Properties

Label 4016.2.a.d.1.4
Level $4016$
Weight $2$
Character 4016.1
Self dual yes
Analytic conductor $32.068$
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] = [4016,2,Mod(1,4016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4016, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4016.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: \(32.0679214517\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.138917.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 6x^{3} - 2x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 502)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.779397\) of defining polynomial
Character \(\chi\) \(=\) 4016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.14049 q^{3} -2.02938 q^{5} +3.81607 q^{7} -1.69928 q^{9} +O(q^{10})\) \(q+1.14049 q^{3} -2.02938 q^{5} +3.81607 q^{7} -1.69928 q^{9} -1.49635 q^{11} +3.70658 q^{13} -2.31449 q^{15} +4.98233 q^{17} +2.85745 q^{19} +4.35219 q^{21} -6.43753 q^{23} -0.881602 q^{25} -5.35948 q^{27} -3.64253 q^{29} +2.05435 q^{31} -1.70658 q^{33} -7.74428 q^{35} +10.4344 q^{37} +4.22731 q^{39} +9.36551 q^{41} +11.0717 q^{43} +3.44850 q^{45} -6.32281 q^{47} +7.56241 q^{49} +5.68229 q^{51} +4.92195 q^{53} +3.03668 q^{55} +3.25889 q^{57} -0.897795 q^{59} -0.628527 q^{61} -6.48459 q^{63} -7.52206 q^{65} -11.6372 q^{67} -7.34194 q^{69} -4.48517 q^{71} +3.44517 q^{73} -1.00546 q^{75} -5.71020 q^{77} +13.5066 q^{79} -1.01458 q^{81} +6.59547 q^{83} -10.1111 q^{85} -4.15426 q^{87} -13.0236 q^{89} +14.1446 q^{91} +2.34296 q^{93} -5.79886 q^{95} -0.00831686 q^{97} +2.54273 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{3} + 7 q^{5} - q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{3} + 7 q^{5} - q^{7} + 7 q^{9} - 7 q^{11} + 4 q^{13} - 3 q^{15} + 6 q^{17} + 10 q^{19} + 2 q^{21} - 13 q^{23} + 6 q^{25} - 8 q^{27} + 4 q^{29} + 13 q^{31} + 6 q^{33} - 13 q^{35} + 16 q^{37} + 16 q^{39} + 6 q^{41} + 8 q^{43} + 26 q^{45} - 29 q^{47} + 4 q^{49} + 7 q^{51} + 25 q^{53} - q^{55} + 19 q^{57} - 11 q^{59} + 11 q^{61} - 15 q^{63} + 7 q^{65} + 6 q^{67} - 12 q^{69} - 15 q^{71} - 9 q^{73} + 25 q^{75} + 12 q^{77} + 29 q^{79} - 7 q^{81} + 9 q^{83} - 7 q^{85} + 11 q^{87} - 9 q^{89} + 34 q^{91} - 8 q^{93} + 13 q^{95} - 6 q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.14049 0.658462 0.329231 0.944249i \(-0.393211\pi\)
0.329231 + 0.944249i \(0.393211\pi\)
\(4\) 0 0
\(5\) −2.02938 −0.907568 −0.453784 0.891112i \(-0.649926\pi\)
−0.453784 + 0.891112i \(0.649926\pi\)
\(6\) 0 0
\(7\) 3.81607 1.44234 0.721170 0.692758i \(-0.243605\pi\)
0.721170 + 0.692758i \(0.243605\pi\)
\(8\) 0 0
\(9\) −1.69928 −0.566428
\(10\) 0 0
\(11\) −1.49635 −0.451168 −0.225584 0.974224i \(-0.572429\pi\)
−0.225584 + 0.974224i \(0.572429\pi\)
\(12\) 0 0
\(13\) 3.70658 1.02802 0.514010 0.857784i \(-0.328160\pi\)
0.514010 + 0.857784i \(0.328160\pi\)
\(14\) 0 0
\(15\) −2.31449 −0.597599
\(16\) 0 0
\(17\) 4.98233 1.20839 0.604196 0.796836i \(-0.293494\pi\)
0.604196 + 0.796836i \(0.293494\pi\)
\(18\) 0 0
\(19\) 2.85745 0.655543 0.327772 0.944757i \(-0.393702\pi\)
0.327772 + 0.944757i \(0.393702\pi\)
\(20\) 0 0
\(21\) 4.35219 0.949726
\(22\) 0 0
\(23\) −6.43753 −1.34232 −0.671159 0.741313i \(-0.734203\pi\)
−0.671159 + 0.741313i \(0.734203\pi\)
\(24\) 0 0
\(25\) −0.881602 −0.176320
\(26\) 0 0
\(27\) −5.35948 −1.03143
\(28\) 0 0
\(29\) −3.64253 −0.676400 −0.338200 0.941074i \(-0.609818\pi\)
−0.338200 + 0.941074i \(0.609818\pi\)
\(30\) 0 0
\(31\) 2.05435 0.368972 0.184486 0.982835i \(-0.440938\pi\)
0.184486 + 0.982835i \(0.440938\pi\)
\(32\) 0 0
\(33\) −1.70658 −0.297077
\(34\) 0 0
\(35\) −7.74428 −1.30902
\(36\) 0 0
\(37\) 10.4344 1.71541 0.857706 0.514140i \(-0.171889\pi\)
0.857706 + 0.514140i \(0.171889\pi\)
\(38\) 0 0
\(39\) 4.22731 0.676912
\(40\) 0 0
\(41\) 9.36551 1.46265 0.731324 0.682031i \(-0.238903\pi\)
0.731324 + 0.682031i \(0.238903\pi\)
\(42\) 0 0
\(43\) 11.0717 1.68843 0.844213 0.536009i \(-0.180069\pi\)
0.844213 + 0.536009i \(0.180069\pi\)
\(44\) 0 0
\(45\) 3.44850 0.514072
\(46\) 0 0
\(47\) −6.32281 −0.922276 −0.461138 0.887328i \(-0.652559\pi\)
−0.461138 + 0.887328i \(0.652559\pi\)
\(48\) 0 0
\(49\) 7.56241 1.08034
\(50\) 0 0
\(51\) 5.68229 0.795680
\(52\) 0 0
