Properties

Label 324.5.d.f.163.14
Level $324$
Weight $5$
Character 324.163
Analytic conductor $33.492$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [324,5,Mod(163,324)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(324, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.163");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 324.d (of order \(2\), degree \(1\), 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: \(33.4918680392\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 163.14
Character \(\chi\) \(=\) 324.163
Dual form 324.5.d.f.163.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.34653 + 3.76654i) q^{2} +(-12.3737 + 10.1435i) q^{4} +11.0316 q^{5} +11.9152i q^{7} +(-54.8676 - 32.9476i) q^{8} +O(q^{10})\) \(q+(1.34653 + 3.76654i) q^{2} +(-12.3737 + 10.1435i) q^{4} +11.0316 q^{5} +11.9152i q^{7} +(-54.8676 - 32.9476i) q^{8} +(14.8544 + 41.5509i) q^{10} +219.387i q^{11} -37.0701 q^{13} +(-44.8790 + 16.0441i) q^{14} +(50.2175 - 251.026i) q^{16} +284.021 q^{17} -45.4901i q^{19} +(-136.502 + 111.899i) q^{20} +(-826.332 + 295.412i) q^{22} -201.286i q^{23} -503.304 q^{25} +(-49.9160 - 139.626i) q^{26} +(-120.862 - 147.435i) q^{28} -1228.31 q^{29} +1514.77i q^{31} +(1013.12 - 148.868i) q^{32} +(382.443 + 1069.78i) q^{34} +131.443i q^{35} -1521.29 q^{37} +(171.340 - 61.2538i) q^{38} +(-605.277 - 363.464i) q^{40} -2633.95 q^{41} -39.9593i q^{43} +(-2225.36 - 2714.63i) q^{44} +(758.153 - 271.038i) q^{46} -2884.86i q^{47} +2259.03 q^{49} +(-677.714 - 1895.72i) q^{50} +(458.694 - 376.021i) q^{52} +1415.13 q^{53} +2420.19i q^{55} +(392.576 - 653.757i) q^{56} +(-1653.95 - 4626.47i) q^{58} -2832.86i q^{59} -5257.27 q^{61} +(-5705.44 + 2039.68i) q^{62} +(1924.92 + 3615.51i) q^{64} -408.941 q^{65} -930.592i q^{67} +(-3514.39 + 2880.97i) q^{68} +(-495.087 + 176.992i) q^{70} -1162.75i q^{71} -2162.87 q^{73} +(-2048.47 - 5730.02i) q^{74} +(461.430 + 562.881i) q^{76} -2614.04 q^{77} +7485.40i q^{79} +(553.978 - 2769.22i) q^{80} +(-3546.69 - 9920.88i) q^{82} -1116.31i q^{83} +3133.20 q^{85} +(150.508 - 53.8064i) q^{86} +(7228.27 - 12037.3i) q^{88} -6739.71 q^{89} -441.696i q^{91} +(2041.75 + 2490.66i) q^{92} +(10865.9 - 3884.55i) q^{94} -501.828i q^{95} +12046.6 q^{97} +(3041.85 + 8508.73i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q + q^{2} + q^{4} + 2 q^{5} + 61 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 22 q + q^{2} + q^{4} + 2 q^{5} + 61 q^{8} + 14 q^{10} + 2 q^{13} - 252 q^{14} + q^{16} - 28 q^{17} + 140 q^{20} + 33 q^{22} + 1752 q^{25} + 548 q^{26} - 258 q^{28} - 526 q^{29} + 121 q^{32} - 385 q^{34} - 4 q^{37} - 1395 q^{38} + 2276 q^{40} + 2762 q^{41} + 3357 q^{44} + 1788 q^{46} - 3428 q^{49} - 6375 q^{50} - 1438 q^{52} - 5044 q^{53} + 7506 q^{56} + 4064 q^{58} + 2 q^{61} - 9162 q^{62} + 4513 q^{64} + 2014 q^{65} + 11405 q^{68} - 3666 q^{70} - 1708 q^{73} - 14620 q^{74} - 1581 q^{76} + 3942 q^{77} + 22760 q^{80} - 4243 q^{82} + 1252 q^{85} - 22113 q^{86} - 1995 q^{88} + 6524 q^{89} + 30294 q^{92} - 7524 q^{94} - 5638 q^{97} - 46469 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.34653 + 3.76654i 0.336633 + 0.941636i
\(3\) 0 0
\(4\) −12.3737 + 10.1435i −0.773357 + 0.633971i
\(5\) 11.0316 0.441263 0.220632 0.975357i \(-0.429188\pi\)
0.220632 + 0.975357i \(0.429188\pi\)
\(6\) 0 0
\(7\) 11.9152i 0.243167i 0.992581 + 0.121583i \(0.0387972\pi\)
−0.992581 + 0.121583i \(0.961203\pi\)
\(8\) −54.8676 32.9476i −0.857307 0.514806i
\(9\) 0 0
\(10\) 14.8544 + 41.5509i 0.148544 + 0.415509i
\(11\) 219.387i 1.81312i 0.422080 + 0.906559i \(0.361300\pi\)
−0.422080 + 0.906559i \(0.638700\pi\)
\(12\) 0 0
\(13\) −37.0701 −0.219349 −0.109675 0.993968i \(-0.534981\pi\)
−0.109675 + 0.993968i \(0.534981\pi\)
\(14\) −44.8790 + 16.0441i −0.228975 + 0.0818579i
\(15\) 0 0
\(16\) 50.2175 251.026i 0.196162 0.980571i
\(17\) 284.021 0.982771 0.491386 0.870942i \(-0.336491\pi\)
0.491386 + 0.870942i \(0.336491\pi\)
\(18\) 0 0
\(19\) 45.4901i 0.126011i −0.998013 0.0630057i \(-0.979931\pi\)
0.998013 0.0630057i \(-0.0200686\pi\)
\(20\) −136.502 + 111.899i −0.341254 + 0.279748i
\(21\) 0 0
\(22\) −826.332 + 295.412i −1.70730 + 0.610354i
\(23\) 201.286i 0.380503i −0.981735 0.190252i \(-0.939070\pi\)
0.981735 0.190252i \(-0.0609304\pi\)
\(24\) 0 0
\(25\) −503.304 −0.805287
\(26\) −49.9160 139.626i −0.0738402 0.206547i
\(27\) 0 0
\(28\) −120.862 147.435i −0.154161 0.188055i
\(29\) −1228.31 −1.46053 −0.730265 0.683164i \(-0.760603\pi\)
−0.730265 + 0.683164i \(0.760603\pi\)
\(30\) 0 0
\(31\) 1514.77i 1.57624i 0.615520 + 0.788121i \(0.288946\pi\)
−0.615520 + 0.788121i \(0.711054\pi\)
\(32\) 1013.12 148.868i 0.989376 0.145379i
\(33\) 0 0
\(34\) 382.443 + 1069.78i 0.330833 + 0.925413i
\(35\) 131.443i 0.107301i
\(36\) 0 0
\(37\) −1521.29 −1.11124 −0.555622 0.831435i \(-0.687520\pi\)
−0.555622 + 0.831435i \(0.687520\pi\)
\(38\) 171.340 61.2538i 0.118657 0.0424195i
\(39\) 0 0
\(40\) −605.277 363.464i −0.378298 0.227165i
\(41\) −2633.95 −1.56689 −0.783447 0.621459i \(-0.786540\pi\)
−0.783447 + 0.621459i \(0.786540\pi\)
\(42\) 0 0
\(43\) 39.9593i 0.0216113i −0.999942 0.0108056i \(-0.996560\pi\)
0.999942 0.0108056i \(-0.00343961\pi\)
\(44\) −2225.36 2714.63i −1.14946 1.40219i
\(45\) 0 0
\(46\) 758.153 271.038i 0.358295 0.128090i
\(47\) 2884.86i 1.30596i −0.757377 0.652978i \(-0.773519\pi\)
0.757377 0.652978i \(-0.226481\pi\)
\(48\) 0 0
\(49\) 2259.03 0.940870
\(50\) −677.714 1895.72i −0.271086 0.758287i
\(51\) 0 0
\(52\) 458.694 376.021i 0.169635 0.139061i
\(53\) 1415.13 0.503783 0.251892 0.967755i \(-0.418947\pi\)
0.251892 + 0.967755i \(0.418947\pi\)
\(54\) 0 0
\(55\) 2420.19i 0.800062i
