Properties

Label 4024.2.a.g.1.17
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 1.17
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)
 
\(f(q)\) \(=\) \(q+0.410265 q^{3} -4.21207 q^{5} +3.36092 q^{7} -2.83168 q^{9} +O(q^{10})\) \(q+0.410265 q^{3} -4.21207 q^{5} +3.36092 q^{7} -2.83168 q^{9} +4.13893 q^{11} -3.22425 q^{13} -1.72807 q^{15} +4.83674 q^{17} -7.13255 q^{19} +1.37887 q^{21} -0.264477 q^{23} +12.7416 q^{25} -2.39254 q^{27} -6.92998 q^{29} +1.07140 q^{31} +1.69806 q^{33} -14.1564 q^{35} +5.79815 q^{37} -1.32280 q^{39} +1.24932 q^{41} +3.97547 q^{43} +11.9273 q^{45} -7.96976 q^{47} +4.29578 q^{49} +1.98435 q^{51} +13.9969 q^{53} -17.4335 q^{55} -2.92624 q^{57} +1.88127 q^{59} -13.0212 q^{61} -9.51706 q^{63} +13.5808 q^{65} +7.34403 q^{67} -0.108506 q^{69} +13.5632 q^{71} -2.49323 q^{73} +5.22742 q^{75} +13.9106 q^{77} -8.07702 q^{79} +7.51347 q^{81} -10.1672 q^{83} -20.3727 q^{85} -2.84313 q^{87} +3.18164 q^{89} -10.8364 q^{91} +0.439559 q^{93} +30.0428 q^{95} +7.00896 q^{97} -11.7201 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) \) 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
\(3\) 0.410265 0.236867 0.118433 0.992962i \(-0.462213\pi\)
0.118433 + 0.992962i \(0.462213\pi\)
\(4\) 0 0
\(5\) −4.21207 −1.88370 −0.941848 0.336038i \(-0.890913\pi\)
−0.941848 + 0.336038i \(0.890913\pi\)
\(6\) 0 0
\(7\) 3.36092 1.27031 0.635154 0.772385i \(-0.280937\pi\)
0.635154 + 0.772385i \(0.280937\pi\)
\(8\) 0 0
\(9\) −2.83168 −0.943894
\(10\) 0 0
\(11\) 4.13893 1.24793 0.623967 0.781451i \(-0.285520\pi\)
0.623967 + 0.781451i \(0.285520\pi\)
\(12\) 0 0
\(13\) −3.22425 −0.894245 −0.447122 0.894473i \(-0.647551\pi\)
−0.447122 + 0.894473i \(0.647551\pi\)
\(14\) 0 0
\(15\) −1.72807 −0.446185
\(16\) 0 0
\(17\) 4.83674 1.17308 0.586541 0.809919i \(-0.300489\pi\)
0.586541 + 0.809919i \(0.300489\pi\)
\(18\) 0 0
\(19\) −7.13255 −1.63632 −0.818159 0.574991i \(-0.805005\pi\)
−0.818159 + 0.574991i \(0.805005\pi\)
\(20\) 0 0
\(21\) 1.37887 0.300894
\(22\) 0 0
\(23\) −0.264477 −0.0551472 −0.0275736 0.999620i \(-0.508778\pi\)
−0.0275736 + 0.999620i \(0.508778\pi\)
\(24\) 0 0
\(25\) 12.7416 2.54831
\(26\) 0 0
\(27\) −2.39254 −0.460444
\(28\) 0 0
\(29\) −6.92998 −1.28686 −0.643432 0.765503i \(-0.722490\pi\)
−0.643432 + 0.765503i \(0.722490\pi\)
\(30\) 0 0
\(31\) 1.07140 0.192429 0.0962147 0.995361i \(-0.469326\pi\)
0.0962147 + 0.995361i \(0.469326\pi\)
\(32\) 0 0
\(33\) 1.69806 0.295594
\(34\) 0 0
\(35\) −14.1564 −2.39288
\(36\) 0 0
\(37\) 5.79815 0.953210 0.476605 0.879117i \(-0.341867\pi\)
0.476605 + 0.879117i \(0.341867\pi\)
\(38\) 0 0
\(39\) −1.32280 −0.211817
\(40\) 0 0
\(41\) 1.24932 0.195111 0.0975553 0.995230i \(-0.468898\pi\)
0.0975553 + 0.995230i \(0.468898\pi\)
\(42\) 0 0
\(43\) 3.97547 0.606253 0.303126 0.952950i \(-0.401970\pi\)
0.303126 + 0.952950i \(0.401970\pi\)
\(44\) 0 0
\(45\) 11.9273 1.77801
\(46\) 0 0
\(47\) −7.96976 −1.16251 −0.581255 0.813722i \(-0.697438\pi\)
−0.581255 + 0.813722i \(0.697438\pi\)
\(48\) 0 0
\(49\) 4.29578 0.613684
\(50\) 0 0
\(51\) 1.98435 0.277864
\(52\) 0 0
\(53\) 13.9969 1.92263 0.961313 0.275460i \(-0.0888301\pi\)
0.961313 + 0.275460i \(0.0888301\pi\)
\(54\) 0 0
\(55\) −17.4335 −2.35073
\(56\) 0 0
\(57\) −2.92624 −0.387590
\(58\) 0 0
\(59\) 1.88127 0.244921 0.122460 0.992473i \(-0.460922\pi\)
0.122460 + 0.992473i \(0.460922\pi\)
\(60\) 0 0
\(61\) −13.0212 −1.66719 −0.833596 0.552374i \(-0.813722\pi\)
−0.833596 + 0.552374i \(0.813722\pi\)
\(62\) 0 0
\(63\) −9.51706 −1.19904
\(64\) 0 0
\(65\) 13.5808 1.68449
\(66\) 0 0
\(67\) 7.34403 0.897215 0.448608 0.893729i \(-0.351920\pi\)
0.448608 + 0.893729i \(0.351920\pi\)
\(68\) 0 0
\(69\) −0.108506 −0.0130625
\(70\) 0 0
\(71\) 13.5632 1.60965 0.804826 0.593510i \(-0.202258\pi\)
0.804826 + 0.593510i \(0.202258\pi\)
\(72\) 0 0
\(73\) −2.49323 −0.291810 −0.145905 0.989299i \(-0.546609\pi\)
−0.145905 + 0.989299i \(0.546609\pi\)
\(74\) 0 0
\(75\) 5.22742 0.603611
\(76\) 0 0
\(77\) 13.9106 1.58526
\(78\) 0 0
\(79\) −8.07702 −0.908736 −0.454368 0.890814i \(-0.650135\pi\)
