Properties

Label 6036.2.a.h.1.17
Level $6036$
Weight $2$
Character 6036.1
Self dual yes
Analytic conductor $48.198$
Analytic rank $0$
Dimension $24$
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] = [6036,2,Mod(1,6036)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6036, 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("6036.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6036.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: \(48.1977026600\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.17
Character \(\chi\) \(=\) 6036.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.44705 q^{5} -3.28775 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +2.44705 q^{5} -3.28775 q^{7} +1.00000 q^{9} +2.45869 q^{11} +3.50225 q^{13} +2.44705 q^{15} +1.87555 q^{17} -6.22829 q^{19} -3.28775 q^{21} +2.41520 q^{23} +0.988047 q^{25} +1.00000 q^{27} -8.78423 q^{29} -0.157121 q^{31} +2.45869 q^{33} -8.04529 q^{35} +8.26847 q^{37} +3.50225 q^{39} +5.46119 q^{41} +12.0118 q^{43} +2.44705 q^{45} +2.17244 q^{47} +3.80931 q^{49} +1.87555 q^{51} +10.4904 q^{53} +6.01653 q^{55} -6.22829 q^{57} -11.3992 q^{59} -5.27697 q^{61} -3.28775 q^{63} +8.57018 q^{65} +12.6870 q^{67} +2.41520 q^{69} +16.3594 q^{71} +6.86626 q^{73} +0.988047 q^{75} -8.08356 q^{77} +7.84100 q^{79} +1.00000 q^{81} -11.3738 q^{83} +4.58957 q^{85} -8.78423 q^{87} +14.4989 q^{89} -11.5145 q^{91} -0.157121 q^{93} -15.2409 q^{95} +13.2629 q^{97} +2.45869 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9} + 9 q^{11} + 7 q^{13} + 18 q^{15} + 12 q^{17} - q^{19} + 9 q^{21} + 22 q^{23} + 48 q^{25} + 24 q^{27} + 44 q^{29} + 15 q^{31} + 9 q^{33} + 3 q^{35} + 18 q^{37} + 7 q^{39} + 37 q^{41} - 8 q^{43} + 18 q^{45} + 24 q^{47} + 63 q^{49} + 12 q^{51} + 59 q^{53} + 20 q^{55} - q^{57} + 44 q^{59} + 33 q^{61} + 9 q^{63} + 15 q^{65} + 13 q^{67} + 22 q^{69} + 47 q^{71} + 28 q^{73} + 48 q^{75} + 19 q^{77} + 12 q^{79} + 24 q^{81} + 17 q^{83} + 31 q^{85} + 44 q^{87} + 41 q^{89} - q^{91} + 15 q^{93} + 58 q^{95} + 51 q^{97} + 9 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 2.44705 1.09435 0.547177 0.837017i \(-0.315703\pi\)
0.547177 + 0.837017i \(0.315703\pi\)
\(6\) 0 0
\(7\) −3.28775 −1.24265 −0.621327 0.783552i \(-0.713406\pi\)
−0.621327 + 0.783552i \(0.713406\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.45869 0.741323 0.370661 0.928768i \(-0.379131\pi\)
0.370661 + 0.928768i \(0.379131\pi\)
\(12\) 0 0
\(13\) 3.50225 0.971350 0.485675 0.874140i \(-0.338574\pi\)
0.485675 + 0.874140i \(0.338574\pi\)
\(14\) 0 0
\(15\) 2.44705 0.631825
\(16\) 0 0
\(17\) 1.87555 0.454889 0.227444 0.973791i \(-0.426963\pi\)
0.227444 + 0.973791i \(0.426963\pi\)
\(18\) 0 0
\(19\) −6.22829 −1.42887 −0.714434 0.699703i \(-0.753316\pi\)
−0.714434 + 0.699703i \(0.753316\pi\)
\(20\) 0 0
\(21\) −3.28775 −0.717446
\(22\) 0 0
\(23\) 2.41520 0.503604 0.251802 0.967779i \(-0.418977\pi\)
0.251802 + 0.967779i \(0.418977\pi\)
\(24\) 0 0
\(25\) 0.988047 0.197609
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −8.78423 −1.63119 −0.815595 0.578623i \(-0.803590\pi\)
−0.815595 + 0.578623i \(0.803590\pi\)
\(30\) 0 0
\(31\) −0.157121 −0.0282197 −0.0141099 0.999900i \(-0.504491\pi\)
−0.0141099 + 0.999900i \(0.504491\pi\)
\(32\) 0 0
\(33\) 2.45869 0.428003
\(34\) 0 0
\(35\) −8.04529 −1.35990
\(36\) 0 0
\(37\) 8.26847 1.35933 0.679664 0.733524i \(-0.262126\pi\)
0.679664 + 0.733524i \(0.262126\pi\)
\(38\) 0 0
\(39\) 3.50225 0.560809
\(40\) 0 0
\(41\) 5.46119 0.852894 0.426447 0.904512i \(-0.359765\pi\)
0.426447 + 0.904512i \(0.359765\pi\)
\(42\) 0 0
\(43\) 12.0118 1.83178 0.915888 0.401434i \(-0.131488\pi\)
0.915888 + 0.401434i \(0.131488\pi\)
\(44\) 0 0
\(45\) 2.44705 0.364784
\(46\) 0 0
\(47\) 2.17244 0.316883 0.158441 0.987368i \(-0.449353\pi\)
0.158441 + 0.987368i \(0.449353\pi\)
\(48\) 0 0
\(49\) 3.80931 0.544188
\(50\) 0 0
\(51\) 1.87555 0.262630
\(52\) 0 0
\(53\) 10.4904 1.44097 0.720486 0.693469i \(-0.243919\pi\)
0.720486 + 0.693469i \(0.243919\pi\)
\(54\) 0 0
\(55\) 6.01653 0.811269
\(56\) 0 0
\(57\) −6.22829 −0.824957
\(58\) 0 0
\(59\) −11.3992 −1.48405 −0.742027 0.670370i \(-0.766135\pi\)
−0.742027 + 0.670370i \(0.766135\pi\)
