Properties

Label 342.5.d.a.37.6
Level $342$
Weight $5$
Character 342.37
Analytic conductor $35.353$
Analytic rank $0$
Dimension $8$
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] = [342,5,Mod(37,342)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(342, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.37");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 342.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: \(35.3525273747\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 450x^{6} + 68229x^{4} + 4001228x^{2} + 77475204 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 38)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 37.6
Root \(-8.07810i\) of defining polynomial
Character \(\chi\) \(=\) 342.37
Dual form 342.5.d.a.37.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.82843i q^{2} -8.00000 q^{4} -26.7027 q^{5} +51.4469 q^{7} -22.6274i q^{8} +O(q^{10})\) \(q+2.82843i q^{2} -8.00000 q^{4} -26.7027 q^{5} +51.4469 q^{7} -22.6274i q^{8} -75.5266i q^{10} +25.0891 q^{11} -53.2531i q^{13} +145.514i q^{14} +64.0000 q^{16} +24.4913 q^{17} +(-78.0101 + 352.470i) q^{19} +213.622 q^{20} +70.9628i q^{22} +612.242 q^{23} +88.0343 q^{25} +150.622 q^{26} -411.576 q^{28} +1345.96i q^{29} -912.680i q^{31} +181.019i q^{32} +69.2720i q^{34} -1373.77 q^{35} +325.059i q^{37} +(-996.937 - 220.646i) q^{38} +604.213i q^{40} +51.7494i q^{41} -2545.26 q^{43} -200.713 q^{44} +1731.68i q^{46} -2998.19 q^{47} +245.788 q^{49} +248.999i q^{50} +426.024i q^{52} +4067.86i q^{53} -669.948 q^{55} -1164.11i q^{56} -3806.96 q^{58} -1078.88i q^{59} -6530.89 q^{61} +2581.45 q^{62} -512.000 q^{64} +1422.00i q^{65} +7386.18i q^{67} -195.931 q^{68} -3885.62i q^{70} -5806.02i q^{71} +3629.56 q^{73} -919.405 q^{74} +(624.081 - 2819.76i) q^{76} +1290.76 q^{77} +3903.16i q^{79} -1708.97 q^{80} -146.369 q^{82} +997.967 q^{83} -653.985 q^{85} -7199.08i q^{86} -567.702i q^{88} +6152.85i q^{89} -2739.71i q^{91} -4897.94 q^{92} -8480.16i q^{94} +(2083.08 - 9411.91i) q^{95} +5023.58i q^{97} +695.195i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 64 q^{4} - 18 q^{5} - 162 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 64 q^{4} - 18 q^{5} - 162 q^{7} + 6 q^{11} + 512 q^{16} - 510 q^{17} - 12 q^{19} + 144 q^{20} + 396 q^{23} + 3458 q^{25} + 192 q^{26} + 1296 q^{28} - 1002 q^{35} + 3216 q^{38} - 8654 q^{43} - 48 q^{44} - 3210 q^{47} + 9222 q^{49} + 17146 q^{55} - 960 q^{58} + 1314 q^{61} + 15168 q^{62} - 4096 q^{64} + 4080 q^{68} + 23398 q^{73} - 13152 q^{74} + 96 q^{76} + 44622 q^{77} - 1152 q^{80} + 16512 q^{82} + 10440 q^{83} + 21274 q^{85} - 3168 q^{92} + 34686 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1\) \(-1\)

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
\(5\) −26.7027 −1.06811 −0.534054 0.845450i \(-0.679332\pi\)
−0.534054 + 0.845450i \(0.679332\pi\)
\(6\) 0 0
\(7\) 51.4469 1.04994 0.524969 0.851121i \(-0.324077\pi\)
0.524969 + 0.851121i \(0.324077\pi\)
\(8\) 22.6274i 0.353553i
\(9\) 0 0
\(10\) 75.5266i 0.755266i
\(11\) 25.0891 0.207348 0.103674 0.994611i \(-0.466940\pi\)
0.103674 + 0.994611i \(0.466940\pi\)
\(12\) 0 0
\(13\) 53.2531i 0.315107i −0.987510 0.157553i \(-0.949639\pi\)
0.987510 0.157553i \(-0.0503606\pi\)
\(14\) 145.514i 0.742418i
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) 24.4913 0.0847451 0.0423726 0.999102i \(-0.486508\pi\)
0.0423726 + 0.999102i \(0.486508\pi\)
\(18\) 0 0
\(19\) −78.0101 + 352.470i −0.216094 + 0.976372i
\(20\) 213.622 0.534054
\(21\) 0 0
\(22\) 70.9628i 0.146617i
\(23\) 612.242 1.15736 0.578679 0.815555i \(-0.303568\pi\)
0.578679 + 0.815555i \(0.303568\pi\)
\(24\) 0 0
\(25\) 88.0343 0.140855
\(26\) 150.622 0.222814
\(27\) 0 0
\(28\) −411.576 −0.524969
\(29\) 1345.96i 1.60043i 0.599713 + 0.800215i \(0.295282\pi\)
−0.599713 + 0.800215i \(0.704718\pi\)
\(30\) 0 0
\(31\) 912.680i 0.949719i −0.880062 0.474860i \(-0.842499\pi\)
0.880062 0.474860i \(-0.157501\pi\)
\(32\) 181.019i 0.176777i
\(33\) 0 0
\(34\) 69.2720i 0.0599238i
\(35\) −1373.77 −1.12145
\(36\) 0 0
\(37\) 325.059i 0.237443i 0.992928 + 0.118721i \(0.0378795\pi\)
−0.992928 + 0.118721i \(0.962120\pi\)
\(38\) −996.937 220.646i −0.690400 0.152802i
\(39\) 0 0
\(40\) 604.213i 0.377633i
\(41\) 51.7494i 0.0307849i 0.999882 + 0.0153924i \(0.00489976\pi\)
−0.999882 + 0.0153924i \(0.995100\pi\)
\(42\) 0 0
\(43\) −2545.26 −1.37656 −0.688280 0.725445i \(-0.741634\pi\)
−0.688280 + 0.725445i \(0.741634\pi\)
\(44\) −200.713 −0.103674
\(45\) 0 0
\(46\) 1731.68i 0.818376i
\(47\) −2998.19 −1.35726 −0.678630 0.734480i \(-0.737426\pi\)
−0.678630 + 0.734480i \(0.737426\pi\)
\(48\) 0 0
\(49\) 245.788 0.102369
\(50\) 248.999i 0.0995994i
\(51\) 0 0
\(52\) 426.024i 0.157553i
\(53\) 4067.86i 1.44815i 0.689720 + 0.724077i \(0.257734\pi\)
