Properties

Label 585.2.h.g.64.2
Level $585$
Weight $2$
Character 585.64
Analytic conductor $4.671$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [585,2,Mod(64,585)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(585, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("585.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.h (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.2593100598870016.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{10} + 9x^{8} - 16x^{6} + 36x^{4} - 64x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 64.2
Root \(-1.37820 + 0.317122i\) of defining polynomial
Character \(\chi\) \(=\) 585.64
Dual form 585.2.h.g.64.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.51912 q^{2} +4.34596 q^{4} +(1.45804 + 1.69532i) q^{5} -1.26849 q^{7} -5.90976 q^{8} +O(q^{10})\) \(q-2.51912 q^{2} +4.34596 q^{4} +(1.45804 + 1.69532i) q^{5} -1.26849 q^{7} -5.90976 q^{8} +(-3.67298 - 4.27072i) q^{10} -2.12216i q^{11} +(-3.15336 - 1.74823i) q^{13} +3.19547 q^{14} +6.19547 q^{16} -7.19547i q^{17} -5.51280i q^{19} +(6.33660 + 7.36780i) q^{20} +5.34596i q^{22} +4.00000i q^{23} +(-0.748228 + 4.94370i) q^{25} +(7.94370 + 4.40400i) q^{26} -5.51280 q^{28} -4.69193 q^{29} -2.97582i q^{31} -3.78761 q^{32} +18.1263i q^{34} +(-1.84951 - 2.15049i) q^{35} +10.5510 q^{37} +13.8874i q^{38} +(-8.61668 - 10.0189i) q^{40} -0.853668i q^{41} -3.19547i q^{43} -9.22281i q^{44} -10.0765i q^{46} +2.12216 q^{47} -5.39094 q^{49} +(1.88488 - 12.4538i) q^{50} +(-13.7044 - 7.59774i) q^{52} -3.49646i q^{53} +(3.59774 - 3.09419i) q^{55} +7.49646 q^{56} +11.8195 q^{58} -8.90344i q^{59} +11.8874 q^{61} +7.49646i q^{62} -2.84951 q^{64} +(-1.63393 - 7.89495i) q^{65} +0.319369 q^{67} -31.2713i q^{68} +(4.65913 + 5.41735i) q^{70} -13.7865i q^{71} -6.30673 q^{73} -26.5793 q^{74} -23.9584i q^{76} +2.69193i q^{77} -0.804530 q^{79} +(9.03325 + 10.5033i) q^{80} +2.15049i q^{82} +1.17304 q^{83} +(12.1986 - 10.4913i) q^{85} +8.04977i q^{86} +12.5414i q^{88} +7.63495i q^{89} +(4.00000 + 2.21760i) q^{91} +17.3839i q^{92} -5.34596 q^{94} +(9.34596 - 8.03789i) q^{95} -5.35761 q^{97} +13.5804 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{4} - 24 q^{10} - 16 q^{14} + 20 q^{16} + 4 q^{25} + 28 q^{26} + 24 q^{29} - 8 q^{35} - 16 q^{40} + 44 q^{49} + 16 q^{55} + 64 q^{56} + 8 q^{61} - 20 q^{64} - 28 q^{65} - 104 q^{74} - 64 q^{79} + 48 q^{91} - 24 q^{94} + 72 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.51912 −1.78129 −0.890643 0.454703i \(-0.849745\pi\)
−0.890643 + 0.454703i \(0.849745\pi\)
\(3\) 0 0
\(4\) 4.34596 2.17298
\(5\) 1.45804 + 1.69532i 0.652056 + 0.758171i
\(6\) 0 0
\(7\) −1.26849 −0.479443 −0.239722 0.970842i \(-0.577056\pi\)
−0.239722 + 0.970842i \(0.577056\pi\)
\(8\) −5.90976 −2.08942
\(9\) 0 0
\(10\) −3.67298 4.27072i −1.16150 1.35052i
\(11\) 2.12216i 0.639854i −0.947442 0.319927i \(-0.896342\pi\)
0.947442 0.319927i \(-0.103658\pi\)
\(12\) 0 0
\(13\) −3.15336 1.74823i −0.874586 0.484871i
\(14\) 3.19547 0.854025
\(15\) 0 0
\(16\) 6.19547 1.54887
\(17\) 7.19547i 1.74516i −0.488473 0.872579i \(-0.662446\pi\)
0.488473 0.872579i \(-0.337554\pi\)
\(18\) 0 0
\(19\) 5.51280i 1.26472i −0.774674 0.632361i \(-0.782086\pi\)
0.774674 0.632361i \(-0.217914\pi\)
\(20\) 6.33660 + 7.36780i 1.41691 + 1.64749i
\(21\) 0 0
\(22\) 5.34596i 1.13976i
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) −0.748228 + 4.94370i −0.149646 + 0.988740i
\(26\) 7.94370 + 4.40400i 1.55789 + 0.863695i
\(27\) 0 0
\(28\) −5.51280 −1.04182
\(29\) −4.69193 −0.871269 −0.435634 0.900124i \(-0.643476\pi\)
−0.435634 + 0.900124i \(0.643476\pi\)
\(30\) 0 0
\(31\) 2.97582i 0.534474i −0.963631 0.267237i \(-0.913889\pi\)
0.963631 0.267237i \(-0.0861106\pi\)
\(32\) −3.78761 −0.669561
\(33\) 0 0
\(34\) 18.1263i 3.10863i
\(35\) −1.84951 2.15049i −0.312624 0.363500i
\(36\) 0 0
\(37\) 10.5510 1.73458 0.867290 0.497803i \(-0.165860\pi\)
0.867290 + 0.497803i \(0.165860\pi\)
\(38\) 13.8874i 2.25283i
\(39\) 0 0
\(40\) −8.61668 10.0189i −1.36242 1.58413i
\(41\) 0.853668i 0.133321i −0.997776 0.0666603i \(-0.978766\pi\)
0.997776 0.0666603i \(-0.0212344\pi\)
\(42\) 0 0
\(43\) 3.19547i 0.487305i −0.969863 0.243652i \(-0.921654\pi\)
0.969863 0.243652i \(-0.0783456\pi\)
\(44\) 9.22281i 1.39039i
\(45\) 0 0
\(46\) 10.0765i 1.48570i
\(47\) 2.12216 0.309548 0.154774 0.987950i \(-0.450535\pi\)
0.154774 + 0.987950i \(0.450535\pi\)
\(48\) 0 0
\(49\) −5.39094 −0.770134
\(50\) 1.88488 12.4538i 0.266562 1.76123i
\(51\) 0 0
\(52\) −13.7044 7.59774i −1.90046 1.05362i
\(53\) 3.49646i 0.480275i −0.970739 0.240138i \(-0.922807\pi\)
0.970739 0.240138i \(-0.0771926\pi\)
\(54\) 0 0
\(55\) 3.59774 3.09419i 0.485119 0.417221i
\(56\) 7.49646 1.00176
\(57\) 0 0
\(58\) 11.8195 1.55198
\(59\) 8.90344i 1.15913i −0.814926 0.579565i \(-0.803223\pi\)
0.814926 0.579565i \(-0.196777\pi\)
\(60\) 0 0
\(61\) 11.8874 1.52203 0.761013 0.648737i \(-0.224703\pi\)
0.761013 + 0.648737i \(0.224703\pi\)
\(62\) 7.49646i 0.952051i
\(63\) 0 0
\(64\) −2.84951 −0.356188
\(65\) −1.63393 7.89495i −0.202664 0.979248i
\(66\) 0 0
\(67\) 0.319369 0.0390172 0.0195086 0.999810i \(-0.493790\pi\)
