107 −0.364127
\(126\) 1.41421 0.125988
\(127\) 20.2426 1.79624 0.898122 0.439746i \(-0.144932\pi\)
0.898122 + 0.439746i \(0.144932\pi\)
\(128\) 11.3137 1.00000
\(129\) −6.48528 −0.570997
\(130\) 0 0
\(131\) 0.414214 0.0361900 0.0180950 0.999836i \(-0.494240\pi\)
0.0180950 + 0.999836i \(0.494240\pi\)
\(132\) 0 0
\(133\) −2.17157 −0.188299
\(134\) −14.1421 −1.22169
\(135\) 0.414214 0.0356498
\(136\) −2.82843 −0.242536
\(137\) 18.5563 1.58538 0.792688 0.609628i \(-0.208681\pi\)
0.792688 + 0.609628i \(0.208681\pi\)
\(138\) 1.75736 0.149596
\(139\) 2.48528 0.210799 0.105399 0.994430i \(-0.466388\pi\)
0.105399 + 0.994430i \(0.466388\pi\)
\(140\) 0 0
\(141\) −10.0711 −0.848137
\(142\) −18.4853 −1.55125
\(143\) 0 0
\(144\) −4.00000 −0.333333
\(145\) 2.00000 0.166091
\(146\) −15.1716 −1.25561
\(147\) −6.00000 −0.494872
\(148\) 0 0
\(149\) −16.0711 −1.31659 −0.658297 0.752759i \(-0.728723\pi\)
−0.658297 + 0.752759i \(0.728723\pi\)
\(150\) −6.82843 −0.557539
\(151\) 0.514719 0.0418872 0.0209436 0.999781i \(-0.493333\pi\)
0.0209436 + 0.999781i \(0.493333\pi\)
\(152\) 6.14214 0.498193
\(153\) 1.00000 0.0808452
\(154\) −8.00000 −0.644658
\(155\) −0.686292 −0.0551243
\(156\) 0 0
\(157\) 10.7279 0.856181 0.428091 0.903736i \(-0.359186\pi\)
0.428091 + 0.903736i \(0.359186\pi\)
\(158\) −1.41421 −0.112509
\(159\) −12.4142 −0.984511
\(160\) 0 0
\(161\) 1.24264 0.0979338
\(162\) 1.41421 0.111111
\(163\) −3.75736 −0.294299 −0.147150 0.989114i \(-0.547010\pi\)
−0.147150 + 0.989114i \(0.547010\pi\)
\(164\) 0 0
\(165\) −2.34315 −0.182414
\(166\) 7.65685 0.594287
\(167\) 10.0711 0.779323 0.389661 0.920958i \(-0.372592\pi\)
0.389661 + 0.920958i \(0.372592\pi\)
\(168\) −2.82843 −0.218218
\(169\) −13.0000 −1.00000
\(170\) 0.585786 0.0449278
\(171\) −2.17157 −0.166064
\(172\) 0 0
\(173\) 5.07107 0.385546 0.192773 0.981243i \(-0.438252\pi\)
0.192773 + 0.981243i \(0.438252\pi\)
\(174\) 6.82843 0.517662
\(175\) −4.82843 −0.364995
\(176\) 22.6274 1.70561
\(177\) −3.58579 −0.269524
\(178\) −8.97056 −0.672372
\(179\) −10.2426 −0.765571 −0.382785 0.923837i \(-0.625035\pi\)
−0.382785 + 0.923837i \(0.625035\pi\)
\(180\) 0 0
\(181\) 0.585786 0.0435412 0.0217706 0.999763i \(-0.493070\pi\)
0.0217706 + 0.999763i \(0.493070\pi\)
\(182\) 0 0
\(183\) −0.656854 −0.0485561
\(184\) −3.51472 −0.259108
\(185\) 0.272078 0.0200036
\(186\) −2.34315 −0.171808
\(187\) −5.65685 −0.413670
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) −1.27208 −0.0922862
\(191\) −19.5858 −1.41718 −0.708589 0.705622i \(-0.750668\pi\)
−0.708589 + 0.705622i \(0.750668\pi\)
\(192\) 8.00000 0.577350
\(193\) 15.4853 1.11465 0.557327 0.830293i \(-0.311827\pi\)
0.557327 + 0.830293i \(0.311827\pi\)
\(194\) 24.6274 1.76815
\(195\) 0 0
\(196\) 0 0
\(197\) 3.51472 0.250413 0.125207 0.992131i \(-0.460041\pi\)
0.125207 + 0.992131i \(0.460041\pi\)
\(198\) −8.00000 −0.568535
\(199\) −1.82843 −0.129614 −0.0648069 0.997898i \(-0.520643\pi\)
−0.0648069 + 0.997898i \(0.520643\pi\)
\(200\) 13.6569 0.965685
\(201\) −10.0000 −0.705346
\(202\) 6.00000 0.422159
\(203\) 4.82843 0.338889
\(204\) 0 0
\(205\) 3.17157 0.221512
\(206\) −3.17157 −0.220974
\(207\) 1.24264 0.0863695
\(208\) 0 0
\(209\) 12.2843 0.849721
\(210\) 0.585786 0.0404231
\(211\) 13.9706 0.961773 0.480887 0.876783i \(-0.340315\pi\)
0.480887 + 0.876783i \(0.340315\pi\)
\(212\) 0 0
\(213\) −13.0711 −0.895615
\(214\) −4.82843 −0.330064
\(215\) −2.68629 −0.183204
\(216\) −2.82843 −0.192450
\(217\) −1.65685 −0.112475
\(218\) 12.0000 0.812743
\(219\) −10.7279 −0.724926
\(220\) 0 0
\(221\) 0 0
\(222\) 0.928932 0.0623458
\(223\) −4.17157 −0.279349 −0.139675 0.990197i \(-0.544606\pi\)
−0.139675 + 0.990197i \(0.544606\pi\)
\(224\) 0 0
\(225\) −4.82843 −0.321895
\(226\) −16.9706 −1.12887
\(227\) −18.8284 −1.24969 −0.624843 0.780750i \(-0.714837\pi\)
−0.624843 + 0.780750i \(0.714837\pi\)
\(228\) 0 0
