Properties

Label 216.3.r.a.43.2
Level $216$
Weight $3$
Character 216.43
Analytic conductor $5.886$
Analytic rank $0$
Dimension $12$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [216,3,Mod(43,216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(216, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 9, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("216.43");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.r (of order \(18\), degree \(6\), 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: \(5.88557371018\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: 12.0.101559956668416.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 8x^{6} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label 43.2
Root \(0.483690 - 1.32893i\) of defining polynomial
Character \(\chi\) \(=\) 216.43
Dual form 216.3.r.a.211.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.53209 - 1.28558i) q^{2} +(1.90707 + 2.31583i) q^{3} +(0.694593 + 3.93923i) q^{4} +(0.0553729 - 5.99974i) q^{6} +(4.00000 - 6.92820i) q^{8} +(-1.72616 + 8.83292i) q^{9} +O(q^{10})\) \(q+(-1.53209 - 1.28558i) q^{2} +(1.90707 + 2.31583i) q^{3} +(0.694593 + 3.93923i) q^{4} +(0.0553729 - 5.99974i) q^{6} +(4.00000 - 6.92820i) q^{8} +(-1.72616 + 8.83292i) q^{9} +(13.4204 - 4.88462i) q^{11} +(-7.79796 + 9.12096i) q^{12} +(-15.0351 + 5.47232i) q^{16} +(10.1424 + 17.5672i) q^{17} +(14.0000 - 11.3137i) q^{18} +(-18.8769 + 32.6958i) q^{19} +(-26.8408 - 9.76924i) q^{22} +(23.6728 - 3.94925i) q^{24} +(19.1511 + 16.0697i) q^{25} +(-23.7474 + 12.8475i) q^{27} +(30.0702 + 10.9446i) q^{32} +(36.9056 + 21.7640i) q^{33} +(7.04485 - 39.9533i) q^{34} +(-35.9939 - 0.664446i) q^{36} +(70.9540 - 25.8251i) q^{38} +(50.8983 - 42.7087i) q^{41} +(41.1752 - 14.9865i) q^{43} +(28.5633 + 49.4731i) q^{44} +(-41.3460 - 24.3826i) q^{48} +(-46.0449 - 16.7590i) q^{49} +(-8.68241 - 49.2404i) q^{50} +(-21.3403 + 56.9900i) q^{51} +(52.8997 + 10.8456i) q^{54} +(-111.718 + 18.6374i) q^{57} +(-45.1367 - 16.4284i) q^{59} +(-32.0000 - 55.4256i) q^{64} +(-28.5633 - 80.7893i) q^{66} +(-81.3715 + 68.2788i) q^{67} +(-62.1564 + 52.1554i) q^{68} +(54.2916 + 47.2908i) q^{72} +(65.2976 - 113.099i) q^{73} +(-0.692161 + 74.9968i) q^{75} +(-141.908 - 51.6503i) q^{76} +(-75.0408 - 30.4940i) q^{81} -132.886 q^{82} +(-26.7420 - 22.4392i) q^{83} +(-82.3504 - 29.9731i) q^{86} +(19.8399 - 112.518i) q^{88} +(80.5908 - 139.587i) q^{89} +(32.0000 + 90.5097i) q^{96} +(172.387 - 62.7437i) q^{97} +(49.0000 + 84.8705i) q^{98} +(19.9798 + 126.973i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 48 q^{8} - 42 q^{11} + 24 q^{12} + 168 q^{18} + 84 q^{22} - 138 q^{27} + 186 q^{33} + 12 q^{34} + 408 q^{38} + 138 q^{41} - 42 q^{43} - 588 q^{51} - 246 q^{57} - 492 q^{59} - 384 q^{64} - 186 q^{67} + 48 q^{68} - 816 q^{76} + 84 q^{86} - 672 q^{88} + 438 q^{89} + 384 q^{96} + 282 q^{97} + 588 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{2}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.53209 1.28558i −0.766044 0.642788i
\(3\) 1.90707 + 2.31583i 0.635691 + 0.771944i
\(4\) 0.694593 + 3.93923i 0.173648 + 0.984808i
\(5\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(6\) 0.0553729 5.99974i 0.00922881 0.999957i
\(7\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(8\) 4.00000 6.92820i 0.500000 0.866025i
\(9\) −1.72616 + 8.83292i −0.191795 + 0.981435i
\(10\) 0 0
\(11\) 13.4204 4.88462i 1.22003 0.444056i 0.349861 0.936802i \(-0.386229\pi\)
0.870173 + 0.492746i \(0.164007\pi\)
\(12\) −7.79796 + 9.12096i −0.649830 + 0.760080i
\(13\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −15.0351 + 5.47232i −0.939693 + 0.342020i
\(17\) 10.1424 + 17.5672i 0.596613 + 1.03336i 0.993317 + 0.115418i \(0.0368206\pi\)
−0.396704 + 0.917947i \(0.629846\pi\)
\(18\) 14.0000 11.3137i 0.777778 0.628539i
\(19\) −18.8769 + 32.6958i −0.993522 + 1.72083i −0.398343 + 0.917236i \(0.630415\pi\)
−0.595178 + 0.803594i \(0.702919\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −26.8408 9.76924i −1.22003 0.444056i
\(23\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(24\) 23.6728 3.94925i 0.986368 0.164552i
\(25\) 19.1511 + 16.0697i 0.766044 + 0.642788i
\(26\) 0 0
\(27\) −23.7474 + 12.8475i −0.879535 + 0.475834i
\(28\) 0 0
\(29\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(30\) 0 0
\(31\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(32\) 30.0702 + 10.9446i 0.939693 + 0.342020i
