Properties

Label 5148.2.a.t.1.1
Level $5148$
Weight $2$
Character 5148.1
Self dual yes
Analytic conductor $41.107$
Analytic rank $0$
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] = [5148,2,Mod(1,5148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5148, 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("5148.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5148.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: \(41.1069869606\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.815952.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 9x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.26478\) of defining polynomial
Character \(\chi\) \(=\) 5148.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.15937 q^{5} +0.571762 q^{7} +O(q^{10})\) \(q-2.15937 q^{5} +0.571762 q^{7} -1.00000 q^{11} -1.00000 q^{13} -2.82226 q^{17} -8.00545 q^{19} +4.07750 q^{23} -0.337115 q^{25} +4.46004 q^{29} -8.10355 q^{31} -1.23465 q^{35} -2.75274 q^{37} +12.5636 q^{41} +10.4953 q^{43} +3.60921 q^{47} -6.67309 q^{49} -4.57722 q^{53} +2.15937 q^{55} +10.7880 q^{59} +0.963556 q^{61} +2.15937 q^{65} +1.35872 q^{67} +7.56601 q^{71} -6.06572 q^{73} -0.571762 q^{77} -5.75496 q^{79} +8.01908 q^{83} +6.09430 q^{85} -5.50194 q^{89} -0.571762 q^{91} +17.2867 q^{95} +14.4017 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2 q^{5} - 2 q^{7} - 5 q^{11} - 5 q^{13} - 2 q^{17} - 2 q^{19} + 12 q^{23} - q^{25} + 4 q^{29} - 2 q^{31} - 2 q^{35} + 6 q^{41} + 14 q^{43} + 14 q^{47} - 7 q^{49} + 20 q^{53} - 2 q^{55} + 20 q^{59} - 2 q^{65} + 10 q^{67} + 26 q^{71} + 16 q^{73} + 2 q^{77} + 2 q^{79} + 16 q^{83} - 18 q^{85} + 24 q^{89} + 2 q^{91} + 22 q^{95}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) −2.15937 −0.965700 −0.482850 0.875703i \(-0.660398\pi\)
−0.482850 + 0.875703i \(0.660398\pi\)
\(6\) 0 0
\(7\) 0.571762 0.216106 0.108053 0.994145i \(-0.465538\pi\)
0.108053 + 0.994145i \(0.465538\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.82226 −0.684498 −0.342249 0.939609i \(-0.611189\pi\)
−0.342249 + 0.939609i \(0.611189\pi\)
\(18\) 0 0
\(19\) −8.00545 −1.83658 −0.918289 0.395912i \(-0.870429\pi\)
−0.918289 + 0.395912i \(0.870429\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.07750 0.850217 0.425109 0.905142i \(-0.360236\pi\)
0.425109 + 0.905142i \(0.360236\pi\)
\(24\) 0 0
\(25\) −0.337115 −0.0674231
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.46004 0.828209 0.414105 0.910229i \(-0.364095\pi\)
0.414105 + 0.910229i \(0.364095\pi\)
\(30\) 0 0
\(31\) −8.10355 −1.45544 −0.727720 0.685874i \(-0.759420\pi\)
−0.727720 + 0.685874i \(0.759420\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.23465 −0.208693
\(36\) 0 0
\(37\) −2.75274 −0.452547 −0.226273 0.974064i \(-0.572654\pi\)
−0.226273 + 0.974064i \(0.572654\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 12.5636 1.96210 0.981052 0.193746i \(-0.0620637\pi\)
0.981052 + 0.193746i \(0.0620637\pi\)
\(42\) 0 0
\(43\) 10.4953 1.60052 0.800262 0.599650i \(-0.204693\pi\)
0.800262 + 0.599650i \(0.204693\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.60921 0.526458 0.263229 0.964733i \(-0.415213\pi\)
0.263229 + 0.964733i \(0.415213\pi\)
\(48\) 0 0
\(49\) −6.67309 −0.953298
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.57722 −0.628729 −0.314364 0.949302i \(-0.601791\pi\)
−0.314364 + 0.949302i \(0.601791\pi\)
\(54\) 0 0
\(55\) 2.15937 0.291170
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.7880 1.40448 0.702241 0.711939i \(-0.252183\pi\)
0.702241 + 0.711939i \(0.252183\pi\)
\(60\) 0 0
\(61\) 0.963556 0.123371 0.0616854 0.998096i \(-0.480352\pi\)
0.0616854 + 0.998096i \(0.480352\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.15937 0.267837
\(66\) 0 0
\(67\) 1.35872 0.165994 0.0829969 0.996550i \(-0.473551\pi\)
0.0829969 + 0.996550i \(0.473551\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.56601 0.897920 0.448960 0.893552i \(-0.351795\pi\)
0.448960 + 0.893552i \(0.351795\pi\)