\(229\) −2.58579 −0.170874 −0.0854368 0.996344i \(-0.527229\pi\)
−0.0854368 + 0.996344i \(0.527229\pi\)
\(230\) 0.727922 0.0479978
\(231\) −5.65685 −0.372194
\(232\) −13.6569 −0.896616
\(233\) 24.3848 1.59750 0.798750 0.601663i \(-0.205495\pi\)
0.798750 + 0.601663i \(0.205495\pi\)
\(234\) 0 0
\(235\) −4.17157 −0.272123
\(236\) 0 0
\(237\) −1.00000 −0.0649570
\(238\) 1.41421 0.0916698
\(239\) −6.72792 −0.435193 −0.217597 0.976039i \(-0.569822\pi\)
−0.217597 + 0.976039i \(0.569822\pi\)
\(240\) −1.65685 −0.106949
\(241\) −2.58579 −0.166565 −0.0832826 0.996526i \(-0.526540\pi\)
−0.0832826 + 0.996526i \(0.526540\pi\)
\(242\) 29.6985 1.90909
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −2.48528 −0.158779
\(246\) 10.8284 0.690395
\(247\) 0 0
\(248\) 4.68629 0.297580
\(249\) 5.41421 0.343112
\(250\) −5.75736 −0.364127
\(251\) 12.5563 0.792550 0.396275 0.918132i \(-0.370303\pi\)
0.396275 + 0.918132i \(0.370303\pi\)
\(252\) 0 0
\(253\) −7.02944 −0.441937
\(254\) 28.6274 1.79624
\(255\) 0.414214 0.0259391
\(256\) 0 0
\(257\) 20.6274 1.28670 0.643351 0.765571i \(-0.277543\pi\)
0.643351 + 0.765571i \(0.277543\pi\)
\(258\) −9.17157 −0.570997
\(259\) 0.656854 0.0408149
\(260\) 0 0
\(261\) 4.82843 0.298872
\(262\) 0.585786 0.0361900
\(263\) −24.3848 −1.50363 −0.751815 0.659374i \(-0.770821\pi\)
−0.751815 + 0.659374i \(0.770821\pi\)
\(264\) 16.0000 0.984732
\(265\) −5.14214 −0.315879
\(266\) −3.07107 −0.188299
\(267\) −6.34315 −0.388194
\(268\) 0 0
\(269\) −14.4853 −0.883183 −0.441592 0.897216i \(-0.645586\pi\)
−0.441592 + 0.897216i \(0.645586\pi\)
\(270\) 0.585786 0.0356498
\(271\) 22.7279 1.38062 0.690311 0.723512i \(-0.257474\pi\)
0.690311 + 0.723512i \(0.257474\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 26.2426 1.58538
\(275\) 27.3137 1.64708
\(276\) 0 0
\(277\) −16.9706 −1.01966 −0.509831 0.860274i \(-0.670292\pi\)
−0.509831 + 0.860274i \(0.670292\pi\)
\(278\) 3.51472 0.210799
\(279\) −1.65685 −0.0991933
\(280\) −1.17157 −0.0700149
\(281\) −6.58579 −0.392875 −0.196438 0.980516i \(-0.562937\pi\)
−0.196438 + 0.980516i \(0.562937\pi\)
\(282\) −14.2426 −0.848137
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 0 0
\(285\) −0.899495 −0.0532815
\(286\) 0 0
\(287\) 7.65685 0.451970
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 2.82843 0.166091
\(291\) 17.4142 1.02084
\(292\) 0 0
\(293\) 23.8701 1.39450 0.697252 0.716826i \(-0.254406\pi\)
0.697252 + 0.716826i \(0.254406\pi\)
\(294\) −8.48528 −0.494872
\(295\) −1.48528 −0.0864764
\(296\) −1.85786 −0.107986
\(297\) −5.65685 −0.328244
\(298\) −22.7279 −1.31659
\(299\) 0 0
\(300\) 0 0
\(301\) −6.48528 −0.373805
\(302\) 0.727922 0.0418872
\(303\) 4.24264 0.243733
\(304\) 8.68629 0.498193
\(305\) −0.272078 −0.0155791
\(306\) 1.41421 0.0808452
\(307\) 29.5563 1.68687 0.843435 0.537231i \(-0.180530\pi\)
0.843435 + 0.537231i \(0.180530\pi\)
\(308\) 0 0
\(309\) −2.24264 −0.127579
\(310\) −0.970563 −0.0551243
\(311\) −13.7990 −0.782469 −0.391234 0.920291i \(-0.627952\pi\)
−0.391234 + 0.920291i \(0.627952\pi\)
\(312\) 0 0
\(313\) −4.68629 −0.264885 −0.132442 0.991191i \(-0.542282\pi\)
−0.132442 + 0.991191i \(0.542282\pi\)
\(314\) 15.1716 0.856181
\(315\) 0.414214 0.0233383
\(316\) 0 0
\(317\) −27.7279 −1.55736 −0.778678 0.627424i \(-0.784109\pi\)
−0.778678 + 0.627424i \(0.784109\pi\)
\(318\) −17.5563 −0.984511
\(319\) −27.3137 −1.52927
\(320\) 3.31371 0.185242
\(321\) −3.41421 −0.190563
\(322\) 1.75736 0.0979338
\(323\) −2.17157 −0.120830
\(324\) 0 0
\(325\) 0 0
\(326\) −5.31371 −0.294299
\(327\) 8.48528 0.469237
\(328\) −21.6569 −1.19580
\(329\) −10.0711 −0.555236
\(330\) −3.31371 −0.182414
\(331\) −13.7574 −0.756173 −0.378086 0.925770i \(-0.623418\pi\)
−0.378086 + 0.925770i \(0.623418\pi\)
\(332\) 0 0
\(333\) 0.656854 0.0359954
\(334\) 14.2426 0.779323
\(335\) −4.14214 −0.226309
\(336\) −4.00000 −0.218218
