Properties

Label 3344.2.a.q.1.3
Level $3344$
Weight $2$
Character 3344.1
Self dual yes
Analytic conductor $26.702$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3344,2,Mod(1,3344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3344, 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("3344.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.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: \(26.7019744359\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.14510\) of defining polynomial
Character \(\chi\) \(=\) 3344.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.14510 q^{3} -1.60147 q^{5} -2.89167 q^{7} +1.60147 q^{9} +O(q^{10})\) \(q+2.14510 q^{3} -1.60147 q^{5} -2.89167 q^{7} +1.60147 q^{9} +1.00000 q^{11} +4.89167 q^{13} -3.43531 q^{15} -5.74657 q^{17} +1.00000 q^{19} -6.20293 q^{21} +7.74657 q^{23} -2.43531 q^{25} -3.00000 q^{27} +5.34803 q^{29} +7.43531 q^{31} +2.14510 q^{33} +4.63091 q^{35} -7.49314 q^{37} +10.4931 q^{39} -2.05783 q^{41} +10.6887 q^{43} -2.56469 q^{45} +7.08727 q^{47} +1.36176 q^{49} -12.3270 q^{51} +8.32698 q^{53} -1.60147 q^{55} +2.14510 q^{57} +2.54364 q^{59} +13.4931 q^{61} -4.63091 q^{63} -7.83384 q^{65} +10.3848 q^{67} +16.6172 q^{69} -14.9789 q^{71} -3.74657 q^{73} -5.22399 q^{75} -2.89167 q^{77} +8.69607 q^{79} -11.2397 q^{81} +6.10833 q^{83} +9.20293 q^{85} +11.4721 q^{87} +3.20293 q^{89} -14.1451 q^{91} +15.9495 q^{93} -1.60147 q^{95} -6.98627 q^{97} +1.60147 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{5} + 6 q^{7} + 3 q^{9} + 3 q^{11} + 9 q^{15} - 9 q^{17} + 3 q^{19} - 15 q^{21} + 15 q^{23} + 12 q^{25} - 9 q^{27} + 6 q^{29} + 3 q^{31} - 6 q^{37} + 15 q^{39} - 9 q^{41} + 21 q^{43} - 27 q^{45} + 12 q^{47} + 27 q^{49} - 3 q^{51} - 9 q^{53} - 3 q^{55} + 3 q^{59} + 24 q^{61} - 6 q^{65} + 3 q^{69} - 21 q^{71} - 3 q^{73} - 36 q^{75} + 6 q^{77} + 6 q^{79} - 9 q^{81} + 33 q^{83} + 24 q^{85} - 6 q^{87} + 6 q^{89} - 36 q^{91} + 36 q^{93} - 3 q^{95} + 12 q^{97} + 3 q^{99}+O(q^{100}) \) 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
\(3\) 2.14510 1.23848 0.619238 0.785204i \(-0.287442\pi\)
0.619238 + 0.785204i \(0.287442\pi\)
\(4\) 0 0
\(5\) −1.60147 −0.716197 −0.358099 0.933684i \(-0.616575\pi\)
−0.358099 + 0.933684i \(0.616575\pi\)
\(6\) 0 0
\(7\) −2.89167 −1.09295 −0.546474 0.837476i \(-0.684030\pi\)
−0.546474 + 0.837476i \(0.684030\pi\)
\(8\) 0 0
\(9\) 1.60147 0.533822
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 4.89167 1.35671 0.678353 0.734736i \(-0.262694\pi\)
0.678353 + 0.734736i \(0.262694\pi\)
\(14\) 0 0
\(15\) −3.43531 −0.886993
\(16\) 0 0
\(17\) −5.74657 −1.39375 −0.696874 0.717194i \(-0.745426\pi\)
−0.696874 + 0.717194i \(0.745426\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −6.20293 −1.35359
\(22\) 0 0
\(23\) 7.74657 1.61527 0.807636 0.589682i \(-0.200747\pi\)
0.807636 + 0.589682i \(0.200747\pi\)
\(24\) 0 0
\(25\) −2.43531 −0.487062
\(26\) 0 0
\(27\) −3.00000 −0.577350
\(28\) 0 0
\(29\) 5.34803 0.993105 0.496552 0.868007i \(-0.334599\pi\)
0.496552 + 0.868007i \(0.334599\pi\)
\(30\) 0 0
\(31\) 7.43531 1.33542 0.667710 0.744421i \(-0.267274\pi\)
0.667710 + 0.744421i \(0.267274\pi\)
\(32\) 0 0
\(33\) 2.14510 0.373414
\(34\) 0 0
\(35\) 4.63091 0.782767
\(36\) 0 0
\(37\) −7.49314 −1.23186 −0.615932 0.787799i \(-0.711220\pi\)
−0.615932 + 0.787799i \(0.711220\pi\)
\(38\) 0 0
\(39\) 10.4931 1.68025
\(40\) 0 0
\(41\) −2.05783 −0.321379 −0.160689 0.987005i \(-0.551372\pi\)
−0.160689 + 0.987005i \(0.551372\pi\)
\(42\) 0 0
\(43\) 10.6887 1.63002 0.815009 0.579449i \(-0.196732\pi\)
0.815009 + 0.579449i \(0.196732\pi\)
\(44\) 0 0
\(45\) −2.56469 −0.382322
\(46\) 0 0
\(47\) 7.08727 1.03379 0.516893 0.856050i \(-0.327089\pi\)
0.516893 + 0.856050i \(0.327089\pi\)
\(48\) 0 0
\(49\) 1.36176 0.194537
\(50\) 0 0
\(51\) −12.3270 −1.72612
\(52\) 0 0
\(53\) 8.32698 1.14380 0.571899 0.820324i \(-0.306207\pi\)
0.571899 + 0.820324i \(0.306207\pi\)
\(54\) 0 0
\(55\) −1.60147 −0.215942
\(56\) 0 0
\(57\) 2.14510 0.284126
\(58\) 0 0
\(59\) 2.54364 0.331153 0.165577 0.986197i \(-0.447051\pi\)
0.165577 + 0.986197i \(0.447051\pi\)
\(60\) 0 0
\(61\) 13.4931 1.72762 0.863810 0.503818i \(-0.168072\pi\)
0.863810 + 0.503818i \(0.168072\pi\)
