Properties

Label 6003.2.a.k.1.1
Level $6003$
Weight $2$
Character 6003.1
Self dual yes
Analytic conductor $47.934$
Analytic rank $1$
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] = [6003,2,Mod(1,6003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6003, 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("6003.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.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: \(47.9341963334\)
Analytic rank: \(1\)
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: \( x^{10} - x^{9} - 17x^{8} + 23x^{7} + 69x^{6} - 88x^{5} - 106x^{4} + 101x^{3} + 60x^{2} - 23x - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2001)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.19874\) of defining polynomial
Character \(\chi\) \(=\) 6003.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.74444 q^{2} +5.53194 q^{4} -0.0839771 q^{5} -2.30146 q^{7} -9.69318 q^{8} +O(q^{10})\) \(q-2.74444 q^{2} +5.53194 q^{4} -0.0839771 q^{5} -2.30146 q^{7} -9.69318 q^{8} +0.230470 q^{10} -4.15428 q^{11} -0.439258 q^{13} +6.31621 q^{14} +15.5385 q^{16} +0.388355 q^{17} +5.29460 q^{19} -0.464556 q^{20} +11.4012 q^{22} +1.00000 q^{23} -4.99295 q^{25} +1.20552 q^{26} -12.7315 q^{28} -1.00000 q^{29} +1.44043 q^{31} -23.2580 q^{32} -1.06582 q^{34} +0.193270 q^{35} +9.04387 q^{37} -14.5307 q^{38} +0.814006 q^{40} +0.159033 q^{41} -6.03876 q^{43} -22.9812 q^{44} -2.74444 q^{46} +0.396132 q^{47} -1.70328 q^{49} +13.7028 q^{50} -2.42995 q^{52} -2.43614 q^{53} +0.348864 q^{55} +22.3085 q^{56} +2.74444 q^{58} +4.08958 q^{59} +9.28497 q^{61} -3.95318 q^{62} +32.7531 q^{64} +0.0368876 q^{65} +4.55339 q^{67} +2.14836 q^{68} -0.530417 q^{70} +3.85137 q^{71} +0.0113549 q^{73} -24.8203 q^{74} +29.2894 q^{76} +9.56090 q^{77} +9.57510 q^{79} -1.30488 q^{80} -0.436455 q^{82} +5.55983 q^{83} -0.0326130 q^{85} +16.5730 q^{86} +40.2682 q^{88} +4.02316 q^{89} +1.01093 q^{91} +5.53194 q^{92} -1.08716 q^{94} -0.444625 q^{95} -19.4508 q^{97} +4.67455 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8} - 4 q^{10} - 9 q^{11} - 16 q^{13} - 16 q^{14} + 27 q^{16} + q^{19} - 21 q^{20} + 17 q^{22} + 10 q^{23} - 4 q^{25} - 28 q^{26} - 14 q^{28} - 10 q^{29} + 17 q^{31} - 21 q^{32} - 3 q^{34} - 29 q^{35} + q^{37} - 32 q^{38} + 13 q^{40} - 5 q^{43} - 33 q^{44} - 3 q^{46} - 15 q^{47} + 31 q^{49} + 22 q^{50} - 21 q^{52} - 35 q^{53} - 20 q^{55} - 18 q^{56} + 3 q^{58} - 49 q^{59} + 8 q^{61} - 15 q^{62} + 12 q^{64} + 3 q^{65} + 35 q^{67} + 18 q^{68} - 16 q^{70} - 30 q^{71} - 15 q^{73} - 23 q^{74} + 10 q^{76} - 23 q^{77} + 24 q^{79} - 23 q^{80} - 5 q^{82} - q^{83} + 10 q^{86} + 18 q^{88} - 15 q^{89} + 26 q^{91} + 17 q^{92} + 3 q^{94} - 7 q^{95} - 35 q^{97} - 3 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.74444 −1.94061 −0.970305 0.241884i \(-0.922235\pi\)
−0.970305 + 0.241884i \(0.922235\pi\)
\(3\) 0 0
\(4\) 5.53194 2.76597
\(5\) −0.0839771 −0.0375557 −0.0187779 0.999824i \(-0.505978\pi\)
−0.0187779 + 0.999824i \(0.505978\pi\)
\(6\) 0 0
\(7\) −2.30146 −0.869870 −0.434935 0.900462i \(-0.643229\pi\)
−0.434935 + 0.900462i \(0.643229\pi\)
\(8\) −9.69318 −3.42706
\(9\) 0 0
\(10\) 0.230470 0.0728810
\(11\) −4.15428 −1.25256 −0.626281 0.779597i \(-0.715424\pi\)
−0.626281 + 0.779597i \(0.715424\pi\)
\(12\) 0 0
\(13\) −0.439258 −0.121828 −0.0609141 0.998143i \(-0.519402\pi\)
−0.0609141 + 0.998143i \(0.519402\pi\)
\(14\) 6.31621 1.68808
\(15\) 0 0
\(16\) 15.5385 3.88462
\(17\) 0.388355 0.0941900 0.0470950 0.998890i \(-0.485004\pi\)
0.0470950 + 0.998890i \(0.485004\pi\)
\(18\) 0 0
\(19\) 5.29460 1.21466 0.607332 0.794448i \(-0.292240\pi\)
0.607332 + 0.794448i \(0.292240\pi\)
\(20\) −0.464556 −0.103878
\(21\) 0 0
\(22\) 11.4012 2.43073
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −4.99295 −0.998590
\(26\) 1.20552 0.236421
\(27\) 0 0
\(28\) −12.7315 −2.40603
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 1.44043 0.258710 0.129355 0.991598i \(-0.458709\pi\)
0.129355 + 0.991598i \(0.458709\pi\)
\(32\) −23.2580 −4.11147
\(33\) 0 0
\(34\) −1.06582 −0.182786
\(35\) 0.193270 0.0326686
\(36\) 0 0
\(37\) 9.04387 1.48680 0.743402 0.668845i \(-0.233211\pi\)
0.743402 + 0.668845i \(0.233211\pi\)
\(38\) −14.5307 −2.35719
\(39\) 0 0
\(40\) 0.814006 0.128706
\(41\) 0.159033 0.0248367 0.0124184 0.999923i \(-0.496047\pi\)
0.0124184 + 0.999923i \(0.496047\pi\)
\(42\) 0 0
\(43\) −6.03876 −0.920902 −0.460451 0.887685i \(-0.652312\pi\)
−0.460451 + 0.887685i \(0.652312\pi\)
\(44\) −22.9812 −3.46455
\(45\) 0 0
\(46\) −2.74444 −0.404645
\(47\) 0.396132 0.0577817 0.0288909 0.999583i \(-0.490802\pi\)