\(337\) −12.7279 −0.693334 −0.346667 0.937988i \(-0.612687\pi\)
−0.346667 + 0.937988i \(0.612687\pi\)
\(338\) −18.3848 −1.00000
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) 9.37258 0.507554
\(342\) −3.07107 −0.166064
\(343\) −13.0000 −0.701934
\(344\) 18.3431 0.988996
\(345\) 0.514719 0.0277115
\(346\) 7.17157 0.385546
\(347\) −5.38478 −0.289070 −0.144535 0.989500i \(-0.546169\pi\)
−0.144535 + 0.989500i \(0.546169\pi\)
\(348\) 0 0
\(349\) −6.24264 −0.334161 −0.167080 0.985943i \(-0.553434\pi\)
−0.167080 + 0.985943i \(0.553434\pi\)
\(350\) −6.82843 −0.364995
\(351\) 0 0
\(352\) 0 0
\(353\) 9.72792 0.517765 0.258883 0.965909i \(-0.416646\pi\)
0.258883 + 0.965909i \(0.416646\pi\)
\(354\) −5.07107 −0.269524
\(355\) −5.41421 −0.287357
\(356\) 0 0
\(357\) 1.00000 0.0529256
\(358\) −14.4853 −0.765571
\(359\) −15.6569 −0.826337 −0.413169 0.910655i \(-0.635578\pi\)
−0.413169 + 0.910655i \(0.635578\pi\)
\(360\) −1.17157 −0.0617473
\(361\) −14.2843 −0.751804
\(362\) 0.828427 0.0435412
\(363\) 21.0000 1.10221
\(364\) 0 0
\(365\) −4.44365 −0.232591
\(366\) −0.928932 −0.0485561
\(367\) −23.5563 −1.22963 −0.614816 0.788671i \(-0.710770\pi\)
−0.614816 + 0.788671i \(0.710770\pi\)
\(368\) −4.97056 −0.259108
\(369\) 7.65685 0.398600
\(370\) 0.384776 0.0200036
\(371\) −12.4142 −0.644514
\(372\) 0 0
\(373\) 14.2426 0.737456 0.368728 0.929537i \(-0.379793\pi\)
0.368728 + 0.929537i \(0.379793\pi\)
\(374\) −8.00000 −0.413670
\(375\) −4.07107 −0.210229
\(376\) 28.4853 1.46902
\(377\) 0 0
\(378\) 1.41421 0.0727393
\(379\) 38.4558 1.97534 0.987672 0.156538i \(-0.0500332\pi\)
0.987672 + 0.156538i \(0.0500332\pi\)
\(380\) 0 0
\(381\) 20.2426 1.03706
\(382\) −27.6985 −1.41718
\(383\) 13.7574 0.702968 0.351484 0.936194i \(-0.385677\pi\)
0.351484 + 0.936194i \(0.385677\pi\)
\(384\) 11.3137 0.577350
\(385\) −2.34315 −0.119418
\(386\) 21.8995 1.11465
\(387\) −6.48528 −0.329665
\(388\) 0 0
\(389\) 17.4142 0.882936 0.441468 0.897277i \(-0.354458\pi\)
0.441468 + 0.897277i \(0.354458\pi\)
\(390\) 0 0
\(391\) 1.24264 0.0628430
\(392\) 16.9706 0.857143
\(393\) 0.414214 0.0208943
\(394\) 4.97056 0.250413
\(395\) −0.414214 −0.0208413
\(396\) 0 0
\(397\) −7.85786 −0.394375 −0.197187 0.980366i \(-0.563181\pi\)
−0.197187 + 0.980366i \(0.563181\pi\)
\(398\) −2.58579 −0.129614
\(399\) −2.17157 −0.108715
\(400\) 19.3137 0.965685
\(401\) 4.24264 0.211867 0.105934 0.994373i \(-0.466217\pi\)
0.105934 + 0.994373i \(0.466217\pi\)
\(402\) −14.1421 −0.705346
\(403\) 0 0
\(404\) 0 0
\(405\) 0.414214 0.0205824
\(406\) 6.82843 0.338889
\(407\) −3.71573 −0.184182
\(408\) −2.82843 −0.140028
\(409\) 18.2843 0.904099 0.452050 0.891993i \(-0.350693\pi\)
0.452050 + 0.891993i \(0.350693\pi\)
\(410\) 4.48528 0.221512
\(411\) 18.5563 0.915317
\(412\) 0 0
\(413\) −3.58579 −0.176445
\(414\) 1.75736 0.0863695
\(415\) 2.24264 0.110087
\(416\) 0 0
\(417\) 2.48528 0.121705
\(418\) 17.3726 0.849721
\(419\) 15.0711 0.736270 0.368135 0.929772i \(-0.379996\pi\)
0.368135 + 0.929772i \(0.379996\pi\)
\(420\) 0 0
\(421\) 11.9706 0.583410 0.291705 0.956508i \(-0.405778\pi\)
0.291705 + 0.956508i \(0.405778\pi\)
\(422\) 19.7574 0.961773
\(423\) −10.0711 −0.489672
\(424\) 35.1127 1.70522
\(425\) −4.82843 −0.234213
\(426\) −18.4853 −0.895615
\(427\) −0.656854 −0.0317874
\(428\) 0 0
\(429\) 0 0
\(430\) −3.79899 −0.183204
\(431\) −39.7279 −1.91363 −0.956813 0.290703i \(-0.906111\pi\)
−0.956813 + 0.290703i \(0.906111\pi\)
\(432\) −4.00000 −0.192450
\(433\) −25.6569 −1.23299 −0.616495 0.787359i \(-0.711448\pi\)
−0.616495 + 0.787359i \(0.711448\pi\)
\(434\) −2.34315 −0.112475
\(435\) 2.00000 0.0958927
\(436\) 0 0
\(437\) −2.69848 −0.129086
\(438\) −15.1716 −0.724926
\(439\) 20.9289 0.998883 0.499442 0.866347i \(-0.333538\pi\)
0.499442 + 0.866347i \(0.333538\pi\)
\(440\) 6.62742 0.315950
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) −32.4142 −1.54005 −0.770023 0.638016i \(-0.779755\pi\)