\(56\) 392.576 653.757i 0.125184 0.208469i
\(57\) 0 0
\(58\) −1653.95 4626.47i −0.491662 1.37529i
\(59\) 2832.86i 0.813808i −0.913471 0.406904i \(-0.866608\pi\)
0.913471 0.406904i \(-0.133392\pi\)
\(60\) 0 0
\(61\) −5257.27 −1.41287 −0.706433 0.707780i \(-0.749697\pi\)
−0.706433 + 0.707780i \(0.749697\pi\)
\(62\) −5705.44 + 2039.68i −1.48425 + 0.530614i
\(63\) 0 0
\(64\) 1924.92 + 3615.51i 0.469950 + 0.882693i
\(65\) −408.941 −0.0967909
\(66\) 0 0
\(67\) 930.592i 0.207305i −0.994614 0.103653i \(-0.966947\pi\)
0.994614 0.103653i \(-0.0330530\pi\)
\(68\) −3514.39 + 2880.97i −0.760033 + 0.623048i
\(69\) 0 0
\(70\) −495.087 + 176.992i −0.101038 + 0.0361209i
\(71\) 1162.75i 0.230659i −0.993327 0.115329i \(-0.963208\pi\)
0.993327 0.115329i \(-0.0367923\pi\)
\(72\) 0 0
\(73\) −2162.87 −0.405867 −0.202934 0.979192i \(-0.565048\pi\)
−0.202934 + 0.979192i \(0.565048\pi\)
\(74\) −2048.47 5730.02i −0.374081 1.04639i
\(75\) 0 0
\(76\) 461.430 + 562.881i 0.0798875 + 0.0974518i
\(77\) −2614.04 −0.440890
\(78\) 0 0
\(79\) 7485.40i 1.19939i 0.800229 + 0.599695i \(0.204711\pi\)
−0.800229 + 0.599695i \(0.795289\pi\)
\(80\) 553.978 2769.22i 0.0865591 0.432690i
\(81\) 0 0
\(82\) −3546.69 9920.88i −0.527467 1.47544i
\(83\) 1116.31i 0.162043i −0.996712 0.0810215i \(-0.974182\pi\)
0.996712 0.0810215i \(-0.0258182\pi\)
\(84\) 0 0
\(85\) 3133.20 0.433661
\(86\) 150.508 53.8064i 0.0203500 0.00727506i
\(87\) 0 0
\(88\) 7228.27 12037.3i 0.933403 1.55440i
\(89\) −6739.71 −0.850866 −0.425433 0.904990i \(-0.639878\pi\)
−0.425433 + 0.904990i \(0.639878\pi\)
\(90\) 0 0
\(91\) 441.696i 0.0533385i
\(92\) 2041.75 + 2490.66i 0.241228 + 0.294265i
\(93\) 0 0
\(94\) 10865.9 3884.55i 1.22974 0.439627i
\(95\) 501.828i 0.0556042i
\(96\) 0 0
\(97\) 12046.6 1.28032 0.640161 0.768240i \(-0.278867\pi\)
0.640161 + 0.768240i \(0.278867\pi\)
\(98\) 3041.85 + 8508.73i 0.316727 + 0.885957i
\(99\) 0 0
\(100\) 6227.74 5105.28i 0.622774 0.510528i
\(101\) −9555.20 −0.936692 −0.468346 0.883545i \(-0.655150\pi\)
−0.468346 + 0.883545i \(0.655150\pi\)
\(102\) 0 0
\(103\) 11503.4i 1.08431i 0.840279 + 0.542154i \(0.182391\pi\)
−0.840279 + 0.542154i \(0.817609\pi\)
\(104\) 2033.95 + 1221.37i 0.188050 + 0.112922i
\(105\) 0 0
\(106\) 1905.51 + 5330.14i 0.169590 + 0.474380i
\(107\) 6602.72i 0.576707i −0.957524 0.288353i \(-0.906892\pi\)
0.957524 0.288353i \(-0.0931078\pi\)
\(108\) 0 0
\(109\) 12045.3 1.01383 0.506913 0.861997i \(-0.330786\pi\)
0.506913 + 0.861997i \(0.330786\pi\)
\(110\) −9115.74 + 3258.86i −0.753367 + 0.269327i
\(111\) 0 0
\(112\) 2991.02 + 598.350i 0.238442 + 0.0477001i
\(113\) 3730.69 0.292168 0.146084 0.989272i \(-0.453333\pi\)
0.146084 + 0.989272i \(0.453333\pi\)
\(114\) 0 0
\(115\) 2220.50i 0.167902i
\(116\) 15198.7 12459.4i 1.12951 0.925933i
\(117\) 0 0
\(118\) 10670.1 3814.54i 0.766311 0.273954i
\(119\) 3384.16i 0.238977i
\(120\) 0 0
\(121\) −33489.7 −2.28739
\(122\) −7079.08 19801.7i −0.475617 1.33041i
\(123\) 0 0
\(124\) −15365.1 18743.3i −0.999291 1.21900i
\(125\) −12447.0 −0.796607
\(126\) 0 0
\(127\) 26549.7i 1.64608i 0.567980 + 0.823042i \(0.307725\pi\)
−0.567980 + 0.823042i \(0.692275\pi\)
\(128\) −11026.0 + 12118.7i −0.672975 + 0.739665i
\(129\) 0 0
\(130\) −550.652 1540.30i −0.0325830 0.0911418i
\(131\) 1319.64i 0.0768978i 0.999261 + 0.0384489i \(0.0122417\pi\)
−0.999261 + 0.0384489i \(0.987758\pi\)
\(132\) 0 0
\(133\) 542.022 0.0306418
\(134\) 3505.12 1253.07i 0.195206 0.0697856i
\(135\) 0 0
\(136\) −15583.6 9357.79i −0.842537 0.505936i
\(137\) 25812.9 1.37530 0.687648 0.726044i \(-0.258643\pi\)
0.687648 + 0.726044i \(0.258643\pi\)
\(138\) 0 0
\(139\) 13138.0i 0.679985i 0.940428 + 0.339993i \(0.110425\pi\)
−0.940428 + 0.339993i \(0.889575\pi\)
\(140\) −1333.30 1626.44i −0.0680254 0.0829816i
\(141\) 0 0
\(142\) 4379.55 1565.68i 0.217197 0.0776472i
\(143\) 8132.70i 0.397706i
\(144\) 0 0
\(145\) −13550.2 −0.644478
\(146\) −2912.37 8146.54i −0.136628 0.382179i
\(147\) 0 0
\(148\) 18824.0 15431.3i 0.859388 0.704496i
\(149\) 25936.4 1.16826 0.584128 0.811662i \(-0.301437\pi\)
0.584128 + 0.811662i \(0.301437\pi\)
\(150\) 0 0
\(151\) 28395.5i 1.24536i 0.782475 + 0.622681i \(0.213957\pi\)
−0.782475 + 0.622681i \(0.786043\pi\)
\(152\) −1498.79 + 2495.93i −0.0648714 + 0.108030i
\(153\) 0 0
\(154\) −3519.88 9845.88i −0.148418 0.415158i
\(155\) 16710.3i 0.695538i
\(156\) 0 0
\(157\) −5520.37 −0.223959 −0.111980 0.993711i \(-0.535719\pi\)
−0.111980 + 0.993711i \(0.535719\pi\)
\(158\) −28194.1 + 10079.3i −1.12939 + 0.403754i
\(159\) 0 0
\(160\) 11176.3 1642.25i 0.436575 0.0641504i
\(161\) 2398.36 0.0925257
\(162\) 0 0
\(163\) 407.281i 0.0153292i −0.999971 0.00766459i \(-0.997560\pi\)
0.999971 0.00766459i \(-0.00243974\pi\)
\(164\) 32591.7 26717.5i 1.21177 0.993365i
\(165\) 0 0
\(166\) 4204.65 1503.15i 0.152586 0.0545490i
\(167\) 23742.9i 0.851334i 0.904880 + 0.425667i \(0.139960\pi\)
−0.904880 + 0.425667i \(0.860040\pi\)
\(168\) 0 0
\(169\) −27186.8 −0.951886
\(170\) 4218.95 + 11801.3i 0.145984 + 0.408351i
\(171\) 0 0
\(172\) 405.328 + 494.444i 0.0137009 + 0.0167132i
\(173\) −2826.49 −0.0944400 −0.0472200 0.998885i \(-0.515036\pi\)
−0.0472200 + 0.998885i \(0.515036\pi\)
\(174\) 0 0
\(175\) 5996.96i 0.195819i
\(176\) 55072.0 + 11017.1i 1.77789 + 0.355665i
\(177\) 0 0
\(178\) −9075.23 25385.4i −0.286429 0.801206i
\(179\) 45743.4i 1.42765i 0.700323 + 0.713826i \(0.253039\pi\)
−0.700323 + 0.713826i \(0.746961\pi\)
\(180\) 0 0
\(181\) −24226.2 −0.739483 −0.369742 0.929135i \(-0.620554\pi\)
−0.369742 + 0.929135i \(0.620554\pi\)
\(182\) 1663.67 594.757i 0.0502255 0.0179555i
\(183\) 0 0
\(184\) −6631.89 + 11044.1i −0.195885 + 0.326208i
