Properties

Label 4005.2.a.r.1.3
Level $4005$
Weight $2$
Character 4005.1
Self dual yes
Analytic conductor $31.980$
Analytic rank $0$
Dimension $10$
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] = [4005,2,Mod(1,4005)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4005, 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("4005.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4005.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: \(31.9800860095\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} - 8x^{8} + 35x^{7} + 29x^{6} - 103x^{5} - 57x^{4} + 106x^{3} + 29x^{2} - 39x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1335)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.32759\) of defining polynomial
Character \(\chi\) \(=\) 4005.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.32759 q^{2} +3.41766 q^{4} -1.00000 q^{5} -0.231668 q^{7} -3.29972 q^{8} +O(q^{10})\) \(q-2.32759 q^{2} +3.41766 q^{4} -1.00000 q^{5} -0.231668 q^{7} -3.29972 q^{8} +2.32759 q^{10} +3.13805 q^{11} +5.17239 q^{13} +0.539228 q^{14} +0.845067 q^{16} +3.12793 q^{17} +8.20035 q^{19} -3.41766 q^{20} -7.30407 q^{22} +2.60339 q^{23} +1.00000 q^{25} -12.0392 q^{26} -0.791763 q^{28} +1.75022 q^{29} +10.4240 q^{31} +4.63247 q^{32} -7.28054 q^{34} +0.231668 q^{35} +0.0476589 q^{37} -19.0870 q^{38} +3.29972 q^{40} +11.0892 q^{41} -5.76699 q^{43} +10.7248 q^{44} -6.05961 q^{46} +1.91824 q^{47} -6.94633 q^{49} -2.32759 q^{50} +17.6775 q^{52} +2.56217 q^{53} -3.13805 q^{55} +0.764441 q^{56} -4.07379 q^{58} +0.116362 q^{59} -0.140106 q^{61} -24.2627 q^{62} -12.4726 q^{64} -5.17239 q^{65} -0.235930 q^{67} +10.6902 q^{68} -0.539228 q^{70} +7.10782 q^{71} -8.81061 q^{73} -0.110930 q^{74} +28.0260 q^{76} -0.726986 q^{77} -10.3149 q^{79} -0.845067 q^{80} -25.8110 q^{82} -8.88288 q^{83} -3.12793 q^{85} +13.4232 q^{86} -10.3547 q^{88} -1.00000 q^{89} -1.19828 q^{91} +8.89749 q^{92} -4.46488 q^{94} -8.20035 q^{95} +17.8901 q^{97} +16.1682 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{2} + 14 q^{4} - 10 q^{5} + 7 q^{7} - 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 6 q^{2} + 14 q^{4} - 10 q^{5} + 7 q^{7} - 15 q^{8} + 6 q^{10} + 10 q^{11} + 7 q^{13} + 7 q^{14} + 22 q^{16} - 11 q^{17} + 10 q^{19} - 14 q^{20} + 8 q^{22} - 6 q^{23} + 10 q^{25} + 14 q^{26} + 16 q^{28} + 3 q^{29} + 12 q^{31} - 21 q^{32} - 6 q^{34} - 7 q^{35} - 19 q^{37} - 6 q^{38} + 15 q^{40} - 13 q^{41} + 9 q^{43} + 26 q^{44} + 8 q^{46} - 21 q^{47} + 53 q^{49} - 6 q^{50} - 43 q^{52} - 7 q^{53} - 10 q^{55} + 53 q^{56} - 42 q^{58} + 19 q^{59} + 4 q^{61} + 28 q^{62} + 5 q^{64} - 7 q^{65} - 6 q^{67} - 2 q^{68} - 7 q^{70} + 6 q^{71} + 6 q^{73} - 2 q^{76} + 40 q^{77} + 25 q^{79} - 22 q^{80} + q^{82} + 22 q^{83} + 11 q^{85} + 2 q^{86} + 20 q^{88} - 10 q^{89} - 10 q^{91} - 10 q^{92} + 25 q^{94} - 10 q^{95} + 40 q^{97} + 56 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.32759 −1.64585 −0.822926 0.568149i \(-0.807660\pi\)
−0.822926 + 0.568149i \(0.807660\pi\)
\(3\) 0 0
\(4\) 3.41766 1.70883
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.231668 −0.0875624 −0.0437812 0.999041i \(-0.513940\pi\)
−0.0437812 + 0.999041i \(0.513940\pi\)
\(8\) −3.29972 −1.16663
\(9\) 0 0
\(10\) 2.32759 0.736047
\(11\) 3.13805 0.946157 0.473078 0.881020i \(-0.343143\pi\)
0.473078 + 0.881020i \(0.343143\pi\)
\(12\) 0 0
\(13\) 5.17239 1.43456 0.717282 0.696783i \(-0.245386\pi\)
0.717282 + 0.696783i \(0.245386\pi\)
\(14\) 0.539228 0.144115
\(15\) 0 0
\(16\) 0.845067 0.211267
\(17\) 3.12793 0.758635 0.379318 0.925266i \(-0.376159\pi\)
0.379318 + 0.925266i \(0.376159\pi\)
\(18\) 0 0
\(19\) 8.20035 1.88129 0.940645 0.339393i \(-0.110222\pi\)
0.940645 + 0.339393i \(0.110222\pi\)
\(20\) −3.41766 −0.764211
\(21\) 0 0
\(22\) −7.30407 −1.55723
\(23\) 2.60339 0.542844 0.271422 0.962460i \(-0.412506\pi\)
0.271422 + 0.962460i \(0.412506\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −12.0392 −2.36108
\(27\) 0 0
\(28\) −0.791763 −0.149629
\(29\) 1.75022 0.325008 0.162504 0.986708i \(-0.448043\pi\)
0.162504 + 0.986708i \(0.448043\pi\)
\(30\) 0 0
\(31\) 10.4240 1.87220 0.936100 0.351734i \(-0.114408\pi\)
0.936100 + 0.351734i \(0.114408\pi\)
\(32\) 4.63247 0.818913
\(33\) 0 0
\(34\) −7.28054 −1.24860
\(35\) 0.231668 0.0391591
\(36\) 0 0
\(37\) 0.0476589 0.00783508 0.00391754 0.999992i \(-0.498753\pi\)
0.00391754 + 0.999992i \(0.498753\pi\)
\(38\) −19.0870 −3.09632
\(39\) 0 0
\(40\) 3.29972 0.521731
\(41\) 11.0892 1.73184 0.865919 0.500184i \(-0.166734\pi\)
0.865919 + 0.500184i \(0.166734\pi\)
\(42\) 0 0
\(43\) −5.76699 −0.879458 −0.439729 0.898131i \(-0.644925\pi\)
