Properties

Label 117.5.j.b.109.3
Level $117$
Weight $5$
Character 117.109
Analytic conductor $12.094$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,5,Mod(73,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.73");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 117.j (of order \(4\), 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: \(12.0942856808\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 5446 x^{16} - 1452 x^{15} + 106320 x^{13} + 8376897 x^{12} - 1643220 x^{11} + 1054152 x^{10} + \cdots + 2103506496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{10} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 109.3
Root \(-2.89776 + 2.89776i\) of defining polynomial
Character \(\chi\) \(=\) 117.109
Dual form 117.5.j.b.73.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.89776 + 2.89776i) q^{2} -0.794047i q^{4} +(-18.6373 + 18.6373i) q^{5} +(-15.2549 - 15.2549i) q^{7} +(-44.0632 - 44.0632i) q^{8} -108.013i q^{10} +(59.2741 + 59.2741i) q^{11} +(-157.156 + 62.1541i) q^{13} +88.4101 q^{14} +268.074 q^{16} -272.906i q^{17} +(-54.5892 + 54.5892i) q^{19} +(14.7989 + 14.7989i) q^{20} -343.524 q^{22} -596.223i q^{23} -69.6962i q^{25} +(275.292 - 635.507i) q^{26} +(-12.1131 + 12.1131i) q^{28} +1314.18 q^{29} +(874.924 - 874.924i) q^{31} +(-71.8036 + 71.8036i) q^{32} +(790.817 + 790.817i) q^{34} +568.619 q^{35} +(-403.847 - 403.847i) q^{37} -316.373i q^{38} +1642.44 q^{40} +(-983.966 + 983.966i) q^{41} -2237.13i q^{43} +(47.0664 - 47.0664i) q^{44} +(1727.71 + 1727.71i) q^{46} +(369.867 + 369.867i) q^{47} -1935.58i q^{49} +(201.963 + 201.963i) q^{50} +(49.3532 + 124.789i) q^{52} -4399.13 q^{53} -2209.42 q^{55} +1344.36i q^{56} +(-3808.17 + 3808.17i) q^{58} +(-976.401 - 976.401i) q^{59} -6163.60 q^{61} +5070.64i q^{62} +3873.05i q^{64} +(1770.57 - 4087.33i) q^{65} +(-1708.57 + 1708.57i) q^{67} -216.700 q^{68} +(-1647.72 + 1647.72i) q^{70} +(-672.586 + 672.586i) q^{71} +(-1169.63 - 1169.63i) q^{73} +2340.51 q^{74} +(43.3464 + 43.3464i) q^{76} -1808.44i q^{77} +8345.50 q^{79} +(-4996.17 + 4996.17i) q^{80} -5702.60i q^{82} +(-6513.68 + 6513.68i) q^{83} +(5086.23 + 5086.23i) q^{85} +(6482.67 + 6482.67i) q^{86} -5223.62i q^{88} +(-8366.65 - 8366.65i) q^{89} +(3345.55 + 1449.24i) q^{91} -473.429 q^{92} -2143.57 q^{94} -2034.79i q^{95} +(2271.58 - 2271.58i) q^{97} +(5608.84 + 5608.84i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 24 q^{5} - 24 q^{7} - 372 q^{11} - 224 q^{13} - 480 q^{14} - 2328 q^{16} - 840 q^{19} - 228 q^{20} + 3536 q^{22} + 828 q^{26} - 1984 q^{28} + 5064 q^{29} + 1712 q^{31} + 7260 q^{32} + 8040 q^{34}+ \cdots - 11544 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.89776 + 2.89776i −0.724440 + 0.724440i −0.969506 0.245066i \(-0.921190\pi\)
0.245066 + 0.969506i \(0.421190\pi\)
\(3\) 0 0
\(4\) 0.794047i 0.0496279i
\(5\) −18.6373 + 18.6373i −0.745491 + 0.745491i −0.973629 0.228138i \(-0.926736\pi\)
0.228138 + 0.973629i \(0.426736\pi\)
\(6\) 0 0
\(7\) −15.2549 15.2549i −0.311324 0.311324i 0.534098 0.845422i \(-0.320651\pi\)
−0.845422 + 0.534098i \(0.820651\pi\)
\(8\) −44.0632 44.0632i −0.688488 0.688488i
\(9\) 0 0
\(10\) 108.013i 1.08013i
\(11\) 59.2741 + 59.2741i 0.489869 + 0.489869i 0.908265 0.418396i \(-0.137408\pi\)
−0.418396 + 0.908265i \(0.637408\pi\)
\(12\) 0 0
\(13\) −157.156 + 62.1541i −0.929915 + 0.367776i
\(14\) 88.4101 0.451072
\(15\) 0 0
\(16\) 268.074 1.04717
\(17\) 272.906i 0.944312i −0.881515 0.472156i \(-0.843476\pi\)
0.881515 0.472156i \(-0.156524\pi\)
\(18\) 0 0
\(19\) −54.5892 + 54.5892i −0.151217 + 0.151217i −0.778661 0.627445i \(-0.784101\pi\)
0.627445 + 0.778661i \(0.284101\pi\)
\(20\) 14.7989 + 14.7989i 0.0369972 + 0.0369972i
\(21\) 0 0
\(22\) −343.524 −0.709761
\(23\) 596.223i 1.12708i −0.826090 0.563538i \(-0.809440\pi\)
0.826090 0.563538i \(-0.190560\pi\)
\(24\) 0 0
\(25\) 69.6962i 0.111514i
\(26\) 275.292 635.507i 0.407236 0.940099i
\(27\) 0 0
\(28\) −12.1131 + 12.1131i −0.0154504 + 0.0154504i
\(29\) 1314.18 1.56263 0.781317 0.624135i \(-0.214548\pi\)
0.781317 + 0.624135i \(0.214548\pi\)
\(30\) 0 0
\(31\) 874.924 874.924i 0.910431 0.910431i −0.0858748 0.996306i \(-0.527368\pi\)
0.996306 + 0.0858748i \(0.0273685\pi\)
\(32\) −71.8036 + 71.8036i −0.0701207 + 0.0701207i
\(33\) 0 0
\(34\) 790.817 + 790.817i 0.684098 + 0.684098i
\(35\) 568.619 0.464179
\(36\) 0 0
\(37\) −403.847 403.847i −0.294994 0.294994i 0.544055 0.839049i \(-0.316888\pi\)
−0.839049 + 0.544055i \(0.816888\pi\)
\(38\) 316.373i 0.219095i
\(39\) 0 0
\(40\) 1642.44 1.02652
\(41\) −983.966 + 983.966i −0.585346 + 0.585346i −0.936367 0.351022i \(-0.885834\pi\)
0.351022 + 0.936367i \(0.385834\pi\)
\(42\) 0 0
\(43\) 2237.13i 1.20991i −0.796258 0.604957i \(-0.793190\pi\)
0.796258 0.604957i \(-0.206810\pi\)
\(44\) 47.0664 47.0664i 0.0243112 0.0243112i
\(45\) 0 0
\(46\) 1727.71 + 1727.71i 0.816500 + 0.816500i
\(47\) 369.867 + 369.867i 0.167437 + 0.167437i 0.785852 0.618415i \(-0.212225\pi\)
−0.618415 + 0.785852i \(0.712225\pi\)
\(48\) 0 0
\(49\) 1935.58i 0.806154i
\(50\) 201.963 + 201.963i 0.0807852 + 0.0807852i
\(51\) 0 0
\(52\) 49.3532 + 124.789i 0.0182519 + 0.0461497i
\(53\) −4399.13 −1.56608 −0.783041 0.621970i \(-0.786333\pi\)
−0.783041 + 0.621970i \(0.786333\pi\)
\(54\) 0 0
\(55\) −2209.42 −0.730385
\(56\) 1344.36i 0.428686i
\(57\) 0 0
\(58\) −3808.17 + 3808.17i −1.13204 + 1.13204i
\(59\) −976.401 976.401i −0.280495 0.280495i 0.552812 0.833306i \(-0.313555\pi\)
−0.833306 + 0.552812i \(0.813555\pi\)
\(60\) 0 0
\(61\) −6163.60 −1.65644 −0.828219 0.560405i \(-0.810646\pi\)
−0.828219 + 0.560405i \(0.810646\pi\)
\(62\) 5070.64i 1.31911i
