Properties

Label 1560.2.bg.g.841.2
Level $1560$
Weight $2$
Character 1560.841
Analytic conductor $12.457$
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] = [1560,2,Mod(601,1560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1560, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1560.601");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1560.bg (of order \(3\), degree \(2\), 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: \(12.4566627153\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 841.2
Root \(-1.63746 + 1.52274i\) of defining polynomial
Character \(\chi\) \(=\) 1560.841
Dual form 1560.2.bg.g.601.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{3} +1.00000 q^{5} +(-0.500000 - 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{3} +1.00000 q^{5} +(-0.500000 - 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(2.13746 - 3.70219i) q^{11} +(2.50000 - 2.59808i) q^{13} +(0.500000 - 0.866025i) q^{15} +(1.13746 + 1.97014i) q^{19} -1.00000 q^{21} +(-3.27492 + 5.67232i) q^{23} +1.00000 q^{25} -1.00000 q^{27} +(5.27492 - 9.13642i) q^{29} -7.27492 q^{31} +(-2.13746 - 3.70219i) q^{33} +(-0.500000 - 0.866025i) q^{35} +(-0.137459 + 0.238085i) q^{37} +(-1.00000 - 3.46410i) q^{39} +(-4.27492 + 7.40437i) q^{41} +(-3.63746 - 6.30026i) q^{43} +(-0.500000 - 0.866025i) q^{45} +4.27492 q^{47} +(3.00000 - 5.19615i) q^{49} +1.72508 q^{53} +(2.13746 - 3.70219i) q^{55} +2.27492 q^{57} +(-7.27492 - 12.6005i) q^{59} +(1.63746 + 2.83616i) q^{61} +(-0.500000 + 0.866025i) q^{63} +(2.50000 - 2.59808i) q^{65} +(2.63746 - 4.56821i) q^{67} +(3.27492 + 5.67232i) q^{69} +(1.00000 + 1.73205i) q^{71} +11.8248 q^{73} +(0.500000 - 0.866025i) q^{75} -4.27492 q^{77} -0.725083 q^{79} +(-0.500000 + 0.866025i) q^{81} +17.0997 q^{83} +(-5.27492 - 9.13642i) q^{87} +(2.86254 - 4.95807i) q^{89} +(-3.50000 - 0.866025i) q^{91} +(-3.63746 + 6.30026i) q^{93} +(1.13746 + 1.97014i) q^{95} +(-3.91238 - 6.77643i) q^{97} -4.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 4 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 4 q^{5} - 2 q^{7} - 2 q^{9} + q^{11} + 10 q^{13} + 2 q^{15} - 3 q^{19} - 4 q^{21} + 2 q^{23} + 4 q^{25} - 4 q^{27} + 6 q^{29} - 14 q^{31} - q^{33} - 2 q^{35} + 7 q^{37} - 4 q^{39} - 2 q^{41} - 7 q^{43} - 2 q^{45} + 2 q^{47} + 12 q^{49} + 22 q^{53} + q^{55} - 6 q^{57} - 14 q^{59} - q^{61} - 2 q^{63} + 10 q^{65} + 3 q^{67} - 2 q^{69} + 4 q^{71} + 2 q^{73} + 2 q^{75} - 2 q^{77} - 18 q^{79} - 2 q^{81} + 8 q^{83} - 6 q^{87} + 19 q^{89} - 14 q^{91} - 7 q^{93} - 3 q^{95} + 7 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times\).

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 0.500000 0.866025i 0.288675 0.500000i
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −0.500000 0.866025i −0.188982 0.327327i 0.755929 0.654654i \(-0.227186\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 0 0
\(9\) −0.500000 0.866025i −0.166667 0.288675i
\(10\) 0 0
\(11\) 2.13746 3.70219i 0.644468 1.11625i −0.339956 0.940441i \(-0.610412\pi\)
0.984424 0.175810i \(-0.0562545\pi\)
\(12\) 0 0
\(13\) 2.50000 2.59808i 0.693375 0.720577i
\(14\) 0 0
\(15\) 0.500000 0.866025i 0.129099 0.223607i
\(16\) 0 0
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) 1.13746 + 1.97014i 0.260951 + 0.451980i 0.966495 0.256686i \(-0.0826306\pi\)
−0.705544 + 0.708666i \(0.749297\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −3.27492 + 5.67232i −0.682867 + 1.18276i 0.291234 + 0.956652i \(0.405934\pi\)
−0.974102 + 0.226109i \(0.927399\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 5.27492 9.13642i 0.979528 1.69659i 0.315424 0.948951i \(-0.397853\pi\)
0.664104 0.747641i \(-0.268813\pi\)
\(30\) 0 0
\(31\) −7.27492 −1.30661 −0.653307 0.757093i \(-0.726619\pi\)
−0.653307 + 0.757093i \(0.726619\pi\)
\(32\) 0 0
\(33\) −2.13746 3.70219i −0.372084 0.644468i
\(34\) 0 0
\(35\) −0.500000 0.866025i −0.0845154 0.146385i
\(36\) 0 0
\(37\) −0.137459 + 0.238085i −0.0225981 + 0.0391410i −0.877103 0.480302i \(-0.840527\pi\)
0.854505 + 0.519443i \(0.173860\pi\)
\(38\) 0 0
\(39\) −1.00000 3.46410i −0.160128 0.554700i
\(40\) 0 0
\(41\) −4.27492 + 7.40437i −0.667630 + 1.15637i 0.310935 + 0.950431i \(0.399358\pi\)
−0.978565 + 0.205938i \(0.933976\pi\)
\(42\) 0 0
\(43\) −3.63746 6.30026i −0.554707 0.960781i −0.997926 0.0643678i \(-0.979497\pi\)
0.443219 0.896413i \(-0.353836\pi\)
\(44\) 0 0
\(45\) −0.500000 0.866025i −0.0745356 0.129099i
\(46\) 0 0
\(47\) 4.27492 0.623561 0.311780 0.950154i \(-0.399075\pi\)
0.311780 + 0.950154i \(0.399075\pi\)
\(48\) 0 0
\(49\) 3.00000 5.19615i 0.428571 0.742307i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.72508 0.236958 0.118479 0.992957i \(-0.462198\pi\)
0.118479 + 0.992957i \(0.462198\pi\)
\(54\) 0 0
\(55\) 2.13746 3.70219i 0.288215 0.499203i
\(56\) 0 0
\(57\) 2.27492 0.301320
\(58\) 0 0
\(59\) −7.27492 12.6005i −0.947114 1.64045i −0.751463 0.659775i \(-0.770652\pi\)
−0.195650 0.980674i \(-0.562682\pi\)
\(60\) 0 0
\(61\) 1.63746 + 2.83616i 0.209655 + 0.363133i 0.951606 0.307321i \(-0.0994326\pi\)
