Properties

Label 3640.2.a.y.1.5
Level $3640$
Weight $2$
Character 3640.1
Self dual yes
Analytic conductor $29.066$
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] = [3640,2,Mod(1,3640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3640, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3640.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3640.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: \(29.0655463357\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.1194649.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 8x^{3} - 3x^{2} + 10x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-2.24233\) of defining polynomial
Character \(\chi\) \(=\) 3640.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.24233 q^{3} -1.00000 q^{5} +1.00000 q^{7} +2.02804 q^{9} +O(q^{10})\) \(q+2.24233 q^{3} -1.00000 q^{5} +1.00000 q^{7} +2.02804 q^{9} -5.09092 q^{11} -1.00000 q^{13} -2.24233 q^{15} -2.63792 q^{17} +6.11896 q^{19} +2.24233 q^{21} -5.44982 q^{23} +1.00000 q^{25} -2.17945 q^{27} -6.47239 q^{29} -3.50723 q^{31} -11.4155 q^{33} -1.00000 q^{35} +0.458471 q^{37} -2.24233 q^{39} -7.15380 q^{41} +2.62019 q^{43} -2.02804 q^{45} -8.09454 q^{47} +1.00000 q^{49} -5.91509 q^{51} +3.74956 q^{53} +5.09092 q^{55} +13.7207 q^{57} +9.17319 q^{59} +9.41552 q^{61} +2.02804 q^{63} +1.00000 q^{65} -5.80828 q^{67} -12.2203 q^{69} -5.83994 q^{71} -8.27638 q^{73} +2.24233 q^{75} -5.09092 q^{77} +3.86437 q^{79} -10.9712 q^{81} -8.61042 q^{83} +2.63792 q^{85} -14.5132 q^{87} -11.7618 q^{89} -1.00000 q^{91} -7.86437 q^{93} -6.11896 q^{95} +8.15062 q^{97} -10.3246 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{5} + 5 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{5} + 5 q^{7} + q^{9} - 7 q^{11} - 5 q^{13} + q^{17} + 3 q^{19} - 5 q^{23} + 5 q^{25} - 9 q^{27} - 9 q^{29} + 6 q^{31} + q^{33} - 5 q^{35} - 10 q^{37} - 8 q^{41} + 6 q^{43} - q^{45} - 13 q^{47} + 5 q^{49} - 4 q^{51} - 16 q^{53} + 7 q^{55} - 10 q^{57} - q^{59} - 11 q^{61} + q^{63} + 5 q^{65} - 30 q^{67} - 15 q^{69} - 12 q^{71} + 23 q^{73} - 7 q^{77} + 2 q^{79} - 11 q^{81} - 2 q^{83} - q^{85} - 5 q^{87} - 25 q^{89} - 5 q^{91} - 22 q^{93} - 3 q^{95} - 5 q^{97} - 12 q^{99}+O(q^{100}) \) 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
\(3\) 2.24233 1.29461 0.647305 0.762231i \(-0.275896\pi\)
0.647305 + 0.762231i \(0.275896\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 2.02804 0.676014
\(10\) 0 0
\(11\) −5.09092 −1.53497 −0.767485 0.641067i \(-0.778492\pi\)
−0.767485 + 0.641067i \(0.778492\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −2.24233 −0.578967
\(16\) 0 0
\(17\) −2.63792 −0.639790 −0.319895 0.947453i \(-0.603648\pi\)
−0.319895 + 0.947453i \(0.603648\pi\)
\(18\) 0 0
\(19\) 6.11896 1.40379 0.701893 0.712282i \(-0.252338\pi\)
0.701893 + 0.712282i \(0.252338\pi\)
\(20\) 0 0
\(21\) 2.24233 0.489316
\(22\) 0 0
\(23\) −5.44982 −1.13637 −0.568183 0.822902i \(-0.692354\pi\)
−0.568183 + 0.822902i \(0.692354\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −2.17945 −0.419435
\(28\) 0 0
\(29\) −6.47239 −1.20189 −0.600947 0.799289i \(-0.705210\pi\)
−0.600947 + 0.799289i \(0.705210\pi\)
\(30\) 0 0
\(31\) −3.50723 −0.629917 −0.314959 0.949105i \(-0.601991\pi\)
−0.314959 + 0.949105i \(0.601991\pi\)
\(32\) 0 0
\(33\) −11.4155 −1.98719
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 0.458471 0.0753721 0.0376860 0.999290i \(-0.488001\pi\)
0.0376860 + 0.999290i \(0.488001\pi\)
\(38\) 0 0
\(39\) −2.24233 −0.359060
\(40\) 0 0
\(41\) −7.15380 −1.11724 −0.558618 0.829425i \(-0.688668\pi\)
−0.558618 + 0.829425i \(0.688668\pi\)
\(42\) 0 0
\(43\) 2.62019 0.399575 0.199787 0.979839i \(-0.435975\pi\)
0.199787 + 0.979839i \(0.435975\pi\)
\(44\) 0 0
\(45\) −2.02804 −0.302323
\(46\) 0 0
\(47\) −8.09454 −1.18071 −0.590355 0.807144i \(-0.701012\pi\)
−0.590355 + 0.807144i \(0.701012\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −5.91509 −0.828278
\(52\) 0 0
\(53\) 3.74956 0.515042 0.257521 0.966273i \(-0.417094\pi\)
0.257521 + 0.966273i \(0.417094\pi\)
\(54\) 0 0
\(55\) 5.09092 0.686460
\(56\) 0 0
\(57\) 13.7207 1.81736
\(58\) 0 0
\(59\) 9.17319 1.19425 0.597124 0.802149i \(-0.296310\pi\)
