Properties

Label 8016.2.a.p.1.5
Level $8016$
Weight $2$
Character 8016.1
Self dual yes
Analytic conductor $64.008$
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] = [8016,2,Mod(1,8016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8016, 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("8016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.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: \(64.0080822603\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.36497.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 3x^{3} + 5x^{2} + x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 501)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.506287\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.21255 q^{5} +3.40756 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.21255 q^{5} +3.40756 q^{7} +1.00000 q^{9} +4.19997 q^{11} -2.51125 q^{13} -1.21255 q^{15} -4.79075 q^{17} -3.24068 q^{19} -3.40756 q^{21} +0.142037 q^{23} -3.52972 q^{25} -1.00000 q^{27} +0.264204 q^{29} +10.0831 q^{31} -4.19997 q^{33} +4.13183 q^{35} +6.54621 q^{37} +2.51125 q^{39} -9.58639 q^{41} -10.2791 q^{43} +1.21255 q^{45} +7.14732 q^{47} +4.61145 q^{49} +4.79075 q^{51} -0.687009 q^{53} +5.09268 q^{55} +3.24068 q^{57} +14.3473 q^{59} +10.7130 q^{61} +3.40756 q^{63} -3.04502 q^{65} +1.19870 q^{67} -0.142037 q^{69} -3.15248 q^{71} +0.153683 q^{73} +3.52972 q^{75} +14.3117 q^{77} +3.65659 q^{79} +1.00000 q^{81} +3.96483 q^{83} -5.80902 q^{85} -0.264204 q^{87} +8.14055 q^{89} -8.55724 q^{91} -10.0831 q^{93} -3.92949 q^{95} +3.44602 q^{97} +4.19997 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{3} - 9 q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{3} - 9 q^{5} + 4 q^{7} + 5 q^{9} + 15 q^{11} + 9 q^{15} - 11 q^{17} + 16 q^{19} - 4 q^{21} + 9 q^{23} + 6 q^{25} - 5 q^{27} - q^{29} + 18 q^{31} - 15 q^{33} + 4 q^{35} + 7 q^{37} - 10 q^{41} - 6 q^{43} - 9 q^{45} + 7 q^{47} + 11 q^{49} + 11 q^{51} - 9 q^{53} - 17 q^{55} - 16 q^{57} + 37 q^{59} - 2 q^{61} + 4 q^{63} - 16 q^{65} - 9 q^{69} - 13 q^{71} - 6 q^{73} - 6 q^{75} - 2 q^{77} - 14 q^{79} + 5 q^{81} + 5 q^{83} + 29 q^{85} + q^{87} - 30 q^{89} + 33 q^{91} - 18 q^{93} - 43 q^{95} - 9 q^{97} + 15 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.21255 0.542268 0.271134 0.962542i \(-0.412601\pi\)
0.271134 + 0.962542i \(0.412601\pi\)
\(6\) 0 0
\(7\) 3.40756 1.28794 0.643968 0.765053i \(-0.277287\pi\)
0.643968 + 0.765053i \(0.277287\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.19997 1.26634 0.633170 0.774013i \(-0.281753\pi\)
0.633170 + 0.774013i \(0.281753\pi\)
\(12\) 0 0
\(13\) −2.51125 −0.696497 −0.348248 0.937402i \(-0.613223\pi\)
−0.348248 + 0.937402i \(0.613223\pi\)
\(14\) 0 0
\(15\) −1.21255 −0.313079
\(16\) 0 0
\(17\) −4.79075 −1.16193 −0.580964 0.813930i \(-0.697324\pi\)
−0.580964 + 0.813930i \(0.697324\pi\)
\(18\) 0 0
\(19\) −3.24068 −0.743464 −0.371732 0.928340i \(-0.621236\pi\)
−0.371732 + 0.928340i \(0.621236\pi\)
\(20\) 0 0
\(21\) −3.40756 −0.743590
\(22\) 0 0
\(23\) 0.142037 0.0296168 0.0148084 0.999890i \(-0.495286\pi\)
0.0148084 + 0.999890i \(0.495286\pi\)
\(24\) 0 0
\(25\) −3.52972 −0.705945
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 0.264204 0.0490615 0.0245308 0.999699i \(-0.492191\pi\)
0.0245308 + 0.999699i \(0.492191\pi\)
\(30\) 0 0
\(31\) 10.0831 1.81098 0.905488 0.424372i \(-0.139505\pi\)
0.905488 + 0.424372i \(0.139505\pi\)
\(32\) 0 0
\(33\) −4.19997 −0.731122
\(34\) 0 0
\(35\) 4.13183 0.698407
\(36\) 0 0
\(37\) 6.54621 1.07619 0.538095 0.842884i \(-0.319144\pi\)
0.538095 + 0.842884i \(0.319144\pi\)
\(38\) 0 0
\(39\) 2.51125 0.402122
\(40\) 0 0
\(41\) −9.58639 −1.49714 −0.748571 0.663054i \(-0.769260\pi\)
−0.748571 + 0.663054i \(0.769260\pi\)
\(42\) 0 0
\(43\) −10.2791 −1.56756 −0.783778 0.621042i \(-0.786710\pi\)
−0.783778 + 0.621042i \(0.786710\pi\)
\(44\) 0 0
\(45\) 1.21255 0.180756
\(46\) 0 0
\(47\) 7.14732 1.04254 0.521272 0.853391i \(-0.325458\pi\)
0.521272 + 0.853391i \(0.325458\pi\)
\(48\) 0 0
\(49\) 4.61145 0.658778
\(50\) 0 0
\(51\) 4.79075 0.670839
\(52\) 0 0
\(53\) −0.687009 −0.0943679 −0.0471840 0.998886i \(-0.515025\pi\)
−0.0471840 + 0.998886i \(0.515025\pi\)
\(54\) 0 0
\(55\) 5.09268 0.686696
\(56\) 0 0
