Properties

Label 3762.2.a.bg.1.1
Level $3762$
Weight $2$
Character 3762.1
Self dual yes
Analytic conductor $30.040$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 3 \) 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 \(-0.523976\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.72545 q^{5} -4.67750 q^{7} +1.00000 q^{8} -2.72545 q^{10} +1.00000 q^{11} -2.67750 q^{13} -4.67750 q^{14} +1.00000 q^{16} -0.201472 q^{17} -1.00000 q^{19} -2.72545 q^{20} +1.00000 q^{22} +1.79853 q^{23} +2.42807 q^{25} -2.67750 q^{26} -4.67750 q^{28} +4.92692 q^{29} -2.57193 q^{31} +1.00000 q^{32} -0.201472 q^{34} +12.7483 q^{35} +4.40294 q^{37} -1.00000 q^{38} -2.72545 q^{40} -4.97487 q^{41} -11.7734 q^{43} +1.00000 q^{44} +1.79853 q^{46} +12.4989 q^{47} +14.8790 q^{49} +2.42807 q^{50} -2.67750 q^{52} +4.10557 q^{53} -2.72545 q^{55} -4.67750 q^{56} +4.92692 q^{58} +5.24943 q^{59} +1.59706 q^{61} -2.57193 q^{62} +1.00000 q^{64} +7.29738 q^{65} +9.08044 q^{67} -0.201472 q^{68} +12.7483 q^{70} -12.8214 q^{71} +2.20147 q^{73} +4.40294 q^{74} -1.00000 q^{76} -4.67750 q^{77} +11.8538 q^{79} -2.72545 q^{80} -4.97487 q^{82} +13.6775 q^{83} +0.549103 q^{85} -11.7734 q^{86} +1.00000 q^{88} +5.45090 q^{89} +12.5240 q^{91} +1.79853 q^{92} +12.4989 q^{94} +2.72545 q^{95} +16.8059 q^{97} +14.8790 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{7} + 3 q^{8} + 3 q^{10} + 3 q^{11} - 6 q^{14} + 3 q^{16} + 9 q^{17} - 3 q^{19} + 3 q^{20} + 3 q^{22} + 15 q^{23} + 12 q^{25} - 6 q^{28} - 6 q^{29} - 3 q^{31} + 3 q^{32}+ \cdots + 27 q^{98}+O(q^{100}) \) 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
\(5\) −2.72545 −1.21886 −0.609429 0.792841i \(-0.708601\pi\)
−0.609429 + 0.792841i \(0.708601\pi\)
\(6\) 0 0
\(7\) −4.67750 −1.76793 −0.883964 0.467556i \(-0.845135\pi\)
−0.883964 + 0.467556i \(0.845135\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −2.72545 −0.861863
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −2.67750 −0.742604 −0.371302 0.928512i \(-0.621089\pi\)
−0.371302 + 0.928512i \(0.621089\pi\)
\(14\) −4.67750 −1.25011
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.201472 −0.0488642 −0.0244321 0.999701i \(-0.507778\pi\)
−0.0244321 + 0.999701i \(0.507778\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) −2.72545 −0.609429
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 1.79853 0.375019 0.187509 0.982263i \(-0.439958\pi\)
0.187509 + 0.982263i \(0.439958\pi\)
\(24\) 0 0
\(25\) 2.42807 0.485614
\(26\) −2.67750 −0.525100
\(27\) 0 0
\(28\) −4.67750 −0.883964
\(29\) 4.92692 0.914906 0.457453 0.889234i \(-0.348762\pi\)
0.457453 + 0.889234i \(0.348762\pi\)
\(30\) 0 0
\(31\) −2.57193 −0.461932 −0.230966 0.972962i \(-0.574189\pi\)
−0.230966 + 0.972962i \(0.574189\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −0.201472 −0.0345522
\(35\) 12.7483 2.15485
\(36\) 0 0
\(37\) 4.40294 0.723840 0.361920 0.932209i \(-0.382121\pi\)
0.361920 + 0.932209i \(0.382121\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) −2.72545 −0.430931
\(41\) −4.97487 −0.776945 −0.388472 0.921460i \(-0.626997\pi\)
−0.388472 + 0.921460i \(0.626997\pi\)
\(42\) 0 0
\(43\) −11.7734 −1.79543 −0.897713 0.440580i \(-0.854773\pi\)
−0.897713 + 0.440580i \(0.854773\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 1.79853 0.265178
\(47\) 12.4989 1.82314 0.911572 0.411140i \(-0.134869\pi\)
0.911572 + 0.411140i \(0.134869\pi\)
\(48\) 0 0
\(49\) 14.8790 2.12557
\(50\) 2.42807 0.343381
\(51\) 0 0
\(52\) −2.67750 −0.371302
\(53\) 4.10557 0.563943 0.281971 0.959423i \(-0.409012\pi\)
0.281971 + 0.959423i \(0.409012\pi\)
\(54\) 0 0
\(55\) −2.72545 −0.367499
\(56\) −4.67750 −0.625057
\(57\) 0 0
\(58\) 4.92692 0.646936
\(59\) 5.24943 0.683417 0.341708 0.939806i \(-0.388994\pi\)
0.341708 + 0.939806i \(0.388994\pi\)
\(60\) 0 0
\(61\) 1.59706 0.204482 0.102241 0.994760i \(-0.467399\pi\)
0.102241 + 0.994760i \(0.467399\pi\)
\(62\) −2.57193 −0.326635
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 7.29738 0.905128
\(66\) 0 0
\(67\) 9.08044 1.10935 0.554676 0.832066i \(-0.312842\pi\)
0.554676 + 0.832066i \(0.312842\pi\)
\(68\) −0.201472 −0.0244321
\(69\) 0 0
\(70\) 12.7483 1.52371
\(71\) −12.8214 −1.52161 −0.760807 0.648978i \(-0.775197\pi\)
−0.760807 + 0.648978i \(0.775197\pi\)
\(72\) 0 0
\(73\) 2.20147 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(74\) 4.40294 0.511832
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −4.67750 −0.533050
\(78\) 0 0