0.0288909 + 0.999583i \(0.490802\pi\)
\(48\) 0 0
\(49\) −1.70328 −0.243326
\(50\) 13.7028 1.93787
\(51\) 0 0
\(52\) −2.42995 −0.336973
\(53\) −2.43614 −0.334630 −0.167315 0.985903i \(-0.553510\pi\)
−0.167315 + 0.985903i \(0.553510\pi\)
\(54\) 0 0
\(55\) 0.348864 0.0470408
\(56\) 22.3085 2.98109
\(57\) 0 0
\(58\) 2.74444 0.360362
\(59\) 4.08958 0.532418 0.266209 0.963915i \(-0.414229\pi\)
0.266209 + 0.963915i \(0.414229\pi\)
\(60\) 0 0
\(61\) 9.28497 1.18882 0.594409 0.804163i \(-0.297386\pi\)
0.594409 + 0.804163i \(0.297386\pi\)
\(62\) −3.95318 −0.502054
\(63\) 0 0
\(64\) 32.7531 4.09414
\(65\) 0.0368876 0.00457534
\(66\) 0 0
\(67\) 4.55339 0.556285 0.278143 0.960540i \(-0.410281\pi\)
0.278143 + 0.960540i \(0.410281\pi\)
\(68\) 2.14836 0.260527
\(69\) 0 0
\(70\) −0.530417 −0.0633970
\(71\) 3.85137 0.457074 0.228537 0.973535i \(-0.426606\pi\)
0.228537 + 0.973535i \(0.426606\pi\)
\(72\) 0 0
\(73\) 0.0113549 0.00132899 0.000664497 1.00000i \(-0.499788\pi\)
0.000664497 1.00000i \(0.499788\pi\)
\(74\) −24.8203 −2.88531
\(75\) 0 0
\(76\) 29.2894 3.35972
\(77\) 9.56090 1.08957
\(78\) 0 0
\(79\) 9.57510 1.07728 0.538641 0.842535i \(-0.318938\pi\)
0.538641 + 0.842535i \(0.318938\pi\)
\(80\) −1.30488 −0.145889
\(81\) 0 0
\(82\) −0.436455 −0.0481984
\(83\) 5.55983 0.610271 0.305136 0.952309i \(-0.401298\pi\)
0.305136 + 0.952309i \(0.401298\pi\)
\(84\) 0 0
\(85\) −0.0326130 −0.00353737
\(86\) 16.5730 1.78711
\(87\) 0 0
\(88\) 40.2682 4.29260
\(89\) 4.02316 0.426454 0.213227 0.977003i \(-0.431603\pi\)
0.213227 + 0.977003i \(0.431603\pi\)
\(90\) 0 0
\(91\) 1.01093 0.105975
\(92\) 5.53194 0.576744
\(93\) 0 0
\(94\) −1.08716 −0.112132
\(95\) −0.444625 −0.0456175
\(96\) 0 0
\(97\) −19.4508 −1.97493 −0.987463 0.157853i \(-0.949543\pi\)
−0.987463 + 0.157853i \(0.949543\pi\)
\(98\) 4.67455 0.472201
\(99\) 0 0
\(100\) −27.6207 −2.76207
\(101\) 14.1145 1.40444 0.702220 0.711960i \(-0.252192\pi\)
0.702220 + 0.711960i \(0.252192\pi\)
\(102\) 0 0
\(103\) −6.05489 −0.596606 −0.298303 0.954471i \(-0.596421\pi\)
−0.298303 + 0.954471i \(0.596421\pi\)
\(104\) 4.25780 0.417512
\(105\) 0 0
\(106\) 6.68584 0.649386
\(107\) 5.56190 0.537689 0.268845 0.963184i \(-0.413358\pi\)
0.268845 + 0.963184i \(0.413358\pi\)
\(108\) 0 0
\(109\) 0.180138 0.0172541 0.00862706 0.999963i \(-0.497254\pi\)
0.00862706 + 0.999963i \(0.497254\pi\)
\(110\) −0.957436 −0.0912879
\(111\) 0 0
\(112\) −35.7611 −3.37911
\(113\) −8.51126 −0.800672 −0.400336 0.916368i \(-0.631107\pi\)
−0.400336 + 0.916368i \(0.631107\pi\)
\(114\) 0 0
\(115\) −0.0839771 −0.00783091
\(116\) −5.53194 −0.513628
\(117\) 0 0
\(118\) −11.2236 −1.03322
\(119\) −0.893784 −0.0819330
\(120\) 0 0
\(121\) 6.25802 0.568911
\(122\) −25.4820 −2.30703
\(123\) 0 0
\(124\) 7.96839 0.715583
\(125\) 0.839179 0.0750584
\(126\) 0 0
\(127\) −16.5892 −1.47205 −0.736025 0.676954i \(-0.763299\pi\)
−0.736025 + 0.676954i \(0.763299\pi\)
\(128\) −43.3730 −3.83367
\(129\) 0 0
\(130\) −0.101236 −0.00887896
\(131\) 7.62105 0.665854 0.332927 0.942953i \(-0.391964\pi\)
0.332927 + 0.942953i \(0.391964\pi\)
\(132\) 0 0
\(133\) −12.1853 −1.05660
\(134\) −12.4965 −1.07953
\(135\) 0 0
\(136\) −3.76440 −0.322795
\(137\) −9.11641 −0.778868 −0.389434 0.921054i \(-0.627329\pi\)
−0.389434 + 0.921054i \(0.627329\pi\)
\(138\) 0 0
\(139\) −0.0164781 −0.00139765 −0.000698827 1.00000i \(-0.500222\pi\)
−0.000698827 1.00000i \(0.500222\pi\)
\(140\) 1.06916 0.0903603
\(141\) 0 0
\(142\) −10.5699 −0.887002
\(143\) 1.82480 0.152597
\(144\) 0 0
\(145\) 0.0839771 0.00697392
\(146\) −0.0311629 −0.00257906
\(147\) 0 0
\(148\) 50.0301 4.11245
\(149\) 15.3748 1.25955 0.629775 0.776777i \(-0.283147\pi\)
0.629775 + 0.776777i \(0.283147\pi\)
\(150\) 0 0
\(151\) −10.3617 −0.843225 −0.421613 0.906776i \(-0.638536\pi\)
−0.421613 + 0.906776i \(0.638536\pi\)
\(152\) −51.3215 −4.16272
\(153\) 0 0
\(154\) −26.2393 −2.11442
\(155\) −0.120963 −0.00971602
\(156\) 0 0
\(157\) −0.319033 −0.0254616 −0.0127308 0.999919i \(-0.504052\pi\)
−0.0127308 + 0.999919i \(0.504052\pi\)
\(158\) −26.2783 −2.09059
\(159\) 0 0
\(160\) 1.95314 0.154409
\(161\) −2.30146 −0.181380
\(162\) 0 0
\(163\) 18.6190 1.45835 0.729176 0.684327i \(-0.239904\pi\)
0.729176 + 0.684327i \(0.239904\pi\)
\(164\) 0.879758 0.0686976
\(165\) 0 0
\(166\) −15.2586 −1.18430
\(167\) 20.0675 1.55287 0.776435 0.630197i \(-0.217026\pi\)
0.776435 + 0.630197i \(0.217026\pi\)
\(168\) 0 0
\(169\) −12.8071 −0.985158
\(170\) 0.0895042 0.00686466
\(171\) 0 0
\(172\) −33.4060 −2.54719
\(173\) 14.5928 1.10947 0.554736 0.832027i \(-0.312819\pi\)
0.554736 + 0.832027i \(0.312819\pi\)
\(174\) 0 0
\(175\) 11.4911 0.868643
