Properties

Label 3762.2.a.y.1.1
Level $3762$
Weight $2$
Character 3762.1
Self dual yes
Analytic conductor $30.040$
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] = [3762,2,Mod(1,3762)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3762, 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("3762.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3762.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: \(30.0397212404\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 3762.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -0.561553 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -0.561553 q^{7} +1.00000 q^{8} -2.00000 q^{10} -1.00000 q^{11} +0.561553 q^{13} -0.561553 q^{14} +1.00000 q^{16} +0.561553 q^{17} -1.00000 q^{19} -2.00000 q^{20} -1.00000 q^{22} -1.43845 q^{23} -1.00000 q^{25} +0.561553 q^{26} -0.561553 q^{28} +5.68466 q^{29} +2.00000 q^{31} +1.00000 q^{32} +0.561553 q^{34} +1.12311 q^{35} -5.12311 q^{37} -1.00000 q^{38} -2.00000 q^{40} -2.00000 q^{41} -1.00000 q^{44} -1.43845 q^{46} -8.00000 q^{47} -6.68466 q^{49} -1.00000 q^{50} +0.561553 q^{52} -12.8078 q^{53} +2.00000 q^{55} -0.561553 q^{56} +5.68466 q^{58} +7.68466 q^{59} +6.24621 q^{61} +2.00000 q^{62} +1.00000 q^{64} -1.12311 q^{65} -7.68466 q^{67} +0.561553 q^{68} +1.12311 q^{70} -6.00000 q^{71} -9.68466 q^{73} -5.12311 q^{74} -1.00000 q^{76} +0.561553 q^{77} -4.00000 q^{79} -2.00000 q^{80} -2.00000 q^{82} -14.2462 q^{83} -1.12311 q^{85} -1.00000 q^{88} +0.876894 q^{89} -0.315342 q^{91} -1.43845 q^{92} -8.00000 q^{94} +2.00000 q^{95} -7.12311 q^{97} -6.68466 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 3 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 3 q^{7} + 2 q^{8} - 4 q^{10} - 2 q^{11} - 3 q^{13} + 3 q^{14} + 2 q^{16} - 3 q^{17} - 2 q^{19} - 4 q^{20} - 2 q^{22} - 7 q^{23} - 2 q^{25} - 3 q^{26} + 3 q^{28} - q^{29} + 4 q^{31} + 2 q^{32} - 3 q^{34} - 6 q^{35} - 2 q^{37} - 2 q^{38} - 4 q^{40} - 4 q^{41} - 2 q^{44} - 7 q^{46} - 16 q^{47} - q^{49} - 2 q^{50} - 3 q^{52} - 5 q^{53} + 4 q^{55} + 3 q^{56} - q^{58} + 3 q^{59} - 4 q^{61} + 4 q^{62} + 2 q^{64} + 6 q^{65} - 3 q^{67} - 3 q^{68} - 6 q^{70} - 12 q^{71} - 7 q^{73} - 2 q^{74} - 2 q^{76} - 3 q^{77} - 8 q^{79} - 4 q^{80} - 4 q^{82} - 12 q^{83} + 6 q^{85} - 2 q^{88} + 10 q^{89} - 13 q^{91} - 7 q^{92} - 16 q^{94} + 4 q^{95} - 6 q^{97} - q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) −0.561553 −0.212247 −0.106124 0.994353i \(-0.533844\pi\)
−0.106124 + 0.994353i \(0.533844\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 0.561553 0.155747 0.0778734 0.996963i \(-0.475187\pi\)
0.0778734 + 0.996963i \(0.475187\pi\)
\(14\) −0.561553 −0.150081
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.561553 0.136197 0.0680983 0.997679i \(-0.478307\pi\)
0.0680983 + 0.997679i \(0.478307\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −1.43845 −0.299937 −0.149968 0.988691i \(-0.547917\pi\)
−0.149968 + 0.988691i \(0.547917\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0.561553 0.110130
\(27\) 0 0
\(28\) −0.561553 −0.106124
\(29\) 5.68466 1.05561 0.527807 0.849364i \(-0.323014\pi\)
0.527807 + 0.849364i \(0.323014\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0.561553 0.0963055
\(35\) 1.12311 0.189839
\(36\) 0 0
\(37\) −5.12311 −0.842233 −0.421117 0.907006i \(-0.638362\pi\)
−0.421117 + 0.907006i \(0.638362\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) −2.00000 −0.316228
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −1.43845 −0.212087
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −6.68466 −0.954951
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 0.561553 0.0778734
\(53\) −12.8078 −1.75928 −0.879641 0.475638i \(-0.842217\pi\)
−0.879641 + 0.475638i \(0.842217\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) −0.561553 −0.0750407
\(57\) 0 0
\(58\) 5.68466 0.746432
\(59\) 7.68466 1.00046 0.500229 0.865893i \(-0.333249\pi\)
0.500229 + 0.865893i \(0.333249\pi\)
\(60\) 0 0
\(61\) 6.24621 0.799745 0.399873 0.916571i \(-0.369054\pi\)
0.399873 + 0.916571i \(0.369054\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.12311 −0.139304
\(66\) 0 0
\(67\) −7.68466 −0.938830 −0.469415 0.882978i \(-0.655535\pi\)
−0.469415 + 0.882978i \(0.655535\pi\)
\(68\) 0.561553 0.0680983
\(69\) 0 0
\(70\) 1.12311 0.134237
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −9.68466 −1.13350 −0.566752 0.823889i \(-0.691800\pi\)
−0.566752 + 0.823889i \(0.691800\pi\)