−0.741951 + 0.670454i \(0.766099\pi\)
\(62\) 0 0
\(63\) −0.500000 + 0.866025i −0.0629941 + 0.109109i
\(64\) 0 0
\(65\) 2.50000 2.59808i 0.310087 0.322252i
\(66\) 0 0
\(67\) 2.63746 4.56821i 0.322217 0.558096i −0.658728 0.752381i \(-0.728905\pi\)
0.980945 + 0.194285i \(0.0622386\pi\)
\(68\) 0 0
\(69\) 3.27492 + 5.67232i 0.394254 + 0.682867i
\(70\) 0 0
\(71\) 1.00000 + 1.73205i 0.118678 + 0.205557i 0.919244 0.393688i \(-0.128801\pi\)
−0.800566 + 0.599245i \(0.795468\pi\)
\(72\) 0 0
\(73\) 11.8248 1.38398 0.691991 0.721906i \(-0.256734\pi\)
0.691991 + 0.721906i \(0.256734\pi\)
\(74\) 0 0
\(75\) 0.500000 0.866025i 0.0577350 0.100000i
\(76\) 0 0
\(77\) −4.27492 −0.487172
\(78\) 0 0
\(79\) −0.725083 −0.0815782 −0.0407891 0.999168i \(-0.512987\pi\)
−0.0407891 + 0.999168i \(0.512987\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 17.0997 1.87693 0.938466 0.345371i \(-0.112247\pi\)
0.938466 + 0.345371i \(0.112247\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −5.27492 9.13642i −0.565530 0.979528i
\(88\) 0 0
\(89\) 2.86254 4.95807i 0.303429 0.525554i −0.673481 0.739204i \(-0.735202\pi\)
0.976910 + 0.213650i \(0.0685352\pi\)
\(90\) 0 0
\(91\) −3.50000 0.866025i −0.366900 0.0907841i
\(92\) 0 0
\(93\) −3.63746 + 6.30026i −0.377187 + 0.653307i
\(94\) 0 0
\(95\) 1.13746 + 1.97014i 0.116701 + 0.202132i
\(96\) 0 0
\(97\) −3.91238 6.77643i −0.397242 0.688043i 0.596143 0.802878i \(-0.296699\pi\)
−0.993384 + 0.114836i \(0.963366\pi\)
\(98\) 0 0
\(99\) −4.27492 −0.429645
\(100\) 0 0
\(101\) 3.27492 5.67232i 0.325866 0.564417i −0.655821 0.754917i \(-0.727677\pi\)
0.981687 + 0.190499i \(0.0610107\pi\)
\(102\) 0 0
\(103\) −15.5498 −1.53217 −0.766085 0.642739i \(-0.777798\pi\)
−0.766085 + 0.642739i \(0.777798\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) −7.54983 + 13.0767i −0.729870 + 1.26417i 0.227068 + 0.973879i \(0.427086\pi\)
−0.956938 + 0.290293i \(0.906247\pi\)
\(108\) 0 0
\(109\) −13.8248 −1.32417 −0.662086 0.749428i \(-0.730328\pi\)
−0.662086 + 0.749428i \(0.730328\pi\)
\(110\) 0 0
\(111\) 0.137459 + 0.238085i 0.0130470 + 0.0225981i
\(112\) 0 0
\(113\) 2.72508 + 4.71998i 0.256354 + 0.444019i 0.965262 0.261282i \(-0.0841453\pi\)
−0.708908 + 0.705301i \(0.750812\pi\)
\(114\) 0 0
\(115\) −3.27492 + 5.67232i −0.305388 + 0.528947i
\(116\) 0 0
\(117\) −3.50000 0.866025i −0.323575 0.0800641i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −3.63746 6.30026i −0.330678 0.572751i
\(122\) 0 0
\(123\) 4.27492 + 7.40437i 0.385456 + 0.667630i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −0.500000 + 0.866025i −0.0443678 + 0.0768473i −0.887357 0.461084i \(-0.847461\pi\)
0.842989 + 0.537931i \(0.180794\pi\)
\(128\) 0 0
\(129\) −7.27492 −0.640521
\(130\) 0 0
\(131\) 3.72508 0.325462 0.162731 0.986670i \(-0.447970\pi\)
0.162731 + 0.986670i \(0.447970\pi\)
\(132\) 0 0
\(133\) 1.13746 1.97014i 0.0986302 0.170832i
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 10.5498 + 18.2728i 0.901333 + 1.56115i 0.825765 + 0.564014i \(0.190744\pi\)
0.0755681 + 0.997141i \(0.475923\pi\)
\(138\) 0 0
\(139\) −5.50000 9.52628i −0.466504 0.808008i 0.532764 0.846264i \(-0.321153\pi\)
−0.999268 + 0.0382553i \(0.987820\pi\)
\(140\) 0 0
\(141\) 2.13746 3.70219i 0.180006 0.311780i
\(142\) 0 0
\(143\) −4.27492 14.8087i −0.357487 1.23837i
\(144\) 0 0
\(145\) 5.27492 9.13642i 0.438058 0.758739i
\(146\) 0 0
\(147\) −3.00000 5.19615i −0.247436 0.428571i
\(148\) 0 0
\(149\) 8.27492 + 14.3326i 0.677908 + 1.17417i 0.975610 + 0.219512i \(0.0704464\pi\)
−0.297702 + 0.954659i \(0.596220\pi\)
\(150\) 0 0
\(151\) −3.45017 −0.280770 −0.140385 0.990097i \(-0.544834\pi\)
−0.140385 + 0.990097i \(0.544834\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −7.27492 −0.584335
\(156\) 0 0
\(157\) 11.5498 0.921777 0.460889 0.887458i \(-0.347531\pi\)
0.460889 + 0.887458i \(0.347531\pi\)
\(158\) 0 0
\(159\) 0.862541 1.49397i 0.0684040 0.118479i
\(160\) 0 0
\(161\) 6.54983 0.516199
\(162\) 0 0
\(163\) −1.36254 2.35999i −0.106722 0.184849i 0.807718 0.589569i \(-0.200702\pi\)
−0.914441 + 0.404720i \(0.867369\pi\)
\(164\) 0 0
\(165\) −2.13746 3.70219i −0.166401 0.288215i
\(166\) 0 0
\(167\) 5.13746 8.89834i 0.397548 0.688574i −0.595874 0.803078i \(-0.703194\pi\)
0.993423 + 0.114503i \(0.0365277\pi\)
\(168\) 0 0
\(169\) −0.500000 12.9904i −0.0384615 0.999260i
\(170\) 0 0
\(171\) 1.13746 1.97014i 0.0869836 0.150660i
\(172\) 0 0
\(173\) 7.86254 + 13.6183i 0.597778 + 1.03538i 0.993148 + 0.116860i \(0.0372829\pi\)
−0.395370 + 0.918522i \(0.629384\pi\)
\(174\) 0 0
\(175\) −0.500000 0.866025i −0.0377964 0.0654654i
\(176\) 0 0
\(177\) −14.5498 −1.09363
\(178\) 0 0
\(179\) 6.00000 10.3923i 0.448461 0.776757i −0.549825 0.835280i \(-0.685306\pi\)
0.998286 + 0.0585225i \(0.0186389\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 3.27492 0.242089
\(184\) 0 0
\(185\) −0.137459 + 0.238085i −0.0101062 + 0.0175044i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0.500000 + 0.866025i 0.0363696 + 0.0629941i
\(190\) 0 0
\(191\) 9.00000 + 15.5885i 0.651217 + 1.12794i 0.982828 + 0.184525i \(0.0590746\pi\)
