Properties

Label 672.2.j.c.239.1
Level $672$
Weight $2$
Character 672.239
Analytic conductor $5.366$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [672,2,Mod(239,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.j (of order \(2\), degree \(1\), not 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: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 239.1
Root \(0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 672.239
Dual form 672.2.j.c.239.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.618034 - 1.61803i) q^{3} -1.23607 q^{5} -1.00000i q^{7} +(-2.23607 + 2.00000i) q^{9} +O(q^{10})\) \(q+(-0.618034 - 1.61803i) q^{3} -1.23607 q^{5} -1.00000i q^{7} +(-2.23607 + 2.00000i) q^{9} -4.47214i q^{11} +3.23607i q^{13} +(0.763932 + 2.00000i) q^{15} -2.00000i q^{17} -2.76393 q^{19} +(-1.61803 + 0.618034i) q^{21} -8.47214 q^{23} -3.47214 q^{25} +(4.61803 + 2.38197i) q^{27} +8.94427i q^{31} +(-7.23607 + 2.76393i) q^{33} +1.23607i q^{35} +2.47214i q^{37} +(5.23607 - 2.00000i) q^{39} -4.47214i q^{41} -8.47214 q^{43} +(2.76393 - 2.47214i) q^{45} +6.47214 q^{47} -1.00000 q^{49} +(-3.23607 + 1.23607i) q^{51} -1.52786 q^{53} +5.52786i q^{55} +(1.70820 + 4.47214i) q^{57} +3.23607i q^{59} -1.70820i q^{61} +(2.00000 + 2.23607i) q^{63} -4.00000i q^{65} -6.94427 q^{67} +(5.23607 + 13.7082i) q^{69} -10.9443 q^{71} +8.47214 q^{73} +(2.14590 + 5.61803i) q^{75} -4.47214 q^{77} -12.9443i q^{79} +(1.00000 - 8.94427i) q^{81} -8.76393i q^{83} +2.47214i q^{85} -6.00000i q^{89} +3.23607 q^{91} +(14.4721 - 5.52786i) q^{93} +3.41641 q^{95} +12.4721 q^{97} +(8.94427 + 10.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 4 q^{5} + 12 q^{15} - 20 q^{19} - 2 q^{21} - 16 q^{23} + 4 q^{25} + 14 q^{27} - 20 q^{33} + 12 q^{39} - 16 q^{43} + 20 q^{45} + 8 q^{47} - 4 q^{49} - 4 q^{51} - 24 q^{53} - 20 q^{57} + 8 q^{63} + 8 q^{67} + 12 q^{69} - 8 q^{71} + 16 q^{73} + 22 q^{75} + 4 q^{81} + 4 q^{91} + 40 q^{93} - 40 q^{95} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.618034 1.61803i −0.356822 0.934172i
\(4\) 0 0
\(5\) −1.23607 −0.552786 −0.276393 0.961045i \(-0.589139\pi\)
−0.276393 + 0.961045i \(0.589139\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −2.23607 + 2.00000i −0.745356 + 0.666667i
\(10\) 0 0
\(11\) 4.47214i 1.34840i −0.738549 0.674200i \(-0.764489\pi\)
0.738549 0.674200i \(-0.235511\pi\)
\(12\) 0 0
\(13\) 3.23607i 0.897524i 0.893651 + 0.448762i \(0.148135\pi\)
−0.893651 + 0.448762i \(0.851865\pi\)
\(14\) 0 0
\(15\) 0.763932 + 2.00000i 0.197246 + 0.516398i
\(16\) 0 0
\(17\) 2.00000i 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 0 0
\(19\) −2.76393 −0.634089 −0.317045 0.948411i \(-0.602691\pi\)
−0.317045 + 0.948411i \(0.602691\pi\)
\(20\) 0 0
\(21\) −1.61803 + 0.618034i −0.353084 + 0.134866i
\(22\) 0 0
\(23\) −8.47214 −1.76656 −0.883281 0.468844i \(-0.844671\pi\)
−0.883281 + 0.468844i \(0.844671\pi\)
\(24\) 0 0
\(25\) −3.47214 −0.694427
\(26\) 0 0
\(27\) 4.61803 + 2.38197i 0.888741 + 0.458410i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 8.94427i 1.60644i 0.595683 + 0.803219i \(0.296881\pi\)
−0.595683 + 0.803219i \(0.703119\pi\)
\(32\) 0 0
\(33\) −7.23607 + 2.76393i −1.25964 + 0.481139i
\(34\) 0 0
\(35\) 1.23607i 0.208934i
\(36\) 0 0
\(37\) 2.47214i 0.406417i 0.979136 + 0.203208i \(0.0651369\pi\)
−0.979136 + 0.203208i \(0.934863\pi\)
\(38\) 0 0
\(39\) 5.23607 2.00000i 0.838442 0.320256i
\(40\) 0 0
\(41\) 4.47214i 0.698430i −0.937043 0.349215i \(-0.886448\pi\)
0.937043 0.349215i \(-0.113552\pi\)
\(42\) 0 0
\(43\) −8.47214 −1.29199 −0.645994 0.763342i \(-0.723557\pi\)
−0.645994 + 0.763342i \(0.723557\pi\)
\(44\) 0 0
\(45\) 2.76393 2.47214i 0.412023 0.368524i
\(46\) 0 0
\(47\) 6.47214 0.944058 0.472029 0.881583i \(-0.343522\pi\)
0.472029 + 0.881583i \(0.343522\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −3.23607 + 1.23607i −0.453140 + 0.173084i
\(52\) 0 0
\(53\) −1.52786 −0.209868 −0.104934 0.994479i \(-0.533463\pi\)
−0.104934 + 0.994479i \(0.533463\pi\)
\(54\) 0 0
\(55\) 5.52786i 0.745377i
\(56\) 0 0
\(57\) 1.70820 + 4.47214i 0.226257 + 0.592349i
\(58\) 0 0
\(59\) 3.23607i 0.421300i 0.977562 + 0.210650i \(0.0675581\pi\)
−0.977562 + 0.210650i \(0.932442\pi\)
\(60\) 0 0
\(61\) 1.70820i 0.218713i −0.994003 0.109357i \(-0.965121\pi\)
0.994003 0.109357i \(-0.0348790\pi\)
\(62\) 0 0
\(63\) 2.00000 + 2.23607i 0.251976 + 0.281718i
\(64\) 0 0
\(65\) 4.00000i 0.496139i
\(66\) 0 0
\(67\) −6.94427 −0.848378 −0.424189 0.905574i \(-0.639441\pi\)
−0.424189 + 0.905574i \(0.639441\pi\)
\(68\) 0 0
\(69\) 5.23607 + 13.7082i 0.630349 + 1.65027i
\(70\) 0 0
\(71\) −10.9443 −1.29885 −0.649423 0.760427i \(-0.724990\pi\)
