Properties

Label 6039.2.a.o.1.19
Level $6039$
Weight $2$
Character 6039.1
Self dual yes
Analytic conductor $48.222$
Analytic rank $0$
Dimension $25$
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] = [6039,2,Mod(1,6039)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6039, 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("6039.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.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: \(48.2216577807\)
Analytic rank: \(0\)
Dimension: \(25\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.19
Character \(\chi\) \(=\) 6039.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.77462 q^{2} +1.14926 q^{4} -1.38341 q^{5} -3.00047 q^{7} -1.50973 q^{8} +O(q^{10})\) \(q+1.77462 q^{2} +1.14926 q^{4} -1.38341 q^{5} -3.00047 q^{7} -1.50973 q^{8} -2.45503 q^{10} +1.00000 q^{11} +1.91001 q^{13} -5.32467 q^{14} -4.97772 q^{16} -5.53610 q^{17} +3.26949 q^{19} -1.58991 q^{20} +1.77462 q^{22} +1.92659 q^{23} -3.08616 q^{25} +3.38954 q^{26} -3.44832 q^{28} +5.32737 q^{29} +6.87230 q^{31} -5.81408 q^{32} -9.82445 q^{34} +4.15089 q^{35} -3.70575 q^{37} +5.80210 q^{38} +2.08859 q^{40} +9.75741 q^{41} -8.90873 q^{43} +1.14926 q^{44} +3.41896 q^{46} -1.73547 q^{47} +2.00280 q^{49} -5.47676 q^{50} +2.19511 q^{52} +3.45951 q^{53} -1.38341 q^{55} +4.52990 q^{56} +9.45404 q^{58} -13.9936 q^{59} -1.00000 q^{61} +12.1957 q^{62} -0.362312 q^{64} -2.64234 q^{65} +1.30997 q^{67} -6.36243 q^{68} +7.36623 q^{70} +14.4465 q^{71} +5.37902 q^{73} -6.57629 q^{74} +3.75751 q^{76} -3.00047 q^{77} +14.7016 q^{79} +6.88625 q^{80} +17.3157 q^{82} -5.37220 q^{83} +7.65872 q^{85} -15.8096 q^{86} -1.50973 q^{88} +5.99196 q^{89} -5.73093 q^{91} +2.21416 q^{92} -3.07979 q^{94} -4.52307 q^{95} +18.5252 q^{97} +3.55419 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8} + 25 q^{11} + 4 q^{13} + 18 q^{14} + 21 q^{16} + 20 q^{17} + 14 q^{19} + 12 q^{20} + 5 q^{22} + 20 q^{23} + 13 q^{25} + 16 q^{26} - 14 q^{28} + 28 q^{29} - 12 q^{31} + 35 q^{32} + 6 q^{34} + 10 q^{35} - 8 q^{37} + 32 q^{38} + 24 q^{40} + 26 q^{41} + 18 q^{43} + 25 q^{44} + 4 q^{46} + 12 q^{47} + 23 q^{49} + 43 q^{50} + 22 q^{52} + 36 q^{53} + 4 q^{55} + 26 q^{56} - 20 q^{58} + 46 q^{59} - 25 q^{61} - 14 q^{62} - 13 q^{64} + 60 q^{65} - 20 q^{67} + 44 q^{68} - 20 q^{70} + 52 q^{71} + 6 q^{73} + 32 q^{74} + 4 q^{77} + 26 q^{79} + 52 q^{80} + 6 q^{82} + 38 q^{83} - 4 q^{85} + 34 q^{86} + 15 q^{88} + 82 q^{89} - 58 q^{91} + 36 q^{92} + 16 q^{94} + 30 q^{95} + 35 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.77462 1.25484 0.627422 0.778680i \(-0.284110\pi\)
0.627422 + 0.778680i \(0.284110\pi\)
\(3\) 0 0
\(4\) 1.14926 0.574631
\(5\) −1.38341 −0.618682 −0.309341 0.950951i \(-0.600108\pi\)
−0.309341 + 0.950951i \(0.600108\pi\)
\(6\) 0 0
\(7\) −3.00047 −1.13407 −0.567035 0.823694i \(-0.691909\pi\)
−0.567035 + 0.823694i \(0.691909\pi\)
\(8\) −1.50973 −0.533771
\(9\) 0 0
\(10\) −2.45503 −0.776349
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 1.91001 0.529743 0.264871 0.964284i \(-0.414671\pi\)
0.264871 + 0.964284i \(0.414671\pi\)
\(14\) −5.32467 −1.42308
\(15\) 0 0
\(16\) −4.97772 −1.24443
\(17\) −5.53610 −1.34270 −0.671351 0.741140i \(-0.734286\pi\)
−0.671351 + 0.741140i \(0.734286\pi\)
\(18\) 0 0
\(19\) 3.26949 0.750074 0.375037 0.927010i \(-0.377630\pi\)
0.375037 + 0.927010i \(0.377630\pi\)
\(20\) −1.58991 −0.355514
\(21\) 0 0
\(22\) 1.77462 0.378349
\(23\) 1.92659 0.401722 0.200861 0.979620i \(-0.435626\pi\)
0.200861 + 0.979620i \(0.435626\pi\)
\(24\) 0 0
\(25\) −3.08616 −0.617233
\(26\) 3.38954 0.664744
\(27\) 0 0
\(28\) −3.44832 −0.651672
\(29\) 5.32737 0.989269 0.494634 0.869101i \(-0.335302\pi\)
0.494634 + 0.869101i \(0.335302\pi\)
\(30\) 0 0
\(31\) 6.87230 1.23430 0.617151 0.786845i \(-0.288287\pi\)
0.617151 + 0.786845i \(0.288287\pi\)
\(32\) −5.81408 −1.02779
\(33\) 0 0
\(34\) −9.82445 −1.68488
\(35\) 4.15089 0.701628
\(36\) 0 0
\(37\) −3.70575 −0.609222 −0.304611 0.952477i \(-0.598527\pi\)
−0.304611 + 0.952477i \(0.598527\pi\)
\(38\) 5.80210 0.941225
\(39\) 0 0
\(40\) 2.08859 0.330235
\(41\) 9.75741 1.52385 0.761926 0.647664i \(-0.224254\pi\)
0.761926 + 0.647664i \(0.224254\pi\)
\(42\) 0 0
\(43\) −8.90873 −1.35857 −0.679285 0.733875i \(-0.737710\pi\)
−0.679285 + 0.733875i \(0.737710\pi\)
\(44\) 1.14926 0.173258
\(45\) 0 0
\(46\) 3.41896 0.504098
\(47\) −1.73547 −0.253144 −0.126572 0.991957i \(-0.540398\pi\)
−0.126572 + 0.991957i \(0.540398\pi\)
\(48\) 0 0
\(49\) 2.00280 0.286114
\(50\) −5.47676 −0.774530
\(51\) 0 0
\(52\) 2.19511 0.304407
