Properties

Label 3040.2.a.s.1.3
Level $3040$
Weight $2$
Character 3040.1
Self dual yes
Analytic conductor $24.275$
Analytic rank $1$
Dimension $4$
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] = [3040,2,Mod(1,3040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, 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("3040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.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: \(24.2745222145\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.78292.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 10x^{2} + 8x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.19719\) of defining polynomial
Character \(\chi\) \(=\) 3040.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.19719 q^{3} -1.00000 q^{5} -0.827631 q^{7} +1.82763 q^{9} +O(q^{10})\) \(q+2.19719 q^{3} -1.00000 q^{5} -0.827631 q^{7} +1.82763 q^{9} +1.74828 q^{11} -3.94547 q^{13} -2.19719 q^{15} -4.92065 q^{17} +1.00000 q^{19} -1.81846 q^{21} -4.82763 q^{23} +1.00000 q^{25} -2.57591 q^{27} -6.97029 q^{29} +5.49657 q^{31} +3.84131 q^{33} +0.827631 q^{35} +0.975182 q^{37} -8.66894 q^{39} +5.15183 q^{41} -6.48740 q^{43} -1.82763 q^{45} -10.0930 q^{47} -6.31503 q^{49} -10.8116 q^{51} -10.3398 q^{53} -1.74828 q^{55} +2.19719 q^{57} +10.4669 q^{59} +1.74828 q^{61} -1.51260 q^{63} +3.94547 q^{65} +14.0385 q^{67} -10.6072 q^{69} +0.344739 q^{71} -12.8116 q^{73} +2.19719 q^{75} -1.44693 q^{77} +4.04964 q^{79} -11.1427 q^{81} -10.4874 q^{83} +4.92065 q^{85} -15.3150 q^{87} +16.2357 q^{89} +3.26539 q^{91} +12.0770 q^{93} -1.00000 q^{95} +15.9996 q^{97} +3.19522 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} - 4 q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{3} - 4 q^{5} - 5 q^{7} + 9 q^{9} - 6 q^{11} + 5 q^{13} - q^{15} - 5 q^{17} + 4 q^{19} - 3 q^{21} - 21 q^{23} + 4 q^{25} + q^{27} - q^{29} - 4 q^{31} - 14 q^{33} + 5 q^{35} + 10 q^{37} - 7 q^{39} - 2 q^{41} + 6 q^{43} - 9 q^{45} - 24 q^{47} + 5 q^{49} + 13 q^{51} - 5 q^{53} + 6 q^{55} + q^{57} - 11 q^{59} - 6 q^{61} - 38 q^{63} - 5 q^{65} + 19 q^{67} - 7 q^{69} - 2 q^{71} + 5 q^{73} + q^{75} + 8 q^{77} + 4 q^{79} - 16 q^{81} - 10 q^{83} + 5 q^{85} - 31 q^{87} + 20 q^{89} - 5 q^{91} - 26 q^{93} - 4 q^{95} - 6 q^{97} - 14 q^{99}+O(q^{100}) \) 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
\(3\) 2.19719 1.26855 0.634273 0.773109i \(-0.281299\pi\)
0.634273 + 0.773109i \(0.281299\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.827631 −0.312815 −0.156407 0.987693i \(-0.549991\pi\)
−0.156407 + 0.987693i \(0.549991\pi\)
\(8\) 0 0
\(9\) 1.82763 0.609210
\(10\) 0 0
\(11\) 1.74828 0.527128 0.263564 0.964642i \(-0.415102\pi\)
0.263564 + 0.964642i \(0.415102\pi\)
\(12\) 0 0
\(13\) −3.94547 −1.09428 −0.547138 0.837042i \(-0.684283\pi\)
−0.547138 + 0.837042i \(0.684283\pi\)
\(14\) 0 0
\(15\) −2.19719 −0.567311
\(16\) 0 0
\(17\) −4.92065 −1.19343 −0.596717 0.802452i \(-0.703528\pi\)
−0.596717 + 0.802452i \(0.703528\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.81846 −0.396820
\(22\) 0 0
\(23\) −4.82763 −1.00663 −0.503315 0.864103i \(-0.667887\pi\)
−0.503315 + 0.864103i \(0.667887\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −2.57591 −0.495735
\(28\) 0 0
\(29\) −6.97029 −1.29435 −0.647175 0.762341i \(-0.724050\pi\)
−0.647175 + 0.762341i \(0.724050\pi\)
\(30\) 0 0
\(31\) 5.49657 0.987213 0.493606 0.869685i \(-0.335678\pi\)
0.493606 + 0.869685i \(0.335678\pi\)
\(32\) 0 0
\(33\) 3.84131 0.668686
\(34\) 0 0
\(35\) 0.827631 0.139895
\(36\) 0 0
\(37\) 0.975182 0.160319 0.0801595 0.996782i \(-0.474457\pi\)
0.0801595 + 0.996782i \(0.474457\pi\)
\(38\) 0 0
\(39\) −8.66894 −1.38814
\(40\) 0 0
\(41\) 5.15183 0.804581 0.402290 0.915512i \(-0.368214\pi\)
0.402290 + 0.915512i \(0.368214\pi\)
\(42\) 0 0
\(43\) −6.48740 −0.989319 −0.494659 0.869087i \(-0.664707\pi\)
−0.494659 + 0.869087i \(0.664707\pi\)
\(44\) 0 0
\(45\) −1.82763 −0.272447
\(46\) 0 0
\(47\) −10.0930 −1.47222 −0.736109 0.676863i \(-0.763339\pi\)
−0.736109 + 0.676863i \(0.763339\pi\)
\(48\) 0 0
\(49\) −6.31503 −0.902147
\(50\) 0 0
\(51\) −10.8116 −1.51393
\(52\) 0 0
\(53\) −10.3398 −1.42029 −0.710143 0.704057i \(-0.751370\pi\)
−0.710143 + 0.704057i \(0.751370\pi\)
\(54\) 0 0