\(62\) 0 0
\(63\) −4.63091 −0.583440
\(64\) 0 0
\(65\) −7.83384 −0.971669
\(66\) 0 0
\(67\) 10.3848 1.26871 0.634353 0.773043i \(-0.281267\pi\)
0.634353 + 0.773043i \(0.281267\pi\)
\(68\) 0 0
\(69\) 16.6172 2.00047
\(70\) 0 0
\(71\) −14.9789 −1.77767 −0.888837 0.458224i \(-0.848486\pi\)
−0.888837 + 0.458224i \(0.848486\pi\)
\(72\) 0 0
\(73\) −3.74657 −0.438503 −0.219251 0.975668i \(-0.570361\pi\)
−0.219251 + 0.975668i \(0.570361\pi\)
\(74\) 0 0
\(75\) −5.22399 −0.603214
\(76\) 0 0
\(77\) −2.89167 −0.329536
\(78\) 0 0
\(79\) 8.69607 0.978384 0.489192 0.872176i \(-0.337292\pi\)
0.489192 + 0.872176i \(0.337292\pi\)
\(80\) 0 0
\(81\) −11.2397 −1.24886
\(82\) 0 0
\(83\) 6.10833 0.670476 0.335238 0.942133i \(-0.391183\pi\)
0.335238 + 0.942133i \(0.391183\pi\)
\(84\) 0 0
\(85\) 9.20293 0.998198
\(86\) 0 0
\(87\) 11.4721 1.22994
\(88\) 0 0
\(89\) 3.20293 0.339510 0.169755 0.985486i \(-0.445702\pi\)
0.169755 + 0.985486i \(0.445702\pi\)
\(90\) 0 0
\(91\) −14.1451 −1.48281
\(92\) 0 0
\(93\) 15.9495 1.65389
\(94\) 0 0
\(95\) −1.60147 −0.164307
\(96\) 0 0
\(97\) −6.98627 −0.709349 −0.354674 0.934990i \(-0.615408\pi\)
−0.354674 + 0.934990i \(0.615408\pi\)
\(98\) 0 0
\(99\) 1.60147 0.160953
\(100\) 0 0
\(101\) −1.20293 −0.119696 −0.0598481 0.998207i \(-0.519062\pi\)
−0.0598481 + 0.998207i \(0.519062\pi\)
\(102\) 0 0
\(103\) −16.8779 −1.66303 −0.831517 0.555500i \(-0.812527\pi\)
−0.831517 + 0.555500i \(0.812527\pi\)
\(104\) 0 0
\(105\) 9.93378 0.969438
\(106\) 0 0
\(107\) 8.03677 0.776944 0.388472 0.921460i \(-0.373003\pi\)
0.388472 + 0.921460i \(0.373003\pi\)
\(108\) 0 0
\(109\) 11.5299 1.10437 0.552183 0.833723i \(-0.313795\pi\)
0.552183 + 0.833723i \(0.313795\pi\)
\(110\) 0 0
\(111\) −16.0735 −1.52563
\(112\) 0 0
\(113\) −8.00000 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(114\) 0 0
\(115\) −12.4059 −1.15685
\(116\) 0 0
\(117\) 7.83384 0.724239
\(118\) 0 0
\(119\) 16.6172 1.52329
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −4.41425 −0.398020
\(124\) 0 0
\(125\) 11.9074 1.06503
\(126\) 0 0
\(127\) 21.2765 1.88798 0.943991 0.329971i \(-0.107039\pi\)
0.943991 + 0.329971i \(0.107039\pi\)
\(128\) 0 0
\(129\) 22.9284 2.01874
\(130\) 0 0
\(131\) 3.43531 0.300144 0.150072 0.988675i \(-0.452049\pi\)
0.150072 + 0.988675i \(0.452049\pi\)
\(132\) 0 0
\(133\) −2.89167 −0.250740
\(134\) 0 0
\(135\) 4.80440 0.413497
\(136\) 0 0
\(137\) 13.5877 1.16088 0.580439 0.814303i \(-0.302881\pi\)
0.580439 + 0.814303i \(0.302881\pi\)
\(138\) 0 0
\(139\) −3.19560 −0.271048 −0.135524 0.990774i \(-0.543272\pi\)
−0.135524 + 0.990774i \(0.543272\pi\)
\(140\) 0 0
\(141\) 15.2029 1.28032
\(142\) 0 0
\(143\) 4.89167 0.409062
\(144\) 0 0
\(145\) −8.56469 −0.711259
\(146\) 0 0
\(147\) 2.92112 0.240930
\(148\) 0 0
\(149\) −12.0735 −0.989104 −0.494552 0.869148i \(-0.664668\pi\)
−0.494552 + 0.869148i \(0.664668\pi\)
\(150\) 0 0
\(151\) 4.91273 0.399792 0.199896 0.979817i \(-0.435940\pi\)
0.199896 + 0.979817i \(0.435940\pi\)
\(152\) 0 0
\(153\) −9.20293 −0.744013
\(154\) 0 0
\(155\) −11.9074 −0.956425
\(156\) 0 0
\(157\) 8.00733 0.639054 0.319527 0.947577i \(-0.396476\pi\)
0.319527 + 0.947577i \(0.396476\pi\)
\(158\) 0 0
\(159\) 17.8622 1.41657
\(160\) 0 0
\(161\) −22.4005 −1.76541
\(162\) 0 0
\(163\) −1.88434 −0.147593 −0.0737966 0.997273i \(-0.523512\pi\)
−0.0737966 + 0.997273i \(0.523512\pi\)
\(164\) 0 0
\(165\) −3.43531 −0.267438
\(166\) 0 0
\(167\) 4.98627 0.385849 0.192925 0.981214i \(-0.438203\pi\)
0.192925 + 0.981214i \(0.438203\pi\)
\(168\) 0 0
\(169\) 10.9284 0.840650
\(170\) 0 0
\(171\) 1.60147 0.122467
\(172\) 0 0
\(173\) 6.51419 0.495265 0.247632 0.968854i \(-0.420347\pi\)
0.247632 + 0.968854i \(0.420347\pi\)
\(174\) 0 0
\(175\) 7.04211 0.532333
\(176\) 0 0
\(177\) 5.45636 0.410125
\(178\) 0 0
\(179\) 5.55096 0.414899 0.207449 0.978246i \(-0.433484\pi\)
0.207449 + 0.978246i \(0.433484\pi\)
\(180\) 0 0
\(181\) 1.88434 0.140062 0.0700311 0.997545i \(-0.477690\pi\)
0.0700311 + 0.997545i \(0.477690\pi\)
\(182\) 0 0
\(183\) 28.9442 2.13961
\(184\) 0 0
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) −5.74657 −0.420231
\(188\) 0 0
\(189\) 8.67501 0.631014
\(190\) 0 0
\(191\) −2.44264 −0.176743 −0.0883715 0.996088i \(-0.528166\pi\)
