Properties

Label 7600.2.a.bj.1.2
Level $7600$
Weight $2$
Character 7600.1
Self dual yes
Analytic conductor $60.686$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7600,2,Mod(1,7600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("7600.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7600.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: \(60.6863055362\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.257.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1900)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.713538\) of defining polynomial
Character \(\chi\) \(=\) 7600.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.286462 q^{3} -0.286462 q^{7} -2.91794 q^{9} +O(q^{10})\) \(q-0.286462 q^{3} -0.286462 q^{7} -2.91794 q^{9} -4.26819 q^{11} +3.20440 q^{13} +0.286462 q^{17} +1.00000 q^{19} +0.0820605 q^{21} -0.936212 q^{23} +1.69527 q^{27} +2.26819 q^{29} +4.18613 q^{31} +1.22267 q^{33} +8.67699 q^{37} -0.917939 q^{39} +1.08206 q^{41} -4.42708 q^{43} +0.759053 q^{47} -6.91794 q^{49} -0.0820605 q^{51} +4.42708 q^{53} -0.286462 q^{57} +4.70050 q^{59} +10.1861 q^{61} +0.835879 q^{63} -9.82284 q^{67} +0.268189 q^{69} -4.83588 q^{71} +10.4726 q^{73} +1.22267 q^{77} -5.10407 q^{79} +8.26819 q^{81} -15.7355 q^{83} -0.649750 q^{87} -9.75382 q^{89} -0.917939 q^{91} -1.19917 q^{93} -9.61320 q^{97} +12.4543 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} - 2 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} - 2 q^{7} + q^{9} + q^{11} + q^{13} + 2 q^{17} + 3 q^{19} + 10 q^{21} - 8 q^{23} - 11 q^{27} - 7 q^{29} - 11 q^{31} + 10 q^{33} - 5 q^{37} + 7 q^{39} + 13 q^{41} - 11 q^{43} - 19 q^{47} - 11 q^{49} - 10 q^{51} + 11 q^{53} - 2 q^{57} + 6 q^{59} + 7 q^{61} - 17 q^{63} - 3 q^{67} - 13 q^{69} + 5 q^{71} + 9 q^{73} + 10 q^{77} + 18 q^{79} + 11 q^{81} - 3 q^{83} - 6 q^{87} + 7 q^{91} + 13 q^{93} - 3 q^{97}+O(q^{100}) \) 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
\(3\) −0.286462 −0.165389 −0.0826945 0.996575i \(-0.526353\pi\)
−0.0826945 + 0.996575i \(0.526353\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −0.286462 −0.108272 −0.0541362 0.998534i \(-0.517241\pi\)
−0.0541362 + 0.998534i \(0.517241\pi\)
\(8\) 0 0
\(9\) −2.91794 −0.972646
\(10\) 0 0
\(11\) −4.26819 −1.28691 −0.643454 0.765485i \(-0.722499\pi\)
−0.643454 + 0.765485i \(0.722499\pi\)
\(12\) 0 0
\(13\) 3.20440 0.888741 0.444371 0.895843i \(-0.353427\pi\)
0.444371 + 0.895843i \(0.353427\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.286462 0.0694773 0.0347386 0.999396i \(-0.488940\pi\)
0.0347386 + 0.999396i \(0.488940\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0.0820605 0.0179071
\(22\) 0 0
\(23\) −0.936212 −0.195214 −0.0976069 0.995225i \(-0.531119\pi\)
−0.0976069 + 0.995225i \(0.531119\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.69527 0.326254
\(28\) 0 0
\(29\) 2.26819 0.421192 0.210596 0.977573i \(-0.432460\pi\)
0.210596 + 0.977573i \(0.432460\pi\)
\(30\) 0 0
\(31\) 4.18613 0.751851 0.375925 0.926650i \(-0.377325\pi\)
0.375925 + 0.926650i \(0.377325\pi\)
\(32\) 0 0
\(33\) 1.22267 0.212840
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.67699 1.42649 0.713244 0.700915i \(-0.247225\pi\)
0.713244 + 0.700915i \(0.247225\pi\)
\(38\) 0 0
\(39\) −0.917939 −0.146988
\(40\) 0 0
\(41\) 1.08206 0.168989 0.0844947 0.996424i \(-0.473072\pi\)
0.0844947 + 0.996424i \(0.473072\pi\)
\(42\) 0 0
\(43\) −4.42708 −0.675123 −0.337561 0.941304i \(-0.609602\pi\)
−0.337561 + 0.941304i \(0.609602\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.759053 0.110719 0.0553596 0.998466i \(-0.482369\pi\)
0.0553596 + 0.998466i \(0.482369\pi\)
\(48\) 0 0
\(49\) −6.91794 −0.988277
\(50\) 0 0
\(51\) −0.0820605 −0.0114908
\(52\) 0 0
\(53\) 4.42708 0.608106 0.304053 0.952655i \(-0.401660\pi\)
0.304053 + 0.952655i \(0.401660\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.286462 −0.0379428
\(58\) 0 0
\(59\) 4.70050 0.611953 0.305976 0.952039i \(-0.401017\pi\)
0.305976 + 0.952039i \(0.401017\pi\)
\(60\) 0 0
\(61\) 10.1861 1.30420 0.652100 0.758133i \(-0.273888\pi\)
0.652100 + 0.758133i \(0.273888\pi\)
\(62\) 0 0
\(63\) 0.835879 0.105311
