Properties

Label 3344.2.a.t.1.5
Level $3344$
Weight $2$
Character 3344.1
Self dual yes
Analytic conductor $26.702$
Analytic rank $1$
Dimension $5$
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: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.246832.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \) 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 209)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.71250\) 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)
 
\(f(q)\) \(=\) \(q+2.98063 q^{3} -3.49235 q^{5} -1.06736 q^{7} +5.88418 q^{9} +O(q^{10})\) \(q+2.98063 q^{3} -3.49235 q^{5} -1.06736 q^{7} +5.88418 q^{9} -1.00000 q^{11} -0.0563258 q^{13} -10.4094 q^{15} -4.53628 q^{17} +1.00000 q^{19} -3.18141 q^{21} +1.07949 q^{23} +7.19651 q^{25} +8.59667 q^{27} +0.299905 q^{29} -9.18548 q^{31} -2.98063 q^{33} +3.72760 q^{35} +4.50448 q^{37} -0.167887 q^{39} +12.0009 q^{41} -10.7260 q^{43} -20.5496 q^{45} -2.89630 q^{47} -5.86074 q^{49} -13.5210 q^{51} -12.3213 q^{53} +3.49235 q^{55} +2.98063 q^{57} +1.14582 q^{59} -8.09599 q^{61} -6.28054 q^{63} +0.196709 q^{65} -11.2733 q^{67} +3.21755 q^{69} -13.4948 q^{71} -11.1470 q^{73} +21.4502 q^{75} +1.06736 q^{77} -11.4250 q^{79} +7.97100 q^{81} +13.9802 q^{83} +15.8423 q^{85} +0.893906 q^{87} -0.183185 q^{89} +0.0601200 q^{91} -27.3785 q^{93} -3.49235 q^{95} +5.66263 q^{97} -5.88418 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} + 10 q^{21} - 3 q^{23} + 6 q^{25} + 11 q^{27} + 10 q^{29} - 11 q^{31} + q^{33} + 8 q^{35} + q^{37} - 2 q^{39} + 2 q^{41} - 20 q^{43} - 28 q^{45} + 20 q^{47} + 3 q^{49} - 24 q^{51} - 14 q^{53} + 5 q^{55} - q^{57} - 3 q^{59} - 10 q^{61} - 24 q^{63} - 9 q^{67} - 5 q^{69} - 23 q^{71} + 18 q^{75} + 6 q^{77} - 44 q^{79} + q^{81} + 14 q^{83} - 12 q^{85} - 28 q^{87} - 27 q^{89} - 24 q^{91} - 27 q^{93} - 5 q^{95} + 15 q^{97} - 4 q^{99}+O(q^{100}) \) 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
\(3\) 2.98063 1.72087 0.860435 0.509561i \(-0.170192\pi\)
0.860435 + 0.509561i \(0.170192\pi\)
\(4\) 0 0
\(5\) −3.49235 −1.56183 −0.780913 0.624639i \(-0.785246\pi\)
−0.780913 + 0.624639i \(0.785246\pi\)
\(6\) 0 0
\(7\) −1.06736 −0.403424 −0.201712 0.979445i \(-0.564651\pi\)
−0.201712 + 0.979445i \(0.564651\pi\)
\(8\) 0 0
\(9\) 5.88418 1.96139
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −0.0563258 −0.0156220 −0.00781098 0.999969i \(-0.502486\pi\)
−0.00781098 + 0.999969i \(0.502486\pi\)
\(14\) 0 0
\(15\) −10.4094 −2.68770
\(16\) 0 0
\(17\) −4.53628 −1.10021 −0.550104 0.835096i \(-0.685412\pi\)
−0.550104 + 0.835096i \(0.685412\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −3.18141 −0.694241
\(22\) 0 0
\(23\) 1.07949 0.225088 0.112544 0.993647i \(-0.464100\pi\)
0.112544 + 0.993647i \(0.464100\pi\)
\(24\) 0 0
\(25\) 7.19651 1.43930
\(26\) 0 0
\(27\) 8.59667 1.65443
\(28\) 0 0
\(29\) 0.299905 0.0556909 0.0278455 0.999612i \(-0.491135\pi\)
0.0278455 + 0.999612i \(0.491135\pi\)
\(30\) 0 0
\(31\) −9.18548 −1.64976 −0.824880 0.565307i \(-0.808758\pi\)
−0.824880 + 0.565307i \(0.808758\pi\)
\(32\) 0 0
\(33\) −2.98063 −0.518862
\(34\) 0 0
\(35\) 3.72760 0.630079
\(36\) 0 0
\(37\) 4.50448 0.740531 0.370266 0.928926i \(-0.379267\pi\)
0.370266 + 0.928926i \(0.379267\pi\)
\(38\) 0 0
\(39\) −0.167887 −0.0268834
\(40\) 0 0
\(41\) 12.0009 1.87423 0.937115 0.349020i \(-0.113486\pi\)
0.937115 + 0.349020i \(0.113486\pi\)
\(42\) 0 0
\(43\) −10.7260 −1.63570 −0.817851 0.575430i \(-0.804835\pi\)
−0.817851 + 0.575430i \(0.804835\pi\)
\(44\) 0 0
\(45\) −20.5496 −3.06335
\(46\) 0 0
\(47\) −2.89630 −0.422469 −0.211235 0.977435i \(-0.567748\pi\)
−0.211235 + 0.977435i \(0.567748\pi\)
\(48\) 0 0
\(49\) −5.86074 −0.837249
\(50\) 0 0
\(51\) −13.5210 −1.89332
\(52\) 0 0
\(53\) −12.3213 −1.69246 −0.846230 0.532818i \(-0.821133\pi\)
−0.846230 + 0.532818i \(0.821133\pi\)
\(54\) 0 0
\(55\) 3.49235 0.470908
\(56\) 0 0
\(57\) 2.98063 0.394795
\(58\) 0 0
\(59\) 1.14582 0.149173 0.0745863 0.997215i \(-0.476236\pi\)
0.0745863 + 0.997215i \(0.476236\pi\)
\(60\) 0 0
\(61\) −8.09599 −1.03659 −0.518293 0.855203i \(-0.673432\pi\)
−0.518293 + 0.855203i \(0.673432\pi\)
\(62\) 0 0
\(63\) −6.28054 −0.791273
\(64\) 0 0
\(65\) 0.196709 0.0243988
