Properties

Label 882.5.b.g.197.6
Level $882$
Weight $5$
Character 882.197
Analytic conductor $91.172$
Analytic rank $0$
Dimension $6$
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] = [882,5,Mod(197,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, 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("882.197");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 882.b (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: \(91.1723074400\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 32x^{4} - 7932x^{3} + 185683x^{2} + 6753864x + 64548144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 197.6
Root \(21.9137 - 13.5986i\) of defining polynomial
Character \(\chi\) \(=\) 882.197
Dual form 882.5.b.g.197.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.82843i q^{2} -8.00000 q^{4} +25.7831i q^{5} -22.6274i q^{8} +O(q^{10})\) \(q+2.82843i q^{2} -8.00000 q^{4} +25.7831i q^{5} -22.6274i q^{8} -72.9255 q^{10} -2.36859i q^{11} +268.521 q^{13} +64.0000 q^{16} -258.563i q^{17} -527.634 q^{19} -206.265i q^{20} +6.69938 q^{22} +128.190i q^{23} -39.7667 q^{25} +759.491i q^{26} +1457.13i q^{29} +792.400 q^{31} +181.019i q^{32} +731.327 q^{34} +2004.12 q^{37} -1492.37i q^{38} +583.404 q^{40} +1014.13i q^{41} +2491.96 q^{43} +18.9487i q^{44} -362.577 q^{46} +655.817i q^{47} -112.477i q^{50} -2148.17 q^{52} -2820.60i q^{53} +61.0695 q^{55} -4121.38 q^{58} +1915.65i q^{59} +3243.35 q^{61} +2241.25i q^{62} -512.000 q^{64} +6923.29i q^{65} -2699.22 q^{67} +2068.51i q^{68} -4680.05i q^{71} +939.107 q^{73} +5668.52i q^{74} +4221.07 q^{76} -5391.03 q^{79} +1650.12i q^{80} -2868.39 q^{82} +7991.44i q^{83} +6666.55 q^{85} +7048.34i q^{86} -53.5951 q^{88} +4128.97i q^{89} -1025.52i q^{92} -1854.93 q^{94} -13604.0i q^{95} -3972.54 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 48 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 48 q^{4} + 40 q^{10} - 278 q^{13} + 384 q^{16} - 654 q^{19} + 808 q^{22} - 1786 q^{25} + 3790 q^{31} + 2768 q^{34} - 4542 q^{37} - 320 q^{40} + 8142 q^{43} - 1136 q^{46} + 2224 q^{52} + 11432 q^{55} - 832 q^{58} + 4024 q^{61} - 3072 q^{64} - 7466 q^{67} - 5506 q^{73} + 5232 q^{76} - 3778 q^{79} - 5496 q^{82} + 37312 q^{85} - 6464 q^{88} - 31224 q^{94} + 30692 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.82843i 0.707107i
\(3\) 0 0
\(4\) −8.00000 −0.500000
\(5\) 25.7831i 1.03132i 0.856792 + 0.515661i \(0.172454\pi\)
−0.856792 + 0.515661i \(0.827546\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) − 22.6274i − 0.353553i
\(9\) 0 0
\(10\) −72.9255 −0.729255
\(11\) − 2.36859i − 0.0195751i −0.999952 0.00978756i \(-0.996884\pi\)
0.999952 0.00978756i \(-0.00311553\pi\)
\(12\) 0 0
\(13\) 268.521 1.58888 0.794440 0.607343i \(-0.207765\pi\)
0.794440 + 0.607343i \(0.207765\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) − 258.563i − 0.894682i −0.894363 0.447341i \(-0.852371\pi\)
0.894363 0.447341i \(-0.147629\pi\)
\(18\) 0 0
\(19\) −527.634 −1.46159 −0.730795 0.682597i \(-0.760850\pi\)
−0.730795 + 0.682597i \(0.760850\pi\)
\(20\) − 206.265i − 0.515661i
\(21\) 0 0
\(22\) 6.69938 0.0138417
\(23\) 128.190i 0.242326i 0.992633 + 0.121163i \(0.0386623\pi\)
−0.992633 + 0.121163i \(0.961338\pi\)
\(24\) 0 0
\(25\) −39.7667 −0.0636267
\(26\) 759.491i 1.12351i
\(27\) 0 0
\(28\) 0 0
\(29\) 1457.13i 1.73261i 0.499514 + 0.866306i \(0.333512\pi\)
−0.499514 + 0.866306i \(0.666488\pi\)
\(30\) 0 0
\(31\) 792.400 0.824558 0.412279 0.911058i \(-0.364733\pi\)
0.412279 + 0.911058i \(0.364733\pi\)
\(32\) 181.019i 0.176777i
\(33\) 0 0
\(34\) 731.327 0.632636
\(35\) 0 0
\(36\) 0 0
\(37\) 2004.12 1.46393 0.731966 0.681341i \(-0.238603\pi\)
0.731966 + 0.681341i \(0.238603\pi\)
\(38\) − 1492.37i − 1.03350i
\(39\) 0 0
\(40\) 583.404 0.364628
\(41\) 1014.13i 0.603289i 0.953420 + 0.301645i \(0.0975356\pi\)
−0.953420 + 0.301645i \(0.902464\pi\)
\(42\) 0 0
\(43\) 2491.96 1.34774 0.673868 0.738852i \(-0.264632\pi\)
0.673868 + 0.738852i \(0.264632\pi\)
\(44\) 18.9487i 0.00978756i
\(45\) 0 0
\(46\) −362.577 −0.171350
\(47\) 655.817i 0.296884i 0.988921 + 0.148442i \(0.0474258\pi\)
−0.988921 + 0.148442i \(0.952574\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) − 112.477i − 0.0449909i
\(51\) 0 0
\(52\) −2148.17 −0.794440
\(53\) − 2820.60i − 1.00413i −0.864830 0.502065i \(-0.832574\pi\)
0.864830 0.502065i \(-0.167426\pi\)
\(54\) 0 0
\(55\) 61.0695 0.0201883
\(56\) 0 0
\(57\) 0 0
\(58\) −4121.38 −1.22514
\(59\) 1915.65i 0.550315i 0.961399 + 0.275158i \(0.0887300\pi\)
−0.961399 + 0.275158i \(0.911270\pi\)
\(60\) 0 0
\(61\) 3243.35 0.871635 0.435818 0.900035i \(-0.356459\pi\)