\(60\) 0 0
\(61\) −5.27697 −0.675647 −0.337823 0.941210i \(-0.609691\pi\)
−0.337823 + 0.941210i \(0.609691\pi\)
\(62\) 0 0
\(63\) −3.28775 −0.414218
\(64\) 0 0
\(65\) 8.57018 1.06300
\(66\) 0 0
\(67\) 12.6870 1.54996 0.774980 0.631986i \(-0.217760\pi\)
0.774980 + 0.631986i \(0.217760\pi\)
\(68\) 0 0
\(69\) 2.41520 0.290756
\(70\) 0 0
\(71\) 16.3594 1.94150 0.970752 0.240084i \(-0.0771749\pi\)
0.970752 + 0.240084i \(0.0771749\pi\)
\(72\) 0 0
\(73\) 6.86626 0.803635 0.401818 0.915720i \(-0.368379\pi\)
0.401818 + 0.915720i \(0.368379\pi\)
\(74\) 0 0
\(75\) 0.988047 0.114090
\(76\) 0 0
\(77\) −8.08356 −0.921208
\(78\) 0 0
\(79\) 7.84100 0.882181 0.441091 0.897463i \(-0.354592\pi\)
0.441091 + 0.897463i \(0.354592\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −11.3738 −1.24844 −0.624220 0.781249i \(-0.714583\pi\)
−0.624220 + 0.781249i \(0.714583\pi\)
\(84\) 0 0
\(85\) 4.58957 0.497809
\(86\) 0 0
\(87\) −8.78423 −0.941768
\(88\) 0 0
\(89\) 14.4989 1.53688 0.768439 0.639923i \(-0.221034\pi\)
0.768439 + 0.639923i \(0.221034\pi\)
\(90\) 0 0
\(91\) −11.5145 −1.20705
\(92\) 0 0
\(93\) −0.157121 −0.0162927
\(94\) 0 0
\(95\) −15.2409 −1.56369
\(96\) 0 0
\(97\) 13.2629 1.34664 0.673320 0.739351i \(-0.264868\pi\)
0.673320 + 0.739351i \(0.264868\pi\)
\(98\) 0 0
\(99\) 2.45869 0.247108
\(100\) 0 0
\(101\) −11.9999 −1.19403 −0.597016 0.802229i \(-0.703647\pi\)
−0.597016 + 0.802229i \(0.703647\pi\)
\(102\) 0 0
\(103\) −7.67378 −0.756120 −0.378060 0.925781i \(-0.623409\pi\)
−0.378060 + 0.925781i \(0.623409\pi\)
\(104\) 0 0
\(105\) −8.04529 −0.785140
\(106\) 0 0
\(107\) −13.4013 −1.29555 −0.647777 0.761830i \(-0.724301\pi\)
−0.647777 + 0.761830i \(0.724301\pi\)
\(108\) 0 0
\(109\) 5.52986 0.529664 0.264832 0.964295i \(-0.414683\pi\)
0.264832 + 0.964295i \(0.414683\pi\)
\(110\) 0 0
\(111\) 8.26847 0.784808
\(112\) 0 0
\(113\) −2.70531 −0.254494 −0.127247 0.991871i \(-0.540614\pi\)
−0.127247 + 0.991871i \(0.540614\pi\)
\(114\) 0 0
\(115\) 5.91011 0.551121
\(116\) 0 0
\(117\) 3.50225 0.323783
\(118\) 0 0
\(119\) −6.16636 −0.565269
\(120\) 0 0
\(121\) −4.95484 −0.450440
\(122\) 0 0
\(123\) 5.46119 0.492419
\(124\) 0 0
\(125\) −9.81744 −0.878099
\(126\) 0 0
\(127\) −7.45150 −0.661214 −0.330607 0.943769i \(-0.607253\pi\)
−0.330607 + 0.943769i \(0.607253\pi\)
\(128\) 0 0
\(129\) 12.0118 1.05758
\(130\) 0 0
\(131\) 16.0072 1.39856 0.699278 0.714849i \(-0.253505\pi\)
0.699278 + 0.714849i \(0.253505\pi\)
\(132\) 0 0
\(133\) 20.4771 1.77559
\(134\) 0 0
\(135\) 2.44705 0.210608
\(136\) 0 0
\(137\) 10.5635 0.902501 0.451250 0.892397i \(-0.350978\pi\)
0.451250 + 0.892397i \(0.350978\pi\)
\(138\) 0 0
\(139\) −13.1827 −1.11815 −0.559073 0.829119i \(-0.688843\pi\)
−0.559073 + 0.829119i \(0.688843\pi\)
\(140\) 0 0
\(141\) 2.17244 0.182952
\(142\) 0 0
\(143\) 8.61095 0.720084
\(144\) 0 0
\(145\) −21.4954 −1.78510
\(146\) 0 0
\(147\) 3.80931 0.314187
\(148\) 0 0
\(149\) −3.95193 −0.323755 −0.161877 0.986811i \(-0.551755\pi\)
−0.161877 + 0.986811i \(0.551755\pi\)
\(150\) 0 0
\(151\) −1.69850 −0.138222 −0.0691111 0.997609i \(-0.522016\pi\)
−0.0691111 + 0.997609i \(0.522016\pi\)
\(152\) 0 0
\(153\) 1.87555 0.151630
\(154\) 0 0
\(155\) −0.384482 −0.0308824
\(156\) 0 0
\(157\) 17.1677 1.37013 0.685065 0.728482i \(-0.259774\pi\)
0.685065 + 0.728482i \(0.259774\pi\)
\(158\) 0 0
\(159\) 10.4904 0.831946
\(160\) 0 0
\(161\) −7.94058 −0.625805
\(162\) 0 0
\(163\) −14.4212 −1.12955 −0.564777 0.825243i \(-0.691038\pi\)
−0.564777 + 0.825243i \(0.691038\pi\)
\(164\) 0 0
\(165\) 6.01653 0.468387
\(166\) 0 0
\(167\) −0.953097 −0.0737528 −0.0368764 0.999320i \(-0.511741\pi\)
−0.0368764 + 0.999320i \(0.511741\pi\)
\(168\) 0 0
\(169\) −0.734237 −0.0564797
\(170\) 0 0
\(171\) −6.22829 −0.476289
\(172\) 0 0
\(173\) −0.689872 −0.0524500 −0.0262250 0.999656i \(-0.508349\pi\)
−0.0262250 + 0.999656i \(0.508349\pi\)
\(174\) 0 0
\(175\) −3.24845 −0.245560
\(176\) 0 0
\(177\) −11.3992 −0.856819
\(178\) 0 0
\(179\) −23.1631 −1.73129 −0.865644 0.500660i \(-0.833091\pi\)
−0.865644 + 0.500660i \(0.833091\pi\)
\(180\) 0 0
\(181\) 20.8009 1.54612 0.773061 0.634332i \(-0.218725\pi\)
0.773061 + 0.634332i \(0.218725\pi\)
\(182\) 0 0