\(74\) −5.12311 −0.595549
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 0.561553 0.0639949
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −2.00000 −0.223607
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) −14.2462 −1.56372 −0.781862 0.623451i \(-0.785730\pi\)
−0.781862 + 0.623451i \(0.785730\pi\)
\(84\) 0 0
\(85\) −1.12311 −0.121818
\(86\) 0 0
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 0.876894 0.0929506 0.0464753 0.998919i \(-0.485201\pi\)
0.0464753 + 0.998919i \(0.485201\pi\)
\(90\) 0 0
\(91\) −0.315342 −0.0330568
\(92\) −1.43845 −0.149968
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) −7.12311 −0.723242 −0.361621 0.932325i \(-0.617777\pi\)
−0.361621 + 0.932325i \(0.617777\pi\)
\(98\) −6.68466 −0.675252
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 6.87689 0.684277 0.342138 0.939650i \(-0.388849\pi\)
0.342138 + 0.939650i \(0.388849\pi\)
\(102\) 0 0
\(103\) −13.3693 −1.31732 −0.658659 0.752442i \(-0.728876\pi\)
−0.658659 + 0.752442i \(0.728876\pi\)
\(104\) 0.561553 0.0550648
\(105\) 0 0
\(106\) −12.8078 −1.24400
\(107\) 9.93087 0.960053 0.480027 0.877254i \(-0.340627\pi\)
0.480027 + 0.877254i \(0.340627\pi\)
\(108\) 0 0
\(109\) −6.31534 −0.604900 −0.302450 0.953165i \(-0.597805\pi\)
−0.302450 + 0.953165i \(0.597805\pi\)
\(110\) 2.00000 0.190693
\(111\) 0 0
\(112\) −0.561553 −0.0530618
\(113\) 3.12311 0.293797 0.146899 0.989152i \(-0.453071\pi\)
0.146899 + 0.989152i \(0.453071\pi\)
\(114\) 0 0
\(115\) 2.87689 0.268272
\(116\) 5.68466 0.527807
\(117\) 0 0
\(118\) 7.68466 0.707430
\(119\) −0.315342 −0.0289073
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 6.24621 0.565505
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 2.24621 0.199319 0.0996595 0.995022i \(-0.468225\pi\)
0.0996595 + 0.995022i \(0.468225\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −1.12311 −0.0985029
\(131\) −1.12311 −0.0981262 −0.0490631 0.998796i \(-0.515624\pi\)
−0.0490631 + 0.998796i \(0.515624\pi\)
\(132\) 0 0
\(133\) 0.561553 0.0486928
\(134\) −7.68466 −0.663853
\(135\) 0 0
\(136\) 0.561553 0.0481528
\(137\) −12.5616 −1.07321 −0.536603 0.843835i \(-0.680293\pi\)
−0.536603 + 0.843835i \(0.680293\pi\)
\(138\) 0 0
\(139\) 7.36932 0.625057 0.312529 0.949908i \(-0.398824\pi\)
0.312529 + 0.949908i \(0.398824\pi\)
\(140\) 1.12311 0.0949197
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) −0.561553 −0.0469594
\(144\) 0 0
\(145\) −11.3693 −0.944170
\(146\) −9.68466 −0.801508
\(147\) 0 0
\(148\) −5.12311 −0.421117
\(149\) 6.87689 0.563377 0.281689 0.959506i \(-0.409106\pi\)
0.281689 + 0.959506i \(0.409106\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) 0.561553 0.0452512
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −14.4924 −1.15662 −0.578311 0.815817i \(-0.696288\pi\)
−0.578311 + 0.815817i \(0.696288\pi\)
\(158\) −4.00000 −0.318223
\(159\) 0 0
\(160\) −2.00000 −0.158114
\(161\) 0.807764 0.0636607
\(162\) 0 0
\(163\) 11.3693 0.890514 0.445257 0.895403i \(-0.353112\pi\)
0.445257 + 0.895403i \(0.353112\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) −14.2462 −1.10572
\(167\) 0.630683 0.0488037 0.0244019 0.999702i \(-0.492232\pi\)
0.0244019 + 0.999702i \(0.492232\pi\)
\(168\) 0 0
\(169\) −12.6847 −0.975743
\(170\) −1.12311 −0.0861383
\(171\) 0 0
\(172\) 0 0
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 0 0
\(175\) 0.561553 0.0424494
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 0.876894 0.0657260
\(179\) −5.75379 −0.430058 −0.215029 0.976608i \(-0.568985\pi\)
−0.215029 + 0.976608i \(0.568985\pi\)
\(180\) 0 0
\(181\) −10.8769 −0.808473 −0.404237 0.914654i \(-0.632463\pi\)
−0.404237 + 0.914654i \(0.632463\pi\)
\(182\) −0.315342 −0.0233747
\(183\) 0 0
\(184\) −1.43845 −0.106044
\(185\) 10.2462 0.753316
\(186\) 0 0
\(187\) −0.561553 −0.0410648
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 2.00000 0.145095
\(191\) 8.31534 0.601677 0.300838 0.953675i \(-0.402733\pi\)
0.300838 + 0.953675i \(0.402733\pi\)
\(192\) 0 0
\(193\) 7.12311 0.512732 0.256366 0.966580i \(-0.417475\pi\)
0.256366 + 0.966580i \(0.417475\pi\)
\(194\) −7.12311 −0.511409
\(195\) 0 0
\(196\) −6.68466 −0.477476
\(197\) −18.2462 −1.29999 −0.649994 0.759939i \(-0.725229\pi\)
−0.649994 + 0.759939i \(0.725229\pi\)
\(198\) 0 0
\(199\) 11.6847 0.828303 0.414152 0.910208i \(-0.364078\pi\)
0.414152 + 0.910208i \(0.364078\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 6.87689 0.483857