\(57\) 3.24068 0.429239
\(58\) 0 0
\(59\) 14.3473 1.86786 0.933929 0.357458i \(-0.116357\pi\)
0.933929 + 0.357458i \(0.116357\pi\)
\(60\) 0 0
\(61\) 10.7130 1.37166 0.685830 0.727762i \(-0.259439\pi\)
0.685830 + 0.727762i \(0.259439\pi\)
\(62\) 0 0
\(63\) 3.40756 0.429312
\(64\) 0 0
\(65\) −3.04502 −0.377688
\(66\) 0 0
\(67\) 1.19870 0.146445 0.0732224 0.997316i \(-0.476672\pi\)
0.0732224 + 0.997316i \(0.476672\pi\)
\(68\) 0 0
\(69\) −0.142037 −0.0170993
\(70\) 0 0
\(71\) −3.15248 −0.374131 −0.187065 0.982347i \(-0.559898\pi\)
−0.187065 + 0.982347i \(0.559898\pi\)
\(72\) 0 0
\(73\) 0.153683 0.0179872 0.00899359 0.999960i \(-0.497137\pi\)
0.00899359 + 0.999960i \(0.497137\pi\)
\(74\) 0 0
\(75\) 3.52972 0.407577
\(76\) 0 0
\(77\) 14.3117 1.63096
\(78\) 0 0
\(79\) 3.65659 0.411398 0.205699 0.978615i \(-0.434053\pi\)
0.205699 + 0.978615i \(0.434053\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.96483 0.435197 0.217599 0.976038i \(-0.430178\pi\)
0.217599 + 0.976038i \(0.430178\pi\)
\(84\) 0 0
\(85\) −5.80902 −0.630076
\(86\) 0 0
\(87\) −0.264204 −0.0283257
\(88\) 0 0
\(89\) 8.14055 0.862897 0.431448 0.902138i \(-0.358003\pi\)
0.431448 + 0.902138i \(0.358003\pi\)
\(90\) 0 0
\(91\) −8.55724 −0.897043
\(92\) 0 0
\(93\) −10.0831 −1.04557
\(94\) 0 0
\(95\) −3.92949 −0.403157
\(96\) 0 0
\(97\) 3.44602 0.349890 0.174945 0.984578i \(-0.444025\pi\)
0.174945 + 0.984578i \(0.444025\pi\)
\(98\) 0 0
\(99\) 4.19997 0.422113
\(100\) 0 0
\(101\) 13.3447 1.32785 0.663923 0.747801i \(-0.268890\pi\)
0.663923 + 0.747801i \(0.268890\pi\)
\(102\) 0 0
\(103\) 15.8434 1.56109 0.780547 0.625097i \(-0.214941\pi\)
0.780547 + 0.625097i \(0.214941\pi\)
\(104\) 0 0
\(105\) −4.13183 −0.403225
\(106\) 0 0
\(107\) −0.0656293 −0.00634462 −0.00317231 0.999995i \(-0.501010\pi\)
−0.00317231 + 0.999995i \(0.501010\pi\)
\(108\) 0 0
\(109\) 17.7446 1.69963 0.849814 0.527082i \(-0.176714\pi\)
0.849814 + 0.527082i \(0.176714\pi\)
\(110\) 0 0
\(111\) −6.54621 −0.621339
\(112\) 0 0
\(113\) 7.48673 0.704292 0.352146 0.935945i \(-0.385452\pi\)
0.352146 + 0.935945i \(0.385452\pi\)
\(114\) 0 0
\(115\) 0.172227 0.0160603
\(116\) 0 0
\(117\) −2.51125 −0.232166
\(118\) 0 0
\(119\) −16.3247 −1.49649
\(120\) 0 0
\(121\) 6.63979 0.603617
\(122\) 0 0
\(123\) 9.58639 0.864376
\(124\) 0 0
\(125\) −10.3427 −0.925080
\(126\) 0 0
\(127\) −20.4833 −1.81760 −0.908802 0.417228i \(-0.863001\pi\)
−0.908802 + 0.417228i \(0.863001\pi\)
\(128\) 0 0
\(129\) 10.2791 0.905028
\(130\) 0 0
\(131\) −10.4722 −0.914963 −0.457482 0.889219i \(-0.651248\pi\)
−0.457482 + 0.889219i \(0.651248\pi\)
\(132\) 0 0
\(133\) −11.0428 −0.957533
\(134\) 0 0
\(135\) −1.21255 −0.104360
\(136\) 0 0
\(137\) −0.157548 −0.0134602 −0.00673010 0.999977i \(-0.502142\pi\)
−0.00673010 + 0.999977i \(0.502142\pi\)
\(138\) 0 0
\(139\) 8.11580 0.688373 0.344187 0.938901i \(-0.388155\pi\)
0.344187 + 0.938901i \(0.388155\pi\)
\(140\) 0 0
\(141\) −7.14732 −0.601913
\(142\) 0 0
\(143\) −10.5472 −0.882001
\(144\) 0 0
\(145\) 0.320361 0.0266045
\(146\) 0 0
\(147\) −4.61145 −0.380346
\(148\) 0 0
\(149\) 16.7804 1.37470 0.687351 0.726325i \(-0.258773\pi\)
0.687351 + 0.726325i \(0.258773\pi\)
\(150\) 0 0
\(151\) 13.1837 1.07288 0.536438 0.843940i \(-0.319770\pi\)
0.536438 + 0.843940i \(0.319770\pi\)
\(152\) 0 0
\(153\) −4.79075 −0.387309
\(154\) 0 0
\(155\) 12.2262 0.982035
\(156\) 0 0
\(157\) 17.2372 1.37568 0.687840 0.725862i \(-0.258559\pi\)
0.687840 + 0.725862i \(0.258559\pi\)
\(158\) 0 0
\(159\) 0.687009 0.0544833
\(160\) 0 0
\(161\) 0.484000 0.0381445
\(162\) 0 0
\(163\) −1.08125 −0.0846901 −0.0423451 0.999103i \(-0.513483\pi\)
−0.0423451 + 0.999103i \(0.513483\pi\)
\(164\) 0 0
\(165\) −5.09268 −0.396464
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) −6.69360 −0.514893
\(170\) 0 0
\(171\) −3.24068 −0.247821
\(172\) 0 0
\(173\) −15.9953 −1.21610 −0.608049 0.793899i \(-0.708048\pi\)
−0.608049 + 0.793899i \(0.708048\pi\)
\(174\) 0 0
\(175\) −12.0277 −0.909212
\(176\) 0 0
\(177\) −14.3473 −1.07841
\(178\) 0 0
\(179\) 19.0256 1.42204 0.711018 0.703173i \(-0.248234\pi\)
0.711018 + 0.703173i \(0.248234\pi\)
\(180\) 0 0
\(181\) −20.3346 −1.51146 −0.755731 0.654883i \(-0.772718\pi\)
−0.755731 + 0.654883i \(0.772718\pi\)