0.597124 + 0.802149i \(0.296310\pi\)
\(60\) 0 0
\(61\) 9.41552 1.20553 0.602767 0.797917i \(-0.294065\pi\)
0.602767 + 0.797917i \(0.294065\pi\)
\(62\) 0 0
\(63\) 2.02804 0.255509
\(64\) 0 0
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) −5.80828 −0.709595 −0.354797 0.934943i \(-0.615450\pi\)
−0.354797 + 0.934943i \(0.615450\pi\)
\(68\) 0 0
\(69\) −12.2203 −1.47115
\(70\) 0 0
\(71\) −5.83994 −0.693074 −0.346537 0.938036i \(-0.612642\pi\)
−0.346537 + 0.938036i \(0.612642\pi\)
\(72\) 0 0
\(73\) −8.27638 −0.968677 −0.484339 0.874881i \(-0.660940\pi\)
−0.484339 + 0.874881i \(0.660940\pi\)
\(74\) 0 0
\(75\) 2.24233 0.258922
\(76\) 0 0
\(77\) −5.09092 −0.580164
\(78\) 0 0
\(79\) 3.86437 0.434775 0.217388 0.976085i \(-0.430246\pi\)
0.217388 + 0.976085i \(0.430246\pi\)
\(80\) 0 0
\(81\) −10.9712 −1.21902
\(82\) 0 0
\(83\) −8.61042 −0.945116 −0.472558 0.881300i \(-0.656669\pi\)
−0.472558 + 0.881300i \(0.656669\pi\)
\(84\) 0 0
\(85\) 2.63792 0.286123
\(86\) 0 0
\(87\) −14.5132 −1.55598
\(88\) 0 0
\(89\) −11.7618 −1.24675 −0.623376 0.781923i \(-0.714239\pi\)
−0.623376 + 0.781923i \(0.714239\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) −7.86437 −0.815497
\(94\) 0 0
\(95\) −6.11896 −0.627792
\(96\) 0 0
\(97\) 8.15062 0.827570 0.413785 0.910375i \(-0.364206\pi\)
0.413785 + 0.910375i \(0.364206\pi\)
\(98\) 0 0
\(99\) −10.3246 −1.03766
\(100\) 0 0
\(101\) −14.9869 −1.49126 −0.745629 0.666362i \(-0.767851\pi\)
−0.745629 + 0.666362i \(0.767851\pi\)
\(102\) 0 0
\(103\) −13.4923 −1.32944 −0.664719 0.747093i \(-0.731449\pi\)
−0.664719 + 0.747093i \(0.731449\pi\)
\(104\) 0 0
\(105\) −2.24233 −0.218829
\(106\) 0 0
\(107\) −5.04876 −0.488082 −0.244041 0.969765i \(-0.578473\pi\)
−0.244041 + 0.969765i \(0.578473\pi\)
\(108\) 0 0
\(109\) 6.11217 0.585440 0.292720 0.956198i \(-0.405440\pi\)
0.292720 + 0.956198i \(0.405440\pi\)
\(110\) 0 0
\(111\) 1.02804 0.0975774
\(112\) 0 0
\(113\) 13.9002 1.30762 0.653810 0.756659i \(-0.273170\pi\)
0.653810 + 0.756659i \(0.273170\pi\)
\(114\) 0 0
\(115\) 5.44982 0.508199
\(116\) 0 0
\(117\) −2.02804 −0.187493
\(118\) 0 0
\(119\) −2.63792 −0.241818
\(120\) 0 0
\(121\) 14.9175 1.35613
\(122\) 0 0
\(123\) −16.0412 −1.44638
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 16.1687 1.43474 0.717370 0.696692i \(-0.245346\pi\)
0.717370 + 0.696692i \(0.245346\pi\)
\(128\) 0 0
\(129\) 5.87532 0.517293
\(130\) 0 0
\(131\) 19.9423 1.74237 0.871185 0.490954i \(-0.163352\pi\)
0.871185 + 0.490954i \(0.163352\pi\)
\(132\) 0 0
\(133\) 6.11896 0.530581
\(134\) 0 0
\(135\) 2.17945 0.187577
\(136\) 0 0
\(137\) 22.5405 1.92576 0.962882 0.269922i \(-0.0869978\pi\)
0.962882 + 0.269922i \(0.0869978\pi\)
\(138\) 0 0
\(139\) −19.4855 −1.65274 −0.826371 0.563127i \(-0.809598\pi\)
−0.826371 + 0.563127i \(0.809598\pi\)
\(140\) 0 0
\(141\) −18.1506 −1.52856
\(142\) 0 0
\(143\) 5.09092 0.425724
\(144\) 0 0
\(145\) 6.47239 0.537503
\(146\) 0 0
\(147\) 2.24233 0.184944
\(148\) 0 0
\(149\) −10.5159 −0.861494 −0.430747 0.902473i \(-0.641750\pi\)
−0.430747 + 0.902473i \(0.641750\pi\)
\(150\) 0 0
\(151\) −14.4613 −1.17684 −0.588422 0.808554i \(-0.700251\pi\)
−0.588422 + 0.808554i \(0.700251\pi\)
\(152\) 0 0
\(153\) −5.34981 −0.432507
\(154\) 0 0
\(155\) 3.50723 0.281708
\(156\) 0 0
\(157\) 1.74790 0.139498 0.0697489 0.997565i \(-0.477780\pi\)
0.0697489 + 0.997565i \(0.477780\pi\)
\(158\) 0 0
\(159\) 8.40775 0.666778
\(160\) 0 0
\(161\) −5.44982 −0.429506
\(162\) 0 0
\(163\) −4.80345 −0.376235 −0.188118 0.982147i \(-0.560239\pi\)
−0.188118 + 0.982147i \(0.560239\pi\)
\(164\) 0 0
\(165\) 11.4155 0.888697
\(166\) 0 0
\(167\) −12.1367 −0.939166 −0.469583 0.882888i \(-0.655596\pi\)
−0.469583 + 0.882888i \(0.655596\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 12.4095 0.948979
\(172\) 0 0
\(173\) 5.45901 0.415041 0.207520 0.978231i \(-0.433461\pi\)
0.207520 + 0.978231i \(0.433461\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 20.5693 1.54609
\(178\) 0 0
\(179\) −12.0323 −0.899335 −0.449668 0.893196i \(-0.648458\pi\)
−0.449668 + 0.893196i \(0.648458\pi\)
\(180\) 0 0
\(181\) −3.50142 −0.260258 −0.130129 0.991497i \(-0.541539\pi\)
−0.130129 + 0.991497i \(0.541539\pi\)
\(182\) 0 0
\(183\) 21.1127 1.56070