\(63\) 0 0
\(64\) 3873.05i 0.945568i
\(65\) 1770.57 4087.33i 0.419070 0.967416i
\(66\) 0 0
\(67\) −1708.57 + 1708.57i −0.380612 + 0.380612i −0.871322 0.490711i \(-0.836737\pi\)
0.490711 + 0.871322i \(0.336737\pi\)
\(68\) −216.700 −0.0468642
\(69\) 0 0
\(70\) −1647.72 + 1647.72i −0.336270 + 0.336270i
\(71\) −672.586 + 672.586i −0.133423 + 0.133423i −0.770664 0.637241i \(-0.780075\pi\)
0.637241 + 0.770664i \(0.280075\pi\)
\(72\) 0 0
\(73\) −1169.63 1169.63i −0.219485 0.219485i 0.588797 0.808281i \(-0.299602\pi\)
−0.808281 + 0.588797i \(0.799602\pi\)
\(74\) 2340.51 0.427412
\(75\) 0 0
\(76\) 43.3464 + 43.3464i 0.00750457 + 0.00750457i
\(77\) 1808.44i 0.305016i
\(78\) 0 0
\(79\) 8345.50 1.33720 0.668602 0.743620i \(-0.266893\pi\)
0.668602 + 0.743620i \(0.266893\pi\)
\(80\) −4996.17 + 4996.17i −0.780652 + 0.780652i
\(81\) 0 0
\(82\) 5702.60i 0.848096i
\(83\) −6513.68 + 6513.68i −0.945519 + 0.945519i −0.998591 0.0530714i \(-0.983099\pi\)
0.0530714 + 0.998591i \(0.483099\pi\)
\(84\) 0 0
\(85\) 5086.23 + 5086.23i 0.703976 + 0.703976i
\(86\) 6482.67 + 6482.67i 0.876510 + 0.876510i
\(87\) 0 0
\(88\) 5223.62i 0.674537i
\(89\) −8366.65 8366.65i −1.05626 1.05626i −0.998320 0.0579425i \(-0.981546\pi\)
−0.0579425 0.998320i \(-0.518454\pi\)
\(90\) 0 0
\(91\) 3345.55 + 1449.24i 0.404003 + 0.175008i
\(92\) −473.429 −0.0559345
\(93\) 0 0
\(94\) −2143.57 −0.242596
\(95\) 2034.79i 0.225461i
\(96\) 0 0
\(97\) 2271.58 2271.58i 0.241426 0.241426i −0.576014 0.817440i \(-0.695393\pi\)
0.817440 + 0.576014i \(0.195393\pi\)
\(98\) 5608.84 + 5608.84i 0.584011 + 0.584011i
\(99\) 0 0
\(100\) −55.3421 −0.00553421
\(101\) 6490.97i 0.636308i −0.948039 0.318154i \(-0.896937\pi\)
0.948039 0.318154i \(-0.103063\pi\)
\(102\) 0 0
\(103\) 5614.96i 0.529264i −0.964350 0.264632i \(-0.914750\pi\)
0.964350 0.264632i \(-0.0852504\pi\)
\(104\) 9663.49 + 4186.07i 0.893444 + 0.387026i
\(105\) 0 0
\(106\) 12747.6 12747.6i 1.13453 1.13453i
\(107\) −9686.14 −0.846025 −0.423012 0.906124i \(-0.639027\pi\)
−0.423012 + 0.906124i \(0.639027\pi\)
\(108\) 0 0
\(109\) −16372.1 + 16372.1i −1.37801 + 1.37801i −0.530029 + 0.847980i \(0.677819\pi\)
−0.847980 + 0.530029i \(0.822181\pi\)
\(110\) 6402.36 6402.36i 0.529121 0.529121i
\(111\) 0 0
\(112\) −4089.44 4089.44i −0.326008 0.326008i
\(113\) 19245.1 1.50717 0.753585 0.657350i \(-0.228323\pi\)
0.753585 + 0.657350i \(0.228323\pi\)
\(114\) 0 0
\(115\) 11112.0 + 11112.0i 0.840225 + 0.840225i
\(116\) 1043.52i 0.0775503i
\(117\) 0 0
\(118\) 5658.76 0.406403
\(119\) −4163.15 + 4163.15i −0.293987 + 0.293987i
\(120\) 0 0
\(121\) 7614.16i 0.520057i
\(122\) 17860.7 17860.7i 1.19999 1.19999i
\(123\) 0 0
\(124\) −694.731 694.731i −0.0451828 0.0451828i
\(125\) −10349.4 10349.4i −0.662358 0.662358i
\(126\) 0 0
\(127\) 26337.0i 1.63290i −0.577417 0.816449i \(-0.695939\pi\)
0.577417 0.816449i \(-0.304061\pi\)
\(128\) −12372.0 12372.0i −0.755129 0.755129i
\(129\) 0 0
\(130\) 6713.43 + 16974.8i 0.397245 + 1.00443i
\(131\) 3094.41 0.180316 0.0901581 0.995927i \(-0.471263\pi\)
0.0901581 + 0.995927i \(0.471263\pi\)
\(132\) 0 0
\(133\) 1665.51 0.0941549
\(134\) 9902.03i 0.551461i
\(135\) 0 0
\(136\) −12025.1 + 12025.1i −0.650147 + 0.650147i
\(137\) 10453.9 + 10453.9i 0.556978 + 0.556978i 0.928446 0.371467i \(-0.121145\pi\)
−0.371467 + 0.928446i \(0.621145\pi\)
\(138\) 0 0
\(139\) 5922.48 0.306531 0.153265 0.988185i \(-0.451021\pi\)
0.153265 + 0.988185i \(0.451021\pi\)
\(140\) 451.511i 0.0230363i
\(141\) 0 0
\(142\) 3897.99i 0.193314i
\(143\) −12999.4 5631.13i −0.635698 0.275374i
\(144\) 0 0
\(145\) −24492.6 + 24492.6i −1.16493 + 1.16493i
\(146\) 6778.64 0.318007
\(147\) 0 0
\(148\) −320.674 + 320.674i −0.0146400 + 0.0146400i
\(149\) 22740.7 22740.7i 1.02431 1.02431i 0.0246105 0.999697i \(-0.492165\pi\)
0.999697 0.0246105i \(-0.00783457\pi\)
\(150\) 0 0
\(151\) −30969.3 30969.3i −1.35824 1.35824i −0.876082 0.482161i \(-0.839852\pi\)
−0.482161 0.876082i \(-0.660148\pi\)
\(152\) 4810.76 0.208222
\(153\) 0 0
\(154\) 5240.43 + 5240.43i 0.220966 + 0.220966i
\(155\) 32612.4i 1.35744i
\(156\) 0 0
\(157\) −16868.7 −0.684358 −0.342179 0.939635i \(-0.611165\pi\)
−0.342179 + 0.939635i \(0.611165\pi\)
\(158\) −24183.3 + 24183.3i −0.968725 + 0.968725i
\(159\) 0 0
\(160\) 2676.45i 0.104549i
\(161\) −9095.32 + 9095.32i −0.350886 + 0.350886i
\(162\) 0 0
\(163\) 23110.4 + 23110.4i 0.869826 + 0.869826i 0.992453 0.122626i \(-0.0391317\pi\)
−0.122626 + 0.992453i \(0.539132\pi\)
\(164\) 781.315 + 781.315i 0.0290495 + 0.0290495i
\(165\) 0 0
\(166\) 37750.2i 1.36994i
\(167\) −16613.8 16613.8i −0.595710 0.595710i 0.343458 0.939168i \(-0.388402\pi\)
−0.939168 + 0.343458i \(0.888402\pi\)
\(168\) 0 0
\(169\) 20834.7 19535.7i 0.729482 0.684000i
\(170\) −29477.3 −1.01998
\(171\) 0 0
\(172\) −1776.39 −0.0600455
\(173\) 9230.43i 0.308411i −0.988039 0.154206i \(-0.950718\pi\)
0.988039 0.154206i \(-0.0492818\pi\)
\(174\) 0 0
\(175\) −1063.21 + 1063.21i −0.0347170 + 0.0347170i
\(176\) 15889.9 + 15889.9i 0.512973 + 0.512973i
\(177\) 0 0
\(178\) 48489.1 1.53040
\(179\) 28771.7i 0.897965i 0.893541 + 0.448982i \(0.148213\pi\)
−0.893541 + 0.448982i \(0.851787\pi\)
\(180\) 0 0
\(181\) 2334.88i 0.0712700i −0.999365 0.0356350i \(-0.988655\pi\)
0.999365 0.0356350i \(-0.0113454\pi\)
\(182\) −13894.1 + 5495.05i −0.419458 + 0.165893i
\(183\) 0 0
\(184\) −26271.5 + 26271.5i −0.775978 + 0.775978i
\(185\) 15053.2 0.439831
\(186\) 0 0
\(187\) 16176.3 16176.3i 0.462589 0.462589i
\(188\) 293.692 293.692i 0.00830953 0.00830953i
\(189\) 0 0
\(190\) 5896.34 + 5896.34i 0.163333 + 0.163333i
\(191\) 11500.7 0.315251 0.157625 0.987499i \(-0.449616\pi\)
