Properties

Label 4024.2.a.f.1.28
Level $4024$
Weight $2$
Character 4024.1
Self dual yes
Analytic conductor $32.132$
Analytic rank $0$
Dimension $33$
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] = [4024,2,Mod(1,4024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4024, 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("4024.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4024 = 2^{3} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4024.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(32.1318017734\)
Analytic rank: \(0\)
Dimension: \(33\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.28
Character \(\chi\) \(=\) 4024.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.45195 q^{3} +0.279836 q^{5} +3.64586 q^{7} +3.01204 q^{9} +O(q^{10})\) \(q+2.45195 q^{3} +0.279836 q^{5} +3.64586 q^{7} +3.01204 q^{9} +4.16835 q^{11} -0.225891 q^{13} +0.686143 q^{15} +1.76597 q^{17} -2.09795 q^{19} +8.93946 q^{21} -2.12140 q^{23} -4.92169 q^{25} +0.0295247 q^{27} +1.16954 q^{29} +7.55159 q^{31} +10.2206 q^{33} +1.02024 q^{35} -2.81161 q^{37} -0.553872 q^{39} +0.812621 q^{41} +7.36747 q^{43} +0.842877 q^{45} +4.04461 q^{47} +6.29232 q^{49} +4.33006 q^{51} +0.886240 q^{53} +1.16645 q^{55} -5.14406 q^{57} +0.749409 q^{59} -7.77767 q^{61} +10.9815 q^{63} -0.0632124 q^{65} +14.9861 q^{67} -5.20157 q^{69} -6.01214 q^{71} -15.3943 q^{73} -12.0677 q^{75} +15.1972 q^{77} +15.4183 q^{79} -8.96373 q^{81} -3.28502 q^{83} +0.494181 q^{85} +2.86765 q^{87} -3.17270 q^{89} -0.823567 q^{91} +18.5161 q^{93} -0.587082 q^{95} -9.57560 q^{97} +12.5552 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9} + 22 q^{11} + 25 q^{13} - 4 q^{15} + 17 q^{17} + 6 q^{19} + 18 q^{21} + 16 q^{23} + 47 q^{25} - 20 q^{27} + 47 q^{29} - 7 q^{31} - 6 q^{33} + 19 q^{35} + 75 q^{37} + 21 q^{39} + 22 q^{41} - 5 q^{43} + 33 q^{45} + 10 q^{47} + 31 q^{49} + 9 q^{51} + 64 q^{53} - 3 q^{55} + 5 q^{57} + 28 q^{59} + 49 q^{61} - 10 q^{63} + 46 q^{65} - 14 q^{67} + 30 q^{69} + 35 q^{71} + 19 q^{73} - 33 q^{75} + 32 q^{77} - 12 q^{79} + 57 q^{81} + 82 q^{85} - 5 q^{87} + 42 q^{89} - 15 q^{91} + 55 q^{93} + 33 q^{95} + 4 q^{97} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.45195 1.41563 0.707816 0.706397i \(-0.249681\pi\)
0.707816 + 0.706397i \(0.249681\pi\)
\(4\) 0 0
\(5\) 0.279836 0.125146 0.0625732 0.998040i \(-0.480069\pi\)
0.0625732 + 0.998040i \(0.480069\pi\)
\(6\) 0 0
\(7\) 3.64586 1.37801 0.689003 0.724758i \(-0.258048\pi\)
0.689003 + 0.724758i \(0.258048\pi\)
\(8\) 0 0
\(9\) 3.01204 1.00401
\(10\) 0 0
\(11\) 4.16835 1.25680 0.628402 0.777889i \(-0.283709\pi\)
0.628402 + 0.777889i \(0.283709\pi\)
\(12\) 0 0
\(13\) −0.225891 −0.0626509 −0.0313254 0.999509i \(-0.509973\pi\)
−0.0313254 + 0.999509i \(0.509973\pi\)
\(14\) 0 0
\(15\) 0.686143 0.177161
\(16\) 0 0
\(17\) 1.76597 0.428310 0.214155 0.976800i \(-0.431300\pi\)
0.214155 + 0.976800i \(0.431300\pi\)
\(18\) 0 0
\(19\) −2.09795 −0.481303 −0.240651 0.970612i \(-0.577361\pi\)
−0.240651 + 0.970612i \(0.577361\pi\)
\(20\) 0 0
\(21\) 8.93946 1.95075
\(22\) 0 0
\(23\) −2.12140 −0.442343 −0.221172 0.975235i \(-0.570988\pi\)
−0.221172 + 0.975235i \(0.570988\pi\)
\(24\) 0 0
\(25\) −4.92169 −0.984338
\(26\) 0 0
\(27\) 0.0295247 0.00568202
\(28\) 0 0
\(29\) 1.16954 0.217178 0.108589 0.994087i \(-0.465367\pi\)
0.108589 + 0.994087i \(0.465367\pi\)
\(30\) 0 0
\(31\) 7.55159 1.35631 0.678153 0.734921i \(-0.262781\pi\)
0.678153 + 0.734921i \(0.262781\pi\)
\(32\) 0 0
\(33\) 10.2206 1.77917
\(34\) 0 0
\(35\) 1.02024 0.172453
\(36\) 0 0
\(37\) −2.81161 −0.462226 −0.231113 0.972927i \(-0.574237\pi\)
−0.231113 + 0.972927i \(0.574237\pi\)
\(38\) 0 0
\(39\) −0.553872 −0.0886906
\(40\) 0 0
\(41\) 0.812621 0.126910 0.0634550 0.997985i \(-0.479788\pi\)
0.0634550 + 0.997985i \(0.479788\pi\)
\(42\) 0 0
\(43\) 7.36747 1.12353 0.561765 0.827297i \(-0.310123\pi\)
0.561765 + 0.827297i \(0.310123\pi\)
\(44\) 0 0
\(45\) 0.842877 0.125649
\(46\) 0 0
\(47\) 4.04461 0.589967 0.294984 0.955502i \(-0.404686\pi\)
0.294984 + 0.955502i \(0.404686\pi\)
\(48\) 0 0
\(49\) 6.29232 0.898903
\(50\) 0 0
\(51\) 4.33006 0.606330
\(52\) 0 0
\(53\) 0.886240 0.121734 0.0608672 0.998146i \(-0.480613\pi\)
0.0608672 + 0.998146i \(0.480613\pi\)