0.0195086 + 0.999810i \(0.493790\pi\)
\(68\) 31.2713i 3.79220i
\(69\) 0 0
\(70\) 4.65913 + 5.41735i 0.556872 + 0.647497i
\(71\) 13.7865i 1.63616i −0.575108 0.818078i \(-0.695040\pi\)
0.575108 0.818078i \(-0.304960\pi\)
\(72\) 0 0
\(73\) −6.30673 −0.738146 −0.369073 0.929400i \(-0.620325\pi\)
−0.369073 + 0.929400i \(0.620325\pi\)
\(74\) −26.5793 −3.08978
\(75\) 0 0
\(76\) 23.9584i 2.74822i
\(77\) 2.69193i 0.306774i
\(78\) 0 0
\(79\) −0.804530 −0.0905167 −0.0452583 0.998975i \(-0.514411\pi\)
−0.0452583 + 0.998975i \(0.514411\pi\)
\(80\) 9.03325 + 10.5033i 1.00995 + 1.17431i
\(81\) 0 0
\(82\) 2.15049i 0.237482i
\(83\) 1.17304 0.128758 0.0643788 0.997926i \(-0.479493\pi\)
0.0643788 + 0.997926i \(0.479493\pi\)
\(84\) 0 0
\(85\) 12.1986 10.4913i 1.32313 1.13794i
\(86\) 8.04977i 0.868029i
\(87\) 0 0
\(88\) 12.5414i 1.33692i
\(89\) 7.63495i 0.809303i 0.914471 + 0.404652i \(0.132607\pi\)
−0.914471 + 0.404652i \(0.867393\pi\)
\(90\) 0 0
\(91\) 4.00000 + 2.21760i 0.419314 + 0.232468i
\(92\) 17.3839i 1.81239i
\(93\) 0 0
\(94\) −5.34596 −0.551394
\(95\) 9.34596 8.03789i 0.958876 0.824670i
\(96\) 0 0
\(97\) −5.35761 −0.543983 −0.271991 0.962300i \(-0.587682\pi\)
−0.271991 + 0.962300i \(0.587682\pi\)
\(98\) 13.5804 1.37183
\(99\) 0 0
\(100\) −3.25177 + 21.4851i −0.325177 + 2.14851i
\(101\) 8.39094 0.834930 0.417465 0.908693i \(-0.362919\pi\)
0.417465 + 0.908693i \(0.362919\pi\)
\(102\) 0 0
\(103\) 18.1884i 1.79215i −0.443898 0.896077i \(-0.646405\pi\)
0.443898 0.896077i \(-0.353595\pi\)
\(104\) 18.6356 + 10.3316i 1.82737 + 1.01310i
\(105\) 0 0
\(106\) 8.80799i 0.855508i
\(107\) 0.300986i 0.0290974i 0.999894 + 0.0145487i \(0.00463116\pi\)
−0.999894 + 0.0145487i \(0.995369\pi\)
\(108\) 0 0
\(109\) 13.3717i 1.28077i 0.768052 + 0.640387i \(0.221226\pi\)
−0.768052 + 0.640387i \(0.778774\pi\)
\(110\) −9.06313 + 7.79464i −0.864135 + 0.743190i
\(111\) 0 0
\(112\) −7.85887 −0.742594
\(113\) 12.5793i 1.18336i 0.806172 + 0.591682i \(0.201536\pi\)
−0.806172 + 0.591682i \(0.798464\pi\)
\(114\) 0 0
\(115\) −6.78128 + 5.83217i −0.632358 + 0.543852i
\(116\) −20.3909 −1.89325
\(117\) 0 0
\(118\) 22.4288i 2.06474i
\(119\) 9.12736i 0.836704i
\(120\) 0 0
\(121\) 6.49646 0.590587
\(122\) −29.9458 −2.71116
\(123\) 0 0
\(124\) 12.9328i 1.16140i
\(125\) −9.47210 + 5.93963i −0.847211 + 0.531257i
\(126\) 0 0
\(127\) 18.3909i 1.63193i −0.578100 0.815966i \(-0.696206\pi\)
0.578100 0.815966i \(-0.303794\pi\)
\(128\) 14.7535 1.30403
\(129\) 0 0
\(130\) 4.11606 + 19.8883i 0.361002 + 1.74432i
\(131\) −17.3839 −1.51883 −0.759417 0.650604i \(-0.774516\pi\)
−0.759417 + 0.650604i \(0.774516\pi\)
\(132\) 0 0
\(133\) 6.99291i 0.606362i
\(134\) −0.804530 −0.0695008
\(135\) 0 0
\(136\) 42.5235i 3.64636i
\(137\) −5.45306 −0.465886 −0.232943 0.972490i \(-0.574836\pi\)
−0.232943 + 0.972490i \(0.574836\pi\)
\(138\) 0 0
\(139\) −17.3839 −1.47448 −0.737240 0.675631i \(-0.763871\pi\)
−0.737240 + 0.675631i \(0.763871\pi\)
\(140\) −8.03789 9.34596i −0.679326 0.789878i
\(141\) 0 0
\(142\) 34.7298i 2.91446i
\(143\) −3.71001 + 6.69193i −0.310247 + 0.559607i
\(144\) 0 0
\(145\) −6.84103 7.95432i −0.568116 0.660571i
\(146\) 15.8874 1.31485
\(147\) 0 0
\(148\) 45.8544 3.76921
\(149\) 3.39064i 0.277772i 0.990308 + 0.138886i \(0.0443522\pi\)
−0.990308 + 0.138886i \(0.955648\pi\)
\(150\) 0 0
\(151\) 2.97582i 0.242169i −0.992642 0.121085i \(-0.961363\pi\)
0.992642 0.121085i \(-0.0386372\pi\)
\(152\) 32.5793i 2.64253i
\(153\) 0 0
\(154\) 6.78128i 0.546452i
\(155\) 5.04498 4.33888i 0.405222 0.348507i
\(156\) 0 0
\(157\) 8.00000i 0.638470i 0.947676 + 0.319235i \(0.103426\pi\)
−0.947676 + 0.319235i \(0.896574\pi\)
\(158\) 2.02671 0.161236
\(159\) 0 0
\(160\) −5.52249 6.42121i −0.436591 0.507641i
\(161\) 5.07395i 0.399883i
\(162\) 0 0
\(163\) −2.02671 −0.158744 −0.0793719 0.996845i \(-0.525291\pi\)
−0.0793719 + 0.996845i \(0.525291\pi\)
\(164\) 3.71001i 0.289703i
\(165\) 0 0
\(166\) −2.95502 −0.229354
\(167\) 6.36647 0.492652 0.246326 0.969187i \(-0.420777\pi\)
0.246326 + 0.969187i \(0.420777\pi\)
\(168\) 0 0
\(169\) 6.88740 + 11.0256i 0.529800 + 0.848123i
\(170\) −30.7298 + 26.4288i −2.35687 + 2.02700i
\(171\) 0 0
\(172\) 13.8874i 1.05890i
\(173\) 16.2783i 1.23762i −0.785541 0.618810i \(-0.787615\pi\)
0.785541 0.618810i \(-0.212385\pi\)
\(174\) 0 0
\(175\) 0.949118 6.27102i 0.0717465 0.474044i
\(176\) 13.1478i 0.991049i
\(177\) 0 0
\(178\) 19.2334i 1.44160i
\(179\) 24.3768 1.82201 0.911003 0.412401i \(-0.135310\pi\)
0.911003 + 0.412401i \(0.135310\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) −10.0765 5.58641i −0.746918 0.414092i
\(183\) 0 0
\(184\) 23.6390i 1.74269i
\(185\) 15.3839 + 17.8874i 1.13104 + 1.31511i
\(186\) 0 0
\(187\) −15.2699 −1.11665
\(188\) 9.22281 0.672643
\(189\) 0 0
\(190\) −23.5436 + 20.2484i −1.70803 + 1.46897i
\(191\) 17.0829 1.23607 0.618036 0.786149i \(-0.287928\pi\)
0.618036 + 0.786149i \(0.287928\pi\)
\(192\) 0 0
\(193\) 2.06242 0.148456 0.0742279 0.997241i \(-0.476351\pi\)
0.0742279 + 0.997241i \(0.476351\pi\)
\(194\) 13.4965 0.968989
\(195\) 0 0
\(196\) −23.4288 −1.67349