\(66\) 0 0
\(67\) −11.2733 −1.37726 −0.688628 0.725115i \(-0.741787\pi\)
−0.688628 + 0.725115i \(0.741787\pi\)
\(68\) 0 0
\(69\) 3.21755 0.387348
\(70\) 0 0
\(71\) −13.4948 −1.60154 −0.800771 0.598970i \(-0.795577\pi\)
−0.800771 + 0.598970i \(0.795577\pi\)
\(72\) 0 0
\(73\) −11.1470 −1.30466 −0.652330 0.757935i \(-0.726208\pi\)
−0.652330 + 0.757935i \(0.726208\pi\)
\(74\) 0 0
\(75\) 21.4502 2.47685
\(76\) 0 0
\(77\) 1.06736 0.121637
\(78\) 0 0
\(79\) −11.4250 −1.28541 −0.642706 0.766113i \(-0.722188\pi\)
−0.642706 + 0.766113i \(0.722188\pi\)
\(80\) 0 0
\(81\) 7.97100 0.885666
\(82\) 0 0
\(83\) 13.9802 1.53452 0.767261 0.641335i \(-0.221619\pi\)
0.767261 + 0.641335i \(0.221619\pi\)
\(84\) 0 0
\(85\) 15.8423 1.71834
\(86\) 0 0
\(87\) 0.893906 0.0958368
\(88\) 0 0
\(89\) −0.183185 −0.0194176 −0.00970878 0.999953i \(-0.503090\pi\)
−0.00970878 + 0.999953i \(0.503090\pi\)
\(90\) 0 0
\(91\) 0.0601200 0.00630228
\(92\) 0 0
\(93\) −27.3785 −2.83902
\(94\) 0 0
\(95\) −3.49235 −0.358308
\(96\) 0 0
\(97\) 5.66263 0.574953 0.287477 0.957788i \(-0.407184\pi\)
0.287477 + 0.957788i \(0.407184\pi\)
\(98\) 0 0
\(99\) −5.88418 −0.591382
\(100\) 0 0
\(101\) 8.00759 0.796785 0.398392 0.917215i \(-0.369568\pi\)
0.398392 + 0.917215i \(0.369568\pi\)
\(102\) 0 0
\(103\) −6.18725 −0.609648 −0.304824 0.952409i \(-0.598598\pi\)
−0.304824 + 0.952409i \(0.598598\pi\)
\(104\) 0 0
\(105\) 11.1106 1.08428
\(106\) 0 0
\(107\) 7.29027 0.704777 0.352388 0.935854i \(-0.385370\pi\)
0.352388 + 0.935854i \(0.385370\pi\)
\(108\) 0 0
\(109\) 7.79895 0.747004 0.373502 0.927629i \(-0.378157\pi\)
0.373502 + 0.927629i \(0.378157\pi\)
\(110\) 0 0
\(111\) 13.4262 1.27436
\(112\) 0 0
\(113\) −0.430558 −0.0405035 −0.0202517 0.999795i \(-0.506447\pi\)
−0.0202517 + 0.999795i \(0.506447\pi\)
\(114\) 0 0
\(115\) −3.76995 −0.351549
\(116\) 0 0
\(117\) −0.331431 −0.0306408
\(118\) 0 0
\(119\) 4.84184 0.443851
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 35.7704 3.22531
\(124\) 0 0
\(125\) −7.67100 −0.686115
\(126\) 0 0
\(127\) 3.13888 0.278531 0.139265 0.990255i \(-0.455526\pi\)
0.139265 + 0.990255i \(0.455526\pi\)
\(128\) 0 0
\(129\) −31.9703 −2.81483
\(130\) 0 0
\(131\) −12.4315 −1.08615 −0.543075 0.839684i \(-0.682740\pi\)
−0.543075 + 0.839684i \(0.682740\pi\)
\(132\) 0 0
\(133\) −1.06736 −0.0925519
\(134\) 0 0
\(135\) −30.0226 −2.58393
\(136\) 0 0
\(137\) 13.8301 1.18159 0.590794 0.806822i \(-0.298814\pi\)
0.590794 + 0.806822i \(0.298814\pi\)
\(138\) 0 0
\(139\) −15.9968 −1.35683 −0.678417 0.734677i \(-0.737334\pi\)
−0.678417 + 0.734677i \(0.737334\pi\)
\(140\) 0 0
\(141\) −8.63281 −0.727014
\(142\) 0 0
\(143\) 0.0563258 0.00471020
\(144\) 0 0
\(145\) −1.04737 −0.0869796
\(146\) 0 0
\(147\) −17.4687 −1.44080
\(148\) 0 0
\(149\) 11.3620 0.930812 0.465406 0.885097i \(-0.345908\pi\)
0.465406 + 0.885097i \(0.345908\pi\)
\(150\) 0 0
\(151\) −4.66341 −0.379503 −0.189751 0.981832i \(-0.560768\pi\)
−0.189751 + 0.981832i \(0.560768\pi\)
\(152\) 0 0
\(153\) −26.6923 −2.15794
\(154\) 0 0
\(155\) 32.0789 2.57664
\(156\) 0 0
\(157\) −9.05206 −0.722433 −0.361217 0.932482i \(-0.617639\pi\)
−0.361217 + 0.932482i \(0.617639\pi\)
\(158\) 0 0
\(159\) −36.7253 −2.91250
\(160\) 0 0
\(161\) −1.15220 −0.0908062
\(162\) 0 0
\(163\) −2.36761 −0.185446 −0.0927229 0.995692i \(-0.529557\pi\)
−0.0927229 + 0.995692i \(0.529557\pi\)
\(164\) 0 0
\(165\) 10.4094 0.810372
\(166\) 0 0
\(167\) 9.27361 0.717613 0.358807 0.933412i \(-0.383184\pi\)
0.358807 + 0.933412i \(0.383184\pi\)
\(168\) 0 0
\(169\) −12.9968 −0.999756
\(170\) 0 0
\(171\) 5.88418 0.449974
\(172\) 0 0
\(173\) 10.7172 0.814812 0.407406 0.913247i \(-0.366433\pi\)
0.407406 + 0.913247i \(0.366433\pi\)
\(174\) 0 0
\(175\) −7.68128 −0.580650
\(176\) 0 0
\(177\) 3.41526 0.256707
\(178\) 0 0
\(179\) 22.9070 1.71215 0.856073 0.516854i \(-0.172897\pi\)
0.856073 + 0.516854i \(0.172897\pi\)
\(180\) 0 0
\(181\) 5.90522 0.438931 0.219466 0.975620i \(-0.429569\pi\)
0.219466 + 0.975620i \(0.429569\pi\)
\(182\) 0 0
\(183\) −24.1312 −1.78383
\(184\) 0 0
\(185\) −15.7312 −1.15658
\(186\) 0 0
\(187\) 4.53628 0.331725
\(188\) 0 0
\(189\) −9.17575 −0.667438
\(190\) 0 0
\(191\) −6.44628 −0.466437 −0.233218 0.972424i \(-0.574926\pi\)