−0.454368 + 0.890814i \(0.650135\pi\)
\(80\) 0 0
\(81\) 7.51347 0.834830
\(82\) 0 0
\(83\) −10.1672 −1.11599 −0.557997 0.829843i \(-0.688430\pi\)
−0.557997 + 0.829843i \(0.688430\pi\)
\(84\) 0 0
\(85\) −20.3727 −2.20973
\(86\) 0 0
\(87\) −2.84313 −0.304816
\(88\) 0 0
\(89\) 3.18164 0.337253 0.168627 0.985680i \(-0.446067\pi\)
0.168627 + 0.985680i \(0.446067\pi\)
\(90\) 0 0
\(91\) −10.8364 −1.13597
\(92\) 0 0
\(93\) 0.439559 0.0455801
\(94\) 0 0
\(95\) 30.0428 3.08233
\(96\) 0 0
\(97\) 7.00896 0.711652 0.355826 0.934552i \(-0.384200\pi\)
0.355826 + 0.934552i \(0.384200\pi\)
\(98\) 0 0
\(99\) −11.7201 −1.17792
\(100\) 0 0
\(101\) 7.13363 0.709823 0.354911 0.934900i \(-0.384511\pi\)
0.354911 + 0.934900i \(0.384511\pi\)
\(102\) 0 0
\(103\) 8.64606 0.851922 0.425961 0.904742i \(-0.359936\pi\)
0.425961 + 0.904742i \(0.359936\pi\)
\(104\) 0 0
\(105\) −5.80790 −0.566793
\(106\) 0 0
\(107\) 2.00657 0.193982 0.0969912 0.995285i \(-0.469078\pi\)
0.0969912 + 0.995285i \(0.469078\pi\)
\(108\) 0 0
\(109\) −3.48178 −0.333494 −0.166747 0.986000i \(-0.553326\pi\)
−0.166747 + 0.986000i \(0.553326\pi\)
\(110\) 0 0
\(111\) 2.37878 0.225784
\(112\) 0 0
\(113\) 17.5080 1.64701 0.823505 0.567308i \(-0.192015\pi\)
0.823505 + 0.567308i \(0.192015\pi\)
\(114\) 0 0
\(115\) 1.11400 0.103881
\(116\) 0 0
\(117\) 9.13004 0.844072
\(118\) 0 0
\(119\) 16.2559 1.49018
\(120\) 0 0
\(121\) 6.13072 0.557338
\(122\) 0 0
\(123\) 0.512552 0.0462152
\(124\) 0 0
\(125\) −32.6081 −2.91655
\(126\) 0 0
\(127\) 2.74768 0.243817 0.121909 0.992541i \(-0.461099\pi\)
0.121909 + 0.992541i \(0.461099\pi\)
\(128\) 0 0
\(129\) 1.63100 0.143601
\(130\) 0 0
\(131\) 9.58654 0.837580 0.418790 0.908083i \(-0.362454\pi\)
0.418790 + 0.908083i \(0.362454\pi\)
\(132\) 0 0
\(133\) −23.9719 −2.07863
\(134\) 0 0
\(135\) 10.0775 0.867337
\(136\) 0 0
\(137\) 14.1551 1.20935 0.604677 0.796471i \(-0.293302\pi\)
0.604677 + 0.796471i \(0.293302\pi\)
\(138\) 0 0
\(139\) 3.03212 0.257181 0.128591 0.991698i \(-0.458955\pi\)
0.128591 + 0.991698i \(0.458955\pi\)
\(140\) 0 0
\(141\) −3.26972 −0.275360
\(142\) 0 0
\(143\) −13.3449 −1.11596
\(144\) 0 0
\(145\) 29.1896 2.42406
\(146\) 0 0
\(147\) 1.76241 0.145361
\(148\) 0 0
\(149\) 9.35096 0.766061 0.383030 0.923736i \(-0.374880\pi\)
0.383030 + 0.923736i \(0.374880\pi\)
\(150\) 0 0
\(151\) −12.0486 −0.980504 −0.490252 0.871581i \(-0.663095\pi\)
−0.490252 + 0.871581i \(0.663095\pi\)
\(152\) 0 0
\(153\) −13.6961 −1.10727
\(154\) 0 0
\(155\) −4.51282 −0.362479
\(156\) 0 0
\(157\) −2.85173 −0.227593 −0.113796 0.993504i \(-0.536301\pi\)
−0.113796 + 0.993504i \(0.536301\pi\)
\(158\) 0 0
\(159\) 5.74245 0.455406
\(160\) 0 0
\(161\) −0.888885 −0.0700539
\(162\) 0 0
\(163\) 12.7867 1.00153 0.500764 0.865584i \(-0.333052\pi\)
0.500764 + 0.865584i \(0.333052\pi\)
\(164\) 0 0
\(165\) −7.15235 −0.556810
\(166\) 0 0
\(167\) 16.3443 1.26476 0.632381 0.774658i \(-0.282078\pi\)
0.632381 + 0.774658i \(0.282078\pi\)
\(168\) 0 0
\(169\) −2.60424 −0.200326
\(170\) 0 0
\(171\) 20.1971 1.54451
\(172\) 0 0
\(173\) 3.23063 0.245620 0.122810 0.992430i \(-0.460809\pi\)
0.122810 + 0.992430i \(0.460809\pi\)
\(174\) 0 0
\(175\) 42.8234 3.23714
\(176\) 0 0
\(177\) 0.771821 0.0580136
\(178\) 0 0
\(179\) −20.9167 −1.56339 −0.781694 0.623662i \(-0.785644\pi\)
−0.781694 + 0.623662i \(0.785644\pi\)
\(180\) 0 0
\(181\) −16.1263 −1.19866 −0.599328 0.800504i \(-0.704565\pi\)
−0.599328 + 0.800504i \(0.704565\pi\)
\(182\) 0 0
\(183\) −5.34214 −0.394903
\(184\) 0 0
\(185\) −24.4222 −1.79556
\(186\) 0 0
\(187\) 20.0189 1.46393
\(188\) 0 0
\(189\) −8.04113 −0.584906
\(190\) 0 0
\(191\) 15.1057 1.09301 0.546506 0.837455i \(-0.315958\pi\)
0.546506 + 0.837455i \(0.315958\pi\)
\(192\) 0 0
\(193\) 16.5469 1.19107 0.595535 0.803329i \(-0.296940\pi\)
0.595535 + 0.803329i \(0.296940\pi\)
\(194\) 0 0
\(195\) 5.57172 0.398999
\(196\) 0 0
\(197\) 15.0779 1.07426 0.537129 0.843500i \(-0.319509\pi\)
0.537129 + 0.843500i \(0.319509\pi\)
\(198\) 0 0
\(199\) 3.05087 0.216271 0.108135 0.994136i \(-0.465512\pi\)
0.108135 + 0.994136i \(0.465512\pi\)
\(200\) 0 0
\(201\) 3.01300 0.212521