−0.439729 + 0.898131i \(0.644925\pi\)
\(44\) 10.7248 1.61682
\(45\) 0 0
\(46\) −6.05961 −0.893441
\(47\) 1.91824 0.279805 0.139902 0.990165i \(-0.455321\pi\)
0.139902 + 0.990165i \(0.455321\pi\)
\(48\) 0 0
\(49\) −6.94633 −0.992333
\(50\) −2.32759 −0.329170
\(51\) 0 0
\(52\) 17.6775 2.45142
\(53\) 2.56217 0.351941 0.175971 0.984395i \(-0.443694\pi\)
0.175971 + 0.984395i \(0.443694\pi\)
\(54\) 0 0
\(55\) −3.13805 −0.423134
\(56\) 0.764441 0.102153
\(57\) 0 0
\(58\) −4.07379 −0.534915
\(59\) 0.116362 0.0151490 0.00757452 0.999971i \(-0.497589\pi\)
0.00757452 + 0.999971i \(0.497589\pi\)
\(60\) 0 0
\(61\) −0.140106 −0.0179388 −0.00896938 0.999960i \(-0.502855\pi\)
−0.00896938 + 0.999960i \(0.502855\pi\)
\(62\) −24.2627 −3.08136
\(63\) 0 0
\(64\) −12.4726 −1.55908
\(65\) −5.17239 −0.641557
\(66\) 0 0
\(67\) −0.235930 −0.0288235 −0.0144117 0.999896i \(-0.504588\pi\)
−0.0144117 + 0.999896i \(0.504588\pi\)
\(68\) 10.6902 1.29638
\(69\) 0 0
\(70\) −0.539228 −0.0644501
\(71\) 7.10782 0.843543 0.421771 0.906702i \(-0.361409\pi\)
0.421771 + 0.906702i \(0.361409\pi\)
\(72\) 0 0
\(73\) −8.81061 −1.03120 −0.515602 0.856828i \(-0.672432\pi\)
−0.515602 + 0.856828i \(0.672432\pi\)
\(74\) −0.110930 −0.0128954
\(75\) 0 0
\(76\) 28.0260 3.21480
\(77\) −0.726986 −0.0828478
\(78\) 0 0
\(79\) −10.3149 −1.16051 −0.580256 0.814434i \(-0.697047\pi\)
−0.580256 + 0.814434i \(0.697047\pi\)
\(80\) −0.845067 −0.0944814
\(81\) 0 0
\(82\) −25.8110 −2.85035
\(83\) −8.88288 −0.975023 −0.487511 0.873117i \(-0.662095\pi\)
−0.487511 + 0.873117i \(0.662095\pi\)
\(84\) 0 0
\(85\) −3.12793 −0.339272
\(86\) 13.4232 1.44746
\(87\) 0 0
\(88\) −10.3547 −1.10381
\(89\) −1.00000 −0.106000
\(90\) 0 0
\(91\) −1.19828 −0.125614
\(92\) 8.89749 0.927627
\(93\) 0 0
\(94\) −4.46488 −0.460517
\(95\) −8.20035 −0.841338
\(96\) 0 0
\(97\) 17.8901 1.81647 0.908233 0.418464i \(-0.137431\pi\)
0.908233 + 0.418464i \(0.137431\pi\)
\(98\) 16.1682 1.63323
\(99\) 0 0
\(100\) 3.41766 0.341766
\(101\) −4.87170 −0.484752 −0.242376 0.970182i \(-0.577927\pi\)
−0.242376 + 0.970182i \(0.577927\pi\)
\(102\) 0 0
\(103\) −16.5885 −1.63451 −0.817255 0.576276i \(-0.804505\pi\)
−0.817255 + 0.576276i \(0.804505\pi\)
\(104\) −17.0675 −1.67360
\(105\) 0 0
\(106\) −5.96367 −0.579243
\(107\) 14.3574 1.38798 0.693990 0.719985i \(-0.255851\pi\)
0.693990 + 0.719985i \(0.255851\pi\)
\(108\) 0 0
\(109\) 14.7030 1.40829 0.704144 0.710057i \(-0.251331\pi\)
0.704144 + 0.710057i \(0.251331\pi\)
\(110\) 7.30407 0.696416
\(111\) 0 0
\(112\) −0.195775 −0.0184990
\(113\) 1.84866 0.173907 0.0869535 0.996212i \(-0.472287\pi\)
0.0869535 + 0.996212i \(0.472287\pi\)
\(114\) 0 0
\(115\) −2.60339 −0.242767
\(116\) 5.98166 0.555383
\(117\) 0 0
\(118\) −0.270843 −0.0249331
\(119\) −0.724643 −0.0664279
\(120\) 0 0
\(121\) −1.15266 −0.104788
\(122\) 0.326109 0.0295245
\(123\) 0 0
\(124\) 35.6255 3.19927
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −3.26036 −0.289311 −0.144655 0.989482i \(-0.546207\pi\)
−0.144655 + 0.989482i \(0.546207\pi\)
\(128\) 19.7661 1.74710
\(129\) 0 0
\(130\) 12.0392 1.05591
\(131\) −6.51671 −0.569368 −0.284684 0.958621i \(-0.591889\pi\)
−0.284684 + 0.958621i \(0.591889\pi\)
\(132\) 0 0
\(133\) −1.89976 −0.164730
\(134\) 0.549148 0.0474392
\(135\) 0 0
\(136\) −10.3213 −0.885045
\(137\) −14.5049 −1.23924 −0.619619 0.784902i \(-0.712713\pi\)
−0.619619 + 0.784902i \(0.712713\pi\)
\(138\) 0 0
\(139\) −8.18687 −0.694401 −0.347201 0.937791i \(-0.612868\pi\)
−0.347201 + 0.937791i \(0.612868\pi\)
\(140\) 0.791763 0.0669162
\(141\) 0 0
\(142\) −16.5441 −1.38835
\(143\) 16.2312 1.35732
\(144\) 0 0
\(145\) −1.75022 −0.145348
\(146\) 20.5074 1.69721
\(147\) 0 0
\(148\) 0.162882 0.0133888
\(149\) 2.00547 0.164295 0.0821474 0.996620i \(-0.473822\pi\)
0.0821474 + 0.996620i \(0.473822\pi\)
\(150\) 0 0
\(151\) 20.5216 1.67002 0.835012 0.550232i \(-0.185461\pi\)
0.835012 + 0.550232i \(0.185461\pi\)
\(152\) −27.0589 −2.19476
\(153\) 0 0
\(154\) 1.69212 0.136355
\(155\) −10.4240 −0.837273
\(156\) 0 0
\(157\) 4.79244 0.382478 0.191239 0.981543i \(-0.438749\pi\)
0.191239 + 0.981543i \(0.438749\pi\)
\(158\) 24.0087 1.91003
\(159\) 0 0
\(160\) −4.63247 −0.366229
\(161\) −0.603123 −0.0475327
\(162\) 0 0
\(163\) −10.4650 −0.819681 −0.409841 0.912157i \(-0.634416\pi\)
−0.409841 + 0.912157i \(0.634416\pi\)
\(164\) 37.8990 2.95941
\(165\) 0 0
\(166\) 20.6757 1.60474
\(167\) −17.8728 −1.38304 −0.691519 0.722358i \(-0.743058\pi\)
−0.691519 + 0.722358i \(0.743058\pi\)
\(168\) 0 0
\(169\) 13.7537 1.05797
\(170\) 7.28054 0.558392
\(171\) 0 0
\(172\) −19.7096 −1.50284