\(72\) 0 0
\(73\) −6.06572 −0.709939 −0.354970 0.934878i \(-0.615509\pi\)
−0.354970 + 0.934878i \(0.615509\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.571762 −0.0651584
\(78\) 0 0
\(79\) −5.75496 −0.647484 −0.323742 0.946145i \(-0.604941\pi\)
−0.323742 + 0.946145i \(0.604941\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.01908 0.880208 0.440104 0.897947i \(-0.354942\pi\)
0.440104 + 0.897947i \(0.354942\pi\)
\(84\) 0 0
\(85\) 6.09430 0.661020
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.50194 −0.583204 −0.291602 0.956540i \(-0.594188\pi\)
−0.291602 + 0.956540i \(0.594188\pi\)
\(90\) 0 0
\(91\) −0.571762 −0.0599370
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 17.2867 1.77358
\(96\) 0 0
\(97\) 14.4017 1.46227 0.731135 0.682232i \(-0.238991\pi\)
0.731135 + 0.682232i \(0.238991\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7.14100 −0.710556 −0.355278 0.934761i \(-0.615614\pi\)
−0.355278 + 0.934761i \(0.615614\pi\)
\(102\) 0 0
\(103\) −1.78918 −0.176293 −0.0881465 0.996108i \(-0.528094\pi\)
−0.0881465 + 0.996108i \(0.528094\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −8.79248 −0.850002 −0.425001 0.905193i \(-0.639726\pi\)
−0.425001 + 0.905193i \(0.639726\pi\)
\(108\) 0 0
\(109\) 20.2684 1.94136 0.970679 0.240379i \(-0.0772717\pi\)
0.970679 + 0.240379i \(0.0772717\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −20.2262 −1.90272 −0.951359 0.308084i \(-0.900312\pi\)
−0.951359 + 0.308084i \(0.900312\pi\)
\(114\) 0 0
\(115\) −8.80484 −0.821055
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.61366 −0.147924
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.5248 1.03081
\(126\) 0 0
\(127\) −17.1763 −1.52415 −0.762075 0.647489i \(-0.775819\pi\)
−0.762075 + 0.647489i \(0.775819\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.83569 −0.422496 −0.211248 0.977432i \(-0.567753\pi\)
−0.211248 + 0.977432i \(0.567753\pi\)
\(132\) 0 0
\(133\) −4.57722 −0.396895
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 15.1032 1.29036 0.645179 0.764032i \(-0.276783\pi\)
0.645179 + 0.764032i \(0.276783\pi\)
\(138\) 0 0
\(139\) 3.21304 0.272527 0.136263 0.990673i \(-0.456491\pi\)
0.136263 + 0.990673i \(0.456491\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) −9.63089 −0.799802
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 23.9973 1.96593 0.982967 0.183782i \(-0.0588340\pi\)
0.982967 + 0.183782i \(0.0588340\pi\)
\(150\) 0 0
\(151\) −8.90958 −0.725051 −0.362526 0.931974i \(-0.618086\pi\)
−0.362526 + 0.931974i \(0.618086\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 17.4986 1.40552
\(156\) 0 0
\(157\) 16.5392 1.31997 0.659986 0.751278i \(-0.270562\pi\)
0.659986 + 0.751278i \(0.270562\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.33136 0.183737
\(162\) 0 0
\(163\) 11.7128 0.917414 0.458707 0.888588i \(-0.348313\pi\)
0.458707 + 0.888588i \(0.348313\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −18.2136 −1.40941 −0.704704 0.709502i \(-0.748920\pi\)
−0.704704 + 0.709502i \(0.748920\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −8.86991 −0.674367 −0.337183 0.941439i \(-0.609474\pi\)
−0.337183 + 0.941439i \(0.609474\pi\)
\(174\) 0 0
\(175\) −0.192750 −0.0145705
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 15.6317 1.16837 0.584185 0.811620i \(-0.301414\pi\)
0.584185 + 0.811620i \(0.301414\pi\)
\(180\) 0 0
\(181\) 1.09772 0.0815928 0.0407964 0.999167i \(-0.487010\pi\)
0.0407964 + 0.999167i \(0.487010\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.94418 0.437025
\(186\) 0 0
\(187\) 2.82226 0.206384
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.6725 −1.20638 −0.603191 0.797597i \(-0.706104\pi\)
−0.603191 + 0.797597i \(0.706104\pi\)
\(192\) 0 0
\(193\) 18.3160 1.31842 0.659208 0.751960i \(-0.270892\pi\)
0.659208 + 0.751960i \(0.270892\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 22.2925 1.58828 0.794138 0.607738i \(-0.207923\pi\)