−0.770023 + 0.638016i \(0.779755\pi\)
\(444\) 0 0
\(445\) −2.62742 −0.124552
\(446\) −5.89949 −0.279349
\(447\) −16.0711 −0.760135
\(448\) 8.00000 0.377964
\(449\) −14.4853 −0.683603 −0.341801 0.939772i \(-0.611037\pi\)
−0.341801 + 0.939772i \(0.611037\pi\)
\(450\) −6.82843 −0.321895
\(451\) −43.3137 −2.03956
\(452\) 0 0
\(453\) 0.514719 0.0241836
\(454\) −26.6274 −1.24969
\(455\) 0 0
\(456\) 6.14214 0.287632
\(457\) 16.4558 0.769772 0.384886 0.922964i \(-0.374241\pi\)
0.384886 + 0.922964i \(0.374241\pi\)
\(458\) −3.65685 −0.170874
\(459\) 1.00000 0.0466760
\(460\) 0 0
\(461\) 31.7990 1.48103 0.740513 0.672042i \(-0.234582\pi\)
0.740513 + 0.672042i \(0.234582\pi\)
\(462\) −8.00000 −0.372194
\(463\) 10.2426 0.476016 0.238008 0.971263i \(-0.423506\pi\)
0.238008 + 0.971263i \(0.423506\pi\)
\(464\) −19.3137 −0.896616
\(465\) −0.686292 −0.0318260
\(466\) 34.4853 1.59750
\(467\) −13.4558 −0.622662 −0.311331 0.950302i \(-0.600775\pi\)
−0.311331 + 0.950302i \(0.600775\pi\)
\(468\) 0 0
\(469\) −10.0000 −0.461757
\(470\) −5.89949 −0.272123
\(471\) 10.7279 0.494317
\(472\) 10.1421 0.466830
\(473\) 36.6863 1.68684
\(474\) −1.41421 −0.0649570
\(475\) 10.4853 0.481098
\(476\) 0 0
\(477\) −12.4142 −0.568408
\(478\) −9.51472 −0.435193
\(479\) 34.8995 1.59460 0.797299 0.603584i \(-0.206261\pi\)
0.797299 + 0.603584i \(0.206261\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −3.65685 −0.166565
\(483\) 1.24264 0.0565421
\(484\) 0 0
\(485\) 7.21320 0.327535
\(486\) 1.41421 0.0641500
\(487\) 41.2132 1.86755 0.933774 0.357863i \(-0.116494\pi\)
0.933774 + 0.357863i \(0.116494\pi\)
\(488\) 1.85786 0.0841016
\(489\) −3.75736 −0.169914
\(490\) −3.51472 −0.158779
\(491\) −11.7990 −0.532481 −0.266240 0.963907i \(-0.585782\pi\)
−0.266240 + 0.963907i \(0.585782\pi\)
\(492\) 0 0
\(493\) 4.82843 0.217461
\(494\) 0 0
\(495\) −2.34315 −0.105317
\(496\) 6.62742 0.297580
\(497\) −13.0711 −0.586318
\(498\) 7.65685 0.343112
\(499\) −27.9411 −1.25082 −0.625408 0.780298i \(-0.715068\pi\)
−0.625408 + 0.780298i \(0.715068\pi\)
\(500\) 0 0
\(501\) 10.0711 0.449942
\(502\) 17.7574 0.792550
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) −2.82843 −0.125988
\(505\) 1.75736 0.0782015
\(506\) −9.94113 −0.441937
\(507\) −13.0000 −0.577350
\(508\) 0 0
\(509\) 25.3848 1.12516 0.562580 0.826743i \(-0.309809\pi\)
0.562580 + 0.826743i \(0.309809\pi\)
\(510\) 0.585786 0.0259391
\(511\) −10.7279 −0.474575
\(512\) −22.6274 −1.00000
\(513\) −2.17157 −0.0958773
\(514\) 29.1716 1.28670
\(515\) −0.928932 −0.0409336
\(516\) 0 0
\(517\) 56.9706 2.50556
\(518\) 0.928932 0.0408149
\(519\) 5.07107 0.222595
\(520\) 0 0
\(521\) 11.3137 0.495663 0.247831 0.968803i \(-0.420282\pi\)
0.247831 + 0.968803i \(0.420282\pi\)
\(522\) 6.82843 0.298872
\(523\) −27.4558 −1.20056 −0.600280 0.799790i \(-0.704944\pi\)
−0.600280 + 0.799790i \(0.704944\pi\)
\(524\) 0 0
\(525\) −4.82843 −0.210730
\(526\) −34.4853 −1.50363
\(527\) −1.65685 −0.0721737
\(528\) 22.6274 0.984732
\(529\) −21.4558 −0.932863
\(530\) −7.27208 −0.315879
\(531\) −3.58579 −0.155610
\(532\) 0 0
\(533\) 0 0
\(534\) −8.97056 −0.388194
\(535\) −1.41421 −0.0611418
\(536\) 28.2843 1.22169
\(537\) −10.2426 −0.442003
\(538\) −20.4853 −0.883183
\(539\) 33.9411 1.46195
\(540\) 0 0
\(541\) −3.27208 −0.140678 −0.0703388 0.997523i \(-0.522408\pi\)
−0.0703388 + 0.997523i \(0.522408\pi\)
\(542\) 32.1421 1.38062
\(543\) 0.585786 0.0251385
\(544\) 0 0
\(545\) 3.51472 0.150554
\(546\) 0 0
\(547\) 43.5563 1.86234 0.931168 0.364592i \(-0.118791\pi\)
0.931168 + 0.364592i \(0.118791\pi\)
\(548\) 0 0
\(549\) −0.656854 −0.0280339
\(550\) 38.6274 1.64708
\(551\) −10.4853 −0.446688
\(552\) −3.51472 −0.149596
\(553\) −1.00000 −0.0425243
\(554\) −24.0000 −1.01966
\(555\) 0.272078 0.0115491
\(556\) 0 0
\(557\) −28.1421 −1.19242 −0.596210 0.802828i \(-0.703328\pi\)