\(197\) 7.99003 0.569266 0.284633 0.958637i \(-0.408128\pi\)
0.284633 + 0.958637i \(0.408128\pi\)
\(198\) 0 0
\(199\) 8.20256 0.581464 0.290732 0.956805i \(-0.406101\pi\)
0.290732 + 0.956805i \(0.406101\pi\)
\(200\) 4.42185 29.2161i 0.312672 2.06589i
\(201\) 0 0
\(202\) −21.1378 −1.48725
\(203\) 5.95165 0.417724
\(204\) 0 0
\(205\) 1.44724 1.24468i 0.101080 0.0869325i
\(206\) 45.8187i 3.19234i
\(207\) 0 0
\(208\) −19.5366 10.8311i −1.35462 0.751001i
\(209\) −11.6990 −0.809238
\(210\) 0 0
\(211\) 3.79744 0.261427 0.130713 0.991420i \(-0.458273\pi\)
0.130713 + 0.991420i \(0.458273\pi\)
\(212\) 15.1955i 1.04363i
\(213\) 0 0
\(214\) 0.758219i 0.0518308i
\(215\) 5.41735 4.65913i 0.369460 0.317750i
\(216\) 0 0
\(217\) 3.77479i 0.256250i
\(218\) 33.6848i 2.28143i
\(219\) 0 0
\(220\) 15.6356 13.4472i 1.05415 0.906613i
\(221\) −12.5793 + 22.6899i −0.846177 + 1.52629i
\(222\) 0 0
\(223\) −19.0754 −1.27738 −0.638691 0.769464i \(-0.720524\pi\)
−0.638691 + 0.769464i \(0.720524\pi\)
\(224\) 4.80453 0.321016
\(225\) 0 0
\(226\) 31.6888i 2.10791i
\(227\) −17.2726 −1.14642 −0.573211 0.819408i \(-0.694302\pi\)
−0.573211 + 0.819408i \(0.694302\pi\)
\(228\) 0 0
\(229\) 18.4456i 1.21892i −0.792816 0.609460i \(-0.791386\pi\)
0.792816 0.609460i \(-0.208614\pi\)
\(230\) 17.0829 14.6919i 1.12641 0.968757i
\(231\) 0 0
\(232\) 27.7282 1.82044
\(233\) 11.4965i 0.753158i 0.926385 + 0.376579i \(0.122900\pi\)
−0.926385 + 0.376579i \(0.877100\pi\)
\(234\) 0 0
\(235\) 3.09419 + 3.59774i 0.201843 + 0.234690i
\(236\) 38.6940i 2.51877i
\(237\) 0 0
\(238\) 22.9929i 1.49041i
\(239\) 7.19610i 0.465477i 0.972539 + 0.232739i \(0.0747687\pi\)
−0.972539 + 0.232739i \(0.925231\pi\)
\(240\) 0 0
\(241\) 15.9086i 1.02477i 0.858757 + 0.512383i \(0.171237\pi\)
−0.858757 + 0.512383i \(0.828763\pi\)
\(242\) −16.3653 −1.05200
\(243\) 0 0
\(244\) 51.6622 3.30733
\(245\) −7.86022 9.13938i −0.502171 0.583893i
\(246\) 0 0
\(247\) −9.63763 + 17.3839i −0.613228 + 1.10611i
\(248\) 17.5864i 1.11674i
\(249\) 0 0
\(250\) 23.8614 14.9626i 1.50913 0.946321i
\(251\) 9.30807 0.587520 0.293760 0.955879i \(-0.405093\pi\)
0.293760 + 0.955879i \(0.405093\pi\)
\(252\) 0 0
\(253\) 8.48862 0.533675
\(254\) 46.3290i 2.90694i
\(255\) 0 0
\(256\) −31.4667 −1.96667
\(257\) 9.88740i 0.616759i −0.951263 0.308379i \(-0.900213\pi\)
0.951263 0.308379i \(-0.0997866\pi\)
\(258\) 0 0
\(259\) −13.3839 −0.831632
\(260\) −7.10099 34.3112i −0.440384 2.12789i
\(261\) 0 0
\(262\) 43.7920 2.70548
\(263\) 6.69193i 0.412642i 0.978484 + 0.206321i \(0.0661491\pi\)
−0.978484 + 0.206321i \(0.933851\pi\)
\(264\) 0 0
\(265\) 5.92762 5.09798i 0.364131 0.313166i
\(266\) 17.6160i 1.08011i
\(267\) 0 0
\(268\) 1.38797 0.0847836
\(269\) −23.6848 −1.44409 −0.722045 0.691846i \(-0.756798\pi\)
−0.722045 + 0.691846i \(0.756798\pi\)
\(270\) 0 0
\(271\) 3.80546i 0.231165i 0.993298 + 0.115583i \(0.0368735\pi\)
−0.993298 + 0.115583i \(0.963126\pi\)
\(272\) 44.5793i 2.70302i
\(273\) 0 0
\(274\) 13.7369 0.829877
\(275\) 10.4913 + 1.58786i 0.632649 + 0.0957513i
\(276\) 0 0
\(277\) 2.61615i 0.157189i 0.996907 + 0.0785945i \(0.0250432\pi\)
−0.996907 + 0.0785945i \(0.974957\pi\)
\(278\) 43.7920 2.62647
\(279\) 0 0
\(280\) 10.9301 + 12.7089i 0.653201 + 0.759502i
\(281\) 14.4162i 0.860001i −0.902829 0.430000i \(-0.858513\pi\)
0.902829 0.430000i \(-0.141487\pi\)
\(282\) 0 0
\(283\) 4.00000i 0.237775i 0.992908 + 0.118888i \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) 59.9156i 3.55534i
\(285\) 0 0
\(286\) 9.34596 16.8578i 0.552638 0.996820i
\(287\) 1.08287i 0.0639196i
\(288\) 0 0
\(289\) −34.7748 −2.04558
\(290\) 17.2334 + 20.0379i 1.01198 + 1.17667i
\(291\) 0 0
\(292\) −27.4088 −1.60398
\(293\) −13.1120 −0.766014 −0.383007 0.923746i \(-0.625111\pi\)
−0.383007 + 0.923746i \(0.625111\pi\)
\(294\) 0 0
\(295\) 15.0942 12.9816i 0.878818 0.755817i
\(296\) −62.3541 −3.62426
\(297\) 0 0
\(298\) 8.54143i 0.494792i
\(299\) 6.99291 12.6135i 0.404411 0.729455i
\(300\) 0 0
\(301\) 4.05341i 0.233635i
\(302\) 7.49646i 0.431373i
\(303\) 0 0
\(304\) 34.1544i 1.95889i
\(305\) 17.3323 + 20.1530i 0.992446 + 1.15395i
\(306\) 0 0
\(307\) 28.3222 1.61643 0.808217 0.588885i \(-0.200433\pi\)
0.808217 + 0.588885i \(0.200433\pi\)
\(308\) 11.6990i 0.666613i
\(309\) 0 0
\(310\) −12.7089 + 10.9301i −0.721817 + 0.620791i
\(311\) 2.08995 0.118510 0.0592552 0.998243i \(-0.481127\pi\)
0.0592552 + 0.998243i \(0.481127\pi\)
\(312\) 0 0
\(313\) 16.8803i 0.954131i 0.878868 + 0.477066i \(0.158300\pi\)
−0.878868 + 0.477066i \(0.841700\pi\)
\(314\) 20.1530i 1.13730i
\(315\) 0 0
\(316\) −3.49646 −0.196691
\(317\) 7.04091 0.395457 0.197729 0.980257i \(-0.436644\pi\)
0.197729 + 0.980257i \(0.436644\pi\)
\(318\) 0 0
\(319\) 9.95700i 0.557485i
\(320\) −4.15470 4.83083i −0.232255 0.270052i
\(321\) 0 0
\(322\) 12.7819i 0.712306i
\(323\) −39.6672 −2.20714
\(324\) 0 0
\(325\) 11.0021 14.2812i 0.610289 0.792179i
\(326\) 5.10552 0.282768
\(327\) 0 0
\(328\) 5.04498i 0.278562i
\(329\) −2.69193 −0.148411
\(330\) 0 0
\(331\) 18.8845i 1.03798i 0.854779 + 0.518992i \(0.173693\pi\)
−0.854779 + 0.518992i \(0.826307\pi\)
\(332\) 5.09798 0.279788
\(333\) 0 0
\(334\) −16.0379 −0.877554