0.794138 + 0.607738i \(0.207923\pi\)
\(198\) 0 0
\(199\) 17.4381 1.23616 0.618078 0.786116i \(-0.287911\pi\)
0.618078 + 0.786116i \(0.287911\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.55008 0.178981
\(204\) 0 0
\(205\) −27.1295 −1.89480
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.00545 0.553749
\(210\) 0 0
\(211\) −16.5430 −1.13887 −0.569433 0.822037i \(-0.692837\pi\)
−0.569433 + 0.822037i \(0.692837\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −22.6633 −1.54563
\(216\) 0 0
\(217\) −4.63330 −0.314529
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.82226 0.189845
\(222\) 0 0
\(223\) 16.5176 1.10610 0.553050 0.833148i \(-0.313464\pi\)
0.553050 + 0.833148i \(0.313464\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 7.83138 0.519787 0.259893 0.965637i \(-0.416313\pi\)
0.259893 + 0.965637i \(0.416313\pi\)
\(228\) 0 0
\(229\) 6.70953 0.443378 0.221689 0.975117i \(-0.428843\pi\)
0.221689 + 0.975117i \(0.428843\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 11.8335 0.775236 0.387618 0.921820i \(-0.373298\pi\)
0.387618 + 0.921820i \(0.373298\pi\)
\(234\) 0 0
\(235\) −7.79363 −0.508400
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 26.6144 1.72154 0.860770 0.508994i \(-0.169982\pi\)
0.860770 + 0.508994i \(0.169982\pi\)
\(240\) 0 0
\(241\) −8.09918 −0.521714 −0.260857 0.965377i \(-0.584005\pi\)
−0.260857 + 0.965377i \(0.584005\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 14.4097 0.920600
\(246\) 0 0
\(247\) 8.00545 0.509375
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6.22678 0.393031 0.196515 0.980501i \(-0.437037\pi\)
0.196515 + 0.980501i \(0.437037\pi\)
\(252\) 0 0
\(253\) −4.07750 −0.256350
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −31.3741 −1.95707 −0.978533 0.206091i \(-0.933926\pi\)
−0.978533 + 0.206091i \(0.933926\pi\)
\(258\) 0 0
\(259\) −1.57391 −0.0977980
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 11.4508 0.706084 0.353042 0.935607i \(-0.385147\pi\)
0.353042 + 0.935607i \(0.385147\pi\)
\(264\) 0 0
\(265\) 9.88391 0.607164
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 18.8833 1.15134 0.575669 0.817683i \(-0.304742\pi\)
0.575669 + 0.817683i \(0.304742\pi\)
\(270\) 0 0
\(271\) 25.1408 1.52720 0.763598 0.645692i \(-0.223431\pi\)
0.763598 + 0.645692i \(0.223431\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.337115 0.0203288
\(276\) 0 0
\(277\) 4.08379 0.245371 0.122686 0.992446i \(-0.460849\pi\)
0.122686 + 0.992446i \(0.460849\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 15.4119 0.919397 0.459698 0.888075i \(-0.347958\pi\)
0.459698 + 0.888075i \(0.347958\pi\)
\(282\) 0 0
\(283\) 32.6178 1.93893 0.969463 0.245238i \(-0.0788661\pi\)
0.969463 + 0.245238i \(0.0788661\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7.18339 0.424022
\(288\) 0 0
\(289\) −9.03487 −0.531463
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −16.3998 −0.958089 −0.479045 0.877790i \(-0.659017\pi\)
−0.479045 + 0.877790i \(0.659017\pi\)
\(294\) 0 0
\(295\) −23.2954 −1.35631
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4.07750 −0.235808
\(300\) 0 0
\(301\) 6.00084 0.345883
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.08068 −0.119139
\(306\) 0 0
\(307\) −14.3286 −0.817779 −0.408889 0.912584i \(-0.634084\pi\)
−0.408889 + 0.912584i \(0.634084\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 19.2414 1.09108 0.545541 0.838084i \(-0.316324\pi\)
0.545541 + 0.838084i \(0.316324\pi\)
\(312\) 0 0
\(313\) 4.35171 0.245973 0.122987 0.992408i \(-0.460753\pi\)
0.122987 + 0.992408i \(0.460753\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −15.8683 −0.891255 −0.445627 0.895219i \(-0.647019\pi\)
−0.445627 + 0.895219i \(0.647019\pi\)
\(318\) 0 0
\(319\) −4.46004 −0.249714
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 22.5934 1.25713
\(324\) 0 0
\(325\) 0.337115 0.0186998
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.06361 0.113771
\(330\) 0 0