\(79\) 11.8538 1.33366 0.666831 0.745209i \(-0.267650\pi\)
0.666831 + 0.745209i \(0.267650\pi\)
\(80\) −2.72545 −0.304714
\(81\) 0 0
\(82\) −4.97487 −0.549383
\(83\) 13.6775 1.50130 0.750650 0.660700i \(-0.229740\pi\)
0.750650 + 0.660700i \(0.229740\pi\)
\(84\) 0 0
\(85\) 0.549103 0.0595585
\(86\) −11.7734 −1.26956
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 5.45090 0.577794 0.288897 0.957360i \(-0.406711\pi\)
0.288897 + 0.957360i \(0.406711\pi\)
\(90\) 0 0
\(91\) 12.5240 1.31287
\(92\) 1.79853 0.187509
\(93\) 0 0
\(94\) 12.4989 1.28916
\(95\) 2.72545 0.279625
\(96\) 0 0
\(97\) 16.8059 1.70638 0.853190 0.521601i \(-0.174665\pi\)
0.853190 + 0.521601i \(0.174665\pi\)
\(98\) 14.8790 1.50300
\(99\) 0 0
\(100\) 2.42807 0.242807
\(101\) −7.45090 −0.741392 −0.370696 0.928754i \(-0.620881\pi\)
−0.370696 + 0.928754i \(0.620881\pi\)
\(102\) 0 0
\(103\) −14.4834 −1.42709 −0.713545 0.700609i \(-0.752912\pi\)
−0.713545 + 0.700609i \(0.752912\pi\)
\(104\) −2.67750 −0.262550
\(105\) 0 0
\(106\) 4.10557 0.398768
\(107\) −1.15352 −0.111515 −0.0557575 0.998444i \(-0.517757\pi\)
−0.0557575 + 0.998444i \(0.517757\pi\)
\(108\) 0 0
\(109\) −9.55646 −0.915343 −0.457672 0.889121i \(-0.651316\pi\)
−0.457672 + 0.889121i \(0.651316\pi\)
\(110\) −2.72545 −0.259861
\(111\) 0 0
\(112\) −4.67750 −0.441982
\(113\) 8.00000 0.752577 0.376288 0.926503i \(-0.377200\pi\)
0.376288 + 0.926503i \(0.377200\pi\)
\(114\) 0 0
\(115\) −4.90179 −0.457095
\(116\) 4.92692 0.457453
\(117\) 0 0
\(118\) 5.24943 0.483249
\(119\) 0.942386 0.0863884
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 1.59706 0.144591
\(123\) 0 0
\(124\) −2.57193 −0.230966
\(125\) 7.00966 0.626963
\(126\) 0 0
\(127\) 5.75794 0.510934 0.255467 0.966818i \(-0.417771\pi\)
0.255467 + 0.966818i \(0.417771\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 7.29738 0.640022
\(131\) −1.42807 −0.124771 −0.0623856 0.998052i \(-0.519871\pi\)
−0.0623856 + 0.998052i \(0.519871\pi\)
\(132\) 0 0
\(133\) 4.67750 0.405590
\(134\) 9.08044 0.784431
\(135\) 0 0
\(136\) −0.201472 −0.0172761
\(137\) 14.5313 1.24150 0.620748 0.784010i \(-0.286829\pi\)
0.620748 + 0.784010i \(0.286829\pi\)
\(138\) 0 0
\(139\) 16.1763 1.37206 0.686030 0.727573i \(-0.259352\pi\)
0.686030 + 0.727573i \(0.259352\pi\)
\(140\) 12.7483 1.07743
\(141\) 0 0
\(142\) −12.8214 −1.07594
\(143\) −2.67750 −0.223903
\(144\) 0 0
\(145\) −13.4281 −1.11514
\(146\) 2.20147 0.182195
\(147\) 0 0
\(148\) 4.40294 0.361920
\(149\) −6.30704 −0.516693 −0.258346 0.966052i \(-0.583178\pi\)
−0.258346 + 0.966052i \(0.583178\pi\)
\(150\) 0 0
\(151\) 0.498850 0.0405959 0.0202979 0.999794i \(-0.493539\pi\)
0.0202979 + 0.999794i \(0.493539\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) −4.67750 −0.376923
\(155\) 7.00966 0.563030
\(156\) 0 0
\(157\) −13.6272 −1.08757 −0.543786 0.839224i \(-0.683010\pi\)
−0.543786 + 0.839224i \(0.683010\pi\)
\(158\) 11.8538 0.943041
\(159\) 0 0
\(160\) −2.72545 −0.215466
\(161\) −8.41261 −0.663006
\(162\) 0 0
\(163\) 15.9497 1.24928 0.624640 0.780913i \(-0.285246\pi\)
0.624640 + 0.780913i \(0.285246\pi\)
\(164\) −4.97487 −0.388472
\(165\) 0 0
\(166\) 13.6775 1.06158
\(167\) −18.8059 −1.45524 −0.727622 0.685979i \(-0.759374\pi\)
−0.727622 + 0.685979i \(0.759374\pi\)
\(168\) 0 0
\(169\) −5.83102 −0.448540
\(170\) 0.549103 0.0421142
\(171\) 0 0
\(172\) −11.7734 −0.897713
\(173\) 3.22430 0.245139 0.122569 0.992460i \(-0.460887\pi\)
0.122569 + 0.992460i \(0.460887\pi\)
\(174\) 0 0
\(175\) −11.3573 −0.858531
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 5.45090 0.408562
\(179\) −13.3778 −0.999905 −0.499953 0.866053i \(-0.666649\pi\)
−0.499953 + 0.866053i \(0.666649\pi\)
\(180\) 0 0
\(181\) 15.9497 1.18554 0.592768 0.805373i \(-0.298035\pi\)
0.592768 + 0.805373i \(0.298035\pi\)
\(182\) 12.5240 0.928339
\(183\) 0 0
\(184\) 1.79853 0.132589
\(185\) −12.0000 −0.882258
\(186\) 0 0
\(187\) −0.201472 −0.0147331
\(188\) 12.4989 0.911572
\(189\) 0 0
\(190\) 2.72545 0.197725
\(191\) 24.0553 1.74058 0.870291 0.492538i \(-0.163931\pi\)
0.870291 + 0.492538i \(0.163931\pi\)
\(192\) 0 0
\(193\) 13.0325 0.938099 0.469050 0.883172i \(-0.344597\pi\)
0.469050 + 0.883172i \(0.344597\pi\)
\(194\) 16.8059 1.20659
\(195\) 0 0
\(196\) 14.8790 1.06278
\(197\) 5.59706 0.398774 0.199387 0.979921i \(-0.436105\pi\)
0.199387 + 0.979921i \(0.436105\pi\)
\(198\) 0 0
\(199\) −7.39558 −0.524259 −0.262129 0.965033i \(-0.584425\pi\)
−0.262129 + 0.965033i \(0.584425\pi\)
\(200\) 2.42807 0.171691