\(55\) −1.74828 −0.235739
\(56\) 0 0
\(57\) 2.19719 0.291025
\(58\) 0 0
\(59\) 10.4669 1.36267 0.681334 0.731972i \(-0.261400\pi\)
0.681334 + 0.731972i \(0.261400\pi\)
\(60\) 0 0
\(61\) 1.74828 0.223845 0.111922 0.993717i \(-0.464299\pi\)
0.111922 + 0.993717i \(0.464299\pi\)
\(62\) 0 0
\(63\) −1.51260 −0.190570
\(64\) 0 0
\(65\) 3.94547 0.489375
\(66\) 0 0
\(67\) 14.0385 1.71508 0.857538 0.514421i \(-0.171993\pi\)
0.857538 + 0.514421i \(0.171993\pi\)
\(68\) 0 0
\(69\) −10.6072 −1.27696
\(70\) 0 0
\(71\) 0.344739 0.0409130 0.0204565 0.999791i \(-0.493488\pi\)
0.0204565 + 0.999791i \(0.493488\pi\)
\(72\) 0 0
\(73\) −12.8116 −1.49948 −0.749742 0.661730i \(-0.769822\pi\)
−0.749742 + 0.661730i \(0.769822\pi\)
\(74\) 0 0
\(75\) 2.19719 0.253709
\(76\) 0 0
\(77\) −1.44693 −0.164893
\(78\) 0 0
\(79\) 4.04964 0.455620 0.227810 0.973706i \(-0.426844\pi\)
0.227810 + 0.973706i \(0.426844\pi\)
\(80\) 0 0
\(81\) −11.1427 −1.23807
\(82\) 0 0
\(83\) −10.4874 −1.15114 −0.575571 0.817752i \(-0.695220\pi\)
−0.575571 + 0.817752i \(0.695220\pi\)
\(84\) 0 0
\(85\) 4.92065 0.533720
\(86\) 0 0
\(87\) −15.3150 −1.64194
\(88\) 0 0
\(89\) 16.2357 1.72098 0.860489 0.509468i \(-0.170158\pi\)
0.860489 + 0.509468i \(0.170158\pi\)
\(90\) 0 0
\(91\) 3.26539 0.342306
\(92\) 0 0
\(93\) 12.0770 1.25233
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 15.9996 1.62451 0.812257 0.583299i \(-0.198239\pi\)
0.812257 + 0.583299i \(0.198239\pi\)
\(98\) 0 0
\(99\) 3.19522 0.321131
\(100\) 0 0
\(101\) 8.48740 0.844528 0.422264 0.906473i \(-0.361236\pi\)
0.422264 + 0.906473i \(0.361236\pi\)
\(102\) 0 0
\(103\) −8.37873 −0.825581 −0.412790 0.910826i \(-0.635446\pi\)
−0.412790 + 0.910826i \(0.635446\pi\)
\(104\) 0 0
\(105\) 1.81846 0.177463
\(106\) 0 0
\(107\) −14.0385 −1.35715 −0.678576 0.734530i \(-0.737403\pi\)
−0.678576 + 0.734530i \(0.737403\pi\)
\(108\) 0 0
\(109\) −5.47372 −0.524287 −0.262144 0.965029i \(-0.584429\pi\)
−0.262144 + 0.965029i \(0.584429\pi\)
\(110\) 0 0
\(111\) 2.14266 0.203372
\(112\) 0 0
\(113\) −7.07445 −0.665509 −0.332754 0.943014i \(-0.607978\pi\)
−0.332754 + 0.943014i \(0.607978\pi\)
\(114\) 0 0
\(115\) 4.82763 0.450179
\(116\) 0 0
\(117\) −7.21086 −0.666645
\(118\) 0 0
\(119\) 4.07248 0.373324
\(120\) 0 0
\(121\) −7.94350 −0.722137
\(122\) 0 0
\(123\) 11.3195 1.02065
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −12.3604 −1.09681 −0.548403 0.836214i \(-0.684764\pi\)
−0.548403 + 0.836214i \(0.684764\pi\)
\(128\) 0 0
\(129\) −14.2540 −1.25500
\(130\) 0 0
\(131\) −4.75746 −0.415661 −0.207830 0.978165i \(-0.566640\pi\)
−0.207830 + 0.978165i \(0.566640\pi\)
\(132\) 0 0
\(133\) −0.827631 −0.0717647
\(134\) 0 0
\(135\) 2.57591 0.221699
\(136\) 0 0
\(137\) 15.3647 1.31269 0.656346 0.754460i \(-0.272101\pi\)
0.656346 + 0.754460i \(0.272101\pi\)
\(138\) 0 0
\(139\) −10.4378 −0.885319 −0.442660 0.896690i \(-0.645965\pi\)
−0.442660 + 0.896690i \(0.645965\pi\)
\(140\) 0 0
\(141\) −22.1763 −1.86758
\(142\) 0 0
\(143\) −6.89781 −0.576823
\(144\) 0 0
\(145\) 6.97029 0.578851
\(146\) 0 0
\(147\) −13.8753 −1.14442
\(148\) 0 0
\(149\) −11.7299 −0.960954 −0.480477 0.877007i \(-0.659536\pi\)
−0.480477 + 0.877007i \(0.659536\pi\)
\(150\) 0 0
\(151\) −8.58042 −0.698265 −0.349132 0.937073i \(-0.613524\pi\)
−0.349132 + 0.937073i \(0.613524\pi\)
\(152\) 0 0
\(153\) −8.99314 −0.727052
\(154\) 0 0
\(155\) −5.49657 −0.441495
\(156\) 0 0
\(157\) 19.5775 1.56245 0.781227 0.624247i \(-0.214594\pi\)
0.781227 + 0.624247i \(0.214594\pi\)
\(158\) 0 0
\(159\) −22.7186 −1.80170
\(160\) 0 0
\(161\) 3.99549 0.314889
\(162\) 0 0
\(163\) 1.00917 0.0790444 0.0395222 0.999219i \(-0.487416\pi\)
0.0395222 + 0.999219i \(0.487416\pi\)
\(164\) 0 0
\(165\) −3.84131 −0.299045
\(166\) 0 0
\(167\) 2.66404 0.206150 0.103075 0.994674i \(-0.467132\pi\)
0.103075 + 0.994674i \(0.467132\pi\)
\(168\) 0 0
\(169\) 2.56674 0.197442
\(170\) 0 0
\(171\) 1.82763 0.139762
\(172\) 0 0
\(173\) 14.9654 1.13780 0.568899 0.822407i \(-0.307370\pi\)
0.568899 + 0.822407i \(0.307370\pi\)
\(174\) 0 0
\(175\) −0.827631 −0.0625630
\(176\) 0 0
\(177\) 22.9976 1.72861
\(178\) 0 0
\(179\) −9.10219 −0.680330 −0.340165 0.940366i \(-0.610483\pi\)
−0.340165 + 0.940366i \(0.610483\pi\)
\(180\) 0 0
\(181\) 7.47823 0.555852 0.277926 0.960602i \(-0.410353\pi\)