\(185\) −16782.3 −0.490351
\(186\) 0 0
\(187\) 62310.5i 1.78188i
\(188\) 29262.6 + 35696.4i 0.827938 + 1.00997i
\(189\) 0 0
\(190\) 1890.16 675.726i 0.0523589 0.0187182i
\(191\) 22522.0i 0.617362i −0.951166 0.308681i \(-0.900112\pi\)
0.951166 0.308681i \(-0.0998875\pi\)
\(192\) 0 0
\(193\) 18288.7 0.490985 0.245493 0.969398i \(-0.421050\pi\)
0.245493 + 0.969398i \(0.421050\pi\)
\(194\) 16221.1 + 45373.9i 0.430998 + 1.20560i
\(195\) 0 0
\(196\) −27952.6 + 22914.5i −0.727628 + 0.596484i
\(197\) 36300.2 0.935356 0.467678 0.883899i \(-0.345091\pi\)
0.467678 + 0.883899i \(0.345091\pi\)
\(198\) 0 0
\(199\) 47336.5i 1.19533i 0.801744 + 0.597667i \(0.203906\pi\)
−0.801744 + 0.597667i \(0.796094\pi\)
\(200\) 27615.1 + 16582.6i 0.690378 + 0.414566i
\(201\) 0 0
\(202\) −12866.4 35990.1i −0.315321 0.882023i
\(203\) 14635.5i 0.355152i
\(204\) 0 0
\(205\) −29056.6 −0.691412
\(206\) −43328.2 + 15489.7i −1.02102 + 0.365014i
\(207\) 0 0
\(208\) −1861.57 + 9305.56i −0.0430280 + 0.215088i
\(209\) 9979.95 0.228473
\(210\) 0 0
\(211\) 46490.2i 1.04423i 0.852875 + 0.522115i \(0.174857\pi\)
−0.852875 + 0.522115i \(0.825143\pi\)
\(212\) −17510.4 + 14354.4i −0.389604 + 0.319384i
\(213\) 0 0
\(214\) 24869.4 8890.76i 0.543048 0.194138i
\(215\) 440.814i 0.00953626i
\(216\) 0 0
\(217\) −18048.7 −0.383290
\(218\) 16219.3 + 45369.1i 0.341287 + 0.954656i
\(219\) 0 0
\(220\) −24549.3 29946.7i −0.507216 0.618734i
\(221\) −10528.7 −0.215570
\(222\) 0 0
\(223\) 82597.9i 1.66096i −0.557048 0.830480i \(-0.688066\pi\)
0.557048 0.830480i \(-0.311934\pi\)
\(224\) 1773.79 + 12071.5i 0.0353514 + 0.240583i
\(225\) 0 0
\(226\) 5023.49 + 14051.8i 0.0983533 + 0.275116i
\(227\) 29701.6i 0.576405i 0.957569 + 0.288203i \(0.0930577\pi\)
−0.957569 + 0.288203i \(0.906942\pi\)
\(228\) 0 0
\(229\) 45335.9 0.864512 0.432256 0.901751i \(-0.357718\pi\)
0.432256 + 0.901751i \(0.357718\pi\)
\(230\) 8363.63 2989.98i 0.158103 0.0565213i
\(231\) 0 0
\(232\) 67394.2 + 40469.7i 1.25212 + 0.751889i
\(233\) 50931.3 0.938151 0.469075 0.883158i \(-0.344587\pi\)
0.469075 + 0.883158i \(0.344587\pi\)
\(234\) 0 0
\(235\) 31824.5i 0.576270i
\(236\) 28735.3 + 35053.1i 0.515930 + 0.629364i
\(237\) 0 0
\(238\) −12746.6 + 4556.87i −0.225030 + 0.0804475i
\(239\) 36888.3i 0.645792i 0.946434 + 0.322896i \(0.104656\pi\)
−0.946434 + 0.322896i \(0.895344\pi\)
\(240\) 0 0
\(241\) 8387.27 0.144407 0.0722033 0.997390i \(-0.476997\pi\)
0.0722033 + 0.997390i \(0.476997\pi\)
\(242\) −45094.9 126141.i −0.770011 2.15389i
\(243\) 0 0
\(244\) 65052.0 53327.3i 1.09265 0.895715i
\(245\) 24920.7 0.415171
\(246\) 0 0
\(247\) 1686.32i 0.0276405i
\(248\) 49907.9 83111.8i 0.811458 1.35132i
\(249\) 0 0
\(250\) −16760.2 46882.1i −0.268164 0.750114i
\(251\) 1475.47i 0.0234198i 0.999931 + 0.0117099i \(0.00372746\pi\)
−0.999931 + 0.0117099i \(0.996273\pi\)
\(252\) 0 0
\(253\) 44159.6 0.689897
\(254\) −100001. + 35750.0i −1.55001 + 0.554126i
\(255\) 0 0
\(256\) −60492.4 25211.8i −0.923041 0.384702i
\(257\) 34673.1 0.524960 0.262480 0.964937i \(-0.415460\pi\)
0.262480 + 0.964937i \(0.415460\pi\)
\(258\) 0 0
\(259\) 18126.5i 0.270218i
\(260\) 5060.12 4148.11i 0.0748539 0.0613626i
\(261\) 0 0
\(262\) −4970.49 + 1776.94i −0.0724097 + 0.0258863i
\(263\) 19032.4i 0.275158i −0.990491 0.137579i \(-0.956068\pi\)
0.990491 0.137579i \(-0.0439321\pi\)
\(264\) 0 0
\(265\) 15611.1 0.222301
\(266\) 729.850 + 2041.55i 0.0103150 + 0.0288534i
\(267\) 0 0
\(268\) 9439.50 + 11514.9i 0.131425 + 0.160321i
\(269\) 58724.1 0.811544 0.405772 0.913974i \(-0.367003\pi\)
0.405772 + 0.913974i \(0.367003\pi\)
\(270\) 0 0
\(271\) 31474.1i 0.428563i −0.976772 0.214281i \(-0.931259\pi\)
0.976772 0.214281i \(-0.0687409\pi\)
\(272\) 14262.8 71296.7i 0.192782 0.963677i
\(273\) 0 0
\(274\) 34757.9 + 97225.5i 0.462969 + 1.29503i
\(275\) 110418.i 1.46008i
\(276\) 0 0
\(277\) 66531.5 0.867098 0.433549 0.901130i \(-0.357261\pi\)
0.433549 + 0.901130i \(0.357261\pi\)
\(278\) −49484.8 + 17690.7i −0.640299 + 0.228905i
\(279\) 0 0
\(280\) 4330.73 7211.98i 0.0552389 0.0919895i
\(281\) −123720. −1.56685 −0.783424 0.621488i \(-0.786529\pi\)
−0.783424 + 0.621488i \(0.786529\pi\)
\(282\) 0 0
\(283\) 54735.0i 0.683427i 0.939804 + 0.341713i \(0.111007\pi\)
−0.939804 + 0.341713i \(0.888993\pi\)
\(284\) 11794.4 + 14387.5i 0.146231 + 0.178382i
\(285\) 0 0
\(286\) 30632.2 10950.9i 0.374495 0.133881i
\(287\) 31383.9i 0.381016i
\(288\) 0 0
\(289\) −2853.15 −0.0341609
\(290\) −18245.7 51037.2i −0.216952 0.606864i
\(291\) 0 0
\(292\) 26762.7 21939.1i 0.313880 0.257308i
\(293\) −88617.9 −1.03225 −0.516127 0.856512i \(-0.672627\pi\)
−0.516127 + 0.856512i \(0.672627\pi\)
\(294\) 0 0
\(295\) 31251.0i 0.359103i
\(296\) 83469.8 + 50122.9i 0.952677 + 0.572075i
\(297\) 0 0
\(298\) 34924.2 + 97690.7i 0.393273 + 1.10007i
\(299\) 7461.69i 0.0834632i
\(300\) 0 0
\(301\) 476.121 0.00525514
\(302\) −106953. + 38235.4i −1.17268 + 0.419230i
\(303\) 0 0
\(304\) −11419.2 2284.40i −0.123563 0.0247186i
\(305\) −57996.0 −0.623446
\(306\) 0 0
\(307\) 123447.i 1.30980i −0.755716 0.654900i \(-0.772711\pi\)
0.755716 0.654900i \(-0.227289\pi\)
\(308\) 32345.3 26515.6i 0.340965 0.279511i
\(309\) 0 0
\(310\) −62940.1 + 22500.9i −0.654943 + 0.234141i
\(311\) 60543.1i 0.625956i 0.949760 + 0.312978i \(0.101327\pi\)
−0.949760 + 0.312978i \(0.898673\pi\)
\(312\) 0 0
\(313\) −52386.2 −0.534722 −0.267361 0.963596i \(-0.586152\pi\)
−0.267361 + 0.963596i \(0.586152\pi\)
\(314\) −7433.35 20792.7i −0.0753920 0.210888i
\(315\) 0 0
\(316\) −75928.4 92622.1i −0.760378 0.927557i
\(317\) −25997.4 −0.258708 −0.129354 0.991598i \(-0.541290\pi\)
−0.129354 + 0.991598i \(0.541290\pi\)
\(318\) 0 0