−0.0883715 + 0.996088i \(0.528166\pi\)
\(192\) 0 0
\(193\) −9.67501 −0.696423 −0.348211 0.937416i \(-0.613211\pi\)
−0.348211 + 0.937416i \(0.613211\pi\)
\(194\) 0 0
\(195\) −16.8044 −1.20339
\(196\) 0 0
\(197\) −17.4931 −1.24633 −0.623167 0.782089i \(-0.714154\pi\)
−0.623167 + 0.782089i \(0.714154\pi\)
\(198\) 0 0
\(199\) 25.2397 1.78920 0.894598 0.446873i \(-0.147462\pi\)
0.894598 + 0.446873i \(0.147462\pi\)
\(200\) 0 0
\(201\) 22.2765 1.57126
\(202\) 0 0
\(203\) −15.4648 −1.08541
\(204\) 0 0
\(205\) 3.29554 0.230171
\(206\) 0 0
\(207\) 12.4059 0.862267
\(208\) 0 0
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 14.8338 1.02120 0.510602 0.859817i \(-0.329423\pi\)
0.510602 + 0.859817i \(0.329423\pi\)
\(212\) 0 0
\(213\) −32.1314 −2.20161
\(214\) 0 0
\(215\) −17.1176 −1.16741
\(216\) 0 0
\(217\) −21.5005 −1.45955
\(218\) 0 0
\(219\) −8.03677 −0.543075
\(220\) 0 0
\(221\) −28.1103 −1.89090
\(222\) 0 0
\(223\) −11.4931 −0.769637 −0.384819 0.922992i \(-0.625736\pi\)
−0.384819 + 0.922992i \(0.625736\pi\)
\(224\) 0 0
\(225\) −3.90006 −0.260004
\(226\) 0 0
\(227\) −8.73284 −0.579619 −0.289810 0.957084i \(-0.593592\pi\)
−0.289810 + 0.957084i \(0.593592\pi\)
\(228\) 0 0
\(229\) 7.84117 0.518159 0.259080 0.965856i \(-0.416581\pi\)
0.259080 + 0.965856i \(0.416581\pi\)
\(230\) 0 0
\(231\) −6.20293 −0.408123
\(232\) 0 0
\(233\) −12.5069 −0.819352 −0.409676 0.912231i \(-0.634358\pi\)
−0.409676 + 0.912231i \(0.634358\pi\)
\(234\) 0 0
\(235\) −11.3500 −0.740394
\(236\) 0 0
\(237\) 18.6540 1.21170
\(238\) 0 0
\(239\) −24.6245 −1.59283 −0.796414 0.604752i \(-0.793272\pi\)
−0.796414 + 0.604752i \(0.793272\pi\)
\(240\) 0 0
\(241\) 16.0809 1.03586 0.517930 0.855423i \(-0.326703\pi\)
0.517930 + 0.855423i \(0.326703\pi\)
\(242\) 0 0
\(243\) −15.1103 −0.969328
\(244\) 0 0
\(245\) −2.18081 −0.139327
\(246\) 0 0
\(247\) 4.89167 0.311250
\(248\) 0 0
\(249\) 13.1030 0.830368
\(250\) 0 0
\(251\) −13.0873 −0.826061 −0.413031 0.910717i \(-0.635530\pi\)
−0.413031 + 0.910717i \(0.635530\pi\)
\(252\) 0 0
\(253\) 7.74657 0.487023
\(254\) 0 0
\(255\) 19.7412 1.23624
\(256\) 0 0
\(257\) 0.723522 0.0451320 0.0225660 0.999745i \(-0.492816\pi\)
0.0225660 + 0.999745i \(0.492816\pi\)
\(258\) 0 0
\(259\) 21.6677 1.34636
\(260\) 0 0
\(261\) 8.56469 0.530141
\(262\) 0 0
\(263\) −13.7760 −0.849465 −0.424733 0.905319i \(-0.639632\pi\)
−0.424733 + 0.905319i \(0.639632\pi\)
\(264\) 0 0
\(265\) −13.3354 −0.819185
\(266\) 0 0
\(267\) 6.87062 0.420475
\(268\) 0 0
\(269\) −19.9579 −1.21685 −0.608427 0.793610i \(-0.708199\pi\)
−0.608427 + 0.793610i \(0.708199\pi\)
\(270\) 0 0
\(271\) −28.7486 −1.74635 −0.873175 0.487406i \(-0.837943\pi\)
−0.873175 + 0.487406i \(0.837943\pi\)
\(272\) 0 0
\(273\) −30.3427 −1.83642
\(274\) 0 0
\(275\) −2.43531 −0.146855
\(276\) 0 0
\(277\) 3.08727 0.185496 0.0927482 0.995690i \(-0.470435\pi\)
0.0927482 + 0.995690i \(0.470435\pi\)
\(278\) 0 0
\(279\) 11.9074 0.712877
\(280\) 0 0
\(281\) 9.09460 0.542538 0.271269 0.962504i \(-0.412557\pi\)
0.271269 + 0.962504i \(0.412557\pi\)
\(282\) 0 0
\(283\) −13.1451 −0.781395 −0.390698 0.920519i \(-0.627766\pi\)
−0.390698 + 0.920519i \(0.627766\pi\)
\(284\) 0 0
\(285\) −3.43531 −0.203490
\(286\) 0 0
\(287\) 5.95056 0.351251
\(288\) 0 0
\(289\) 16.0230 0.942532
\(290\) 0 0
\(291\) −14.9863 −0.878511
\(292\) 0 0
\(293\) 3.10833 0.181591 0.0907953 0.995870i \(-0.471059\pi\)
0.0907953 + 0.995870i \(0.471059\pi\)
\(294\) 0 0
\(295\) −4.07355 −0.237171
\(296\) 0 0
\(297\) −3.00000 −0.174078
\(298\) 0 0
\(299\) 37.8937 2.19145
\(300\) 0 0
\(301\) −30.9083 −1.78153
\(302\) 0 0
\(303\) −2.58041 −0.148241
\(304\) 0 0
\(305\) −21.6088 −1.23732
\(306\) 0 0
\(307\) −2.18920 −0.124944 −0.0624722 0.998047i \(-0.519898\pi\)
−0.0624722 + 0.998047i \(0.519898\pi\)
\(308\) 0 0
\(309\) −36.2049 −2.05963
\(310\) 0 0
\(311\) −3.92112 −0.222346 −0.111173 0.993801i \(-0.535461\pi\)
−0.111173 + 0.993801i \(0.535461\pi\)
\(312\) 0 0
\(313\) −21.7118 −1.22722 −0.613611 0.789608i \(-0.710284\pi\)
−0.613611 + 0.789608i \(0.710284\pi\)
\(314\) 0 0
\(315\) 7.41625 0.417858
\(316\) 0 0
\(317\) 6.90739 0.387958 0.193979 0.981006i \(-0.437861\pi\)
0.193979 + 0.981006i \(0.437861\pi\)