\(203\) −3.19224 −0.224051
\(204\) 0 0
\(205\) 4.00000 0.279372
\(206\) −13.3693 −0.931484
\(207\) 0 0
\(208\) 0.561553 0.0389367
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) −0.315342 −0.0217090 −0.0108545 0.999941i \(-0.503455\pi\)
−0.0108545 + 0.999941i \(0.503455\pi\)
\(212\) −12.8078 −0.879641
\(213\) 0 0
\(214\) 9.93087 0.678860
\(215\) 0 0
\(216\) 0 0
\(217\) −1.12311 −0.0762414
\(218\) −6.31534 −0.427729
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) 0.315342 0.0212122
\(222\) 0 0
\(223\) 24.7386 1.65662 0.828311 0.560269i \(-0.189302\pi\)
0.828311 + 0.560269i \(0.189302\pi\)
\(224\) −0.561553 −0.0375203
\(225\) 0 0
\(226\) 3.12311 0.207746
\(227\) −28.1771 −1.87018 −0.935089 0.354412i \(-0.884681\pi\)
−0.935089 + 0.354412i \(0.884681\pi\)
\(228\) 0 0
\(229\) 12.8769 0.850929 0.425465 0.904975i \(-0.360111\pi\)
0.425465 + 0.904975i \(0.360111\pi\)
\(230\) 2.87689 0.189697
\(231\) 0 0
\(232\) 5.68466 0.373216
\(233\) 28.7386 1.88273 0.941365 0.337389i \(-0.109544\pi\)
0.941365 + 0.337389i \(0.109544\pi\)
\(234\) 0 0
\(235\) 16.0000 1.04372
\(236\) 7.68466 0.500229
\(237\) 0 0
\(238\) −0.315342 −0.0204406
\(239\) −23.9309 −1.54796 −0.773980 0.633210i \(-0.781737\pi\)
−0.773980 + 0.633210i \(0.781737\pi\)
\(240\) 0 0
\(241\) −19.6155 −1.26355 −0.631774 0.775153i \(-0.717673\pi\)
−0.631774 + 0.775153i \(0.717673\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 6.24621 0.399873
\(245\) 13.3693 0.854134
\(246\) 0 0
\(247\) −0.561553 −0.0357307
\(248\) 2.00000 0.127000
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) −17.1231 −1.08080 −0.540400 0.841408i \(-0.681727\pi\)
−0.540400 + 0.841408i \(0.681727\pi\)
\(252\) 0 0
\(253\) 1.43845 0.0904344
\(254\) 2.24621 0.140940
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 22.0000 1.37232 0.686161 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 0 0
\(259\) 2.87689 0.178762
\(260\) −1.12311 −0.0696521
\(261\) 0 0
\(262\) −1.12311 −0.0693857
\(263\) −8.87689 −0.547373 −0.273686 0.961819i \(-0.588243\pi\)
−0.273686 + 0.961819i \(0.588243\pi\)
\(264\) 0 0
\(265\) 25.6155 1.57355
\(266\) 0.561553 0.0344310
\(267\) 0 0
\(268\) −7.68466 −0.469415
\(269\) −2.87689 −0.175407 −0.0877037 0.996147i \(-0.527953\pi\)
−0.0877037 + 0.996147i \(0.527953\pi\)
\(270\) 0 0
\(271\) 25.6847 1.56023 0.780116 0.625635i \(-0.215160\pi\)
0.780116 + 0.625635i \(0.215160\pi\)
\(272\) 0.561553 0.0340491
\(273\) 0 0
\(274\) −12.5616 −0.758871
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 8.49242 0.510260 0.255130 0.966907i \(-0.417882\pi\)
0.255130 + 0.966907i \(0.417882\pi\)
\(278\) 7.36932 0.441982
\(279\) 0 0
\(280\) 1.12311 0.0671184
\(281\) 25.3693 1.51341 0.756703 0.653758i \(-0.226809\pi\)
0.756703 + 0.653758i \(0.226809\pi\)
\(282\) 0 0
\(283\) 11.3693 0.675836 0.337918 0.941176i \(-0.390277\pi\)
0.337918 + 0.941176i \(0.390277\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) −0.561553 −0.0332053
\(287\) 1.12311 0.0662948
\(288\) 0 0
\(289\) −16.6847 −0.981450
\(290\) −11.3693 −0.667629
\(291\) 0 0
\(292\) −9.68466 −0.566752
\(293\) −3.93087 −0.229644 −0.114822 0.993386i \(-0.536630\pi\)
−0.114822 + 0.993386i \(0.536630\pi\)
\(294\) 0 0
\(295\) −15.3693 −0.894836
\(296\) −5.12311 −0.297774
\(297\) 0 0
\(298\) 6.87689 0.398368
\(299\) −0.807764 −0.0467142
\(300\) 0 0
\(301\) 0 0
\(302\) 4.00000 0.230174
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) −12.4924 −0.715314
\(306\) 0 0
\(307\) −22.2462 −1.26966 −0.634829 0.772653i \(-0.718930\pi\)
−0.634829 + 0.772653i \(0.718930\pi\)
\(308\) 0.561553 0.0319974
\(309\) 0 0
\(310\) −4.00000 −0.227185
\(311\) 4.80776 0.272623 0.136312 0.990666i \(-0.456475\pi\)
0.136312 + 0.990666i \(0.456475\pi\)
\(312\) 0 0
\(313\) −0.561553 −0.0317408 −0.0158704 0.999874i \(-0.505052\pi\)
−0.0158704 + 0.999874i \(0.505052\pi\)
\(314\) −14.4924 −0.817855
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 7.05398 0.396191 0.198095 0.980183i \(-0.436524\pi\)
0.198095 + 0.980183i \(0.436524\pi\)
\(318\) 0 0
\(319\) −5.68466 −0.318280
\(320\) −2.00000 −0.111803
\(321\) 0 0
\(322\) 0.807764 0.0450149
\(323\) −0.561553 −0.0312456
\(324\) 0 0
\(325\) −0.561553 −0.0311493
\(326\) 11.3693 0.629688
\(327\) 0 0
\(328\) −2.00000 −0.110432
\(329\) 4.49242 0.247675
\(330\) 0 0
\(331\) −14.5616 −0.800375 −0.400188 0.916433i \(-0.631055\pi\)
−0.400188 + 0.916433i \(0.631055\pi\)
\(332\) −14.2462 −0.781862
\(333\) 0 0