−0.689720 + 0.724077i \(0.742266\pi\)
\(54\) 0 0
\(55\) −669.948 −0.221470
\(56\) 1164.11i 0.371209i
\(57\) 0 0
\(58\) −3806.96 −1.13168
\(59\) 1078.88i 0.309933i −0.987920 0.154966i \(-0.950473\pi\)
0.987920 0.154966i \(-0.0495269\pi\)
\(60\) 0 0
\(61\) −6530.89 −1.75514 −0.877572 0.479445i \(-0.840838\pi\)
−0.877572 + 0.479445i \(0.840838\pi\)
\(62\) 2581.45 0.671553
\(63\) 0 0
\(64\) −512.000 −0.125000
\(65\) 1422.00i 0.336568i
\(66\) 0 0
\(67\) 7386.18i 1.64540i 0.568478 + 0.822698i \(0.307532\pi\)
−0.568478 + 0.822698i \(0.692468\pi\)
\(68\) −195.931 −0.0423726
\(69\) 0 0
\(70\) 3885.62i 0.792983i
\(71\) 5806.02i 1.15176i −0.817535 0.575879i \(-0.804660\pi\)
0.817535 0.575879i \(-0.195340\pi\)
\(72\) 0 0
\(73\) 3629.56 0.681096 0.340548 0.940227i \(-0.389387\pi\)
0.340548 + 0.940227i \(0.389387\pi\)
\(74\) −919.405 −0.167897
\(75\) 0 0
\(76\) 624.081 2819.76i 0.108047 0.488186i
\(77\) 1290.76 0.217703
\(78\) 0 0
\(79\) 3903.16i 0.625407i 0.949851 + 0.312703i \(0.101235\pi\)
−0.949851 + 0.312703i \(0.898765\pi\)
\(80\) −1708.97 −0.267027
\(81\) 0 0
\(82\) −146.369 −0.0217682
\(83\) 997.967 0.144864 0.0724319 0.997373i \(-0.476924\pi\)
0.0724319 + 0.997373i \(0.476924\pi\)
\(84\) 0 0
\(85\) −653.985 −0.0905169
\(86\) 7199.08i 0.973375i
\(87\) 0 0
\(88\) 567.702i 0.0733087i
\(89\) 6152.85i 0.776777i 0.921496 + 0.388389i \(0.126968\pi\)
−0.921496 + 0.388389i \(0.873032\pi\)
\(90\) 0 0
\(91\) 2739.71i 0.330843i
\(92\) −4897.94 −0.578679
\(93\) 0 0
\(94\) 8480.16i 0.959728i
\(95\) 2083.08 9411.91i 0.230812 1.04287i
\(96\) 0 0
\(97\) 5023.58i 0.533912i 0.963709 + 0.266956i \(0.0860178\pi\)
−0.963709 + 0.266956i \(0.913982\pi\)
\(98\) 695.195i 0.0723860i
\(99\) 0 0
\(100\) −704.274 −0.0704274
\(101\) −15793.7 −1.54825 −0.774123 0.633035i \(-0.781809\pi\)
−0.774123 + 0.633035i \(0.781809\pi\)
\(102\) 0 0
\(103\) 13068.7i 1.23185i −0.787806 0.615924i \(-0.788783\pi\)
0.787806 0.615924i \(-0.211217\pi\)
\(104\) −1204.98 −0.111407
\(105\) 0 0
\(106\) −11505.7 −1.02400
\(107\) 8273.09i 0.722604i 0.932449 + 0.361302i \(0.117668\pi\)
−0.932449 + 0.361302i \(0.882332\pi\)
\(108\) 0 0
\(109\) 14219.6i 1.19683i 0.801186 + 0.598416i \(0.204203\pi\)
−0.801186 + 0.598416i \(0.795797\pi\)
\(110\) 1894.90i 0.156603i
\(111\) 0 0
\(112\) 3292.60 0.262484
\(113\) 20747.4i 1.62483i 0.583080 + 0.812414i \(0.301847\pi\)
−0.583080 + 0.812414i \(0.698153\pi\)
\(114\) 0 0
\(115\) −16348.5 −1.23618
\(116\) 10767.7i 0.800215i
\(117\) 0 0
\(118\) 3051.52 0.219155
\(119\) 1260.00 0.0889771
\(120\) 0 0
\(121\) −14011.5 −0.957007
\(122\) 18472.1i 1.24107i
\(123\) 0 0
\(124\) 7301.44i 0.474860i
\(125\) 14338.4 0.917660
\(126\) 0 0
\(127\) 11117.8i 0.689305i −0.938730 0.344653i \(-0.887997\pi\)
0.938730 0.344653i \(-0.112003\pi\)
\(128\) 1448.15i 0.0883883i
\(129\) 0 0
\(130\) −4022.02 −0.237990
\(131\) −223.317 −0.0130130 −0.00650652 0.999979i \(-0.502071\pi\)
−0.00650652 + 0.999979i \(0.502071\pi\)
\(132\) 0 0
\(133\) −4013.38 + 18133.5i −0.226886 + 1.02513i
\(134\) −20891.3 −1.16347
\(135\) 0 0
\(136\) 554.176i 0.0299619i
\(137\) −18953.1 −1.00981 −0.504903 0.863176i \(-0.668472\pi\)
−0.504903 + 0.863176i \(0.668472\pi\)
\(138\) 0 0
\(139\) −11115.5 −0.575309 −0.287655 0.957734i \(-0.592875\pi\)
−0.287655 + 0.957734i \(0.592875\pi\)
\(140\) 10990.2 0.560723
\(141\) 0 0
\(142\) 16421.9 0.814416
\(143\) 1336.07i 0.0653368i
\(144\) 0 0
\(145\) 35940.8i 1.70943i
\(146\) 10265.9i 0.481608i
\(147\) 0 0
\(148\) 2600.47i 0.118721i
\(149\) 19727.7 0.888597 0.444299 0.895879i \(-0.353453\pi\)
0.444299 + 0.895879i \(0.353453\pi\)
\(150\) 0 0
\(151\) 770.565i 0.0337952i 0.999857 + 0.0168976i \(0.00537894\pi\)
−0.999857 + 0.0168976i \(0.994621\pi\)
\(152\) 7975.50 + 1765.17i 0.345200 + 0.0764009i
\(153\) 0 0
\(154\) 3650.82i 0.153939i
\(155\) 24371.0i 1.01440i
\(156\) 0 0
\(157\) 46244.5 1.87612 0.938061 0.346470i \(-0.112620\pi\)
0.938061 + 0.346470i \(0.112620\pi\)
\(158\) −11039.8 −0.442229
\(159\) 0 0
\(160\) 4833.71i 0.188817i
\(161\) 31498.0 1.21515
\(162\) 0 0
\(163\) 32951.9 1.24024 0.620119 0.784507i \(-0.287084\pi\)
0.620119 + 0.784507i \(0.287084\pi\)
\(164\) 413.995i 0.0153924i
\(165\) 0 0
\(166\) 2822.68i 0.102434i
\(167\) 48049.1i 1.72287i 0.507870 + 0.861434i \(0.330433\pi\)
−0.507870 + 0.861434i \(0.669567\pi\)
\(168\) 0 0
\(169\) 25725.1 0.900708
\(170\) 1849.75i 0.0640051i
\(171\) 0 0
\(172\) 20362.1 0.688280
\(173\) 46219.6i 1.54431i −0.635436 0.772153i \(-0.719180\pi\)
0.635436 0.772153i \(-0.280820\pi\)
\(174\) 0 0
\(175\) 4529.10 0.147889
\(176\) 1605.70 0.0518371
\(177\) 0 0
\(178\) −17402.9 −0.549264
\(179\) 39826.5i 1.24299i −0.783420 0.621493i \(-0.786527\pi\)
0.783420 0.621493i \(-0.213473\pi\)
\(180\) 0 0