0.277926 + 0.960602i \(0.410353\pi\)
\(182\) 0 0
\(183\) 3.84131 0.283958
\(184\) 0 0
\(185\) −0.975182 −0.0716968
\(186\) 0 0
\(187\) −8.60270 −0.629092
\(188\) 0 0
\(189\) 2.13191 0.155073
\(190\) 0 0
\(191\) 7.70940 0.557833 0.278916 0.960315i \(-0.410025\pi\)
0.278916 + 0.960315i \(0.410025\pi\)
\(192\) 0 0
\(193\) −2.26736 −0.163208 −0.0816041 0.996665i \(-0.526004\pi\)
−0.0816041 + 0.996665i \(0.526004\pi\)
\(194\) 0 0
\(195\) 8.66894 0.620796
\(196\) 0 0
\(197\) 11.3602 0.809378 0.404689 0.914454i \(-0.367380\pi\)
0.404689 + 0.914454i \(0.367380\pi\)
\(198\) 0 0
\(199\) 0.113563 0.00805026 0.00402513 0.999992i \(-0.498719\pi\)
0.00402513 + 0.999992i \(0.498719\pi\)
\(200\) 0 0
\(201\) 30.8452 2.17565
\(202\) 0 0
\(203\) 5.76882 0.404892
\(204\) 0 0
\(205\) −5.15183 −0.359819
\(206\) 0 0
\(207\) −8.82313 −0.613250
\(208\) 0 0
\(209\) 1.74828 0.120931
\(210\) 0 0
\(211\) 16.1038 1.10863 0.554315 0.832307i \(-0.312980\pi\)
0.554315 + 0.832307i \(0.312980\pi\)
\(212\) 0 0
\(213\) 0.757456 0.0519000
\(214\) 0 0
\(215\) 6.48740 0.442437
\(216\) 0 0
\(217\) −4.54913 −0.308815
\(218\) 0 0
\(219\) −28.1495 −1.90217
\(220\) 0 0
\(221\) 19.4143 1.30595
\(222\) 0 0
\(223\) −28.3963 −1.90156 −0.950778 0.309872i \(-0.899714\pi\)
−0.950778 + 0.309872i \(0.899714\pi\)
\(224\) 0 0
\(225\) 1.82763 0.121842
\(226\) 0 0
\(227\) −5.39010 −0.357753 −0.178877 0.983872i \(-0.557246\pi\)
−0.178877 + 0.983872i \(0.557246\pi\)
\(228\) 0 0
\(229\) −7.33557 −0.484748 −0.242374 0.970183i \(-0.577926\pi\)
−0.242374 + 0.970183i \(0.577926\pi\)
\(230\) 0 0
\(231\) −3.17918 −0.209175
\(232\) 0 0
\(233\) −3.26089 −0.213628 −0.106814 0.994279i \(-0.534065\pi\)
−0.106814 + 0.994279i \(0.534065\pi\)
\(234\) 0 0
\(235\) 10.0930 0.658396
\(236\) 0 0
\(237\) 8.89781 0.577975
\(238\) 0 0
\(239\) −27.0969 −1.75275 −0.876377 0.481626i \(-0.840046\pi\)
−0.876377 + 0.481626i \(0.840046\pi\)
\(240\) 0 0
\(241\) 6.70782 0.432089 0.216044 0.976384i \(-0.430684\pi\)
0.216044 + 0.976384i \(0.430684\pi\)
\(242\) 0 0
\(243\) −16.7548 −1.07482
\(244\) 0 0
\(245\) 6.31503 0.403452
\(246\) 0 0
\(247\) −3.94547 −0.251044
\(248\) 0 0
\(249\) −23.0428 −1.46028
\(250\) 0 0
\(251\) −17.0741 −1.07771 −0.538853 0.842400i \(-0.681142\pi\)
−0.538853 + 0.842400i \(0.681142\pi\)
\(252\) 0 0
\(253\) −8.44007 −0.530623
\(254\) 0 0
\(255\) 10.8116 0.677048
\(256\) 0 0
\(257\) −2.92555 −0.182491 −0.0912453 0.995828i \(-0.529085\pi\)
−0.0912453 + 0.995828i \(0.529085\pi\)
\(258\) 0 0
\(259\) −0.807091 −0.0501502
\(260\) 0 0
\(261\) −12.7391 −0.788531
\(262\) 0 0
\(263\) 14.1923 0.875134 0.437567 0.899186i \(-0.355840\pi\)
0.437567 + 0.899186i \(0.355840\pi\)
\(264\) 0 0
\(265\) 10.3398 0.635172
\(266\) 0 0
\(267\) 35.6728 2.18314
\(268\) 0 0
\(269\) 13.0839 0.797737 0.398868 0.917008i \(-0.369403\pi\)
0.398868 + 0.917008i \(0.369403\pi\)
\(270\) 0 0
\(271\) −3.51711 −0.213649 −0.106825 0.994278i \(-0.534068\pi\)
−0.106825 + 0.994278i \(0.534068\pi\)
\(272\) 0 0
\(273\) 7.17468 0.434231
\(274\) 0 0
\(275\) 1.74828 0.105426
\(276\) 0 0
\(277\) 0.630055 0.0378564 0.0189282 0.999821i \(-0.493975\pi\)
0.0189282 + 0.999821i \(0.493975\pi\)
\(278\) 0 0
\(279\) 10.0457 0.601420
\(280\) 0 0
\(281\) −2.15869 −0.128777 −0.0643884 0.997925i \(-0.520510\pi\)
−0.0643884 + 0.997925i \(0.520510\pi\)
\(282\) 0 0
\(283\) 26.5867 1.58041 0.790207 0.612840i \(-0.209973\pi\)
0.790207 + 0.612840i \(0.209973\pi\)
\(284\) 0 0
\(285\) −2.19719 −0.130150
\(286\) 0 0
\(287\) −4.26381 −0.251685
\(288\) 0 0
\(289\) 7.21283 0.424284
\(290\) 0 0
\(291\) 35.1541 2.06077
\(292\) 0 0
\(293\) 18.2719 1.06745 0.533727 0.845657i \(-0.320791\pi\)
0.533727 + 0.845657i \(0.320791\pi\)
\(294\) 0 0
\(295\) −10.4669 −0.609404
\(296\) 0 0
\(297\) −4.50343 −0.261316
\(298\) 0 0
\(299\) 19.0473 1.10153
\(300\) 0 0
\(301\) 5.36917 0.309474
\(302\) 0 0
\(303\) 18.6484 1.07132
\(304\) 0 0
\(305\) −1.74828 −0.100106
\(306\) 0 0
\(307\) 25.2667 1.44205 0.721025 0.692909i \(-0.243671\pi\)
0.721025 + 0.692909i \(0.243671\pi\)
\(308\) 0 0
\(309\) −18.4096 −1.04729
\(310\) 0 0
\(311\) −8.40513 −0.476611 −0.238305 0.971190i \(-0.576592\pi\)
−0.238305 + 0.971190i \(0.576592\pi\)
\(312\) 0 0