\(53\) 4.92195 0.676082 0.338041 0.941131i \(-0.390236\pi\)
0.338041 + 0.941131i \(0.390236\pi\)
\(54\) 0 0
\(55\) 3.03668 0.409465
\(56\) 0 0
\(57\) 3.25889 0.431650
\(58\) 0 0
\(59\) −0.897795 −0.116883 −0.0584415 0.998291i \(-0.518613\pi\)
−0.0584415 + 0.998291i \(0.518613\pi\)
\(60\) 0 0
\(61\) −0.628527 −0.0804746 −0.0402373 0.999190i \(-0.512811\pi\)
−0.0402373 + 0.999190i \(0.512811\pi\)
\(62\) 0 0
\(63\) −6.48459 −0.816982
\(64\) 0 0
\(65\) −7.52206 −0.932997
\(66\) 0 0
\(67\) −11.6372 −1.42172 −0.710858 0.703336i \(-0.751693\pi\)
−0.710858 + 0.703336i \(0.751693\pi\)
\(68\) 0 0
\(69\) −7.34194 −0.883866
\(70\) 0 0
\(71\) −4.48517 −0.532292 −0.266146 0.963933i \(-0.585750\pi\)
−0.266146 + 0.963933i \(0.585750\pi\)
\(72\) 0 0
\(73\) 3.44517 0.403227 0.201613 0.979465i \(-0.435382\pi\)
0.201613 + 0.979465i \(0.435382\pi\)
\(74\) 0 0
\(75\) −1.00546 −0.116100
\(76\) 0 0
\(77\) −5.71020 −0.650737
\(78\) 0 0
\(79\) 13.5066 1.51961 0.759805 0.650151i \(-0.225295\pi\)
0.759805 + 0.650151i \(0.225295\pi\)
\(80\) 0 0
\(81\) −1.01458 −0.112732
\(82\) 0 0
\(83\) 6.59547 0.723947 0.361973 0.932188i \(-0.382103\pi\)
0.361973 + 0.932188i \(0.382103\pi\)
\(84\) 0 0
\(85\) −10.1111 −1.09670
\(86\) 0 0
\(87\) −4.15426 −0.445384
\(88\) 0 0
\(89\) −13.0236 −1.38050 −0.690252 0.723569i \(-0.742500\pi\)
−0.690252 + 0.723569i \(0.742500\pi\)
\(90\) 0 0
\(91\) 14.1446 1.48275
\(92\) 0 0
\(93\) 2.34296 0.242954
\(94\) 0 0
\(95\) −5.79886 −0.594950
\(96\) 0 0
\(97\) −0.00831686 −0.000844449 0 −0.000422224 1.00000i \(-0.500134\pi\)
−0.000422224 1.00000i \(0.500134\pi\)
\(98\) 0 0
\(99\) 2.54273 0.255554
\(100\) 0 0
\(101\) 13.3535 1.32873 0.664363 0.747411i \(-0.268703\pi\)
0.664363 + 0.747411i \(0.268703\pi\)
\(102\) 0 0
\(103\) 5.93777 0.585066 0.292533 0.956255i \(-0.405502\pi\)
0.292533 + 0.956255i \(0.405502\pi\)
\(104\) 0 0
\(105\) −8.83227 −0.861941
\(106\) 0 0
\(107\) −12.9211 −1.24913 −0.624565 0.780973i \(-0.714724\pi\)
−0.624565 + 0.780973i \(0.714724\pi\)
\(108\) 0 0
\(109\) 11.6838 1.11911 0.559555 0.828793i \(-0.310972\pi\)
0.559555 + 0.828793i \(0.310972\pi\)
\(110\) 0 0
\(111\) 11.9004 1.12953
\(112\) 0 0
\(113\) −2.88626 −0.271517 −0.135758 0.990742i \(-0.543347\pi\)
−0.135758 + 0.990742i \(0.543347\pi\)
\(114\) 0 0
\(115\) 13.0642 1.21825
\(116\) 0 0
\(117\) −6.29852 −0.582299
\(118\) 0 0
\(119\) 19.0129 1.74291
\(120\) 0 0
\(121\) −8.76093 −0.796448
\(122\) 0 0
\(123\) 10.6813 0.963097
\(124\) 0 0
\(125\) 11.9360 1.06759
\(126\) 0 0
\(127\) −4.94904 −0.439157 −0.219578 0.975595i \(-0.570468\pi\)
−0.219578 + 0.975595i \(0.570468\pi\)
\(128\) 0 0
\(129\) 12.6272 1.11176
\(130\) 0 0
\(131\) 15.1764 1.32596 0.662982 0.748635i \(-0.269291\pi\)
0.662982 + 0.748635i \(0.269291\pi\)
\(132\) 0 0
\(133\) 10.9042 0.945516
\(134\) 0 0
\(135\) 10.8764 0.936096
\(136\) 0 0
\(137\) 13.0367 1.11380 0.556899 0.830580i \(-0.311991\pi\)
0.556899 + 0.830580i \(0.311991\pi\)
\(138\) 0 0
\(139\) 7.63891 0.647924 0.323962 0.946070i \(-0.394985\pi\)
0.323962 + 0.946070i \(0.394985\pi\)
\(140\) 0 0
\(141\) −7.21110 −0.607284
\(142\) 0 0
\(143\) −5.54635 −0.463809
\(144\) 0 0
\(145\) 7.39208 0.613879
\(146\) 0 0
\(147\) 8.62485 0.711366
\(148\) 0 0
\(149\) 11.7367 0.961509 0.480755 0.876855i \(-0.340363\pi\)
0.480755 + 0.876855i \(0.340363\pi\)
\(150\) 0 0
\(151\) −17.7106 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(152\) 0 0
\(153\) −8.46639 −0.684467
\(154\) 0 0
\(155\) −4.16906 −0.334867
\(156\) 0 0
\(157\) 3.93777 0.314269 0.157134 0.987577i \(-0.449774\pi\)
0.157134 + 0.987577i \(0.449774\pi\)
\(158\) 0 0
\(159\) 5.61343 0.445174
\(160\) 0 0
\(161\) −24.5661 −1.93608
\(162\) 0 0
\(163\) 11.5452 0.904290 0.452145 0.891945i \(-0.350659\pi\)
0.452145 + 0.891945i \(0.350659\pi\)
\(164\) 0 0
\(165\) 3.46330 0.269617
\(166\) 0 0
\(167\) 23.4435 1.81411 0.907057 0.421009i \(-0.138324\pi\)
0.907057 + 0.421009i \(0.138324\pi\)
\(168\) 0 0
\(169\) 0.738705 0.0568235
\(170\) 0 0
\(171\) −4.85561 −0.371318
\(172\) 0 0
\(173\) 21.8265 1.65944 0.829718 0.558183i \(-0.188501\pi\)
0.829718 + 0.558183i \(0.188501\pi\)
\(174\) 0 0
\(175\) −3.36426 −0.254314
\(176\) 0 0
\(177\) −1.02393 −0.0769630
\(178\) 0 0
\(179\) −13.0480 −0.975257 −0.487628 0.873051i \(-0.662138\pi\)