\(318\) 0 0
\(319\) 5.34803 0.299432
\(320\) 0 0
\(321\) 17.2397 0.962226
\(322\) 0 0
\(323\) −5.74657 −0.319748
\(324\) 0 0
\(325\) −11.9127 −0.660799
\(326\) 0 0
\(327\) 24.7328 1.36773
\(328\) 0 0
\(329\) −20.4941 −1.12987
\(330\) 0 0
\(331\) −32.1030 −1.76454 −0.882270 0.470744i \(-0.843986\pi\)
−0.882270 + 0.470744i \(0.843986\pi\)
\(332\) 0 0
\(333\) −12.0000 −0.657596
\(334\) 0 0
\(335\) −16.6309 −0.908644
\(336\) 0 0
\(337\) −1.55096 −0.0844864 −0.0422432 0.999107i \(-0.513450\pi\)
−0.0422432 + 0.999107i \(0.513450\pi\)
\(338\) 0 0
\(339\) −17.1608 −0.932048
\(340\) 0 0
\(341\) 7.43531 0.402645
\(342\) 0 0
\(343\) 16.3039 0.880330
\(344\) 0 0
\(345\) −26.6118 −1.43273
\(346\) 0 0
\(347\) 0.506864 0.0272099 0.0136049 0.999907i \(-0.495669\pi\)
0.0136049 + 0.999907i \(0.495669\pi\)
\(348\) 0 0
\(349\) 10.8117 0.578738 0.289369 0.957218i \(-0.406554\pi\)
0.289369 + 0.957218i \(0.406554\pi\)
\(350\) 0 0
\(351\) −14.6750 −0.783294
\(352\) 0 0
\(353\) 27.0966 1.44221 0.721103 0.692828i \(-0.243635\pi\)
0.721103 + 0.692828i \(0.243635\pi\)
\(354\) 0 0
\(355\) 23.9883 1.27316
\(356\) 0 0
\(357\) 35.6456 1.88656
\(358\) 0 0
\(359\) 20.7486 1.09507 0.547534 0.836784i \(-0.315567\pi\)
0.547534 + 0.836784i \(0.315567\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 2.14510 0.112589
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) −25.6402 −1.33841 −0.669205 0.743078i \(-0.733365\pi\)
−0.669205 + 0.743078i \(0.733365\pi\)
\(368\) 0 0
\(369\) −3.29554 −0.171559
\(370\) 0 0
\(371\) −24.0789 −1.25011
\(372\) 0 0
\(373\) 25.2049 1.30506 0.652531 0.757762i \(-0.273707\pi\)
0.652531 + 0.757762i \(0.273707\pi\)
\(374\) 0 0
\(375\) 25.5426 1.31901
\(376\) 0 0
\(377\) 26.1608 1.34735
\(378\) 0 0
\(379\) −3.47208 −0.178349 −0.0891744 0.996016i \(-0.528423\pi\)
−0.0891744 + 0.996016i \(0.528423\pi\)
\(380\) 0 0
\(381\) 45.6402 2.33822
\(382\) 0 0
\(383\) −23.1176 −1.18126 −0.590628 0.806944i \(-0.701120\pi\)
−0.590628 + 0.806944i \(0.701120\pi\)
\(384\) 0 0
\(385\) 4.63091 0.236013
\(386\) 0 0
\(387\) 17.1176 0.870139
\(388\) 0 0
\(389\) −26.7623 −1.35690 −0.678451 0.734646i \(-0.737348\pi\)
−0.678451 + 0.734646i \(0.737348\pi\)
\(390\) 0 0
\(391\) −44.5162 −2.25128
\(392\) 0 0
\(393\) 7.36909 0.371721
\(394\) 0 0
\(395\) −13.9265 −0.700716
\(396\) 0 0
\(397\) −31.1030 −1.56101 −0.780507 0.625147i \(-0.785039\pi\)
−0.780507 + 0.625147i \(0.785039\pi\)
\(398\) 0 0
\(399\) −6.20293 −0.310535
\(400\) 0 0
\(401\) 29.3500 1.46567 0.732835 0.680406i \(-0.238197\pi\)
0.732835 + 0.680406i \(0.238197\pi\)
\(402\) 0 0
\(403\) 36.3711 1.81177
\(404\) 0 0
\(405\) 18.0000 0.894427
\(406\) 0 0
\(407\) −7.49314 −0.371421
\(408\) 0 0
\(409\) 8.58774 0.424636 0.212318 0.977201i \(-0.431899\pi\)
0.212318 + 0.977201i \(0.431899\pi\)
\(410\) 0 0
\(411\) 29.1471 1.43772
\(412\) 0 0
\(413\) −7.35536 −0.361934
\(414\) 0 0
\(415\) −9.78228 −0.480193
\(416\) 0 0
\(417\) −6.85490 −0.335686
\(418\) 0 0
\(419\) −7.59414 −0.370998 −0.185499 0.982644i \(-0.559390\pi\)
−0.185499 + 0.982644i \(0.559390\pi\)
\(420\) 0 0
\(421\) 8.76030 0.426951 0.213475 0.976948i \(-0.431522\pi\)
0.213475 + 0.976948i \(0.431522\pi\)
\(422\) 0 0
\(423\) 11.3500 0.551857
\(424\) 0 0
\(425\) 13.9947 0.678841
\(426\) 0 0
\(427\) −39.0177 −1.88820
\(428\) 0 0
\(429\) 10.4931 0.506613
\(430\) 0 0
\(431\) 33.6677 1.62172 0.810858 0.585243i \(-0.199001\pi\)
0.810858 + 0.585243i \(0.199001\pi\)
\(432\) 0 0
\(433\) 30.5383 1.46758 0.733789 0.679378i \(-0.237750\pi\)
0.733789 + 0.679378i \(0.237750\pi\)
\(434\) 0 0
\(435\) −18.3721 −0.880877
\(436\) 0 0
\(437\) 7.74657 0.370569
\(438\) 0 0
\(439\) 12.5069 0.596920 0.298460 0.954422i \(-0.403527\pi\)
0.298460 + 0.954422i \(0.403527\pi\)
\(440\) 0 0
\(441\) 2.18081 0.103848
\(442\) 0 0
\(443\) 17.6677 0.839417 0.419709 0.907659i \(-0.362132\pi\)
0.419709 + 0.907659i \(0.362132\pi\)
\(444\) 0 0
\(445\) −5.12938 −0.243156
\(446\) 0 0
\(447\) −25.8990 −1.22498
\(448\) 0 0
\(449\) −17.7098 −0.835777 −0.417888 0.908498i \(-0.637230\pi\)
−0.417888 + 0.908498i \(0.637230\pi\)
\(450\) 0 0
\(451\) −2.05783 −0.0968994
\(452\) 0 0
\(453\) 10.5383 0.495133
\(454\) 0 0
\(455\) 22.6529 1.06198
\(456\) 0 0