\(181\) 51844.5i 1.58251i 0.611488 + 0.791254i \(0.290571\pi\)
−0.611488 + 0.791254i \(0.709429\pi\)
\(182\) 7749.06 0.233941
\(183\) 0 0
\(184\) 13853.5i 0.409188i
\(185\) 8679.95i 0.253614i
\(186\) 0 0
\(187\) 614.467 0.0175718
\(188\) 23985.5 0.678630
\(189\) 0 0
\(190\) 26620.9 + 5891.84i 0.737421 + 0.163209i
\(191\) 38958.8 1.06792 0.533960 0.845509i \(-0.320703\pi\)
0.533960 + 0.845509i \(0.320703\pi\)
\(192\) 0 0
\(193\) 43058.1i 1.15595i 0.816053 + 0.577977i \(0.196158\pi\)
−0.816053 + 0.577977i \(0.803842\pi\)
\(194\) −14208.8 −0.377533
\(195\) 0 0
\(196\) −1966.31 −0.0511846
\(197\) −26311.0 −0.677962 −0.338981 0.940793i \(-0.610082\pi\)
−0.338981 + 0.940793i \(0.610082\pi\)
\(198\) 0 0
\(199\) −11773.1 −0.297293 −0.148647 0.988890i \(-0.547492\pi\)
−0.148647 + 0.988890i \(0.547492\pi\)
\(200\) 1991.99i 0.0497997i
\(201\) 0 0
\(202\) 44671.2i 1.09478i
\(203\) 69245.7i 1.68035i
\(204\) 0 0
\(205\) 1381.85i 0.0328816i
\(206\) 36963.8 0.871048
\(207\) 0 0
\(208\) 3408.20i 0.0787767i
\(209\) −1957.21 + 8843.18i −0.0448068 + 0.202449i
\(210\) 0 0
\(211\) 12495.7i 0.280670i −0.990104 0.140335i \(-0.955182\pi\)
0.990104 0.140335i \(-0.0448179\pi\)
\(212\) 32542.9i 0.724077i
\(213\) 0 0
\(214\) −23399.8 −0.510958
\(215\) 67965.3 1.47032
\(216\) 0 0
\(217\) 46954.6i 0.997146i
\(218\) −40219.0 −0.846288
\(219\) 0 0
\(220\) 5359.58 0.110735
\(221\) 1304.24i 0.0267038i
\(222\) 0 0
\(223\) 59706.5i 1.20064i 0.799761 + 0.600319i \(0.204960\pi\)
−0.799761 + 0.600319i \(0.795040\pi\)
\(224\) 9312.89i 0.185605i
\(225\) 0 0
\(226\) −58682.6 −1.14893
\(227\) 57964.2i 1.12489i 0.826836 + 0.562443i \(0.190138\pi\)
−0.826836 + 0.562443i \(0.809862\pi\)
\(228\) 0 0
\(229\) 40157.9 0.765773 0.382886 0.923795i \(-0.374930\pi\)
0.382886 + 0.923795i \(0.374930\pi\)
\(230\) 46240.6i 0.874114i
\(231\) 0 0
\(232\) 30455.6 0.565838
\(233\) −71740.1 −1.32145 −0.660724 0.750629i \(-0.729751\pi\)
−0.660724 + 0.750629i \(0.729751\pi\)
\(234\) 0 0
\(235\) 80059.7 1.44970
\(236\) 8631.00i 0.154966i
\(237\) 0 0
\(238\) 3563.83i 0.0629163i
\(239\) −44178.4 −0.773417 −0.386709 0.922202i \(-0.626388\pi\)
−0.386709 + 0.922202i \(0.626388\pi\)
\(240\) 0 0
\(241\) 33823.7i 0.582353i −0.956669 0.291177i \(-0.905953\pi\)
0.956669 0.291177i \(-0.0940467\pi\)
\(242\) 39630.6i 0.676706i
\(243\) 0 0
\(244\) 52247.1 0.877572
\(245\) −6563.22 −0.109341
\(246\) 0 0
\(247\) 18770.1 + 4154.27i 0.307662 + 0.0680928i
\(248\) −20651.6 −0.335777
\(249\) 0 0
\(250\) 40555.2i 0.648884i
\(251\) 64952.8 1.03098 0.515490 0.856896i \(-0.327610\pi\)
0.515490 + 0.856896i \(0.327610\pi\)
\(252\) 0 0
\(253\) 15360.6 0.239976
\(254\) 31445.9 0.487412
\(255\) 0 0
\(256\) 4096.00 0.0625000
\(257\) 1155.59i 0.0174959i −0.999962 0.00874796i \(-0.997215\pi\)
0.999962 0.00874796i \(-0.00278460\pi\)
\(258\) 0 0
\(259\) 16723.3i 0.249300i
\(260\) 11376.0i 0.168284i
\(261\) 0 0
\(262\) 631.635i 0.00920161i
\(263\) −6732.94 −0.0973404 −0.0486702 0.998815i \(-0.515498\pi\)
−0.0486702 + 0.998815i \(0.515498\pi\)
\(264\) 0 0
\(265\) 108623.i 1.54678i
\(266\) −51289.4 11351.6i −0.724877 0.160432i
\(267\) 0 0
\(268\) 59089.5i 0.822698i
\(269\) 22231.5i 0.307230i 0.988131 + 0.153615i \(0.0490915\pi\)
−0.988131 + 0.153615i \(0.950908\pi\)
\(270\) 0 0
\(271\) −46597.6 −0.634490 −0.317245 0.948344i \(-0.602758\pi\)
−0.317245 + 0.948344i \(0.602758\pi\)
\(272\) 1567.45 0.0211863
\(273\) 0 0
\(274\) 53607.3i 0.714041i
\(275\) 2208.70 0.0292060
\(276\) 0 0
\(277\) 105817. 1.37910 0.689548 0.724240i \(-0.257809\pi\)
0.689548 + 0.724240i \(0.257809\pi\)
\(278\) 31439.5i 0.406805i
\(279\) 0 0
\(280\) 31084.9i 0.396491i
\(281\) 20298.2i 0.257067i −0.991705 0.128533i \(-0.958973\pi\)
0.991705 0.128533i \(-0.0410269\pi\)
\(282\) 0 0
\(283\) −55710.6 −0.695609 −0.347804 0.937567i \(-0.613073\pi\)
−0.347804 + 0.937567i \(0.613073\pi\)
\(284\) 46448.1i 0.575879i
\(285\) 0 0
\(286\) 3778.99 0.0462001
\(287\) 2662.35i 0.0323222i
\(288\) 0 0
\(289\) −82921.2 −0.992818
\(290\) 101656. 1.20875
\(291\) 0 0
\(292\) −29036.5 −0.340548
\(293\) 84053.1i 0.979081i −0.871980 0.489541i \(-0.837164\pi\)
0.871980 0.489541i \(-0.162836\pi\)
\(294\) 0 0
\(295\) 28808.9i 0.331042i
\(296\) 7355.24 0.0839486
\(297\) 0 0
\(298\) 55798.5i 0.628333i
\(299\) 32603.8i 0.364691i
\(300\) 0 0
\(301\) −130946. −1.44530
\(302\) −2179.49 −0.0238968
\(303\) 0 0
\(304\) −4992.64 + 22558.1i −0.0540236 + 0.244093i
\(305\) 174392. 1.87468
\(306\) 0 0
\(307\) 10396.7i 0.110311i −0.998478 0.0551555i \(-0.982435\pi\)
0.998478 0.0551555i \(-0.0175655\pi\)
\(308\) −10326.1 −0.108851
\(309\) 0 0
\(310\) −68931.7 −0.717291
\(311\) −118107. −1.22111 −0.610557 0.791973i \(-0.709054\pi\)
−0.610557 + 0.791973i \(0.709054\pi\)
\(312\) 0 0
\(313\) −8391.17 −0.0856513 −0.0428257 0.999083i \(-0.513636\pi\)