\(319\) 269474.i 2.64811i
\(320\) 21234.9 + 39884.8i 0.207372 + 0.389500i
\(321\) 0 0
\(322\) 3229.46 + 9033.52i 0.0311472 + 0.0871255i
\(323\) 12920.1i 0.123840i
\(324\) 0 0
\(325\) 18657.5 0.176639
\(326\) 1534.04 548.416i 0.0144345 0.00516030i
\(327\) 0 0
\(328\) 144518. + 86782.1i 1.34331 + 0.806645i
\(329\) 34373.6 0.317565
\(330\) 0 0
\(331\) 43231.1i 0.394584i 0.980345 + 0.197292i \(0.0632148\pi\)
−0.980345 + 0.197292i \(0.936785\pi\)
\(332\) 11323.4 + 13812.9i 0.102731 + 0.125317i
\(333\) 0 0
\(334\) −89428.5 + 31970.5i −0.801647 + 0.286587i
\(335\) 10265.9i 0.0914761i
\(336\) 0 0
\(337\) 11852.8 0.104366 0.0521831 0.998638i \(-0.483382\pi\)
0.0521831 + 0.998638i \(0.483382\pi\)
\(338\) −36607.9 102400.i −0.320436 0.896330i
\(339\) 0 0
\(340\) −38769.3 + 31781.7i −0.335375 + 0.274928i
\(341\) −332321. −2.85791
\(342\) 0 0
\(343\) 55525.0i 0.471955i
\(344\) −1316.56 + 2192.47i −0.0111256 + 0.0185275i
\(345\) 0 0
\(346\) −3805.96 10646.1i −0.0317916 0.0889281i
\(347\) 29124.4i 0.241879i −0.992660 0.120940i \(-0.961409\pi\)
0.992660 0.120940i \(-0.0385907\pi\)
\(348\) 0 0
\(349\) 119977. 0.985023 0.492512 0.870306i \(-0.336079\pi\)
0.492512 + 0.870306i \(0.336079\pi\)
\(350\) 22587.8 8075.08i 0.184390 0.0659191i
\(351\) 0 0
\(352\) 32659.8 + 222266.i 0.263589 + 1.79385i
\(353\) 154758. 1.24195 0.620975 0.783831i \(-0.286737\pi\)
0.620975 + 0.783831i \(0.286737\pi\)
\(354\) 0 0
\(355\) 12827.0i 0.101781i
\(356\) 83395.3 68364.5i 0.658023 0.539424i
\(357\) 0 0
\(358\) −172294. + 61594.9i −1.34433 + 0.480594i
\(359\) 66839.6i 0.518615i −0.965795 0.259307i \(-0.916506\pi\)
0.965795 0.259307i \(-0.0834942\pi\)
\(360\) 0 0
\(361\) 128252. 0.984121
\(362\) −32621.3 91249.1i −0.248934 0.696324i
\(363\) 0 0
\(364\) 4480.36 + 5465.42i 0.0338151 + 0.0412497i
\(365\) −23859.8 −0.179094
\(366\) 0 0
\(367\) 94571.4i 0.702147i −0.936348 0.351073i \(-0.885817\pi\)
0.936348 0.351073i \(-0.114183\pi\)
\(368\) −50528.1 10108.1i −0.373110 0.0746403i
\(369\) 0 0
\(370\) −22597.8 63211.2i −0.165068 0.461732i
\(371\) 16861.5i 0.122503i
\(372\) 0 0
\(373\) −181061. −1.30139 −0.650694 0.759340i \(-0.725522\pi\)
−0.650694 + 0.759340i \(0.725522\pi\)
\(374\) −234695. + 83903.0i −1.67788 + 0.599839i
\(375\) 0 0
\(376\) −95049.0 + 158285.i −0.672313 + 1.11960i
\(377\) 45533.4 0.320366
\(378\) 0 0
\(379\) 51191.3i 0.356384i 0.983996 + 0.178192i \(0.0570248\pi\)
−0.983996 + 0.178192i \(0.942975\pi\)
\(380\) 5090.31 + 6209.47i 0.0352514 + 0.0430019i
\(381\) 0 0
\(382\) 84830.0 30326.5i 0.581330 0.207824i
\(383\) 152568.i 1.04008i 0.854143 + 0.520039i \(0.174083\pi\)
−0.854143 + 0.520039i \(0.825917\pi\)
\(384\) 0 0
\(385\) −28836.9 −0.194548
\(386\) 24626.3 + 68885.2i 0.165282 + 0.462329i
\(387\) 0 0
\(388\) −149061. + 122195.i −0.990147 + 0.811687i
\(389\) 67896.8 0.448694 0.224347 0.974509i \(-0.427975\pi\)
0.224347 + 0.974509i \(0.427975\pi\)
\(390\) 0 0
\(391\) 57169.5i 0.373947i
\(392\) −123948. 74429.5i −0.806614 0.484365i
\(393\) 0 0
\(394\) 48879.4 + 136726.i 0.314871 + 0.880765i
\(395\) 82575.7i 0.529247i
\(396\) 0 0
\(397\) −126087. −0.799998 −0.399999 0.916516i \(-0.630990\pi\)
−0.399999 + 0.916516i \(0.630990\pi\)
\(398\) −178295. + 63740.0i −1.12557 + 0.402389i
\(399\) 0 0
\(400\) −25274.7 + 126343.i −0.157967 + 0.789641i
\(401\) 70890.8 0.440860 0.220430 0.975403i \(-0.429254\pi\)
0.220430 + 0.975403i \(0.429254\pi\)
\(402\) 0 0
\(403\) 56152.6i 0.345748i
\(404\) 118233. 96923.5i 0.724398 0.593836i
\(405\) 0 0
\(406\) 55125.1 19707.1i 0.334424 0.119556i
\(407\) 333752.i 2.01482i
\(408\) 0 0
\(409\) 43544.2 0.260306 0.130153 0.991494i \(-0.458453\pi\)
0.130153 + 0.991494i \(0.458453\pi\)
\(410\) −39125.6 109443.i −0.232752 0.651059i
\(411\) 0 0
\(412\) −116685. 142340.i −0.687420 0.838557i
\(413\) 33754.1 0.197891
\(414\) 0 0
\(415\) 12314.7i 0.0715036i
\(416\) −37556.5 + 5518.55i −0.217019 + 0.0318888i
\(417\) 0 0
\(418\) 13438.3 + 37589.9i 0.0769116 + 0.215139i
\(419\) 210600.i 1.19958i −0.800156 0.599792i \(-0.795250\pi\)
0.800156 0.599792i \(-0.204750\pi\)
\(420\) 0 0
\(421\) 84126.0 0.474642 0.237321 0.971431i \(-0.423731\pi\)
0.237321 + 0.971431i \(0.423731\pi\)
\(422\) −175107. + 62600.4i −0.983285 + 0.351522i
\(423\) 0 0
\(424\) −77644.7 46625.0i −0.431897 0.259350i
\(425\) −142949. −0.791413
\(426\) 0 0
\(427\) 62641.3i 0.343562i
\(428\) 66974.9 + 81700.1i 0.365615 + 0.446000i
\(429\) 0 0
\(430\) 1660.34 593.569i 0.00897969 0.00321022i
\(431\) 150083.i 0.807933i −0.914774 0.403967i \(-0.867631\pi\)
0.914774 0.403967i \(-0.132369\pi\)
\(432\) 0 0
\(433\) 71221.5 0.379871 0.189935 0.981797i \(-0.439172\pi\)
0.189935 + 0.981797i \(0.439172\pi\)
\(434\) −24303.2 67981.3i −0.129028 0.360919i
\(435\) 0 0
\(436\) −149045. + 122182.i −0.784050 + 0.642736i
\(437\) −9156.53 −0.0479477
\(438\) 0 0
\(439\) 109245.i 0.566858i −0.958993 0.283429i \(-0.908528\pi\)
0.958993 0.283429i \(-0.0914720\pi\)
\(440\) 79739.3 132790.i 0.411876 0.685899i
\(441\) 0 0
\(442\) −14177.2 39656.7i −0.0725680 0.202989i
\(443\) 230658.i 1.17533i 0.809103 + 0.587666i \(0.199953\pi\)
−0.809103 + 0.587666i \(0.800047\pi\)
\(444\) 0 0
\(445\) −74349.7 −0.375456
\(446\) 311109. 111221.i 1.56402 0.559133i
\(447\) 0 0
\(448\) −43079.4 + 22935.7i −0.214642 + 0.114276i
\(449\) −9751.64 −0.0483710 −0.0241855 0.999707i \(-0.507699\pi\)
−0.0241855 + 0.999707i \(0.507699\pi\)
\(450\) 0 0
\(451\) 577854.i 2.84096i
\(452\) −46162.5 + 37842.4i −0.225950 + 0.185226i
\(453\) 0 0
\(454\) −111872. + 39994.1i −0.542764 + 0.194037i
\(455\) 4872.61i 0.0235363i
\(456\) 0 0
\(457\) 66139.1 0.316684 0.158342 0.987384i \(-0.449385\pi\)