\(183\) −5.27697 −0.390085
\(184\) 0 0
\(185\) 20.2333 1.48758
\(186\) 0 0
\(187\) 4.61140 0.337219
\(188\) 0 0
\(189\) −3.28775 −0.239149
\(190\) 0 0
\(191\) 1.18366 0.0856467 0.0428233 0.999083i \(-0.486365\pi\)
0.0428233 + 0.999083i \(0.486365\pi\)
\(192\) 0 0
\(193\) 24.6980 1.77780 0.888902 0.458098i \(-0.151469\pi\)
0.888902 + 0.458098i \(0.151469\pi\)
\(194\) 0 0
\(195\) 8.57018 0.613723
\(196\) 0 0
\(197\) −22.9253 −1.63336 −0.816678 0.577093i \(-0.804187\pi\)
−0.816678 + 0.577093i \(0.804187\pi\)
\(198\) 0 0
\(199\) 5.39101 0.382159 0.191079 0.981575i \(-0.438801\pi\)
0.191079 + 0.981575i \(0.438801\pi\)
\(200\) 0 0
\(201\) 12.6870 0.894870
\(202\) 0 0
\(203\) 28.8804 2.02700
\(204\) 0 0
\(205\) 13.3638 0.933368
\(206\) 0 0
\(207\) 2.41520 0.167868
\(208\) 0 0
\(209\) −15.3134 −1.05925
\(210\) 0 0
\(211\) −10.5456 −0.725987 −0.362994 0.931792i \(-0.618245\pi\)
−0.362994 + 0.931792i \(0.618245\pi\)
\(212\) 0 0
\(213\) 16.3594 1.12093
\(214\) 0 0
\(215\) 29.3934 2.00461
\(216\) 0 0
\(217\) 0.516574 0.0350674
\(218\) 0 0
\(219\) 6.86626 0.463979
\(220\) 0 0
\(221\) 6.56866 0.441856
\(222\) 0 0
\(223\) 3.05644 0.204674 0.102337 0.994750i \(-0.467368\pi\)
0.102337 + 0.994750i \(0.467368\pi\)
\(224\) 0 0
\(225\) 0.988047 0.0658698
\(226\) 0 0
\(227\) 17.4491 1.15814 0.579068 0.815279i \(-0.303416\pi\)
0.579068 + 0.815279i \(0.303416\pi\)
\(228\) 0 0
\(229\) 12.0819 0.798392 0.399196 0.916866i \(-0.369289\pi\)
0.399196 + 0.916866i \(0.369289\pi\)
\(230\) 0 0
\(231\) −8.08356 −0.531859
\(232\) 0 0
\(233\) 12.3600 0.809733 0.404867 0.914376i \(-0.367318\pi\)
0.404867 + 0.914376i \(0.367318\pi\)
\(234\) 0 0
\(235\) 5.31606 0.346782
\(236\) 0 0
\(237\) 7.84100 0.509328
\(238\) 0 0
\(239\) 14.2718 0.923169 0.461585 0.887096i \(-0.347281\pi\)
0.461585 + 0.887096i \(0.347281\pi\)
\(240\) 0 0
\(241\) −4.58419 −0.295294 −0.147647 0.989040i \(-0.547170\pi\)
−0.147647 + 0.989040i \(0.547170\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 9.32158 0.595534
\(246\) 0 0
\(247\) −21.8130 −1.38793
\(248\) 0 0
\(249\) −11.3738 −0.720787
\(250\) 0 0
\(251\) 16.0666 1.01411 0.507056 0.861913i \(-0.330734\pi\)
0.507056 + 0.861913i \(0.330734\pi\)
\(252\) 0 0
\(253\) 5.93823 0.373333
\(254\) 0 0
\(255\) 4.58957 0.287410
\(256\) 0 0
\(257\) 13.3839 0.834863 0.417432 0.908708i \(-0.362930\pi\)
0.417432 + 0.908708i \(0.362930\pi\)
\(258\) 0 0
\(259\) −27.1847 −1.68917
\(260\) 0 0
\(261\) −8.78423 −0.543730
\(262\) 0 0
\(263\) 10.4612 0.645067 0.322534 0.946558i \(-0.395465\pi\)
0.322534 + 0.946558i \(0.395465\pi\)
\(264\) 0 0
\(265\) 25.6706 1.57693
\(266\) 0 0
\(267\) 14.4989 0.887317
\(268\) 0 0
\(269\) −5.65825 −0.344990 −0.172495 0.985010i \(-0.555183\pi\)
−0.172495 + 0.985010i \(0.555183\pi\)
\(270\) 0 0
\(271\) −14.9392 −0.907494 −0.453747 0.891131i \(-0.649913\pi\)
−0.453747 + 0.891131i \(0.649913\pi\)
\(272\) 0 0
\(273\) −11.5145 −0.696891
\(274\) 0 0
\(275\) 2.42930 0.146492
\(276\) 0 0
\(277\) −18.1887 −1.09285 −0.546427 0.837506i \(-0.684013\pi\)
−0.546427 + 0.837506i \(0.684013\pi\)
\(278\) 0 0
\(279\) −0.157121 −0.00940658
\(280\) 0 0
\(281\) 14.8090 0.883430 0.441715 0.897155i \(-0.354370\pi\)
0.441715 + 0.897155i \(0.354370\pi\)
\(282\) 0 0
\(283\) 0.132035 0.00784866 0.00392433 0.999992i \(-0.498751\pi\)
0.00392433 + 0.999992i \(0.498751\pi\)
\(284\) 0 0
\(285\) −15.2409 −0.902794
\(286\) 0 0
\(287\) −17.9550 −1.05985
\(288\) 0 0
\(289\) −13.4823 −0.793076
\(290\) 0 0
\(291\) 13.2629 0.777483
\(292\) 0 0
\(293\) 22.4272 1.31021 0.655104 0.755538i \(-0.272625\pi\)
0.655104 + 0.755538i \(0.272625\pi\)
\(294\) 0 0
\(295\) −27.8945 −1.62408
\(296\) 0 0
\(297\) 2.45869 0.142668
\(298\) 0 0
\(299\) 8.45864 0.489176
\(300\) 0 0
\(301\) −39.4917 −2.27626
\(302\) 0 0
\(303\) −11.9999 −0.689375
\(304\) 0 0
\(305\) −12.9130 −0.739396
\(306\) 0 0
\(307\) 17.8792 1.02042 0.510210 0.860050i \(-0.329568\pi\)
0.510210 + 0.860050i \(0.329568\pi\)
\(308\) 0 0
\(309\) −7.67378 −0.436546
\(310\) 0 0
\(311\) −4.18402 −0.237254 −0.118627 0.992939i \(-0.537849\pi\)
−0.118627 + 0.992939i \(0.537849\pi\)
\(312\) 0 0
\(313\) 26.1802 1.47979 0.739896 0.672722i \(-0.234875\pi\)
