Properties

Label 117.5.j.b.73.3
Level $117$
Weight $5$
Character 117.73
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 73.3
Root \(-2.89776 - 2.89776i\) of defining polynomial
Character \(\chi\) \(=\) 117.73
Dual form 117.5.j.b.109.3

$q$-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 \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

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

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 0.245066 0.969506i \(-0.421190\pi\)
−0.969506 + 0.245066i \(0.921190\pi\)
\(3\) 0 0
\(4\) 0.794047i 0.0496279i
\(5\) −18.6373 18.6373i −0.745491 0.745491i 0.228138 0.973629i \(-0.426736\pi\)
−0.973629 + 0.228138i \(0.926736\pi\)
\(6\) 0 0
\(7\) −15.2549 + 15.2549i −0.311324 + 0.311324i −0.845422 0.534098i \(-0.820651\pi\)
0.534098 + 0.845422i \(0.320651\pi\)
\(8\) −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 −0.418396 0.908265i \(-0.637408\pi\)
0.908265 + 0.418396i \(0.137408\pi\)
\(12\) 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 \(0.156524\pi\)
−0.881515 + 0.472156i \(0.843476\pi\)
\(18\) 0 0
\(19\) −54.5892 54.5892i −0.151217 0.151217i 0.627445 0.778661i \(-0.284101\pi\)
−0.778661 + 0.627445i \(0.784101\pi\)
\(20\) 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 \(0.190560\pi\)
−0.826090 + 0.563538i \(0.809440\pi\)
\(24\) 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 0.996306 0.0858748i \(-0.0273685\pi\)
−0.0858748 + 0.996306i \(0.527368\pi\)
\(32\) −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 −0.839049 0.544055i \(-0.816888\pi\)
0.544055 + 0.839049i \(0.316888\pi\)
\(38\) 316.373i 0.219095i
\(39\) 0 0
\(40\) 1642.44 1.02652
\(41\) −983.966 983.966i −0.585346 0.585346i 0.351022 0.936367i \(-0.385834\pi\)
−0.936367 + 0.351022i \(0.885834\pi\)
\(42\) 0 0
\(43\) 2237.13i 1.20991i 0.796258 + 0.604957i \(0.206810\pi\)
−0.796258 + 0.604957i \(0.793190\pi\)
\(44\) 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 −0.618415 0.785852i \(-0.712225\pi\)
0.785852 + 0.618415i \(0.212225\pi\)
\(48\) 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 −0.833306 0.552812i \(-0.813555\pi\)
0.552812 + 0.833306i \(0.313555\pi\)
\(60\) 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 0.490711 0.871322i \(-0.336737\pi\)
−0.871322 + 0.490711i \(0.836737\pi\)
\(68\) −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 0.637241 0.770664i \(-0.280075\pi\)
−0.770664 + 0.637241i \(0.780075\pi\)
\(72\) 0 0
\(73\) −1169.63 + 1169.63i −0.219485 + 0.219485i −0.808281 0.588797i \(-0.799602\pi\)
0.588797 + 0.808281i \(0.299602\pi\)
\(74\) 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 0.0530714 0.998591i \(-0.483099\pi\)
−0.998591 + 0.0530714i \(0.983099\pi\)
\(84\) 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 −0.0579425 + 0.998320i \(0.518454\pi\)
−0.998320 + 0.0579425i \(0.981546\pi\)
\(90\) 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 0.817440 0.576014i \(-0.195393\pi\)
−0.576014 + 0.817440i \(0.695393\pi\)
\(98\) 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 \(0.103063\pi\)
−0.948039 + 0.318154i \(0.896937\pi\)
\(102\) 0 0
\(103\) 5614.96i 0.529264i 0.964350 + 0.264632i \(0.0852504\pi\)
−0.964350 + 0.264632i \(0.914750\pi\)
\(104\) 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 −0.847980 0.530029i \(-0.822181\pi\)
−0.530029 0.847980i \(-0.677819\pi\)
\(110\) 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 \(0.304061\pi\)
−0.577417 + 0.816449i \(0.695939\pi\)
\(128\) −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 −0.371467 0.928446i \(-0.621145\pi\)
0.928446 + 0.371467i \(0.121145\pi\)