−0.331611 + 0.943416i \(0.607592\pi\)
\(192\) 0 0
\(193\) −7.18729 + 12.4488i −0.517353 + 0.896081i 0.482444 + 0.875927i \(0.339749\pi\)
−0.999797 + 0.0201544i \(0.993584\pi\)
\(194\) 0 0
\(195\) −1.00000 3.46410i −0.0716115 0.248069i
\(196\) 0 0
\(197\) 1.41238 2.44631i 0.100628 0.174292i −0.811316 0.584608i \(-0.801248\pi\)
0.911943 + 0.410316i \(0.134582\pi\)
\(198\) 0 0
\(199\) −2.36254 4.09204i −0.167476 0.290077i 0.770056 0.637977i \(-0.220228\pi\)
−0.937532 + 0.347899i \(0.886895\pi\)
\(200\) 0 0
\(201\) −2.63746 4.56821i −0.186032 0.322217i
\(202\) 0 0
\(203\) −10.5498 −0.740453
\(204\) 0 0
\(205\) −4.27492 + 7.40437i −0.298573 + 0.517144i
\(206\) 0 0
\(207\) 6.54983 0.455245
\(208\) 0 0
\(209\) 9.72508 0.672698
\(210\) 0 0
\(211\) −2.50000 + 4.33013i −0.172107 + 0.298098i −0.939156 0.343490i \(-0.888391\pi\)
0.767049 + 0.641588i \(0.221724\pi\)
\(212\) 0 0
\(213\) 2.00000 0.137038
\(214\) 0 0
\(215\) −3.63746 6.30026i −0.248073 0.429674i
\(216\) 0 0
\(217\) 3.63746 + 6.30026i 0.246927 + 0.427690i
\(218\) 0 0
\(219\) 5.91238 10.2405i 0.399521 0.691991i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −13.4124 + 23.2309i −0.898159 + 1.55566i −0.0683137 + 0.997664i \(0.521762\pi\)
−0.829845 + 0.557993i \(0.811571\pi\)
\(224\) 0 0
\(225\) −0.500000 0.866025i −0.0333333 0.0577350i
\(226\) 0 0
\(227\) −4.54983 7.88054i −0.301983 0.523050i 0.674602 0.738182i \(-0.264315\pi\)
−0.976585 + 0.215132i \(0.930982\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) −2.13746 + 3.70219i −0.140634 + 0.243586i
\(232\) 0 0
\(233\) 23.0997 1.51331 0.756655 0.653815i \(-0.226832\pi\)
0.756655 + 0.653815i \(0.226832\pi\)
\(234\) 0 0
\(235\) 4.27492 0.278865
\(236\) 0 0
\(237\) −0.362541 + 0.627940i −0.0235496 + 0.0407891i
\(238\) 0 0
\(239\) −4.54983 −0.294304 −0.147152 0.989114i \(-0.547011\pi\)
−0.147152 + 0.989114i \(0.547011\pi\)
\(240\) 0 0
\(241\) −4.13746 7.16629i −0.266517 0.461621i 0.701443 0.712726i \(-0.252540\pi\)
−0.967960 + 0.251104i \(0.919206\pi\)
\(242\) 0 0
\(243\) 0.500000 + 0.866025i 0.0320750 + 0.0555556i
\(244\) 0 0
\(245\) 3.00000 5.19615i 0.191663 0.331970i
\(246\) 0 0
\(247\) 7.96221 + 1.97014i 0.506623 + 0.125357i
\(248\) 0 0
\(249\) 8.54983 14.8087i 0.541824 0.938466i
\(250\) 0 0
\(251\) 13.1375 + 22.7547i 0.829229 + 1.43627i 0.898644 + 0.438679i \(0.144553\pi\)
−0.0694148 + 0.997588i \(0.522113\pi\)
\(252\) 0 0
\(253\) 14.0000 + 24.2487i 0.880172 + 1.52450i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −4.00000 + 6.92820i −0.249513 + 0.432169i −0.963391 0.268101i \(-0.913604\pi\)
0.713878 + 0.700270i \(0.246937\pi\)
\(258\) 0 0
\(259\) 0.274917 0.0170825
\(260\) 0 0
\(261\) −10.5498 −0.653018
\(262\) 0 0
\(263\) 1.13746 1.97014i 0.0701387 0.121484i −0.828823 0.559511i \(-0.810989\pi\)
0.898962 + 0.438027i \(0.144322\pi\)
\(264\) 0 0
\(265\) 1.72508 0.105971
\(266\) 0 0
\(267\) −2.86254 4.95807i −0.175185 0.303429i
\(268\) 0 0
\(269\) 6.00000 + 10.3923i 0.365826 + 0.633630i 0.988908 0.148527i \(-0.0474530\pi\)
−0.623082 + 0.782157i \(0.714120\pi\)
\(270\) 0 0
\(271\) −12.6375 + 21.8887i −0.767671 + 1.32965i 0.171152 + 0.985245i \(0.445251\pi\)
−0.938823 + 0.344400i \(0.888082\pi\)
\(272\) 0 0
\(273\) −2.50000 + 2.59808i −0.151307 + 0.157243i
\(274\) 0 0
\(275\) 2.13746 3.70219i 0.128894 0.223250i
\(276\) 0 0
\(277\) 14.4124 + 24.9630i 0.865956 + 1.49988i 0.866095 + 0.499879i \(0.166622\pi\)
−0.000139316 1.00000i \(0.500044\pi\)
\(278\) 0 0
\(279\) 3.63746 + 6.30026i 0.217769 + 0.377187i
\(280\) 0 0
\(281\) −27.0997 −1.61663 −0.808315 0.588750i \(-0.799620\pi\)
−0.808315 + 0.588750i \(0.799620\pi\)
\(282\) 0 0
\(283\) 11.4622 19.8531i 0.681358 1.18015i −0.293209 0.956048i \(-0.594723\pi\)
0.974567 0.224098i \(-0.0719435\pi\)
\(284\) 0 0
\(285\) 2.27492 0.134754
\(286\) 0 0
\(287\) 8.54983 0.504681
\(288\) 0 0
\(289\) 8.50000 14.7224i 0.500000 0.866025i
\(290\) 0 0
\(291\) −7.82475 −0.458695
\(292\) 0 0
\(293\) 14.1375 + 24.4868i 0.825919 + 1.43053i 0.901215 + 0.433373i \(0.142677\pi\)
−0.0752957 + 0.997161i \(0.523990\pi\)
\(294\) 0 0
\(295\) −7.27492 12.6005i −0.423562 0.733631i
\(296\) 0 0
\(297\) −2.13746 + 3.70219i −0.124028 + 0.214823i
\(298\) 0 0
\(299\) 6.54983 + 22.6893i 0.378787 + 1.31216i
\(300\) 0 0
\(301\) −3.63746 + 6.30026i −0.209660 + 0.363141i
\(302\) 0 0
\(303\) −3.27492 5.67232i −0.188139 0.325866i
\(304\) 0 0
\(305\) 1.63746 + 2.83616i 0.0937606 + 0.162398i
\(306\) 0 0
\(307\) −22.9244 −1.30837 −0.654183 0.756336i \(-0.726987\pi\)
−0.654183 + 0.756336i \(0.726987\pi\)
\(308\) 0 0
\(309\) −7.77492 + 13.4666i −0.442300 + 0.766085i
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 4.72508 0.267077 0.133539 0.991044i \(-0.457366\pi\)
0.133539 + 0.991044i \(0.457366\pi\)
\(314\) 0 0
\(315\) −0.500000 + 0.866025i −0.0281718 + 0.0487950i
\(316\) 0 0
\(317\) −19.3746 −1.08819 −0.544093 0.839025i \(-0.683126\pi\)
−0.544093 + 0.839025i \(0.683126\pi\)
\(318\) 0 0
\(319\) −22.5498 39.0575i −1.26255 2.18680i
\(320\) 0 0
\(321\) 7.54983 + 13.0767i 0.421391 + 0.729870i