\(182\) 0 0
\(183\) −10.7130 −0.791928
\(184\) 0 0
\(185\) 7.93761 0.583584
\(186\) 0 0
\(187\) −20.1210 −1.47139
\(188\) 0 0
\(189\) −3.40756 −0.247863
\(190\) 0 0
\(191\) 16.4252 1.18849 0.594243 0.804286i \(-0.297452\pi\)
0.594243 + 0.804286i \(0.297452\pi\)
\(192\) 0 0
\(193\) 6.63951 0.477922 0.238961 0.971029i \(-0.423193\pi\)
0.238961 + 0.971029i \(0.423193\pi\)
\(194\) 0 0
\(195\) 3.04502 0.218058
\(196\) 0 0
\(197\) 7.07078 0.503772 0.251886 0.967757i \(-0.418949\pi\)
0.251886 + 0.967757i \(0.418949\pi\)
\(198\) 0 0
\(199\) 0.717075 0.0508321 0.0254161 0.999677i \(-0.491909\pi\)
0.0254161 + 0.999677i \(0.491909\pi\)
\(200\) 0 0
\(201\) −1.19870 −0.0845499
\(202\) 0 0
\(203\) 0.900292 0.0631881
\(204\) 0 0
\(205\) −11.6240 −0.811853
\(206\) 0 0
\(207\) 0.142037 0.00987227
\(208\) 0 0
\(209\) −13.6108 −0.941478
\(210\) 0 0
\(211\) −12.1736 −0.838066 −0.419033 0.907971i \(-0.637631\pi\)
−0.419033 + 0.907971i \(0.637631\pi\)
\(212\) 0 0
\(213\) 3.15248 0.216005
\(214\) 0 0
\(215\) −12.4640 −0.850036
\(216\) 0 0
\(217\) 34.3587 2.33242
\(218\) 0 0
\(219\) −0.153683 −0.0103849
\(220\) 0 0
\(221\) 12.0308 0.809278
\(222\) 0 0
\(223\) −6.72478 −0.450325 −0.225162 0.974321i \(-0.572291\pi\)
−0.225162 + 0.974321i \(0.572291\pi\)
\(224\) 0 0
\(225\) −3.52972 −0.235315
\(226\) 0 0
\(227\) 11.2701 0.748023 0.374011 0.927424i \(-0.377982\pi\)
0.374011 + 0.927424i \(0.377982\pi\)
\(228\) 0 0
\(229\) −6.52089 −0.430912 −0.215456 0.976514i \(-0.569124\pi\)
−0.215456 + 0.976514i \(0.569124\pi\)
\(230\) 0 0
\(231\) −14.3117 −0.941638
\(232\) 0 0
\(233\) −22.0249 −1.44290 −0.721451 0.692466i \(-0.756524\pi\)
−0.721451 + 0.692466i \(0.756524\pi\)
\(234\) 0 0
\(235\) 8.66647 0.565338
\(236\) 0 0
\(237\) −3.65659 −0.237521
\(238\) 0 0
\(239\) −4.88322 −0.315869 −0.157934 0.987450i \(-0.550483\pi\)
−0.157934 + 0.987450i \(0.550483\pi\)
\(240\) 0 0
\(241\) −14.3294 −0.923039 −0.461519 0.887130i \(-0.652696\pi\)
−0.461519 + 0.887130i \(0.652696\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 5.59161 0.357235
\(246\) 0 0
\(247\) 8.13818 0.517820
\(248\) 0 0
\(249\) −3.96483 −0.251261
\(250\) 0 0
\(251\) −2.59502 −0.163796 −0.0818981 0.996641i \(-0.526098\pi\)
−0.0818981 + 0.996641i \(0.526098\pi\)
\(252\) 0 0
\(253\) 0.596553 0.0375049
\(254\) 0 0
\(255\) 5.80902 0.363775
\(256\) 0 0
\(257\) −29.6878 −1.85187 −0.925937 0.377679i \(-0.876722\pi\)
−0.925937 + 0.377679i \(0.876722\pi\)
\(258\) 0 0
\(259\) 22.3066 1.38606
\(260\) 0 0
\(261\) 0.264204 0.0163538
\(262\) 0 0
\(263\) 4.24468 0.261738 0.130869 0.991400i \(-0.458223\pi\)
0.130869 + 0.991400i \(0.458223\pi\)
\(264\) 0 0
\(265\) −0.833032 −0.0511727
\(266\) 0 0
\(267\) −8.14055 −0.498194
\(268\) 0 0
\(269\) −29.8482 −1.81988 −0.909938 0.414744i \(-0.863871\pi\)
−0.909938 + 0.414744i \(0.863871\pi\)
\(270\) 0 0
\(271\) 31.6214 1.92086 0.960432 0.278516i \(-0.0898425\pi\)
0.960432 + 0.278516i \(0.0898425\pi\)
\(272\) 0 0
\(273\) 8.55724 0.517908
\(274\) 0 0
\(275\) −14.8248 −0.893966
\(276\) 0 0
\(277\) 20.4671 1.22975 0.614874 0.788625i \(-0.289207\pi\)
0.614874 + 0.788625i \(0.289207\pi\)
\(278\) 0 0
\(279\) 10.0831 0.603659
\(280\) 0 0
\(281\) −22.8045 −1.36040 −0.680202 0.733025i \(-0.738108\pi\)
−0.680202 + 0.733025i \(0.738108\pi\)
\(282\) 0 0
\(283\) 6.36109 0.378128 0.189064 0.981965i \(-0.439455\pi\)
0.189064 + 0.981965i \(0.439455\pi\)
\(284\) 0 0
\(285\) 3.92949 0.232763
\(286\) 0 0
\(287\) −32.6662 −1.92822
\(288\) 0 0
\(289\) 5.95127 0.350075
\(290\) 0 0
\(291\) −3.44602 −0.202009
\(292\) 0 0
\(293\) −3.49011 −0.203894 −0.101947 0.994790i \(-0.532507\pi\)
−0.101947 + 0.994790i \(0.532507\pi\)
\(294\) 0 0
\(295\) 17.3968 1.01288
\(296\) 0 0
\(297\) −4.19997 −0.243707
\(298\) 0 0
\(299\) −0.356691 −0.0206280
\(300\) 0 0
\(301\) −35.0268 −2.01891
\(302\) 0 0
\(303\) −13.3447 −0.766633
\(304\) 0 0
\(305\) 12.9900 0.743808
\(306\) 0 0
\(307\) 3.64448 0.208002 0.104001 0.994577i \(-0.466836\pi\)
0.104001 + 0.994577i \(0.466836\pi\)
\(308\) 0 0
\(309\) −15.8434 −0.901298
\(310\) 0 0
\(311\) −10.9171 −0.619049 −0.309525 0.950891i \(-0.600170\pi\)
−0.309525 + 0.950891i \(0.600170\pi\)
\(312\) 0 0
\(313\) −2.69366 −0.152254 −0.0761272 0.997098i \(-0.524256\pi\)