−0.596210 + 0.802828i \(0.703328\pi\)
\(558\) −2.34315 −0.0991933
\(559\) 0 0
\(560\) −1.65685 −0.0700149
\(561\) −5.65685 −0.238833
\(562\) −9.31371 −0.392875
\(563\) −34.5858 −1.45762 −0.728809 0.684718i \(-0.759926\pi\)
−0.728809 + 0.684718i \(0.759926\pi\)
\(564\) 0 0
\(565\) −4.97056 −0.209113
\(566\) 16.9706 0.713326
\(567\) 1.00000 0.0419961
\(568\) 36.9706 1.55125
\(569\) 38.3848 1.60917 0.804587 0.593835i \(-0.202387\pi\)
0.804587 + 0.593835i \(0.202387\pi\)
\(570\) −1.27208 −0.0532815
\(571\) 20.6274 0.863231 0.431615 0.902058i \(-0.357944\pi\)
0.431615 + 0.902058i \(0.357944\pi\)
\(572\) 0 0
\(573\) −19.5858 −0.818208
\(574\) 10.8284 0.451970
\(575\) −6.00000 −0.250217
\(576\) 8.00000 0.333333
\(577\) −6.72792 −0.280087 −0.140044 0.990145i \(-0.544724\pi\)
−0.140044 + 0.990145i \(0.544724\pi\)
\(578\) 1.41421 0.0588235
\(579\) 15.4853 0.643546
\(580\) 0 0
\(581\) 5.41421 0.224619
\(582\) 24.6274 1.02084
\(583\) 70.2254 2.90844
\(584\) 30.3431 1.25561
\(585\) 0 0
\(586\) 33.7574 1.39450
\(587\) −6.89949 −0.284773 −0.142386 0.989811i \(-0.545478\pi\)
−0.142386 + 0.989811i \(0.545478\pi\)
\(588\) 0 0
\(589\) 3.59798 0.148252
\(590\) −2.10051 −0.0864764
\(591\) 3.51472 0.144576
\(592\) −2.62742 −0.107986
\(593\) −35.8995 −1.47422 −0.737108 0.675775i \(-0.763809\pi\)
−0.737108 + 0.675775i \(0.763809\pi\)
\(594\) −8.00000 −0.328244
\(595\) 0.414214 0.0169811
\(596\) 0 0
\(597\) −1.82843 −0.0748325
\(598\) 0 0
\(599\) −16.4437 −0.671869 −0.335935 0.941885i \(-0.609052\pi\)
−0.335935 + 0.941885i \(0.609052\pi\)
\(600\) 13.6569 0.557539
\(601\) 31.9706 1.30411 0.652053 0.758173i \(-0.273908\pi\)
0.652053 + 0.758173i \(0.273908\pi\)
\(602\) −9.17157 −0.373805
\(603\) −10.0000 −0.407231
\(604\) 0 0
\(605\) 8.69848 0.353644
\(606\) 6.00000 0.243733
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) 0 0
\(609\) 4.82843 0.195658
\(610\) −0.384776 −0.0155791
\(611\) 0 0
\(612\) 0 0
\(613\) 1.51472 0.0611789 0.0305895 0.999532i \(-0.490262\pi\)
0.0305895 + 0.999532i \(0.490262\pi\)
\(614\) 41.7990 1.68687
\(615\) 3.17157 0.127890
\(616\) 16.0000 0.644658
\(617\) −9.31371 −0.374956 −0.187478 0.982269i \(-0.560031\pi\)
−0.187478 + 0.982269i \(0.560031\pi\)
\(618\) −3.17157 −0.127579
\(619\) 21.0000 0.844061 0.422031 0.906582i \(-0.361317\pi\)
0.422031 + 0.906582i \(0.361317\pi\)
\(620\) 0 0
\(621\) 1.24264 0.0498655
\(622\) −19.5147 −0.782469
\(623\) −6.34315 −0.254133
\(624\) 0 0
\(625\) 22.4558 0.898234
\(626\) −6.62742 −0.264885
\(627\) 12.2843 0.490587
\(628\) 0 0
\(629\) 0.656854 0.0261905
\(630\) 0.585786 0.0233383
\(631\) 24.0000 0.955425 0.477712 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(632\) 2.82843 0.112509
\(633\) 13.9706 0.555280
\(634\) −39.2132 −1.55736
\(635\) 8.38478 0.332740
\(636\) 0 0
\(637\) 0 0
\(638\) −38.6274 −1.52927
\(639\) −13.0711 −0.517083
\(640\) 4.68629 0.185242
\(641\) 6.55635 0.258960 0.129480 0.991582i \(-0.458669\pi\)
0.129480 + 0.991582i \(0.458669\pi\)
\(642\) −4.82843 −0.190563
\(643\) −33.2132 −1.30980 −0.654900 0.755715i \(-0.727289\pi\)
−0.654900 + 0.755715i \(0.727289\pi\)
\(644\) 0 0
\(645\) −2.68629 −0.105773
\(646\) −3.07107 −0.120830
\(647\) −1.58579 −0.0623437 −0.0311718 0.999514i \(-0.509924\pi\)
−0.0311718 + 0.999514i \(0.509924\pi\)
\(648\) −2.82843 −0.111111
\(649\) 20.2843 0.796227
\(650\) 0 0
\(651\) −1.65685 −0.0649372
\(652\) 0 0
\(653\) −32.0711 −1.25504 −0.627519 0.778601i \(-0.715930\pi\)
−0.627519 + 0.778601i \(0.715930\pi\)
\(654\) 12.0000 0.469237
\(655\) 0.171573 0.00670391
\(656\) −30.6274 −1.19580
\(657\) −10.7279 −0.418536
\(658\) −14.2426 −0.555236
\(659\) 3.72792 0.145219 0.0726096 0.997360i \(-0.476867\pi\)
0.0726096 + 0.997360i \(0.476867\pi\)
\(660\) 0 0
\(661\) 45.6985 1.77747 0.888733 0.458426i \(-0.151587\pi\)
0.888733 + 0.458426i \(0.151587\pi\)
\(662\) −19.4558 −0.756173
\(663\) 0 0