\(173\) −17.7804 −1.35182 −0.675909 0.736985i \(-0.736249\pi\)
−0.675909 + 0.736985i \(0.736249\pi\)
\(174\) 0 0
\(175\) −0.231668 −0.0175125
\(176\) 2.65186 0.199891
\(177\) 0 0
\(178\) 2.32759 0.174460
\(179\) 17.8343 1.33300 0.666500 0.745505i \(-0.267791\pi\)
0.666500 + 0.745505i \(0.267791\pi\)
\(180\) 0 0
\(181\) 12.3953 0.921332 0.460666 0.887574i \(-0.347611\pi\)
0.460666 + 0.887574i \(0.347611\pi\)
\(182\) 2.78910 0.206742
\(183\) 0 0
\(184\) −8.59045 −0.633296
\(185\) −0.0476589 −0.00350395
\(186\) 0 0
\(187\) 9.81560 0.717788
\(188\) 6.55590 0.478138
\(189\) 0 0
\(190\) 19.0870 1.38472
\(191\) 2.95826 0.214052 0.107026 0.994256i \(-0.465867\pi\)
0.107026 + 0.994256i \(0.465867\pi\)
\(192\) 0 0
\(193\) −16.8981 −1.21635 −0.608177 0.793801i \(-0.708099\pi\)
−0.608177 + 0.793801i \(0.708099\pi\)
\(194\) −41.6408 −2.98964
\(195\) 0 0
\(196\) −23.7402 −1.69573
\(197\) −21.3318 −1.51983 −0.759915 0.650022i \(-0.774760\pi\)
−0.759915 + 0.650022i \(0.774760\pi\)
\(198\) 0 0
\(199\) −19.6380 −1.39210 −0.696051 0.717993i \(-0.745061\pi\)
−0.696051 + 0.717993i \(0.745061\pi\)
\(200\) −3.29972 −0.233325
\(201\) 0 0
\(202\) 11.3393 0.797831
\(203\) −0.405471 −0.0284585
\(204\) 0 0
\(205\) −11.0892 −0.774501
\(206\) 38.6111 2.69016
\(207\) 0 0
\(208\) 4.37102 0.303076
\(209\) 25.7331 1.77999
\(210\) 0 0
\(211\) −2.32091 −0.159778 −0.0798889 0.996804i \(-0.525457\pi\)
−0.0798889 + 0.996804i \(0.525457\pi\)
\(212\) 8.75662 0.601407
\(213\) 0 0
\(214\) −33.4180 −2.28441
\(215\) 5.76699 0.393306
\(216\) 0 0
\(217\) −2.41490 −0.163934
\(218\) −34.2224 −2.31783
\(219\) 0 0
\(220\) −10.7248 −0.723064
\(221\) 16.1789 1.08831
\(222\) 0 0
\(223\) −6.41824 −0.429797 −0.214899 0.976636i \(-0.568942\pi\)
−0.214899 + 0.976636i \(0.568942\pi\)
\(224\) −1.07320 −0.0717060
\(225\) 0 0
\(226\) −4.30291 −0.286225
\(227\) −9.48723 −0.629690 −0.314845 0.949143i \(-0.601953\pi\)
−0.314845 + 0.949143i \(0.601953\pi\)
\(228\) 0 0
\(229\) 29.9079 1.97637 0.988185 0.153263i \(-0.0489781\pi\)
0.988185 + 0.153263i \(0.0489781\pi\)
\(230\) 6.05961 0.399559
\(231\) 0 0
\(232\) −5.77524 −0.379163
\(233\) −18.6116 −1.21929 −0.609643 0.792676i \(-0.708687\pi\)
−0.609643 + 0.792676i \(0.708687\pi\)
\(234\) 0 0
\(235\) −1.91824 −0.125132
\(236\) 0.397686 0.0258871
\(237\) 0 0
\(238\) 1.68667 0.109331
\(239\) 7.65622 0.495240 0.247620 0.968857i \(-0.420352\pi\)
0.247620 + 0.968857i \(0.420352\pi\)
\(240\) 0 0
\(241\) 9.09143 0.585631 0.292815 0.956169i \(-0.405408\pi\)
0.292815 + 0.956169i \(0.405408\pi\)
\(242\) 2.68292 0.172465
\(243\) 0 0
\(244\) −0.478835 −0.0306543
\(245\) 6.94633 0.443785
\(246\) 0 0
\(247\) 42.4154 2.69883
\(248\) −34.3962 −2.18416
\(249\) 0 0
\(250\) 2.32759 0.147209
\(251\) −23.8649 −1.50634 −0.753171 0.657825i \(-0.771477\pi\)
−0.753171 + 0.657825i \(0.771477\pi\)
\(252\) 0 0
\(253\) 8.16955 0.513615
\(254\) 7.58878 0.476162
\(255\) 0 0
\(256\) −21.0622 −1.31639
\(257\) 17.8544 1.11373 0.556863 0.830605i \(-0.312005\pi\)
0.556863 + 0.830605i \(0.312005\pi\)
\(258\) 0 0
\(259\) −0.0110411 −0.000686058 0
\(260\) −17.6775 −1.09631
\(261\) 0 0
\(262\) 15.1682 0.937095
\(263\) 6.89818 0.425360 0.212680 0.977122i \(-0.431781\pi\)
0.212680 + 0.977122i \(0.431781\pi\)
\(264\) 0 0
\(265\) −2.56217 −0.157393
\(266\) 4.42186 0.271122
\(267\) 0 0
\(268\) −0.806329 −0.0492544
\(269\) −11.5710 −0.705495 −0.352747 0.935719i \(-0.614753\pi\)
−0.352747 + 0.935719i \(0.614753\pi\)
\(270\) 0 0
\(271\) −27.6367 −1.67881 −0.839406 0.543504i \(-0.817097\pi\)
−0.839406 + 0.543504i \(0.817097\pi\)
\(272\) 2.64331 0.160274
\(273\) 0 0
\(274\) 33.7614 2.03960
\(275\) 3.13805 0.189231
\(276\) 0 0
\(277\) −7.98870 −0.479995 −0.239997 0.970774i \(-0.577147\pi\)
−0.239997 + 0.970774i \(0.577147\pi\)
\(278\) 19.0556 1.14288
\(279\) 0 0
\(280\) −0.764441 −0.0456841
\(281\) −16.6642 −0.994104 −0.497052 0.867721i \(-0.665584\pi\)
−0.497052 + 0.867721i \(0.665584\pi\)
\(282\) 0 0
\(283\) −10.0186 −0.595541 −0.297771 0.954637i \(-0.596243\pi\)
−0.297771 + 0.954637i \(0.596243\pi\)
\(284\) 24.2921 1.44147
\(285\) 0 0
\(286\) −37.7795 −2.23395
\(287\) −2.56901 −0.151644
\(288\) 0 0
\(289\) −7.21603 −0.424472
\(290\) 4.07379 0.239221
\(291\) 0 0
\(292\) −30.1116 −1.76215
\(293\) 0.952327 0.0556356 0.0278178 0.999613i \(-0.491144\pi\)
0.0278178 + 0.999613i \(0.491144\pi\)
\(294\) 0 0
\(295\) −0.116362 −0.00677486
\(296\) −0.157261 −0.00914061
\(297\) 0 0
\(298\) −4.66791 −0.270405
\(299\) 13.4657 0.778744
\(300\) 0 0
\(301\) 1.33603 0.0770075
\(302\) −47.7658 −2.74861
\(303\) 0 0
\(304\) 6.92984 0.397454
\(305\) 0.140106 0.00802245
\(306\) 0 0