−0.487628 + 0.873051i \(0.662138\pi\)
\(180\) 0 0
\(181\) −15.2566 −1.13402 −0.567009 0.823712i \(-0.691899\pi\)
−0.567009 + 0.823712i \(0.691899\pi\)
\(182\) 0 0
\(183\) −0.716828 −0.0529894
\(184\) 0 0
\(185\) −21.1755 −1.55685
\(186\) 0 0
\(187\) −7.45532 −0.545187
\(188\) 0 0
\(189\) −20.4522 −1.48768
\(190\) 0 0
\(191\) −9.41734 −0.681415 −0.340707 0.940169i \(-0.610667\pi\)
−0.340707 + 0.940169i \(0.610667\pi\)
\(192\) 0 0
\(193\) 21.1899 1.52528 0.762640 0.646823i \(-0.223903\pi\)
0.762640 + 0.646823i \(0.223903\pi\)
\(194\) 0 0
\(195\) −8.57884 −0.614343
\(196\) 0 0
\(197\) 11.4445 0.815389 0.407694 0.913118i \(-0.366333\pi\)
0.407694 + 0.913118i \(0.366333\pi\)
\(198\) 0 0
\(199\) 9.06949 0.642919 0.321460 0.946923i \(-0.395827\pi\)
0.321460 + 0.946923i \(0.395827\pi\)
\(200\) 0 0
\(201\) −13.2722 −0.936146
\(202\) 0 0
\(203\) −13.9001 −0.975599
\(204\) 0 0
\(205\) −19.0062 −1.32745
\(206\) 0 0
\(207\) 10.9392 0.760327
\(208\) 0 0
\(209\) −4.27575 −0.295760
\(210\) 0 0
\(211\) −13.4721 −0.927460 −0.463730 0.885977i \(-0.653489\pi\)
−0.463730 + 0.885977i \(0.653489\pi\)
\(212\) 0 0
\(213\) −5.11529 −0.350494
\(214\) 0 0
\(215\) −22.4688 −1.53236
\(216\) 0 0
\(217\) 7.83955 0.532183
\(218\) 0 0
\(219\) 3.92918 0.265509
\(220\) 0 0
\(221\) 18.4674 1.24225
\(222\) 0 0
\(223\) 2.05994 0.137944 0.0689718 0.997619i \(-0.478028\pi\)
0.0689718 + 0.997619i \(0.478028\pi\)
\(224\) 0 0
\(225\) 1.49809 0.0998728
\(226\) 0 0
\(227\) 9.76947 0.648423 0.324211 0.945985i \(-0.394901\pi\)
0.324211 + 0.945985i \(0.394901\pi\)
\(228\) 0 0
\(229\) 22.0709 1.45849 0.729244 0.684254i \(-0.239872\pi\)
0.729244 + 0.684254i \(0.239872\pi\)
\(230\) 0 0
\(231\) −6.51242 −0.428486
\(232\) 0 0
\(233\) −16.1694 −1.05929 −0.529645 0.848219i \(-0.677675\pi\)
−0.529645 + 0.848219i \(0.677675\pi\)
\(234\) 0 0
\(235\) 12.8314 0.837028
\(236\) 0 0
\(237\) 15.4041 1.00061
\(238\) 0 0
\(239\) −21.6213 −1.39857 −0.699283 0.714845i \(-0.746497\pi\)
−0.699283 + 0.714845i \(0.746497\pi\)
\(240\) 0 0
\(241\) 19.8537 1.27889 0.639446 0.768836i \(-0.279164\pi\)
0.639446 + 0.768836i \(0.279164\pi\)
\(242\) 0 0
\(243\) 14.9213 0.957204
\(244\) 0 0
\(245\) −15.3470 −0.980486
\(246\) 0 0
\(247\) 10.5913 0.673911
\(248\) 0 0
\(249\) 7.52206 0.476691
\(250\) 0 0
\(251\) −1.00000 −0.0631194
\(252\) 0 0
\(253\) 9.63283 0.605611
\(254\) 0 0
\(255\) −11.5316 −0.722134
\(256\) 0 0
\(257\) −1.49666 −0.0933588 −0.0466794 0.998910i \(-0.514864\pi\)
−0.0466794 + 0.998910i \(0.514864\pi\)
\(258\) 0 0
\(259\) 39.8186 2.47421
\(260\) 0 0
\(261\) 6.18969 0.383132
\(262\) 0 0
\(263\) −0.639363 −0.0394248 −0.0197124 0.999806i \(-0.506275\pi\)
−0.0197124 + 0.999806i \(0.506275\pi\)
\(264\) 0 0
\(265\) −9.98853 −0.613590
\(266\) 0 0
\(267\) −14.8533 −0.909009
\(268\) 0 0
\(269\) 5.15667 0.314408 0.157204 0.987566i \(-0.449752\pi\)
0.157204 + 0.987566i \(0.449752\pi\)
\(270\) 0 0
\(271\) −18.1081 −1.09999 −0.549994 0.835169i \(-0.685370\pi\)
−0.549994 + 0.835169i \(0.685370\pi\)
\(272\) 0 0
\(273\) 16.1317 0.976337
\(274\) 0 0
\(275\) 1.31919 0.0795500
\(276\) 0 0
\(277\) −22.7214 −1.36520 −0.682598 0.730794i \(-0.739150\pi\)
−0.682598 + 0.730794i \(0.739150\pi\)
\(278\) 0 0
\(279\) −3.49092 −0.208996
\(280\) 0 0
\(281\) −13.4504 −0.802384 −0.401192 0.915994i \(-0.631404\pi\)
−0.401192 + 0.915994i \(0.631404\pi\)
\(282\) 0 0
\(283\) −20.4980 −1.21848 −0.609239 0.792986i \(-0.708525\pi\)
−0.609239 + 0.792986i \(0.708525\pi\)
\(284\) 0 0
\(285\) −6.61353 −0.391752
\(286\) 0 0
\(287\) 35.7395 2.10963
\(288\) 0 0
\(289\) 7.82358 0.460211
\(290\) 0 0
\(291\) −0.00948529 −0.000556037 0
\(292\) 0 0
\(293\) 10.4624 0.611219 0.305610 0.952157i \(-0.401140\pi\)
0.305610 + 0.952157i \(0.401140\pi\)
\(294\) 0 0
\(295\) 1.82197 0.106079
\(296\) 0 0
\(297\) 8.01968 0.465349
\(298\) 0 0
\(299\) −23.8612 −1.37993
\(300\) 0 0
\(301\) 42.2506 2.43528
\(302\) 0 0
\(303\) 15.2296 0.874915
\(304\) 0 0
\(305\) 1.27552 0.0730361
\(306\) 0 0
\(307\) 4.32266 0.246707 0.123354 0.992363i \(-0.460635\pi\)
0.123354 + 0.992363i \(0.460635\pi\)
\(308\) 0 0
\(309\) 6.77197 0.385244
\(310\) 0 0
\(311\) 2.55444 0.144849 0.0724245 0.997374i \(-0.476926\pi\)
0.0724245 + 0.997374i \(0.476926\pi\)