\(54\) 0 0
\(55\) 1.16645 0.157285
\(56\) 0 0
\(57\) −5.14406 −0.681348
\(58\) 0 0
\(59\) 0.749409 0.0975647 0.0487824 0.998809i \(-0.484466\pi\)
0.0487824 + 0.998809i \(0.484466\pi\)
\(60\) 0 0
\(61\) −7.77767 −0.995829 −0.497914 0.867226i \(-0.665901\pi\)
−0.497914 + 0.867226i \(0.665901\pi\)
\(62\) 0 0
\(63\) 10.9815 1.38354
\(64\) 0 0
\(65\) −0.0632124 −0.00784053
\(66\) 0 0
\(67\) 14.9861 1.83084 0.915420 0.402500i \(-0.131859\pi\)
0.915420 + 0.402500i \(0.131859\pi\)
\(68\) 0 0
\(69\) −5.20157 −0.626195
\(70\) 0 0
\(71\) −6.01214 −0.713510 −0.356755 0.934198i \(-0.616117\pi\)
−0.356755 + 0.934198i \(0.616117\pi\)
\(72\) 0 0
\(73\) −15.3943 −1.80177 −0.900885 0.434059i \(-0.857081\pi\)
−0.900885 + 0.434059i \(0.857081\pi\)
\(74\) 0 0
\(75\) −12.0677 −1.39346
\(76\) 0 0
\(77\) 15.1972 1.73189
\(78\) 0 0
\(79\) 15.4183 1.73469 0.867346 0.497705i \(-0.165824\pi\)
0.867346 + 0.497705i \(0.165824\pi\)
\(80\) 0 0
\(81\) −8.96373 −0.995970
\(82\) 0 0
\(83\) −3.28502 −0.360578 −0.180289 0.983614i \(-0.557703\pi\)
−0.180289 + 0.983614i \(0.557703\pi\)
\(84\) 0 0
\(85\) 0.494181 0.0536015
\(86\) 0 0
\(87\) 2.86765 0.307445
\(88\) 0 0
\(89\) −3.17270 −0.336306 −0.168153 0.985761i \(-0.553780\pi\)
−0.168153 + 0.985761i \(0.553780\pi\)
\(90\) 0 0
\(91\) −0.823567 −0.0863333
\(92\) 0 0
\(93\) 18.5161 1.92003
\(94\) 0 0
\(95\) −0.587082 −0.0602333
\(96\) 0 0
\(97\) −9.57560 −0.972255 −0.486128 0.873888i \(-0.661591\pi\)
−0.486128 + 0.873888i \(0.661591\pi\)
\(98\) 0 0
\(99\) 12.5552 1.26185
\(100\) 0 0
\(101\) −13.1773 −1.31119 −0.655594 0.755113i \(-0.727582\pi\)
−0.655594 + 0.755113i \(0.727582\pi\)
\(102\) 0 0
\(103\) −15.0313 −1.48108 −0.740541 0.672012i \(-0.765431\pi\)
−0.740541 + 0.672012i \(0.765431\pi\)
\(104\) 0 0
\(105\) 2.50158 0.244129
\(106\) 0 0
\(107\) −9.74626 −0.942206 −0.471103 0.882078i \(-0.656144\pi\)
−0.471103 + 0.882078i \(0.656144\pi\)
\(108\) 0 0
\(109\) −7.14740 −0.684597 −0.342299 0.939591i \(-0.611205\pi\)
−0.342299 + 0.939591i \(0.611205\pi\)
\(110\) 0 0
\(111\) −6.89392 −0.654342
\(112\) 0 0
\(113\) 14.8344 1.39550 0.697751 0.716340i \(-0.254184\pi\)
0.697751 + 0.716340i \(0.254184\pi\)
\(114\) 0 0
\(115\) −0.593645 −0.0553577
\(116\) 0 0
\(117\) −0.680393 −0.0629023
\(118\) 0 0
\(119\) 6.43848 0.590214
\(120\) 0 0
\(121\) 6.37513 0.579557
\(122\) 0 0
\(123\) 1.99250 0.179658
\(124\) 0 0
\(125\) −2.77645 −0.248333
\(126\) 0 0
\(127\) 9.44701 0.838287 0.419143 0.907920i \(-0.362330\pi\)
0.419143 + 0.907920i \(0.362330\pi\)
\(128\) 0 0
\(129\) 18.0646 1.59050
\(130\) 0 0
\(131\) −9.27582 −0.810433 −0.405216 0.914221i \(-0.632804\pi\)
−0.405216 + 0.914221i \(0.632804\pi\)
\(132\) 0 0
\(133\) −7.64884 −0.663239
\(134\) 0 0
\(135\) 0.00826206 0.000711085 0
\(136\) 0 0
\(137\) −13.6122 −1.16297 −0.581485 0.813557i \(-0.697528\pi\)
−0.581485 + 0.813557i \(0.697528\pi\)
\(138\) 0 0
\(139\) −11.5093 −0.976205 −0.488102 0.872786i \(-0.662311\pi\)
−0.488102 + 0.872786i \(0.662311\pi\)
\(140\) 0 0
\(141\) 9.91717 0.835177
\(142\) 0 0
\(143\) −0.941592 −0.0787399
\(144\) 0 0
\(145\) 0.327280 0.0271791
\(146\) 0 0
\(147\) 15.4284 1.27252
\(148\) 0 0
\(149\) 14.2726 1.16925 0.584627 0.811302i \(-0.301241\pi\)
0.584627 + 0.811302i \(0.301241\pi\)
\(150\) 0 0
\(151\) 13.5122 1.09960 0.549802 0.835295i \(-0.314703\pi\)
0.549802 + 0.835295i \(0.314703\pi\)
\(152\) 0 0
\(153\) 5.31917 0.430029
\(154\) 0 0
\(155\) 2.11321 0.169737
\(156\) 0 0
\(157\) 19.1266 1.52647 0.763233 0.646123i \(-0.223611\pi\)
0.763233 + 0.646123i \(0.223611\pi\)
\(158\) 0 0
\(159\) 2.17301 0.172331
\(160\) 0 0
\(161\) −7.73435 −0.609552
\(162\) 0 0
\(163\) −10.7533 −0.842262 −0.421131 0.907000i \(-0.638367\pi\)
−0.421131 + 0.907000i \(0.638367\pi\)
\(164\) 0 0
\(165\) 2.86008 0.222657
\(166\) 0 0
\(167\) 2.46576 0.190807 0.0954033 0.995439i \(-0.469586\pi\)
0.0954033 + 0.995439i \(0.469586\pi\)
\(168\) 0 0
\(169\) −12.9490 −0.996075
\(170\) 0 0
\(171\) −6.31911 −0.483235
\(172\) 0 0
\(173\) −22.4963 −1.71036 −0.855179 0.518333i \(-0.826553\pi\)
−0.855179 + 0.518333i \(0.826553\pi\)
\(174\) 0 0
\(175\) −17.9438 −1.35643
\(176\) 0 0
\(177\) 1.83751 0.138116
\(178\) 0 0
\(179\) 13.7480 1.02757 0.513787 0.857918i \(-0.328242\pi\)
0.513787 + 0.857918i \(0.328242\pi\)