−0.233218 + 0.972424i \(0.574926\pi\)
\(192\) 0 0
\(193\) 1.43606 0.103370 0.0516849 0.998663i \(-0.483541\pi\)
0.0516849 + 0.998663i \(0.483541\pi\)
\(194\) 0 0
\(195\) 0.586319 0.0419872
\(196\) 0 0
\(197\) −4.75399 −0.338708 −0.169354 0.985555i \(-0.554168\pi\)
−0.169354 + 0.985555i \(0.554168\pi\)
\(198\) 0 0
\(199\) 2.36002 0.167298 0.0836489 0.996495i \(-0.473343\pi\)
0.0836489 + 0.996495i \(0.473343\pi\)
\(200\) 0 0
\(201\) −33.6017 −2.37008
\(202\) 0 0
\(203\) −0.320107 −0.0224671
\(204\) 0 0
\(205\) −41.9115 −2.92722
\(206\) 0 0
\(207\) 6.35189 0.441487
\(208\) 0 0
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 24.5133 1.68756 0.843781 0.536687i \(-0.180324\pi\)
0.843781 + 0.536687i \(0.180324\pi\)
\(212\) 0 0
\(213\) −40.2232 −2.75605
\(214\) 0 0
\(215\) 37.4590 2.55468
\(216\) 0 0
\(217\) 9.80422 0.665554
\(218\) 0 0
\(219\) −33.2252 −2.24515
\(220\) 0 0
\(221\) 0.255509 0.0171874
\(222\) 0 0
\(223\) −8.47427 −0.567479 −0.283740 0.958901i \(-0.591575\pi\)
−0.283740 + 0.958901i \(0.591575\pi\)
\(224\) 0 0
\(225\) 42.3456 2.82304
\(226\) 0 0
\(227\) −13.7491 −0.912558 −0.456279 0.889837i \(-0.650818\pi\)
−0.456279 + 0.889837i \(0.650818\pi\)
\(228\) 0 0
\(229\) −1.99640 −0.131926 −0.0659629 0.997822i \(-0.521012\pi\)
−0.0659629 + 0.997822i \(0.521012\pi\)
\(230\) 0 0
\(231\) 3.18141 0.209321
\(232\) 0 0
\(233\) 11.3481 0.743437 0.371718 0.928346i \(-0.378769\pi\)
0.371718 + 0.928346i \(0.378769\pi\)
\(234\) 0 0
\(235\) 10.1149 0.659823
\(236\) 0 0
\(237\) −34.0537 −2.21203
\(238\) 0 0
\(239\) −5.21045 −0.337036 −0.168518 0.985699i \(-0.553898\pi\)
−0.168518 + 0.985699i \(0.553898\pi\)
\(240\) 0 0
\(241\) 6.60827 0.425676 0.212838 0.977087i \(-0.431729\pi\)
0.212838 + 0.977087i \(0.431729\pi\)
\(242\) 0 0
\(243\) −2.03139 −0.130314
\(244\) 0 0
\(245\) 20.4678 1.30764
\(246\) 0 0
\(247\) −0.0563258 −0.00358393
\(248\) 0 0
\(249\) 41.6697 2.64071
\(250\) 0 0
\(251\) −24.0024 −1.51502 −0.757510 0.652823i \(-0.773585\pi\)
−0.757510 + 0.652823i \(0.773585\pi\)
\(252\) 0 0
\(253\) −1.07949 −0.0678667
\(254\) 0 0
\(255\) 47.2200 2.95703
\(256\) 0 0
\(257\) −5.68903 −0.354872 −0.177436 0.984132i \(-0.556780\pi\)
−0.177436 + 0.984132i \(0.556780\pi\)
\(258\) 0 0
\(259\) −4.80790 −0.298748
\(260\) 0 0
\(261\) 1.76469 0.109232
\(262\) 0 0
\(263\) −13.9857 −0.862395 −0.431197 0.902258i \(-0.641909\pi\)
−0.431197 + 0.902258i \(0.641909\pi\)
\(264\) 0 0
\(265\) 43.0303 2.64333
\(266\) 0 0
\(267\) −0.546007 −0.0334151
\(268\) 0 0
\(269\) 15.4020 0.939075 0.469538 0.882912i \(-0.344421\pi\)
0.469538 + 0.882912i \(0.344421\pi\)
\(270\) 0 0
\(271\) 25.3911 1.54240 0.771199 0.636595i \(-0.219658\pi\)
0.771199 + 0.636595i \(0.219658\pi\)
\(272\) 0 0
\(273\) 0.179196 0.0108454
\(274\) 0 0
\(275\) −7.19651 −0.433966
\(276\) 0 0
\(277\) 19.9798 1.20047 0.600235 0.799824i \(-0.295074\pi\)
0.600235 + 0.799824i \(0.295074\pi\)
\(278\) 0 0
\(279\) −54.0490 −3.23583
\(280\) 0 0
\(281\) 6.18130 0.368745 0.184373 0.982856i \(-0.440975\pi\)
0.184373 + 0.982856i \(0.440975\pi\)
\(282\) 0 0
\(283\) 7.12127 0.423316 0.211658 0.977344i \(-0.432114\pi\)
0.211658 + 0.977344i \(0.432114\pi\)
\(284\) 0 0
\(285\) −10.4094 −0.616601
\(286\) 0 0
\(287\) −12.8093 −0.756110
\(288\) 0 0
\(289\) 3.57781 0.210459
\(290\) 0 0
\(291\) 16.8782 0.989420
\(292\) 0 0
\(293\) 19.1342 1.11783 0.558916 0.829224i \(-0.311217\pi\)
0.558916 + 0.829224i \(0.311217\pi\)
\(294\) 0 0
\(295\) −4.00159 −0.232982
\(296\) 0 0
\(297\) −8.59667 −0.498829
\(298\) 0 0
\(299\) −0.0608030 −0.00351633
\(300\) 0 0
\(301\) 11.4485 0.659882
\(302\) 0 0
\(303\) 23.8677 1.37116
\(304\) 0 0
\(305\) 28.2740 1.61897
\(306\) 0 0
\(307\) −6.82573 −0.389565 −0.194782 0.980846i \(-0.562400\pi\)
−0.194782 + 0.980846i \(0.562400\pi\)
\(308\) 0 0
\(309\) −18.4419 −1.04912
\(310\) 0 0
\(311\) 12.8609 0.729276 0.364638 0.931149i \(-0.381193\pi\)
0.364638 + 0.931149i \(0.381193\pi\)
\(312\) 0 0
\(313\) −3.01707 −0.170535 −0.0852673 0.996358i \(-0.527174\pi\)
−0.0852673 + 0.996358i \(0.527174\pi\)
\(314\) 0 0
\(315\) 21.9338 1.23583
\(316\) 0 0
\(317\) 17.7857 0.998943 0.499471 0.866330i \(-0.333528\pi\)
0.499471 + 0.866330i \(0.333528\pi\)
\(318\) 0 0
\(319\) −0.299905 −0.0167914