\(307\) 2.68360 0.153161 0.0765806 0.997063i \(-0.475600\pi\)
0.0765806 + 0.997063i \(0.475600\pi\)
\(308\) −2.48459 −0.141573
\(309\) 0 0
\(310\) 24.2627 1.37803
\(311\) 24.5758 1.39357 0.696783 0.717282i \(-0.254614\pi\)
0.696783 + 0.717282i \(0.254614\pi\)
\(312\) 0 0
\(313\) −11.9809 −0.677199 −0.338599 0.940931i \(-0.609953\pi\)
−0.338599 + 0.940931i \(0.609953\pi\)
\(314\) −11.1548 −0.629503
\(315\) 0 0
\(316\) −35.2526 −1.98312
\(317\) −27.1918 −1.52725 −0.763623 0.645663i \(-0.776581\pi\)
−0.763623 + 0.645663i \(0.776581\pi\)
\(318\) 0 0
\(319\) 5.49228 0.307509
\(320\) 12.4726 0.697240
\(321\) 0 0
\(322\) 1.40382 0.0782318
\(323\) 25.6501 1.42721
\(324\) 0 0
\(325\) 5.17239 0.286913
\(326\) 24.3582 1.34907
\(327\) 0 0
\(328\) −36.5912 −2.02041
\(329\) −0.444396 −0.0245004
\(330\) 0 0
\(331\) 30.1373 1.65650 0.828248 0.560361i \(-0.189338\pi\)
0.828248 + 0.560361i \(0.189338\pi\)
\(332\) −30.3586 −1.66615
\(333\) 0 0
\(334\) 41.6005 2.27628
\(335\) 0.235930 0.0128903
\(336\) 0 0
\(337\) 2.13493 0.116297 0.0581486 0.998308i \(-0.481480\pi\)
0.0581486 + 0.998308i \(0.481480\pi\)
\(338\) −32.0128 −1.74127
\(339\) 0 0
\(340\) −10.6902 −0.579758
\(341\) 32.7109 1.77139
\(342\) 0 0
\(343\) 3.23092 0.174453
\(344\) 19.0295 1.02600
\(345\) 0 0
\(346\) 41.3854 2.22489
\(347\) 33.8723 1.81836 0.909180 0.416403i \(-0.136710\pi\)
0.909180 + 0.416403i \(0.136710\pi\)
\(348\) 0 0
\(349\) −17.9187 −0.959168 −0.479584 0.877496i \(-0.659212\pi\)
−0.479584 + 0.877496i \(0.659212\pi\)
\(350\) 0.539228 0.0288230
\(351\) 0 0
\(352\) 14.5369 0.774820
\(353\) 17.0627 0.908158 0.454079 0.890961i \(-0.349968\pi\)
0.454079 + 0.890961i \(0.349968\pi\)
\(354\) 0 0
\(355\) −7.10782 −0.377244
\(356\) −3.41766 −0.181135
\(357\) 0 0
\(358\) −41.5110 −2.19392
\(359\) −35.0326 −1.84895 −0.924474 0.381245i \(-0.875495\pi\)
−0.924474 + 0.381245i \(0.875495\pi\)
\(360\) 0 0
\(361\) 48.2457 2.53925
\(362\) −28.8510 −1.51638
\(363\) 0 0
\(364\) −4.09531 −0.214653
\(365\) 8.81061 0.461168
\(366\) 0 0
\(367\) 7.12479 0.371911 0.185956 0.982558i \(-0.440462\pi\)
0.185956 + 0.982558i \(0.440462\pi\)
\(368\) 2.20004 0.114685
\(369\) 0 0
\(370\) 0.110930 0.00576699
\(371\) −0.593574 −0.0308168
\(372\) 0 0
\(373\) 23.9155 1.23830 0.619148 0.785274i \(-0.287478\pi\)
0.619148 + 0.785274i \(0.287478\pi\)
\(374\) −22.8467 −1.18137
\(375\) 0 0
\(376\) −6.32967 −0.326428
\(377\) 9.05284 0.466245
\(378\) 0 0
\(379\) 4.25093 0.218355 0.109178 0.994022i \(-0.465178\pi\)
0.109178 + 0.994022i \(0.465178\pi\)
\(380\) −28.0260 −1.43770
\(381\) 0 0
\(382\) −6.88560 −0.352298
\(383\) 7.11909 0.363769 0.181884 0.983320i \(-0.441780\pi\)
0.181884 + 0.983320i \(0.441780\pi\)
\(384\) 0 0
\(385\) 0.726986 0.0370506
\(386\) 39.3319 2.00194
\(387\) 0 0
\(388\) 61.1423 3.10403
\(389\) −22.6491 −1.14835 −0.574177 0.818731i \(-0.694678\pi\)
−0.574177 + 0.818731i \(0.694678\pi\)
\(390\) 0 0
\(391\) 8.14323 0.411821
\(392\) 22.9209 1.15768
\(393\) 0 0
\(394\) 49.6517 2.50142
\(395\) 10.3149 0.518996
\(396\) 0 0
\(397\) −22.4196 −1.12521 −0.562605 0.826726i \(-0.690201\pi\)
−0.562605 + 0.826726i \(0.690201\pi\)
\(398\) 45.7092 2.29119
\(399\) 0 0
\(400\) 0.845067 0.0422534
\(401\) 28.6313 1.42978 0.714889 0.699238i \(-0.246477\pi\)
0.714889 + 0.699238i \(0.246477\pi\)
\(402\) 0 0
\(403\) 53.9169 2.68579
\(404\) −16.6498 −0.828359
\(405\) 0 0
\(406\) 0.943769 0.0468385
\(407\) 0.149556 0.00741321
\(408\) 0 0
\(409\) 30.5726 1.51172 0.755859 0.654734i \(-0.227219\pi\)
0.755859 + 0.654734i \(0.227219\pi\)
\(410\) 25.8110 1.27471
\(411\) 0 0
\(412\) −56.6937 −2.79310
\(413\) −0.0269574 −0.00132649
\(414\) 0 0
\(415\) 8.88288 0.436043
\(416\) 23.9610 1.17478
\(417\) 0 0
\(418\) −59.8960 −2.92961
\(419\) −10.4624 −0.511124 −0.255562 0.966793i \(-0.582260\pi\)
−0.255562 + 0.966793i \(0.582260\pi\)
\(420\) 0 0
\(421\) −9.28945 −0.452740 −0.226370 0.974041i \(-0.572686\pi\)
−0.226370 + 0.974041i \(0.572686\pi\)
\(422\) 5.40211 0.262970
\(423\) 0 0
\(424\) −8.45444 −0.410584
\(425\) 3.12793 0.151727
\(426\) 0 0
\(427\) 0.0324582 0.00157076
\(428\) 49.0686 2.37182
\(429\) 0 0
\(430\) −13.4232 −0.647323
\(431\) 17.0955 0.823462 0.411731 0.911305i \(-0.364924\pi\)
0.411731 + 0.911305i \(0.364924\pi\)
\(432\) 0 0
\(433\) −20.6947 −0.994523 −0.497261 0.867601i \(-0.665661\pi\)
−0.497261 + 0.867601i \(0.665661\pi\)
\(434\) 5.62090 0.269812
\(435\) 0 0
\(436\) 50.2497 2.40652
\(437\) 21.3487 1.02125
\(438\) 0 0
\(439\) 1.07115 0.0511230 0.0255615 0.999673i \(-0.491863\pi\)
0.0255615 + 0.999673i \(0.491863\pi\)
\(440\) 10.3547 0.493640
\(441\) 0 0
\(442\) −37.6578 −1.79120