\(457\) −24.8064 −1.16039 −0.580197 0.814476i \(-0.697024\pi\)
−0.580197 + 0.814476i \(0.697024\pi\)
\(458\) 0 0
\(459\) 17.2397 0.804681
\(460\) 0 0
\(461\) −2.98627 −0.139085 −0.0695423 0.997579i \(-0.522154\pi\)
−0.0695423 + 0.997579i \(0.522154\pi\)
\(462\) 0 0
\(463\) 39.5667 1.83882 0.919410 0.393301i \(-0.128667\pi\)
0.919410 + 0.393301i \(0.128667\pi\)
\(464\) 0 0
\(465\) −25.5426 −1.18451
\(466\) 0 0
\(467\) −4.46475 −0.206604 −0.103302 0.994650i \(-0.532941\pi\)
−0.103302 + 0.994650i \(0.532941\pi\)
\(468\) 0 0
\(469\) −30.0294 −1.38663
\(470\) 0 0
\(471\) 17.1765 0.791453
\(472\) 0 0
\(473\) 10.6887 0.491469
\(474\) 0 0
\(475\) −2.43531 −0.111740
\(476\) 0 0
\(477\) 13.3354 0.610585
\(478\) 0 0
\(479\) 20.8863 0.954321 0.477161 0.878816i \(-0.341666\pi\)
0.477161 + 0.878816i \(0.341666\pi\)
\(480\) 0 0
\(481\) −36.6540 −1.67128
\(482\) 0 0
\(483\) −48.0514 −2.18642
\(484\) 0 0
\(485\) 11.1883 0.508033
\(486\) 0 0
\(487\) −15.2838 −0.692575 −0.346288 0.938128i \(-0.612558\pi\)
−0.346288 + 0.938128i \(0.612558\pi\)
\(488\) 0 0
\(489\) −4.04211 −0.182791
\(490\) 0 0
\(491\) −3.31965 −0.149814 −0.0749069 0.997191i \(-0.523866\pi\)
−0.0749069 + 0.997191i \(0.523866\pi\)
\(492\) 0 0
\(493\) −30.7328 −1.38414
\(494\) 0 0
\(495\) −2.56469 −0.115274
\(496\) 0 0
\(497\) 43.3142 1.94291
\(498\) 0 0
\(499\) 2.07355 0.0928247 0.0464124 0.998922i \(-0.485221\pi\)
0.0464124 + 0.998922i \(0.485221\pi\)
\(500\) 0 0
\(501\) 10.6961 0.477865
\(502\) 0 0
\(503\) 1.11672 0.0497921 0.0248960 0.999690i \(-0.492075\pi\)
0.0248960 + 0.999690i \(0.492075\pi\)
\(504\) 0 0
\(505\) 1.92645 0.0857260
\(506\) 0 0
\(507\) 23.4426 1.04112
\(508\) 0 0
\(509\) −38.2481 −1.69532 −0.847659 0.530542i \(-0.821988\pi\)
−0.847659 + 0.530542i \(0.821988\pi\)
\(510\) 0 0
\(511\) 10.8338 0.479261
\(512\) 0 0
\(513\) −3.00000 −0.132453
\(514\) 0 0
\(515\) 27.0294 1.19106
\(516\) 0 0
\(517\) 7.08727 0.311698
\(518\) 0 0
\(519\) 13.9736 0.613373
\(520\) 0 0
\(521\) 10.8853 0.476892 0.238446 0.971156i \(-0.423362\pi\)
0.238446 + 0.971156i \(0.423362\pi\)
\(522\) 0 0
\(523\) 15.7191 0.687349 0.343674 0.939089i \(-0.388328\pi\)
0.343674 + 0.939089i \(0.388328\pi\)
\(524\) 0 0
\(525\) 15.1060 0.659282
\(526\) 0 0
\(527\) −42.7275 −1.86124
\(528\) 0 0
\(529\) 37.0093 1.60910
\(530\) 0 0
\(531\) 4.07355 0.176777
\(532\) 0 0
\(533\) −10.0662 −0.436016
\(534\) 0 0
\(535\) −12.8706 −0.556445
\(536\) 0 0
\(537\) 11.9074 0.513842
\(538\) 0 0
\(539\) 1.36176 0.0586552
\(540\) 0 0
\(541\) −26.4373 −1.13663 −0.568314 0.822812i \(-0.692404\pi\)
−0.568314 + 0.822812i \(0.692404\pi\)
\(542\) 0 0
\(543\) 4.04211 0.173464
\(544\) 0 0
\(545\) −18.4648 −0.790943
\(546\) 0 0
\(547\) 24.9588 1.06716 0.533581 0.845749i \(-0.320846\pi\)
0.533581 + 0.845749i \(0.320846\pi\)
\(548\) 0 0
\(549\) 21.6088 0.922241
\(550\) 0 0
\(551\) 5.34803 0.227834
\(552\) 0 0
\(553\) −25.1462 −1.06932
\(554\) 0 0
\(555\) 25.7412 1.09265
\(556\) 0 0
\(557\) 8.30486 0.351888 0.175944 0.984400i \(-0.443702\pi\)
0.175944 + 0.984400i \(0.443702\pi\)
\(558\) 0 0
\(559\) 52.2858 2.21145
\(560\) 0 0
\(561\) −12.3270 −0.520445
\(562\) 0 0
\(563\) −41.6402 −1.75493 −0.877463 0.479644i \(-0.840766\pi\)
−0.877463 + 0.479644i \(0.840766\pi\)
\(564\) 0 0
\(565\) 12.8117 0.538993
\(566\) 0 0
\(567\) 32.5015 1.36494
\(568\) 0 0
\(569\) −14.9138 −0.625219 −0.312609 0.949882i \(-0.601203\pi\)
−0.312609 + 0.949882i \(0.601203\pi\)
\(570\) 0 0
\(571\) 23.0020 0.962603 0.481302 0.876555i \(-0.340164\pi\)
0.481302 + 0.876555i \(0.340164\pi\)
\(572\) 0 0
\(573\) −5.23970 −0.218892
\(574\) 0 0
\(575\) −18.8653 −0.786737
\(576\) 0 0
\(577\) −33.6613 −1.40134 −0.700669 0.713487i \(-0.747115\pi\)
−0.700669 + 0.713487i \(0.747115\pi\)
\(578\) 0 0
\(579\) −20.7539 −0.862502
\(580\) 0 0
\(581\) −17.6633 −0.732796
\(582\) 0 0
\(583\) 8.32698 0.344868
\(584\) 0 0
\(585\) −12.5456 −0.518698
\(586\) 0 0
\(587\) −4.00000 −0.165098 −0.0825488 0.996587i \(-0.526306\pi\)
−0.0825488 + 0.996587i \(0.526306\pi\)
\(588\) 0 0
\(589\) 7.43531 0.306367
\(590\) 0 0
\(591\) −37.5246 −1.54355
\(592\) 0 0
\(593\) −30.9588 −1.27133 −0.635663 0.771967i \(-0.719273\pi\)
−0.635663 + 0.771967i \(0.719273\pi\)
\(594\) 0 0