\(201\) 0 0
\(202\) −7.45090 −0.524243
\(203\) −23.0457 −1.61749
\(204\) 0 0
\(205\) 13.5588 0.946985
\(206\) −14.4834 −1.00911
\(207\) 0 0
\(208\) −2.67750 −0.185651
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −14.2974 −0.984272 −0.492136 0.870518i \(-0.663784\pi\)
−0.492136 + 0.870518i \(0.663784\pi\)
\(212\) 4.10557 0.281971
\(213\) 0 0
\(214\) −1.15352 −0.0788530
\(215\) 32.0878 2.18837
\(216\) 0 0
\(217\) 12.0302 0.816662
\(218\) −9.55646 −0.647245
\(219\) 0 0
\(220\) −2.72545 −0.183750
\(221\) 0.539441 0.0362868
\(222\) 0 0
\(223\) −0.402945 −0.0269832 −0.0134916 0.999909i \(-0.504295\pi\)
−0.0134916 + 0.999909i \(0.504295\pi\)
\(224\) −4.67750 −0.312528
\(225\) 0 0
\(226\) 8.00000 0.532152
\(227\) 21.0074 1.39431 0.697154 0.716922i \(-0.254449\pi\)
0.697154 + 0.716922i \(0.254449\pi\)
\(228\) 0 0
\(229\) −14.3299 −0.946944 −0.473472 0.880809i \(-0.657000\pi\)
−0.473472 + 0.880809i \(0.657000\pi\)
\(230\) −4.90179 −0.323215
\(231\) 0 0
\(232\) 4.92692 0.323468
\(233\) 24.4029 1.59869 0.799345 0.600872i \(-0.205180\pi\)
0.799345 + 0.600872i \(0.205180\pi\)
\(234\) 0 0
\(235\) −34.0650 −2.22215
\(236\) 5.24943 0.341708
\(237\) 0 0
\(238\) 0.942386 0.0610858
\(239\) 12.6849 0.820515 0.410258 0.911970i \(-0.365439\pi\)
0.410258 + 0.911970i \(0.365439\pi\)
\(240\) 0 0
\(241\) −23.9343 −1.54174 −0.770871 0.636991i \(-0.780179\pi\)
−0.770871 + 0.636991i \(0.780179\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 1.59706 0.102241
\(245\) −40.5519 −2.59076
\(246\) 0 0
\(247\) 2.67750 0.170365
\(248\) −2.57193 −0.163318
\(249\) 0 0
\(250\) 7.00966 0.443330
\(251\) −18.4989 −1.16764 −0.583819 0.811884i \(-0.698442\pi\)
−0.583819 + 0.811884i \(0.698442\pi\)
\(252\) 0 0
\(253\) 1.79853 0.113072
\(254\) 5.75794 0.361285
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −27.7579 −1.73149 −0.865746 0.500483i \(-0.833156\pi\)
−0.865746 + 0.500483i \(0.833156\pi\)
\(258\) 0 0
\(259\) −20.5948 −1.27970
\(260\) 7.29738 0.452564
\(261\) 0 0
\(262\) −1.42807 −0.0882265
\(263\) −20.2723 −1.25004 −0.625020 0.780608i \(-0.714909\pi\)
−0.625020 + 0.780608i \(0.714909\pi\)
\(264\) 0 0
\(265\) −11.1895 −0.687366
\(266\) 4.67750 0.286796
\(267\) 0 0
\(268\) 9.08044 0.554676
\(269\) 15.6427 0.953753 0.476876 0.878970i \(-0.341769\pi\)
0.476876 + 0.878970i \(0.341769\pi\)
\(270\) 0 0
\(271\) −12.3395 −0.749573 −0.374786 0.927111i \(-0.622284\pi\)
−0.374786 + 0.927111i \(0.622284\pi\)
\(272\) −0.201472 −0.0122161
\(273\) 0 0
\(274\) 14.5313 0.877870
\(275\) 2.42807 0.146418
\(276\) 0 0
\(277\) 8.49885 0.510646 0.255323 0.966856i \(-0.417818\pi\)
0.255323 + 0.966856i \(0.417818\pi\)
\(278\) 16.1763 0.970193
\(279\) 0 0
\(280\) 12.7483 0.761855
\(281\) 7.12839 0.425244 0.212622 0.977134i \(-0.431800\pi\)
0.212622 + 0.977134i \(0.431800\pi\)
\(282\) 0 0
\(283\) 11.5240 0.685029 0.342515 0.939512i \(-0.388721\pi\)
0.342515 + 0.939512i \(0.388721\pi\)
\(284\) −12.8214 −0.760807
\(285\) 0 0
\(286\) −2.67750 −0.158324
\(287\) 23.2700 1.37358
\(288\) 0 0
\(289\) −16.9594 −0.997612
\(290\) −13.4281 −0.788523
\(291\) 0 0
\(292\) 2.20147 0.128831
\(293\) −10.6775 −0.623786 −0.311893 0.950117i \(-0.600963\pi\)
−0.311893 + 0.950117i \(0.600963\pi\)
\(294\) 0 0
\(295\) −14.3070 −0.832988
\(296\) 4.40294 0.255916
\(297\) 0 0
\(298\) −6.30704 −0.365357
\(299\) −4.81555 −0.278490
\(300\) 0 0
\(301\) 55.0700 3.17418
\(302\) 0.498850 0.0287056
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) −4.35269 −0.249234
\(306\) 0 0
\(307\) −30.2568 −1.72685 −0.863423 0.504481i \(-0.831684\pi\)
−0.863423 + 0.504481i \(0.831684\pi\)
\(308\) −4.67750 −0.266525
\(309\) 0 0
\(310\) 7.00966 0.398122
\(311\) −8.79623 −0.498788 −0.249394 0.968402i \(-0.580231\pi\)
−0.249394 + 0.968402i \(0.580231\pi\)
\(312\) 0 0
\(313\) 10.1860 0.575747 0.287874 0.957668i \(-0.407052\pi\)
0.287874 + 0.957668i \(0.407052\pi\)
\(314\) −13.6272 −0.769030
\(315\) 0 0
\(316\) 11.8538 0.666831
\(317\) 12.0097 0.674530 0.337265 0.941410i \(-0.390498\pi\)
0.337265 + 0.941410i \(0.390498\pi\)
\(318\) 0 0
\(319\) 4.92692 0.275855
\(320\) −2.72545 −0.152357
\(321\) 0 0
\(322\) −8.41261 −0.468816
\(323\) 0.201472 0.0112102
\(324\) 0 0
\(325\) −6.50115 −0.360619
\(326\) 15.9497 0.883375
\(327\) 0 0
\(328\) −4.97487 −0.274691
\(329\) −58.4633 −3.22319
\(330\) 0 0
\(331\) 26.1667 1.43825 0.719126 0.694880i \(-0.244543\pi\)
0.719126 + 0.694880i \(0.244543\pi\)
\(332\) 13.6775 0.750650
\(333\) 0 0