\(313\) 17.6781 0.999226 0.499613 0.866249i \(-0.333476\pi\)
0.499613 + 0.866249i \(0.333476\pi\)
\(314\) 0 0
\(315\) 1.51260 0.0852255
\(316\) 0 0
\(317\) 27.0692 1.52036 0.760178 0.649715i \(-0.225112\pi\)
0.760178 + 0.649715i \(0.225112\pi\)
\(318\) 0 0
\(319\) −12.1860 −0.682288
\(320\) 0 0
\(321\) −30.8452 −1.72161
\(322\) 0 0
\(323\) −4.92065 −0.273792
\(324\) 0 0
\(325\) −3.94547 −0.218855
\(326\) 0 0
\(327\) −12.0268 −0.665083
\(328\) 0 0
\(329\) 8.35329 0.460532
\(330\) 0 0
\(331\) 32.9703 1.81221 0.906105 0.423052i \(-0.139041\pi\)
0.906105 + 0.423052i \(0.139041\pi\)
\(332\) 0 0
\(333\) 1.78227 0.0976680
\(334\) 0 0
\(335\) −14.0385 −0.767005
\(336\) 0 0
\(337\) −31.7229 −1.72805 −0.864027 0.503446i \(-0.832065\pi\)
−0.864027 + 0.503446i \(0.832065\pi\)
\(338\) 0 0
\(339\) −15.5439 −0.844228
\(340\) 0 0
\(341\) 9.60956 0.520387
\(342\) 0 0
\(343\) 11.0199 0.595020
\(344\) 0 0
\(345\) 10.6072 0.571073
\(346\) 0 0
\(347\) −0.350986 −0.0188419 −0.00942095 0.999956i \(-0.502999\pi\)
−0.00942095 + 0.999956i \(0.502999\pi\)
\(348\) 0 0
\(349\) 11.3602 0.608095 0.304048 0.952657i \(-0.401662\pi\)
0.304048 + 0.952657i \(0.401662\pi\)
\(350\) 0 0
\(351\) 10.1632 0.542471
\(352\) 0 0
\(353\) −30.6431 −1.63097 −0.815484 0.578779i \(-0.803529\pi\)
−0.815484 + 0.578779i \(0.803529\pi\)
\(354\) 0 0
\(355\) −0.344739 −0.0182968
\(356\) 0 0
\(357\) 8.94801 0.473579
\(358\) 0 0
\(359\) 0.305856 0.0161425 0.00807124 0.999967i \(-0.497431\pi\)
0.00807124 + 0.999967i \(0.497431\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −17.4534 −0.916064
\(364\) 0 0
\(365\) 12.8116 0.670590
\(366\) 0 0
\(367\) −22.3470 −1.16651 −0.583253 0.812290i \(-0.698220\pi\)
−0.583253 + 0.812290i \(0.698220\pi\)
\(368\) 0 0
\(369\) 9.41564 0.490159
\(370\) 0 0
\(371\) 8.55757 0.444287
\(372\) 0 0
\(373\) 0.793642 0.0410932 0.0205466 0.999789i \(-0.493459\pi\)
0.0205466 + 0.999789i \(0.493459\pi\)
\(374\) 0 0
\(375\) −2.19719 −0.113462
\(376\) 0 0
\(377\) 27.5011 1.41638
\(378\) 0 0
\(379\) −1.03827 −0.0533322 −0.0266661 0.999644i \(-0.508489\pi\)
−0.0266661 + 0.999644i \(0.508489\pi\)
\(380\) 0 0
\(381\) −27.1581 −1.39135
\(382\) 0 0
\(383\) 16.9494 0.866072 0.433036 0.901377i \(-0.357442\pi\)
0.433036 + 0.901377i \(0.357442\pi\)
\(384\) 0 0
\(385\) 1.44693 0.0737426
\(386\) 0 0
\(387\) −11.8566 −0.602703
\(388\) 0 0
\(389\) −9.99021 −0.506524 −0.253262 0.967398i \(-0.581503\pi\)
−0.253262 + 0.967398i \(0.581503\pi\)
\(390\) 0 0
\(391\) 23.7551 1.20135
\(392\) 0 0
\(393\) −10.4530 −0.527285
\(394\) 0 0
\(395\) −4.04964 −0.203759
\(396\) 0 0
\(397\) 11.3692 0.570602 0.285301 0.958438i \(-0.407906\pi\)
0.285301 + 0.958438i \(0.407906\pi\)
\(398\) 0 0
\(399\) −1.81846 −0.0910368
\(400\) 0 0
\(401\) −38.0672 −1.90099 −0.950493 0.310747i \(-0.899421\pi\)
−0.950493 + 0.310747i \(0.899421\pi\)
\(402\) 0 0
\(403\) −21.6866 −1.08028
\(404\) 0 0
\(405\) 11.1427 0.553683
\(406\) 0 0
\(407\) 1.70490 0.0845086
\(408\) 0 0
\(409\) 15.2882 0.755955 0.377977 0.925815i \(-0.376620\pi\)
0.377977 + 0.925815i \(0.376620\pi\)
\(410\) 0 0
\(411\) 33.7590 1.66521
\(412\) 0 0
\(413\) −8.66269 −0.426263
\(414\) 0 0
\(415\) 10.4874 0.514806
\(416\) 0 0
\(417\) −22.9337 −1.12307
\(418\) 0 0
\(419\) −17.4059 −0.850332 −0.425166 0.905115i \(-0.639784\pi\)
−0.425166 + 0.905115i \(0.639784\pi\)
\(420\) 0 0
\(421\) 8.39888 0.409336 0.204668 0.978831i \(-0.434388\pi\)
0.204668 + 0.978831i \(0.434388\pi\)
\(422\) 0 0
\(423\) −18.4463 −0.896891
\(424\) 0 0
\(425\) −4.92065 −0.238687
\(426\) 0 0
\(427\) −1.44693 −0.0700220
\(428\) 0 0
\(429\) −15.1558 −0.731727
\(430\) 0 0
\(431\) −23.8132 −1.14704 −0.573520 0.819191i \(-0.694423\pi\)
−0.573520 + 0.819191i \(0.694423\pi\)
\(432\) 0 0
\(433\) −14.2490 −0.684764 −0.342382 0.939561i \(-0.611234\pi\)
−0.342382 + 0.939561i \(0.611234\pi\)
\(434\) 0 0
\(435\) 15.3150 0.734299
\(436\) 0 0
\(437\) −4.82763 −0.230937
\(438\) 0 0
\(439\) 36.8063 1.75667 0.878335 0.478046i \(-0.158655\pi\)
0.878335 + 0.478046i \(0.158655\pi\)
\(440\) 0 0
\(441\) −11.5415 −0.549597
\(442\) 0 0
\(443\) 32.6820 1.55277 0.776384 0.630260i \(-0.217052\pi\)
0.776384 + 0.630260i \(0.217052\pi\)
\(444\) 0 0
\(445\) −16.2357 −0.769645
\(446\) 0 0
\(447\) −25.7729 −1.21902
\(448\) 0 0