\(664\) −15.3137 −0.594287
\(665\) −0.899495 −0.0348809
\(666\) 0.928932 0.0359954
\(667\) 6.00000 0.232321
\(668\) 0 0
\(669\) −4.17157 −0.161282
\(670\) −5.85786 −0.226309
\(671\) 3.71573 0.143444
\(672\) 0 0
\(673\) 16.9706 0.654167 0.327084 0.944995i \(-0.393934\pi\)
0.327084 + 0.944995i \(0.393934\pi\)
\(674\) −18.0000 −0.693334
\(675\) −4.82843 −0.185846
\(676\) 0 0
\(677\) −21.5858 −0.829609 −0.414805 0.909911i \(-0.636150\pi\)
−0.414805 + 0.909911i \(0.636150\pi\)
\(678\) −16.9706 −0.651751
\(679\) 17.4142 0.668296
\(680\) −1.17157 −0.0449278
\(681\) −18.8284 −0.721507
\(682\) 13.2548 0.507554
\(683\) 25.8701 0.989890 0.494945 0.868924i \(-0.335188\pi\)
0.494945 + 0.868924i \(0.335188\pi\)
\(684\) 0 0
\(685\) 7.68629 0.293678
\(686\) −18.3848 −0.701934
\(687\) −2.58579 −0.0986539
\(688\) 25.9411 0.988996
\(689\) 0 0
\(690\) 0.727922 0.0277115
\(691\) −25.4853 −0.969506 −0.484753 0.874651i \(-0.661090\pi\)
−0.484753 + 0.874651i \(0.661090\pi\)
\(692\) 0 0
\(693\) −5.65685 −0.214886
\(694\) −7.61522 −0.289070
\(695\) 1.02944 0.0390488
\(696\) −13.6569 −0.517662
\(697\) 7.65685 0.290024
\(698\) −8.82843 −0.334161
\(699\) 24.3848 0.922317
\(700\) 0 0
\(701\) −10.4142 −0.393339 −0.196670 0.980470i \(-0.563013\pi\)
−0.196670 + 0.980470i \(0.563013\pi\)
\(702\) 0 0
\(703\) −1.42641 −0.0537980
\(704\) −45.2548 −1.70561
\(705\) −4.17157 −0.157111
\(706\) 13.7574 0.517765
\(707\) 4.24264 0.159561
\(708\) 0 0
\(709\) −44.7990 −1.68246 −0.841231 0.540676i \(-0.818168\pi\)
−0.841231 + 0.540676i \(0.818168\pi\)
\(710\) −7.65685 −0.287357
\(711\) −1.00000 −0.0375029
\(712\) 17.9411 0.672372
\(713\) −2.05887 −0.0771055
\(714\) 1.41421 0.0529256
\(715\) 0 0
\(716\) 0 0
\(717\) −6.72792 −0.251259
\(718\) −22.1421 −0.826337
\(719\) 15.3137 0.571105 0.285552 0.958363i \(-0.407823\pi\)
0.285552 + 0.958363i \(0.407823\pi\)
\(720\) −1.65685 −0.0617473
\(721\) −2.24264 −0.0835203
\(722\) −20.2010 −0.751804
\(723\) −2.58579 −0.0961664
\(724\) 0 0
\(725\) −23.3137 −0.865849
\(726\) 29.6985 1.10221
\(727\) 5.00000 0.185440 0.0927199 0.995692i \(-0.470444\pi\)
0.0927199 + 0.995692i \(0.470444\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −6.28427 −0.232591
\(731\) −6.48528 −0.239867
\(732\) 0 0
\(733\) −41.2843 −1.52487 −0.762435 0.647065i \(-0.775996\pi\)
−0.762435 + 0.647065i \(0.775996\pi\)
\(734\) −33.3137 −1.22963
\(735\) −2.48528 −0.0916710
\(736\) 0 0
\(737\) 56.5685 2.08373
\(738\) 10.8284 0.398600
\(739\) 3.37258 0.124062 0.0620312 0.998074i \(-0.480242\pi\)
0.0620312 + 0.998074i \(0.480242\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −17.5563 −0.644514
\(743\) −42.2132 −1.54865 −0.774326 0.632787i \(-0.781911\pi\)
−0.774326 + 0.632787i \(0.781911\pi\)
\(744\) 4.68629 0.171808
\(745\) −6.65685 −0.243888
\(746\) 20.1421 0.737456
\(747\) 5.41421 0.198096
\(748\) 0 0
\(749\) −3.41421 −0.124753
\(750\) −5.75736 −0.210229
\(751\) −32.2843 −1.17807 −0.589035 0.808108i \(-0.700492\pi\)
−0.589035 + 0.808108i \(0.700492\pi\)
\(752\) 40.2843 1.46902
\(753\) 12.5563 0.457579
\(754\) 0 0
\(755\) 0.213203 0.00775927
\(756\) 0 0
\(757\) −11.9706 −0.435078 −0.217539 0.976052i \(-0.569803\pi\)
−0.217539 + 0.976052i \(0.569803\pi\)
\(758\) 54.3848 1.97534
\(759\) −7.02944 −0.255152
\(760\) 2.54416 0.0922862
\(761\) −3.55635 −0.128918 −0.0644588 0.997920i \(-0.520532\pi\)
−0.0644588 + 0.997920i \(0.520532\pi\)
\(762\) 28.6274 1.03706
\(763\) 8.48528 0.307188
\(764\) 0 0
\(765\) 0.414214 0.0149759
\(766\) 19.4558 0.702968
\(767\) 0 0
\(768\) 0 0
\(769\) 40.4853 1.45994 0.729968 0.683481i \(-0.239535\pi\)
0.729968 + 0.683481i \(0.239535\pi\)
\(770\) −3.31371 −0.119418
\(771\) 20.6274 0.742878
\(772\) 0 0
\(773\) −17.6569 −0.635073 −0.317536 0.948246i \(-0.602856\pi\)
−0.317536 + 0.948246i \(0.602856\pi\)
\(774\) −9.17157 −0.329665
\(775\) 8.00000 0.287368
\(776\) −49.2548 −1.76815