\(33\) 36.9056 + 21.7640i 1.11835 + 0.659516i
\(34\) 7.04485 39.9533i 0.207202 1.17510i
\(35\) 0 0
\(36\) −35.9939 0.664446i −0.999830 0.0184568i
\(37\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(38\) 70.9540 25.8251i 1.86721 0.679609i
\(39\) 0 0
\(40\) 0 0
\(41\) 50.8983 42.7087i 1.24142 1.04168i 0.244009 0.969773i \(-0.421537\pi\)
0.997412 0.0719030i \(-0.0229072\pi\)
\(42\) 0 0
\(43\) 41.1752 14.9865i 0.957562 0.348524i 0.184485 0.982835i \(-0.440938\pi\)
0.773078 + 0.634311i \(0.218716\pi\)
\(44\) 28.5633 + 49.4731i 0.649167 + 1.12439i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(48\) −41.3460 24.3826i −0.861374 0.507971i
\(49\) −46.0449 16.7590i −0.939693 0.342020i
\(50\) −8.68241 49.2404i −0.173648 0.984808i
\(51\) −21.3403 + 56.9900i −0.418438 + 1.11745i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 52.8997 + 10.8456i 0.979623 + 0.200844i
\(55\) 0 0
\(56\) 0 0
\(57\) −111.718 + 18.6374i −1.95996 + 0.326972i
\(58\) 0 0
\(59\) −45.1367 16.4284i −0.765028 0.278447i −0.0701128 0.997539i \(-0.522336\pi\)
−0.694915 + 0.719092i \(0.744558\pi\)
\(60\) 0 0
\(61\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −32.0000 55.4256i −0.500000 0.866025i
\(65\) 0 0
\(66\) −28.5633 80.7893i −0.432778 1.22408i
\(67\) −81.3715 + 68.2788i −1.21450 + 1.01909i −0.215407 + 0.976524i \(0.569108\pi\)
−0.999094 + 0.0425626i \(0.986448\pi\)
\(68\) −62.1564 + 52.1554i −0.914064 + 0.766991i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(72\) 54.2916 + 47.2908i 0.754050 + 0.656817i
\(73\) 65.2976 113.099i 0.894488 1.54930i 0.0600510 0.998195i \(-0.480874\pi\)
0.834437 0.551103i \(-0.185793\pi\)
\(74\) 0 0
\(75\) −0.692161 + 74.9968i −0.00922881 + 0.999957i
\(76\) −141.908 51.6503i −1.86721 0.679609i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(80\) 0 0
\(81\) −75.0408 30.4940i −0.926429 0.376469i
\(82\) −132.886 −1.62056
\(83\) −26.7420 22.4392i −0.322193 0.270352i 0.467317 0.884090i \(-0.345221\pi\)
−0.789510 + 0.613738i \(0.789665\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −82.3504 29.9731i −0.957562 0.348524i
\(87\) 0 0
\(88\) 19.8399 112.518i 0.225453 1.27861i
\(89\) 80.5908 139.587i 0.905515 1.56840i 0.0852901 0.996356i \(-0.472818\pi\)
0.820225 0.572041i \(-0.193848\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 32.0000 + 90.5097i 0.333333 + 0.942809i
\(97\) 172.387 62.7437i 1.77718 0.646842i 0.777343 0.629077i \(-0.216567\pi\)
0.999842 0.0177651i \(-0.00565510\pi\)
\(98\) 49.0000 + 84.8705i 0.500000 + 0.866025i
\(99\) 19.9798 + 126.973i 0.201816 + 1.28255i
\(100\) −50.0000 + 86.6025i −0.500000 + 0.866025i
\(101\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(102\) 105.960 59.8792i 1.03883 0.587051i
\(103\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −59.9966 −0.560716 −0.280358 0.959895i \(-0.590453\pi\)
−0.280358 + 0.959895i \(0.590453\pi\)
\(108\) −67.1041 84.6229i −0.621335 0.783545i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −119.682 43.5608i −1.05914 0.385494i −0.247032 0.969007i \(-0.579455\pi\)
−0.812103 + 0.583513i \(0.801678\pi\)
\(114\) 195.121 + 115.067i 1.71159 + 1.00936i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 48.0334 + 83.1963i 0.407063 + 0.705054i
\(119\) 0 0
\(120\) 0 0
\(121\) 63.5557 53.3295i 0.525253 0.440740i
\(122\) 0 0
\(123\) 195.973 + 36.4232i 1.59327 + 0.296124i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(128\) −22.2270 + 126.055i −0.173648 + 0.984808i
\(129\) 113.230 + 66.7744i 0.877755 + 0.517631i
\(130\) 0 0
\(131\) −43.6645 247.634i −0.333317 1.89034i −0.443253 0.896397i \(-0.646176\pi\)
0.109936 0.993939i \(-0.464935\pi\)
\(132\) −60.0992 + 160.497i −0.455297 + 1.21588i
\(133\) 0 0
\(134\) 212.446 1.58542
\(135\) 0 0
\(136\) 162.279 1.19323
\(137\) 70.7623 + 59.3766i 0.516513 + 0.433406i 0.863414 0.504496i \(-0.168322\pi\)
−0.346901 + 0.937902i \(0.612766\pi\)
\(138\) 0 0
\(139\) 38.1352 + 216.275i 0.274354 + 1.55594i 0.741007 + 0.671497i \(0.234348\pi\)
−0.466653 + 0.884440i \(0.654540\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −22.3837 142.250i −0.155442 0.987845i
\(145\) 0 0
\(146\) −245.439 + 89.3324i −1.68109 + 0.611866i
\(147\) −49.0000 138.593i −0.333333 0.942809i
\(148\) 0 0
\(149\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(150\) 97.4745 114.012i 0.649830 0.760080i