\(312\) 0 0
\(313\) −20.5974 −1.16424 −0.582118 0.813104i \(-0.697776\pi\)
−0.582118 + 0.813104i \(0.697776\pi\)
\(314\) 0 0
\(315\) 13.1597 0.741466
\(316\) 0 0
\(317\) 28.4867 1.59997 0.799985 0.600020i \(-0.204841\pi\)
0.799985 + 0.600020i \(0.204841\pi\)
\(318\) 0 0
\(319\) 5.45051 0.305170
\(320\) 0 0
\(321\) −14.7364 −0.822504
\(322\) 0 0
\(323\) 14.2367 0.792153
\(324\) 0 0
\(325\) −3.26772 −0.181261
\(326\) 0 0
\(327\) 13.3253 0.736891
\(328\) 0 0
\(329\) −24.1283 −1.33024
\(330\) 0 0
\(331\) −29.7471 −1.63505 −0.817526 0.575892i \(-0.804655\pi\)
−0.817526 + 0.575892i \(0.804655\pi\)
\(332\) 0 0
\(333\) −17.7311 −0.971657
\(334\) 0 0
\(335\) 23.6164 1.29030
\(336\) 0 0
\(337\) 6.39501 0.348358 0.174179 0.984714i \(-0.444273\pi\)
0.174179 + 0.984714i \(0.444273\pi\)
\(338\) 0 0
\(339\) −3.29175 −0.178783
\(340\) 0 0
\(341\) −3.07403 −0.166468
\(342\) 0 0
\(343\) 2.14621 0.115885
\(344\) 0 0
\(345\) 14.8996 0.802168
\(346\) 0 0
\(347\) 2.73435 0.146788 0.0733938 0.997303i \(-0.476617\pi\)
0.0733938 + 0.997303i \(0.476617\pi\)
\(348\) 0 0
\(349\) 7.63841 0.408875 0.204437 0.978880i \(-0.434463\pi\)
0.204437 + 0.978880i \(0.434463\pi\)
\(350\) 0 0
\(351\) −19.8653 −1.06033
\(352\) 0 0
\(353\) 10.8175 0.575758 0.287879 0.957667i \(-0.407050\pi\)
0.287879 + 0.957667i \(0.407050\pi\)
\(354\) 0 0
\(355\) 9.10214 0.483091
\(356\) 0 0
\(357\) 21.6840 1.14764
\(358\) 0 0
\(359\) 7.26766 0.383572 0.191786 0.981437i \(-0.438572\pi\)
0.191786 + 0.981437i \(0.438572\pi\)
\(360\) 0 0
\(361\) −10.8350 −0.570263
\(362\) 0 0
\(363\) −9.99174 −0.524431
\(364\) 0 0
\(365\) −6.99157 −0.365955
\(366\) 0 0
\(367\) 25.7296 1.34307 0.671537 0.740971i \(-0.265635\pi\)
0.671537 + 0.740971i \(0.265635\pi\)
\(368\) 0 0
\(369\) −15.9147 −0.828484
\(370\) 0 0
\(371\) 18.7825 0.975140
\(372\) 0 0
\(373\) 25.2892 1.30943 0.654714 0.755877i \(-0.272789\pi\)
0.654714 + 0.755877i \(0.272789\pi\)
\(374\) 0 0
\(375\) 13.6129 0.702968
\(376\) 0 0
\(377\) −13.5013 −0.695352
\(378\) 0 0
\(379\) −29.2568 −1.50282 −0.751410 0.659835i \(-0.770626\pi\)
−0.751410 + 0.659835i \(0.770626\pi\)
\(380\) 0 0
\(381\) −5.64433 −0.289168
\(382\) 0 0
\(383\) −17.1697 −0.877331 −0.438666 0.898650i \(-0.644549\pi\)
−0.438666 + 0.898650i \(0.644549\pi\)
\(384\) 0 0
\(385\) 11.5882 0.590588
\(386\) 0 0
\(387\) −18.8140 −0.956371
\(388\) 0 0
\(389\) 1.28895 0.0653525 0.0326763 0.999466i \(-0.489597\pi\)
0.0326763 + 0.999466i \(0.489597\pi\)
\(390\) 0 0
\(391\) −32.0739 −1.62205
\(392\) 0 0
\(393\) 17.3085 0.873097
\(394\) 0 0
\(395\) −27.4100 −1.37915
\(396\) 0 0
\(397\) −0.194786 −0.00977601 −0.00488801 0.999988i \(-0.501556\pi\)
−0.00488801 + 0.999988i \(0.501556\pi\)
\(398\) 0 0
\(399\) 12.4362 0.622586
\(400\) 0 0
\(401\) −22.4677 −1.12198 −0.560991 0.827822i \(-0.689580\pi\)
−0.560991 + 0.827822i \(0.689580\pi\)
\(402\) 0 0
\(403\) 7.61460 0.379310
\(404\) 0 0
\(405\) 2.05898 0.102312
\(406\) 0 0
\(407\) −15.6136 −0.773938
\(408\) 0 0
\(409\) −14.5990 −0.721873 −0.360936 0.932590i \(-0.617543\pi\)
−0.360936 + 0.932590i \(0.617543\pi\)
\(410\) 0 0
\(411\) 14.8682 0.733394
\(412\) 0 0
\(413\) −3.42605 −0.168585
\(414\) 0 0
\(415\) −13.3847 −0.657031
\(416\) 0 0
\(417\) 8.71209 0.426633
\(418\) 0 0
\(419\) 26.0017 1.27027 0.635134 0.772402i \(-0.280945\pi\)
0.635134 + 0.772402i \(0.280945\pi\)
\(420\) 0 0
\(421\) 13.7755 0.671377 0.335689 0.941973i \(-0.391031\pi\)
0.335689 + 0.941973i \(0.391031\pi\)
\(422\) 0 0
\(423\) 10.7442 0.522403
\(424\) 0 0
\(425\) −4.39243 −0.213064
\(426\) 0 0
\(427\) −2.39850 −0.116072
\(428\) 0 0
\(429\) −6.32555 −0.305401
\(430\) 0 0
\(431\) 8.41827 0.405494 0.202747 0.979231i \(-0.435013\pi\)
0.202747 + 0.979231i \(0.435013\pi\)
\(432\) 0 0
\(433\) −17.0471 −0.819233 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(434\) 0 0
\(435\) 8.43060 0.404216
\(436\) 0 0
\(437\) −18.3949 −0.879948
\(438\) 0 0
\(439\) 24.7138 1.17952 0.589762 0.807577i \(-0.299222\pi\)
0.589762 + 0.807577i \(0.299222\pi\)
\(440\) 0 0
\(441\) −12.8507 −0.611937
\(442\) 0 0
\(443\) −30.4345 −1.44599 −0.722993 0.690856i \(-0.757234\pi\)
−0.722993 + 0.690856i \(0.757234\pi\)
\(444\) 0 0
\(445\) 26.4300 1.25290
\(446\) 0 0
\(447\) 13.3856 0.633117
\(448\) 0 0