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −9.82284 −1.20005 −0.600025 0.799981i \(-0.704843\pi\)
−0.600025 + 0.799981i \(0.704843\pi\)
\(68\) 0 0
\(69\) 0.268189 0.0322862
\(70\) 0 0
\(71\) −4.83588 −0.573913 −0.286957 0.957944i \(-0.592644\pi\)
−0.286957 + 0.957944i \(0.592644\pi\)
\(72\) 0 0
\(73\) 10.4726 1.22572 0.612862 0.790190i \(-0.290018\pi\)
0.612862 + 0.790190i \(0.290018\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.22267 0.139337
\(78\) 0 0
\(79\) −5.10407 −0.574253 −0.287126 0.957893i \(-0.592700\pi\)
−0.287126 + 0.957893i \(0.592700\pi\)
\(80\) 0 0
\(81\) 8.26819 0.918688
\(82\) 0 0
\(83\) −15.7355 −1.72720 −0.863600 0.504177i \(-0.831796\pi\)
−0.863600 + 0.504177i \(0.831796\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −0.649750 −0.0696605
\(88\) 0 0
\(89\) −9.75382 −1.03390 −0.516951 0.856015i \(-0.672933\pi\)
−0.516951 + 0.856015i \(0.672933\pi\)
\(90\) 0 0
\(91\) −0.917939 −0.0962262
\(92\) 0 0
\(93\) −1.19917 −0.124348
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.61320 −0.976073 −0.488037 0.872823i \(-0.662287\pi\)
−0.488037 + 0.872823i \(0.662287\pi\)
\(98\) 0 0
\(99\) 12.4543 1.25171
\(100\) 0 0
\(101\) 0.731811 0.0728179 0.0364089 0.999337i \(-0.488408\pi\)
0.0364089 + 0.999337i \(0.488408\pi\)
\(102\) 0 0
\(103\) 3.10407 0.305853 0.152926 0.988238i \(-0.451130\pi\)
0.152926 + 0.988238i \(0.451130\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −14.1731 −1.37016 −0.685082 0.728466i \(-0.740234\pi\)
−0.685082 + 0.728466i \(0.740234\pi\)
\(108\) 0 0
\(109\) −9.61844 −0.921279 −0.460640 0.887587i \(-0.652380\pi\)
−0.460640 + 0.887587i \(0.652380\pi\)
\(110\) 0 0
\(111\) −2.48563 −0.235925
\(112\) 0 0
\(113\) 8.77733 0.825701 0.412851 0.910799i \(-0.364533\pi\)
0.412851 + 0.910799i \(0.364533\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −9.35025 −0.864431
\(118\) 0 0
\(119\) −0.0820605 −0.00752248
\(120\) 0 0
\(121\) 7.21744 0.656131
\(122\) 0 0
\(123\) −0.309969 −0.0279490
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −6.77209 −0.600926 −0.300463 0.953793i \(-0.597141\pi\)
−0.300463 + 0.953793i \(0.597141\pi\)
\(128\) 0 0
\(129\) 1.26819 0.111658
\(130\) 0 0
\(131\) 2.91794 0.254942 0.127471 0.991842i \(-0.459314\pi\)
0.127471 + 0.991842i \(0.459314\pi\)
\(132\) 0 0
\(133\) −0.286462 −0.0248394
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −14.4360 −1.23335 −0.616677 0.787216i \(-0.711522\pi\)
−0.616677 + 0.787216i \(0.711522\pi\)
\(138\) 0 0
\(139\) 2.29950 0.195041 0.0975205 0.995234i \(-0.468909\pi\)
0.0975205 + 0.995234i \(0.468909\pi\)
\(140\) 0 0
\(141\) −0.217440 −0.0183117
\(142\) 0 0
\(143\) −13.6770 −1.14373
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.98173 0.163450
\(148\) 0 0
\(149\) 6.75382 0.553294 0.276647 0.960972i \(-0.410777\pi\)
0.276647 + 0.960972i \(0.410777\pi\)
\(150\) 0 0
\(151\) 11.8866 0.967320 0.483660 0.875256i \(-0.339307\pi\)
0.483660 + 0.875256i \(0.339307\pi\)
\(152\) 0 0
\(153\) −0.835879 −0.0675768
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −14.1731 −1.13114 −0.565568 0.824702i \(-0.691343\pi\)
−0.565568 + 0.824702i \(0.691343\pi\)
\(158\) 0 0
\(159\) −1.26819 −0.100574
\(160\) 0 0
\(161\) 0.268189 0.0211363
\(162\) 0 0
\(163\) −7.89443 −0.618340 −0.309170 0.951007i \(-0.600051\pi\)
−0.309170 + 0.951007i \(0.600051\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −20.4726 −1.58422 −0.792108 0.610381i \(-0.791017\pi\)
−0.792108 + 0.610381i \(0.791017\pi\)
\(168\) 0 0
\(169\) −2.73181 −0.210139
\(170\) 0 0
\(171\) −2.91794 −0.223140
\(172\) 0 0
\(173\) −24.0492 −1.82843 −0.914215 0.405229i \(-0.867192\pi\)
−0.914215 + 0.405229i \(0.867192\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1.34651 −0.101210
\(178\) 0 0
\(179\) −18.6718 −1.39559 −0.697796 0.716296i \(-0.745836\pi\)
−0.697796 + 0.716296i \(0.745836\pi\)
\(180\) 0 0
\(181\) −3.53638 −0.262857 −0.131428 0.991326i \(-0.541956\pi\)
−0.131428 + 0.991326i \(0.541956\pi\)
\(182\) 0 0
\(183\) −2.91794 −0.215700
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1.22267 −0.0894108
\(188\) 0 0
\(189\) −0.485629 −0.0353243
\(190\) 0 0