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 2.50000 2.59808i 0.138675 0.144115i
\(326\) 0 0
\(327\) −6.91238 + 11.9726i −0.382255 + 0.662086i
\(328\) 0 0
\(329\) −2.13746 3.70219i −0.117842 0.204108i
\(330\) 0 0
\(331\) 9.36254 + 16.2164i 0.514612 + 0.891334i 0.999856 + 0.0169553i \(0.00539729\pi\)
−0.485244 + 0.874379i \(0.661269\pi\)
\(332\) 0 0
\(333\) 0.274917 0.0150654
\(334\) 0 0
\(335\) 2.63746 4.56821i 0.144100 0.249588i
\(336\) 0 0
\(337\) 19.2749 1.04997 0.524986 0.851111i \(-0.324071\pi\)
0.524986 + 0.851111i \(0.324071\pi\)
\(338\) 0 0
\(339\) 5.45017 0.296012
\(340\) 0 0
\(341\) −15.5498 + 26.9331i −0.842071 + 1.45851i
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 0 0
\(345\) 3.27492 + 5.67232i 0.176316 + 0.305388i
\(346\) 0 0
\(347\) 8.54983 + 14.8087i 0.458979 + 0.794975i 0.998907 0.0467359i \(-0.0148819\pi\)
−0.539928 + 0.841711i \(0.681549\pi\)
\(348\) 0 0
\(349\) 10.4622 18.1211i 0.560029 0.969999i −0.437464 0.899236i \(-0.644123\pi\)
0.997493 0.0707632i \(-0.0225435\pi\)
\(350\) 0 0
\(351\) −2.50000 + 2.59808i −0.133440 + 0.138675i
\(352\) 0 0
\(353\) −15.2749 + 26.4569i −0.813002 + 1.40816i 0.0977520 + 0.995211i \(0.468835\pi\)
−0.910754 + 0.412950i \(0.864499\pi\)
\(354\) 0 0
\(355\) 1.00000 + 1.73205i 0.0530745 + 0.0919277i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 28.5498 1.50680 0.753401 0.657561i \(-0.228412\pi\)
0.753401 + 0.657561i \(0.228412\pi\)
\(360\) 0 0
\(361\) 6.91238 11.9726i 0.363809 0.630136i
\(362\) 0 0
\(363\) −7.27492 −0.381834
\(364\) 0 0
\(365\) 11.8248 0.618936
\(366\) 0 0
\(367\) −8.63746 + 14.9605i −0.450872 + 0.780933i −0.998440 0.0558281i \(-0.982220\pi\)
0.547569 + 0.836761i \(0.315553\pi\)
\(368\) 0 0
\(369\) 8.54983 0.445087
\(370\) 0 0
\(371\) −0.862541 1.49397i −0.0447809 0.0775628i
\(372\) 0 0
\(373\) 7.63746 + 13.2285i 0.395453 + 0.684944i 0.993159 0.116771i \(-0.0372544\pi\)
−0.597706 + 0.801715i \(0.703921\pi\)
\(374\) 0 0
\(375\) 0.500000 0.866025i 0.0258199 0.0447214i
\(376\) 0 0
\(377\) −10.5498 36.5457i −0.543344 1.88220i
\(378\) 0 0
\(379\) −10.7749 + 18.6627i −0.553470 + 0.958639i 0.444550 + 0.895754i \(0.353363\pi\)
−0.998021 + 0.0628849i \(0.979970\pi\)
\(380\) 0 0
\(381\) 0.500000 + 0.866025i 0.0256158 + 0.0443678i
\(382\) 0 0
\(383\) 12.5498 + 21.7370i 0.641267 + 1.11071i 0.985150 + 0.171694i \(0.0549241\pi\)
−0.343884 + 0.939012i \(0.611743\pi\)
\(384\) 0 0
\(385\) −4.27492 −0.217870
\(386\) 0 0
\(387\) −3.63746 + 6.30026i −0.184902 + 0.320260i
\(388\) 0 0
\(389\) −8.00000 −0.405616 −0.202808 0.979219i \(-0.565007\pi\)
−0.202808 + 0.979219i \(0.565007\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 1.86254 3.22602i 0.0939528 0.162731i
\(394\) 0 0
\(395\) −0.725083 −0.0364829
\(396\) 0 0
\(397\) −2.04983 3.55042i −0.102878 0.178190i 0.809991 0.586442i \(-0.199472\pi\)
−0.912869 + 0.408252i \(0.866139\pi\)
\(398\) 0 0
\(399\) −1.13746 1.97014i −0.0569442 0.0986302i
\(400\) 0 0
\(401\) 19.9622 34.5756i 0.996865 1.72662i 0.429914 0.902870i \(-0.358544\pi\)
0.566951 0.823751i \(-0.308123\pi\)
\(402\) 0 0
\(403\) −18.1873 + 18.9008i −0.905974 + 0.941515i
\(404\) 0 0
\(405\) −0.500000 + 0.866025i −0.0248452 + 0.0430331i
\(406\) 0 0
\(407\) 0.587624 + 1.01779i 0.0291274 + 0.0504502i
\(408\) 0 0
\(409\) −1.95017 3.37779i −0.0964295 0.167021i 0.813775 0.581180i \(-0.197409\pi\)
−0.910204 + 0.414160i \(0.864076\pi\)
\(410\) 0 0
\(411\) 21.0997 1.04077
\(412\) 0 0
\(413\) −7.27492 + 12.6005i −0.357975 + 0.620031i
\(414\) 0 0
\(415\) 17.0997 0.839390
\(416\) 0 0
\(417\) −11.0000 −0.538672
\(418\) 0 0
\(419\) 15.0997 26.1534i 0.737667 1.27768i −0.215876 0.976421i \(-0.569261\pi\)
0.953543 0.301256i \(-0.0974059\pi\)
\(420\) 0 0
\(421\) 15.8248 0.771251 0.385626 0.922655i \(-0.373986\pi\)
0.385626 + 0.922655i \(0.373986\pi\)
\(422\) 0 0
\(423\) −2.13746 3.70219i −0.103927 0.180006i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.63746 2.83616i 0.0792422 0.137251i
\(428\) 0 0
\(429\) −14.9622 3.70219i −0.722382 0.178743i
\(430\) 0 0
\(431\) −6.00000 + 10.3923i −0.289010 + 0.500580i −0.973574 0.228373i \(-0.926659\pi\)
0.684564 + 0.728953i \(0.259993\pi\)
\(432\) 0 0
\(433\) −7.18729 12.4488i −0.345399 0.598249i 0.640027 0.768352i \(-0.278923\pi\)
−0.985426 + 0.170103i \(0.945590\pi\)
\(434\) 0 0
\(435\) −5.27492 9.13642i −0.252913 0.438058i
\(436\) 0 0
\(437\) −14.9003 −0.712780
\(438\) 0 0
\(439\) 12.3625 21.4125i 0.590032 1.02197i −0.404196 0.914673i \(-0.632448\pi\)
0.994227 0.107293i \(-0.0342182\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) −17.6495 −0.838553 −0.419277 0.907859i \(-0.637716\pi\)
−0.419277 + 0.907859i \(0.637716\pi\)
\(444\) 0 0
\(445\) 2.86254 4.95807i 0.135697 0.235035i
\(446\) 0 0
\(447\) 16.5498 0.782780
\(448\) 0 0
\(449\) −7.86254 13.6183i −0.371056 0.642688i 0.618672 0.785649i \(-0.287671\pi\)
−0.989728 + 0.142961i \(0.954338\pi\)
\(450\) 0 0
\(451\) 18.2749 + 31.6531i 0.860532 + 1.49049i
\(452\) 0 0
\(453\) −1.72508 + 2.98793i −0.0810515 + 0.140385i
\(454\) 0 0