\(777\) 0.656854 0.0235645
\(778\) 24.6274 0.882936
\(779\) −16.6274 −0.595739
\(780\) 0 0
\(781\) 73.9411 2.64582
\(782\) 1.75736 0.0628430
\(783\) 4.82843 0.172554
\(784\) 24.0000 0.857143
\(785\) 4.44365 0.158601
\(786\) 0.585786 0.0208943
\(787\) −3.27208 −0.116637 −0.0583185 0.998298i \(-0.518574\pi\)
−0.0583185 + 0.998298i \(0.518574\pi\)
\(788\) 0 0
\(789\) −24.3848 −0.868121
\(790\) −0.585786 −0.0208413
\(791\) −12.0000 −0.426671
\(792\) 16.0000 0.568535
\(793\) 0 0
\(794\) −11.1127 −0.394375
\(795\) −5.14214 −0.182373
\(796\) 0 0
\(797\) −33.5858 −1.18967 −0.594835 0.803848i \(-0.702783\pi\)
−0.594835 + 0.803848i \(0.702783\pi\)
\(798\) −3.07107 −0.108715
\(799\) −10.0711 −0.356289
\(800\) 0 0
\(801\) −6.34315 −0.224124
\(802\) 6.00000 0.211867
\(803\) 60.6863 2.14157
\(804\) 0 0
\(805\) 0.514719 0.0181414
\(806\) 0 0
\(807\) −14.4853 −0.509906
\(808\) −12.0000 −0.422159
\(809\) −3.65685 −0.128568 −0.0642841 0.997932i \(-0.520476\pi\)
−0.0642841 + 0.997932i \(0.520476\pi\)
\(810\) 0.585786 0.0205824
\(811\) 2.72792 0.0957903 0.0478951 0.998852i \(-0.484749\pi\)
0.0478951 + 0.998852i \(0.484749\pi\)
\(812\) 0 0
\(813\) 22.7279 0.797103
\(814\) −5.25483 −0.184182
\(815\) −1.55635 −0.0545165
\(816\) −4.00000 −0.140028
\(817\) 14.0833 0.492711
\(818\) 25.8579 0.904099
\(819\) 0 0
\(820\) 0 0
\(821\) −43.1127 −1.50464 −0.752322 0.658796i \(-0.771066\pi\)
−0.752322 + 0.658796i \(0.771066\pi\)
\(822\) 26.2426 0.915317
\(823\) −31.4853 −1.09751 −0.548754 0.835984i \(-0.684898\pi\)
−0.548754 + 0.835984i \(0.684898\pi\)
\(824\) 6.34315 0.220974
\(825\) 27.3137 0.950941
\(826\) −5.07107 −0.176445
\(827\) 19.7574 0.687031 0.343515 0.939147i \(-0.388382\pi\)
0.343515 + 0.939147i \(0.388382\pi\)
\(828\) 0 0
\(829\) −21.3137 −0.740256 −0.370128 0.928981i \(-0.620686\pi\)
−0.370128 + 0.928981i \(0.620686\pi\)
\(830\) 3.17157 0.110087
\(831\) −16.9706 −0.588702
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) 3.51472 0.121705
\(835\) 4.17157 0.144363
\(836\) 0 0
\(837\) −1.65685 −0.0572693
\(838\) 21.3137 0.736270
\(839\) 47.5269 1.64081 0.820406 0.571782i \(-0.193748\pi\)
0.820406 + 0.571782i \(0.193748\pi\)
\(840\) −1.17157 −0.0404231
\(841\) −5.68629 −0.196079
\(842\) 16.9289 0.583410
\(843\) −6.58579 −0.226827
\(844\) 0 0
\(845\) −5.38478 −0.185242
\(846\) −14.2426 −0.489672
\(847\) 21.0000 0.721569
\(848\) 49.6569 1.70522
\(849\) 12.0000 0.411839
\(850\) −6.82843 −0.234213
\(851\) 0.816234 0.0279801
\(852\) 0 0
\(853\) 10.0294 0.343401 0.171701 0.985149i \(-0.445074\pi\)
0.171701 + 0.985149i \(0.445074\pi\)
\(854\) −0.928932 −0.0317874
\(855\) −0.899495 −0.0307621
\(856\) 9.65685 0.330064
\(857\) 9.31371 0.318150 0.159075 0.987266i \(-0.449149\pi\)
0.159075 + 0.987266i \(0.449149\pi\)
\(858\) 0 0
\(859\) −21.4558 −0.732064 −0.366032 0.930602i \(-0.619284\pi\)
−0.366032 + 0.930602i \(0.619284\pi\)
\(860\) 0 0
\(861\) 7.65685 0.260945
\(862\) −56.1838 −1.91363
\(863\) −30.6274 −1.04257 −0.521285 0.853383i \(-0.674547\pi\)
−0.521285 + 0.853383i \(0.674547\pi\)
\(864\) 0 0
\(865\) 2.10051 0.0714193
\(866\) −36.2843 −1.23299
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) 5.65685 0.191896
\(870\) 2.82843 0.0958927
\(871\) 0 0
\(872\) −24.0000 −0.812743
\(873\) 17.4142 0.589382
\(874\) −3.81623 −0.129086
\(875\) −4.07107 −0.137627
\(876\) 0 0
\(877\) −42.7279 −1.44282 −0.721410 0.692509i \(-0.756506\pi\)
−0.721410 + 0.692509i \(0.756506\pi\)
\(878\) 29.5980 0.998883
\(879\) 23.8701 0.805117
\(880\) 9.37258 0.315950
\(881\) 33.9411 1.14351 0.571753 0.820426i \(-0.306264\pi\)
0.571753 + 0.820426i \(0.306264\pi\)
\(882\) −8.48528 −0.285714
\(883\) −25.0294 −0.842308 −0.421154 0.906989i \(-0.638375\pi\)
−0.421154 + 0.906989i \(0.638375\pi\)
\(884\) 0 0
\(885\) −1.48528 −0.0499272
\(886\) −45.8406 −1.54005
\(887\) 24.4142 0.819749 0.409875 0.912142i \(-0.365572\pi\)