\(449\) −0.109058 −0.00514674 −0.00257337 0.999997i \(-0.500819\pi\)
−0.00257337 + 0.999997i \(0.500819\pi\)
\(450\) 0 0
\(451\) 9.00686 0.424117
\(452\) 0 0
\(453\) −18.8528 −0.885781
\(454\) 0 0
\(455\) −3.26539 −0.153084
\(456\) 0 0
\(457\) 14.7209 0.688614 0.344307 0.938857i \(-0.388114\pi\)
0.344307 + 0.938857i \(0.388114\pi\)
\(458\) 0 0
\(459\) 12.6752 0.591627
\(460\) 0 0
\(461\) −15.2968 −0.712443 −0.356221 0.934402i \(-0.615935\pi\)
−0.356221 + 0.934402i \(0.615935\pi\)
\(462\) 0 0
\(463\) −10.9810 −0.510332 −0.255166 0.966897i \(-0.582130\pi\)
−0.255166 + 0.966897i \(0.582130\pi\)
\(464\) 0 0
\(465\) −12.0770 −0.560057
\(466\) 0 0
\(467\) −31.3898 −1.45255 −0.726274 0.687406i \(-0.758750\pi\)
−0.726274 + 0.687406i \(0.758750\pi\)
\(468\) 0 0
\(469\) −11.6187 −0.536501
\(470\) 0 0
\(471\) 43.0154 1.98205
\(472\) 0 0
\(473\) −11.3418 −0.521497
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) −18.8974 −0.865253
\(478\) 0 0
\(479\) −2.75515 −0.125886 −0.0629429 0.998017i \(-0.520049\pi\)
−0.0629429 + 0.998017i \(0.520049\pi\)
\(480\) 0 0
\(481\) −3.84755 −0.175433
\(482\) 0 0
\(483\) 8.77885 0.399451
\(484\) 0 0
\(485\) −15.9996 −0.726505
\(486\) 0 0
\(487\) 21.0584 0.954248 0.477124 0.878836i \(-0.341679\pi\)
0.477124 + 0.878836i \(0.341679\pi\)
\(488\) 0 0
\(489\) 2.21734 0.100272
\(490\) 0 0
\(491\) −16.0215 −0.723040 −0.361520 0.932364i \(-0.617742\pi\)
−0.361520 + 0.932364i \(0.617742\pi\)
\(492\) 0 0
\(493\) 34.2984 1.54472
\(494\) 0 0
\(495\) −3.19522 −0.143614
\(496\) 0 0
\(497\) −0.285316 −0.0127982
\(498\) 0 0
\(499\) −20.1923 −0.903931 −0.451966 0.892035i \(-0.649277\pi\)
−0.451966 + 0.892035i \(0.649277\pi\)
\(500\) 0 0
\(501\) 5.85340 0.261511
\(502\) 0 0
\(503\) −40.3914 −1.80096 −0.900482 0.434894i \(-0.856786\pi\)
−0.900482 + 0.434894i \(0.856786\pi\)
\(504\) 0 0
\(505\) −8.48740 −0.377684
\(506\) 0 0
\(507\) 5.63962 0.250464
\(508\) 0 0
\(509\) −17.5645 −0.778535 −0.389267 0.921125i \(-0.627272\pi\)
−0.389267 + 0.921125i \(0.627272\pi\)
\(510\) 0 0
\(511\) 10.6033 0.469061
\(512\) 0 0
\(513\) −2.57591 −0.113729
\(514\) 0 0
\(515\) 8.37873 0.369211
\(516\) 0 0
\(517\) −17.6455 −0.776047
\(518\) 0 0
\(519\) 32.8818 1.44335
\(520\) 0 0
\(521\) 10.5124 0.460558 0.230279 0.973125i \(-0.426036\pi\)
0.230279 + 0.973125i \(0.426036\pi\)
\(522\) 0 0
\(523\) 4.69668 0.205371 0.102686 0.994714i \(-0.467256\pi\)
0.102686 + 0.994714i \(0.467256\pi\)
\(524\) 0 0
\(525\) −1.81846 −0.0793641
\(526\) 0 0
\(527\) −27.0467 −1.17817
\(528\) 0 0
\(529\) 0.306017 0.0133051
\(530\) 0 0
\(531\) 19.1295 0.830152
\(532\) 0 0
\(533\) −20.3264 −0.880434
\(534\) 0 0
\(535\) 14.0385 0.606937
\(536\) 0 0
\(537\) −19.9992 −0.863030
\(538\) 0 0
\(539\) −11.0405 −0.475546
\(540\) 0 0
\(541\) 35.0764 1.50805 0.754026 0.656845i \(-0.228109\pi\)
0.754026 + 0.656845i \(0.228109\pi\)
\(542\) 0 0
\(543\) 16.4311 0.705124
\(544\) 0 0
\(545\) 5.47372 0.234468
\(546\) 0 0
\(547\) 1.88508 0.0806004 0.0403002 0.999188i \(-0.487169\pi\)
0.0403002 + 0.999188i \(0.487169\pi\)
\(548\) 0 0
\(549\) 3.19522 0.136369
\(550\) 0 0
\(551\) −6.97029 −0.296944
\(552\) 0 0
\(553\) −3.35160 −0.142525
\(554\) 0 0
\(555\) −2.14266 −0.0909508
\(556\) 0 0
\(557\) 27.1518 1.15046 0.575230 0.817992i \(-0.304913\pi\)
0.575230 + 0.817992i \(0.304913\pi\)
\(558\) 0 0
\(559\) 25.5958 1.08259
\(560\) 0 0
\(561\) −18.9017 −0.798032
\(562\) 0 0
\(563\) −11.7800 −0.496466 −0.248233 0.968700i \(-0.579850\pi\)
−0.248233 + 0.968700i \(0.579850\pi\)
\(564\) 0 0
\(565\) 7.07445 0.297624
\(566\) 0 0
\(567\) 9.22200 0.387288
\(568\) 0 0
\(569\) −41.2288 −1.72840 −0.864201 0.503147i \(-0.832176\pi\)
−0.864201 + 0.503147i \(0.832176\pi\)
\(570\) 0 0
\(571\) −0.881771 −0.0369010 −0.0184505 0.999830i \(-0.505873\pi\)
−0.0184505 + 0.999830i \(0.505873\pi\)
\(572\) 0 0
\(573\) 16.9390 0.707637
\(574\) 0 0
\(575\) −4.82763 −0.201326
\(576\) 0 0
\(577\) −11.0571 −0.460312 −0.230156 0.973154i \(-0.573924\pi\)
−0.230156 + 0.973154i \(0.573924\pi\)
\(578\) 0 0
\(579\) −4.98182 −0.207037
\(580\) 0 0
\(581\) 8.67969 0.360094
\(582\) 0 0
\(583\) −18.0770 −0.748672
\(584\) 0 0
\(585\) 7.21086 0.298133
\(586\) 0 0
\(587\) 0.972486 0.0401388 0.0200694 0.999799i \(-0.493611\pi\)
0.0200694 + 0.999799i \(0.493611\pi\)