−0.649423 + 0.760427i \(0.724990\pi\)
\(72\) 0 0
\(73\) 8.47214 0.991589 0.495794 0.868440i \(-0.334877\pi\)
0.495794 + 0.868440i \(0.334877\pi\)
\(74\) 0 0
\(75\) 2.14590 + 5.61803i 0.247787 + 0.648715i
\(76\) 0 0
\(77\) −4.47214 −0.509647
\(78\) 0 0
\(79\) 12.9443i 1.45634i −0.685394 0.728172i \(-0.740370\pi\)
0.685394 0.728172i \(-0.259630\pi\)
\(80\) 0 0
\(81\) 1.00000 8.94427i 0.111111 0.993808i
\(82\) 0 0
\(83\) 8.76393i 0.961967i −0.876730 0.480983i \(-0.840280\pi\)
0.876730 0.480983i \(-0.159720\pi\)
\(84\) 0 0
\(85\) 2.47214i 0.268141i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000i 0.635999i −0.948091 0.317999i \(-0.896989\pi\)
0.948091 0.317999i \(-0.103011\pi\)
\(90\) 0 0
\(91\) 3.23607 0.339232
\(92\) 0 0
\(93\) 14.4721 5.52786i 1.50069 0.573213i
\(94\) 0 0
\(95\) 3.41641 0.350516
\(96\) 0 0
\(97\) 12.4721 1.26635 0.633177 0.774007i \(-0.281751\pi\)
0.633177 + 0.774007i \(0.281751\pi\)
\(98\) 0 0
\(99\) 8.94427 + 10.0000i 0.898933 + 1.00504i
\(100\) 0 0
\(101\) −19.7082 −1.96104 −0.980520 0.196420i \(-0.937068\pi\)
−0.980520 + 0.196420i \(0.937068\pi\)
\(102\) 0 0
\(103\) 10.4721i 1.03185i −0.856634 0.515925i \(-0.827448\pi\)
0.856634 0.515925i \(-0.172552\pi\)
\(104\) 0 0
\(105\) 2.00000 0.763932i 0.195180 0.0745521i
\(106\) 0 0
\(107\) 2.00000i 0.193347i 0.995316 + 0.0966736i \(0.0308203\pi\)
−0.995316 + 0.0966736i \(0.969180\pi\)
\(108\) 0 0
\(109\) 19.4164i 1.85975i −0.367870 0.929877i \(-0.619913\pi\)
0.367870 0.929877i \(-0.380087\pi\)
\(110\) 0 0
\(111\) 4.00000 1.52786i 0.379663 0.145018i
\(112\) 0 0
\(113\) 4.94427i 0.465118i 0.972582 + 0.232559i \(0.0747099\pi\)
−0.972582 + 0.232559i \(0.925290\pi\)
\(114\) 0 0
\(115\) 10.4721 0.976532
\(116\) 0 0
\(117\) −6.47214 7.23607i −0.598349 0.668975i
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) −7.23607 + 2.76393i −0.652454 + 0.249215i
\(124\) 0 0
\(125\) 10.4721 0.936656
\(126\) 0 0
\(127\) 0.944272i 0.0837906i 0.999122 + 0.0418953i \(0.0133396\pi\)
−0.999122 + 0.0418953i \(0.986660\pi\)
\(128\) 0 0
\(129\) 5.23607 + 13.7082i 0.461010 + 1.20694i
\(130\) 0 0
\(131\) 7.23607i 0.632218i −0.948723 0.316109i \(-0.897623\pi\)
0.948723 0.316109i \(-0.102377\pi\)
\(132\) 0 0
\(133\) 2.76393i 0.239663i
\(134\) 0 0
\(135\) −5.70820 2.94427i −0.491284 0.253403i
\(136\) 0 0
\(137\) 0.944272i 0.0806746i −0.999186 0.0403373i \(-0.987157\pi\)
0.999186 0.0403373i \(-0.0128432\pi\)
\(138\) 0 0
\(139\) 20.6525 1.75172 0.875860 0.482565i \(-0.160295\pi\)
0.875860 + 0.482565i \(0.160295\pi\)
\(140\) 0 0
\(141\) −4.00000 10.4721i −0.336861 0.881913i
\(142\) 0 0
\(143\) 14.4721 1.21022
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0.618034 + 1.61803i 0.0509746 + 0.133453i
\(148\) 0 0
\(149\) 3.41641 0.279883 0.139942 0.990160i \(-0.455309\pi\)
0.139942 + 0.990160i \(0.455309\pi\)
\(150\) 0 0
\(151\) 8.94427i 0.727875i −0.931423 0.363937i \(-0.881432\pi\)
0.931423 0.363937i \(-0.118568\pi\)
\(152\) 0 0
\(153\) 4.00000 + 4.47214i 0.323381 + 0.361551i
\(154\) 0 0
\(155\) 11.0557i 0.888017i
\(156\) 0 0
\(157\) 9.70820i 0.774799i 0.921912 + 0.387400i \(0.126627\pi\)
−0.921912 + 0.387400i \(0.873373\pi\)
\(158\) 0 0
\(159\) 0.944272 + 2.47214i 0.0748856 + 0.196053i
\(160\) 0 0
\(161\) 8.47214i 0.667698i
\(162\) 0 0
\(163\) 0.472136 0.0369805 0.0184903 0.999829i \(-0.494114\pi\)
0.0184903 + 0.999829i \(0.494114\pi\)
\(164\) 0 0
\(165\) 8.94427 3.41641i 0.696311 0.265967i
\(166\) 0 0
\(167\) −16.9443 −1.31119 −0.655594 0.755114i \(-0.727582\pi\)
−0.655594 + 0.755114i \(0.727582\pi\)
\(168\) 0 0
\(169\) 2.52786 0.194451
\(170\) 0 0
\(171\) 6.18034 5.52786i 0.472622 0.422726i
\(172\) 0 0
\(173\) −4.29180 −0.326299 −0.163150 0.986601i \(-0.552165\pi\)
−0.163150 + 0.986601i \(0.552165\pi\)
\(174\) 0 0
\(175\) 3.47214i 0.262469i
\(176\) 0 0
\(177\) 5.23607 2.00000i 0.393567 0.150329i
\(178\) 0 0
\(179\) 2.94427i 0.220065i −0.993928 0.110033i \(-0.964904\pi\)
0.993928 0.110033i \(-0.0350955\pi\)
\(180\) 0 0
\(181\) 12.7639i 0.948736i −0.880327 0.474368i \(-0.842677\pi\)
0.880327 0.474368i \(-0.157323\pi\)
\(182\) 0 0
\(183\) −2.76393 + 1.05573i −0.204316 + 0.0780417i
\(184\) 0 0
\(185\) 3.05573i 0.224662i
\(186\) 0 0
\(187\) −8.94427 −0.654070
\(188\) 0 0
\(189\) 2.38197 4.61803i 0.173263 0.335913i
\(190\) 0 0
\(191\) −2.00000 −0.144715 −0.0723575 0.997379i \(-0.523052\pi\)
−0.0723575 + 0.997379i \(0.523052\pi\)
\(192\) 0 0
\(193\) −20.4721 −1.47362 −0.736808 0.676102i \(-0.763668\pi\)
−0.736808 + 0.676102i \(0.763668\pi\)
\(194\) 0 0
\(195\) −6.47214 + 2.47214i −0.463479 + 0.177033i
\(196\) 0 0
\(197\) 13.5279 0.963820 0.481910 0.876221i \(-0.339943\pi\)