0.435818 + 0.900035i \(0.356459\pi\)
\(62\) 2241.25i 0.583051i
\(63\) 0 0
\(64\) −512.000 −0.125000
\(65\) 6923.29i 1.63865i
\(66\) 0 0
\(67\) −2699.22 −0.601295 −0.300648 0.953735i \(-0.597203\pi\)
−0.300648 + 0.953735i \(0.597203\pi\)
\(68\) 2068.51i 0.447341i
\(69\) 0 0
\(70\) 0 0
\(71\) − 4680.05i − 0.928397i −0.885731 0.464199i \(-0.846342\pi\)
0.885731 0.464199i \(-0.153658\pi\)
\(72\) 0 0
\(73\) 939.107 0.176226 0.0881129 0.996110i \(-0.471916\pi\)
0.0881129 + 0.996110i \(0.471916\pi\)
\(74\) 5668.52i 1.03516i
\(75\) 0 0
\(76\) 4221.07 0.730795
\(77\) 0 0
\(78\) 0 0
\(79\) −5391.03 −0.863808 −0.431904 0.901920i \(-0.642158\pi\)
−0.431904 + 0.901920i \(0.642158\pi\)
\(80\) 1650.12i 0.257831i
\(81\) 0 0
\(82\) −2868.39 −0.426590
\(83\) 7991.44i 1.16003i 0.814606 + 0.580014i \(0.196953\pi\)
−0.814606 + 0.580014i \(0.803047\pi\)
\(84\) 0 0
\(85\) 6666.55 0.922706
\(86\) 7048.34i 0.952993i
\(87\) 0 0
\(88\) −53.5951 −0.00692085
\(89\) 4128.97i 0.521269i 0.965438 + 0.260634i \(0.0839317\pi\)
−0.965438 + 0.260634i \(0.916068\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 1025.52i − 0.121163i
\(93\) 0 0
\(94\) −1854.93 −0.209929
\(95\) − 13604.0i − 1.50737i
\(96\) 0 0
\(97\) −3972.54 −0.422206 −0.211103 0.977464i \(-0.567706\pi\)
−0.211103 + 0.977464i \(0.567706\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 318.134 0.0318134
\(101\) 13514.6i 1.32483i 0.749136 + 0.662416i \(0.230469\pi\)
−0.749136 + 0.662416i \(0.769531\pi\)
\(102\) 0 0
\(103\) −10215.6 −0.962918 −0.481459 0.876469i \(-0.659893\pi\)
−0.481459 + 0.876469i \(0.659893\pi\)
\(104\) − 6075.93i − 0.561754i
\(105\) 0 0
\(106\) 7977.87 0.710027
\(107\) − 21890.2i − 1.91197i −0.293410 0.955987i \(-0.594790\pi\)
0.293410 0.955987i \(-0.405210\pi\)
\(108\) 0 0
\(109\) −8385.96 −0.705830 −0.352915 0.935655i \(-0.614809\pi\)
−0.352915 + 0.935655i \(0.614809\pi\)
\(110\) 172.731i 0.0142753i
\(111\) 0 0
\(112\) 0 0
\(113\) − 8691.80i − 0.680695i −0.940300 0.340348i \(-0.889455\pi\)
0.940300 0.340348i \(-0.110545\pi\)
\(114\) 0 0
\(115\) −3305.14 −0.249916
\(116\) − 11657.0i − 0.866306i
\(117\) 0 0
\(118\) −5418.27 −0.389132
\(119\) 0 0
\(120\) 0 0
\(121\) 14635.4 0.999617
\(122\) 9173.59i 0.616339i
\(123\) 0 0
\(124\) −6339.20 −0.412279
\(125\) 15089.1i 0.965703i
\(126\) 0 0
\(127\) −238.060 −0.0147598 −0.00737989 0.999973i \(-0.502349\pi\)
−0.00737989 + 0.999973i \(0.502349\pi\)
\(128\) − 1448.15i − 0.0883883i
\(129\) 0 0
\(130\) −19582.0 −1.15870
\(131\) 32201.2i 1.87642i 0.346073 + 0.938208i \(0.387515\pi\)
−0.346073 + 0.938208i \(0.612485\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) − 7634.53i − 0.425180i
\(135\) 0 0
\(136\) −5850.62 −0.316318
\(137\) 35602.4i 1.89687i 0.316970 + 0.948436i \(0.397335\pi\)
−0.316970 + 0.948436i \(0.602665\pi\)
\(138\) 0 0
\(139\) 11433.9 0.591786 0.295893 0.955221i \(-0.404383\pi\)
0.295893 + 0.955221i \(0.404383\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 13237.2 0.656476
\(143\) − 636.015i − 0.0311025i
\(144\) 0 0
\(145\) −37569.2 −1.78688
\(146\) 2656.20i 0.124610i
\(147\) 0 0
\(148\) −16033.0 −0.731966
\(149\) 21519.8i 0.969318i 0.874703 + 0.484659i \(0.161056\pi\)
−0.874703 + 0.484659i \(0.838944\pi\)
\(150\) 0 0
\(151\) −9444.26 −0.414204 −0.207102 0.978319i \(-0.566403\pi\)
−0.207102 + 0.978319i \(0.566403\pi\)
\(152\) 11939.0i 0.516750i
\(153\) 0 0
\(154\) 0 0
\(155\) 20430.5i 0.850386i
\(156\) 0 0
\(157\) −33876.2 −1.37435 −0.687173 0.726494i \(-0.741148\pi\)
−0.687173 + 0.726494i \(0.741148\pi\)
\(158\) − 15248.1i − 0.610804i
\(159\) 0 0
\(160\) −4667.23 −0.182314
\(161\) 0 0
\(162\) 0 0
\(163\) −40735.0 −1.53318 −0.766589 0.642138i \(-0.778048\pi\)
−0.766589 + 0.642138i \(0.778048\pi\)
\(164\) − 8113.03i − 0.301645i
\(165\) 0 0
\(166\) −22603.2 −0.820264
\(167\) 8580.60i 0.307670i 0.988097 + 0.153835i \(0.0491624\pi\)
−0.988097 + 0.153835i \(0.950838\pi\)
\(168\) 0 0
\(169\) 43542.4 1.52454
\(170\) 18855.9i 0.652452i
\(171\) 0 0
\(172\) −19935.7 −0.673868
\(173\) 18416.1i 0.615326i 0.951495 + 0.307663i \(0.0995469\pi\)
−0.951495 + 0.307663i \(0.900453\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) − 151.590i − 0.00489378i
\(177\) 0 0
\(178\) −11678.5 −0.368593
\(179\) 53015.7i 1.65462i 0.561744 + 0.827311i \(0.310130\pi\)
−0.561744 + 0.827311i \(0.689870\pi\)
\(180\) 0 0
\(181\) 23866.0 0.728487 0.364243 0.931304i \(-0.381328\pi\)
0.364243 + 0.931304i \(0.381328\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 2900.62 0.0856751
\(185\) 51672.5i 1.50979i
\(186\) 0 0
\(187\) −612.430 −0.0175135