\(334\) 0.630683 0.0345094
\(335\) 15.3693 0.839715
\(336\) 0 0
\(337\) 11.1231 0.605914 0.302957 0.953004i \(-0.402026\pi\)
0.302957 + 0.953004i \(0.402026\pi\)
\(338\) −12.6847 −0.689954
\(339\) 0 0
\(340\) −1.12311 −0.0609090
\(341\) −2.00000 −0.108306
\(342\) 0 0
\(343\) 7.68466 0.414933
\(344\) 0 0
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) −9.61553 −0.516189 −0.258094 0.966120i \(-0.583095\pi\)
−0.258094 + 0.966120i \(0.583095\pi\)
\(348\) 0 0
\(349\) 21.6155 1.15705 0.578526 0.815664i \(-0.303628\pi\)
0.578526 + 0.815664i \(0.303628\pi\)
\(350\) 0.561553 0.0300163
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −22.1771 −1.18037 −0.590183 0.807269i \(-0.700945\pi\)
−0.590183 + 0.807269i \(0.700945\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) 0.876894 0.0464753
\(357\) 0 0
\(358\) −5.75379 −0.304097
\(359\) 15.9309 0.840799 0.420400 0.907339i \(-0.361890\pi\)
0.420400 + 0.907339i \(0.361890\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −10.8769 −0.571677
\(363\) 0 0
\(364\) −0.315342 −0.0165284
\(365\) 19.3693 1.01384
\(366\) 0 0
\(367\) 30.2462 1.57884 0.789420 0.613854i \(-0.210382\pi\)
0.789420 + 0.613854i \(0.210382\pi\)
\(368\) −1.43845 −0.0749842
\(369\) 0 0
\(370\) 10.2462 0.532675
\(371\) 7.19224 0.373402
\(372\) 0 0
\(373\) 3.43845 0.178036 0.0890180 0.996030i \(-0.471627\pi\)
0.0890180 + 0.996030i \(0.471627\pi\)
\(374\) −0.561553 −0.0290372
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) 3.19224 0.164409
\(378\) 0 0
\(379\) 6.56155 0.337044 0.168522 0.985698i \(-0.446101\pi\)
0.168522 + 0.985698i \(0.446101\pi\)
\(380\) 2.00000 0.102598
\(381\) 0 0
\(382\) 8.31534 0.425450
\(383\) 2.00000 0.102195 0.0510976 0.998694i \(-0.483728\pi\)
0.0510976 + 0.998694i \(0.483728\pi\)
\(384\) 0 0
\(385\) −1.12311 −0.0572388
\(386\) 7.12311 0.362557
\(387\) 0 0
\(388\) −7.12311 −0.361621
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) −0.807764 −0.0408504
\(392\) −6.68466 −0.337626
\(393\) 0 0
\(394\) −18.2462 −0.919231
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) 38.9848 1.95659 0.978297 0.207209i \(-0.0664381\pi\)
0.978297 + 0.207209i \(0.0664381\pi\)
\(398\) 11.6847 0.585699
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 7.75379 0.387206 0.193603 0.981080i \(-0.437983\pi\)
0.193603 + 0.981080i \(0.437983\pi\)
\(402\) 0 0
\(403\) 1.12311 0.0559459
\(404\) 6.87689 0.342138
\(405\) 0 0
\(406\) −3.19224 −0.158428
\(407\) 5.12311 0.253943
\(408\) 0 0
\(409\) 16.2462 0.803323 0.401662 0.915788i \(-0.368433\pi\)
0.401662 + 0.915788i \(0.368433\pi\)
\(410\) 4.00000 0.197546
\(411\) 0 0
\(412\) −13.3693 −0.658659
\(413\) −4.31534 −0.212344
\(414\) 0 0
\(415\) 28.4924 1.39864
\(416\) 0.561553 0.0275324
\(417\) 0 0
\(418\) 1.00000 0.0489116
\(419\) 14.8769 0.726784 0.363392 0.931636i \(-0.381619\pi\)
0.363392 + 0.931636i \(0.381619\pi\)
\(420\) 0 0
\(421\) −29.9309 −1.45874 −0.729371 0.684119i \(-0.760187\pi\)
−0.729371 + 0.684119i \(0.760187\pi\)
\(422\) −0.315342 −0.0153506
\(423\) 0 0
\(424\) −12.8078 −0.622000
\(425\) −0.561553 −0.0272393
\(426\) 0 0
\(427\) −3.50758 −0.169744
\(428\) 9.93087 0.480027
\(429\) 0 0
\(430\) 0 0
\(431\) −31.8617 −1.53473 −0.767363 0.641213i \(-0.778432\pi\)
−0.767363 + 0.641213i \(0.778432\pi\)
\(432\) 0 0
\(433\) 6.63068 0.318650 0.159325 0.987226i \(-0.449068\pi\)
0.159325 + 0.987226i \(0.449068\pi\)
\(434\) −1.12311 −0.0539108
\(435\) 0 0
\(436\) −6.31534 −0.302450
\(437\) 1.43845 0.0688103
\(438\) 0 0
\(439\) −9.61553 −0.458924 −0.229462 0.973318i \(-0.573697\pi\)
−0.229462 + 0.973318i \(0.573697\pi\)
\(440\) 2.00000 0.0953463
\(441\) 0 0
\(442\) 0.315342 0.0149993
\(443\) −39.8617 −1.89389 −0.946944 0.321398i \(-0.895847\pi\)
−0.946944 + 0.321398i \(0.895847\pi\)
\(444\) 0 0
\(445\) −1.75379 −0.0831376
\(446\) 24.7386 1.17141
\(447\) 0 0
\(448\) −0.561553 −0.0265309
\(449\) −23.6155 −1.11449 −0.557243 0.830350i \(-0.688141\pi\)
−0.557243 + 0.830350i \(0.688141\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) 3.12311 0.146899
\(453\) 0 0
\(454\) −28.1771 −1.32242
\(455\) 0.630683 0.0295669
\(456\) 0 0
\(457\) 36.5616 1.71028 0.855139 0.518399i \(-0.173472\pi\)
0.855139 + 0.518399i \(0.173472\pi\)
\(458\) 12.8769 0.601698
\(459\) 0 0
\(460\) 2.87689 0.134136
\(461\) −26.7386 −1.24534 −0.622671 0.782484i \(-0.713953\pi\)
−0.622671 + 0.782484i \(0.713953\pi\)
\(462\) 0 0
\(463\) 3.50758 0.163011 0.0815055 0.996673i \(-0.474027\pi\)