\(184\) 0 0
\(185\) −0.458471 −0.0337074
\(186\) 0 0
\(187\) 13.4294 0.982058
\(188\) 0 0
\(189\) −2.17945 −0.158532
\(190\) 0 0
\(191\) −0.371166 −0.0268566 −0.0134283 0.999910i \(-0.504274\pi\)
−0.0134283 + 0.999910i \(0.504274\pi\)
\(192\) 0 0
\(193\) −14.3314 −1.03160 −0.515798 0.856710i \(-0.672505\pi\)
−0.515798 + 0.856710i \(0.672505\pi\)
\(194\) 0 0
\(195\) 2.24233 0.160577
\(196\) 0 0
\(197\) −18.6403 −1.32807 −0.664034 0.747703i \(-0.731157\pi\)
−0.664034 + 0.747703i \(0.731157\pi\)
\(198\) 0 0
\(199\) 8.80477 0.624154 0.312077 0.950057i \(-0.398975\pi\)
0.312077 + 0.950057i \(0.398975\pi\)
\(200\) 0 0
\(201\) −13.0241 −0.918648
\(202\) 0 0
\(203\) −6.47239 −0.454273
\(204\) 0 0
\(205\) 7.15380 0.499643
\(206\) 0 0
\(207\) −11.0525 −0.768200
\(208\) 0 0
\(209\) −31.1512 −2.15477
\(210\) 0 0
\(211\) 16.8270 1.15842 0.579209 0.815179i \(-0.303361\pi\)
0.579209 + 0.815179i \(0.303361\pi\)
\(212\) 0 0
\(213\) −13.0951 −0.897260
\(214\) 0 0
\(215\) −2.62019 −0.178695
\(216\) 0 0
\(217\) −3.50723 −0.238086
\(218\) 0 0
\(219\) −18.5584 −1.25406
\(220\) 0 0
\(221\) 2.63792 0.177446
\(222\) 0 0
\(223\) 14.7610 0.988472 0.494236 0.869328i \(-0.335448\pi\)
0.494236 + 0.869328i \(0.335448\pi\)
\(224\) 0 0
\(225\) 2.02804 0.135203
\(226\) 0 0
\(227\) 21.2596 1.41105 0.705525 0.708685i \(-0.250711\pi\)
0.705525 + 0.708685i \(0.250711\pi\)
\(228\) 0 0
\(229\) −19.6072 −1.29568 −0.647841 0.761775i \(-0.724328\pi\)
−0.647841 + 0.761775i \(0.724328\pi\)
\(230\) 0 0
\(231\) −11.4155 −0.751086
\(232\) 0 0
\(233\) 4.45714 0.291997 0.145999 0.989285i \(-0.453360\pi\)
0.145999 + 0.989285i \(0.453360\pi\)
\(234\) 0 0
\(235\) 8.09454 0.528029
\(236\) 0 0
\(237\) 8.66519 0.562864
\(238\) 0 0
\(239\) 14.5871 0.943558 0.471779 0.881717i \(-0.343612\pi\)
0.471779 + 0.881717i \(0.343612\pi\)
\(240\) 0 0
\(241\) 23.8588 1.53688 0.768439 0.639923i \(-0.221034\pi\)
0.768439 + 0.639923i \(0.221034\pi\)
\(242\) 0 0
\(243\) −18.0626 −1.15872
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −6.11896 −0.389340
\(248\) 0 0
\(249\) −19.3074 −1.22356
\(250\) 0 0
\(251\) 2.99946 0.189324 0.0946621 0.995509i \(-0.469823\pi\)
0.0946621 + 0.995509i \(0.469823\pi\)
\(252\) 0 0
\(253\) 27.7446 1.74429
\(254\) 0 0
\(255\) 5.91509 0.370417
\(256\) 0 0
\(257\) 12.4674 0.777693 0.388846 0.921303i \(-0.372874\pi\)
0.388846 + 0.921303i \(0.372874\pi\)
\(258\) 0 0
\(259\) 0.458471 0.0284880
\(260\) 0 0
\(261\) −13.1263 −0.812497
\(262\) 0 0
\(263\) 13.3601 0.823817 0.411909 0.911225i \(-0.364862\pi\)
0.411909 + 0.911225i \(0.364862\pi\)
\(264\) 0 0
\(265\) −3.74956 −0.230334
\(266\) 0 0
\(267\) −26.3739 −1.61406
\(268\) 0 0
\(269\) 2.98695 0.182117 0.0910587 0.995846i \(-0.470975\pi\)
0.0910587 + 0.995846i \(0.470975\pi\)
\(270\) 0 0
\(271\) −15.0296 −0.912981 −0.456490 0.889728i \(-0.650894\pi\)
−0.456490 + 0.889728i \(0.650894\pi\)
\(272\) 0 0
\(273\) −2.24233 −0.135712
\(274\) 0 0
\(275\) −5.09092 −0.306994
\(276\) 0 0
\(277\) −28.5000 −1.71240 −0.856199 0.516646i \(-0.827180\pi\)
−0.856199 + 0.516646i \(0.827180\pi\)
\(278\) 0 0
\(279\) −7.11281 −0.425833
\(280\) 0 0
\(281\) 21.2145 1.26555 0.632775 0.774336i \(-0.281916\pi\)
0.632775 + 0.774336i \(0.281916\pi\)
\(282\) 0 0
\(283\) 10.9340 0.649956 0.324978 0.945722i \(-0.394643\pi\)
0.324978 + 0.945722i \(0.394643\pi\)
\(284\) 0 0
\(285\) −13.7207 −0.812746
\(286\) 0 0
\(287\) −7.15380 −0.422276
\(288\) 0 0
\(289\) −10.0414 −0.590669
\(290\) 0 0
\(291\) 18.2764 1.07138
\(292\) 0 0
\(293\) −28.5931 −1.67043 −0.835214 0.549925i \(-0.814656\pi\)
−0.835214 + 0.549925i \(0.814656\pi\)
\(294\) 0 0
\(295\) −9.17319 −0.534084
\(296\) 0 0
\(297\) 11.0954 0.643821
\(298\) 0 0
\(299\) 5.44982 0.315171
\(300\) 0 0
\(301\) 2.62019 0.151025
\(302\) 0 0
\(303\) −33.6057 −1.93060
\(304\) 0 0
\(305\) −9.41552 −0.539131
\(306\) 0 0
\(307\) −23.1899 −1.32352 −0.661760 0.749716i \(-0.730190\pi\)
−0.661760 + 0.749716i \(0.730190\pi\)
\(308\) 0 0
\(309\) −30.2542 −1.72110
\(310\) 0 0
\(311\) 31.7370 1.79964 0.899821 0.436258i \(-0.143697\pi\)
0.899821 + 0.436258i \(0.143697\pi\)
\(312\) 0 0
\(313\) 20.8433 1.17813 0.589067 0.808084i \(-0.299495\pi\)
0.589067 + 0.808084i \(0.299495\pi\)
