Properties

Label 531.5.b.a.296.65
Level $531$
Weight $5$
Character 531.296
Analytic conductor $54.889$
Analytic rank $0$
Dimension $76$
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] = [531,5,Mod(296,531)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(531, 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("531.296");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 531.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: \(54.8894503975\)
Analytic rank: \(0\)
Dimension: \(76\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 296.65
Character \(\chi\) \(=\) 531.296
Dual form 531.5.b.a.296.12

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+6.05329i q^{2} -20.6423 q^{4} +34.3242i q^{5} -5.09902 q^{7} -28.1013i q^{8} +O(q^{10})\) \(q+6.05329i q^{2} -20.6423 q^{4} +34.3242i q^{5} -5.09902 q^{7} -28.1013i q^{8} -207.774 q^{10} -148.668i q^{11} +69.3021 q^{13} -30.8658i q^{14} -160.172 q^{16} -321.266i q^{17} +76.0665 q^{19} -708.530i q^{20} +899.929 q^{22} -232.292i q^{23} -553.149 q^{25} +419.506i q^{26} +105.256 q^{28} -806.320i q^{29} +882.027 q^{31} -1419.19i q^{32} +1944.72 q^{34} -175.020i q^{35} -1272.10 q^{37} +460.452i q^{38} +964.553 q^{40} -2407.30i q^{41} +358.183 q^{43} +3068.85i q^{44} +1406.13 q^{46} -1990.82i q^{47} -2375.00 q^{49} -3348.37i q^{50} -1430.56 q^{52} -1991.68i q^{53} +5102.90 q^{55} +143.289i q^{56} +4880.89 q^{58} -453.188i q^{59} -6446.83 q^{61} +5339.17i q^{62} +6028.00 q^{64} +2378.74i q^{65} -2781.90 q^{67} +6631.68i q^{68} +1059.44 q^{70} +6198.13i q^{71} +5122.02 q^{73} -7700.41i q^{74} -1570.19 q^{76} +758.060i q^{77} +4187.53 q^{79} -5497.77i q^{80} +14572.1 q^{82} -3309.98i q^{83} +11027.2 q^{85} +2168.18i q^{86} -4177.75 q^{88} +4942.81i q^{89} -353.372 q^{91} +4795.04i q^{92} +12051.0 q^{94} +2610.92i q^{95} +16251.7 q^{97} -14376.6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 76 q - 632 q^{4} - 48 q^{7} - 72 q^{10} + 448 q^{13} + 5696 q^{16} - 2240 q^{19} + 1008 q^{22} - 11180 q^{25} + 5376 q^{28} - 3440 q^{31} - 7112 q^{34} + 7016 q^{37} - 832 q^{40} - 608 q^{43} + 17728 q^{46} + 30628 q^{49} - 27056 q^{52} - 9312 q^{55} + 4616 q^{58} - 10056 q^{61} - 42864 q^{64} + 31160 q^{67} + 7544 q^{70} + 4488 q^{73} - 15632 q^{76} + 19208 q^{79} + 9768 q^{82} - 7568 q^{85} - 41736 q^{88} - 31240 q^{91} + 27360 q^{94} + 22216 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/531\mathbb{Z}\right)^\times\).

\(n\) \(119\) \(415\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.05329i 1.51332i 0.653807 + 0.756661i \(0.273171\pi\)
−0.653807 + 0.756661i \(0.726829\pi\)
\(3\) 0 0
\(4\) −20.6423 −1.29014
\(5\) 34.3242i 1.37297i 0.727145 + 0.686484i \(0.240847\pi\)
−0.727145 + 0.686484i \(0.759153\pi\)
\(6\) 0 0
\(7\) −5.09902 −0.104062 −0.0520308 0.998645i \(-0.516569\pi\)
−0.0520308 + 0.998645i \(0.516569\pi\)
\(8\) − 28.1013i − 0.439082i
\(9\) 0 0
\(10\) −207.774 −2.07774
\(11\) − 148.668i − 1.22866i −0.789049 0.614330i \(-0.789426\pi\)
0.789049 0.614330i \(-0.210574\pi\)
\(12\) 0 0
\(13\) 69.3021 0.410071 0.205036 0.978754i \(-0.434269\pi\)
0.205036 + 0.978754i \(0.434269\pi\)
\(14\) − 30.8658i − 0.157479i
\(15\) 0 0
\(16\) −160.172 −0.625671
\(17\) − 321.266i − 1.11165i −0.831300 0.555824i \(-0.812403\pi\)
0.831300 0.555824i \(-0.187597\pi\)
\(18\) 0 0
\(19\) 76.0665 0.210710 0.105355 0.994435i \(-0.466402\pi\)
0.105355 + 0.994435i \(0.466402\pi\)
\(20\) − 708.530i − 1.77133i
\(21\) 0 0
\(22\) 899.929 1.85936
\(23\) − 232.292i − 0.439115i −0.975600 0.219558i \(-0.929539\pi\)
0.975600 0.219558i \(-0.0704614\pi\)
\(24\) 0 0
\(25\) −553.149 −0.885039
\(26\) 419.506i 0.620570i
\(27\) 0 0
\(28\) 105.256 0.134254
\(29\) − 806.320i − 0.958763i −0.877607 0.479382i \(-0.840861\pi\)
0.877607 0.479382i \(-0.159139\pi\)
\(30\) 0 0
\(31\) 882.027 0.917823 0.458911 0.888482i \(-0.348240\pi\)
0.458911 + 0.888482i \(0.348240\pi\)
\(32\) − 1419.19i − 1.38592i
\(33\) 0 0
\(34\) 1944.72 1.68228
\(35\) − 175.020i − 0.142873i
\(36\) 0 0
\(37\) −1272.10 −0.929221 −0.464610 0.885515i \(-0.653806\pi\)
−0.464610 + 0.885515i \(0.653806\pi\)
\(38\) 460.452i 0.318873i
\(39\) 0 0
\(40\) 964.553 0.602846
\(41\) − 2407.30i − 1.43206i −0.698069 0.716031i \(-0.745957\pi\)
0.698069 0.716031i \(-0.254043\pi\)
\(42\) 0 0
\(43\) 358.183 0.193717 0.0968584 0.995298i \(-0.469121\pi\)
0.0968584 + 0.995298i \(0.469121\pi\)
\(44\) 3068.85i 1.58515i
\(45\) 0 0
\(46\) 1406.13 0.664523
\(47\) − 1990.82i − 0.901233i −0.892718 0.450616i \(-0.851204\pi\)
0.892718 0.450616i \(-0.148796\pi\)
\(48\) 0 0
\(49\) −2375.00 −0.989171
\(50\) − 3348.37i − 1.33935i
\(51\) 0 0
\(52\) −1430.56 −0.529051
\(53\) − 1991.68i − 0.709036i −0.935049 0.354518i \(-0.884645\pi\)