0.481910 + 0.876221i \(0.339943\pi\)
\(198\) 0 0
\(199\) 1.52786i 0.108307i 0.998533 + 0.0541537i \(0.0172461\pi\)
−0.998533 + 0.0541537i \(0.982754\pi\)
\(200\) 0 0
\(201\) 4.29180 + 11.2361i 0.302720 + 0.792531i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 5.52786i 0.386083i
\(206\) 0 0
\(207\) 18.9443 16.9443i 1.31672 1.17771i
\(208\) 0 0
\(209\) 12.3607i 0.855006i
\(210\) 0 0
\(211\) 26.9443 1.85492 0.927460 0.373922i \(-0.121987\pi\)
0.927460 + 0.373922i \(0.121987\pi\)
\(212\) 0 0
\(213\) 6.76393 + 17.7082i 0.463457 + 1.21335i
\(214\) 0 0
\(215\) 10.4721 0.714194
\(216\) 0 0
\(217\) 8.94427 0.607177
\(218\) 0 0
\(219\) −5.23607 13.7082i −0.353821 0.926315i
\(220\) 0 0
\(221\) 6.47214 0.435363
\(222\) 0 0
\(223\) 27.4164i 1.83594i 0.396652 + 0.917969i \(0.370172\pi\)
−0.396652 + 0.917969i \(0.629828\pi\)
\(224\) 0 0
\(225\) 7.76393 6.94427i 0.517595 0.462951i
\(226\) 0 0
\(227\) 13.7082i 0.909845i 0.890531 + 0.454923i \(0.150333\pi\)
−0.890531 + 0.454923i \(0.849667\pi\)
\(228\) 0 0
\(229\) 17.7082i 1.17019i −0.810964 0.585096i \(-0.801057\pi\)
0.810964 0.585096i \(-0.198943\pi\)
\(230\) 0 0
\(231\) 2.76393 + 7.23607i 0.181853 + 0.476098i
\(232\) 0 0
\(233\) 16.0000i 1.04819i 0.851658 + 0.524097i \(0.175597\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) 0 0
\(237\) −20.9443 + 8.00000i −1.36048 + 0.519656i
\(238\) 0 0
\(239\) 13.4164 0.867835 0.433918 0.900953i \(-0.357131\pi\)
0.433918 + 0.900953i \(0.357131\pi\)
\(240\) 0 0
\(241\) −12.4721 −0.803401 −0.401700 0.915771i \(-0.631581\pi\)
−0.401700 + 0.915771i \(0.631581\pi\)
\(242\) 0 0
\(243\) −15.0902 + 3.90983i −0.968035 + 0.250816i
\(244\) 0 0
\(245\) 1.23607 0.0789695
\(246\) 0 0
\(247\) 8.94427i 0.569110i
\(248\) 0 0
\(249\) −14.1803 + 5.41641i −0.898643 + 0.343251i
\(250\) 0 0
\(251\) 21.7082i 1.37021i 0.728444 + 0.685105i \(0.240244\pi\)
−0.728444 + 0.685105i \(0.759756\pi\)
\(252\) 0 0
\(253\) 37.8885i 2.38203i
\(254\) 0 0
\(255\) 4.00000 1.52786i 0.250490 0.0956786i
\(256\) 0 0
\(257\) 25.4164i 1.58543i −0.609591 0.792716i \(-0.708666\pi\)
0.609591 0.792716i \(-0.291334\pi\)
\(258\) 0 0
\(259\) 2.47214 0.153611
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −14.0000 −0.863277 −0.431638 0.902047i \(-0.642064\pi\)
−0.431638 + 0.902047i \(0.642064\pi\)
\(264\) 0 0
\(265\) 1.88854 0.116012
\(266\) 0 0
\(267\) −9.70820 + 3.70820i −0.594132 + 0.226938i
\(268\) 0 0
\(269\) −6.18034 −0.376822 −0.188411 0.982090i \(-0.560334\pi\)
−0.188411 + 0.982090i \(0.560334\pi\)
\(270\) 0 0
\(271\) 5.52786i 0.335794i −0.985805 0.167897i \(-0.946302\pi\)
0.985805 0.167897i \(-0.0536976\pi\)
\(272\) 0 0
\(273\) −2.00000 5.23607i −0.121046 0.316901i
\(274\) 0 0
\(275\) 15.5279i 0.936365i
\(276\) 0 0
\(277\) 12.0000i 0.721010i −0.932757 0.360505i \(-0.882604\pi\)
0.932757 0.360505i \(-0.117396\pi\)
\(278\) 0 0
\(279\) −17.8885 20.0000i −1.07096 1.19737i
\(280\) 0 0
\(281\) 17.8885i 1.06714i −0.845756 0.533571i \(-0.820850\pi\)
0.845756 0.533571i \(-0.179150\pi\)
\(282\) 0 0
\(283\) −1.23607 −0.0734766 −0.0367383 0.999325i \(-0.511697\pi\)
−0.0367383 + 0.999325i \(0.511697\pi\)
\(284\) 0 0
\(285\) −2.11146 5.52786i −0.125072 0.327442i
\(286\) 0 0
\(287\) −4.47214 −0.263982
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) −7.70820 20.1803i −0.451863 1.18299i
\(292\) 0 0
\(293\) −11.1246 −0.649907 −0.324953 0.945730i \(-0.605349\pi\)
−0.324953 + 0.945730i \(0.605349\pi\)
\(294\) 0 0
\(295\) 4.00000i 0.232889i
\(296\) 0 0
\(297\) 10.6525 20.6525i 0.618119 1.19838i
\(298\) 0 0
\(299\) 27.4164i 1.58553i
\(300\) 0 0
\(301\) 8.47214i 0.488326i
\(302\) 0 0
\(303\) 12.1803 + 31.8885i 0.699742 + 1.83195i
\(304\) 0 0
\(305\) 2.11146i 0.120902i
\(306\) 0 0
\(307\) −23.1246 −1.31979 −0.659896 0.751357i \(-0.729400\pi\)
−0.659896 + 0.751357i \(0.729400\pi\)
\(308\) 0 0
\(309\) −16.9443 + 6.47214i −0.963926 + 0.368187i
\(310\) 0 0
\(311\) −24.3607 −1.38137 −0.690684 0.723157i \(-0.742690\pi\)
−0.690684 + 0.723157i \(0.742690\pi\)
\(312\) 0 0
\(313\) −14.9443 −0.844700 −0.422350 0.906433i \(-0.638795\pi\)
−0.422350 + 0.906433i \(0.638795\pi\)
\(314\) 0 0
\(315\) −2.47214 2.76393i −0.139289 0.155730i
\(316\) 0 0
\(317\) −6.47214 −0.363511 −0.181756 0.983344i \(-0.558178\pi\)
−0.181756 + 0.983344i \(0.558178\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 3.23607 1.23607i 0.180620 0.0689906i
\(322\) 0 0
\(323\) 5.52786i 0.307579i
\(324\) 0 0
\(325\) 11.2361i 0.623265i
\(326\) 0 0
\(327\) −31.4164 + 12.0000i −1.73733 + 0.663602i
\(328\) 0 0
\(329\) 6.47214i 0.356820i
\(330\) 0 0
\(331\) −14.3607 −0.789334 −0.394667 0.918824i \(-0.629140\pi\)
