Properties

Label 4028.2.c.a.3497.7
Level $4028$
Weight $2$
Character 4028.3497
Analytic conductor $32.164$
Analytic rank $0$
Dimension $82$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4028,2,Mod(3497,4028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4028, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4028.3497");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4028.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(32.1637419342\)
Analytic rank: \(0\)
Dimension: \(82\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3497.7
Character \(\chi\) \(=\) 4028.3497
Dual form 4028.2.c.a.3497.76

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.92218i q^{3} +2.97792i q^{5} +2.58698 q^{7} -5.53915 q^{9} +O(q^{10})\) \(q-2.92218i q^{3} +2.97792i q^{5} +2.58698 q^{7} -5.53915 q^{9} -5.66211 q^{11} -0.873190 q^{13} +8.70203 q^{15} +2.71791 q^{17} -1.00000i q^{19} -7.55964i q^{21} +3.52478i q^{23} -3.86801 q^{25} +7.41985i q^{27} +5.84976 q^{29} -4.81613i q^{31} +16.5457i q^{33} +7.70383i q^{35} -9.31304 q^{37} +2.55162i q^{39} +8.85910i q^{41} +9.93035 q^{43} -16.4951i q^{45} +7.77941 q^{47} -0.307519 q^{49} -7.94222i q^{51} +(3.46126 - 6.40466i) q^{53} -16.8613i q^{55} -2.92218 q^{57} +10.6525 q^{59} +4.91344i q^{61} -14.3297 q^{63} -2.60029i q^{65} -4.27118i q^{67} +10.3000 q^{69} +12.4279i q^{71} -15.3702i q^{73} +11.3030i q^{75} -14.6478 q^{77} +8.78273i q^{79} +5.06472 q^{81} -8.63091i q^{83} +8.09371i q^{85} -17.0941i q^{87} +14.0026 q^{89} -2.25893 q^{91} -14.0736 q^{93} +2.97792 q^{95} +17.5439 q^{97} +31.3633 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 82 q - 8 q^{7} - 82 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 82 q - 8 q^{7} - 82 q^{9} + 4 q^{13} + 4 q^{15} - 4 q^{17} - 58 q^{25} - 16 q^{29} - 12 q^{37} - 32 q^{43} + 8 q^{47} + 98 q^{49} + 6 q^{53} - 4 q^{57} + 4 q^{59} + 8 q^{63} + 28 q^{69} - 8 q^{77} + 154 q^{81} - 20 q^{89} + 48 q^{91} - 56 q^{93} - 44 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4028\mathbb{Z}\right)^\times\).

\(n\) \(2015\) \(2281\) \(2757\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.92218i 1.68712i −0.537033 0.843561i \(-0.680455\pi\)
0.537033 0.843561i \(-0.319545\pi\)
\(4\) 0 0
\(5\) 2.97792i 1.33177i 0.746056 + 0.665883i \(0.231945\pi\)
−0.746056 + 0.665883i \(0.768055\pi\)
\(6\) 0 0
\(7\) 2.58698 0.977788 0.488894 0.872343i \(-0.337401\pi\)
0.488894 + 0.872343i \(0.337401\pi\)
\(8\) 0 0
\(9\) −5.53915 −1.84638
\(10\) 0 0
\(11\) −5.66211 −1.70719 −0.853596 0.520936i \(-0.825583\pi\)
−0.853596 + 0.520936i \(0.825583\pi\)
\(12\) 0 0
\(13\) −0.873190 −0.242179 −0.121090 0.992642i \(-0.538639\pi\)
−0.121090 + 0.992642i \(0.538639\pi\)
\(14\) 0 0
\(15\) 8.70203 2.24685
\(16\) 0 0
\(17\) 2.71791 0.659189 0.329594 0.944123i \(-0.393088\pi\)
0.329594 + 0.944123i \(0.393088\pi\)
\(18\) 0 0
\(19\) 1.00000i 0.229416i
\(20\) 0 0
\(21\) 7.55964i 1.64965i
\(22\) 0 0
\(23\) 3.52478i 0.734967i 0.930030 + 0.367483i \(0.119781\pi\)
−0.930030 + 0.367483i \(0.880219\pi\)
\(24\) 0 0
\(25\) −3.86801 −0.773602
\(26\) 0 0
\(27\) 7.41985i 1.42795i
\(28\) 0 0
\(29\) 5.84976 1.08627 0.543137 0.839644i \(-0.317237\pi\)
0.543137 + 0.839644i \(0.317237\pi\)
\(30\) 0 0
\(31\) 4.81613i 0.865003i −0.901633 0.432501i \(-0.857631\pi\)
0.901633 0.432501i \(-0.142369\pi\)
\(32\) 0 0
\(33\) 16.5457i 2.88024i
\(34\) 0 0
\(35\) 7.70383i 1.30219i
\(36\) 0 0
\(37\) −9.31304 −1.53105 −0.765527 0.643404i \(-0.777522\pi\)
−0.765527 + 0.643404i \(0.777522\pi\)
\(38\) 0 0
\(39\) 2.55162i 0.408586i
\(40\) 0 0
\(41\) 8.85910i 1.38356i 0.722109 + 0.691779i \(0.243173\pi\)
−0.722109 + 0.691779i \(0.756827\pi\)
\(42\) 0 0
\(43\) 9.93035 1.51436 0.757182 0.653204i \(-0.226575\pi\)
0.757182 + 0.653204i \(0.226575\pi\)
\(44\) 0 0
\(45\) 16.4951i 2.45895i
\(46\) 0 0
\(47\) 7.77941 1.13474 0.567372 0.823462i \(-0.307960\pi\)
0.567372 + 0.823462i \(0.307960\pi\)
\(48\) 0 0
\(49\) −0.307519 −0.0439312
\(50\) 0 0
\(51\) 7.94222i 1.11213i
\(52\) 0 0
\(53\) 3.46126 6.40466i 0.475440 0.879748i
\(54\) 0 0
\(55\) 16.8613i 2.27358i
\(56\) 0 0
\(57\) −2.92218 −0.387052
\(58\) 0 0
\(59\) 10.6525 1.38683 0.693417 0.720537i \(-0.256105\pi\)
0.693417 + 0.720537i \(0.256105\pi\)
\(60\) 0 0