\(176\) −64.5511 −4.86572
\(177\) 0 0
\(178\) −11.0413 −0.827581
\(179\) −20.7807 −1.55322 −0.776612 0.629979i \(-0.783063\pi\)
−0.776612 + 0.629979i \(0.783063\pi\)
\(180\) 0 0
\(181\) −4.46140 −0.331613 −0.165806 0.986158i \(-0.553023\pi\)
−0.165806 + 0.986158i \(0.553023\pi\)
\(182\) −2.77444 −0.205655
\(183\) 0 0
\(184\) −9.69318 −0.714591
\(185\) −0.759478 −0.0558379
\(186\) 0 0
\(187\) −1.61334 −0.117979
\(188\) 2.19137 0.159822
\(189\) 0 0
\(190\) 1.22025 0.0885259
\(191\) −16.4833 −1.19269 −0.596346 0.802727i \(-0.703382\pi\)
−0.596346 + 0.802727i \(0.703382\pi\)
\(192\) 0 0
\(193\) −6.03329 −0.434286 −0.217143 0.976140i \(-0.569674\pi\)
−0.217143 + 0.976140i \(0.569674\pi\)
\(194\) 53.3814 3.83256
\(195\) 0 0
\(196\) −9.42246 −0.673033
\(197\) −18.4203 −1.31239 −0.656197 0.754590i \(-0.727836\pi\)
−0.656197 + 0.754590i \(0.727836\pi\)
\(198\) 0 0
\(199\) −22.7882 −1.61542 −0.807708 0.589583i \(-0.799292\pi\)
−0.807708 + 0.589583i \(0.799292\pi\)
\(200\) 48.3976 3.42222
\(201\) 0 0
\(202\) −38.7362 −2.72547
\(203\) 2.30146 0.161531
\(204\) 0 0
\(205\) −0.0133551 −0.000932760 0
\(206\) 16.6173 1.15778
\(207\) 0 0
\(208\) −6.82539 −0.473255
\(209\) −21.9952 −1.52144
\(210\) 0 0
\(211\) 1.48145 0.101987 0.0509934 0.998699i \(-0.483761\pi\)
0.0509934 + 0.998699i \(0.483761\pi\)
\(212\) −13.4766 −0.925576
\(213\) 0 0
\(214\) −15.2643 −1.04345
\(215\) 0.507117 0.0345851
\(216\) 0 0
\(217\) −3.31510 −0.225044
\(218\) −0.494378 −0.0334835
\(219\) 0 0
\(220\) 1.92990 0.130114
\(221\) −0.170588 −0.0114750
\(222\) 0 0
\(223\) −23.0654 −1.54457 −0.772287 0.635274i \(-0.780887\pi\)
−0.772287 + 0.635274i \(0.780887\pi\)
\(224\) 53.5273 3.57644
\(225\) 0 0
\(226\) 23.3586 1.55379
\(227\) −13.2378 −0.878621 −0.439311 0.898335i \(-0.644777\pi\)
−0.439311 + 0.898335i \(0.644777\pi\)
\(228\) 0 0
\(229\) 16.6610 1.10099 0.550496 0.834838i \(-0.314439\pi\)
0.550496 + 0.834838i \(0.314439\pi\)
\(230\) 0.230470 0.0151967
\(231\) 0 0
\(232\) 9.69318 0.636389
\(233\) 13.4436 0.880717 0.440359 0.897822i \(-0.354851\pi\)
0.440359 + 0.897822i \(0.354851\pi\)
\(234\) 0 0
\(235\) −0.0332660 −0.00217003
\(236\) 22.6233 1.47265
\(237\) 0 0
\(238\) 2.45293 0.159000
\(239\) 7.56856 0.489570 0.244785 0.969577i \(-0.421283\pi\)
0.244785 + 0.969577i \(0.421283\pi\)
\(240\) 0 0
\(241\) −18.2121 −1.17314 −0.586572 0.809897i \(-0.699523\pi\)
−0.586572 + 0.809897i \(0.699523\pi\)
\(242\) −17.1748 −1.10403
\(243\) 0 0
\(244\) 51.3639 3.28823
\(245\) 0.143037 0.00913829
\(246\) 0 0
\(247\) −2.32569 −0.147980
\(248\) −13.9624 −0.886613
\(249\) 0 0
\(250\) −2.30307 −0.145659
\(251\) −19.0091 −1.19984 −0.599921 0.800059i \(-0.704801\pi\)
−0.599921 + 0.800059i \(0.704801\pi\)
\(252\) 0 0
\(253\) −4.15428 −0.261177
\(254\) 45.5279 2.85668
\(255\) 0 0
\(256\) 53.5282 3.34551
\(257\) 22.6564 1.41327 0.706635 0.707579i \(-0.250212\pi\)
0.706635 + 0.707579i \(0.250212\pi\)
\(258\) 0 0
\(259\) −20.8141 −1.29333
\(260\) 0.204060 0.0126553
\(261\) 0 0
\(262\) −20.9155 −1.29216
\(263\) 0.221663 0.0136683 0.00683416 0.999977i \(-0.497825\pi\)
0.00683416 + 0.999977i \(0.497825\pi\)
\(264\) 0 0
\(265\) 0.204580 0.0125673
\(266\) 33.4418 2.05045
\(267\) 0 0
\(268\) 25.1891 1.53867
\(269\) −6.77287 −0.412949 −0.206475 0.978452i \(-0.566199\pi\)
−0.206475 + 0.978452i \(0.566199\pi\)
\(270\) 0 0
\(271\) 31.0475 1.88600 0.943001 0.332790i \(-0.107990\pi\)
0.943001 + 0.332790i \(0.107990\pi\)
\(272\) 6.03444 0.365892
\(273\) 0 0
\(274\) 25.0194 1.51148
\(275\) 20.7421 1.25080
\(276\) 0 0
\(277\) −15.8947 −0.955018 −0.477509 0.878627i \(-0.658460\pi\)
−0.477509 + 0.878627i \(0.658460\pi\)
\(278\) 0.0452231 0.00271230
\(279\) 0 0
\(280\) −1.87340 −0.111957
\(281\) −4.24811 −0.253421 −0.126711 0.991940i \(-0.540442\pi\)
−0.126711 + 0.991940i \(0.540442\pi\)
\(282\) 0 0
\(283\) 25.7398 1.53007 0.765036 0.643987i \(-0.222721\pi\)
0.765036 + 0.643987i \(0.222721\pi\)
\(284\) 21.3056 1.26425
\(285\) 0 0
\(286\) −5.00804 −0.296132
\(287\) −0.366007 −0.0216047
\(288\) 0 0
\(289\) −16.8492 −0.991128
\(290\) −0.230470 −0.0135337
\(291\) 0 0
\(292\) 0.0628148 0.00367596
\(293\) 6.52744 0.381337 0.190669 0.981654i \(-0.438934\pi\)
0.190669 + 0.981654i \(0.438934\pi\)
\(294\) 0 0
\(295\) −0.343431 −0.0199953
\(296\) −87.6639 −5.09536
\(297\) 0 0
\(298\) −42.1951 −2.44430
\(299\) −0.439258 −0.0254029
\(300\) 0 0
\(301\) 13.8980 0.801065
\(302\) 28.4371 1.63637
\(303\) 0 0
\(304\) 82.2699 4.71850
\(305\) −0.779725 −0.0446469
\(306\) 0 0
\(307\) 16.3947 0.935695 0.467847 0.883809i \(-0.345030\pi\)
0.467847 + 0.883809i \(0.345030\pi\)
\(308\) 52.8903 3.01371