0.158342 + 0.987384i \(0.449385\pi\)
\(458\) 61046.1 + 170760.i 0.291023 + 0.814056i
\(459\) 0 0
\(460\) 22523.8 + 27475.9i 0.106445 + 0.129848i
\(461\) −173989. −0.818692 −0.409346 0.912379i \(-0.634243\pi\)
−0.409346 + 0.912379i \(0.634243\pi\)
\(462\) 0 0
\(463\) 173993.i 0.811650i 0.913951 + 0.405825i \(0.133016\pi\)
−0.913951 + 0.405825i \(0.866984\pi\)
\(464\) −61682.4 + 308337.i −0.286500 + 1.43215i
\(465\) 0 0
\(466\) 68580.5 + 191835.i 0.315812 + 0.883397i
\(467\) 172090.i 0.789080i −0.918879 0.394540i \(-0.870904\pi\)
0.918879 0.394540i \(-0.129096\pi\)
\(468\) 0 0
\(469\) 11088.2 0.0504097
\(470\) 119868. 42852.7i 0.542637 0.193991i
\(471\) 0 0
\(472\) −93336.0 + 155433.i −0.418953 + 0.697683i
\(473\) 8766.55 0.0391838
\(474\) 0 0
\(475\) 22895.4i 0.101475i
\(476\) −34327.3 41874.6i −0.151505 0.184815i
\(477\) 0 0
\(478\) −138941. + 49671.2i −0.608101 + 0.217395i
\(479\) 218398.i 0.951869i 0.879481 + 0.475935i \(0.157890\pi\)
−0.879481 + 0.475935i \(0.842110\pi\)
\(480\) 0 0
\(481\) 56394.4 0.243751
\(482\) 11293.7 + 31591.0i 0.0486119 + 0.135978i
\(483\) 0 0
\(484\) 414392. 339704.i 1.76897 1.45014i
\(485\) 132893. 0.564959
\(486\) 0 0
\(487\) 392112.i 1.65330i 0.562716 + 0.826650i \(0.309757\pi\)
−0.562716 + 0.826650i \(0.690243\pi\)
\(488\) 288454. + 173214.i 1.21126 + 0.727351i
\(489\) 0 0
\(490\) 33556.4 + 93864.8i 0.139760 + 0.390940i
\(491\) 209717.i 0.869903i −0.900454 0.434951i \(-0.856766\pi\)
0.900454 0.434951i \(-0.143234\pi\)
\(492\) 0 0
\(493\) −348864. −1.43537
\(494\) −6351.60 + 2270.68i −0.0260273 + 0.00930470i
\(495\) 0 0
\(496\) 380247. + 76067.9i 1.54562 + 0.309199i
\(497\) 13854.4 0.0560885
\(498\) 0 0
\(499\) 135886.i 0.545725i −0.962053 0.272862i \(-0.912030\pi\)
0.962053 0.272862i \(-0.0879704\pi\)
\(500\) 154015. 126256.i 0.616061 0.505025i
\(501\) 0 0
\(502\) −5557.42 + 1986.76i −0.0220529 + 0.00788385i
\(503\) 232332.i 0.918275i 0.888365 + 0.459137i \(0.151841\pi\)
−0.888365 + 0.459137i \(0.848159\pi\)
\(504\) 0 0
\(505\) −105409. −0.413328
\(506\) 59462.2 + 166329.i 0.232242 + 0.649632i
\(507\) 0 0
\(508\) −269308. 328518.i −1.04357 1.27301i
\(509\) −145319. −0.560901 −0.280451 0.959868i \(-0.590484\pi\)
−0.280451 + 0.959868i \(0.590484\pi\)
\(510\) 0 0
\(511\) 25770.9i 0.0986935i
\(512\) 13506.6 261796.i 0.0515234 0.998672i
\(513\) 0 0
\(514\) 46688.3 + 130598.i 0.176719 + 0.494321i
\(515\) 126901.i 0.478465i
\(516\) 0 0
\(517\) 632900. 2.36785
\(518\) 68274.1 24407.8i 0.254447 0.0909641i
\(519\) 0 0
\(520\) 22437.7 + 13473.6i 0.0829795 + 0.0498285i
\(521\) 443088. 1.63235 0.816177 0.577803i \(-0.196090\pi\)
0.816177 + 0.577803i \(0.196090\pi\)
\(522\) 0 0
\(523\) 73202.4i 0.267622i −0.991007 0.133811i \(-0.957278\pi\)
0.991007 0.133811i \(-0.0427215\pi\)
\(524\) −13385.8 16328.9i −0.0487509 0.0594694i
\(525\) 0 0
\(526\) 71686.3 25627.7i 0.259098 0.0926271i
\(527\) 430226.i 1.54909i
\(528\) 0 0
\(529\) 239325. 0.855217
\(530\) 21020.8 + 58799.9i 0.0748338 + 0.209327i
\(531\) 0 0
\(532\) −6706.83 + 5498.02i −0.0236970 + 0.0194260i
\(533\) 97640.6 0.343697
\(534\) 0 0
\(535\) 72838.4i 0.254480i
\(536\) −30660.7 + 51059.4i −0.106722 + 0.177724i
\(537\) 0 0
\(538\) 79073.8 + 221187.i 0.273192 + 0.764179i
\(539\) 495602.i 1.70591i
\(540\) 0 0
\(541\) 438165. 1.49707 0.748536 0.663094i \(-0.230757\pi\)
0.748536 + 0.663094i \(0.230757\pi\)
\(542\) 118548. 42380.8i 0.403550 0.144268i
\(543\) 0 0
\(544\) 287748. 42281.7i 0.972330 0.142874i
\(545\) 132878. 0.447364
\(546\) 0 0
\(547\) 312024.i 1.04283i −0.853303 0.521416i \(-0.825404\pi\)
0.853303 0.521416i \(-0.174596\pi\)
\(548\) −319402. + 261834.i −1.06359 + 0.871898i
\(549\) 0 0
\(550\) 415896. 148682.i 1.37486 0.491510i
\(551\) 55875.7i 0.184043i
\(552\) 0 0
\(553\) −89189.8 −0.291652
\(554\) 89586.7 + 250594.i 0.291893 + 0.816490i
\(555\) 0 0
\(556\) −133266. 162566.i −0.431091 0.525871i
\(557\) 312549. 1.00741 0.503707 0.863874i \(-0.331969\pi\)
0.503707 + 0.863874i \(0.331969\pi\)
\(558\) 0 0
\(559\) 1481.29i 0.00474042i
\(560\) 32995.7 + 6600.75i 0.105216 + 0.0210483i
\(561\) 0 0
\(562\) −166593. 465996.i −0.527452 1.47540i
\(563\) 86349.2i 0.272422i 0.990680 + 0.136211i \(0.0434925\pi\)
−0.990680 + 0.136211i \(0.956508\pi\)
\(564\) 0 0
\(565\) 41155.5 0.128923
\(566\) −206162. + 73702.3i −0.643539 + 0.230064i
\(567\) 0 0
\(568\) −38309.8 + 63797.4i −0.118744 + 0.197745i
\(569\) 554958. 1.71410 0.857049 0.515235i \(-0.172295\pi\)
0.857049 + 0.515235i \(0.172295\pi\)
\(570\) 0 0
\(571\) 303511.i 0.930898i 0.885075 + 0.465449i \(0.154107\pi\)
−0.885075 + 0.465449i \(0.845893\pi\)
\(572\) 82494.3 + 100632.i 0.252134 + 0.307569i
\(573\) 0 0
\(574\) 118209. 42259.4i 0.358779 0.128263i
\(575\) 101308.i 0.306414i
\(576\) 0 0
\(577\) −500884. −1.50448 −0.752238 0.658892i \(-0.771026\pi\)
−0.752238 + 0.658892i \(0.771026\pi\)
\(578\) −3841.86 10746.5i −0.0114997 0.0321671i
\(579\) 0 0
\(580\) 167666. 137446.i 0.498412 0.408580i
\(581\) 13301.1 0.0394035
\(582\) 0 0
\(583\) 310461.i 0.913418i
\(584\) 118671. + 71261.2i 0.347953 + 0.208943i
\(585\) 0 0
\(586\) −119327. 333783.i −0.347490 0.972007i
\(587\) 575710.i 1.67081i −0.549633 0.835406i \(-0.685233\pi\)
0.549633 0.835406i \(-0.314767\pi\)
\(588\) 0 0
\(589\) 68907.0 0.198624
\(590\) 117708. 42080.4i 0.338145 0.120886i
\(591\) 0 0
\(592\) −76395.5 + 381885.i −0.217984 + 1.08965i
\(593\) −138143. −0.392843 −0.196421 0.980520i \(-0.562932\pi\)
−0.196421 + 0.980520i \(0.562932\pi\)
\(594\) 0 0
\(595\) 37332.6i 0.105452i
\(596\) −320930. + 263087.i −0.903478 + 0.740640i
\(597\) 0 0
\(598\) −28104.8 + 10047.4i −0.0785919 + 0.0280964i
\(599\) 191294.i 0.533147i 0.963815 + 0.266574i \(0.0858916\pi\)