\(314\) 0 0
\(315\) −2.02804 −0.114267
\(316\) 0 0
\(317\) 2.80140 0.157342 0.0786711 0.996901i \(-0.474932\pi\)
0.0786711 + 0.996901i \(0.474932\pi\)
\(318\) 0 0
\(319\) 32.9504 1.84487
\(320\) 0 0
\(321\) −11.3210 −0.631876
\(322\) 0 0
\(323\) −16.1413 −0.898128
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) 13.7055 0.757916
\(328\) 0 0
\(329\) −8.09454 −0.446266
\(330\) 0 0
\(331\) 0.789678 0.0434046 0.0217023 0.999764i \(-0.493091\pi\)
0.0217023 + 0.999764i \(0.493091\pi\)
\(332\) 0 0
\(333\) 0.929797 0.0509526
\(334\) 0 0
\(335\) 5.80828 0.317340
\(336\) 0 0
\(337\) −15.4083 −0.839343 −0.419672 0.907676i \(-0.637855\pi\)
−0.419672 + 0.907676i \(0.637855\pi\)
\(338\) 0 0
\(339\) 31.1688 1.69286
\(340\) 0 0
\(341\) 17.8550 0.966905
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 12.2203 0.657919
\(346\) 0 0
\(347\) 29.2569 1.57059 0.785296 0.619121i \(-0.212511\pi\)
0.785296 + 0.619121i \(0.212511\pi\)
\(348\) 0 0
\(349\) −32.5918 −1.74460 −0.872299 0.488972i \(-0.837372\pi\)
−0.872299 + 0.488972i \(0.837372\pi\)
\(350\) 0 0
\(351\) 2.17945 0.116330
\(352\) 0 0
\(353\) 4.27160 0.227354 0.113677 0.993518i \(-0.463737\pi\)
0.113677 + 0.993518i \(0.463737\pi\)
\(354\) 0 0
\(355\) 5.83994 0.309952
\(356\) 0 0
\(357\) −5.91509 −0.313060
\(358\) 0 0
\(359\) −20.0108 −1.05613 −0.528066 0.849203i \(-0.677083\pi\)
−0.528066 + 0.849203i \(0.677083\pi\)
\(360\) 0 0
\(361\) 18.4417 0.970616
\(362\) 0 0
\(363\) 33.4499 1.75566
\(364\) 0 0
\(365\) 8.27638 0.433206
\(366\) 0 0
\(367\) 22.6944 1.18464 0.592320 0.805703i \(-0.298212\pi\)
0.592320 + 0.805703i \(0.298212\pi\)
\(368\) 0 0
\(369\) −14.5082 −0.755267
\(370\) 0 0
\(371\) 3.74956 0.194668
\(372\) 0 0
\(373\) −22.2489 −1.15200 −0.576001 0.817449i \(-0.695388\pi\)
−0.576001 + 0.817449i \(0.695388\pi\)
\(374\) 0 0
\(375\) −2.24233 −0.115793
\(376\) 0 0
\(377\) 6.47239 0.333345
\(378\) 0 0
\(379\) 5.53342 0.284233 0.142116 0.989850i \(-0.454609\pi\)
0.142116 + 0.989850i \(0.454609\pi\)
\(380\) 0 0
\(381\) 36.2556 1.85743
\(382\) 0 0
\(383\) 37.6545 1.92406 0.962028 0.272950i \(-0.0879993\pi\)
0.962028 + 0.272950i \(0.0879993\pi\)
\(384\) 0 0
\(385\) 5.09092 0.259457
\(386\) 0 0
\(387\) 5.31385 0.270118
\(388\) 0 0
\(389\) 13.3921 0.679004 0.339502 0.940605i \(-0.389741\pi\)
0.339502 + 0.940605i \(0.389741\pi\)
\(390\) 0 0
\(391\) 14.3762 0.727036
\(392\) 0 0
\(393\) 44.7173 2.25569
\(394\) 0 0
\(395\) −3.86437 −0.194437
\(396\) 0 0
\(397\) −33.7589 −1.69431 −0.847156 0.531344i \(-0.821687\pi\)
−0.847156 + 0.531344i \(0.821687\pi\)
\(398\) 0 0
\(399\) 13.7207 0.686896
\(400\) 0 0
\(401\) −28.5245 −1.42445 −0.712223 0.701953i \(-0.752312\pi\)
−0.712223 + 0.701953i \(0.752312\pi\)
\(402\) 0 0
\(403\) 3.50723 0.174708
\(404\) 0 0
\(405\) 10.9712 0.545162
\(406\) 0 0
\(407\) −2.33404 −0.115694
\(408\) 0 0
\(409\) −13.6113 −0.673035 −0.336517 0.941677i \(-0.609249\pi\)
−0.336517 + 0.941677i \(0.609249\pi\)
\(410\) 0 0
\(411\) 50.5432 2.49311
\(412\) 0 0
\(413\) 9.17319 0.451383
\(414\) 0 0
\(415\) 8.61042 0.422669
\(416\) 0 0
\(417\) −43.6930 −2.13965
\(418\) 0 0
\(419\) 19.8892 0.971653 0.485827 0.874055i \(-0.338519\pi\)
0.485827 + 0.874055i \(0.338519\pi\)
\(420\) 0 0
\(421\) −4.98324 −0.242868 −0.121434 0.992599i \(-0.538749\pi\)
−0.121434 + 0.992599i \(0.538749\pi\)
\(422\) 0 0
\(423\) −16.4161 −0.798176
\(424\) 0 0
\(425\) −2.63792 −0.127958
\(426\) 0 0
\(427\) 9.41552 0.455649
\(428\) 0 0
\(429\) 11.4155 0.551147
\(430\) 0 0
\(431\) 30.2146 1.45538 0.727692 0.685904i \(-0.240593\pi\)
0.727692 + 0.685904i \(0.240593\pi\)
\(432\) 0 0
\(433\) −7.49507 −0.360190 −0.180095 0.983649i \(-0.557641\pi\)
−0.180095 + 0.983649i \(0.557641\pi\)
\(434\) 0 0
\(435\) 14.5132 0.695857
\(436\) 0 0
\(437\) −33.3473 −1.59522
\(438\) 0 0
\(439\) −9.97390 −0.476028 −0.238014 0.971262i \(-0.576496\pi\)
−0.238014 + 0.971262i \(0.576496\pi\)
\(440\) 0 0
\(441\) 2.02804 0.0965734
\(442\) 0 0
\(443\) −19.5073 −0.926821 −0.463410 0.886144i \(-0.653374\pi\)
−0.463410 + 0.886144i \(0.653374\pi\)
\(444\) 0 0
\(445\) 11.7618 0.557564
\(446\) 0 0
\(447\) −23.5801 −1.11530
\(448\) 0 0
\(449\) −38.6895 −1.82587 −0.912934 0.408107i \(-0.866189\pi\)