\(191\) 14.2682 1.03241 0.516205 0.856465i \(-0.327344\pi\)
0.516205 + 0.856465i \(0.327344\pi\)
\(192\) 0 0
\(193\) −8.18089 −0.588874 −0.294437 0.955671i \(-0.595132\pi\)
−0.294437 + 0.955671i \(0.595132\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.6404 0.900595 0.450297 0.892879i \(-0.351318\pi\)
0.450297 + 0.892879i \(0.351318\pi\)
\(198\) 0 0
\(199\) 4.10407 0.290930 0.145465 0.989363i \(-0.453532\pi\)
0.145465 + 0.989363i \(0.453532\pi\)
\(200\) 0 0
\(201\) 2.81387 0.198475
\(202\) 0 0
\(203\) −0.649750 −0.0456035
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.73181 0.189874
\(208\) 0 0
\(209\) −4.26819 −0.295237
\(210\) 0 0
\(211\) −1.53638 −0.105769 −0.0528843 0.998601i \(-0.516841\pi\)
−0.0528843 + 0.998601i \(0.516841\pi\)
\(212\) 0 0
\(213\) 1.38530 0.0949189
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1.19917 −0.0814048
\(218\) 0 0
\(219\) −3.00000 −0.202721
\(220\) 0 0
\(221\) 0.917939 0.0617473
\(222\) 0 0
\(223\) −1.40880 −0.0943404 −0.0471702 0.998887i \(-0.515020\pi\)
−0.0471702 + 0.998887i \(0.515020\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −22.9179 −1.52112 −0.760559 0.649269i \(-0.775075\pi\)
−0.760559 + 0.649269i \(0.775075\pi\)
\(228\) 0 0
\(229\) 3.78513 0.250128 0.125064 0.992149i \(-0.460086\pi\)
0.125064 + 0.992149i \(0.460086\pi\)
\(230\) 0 0
\(231\) −0.350250 −0.0230447
\(232\) 0 0
\(233\) 2.26819 0.148594 0.0742970 0.997236i \(-0.476329\pi\)
0.0742970 + 0.997236i \(0.476329\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.46212 0.0949750
\(238\) 0 0
\(239\) 2.18613 0.141409 0.0707045 0.997497i \(-0.477475\pi\)
0.0707045 + 0.997497i \(0.477475\pi\)
\(240\) 0 0
\(241\) 19.9907 1.28771 0.643857 0.765146i \(-0.277333\pi\)
0.643857 + 0.765146i \(0.277333\pi\)
\(242\) 0 0
\(243\) −7.45432 −0.478195
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.20440 0.203891
\(248\) 0 0
\(249\) 4.50764 0.285660
\(250\) 0 0
\(251\) −1.35025 −0.0852270 −0.0426135 0.999092i \(-0.513568\pi\)
−0.0426135 + 0.999092i \(0.513568\pi\)
\(252\) 0 0
\(253\) 3.99593 0.251222
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −11.9582 −0.745933 −0.372967 0.927845i \(-0.621659\pi\)
−0.372967 + 0.927845i \(0.621659\pi\)
\(258\) 0 0
\(259\) −2.48563 −0.154449
\(260\) 0 0
\(261\) −6.61844 −0.409671
\(262\) 0 0
\(263\) 8.31370 0.512645 0.256322 0.966591i \(-0.417489\pi\)
0.256322 + 0.966591i \(0.417489\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 2.79410 0.170996
\(268\) 0 0
\(269\) −5.02201 −0.306197 −0.153099 0.988211i \(-0.548925\pi\)
−0.153099 + 0.988211i \(0.548925\pi\)
\(270\) 0 0
\(271\) −6.96869 −0.423318 −0.211659 0.977344i \(-0.567887\pi\)
−0.211659 + 0.977344i \(0.567887\pi\)
\(272\) 0 0
\(273\) 0.262955 0.0159148
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −12.7083 −0.763568 −0.381784 0.924252i \(-0.624690\pi\)
−0.381784 + 0.924252i \(0.624690\pi\)
\(278\) 0 0
\(279\) −12.2149 −0.731285
\(280\) 0 0
\(281\) −16.5364 −0.986478 −0.493239 0.869894i \(-0.664187\pi\)
−0.493239 + 0.869894i \(0.664187\pi\)
\(282\) 0 0
\(283\) 26.1548 1.55474 0.777371 0.629042i \(-0.216553\pi\)
0.777371 + 0.629042i \(0.216553\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.309969 −0.0182969
\(288\) 0 0
\(289\) −16.9179 −0.995173
\(290\) 0 0
\(291\) 2.75382 0.161432
\(292\) 0 0
\(293\) −3.95449 −0.231023 −0.115512 0.993306i \(-0.536851\pi\)
−0.115512 + 0.993306i \(0.536851\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −7.23571 −0.419859
\(298\) 0 0
\(299\) −3.00000 −0.173494
\(300\) 0 0
\(301\) 1.26819 0.0730972
\(302\) 0 0
\(303\) −0.209636 −0.0120433
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 20.7355 1.18344 0.591720 0.806144i \(-0.298449\pi\)
0.591720 + 0.806144i \(0.298449\pi\)
\(308\) 0 0
\(309\) −0.889198 −0.0505847
\(310\) 0 0
\(311\) 1.97799 0.112162 0.0560808 0.998426i \(-0.482140\pi\)
0.0560808 + 0.998426i \(0.482140\pi\)
\(312\) 0 0
\(313\) −8.67699 −0.490453 −0.245226 0.969466i \(-0.578862\pi\)
−0.245226 + 0.969466i \(0.578862\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.34502 −0.131709 −0.0658546 0.997829i \(-0.520977\pi\)
−0.0658546 + 0.997829i \(0.520977\pi\)
\(318\) 0 0