\(334\) −18.8059 −1.02901
\(335\) −24.7483 −1.35214
\(336\) 0 0
\(337\) 17.3778 0.946630 0.473315 0.880893i \(-0.343057\pi\)
0.473315 + 0.880893i \(0.343057\pi\)
\(338\) −5.83102 −0.317165
\(339\) 0 0
\(340\) 0.549103 0.0297793
\(341\) −2.57193 −0.139278
\(342\) 0 0
\(343\) −36.8538 −1.98992
\(344\) −11.7734 −0.634779
\(345\) 0 0
\(346\) 3.22430 0.173339
\(347\) 12.4029 0.665825 0.332912 0.942958i \(-0.391969\pi\)
0.332912 + 0.942958i \(0.391969\pi\)
\(348\) 0 0
\(349\) −23.8036 −1.27418 −0.637088 0.770791i \(-0.719861\pi\)
−0.637088 + 0.770791i \(0.719861\pi\)
\(350\) −11.3573 −0.607073
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 24.2664 1.29157 0.645786 0.763518i \(-0.276530\pi\)
0.645786 + 0.763518i \(0.276530\pi\)
\(354\) 0 0
\(355\) 34.9439 1.85463
\(356\) 5.45090 0.288897
\(357\) 0 0
\(358\) −13.3778 −0.707040
\(359\) −20.3395 −1.07348 −0.536740 0.843748i \(-0.680344\pi\)
−0.536740 + 0.843748i \(0.680344\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 15.9497 0.838300
\(363\) 0 0
\(364\) 12.5240 0.656435
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) −23.0170 −1.20148 −0.600739 0.799445i \(-0.705127\pi\)
−0.600739 + 0.799445i \(0.705127\pi\)
\(368\) 1.79853 0.0937547
\(369\) 0 0
\(370\) −12.0000 −0.623850
\(371\) −19.2038 −0.997010
\(372\) 0 0
\(373\) −18.5890 −0.962499 −0.481250 0.876584i \(-0.659817\pi\)
−0.481250 + 0.876584i \(0.659817\pi\)
\(374\) −0.201472 −0.0104179
\(375\) 0 0
\(376\) 12.4989 0.644579
\(377\) −13.1918 −0.679413
\(378\) 0 0
\(379\) −10.5816 −0.543540 −0.271770 0.962362i \(-0.587609\pi\)
−0.271770 + 0.962362i \(0.587609\pi\)
\(380\) 2.72545 0.139813
\(381\) 0 0
\(382\) 24.0553 1.23078
\(383\) 26.0878 1.33302 0.666512 0.745494i \(-0.267786\pi\)
0.666512 + 0.745494i \(0.267786\pi\)
\(384\) 0 0
\(385\) 12.7483 0.649712
\(386\) 13.0325 0.663336
\(387\) 0 0
\(388\) 16.8059 0.853190
\(389\) 9.46636 0.479964 0.239982 0.970777i \(-0.422859\pi\)
0.239982 + 0.970777i \(0.422859\pi\)
\(390\) 0 0
\(391\) −0.362354 −0.0183250
\(392\) 14.8790 0.751501
\(393\) 0 0
\(394\) 5.59706 0.281976
\(395\) −32.3070 −1.62554
\(396\) 0 0
\(397\) −25.1667 −1.26308 −0.631540 0.775343i \(-0.717577\pi\)
−0.631540 + 0.775343i \(0.717577\pi\)
\(398\) −7.39558 −0.370707
\(399\) 0 0
\(400\) 2.42807 0.121404
\(401\) 16.0650 0.802247 0.401123 0.916024i \(-0.368620\pi\)
0.401123 + 0.916024i \(0.368620\pi\)
\(402\) 0 0
\(403\) 6.88633 0.343033
\(404\) −7.45090 −0.370696
\(405\) 0 0
\(406\) −23.0457 −1.14374
\(407\) 4.40294 0.218246
\(408\) 0 0
\(409\) −19.5313 −0.965763 −0.482881 0.875686i \(-0.660410\pi\)
−0.482881 + 0.875686i \(0.660410\pi\)
\(410\) 13.5588 0.669620
\(411\) 0 0
\(412\) −14.4834 −0.713545
\(413\) −24.5542 −1.20823
\(414\) 0 0
\(415\) −37.2773 −1.82987
\(416\) −2.67750 −0.131275
\(417\) 0 0
\(418\) −1.00000 −0.0489116
\(419\) −24.9018 −1.21653 −0.608266 0.793733i \(-0.708135\pi\)
−0.608266 + 0.793733i \(0.708135\pi\)
\(420\) 0 0
\(421\) 26.6044 1.29662 0.648310 0.761377i \(-0.275476\pi\)
0.648310 + 0.761377i \(0.275476\pi\)
\(422\) −14.2974 −0.695985
\(423\) 0 0
\(424\) 4.10557 0.199384
\(425\) −0.489189 −0.0237292
\(426\) 0 0
\(427\) −7.47022 −0.361509
\(428\) −1.15352 −0.0557575
\(429\) 0 0
\(430\) 32.0878 1.54741
\(431\) 32.5948 1.57003 0.785017 0.619474i \(-0.212654\pi\)
0.785017 + 0.619474i \(0.212654\pi\)
\(432\) 0 0
\(433\) 19.7386 0.948577 0.474289 0.880369i \(-0.342705\pi\)
0.474289 + 0.880369i \(0.342705\pi\)
\(434\) 12.0302 0.577468
\(435\) 0 0
\(436\) −9.55646 −0.457672
\(437\) −1.79853 −0.0860352
\(438\) 0 0
\(439\) −24.4029 −1.16469 −0.582345 0.812942i \(-0.697865\pi\)
−0.582345 + 0.812942i \(0.697865\pi\)
\(440\) −2.72545 −0.129931
\(441\) 0 0
\(442\) 0.539441 0.0256586
\(443\) 16.5948 0.788441 0.394220 0.919016i \(-0.371015\pi\)
0.394220 + 0.919016i \(0.371015\pi\)
\(444\) 0 0
\(445\) −14.8561 −0.704249
\(446\) −0.402945 −0.0190800
\(447\) 0 0
\(448\) −4.67750 −0.220991
\(449\) 20.9520 0.988788 0.494394 0.869238i \(-0.335390\pi\)
0.494394 + 0.869238i \(0.335390\pi\)
\(450\) 0 0
\(451\) −4.97487 −0.234258
\(452\) 8.00000 0.376288
\(453\) 0 0
\(454\) 21.0074 0.985924
\(455\) −34.1335 −1.60020
\(456\) 0 0
\(457\) 23.3144 1.09060 0.545301 0.838240i \(-0.316415\pi\)
0.545301 + 0.838240i \(0.316415\pi\)
\(458\) −14.3299 −0.669591
\(459\) 0 0
\(460\) −4.90179 −0.228547
\(461\) −20.8059 −0.969027 −0.484513 0.874784i \(-0.661003\pi\)
−0.484513 + 0.874784i \(0.661003\pi\)
\(462\) 0 0
\(463\) −9.29002 −0.431744 −0.215872 0.976422i \(-0.569259\pi\)