\(309\) 0 0
\(310\) 0.331977 0.0188550
\(311\) 6.55243 0.371554 0.185777 0.982592i \(-0.440520\pi\)
0.185777 + 0.982592i \(0.440520\pi\)
\(312\) 0 0
\(313\) −25.1852 −1.42355 −0.711776 0.702406i \(-0.752109\pi\)
−0.711776 + 0.702406i \(0.752109\pi\)
\(314\) 0.875566 0.0494111
\(315\) 0 0
\(316\) 52.9689 2.97973
\(317\) −30.9737 −1.73965 −0.869827 0.493357i \(-0.835770\pi\)
−0.869827 + 0.493357i \(0.835770\pi\)
\(318\) 0 0
\(319\) 4.15428 0.232595
\(320\) −2.75051 −0.153758
\(321\) 0 0
\(322\) 6.31621 0.351989
\(323\) 2.05618 0.114409
\(324\) 0 0
\(325\) 2.19319 0.121656
\(326\) −51.0986 −2.83009
\(327\) 0 0
\(328\) −1.54153 −0.0851169
\(329\) −0.911681 −0.0502626
\(330\) 0 0
\(331\) 20.0846 1.10395 0.551973 0.833862i \(-0.313875\pi\)
0.551973 + 0.833862i \(0.313875\pi\)
\(332\) 30.7567 1.68799
\(333\) 0 0
\(334\) −55.0740 −3.01352
\(335\) −0.382381 −0.0208917
\(336\) 0 0
\(337\) −11.5444 −0.628864 −0.314432 0.949280i \(-0.601814\pi\)
−0.314432 + 0.949280i \(0.601814\pi\)
\(338\) 35.1482 1.91181
\(339\) 0 0
\(340\) −0.180413 −0.00978426
\(341\) −5.98396 −0.324050
\(342\) 0 0
\(343\) 20.0303 1.08153
\(344\) 58.5348 3.15598
\(345\) 0 0
\(346\) −40.0491 −2.15305
\(347\) −26.3029 −1.41202 −0.706008 0.708204i \(-0.749506\pi\)
−0.706008 + 0.708204i \(0.749506\pi\)
\(348\) 0 0
\(349\) −1.41214 −0.0755899 −0.0377949 0.999286i \(-0.512033\pi\)
−0.0377949 + 0.999286i \(0.512033\pi\)
\(350\) −31.5365 −1.68570
\(351\) 0 0
\(352\) 96.6201 5.14987
\(353\) 14.2825 0.760182 0.380091 0.924949i \(-0.375893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(354\) 0 0
\(355\) −0.323427 −0.0171657
\(356\) 22.2559 1.17956
\(357\) 0 0
\(358\) 57.0314 3.01420
\(359\) −12.7403 −0.672408 −0.336204 0.941789i \(-0.609143\pi\)
−0.336204 + 0.941789i \(0.609143\pi\)
\(360\) 0 0
\(361\) 9.03274 0.475407
\(362\) 12.2440 0.643532
\(363\) 0 0
\(364\) 5.59242 0.293123
\(365\) −0.000953555 0 −4.99113e−5 0
\(366\) 0 0
\(367\) 19.7406 1.03045 0.515224 0.857055i \(-0.327709\pi\)
0.515224 + 0.857055i \(0.327709\pi\)
\(368\) 15.5385 0.809998
\(369\) 0 0
\(370\) 2.08434 0.108360
\(371\) 5.60668 0.291085
\(372\) 0 0
\(373\) −12.0840 −0.625685 −0.312843 0.949805i \(-0.601281\pi\)
−0.312843 + 0.949805i \(0.601281\pi\)
\(374\) 4.42770 0.228951
\(375\) 0 0
\(376\) −3.83978 −0.198021
\(377\) 0.439258 0.0226229
\(378\) 0 0
\(379\) 9.89440 0.508241 0.254121 0.967173i \(-0.418214\pi\)
0.254121 + 0.967173i \(0.418214\pi\)
\(380\) −2.45964 −0.126177
\(381\) 0 0
\(382\) 45.2375 2.31455
\(383\) −14.5063 −0.741238 −0.370619 0.928785i \(-0.620854\pi\)
−0.370619 + 0.928785i \(0.620854\pi\)
\(384\) 0 0
\(385\) −0.802897 −0.0409194
\(386\) 16.5580 0.842780
\(387\) 0 0
\(388\) −107.600 −5.46258
\(389\) 32.8023 1.66314 0.831571 0.555418i \(-0.187442\pi\)
0.831571 + 0.555418i \(0.187442\pi\)
\(390\) 0 0
\(391\) 0.388355 0.0196400
\(392\) 16.5102 0.833893
\(393\) 0 0
\(394\) 50.5534 2.54685
\(395\) −0.804089 −0.0404581
\(396\) 0 0
\(397\) −27.4361 −1.37698 −0.688489 0.725247i \(-0.741726\pi\)
−0.688489 + 0.725247i \(0.741726\pi\)
\(398\) 62.5409 3.13489
\(399\) 0 0
\(400\) −77.5827 −3.87914
\(401\) 2.77335 0.138495 0.0692473 0.997600i \(-0.477940\pi\)
0.0692473 + 0.997600i \(0.477940\pi\)
\(402\) 0 0
\(403\) −0.632721 −0.0315181
\(404\) 78.0803 3.88464
\(405\) 0 0
\(406\) −6.31621 −0.313468
\(407\) −37.5707 −1.86231
\(408\) 0 0
\(409\) −3.98777 −0.197183 −0.0985914 0.995128i \(-0.531434\pi\)
−0.0985914 + 0.995128i \(0.531434\pi\)
\(410\) 0.0366522 0.00181012
\(411\) 0 0
\(412\) −33.4953 −1.65019
\(413\) −9.41200 −0.463134
\(414\) 0 0
\(415\) −0.466899 −0.0229192
\(416\) 10.2162 0.500892
\(417\) 0 0
\(418\) 60.3645 2.95252
\(419\) −26.5230 −1.29573 −0.647867 0.761753i \(-0.724339\pi\)
−0.647867 + 0.761753i \(0.724339\pi\)
\(420\) 0 0
\(421\) 34.1443 1.66409 0.832046 0.554706i \(-0.187169\pi\)
0.832046 + 0.554706i \(0.187169\pi\)
\(422\) −4.06573 −0.197917
\(423\) 0 0
\(424\) 23.6140 1.14680
\(425\) −1.93904 −0.0940571
\(426\) 0 0
\(427\) −21.3690 −1.03412
\(428\) 30.7681 1.48723
\(429\) 0 0
\(430\) −1.39175 −0.0671162
\(431\) −32.0693 −1.54472 −0.772362 0.635183i \(-0.780925\pi\)
−0.772362 + 0.635183i \(0.780925\pi\)
\(432\) 0 0
\(433\) −18.9769 −0.911971 −0.455985 0.889987i \(-0.650713\pi\)
−0.455985 + 0.889987i \(0.650713\pi\)
\(434\) 9.09809 0.436722
\(435\) 0 0
\(436\) 0.996514 0.0477244
\(437\) 5.29460 0.253275
\(438\) 0 0
\(439\) 32.8955 1.57002 0.785008 0.619485i \(-0.212659\pi\)
0.785008 + 0.619485i \(0.212659\pi\)
\(440\) −3.38160 −0.161212
\(441\) 0 0
\(442\) 0.468168 0.0222685
\(443\) −34.0029 −1.61553 −0.807763 0.589507i \(-0.799322\pi\)