−0.394667 + 0.918824i \(0.629140\pi\)
\(332\) 0 0
\(333\) −4.94427 5.52786i −0.270944 0.302925i
\(334\) 0 0
\(335\) 8.58359 0.468972
\(336\) 0 0
\(337\) −7.52786 −0.410069 −0.205034 0.978755i \(-0.565731\pi\)
−0.205034 + 0.978755i \(0.565731\pi\)
\(338\) 0 0
\(339\) 8.00000 3.05573i 0.434500 0.165964i
\(340\) 0 0
\(341\) 40.0000 2.16612
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) −6.47214 16.9443i −0.348448 0.912249i
\(346\) 0 0
\(347\) 16.4721i 0.884271i 0.896948 + 0.442135i \(0.145779\pi\)
−0.896948 + 0.442135i \(0.854221\pi\)
\(348\) 0 0
\(349\) 0.180340i 0.00965337i 0.999988 + 0.00482669i \(0.00153639\pi\)
−0.999988 + 0.00482669i \(0.998464\pi\)
\(350\) 0 0
\(351\) −7.70820 + 14.9443i −0.411433 + 0.797666i
\(352\) 0 0
\(353\) 11.5279i 0.613566i 0.951779 + 0.306783i \(0.0992526\pi\)
−0.951779 + 0.306783i \(0.900747\pi\)
\(354\) 0 0
\(355\) 13.5279 0.717984
\(356\) 0 0
\(357\) 1.23607 + 3.23607i 0.0654197 + 0.171271i
\(358\) 0 0
\(359\) 13.4164 0.708091 0.354045 0.935228i \(-0.384806\pi\)
0.354045 + 0.935228i \(0.384806\pi\)
\(360\) 0 0
\(361\) −11.3607 −0.597931
\(362\) 0 0
\(363\) 5.56231 + 14.5623i 0.291945 + 0.764323i
\(364\) 0 0
\(365\) −10.4721 −0.548137
\(366\) 0 0
\(367\) 6.47214i 0.337843i 0.985630 + 0.168921i \(0.0540284\pi\)
−0.985630 + 0.168921i \(0.945972\pi\)
\(368\) 0 0
\(369\) 8.94427 + 10.0000i 0.465620 + 0.520579i
\(370\) 0 0
\(371\) 1.52786i 0.0793227i
\(372\) 0 0
\(373\) 1.88854i 0.0977851i −0.998804 0.0488925i \(-0.984431\pi\)
0.998804 0.0488925i \(-0.0155692\pi\)
\(374\) 0 0
\(375\) −6.47214 16.9443i −0.334220 0.874998i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 4.47214 0.229718 0.114859 0.993382i \(-0.463358\pi\)
0.114859 + 0.993382i \(0.463358\pi\)
\(380\) 0 0
\(381\) 1.52786 0.583592i 0.0782748 0.0298983i
\(382\) 0 0
\(383\) 21.5279 1.10002 0.550011 0.835157i \(-0.314623\pi\)
0.550011 + 0.835157i \(0.314623\pi\)
\(384\) 0 0
\(385\) 5.52786 0.281726
\(386\) 0 0
\(387\) 18.9443 16.9443i 0.962991 0.861326i
\(388\) 0 0
\(389\) −11.0557 −0.560548 −0.280274 0.959920i \(-0.590425\pi\)
−0.280274 + 0.959920i \(0.590425\pi\)
\(390\) 0 0
\(391\) 16.9443i 0.856909i
\(392\) 0 0
\(393\) −11.7082 + 4.47214i −0.590601 + 0.225589i
\(394\) 0 0
\(395\) 16.0000i 0.805047i
\(396\) 0 0
\(397\) 35.2361i 1.76845i 0.467063 + 0.884224i \(0.345312\pi\)
−0.467063 + 0.884224i \(0.654688\pi\)
\(398\) 0 0
\(399\) 4.47214 1.70820i 0.223887 0.0855172i
\(400\) 0 0
\(401\) 17.8885i 0.893311i −0.894706 0.446656i \(-0.852615\pi\)
0.894706 0.446656i \(-0.147385\pi\)
\(402\) 0 0
\(403\) −28.9443 −1.44182
\(404\) 0 0
\(405\) −1.23607 + 11.0557i −0.0614207 + 0.549364i
\(406\) 0 0
\(407\) 11.0557 0.548012
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) −1.52786 + 0.583592i −0.0753640 + 0.0287865i
\(412\) 0 0
\(413\) 3.23607 0.159236
\(414\) 0 0
\(415\) 10.8328i 0.531762i
\(416\) 0 0
\(417\) −12.7639 33.4164i −0.625052 1.63641i
\(418\) 0 0
\(419\) 22.2918i 1.08903i −0.838753 0.544513i \(-0.816715\pi\)
0.838753 0.544513i \(-0.183285\pi\)
\(420\) 0 0
\(421\) 8.94427i 0.435917i −0.975958 0.217959i \(-0.930060\pi\)
0.975958 0.217959i \(-0.0699398\pi\)
\(422\) 0 0
\(423\) −14.4721 + 12.9443i −0.703659 + 0.629372i
\(424\) 0 0
\(425\) 6.94427i 0.336847i
\(426\) 0 0
\(427\) −1.70820 −0.0826658
\(428\) 0 0
\(429\) −8.94427 23.4164i −0.431834 1.13055i
\(430\) 0 0
\(431\) 3.52786 0.169931 0.0849656 0.996384i \(-0.472922\pi\)
0.0849656 + 0.996384i \(0.472922\pi\)
\(432\) 0 0
\(433\) −18.3607 −0.882358 −0.441179 0.897419i \(-0.645440\pi\)
−0.441179 + 0.897419i \(0.645440\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 23.4164 1.12016
\(438\) 0 0
\(439\) 1.88854i 0.0901353i −0.998984 0.0450676i \(-0.985650\pi\)
0.998984 0.0450676i \(-0.0143503\pi\)
\(440\) 0 0
\(441\) 2.23607 2.00000i 0.106479 0.0952381i
\(442\) 0 0
\(443\) 14.9443i 0.710024i −0.934862 0.355012i \(-0.884477\pi\)
0.934862 0.355012i \(-0.115523\pi\)
\(444\) 0 0
\(445\) 7.41641i 0.351571i
\(446\) 0 0
\(447\) −2.11146 5.52786i −0.0998685 0.261459i
\(448\) 0 0
\(449\) 21.8885i 1.03298i 0.856292 + 0.516492i \(0.172762\pi\)
−0.856292 + 0.516492i \(0.827238\pi\)
\(450\) 0 0
\(451\) −20.0000 −0.941763
\(452\) 0 0
\(453\) −14.4721 + 5.52786i −0.679960 + 0.259722i
\(454\) 0 0
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) −5.41641 −0.253369 −0.126684 0.991943i \(-0.540434\pi\)
−0.126684 + 0.991943i \(0.540434\pi\)
\(458\) 0 0
\(459\) 4.76393 9.23607i 0.222361 0.431103i
\(460\) 0 0
\(461\) −14.1803 −0.660444 −0.330222 0.943903i \(-0.607124\pi\)
−0.330222 + 0.943903i \(0.607124\pi\)
\(462\) 0 0
