Properties

Label 684.5.y.f.145.12
Level $684$
Weight $5$
Character 684.145
Analytic conductor $70.705$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [684,5,Mod(145,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.145");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 684.y (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(70.7050547493\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 145.12
Character \(\chi\) \(=\) 684.145
Dual form 684.5.y.f.217.12

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(22.6095 + 39.1607i) q^{5} +80.9352 q^{7} +O(q^{10})\) \(q+(22.6095 + 39.1607i) q^{5} +80.9352 q^{7} -74.1039 q^{11} +(-249.736 - 144.185i) q^{13} +(255.062 + 441.781i) q^{17} +(-356.260 + 58.3104i) q^{19} +(-122.424 + 212.044i) q^{23} +(-709.876 + 1229.54i) q^{25} +(-932.845 - 538.578i) q^{29} -96.0919i q^{31} +(1829.90 + 3169.48i) q^{35} +1334.85i q^{37} +(-1755.03 + 1013.27i) q^{41} +(-1407.13 - 2437.22i) q^{43} +(-1409.02 + 2440.49i) q^{47} +4149.51 q^{49} +(-754.360 - 435.530i) q^{53} +(-1675.45 - 2901.96i) q^{55} +(5311.50 - 3066.60i) q^{59} +(-14.3481 + 24.8516i) q^{61} -13039.8i q^{65} +(3618.21 + 2088.97i) q^{67} +(2892.44 - 1669.95i) q^{71} +(-457.115 - 791.747i) q^{73} -5997.61 q^{77} +(413.941 - 238.989i) q^{79} -1080.06 q^{83} +(-11533.6 + 19976.9i) q^{85} +(10502.6 + 6063.65i) q^{89} +(-20212.5 - 11669.7i) q^{91} +(-10338.3 - 12633.0i) q^{95} +(-10463.6 + 6041.19i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 304 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 304 q^{7} + 516 q^{13} + 416 q^{19} - 5108 q^{25} - 720 q^{43} + 25440 q^{49} - 12536 q^{55} + 5020 q^{61} - 13080 q^{67} + 22572 q^{73} + 63096 q^{79} - 32624 q^{85} - 89568 q^{91} - 3888 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 22.6095 + 39.1607i 0.904379 + 1.56643i 0.821749 + 0.569850i \(0.192999\pi\)
0.0826297 + 0.996580i \(0.473668\pi\)
\(6\) 0 0
\(7\) 80.9352 1.65174 0.825870 0.563861i \(-0.190685\pi\)
0.825870 + 0.563861i \(0.190685\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −74.1039 −0.612429 −0.306214 0.951963i \(-0.599062\pi\)
−0.306214 + 0.951963i \(0.599062\pi\)
\(12\) 0 0
\(13\) −249.736 144.185i −1.47773 0.853168i −0.478047 0.878335i \(-0.658655\pi\)
−0.999683 + 0.0251668i \(0.991988\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 255.062 + 441.781i 0.882568 + 1.52865i 0.848475 + 0.529235i \(0.177521\pi\)
0.0340930 + 0.999419i \(0.489146\pi\)
\(18\) 0 0
\(19\) −356.260 + 58.3104i −0.986869 + 0.161525i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −122.424 + 212.044i −0.231425 + 0.400839i −0.958228 0.286007i \(-0.907672\pi\)
0.726803 + 0.686846i \(0.241005\pi\)
\(24\) 0 0
\(25\) −709.876 + 1229.54i −1.13580 + 1.96727i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −932.845 538.578i −1.10921 0.640402i −0.170585 0.985343i \(-0.554566\pi\)
−0.938624 + 0.344941i \(0.887899\pi\)
\(30\) 0 0
\(31\) 96.0919i 0.0999915i −0.998749 0.0499958i \(-0.984079\pi\)
0.998749 0.0499958i \(-0.0159208\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1829.90 + 3169.48i 1.49380 + 2.58733i
\(36\) 0 0
\(37\) 1334.85i 0.975058i 0.873107 + 0.487529i \(0.162102\pi\)
−0.873107 + 0.487529i \(0.837898\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1755.03 + 1013.27i −1.04404 + 0.602778i −0.920975 0.389621i \(-0.872606\pi\)
−0.123066 + 0.992398i \(0.539273\pi\)
\(42\) 0 0
\(43\) −1407.13 2437.22i −0.761020 1.31813i −0.942325 0.334699i \(-0.891365\pi\)