−0.0761272 + 0.997098i \(0.524256\pi\)
\(314\) 0 0
\(315\) 4.13183 0.232802
\(316\) 0 0
\(317\) −11.6786 −0.655937 −0.327968 0.944689i \(-0.606364\pi\)
−0.327968 + 0.944689i \(0.606364\pi\)
\(318\) 0 0
\(319\) 1.10965 0.0621286
\(320\) 0 0
\(321\) 0.0656293 0.00366307
\(322\) 0 0
\(323\) 15.5253 0.863851
\(324\) 0 0
\(325\) 8.86403 0.491688
\(326\) 0 0
\(327\) −17.7446 −0.981281
\(328\) 0 0
\(329\) 24.3549 1.34273
\(330\) 0 0
\(331\) 4.93784 0.271408 0.135704 0.990749i \(-0.456670\pi\)
0.135704 + 0.990749i \(0.456670\pi\)
\(332\) 0 0
\(333\) 6.54621 0.358730
\(334\) 0 0
\(335\) 1.45348 0.0794124
\(336\) 0 0
\(337\) 1.78104 0.0970192 0.0485096 0.998823i \(-0.484553\pi\)
0.0485096 + 0.998823i \(0.484553\pi\)
\(338\) 0 0
\(339\) −7.48673 −0.406623
\(340\) 0 0
\(341\) 42.3487 2.29331
\(342\) 0 0
\(343\) −8.13913 −0.439472
\(344\) 0 0
\(345\) −0.172227 −0.00927240
\(346\) 0 0
\(347\) −0.767188 −0.0411848 −0.0205924 0.999788i \(-0.506555\pi\)
−0.0205924 + 0.999788i \(0.506555\pi\)
\(348\) 0 0
\(349\) 21.8125 1.16760 0.583798 0.811899i \(-0.301566\pi\)
0.583798 + 0.811899i \(0.301566\pi\)
\(350\) 0 0
\(351\) 2.51125 0.134041
\(352\) 0 0
\(353\) 22.7534 1.21104 0.605521 0.795829i \(-0.292965\pi\)
0.605521 + 0.795829i \(0.292965\pi\)
\(354\) 0 0
\(355\) −3.82254 −0.202879
\(356\) 0 0
\(357\) 16.3247 0.863997
\(358\) 0 0
\(359\) 30.8595 1.62870 0.814350 0.580374i \(-0.197093\pi\)
0.814350 + 0.580374i \(0.197093\pi\)
\(360\) 0 0
\(361\) −8.49797 −0.447262
\(362\) 0 0
\(363\) −6.63979 −0.348498
\(364\) 0 0
\(365\) 0.186348 0.00975388
\(366\) 0 0
\(367\) −5.53204 −0.288770 −0.144385 0.989522i \(-0.546120\pi\)
−0.144385 + 0.989522i \(0.546120\pi\)
\(368\) 0 0
\(369\) −9.58639 −0.499047
\(370\) 0 0
\(371\) −2.34102 −0.121540
\(372\) 0 0
\(373\) 32.5894 1.68742 0.843708 0.536802i \(-0.180368\pi\)
0.843708 + 0.536802i \(0.180368\pi\)
\(374\) 0 0
\(375\) 10.3427 0.534095
\(376\) 0 0
\(377\) −0.663484 −0.0341712
\(378\) 0 0
\(379\) 34.8665 1.79097 0.895487 0.445088i \(-0.146828\pi\)
0.895487 + 0.445088i \(0.146828\pi\)
\(380\) 0 0
\(381\) 20.4833 1.04939
\(382\) 0 0
\(383\) −2.30034 −0.117542 −0.0587709 0.998271i \(-0.518718\pi\)
−0.0587709 + 0.998271i \(0.518718\pi\)
\(384\) 0 0
\(385\) 17.3536 0.884421
\(386\) 0 0
\(387\) −10.2791 −0.522518
\(388\) 0 0
\(389\) 0.332692 0.0168682 0.00843408 0.999964i \(-0.497315\pi\)
0.00843408 + 0.999964i \(0.497315\pi\)
\(390\) 0 0
\(391\) −0.680464 −0.0344126
\(392\) 0 0
\(393\) 10.4722 0.528254
\(394\) 0 0
\(395\) 4.43379 0.223088
\(396\) 0 0
\(397\) 27.8441 1.39745 0.698727 0.715388i \(-0.253750\pi\)
0.698727 + 0.715388i \(0.253750\pi\)
\(398\) 0 0
\(399\) 11.0428 0.552832
\(400\) 0 0
\(401\) −9.63219 −0.481009 −0.240504 0.970648i \(-0.577313\pi\)
−0.240504 + 0.970648i \(0.577313\pi\)
\(402\) 0 0
\(403\) −25.3212 −1.26134
\(404\) 0 0
\(405\) 1.21255 0.0602521
\(406\) 0 0
\(407\) 27.4939 1.36282
\(408\) 0 0
\(409\) 37.0060 1.82983 0.914915 0.403647i \(-0.132258\pi\)
0.914915 + 0.403647i \(0.132258\pi\)
\(410\) 0 0
\(411\) 0.157548 0.00777125
\(412\) 0 0
\(413\) 48.8892 2.40568
\(414\) 0 0
\(415\) 4.80756 0.235994
\(416\) 0 0
\(417\) −8.11580 −0.397432
\(418\) 0 0
\(419\) 6.04230 0.295186 0.147593 0.989048i \(-0.452848\pi\)
0.147593 + 0.989048i \(0.452848\pi\)
\(420\) 0 0
\(421\) 14.4831 0.705862 0.352931 0.935649i \(-0.385185\pi\)
0.352931 + 0.935649i \(0.385185\pi\)
\(422\) 0 0
\(423\) 7.14732 0.347514
\(424\) 0 0
\(425\) 16.9100 0.820256
\(426\) 0 0
\(427\) 36.5052 1.76661
\(428\) 0 0
\(429\) 10.5472 0.509224
\(430\) 0 0
\(431\) 3.16758 0.152577 0.0762885 0.997086i \(-0.475693\pi\)
0.0762885 + 0.997086i \(0.475693\pi\)
\(432\) 0 0
\(433\) 26.8759 1.29157 0.645787 0.763518i \(-0.276529\pi\)
0.645787 + 0.763518i \(0.276529\pi\)
\(434\) 0 0
\(435\) −0.320361 −0.0153601
\(436\) 0 0
\(437\) −0.460298 −0.0220190
\(438\) 0 0
\(439\) −14.6647 −0.699909 −0.349955 0.936767i \(-0.613803\pi\)
−0.349955 + 0.936767i \(0.613803\pi\)
\(440\) 0 0
\(441\) 4.61145 0.219593
\(442\) 0 0
\(443\) −2.62513 −0.124724 −0.0623618 0.998054i \(-0.519863\pi\)
−0.0623618 + 0.998054i \(0.519863\pi\)
\(444\) 0 0
\(445\) 9.87082 0.467922
\(446\) 0 0
\(447\) −16.7804 −0.793685
\(448\) 0 0
\(449\) 2.26746 0.107008 0.0535041 0.998568i \(-0.482961\pi\)