−0.912934 + 0.408107i \(0.866189\pi\)
\(450\) 0 0
\(451\) 36.4194 1.71492
\(452\) 0 0
\(453\) −32.4270 −1.52355
\(454\) 0 0
\(455\) 1.00000 0.0468807
\(456\) 0 0
\(457\) 33.6615 1.57462 0.787308 0.616559i \(-0.211474\pi\)
0.787308 + 0.616559i \(0.211474\pi\)
\(458\) 0 0
\(459\) 5.74922 0.268350
\(460\) 0 0
\(461\) 6.26378 0.291733 0.145867 0.989304i \(-0.453403\pi\)
0.145867 + 0.989304i \(0.453403\pi\)
\(462\) 0 0
\(463\) −36.6265 −1.70218 −0.851089 0.525022i \(-0.824057\pi\)
−0.851089 + 0.525022i \(0.824057\pi\)
\(464\) 0 0
\(465\) 7.86437 0.364701
\(466\) 0 0
\(467\) −19.6170 −0.907767 −0.453883 0.891061i \(-0.649962\pi\)
−0.453883 + 0.891061i \(0.649962\pi\)
\(468\) 0 0
\(469\) −5.80828 −0.268202
\(470\) 0 0
\(471\) 3.91937 0.180595
\(472\) 0 0
\(473\) −13.3392 −0.613335
\(474\) 0 0
\(475\) 6.11896 0.280757
\(476\) 0 0
\(477\) 7.60427 0.348175
\(478\) 0 0
\(479\) −4.13528 −0.188946 −0.0944729 0.995527i \(-0.530117\pi\)
−0.0944729 + 0.995527i \(0.530117\pi\)
\(480\) 0 0
\(481\) −0.458471 −0.0209045
\(482\) 0 0
\(483\) −12.2203 −0.556043
\(484\) 0 0
\(485\) −8.15062 −0.370101
\(486\) 0 0
\(487\) 6.35387 0.287921 0.143961 0.989583i \(-0.454016\pi\)
0.143961 + 0.989583i \(0.454016\pi\)
\(488\) 0 0
\(489\) −10.7709 −0.487078
\(490\) 0 0
\(491\) −7.82665 −0.353212 −0.176606 0.984282i \(-0.556512\pi\)
−0.176606 + 0.984282i \(0.556512\pi\)
\(492\) 0 0
\(493\) 17.0737 0.768959
\(494\) 0 0
\(495\) 10.3246 0.464056
\(496\) 0 0
\(497\) −5.83994 −0.261957
\(498\) 0 0
\(499\) 8.51929 0.381376 0.190688 0.981651i \(-0.438928\pi\)
0.190688 + 0.981651i \(0.438928\pi\)
\(500\) 0 0
\(501\) −27.2145 −1.21585
\(502\) 0 0
\(503\) −23.1975 −1.03432 −0.517162 0.855887i \(-0.673012\pi\)
−0.517162 + 0.855887i \(0.673012\pi\)
\(504\) 0 0
\(505\) 14.9869 0.666910
\(506\) 0 0
\(507\) 2.24233 0.0995854
\(508\) 0 0
\(509\) −20.4922 −0.908302 −0.454151 0.890925i \(-0.650057\pi\)
−0.454151 + 0.890925i \(0.650057\pi\)
\(510\) 0 0
\(511\) −8.27638 −0.366126
\(512\) 0 0
\(513\) −13.3360 −0.588798
\(514\) 0 0
\(515\) 13.4923 0.594543
\(516\) 0 0
\(517\) 41.2087 1.81235
\(518\) 0 0
\(519\) 12.2409 0.537316
\(520\) 0 0
\(521\) 9.14638 0.400710 0.200355 0.979723i \(-0.435790\pi\)
0.200355 + 0.979723i \(0.435790\pi\)
\(522\) 0 0
\(523\) 3.33395 0.145783 0.0728917 0.997340i \(-0.476777\pi\)
0.0728917 + 0.997340i \(0.476777\pi\)
\(524\) 0 0
\(525\) 2.24233 0.0978633
\(526\) 0 0
\(527\) 9.25180 0.403015
\(528\) 0 0
\(529\) 6.70056 0.291329
\(530\) 0 0
\(531\) 18.6036 0.807328
\(532\) 0 0
\(533\) 7.15380 0.309866
\(534\) 0 0
\(535\) 5.04876 0.218277
\(536\) 0 0
\(537\) −26.9804 −1.16429
\(538\) 0 0
\(539\) −5.09092 −0.219282
\(540\) 0 0
\(541\) 19.9936 0.859594 0.429797 0.902926i \(-0.358585\pi\)
0.429797 + 0.902926i \(0.358585\pi\)
\(542\) 0 0
\(543\) −7.85133 −0.336933
\(544\) 0 0
\(545\) −6.11217 −0.261817
\(546\) 0 0
\(547\) 26.7271 1.14277 0.571384 0.820683i \(-0.306407\pi\)
0.571384 + 0.820683i \(0.306407\pi\)
\(548\) 0 0
\(549\) 19.0951 0.814958
\(550\) 0 0
\(551\) −39.6043 −1.68720
\(552\) 0 0
\(553\) 3.86437 0.164330
\(554\) 0 0
\(555\) −1.02804 −0.0436380
\(556\) 0 0
\(557\) −1.05415 −0.0446656 −0.0223328 0.999751i \(-0.507109\pi\)
−0.0223328 + 0.999751i \(0.507109\pi\)
\(558\) 0 0
\(559\) −2.62019 −0.110822
\(560\) 0 0
\(561\) 30.1132 1.27138
\(562\) 0 0
\(563\) 15.3526 0.647037 0.323518 0.946222i \(-0.395134\pi\)
0.323518 + 0.946222i \(0.395134\pi\)
\(564\) 0 0
\(565\) −13.9002 −0.584785
\(566\) 0 0
\(567\) −10.9712 −0.460746
\(568\) 0 0
\(569\) 37.9806 1.59223 0.796114 0.605147i \(-0.206886\pi\)
0.796114 + 0.605147i \(0.206886\pi\)
\(570\) 0 0
\(571\) 15.5848 0.652202 0.326101 0.945335i \(-0.394265\pi\)
0.326101 + 0.945335i \(0.394265\pi\)
\(572\) 0 0
\(573\) −0.832276 −0.0347688
\(574\) 0 0
\(575\) −5.44982 −0.227273
\(576\) 0 0
\(577\) −20.0473 −0.834579 −0.417290 0.908774i \(-0.637020\pi\)
−0.417290 + 0.908774i \(0.637020\pi\)
\(578\) 0 0
\(579\) −32.1357 −1.33551
\(580\) 0 0
\(581\) −8.61042 −0.357220
\(582\) 0 0
\(583\) −19.0887 −0.790574
\(584\) 0 0
\(585\) 2.02804 0.0838492
\(586\) 0 0
\(587\) 39.1180 1.61457 0.807287 0.590159i \(-0.200935\pi\)
0.807287 + 0.590159i \(0.200935\pi\)
\(588\) 0 0
\(589\) −21.4606 −0.884269