\(455\) −3.50000 0.866025i −0.164083 0.0405999i
\(456\) 0 0
\(457\) −14.1873 + 24.5731i −0.663654 + 1.14948i 0.315995 + 0.948761i \(0.397662\pi\)
−0.979648 + 0.200721i \(0.935672\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.00000 + 10.3923i 0.279448 + 0.484018i 0.971248 0.238071i \(-0.0765153\pi\)
−0.691800 + 0.722089i \(0.743182\pi\)
\(462\) 0 0
\(463\) −30.7251 −1.42792 −0.713958 0.700189i \(-0.753099\pi\)
−0.713958 + 0.700189i \(0.753099\pi\)
\(464\) 0 0
\(465\) −3.63746 + 6.30026i −0.168683 + 0.292168i
\(466\) 0 0
\(467\) 14.5498 0.673286 0.336643 0.941632i \(-0.390708\pi\)
0.336643 + 0.941632i \(0.390708\pi\)
\(468\) 0 0
\(469\) −5.27492 −0.243573
\(470\) 0 0
\(471\) 5.77492 10.0025i 0.266094 0.460889i
\(472\) 0 0
\(473\) −31.0997 −1.42996
\(474\) 0 0
\(475\) 1.13746 + 1.97014i 0.0521902 + 0.0903960i
\(476\) 0 0
\(477\) −0.862541 1.49397i −0.0394931 0.0684040i
\(478\) 0 0
\(479\) −14.5498 + 25.2011i −0.664799 + 1.15147i 0.314541 + 0.949244i \(0.398149\pi\)
−0.979340 + 0.202222i \(0.935184\pi\)
\(480\) 0 0
\(481\) 0.274917 + 0.952341i 0.0125351 + 0.0434230i
\(482\) 0 0
\(483\) 3.27492 5.67232i 0.149014 0.258100i
\(484\) 0 0
\(485\) −3.91238 6.77643i −0.177652 0.307702i
\(486\) 0 0
\(487\) 17.1375 + 29.6829i 0.776572 + 1.34506i 0.933906 + 0.357517i \(0.116377\pi\)
−0.157334 + 0.987545i \(0.550290\pi\)
\(488\) 0 0
\(489\) −2.72508 −0.123233
\(490\) 0 0
\(491\) 3.58762 6.21395i 0.161907 0.280432i −0.773645 0.633619i \(-0.781569\pi\)
0.935553 + 0.353187i \(0.114902\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −4.27492 −0.192143
\(496\) 0 0
\(497\) 1.00000 1.73205i 0.0448561 0.0776931i
\(498\) 0 0
\(499\) 17.0997 0.765486 0.382743 0.923855i \(-0.374979\pi\)
0.382743 + 0.923855i \(0.374979\pi\)
\(500\) 0 0
\(501\) −5.13746 8.89834i −0.229525 0.397548i
\(502\) 0 0
\(503\) 12.4124 + 21.4989i 0.553440 + 0.958587i 0.998023 + 0.0628492i \(0.0200187\pi\)
−0.444583 + 0.895738i \(0.646648\pi\)
\(504\) 0 0
\(505\) 3.27492 5.67232i 0.145732 0.252415i
\(506\) 0 0
\(507\) −11.5000 6.06218i −0.510733 0.269231i
\(508\) 0 0
\(509\) −11.5498 + 20.0049i −0.511937 + 0.886702i 0.487967 + 0.872862i \(0.337739\pi\)
−0.999904 + 0.0138394i \(0.995595\pi\)
\(510\) 0 0
\(511\) −5.91238 10.2405i −0.261548 0.453015i
\(512\) 0 0
\(513\) −1.13746 1.97014i −0.0502200 0.0869836i
\(514\) 0 0
\(515\) −15.5498 −0.685208
\(516\) 0 0
\(517\) 9.13746 15.8265i 0.401865 0.696051i
\(518\) 0 0
\(519\) 15.7251 0.690255
\(520\) 0 0
\(521\) −13.7251 −0.601307 −0.300653 0.953733i \(-0.597205\pi\)
−0.300653 + 0.953733i \(0.597205\pi\)
\(522\) 0 0
\(523\) −16.2749 + 28.1890i −0.711652 + 1.23262i 0.252584 + 0.967575i \(0.418720\pi\)
−0.964237 + 0.265043i \(0.914614\pi\)
\(524\) 0 0
\(525\) −1.00000 −0.0436436
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −9.95017 17.2342i −0.432616 0.749313i
\(530\) 0 0
\(531\) −7.27492 + 12.6005i −0.315705 + 0.546816i
\(532\) 0 0
\(533\) 8.54983 + 29.6175i 0.370334 + 1.28288i
\(534\) 0 0
\(535\) −7.54983 + 13.0767i −0.326408 + 0.565355i
\(536\) 0 0
\(537\) −6.00000 10.3923i −0.258919 0.448461i
\(538\) 0 0
\(539\) −12.8248 22.2131i −0.552401 0.956787i
\(540\) 0 0
\(541\) −29.8248 −1.28227 −0.641133 0.767430i \(-0.721535\pi\)
−0.641133 + 0.767430i \(0.721535\pi\)
\(542\) 0 0
\(543\) −1.00000 + 1.73205i −0.0429141 + 0.0743294i
\(544\) 0 0
\(545\) −13.8248 −0.592187
\(546\) 0 0
\(547\) 29.8248 1.27521 0.637607 0.770362i \(-0.279924\pi\)
0.637607 + 0.770362i \(0.279924\pi\)
\(548\) 0 0
\(549\) 1.63746 2.83616i 0.0698850 0.121044i
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) 0.362541 + 0.627940i 0.0154168 + 0.0267027i
\(554\) 0 0
\(555\) 0.137459 + 0.238085i 0.00583479 + 0.0101062i
\(556\) 0 0
\(557\) 2.31271 4.00573i 0.0979925 0.169728i −0.812861 0.582458i \(-0.802091\pi\)
0.910854 + 0.412730i \(0.135425\pi\)
\(558\) 0 0
\(559\) −25.4622 6.30026i −1.07694 0.266473i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9.27492 + 16.0646i 0.390891 + 0.677043i 0.992567 0.121697i \(-0.0388335\pi\)
−0.601676 + 0.798740i \(0.705500\pi\)
\(564\) 0 0
\(565\) 2.72508 + 4.71998i 0.114645 + 0.198571i
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 20.6873 35.8314i 0.867256 1.50213i 0.00246734 0.999997i \(-0.499215\pi\)
0.864789 0.502135i \(-0.167452\pi\)
\(570\) 0 0
\(571\) 4.62541 0.193568 0.0967838 0.995305i \(-0.469144\pi\)
0.0967838 + 0.995305i \(0.469144\pi\)
\(572\) 0 0
\(573\) 18.0000 0.751961
\(574\) 0 0
\(575\) −3.27492 + 5.67232i −0.136573 + 0.236552i
\(576\) 0 0
\(577\) 31.0997 1.29470 0.647348 0.762195i \(-0.275878\pi\)
0.647348 + 0.762195i \(0.275878\pi\)
\(578\) 0 0
\(579\) 7.18729 + 12.4488i 0.298694 + 0.517353i
\(580\) 0 0
\(581\) −8.54983 14.8087i −0.354707 0.614370i
\(582\) 0 0
\(583\) 3.68729 6.38658i 0.152712 0.264505i
\(584\) 0 0
\(585\) −3.50000 0.866025i −0.144707 0.0358057i
\(586\) 0 0
\(587\) 10.0997 17.4931i 0.416858 0.722019i −0.578764 0.815495i \(-0.696465\pi\)
0.995622 + 0.0934763i \(0.0297979\pi\)
\(588\) 0 0
\(589\) −8.27492 14.3326i −0.340962 0.590564i
\(590\) 0 0
\(591\) −1.41238 2.44631i −0.0580974 0.100628i