\(138\) 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 0.999697 + 0.0246105i \(0.00783457\pi\)
0.0246105 + 0.999697i \(0.492165\pi\)
\(150\) 0 0
\(151\) −30969.3 + 30969.3i −1.35824 + 1.35824i −0.482161 + 0.876082i \(0.660148\pi\)
−0.876082 + 0.482161i \(0.839852\pi\)
\(152\) 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 −0.122626 0.992453i \(-0.539132\pi\)
0.992453 + 0.122626i \(0.0391317\pi\)
\(164\) 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 −0.939168 0.343458i \(-0.888402\pi\)
0.343458 + 0.939168i \(0.388402\pi\)
\(168\) 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 \(0.0492818\pi\)
−0.988039 + 0.154206i \(0.950718\pi\)
\(174\) 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 \(-0.851787\pi\)
0.893541 0.448982i \(-0.148213\pi\)
\(180\) 0 0
\(181\) 2334.88i 0.0712700i 0.999365 + 0.0356350i \(0.0113454\pi\)
−0.999365 + 0.0356350i \(0.988655\pi\)
\(182\) −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 −0.0148869 0.999889i \(-0.504739\pi\)
0.999889 + 0.0148869i \(0.00473884\pi\)
\(194\) 13165.0i 0.349797i
\(195\) 0 0
\(196\) −1536.94 −0.0400078
\(197\) −3775.20 3775.20i −0.0972763 0.0972763i 0.656794 0.754070i \(-0.271912\pi\)
−0.754070 + 0.656794i \(0.771912\pi\)
\(198\) 0 0
\(199\) 47612.3i 1.20230i −0.799136 0.601151i \(-0.794709\pi\)
0.799136 0.601151i \(-0.205291\pi\)
\(200\) −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 0.697458 0.716626i \(-0.254314\pi\)
−0.716626 + 0.697458i \(0.754314\pi\)
\(224\) 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 0.645329 0.763905i \(-0.276720\pi\)
−0.763905 + 0.645329i \(0.776720\pi\)
\(228\) 0 0
\(229\) 9576.99 9576.99i 0.182624 0.182624i −0.609874 0.792498i \(-0.708780\pi\)
0.792498 + 0.609874i \(0.208780\pi\)
\(230\) −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 \(0.233920\pi\)
−0.741910 + 0.670500i \(0.766080\pi\)
\(234\) 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 −0.945170 0.326578i \(-0.894105\pi\)
−0.326578 0.945170i \(-0.605895\pi\)
\(240\) 0 0
\(241\) 64322.5 64322.5i 1.10746 1.10746i 0.113978 0.993483i \(-0.463641\pi\)
0.993483 0.113978i \(-0.0363593\pi\)
\(242\) 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 \(0.127945\pi\)
−0.920300 + 0.391213i \(0.872055\pi\)
\(252\) 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 \(0.214479\pi\)
−0.781453 + 0.623964i \(0.785521\pi\)
\(258\) 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 −0.0326374 + 0.999467i \(0.510391\pi\)
−0.999467 + 0.0326374i \(0.989609\pi\)
\(272\) 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 \(0.338501\pi\)
−0.485876 + 0.874028i \(0.661499\pi\)
\(278\) −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 −0.569268 0.822152i \(-0.692773\pi\)
0.822152 + 0.569268i \(0.192773\pi\)
\(282\) 0 0
\(283\) 151960.i 1.89739i −0.316186 0.948697i \(-0.602402\pi\)
0.316186 0.948697i \(-0.397598\pi\)
\(284\) 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 −0.825423 0.564514i \(-0.809063\pi\)
0.564514 + 0.825423i \(0.309063\pi\)
\(294\) 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 −0.484469 0.874808i \(-0.660987\pi\)
0.874808 + 0.484469i \(0.160987\pi\)
\(308\) −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 \(0.133633\pi\)
−0.913162 + 0.407597i \(0.866367\pi\)
\(312\) 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 0.465699 0.884943i \(-0.345803\pi\)
−0.884943 + 0.465699i \(0.845803\pi\)
\(318\) 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 0.946418 0.322943i \(-0.104672\pi\)
−0.322943 + 0.946418i \(0.604672\pi\)
\(332\) 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 \(0.126387\pi\)