0.157625 + 0.987499i \(0.449616\pi\)
\(192\) 0 0
\(193\) 36690.3 + 36690.3i 0.985002 + 0.985002i 0.999889 0.0148869i \(-0.00473884\pi\)
−0.0148869 + 0.999889i \(0.504739\pi\)
\(194\) 13165.0i 0.349797i
\(195\) 0 0
\(196\) −1536.94 −0.0400078
\(197\) −3775.20 + 3775.20i −0.0972763 + 0.0972763i −0.754070 0.656794i \(-0.771912\pi\)
0.656794 + 0.754070i \(0.271912\pi\)
\(198\) 0 0
\(199\) 47612.3i 1.20230i 0.799136 + 0.601151i \(0.205291\pi\)
−0.799136 + 0.601151i \(0.794709\pi\)
\(200\) −3071.04 + 3071.04i −0.0767760 + 0.0767760i
\(201\) 0 0
\(202\) 18809.3 + 18809.3i 0.460967 + 0.460967i
\(203\) −20047.6 20047.6i −0.486486 0.486486i
\(204\) 0 0
\(205\) 36676.9i 0.872740i
\(206\) 16270.8 + 16270.8i 0.383420 + 0.383420i
\(207\) 0 0
\(208\) −42129.4 + 16661.9i −0.973774 + 0.385122i
\(209\) −6471.46 −0.148153
\(210\) 0 0
\(211\) −75668.4 −1.69961 −0.849806 0.527096i \(-0.823281\pi\)
−0.849806 + 0.527096i \(0.823281\pi\)
\(212\) 3493.11i 0.0777215i
\(213\) 0 0
\(214\) 28068.1 28068.1i 0.612894 0.612894i
\(215\) 41694.0 + 41694.0i 0.901980 + 0.901980i
\(216\) 0 0
\(217\) −26693.8 −0.566879
\(218\) 94885.0i 1.99657i
\(219\) 0 0
\(220\) 1754.38i 0.0362475i
\(221\) 16962.2 + 42888.7i 0.347295 + 0.878129i
\(222\) 0 0
\(223\) −953.201 + 953.201i −0.0191679 + 0.0191679i −0.716626 0.697458i \(-0.754314\pi\)
0.697458 + 0.716626i \(0.254314\pi\)
\(224\) 2190.71 0.0436606
\(225\) 0 0
\(226\) −55767.6 + 55767.6i −1.09186 + 1.09186i
\(227\) −6110.06 + 6110.06i −0.118575 + 0.118575i −0.763905 0.645329i \(-0.776720\pi\)
0.645329 + 0.763905i \(0.276720\pi\)
\(228\) 0 0
\(229\) 9576.99 + 9576.99i 0.182624 + 0.182624i 0.792498 0.609874i \(-0.208780\pi\)
−0.609874 + 0.792498i \(0.708780\pi\)
\(230\) −64399.7 −1.21739
\(231\) 0 0
\(232\) −57906.8 57906.8i −1.07585 1.07585i
\(233\) 72801.5i 1.34100i −0.741910 0.670500i \(-0.766080\pi\)
0.741910 0.670500i \(-0.233920\pi\)
\(234\) 0 0
\(235\) −13786.6 −0.249645
\(236\) −775.309 + 775.309i −0.0139204 + 0.0139204i
\(237\) 0 0
\(238\) 24127.7i 0.425952i
\(239\) −72643.5 + 72643.5i −1.27175 + 1.27175i −0.326578 + 0.945170i \(0.605895\pi\)
−0.945170 + 0.326578i \(0.894105\pi\)
\(240\) 0 0
\(241\) 64322.5 + 64322.5i 1.10746 + 1.10746i 0.993483 + 0.113978i \(0.0363593\pi\)
0.113978 + 0.993483i \(0.463641\pi\)
\(242\) 22064.0 + 22064.0i 0.376751 + 0.376751i
\(243\) 0 0
\(244\) 4894.19i 0.0822056i
\(245\) 36073.9 + 36073.9i 0.600981 + 0.600981i
\(246\) 0 0
\(247\) 5186.06 11971.9i 0.0850048 0.196232i
\(248\) −77104.0 −1.25364
\(249\) 0 0
\(250\) 59979.9 0.959678
\(251\) 49293.7i 0.782427i −0.920300 0.391213i \(-0.872055\pi\)
0.920300 0.391213i \(-0.127945\pi\)
\(252\) 0 0
\(253\) 35340.6 35340.6i 0.552119 0.552119i
\(254\) 76318.4 + 76318.4i 1.18294 + 1.18294i
\(255\) 0 0
\(256\) 9733.62 0.148523
\(257\) 82424.4i 1.24793i −0.781453 0.623964i \(-0.785521\pi\)
0.781453 0.623964i \(-0.214479\pi\)
\(258\) 0 0
\(259\) 12321.3i 0.183678i
\(260\) −3245.54 1405.92i −0.0480109 0.0207976i
\(261\) 0 0
\(262\) −8966.85 + 8966.85i −0.130628 + 0.130628i
\(263\) −41367.3 −0.598062 −0.299031 0.954243i \(-0.596663\pi\)
−0.299031 + 0.954243i \(0.596663\pi\)
\(264\) 0 0
\(265\) 81987.7 81987.7i 1.16750 1.16750i
\(266\) −4826.24 + 4826.24i −0.0682096 + 0.0682096i
\(267\) 0 0
\(268\) 1356.68 + 1356.68i 0.0188890 + 0.0188890i
\(269\) 59626.2 0.824011 0.412005 0.911181i \(-0.364828\pi\)
0.412005 + 0.911181i \(0.364828\pi\)
\(270\) 0 0
\(271\) −75798.8 75798.8i −1.03210 1.03210i −0.999467 0.0326374i \(-0.989609\pi\)
−0.0326374 0.999467i \(-0.510391\pi\)
\(272\) 73159.1i 0.988850i
\(273\) 0 0
\(274\) −60586.0 −0.806996
\(275\) 4131.18 4131.18i 0.0546272 0.0546272i
\(276\) 0 0
\(277\) 134127.i 1.74806i −0.485876 0.874028i \(-0.661499\pi\)
0.485876 0.874028i \(-0.338501\pi\)
\(278\) −17161.9 + 17161.9i −0.222063 + 0.222063i
\(279\) 0 0
\(280\) −25055.2 25055.2i −0.319582 0.319582i
\(281\) 19968.0 + 19968.0i 0.252885 + 0.252885i 0.822152 0.569268i \(-0.192773\pi\)
−0.569268 + 0.822152i \(0.692773\pi\)
\(282\) 0 0
\(283\) 151960.i 1.89739i 0.316186 + 0.948697i \(0.397598\pi\)
−0.316186 + 0.948697i \(0.602402\pi\)
\(284\) 534.065 + 534.065i 0.00662151 + 0.00662151i
\(285\) 0 0
\(286\) 53986.8 21351.4i 0.660017 0.261033i
\(287\) 30020.6 0.364465
\(288\) 0 0
\(289\) 9043.27 0.108275
\(290\) 141948.i 1.68784i
\(291\) 0 0
\(292\) −928.744 + 928.744i −0.0108926 + 0.0108926i
\(293\) −22398.8 22398.8i −0.260909 0.260909i 0.564514 0.825423i \(-0.309063\pi\)
−0.825423 + 0.564514i \(0.809063\pi\)
\(294\) 0 0
\(295\) 36394.9 0.418212
\(296\) 35589.6i 0.406200i
\(297\) 0 0
\(298\) 131794.i 1.48410i
\(299\) 37057.7 + 93699.8i 0.414511 + 1.04808i
\(300\) 0 0
\(301\) −34127.2 + 34127.2i −0.376676 + 0.376676i
\(302\) 179483. 1.96793
\(303\) 0 0
\(304\) −14634.0 + 14634.0i −0.158349 + 0.158349i
\(305\) 114873. 114873.i 1.23486 1.23486i
\(306\) 0 0
\(307\) 36789.1 + 36789.1i 0.390339 + 0.390339i 0.874808 0.484469i \(-0.160987\pi\)
−0.484469 + 0.874808i \(0.660987\pi\)
\(308\) −1435.99 −0.0151373
\(309\) 0 0
\(310\) −94503.0 94503.0i −0.983382 0.983382i
\(311\) 78846.4i 0.815194i −0.913162 0.407597i \(-0.866367\pi\)
0.913162 0.407597i \(-0.133633\pi\)
\(312\) 0 0
\(313\) 20618.6 0.210460 0.105230 0.994448i \(-0.466442\pi\)
0.105230 + 0.994448i \(0.466442\pi\)
\(314\) 48881.6 48881.6i 0.495777 0.495777i
\(315\) 0 0
\(316\) 6626.72i 0.0663627i
\(317\) −42129.4 + 42129.4i −0.419244 + 0.419244i −0.884943 0.465699i \(-0.845803\pi\)
0.465699 + 0.884943i \(0.345803\pi\)
\(318\) 0 0
\(319\) 77896.6 + 77896.6i 0.765485 + 0.765485i
\(320\) −72183.1 72183.1i −0.704913 0.704913i