\(320\) 0 0
\(321\) 21.7296 1.21283
\(322\) 0 0
\(323\) −4.53628 −0.252405
\(324\) 0 0
\(325\) −0.405349 −0.0224847
\(326\) 0 0
\(327\) 23.2458 1.28550
\(328\) 0 0
\(329\) 3.09140 0.170434
\(330\) 0 0
\(331\) 26.3860 1.45030 0.725152 0.688589i \(-0.241770\pi\)
0.725152 + 0.688589i \(0.241770\pi\)
\(332\) 0 0
\(333\) 26.5051 1.45247
\(334\) 0 0
\(335\) 39.3704 2.15104
\(336\) 0 0
\(337\) −13.2024 −0.719181 −0.359590 0.933110i \(-0.617084\pi\)
−0.359590 + 0.933110i \(0.617084\pi\)
\(338\) 0 0
\(339\) −1.28334 −0.0697012
\(340\) 0 0
\(341\) 9.18548 0.497422
\(342\) 0 0
\(343\) 13.7271 0.741191
\(344\) 0 0
\(345\) −11.2368 −0.604970
\(346\) 0 0
\(347\) −16.4809 −0.884740 −0.442370 0.896833i \(-0.645862\pi\)
−0.442370 + 0.896833i \(0.645862\pi\)
\(348\) 0 0
\(349\) −19.0165 −1.01793 −0.508966 0.860787i \(-0.669972\pi\)
−0.508966 + 0.860787i \(0.669972\pi\)
\(350\) 0 0
\(351\) −0.484214 −0.0258455
\(352\) 0 0
\(353\) 13.1955 0.702325 0.351162 0.936315i \(-0.385787\pi\)
0.351162 + 0.936315i \(0.385787\pi\)
\(354\) 0 0
\(355\) 47.1287 2.50133
\(356\) 0 0
\(357\) 14.4318 0.763810
\(358\) 0 0
\(359\) 2.88756 0.152400 0.0761998 0.997093i \(-0.475721\pi\)
0.0761998 + 0.997093i \(0.475721\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 2.98063 0.156443
\(364\) 0 0
\(365\) 38.9293 2.03765
\(366\) 0 0
\(367\) 2.10833 0.110054 0.0550269 0.998485i \(-0.482476\pi\)
0.0550269 + 0.998485i \(0.482476\pi\)
\(368\) 0 0
\(369\) 70.6156 3.67610
\(370\) 0 0
\(371\) 13.1513 0.682780
\(372\) 0 0
\(373\) 8.07305 0.418007 0.209003 0.977915i \(-0.432978\pi\)
0.209003 + 0.977915i \(0.432978\pi\)
\(374\) 0 0
\(375\) −22.8644 −1.18071
\(376\) 0 0
\(377\) −0.0168924 −0.000870002 0
\(378\) 0 0
\(379\) −14.3355 −0.736365 −0.368183 0.929754i \(-0.620020\pi\)
−0.368183 + 0.929754i \(0.620020\pi\)
\(380\) 0 0
\(381\) 9.35586 0.479315
\(382\) 0 0
\(383\) 9.79867 0.500689 0.250344 0.968157i \(-0.419456\pi\)
0.250344 + 0.968157i \(0.419456\pi\)
\(384\) 0 0
\(385\) −3.72760 −0.189976
\(386\) 0 0
\(387\) −63.1138 −3.20825
\(388\) 0 0
\(389\) −19.9236 −1.01016 −0.505082 0.863071i \(-0.668538\pi\)
−0.505082 + 0.863071i \(0.668538\pi\)
\(390\) 0 0
\(391\) −4.89685 −0.247644
\(392\) 0 0
\(393\) −37.0539 −1.86912
\(394\) 0 0
\(395\) 39.9001 2.00759
\(396\) 0 0
\(397\) 13.9941 0.702342 0.351171 0.936311i \(-0.385784\pi\)
0.351171 + 0.936311i \(0.385784\pi\)
\(398\) 0 0
\(399\) −3.18141 −0.159270
\(400\) 0 0
\(401\) −18.6293 −0.930301 −0.465151 0.885232i \(-0.654000\pi\)
−0.465151 + 0.885232i \(0.654000\pi\)
\(402\) 0 0
\(403\) 0.517380 0.0257725
\(404\) 0 0
\(405\) −27.8375 −1.38326
\(406\) 0 0
\(407\) −4.50448 −0.223279
\(408\) 0 0
\(409\) −30.8428 −1.52508 −0.762540 0.646941i \(-0.776048\pi\)
−0.762540 + 0.646941i \(0.776048\pi\)
\(410\) 0 0
\(411\) 41.2226 2.03336
\(412\) 0 0
\(413\) −1.22300 −0.0601799
\(414\) 0 0
\(415\) −48.8236 −2.39666
\(416\) 0 0
\(417\) −47.6807 −2.33493
\(418\) 0 0
\(419\) −37.9347 −1.85323 −0.926616 0.376009i \(-0.877296\pi\)
−0.926616 + 0.376009i \(0.877296\pi\)
\(420\) 0 0
\(421\) −18.4642 −0.899891 −0.449945 0.893056i \(-0.648557\pi\)
−0.449945 + 0.893056i \(0.648557\pi\)
\(422\) 0 0
\(423\) −17.0423 −0.828627
\(424\) 0 0
\(425\) −32.6454 −1.58353
\(426\) 0 0
\(427\) 8.64134 0.418184
\(428\) 0 0
\(429\) 0.167887 0.00810564
\(430\) 0 0
\(431\) −33.6742 −1.62203 −0.811013 0.585027i \(-0.801084\pi\)
−0.811013 + 0.585027i \(0.801084\pi\)
\(432\) 0 0
\(433\) 11.4041 0.548047 0.274024 0.961723i \(-0.411645\pi\)
0.274024 + 0.961723i \(0.411645\pi\)
\(434\) 0 0
\(435\) −3.12183 −0.149680
\(436\) 0 0
\(437\) 1.07949 0.0516388
\(438\) 0 0
\(439\) 3.76890 0.179880 0.0899399 0.995947i \(-0.471333\pi\)
0.0899399 + 0.995947i \(0.471333\pi\)
\(440\) 0 0
\(441\) −34.4856 −1.64217
\(442\) 0 0
\(443\) −4.51841 −0.214676 −0.107338 0.994223i \(-0.534233\pi\)
−0.107338 + 0.994223i \(0.534233\pi\)
\(444\) 0 0
\(445\) 0.639746 0.0303269
\(446\) 0 0
\(447\) 33.8660 1.60181
\(448\) 0 0
\(449\) −37.3611 −1.76318 −0.881590 0.472016i \(-0.843526\pi\)
−0.881590 + 0.472016i \(0.843526\pi\)
\(450\) 0 0
\(451\) −12.0009 −0.565102
\(452\) 0 0
\(453\) −13.8999 −0.653075
\(454\) 0 0
\(455\) −0.209960 −0.00984308
\(456\) 0 0