\(331\) 18.0430 0.991734 0.495867 0.868399i \(-0.334850\pi\)
0.495867 + 0.868399i \(0.334850\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2.93398 −0.160300
\(336\) 0 0
\(337\) −2.74038 −0.149278 −0.0746391 0.997211i \(-0.523780\pi\)
−0.0746391 + 0.997211i \(0.523780\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8.10355 0.438832
\(342\) 0 0
\(343\) −7.81775 −0.422119
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 30.1981 1.62112 0.810559 0.585657i \(-0.199163\pi\)
0.810559 + 0.585657i \(0.199163\pi\)
\(348\) 0 0
\(349\) −13.1511 −0.703962 −0.351981 0.936007i \(-0.614492\pi\)
−0.351981 + 0.936007i \(0.614492\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −19.4960 −1.03767 −0.518835 0.854874i \(-0.673634\pi\)
−0.518835 + 0.854874i \(0.673634\pi\)
\(354\) 0 0
\(355\) −16.3378 −0.867121
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.2048 1.06637 0.533183 0.846000i \(-0.320996\pi\)
0.533183 + 0.846000i \(0.320996\pi\)
\(360\) 0 0
\(361\) 45.0873 2.37302
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 13.0981 0.685588
\(366\) 0 0
\(367\) −2.16762 −0.113149 −0.0565743 0.998398i \(-0.518018\pi\)
−0.0565743 + 0.998398i \(0.518018\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2.61708 −0.135872
\(372\) 0 0
\(373\) −6.65344 −0.344502 −0.172251 0.985053i \(-0.555104\pi\)
−0.172251 + 0.985053i \(0.555104\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.46004 −0.229704
\(378\) 0 0
\(379\) 31.4982 1.61795 0.808977 0.587840i \(-0.200021\pi\)
0.808977 + 0.587840i \(0.200021\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −0.553088 −0.0282615 −0.0141307 0.999900i \(-0.504498\pi\)
−0.0141307 + 0.999900i \(0.504498\pi\)
\(384\) 0 0
\(385\) 1.23465 0.0629234
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −8.26159 −0.418879 −0.209440 0.977822i \(-0.567164\pi\)
−0.209440 + 0.977822i \(0.567164\pi\)
\(390\) 0 0
\(391\) −11.5077 −0.581972
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 12.4271 0.625275
\(396\) 0 0
\(397\) −0.679848 −0.0341206 −0.0170603 0.999854i \(-0.505431\pi\)
−0.0170603 + 0.999854i \(0.505431\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −0.378361 −0.0188945 −0.00944723 0.999955i \(-0.503007\pi\)
−0.00944723 + 0.999955i \(0.503007\pi\)
\(402\) 0 0
\(403\) 8.10355 0.403667
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.75274 0.136448
\(408\) 0 0
\(409\) 3.54649 0.175363 0.0876814 0.996149i \(-0.472054\pi\)
0.0876814 + 0.996149i \(0.472054\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 6.16819 0.303517
\(414\) 0 0
\(415\) −17.3162 −0.850017
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0.823510 0.0402311 0.0201155 0.999798i \(-0.493597\pi\)
0.0201155 + 0.999798i \(0.493597\pi\)
\(420\) 0 0
\(421\) −6.84698 −0.333701 −0.166851 0.985982i \(-0.553360\pi\)
−0.166851 + 0.985982i \(0.553360\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0.951426 0.0461509
\(426\) 0 0
\(427\) 0.550925 0.0266611
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −7.40956 −0.356906 −0.178453 0.983948i \(-0.557109\pi\)
−0.178453 + 0.983948i \(0.557109\pi\)
\(432\) 0 0
\(433\) −21.1774 −1.01772 −0.508860 0.860849i \(-0.669933\pi\)
−0.508860 + 0.860849i \(0.669933\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −32.6422 −1.56149
\(438\) 0 0
\(439\) −32.7862 −1.56480 −0.782401 0.622775i \(-0.786005\pi\)
−0.782401 + 0.622775i \(0.786005\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 8.96382 0.425884 0.212942 0.977065i \(-0.431695\pi\)
0.212942 + 0.977065i \(0.431695\pi\)
\(444\) 0 0
\(445\) 11.8807 0.563201
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −27.9476 −1.31893 −0.659464 0.751736i \(-0.729217\pi\)
−0.659464 + 0.751736i \(0.729217\pi\)
\(450\) 0 0
\(451\) −12.5636 −0.591596
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.23465 0.0578811
\(456\) 0 0
\(457\) −37.4179 −1.75034 −0.875168 0.483818i \(-0.839250\pi\)
−0.875168 + 0.483818i \(0.839250\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −6.15399 −0.286620 −0.143310 0.989678i \(-0.545775\pi\)