−0.215872 + 0.976422i \(0.569259\pi\)
\(464\) 4.92692 0.228727
\(465\) 0 0
\(466\) 24.4029 1.13044
\(467\) −12.0457 −0.557406 −0.278703 0.960377i \(-0.589905\pi\)
−0.278703 + 0.960377i \(0.589905\pi\)
\(468\) 0 0
\(469\) −42.4737 −1.96125
\(470\) −34.0650 −1.57130
\(471\) 0 0
\(472\) 5.24943 0.241624
\(473\) −11.7734 −0.541342
\(474\) 0 0
\(475\) −2.42807 −0.111408
\(476\) 0.942386 0.0431942
\(477\) 0 0
\(478\) 12.6849 0.580192
\(479\) −0.188307 −0.00860396 −0.00430198 0.999991i \(-0.501369\pi\)
−0.00430198 + 0.999991i \(0.501369\pi\)
\(480\) 0 0
\(481\) −11.7889 −0.537526
\(482\) −23.9343 −1.09018
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −45.8036 −2.07983
\(486\) 0 0
\(487\) −33.3852 −1.51283 −0.756413 0.654094i \(-0.773050\pi\)
−0.756413 + 0.654094i \(0.773050\pi\)
\(488\) 1.59706 0.0722953
\(489\) 0 0
\(490\) −40.5519 −1.83195
\(491\) −12.5217 −0.565095 −0.282548 0.959253i \(-0.591180\pi\)
−0.282548 + 0.959253i \(0.591180\pi\)
\(492\) 0 0
\(493\) −0.992638 −0.0447062
\(494\) 2.67750 0.120466
\(495\) 0 0
\(496\) −2.57193 −0.115483
\(497\) 59.9718 2.69010
\(498\) 0 0
\(499\) 16.3070 0.730003 0.365002 0.931007i \(-0.381068\pi\)
0.365002 + 0.931007i \(0.381068\pi\)
\(500\) 7.00966 0.313482
\(501\) 0 0
\(502\) −18.4989 −0.825644
\(503\) 18.9726 0.845945 0.422973 0.906142i \(-0.360987\pi\)
0.422973 + 0.906142i \(0.360987\pi\)
\(504\) 0 0
\(505\) 20.3070 0.903651
\(506\) 1.79853 0.0799543
\(507\) 0 0
\(508\) 5.75794 0.255467
\(509\) 30.6907 1.36034 0.680170 0.733055i \(-0.261906\pi\)
0.680170 + 0.733055i \(0.261906\pi\)
\(510\) 0 0
\(511\) −10.2974 −0.455529
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −27.7579 −1.22435
\(515\) 39.4737 1.73942
\(516\) 0 0
\(517\) 12.4989 0.549699
\(518\) −20.5948 −0.904882
\(519\) 0 0
\(520\) 7.29738 0.320011
\(521\) 42.1106 1.84490 0.922450 0.386116i \(-0.126184\pi\)
0.922450 + 0.386116i \(0.126184\pi\)
\(522\) 0 0
\(523\) 37.8133 1.65346 0.826729 0.562600i \(-0.190199\pi\)
0.826729 + 0.562600i \(0.190199\pi\)
\(524\) −1.42807 −0.0623856
\(525\) 0 0
\(526\) −20.2723 −0.883912
\(527\) 0.518173 0.0225720
\(528\) 0 0
\(529\) −19.7653 −0.859361
\(530\) −11.1895 −0.486041
\(531\) 0 0
\(532\) 4.67750 0.202795
\(533\) 13.3202 0.576962
\(534\) 0 0
\(535\) 3.14386 0.135921
\(536\) 9.08044 0.392215
\(537\) 0 0
\(538\) 15.6427 0.674405
\(539\) 14.8790 0.640883
\(540\) 0 0
\(541\) 13.5661 0.583253 0.291627 0.956532i \(-0.405804\pi\)
0.291627 + 0.956532i \(0.405804\pi\)
\(542\) −12.3395 −0.530028
\(543\) 0 0
\(544\) −0.201472 −0.00863806
\(545\) 26.0457 1.11567
\(546\) 0 0
\(547\) 46.4177 1.98468 0.992338 0.123552i \(-0.0394286\pi\)
0.992338 + 0.123552i \(0.0394286\pi\)
\(548\) 14.5313 0.620748
\(549\) 0 0
\(550\) 2.42807 0.103533
\(551\) −4.92692 −0.209894
\(552\) 0 0
\(553\) −55.4463 −2.35782
\(554\) 8.49885 0.361082
\(555\) 0 0
\(556\) 16.1763 0.686030
\(557\) 38.2065 1.61886 0.809431 0.587214i \(-0.199775\pi\)
0.809431 + 0.587214i \(0.199775\pi\)
\(558\) 0 0
\(559\) 31.5232 1.33329
\(560\) 12.7483 0.538713
\(561\) 0 0
\(562\) 7.12839 0.300693
\(563\) 7.01702 0.295732 0.147866 0.989007i \(-0.452760\pi\)
0.147866 + 0.989007i \(0.452760\pi\)
\(564\) 0 0
\(565\) −21.8036 −0.917284
\(566\) 11.5240 0.484389
\(567\) 0 0
\(568\) −12.8214 −0.537972
\(569\) 41.4235 1.73656 0.868281 0.496072i \(-0.165225\pi\)
0.868281 + 0.496072i \(0.165225\pi\)
\(570\) 0 0
\(571\) 12.1381 0.507962 0.253981 0.967209i \(-0.418260\pi\)
0.253981 + 0.967209i \(0.418260\pi\)
\(572\) −2.67750 −0.111952
\(573\) 0 0
\(574\) 23.2700 0.971269
\(575\) 4.36695 0.182115
\(576\) 0 0
\(577\) 12.8384 0.534469 0.267234 0.963632i \(-0.413890\pi\)
0.267234 + 0.963632i \(0.413890\pi\)
\(578\) −16.9594 −0.705418
\(579\) 0 0
\(580\) −13.4281 −0.557570
\(581\) −63.9764 −2.65419
\(582\) 0 0
\(583\) 4.10557 0.170035
\(584\) 2.20147 0.0910976
\(585\) 0 0
\(586\) −10.6775 −0.441083
\(587\) −4.00000 −0.165098 −0.0825488 0.996587i \(-0.526306\pi\)
−0.0825488 + 0.996587i \(0.526306\pi\)
\(588\) 0 0
\(589\) 2.57193 0.105974
\(590\) −14.3070 −0.589011
\(591\) 0 0
\(592\) 4.40294 0.180960
\(593\) −40.4177 −1.65975 −0.829877 0.557946i \(-0.811590\pi\)
−0.829877 + 0.557946i \(0.811590\pi\)
\(594\) 0 0
\(595\) −2.56842 −0.105295
\(596\) −6.30704 −0.258346
\(597\) 0 0
\(598\) −4.81555 −0.196923
\(599\) −0.679795 −0.0277757 −0.0138878 0.999904i \(-0.504421\pi\)
−0.0138878 + 0.999904i \(0.504421\pi\)
\(600\) 0 0
\(601\) 21.5240 0.877981 0.438991 0.898492i \(-0.355336\pi\)