\(188\) − 5246.53i − 0.148442i
\(189\) 0 0
\(190\) 38478.0 1.06587
\(191\) 5915.71i 0.162159i 0.996708 + 0.0810793i \(0.0258367\pi\)
−0.996708 + 0.0810793i \(0.974163\pi\)
\(192\) 0 0
\(193\) 2029.50 0.0544847 0.0272424 0.999629i \(-0.491327\pi\)
0.0272424 + 0.999629i \(0.491327\pi\)
\(194\) − 11236.0i − 0.298545i
\(195\) 0 0
\(196\) 0 0
\(197\) − 65625.9i − 1.69100i −0.533978 0.845499i \(-0.679303\pi\)
0.533978 0.845499i \(-0.320697\pi\)
\(198\) 0 0
\(199\) −40607.3 −1.02541 −0.512705 0.858565i \(-0.671357\pi\)
−0.512705 + 0.858565i \(0.671357\pi\)
\(200\) 899.818i 0.0224954i
\(201\) 0 0
\(202\) −38225.1 −0.936798
\(203\) 0 0
\(204\) 0 0
\(205\) −26147.4 −0.622186
\(206\) − 28894.1i − 0.680886i
\(207\) 0 0
\(208\) 17185.3 0.397220
\(209\) 1249.75i 0.0286108i
\(210\) 0 0
\(211\) 79629.9 1.78859 0.894295 0.447477i \(-0.147677\pi\)
0.894295 + 0.447477i \(0.147677\pi\)
\(212\) 22564.8i 0.502065i
\(213\) 0 0
\(214\) 61914.8 1.35197
\(215\) 64250.5i 1.38995i
\(216\) 0 0
\(217\) 0 0
\(218\) − 23719.1i − 0.499097i
\(219\) 0 0
\(220\) −488.556 −0.0100941
\(221\) − 69429.6i − 1.42154i
\(222\) 0 0
\(223\) −76804.2 −1.54446 −0.772228 0.635346i \(-0.780858\pi\)
−0.772228 + 0.635346i \(0.780858\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 24584.1 0.481324
\(227\) − 21098.1i − 0.409441i −0.978820 0.204720i \(-0.934371\pi\)
0.978820 0.204720i \(-0.0656285\pi\)
\(228\) 0 0
\(229\) −45785.6 −0.873088 −0.436544 0.899683i \(-0.643798\pi\)
−0.436544 + 0.899683i \(0.643798\pi\)
\(230\) − 9348.35i − 0.176717i
\(231\) 0 0
\(232\) 32971.0 0.612571
\(233\) 4191.34i 0.0772042i 0.999255 + 0.0386021i \(0.0122905\pi\)
−0.999255 + 0.0386021i \(0.987710\pi\)
\(234\) 0 0
\(235\) −16909.0 −0.306183
\(236\) − 15325.2i − 0.275158i
\(237\) 0 0
\(238\) 0 0
\(239\) 30870.6i 0.540442i 0.962798 + 0.270221i \(0.0870967\pi\)
−0.962798 + 0.270221i \(0.912903\pi\)
\(240\) 0 0
\(241\) 33963.1 0.584754 0.292377 0.956303i \(-0.405554\pi\)
0.292377 + 0.956303i \(0.405554\pi\)
\(242\) 41395.1i 0.706836i
\(243\) 0 0
\(244\) −25946.8 −0.435818
\(245\) 0 0
\(246\) 0 0
\(247\) −141681. −2.32229
\(248\) − 17930.0i − 0.291525i
\(249\) 0 0
\(250\) −42678.5 −0.682855
\(251\) − 32972.4i − 0.523364i −0.965154 0.261682i \(-0.915723\pi\)
0.965154 0.261682i \(-0.0842771\pi\)
\(252\) 0 0
\(253\) 303.630 0.00474356
\(254\) − 673.336i − 0.0104367i
\(255\) 0 0
\(256\) 4096.00 0.0625000
\(257\) − 84545.3i − 1.28004i −0.768359 0.640019i \(-0.778926\pi\)
0.768359 0.640019i \(-0.221074\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) − 55386.3i − 0.819324i
\(261\) 0 0
\(262\) −91078.6 −1.32683
\(263\) 103271.i 1.49303i 0.665371 + 0.746513i \(0.268273\pi\)
−0.665371 + 0.746513i \(0.731727\pi\)
\(264\) 0 0
\(265\) 72723.8 1.03558
\(266\) 0 0
\(267\) 0 0
\(268\) 21593.7 0.300648
\(269\) 115242.i 1.59259i 0.604907 + 0.796296i \(0.293210\pi\)
−0.604907 + 0.796296i \(0.706790\pi\)
\(270\) 0 0
\(271\) −119513. −1.62734 −0.813668 0.581330i \(-0.802533\pi\)
−0.813668 + 0.581330i \(0.802533\pi\)
\(272\) − 16548.0i − 0.223671i
\(273\) 0 0
\(274\) −100699. −1.34129
\(275\) 94.1910i 0.00124550i
\(276\) 0 0
\(277\) −35918.2 −0.468117 −0.234059 0.972222i \(-0.575201\pi\)
−0.234059 + 0.972222i \(0.575201\pi\)
\(278\) 32339.9i 0.418456i
\(279\) 0 0
\(280\) 0 0
\(281\) − 107567.i − 1.36227i −0.732156 0.681137i \(-0.761486\pi\)
0.732156 0.681137i \(-0.238514\pi\)
\(282\) 0 0
\(283\) 74573.0 0.931127 0.465563 0.885015i \(-0.345852\pi\)
0.465563 + 0.885015i \(0.345852\pi\)
\(284\) 37440.4i 0.464199i
\(285\) 0 0
\(286\) 1798.92 0.0219928
\(287\) 0 0
\(288\) 0 0
\(289\) 16666.1 0.199544
\(290\) − 106262.i − 1.26352i
\(291\) 0 0
\(292\) −7512.86 −0.0881129
\(293\) − 31081.0i − 0.362042i −0.983479 0.181021i \(-0.942060\pi\)
0.983479 0.181021i \(-0.0579403\pi\)
\(294\) 0 0
\(295\) −49391.3 −0.567552
\(296\) − 45348.1i − 0.517578i
\(297\) 0 0
\(298\) −60867.3 −0.685411
\(299\) 34421.7i 0.385026i
\(300\) 0 0
\(301\) 0 0
\(302\) − 26712.4i − 0.292886i
\(303\) 0 0
\(304\) −33768.6 −0.365397
\(305\) 83623.6i 0.898937i
\(306\) 0 0
\(307\) −19922.2 −0.211378 −0.105689 0.994399i \(-0.533705\pi\)
−0.105689 + 0.994399i \(0.533705\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −57786.2 −0.601314
\(311\) 174219.i 1.80125i 0.434596 + 0.900626i \(0.356891\pi\)
−0.434596 + 0.900626i \(0.643109\pi\)
\(312\) 0 0
\(313\) −85378.5 −0.871485 −0.435743 0.900071i \(-0.643514\pi\)
−0.435743 + 0.900071i \(0.643514\pi\)
\(314\) − 95816.5i − 0.971809i
\(315\) 0 0
\(316\) 43128.2 0.431904
\(317\) − 177007.i − 1.76146i −0.473623 0.880728i \(-0.657054\pi\)
0.473623 0.880728i \(-0.342946\pi\)