\(53\) 3.45951 0.475200 0.237600 0.971363i \(-0.423639\pi\)
0.237600 + 0.971363i \(0.423639\pi\)
\(54\) 0 0
\(55\) −1.38341 −0.186540
\(56\) 4.52990 0.605334
\(57\) 0 0
\(58\) 9.45404 1.24138
\(59\) −13.9936 −1.82182 −0.910909 0.412608i \(-0.864618\pi\)
−0.910909 + 0.412608i \(0.864618\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037
\(62\) 12.1957 1.54886
\(63\) 0 0
\(64\) −0.362312 −0.0452890
\(65\) −2.64234 −0.327742
\(66\) 0 0
\(67\) 1.30997 0.160038 0.0800189 0.996793i \(-0.474502\pi\)
0.0800189 + 0.996793i \(0.474502\pi\)
\(68\) −6.36243 −0.771558
\(69\) 0 0
\(70\) 7.36623 0.880433
\(71\) 14.4465 1.71449 0.857244 0.514911i \(-0.172175\pi\)
0.857244 + 0.514911i \(0.172175\pi\)
\(72\) 0 0
\(73\) 5.37902 0.629567 0.314783 0.949164i \(-0.398068\pi\)
0.314783 + 0.949164i \(0.398068\pi\)
\(74\) −6.57629 −0.764478
\(75\) 0 0
\(76\) 3.75751 0.431016
\(77\) −3.00047 −0.341935
\(78\) 0 0
\(79\) 14.7016 1.65406 0.827029 0.562159i \(-0.190029\pi\)
0.827029 + 0.562159i \(0.190029\pi\)
\(80\) 6.88625 0.769906
\(81\) 0 0
\(82\) 17.3157 1.91219
\(83\) −5.37220 −0.589676 −0.294838 0.955547i \(-0.595266\pi\)
−0.294838 + 0.955547i \(0.595266\pi\)
\(84\) 0 0
\(85\) 7.65872 0.830705
\(86\) −15.8096 −1.70479
\(87\) 0 0
\(88\) −1.50973 −0.160938
\(89\) 5.99196 0.635146 0.317573 0.948234i \(-0.397132\pi\)
0.317573 + 0.948234i \(0.397132\pi\)
\(90\) 0 0
\(91\) −5.73093 −0.600765
\(92\) 2.21416 0.230842
\(93\) 0 0
\(94\) −3.07979 −0.317657
\(95\) −4.52307 −0.464057
\(96\) 0 0
\(97\) 18.5252 1.88095 0.940473 0.339867i \(-0.110382\pi\)
0.940473 + 0.339867i \(0.110382\pi\)
\(98\) 3.55419 0.359028
\(99\) 0 0
\(100\) −3.54681 −0.354681
\(101\) 10.5775 1.05251 0.526253 0.850328i \(-0.323597\pi\)
0.526253 + 0.850328i \(0.323597\pi\)
\(102\) 0 0
\(103\) −0.331727 −0.0326861 −0.0163430 0.999866i \(-0.505202\pi\)
−0.0163430 + 0.999866i \(0.505202\pi\)
\(104\) −2.88361 −0.282761
\(105\) 0 0
\(106\) 6.13930 0.596302
\(107\) 20.2776 1.96031 0.980153 0.198240i \(-0.0635226\pi\)
0.980153 + 0.198240i \(0.0635226\pi\)
\(108\) 0 0
\(109\) 1.79571 0.171998 0.0859991 0.996295i \(-0.472592\pi\)
0.0859991 + 0.996295i \(0.472592\pi\)
\(110\) −2.45503 −0.234078
\(111\) 0 0
\(112\) 14.9355 1.41127
\(113\) 12.2151 1.14910 0.574548 0.818471i \(-0.305178\pi\)
0.574548 + 0.818471i \(0.305178\pi\)
\(114\) 0 0
\(115\) −2.66527 −0.248538
\(116\) 6.12255 0.568464
\(117\) 0 0
\(118\) −24.8333 −2.28610
\(119\) 16.6109 1.52272
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −1.77462 −0.160666
\(123\) 0 0
\(124\) 7.89808 0.709268
\(125\) 11.1865 1.00055
\(126\) 0 0
\(127\) −1.31827 −0.116978 −0.0584888 0.998288i \(-0.518628\pi\)
−0.0584888 + 0.998288i \(0.518628\pi\)
\(128\) 10.9852 0.970963
\(129\) 0 0
\(130\) −4.68914 −0.411265
\(131\) −4.60086 −0.401979 −0.200990 0.979593i \(-0.564416\pi\)
−0.200990 + 0.979593i \(0.564416\pi\)
\(132\) 0 0
\(133\) −9.81001 −0.850636
\(134\) 2.32469 0.200822
\(135\) 0 0
\(136\) 8.35803 0.716696
\(137\) 1.93120 0.164994 0.0824969 0.996591i \(-0.473711\pi\)
0.0824969 + 0.996591i \(0.473711\pi\)
\(138\) 0 0
\(139\) 0.524643 0.0444996 0.0222498 0.999752i \(-0.492917\pi\)
0.0222498 + 0.999752i \(0.492917\pi\)
\(140\) 4.77046 0.403177
\(141\) 0 0
\(142\) 25.6370 2.15141
\(143\) 1.91001 0.159723
\(144\) 0 0
\(145\) −7.36997 −0.612042
\(146\) 9.54569 0.790007
\(147\) 0 0
\(148\) −4.25888 −0.350078
\(149\) −20.5293 −1.68182 −0.840912 0.541171i \(-0.817981\pi\)
−0.840912 + 0.541171i \(0.817981\pi\)
\(150\) 0 0
\(151\) 16.2387 1.32149 0.660743 0.750612i \(-0.270241\pi\)
0.660743 + 0.750612i \(0.270241\pi\)
\(152\) −4.93606 −0.400368
\(153\) 0 0
\(154\) −5.32467 −0.429075
\(155\) −9.50724 −0.763640
\(156\) 0 0
\(157\) 11.0781 0.884127 0.442063 0.896984i \(-0.354247\pi\)
0.442063 + 0.896984i \(0.354247\pi\)
\(158\) 26.0897 2.07558
\(159\) 0 0
\(160\) 8.04328 0.635877
\(161\) −5.78067 −0.455580
\(162\) 0 0
\(163\) 0.840249 0.0658134 0.0329067 0.999458i \(-0.489524\pi\)
0.0329067 + 0.999458i \(0.489524\pi\)
\(164\) 11.2138 0.875652
\(165\) 0 0
\(166\) −9.53359 −0.739950
\(167\) 2.15956 0.167112 0.0835559 0.996503i \(-0.473372\pi\)
0.0835559 + 0.996503i \(0.473372\pi\)
\(168\) 0 0
\(169\) −9.35184 −0.719373
\(170\) 13.5913 1.04240
\(171\) 0 0
\(172\) −10.2385 −0.780676
\(173\) 8.83539 0.671742 0.335871 0.941908i \(-0.390969\pi\)
0.335871 + 0.941908i \(0.390969\pi\)
\(174\) 0 0
\(175\) 9.25993 0.699985
\(176\) −4.97772 −0.375210
\(177\) 0 0
\(178\) 10.6334 0.797008
\(179\) −2.24198 −0.167574 −0.0837868 0.996484i \(-0.526701\pi\)