−0.0428257 + 0.999083i \(0.513636\pi\)
\(314\) 130799.i 1.32662i
\(315\) 0 0
\(316\) 31225.3i 0.312703i
\(317\) 143474.i 1.42776i −0.700270 0.713878i \(-0.746937\pi\)
0.700270 0.713878i \(-0.253063\pi\)
\(318\) 0 0
\(319\) 33769.0i 0.331847i
\(320\) 13671.8 0.133514
\(321\) 0 0
\(322\) 89089.8i 0.859244i
\(323\) −1910.57 + 8632.47i −0.0183129 + 0.0827428i
\(324\) 0 0
\(325\) 4688.09i 0.0443843i
\(326\) 93202.1i 0.876981i
\(327\) 0 0
\(328\) 1170.96 0.0108841
\(329\) −154248. −1.42504
\(330\) 0 0
\(331\) 16943.3i 0.154647i −0.997006 0.0773237i \(-0.975363\pi\)
0.997006 0.0773237i \(-0.0246375\pi\)
\(332\) −7983.73 −0.0724319
\(333\) 0 0
\(334\) −135903. −1.21825
\(335\) 197231.i 1.75746i
\(336\) 0 0
\(337\) 217168.i 1.91221i −0.293021 0.956106i \(-0.594661\pi\)
0.293021 0.956106i \(-0.405339\pi\)
\(338\) 72761.6i 0.636897i
\(339\) 0 0
\(340\) 5231.88 0.0452585
\(341\) 22898.4i 0.196923i
\(342\) 0 0
\(343\) −110879. −0.942456
\(344\) 57592.7i 0.486688i
\(345\) 0 0
\(346\) 130729. 1.09199
\(347\) −64555.0 −0.536131 −0.268066 0.963401i \(-0.586384\pi\)
−0.268066 + 0.963401i \(0.586384\pi\)
\(348\) 0 0
\(349\) −113275. −0.930001 −0.465001 0.885310i \(-0.653946\pi\)
−0.465001 + 0.885310i \(0.653946\pi\)
\(350\) 12810.2i 0.104573i
\(351\) 0 0
\(352\) 4541.62i 0.0366543i
\(353\) −27685.3 −0.222178 −0.111089 0.993810i \(-0.535434\pi\)
−0.111089 + 0.993810i \(0.535434\pi\)
\(354\) 0 0
\(355\) 155036.i 1.23020i
\(356\) 49222.8i 0.388389i
\(357\) 0 0
\(358\) 112646. 0.878923
\(359\) −65336.2 −0.506950 −0.253475 0.967342i \(-0.581574\pi\)
−0.253475 + 0.967342i \(0.581574\pi\)
\(360\) 0 0
\(361\) −118150. 54992.5i −0.906606 0.421977i
\(362\) −146638. −1.11900
\(363\) 0 0
\(364\) 21917.7i 0.165421i
\(365\) −96919.1 −0.727484
\(366\) 0 0
\(367\) 177298. 1.31635 0.658176 0.752864i \(-0.271328\pi\)
0.658176 + 0.752864i \(0.271328\pi\)
\(368\) 39183.5 0.289340
\(369\) 0 0
\(370\) 24550.6 0.179332
\(371\) 209279.i 1.52047i
\(372\) 0 0
\(373\) 15381.6i 0.110557i 0.998471 + 0.0552783i \(0.0176046\pi\)
−0.998471 + 0.0552783i \(0.982395\pi\)
\(374\) 1737.97i 0.0124251i
\(375\) 0 0
\(376\) 67841.3i 0.479864i
\(377\) 71676.6 0.504307
\(378\) 0 0
\(379\) 232626.i 1.61949i 0.586780 + 0.809746i \(0.300395\pi\)
−0.586780 + 0.809746i \(0.699605\pi\)
\(380\) −16664.6 + 75295.3i −0.115406 + 0.521436i
\(381\) 0 0
\(382\) 110192.i 0.755134i
\(383\) 52033.6i 0.354721i 0.984146 + 0.177360i \(0.0567558\pi\)
−0.984146 + 0.177360i \(0.943244\pi\)
\(384\) 0 0
\(385\) −34466.8 −0.232530
\(386\) −121787. −0.817383
\(387\) 0 0
\(388\) 40188.6i 0.266956i
\(389\) −57687.6 −0.381227 −0.190613 0.981665i \(-0.561048\pi\)
−0.190613 + 0.981665i \(0.561048\pi\)
\(390\) 0 0
\(391\) 14994.6 0.0980804
\(392\) 5561.56i 0.0361930i
\(393\) 0 0
\(394\) 74418.9i 0.479392i
\(395\) 104225.i 0.668002i
\(396\) 0 0
\(397\) 16914.2 0.107317 0.0536586 0.998559i \(-0.482912\pi\)
0.0536586 + 0.998559i \(0.482912\pi\)
\(398\) 33299.4i 0.210218i
\(399\) 0 0
\(400\) 5634.19 0.0352137
\(401\) 23481.6i 0.146029i 0.997331 + 0.0730144i \(0.0232619\pi\)
−0.997331 + 0.0730144i \(0.976738\pi\)
\(402\) 0 0
\(403\) −48603.0 −0.299263
\(404\) 126349. 0.774123
\(405\) 0 0
\(406\) −195856. −1.18819
\(407\) 8155.45i 0.0492333i
\(408\) 0 0
\(409\) 310776.i 1.85781i −0.370321 0.928904i \(-0.620752\pi\)
0.370321 0.928904i \(-0.379248\pi\)
\(410\) 3908.46 0.0232508
\(411\) 0 0
\(412\) 104549.i 0.615924i
\(413\) 55504.9i 0.325410i
\(414\) 0 0
\(415\) −26648.4 −0.154730
\(416\) 9639.83 0.0557035
\(417\) 0 0
\(418\) −25012.3 5535.81i −0.143153 0.0316832i
\(419\) 271562. 1.54682 0.773411 0.633905i \(-0.218549\pi\)
0.773411 + 0.633905i \(0.218549\pi\)
\(420\) 0 0
\(421\) 322467.i 1.81937i 0.415299 + 0.909685i \(0.363677\pi\)
−0.415299 + 0.909685i \(0.636323\pi\)
\(422\) 35343.2 0.198464
\(423\) 0 0
\(424\) 92045.2 0.512000
\(425\) 2156.08 0.0119368
\(426\) 0 0
\(427\) −335994. −1.84279
\(428\) 66184.7i 0.361302i
\(429\) 0 0
\(430\) 192235.i 1.03967i
\(431\) 157951.i 0.850290i 0.905125 + 0.425145i \(0.139777\pi\)
−0.905125 + 0.425145i \(0.860223\pi\)
\(432\) 0 0
\(433\) 36946.7i 0.197061i 0.995134 + 0.0985304i \(0.0314142\pi\)
−0.995134 + 0.0985304i \(0.968586\pi\)
\(434\) 132808. 0.705089
\(435\) 0 0
\(436\) 113756.i 0.598416i
\(437\) −47761.1 + 215797.i −0.250099 + 1.13001i
\(438\) 0 0
\(439\) 194377.i 1.00859i −0.863531 0.504296i \(-0.831752\pi\)
0.863531 0.504296i \(-0.168248\pi\)
\(440\) 15159.2i 0.0783016i
\(441\) 0 0
\(442\) 3688.94 0.0188824
\(443\) −30671.7 −0.156290 −0.0781449 0.996942i \(-0.524900\pi\)
−0.0781449 + 0.996942i \(0.524900\pi\)
\(444\) 0 0
\(445\) 164298.i 0.829682i
\(446\) −168875. −0.848979
\(447\) 0 0
\(448\) −26340.8 −0.131242
\(449\) 236846.i 1.17483i −0.809287 0.587413i \(-0.800146\pi\)