\(318\) 0 0
\(319\) 3451.33 0.0339161
\(320\) − 13200.9i − 0.128915i
\(321\) 0 0
\(322\) 0 0
\(323\) 136427.i 1.30766i
\(324\) 0 0
\(325\) −10678.2 −0.101095
\(326\) − 115216.i − 1.08412i
\(327\) 0 0
\(328\) 22947.1 0.213295
\(329\) 0 0
\(330\) 0 0
\(331\) 137003. 1.25047 0.625234 0.780437i \(-0.285003\pi\)
0.625234 + 0.780437i \(0.285003\pi\)
\(332\) − 63931.5i − 0.580014i
\(333\) 0 0
\(334\) −24269.6 −0.217555
\(335\) − 69594.1i − 0.620130i
\(336\) 0 0
\(337\) 56842.6 0.500511 0.250256 0.968180i \(-0.419485\pi\)
0.250256 + 0.968180i \(0.419485\pi\)
\(338\) 123156.i 1.07801i
\(339\) 0 0
\(340\) −53332.4 −0.461353
\(341\) − 1876.87i − 0.0161408i
\(342\) 0 0
\(343\) 0 0
\(344\) − 56386.7i − 0.476497i
\(345\) 0 0
\(346\) −52088.6 −0.435101
\(347\) − 51452.6i − 0.427315i −0.976909 0.213658i \(-0.931462\pi\)
0.976909 0.213658i \(-0.0685377\pi\)
\(348\) 0 0
\(349\) −69068.5 −0.567060 −0.283530 0.958963i \(-0.591506\pi\)
−0.283530 + 0.958963i \(0.591506\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 428.761 0.00346043
\(353\) 20365.1i 0.163432i 0.996656 + 0.0817161i \(0.0260401\pi\)
−0.996656 + 0.0817161i \(0.973960\pi\)
\(354\) 0 0
\(355\) 120666. 0.957477
\(356\) − 33031.8i − 0.260634i
\(357\) 0 0
\(358\) −149951. −1.16999
\(359\) − 134562.i − 1.04408i −0.852922 0.522039i \(-0.825172\pi\)
0.852922 0.522039i \(-0.174828\pi\)
\(360\) 0 0
\(361\) 148076. 1.13624
\(362\) 67503.1i 0.515118i
\(363\) 0 0
\(364\) 0 0
\(365\) 24213.1i 0.181746i
\(366\) 0 0
\(367\) 199428. 1.48066 0.740329 0.672244i \(-0.234670\pi\)
0.740329 + 0.672244i \(0.234670\pi\)
\(368\) 8204.18i 0.0605814i
\(369\) 0 0
\(370\) −146152. −1.06758
\(371\) 0 0
\(372\) 0 0
\(373\) 12186.3 0.0875898 0.0437949 0.999041i \(-0.486055\pi\)
0.0437949 + 0.999041i \(0.486055\pi\)
\(374\) − 1732.21i − 0.0123839i
\(375\) 0 0
\(376\) 14839.4 0.104964
\(377\) 391269.i 2.75291i
\(378\) 0 0
\(379\) −47966.9 −0.333936 −0.166968 0.985962i \(-0.553398\pi\)
−0.166968 + 0.985962i \(0.553398\pi\)
\(380\) 108832.i 0.753685i
\(381\) 0 0
\(382\) −16732.1 −0.114663
\(383\) − 7540.04i − 0.0514015i −0.999670 0.0257008i \(-0.991818\pi\)
0.999670 0.0257008i \(-0.00818170\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 5740.30i 0.0385265i
\(387\) 0 0
\(388\) 31780.3 0.211103
\(389\) − 30745.8i − 0.203183i −0.994826 0.101591i \(-0.967607\pi\)
0.994826 0.101591i \(-0.0323935\pi\)
\(390\) 0 0
\(391\) 33145.3 0.216804
\(392\) 0 0
\(393\) 0 0
\(394\) 185618. 1.19572
\(395\) − 138997.i − 0.890865i
\(396\) 0 0
\(397\) 261023. 1.65614 0.828070 0.560625i \(-0.189439\pi\)
0.828070 + 0.560625i \(0.189439\pi\)
\(398\) − 114855.i − 0.725074i
\(399\) 0 0
\(400\) −2545.07 −0.0159067
\(401\) − 10220.1i − 0.0635574i −0.999495 0.0317787i \(-0.989883\pi\)
0.999495 0.0317787i \(-0.0101172\pi\)
\(402\) 0 0
\(403\) 212776. 1.31012
\(404\) − 108117.i − 0.662416i
\(405\) 0 0
\(406\) 0 0
\(407\) − 4746.95i − 0.0286567i
\(408\) 0 0
\(409\) −135962. −0.812776 −0.406388 0.913701i \(-0.633212\pi\)
−0.406388 + 0.913701i \(0.633212\pi\)
\(410\) − 73955.9i − 0.439952i
\(411\) 0 0
\(412\) 81724.8 0.481459
\(413\) 0 0
\(414\) 0 0
\(415\) −206044. −1.19636
\(416\) 48607.4i 0.280877i
\(417\) 0 0
\(418\) −3534.82 −0.0202309
\(419\) 239307.i 1.36310i 0.731773 + 0.681549i \(0.238693\pi\)
−0.731773 + 0.681549i \(0.761307\pi\)
\(420\) 0 0
\(421\) 221785. 1.25132 0.625659 0.780097i \(-0.284830\pi\)
0.625659 + 0.780097i \(0.284830\pi\)
\(422\) 225227.i 1.26472i
\(423\) 0 0
\(424\) −63822.9 −0.355014
\(425\) 10282.2i 0.0569257i
\(426\) 0 0
\(427\) 0 0
\(428\) 175121.i 0.955987i
\(429\) 0 0
\(430\) −181728. −0.982844
\(431\) − 25402.7i − 0.136750i −0.997660 0.0683748i \(-0.978219\pi\)
0.997660 0.0683748i \(-0.0217814\pi\)
\(432\) 0 0
\(433\) 277393. 1.47952 0.739759 0.672872i \(-0.234939\pi\)
0.739759 + 0.672872i \(0.234939\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 67087.7 0.352915
\(437\) − 67637.5i − 0.354181i
\(438\) 0 0
\(439\) 245813. 1.27549 0.637743 0.770249i \(-0.279868\pi\)
0.637743 + 0.770249i \(0.279868\pi\)
\(440\) − 1381.85i − 0.00713763i
\(441\) 0 0
\(442\) 196376. 1.00518
\(443\) 49514.3i 0.252304i 0.992011 + 0.126152i \(0.0402626\pi\)
−0.992011 + 0.126152i \(0.959737\pi\)
\(444\) 0 0
\(445\) −106458. −0.537596
\(446\) − 217235.i − 1.09210i
\(447\) 0 0
\(448\) 0 0
\(449\) 315550.i 1.56522i 0.622512 + 0.782610i \(0.286112\pi\)
−0.622512 + 0.782610i \(0.713888\pi\)
\(450\) 0 0
\(451\) 2402.06 0.0118095
\(452\) 69534.4i 0.340348i
\(453\) 0 0
\(454\) 59674.4 0.289518
\(455\) 0 0
\(456\) 0 0
\(457\) 47052.2 0.225293 0.112646 0.993635i \(-0.464067\pi\)