\(463\) 4.94427i 0.229780i −0.993378 0.114890i \(-0.963348\pi\)
0.993378 0.114890i \(-0.0366515\pi\)
\(464\) 0 0
\(465\) −17.8885 + 6.83282i −0.829561 + 0.316864i
\(466\) 0 0
\(467\) 13.7082i 0.634340i 0.948369 + 0.317170i \(0.102733\pi\)
−0.948369 + 0.317170i \(0.897267\pi\)
\(468\) 0 0
\(469\) 6.94427i 0.320657i
\(470\) 0 0
\(471\) 15.7082 6.00000i 0.723796 0.276465i
\(472\) 0 0
\(473\) 37.8885i 1.74212i
\(474\) 0 0
\(475\) 9.59675 0.440329
\(476\) 0 0
\(477\) 3.41641 3.05573i 0.156427 0.139912i
\(478\) 0 0
\(479\) −25.5279 −1.16640 −0.583199 0.812329i \(-0.698199\pi\)
−0.583199 + 0.812329i \(0.698199\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 0 0
\(483\) 13.7082 5.23607i 0.623745 0.238249i
\(484\) 0 0
\(485\) −15.4164 −0.700023
\(486\) 0 0
\(487\) 5.88854i 0.266835i −0.991060 0.133418i \(-0.957405\pi\)
0.991060 0.133418i \(-0.0425952\pi\)
\(488\) 0 0
\(489\) −0.291796 0.763932i −0.0131955 0.0345462i
\(490\) 0 0
\(491\) 7.88854i 0.356005i −0.984030 0.178002i \(-0.943036\pi\)
0.984030 0.178002i \(-0.0569635\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −11.0557 12.3607i −0.496918 0.555571i
\(496\) 0 0
\(497\) 10.9443i 0.490918i
\(498\) 0 0
\(499\) 7.88854 0.353140 0.176570 0.984288i \(-0.443500\pi\)
0.176570 + 0.984288i \(0.443500\pi\)
\(500\) 0 0
\(501\) 10.4721 + 27.4164i 0.467861 + 1.22487i
\(502\) 0 0
\(503\) 39.4164 1.75749 0.878745 0.477291i \(-0.158381\pi\)
0.878745 + 0.477291i \(0.158381\pi\)
\(504\) 0 0
\(505\) 24.3607 1.08404
\(506\) 0 0
\(507\) −1.56231 4.09017i −0.0693844 0.181651i
\(508\) 0 0
\(509\) 35.1246 1.55687 0.778436 0.627725i \(-0.216014\pi\)
0.778436 + 0.627725i \(0.216014\pi\)
\(510\) 0 0
\(511\) 8.47214i 0.374785i
\(512\) 0 0
\(513\) −12.7639 6.58359i −0.563541 0.290673i
\(514\) 0 0
\(515\) 12.9443i 0.570393i
\(516\) 0 0
\(517\) 28.9443i 1.27297i
\(518\) 0 0
\(519\) 2.65248 + 6.94427i 0.116431 + 0.304820i
\(520\) 0 0
\(521\) 33.4164i 1.46400i 0.681305 + 0.732000i \(0.261413\pi\)
−0.681305 + 0.732000i \(0.738587\pi\)
\(522\) 0 0
\(523\) −12.2918 −0.537483 −0.268741 0.963212i \(-0.586608\pi\)
−0.268741 + 0.963212i \(0.586608\pi\)
\(524\) 0 0
\(525\) 5.61803 2.14590i 0.245191 0.0936547i
\(526\) 0 0
\(527\) 17.8885 0.779237
\(528\) 0 0
\(529\) 48.7771 2.12074
\(530\) 0 0
\(531\) −6.47214 7.23607i −0.280867 0.314019i
\(532\) 0 0
\(533\) 14.4721 0.626858
\(534\) 0 0
\(535\) 2.47214i 0.106880i
\(536\) 0 0
\(537\) −4.76393 + 1.81966i −0.205579 + 0.0785241i
\(538\) 0 0
\(539\) 4.47214i 0.192629i
\(540\) 0 0
\(541\) 8.94427i 0.384544i 0.981342 + 0.192272i \(0.0615856\pi\)
−0.981342 + 0.192272i \(0.938414\pi\)
\(542\) 0 0
\(543\) −20.6525 + 7.88854i −0.886283 + 0.338530i
\(544\) 0 0
\(545\) 24.0000i 1.02805i
\(546\) 0 0
\(547\) −23.5279 −1.00598 −0.502989 0.864293i \(-0.667767\pi\)
−0.502989 + 0.864293i \(0.667767\pi\)
\(548\) 0 0
\(549\) 3.41641 + 3.81966i 0.145809 + 0.163019i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −12.9443 −0.550446
\(554\) 0 0
\(555\) −4.94427 + 1.88854i −0.209873 + 0.0801642i
\(556\) 0 0
\(557\) −6.47214 −0.274233 −0.137117 0.990555i \(-0.543784\pi\)
−0.137117 + 0.990555i \(0.543784\pi\)
\(558\) 0 0
\(559\) 27.4164i 1.15959i
\(560\) 0 0
\(561\) 5.52786 + 14.4721i 0.233387 + 0.611014i
\(562\) 0 0
\(563\) 32.1803i 1.35624i −0.734951 0.678120i \(-0.762795\pi\)
0.734951 0.678120i \(-0.237205\pi\)
\(564\) 0 0
\(565\) 6.11146i 0.257111i
\(566\) 0 0
\(567\) −8.94427 1.00000i −0.375624 0.0419961i
\(568\) 0 0
\(569\) 21.8885i 0.917615i 0.888536 + 0.458808i \(0.151723\pi\)
−0.888536 + 0.458808i \(0.848277\pi\)
\(570\) 0 0
\(571\) 32.4721 1.35892 0.679458 0.733714i \(-0.262215\pi\)
0.679458 + 0.733714i \(0.262215\pi\)
\(572\) 0 0
\(573\) 1.23607 + 3.23607i 0.0516375 + 0.135189i
\(574\) 0 0
\(575\) 29.4164 1.22675
\(576\) 0 0
\(577\) 26.9443 1.12170 0.560852 0.827916i \(-0.310474\pi\)
0.560852 + 0.827916i \(0.310474\pi\)
\(578\) 0 0
\(579\) 12.6525 + 33.1246i 0.525819 + 1.37661i
\(580\) 0 0
\(581\) −8.76393 −0.363589
\(582\) 0 0
\(583\) 6.83282i 0.282986i
\(584\) 0 0
\(585\) 8.00000 + 8.94427i 0.330759 + 0.369800i
\(586\) 0 0
\(587\) 10.2918i 0.424788i 0.977184 + 0.212394i \(0.0681260\pi\)
−0.977184 + 0.212394i \(0.931874\pi\)
\(588\) 0 0
\(589\) 24.7214i 1.01863i
\(590\) 0 0
\(591\) −8.36068 21.8885i −0.343912 0.900374i
\(592\) 0 0
\(593\) 17.4164i 0.715206i −0.933874 0.357603i \(-0.883594\pi\)
0.933874 0.357603i \(-0.116406\pi\)
\(594\) 0 0
\(595\) 2.47214 0.101348
\(596\) 0 0
\(597\) 2.47214 0.944272i 0.101178 0.0386465i
\(598\) 0 0
\(599\) −18.9443 −0.774042 −0.387021 0.922071i \(-0.626496\pi\)
−0.387021 + 0.922071i \(0.626496\pi\)
\(600\) 0 0