\(321\) 0 0
\(322\) 52712.2i 0.508392i
\(323\) 14897.7 + 14897.7i 0.142796 + 0.142796i
\(324\) 0 0
\(325\) 4331.90 + 10953.2i 0.0410121 + 0.103698i
\(326\) −133937. −1.26027
\(327\) 0 0
\(328\) 86713.5 0.806007
\(329\) 11284.6i 0.104254i
\(330\) 0 0
\(331\) 68308.6 68308.6i 0.623476 0.623476i −0.322943 0.946418i \(-0.604672\pi\)
0.946418 + 0.322943i \(0.104672\pi\)
\(332\) 5172.17 + 5172.17i 0.0469242 + 0.0469242i
\(333\) 0 0
\(334\) 96285.5 0.863113
\(335\) 63686.0i 0.567485i
\(336\) 0 0
\(337\) 87835.7i 0.773412i −0.922203 0.386706i \(-0.873613\pi\)
0.922203 0.386706i \(-0.126387\pi\)
\(338\) −3764.29 + 116984.i −0.0329495 + 1.02398i
\(339\) 0 0
\(340\) 4038.70 4038.70i 0.0349369 0.0349369i
\(341\) 103721. 0.891983
\(342\) 0 0
\(343\) −66154.0 + 66154.0i −0.562300 + 0.562300i
\(344\) −98575.2 + 98575.2i −0.833011 + 0.833011i
\(345\) 0 0
\(346\) 26747.6 + 26747.6i 0.223425 + 0.223425i
\(347\) −98270.4 −0.816138 −0.408069 0.912951i \(-0.633798\pi\)
−0.408069 + 0.912951i \(0.633798\pi\)
\(348\) 0 0
\(349\) −17440.9 17440.9i −0.143192 0.143192i 0.631877 0.775069i \(-0.282285\pi\)
−0.775069 + 0.631877i \(0.782285\pi\)
\(350\) 6161.85i 0.0503008i
\(351\) 0 0
\(352\) −8512.19 −0.0686999
\(353\) −63740.5 + 63740.5i −0.511524 + 0.511524i −0.914993 0.403469i \(-0.867804\pi\)
0.403469 + 0.914993i \(0.367804\pi\)
\(354\) 0 0
\(355\) 25070.3i 0.198931i
\(356\) −6643.52 + 6643.52i −0.0524201 + 0.0524201i
\(357\) 0 0
\(358\) −83373.5 83373.5i −0.650522 0.650522i
\(359\) 168217. + 168217.i 1.30521 + 1.30521i 0.924831 + 0.380377i \(0.124206\pi\)
0.380377 + 0.924831i \(0.375794\pi\)
\(360\) 0 0
\(361\) 124361.i 0.954267i
\(362\) 6765.91 + 6765.91i 0.0516308 + 0.0516308i
\(363\) 0 0
\(364\) 1150.76 2656.52i 0.00868527 0.0200498i
\(365\) 43597.6 0.327248
\(366\) 0 0
\(367\) 4967.29 0.0368797 0.0184399 0.999830i \(-0.494130\pi\)
0.0184399 + 0.999830i \(0.494130\pi\)
\(368\) 159832.i 1.18023i
\(369\) 0 0
\(370\) −43620.7 + 43620.7i −0.318632 + 0.318632i
\(371\) 67108.2 + 67108.2i 0.487560 + 0.487560i
\(372\) 0 0
\(373\) −115465. −0.829914 −0.414957 0.909841i \(-0.636203\pi\)
−0.414957 + 0.909841i \(0.636203\pi\)
\(374\) 93749.9i 0.670236i
\(375\) 0 0
\(376\) 32595.1i 0.230556i
\(377\) −206530. + 81681.3i −1.45312 + 0.574698i
\(378\) 0 0
\(379\) −79418.8 + 79418.8i −0.552898 + 0.552898i −0.927276 0.374378i \(-0.877856\pi\)
0.374378 + 0.927276i \(0.377856\pi\)
\(380\) −1615.72 −0.0111892
\(381\) 0 0
\(382\) −33326.2 + 33326.2i −0.228380 + 0.228380i
\(383\) −30297.2 + 30297.2i −0.206540 + 0.206540i −0.802795 0.596255i \(-0.796655\pi\)
0.596255 + 0.802795i \(0.296655\pi\)
\(384\) 0 0
\(385\) 33704.4 + 33704.4i 0.227387 + 0.227387i
\(386\) −212640. −1.42715
\(387\) 0 0
\(388\) −1803.74 1803.74i −0.0119815 0.0119815i
\(389\) 135764.i 0.897193i 0.893734 + 0.448597i \(0.148076\pi\)
−0.893734 + 0.448597i \(0.851924\pi\)
\(390\) 0 0
\(391\) −162713. −1.06431
\(392\) −85287.7 + 85287.7i −0.555028 + 0.555028i
\(393\) 0 0
\(394\) 21879.2i 0.140942i
\(395\) −155537. + 155537.i −0.996874 + 0.996874i
\(396\) 0 0
\(397\) −122596. 122596.i −0.777851 0.777851i 0.201614 0.979465i \(-0.435381\pi\)
−0.979465 + 0.201614i \(0.935381\pi\)
\(398\) −137969. 137969.i −0.870996 0.870996i
\(399\) 0 0
\(400\) 18683.8i 0.116774i
\(401\) −132123. 132123.i −0.821653 0.821653i 0.164692 0.986345i \(-0.447337\pi\)
−0.986345 + 0.164692i \(0.947337\pi\)
\(402\) 0 0
\(403\) −83119.1 + 191879.i −0.511789 + 1.18146i
\(404\) −5154.14 −0.0315786
\(405\) 0 0
\(406\) 116186. 0.704860
\(407\) 47875.4i 0.289017i
\(408\) 0 0
\(409\) 2700.68 2700.68i 0.0161445 0.0161445i −0.698988 0.715133i \(-0.746366\pi\)
0.715133 + 0.698988i \(0.246366\pi\)
\(410\) 106281. + 106281.i 0.632248 + 0.632248i
\(411\) 0 0
\(412\) −4458.54 −0.0262663
\(413\) 29789.8i 0.174650i
\(414\) 0 0
\(415\) 242795.i 1.40975i
\(416\) 6821.45 15747.2i 0.0394176 0.0909950i
\(417\) 0 0
\(418\) 18752.7 18752.7i 0.107328 0.107328i
\(419\) 94447.5 0.537975 0.268988 0.963144i \(-0.413311\pi\)
0.268988 + 0.963144i \(0.413311\pi\)
\(420\) 0 0
\(421\) 15371.4 15371.4i 0.0867262 0.0867262i −0.662413 0.749139i \(-0.730468\pi\)
0.749139 + 0.662413i \(0.230468\pi\)
\(422\) 219269. 219269.i 1.23127 1.23127i
\(423\) 0 0
\(424\) 193840. + 193840.i 1.07823 + 1.07823i
\(425\) −19020.5 −0.105304
\(426\) 0 0
\(427\) 94025.1 + 94025.1i 0.515689 + 0.515689i
\(428\) 7691.25i 0.0419865i
\(429\) 0 0
\(430\) −241639. −1.30686
\(431\) 42041.0 42041.0i 0.226317 0.226317i −0.584835 0.811152i \(-0.698841\pi\)
0.811152 + 0.584835i \(0.198841\pi\)
\(432\) 0 0
\(433\) 29381.2i 0.156709i 0.996926 + 0.0783545i \(0.0249666\pi\)
−0.996926 + 0.0783545i \(0.975033\pi\)
\(434\) 77352.1 77352.1i 0.410670 0.410670i
\(435\) 0 0
\(436\) 13000.2 + 13000.2i 0.0683877 + 0.0683877i
\(437\) 32547.4 + 32547.4i 0.170433 + 0.170433i
\(438\) 0 0
\(439\) 354668.i 1.84032i 0.391542 + 0.920160i \(0.371942\pi\)
−0.391542 + 0.920160i \(0.628058\pi\)
\(440\) 97354.0 + 97354.0i 0.502862 + 0.502862i
\(441\) 0 0
\(442\) −173434. 75128.8i −0.887747 0.384558i
\(443\) 22527.8 0.114792 0.0573960 0.998351i \(-0.481720\pi\)
0.0573960 + 0.998351i \(0.481720\pi\)
\(444\) 0 0
\(445\) 311863. 1.57487
\(446\) 5524.30i 0.0277720i
\(447\) 0 0
\(448\) 59082.9 59082.9i 0.294378 0.294378i
\(449\) 53263.7 + 53263.7i 0.264204 + 0.264204i 0.826759 0.562556i \(-0.190182\pi\)
−0.562556 + 0.826759i \(0.690182\pi\)
\(450\) 0 0
\(451\) −116647. −0.573485
\(452\) 15281.5i 0.0747978i
\(453\) 0 0
\(454\) 35411.0i 0.171801i
\(455\) −89361.7 + 35342.0i −0.431647 + 0.170714i
\(456\) 0 0
\(457\) 94033.6 94033.6i 0.450247 0.450247i −0.445189 0.895436i \(-0.646864\pi\)