\(335\) 0.465654 + 0.541434i 0.0254414 + 0.0295817i
\(336\) 0 0
\(337\) 15.2713i 0.831878i −0.909393 0.415939i \(-0.863453\pi\)
0.909393 0.415939i \(-0.136547\pi\)
\(338\) −17.3502 27.7748i −0.943725 1.51075i
\(339\) 0 0
\(340\) 53.0148 45.5948i 2.87513 2.47272i
\(341\) −6.31516 −0.341985
\(342\) 0 0
\(343\) 15.7177 0.848679
\(344\) 18.8845i 1.01818i
\(345\) 0 0
\(346\) 41.0071i 2.20455i
\(347\) 6.61615i 0.355173i 0.984105 + 0.177587i \(0.0568290\pi\)
−0.984105 + 0.177587i \(0.943171\pi\)
\(348\) 0 0
\(349\) 20.1530i 1.07876i 0.842062 + 0.539382i \(0.181342\pi\)
−0.842062 + 0.539382i \(0.818658\pi\)
\(350\) −2.39094 + 15.7974i −0.127801 + 0.844409i
\(351\) 0 0
\(352\) 8.03789i 0.428421i
\(353\) −18.0665 −0.961583 −0.480792 0.876835i \(-0.659651\pi\)
−0.480792 + 0.876835i \(0.659651\pi\)
\(354\) 0 0
\(355\) 23.3725 20.1013i 1.24049 1.06687i
\(356\) 33.1812i 1.75860i
\(357\) 0 0
\(358\) −61.4080 −3.24551
\(359\) 29.8860i 1.57732i 0.614827 + 0.788662i \(0.289226\pi\)
−0.614827 + 0.788662i \(0.710774\pi\)
\(360\) 0 0
\(361\) −11.3909 −0.599523
\(362\) 15.1147 0.794412
\(363\) 0 0
\(364\) 17.3839 + 9.63763i 0.911161 + 0.505149i
\(365\) −9.19547 10.6919i −0.481313 0.559641i
\(366\) 0 0
\(367\) 11.1955i 0.584399i −0.956357 0.292199i \(-0.905613\pi\)
0.956357 0.292199i \(-0.0943871\pi\)
\(368\) 24.7819i 1.29184i
\(369\) 0 0
\(370\) −38.7538 45.0605i −2.01471 2.34258i
\(371\) 4.43521i 0.230265i
\(372\) 0 0
\(373\) 26.8945i 1.39254i 0.717778 + 0.696272i \(0.245159\pi\)
−0.717778 + 0.696272i \(0.754841\pi\)
\(374\) 38.4667 1.98907
\(375\) 0 0
\(376\) −12.5414 −0.646775
\(377\) 14.7953 + 8.20256i 0.761999 + 0.422453i
\(378\) 0 0
\(379\) 28.2027i 1.44868i −0.689445 0.724338i \(-0.742145\pi\)
0.689445 0.724338i \(-0.257855\pi\)
\(380\) 40.6172 34.9324i 2.08362 1.79199i
\(381\) 0 0
\(382\) −43.0338 −2.20180
\(383\) −13.7865 −0.704457 −0.352228 0.935914i \(-0.614576\pi\)
−0.352228 + 0.935914i \(0.614576\pi\)
\(384\) 0 0
\(385\) −4.56368 + 3.92494i −0.232587 + 0.200034i
\(386\) −5.19547 −0.264442
\(387\) 0 0
\(388\) −23.2840 −1.18206
\(389\) −15.6848 −0.795253 −0.397626 0.917547i \(-0.630166\pi\)
−0.397626 + 0.917547i \(0.630166\pi\)
\(390\) 0 0
\(391\) 28.7819 1.45556
\(392\) 31.8592 1.60913
\(393\) 0 0
\(394\) −20.1278 −1.01403
\(395\) −1.17304 1.36394i −0.0590219 0.0686271i
\(396\) 0 0
\(397\) 4.59939 0.230837 0.115418 0.993317i \(-0.463179\pi\)
0.115418 + 0.993317i \(0.463179\pi\)
\(398\) −20.6632 −1.03575
\(399\) 0 0
\(400\) −4.63563 + 30.6285i −0.231781 + 1.53143i
\(401\) 11.0016i 0.549392i −0.961531 0.274696i \(-0.911423\pi\)
0.961531 0.274696i \(-0.0885772\pi\)
\(402\) 0 0
\(403\) −5.20242 + 9.38385i −0.259151 + 0.467443i
\(404\) 36.4667 1.81429
\(405\) 0 0
\(406\) −14.9929 −0.744086
\(407\) 22.3909i 1.10988i
\(408\) 0 0
\(409\) 28.8325i 1.42567i 0.701330 + 0.712837i \(0.252590\pi\)
−0.701330 + 0.712837i \(0.747410\pi\)
\(410\) −3.64578 + 3.13551i −0.180052 + 0.154852i
\(411\) 0 0
\(412\) 79.0460i 3.89432i
\(413\) 11.2939i 0.555736i
\(414\) 0 0
\(415\) 1.71034 + 1.98868i 0.0839572 + 0.0976203i
\(416\) 11.9437 + 6.62160i 0.585588 + 0.324651i
\(417\) 0 0
\(418\) 29.4712 1.44148
\(419\) −24.3768 −1.19088 −0.595441 0.803399i \(-0.703023\pi\)
−0.595441 + 0.803399i \(0.703023\pi\)
\(420\) 0 0
\(421\) 9.95700i 0.485274i −0.970117 0.242637i \(-0.921988\pi\)
0.970117 0.242637i \(-0.0780125\pi\)
\(422\) −9.56621 −0.465676
\(423\) 0 0
\(424\) 20.6632i 1.00349i
\(425\) 35.5722 + 5.38385i 1.72551 + 0.261155i
\(426\) 0 0
\(427\) −15.0790 −0.729724
\(428\) 1.30807i 0.0632281i
\(429\) 0 0
\(430\) −13.6469 + 11.7369i −0.658114 + 0.566004i
\(431\) 4.65913i 0.224422i −0.993684 0.112211i \(-0.964207\pi\)
0.993684 0.112211i \(-0.0357933\pi\)
\(432\) 0 0
\(433\) 2.89448i 0.139100i −0.997578 0.0695500i \(-0.977844\pi\)
0.997578 0.0695500i \(-0.0221563\pi\)
\(434\) 9.50916i 0.456454i
\(435\) 0 0
\(436\) 58.1128i 2.78310i
\(437\) 22.0512 1.05485
\(438\) 0 0
\(439\) 5.38385 0.256957 0.128479 0.991712i \(-0.458991\pi\)
0.128479 + 0.991712i \(0.458991\pi\)
\(440\) −21.2618 + 18.2859i −1.01361 + 0.871748i
\(441\) 0 0
\(442\) 31.6888 57.1586i 1.50728 2.71876i
\(443\) 7.29390i 0.346544i 0.984874 + 0.173272i \(0.0554339\pi\)
−0.984874 + 0.173272i \(0.944566\pi\)
\(444\) 0 0
\(445\) −12.9437 + 11.1321i −0.613590 + 0.527711i
\(446\) 48.0531 2.27538
\(447\) 0 0
\(448\) 3.61456 0.170772
\(449\) 22.4750i 1.06066i −0.847791 0.530330i \(-0.822068\pi\)
0.847791 0.530330i \(-0.177932\pi\)
\(450\) 0 0
\(451\) −1.81162 −0.0853057
\(452\) 54.6693i 2.57143i
\(453\) 0 0
\(454\) 43.5117 2.04211
\(455\) 2.07262 + 10.0146i 0.0971657 + 0.469494i
\(456\) 0 0
\(457\) 20.5080 0.959325 0.479663 0.877453i \(-0.340759\pi\)
0.479663 + 0.877453i \(0.340759\pi\)
\(458\) 46.4667i 2.17125i
\(459\) 0 0
\(460\) −29.4712 + 25.3464i −1.37410 + 1.18178i
\(461\) 1.68331i 0.0783994i 0.999231 + 0.0391997i \(0.0124808\pi\)
−0.999231 + 0.0391997i \(0.987519\pi\)
\(462\) 0 0
\(463\) −3.80546 −0.176855 −0.0884274 0.996083i \(-0.528184\pi\)
−0.0884274 + 0.996083i \(0.528184\pi\)
\(464\) −29.0687 −1.34948
\(465\) 0 0
\(466\) 28.9609i 1.34159i