0.181305 0.983427i \(-0.441968\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1409.02 + 2440.49i −0.637854 + 1.10479i 0.348049 + 0.937476i \(0.386844\pi\)
−0.985903 + 0.167319i \(0.946489\pi\)
\(48\) 0 0
\(49\) 4149.51 1.72824
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −754.360 435.530i −0.268551 0.155048i 0.359678 0.933077i \(-0.382887\pi\)
−0.628229 + 0.778028i \(0.716220\pi\)
\(54\) 0 0
\(55\) −1675.45 2901.96i −0.553867 0.959326i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5311.50 3066.60i 1.52586 0.880953i 0.526326 0.850283i \(-0.323569\pi\)
0.999530 0.0306702i \(-0.00976415\pi\)
\(60\) 0 0
\(61\) −14.3481 + 24.8516i −0.00385598 + 0.00667875i −0.867947 0.496657i \(-0.834561\pi\)
0.864091 + 0.503336i \(0.167894\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 13039.8i 3.08635i
\(66\) 0 0
\(67\) 3618.21 + 2088.97i 0.806016 + 0.465354i 0.845571 0.533864i \(-0.179260\pi\)
−0.0395543 + 0.999217i \(0.512594\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2892.44 1669.95i 0.573783 0.331274i −0.184876 0.982762i \(-0.559188\pi\)
0.758659 + 0.651488i \(0.225855\pi\)
\(72\) 0 0
\(73\) −457.115 791.747i −0.0857788 0.148573i 0.819944 0.572444i \(-0.194005\pi\)
−0.905723 + 0.423870i \(0.860671\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5997.61 −1.01157
\(78\) 0 0
\(79\) 413.941 238.989i 0.0663261 0.0382934i −0.466470 0.884537i \(-0.654475\pi\)
0.532796 + 0.846244i \(0.321141\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1080.06 −0.156781 −0.0783903 0.996923i \(-0.524978\pi\)
−0.0783903 + 0.996923i \(0.524978\pi\)
\(84\) 0 0
\(85\) −11533.6 + 19976.9i −1.59635 + 2.76496i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10502.6 + 6063.65i 1.32591 + 0.765516i 0.984665 0.174457i \(-0.0558171\pi\)
0.341248 + 0.939973i \(0.389150\pi\)
\(90\) 0 0
\(91\) −20212.5 11669.7i −2.44082 1.40921i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −10338.3 12633.0i −1.14552 1.39978i
\(96\) 0 0
\(97\) −10463.6 + 6041.19i −1.11209 + 0.642065i −0.939370 0.342906i \(-0.888589\pi\)
−0.172719 + 0.984971i \(0.555255\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1988.34 + 3443.91i −0.194917 + 0.337606i −0.946873 0.321607i \(-0.895777\pi\)
0.751957 + 0.659213i \(0.229110\pi\)
\(102\) 0 0
\(103\) 11605.7i 1.09395i 0.837149 + 0.546975i \(0.184221\pi\)
−0.837149 + 0.546975i \(0.815779\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8776.07i 0.766536i −0.923637 0.383268i \(-0.874799\pi\)
0.923637 0.383268i \(-0.125201\pi\)
\(108\) 0 0
\(109\) −7467.61 + 4311.42i −0.628533 + 0.362884i −0.780184 0.625550i \(-0.784875\pi\)
0.151650 + 0.988434i \(0.451541\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 10759.3i 0.842609i −0.906919 0.421305i \(-0.861572\pi\)
0.906919 0.421305i \(-0.138428\pi\)
\(114\) 0 0
\(115\) −11071.7 −0.837182
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 20643.5 + 35755.6i 1.45777 + 2.52494i
\(120\) 0 0
\(121\) −9149.62 −0.624931
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −35937.8 −2.30002
\(126\) 0 0
\(127\) 17433.4 + 10065.2i 1.08087 + 0.624042i 0.931132 0.364683i \(-0.118823\pi\)
0.149741 + 0.988725i \(0.452156\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8527.03 + 14769.2i 0.496884 + 0.860629i 0.999994 0.00359405i \(-0.00114402\pi\)
−0.503109 + 0.864223i \(0.667811\pi\)
\(132\) 0 0
\(133\) −28833.9 + 4719.36i −1.63005 + 0.266796i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1504.38 2605.67i 0.0801526 0.138828i −0.823163 0.567805i \(-0.807793\pi\)