\(601\) 16.4721 0.671912 0.335956 0.941878i \(-0.390941\pi\)
0.335956 + 0.941878i \(0.390941\pi\)
\(602\) 0 0
\(603\) 15.5279 13.8885i 0.632344 0.565585i
\(604\) 0 0
\(605\) 11.1246 0.452280
\(606\) 0 0
\(607\) 6.47214i 0.262696i 0.991336 + 0.131348i \(0.0419305\pi\)
−0.991336 + 0.131348i \(0.958069\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 20.9443i 0.847315i
\(612\) 0 0
\(613\) 27.4164i 1.10734i −0.832737 0.553669i \(-0.813227\pi\)
0.832737 0.553669i \(-0.186773\pi\)
\(614\) 0 0
\(615\) 8.94427 3.41641i 0.360668 0.137763i
\(616\) 0 0
\(617\) 32.0000i 1.28827i −0.764911 0.644136i \(-0.777217\pi\)
0.764911 0.644136i \(-0.222783\pi\)
\(618\) 0 0
\(619\) 35.1246 1.41178 0.705889 0.708323i \(-0.250548\pi\)
0.705889 + 0.708323i \(0.250548\pi\)
\(620\) 0 0
\(621\) −39.1246 20.1803i −1.57002 0.809809i
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 4.41641 0.176656
\(626\) 0 0
\(627\) 20.0000 7.63932i 0.798723 0.305085i
\(628\) 0 0
\(629\) 4.94427 0.197141
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) −16.6525 43.5967i −0.661876 1.73282i
\(634\) 0 0
\(635\) 1.16718i 0.0463183i
\(636\) 0 0
\(637\) 3.23607i 0.128218i
\(638\) 0 0
\(639\) 24.4721 21.8885i 0.968103 0.865897i
\(640\) 0 0
\(641\) 40.0000i 1.57991i −0.613168 0.789953i \(-0.710105\pi\)
0.613168 0.789953i \(-0.289895\pi\)
\(642\) 0 0
\(643\) 9.81966 0.387250 0.193625 0.981076i \(-0.437976\pi\)
0.193625 + 0.981076i \(0.437976\pi\)
\(644\) 0 0
\(645\) −6.47214 16.9443i −0.254840 0.667180i
\(646\) 0 0
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) 14.4721 0.568081
\(650\) 0 0
\(651\) −5.52786 14.4721i −0.216654 0.567208i
\(652\) 0 0
\(653\) −42.8328 −1.67618 −0.838089 0.545534i \(-0.816327\pi\)
−0.838089 + 0.545534i \(0.816327\pi\)
\(654\) 0 0
\(655\) 8.94427i 0.349482i
\(656\) 0 0
\(657\) −18.9443 + 16.9443i −0.739086 + 0.661059i
\(658\) 0 0
\(659\) 1.63932i 0.0638588i −0.999490 0.0319294i \(-0.989835\pi\)
0.999490 0.0319294i \(-0.0101652\pi\)
\(660\) 0 0
\(661\) 5.12461i 0.199324i 0.995021 + 0.0996621i \(0.0317762\pi\)
−0.995021 + 0.0996621i \(0.968224\pi\)
\(662\) 0 0
\(663\) −4.00000 10.4721i −0.155347 0.406704i
\(664\) 0 0
\(665\) 3.41641i 0.132483i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 44.3607 16.9443i 1.71508 0.655103i
\(670\) 0 0
\(671\) −7.63932 −0.294913
\(672\) 0 0
\(673\) 5.05573 0.194884 0.0974420 0.995241i \(-0.468934\pi\)
0.0974420 + 0.995241i \(0.468934\pi\)
\(674\) 0 0
\(675\) −16.0344 8.27051i −0.617166 0.318332i
\(676\) 0 0
\(677\) 1.81966 0.0699352 0.0349676 0.999388i \(-0.488867\pi\)
0.0349676 + 0.999388i \(0.488867\pi\)
\(678\) 0 0
\(679\) 12.4721i 0.478637i
\(680\) 0 0
\(681\) 22.1803 8.47214i 0.849952 0.324653i
\(682\) 0 0
\(683\) 25.0557i 0.958731i 0.877615 + 0.479365i \(0.159133\pi\)
−0.877615 + 0.479365i \(0.840867\pi\)
\(684\) 0 0
\(685\) 1.16718i 0.0445958i
\(686\) 0 0
\(687\) −28.6525 + 10.9443i −1.09316 + 0.417550i
\(688\) 0 0
\(689\) 4.94427i 0.188362i
\(690\) 0 0
\(691\) 28.6525 1.08999 0.544996 0.838439i \(-0.316531\pi\)
0.544996 + 0.838439i \(0.316531\pi\)
\(692\) 0 0
\(693\) 10.0000 8.94427i 0.379869 0.339765i
\(694\) 0 0
\(695\) −25.5279 −0.968327
\(696\) 0 0
\(697\) −8.94427 −0.338788
\(698\) 0 0
\(699\) 25.8885 9.88854i 0.979195 0.374019i
\(700\) 0 0
\(701\) 29.8885 1.12887 0.564437 0.825476i \(-0.309093\pi\)
0.564437 + 0.825476i \(0.309093\pi\)
\(702\) 0 0
\(703\) 6.83282i 0.257705i
\(704\) 0 0
\(705\) 4.94427 + 12.9443i 0.186212 + 0.487509i
\(706\) 0 0
\(707\) 19.7082i 0.741203i
\(708\) 0 0
\(709\) 1.52786i 0.0573801i −0.999588 0.0286901i \(-0.990866\pi\)
0.999588 0.0286901i \(-0.00913359\pi\)
\(710\) 0 0
\(711\) 25.8885 + 28.9443i 0.970896 + 1.08550i
\(712\) 0 0
\(713\) 75.7771i 2.83787i
\(714\) 0 0
\(715\) −17.8885 −0.668994
\(716\) 0 0
\(717\) −8.29180 21.7082i −0.309663 0.810708i
\(718\) 0 0
\(719\) −12.3607 −0.460976 −0.230488 0.973075i \(-0.574032\pi\)
−0.230488 + 0.973075i \(0.574032\pi\)
\(720\) 0 0
\(721\) −10.4721 −0.390003
\(722\) 0 0
\(723\) 7.70820 + 20.1803i 0.286671 + 0.750515i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 15.4164i 0.571763i 0.958265 + 0.285881i \(0.0922864\pi\)
−0.958265 + 0.285881i \(0.907714\pi\)
\(728\) 0 0
\(729\) 15.6525 + 22.0000i 0.579721 + 0.814815i
\(730\) 0 0
\(731\) 16.9443i 0.626707i
\(732\) 0 0
\(733\) 24.5410i 0.906443i 0.891398 + 0.453222i \(0.149725\pi\)
−0.891398 + 0.453222i \(0.850275\pi\)
\(734\) 0 0
\(735\) −0.763932 2.00000i −0.0281781 0.0737711i
\(736\) 0 0
\(737\) 31.0557i 1.14395i
\(738\) 0 0
\(739\) 4.47214 0.164510 0.0822551 0.996611i \(-0.473788\pi\)
0.0822551 + 0.996611i \(0.473788\pi\)
\(740\) 0 0
\(741\) −14.4721 + 5.52786i −0.531647 + 0.203071i