0.935049 0.354518i \(-0.115355\pi\)
\(54\) 0 0
\(55\) 5102.90 1.68691
\(56\) 143.289i 0.0456916i
\(57\) 0 0
\(58\) 4880.89 1.45092
\(59\) − 453.188i − 0.130189i
\(60\) 0 0
\(61\) −6446.83 −1.73255 −0.866276 0.499565i \(-0.833493\pi\)
−0.866276 + 0.499565i \(0.833493\pi\)
\(62\) 5339.17i 1.38896i
\(63\) 0 0
\(64\) 6028.00 1.47168
\(65\) 2378.74i 0.563015i
\(66\) 0 0
\(67\) −2781.90 −0.619715 −0.309858 0.950783i \(-0.600281\pi\)
−0.309858 + 0.950783i \(0.600281\pi\)
\(68\) 6631.68i 1.43419i
\(69\) 0 0
\(70\) 1059.44 0.216213
\(71\) 6198.13i 1.22954i 0.788705 + 0.614771i \(0.210752\pi\)
−0.788705 + 0.614771i \(0.789248\pi\)
\(72\) 0 0
\(73\) 5122.02 0.961159 0.480579 0.876951i \(-0.340426\pi\)
0.480579 + 0.876951i \(0.340426\pi\)
\(74\) − 7700.41i − 1.40621i
\(75\) 0 0
\(76\) −1570.19 −0.271847
\(77\) 758.060i 0.127856i
\(78\) 0 0
\(79\) 4187.53 0.670972 0.335486 0.942045i \(-0.391100\pi\)
0.335486 + 0.942045i \(0.391100\pi\)
\(80\) − 5497.77i − 0.859026i
\(81\) 0 0
\(82\) 14572.1 2.16717
\(83\) − 3309.98i − 0.480474i −0.970714 0.240237i \(-0.922775\pi\)
0.970714 0.240237i \(-0.0772251\pi\)
\(84\) 0 0
\(85\) 11027.2 1.52626
\(86\) 2168.18i 0.293156i
\(87\) 0 0
\(88\) −4177.75 −0.539483
\(89\) 4942.81i 0.624013i 0.950080 + 0.312007i \(0.101001\pi\)
−0.950080 + 0.312007i \(0.898999\pi\)
\(90\) 0 0
\(91\) −353.372 −0.0426727
\(92\) 4795.04i 0.566522i
\(93\) 0 0
\(94\) 12051.0 1.36386
\(95\) 2610.92i 0.289299i
\(96\) 0 0
\(97\) 16251.7 1.72725 0.863624 0.504137i \(-0.168189\pi\)
0.863624 + 0.504137i \(0.168189\pi\)
\(98\) − 14376.6i − 1.49693i
\(99\) 0 0
\(100\) 11418.3 1.14183
\(101\) 2394.47i 0.234729i 0.993089 + 0.117365i \(0.0374446\pi\)
−0.993089 + 0.117365i \(0.962555\pi\)
\(102\) 0 0
\(103\) −6511.44 −0.613766 −0.306883 0.951747i \(-0.599286\pi\)
−0.306883 + 0.951747i \(0.599286\pi\)
\(104\) − 1947.48i − 0.180055i
\(105\) 0 0
\(106\) 12056.2 1.07300
\(107\) 5817.11i 0.508089i 0.967193 + 0.254044i \(0.0817609\pi\)
−0.967193 + 0.254044i \(0.918239\pi\)
\(108\) 0 0
\(109\) −659.114 −0.0554763 −0.0277382 0.999615i \(-0.508830\pi\)
−0.0277382 + 0.999615i \(0.508830\pi\)
\(110\) 30889.3i 2.55284i
\(111\) 0 0
\(112\) 816.719 0.0651084
\(113\) 4935.39i 0.386513i 0.981148 + 0.193257i \(0.0619050\pi\)
−0.981148 + 0.193257i \(0.938095\pi\)
\(114\) 0 0
\(115\) 7973.23 0.602890
\(116\) 16644.3i 1.23694i
\(117\) 0 0
\(118\) 2743.28 0.197018
\(119\) 1638.14i 0.115680i
\(120\) 0 0
\(121\) −7461.11 −0.509604
\(122\) − 39024.5i − 2.62191i
\(123\) 0 0
\(124\) −18207.1 −1.18412
\(125\) 2466.22i 0.157838i
\(126\) 0 0
\(127\) −10916.6 −0.676828 −0.338414 0.940997i \(-0.609890\pi\)
−0.338414 + 0.940997i \(0.609890\pi\)
\(128\) 13782.2i 0.841201i
\(129\) 0 0
\(130\) −14399.2 −0.852023
\(131\) − 4652.94i − 0.271135i −0.990768 0.135567i \(-0.956714\pi\)
0.990768 0.135567i \(-0.0432857\pi\)
\(132\) 0 0
\(133\) −387.864 −0.0219269
\(134\) − 16839.7i − 0.937829i
\(135\) 0 0
\(136\) −9027.99 −0.488105
\(137\) − 8997.95i − 0.479405i −0.970846 0.239702i \(-0.922950\pi\)
0.970846 0.239702i \(-0.0770499\pi\)
\(138\) 0 0
\(139\) −32517.1 −1.68299 −0.841496 0.540263i \(-0.818325\pi\)
−0.841496 + 0.540263i \(0.818325\pi\)
\(140\) 3612.81i 0.184327i
\(141\) 0 0
\(142\) −37519.0 −1.86069
\(143\) − 10303.0i − 0.503838i
\(144\) 0 0
\(145\) 27676.3 1.31635
\(146\) 31005.0i 1.45454i
\(147\) 0 0
\(148\) 26259.2 1.19883
\(149\) − 18444.7i − 0.830805i −0.909638 0.415402i \(-0.863641\pi\)
0.909638 0.415402i \(-0.136359\pi\)
\(150\) 0 0
\(151\) 10564.8 0.463348 0.231674 0.972793i \(-0.425580\pi\)
0.231674 + 0.972793i \(0.425580\pi\)
\(152\) − 2137.56i − 0.0925193i
\(153\) 0 0
\(154\) −4588.75 −0.193488
\(155\) 30274.9i 1.26014i
\(156\) 0 0
\(157\) −15812.9 −0.641523 −0.320761 0.947160i \(-0.603939\pi\)
−0.320761 + 0.947160i \(0.603939\pi\)
\(158\) 25348.4i 1.01540i
\(159\) 0 0
\(160\) 48712.4 1.90283
\(161\) 1184.46i 0.0456950i
\(162\) 0 0
\(163\) −7471.75 −0.281221 −0.140610 0.990065i \(-0.544906\pi\)
−0.140610 + 0.990065i \(0.544906\pi\)
\(164\) 49692.1i 1.84757i
\(165\) 0 0
\(166\) 20036.3 0.727111
\(167\) 38695.1i 1.38747i 0.720232 + 0.693733i \(0.244035\pi\)
−0.720232 + 0.693733i \(0.755965\pi\)
\(168\) 0 0
\(169\) −23758.2 −0.831841
\(170\) 66750.9i 2.30972i
\(171\) 0 0
\(172\) −7393.72 −0.249923
\(173\) 7440.75i 0.248613i 0.992244 + 0.124307i \(0.0396706\pi\)
−0.992244 + 0.124307i \(0.960329\pi\)
\(174\) 0 0
\(175\) 2820.52 0.0920985
\(176\) 23812.4i 0.768737i
\(177\) 0 0
\(178\) −29920.2 −0.944333
\(179\) − 32354.8i − 1.00979i −0.863180 0.504897i \(-0.831531\pi\)
0.863180 0.504897i \(-0.168469\pi\)
\(180\) 0 0
\(181\) 18355.0 0.560270 0.280135 0.959961i \(-0.409621\pi\)
0.280135 + 0.959961i \(0.409621\pi\)