\(202\) 0 0
\(203\) −23.2911 −1.63472
\(204\) 0 0
\(205\) −5.26222 −0.367529
\(206\) 0 0
\(207\) 0.748914 0.0520531
\(208\) 0 0
\(209\) −29.5211 −2.04202
\(210\) 0 0
\(211\) 22.4233 1.54368 0.771840 0.635816i \(-0.219336\pi\)
0.771840 + 0.635816i \(0.219336\pi\)
\(212\) 0 0
\(213\) 5.56450 0.381273
\(214\) 0 0
\(215\) −16.7450 −1.14200
\(216\) 0 0
\(217\) 3.60089 0.244445
\(218\) 0 0
\(219\) −1.02289 −0.0691202
\(220\) 0 0
\(221\) −15.5949 −1.04902
\(222\) 0 0
\(223\) −3.37993 −0.226337 −0.113169 0.993576i \(-0.536100\pi\)
−0.113169 + 0.993576i \(0.536100\pi\)
\(224\) 0 0
\(225\) −36.0801 −2.40534
\(226\) 0 0
\(227\) 24.3974 1.61931 0.809655 0.586907i \(-0.199654\pi\)
0.809655 + 0.586907i \(0.199654\pi\)
\(228\) 0 0
\(229\) 22.6787 1.49865 0.749326 0.662202i \(-0.230378\pi\)
0.749326 + 0.662202i \(0.230378\pi\)
\(230\) 0 0
\(231\) 5.70704 0.375496
\(232\) 0 0
\(233\) 24.2114 1.58614 0.793070 0.609130i \(-0.208481\pi\)
0.793070 + 0.609130i \(0.208481\pi\)
\(234\) 0 0
\(235\) 33.5692 2.18982
\(236\) 0 0
\(237\) −3.31372 −0.215249
\(238\) 0 0
\(239\) −1.85452 −0.119959 −0.0599795 0.998200i \(-0.519104\pi\)
−0.0599795 + 0.998200i \(0.519104\pi\)
\(240\) 0 0
\(241\) 10.1215 0.651985 0.325993 0.945372i \(-0.394302\pi\)
0.325993 + 0.945372i \(0.394302\pi\)
\(242\) 0 0
\(243\) 10.2601 0.658188
\(244\) 0 0
\(245\) −18.0942 −1.15599
\(246\) 0 0
\(247\) 22.9971 1.46327
\(248\) 0 0
\(249\) −4.17125 −0.264342
\(250\) 0 0
\(251\) 4.41727 0.278816 0.139408 0.990235i \(-0.455480\pi\)
0.139408 + 0.990235i \(0.455480\pi\)
\(252\) 0 0
\(253\) −1.09465 −0.0688200
\(254\) 0 0
\(255\) −8.35822 −0.523412
\(256\) 0 0
\(257\) 19.8163 1.23611 0.618055 0.786135i \(-0.287921\pi\)
0.618055 + 0.786135i \(0.287921\pi\)
\(258\) 0 0
\(259\) 19.4871 1.21087
\(260\) 0 0
\(261\) 19.6235 1.21466
\(262\) 0 0
\(263\) 28.1818 1.73776 0.868882 0.495019i \(-0.164839\pi\)
0.868882 + 0.495019i \(0.164839\pi\)
\(264\) 0 0
\(265\) −58.9561 −3.62164
\(266\) 0 0
\(267\) 1.30532 0.0798842
\(268\) 0 0
\(269\) 6.37959 0.388971 0.194485 0.980905i \(-0.437696\pi\)
0.194485 + 0.980905i \(0.437696\pi\)
\(270\) 0 0
\(271\) 11.1955 0.680076 0.340038 0.940412i \(-0.389560\pi\)
0.340038 + 0.940412i \(0.389560\pi\)
\(272\) 0 0
\(273\) −4.44581 −0.269073
\(274\) 0 0
\(275\) 52.7364 3.18013
\(276\) 0 0
\(277\) −26.1314 −1.57008 −0.785042 0.619443i \(-0.787359\pi\)
−0.785042 + 0.619443i \(0.787359\pi\)
\(278\) 0 0
\(279\) −3.03387 −0.181633
\(280\) 0 0
\(281\) −4.54850 −0.271341 −0.135670 0.990754i \(-0.543319\pi\)
−0.135670 + 0.990754i \(0.543319\pi\)
\(282\) 0 0
\(283\) 19.4187 1.15432 0.577162 0.816630i \(-0.304160\pi\)
0.577162 + 0.816630i \(0.304160\pi\)
\(284\) 0 0
\(285\) 12.3255 0.730101
\(286\) 0 0
\(287\) 4.19886 0.247851
\(288\) 0 0
\(289\) 6.39410 0.376123
\(290\) 0 0
\(291\) 2.87553 0.168567
\(292\) 0 0
\(293\) 8.94508 0.522577 0.261289 0.965261i \(-0.415853\pi\)
0.261289 + 0.965261i \(0.415853\pi\)
\(294\) 0 0
\(295\) −7.92406 −0.461357
\(296\) 0 0
\(297\) −9.90254 −0.574604
\(298\) 0 0
\(299\) 0.852738 0.0493151
\(300\) 0 0
\(301\) 13.3612 0.770128
\(302\) 0 0
\(303\) 2.92668 0.168133
\(304\) 0 0
\(305\) 54.8462 3.14049
\(306\) 0 0
\(307\) −22.9511 −1.30989 −0.654943 0.755678i \(-0.727307\pi\)
−0.654943 + 0.755678i \(0.727307\pi\)
\(308\) 0 0
\(309\) 3.54718 0.201792
\(310\) 0 0
\(311\) −33.3402 −1.89055 −0.945276 0.326272i \(-0.894207\pi\)
−0.945276 + 0.326272i \(0.894207\pi\)
\(312\) 0 0
\(313\) −14.3601 −0.811680 −0.405840 0.913944i \(-0.633021\pi\)
−0.405840 + 0.913944i \(0.633021\pi\)
\(314\) 0 0
\(315\) 40.0866 2.25862
\(316\) 0 0
\(317\) −1.42229 −0.0798840 −0.0399420 0.999202i \(-0.512717\pi\)
−0.0399420 + 0.999202i \(0.512717\pi\)
\(318\) 0 0
\(319\) −28.6827 −1.60592
\(320\) 0 0
\(321\) 0.823226 0.0459480
\(322\) 0 0
\(323\) −34.4983 −1.91954
\(324\) 0 0
\(325\) −41.0820 −2.27882
\(326\) 0 0
\(327\) −1.42845 −0.0789936
\(328\) 0 0
\(329\) −26.7857 −1.47675
\(330\) 0 0
\(331\) −0.937586 −0.0515344 −0.0257672 0.999668i \(-0.508203\pi\)
−0.0257672 + 0.999668i \(0.508203\pi\)
\(332\) 0 0
\(333\) −16.4185 −0.899730