0.438991 + 0.898492i \(0.355336\pi\)
\(602\) 55.0700 2.24449
\(603\) 0 0
\(604\) 0.498850 0.0202979
\(605\) −2.72545 −0.110805
\(606\) 0 0
\(607\) 13.3550 0.542062 0.271031 0.962571i \(-0.412635\pi\)
0.271031 + 0.962571i \(0.412635\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) −4.35269 −0.176235
\(611\) −33.4656 −1.35387
\(612\) 0 0
\(613\) 9.11293 0.368068 0.184034 0.982920i \(-0.441084\pi\)
0.184034 + 0.982920i \(0.441084\pi\)
\(614\) −30.2568 −1.22106
\(615\) 0 0
\(616\) −4.67750 −0.188462
\(617\) −0.329866 −0.0132799 −0.00663995 0.999978i \(-0.502114\pi\)
−0.00663995 + 0.999978i \(0.502114\pi\)
\(618\) 0 0
\(619\) 8.92112 0.358570 0.179285 0.983797i \(-0.442622\pi\)
0.179285 + 0.983797i \(0.442622\pi\)
\(620\) 7.00966 0.281515
\(621\) 0 0
\(622\) −8.79623 −0.352697
\(623\) −25.4966 −1.02150
\(624\) 0 0
\(625\) −31.2448 −1.24979
\(626\) 10.1860 0.407115
\(627\) 0 0
\(628\) −13.6272 −0.543786
\(629\) −0.887072 −0.0353699
\(630\) 0 0
\(631\) 44.6597 1.77788 0.888938 0.458028i \(-0.151444\pi\)
0.888938 + 0.458028i \(0.151444\pi\)
\(632\) 11.8538 0.471521
\(633\) 0 0
\(634\) 12.0097 0.476965
\(635\) −15.6930 −0.622756
\(636\) 0 0
\(637\) −39.8384 −1.57845
\(638\) 4.92692 0.195059
\(639\) 0 0
\(640\) −2.72545 −0.107733
\(641\) 6.93272 0.273826 0.136913 0.990583i \(-0.456282\pi\)
0.136913 + 0.990583i \(0.456282\pi\)
\(642\) 0 0
\(643\) 23.4966 0.926614 0.463307 0.886198i \(-0.346663\pi\)
0.463307 + 0.886198i \(0.346663\pi\)
\(644\) −8.41261 −0.331503
\(645\) 0 0
\(646\) 0.201472 0.00792682
\(647\) −20.9880 −0.825125 −0.412562 0.910929i \(-0.635366\pi\)
−0.412562 + 0.910929i \(0.635366\pi\)
\(648\) 0 0
\(649\) 5.24943 0.206058
\(650\) −6.50115 −0.254996
\(651\) 0 0
\(652\) 15.9497 0.624640
\(653\) 44.1335 1.72708 0.863538 0.504284i \(-0.168244\pi\)
0.863538 + 0.504284i \(0.168244\pi\)
\(654\) 0 0
\(655\) 3.89213 0.152078
\(656\) −4.97487 −0.194236
\(657\) 0 0
\(658\) −58.4633 −2.27914
\(659\) −24.0862 −0.938267 −0.469133 0.883127i \(-0.655434\pi\)
−0.469133 + 0.883127i \(0.655434\pi\)
\(660\) 0 0
\(661\) −9.65237 −0.375434 −0.187717 0.982223i \(-0.560109\pi\)
−0.187717 + 0.982223i \(0.560109\pi\)
\(662\) 26.1667 1.01700
\(663\) 0 0
\(664\) 13.6775 0.530790
\(665\) −12.7483 −0.494357
\(666\) 0 0
\(667\) 8.86120 0.343107
\(668\) −18.8059 −0.727622
\(669\) 0 0
\(670\) −24.7483 −0.956109
\(671\) 1.59706 0.0616536
\(672\) 0 0
\(673\) 24.6176 0.948938 0.474469 0.880272i \(-0.342640\pi\)
0.474469 + 0.880272i \(0.342640\pi\)
\(674\) 17.3778 0.669369
\(675\) 0 0
\(676\) −5.83102 −0.224270
\(677\) −21.5719 −0.829077 −0.414538 0.910032i \(-0.636057\pi\)
−0.414538 + 0.910032i \(0.636057\pi\)
\(678\) 0 0
\(679\) −78.6095 −3.01675
\(680\) 0.549103 0.0210571
\(681\) 0 0
\(682\) −2.57193 −0.0984843
\(683\) 0.997701 0.0381759 0.0190880 0.999818i \(-0.493924\pi\)
0.0190880 + 0.999818i \(0.493924\pi\)
\(684\) 0 0
\(685\) −39.6044 −1.51321
\(686\) −36.8538 −1.40709
\(687\) 0 0
\(688\) −11.7734 −0.448857
\(689\) −10.9926 −0.418786
\(690\) 0 0
\(691\) −9.00230 −0.342464 −0.171232 0.985231i \(-0.554775\pi\)
−0.171232 + 0.985231i \(0.554775\pi\)
\(692\) 3.22430 0.122569
\(693\) 0 0
\(694\) 12.4029 0.470809
\(695\) −44.0878 −1.67235
\(696\) 0 0
\(697\) 1.00230 0.0379648
\(698\) −23.8036 −0.900979
\(699\) 0 0
\(700\) −11.3573 −0.429265
\(701\) 15.0936 0.570078 0.285039 0.958516i \(-0.407994\pi\)
0.285039 + 0.958516i \(0.407994\pi\)
\(702\) 0 0
\(703\) −4.40294 −0.166060
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 24.2664 0.913280
\(707\) 34.8515 1.31073
\(708\) 0 0
\(709\) −22.9246 −0.860952 −0.430476 0.902602i \(-0.641654\pi\)
−0.430476 + 0.902602i \(0.641654\pi\)
\(710\) 34.9439 1.31142
\(711\) 0 0
\(712\) 5.45090 0.204281
\(713\) −4.62569 −0.173233
\(714\) 0 0
\(715\) 7.29738 0.272906
\(716\) −13.3778 −0.499953
\(717\) 0 0
\(718\) −20.3395 −0.759064
\(719\) −6.70032 −0.249880 −0.124940 0.992164i \(-0.539874\pi\)
−0.124940 + 0.992164i \(0.539874\pi\)
\(720\) 0 0
\(721\) 67.7460 2.52299
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) 15.9497 0.592768
\(725\) 11.9629 0.444291
\(726\) 0 0
\(727\) −18.4583 −0.684579 −0.342289 0.939595i \(-0.611202\pi\)
−0.342289 + 0.939595i \(0.611202\pi\)
\(728\) 12.5240 0.464169
\(729\) 0 0
\(730\) −6.00000 −0.222070
\(731\) 2.37201 0.0877321
\(732\) 0 0
\(733\) 41.8995 1.54759 0.773797 0.633434i \(-0.218355\pi\)
0.773797 + 0.633434i \(0.218355\pi\)
\(734\) −23.0170 −0.849574
\(735\) 0 0