\(592\) 0 0
\(593\) −17.4502 −0.716592 −0.358296 0.933608i \(-0.616642\pi\)
−0.358296 + 0.933608i \(0.616642\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −4.72508 −0.193385
\(598\) 0 0
\(599\) 4.90033 0.200222 0.100111 0.994976i \(-0.468080\pi\)
0.100111 + 0.994976i \(0.468080\pi\)
\(600\) 0 0
\(601\) 18.1375 31.4150i 0.739842 1.28144i −0.212724 0.977112i \(-0.568233\pi\)
0.952566 0.304332i \(-0.0984333\pi\)
\(602\) 0 0
\(603\) −5.27492 −0.214811
\(604\) 0 0
\(605\) −3.63746 6.30026i −0.147884 0.256142i
\(606\) 0 0
\(607\) −7.68729 13.3148i −0.312018 0.540430i 0.666781 0.745253i \(-0.267671\pi\)
−0.978799 + 0.204823i \(0.934338\pi\)
\(608\) 0 0
\(609\) −5.27492 + 9.13642i −0.213750 + 0.370227i
\(610\) 0 0
\(611\) 10.6873 11.1066i 0.432362 0.449323i
\(612\) 0 0
\(613\) 9.59967 16.6271i 0.387727 0.671563i −0.604416 0.796669i \(-0.706594\pi\)
0.992143 + 0.125106i \(0.0399270\pi\)
\(614\) 0 0
\(615\) 4.27492 + 7.40437i 0.172381 + 0.298573i
\(616\) 0 0
\(617\) −2.45017 4.24381i −0.0986400 0.170849i 0.812482 0.582986i \(-0.198116\pi\)
−0.911122 + 0.412137i \(0.864783\pi\)
\(618\) 0 0
\(619\) 7.54983 0.303453 0.151727 0.988422i \(-0.451517\pi\)
0.151727 + 0.988422i \(0.451517\pi\)
\(620\) 0 0
\(621\) 3.27492 5.67232i 0.131418 0.227622i
\(622\) 0 0
\(623\) −5.72508 −0.229371
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 4.86254 8.42217i 0.194191 0.336349i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1.36254 2.35999i −0.0542419 0.0939498i 0.837630 0.546239i \(-0.183941\pi\)
−0.891871 + 0.452289i \(0.850608\pi\)
\(632\) 0 0
\(633\) 2.50000 + 4.33013i 0.0993661 + 0.172107i
\(634\) 0 0
\(635\) −0.500000 + 0.866025i −0.0198419 + 0.0343672i
\(636\) 0 0
\(637\) −6.00000 20.7846i −0.237729 0.823516i
\(638\) 0 0
\(639\) 1.00000 1.73205i 0.0395594 0.0685189i
\(640\) 0 0
\(641\) 16.9622 + 29.3794i 0.669967 + 1.16042i 0.977913 + 0.209013i \(0.0670251\pi\)
−0.307946 + 0.951404i \(0.599642\pi\)
\(642\) 0 0
\(643\) −1.36254 2.35999i −0.0537334 0.0930690i 0.837908 0.545812i \(-0.183779\pi\)
−0.891641 + 0.452743i \(0.850445\pi\)
\(644\) 0 0
\(645\) −7.27492 −0.286450
\(646\) 0 0
\(647\) 7.58762 13.1422i 0.298300 0.516671i −0.677447 0.735572i \(-0.736914\pi\)
0.975747 + 0.218900i \(0.0702470\pi\)
\(648\) 0 0
\(649\) −62.1993 −2.44154
\(650\) 0 0
\(651\) 7.27492 0.285126
\(652\) 0 0
\(653\) −2.58762 + 4.48190i −0.101262 + 0.175390i −0.912205 0.409735i \(-0.865621\pi\)
0.810943 + 0.585125i \(0.198955\pi\)
\(654\) 0 0
\(655\) 3.72508 0.145551
\(656\) 0 0
\(657\) −5.91238 10.2405i −0.230664 0.399521i
\(658\) 0 0
\(659\) −6.00000 10.3923i −0.233727 0.404827i 0.725175 0.688565i \(-0.241759\pi\)
−0.958902 + 0.283738i \(0.908425\pi\)
\(660\) 0 0
\(661\) −11.4622 + 19.8531i −0.445828 + 0.772197i −0.998110 0.0614606i \(-0.980424\pi\)
0.552281 + 0.833658i \(0.313757\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.13746 1.97014i 0.0441088 0.0763986i
\(666\) 0 0
\(667\) 34.5498 + 59.8421i 1.33777 + 2.31709i
\(668\) 0 0
\(669\) 13.4124 + 23.2309i 0.518552 + 0.898159i
\(670\) 0 0
\(671\) 14.0000 0.540464
\(672\) 0 0
\(673\) 4.81271 8.33585i 0.185516 0.321324i −0.758234 0.651982i \(-0.773938\pi\)
0.943750 + 0.330659i \(0.107271\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −7.45017 −0.286333 −0.143167 0.989699i \(-0.545728\pi\)
−0.143167 + 0.989699i \(0.545728\pi\)
\(678\) 0 0
\(679\) −3.91238 + 6.77643i −0.150143 + 0.260056i
\(680\) 0 0
\(681\) −9.09967 −0.348700
\(682\) 0 0
\(683\) −15.0000 25.9808i −0.573959 0.994126i −0.996154 0.0876211i \(-0.972074\pi\)
0.422195 0.906505i \(-0.361260\pi\)
\(684\) 0 0
\(685\) 10.5498 + 18.2728i 0.403088 + 0.698170i
\(686\) 0 0
\(687\) −5.00000 + 8.66025i −0.190762 + 0.330409i
\(688\) 0 0
\(689\) 4.31271 4.48190i 0.164301 0.170747i
\(690\) 0 0
\(691\) 17.3248 30.0074i 0.659065 1.14153i −0.321794 0.946810i \(-0.604286\pi\)
0.980858 0.194724i \(-0.0623810\pi\)
\(692\) 0 0
\(693\) 2.13746 + 3.70219i 0.0811953 + 0.140634i
\(694\) 0 0
\(695\) −5.50000 9.52628i −0.208627 0.361352i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 11.5498 20.0049i 0.436855 0.756655i
\(700\) 0 0
\(701\) −28.0000 −1.05755 −0.528773 0.848763i \(-0.677348\pi\)
−0.528773 + 0.848763i \(0.677348\pi\)
\(702\) 0 0
\(703\) −0.625414 −0.0235879
\(704\) 0 0
\(705\) 2.13746 3.70219i 0.0805013 0.139432i
\(706\) 0 0
\(707\) −6.54983 −0.246332
\(708\) 0 0
\(709\) −5.36254 9.28819i −0.201394 0.348825i 0.747583 0.664168i \(-0.231214\pi\)
−0.948978 + 0.315342i \(0.897881\pi\)
\(710\) 0 0
\(711\) 0.362541 + 0.627940i 0.0135964 + 0.0235496i
\(712\) 0 0
\(713\) 23.8248 41.2657i 0.892244 1.54541i
\(714\) 0 0
\(715\) −4.27492 14.8087i −0.159873 0.553816i
\(716\) 0 0
\(717\) −2.27492 + 3.94027i −0.0849583 + 0.147152i
\(718\) 0 0
\(719\) 11.2749 + 19.5287i 0.420483 + 0.728299i 0.995987 0.0895007i \(-0.0285271\pi\)
−0.575503 + 0.817799i \(0.695194\pi\)
\(720\) 0 0
\(721\) 7.77492 + 13.4666i 0.289553 + 0.501521i
\(722\) 0 0
\(723\) −8.27492 −0.307747
\(724\) 0 0
\(725\) 5.27492 9.13642i 0.195906 0.339318i
\(726\) 0 0