\(334\) 0 0
\(335\) −30.9336 −1.69008
\(336\) 0 0
\(337\) 16.3871 0.892662 0.446331 0.894868i \(-0.352730\pi\)
0.446331 + 0.894868i \(0.352730\pi\)
\(338\) 0 0
\(339\) 7.18291 0.390122
\(340\) 0 0
\(341\) 4.43445 0.240139
\(342\) 0 0
\(343\) −9.08865 −0.490741
\(344\) 0 0
\(345\) 0.457034 0.0246059
\(346\) 0 0
\(347\) −21.5142 −1.15494 −0.577470 0.816412i \(-0.695960\pi\)
−0.577470 + 0.816412i \(0.695960\pi\)
\(348\) 0 0
\(349\) −35.9721 −1.92554 −0.962770 0.270320i \(-0.912870\pi\)
−0.962770 + 0.270320i \(0.912870\pi\)
\(350\) 0 0
\(351\) 7.71413 0.411750
\(352\) 0 0
\(353\) −31.5436 −1.67890 −0.839449 0.543438i \(-0.817122\pi\)
−0.839449 + 0.543438i \(0.817122\pi\)
\(354\) 0 0
\(355\) −57.1291 −3.03210
\(356\) 0 0
\(357\) 6.66924 0.352973
\(358\) 0 0
\(359\) −36.2750 −1.91452 −0.957261 0.289227i \(-0.906602\pi\)
−0.957261 + 0.289227i \(0.906602\pi\)
\(360\) 0 0
\(361\) 31.8732 1.67754
\(362\) 0 0
\(363\) 2.51522 0.132015
\(364\) 0 0
\(365\) 10.5017 0.549682
\(366\) 0 0
\(367\) 18.3867 0.959776 0.479888 0.877330i \(-0.340677\pi\)
0.479888 + 0.877330i \(0.340677\pi\)
\(368\) 0 0
\(369\) −3.53767 −0.184164
\(370\) 0 0
\(371\) 47.0425 2.44233
\(372\) 0 0
\(373\) 24.2082 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(374\) 0 0
\(375\) −13.3780 −0.690835
\(376\) 0 0
\(377\) 22.3440 1.15077
\(378\) 0 0
\(379\) −1.23158 −0.0632622 −0.0316311 0.999500i \(-0.510070\pi\)
−0.0316311 + 0.999500i \(0.510070\pi\)
\(380\) 0 0
\(381\) 1.12728 0.0577522
\(382\) 0 0
\(383\) −22.7396 −1.16194 −0.580969 0.813926i \(-0.697326\pi\)
−0.580969 + 0.813926i \(0.697326\pi\)
\(384\) 0 0
\(385\) −58.5925 −2.98615
\(386\) 0 0
\(387\) −11.2573 −0.572239
\(388\) 0 0
\(389\) −15.0769 −0.764429 −0.382214 0.924074i \(-0.624838\pi\)
−0.382214 + 0.924074i \(0.624838\pi\)
\(390\) 0 0
\(391\) −1.27921 −0.0646922
\(392\) 0 0
\(393\) 3.93302 0.198395
\(394\) 0 0
\(395\) 34.0210 1.71178
\(396\) 0 0
\(397\) 10.8639 0.545246 0.272623 0.962121i \(-0.412109\pi\)
0.272623 + 0.962121i \(0.412109\pi\)
\(398\) 0 0
\(399\) −9.83485 −0.492358
\(400\) 0 0
\(401\) −11.8865 −0.593581 −0.296791 0.954943i \(-0.595916\pi\)
−0.296791 + 0.954943i \(0.595916\pi\)
\(402\) 0 0
\(403\) −3.45446 −0.172079
\(404\) 0 0
\(405\) −31.6473 −1.57257
\(406\) 0 0
\(407\) 23.9981 1.18954
\(408\) 0 0
\(409\) −20.2652 −1.00205 −0.501026 0.865432i \(-0.667044\pi\)
−0.501026 + 0.865432i \(0.667044\pi\)
\(410\) 0 0
\(411\) 5.80736 0.286456
\(412\) 0 0
\(413\) 6.32281 0.311125
\(414\) 0 0
\(415\) 42.8250 2.10219
\(416\) 0 0
\(417\) 1.24398 0.0609177
\(418\) 0 0
\(419\) −5.46963 −0.267209 −0.133604 0.991035i \(-0.542655\pi\)
−0.133604 + 0.991035i \(0.542655\pi\)
\(420\) 0 0
\(421\) 34.3527 1.67425 0.837123 0.547015i \(-0.184236\pi\)
0.837123 + 0.547015i \(0.184236\pi\)
\(422\) 0 0
\(423\) 22.5678 1.09729
\(424\) 0 0
\(425\) 61.6277 2.98938
\(426\) 0 0
\(427\) −43.7632 −2.11785
\(428\) 0 0
\(429\) −5.47496 −0.264333
\(430\) 0 0
\(431\) −5.85713 −0.282128 −0.141064 0.990000i \(-0.545052\pi\)
−0.141064 + 0.990000i \(0.545052\pi\)
\(432\) 0 0
\(433\) 2.13150 0.102434 0.0512168 0.998688i \(-0.483690\pi\)
0.0512168 + 0.998688i \(0.483690\pi\)
\(434\) 0 0
\(435\) 11.9755 0.574180
\(436\) 0 0
\(437\) 1.88639 0.0902384
\(438\) 0 0
\(439\) −25.8927 −1.23579 −0.617896 0.786259i \(-0.712015\pi\)
−0.617896 + 0.786259i \(0.712015\pi\)
\(440\) 0 0
\(441\) −12.1643 −0.579252
\(442\) 0 0
\(443\) −11.3441 −0.538974 −0.269487 0.963004i \(-0.586854\pi\)
−0.269487 + 0.963004i \(0.586854\pi\)
\(444\) 0 0
\(445\) −13.4013 −0.635283
\(446\) 0 0
\(447\) 3.83638 0.181454
\(448\) 0 0
\(449\) 29.4795 1.39123 0.695613 0.718417i \(-0.255133\pi\)
0.695613 + 0.718417i \(0.255133\pi\)
\(450\) 0 0
\(451\) 5.17084 0.243485
\(452\) 0 0
\(453\) −4.94314 −0.232249
\(454\) 0 0
\(455\) 45.6439 2.13982
\(456\) 0 0
\(457\) −36.6285 −1.71341 −0.856705 0.515806i \(-0.827492\pi\)
−0.856705 + 0.515806i \(0.827492\pi\)
\(458\) 0 0
\(459\) −11.5721 −0.540139
\(460\) 0 0
\(461\) 25.4126 1.18358 0.591791 0.806091i \(-0.298421\pi\)
0.591791 + 0.806091i \(0.298421\pi\)
\(462\) 0 0