\(457\) 9.37773 0.438672 0.219336 0.975649i \(-0.429611\pi\)
0.219336 + 0.975649i \(0.429611\pi\)
\(458\) 0 0
\(459\) −38.9969 −1.82022
\(460\) 0 0
\(461\) 31.6785 1.47542 0.737708 0.675120i \(-0.235908\pi\)
0.737708 + 0.675120i \(0.235908\pi\)
\(462\) 0 0
\(463\) 37.1284 1.72550 0.862752 0.505628i \(-0.168739\pi\)
0.862752 + 0.505628i \(0.168739\pi\)
\(464\) 0 0
\(465\) 95.6155 4.43406
\(466\) 0 0
\(467\) 15.9678 0.738902 0.369451 0.929250i \(-0.379546\pi\)
0.369451 + 0.929250i \(0.379546\pi\)
\(468\) 0 0
\(469\) 12.0327 0.555619
\(470\) 0 0
\(471\) −26.9809 −1.24321
\(472\) 0 0
\(473\) 10.7260 0.493183
\(474\) 0 0
\(475\) 7.19651 0.330199
\(476\) 0 0
\(477\) −72.5006 −3.31958
\(478\) 0 0
\(479\) −28.1729 −1.28725 −0.643627 0.765340i \(-0.722571\pi\)
−0.643627 + 0.765340i \(0.722571\pi\)
\(480\) 0 0
\(481\) −0.253718 −0.0115686
\(482\) 0 0
\(483\) −3.43429 −0.156266
\(484\) 0 0
\(485\) −19.7759 −0.897977
\(486\) 0 0
\(487\) −24.0613 −1.09032 −0.545161 0.838331i \(-0.683532\pi\)
−0.545161 + 0.838331i \(0.683532\pi\)
\(488\) 0 0
\(489\) −7.05699 −0.319128
\(490\) 0 0
\(491\) 29.0197 1.30964 0.654821 0.755784i \(-0.272744\pi\)
0.654821 + 0.755784i \(0.272744\pi\)
\(492\) 0 0
\(493\) −1.36045 −0.0612716
\(494\) 0 0
\(495\) 20.5496 0.923636
\(496\) 0 0
\(497\) 14.4039 0.646102
\(498\) 0 0
\(499\) 15.0257 0.672642 0.336321 0.941747i \(-0.390817\pi\)
0.336321 + 0.941747i \(0.390817\pi\)
\(500\) 0 0
\(501\) 27.6412 1.23492
\(502\) 0 0
\(503\) 28.7785 1.28317 0.641585 0.767052i \(-0.278277\pi\)
0.641585 + 0.767052i \(0.278277\pi\)
\(504\) 0 0
\(505\) −27.9653 −1.24444
\(506\) 0 0
\(507\) −38.7388 −1.72045
\(508\) 0 0
\(509\) −10.4772 −0.464395 −0.232197 0.972669i \(-0.574592\pi\)
−0.232197 + 0.972669i \(0.574592\pi\)
\(510\) 0 0
\(511\) 11.8979 0.526332
\(512\) 0 0
\(513\) 8.59667 0.379552
\(514\) 0 0
\(515\) 21.6081 0.952165
\(516\) 0 0
\(517\) 2.89630 0.127379
\(518\) 0 0
\(519\) 31.9440 1.40219
\(520\) 0 0
\(521\) −24.7590 −1.08471 −0.542357 0.840148i \(-0.682468\pi\)
−0.542357 + 0.840148i \(0.682468\pi\)
\(522\) 0 0
\(523\) −14.8566 −0.649635 −0.324818 0.945777i \(-0.605303\pi\)
−0.324818 + 0.945777i \(0.605303\pi\)
\(524\) 0 0
\(525\) −22.8951 −0.999223
\(526\) 0 0
\(527\) 41.6679 1.81508
\(528\) 0 0
\(529\) −21.8347 −0.949335
\(530\) 0 0
\(531\) 6.74219 0.292586
\(532\) 0 0
\(533\) −0.675962 −0.0292792
\(534\) 0 0
\(535\) −25.4602 −1.10074
\(536\) 0 0
\(537\) 68.2773 2.94638
\(538\) 0 0
\(539\) 5.86074 0.252440
\(540\) 0 0
\(541\) 3.88960 0.167227 0.0836134 0.996498i \(-0.473354\pi\)
0.0836134 + 0.996498i \(0.473354\pi\)
\(542\) 0 0
\(543\) 17.6013 0.755343
\(544\) 0 0
\(545\) −27.2367 −1.16669
\(546\) 0 0
\(547\) −17.5180 −0.749015 −0.374507 0.927224i \(-0.622188\pi\)
−0.374507 + 0.927224i \(0.622188\pi\)
\(548\) 0 0
\(549\) −47.6382 −2.03315
\(550\) 0 0
\(551\) 0.299905 0.0127764
\(552\) 0 0
\(553\) 12.1946 0.518567
\(554\) 0 0
\(555\) −46.8890 −1.99033
\(556\) 0 0
\(557\) 28.9860 1.22818 0.614088 0.789237i \(-0.289524\pi\)
0.614088 + 0.789237i \(0.289524\pi\)
\(558\) 0 0
\(559\) 0.604152 0.0255529
\(560\) 0 0
\(561\) 13.5210 0.570856
\(562\) 0 0
\(563\) −29.6112 −1.24796 −0.623982 0.781438i \(-0.714486\pi\)
−0.623982 + 0.781438i \(0.714486\pi\)
\(564\) 0 0
\(565\) 1.50366 0.0632594
\(566\) 0 0
\(567\) −8.50793 −0.357299
\(568\) 0 0
\(569\) −4.05818 −0.170128 −0.0850640 0.996375i \(-0.527109\pi\)
−0.0850640 + 0.996375i \(0.527109\pi\)
\(570\) 0 0
\(571\) −20.3380 −0.851117 −0.425559 0.904931i \(-0.639922\pi\)
−0.425559 + 0.904931i \(0.639922\pi\)
\(572\) 0 0
\(573\) −19.2140 −0.802677
\(574\) 0 0
\(575\) 7.76854 0.323971
\(576\) 0 0
\(577\) 24.4768 1.01898 0.509492 0.860476i \(-0.329833\pi\)
0.509492 + 0.860476i \(0.329833\pi\)
\(578\) 0 0
\(579\) 4.28037 0.177886
\(580\) 0 0
\(581\) −14.9219 −0.619064
\(582\) 0 0
\(583\) 12.3213 0.510296
\(584\) 0 0
\(585\) 1.15747 0.0478556
\(586\) 0 0
\(587\) 3.34628 0.138116 0.0690579 0.997613i \(-0.478001\pi\)
0.0690579 + 0.997613i \(0.478001\pi\)
\(588\) 0 0
\(589\) −9.18548 −0.378481
\(590\) 0 0
\(591\) −14.1699 −0.582872
\(592\) 0 0
\(593\) −39.2063 −1.61001 −0.805006 0.593267i \(-0.797838\pi\)
−0.805006 + 0.593267i \(0.797838\pi\)
\(594\) 0 0
\(595\) −16.9094 −0.693219
\(596\) 0 0