\(727\) 28.4502 1.05516 0.527579 0.849506i \(-0.323100\pi\)
0.527579 + 0.849506i \(0.323100\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 26.6495 0.984322 0.492161 0.870504i \(-0.336207\pi\)
0.492161 + 0.870504i \(0.336207\pi\)
\(734\) 0 0
\(735\) −3.00000 5.19615i −0.110657 0.191663i
\(736\) 0 0
\(737\) −11.2749 19.5287i −0.415317 0.719350i
\(738\) 0 0
\(739\) −2.86254 + 4.95807i −0.105300 + 0.182385i −0.913861 0.406028i \(-0.866914\pi\)
0.808561 + 0.588413i \(0.200247\pi\)
\(740\) 0 0
\(741\) 5.68729 5.91041i 0.208928 0.217124i
\(742\) 0 0
\(743\) 11.8248 20.4811i 0.433808 0.751378i −0.563389 0.826191i \(-0.690503\pi\)
0.997198 + 0.0748138i \(0.0238363\pi\)
\(744\) 0 0
\(745\) 8.27492 + 14.3326i 0.303170 + 0.525105i
\(746\) 0 0
\(747\) −8.54983 14.8087i −0.312822 0.541824i
\(748\) 0 0
\(749\) 15.0997 0.551730
\(750\) 0 0
\(751\) 3.45017 5.97586i 0.125898 0.218062i −0.796185 0.605053i \(-0.793152\pi\)
0.922084 + 0.386990i \(0.126485\pi\)
\(752\) 0 0
\(753\) 26.2749 0.957511
\(754\) 0 0
\(755\) −3.45017 −0.125564
\(756\) 0 0
\(757\) −6.68729 + 11.5827i −0.243054 + 0.420982i −0.961583 0.274516i \(-0.911482\pi\)
0.718529 + 0.695497i \(0.244816\pi\)
\(758\) 0 0
\(759\) 28.0000 1.01634
\(760\) 0 0
\(761\) −25.6873 44.4917i −0.931164 1.61282i −0.781336 0.624111i \(-0.785461\pi\)
−0.149828 0.988712i \(-0.547872\pi\)
\(762\) 0 0
\(763\) 6.91238 + 11.9726i 0.250245 + 0.433437i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −50.9244 12.6005i −1.83877 0.454979i
\(768\) 0 0
\(769\) 5.00000 8.66025i 0.180305 0.312297i −0.761680 0.647954i \(-0.775625\pi\)
0.941984 + 0.335657i \(0.108958\pi\)
\(770\) 0 0
\(771\) 4.00000 + 6.92820i 0.144056 + 0.249513i
\(772\) 0 0
\(773\) −19.2371 33.3197i −0.691911 1.19843i −0.971211 0.238222i \(-0.923436\pi\)
0.279299 0.960204i \(-0.409898\pi\)
\(774\) 0 0
\(775\) −7.27492 −0.261323
\(776\) 0 0
\(777\) 0.137459 0.238085i 0.00493130 0.00854126i
\(778\) 0 0
\(779\) −19.4502 −0.696875
\(780\) 0 0
\(781\) 8.54983 0.305937
\(782\) 0 0
\(783\) −5.27492 + 9.13642i −0.188510 + 0.326509i
\(784\) 0 0
\(785\) 11.5498 0.412231
\(786\) 0 0
\(787\) 11.1873 + 19.3770i 0.398784 + 0.690714i 0.993576 0.113165i \(-0.0360990\pi\)
−0.594792 + 0.803879i \(0.702766\pi\)
\(788\) 0 0
\(789\) −1.13746 1.97014i −0.0404946 0.0701387i
\(790\) 0 0
\(791\) 2.72508 4.71998i 0.0968928 0.167823i
\(792\) 0 0
\(793\) 11.4622 + 2.83616i 0.407035 + 0.100715i
\(794\) 0 0
\(795\) 0.862541 1.49397i 0.0305912 0.0529855i
\(796\) 0 0
\(797\) 0.274917 + 0.476171i 0.00973807 + 0.0168668i 0.870853 0.491543i \(-0.163567\pi\)
−0.861115 + 0.508410i \(0.830234\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −5.72508 −0.202286
\(802\) 0 0
\(803\) 25.2749 43.7774i 0.891932 1.54487i
\(804\) 0 0
\(805\) 6.54983 0.230851
\(806\) 0 0
\(807\) 12.0000 0.422420
\(808\) 0 0
\(809\) 15.7251 27.2366i 0.552864 0.957589i −0.445202 0.895430i \(-0.646868\pi\)
0.998066 0.0621591i \(-0.0197986\pi\)
\(810\) 0 0
\(811\) 15.5498 0.546029 0.273014 0.962010i \(-0.411979\pi\)
0.273014 + 0.962010i \(0.411979\pi\)
\(812\) 0 0
\(813\) 12.6375 + 21.8887i 0.443215 + 0.767671i
\(814\) 0 0
\(815\) −1.36254 2.35999i −0.0477277 0.0826669i
\(816\) 0 0
\(817\) 8.27492 14.3326i 0.289503 0.501433i
\(818\) 0 0
\(819\) 1.00000 + 3.46410i 0.0349428 + 0.121046i
\(820\) 0 0
\(821\) −9.00000 + 15.5885i −0.314102 + 0.544041i −0.979246 0.202674i \(-0.935037\pi\)
0.665144 + 0.746715i \(0.268370\pi\)
\(822\) 0 0
\(823\) −4.31271 7.46983i −0.150332 0.260382i 0.781018 0.624509i \(-0.214701\pi\)
−0.931349 + 0.364127i \(0.881367\pi\)
\(824\) 0 0
\(825\) −2.13746 3.70219i −0.0744168 0.128894i
\(826\) 0 0
\(827\) −16.5498 −0.575494 −0.287747 0.957706i \(-0.592906\pi\)
−0.287747 + 0.957706i \(0.592906\pi\)
\(828\) 0 0
\(829\) −11.9124 + 20.6328i −0.413734 + 0.716608i −0.995295 0.0968948i \(-0.969109\pi\)
0.581561 + 0.813503i \(0.302442\pi\)
\(830\) 0 0
\(831\) 28.8248 0.999920
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 5.13746 8.89834i 0.177789 0.307940i
\(836\) 0 0
\(837\) 7.27492 0.251458
\(838\) 0 0
\(839\) 1.54983 + 2.68439i 0.0535062 + 0.0926755i 0.891538 0.452946i \(-0.149627\pi\)
−0.838032 + 0.545622i \(0.816294\pi\)
\(840\) 0 0
\(841\) −41.1495 71.2730i −1.41895 2.45769i
\(842\) 0 0
\(843\) −13.5498 + 23.4690i −0.466681 + 0.808315i
\(844\) 0 0
\(845\) −0.500000 12.9904i −0.0172005 0.446883i
\(846\) 0 0
\(847\) −3.63746 + 6.30026i −0.124985 + 0.216480i
\(848\) 0 0
\(849\) −11.4622 19.8531i −0.393382 0.681358i
\(850\) 0 0
\(851\) −0.900331 1.55942i −0.0308630 0.0534562i
\(852\) 0 0
\(853\) −54.5739 −1.86858 −0.934288 0.356520i \(-0.883963\pi\)
−0.934288 + 0.356520i \(0.883963\pi\)
\(854\) 0 0
\(855\) 1.13746 1.97014i 0.0389003 0.0673772i
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) −16.6495 −0.568074 −0.284037 0.958813i \(-0.591674\pi\)
−0.284037 + 0.958813i \(0.591674\pi\)
\(860\) 0 0
\(861\) 4.27492 7.40437i 0.145689 0.252340i
\(862\) 0 0
\(863\) 0.350497 0.0119310 0.00596552 0.999982i \(-0.498101\pi\)
0.00596552 + 0.999982i \(0.498101\pi\)
\(864\) 0 0