\(443\) 11.4820 0.545529 0.272764 0.962081i \(-0.412062\pi\)
0.272764 + 0.962081i \(0.412062\pi\)
\(444\) 0 0
\(445\) 1.00000 0.0474045
\(446\) 14.9390 0.707383
\(447\) 0 0
\(448\) 2.88951 0.136517
\(449\) −13.8252 −0.652451 −0.326225 0.945292i \(-0.605777\pi\)
−0.326225 + 0.945292i \(0.605777\pi\)
\(450\) 0 0
\(451\) 34.7983 1.63859
\(452\) 6.31808 0.297177
\(453\) 0 0
\(454\) 22.0824 1.03638
\(455\) 1.19828 0.0561762
\(456\) 0 0
\(457\) 30.7435 1.43812 0.719059 0.694949i \(-0.244573\pi\)
0.719059 + 0.694949i \(0.244573\pi\)
\(458\) −69.6133 −3.25281
\(459\) 0 0
\(460\) −8.89749 −0.414848
\(461\) 14.2580 0.664062 0.332031 0.943268i \(-0.392266\pi\)
0.332031 + 0.943268i \(0.392266\pi\)
\(462\) 0 0
\(463\) 38.2904 1.77951 0.889753 0.456442i \(-0.150876\pi\)
0.889753 + 0.456442i \(0.150876\pi\)
\(464\) 1.47906 0.0686634
\(465\) 0 0
\(466\) 43.3201 2.00676
\(467\) −30.6489 −1.41826 −0.709132 0.705076i \(-0.750913\pi\)
−0.709132 + 0.705076i \(0.750913\pi\)
\(468\) 0 0
\(469\) 0.0546576 0.00252385
\(470\) 4.46488 0.205949
\(471\) 0 0
\(472\) −0.383962 −0.0176733
\(473\) −18.0971 −0.832105
\(474\) 0 0
\(475\) 8.20035 0.376258
\(476\) −2.47658 −0.113514
\(477\) 0 0
\(478\) −17.8205 −0.815092
\(479\) −4.85526 −0.221842 −0.110921 0.993829i \(-0.535380\pi\)
−0.110921 + 0.993829i \(0.535380\pi\)
\(480\) 0 0
\(481\) 0.246511 0.0112399
\(482\) −21.1611 −0.963861
\(483\) 0 0
\(484\) −3.93941 −0.179064
\(485\) −17.8901 −0.812349
\(486\) 0 0
\(487\) −33.2097 −1.50488 −0.752438 0.658663i \(-0.771122\pi\)
−0.752438 + 0.658663i \(0.771122\pi\)
\(488\) 0.462311 0.0209278
\(489\) 0 0
\(490\) −16.1682 −0.730404
\(491\) −17.7104 −0.799260 −0.399630 0.916677i \(-0.630861\pi\)
−0.399630 + 0.916677i \(0.630861\pi\)
\(492\) 0 0
\(493\) 5.47458 0.246563
\(494\) −98.7256 −4.44187
\(495\) 0 0
\(496\) 8.80895 0.395534
\(497\) −1.64666 −0.0738626
\(498\) 0 0
\(499\) 5.20086 0.232823 0.116411 0.993201i \(-0.462861\pi\)
0.116411 + 0.993201i \(0.462861\pi\)
\(500\) −3.41766 −0.152842
\(501\) 0 0
\(502\) 55.5477 2.47922
\(503\) 35.5133 1.58346 0.791729 0.610873i \(-0.209181\pi\)
0.791729 + 0.610873i \(0.209181\pi\)
\(504\) 0 0
\(505\) 4.87170 0.216788
\(506\) −19.0153 −0.845335
\(507\) 0 0
\(508\) −11.1428 −0.494382
\(509\) −35.3342 −1.56616 −0.783081 0.621919i \(-0.786353\pi\)
−0.783081 + 0.621919i \(0.786353\pi\)
\(510\) 0 0
\(511\) 2.04114 0.0902947
\(512\) 9.49172 0.419479
\(513\) 0 0
\(514\) −41.5576 −1.83303
\(515\) 16.5885 0.730975
\(516\) 0 0
\(517\) 6.01954 0.264739
\(518\) 0.0256990 0.00112915
\(519\) 0 0
\(520\) 17.0675 0.748457
\(521\) 1.65866 0.0726674 0.0363337 0.999340i \(-0.488432\pi\)
0.0363337 + 0.999340i \(0.488432\pi\)
\(522\) 0 0
\(523\) 31.5405 1.37917 0.689584 0.724205i \(-0.257793\pi\)
0.689584 + 0.724205i \(0.257793\pi\)
\(524\) −22.2719 −0.972952
\(525\) 0 0
\(526\) −16.0561 −0.700080
\(527\) 32.6055 1.42032
\(528\) 0 0
\(529\) −16.2224 −0.705320
\(530\) 5.96367 0.259045
\(531\) 0 0
\(532\) −6.49273 −0.281496
\(533\) 57.3576 2.48443
\(534\) 0 0
\(535\) −14.3574 −0.620723
\(536\) 0.778504 0.0336263
\(537\) 0 0
\(538\) 26.9324 1.16114
\(539\) −21.7979 −0.938902
\(540\) 0 0
\(541\) 28.4402 1.22274 0.611369 0.791346i \(-0.290619\pi\)
0.611369 + 0.791346i \(0.290619\pi\)
\(542\) 64.3269 2.76308
\(543\) 0 0
\(544\) 14.4901 0.621257
\(545\) −14.7030 −0.629806
\(546\) 0 0
\(547\) −3.21137 −0.137308 −0.0686541 0.997641i \(-0.521870\pi\)
−0.0686541 + 0.997641i \(0.521870\pi\)
\(548\) −49.5728 −2.11765
\(549\) 0 0
\(550\) −7.30407 −0.311447
\(551\) 14.3524 0.611434
\(552\) 0 0
\(553\) 2.38962 0.101617
\(554\) 18.5944 0.790000
\(555\) 0 0
\(556\) −27.9799 −1.18661
\(557\) 7.74724 0.328261 0.164131 0.986439i \(-0.447518\pi\)
0.164131 + 0.986439i \(0.447518\pi\)
\(558\) 0 0
\(559\) −29.8291 −1.26164
\(560\) 0.195775 0.00827302
\(561\) 0 0
\(562\) 38.7874 1.63615
\(563\) −6.11227 −0.257602 −0.128801 0.991670i \(-0.541113\pi\)
−0.128801 + 0.991670i \(0.541113\pi\)
\(564\) 0 0
\(565\) −1.84866 −0.0777736
\(566\) 23.3190 0.980172
\(567\) 0 0
\(568\) −23.4538 −0.984100
\(569\) −17.8784 −0.749502 −0.374751 0.927126i \(-0.622272\pi\)
−0.374751 + 0.927126i \(0.622272\pi\)
\(570\) 0 0
\(571\) −1.88499 −0.0788844 −0.0394422 0.999222i \(-0.512558\pi\)
−0.0394422 + 0.999222i \(0.512558\pi\)
\(572\) 55.4727 2.31943
\(573\) 0 0
\(574\) 5.97959 0.249583
\(575\) 2.60339 0.108569
\(576\) 0 0
\(577\) 20.5466 0.855368 0.427684 0.903928i \(-0.359330\pi\)
0.427684 + 0.903928i \(0.359330\pi\)
\(578\) 16.7959 0.698619
\(579\) 0 0
\(580\) −5.98166 −0.248375
\(581\) 2.05788 0.0853754
\(582\) 0 0
\(583\) 8.04021 0.332991
\(584\) 29.0725 1.20303
\(585\) 0 0
\(586\) −2.21662 −0.0915679