\(597\) 7.03437 0.287898
\(598\) 0 0
\(599\) −5.72987 −0.234116 −0.117058 0.993125i \(-0.537346\pi\)
−0.117058 + 0.993125i \(0.537346\pi\)
\(600\) 0 0
\(601\) 13.9163 0.567656 0.283828 0.958875i \(-0.408395\pi\)
0.283828 + 0.958875i \(0.408395\pi\)
\(602\) 0 0
\(603\) −66.3343 −2.70134
\(604\) 0 0
\(605\) −3.49235 −0.141984
\(606\) 0 0
\(607\) −0.156175 −0.00633897 −0.00316948 0.999995i \(-0.501009\pi\)
−0.00316948 + 0.999995i \(0.501009\pi\)
\(608\) 0 0
\(609\) −0.954120 −0.0386629
\(610\) 0 0
\(611\) 0.163137 0.00659980
\(612\) 0 0
\(613\) 40.1902 1.62327 0.811634 0.584166i \(-0.198578\pi\)
0.811634 + 0.584166i \(0.198578\pi\)
\(614\) 0 0
\(615\) −124.923 −5.03737
\(616\) 0 0
\(617\) 12.8606 0.517747 0.258874 0.965911i \(-0.416649\pi\)
0.258874 + 0.965911i \(0.416649\pi\)
\(618\) 0 0
\(619\) 2.03398 0.0817526 0.0408763 0.999164i \(-0.486985\pi\)
0.0408763 + 0.999164i \(0.486985\pi\)
\(620\) 0 0
\(621\) 9.27999 0.372393
\(622\) 0 0
\(623\) 0.195524 0.00783352
\(624\) 0 0
\(625\) −9.19275 −0.367710
\(626\) 0 0
\(627\) −2.98063 −0.119035
\(628\) 0 0
\(629\) −20.4336 −0.814739
\(630\) 0 0
\(631\) −37.6984 −1.50075 −0.750375 0.661012i \(-0.770127\pi\)
−0.750375 + 0.661012i \(0.770127\pi\)
\(632\) 0 0
\(633\) 73.0651 2.90408
\(634\) 0 0
\(635\) −10.9621 −0.435017
\(636\) 0 0
\(637\) 0.330111 0.0130795
\(638\) 0 0
\(639\) −79.4060 −3.14125
\(640\) 0 0
\(641\) 2.84360 0.112315 0.0561576 0.998422i \(-0.482115\pi\)
0.0561576 + 0.998422i \(0.482115\pi\)
\(642\) 0 0
\(643\) −30.6389 −1.20828 −0.604140 0.796878i \(-0.706483\pi\)
−0.604140 + 0.796878i \(0.706483\pi\)
\(644\) 0 0
\(645\) 111.652 4.39628
\(646\) 0 0
\(647\) 6.76617 0.266006 0.133003 0.991116i \(-0.457538\pi\)
0.133003 + 0.991116i \(0.457538\pi\)
\(648\) 0 0
\(649\) −1.14582 −0.0449772
\(650\) 0 0
\(651\) 29.2228 1.14533
\(652\) 0 0
\(653\) 11.9015 0.465743 0.232872 0.972508i \(-0.425188\pi\)
0.232872 + 0.972508i \(0.425188\pi\)
\(654\) 0 0
\(655\) 43.4153 1.69638
\(656\) 0 0
\(657\) −65.5910 −2.55895
\(658\) 0 0
\(659\) 11.2353 0.437664 0.218832 0.975763i \(-0.429775\pi\)
0.218832 + 0.975763i \(0.429775\pi\)
\(660\) 0 0
\(661\) −22.9273 −0.891768 −0.445884 0.895091i \(-0.647111\pi\)
−0.445884 + 0.895091i \(0.647111\pi\)
\(662\) 0 0
\(663\) 0.761580 0.0295773
\(664\) 0 0
\(665\) 3.72760 0.144550
\(666\) 0 0
\(667\) 0.323743 0.0125354
\(668\) 0 0
\(669\) −25.2587 −0.976558
\(670\) 0 0
\(671\) 8.09599 0.312542
\(672\) 0 0
\(673\) −3.09828 −0.119430 −0.0597150 0.998215i \(-0.519019\pi\)
−0.0597150 + 0.998215i \(0.519019\pi\)
\(674\) 0 0
\(675\) 61.8661 2.38123
\(676\) 0 0
\(677\) 33.3985 1.28361 0.641804 0.766868i \(-0.278186\pi\)
0.641804 + 0.766868i \(0.278186\pi\)
\(678\) 0 0
\(679\) −6.04407 −0.231950
\(680\) 0 0
\(681\) −40.9810 −1.57039
\(682\) 0 0
\(683\) −6.01634 −0.230209 −0.115104 0.993353i \(-0.536720\pi\)
−0.115104 + 0.993353i \(0.536720\pi\)
\(684\) 0 0
\(685\) −48.2997 −1.84544
\(686\) 0 0
\(687\) −5.95054 −0.227027
\(688\) 0 0
\(689\) 0.694007 0.0264396
\(690\) 0 0
\(691\) 15.6730 0.596227 0.298114 0.954530i \(-0.403643\pi\)
0.298114 + 0.954530i \(0.403643\pi\)
\(692\) 0 0
\(693\) 6.28054 0.238578
\(694\) 0 0
\(695\) 55.8665 2.11914
\(696\) 0 0
\(697\) −54.4395 −2.06204
\(698\) 0 0
\(699\) 33.8244 1.27936
\(700\) 0 0
\(701\) 18.0567 0.681992 0.340996 0.940065i \(-0.389236\pi\)
0.340996 + 0.940065i \(0.389236\pi\)
\(702\) 0 0
\(703\) 4.50448 0.169890
\(704\) 0 0
\(705\) 30.1488 1.13547
\(706\) 0 0
\(707\) −8.54699 −0.321442
\(708\) 0 0
\(709\) 16.2048 0.608586 0.304293 0.952579i \(-0.401580\pi\)
0.304293 + 0.952579i \(0.401580\pi\)
\(710\) 0 0
\(711\) −67.2266 −2.52120
\(712\) 0 0
\(713\) −9.91560 −0.371342
\(714\) 0 0
\(715\) −0.196709 −0.00735652
\(716\) 0 0
\(717\) −15.5304 −0.579995
\(718\) 0 0
\(719\) 27.9403 1.04200 0.520998 0.853558i \(-0.325560\pi\)
0.520998 + 0.853558i \(0.325560\pi\)
\(720\) 0 0
\(721\) 6.60403 0.245947
\(722\) 0 0
\(723\) 19.6968 0.732533
\(724\) 0 0
\(725\) 2.15827 0.0801561
\(726\) 0 0
\(727\) 30.7020 1.13867 0.569337 0.822104i \(-0.307200\pi\)
0.569337 + 0.822104i \(0.307200\pi\)
\(728\) 0 0
\(729\) −29.9678 −1.10992
\(730\) 0 0
\(731\) 48.6562 1.79961
\(732\) 0 0
\(733\) 44.9333 1.65965 0.829826 0.558023i \(-0.188440\pi\)