\(151\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(152\) 151.015 + 261.566i 0.993522 + 1.72083i
\(153\) −172.677 + 59.2635i −1.12861 + 0.387343i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 75.7669 + 143.190i 0.467697 + 0.883889i
\(163\) 205.091 1.25823 0.629113 0.777314i \(-0.283418\pi\)
0.629113 + 0.777314i \(0.283418\pi\)
\(164\) 203.593 + 170.835i 1.24142 + 1.04168i
\(165\) 0 0
\(166\) 12.1238 + 68.7577i 0.0730351 + 0.414203i
\(167\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(168\) 0 0
\(169\) 29.3465 166.433i 0.173648 0.984808i
\(170\) 0 0
\(171\) −256.214 223.176i −1.49833 1.30512i
\(172\) 87.6354 + 151.789i 0.509508 + 0.882494i
\(173\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −175.046 + 146.881i −0.994581 + 0.834553i
\(177\) −48.0334 135.859i −0.271375 0.767565i
\(178\) −302.922 + 110.255i −1.70181 + 0.619409i
\(179\) −162.818 282.009i −0.909597 1.57547i −0.814625 0.579988i \(-0.803057\pi\)
−0.0949721 0.995480i \(-0.530276\pi\)
\(180\) 0 0
\(181\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 221.924 + 186.216i 1.18676 + 0.995810i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(192\) 67.3301 179.807i 0.350678 0.936496i
\(193\) 6.18260 + 35.0633i 0.0320342 + 0.181675i 0.996627 0.0820705i \(-0.0261533\pi\)
−0.964592 + 0.263745i \(0.915042\pi\)
\(194\) −344.774 125.487i −1.77718 0.646842i
\(195\) 0 0
\(196\) 34.0350 193.022i 0.173648 0.984808i
\(197\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(198\) 132.622 220.219i 0.669809 1.11222i
\(199\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(200\) 187.939 68.4040i 0.939693 0.342020i
\(201\) −313.304 58.2302i −1.55872 0.289702i
\(202\) 0 0
\(203\) 0 0
\(204\) −239.320 44.4796i −1.17314 0.218037i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −93.6289 + 530.996i −0.447985 + 2.54065i
\(210\) 0 0
\(211\) −396.208 144.208i −1.87776 0.683450i −0.953090 0.302688i \(-0.902116\pi\)
−0.924673 0.380762i \(-0.875662\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 91.9202 + 77.1302i 0.429534 + 0.360421i
\(215\) 0 0
\(216\) −5.97959 + 215.917i −0.0276833 + 0.999617i
\(217\) 0 0
\(218\) 0 0
\(219\) 386.445 64.4692i 1.76459 0.294380i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(224\) 0 0
\(225\) −175.000 + 141.421i −0.777778 + 0.628539i
\(226\) 127.363 + 220.600i 0.563554 + 0.976105i
\(227\) −269.798 + 98.1986i −1.18854 + 0.432593i −0.859210 0.511622i \(-0.829045\pi\)
−0.329329 + 0.944215i \(0.606822\pi\)
\(228\) −151.015 427.136i −0.662348 1.87340i
\(229\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 174.892 + 302.922i 0.750609 + 1.30009i 0.947528 + 0.319673i \(0.103573\pi\)
−0.196919 + 0.980420i \(0.563093\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 33.3637 189.215i 0.141371 0.801757i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(240\) 0 0
\(241\) 358.677 + 300.965i 1.48828 + 1.24882i 0.896748 + 0.442541i \(0.145923\pi\)
0.591536 + 0.806279i \(0.298522\pi\)
\(242\) −165.932 −0.685670
\(243\) −72.4892 231.936i −0.298310 0.954469i
\(244\) 0 0
\(245\) 0 0
\(246\) −253.423 307.741i −1.03017 1.25098i
\(247\) 0 0
\(248\) 0 0
\(249\) 0.966511 104.723i 0.00388157 0.420575i
\(250\) 0 0
\(251\) 238.485 413.068i 0.950139 1.64569i 0.205020 0.978758i \(-0.434274\pi\)
0.745119 0.666932i \(-0.232393\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 196.107 164.554i 0.766044 0.642788i
\(257\) 307.401 257.940i 1.19611 1.00366i 0.196379 0.980528i \(-0.437082\pi\)
0.999732 0.0231286i \(-0.00736271\pi\)
\(258\) −87.6354 247.870i −0.339672 0.960738i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −251.454 + 435.531i −0.959748 + 1.66233i
\(263\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(264\) 298.408 168.633i 1.13033 0.638762i
\(265\) 0 0
\(266\) 0 0
\(267\) 476.953 79.5684i 1.78634 0.298009i
\(268\) −325.486 273.115i −1.21450 1.01909i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −248.625 208.622i −0.914064 0.766991i
\(273\) 0 0
\(274\) −32.0810 181.941i −0.117084 0.664017i
\(275\) 335.509 + 122.115i 1.22003 + 0.444056i
\(276\) 0 0
\(277\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(278\) 219.612 380.379i 0.789970 1.36827i
\(279\) 0 0
\(280\) 0 0
\(281\) −526.142 + 191.500i −1.87239 + 0.681494i −0.906713 + 0.421749i \(0.861416\pi\)
−0.965677 + 0.259746i \(0.916361\pi\)
\(282\) 0 0
\(283\) −340.123 + 285.397i −1.20185 + 1.00847i −0.202273 + 0.979329i \(0.564833\pi\)