0.903315 + 0.428977i \(0.141126\pi\)
\(138\) 0 0
\(139\) −14478.5 + 25077.6i −0.749368 + 1.29794i 0.198758 + 0.980049i \(0.436309\pi\)
−0.948126 + 0.317895i \(0.897024\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 18506.4 + 10684.7i 0.905004 + 0.522504i
\(144\) 0 0
\(145\) 48707.9i 2.31666i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2948.53 5107.00i −0.132811 0.230035i 0.791948 0.610588i \(-0.209067\pi\)
−0.924759 + 0.380553i \(0.875734\pi\)
\(150\) 0 0
\(151\) 17146.3i 0.752000i −0.926620 0.376000i \(-0.877299\pi\)
0.926620 0.376000i \(-0.122701\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3763.03 2172.59i 0.156630 0.0904302i
\(156\) 0 0
\(157\) −5645.97 9779.11i −0.229055 0.396735i 0.728473 0.685074i \(-0.240230\pi\)
−0.957528 + 0.288340i \(0.906897\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −9908.39 + 17161.8i −0.382253 + 0.662082i
\(162\) 0 0
\(163\) 7367.31 0.277290 0.138645 0.990342i \(-0.455725\pi\)
0.138645 + 0.990342i \(0.455725\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11698.1 + 6753.93i 0.419454 + 0.242172i 0.694844 0.719161i \(-0.255474\pi\)
−0.275390 + 0.961333i \(0.588807\pi\)
\(168\) 0 0
\(169\) 27298.3 + 47282.1i 0.955790 + 1.65548i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3410.27 1968.92i 0.113946 0.0657865i −0.441944 0.897043i \(-0.645711\pi\)
0.555890 + 0.831256i \(0.312378\pi\)
\(174\) 0 0
\(175\) −57454.0 + 99513.2i −1.87605 + 3.24941i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 54947.2i 1.71490i 0.514565 + 0.857451i \(0.327953\pi\)
−0.514565 + 0.857451i \(0.672047\pi\)
\(180\) 0 0
\(181\) 6115.34 + 3530.69i 0.186665 + 0.107771i 0.590420 0.807096i \(-0.298962\pi\)
−0.403755 + 0.914867i \(0.632295\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −52273.9 + 30180.4i −1.52736 + 0.881822i
\(186\) 0 0
\(187\) −18901.1 32737.7i −0.540510 0.936191i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 25754.7 0.705975 0.352988 0.935628i \(-0.385166\pi\)
0.352988 + 0.935628i \(0.385166\pi\)
\(192\) 0 0
\(193\) 14871.2 8585.90i 0.399238 0.230500i −0.286917 0.957955i \(-0.592630\pi\)
0.686155 + 0.727455i \(0.259297\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 22172.8 0.571331 0.285666 0.958329i \(-0.407785\pi\)
0.285666 + 0.958329i \(0.407785\pi\)
\(198\) 0 0
\(199\) −5574.48 + 9655.29i −0.140766 + 0.243814i −0.927785 0.373114i \(-0.878290\pi\)
0.787019 + 0.616929i \(0.211623\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −75500.0 43590.0i −1.83212 1.05778i
\(204\) 0 0
\(205\) −79360.7 45818.9i −1.88842 1.09028i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 26400.2 4321.02i 0.604387 0.0989222i
\(210\) 0 0
\(211\) −6725.42 + 3882.93i −0.151062 + 0.0872156i −0.573626 0.819118i \(-0.694464\pi\)
0.422564 + 0.906333i \(0.361130\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 63628.8 110208.i 1.37650 2.38417i
\(216\) 0 0
\(217\) 7777.22i 0.165160i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 147105.i 3.01192i
\(222\) 0 0
\(223\) −9431.33 + 5445.18i −0.189654 + 0.109497i −0.591821 0.806070i \(-0.701591\pi\)
0.402166 + 0.915567i \(0.368257\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 36261.1i 0.703703i −0.936056 0.351852i \(-0.885552\pi\)
0.936056 0.351852i \(-0.114448\pi\)
\(228\) 0 0
\(229\) −38554.9 −0.735205 −0.367603 0.929983i \(-0.619821\pi\)
−0.367603 + 0.929983i \(0.619821\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −22940.1 39733.4i −0.422555 0.731886i 0.573634 0.819112i \(-0.305533\pi\)