0.739896 + 0.672722i \(0.234875\pi\)
\(314\) 0 0
\(315\) −8.04529 −0.453301
\(316\) 0 0
\(317\) 21.8635 1.22798 0.613989 0.789315i \(-0.289564\pi\)
0.613989 + 0.789315i \(0.289564\pi\)
\(318\) 0 0
\(319\) −21.5977 −1.20924
\(320\) 0 0
\(321\) −13.4013 −0.747989
\(322\) 0 0
\(323\) −11.6815 −0.649975
\(324\) 0 0
\(325\) 3.46039 0.191948
\(326\) 0 0
\(327\) 5.52986 0.305802
\(328\) 0 0
\(329\) −7.14244 −0.393776
\(330\) 0 0
\(331\) −24.3232 −1.33692 −0.668462 0.743746i \(-0.733047\pi\)
−0.668462 + 0.743746i \(0.733047\pi\)
\(332\) 0 0
\(333\) 8.26847 0.453109
\(334\) 0 0
\(335\) 31.0456 1.69620
\(336\) 0 0
\(337\) −12.4492 −0.678151 −0.339075 0.940759i \(-0.610114\pi\)
−0.339075 + 0.940759i \(0.610114\pi\)
\(338\) 0 0
\(339\) −2.70531 −0.146932
\(340\) 0 0
\(341\) −0.386311 −0.0209199
\(342\) 0 0
\(343\) 10.4902 0.566417
\(344\) 0 0
\(345\) 5.91011 0.318190
\(346\) 0 0
\(347\) −24.4150 −1.31067 −0.655334 0.755339i \(-0.727472\pi\)
−0.655334 + 0.755339i \(0.727472\pi\)
\(348\) 0 0
\(349\) 16.9424 0.906908 0.453454 0.891280i \(-0.350192\pi\)
0.453454 + 0.891280i \(0.350192\pi\)
\(350\) 0 0
\(351\) 3.50225 0.186936
\(352\) 0 0
\(353\) −29.4654 −1.56829 −0.784144 0.620579i \(-0.786898\pi\)
−0.784144 + 0.620579i \(0.786898\pi\)
\(354\) 0 0
\(355\) 40.0323 2.12469
\(356\) 0 0
\(357\) −6.16636 −0.326358
\(358\) 0 0
\(359\) −7.70423 −0.406614 −0.203307 0.979115i \(-0.565169\pi\)
−0.203307 + 0.979115i \(0.565169\pi\)
\(360\) 0 0
\(361\) 19.7916 1.04166
\(362\) 0 0
\(363\) −4.95484 −0.260062
\(364\) 0 0
\(365\) 16.8021 0.879461
\(366\) 0 0
\(367\) −37.8619 −1.97637 −0.988186 0.153258i \(-0.951024\pi\)
−0.988186 + 0.153258i \(0.951024\pi\)
\(368\) 0 0
\(369\) 5.46119 0.284298
\(370\) 0 0
\(371\) −34.4900 −1.79063
\(372\) 0 0
\(373\) 23.6981 1.22704 0.613520 0.789679i \(-0.289753\pi\)
0.613520 + 0.789679i \(0.289753\pi\)
\(374\) 0 0
\(375\) −9.81744 −0.506971
\(376\) 0 0
\(377\) −30.7646 −1.58446
\(378\) 0 0
\(379\) −10.0897 −0.518275 −0.259137 0.965840i \(-0.583438\pi\)
−0.259137 + 0.965840i \(0.583438\pi\)
\(380\) 0 0
\(381\) −7.45150 −0.381752
\(382\) 0 0
\(383\) 1.18569 0.0605860 0.0302930 0.999541i \(-0.490356\pi\)
0.0302930 + 0.999541i \(0.490356\pi\)
\(384\) 0 0
\(385\) −19.7809 −1.00813
\(386\) 0 0
\(387\) 12.0118 0.610592
\(388\) 0 0
\(389\) 3.55677 0.180335 0.0901677 0.995927i \(-0.471260\pi\)
0.0901677 + 0.995927i \(0.471260\pi\)
\(390\) 0 0
\(391\) 4.52984 0.229084
\(392\) 0 0
\(393\) 16.0072 0.807457
\(394\) 0 0
\(395\) 19.1873 0.965418
\(396\) 0 0
\(397\) 11.7615 0.590292 0.295146 0.955452i \(-0.404632\pi\)
0.295146 + 0.955452i \(0.404632\pi\)
\(398\) 0 0
\(399\) 20.4771 1.02514
\(400\) 0 0
\(401\) 26.8739 1.34202 0.671008 0.741450i \(-0.265862\pi\)
0.671008 + 0.741450i \(0.265862\pi\)
\(402\) 0 0
\(403\) −0.550277 −0.0274112
\(404\) 0 0
\(405\) 2.44705 0.121595
\(406\) 0 0
\(407\) 20.3296 1.00770
\(408\) 0 0
\(409\) 18.8238 0.930778 0.465389 0.885106i \(-0.345914\pi\)
0.465389 + 0.885106i \(0.345914\pi\)
\(410\) 0 0
\(411\) 10.5635 0.521059
\(412\) 0 0
\(413\) 37.4779 1.84416
\(414\) 0 0
\(415\) −27.8323 −1.36623
\(416\) 0 0
\(417\) −13.1827 −0.645562
\(418\) 0 0
\(419\) −20.4358 −0.998354 −0.499177 0.866500i \(-0.666364\pi\)
−0.499177 + 0.866500i \(0.666364\pi\)
\(420\) 0 0
\(421\) −26.8877 −1.31043 −0.655213 0.755445i \(-0.727421\pi\)
−0.655213 + 0.755445i \(0.727421\pi\)
\(422\) 0 0
\(423\) 2.17244 0.105628
\(424\) 0 0
\(425\) 1.85313 0.0898902
\(426\) 0 0
\(427\) 17.3494 0.839595
\(428\) 0 0
\(429\) 8.61095 0.415741
\(430\) 0 0
\(431\) 2.54022 0.122358 0.0611791 0.998127i \(-0.480514\pi\)
0.0611791 + 0.998127i \(0.480514\pi\)
\(432\) 0 0
\(433\) 1.93837 0.0931523 0.0465761 0.998915i \(-0.485169\pi\)
0.0465761 + 0.998915i \(0.485169\pi\)
\(434\) 0 0
\(435\) −21.4954 −1.03063
\(436\) 0 0
\(437\) −15.0426 −0.719583
\(438\) 0 0
\(439\) −15.5381 −0.741595 −0.370797 0.928714i \(-0.620916\pi\)
−0.370797 + 0.928714i \(0.620916\pi\)
\(440\) 0 0
\(441\) 3.80931 0.181396
\(442\) 0 0
\(443\) −9.25575 −0.439754 −0.219877 0.975528i \(-0.570566\pi\)
−0.219877 + 0.975528i \(0.570566\pi\)
\(444\) 0 0
\(445\) 35.4795 1.68189
\(446\) 0 0
\(447\) −3.95193 −0.186920
\(448\) 0 0