0.112646 + 0.993635i \(0.464067\pi\)
\(458\) − 129501.i − 0.617367i
\(459\) 0 0
\(460\) 26441.1 0.124958
\(461\) − 144717.i − 0.680956i −0.940252 0.340478i \(-0.889411\pi\)
0.940252 0.340478i \(-0.110589\pi\)
\(462\) 0 0
\(463\) −314156. −1.46549 −0.732746 0.680502i \(-0.761762\pi\)
−0.732746 + 0.680502i \(0.761762\pi\)
\(464\) 93256.1i 0.433153i
\(465\) 0 0
\(466\) −11854.9 −0.0545916
\(467\) − 71579.3i − 0.328211i −0.986443 0.164106i \(-0.947526\pi\)
0.986443 0.164106i \(-0.0524738\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) − 47825.8i − 0.216504i
\(471\) 0 0
\(472\) 43346.1 0.194566
\(473\) − 5902.44i − 0.0263821i
\(474\) 0 0
\(475\) 20982.3 0.0929962
\(476\) 0 0
\(477\) 0 0
\(478\) −87315.2 −0.382150
\(479\) − 31394.1i − 0.136829i −0.997657 0.0684144i \(-0.978206\pi\)
0.997657 0.0684144i \(-0.0217940\pi\)
\(480\) 0 0
\(481\) 538149. 2.32601
\(482\) 96062.2i 0.413484i
\(483\) 0 0
\(484\) −117083. −0.499808
\(485\) − 102424.i − 0.435431i
\(486\) 0 0
\(487\) −162986. −0.687214 −0.343607 0.939114i \(-0.611649\pi\)
−0.343607 + 0.939114i \(0.611649\pi\)
\(488\) − 73388.7i − 0.308170i
\(489\) 0 0
\(490\) 0 0
\(491\) 213254.i 0.884573i 0.896874 + 0.442286i \(0.145833\pi\)
−0.896874 + 0.442286i \(0.854167\pi\)
\(492\) 0 0
\(493\) 376759. 1.55014
\(494\) − 400733.i − 1.64211i
\(495\) 0 0
\(496\) 50713.6 0.206140
\(497\) 0 0
\(498\) 0 0
\(499\) 170422. 0.684421 0.342210 0.939623i \(-0.388824\pi\)
0.342210 + 0.939623i \(0.388824\pi\)
\(500\) − 120713.i − 0.482852i
\(501\) 0 0
\(502\) 93260.1 0.370074
\(503\) 144506.i 0.571151i 0.958356 + 0.285575i \(0.0921847\pi\)
−0.958356 + 0.285575i \(0.907815\pi\)
\(504\) 0 0
\(505\) −348448. −1.36633
\(506\) 858.796i 0.00335420i
\(507\) 0 0
\(508\) 1904.48 0.00737989
\(509\) 423335.i 1.63399i 0.576647 + 0.816993i \(0.304361\pi\)
−0.576647 + 0.816993i \(0.695639\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11585.2i 0.0441942i
\(513\) 0 0
\(514\) 239130. 0.905124
\(515\) − 263390.i − 0.993080i
\(516\) 0 0
\(517\) 1553.36 0.00581154
\(518\) 0 0
\(519\) 0 0
\(520\) 156656. 0.579350
\(521\) 221150.i 0.814726i 0.913266 + 0.407363i \(0.133552\pi\)
−0.913266 + 0.407363i \(0.866448\pi\)
\(522\) 0 0
\(523\) −377707. −1.38087 −0.690434 0.723396i \(-0.742580\pi\)
−0.690434 + 0.723396i \(0.742580\pi\)
\(524\) − 257609.i − 0.938208i
\(525\) 0 0
\(526\) −292095. −1.05573
\(527\) − 204886.i − 0.737718i
\(528\) 0 0
\(529\) 263408. 0.941278
\(530\) 205694.i 0.732267i
\(531\) 0 0
\(532\) 0 0
\(533\) 272315.i 0.958554i
\(534\) 0 0
\(535\) 564396. 1.97186
\(536\) 61076.3i 0.212590i
\(537\) 0 0
\(538\) −325952. −1.12613
\(539\) 0 0
\(540\) 0 0
\(541\) 135542. 0.463105 0.231553 0.972822i \(-0.425619\pi\)
0.231553 + 0.972822i \(0.425619\pi\)
\(542\) − 338034.i − 1.15070i
\(543\) 0 0
\(544\) 46804.9 0.158159
\(545\) − 216216.i − 0.727938i
\(546\) 0 0
\(547\) 314274. 1.05035 0.525175 0.850994i \(-0.324000\pi\)
0.525175 + 0.850994i \(0.324000\pi\)
\(548\) − 284819.i − 0.948436i
\(549\) 0 0
\(550\) −266.412 −0.000880702 0
\(551\) − 768829.i − 2.53237i
\(552\) 0 0
\(553\) 0 0
\(554\) − 101592.i − 0.331009i
\(555\) 0 0
\(556\) −91471.2 −0.295893
\(557\) 59258.7i 0.191004i 0.995429 + 0.0955019i \(0.0304456\pi\)
−0.995429 + 0.0955019i \(0.969554\pi\)
\(558\) 0 0
\(559\) 669144. 2.14139
\(560\) 0 0
\(561\) 0 0
\(562\) 304244. 0.963273
\(563\) − 214575.i − 0.676958i −0.940974 0.338479i \(-0.890088\pi\)
0.940974 0.338479i \(-0.109912\pi\)
\(564\) 0 0
\(565\) 224101. 0.702016
\(566\) 210924.i 0.658406i
\(567\) 0 0
\(568\) −105897. −0.328238
\(569\) − 123955.i − 0.382861i −0.981506 0.191430i \(-0.938687\pi\)
0.981506 0.191430i \(-0.0613126\pi\)
\(570\) 0 0
\(571\) 169974. 0.521327 0.260664 0.965430i \(-0.416059\pi\)
0.260664 + 0.965430i \(0.416059\pi\)
\(572\) 5088.12i 0.0155513i
\(573\) 0 0
\(574\) 0 0
\(575\) − 5097.71i − 0.0154184i
\(576\) 0 0
\(577\) 336708. 1.01135 0.505675 0.862724i \(-0.331243\pi\)
0.505675 + 0.862724i \(0.331243\pi\)
\(578\) 47138.8i 0.141099i
\(579\) 0 0
\(580\) 300554. 0.893441
\(581\) 0 0
\(582\) 0 0
\(583\) −6680.85 −0.0196560
\(584\) − 21249.6i − 0.0623052i
\(585\) 0 0
\(586\) 87910.3 0.256003
\(587\) 216616.i 0.628658i 0.949314 + 0.314329i \(0.101780\pi\)
−0.949314 + 0.314329i \(0.898220\pi\)
\(588\) 0 0
\(589\) −418097. −1.20517
\(590\) − 139700.i − 0.401320i
\(591\) 0 0
\(592\) 128264. 0.365983
\(593\) − 186592.i − 0.530619i −0.964163 0.265310i \(-0.914526\pi\)
0.964163 0.265310i \(-0.0854742\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) − 172159.i − 0.484659i
\(597\) 0 0
\(598\) −97359.4 −0.272255