−0.963815 + 0.266574i \(0.914108\pi\)
\(600\) 0 0
\(601\) −406240. −1.12469 −0.562346 0.826902i \(-0.690101\pi\)
−0.562346 + 0.826902i \(0.690101\pi\)
\(602\) 641.112 + 1793.33i 0.00176905 + 0.00494843i
\(603\) 0 0
\(604\) −288031. 351358.i −0.789524 0.963110i
\(605\) −369445. −1.00934
\(606\) 0 0
\(607\) 68666.5i 0.186366i −0.995649 0.0931831i \(-0.970296\pi\)
0.995649 0.0931831i \(-0.0297042\pi\)
\(608\) −6772.03 46087.0i −0.0183194 0.124673i
\(609\) 0 0
\(610\) −78093.4 218445.i −0.209872 0.587059i
\(611\) 106942.i 0.286461i
\(612\) 0 0
\(613\) −572114. −1.52252 −0.761258 0.648450i \(-0.775418\pi\)
−0.761258 + 0.648450i \(0.775418\pi\)
\(614\) 464970. 166225.i 1.23335 0.440921i
\(615\) 0 0
\(616\) 143426. + 86126.1i 0.377978 + 0.226973i
\(617\) 20380.8 0.0535367 0.0267684 0.999642i \(-0.491478\pi\)
0.0267684 + 0.999642i \(0.491478\pi\)
\(618\) 0 0
\(619\) 448721.i 1.17110i 0.810635 + 0.585552i \(0.199122\pi\)
−0.810635 + 0.585552i \(0.800878\pi\)
\(620\) −169501. 206768.i −0.440951 0.537899i
\(621\) 0 0
\(622\) −228038. + 81523.2i −0.589423 + 0.210717i
\(623\) 80304.8i 0.206902i
\(624\) 0 0
\(625\) 177255. 0.453773
\(626\) −70539.6 197315.i −0.180005 0.503514i
\(627\) 0 0
\(628\) 68307.5 55996.1i 0.173200 0.141984i
\(629\) −432079. −1.09210
\(630\) 0 0
\(631\) 402529.i 1.01097i 0.862835 + 0.505486i \(0.168687\pi\)
−0.862835 + 0.505486i \(0.831313\pi\)
\(632\) 246625. 410706.i 0.617453 1.02825i
\(633\) 0 0
\(634\) −35006.2 97920.2i −0.0870897 0.243609i
\(635\) 292885.i 0.726357i
\(636\) 0 0
\(637\) −83742.3 −0.206379
\(638\) 1.01499e6 362856.i 2.49356 0.891441i
\(639\) 0 0
\(640\) −121634. + 133688.i −0.296959 + 0.326387i
\(641\) 353813. 0.861108 0.430554 0.902565i \(-0.358318\pi\)
0.430554 + 0.902565i \(0.358318\pi\)
\(642\) 0 0
\(643\) 263141.i 0.636452i 0.948015 + 0.318226i \(0.103087\pi\)
−0.948015 + 0.318226i \(0.896913\pi\)
\(644\) −29676.6 + 24327.8i −0.0715554 + 0.0586586i
\(645\) 0 0
\(646\) 48664.3 17397.4i 0.116613 0.0416887i
\(647\) 265174.i 0.633464i −0.948515 0.316732i \(-0.897414\pi\)
0.948515 0.316732i \(-0.102586\pi\)
\(648\) 0 0
\(649\) 621494. 1.47553
\(650\) 25122.9 + 70274.4i 0.0594625 + 0.166330i
\(651\) 0 0
\(652\) 4131.27 + 5039.58i 0.00971825 + 0.0118549i
\(653\) −5877.92 −0.0137847 −0.00689235 0.999976i \(-0.502194\pi\)
−0.00689235 + 0.999976i \(0.502194\pi\)
\(654\) 0 0
\(655\) 14557.7i 0.0339322i
\(656\) −132270. + 661190.i −0.307365 + 1.53645i
\(657\) 0 0
\(658\) 46285.0 + 129470.i 0.106903 + 0.299031i
\(659\) 554642.i 1.27715i 0.769560 + 0.638575i \(0.220476\pi\)
−0.769560 + 0.638575i \(0.779524\pi\)
\(660\) 0 0
\(661\) −821227. −1.87958 −0.939789 0.341755i \(-0.888979\pi\)
−0.939789 + 0.341755i \(0.888979\pi\)
\(662\) −162832. + 58212.0i −0.371555 + 0.132830i
\(663\) 0 0
\(664\) −36779.8 + 61249.5i −0.0834206 + 0.138921i
\(665\) 5979.36 0.0135211
\(666\) 0 0
\(667\) 247241.i 0.555736i
\(668\) −240836. 293787.i −0.539721 0.658385i
\(669\) 0 0
\(670\) 38667.0 13823.4i 0.0861372 0.0307938i
\(671\) 1.15338e6i 2.56169i
\(672\) 0 0
\(673\) 516865. 1.14116 0.570580 0.821242i \(-0.306718\pi\)
0.570580 + 0.821242i \(0.306718\pi\)
\(674\) 15960.1 + 44643.9i 0.0351330 + 0.0982749i
\(675\) 0 0
\(676\) 336402. 275770.i 0.736148 0.603468i
\(677\) −690686. −1.50697 −0.753483 0.657468i \(-0.771628\pi\)
−0.753483 + 0.657468i \(0.771628\pi\)
\(678\) 0 0
\(679\) 143537.i 0.311332i
\(680\) −171911. 103231.i −0.371780 0.223251i
\(681\) 0 0
\(682\) −447480. 1.25170e6i −0.962066 2.69111i
\(683\) 764551.i 1.63895i −0.573117 0.819474i \(-0.694266\pi\)
0.573117 0.819474i \(-0.305734\pi\)
\(684\) 0 0
\(685\) 284757. 0.606868
\(686\) −209138. + 74766.2i −0.444410 + 0.158875i
\(687\) 0 0
\(688\) −10030.8 2006.65i −0.0211914 0.00423931i
\(689\) −52458.9 −0.110505
\(690\) 0 0
\(691\) 321130.i 0.672550i 0.941764 + 0.336275i \(0.109167\pi\)
−0.941764 + 0.336275i \(0.890833\pi\)
\(692\) 34974.2 28670.6i 0.0730358 0.0598722i
\(693\) 0 0
\(694\) 109698. 39216.9i 0.227762 0.0814244i
\(695\) 144933.i 0.300053i
\(696\) 0 0
\(697\) −748096. −1.53990
\(698\) 161552. + 451898.i 0.331591 + 0.927534i
\(699\) 0 0
\(700\) 60830.3 + 74204.6i 0.124144 + 0.151438i
\(701\) −621799. −1.26536 −0.632680 0.774413i \(-0.718045\pi\)
−0.632680 + 0.774413i \(0.718045\pi\)
\(702\) 0 0
\(703\) 69203.8i 0.140029i
\(704\) −793196. + 422302.i −1.60043 + 0.852075i
\(705\) 0 0
\(706\) 208386. + 582903.i 0.418081 + 1.16946i
\(707\) 113852.i 0.227772i
\(708\) 0 0
\(709\) −524870. −1.04414 −0.522070 0.852903i \(-0.674840\pi\)
−0.522070 + 0.852903i \(0.674840\pi\)
\(710\) 48313.4 17271.9i 0.0958409 0.0342629i
\(711\) 0 0
\(712\) 369792. + 222057.i 0.729454 + 0.438031i
\(713\) 304902. 0.599765
\(714\) 0 0
\(715\) 89716.5i 0.175493i
\(716\) −463999. 566015.i −0.905089 1.10408i
\(717\) 0 0
\(718\) 251754. 90001.5i 0.488346 0.174583i
\(719\) 348658.i 0.674438i −0.941426 0.337219i \(-0.890514\pi\)
0.941426 0.337219i \(-0.109486\pi\)
\(720\) 0 0
\(721\) −137065. −0.263668
\(722\) 172695. + 483066.i 0.331287 + 0.926684i
\(723\) 0 0
\(724\) 299768. 245739.i 0.571885 0.468811i
\(725\) 618211. 1.17615
\(726\) 0 0
\(727\) 604820.i 1.14435i 0.820133 + 0.572173i \(0.193899\pi\)
−0.820133 + 0.572173i \(0.806101\pi\)
\(728\) −14552.8 + 24234.8i −0.0274590 + 0.0457275i
\(729\) 0 0
\(730\) −32128.0 89869.2i −0.0602890 0.168642i
\(731\) 11349.3i 0.0212389i
\(732\) 0 0
\(733\) −475610. −0.885204 −0.442602 0.896718i \(-0.645945\pi\)
−0.442602 + 0.896718i \(0.645945\pi\)
\(734\) 356208. 127343.i 0.661167 0.236365i
\(735\) 0 0
\(736\) −29965.1 203927.i −0.0553172 0.376461i
\(737\) 204160. 0.375868
\(738\) 0 0