\(319\) −9.68106 −0.542035
\(320\) 0 0
\(321\) 4.06005 0.226610
\(322\) 0 0
\(323\) 0.286462 0.0159392
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2.75532 0.152369
\(328\) 0 0
\(329\) −0.217440 −0.0119878
\(330\) 0 0
\(331\) 22.8579 1.25638 0.628192 0.778059i \(-0.283795\pi\)
0.628192 + 0.778059i \(0.283795\pi\)
\(332\) 0 0
\(333\) −25.3189 −1.38747
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −0.386795 −0.0210701 −0.0105350 0.999945i \(-0.503353\pi\)
−0.0105350 + 0.999945i \(0.503353\pi\)
\(338\) 0 0
\(339\) −2.51437 −0.136562
\(340\) 0 0
\(341\) −17.8672 −0.967563
\(342\) 0 0
\(343\) 3.98696 0.215276
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −14.1406 −0.759108 −0.379554 0.925170i \(-0.623923\pi\)
−0.379554 + 0.925170i \(0.623923\pi\)
\(348\) 0 0
\(349\) −2.81387 −0.150623 −0.0753115 0.997160i \(-0.523995\pi\)
−0.0753115 + 0.997160i \(0.523995\pi\)
\(350\) 0 0
\(351\) 5.43231 0.289955
\(352\) 0 0
\(353\) −13.3137 −0.708617 −0.354308 0.935129i \(-0.615284\pi\)
−0.354308 + 0.935129i \(0.615284\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0.0235072 0.00124413
\(358\) 0 0
\(359\) −14.1861 −0.748715 −0.374358 0.927284i \(-0.622137\pi\)
−0.374358 + 0.927284i \(0.622137\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −2.06752 −0.108517
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 28.0272 1.46301 0.731505 0.681836i \(-0.238818\pi\)
0.731505 + 0.681836i \(0.238818\pi\)
\(368\) 0 0
\(369\) −3.15739 −0.164367
\(370\) 0 0
\(371\) −1.26819 −0.0658411
\(372\) 0 0
\(373\) −24.3502 −1.26081 −0.630404 0.776267i \(-0.717111\pi\)
−0.630404 + 0.776267i \(0.717111\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7.26819 0.374331
\(378\) 0 0
\(379\) −28.8866 −1.48381 −0.741903 0.670507i \(-0.766077\pi\)
−0.741903 + 0.670507i \(0.766077\pi\)
\(380\) 0 0
\(381\) 1.93995 0.0993865
\(382\) 0 0
\(383\) 10.3398 0.528338 0.264169 0.964476i \(-0.414902\pi\)
0.264169 + 0.964476i \(0.414902\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 12.9179 0.656656
\(388\) 0 0
\(389\) 33.9086 1.71924 0.859618 0.510937i \(-0.170702\pi\)
0.859618 + 0.510937i \(0.170702\pi\)
\(390\) 0 0
\(391\) −0.268189 −0.0135629
\(392\) 0 0
\(393\) −0.835879 −0.0421645
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −33.8866 −1.70072 −0.850361 0.526200i \(-0.823616\pi\)
−0.850361 + 0.526200i \(0.823616\pi\)
\(398\) 0 0
\(399\) 0.0820605 0.00410816
\(400\) 0 0
\(401\) −10.5051 −0.524598 −0.262299 0.964987i \(-0.584481\pi\)
−0.262299 + 0.964987i \(0.584481\pi\)
\(402\) 0 0
\(403\) 13.4140 0.668201
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −37.0350 −1.83576
\(408\) 0 0
\(409\) −6.78256 −0.335376 −0.167688 0.985840i \(-0.553630\pi\)
−0.167688 + 0.985840i \(0.553630\pi\)
\(410\) 0 0
\(411\) 4.13538 0.203983
\(412\) 0 0
\(413\) −1.34651 −0.0662577
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −0.658720 −0.0322576
\(418\) 0 0
\(419\) −18.6498 −0.911100 −0.455550 0.890210i \(-0.650557\pi\)
−0.455550 + 0.890210i \(0.650557\pi\)
\(420\) 0 0
\(421\) 9.64975 0.470300 0.235150 0.971959i \(-0.424442\pi\)
0.235150 + 0.971959i \(0.424442\pi\)
\(422\) 0 0
\(423\) −2.21487 −0.107691
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −2.91794 −0.141209
\(428\) 0 0
\(429\) 3.91794 0.189160
\(430\) 0 0
\(431\) 33.7758 1.62692 0.813462 0.581618i \(-0.197580\pi\)
0.813462 + 0.581618i \(0.197580\pi\)
\(432\) 0 0
\(433\) 1.08986 0.0523755 0.0261878 0.999657i \(-0.491663\pi\)
0.0261878 + 0.999657i \(0.491663\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −0.936212 −0.0447851
\(438\) 0 0
\(439\) 12.9687 0.618962 0.309481 0.950906i \(-0.399845\pi\)
0.309481 + 0.950906i \(0.399845\pi\)
\(440\) 0 0
\(441\) 20.1861 0.961244
\(442\) 0 0
\(443\) 31.9437 1.51769 0.758845 0.651271i \(-0.225764\pi\)
0.758845 + 0.651271i \(0.225764\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −1.93471 −0.0915088
\(448\) 0 0
\(449\) −5.05075 −0.238360 −0.119180 0.992873i \(-0.538026\pi\)
−0.119180 + 0.992873i \(0.538026\pi\)
\(450\) 0 0
\(451\) −4.61844 −0.217474
\(452\) 0 0
\(453\) −3.40507 −0.159984
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 35.0948 1.64166 0.820832 0.571170i \(-0.193510\pi\)