0.0535041 + 0.998568i \(0.482961\pi\)
\(450\) 0 0
\(451\) −40.2626 −1.89589
\(452\) 0 0
\(453\) −13.1837 −0.619425
\(454\) 0 0
\(455\) −10.3761 −0.486438
\(456\) 0 0
\(457\) −13.2244 −0.618610 −0.309305 0.950963i \(-0.600096\pi\)
−0.309305 + 0.950963i \(0.600096\pi\)
\(458\) 0 0
\(459\) 4.79075 0.223613
\(460\) 0 0
\(461\) −33.5434 −1.56227 −0.781135 0.624362i \(-0.785359\pi\)
−0.781135 + 0.624362i \(0.785359\pi\)
\(462\) 0 0
\(463\) 22.8174 1.06042 0.530208 0.847867i \(-0.322114\pi\)
0.530208 + 0.847867i \(0.322114\pi\)
\(464\) 0 0
\(465\) −12.2262 −0.566978
\(466\) 0 0
\(467\) −12.1148 −0.560605 −0.280303 0.959912i \(-0.590435\pi\)
−0.280303 + 0.959912i \(0.590435\pi\)
\(468\) 0 0
\(469\) 4.08464 0.188611
\(470\) 0 0
\(471\) −17.2372 −0.794250
\(472\) 0 0
\(473\) −43.1722 −1.98506
\(474\) 0 0
\(475\) 11.4387 0.524844
\(476\) 0 0
\(477\) −0.687009 −0.0314560
\(478\) 0 0
\(479\) 2.35149 0.107442 0.0537211 0.998556i \(-0.482892\pi\)
0.0537211 + 0.998556i \(0.482892\pi\)
\(480\) 0 0
\(481\) −16.4392 −0.749563
\(482\) 0 0
\(483\) −0.484000 −0.0220228
\(484\) 0 0
\(485\) 4.17847 0.189735
\(486\) 0 0
\(487\) 15.3635 0.696187 0.348094 0.937460i \(-0.386829\pi\)
0.348094 + 0.937460i \(0.386829\pi\)
\(488\) 0 0
\(489\) 1.08125 0.0488959
\(490\) 0 0
\(491\) 20.0376 0.904283 0.452141 0.891946i \(-0.350660\pi\)
0.452141 + 0.891946i \(0.350660\pi\)
\(492\) 0 0
\(493\) −1.26574 −0.0570059
\(494\) 0 0
\(495\) 5.09268 0.228899
\(496\) 0 0
\(497\) −10.7423 −0.481857
\(498\) 0 0
\(499\) 1.87902 0.0841166 0.0420583 0.999115i \(-0.486608\pi\)
0.0420583 + 0.999115i \(0.486608\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) −18.7896 −0.837787 −0.418893 0.908035i \(-0.637582\pi\)
−0.418893 + 0.908035i \(0.637582\pi\)
\(504\) 0 0
\(505\) 16.1811 0.720049
\(506\) 0 0
\(507\) 6.69360 0.297273
\(508\) 0 0
\(509\) 40.2594 1.78447 0.892233 0.451576i \(-0.149138\pi\)
0.892233 + 0.451576i \(0.149138\pi\)
\(510\) 0 0
\(511\) 0.523682 0.0231663
\(512\) 0 0
\(513\) 3.24068 0.143080
\(514\) 0 0
\(515\) 19.2109 0.846532
\(516\) 0 0
\(517\) 30.0185 1.32021
\(518\) 0 0
\(519\) 15.9953 0.702115
\(520\) 0 0
\(521\) 10.1422 0.444337 0.222168 0.975008i \(-0.428687\pi\)
0.222168 + 0.975008i \(0.428687\pi\)
\(522\) 0 0
\(523\) 32.5975 1.42539 0.712694 0.701475i \(-0.247475\pi\)
0.712694 + 0.701475i \(0.247475\pi\)
\(524\) 0 0
\(525\) 12.0277 0.524934
\(526\) 0 0
\(527\) −48.3055 −2.10422
\(528\) 0 0
\(529\) −22.9798 −0.999123
\(530\) 0 0
\(531\) 14.3473 0.622619
\(532\) 0 0
\(533\) 24.0739 1.04275
\(534\) 0 0
\(535\) −0.0795787 −0.00344049
\(536\) 0 0
\(537\) −19.0256 −0.821013
\(538\) 0 0
\(539\) 19.3680 0.834237
\(540\) 0 0
\(541\) −15.3949 −0.661880 −0.330940 0.943652i \(-0.607366\pi\)
−0.330940 + 0.943652i \(0.607366\pi\)
\(542\) 0 0
\(543\) 20.3346 0.872643
\(544\) 0 0
\(545\) 21.5163 0.921655
\(546\) 0 0
\(547\) −41.5584 −1.77691 −0.888455 0.458964i \(-0.848221\pi\)
−0.888455 + 0.458964i \(0.848221\pi\)
\(548\) 0 0
\(549\) 10.7130 0.457220
\(550\) 0 0
\(551\) −0.856203 −0.0364755
\(552\) 0 0
\(553\) 12.4600 0.529855
\(554\) 0 0
\(555\) −7.93761 −0.336933
\(556\) 0 0
\(557\) 18.5224 0.784817 0.392409 0.919791i \(-0.371642\pi\)
0.392409 + 0.919791i \(0.371642\pi\)
\(558\) 0 0
\(559\) 25.8135 1.09180
\(560\) 0 0
\(561\) 20.1210 0.849510
\(562\) 0 0
\(563\) −31.9142 −1.34502 −0.672511 0.740087i \(-0.734784\pi\)
−0.672511 + 0.740087i \(0.734784\pi\)
\(564\) 0 0
\(565\) 9.07803 0.381916
\(566\) 0 0
\(567\) 3.40756 0.143104
\(568\) 0 0
\(569\) −27.6777 −1.16031 −0.580156 0.814506i \(-0.697008\pi\)
−0.580156 + 0.814506i \(0.697008\pi\)
\(570\) 0 0
\(571\) 32.6675 1.36709 0.683546 0.729907i \(-0.260437\pi\)
0.683546 + 0.729907i \(0.260437\pi\)
\(572\) 0 0
\(573\) −16.4252 −0.686172
\(574\) 0 0
\(575\) −0.501352 −0.0209078
\(576\) 0 0
\(577\) −32.7241 −1.36232 −0.681162 0.732133i \(-0.738525\pi\)
−0.681162 + 0.732133i \(0.738525\pi\)
\(578\) 0 0
\(579\) −6.63951 −0.275928
\(580\) 0 0
\(581\) 13.5104 0.560506
\(582\) 0 0
\(583\) −2.88542 −0.119502
\(584\) 0 0
\(585\) −3.04502 −0.125896
\(586\) 0 0
\(587\) 17.2455 0.711798 0.355899 0.934524i \(-0.384175\pi\)
0.355899 + 0.934524i \(0.384175\pi\)
\(588\) 0 0
\(589\) −32.6761 −1.34639
\(590\) 0 0
\(591\) −7.07078 −0.290853