0.829826 + 0.558023i \(0.188440\pi\)
\(734\) 0 0
\(735\) 61.0069 2.25027
\(736\) 0 0
\(737\) 11.2733 0.415258
\(738\) 0 0
\(739\) 3.19172 0.117409 0.0587047 0.998275i \(-0.481303\pi\)
0.0587047 + 0.998275i \(0.481303\pi\)
\(740\) 0 0
\(741\) −0.167887 −0.00616747
\(742\) 0 0
\(743\) −17.4348 −0.639619 −0.319810 0.947482i \(-0.603619\pi\)
−0.319810 + 0.947482i \(0.603619\pi\)
\(744\) 0 0
\(745\) −39.6801 −1.45377
\(746\) 0 0
\(747\) 82.2617 3.00980
\(748\) 0 0
\(749\) −7.78135 −0.284324
\(750\) 0 0
\(751\) −9.55633 −0.348715 −0.174358 0.984682i \(-0.555785\pi\)
−0.174358 + 0.984682i \(0.555785\pi\)
\(752\) 0 0
\(753\) −71.5425 −2.60715
\(754\) 0 0
\(755\) 16.2863 0.592718
\(756\) 0 0
\(757\) −48.5259 −1.76370 −0.881852 0.471527i \(-0.843703\pi\)
−0.881852 + 0.471527i \(0.843703\pi\)
\(758\) 0 0
\(759\) −3.21755 −0.116790
\(760\) 0 0
\(761\) 38.6035 1.39937 0.699687 0.714449i \(-0.253323\pi\)
0.699687 + 0.714449i \(0.253323\pi\)
\(762\) 0 0
\(763\) −8.32429 −0.301360
\(764\) 0 0
\(765\) 93.2187 3.37033
\(766\) 0 0
\(767\) −0.0645391 −0.00233037
\(768\) 0 0
\(769\) 53.2658 1.92081 0.960406 0.278603i \(-0.0898712\pi\)
0.960406 + 0.278603i \(0.0898712\pi\)
\(770\) 0 0
\(771\) −16.9569 −0.610688
\(772\) 0 0
\(773\) 9.31748 0.335126 0.167563 0.985861i \(-0.446410\pi\)
0.167563 + 0.985861i \(0.446410\pi\)
\(774\) 0 0
\(775\) −66.1034 −2.37451
\(776\) 0 0
\(777\) −14.3306 −0.514107
\(778\) 0 0
\(779\) 12.0009 0.429978
\(780\) 0 0
\(781\) 13.4948 0.482883
\(782\) 0 0
\(783\) 2.57818 0.0921367
\(784\) 0 0
\(785\) 31.6130 1.12832
\(786\) 0 0
\(787\) 15.0616 0.536889 0.268444 0.963295i \(-0.413490\pi\)
0.268444 + 0.963295i \(0.413490\pi\)
\(788\) 0 0
\(789\) −41.6862 −1.48407
\(790\) 0 0
\(791\) 0.459561 0.0163401
\(792\) 0 0
\(793\) 0.456013 0.0161935
\(794\) 0 0
\(795\) 128.257 4.54882
\(796\) 0 0
\(797\) −19.0593 −0.675114 −0.337557 0.941305i \(-0.609601\pi\)
−0.337557 + 0.941305i \(0.609601\pi\)
\(798\) 0 0
\(799\) 13.1384 0.464804
\(800\) 0 0
\(801\) −1.07789 −0.0380855
\(802\) 0 0
\(803\) 11.1470 0.393370
\(804\) 0 0
\(805\) 4.02389 0.141824
\(806\) 0 0
\(807\) 45.9077 1.61603
\(808\) 0 0
\(809\) −15.2273 −0.535363 −0.267681 0.963507i \(-0.586257\pi\)
−0.267681 + 0.963507i \(0.586257\pi\)
\(810\) 0 0
\(811\) 43.3354 1.52171 0.760857 0.648920i \(-0.224779\pi\)
0.760857 + 0.648920i \(0.224779\pi\)
\(812\) 0 0
\(813\) 75.6814 2.65426
\(814\) 0 0
\(815\) 8.26854 0.289634
\(816\) 0 0
\(817\) −10.7260 −0.375256
\(818\) 0 0
\(819\) 0.353756 0.0123612
\(820\) 0 0
\(821\) 19.4279 0.678040 0.339020 0.940779i \(-0.389905\pi\)
0.339020 + 0.940779i \(0.389905\pi\)
\(822\) 0 0
\(823\) −2.35683 −0.0821539 −0.0410769 0.999156i \(-0.513079\pi\)
−0.0410769 + 0.999156i \(0.513079\pi\)
\(824\) 0 0
\(825\) −21.4502 −0.746799
\(826\) 0 0
\(827\) 29.8127 1.03669 0.518344 0.855172i \(-0.326549\pi\)
0.518344 + 0.855172i \(0.326549\pi\)
\(828\) 0 0
\(829\) 52.6510 1.82864 0.914322 0.404988i \(-0.132724\pi\)
0.914322 + 0.404988i \(0.132724\pi\)
\(830\) 0 0
\(831\) 59.5524 2.06585
\(832\) 0 0
\(833\) 26.5859 0.921148
\(834\) 0 0
\(835\) −32.3867 −1.12079
\(836\) 0 0
\(837\) −78.9645 −2.72941
\(838\) 0 0
\(839\) −29.7892 −1.02844 −0.514218 0.857660i \(-0.671918\pi\)
−0.514218 + 0.857660i \(0.671918\pi\)
\(840\) 0 0
\(841\) −28.9101 −0.996899
\(842\) 0 0
\(843\) 18.4242 0.634563
\(844\) 0 0
\(845\) 45.3895 1.56145
\(846\) 0 0
\(847\) −1.06736 −0.0366749
\(848\) 0 0
\(849\) 21.2259 0.728471
\(850\) 0 0
\(851\) 4.86252 0.166685
\(852\) 0 0
\(853\) −35.2393 −1.20657 −0.603285 0.797526i \(-0.706142\pi\)
−0.603285 + 0.797526i \(0.706142\pi\)
\(854\) 0 0
\(855\) −20.5496 −0.702782
\(856\) 0 0
\(857\) −21.5877 −0.737421 −0.368711 0.929544i \(-0.620201\pi\)
−0.368711 + 0.929544i \(0.620201\pi\)
\(858\) 0 0
\(859\) −27.7669 −0.947396 −0.473698 0.880687i \(-0.657081\pi\)
−0.473698 + 0.880687i \(0.657081\pi\)
\(860\) 0 0
\(861\) −38.1799 −1.30117
\(862\) 0 0
\(863\) −7.11774 −0.242291 −0.121145 0.992635i \(-0.538657\pi\)
−0.121145 + 0.992635i \(0.538657\pi\)
\(864\) 0 0
\(865\) −37.4282 −1.27260
\(866\) 0 0
\(867\) 10.6641 0.362173
\(868\) 0 0
\(869\) 11.4250 0.387566
\(870\) 0 0
\(871\) 0.634980 0.0215155
\(872\) 0 0