\(463\) −24.3241 −1.13044 −0.565218 0.824942i \(-0.691208\pi\)
−0.565218 + 0.824942i \(0.691208\pi\)
\(464\) 0 0
\(465\) −1.85145 −0.0858592
\(466\) 0 0
\(467\) 27.9330 1.29258 0.646292 0.763090i \(-0.276319\pi\)
0.646292 + 0.763090i \(0.276319\pi\)
\(468\) 0 0
\(469\) 24.6827 1.13974
\(470\) 0 0
\(471\) −1.16997 −0.0539092
\(472\) 0 0
\(473\) 16.4542 0.756563
\(474\) 0 0
\(475\) −90.8799 −4.16985
\(476\) 0 0
\(477\) −39.6348 −1.81475
\(478\) 0 0
\(479\) 31.8937 1.45726 0.728631 0.684907i \(-0.240157\pi\)
0.728631 + 0.684907i \(0.240157\pi\)
\(480\) 0 0
\(481\) −18.6947 −0.852403
\(482\) 0 0
\(483\) −0.364679 −0.0165935
\(484\) 0 0
\(485\) −29.5223 −1.34054
\(486\) 0 0
\(487\) 9.97200 0.451875 0.225937 0.974142i \(-0.427456\pi\)
0.225937 + 0.974142i \(0.427456\pi\)
\(488\) 0 0
\(489\) 5.24592 0.237229
\(490\) 0 0
\(491\) −12.5310 −0.565515 −0.282758 0.959191i \(-0.591249\pi\)
−0.282758 + 0.959191i \(0.591249\pi\)
\(492\) 0 0
\(493\) −33.5185 −1.50960
\(494\) 0 0
\(495\) 49.3660 2.21884
\(496\) 0 0
\(497\) 45.5847 2.04476
\(498\) 0 0
\(499\) −24.7196 −1.10660 −0.553301 0.832981i \(-0.686632\pi\)
−0.553301 + 0.832981i \(0.686632\pi\)
\(500\) 0 0
\(501\) 6.70551 0.299580
\(502\) 0 0
\(503\) 1.00000 0.0445878
\(504\) 0 0
\(505\) −30.0474 −1.33709
\(506\) 0 0
\(507\) −1.06843 −0.0474506
\(508\) 0 0
\(509\) −13.3757 −0.592866 −0.296433 0.955054i \(-0.595797\pi\)
−0.296433 + 0.955054i \(0.595797\pi\)
\(510\) 0 0
\(511\) −8.37954 −0.370689
\(512\) 0 0
\(513\) 17.0649 0.753433
\(514\) 0 0
\(515\) −36.4178 −1.60476
\(516\) 0 0
\(517\) −32.9863 −1.45073
\(518\) 0 0
\(519\) 1.32541 0.0581792
\(520\) 0 0
\(521\) −3.01024 −0.131881 −0.0659406 0.997824i \(-0.521005\pi\)
−0.0659406 + 0.997824i \(0.521005\pi\)
\(522\) 0 0
\(523\) 0.371149 0.0162292 0.00811462 0.999967i \(-0.497417\pi\)
0.00811462 + 0.999967i \(0.497417\pi\)
\(524\) 0 0
\(525\) 17.5690 0.766772
\(526\) 0 0
\(527\) 5.18209 0.225736
\(528\) 0 0
\(529\) −22.9301 −0.996959
\(530\) 0 0
\(531\) −5.32717 −0.231179
\(532\) 0 0
\(533\) −4.02811 −0.174477
\(534\) 0 0
\(535\) −8.45182 −0.365404
\(536\) 0 0
\(537\) −8.58140 −0.370315
\(538\) 0 0
\(539\) 17.7799 0.765836
\(540\) 0 0
\(541\) −12.2639 −0.527266 −0.263633 0.964623i \(-0.584921\pi\)
−0.263633 + 0.964623i \(0.584921\pi\)
\(542\) 0 0
\(543\) −6.61605 −0.283922
\(544\) 0 0
\(545\) 14.6655 0.628201
\(546\) 0 0
\(547\) −36.4868 −1.56006 −0.780031 0.625741i \(-0.784797\pi\)
−0.780031 + 0.625741i \(0.784797\pi\)
\(548\) 0 0
\(549\) 36.8719 1.57365
\(550\) 0 0
\(551\) 49.4284 2.10572
\(552\) 0 0
\(553\) −27.1462 −1.15437
\(554\) 0 0
\(555\) −10.0196 −0.425308
\(556\) 0 0
\(557\) 26.0934 1.10561 0.552807 0.833310i \(-0.313557\pi\)
0.552807 + 0.833310i \(0.313557\pi\)
\(558\) 0 0
\(559\) −12.8179 −0.542139
\(560\) 0 0
\(561\) 8.21308 0.346756
\(562\) 0 0
\(563\) 12.5584 0.529274 0.264637 0.964348i \(-0.414748\pi\)
0.264637 + 0.964348i \(0.414748\pi\)
\(564\) 0 0
\(565\) −73.7448 −3.10247
\(566\) 0 0
\(567\) 25.2522 1.06049
\(568\) 0 0
\(569\) −13.7961 −0.578364 −0.289182 0.957274i \(-0.593383\pi\)
−0.289182 + 0.957274i \(0.593383\pi\)
\(570\) 0 0
\(571\) 12.9291 0.541067 0.270534 0.962711i \(-0.412800\pi\)
0.270534 + 0.962711i \(0.412800\pi\)
\(572\) 0 0
\(573\) 6.19736 0.258898
\(574\) 0 0
\(575\) −3.36985 −0.140532
\(576\) 0 0
\(577\) 38.2876 1.59393 0.796967 0.604023i \(-0.206436\pi\)
0.796967 + 0.604023i \(0.206436\pi\)
\(578\) 0 0
\(579\) 6.78861 0.282125
\(580\) 0 0
\(581\) −34.1711 −1.41766
\(582\) 0 0
\(583\) 57.9322 2.39931
\(584\) 0 0
\(585\) −38.4564 −1.58998
\(586\) 0 0
\(587\) 41.9521 1.73155 0.865775 0.500434i \(-0.166826\pi\)
0.865775 + 0.500434i \(0.166826\pi\)
\(588\) 0 0
\(589\) −7.64182 −0.314876
\(590\) 0 0
\(591\) 6.18595 0.254456
\(592\) 0 0
\(593\) 2.09406 0.0859928 0.0429964 0.999075i \(-0.486310\pi\)
0.0429964 + 0.999075i \(0.486310\pi\)
\(594\) 0 0
\(595\) −68.4711 −2.80704
\(596\) 0 0
\(597\) 1.25167 0.0512274
\(598\) 0 0
\(599\) 9.46204 0.386608 0.193304 0.981139i \(-0.438080\pi\)
0.193304 + 0.981139i \(0.438080\pi\)
\(600\) 0 0