\(742\) 0 0
\(743\) −15.3050 −0.561484 −0.280742 0.959783i \(-0.590581\pi\)
−0.280742 + 0.959783i \(0.590581\pi\)
\(744\) 0 0
\(745\) −4.22291 −0.154716
\(746\) 0 0
\(747\) 17.5279 + 19.5967i 0.641311 + 0.717008i
\(748\) 0 0
\(749\) 2.00000 0.0730784
\(750\) 0 0
\(751\) 26.8328i 0.979143i −0.871963 0.489572i \(-0.837153\pi\)
0.871963 0.489572i \(-0.162847\pi\)
\(752\) 0 0
\(753\) 35.1246 13.4164i 1.28001 0.488921i
\(754\) 0 0
\(755\) 11.0557i 0.402359i
\(756\) 0 0
\(757\) 35.4164i 1.28723i −0.765349 0.643616i \(-0.777434\pi\)
0.765349 0.643616i \(-0.222566\pi\)
\(758\) 0 0
\(759\) 61.3050 23.4164i 2.22523 0.849962i
\(760\) 0 0
\(761\) 11.3050i 0.409804i 0.978782 + 0.204902i \(0.0656876\pi\)
−0.978782 + 0.204902i \(0.934312\pi\)
\(762\) 0 0
\(763\) −19.4164 −0.702921
\(764\) 0 0
\(765\) −4.94427 5.52786i −0.178761 0.199860i
\(766\) 0 0
\(767\) −10.4721 −0.378127
\(768\) 0 0
\(769\) −22.3607 −0.806347 −0.403173 0.915124i \(-0.632093\pi\)
−0.403173 + 0.915124i \(0.632093\pi\)
\(770\) 0 0
\(771\) −41.1246 + 15.7082i −1.48107 + 0.565717i
\(772\) 0 0
\(773\) 24.6525 0.886688 0.443344 0.896352i \(-0.353792\pi\)
0.443344 + 0.896352i \(0.353792\pi\)
\(774\) 0 0
\(775\) 31.0557i 1.11555i
\(776\) 0 0
\(777\) −1.52786 4.00000i −0.0548118 0.143499i
\(778\) 0 0
\(779\) 12.3607i 0.442867i
\(780\) 0 0
\(781\) 48.9443i 1.75136i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 12.0000i 0.428298i
\(786\) 0 0
\(787\) −23.1246 −0.824303 −0.412152 0.911115i \(-0.635223\pi\)
−0.412152 + 0.911115i \(0.635223\pi\)
\(788\) 0 0
\(789\) 8.65248 + 22.6525i 0.308036 + 0.806449i
\(790\) 0 0
\(791\) 4.94427 0.175798
\(792\) 0 0
\(793\) 5.52786 0.196300
\(794\) 0 0
\(795\) −1.16718 3.05573i −0.0413958 0.108376i
\(796\) 0 0
\(797\) 32.0689 1.13594 0.567969 0.823050i \(-0.307729\pi\)
0.567969 + 0.823050i \(0.307729\pi\)
\(798\) 0 0
\(799\) 12.9443i 0.457935i
\(800\) 0 0
\(801\) 12.0000 + 13.4164i 0.423999 + 0.474045i
\(802\) 0 0
\(803\) 37.8885i 1.33706i
\(804\) 0 0
\(805\) 10.4721i 0.369094i
\(806\) 0 0
\(807\) 3.81966 + 10.0000i 0.134458 + 0.352017i
\(808\) 0 0
\(809\) 48.7214i 1.71295i 0.516187 + 0.856476i \(0.327351\pi\)
−0.516187 + 0.856476i \(0.672649\pi\)
\(810\) 0 0
\(811\) −16.0689 −0.564255 −0.282127 0.959377i \(-0.591040\pi\)
−0.282127 + 0.959377i \(0.591040\pi\)
\(812\) 0 0
\(813\) −8.94427 + 3.41641i −0.313689 + 0.119819i
\(814\) 0 0
\(815\) −0.583592 −0.0204423
\(816\) 0 0
\(817\) 23.4164 0.819236
\(818\) 0 0
\(819\) −7.23607 + 6.47214i −0.252849 + 0.226155i
\(820\) 0 0
\(821\) −22.4721 −0.784283 −0.392141 0.919905i \(-0.628266\pi\)
−0.392141 + 0.919905i \(0.628266\pi\)
\(822\) 0 0
\(823\) 30.8328i 1.07476i 0.843339 + 0.537382i \(0.180587\pi\)
−0.843339 + 0.537382i \(0.819413\pi\)
\(824\) 0 0
\(825\) 25.1246 9.59675i 0.874727 0.334116i
\(826\) 0 0
\(827\) 2.00000i 0.0695468i 0.999395 + 0.0347734i \(0.0110710\pi\)
−0.999395 + 0.0347734i \(0.988929\pi\)
\(828\) 0 0
\(829\) 6.65248i 0.231050i −0.993305 0.115525i \(-0.963145\pi\)
0.993305 0.115525i \(-0.0368550\pi\)
\(830\) 0 0
\(831\) −19.4164 + 7.41641i −0.673548 + 0.257272i
\(832\) 0 0
\(833\) 2.00000i 0.0692959i
\(834\) 0 0
\(835\) 20.9443 0.724806
\(836\) 0 0
\(837\) −21.3050 + 41.3050i −0.736407 + 1.42771i
\(838\) 0 0
\(839\) 15.7771 0.544686 0.272343 0.962200i \(-0.412202\pi\)
0.272343 + 0.962200i \(0.412202\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) −28.9443 + 11.0557i −0.996894 + 0.380780i
\(844\) 0 0
\(845\) −3.12461 −0.107490
\(846\) 0 0
\(847\) 9.00000i 0.309244i
\(848\) 0 0
\(849\) 0.763932 + 2.00000i 0.0262181 + 0.0686398i
\(850\) 0 0
\(851\) 20.9443i 0.717960i
\(852\) 0 0
\(853\) 21.1246i 0.723293i 0.932315 + 0.361646i \(0.117785\pi\)
−0.932315 + 0.361646i \(0.882215\pi\)
\(854\) 0 0
\(855\) −7.63932 + 6.83282i −0.261259 + 0.233677i
\(856\) 0 0
\(857\) 23.3050i 0.796082i −0.917368 0.398041i \(-0.869690\pi\)
0.917368 0.398041i \(-0.130310\pi\)
\(858\) 0 0
\(859\) −44.0689 −1.50361 −0.751805 0.659385i \(-0.770817\pi\)
−0.751805 + 0.659385i \(0.770817\pi\)
\(860\) 0 0
\(861\) 2.76393 + 7.23607i 0.0941946 + 0.246605i
\(862\) 0 0
\(863\) −38.7214 −1.31809 −0.659045 0.752103i \(-0.729039\pi\)
−0.659045 + 0.752103i \(0.729039\pi\)
\(864\) 0 0
\(865\) 5.30495 0.180374
\(866\) 0 0
\(867\) −8.03444 21.0344i −0.272864 0.714367i
\(868\) 0 0
\(869\) −57.8885 −1.96373
\(870\) 0 0
\(871\) 22.4721i 0.761439i
\(872\) 0 0
\(873\) −27.8885 + 24.9443i −0.943884 + 0.844236i
\(874\) 0 0
\(875\) 10.4721i 0.354023i
\(876\) 0 0
\(877\) 4.58359i 0.154777i 0.997001 + 0.0773885i \(0.0246582\pi\)
−0.997001 + 0.0773885i \(0.975342\pi\)
\(878\) 0 0