−0.996189 + 0.0872256i \(0.972200\pi\)
\(234\) 0 0
\(235\) −127429. −2.30744
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −44600.7 −0.780812 −0.390406 0.920643i \(-0.627665\pi\)
−0.390406 + 0.920643i \(0.627665\pi\)
\(240\) 0 0
\(241\) 64615.1 + 37305.6i 1.11250 + 0.642302i 0.939476 0.342615i \(-0.111313\pi\)
0.173024 + 0.984918i \(0.444646\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 93818.2 + 162498.i 1.56299 + 2.70717i
\(246\) 0 0
\(247\) 97378.5 + 36805.2i 1.59613 + 0.603275i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −49554.2 + 85830.3i −0.786562 + 1.36236i 0.141500 + 0.989938i \(0.454807\pi\)
−0.928062 + 0.372426i \(0.878526\pi\)
\(252\) 0 0
\(253\) 9072.07 15713.3i 0.141731 0.245485i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −79479.8 45887.7i −1.20335 0.694752i −0.242048 0.970264i \(-0.577819\pi\)
−0.961298 + 0.275512i \(0.911153\pi\)
\(258\) 0 0
\(259\) 108037.i 1.61054i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1387.54 2403.29i −0.0200602 0.0347452i 0.855821 0.517272i \(-0.173052\pi\)
−0.875881 + 0.482527i \(0.839719\pi\)
\(264\) 0 0
\(265\) 39388.4i 0.560889i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 109424. 63175.9i 1.51219 0.873065i 0.512295 0.858810i \(-0.328796\pi\)
0.999898 0.0142555i \(-0.00453783\pi\)
\(270\) 0 0
\(271\) −61390.2 106331.i −0.835911 1.44784i −0.893286 0.449488i \(-0.851606\pi\)
0.0573751 0.998353i \(-0.481727\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 52604.6 91113.8i 0.695597 1.20481i
\(276\) 0 0
\(277\) −32054.0 −0.417756 −0.208878 0.977942i \(-0.566981\pi\)
−0.208878 + 0.977942i \(0.566981\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 66717.7 + 38519.5i 0.844945 + 0.487829i 0.858942 0.512073i \(-0.171122\pi\)
−0.0139972 + 0.999902i \(0.504456\pi\)
\(282\) 0 0
\(283\) −9079.82 15726.7i −0.113372 0.196365i 0.803756 0.594959i \(-0.202832\pi\)
−0.917128 + 0.398594i \(0.869498\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −142044. + 82009.2i −1.72448 + 0.995631i
\(288\) 0 0
\(289\) −88353.0 + 153032.i −1.05785 + 1.83226i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 63343.7i 0.737850i 0.929459 + 0.368925i \(0.120274\pi\)
−0.929459 + 0.368925i \(0.879726\pi\)
\(294\) 0 0
\(295\) 240181. + 138668.i 2.75990 + 1.59343i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 61147.3 35303.4i 0.683966 0.394888i
\(300\) 0 0
\(301\) −113886. 197257.i −1.25701 2.17720i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1297.61 −0.0139491
\(306\) 0 0
\(307\) −43757.1 + 25263.2i −0.464271 + 0.268047i −0.713839 0.700310i \(-0.753045\pi\)
0.249567 + 0.968357i \(0.419712\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −87150.9 −0.901055 −0.450527 0.892763i \(-0.648764\pi\)
−0.450527 + 0.892763i \(0.648764\pi\)
\(312\) 0 0
\(313\) 43099.7 74650.9i 0.439932 0.761985i −0.557752 0.830008i \(-0.688336\pi\)
0.997684 + 0.0680232i \(0.0216692\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 150310. + 86781.6i 1.49579 + 0.863593i 0.999988 0.00484511i \(-0.00154225\pi\)
0.495798 + 0.868438i \(0.334876\pi\)
\(318\) 0 0
\(319\) 69127.4 + 39910.7i 0.679311 + 0.392201i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −116629. 142516.i −1.11789 1.36602i
\(324\) 0 0
\(325\) 354564. 204707.i 3.35682 1.93806i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −114039. + 197522.i −1.05357 + 1.82483i
\(330\) 0 0
\(331\) 185379.i 1.69201i −0.533173 0.846006i \(-0.679001\pi\)