0.409875 + 0.912142i \(0.365572\pi\)
\(888\) −1.85786 −0.0623458
\(889\) 20.2426 0.678916
\(890\) −3.71573 −0.124552
\(891\) −5.65685 −0.189512
\(892\) 0 0
\(893\) 21.8701 0.731854
\(894\) −22.7279 −0.760135
\(895\) −4.24264 −0.141816
\(896\) 11.3137 0.377964
\(897\) 0 0
\(898\) −20.4853 −0.683603
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) −12.4142 −0.413577
\(902\) −61.2548 −2.03956
\(903\) −6.48528 −0.215817
\(904\) 33.9411 1.12887
\(905\) 0.242641 0.00806565
\(906\) 0.727922 0.0241836
\(907\) −22.1005 −0.733835 −0.366918 0.930253i \(-0.619587\pi\)
−0.366918 + 0.930253i \(0.619587\pi\)
\(908\) 0 0
\(909\) 4.24264 0.140720
\(910\) 0 0
\(911\) −33.3137 −1.10373 −0.551866 0.833933i \(-0.686084\pi\)
−0.551866 + 0.833933i \(0.686084\pi\)
\(912\) 8.68629 0.287632
\(913\) −30.6274 −1.01362
\(914\) 23.2721 0.769772
\(915\) −0.272078 −0.00899462
\(916\) 0 0
\(917\) 0.414214 0.0136785
\(918\) 1.41421 0.0466760
\(919\) 6.97056 0.229938 0.114969 0.993369i \(-0.463323\pi\)
0.114969 + 0.993369i \(0.463323\pi\)
\(920\) −1.45584 −0.0479978
\(921\) 29.5563 0.973915
\(922\) 44.9706 1.48103
\(923\) 0 0
\(924\) 0 0
\(925\) −3.17157 −0.104281
\(926\) 14.4853 0.476016
\(927\) −2.24264 −0.0736580
\(928\) 0 0
\(929\) −28.9289 −0.949127 −0.474564 0.880221i \(-0.657394\pi\)
−0.474564 + 0.880221i \(0.657394\pi\)
\(930\) −0.970563 −0.0318260
\(931\) 13.0294 0.427023
\(932\) 0 0
\(933\) −13.7990 −0.451759
\(934\) −19.0294 −0.622662
\(935\) −2.34315 −0.0766291
\(936\) 0 0
\(937\) 23.2721 0.760266 0.380133 0.924932i \(-0.375878\pi\)
0.380133 + 0.924932i \(0.375878\pi\)
\(938\) −14.1421 −0.461757
\(939\) −4.68629 −0.152931
\(940\) 0 0
\(941\) 50.4975 1.64617 0.823085 0.567918i \(-0.192251\pi\)
0.823085 + 0.567918i \(0.192251\pi\)
\(942\) 15.1716 0.494317
\(943\) 9.51472 0.309842
\(944\) 14.3431 0.466830
\(945\) 0.414214 0.0134744
\(946\) 51.8823 1.68684
\(947\) 6.68629 0.217275 0.108638 0.994081i \(-0.465351\pi\)
0.108638 + 0.994081i \(0.465351\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 14.8284 0.481098
\(951\) −27.7279 −0.899139
\(952\) −2.82843 −0.0916698
\(953\) 12.3431 0.399834 0.199917 0.979813i \(-0.435933\pi\)
0.199917 + 0.979813i \(0.435933\pi\)
\(954\) −17.5563 −0.568408
\(955\) −8.11270 −0.262521
\(956\) 0 0
\(957\) −27.3137 −0.882927
\(958\) 49.3553 1.59460
\(959\) 18.5563 0.599216
\(960\) 3.31371 0.106949
\(961\) −28.2548 −0.911446
\(962\) 0 0
\(963\) −3.41421 −0.110021
\(964\) 0 0
\(965\) 6.41421 0.206481
\(966\) 1.75736 0.0565421
\(967\) −34.3137 −1.10345 −0.551727 0.834025i \(-0.686031\pi\)
−0.551727 + 0.834025i \(0.686031\pi\)
\(968\) −59.3970 −1.90909
\(969\) −2.17157 −0.0697610
\(970\) 10.2010 0.327535
\(971\) −48.2843 −1.54952 −0.774758 0.632258i \(-0.782128\pi\)
−0.774758 + 0.632258i \(0.782128\pi\)
\(972\) 0 0
\(973\) 2.48528 0.0796745
\(974\) 58.2843 1.86755
\(975\) 0 0
\(976\) 2.62742 0.0841016
\(977\) −28.4142 −0.909051 −0.454526 0.890734i \(-0.650191\pi\)
−0.454526 + 0.890734i \(0.650191\pi\)
\(978\) −5.31371 −0.169914
\(979\) 35.8823 1.14680
\(980\) 0 0
\(981\) 8.48528 0.270914
\(982\) −16.6863 −0.532481
\(983\) 11.2721 0.359523 0.179762 0.983710i \(-0.442467\pi\)
0.179762 + 0.983710i \(0.442467\pi\)
\(984\) −21.6569 −0.690395
\(985\) 1.45584 0.0463871
\(986\) 6.82843 0.217461
\(987\) −10.0711 −0.320566
\(988\) 0 0
\(989\) −8.05887 −0.256257
\(990\) −3.31371 −0.105317
\(991\) −25.4853 −0.809567 −0.404783 0.914413i \(-0.632653\pi\)
−0.404783 + 0.914413i \(0.632653\pi\)
\(992\) 0 0
\(993\) −13.7574 −0.436577
\(994\) −18.4853 −0.586318
\(995\) −0.757359 −0.0240099
\(996\) 0 0
\(997\) 42.4853 1.34552 0.672761 0.739860i \(-0.265108\pi\)
0.672761 + 0.739860i \(0.265108\pi\)
\(998\) −39.5147 −1.25082
\(999\) 0.656854 0.0207819
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 4029.2.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4029.2.a.c.1.2 2 1.1 even 1 trivial