\(449\) −18.6290 −0.879155 −0.439578 0.898205i \(-0.644872\pi\)
−0.439578 + 0.898205i \(0.644872\pi\)
\(450\) 0 0
\(451\) −14.0141 −0.659899
\(452\) 0 0
\(453\) −20.1988 −0.949020
\(454\) 0 0
\(455\) −28.7047 −1.34570
\(456\) 0 0
\(457\) −4.74271 −0.221855 −0.110927 0.993829i \(-0.535382\pi\)
−0.110927 + 0.993829i \(0.535382\pi\)
\(458\) 0 0
\(459\) −26.7027 −1.24638
\(460\) 0 0
\(461\) −27.7318 −1.29160 −0.645800 0.763506i \(-0.723476\pi\)
−0.645800 + 0.763506i \(0.723476\pi\)
\(462\) 0 0
\(463\) 26.5182 1.23240 0.616202 0.787588i \(-0.288670\pi\)
0.616202 + 0.787588i \(0.288670\pi\)
\(464\) 0 0
\(465\) −4.75477 −0.220497
\(466\) 0 0
\(467\) −19.5357 −0.904005 −0.452002 0.892017i \(-0.649290\pi\)
−0.452002 + 0.892017i \(0.649290\pi\)
\(468\) 0 0
\(469\) −44.4086 −2.05060
\(470\) 0 0
\(471\) 4.49099 0.206934
\(472\) 0 0
\(473\) −16.5672 −0.761763
\(474\) 0 0
\(475\) −2.51913 −0.115586
\(476\) 0 0
\(477\) −8.36379 −0.382952
\(478\) 0 0
\(479\) 14.9921 0.685007 0.342503 0.939517i \(-0.388725\pi\)
0.342503 + 0.939517i \(0.388725\pi\)
\(480\) 0 0
\(481\) 38.6761 1.76348
\(482\) 0 0
\(483\) −28.0174 −1.27483
\(484\) 0 0
\(485\) 0.0168781 0.000766395 0
\(486\) 0 0
\(487\) 6.27755 0.284463 0.142231 0.989833i \(-0.454572\pi\)
0.142231 + 0.989833i \(0.454572\pi\)
\(488\) 0 0
\(489\) 13.1672 0.595440
\(490\) 0 0
\(491\) −17.3489 −0.782946 −0.391473 0.920190i \(-0.628034\pi\)
−0.391473 + 0.920190i \(0.628034\pi\)
\(492\) 0 0
\(493\) −18.1483 −0.817357
\(494\) 0 0
\(495\) −5.16017 −0.231933
\(496\) 0 0
\(497\) −17.1158 −0.767746
\(498\) 0 0
\(499\) −12.3577 −0.553209 −0.276604 0.960984i \(-0.589209\pi\)
−0.276604 + 0.960984i \(0.589209\pi\)
\(500\) 0 0
\(501\) 26.7371 1.19452
\(502\) 0 0
\(503\) 26.5532 1.18395 0.591975 0.805956i \(-0.298348\pi\)
0.591975 + 0.805956i \(0.298348\pi\)
\(504\) 0 0
\(505\) −27.0994 −1.20591
\(506\) 0 0
\(507\) 0.842485 0.0374161
\(508\) 0 0
\(509\) 6.23099 0.276184 0.138092 0.990419i \(-0.455903\pi\)
0.138092 + 0.990419i \(0.455903\pi\)
\(510\) 0 0
\(511\) 13.1470 0.581590
\(512\) 0 0
\(513\) −15.3144 −0.676149
\(514\) 0 0
\(515\) −12.0500 −0.530987
\(516\) 0 0
\(517\) 9.46116 0.416101
\(518\) 0 0
\(519\) 24.8929 1.09268
\(520\) 0 0
\(521\) 25.6417 1.12338 0.561692 0.827346i \(-0.310150\pi\)
0.561692 + 0.827346i \(0.310150\pi\)
\(522\) 0 0
\(523\) 21.9192 0.958460 0.479230 0.877689i \(-0.340916\pi\)
0.479230 + 0.877689i \(0.340916\pi\)
\(524\) 0 0
\(525\) −3.83690 −0.167456
\(526\) 0 0
\(527\) 10.2354 0.445863
\(528\) 0 0
\(529\) 18.4418 0.801819
\(530\) 0 0
\(531\) 1.52561 0.0662058
\(532\) 0 0
\(533\) 34.7140 1.50363
\(534\) 0 0
\(535\) 26.2219 1.13367
\(536\) 0 0
\(537\) −14.8812 −0.642169
\(538\) 0 0
\(539\) −11.3160 −0.487417
\(540\) 0 0
\(541\) −20.5507 −0.883545 −0.441772 0.897127i \(-0.645650\pi\)
−0.441772 + 0.897127i \(0.645650\pi\)
\(542\) 0 0
\(543\) −17.4000 −0.746707
\(544\) 0 0
\(545\) −23.7110 −1.01567
\(546\) 0 0
\(547\) −19.1703 −0.819665 −0.409832 0.912161i \(-0.634413\pi\)
−0.409832 + 0.912161i \(0.634413\pi\)
\(548\) 0 0
\(549\) 1.06804 0.0455830
\(550\) 0 0
\(551\) −10.4083 −0.443410
\(552\) 0 0
\(553\) 51.5421 2.19179
\(554\) 0 0
\(555\) −24.1504 −1.02513
\(556\) 0 0
\(557\) −32.5259 −1.37817 −0.689083 0.724682i \(-0.741986\pi\)
−0.689083 + 0.724682i \(0.741986\pi\)
\(558\) 0 0
\(559\) 41.0383 1.73573
\(560\) 0 0
\(561\) −8.50272 −0.358985
\(562\) 0 0
\(563\) 9.87879 0.416341 0.208171 0.978093i \(-0.433249\pi\)
0.208171 + 0.978093i \(0.433249\pi\)
\(564\) 0 0
\(565\) 5.85733 0.246420
\(566\) 0 0
\(567\) −3.87173 −0.162597
\(568\) 0 0
\(569\) −46.9242 −1.96717 −0.983583 0.180455i \(-0.942243\pi\)
−0.983583 + 0.180455i \(0.942243\pi\)
\(570\) 0 0
\(571\) 2.80202 0.117261 0.0586304 0.998280i \(-0.481327\pi\)
0.0586304 + 0.998280i \(0.481327\pi\)
\(572\) 0 0
\(573\) −10.7404 −0.448686
\(574\) 0 0
\(575\) 5.67534 0.236678
\(576\) 0 0
\(577\) 4.53373 0.188741 0.0943707 0.995537i \(-0.469916\pi\)
0.0943707 + 0.995537i \(0.469916\pi\)
\(578\) 0 0
\(579\) 24.1668 1.00434
\(580\) 0 0
\(581\) 25.1688 1.04418
\(582\) 0 0
\(583\) −7.36498 −0.305026
\(584\) 0 0
\(585\) 12.7821 0.528476
\(586\) 0 0
\(587\) −20.8948 −0.862422 −0.431211 0.902251i \(-0.641914\pi\)
−0.431211 + 0.902251i \(0.641914\pi\)