−0.807763 + 0.589507i \(0.799322\pi\)
\(444\) 0 0
\(445\) −0.337853 −0.0160158
\(446\) 63.3015 2.99741
\(447\) 0 0
\(448\) −75.3800 −3.56137
\(449\) 13.9903 0.660244 0.330122 0.943938i \(-0.392910\pi\)
0.330122 + 0.943938i \(0.392910\pi\)
\(450\) 0 0
\(451\) −0.660665 −0.0311095
\(452\) −47.0838 −2.21463
\(453\) 0 0
\(454\) 36.3302 1.70506
\(455\) −0.0848953 −0.00397995
\(456\) 0 0
\(457\) −26.2419 −1.22755 −0.613773 0.789483i \(-0.710349\pi\)
−0.613773 + 0.789483i \(0.710349\pi\)
\(458\) −45.7251 −2.13660
\(459\) 0 0
\(460\) −0.464556 −0.0216600
\(461\) −4.16446 −0.193958 −0.0969792 0.995286i \(-0.530918\pi\)
−0.0969792 + 0.995286i \(0.530918\pi\)
\(462\) 0 0
\(463\) −24.8490 −1.15483 −0.577415 0.816451i \(-0.695939\pi\)
−0.577415 + 0.816451i \(0.695939\pi\)
\(464\) −15.5385 −0.721355
\(465\) 0 0
\(466\) −36.8950 −1.70913
\(467\) 2.92425 0.135318 0.0676590 0.997709i \(-0.478447\pi\)
0.0676590 + 0.997709i \(0.478447\pi\)
\(468\) 0 0
\(469\) −10.4794 −0.483896
\(470\) 0.0912964 0.00421119
\(471\) 0 0
\(472\) −39.6410 −1.82463
\(473\) 25.0867 1.15349
\(474\) 0 0
\(475\) −26.4356 −1.21295
\(476\) −4.94436 −0.226624
\(477\) 0 0
\(478\) −20.7714 −0.950064
\(479\) −6.71092 −0.306630 −0.153315 0.988177i \(-0.548995\pi\)
−0.153315 + 0.988177i \(0.548995\pi\)
\(480\) 0 0
\(481\) −3.97259 −0.181134
\(482\) 49.9820 2.27662
\(483\) 0 0
\(484\) 34.6190 1.57359
\(485\) 1.63342 0.0741697
\(486\) 0 0
\(487\) 15.1508 0.686550 0.343275 0.939235i \(-0.388464\pi\)
0.343275 + 0.939235i \(0.388464\pi\)
\(488\) −90.0009 −4.07415
\(489\) 0 0
\(490\) −0.392556 −0.0177339
\(491\) 14.6498 0.661136 0.330568 0.943782i \(-0.392760\pi\)
0.330568 + 0.943782i \(0.392760\pi\)
\(492\) 0 0
\(493\) −0.388355 −0.0174906
\(494\) 6.38271 0.287172
\(495\) 0 0
\(496\) 22.3821 1.00499
\(497\) −8.86378 −0.397595
\(498\) 0 0
\(499\) −12.3563 −0.553143 −0.276572 0.960993i \(-0.589198\pi\)
−0.276572 + 0.960993i \(0.589198\pi\)
\(500\) 4.64229 0.207609
\(501\) 0 0
\(502\) 52.1692 2.32843
\(503\) 15.4559 0.689144 0.344572 0.938760i \(-0.388024\pi\)
0.344572 + 0.938760i \(0.388024\pi\)
\(504\) 0 0
\(505\) −1.18529 −0.0527448
\(506\) 11.4012 0.506843
\(507\) 0 0
\(508\) −91.7702 −4.07164
\(509\) −38.3172 −1.69838 −0.849190 0.528087i \(-0.822909\pi\)
−0.849190 + 0.528087i \(0.822909\pi\)
\(510\) 0 0
\(511\) −0.0261329 −0.00115605
\(512\) −60.1588 −2.65867
\(513\) 0 0
\(514\) −62.1792 −2.74261
\(515\) 0.508472 0.0224059
\(516\) 0 0
\(517\) −1.64564 −0.0723752
\(518\) 57.1230 2.50984
\(519\) 0 0
\(520\) −0.357558 −0.0156800
\(521\) −33.0818 −1.44934 −0.724670 0.689096i \(-0.758008\pi\)
−0.724670 + 0.689096i \(0.758008\pi\)
\(522\) 0 0
\(523\) −33.4006 −1.46051 −0.730253 0.683176i \(-0.760598\pi\)
−0.730253 + 0.683176i \(0.760598\pi\)
\(524\) 42.1592 1.84173
\(525\) 0 0
\(526\) −0.608340 −0.0265249
\(527\) 0.559400 0.0243679
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −0.561458 −0.0243882
\(531\) 0 0
\(532\) −67.4083 −2.92252
\(533\) −0.0698563 −0.00302581
\(534\) 0 0
\(535\) −0.467072 −0.0201933
\(536\) −44.1369 −1.90642
\(537\) 0 0
\(538\) 18.5877 0.801374
\(539\) 7.07591 0.304781
\(540\) 0 0
\(541\) −10.7021 −0.460117 −0.230059 0.973177i \(-0.573892\pi\)
−0.230059 + 0.973177i \(0.573892\pi\)
\(542\) −85.2080 −3.66000
\(543\) 0 0
\(544\) −9.03236 −0.387259
\(545\) −0.0151275 −0.000647991 0
\(546\) 0 0
\(547\) −35.8489 −1.53279 −0.766395 0.642370i \(-0.777951\pi\)
−0.766395 + 0.642370i \(0.777951\pi\)
\(548\) −50.4314 −2.15432
\(549\) 0 0
\(550\) −56.9254 −2.42731
\(551\) −5.29460 −0.225557
\(552\) 0 0
\(553\) −22.0367 −0.937096
\(554\) 43.6219 1.85332
\(555\) 0 0
\(556\) −0.0911559 −0.00386587
\(557\) 25.0458 1.06123 0.530613 0.847614i \(-0.321962\pi\)
0.530613 + 0.847614i \(0.321962\pi\)
\(558\) 0 0
\(559\) 2.65257 0.112192
\(560\) 3.00312 0.126905
\(561\) 0 0
\(562\) 11.6587 0.491792
\(563\) 20.8013 0.876668 0.438334 0.898812i \(-0.355569\pi\)
0.438334 + 0.898812i \(0.355569\pi\)
\(564\) 0 0
\(565\) 0.714751 0.0300698
\(566\) −70.6413 −2.96927
\(567\) 0 0
\(568\) −37.3321 −1.56642
\(569\) −21.0178 −0.881111 −0.440556 0.897725i \(-0.645219\pi\)
−0.440556 + 0.897725i \(0.645219\pi\)
\(570\) 0 0
\(571\) −8.08678 −0.338421 −0.169211 0.985580i \(-0.554122\pi\)
−0.169211 + 0.985580i \(0.554122\pi\)
\(572\) 10.0947 0.422079
\(573\) 0 0
\(574\) 1.00448 0.0419263
\(575\) −4.99295 −0.208220
\(576\) 0 0
\(577\) −5.09269 −0.212012 −0.106006 0.994366i \(-0.533806\pi\)
−0.106006 + 0.994366i \(0.533806\pi\)
\(578\) 46.2415 1.92339
\(579\) 0 0
\(580\) 0.464556 0.0192896
\(581\) −12.7957 −0.530856
\(582\) 0 0
\(583\) 10.1204 0.419145