\(595\) −26.6118 −1.09098
\(596\) 0 0
\(597\) 54.1418 2.21587
\(598\) 0 0
\(599\) −3.93378 −0.160730 −0.0803650 0.996766i \(-0.525609\pi\)
−0.0803650 + 0.996766i \(0.525609\pi\)
\(600\) 0 0
\(601\) 23.1451 0.944108 0.472054 0.881570i \(-0.343513\pi\)
0.472054 + 0.881570i \(0.343513\pi\)
\(602\) 0 0
\(603\) 16.6309 0.677263
\(604\) 0 0
\(605\) −1.60147 −0.0651088
\(606\) 0 0
\(607\) 1.78334 0.0723836 0.0361918 0.999345i \(-0.488477\pi\)
0.0361918 + 0.999345i \(0.488477\pi\)
\(608\) 0 0
\(609\) −33.1735 −1.34426
\(610\) 0 0
\(611\) 34.6686 1.40254
\(612\) 0 0
\(613\) −33.0598 −1.33527 −0.667637 0.744487i \(-0.732694\pi\)
−0.667637 + 0.744487i \(0.732694\pi\)
\(614\) 0 0
\(615\) 7.06927 0.285061
\(616\) 0 0
\(617\) −21.8412 −0.879292 −0.439646 0.898171i \(-0.644896\pi\)
−0.439646 + 0.898171i \(0.644896\pi\)
\(618\) 0 0
\(619\) 46.2206 1.85776 0.928882 0.370375i \(-0.120771\pi\)
0.928882 + 0.370375i \(0.120771\pi\)
\(620\) 0 0
\(621\) −23.2397 −0.932577
\(622\) 0 0
\(623\) −9.26182 −0.371067
\(624\) 0 0
\(625\) −6.89273 −0.275709
\(626\) 0 0
\(627\) 2.14510 0.0856671
\(628\) 0 0
\(629\) 43.0598 1.71691
\(630\) 0 0
\(631\) −0.317659 −0.0126458 −0.00632291 0.999980i \(-0.502013\pi\)
−0.00632291 + 0.999980i \(0.502013\pi\)
\(632\) 0 0
\(633\) 31.8201 1.26474
\(634\) 0 0
\(635\) −34.0735 −1.35217
\(636\) 0 0
\(637\) 6.66129 0.263930
\(638\) 0 0
\(639\) −23.9883 −0.948961
\(640\) 0 0
\(641\) −41.5246 −1.64012 −0.820061 0.572276i \(-0.806061\pi\)
−0.820061 + 0.572276i \(0.806061\pi\)
\(642\) 0 0
\(643\) −7.26182 −0.286378 −0.143189 0.989695i \(-0.545736\pi\)
−0.143189 + 0.989695i \(0.545736\pi\)
\(644\) 0 0
\(645\) −36.7191 −1.44581
\(646\) 0 0
\(647\) −29.0819 −1.14333 −0.571664 0.820487i \(-0.693702\pi\)
−0.571664 + 0.820487i \(0.693702\pi\)
\(648\) 0 0
\(649\) 2.54364 0.0998465
\(650\) 0 0
\(651\) −46.1207 −1.80761
\(652\) 0 0
\(653\) 12.6529 0.495146 0.247573 0.968869i \(-0.420367\pi\)
0.247573 + 0.968869i \(0.420367\pi\)
\(654\) 0 0
\(655\) −5.50153 −0.214962
\(656\) 0 0
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) −49.4878 −1.92777 −0.963886 0.266317i \(-0.914193\pi\)
−0.963886 + 0.266317i \(0.914193\pi\)
\(660\) 0 0
\(661\) 4.94950 0.192513 0.0962566 0.995357i \(-0.469313\pi\)
0.0962566 + 0.995357i \(0.469313\pi\)
\(662\) 0 0
\(663\) −60.2995 −2.34184
\(664\) 0 0
\(665\) 4.63091 0.179579
\(666\) 0 0
\(667\) 41.4289 1.60413
\(668\) 0 0
\(669\) −24.6540 −0.953177
\(670\) 0 0
\(671\) 13.4931 0.520897
\(672\) 0 0
\(673\) 21.9001 0.844185 0.422093 0.906553i \(-0.361296\pi\)
0.422093 + 0.906553i \(0.361296\pi\)
\(674\) 0 0
\(675\) 7.30592 0.281205
\(676\) 0 0
\(677\) 26.4353 1.01599 0.507996 0.861360i \(-0.330387\pi\)
0.507996 + 0.861360i \(0.330387\pi\)
\(678\) 0 0
\(679\) 20.2020 0.775282
\(680\) 0 0
\(681\) −18.7328 −0.717844
\(682\) 0 0
\(683\) −9.82545 −0.375960 −0.187980 0.982173i \(-0.560194\pi\)
−0.187980 + 0.982173i \(0.560194\pi\)
\(684\) 0 0
\(685\) −21.7603 −0.831418
\(686\) 0 0
\(687\) 16.8201 0.641727
\(688\) 0 0
\(689\) 40.7328 1.55180
\(690\) 0 0
\(691\) 19.8255 0.754196 0.377098 0.926173i \(-0.376922\pi\)
0.377098 + 0.926173i \(0.376922\pi\)
\(692\) 0 0
\(693\) −4.63091 −0.175914
\(694\) 0 0
\(695\) 5.11765 0.194123
\(696\) 0 0
\(697\) 11.8255 0.447921
\(698\) 0 0
\(699\) −26.8285 −1.01475
\(700\) 0 0
\(701\) −10.7550 −0.406209 −0.203105 0.979157i \(-0.565103\pi\)
−0.203105 + 0.979157i \(0.565103\pi\)
\(702\) 0 0
\(703\) −7.49314 −0.282609
\(704\) 0 0
\(705\) −24.3470 −0.916960
\(706\) 0 0
\(707\) 3.47848 0.130822
\(708\) 0 0
\(709\) −1.82651 −0.0685962 −0.0342981 0.999412i \(-0.510920\pi\)
−0.0342981 + 0.999412i \(0.510920\pi\)
\(710\) 0 0
\(711\) 13.9265 0.522283
\(712\) 0 0
\(713\) 57.5981 2.15707
\(714\) 0 0
\(715\) −7.83384 −0.292969
\(716\) 0 0
\(717\) −52.8221 −1.97268
\(718\) 0 0
\(719\) 4.65929 0.173762 0.0868812 0.996219i \(-0.472310\pi\)
0.0868812 + 0.996219i \(0.472310\pi\)
\(720\) 0 0
\(721\) 48.8055 1.81761
\(722\) 0 0
\(723\) 34.4951 1.28289
\(724\) 0 0
\(725\) −13.0241 −0.483703
\(726\) 0 0
\(727\) −19.9358 −0.739377 −0.369688 0.929156i \(-0.620536\pi\)
−0.369688 + 0.929156i \(0.620536\pi\)
\(728\) 0 0
\(729\) 1.30592 0.0483676
\(730\) 0 0
\(731\) −61.4236 −2.27183