−0.922203 + 0.386706i \(0.873613\pi\)
\(338\) −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 −0.775069 0.631877i \(-0.782285\pi\)
0.631877 + 0.775069i \(0.282285\pi\)
\(350\) 6161.85i 0.0503008i
\(351\) 0 0
\(352\) −8512.19 −0.0686999
\(353\) −63740.5 63740.5i −0.511524 0.511524i 0.403469 0.914993i \(-0.367804\pi\)
−0.914993 + 0.403469i \(0.867804\pi\)
\(354\) 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 0.380377 0.924831i \(-0.375794\pi\)
0.924831 0.380377i \(-0.124206\pi\)
\(360\) 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 0.374378 0.927276i \(-0.377856\pi\)
−0.927276 + 0.374378i \(0.877856\pi\)
\(380\) −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 0.596255 0.802795i \(-0.296655\pi\)
−0.802795 + 0.596255i \(0.796655\pi\)
\(384\) 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 \(-0.851924\pi\)
0.893734 0.448597i \(-0.148076\pi\)
\(390\) 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 −0.979465 0.201614i \(-0.935381\pi\)
0.201614 + 0.979465i \(0.435381\pi\)
\(398\) −137969. + 137969.i −0.870996 + 0.870996i
\(399\) 0 0
\(400\) 18683.8i 0.116774i
\(401\) −132123. + 132123.i −0.821653 + 0.821653i −0.986345 0.164692i \(-0.947337\pi\)
0.164692 + 0.986345i \(0.447337\pi\)
\(402\) 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 0.715133 0.698988i \(-0.246366\pi\)
−0.698988 + 0.715133i \(0.746366\pi\)
\(410\) 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 0.749139 0.662413i \(-0.230468\pi\)
−0.662413 + 0.749139i \(0.730468\pi\)
\(422\) 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 0.811152 0.584835i \(-0.198841\pi\)
−0.584835 + 0.811152i \(0.698841\pi\)
\(432\) 0 0
\(433\) 29381.2i 0.156709i −0.996926 0.0783545i \(-0.975033\pi\)
0.996926 0.0783545i \(-0.0249666\pi\)
\(434\) 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 \(-0.628058\pi\)
0.391542 0.920160i \(-0.371942\pi\)
\(440\) 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 −0.562556 0.826759i \(-0.690182\pi\)
0.826759 + 0.562556i \(0.190182\pi\)
\(450\) 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 0.895436 0.445189i \(-0.146864\pi\)
−0.445189 + 0.895436i \(0.646864\pi\)
\(458\) −55503.7 −0.264601
\(459\) 0 0
\(460\) 8823.43 + 8823.43i 0.0416986 + 0.0416986i
\(461\) 220771. + 220771.i 1.03882 + 1.03882i 0.999215 + 0.0396051i \(0.0126100\pi\)
0.0396051 + 0.999215i \(0.487390\pi\)
\(462\) 0 0
\(463\) −148393. + 148393.i −0.692231 + 0.692231i −0.962722 0.270491i \(-0.912814\pi\)
0.270491 + 0.962722i \(0.412814\pi\)
\(464\) 352296. 1.63634
\(465\) 0 0
\(466\) 210962. 210962.i 0.971474 0.971474i
\(467\) 89596.5i 0.410826i 0.978675 + 0.205413i \(0.0658537\pi\)
−0.978675 + 0.205413i \(0.934146\pi\)
\(468\) 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 −0.318023 + 0.948083i \(0.603019\pi\)
−0.948083 + 0.318023i \(0.896981\pi\)
\(480\) 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 0.981065 0.193677i \(-0.0620414\pi\)
−0.193677 + 0.981065i \(0.562041\pi\)
\(488\) 271588. 271588.i 1.14044 1.14044i
\(489\) 0 0
\(490\) −209067. −0.870750
\(491\) 26475.0i 0.109818i 0.998491 + 0.0549088i \(0.0174868\pi\)
−0.998491 + 0.0549088i \(0.982513\pi\)
\(492\) 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 0.837862 0.545883i \(-0.183806\pi\)
−0.545883 + 0.837862i \(0.683806\pi\)
\(500\) −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 0.881428 0.472319i \(-0.156583\pi\)
−0.472319 + 0.881428i \(0.656583\pi\)
\(510\) 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 −0.347823 + 0.937560i \(0.613079\pi\)
−0.937560 + 0.347823i \(0.886921\pi\)
\(542\) 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 −0.897569 0.440873i \(-0.854669\pi\)