\(739\) 844449.i 1.54627i 0.634244 + 0.773133i \(0.281312\pi\)
−0.634244 + 0.773133i \(0.718688\pi\)
\(740\) 207659. 170231.i 0.379216 0.310868i
\(741\) 0 0
\(742\) −63509.5 + 22704.5i −0.115354 + 0.0412386i
\(743\) 620323.i 1.12367i 0.827248 + 0.561837i \(0.189905\pi\)
−0.827248 + 0.561837i \(0.810095\pi\)
\(744\) 0 0
\(745\) 286120. 0.515508
\(746\) −243804. 681974.i −0.438090 1.22543i
\(747\) 0 0
\(748\) −632049. 771013.i −1.12966 1.37803i
\(749\) 78672.5 0.140236
\(750\) 0 0
\(751\) 730748.i 1.29565i 0.761789 + 0.647825i \(0.224321\pi\)
−0.761789 + 0.647825i \(0.775679\pi\)
\(752\) −724175. 144870.i −1.28058 0.256179i
\(753\) 0 0
\(754\) 61312.0 + 171503.i 0.107846 + 0.301669i
\(755\) 313247.i 0.549533i
\(756\) 0 0
\(757\) −769023. −1.34198 −0.670992 0.741465i \(-0.734132\pi\)
−0.670992 + 0.741465i \(0.734132\pi\)
\(758\) −192814. + 68930.6i −0.335584 + 0.119970i
\(759\) 0 0
\(760\) −16534.0 + 27534.1i −0.0286253 + 0.0476699i
\(761\) 275394. 0.475538 0.237769 0.971322i \(-0.423584\pi\)
0.237769 + 0.971322i \(0.423584\pi\)
\(762\) 0 0
\(763\) 143522.i 0.246529i
\(764\) 228452. + 278680.i 0.391389 + 0.477441i
\(765\) 0 0
\(766\) −574654. + 205437.i −0.979375 + 0.350124i
\(767\) 105014.i 0.178508i
\(768\) 0 0
\(769\) −52842.1 −0.0893568 −0.0446784 0.999001i \(-0.514226\pi\)
−0.0446784 + 0.999001i \(0.514226\pi\)
\(770\) −38829.8 108616.i −0.0654914 0.183194i
\(771\) 0 0
\(772\) −226299. + 185512.i −0.379707 + 0.311270i
\(773\) −195309. −0.326861 −0.163430 0.986555i \(-0.552256\pi\)
−0.163430 + 0.986555i \(0.552256\pi\)
\(774\) 0 0
\(775\) 762389.i 1.26933i
\(776\) −660966. 396905.i −1.09763 0.659117i
\(777\) 0 0
\(778\) 91425.1 + 255736.i 0.151045 + 0.422506i
\(779\) 119819.i 0.197446i
\(780\) 0 0
\(781\) 255093. 0.418211
\(782\) 215331. 76980.4i 0.352122 0.125883i
\(783\) 0 0
\(784\) 113443. 567076.i 0.184563 0.922590i
\(785\) −60898.4 −0.0988250
\(786\) 0 0
\(787\) 782238.i 1.26296i 0.775393 + 0.631480i \(0.217552\pi\)
−0.775393 + 0.631480i \(0.782448\pi\)
\(788\) −449169. + 368213.i −0.723364 + 0.592989i
\(789\) 0 0
\(790\) −311025. + 111191.i −0.498358 + 0.178162i
\(791\) 44451.9i 0.0710456i
\(792\) 0 0
\(793\) 194887. 0.309911
\(794\) −169780. 474912.i −0.269305 0.753307i
\(795\) 0 0
\(796\) −480159. 585728.i −0.757807 0.924421i
\(797\) −1.17343e6 −1.84731 −0.923655 0.383226i \(-0.874813\pi\)
−0.923655 + 0.383226i \(0.874813\pi\)
\(798\) 0 0
\(799\) 819359.i 1.28346i
\(800\) −509908. + 74926.0i −0.796731 + 0.117072i
\(801\) 0 0
\(802\) 95456.6 + 267013.i 0.148408 + 0.415130i
\(803\) 474505.i 0.735885i
\(804\) 0 0
\(805\) 26457.7 0.0408282
\(806\) 211501. 75611.1i 0.325569 0.116390i
\(807\) 0 0
\(808\) 524271. + 314820.i 0.803033 + 0.482214i
\(809\) −421614. −0.644196 −0.322098 0.946706i \(-0.604388\pi\)
−0.322098 + 0.946706i \(0.604388\pi\)
\(810\) 0 0
\(811\) 504037.i 0.766339i 0.923678 + 0.383169i \(0.125167\pi\)
−0.923678 + 0.383169i \(0.874833\pi\)
\(812\) 148455. + 181095.i 0.225156 + 0.274659i
\(813\) 0 0
\(814\) 1.25709e6 449407.i 1.89722 0.678253i
\(815\) 4492.95i 0.00676420i
\(816\) 0 0
\(817\) −1817.75 −0.00272327
\(818\) 58633.6 + 164011.i 0.0876275 + 0.245113i
\(819\) 0 0
\(820\) 359538. 294737.i 0.534709 0.438335i
\(821\) −71554.1 −0.106157 −0.0530785 0.998590i \(-0.516903\pi\)
−0.0530785 + 0.998590i \(0.516903\pi\)
\(822\) 0 0
\(823\) 812709.i 1.19987i −0.800048 0.599936i \(-0.795193\pi\)
0.800048 0.599936i \(-0.204807\pi\)
\(824\) 379010. 631166.i 0.558208 0.929585i
\(825\) 0 0
\(826\) 45450.9 + 127136.i 0.0666166 + 0.186341i
\(827\) 375951.i 0.549693i 0.961488 + 0.274847i \(0.0886271\pi\)
−0.961488 + 0.274847i \(0.911373\pi\)
\(828\) 0 0
\(829\) 423699. 0.616522 0.308261 0.951302i \(-0.400253\pi\)
0.308261 + 0.951302i \(0.400253\pi\)
\(830\) 46383.9 16582.1i 0.0673304 0.0240704i
\(831\) 0 0
\(832\) −71356.8 134027.i −0.103083 0.193618i
\(833\) 641611. 0.924660
\(834\) 0 0
\(835\) 261921.i 0.375662i
\(836\) −123489. + 101232.i −0.176691 + 0.144845i
\(837\) 0 0
\(838\) 793234. 283579.i 1.12957 0.403819i
\(839\) 1.10840e6i 1.57460i −0.616568 0.787302i \(-0.711477\pi\)
0.616568 0.787302i \(-0.288523\pi\)
\(840\) 0 0
\(841\) 801453. 1.13315
\(842\) 113278. + 316864.i 0.159780 + 0.446940i
\(843\) 0 0
\(844\) −471575. 575256.i −0.662012 0.807563i
\(845\) −299914. −0.420032
\(846\) 0 0
\(847\) 399036.i 0.556218i
\(848\) 71064.1 355234.i 0.0988231 0.493995i
\(849\) 0 0
\(850\) −192485. 538423.i −0.266415 0.745223i
\(851\) 306215.i 0.422832i
\(852\) 0 0
\(853\) −509319. −0.699989 −0.349995 0.936752i \(-0.613817\pi\)
−0.349995 + 0.936752i \(0.613817\pi\)
\(854\) 235941. 84348.4i 0.323510 0.115654i
\(855\) 0 0
\(856\) −217543. + 362276.i −0.296892 + 0.494415i
\(857\) −656788. −0.894260 −0.447130 0.894469i \(-0.647554\pi\)
−0.447130 + 0.894469i \(0.647554\pi\)
\(858\) 0 0
\(859\) 1.05866e6i 1.43474i −0.696694 0.717368i \(-0.745347\pi\)
0.696694 0.717368i \(-0.254653\pi\)
\(860\) 4471.41 + 5454.50i 0.00604571 + 0.00737494i
\(861\) 0 0
\(862\) 565292. 202091.i 0.760779 0.271977i
\(863\) 676878.i 0.908844i 0.890787 + 0.454422i \(0.150154\pi\)
−0.890787 + 0.454422i \(0.849846\pi\)
\(864\) 0 0
\(865\) −31180.7 −0.0416729
\(866\) 95902.0 + 268259.i 0.127877 + 0.357700i
\(867\) 0 0
\(868\) 223330. 183078.i 0.296420 0.242994i
\(869\) −1.64220e6 −2.17464
\(870\) 0 0
\(871\) 34497.1i 0.0454723i
\(872\) −660896. 396862.i −0.869161 0.521924i
\(873\) 0 0
\(874\) −12329.5 34488.5i −0.0161408 0.0451493i
\(875\) 148308.i 0.193708i
\(876\) 0 0
\(877\) 1.47583e6 1.91883 0.959415 0.281998i \(-0.0909973\pi\)
0.959415 + 0.281998i \(0.0909973\pi\)
\(878\) 411478. 147102.i 0.533774 0.190823i
\(879\) 0 0