\(584\) −0.110065 −0.00455454
\(585\) 0 0
\(586\) −17.9142 −0.740027
\(587\) 21.0953 0.870696 0.435348 0.900262i \(-0.356625\pi\)
0.435348 + 0.900262i \(0.356625\pi\)
\(588\) 0 0
\(589\) 7.62651 0.314245
\(590\) 0.942525 0.0388031
\(591\) 0 0
\(592\) 140.528 5.77566
\(593\) −16.2516 −0.667372 −0.333686 0.942684i \(-0.608293\pi\)
−0.333686 + 0.942684i \(0.608293\pi\)
\(594\) 0 0
\(595\) 0.0750574 0.00307705
\(596\) 85.0523 3.48388
\(597\) 0 0
\(598\) 1.20552 0.0492972
\(599\) 22.2935 0.910889 0.455445 0.890264i \(-0.349480\pi\)
0.455445 + 0.890264i \(0.349480\pi\)
\(600\) 0 0
\(601\) 33.0543 1.34831 0.674156 0.738589i \(-0.264508\pi\)
0.674156 + 0.738589i \(0.264508\pi\)
\(602\) −38.1421 −1.55455
\(603\) 0 0
\(604\) −57.3204 −2.33234
\(605\) −0.525531 −0.0213659
\(606\) 0 0
\(607\) 15.8841 0.644716 0.322358 0.946618i \(-0.395525\pi\)
0.322358 + 0.946618i \(0.395525\pi\)
\(608\) −123.142 −4.99405
\(609\) 0 0
\(610\) 2.13991 0.0866423
\(611\) −0.174004 −0.00703944
\(612\) 0 0
\(613\) −13.7622 −0.555851 −0.277925 0.960603i \(-0.589647\pi\)
−0.277925 + 0.960603i \(0.589647\pi\)
\(614\) −44.9942 −1.81582
\(615\) 0 0
\(616\) −92.6756 −3.73401
\(617\) −19.1332 −0.770274 −0.385137 0.922859i \(-0.625846\pi\)
−0.385137 + 0.922859i \(0.625846\pi\)
\(618\) 0 0
\(619\) 18.0073 0.723776 0.361888 0.932222i \(-0.382132\pi\)
0.361888 + 0.932222i \(0.382132\pi\)
\(620\) −0.669162 −0.0268742
\(621\) 0 0
\(622\) −17.9827 −0.721042
\(623\) −9.25914 −0.370959
\(624\) 0 0
\(625\) 24.8943 0.995771
\(626\) 69.1193 2.76256
\(627\) 0 0
\(628\) −1.76487 −0.0704260
\(629\) 3.51223 0.140042
\(630\) 0 0
\(631\) 19.2913 0.767973 0.383987 0.923339i \(-0.374551\pi\)
0.383987 + 0.923339i \(0.374551\pi\)
\(632\) −92.8132 −3.69191
\(633\) 0 0
\(634\) 85.0053 3.37599
\(635\) 1.39311 0.0552839
\(636\) 0 0
\(637\) 0.748180 0.0296440
\(638\) −11.4012 −0.451376
\(639\) 0 0
\(640\) 3.64234 0.143976
\(641\) 20.8851 0.824913 0.412457 0.910977i \(-0.364671\pi\)
0.412457 + 0.910977i \(0.364671\pi\)
\(642\) 0 0
\(643\) 34.3708 1.35545 0.677725 0.735315i \(-0.262966\pi\)
0.677725 + 0.735315i \(0.262966\pi\)
\(644\) −12.7315 −0.501693
\(645\) 0 0
\(646\) −5.64307 −0.222024
\(647\) 2.96720 0.116653 0.0583263 0.998298i \(-0.481424\pi\)
0.0583263 + 0.998298i \(0.481424\pi\)
\(648\) 0 0
\(649\) −16.9892 −0.666886
\(650\) −6.01907 −0.236088
\(651\) 0 0
\(652\) 102.999 4.03375
\(653\) −47.8737 −1.87344 −0.936722 0.350074i \(-0.886156\pi\)
−0.936722 + 0.350074i \(0.886156\pi\)
\(654\) 0 0
\(655\) −0.639994 −0.0250066
\(656\) 2.47112 0.0964811
\(657\) 0 0
\(658\) 2.50205 0.0975401
\(659\) 7.75068 0.301923 0.150962 0.988540i \(-0.451763\pi\)
0.150962 + 0.988540i \(0.451763\pi\)
\(660\) 0 0
\(661\) −31.9075 −1.24106 −0.620529 0.784183i \(-0.713082\pi\)
−0.620529 + 0.784183i \(0.713082\pi\)
\(662\) −55.1208 −2.14233
\(663\) 0 0
\(664\) −53.8925 −2.09143
\(665\) 1.02329 0.0396813
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 111.012 4.29519
\(669\) 0 0
\(670\) 1.04942 0.0405426
\(671\) −38.5723 −1.48907
\(672\) 0 0
\(673\) −41.8507 −1.61323 −0.806613 0.591081i \(-0.798702\pi\)
−0.806613 + 0.591081i \(0.798702\pi\)
\(674\) 31.6829 1.22038
\(675\) 0 0
\(676\) −70.8478 −2.72492
\(677\) −45.1097 −1.73371 −0.866854 0.498563i \(-0.833861\pi\)
−0.866854 + 0.498563i \(0.833861\pi\)
\(678\) 0 0
\(679\) 44.7651 1.71793
\(680\) 0.316123 0.0121228
\(681\) 0 0
\(682\) 16.4226 0.628854
\(683\) 3.91287 0.149722 0.0748609 0.997194i \(-0.476149\pi\)
0.0748609 + 0.997194i \(0.476149\pi\)
\(684\) 0 0
\(685\) 0.765570 0.0292509
\(686\) −54.9718 −2.09883
\(687\) 0 0
\(688\) −93.8330 −3.57735
\(689\) 1.07009 0.0407673
\(690\) 0 0
\(691\) 12.2317 0.465317 0.232659 0.972558i \(-0.425258\pi\)
0.232659 + 0.972558i \(0.425258\pi\)
\(692\) 80.7265 3.06876
\(693\) 0 0
\(694\) 72.1868 2.74017
\(695\) 0.00138378 5.24899e−5 0
\(696\) 0 0
\(697\) 0.0617611 0.00233937
\(698\) 3.87552 0.146691
\(699\) 0 0
\(700\) 63.5679 2.40264
\(701\) −21.6681 −0.818391 −0.409195 0.912447i \(-0.634191\pi\)
−0.409195 + 0.912447i \(0.634191\pi\)
\(702\) 0 0
\(703\) 47.8836 1.80597
\(704\) −136.066 −5.12817
\(705\) 0 0
\(706\) −39.1975 −1.47522
\(707\) −32.4838 −1.22168
\(708\) 0 0
\(709\) −6.77011 −0.254257 −0.127128 0.991886i \(-0.540576\pi\)
−0.127128 + 0.991886i \(0.540576\pi\)
\(710\) 0.887625 0.0333120
\(711\) 0 0
\(712\) −38.9972 −1.46148
\(713\) 1.44043 0.0539447
\(714\) 0 0
\(715\) −0.153241 −0.00573090
\(716\) −114.958 −4.29617
\(717\) 0 0
\(718\) 34.9650 1.30488
\(719\) −36.9817 −1.37919 −0.689593 0.724197i \(-0.742211\pi\)
−0.689593 + 0.724197i \(0.742211\pi\)
\(720\) 0 0
\(721\) 13.9351 0.518969