\(601\) 34.1000 1.39097 0.695484 0.718542i \(-0.255190\pi\)
0.695484 + 0.718542i \(0.255190\pi\)
\(602\) 0 0
\(603\) −20.7959 −0.846876
\(604\) 0 0
\(605\) −25.8230 −1.04986
\(606\) 0 0
\(607\) −34.3332 −1.39354 −0.696770 0.717294i \(-0.745380\pi\)
−0.696770 + 0.717294i \(0.745380\pi\)
\(608\) 0 0
\(609\) −9.55554 −0.387210
\(610\) 0 0
\(611\) 25.6965 1.03957
\(612\) 0 0
\(613\) −10.9176 −0.440957 −0.220479 0.975392i \(-0.570762\pi\)
−0.220479 + 0.975392i \(0.570762\pi\)
\(614\) 0 0
\(615\) −2.15891 −0.0870555
\(616\) 0 0
\(617\) −37.3923 −1.50536 −0.752678 0.658389i \(-0.771238\pi\)
−0.752678 + 0.658389i \(0.771238\pi\)
\(618\) 0 0
\(619\) −10.0717 −0.404817 −0.202409 0.979301i \(-0.564877\pi\)
−0.202409 + 0.979301i \(0.564877\pi\)
\(620\) 0 0
\(621\) 0.632770 0.0253922
\(622\) 0 0
\(623\) 10.6932 0.428416
\(624\) 0 0
\(625\) 73.6397 2.94559
\(626\) 0 0
\(627\) −12.1115 −0.483686
\(628\) 0 0
\(629\) 28.0442 1.11819
\(630\) 0 0
\(631\) −2.11890 −0.0843520 −0.0421760 0.999110i \(-0.513429\pi\)
−0.0421760 + 0.999110i \(0.513429\pi\)
\(632\) 0 0
\(633\) 9.19949 0.365647
\(634\) 0 0
\(635\) −11.5734 −0.459277
\(636\) 0 0
\(637\) −13.8507 −0.548783
\(638\) 0 0
\(639\) −38.4066 −1.51934
\(640\) 0 0
\(641\) −4.78686 −0.189070 −0.0945348 0.995522i \(-0.530136\pi\)
−0.0945348 + 0.995522i \(0.530136\pi\)
\(642\) 0 0
\(643\) 31.6313 1.24742 0.623709 0.781656i \(-0.285625\pi\)
0.623709 + 0.781656i \(0.285625\pi\)
\(644\) 0 0
\(645\) −6.86988 −0.270501
\(646\) 0 0
\(647\) −13.7434 −0.540310 −0.270155 0.962817i \(-0.587075\pi\)
−0.270155 + 0.962817i \(0.587075\pi\)
\(648\) 0 0
\(649\) 7.78645 0.305645
\(650\) 0 0
\(651\) 1.47732 0.0579008
\(652\) 0 0
\(653\) −0.0587916 −0.00230069 −0.00115035 0.999999i \(-0.500366\pi\)
−0.00115035 + 0.999999i \(0.500366\pi\)
\(654\) 0 0
\(655\) −40.3792 −1.57775
\(656\) 0 0
\(657\) 7.06003 0.275438
\(658\) 0 0
\(659\) 14.1013 0.549308 0.274654 0.961543i \(-0.411437\pi\)
0.274654 + 0.961543i \(0.411437\pi\)
\(660\) 0 0
\(661\) 2.47912 0.0964265 0.0482133 0.998837i \(-0.484647\pi\)
0.0482133 + 0.998837i \(0.484647\pi\)
\(662\) 0 0
\(663\) −6.39803 −0.248479
\(664\) 0 0
\(665\) 100.972 3.91551
\(666\) 0 0
\(667\) 1.83282 0.0709670
\(668\) 0 0
\(669\) −1.38667 −0.0536117
\(670\) 0 0
\(671\) −53.8938 −2.08055
\(672\) 0 0
\(673\) 1.88041 0.0724845 0.0362423 0.999343i \(-0.488461\pi\)
0.0362423 + 0.999343i \(0.488461\pi\)
\(674\) 0 0
\(675\) −30.4847 −1.17336
\(676\) 0 0
\(677\) 11.6258 0.446815 0.223407 0.974725i \(-0.428282\pi\)
0.223407 + 0.974725i \(0.428282\pi\)
\(678\) 0 0
\(679\) 23.5565 0.904017
\(680\) 0 0
\(681\) 10.0094 0.383561
\(682\) 0 0
\(683\) 0.350145 0.0133979 0.00669897 0.999978i \(-0.497868\pi\)
0.00669897 + 0.999978i \(0.497868\pi\)
\(684\) 0 0
\(685\) −59.6225 −2.27806
\(686\) 0 0
\(687\) 9.30429 0.354981
\(688\) 0 0
\(689\) −45.1295 −1.71930
\(690\) 0 0
\(691\) −12.9419 −0.492335 −0.246167 0.969227i \(-0.579171\pi\)
−0.246167 + 0.969227i \(0.579171\pi\)
\(692\) 0 0
\(693\) −39.3904 −1.49632
\(694\) 0 0
\(695\) −12.7715 −0.484452
\(696\) 0 0
\(697\) 6.04263 0.228881
\(698\) 0 0
\(699\) 9.93309 0.375704
\(700\) 0 0
\(701\) −42.8216 −1.61735 −0.808674 0.588256i \(-0.799815\pi\)
−0.808674 + 0.588256i \(0.799815\pi\)
\(702\) 0 0
\(703\) −41.3556 −1.55976
\(704\) 0 0
\(705\) 13.7723 0.518695
\(706\) 0 0
\(707\) 23.9756 0.901694
\(708\) 0 0
\(709\) −47.8206 −1.79594 −0.897971 0.440055i \(-0.854959\pi\)
−0.897971 + 0.440055i \(0.854959\pi\)
\(710\) 0 0
\(711\) 22.8716 0.857751
\(712\) 0 0
\(713\) −0.283361 −0.0106119
\(714\) 0 0
\(715\) 56.2098 2.10213
\(716\) 0 0
\(717\) −0.760845 −0.0284143
\(718\) 0 0
\(719\) −1.69842 −0.0633402 −0.0316701 0.999498i \(-0.510083\pi\)
−0.0316701 + 0.999498i \(0.510083\pi\)
\(720\) 0 0
\(721\) 29.0587 1.08220
\(722\) 0 0
\(723\) 4.15252 0.154434
\(724\) 0 0
\(725\) −88.2988 −3.27934
\(726\) 0 0
\(727\) 50.3164 1.86613 0.933066 0.359706i \(-0.117123\pi\)
0.933066 + 0.359706i \(0.117123\pi\)
\(728\) 0 0
\(729\) −18.3310 −0.678927
\(730\) 0 0
\(731\) 19.2283 0.711185
\(732\) 0 0
\(733\) 4.24192 0.156679 0.0783395 0.996927i \(-0.475038\pi\)