\(590\) 0 0
\(591\) −41.7977 −1.71933
\(592\) 0 0
\(593\) 6.30917 0.259087 0.129543 0.991574i \(-0.458649\pi\)
0.129543 + 0.991574i \(0.458649\pi\)
\(594\) 0 0
\(595\) 2.63792 0.108144
\(596\) 0 0
\(597\) 19.7432 0.808036
\(598\) 0 0
\(599\) 30.9916 1.26628 0.633140 0.774037i \(-0.281766\pi\)
0.633140 + 0.774037i \(0.281766\pi\)
\(600\) 0 0
\(601\) 2.65801 0.108423 0.0542113 0.998529i \(-0.482736\pi\)
0.0542113 + 0.998529i \(0.482736\pi\)
\(602\) 0 0
\(603\) −11.7794 −0.479696
\(604\) 0 0
\(605\) −14.9175 −0.606482
\(606\) 0 0
\(607\) −3.48607 −0.141495 −0.0707477 0.997494i \(-0.522539\pi\)
−0.0707477 + 0.997494i \(0.522539\pi\)
\(608\) 0 0
\(609\) −14.5132 −0.588106
\(610\) 0 0
\(611\) 8.09454 0.327470
\(612\) 0 0
\(613\) −41.1579 −1.66235 −0.831175 0.556011i \(-0.812331\pi\)
−0.831175 + 0.556011i \(0.812331\pi\)
\(614\) 0 0
\(615\) 16.0412 0.646843
\(616\) 0 0
\(617\) −6.57548 −0.264719 −0.132359 0.991202i \(-0.542255\pi\)
−0.132359 + 0.991202i \(0.542255\pi\)
\(618\) 0 0
\(619\) 12.0917 0.486007 0.243003 0.970025i \(-0.421867\pi\)
0.243003 + 0.970025i \(0.421867\pi\)
\(620\) 0 0
\(621\) 11.8776 0.476632
\(622\) 0 0
\(623\) −11.7618 −0.471228
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −69.8512 −2.78959
\(628\) 0 0
\(629\) −1.20941 −0.0482223
\(630\) 0 0
\(631\) −8.45054 −0.336411 −0.168205 0.985752i \(-0.553797\pi\)
−0.168205 + 0.985752i \(0.553797\pi\)
\(632\) 0 0
\(633\) 37.7317 1.49970
\(634\) 0 0
\(635\) −16.1687 −0.641635
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) −11.8436 −0.468528
\(640\) 0 0
\(641\) 7.16904 0.283160 0.141580 0.989927i \(-0.454782\pi\)
0.141580 + 0.989927i \(0.454782\pi\)
\(642\) 0 0
\(643\) −29.0789 −1.14676 −0.573379 0.819290i \(-0.694368\pi\)
−0.573379 + 0.819290i \(0.694368\pi\)
\(644\) 0 0
\(645\) −5.87532 −0.231340
\(646\) 0 0
\(647\) 28.9981 1.14003 0.570016 0.821633i \(-0.306937\pi\)
0.570016 + 0.821633i \(0.306937\pi\)
\(648\) 0 0
\(649\) −46.7000 −1.83314
\(650\) 0 0
\(651\) −7.86437 −0.308229
\(652\) 0 0
\(653\) −5.94499 −0.232646 −0.116323 0.993211i \(-0.537111\pi\)
−0.116323 + 0.993211i \(0.537111\pi\)
\(654\) 0 0
\(655\) −19.9423 −0.779212
\(656\) 0 0
\(657\) −16.7848 −0.654839
\(658\) 0 0
\(659\) −14.2339 −0.554473 −0.277237 0.960802i \(-0.589419\pi\)
−0.277237 + 0.960802i \(0.589419\pi\)
\(660\) 0 0
\(661\) −2.19138 −0.0852349 −0.0426175 0.999091i \(-0.513570\pi\)
−0.0426175 + 0.999091i \(0.513570\pi\)
\(662\) 0 0
\(663\) 5.91509 0.229723
\(664\) 0 0
\(665\) −6.11896 −0.237283
\(666\) 0 0
\(667\) 35.2734 1.36579
\(668\) 0 0
\(669\) 33.0991 1.27969
\(670\) 0 0
\(671\) −47.9337 −1.85046
\(672\) 0 0
\(673\) 9.31430 0.359040 0.179520 0.983754i \(-0.442546\pi\)
0.179520 + 0.983754i \(0.442546\pi\)
\(674\) 0 0
\(675\) −2.17945 −0.0838871
\(676\) 0 0
\(677\) 1.32289 0.0508428 0.0254214 0.999677i \(-0.491907\pi\)
0.0254214 + 0.999677i \(0.491907\pi\)
\(678\) 0 0
\(679\) 8.15062 0.312792
\(680\) 0 0
\(681\) 47.6711 1.82676
\(682\) 0 0
\(683\) −16.3381 −0.625160 −0.312580 0.949892i \(-0.601193\pi\)
−0.312580 + 0.949892i \(0.601193\pi\)
\(684\) 0 0
\(685\) −22.5405 −0.861228
\(686\) 0 0
\(687\) −43.9659 −1.67740
\(688\) 0 0
\(689\) −3.74956 −0.142847
\(690\) 0 0
\(691\) 28.7187 1.09251 0.546256 0.837618i \(-0.316053\pi\)
0.546256 + 0.837618i \(0.316053\pi\)
\(692\) 0 0
\(693\) −10.3246 −0.392199
\(694\) 0 0
\(695\) 19.4855 0.739128
\(696\) 0 0
\(697\) 18.8712 0.714796
\(698\) 0 0
\(699\) 9.99439 0.378022
\(700\) 0 0
\(701\) 2.41572 0.0912404 0.0456202 0.998959i \(-0.485474\pi\)
0.0456202 + 0.998959i \(0.485474\pi\)
\(702\) 0 0
\(703\) 2.80536 0.105806
\(704\) 0 0
\(705\) 18.1506 0.683592
\(706\) 0 0
\(707\) −14.9869 −0.563642
\(708\) 0 0
\(709\) −19.2385 −0.722516 −0.361258 0.932466i \(-0.617653\pi\)
−0.361258 + 0.932466i \(0.617653\pi\)
\(710\) 0 0
\(711\) 7.83710 0.293914
\(712\) 0 0
\(713\) 19.1138 0.715817
\(714\) 0 0
\(715\) −5.09092 −0.190390
\(716\) 0 0
\(717\) 32.7090 1.22154
\(718\) 0 0
\(719\) −40.1912 −1.49888 −0.749440 0.662073i \(-0.769677\pi\)
−0.749440 + 0.662073i \(0.769677\pi\)
\(720\) 0 0
\(721\) −13.4923 −0.502480
\(722\) 0 0
\(723\) 53.4992 1.98966
\(724\) 0 0
\(725\) −6.47239 −0.240379
\(726\) 0 0