\(449\) −28.8498 −1.36150 −0.680752 0.732514i \(-0.738347\pi\)
−0.680752 + 0.732514i \(0.738347\pi\)
\(450\) 0 0
\(451\) 13.4274 0.632270
\(452\) 0 0
\(453\) −1.69850 −0.0798026
\(454\) 0 0
\(455\) −28.1766 −1.32094
\(456\) 0 0
\(457\) −32.3694 −1.51418 −0.757089 0.653312i \(-0.773379\pi\)
−0.757089 + 0.653312i \(0.773379\pi\)
\(458\) 0 0
\(459\) 1.87555 0.0875433
\(460\) 0 0
\(461\) −28.6695 −1.33527 −0.667635 0.744489i \(-0.732693\pi\)
−0.667635 + 0.744489i \(0.732693\pi\)
\(462\) 0 0
\(463\) 2.40456 0.111749 0.0558747 0.998438i \(-0.482205\pi\)
0.0558747 + 0.998438i \(0.482205\pi\)
\(464\) 0 0
\(465\) −0.384482 −0.0178299
\(466\) 0 0
\(467\) 7.79123 0.360535 0.180268 0.983618i \(-0.442304\pi\)
0.180268 + 0.983618i \(0.442304\pi\)
\(468\) 0 0
\(469\) −41.7116 −1.92606
\(470\) 0 0
\(471\) 17.1677 0.791045
\(472\) 0 0
\(473\) 29.5332 1.35794
\(474\) 0 0
\(475\) −6.15384 −0.282358
\(476\) 0 0
\(477\) 10.4904 0.480324
\(478\) 0 0
\(479\) −0.858965 −0.0392471 −0.0196235 0.999807i \(-0.506247\pi\)
−0.0196235 + 0.999807i \(0.506247\pi\)
\(480\) 0 0
\(481\) 28.9582 1.32038
\(482\) 0 0
\(483\) −7.94058 −0.361309
\(484\) 0 0
\(485\) 32.4549 1.47370
\(486\) 0 0
\(487\) 12.5179 0.567241 0.283621 0.958937i \(-0.408464\pi\)
0.283621 + 0.958937i \(0.408464\pi\)
\(488\) 0 0
\(489\) −14.4212 −0.652148
\(490\) 0 0
\(491\) 25.4257 1.14745 0.573723 0.819049i \(-0.305499\pi\)
0.573723 + 0.819049i \(0.305499\pi\)
\(492\) 0 0
\(493\) −16.4753 −0.742010
\(494\) 0 0
\(495\) 6.01653 0.270423
\(496\) 0 0
\(497\) −53.7857 −2.41262
\(498\) 0 0
\(499\) −7.10065 −0.317869 −0.158934 0.987289i \(-0.550806\pi\)
−0.158934 + 0.987289i \(0.550806\pi\)
\(500\) 0 0
\(501\) −0.953097 −0.0425812
\(502\) 0 0
\(503\) 1.00000 0.0445878
\(504\) 0 0
\(505\) −29.3643 −1.30669
\(506\) 0 0
\(507\) −0.734237 −0.0326086
\(508\) 0 0
\(509\) 8.72869 0.386892 0.193446 0.981111i \(-0.438034\pi\)
0.193446 + 0.981111i \(0.438034\pi\)
\(510\) 0 0
\(511\) −22.5746 −0.998640
\(512\) 0 0
\(513\) −6.22829 −0.274986
\(514\) 0 0
\(515\) −18.7781 −0.827462
\(516\) 0 0
\(517\) 5.34135 0.234912
\(518\) 0 0
\(519\) −0.689872 −0.0302820
\(520\) 0 0
\(521\) −15.5055 −0.679307 −0.339653 0.940551i \(-0.610310\pi\)
−0.339653 + 0.940551i \(0.610310\pi\)
\(522\) 0 0
\(523\) −15.2909 −0.668623 −0.334311 0.942463i \(-0.608504\pi\)
−0.334311 + 0.942463i \(0.608504\pi\)
\(524\) 0 0
\(525\) −3.24845 −0.141774
\(526\) 0 0
\(527\) −0.294689 −0.0128368
\(528\) 0 0
\(529\) −17.1668 −0.746383
\(530\) 0 0
\(531\) −11.3992 −0.494685
\(532\) 0 0
\(533\) 19.1265 0.828459
\(534\) 0 0
\(535\) −32.7937 −1.41779
\(536\) 0 0
\(537\) −23.1631 −0.999560
\(538\) 0 0
\(539\) 9.36592 0.403419
\(540\) 0 0
\(541\) −41.0196 −1.76357 −0.881786 0.471650i \(-0.843659\pi\)
−0.881786 + 0.471650i \(0.843659\pi\)
\(542\) 0 0
\(543\) 20.8009 0.892654
\(544\) 0 0
\(545\) 13.5318 0.579640
\(546\) 0 0
\(547\) −30.1413 −1.28875 −0.644374 0.764711i \(-0.722882\pi\)
−0.644374 + 0.764711i \(0.722882\pi\)
\(548\) 0 0
\(549\) −5.27697 −0.225216
\(550\) 0 0
\(551\) 54.7107 2.33076
\(552\) 0 0
\(553\) −25.7793 −1.09625
\(554\) 0 0
\(555\) 20.2333 0.858857
\(556\) 0 0
\(557\) −29.4608 −1.24829 −0.624147 0.781307i \(-0.714553\pi\)
−0.624147 + 0.781307i \(0.714553\pi\)
\(558\) 0 0
\(559\) 42.0682 1.77929
\(560\) 0 0
\(561\) 4.61140 0.194694
\(562\) 0 0
\(563\) 5.51327 0.232357 0.116178 0.993228i \(-0.462936\pi\)
0.116178 + 0.993228i \(0.462936\pi\)
\(564\) 0 0
\(565\) −6.62002 −0.278506
\(566\) 0 0
\(567\) −3.28775 −0.138073
\(568\) 0 0
\(569\) −15.2813 −0.640624 −0.320312 0.947312i \(-0.603788\pi\)
−0.320312 + 0.947312i \(0.603788\pi\)
\(570\) 0 0
\(571\) 10.3609 0.433591 0.216795 0.976217i \(-0.430440\pi\)
0.216795 + 0.976217i \(0.430440\pi\)
\(572\) 0 0
\(573\) 1.18366 0.0494481
\(574\) 0 0
\(575\) 2.38633 0.0995168
\(576\) 0 0
\(577\) 36.9287 1.53736 0.768680 0.639633i \(-0.220914\pi\)
0.768680 + 0.639633i \(0.220914\pi\)
\(578\) 0 0
\(579\) 24.6980 1.02642
\(580\) 0 0
\(581\) 37.3943 1.55138
\(582\) 0 0
\(583\) 25.7927 1.06823
\(584\) 0 0
\(585\) 8.57018 0.354333
\(586\) 0 0
\(587\) −35.3442 −1.45881 −0.729405 0.684082i \(-0.760203\pi\)
−0.729405 + 0.684082i \(0.760203\pi\)
\(588\) 0 0
\(589\) 0.978594 0.0403223