\(180\) 0 0
\(181\) 17.4413 1.29640 0.648202 0.761469i \(-0.275521\pi\)
0.648202 + 0.761469i \(0.275521\pi\)
\(182\) 0 0
\(183\) −19.0704 −1.40973
\(184\) 0 0
\(185\) −0.786790 −0.0578460
\(186\) 0 0
\(187\) 7.36117 0.538302
\(188\) 0 0
\(189\) 0.107643 0.00782987
\(190\) 0 0
\(191\) 24.4928 1.77224 0.886118 0.463459i \(-0.153392\pi\)
0.886118 + 0.463459i \(0.153392\pi\)
\(192\) 0 0
\(193\) 13.7880 0.992484 0.496242 0.868184i \(-0.334713\pi\)
0.496242 + 0.868184i \(0.334713\pi\)
\(194\) 0 0
\(195\) −0.154993 −0.0110993
\(196\) 0 0
\(197\) −1.93368 −0.137769 −0.0688846 0.997625i \(-0.521944\pi\)
−0.0688846 + 0.997625i \(0.521944\pi\)
\(198\) 0 0
\(199\) −13.3959 −0.949613 −0.474807 0.880090i \(-0.657482\pi\)
−0.474807 + 0.880090i \(0.657482\pi\)
\(200\) 0 0
\(201\) 36.7450 2.59180
\(202\) 0 0
\(203\) 4.26399 0.299273
\(204\) 0 0
\(205\) 0.227401 0.0158823
\(206\) 0 0
\(207\) −6.38975 −0.444119
\(208\) 0 0
\(209\) −8.74499 −0.604903
\(210\) 0 0
\(211\) 11.2044 0.771342 0.385671 0.922636i \(-0.373970\pi\)
0.385671 + 0.922636i \(0.373970\pi\)
\(212\) 0 0
\(213\) −14.7415 −1.01007
\(214\) 0 0
\(215\) 2.06168 0.140606
\(216\) 0 0
\(217\) 27.5321 1.86900
\(218\) 0 0
\(219\) −37.7461 −2.55064
\(220\) 0 0
\(221\) −0.398916 −0.0268340
\(222\) 0 0
\(223\) −23.7475 −1.59025 −0.795124 0.606447i \(-0.792594\pi\)
−0.795124 + 0.606447i \(0.792594\pi\)
\(224\) 0 0
\(225\) −14.8243 −0.988289
\(226\) 0 0
\(227\) 22.0382 1.46273 0.731364 0.681987i \(-0.238884\pi\)
0.731364 + 0.681987i \(0.238884\pi\)
\(228\) 0 0
\(229\) 10.5814 0.699238 0.349619 0.936892i \(-0.386311\pi\)
0.349619 + 0.936892i \(0.386311\pi\)
\(230\) 0 0
\(231\) 37.2628 2.45171
\(232\) 0 0
\(233\) 1.44152 0.0944370 0.0472185 0.998885i \(-0.484964\pi\)
0.0472185 + 0.998885i \(0.484964\pi\)
\(234\) 0 0
\(235\) 1.13183 0.0738323
\(236\) 0 0
\(237\) 37.8048 2.45569
\(238\) 0 0
\(239\) 4.72999 0.305957 0.152979 0.988229i \(-0.451113\pi\)
0.152979 + 0.988229i \(0.451113\pi\)
\(240\) 0 0
\(241\) −7.11002 −0.457997 −0.228998 0.973427i \(-0.573545\pi\)
−0.228998 + 0.973427i \(0.573545\pi\)
\(242\) 0 0
\(243\) −22.0672 −1.41561
\(244\) 0 0
\(245\) 1.76082 0.112495
\(246\) 0 0
\(247\) 0.473908 0.0301540
\(248\) 0 0
\(249\) −8.05470 −0.510446
\(250\) 0 0
\(251\) −10.0189 −0.632387 −0.316193 0.948695i \(-0.602405\pi\)
−0.316193 + 0.948695i \(0.602405\pi\)
\(252\) 0 0
\(253\) −8.84275 −0.555939
\(254\) 0 0
\(255\) 1.21171 0.0758800
\(256\) 0 0
\(257\) −1.56785 −0.0977998 −0.0488999 0.998804i \(-0.515572\pi\)
−0.0488999 + 0.998804i \(0.515572\pi\)
\(258\) 0 0
\(259\) −10.2508 −0.636951
\(260\) 0 0
\(261\) 3.52271 0.218050
\(262\) 0 0
\(263\) −0.115444 −0.00711857 −0.00355929 0.999994i \(-0.501133\pi\)
−0.00355929 + 0.999994i \(0.501133\pi\)
\(264\) 0 0
\(265\) 0.248002 0.0152346
\(266\) 0 0
\(267\) −7.77930 −0.476085
\(268\) 0 0
\(269\) 10.6495 0.649311 0.324656 0.945832i \(-0.394752\pi\)
0.324656 + 0.945832i \(0.394752\pi\)
\(270\) 0 0
\(271\) −4.22698 −0.256771 −0.128385 0.991724i \(-0.540979\pi\)
−0.128385 + 0.991724i \(0.540979\pi\)
\(272\) 0 0
\(273\) −2.01934 −0.122216
\(274\) 0 0
\(275\) −20.5153 −1.23712
\(276\) 0 0
\(277\) −2.26245 −0.135938 −0.0679688 0.997687i \(-0.521652\pi\)
−0.0679688 + 0.997687i \(0.521652\pi\)
\(278\) 0 0
\(279\) 22.7457 1.36175
\(280\) 0 0
\(281\) 21.5615 1.28625 0.643125 0.765761i \(-0.277638\pi\)
0.643125 + 0.765761i \(0.277638\pi\)
\(282\) 0 0
\(283\) 12.8384 0.763164 0.381582 0.924335i \(-0.375379\pi\)
0.381582 + 0.924335i \(0.375379\pi\)
\(284\) 0 0
\(285\) −1.43949 −0.0852682
\(286\) 0 0
\(287\) 2.96271 0.174883
\(288\) 0 0
\(289\) −13.8814 −0.816550
\(290\) 0 0
\(291\) −23.4789 −1.37636
\(292\) 0 0
\(293\) 19.8753 1.16112 0.580562 0.814216i \(-0.302833\pi\)
0.580562 + 0.814216i \(0.302833\pi\)
\(294\) 0 0
\(295\) 0.209712 0.0122099
\(296\) 0 0
\(297\) 0.123069 0.00714119
\(298\) 0 0
\(299\) 0.479206 0.0277132
\(300\) 0 0
\(301\) 26.8608 1.54823
\(302\) 0 0
\(303\) −32.3100 −1.85616
\(304\) 0 0
\(305\) −2.17647 −0.124624
\(306\) 0 0
\(307\) −10.5054 −0.599575 −0.299787 0.954006i \(-0.596916\pi\)
−0.299787 + 0.954006i \(0.596916\pi\)
\(308\) 0 0
\(309\) −36.8560 −2.09667
\(310\) 0 0
\(311\) 10.5735 0.599566 0.299783 0.954007i \(-0.403086\pi\)