−0.999575 + 0.0291408i \(0.990723\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −148.579 + 246.715i −0.515899 + 0.856649i
\(289\) −61.2375 + 106.066i −0.211894 + 0.367012i
\(290\) 0 0
\(291\) 474.058 + 279.562i 1.62907 + 0.960696i
\(292\) 490.878 + 178.665i 1.68109 + 0.611866i
\(293\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(294\) −103.099 + 275.330i −0.350678 + 0.936496i
\(295\) 0 0
\(296\) 0 0
\(297\) −255.945 + 288.416i −0.861766 + 0.971097i
\(298\) 0 0
\(299\) 0 0
\(300\) −295.911 + 49.3657i −0.986368 + 0.164552i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 104.894 594.884i 0.345046 1.95686i
\(305\) 0 0
\(306\) 340.744 + 131.192i 1.11354 + 0.428733i
\(307\) 303.993 + 526.531i 0.990205 + 1.71509i 0.616016 + 0.787734i \(0.288746\pi\)
0.374189 + 0.927352i \(0.377921\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(312\) 0 0
\(313\) −355.385 + 129.350i −1.13542 + 0.413257i −0.840256 0.542191i \(-0.817595\pi\)
−0.295160 + 0.955448i \(0.595373\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −114.418 138.942i −0.356442 0.432842i
\(322\) 0 0
\(323\) −765.830 −2.37099
\(324\) 68.0000 316.784i 0.209877 0.977728i
\(325\) 0 0
\(326\) −314.217 263.660i −0.963857 0.808772i
\(327\) 0 0
\(328\) −92.3016 523.468i −0.281407 1.59594i
\(329\) 0 0
\(330\) 0 0
\(331\) 98.3162 557.579i 0.297028 1.68453i −0.361816 0.932249i \(-0.617843\pi\)
0.658844 0.752279i \(-0.271046\pi\)
\(332\) 69.8184 120.929i 0.210296 0.364244i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −46.5235 + 39.0379i −0.138052 + 0.115839i −0.709199 0.705009i \(-0.750943\pi\)
0.571147 + 0.820848i \(0.306499\pi\)
\(338\) −258.923 + 217.262i −0.766044 + 0.642788i
\(339\) −127.363 360.238i −0.375703 1.06265i
\(340\) 0 0
\(341\) 0 0
\(342\) 105.634 + 671.309i 0.308870 + 1.96289i
\(343\) 0 0
\(344\) 60.8709 345.216i 0.176950 1.00354i
\(345\) 0 0
\(346\) 0 0
\(347\) 120.472 + 683.233i 0.347183 + 1.96897i 0.198761 + 0.980048i \(0.436308\pi\)
0.148422 + 0.988924i \(0.452580\pi\)
\(348\) 0 0
\(349\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 457.013 1.29833
\(353\) 448.099 + 375.999i 1.26940 + 1.06515i 0.994614 + 0.103649i \(0.0330520\pi\)
0.274788 + 0.961505i \(0.411392\pi\)
\(354\) −101.066 + 269.899i −0.285496 + 0.762426i
\(355\) 0 0
\(356\) 605.845 + 220.509i 1.70181 + 0.619409i
\(357\) 0 0
\(358\) −113.092 + 641.377i −0.315900 + 1.79156i
\(359\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(360\) 0 0
\(361\) −532.176 921.755i −1.47417 2.55334i
\(362\) 0 0
\(363\) 244.707 + 45.4810i 0.674125 + 0.125292i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(368\) 0 0
\(369\) 289.384 + 523.302i 0.784239 + 1.41816i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(374\) −100.612 570.600i −0.269017 1.52567i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −534.348 −1.40989 −0.704944 0.709262i \(-0.749028\pi\)
−0.704944 + 0.709262i \(0.749028\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(384\) −334.312 + 188.923i −0.870603 + 0.491986i
\(385\) 0 0
\(386\) 35.6042 61.6682i 0.0922388 0.159762i
\(387\) 61.3001 + 389.566i 0.158398 + 1.00663i
\(388\) 366.901 + 635.491i 0.945620 + 1.63786i
\(389\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −300.289 + 251.973i −0.766044 + 0.642788i
\(393\) 490.207 573.375i 1.24735 1.45897i
\(394\) 0 0
\(395\) 0 0
\(396\) −486.297 + 166.899i −1.22802 + 0.421463i
\(397\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −375.877 136.808i −0.939693 0.342020i
\(401\) 110.882 + 628.844i 0.276514 + 1.56819i 0.734111 + 0.679029i \(0.237599\pi\)
−0.457597 + 0.889160i \(0.651290\pi\)
\(402\) 405.150 + 491.989i 1.00784 + 1.22385i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 309.477 + 375.810i 0.758523 + 0.921104i
\(409\) −127.776 724.652i −0.312410 1.77177i −0.586388 0.810030i \(-0.699451\pi\)
0.273978 0.961736i \(-0.411661\pi\)
\(410\) 0 0
\(411\) −2.55750 + 277.109i −0.00622262 + 0.674232i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −428.131 + 500.767i −1.02669 + 1.20088i
\(418\) 826.083 693.166i 1.97628 1.65829i
\(419\) 635.955 533.630i 1.51779 1.27358i 0.671291 0.741194i \(-0.265740\pi\)
0.846502 0.532385i \(-0.178704\pi\)
\(420\) 0 0
\(421\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(422\) 421.636 + 730.294i 0.999137 + 1.73056i
\(423\) 0 0
\(424\) 0 0