\(587\) −14.3763 −0.593372 −0.296686 0.954975i \(-0.595882\pi\)
−0.296686 + 0.954975i \(0.595882\pi\)
\(588\) 0 0
\(589\) 85.4802 3.52215
\(590\) 0.270843 0.0111504
\(591\) 0 0
\(592\) 0.0402750 0.00165529
\(593\) −4.67911 −0.192148 −0.0960741 0.995374i \(-0.530629\pi\)
−0.0960741 + 0.995374i \(0.530629\pi\)
\(594\) 0 0
\(595\) 0.724643 0.0297075
\(596\) 6.85402 0.280752
\(597\) 0 0
\(598\) −31.3427 −1.28170
\(599\) 38.8668 1.58806 0.794028 0.607881i \(-0.207980\pi\)
0.794028 + 0.607881i \(0.207980\pi\)
\(600\) 0 0
\(601\) 19.2250 0.784202 0.392101 0.919922i \(-0.371748\pi\)
0.392101 + 0.919922i \(0.371748\pi\)
\(602\) −3.10972 −0.126743
\(603\) 0 0
\(604\) 70.1358 2.85378
\(605\) 1.15266 0.0468624
\(606\) 0 0
\(607\) −2.77256 −0.112535 −0.0562673 0.998416i \(-0.517920\pi\)
−0.0562673 + 0.998416i \(0.517920\pi\)
\(608\) 37.9879 1.54061
\(609\) 0 0
\(610\) −0.326109 −0.0132038
\(611\) 9.92191 0.401398
\(612\) 0 0
\(613\) −25.7952 −1.04186 −0.520930 0.853599i \(-0.674415\pi\)
−0.520930 + 0.853599i \(0.674415\pi\)
\(614\) −6.24631 −0.252081
\(615\) 0 0
\(616\) 2.39885 0.0966524
\(617\) 6.65304 0.267841 0.133921 0.990992i \(-0.457243\pi\)
0.133921 + 0.990992i \(0.457243\pi\)
\(618\) 0 0
\(619\) −36.4261 −1.46409 −0.732044 0.681258i \(-0.761433\pi\)
−0.732044 + 0.681258i \(0.761433\pi\)
\(620\) −35.6255 −1.43076
\(621\) 0 0
\(622\) −57.2023 −2.29360
\(623\) 0.231668 0.00928160
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 27.8865 1.11457
\(627\) 0 0
\(628\) 16.3789 0.653590
\(629\) 0.149074 0.00594397
\(630\) 0 0
\(631\) −15.4933 −0.616780 −0.308390 0.951260i \(-0.599790\pi\)
−0.308390 + 0.951260i \(0.599790\pi\)
\(632\) 34.0361 1.35388
\(633\) 0 0
\(634\) 63.2913 2.51362
\(635\) 3.26036 0.129384
\(636\) 0 0
\(637\) −35.9292 −1.42356
\(638\) −12.7838 −0.506114
\(639\) 0 0
\(640\) −19.7661 −0.781325
\(641\) −21.3356 −0.842704 −0.421352 0.906897i \(-0.638444\pi\)
−0.421352 + 0.906897i \(0.638444\pi\)
\(642\) 0 0
\(643\) 2.08343 0.0821626 0.0410813 0.999156i \(-0.486920\pi\)
0.0410813 + 0.999156i \(0.486920\pi\)
\(644\) −2.06127 −0.0812253
\(645\) 0 0
\(646\) −59.7029 −2.34898
\(647\) −2.30024 −0.0904318 −0.0452159 0.998977i \(-0.514398\pi\)
−0.0452159 + 0.998977i \(0.514398\pi\)
\(648\) 0 0
\(649\) 0.365149 0.0143334
\(650\) −12.0392 −0.472216
\(651\) 0 0
\(652\) −35.7658 −1.40070
\(653\) 23.5469 0.921463 0.460731 0.887540i \(-0.347587\pi\)
0.460731 + 0.887540i \(0.347587\pi\)
\(654\) 0 0
\(655\) 6.51671 0.254629
\(656\) 9.37110 0.365880
\(657\) 0 0
\(658\) 1.03437 0.0403240
\(659\) −20.0639 −0.781580 −0.390790 0.920480i \(-0.627798\pi\)
−0.390790 + 0.920480i \(0.627798\pi\)
\(660\) 0 0
\(661\) 9.43421 0.366948 0.183474 0.983025i \(-0.441266\pi\)
0.183474 + 0.983025i \(0.441266\pi\)
\(662\) −70.1472 −2.72635
\(663\) 0 0
\(664\) 29.3110 1.13749
\(665\) 1.89976 0.0736696
\(666\) 0 0
\(667\) 4.55651 0.176429
\(668\) −61.0831 −2.36338
\(669\) 0 0
\(670\) −0.549148 −0.0212154
\(671\) −0.439660 −0.0169729
\(672\) 0 0
\(673\) −32.9917 −1.27173 −0.635867 0.771798i \(-0.719357\pi\)
−0.635867 + 0.771798i \(0.719357\pi\)
\(674\) −4.96924 −0.191408
\(675\) 0 0
\(676\) 47.0053 1.80790
\(677\) 15.0572 0.578695 0.289347 0.957224i \(-0.406562\pi\)
0.289347 + 0.957224i \(0.406562\pi\)
\(678\) 0 0
\(679\) −4.14458 −0.159054
\(680\) 10.3213 0.395804
\(681\) 0 0
\(682\) −76.1374 −2.91545
\(683\) 42.2040 1.61489 0.807446 0.589942i \(-0.200849\pi\)
0.807446 + 0.589942i \(0.200849\pi\)
\(684\) 0 0
\(685\) 14.5049 0.554204
\(686\) −7.52025 −0.287125
\(687\) 0 0
\(688\) −4.87349 −0.185800
\(689\) 13.2526 0.504882
\(690\) 0 0
\(691\) 15.3797 0.585073 0.292536 0.956254i \(-0.405501\pi\)
0.292536 + 0.956254i \(0.405501\pi\)
\(692\) −60.7672 −2.31002
\(693\) 0 0
\(694\) −78.8407 −2.99275
\(695\) 8.18687 0.310546
\(696\) 0 0
\(697\) 34.6862 1.31383
\(698\) 41.7074 1.57865
\(699\) 0 0
\(700\) −0.791763 −0.0299258
\(701\) −18.5351 −0.700063 −0.350031 0.936738i \(-0.613829\pi\)
−0.350031 + 0.936738i \(0.613829\pi\)
\(702\) 0 0
\(703\) 0.390820 0.0147400
\(704\) −39.1396 −1.47513
\(705\) 0 0
\(706\) −39.7150 −1.49469
\(707\) 1.12862 0.0424461
\(708\) 0 0
\(709\) −5.77286 −0.216804 −0.108402 0.994107i \(-0.534573\pi\)
−0.108402 + 0.994107i \(0.534573\pi\)
\(710\) 16.5441 0.620887
\(711\) 0 0
\(712\) 3.29972 0.123662
\(713\) 27.1376 1.01631
\(714\) 0 0
\(715\) −16.2312 −0.607013
\(716\) 60.9516 2.27787
\(717\) 0 0
\(718\) 81.5413 3.04309
\(719\) 14.0660 0.524573 0.262287 0.964990i \(-0.415523\pi\)
0.262287 + 0.964990i \(0.415523\pi\)
\(720\) 0 0
\(721\) 3.84302 0.143122
\(722\) −112.296 −4.17923
\(723\) 0 0