0.299783 + 0.954007i \(0.403086\pi\)
\(312\) 0 0
\(313\) −0.370212 −0.0209256 −0.0104628 0.999945i \(-0.503330\pi\)
−0.0104628 + 0.999945i \(0.503330\pi\)
\(314\) 0 0
\(315\) 3.07302 0.173145
\(316\) 0 0
\(317\) 10.9384 0.614362 0.307181 0.951651i \(-0.400614\pi\)
0.307181 + 0.951651i \(0.400614\pi\)
\(318\) 0 0
\(319\) 4.87505 0.272951
\(320\) 0 0
\(321\) −23.8973 −1.33382
\(322\) 0 0
\(323\) −3.70491 −0.206147
\(324\) 0 0
\(325\) 1.11177 0.0616696
\(326\) 0 0
\(327\) −17.5250 −0.969137
\(328\) 0 0
\(329\) 14.7461 0.812979
\(330\) 0 0
\(331\) 15.4728 0.850460 0.425230 0.905085i \(-0.360193\pi\)
0.425230 + 0.905085i \(0.360193\pi\)
\(332\) 0 0
\(333\) −8.46869 −0.464081
\(334\) 0 0
\(335\) 4.19364 0.229123
\(336\) 0 0
\(337\) −6.67136 −0.363412 −0.181706 0.983353i \(-0.558162\pi\)
−0.181706 + 0.983353i \(0.558162\pi\)
\(338\) 0 0
\(339\) 36.3731 1.97552
\(340\) 0 0
\(341\) 31.4776 1.70461
\(342\) 0 0
\(343\) −2.58010 −0.139312
\(344\) 0 0
\(345\) −1.45559 −0.0783661
\(346\) 0 0
\(347\) −18.5236 −0.994400 −0.497200 0.867636i \(-0.665638\pi\)
−0.497200 + 0.867636i \(0.665638\pi\)
\(348\) 0 0
\(349\) −12.7643 −0.683256 −0.341628 0.939835i \(-0.610978\pi\)
−0.341628 + 0.939835i \(0.610978\pi\)
\(350\) 0 0
\(351\) −0.00666935 −0.000355984 0
\(352\) 0 0
\(353\) −17.2783 −0.919633 −0.459816 0.888014i \(-0.652085\pi\)
−0.459816 + 0.888014i \(0.652085\pi\)
\(354\) 0 0
\(355\) −1.68241 −0.0892932
\(356\) 0 0
\(357\) 15.7868 0.835526
\(358\) 0 0
\(359\) 19.2555 1.01627 0.508133 0.861279i \(-0.330336\pi\)
0.508133 + 0.861279i \(0.330336\pi\)
\(360\) 0 0
\(361\) −14.5986 −0.768348
\(362\) 0 0
\(363\) 15.6315 0.820440
\(364\) 0 0
\(365\) −4.30788 −0.225485
\(366\) 0 0
\(367\) −13.5451 −0.707047 −0.353524 0.935426i \(-0.615017\pi\)
−0.353524 + 0.935426i \(0.615017\pi\)
\(368\) 0 0
\(369\) 2.44765 0.127419
\(370\) 0 0
\(371\) 3.23111 0.167751
\(372\) 0 0
\(373\) 11.9860 0.620614 0.310307 0.950636i \(-0.399568\pi\)
0.310307 + 0.950636i \(0.399568\pi\)
\(374\) 0 0
\(375\) −6.80770 −0.351548
\(376\) 0 0
\(377\) −0.264189 −0.0136064
\(378\) 0 0
\(379\) −22.3111 −1.14604 −0.573021 0.819540i \(-0.694229\pi\)
−0.573021 + 0.819540i \(0.694229\pi\)
\(380\) 0 0
\(381\) 23.1636 1.18671
\(382\) 0 0
\(383\) 11.9953 0.612929 0.306464 0.951882i \(-0.400854\pi\)
0.306464 + 0.951882i \(0.400854\pi\)
\(384\) 0 0
\(385\) 4.25273 0.216739
\(386\) 0 0
\(387\) 22.1911 1.12804
\(388\) 0 0
\(389\) 26.1536 1.32604 0.663019 0.748602i \(-0.269275\pi\)
0.663019 + 0.748602i \(0.269275\pi\)
\(390\) 0 0
\(391\) −3.74633 −0.189460
\(392\) 0 0
\(393\) −22.7438 −1.14727
\(394\) 0 0
\(395\) 4.31459 0.217091
\(396\) 0 0
\(397\) 0.895828 0.0449603 0.0224802 0.999747i \(-0.492844\pi\)
0.0224802 + 0.999747i \(0.492844\pi\)
\(398\) 0 0
\(399\) −18.7545 −0.938902
\(400\) 0 0
\(401\) −15.1250 −0.755307 −0.377654 0.925947i \(-0.623269\pi\)
−0.377654 + 0.925947i \(0.623269\pi\)
\(402\) 0 0
\(403\) −1.70583 −0.0849737
\(404\) 0 0
\(405\) −2.50837 −0.124642
\(406\) 0 0
\(407\) −11.7198 −0.580928
\(408\) 0 0
\(409\) −29.6119 −1.46421 −0.732107 0.681190i \(-0.761463\pi\)
−0.732107 + 0.681190i \(0.761463\pi\)
\(410\) 0 0
\(411\) −33.3764 −1.64634
\(412\) 0 0
\(413\) 2.73224 0.134445
\(414\) 0 0
\(415\) −0.919267 −0.0451250
\(416\) 0 0
\(417\) −28.2201 −1.38195
\(418\) 0 0
\(419\) −6.05052 −0.295587 −0.147794 0.989018i \(-0.547217\pi\)
−0.147794 + 0.989018i \(0.547217\pi\)
\(420\) 0 0
\(421\) −24.0551 −1.17237 −0.586186 0.810176i \(-0.699371\pi\)
−0.586186 + 0.810176i \(0.699371\pi\)
\(422\) 0 0
\(423\) 12.1825 0.592335
\(424\) 0 0
\(425\) −8.69155 −0.421602
\(426\) 0 0
\(427\) −28.3563 −1.37226
\(428\) 0 0
\(429\) −2.30873 −0.111467
\(430\) 0 0
\(431\) −23.4569 −1.12988 −0.564940 0.825132i \(-0.691101\pi\)
−0.564940 + 0.825132i \(0.691101\pi\)
\(432\) 0 0
\(433\) −18.3380 −0.881266 −0.440633 0.897687i \(-0.645246\pi\)
−0.440633 + 0.897687i \(0.645246\pi\)
\(434\) 0 0
\(435\) 0.802472 0.0384756
\(436\) 0 0
\(437\) 4.45060 0.212901
\(438\) 0 0
\(439\) 21.7869 1.03983 0.519917 0.854217i \(-0.325963\pi\)
0.519917 + 0.854217i \(0.325963\pi\)
\(440\) 0 0
\(441\) 18.9527 0.902511
\(442\) 0 0
\(443\) −34.8636 −1.65642 −0.828210 0.560418i \(-0.810641\pi\)
−0.828210 + 0.560418i \(0.810641\pi\)
\(444\) 0 0