\(588\) 0 0
\(589\) 5.49657 0.226482
\(590\) 0 0
\(591\) 24.9604 1.02673
\(592\) 0 0
\(593\) −26.8880 −1.10416 −0.552079 0.833792i \(-0.686165\pi\)
−0.552079 + 0.833792i \(0.686165\pi\)
\(594\) 0 0
\(595\) −4.07248 −0.166956
\(596\) 0 0
\(597\) 0.249519 0.0102121
\(598\) 0 0
\(599\) 16.9252 0.691543 0.345772 0.938319i \(-0.387617\pi\)
0.345772 + 0.938319i \(0.387617\pi\)
\(600\) 0 0
\(601\) −13.9281 −0.568138 −0.284069 0.958804i \(-0.591685\pi\)
−0.284069 + 0.958804i \(0.591685\pi\)
\(602\) 0 0
\(603\) 25.6572 1.04484
\(604\) 0 0
\(605\) 7.94350 0.322949
\(606\) 0 0
\(607\) −33.4939 −1.35947 −0.679737 0.733456i \(-0.737906\pi\)
−0.679737 + 0.733456i \(0.737906\pi\)
\(608\) 0 0
\(609\) 12.6752 0.513624
\(610\) 0 0
\(611\) 39.8217 1.61101
\(612\) 0 0
\(613\) 41.8589 1.69066 0.845332 0.534241i \(-0.179403\pi\)
0.845332 + 0.534241i \(0.179403\pi\)
\(614\) 0 0
\(615\) −11.3195 −0.456448
\(616\) 0 0
\(617\) 8.13641 0.327560 0.163780 0.986497i \(-0.447631\pi\)
0.163780 + 0.986497i \(0.447631\pi\)
\(618\) 0 0
\(619\) 8.18328 0.328914 0.164457 0.986384i \(-0.447413\pi\)
0.164457 + 0.986384i \(0.447413\pi\)
\(620\) 0 0
\(621\) 12.4356 0.499022
\(622\) 0 0
\(623\) −13.4371 −0.538348
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 3.84131 0.153407
\(628\) 0 0
\(629\) −4.79854 −0.191330
\(630\) 0 0
\(631\) 43.6979 1.73958 0.869792 0.493418i \(-0.164253\pi\)
0.869792 + 0.493418i \(0.164253\pi\)
\(632\) 0 0
\(633\) 35.3830 1.40635
\(634\) 0 0
\(635\) 12.3604 0.490507
\(636\) 0 0
\(637\) 24.9158 0.987198
\(638\) 0 0
\(639\) 0.630055 0.0249246
\(640\) 0 0
\(641\) 25.3469 1.00114 0.500571 0.865696i \(-0.333123\pi\)
0.500571 + 0.865696i \(0.333123\pi\)
\(642\) 0 0
\(643\) 4.94119 0.194862 0.0974308 0.995242i \(-0.468938\pi\)
0.0974308 + 0.995242i \(0.468938\pi\)
\(644\) 0 0
\(645\) 14.2540 0.561252
\(646\) 0 0
\(647\) −29.9205 −1.17630 −0.588148 0.808753i \(-0.700143\pi\)
−0.588148 + 0.808753i \(0.700143\pi\)
\(648\) 0 0
\(649\) 18.2990 0.718300
\(650\) 0 0
\(651\) −9.99529 −0.391746
\(652\) 0 0
\(653\) 33.3875 1.30655 0.653277 0.757119i \(-0.273394\pi\)
0.653277 + 0.757119i \(0.273394\pi\)
\(654\) 0 0
\(655\) 4.75746 0.185889
\(656\) 0 0
\(657\) −23.4149 −0.913501
\(658\) 0 0
\(659\) −32.9519 −1.28363 −0.641813 0.766861i \(-0.721817\pi\)
−0.641813 + 0.766861i \(0.721817\pi\)
\(660\) 0 0
\(661\) −17.0884 −0.664660 −0.332330 0.943163i \(-0.607835\pi\)
−0.332330 + 0.943163i \(0.607835\pi\)
\(662\) 0 0
\(663\) 42.6568 1.65665
\(664\) 0 0
\(665\) 0.827631 0.0320941
\(666\) 0 0
\(667\) 33.6500 1.30293
\(668\) 0 0
\(669\) −62.3920 −2.41221
\(670\) 0 0
\(671\) 3.05650 0.117995
\(672\) 0 0
\(673\) 8.99353 0.346675 0.173337 0.984862i \(-0.444545\pi\)
0.173337 + 0.984862i \(0.444545\pi\)
\(674\) 0 0
\(675\) −2.57591 −0.0991470
\(676\) 0 0
\(677\) −41.2919 −1.58698 −0.793488 0.608585i \(-0.791737\pi\)
−0.793488 + 0.608585i \(0.791737\pi\)
\(678\) 0 0
\(679\) −13.2418 −0.508172
\(680\) 0 0
\(681\) −11.8430 −0.453827
\(682\) 0 0
\(683\) −3.46258 −0.132492 −0.0662460 0.997803i \(-0.521102\pi\)
−0.0662460 + 0.997803i \(0.521102\pi\)
\(684\) 0 0
\(685\) −15.3647 −0.587054
\(686\) 0 0
\(687\) −16.1176 −0.614925
\(688\) 0 0
\(689\) 40.7956 1.55419
\(690\) 0 0
\(691\) 19.5896 0.745223 0.372612 0.927987i \(-0.378462\pi\)
0.372612 + 0.927987i \(0.378462\pi\)
\(692\) 0 0
\(693\) −2.64446 −0.100455
\(694\) 0 0
\(695\) 10.4378 0.395927
\(696\) 0 0
\(697\) −25.3504 −0.960214
\(698\) 0 0
\(699\) −7.16478 −0.270997
\(700\) 0 0
\(701\) 16.6546 0.629037 0.314519 0.949251i \(-0.398157\pi\)
0.314519 + 0.949251i \(0.398157\pi\)
\(702\) 0 0
\(703\) 0.975182 0.0367797
\(704\) 0 0
\(705\) 22.1763 0.835206
\(706\) 0 0
\(707\) −7.02443 −0.264181
\(708\) 0 0
\(709\) −32.6203 −1.22508 −0.612540 0.790440i \(-0.709852\pi\)
−0.612540 + 0.790440i \(0.709852\pi\)
\(710\) 0 0
\(711\) 7.40124 0.277568
\(712\) 0 0
\(713\) −26.5354 −0.993759
\(714\) 0 0
\(715\) 6.89781 0.257963
\(716\) 0 0
\(717\) −59.5370 −2.22345
\(718\) 0 0
\(719\) −5.03202 −0.187663 −0.0938313 0.995588i \(-0.529911\pi\)
−0.0938313 + 0.995588i \(0.529911\pi\)
\(720\) 0 0
\(721\) 6.93449 0.258254
\(722\) 0 0
\(723\) 14.7383 0.548125
\(724\) 0 0
\(725\) −6.97029 −0.258870
\(726\) 0 0
\(727\) −7.56674 −0.280635 −0.140317 0.990107i \(-0.544812\pi\)