\(599\) − 239406.i − 0.667238i −0.942708 0.333619i \(-0.891730\pi\)
0.942708 0.333619i \(-0.108270\pi\)
\(600\) 0 0
\(601\) 51508.8 0.142604 0.0713021 0.997455i \(-0.477285\pi\)
0.0713021 + 0.997455i \(0.477285\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 75554.1 0.207102
\(605\) 377345.i 1.03093i
\(606\) 0 0
\(607\) −24554.5 −0.0666429 −0.0333214 0.999445i \(-0.510609\pi\)
−0.0333214 + 0.999445i \(0.510609\pi\)
\(608\) − 95511.9i − 0.258375i
\(609\) 0 0
\(610\) −236523. −0.635645
\(611\) 176100.i 0.471713i
\(612\) 0 0
\(613\) 169561. 0.451237 0.225618 0.974216i \(-0.427560\pi\)
0.225618 + 0.974216i \(0.427560\pi\)
\(614\) − 56348.5i − 0.149467i
\(615\) 0 0
\(616\) 0 0
\(617\) − 624837.i − 1.64133i −0.571409 0.820665i \(-0.693603\pi\)
0.571409 0.820665i \(-0.306397\pi\)
\(618\) 0 0
\(619\) −334855. −0.873928 −0.436964 0.899479i \(-0.643946\pi\)
−0.436964 + 0.899479i \(0.643946\pi\)
\(620\) − 163444.i − 0.425193i
\(621\) 0 0
\(622\) −492765. −1.27368
\(623\) 0 0
\(624\) 0 0
\(625\) −413898. −1.05958
\(626\) − 241487.i − 0.616233i
\(627\) 0 0
\(628\) 271010. 0.687173
\(629\) − 518193.i − 1.30975i
\(630\) 0 0
\(631\) −154411. −0.387809 −0.193905 0.981020i \(-0.562115\pi\)
−0.193905 + 0.981020i \(0.562115\pi\)
\(632\) 121985.i 0.305402i
\(633\) 0 0
\(634\) 500651. 1.24554
\(635\) − 6137.93i − 0.0152221i
\(636\) 0 0
\(637\) 0 0
\(638\) 9761.85i 0.0239823i
\(639\) 0 0
\(640\) 37337.9 0.0911569
\(641\) 350351.i 0.852682i 0.904562 + 0.426341i \(0.140198\pi\)
−0.904562 + 0.426341i \(0.859802\pi\)
\(642\) 0 0
\(643\) 79292.9 0.191784 0.0958920 0.995392i \(-0.469430\pi\)
0.0958920 + 0.995392i \(0.469430\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −385873. −0.924654
\(647\) 711211.i 1.69899i 0.527600 + 0.849493i \(0.323092\pi\)
−0.527600 + 0.849493i \(0.676908\pi\)
\(648\) 0 0
\(649\) 4537.38 0.0107725
\(650\) − 30202.5i − 0.0714851i
\(651\) 0 0
\(652\) 325880. 0.766589
\(653\) 33637.9i 0.0788864i 0.999222 + 0.0394432i \(0.0125584\pi\)
−0.999222 + 0.0394432i \(0.987442\pi\)
\(654\) 0 0
\(655\) −830245. −1.93519
\(656\) 64904.3i 0.150822i
\(657\) 0 0
\(658\) 0 0
\(659\) 358536.i 0.825585i 0.910825 + 0.412793i \(0.135447\pi\)
−0.910825 + 0.412793i \(0.864553\pi\)
\(660\) 0 0
\(661\) 584224. 1.33714 0.668569 0.743650i \(-0.266907\pi\)
0.668569 + 0.743650i \(0.266907\pi\)
\(662\) 387502.i 0.884215i
\(663\) 0 0
\(664\) 180826. 0.410132
\(665\) 0 0
\(666\) 0 0
\(667\) −186789. −0.419856
\(668\) − 68644.8i − 0.153835i
\(669\) 0 0
\(670\) 196842. 0.438498
\(671\) − 7682.18i − 0.0170624i
\(672\) 0 0
\(673\) −175181. −0.386773 −0.193387 0.981123i \(-0.561947\pi\)
−0.193387 + 0.981123i \(0.561947\pi\)
\(674\) 160775.i 0.353915i
\(675\) 0 0
\(676\) −348339. −0.762269
\(677\) − 61931.0i − 0.135124i −0.997715 0.0675618i \(-0.978478\pi\)
0.997715 0.0675618i \(-0.0215220\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) − 150847.i − 0.326226i
\(681\) 0 0
\(682\) 5308.60 0.0114133
\(683\) − 206583.i − 0.442846i −0.975178 0.221423i \(-0.928930\pi\)
0.975178 0.221423i \(-0.0710702\pi\)
\(684\) 0 0
\(685\) −917939. −1.95629
\(686\) 0 0
\(687\) 0 0
\(688\) 159486. 0.336934
\(689\) − 757390.i − 1.59544i
\(690\) 0 0
\(691\) 140378. 0.293997 0.146998 0.989137i \(-0.453039\pi\)
0.146998 + 0.989137i \(0.453039\pi\)
\(692\) − 147329.i − 0.307663i
\(693\) 0 0
\(694\) 145530. 0.302157
\(695\) 294801.i 0.610322i
\(696\) 0 0
\(697\) 262216. 0.539752
\(698\) − 195355.i − 0.400972i
\(699\) 0 0
\(700\) 0 0
\(701\) 798257.i 1.62445i 0.583343 + 0.812226i \(0.301744\pi\)
−0.583343 + 0.812226i \(0.698256\pi\)
\(702\) 0 0
\(703\) −1.05744e6 −2.13967
\(704\) 1212.72i 0.00244689i
\(705\) 0 0
\(706\) −57601.3 −0.115564
\(707\) 0 0
\(708\) 0 0
\(709\) −409921. −0.815470 −0.407735 0.913100i \(-0.633681\pi\)
−0.407735 + 0.913100i \(0.633681\pi\)
\(710\) 341295.i 0.677039i
\(711\) 0 0
\(712\) 93427.9 0.184296
\(713\) 101578.i 0.199812i
\(714\) 0 0
\(715\) 16398.4 0.0320767
\(716\) − 424126.i − 0.827311i
\(717\) 0 0
\(718\) 380598. 0.738274
\(719\) − 721642.i − 1.39593i −0.716131 0.697966i \(-0.754089\pi\)
0.716131 0.697966i \(-0.245911\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 418823.i 0.803445i
\(723\) 0 0
\(724\) −190928. −0.364243
\(725\) − 57945.1i − 0.110240i
\(726\) 0 0
\(727\) −695893. −1.31666 −0.658330 0.752729i \(-0.728737\pi\)
−0.658330 + 0.752729i \(0.728737\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −68484.9 −0.128514
\(731\) − 644330.i − 1.20580i
\(732\) 0 0
\(733\) 344991. 0.642096 0.321048 0.947063i \(-0.395965\pi\)
0.321048 + 0.947063i \(0.395965\pi\)
\(734\) 564069.i 1.04698i