\(592\) 0 0
\(593\) −44.0912 −1.81061 −0.905305 0.424762i \(-0.860358\pi\)
−0.905305 + 0.424762i \(0.860358\pi\)
\(594\) 0 0
\(595\) −19.7946 −0.811498
\(596\) 0 0
\(597\) −0.717075 −0.0293479
\(598\) 0 0
\(599\) −4.74123 −0.193721 −0.0968606 0.995298i \(-0.530880\pi\)
−0.0968606 + 0.995298i \(0.530880\pi\)
\(600\) 0 0
\(601\) 3.28382 0.133950 0.0669750 0.997755i \(-0.478665\pi\)
0.0669750 + 0.997755i \(0.478665\pi\)
\(602\) 0 0
\(603\) 1.19870 0.0488149
\(604\) 0 0
\(605\) 8.05107 0.327323
\(606\) 0 0
\(607\) 14.4426 0.586206 0.293103 0.956081i \(-0.405312\pi\)
0.293103 + 0.956081i \(0.405312\pi\)
\(608\) 0 0
\(609\) −0.900292 −0.0364817
\(610\) 0 0
\(611\) −17.9487 −0.726128
\(612\) 0 0
\(613\) 1.58209 0.0639000 0.0319500 0.999489i \(-0.489828\pi\)
0.0319500 + 0.999489i \(0.489828\pi\)
\(614\) 0 0
\(615\) 11.6240 0.468724
\(616\) 0 0
\(617\) −11.2773 −0.454008 −0.227004 0.973894i \(-0.572893\pi\)
−0.227004 + 0.973894i \(0.572893\pi\)
\(618\) 0 0
\(619\) 28.9029 1.16171 0.580854 0.814008i \(-0.302719\pi\)
0.580854 + 0.814008i \(0.302719\pi\)
\(620\) 0 0
\(621\) −0.142037 −0.00569976
\(622\) 0 0
\(623\) 27.7394 1.11136
\(624\) 0 0
\(625\) 5.10758 0.204303
\(626\) 0 0
\(627\) 13.6108 0.543563
\(628\) 0 0
\(629\) −31.3613 −1.25046
\(630\) 0 0
\(631\) 3.50507 0.139534 0.0697672 0.997563i \(-0.477774\pi\)
0.0697672 + 0.997563i \(0.477774\pi\)
\(632\) 0 0
\(633\) 12.1736 0.483857
\(634\) 0 0
\(635\) −24.8371 −0.985629
\(636\) 0 0
\(637\) −11.5805 −0.458837
\(638\) 0 0
\(639\) −3.15248 −0.124710
\(640\) 0 0
\(641\) −4.89788 −0.193454 −0.0967272 0.995311i \(-0.530837\pi\)
−0.0967272 + 0.995311i \(0.530837\pi\)
\(642\) 0 0
\(643\) −31.2912 −1.23400 −0.617001 0.786962i \(-0.711653\pi\)
−0.617001 + 0.786962i \(0.711653\pi\)
\(644\) 0 0
\(645\) 12.4640 0.490768
\(646\) 0 0
\(647\) −30.8557 −1.21306 −0.606532 0.795059i \(-0.707440\pi\)
−0.606532 + 0.795059i \(0.707440\pi\)
\(648\) 0 0
\(649\) 60.2583 2.36534
\(650\) 0 0
\(651\) −34.3587 −1.34662
\(652\) 0 0
\(653\) 29.1220 1.13963 0.569815 0.821773i \(-0.307015\pi\)
0.569815 + 0.821773i \(0.307015\pi\)
\(654\) 0 0
\(655\) −12.6981 −0.496156
\(656\) 0 0
\(657\) 0.153683 0.00599573
\(658\) 0 0
\(659\) −21.0879 −0.821469 −0.410734 0.911755i \(-0.634728\pi\)
−0.410734 + 0.911755i \(0.634728\pi\)
\(660\) 0 0
\(661\) −17.9637 −0.698709 −0.349354 0.936991i \(-0.613599\pi\)
−0.349354 + 0.936991i \(0.613599\pi\)
\(662\) 0 0
\(663\) −12.0308 −0.467237
\(664\) 0 0
\(665\) −13.3900 −0.519240
\(666\) 0 0
\(667\) 0.0375269 0.00145305
\(668\) 0 0
\(669\) 6.72478 0.259995
\(670\) 0 0
\(671\) 44.9943 1.73699
\(672\) 0 0
\(673\) −16.9700 −0.654146 −0.327073 0.944999i \(-0.606062\pi\)
−0.327073 + 0.944999i \(0.606062\pi\)
\(674\) 0 0
\(675\) 3.52972 0.135859
\(676\) 0 0
\(677\) −21.1696 −0.813614 −0.406807 0.913514i \(-0.633358\pi\)
−0.406807 + 0.913514i \(0.633358\pi\)
\(678\) 0 0
\(679\) 11.7425 0.450636
\(680\) 0 0
\(681\) −11.2701 −0.431871
\(682\) 0 0
\(683\) 23.8834 0.913874 0.456937 0.889499i \(-0.348946\pi\)
0.456937 + 0.889499i \(0.348946\pi\)
\(684\) 0 0
\(685\) −0.191034 −0.00729904
\(686\) 0 0
\(687\) 6.52089 0.248787
\(688\) 0 0
\(689\) 1.72525 0.0657269
\(690\) 0 0
\(691\) 7.70109 0.292963 0.146482 0.989213i \(-0.453205\pi\)
0.146482 + 0.989213i \(0.453205\pi\)
\(692\) 0 0
\(693\) 14.3117 0.543655
\(694\) 0 0
\(695\) 9.84081 0.373283
\(696\) 0 0
\(697\) 45.9260 1.73957
\(698\) 0 0
\(699\) 22.0249 0.833060
\(700\) 0 0
\(701\) −27.4380 −1.03632 −0.518159 0.855284i \(-0.673382\pi\)
−0.518159 + 0.855284i \(0.673382\pi\)
\(702\) 0 0
\(703\) −21.2142 −0.800109
\(704\) 0 0
\(705\) −8.66647 −0.326398
\(706\) 0 0
\(707\) 45.4728 1.71018
\(708\) 0 0
\(709\) 13.6483 0.512572 0.256286 0.966601i \(-0.417501\pi\)
0.256286 + 0.966601i \(0.417501\pi\)
\(710\) 0 0
\(711\) 3.65659 0.137133
\(712\) 0 0
\(713\) 1.43217 0.0536353
\(714\) 0 0
\(715\) −12.7890 −0.478282
\(716\) 0 0
\(717\) 4.88322 0.182367
\(718\) 0 0
\(719\) −12.2274 −0.456007 −0.228003 0.973660i \(-0.573220\pi\)
−0.228003 + 0.973660i \(0.573220\pi\)
\(720\) 0 0
\(721\) 53.9872 2.01059
\(722\) 0 0
\(723\) 14.3294 0.532917
\(724\) 0 0
\(725\) −0.932569 −0.0346347
\(726\) 0 0
\(727\) 16.9710 0.629420 0.314710 0.949188i \(-0.398093\pi\)