−0.143310 + 0.989678i \(0.545775\pi\)
\(462\) 0 0
\(463\) 6.69684 0.311228 0.155614 0.987818i \(-0.450264\pi\)
0.155614 + 0.987818i \(0.450264\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 33.7716 1.56276 0.781381 0.624054i \(-0.214516\pi\)
0.781381 + 0.624054i \(0.214516\pi\)
\(468\) 0 0
\(469\) 0.776863 0.0358722
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −10.4953 −0.482576
\(474\) 0 0
\(475\) 2.69876 0.123828
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −30.5346 −1.39516 −0.697582 0.716505i \(-0.745741\pi\)
−0.697582 + 0.716505i \(0.745741\pi\)
\(480\) 0 0
\(481\) 2.75274 0.125514
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −31.0986 −1.41212
\(486\) 0 0
\(487\) −24.4531 −1.10808 −0.554039 0.832491i \(-0.686914\pi\)
−0.554039 + 0.832491i \(0.686914\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7.79925 0.351975 0.175988 0.984392i \(-0.443688\pi\)
0.175988 + 0.984392i \(0.443688\pi\)
\(492\) 0 0
\(493\) −12.5874 −0.566907
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.32596 0.194046
\(498\) 0 0
\(499\) −17.7279 −0.793609 −0.396804 0.917903i \(-0.629881\pi\)
−0.396804 + 0.917903i \(0.629881\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 2.73089 0.121764 0.0608821 0.998145i \(-0.480609\pi\)
0.0608821 + 0.998145i \(0.480609\pi\)
\(504\) 0 0
\(505\) 15.4201 0.686184
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 11.9965 0.531734 0.265867 0.964010i \(-0.414342\pi\)
0.265867 + 0.964010i \(0.414342\pi\)
\(510\) 0 0
\(511\) −3.46815 −0.153422
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.86350 0.170246
\(516\) 0 0
\(517\) −3.60921 −0.158733
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4.74913 0.208063 0.104032 0.994574i \(-0.466826\pi\)
0.104032 + 0.994574i \(0.466826\pi\)
\(522\) 0 0
\(523\) −20.0210 −0.875457 −0.437729 0.899107i \(-0.644217\pi\)
−0.437729 + 0.899107i \(0.644217\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 22.8703 0.996246
\(528\) 0 0
\(529\) −6.37400 −0.277130
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −12.5636 −0.544190
\(534\) 0 0
\(535\) 18.9862 0.820847
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 6.67309 0.287430
\(540\) 0 0
\(541\) −41.5348 −1.78572 −0.892860 0.450335i \(-0.851305\pi\)
−0.892860 + 0.450335i \(0.851305\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −43.7669 −1.87477
\(546\) 0 0
\(547\) −1.14016 −0.0487496 −0.0243748 0.999703i \(-0.507760\pi\)
−0.0243748 + 0.999703i \(0.507760\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −35.7047 −1.52107
\(552\) 0 0
\(553\) −3.29047 −0.139925
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 37.7211 1.59830 0.799148 0.601134i \(-0.205284\pi\)
0.799148 + 0.601134i \(0.205284\pi\)
\(558\) 0 0
\(559\) −10.4953 −0.443906
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 32.3209 1.36216 0.681082 0.732207i \(-0.261510\pi\)
0.681082 + 0.732207i \(0.261510\pi\)
\(564\) 0 0
\(565\) 43.6758 1.83746
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −19.9313 −0.835564 −0.417782 0.908547i \(-0.637192\pi\)
−0.417782 + 0.908547i \(0.637192\pi\)
\(570\) 0 0
\(571\) 10.2338 0.428269 0.214135 0.976804i \(-0.431307\pi\)
0.214135 + 0.976804i \(0.431307\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.37459 −0.0573243
\(576\) 0 0
\(577\) 2.46485 0.102613 0.0513064 0.998683i \(-0.483661\pi\)
0.0513064 + 0.998683i \(0.483661\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4.58500 0.190218
\(582\) 0 0
\(583\) 4.57722 0.189569
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 31.9584 1.31907 0.659533 0.751676i \(-0.270754\pi\)
0.659533 + 0.751676i \(0.270754\pi\)
\(588\) 0 0
\(589\) 64.8726 2.67303
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.99394 0.0818814 0.0409407 0.999162i \(-0.486965\pi\)
0.0409407 + 0.999162i \(0.486965\pi\)
\(594\) 0 0
\(595\) 3.48449 0.142850