\(735\) 0 0
\(736\) −23204.9 −0.0428375
\(737\) 6393.33i 0.0117704i
\(738\) 0 0
\(739\) 553173. 1.01291 0.506457 0.862266i \(-0.330955\pi\)
0.506457 + 0.862266i \(0.330955\pi\)
\(740\) − 413380.i − 0.754893i
\(741\) 0 0
\(742\) 0 0
\(743\) − 331872.i − 0.601163i −0.953756 0.300582i \(-0.902819\pi\)
0.953756 0.300582i \(-0.0971808\pi\)
\(744\) 0 0
\(745\) −554847. −0.999680
\(746\) 34468.0i 0.0619353i
\(747\) 0 0
\(748\) 4899.44 0.00875676
\(749\) 0 0
\(750\) 0 0
\(751\) 908007. 1.60994 0.804969 0.593316i \(-0.202182\pi\)
0.804969 + 0.593316i \(0.202182\pi\)
\(752\) 41972.3i 0.0742210i
\(753\) 0 0
\(754\) −1.10667e6 −1.94660
\(755\) − 243502.i − 0.427178i
\(756\) 0 0
\(757\) 906297. 1.58153 0.790767 0.612117i \(-0.209682\pi\)
0.790767 + 0.612117i \(0.209682\pi\)
\(758\) − 135671.i − 0.236129i
\(759\) 0 0
\(760\) −307824. −0.532936
\(761\) 972723.i 1.67966i 0.542853 + 0.839828i \(0.317344\pi\)
−0.542853 + 0.839828i \(0.682656\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) − 47325.7i − 0.0810793i
\(765\) 0 0
\(766\) 21326.4 0.0363464
\(767\) 514391.i 0.874384i
\(768\) 0 0
\(769\) −91810.6 −0.155253 −0.0776265 0.996983i \(-0.524734\pi\)
−0.0776265 + 0.996983i \(0.524734\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −16236.0 −0.0272424
\(773\) 493582.i 0.826039i 0.910722 + 0.413020i \(0.135526\pi\)
−0.910722 + 0.413020i \(0.864474\pi\)
\(774\) 0 0
\(775\) −31511.2 −0.0524639
\(776\) 89888.3i 0.149272i
\(777\) 0 0
\(778\) 86962.4 0.143672
\(779\) − 535089.i − 0.881761i
\(780\) 0 0
\(781\) −11085.1 −0.0181735
\(782\) 93749.0i 0.153304i
\(783\) 0 0
\(784\) 0 0
\(785\) − 873433.i − 1.41739i
\(786\) 0 0
\(787\) 172472. 0.278465 0.139232 0.990260i \(-0.455537\pi\)
0.139232 + 0.990260i \(0.455537\pi\)
\(788\) 525007.i 0.845499i
\(789\) 0 0
\(790\) 393143. 0.629937
\(791\) 0 0
\(792\) 0 0
\(793\) 870908. 1.38492
\(794\) 738283.i 1.17107i
\(795\) 0 0
\(796\) 324858. 0.512705
\(797\) − 48077.3i − 0.0756873i −0.999284 0.0378437i \(-0.987951\pi\)
0.999284 0.0378437i \(-0.0120489\pi\)
\(798\) 0 0
\(799\) 169570. 0.265617
\(800\) − 7198.54i − 0.0112477i
\(801\) 0 0
\(802\) 28906.8 0.0449419
\(803\) − 2224.36i − 0.00344964i
\(804\) 0 0
\(805\) 0 0
\(806\) 601821.i 0.926398i
\(807\) 0 0
\(808\) 305801. 0.468399
\(809\) 692003.i 1.05733i 0.848830 + 0.528666i \(0.177307\pi\)
−0.848830 + 0.528666i \(0.822693\pi\)
\(810\) 0 0
\(811\) 327815. 0.498410 0.249205 0.968451i \(-0.419831\pi\)
0.249205 + 0.968451i \(0.419831\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 13426.4 0.0202633
\(815\) − 1.05027e6i − 1.58120i
\(816\) 0 0
\(817\) −1.31484e6 −1.96984
\(818\) − 384559.i − 0.574720i
\(819\) 0 0
\(820\) 209179. 0.311093
\(821\) 327066.i 0.485232i 0.970122 + 0.242616i \(0.0780055\pi\)
−0.970122 + 0.242616i \(0.921995\pi\)
\(822\) 0 0
\(823\) 378351. 0.558593 0.279296 0.960205i \(-0.409899\pi\)
0.279296 + 0.960205i \(0.409899\pi\)
\(824\) 231153.i 0.340443i
\(825\) 0 0
\(826\) 0 0
\(827\) − 555342.i − 0.811988i −0.913876 0.405994i \(-0.866925\pi\)
0.913876 0.405994i \(-0.133075\pi\)
\(828\) 0 0
\(829\) −519799. −0.756356 −0.378178 0.925733i \(-0.623449\pi\)
−0.378178 + 0.925733i \(0.623449\pi\)
\(830\) − 582780.i − 0.845957i
\(831\) 0 0
\(832\) −137483. −0.198610
\(833\) 0 0
\(834\) 0 0
\(835\) −221234. −0.317307
\(836\) − 9997.98i − 0.0143054i
\(837\) 0 0
\(838\) −676862. −0.963855
\(839\) − 836509.i − 1.18836i −0.804333 0.594178i \(-0.797477\pi\)
0.804333 0.594178i \(-0.202523\pi\)
\(840\) 0 0
\(841\) −1.41594e6 −2.00194
\(842\) 627302.i 0.884816i
\(843\) 0 0
\(844\) −637039. −0.894295
\(845\) 1.12266e6i 1.57229i
\(846\) 0 0
\(847\) 0 0
\(848\) − 180518.i − 0.251033i
\(849\) 0 0
\(850\) −29082.5 −0.0402526
\(851\) 256909.i 0.354748i
\(852\) 0 0
\(853\) 1.03133e6 1.41742 0.708711 0.705499i \(-0.249277\pi\)
0.708711 + 0.705499i \(0.249277\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −495318. −0.675985
\(857\) − 781985.i − 1.06472i −0.846517 0.532362i \(-0.821305\pi\)
0.846517 0.532362i \(-0.178695\pi\)
\(858\) 0 0
\(859\) −1.06243e6 −1.43984 −0.719921 0.694056i \(-0.755822\pi\)
−0.719921 + 0.694056i \(0.755822\pi\)
\(860\) − 514004.i − 0.694975i
\(861\) 0 0
\(862\) 71849.8 0.0966966
\(863\) − 839858.i − 1.12768i −0.825886 0.563838i \(-0.809324\pi\)
0.825886 0.563838i \(-0.190676\pi\)
\(864\) 0 0
\(865\) −474823. −0.634600
\(866\) 784587.i 1.04618i
\(867\) 0 0
\(868\) 0 0
\(869\) 12769.1i 0.0169091i
\(870\) 0 0
\(871\) −724795. −0.955386
\(872\) 189753.i 0.249548i
\(873\) 0 0
\(874\) 191308. 0.250444
\(875\) 0 0
\(876\) 0 0
\(877\) −532982. −0.692968 −0.346484 0.938056i \(-0.612625\pi\)