0.820832 + 0.571170i \(0.193510\pi\)
\(458\) 0 0
\(459\) 0.485629 0.0226672
\(460\) 0 0
\(461\) −27.8866 −1.29881 −0.649405 0.760443i \(-0.724982\pi\)
−0.649405 + 0.760443i \(0.724982\pi\)
\(462\) 0 0
\(463\) 28.0037 1.30144 0.650722 0.759316i \(-0.274466\pi\)
0.650722 + 0.759316i \(0.274466\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −17.4543 −0.807690 −0.403845 0.914828i \(-0.632326\pi\)
−0.403845 + 0.914828i \(0.632326\pi\)
\(468\) 0 0
\(469\) 2.81387 0.129933
\(470\) 0 0
\(471\) 4.06005 0.187077
\(472\) 0 0
\(473\) 18.8956 0.868821
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −12.9179 −0.591472
\(478\) 0 0
\(479\) −26.0440 −1.18998 −0.594991 0.803733i \(-0.702844\pi\)
−0.594991 + 0.803733i \(0.702844\pi\)
\(480\) 0 0
\(481\) 27.8046 1.26778
\(482\) 0 0
\(483\) −0.0768261 −0.00349571
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 21.9907 0.996494 0.498247 0.867035i \(-0.333977\pi\)
0.498247 + 0.867035i \(0.333977\pi\)
\(488\) 0 0
\(489\) 2.26146 0.102267
\(490\) 0 0
\(491\) 33.9907 1.53398 0.766989 0.641660i \(-0.221754\pi\)
0.766989 + 0.641660i \(0.221754\pi\)
\(492\) 0 0
\(493\) 0.649750 0.0292633
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.38530 0.0621390
\(498\) 0 0
\(499\) −6.83331 −0.305901 −0.152950 0.988234i \(-0.548877\pi\)
−0.152950 + 0.988234i \(0.548877\pi\)
\(500\) 0 0
\(501\) 5.86462 0.262012
\(502\) 0 0
\(503\) −34.4685 −1.53688 −0.768438 0.639925i \(-0.778966\pi\)
−0.768438 + 0.639925i \(0.778966\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.782560 0.0347547
\(508\) 0 0
\(509\) 18.9179 0.838523 0.419261 0.907866i \(-0.362289\pi\)
0.419261 + 0.907866i \(0.362289\pi\)
\(510\) 0 0
\(511\) −3.00000 −0.132712
\(512\) 0 0
\(513\) 1.69527 0.0748478
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −3.23978 −0.142485
\(518\) 0 0
\(519\) 6.88920 0.302402
\(520\) 0 0
\(521\) −1.62774 −0.0713127 −0.0356563 0.999364i \(-0.511352\pi\)
−0.0356563 + 0.999364i \(0.511352\pi\)
\(522\) 0 0
\(523\) 1.16669 0.0510158 0.0255079 0.999675i \(-0.491880\pi\)
0.0255079 + 0.999675i \(0.491880\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.19917 0.0522365
\(528\) 0 0
\(529\) −22.1235 −0.961892
\(530\) 0 0
\(531\) −13.7158 −0.595214
\(532\) 0 0
\(533\) 3.46736 0.150188
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 5.34875 0.230816
\(538\) 0 0
\(539\) 29.5271 1.27182
\(540\) 0 0
\(541\) −23.9399 −1.02926 −0.514629 0.857413i \(-0.672070\pi\)
−0.514629 + 0.857413i \(0.672070\pi\)
\(542\) 0 0
\(543\) 1.01304 0.0434736
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −14.8139 −0.633395 −0.316698 0.948527i \(-0.602574\pi\)
−0.316698 + 0.948527i \(0.602574\pi\)
\(548\) 0 0
\(549\) −29.7225 −1.26853
\(550\) 0 0
\(551\) 2.26819 0.0966281
\(552\) 0 0
\(553\) 1.46212 0.0621757
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −39.3592 −1.66770 −0.833852 0.551988i \(-0.813869\pi\)
−0.833852 + 0.551988i \(0.813869\pi\)
\(558\) 0 0
\(559\) −14.1861 −0.600009
\(560\) 0 0
\(561\) 0.350250 0.0147876
\(562\) 0 0
\(563\) −9.55722 −0.402789 −0.201394 0.979510i \(-0.564547\pi\)
−0.201394 + 0.979510i \(0.564547\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −2.36852 −0.0994686
\(568\) 0 0
\(569\) 21.2995 0.892922 0.446461 0.894803i \(-0.352684\pi\)
0.446461 + 0.894803i \(0.352684\pi\)
\(570\) 0 0
\(571\) 17.4543 0.730440 0.365220 0.930921i \(-0.380994\pi\)
0.365220 + 0.930921i \(0.380994\pi\)
\(572\) 0 0
\(573\) −4.08729 −0.170749
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −37.2861 −1.55224 −0.776121 0.630584i \(-0.782815\pi\)
−0.776121 + 0.630584i \(0.782815\pi\)
\(578\) 0 0
\(579\) 2.34352 0.0973932
\(580\) 0 0
\(581\) 4.50764 0.187008
\(582\) 0 0
\(583\) −18.8956 −0.782576
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 7.73181 0.319126 0.159563 0.987188i \(-0.448991\pi\)
0.159563 + 0.987188i \(0.448991\pi\)
\(588\) 0 0
\(589\) 4.18613 0.172486
\(590\) 0 0
\(591\) −3.62101 −0.148948
\(592\) 0 0
\(593\) −6.74858 −0.277131 −0.138566 0.990353i \(-0.544249\pi\)
−0.138566 + 0.990353i \(0.544249\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.17566 −0.0481166