0.314710 + 0.949188i \(0.398093\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 49.2448 1.82138
\(732\) 0 0
\(733\) −24.4232 −0.902092 −0.451046 0.892501i \(-0.648949\pi\)
−0.451046 + 0.892501i \(0.648949\pi\)
\(734\) 0 0
\(735\) −5.59161 −0.206249
\(736\) 0 0
\(737\) 5.03452 0.185449
\(738\) 0 0
\(739\) −5.37262 −0.197635 −0.0988175 0.995106i \(-0.531506\pi\)
−0.0988175 + 0.995106i \(0.531506\pi\)
\(740\) 0 0
\(741\) −8.13818 −0.298963
\(742\) 0 0
\(743\) 16.3799 0.600921 0.300461 0.953794i \(-0.402860\pi\)
0.300461 + 0.953794i \(0.402860\pi\)
\(744\) 0 0
\(745\) 20.3470 0.745458
\(746\) 0 0
\(747\) 3.96483 0.145066
\(748\) 0 0
\(749\) −0.223636 −0.00817146
\(750\) 0 0
\(751\) −5.00958 −0.182802 −0.0914011 0.995814i \(-0.529135\pi\)
−0.0914011 + 0.995814i \(0.529135\pi\)
\(752\) 0 0
\(753\) 2.59502 0.0945678
\(754\) 0 0
\(755\) 15.9859 0.581787
\(756\) 0 0
\(757\) −19.0233 −0.691415 −0.345708 0.938342i \(-0.612361\pi\)
−0.345708 + 0.938342i \(0.612361\pi\)
\(758\) 0 0
\(759\) −0.596553 −0.0216535
\(760\) 0 0
\(761\) 22.5846 0.818690 0.409345 0.912380i \(-0.365757\pi\)
0.409345 + 0.912380i \(0.365757\pi\)
\(762\) 0 0
\(763\) 60.4659 2.18901
\(764\) 0 0
\(765\) −5.80902 −0.210025
\(766\) 0 0
\(767\) −36.0297 −1.30096
\(768\) 0 0
\(769\) −4.96145 −0.178915 −0.0894573 0.995991i \(-0.528513\pi\)
−0.0894573 + 0.995991i \(0.528513\pi\)
\(770\) 0 0
\(771\) 29.6878 1.06918
\(772\) 0 0
\(773\) 50.0966 1.80185 0.900925 0.433974i \(-0.142889\pi\)
0.900925 + 0.433974i \(0.142889\pi\)
\(774\) 0 0
\(775\) −35.5905 −1.27845
\(776\) 0 0
\(777\) −22.3066 −0.800245
\(778\) 0 0
\(779\) 31.0664 1.11307
\(780\) 0 0
\(781\) −13.2404 −0.473777
\(782\) 0 0
\(783\) −0.264204 −0.00944189
\(784\) 0 0
\(785\) 20.9010 0.745988
\(786\) 0 0
\(787\) −6.00849 −0.214180 −0.107090 0.994249i \(-0.534153\pi\)
−0.107090 + 0.994249i \(0.534153\pi\)
\(788\) 0 0
\(789\) −4.24468 −0.151115
\(790\) 0 0
\(791\) 25.5115 0.907083
\(792\) 0 0
\(793\) −26.9031 −0.955356
\(794\) 0 0
\(795\) 0.833032 0.0295446
\(796\) 0 0
\(797\) −18.3258 −0.649134 −0.324567 0.945863i \(-0.605219\pi\)
−0.324567 + 0.945863i \(0.605219\pi\)
\(798\) 0 0
\(799\) −34.2410 −1.21136
\(800\) 0 0
\(801\) 8.14055 0.287632
\(802\) 0 0
\(803\) 0.645463 0.0227779
\(804\) 0 0
\(805\) 0.586874 0.0206846
\(806\) 0 0
\(807\) 29.8482 1.05071
\(808\) 0 0
\(809\) 10.5235 0.369988 0.184994 0.982740i \(-0.440773\pi\)
0.184994 + 0.982740i \(0.440773\pi\)
\(810\) 0 0
\(811\) −2.86669 −0.100663 −0.0503315 0.998733i \(-0.516028\pi\)
−0.0503315 + 0.998733i \(0.516028\pi\)
\(812\) 0 0
\(813\) −31.6214 −1.10901
\(814\) 0 0
\(815\) −1.31107 −0.0459248
\(816\) 0 0
\(817\) 33.3115 1.16542
\(818\) 0 0
\(819\) −8.55724 −0.299014
\(820\) 0 0
\(821\) −24.3763 −0.850738 −0.425369 0.905020i \(-0.639856\pi\)
−0.425369 + 0.905020i \(0.639856\pi\)
\(822\) 0 0
\(823\) −43.9211 −1.53099 −0.765497 0.643440i \(-0.777507\pi\)
−0.765497 + 0.643440i \(0.777507\pi\)
\(824\) 0 0
\(825\) 14.8248 0.516132
\(826\) 0 0
\(827\) −45.4139 −1.57920 −0.789598 0.613625i \(-0.789711\pi\)
−0.789598 + 0.613625i \(0.789711\pi\)
\(828\) 0 0
\(829\) 28.6971 0.996691 0.498346 0.866978i \(-0.333941\pi\)
0.498346 + 0.866978i \(0.333941\pi\)
\(830\) 0 0
\(831\) −20.4671 −0.709995
\(832\) 0 0
\(833\) −22.0923 −0.765452
\(834\) 0 0
\(835\) −1.21255 −0.0419620
\(836\) 0 0
\(837\) −10.0831 −0.348522
\(838\) 0 0
\(839\) −55.5225 −1.91685 −0.958425 0.285344i \(-0.907892\pi\)
−0.958425 + 0.285344i \(0.907892\pi\)
\(840\) 0 0
\(841\) −28.9302 −0.997593
\(842\) 0 0
\(843\) 22.8045 0.785429
\(844\) 0 0
\(845\) −8.11632 −0.279210
\(846\) 0 0
\(847\) 22.6255 0.777420
\(848\) 0 0
\(849\) −6.36109 −0.218312
\(850\) 0 0
\(851\) 0.929806 0.0318733
\(852\) 0 0
\(853\) −25.5294 −0.874109 −0.437054 0.899435i \(-0.643978\pi\)
−0.437054 + 0.899435i \(0.643978\pi\)
\(854\) 0 0
\(855\) −3.92949 −0.134386
\(856\) 0 0
\(857\) −28.2575 −0.965258 −0.482629 0.875825i \(-0.660318\pi\)
−0.482629 + 0.875825i \(0.660318\pi\)
\(858\) 0 0
\(859\) 38.5020 1.31367 0.656835 0.754034i \(-0.271895\pi\)
0.656835 + 0.754034i \(0.271895\pi\)
\(860\) 0 0
\(861\) 32.6662 1.11326
\(862\) 0 0
\(863\) 10.3942 0.353824 0.176912 0.984227i \(-0.443389\pi\)
0.176912 + 0.984227i \(0.443389\pi\)