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 40.8312 1.66832 0.834159 0.551524i \(-0.185953\pi\)
0.834159 + 0.551524i \(0.185953\pi\)
\(600\) 0 0
\(601\) 21.6994 0.885136 0.442568 0.896735i \(-0.354068\pi\)
0.442568 + 0.896735i \(0.354068\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −2.15937 −0.0877909
\(606\) 0 0
\(607\) −3.80717 −0.154528 −0.0772642 0.997011i \(-0.524618\pi\)
−0.0772642 + 0.997011i \(0.524618\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −3.60921 −0.146013
\(612\) 0 0
\(613\) 38.8192 1.56789 0.783946 0.620829i \(-0.213204\pi\)
0.783946 + 0.620829i \(0.213204\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 43.3000 1.74319 0.871596 0.490224i \(-0.163085\pi\)
0.871596 + 0.490224i \(0.163085\pi\)
\(618\) 0 0
\(619\) −46.2723 −1.85984 −0.929921 0.367760i \(-0.880125\pi\)
−0.929921 + 0.367760i \(0.880125\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.14580 −0.126034
\(624\) 0 0
\(625\) −23.2008 −0.928031
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 7.76892 0.309767
\(630\) 0 0
\(631\) 27.7943 1.10647 0.553236 0.833024i \(-0.313393\pi\)
0.553236 + 0.833024i \(0.313393\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 37.0900 1.47187
\(636\) 0 0
\(637\) 6.67309 0.264397
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −49.0927 −1.93905 −0.969523 0.245000i \(-0.921212\pi\)
−0.969523 + 0.245000i \(0.921212\pi\)
\(642\) 0 0
\(643\) −43.7541 −1.72549 −0.862747 0.505635i \(-0.831258\pi\)
−0.862747 + 0.505635i \(0.831258\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.7322 0.500555 0.250278 0.968174i \(-0.419478\pi\)
0.250278 + 0.968174i \(0.419478\pi\)
\(648\) 0 0
\(649\) −10.7880 −0.423467
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −23.7499 −0.929405 −0.464703 0.885467i \(-0.653839\pi\)
−0.464703 + 0.885467i \(0.653839\pi\)
\(654\) 0 0
\(655\) 10.4420 0.408005
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −37.6374 −1.46615 −0.733073 0.680150i \(-0.761914\pi\)
−0.733073 + 0.680150i \(0.761914\pi\)
\(660\) 0 0
\(661\) −10.6102 −0.412690 −0.206345 0.978479i \(-0.566157\pi\)
−0.206345 + 0.978479i \(0.566157\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 9.88391 0.383282
\(666\) 0 0
\(667\) 18.1858 0.704158
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −0.963556 −0.0371977
\(672\) 0 0
\(673\) −22.7426 −0.876661 −0.438331 0.898814i \(-0.644430\pi\)
−0.438331 + 0.898814i \(0.644430\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −4.57385 −0.175787 −0.0878937 0.996130i \(-0.528014\pi\)
−0.0878937 + 0.996130i \(0.528014\pi\)
\(678\) 0 0
\(679\) 8.23435 0.316005
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −46.9411 −1.79615 −0.898075 0.439843i \(-0.855034\pi\)
−0.898075 + 0.439843i \(0.855034\pi\)
\(684\) 0 0
\(685\) −32.6135 −1.24610
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4.57722 0.174378
\(690\) 0 0
\(691\) −12.4711 −0.474424 −0.237212 0.971458i \(-0.576234\pi\)
−0.237212 + 0.971458i \(0.576234\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.93816 −0.263179
\(696\) 0 0
\(697\) −35.4577 −1.34306
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 25.0770 0.947146 0.473573 0.880755i \(-0.342964\pi\)
0.473573 + 0.880755i \(0.342964\pi\)
\(702\) 0 0
\(703\) 22.0369 0.831137
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4.08295 −0.153555
\(708\) 0 0
\(709\) 35.5561 1.33534 0.667669 0.744458i \(-0.267292\pi\)
0.667669 + 0.744458i \(0.267292\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −33.0422 −1.23744
\(714\) 0 0
\(715\) −2.15937 −0.0807559
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −17.9113 −0.667979 −0.333990 0.942577i \(-0.608395\pi\)
−0.333990 + 0.942577i \(0.608395\pi\)
\(720\) 0 0
\(721\) −1.02299 −0.0380980
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.50355 −0.0558404
\(726\) 0 0
\(727\) 43.8561 1.62653 0.813266 0.581893i \(-0.197688\pi\)
0.813266 + 0.581893i \(0.197688\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −29.6205 −1.09556