0.533173 0.846006i \(-0.320999\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 188922.i 1.68342i
\(336\) 0 0
\(337\) 117923. 68083.2i 1.03834 0.599487i 0.118979 0.992897i \(-0.462038\pi\)
0.919363 + 0.393410i \(0.128705\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 7120.78i 0.0612377i
\(342\) 0 0
\(343\) 141516. 1.20287
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 33006.4 + 57168.7i 0.274119 + 0.474788i 0.969912 0.243454i \(-0.0782805\pi\)
−0.695794 + 0.718242i \(0.744947\pi\)
\(348\) 0 0
\(349\) 88069.6 0.723061 0.361531 0.932360i \(-0.382254\pi\)
0.361531 + 0.932360i \(0.382254\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −46526.1 −0.373377 −0.186688 0.982419i \(-0.559775\pi\)
−0.186688 + 0.982419i \(0.559775\pi\)
\(354\) 0 0
\(355\) 130793. + 75513.4i 1.03783 + 0.599194i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1061.12 1837.91i −0.00823333 0.0142606i 0.861879 0.507113i \(-0.169287\pi\)
−0.870113 + 0.492853i \(0.835954\pi\)
\(360\) 0 0
\(361\) 123521. 41547.2i 0.947820 0.318807i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 20670.3 35802.0i 0.155153 0.268733i
\(366\) 0 0
\(367\) −85852.7 + 148701.i −0.637415 + 1.10403i 0.348583 + 0.937278i \(0.386663\pi\)
−0.985998 + 0.166757i \(0.946671\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −61054.3 35249.7i −0.443576 0.256099i
\(372\) 0 0
\(373\) 197021.i 1.41611i −0.706160 0.708053i \(-0.749574\pi\)
0.706160 0.708053i \(-0.250426\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 155310. + 269005.i 1.09274 + 1.89268i
\(378\) 0 0
\(379\) 77106.2i 0.536798i 0.963308 + 0.268399i \(0.0864946\pi\)
−0.963308 + 0.268399i \(0.913505\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −78369.6 + 45246.7i −0.534257 + 0.308453i −0.742748 0.669571i \(-0.766478\pi\)
0.208491 + 0.978024i \(0.433145\pi\)
\(384\) 0 0
\(385\) −135603. 234871.i −0.914844 1.58456i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 146861. 254371.i 0.970525 1.68100i 0.276552 0.960999i \(-0.410808\pi\)
0.693974 0.720000i \(-0.255858\pi\)
\(390\) 0 0
\(391\) −124903. −0.816993
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 18718.0 + 10806.8i 0.119968 + 0.0692635i
\(396\) 0 0
\(397\) 150469. + 260620.i 0.954700 + 1.65359i 0.735054 + 0.678009i \(0.237157\pi\)
0.219646 + 0.975580i \(0.429510\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −183175. + 105756.i −1.13914 + 0.657682i −0.946217 0.323532i \(-0.895130\pi\)
−0.192921 + 0.981214i \(0.561796\pi\)
\(402\) 0 0
\(403\) −13855.0 + 23997.6i −0.0853095 + 0.147760i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 98917.9i 0.597154i
\(408\) 0 0
\(409\) −104736. 60469.2i −0.626107 0.361483i 0.153136 0.988205i \(-0.451063\pi\)
−0.779243 + 0.626722i \(0.784396\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 429888. 248196.i 2.52032 1.45510i
\(414\) 0 0
\(415\) −24419.6 42296.0i −0.141789 0.245586i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 63415.7 0.361218 0.180609 0.983555i \(-0.442193\pi\)
0.180609 + 0.983555i \(0.442193\pi\)
\(420\) 0 0
\(421\) −138630. + 80038.0i −0.782155 + 0.451577i −0.837194 0.546907i \(-0.815805\pi\)
0.0550384 + 0.998484i \(0.482472\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −724250. −4.00969
\(426\) 0 0
\(427\) −1161.27 + 2011.37i −0.00636907 + 0.0110316i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 281898. + 162754.i 1.51753 + 0.876147i 0.999788 + 0.0206104i \(0.00656095\pi\)
0.517743 + 0.855536i \(0.326772\pi\)
\(432\) 0 0