−0.0837868 + 0.996484i \(0.526701\pi\)
\(180\) 0 0
\(181\) −3.43057 −0.254992 −0.127496 0.991839i \(-0.540694\pi\)
−0.127496 + 0.991839i \(0.540694\pi\)
\(182\) −10.1702 −0.753866
\(183\) 0 0
\(184\) −2.90864 −0.214428
\(185\) 5.12659 0.376915
\(186\) 0 0
\(187\) −5.53610 −0.404840
\(188\) −1.99451 −0.145465
\(189\) 0 0
\(190\) −8.02671 −0.582319
\(191\) 11.0429 0.799036 0.399518 0.916725i \(-0.369177\pi\)
0.399518 + 0.916725i \(0.369177\pi\)
\(192\) 0 0
\(193\) −10.8007 −0.777453 −0.388726 0.921353i \(-0.627085\pi\)
−0.388726 + 0.921353i \(0.627085\pi\)
\(194\) 32.8751 2.36029
\(195\) 0 0
\(196\) 2.30174 0.164410
\(197\) 2.66347 0.189764 0.0948822 0.995489i \(-0.469753\pi\)
0.0948822 + 0.995489i \(0.469753\pi\)
\(198\) 0 0
\(199\) −7.50042 −0.531691 −0.265845 0.964016i \(-0.585651\pi\)
−0.265845 + 0.964016i \(0.585651\pi\)
\(200\) 4.65928 0.329461
\(201\) 0 0
\(202\) 18.7711 1.32073
\(203\) −15.9846 −1.12190
\(204\) 0 0
\(205\) −13.4985 −0.942779
\(206\) −0.588689 −0.0410159
\(207\) 0 0
\(208\) −9.50752 −0.659228
\(209\) 3.26949 0.226156
\(210\) 0 0
\(211\) −16.6852 −1.14866 −0.574328 0.818625i \(-0.694737\pi\)
−0.574328 + 0.818625i \(0.694737\pi\)
\(212\) 3.97588 0.273065
\(213\) 0 0
\(214\) 35.9849 2.45988
\(215\) 12.3245 0.840522
\(216\) 0 0
\(217\) −20.6201 −1.39978
\(218\) 3.18670 0.215831
\(219\) 0 0
\(220\) −1.58991 −0.107191
\(221\) −10.5740 −0.711286
\(222\) 0 0
\(223\) −13.5951 −0.910397 −0.455198 0.890390i \(-0.650432\pi\)
−0.455198 + 0.890390i \(0.650432\pi\)
\(224\) 17.4449 1.16559
\(225\) 0 0
\(226\) 21.6770 1.44194
\(227\) 0.528432 0.0350733 0.0175366 0.999846i \(-0.494418\pi\)
0.0175366 + 0.999846i \(0.494418\pi\)
\(228\) 0 0
\(229\) 2.67417 0.176714 0.0883571 0.996089i \(-0.471838\pi\)
0.0883571 + 0.996089i \(0.471838\pi\)
\(230\) −4.72984 −0.311876
\(231\) 0 0
\(232\) −8.04291 −0.528043
\(233\) −27.0368 −1.77124 −0.885621 0.464408i \(-0.846267\pi\)
−0.885621 + 0.464408i \(0.846267\pi\)
\(234\) 0 0
\(235\) 2.40088 0.156616
\(236\) −16.0824 −1.04687
\(237\) 0 0
\(238\) 29.4779 1.91077
\(239\) 12.4532 0.805533 0.402767 0.915303i \(-0.368049\pi\)
0.402767 + 0.915303i \(0.368049\pi\)
\(240\) 0 0
\(241\) −8.41293 −0.541924 −0.270962 0.962590i \(-0.587342\pi\)
−0.270962 + 0.962590i \(0.587342\pi\)
\(242\) 1.77462 0.114077
\(243\) 0 0
\(244\) −1.14926 −0.0735740
\(245\) −2.77070 −0.177013
\(246\) 0 0
\(247\) 6.24478 0.397346
\(248\) −10.3753 −0.658835
\(249\) 0 0
\(250\) 19.8518 1.25554
\(251\) 15.1182 0.954254 0.477127 0.878834i \(-0.341678\pi\)
0.477127 + 0.878834i \(0.341678\pi\)
\(252\) 0 0
\(253\) 1.92659 0.121124
\(254\) −2.33943 −0.146789
\(255\) 0 0
\(256\) 20.2191 1.26369
\(257\) 18.1387 1.13146 0.565732 0.824589i \(-0.308594\pi\)
0.565732 + 0.824589i \(0.308594\pi\)
\(258\) 0 0
\(259\) 11.1190 0.690900
\(260\) −3.03674 −0.188331
\(261\) 0 0
\(262\) −8.16477 −0.504421
\(263\) −11.1417 −0.687027 −0.343514 0.939148i \(-0.611617\pi\)
−0.343514 + 0.939148i \(0.611617\pi\)
\(264\) 0 0
\(265\) −4.78594 −0.293998
\(266\) −17.4090 −1.06741
\(267\) 0 0
\(268\) 1.50549 0.0919627
\(269\) 4.30607 0.262546 0.131273 0.991346i \(-0.458094\pi\)
0.131273 + 0.991346i \(0.458094\pi\)
\(270\) 0 0
\(271\) 10.9322 0.664083 0.332042 0.943265i \(-0.392263\pi\)
0.332042 + 0.943265i \(0.392263\pi\)
\(272\) 27.5572 1.67090
\(273\) 0 0
\(274\) 3.42714 0.207041
\(275\) −3.08616 −0.186103
\(276\) 0 0
\(277\) −24.4176 −1.46711 −0.733557 0.679628i \(-0.762141\pi\)
−0.733557 + 0.679628i \(0.762141\pi\)
\(278\) 0.931039 0.0558400
\(279\) 0 0
\(280\) −6.26673 −0.374509
\(281\) −0.333656 −0.0199042 −0.00995212 0.999950i \(-0.503168\pi\)
−0.00995212 + 0.999950i \(0.503168\pi\)
\(282\) 0 0
\(283\) 4.36506 0.259476 0.129738 0.991548i \(-0.458586\pi\)
0.129738 + 0.991548i \(0.458586\pi\)
\(284\) 16.6028 0.985198
\(285\) 0 0
\(286\) 3.38954 0.200428
\(287\) −29.2768 −1.72815
\(288\) 0 0
\(289\) 13.6484 0.802848
\(290\) −13.0789 −0.768017
\(291\) 0 0
\(292\) 6.18190 0.361768
\(293\) −23.9647 −1.40003 −0.700016 0.714127i \(-0.746824\pi\)
−0.700016 + 0.714127i \(0.746824\pi\)
\(294\) 0 0
\(295\) 19.3590 1.12713
\(296\) 5.59470 0.325185
\(297\) 0 0
\(298\) −36.4316 −2.11043
\(299\) 3.67982 0.212809
\(300\) 0 0
\(301\) 26.7303 1.54071
\(302\) 28.8174 1.65826
\(303\) 0 0
\(304\) −16.2746 −0.933414
\(305\) 1.38341 0.0792141
\(306\) 0 0
\(307\) 22.0771 1.26000 0.630002 0.776593i \(-0.283054\pi\)
0.630002 + 0.776593i \(0.283054\pi\)
\(308\) −3.44832 −0.196486
\(309\) 0 0
\(310\) −16.8717 −0.958249
\(311\) 4.88561 0.277037 0.138519 0.990360i \(-0.455766\pi\)
0.138519 + 0.990360i \(0.455766\pi\)