\(61\) 4.91344i 0.629101i 0.949241 + 0.314550i \(0.101854\pi\)
−0.949241 + 0.314550i \(0.898146\pi\)
\(62\) 0 0
\(63\) −14.3297 −1.80537
\(64\) 0 0
\(65\) 2.60029i 0.322526i
\(66\) 0 0
\(67\) 4.27118i 0.521808i −0.965365 0.260904i \(-0.915979\pi\)
0.965365 0.260904i \(-0.0840205\pi\)
\(68\) 0 0
\(69\) 10.3000 1.23998
\(70\) 0 0
\(71\) 12.4279i 1.47492i 0.675391 + 0.737460i \(0.263975\pi\)
−0.675391 + 0.737460i \(0.736025\pi\)
\(72\) 0 0
\(73\) 15.3702i 1.79894i −0.436979 0.899471i \(-0.643952\pi\)
0.436979 0.899471i \(-0.356048\pi\)
\(74\) 0 0
\(75\) 11.3030i 1.30516i
\(76\) 0 0
\(77\) −14.6478 −1.66927
\(78\) 0 0
\(79\) 8.78273i 0.988135i 0.869424 + 0.494067i \(0.164490\pi\)
−0.869424 + 0.494067i \(0.835510\pi\)
\(80\) 0 0
\(81\) 5.06472 0.562747
\(82\) 0 0
\(83\) 8.63091i 0.947365i −0.880696 0.473682i \(-0.842924\pi\)
0.880696 0.473682i \(-0.157076\pi\)
\(84\) 0 0
\(85\) 8.09371i 0.877886i
\(86\) 0 0
\(87\) 17.0941i 1.83268i
\(88\) 0 0
\(89\) 14.0026 1.48427 0.742135 0.670250i \(-0.233813\pi\)
0.742135 + 0.670250i \(0.233813\pi\)
\(90\) 0 0
\(91\) −2.25893 −0.236800
\(92\) 0 0
\(93\) −14.0736 −1.45937
\(94\) 0 0
\(95\) 2.97792 0.305528
\(96\) 0 0
\(97\) 17.5439 1.78131 0.890657 0.454677i \(-0.150245\pi\)
0.890657 + 0.454677i \(0.150245\pi\)
\(98\) 0 0
\(99\) 31.3633 3.15213
\(100\) 0 0
\(101\) 5.35146i 0.532490i 0.963905 + 0.266245i \(0.0857830\pi\)
−0.963905 + 0.266245i \(0.914217\pi\)
\(102\) 0 0
\(103\) 18.1784i 1.79118i 0.444885 + 0.895588i \(0.353244\pi\)
−0.444885 + 0.895588i \(0.646756\pi\)
\(104\) 0 0
\(105\) 22.5120 2.19695
\(106\) 0 0
\(107\) 14.7482 1.42576 0.712880 0.701286i \(-0.247390\pi\)
0.712880 + 0.701286i \(0.247390\pi\)
\(108\) 0 0
\(109\) 8.50694i 0.814817i −0.913246 0.407409i \(-0.866433\pi\)
0.913246 0.407409i \(-0.133567\pi\)
\(110\) 0 0
\(111\) 27.2144i 2.58308i
\(112\) 0 0
\(113\) −5.62454 −0.529112 −0.264556 0.964370i \(-0.585225\pi\)
−0.264556 + 0.964370i \(0.585225\pi\)
\(114\) 0 0
\(115\) −10.4965 −0.978804
\(116\) 0 0
\(117\) 4.83673 0.447156
\(118\) 0 0
\(119\) 7.03118 0.644547
\(120\) 0 0
\(121\) 21.0595 1.91450
\(122\) 0 0
\(123\) 25.8879 2.33423
\(124\) 0 0
\(125\) 3.37097i 0.301509i
\(126\) 0 0
\(127\) 21.4678i 1.90496i 0.304606 + 0.952478i \(0.401475\pi\)
−0.304606 + 0.952478i \(0.598525\pi\)
\(128\) 0 0
\(129\) 29.0183i 2.55492i
\(130\) 0 0
\(131\) 20.0130 1.74854 0.874271 0.485437i \(-0.161340\pi\)
0.874271 + 0.485437i \(0.161340\pi\)
\(132\) 0 0
\(133\) 2.58698i 0.224320i
\(134\) 0 0
\(135\) −22.0957 −1.90170
\(136\) 0 0
\(137\) 13.4502i 1.14912i 0.818461 + 0.574562i \(0.194828\pi\)
−0.818461 + 0.574562i \(0.805172\pi\)
\(138\) 0 0
\(139\) 4.59168i 0.389461i 0.980857 + 0.194730i \(0.0623832\pi\)
−0.980857 + 0.194730i \(0.937617\pi\)
\(140\) 0 0
\(141\) 22.7329i 1.91445i
\(142\) 0 0
\(143\) 4.94410 0.413446
\(144\) 0 0
\(145\) 17.4201i 1.44666i
\(146\) 0 0
\(147\) 0.898626i 0.0741174i
\(148\) 0 0
\(149\) 1.87188 0.153350 0.0766751 0.997056i \(-0.475570\pi\)
0.0766751 + 0.997056i \(0.475570\pi\)
\(150\) 0 0
\(151\) 6.10740i 0.497013i 0.968630 + 0.248507i \(0.0799398\pi\)
−0.968630 + 0.248507i \(0.920060\pi\)
\(152\) 0 0
\(153\) −15.0549 −1.21712
\(154\) 0 0
\(155\) 14.3421 1.15198
\(156\) 0 0
\(157\) 17.7536i 1.41689i −0.705767 0.708444i \(-0.749397\pi\)
0.705767 0.708444i \(-0.250603\pi\)
\(158\) 0 0
\(159\) −18.7156 10.1144i −1.48424 0.802126i
\(160\) 0 0
\(161\) 9.11854i 0.718641i
\(162\) 0 0
\(163\) 23.5344 1.84335 0.921677 0.387959i \(-0.126820\pi\)
0.921677 + 0.387959i \(0.126820\pi\)
\(164\) 0 0
\(165\) −49.2719 −3.83581
\(166\) 0 0
\(167\) 3.03964i 0.235215i −0.993060 0.117607i \(-0.962478\pi\)
0.993060 0.117607i \(-0.0375224\pi\)
\(168\) 0 0
\(169\) −12.2375 −0.941349
\(170\) 0 0
\(171\) 5.53915i 0.423589i
\(172\) 0 0
\(173\) 20.7816i 1.58000i −0.613108 0.789999i \(-0.710081\pi\)
0.613108 0.789999i \(-0.289919\pi\)
\(174\) 0 0
\(175\) −10.0065 −0.756419
\(176\) 0 0
\(177\) 31.1285i 2.33976i
\(178\) 0 0
\(179\) 9.41793i 0.703930i −0.936013 0.351965i \(-0.885514\pi\)
0.936013 0.351965i \(-0.114486\pi\)
\(180\) 0 0
\(181\) 9.29826i 0.691134i 0.938394 + 0.345567i \(0.112313\pi\)
−0.938394 + 0.345567i \(0.887687\pi\)
\(182\) 0 0
\(183\) 14.3580 1.06137
\(184\) 0 0
\(185\) 27.7335i 2.03901i
\(186\) 0 0
\(187\) −15.3891 −1.12536