\(182\) − 2139.07i − 0.0645775i
\(183\) 0 0
\(184\) −6527.70 −0.192808
\(185\) − 43663.9i − 1.27579i
\(186\) 0 0
\(187\) −47762.0 −1.36584
\(188\) 41095.2i 1.16272i
\(189\) 0 0
\(190\) −15804.7 −0.437802
\(191\) − 9333.68i − 0.255850i −0.991784 0.127925i \(-0.959168\pi\)
0.991784 0.127925i \(-0.0408317\pi\)
\(192\) 0 0
\(193\) −28456.8 −0.763961 −0.381981 0.924170i \(-0.624758\pi\)
−0.381981 + 0.924170i \(0.624758\pi\)
\(194\) 98376.0i 2.61388i
\(195\) 0 0
\(196\) 49025.5 1.27617
\(197\) − 36243.1i − 0.933885i −0.884288 0.466942i \(-0.845356\pi\)
0.884288 0.466942i \(-0.154644\pi\)
\(198\) 0 0
\(199\) 51882.7 1.31014 0.655068 0.755570i \(-0.272640\pi\)
0.655068 + 0.755570i \(0.272640\pi\)
\(200\) 15544.2i 0.388605i
\(201\) 0 0
\(202\) −14494.4 −0.355221
\(203\) 4111.44i 0.0997704i
\(204\) 0 0
\(205\) 82628.4 1.96617
\(206\) − 39415.6i − 0.928825i
\(207\) 0 0
\(208\) −11100.2 −0.256570
\(209\) − 11308.6i − 0.258891i
\(210\) 0 0
\(211\) −33215.6 −0.746066 −0.373033 0.927818i \(-0.621682\pi\)
−0.373033 + 0.927818i \(0.621682\pi\)
\(212\) 41112.9i 0.914759i
\(213\) 0 0
\(214\) −35212.6 −0.768902
\(215\) 12294.3i 0.265967i
\(216\) 0 0
\(217\) −4497.47 −0.0955101
\(218\) − 3989.81i − 0.0839536i
\(219\) 0 0
\(220\) −105336. −2.17636
\(221\) − 22264.4i − 0.455855i
\(222\) 0 0
\(223\) −38862.5 −0.781486 −0.390743 0.920500i \(-0.627782\pi\)
−0.390743 + 0.920500i \(0.627782\pi\)
\(224\) 7236.46i 0.144222i
\(225\) 0 0
\(226\) −29875.3 −0.584919
\(227\) − 75821.4i − 1.47143i −0.677290 0.735716i \(-0.736846\pi\)
0.677290 0.735716i \(-0.263154\pi\)
\(228\) 0 0
\(229\) 93042.1 1.77422 0.887112 0.461555i \(-0.152708\pi\)
0.887112 + 0.461555i \(0.152708\pi\)
\(230\) 48264.2i 0.912368i
\(231\) 0 0
\(232\) −22658.6 −0.420976
\(233\) − 68111.6i − 1.25461i −0.778773 0.627306i \(-0.784158\pi\)
0.778773 0.627306i \(-0.215842\pi\)
\(234\) 0 0
\(235\) 68333.4 1.23736
\(236\) 9354.84i 0.167963i
\(237\) 0 0
\(238\) −9916.15 −0.175061
\(239\) 22873.1i 0.400433i 0.979752 + 0.200216i \(0.0641645\pi\)
−0.979752 + 0.200216i \(0.935836\pi\)
\(240\) 0 0
\(241\) 54776.3 0.943102 0.471551 0.881839i \(-0.343694\pi\)
0.471551 + 0.881839i \(0.343694\pi\)
\(242\) − 45164.3i − 0.771195i
\(243\) 0 0
\(244\) 133077. 2.23524
\(245\) − 81519.9i − 1.35810i
\(246\) 0 0
\(247\) 5271.56 0.0864063
\(248\) − 24786.1i − 0.403000i
\(249\) 0 0
\(250\) −14928.8 −0.238860
\(251\) − 109451.i − 1.73729i −0.495438 0.868644i \(-0.664992\pi\)
0.495438 0.868644i \(-0.335008\pi\)
\(252\) 0 0
\(253\) −34534.3 −0.539523
\(254\) − 66081.1i − 1.02426i
\(255\) 0 0
\(256\) 13020.1 0.198671
\(257\) 42118.9i 0.637691i 0.947807 + 0.318846i \(0.103295\pi\)
−0.947807 + 0.318846i \(0.896705\pi\)
\(258\) 0 0
\(259\) 6486.48 0.0966962
\(260\) − 49102.6i − 0.726370i
\(261\) 0 0
\(262\) 28165.6 0.410314
\(263\) − 81685.9i − 1.18096i −0.807052 0.590481i \(-0.798938\pi\)
0.807052 0.590481i \(-0.201062\pi\)
\(264\) 0 0
\(265\) 68362.9 0.973483
\(266\) − 2347.85i − 0.0331824i
\(267\) 0 0
\(268\) 57424.9 0.799522
\(269\) − 26304.9i − 0.363523i −0.983343 0.181762i \(-0.941820\pi\)
0.983343 0.181762i \(-0.0581799\pi\)
\(270\) 0 0
\(271\) −131117. −1.78534 −0.892671 0.450709i \(-0.851171\pi\)
−0.892671 + 0.450709i \(0.851171\pi\)
\(272\) 51457.8i 0.695527i
\(273\) 0 0
\(274\) 54467.2 0.725494
\(275\) 82235.4i 1.08741i
\(276\) 0 0
\(277\) 20798.7 0.271067 0.135534 0.990773i \(-0.456725\pi\)
0.135534 + 0.990773i \(0.456725\pi\)
\(278\) − 196835.i − 2.54691i
\(279\) 0 0
\(280\) −4918.27 −0.0627331
\(281\) 30019.1i 0.380176i 0.981767 + 0.190088i \(0.0608774\pi\)
−0.981767 + 0.190088i \(0.939123\pi\)
\(282\) 0 0
\(283\) −72440.4 −0.904499 −0.452249 0.891891i \(-0.649378\pi\)
−0.452249 + 0.891891i \(0.649378\pi\)
\(284\) − 127944.i − 1.58629i
\(285\) 0 0
\(286\) 62367.0 0.762470
\(287\) 12274.8i 0.149023i
\(288\) 0 0
\(289\) −19691.1 −0.235762
\(290\) 167532.i 1.99206i
\(291\) 0 0
\(292\) −105730. −1.24003
\(293\) − 142815.i − 1.66357i −0.555101 0.831783i \(-0.687320\pi\)
0.555101 0.831783i \(-0.312680\pi\)
\(294\) 0 0
\(295\) 15555.3 0.178745
\(296\) 35747.7i 0.408004i
\(297\) 0 0
\(298\) 111651. 1.25728
\(299\) − 16098.3i − 0.180069i
\(300\) 0 0
\(301\) −1826.38 −0.0201585
\(302\) 63951.8i 0.701196i
\(303\) 0 0
\(304\) −12183.7 −0.131836
\(305\) − 221282.i − 2.37874i
\(306\) 0 0
\(307\) 129232. 1.37117 0.685586 0.727992i \(-0.259546\pi\)
0.685586 + 0.727992i \(0.259546\pi\)
\(308\) − 15648.1i − 0.164953i
\(309\) 0 0
\(310\) −183263. −1.90700
\(311\) − 46811.4i − 0.483984i −0.970278 0.241992i \(-0.922199\pi\)
0.970278 0.241992i \(-0.0778008\pi\)
\(312\) 0 0
\(313\) −22268.0 −0.227297 −0.113648 0.993521i \(-0.536254\pi\)
−0.113648 + 0.993521i \(0.536254\pi\)