\(879\) 6.87539 + 18.0000i 0.231901 + 0.607125i
\(880\) 0 0
\(881\) 13.4164i 0.452010i −0.974126 0.226005i \(-0.927433\pi\)
0.974126 0.226005i \(-0.0725666\pi\)
\(882\) 0 0
\(883\) 14.9443 0.502915 0.251457 0.967868i \(-0.419090\pi\)
0.251457 + 0.967868i \(0.419090\pi\)
\(884\) 0 0
\(885\) −6.47214 + 2.47214i −0.217558 + 0.0830999i
\(886\) 0 0
\(887\) 29.8885 1.00356 0.501780 0.864996i \(-0.332679\pi\)
0.501780 + 0.864996i \(0.332679\pi\)
\(888\) 0 0
\(889\) 0.944272 0.0316699
\(890\) 0 0
\(891\) −40.0000 4.47214i −1.34005 0.149822i
\(892\) 0 0
\(893\) −17.8885 −0.598617
\(894\) 0 0
\(895\) 3.63932i 0.121649i
\(896\) 0 0
\(897\) −44.3607 + 16.9443i −1.48116 + 0.565753i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 3.05573i 0.101801i
\(902\) 0 0
\(903\) 13.7082 5.23607i 0.456180 0.174245i
\(904\) 0 0
\(905\) 15.7771i 0.524448i
\(906\) 0 0
\(907\) 46.7214 1.55136 0.775679 0.631128i \(-0.217408\pi\)
0.775679 + 0.631128i \(0.217408\pi\)
\(908\) 0 0
\(909\) 44.0689 39.4164i 1.46167 1.30736i
\(910\) 0 0
\(911\) −9.63932 −0.319365 −0.159682 0.987168i \(-0.551047\pi\)
−0.159682 + 0.987168i \(0.551047\pi\)
\(912\) 0 0
\(913\) −39.1935 −1.29712
\(914\) 0 0
\(915\) 3.41641 1.30495i 0.112943 0.0431404i
\(916\) 0 0
\(917\) −7.23607 −0.238956
\(918\) 0 0
\(919\) 44.9443i 1.48257i 0.671188 + 0.741287i \(0.265784\pi\)
−0.671188 + 0.741287i \(0.734216\pi\)
\(920\) 0 0
\(921\) 14.2918 + 37.4164i 0.470931 + 1.23291i
\(922\) 0 0
\(923\) 35.4164i 1.16575i
\(924\) 0 0
\(925\) 8.58359i 0.282227i
\(926\) 0 0
\(927\) 20.9443 + 23.4164i 0.687900 + 0.769096i
\(928\) 0 0
\(929\) 37.0557i 1.21576i −0.794029 0.607880i \(-0.792020\pi\)
0.794029 0.607880i \(-0.207980\pi\)
\(930\) 0 0
\(931\) 2.76393 0.0905842
\(932\) 0 0
\(933\) 15.0557 + 39.4164i 0.492902 + 1.29044i
\(934\) 0 0
\(935\) 11.0557 0.361561
\(936\) 0 0
\(937\) −5.41641 −0.176946 −0.0884732 0.996079i \(-0.528199\pi\)
−0.0884732 + 0.996079i \(0.528199\pi\)
\(938\) 0 0
\(939\) 9.23607 + 24.1803i 0.301408 + 0.789096i
\(940\) 0 0
\(941\) −32.0689 −1.04542 −0.522708 0.852512i \(-0.675078\pi\)
−0.522708 + 0.852512i \(0.675078\pi\)
\(942\) 0 0
\(943\) 37.8885i 1.23382i
\(944\) 0 0
\(945\) −2.94427 + 5.70820i −0.0957772 + 0.185688i
\(946\) 0 0
\(947\) 7.52786i 0.244623i 0.992492 + 0.122311i \(0.0390306\pi\)
−0.992492 + 0.122311i \(0.960969\pi\)
\(948\) 0 0
\(949\) 27.4164i 0.889974i
\(950\) 0 0
\(951\) 4.00000 + 10.4721i 0.129709 + 0.339582i
\(952\) 0 0
\(953\) 1.88854i 0.0611759i −0.999532 0.0305880i \(-0.990262\pi\)
0.999532 0.0305880i \(-0.00973797\pi\)
\(954\) 0 0
\(955\) 2.47214 0.0799964
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −0.944272 −0.0304921
\(960\) 0 0
\(961\) −49.0000 −1.58065
\(962\) 0 0
\(963\) −4.00000 4.47214i −0.128898 0.144113i
\(964\) 0 0
\(965\) 25.3050 0.814595
\(966\) 0 0
\(967\) 16.9443i 0.544891i −0.962171 0.272446i \(-0.912168\pi\)
0.962171 0.272446i \(-0.0878325\pi\)
\(968\) 0 0
\(969\) 8.94427 3.41641i 0.287331 0.109751i
\(970\) 0 0
\(971\) 41.7082i 1.33848i −0.743046 0.669240i \(-0.766620\pi\)
0.743046 0.669240i \(-0.233380\pi\)
\(972\) 0 0
\(973\) 20.6525i 0.662088i
\(974\) 0 0
\(975\) −18.1803 + 6.94427i −0.582237 + 0.222395i
\(976\) 0 0
\(977\) 9.88854i 0.316363i −0.987410 0.158181i \(-0.949437\pi\)
0.987410 0.158181i \(-0.0505630\pi\)
\(978\) 0 0
\(979\) −26.8328 −0.857581
\(980\) 0 0
\(981\) 38.8328 + 43.4164i 1.23984 + 1.38618i
\(982\) 0 0
\(983\) −35.0557 −1.11810 −0.559052 0.829133i \(-0.688835\pi\)
−0.559052 + 0.829133i \(0.688835\pi\)
\(984\) 0 0
\(985\) −16.7214 −0.532787
\(986\) 0 0
\(987\) −10.4721 + 4.00000i −0.333332 + 0.127321i
\(988\) 0 0
\(989\) 71.7771 2.28238
\(990\) 0 0
\(991\) 40.0000i 1.27064i 0.772248 + 0.635321i \(0.219132\pi\)
−0.772248 + 0.635321i \(0.780868\pi\)
\(992\) 0 0
\(993\) 8.87539 + 23.2361i 0.281652 + 0.737374i
\(994\) 0 0
\(995\) 1.88854i 0.0598709i
\(996\) 0 0
\(997\) 2.87539i 0.0910645i 0.998963 + 0.0455322i \(0.0144984\pi\)
−0.998963 + 0.0455322i \(0.985502\pi\)
\(998\) 0 0
\(999\) −5.88854 + 11.4164i −0.186305 + 0.361199i
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 672.2.j.c.239.1 4
3.2 odd 2 672.2.j.b.239.2 4
4.3 odd 2 168.2.j.a.155.2 4
8.3 odd 2 672.2.j.b.239.1 4
8.5 even 2 168.2.j.c.155.2 yes 4
12.11 even 2 168.2.j.c.155.4 yes 4
24.5 odd 2 168.2.j.a.155.4 yes 4
24.11 even 2 inner 672.2.j.c.239.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.j.a.155.2 4 4.3 odd 2
168.2.j.a.155.4 yes 4 24.5 odd 2
168.2.j.c.155.2 yes 4 8.5 even 2
168.2.j.c.155.4 yes 4 12.11 even 2
672.2.j.b.239.1 4 8.3 odd 2
672.2.j.b.239.2 4 3.2 odd 2
672.2.j.c.239.1 4 1.1 even 1 trivial
672.2.j.c.239.2 4 24.11 even 2 inner