\(724\) 42.3627 1.57440
\(725\) 1.75022 0.0650016
\(726\) 0 0
\(727\) 19.3160 0.716390 0.358195 0.933647i \(-0.383392\pi\)
0.358195 + 0.933647i \(0.383392\pi\)
\(728\) 3.95399 0.146545
\(729\) 0 0
\(730\) −20.5074 −0.759015
\(731\) −18.0388 −0.667188
\(732\) 0 0
\(733\) −46.1230 −1.70359 −0.851795 0.523875i \(-0.824486\pi\)
−0.851795 + 0.523875i \(0.824486\pi\)
\(734\) −16.5836 −0.612111
\(735\) 0 0
\(736\) 12.0601 0.444542
\(737\) −0.740360 −0.0272715
\(738\) 0 0
\(739\) −1.81531 −0.0667771 −0.0333885 0.999442i \(-0.510630\pi\)
−0.0333885 + 0.999442i \(0.510630\pi\)
\(740\) −0.162882 −0.00598766
\(741\) 0 0
\(742\) 1.38159 0.0507199
\(743\) −20.0513 −0.735611 −0.367805 0.929903i \(-0.619891\pi\)
−0.367805 + 0.929903i \(0.619891\pi\)
\(744\) 0 0
\(745\) −2.00547 −0.0734749
\(746\) −55.6653 −2.03805
\(747\) 0 0
\(748\) 33.5464 1.22658
\(749\) −3.32615 −0.121535
\(750\) 0 0
\(751\) 32.4991 1.18591 0.592955 0.805235i \(-0.297961\pi\)
0.592955 + 0.805235i \(0.297961\pi\)
\(752\) 1.62104 0.0591134
\(753\) 0 0
\(754\) −21.0713 −0.767370
\(755\) −20.5216 −0.746857
\(756\) 0 0
\(757\) −3.19706 −0.116199 −0.0580995 0.998311i \(-0.518504\pi\)
−0.0580995 + 0.998311i \(0.518504\pi\)
\(758\) −9.89439 −0.359381
\(759\) 0 0
\(760\) 27.0589 0.981528
\(761\) 1.78809 0.0648183 0.0324091 0.999475i \(-0.489682\pi\)
0.0324091 + 0.999475i \(0.489682\pi\)
\(762\) 0 0
\(763\) −3.40621 −0.123313
\(764\) 10.1103 0.365778
\(765\) 0 0
\(766\) −16.5703 −0.598709
\(767\) 0.601870 0.0217323
\(768\) 0 0
\(769\) 11.4774 0.413884 0.206942 0.978353i \(-0.433649\pi\)
0.206942 + 0.978353i \(0.433649\pi\)
\(770\) −1.69212 −0.0609799
\(771\) 0 0
\(772\) −57.7520 −2.07854
\(773\) 33.9319 1.22045 0.610223 0.792230i \(-0.291080\pi\)
0.610223 + 0.792230i \(0.291080\pi\)
\(774\) 0 0
\(775\) 10.4240 0.374440
\(776\) −59.0324 −2.11914
\(777\) 0 0
\(778\) 52.7177 1.89002
\(779\) 90.9351 3.25809
\(780\) 0 0
\(781\) 22.3047 0.798123
\(782\) −18.9541 −0.677796
\(783\) 0 0
\(784\) −5.87011 −0.209647
\(785\) −4.79244 −0.171049
\(786\) 0 0
\(787\) −12.6811 −0.452034 −0.226017 0.974123i \(-0.572570\pi\)
−0.226017 + 0.974123i \(0.572570\pi\)
\(788\) −72.9049 −2.59713
\(789\) 0 0
\(790\) −24.0087 −0.854191
\(791\) −0.428275 −0.0152277
\(792\) 0 0
\(793\) −0.724684 −0.0257343
\(794\) 52.1836 1.85193
\(795\) 0 0
\(796\) −67.1160 −2.37886
\(797\) −49.4504 −1.75162 −0.875812 0.482653i \(-0.839673\pi\)
−0.875812 + 0.482653i \(0.839673\pi\)
\(798\) 0 0
\(799\) 6.00014 0.212270
\(800\) 4.63247 0.163783
\(801\) 0 0
\(802\) −66.6418 −2.35320
\(803\) −27.6481 −0.975680
\(804\) 0 0
\(805\) 0.603123 0.0212573
\(806\) −125.496 −4.42041
\(807\) 0 0
\(808\) 16.0752 0.565525
\(809\) −48.8608 −1.71785 −0.858926 0.512099i \(-0.828868\pi\)
−0.858926 + 0.512099i \(0.828868\pi\)
\(810\) 0 0
\(811\) 23.1735 0.813731 0.406866 0.913488i \(-0.366622\pi\)
0.406866 + 0.913488i \(0.366622\pi\)
\(812\) −1.38576 −0.0486307
\(813\) 0 0
\(814\) −0.348104 −0.0122010
\(815\) 10.4650 0.366573
\(816\) 0 0
\(817\) −47.2913 −1.65451
\(818\) −71.1604 −2.48806
\(819\) 0 0
\(820\) −37.8990 −1.32349
\(821\) −51.1545 −1.78530 −0.892652 0.450747i \(-0.851158\pi\)
−0.892652 + 0.450747i \(0.851158\pi\)
\(822\) 0 0
\(823\) −23.0988 −0.805175 −0.402587 0.915382i \(-0.631889\pi\)
−0.402587 + 0.915382i \(0.631889\pi\)
\(824\) 54.7373 1.90686
\(825\) 0 0
\(826\) 0.0627457 0.00218320
\(827\) 9.60440 0.333978 0.166989 0.985959i \(-0.446596\pi\)
0.166989 + 0.985959i \(0.446596\pi\)
\(828\) 0 0
\(829\) −16.6462 −0.578148 −0.289074 0.957307i \(-0.593347\pi\)
−0.289074 + 0.957307i \(0.593347\pi\)
\(830\) −20.6757 −0.717663
\(831\) 0 0
\(832\) −64.5133 −2.23660
\(833\) −21.7277 −0.752819
\(834\) 0 0
\(835\) 17.8728 0.618514
\(836\) 87.9468 3.04171
\(837\) 0 0
\(838\) 24.3522 0.841234
\(839\) −51.5325 −1.77910 −0.889550 0.456839i \(-0.848982\pi\)
−0.889550 + 0.456839i \(0.848982\pi\)
\(840\) 0 0
\(841\) −25.9367 −0.894370
\(842\) 21.6220 0.745143
\(843\) 0 0
\(844\) −7.93206 −0.273033
\(845\) −13.7537 −0.473140
\(846\) 0 0
\(847\) 0.267035 0.00917545
\(848\) 2.16521 0.0743535
\(849\) 0 0
\(850\) −7.28054 −0.249720
\(851\) 0.124075 0.00425322
\(852\) 0 0
\(853\) −9.83286 −0.336671 −0.168335 0.985730i \(-0.553839\pi\)
−0.168335 + 0.985730i \(0.553839\pi\)
\(854\) −0.0755492 −0.00258524
\(855\) 0 0
\(856\) −47.3753 −1.61925
\(857\) 17.5102 0.598138 0.299069 0.954232i \(-0.403324\pi\)
0.299069 + 0.954232i \(0.403324\pi\)
\(858\) 0 0
\(859\) −4.18058 −0.142640 −0.0713198 0.997453i \(-0.522721\pi\)
−0.0713198 + 0.997453i \(0.522721\pi\)
\(860\) 19.7096 0.672092
\(861\) 0 0
\(862\) −39.7913 −1.35530