\(188\) 0 0
\(189\) 19.1950i 1.39623i
\(190\) 0 0
\(191\) 11.4023i 0.825040i −0.910949 0.412520i \(-0.864649\pi\)
0.910949 0.412520i \(-0.135351\pi\)
\(192\) 0 0
\(193\) 19.7323i 1.42036i 0.704020 + 0.710181i \(0.251387\pi\)
−0.704020 + 0.710181i \(0.748613\pi\)
\(194\) 0 0
\(195\) −7.59852 −0.544141
\(196\) 0 0
\(197\) 3.38501 0.241172 0.120586 0.992703i \(-0.461523\pi\)
0.120586 + 0.992703i \(0.461523\pi\)
\(198\) 0 0
\(199\) −17.6520 −1.25132 −0.625660 0.780096i \(-0.715170\pi\)
−0.625660 + 0.780096i \(0.715170\pi\)
\(200\) 0 0
\(201\) −12.4812 −0.880353
\(202\) 0 0
\(203\) 15.1332 1.06214
\(204\) 0 0
\(205\) −26.3817 −1.84258
\(206\) 0 0
\(207\) 19.5243i 1.35703i
\(208\) 0 0
\(209\) 5.66211i 0.391657i
\(210\) 0 0
\(211\) 0.0167637 0.00115406 0.000577030 1.00000i \(-0.499816\pi\)
0.000577030 1.00000i \(0.499816\pi\)
\(212\) 0 0
\(213\) 36.3166 2.48837
\(214\) 0 0
\(215\) 29.5718i 2.01678i
\(216\) 0 0
\(217\) 12.4592i 0.845789i
\(218\) 0 0
\(219\) −44.9145 −3.03504
\(220\) 0 0
\(221\) −2.37325 −0.159642
\(222\) 0 0
\(223\) −14.4857 −0.970035 −0.485018 0.874504i \(-0.661187\pi\)
−0.485018 + 0.874504i \(0.661187\pi\)
\(224\) 0 0
\(225\) 21.4255 1.42837
\(226\) 0 0
\(227\) 26.7948 1.77843 0.889217 0.457485i \(-0.151250\pi\)
0.889217 + 0.457485i \(0.151250\pi\)
\(228\) 0 0
\(229\) 6.31143 0.417071 0.208536 0.978015i \(-0.433130\pi\)
0.208536 + 0.978015i \(0.433130\pi\)
\(230\) 0 0
\(231\) 42.8035i 2.81626i
\(232\) 0 0
\(233\) 17.1969i 1.12661i −0.826250 0.563304i \(-0.809530\pi\)
0.826250 0.563304i \(-0.190470\pi\)
\(234\) 0 0
\(235\) 23.1665i 1.51121i
\(236\) 0 0
\(237\) 25.6648 1.66710
\(238\) 0 0
\(239\) 16.3930i 1.06038i 0.847880 + 0.530188i \(0.177879\pi\)
−0.847880 + 0.530188i \(0.822121\pi\)
\(240\) 0 0
\(241\) −1.43061 −0.0921535 −0.0460768 0.998938i \(-0.514672\pi\)
−0.0460768 + 0.998938i \(0.514672\pi\)
\(242\) 0 0
\(243\) 7.45953i 0.478529i
\(244\) 0 0
\(245\) 0.915766i 0.0585062i
\(246\) 0 0
\(247\) 0.873190i 0.0555597i
\(248\) 0 0
\(249\) −25.2211 −1.59832
\(250\) 0 0
\(251\) 21.8073i 1.37647i 0.725489 + 0.688233i \(0.241614\pi\)
−0.725489 + 0.688233i \(0.758386\pi\)
\(252\) 0 0
\(253\) 19.9577i 1.25473i
\(254\) 0 0
\(255\) 23.6513 1.48110
\(256\) 0 0
\(257\) 19.4605i 1.21392i −0.794734 0.606958i \(-0.792390\pi\)
0.794734 0.606958i \(-0.207610\pi\)
\(258\) 0 0
\(259\) −24.0927 −1.49705
\(260\) 0 0
\(261\) −32.4027 −2.00568
\(262\) 0 0
\(263\) 29.3106i 1.80737i −0.428201 0.903683i \(-0.640853\pi\)
0.428201 0.903683i \(-0.359147\pi\)
\(264\) 0 0
\(265\) 19.0726 + 10.3074i 1.17162 + 0.633176i
\(266\) 0 0
\(267\) 40.9181i 2.50415i
\(268\) 0 0
\(269\) −22.5932 −1.37753 −0.688767 0.724983i \(-0.741848\pi\)
−0.688767 + 0.724983i \(0.741848\pi\)
\(270\) 0 0
\(271\) 7.78178 0.472709 0.236355 0.971667i \(-0.424047\pi\)
0.236355 + 0.971667i \(0.424047\pi\)
\(272\) 0 0
\(273\) 6.60100i 0.399510i
\(274\) 0 0
\(275\) 21.9011 1.32069
\(276\) 0 0
\(277\) 8.46870i 0.508835i 0.967095 + 0.254417i \(0.0818837\pi\)
−0.967095 + 0.254417i \(0.918116\pi\)
\(278\) 0 0
\(279\) 26.6773i 1.59713i
\(280\) 0 0
\(281\) −16.1573 −0.963861 −0.481930 0.876209i \(-0.660064\pi\)
−0.481930 + 0.876209i \(0.660064\pi\)
\(282\) 0 0
\(283\) 13.1725i 0.783023i −0.920173 0.391512i \(-0.871952\pi\)
0.920173 0.391512i \(-0.128048\pi\)
\(284\) 0 0
\(285\) 8.70203i 0.515464i
\(286\) 0 0
\(287\) 22.9183i 1.35283i
\(288\) 0 0
\(289\) −9.61299 −0.565470
\(290\) 0 0
\(291\) 51.2665i 3.00529i
\(292\) 0 0
\(293\) 2.19643 0.128317 0.0641584 0.997940i \(-0.479564\pi\)
0.0641584 + 0.997940i \(0.479564\pi\)
\(294\) 0 0
\(295\) 31.7222i 1.84694i
\(296\) 0 0
\(297\) 42.0121i 2.43779i
\(298\) 0 0
\(299\) 3.07780i 0.177994i
\(300\) 0 0
\(301\) 25.6897 1.48073
\(302\) 0 0
\(303\) 15.6379 0.898376
\(304\) 0 0
\(305\) −14.6318 −0.837816
\(306\) 0 0
\(307\) 2.24976 0.128400 0.0642002 0.997937i \(-0.479550\pi\)
0.0642002 + 0.997937i \(0.479550\pi\)
\(308\) 0 0
\(309\) 53.1207 3.02193
\(310\) 0 0
\(311\) −6.02059 −0.341397 −0.170698 0.985323i \(-0.554602\pi\)
−0.170698 + 0.985323i \(0.554602\pi\)
\(312\) 0 0
\(313\) 25.8826i 1.46297i −0.681857 0.731486i \(-0.738827\pi\)
0.681857 0.731486i \(-0.261173\pi\)
\(314\) 0 0
\(315\) 42.6727i 2.40433i
\(316\) 0 0
\(317\) −19.1795 −1.07723 −0.538613 0.842553i \(-0.681052\pi\)