0.0783395 + 0.996927i \(0.475038\pi\)
\(734\) 0 0
\(735\) −7.42341 −0.273817
\(736\) 0 0
\(737\) 30.3964 1.11967
\(738\) 0 0
\(739\) 19.1496 0.704428 0.352214 0.935920i \(-0.385429\pi\)
0.352214 + 0.935920i \(0.385429\pi\)
\(740\) 0 0
\(741\) 9.43491 0.346600
\(742\) 0 0
\(743\) 42.3558 1.55388 0.776942 0.629572i \(-0.216770\pi\)
0.776942 + 0.629572i \(0.216770\pi\)
\(744\) 0 0
\(745\) −39.3870 −1.44303
\(746\) 0 0
\(747\) 28.7903 1.05338
\(748\) 0 0
\(749\) 6.74392 0.246417
\(750\) 0 0
\(751\) −27.0770 −0.988053 −0.494027 0.869447i \(-0.664475\pi\)
−0.494027 + 0.869447i \(0.664475\pi\)
\(752\) 0 0
\(753\) 1.81225 0.0660422
\(754\) 0 0
\(755\) 50.7497 1.84697
\(756\) 0 0
\(757\) −37.8455 −1.37552 −0.687759 0.725939i \(-0.741405\pi\)
−0.687759 + 0.725939i \(0.741405\pi\)
\(758\) 0 0
\(759\) −0.449097 −0.0163012
\(760\) 0 0
\(761\) 35.5873 1.29004 0.645019 0.764166i \(-0.276849\pi\)
0.645019 + 0.764166i \(0.276849\pi\)
\(762\) 0 0
\(763\) −11.7020 −0.423640
\(764\) 0 0
\(765\) 57.6891 2.08575
\(766\) 0 0
\(767\) −6.06569 −0.219019
\(768\) 0 0
\(769\) 37.9803 1.36960 0.684802 0.728729i \(-0.259889\pi\)
0.684802 + 0.728729i \(0.259889\pi\)
\(770\) 0 0
\(771\) 8.12996 0.292793
\(772\) 0 0
\(773\) 16.7036 0.600786 0.300393 0.953816i \(-0.402882\pi\)
0.300393 + 0.953816i \(0.402882\pi\)
\(774\) 0 0
\(775\) 13.6513 0.490370
\(776\) 0 0
\(777\) 7.99489 0.286815
\(778\) 0 0
\(779\) −8.91082 −0.319263
\(780\) 0 0
\(781\) 56.1370 2.00874
\(782\) 0 0
\(783\) 16.5802 0.592529
\(784\) 0 0
\(785\) 12.0117 0.428716
\(786\) 0 0
\(787\) 7.94706 0.283282 0.141641 0.989918i \(-0.454762\pi\)
0.141641 + 0.989918i \(0.454762\pi\)
\(788\) 0 0
\(789\) 11.5620 0.411619
\(790\) 0 0
\(791\) 58.8429 2.09221
\(792\) 0 0
\(793\) 41.9835 1.49088
\(794\) 0 0
\(795\) −24.1876 −0.857847
\(796\) 0 0
\(797\) −8.76611 −0.310511 −0.155256 0.987874i \(-0.549620\pi\)
−0.155256 + 0.987874i \(0.549620\pi\)
\(798\) 0 0
\(799\) −38.5477 −1.36372
\(800\) 0 0
\(801\) −9.00940 −0.318332
\(802\) 0 0
\(803\) −10.3193 −0.364160
\(804\) 0 0
\(805\) 3.74405 0.131960
\(806\) 0 0
\(807\) 2.61733 0.0921343
\(808\) 0 0
\(809\) 7.02651 0.247039 0.123519 0.992342i \(-0.460582\pi\)
0.123519 + 0.992342i \(0.460582\pi\)
\(810\) 0 0
\(811\) 9.74637 0.342241 0.171121 0.985250i \(-0.445261\pi\)
0.171121 + 0.985250i \(0.445261\pi\)
\(812\) 0 0
\(813\) 4.59311 0.161087
\(814\) 0 0
\(815\) −53.8584 −1.88658
\(816\) 0 0
\(817\) −28.3552 −0.992023
\(818\) 0 0
\(819\) 30.6853 1.07223
\(820\) 0 0
\(821\) −3.40971 −0.119000 −0.0594998 0.998228i \(-0.518951\pi\)
−0.0594998 + 0.998228i \(0.518951\pi\)
\(822\) 0 0
\(823\) −6.84363 −0.238554 −0.119277 0.992861i \(-0.538058\pi\)
−0.119277 + 0.992861i \(0.538058\pi\)
\(824\) 0 0
\(825\) 21.6359 0.753266
\(826\) 0 0
\(827\) 34.8563 1.21207 0.606036 0.795437i \(-0.292759\pi\)
0.606036 + 0.795437i \(0.292759\pi\)
\(828\) 0 0
\(829\) 27.9587 0.971047 0.485523 0.874224i \(-0.338629\pi\)
0.485523 + 0.874224i \(0.338629\pi\)
\(830\) 0 0
\(831\) −10.7208 −0.371901
\(832\) 0 0
\(833\) 20.7776 0.719902
\(834\) 0 0
\(835\) −68.8435 −2.38243
\(836\) 0 0
\(837\) −2.56337 −0.0886030
\(838\) 0 0
\(839\) −51.8984 −1.79173 −0.895865 0.444326i \(-0.853443\pi\)
−0.895865 + 0.444326i \(0.853443\pi\)
\(840\) 0 0
\(841\) 19.0246 0.656021
\(842\) 0 0
\(843\) −1.86609 −0.0642716
\(844\) 0 0
\(845\) 10.9693 0.377354
\(846\) 0 0
\(847\) 20.6049 0.707991
\(848\) 0 0
\(849\) 7.96683 0.273421
\(850\) 0 0
\(851\) −1.53348 −0.0525669
\(852\) 0 0
\(853\) −14.5182 −0.497094 −0.248547 0.968620i \(-0.579953\pi\)
−0.248547 + 0.968620i \(0.579953\pi\)
\(854\) 0 0
\(855\) −85.0717 −2.90939
\(856\) 0 0
\(857\) −6.51805 −0.222652 −0.111326 0.993784i \(-0.535510\pi\)
−0.111326 + 0.993784i \(0.535510\pi\)
\(858\) 0 0
\(859\) −18.9016 −0.644913 −0.322456 0.946584i \(-0.604509\pi\)
−0.322456 + 0.946584i \(0.604509\pi\)
\(860\) 0 0
\(861\) 1.72265 0.0587076
\(862\) 0 0
\(863\) −40.4662 −1.37749 −0.688743 0.725005i \(-0.741837\pi\)
−0.688743 + 0.725005i \(0.741837\pi\)
\(864\) 0 0
\(865\) −13.6076 −0.462674
\(866\) 0 0
\(867\) 2.62328 0.0890911