\(722\) −24.7898 −0.922580
\(723\) 0 0
\(724\) −24.6802 −0.917231
\(725\) 4.99295 0.185433
\(726\) 0 0
\(727\) −11.5276 −0.427536 −0.213768 0.976884i \(-0.568574\pi\)
−0.213768 + 0.976884i \(0.568574\pi\)
\(728\) −9.79917 −0.363181
\(729\) 0 0
\(730\) 0.00261697 9.68585e−5 0
\(731\) −2.34518 −0.0867397
\(732\) 0 0
\(733\) −34.5558 −1.27635 −0.638174 0.769892i \(-0.720310\pi\)
−0.638174 + 0.769892i \(0.720310\pi\)
\(734\) −54.1767 −1.99970
\(735\) 0 0
\(736\) −23.2580 −0.857300
\(737\) −18.9160 −0.696782
\(738\) 0 0
\(739\) −11.0597 −0.406838 −0.203419 0.979092i \(-0.565205\pi\)
−0.203419 + 0.979092i \(0.565205\pi\)
\(740\) −4.20139 −0.154446
\(741\) 0 0
\(742\) −15.3872 −0.564882
\(743\) −37.7977 −1.38666 −0.693331 0.720619i \(-0.743858\pi\)
−0.693331 + 0.720619i \(0.743858\pi\)
\(744\) 0 0
\(745\) −1.29113 −0.0473033
\(746\) 33.1638 1.21421
\(747\) 0 0
\(748\) −8.92487 −0.326326
\(749\) −12.8005 −0.467720
\(750\) 0 0
\(751\) −34.4216 −1.25606 −0.628032 0.778188i \(-0.716139\pi\)
−0.628032 + 0.778188i \(0.716139\pi\)
\(752\) 6.15527 0.224460
\(753\) 0 0
\(754\) −1.20552 −0.0439023
\(755\) 0.870148 0.0316679
\(756\) 0 0
\(757\) −31.9962 −1.16292 −0.581461 0.813574i \(-0.697519\pi\)
−0.581461 + 0.813574i \(0.697519\pi\)
\(758\) −27.1546 −0.986298
\(759\) 0 0
\(760\) 4.30983 0.156334
\(761\) 14.0910 0.510800 0.255400 0.966836i \(-0.417793\pi\)
0.255400 + 0.966836i \(0.417793\pi\)
\(762\) 0 0
\(763\) −0.414581 −0.0150088
\(764\) −91.1848 −3.29895
\(765\) 0 0
\(766\) 39.8117 1.43845
\(767\) −1.79638 −0.0648635
\(768\) 0 0
\(769\) 41.1134 1.48259 0.741294 0.671181i \(-0.234213\pi\)
0.741294 + 0.671181i \(0.234213\pi\)
\(770\) 2.20350 0.0794086
\(771\) 0 0
\(772\) −33.3758 −1.20122
\(773\) 21.0358 0.756604 0.378302 0.925682i \(-0.376508\pi\)
0.378302 + 0.925682i \(0.376508\pi\)
\(774\) 0 0
\(775\) −7.19201 −0.258345
\(776\) 188.540 6.76818
\(777\) 0 0
\(778\) −90.0239 −3.22751
\(779\) 0.842013 0.0301682
\(780\) 0 0
\(781\) −15.9997 −0.572513
\(782\) −1.06582 −0.0381135
\(783\) 0 0
\(784\) −26.4664 −0.945229
\(785\) 0.0267915 0.000956229 0
\(786\) 0 0
\(787\) 25.6134 0.913019 0.456510 0.889719i \(-0.349099\pi\)
0.456510 + 0.889719i \(0.349099\pi\)
\(788\) −101.900 −3.63004
\(789\) 0 0
\(790\) 2.20677 0.0785134
\(791\) 19.5883 0.696481
\(792\) 0 0
\(793\) −4.07849 −0.144832
\(794\) 75.2967 2.67218
\(795\) 0 0
\(796\) −126.063 −4.46819
\(797\) −32.8527 −1.16370 −0.581851 0.813296i \(-0.697671\pi\)
−0.581851 + 0.813296i \(0.697671\pi\)
\(798\) 0 0
\(799\) 0.153840 0.00544246
\(800\) 116.126 4.10567
\(801\) 0 0
\(802\) −7.61130 −0.268764
\(803\) −0.0471716 −0.00166465
\(804\) 0 0
\(805\) 0.193270 0.00681187
\(806\) 1.73646 0.0611644
\(807\) 0 0
\(808\) −136.814 −4.81310
\(809\) −9.76643 −0.343369 −0.171685 0.985152i \(-0.554921\pi\)
−0.171685 + 0.985152i \(0.554921\pi\)
\(810\) 0 0
\(811\) 35.6839 1.25303 0.626516 0.779408i \(-0.284480\pi\)
0.626516 + 0.779408i \(0.284480\pi\)
\(812\) 12.7315 0.446789
\(813\) 0 0
\(814\) 103.111 3.61402
\(815\) −1.56357 −0.0547694
\(816\) 0 0
\(817\) −31.9728 −1.11859
\(818\) 10.9442 0.382655
\(819\) 0 0
\(820\) −0.0738796 −0.00257999
\(821\) 49.2275 1.71805 0.859026 0.511931i \(-0.171070\pi\)
0.859026 + 0.511931i \(0.171070\pi\)
\(822\) 0 0
\(823\) 30.5680 1.06553 0.532766 0.846263i \(-0.321153\pi\)
0.532766 + 0.846263i \(0.321153\pi\)
\(824\) 58.6911 2.04460
\(825\) 0 0
\(826\) 25.8307 0.898763
\(827\) −26.6441 −0.926505 −0.463253 0.886226i \(-0.653318\pi\)
−0.463253 + 0.886226i \(0.653318\pi\)
\(828\) 0 0
\(829\) 15.9172 0.552828 0.276414 0.961039i \(-0.410854\pi\)
0.276414 + 0.961039i \(0.410854\pi\)
\(830\) 1.28137 0.0444772
\(831\) 0 0
\(832\) −14.3871 −0.498782
\(833\) −0.661479 −0.0229189
\(834\) 0 0
\(835\) −1.68521 −0.0583191
\(836\) −121.676 −4.20826
\(837\) 0 0
\(838\) 72.7908 2.51452
\(839\) −17.3019 −0.597328 −0.298664 0.954358i \(-0.596541\pi\)
−0.298664 + 0.954358i \(0.596541\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −93.7070 −3.22936
\(843\) 0 0
\(844\) 8.19526 0.282093
\(845\) 1.07550 0.0369983
\(846\) 0 0
\(847\) −14.4026 −0.494879
\(848\) −37.8539 −1.29991
\(849\) 0 0
\(850\) 5.32157 0.182528
\(851\) 9.04387 0.310020
\(852\) 0 0
\(853\) 49.6636 1.70045 0.850224 0.526421i \(-0.176466\pi\)
0.850224 + 0.526421i \(0.176466\pi\)
\(854\) 58.6458 2.00682
\(855\) 0 0
\(856\) −53.9125 −1.84269
\(857\) −48.7452 −1.66511 −0.832553 0.553946i \(-0.813121\pi\)
−0.832553 + 0.553946i \(0.813121\pi\)
\(858\) 0 0
\(859\) −27.2365 −0.929298 −0.464649 0.885495i \(-0.653820\pi\)
−0.464649 + 0.885495i \(0.653820\pi\)
\(860\) 2.80534 0.0956614
\(861\) 0 0