\(467\) 15.7748i 0.729970i 0.931013 + 0.364985i \(0.118926\pi\)
−0.931013 + 0.364985i \(0.881074\pi\)
\(468\) 0 0
\(469\) −0.405116 −0.0187065
\(470\) −7.79464 9.06313i −0.359540 0.418051i
\(471\) 0 0
\(472\) 52.6172i 2.42190i
\(473\) −6.78128 −0.311804
\(474\) 0 0
\(475\) 27.2536 + 4.12483i 1.25048 + 0.189260i
\(476\) 39.6672i 1.81814i
\(477\) 0 0
\(478\) 18.1278i 0.829148i
\(479\) 26.5194i 1.21170i 0.795578 + 0.605852i \(0.207168\pi\)
−0.795578 + 0.605852i \(0.792832\pi\)
\(480\) 0 0
\(481\) −33.2713 18.4456i −1.51704 0.841048i
\(482\) 40.0758i 1.82540i
\(483\) 0 0
\(484\) 28.2334 1.28333
\(485\) −7.81162 9.08287i −0.354707 0.412432i
\(486\) 0 0
\(487\) −17.1771 −0.778370 −0.389185 0.921160i \(-0.627243\pi\)
−0.389185 + 0.921160i \(0.627243\pi\)
\(488\) −70.2517 −3.18014
\(489\) 0 0
\(490\) 19.8008 + 23.0232i 0.894510 + 1.04008i
\(491\) −16.7061 −0.753936 −0.376968 0.926226i \(-0.623033\pi\)
−0.376968 + 0.926226i \(0.623033\pi\)
\(492\) 0 0
\(493\) 33.7606i 1.52050i
\(494\) 24.2783 43.7920i 1.09233 1.97030i
\(495\) 0 0
\(496\) 18.4366i 0.827829i
\(497\) 17.4880i 0.784443i
\(498\) 0 0
\(499\) 0.390787i 0.0174940i −0.999962 0.00874702i \(-0.997216\pi\)
0.999962 0.00874702i \(-0.00278430\pi\)
\(500\) −41.1654 + 25.8134i −1.84097 + 1.15441i
\(501\) 0 0
\(502\) −23.4481 −1.04654
\(503\) 43.4738i 1.93840i −0.246272 0.969201i \(-0.579206\pi\)
0.246272 0.969201i \(-0.420794\pi\)
\(504\) 0 0
\(505\) 12.2343 + 14.2253i 0.544421 + 0.633019i
\(506\) −21.3839 −0.950628
\(507\) 0 0
\(508\) 79.9264i 3.54616i
\(509\) 6.80532i 0.301640i 0.988561 + 0.150820i \(0.0481914\pi\)
−0.988561 + 0.150820i \(0.951809\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) 49.7615 2.19917
\(513\) 0 0
\(514\) 24.9075i 1.09862i
\(515\) 30.8352 26.5194i 1.35876 1.16859i
\(516\) 0 0
\(517\) 4.50354i 0.198066i
\(518\) 33.7155 1.48138
\(519\) 0 0
\(520\) 9.65612 + 46.6573i 0.423449 + 2.04606i
\(521\) 14.7061 0.644286 0.322143 0.946691i \(-0.395597\pi\)
0.322143 + 0.946691i \(0.395597\pi\)
\(522\) 0 0
\(523\) 18.5935i 0.813037i 0.913643 + 0.406518i \(0.133257\pi\)
−0.913643 + 0.406518i \(0.866743\pi\)
\(524\) −75.5496 −3.30040
\(525\) 0 0
\(526\) 16.8578i 0.735033i
\(527\) −21.4125 −0.932741
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) −14.9324 + 12.8424i −0.648621 + 0.557839i
\(531\) 0 0
\(532\) 30.3909i 1.31761i
\(533\) −1.49241 + 2.69193i −0.0646433 + 0.116600i
\(534\) 0 0
\(535\) −0.510268 + 0.438850i −0.0220608 + 0.0189731i
\(536\) −1.88740 −0.0815231
\(537\) 0 0
\(538\) 59.6649 2.57234
\(539\) 11.4404i 0.492774i
\(540\) 0 0
\(541\) 32.6469i 1.40360i −0.712374 0.701801i \(-0.752380\pi\)
0.712374 0.701801i \(-0.247620\pi\)
\(542\) 9.58641i 0.411772i
\(543\) 0 0
\(544\) 27.2536i 1.16849i
\(545\) −22.6693 + 19.4965i −0.971045 + 0.835136i
\(546\) 0 0
\(547\) 18.5935i 0.795001i 0.917602 + 0.397500i \(0.130122\pi\)
−0.917602 + 0.397500i \(0.869878\pi\)
\(548\) −23.6988 −1.01236
\(549\) 0 0
\(550\) −26.4288 4.00000i −1.12693 0.170561i
\(551\) 25.8656i 1.10191i
\(552\) 0 0
\(553\) 1.02054 0.0433976
\(554\) 6.59039i 0.279999i
\(555\) 0 0
\(556\) −75.5496 −3.20402
\(557\) −8.22899 −0.348674 −0.174337 0.984686i \(-0.555778\pi\)
−0.174337 + 0.984686i \(0.555778\pi\)
\(558\) 0 0
\(559\) −5.58641 + 10.0765i −0.236280 + 0.426190i
\(560\) −11.4586 13.3233i −0.484213 0.563013i
\(561\) 0 0
\(562\) 36.3162i 1.53191i
\(563\) 21.0829i 0.888537i −0.895894 0.444268i \(-0.853464\pi\)
0.895894 0.444268i \(-0.146536\pi\)
\(564\) 0 0
\(565\) −21.3260 + 18.3412i −0.897191 + 0.771619i
\(566\) 10.0765i 0.423546i
\(567\) 0 0
\(568\) 81.4749i 3.41861i
\(569\) 24.3909 1.02252 0.511261 0.859426i \(-0.329179\pi\)
0.511261 + 0.859426i \(0.329179\pi\)
\(570\) 0 0
\(571\) −11.7974 −0.493708 −0.246854 0.969053i \(-0.579397\pi\)
−0.246854 + 0.969053i \(0.579397\pi\)
\(572\) −16.1236 + 29.0829i −0.674161 + 1.21602i
\(573\) 0 0
\(574\) 2.72787i 0.113859i
\(575\) −19.7748 2.99291i −0.824666 0.124813i
\(576\) 0 0
\(577\) 22.2868 0.927811 0.463906 0.885885i \(-0.346448\pi\)
0.463906 + 0.885885i \(0.346448\pi\)
\(578\) 87.6019 3.64376
\(579\) 0 0
\(580\) −29.7308 34.5692i −1.23451 1.43541i
\(581\) −1.48798 −0.0617319
\(582\) 0 0
\(583\) −7.42002 −0.307306
\(584\) 37.2713 1.54230
\(585\) 0 0
\(586\) 33.0308 1.36449
\(587\) 39.9625 1.64943 0.824715 0.565549i \(-0.191336\pi\)
0.824715 + 0.565549i \(0.191336\pi\)
\(588\) 0 0
\(589\) −16.4051 −0.675961
\(590\) −38.0241 + 32.7022i −1.56543 + 1.34633i
\(591\) 0 0
\(592\) 65.3686 2.68663
\(593\) 31.6291 1.29885 0.649425 0.760425i \(-0.275010\pi\)
0.649425 + 0.760425i \(0.275010\pi\)
\(594\) 0 0
\(595\) −15.4738 + 13.3081i −0.634364 + 0.545578i
\(596\) 14.7356i 0.603594i
\(597\) 0 0
\(598\) −17.6160 + 31.7748i −0.720371 + 1.29937i
\(599\) 20.3768 0.832572 0.416286 0.909234i \(-0.363331\pi\)
0.416286 + 0.909234i \(0.363331\pi\)
\(600\) 0 0
\(601\) −1.49646 −0.0610417 −0.0305209 0.999534i \(-0.509717\pi\)
−0.0305209 + 0.999534i \(0.509717\pi\)
\(602\) 10.2110i 0.416171i
\(603\) 0 0
\(604\) 12.9328i 0.526229i
\(605\) 9.47210 + 11.0136i 0.385096 + 0.447766i
\(606\) 0 0
\(607\) 4.20256i 0.170577i −0.996356 0.0852883i \(-0.972819\pi\)