0.0815055 + 0.996673i \(0.474027\pi\)
\(464\) 5.68466 0.263904
\(465\) 0 0
\(466\) 28.7386 1.33129
\(467\) −9.75379 −0.451352 −0.225676 0.974202i \(-0.572459\pi\)
−0.225676 + 0.974202i \(0.572459\pi\)
\(468\) 0 0
\(469\) 4.31534 0.199264
\(470\) 16.0000 0.738025
\(471\) 0 0
\(472\) 7.68466 0.353715
\(473\) 0 0
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) −0.315342 −0.0144537
\(477\) 0 0
\(478\) −23.9309 −1.09457
\(479\) 13.3693 0.610860 0.305430 0.952215i \(-0.401200\pi\)
0.305430 + 0.952215i \(0.401200\pi\)
\(480\) 0 0
\(481\) −2.87689 −0.131175
\(482\) −19.6155 −0.893463
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 14.2462 0.646887
\(486\) 0 0
\(487\) 31.1231 1.41032 0.705161 0.709047i \(-0.250875\pi\)
0.705161 + 0.709047i \(0.250875\pi\)
\(488\) 6.24621 0.282753
\(489\) 0 0
\(490\) 13.3693 0.603964
\(491\) 39.3693 1.77671 0.888356 0.459155i \(-0.151848\pi\)
0.888356 + 0.459155i \(0.151848\pi\)
\(492\) 0 0
\(493\) 3.19224 0.143771
\(494\) −0.561553 −0.0252655
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) 3.36932 0.151135
\(498\) 0 0
\(499\) 9.75379 0.436640 0.218320 0.975877i \(-0.429942\pi\)
0.218320 + 0.975877i \(0.429942\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) −17.1231 −0.764242
\(503\) −21.6847 −0.966871 −0.483436 0.875380i \(-0.660611\pi\)
−0.483436 + 0.875380i \(0.660611\pi\)
\(504\) 0 0
\(505\) −13.7538 −0.612036
\(506\) 1.43845 0.0639468
\(507\) 0 0
\(508\) 2.24621 0.0996595
\(509\) 31.3693 1.39042 0.695210 0.718806i \(-0.255311\pi\)
0.695210 + 0.718806i \(0.255311\pi\)
\(510\) 0 0
\(511\) 5.43845 0.240583
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 22.0000 0.970378
\(515\) 26.7386 1.17824
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) 2.87689 0.126403
\(519\) 0 0
\(520\) −1.12311 −0.0492514
\(521\) 7.12311 0.312069 0.156034 0.987752i \(-0.450129\pi\)
0.156034 + 0.987752i \(0.450129\pi\)
\(522\) 0 0
\(523\) 31.0540 1.35790 0.678948 0.734187i \(-0.262436\pi\)
0.678948 + 0.734187i \(0.262436\pi\)
\(524\) −1.12311 −0.0490631
\(525\) 0 0
\(526\) −8.87689 −0.387051
\(527\) 1.12311 0.0489232
\(528\) 0 0
\(529\) −20.9309 −0.910038
\(530\) 25.6155 1.11267
\(531\) 0 0
\(532\) 0.561553 0.0243464
\(533\) −1.12311 −0.0486471
\(534\) 0 0
\(535\) −19.8617 −0.858698
\(536\) −7.68466 −0.331927
\(537\) 0 0
\(538\) −2.87689 −0.124032
\(539\) 6.68466 0.287929
\(540\) 0 0
\(541\) −39.2311 −1.68667 −0.843337 0.537384i \(-0.819412\pi\)
−0.843337 + 0.537384i \(0.819412\pi\)
\(542\) 25.6847 1.10325
\(543\) 0 0
\(544\) 0.561553 0.0240764
\(545\) 12.6307 0.541039
\(546\) 0 0
\(547\) −42.7386 −1.82737 −0.913686 0.406421i \(-0.866777\pi\)
−0.913686 + 0.406421i \(0.866777\pi\)
\(548\) −12.5616 −0.536603
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) −5.68466 −0.242175
\(552\) 0 0
\(553\) 2.24621 0.0955186
\(554\) 8.49242 0.360808
\(555\) 0 0
\(556\) 7.36932 0.312529
\(557\) −17.1231 −0.725529 −0.362765 0.931881i \(-0.618167\pi\)
−0.362765 + 0.931881i \(0.618167\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 1.12311 0.0474599
\(561\) 0 0
\(562\) 25.3693 1.07014
\(563\) 42.7386 1.80122 0.900609 0.434630i \(-0.143121\pi\)
0.900609 + 0.434630i \(0.143121\pi\)
\(564\) 0 0
\(565\) −6.24621 −0.262780
\(566\) 11.3693 0.477888
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) 21.3693 0.895848 0.447924 0.894072i \(-0.352163\pi\)
0.447924 + 0.894072i \(0.352163\pi\)
\(570\) 0 0
\(571\) −5.61553 −0.235003 −0.117501 0.993073i \(-0.537488\pi\)
−0.117501 + 0.993073i \(0.537488\pi\)
\(572\) −0.561553 −0.0234797
\(573\) 0 0
\(574\) 1.12311 0.0468775
\(575\) 1.43845 0.0599874
\(576\) 0 0
\(577\) 41.0540 1.70910 0.854550 0.519370i \(-0.173833\pi\)
0.854550 + 0.519370i \(0.173833\pi\)
\(578\) −16.6847 −0.693990
\(579\) 0 0
\(580\) −11.3693 −0.472085
\(581\) 8.00000 0.331896
\(582\) 0 0
\(583\) 12.8078 0.530443
\(584\) −9.68466 −0.400754
\(585\) 0 0
\(586\) −3.93087 −0.162383
\(587\) 40.4924 1.67130 0.835651 0.549261i \(-0.185091\pi\)
0.835651 + 0.549261i \(0.185091\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) −15.3693 −0.632745
\(591\) 0 0
\(592\) −5.12311 −0.210558
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 0 0
\(595\) 0.630683 0.0258555
\(596\) 6.87689 0.281689
\(597\) 0 0
\(598\) −0.807764 −0.0330319
\(599\) 17.8617 0.729811 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(600\) 0 0
\(601\) 21.3693 0.871673 0.435836 0.900026i \(-0.356453\pi\)