−0.538613 + 0.842553i \(0.681052\pi\)
\(318\) 0 0
\(319\) −33.1220 −1.85448
\(320\) 0 0
\(321\) 43.0969i 2.40543i
\(322\) 0 0
\(323\) 2.71791i 0.151228i
\(324\) 0 0
\(325\) 3.37751 0.187350
\(326\) 0 0
\(327\) −24.8588 −1.37470
\(328\) 0 0
\(329\) 20.1252 1.10954
\(330\) 0 0
\(331\) −8.77508 −0.482322 −0.241161 0.970485i \(-0.577528\pi\)
−0.241161 + 0.970485i \(0.577528\pi\)
\(332\) 0 0
\(333\) 51.5863 2.82691
\(334\) 0 0
\(335\) 12.7192 0.694926
\(336\) 0 0
\(337\) 18.2465i 0.993952i 0.867764 + 0.496976i \(0.165556\pi\)
−0.867764 + 0.496976i \(0.834444\pi\)
\(338\) 0 0
\(339\) 16.4359i 0.892677i
\(340\) 0 0
\(341\) 27.2695i 1.47673i
\(342\) 0 0
\(343\) −18.9044 −1.02074
\(344\) 0 0
\(345\) 30.6727i 1.65136i
\(346\) 0 0
\(347\) 14.9575 0.802961 0.401480 0.915868i \(-0.368496\pi\)
0.401480 + 0.915868i \(0.368496\pi\)
\(348\) 0 0
\(349\) 0.432682i 0.0231609i −0.999933 0.0115805i \(-0.996314\pi\)
0.999933 0.0115805i \(-0.00368626\pi\)
\(350\) 0 0
\(351\) 6.47894i 0.345820i
\(352\) 0 0
\(353\) 7.26390i 0.386618i −0.981138 0.193309i \(-0.938078\pi\)
0.981138 0.193309i \(-0.0619220\pi\)
\(354\) 0 0
\(355\) −37.0093 −1.96425
\(356\) 0 0
\(357\) 20.5464i 1.08743i
\(358\) 0 0
\(359\) 18.6505i 0.984333i 0.870501 + 0.492167i \(0.163795\pi\)
−0.870501 + 0.492167i \(0.836205\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 61.5398i 3.23000i
\(364\) 0 0
\(365\) 45.7712 2.39577
\(366\) 0 0
\(367\) −1.60750 −0.0839110 −0.0419555 0.999119i \(-0.513359\pi\)
−0.0419555 + 0.999119i \(0.513359\pi\)
\(368\) 0 0
\(369\) 49.0719i 2.55458i
\(370\) 0 0
\(371\) 8.95422 16.5688i 0.464880 0.860207i
\(372\) 0 0
\(373\) 1.52528i 0.0789761i 0.999220 + 0.0394881i \(0.0125727\pi\)
−0.999220 + 0.0394881i \(0.987427\pi\)
\(374\) 0 0
\(375\) 9.85059 0.508682
\(376\) 0 0
\(377\) −5.10795 −0.263073
\(378\) 0 0
\(379\) 1.62205i 0.0833194i 0.999132 + 0.0416597i \(0.0132645\pi\)
−0.999132 + 0.0416597i \(0.986735\pi\)
\(380\) 0 0
\(381\) 62.7327 3.21390
\(382\) 0 0
\(383\) 14.6255i 0.747327i −0.927564 0.373663i \(-0.878102\pi\)
0.927564 0.373663i \(-0.121898\pi\)
\(384\) 0 0
\(385\) 43.6200i 2.22308i
\(386\) 0 0
\(387\) −55.0057 −2.79610
\(388\) 0 0
\(389\) 8.86911i 0.449682i −0.974396 0.224841i \(-0.927814\pi\)
0.974396 0.224841i \(-0.0721862\pi\)
\(390\) 0 0
\(391\) 9.58001i 0.484482i
\(392\) 0 0
\(393\) 58.4816i 2.95001i
\(394\) 0 0
\(395\) −26.1543 −1.31597
\(396\) 0 0
\(397\) 3.58643i 0.179998i −0.995942 0.0899988i \(-0.971314\pi\)
0.995942 0.0899988i \(-0.0286863\pi\)
\(398\) 0 0
\(399\) −7.55964 −0.378455
\(400\) 0 0
\(401\) 34.9600i 1.74582i 0.487881 + 0.872910i \(0.337770\pi\)
−0.487881 + 0.872910i \(0.662230\pi\)
\(402\) 0 0
\(403\) 4.20540i 0.209486i
\(404\) 0 0
\(405\) 15.0823i 0.749447i
\(406\) 0 0
\(407\) 52.7315 2.61380
\(408\) 0 0
\(409\) 25.1962 1.24587 0.622935 0.782273i \(-0.285940\pi\)
0.622935 + 0.782273i \(0.285940\pi\)
\(410\) 0 0
\(411\) 39.3038 1.93871
\(412\) 0 0
\(413\) 27.5578 1.35603
\(414\) 0 0
\(415\) 25.7022 1.26167
\(416\) 0 0
\(417\) 13.4177 0.657068
\(418\) 0 0
\(419\) 18.9135i 0.923984i −0.886884 0.461992i \(-0.847135\pi\)
0.886884 0.461992i \(-0.152865\pi\)
\(420\) 0 0
\(421\) 5.46033i 0.266120i 0.991108 + 0.133060i \(0.0424803\pi\)
−0.991108 + 0.133060i \(0.957520\pi\)
\(422\) 0 0
\(423\) −43.0913 −2.09517
\(424\) 0 0
\(425\) −10.5129 −0.509950
\(426\) 0 0
\(427\) 12.7110i 0.615127i
\(428\) 0 0
\(429\) 14.4476i 0.697535i
\(430\) 0 0
\(431\) −4.82835 −0.232573 −0.116287 0.993216i \(-0.537099\pi\)
−0.116287 + 0.993216i \(0.537099\pi\)
\(432\) 0 0
\(433\) −29.2901 −1.40759 −0.703796 0.710402i \(-0.748513\pi\)
−0.703796 + 0.710402i \(0.748513\pi\)
\(434\) 0 0
\(435\) 50.9048 2.44070
\(436\) 0 0
\(437\) 3.52478 0.168613
\(438\) 0 0
\(439\) −8.47016 −0.404259 −0.202129 0.979359i \(-0.564786\pi\)
−0.202129 + 0.979359i \(0.564786\pi\)
\(440\) 0 0
\(441\) 1.70339 0.0811139
\(442\) 0 0
\(443\) 3.83398i 0.182158i −0.995844 0.0910789i \(-0.970968\pi\)
0.995844 0.0910789i \(-0.0290316\pi\)
\(444\) 0 0
\(445\) 41.6986i 1.97670i
\(446\) 0 0
\(447\) 5.46997i 0.258721i
\(448\) 0 0
\(449\) 3.02046 0.142544 0.0712721 0.997457i \(-0.477294\pi\)
0.0712721 + 0.997457i \(0.477294\pi\)
\(450\) 0 0
\(451\) 50.1612i 2.36200i
\(452\) 0 0