\(736\) 1.79853 0.0662946
\(737\) 9.08044 0.334482
\(738\) 0 0
\(739\) −34.8361 −1.28147 −0.640733 0.767764i \(-0.721369\pi\)
−0.640733 + 0.767764i \(0.721369\pi\)
\(740\) −12.0000 −0.441129
\(741\) 0 0
\(742\) −19.2038 −0.704993
\(743\) 25.7889 0.946102 0.473051 0.881035i \(-0.343153\pi\)
0.473051 + 0.881035i \(0.343153\pi\)
\(744\) 0 0
\(745\) 17.1895 0.629775
\(746\) −18.5890 −0.680590
\(747\) 0 0
\(748\) −0.201472 −0.00736656
\(749\) 5.39558 0.197150
\(750\) 0 0
\(751\) −33.5011 −1.22247 −0.611237 0.791447i \(-0.709328\pi\)
−0.611237 + 0.791447i \(0.709328\pi\)
\(752\) 12.4989 0.455786
\(753\) 0 0
\(754\) −13.1918 −0.480417
\(755\) −1.35959 −0.0494806
\(756\) 0 0
\(757\) 7.53134 0.273731 0.136866 0.990590i \(-0.456297\pi\)
0.136866 + 0.990590i \(0.456297\pi\)
\(758\) −10.5816 −0.384341
\(759\) 0 0
\(760\) 2.72545 0.0988624
\(761\) −38.1969 −1.38464 −0.692318 0.721593i \(-0.743410\pi\)
−0.692318 + 0.721593i \(0.743410\pi\)
\(762\) 0 0
\(763\) 44.7003 1.61826
\(764\) 24.0553 0.870291
\(765\) 0 0
\(766\) 26.0878 0.942591
\(767\) −14.0553 −0.507508
\(768\) 0 0
\(769\) 16.6501 0.600417 0.300208 0.953874i \(-0.402944\pi\)
0.300208 + 0.953874i \(0.402944\pi\)
\(770\) 12.7483 0.459416
\(771\) 0 0
\(772\) 13.0325 0.469050
\(773\) 23.6330 0.850022 0.425011 0.905188i \(-0.360270\pi\)
0.425011 + 0.905188i \(0.360270\pi\)
\(774\) 0 0
\(775\) −6.24483 −0.224321
\(776\) 16.8059 0.603296
\(777\) 0 0
\(778\) 9.46636 0.339386
\(779\) 4.97487 0.178243
\(780\) 0 0
\(781\) −12.8214 −0.458784
\(782\) −0.362354 −0.0129577
\(783\) 0 0
\(784\) 14.8790 0.531392
\(785\) 37.1404 1.32560
\(786\) 0 0
\(787\) −20.3933 −0.726942 −0.363471 0.931606i \(-0.618408\pi\)
−0.363471 + 0.931606i \(0.618408\pi\)
\(788\) 5.59706 0.199387
\(789\) 0 0
\(790\) −32.3070 −1.14943
\(791\) −37.4200 −1.33050
\(792\) 0 0
\(793\) −4.27611 −0.151849
\(794\) −25.1667 −0.893132
\(795\) 0 0
\(796\) −7.39558 −0.262129
\(797\) 9.70262 0.343685 0.171842 0.985124i \(-0.445028\pi\)
0.171842 + 0.985124i \(0.445028\pi\)
\(798\) 0 0
\(799\) −2.51817 −0.0890865
\(800\) 2.42807 0.0858453
\(801\) 0 0
\(802\) 16.0650 0.567274
\(803\) 2.20147 0.0776883
\(804\) 0 0
\(805\) 22.9281 0.808110
\(806\) 6.88633 0.242561
\(807\) 0 0
\(808\) −7.45090 −0.262122
\(809\) 27.0723 0.951813 0.475906 0.879496i \(-0.342120\pi\)
0.475906 + 0.879496i \(0.342120\pi\)
\(810\) 0 0
\(811\) 51.9548 1.82438 0.912190 0.409767i \(-0.134390\pi\)
0.912190 + 0.409767i \(0.134390\pi\)
\(812\) −23.0457 −0.808744
\(813\) 0 0
\(814\) 4.40294 0.154323
\(815\) −43.4702 −1.52270
\(816\) 0 0
\(817\) 11.7734 0.411899
\(818\) −19.5313 −0.682897
\(819\) 0 0
\(820\) 13.5588 0.473493
\(821\) 25.6574 0.895451 0.447725 0.894171i \(-0.352234\pi\)
0.447725 + 0.894171i \(0.352234\pi\)
\(822\) 0 0
\(823\) −41.8635 −1.45927 −0.729635 0.683837i \(-0.760310\pi\)
−0.729635 + 0.683837i \(0.760310\pi\)
\(824\) −14.4834 −0.504553
\(825\) 0 0
\(826\) −24.5542 −0.854349
\(827\) −6.15582 −0.214059 −0.107029 0.994256i \(-0.534134\pi\)
−0.107029 + 0.994256i \(0.534134\pi\)
\(828\) 0 0
\(829\) −0.247126 −0.00858303 −0.00429151 0.999991i \(-0.501366\pi\)
−0.00429151 + 0.999991i \(0.501366\pi\)
\(830\) −37.2773 −1.29391
\(831\) 0 0
\(832\) −2.67750 −0.0928255
\(833\) −2.99770 −0.103864
\(834\) 0 0
\(835\) 51.2545 1.77373
\(836\) −1.00000 −0.0345857
\(837\) 0 0
\(838\) −24.9018 −0.860218
\(839\) −16.9320 −0.584557 −0.292278 0.956333i \(-0.594413\pi\)
−0.292278 + 0.956333i \(0.594413\pi\)
\(840\) 0 0
\(841\) −4.72545 −0.162947
\(842\) 26.6044 0.916849
\(843\) 0 0
\(844\) −14.2974 −0.492136
\(845\) 15.8921 0.546706
\(846\) 0 0
\(847\) −4.67750 −0.160721
\(848\) 4.10557 0.140986
\(849\) 0 0
\(850\) −0.489189 −0.0167790
\(851\) 7.91882 0.271454
\(852\) 0 0
\(853\) 21.6930 0.742753 0.371376 0.928482i \(-0.378886\pi\)
0.371376 + 0.928482i \(0.378886\pi\)
\(854\) −7.47022 −0.255626
\(855\) 0 0
\(856\) −1.15352 −0.0394265
\(857\) −34.7328 −1.18645 −0.593225 0.805037i \(-0.702146\pi\)
−0.593225 + 0.805037i \(0.702146\pi\)
\(858\) 0 0
\(859\) −15.3047 −0.522191 −0.261095 0.965313i \(-0.584084\pi\)
−0.261095 + 0.965313i \(0.584084\pi\)
\(860\) 32.0878 1.09418
\(861\) 0 0
\(862\) 32.5948 1.11018
\(863\) −35.8457 −1.22020 −0.610102 0.792323i \(-0.708871\pi\)
−0.610102 + 0.792323i \(0.708871\pi\)
\(864\) 0 0
\(865\) −8.78766 −0.298789
\(866\) 19.7386 0.670745
\(867\) 0 0
\(868\) 12.0302 0.408331
\(869\) 11.8538 0.402114
\(870\) 0 0
\(871\) −24.3128 −0.823809
\(872\) −9.55646 −0.323623