\(598\) 0 0
\(599\) 16.9687 0.693322 0.346661 0.937991i \(-0.387315\pi\)
0.346661 + 0.937991i \(0.387315\pi\)
\(600\) 0 0
\(601\) 31.7251 1.29409 0.647046 0.762451i \(-0.276004\pi\)
0.647046 + 0.762451i \(0.276004\pi\)
\(602\) 0 0
\(603\) 28.6625 1.16723
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −3.84518 −0.156071 −0.0780356 0.996951i \(-0.524865\pi\)
−0.0780356 + 0.996951i \(0.524865\pi\)
\(608\) 0 0
\(609\) 0.186129 0.00754232
\(610\) 0 0
\(611\) 2.43231 0.0984007
\(612\) 0 0
\(613\) −32.1208 −1.29735 −0.648674 0.761066i \(-0.724676\pi\)
−0.648674 + 0.761066i \(0.724676\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 35.2630 1.41963 0.709817 0.704387i \(-0.248778\pi\)
0.709817 + 0.704387i \(0.248778\pi\)
\(618\) 0 0
\(619\) −9.29020 −0.373405 −0.186702 0.982417i \(-0.559780\pi\)
−0.186702 + 0.982417i \(0.559780\pi\)
\(620\) 0 0
\(621\) −1.58713 −0.0636893
\(622\) 0 0
\(623\) 2.79410 0.111943
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 1.22267 0.0488289
\(628\) 0 0
\(629\) 2.48563 0.0991085
\(630\) 0 0
\(631\) 26.5271 1.05603 0.528013 0.849236i \(-0.322937\pi\)
0.528013 + 0.849236i \(0.322937\pi\)
\(632\) 0 0
\(633\) 0.440114 0.0174930
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −22.1679 −0.878322
\(638\) 0 0
\(639\) 14.1108 0.558215
\(640\) 0 0
\(641\) −7.86462 −0.310634 −0.155317 0.987865i \(-0.549640\pi\)
−0.155317 + 0.987865i \(0.549640\pi\)
\(642\) 0 0
\(643\) 1.67176 0.0659277 0.0329638 0.999457i \(-0.489505\pi\)
0.0329638 + 0.999457i \(0.489505\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −38.4909 −1.51323 −0.756616 0.653859i \(-0.773149\pi\)
−0.756616 + 0.653859i \(0.773149\pi\)
\(648\) 0 0
\(649\) −20.0626 −0.787527
\(650\) 0 0
\(651\) 0.343516 0.0134634
\(652\) 0 0
\(653\) −25.9985 −1.01740 −0.508700 0.860944i \(-0.669874\pi\)
−0.508700 + 0.860944i \(0.669874\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −30.5584 −1.19220
\(658\) 0 0
\(659\) −46.9594 −1.82928 −0.914639 0.404272i \(-0.867525\pi\)
−0.914639 + 0.404272i \(0.867525\pi\)
\(660\) 0 0
\(661\) −35.7225 −1.38944 −0.694722 0.719278i \(-0.744473\pi\)
−0.694722 + 0.719278i \(0.744473\pi\)
\(662\) 0 0
\(663\) −0.262955 −0.0102123
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −2.12351 −0.0822225
\(668\) 0 0
\(669\) 0.403569 0.0156029
\(670\) 0 0
\(671\) −43.4763 −1.67838
\(672\) 0 0
\(673\) 27.3450 1.05407 0.527036 0.849843i \(-0.323303\pi\)
0.527036 + 0.849843i \(0.323303\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −39.9944 −1.53711 −0.768555 0.639783i \(-0.779024\pi\)
−0.768555 + 0.639783i \(0.779024\pi\)
\(678\) 0 0
\(679\) 2.75382 0.105682
\(680\) 0 0
\(681\) 6.56512 0.251576
\(682\) 0 0
\(683\) −21.0130 −0.804042 −0.402021 0.915631i \(-0.631692\pi\)
−0.402021 + 0.915631i \(0.631692\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −1.08430 −0.0413685
\(688\) 0 0
\(689\) 14.1861 0.540448
\(690\) 0 0
\(691\) 10.7445 0.408741 0.204370 0.978894i \(-0.434485\pi\)
0.204370 + 0.978894i \(0.434485\pi\)
\(692\) 0 0
\(693\) −3.56769 −0.135525
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0.309969 0.0117409
\(698\) 0 0
\(699\) −0.649750 −0.0245758
\(700\) 0 0
\(701\) 42.3029 1.59776 0.798879 0.601491i \(-0.205427\pi\)
0.798879 + 0.601491i \(0.205427\pi\)
\(702\) 0 0
\(703\) 8.67699 0.327259
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −0.209636 −0.00788417
\(708\) 0 0
\(709\) −40.5584 −1.52320 −0.761601 0.648046i \(-0.775586\pi\)
−0.761601 + 0.648046i \(0.775586\pi\)
\(710\) 0 0
\(711\) 14.8934 0.558545
\(712\) 0 0
\(713\) −3.91911 −0.146772
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −0.626243 −0.0233875
\(718\) 0 0
\(719\) −16.8866 −0.629765 −0.314882 0.949131i \(-0.601965\pi\)
−0.314882 + 0.949131i \(0.601965\pi\)
\(720\) 0 0
\(721\) −0.889198 −0.0331155
\(722\) 0 0
\(723\) −5.72658 −0.212974
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −27.4398 −1.01769 −0.508843 0.860860i \(-0.669926\pi\)
−0.508843 + 0.860860i \(0.669926\pi\)
\(728\) 0 0
\(729\) −22.6692 −0.839600
\(730\) 0 0
\(731\) −1.26819 −0.0469057
\(732\) 0 0
\(733\) −14.0078 −0.517390 −0.258695 0.965959i \(-0.583292\pi\)