0.895436 + 0.445189i \(0.146864\pi\)
\(458\) −55503.7 −0.264601
\(459\) 0 0
\(460\) 8823.43 8823.43i 0.0416986 0.0416986i
\(461\) 220771. 220771.i 1.03882 1.03882i 0.0396051 0.999215i \(-0.487390\pi\)
0.999215 0.0396051i \(-0.0126100\pi\)
\(462\) 0 0
\(463\) −148393. 148393.i −0.692231 0.692231i 0.270491 0.962722i \(-0.412814\pi\)
−0.962722 + 0.270491i \(0.912814\pi\)
\(464\) 352296. 1.63634
\(465\) 0 0
\(466\) 210962. + 210962.i 0.971474 + 0.971474i
\(467\) 89596.5i 0.410826i −0.978675 0.205413i \(-0.934146\pi\)
0.978675 0.205413i \(-0.0658537\pi\)
\(468\) 0 0
\(469\) 52128.0 0.236987
\(470\) 39950.4 39950.4i 0.180853 0.180853i
\(471\) 0 0
\(472\) 86046.8i 0.386234i
\(473\) 132604. 132604.i 0.592699 0.592699i
\(474\) 0 0
\(475\) 3804.66 + 3804.66i 0.0168628 + 0.0168628i
\(476\) 3305.74 + 3305.74i 0.0145900 + 0.0145900i
\(477\) 0 0
\(478\) 421007.i 1.84261i
\(479\) −290497. 290497.i −1.26611 1.26611i −0.948083 0.318023i \(-0.896981\pi\)
−0.318023 0.948083i \(-0.603019\pi\)
\(480\) 0 0
\(481\) 88567.6 + 38366.1i 0.382811 + 0.165828i
\(482\) −372782. −1.60458
\(483\) 0 0
\(484\) −6046.00 −0.0258094
\(485\) 84672.0i 0.359962i
\(486\) 0 0
\(487\) 186744. 186744.i 0.787388 0.787388i −0.193677 0.981065i \(-0.562041\pi\)
0.981065 + 0.193677i \(0.0620414\pi\)
\(488\) 271588. + 271588.i 1.14044 + 1.14044i
\(489\) 0 0
\(490\) −209067. −0.870750
\(491\) 26475.0i 0.109818i −0.998491 0.0549088i \(-0.982513\pi\)
0.998491 0.0549088i \(-0.0174868\pi\)
\(492\) 0 0
\(493\) 358646.i 1.47561i
\(494\) 19663.9 + 49719.8i 0.0805778 + 0.203740i
\(495\) 0 0
\(496\) 234545. 234545.i 0.953372 0.953372i
\(497\) 20520.4 0.0830757
\(498\) 0 0
\(499\) 72703.0 72703.0i 0.291979 0.291979i −0.545883 0.837862i \(-0.683806\pi\)
0.837862 + 0.545883i \(0.183806\pi\)
\(500\) −8217.87 + 8217.87i −0.0328715 + 0.0328715i
\(501\) 0 0
\(502\) 142841. + 142841.i 0.566822 + 0.566822i
\(503\) −245797. −0.971496 −0.485748 0.874099i \(-0.661453\pi\)
−0.485748 + 0.874099i \(0.661453\pi\)
\(504\) 0 0
\(505\) 120974. + 120974.i 0.474362 + 0.474362i
\(506\) 204817.i 0.799955i
\(507\) 0 0
\(508\) −20912.8 −0.0810374
\(509\) 105992. 105992.i 0.409109 0.409109i −0.472319 0.881428i \(-0.656583\pi\)
0.881428 + 0.472319i \(0.156583\pi\)
\(510\) 0 0
\(511\) 35685.3i 0.136662i
\(512\) 169747. 169747.i 0.647533 0.647533i
\(513\) 0 0
\(514\) 238846. + 238846.i 0.904049 + 0.904049i
\(515\) 104648. + 104648.i 0.394561 + 0.394561i
\(516\) 0 0
\(517\) 43847.1i 0.164044i
\(518\) −35704.2 35704.2i −0.133064 0.133064i
\(519\) 0 0
\(520\) −258118. + 102084.i −0.954579 + 0.377530i
\(521\) −408225. −1.50392 −0.751959 0.659209i \(-0.770891\pi\)
−0.751959 + 0.659209i \(0.770891\pi\)
\(522\) 0 0
\(523\) −145126. −0.530567 −0.265284 0.964170i \(-0.585466\pi\)
−0.265284 + 0.964170i \(0.585466\pi\)
\(524\) 2457.10i 0.00894872i
\(525\) 0 0
\(526\) 119873. 119873.i 0.433260 0.433260i
\(527\) −238772. 238772.i −0.859731 0.859731i
\(528\) 0 0
\(529\) −75641.2 −0.270300
\(530\) 475162.i 1.69157i
\(531\) 0 0
\(532\) 1322.49i 0.00467271i
\(533\) 93478.3 215793.i 0.329046 0.759597i
\(534\) 0 0
\(535\) 180523. 180523.i 0.630704 0.630704i
\(536\) 150570. 0.524093
\(537\) 0 0
\(538\) −172783. + 172783.i −0.596947 + 0.596947i
\(539\) 114730. 114730.i 0.394910 0.394910i
\(540\) 0 0
\(541\) −376207. 376207.i −1.28538 1.28538i −0.937560 0.347823i \(-0.886921\pi\)
−0.347823 0.937560i \(-0.613079\pi\)
\(542\) 439294. 1.49540
\(543\) 0 0
\(544\) 19595.6 + 19595.6i 0.0662158 + 0.0662158i
\(545\) 610263.i 2.05459i
\(546\) 0 0
\(547\) −110782. −0.370249 −0.185124 0.982715i \(-0.559269\pi\)
−0.185124 + 0.982715i \(0.559269\pi\)
\(548\) 8300.91 8300.91i 0.0276417 0.0276417i
\(549\) 0 0
\(550\) 23942.4i 0.0791483i
\(551\) −71739.8 + 71739.8i −0.236296 + 0.236296i
\(552\) 0 0
\(553\) −127310. 127310.i −0.416304 0.416304i
\(554\) 388667. + 388667.i 1.26636 + 1.26636i
\(555\) 0 0
\(556\) 4702.73i 0.0152125i
\(557\) −141690. 141690.i −0.456696 0.456696i 0.440873 0.897569i \(-0.354669\pi\)
−0.897569 + 0.440873i \(0.854669\pi\)
\(558\) 0 0
\(559\) 139047. + 351577.i 0.444977 + 1.12512i
\(560\) 152432. 0.486072
\(561\) 0 0
\(562\) −115725. −0.366400
\(563\) 301845.i 0.952287i 0.879368 + 0.476143i \(0.157966\pi\)
−0.879368 + 0.476143i \(0.842034\pi\)
\(564\) 0 0
\(565\) −358675. + 358675.i −1.12358 + 1.12358i
\(566\) −440345. 440345.i −1.37455 1.37455i
\(567\) 0 0
\(568\) 59272.6 0.183720
\(569\) 149750.i 0.462532i −0.972891 0.231266i \(-0.925713\pi\)
0.972891 0.231266i \(-0.0742867\pi\)
\(570\) 0 0
\(571\) 477531.i 1.46463i 0.680964 + 0.732317i \(0.261561\pi\)
−0.680964 + 0.732317i \(0.738439\pi\)
\(572\) −4471.38 + 10322.1i −0.0136663 + 0.0315484i
\(573\) 0 0
\(574\) −86992.5 + 86992.5i −0.264033 + 0.264033i
\(575\) −41554.5 −0.125685
\(576\) 0 0
\(577\) 108891. 108891.i 0.327069 0.327069i −0.524402 0.851471i \(-0.675711\pi\)
0.851471 + 0.524402i \(0.175711\pi\)
\(578\) −26205.2 + 26205.2i −0.0784391 + 0.0784391i
\(579\) 0 0
\(580\) 19448.3 + 19448.3i 0.0578131 + 0.0578131i
\(581\) 198731. 0.588726
\(582\) 0 0
\(583\) −260754. 260754.i −0.767175 0.767175i
\(584\) 103076.i 0.302225i
\(585\) 0 0
\(586\) 129813. 0.378026
\(587\) −379654. + 379654.i −1.10182 + 1.10182i −0.107632 + 0.994191i \(0.534327\pi\)
−0.994191 + 0.107632i \(0.965673\pi\)
\(588\) 0 0
\(589\) 95522.9i 0.275345i
\(590\) −105464. + 105464.i −0.302970 + 0.302970i
\(591\) 0 0
\(592\) −108261. 108261.i −0.308908 0.308908i
\(593\) −20445.7 20445.7i −0.0581423 0.0581423i 0.677438 0.735580i \(-0.263090\pi\)
−0.735580 + 0.677438i \(0.763090\pi\)
\(594\) 0 0
\(595\) 155180.i 0.438330i