\(433\) −225615. 130259.i −1.20335 0.694755i −0.242053 0.970263i \(-0.577821\pi\)
−0.961299 + 0.275508i \(0.911154\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 31250.2 82681.3i 0.163640 0.432957i
\(438\) 0 0
\(439\) 156044. 90092.1i 0.809689 0.467474i −0.0371590 0.999309i \(-0.511831\pi\)
0.846848 + 0.531835i \(0.178497\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 121785. 210938.i 0.620564 1.07485i −0.368817 0.929502i \(-0.620237\pi\)
0.989381 0.145346i \(-0.0464297\pi\)
\(444\) 0 0
\(445\) 548384.i 2.76927i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 108921.i 0.540280i −0.962821 0.270140i \(-0.912930\pi\)
0.962821 0.270140i \(-0.0870699\pi\)
\(450\) 0 0
\(451\) 130055. 75087.1i 0.639401 0.369158i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.05538e6i 5.09784i
\(456\) 0 0
\(457\) 243509. 1.16596 0.582979 0.812488i \(-0.301887\pi\)
0.582979 + 0.812488i \(0.301887\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −15578.6 26982.9i −0.0733038 0.126966i 0.827044 0.562138i \(-0.190021\pi\)
−0.900347 + 0.435172i \(0.856688\pi\)
\(462\) 0 0
\(463\) 281050. 1.31106 0.655529 0.755170i \(-0.272446\pi\)
0.655529 + 0.755170i \(0.272446\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 69672.6 0.319469 0.159734 0.987160i \(-0.448936\pi\)
0.159734 + 0.987160i \(0.448936\pi\)
\(468\) 0 0
\(469\) 292840. + 169071.i 1.33133 + 0.768643i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 104274. + 180607.i 0.466071 + 0.807258i
\(474\) 0 0
\(475\) 181205. 479429.i 0.803125 2.12489i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −139152. + 241019.i −0.606485 + 1.05046i 0.385330 + 0.922779i \(0.374088\pi\)
−0.991815 + 0.127684i \(0.959246\pi\)
\(480\) 0 0
\(481\) 192466. 333362.i 0.831888 1.44087i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −473155. 273176.i −2.01150 1.16134i
\(486\) 0 0
\(487\) 189370.i 0.798460i 0.916851 + 0.399230i \(0.130723\pi\)
−0.916851 + 0.399230i \(0.869277\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 201087. + 348293.i 0.834105 + 1.44471i 0.894757 + 0.446554i \(0.147349\pi\)
−0.0606517 + 0.998159i \(0.519318\pi\)
\(492\) 0 0
\(493\) 549484.i 2.26080i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 234100. 135158.i 0.947740 0.547178i
\(498\) 0 0
\(499\) 16839.7 + 29167.2i 0.0676290 + 0.117137i 0.897857 0.440287i \(-0.145123\pi\)
−0.830228 + 0.557424i \(0.811790\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −186494. + 323016.i −0.737102 + 1.27670i 0.216692 + 0.976240i \(0.430473\pi\)
−0.953795 + 0.300459i \(0.902860\pi\)
\(504\) 0 0
\(505\) −179822. −0.705114
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 322491. + 186190.i 1.24475 + 0.718656i 0.970057 0.242877i \(-0.0780912\pi\)
0.274691 + 0.961533i \(0.411425\pi\)
\(510\) 0 0
\(511\) −36996.7 64080.2i −0.141684 0.245404i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −454489. + 262399.i −1.71360 + 0.989345i
\(516\) 0 0
\(517\) 104414. 180850.i 0.390640 0.676608i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 158791.i 0.584994i 0.956266 + 0.292497i \(0.0944861\pi\)
−0.956266 + 0.292497i \(0.905514\pi\)
\(522\) 0 0
\(523\) −87644.1 50601.4i −0.320420 0.184994i 0.331160 0.943575i \(-0.392560\pi\)
−0.651580 + 0.758580i \(0.725893\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 42451.5 24509.4i 0.152852 0.0882494i
\(528\) 0 0
\(529\) 109945. + 190431.i 0.392885 + 0.680497i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 584394. 2.05708
\(534\) 0 0