0.996356 0.0852883i \(-0.0271811\pi\)
\(608\) 20.8803i 0.846808i
\(609\) 0 0
\(610\) −43.6622 50.7677i −1.76783 2.05552i
\(611\) −6.69193 3.71001i −0.270726 0.150091i
\(612\) 0 0
\(613\) −6.11583 −0.247016 −0.123508 0.992344i \(-0.539414\pi\)
−0.123508 + 0.992344i \(0.539414\pi\)
\(614\) −71.3470 −2.87933
\(615\) 0 0
\(616\) 15.9086i 0.640978i
\(617\) 21.6721 0.872485 0.436243 0.899829i \(-0.356309\pi\)
0.436243 + 0.899829i \(0.356309\pi\)
\(618\) 0 0
\(619\) 21.4214i 0.861001i −0.902590 0.430500i \(-0.858337\pi\)
0.902590 0.430500i \(-0.141663\pi\)
\(620\) 21.9253 18.8566i 0.880541 0.757299i
\(621\) 0 0
\(622\) −5.26485 −0.211101
\(623\) 9.68484i 0.388015i
\(624\) 0 0
\(625\) −23.8803 7.39803i −0.955212 0.295921i
\(626\) 42.5235i 1.69958i
\(627\) 0 0
\(628\) 34.7677i 1.38738i
\(629\) 75.9197i 3.02712i
\(630\) 0 0
\(631\) 21.6123i 0.860374i −0.902740 0.430187i \(-0.858448\pi\)
0.902740 0.430187i \(-0.141552\pi\)
\(632\) 4.75458 0.189127
\(633\) 0 0
\(634\) −17.7369 −0.704422
\(635\) 31.1786 26.8148i 1.23728 1.06411i
\(636\) 0 0
\(637\) 16.9996 + 9.42459i 0.673548 + 0.373416i
\(638\) 25.0829i 0.993040i
\(639\) 0 0
\(640\) 21.5112 + 25.0119i 0.850303 + 0.988680i
\(641\) 3.91005 0.154437 0.0772187 0.997014i \(-0.475396\pi\)
0.0772187 + 0.997014i \(0.475396\pi\)
\(642\) 0 0
\(643\) 28.9609 1.14211 0.571054 0.820912i \(-0.306535\pi\)
0.571054 + 0.820912i \(0.306535\pi\)
\(644\) 22.0512i 0.868939i
\(645\) 0 0
\(646\) 99.9264 3.93155
\(647\) 18.4667i 0.726002i −0.931789 0.363001i \(-0.881752\pi\)
0.931789 0.363001i \(-0.118248\pi\)
\(648\) 0 0
\(649\) −18.8945 −0.741673
\(650\) −27.7157 + 35.9761i −1.08710 + 1.41110i
\(651\) 0 0
\(652\) −8.80799 −0.344948
\(653\) 31.3470i 1.22670i 0.789810 + 0.613352i \(0.210179\pi\)
−0.789810 + 0.613352i \(0.789821\pi\)
\(654\) 0 0
\(655\) −25.3464 29.4712i −0.990365 1.15154i
\(656\) 5.28888i 0.206496i
\(657\) 0 0
\(658\) 6.78128 0.264362
\(659\) −32.8577 −1.27995 −0.639976 0.768395i \(-0.721056\pi\)
−0.639976 + 0.768395i \(0.721056\pi\)
\(660\) 0 0
\(661\) 3.17571i 0.123521i 0.998091 + 0.0617605i \(0.0196715\pi\)
−0.998091 + 0.0617605i \(0.980329\pi\)
\(662\) 47.5722i 1.84895i
\(663\) 0 0
\(664\) −6.93237 −0.269028
\(665\) −11.8552 + 10.1960i −0.459726 + 0.395382i
\(666\) 0 0
\(667\) 18.7677i 0.726688i
\(668\) 27.6684 1.07052
\(669\) 0 0
\(670\) −1.17304 1.36394i −0.0453184 0.0526935i
\(671\) 25.2269i 0.973874i
\(672\) 0 0
\(673\) 30.3909i 1.17148i −0.810497 0.585742i \(-0.800803\pi\)
0.810497 0.585742i \(-0.199197\pi\)
\(674\) 38.4701i 1.48181i
\(675\) 0 0
\(676\) 29.9324 + 47.9168i 1.15125 + 1.84296i
\(677\) 14.2642i 0.548216i −0.961699 0.274108i \(-0.911617\pi\)
0.961699 0.274108i \(-0.0883826\pi\)
\(678\) 0 0
\(679\) 6.79606 0.260809
\(680\) −72.0910 + 62.0011i −2.76456 + 2.37763i
\(681\) 0 0
\(682\) 15.9086 0.609174
\(683\) 15.6133 0.597427 0.298713 0.954343i \(-0.403443\pi\)
0.298713 + 0.954343i \(0.403443\pi\)
\(684\) 0 0
\(685\) −7.95079 9.24468i −0.303784 0.353221i
\(686\) −39.5949 −1.51174
\(687\) 0 0
\(688\) 19.7974i 0.754770i
\(689\) −6.11260 + 11.0256i −0.232872 + 0.420042i
\(690\) 0 0
\(691\) 5.70370i 0.216979i 0.994098 + 0.108489i \(0.0346014\pi\)
−0.994098 + 0.108489i \(0.965399\pi\)
\(692\) 70.7451i 2.68932i
\(693\) 0 0
\(694\) 16.6669i 0.632666i
\(695\) −25.3464 29.4712i −0.961443 1.11791i
\(696\) 0 0
\(697\) −6.14255 −0.232666
\(698\) 50.7677i 1.92159i
\(699\) 0 0
\(700\) 4.12483 27.2536i 0.155904 1.03009i
\(701\) 17.0687 0.644676 0.322338 0.946625i \(-0.395531\pi\)
0.322338 + 0.946625i \(0.395531\pi\)
\(702\) 0 0
\(703\) 58.1657i 2.19376i
\(704\) 6.04710i 0.227909i
\(705\) 0 0
\(706\) 45.5117 1.71286
\(707\) −10.6438 −0.400301
\(708\) 0 0
\(709\) 22.4990i 0.844969i 0.906370 + 0.422484i \(0.138842\pi\)
−0.906370 + 0.422484i \(0.861158\pi\)
\(710\) −58.8782 + 50.6375i −2.20966 + 1.90039i
\(711\) 0 0
\(712\) 45.1208i 1.69097i
\(713\) 11.9033 0.445782
\(714\) 0 0
\(715\) −16.7543 + 3.46745i −0.626576 + 0.129675i
\(716\) 105.941 3.95918
\(717\) 0 0
\(718\) 75.2865i 2.80967i
\(719\) 25.0829 0.935433 0.467717 0.883879i \(-0.345077\pi\)
0.467717 + 0.883879i \(0.345077\pi\)
\(720\) 0 0
\(721\) 23.0717i 0.859236i
\(722\) 28.6951 1.06792
\(723\) 0 0
\(724\) −26.0758 −0.969099
\(725\) 3.51063 23.1955i 0.130382 0.861458i
\(726\) 0 0
\(727\) 0.376766i 0.0139735i −0.999976 0.00698673i \(-0.997776\pi\)
0.999976 0.00698673i \(-0.00222396\pi\)
\(728\) −23.6390 13.1055i −0.876121 0.485723i
\(729\) 0 0
\(730\) 23.1645 + 26.9342i 0.857356 + 0.996881i
\(731\) −22.9929 −0.850424
\(732\) 0 0
\(733\) 3.45937 0.127775 0.0638874 0.997957i \(-0.479650\pi\)
0.0638874 + 0.997957i \(0.479650\pi\)
\(734\) 28.2027i 1.04098i
\(735\) 0 0
\(736\) 15.1504i 0.558452i
\(737\) 0.677751i 0.0249653i
\(738\) 0 0
\(739\) 30.5488i 1.12376i −0.827220 0.561878i \(-0.810079\pi\)
0.827220 0.561878i \(-0.189921\pi\)
\(740\) 66.8577 + 77.7380i 2.45774 + 2.85770i
\(741\) 0 0
\(742\) 11.1728i 0.410167i
\(743\) 9.73308 0.357072 0.178536 0.983933i \(-0.442864\pi\)
0.178536 + 0.983933i \(0.442864\pi\)
\(744\) 0 0
\(745\) −5.74823 + 4.94370i −0.210599 + 0.181123i