\(445\) −0.887836 −0.0420875
\(446\) 0 0
\(447\) 34.9956 1.65523
\(448\) 0 0
\(449\) 34.5288 1.62951 0.814757 0.579802i \(-0.196870\pi\)
0.814757 + 0.579802i \(0.196870\pi\)
\(450\) 0 0
\(451\) 3.38729 0.159501
\(452\) 0 0
\(453\) 33.1311 1.55664
\(454\) 0 0
\(455\) −0.230464 −0.0108043
\(456\) 0 0
\(457\) −27.0723 −1.26639 −0.633195 0.773993i \(-0.718257\pi\)
−0.633195 + 0.773993i \(0.718257\pi\)
\(458\) 0 0
\(459\) 0.0521396 0.00243367
\(460\) 0 0
\(461\) −9.06196 −0.422058 −0.211029 0.977480i \(-0.567681\pi\)
−0.211029 + 0.977480i \(0.567681\pi\)
\(462\) 0 0
\(463\) −19.9997 −0.929466 −0.464733 0.885451i \(-0.653850\pi\)
−0.464733 + 0.885451i \(0.653850\pi\)
\(464\) 0 0
\(465\) 5.18147 0.240285
\(466\) 0 0
\(467\) 9.34435 0.432405 0.216202 0.976349i \(-0.430633\pi\)
0.216202 + 0.976349i \(0.430633\pi\)
\(468\) 0 0
\(469\) 54.6372 2.52291
\(470\) 0 0
\(471\) 46.8973 2.16092
\(472\) 0 0
\(473\) 30.7102 1.41206
\(474\) 0 0
\(475\) 10.3255 0.473765
\(476\) 0 0
\(477\) 2.66939 0.122223
\(478\) 0 0
\(479\) 1.81702 0.0830215 0.0415108 0.999138i \(-0.486783\pi\)
0.0415108 + 0.999138i \(0.486783\pi\)
\(480\) 0 0
\(481\) 0.635118 0.0289589
\(482\) 0 0
\(483\) −18.9642 −0.862901
\(484\) 0 0
\(485\) −2.67960 −0.121674
\(486\) 0 0
\(487\) −18.0300 −0.817019 −0.408510 0.912754i \(-0.633951\pi\)
−0.408510 + 0.912754i \(0.633951\pi\)
\(488\) 0 0
\(489\) −26.3665 −1.19233
\(490\) 0 0
\(491\) −34.2552 −1.54591 −0.772957 0.634458i \(-0.781223\pi\)
−0.772957 + 0.634458i \(0.781223\pi\)
\(492\) 0 0
\(493\) 2.06537 0.0930197
\(494\) 0 0
\(495\) 3.51341 0.157916
\(496\) 0 0
\(497\) −21.9195 −0.983222
\(498\) 0 0
\(499\) −9.11779 −0.408168 −0.204084 0.978953i \(-0.565422\pi\)
−0.204084 + 0.978953i \(0.565422\pi\)
\(500\) 0 0
\(501\) 6.04592 0.270112
\(502\) 0 0
\(503\) −1.00000 −0.0445878
\(504\) 0 0
\(505\) −3.68748 −0.164091
\(506\) 0 0
\(507\) −31.7502 −1.41008
\(508\) 0 0
\(509\) −10.6683 −0.472862 −0.236431 0.971648i \(-0.575978\pi\)
−0.236431 + 0.971648i \(0.575978\pi\)
\(510\) 0 0
\(511\) −56.1256 −2.48285
\(512\) 0 0
\(513\) −0.0619413 −0.00273477
\(514\) 0 0
\(515\) −4.20631 −0.185352
\(516\) 0 0
\(517\) 16.8594 0.741473
\(518\) 0 0
\(519\) −55.1596 −2.42124
\(520\) 0 0
\(521\) −9.57053 −0.419293 −0.209646 0.977777i \(-0.567231\pi\)
−0.209646 + 0.977777i \(0.567231\pi\)
\(522\) 0 0
\(523\) −20.3693 −0.890689 −0.445344 0.895359i \(-0.646919\pi\)
−0.445344 + 0.895359i \(0.646919\pi\)
\(524\) 0 0
\(525\) −43.9973 −1.92020
\(526\) 0 0
\(527\) 13.3359 0.580919
\(528\) 0 0
\(529\) −18.4996 −0.804333
\(530\) 0 0
\(531\) 2.25725 0.0979563
\(532\) 0 0
\(533\) −0.183564 −0.00795103
\(534\) 0 0
\(535\) −2.72735 −0.117914
\(536\) 0 0
\(537\) 33.7094 1.45467
\(538\) 0 0
\(539\) 26.2286 1.12975
\(540\) 0 0
\(541\) −31.8467 −1.36920 −0.684599 0.728920i \(-0.740023\pi\)
−0.684599 + 0.728920i \(0.740023\pi\)
\(542\) 0 0
\(543\) 42.7652 1.83523
\(544\) 0 0
\(545\) −2.00010 −0.0856749
\(546\) 0 0
\(547\) −27.4629 −1.17423 −0.587114 0.809505i \(-0.699736\pi\)
−0.587114 + 0.809505i \(0.699736\pi\)
\(548\) 0 0
\(549\) −23.4267 −0.999826
\(550\) 0 0
\(551\) −2.45364 −0.104529
\(552\) 0 0
\(553\) 56.2130 2.39042
\(554\) 0 0
\(555\) −1.92917 −0.0818886
\(556\) 0 0
\(557\) 35.3783 1.49903 0.749514 0.661989i \(-0.230287\pi\)
0.749514 + 0.661989i \(0.230287\pi\)
\(558\) 0 0
\(559\) −1.66425 −0.0703901
\(560\) 0 0
\(561\) 18.0492 0.762038
\(562\) 0 0
\(563\) −25.1900 −1.06163 −0.530816 0.847487i \(-0.678115\pi\)
−0.530816 + 0.847487i \(0.678115\pi\)
\(564\) 0 0
\(565\) 4.15120 0.174642
\(566\) 0 0
\(567\) −32.6805 −1.37245
\(568\) 0 0
\(569\) 13.1636 0.551846 0.275923 0.961180i \(-0.411017\pi\)
0.275923 + 0.961180i \(0.411017\pi\)
\(570\) 0 0
\(571\) 9.95587 0.416640 0.208320 0.978061i \(-0.433200\pi\)
0.208320 + 0.978061i \(0.433200\pi\)
\(572\) 0 0
\(573\) 60.0550 2.50884
\(574\) 0 0
\(575\) 10.4409 0.435415
\(576\) 0 0
\(577\) 21.1666 0.881176 0.440588 0.897710i \(-0.354770\pi\)
0.440588 + 0.897710i \(0.354770\pi\)
\(578\) 0 0
\(579\) 33.8075 1.40499
\(580\) 0 0
\(581\) −11.9767 −0.496879
\(582\) 0 0
\(583\) 3.69416 0.152996
\(584\) 0 0
\(585\) −0.190398 −0.00787200
\(586\) 0 0
\(587\) 13.3303 0.550202 0.275101 0.961415i \(-0.411289\pi\)
0.275101 + 0.961415i \(0.411289\pi\)