\(314\) − 95720.0i − 0.970831i
\(315\) 0 0
\(316\) −86440.4 −0.865650
\(317\) − 33901.4i − 0.337365i −0.985670 0.168682i \(-0.946049\pi\)
0.985670 0.168682i \(-0.0539512\pi\)
\(318\) 0 0
\(319\) −119874. −1.17799
\(320\) 206906.i 2.02057i
\(321\) 0 0
\(322\) −7169.88 −0.0691513
\(323\) − 24437.6i − 0.234236i
\(324\) 0 0
\(325\) −38334.4 −0.362929
\(326\) − 45228.7i − 0.425577i
\(327\) 0 0
\(328\) −67648.1 −0.628793
\(329\) 10151.2i 0.0937837i
\(330\) 0 0
\(331\) 151611. 1.38381 0.691904 0.721989i \(-0.256772\pi\)
0.691904 + 0.721989i \(0.256772\pi\)
\(332\) 68325.7i 0.619880i
\(333\) 0 0
\(334\) −234232. −2.09968
\(335\) − 95486.5i − 0.850848i
\(336\) 0 0
\(337\) −123967. −1.09156 −0.545780 0.837928i \(-0.683767\pi\)
−0.545780 + 0.837928i \(0.683767\pi\)
\(338\) − 143815.i − 1.25884i
\(339\) 0 0
\(340\) −227627. −1.96909
\(341\) − 131129.i − 1.12769i
\(342\) 0 0
\(343\) 24352.9 0.206996
\(344\) − 10065.4i − 0.0850577i
\(345\) 0 0
\(346\) −45041.0 −0.376232
\(347\) 188629.i 1.56657i 0.621662 + 0.783286i \(0.286458\pi\)
−0.621662 + 0.783286i \(0.713542\pi\)
\(348\) 0 0
\(349\) 28962.2 0.237783 0.118892 0.992907i \(-0.462066\pi\)
0.118892 + 0.992907i \(0.462066\pi\)
\(350\) 17073.4i 0.139375i
\(351\) 0 0
\(352\) −210987. −1.70283
\(353\) − 217090.i − 1.74217i −0.491135 0.871083i \(-0.663418\pi\)
0.491135 0.871083i \(-0.336582\pi\)
\(354\) 0 0
\(355\) −212746. −1.68812
\(356\) − 102031.i − 0.805067i
\(357\) 0 0
\(358\) 195853. 1.52814
\(359\) 20152.3i 0.156364i 0.996939 + 0.0781818i \(0.0249115\pi\)
−0.996939 + 0.0781818i \(0.975089\pi\)
\(360\) 0 0
\(361\) −124535. −0.955601
\(362\) 111108.i 0.847870i
\(363\) 0 0
\(364\) 7294.42 0.0550539
\(365\) 175809.i 1.31964i
\(366\) 0 0
\(367\) 32590.5 0.241969 0.120984 0.992654i \(-0.461395\pi\)
0.120984 + 0.992654i \(0.461395\pi\)
\(368\) 37206.6i 0.274742i
\(369\) 0 0
\(370\) 264310. 1.93068
\(371\) 10155.6i 0.0737834i
\(372\) 0 0
\(373\) −25683.9 −0.184605 −0.0923024 0.995731i \(-0.529423\pi\)
−0.0923024 + 0.995731i \(0.529423\pi\)
\(374\) − 289117.i − 2.06695i
\(375\) 0 0
\(376\) −55944.7 −0.395715
\(377\) − 55879.6i − 0.393161i
\(378\) 0 0
\(379\) −103918. −0.723459 −0.361730 0.932283i \(-0.617814\pi\)
−0.361730 + 0.932283i \(0.617814\pi\)
\(380\) − 53895.4i − 0.373237i
\(381\) 0 0
\(382\) 56499.4 0.387184
\(383\) − 143133.i − 0.975759i −0.872911 0.487879i \(-0.837771\pi\)
0.872911 0.487879i \(-0.162229\pi\)
\(384\) 0 0
\(385\) −26019.8 −0.175542
\(386\) − 172257.i − 1.15612i
\(387\) 0 0
\(388\) −335472. −2.22840
\(389\) 37268.5i 0.246288i 0.992389 + 0.123144i \(0.0392977\pi\)
−0.992389 + 0.123144i \(0.960702\pi\)
\(390\) 0 0
\(391\) −74627.6 −0.488142
\(392\) 66740.5i 0.434328i
\(393\) 0 0
\(394\) 219390. 1.41327
\(395\) 143734.i 0.921222i
\(396\) 0 0
\(397\) 228939. 1.45257 0.726287 0.687391i \(-0.241244\pi\)
0.726287 + 0.687391i \(0.241244\pi\)
\(398\) 314061.i 1.98266i
\(399\) 0 0
\(400\) 88598.9 0.553743
\(401\) 290022.i 1.80361i 0.432142 + 0.901805i \(0.357758\pi\)
−0.432142 + 0.901805i \(0.642242\pi\)
\(402\) 0 0
\(403\) 61126.3 0.376373
\(404\) − 49427.4i − 0.302834i
\(405\) 0 0
\(406\) −24887.7 −0.150985
\(407\) 189121.i 1.14170i
\(408\) 0 0
\(409\) 62204.2 0.371854 0.185927 0.982564i \(-0.440471\pi\)
0.185927 + 0.982564i \(0.440471\pi\)
\(410\) 500174.i 2.97545i
\(411\) 0 0
\(412\) 134411. 0.791846
\(413\) 2310.81i 0.0135477i
\(414\) 0 0
\(415\) 113612. 0.659674
\(416\) − 98352.6i − 0.568328i
\(417\) 0 0
\(418\) 68454.4 0.391786
\(419\) 227802.i 1.29756i 0.760974 + 0.648782i \(0.224721\pi\)
−0.760974 + 0.648782i \(0.775279\pi\)
\(420\) 0 0
\(421\) 296676. 1.67385 0.836927 0.547314i \(-0.184350\pi\)
0.836927 + 0.547314i \(0.184350\pi\)
\(422\) − 201064.i − 1.12904i
\(423\) 0 0
\(424\) −55968.8 −0.311325
\(425\) 177708.i 0.983852i
\(426\) 0 0
\(427\) 32872.5 0.180292
\(428\) − 120079.i − 0.655508i
\(429\) 0 0
\(430\) −74421.1 −0.402494
\(431\) 149726.i 0.806014i 0.915197 + 0.403007i \(0.132035\pi\)
−0.915197 + 0.403007i \(0.867965\pi\)
\(432\) 0 0
\(433\) −125543. −0.669602 −0.334801 0.942289i \(-0.608669\pi\)
−0.334801 + 0.942289i \(0.608669\pi\)
\(434\) − 27224.5i − 0.144537i
\(435\) 0 0
\(436\) 13605.6 0.0715725
\(437\) − 17669.6i − 0.0925261i
\(438\) 0 0
\(439\) −204301. −1.06009 −0.530044 0.847970i \(-0.677825\pi\)
−0.530044 + 0.847970i \(0.677825\pi\)
\(440\) − 143398.i − 0.740692i
\(441\) 0 0
\(442\) 134773. 0.689856
\(443\) 132125.i 0.673251i 0.941639 + 0.336625i \(0.109286\pi\)
−0.941639 + 0.336625i \(0.890714\pi\)
\(444\) 0 0
\(445\) −169658. −0.856749
\(446\) − 235246.i − 1.18264i
\(447\) 0 0
\(448\) −30736.9 −0.153145
\(449\) − 57959.1i − 0.287494i −0.989614 0.143747i \(-0.954085\pi\)