\(746\) 67.7504i 2.48052i
\(747\) 0 0
\(748\) −66.3625 −2.42645
\(749\) 0.381797i 0.0139505i
\(750\) 0 0
\(751\) 34.7677 1.26869 0.634346 0.773049i \(-0.281270\pi\)
0.634346 + 0.773049i \(0.281270\pi\)
\(752\) 13.1478 0.479449
\(753\) 0 0
\(754\) −37.2713 20.6632i −1.35734 0.752510i
\(755\) 5.04498 4.33888i 0.183606 0.157908i
\(756\) 0 0
\(757\) 30.6693i 1.11469i 0.830280 + 0.557347i \(0.188181\pi\)
−0.830280 + 0.557347i \(0.811819\pi\)
\(758\) 71.0460i 2.58051i
\(759\) 0 0
\(760\) −55.2324 + 47.5020i −2.00349 + 1.72308i
\(761\) 2.51294i 0.0910941i −0.998962 0.0455470i \(-0.985497\pi\)
0.998962 0.0455470i \(-0.0145031\pi\)
\(762\) 0 0
\(763\) 16.9618i 0.614058i
\(764\) 74.2415 2.68596
\(765\) 0 0
\(766\) 34.7298 1.25484
\(767\) −15.5652 + 28.0758i −0.562028 + 1.01376i
\(768\) 0 0
\(769\) 32.8378i 1.18416i 0.805878 + 0.592081i \(0.201694\pi\)
−0.805878 + 0.592081i \(0.798306\pi\)
\(770\) 11.4965 9.88740i 0.414304 0.356317i
\(771\) 0 0
\(772\) 8.96318 0.322592
\(773\) −28.3339 −1.01910 −0.509550 0.860441i \(-0.670188\pi\)
−0.509550 + 0.860441i \(0.670188\pi\)
\(774\) 0 0
\(775\) 14.7116 + 2.22660i 0.528455 + 0.0799817i
\(776\) 31.6622 1.13661
\(777\) 0 0
\(778\) 39.5120 1.41657
\(779\) −4.70610 −0.168614
\(780\) 0 0
\(781\) −29.2571 −1.04690
\(782\) −72.5050 −2.59277
\(783\) 0 0
\(784\) −33.3994 −1.19284
\(785\) −13.5626 + 11.6643i −0.484069 + 0.416318i
\(786\) 0 0
\(787\) 19.8336 0.706991 0.353496 0.935436i \(-0.384993\pi\)
0.353496 + 0.935436i \(0.384993\pi\)
\(788\) 34.7244 1.23700
\(789\) 0 0
\(790\) 2.95502 + 3.43592i 0.105135 + 0.122244i
\(791\) 15.9567i 0.567355i
\(792\) 0 0
\(793\) −37.4853 20.7819i −1.33114 0.737986i
\(794\) −11.5864 −0.411186
\(795\) 0 0
\(796\) 35.6480 1.26351
\(797\) 26.3683i 0.934013i 0.884254 + 0.467006i \(0.154668\pi\)
−0.884254 + 0.467006i \(0.845332\pi\)
\(798\) 0 0
\(799\) 15.2699i 0.540210i
\(800\) 2.83399 18.7248i 0.100197 0.662021i
\(801\) 0 0
\(802\) 27.7143i 0.978624i
\(803\) 13.3839i 0.472306i
\(804\) 0 0
\(805\) 8.60197 7.39803i 0.303180 0.260746i
\(806\) 13.1055 23.6390i 0.461622 0.832650i
\(807\) 0 0
\(808\) −49.5885 −1.74452
\(809\) −15.6848 −0.551450 −0.275725 0.961237i \(-0.588918\pi\)
−0.275725 + 0.961237i \(0.588918\pi\)
\(810\) 0 0
\(811\) 31.6174i 1.11024i −0.831771 0.555119i \(-0.812673\pi\)
0.831771 0.555119i \(-0.187327\pi\)
\(812\) 25.8656 0.907706
\(813\) 0 0
\(814\) 56.4055i 1.97701i
\(815\) −2.95502 3.43592i −0.103510 0.120355i
\(816\) 0 0
\(817\) −17.6160 −0.616305
\(818\) 72.6325i 2.53953i
\(819\) 0 0
\(820\) 6.28966 5.40935i 0.219645 0.188903i
\(821\) 13.5385i 0.472498i 0.971693 + 0.236249i \(0.0759181\pi\)
−0.971693 + 0.236249i \(0.924082\pi\)
\(822\) 0 0
\(823\) 30.3683i 1.05857i 0.848443 + 0.529286i \(0.177540\pi\)
−0.848443 + 0.529286i \(0.822460\pi\)
\(824\) 107.489i 3.74456i
\(825\) 0 0
\(826\) 28.4507i 0.989926i
\(827\) −36.9777 −1.28584 −0.642920 0.765933i \(-0.722277\pi\)
−0.642920 + 0.765933i \(0.722277\pi\)
\(828\) 0 0
\(829\) 49.6764 1.72533 0.862666 0.505774i \(-0.168793\pi\)
0.862666 + 0.505774i \(0.168793\pi\)
\(830\) −4.30855 5.00971i −0.149552 0.173890i
\(831\) 0 0
\(832\) 8.98553 + 4.98159i 0.311517 + 0.172706i
\(833\) 38.7904i 1.34401i
\(834\) 0 0
\(835\) 9.28257 + 10.7932i 0.321237 + 0.373514i
\(836\) −50.8435 −1.75846
\(837\) 0 0
\(838\) 61.4080 2.12130
\(839\) 35.8377i 1.23725i −0.785685 0.618627i \(-0.787689\pi\)
0.785685 0.618627i \(-0.212311\pi\)
\(840\) 0 0
\(841\) −6.98582 −0.240891
\(842\) 25.0829i 0.864413i
\(843\) 0 0
\(844\) 16.5035 0.568075
\(845\) −8.64981 + 27.7521i −0.297563 + 0.954702i
\(846\) 0 0
\(847\) −8.24067 −0.283153
\(848\) 21.6622i 0.743883i
\(849\) 0 0
\(850\) −89.6107 13.5626i −3.07362 0.465192i
\(851\) 42.2041i 1.44674i
\(852\) 0 0
\(853\) 11.1898 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(854\) 37.9858 1.29985
\(855\) 0 0
\(856\) 1.77875i 0.0607966i
\(857\) 15.1197i 0.516479i −0.966081 0.258239i \(-0.916858\pi\)
0.966081 0.258239i \(-0.0831423\pi\)
\(858\) 0 0
\(859\) −23.5722 −0.804274 −0.402137 0.915579i \(-0.631733\pi\)
−0.402137 + 0.915579i \(0.631733\pi\)
\(860\) 23.5436 20.2484i 0.802830 0.690465i
\(861\) 0 0
\(862\) 11.7369i 0.399761i
\(863\) 39.2524 1.33617 0.668083 0.744087i \(-0.267115\pi\)
0.668083 + 0.744087i \(0.267115\pi\)
\(864\) 0 0
\(865\) 27.5970 23.7345i 0.938327 0.806997i
\(866\) 7.29155i 0.247777i
\(867\) 0 0
\(868\) 16.4051i 0.556826i
\(869\) 1.70734i 0.0579174i
\(870\) 0 0
\(871\) −1.00709 0.558331i −0.0341239 0.0189183i
\(872\) 79.0234i 2.67607i
\(873\) 0 0
\(874\) −55.5496 −1.87899
\(875\) 12.0152 7.53435i 0.406189 0.254707i
\(876\) 0 0
\(877\) −22.8541 −0.771728 −0.385864 0.922556i \(-0.626097\pi\)
−0.385864 + 0.922556i \(0.626097\pi\)
\(878\) −13.5626 −0.457715
\(879\) 0 0
\(880\) 22.2897 19.1700i 0.751384 0.646220i
\(881\) 27.1586 0.914998 0.457499 0.889210i \(-0.348745\pi\)
0.457499 + 0.889210i \(0.348745\pi\)
\(882\) 0 0
\(883\) 36.1516i 1.21660i 0.793708 + 0.608298i \(0.208148\pi\)
−0.793708 + 0.608298i \(0.791852\pi\)
\(884\) −54.6693 + 98.6096i −1.83873 + 3.31660i
\(885\) 0 0
\(886\) 18.3742i 0.617293i