−0.258695 + 0.965959i \(0.583292\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 41.9257 1.54435
\(738\) 0 0
\(739\) 19.7851 0.727808 0.363904 0.931437i \(-0.381444\pi\)
0.363904 + 0.931437i \(0.381444\pi\)
\(740\) 0 0
\(741\) −0.917939 −0.0337213
\(742\) 0 0
\(743\) −41.8344 −1.53475 −0.767377 0.641196i \(-0.778439\pi\)
−0.767377 + 0.641196i \(0.778439\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 45.9154 1.67996
\(748\) 0 0
\(749\) 4.06005 0.148351
\(750\) 0 0
\(751\) −48.4983 −1.76973 −0.884865 0.465848i \(-0.845749\pi\)
−0.884865 + 0.465848i \(0.845749\pi\)
\(752\) 0 0
\(753\) 0.386795 0.0140956
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 32.2537 1.17228 0.586139 0.810210i \(-0.300647\pi\)
0.586139 + 0.810210i \(0.300647\pi\)
\(758\) 0 0
\(759\) −1.14468 −0.0415493
\(760\) 0 0
\(761\) −25.6184 −0.928668 −0.464334 0.885660i \(-0.653706\pi\)
−0.464334 + 0.885660i \(0.653706\pi\)
\(762\) 0 0
\(763\) 2.75532 0.0997492
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 15.0623 0.543868
\(768\) 0 0
\(769\) 42.3096 1.52572 0.762862 0.646561i \(-0.223793\pi\)
0.762862 + 0.646561i \(0.223793\pi\)
\(770\) 0 0
\(771\) 3.42558 0.123369
\(772\) 0 0
\(773\) −0.889198 −0.0319822 −0.0159911 0.999872i \(-0.505090\pi\)
−0.0159911 + 0.999872i \(0.505090\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0.712038 0.0255442
\(778\) 0 0
\(779\) 1.08206 0.0387688
\(780\) 0 0
\(781\) 20.6404 0.738573
\(782\) 0 0
\(783\) 3.84518 0.137416
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −50.6650 −1.80601 −0.903007 0.429627i \(-0.858645\pi\)
−0.903007 + 0.429627i \(0.858645\pi\)
\(788\) 0 0
\(789\) −2.38156 −0.0847858
\(790\) 0 0
\(791\) −2.51437 −0.0894007
\(792\) 0 0
\(793\) 32.6404 1.15910
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 33.4909 1.18631 0.593154 0.805089i \(-0.297883\pi\)
0.593154 + 0.805089i \(0.297883\pi\)
\(798\) 0 0
\(799\) 0.217440 0.00769247
\(800\) 0 0
\(801\) 28.4611 1.00562
\(802\) 0 0
\(803\) −44.6990 −1.57739
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 1.43861 0.0506416
\(808\) 0 0
\(809\) −6.35025 −0.223263 −0.111631 0.993750i \(-0.535608\pi\)
−0.111631 + 0.993750i \(0.535608\pi\)
\(810\) 0 0
\(811\) 1.08463 0.0380865 0.0190433 0.999819i \(-0.493938\pi\)
0.0190433 + 0.999819i \(0.493938\pi\)
\(812\) 0 0
\(813\) 1.99627 0.0700121
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −4.42708 −0.154884
\(818\) 0 0
\(819\) 2.67849 0.0935941
\(820\) 0 0
\(821\) −12.3630 −0.431470 −0.215735 0.976452i \(-0.569215\pi\)
−0.215735 + 0.976452i \(0.569215\pi\)
\(822\) 0 0
\(823\) −27.0765 −0.943827 −0.471914 0.881645i \(-0.656437\pi\)
−0.471914 + 0.881645i \(0.656437\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 17.0765 0.593808 0.296904 0.954907i \(-0.404046\pi\)
0.296904 + 0.954907i \(0.404046\pi\)
\(828\) 0 0
\(829\) −28.5804 −0.992638 −0.496319 0.868140i \(-0.665315\pi\)
−0.496319 + 0.868140i \(0.665315\pi\)
\(830\) 0 0
\(831\) 3.64045 0.126286
\(832\) 0 0
\(833\) −1.98173 −0.0686628
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 7.09660 0.245294
\(838\) 0 0
\(839\) 55.9019 1.92995 0.964974 0.262346i \(-0.0844960\pi\)
0.964974 + 0.262346i \(0.0844960\pi\)
\(840\) 0 0
\(841\) −23.8553 −0.822597
\(842\) 0 0
\(843\) 4.73705 0.163153
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −2.06752 −0.0710409
\(848\) 0 0
\(849\) −7.49236 −0.257137
\(850\) 0 0
\(851\) −8.12351 −0.278470
\(852\) 0 0
\(853\) −4.93214 −0.168873 −0.0844367 0.996429i \(-0.526909\pi\)
−0.0844367 + 0.996429i \(0.526909\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 39.9411 1.36436 0.682181 0.731183i \(-0.261032\pi\)
0.682181 + 0.731183i \(0.261032\pi\)
\(858\) 0 0
\(859\) −1.99070 −0.0679217 −0.0339608 0.999423i \(-0.510812\pi\)
−0.0339608 + 0.999423i \(0.510812\pi\)
\(860\) 0 0
\(861\) 0.0887944 0.00302611
\(862\) 0 0
\(863\) −31.9817 −1.08867 −0.544335 0.838868i \(-0.683218\pi\)
−0.544335 + 0.838868i \(0.683218\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 4.84635 0.164591
\(868\) 0 0
\(869\) 21.7851 0.739010
\(870\) 0 0
\(871\) −31.4763 −1.06653
\(872\) 0 0
\(873\) 28.0507 0.949374