\(588\) 0 0
\(589\) −15.8429 −0.652794
\(590\) 0 0
\(591\) −4.74129 −0.195030
\(592\) 0 0
\(593\) 9.44323 0.387787 0.193893 0.981023i \(-0.437888\pi\)
0.193893 + 0.981023i \(0.437888\pi\)
\(594\) 0 0
\(595\) 1.80172 0.0738632
\(596\) 0 0
\(597\) −32.8461 −1.34430
\(598\) 0 0
\(599\) 24.0118 0.981097 0.490549 0.871414i \(-0.336796\pi\)
0.490549 + 0.871414i \(0.336796\pi\)
\(600\) 0 0
\(601\) 45.0203 1.83641 0.918207 0.396100i \(-0.129637\pi\)
0.918207 + 0.396100i \(0.129637\pi\)
\(602\) 0 0
\(603\) 45.1387 1.83819
\(604\) 0 0
\(605\) 1.78399 0.0725295
\(606\) 0 0
\(607\) 7.44273 0.302091 0.151046 0.988527i \(-0.451736\pi\)
0.151046 + 0.988527i \(0.451736\pi\)
\(608\) 0 0
\(609\) 10.4551 0.423661
\(610\) 0 0
\(611\) −0.913641 −0.0369620
\(612\) 0 0
\(613\) −28.4172 −1.14776 −0.573879 0.818940i \(-0.694562\pi\)
−0.573879 + 0.818940i \(0.694562\pi\)
\(614\) 0 0
\(615\) 0.557574 0.0224836
\(616\) 0 0
\(617\) 26.2593 1.05716 0.528579 0.848884i \(-0.322725\pi\)
0.528579 + 0.848884i \(0.322725\pi\)
\(618\) 0 0
\(619\) 18.0880 0.727019 0.363509 0.931590i \(-0.381578\pi\)
0.363509 + 0.931590i \(0.381578\pi\)
\(620\) 0 0
\(621\) −0.0626337 −0.00251340
\(622\) 0 0
\(623\) −11.5672 −0.463432
\(624\) 0 0
\(625\) 23.8315 0.953260
\(626\) 0 0
\(627\) −21.4422 −0.856321
\(628\) 0 0
\(629\) −4.96522 −0.197976
\(630\) 0 0
\(631\) −26.4988 −1.05490 −0.527451 0.849586i \(-0.676852\pi\)
−0.527451 + 0.849586i \(0.676852\pi\)
\(632\) 0 0
\(633\) 27.4726 1.09194
\(634\) 0 0
\(635\) 2.64361 0.104909
\(636\) 0 0
\(637\) −1.42138 −0.0563171
\(638\) 0 0
\(639\) −18.1088 −0.716374
\(640\) 0 0
\(641\) 36.9823 1.46071 0.730357 0.683066i \(-0.239354\pi\)
0.730357 + 0.683066i \(0.239354\pi\)
\(642\) 0 0
\(643\) −8.21713 −0.324052 −0.162026 0.986787i \(-0.551803\pi\)
−0.162026 + 0.986787i \(0.551803\pi\)
\(644\) 0 0
\(645\) 5.05514 0.199046
\(646\) 0 0
\(647\) 37.1321 1.45982 0.729908 0.683546i \(-0.239563\pi\)
0.729908 + 0.683546i \(0.239563\pi\)
\(648\) 0 0
\(649\) 3.12380 0.122620
\(650\) 0 0
\(651\) 67.5071 2.64581
\(652\) 0 0
\(653\) 10.4087 0.407325 0.203662 0.979041i \(-0.434716\pi\)
0.203662 + 0.979041i \(0.434716\pi\)
\(654\) 0 0
\(655\) −2.59571 −0.101423
\(656\) 0 0
\(657\) −46.3683 −1.80900
\(658\) 0 0
\(659\) −4.11547 −0.160316 −0.0801579 0.996782i \(-0.525542\pi\)
−0.0801579 + 0.996782i \(0.525542\pi\)
\(660\) 0 0
\(661\) −6.06571 −0.235929 −0.117964 0.993018i \(-0.537637\pi\)
−0.117964 + 0.993018i \(0.537637\pi\)
\(662\) 0 0
\(663\) −0.978121 −0.0379871
\(664\) 0 0
\(665\) −2.14042 −0.0830019
\(666\) 0 0
\(667\) −2.48107 −0.0960673
\(668\) 0 0
\(669\) −58.2275 −2.25121
\(670\) 0 0
\(671\) −32.4200 −1.25156
\(672\) 0 0
\(673\) 28.4558 1.09689 0.548445 0.836187i \(-0.315220\pi\)
0.548445 + 0.836187i \(0.315220\pi\)
\(674\) 0 0
\(675\) −0.145311 −0.00559304
\(676\) 0 0
\(677\) −7.44305 −0.286060 −0.143030 0.989718i \(-0.545684\pi\)
−0.143030 + 0.989718i \(0.545684\pi\)
\(678\) 0 0
\(679\) −34.9113 −1.33977
\(680\) 0 0
\(681\) 54.0365 2.07068
\(682\) 0 0
\(683\) −0.0920853 −0.00352354 −0.00176177 0.999998i \(-0.500561\pi\)
−0.00176177 + 0.999998i \(0.500561\pi\)
\(684\) 0 0
\(685\) −3.80919 −0.145542
\(686\) 0 0
\(687\) 25.9450 0.989863
\(688\) 0 0
\(689\) −0.200194 −0.00762677
\(690\) 0 0
\(691\) −25.4439 −0.967932 −0.483966 0.875087i \(-0.660804\pi\)
−0.483966 + 0.875087i \(0.660804\pi\)
\(692\) 0 0
\(693\) 45.7747 1.73884
\(694\) 0 0
\(695\) −3.22071 −0.122169
\(696\) 0 0
\(697\) 1.43506 0.0543569
\(698\) 0 0
\(699\) 3.53452 0.133688
\(700\) 0 0
\(701\) −5.52065 −0.208512 −0.104256 0.994550i \(-0.533246\pi\)
−0.104256 + 0.994550i \(0.533246\pi\)
\(702\) 0 0
\(703\) 5.89862 0.222471
\(704\) 0 0
\(705\) 2.77518 0.104519
\(706\) 0 0
\(707\) −48.0426 −1.80683
\(708\) 0 0
\(709\) 25.5373 0.959074 0.479537 0.877522i \(-0.340805\pi\)
0.479537 + 0.877522i \(0.340805\pi\)
\(710\) 0 0
\(711\) 46.4405 1.74166
\(712\) 0 0
\(713\) −16.0200 −0.599952
\(714\) 0 0
\(715\) −0.263491 −0.00985401
\(716\) 0 0
\(717\) 11.5977 0.433123
\(718\) 0 0
\(719\) −28.5238 −1.06376 −0.531880 0.846820i \(-0.678514\pi\)
−0.531880 + 0.846820i \(0.678514\pi\)
\(720\) 0 0
\(721\) −54.8022 −2.04094
\(722\) 0 0
\(723\) −17.4334 −0.648355
\(724\) 0 0
\(725\) −5.75612 −0.213777
\(726\) 0 0