\(596\) −18057.1 18057.1i −0.0508343 0.0508343i
\(597\) 0 0
\(598\) −378904. 164135.i −1.05956 0.458986i
\(599\) −45314.1 −0.126293 −0.0631466 0.998004i \(-0.520114\pi\)
−0.0631466 + 0.998004i \(0.520114\pi\)
\(600\) 0 0
\(601\) −71319.6 −0.197451 −0.0987256 0.995115i \(-0.531477\pi\)
−0.0987256 + 0.995115i \(0.531477\pi\)
\(602\) 197785.i 0.545758i
\(603\) 0 0
\(604\) −24591.1 + 24591.1i −0.0674068 + 0.0674068i
\(605\) 141907. + 141907.i 0.387698 + 0.387698i
\(606\) 0 0
\(607\) 37835.4 0.102688 0.0513441 0.998681i \(-0.483649\pi\)
0.0513441 + 0.998681i \(0.483649\pi\)
\(608\) 7839.41i 0.0212069i
\(609\) 0 0
\(610\) 665748.i 1.78916i
\(611\) −81115.5 35138.0i −0.217281 0.0941226i
\(612\) 0 0
\(613\) 269915. 269915.i 0.718301 0.718301i −0.249956 0.968257i \(-0.580416\pi\)
0.968257 + 0.249956i \(0.0804162\pi\)
\(614\) −213212. −0.565555
\(615\) 0 0
\(616\) −79685.7 + 79685.7i −0.210000 + 0.210000i
\(617\) −128460. + 128460.i −0.337440 + 0.337440i −0.855403 0.517963i \(-0.826690\pi\)
0.517963 + 0.855403i \(0.326690\pi\)
\(618\) 0 0
\(619\) −78942.1 78942.1i −0.206029 0.206029i 0.596548 0.802577i \(-0.296538\pi\)
−0.802577 + 0.596548i \(0.796538\pi\)
\(620\) 25895.8 0.0673668
\(621\) 0 0
\(622\) 228478. + 228478.i 0.590560 + 0.590560i
\(623\) 255265.i 0.657680i
\(624\) 0 0
\(625\) 429328. 1.09908
\(626\) −59747.7 + 59747.7i −0.152466 + 0.152466i
\(627\) 0 0
\(628\) 13394.6i 0.0339633i
\(629\) −110212. + 110212.i −0.278567 + 0.278567i
\(630\) 0 0
\(631\) −138220. 138220.i −0.347146 0.347146i 0.511899 0.859045i \(-0.328942\pi\)
−0.859045 + 0.511899i \(0.828942\pi\)
\(632\) −367729. 367729.i −0.920649 0.920649i
\(633\) 0 0
\(634\) 244162.i 0.607435i
\(635\) 490850. + 490850.i 1.21731 + 1.21731i
\(636\) 0 0
\(637\) 120304. + 304187.i 0.296484 + 0.749655i
\(638\) −451451. −1.10910
\(639\) 0 0
\(640\) 461162. 1.12588
\(641\) 369011.i 0.898098i 0.893507 + 0.449049i \(0.148237\pi\)
−0.893507 + 0.449049i \(0.851763\pi\)
\(642\) 0 0
\(643\) 111450. 111450.i 0.269561 0.269561i −0.559362 0.828923i \(-0.688954\pi\)
0.828923 + 0.559362i \(0.188954\pi\)
\(644\) 7222.11 + 7222.11i 0.0174138 + 0.0174138i
\(645\) 0 0
\(646\) −86340.2 −0.206894
\(647\) 401104.i 0.958184i 0.877765 + 0.479092i \(0.159034\pi\)
−0.877765 + 0.479092i \(0.840966\pi\)
\(648\) 0 0
\(649\) 115751.i 0.274811i
\(650\) −44292.5 19186.8i −0.104834 0.0454125i
\(651\) 0 0
\(652\) 18350.8 18350.8i 0.0431677 0.0431677i
\(653\) 341135. 0.800018 0.400009 0.916511i \(-0.369007\pi\)
0.400009 + 0.916511i \(0.369007\pi\)
\(654\) 0 0
\(655\) −57671.3 + 57671.3i −0.134424 + 0.134424i
\(656\) −263776. + 263776.i −0.612954 + 0.612954i
\(657\) 0 0
\(658\) 32700.0 + 32700.0i 0.0755259 + 0.0755259i
\(659\) −32739.6 −0.0753881 −0.0376941 0.999289i \(-0.512001\pi\)
−0.0376941 + 0.999289i \(0.512001\pi\)
\(660\) 0 0
\(661\) 20237.5 + 20237.5i 0.0463184 + 0.0463184i 0.729887 0.683568i \(-0.239573\pi\)
−0.683568 + 0.729887i \(0.739573\pi\)
\(662\) 395884.i 0.903342i
\(663\) 0 0
\(664\) 574028. 1.30196
\(665\) −31040.5 + 31040.5i −0.0701916 + 0.0701916i
\(666\) 0 0
\(667\) 783542.i 1.76121i
\(668\) −13192.1 + 13192.1i −0.0295639 + 0.0295639i
\(669\) 0 0
\(670\) 184547. + 184547.i 0.411109 + 0.411109i
\(671\) −365342. 365342.i −0.811437 0.811437i
\(672\) 0 0
\(673\) 151692.i 0.334914i 0.985879 + 0.167457i \(0.0535556\pi\)
−0.985879 + 0.167457i \(0.946444\pi\)
\(674\) 254527. + 254527.i 0.560291 + 0.560291i
\(675\) 0 0
\(676\) −15512.3 16543.8i −0.0339455 0.0362027i
\(677\) 753102. 1.64315 0.821574 0.570102i \(-0.193096\pi\)
0.821574 + 0.570102i \(0.193096\pi\)
\(678\) 0 0
\(679\) −69305.3 −0.150323
\(680\) 448231.i 0.969358i
\(681\) 0 0
\(682\) −300558. + 300558.i −0.646189 + 0.646189i
\(683\) −50909.1 50909.1i −0.109132 0.109132i 0.650432 0.759564i \(-0.274588\pi\)
−0.759564 + 0.650432i \(0.774588\pi\)
\(684\) 0 0
\(685\) −389666. −0.830445
\(686\) 383397.i 0.814705i
\(687\) 0 0
\(688\) 599717.i 1.26698i
\(689\) 691347. 273424.i 1.45632 0.575967i
\(690\) 0 0
\(691\) −281346. + 281346.i −0.589230 + 0.589230i −0.937423 0.348193i \(-0.886795\pi\)
0.348193 + 0.937423i \(0.386795\pi\)
\(692\) −7329.40 −0.0153058
\(693\) 0 0
\(694\) 284764. 284764.i 0.591243 0.591243i
\(695\) −110379. + 110379.i −0.228516 + 0.228516i
\(696\) 0 0
\(697\) 268530. + 268530.i 0.552749 + 0.552749i
\(698\) 101079. 0.207468
\(699\) 0 0
\(700\) 844.238 + 844.238i 0.00172293 + 0.00172293i
\(701\) 212436.i 0.432307i −0.976359 0.216154i \(-0.930649\pi\)
0.976359 0.216154i \(-0.0693512\pi\)
\(702\) 0 0
\(703\) 44091.4 0.0892161
\(704\) −229571. + 229571.i −0.463204 + 0.463204i
\(705\) 0 0
\(706\) 369410.i 0.741138i
\(707\) −99019.1 + 99019.1i −0.198098 + 0.198098i
\(708\) 0 0
\(709\) 27433.3 + 27433.3i 0.0545741 + 0.0545741i 0.733867 0.679293i \(-0.237713\pi\)
−0.679293 + 0.733867i \(0.737713\pi\)
\(710\) 72647.8 + 72647.8i 0.144114 + 0.144114i
\(711\) 0 0
\(712\) 737324.i 1.45445i
\(713\) −521650. 521650.i −1.02613 1.02613i
\(714\) 0 0
\(715\) 347222. 137324.i 0.679196 0.268618i
\(716\) 22846.1 0.0445641
\(717\) 0 0
\(718\) −974903. −1.89109
\(719\) 537471.i 1.03967i 0.854265 + 0.519837i \(0.174007\pi\)
−0.854265 + 0.519837i \(0.825993\pi\)
\(720\) 0 0
\(721\) −85655.6 + 85655.6i −0.164773 + 0.164773i
\(722\) −360369. 360369.i −0.691310 0.691310i
\(723\) 0 0
\(724\) −1854.00 −0.00353698
\(725\) 91593.1i 0.174256i
\(726\) 0 0
\(727\) 330967.i 0.626204i 0.949720 + 0.313102i \(0.101368\pi\)
−0.949720 + 0.313102i \(0.898632\pi\)
\(728\) −83557.4 211274.i −0.157660 0.398642i
\(729\) 0 0
\(730\) −126335. + 126335.i −0.237071 + 0.237071i
\(731\) −610526. −1.14254