\(590\) 0 0
\(591\) −22.9253 −0.943019
\(592\) 0 0
\(593\) −38.2267 −1.56978 −0.784890 0.619635i \(-0.787281\pi\)
−0.784890 + 0.619635i \(0.787281\pi\)
\(594\) 0 0
\(595\) −15.0894 −0.618604
\(596\) 0 0
\(597\) 5.39101 0.220639
\(598\) 0 0
\(599\) −3.24011 −0.132388 −0.0661938 0.997807i \(-0.521086\pi\)
−0.0661938 + 0.997807i \(0.521086\pi\)
\(600\) 0 0
\(601\) −40.3509 −1.64595 −0.822973 0.568081i \(-0.807686\pi\)
−0.822973 + 0.568081i \(0.807686\pi\)
\(602\) 0 0
\(603\) 12.6870 0.516653
\(604\) 0 0
\(605\) −12.1247 −0.492941
\(606\) 0 0
\(607\) 33.3944 1.35544 0.677719 0.735321i \(-0.262969\pi\)
0.677719 + 0.735321i \(0.262969\pi\)
\(608\) 0 0
\(609\) 28.8804 1.17029
\(610\) 0 0
\(611\) 7.60843 0.307804
\(612\) 0 0
\(613\) 29.2497 1.18138 0.590692 0.806897i \(-0.298855\pi\)
0.590692 + 0.806897i \(0.298855\pi\)
\(614\) 0 0
\(615\) 13.3638 0.538880
\(616\) 0 0
\(617\) 14.9818 0.603144 0.301572 0.953443i \(-0.402489\pi\)
0.301572 + 0.953443i \(0.402489\pi\)
\(618\) 0 0
\(619\) 37.7238 1.51625 0.758124 0.652111i \(-0.226116\pi\)
0.758124 + 0.652111i \(0.226116\pi\)
\(620\) 0 0
\(621\) 2.41520 0.0969186
\(622\) 0 0
\(623\) −47.6687 −1.90981
\(624\) 0 0
\(625\) −28.9640 −1.15856
\(626\) 0 0
\(627\) −15.3134 −0.611560
\(628\) 0 0
\(629\) 15.5080 0.618343
\(630\) 0 0
\(631\) 41.0976 1.63607 0.818036 0.575167i \(-0.195063\pi\)
0.818036 + 0.575167i \(0.195063\pi\)
\(632\) 0 0
\(633\) −10.5456 −0.419149
\(634\) 0 0
\(635\) −18.2342 −0.723601
\(636\) 0 0
\(637\) 13.3412 0.528597
\(638\) 0 0
\(639\) 16.3594 0.647168
\(640\) 0 0
\(641\) 41.4502 1.63718 0.818592 0.574376i \(-0.194755\pi\)
0.818592 + 0.574376i \(0.194755\pi\)
\(642\) 0 0
\(643\) −19.7449 −0.778661 −0.389331 0.921098i \(-0.627294\pi\)
−0.389331 + 0.921098i \(0.627294\pi\)
\(644\) 0 0
\(645\) 29.3934 1.15736
\(646\) 0 0
\(647\) 32.0948 1.26178 0.630889 0.775873i \(-0.282690\pi\)
0.630889 + 0.775873i \(0.282690\pi\)
\(648\) 0 0
\(649\) −28.0272 −1.10016
\(650\) 0 0
\(651\) 0.516574 0.0202461
\(652\) 0 0
\(653\) 24.0025 0.939288 0.469644 0.882856i \(-0.344382\pi\)
0.469644 + 0.882856i \(0.344382\pi\)
\(654\) 0 0
\(655\) 39.1704 1.53052
\(656\) 0 0
\(657\) 6.86626 0.267878
\(658\) 0 0
\(659\) 32.3326 1.25950 0.629751 0.776797i \(-0.283157\pi\)
0.629751 + 0.776797i \(0.283157\pi\)
\(660\) 0 0
\(661\) 12.0304 0.467928 0.233964 0.972245i \(-0.424830\pi\)
0.233964 + 0.972245i \(0.424830\pi\)
\(662\) 0 0
\(663\) 6.56866 0.255106
\(664\) 0 0
\(665\) 50.1084 1.94312
\(666\) 0 0
\(667\) −21.2157 −0.821474
\(668\) 0 0
\(669\) 3.05644 0.118169
\(670\) 0 0
\(671\) −12.9744 −0.500872
\(672\) 0 0
\(673\) −40.9180 −1.57727 −0.788637 0.614859i \(-0.789213\pi\)
−0.788637 + 0.614859i \(0.789213\pi\)
\(674\) 0 0
\(675\) 0.988047 0.0380299
\(676\) 0 0
\(677\) 42.2119 1.62234 0.811168 0.584814i \(-0.198832\pi\)
0.811168 + 0.584814i \(0.198832\pi\)
\(678\) 0 0
\(679\) −43.6050 −1.67341
\(680\) 0 0
\(681\) 17.4491 0.668651
\(682\) 0 0
\(683\) 45.8188 1.75321 0.876603 0.481213i \(-0.159804\pi\)
0.876603 + 0.481213i \(0.159804\pi\)
\(684\) 0 0
\(685\) 25.8494 0.987655
\(686\) 0 0
\(687\) 12.0819 0.460952
\(688\) 0 0
\(689\) 36.7401 1.39969
\(690\) 0 0
\(691\) −15.9383 −0.606320 −0.303160 0.952940i \(-0.598042\pi\)
−0.303160 + 0.952940i \(0.598042\pi\)
\(692\) 0 0
\(693\) −8.08356 −0.307069
\(694\) 0 0
\(695\) −32.2588 −1.22365
\(696\) 0 0
\(697\) 10.2428 0.387972
\(698\) 0 0
\(699\) 12.3600 0.467500
\(700\) 0 0
\(701\) −19.2437 −0.726823 −0.363412 0.931629i \(-0.618388\pi\)
−0.363412 + 0.931629i \(0.618388\pi\)
\(702\) 0 0
\(703\) −51.4984 −1.94230
\(704\) 0 0
\(705\) 5.31606 0.200215
\(706\) 0 0
\(707\) 39.4526 1.48377
\(708\) 0 0
\(709\) −15.5594 −0.584346 −0.292173 0.956366i \(-0.594378\pi\)
−0.292173 + 0.956366i \(0.594378\pi\)
\(710\) 0 0
\(711\) 7.84100 0.294060
\(712\) 0 0
\(713\) −0.379478 −0.0142116
\(714\) 0 0
\(715\) 21.0714 0.788026
\(716\) 0 0
\(717\) 14.2718 0.532992
\(718\) 0 0
\(719\) −13.3791 −0.498955 −0.249478 0.968381i \(-0.580259\pi\)
−0.249478 + 0.968381i \(0.580259\pi\)
\(720\) 0 0
\(721\) 25.2295 0.939595
\(722\) 0 0
\(723\) −4.58419 −0.170488
\(724\) 0 0
\(725\) −8.67923 −0.322338
\(726\) 0 0