\(863\) −40.8099 −1.38918 −0.694592 0.719404i \(-0.744415\pi\)
−0.694592 + 0.719404i \(0.744415\pi\)
\(864\) 0 0
\(865\) 17.7804 0.604551
\(866\) 48.1687 1.63684
\(867\) 0 0
\(868\) −8.25331 −0.280136
\(869\) −32.3685 −1.09803
\(870\) 0 0
\(871\) −1.22032 −0.0413491
\(872\) −48.5157 −1.64295
\(873\) 0 0
\(874\) −49.6909 −1.68082
\(875\) 0.231668 0.00783182
\(876\) 0 0
\(877\) −46.1475 −1.55829 −0.779145 0.626844i \(-0.784346\pi\)
−0.779145 + 0.626844i \(0.784346\pi\)
\(878\) −2.49318 −0.0841408
\(879\) 0 0
\(880\) −2.65186 −0.0893942
\(881\) −29.8832 −1.00679 −0.503396 0.864056i \(-0.667916\pi\)
−0.503396 + 0.864056i \(0.667916\pi\)
\(882\) 0 0
\(883\) 44.2977 1.49073 0.745367 0.666654i \(-0.232274\pi\)
0.745367 + 0.666654i \(0.232274\pi\)
\(884\) 55.2940 1.85974
\(885\) 0 0
\(886\) −26.7255 −0.897859
\(887\) 52.5198 1.76344 0.881721 0.471770i \(-0.156385\pi\)
0.881721 + 0.471770i \(0.156385\pi\)
\(888\) 0 0
\(889\) 0.755323 0.0253327
\(890\) −2.32759 −0.0780209
\(891\) 0 0
\(892\) −21.9354 −0.734450
\(893\) 15.7303 0.526393
\(894\) 0 0
\(895\) −17.8343 −0.596136
\(896\) −4.57919 −0.152980
\(897\) 0 0
\(898\) 32.1793 1.07384
\(899\) 18.2443 0.608480
\(900\) 0 0
\(901\) 8.01430 0.266995
\(902\) −80.9961 −2.69688
\(903\) 0 0
\(904\) −6.10005 −0.202885
\(905\) −12.3953 −0.412032
\(906\) 0 0
\(907\) −13.7670 −0.457124 −0.228562 0.973529i \(-0.573402\pi\)
−0.228562 + 0.973529i \(0.573402\pi\)
\(908\) −32.4241 −1.07603
\(909\) 0 0
\(910\) −2.78910 −0.0924578
\(911\) −24.4481 −0.810001 −0.405001 0.914316i \(-0.632729\pi\)
−0.405001 + 0.914316i \(0.632729\pi\)
\(912\) 0 0
\(913\) −27.8749 −0.922524
\(914\) −71.5580 −2.36693
\(915\) 0 0
\(916\) 102.215 3.37728
\(917\) 1.50972 0.0498552
\(918\) 0 0
\(919\) −28.9181 −0.953920 −0.476960 0.878925i \(-0.658261\pi\)
−0.476960 + 0.878925i \(0.658261\pi\)
\(920\) 8.59045 0.283219
\(921\) 0 0
\(922\) −33.1868 −1.09295
\(923\) 36.7644 1.21012
\(924\) 0 0
\(925\) 0.0476589 0.00156702
\(926\) −89.1243 −2.92880
\(927\) 0 0
\(928\) 8.10786 0.266154
\(929\) −21.1894 −0.695201 −0.347601 0.937643i \(-0.613003\pi\)
−0.347601 + 0.937643i \(0.613003\pi\)
\(930\) 0 0
\(931\) −56.9623 −1.86686
\(932\) −63.6081 −2.08355
\(933\) 0 0
\(934\) 71.3381 2.33425
\(935\) −9.81560 −0.321005
\(936\) 0 0
\(937\) 33.8348 1.10534 0.552668 0.833401i \(-0.313610\pi\)
0.552668 + 0.833401i \(0.313610\pi\)
\(938\) −0.127220 −0.00415389
\(939\) 0 0
\(940\) −6.55590 −0.213830
\(941\) −28.9054 −0.942288 −0.471144 0.882056i \(-0.656159\pi\)
−0.471144 + 0.882056i \(0.656159\pi\)
\(942\) 0 0
\(943\) 28.8694 0.940118
\(944\) 0.0983337 0.00320049
\(945\) 0 0
\(946\) 42.1225 1.36952
\(947\) 13.3591 0.434114 0.217057 0.976159i \(-0.430354\pi\)
0.217057 + 0.976159i \(0.430354\pi\)
\(948\) 0 0
\(949\) −45.5719 −1.47933
\(950\) −19.0870 −0.619265
\(951\) 0 0
\(952\) 2.39112 0.0774966
\(953\) 12.6493 0.409751 0.204875 0.978788i \(-0.434321\pi\)
0.204875 + 0.978788i \(0.434321\pi\)
\(954\) 0 0
\(955\) −2.95826 −0.0957270
\(956\) 26.1664 0.846280
\(957\) 0 0
\(958\) 11.3010 0.365120
\(959\) 3.36033 0.108511
\(960\) 0 0
\(961\) 77.6591 2.50513
\(962\) −0.573775 −0.0184992
\(963\) 0 0
\(964\) 31.0714 1.00074
\(965\) 16.8981 0.543970
\(966\) 0 0
\(967\) −12.6366 −0.406367 −0.203183 0.979141i \(-0.565129\pi\)
−0.203183 + 0.979141i \(0.565129\pi\)
\(968\) 3.80346 0.122248
\(969\) 0 0
\(970\) 41.6408 1.33701
\(971\) −54.6786 −1.75472 −0.877360 0.479832i \(-0.840697\pi\)
−0.877360 + 0.479832i \(0.840697\pi\)
\(972\) 0 0
\(973\) 1.89664 0.0608034
\(974\) 77.2985 2.47680
\(975\) 0 0
\(976\) −0.118399 −0.00378986
\(977\) −26.1759 −0.837442 −0.418721 0.908115i \(-0.637521\pi\)
−0.418721 + 0.908115i \(0.637521\pi\)
\(978\) 0 0
\(979\) −3.13805 −0.100292
\(980\) 23.7402 0.758352
\(981\) 0 0
\(982\) 41.2225 1.31546
\(983\) 9.12819 0.291144 0.145572 0.989348i \(-0.453498\pi\)
0.145572 + 0.989348i \(0.453498\pi\)
\(984\) 0 0
\(985\) 21.3318 0.679689
\(986\) −12.7426 −0.405806
\(987\) 0 0
\(988\) 144.961 4.61184
\(989\) −15.0137 −0.477408
\(990\) 0 0
\(991\) −21.6993 −0.689301 −0.344650 0.938731i \(-0.612003\pi\)
−0.344650 + 0.938731i \(0.612003\pi\)
\(992\) 48.2887 1.53317
\(993\) 0 0
\(994\) 3.83273 0.121567
\(995\) 19.6380 0.622567
\(996\) 0 0
\(997\) 7.02812 0.222583 0.111291 0.993788i \(-0.464501\pi\)
0.111291 + 0.993788i \(0.464501\pi\)
\(998\) −12.1055 −0.383192
\(999\) 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 4005.2.a.r.1.3 10
3.2 odd 2 1335.2.a.k.1.8 10
15.14 odd 2 6675.2.a.y.1.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1335.2.a.k.1.8 10 3.2 odd 2
4005.2.a.r.1.3 10 1.1 even 1 trivial
6675.2.a.y.1.3 10 15.14 odd 2