\(425\) −88.0607 + 499.417i −0.207202 + 1.17510i
\(426\) 0 0
\(427\) 0 0
\(428\) −41.6732 236.341i −0.0973674 0.552198i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 286.739 323.117i 0.663748 0.747956i
\(433\) 28.2519 0.0652469 0.0326235 0.999468i \(-0.489614\pi\)
0.0326235 + 0.999468i \(0.489614\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −674.948 398.032i −1.54098 0.908748i
\(439\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(440\) 0 0
\(441\) 227.511 377.782i 0.515899 0.856649i
\(442\) 0 0
\(443\) 737.114 268.288i 1.66391 0.605615i 0.672943 0.739694i \(-0.265030\pi\)
0.990971 + 0.134079i \(0.0428076\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −192.179 332.864i −0.428015 0.741344i 0.568681 0.822558i \(-0.307454\pi\)
−0.996697 + 0.0812137i \(0.974120\pi\)
\(450\) 449.923 + 8.30558i 0.999830 + 0.0184568i
\(451\) 474.458 821.786i 1.05201 1.82214i
\(452\) 88.4656 501.713i 0.195720 1.10999i
\(453\) 0 0
\(454\) 539.597 + 196.397i 1.18854 + 0.432593i
\(455\) 0 0
\(456\) −317.746 + 848.551i −0.696812 + 1.86086i
\(457\) 402.533 + 337.765i 0.880816 + 0.739092i 0.966347 0.257244i \(-0.0828143\pi\)
−0.0855311 + 0.996336i \(0.527259\pi\)
\(458\) 0 0
\(459\) −466.551 286.871i −1.01645 0.624991i
\(460\) 0 0
\(461\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(462\) 0 0
\(463\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 121.479 688.940i 0.260684 1.47841i
\(467\) −175.592 + 304.135i −0.376001 + 0.651252i −0.990476 0.137684i \(-0.956034\pi\)
0.614476 + 0.788936i \(0.289368\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −294.366 + 247.002i −0.623657 + 0.523310i
\(473\) 479.383 402.250i 1.01349 0.850423i
\(474\) 0 0
\(475\) −886.925 + 322.814i −1.86721 + 0.679609i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −162.611 922.211i −0.337367 1.91330i
\(483\) 0 0
\(484\) 254.223 + 213.318i 0.525253 + 0.440740i
\(485\) 0 0
\(486\) −187.111 + 448.537i −0.385003 + 0.922916i
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 391.123 + 474.956i 0.799842 + 0.971280i
\(490\) 0 0
\(491\) −204.149 74.3040i −0.415781 0.151332i 0.125655 0.992074i \(-0.459897\pi\)
−0.541436 + 0.840742i \(0.682119\pi\)
\(492\) −7.35827 + 797.282i −0.0149558 + 1.62049i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −136.110 + 159.203i −0.273314 + 0.319684i
\(499\) −763.106 + 640.322i −1.52927 + 1.28321i −0.725663 + 0.688050i \(0.758467\pi\)
−0.803607 + 0.595160i \(0.797089\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −896.410 + 326.267i −1.78568 + 0.649933i
\(503\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 441.396 249.437i 0.870603 0.491986i
\(508\) 0 0
\(509\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −512.000 −1.00000
\(513\) 28.2190 1018.96i 0.0550079 1.98628i
\(514\) −802.566 −1.56141
\(515\) 0 0
\(516\) −184.391 + 492.421i −0.357347 + 0.954305i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 298.053 516.242i 0.572078 0.990868i −0.424274 0.905534i \(-0.639471\pi\)
0.996352 0.0853344i \(-0.0271959\pi\)
\(522\) 0 0
\(523\) −319.363 553.153i −0.610636 1.05765i −0.991133 0.132871i \(-0.957580\pi\)
0.380497 0.924782i \(-0.375753\pi\)
\(524\) 945.158 344.009i 1.80374 0.656507i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −673.978 125.265i −1.27647 0.237244i
\(529\) −497.097 + 180.929i −0.939693 + 0.342020i
\(530\) 0 0
\(531\) 223.024 370.330i 0.420007 0.697420i
\(532\) 0 0
\(533\) 0 0
\(534\) −833.026 491.254i −1.55997 0.919951i
\(535\) 0 0
\(536\) 147.563 + 836.874i 0.275305 + 1.56133i
\(537\) 342.580 914.870i 0.637951 1.70367i
\(538\) 0 0
\(539\) −699.802 −1.29833
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 112.718 + 639.254i 0.207202 + 1.17510i
\(545\) 0 0
\(546\) 0 0
\(547\) 92.3813 523.920i 0.168887 0.957807i −0.776078 0.630637i \(-0.782794\pi\)
0.944965 0.327170i \(-0.106095\pi\)
\(548\) −184.747 + 319.992i −0.337130 + 0.583927i
\(549\) 0 0
\(550\) −357.042 618.414i −0.649167 1.12439i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −825.470 + 300.446i −1.48466 + 0.540371i
\(557\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −8.02080 + 869.067i −0.0142973 + 1.54914i
\(562\) 1052.28 + 383.000i 1.87239 + 0.681494i
\(563\) 181.824 + 1031.18i 0.322956 + 1.83157i 0.523666 + 0.851923i \(0.324564\pi\)
−0.200710 + 0.979651i \(0.564325\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 887.998 1.56890
\(567\) 0 0