\(312\) 0 0
\(313\) −15.8503 −0.895914 −0.447957 0.894055i \(-0.647848\pi\)
−0.447957 + 0.894055i \(0.647848\pi\)
\(314\) 19.6593 1.10944
\(315\) 0 0
\(316\) 16.8960 0.950473
\(317\) −12.4684 −0.700295 −0.350148 0.936694i \(-0.613869\pi\)
−0.350148 + 0.936694i \(0.613869\pi\)
\(318\) 0 0
\(319\) 5.32737 0.298276
\(320\) 0.501228 0.0280195
\(321\) 0 0
\(322\) −10.2585 −0.571682
\(323\) −18.1003 −1.00712
\(324\) 0 0
\(325\) −5.89462 −0.326975
\(326\) 1.49112 0.0825855
\(327\) 0 0
\(328\) −14.7311 −0.813388
\(329\) 5.20722 0.287083
\(330\) 0 0
\(331\) 1.37223 0.0754245 0.0377123 0.999289i \(-0.487993\pi\)
0.0377123 + 0.999289i \(0.487993\pi\)
\(332\) −6.17407 −0.338846
\(333\) 0 0
\(334\) 3.83239 0.209699
\(335\) −1.81223 −0.0990125
\(336\) 0 0
\(337\) 3.72751 0.203051 0.101525 0.994833i \(-0.467628\pi\)
0.101525 + 0.994833i \(0.467628\pi\)
\(338\) −16.5959 −0.902700
\(339\) 0 0
\(340\) 8.80188 0.477349
\(341\) 6.87230 0.372156
\(342\) 0 0
\(343\) 14.9939 0.809597
\(344\) 13.4498 0.725165
\(345\) 0 0
\(346\) 15.6794 0.842931
\(347\) 9.56283 0.513360 0.256680 0.966497i \(-0.417371\pi\)
0.256680 + 0.966497i \(0.417371\pi\)
\(348\) 0 0
\(349\) −18.5888 −0.995038 −0.497519 0.867453i \(-0.665756\pi\)
−0.497519 + 0.867453i \(0.665756\pi\)
\(350\) 16.4328 0.878371
\(351\) 0 0
\(352\) −5.81408 −0.309891
\(353\) 7.34062 0.390702 0.195351 0.980733i \(-0.437415\pi\)
0.195351 + 0.980733i \(0.437415\pi\)
\(354\) 0 0
\(355\) −19.9855 −1.06072
\(356\) 6.88633 0.364975
\(357\) 0 0
\(358\) −3.97866 −0.210278
\(359\) 7.51165 0.396450 0.198225 0.980157i \(-0.436482\pi\)
0.198225 + 0.980157i \(0.436482\pi\)
\(360\) 0 0
\(361\) −8.31040 −0.437390
\(362\) −6.08795 −0.319975
\(363\) 0 0
\(364\) −6.58634 −0.345218
\(365\) −7.44141 −0.389501
\(366\) 0 0
\(367\) −35.0261 −1.82835 −0.914175 0.405320i \(-0.867160\pi\)
−0.914175 + 0.405320i \(0.867160\pi\)
\(368\) −9.59003 −0.499915
\(369\) 0 0
\(370\) 9.09774 0.472969
\(371\) −10.3801 −0.538910
\(372\) 0 0
\(373\) 32.0943 1.66178 0.830891 0.556436i \(-0.187832\pi\)
0.830891 + 0.556436i \(0.187832\pi\)
\(374\) −9.82445 −0.508010
\(375\) 0 0
\(376\) 2.62010 0.135121
\(377\) 10.1754 0.524058
\(378\) 0 0
\(379\) 12.9206 0.663688 0.331844 0.943334i \(-0.392329\pi\)
0.331844 + 0.943334i \(0.392329\pi\)
\(380\) −5.19819 −0.266661
\(381\) 0 0
\(382\) 19.5969 1.00267
\(383\) 11.4054 0.582789 0.291395 0.956603i \(-0.405881\pi\)
0.291395 + 0.956603i \(0.405881\pi\)
\(384\) 0 0
\(385\) 4.15089 0.211549
\(386\) −19.1671 −0.975581
\(387\) 0 0
\(388\) 21.2903 1.08085
\(389\) 21.1661 1.07316 0.536582 0.843848i \(-0.319715\pi\)
0.536582 + 0.843848i \(0.319715\pi\)
\(390\) 0 0
\(391\) −10.6658 −0.539393
\(392\) −3.02369 −0.152719
\(393\) 0 0
\(394\) 4.72664 0.238125
\(395\) −20.3384 −1.02334
\(396\) 0 0
\(397\) 2.05003 0.102888 0.0514441 0.998676i \(-0.483618\pi\)
0.0514441 + 0.998676i \(0.483618\pi\)
\(398\) −13.3104 −0.667188
\(399\) 0 0
\(400\) 15.3621 0.768103
\(401\) 28.6494 1.43068 0.715342 0.698774i \(-0.246271\pi\)
0.715342 + 0.698774i \(0.246271\pi\)
\(402\) 0 0
\(403\) 13.1262 0.653863
\(404\) 12.1564 0.604802
\(405\) 0 0
\(406\) −28.3665 −1.40781
\(407\) −3.70575 −0.183687
\(408\) 0 0
\(409\) 1.34125 0.0663204 0.0331602 0.999450i \(-0.489443\pi\)
0.0331602 + 0.999450i \(0.489443\pi\)
\(410\) −23.9547 −1.18304
\(411\) 0 0
\(412\) −0.381242 −0.0187824
\(413\) 41.9875 2.06607
\(414\) 0 0
\(415\) 7.43198 0.364822
\(416\) −11.1050 −0.544466
\(417\) 0 0
\(418\) 5.80210 0.283790
\(419\) 16.6723 0.814495 0.407247 0.913318i \(-0.366489\pi\)
0.407247 + 0.913318i \(0.366489\pi\)
\(420\) 0 0
\(421\) −34.9609 −1.70389 −0.851946 0.523630i \(-0.824577\pi\)
−0.851946 + 0.523630i \(0.824577\pi\)
\(422\) −29.6098 −1.44138
\(423\) 0 0
\(424\) −5.22294 −0.253648
\(425\) 17.0853 0.828760
\(426\) 0 0
\(427\) 3.00047 0.145203
\(428\) 23.3042 1.12645
\(429\) 0 0
\(430\) 21.8712 1.05472
\(431\) 1.32377 0.0637638 0.0318819 0.999492i \(-0.489850\pi\)
0.0318819 + 0.999492i \(0.489850\pi\)
\(432\) 0 0
\(433\) −25.4879 −1.22487 −0.612436 0.790520i \(-0.709810\pi\)
−0.612436 + 0.790520i \(0.709810\pi\)
\(434\) −36.5928 −1.75651
\(435\) 0 0
\(436\) 2.06375 0.0988356
\(437\) 6.29898 0.301321
\(438\) 0 0
\(439\) 36.3742 1.73605 0.868023 0.496525i \(-0.165391\pi\)
0.868023 + 0.496525i \(0.165391\pi\)
\(440\) 2.08859 0.0995695
\(441\) 0 0
\(442\) −18.7648 −0.892553
\(443\) 11.7349 0.557542 0.278771 0.960358i \(-0.410073\pi\)
0.278771 + 0.960358i \(0.410073\pi\)
\(444\) 0 0
\(445\) −8.28936 −0.392953
\(446\) −24.1261 −1.14240
\(447\) 0 0