0.440873 + 0.897569i \(0.354669\pi\)
\(558\) 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 \(-0.842034\pi\)
0.879368 0.476143i \(-0.157966\pi\)
\(564\) 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 \(0.0742867\pi\)
−0.972891 + 0.231266i \(0.925713\pi\)
\(570\) 0 0
\(571\) 477531.i 1.46463i −0.680964 0.732317i \(-0.738439\pi\)
0.680964 0.732317i \(-0.261561\pi\)
\(572\) −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 0.851471 0.524402i \(-0.175711\pi\)
−0.524402 + 0.851471i \(0.675711\pi\)
\(578\) −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 −0.994191 0.107632i \(-0.965673\pi\)
−0.107632 0.994191i \(-0.534327\pi\)
\(588\) 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 −0.735580 0.677438i \(-0.763090\pi\)
0.677438 + 0.735580i \(0.263090\pi\)
\(594\) 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 0.968257 0.249956i \(-0.0804162\pi\)
−0.249956 + 0.968257i \(0.580416\pi\)
\(614\) −213212. −0.565555
\(615\) 0 0
\(616\) −79685.7 79685.7i −0.210000 0.210000i
\(617\) −128460. 128460.i −0.337440 0.337440i 0.517963 0.855403i \(-0.326690\pi\)
−0.855403 + 0.517963i \(0.826690\pi\)
\(618\) 0 0
\(619\) −78942.1 + 78942.1i −0.206029 + 0.206029i −0.802577 0.596548i \(-0.796538\pi\)
0.596548 + 0.802577i \(0.296538\pi\)
\(620\) 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 −0.859045 0.511899i \(-0.828942\pi\)
0.511899 + 0.859045i \(0.328942\pi\)
\(632\) −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 \(-0.851763\pi\)
0.893507 0.449049i \(-0.148237\pi\)
\(642\) 0 0
\(643\) 111450. + 111450.i 0.269561 + 0.269561i 0.828923 0.559362i \(-0.188954\pi\)
−0.559362 + 0.828923i \(0.688954\pi\)
\(644\) 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 \(-0.840966\pi\)
0.877765 0.479092i \(-0.159034\pi\)
\(648\) 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 −0.683568 0.729887i \(-0.739573\pi\)
0.729887 + 0.683568i \(0.239573\pi\)
\(662\) 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 \(-0.946444\pi\)
0.985879 0.167457i \(-0.0535556\pi\)
\(674\) 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 −0.759564 0.650432i \(-0.774588\pi\)
0.650432 + 0.759564i \(0.274588\pi\)
\(684\) 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 0.348193 0.937423i \(-0.386795\pi\)
−0.937423 + 0.348193i \(0.886795\pi\)
\(692\) −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 \(0.0693512\pi\)
−0.976359 + 0.216154i \(0.930649\pi\)
\(702\) 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 −0.679293 0.733867i \(-0.737713\pi\)
0.733867 + 0.679293i \(0.237713\pi\)
\(710\) 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 \(-0.825993\pi\)
0.854265 0.519837i \(-0.174007\pi\)
\(720\) 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 \(-0.898632\pi\)
0.949720 0.313102i \(-0.101368\pi\)
\(728\) −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 0.537648 0.843170i \(-0.319313\pi\)
−0.843170 + 0.537648i \(0.819313\pi\)
\(734\) −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 −0.909058 0.416670i \(-0.863197\pi\)
0.416670 + 0.909058i \(0.363197\pi\)
\(740\) 11953.0i 0.0218279i
\(741\) 0 0
\(742\) −388927. −0.706416
\(743\) 397022. + 397022.i 0.719180 + 0.719180i 0.968437 0.249258i \(-0.0801866\pi\)
−0.249258 + 0.968437i \(0.580187\pi\)
\(744\) 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 \(0.0285231\pi\)
−0.995988 + 0.0894882i \(0.971477\pi\)
\(752\) 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 −0.643505 0.765442i \(-0.722520\pi\)
0.765442 + 0.643505i \(0.222520\pi\)
\(762\) 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 −0.995873 0.0907545i \(-0.971072\pi\)