\(732\) 0 0
\(733\) −164154. + 164154.i −0.305522 + 0.305522i −0.843170 0.537648i \(-0.819313\pi\)
0.537648 + 0.843170i \(0.319313\pi\)
\(734\) −14394.0 + 14394.0i −0.0267171 + 0.0267171i
\(735\) 0 0
\(736\) 42811.0 + 42811.0i 0.0790314 + 0.0790314i
\(737\) −202547. −0.372899
\(738\) 0 0
\(739\) −268903. 268903.i −0.492387 0.492387i 0.416670 0.909058i \(-0.363197\pi\)
−0.909058 + 0.416670i \(0.863197\pi\)
\(740\) 11953.0i 0.0218279i
\(741\) 0 0
\(742\) −388927. −0.706416
\(743\) 397022. 397022.i 0.719180 0.719180i −0.249258 0.968437i \(-0.580187\pi\)
0.968437 + 0.249258i \(0.0801866\pi\)
\(744\) 0 0
\(745\) 847648.i 1.52722i
\(746\) 334590. 334590.i 0.601223 0.601223i
\(747\) 0 0
\(748\) −12844.7 12844.7i −0.0229573 0.0229573i
\(749\) 147761. + 147761.i 0.263388 + 0.263388i
\(750\) 0 0
\(751\) 100943.i 0.178976i −0.995988 0.0894882i \(-0.971477\pi\)
0.995988 0.0894882i \(-0.0285231\pi\)
\(752\) 99151.9 + 99151.9i 0.175334 + 0.175334i
\(753\) 0 0
\(754\) 361782. 835168.i 0.636361 1.46903i
\(755\) 1.15437e6 2.02512
\(756\) 0 0
\(757\) −418346. −0.730035 −0.365018 0.931001i \(-0.618937\pi\)
−0.365018 + 0.931001i \(0.618937\pi\)
\(758\) 460273.i 0.801083i
\(759\) 0 0
\(760\) −89659.4 + 89659.4i −0.155227 + 0.155227i
\(761\) 70616.1 + 70616.1i 0.121937 + 0.121937i 0.765442 0.643505i \(-0.222520\pi\)
−0.643505 + 0.765442i \(0.722520\pi\)
\(762\) 0 0
\(763\) 499510. 0.858015
\(764\) 9132.06i 0.0156452i
\(765\) 0 0
\(766\) 175588.i 0.299253i
\(767\) 214134. + 92759.6i 0.363995 + 0.157677i
\(768\) 0 0
\(769\) −642589. + 642589.i −1.08663 + 1.08663i −0.0907545 + 0.995873i \(0.528928\pi\)
−0.995873 + 0.0907545i \(0.971072\pi\)
\(770\) −195335. −0.329456
\(771\) 0 0
\(772\) 29133.9 29133.9i 0.0488836 0.0488836i
\(773\) −152962. + 152962.i −0.255990 + 0.255990i −0.823421 0.567431i \(-0.807937\pi\)
0.567431 + 0.823421i \(0.307937\pi\)
\(774\) 0 0
\(775\) −60978.9 60978.9i −0.101526 0.101526i
\(776\) −200186. −0.332438
\(777\) 0 0
\(778\) −393412. 393412.i −0.649963 0.649963i
\(779\) 107428.i 0.177028i
\(780\) 0 0
\(781\) −79733.8 −0.130720
\(782\) 471503. 471503.i 0.771030 0.771030i
\(783\) 0 0
\(784\) 518878.i 0.844177i
\(785\) 314387. 314387.i 0.510183 0.510183i
\(786\) 0 0
\(787\) 210034. + 210034.i 0.339110 + 0.339110i 0.856032 0.516922i \(-0.172922\pi\)
−0.516922 + 0.856032i \(0.672922\pi\)
\(788\) 2997.68 + 2997.68i 0.00482762 + 0.00482762i
\(789\) 0 0
\(790\) 901420.i 1.44435i
\(791\) −293581. 293581.i −0.469219 0.469219i
\(792\) 0 0
\(793\) 968645. 383093.i 1.54035 0.609197i
\(794\) 710510. 1.12701
\(795\) 0 0
\(796\) 37806.4 0.0596677
\(797\) 508211.i 0.800069i 0.916500 + 0.400035i \(0.131002\pi\)
−0.916500 + 0.400035i \(0.868998\pi\)
\(798\) 0 0
\(799\) 100939. 100939.i 0.158112 0.158112i
\(800\) 5004.44 + 5004.44i 0.00781944 + 0.00781944i
\(801\) 0 0
\(802\) 765719. 1.19048
\(803\) 138658.i 0.215037i
\(804\) 0 0
\(805\) 339024.i 0.523165i
\(806\) −315161. 796880.i −0.485135 1.22666i
\(807\) 0 0
\(808\) −286013. + 286013.i −0.438090 + 0.438090i
\(809\) 302513. 0.462218 0.231109 0.972928i \(-0.425765\pi\)
0.231109 + 0.972928i \(0.425765\pi\)
\(810\) 0 0
\(811\) 261828. 261828.i 0.398083 0.398083i −0.479473 0.877557i \(-0.659172\pi\)
0.877557 + 0.479473i \(0.159172\pi\)
\(812\) −15918.7 + 15918.7i −0.0241433 + 0.0241433i
\(813\) 0 0
\(814\) 138731. + 138731.i 0.209376 + 0.209376i
\(815\) −861431. −1.29690
\(816\) 0 0
\(817\) 122123. + 122123.i 0.182959 + 0.182959i
\(818\) 15651.8i 0.0233915i
\(819\) 0 0
\(820\) −29123.2 −0.0433123
\(821\) 141594. 141594.i 0.210068 0.210068i −0.594228 0.804296i \(-0.702542\pi\)
0.804296 + 0.594228i \(0.202542\pi\)
\(822\) 0 0
\(823\) 1.05798e6i 1.56199i 0.624540 + 0.780993i \(0.285287\pi\)
−0.624540 + 0.780993i \(0.714713\pi\)
\(824\) −247413. + 247413.i −0.364392 + 0.364392i
\(825\) 0 0
\(826\) −86323.7 86323.7i −0.126523 0.126523i
\(827\) −95483.6 95483.6i −0.139610 0.139610i 0.633848 0.773458i \(-0.281475\pi\)
−0.773458 + 0.633848i \(0.781475\pi\)
\(828\) 0 0
\(829\) 623513.i 0.907270i −0.891188 0.453635i \(-0.850127\pi\)
0.891188 0.453635i \(-0.149873\pi\)
\(830\) 703561. + 703561.i 1.02128 + 1.02128i
\(831\) 0 0
\(832\) −240726. 608671.i −0.347757 0.879298i
\(833\) −528231. −0.761261
\(834\) 0 0
\(835\) 619271. 0.888193
\(836\) 5138.64i 0.00735251i
\(837\) 0 0
\(838\) −273686. + 273686.i −0.389731 + 0.389731i
\(839\) 666830. + 666830.i 0.947308 + 0.947308i 0.998680 0.0513717i \(-0.0163593\pi\)
−0.0513717 + 0.998680i \(0.516359\pi\)
\(840\) 0 0
\(841\) 1.01978e6 1.44182
\(842\) 89085.5i 0.125656i
\(843\) 0 0
\(844\) 60084.3i 0.0843482i
\(845\) −24210.4 + 752395.i −0.0339070 + 1.05374i
\(846\) 0 0
\(847\) −116153. + 116153.i −0.161907 + 0.161907i
\(848\) −1.17929e6 −1.63995
\(849\) 0 0
\(850\) 55117.0 55117.0i 0.0762864 0.0762864i
\(851\) −240783. + 240783.i −0.332481 + 0.332481i
\(852\) 0 0
\(853\) −515777. 515777.i −0.708866 0.708866i 0.257431 0.966297i \(-0.417124\pi\)
−0.966297 + 0.257431i \(0.917124\pi\)
\(854\) −544925. −0.747172
\(855\) 0 0
\(856\) 426802. + 426802.i 0.582478 + 0.582478i
\(857\) 489846.i 0.666957i 0.942758 + 0.333479i \(0.108222\pi\)
−0.942758 + 0.333479i \(0.891778\pi\)
\(858\) 0 0
\(859\) 1.37694e6 1.86607 0.933037 0.359780i \(-0.117148\pi\)
0.933037 + 0.359780i \(0.117148\pi\)
\(860\) 33107.0 33107.0i 0.0447634 0.0447634i
\(861\) 0 0
\(862\) 243649.i 0.327907i
\(863\) 715898. 715898.i 0.961235 0.961235i −0.0380415 0.999276i \(-0.512112\pi\)
0.999276 + 0.0380415i \(0.0121119\pi\)
\(864\) 0 0
\(865\) 172030. + 172030.i 0.229918 + 0.229918i
\(866\) −85139.8 85139.8i −0.113526 0.113526i
\(867\) 0 0