\(732\) 0 0
\(733\) 13.7687 0.508558 0.254279 0.967131i \(-0.418162\pi\)
0.254279 + 0.967131i \(0.418162\pi\)
\(734\) 0 0
\(735\) −4.67807 −0.172553
\(736\) 0 0
\(737\) 10.3848 0.382529
\(738\) 0 0
\(739\) −22.4867 −0.827188 −0.413594 0.910461i \(-0.635727\pi\)
−0.413594 + 0.910461i \(0.635727\pi\)
\(740\) 0 0
\(741\) 10.4931 0.385475
\(742\) 0 0
\(743\) 50.6540 1.85831 0.929157 0.369686i \(-0.120535\pi\)
0.929157 + 0.369686i \(0.120535\pi\)
\(744\) 0 0
\(745\) 19.3354 0.708393
\(746\) 0 0
\(747\) 9.78228 0.357915
\(748\) 0 0
\(749\) −23.2397 −0.849160
\(750\) 0 0
\(751\) 38.9127 1.41995 0.709973 0.704229i \(-0.248707\pi\)
0.709973 + 0.704229i \(0.248707\pi\)
\(752\) 0 0
\(753\) −28.0735 −1.02306
\(754\) 0 0
\(755\) −7.86756 −0.286330
\(756\) 0 0
\(757\) −20.5877 −0.748274 −0.374137 0.927373i \(-0.622061\pi\)
−0.374137 + 0.927373i \(0.622061\pi\)
\(758\) 0 0
\(759\) 16.6172 0.603166
\(760\) 0 0
\(761\) 10.6025 0.384341 0.192171 0.981362i \(-0.438447\pi\)
0.192171 + 0.981362i \(0.438447\pi\)
\(762\) 0 0
\(763\) −33.3407 −1.20701
\(764\) 0 0
\(765\) 14.7382 0.532860
\(766\) 0 0
\(767\) 12.4426 0.449278
\(768\) 0 0
\(769\) −8.77495 −0.316433 −0.158216 0.987404i \(-0.550574\pi\)
−0.158216 + 0.987404i \(0.550574\pi\)
\(770\) 0 0
\(771\) 1.55203 0.0558949
\(772\) 0 0
\(773\) −46.8653 −1.68563 −0.842813 0.538206i \(-0.819102\pi\)
−0.842813 + 0.538206i \(0.819102\pi\)
\(774\) 0 0
\(775\) −18.1073 −0.650432
\(776\) 0 0
\(777\) 46.4794 1.66744
\(778\) 0 0
\(779\) −2.05783 −0.0737294
\(780\) 0 0
\(781\) −14.9789 −0.535989
\(782\) 0 0
\(783\) −16.0441 −0.573369
\(784\) 0 0
\(785\) −12.8235 −0.457689
\(786\) 0 0
\(787\) 27.4143 0.977213 0.488606 0.872504i \(-0.337505\pi\)
0.488606 + 0.872504i \(0.337505\pi\)
\(788\) 0 0
\(789\) −29.5510 −1.05204
\(790\) 0 0
\(791\) 23.1334 0.822528
\(792\) 0 0
\(793\) 66.0040 2.34387
\(794\) 0 0
\(795\) −28.6057 −1.01454
\(796\) 0 0
\(797\) −9.16616 −0.324682 −0.162341 0.986735i \(-0.551904\pi\)
−0.162341 + 0.986735i \(0.551904\pi\)
\(798\) 0 0
\(799\) −40.7275 −1.44084
\(800\) 0 0
\(801\) 5.12938 0.181238
\(802\) 0 0
\(803\) −3.74657 −0.132214
\(804\) 0 0
\(805\) 35.8737 1.26438
\(806\) 0 0
\(807\) −42.8117 −1.50704
\(808\) 0 0
\(809\) 48.0829 1.69050 0.845252 0.534368i \(-0.179450\pi\)
0.845252 + 0.534368i \(0.179450\pi\)
\(810\) 0 0
\(811\) 2.67395 0.0938951 0.0469475 0.998897i \(-0.485051\pi\)
0.0469475 + 0.998897i \(0.485051\pi\)
\(812\) 0 0
\(813\) −61.6686 −2.16281
\(814\) 0 0
\(815\) 3.01771 0.105706
\(816\) 0 0
\(817\) 10.6887 0.373952
\(818\) 0 0
\(819\) −22.6529 −0.791556
\(820\) 0 0
\(821\) 29.5078 1.02983 0.514915 0.857242i \(-0.327824\pi\)
0.514915 + 0.857242i \(0.327824\pi\)
\(822\) 0 0
\(823\) 2.39654 0.0835382 0.0417691 0.999127i \(-0.486701\pi\)
0.0417691 + 0.999127i \(0.486701\pi\)
\(824\) 0 0
\(825\) −5.22399 −0.181876
\(826\) 0 0
\(827\) −7.78868 −0.270839 −0.135419 0.990788i \(-0.543238\pi\)
−0.135419 + 0.990788i \(0.543238\pi\)
\(828\) 0 0
\(829\) 13.2818 0.461296 0.230648 0.973037i \(-0.425915\pi\)
0.230648 + 0.973037i \(0.425915\pi\)
\(830\) 0 0
\(831\) 6.62252 0.229733
\(832\) 0 0
\(833\) −7.82545 −0.271136
\(834\) 0 0
\(835\) −7.98534 −0.276344
\(836\) 0 0
\(837\) −22.3059 −0.771006
\(838\) 0 0
\(839\) 33.9063 1.17058 0.585288 0.810825i \(-0.300981\pi\)
0.585288 + 0.810825i \(0.300981\pi\)
\(840\) 0 0
\(841\) −0.398534 −0.0137426
\(842\) 0 0
\(843\) 19.5089 0.671921
\(844\) 0 0
\(845\) −17.5015 −0.602071
\(846\) 0 0
\(847\) −2.89167 −0.0993590
\(848\) 0 0
\(849\) −28.1976 −0.967739
\(850\) 0 0
\(851\) −58.0461 −1.98979
\(852\) 0 0
\(853\) 40.0735 1.37209 0.686046 0.727558i \(-0.259345\pi\)
0.686046 + 0.727558i \(0.259345\pi\)
\(854\) 0 0
\(855\) −2.56469 −0.0877106
\(856\) 0 0
\(857\) 0.665693 0.0227396 0.0113698 0.999935i \(-0.496381\pi\)
0.0113698 + 0.999935i \(0.496381\pi\)
\(858\) 0 0
\(859\) −13.8990 −0.474228 −0.237114 0.971482i \(-0.576201\pi\)
−0.237114 + 0.971482i \(0.576201\pi\)
\(860\) 0 0
\(861\) 12.7646 0.435015
\(862\) 0 0
\(863\) 40.3941 1.37503 0.687516 0.726169i \(-0.258701\pi\)
0.687516 + 0.726169i \(0.258701\pi\)
\(864\) 0 0
\(865\) −10.4323 −0.354707
\(866\) 0 0
\(867\) 34.3711 1.16730
\(868\) 0 0
\(869\) 8.69607 0.294994
\(870\) 0 0