\(864\) 0 0
\(865\) −19.3951 −0.659452
\(866\) 0 0
\(867\) −5.95127 −0.202116
\(868\) 0 0
\(869\) 15.3576 0.520970
\(870\) 0 0
\(871\) −3.01024 −0.101998
\(872\) 0 0
\(873\) 3.44602 0.116630
\(874\) 0 0
\(875\) −35.2434 −1.19144
\(876\) 0 0
\(877\) 36.3335 1.22689 0.613447 0.789736i \(-0.289782\pi\)
0.613447 + 0.789736i \(0.289782\pi\)
\(878\) 0 0
\(879\) 3.49011 0.117719
\(880\) 0 0
\(881\) 21.9157 0.738360 0.369180 0.929358i \(-0.379639\pi\)
0.369180 + 0.929358i \(0.379639\pi\)
\(882\) 0 0
\(883\) −22.7689 −0.766233 −0.383117 0.923700i \(-0.625149\pi\)
−0.383117 + 0.923700i \(0.625149\pi\)
\(884\) 0 0
\(885\) −17.3968 −0.584787
\(886\) 0 0
\(887\) −3.73121 −0.125282 −0.0626409 0.998036i \(-0.519952\pi\)
−0.0626409 + 0.998036i \(0.519952\pi\)
\(888\) 0 0
\(889\) −69.7982 −2.34096
\(890\) 0 0
\(891\) 4.19997 0.140704
\(892\) 0 0
\(893\) −23.1622 −0.775093
\(894\) 0 0
\(895\) 23.0694 0.771126
\(896\) 0 0
\(897\) 0.356691 0.0119096
\(898\) 0 0
\(899\) 2.66400 0.0888492
\(900\) 0 0
\(901\) 3.29129 0.109649
\(902\) 0 0
\(903\) 35.0268 1.16562
\(904\) 0 0
\(905\) −24.6567 −0.819618
\(906\) 0 0
\(907\) −57.8699 −1.92154 −0.960770 0.277347i \(-0.910545\pi\)
−0.960770 + 0.277347i \(0.910545\pi\)
\(908\) 0 0
\(909\) 13.3447 0.442616
\(910\) 0 0
\(911\) −47.6341 −1.57819 −0.789095 0.614272i \(-0.789450\pi\)
−0.789095 + 0.614272i \(0.789450\pi\)
\(912\) 0 0
\(913\) 16.6522 0.551107
\(914\) 0 0
\(915\) −12.9900 −0.429438
\(916\) 0 0
\(917\) −35.6847 −1.17841
\(918\) 0 0
\(919\) −51.2555 −1.69076 −0.845381 0.534164i \(-0.820626\pi\)
−0.845381 + 0.534164i \(0.820626\pi\)
\(920\) 0 0
\(921\) −3.64448 −0.120090
\(922\) 0 0
\(923\) 7.91669 0.260581
\(924\) 0 0
\(925\) −23.1063 −0.759731
\(926\) 0 0
\(927\) 15.8434 0.520365
\(928\) 0 0
\(929\) 14.4121 0.472847 0.236424 0.971650i \(-0.424025\pi\)
0.236424 + 0.971650i \(0.424025\pi\)
\(930\) 0 0
\(931\) −14.9442 −0.489778
\(932\) 0 0
\(933\) 10.9171 0.357408
\(934\) 0 0
\(935\) −24.3977 −0.797891
\(936\) 0 0
\(937\) 3.65491 0.119401 0.0597004 0.998216i \(-0.480985\pi\)
0.0597004 + 0.998216i \(0.480985\pi\)
\(938\) 0 0
\(939\) 2.69366 0.0879041
\(940\) 0 0
\(941\) −31.3761 −1.02283 −0.511416 0.859333i \(-0.670879\pi\)
−0.511416 + 0.859333i \(0.670879\pi\)
\(942\) 0 0
\(943\) −1.36162 −0.0443406
\(944\) 0 0
\(945\) −4.13183 −0.134408
\(946\) 0 0
\(947\) 2.52840 0.0821620 0.0410810 0.999156i \(-0.486920\pi\)
0.0410810 + 0.999156i \(0.486920\pi\)
\(948\) 0 0
\(949\) −0.385936 −0.0125280
\(950\) 0 0
\(951\) 11.6786 0.378705
\(952\) 0 0
\(953\) 25.2734 0.818687 0.409344 0.912380i \(-0.365758\pi\)
0.409344 + 0.912380i \(0.365758\pi\)
\(954\) 0 0
\(955\) 19.9164 0.644478
\(956\) 0 0
\(957\) −1.10965 −0.0358699
\(958\) 0 0
\(959\) −0.536852 −0.0173359
\(960\) 0 0
\(961\) 70.6686 2.27963
\(962\) 0 0
\(963\) −0.0656293 −0.00211487
\(964\) 0 0
\(965\) 8.05073 0.259162
\(966\) 0 0
\(967\) −0.518625 −0.0166779 −0.00833893 0.999965i \(-0.502654\pi\)
−0.00833893 + 0.999965i \(0.502654\pi\)
\(968\) 0 0
\(969\) −15.5253 −0.498744
\(970\) 0 0
\(971\) 47.8549 1.53574 0.767869 0.640607i \(-0.221317\pi\)
0.767869 + 0.640607i \(0.221317\pi\)
\(972\) 0 0
\(973\) 27.6551 0.886580
\(974\) 0 0
\(975\) −8.86403 −0.283876
\(976\) 0 0
\(977\) −45.8694 −1.46749 −0.733746 0.679423i \(-0.762230\pi\)
−0.733746 + 0.679423i \(0.762230\pi\)
\(978\) 0 0
\(979\) 34.1901 1.09272
\(980\) 0 0
\(981\) 17.7446 0.566543
\(982\) 0 0
\(983\) 27.4603 0.875848 0.437924 0.899012i \(-0.355714\pi\)
0.437924 + 0.899012i \(0.355714\pi\)
\(984\) 0 0
\(985\) 8.57367 0.273180
\(986\) 0 0
\(987\) −24.3549 −0.775225
\(988\) 0 0
\(989\) −1.46002 −0.0464260
\(990\) 0 0
\(991\) 8.77212 0.278656 0.139328 0.990246i \(-0.455506\pi\)
0.139328 + 0.990246i \(0.455506\pi\)
\(992\) 0 0
\(993\) −4.93784 −0.156697
\(994\) 0 0
\(995\) 0.869489 0.0275647
\(996\) 0 0
\(997\) −7.87335 −0.249351 −0.124676 0.992198i \(-0.539789\pi\)
−0.124676 + 0.992198i \(0.539789\pi\)
\(998\) 0 0
\(999\) −6.54621 −0.207113
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 8016.2.a.p.1.5 5
4.3 odd 2 501.2.a.b.1.3 5
12.11 even 2 1503.2.a.d.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
501.2.a.b.1.3 5 4.3 odd 2
1503.2.a.d.1.3 5 12.11 even 2
8016.2.a.p.1.5 5 1.1 even 1 trivial