\(588\) 0 0
\(589\) 5.87019 0.241877
\(590\) 0 0
\(591\) 13.0524 0.536902
\(592\) 0 0
\(593\) 8.93943 0.367098 0.183549 0.983011i \(-0.441241\pi\)
0.183549 + 0.983011i \(0.441241\pi\)
\(594\) 0 0
\(595\) −38.5845 −1.58181
\(596\) 0 0
\(597\) 10.3437 0.423338
\(598\) 0 0
\(599\) −38.0894 −1.55629 −0.778146 0.628084i \(-0.783840\pi\)
−0.778146 + 0.628084i \(0.783840\pi\)
\(600\) 0 0
\(601\) −48.0421 −1.95968 −0.979839 0.199790i \(-0.935974\pi\)
−0.979839 + 0.199790i \(0.935974\pi\)
\(602\) 0 0
\(603\) 19.7750 0.805299
\(604\) 0 0
\(605\) 17.7793 0.722830
\(606\) 0 0
\(607\) −34.7009 −1.40846 −0.704232 0.709970i \(-0.748709\pi\)
−0.704232 + 0.709970i \(0.748709\pi\)
\(608\) 0 0
\(609\) −15.8530 −0.642395
\(610\) 0 0
\(611\) −23.4360 −0.948118
\(612\) 0 0
\(613\) −3.07375 −0.124148 −0.0620739 0.998072i \(-0.519771\pi\)
−0.0620739 + 0.998072i \(0.519771\pi\)
\(614\) 0 0
\(615\) −21.6764 −0.874076
\(616\) 0 0
\(617\) −7.75817 −0.312332 −0.156166 0.987731i \(-0.549913\pi\)
−0.156166 + 0.987731i \(0.549913\pi\)
\(618\) 0 0
\(619\) −8.20212 −0.329671 −0.164835 0.986321i \(-0.552709\pi\)
−0.164835 + 0.986321i \(0.552709\pi\)
\(620\) 0 0
\(621\) 34.5019 1.38451
\(622\) 0 0
\(623\) −49.6992 −1.99116
\(624\) 0 0
\(625\) −19.8148 −0.792591
\(626\) 0 0
\(627\) −4.87645 −0.194747
\(628\) 0 0
\(629\) 51.9878 2.07289
\(630\) 0 0
\(631\) −36.5135 −1.45358 −0.726790 0.686860i \(-0.758989\pi\)
−0.726790 + 0.686860i \(0.758989\pi\)
\(632\) 0 0
\(633\) −15.3648 −0.610697
\(634\) 0 0
\(635\) 10.0435 0.398565
\(636\) 0 0
\(637\) 28.0307 1.11062
\(638\) 0 0
\(639\) 7.62158 0.301505
\(640\) 0 0
\(641\) 48.4002 1.91169 0.955846 0.293869i \(-0.0949430\pi\)
0.955846 + 0.293869i \(0.0949430\pi\)
\(642\) 0 0
\(643\) 17.1467 0.676198 0.338099 0.941111i \(-0.390216\pi\)
0.338099 + 0.941111i \(0.390216\pi\)
\(644\) 0 0
\(645\) −25.6254 −1.00900
\(646\) 0 0
\(647\) −32.9336 −1.29475 −0.647377 0.762170i \(-0.724134\pi\)
−0.647377 + 0.762170i \(0.724134\pi\)
\(648\) 0 0
\(649\) 1.34342 0.0527338
\(650\) 0 0
\(651\) 8.94092 0.350422
\(652\) 0 0
\(653\) 5.23931 0.205030 0.102515 0.994731i \(-0.467311\pi\)
0.102515 + 0.994731i \(0.467311\pi\)
\(654\) 0 0
\(655\) −30.7986 −1.20340
\(656\) 0 0
\(657\) −5.85432 −0.228399
\(658\) 0 0
\(659\) −10.1470 −0.395271 −0.197635 0.980276i \(-0.563326\pi\)
−0.197635 + 0.980276i \(0.563326\pi\)
\(660\) 0 0
\(661\) 17.5201 0.681453 0.340727 0.940162i \(-0.389327\pi\)
0.340727 + 0.940162i \(0.389327\pi\)
\(662\) 0 0
\(663\) 21.0618 0.817974
\(664\) 0 0
\(665\) −22.1289 −0.858120
\(666\) 0 0
\(667\) 23.4489 0.907945
\(668\) 0 0
\(669\) 2.34934 0.0908306
\(670\) 0 0
\(671\) 0.940498 0.0363075
\(672\) 0 0
\(673\) 48.1182 1.85482 0.927410 0.374046i \(-0.122030\pi\)
0.927410 + 0.374046i \(0.122030\pi\)
\(674\) 0 0
\(675\) 4.72493 0.181863
\(676\) 0 0
\(677\) −11.8423 −0.455137 −0.227568 0.973762i \(-0.573078\pi\)
−0.227568 + 0.973762i \(0.573078\pi\)
\(678\) 0 0
\(679\) −0.0317377 −0.00121798
\(680\) 0 0
\(681\) 11.1420 0.426962
\(682\) 0 0
\(683\) −28.2604 −1.08136 −0.540678 0.841230i \(-0.681832\pi\)
−0.540678 + 0.841230i \(0.681832\pi\)
\(684\) 0 0
\(685\) −26.4564 −1.01085
\(686\) 0 0
\(687\) 25.1717 0.960359
\(688\) 0 0
\(689\) 18.2436 0.695025
\(690\) 0 0
\(691\) 16.4838 0.627073 0.313536 0.949576i \(-0.398486\pi\)
0.313536 + 0.949576i \(0.398486\pi\)
\(692\) 0 0
\(693\) 9.70324 0.368596
\(694\) 0 0
\(695\) −15.5023 −0.588035
\(696\) 0 0
\(697\) 46.6620 1.76745
\(698\) 0 0
\(699\) −18.4410 −0.697502
\(700\) 0 0
\(701\) −50.3831 −1.90294 −0.951471 0.307738i \(-0.900428\pi\)
−0.951471 + 0.307738i \(0.900428\pi\)
\(702\) 0 0
\(703\) 29.8159 1.12453
\(704\) 0 0
\(705\) 14.6341 0.551151
\(706\) 0 0
\(707\) 50.9580 1.91647
\(708\) 0 0
\(709\) −33.2434 −1.24848 −0.624241 0.781231i \(-0.714592\pi\)
−0.624241 + 0.781231i \(0.714592\pi\)
\(710\) 0 0
\(711\) −22.9515 −0.860749
\(712\) 0 0
\(713\) −13.2249 −0.495278
\(714\) 0 0
\(715\) 11.2557 0.420938
\(716\) 0 0
\(717\) −24.6589 −0.920902
\(718\) 0 0
\(719\) 10.4707 0.390492 0.195246 0.980754i \(-0.437449\pi\)
0.195246 + 0.980754i \(0.437449\pi\)
\(720\) 0 0
\(721\) 22.6590 0.843865
\(722\) 0 0
\(723\) 22.6430 0.842101
\(724\) 0 0
\(725\) 3.21126 0.119263
\(726\) 0 0