\(448\) 1.08711 0.0513609
\(449\) 5.50937 0.260003 0.130002 0.991514i \(-0.458502\pi\)
0.130002 + 0.991514i \(0.458502\pi\)
\(450\) 0 0
\(451\) 9.75741 0.459458
\(452\) 14.0383 0.660306
\(453\) 0 0
\(454\) 0.937764 0.0440114
\(455\) 7.92826 0.371682
\(456\) 0 0
\(457\) −0.412085 −0.0192765 −0.00963827 0.999954i \(-0.503068\pi\)
−0.00963827 + 0.999954i \(0.503068\pi\)
\(458\) 4.74563 0.221748
\(459\) 0 0
\(460\) −3.06310 −0.142818
\(461\) −20.4352 −0.951764 −0.475882 0.879509i \(-0.657871\pi\)
−0.475882 + 0.879509i \(0.657871\pi\)
\(462\) 0 0
\(463\) −29.9214 −1.39057 −0.695283 0.718736i \(-0.744721\pi\)
−0.695283 + 0.718736i \(0.744721\pi\)
\(464\) −26.5182 −1.23108
\(465\) 0 0
\(466\) −47.9800 −2.22263
\(467\) −8.83738 −0.408945 −0.204473 0.978872i \(-0.565548\pi\)
−0.204473 + 0.978872i \(0.565548\pi\)
\(468\) 0 0
\(469\) −3.93051 −0.181494
\(470\) 4.26063 0.196528
\(471\) 0 0
\(472\) 21.1267 0.972434
\(473\) −8.90873 −0.409624
\(474\) 0 0
\(475\) −10.0902 −0.462970
\(476\) 19.0903 0.875000
\(477\) 0 0
\(478\) 22.0997 1.01082
\(479\) −13.5867 −0.620793 −0.310397 0.950607i \(-0.600462\pi\)
−0.310397 + 0.950607i \(0.600462\pi\)
\(480\) 0 0
\(481\) −7.07805 −0.322731
\(482\) −14.9297 −0.680030
\(483\) 0 0
\(484\) 1.14926 0.0522392
\(485\) −25.6280 −1.16371
\(486\) 0 0
\(487\) −4.43282 −0.200870 −0.100435 0.994944i \(-0.532024\pi\)
−0.100435 + 0.994944i \(0.532024\pi\)
\(488\) 1.50973 0.0683424
\(489\) 0 0
\(490\) −4.91692 −0.222124
\(491\) −11.8780 −0.536049 −0.268024 0.963412i \(-0.586371\pi\)
−0.268024 + 0.963412i \(0.586371\pi\)
\(492\) 0 0
\(493\) −29.4929 −1.32829
\(494\) 11.0821 0.498607
\(495\) 0 0
\(496\) −34.2084 −1.53600
\(497\) −43.3463 −1.94435
\(498\) 0 0
\(499\) −16.8417 −0.753940 −0.376970 0.926225i \(-0.623034\pi\)
−0.376970 + 0.926225i \(0.623034\pi\)
\(500\) 12.8562 0.574949
\(501\) 0 0
\(502\) 26.8290 1.19744
\(503\) −28.5265 −1.27193 −0.635967 0.771716i \(-0.719399\pi\)
−0.635967 + 0.771716i \(0.719399\pi\)
\(504\) 0 0
\(505\) −14.6331 −0.651166
\(506\) 3.41896 0.151991
\(507\) 0 0
\(508\) −1.51504 −0.0672190
\(509\) 32.0469 1.42045 0.710227 0.703973i \(-0.248592\pi\)
0.710227 + 0.703973i \(0.248592\pi\)
\(510\) 0 0
\(511\) −16.1396 −0.713972
\(512\) 13.9108 0.614776
\(513\) 0 0
\(514\) 32.1893 1.41981
\(515\) 0.458916 0.0202223
\(516\) 0 0
\(517\) −1.73547 −0.0763259
\(518\) 19.7319 0.866972
\(519\) 0 0
\(520\) 3.98923 0.174939
\(521\) 29.7095 1.30159 0.650797 0.759251i \(-0.274435\pi\)
0.650797 + 0.759251i \(0.274435\pi\)
\(522\) 0 0
\(523\) −30.6258 −1.33917 −0.669587 0.742734i \(-0.733529\pi\)
−0.669587 + 0.742734i \(0.733529\pi\)
\(524\) −5.28760 −0.230990
\(525\) 0 0
\(526\) −19.7723 −0.862112
\(527\) −38.0458 −1.65730
\(528\) 0 0
\(529\) −19.2883 −0.838620
\(530\) −8.49320 −0.368921
\(531\) 0 0
\(532\) −11.2743 −0.488802
\(533\) 18.6368 0.807249
\(534\) 0 0
\(535\) −28.0523 −1.21281
\(536\) −1.97770 −0.0854236
\(537\) 0 0
\(538\) 7.64162 0.329454
\(539\) 2.00280 0.0862665
\(540\) 0 0
\(541\) −12.0599 −0.518494 −0.259247 0.965811i \(-0.583474\pi\)
−0.259247 + 0.965811i \(0.583474\pi\)
\(542\) 19.4004 0.833320
\(543\) 0 0
\(544\) 32.1873 1.38002
\(545\) −2.48422 −0.106412
\(546\) 0 0
\(547\) 15.6713 0.670056 0.335028 0.942208i \(-0.391254\pi\)
0.335028 + 0.942208i \(0.391254\pi\)
\(548\) 2.21946 0.0948106
\(549\) 0 0
\(550\) −5.47676 −0.233530
\(551\) 17.4178 0.742024
\(552\) 0 0
\(553\) −44.1116 −1.87582
\(554\) −43.3319 −1.84100
\(555\) 0 0
\(556\) 0.602952 0.0255709
\(557\) 21.7950 0.923485 0.461742 0.887014i \(-0.347224\pi\)
0.461742 + 0.887014i \(0.347224\pi\)
\(558\) 0 0
\(559\) −17.0158 −0.719692
\(560\) −20.6620 −0.873127
\(561\) 0 0
\(562\) −0.592111 −0.0249767
\(563\) 40.1440 1.69187 0.845933 0.533289i \(-0.179044\pi\)
0.845933 + 0.533289i \(0.179044\pi\)
\(564\) 0 0
\(565\) −16.8985 −0.710925
\(566\) 7.74631 0.325602
\(567\) 0 0
\(568\) −21.8104 −0.915144
\(569\) 27.1384 1.13770 0.568850 0.822442i \(-0.307389\pi\)
0.568850 + 0.822442i \(0.307389\pi\)
\(570\) 0 0
\(571\) 19.8310 0.829901 0.414950 0.909844i \(-0.363799\pi\)
0.414950 + 0.909844i \(0.363799\pi\)
\(572\) 2.19511 0.0917820
\(573\) 0 0
\(574\) −51.9550 −2.16856
\(575\) −5.94577 −0.247956
\(576\) 0 0
\(577\) −36.3627 −1.51380 −0.756899 0.653532i \(-0.773287\pi\)
−0.756899 + 0.653532i \(0.773287\pi\)
\(578\) 24.2207 1.00745
\(579\) 0 0
\(580\) −8.47002 −0.351699
\(581\) 16.1191 0.668733
\(582\) 0 0
\(583\) 3.45951 0.143278
\(584\) −8.12088 −0.336045
\(585\) 0 0
\(586\) −42.5281 −1.75682
\(587\) −18.8007 −0.775990 −0.387995 0.921661i \(-0.626832\pi\)