0.809287 0.587413i \(-0.199854\pi\)
\(450\) 0 0
\(451\) 1298.35i 0.00638319i
\(452\) 165980.i 0.812414i
\(453\) 0 0
\(454\) −163948. −0.795414
\(455\) 73157.6i 0.353376i
\(456\) 0 0
\(457\) −82060.7 −0.392919 −0.196459 0.980512i \(-0.562944\pi\)
−0.196459 + 0.980512i \(0.562944\pi\)
\(458\) 113584.i 0.541483i
\(459\) 0 0
\(460\) 130788. 0.618092
\(461\) 16781.0 0.0789614 0.0394807 0.999220i \(-0.487430\pi\)
0.0394807 + 0.999220i \(0.487430\pi\)
\(462\) 0 0
\(463\) 100169. 0.467274 0.233637 0.972324i \(-0.424937\pi\)
0.233637 + 0.972324i \(0.424937\pi\)
\(464\) 86141.6i 0.400108i
\(465\) 0 0
\(466\) 202912.i 0.934405i
\(467\) 216772. 0.993960 0.496980 0.867762i \(-0.334442\pi\)
0.496980 + 0.867762i \(0.334442\pi\)
\(468\) 0 0
\(469\) 379997.i 1.72756i
\(470\) 226443.i 1.02509i
\(471\) 0 0
\(472\) −24412.2 −0.109578
\(473\) −63858.4 −0.285427
\(474\) 0 0
\(475\) −6867.56 + 31029.5i −0.0304379 + 0.137527i
\(476\) −10080.0 −0.0444885
\(477\) 0 0
\(478\) 124955.i 0.546889i
\(479\) −91676.2 −0.399563 −0.199782 0.979840i \(-0.564023\pi\)
−0.199782 + 0.979840i \(0.564023\pi\)
\(480\) 0 0
\(481\) 17310.4 0.0748198
\(482\) 95667.8 0.411786
\(483\) 0 0
\(484\) 112092. 0.478503
\(485\) 134143.i 0.570276i
\(486\) 0 0
\(487\) 107326.i 0.452528i −0.974066 0.226264i \(-0.927349\pi\)
0.974066 0.226264i \(-0.0726512\pi\)
\(488\) 147777.i 0.620537i
\(489\) 0 0
\(490\) 18563.6i 0.0773160i
\(491\) −211818. −0.878619 −0.439310 0.898336i \(-0.644777\pi\)
−0.439310 + 0.898336i \(0.644777\pi\)
\(492\) 0 0
\(493\) 32964.4i 0.135629i
\(494\) −11750.1 + 53089.9i −0.0481489 + 0.217550i
\(495\) 0 0
\(496\) 58411.5i 0.237430i
\(497\) 298702.i 1.20928i
\(498\) 0 0
\(499\) 145489. 0.584291 0.292146 0.956374i \(-0.405631\pi\)
0.292146 + 0.956374i \(0.405631\pi\)
\(500\) −114707. −0.458830
\(501\) 0 0
\(502\) 183714.i 0.729013i
\(503\) 33735.6 0.133338 0.0666689 0.997775i \(-0.478763\pi\)
0.0666689 + 0.997775i \(0.478763\pi\)
\(504\) 0 0
\(505\) 421733. 1.65369
\(506\) 43446.4i 0.169689i
\(507\) 0 0
\(508\) 88942.4i 0.344653i
\(509\) 128247.i 0.495009i −0.968887 0.247505i \(-0.920389\pi\)
0.968887 0.247505i \(-0.0796105\pi\)
\(510\) 0 0
\(511\) 186730. 0.715108
\(512\) 11585.2i 0.0441942i
\(513\) 0 0
\(514\) 3268.50 0.0123715
\(515\) 348969.i 1.31575i
\(516\) 0 0
\(517\) −75222.0 −0.281426
\(518\) −47300.6 −0.176282
\(519\) 0 0
\(520\) 32176.2 0.118995
\(521\) 38092.9i 0.140336i −0.997535 0.0701680i \(-0.977646\pi\)
0.997535 0.0701680i \(-0.0223535\pi\)
\(522\) 0 0
\(523\) 96329.4i 0.352173i −0.984375 0.176086i \(-0.943656\pi\)
0.984375 0.176086i \(-0.0563438\pi\)
\(524\) 1786.53 0.00650652
\(525\) 0 0
\(526\) 19043.6i 0.0688301i
\(527\) 22352.8i 0.0804841i
\(528\) 0 0
\(529\) 94999.8 0.339478
\(530\) 307232. 1.09374
\(531\) 0 0
\(532\) 32107.0 145068.i 0.113443 0.512565i
\(533\) 2755.81 0.00970053
\(534\) 0 0
\(535\) 220914.i 0.771819i
\(536\) 167130. 0.581735
\(537\) 0 0
\(538\) −62880.1 −0.217244
\(539\) 6166.62 0.0212261
\(540\) 0 0
\(541\) −163021. −0.556993 −0.278496 0.960437i \(-0.589836\pi\)
−0.278496 + 0.960437i \(0.589836\pi\)
\(542\) 131798.i 0.448652i
\(543\) 0 0
\(544\) 4433.41i 0.0149810i
\(545\) 379701.i 1.27835i
\(546\) 0 0
\(547\) 450817.i 1.50670i −0.657621 0.753349i \(-0.728437\pi\)
0.657621 0.753349i \(-0.271563\pi\)
\(548\) 151624. 0.504903
\(549\) 0 0
\(550\) 6247.16i 0.0206518i
\(551\) −474412. 104999.i −1.56262 0.345844i
\(552\) 0 0
\(553\) 200806.i 0.656638i
\(554\) 299295.i 0.975168i
\(555\) 0 0
\(556\) 88924.4 0.287655
\(557\) 270930. 0.873268 0.436634 0.899639i \(-0.356171\pi\)
0.436634 + 0.899639i \(0.356171\pi\)
\(558\) 0 0
\(559\) 135543.i 0.433764i
\(560\) −87921.4 −0.280362
\(561\) 0 0
\(562\) 57412.1 0.181774
\(563\) 59429.1i 0.187492i −0.995596 0.0937459i \(-0.970116\pi\)
0.995596 0.0937459i \(-0.0298841\pi\)
\(564\) 0 0
\(565\) 554013.i 1.73549i
\(566\) 157573.i 0.491870i
\(567\) 0 0
\(568\) −131375. −0.407208
\(569\) 364145.i 1.12473i −0.826888 0.562367i \(-0.809891\pi\)
0.826888 0.562367i \(-0.190109\pi\)
\(570\) 0 0
\(571\) 410057. 1.25769 0.628843 0.777532i \(-0.283529\pi\)
0.628843 + 0.777532i \(0.283529\pi\)
\(572\) 10688.6i 0.0326684i
\(573\) 0 0
\(574\) −7530.26 −0.0228553
\(575\) 53898.3 0.163020
\(576\) 0 0
\(577\) −44617.5 −0.134015 −0.0670075 0.997752i \(-0.521345\pi\)
−0.0670075 + 0.997752i \(0.521345\pi\)
\(578\) 234536.i 0.702029i
\(579\) 0 0
\(580\) 287527.i 0.854717i
\(581\) 51342.3 0.152098
\(582\) 0 0
\(583\) 102059.i 0.300272i
\(584\) 82127.6i 0.240804i
\(585\) 0 0
\(586\) 237738. 0.692315
\(587\) 371422. 1.07793 0.538966 0.842328i \(-0.318815\pi\)
0.538966 + 0.842328i \(0.318815\pi\)
\(588\) 0 0
\(589\) 321693. + 71198.3i 0.927280 + 0.205229i
\(590\) −81483.9 −0.234082
\(591\) 0 0
\(592\) 20803.8i 0.0593606i