0.989614 0.143747i \(-0.0459152\pi\)
\(450\) 0 0
\(451\) −357887. −1.75952
\(452\) − 101878.i − 0.498658i
\(453\) 0 0
\(454\) 458969. 2.22675
\(455\) − 12129.2i − 0.0585882i
\(456\) 0 0
\(457\) −297632. −1.42511 −0.712553 0.701618i \(-0.752461\pi\)
−0.712553 + 0.701618i \(0.752461\pi\)
\(458\) 563211.i 2.68497i
\(459\) 0 0
\(460\) −164586. −0.777816
\(461\) − 236589.i − 1.11325i −0.830764 0.556625i \(-0.812096\pi\)
0.830764 0.556625i \(-0.187904\pi\)
\(462\) 0 0
\(463\) −43406.7 −0.202486 −0.101243 0.994862i \(-0.532282\pi\)
−0.101243 + 0.994862i \(0.532282\pi\)
\(464\) 129150.i 0.599871i
\(465\) 0 0
\(466\) 412299. 1.89863
\(467\) − 34421.1i − 0.157831i −0.996881 0.0789153i \(-0.974854\pi\)
0.996881 0.0789153i \(-0.0251457\pi\)
\(468\) 0 0
\(469\) 14185.0 0.0644885
\(470\) 413642.i 1.87253i
\(471\) 0 0
\(472\) −12735.1 −0.0571637
\(473\) − 53250.2i − 0.238012i
\(474\) 0 0
\(475\) −42076.1 −0.186487
\(476\) − 33815.1i − 0.149244i
\(477\) 0 0
\(478\) −138458. −0.605984
\(479\) − 63497.4i − 0.276748i −0.990380 0.138374i \(-0.955812\pi\)
0.990380 0.138374i \(-0.0441877\pi\)
\(480\) 0 0
\(481\) −88159.4 −0.381047
\(482\) 331577.i 1.42722i
\(483\) 0 0
\(484\) 154015. 0.657463
\(485\) 557825.i 2.37145i
\(486\) 0 0
\(487\) 245908. 1.03685 0.518423 0.855125i \(-0.326519\pi\)
0.518423 + 0.855125i \(0.326519\pi\)
\(488\) 181164.i 0.760733i
\(489\) 0 0
\(490\) 493464. 2.05524
\(491\) − 233648.i − 0.969170i −0.874744 0.484585i \(-0.838971\pi\)
0.874744 0.484585i \(-0.161029\pi\)
\(492\) 0 0
\(493\) −259043. −1.06581
\(494\) 31910.3i 0.130761i
\(495\) 0 0
\(496\) −141276. −0.574255
\(497\) − 31604.3i − 0.127948i
\(498\) 0 0
\(499\) −372352. −1.49539 −0.747693 0.664045i \(-0.768838\pi\)
−0.747693 + 0.664045i \(0.768838\pi\)
\(500\) − 50908.6i − 0.203634i
\(501\) 0 0
\(502\) 662537. 2.62908
\(503\) − 401246.i − 1.58589i −0.609290 0.792947i \(-0.708546\pi\)
0.609290 0.792947i \(-0.291454\pi\)
\(504\) 0 0
\(505\) −82188.3 −0.322275
\(506\) − 209046.i − 0.816472i
\(507\) 0 0
\(508\) 225343. 0.873207
\(509\) − 302843.i − 1.16891i −0.811425 0.584456i \(-0.801308\pi\)
0.811425 0.584456i \(-0.198692\pi\)
\(510\) 0 0
\(511\) −26117.2 −0.100020
\(512\) 299330.i 1.14186i
\(513\) 0 0
\(514\) −254958. −0.965033
\(515\) − 223500.i − 0.842680i
\(516\) 0 0
\(517\) −295971. −1.10731
\(518\) 39264.5i 0.146332i
\(519\) 0 0
\(520\) 66845.5 0.247210
\(521\) 113627.i 0.418608i 0.977851 + 0.209304i \(0.0671197\pi\)
−0.977851 + 0.209304i \(0.932880\pi\)
\(522\) 0 0
\(523\) −333102. −1.21779 −0.608896 0.793250i \(-0.708388\pi\)
−0.608896 + 0.793250i \(0.708388\pi\)
\(524\) 96047.5i 0.349803i
\(525\) 0 0
\(526\) 494469. 1.78718
\(527\) − 283366.i − 1.02030i
\(528\) 0 0
\(529\) 225881. 0.807178
\(530\) 413820.i 1.47319i
\(531\) 0 0
\(532\) 8006.42 0.0282888
\(533\) − 166831.i − 0.587247i
\(534\) 0 0
\(535\) −199667. −0.697589
\(536\) 78175.0i 0.272106i
\(537\) 0 0
\(538\) 159231. 0.550128
\(539\) 353086.i 1.21535i
\(540\) 0 0
\(541\) 20526.2 0.0701317 0.0350658 0.999385i \(-0.488836\pi\)
0.0350658 + 0.999385i \(0.488836\pi\)
\(542\) − 793691.i − 2.70180i
\(543\) 0 0
\(544\) −455937. −1.54066
\(545\) − 22623.6i − 0.0761672i
\(546\) 0 0
\(547\) −23527.0 −0.0786305 −0.0393153 0.999227i \(-0.512518\pi\)
−0.0393153 + 0.999227i \(0.512518\pi\)
\(548\) 185739.i 0.618502i
\(549\) 0 0
\(550\) −497795. −1.64560
\(551\) − 61333.9i − 0.202021i
\(552\) 0 0
\(553\) −21352.3 −0.0698224
\(554\) 125901.i 0.410212i
\(555\) 0 0
\(556\) 671228. 2.17130
\(557\) 273042.i 0.880075i 0.897980 + 0.440037i \(0.145035\pi\)
−0.897980 + 0.440037i \(0.854965\pi\)
\(558\) 0 0
\(559\) 24822.8 0.0794378
\(560\) 28033.2i 0.0893916i
\(561\) 0 0
\(562\) −181714. −0.575329
\(563\) − 136595.i − 0.430942i −0.976510 0.215471i \(-0.930871\pi\)
0.976510 0.215471i \(-0.0691287\pi\)
\(564\) 0 0
\(565\) −169403. −0.530670
\(566\) − 438503.i − 1.36880i
\(567\) 0 0
\(568\) 174175. 0.539871
\(569\) 536600.i 1.65740i 0.559695 + 0.828698i \(0.310918\pi\)
−0.559695 + 0.828698i \(0.689082\pi\)
\(570\) 0 0
\(571\) 254762. 0.781380 0.390690 0.920522i \(-0.372236\pi\)
0.390690 + 0.920522i \(0.372236\pi\)
\(572\) 212677.i 0.650024i
\(573\) 0 0
\(574\) −74303.1 −0.225519
\(575\) 128492.i 0.388634i
\(576\) 0 0
\(577\) 117827. 0.353909 0.176954 0.984219i \(-0.443375\pi\)
0.176954 + 0.984219i \(0.443375\pi\)
\(578\) − 119196.i − 0.356784i
\(579\) 0 0
\(580\) −571302. −1.69828
\(581\) 16877.7i 0.0499988i
\(582\) 0 0
\(583\) −296099. −0.871164
\(584\) − 143935.i − 0.422028i
\(585\) 0 0
\(586\) 864503. 2.51751
\(587\) 342790.i 0.994837i 0.867511 + 0.497419i \(0.165719\pi\)
−0.867511 + 0.497419i \(0.834281\pi\)
\(588\) 0 0
\(589\) 67092.7 0.193395
\(590\) 94160.7i 0.270499i
\(591\) 0 0