\(727\) −18.4473 −0.684171 −0.342086 0.939669i \(-0.611133\pi\)
−0.342086 + 0.939669i \(0.611133\pi\)
\(728\) 0 0
\(729\) −27.2163 −1.00801
\(730\) 0 0
\(731\) 13.0107 0.481219
\(732\) 0 0
\(733\) 11.9309 0.440677 0.220338 0.975424i \(-0.429284\pi\)
0.220338 + 0.975424i \(0.429284\pi\)
\(734\) 0 0
\(735\) 4.31743 0.159251
\(736\) 0 0
\(737\) 62.4672 2.30101
\(738\) 0 0
\(739\) −14.6370 −0.538431 −0.269215 0.963080i \(-0.586764\pi\)
−0.269215 + 0.963080i \(0.586764\pi\)
\(740\) 0 0
\(741\) 1.16200 0.0426870
\(742\) 0 0
\(743\) −21.3801 −0.784360 −0.392180 0.919889i \(-0.628279\pi\)
−0.392180 + 0.919889i \(0.628279\pi\)
\(744\) 0 0
\(745\) 3.99398 0.146328
\(746\) 0 0
\(747\) −9.89462 −0.362025
\(748\) 0 0
\(749\) −35.5335 −1.29837
\(750\) 0 0
\(751\) −3.67550 −0.134121 −0.0670605 0.997749i \(-0.521362\pi\)
−0.0670605 + 0.997749i \(0.521362\pi\)
\(752\) 0 0
\(753\) −24.5658 −0.895227
\(754\) 0 0
\(755\) 3.78119 0.137612
\(756\) 0 0
\(757\) 32.4950 1.18105 0.590525 0.807019i \(-0.298921\pi\)
0.590525 + 0.807019i \(0.298921\pi\)
\(758\) 0 0
\(759\) −21.6819 −0.787005
\(760\) 0 0
\(761\) −30.0784 −1.09034 −0.545170 0.838326i \(-0.683535\pi\)
−0.545170 + 0.838326i \(0.683535\pi\)
\(762\) 0 0
\(763\) −26.0585 −0.943380
\(764\) 0 0
\(765\) 1.48849 0.0538166
\(766\) 0 0
\(767\) −0.169285 −0.00611251
\(768\) 0 0
\(769\) −41.8504 −1.50917 −0.754583 0.656205i \(-0.772161\pi\)
−0.754583 + 0.656205i \(0.772161\pi\)
\(770\) 0 0
\(771\) −3.84429 −0.138449
\(772\) 0 0
\(773\) 41.1452 1.47989 0.739945 0.672668i \(-0.234852\pi\)
0.739945 + 0.672668i \(0.234852\pi\)
\(774\) 0 0
\(775\) −37.1666 −1.33506
\(776\) 0 0
\(777\) −25.1343 −0.901688
\(778\) 0 0
\(779\) −1.70484 −0.0610822
\(780\) 0 0
\(781\) −25.0607 −0.896743
\(782\) 0 0
\(783\) 0.0345303 0.00123401
\(784\) 0 0
\(785\) 5.35230 0.191032
\(786\) 0 0
\(787\) 27.9028 0.994627 0.497313 0.867571i \(-0.334320\pi\)
0.497313 + 0.867571i \(0.334320\pi\)
\(788\) 0 0
\(789\) −0.283062 −0.0100773
\(790\) 0 0
\(791\) 54.0842 1.92301
\(792\) 0 0
\(793\) 1.75691 0.0623895
\(794\) 0 0
\(795\) 0.608087 0.0215666
\(796\) 0 0
\(797\) −3.48784 −0.123545 −0.0617727 0.998090i \(-0.519675\pi\)
−0.0617727 + 0.998090i \(0.519675\pi\)
\(798\) 0 0
\(799\) 7.14266 0.252689
\(800\) 0 0
\(801\) −9.55631 −0.337656
\(802\) 0 0
\(803\) −64.1689 −2.26447
\(804\) 0 0
\(805\) −2.16435 −0.0762832
\(806\) 0 0
\(807\) 26.1120 0.919186
\(808\) 0 0
\(809\) −22.9333 −0.806291 −0.403146 0.915136i \(-0.632083\pi\)
−0.403146 + 0.915136i \(0.632083\pi\)
\(810\) 0 0
\(811\) −15.1255 −0.531128 −0.265564 0.964093i \(-0.585558\pi\)
−0.265564 + 0.964093i \(0.585558\pi\)
\(812\) 0 0
\(813\) −10.3643 −0.363493
\(814\) 0 0
\(815\) −3.00916 −0.105406
\(816\) 0 0
\(817\) −15.4566 −0.540758
\(818\) 0 0
\(819\) −2.48062 −0.0866798
\(820\) 0 0
\(821\) 5.73331 0.200094 0.100047 0.994983i \(-0.468101\pi\)
0.100047 + 0.994983i \(0.468101\pi\)
\(822\) 0 0
\(823\) −6.64174 −0.231516 −0.115758 0.993277i \(-0.536930\pi\)
−0.115758 + 0.993277i \(0.536930\pi\)
\(824\) 0 0
\(825\) −50.3025 −1.75131
\(826\) 0 0
\(827\) −28.9616 −1.00709 −0.503547 0.863968i \(-0.667972\pi\)
−0.503547 + 0.863968i \(0.667972\pi\)
\(828\) 0 0
\(829\) 22.9043 0.795498 0.397749 0.917494i \(-0.369791\pi\)
0.397749 + 0.917494i \(0.369791\pi\)
\(830\) 0 0
\(831\) −5.54741 −0.192438
\(832\) 0 0
\(833\) 11.1120 0.385009
\(834\) 0 0
\(835\) 0.690009 0.0238788
\(836\) 0 0
\(837\) 0.222958 0.00770656
\(838\) 0 0
\(839\) −20.7724 −0.717144 −0.358572 0.933502i \(-0.616736\pi\)
−0.358572 + 0.933502i \(0.616736\pi\)
\(840\) 0 0
\(841\) −27.6322 −0.952834
\(842\) 0 0
\(843\) 52.8676 1.82086
\(844\) 0 0
\(845\) −3.62359 −0.124655
\(846\) 0 0
\(847\) 23.2429 0.798634
\(848\) 0 0
\(849\) 31.4791 1.08036
\(850\) 0 0
\(851\) 5.96456 0.204463
\(852\) 0 0
\(853\) −6.17843 −0.211546 −0.105773 0.994390i \(-0.533732\pi\)
−0.105773 + 0.994390i \(0.533732\pi\)
\(854\) 0 0
\(855\) −1.76831 −0.0604751
\(856\) 0 0
\(857\) −30.2746 −1.03416 −0.517080 0.855937i \(-0.672981\pi\)
−0.517080 + 0.855937i \(0.672981\pi\)
\(858\) 0 0
\(859\) 15.8620 0.541206 0.270603 0.962691i \(-0.412777\pi\)
0.270603 + 0.962691i \(0.412777\pi\)
\(860\) 0 0
\(861\) 7.26440 0.247570
\(862\) 0 0
\(863\) 26.0206 0.885753 0.442877 0.896583i \(-0.353958\pi\)