\(568\) 0 0
\(569\) −699.618 587.050i −1.22956 1.03172i −0.998268 0.0588367i \(-0.981261\pi\)
−0.231290 0.972885i \(-0.574295\pi\)
\(570\) 0 0
\(571\) 16.6543 + 94.4514i 0.0291670 + 0.165414i 0.995912 0.0903277i \(-0.0287914\pi\)
−0.966745 + 0.255742i \(0.917680\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 544.807 186.980i 0.945845 0.324618i
\(577\) −568.407 984.510i −0.985107 1.70626i −0.641459 0.767157i \(-0.721671\pi\)
−0.343648 0.939098i \(-0.611663\pi\)
\(578\) 230.178 83.7778i 0.398231 0.144944i
\(579\) −69.4099 + 81.1860i −0.119879 + 0.140218i
\(580\) 0 0
\(581\) 0 0
\(582\) −366.901 1037.75i −0.630414 1.78308i
\(583\) 0 0
\(584\) −522.381 904.790i −0.894488 1.54930i
\(585\) 0 0
\(586\) 0 0
\(587\) 15.0214 85.1908i 0.0255902 0.145129i −0.969336 0.245741i \(-0.920969\pi\)
0.994926 + 0.100612i \(0.0320799\pi\)
\(588\) 511.915 289.288i 0.870603 0.491986i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1136.45 1.91645 0.958223 0.286021i \(-0.0923326\pi\)
0.958223 + 0.286021i \(0.0923326\pi\)
\(594\) 762.910 112.843i 1.28436 0.189971i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(600\) 516.824 + 304.783i 0.861374 + 0.507971i
\(601\) −105.938 + 600.803i −0.176269 + 0.999673i 0.760399 + 0.649456i \(0.225003\pi\)
−0.936669 + 0.350217i \(0.886108\pi\)
\(602\) 0 0
\(603\) −462.641 836.608i −0.767233 1.38741i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(608\) −925.475 + 776.566i −1.52216 + 1.27725i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −353.393 639.050i −0.577439 1.04420i
\(613\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(614\) 211.151 1197.50i 0.343895 1.95032i
\(615\) 0 0
\(616\) 0 0
\(617\) −213.221 1209.24i −0.345577 1.95986i −0.270665 0.962674i \(-0.587243\pi\)
−0.0749124 0.997190i \(-0.523868\pi\)
\(618\) 0 0
\(619\) 115.542 + 96.9513i 0.186659 + 0.156626i 0.731329 0.682025i \(-0.238901\pi\)
−0.544670 + 0.838651i \(0.683345\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 108.530 + 615.505i 0.173648 + 0.984808i
\(626\) 710.770 + 258.699i 1.13542 + 0.413257i
\(627\) −1408.25 + 795.819i −2.24602 + 1.26925i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(632\) 0 0
\(633\) −421.636 1192.57i −0.666091 1.88399i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −8.09767 + 45.9242i −0.0126329 + 0.0716446i −0.990472 0.137711i \(-0.956025\pi\)
0.977840 + 0.209356i \(0.0671366\pi\)
\(642\) −3.32219 + 359.965i −0.00517474 + 0.560692i
\(643\) 194.525 + 70.8015i 0.302528 + 0.110111i 0.488824 0.872383i \(-0.337426\pi\)
−0.186296 + 0.982494i \(0.559648\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 1173.32 + 984.533i 1.81629 + 1.52404i
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) −511.431 + 397.922i −0.789246 + 0.614077i
\(649\) −685.997 −1.05701
\(650\) 0 0
\(651\) 0 0
\(652\) 142.455 + 807.900i 0.218489 + 1.23911i
\(653\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −531.544 + 920.661i −0.810280 + 1.40345i
\(657\) 886.278 + 771.994i 1.34898 + 1.17503i
\(658\) 0 0
\(659\) 237.314 86.3751i 0.360112 0.131070i −0.155627 0.987816i \(-0.549740\pi\)
0.515739 + 0.856746i \(0.327518\pi\)
\(660\) 0 0
\(661\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(662\) −867.439 + 727.868i −1.31033 + 1.09950i
\(663\) 0 0
\(664\) −262.431 + 95.5172i −0.395228 + 0.143851i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 815.001 + 683.867i 1.21100 + 1.01615i 0.999246 + 0.0388234i \(0.0123610\pi\)
0.211751 + 0.977324i \(0.432083\pi\)
\(674\) 121.464 0.180214
\(675\) −661.246 135.570i −0.979623 0.200844i
\(676\) 676.000 1.00000
\(677\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(678\) −267.981 + 715.651i −0.395252 + 1.05553i
\(679\) 0 0
\(680\) 0 0
\(681\) −741.936 437.536i −1.08948 0.642491i
\(682\) 0 0
\(683\) −677.997 + 1174.32i −0.992674 + 1.71936i −0.391703 + 0.920092i \(0.628114\pi\)
−0.600971 + 0.799271i \(0.705219\pi\)
\(684\) 701.178 1164.30i 1.02511 1.70220i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −537.061 + 450.648i −0.780612 + 0.655011i
\(689\) 0 0
\(690\) 0 0
\(691\) 1297.80 472.360i 1.87815 0.683589i 0.926655 0.375913i \(-0.122671\pi\)
0.951491 0.307676i \(-0.0995512\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 693.773 1201.65i 0.999673 1.73149i
\(695\) 0 0
\(696\) 0 0
\(697\) 1266.50 + 460.970i 1.81708 + 0.661363i
\(698\) 0 0
\(699\) −367.984 + 982.714i −0.526444 + 1.40589i