\(593\) 327963. 0.932644 0.466322 0.884615i \(-0.345579\pi\)
0.466322 + 0.884615i \(0.345579\pi\)
\(594\) 0 0
\(595\) −33645.5 −0.0950372
\(596\) −157822. −0.444299
\(597\) 0 0
\(598\) 92217.4 0.257876
\(599\) 81737.2i 0.227806i 0.993492 + 0.113903i \(0.0363354\pi\)
−0.993492 + 0.113903i \(0.963665\pi\)
\(600\) 0 0
\(601\) 75524.8i 0.209093i −0.994520 0.104547i \(-0.966661\pi\)
0.994520 0.104547i \(-0.0333392\pi\)
\(602\) 370371.i 1.02198i
\(603\) 0 0
\(604\) 6164.52i 0.0168976i
\(605\) 374146. 1.02219
\(606\) 0 0
\(607\) 97397.5i 0.264345i −0.991227 0.132172i \(-0.957805\pi\)
0.991227 0.132172i \(-0.0421952\pi\)
\(608\) −63804.0 14121.3i −0.172600 0.0382005i
\(609\) 0 0
\(610\) 493256.i 1.32560i
\(611\) 159663.i 0.427682i
\(612\) 0 0
\(613\) 185283. 0.493078 0.246539 0.969133i \(-0.420707\pi\)
0.246539 + 0.969133i \(0.420707\pi\)
\(614\) 29406.3 0.0780017
\(615\) 0 0
\(616\) 29206.6i 0.0769695i
\(617\) −628408. −1.65071 −0.825357 0.564612i \(-0.809026\pi\)
−0.825357 + 0.564612i \(0.809026\pi\)
\(618\) 0 0
\(619\) 203791. 0.531867 0.265933 0.963991i \(-0.414320\pi\)
0.265933 + 0.963991i \(0.414320\pi\)
\(620\) 194968.i 0.507201i
\(621\) 0 0
\(622\) 334058.i 0.863458i
\(623\) 316545.i 0.815568i
\(624\) 0 0
\(625\) −437896. −1.12101
\(626\) 23733.8i 0.0605646i
\(627\) 0 0
\(628\) −369956. −0.938061
\(629\) 7961.13i 0.0201221i
\(630\) 0 0
\(631\) 496722. 1.24754 0.623771 0.781607i \(-0.285600\pi\)
0.623771 + 0.781607i \(0.285600\pi\)
\(632\) 88318.5 0.221115
\(633\) 0 0
\(634\) 405805. 1.00958
\(635\) 296875.i 0.736252i
\(636\) 0 0
\(637\) 13089.0i 0.0322572i
\(638\) −95513.2 −0.234651
\(639\) 0 0
\(640\) 38669.6i 0.0944083i
\(641\) 44345.5i 0.107928i 0.998543 + 0.0539639i \(0.0171856\pi\)
−0.998543 + 0.0539639i \(0.982814\pi\)
\(642\) 0 0
\(643\) 139327. 0.336987 0.168494 0.985703i \(-0.446110\pi\)
0.168494 + 0.985703i \(0.446110\pi\)
\(644\) −251984. −0.607577
\(645\) 0 0
\(646\) −24416.3 5403.91i −0.0585080 0.0129492i
\(647\) 617612. 1.47539 0.737696 0.675133i \(-0.235914\pi\)
0.737696 + 0.675133i \(0.235914\pi\)
\(648\) 0 0
\(649\) 27068.1i 0.0642640i
\(650\) 13259.9 0.0313845
\(651\) 0 0
\(652\) −263615. −0.620119
\(653\) −570062. −1.33689 −0.668446 0.743761i \(-0.733040\pi\)
−0.668446 + 0.743761i \(0.733040\pi\)
\(654\) 0 0
\(655\) 5963.16 0.0138993
\(656\) 3311.96i 0.00769622i
\(657\) 0 0
\(658\) 436278.i 1.00765i
\(659\) 716546.i 1.64996i 0.565162 + 0.824980i \(0.308814\pi\)
−0.565162 + 0.824980i \(0.691186\pi\)
\(660\) 0 0
\(661\) 187692.i 0.429579i −0.976660 0.214790i \(-0.931093\pi\)
0.976660 0.214790i \(-0.0689066\pi\)
\(662\) 47923.0 0.109352
\(663\) 0 0
\(664\) 22581.4i 0.0512171i
\(665\) 107168. 484214.i 0.242338 1.09495i
\(666\) 0 0
\(667\) 824055.i 1.85227i
\(668\) 384393.i 0.861434i
\(669\) 0 0
\(670\) 557854. 1.24271
\(671\) −163854. −0.363926
\(672\) 0 0
\(673\) 504263.i 1.11334i 0.830734 + 0.556669i \(0.187921\pi\)
−0.830734 + 0.556669i \(0.812079\pi\)
\(674\) 614244. 1.35214
\(675\) 0 0
\(676\) −205801. −0.450354
\(677\) 689926.i 1.50531i 0.658417 + 0.752653i \(0.271226\pi\)
−0.658417 + 0.752653i \(0.728774\pi\)
\(678\) 0 0
\(679\) 258448.i 0.560574i
\(680\) 14798.0i 0.0320026i
\(681\) 0 0
\(682\) 64766.3 0.139245
\(683\) 226300.i 0.485113i −0.970137 0.242556i \(-0.922014\pi\)
0.970137 0.242556i \(-0.0779859\pi\)
\(684\) 0 0
\(685\) 506098. 1.07858
\(686\) 313613.i 0.666417i
\(687\) 0 0
\(688\) −162897. −0.344140
\(689\) 216626. 0.456323
\(690\) 0 0
\(691\) −921891. −1.93074 −0.965370 0.260886i \(-0.915985\pi\)
−0.965370 + 0.260886i \(0.915985\pi\)
\(692\) 369756.i 0.772153i
\(693\) 0 0
\(694\) 182589.i 0.379102i
\(695\) 296815. 0.614492
\(696\) 0 0
\(697\) 1267.41i 0.00260887i
\(698\) 320390.i 0.657610i
\(699\) 0 0
\(700\) −36232.8 −0.0739444
\(701\) 561368. 1.14238 0.571191 0.820817i \(-0.306481\pi\)
0.571191 + 0.820817i \(0.306481\pi\)
\(702\) 0 0
\(703\) −114574. 25357.9i −0.231832 0.0513100i
\(704\) −12845.6 −0.0259185
\(705\) 0 0
\(706\) 78306.0i 0.157103i
\(707\) −812536. −1.62556
\(708\) 0 0
\(709\) −287728. −0.572386 −0.286193 0.958172i \(-0.592390\pi\)
−0.286193 + 0.958172i \(0.592390\pi\)
\(710\) −438509. −0.869885
\(711\) 0 0
\(712\) 139223. 0.274632
\(713\) 558782.i 1.09917i
\(714\) 0 0
\(715\) 35676.8i 0.0697868i
\(716\) 318612.i 0.621493i
\(717\) 0 0
\(718\) 184799.i 0.358468i
\(719\) 805860. 1.55884 0.779420 0.626502i \(-0.215514\pi\)
0.779420 + 0.626502i \(0.215514\pi\)
\(720\) 0 0
\(721\) 672344.i 1.29336i
\(722\) 155542. 334178.i 0.298383 0.641068i
\(723\) 0 0
\(724\) 414756.i 0.791254i
\(725\) 118491.i 0.225428i
\(726\) 0 0
\(727\) −348604. −0.659575 −0.329787 0.944055i \(-0.606977\pi\)
−0.329787 + 0.944055i \(0.606977\pi\)
\(728\) −61992.5 −0.116970
\(729\) 0 0
\(730\) 274129.i 0.514409i