−0.0907545 0.995873i \(-0.528928\pi\)
\(770\) −195335. −0.329456
\(771\) 0 0
\(772\) 29133.9 + 29133.9i 0.0488836 + 0.0488836i
\(773\) −152962. 152962.i −0.255990 0.255990i 0.567431 0.823421i \(-0.307937\pi\)
−0.823421 + 0.567431i \(0.807937\pi\)
\(774\) 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 −0.516922 0.856032i \(-0.672922\pi\)
0.856032 + 0.516922i \(0.172922\pi\)
\(788\) 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 \(-0.868998\pi\)
0.916500 0.400035i \(-0.131002\pi\)
\(798\) 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 0.877557 0.479473i \(-0.159172\pi\)
−0.479473 + 0.877557i \(0.659172\pi\)
\(812\) −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 0.804296 0.594228i \(-0.202542\pi\)
−0.594228 + 0.804296i \(0.702542\pi\)
\(822\) 0 0
\(823\) 1.05798e6i 1.56199i −0.624540 0.780993i \(-0.714713\pi\)
0.624540 0.780993i \(-0.285287\pi\)
\(824\) −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 −0.773458 0.633848i \(-0.781475\pi\)
0.633848 + 0.773458i \(0.281475\pi\)
\(828\) 0 0
\(829\) 623513.i 0.907270i 0.891188 + 0.453635i \(0.149873\pi\)
−0.891188 + 0.453635i \(0.850127\pi\)
\(830\) 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 −0.0513717 0.998680i \(-0.516359\pi\)
0.998680 + 0.0513717i \(0.0163593\pi\)
\(840\) 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 −0.966297 0.257431i \(-0.917124\pi\)
0.257431 + 0.966297i \(0.417124\pi\)
\(854\) −544925. −0.747172
\(855\) 0 0
\(856\) 426802. 426802.i 0.582478 0.582478i
\(857\) 489846.i 0.666957i −0.942758 0.333479i \(-0.891778\pi\)
0.942758 0.333479i \(-0.108222\pi\)
\(858\) 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 0.999276 0.0380415i \(-0.0121119\pi\)
−0.0380415 + 0.999276i \(0.512112\pi\)
\(864\) 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 0.622454 0.782657i \(-0.286136\pi\)
−0.782657 + 0.622454i \(0.786136\pi\)
\(878\) −1.02774e6 + 1.02774e6i −1.33320 + 1.33320i
\(879\) 0 0
\(880\) −592288. −0.764834
\(881\) 863892.i 1.11303i 0.830837 + 0.556516i \(0.187862\pi\)
−0.830837 + 0.556516i \(0.812138\pi\)
\(882\) 0 0
\(883\) 268829.i 0.344790i −0.985028 0.172395i \(-0.944850\pi\)
0.985028 0.172395i \(-0.0551505\pi\)
\(884\) 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 \(-0.942995\pi\)
0.984007 0.178130i \(-0.0570047\pi\)
\(908\) 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 −0.999997 0.00258353i \(-0.999178\pi\)
−0.00258353 0.999997i \(-0.500822\pi\)
\(930\) 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 −0.897618 0.440774i \(-0.854704\pi\)
−0.440774 0.897618i \(-0.645296\pi\)
\(942\) 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 −0.758014 0.652238i \(-0.773830\pi\)
0.652238 + 0.758014i \(0.273830\pi\)
\(948\) 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 \(-0.944860\pi\)
0.985034 0.172361i \(-0.0551397\pi\)
\(954\) 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 0.872590 0.488454i \(-0.162439\pi\)
−0.488454 + 0.872590i \(0.662439\pi\)
\(968\) −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 0.0972150 0.995263i \(-0.469007\pi\)
−0.995263 + 0.0972150i \(0.969007\pi\)
\(978\) 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 −0.990136 0.140107i \(-0.955255\pi\)
0.140107 + 0.990136i \(0.455255\pi\)
\(984\) 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.73.3 20
3.2 odd 2 39.5.g.a.34.8 yes 20
13.5 odd 4 inner 117.5.j.b.109.3 20
39.5 even 4 39.5.g.a.31.8 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.5.g.a.31.8 20 39.5 even 4
39.5.g.a.34.8 yes 20 3.2 odd 2
117.5.j.b.73.3 20 1.1 even 1 trivial
117.5.j.b.109.3 20 13.5 odd 4 inner