−0.387995 + 0.921661i \(0.626832\pi\)
\(588\) 0 0
\(589\) 22.4690 0.925817
\(590\) 34.3548 1.41437
\(591\) 0 0
\(592\) 18.4462 0.758135
\(593\) −9.65046 −0.396297 −0.198148 0.980172i \(-0.563493\pi\)
−0.198148 + 0.980172i \(0.563493\pi\)
\(594\) 0 0
\(595\) −22.9797 −0.942077
\(596\) −23.5935 −0.966429
\(597\) 0 0
\(598\) 6.53026 0.267042
\(599\) −3.84944 −0.157284 −0.0786420 0.996903i \(-0.525058\pi\)
−0.0786420 + 0.996903i \(0.525058\pi\)
\(600\) 0 0
\(601\) 3.79246 0.154698 0.0773488 0.997004i \(-0.475355\pi\)
0.0773488 + 0.997004i \(0.475355\pi\)
\(602\) 47.4361 1.93335
\(603\) 0 0
\(604\) 18.6625 0.759367
\(605\) −1.38341 −0.0562438
\(606\) 0 0
\(607\) −8.42952 −0.342144 −0.171072 0.985259i \(-0.554723\pi\)
−0.171072 + 0.985259i \(0.554723\pi\)
\(608\) −19.0091 −0.770921
\(609\) 0 0
\(610\) 2.45503 0.0994012
\(611\) −3.31477 −0.134101
\(612\) 0 0
\(613\) −33.9296 −1.37040 −0.685202 0.728353i \(-0.740286\pi\)
−0.685202 + 0.728353i \(0.740286\pi\)
\(614\) 39.1783 1.58111
\(615\) 0 0
\(616\) 4.52990 0.182515
\(617\) 33.4620 1.34713 0.673564 0.739129i \(-0.264762\pi\)
0.673564 + 0.739129i \(0.264762\pi\)
\(618\) 0 0
\(619\) −44.1918 −1.77622 −0.888109 0.459633i \(-0.847981\pi\)
−0.888109 + 0.459633i \(0.847981\pi\)
\(620\) −10.9263 −0.438811
\(621\) 0 0
\(622\) 8.67008 0.347638
\(623\) −17.9787 −0.720300
\(624\) 0 0
\(625\) −0.0447694 −0.00179078
\(626\) −28.1283 −1.12423
\(627\) 0 0
\(628\) 12.7316 0.508047
\(629\) 20.5154 0.818004
\(630\) 0 0
\(631\) 44.2349 1.76096 0.880482 0.474079i \(-0.157219\pi\)
0.880482 + 0.474079i \(0.157219\pi\)
\(632\) −22.1955 −0.882889
\(633\) 0 0
\(634\) −22.1266 −0.878761
\(635\) 1.82372 0.0723720
\(636\) 0 0
\(637\) 3.82537 0.151567
\(638\) 9.45404 0.374289
\(639\) 0 0
\(640\) −15.1971 −0.600717
\(641\) 7.36324 0.290831 0.145415 0.989371i \(-0.453548\pi\)
0.145415 + 0.989371i \(0.453548\pi\)
\(642\) 0 0
\(643\) 38.6344 1.52359 0.761795 0.647818i \(-0.224318\pi\)
0.761795 + 0.647818i \(0.224318\pi\)
\(644\) −6.64350 −0.261791
\(645\) 0 0
\(646\) −32.1210 −1.26378
\(647\) −16.0672 −0.631667 −0.315834 0.948815i \(-0.602284\pi\)
−0.315834 + 0.948815i \(0.602284\pi\)
\(648\) 0 0
\(649\) −13.9936 −0.549299
\(650\) −10.4607 −0.410302
\(651\) 0 0
\(652\) 0.965667 0.0378184
\(653\) −29.4442 −1.15224 −0.576120 0.817365i \(-0.695434\pi\)
−0.576120 + 0.817365i \(0.695434\pi\)
\(654\) 0 0
\(655\) 6.36490 0.248697
\(656\) −48.5697 −1.89633
\(657\) 0 0
\(658\) 9.24082 0.360245
\(659\) −2.22704 −0.0867532 −0.0433766 0.999059i \(-0.513812\pi\)
−0.0433766 + 0.999059i \(0.513812\pi\)
\(660\) 0 0
\(661\) −4.78984 −0.186303 −0.0931515 0.995652i \(-0.529694\pi\)
−0.0931515 + 0.995652i \(0.529694\pi\)
\(662\) 2.43518 0.0946459
\(663\) 0 0
\(664\) 8.11059 0.314752
\(665\) 13.5713 0.526273
\(666\) 0 0
\(667\) 10.2637 0.397411
\(668\) 2.48190 0.0960276
\(669\) 0 0
\(670\) −3.21601 −0.124245
\(671\) −1.00000 −0.0386046
\(672\) 0 0
\(673\) 37.9084 1.46126 0.730630 0.682773i \(-0.239226\pi\)
0.730630 + 0.682773i \(0.239226\pi\)
\(674\) 6.61491 0.254797
\(675\) 0 0
\(676\) −10.7477 −0.413374
\(677\) 46.7699 1.79751 0.898756 0.438448i \(-0.144472\pi\)
0.898756 + 0.438448i \(0.144472\pi\)
\(678\) 0 0
\(679\) −55.5842 −2.13312
\(680\) −11.5626 −0.443406
\(681\) 0 0
\(682\) 12.1957 0.466997
\(683\) −20.7596 −0.794343 −0.397171 0.917744i \(-0.630008\pi\)
−0.397171 + 0.917744i \(0.630008\pi\)
\(684\) 0 0
\(685\) −2.67165 −0.102079
\(686\) 26.6085 1.01592
\(687\) 0 0
\(688\) 44.3452 1.69064
\(689\) 6.60771 0.251734
\(690\) 0 0
\(691\) 30.0782 1.14423 0.572115 0.820173i \(-0.306123\pi\)
0.572115 + 0.820173i \(0.306123\pi\)
\(692\) 10.1542 0.386004
\(693\) 0 0
\(694\) 16.9703 0.644186
\(695\) −0.725798 −0.0275311
\(696\) 0 0
\(697\) −54.0180 −2.04608
\(698\) −32.9881 −1.24862
\(699\) 0 0
\(700\) 10.6421 0.402233
\(701\) 19.9910 0.755048 0.377524 0.926000i \(-0.376776\pi\)
0.377524 + 0.926000i \(0.376776\pi\)
\(702\) 0 0
\(703\) −12.1159 −0.456962
\(704\) −0.362312 −0.0136552
\(705\) 0 0
\(706\) 13.0268 0.490269
\(707\) −31.7376 −1.19361
\(708\) 0 0
\(709\) −38.8223 −1.45800 −0.729001 0.684513i \(-0.760015\pi\)
−0.729001 + 0.684513i \(0.760015\pi\)
\(710\) −35.4667 −1.33104
\(711\) 0 0
\(712\) −9.04625 −0.339023
\(713\) 13.2401 0.495846
\(714\) 0 0
\(715\) −2.64234 −0.0988180
\(716\) −2.57662 −0.0962930
\(717\) 0 0
\(718\) 13.3303 0.497482
\(719\) 39.1271 1.45919 0.729597 0.683877i \(-0.239708\pi\)
0.729597 + 0.683877i \(0.239708\pi\)
\(720\) 0 0
\(721\) 0.995336 0.0370683
\(722\) −14.7478 −0.548855
\(723\) 0 0
\(724\) −3.94263 −0.146527