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −22.2917 −0.752737 −0.376369 0.926470i \(-0.622827\pi\)
−0.376369 + 0.926470i \(0.622827\pi\)
\(878\) 0 0
\(879\) 1.13281 0.0382087
\(880\) 0 0
\(881\) 35.6912 1.20247 0.601233 0.799073i \(-0.294676\pi\)
0.601233 + 0.799073i \(0.294676\pi\)
\(882\) 0 0
\(883\) −17.9948 −0.605572 −0.302786 0.953059i \(-0.597917\pi\)
−0.302786 + 0.953059i \(0.597917\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −24.3566 −0.817813 −0.408907 0.912576i \(-0.634090\pi\)
−0.408907 + 0.912576i \(0.634090\pi\)
\(888\) 0 0
\(889\) 1.93995 0.0650637
\(890\) 0 0
\(891\) −35.2902 −1.18227
\(892\) 0 0
\(893\) 0.759053 0.0254007
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0.859386 0.0286941
\(898\) 0 0
\(899\) 9.49493 0.316674
\(900\) 0 0
\(901\) 1.26819 0.0422495
\(902\) 0 0
\(903\) −0.363288 −0.0120895
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −5.73821 −0.190534 −0.0952671 0.995452i \(-0.530371\pi\)
−0.0952671 + 0.995452i \(0.530371\pi\)
\(908\) 0 0
\(909\) −2.13538 −0.0708261
\(910\) 0 0
\(911\) 46.8291 1.55152 0.775759 0.631029i \(-0.217367\pi\)
0.775759 + 0.631029i \(0.217367\pi\)
\(912\) 0 0
\(913\) 67.1623 2.22275
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −0.835879 −0.0276032
\(918\) 0 0
\(919\) −2.32151 −0.0765795 −0.0382897 0.999267i \(-0.512191\pi\)
−0.0382897 + 0.999267i \(0.512191\pi\)
\(920\) 0 0
\(921\) −5.93995 −0.195728
\(922\) 0 0
\(923\) −15.4961 −0.510060
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −9.05748 −0.297487
\(928\) 0 0
\(929\) −25.5610 −0.838628 −0.419314 0.907841i \(-0.637729\pi\)
−0.419314 + 0.907841i \(0.637729\pi\)
\(930\) 0 0
\(931\) −6.91794 −0.226726
\(932\) 0 0
\(933\) −0.566620 −0.0185503
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −25.7501 −0.841219 −0.420609 0.907242i \(-0.638184\pi\)
−0.420609 + 0.907242i \(0.638184\pi\)
\(938\) 0 0
\(939\) 2.48563 0.0811154
\(940\) 0 0
\(941\) −29.1261 −0.949483 −0.474741 0.880125i \(-0.657458\pi\)
−0.474741 + 0.880125i \(0.657458\pi\)
\(942\) 0 0
\(943\) −1.01304 −0.0329891
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 6.21711 0.202029 0.101014 0.994885i \(-0.467791\pi\)
0.101014 + 0.994885i \(0.467791\pi\)
\(948\) 0 0
\(949\) 33.5584 1.08935
\(950\) 0 0
\(951\) 0.671758 0.0217832
\(952\) 0 0
\(953\) 47.8266 1.54925 0.774627 0.632418i \(-0.217937\pi\)
0.774627 + 0.632418i \(0.217937\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 2.77326 0.0896467
\(958\) 0 0
\(959\) 4.13538 0.133538
\(960\) 0 0
\(961\) −13.4763 −0.434720
\(962\) 0 0
\(963\) 41.3562 1.33269
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 37.2301 1.19724 0.598620 0.801033i \(-0.295716\pi\)
0.598620 + 0.801033i \(0.295716\pi\)
\(968\) 0 0
\(969\) −0.0820605 −0.00263616
\(970\) 0 0
\(971\) −37.8359 −1.21421 −0.607106 0.794621i \(-0.707669\pi\)
−0.607106 + 0.794621i \(0.707669\pi\)
\(972\) 0 0
\(973\) −0.658720 −0.0211176
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 55.8684 1.78739 0.893694 0.448678i \(-0.148105\pi\)
0.893694 + 0.448678i \(0.148105\pi\)
\(978\) 0 0
\(979\) 41.6311 1.33054
\(980\) 0 0
\(981\) 28.0660 0.896079
\(982\) 0 0
\(983\) 6.93738 0.221268 0.110634 0.993861i \(-0.464712\pi\)
0.110634 + 0.993861i \(0.464712\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0.0622883 0.00198266
\(988\) 0 0
\(989\) 4.14468 0.131793
\(990\) 0 0
\(991\) −36.5897 −1.16231 −0.581155 0.813793i \(-0.697399\pi\)
−0.581155 + 0.813793i \(0.697399\pi\)
\(992\) 0 0
\(993\) −6.54792 −0.207792
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 33.7538 1.06899 0.534497 0.845170i \(-0.320501\pi\)
0.534497 + 0.845170i \(0.320501\pi\)
\(998\) 0 0
\(999\) 14.7098 0.465398
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 7600.2.a.bj.1.2 3
4.3 odd 2 1900.2.a.h.1.2 yes 3
5.4 even 2 7600.2.a.by.1.2 3
20.3 even 4 1900.2.c.g.1749.4 6
20.7 even 4 1900.2.c.g.1749.3 6
20.19 odd 2 1900.2.a.f.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1900.2.a.f.1.2 3 20.19 odd 2
1900.2.a.h.1.2 yes 3 4.3 odd 2
1900.2.c.g.1749.3 6 20.7 even 4
1900.2.c.g.1749.4 6 20.3 even 4
7600.2.a.bj.1.2 3 1.1 even 1 trivial
7600.2.a.by.1.2 3 5.4 even 2