−0.346484 + 0.938056i \(0.612625\pi\)
\(878\) 695264.i 0.901905i
\(879\) 0 0
\(880\) 3908.45 0.00504707
\(881\) 1.36059e6i 1.75297i 0.481430 + 0.876484i \(0.340117\pi\)
−0.481430 + 0.876484i \(0.659883\pi\)
\(882\) 0 0
\(883\) 1.42318e6 1.82531 0.912657 0.408727i \(-0.134027\pi\)
0.912657 + 0.408727i \(0.134027\pi\)
\(884\) 555436.i 0.710771i
\(885\) 0 0
\(886\) −140048. −0.178406
\(887\) − 158940.i − 0.202016i −0.994886 0.101008i \(-0.967793\pi\)
0.994886 0.101008i \(-0.0322068\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) − 301107.i − 0.380138i
\(891\) 0 0
\(892\) 614434. 0.772228
\(893\) − 346031.i − 0.433923i
\(894\) 0 0
\(895\) −1.36691e6 −1.70645
\(896\) 0 0
\(897\) 0 0
\(898\) −892510. −1.10678
\(899\) 1.15463e6i 1.42864i
\(900\) 0 0
\(901\) −729304. −0.898377
\(902\) 6794.04i 0.00835055i
\(903\) 0 0
\(904\) −196673. −0.240662
\(905\) 615338.i 0.751305i
\(906\) 0 0
\(907\) −683535. −0.830895 −0.415447 0.909617i \(-0.636375\pi\)
−0.415447 + 0.909617i \(0.636375\pi\)
\(908\) 168785.i 0.204720i
\(909\) 0 0
\(910\) 0 0
\(911\) 16125.4i 0.0194300i 0.999953 + 0.00971500i \(0.00309243\pi\)
−0.999953 + 0.00971500i \(0.996908\pi\)
\(912\) 0 0
\(913\) 18928.4 0.0227077
\(914\) 133084.i 0.159306i
\(915\) 0 0
\(916\) 366285. 0.436544
\(917\) 0 0
\(918\) 0 0
\(919\) −418627. −0.495674 −0.247837 0.968802i \(-0.579720\pi\)
−0.247837 + 0.968802i \(0.579720\pi\)
\(920\) 74786.8i 0.0883587i
\(921\) 0 0
\(922\) 409323. 0.481509
\(923\) − 1.25669e6i − 1.47511i
\(924\) 0 0
\(925\) −79697.4 −0.0931452
\(926\) − 888567.i − 1.03626i
\(927\) 0 0
\(928\) −263768. −0.306285
\(929\) 268591.i 0.311214i 0.987819 + 0.155607i \(0.0497334\pi\)
−0.987819 + 0.155607i \(0.950267\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 33530.7i − 0.0386021i
\(933\) 0 0
\(934\) 202457. 0.232080
\(935\) − 15790.3i − 0.0180621i
\(936\) 0 0
\(937\) −543470. −0.619008 −0.309504 0.950898i \(-0.600163\pi\)
−0.309504 + 0.950898i \(0.600163\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 135272. 0.153092
\(941\) 434266.i 0.490430i 0.969469 + 0.245215i \(0.0788585\pi\)
−0.969469 + 0.245215i \(0.921141\pi\)
\(942\) 0 0
\(943\) −130002. −0.146192
\(944\) 122601.i 0.137579i
\(945\) 0 0
\(946\) 16694.6 0.0186550
\(947\) 873995.i 0.974560i 0.873246 + 0.487280i \(0.162011\pi\)
−0.873246 + 0.487280i \(0.837989\pi\)
\(948\) 0 0
\(949\) 252170. 0.280001
\(950\) 59346.8i 0.0657582i
\(951\) 0 0
\(952\) 0 0
\(953\) − 1.51221e6i − 1.66504i −0.553994 0.832521i \(-0.686897\pi\)
0.553994 0.832521i \(-0.313103\pi\)
\(954\) 0 0
\(955\) −152525. −0.167238
\(956\) − 246965.i − 0.270221i
\(957\) 0 0
\(958\) 88796.1 0.0967526
\(959\) 0 0
\(960\) 0 0
\(961\) −295622. −0.320104
\(962\) 1.52211e6i 1.64474i
\(963\) 0 0
\(964\) −271705. −0.292377
\(965\) 52326.8i 0.0561913i
\(966\) 0 0
\(967\) 1.49916e6 1.60322 0.801612 0.597844i \(-0.203976\pi\)
0.801612 + 0.597844i \(0.203976\pi\)
\(968\) − 331161.i − 0.353418i
\(969\) 0 0
\(970\) 289699. 0.307896
\(971\) − 470558.i − 0.499085i −0.968364 0.249543i \(-0.919720\pi\)
0.968364 0.249543i \(-0.0802803\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) − 460994.i − 0.485934i
\(975\) 0 0
\(976\) 207575. 0.217909
\(977\) − 126944.i − 0.132992i −0.997787 0.0664959i \(-0.978818\pi\)
0.997787 0.0664959i \(-0.0211819\pi\)
\(978\) 0 0
\(979\) 9779.84 0.0102039
\(980\) 0 0
\(981\) 0 0
\(982\) −603173. −0.625487
\(983\) − 218289.i − 0.225904i −0.993600 0.112952i \(-0.963969\pi\)
0.993600 0.112952i \(-0.0360306\pi\)
\(984\) 0 0
\(985\) 1.69204e6 1.74396
\(986\) 1.06564e6i 1.09611i
\(987\) 0 0
\(988\) 1.13344e6 1.16114
\(989\) 319446.i 0.326591i
\(990\) 0 0
\(991\) −203403. −0.207115 −0.103557 0.994623i \(-0.533023\pi\)
−0.103557 + 0.994623i \(0.533023\pi\)
\(992\) 143440.i 0.145763i
\(993\) 0 0
\(994\) 0 0
\(995\) − 1.04698e6i − 1.05753i
\(996\) 0 0
\(997\) 461568. 0.464350 0.232175 0.972674i \(-0.425416\pi\)
0.232175 + 0.972674i \(0.425416\pi\)
\(998\) 482025.i 0.483959i
\(999\) 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 882.5.b.g.197.6 6
3.2 odd 2 inner 882.5.b.g.197.1 6
7.3 odd 6 126.5.s.a.107.3 yes 12
7.5 odd 6 126.5.s.a.53.4 yes 12
7.6 odd 2 882.5.b.f.197.4 6
21.5 even 6 126.5.s.a.53.3 12
21.17 even 6 126.5.s.a.107.4 yes 12
21.20 even 2 882.5.b.f.197.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.5.s.a.53.3 12 21.5 even 6
126.5.s.a.53.4 yes 12 7.5 odd 6
126.5.s.a.107.3 yes 12 7.3 odd 6
126.5.s.a.107.4 yes 12 21.17 even 6
882.5.b.f.197.3 6 21.20 even 2
882.5.b.f.197.4 6 7.6 odd 2
882.5.b.g.197.1 6 3.2 odd 2 inner
882.5.b.g.197.6 6 1.1 even 1 trivial