0.435836 + 0.900026i \(0.356453\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 4.00000 0.162758
\(605\) −2.00000 −0.0813116
\(606\) 0 0
\(607\) −13.6155 −0.552637 −0.276319 0.961066i \(-0.589115\pi\)
−0.276319 + 0.961066i \(0.589115\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) −12.4924 −0.505803
\(611\) −4.49242 −0.181744
\(612\) 0 0
\(613\) 18.2462 0.736958 0.368479 0.929636i \(-0.379879\pi\)
0.368479 + 0.929636i \(0.379879\pi\)
\(614\) −22.2462 −0.897784
\(615\) 0 0
\(616\) 0.561553 0.0226256
\(617\) −44.7386 −1.80111 −0.900555 0.434743i \(-0.856839\pi\)
−0.900555 + 0.434743i \(0.856839\pi\)
\(618\) 0 0
\(619\) −9.75379 −0.392038 −0.196019 0.980600i \(-0.562801\pi\)
−0.196019 + 0.980600i \(0.562801\pi\)
\(620\) −4.00000 −0.160644
\(621\) 0 0
\(622\) 4.80776 0.192774
\(623\) −0.492423 −0.0197285
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) −0.561553 −0.0224442
\(627\) 0 0
\(628\) −14.4924 −0.578311
\(629\) −2.87689 −0.114709
\(630\) 0 0
\(631\) 3.50758 0.139634 0.0698172 0.997560i \(-0.477758\pi\)
0.0698172 + 0.997560i \(0.477758\pi\)
\(632\) −4.00000 −0.159111
\(633\) 0 0
\(634\) 7.05398 0.280149
\(635\) −4.49242 −0.178276
\(636\) 0 0
\(637\) −3.75379 −0.148731
\(638\) −5.68466 −0.225058
\(639\) 0 0
\(640\) −2.00000 −0.0790569
\(641\) −31.6155 −1.24874 −0.624369 0.781129i \(-0.714644\pi\)
−0.624369 + 0.781129i \(0.714644\pi\)
\(642\) 0 0
\(643\) −12.6307 −0.498106 −0.249053 0.968490i \(-0.580119\pi\)
−0.249053 + 0.968490i \(0.580119\pi\)
\(644\) 0.807764 0.0318304
\(645\) 0 0
\(646\) −0.561553 −0.0220940
\(647\) 47.5464 1.86924 0.934621 0.355646i \(-0.115739\pi\)
0.934621 + 0.355646i \(0.115739\pi\)
\(648\) 0 0
\(649\) −7.68466 −0.301649
\(650\) −0.561553 −0.0220259
\(651\) 0 0
\(652\) 11.3693 0.445257
\(653\) 15.6155 0.611083 0.305541 0.952179i \(-0.401163\pi\)
0.305541 + 0.952179i \(0.401163\pi\)
\(654\) 0 0
\(655\) 2.24621 0.0877667
\(656\) −2.00000 −0.0780869
\(657\) 0 0
\(658\) 4.49242 0.175133
\(659\) 14.4233 0.561852 0.280926 0.959729i \(-0.409359\pi\)
0.280926 + 0.959729i \(0.409359\pi\)
\(660\) 0 0
\(661\) −26.5616 −1.03312 −0.516562 0.856250i \(-0.672789\pi\)
−0.516562 + 0.856250i \(0.672789\pi\)
\(662\) −14.5616 −0.565951
\(663\) 0 0
\(664\) −14.2462 −0.552860
\(665\) −1.12311 −0.0435522
\(666\) 0 0
\(667\) −8.17708 −0.316618
\(668\) 0.630683 0.0244019
\(669\) 0 0
\(670\) 15.3693 0.593769
\(671\) −6.24621 −0.241132
\(672\) 0 0
\(673\) −27.1231 −1.04552 −0.522759 0.852480i \(-0.675097\pi\)
−0.522759 + 0.852480i \(0.675097\pi\)
\(674\) 11.1231 0.428446
\(675\) 0 0
\(676\) −12.6847 −0.487871
\(677\) −6.17708 −0.237405 −0.118702 0.992930i \(-0.537873\pi\)
−0.118702 + 0.992930i \(0.537873\pi\)
\(678\) 0 0
\(679\) 4.00000 0.153506
\(680\) −1.12311 −0.0430691
\(681\) 0 0
\(682\) −2.00000 −0.0765840
\(683\) −28.4924 −1.09023 −0.545116 0.838361i \(-0.683514\pi\)
−0.545116 + 0.838361i \(0.683514\pi\)
\(684\) 0 0
\(685\) 25.1231 0.959905
\(686\) 7.68466 0.293402
\(687\) 0 0
\(688\) 0 0
\(689\) −7.19224 −0.274002
\(690\) 0 0
\(691\) −2.38447 −0.0907096 −0.0453548 0.998971i \(-0.514442\pi\)
−0.0453548 + 0.998971i \(0.514442\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) −9.61553 −0.365000
\(695\) −14.7386 −0.559068
\(696\) 0 0
\(697\) −1.12311 −0.0425407
\(698\) 21.6155 0.818160
\(699\) 0 0
\(700\) 0.561553 0.0212247
\(701\) 40.9848 1.54798 0.773988 0.633200i \(-0.218259\pi\)
0.773988 + 0.633200i \(0.218259\pi\)
\(702\) 0 0
\(703\) 5.12311 0.193222
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −22.1771 −0.834645
\(707\) −3.86174 −0.145236
\(708\) 0 0
\(709\) −42.4924 −1.59584 −0.797918 0.602766i \(-0.794065\pi\)
−0.797918 + 0.602766i \(0.794065\pi\)
\(710\) 12.0000 0.450352
\(711\) 0 0
\(712\) 0.876894 0.0328630
\(713\) −2.87689 −0.107741
\(714\) 0 0
\(715\) 1.12311 0.0420018
\(716\) −5.75379 −0.215029
\(717\) 0 0
\(718\) 15.9309 0.594535
\(719\) 10.5616 0.393879 0.196940 0.980416i \(-0.436900\pi\)
0.196940 + 0.980416i \(0.436900\pi\)
\(720\) 0 0
\(721\) 7.50758 0.279597
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) −10.8769 −0.404237
\(725\) −5.68466 −0.211123
\(726\) 0 0
\(727\) 22.4233 0.831634 0.415817 0.909448i \(-0.363496\pi\)
0.415817 + 0.909448i \(0.363496\pi\)
\(728\) −0.315342 −0.0116873
\(729\) 0 0
\(730\) 19.3693 0.716891
\(731\) 0 0
\(732\) 0 0
\(733\) 26.1080 0.964319 0.482160 0.876083i \(-0.339853\pi\)
0.482160 + 0.876083i \(0.339853\pi\)
\(734\) 30.2462 1.11641
\(735\) 0 0
\(736\) −1.43845 −0.0530219