\(453\) 17.8469 0.838522
\(454\) 0 0
\(455\) 6.72691i 0.315362i
\(456\) 0 0
\(457\) 4.32553i 0.202340i −0.994869 0.101170i \(-0.967741\pi\)
0.994869 0.101170i \(-0.0322586\pi\)
\(458\) 0 0
\(459\) 20.1665i 0.941290i
\(460\) 0 0
\(461\) −22.4438 −1.04531 −0.522657 0.852543i \(-0.675059\pi\)
−0.522657 + 0.852543i \(0.675059\pi\)
\(462\) 0 0
\(463\) 25.2974i 1.17567i −0.808981 0.587835i \(-0.799981\pi\)
0.808981 0.587835i \(-0.200019\pi\)
\(464\) 0 0
\(465\) 41.9101i 1.94353i
\(466\) 0 0
\(467\) 12.6679 0.586199 0.293099 0.956082i \(-0.405313\pi\)
0.293099 + 0.956082i \(0.405313\pi\)
\(468\) 0 0
\(469\) 11.0495i 0.510217i
\(470\) 0 0
\(471\) −51.8792 −2.39047
\(472\) 0 0
\(473\) −56.2268 −2.58531
\(474\) 0 0
\(475\) 3.86801i 0.177477i
\(476\) 0 0
\(477\) −19.1724 + 35.4764i −0.877845 + 1.62435i
\(478\) 0 0
\(479\) 2.71881i 0.124225i −0.998069 0.0621127i \(-0.980216\pi\)
0.998069 0.0621127i \(-0.0197838\pi\)
\(480\) 0 0
\(481\) 8.13205 0.370790
\(482\) 0 0
\(483\) 26.6460 1.21244
\(484\) 0 0
\(485\) 52.2443i 2.37229i
\(486\) 0 0
\(487\) −18.5719 −0.841571 −0.420786 0.907160i \(-0.638246\pi\)
−0.420786 + 0.907160i \(0.638246\pi\)
\(488\) 0 0
\(489\) 68.7717i 3.10996i
\(490\) 0 0
\(491\) 0.989061i 0.0446357i 0.999751 + 0.0223178i \(0.00710458\pi\)
−0.999751 + 0.0223178i \(0.992895\pi\)
\(492\) 0 0
\(493\) 15.8991 0.716059
\(494\) 0 0
\(495\) 93.3974i 4.19790i
\(496\) 0 0
\(497\) 32.1508i 1.44216i
\(498\) 0 0
\(499\) 2.66523i 0.119312i 0.998219 + 0.0596560i \(0.0190004\pi\)
−0.998219 + 0.0596560i \(0.981000\pi\)
\(500\) 0 0
\(501\) −8.88239 −0.396836
\(502\) 0 0
\(503\) 0.147697i 0.00658547i −0.999995 0.00329273i \(-0.998952\pi\)
0.999995 0.00329273i \(-0.00104811\pi\)
\(504\) 0 0
\(505\) −15.9362 −0.709152
\(506\) 0 0
\(507\) 35.7603i 1.58817i
\(508\) 0 0
\(509\) 32.9681i 1.46129i 0.682760 + 0.730643i \(0.260780\pi\)
−0.682760 + 0.730643i \(0.739220\pi\)
\(510\) 0 0
\(511\) 39.7624i 1.75898i
\(512\) 0 0
\(513\) 7.41985 0.327595
\(514\) 0 0
\(515\) −54.1340 −2.38543
\(516\) 0 0
\(517\) −44.0479 −1.93723
\(518\) 0 0
\(519\) −60.7277 −2.66565
\(520\) 0 0
\(521\) −27.1515 −1.18953 −0.594764 0.803900i \(-0.702755\pi\)
−0.594764 + 0.803900i \(0.702755\pi\)
\(522\) 0 0
\(523\) 34.3372 1.50146 0.750730 0.660609i \(-0.229702\pi\)
0.750730 + 0.660609i \(0.229702\pi\)
\(524\) 0 0
\(525\) 29.2408i 1.27617i
\(526\) 0 0
\(527\) 13.0898i 0.570200i
\(528\) 0 0
\(529\) 10.5759 0.459824
\(530\) 0 0
\(531\) −59.0056 −2.56063
\(532\) 0 0
\(533\) 7.73567i 0.335069i
\(534\) 0 0
\(535\) 43.9189i 1.89878i
\(536\) 0 0
\(537\) −27.5209 −1.18762
\(538\) 0 0
\(539\) 1.74121 0.0749990
\(540\) 0 0
\(541\) −29.2077 −1.25574 −0.627869 0.778319i \(-0.716073\pi\)
−0.627869 + 0.778319i \(0.716073\pi\)
\(542\) 0 0
\(543\) 27.1712 1.16603
\(544\) 0 0
\(545\) 25.3330 1.08515
\(546\) 0 0
\(547\) −0.0988080 −0.00422472 −0.00211236 0.999998i \(-0.500672\pi\)
−0.00211236 + 0.999998i \(0.500672\pi\)
\(548\) 0 0
\(549\) 27.2162i 1.16156i
\(550\) 0 0
\(551\) 5.84976i 0.249208i
\(552\) 0 0
\(553\) 22.7208i 0.966186i
\(554\) 0 0
\(555\) −81.0423 −3.44006
\(556\) 0 0
\(557\) 13.3316i 0.564878i 0.959285 + 0.282439i \(0.0911434\pi\)
−0.959285 + 0.282439i \(0.908857\pi\)
\(558\) 0 0
\(559\) −8.67108 −0.366748
\(560\) 0 0
\(561\) 44.9697i 1.89862i
\(562\) 0 0
\(563\) 3.76092i 0.158504i 0.996855 + 0.0792518i \(0.0252531\pi\)
−0.996855 + 0.0792518i \(0.974747\pi\)
\(564\) 0 0
\(565\) 16.7494i 0.704654i
\(566\) 0 0
\(567\) 13.1023 0.550247
\(568\) 0 0
\(569\) 4.80431i 0.201407i 0.994916 + 0.100704i \(0.0321094\pi\)
−0.994916 + 0.100704i \(0.967891\pi\)
\(570\) 0 0
\(571\) 25.1843i 1.05393i 0.849887 + 0.526966i \(0.176670\pi\)
−0.849887 + 0.526966i \(0.823330\pi\)
\(572\) 0 0
\(573\) −33.3195 −1.39194
\(574\) 0 0
\(575\) 13.6339i 0.568572i
\(576\) 0 0
\(577\) 39.2955 1.63589 0.817946 0.575294i \(-0.195112\pi\)
0.817946 + 0.575294i \(0.195112\pi\)
\(578\) 0 0
\(579\) 57.6613 2.39632
\(580\) 0 0
\(581\) 22.3280i 0.926322i
\(582\) 0 0
\(583\) −19.5980 + 36.2639i −0.811668 + 1.50190i
\(584\) 0 0
\(585\) 14.4034i 0.595507i
\(586\) 0 0
\(587\) −26.4168 −1.09034 −0.545169 0.838326i \(-0.683535\pi\)
−0.545169 + 0.838326i \(0.683535\pi\)
\(588\) 0 0
\(589\) −4.81613 −0.198445
\(590\) 0 0
\(591\) 9.89162i 0.406887i