\(727\) −28.5204 −1.05776 −0.528881 0.848696i \(-0.677388\pi\)
−0.528881 + 0.848696i \(0.677388\pi\)
\(728\) 0 0
\(729\) 20.0614 0.743014
\(730\) 0 0
\(731\) 55.1630 2.04028
\(732\) 0 0
\(733\) 3.63254 0.134171 0.0670855 0.997747i \(-0.478630\pi\)
0.0670855 + 0.997747i \(0.478630\pi\)
\(734\) 0 0
\(735\) −17.5031 −0.645613
\(736\) 0 0
\(737\) 17.4134 0.641432
\(738\) 0 0
\(739\) 24.0059 0.883072 0.441536 0.897244i \(-0.354434\pi\)
0.441536 + 0.897244i \(0.354434\pi\)
\(740\) 0 0
\(741\) 12.0793 0.443745
\(742\) 0 0
\(743\) 20.4194 0.749113 0.374557 0.927204i \(-0.377795\pi\)
0.374557 + 0.927204i \(0.377795\pi\)
\(744\) 0 0
\(745\) −23.8183 −0.872635
\(746\) 0 0
\(747\) −11.2076 −0.410064
\(748\) 0 0
\(749\) −49.3079 −1.80167
\(750\) 0 0
\(751\) −44.8862 −1.63792 −0.818961 0.573849i \(-0.805450\pi\)
−0.818961 + 0.573849i \(0.805450\pi\)
\(752\) 0 0
\(753\) −1.14049 −0.0415618
\(754\) 0 0
\(755\) 35.9416 1.30805
\(756\) 0 0
\(757\) −29.9782 −1.08957 −0.544787 0.838574i \(-0.683390\pi\)
−0.544787 + 0.838574i \(0.683390\pi\)
\(758\) 0 0
\(759\) 10.9861 0.398772
\(760\) 0 0
\(761\) −36.1087 −1.30894 −0.654469 0.756089i \(-0.727108\pi\)
−0.654469 + 0.756089i \(0.727108\pi\)
\(762\) 0 0
\(763\) 44.5864 1.61414
\(764\) 0 0
\(765\) 17.1815 0.621200
\(766\) 0 0
\(767\) −3.32775 −0.120158
\(768\) 0 0
\(769\) −48.6196 −1.75327 −0.876633 0.481160i \(-0.840216\pi\)
−0.876633 + 0.481160i \(0.840216\pi\)
\(770\) 0 0
\(771\) −1.70692 −0.0614732
\(772\) 0 0
\(773\) 18.0205 0.648151 0.324075 0.946031i \(-0.394947\pi\)
0.324075 + 0.946031i \(0.394947\pi\)
\(774\) 0 0
\(775\) −1.81112 −0.0650573
\(776\) 0 0
\(777\) 45.4127 1.62917
\(778\) 0 0
\(779\) 26.7614 0.958828
\(780\) 0 0
\(781\) 6.71141 0.240153
\(782\) 0 0
\(783\) 19.5221 0.697662
\(784\) 0 0
\(785\) −7.99125 −0.285220
\(786\) 0 0
\(787\) 37.4092 1.33350 0.666748 0.745283i \(-0.267686\pi\)
0.666748 + 0.745283i \(0.267686\pi\)
\(788\) 0 0
\(789\) −0.729187 −0.0259597
\(790\) 0 0
\(791\) −11.0142 −0.391619
\(792\) 0 0
\(793\) −2.32968 −0.0827294
\(794\) 0 0
\(795\) −11.3918 −0.404026
\(796\) 0 0
\(797\) 38.7582 1.37288 0.686442 0.727184i \(-0.259171\pi\)
0.686442 + 0.727184i \(0.259171\pi\)
\(798\) 0 0
\(799\) −31.5023 −1.11447
\(800\) 0 0
\(801\) 22.1309 0.781956
\(802\) 0 0
\(803\) −5.15519 −0.181923
\(804\) 0 0
\(805\) 49.8540 1.75712
\(806\) 0 0
\(807\) 5.88113 0.207026
\(808\) 0 0
\(809\) 5.29507 0.186165 0.0930824 0.995658i \(-0.470328\pi\)
0.0930824 + 0.995658i \(0.470328\pi\)
\(810\) 0 0
\(811\) −20.7038 −0.727009 −0.363504 0.931592i \(-0.618420\pi\)
−0.363504 + 0.931592i \(0.618420\pi\)
\(812\) 0 0
\(813\) −20.6521 −0.724300
\(814\) 0 0
\(815\) −23.4296 −0.820704
\(816\) 0 0
\(817\) 31.6369 1.10684
\(818\) 0 0
\(819\) −24.0356 −0.839873
\(820\) 0 0
\(821\) −43.1154 −1.50474 −0.752369 0.658741i \(-0.771089\pi\)
−0.752369 + 0.658741i \(0.771089\pi\)
\(822\) 0 0
\(823\) −26.6434 −0.928729 −0.464365 0.885644i \(-0.653717\pi\)
−0.464365 + 0.885644i \(0.653717\pi\)
\(824\) 0 0
\(825\) 1.50452 0.0523807
\(826\) 0 0
\(827\) 14.3708 0.499722 0.249861 0.968282i \(-0.419615\pi\)
0.249861 + 0.968282i \(0.419615\pi\)
\(828\) 0 0
\(829\) −33.9001 −1.17740 −0.588700 0.808351i \(-0.700360\pi\)
−0.588700 + 0.808351i \(0.700360\pi\)
\(830\) 0 0
\(831\) −25.9135 −0.898930
\(832\) 0 0
\(833\) 37.6784 1.30548
\(834\) 0 0
\(835\) −47.5759 −1.64643
\(836\) 0 0
\(837\) −11.0103 −0.380570
\(838\) 0 0
\(839\) 52.1634 1.80088 0.900441 0.434978i \(-0.143244\pi\)
0.900441 + 0.434978i \(0.143244\pi\)
\(840\) 0 0
\(841\) −15.7320 −0.542483
\(842\) 0 0
\(843\) −15.3400 −0.528339
\(844\) 0 0
\(845\) −1.49912 −0.0515712
\(846\) 0 0
\(847\) −33.4323 −1.14875
\(848\) 0 0
\(849\) −23.3777 −0.802322
\(850\) 0 0
\(851\) −67.1721 −2.30263
\(852\) 0 0
\(853\) 35.3930 1.21183 0.605917 0.795528i \(-0.292806\pi\)
0.605917 + 0.795528i \(0.292806\pi\)
\(854\) 0 0
\(855\) 9.85390 0.336996
\(856\) 0 0
\(857\) −50.1472 −1.71300 −0.856498 0.516151i \(-0.827364\pi\)
−0.856498 + 0.516151i \(0.827364\pi\)
\(858\) 0 0
\(859\) −41.2759 −1.40832 −0.704158 0.710043i \(-0.748675\pi\)
−0.704158 + 0.710043i \(0.748675\pi\)
\(860\) 0 0
\(861\) 40.7605 1.38911
\(862\) 0 0
\(863\) 13.8787 0.472436 0.236218 0.971700i \(-0.424092\pi\)