\(737\) 7.68466 0.283068
\(738\) 0 0
\(739\) −44.9848 −1.65479 −0.827397 0.561617i \(-0.810179\pi\)
−0.827397 + 0.561617i \(0.810179\pi\)
\(740\) 10.2462 0.376658
\(741\) 0 0
\(742\) 7.19224 0.264035
\(743\) 42.2462 1.54986 0.774932 0.632045i \(-0.217784\pi\)
0.774932 + 0.632045i \(0.217784\pi\)
\(744\) 0 0
\(745\) −13.7538 −0.503900
\(746\) 3.43845 0.125890
\(747\) 0 0
\(748\) −0.561553 −0.0205324
\(749\) −5.57671 −0.203768
\(750\) 0 0
\(751\) 41.2311 1.50454 0.752271 0.658853i \(-0.228958\pi\)
0.752271 + 0.658853i \(0.228958\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) 3.19224 0.116254
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) −42.4924 −1.54441 −0.772207 0.635371i \(-0.780847\pi\)
−0.772207 + 0.635371i \(0.780847\pi\)
\(758\) 6.56155 0.238326
\(759\) 0 0
\(760\) 2.00000 0.0725476
\(761\) 42.8078 1.55178 0.775890 0.630868i \(-0.217301\pi\)
0.775890 + 0.630868i \(0.217301\pi\)
\(762\) 0 0
\(763\) 3.54640 0.128388
\(764\) 8.31534 0.300838
\(765\) 0 0
\(766\) 2.00000 0.0722629
\(767\) 4.31534 0.155818
\(768\) 0 0
\(769\) −13.6847 −0.493481 −0.246741 0.969082i \(-0.579360\pi\)
−0.246741 + 0.969082i \(0.579360\pi\)
\(770\) −1.12311 −0.0404739
\(771\) 0 0
\(772\) 7.12311 0.256366
\(773\) −37.9309 −1.36428 −0.682139 0.731222i \(-0.738950\pi\)
−0.682139 + 0.731222i \(0.738950\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) −7.12311 −0.255705
\(777\) 0 0
\(778\) −18.0000 −0.645331
\(779\) 2.00000 0.0716574
\(780\) 0 0
\(781\) 6.00000 0.214697
\(782\) −0.807764 −0.0288856
\(783\) 0 0
\(784\) −6.68466 −0.238738
\(785\) 28.9848 1.03451
\(786\) 0 0
\(787\) 15.6847 0.559098 0.279549 0.960131i \(-0.409815\pi\)
0.279549 + 0.960131i \(0.409815\pi\)
\(788\) −18.2462 −0.649994
\(789\) 0 0
\(790\) 8.00000 0.284627
\(791\) −1.75379 −0.0623575
\(792\) 0 0
\(793\) 3.50758 0.124558
\(794\) 38.9848 1.38352
\(795\) 0 0
\(796\) 11.6847 0.414152
\(797\) 15.1922 0.538137 0.269068 0.963121i \(-0.413284\pi\)
0.269068 + 0.963121i \(0.413284\pi\)
\(798\) 0 0
\(799\) −4.49242 −0.158930
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 7.75379 0.273796
\(803\) 9.68466 0.341764
\(804\) 0 0
\(805\) −1.61553 −0.0569399
\(806\) 1.12311 0.0395597
\(807\) 0 0
\(808\) 6.87689 0.241928
\(809\) −8.06913 −0.283696 −0.141848 0.989888i \(-0.545304\pi\)
−0.141848 + 0.989888i \(0.545304\pi\)
\(810\) 0 0
\(811\) 8.31534 0.291991 0.145996 0.989285i \(-0.453361\pi\)
0.145996 + 0.989285i \(0.453361\pi\)
\(812\) −3.19224 −0.112026
\(813\) 0 0
\(814\) 5.12311 0.179565
\(815\) −22.7386 −0.796500
\(816\) 0 0
\(817\) 0 0
\(818\) 16.2462 0.568035
\(819\) 0 0
\(820\) 4.00000 0.139686
\(821\) 20.9848 0.732376 0.366188 0.930541i \(-0.380663\pi\)
0.366188 + 0.930541i \(0.380663\pi\)
\(822\) 0 0
\(823\) −13.9309 −0.485600 −0.242800 0.970076i \(-0.578066\pi\)
−0.242800 + 0.970076i \(0.578066\pi\)
\(824\) −13.3693 −0.465742
\(825\) 0 0
\(826\) −4.31534 −0.150150
\(827\) −49.9309 −1.73627 −0.868133 0.496331i \(-0.834680\pi\)
−0.868133 + 0.496331i \(0.834680\pi\)
\(828\) 0 0
\(829\) 12.1771 0.422928 0.211464 0.977386i \(-0.432177\pi\)
0.211464 + 0.977386i \(0.432177\pi\)
\(830\) 28.4924 0.988986
\(831\) 0 0
\(832\) 0.561553 0.0194683
\(833\) −3.75379 −0.130061
\(834\) 0 0
\(835\) −1.26137 −0.0436514
\(836\) 1.00000 0.0345857
\(837\) 0 0
\(838\) 14.8769 0.513914
\(839\) 7.12311 0.245917 0.122958 0.992412i \(-0.460762\pi\)
0.122958 + 0.992412i \(0.460762\pi\)
\(840\) 0 0
\(841\) 3.31534 0.114322
\(842\) −29.9309 −1.03149
\(843\) 0 0
\(844\) −0.315342 −0.0108545
\(845\) 25.3693 0.872731
\(846\) 0 0
\(847\) −0.561553 −0.0192952
\(848\) −12.8078 −0.439820
\(849\) 0 0
\(850\) −0.561553 −0.0192611
\(851\) 7.36932 0.252617
\(852\) 0 0
\(853\) −52.3542 −1.79257 −0.896286 0.443476i \(-0.853745\pi\)
−0.896286 + 0.443476i \(0.853745\pi\)
\(854\) −3.50758 −0.120027
\(855\) 0 0
\(856\) 9.93087 0.339430
\(857\) 10.0000 0.341593 0.170797 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) 0 0
\(859\) 15.8617 0.541196 0.270598 0.962692i \(-0.412779\pi\)
0.270598 + 0.962692i \(0.412779\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −31.8617 −1.08522
\(863\) −0.246211 −0.00838113 −0.00419056 0.999991i \(-0.501334\pi\)
−0.00419056 + 0.999991i \(0.501334\pi\)
\(864\) 0 0
\(865\) 36.0000 1.22404
\(866\) 6.63068 0.225320
\(867\) 0 0
\(868\) −1.12311 −0.0381207
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) −4.31534 −0.146220
\(872\) −6.31534 −0.213864
\(873\) 0 0