\(732\) 0 0
\(733\) −3.74975 −0.138500 −0.0692500 0.997599i \(-0.522061\pi\)
−0.0692500 + 0.997599i \(0.522061\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.35872 −0.0500490
\(738\) 0 0
\(739\) 17.6665 0.649873 0.324936 0.945736i \(-0.394657\pi\)
0.324936 + 0.945736i \(0.394657\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 25.3176 0.928813 0.464406 0.885622i \(-0.346268\pi\)
0.464406 + 0.885622i \(0.346268\pi\)
\(744\) 0 0
\(745\) −51.8190 −1.89850
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −5.02721 −0.183690
\(750\) 0 0
\(751\) 32.3921 1.18200 0.591002 0.806670i \(-0.298733\pi\)
0.591002 + 0.806670i \(0.298733\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 19.2391 0.700182
\(756\) 0 0
\(757\) −34.8397 −1.26627 −0.633135 0.774041i \(-0.718232\pi\)
−0.633135 + 0.774041i \(0.718232\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −37.0873 −1.34441 −0.672206 0.740364i \(-0.734653\pi\)
−0.672206 + 0.740364i \(0.734653\pi\)
\(762\) 0 0
\(763\) 11.5887 0.419539
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −10.7880 −0.389533
\(768\) 0 0
\(769\) 4.28238 0.154426 0.0772132 0.997015i \(-0.475398\pi\)
0.0772132 + 0.997015i \(0.475398\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −36.3602 −1.30779 −0.653893 0.756587i \(-0.726865\pi\)
−0.653893 + 0.756587i \(0.726865\pi\)
\(774\) 0 0
\(775\) 2.73183 0.0981302
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −100.577 −3.60355
\(780\) 0 0
\(781\) −7.56601 −0.270733
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −35.7143 −1.27470
\(786\) 0 0
\(787\) −11.5241 −0.410789 −0.205394 0.978679i \(-0.565848\pi\)
−0.205394 + 0.978679i \(0.565848\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −11.5646 −0.411189
\(792\) 0 0
\(793\) −0.963556 −0.0342169
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −33.5889 −1.18978 −0.594890 0.803807i \(-0.702804\pi\)
−0.594890 + 0.803807i \(0.702804\pi\)
\(798\) 0 0
\(799\) −10.1861 −0.360359
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 6.06572 0.214055
\(804\) 0 0
\(805\) −5.03427 −0.177435
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 30.2200 1.06248 0.531239 0.847222i \(-0.321727\pi\)
0.531239 + 0.847222i \(0.321727\pi\)
\(810\) 0 0
\(811\) −37.6846 −1.32328 −0.661642 0.749820i \(-0.730140\pi\)
−0.661642 + 0.749820i \(0.730140\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −25.2922 −0.885947
\(816\) 0 0
\(817\) −84.0200 −2.93949
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 6.98136 0.243651 0.121826 0.992552i \(-0.461125\pi\)
0.121826 + 0.992552i \(0.461125\pi\)
\(822\) 0 0
\(823\) 48.7025 1.69766 0.848831 0.528664i \(-0.177307\pi\)
0.848831 + 0.528664i \(0.177307\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.22517 −0.0773768 −0.0386884 0.999251i \(-0.512318\pi\)
−0.0386884 + 0.999251i \(0.512318\pi\)
\(828\) 0 0
\(829\) 30.5640 1.06153 0.530766 0.847519i \(-0.321904\pi\)
0.530766 + 0.847519i \(0.321904\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 18.8332 0.652530
\(834\) 0 0
\(835\) 39.3298 1.36107
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 41.2608 1.42448 0.712240 0.701936i \(-0.247681\pi\)
0.712240 + 0.701936i \(0.247681\pi\)
\(840\) 0 0
\(841\) −9.10802 −0.314070
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −2.15937 −0.0742846
\(846\) 0 0
\(847\) 0.571762 0.0196460
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −11.2243 −0.384763
\(852\) 0 0
\(853\) 8.35245 0.285982 0.142991 0.989724i \(-0.454328\pi\)
0.142991 + 0.989724i \(0.454328\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −23.6353 −0.807365 −0.403683 0.914899i \(-0.632270\pi\)
−0.403683 + 0.914899i \(0.632270\pi\)
\(858\) 0 0
\(859\) 37.6529 1.28470 0.642350 0.766412i \(-0.277960\pi\)
0.642350 + 0.766412i \(0.277960\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 18.7537 0.638384 0.319192 0.947690i \(-0.396588\pi\)
0.319192 + 0.947690i \(0.396588\pi\)
\(864\) 0 0
\(865\) 19.1534 0.651236