\(865\) 7.86254 + 13.6183i 0.267334 + 0.463037i
\(866\) 0 0
\(867\) −8.50000 14.7224i −0.288675 0.500000i
\(868\) 0 0
\(869\) −1.54983 + 2.68439i −0.0525745 + 0.0910618i
\(870\) 0 0
\(871\) −5.27492 18.2728i −0.178734 0.619152i
\(872\) 0 0
\(873\) −3.91238 + 6.77643i −0.132414 + 0.229348i
\(874\) 0 0
\(875\) −0.500000 0.866025i −0.0169031 0.0292770i
\(876\) 0 0
\(877\) 2.45017 + 4.24381i 0.0827362 + 0.143303i 0.904424 0.426634i \(-0.140301\pi\)
−0.821688 + 0.569937i \(0.806967\pi\)
\(878\) 0 0
\(879\) 28.2749 0.953689
\(880\) 0 0
\(881\) −12.2371 + 21.1953i −0.412279 + 0.714089i −0.995139 0.0984844i \(-0.968601\pi\)
0.582859 + 0.812573i \(0.301934\pi\)
\(882\) 0 0
\(883\) −4.37459 −0.147217 −0.0736083 0.997287i \(-0.523451\pi\)
−0.0736083 + 0.997287i \(0.523451\pi\)
\(884\) 0 0
\(885\) −14.5498 −0.489087
\(886\) 0 0
\(887\) 16.9622 29.3794i 0.569535 0.986464i −0.427077 0.904215i \(-0.640457\pi\)
0.996612 0.0822486i \(-0.0262101\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) 0 0
\(891\) 2.13746 + 3.70219i 0.0716076 + 0.124028i
\(892\) 0 0
\(893\) 4.86254 + 8.42217i 0.162719 + 0.281837i
\(894\) 0 0
\(895\) 6.00000 10.3923i 0.200558 0.347376i
\(896\) 0 0
\(897\) 22.9244 + 5.67232i 0.765424 + 0.189393i
\(898\) 0 0
\(899\) −38.3746 + 66.4667i −1.27986 + 2.21679i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 3.63746 + 6.30026i 0.121047 + 0.209660i
\(904\) 0 0
\(905\) −2.00000 −0.0664822
\(906\) 0 0
\(907\) 12.2749 21.2608i 0.407582 0.705953i −0.587036 0.809561i \(-0.699705\pi\)
0.994618 + 0.103608i \(0.0330387\pi\)
\(908\) 0 0
\(909\) −6.54983 −0.217244
\(910\) 0 0
\(911\) 9.09967 0.301485 0.150743 0.988573i \(-0.451833\pi\)
0.150743 + 0.988573i \(0.451833\pi\)
\(912\) 0 0
\(913\) 36.5498 63.3062i 1.20962 2.09513i
\(914\) 0 0
\(915\) 3.27492 0.108265
\(916\) 0 0
\(917\) −1.86254 3.22602i −0.0615065 0.106532i
\(918\) 0 0
\(919\) −6.27492 10.8685i −0.206990 0.358518i 0.743775 0.668430i \(-0.233034\pi\)
−0.950765 + 0.309913i \(0.899700\pi\)
\(920\) 0 0
\(921\) −11.4622 + 19.8531i −0.377693 + 0.654183i
\(922\) 0 0
\(923\) 7.00000 + 1.73205i 0.230408 + 0.0570111i
\(924\) 0 0
\(925\) −0.137459 + 0.238085i −0.00451961 + 0.00782820i
\(926\) 0 0
\(927\) 7.77492 + 13.4666i 0.255362 + 0.442300i
\(928\) 0 0
\(929\) −20.8248 36.0695i −0.683238 1.18340i −0.973987 0.226604i \(-0.927238\pi\)
0.290749 0.956799i \(-0.406095\pi\)
\(930\) 0 0
\(931\) 13.6495 0.447344
\(932\) 0 0
\(933\) 6.00000 10.3923i 0.196431 0.340229i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) 2.36254 4.09204i 0.0770986 0.133539i
\(940\) 0 0
\(941\) −0.350497 −0.0114259 −0.00571293 0.999984i \(-0.501818\pi\)
−0.00571293 + 0.999984i \(0.501818\pi\)
\(942\) 0 0
\(943\) −28.0000 48.4974i −0.911805 1.57929i
\(944\) 0 0
\(945\) 0.500000 + 0.866025i 0.0162650 + 0.0281718i
\(946\) 0 0
\(947\) 3.17525 5.49969i 0.103182 0.178716i −0.809812 0.586689i \(-0.800431\pi\)
0.912994 + 0.407973i \(0.133764\pi\)
\(948\) 0 0
\(949\) 29.5619 30.7216i 0.959619 0.997266i
\(950\) 0 0
\(951\) −9.68729 + 16.7789i −0.314132 + 0.544093i
\(952\) 0 0
\(953\) 10.0997 + 17.4931i 0.327160 + 0.566658i 0.981947 0.189155i \(-0.0605749\pi\)
−0.654787 + 0.755814i \(0.727242\pi\)
\(954\) 0 0
\(955\) 9.00000 + 15.5885i 0.291233 + 0.504431i
\(956\) 0 0
\(957\) −45.0997 −1.45787
\(958\) 0 0
\(959\) 10.5498 18.2728i 0.340672 0.590061i
\(960\) 0 0
\(961\) 21.9244 0.707239
\(962\) 0 0
\(963\) 15.0997 0.486580
\(964\) 0 0
\(965\) −7.18729 + 12.4488i −0.231367 + 0.400740i
\(966\) 0 0
\(967\) 12.6254 0.406006 0.203003 0.979178i \(-0.434930\pi\)
0.203003 + 0.979178i \(0.434930\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −18.5120 32.0638i −0.594080 1.02898i −0.993676 0.112285i \(-0.964183\pi\)
0.399596 0.916691i \(-0.369150\pi\)
\(972\) 0 0
\(973\) −5.50000 + 9.52628i −0.176322 + 0.305398i
\(974\) 0 0
\(975\) −1.00000 3.46410i −0.0320256 0.110940i
\(976\) 0 0
\(977\) −16.0000 + 27.7128i −0.511885 + 0.886611i 0.488020 + 0.872833i \(0.337719\pi\)
−0.999905 + 0.0137788i \(0.995614\pi\)
\(978\) 0 0
\(979\) −12.2371 21.1953i −0.391100 0.677406i
\(980\) 0 0
\(981\) 6.91238 + 11.9726i 0.220695 + 0.382255i
\(982\) 0 0
\(983\) −37.3746 −1.19206 −0.596032 0.802961i \(-0.703257\pi\)
−0.596032 + 0.802961i \(0.703257\pi\)
\(984\) 0 0
\(985\) 1.41238 2.44631i 0.0450020 0.0779458i
\(986\) 0 0
\(987\) −4.27492 −0.136072
\(988\) 0 0
\(989\) 47.6495 1.51517
\(990\) 0 0
\(991\) −5.72508 + 9.91613i −0.181863 + 0.314996i −0.942515 0.334164i \(-0.891546\pi\)
0.760652 + 0.649160i \(0.224879\pi\)
\(992\) 0 0
\(993\) 18.7251 0.594223
\(994\) 0 0
\(995\) −2.36254 4.09204i −0.0748976 0.129726i
\(996\) 0 0
\(997\) −26.8746 46.5481i −0.851127 1.47419i −0.880192 0.474617i \(-0.842587\pi\)
0.0290657 0.999578i \(-0.490747\pi\)
\(998\) 0 0
\(999\) 0.137459 0.238085i 0.00434900 0.00753269i
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 1560.2.bg.g.841.2 yes 4
13.3 even 3 inner 1560.2.bg.g.601.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.bg.g.601.2 4 13.3 even 3 inner
1560.2.bg.g.841.2 yes 4 1.1 even 1 trivial