\(592\) 203755. 0.581387
\(593\) 37979.0i 0.108003i 0.998541 + 0.0540013i \(0.0171975\pi\)
−0.998541 + 0.0540013i \(0.982802\pi\)
\(594\) 0 0
\(595\) −56227.9 −0.158825
\(596\) 380741.i 1.07186i
\(597\) 0 0
\(598\) 97447.7 0.272502
\(599\) − 68518.6i − 0.190965i −0.995431 0.0954827i \(-0.969561\pi\)
0.995431 0.0954827i \(-0.0304395\pi\)
\(600\) 0 0
\(601\) 322023. 0.891533 0.445767 0.895149i \(-0.352931\pi\)
0.445767 + 0.895149i \(0.352931\pi\)
\(602\) − 11055.6i − 0.0305063i
\(603\) 0 0
\(604\) −218082. −0.597786
\(605\) − 256096.i − 0.699669i
\(606\) 0 0
\(607\) 66326.6 0.180016 0.0900079 0.995941i \(-0.471311\pi\)
0.0900079 + 0.995941i \(0.471311\pi\)
\(608\) − 107953.i − 0.292029i
\(609\) 0 0
\(610\) 1.33948e6 3.59980
\(611\) − 137968.i − 0.369570i
\(612\) 0 0
\(613\) 153314. 0.408001 0.204000 0.978971i \(-0.434606\pi\)
0.204000 + 0.978971i \(0.434606\pi\)
\(614\) 782276.i 2.07503i
\(615\) 0 0
\(616\) 21302.4 0.0561394
\(617\) 17700.3i 0.0464956i 0.999730 + 0.0232478i \(0.00740066\pi\)
−0.999730 + 0.0232478i \(0.992599\pi\)
\(618\) 0 0
\(619\) −28034.0 −0.0731651 −0.0365826 0.999331i \(-0.511647\pi\)
−0.0365826 + 0.999331i \(0.511647\pi\)
\(620\) − 624943.i − 1.62576i
\(621\) 0 0
\(622\) 283363. 0.732424
\(623\) − 25203.5i − 0.0649358i
\(624\) 0 0
\(625\) −430369. −1.10175
\(626\) − 134795.i − 0.343973i
\(627\) 0 0
\(628\) 326415. 0.827657
\(629\) 408684.i 1.03297i
\(630\) 0 0
\(631\) 325646. 0.817875 0.408937 0.912562i \(-0.365899\pi\)
0.408937 + 0.912562i \(0.365899\pi\)
\(632\) − 117675.i − 0.294612i
\(633\) 0 0
\(634\) 205215. 0.510542
\(635\) − 374702.i − 0.929263i
\(636\) 0 0
\(637\) −164592. −0.405631
\(638\) − 725631.i − 1.78268i
\(639\) 0 0
\(640\) −473064. −1.15494
\(641\) 314713.i 0.765947i 0.923759 + 0.382974i \(0.125100\pi\)
−0.923759 + 0.382974i \(0.874900\pi\)
\(642\) 0 0
\(643\) −464152. −1.12263 −0.561317 0.827601i \(-0.689705\pi\)
−0.561317 + 0.827601i \(0.689705\pi\)
\(644\) − 24450.0i − 0.0589532i
\(645\) 0 0
\(646\) 147928. 0.354475
\(647\) 27753.6i 0.0662996i 0.999450 + 0.0331498i \(0.0105538\pi\)
−0.999450 + 0.0331498i \(0.989446\pi\)
\(648\) 0 0
\(649\) −67374.4 −0.159958
\(650\) − 232049.i − 0.549229i
\(651\) 0 0
\(652\) 154234. 0.362815
\(653\) − 594080.i − 1.39322i −0.717451 0.696609i \(-0.754691\pi\)
0.717451 0.696609i \(-0.245309\pi\)
\(654\) 0 0
\(655\) 159708. 0.372259
\(656\) 385581.i 0.896000i
\(657\) 0 0
\(658\) −61448.4 −0.141925
\(659\) 373514.i 0.860075i 0.902811 + 0.430037i \(0.141500\pi\)
−0.902811 + 0.430037i \(0.858500\pi\)
\(660\) 0 0
\(661\) 179178. 0.410092 0.205046 0.978752i \(-0.434266\pi\)
0.205046 + 0.978752i \(0.434266\pi\)
\(662\) 917748.i 2.09415i
\(663\) 0 0
\(664\) −93014.7 −0.210967
\(665\) − 13313.1i − 0.0301049i
\(666\) 0 0
\(667\) −187302. −0.421007
\(668\) − 798755.i − 1.79003i
\(669\) 0 0
\(670\) 578007. 1.28761
\(671\) 958436.i 2.12872i
\(672\) 0 0
\(673\) −293020. −0.646945 −0.323472 0.946238i \(-0.604850\pi\)
−0.323472 + 0.946238i \(0.604850\pi\)
\(674\) − 750411.i − 1.65188i
\(675\) 0 0
\(676\) 490425. 1.07320
\(677\) − 631124.i − 1.37701i −0.725232 0.688505i \(-0.758267\pi\)
0.725232 0.688505i \(-0.241733\pi\)
\(678\) 0 0
\(679\) −82867.5 −0.179740
\(680\) − 309878.i − 0.670152i
\(681\) 0 0
\(682\) 793762. 1.70656
\(683\) 476419.i 1.02129i 0.859793 + 0.510643i \(0.170593\pi\)
−0.859793 + 0.510643i \(0.829407\pi\)
\(684\) 0 0
\(685\) 308847. 0.658207
\(686\) 147415.i 0.313252i
\(687\) 0 0
\(688\) −57370.8 −0.121203
\(689\) − 138028.i − 0.290755i
\(690\) 0 0
\(691\) 311263. 0.651886 0.325943 0.945389i \(-0.394318\pi\)
0.325943 + 0.945389i \(0.394318\pi\)
\(692\) − 153594.i − 0.320747i
\(693\) 0 0
\(694\) −1.14183e6 −2.37073
\(695\) − 1.11612e6i − 2.31069i
\(696\) 0 0
\(697\) −773383. −1.59195
\(698\) 175317.i 0.359843i
\(699\) 0 0
\(700\) −58222.0 −0.118820
\(701\) 14396.4i 0.0292967i 0.999893 + 0.0146484i \(0.00466289\pi\)
−0.999893 + 0.0146484i \(0.995337\pi\)
\(702\) 0 0
\(703\) −96764.4 −0.195797
\(704\) − 896170.i − 1.80819i
\(705\) 0 0
\(706\) 1.31411e6 2.63646
\(707\) − 12209.5i − 0.0244263i
\(708\) 0 0
\(709\) 868112. 1.72696 0.863482 0.504379i \(-0.168279\pi\)
0.863482 + 0.504379i \(0.168279\pi\)
\(710\) − 1.28781e6i − 2.55467i
\(711\) 0 0
\(712\) 138899. 0.273993
\(713\) − 204888.i − 0.403030i
\(714\) 0 0
\(715\) 353642. 0.691753
\(716\) 667878.i 1.30278i
\(717\) 0 0
\(718\) −121988. −0.236629
\(719\) − 398354.i − 0.770570i −0.922798 0.385285i \(-0.874103\pi\)
0.922798 0.385285i \(-0.125897\pi\)
\(720\) 0 0
\(721\) 33201.9 0.0638694
\(722\) − 753846.i − 1.44613i
\(723\) 0 0
\(724\) −378890. −0.722830
\(725\) 446015.i 0.848542i
\(726\) 0 0
\(727\) 836262. 1.58224 0.791122 0.611659i \(-0.209497\pi\)