\(725\) −16.4412 −0.610609
\(726\) 0 0
\(727\) −16.6521 −0.617592 −0.308796 0.951128i \(-0.599926\pi\)
−0.308796 + 0.951128i \(0.599926\pi\)
\(728\) 8.65218 0.320671
\(729\) 0 0
\(730\) −13.2056 −0.488763
\(731\) 49.3196 1.82415
\(732\) 0 0
\(733\) 31.5747 1.16624 0.583119 0.812387i \(-0.301832\pi\)
0.583119 + 0.812387i \(0.301832\pi\)
\(734\) −62.1580 −2.29429
\(735\) 0 0
\(736\) −11.2013 −0.412887
\(737\) 1.30997 0.0482532
\(738\) 0 0
\(739\) −35.2375 −1.29623 −0.648117 0.761541i \(-0.724443\pi\)
−0.648117 + 0.761541i \(0.724443\pi\)
\(740\) 5.89180 0.216587
\(741\) 0 0
\(742\) −18.4208 −0.676247
\(743\) 22.9658 0.842534 0.421267 0.906937i \(-0.361585\pi\)
0.421267 + 0.906937i \(0.361585\pi\)
\(744\) 0 0
\(745\) 28.4005 1.04051
\(746\) 56.9551 2.08527
\(747\) 0 0
\(748\) −6.36243 −0.232633
\(749\) −60.8422 −2.22312
\(750\) 0 0
\(751\) 45.4722 1.65930 0.829652 0.558281i \(-0.188539\pi\)
0.829652 + 0.558281i \(0.188539\pi\)
\(752\) 8.63869 0.315021
\(753\) 0 0
\(754\) 18.0574 0.657610
\(755\) −22.4648 −0.817579
\(756\) 0 0
\(757\) 14.4333 0.524588 0.262294 0.964988i \(-0.415521\pi\)
0.262294 + 0.964988i \(0.415521\pi\)
\(758\) 22.9291 0.832824
\(759\) 0 0
\(760\) 6.82862 0.247700
\(761\) 47.1071 1.70763 0.853815 0.520576i \(-0.174283\pi\)
0.853815 + 0.520576i \(0.174283\pi\)
\(762\) 0 0
\(763\) −5.38798 −0.195058
\(764\) 12.6912 0.459151
\(765\) 0 0
\(766\) 20.2402 0.731309
\(767\) −26.7281 −0.965095
\(768\) 0 0
\(769\) 42.9030 1.54712 0.773561 0.633721i \(-0.218473\pi\)
0.773561 + 0.633721i \(0.218473\pi\)
\(770\) 7.36623 0.265461
\(771\) 0 0
\(772\) −12.4129 −0.446748
\(773\) 44.2757 1.59249 0.796243 0.604976i \(-0.206817\pi\)
0.796243 + 0.604976i \(0.206817\pi\)
\(774\) 0 0
\(775\) −21.2091 −0.761852
\(776\) −27.9681 −1.00400
\(777\) 0 0
\(778\) 37.5617 1.34665
\(779\) 31.9018 1.14300
\(780\) 0 0
\(781\) 14.4465 0.516937
\(782\) −18.9277 −0.676853
\(783\) 0 0
\(784\) −9.96936 −0.356048
\(785\) −15.3256 −0.546993
\(786\) 0 0
\(787\) −14.5288 −0.517896 −0.258948 0.965891i \(-0.583376\pi\)
−0.258948 + 0.965891i \(0.583376\pi\)
\(788\) 3.06103 0.109045
\(789\) 0 0
\(790\) −36.0928 −1.28413
\(791\) −36.6509 −1.30315
\(792\) 0 0
\(793\) −1.91001 −0.0678266
\(794\) 3.63802 0.129109
\(795\) 0 0
\(796\) −8.61995 −0.305526
\(797\) 41.9535 1.48607 0.743034 0.669253i \(-0.233386\pi\)
0.743034 + 0.669253i \(0.233386\pi\)
\(798\) 0 0
\(799\) 9.60774 0.339897
\(800\) 17.9432 0.634388
\(801\) 0 0
\(802\) 50.8418 1.79528
\(803\) 5.37902 0.189821
\(804\) 0 0
\(805\) 7.99706 0.281859
\(806\) 23.2940 0.820495
\(807\) 0 0
\(808\) −15.9693 −0.561797
\(809\) 26.4843 0.931140 0.465570 0.885011i \(-0.345849\pi\)
0.465570 + 0.885011i \(0.345849\pi\)
\(810\) 0 0
\(811\) 21.8378 0.766828 0.383414 0.923577i \(-0.374748\pi\)
0.383414 + 0.923577i \(0.374748\pi\)
\(812\) −18.3705 −0.644678
\(813\) 0 0
\(814\) −6.57629 −0.230499
\(815\) −1.16241 −0.0407176
\(816\) 0 0
\(817\) −29.1271 −1.01903
\(818\) 2.38020 0.0832217
\(819\) 0 0
\(820\) −15.5134 −0.541750
\(821\) 47.3362 1.65205 0.826023 0.563636i \(-0.190598\pi\)
0.826023 + 0.563636i \(0.190598\pi\)
\(822\) 0 0
\(823\) −15.0471 −0.524508 −0.262254 0.964999i \(-0.584466\pi\)
−0.262254 + 0.964999i \(0.584466\pi\)
\(824\) 0.500820 0.0174469
\(825\) 0 0
\(826\) 74.5116 2.59259
\(827\) 14.8054 0.514835 0.257418 0.966300i \(-0.417128\pi\)
0.257418 + 0.966300i \(0.417128\pi\)
\(828\) 0 0
\(829\) −29.6278 −1.02901 −0.514507 0.857486i \(-0.672025\pi\)
−0.514507 + 0.857486i \(0.672025\pi\)
\(830\) 13.1889 0.457794
\(831\) 0 0
\(832\) −0.692022 −0.0239915
\(833\) −11.0877 −0.384165
\(834\) 0 0
\(835\) −2.98757 −0.103389
\(836\) 3.75751 0.129956
\(837\) 0 0
\(838\) 29.5869 1.02206
\(839\) 23.2760 0.803578 0.401789 0.915732i \(-0.368389\pi\)
0.401789 + 0.915732i \(0.368389\pi\)
\(840\) 0 0
\(841\) −0.619079 −0.0213475
\(842\) −62.0422 −2.13812
\(843\) 0 0
\(844\) −19.1757 −0.660054
\(845\) 12.9375 0.445063
\(846\) 0 0
\(847\) −3.00047 −0.103097
\(848\) −17.2205 −0.591353
\(849\) 0 0
\(850\) 30.3199 1.03996
\(851\) −7.13947 −0.244738
\(852\) 0 0
\(853\) 1.24162 0.0425122 0.0212561 0.999774i \(-0.493233\pi\)
0.0212561 + 0.999774i \(0.493233\pi\)
\(854\) 5.32467 0.182207
\(855\) 0 0
\(856\) −30.6137 −1.04636
\(857\) −6.05692 −0.206900 −0.103450 0.994635i \(-0.532988\pi\)
−0.103450 + 0.994635i \(0.532988\pi\)
\(858\) 0 0
\(859\) −54.5519 −1.86129 −0.930643 0.365928i \(-0.880752\pi\)
−0.930643 + 0.365928i \(0.880752\pi\)
\(860\) 14.1640 0.482990
\(861\) 0 0
\(862\) 2.34919 0.0800136
\(863\) −29.1925 −0.993726 −0.496863 0.867829i \(-0.665515\pi\)