\(592\) 0 0
\(593\) 34.0602 1.39868 0.699342 0.714787i \(-0.253476\pi\)
0.699342 + 0.714787i \(0.253476\pi\)
\(594\) 0 0
\(595\) 20.9383i 0.858386i
\(596\) 0 0
\(597\) 51.5824i 2.11113i
\(598\) 0 0
\(599\) 8.59920 0.351354 0.175677 0.984448i \(-0.443789\pi\)
0.175677 + 0.984448i \(0.443789\pi\)
\(600\) 0 0
\(601\) 28.8138i 1.17534i 0.809101 + 0.587670i \(0.199955\pi\)
−0.809101 + 0.587670i \(0.800045\pi\)
\(602\) 0 0
\(603\) 23.6587i 0.963457i
\(604\) 0 0
\(605\) 62.7136i 2.54967i
\(606\) 0 0
\(607\) 0.0936573 0.00380143 0.00190072 0.999998i \(-0.499395\pi\)
0.00190072 + 0.999998i \(0.499395\pi\)
\(608\) 0 0
\(609\) 44.2221i 1.79197i
\(610\) 0 0
\(611\) −6.79290 −0.274812
\(612\) 0 0
\(613\) 39.7266i 1.60454i −0.596959 0.802272i \(-0.703625\pi\)
0.596959 0.802272i \(-0.296375\pi\)
\(614\) 0 0
\(615\) 77.0921i 3.10865i
\(616\) 0 0
\(617\) 16.5087i 0.664614i 0.943171 + 0.332307i \(0.107827\pi\)
−0.943171 + 0.332307i \(0.892173\pi\)
\(618\) 0 0
\(619\) −12.6618 −0.508920 −0.254460 0.967083i \(-0.581898\pi\)
−0.254460 + 0.967083i \(0.581898\pi\)
\(620\) 0 0
\(621\) −26.1533 −1.04950
\(622\) 0 0
\(623\) 36.2244 1.45130
\(624\) 0 0
\(625\) −29.3785 −1.17514
\(626\) 0 0
\(627\) 16.5457 0.660773
\(628\) 0 0
\(629\) −25.3120 −1.00925
\(630\) 0 0
\(631\) 33.0876i 1.31720i 0.752495 + 0.658598i \(0.228850\pi\)
−0.752495 + 0.658598i \(0.771150\pi\)
\(632\) 0 0
\(633\) 0.0489865i 0.00194704i
\(634\) 0 0
\(635\) −63.9293 −2.53696
\(636\) 0 0
\(637\) 0.268522 0.0106392
\(638\) 0 0
\(639\) 68.8400i 2.72327i
\(640\) 0 0
\(641\) 16.8134i 0.664091i 0.943263 + 0.332046i \(0.107739\pi\)
−0.943263 + 0.332046i \(0.892261\pi\)
\(642\) 0 0
\(643\) 15.2951 0.603179 0.301589 0.953438i \(-0.402483\pi\)
0.301589 + 0.953438i \(0.402483\pi\)
\(644\) 0 0
\(645\) 86.4142 3.40256
\(646\) 0 0
\(647\) 31.9648 1.25667 0.628333 0.777944i \(-0.283737\pi\)
0.628333 + 0.777944i \(0.283737\pi\)
\(648\) 0 0
\(649\) −60.3155 −2.36759
\(650\) 0 0
\(651\) −36.4082 −1.42695
\(652\) 0 0
\(653\) 9.95730 0.389659 0.194830 0.980837i \(-0.437585\pi\)
0.194830 + 0.980837i \(0.437585\pi\)
\(654\) 0 0
\(655\) 59.5971i 2.32865i
\(656\) 0 0
\(657\) 85.1377i 3.32154i
\(658\) 0 0
\(659\) 29.5258i 1.15016i 0.818097 + 0.575081i \(0.195029\pi\)
−0.818097 + 0.575081i \(0.804971\pi\)
\(660\) 0 0
\(661\) −1.51958 −0.0591048 −0.0295524 0.999563i \(-0.509408\pi\)
−0.0295524 + 0.999563i \(0.509408\pi\)
\(662\) 0 0
\(663\) 6.93506i 0.269335i
\(664\) 0 0
\(665\) 7.70383 0.298742
\(666\) 0 0
\(667\) 20.6191i 0.798375i
\(668\) 0 0
\(669\) 42.3299i 1.63657i
\(670\) 0 0
\(671\) 27.8204i 1.07400i
\(672\) 0 0
\(673\) 24.2225 0.933710 0.466855 0.884334i \(-0.345387\pi\)
0.466855 + 0.884334i \(0.345387\pi\)
\(674\) 0 0
\(675\) 28.7001i 1.10467i
\(676\) 0 0
\(677\) 7.79160i 0.299455i 0.988727 + 0.149728i \(0.0478397\pi\)
−0.988727 + 0.149728i \(0.952160\pi\)
\(678\) 0 0
\(679\) 45.3858 1.74175
\(680\) 0 0
\(681\) 78.2993i 3.00044i
\(682\) 0 0
\(683\) −27.3274 −1.04565 −0.522827 0.852439i \(-0.675123\pi\)
−0.522827 + 0.852439i \(0.675123\pi\)
\(684\) 0 0
\(685\) −40.0535 −1.53037
\(686\) 0 0
\(687\) 18.4432i 0.703650i
\(688\) 0 0
\(689\) −3.02234 + 5.59248i −0.115142 + 0.213057i
\(690\) 0 0
\(691\) 44.6616i 1.69901i −0.527582 0.849504i \(-0.676901\pi\)
0.527582 0.849504i \(-0.323099\pi\)
\(692\) 0 0
\(693\) 81.1363 3.08211
\(694\) 0 0
\(695\) −13.6736 −0.518671
\(696\) 0 0
\(697\) 24.0782i 0.912027i
\(698\) 0 0
\(699\) −50.2526 −1.90073
\(700\) 0 0
\(701\) 8.84731i 0.334158i 0.985943 + 0.167079i \(0.0534336\pi\)
−0.985943 + 0.167079i \(0.946566\pi\)
\(702\) 0 0
\(703\) 9.31304i 0.351248i
\(704\) 0 0
\(705\) 67.6967 2.54960
\(706\) 0 0
\(707\) 13.8441i 0.520662i
\(708\) 0 0
\(709\) 5.15654i 0.193658i −0.995301 0.0968289i \(-0.969130\pi\)
0.995301 0.0968289i \(-0.0308700\pi\)
\(710\) 0 0
\(711\) 48.6489i 1.82448i
\(712\) 0 0
\(713\) 16.9758 0.635748
\(714\) 0 0
\(715\) 14.7231i 0.550614i
\(716\) 0 0
\(717\) 47.9034 1.78898
\(718\) 0 0
\(719\) 19.0114i 0.709004i −0.935055 0.354502i \(-0.884650\pi\)
0.935055 0.354502i \(-0.115350\pi\)
\(720\) 0 0
\(721\) 47.0273i 1.75139i
\(722\) 0 0
\(723\) 4.18050i 0.155474i
\(724\) 0 0
\(725\) −22.6269 −0.840344
\(726\) 0 0
\(727\) 17.9647 0.666274 0.333137 0.942879i \(-0.391893\pi\)