\(727\) 9.35115 0.346815 0.173408 0.984850i \(-0.444522\pi\)
0.173408 + 0.984850i \(0.444522\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 22.5287 0.833254
\(732\) 0 0
\(733\) −32.1734 −1.18835 −0.594176 0.804335i \(-0.702522\pi\)
−0.594176 + 0.804335i \(0.702522\pi\)
\(734\) 0 0
\(735\) 9.32158 0.343832
\(736\) 0 0
\(737\) 31.1933 1.14902
\(738\) 0 0
\(739\) −37.1273 −1.36575 −0.682874 0.730536i \(-0.739270\pi\)
−0.682874 + 0.730536i \(0.739270\pi\)
\(740\) 0 0
\(741\) −21.8130 −0.801322
\(742\) 0 0
\(743\) 18.7787 0.688923 0.344462 0.938800i \(-0.388061\pi\)
0.344462 + 0.938800i \(0.388061\pi\)
\(744\) 0 0
\(745\) −9.67057 −0.354302
\(746\) 0 0
\(747\) −11.3738 −0.416147
\(748\) 0 0
\(749\) 44.0602 1.60993
\(750\) 0 0
\(751\) −24.2143 −0.883591 −0.441796 0.897116i \(-0.645658\pi\)
−0.441796 + 0.897116i \(0.645658\pi\)
\(752\) 0 0
\(753\) 16.0666 0.585498
\(754\) 0 0
\(755\) −4.15632 −0.151264
\(756\) 0 0
\(757\) −15.7002 −0.570632 −0.285316 0.958433i \(-0.592099\pi\)
−0.285316 + 0.958433i \(0.592099\pi\)
\(758\) 0 0
\(759\) 5.93823 0.215544
\(760\) 0 0
\(761\) −28.4105 −1.02988 −0.514940 0.857226i \(-0.672186\pi\)
−0.514940 + 0.857226i \(0.672186\pi\)
\(762\) 0 0
\(763\) −18.1808 −0.658189
\(764\) 0 0
\(765\) 4.58957 0.165936
\(766\) 0 0
\(767\) −39.9230 −1.44154
\(768\) 0 0
\(769\) −0.929133 −0.0335054 −0.0167527 0.999860i \(-0.505333\pi\)
−0.0167527 + 0.999860i \(0.505333\pi\)
\(770\) 0 0
\(771\) 13.3839 0.482009
\(772\) 0 0
\(773\) 26.2797 0.945216 0.472608 0.881273i \(-0.343313\pi\)
0.472608 + 0.881273i \(0.343313\pi\)
\(774\) 0 0
\(775\) −0.155243 −0.00557648
\(776\) 0 0
\(777\) −27.1847 −0.975245
\(778\) 0 0
\(779\) −34.0139 −1.21867
\(780\) 0 0
\(781\) 40.2227 1.43928
\(782\) 0 0
\(783\) −8.78423 −0.313923
\(784\) 0 0
\(785\) 42.0101 1.49941
\(786\) 0 0
\(787\) −2.37433 −0.0846358 −0.0423179 0.999104i \(-0.513474\pi\)
−0.0423179 + 0.999104i \(0.513474\pi\)
\(788\) 0 0
\(789\) 10.4612 0.372430
\(790\) 0 0
\(791\) 8.89439 0.316248
\(792\) 0 0
\(793\) −18.4813 −0.656289
\(794\) 0 0
\(795\) 25.6706 0.910443
\(796\) 0 0
\(797\) −10.3670 −0.367219 −0.183610 0.982999i \(-0.558778\pi\)
−0.183610 + 0.982999i \(0.558778\pi\)
\(798\) 0 0
\(799\) 4.07453 0.144146
\(800\) 0 0
\(801\) 14.4989 0.512293
\(802\) 0 0
\(803\) 16.8820 0.595753
\(804\) 0 0
\(805\) −19.4310 −0.684852
\(806\) 0 0
\(807\) −5.65825 −0.199180
\(808\) 0 0
\(809\) 20.8730 0.733855 0.366928 0.930249i \(-0.380410\pi\)
0.366928 + 0.930249i \(0.380410\pi\)
\(810\) 0 0
\(811\) 46.9934 1.65016 0.825081 0.565014i \(-0.191129\pi\)
0.825081 + 0.565014i \(0.191129\pi\)
\(812\) 0 0
\(813\) −14.9392 −0.523942
\(814\) 0 0
\(815\) −35.2893 −1.23613
\(816\) 0 0
\(817\) −74.8127 −2.61736
\(818\) 0 0
\(819\) −11.5145 −0.402350
\(820\) 0 0
\(821\) 38.5702 1.34611 0.673055 0.739592i \(-0.264982\pi\)
0.673055 + 0.739592i \(0.264982\pi\)
\(822\) 0 0
\(823\) −25.8794 −0.902100 −0.451050 0.892499i \(-0.648950\pi\)
−0.451050 + 0.892499i \(0.648950\pi\)
\(824\) 0 0
\(825\) 2.42930 0.0845774
\(826\) 0 0
\(827\) 17.8780 0.621679 0.310839 0.950462i \(-0.399390\pi\)
0.310839 + 0.950462i \(0.399390\pi\)
\(828\) 0 0
\(829\) −32.0345 −1.11261 −0.556303 0.830980i \(-0.687781\pi\)
−0.556303 + 0.830980i \(0.687781\pi\)
\(830\) 0 0
\(831\) −18.1887 −0.630960
\(832\) 0 0
\(833\) 7.14457 0.247545
\(834\) 0 0
\(835\) −2.33227 −0.0807117
\(836\) 0 0
\(837\) −0.157121 −0.00543089
\(838\) 0 0
\(839\) 17.0182 0.587532 0.293766 0.955877i \(-0.405091\pi\)
0.293766 + 0.955877i \(0.405091\pi\)
\(840\) 0 0
\(841\) 48.1627 1.66078
\(842\) 0 0
\(843\) 14.8090 0.510049
\(844\) 0 0
\(845\) −1.79671 −0.0618088
\(846\) 0 0
\(847\) 16.2903 0.559741
\(848\) 0 0
\(849\) 0.132035 0.00453143
\(850\) 0 0
\(851\) 19.9700 0.684563
\(852\) 0 0
\(853\) −43.3715 −1.48501 −0.742506 0.669840i \(-0.766363\pi\)
−0.742506 + 0.669840i \(0.766363\pi\)
\(854\) 0 0
\(855\) −15.2409 −0.521229
\(856\) 0 0
\(857\) 24.8892 0.850198 0.425099 0.905147i \(-0.360239\pi\)
0.425099 + 0.905147i \(0.360239\pi\)
\(858\) 0 0
\(859\) −7.04514 −0.240377 −0.120188 0.992751i \(-0.538350\pi\)
−0.120188 + 0.992751i \(0.538350\pi\)
\(860\) 0 0
\(861\) −17.9550 −0.611906
\(862\) 0 0