0.791122 + 0.611659i \(0.209497\pi\)
\(728\) 9930.21i 0.0187368i
\(729\) 0 0
\(730\) −1.06422e6 −1.99704
\(731\) − 115072.i − 0.215345i
\(732\) 0 0
\(733\) −411743. −0.766334 −0.383167 0.923679i \(-0.625167\pi\)
−0.383167 + 0.923679i \(0.625167\pi\)
\(734\) 197280.i 0.366176i
\(735\) 0 0
\(736\) −329666. −0.608581
\(737\) 413579.i 0.761419i
\(738\) 0 0
\(739\) 759107. 1.39000 0.694999 0.719011i \(-0.255405\pi\)
0.694999 + 0.719011i \(0.255405\pi\)
\(740\) 901324.i 1.64595i
\(741\) 0 0
\(742\) −61474.9 −0.111658
\(743\) 127989.i 0.231844i 0.993258 + 0.115922i \(0.0369823\pi\)
−0.993258 + 0.115922i \(0.963018\pi\)
\(744\) 0 0
\(745\) 633099. 1.14067
\(746\) − 155472.i − 0.279367i
\(747\) 0 0
\(748\) 985917. 1.76213
\(749\) − 29661.5i − 0.0528725i
\(750\) 0 0
\(751\) −556352. −0.986439 −0.493219 0.869905i \(-0.664180\pi\)
−0.493219 + 0.869905i \(0.664180\pi\)
\(752\) 318874.i 0.563876i
\(753\) 0 0
\(754\) 338256. 0.594980
\(755\) 362628.i 0.636162i
\(756\) 0 0
\(757\) 571874. 0.997950 0.498975 0.866616i \(-0.333710\pi\)
0.498975 + 0.866616i \(0.333710\pi\)
\(758\) − 629048.i − 1.09483i
\(759\) 0 0
\(760\) 73370.2 0.127026
\(761\) 625719.i 1.08046i 0.841516 + 0.540232i \(0.181663\pi\)
−0.841516 + 0.540232i \(0.818337\pi\)
\(762\) 0 0
\(763\) 3360.83 0.00577295
\(764\) 192669.i 0.330084i
\(765\) 0 0
\(766\) 866426. 1.47664
\(767\) − 31406.8i − 0.0533868i
\(768\) 0 0
\(769\) 511008. 0.864122 0.432061 0.901844i \(-0.357786\pi\)
0.432061 + 0.901844i \(0.357786\pi\)
\(770\) − 157505.i − 0.265652i
\(771\) 0 0
\(772\) 587414. 0.985620
\(773\) − 102844.i − 0.172116i −0.996290 0.0860578i \(-0.972573\pi\)
0.996290 0.0860578i \(-0.0274270\pi\)
\(774\) 0 0
\(775\) −487893. −0.812308
\(776\) − 456693.i − 0.758404i
\(777\) 0 0
\(778\) −225597. −0.372713
\(779\) − 183114.i − 0.301750i
\(780\) 0 0
\(781\) 921462. 1.51069
\(782\) − 451742.i − 0.738716i
\(783\) 0 0
\(784\) 380408. 0.618896
\(785\) − 542764.i − 0.880789i
\(786\) 0 0
\(787\) −184960. −0.298626 −0.149313 0.988790i \(-0.547706\pi\)
−0.149313 + 0.988790i \(0.547706\pi\)
\(788\) 748142.i 1.20485i
\(789\) 0 0
\(790\) −870061. −1.39411
\(791\) − 25165.6i − 0.0402212i
\(792\) 0 0
\(793\) −446779. −0.710470
\(794\) 1.38583e6i 2.19821i
\(795\) 0 0
\(796\) −1.07098e6 −1.69027
\(797\) 1.14811e6i 1.80745i 0.428108 + 0.903727i \(0.359180\pi\)
−0.428108 + 0.903727i \(0.640820\pi\)
\(798\) 0 0
\(799\) −639585. −1.00185
\(800\) 785022.i 1.22660i
\(801\) 0 0
\(802\) −1.75559e6 −2.72944
\(803\) − 761479.i − 1.18094i
\(804\) 0 0
\(805\) −40655.6 −0.0627377
\(806\) 370015.i 0.569573i
\(807\) 0 0
\(808\) 67287.7 0.103065
\(809\) 355139.i 0.542627i 0.962491 + 0.271313i \(0.0874580\pi\)
−0.962491 + 0.271313i \(0.912542\pi\)
\(810\) 0 0
\(811\) −524603. −0.797608 −0.398804 0.917036i \(-0.630575\pi\)
−0.398804 + 0.917036i \(0.630575\pi\)
\(812\) − 84869.6i − 0.128718i
\(813\) 0 0
\(814\) −1.14480e6 −1.72775
\(815\) − 256462.i − 0.386106i
\(816\) 0 0
\(817\) 27245.7 0.0408182
\(818\) 376540.i 0.562736i
\(819\) 0 0
\(820\) −1.70564e6 −2.53665
\(821\) − 808509.i − 1.19950i −0.800189 0.599748i \(-0.795268\pi\)
0.800189 0.599748i \(-0.204732\pi\)
\(822\) 0 0
\(823\) −1.00364e6 −1.48176 −0.740880 0.671638i \(-0.765591\pi\)
−0.740880 + 0.671638i \(0.765591\pi\)
\(824\) 182980.i 0.269494i
\(825\) 0 0
\(826\) −13988.0 −0.0205020
\(827\) 298747.i 0.436811i 0.975858 + 0.218405i \(0.0700855\pi\)
−0.975858 + 0.218405i \(0.929915\pi\)
\(828\) 0 0
\(829\) 168772. 0.245579 0.122789 0.992433i \(-0.460816\pi\)
0.122789 + 0.992433i \(0.460816\pi\)
\(830\) 687729.i 0.998300i
\(831\) 0 0
\(832\) 417753. 0.603494
\(833\) 763008.i 1.09961i
\(834\) 0 0
\(835\) −1.32818e6 −1.90495
\(836\) 233436.i 0.334007i
\(837\) 0 0
\(838\) −1.37895e6 −1.96363
\(839\) − 546526.i − 0.776402i −0.921575 0.388201i \(-0.873097\pi\)
0.921575 0.388201i \(-0.126903\pi\)
\(840\) 0 0
\(841\) 57129.4 0.0807733
\(842\) 1.79586e6i 2.53308i
\(843\) 0 0
\(844\) 685647. 0.962533
\(845\) − 815481.i − 1.14209i
\(846\) 0 0
\(847\) 38044.3 0.0530302
\(848\) 319012.i 0.443624i
\(849\) 0 0
\(850\) −1.07572e6 −1.48888
\(851\) 295499.i 0.408035i
\(852\) 0 0
\(853\) −67958.4 −0.0933997 −0.0466998 0.998909i \(-0.514870\pi\)
−0.0466998 + 0.998909i \(0.514870\pi\)
\(854\) 198987.i 0.272840i
\(855\) 0 0
\(856\) 163468. 0.223093
\(857\) 789404.i 1.07483i 0.843319 + 0.537413i \(0.180598\pi\)
−0.843319 + 0.537413i \(0.819402\pi\)
\(858\) 0 0
\(859\) −885103. −1.19952 −0.599760 0.800180i \(-0.704737\pi\)
−0.599760 + 0.800180i \(0.704737\pi\)
\(860\) − 253783.i − 0.343136i
\(861\) 0 0
\(862\) −906335. −1.21976
\(863\) 1.06409e6i 1.42876i 0.699760 + 0.714378i \(0.253290\pi\)
−0.699760 + 0.714378i \(0.746710\pi\)
\(864\) 0 0