\(871\) 50.7991 1.72126
\(872\) 0 0
\(873\) −11.1883 −0.378666
\(874\) 0 0
\(875\) −34.4323 −1.16402
\(876\) 0 0
\(877\) 15.6108 0.527139 0.263569 0.964640i \(-0.415100\pi\)
0.263569 + 0.964640i \(0.415100\pi\)
\(878\) 0 0
\(879\) 6.66769 0.224895
\(880\) 0 0
\(881\) 19.9843 0.673288 0.336644 0.941632i \(-0.390708\pi\)
0.336644 + 0.941632i \(0.390708\pi\)
\(882\) 0 0
\(883\) −20.2167 −0.680345 −0.340172 0.940363i \(-0.610485\pi\)
−0.340172 + 0.940363i \(0.610485\pi\)
\(884\) 0 0
\(885\) −8.73818 −0.293731
\(886\) 0 0
\(887\) −20.1471 −0.676473 −0.338237 0.941061i \(-0.609830\pi\)
−0.338237 + 0.941061i \(0.609830\pi\)
\(888\) 0 0
\(889\) −61.5246 −2.06347
\(890\) 0 0
\(891\) −11.2397 −0.376544
\(892\) 0 0
\(893\) 7.08727 0.237167
\(894\) 0 0
\(895\) −8.88968 −0.297149
\(896\) 0 0
\(897\) 81.2858 2.71405
\(898\) 0 0
\(899\) 39.7643 1.32621
\(900\) 0 0
\(901\) −47.8516 −1.59417
\(902\) 0 0
\(903\) −66.3015 −2.20638
\(904\) 0 0
\(905\) −3.01771 −0.100312
\(906\) 0 0
\(907\) −4.57109 −0.151781 −0.0758903 0.997116i \(-0.524180\pi\)
−0.0758903 + 0.997116i \(0.524180\pi\)
\(908\) 0 0
\(909\) −1.92645 −0.0638964
\(910\) 0 0
\(911\) −51.6402 −1.71092 −0.855459 0.517871i \(-0.826725\pi\)
−0.855459 + 0.517871i \(0.826725\pi\)
\(912\) 0 0
\(913\) 6.10833 0.202156
\(914\) 0 0
\(915\) −46.3531 −1.53239
\(916\) 0 0
\(917\) −9.93378 −0.328042
\(918\) 0 0
\(919\) 48.8412 1.61112 0.805561 0.592513i \(-0.201864\pi\)
0.805561 + 0.592513i \(0.201864\pi\)
\(920\) 0 0
\(921\) −4.69607 −0.154741
\(922\) 0 0
\(923\) −73.2721 −2.41178
\(924\) 0 0
\(925\) 18.2481 0.599994
\(926\) 0 0
\(927\) −27.0294 −0.887763
\(928\) 0 0
\(929\) −57.8819 −1.89904 −0.949522 0.313700i \(-0.898431\pi\)
−0.949522 + 0.313700i \(0.898431\pi\)
\(930\) 0 0
\(931\) 1.36176 0.0446299
\(932\) 0 0
\(933\) −8.41120 −0.275370
\(934\) 0 0
\(935\) 9.20293 0.300968
\(936\) 0 0
\(937\) −57.6496 −1.88333 −0.941664 0.336553i \(-0.890739\pi\)
−0.941664 + 0.336553i \(0.890739\pi\)
\(938\) 0 0
\(939\) −46.5740 −1.51989
\(940\) 0 0
\(941\) −15.0966 −0.492135 −0.246067 0.969253i \(-0.579138\pi\)
−0.246067 + 0.969253i \(0.579138\pi\)
\(942\) 0 0
\(943\) −15.9411 −0.519114
\(944\) 0 0
\(945\) −13.8927 −0.451931
\(946\) 0 0
\(947\) −33.3775 −1.08462 −0.542311 0.840178i \(-0.682451\pi\)
−0.542311 + 0.840178i \(0.682451\pi\)
\(948\) 0 0
\(949\) −18.3270 −0.594919
\(950\) 0 0
\(951\) 14.8171 0.480476
\(952\) 0 0
\(953\) −12.4794 −0.404248 −0.202124 0.979360i \(-0.564784\pi\)
−0.202124 + 0.979360i \(0.564784\pi\)
\(954\) 0 0
\(955\) 3.91180 0.126583
\(956\) 0 0
\(957\) 11.4721 0.370840
\(958\) 0 0
\(959\) −39.2913 −1.26878
\(960\) 0 0
\(961\) 24.2838 0.783349
\(962\) 0 0
\(963\) 12.8706 0.414750
\(964\) 0 0
\(965\) 15.4942 0.498776
\(966\) 0 0
\(967\) 12.5804 0.404559 0.202279 0.979328i \(-0.435165\pi\)
0.202279 + 0.979328i \(0.435165\pi\)
\(968\) 0 0
\(969\) −12.3270 −0.396000
\(970\) 0 0
\(971\) 1.79067 0.0574653 0.0287327 0.999587i \(-0.490853\pi\)
0.0287327 + 0.999587i \(0.490853\pi\)
\(972\) 0 0
\(973\) 9.24063 0.296241
\(974\) 0 0
\(975\) −25.5540 −0.818384
\(976\) 0 0
\(977\) 6.40586 0.204942 0.102471 0.994736i \(-0.467325\pi\)
0.102471 + 0.994736i \(0.467325\pi\)
\(978\) 0 0
\(979\) 3.20293 0.102366
\(980\) 0 0
\(981\) 18.4648 0.589534
\(982\) 0 0
\(983\) 20.5647 0.655912 0.327956 0.944693i \(-0.393640\pi\)
0.327956 + 0.944693i \(0.393640\pi\)
\(984\) 0 0
\(985\) 28.0147 0.892621
\(986\) 0 0
\(987\) −43.9619 −1.39932
\(988\) 0 0
\(989\) 82.8011 2.63292
\(990\) 0 0
\(991\) 29.0166 0.921744 0.460872 0.887467i \(-0.347537\pi\)
0.460872 + 0.887467i \(0.347537\pi\)
\(992\) 0 0
\(993\) −68.8642 −2.18534
\(994\) 0 0
\(995\) −40.4205 −1.28142
\(996\) 0 0
\(997\) 42.3638 1.34167 0.670837 0.741605i \(-0.265935\pi\)
0.670837 + 0.741605i \(0.265935\pi\)
\(998\) 0 0
\(999\) 22.4794 0.711217
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 3344.2.a.q.1.3 3
4.3 odd 2 418.2.a.g.1.1 3
12.11 even 2 3762.2.a.bg.1.2 3
44.43 even 2 4598.2.a.bo.1.1 3
76.75 even 2 7942.2.a.bi.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.g.1.1 3 4.3 odd 2
3344.2.a.q.1.3 3 1.1 even 1 trivial
3762.2.a.bg.1.2 3 12.11 even 2
4598.2.a.bo.1.1 3 44.43 even 2
7942.2.a.bi.1.3 3 76.75 even 2