\(873\) 0 0
\(874\) −1.79853 −0.0608361
\(875\) −32.7877 −1.10843
\(876\) 0 0
\(877\) −45.4907 −1.53611 −0.768057 0.640382i \(-0.778776\pi\)
−0.768057 + 0.640382i \(0.778776\pi\)
\(878\) −24.4029 −0.823559
\(879\) 0 0
\(880\) −2.72545 −0.0918749
\(881\) −31.3322 −1.05561 −0.527804 0.849366i \(-0.676984\pi\)
−0.527804 + 0.849366i \(0.676984\pi\)
\(882\) 0 0
\(883\) 35.3550 1.18979 0.594895 0.803803i \(-0.297194\pi\)
0.594895 + 0.803803i \(0.297194\pi\)
\(884\) 0.539441 0.0181434
\(885\) 0 0
\(886\) 16.5948 0.557512
\(887\) 16.6141 0.557846 0.278923 0.960313i \(-0.410023\pi\)
0.278923 + 0.960313i \(0.410023\pi\)
\(888\) 0 0
\(889\) −26.9327 −0.903295
\(890\) −14.8561 −0.497979
\(891\) 0 0
\(892\) −0.402945 −0.0134916
\(893\) −12.4989 −0.418258
\(894\) 0 0
\(895\) 36.4606 1.21874
\(896\) −4.67750 −0.156264
\(897\) 0 0
\(898\) 20.9520 0.699179
\(899\) −12.6717 −0.422625
\(900\) 0 0
\(901\) −0.827158 −0.0275566
\(902\) −4.97487 −0.165645
\(903\) 0 0
\(904\) 8.00000 0.266076
\(905\) −43.4702 −1.44500
\(906\) 0 0
\(907\) 54.8612 1.82164 0.910818 0.412808i \(-0.135452\pi\)
0.910818 + 0.412808i \(0.135452\pi\)
\(908\) 21.0074 0.697154
\(909\) 0 0
\(910\) −34.1335 −1.13151
\(911\) −2.98298 −0.0988304 −0.0494152 0.998778i \(-0.515736\pi\)
−0.0494152 + 0.998778i \(0.515736\pi\)
\(912\) 0 0
\(913\) 13.6775 0.452659
\(914\) 23.3144 0.771172
\(915\) 0 0
\(916\) −14.3299 −0.473472
\(917\) 6.67980 0.220586
\(918\) 0 0
\(919\) −26.6701 −0.879767 −0.439883 0.898055i \(-0.644980\pi\)
−0.439883 + 0.898055i \(0.644980\pi\)
\(920\) −4.90179 −0.161607
\(921\) 0 0
\(922\) −20.8059 −0.685205
\(923\) 34.3291 1.12996
\(924\) 0 0
\(925\) 10.6907 0.351507
\(926\) −9.29002 −0.305289
\(927\) 0 0
\(928\) 4.92692 0.161734
\(929\) −43.7595 −1.43570 −0.717851 0.696197i \(-0.754874\pi\)
−0.717851 + 0.696197i \(0.754874\pi\)
\(930\) 0 0
\(931\) −14.8790 −0.487638
\(932\) 24.4029 0.799345
\(933\) 0 0
\(934\) −12.0457 −0.394146
\(935\) 0.549103 0.0179576
\(936\) 0 0
\(937\) 47.7823 1.56098 0.780490 0.625168i \(-0.214970\pi\)
0.780490 + 0.625168i \(0.214970\pi\)
\(938\) −42.4737 −1.38682
\(939\) 0 0
\(940\) −34.0650 −1.11108
\(941\) −36.2664 −1.18225 −0.591126 0.806579i \(-0.701316\pi\)
−0.591126 + 0.806579i \(0.701316\pi\)
\(942\) 0 0
\(943\) −8.94745 −0.291369
\(944\) 5.24943 0.170854
\(945\) 0 0
\(946\) −11.7734 −0.382786
\(947\) −35.5468 −1.15512 −0.577558 0.816350i \(-0.695994\pi\)
−0.577558 + 0.816350i \(0.695994\pi\)
\(948\) 0 0
\(949\) −5.89443 −0.191341
\(950\) −2.42807 −0.0787770
\(951\) 0 0
\(952\) 0.942386 0.0305429
\(953\) −23.2088 −0.751808 −0.375904 0.926659i \(-0.622668\pi\)
−0.375904 + 0.926659i \(0.622668\pi\)
\(954\) 0 0
\(955\) −65.5615 −2.12152
\(956\) 12.6849 0.410258
\(957\) 0 0
\(958\) −0.188307 −0.00608392
\(959\) −67.9703 −2.19487
\(960\) 0 0
\(961\) −24.3852 −0.786619
\(962\) −11.7889 −0.380088
\(963\) 0 0
\(964\) −23.9343 −0.770871
\(965\) −35.5194 −1.14341
\(966\) 0 0
\(967\) −6.09591 −0.196031 −0.0980156 0.995185i \(-0.531249\pi\)
−0.0980156 + 0.995185i \(0.531249\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −45.8036 −1.47066
\(971\) −34.9822 −1.12263 −0.561317 0.827601i \(-0.689705\pi\)
−0.561317 + 0.827601i \(0.689705\pi\)
\(972\) 0 0
\(973\) −75.6648 −2.42570
\(974\) −33.3852 −1.06973
\(975\) 0 0
\(976\) 1.59706 0.0511205
\(977\) 10.9018 0.348779 0.174390 0.984677i \(-0.444205\pi\)
0.174390 + 0.984677i \(0.444205\pi\)
\(978\) 0 0
\(979\) 5.45090 0.174211
\(980\) −40.5519 −1.29538
\(981\) 0 0
\(982\) −12.5217 −0.399583
\(983\) 25.4281 0.811030 0.405515 0.914089i \(-0.367092\pi\)
0.405515 + 0.914089i \(0.367092\pi\)
\(984\) 0 0
\(985\) −15.2545 −0.486048
\(986\) −0.992638 −0.0316120
\(987\) 0 0
\(988\) 2.67750 0.0851825
\(989\) −21.1748 −0.673319
\(990\) 0 0
\(991\) 49.3925 1.56901 0.784503 0.620125i \(-0.212918\pi\)
0.784503 + 0.620125i \(0.212918\pi\)
\(992\) −2.57193 −0.0816588
\(993\) 0 0
\(994\) 59.9718 1.90219
\(995\) 20.1563 0.638997
\(996\) 0 0
\(997\) 20.7409 0.656871 0.328436 0.944526i \(-0.393479\pi\)
0.328436 + 0.944526i \(0.393479\pi\)
\(998\) 16.3070 0.516190
\(999\) 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.bg.1.1 3
3.2 odd 2 418.2.a.g.1.2 3
12.11 even 2 3344.2.a.q.1.2 3
33.32 even 2 4598.2.a.bo.1.2 3
57.56 even 2 7942.2.a.bi.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.g.1.2 3 3.2 odd 2
3344.2.a.q.1.2 3 12.11 even 2
3762.2.a.bg.1.1 3 1.1 even 1 trivial
4598.2.a.bo.1.2 3 33.32 even 2
7942.2.a.bi.1.2 3 57.56 even 2