\(727\) −38.8861 −1.44221 −0.721103 0.692828i \(-0.756364\pi\)
−0.721103 + 0.692828i \(0.756364\pi\)
\(728\) 0 0
\(729\) −7.58886 −0.281069
\(730\) 0 0
\(731\) −6.91184 −0.255644
\(732\) 0 0
\(733\) −32.8166 −1.21211 −0.606054 0.795423i \(-0.707248\pi\)
−0.606054 + 0.795423i \(0.707248\pi\)
\(734\) 0 0
\(735\) −2.24233 −0.0827096
\(736\) 0 0
\(737\) 29.5695 1.08921
\(738\) 0 0
\(739\) 53.2128 1.95746 0.978732 0.205141i \(-0.0657653\pi\)
0.978732 + 0.205141i \(0.0657653\pi\)
\(740\) 0 0
\(741\) −13.7207 −0.504044
\(742\) 0 0
\(743\) −10.5818 −0.388210 −0.194105 0.980981i \(-0.562180\pi\)
−0.194105 + 0.980981i \(0.562180\pi\)
\(744\) 0 0
\(745\) 10.5159 0.385272
\(746\) 0 0
\(747\) −17.4623 −0.638912
\(748\) 0 0
\(749\) −5.04876 −0.184478
\(750\) 0 0
\(751\) −19.8391 −0.723938 −0.361969 0.932190i \(-0.617895\pi\)
−0.361969 + 0.932190i \(0.617895\pi\)
\(752\) 0 0
\(753\) 6.72578 0.245101
\(754\) 0 0
\(755\) 14.4613 0.526301
\(756\) 0 0
\(757\) −26.3012 −0.955935 −0.477967 0.878378i \(-0.658626\pi\)
−0.477967 + 0.878378i \(0.658626\pi\)
\(758\) 0 0
\(759\) 62.2126 2.25817
\(760\) 0 0
\(761\) 45.4116 1.64617 0.823084 0.567920i \(-0.192252\pi\)
0.823084 + 0.567920i \(0.192252\pi\)
\(762\) 0 0
\(763\) 6.11217 0.221275
\(764\) 0 0
\(765\) 5.34981 0.193423
\(766\) 0 0
\(767\) −9.17319 −0.331225
\(768\) 0 0
\(769\) 6.40438 0.230948 0.115474 0.993311i \(-0.463161\pi\)
0.115474 + 0.993311i \(0.463161\pi\)
\(770\) 0 0
\(771\) 27.9559 1.00681
\(772\) 0 0
\(773\) −15.4161 −0.554477 −0.277239 0.960801i \(-0.589419\pi\)
−0.277239 + 0.960801i \(0.589419\pi\)
\(774\) 0 0
\(775\) −3.50723 −0.125983
\(776\) 0 0
\(777\) 1.02804 0.0368808
\(778\) 0 0
\(779\) −43.7738 −1.56836
\(780\) 0 0
\(781\) 29.7307 1.06385
\(782\) 0 0
\(783\) 14.1063 0.504117
\(784\) 0 0
\(785\) −1.74790 −0.0623853
\(786\) 0 0
\(787\) −11.4713 −0.408907 −0.204453 0.978876i \(-0.565542\pi\)
−0.204453 + 0.978876i \(0.565542\pi\)
\(788\) 0 0
\(789\) 29.9577 1.06652
\(790\) 0 0
\(791\) 13.9002 0.494234
\(792\) 0 0
\(793\) −9.41552 −0.334355
\(794\) 0 0
\(795\) −8.40775 −0.298192
\(796\) 0 0
\(797\) 38.9694 1.38037 0.690184 0.723634i \(-0.257530\pi\)
0.690184 + 0.723634i \(0.257530\pi\)
\(798\) 0 0
\(799\) 21.3527 0.755406
\(800\) 0 0
\(801\) −23.8535 −0.842821
\(802\) 0 0
\(803\) 42.1344 1.48689
\(804\) 0 0
\(805\) 5.44982 0.192081
\(806\) 0 0
\(807\) 6.69772 0.235771
\(808\) 0 0
\(809\) −19.1824 −0.674416 −0.337208 0.941430i \(-0.609483\pi\)
−0.337208 + 0.941430i \(0.609483\pi\)
\(810\) 0 0
\(811\) 8.52598 0.299388 0.149694 0.988732i \(-0.452171\pi\)
0.149694 + 0.988732i \(0.452171\pi\)
\(812\) 0 0
\(813\) −33.7012 −1.18195
\(814\) 0 0
\(815\) 4.80345 0.168257
\(816\) 0 0
\(817\) 16.0328 0.560917
\(818\) 0 0
\(819\) −2.02804 −0.0708655
\(820\) 0 0
\(821\) 18.5580 0.647680 0.323840 0.946112i \(-0.395026\pi\)
0.323840 + 0.946112i \(0.395026\pi\)
\(822\) 0 0
\(823\) 31.6231 1.10231 0.551155 0.834403i \(-0.314187\pi\)
0.551155 + 0.834403i \(0.314187\pi\)
\(824\) 0 0
\(825\) −11.4155 −0.397438
\(826\) 0 0
\(827\) −40.8572 −1.42074 −0.710371 0.703827i \(-0.751473\pi\)
−0.710371 + 0.703827i \(0.751473\pi\)
\(828\) 0 0
\(829\) 16.4917 0.572780 0.286390 0.958113i \(-0.407545\pi\)
0.286390 + 0.958113i \(0.407545\pi\)
\(830\) 0 0
\(831\) −63.9064 −2.21689
\(832\) 0 0
\(833\) −2.63792 −0.0913985
\(834\) 0 0
\(835\) 12.1367 0.420008
\(836\) 0 0
\(837\) 7.64384 0.264210
\(838\) 0 0
\(839\) 2.49868 0.0862639 0.0431319 0.999069i \(-0.486266\pi\)
0.0431319 + 0.999069i \(0.486266\pi\)
\(840\) 0 0
\(841\) 12.8919 0.444548
\(842\) 0 0
\(843\) 47.5699 1.63839
\(844\) 0 0
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) 14.9175 0.512571
\(848\) 0 0
\(849\) 24.5175 0.841439
\(850\) 0 0
\(851\) −2.49858 −0.0856503
\(852\) 0 0
\(853\) 20.9813 0.718387 0.359194 0.933263i \(-0.383052\pi\)
0.359194 + 0.933263i \(0.383052\pi\)
\(854\) 0 0
\(855\) −12.4095 −0.424396
\(856\) 0 0
\(857\) −40.0604 −1.36844 −0.684219 0.729277i \(-0.739857\pi\)
−0.684219 + 0.729277i \(0.739857\pi\)
\(858\) 0 0
\(859\) 1.19224 0.0406788 0.0203394 0.999793i \(-0.493525\pi\)
0.0203394 + 0.999793i \(0.493525\pi\)
\(860\) 0 0
\(861\) −16.0412 −0.546682
\(862\) 0 0
\(863\) 17.4481 0.593942 0.296971 0.954887i \(-0.404024\pi\)