\(874\) 1.43845 0.0486562
\(875\) −6.73863 −0.227807
\(876\) 0 0
\(877\) −44.5616 −1.50474 −0.752368 0.658743i \(-0.771089\pi\)
−0.752368 + 0.658743i \(0.771089\pi\)
\(878\) −9.61553 −0.324508
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) −20.7386 −0.698702 −0.349351 0.936992i \(-0.613598\pi\)
−0.349351 + 0.936992i \(0.613598\pi\)
\(882\) 0 0
\(883\) −5.26137 −0.177059 −0.0885295 0.996074i \(-0.528217\pi\)
−0.0885295 + 0.996074i \(0.528217\pi\)
\(884\) 0.315342 0.0106061
\(885\) 0 0
\(886\) −39.8617 −1.33918
\(887\) 32.0000 1.07445 0.537227 0.843437i \(-0.319472\pi\)
0.537227 + 0.843437i \(0.319472\pi\)
\(888\) 0 0
\(889\) −1.26137 −0.0423049
\(890\) −1.75379 −0.0587871
\(891\) 0 0
\(892\) 24.7386 0.828311
\(893\) 8.00000 0.267710
\(894\) 0 0
\(895\) 11.5076 0.384656
\(896\) −0.561553 −0.0187602
\(897\) 0 0
\(898\) −23.6155 −0.788060
\(899\) 11.3693 0.379188
\(900\) 0 0
\(901\) −7.19224 −0.239608
\(902\) 2.00000 0.0665927
\(903\) 0 0
\(904\) 3.12311 0.103873
\(905\) 21.7538 0.723120
\(906\) 0 0
\(907\) −29.7926 −0.989247 −0.494624 0.869107i \(-0.664694\pi\)
−0.494624 + 0.869107i \(0.664694\pi\)
\(908\) −28.1771 −0.935089
\(909\) 0 0
\(910\) 0.630683 0.0209069
\(911\) −18.9848 −0.628996 −0.314498 0.949258i \(-0.601836\pi\)
−0.314498 + 0.949258i \(0.601836\pi\)
\(912\) 0 0
\(913\) 14.2462 0.471481
\(914\) 36.5616 1.20935
\(915\) 0 0
\(916\) 12.8769 0.425465
\(917\) 0.630683 0.0208270
\(918\) 0 0
\(919\) 32.5616 1.07411 0.537053 0.843548i \(-0.319537\pi\)
0.537053 + 0.843548i \(0.319537\pi\)
\(920\) 2.87689 0.0948484
\(921\) 0 0
\(922\) −26.7386 −0.880590
\(923\) −3.36932 −0.110902
\(924\) 0 0
\(925\) 5.12311 0.168447
\(926\) 3.50758 0.115266
\(927\) 0 0
\(928\) 5.68466 0.186608
\(929\) −11.9309 −0.391439 −0.195720 0.980660i \(-0.562704\pi\)
−0.195720 + 0.980660i \(0.562704\pi\)
\(930\) 0 0
\(931\) 6.68466 0.219081
\(932\) 28.7386 0.941365
\(933\) 0 0
\(934\) −9.75379 −0.319154
\(935\) 1.12311 0.0367295
\(936\) 0 0
\(937\) −0.699813 −0.0228619 −0.0114310 0.999935i \(-0.503639\pi\)
−0.0114310 + 0.999935i \(0.503639\pi\)
\(938\) 4.31534 0.140901
\(939\) 0 0
\(940\) 16.0000 0.521862
\(941\) −24.5616 −0.800684 −0.400342 0.916366i \(-0.631109\pi\)
−0.400342 + 0.916366i \(0.631109\pi\)
\(942\) 0 0
\(943\) 2.87689 0.0936846
\(944\) 7.68466 0.250114
\(945\) 0 0
\(946\) 0 0
\(947\) 36.9848 1.20185 0.600923 0.799307i \(-0.294800\pi\)
0.600923 + 0.799307i \(0.294800\pi\)
\(948\) 0 0
\(949\) −5.43845 −0.176539
\(950\) 1.00000 0.0324443
\(951\) 0 0
\(952\) −0.315342 −0.0102203
\(953\) 0.876894 0.0284054 0.0142027 0.999899i \(-0.495479\pi\)
0.0142027 + 0.999899i \(0.495479\pi\)
\(954\) 0 0
\(955\) −16.6307 −0.538156
\(956\) −23.9309 −0.773980
\(957\) 0 0
\(958\) 13.3693 0.431943
\(959\) 7.05398 0.227785
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −2.87689 −0.0927548
\(963\) 0 0
\(964\) −19.6155 −0.631774
\(965\) −14.2462 −0.458602
\(966\) 0 0
\(967\) 53.8617 1.73208 0.866038 0.499978i \(-0.166658\pi\)
0.866038 + 0.499978i \(0.166658\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 14.2462 0.457418
\(971\) −2.24621 −0.0720843 −0.0360422 0.999350i \(-0.511475\pi\)
−0.0360422 + 0.999350i \(0.511475\pi\)
\(972\) 0 0
\(973\) −4.13826 −0.132667
\(974\) 31.1231 0.997249
\(975\) 0 0
\(976\) 6.24621 0.199936
\(977\) −30.6307 −0.979962 −0.489981 0.871733i \(-0.662996\pi\)
−0.489981 + 0.871733i \(0.662996\pi\)
\(978\) 0 0
\(979\) −0.876894 −0.0280257
\(980\) 13.3693 0.427067
\(981\) 0 0
\(982\) 39.3693 1.25633
\(983\) 31.7538 1.01279 0.506394 0.862302i \(-0.330978\pi\)
0.506394 + 0.862302i \(0.330978\pi\)
\(984\) 0 0
\(985\) 36.4924 1.16275
\(986\) 3.19224 0.101662
\(987\) 0 0
\(988\) −0.561553 −0.0178654
\(989\) 0 0
\(990\) 0 0
\(991\) 12.8769 0.409048 0.204524 0.978862i \(-0.434435\pi\)
0.204524 + 0.978862i \(0.434435\pi\)
\(992\) 2.00000 0.0635001
\(993\) 0 0
\(994\) 3.36932 0.106868
\(995\) −23.3693 −0.740857
\(996\) 0 0
\(997\) −28.6307 −0.906743 −0.453371 0.891322i \(-0.649779\pi\)
−0.453371 + 0.891322i \(0.649779\pi\)
\(998\) 9.75379 0.308751
\(999\) 0 0
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 3762.2.a.y.1.1 2
3.2 odd 2 418.2.a.e.1.2 2
12.11 even 2 3344.2.a.k.1.1 2
33.32 even 2 4598.2.a.bj.1.2 2
57.56 even 2 7942.2.a.x.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.e.1.2 2 3.2 odd 2
3344.2.a.k.1.1 2 12.11 even 2
3762.2.a.y.1.1 2 1.1 even 1 trivial
4598.2.a.bj.1.2 2 33.32 even 2
7942.2.a.x.1.1 2 57.56 even 2