0.236218 + 0.971700i \(0.424092\pi\)
\(864\) 0 0
\(865\) −44.2943 −1.50605
\(866\) 0 0
\(867\) 8.92271 0.303031
\(868\) 0 0
\(869\) −20.2106 −0.685599
\(870\) 0 0
\(871\) −43.1343 −1.46155
\(872\) 0 0
\(873\) 0.0141327 0.000478319 0
\(874\) 0 0
\(875\) 45.5488 1.53983
\(876\) 0 0
\(877\) 10.1481 0.342676 0.171338 0.985212i \(-0.445191\pi\)
0.171338 + 0.985212i \(0.445191\pi\)
\(878\) 0 0
\(879\) 11.9322 0.402464
\(880\) 0 0
\(881\) 16.3824 0.551937 0.275968 0.961167i \(-0.411001\pi\)
0.275968 + 0.961167i \(0.411001\pi\)
\(882\) 0 0
\(883\) −15.7048 −0.528508 −0.264254 0.964453i \(-0.585126\pi\)
−0.264254 + 0.964453i \(0.585126\pi\)
\(884\) 0 0
\(885\) 2.07794 0.0698492
\(886\) 0 0
\(887\) 5.68720 0.190957 0.0954787 0.995431i \(-0.469562\pi\)
0.0954787 + 0.995431i \(0.469562\pi\)
\(888\) 0 0
\(889\) −18.8859 −0.633413
\(890\) 0 0
\(891\) 1.51818 0.0508609
\(892\) 0 0
\(893\) −18.0671 −0.604592
\(894\) 0 0
\(895\) 26.4795 0.885112
\(896\) 0 0
\(897\) −27.2135 −0.908631
\(898\) 0 0
\(899\) −7.48302 −0.249573
\(900\) 0 0
\(901\) 24.5228 0.816972
\(902\) 0 0
\(903\) 48.1863 1.60354
\(904\) 0 0
\(905\) 30.9616 1.02920
\(906\) 0 0
\(907\) 43.7121 1.45144 0.725718 0.687992i \(-0.241508\pi\)
0.725718 + 0.687992i \(0.241508\pi\)
\(908\) 0 0
\(909\) −22.6914 −0.752627
\(910\) 0 0
\(911\) −8.68438 −0.287726 −0.143863 0.989598i \(-0.545953\pi\)
−0.143863 + 0.989598i \(0.545953\pi\)
\(912\) 0 0
\(913\) −9.86916 −0.326621
\(914\) 0 0
\(915\) 1.45472 0.0480915
\(916\) 0 0
\(917\) 57.9141 1.91249
\(918\) 0 0
\(919\) −6.65811 −0.219631 −0.109815 0.993952i \(-0.535026\pi\)
−0.109815 + 0.993952i \(0.535026\pi\)
\(920\) 0 0
\(921\) 4.92995 0.162447
\(922\) 0 0
\(923\) −16.6246 −0.547207
\(924\) 0 0
\(925\) −9.19903 −0.302462
\(926\) 0 0
\(927\) −10.0900 −0.331398
\(928\) 0 0
\(929\) −27.5701 −0.904544 −0.452272 0.891880i \(-0.649386\pi\)
−0.452272 + 0.891880i \(0.649386\pi\)
\(930\) 0 0
\(931\) 21.6092 0.708213
\(932\) 0 0
\(933\) 2.91331 0.0953776
\(934\) 0 0
\(935\) 15.1297 0.494795
\(936\) 0 0
\(937\) 40.4470 1.32135 0.660673 0.750674i \(-0.270271\pi\)
0.660673 + 0.750674i \(0.270271\pi\)
\(938\) 0 0
\(939\) −23.4912 −0.766605
\(940\) 0 0
\(941\) 28.6692 0.934588 0.467294 0.884102i \(-0.345229\pi\)
0.467294 + 0.884102i \(0.345229\pi\)
\(942\) 0 0
\(943\) −60.2908 −1.96334
\(944\) 0 0
\(945\) 41.5053 1.35017
\(946\) 0 0
\(947\) 15.0634 0.489496 0.244748 0.969587i \(-0.421295\pi\)
0.244748 + 0.969587i \(0.421295\pi\)
\(948\) 0 0
\(949\) 12.7698 0.414525
\(950\) 0 0
\(951\) 32.4887 1.05352
\(952\) 0 0
\(953\) 3.82793 0.123999 0.0619995 0.998076i \(-0.480252\pi\)
0.0619995 + 0.998076i \(0.480252\pi\)
\(954\) 0 0
\(955\) 19.1114 0.618430
\(956\) 0 0
\(957\) 6.21625 0.200943
\(958\) 0 0
\(959\) 49.7489 1.60648
\(960\) 0 0
\(961\) −26.7796 −0.863860
\(962\) 0 0
\(963\) 21.9566 0.707542
\(964\) 0 0
\(965\) −43.0024 −1.38429
\(966\) 0 0
\(967\) 0.660265 0.0212327 0.0106163 0.999944i \(-0.496621\pi\)
0.0106163 + 0.999944i \(0.496621\pi\)
\(968\) 0 0
\(969\) 16.2368 0.521603
\(970\) 0 0
\(971\) 25.2503 0.810320 0.405160 0.914246i \(-0.367216\pi\)
0.405160 + 0.914246i \(0.367216\pi\)
\(972\) 0 0
\(973\) 29.1506 0.934526
\(974\) 0 0
\(975\) −3.72681 −0.119353
\(976\) 0 0
\(977\) 14.3327 0.458544 0.229272 0.973362i \(-0.426365\pi\)
0.229272 + 0.973362i \(0.426365\pi\)
\(978\) 0 0
\(979\) 19.4880 0.622839
\(980\) 0 0
\(981\) −19.8542 −0.633895
\(982\) 0 0
\(983\) 54.6511 1.74310 0.871550 0.490306i \(-0.163115\pi\)
0.871550 + 0.490306i \(0.163115\pi\)
\(984\) 0 0
\(985\) −23.2253 −0.740021
\(986\) 0 0
\(987\) −27.5181 −0.875910
\(988\) 0 0
\(989\) −71.2747 −2.26640
\(990\) 0 0
\(991\) −50.0085 −1.58857 −0.794285 0.607545i \(-0.792155\pi\)
−0.794285 + 0.607545i \(0.792155\pi\)
\(992\) 0 0
\(993\) −33.9263 −1.07662
\(994\) 0 0
\(995\) −18.4055 −0.583493
\(996\) 0 0
\(997\) −28.7296 −0.909876 −0.454938 0.890523i \(-0.650339\pi\)
−0.454938 + 0.890523i \(0.650339\pi\)
\(998\) 0 0
\(999\) −55.9232 −1.76933
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 4016.2.a.d.1.4 5
4.3 odd 2 502.2.a.d.1.2 5
12.11 even 2 4518.2.a.t.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
502.2.a.d.1.2 5 4.3 odd 2
4016.2.a.d.1.4 5 1.1 even 1 trivial
4518.2.a.t.1.5 5 12.11 even 2