−0.496863 + 0.867829i \(0.665515\pi\)
\(864\) 0 0
\(865\) −12.2230 −0.415595
\(866\) −45.2313 −1.53702
\(867\) 0 0
\(868\) −23.6979 −0.804360
\(869\) 14.7016 0.498717
\(870\) 0 0
\(871\) 2.50205 0.0847789
\(872\) −2.71105 −0.0918078
\(873\) 0 0
\(874\) 11.1783 0.378110
\(875\) −33.5648 −1.13470
\(876\) 0 0
\(877\) 32.8509 1.10930 0.554649 0.832085i \(-0.312853\pi\)
0.554649 + 0.832085i \(0.312853\pi\)
\(878\) 64.5502 2.17846
\(879\) 0 0
\(880\) 6.88625 0.232135
\(881\) −36.5828 −1.23251 −0.616254 0.787548i \(-0.711350\pi\)
−0.616254 + 0.787548i \(0.711350\pi\)
\(882\) 0 0
\(883\) 29.2412 0.984045 0.492022 0.870583i \(-0.336258\pi\)
0.492022 + 0.870583i \(0.336258\pi\)
\(884\) −12.1523 −0.408727
\(885\) 0 0
\(886\) 20.8250 0.699628
\(887\) −34.0633 −1.14373 −0.571866 0.820347i \(-0.693780\pi\)
−0.571866 + 0.820347i \(0.693780\pi\)
\(888\) 0 0
\(889\) 3.95543 0.132661
\(890\) −14.7104 −0.493095
\(891\) 0 0
\(892\) −15.6244 −0.523142
\(893\) −5.67411 −0.189877
\(894\) 0 0
\(895\) 3.10159 0.103675
\(896\) −32.9607 −1.10114
\(897\) 0 0
\(898\) 9.77702 0.326263
\(899\) 36.6113 1.22106
\(900\) 0 0
\(901\) −19.1522 −0.638052
\(902\) 17.3157 0.576548
\(903\) 0 0
\(904\) −18.4415 −0.613354
\(905\) 4.74590 0.157759
\(906\) 0 0
\(907\) 9.07580 0.301357 0.150679 0.988583i \(-0.451854\pi\)
0.150679 + 0.988583i \(0.451854\pi\)
\(908\) 0.607307 0.0201542
\(909\) 0 0
\(910\) 14.0696 0.466403
\(911\) 31.1996 1.03369 0.516845 0.856079i \(-0.327106\pi\)
0.516845 + 0.856079i \(0.327106\pi\)
\(912\) 0 0
\(913\) −5.37220 −0.177794
\(914\) −0.731293 −0.0241890
\(915\) 0 0
\(916\) 3.07332 0.101545
\(917\) 13.8047 0.455872
\(918\) 0 0
\(919\) −53.8654 −1.77686 −0.888429 0.459015i \(-0.848202\pi\)
−0.888429 + 0.459015i \(0.848202\pi\)
\(920\) 4.02385 0.132662
\(921\) 0 0
\(922\) −36.2647 −1.19431
\(923\) 27.5931 0.908237
\(924\) 0 0
\(925\) 11.4366 0.376032
\(926\) −53.0990 −1.74494
\(927\) 0 0
\(928\) −30.9738 −1.01676
\(929\) −3.87844 −0.127248 −0.0636238 0.997974i \(-0.520266\pi\)
−0.0636238 + 0.997974i \(0.520266\pi\)
\(930\) 0 0
\(931\) 6.54813 0.214606
\(932\) −31.0724 −1.01781
\(933\) 0 0
\(934\) −15.6830 −0.513162
\(935\) 7.65872 0.250467
\(936\) 0 0
\(937\) 29.9089 0.977083 0.488541 0.872541i \(-0.337529\pi\)
0.488541 + 0.872541i \(0.337529\pi\)
\(938\) −6.97514 −0.227747
\(939\) 0 0
\(940\) 2.75924 0.0899963
\(941\) 31.0364 1.01176 0.505878 0.862605i \(-0.331169\pi\)
0.505878 + 0.862605i \(0.331169\pi\)
\(942\) 0 0
\(943\) 18.7985 0.612164
\(944\) 69.6565 2.26712
\(945\) 0 0
\(946\) −15.8096 −0.514014
\(947\) −24.5563 −0.797974 −0.398987 0.916957i \(-0.630638\pi\)
−0.398987 + 0.916957i \(0.630638\pi\)
\(948\) 0 0
\(949\) 10.2740 0.333508
\(950\) −17.9062 −0.580955
\(951\) 0 0
\(952\) −25.0780 −0.812783
\(953\) 16.9959 0.550551 0.275276 0.961365i \(-0.411231\pi\)
0.275276 + 0.961365i \(0.411231\pi\)
\(954\) 0 0
\(955\) −15.2769 −0.494349
\(956\) 14.3120 0.462885
\(957\) 0 0
\(958\) −24.1112 −0.778998
\(959\) −5.79451 −0.187114
\(960\) 0 0
\(961\) 16.2286 0.523502
\(962\) −12.5608 −0.404977
\(963\) 0 0
\(964\) −9.66866 −0.311407
\(965\) 14.9419 0.480996
\(966\) 0 0
\(967\) −24.6372 −0.792278 −0.396139 0.918191i \(-0.629650\pi\)
−0.396139 + 0.918191i \(0.629650\pi\)
\(968\) −1.50973 −0.0485247
\(969\) 0 0
\(970\) −45.4799 −1.46027
\(971\) −23.2628 −0.746539 −0.373270 0.927723i \(-0.621763\pi\)
−0.373270 + 0.927723i \(0.621763\pi\)
\(972\) 0 0
\(973\) −1.57417 −0.0504656
\(974\) −7.86656 −0.252061
\(975\) 0 0
\(976\) 4.97772 0.159333
\(977\) −3.96648 −0.126899 −0.0634494 0.997985i \(-0.520210\pi\)
−0.0634494 + 0.997985i \(0.520210\pi\)
\(978\) 0 0
\(979\) 5.99196 0.191504
\(980\) −3.18426 −0.101717
\(981\) 0 0
\(982\) −21.0790 −0.672657
\(983\) −28.3662 −0.904742 −0.452371 0.891830i \(-0.649422\pi\)
−0.452371 + 0.891830i \(0.649422\pi\)
\(984\) 0 0
\(985\) −3.68468 −0.117404
\(986\) −52.3385 −1.66680
\(987\) 0 0
\(988\) 7.17689 0.228327
\(989\) −17.1635 −0.545767
\(990\) 0 0
\(991\) 56.0031 1.77900 0.889499 0.456937i \(-0.151053\pi\)
0.889499 + 0.456937i \(0.151053\pi\)
\(992\) −39.9561 −1.26861
\(993\) 0 0
\(994\) −76.9231 −2.43985
\(995\) 10.3762 0.328947
\(996\) 0 0
\(997\) 11.2007 0.354729 0.177365 0.984145i \(-0.443243\pi\)
0.177365 + 0.984145i \(0.443243\pi\)
\(998\) −29.8876 −0.946076
\(999\) 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 6039.2.a.o.1.19 yes 25
3.2 odd 2 6039.2.a.n.1.7 25
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6039.2.a.n.1.7 25 3.2 odd 2
6039.2.a.o.1.19 yes 25 1.1 even 1 trivial