0.333137 + 0.942879i \(0.391893\pi\)
\(728\) 0 0
\(729\) 36.9923 1.37008
\(730\) 0 0
\(731\) 26.9898 0.998252
\(732\) 0 0
\(733\) −37.1881 −1.37358 −0.686788 0.726858i \(-0.740980\pi\)
−0.686788 + 0.726858i \(0.740980\pi\)
\(734\) 0 0
\(735\) −2.67604 −0.0987071
\(736\) 0 0
\(737\) 24.1839i 0.890825i
\(738\) 0 0
\(739\) 27.4155i 1.00850i −0.863559 0.504248i \(-0.831770\pi\)
0.863559 0.504248i \(-0.168230\pi\)
\(740\) 0 0
\(741\) 2.55162 0.0937361
\(742\) 0 0
\(743\) −2.06951 −0.0759230 −0.0379615 0.999279i \(-0.512086\pi\)
−0.0379615 + 0.999279i \(0.512086\pi\)
\(744\) 0 0
\(745\) 5.57430i 0.204227i
\(746\) 0 0
\(747\) 47.8079i 1.74920i
\(748\) 0 0
\(749\) 38.1533 1.39409
\(750\) 0 0
\(751\) 52.1874 1.90435 0.952173 0.305561i \(-0.0988440\pi\)
0.952173 + 0.305561i \(0.0988440\pi\)
\(752\) 0 0
\(753\) 63.7250 2.32227
\(754\) 0 0
\(755\) −18.1874 −0.661906
\(756\) 0 0
\(757\) −17.9213 −0.651360 −0.325680 0.945480i \(-0.605593\pi\)
−0.325680 + 0.945480i \(0.605593\pi\)
\(758\) 0 0
\(759\) −58.3200 −2.11688
\(760\) 0 0
\(761\) 14.2321i 0.515912i 0.966157 + 0.257956i \(0.0830489\pi\)
−0.966157 + 0.257956i \(0.916951\pi\)
\(762\) 0 0
\(763\) 22.0073i 0.796718i
\(764\) 0 0
\(765\) 44.8323i 1.62091i
\(766\) 0 0
\(767\) −9.30163 −0.335862
\(768\) 0 0
\(769\) 38.0329i 1.37150i −0.727836 0.685751i \(-0.759474\pi\)
0.727836 0.685751i \(-0.240526\pi\)
\(770\) 0 0
\(771\) −56.8673 −2.04802
\(772\) 0 0
\(773\) 2.06510i 0.0742766i 0.999310 + 0.0371383i \(0.0118242\pi\)
−0.999310 + 0.0371383i \(0.988176\pi\)
\(774\) 0 0
\(775\) 18.6289i 0.669168i
\(776\) 0 0
\(777\) 70.4032i 2.52570i
\(778\) 0 0
\(779\) 8.85910 0.317410
\(780\) 0 0
\(781\) 70.3682i 2.51797i
\(782\) 0 0
\(783\) 43.4044i 1.55115i
\(784\) 0 0
\(785\) 52.8687 1.88697
\(786\) 0 0
\(787\) 0.607130i 0.0216418i −0.999941 0.0108209i \(-0.996556\pi\)
0.999941 0.0108209i \(-0.00344447\pi\)
\(788\) 0 0
\(789\) −85.6508 −3.04925
\(790\) 0 0
\(791\) −14.5506 −0.517359
\(792\) 0 0
\(793\) 4.29036i 0.152355i
\(794\) 0 0
\(795\) 30.1200 55.7335i 1.06824 1.97666i
\(796\) 0 0
\(797\) 12.6605i 0.448457i 0.974537 + 0.224228i \(0.0719861\pi\)
−0.974537 + 0.224228i \(0.928014\pi\)
\(798\) 0 0
\(799\) 21.1437 0.748011
\(800\) 0 0
\(801\) −77.5624 −2.74053
\(802\) 0 0
\(803\) 87.0277i 3.07114i
\(804\) 0 0
\(805\) −27.1543 −0.957063
\(806\) 0 0
\(807\) 66.0215i 2.32407i
\(808\) 0 0
\(809\) 24.2797i 0.853629i 0.904339 + 0.426815i \(0.140364\pi\)
−0.904339 + 0.426815i \(0.859636\pi\)
\(810\) 0 0
\(811\) 35.0449 1.23059 0.615296 0.788296i \(-0.289036\pi\)
0.615296 + 0.788296i \(0.289036\pi\)
\(812\) 0 0
\(813\) 22.7398i 0.797519i
\(814\) 0 0
\(815\) 70.0835i 2.45492i
\(816\) 0 0
\(817\) 9.93035i 0.347419i
\(818\) 0 0
\(819\) 12.5125 0.437223
\(820\) 0 0
\(821\) 4.54165i 0.158505i 0.996855 + 0.0792523i \(0.0252533\pi\)
−0.996855 + 0.0792523i \(0.974747\pi\)
\(822\) 0 0
\(823\) −24.6886 −0.860592 −0.430296 0.902688i \(-0.641591\pi\)
−0.430296 + 0.902688i \(0.641591\pi\)
\(824\) 0 0
\(825\) 63.9991i 2.22816i
\(826\) 0 0
\(827\) 17.2861i 0.601096i −0.953767 0.300548i \(-0.902830\pi\)
0.953767 0.300548i \(-0.0971695\pi\)
\(828\) 0 0
\(829\) 8.94233i 0.310580i −0.987869 0.155290i \(-0.950369\pi\)
0.987869 0.155290i \(-0.0496312\pi\)
\(830\) 0 0
\(831\) 24.7471 0.858467
\(832\) 0 0
\(833\) −0.835807 −0.0289590
\(834\) 0 0
\(835\) 9.05182 0.313251
\(836\) 0 0
\(837\) 35.7350 1.23518
\(838\) 0 0
\(839\) −33.5160 −1.15710 −0.578550 0.815647i \(-0.696381\pi\)
−0.578550 + 0.815647i \(0.696381\pi\)
\(840\) 0 0
\(841\) 5.21971 0.179990
\(842\) 0 0
\(843\) 47.2144i 1.62615i
\(844\) 0 0
\(845\) 36.4424i 1.25366i
\(846\) 0 0
\(847\) 54.4807 1.87198
\(848\) 0 0
\(849\) −38.4924 −1.32106
\(850\) 0 0
\(851\) 32.8264i 1.12527i
\(852\) 0 0
\(853\) 14.2734i 0.488713i 0.969685 + 0.244356i \(0.0785767\pi\)
−0.969685 + 0.244356i \(0.921423\pi\)
\(854\) 0 0
\(855\) −16.4951 −0.564122
\(856\) 0 0
\(857\) −10.9870 −0.375307 −0.187654 0.982235i \(-0.560088\pi\)
−0.187654 + 0.982235i \(0.560088\pi\)
\(858\) 0 0
\(859\) −2.00168 −0.0682964 −0.0341482 0.999417i \(-0.510872\pi\)
−0.0341482 + 0.999417i \(0.510872\pi\)
\(860\) 0 0
\(861\) 66.9716 2.28238
\(862\) 0 0
\(863\) 31.1696 1.06102 0.530512 0.847677i \(-0.322000\pi\)
0.530512 + 0.847677i \(0.322000\pi\)