\(868\) 0 0
\(869\) −33.4302 −1.13404
\(870\) 0 0
\(871\) −23.6789 −0.802330
\(872\) 0 0
\(873\) −19.8471 −0.671724
\(874\) 0 0
\(875\) −109.593 −3.70492
\(876\) 0 0
\(877\) −38.3146 −1.29379 −0.646896 0.762578i \(-0.723933\pi\)
−0.646896 + 0.762578i \(0.723933\pi\)
\(878\) 0 0
\(879\) 3.66986 0.123781
\(880\) 0 0
\(881\) −48.1919 −1.62363 −0.811813 0.583917i \(-0.801519\pi\)
−0.811813 + 0.583917i \(0.801519\pi\)
\(882\) 0 0
\(883\) 24.0449 0.809176 0.404588 0.914499i \(-0.367415\pi\)
0.404588 + 0.914499i \(0.367415\pi\)
\(884\) 0 0
\(885\) −3.25097 −0.109280
\(886\) 0 0
\(887\) 3.95712 0.132867 0.0664336 0.997791i \(-0.478838\pi\)
0.0664336 + 0.997791i \(0.478838\pi\)
\(888\) 0 0
\(889\) 9.23473 0.309723
\(890\) 0 0
\(891\) 31.0977 1.04181
\(892\) 0 0
\(893\) 56.8447 1.90224
\(894\) 0 0
\(895\) 88.1027 2.94495
\(896\) 0 0
\(897\) 0.349849 0.0116811
\(898\) 0 0
\(899\) −7.42479 −0.247631
\(900\) 0 0
\(901\) 67.6995 2.25540
\(902\) 0 0
\(903\) 5.48165 0.182418
\(904\) 0 0
\(905\) 67.9250 2.25790
\(906\) 0 0
\(907\) 46.3721 1.53976 0.769880 0.638189i \(-0.220316\pi\)
0.769880 + 0.638189i \(0.220316\pi\)
\(908\) 0 0
\(909\) −20.2002 −0.669998
\(910\) 0 0
\(911\) −13.0731 −0.433133 −0.216566 0.976268i \(-0.569486\pi\)
−0.216566 + 0.976268i \(0.569486\pi\)
\(912\) 0 0
\(913\) −42.0813 −1.39269
\(914\) 0 0
\(915\) 22.5015 0.743877
\(916\) 0 0
\(917\) 32.2196 1.06398
\(918\) 0 0
\(919\) 55.9346 1.84511 0.922556 0.385864i \(-0.126097\pi\)
0.922556 + 0.385864i \(0.126097\pi\)
\(920\) 0 0
\(921\) −9.41603 −0.310269
\(922\) 0 0
\(923\) −43.7310 −1.43942
\(924\) 0 0
\(925\) 73.8776 2.42908
\(926\) 0 0
\(927\) −24.4829 −0.804124
\(928\) 0 0
\(929\) 20.7469 0.680684 0.340342 0.940302i \(-0.389457\pi\)
0.340342 + 0.940302i \(0.389457\pi\)
\(930\) 0 0
\(931\) −30.6399 −1.00418
\(932\) 0 0
\(933\) −13.6783 −0.447809
\(934\) 0 0
\(935\) −84.3212 −2.75760
\(936\) 0 0
\(937\) −21.5805 −0.705006 −0.352503 0.935811i \(-0.614669\pi\)
−0.352503 + 0.935811i \(0.614669\pi\)
\(938\) 0 0
\(939\) −5.89145 −0.192260
\(940\) 0 0
\(941\) 53.5707 1.74636 0.873178 0.487401i \(-0.162055\pi\)
0.873178 + 0.487401i \(0.162055\pi\)
\(942\) 0 0
\(943\) −0.330415 −0.0107598
\(944\) 0 0
\(945\) 33.8698 1.10179
\(946\) 0 0
\(947\) 7.27841 0.236516 0.118258 0.992983i \(-0.462269\pi\)
0.118258 + 0.992983i \(0.462269\pi\)
\(948\) 0 0
\(949\) 8.03878 0.260950
\(950\) 0 0
\(951\) −0.583518 −0.0189219
\(952\) 0 0
\(953\) 33.6483 1.08998 0.544988 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(954\) 0 0
\(955\) −63.6265 −2.05890
\(956\) 0 0
\(957\) −11.7675 −0.380390
\(958\) 0 0
\(959\) 47.5743 1.53625
\(960\) 0 0
\(961\) −29.8521 −0.962971
\(962\) 0 0
\(963\) −5.68196 −0.183099
\(964\) 0 0
\(965\) −69.6966 −2.24361
\(966\) 0 0
\(967\) 12.9918 0.417788 0.208894 0.977938i \(-0.433014\pi\)
0.208894 + 0.977938i \(0.433014\pi\)
\(968\) 0 0
\(969\) −14.1535 −0.454675
\(970\) 0 0
\(971\) 6.30177 0.202233 0.101117 0.994875i \(-0.467758\pi\)
0.101117 + 0.994875i \(0.467758\pi\)
\(972\) 0 0
\(973\) 10.1907 0.326700
\(974\) 0 0
\(975\) −16.8545 −0.539776
\(976\) 0 0
\(977\) −42.7142 −1.36655 −0.683275 0.730161i \(-0.739445\pi\)
−0.683275 + 0.730161i \(0.739445\pi\)
\(978\) 0 0
\(979\) 13.1686 0.420870
\(980\) 0 0
\(981\) 9.85929 0.314783
\(982\) 0 0
\(983\) −0.161818 −0.00516121 −0.00258060 0.999997i \(-0.500821\pi\)
−0.00258060 + 0.999997i \(0.500821\pi\)
\(984\) 0 0
\(985\) −63.5093 −2.02358
\(986\) 0 0
\(987\) −10.9893 −0.349792
\(988\) 0 0
\(989\) −1.05142 −0.0334331
\(990\) 0 0
\(991\) 8.47381 0.269179 0.134590 0.990901i \(-0.457028\pi\)
0.134590 + 0.990901i \(0.457028\pi\)
\(992\) 0 0
\(993\) −0.384659 −0.0122068
\(994\) 0 0
\(995\) −12.8505 −0.407389
\(996\) 0 0
\(997\) −33.6467 −1.06560 −0.532800 0.846241i \(-0.678860\pi\)
−0.532800 + 0.846241i \(0.678860\pi\)
\(998\) 0 0
\(999\) −13.8723 −0.438900
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.g.1.17 33
4.3 odd 2 8048.2.a.x.1.17 33
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4024.2.a.g.1.17 33 1.1 even 1 trivial
8048.2.a.x.1.17 33 4.3 odd 2