−0.140317 + 0.990107i \(0.544812\pi\)
\(728\) 0 0
\(729\) −3.38536 −0.125384
\(730\) 0 0
\(731\) 31.9222 1.18069
\(732\) 0 0
\(733\) 22.3577 0.825800 0.412900 0.910776i \(-0.364516\pi\)
0.412900 + 0.910776i \(0.364516\pi\)
\(734\) 0 0
\(735\) 13.8753 0.511798
\(736\) 0 0
\(737\) 24.5433 0.904063
\(738\) 0 0
\(739\) 36.3435 1.33692 0.668459 0.743749i \(-0.266954\pi\)
0.668459 + 0.743749i \(0.266954\pi\)
\(740\) 0 0
\(741\) −8.66894 −0.318461
\(742\) 0 0
\(743\) 36.2102 1.32843 0.664213 0.747544i \(-0.268767\pi\)
0.664213 + 0.747544i \(0.268767\pi\)
\(744\) 0 0
\(745\) 11.7299 0.429752
\(746\) 0 0
\(747\) −19.1671 −0.701287
\(748\) 0 0
\(749\) 11.6187 0.424538
\(750\) 0 0
\(751\) 12.4936 0.455900 0.227950 0.973673i \(-0.426798\pi\)
0.227950 + 0.973673i \(0.426798\pi\)
\(752\) 0 0
\(753\) −37.5149 −1.36712
\(754\) 0 0
\(755\) 8.58042 0.312273
\(756\) 0 0
\(757\) 18.1548 0.659846 0.329923 0.944008i \(-0.392977\pi\)
0.329923 + 0.944008i \(0.392977\pi\)
\(758\) 0 0
\(759\) −18.5444 −0.673120
\(760\) 0 0
\(761\) −10.1113 −0.366533 −0.183266 0.983063i \(-0.558667\pi\)
−0.183266 + 0.983063i \(0.558667\pi\)
\(762\) 0 0
\(763\) 4.53022 0.164005
\(764\) 0 0
\(765\) 8.99314 0.325148
\(766\) 0 0
\(767\) −41.2967 −1.49114
\(768\) 0 0
\(769\) 23.1032 0.833121 0.416561 0.909108i \(-0.363235\pi\)
0.416561 + 0.909108i \(0.363235\pi\)
\(770\) 0 0
\(771\) −6.42797 −0.231498
\(772\) 0 0
\(773\) −10.3895 −0.373684 −0.186842 0.982390i \(-0.559825\pi\)
−0.186842 + 0.982390i \(0.559825\pi\)
\(774\) 0 0
\(775\) 5.49657 0.197443
\(776\) 0 0
\(777\) −1.77333 −0.0636178
\(778\) 0 0
\(779\) 5.15183 0.184583
\(780\) 0 0
\(781\) 0.602702 0.0215664
\(782\) 0 0
\(783\) 17.9549 0.641655
\(784\) 0 0
\(785\) −19.5775 −0.698751
\(786\) 0 0
\(787\) 36.9355 1.31661 0.658305 0.752752i \(-0.271274\pi\)
0.658305 + 0.752752i \(0.271274\pi\)
\(788\) 0 0
\(789\) 31.1831 1.11015
\(790\) 0 0
\(791\) 5.85503 0.208181
\(792\) 0 0
\(793\) −6.89781 −0.244948
\(794\) 0 0
\(795\) 22.7186 0.805745
\(796\) 0 0
\(797\) 3.60974 0.127864 0.0639318 0.997954i \(-0.479636\pi\)
0.0639318 + 0.997954i \(0.479636\pi\)
\(798\) 0 0
\(799\) 49.6643 1.75700
\(800\) 0 0
\(801\) 29.6728 1.04844
\(802\) 0 0
\(803\) −22.3983 −0.790419
\(804\) 0 0
\(805\) −3.99549 −0.140823
\(806\) 0 0
\(807\) 28.7477 1.01197
\(808\) 0 0
\(809\) 24.3082 0.854630 0.427315 0.904103i \(-0.359460\pi\)
0.427315 + 0.904103i \(0.359460\pi\)
\(810\) 0 0
\(811\) 13.2470 0.465167 0.232583 0.972576i \(-0.425282\pi\)
0.232583 + 0.972576i \(0.425282\pi\)
\(812\) 0 0
\(813\) −7.72774 −0.271024
\(814\) 0 0
\(815\) −1.00917 −0.0353497
\(816\) 0 0
\(817\) −6.48740 −0.226965
\(818\) 0 0
\(819\) 5.96793 0.208536
\(820\) 0 0
\(821\) 17.2640 0.602519 0.301260 0.953542i \(-0.402593\pi\)
0.301260 + 0.953542i \(0.402593\pi\)
\(822\) 0 0
\(823\) 21.1442 0.737042 0.368521 0.929619i \(-0.379864\pi\)
0.368521 + 0.929619i \(0.379864\pi\)
\(824\) 0 0
\(825\) 3.84131 0.133737
\(826\) 0 0
\(827\) −0.256609 −0.00892318 −0.00446159 0.999990i \(-0.501420\pi\)
−0.00446159 + 0.999990i \(0.501420\pi\)
\(828\) 0 0
\(829\) −15.3607 −0.533500 −0.266750 0.963766i \(-0.585950\pi\)
−0.266750 + 0.963766i \(0.585950\pi\)
\(830\) 0 0
\(831\) 1.38435 0.0480225
\(832\) 0 0
\(833\) 31.0741 1.07665
\(834\) 0 0
\(835\) −2.66404 −0.0921931
\(836\) 0 0
\(837\) −14.1587 −0.489396
\(838\) 0 0
\(839\) −11.9093 −0.411154 −0.205577 0.978641i \(-0.565907\pi\)
−0.205577 + 0.978641i \(0.565907\pi\)
\(840\) 0 0
\(841\) 19.5849 0.675342
\(842\) 0 0
\(843\) −4.74305 −0.163359
\(844\) 0 0
\(845\) −2.56674 −0.0882987
\(846\) 0 0
\(847\) 6.57429 0.225895
\(848\) 0 0
\(849\) 58.4159 2.00483
\(850\) 0 0
\(851\) −4.70782 −0.161382
\(852\) 0 0
\(853\) −51.2280 −1.75401 −0.877007 0.480477i \(-0.840464\pi\)
−0.877007 + 0.480477i \(0.840464\pi\)
\(854\) 0 0
\(855\) −1.82763 −0.0625036
\(856\) 0 0
\(857\) −36.9654 −1.26271 −0.631357 0.775492i \(-0.717502\pi\)
−0.631357 + 0.775492i \(0.717502\pi\)
\(858\) 0 0
\(859\) −36.6065 −1.24900 −0.624500 0.781025i \(-0.714697\pi\)
−0.624500 + 0.781025i \(0.714697\pi\)
\(860\) 0 0
\(861\) −9.36839 −0.319274
\(862\) 0 0
\(863\) 57.6474 1.96234 0.981170 0.193146i \(-0.0618691\pi\)
0.981170 + 0.193146i \(0.0618691\pi\)
\(864\) 0 0