\(864\) 0 0
\(865\) 61.8860 2.10419
\(866\) 0 0
\(867\) 28.0909i 0.954017i
\(868\) 0 0
\(869\) 49.7288i 1.68694i
\(870\) 0 0
\(871\) 3.72955i 0.126371i
\(872\) 0 0
\(873\) −97.1783 −3.28899
\(874\) 0 0
\(875\) 8.72065i 0.294812i
\(876\) 0 0
\(877\) −14.8240 −0.500571 −0.250285 0.968172i \(-0.580524\pi\)
−0.250285 + 0.968172i \(0.580524\pi\)
\(878\) 0 0
\(879\) 6.41836i 0.216486i
\(880\) 0 0
\(881\) 3.61644i 0.121841i −0.998143 0.0609204i \(-0.980596\pi\)
0.998143 0.0609204i \(-0.0194036\pi\)
\(882\) 0 0
\(883\) 25.4476i 0.856381i −0.903689 0.428190i \(-0.859151\pi\)
0.903689 0.428190i \(-0.140849\pi\)
\(884\) 0 0
\(885\) 92.6981 3.11601
\(886\) 0 0
\(887\) 1.30599i 0.0438507i 0.999760 + 0.0219254i \(0.00697962\pi\)
−0.999760 + 0.0219254i \(0.993020\pi\)
\(888\) 0 0
\(889\) 55.5368i 1.86264i
\(890\) 0 0
\(891\) −28.6770 −0.960717
\(892\) 0 0
\(893\) 7.77941i 0.260328i
\(894\) 0 0
\(895\) 28.0459 0.937470
\(896\) 0 0
\(897\) −8.99389 −0.300297
\(898\) 0 0
\(899\) 28.1732i 0.939629i
\(900\) 0 0
\(901\) 9.40737 17.4073i 0.313405 0.579920i
\(902\) 0 0
\(903\) 75.0698i 2.49817i
\(904\) 0 0
\(905\) −27.6895 −0.920429
\(906\) 0 0
\(907\) −30.9488 −1.02764 −0.513820 0.857898i \(-0.671770\pi\)
−0.513820 + 0.857898i \(0.671770\pi\)
\(908\) 0 0
\(909\) 29.6425i 0.983180i
\(910\) 0 0
\(911\) −2.79189 −0.0924995 −0.0462497 0.998930i \(-0.514727\pi\)
−0.0462497 + 0.998930i \(0.514727\pi\)
\(912\) 0 0
\(913\) 48.8692i 1.61733i
\(914\) 0 0
\(915\) 42.7568i 1.41350i
\(916\) 0 0
\(917\) 51.7733 1.70970
\(918\) 0 0
\(919\) 31.8280i 1.04991i 0.851130 + 0.524954i \(0.175918\pi\)
−0.851130 + 0.524954i \(0.824082\pi\)
\(920\) 0 0
\(921\) 6.57420i 0.216627i
\(922\) 0 0
\(923\) 10.8519i 0.357195i
\(924\) 0 0
\(925\) 36.0229 1.18443
\(926\) 0 0
\(927\) 100.693i 3.30720i
\(928\) 0 0
\(929\) 38.3188 1.25720 0.628600 0.777729i \(-0.283628\pi\)
0.628600 + 0.777729i \(0.283628\pi\)
\(930\) 0 0
\(931\) 0.307519i 0.0100785i
\(932\) 0 0
\(933\) 17.5933i 0.575978i
\(934\) 0 0
\(935\) 45.8275i 1.49872i
\(936\) 0 0
\(937\) 51.1830 1.67208 0.836038 0.548672i \(-0.184866\pi\)
0.836038 + 0.548672i \(0.184866\pi\)
\(938\) 0 0
\(939\) −75.6337 −2.46821
\(940\) 0 0
\(941\) 1.87662 0.0611759 0.0305880 0.999532i \(-0.490262\pi\)
0.0305880 + 0.999532i \(0.490262\pi\)
\(942\) 0 0
\(943\) −31.2264 −1.01687
\(944\) 0 0
\(945\) −57.1613 −1.85946
\(946\) 0 0
\(947\) 4.86890 0.158218 0.0791090 0.996866i \(-0.474792\pi\)
0.0791090 + 0.996866i \(0.474792\pi\)
\(948\) 0 0
\(949\) 13.4211i 0.435667i
\(950\) 0 0
\(951\) 56.0459i 1.81741i
\(952\) 0 0
\(953\) −9.45162 −0.306168 −0.153084 0.988213i \(-0.548920\pi\)
−0.153084 + 0.988213i \(0.548920\pi\)
\(954\) 0 0
\(955\) 33.9551 1.09876
\(956\) 0 0
\(957\) 96.7886i 3.12873i
\(958\) 0 0
\(959\) 34.7953i 1.12360i
\(960\) 0 0
\(961\) 7.80488 0.251770
\(962\) 0 0
\(963\) −81.6923 −2.63250
\(964\) 0 0
\(965\) −58.7612 −1.89159
\(966\) 0 0
\(967\) −4.39579 −0.141359 −0.0706796 0.997499i \(-0.522517\pi\)
−0.0706796 + 0.997499i \(0.522517\pi\)
\(968\) 0 0
\(969\) −7.94222 −0.255141
\(970\) 0 0
\(971\) 57.0827 1.83187 0.915935 0.401326i \(-0.131450\pi\)
0.915935 + 0.401326i \(0.131450\pi\)
\(972\) 0 0
\(973\) 11.8786i 0.380810i
\(974\) 0 0
\(975\) 9.86969i 0.316083i
\(976\) 0 0
\(977\) 38.1020i 1.21899i 0.792790 + 0.609495i \(0.208628\pi\)
−0.792790 + 0.609495i \(0.791372\pi\)
\(978\) 0 0
\(979\) −79.2842 −2.53393
\(980\) 0 0
\(981\) 47.1212i 1.50446i
\(982\) 0 0
\(983\) −58.2692 −1.85850 −0.929250 0.369452i \(-0.879546\pi\)
−0.929250 + 0.369452i \(0.879546\pi\)
\(984\) 0 0
\(985\) 10.0803i 0.321185i
\(986\) 0 0
\(987\) 58.8095i 1.87193i
\(988\) 0 0
\(989\) 35.0023i 1.11301i
\(990\) 0 0
\(991\) −0.404629 −0.0128535 −0.00642673 0.999979i \(-0.502046\pi\)
−0.00642673 + 0.999979i \(0.502046\pi\)
\(992\) 0 0
\(993\) 25.6424i 0.813737i
\(994\) 0 0
\(995\) 52.5663i 1.66647i
\(996\) 0 0
\(997\) 33.5759 1.06336 0.531679 0.846946i \(-0.321561\pi\)
0.531679 + 0.846946i \(0.321561\pi\)
\(998\) 0 0
\(999\) 69.1014i 2.18627i
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 4028.2.c.a.3497.7 82
53.52 even 2 inner 4028.2.c.a.3497.76 yes 82
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4028.2.c.a.3497.7 82 1.1 even 1 trivial
4028.2.c.a.3497.76 yes 82 53.52 even 2 inner