\(863\) −44.8097 −1.52534 −0.762669 0.646789i \(-0.776112\pi\)
−0.762669 + 0.646789i \(0.776112\pi\)
\(864\) 0 0
\(865\) −1.68815 −0.0573988
\(866\) 0 0
\(867\) −13.4823 −0.457883
\(868\) 0 0
\(869\) 19.2786 0.653981
\(870\) 0 0
\(871\) 44.4330 1.50555
\(872\) 0 0
\(873\) 13.2629 0.448880
\(874\) 0 0
\(875\) 32.2773 1.09117
\(876\) 0 0
\(877\) 48.4494 1.63602 0.818009 0.575205i \(-0.195078\pi\)
0.818009 + 0.575205i \(0.195078\pi\)
\(878\) 0 0
\(879\) 22.4272 0.756449
\(880\) 0 0
\(881\) −10.2191 −0.344291 −0.172145 0.985072i \(-0.555070\pi\)
−0.172145 + 0.985072i \(0.555070\pi\)
\(882\) 0 0
\(883\) −21.4398 −0.721508 −0.360754 0.932661i \(-0.617481\pi\)
−0.360754 + 0.932661i \(0.617481\pi\)
\(884\) 0 0
\(885\) −27.8945 −0.937663
\(886\) 0 0
\(887\) 31.2685 1.04989 0.524946 0.851135i \(-0.324085\pi\)
0.524946 + 0.851135i \(0.324085\pi\)
\(888\) 0 0
\(889\) 24.4987 0.821659
\(890\) 0 0
\(891\) 2.45869 0.0823692
\(892\) 0 0
\(893\) −13.5306 −0.452784
\(894\) 0 0
\(895\) −56.6811 −1.89464
\(896\) 0 0
\(897\) 8.45864 0.282426
\(898\) 0 0
\(899\) 1.38019 0.0460318
\(900\) 0 0
\(901\) 19.6754 0.655482
\(902\) 0 0
\(903\) −39.4917 −1.31420
\(904\) 0 0
\(905\) 50.9009 1.69200
\(906\) 0 0
\(907\) −48.9054 −1.62388 −0.811939 0.583742i \(-0.801588\pi\)
−0.811939 + 0.583742i \(0.801588\pi\)
\(908\) 0 0
\(909\) −11.9999 −0.398011
\(910\) 0 0
\(911\) 15.9314 0.527831 0.263916 0.964546i \(-0.414986\pi\)
0.263916 + 0.964546i \(0.414986\pi\)
\(912\) 0 0
\(913\) −27.9647 −0.925497
\(914\) 0 0
\(915\) −12.9130 −0.426891
\(916\) 0 0
\(917\) −52.6278 −1.73792
\(918\) 0 0
\(919\) 15.4425 0.509402 0.254701 0.967020i \(-0.418023\pi\)
0.254701 + 0.967020i \(0.418023\pi\)
\(920\) 0 0
\(921\) 17.8792 0.589140
\(922\) 0 0
\(923\) 57.2948 1.88588
\(924\) 0 0
\(925\) 8.16963 0.268616
\(926\) 0 0
\(927\) −7.67378 −0.252040
\(928\) 0 0
\(929\) −50.4489 −1.65517 −0.827587 0.561337i \(-0.810287\pi\)
−0.827587 + 0.561337i \(0.810287\pi\)
\(930\) 0 0
\(931\) −23.7255 −0.777572
\(932\) 0 0
\(933\) −4.18402 −0.136979
\(934\) 0 0
\(935\) 11.2843 0.369037
\(936\) 0 0
\(937\) −48.0069 −1.56832 −0.784158 0.620561i \(-0.786905\pi\)
−0.784158 + 0.620561i \(0.786905\pi\)
\(938\) 0 0
\(939\) 26.1802 0.854358
\(940\) 0 0
\(941\) −35.4091 −1.15430 −0.577152 0.816636i \(-0.695836\pi\)
−0.577152 + 0.816636i \(0.695836\pi\)
\(942\) 0 0
\(943\) 13.1899 0.429521
\(944\) 0 0
\(945\) −8.04529 −0.261713
\(946\) 0 0
\(947\) −41.7519 −1.35676 −0.678378 0.734713i \(-0.737317\pi\)
−0.678378 + 0.734713i \(0.737317\pi\)
\(948\) 0 0
\(949\) 24.0474 0.780611
\(950\) 0 0
\(951\) 21.8635 0.708973
\(952\) 0 0
\(953\) −22.6918 −0.735061 −0.367530 0.930012i \(-0.619797\pi\)
−0.367530 + 0.930012i \(0.619797\pi\)
\(954\) 0 0
\(955\) 2.89648 0.0937277
\(956\) 0 0
\(957\) −21.5977 −0.698154
\(958\) 0 0
\(959\) −34.7302 −1.12150
\(960\) 0 0
\(961\) −30.9753 −0.999204
\(962\) 0 0
\(963\) −13.4013 −0.431852
\(964\) 0 0
\(965\) 60.4373 1.94555
\(966\) 0 0
\(967\) −49.3746 −1.58778 −0.793891 0.608060i \(-0.791948\pi\)
−0.793891 + 0.608060i \(0.791948\pi\)
\(968\) 0 0
\(969\) −11.6815 −0.375264
\(970\) 0 0
\(971\) −26.4415 −0.848547 −0.424273 0.905534i \(-0.639470\pi\)
−0.424273 + 0.905534i \(0.639470\pi\)
\(972\) 0 0
\(973\) 43.3416 1.38947
\(974\) 0 0
\(975\) 3.46039 0.110821
\(976\) 0 0
\(977\) −46.0193 −1.47229 −0.736144 0.676825i \(-0.763356\pi\)
−0.736144 + 0.676825i \(0.763356\pi\)
\(978\) 0 0
\(979\) 35.6483 1.13932
\(980\) 0 0
\(981\) 5.52986 0.176555
\(982\) 0 0
\(983\) 12.5448 0.400118 0.200059 0.979784i \(-0.435887\pi\)
0.200059 + 0.979784i \(0.435887\pi\)
\(984\) 0 0
\(985\) −56.0992 −1.78747
\(986\) 0 0
\(987\) −7.14244 −0.227346
\(988\) 0 0
\(989\) 29.0108 0.922489
\(990\) 0 0
\(991\) −61.3754 −1.94965 −0.974827 0.222963i \(-0.928427\pi\)
−0.974827 + 0.222963i \(0.928427\pi\)
\(992\) 0 0
\(993\) −24.3232 −0.771874
\(994\) 0 0
\(995\) 13.1921 0.418217
\(996\) 0 0
\(997\) 18.2665 0.578504 0.289252 0.957253i \(-0.406593\pi\)
0.289252 + 0.957253i \(0.406593\pi\)
\(998\) 0 0
\(999\) 8.26847 0.261603
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 6036.2.a.h.1.17 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6036.2.a.h.1.17 24 1.1 even 1 trivial