\(731\) −62336.8 −0.116657
\(732\) 0 0
\(733\) 238795. 0.444443 0.222222 0.974996i \(-0.428669\pi\)
0.222222 + 0.974996i \(0.428669\pi\)
\(734\) 501475.i 0.930801i
\(735\) 0 0
\(736\) 110828.i 0.204594i
\(737\) 185313.i 0.341170i
\(738\) 0 0
\(739\) 502122. 0.919433 0.459717 0.888066i \(-0.347951\pi\)
0.459717 + 0.888066i \(0.347951\pi\)
\(740\) 69439.6i 0.126807i
\(741\) 0 0
\(742\) −591931. −1.07514
\(743\) 411861.i 0.746058i 0.927820 + 0.373029i \(0.121681\pi\)
−0.927820 + 0.373029i \(0.878319\pi\)
\(744\) 0 0
\(745\) −526784. −0.949118
\(746\) −43505.8 −0.0781753
\(747\) 0 0
\(748\) −4915.73 −0.00878588
\(749\) 425625.i 0.758689i
\(750\) 0 0
\(751\) 876407.i 1.55391i 0.629556 + 0.776955i \(0.283237\pi\)
−0.629556 + 0.776955i \(0.716763\pi\)
\(752\) −191884. −0.339315
\(753\) 0 0
\(754\) 202732.i 0.356599i
\(755\) 20576.2i 0.0360970i
\(756\) 0 0
\(757\) −447975. −0.781739 −0.390870 0.920446i \(-0.627826\pi\)
−0.390870 + 0.920446i \(0.627826\pi\)
\(758\) −657964. −1.14515
\(759\) 0 0
\(760\) −212967. 47134.7i −0.368711 0.0816044i
\(761\) −527400. −0.910690 −0.455345 0.890315i \(-0.650484\pi\)
−0.455345 + 0.890315i \(0.650484\pi\)
\(762\) 0 0
\(763\) 731553.i 1.25660i
\(764\) −311671. −0.533960
\(765\) 0 0
\(766\) −147173. −0.250825
\(767\) −57453.4 −0.0976619
\(768\) 0 0
\(769\) 549287. 0.928852 0.464426 0.885612i \(-0.346261\pi\)
0.464426 + 0.885612i \(0.346261\pi\)
\(770\) 97486.7i 0.164424i
\(771\) 0 0
\(772\) 344465.i 0.577977i
\(773\) 1.03569e6i 1.73329i 0.498927 + 0.866644i \(0.333728\pi\)
−0.498927 + 0.866644i \(0.666272\pi\)
\(774\) 0 0
\(775\) 80347.2i 0.133773i
\(776\) 113671. 0.188766
\(777\) 0 0
\(778\) 163165.i 0.269568i
\(779\) −18240.1 4036.98i −0.0300575 0.00665244i
\(780\) 0 0
\(781\) 145668.i 0.238815i
\(782\) 42411.2i 0.0693533i
\(783\) 0 0
\(784\) 15730.5 0.0255923
\(785\) −1.23485e6 −2.00390
\(786\) 0 0
\(787\) 1.04115e6i 1.68098i 0.541824 + 0.840492i \(0.317734\pi\)
−0.541824 + 0.840492i \(0.682266\pi\)
\(788\) 210488. 0.338981
\(789\) 0 0
\(790\) 294793. 0.472349
\(791\) 1.06739e6i 1.70597i
\(792\) 0 0
\(793\) 347790.i 0.553058i
\(794\) 47840.5i 0.0758847i
\(795\) 0 0
\(796\) 94184.8 0.148647
\(797\) 476576.i 0.750266i −0.926971 0.375133i \(-0.877597\pi\)
0.926971 0.375133i \(-0.122403\pi\)
\(798\) 0 0
\(799\) −73429.7 −0.115021
\(800\) 15935.9i 0.0248999i
\(801\) 0 0
\(802\) −66415.9 −0.103258
\(803\) 91062.5 0.141224
\(804\) 0 0
\(805\) −841082. −1.29792
\(806\) 137470.i 0.211611i
\(807\) 0 0
\(808\) 357370.i 0.547388i
\(809\) 1.00569e6 1.53663 0.768314 0.640074i \(-0.221096\pi\)
0.768314 + 0.640074i \(0.221096\pi\)
\(810\) 0 0
\(811\) 257961.i 0.392205i −0.980583 0.196102i \(-0.937171\pi\)
0.980583 0.196102i \(-0.0628285\pi\)
\(812\) 553965.i 0.840176i
\(813\) 0 0
\(814\) −23067.1 −0.0348132
\(815\) −879905. −1.32471
\(816\) 0 0
\(817\) 198556. 897129.i 0.297467 1.34404i
\(818\) 879007. 1.31367
\(819\) 0 0
\(820\) 11054.8i 0.0164408i
\(821\) 65320.9 0.0969094 0.0484547 0.998825i \(-0.484570\pi\)
0.0484547 + 0.998825i \(0.484570\pi\)
\(822\) 0 0
\(823\) −109324. −0.161404 −0.0807020 0.996738i \(-0.525716\pi\)
−0.0807020 + 0.996738i \(0.525716\pi\)
\(824\) −295710. −0.435524
\(825\) 0 0
\(826\) 156991. 0.230100
\(827\) 988603.i 1.44548i −0.691122 0.722738i \(-0.742883\pi\)
0.691122 0.722738i \(-0.257117\pi\)
\(828\) 0 0
\(829\) 467829.i 0.680735i −0.940292 0.340368i \(-0.889448\pi\)
0.940292 0.340368i \(-0.110552\pi\)
\(830\) 75373.1i 0.109411i
\(831\) 0 0
\(832\) 27265.6i 0.0393884i
\(833\) 6019.69 0.00867529
\(834\) 0 0
\(835\) 1.28304e6i 1.84021i
\(836\) 15657.6 70745.4i 0.0224034 0.101225i
\(837\) 0 0
\(838\) 768092.i 1.09377i
\(839\) 657894.i 0.934613i 0.884095 + 0.467306i \(0.154776\pi\)
−0.884095 + 0.467306i \(0.845224\pi\)
\(840\) 0 0
\(841\) −1.10433e6 −1.56138
\(842\) −912074. −1.28649
\(843\) 0 0
\(844\) 99965.6i 0.140335i
\(845\) −686930. −0.962053
\(846\) 0 0
\(847\) −720851. −1.00480
\(848\) 260343.i 0.362038i
\(849\) 0 0
\(850\) 6098.31i 0.00844057i
\(851\) 199015.i 0.274806i
\(852\) 0 0
\(853\) −226576. −0.311398 −0.155699 0.987805i \(-0.549763\pi\)
−0.155699 + 0.987805i \(0.549763\pi\)
\(854\) 950336.i 1.30305i
\(855\) 0 0
\(856\) 187199. 0.255479
\(857\) 183273.i 0.249539i 0.992186 + 0.124769i \(0.0398191\pi\)
−0.992186 + 0.124769i \(0.960181\pi\)
\(858\) 0 0
\(859\) 565839. 0.766843 0.383421 0.923574i \(-0.374746\pi\)
0.383421 + 0.923574i \(0.374746\pi\)
\(860\) −543723. −0.735158
\(861\) 0 0
\(862\) −446752. −0.601246
\(863\) 247431.i 0.332225i −0.986107 0.166113i \(-0.946878\pi\)
0.986107 0.166113i \(-0.0531216\pi\)
\(864\) 0 0
\(865\) 1.23419e6i 1.64949i
\(866\) −104501. −0.139343
\(867\) 0 0
\(868\) 375637.i 0.498573i
\(869\) 97927.0i 0.129677i
\(870\) 0 0
\(871\) 393337. 0.518476