\(862\) 88.0122 2.99771
\(863\) −5.96572 −0.203076 −0.101538 0.994832i \(-0.532376\pi\)
−0.101538 + 0.994832i \(0.532376\pi\)
\(864\) 0 0
\(865\) −1.22546 −0.0416670
\(866\) 52.0809 1.76978
\(867\) 0 0
\(868\) −18.3389 −0.622464
\(869\) −39.7776 −1.34936
\(870\) 0 0
\(871\) −2.00011 −0.0677712
\(872\) −1.74611 −0.0591309
\(873\) 0 0
\(874\) −14.5307 −0.491508
\(875\) −1.93134 −0.0652911
\(876\) 0 0
\(877\) 38.5251 1.30090 0.650450 0.759549i \(-0.274580\pi\)
0.650450 + 0.759549i \(0.274580\pi\)
\(878\) −90.2797 −3.04679
\(879\) 0 0
\(880\) 5.42081 0.182736
\(881\) 1.78900 0.0602730 0.0301365 0.999546i \(-0.490406\pi\)
0.0301365 + 0.999546i \(0.490406\pi\)
\(882\) 0 0
\(883\) −44.6128 −1.50134 −0.750669 0.660678i \(-0.770269\pi\)
−0.750669 + 0.660678i \(0.770269\pi\)
\(884\) −0.943682 −0.0317395
\(885\) 0 0
\(886\) 93.3189 3.13511
\(887\) −14.6813 −0.492950 −0.246475 0.969149i \(-0.579272\pi\)
−0.246475 + 0.969149i \(0.579272\pi\)
\(888\) 0 0
\(889\) 38.1793 1.28049
\(890\) 0.927217 0.0310804
\(891\) 0 0
\(892\) −127.596 −4.27224
\(893\) 2.09736 0.0701853
\(894\) 0 0
\(895\) 1.74510 0.0583324
\(896\) 99.8212 3.33479
\(897\) 0 0
\(898\) −38.3956 −1.28128
\(899\) −1.44043 −0.0480412
\(900\) 0 0
\(901\) −0.946089 −0.0315188
\(902\) 1.81316 0.0603715
\(903\) 0 0
\(904\) 82.5012 2.74395
\(905\) 0.374655 0.0124540
\(906\) 0 0
\(907\) −29.0029 −0.963027 −0.481513 0.876439i \(-0.659913\pi\)
−0.481513 + 0.876439i \(0.659913\pi\)
\(908\) −73.2305 −2.43024
\(909\) 0 0
\(910\) 0.232990 0.00772354
\(911\) 32.3586 1.07209 0.536044 0.844190i \(-0.319918\pi\)
0.536044 + 0.844190i \(0.319918\pi\)
\(912\) 0 0
\(913\) −23.0971 −0.764402
\(914\) 72.0193 2.38219
\(915\) 0 0
\(916\) 92.1678 3.04531
\(917\) −17.5395 −0.579207
\(918\) 0 0
\(919\) 51.9475 1.71359 0.856795 0.515656i \(-0.172452\pi\)
0.856795 + 0.515656i \(0.172452\pi\)
\(920\) 0.814006 0.0268370
\(921\) 0 0
\(922\) 11.4291 0.376398
\(923\) −1.69174 −0.0556844
\(924\) 0 0
\(925\) −45.1556 −1.48471
\(926\) 68.1965 2.24108
\(927\) 0 0
\(928\) 23.2580 0.763480
\(929\) −13.1004 −0.429809 −0.214904 0.976635i \(-0.568944\pi\)
−0.214904 + 0.976635i \(0.568944\pi\)
\(930\) 0 0
\(931\) −9.01820 −0.295559
\(932\) 74.3690 2.43604
\(933\) 0 0
\(934\) −8.02541 −0.262599
\(935\) 0.135483 0.00443078
\(936\) 0 0
\(937\) 13.1382 0.429205 0.214602 0.976701i \(-0.431154\pi\)
0.214602 + 0.976701i \(0.431154\pi\)
\(938\) 28.7602 0.939053
\(939\) 0 0
\(940\) −0.184025 −0.00600224
\(941\) −34.5161 −1.12519 −0.562596 0.826732i \(-0.690197\pi\)
−0.562596 + 0.826732i \(0.690197\pi\)
\(942\) 0 0
\(943\) 0.159033 0.00517881
\(944\) 63.5458 2.06824
\(945\) 0 0
\(946\) −68.8488 −2.23847
\(947\) −1.49916 −0.0487161 −0.0243581 0.999703i \(-0.507754\pi\)
−0.0243581 + 0.999703i \(0.507754\pi\)
\(948\) 0 0
\(949\) −0.00498774 −0.000161909 0
\(950\) 72.5510 2.35386
\(951\) 0 0
\(952\) 8.66361 0.280789
\(953\) 34.1120 1.10499 0.552497 0.833515i \(-0.313675\pi\)
0.552497 + 0.833515i \(0.313675\pi\)
\(954\) 0 0
\(955\) 1.38422 0.0447924
\(956\) 41.8688 1.35413
\(957\) 0 0
\(958\) 18.4177 0.595049
\(959\) 20.9811 0.677514
\(960\) 0 0
\(961\) −28.9252 −0.933069
\(962\) 10.9025 0.351511
\(963\) 0 0
\(964\) −100.748 −3.24488
\(965\) 0.506658 0.0163099
\(966\) 0 0
\(967\) 5.38589 0.173199 0.0865993 0.996243i \(-0.472400\pi\)
0.0865993 + 0.996243i \(0.472400\pi\)
\(968\) −60.6602 −1.94969
\(969\) 0 0
\(970\) −4.48281 −0.143935
\(971\) 28.2316 0.905994 0.452997 0.891512i \(-0.350355\pi\)
0.452997 + 0.891512i \(0.350355\pi\)
\(972\) 0 0
\(973\) 0.0379237 0.00121578
\(974\) −41.5805 −1.33233
\(975\) 0 0
\(976\) 144.274 4.61810
\(977\) −44.7588 −1.43196 −0.715981 0.698120i \(-0.754020\pi\)
−0.715981 + 0.698120i \(0.754020\pi\)
\(978\) 0 0
\(979\) −16.7133 −0.534160
\(980\) 0.791271 0.0252762
\(981\) 0 0
\(982\) −40.2054 −1.28301
\(983\) −23.6003 −0.752732 −0.376366 0.926471i \(-0.622827\pi\)
−0.376366 + 0.926471i \(0.622827\pi\)
\(984\) 0 0
\(985\) 1.54689 0.0492879
\(986\) 1.06582 0.0339425
\(987\) 0 0
\(988\) −12.8656 −0.409309
\(989\) −6.03876 −0.192021
\(990\) 0 0
\(991\) −51.8865 −1.64823 −0.824114 0.566423i \(-0.808327\pi\)
−0.824114 + 0.566423i \(0.808327\pi\)
\(992\) −33.5016 −1.06368
\(993\) 0 0
\(994\) 24.3261 0.771577
\(995\) 1.91369 0.0606681
\(996\) 0 0
\(997\) −14.0976 −0.446476 −0.223238 0.974764i \(-0.571663\pi\)
−0.223238 + 0.974764i \(0.571663\pi\)
\(998\) 33.9111 1.07344
\(999\) 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 6003.2.a.k.1.1 10
3.2 odd 2 2001.2.a.k.1.10 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.k.1.10 10 3.2 odd 2
6003.2.a.k.1.1 10 1.1 even 1 trivial