\(873\) 33.3199 1.12771
\(874\) 0 0
\(875\) 8.18772 0.276796
\(876\) 0 0
\(877\) −44.1031 −1.48925 −0.744627 0.667481i \(-0.767373\pi\)
−0.744627 + 0.667481i \(0.767373\pi\)
\(878\) 0 0
\(879\) 57.0321 1.92364
\(880\) 0 0
\(881\) −37.9204 −1.27757 −0.638785 0.769385i \(-0.720563\pi\)
−0.638785 + 0.769385i \(0.720563\pi\)
\(882\) 0 0
\(883\) −4.85252 −0.163300 −0.0816502 0.996661i \(-0.526019\pi\)
−0.0816502 + 0.996661i \(0.526019\pi\)
\(884\) 0 0
\(885\) −11.9273 −0.400931
\(886\) 0 0
\(887\) −11.2754 −0.378591 −0.189295 0.981920i \(-0.560620\pi\)
−0.189295 + 0.981920i \(0.560620\pi\)
\(888\) 0 0
\(889\) −3.35032 −0.112366
\(890\) 0 0
\(891\) −7.97100 −0.267038
\(892\) 0 0
\(893\) −2.89630 −0.0969210
\(894\) 0 0
\(895\) −79.9991 −2.67408
\(896\) 0 0
\(897\) −0.181231 −0.00605114
\(898\) 0 0
\(899\) −2.75477 −0.0918767
\(900\) 0 0
\(901\) 55.8928 1.86206
\(902\) 0 0
\(903\) 34.1239 1.13557
\(904\) 0 0
\(905\) −20.6231 −0.685534
\(906\) 0 0
\(907\) −24.0191 −0.797540 −0.398770 0.917051i \(-0.630563\pi\)
−0.398770 + 0.917051i \(0.630563\pi\)
\(908\) 0 0
\(909\) 47.1181 1.56281
\(910\) 0 0
\(911\) −7.48540 −0.248002 −0.124001 0.992282i \(-0.539573\pi\)
−0.124001 + 0.992282i \(0.539573\pi\)
\(912\) 0 0
\(913\) −13.9802 −0.462676
\(914\) 0 0
\(915\) 84.2745 2.78603
\(916\) 0 0
\(917\) 13.2689 0.438179
\(918\) 0 0
\(919\) −40.0187 −1.32009 −0.660047 0.751224i \(-0.729464\pi\)
−0.660047 + 0.751224i \(0.729464\pi\)
\(920\) 0 0
\(921\) −20.3450 −0.670390
\(922\) 0 0
\(923\) 0.760108 0.0250193
\(924\) 0 0
\(925\) 32.4165 1.06585
\(926\) 0 0
\(927\) −36.4069 −1.19576
\(928\) 0 0
\(929\) −32.3099 −1.06005 −0.530027 0.847981i \(-0.677818\pi\)
−0.530027 + 0.847981i \(0.677818\pi\)
\(930\) 0 0
\(931\) −5.86074 −0.192078
\(932\) 0 0
\(933\) 38.3337 1.25499
\(934\) 0 0
\(935\) −15.8423 −0.518098
\(936\) 0 0
\(937\) −54.1000 −1.76737 −0.883684 0.468083i \(-0.844945\pi\)
−0.883684 + 0.468083i \(0.844945\pi\)
\(938\) 0 0
\(939\) −8.99277 −0.293468
\(940\) 0 0
\(941\) 26.6955 0.870248 0.435124 0.900371i \(-0.356705\pi\)
0.435124 + 0.900371i \(0.356705\pi\)
\(942\) 0 0
\(943\) 12.9548 0.421868
\(944\) 0 0
\(945\) 32.0449 1.04242
\(946\) 0 0
\(947\) 39.4463 1.28183 0.640916 0.767611i \(-0.278555\pi\)
0.640916 + 0.767611i \(0.278555\pi\)
\(948\) 0 0
\(949\) 0.627865 0.0203814
\(950\) 0 0
\(951\) 53.0126 1.71905
\(952\) 0 0
\(953\) −48.7523 −1.57924 −0.789621 0.613595i \(-0.789723\pi\)
−0.789621 + 0.613595i \(0.789723\pi\)
\(954\) 0 0
\(955\) 22.5127 0.728493
\(956\) 0 0
\(957\) −0.893906 −0.0288959
\(958\) 0 0
\(959\) −14.7618 −0.476682
\(960\) 0 0
\(961\) 53.3730 1.72171
\(962\) 0 0
\(963\) 42.8972 1.38234
\(964\) 0 0
\(965\) −5.01522 −0.161446
\(966\) 0 0
\(967\) 1.39478 0.0448532 0.0224266 0.999748i \(-0.492861\pi\)
0.0224266 + 0.999748i \(0.492861\pi\)
\(968\) 0 0
\(969\) −13.5210 −0.434356
\(970\) 0 0
\(971\) 3.11733 0.100040 0.0500200 0.998748i \(-0.484071\pi\)
0.0500200 + 0.998748i \(0.484071\pi\)
\(972\) 0 0
\(973\) 17.0744 0.547380
\(974\) 0 0
\(975\) −1.20820 −0.0386933
\(976\) 0 0
\(977\) −22.3690 −0.715647 −0.357823 0.933789i \(-0.616481\pi\)
−0.357823 + 0.933789i \(0.616481\pi\)
\(978\) 0 0
\(979\) 0.183185 0.00585462
\(980\) 0 0
\(981\) 45.8904 1.46517
\(982\) 0 0
\(983\) −42.2421 −1.34731 −0.673657 0.739044i \(-0.735277\pi\)
−0.673657 + 0.739044i \(0.735277\pi\)
\(984\) 0 0
\(985\) 16.6026 0.529003
\(986\) 0 0
\(987\) 9.21433 0.293295
\(988\) 0 0
\(989\) −11.5786 −0.368178
\(990\) 0 0
\(991\) 1.69828 0.0539477 0.0269738 0.999636i \(-0.491413\pi\)
0.0269738 + 0.999636i \(0.491413\pi\)
\(992\) 0 0
\(993\) 78.6469 2.49578
\(994\) 0 0
\(995\) −8.24203 −0.261290
\(996\) 0 0
\(997\) 18.9376 0.599759 0.299879 0.953977i \(-0.403054\pi\)
0.299879 + 0.953977i \(0.403054\pi\)
\(998\) 0 0
\(999\) 38.7235 1.22516
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.t.1.5 5
4.3 odd 2 209.2.a.c.1.2 5
12.11 even 2 1881.2.a.k.1.4 5
20.19 odd 2 5225.2.a.h.1.4 5
44.43 even 2 2299.2.a.n.1.4 5
76.75 even 2 3971.2.a.h.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.c.1.2 5 4.3 odd 2
1881.2.a.k.1.4 5 12.11 even 2
2299.2.a.n.1.4 5 44.43 even 2
3344.2.a.t.1.5 5 1.1 even 1 trivial
3971.2.a.h.1.4 5 76.75 even 2
5225.2.a.h.1.4 5 20.19 odd 2