\(872\) 321752. 0.423144
\(873\) 0 0
\(874\) −610367. 135089.i −0.799040 0.176846i
\(875\) 737669. 0.963486
\(876\) 0 0
\(877\) 58891.6i 0.0765692i 0.999267 + 0.0382846i \(0.0121893\pi\)
−0.999267 + 0.0382846i \(0.987811\pi\)
\(878\) 549780. 0.713182
\(879\) 0 0
\(880\) −42876.7 −0.0553676
\(881\) 844161. 1.08761 0.543806 0.839211i \(-0.316983\pi\)
0.543806 + 0.839211i \(0.316983\pi\)
\(882\) 0 0
\(883\) −1.34541e6 −1.72558 −0.862788 0.505565i \(-0.831284\pi\)
−0.862788 + 0.505565i \(0.831284\pi\)
\(884\) 10433.9i 0.0133519i
\(885\) 0 0
\(886\) 86752.7i 0.110514i
\(887\) 268542.i 0.341323i 0.985330 + 0.170661i \(0.0545904\pi\)
−0.985330 + 0.170661i \(0.945410\pi\)
\(888\) 0 0
\(889\) 571977.i 0.723727i
\(890\) 464704. 0.586674
\(891\) 0 0
\(892\) 477652.i 0.600319i
\(893\) 233889. 1.05677e6i 0.293296 1.32519i
\(894\) 0 0
\(895\) 1.06348e6i 1.32764i
\(896\) 74503.1i 0.0928023i
\(897\) 0 0
\(898\) 669902. 0.830728
\(899\) 1.22843e6 1.51996
\(900\) 0 0
\(901\) 99627.4i 0.122724i
\(902\) −3672.28 −0.00451360
\(903\) 0 0
\(904\) 469461. 0.574464
\(905\) 1.38439e6i 1.69029i
\(906\) 0 0
\(907\) 1.24035e6i 1.50775i −0.657017 0.753876i \(-0.728182\pi\)
0.657017 0.753876i \(-0.271818\pi\)
\(908\) 463714.i 0.562443i
\(909\) 0 0
\(910\) −206921. −0.249874
\(911\) 191069.i 0.230226i −0.993352 0.115113i \(-0.963277\pi\)
0.993352 0.115113i \(-0.0367229\pi\)
\(912\) 0 0
\(913\) 25038.1 0.0300372
\(914\) 232103.i 0.277836i
\(915\) 0 0
\(916\) −321263. −0.382886
\(917\) −11489.0 −0.0136629
\(918\) 0 0
\(919\) 830092. 0.982868 0.491434 0.870915i \(-0.336473\pi\)
0.491434 + 0.870915i \(0.336473\pi\)
\(920\) 369925.i 0.437057i
\(921\) 0 0
\(922\) 47463.7i 0.0558342i
\(923\) −309188. −0.362927
\(924\) 0 0
\(925\) 28616.3i 0.0334449i
\(926\) 283321.i 0.330413i
\(927\) 0 0
\(928\) −243645. −0.282919
\(929\) −1.18305e6 −1.37080 −0.685399 0.728168i \(-0.740372\pi\)
−0.685399 + 0.728168i \(0.740372\pi\)
\(930\) 0 0
\(931\) −19174.0 + 86633.2i −0.0221214 + 0.0999505i
\(932\) 573921. 0.660724
\(933\) 0 0
\(934\) 613123.i 0.702836i
\(935\) −16407.9 −0.0187685
\(936\) 0 0
\(937\) 1.14337e6 1.30229 0.651144 0.758954i \(-0.274289\pi\)
0.651144 + 0.758954i \(0.274289\pi\)
\(938\) −1.07479e6 −1.22157
\(939\) 0 0
\(940\) −640478. −0.724851
\(941\) 1.12716e6i 1.27294i 0.771302 + 0.636469i \(0.219606\pi\)
−0.771302 + 0.636469i \(0.780394\pi\)
\(942\) 0 0
\(943\) 31683.2i 0.0356291i
\(944\) 69048.0i 0.0774832i
\(945\) 0 0
\(946\) 180619.i 0.201828i
\(947\) −1.01859e6 −1.13579 −0.567897 0.823099i \(-0.692243\pi\)
−0.567897 + 0.823099i \(0.692243\pi\)
\(948\) 0 0
\(949\) 193285.i 0.214618i
\(950\) −87764.6 19424.4i −0.0972461 0.0215229i
\(951\) 0 0
\(952\) 28510.6i 0.0314582i
\(953\) 18352.7i 0.0202076i 0.999949 + 0.0101038i \(0.00321619\pi\)
−0.999949 + 0.0101038i \(0.996784\pi\)
\(954\) 0 0
\(955\) −1.04031e6 −1.14066
\(956\) 353427. 0.386709
\(957\) 0 0
\(958\) 259299.i 0.282534i
\(959\) −975077. −1.06023
\(960\) 0 0
\(961\) 90535.6 0.0980331
\(962\) 48961.1i 0.0529056i
\(963\) 0 0
\(964\) 270589.i 0.291177i
\(965\) 1.14977e6i 1.23468i
\(966\) 0 0
\(967\) 301406. 0.322329 0.161164 0.986928i \(-0.448475\pi\)
0.161164 + 0.986928i \(0.448475\pi\)
\(968\) 317045.i 0.338353i
\(969\) 0 0
\(970\) 379414. 0.403246
\(971\) 796081.i 0.844342i −0.906516 0.422171i \(-0.861268\pi\)
0.906516 0.422171i \(-0.138732\pi\)
\(972\) 0 0
\(973\) −571861. −0.604039
\(974\) 303562. 0.319985
\(975\) 0 0
\(976\) −417977. −0.438786
\(977\) 664866.i 0.696538i −0.937395 0.348269i \(-0.886770\pi\)
0.937395 0.348269i \(-0.113230\pi\)
\(978\) 0 0
\(979\) 154370.i 0.161063i
\(980\) 52505.7 0.0546707
\(981\) 0 0
\(982\) 599113.i 0.621278i
\(983\) 1.05287e6i 1.08960i 0.838566 + 0.544801i \(0.183395\pi\)
−0.838566 + 0.544801i \(0.816605\pi\)
\(984\) 0 0
\(985\) 702576. 0.724137
\(986\) −93237.5 −0.0959040
\(987\) 0 0
\(988\) −150161. 33234.2i −0.153831 0.0340464i
\(989\) −1.55832e6 −1.59317
\(990\) 0 0
\(991\) 1.85377e6i 1.88759i 0.330529 + 0.943796i \(0.392773\pi\)
−0.330529 + 0.943796i \(0.607227\pi\)
\(992\) 165213. 0.167888
\(993\) 0 0
\(994\) 844856. 0.855087
\(995\) 314374. 0.317541
\(996\) 0 0
\(997\) 494751. 0.497733 0.248867 0.968538i \(-0.419942\pi\)
0.248867 + 0.968538i \(0.419942\pi\)
\(998\) 411505.i 0.413156i
\(999\) 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 342.5.d.a.37.6 8
3.2 odd 2 38.5.b.a.37.3 8
12.11 even 2 304.5.e.e.113.4 8
19.18 odd 2 inner 342.5.d.a.37.2 8
57.56 even 2 38.5.b.a.37.6 yes 8
228.227 odd 2 304.5.e.e.113.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.5.b.a.37.3 8 3.2 odd 2
38.5.b.a.37.6 yes 8 57.56 even 2
304.5.e.e.113.4 8 12.11 even 2
304.5.e.e.113.5 8 228.227 odd 2
342.5.d.a.37.2 8 19.18 odd 2 inner
342.5.d.a.37.6 8 1.1 even 1 trivial