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 5.75496 0.195224
\(870\) 0 0
\(871\) −1.35872 −0.0460384
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 6.58945 0.222764
\(876\) 0 0
\(877\) 45.3578 1.53162 0.765812 0.643064i \(-0.222337\pi\)
0.765812 + 0.643064i \(0.222337\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.37370 −0.0462810 −0.0231405 0.999732i \(-0.507367\pi\)
−0.0231405 + 0.999732i \(0.507367\pi\)
\(882\) 0 0
\(883\) −45.7456 −1.53946 −0.769731 0.638368i \(-0.779610\pi\)
−0.769731 + 0.638368i \(0.779610\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 36.1835 1.21492 0.607462 0.794349i \(-0.292188\pi\)
0.607462 + 0.794349i \(0.292188\pi\)
\(888\) 0 0
\(889\) −9.82076 −0.329378
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −28.8934 −0.966880
\(894\) 0 0
\(895\) −33.7547 −1.12830
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −36.1422 −1.20541
\(900\) 0 0
\(901\) 12.9181 0.430363
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.37038 −0.0787942
\(906\) 0 0
\(907\) 44.2643 1.46977 0.734886 0.678190i \(-0.237235\pi\)
0.734886 + 0.678190i \(0.237235\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 45.2601 1.49953 0.749766 0.661703i \(-0.230166\pi\)
0.749766 + 0.661703i \(0.230166\pi\)
\(912\) 0 0
\(913\) −8.01908 −0.265393
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2.76486 −0.0913039
\(918\) 0 0
\(919\) −22.6067 −0.745725 −0.372862 0.927887i \(-0.621624\pi\)
−0.372862 + 0.927887i \(0.621624\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −7.56601 −0.249038
\(924\) 0 0
\(925\) 0.927989 0.0305121
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −36.2462 −1.18920 −0.594600 0.804021i \(-0.702690\pi\)
−0.594600 + 0.804021i \(0.702690\pi\)
\(930\) 0 0
\(931\) 53.4211 1.75081
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −6.09430 −0.199305
\(936\) 0 0
\(937\) 20.6755 0.675440 0.337720 0.941247i \(-0.390344\pi\)
0.337720 + 0.941247i \(0.390344\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −56.1354 −1.82996 −0.914980 0.403498i \(-0.867794\pi\)
−0.914980 + 0.403498i \(0.867794\pi\)
\(942\) 0 0
\(943\) 51.2280 1.66821
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 46.7341 1.51865 0.759327 0.650709i \(-0.225528\pi\)
0.759327 + 0.650709i \(0.225528\pi\)
\(948\) 0 0
\(949\) 6.06572 0.196902
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −0.598932 −0.0194013 −0.00970065 0.999953i \(-0.503088\pi\)
−0.00970065 + 0.999953i \(0.503088\pi\)
\(954\) 0 0
\(955\) 36.0022 1.16500
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 8.63547 0.278854
\(960\) 0 0
\(961\) 34.6675 1.11831
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −39.5511 −1.27320
\(966\) 0 0
\(967\) 26.5248 0.852981 0.426491 0.904492i \(-0.359750\pi\)
0.426491 + 0.904492i \(0.359750\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −4.59988 −0.147617 −0.0738085 0.997272i \(-0.523515\pi\)
−0.0738085 + 0.997272i \(0.523515\pi\)
\(972\) 0 0
\(973\) 1.83710 0.0588946
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 36.1212 1.15562 0.577810 0.816172i \(-0.303908\pi\)
0.577810 + 0.816172i \(0.303908\pi\)
\(978\) 0 0
\(979\) 5.50194 0.175843
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −47.3814 −1.51123 −0.755616 0.655015i \(-0.772662\pi\)
−0.755616 + 0.655015i \(0.772662\pi\)
\(984\) 0 0
\(985\) −48.1378 −1.53380
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 42.7948 1.36079
\(990\) 0 0
\(991\) −20.4322 −0.649049 −0.324525 0.945877i \(-0.605204\pi\)
−0.324525 + 0.945877i \(0.605204\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −37.6554 −1.19376
\(996\) 0 0
\(997\) −33.0307 −1.04609 −0.523047 0.852304i \(-0.675205\pi\)
−0.523047 + 0.852304i \(0.675205\pi\)
\(998\) 0 0
\(999\) 0 0
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 5148.2.a.t.1.1 yes 5
3.2 odd 2 5148.2.a.s.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5148.2.a.s.1.5 5 3.2 odd 2
5148.2.a.t.1.1 yes 5 1.1 even 1 trivial