\(880\) 607531. + 121536.i 0.784518 + 0.156942i
\(881\) 599513. 0.772408 0.386204 0.922413i \(-0.373786\pi\)
0.386204 + 0.922413i \(0.373786\pi\)
\(882\) 0 0
\(883\) 729187.i 0.935229i 0.883933 + 0.467614i \(0.154886\pi\)
−0.883933 + 0.467614i \(0.845114\pi\)
\(884\) 130279. 106798.i 0.166713 0.136665i
\(885\) 0 0
\(886\) −868783. + 310588.i −1.10674 + 0.395655i
\(887\) 1.45744e6i 1.85244i −0.376981 0.926221i \(-0.623038\pi\)
0.376981 0.926221i \(-0.376962\pi\)
\(888\) 0 0
\(889\) −316344. −0.400273
\(890\) −100114. 280041.i −0.126391 0.353543i
\(891\) 0 0
\(892\) 837834. + 1.02204e6i 1.05300 + 1.28452i
\(893\) −131232. −0.164565
\(894\) 0 0
\(895\) 504622.i 0.629970i
\(896\) −144396. 131377.i −0.179862 0.163645i
\(897\) 0 0
\(898\) −13130.9 36730.0i −0.0162833 0.0455479i
\(899\) 1.86060e6i 2.30215i
\(900\) 0 0
\(901\) 401926. 0.495104
\(902\) 2.17651e6 778098.i 2.67515 0.956360i
\(903\) 0 0
\(904\) −204694. 122917.i −0.250478 0.150410i
\(905\) −267253. −0.326307
\(906\) 0 0
\(907\) 3476.53i 0.00422602i 0.999998 + 0.00211301i \(0.000672593\pi\)
−0.999998 + 0.00211301i \(0.999327\pi\)
\(908\) −301279. 367519.i −0.365424 0.445767i
\(909\) 0 0
\(910\) 18352.9 6561.11i 0.0221627 0.00792309i
\(911\) 822935.i 0.991583i 0.868442 + 0.495791i \(0.165122\pi\)
−0.868442 + 0.495791i \(0.834878\pi\)
\(912\) 0 0
\(913\) 244905. 0.293803
\(914\) 89058.3 + 249116.i 0.106606 + 0.298201i
\(915\) 0 0
\(916\) −560973. + 459866.i −0.668577 + 0.548076i
\(917\) −15723.8 −0.0186990
\(918\) 0 0
\(919\) 873279.i 1.03400i −0.855984 0.517002i \(-0.827048\pi\)
0.855984 0.517002i \(-0.172952\pi\)
\(920\) −73160.2 + 121834.i −0.0864369 + 0.143944i
\(921\) 0 0
\(922\) −234282. 655338.i −0.275598 0.770910i
\(923\) 43103.2i 0.0505949i
\(924\) 0 0
\(925\) 765673. 0.894870
\(926\) −655351. + 234286.i −0.764279 + 0.273228i
\(927\) 0 0
\(928\) −1.24442e6 + 182856.i −1.44501 + 0.212330i
\(929\) 345961. 0.400863 0.200431 0.979708i \(-0.435766\pi\)
0.200431 + 0.979708i \(0.435766\pi\)
\(930\) 0 0
\(931\) 102763.i 0.118560i
\(932\) −630209. + 516623.i −0.725526 + 0.594760i
\(933\) 0 0
\(934\) 648183. 231724.i 0.743026 0.265630i
\(935\) 687384.i 0.786278i
\(936\) 0 0
\(937\) −1.10958e6 −1.26380 −0.631901 0.775049i \(-0.717725\pi\)
−0.631901 + 0.775049i \(0.717725\pi\)
\(938\) 14930.6 + 41764.1i 0.0169696 + 0.0474676i
\(939\) 0 0
\(940\) 322813. + 393788.i 0.365339 + 0.445663i
\(941\) 322543. 0.364257 0.182129 0.983275i \(-0.441701\pi\)
0.182129 + 0.983275i \(0.441701\pi\)
\(942\) 0 0
\(943\) 530177.i 0.596208i
\(944\) −711124. 142259.i −0.797997 0.159638i
\(945\) 0 0
\(946\) 11804.4 + 33019.6i 0.0131905 + 0.0368969i
\(947\) 139420.i 0.155463i 0.996974 + 0.0777314i \(0.0247676\pi\)
−0.996974 + 0.0777314i \(0.975232\pi\)
\(948\) 0 0
\(949\) 80177.6 0.0890268
\(950\) −86236.4 + 30829.3i −0.0955528 + 0.0341599i
\(951\) 0 0
\(952\) 111500. 185681.i 0.123027 0.204877i
\(953\) 1.68438e6 1.85462 0.927308 0.374300i \(-0.122117\pi\)
0.927308 + 0.374300i \(0.122117\pi\)
\(954\) 0 0
\(955\) 248453.i 0.272419i
\(956\) −374178. 456445.i −0.409413 0.499428i
\(957\) 0 0
\(958\) −822605. + 294079.i −0.896314 + 0.320430i
\(959\) 307566.i 0.334426i
\(960\) 0 0
\(961\) −1.37100e6 −1.48454
\(962\) 75936.8 + 212412.i 0.0820545 + 0.229525i
\(963\) 0 0
\(964\) −103782. + 85076.6i −0.111678 + 0.0915495i
\(965\) 201753. 0.216654
\(966\) 0 0
\(967\) 45357.5i 0.0485061i −0.999706 0.0242530i \(-0.992279\pi\)
0.999706 0.0242530i \(-0.00772074\pi\)
\(968\) 1.83750e6 + 1.10340e6i 1.96100 + 1.17756i
\(969\) 0 0
\(970\) 178944. + 500546.i 0.190184 + 0.531986i
\(971\) 1.18319e6i 1.25492i 0.778648 + 0.627461i \(0.215906\pi\)
−0.778648 + 0.627461i \(0.784094\pi\)
\(972\) 0 0
\(973\) −156541. −0.165350
\(974\) −1.47691e6 + 527990.i −1.55681 + 0.556555i
\(975\) 0 0
\(976\) −264007. + 1.31971e6i −0.277151 + 1.38542i
\(977\) 1.12593e6 1.17957 0.589783 0.807562i \(-0.299213\pi\)
0.589783 + 0.807562i \(0.299213\pi\)
\(978\) 0 0
\(979\) 1.47861e6i 1.54272i
\(980\) −308361. + 252784.i −0.321076 + 0.263206i
\(981\) 0 0
\(982\) 789909. 282390.i 0.819132 0.292838i
\(983\) 1.35328e6i 1.40049i 0.713904 + 0.700243i \(0.246925\pi\)
−0.713904 + 0.700243i \(0.753075\pi\)
\(984\) 0 0
\(985\) 400449. 0.412738
\(986\) −469756. 1.31401e6i −0.483191 1.35159i
\(987\) 0 0
\(988\) −17105.3 20866.0i −0.0175233 0.0213760i
\(989\) −8043.24 −0.00822316
\(990\) 0 0
\(991\) 1.02847e6i 1.04724i −0.851953 0.523619i \(-0.824582\pi\)
0.851953 0.523619i \(-0.175418\pi\)
\(992\) 225501. + 1.53464e6i 0.229153 + 1.55950i
\(993\) 0 0
\(994\) 18655.3 + 52183.1i 0.0188812 + 0.0528150i
\(995\) 522196.i 0.527457i
\(996\) 0 0
\(997\) 1.12806e6 1.13486 0.567430 0.823422i \(-0.307938\pi\)
0.567430 + 0.823422i \(0.307938\pi\)
\(998\) 511821. 182975.i 0.513874 0.183709i
\(999\) 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 324.5.d.f.163.14 22
3.2 odd 2 324.5.d.e.163.9 22
4.3 odd 2 inner 324.5.d.f.163.13 22
9.2 odd 6 108.5.f.a.91.7 44
9.4 even 3 36.5.f.a.7.2 44
9.5 odd 6 108.5.f.a.19.21 44
9.7 even 3 36.5.f.a.31.16 yes 44
12.11 even 2 324.5.d.e.163.10 22
36.7 odd 6 36.5.f.a.31.2 yes 44
36.11 even 6 108.5.f.a.91.21 44
36.23 even 6 108.5.f.a.19.7 44
36.31 odd 6 36.5.f.a.7.16 yes 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.5.f.a.7.2 44 9.4 even 3
36.5.f.a.7.16 yes 44 36.31 odd 6
36.5.f.a.31.2 yes 44 36.7 odd 6
36.5.f.a.31.16 yes 44 9.7 even 3
108.5.f.a.19.7 44 36.23 even 6
108.5.f.a.19.21 44 9.5 odd 6
108.5.f.a.91.7 44 9.2 odd 6
108.5.f.a.91.21 44 36.11 even 6
324.5.d.e.163.9 22 3.2 odd 2
324.5.d.e.163.10 22 12.11 even 2
324.5.d.f.163.13 22 4.3 odd 2 inner
324.5.d.f.163.14 22 1.1 even 1 trivial