\(865\) −14.9654 −0.508839
\(866\) 0 0
\(867\) 15.8479 0.538224
\(868\) 0 0
\(869\) 7.07991 0.240170
\(870\) 0 0
\(871\) −55.3885 −1.87677
\(872\) 0 0
\(873\) 29.2414 0.989671
\(874\) 0 0
\(875\) 0.827631 0.0279790
\(876\) 0 0
\(877\) 28.5755 0.964927 0.482463 0.875916i \(-0.339742\pi\)
0.482463 + 0.875916i \(0.339742\pi\)
\(878\) 0 0
\(879\) 40.1467 1.35411
\(880\) 0 0
\(881\) −57.4023 −1.93393 −0.966967 0.254903i \(-0.917956\pi\)
−0.966967 + 0.254903i \(0.917956\pi\)
\(882\) 0 0
\(883\) −41.4251 −1.39406 −0.697032 0.717040i \(-0.745496\pi\)
−0.697032 + 0.717040i \(0.745496\pi\)
\(884\) 0 0
\(885\) −22.9976 −0.773057
\(886\) 0 0
\(887\) −30.1110 −1.01103 −0.505514 0.862818i \(-0.668697\pi\)
−0.505514 + 0.862818i \(0.668697\pi\)
\(888\) 0 0
\(889\) 10.2298 0.343098
\(890\) 0 0
\(891\) −19.4805 −0.652622
\(892\) 0 0
\(893\) −10.0930 −0.337750
\(894\) 0 0
\(895\) 9.10219 0.304253
\(896\) 0 0
\(897\) 41.8504 1.39735
\(898\) 0 0
\(899\) −38.3127 −1.27780
\(900\) 0 0
\(901\) 50.8788 1.69502
\(902\) 0 0
\(903\) 11.7971 0.392582
\(904\) 0 0
\(905\) −7.47823 −0.248585
\(906\) 0 0
\(907\) 34.4329 1.14332 0.571662 0.820489i \(-0.306299\pi\)
0.571662 + 0.820489i \(0.306299\pi\)
\(908\) 0 0
\(909\) 15.5118 0.514495
\(910\) 0 0
\(911\) −3.34181 −0.110719 −0.0553596 0.998466i \(-0.517631\pi\)
−0.0553596 + 0.998466i \(0.517631\pi\)
\(912\) 0 0
\(913\) −18.3350 −0.606798
\(914\) 0 0
\(915\) −3.84131 −0.126990
\(916\) 0 0
\(917\) 3.93742 0.130025
\(918\) 0 0
\(919\) −15.9392 −0.525787 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(920\) 0 0
\(921\) 55.5158 1.82931
\(922\) 0 0
\(923\) −1.36016 −0.0447701
\(924\) 0 0
\(925\) 0.975182 0.0320638
\(926\) 0 0
\(927\) −15.3132 −0.502952
\(928\) 0 0
\(929\) 18.6385 0.611509 0.305755 0.952110i \(-0.401091\pi\)
0.305755 + 0.952110i \(0.401091\pi\)
\(930\) 0 0
\(931\) −6.31503 −0.206967
\(932\) 0 0
\(933\) −18.4676 −0.604603
\(934\) 0 0
\(935\) 8.60270 0.281338
\(936\) 0 0
\(937\) −21.4189 −0.699725 −0.349863 0.936801i \(-0.613772\pi\)
−0.349863 + 0.936801i \(0.613772\pi\)
\(938\) 0 0
\(939\) 38.8421 1.26756
\(940\) 0 0
\(941\) 11.7688 0.383653 0.191826 0.981429i \(-0.438559\pi\)
0.191826 + 0.981429i \(0.438559\pi\)
\(942\) 0 0
\(943\) −24.8711 −0.809915
\(944\) 0 0
\(945\) −2.13191 −0.0693509
\(946\) 0 0
\(947\) −47.9558 −1.55836 −0.779178 0.626803i \(-0.784363\pi\)
−0.779178 + 0.626803i \(0.784363\pi\)
\(948\) 0 0
\(949\) 50.5478 1.64085
\(950\) 0 0
\(951\) 59.4760 1.92864
\(952\) 0 0
\(953\) −3.67322 −0.118987 −0.0594936 0.998229i \(-0.518949\pi\)
−0.0594936 + 0.998229i \(0.518949\pi\)
\(954\) 0 0
\(955\) −7.70940 −0.249470
\(956\) 0 0
\(957\) −26.7750 −0.865514
\(958\) 0 0
\(959\) −12.7163 −0.410630
\(960\) 0 0
\(961\) −0.787734 −0.0254108
\(962\) 0 0
\(963\) −25.6572 −0.826791
\(964\) 0 0
\(965\) 2.26736 0.0729890
\(966\) 0 0
\(967\) −33.3402 −1.07215 −0.536074 0.844171i \(-0.680093\pi\)
−0.536074 + 0.844171i \(0.680093\pi\)
\(968\) 0 0
\(969\) −10.8116 −0.347319
\(970\) 0 0
\(971\) 48.6105 1.55998 0.779992 0.625789i \(-0.215223\pi\)
0.779992 + 0.625789i \(0.215223\pi\)
\(972\) 0 0
\(973\) 8.63861 0.276941
\(974\) 0 0
\(975\) −8.66894 −0.277628
\(976\) 0 0
\(977\) 28.7297 0.919145 0.459572 0.888140i \(-0.348003\pi\)
0.459572 + 0.888140i \(0.348003\pi\)
\(978\) 0 0
\(979\) 28.3846 0.907175
\(980\) 0 0
\(981\) −10.0039 −0.319401
\(982\) 0 0
\(983\) −27.8976 −0.889795 −0.444897 0.895582i \(-0.646760\pi\)
−0.444897 + 0.895582i \(0.646760\pi\)
\(984\) 0 0
\(985\) −11.3602 −0.361965
\(986\) 0 0
\(987\) 18.3537 0.584206
\(988\) 0 0
\(989\) 31.3188 0.995879
\(990\) 0 0
\(991\) 38.1579 1.21213 0.606063 0.795417i \(-0.292748\pi\)
0.606063 + 0.795417i \(0.292748\pi\)
\(992\) 0 0
\(993\) 72.4419 2.29887
\(994\) 0 0
\(995\) −0.113563 −0.00360019
\(996\) 0 0
\(997\) −43.7635 −1.38601 −0.693003 0.720935i \(-0.743713\pi\)
−0.693003 + 0.720935i \(0.743713\pi\)
\(998\) 0 0
\(999\) −2.51199 −0.0794758
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 3040.2.a.s.1.3 yes 4
4.3 odd 2 3040.2.a.q.1.2 4
8.3 odd 2 6080.2.a.cg.1.3 4
8.5 even 2 6080.2.a.ce.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.q.1.2 4 4.3 odd 2
3040.2.a.s.1.3 yes 4 1.1 even 1 trivial
6080.2.a.ce.1.2 4 8.5 even 2
6080.2.a.cg.1.3 4 8.3 odd 2