\(865\) −255397. −0.341338
\(866\) − 759948.i − 1.01332i
\(867\) 0 0
\(868\) 92838.2 0.123222
\(869\) − 622551.i − 0.824396i
\(870\) 0 0
\(871\) −192792. −0.254127
\(872\) 18521.9i 0.0243587i
\(873\) 0 0
\(874\) 106959. 0.140022
\(875\) − 12575.3i − 0.0164249i
\(876\) 0 0
\(877\) −1.18410e6 −1.53954 −0.769770 0.638321i \(-0.779629\pi\)
−0.769770 + 0.638321i \(0.779629\pi\)
\(878\) − 1.23669e6i − 1.60425i
\(879\) 0 0
\(880\) −817341. −1.05545
\(881\) − 1.43432e6i − 1.84796i −0.382438 0.923981i \(-0.624916\pi\)
0.382438 0.923981i \(-0.375084\pi\)
\(882\) 0 0
\(883\) 1.47737e6 1.89481 0.947407 0.320032i \(-0.103694\pi\)
0.947407 + 0.320032i \(0.103694\pi\)
\(884\) 459589.i 0.588119i
\(885\) 0 0
\(886\) −799790. −1.01885
\(887\) − 109487.i − 0.139161i −0.997576 0.0695804i \(-0.977834\pi\)
0.997576 0.0695804i \(-0.0221660\pi\)
\(888\) 0 0
\(889\) 55663.8 0.0704318
\(890\) − 1.02699e6i − 1.29654i
\(891\) 0 0
\(892\) 802212. 1.00823
\(893\) − 151435.i − 0.189899i
\(894\) 0 0
\(895\) 1.11055e6 1.38641
\(896\) − 70275.9i − 0.0875367i
\(897\) 0 0
\(898\) 350843. 0.435071
\(899\) − 711196.i − 0.879974i
\(900\) 0 0
\(901\) −639861. −0.788199
\(902\) − 2.16640e6i − 2.66271i
\(903\) 0 0
\(904\) 138691. 0.169711
\(905\) 630021.i 0.769233i
\(906\) 0 0
\(907\) −1.22088e6 −1.48409 −0.742043 0.670352i \(-0.766143\pi\)
−0.742043 + 0.670352i \(0.766143\pi\)
\(908\) 1.56513e6i 1.89836i
\(909\) 0 0
\(910\) 73421.7 0.0886628
\(911\) − 439298.i − 0.529326i −0.964341 0.264663i \(-0.914739\pi\)
0.964341 0.264663i \(-0.0852607\pi\)
\(912\) 0 0
\(913\) −492088. −0.590338
\(914\) − 1.80165e6i − 2.15664i
\(915\) 0 0
\(916\) −1.92060e6 −2.28901
\(917\) 23725.4i 0.0282147i
\(918\) 0 0
\(919\) 4513.39 0.00534406 0.00267203 0.999996i \(-0.499149\pi\)
0.00267203 + 0.999996i \(0.499149\pi\)
\(920\) − 224058.i − 0.264719i
\(921\) 0 0
\(922\) 1.43214e6 1.68471
\(923\) 429543.i 0.504200i
\(924\) 0 0
\(925\) 703663. 0.822396
\(926\) − 262753.i − 0.306426i
\(927\) 0 0
\(928\) −1.14432e6 −1.32877
\(929\) 1.54617e6i 1.79154i 0.444519 + 0.895769i \(0.353375\pi\)
−0.444519 + 0.895769i \(0.646625\pi\)
\(930\) 0 0
\(931\) −180658. −0.208429
\(932\) 1.40598e6i 1.61863i
\(933\) 0 0
\(934\) 208361. 0.238849
\(935\) − 1.63939e6i − 1.87525i
\(936\) 0 0
\(937\) −1.09079e6 −1.24240 −0.621201 0.783652i \(-0.713355\pi\)
−0.621201 + 0.783652i \(0.713355\pi\)
\(938\) 85865.7i 0.0975919i
\(939\) 0 0
\(940\) −1.41056e6 −1.59638
\(941\) − 129887.i − 0.146685i −0.997307 0.0733425i \(-0.976633\pi\)
0.997307 0.0733425i \(-0.0233666\pi\)
\(942\) 0 0
\(943\) −559195. −0.628840
\(944\) 72587.9i 0.0814555i
\(945\) 0 0
\(946\) 322339. 0.360189
\(947\) 1.00321e6i 1.11864i 0.828952 + 0.559320i \(0.188938\pi\)
−0.828952 + 0.559320i \(0.811062\pi\)
\(948\) 0 0
\(949\) 354966. 0.394144
\(950\) − 254699.i − 0.282215i
\(951\) 0 0
\(952\) 46033.9 0.0507930
\(953\) − 529351.i − 0.582851i −0.956594 0.291426i \(-0.905870\pi\)
0.956594 0.291426i \(-0.0941296\pi\)
\(954\) 0 0
\(955\) 320371. 0.351274
\(956\) − 472154.i − 0.516616i
\(957\) 0 0
\(958\) 384368. 0.418810
\(959\) 45880.7i 0.0498876i
\(960\) 0 0
\(961\) −145549. −0.157602
\(962\) − 533654.i − 0.576647i
\(963\) 0 0
\(964\) −1.13071e6 −1.21674
\(965\) − 976756.i − 1.04889i
\(966\) 0 0
\(967\) −455368. −0.486979 −0.243489 0.969904i \(-0.578292\pi\)
−0.243489 + 0.969904i \(0.578292\pi\)
\(968\) 209667.i 0.223758i
\(969\) 0 0
\(970\) −3.37668e6 −3.58877
\(971\) 100791.i 0.106901i 0.998570 + 0.0534506i \(0.0170220\pi\)
−0.998570 + 0.0534506i \(0.982978\pi\)
\(972\) 0 0
\(973\) 165805. 0.175135
\(974\) 1.48855e6i 1.56908i
\(975\) 0 0
\(976\) 1.03260e6 1.08401
\(977\) 429967.i 0.450449i 0.974307 + 0.225224i \(0.0723116\pi\)
−0.974307 + 0.225224i \(0.927688\pi\)
\(978\) 0 0
\(979\) 734836. 0.766700
\(980\) 1.68276e6i 1.75214i
\(981\) 0 0
\(982\) 1.41434e6 1.46667
\(983\) 1.10232e6i 1.14077i 0.821377 + 0.570386i \(0.193206\pi\)
−0.821377 + 0.570386i \(0.806794\pi\)
\(984\) 0 0
\(985\) 1.24402e6 1.28219
\(986\) − 1.56807e6i − 1.61291i
\(987\) 0 0
\(988\) −108817. −0.111477
\(989\) − 83202.9i − 0.0850640i
\(990\) 0 0
\(991\) −223311. −0.227385 −0.113693 0.993516i \(-0.536268\pi\)
−0.113693 + 0.993516i \(0.536268\pi\)
\(992\) − 1.25176e6i − 1.27203i
\(993\) 0 0
\(994\) 191310. 0.193627
\(995\) 1.78083e6i 1.79877i
\(996\) 0 0
\(997\) −222839. −0.224182 −0.112091 0.993698i \(-0.535755\pi\)
−0.112091 + 0.993698i \(0.535755\pi\)
\(998\) − 2.25396e6i − 2.26300i
\(999\) 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 531.5.b.a.296.65 yes 76
3.2 odd 2 inner 531.5.b.a.296.12 76
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
531.5.b.a.296.12 76 3.2 odd 2 inner
531.5.b.a.296.65 yes 76 1.1 even 1 trivial