0.442877 + 0.896583i \(0.353958\pi\)
\(864\) 0 0
\(865\) −6.29526 −0.214045
\(866\) 0 0
\(867\) −34.0363 −1.15593
\(868\) 0 0
\(869\) 64.2688 2.18017
\(870\) 0 0
\(871\) −3.38522 −0.114704
\(872\) 0 0
\(873\) −28.8421 −0.976158
\(874\) 0 0
\(875\) −10.1225 −0.342204
\(876\) 0 0
\(877\) −28.1487 −0.950515 −0.475257 0.879847i \(-0.657645\pi\)
−0.475257 + 0.879847i \(0.657645\pi\)
\(878\) 0 0
\(879\) 48.7331 1.64373
\(880\) 0 0
\(881\) 28.2636 0.952226 0.476113 0.879384i \(-0.342045\pi\)
0.476113 + 0.879384i \(0.342045\pi\)
\(882\) 0 0
\(883\) 36.1411 1.21625 0.608123 0.793843i \(-0.291923\pi\)
0.608123 + 0.793843i \(0.291923\pi\)
\(884\) 0 0
\(885\) 0.514201 0.0172847
\(886\) 0 0
\(887\) 58.0021 1.94752 0.973760 0.227579i \(-0.0730810\pi\)
0.973760 + 0.227579i \(0.0730810\pi\)
\(888\) 0 0
\(889\) 34.4425 1.15517
\(890\) 0 0
\(891\) −37.3640 −1.25174
\(892\) 0 0
\(893\) −8.48540 −0.283953
\(894\) 0 0
\(895\) 3.84718 0.128597
\(896\) 0 0
\(897\) 1.17499 0.0392317
\(898\) 0 0
\(899\) 8.83189 0.294560
\(900\) 0 0
\(901\) 1.56507 0.0521401
\(902\) 0 0
\(903\) 65.8612 2.19173
\(904\) 0 0
\(905\) 4.88071 0.162240
\(906\) 0 0
\(907\) −0.593648 −0.0197117 −0.00985587 0.999951i \(-0.503137\pi\)
−0.00985587 + 0.999951i \(0.503137\pi\)
\(908\) 0 0
\(909\) −39.6905 −1.31645
\(910\) 0 0
\(911\) 49.4928 1.63977 0.819885 0.572529i \(-0.194037\pi\)
0.819885 + 0.572529i \(0.194037\pi\)
\(912\) 0 0
\(913\) −13.6931 −0.453176
\(914\) 0 0
\(915\) −5.33659 −0.176422
\(916\) 0 0
\(917\) −33.8184 −1.11678
\(918\) 0 0
\(919\) 28.9431 0.954746 0.477373 0.878701i \(-0.341589\pi\)
0.477373 + 0.878701i \(0.341589\pi\)
\(920\) 0 0
\(921\) −25.7587 −0.848777
\(922\) 0 0
\(923\) 1.35809 0.0447020
\(924\) 0 0
\(925\) 13.8379 0.454987
\(926\) 0 0
\(927\) −45.2750 −1.48703
\(928\) 0 0
\(929\) −2.16662 −0.0710845 −0.0355423 0.999368i \(-0.511316\pi\)
−0.0355423 + 0.999368i \(0.511316\pi\)
\(930\) 0 0
\(931\) −13.2010 −0.432645
\(932\) 0 0
\(933\) 25.9255 0.848764
\(934\) 0 0
\(935\) 2.05992 0.0673666
\(936\) 0 0
\(937\) 48.0007 1.56811 0.784057 0.620689i \(-0.213147\pi\)
0.784057 + 0.620689i \(0.213147\pi\)
\(938\) 0 0
\(939\) −0.907741 −0.0296230
\(940\) 0 0
\(941\) −35.4555 −1.15581 −0.577907 0.816102i \(-0.696131\pi\)
−0.577907 + 0.816102i \(0.696131\pi\)
\(942\) 0 0
\(943\) −1.72390 −0.0561378
\(944\) 0 0
\(945\) 0.0301224 0.000979880 0
\(946\) 0 0
\(947\) 26.4416 0.859236 0.429618 0.903011i \(-0.358648\pi\)
0.429618 + 0.903011i \(0.358648\pi\)
\(948\) 0 0
\(949\) 3.47744 0.112882
\(950\) 0 0
\(951\) 26.8204 0.869711
\(952\) 0 0
\(953\) −55.7670 −1.80647 −0.903236 0.429144i \(-0.858815\pi\)
−0.903236 + 0.429144i \(0.858815\pi\)
\(954\) 0 0
\(955\) 6.85397 0.221789
\(956\) 0 0
\(957\) 11.9534 0.386398
\(958\) 0 0
\(959\) −49.6283 −1.60258
\(960\) 0 0
\(961\) 26.0265 0.839564
\(962\) 0 0
\(963\) −29.3561 −0.945988
\(964\) 0 0
\(965\) 3.85839 0.124206
\(966\) 0 0
\(967\) −7.42604 −0.238805 −0.119403 0.992846i \(-0.538098\pi\)
−0.119403 + 0.992846i \(0.538098\pi\)
\(968\) 0 0
\(969\) −9.08425 −0.291828
\(970\) 0 0
\(971\) 0.333254 0.0106946 0.00534731 0.999986i \(-0.498298\pi\)
0.00534731 + 0.999986i \(0.498298\pi\)
\(972\) 0 0
\(973\) −41.9613 −1.34522
\(974\) 0 0
\(975\) 2.72599 0.0873015
\(976\) 0 0
\(977\) −45.3521 −1.45094 −0.725470 0.688253i \(-0.758378\pi\)
−0.725470 + 0.688253i \(0.758378\pi\)
\(978\) 0 0
\(979\) −13.2249 −0.422671
\(980\) 0 0
\(981\) −21.5283 −0.687345
\(982\) 0 0
\(983\) 21.6324 0.689965 0.344983 0.938609i \(-0.387885\pi\)
0.344983 + 0.938609i \(0.387885\pi\)
\(984\) 0 0
\(985\) −0.541114 −0.0172413
\(986\) 0 0
\(987\) 36.1567 1.15088
\(988\) 0 0
\(989\) −15.6294 −0.496985
\(990\) 0 0
\(991\) −41.5815 −1.32088 −0.660441 0.750878i \(-0.729630\pi\)
−0.660441 + 0.750878i \(0.729630\pi\)
\(992\) 0 0
\(993\) 37.9384 1.20394
\(994\) 0 0
\(995\) −3.74867 −0.118841
\(996\) 0 0
\(997\) 22.3236 0.706995 0.353498 0.935435i \(-0.384992\pi\)
0.353498 + 0.935435i \(0.384992\pi\)
\(998\) 0 0
\(999\) −0.0830119 −0.00262638
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 4024.2.a.f.1.28 33
4.3 odd 2 8048.2.a.y.1.6 33
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4024.2.a.f.1.28 33 1.1 even 1 trivial
8048.2.a.y.1.6 33 4.3 odd 2