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −700.185 587.525i −0.994581 0.834553i
\(705\) 0 0
\(706\) −203.151 1152.13i −0.287750 1.63191i
\(707\) 0 0
\(708\) 501.816 283.581i 0.708780 0.400539i
\(709\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −644.727 1116.70i −0.905515 1.56840i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 997.806 837.258i 1.39358 1.16936i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −369.645 + 2096.36i −0.511974 + 2.90355i
\(723\) −12.9633 + 1404.60i −0.0179299 + 1.94273i
\(724\) 0 0
\(725\) 0 0
\(726\) −316.444 384.271i −0.435874 0.529299i
\(727\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(728\) 0 0
\(729\) 398.883 610.191i 0.547164 0.837025i
\(730\) 0 0
\(731\) 680.888 + 571.333i 0.931447 + 0.781577i
\(732\) 0 0
\(733\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −758.521 + 1313.80i −1.02920 + 1.78263i
\(738\) 229.382 1173.77i 0.310815 1.59047i
\(739\) 738.249 + 1278.68i 0.998984 + 1.73029i 0.538528 + 0.842608i \(0.318981\pi\)
0.460456 + 0.887683i \(0.347686\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 244.364 197.476i 0.327128 0.264359i
\(748\) −579.403 + 1003.56i −0.774603 + 1.34165i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(752\) 0 0
\(753\) 1411.40 235.459i 1.87437 0.312695i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 818.669 + 686.944i 1.08004 + 0.906259i
\(759\) 0 0
\(760\) 0 0
\(761\) −1309.93 476.776i −1.72133 0.626513i −0.723376 0.690455i \(-0.757410\pi\)
−0.997954 + 0.0639422i \(0.979633\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 755.069 + 140.336i 0.983163 + 0.182729i
\(769\) 1146.77 962.252i 1.49124 1.25130i 0.598190 0.801355i \(-0.295887\pi\)
0.893055 0.449948i \(-0.148558\pi\)
\(770\) 0 0
\(771\) 1183.58 + 219.979i 1.53512 + 0.285316i
\(772\) −133.828 + 48.7094i −0.173352 + 0.0630950i
\(773\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(774\) 406.899 675.656i 0.525710 0.872940i
\(775\) 0 0
\(776\) 254.847 1445.31i 0.328410 1.86251i
\(777\) 0 0
\(778\) 0 0
\(779\) 435.592 + 2470.37i 0.559168 + 3.17120i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 784.000 1.00000
\(785\) 0 0
\(786\) −1488.16 + 248.264i −1.89333 + 0.315857i
\(787\) 111.008 + 629.558i 0.141052 + 0.799947i 0.970452 + 0.241292i \(0.0775713\pi\)
−0.829400 + 0.558655i \(0.811318\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 959.611 + 369.467i 1.21163 + 0.466499i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 400.000 + 692.820i 0.500000 + 0.866025i
\(801\) 1093.85 + 952.801i 1.36561 + 1.18951i
\(802\) 638.545 1105.99i 0.796190 1.37904i
\(803\) 323.874 1836.78i 0.403330 2.28740i
\(804\) 11.7637 1274.62i 0.0146315 1.58535i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1430.05 −1.76768 −0.883839 0.467791i \(-0.845050\pi\)
−0.883839 + 0.467791i \(0.845050\pi\)
\(810\) 0 0
\(811\) 1556.80 1.91961 0.959803 0.280673i \(-0.0905577\pi\)
0.959803 + 0.280673i \(0.0905577\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 8.98584 973.631i 0.0110121 1.19318i
\(817\) −287.264 + 1629.15i −0.351608 + 1.99407i
\(818\) −735.831 + 1274.50i −0.899549 + 1.55807i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(822\) 360.163 421.268i 0.438155 0.512492i
\(823\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(824\) 0 0
\(825\) 357.042 + 1009.87i 0.432778 + 1.22408i
\(826\) 0 0
\(827\) −631.000 1092.92i −0.762999 1.32155i −0.941298 0.337576i \(-0.890393\pi\)
0.178299 0.983976i \(-0.442940\pi\)
\(828\) 0 0
\(829\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −172.599 978.857i −0.207202 1.17510i
\(834\) 1299.71 216.825i 1.55840 0.259983i
\(835\) 0 0
\(836\) −2156.75 −2.57984
\(837\) 0 0
\(838\) −1660.36 −1.98134
\(839\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(840\) 0 0
\(841\) 146.038 + 828.223i 0.173648 + 0.984808i
\(842\) 0 0
\(843\) −1446.87 853.251i −1.71634 1.01216i
\(844\) 292.865 1660.92i 0.346997 1.96792i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −1309.57 243.395i −1.54249 0.286684i
\(850\) 776.955 651.942i 0.914064 0.766991i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −239.987 + 415.669i −0.280358 + 0.485595i
\(857\) 208.725 1183.74i 0.243553 1.38126i −0.580276 0.814420i \(-0.697055\pi\)
0.823829 0.566839i \(-0.191834\pi\)
\(858\) 0 0
\(859\) −724.028 263.525i −0.842873 0.306781i −0.115742 0.993279i \(-0.536925\pi\)