\(887\) 2.99291i 0.100492i −0.998737 0.0502461i \(-0.983999\pi\)
0.998737 0.0502461i \(-0.0160006\pi\)
\(888\) 0 0
\(889\) 23.3287i 0.782418i
\(890\) 32.6067 28.0430i 1.09298 0.940005i
\(891\) 0 0
\(892\) −82.9008 −2.77573
\(893\) 11.6990i 0.391493i
\(894\) 0 0
\(895\) 35.5423 + 41.3264i 1.18805 + 1.38139i
\(896\) −18.7146 −0.625210
\(897\) 0 0
\(898\) 56.6172i 1.88934i
\(899\) 13.9623i 0.465670i
\(900\) 0 0
\(901\) −25.1586 −0.838156
\(902\) 4.56368 0.151954
\(903\) 0 0
\(904\) 74.3408i 2.47254i
\(905\) −8.74825 10.1719i −0.290802 0.338126i
\(906\) 0 0
\(907\) 22.1657i 0.736001i 0.929826 + 0.368001i \(0.119958\pi\)
−0.929826 + 0.368001i \(0.880042\pi\)
\(908\) −75.0660 −2.49115
\(909\) 0 0
\(910\) −5.22117 25.2281i −0.173080 0.836303i
\(911\) 14.9171 0.494227 0.247113 0.968987i \(-0.420518\pi\)
0.247113 + 0.968987i \(0.420518\pi\)
\(912\) 0 0
\(913\) 2.48937i 0.0823861i
\(914\) −51.6622 −1.70883
\(915\) 0 0
\(916\) 80.1640i 2.64869i
\(917\) 22.0512 0.728195
\(918\) 0 0
\(919\) −0.804530 −0.0265390 −0.0132695 0.999912i \(-0.504224\pi\)
−0.0132695 + 0.999912i \(0.504224\pi\)
\(920\) 40.0758 34.4667i 1.32126 1.13633i
\(921\) 0 0
\(922\) 4.24045i 0.139652i
\(923\) −24.1019 + 43.4738i −0.793325 + 1.43096i
\(924\) 0 0
\(925\) −7.89458 + 52.1611i −0.259572 + 1.71505i
\(926\) 9.58641 0.315029
\(927\) 0 0
\(928\) 17.7712 0.583367
\(929\) 22.9049i 0.751484i −0.926724 0.375742i \(-0.877388\pi\)
0.926724 0.375742i \(-0.122612\pi\)
\(930\) 0 0
\(931\) 29.7192i 0.974006i
\(932\) 49.9632i 1.63660i
\(933\) 0 0
\(934\) 39.7386i 1.30029i
\(935\) −22.2642 25.8874i −0.728116 0.846608i
\(936\) 0 0
\(937\) 29.6622i 0.969022i −0.874785 0.484511i \(-0.838998\pi\)
0.874785 0.484511i \(-0.161002\pi\)
\(938\) 1.02054 0.0333217
\(939\) 0 0
\(940\) 13.4472 + 15.6356i 0.438601 + 0.509978i
\(941\) 49.2004i 1.60389i 0.597401 + 0.801943i \(0.296200\pi\)
−0.597401 + 0.801943i \(0.703800\pi\)
\(942\) 0 0
\(943\) 3.41467 0.111197
\(944\) 55.1610i 1.79534i
\(945\) 0 0
\(946\) 17.0829 0.555412
\(947\) −46.5529 −1.51277 −0.756383 0.654129i \(-0.773035\pi\)
−0.756383 + 0.654129i \(0.773035\pi\)
\(948\) 0 0
\(949\) 19.8874 + 11.0256i 0.645572 + 0.357906i
\(950\) −68.6551 10.3909i −2.22747 0.337127i
\(951\) 0 0
\(952\) 53.9405i 1.74822i
\(953\) 16.8803i 0.546807i 0.961899 + 0.273403i \(0.0881494\pi\)
−0.961899 + 0.273403i \(0.911851\pi\)
\(954\) 0 0
\(955\) 24.9075 + 28.9609i 0.805989 + 0.937154i
\(956\) 31.2740i 1.01147i
\(957\) 0 0
\(958\) 66.8056i 2.15839i
\(959\) 6.91713 0.223366
\(960\) 0 0
\(961\) 22.1445 0.714338
\(962\) 83.8143 + 46.4667i 2.70228 + 1.49815i
\(963\) 0 0
\(964\) 69.1384i 2.22680i
\(965\) 3.00709 + 3.49646i 0.0968016 + 0.112555i
\(966\) 0 0
\(967\) 19.9531 0.641647 0.320824 0.947139i \(-0.396040\pi\)
0.320824 + 0.947139i \(0.396040\pi\)
\(968\) −38.3925 −1.23398
\(969\) 0 0
\(970\) 19.6784 + 22.8808i 0.631835 + 0.734659i
\(971\) 44.4809 1.42746 0.713730 0.700421i \(-0.247004\pi\)
0.713730 + 0.700421i \(0.247004\pi\)
\(972\) 0 0
\(973\) 22.0512 0.706929
\(974\) 43.2713 1.38650
\(975\) 0 0
\(976\) 73.6480 2.35742
\(977\) 17.1174 0.547634 0.273817 0.961782i \(-0.411714\pi\)
0.273817 + 0.961782i \(0.411714\pi\)
\(978\) 0 0
\(979\) 16.2026 0.517836
\(980\) −34.1602 39.7194i −1.09121 1.26879i
\(981\) 0 0
\(982\) 42.0847 1.34298
\(983\) −57.2501 −1.82600 −0.912998 0.407964i \(-0.866239\pi\)
−0.912998 + 0.407964i \(0.866239\pi\)
\(984\) 0 0
\(985\) 11.6498 + 13.5457i 0.371193 + 0.431601i
\(986\) 85.0470i 2.70845i
\(987\) 0 0
\(988\) −41.8848 + 75.5496i −1.33253 + 2.40355i
\(989\) 12.7819 0.406440
\(990\) 0 0
\(991\) −21.9858 −0.698403 −0.349201 0.937048i \(-0.613547\pi\)
−0.349201 + 0.937048i \(0.613547\pi\)
\(992\) 11.2713i 0.357863i
\(993\) 0 0
\(994\) 44.0543i 1.39732i
\(995\) 11.9597 + 13.9060i 0.379147 + 0.440849i
\(996\) 0 0
\(997\) 49.1586i 1.55687i −0.627725 0.778435i \(-0.716014\pi\)
0.627725 0.778435i \(-0.283986\pi\)
\(998\) 0.984439i 0.0311619i
\(999\) 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 585.2.h.g.64.2 12
3.2 odd 2 195.2.h.c.64.12 yes 12
5.4 even 2 inner 585.2.h.g.64.12 12
12.11 even 2 3120.2.r.n.2209.2 12
13.12 even 2 inner 585.2.h.g.64.11 12
15.2 even 4 975.2.b.l.376.6 6
15.8 even 4 975.2.b.j.376.1 6
15.14 odd 2 195.2.h.c.64.1 12
39.38 odd 2 195.2.h.c.64.2 yes 12
60.59 even 2 3120.2.r.n.2209.11 12
65.64 even 2 inner 585.2.h.g.64.1 12
156.155 even 2 3120.2.r.n.2209.5 12
195.38 even 4 975.2.b.j.376.6 6
195.77 even 4 975.2.b.l.376.1 6
195.194 odd 2 195.2.h.c.64.11 yes 12
780.779 even 2 3120.2.r.n.2209.8 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.h.c.64.1 12 15.14 odd 2
195.2.h.c.64.2 yes 12 39.38 odd 2
195.2.h.c.64.11 yes 12 195.194 odd 2
195.2.h.c.64.12 yes 12 3.2 odd 2
585.2.h.g.64.1 12 65.64 even 2 inner
585.2.h.g.64.2 12 1.1 even 1 trivial
585.2.h.g.64.11 12 13.12 even 2 inner
585.2.h.g.64.12 12 5.4 even 2 inner
975.2.b.j.376.1 6 15.8 even 4
975.2.b.j.376.6 6 195.38 even 4
975.2.b.l.376.1 6 195.77 even 4
975.2.b.l.376.6 6 15.2 even 4
3120.2.r.n.2209.2 12 12.11 even 2
3120.2.r.n.2209.5 12 156.155 even 2
3120.2.r.n.2209.8 12 780.779 even 2
3120.2.r.n.2209.11 12 60.59 even 2