\(868\) 21196.1i 0.0281330i
\(869\) 494672. + 494672.i 0.655055 + 0.655055i
\(870\) 0 0
\(871\) 162316. 374705.i 0.213957 0.493916i
\(872\) 1.44282e6 1.89748
\(873\) 0 0
\(874\) −188629. −0.246937
\(875\) 315756.i 0.412417i
\(876\) 0 0
\(877\) −123217. + 123217.i −0.160203 + 0.160203i −0.782657 0.622454i \(-0.786136\pi\)
0.622454 + 0.782657i \(0.286136\pi\)
\(878\) −1.02774e6 1.02774e6i −1.33320 1.33320i
\(879\) 0 0
\(880\) −592288. −0.764834
\(881\) 863892.i 1.11303i −0.830837 0.556516i \(-0.812138\pi\)
0.830837 0.556516i \(-0.187862\pi\)
\(882\) 0 0
\(883\) 268829.i 0.344790i 0.985028 + 0.172395i \(0.0551505\pi\)
−0.985028 + 0.172395i \(0.944850\pi\)
\(884\) 34055.7 13468.8i 0.0435797 0.0172355i
\(885\) 0 0
\(886\) −65280.2 + 65280.2i −0.0831599 + 0.0831599i
\(887\) −1.04691e6 −1.33065 −0.665323 0.746556i \(-0.731706\pi\)
−0.665323 + 0.746556i \(0.731706\pi\)
\(888\) 0 0
\(889\) −401768. + 401768.i −0.508361 + 0.508361i
\(890\) −903706. + 903706.i −1.14090 + 1.14090i
\(891\) 0 0
\(892\) 756.886 + 756.886i 0.000951263 + 0.000951263i
\(893\) −40381.5 −0.0506384
\(894\) 0 0
\(895\) −536226. 536226.i −0.669425 0.669425i
\(896\) 377468.i 0.470180i
\(897\) 0 0
\(898\) −308691. −0.382800
\(899\) 1.14980e6 1.14980e6i 1.42267 1.42267i
\(900\) 0 0
\(901\) 1.20055e6i 1.47887i
\(902\) 338016. 338016.i 0.415456 0.415456i
\(903\) 0 0
\(904\) −847999. 847999.i −1.03767 1.03767i
\(905\) 43515.7 + 43515.7i 0.0531311 + 0.0531311i
\(906\) 0 0
\(907\) 293077.i 0.356260i 0.984007 + 0.178130i \(0.0570047\pi\)
−0.984007 + 0.178130i \(0.942995\pi\)
\(908\) 4851.68 + 4851.68i 0.00588464 + 0.00588464i
\(909\) 0 0
\(910\) 156536. 361362.i 0.189031 0.436374i
\(911\) 1.22775e6 1.47936 0.739681 0.672958i \(-0.234976\pi\)
0.739681 + 0.672958i \(0.234976\pi\)
\(912\) 0 0
\(913\) −772185. −0.926361
\(914\) 544974.i 0.652354i
\(915\) 0 0
\(916\) 7604.58 7604.58i 0.00906326 0.00906326i
\(917\) −47204.8 47204.8i −0.0561368 0.0561368i
\(918\) 0 0
\(919\) −952406. −1.12769 −0.563847 0.825879i \(-0.690679\pi\)
−0.563847 + 0.825879i \(0.690679\pi\)
\(920\) 979259.i 1.15697i
\(921\) 0 0
\(922\) 1.27948e6i 1.50513i
\(923\) 63896.6 147504.i 0.0750023 0.173142i
\(924\) 0 0
\(925\) −28146.6 + 28146.6i −0.0328960 + 0.0328960i
\(926\) 860015. 1.00296
\(927\) 0 0
\(928\) −94362.5 + 94362.5i −0.109573 + 0.109573i
\(929\) −865268. + 865268.i −1.00258 + 1.00258i −0.00258353 + 0.999997i \(0.500822\pi\)
−0.999997 + 0.00258353i \(0.999178\pi\)
\(930\) 0 0
\(931\) 105662. + 105662.i 0.121904 + 0.121904i
\(932\) −57807.8 −0.0665511
\(933\) 0 0
\(934\) 259629. + 259629.i 0.297619 + 0.297619i
\(935\) 602963.i 0.689712i
\(936\) 0 0
\(937\) −829106. −0.944345 −0.472173 0.881506i \(-0.656530\pi\)
−0.472173 + 0.881506i \(0.656530\pi\)
\(938\) −151054. + 151054.i −0.171683 + 0.171683i
\(939\) 0 0
\(940\) 10947.2i 0.0123894i
\(941\) −1.18512e6 + 1.18512e6i −1.33839 + 1.33839i −0.440774 + 0.897618i \(0.645296\pi\)
−0.897618 + 0.440774i \(0.854704\pi\)
\(942\) 0 0
\(943\) 586664. + 586664.i 0.659729 + 0.659729i
\(944\) −261748. 261748.i −0.293724 0.293724i
\(945\) 0 0
\(946\) 768509.i 0.858750i
\(947\) −94860.4 94860.4i −0.105775 0.105775i 0.652238 0.758014i \(-0.273830\pi\)
−0.758014 + 0.652238i \(0.773830\pi\)
\(948\) 0 0
\(949\) 256512. + 111117.i 0.284823 + 0.123381i
\(950\) −22050.0 −0.0244322
\(951\) 0 0
\(952\) 366884. 0.404813
\(953\) 313080.i 0.344723i 0.985034 + 0.172361i \(0.0551397\pi\)
−0.985034 + 0.172361i \(0.944860\pi\)
\(954\) 0 0
\(955\) −214341. + 214341.i −0.235016 + 0.235016i
\(956\) 57682.4 + 57682.4i 0.0631142 + 0.0631142i
\(957\) 0 0
\(958\) 1.68358e6 1.83444
\(959\) 318947.i 0.346802i
\(960\) 0 0
\(961\) 607464.i 0.657770i
\(962\) −367824. + 145472.i −0.397456 + 0.157192i
\(963\) 0 0
\(964\) 51075.1 51075.1i 0.0549610 0.0549610i
\(965\) −1.36762e6 −1.46862
\(966\) 0 0
\(967\) 359201. 359201.i 0.384136 0.384136i −0.488454 0.872590i \(-0.662439\pi\)
0.872590 + 0.488454i \(0.162439\pi\)
\(968\) −335504. + 335504.i −0.358053 + 0.358053i
\(969\) 0 0
\(970\) −245359. 245359.i −0.260771 0.260771i
\(971\) 1.68455e6 1.78668 0.893338 0.449385i \(-0.148357\pi\)
0.893338 + 0.449385i \(0.148357\pi\)
\(972\) 0 0
\(973\) −90346.8 90346.8i −0.0954305 0.0954305i
\(974\) 1.08228e6i 1.14083i
\(975\) 0 0
\(976\) −1.65230e6 −1.73456
\(977\) −857213. + 857213.i −0.898048 + 0.898048i −0.995263 0.0972150i \(-0.969007\pi\)
0.0972150 + 0.995263i \(0.469007\pi\)
\(978\) 0 0
\(979\) 991852.i 1.03486i
\(980\) 28644.4 28644.4i 0.0298254 0.0298254i
\(981\) 0 0
\(982\) 76718.1 + 76718.1i 0.0795564 + 0.0795564i
\(983\) −821373. 821373.i −0.850029 0.850029i 0.140107 0.990136i \(-0.455255\pi\)
−0.990136 + 0.140107i \(0.955255\pi\)
\(984\) 0 0
\(985\) 140719.i 0.145037i
\(986\) 1.03927e6 + 1.03927e6i 1.06899 + 1.06899i
\(987\) 0 0
\(988\) −9506.29 4117.98i −0.00973861 0.00421861i
\(989\) −1.33383e6 −1.36366
\(990\) 0 0
\(991\) −1.01521e6 −1.03374 −0.516868 0.856065i \(-0.672902\pi\)
−0.516868 + 0.856065i \(0.672902\pi\)
\(992\) 125645.i 0.127680i
\(993\) 0 0
\(994\) −59463.3 + 59463.3i −0.0601834 + 0.0601834i
\(995\) −887364. 887364.i −0.896305 0.896305i
\(996\) 0 0
\(997\) 590999. 0.594561 0.297280 0.954790i \(-0.403920\pi\)
0.297280 + 0.954790i \(0.403920\pi\)
\(998\) 421352.i 0.423043i
\(999\) 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 117.5.j.b.109.3 20
3.2 odd 2 39.5.g.a.31.8 20
13.8 odd 4 inner 117.5.j.b.73.3 20
39.8 even 4 39.5.g.a.34.8 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.5.g.a.31.8 20 3.2 odd 2
39.5.g.a.34.8 yes 20 39.8 even 4
117.5.j.b.73.3 20 13.8 odd 4 inner
117.5.j.b.109.3 20 1.1 even 1 trivial