−0.727131 + 0.686499i \(0.759147\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) −854.701 + 126.420i −0.989237 + 0.146319i
\(865\) 0 0
\(866\) −43.2845 36.3200i −0.0499821 0.0419399i
\(867\) −362.416 + 60.4606i −0.418012 + 0.0697354i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 256.643 + 1630.98i 0.293979 + 1.86825i
\(874\) 0 0
\(875\) 0 0
\(876\) 522.381 + 1477.52i 0.596325 + 1.68666i
\(877\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 81.4082 + 141.003i 0.0924043 + 0.160049i 0.908522 0.417836i \(-0.137211\pi\)
−0.816118 + 0.577885i \(0.803878\pi\)
\(882\) −834.235 + 286.313i −0.945845 + 0.324618i
\(883\) 636.706 1102.81i 0.721071 1.24893i −0.239500 0.970896i \(-0.576984\pi\)
0.960571 0.278035i \(-0.0896831\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −1474.23 536.575i −1.66391 0.605615i
\(887\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1156.03 42.6950i −1.29745 0.0479181i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −133.486 + 757.037i −0.148648 + 0.843025i
\(899\) 0 0
\(900\) −678.645 591.135i −0.754050 0.656817i
\(901\) 0 0
\(902\) −1783.38 + 649.097i −1.97714 + 0.719620i
\(903\) 0 0
\(904\) −780.527 + 654.940i −0.863415 + 0.724491i
\(905\) 0 0
\(906\) 0 0
\(907\) −269.986 + 98.2668i −0.297669 + 0.108343i −0.486537 0.873660i \(-0.661740\pi\)
0.188868 + 0.982002i \(0.439518\pi\)
\(908\) −574.227 994.590i −0.632409 1.09536i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(912\) 1577.69 891.570i 1.72993 0.977598i
\(913\) −468.494 170.518i −0.513137 0.186767i
\(914\) −182.494 1034.97i −0.199665 1.13235i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 346.004 + 1039.30i 0.376911 + 1.13213i
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) −639.622 + 1708.13i −0.694486 + 1.85465i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −994.195 + 361.857i −1.07018 + 0.389513i −0.816243 0.577709i \(-0.803947\pi\)
−0.253934 + 0.967221i \(0.581725\pi\)
\(930\) 0 0
\(931\) 1417.13 1189.12i 1.52216 1.27725i
\(932\) −1071.80 + 899.347i −1.15000 + 0.964964i
\(933\) 0 0
\(934\) 660.011 240.224i 0.706650 0.257200i
\(935\) 0 0
\(936\) 0 0
\(937\) −929.044 + 1609.15i −0.991509 + 1.71734i −0.383138 + 0.923691i \(0.625157\pi\)
−0.608371 + 0.793653i \(0.708177\pi\)
\(938\) 0 0
\(939\) −977.297 576.333i −1.04078 0.613773i
\(940\) 0 0
\(941\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 768.535 0.814126
\(945\) 0 0
\(946\) −1251.58 −1.32302
\(947\) −1377.16 1155.57i −1.45423 1.22025i −0.929418 0.369030i \(-0.879690\pi\)
−0.524815 0.851216i \(-0.675866\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 1773.85 + 645.628i 1.86721 + 0.679609i
\(951\) 0 0
\(952\) 0 0
\(953\) 948.243 1642.40i 0.995008 1.72340i 0.411078 0.911600i \(-0.365152\pi\)
0.583930 0.811804i \(-0.301514\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −903.045 + 328.681i −0.939693 + 0.342020i
\(962\) 0 0
\(963\) 103.564 529.945i 0.107543 0.550307i
\(964\) −936.438 + 1621.96i −0.971409 + 1.68253i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(968\) −115.255 653.645i −0.119065 0.675253i
\(969\) −1460.49 1773.53i −1.50722 1.83027i
\(970\) 0 0
\(971\) 974.000 1.00309 0.501545 0.865132i \(-0.332765\pi\)
0.501545 + 0.865132i \(0.332765\pi\)
\(972\) 863.299 446.653i 0.888168 0.459519i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1813.63 660.107i −1.85633 0.675647i −0.981576 0.191071i \(-0.938804\pi\)
−0.874749 0.484576i \(-0.838974\pi\)
\(978\) 11.3565 1230.49i 0.0116119 1.25817i
\(979\) 399.728 2266.97i 0.408302 2.31560i
\(980\) 0 0
\(981\) 0 0
\(982\) 217.250 + 376.289i 0.221233 + 0.383186i
\(983\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(984\) 1036.24 1212.05i 1.05309 1.23175i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(992\) 0 0
\(993\) 1478.76 835.659i 1.48918 0.841550i
\(994\) 0 0
\(995\) 0 0
\(996\) 413.200 68.9326i 0.414859 0.0692094i
\(997\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(998\) 1992.33 1.99632
\(999\) 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 216.3.r.a.43.2 12
8.3 odd 2 CM 216.3.r.a.43.2 12
27.22 even 9 inner 216.3.r.a.211.2 yes 12
216.211 odd 18 inner 216.3.r.a.211.2 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.3.r.a.43.2 12 1.1 even 1 trivial
216.3.r.a.43.2 12 8.3 odd 2 CM
216.3.r.a.211.2 yes 12 27.22 even 9 inner
216.3.r.a.211.2 yes 12 216.211 odd 18 inner