0.296971 + 0.954887i \(0.404024\pi\)
\(864\) 0 0
\(865\) −5.45901 −0.185612
\(866\) 0 0
\(867\) −22.5161 −0.764686
\(868\) 0 0
\(869\) −19.6732 −0.667367
\(870\) 0 0
\(871\) 5.80828 0.196806
\(872\) 0 0
\(873\) 16.5298 0.559449
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −40.9281 −1.38204 −0.691021 0.722834i \(-0.742839\pi\)
−0.691021 + 0.722834i \(0.742839\pi\)
\(878\) 0 0
\(879\) −64.1152 −2.16255
\(880\) 0 0
\(881\) 13.0769 0.440572 0.220286 0.975435i \(-0.429301\pi\)
0.220286 + 0.975435i \(0.429301\pi\)
\(882\) 0 0
\(883\) 9.18193 0.308997 0.154498 0.987993i \(-0.450624\pi\)
0.154498 + 0.987993i \(0.450624\pi\)
\(884\) 0 0
\(885\) −20.5693 −0.691430
\(886\) 0 0
\(887\) −27.0349 −0.907744 −0.453872 0.891067i \(-0.649958\pi\)
−0.453872 + 0.891067i \(0.649958\pi\)
\(888\) 0 0
\(889\) 16.1687 0.542281
\(890\) 0 0
\(891\) 55.8534 1.87116
\(892\) 0 0
\(893\) −49.5302 −1.65746
\(894\) 0 0
\(895\) 12.0323 0.402195
\(896\) 0 0
\(897\) 12.2203 0.408024
\(898\) 0 0
\(899\) 22.7002 0.757093
\(900\) 0 0
\(901\) −9.89104 −0.329518
\(902\) 0 0
\(903\) 5.87532 0.195518
\(904\) 0 0
\(905\) 3.50142 0.116391
\(906\) 0 0
\(907\) 49.5193 1.64426 0.822131 0.569298i \(-0.192785\pi\)
0.822131 + 0.569298i \(0.192785\pi\)
\(908\) 0 0
\(909\) −30.3942 −1.00811
\(910\) 0 0
\(911\) 24.5483 0.813322 0.406661 0.913579i \(-0.366693\pi\)
0.406661 + 0.913579i \(0.366693\pi\)
\(912\) 0 0
\(913\) 43.8350 1.45073
\(914\) 0 0
\(915\) −21.1127 −0.697965
\(916\) 0 0
\(917\) 19.9423 0.658554
\(918\) 0 0
\(919\) −10.5408 −0.347710 −0.173855 0.984771i \(-0.555622\pi\)
−0.173855 + 0.984771i \(0.555622\pi\)
\(920\) 0 0
\(921\) −51.9995 −1.71344
\(922\) 0 0
\(923\) 5.83994 0.192224
\(924\) 0 0
\(925\) 0.458471 0.0150744
\(926\) 0 0
\(927\) −27.3630 −0.898719
\(928\) 0 0
\(929\) 49.6214 1.62803 0.814013 0.580846i \(-0.197278\pi\)
0.814013 + 0.580846i \(0.197278\pi\)
\(930\) 0 0
\(931\) 6.11896 0.200541
\(932\) 0 0
\(933\) 71.1649 2.32983
\(934\) 0 0
\(935\) −13.4294 −0.439190
\(936\) 0 0
\(937\) −42.4164 −1.38568 −0.692842 0.721090i \(-0.743642\pi\)
−0.692842 + 0.721090i \(0.743642\pi\)
\(938\) 0 0
\(939\) 46.7376 1.52522
\(940\) 0 0
\(941\) 29.0006 0.945391 0.472695 0.881226i \(-0.343281\pi\)
0.472695 + 0.881226i \(0.343281\pi\)
\(942\) 0 0
\(943\) 38.9869 1.26959
\(944\) 0 0
\(945\) 2.17945 0.0708975
\(946\) 0 0
\(947\) 4.42169 0.143685 0.0718427 0.997416i \(-0.477112\pi\)
0.0718427 + 0.997416i \(0.477112\pi\)
\(948\) 0 0
\(949\) 8.27638 0.268663
\(950\) 0 0
\(951\) 6.28166 0.203697
\(952\) 0 0
\(953\) −21.5536 −0.698191 −0.349095 0.937087i \(-0.613511\pi\)
−0.349095 + 0.937087i \(0.613511\pi\)
\(954\) 0 0
\(955\) 0.371166 0.0120106
\(956\) 0 0
\(957\) 73.8858 2.38839
\(958\) 0 0
\(959\) 22.5405 0.727871
\(960\) 0 0
\(961\) −18.6993 −0.603204
\(962\) 0 0
\(963\) −10.2391 −0.329950
\(964\) 0 0
\(965\) 14.3314 0.461344
\(966\) 0 0
\(967\) 14.9791 0.481697 0.240848 0.970563i \(-0.422574\pi\)
0.240848 + 0.970563i \(0.422574\pi\)
\(968\) 0 0
\(969\) −36.1942 −1.16273
\(970\) 0 0
\(971\) 9.31800 0.299029 0.149514 0.988760i \(-0.452229\pi\)
0.149514 + 0.988760i \(0.452229\pi\)
\(972\) 0 0
\(973\) −19.4855 −0.624677
\(974\) 0 0
\(975\) −2.24233 −0.0718120
\(976\) 0 0
\(977\) −18.9661 −0.606780 −0.303390 0.952866i \(-0.598119\pi\)
−0.303390 + 0.952866i \(0.598119\pi\)
\(978\) 0 0
\(979\) 59.8785 1.91373
\(980\) 0 0
\(981\) 12.3957 0.395765
\(982\) 0 0
\(983\) 41.5464 1.32512 0.662562 0.749007i \(-0.269469\pi\)
0.662562 + 0.749007i \(0.269469\pi\)
\(984\) 0 0
\(985\) 18.6403 0.593930
\(986\) 0 0
\(987\) −18.1506 −0.577741
\(988\) 0 0
\(989\) −14.2795 −0.454063
\(990\) 0 0
\(991\) 6.23529 0.198070 0.0990352 0.995084i \(-0.468424\pi\)
0.0990352 + 0.995084i \(0.468424\pi\)
\(992\) 0 0
\(993\) 1.77072 0.0561920
\(994\) 0 0
\(995\) −8.80477 −0.279130
\(996\) 0 0
\(997\) 28.6321 0.906788 0.453394 0.891310i \(-0.350213\pi\)
0.453394 + 0.891310i \(0.350213\pi\)
\(998\) 0 0
\(999\) −0.999214 −0.0316137
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 3640.2.a.y.1.5 5
4.3 odd 2 7280.2.a.cc.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3640.2.a.y.1.5 5 1.1 even 1 trivial
7280.2.a.cc.1.1 5 4.3 odd 2