\(535\) 343678. 198422.i 1.20073 0.693239i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −307495. −1.05842
\(540\) 0 0
\(541\) −16520.2 + 28613.8i −0.0564443 + 0.0977644i −0.892867 0.450321i \(-0.851310\pi\)
0.836423 + 0.548085i \(0.184643\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −337677. 194958.i −1.13686 0.656369i
\(546\) 0 0
\(547\) −346166. 199859.i −1.15694 0.667959i −0.206370 0.978474i \(-0.566165\pi\)
−0.950568 + 0.310515i \(0.899498\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 363740. + 137479.i 1.19808 + 0.452828i
\(552\) 0 0
\(553\) 33502.4 19342.6i 0.109553 0.0632507i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −154975. + 268425.i −0.499520 + 0.865193i −1.00000 0.000554526i \(-0.999823\pi\)
0.500480 + 0.865748i \(0.333157\pi\)
\(558\) 0 0
\(559\) 811548.i 2.59711i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 333661.i 1.05266i 0.850280 + 0.526330i \(0.176432\pi\)
−0.850280 + 0.526330i \(0.823568\pi\)
\(564\) 0 0
\(565\) 421341. 243262.i 1.31989 0.762038i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 199994.i 0.617720i −0.951107 0.308860i \(-0.900053\pi\)
0.951107 0.308860i \(-0.0999475\pi\)
\(570\) 0 0
\(571\) −57688.4 −0.176936 −0.0884680 0.996079i \(-0.528197\pi\)
−0.0884680 + 0.996079i \(0.528197\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −173811. 301050.i −0.525705 0.910548i
\(576\) 0 0
\(577\) 309540. 0.929748 0.464874 0.885377i \(-0.346100\pi\)
0.464874 + 0.885377i \(0.346100\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −87415.0 −0.258961
\(582\) 0 0
\(583\) 55901.0 + 32274.5i 0.164468 + 0.0949559i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 135516. + 234721.i 0.393292 + 0.681201i 0.992882 0.119106i \(-0.0380029\pi\)
−0.599590 + 0.800307i \(0.704670\pi\)
\(588\) 0 0
\(589\) 5603.15 + 34233.6i 0.0161511 + 0.0986785i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −237698. + 411705.i −0.675952 + 1.17078i 0.300237 + 0.953865i \(0.402934\pi\)
−0.976190 + 0.216919i \(0.930399\pi\)
\(594\) 0 0
\(595\) −933478. + 1.61683e6i −2.63676 + 4.56700i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 107867. + 62276.8i 0.300631 + 0.173569i 0.642726 0.766096i \(-0.277803\pi\)
−0.342095 + 0.939665i \(0.611137\pi\)
\(600\) 0 0
\(601\) 227894.i 0.630934i −0.948937 0.315467i \(-0.897839\pi\)
0.948937 0.315467i \(-0.102161\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −206868. 358306.i −0.565175 0.978911i
\(606\) 0 0
\(607\) 16725.2i 0.0453936i 0.999742 + 0.0226968i \(0.00722523\pi\)
−0.999742 + 0.0226968i \(0.992775\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 703766. 406320.i 1.88515 1.08839i
\(612\) 0 0
\(613\) −265181. 459308.i −0.705703 1.22231i −0.966437 0.256903i \(-0.917298\pi\)
0.260734 0.965411i \(-0.416036\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 88813.7 153830.i 0.233297 0.404083i −0.725479 0.688244i \(-0.758382\pi\)
0.958776 + 0.284161i \(0.0917152\pi\)
\(618\) 0 0
\(619\) −152148. −0.397086 −0.198543 0.980092i \(-0.563621\pi\)
−0.198543 + 0.980092i \(0.563621\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 850026. + 490763.i 2.19006 + 1.26443i
\(624\) 0 0
\(625\) −368863. 638890.i −0.944289 1.63556i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −589713. + 340471.i −1.49053 + 0.860556i
\(630\) 0 0
\(631\) −124898. + 216329.i −0.313686 + 0.543320i −0.979157 0.203103i \(-0.934897\pi\)
0.665471 + 0.746423i \(0.268231\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 910273.i 2.25748i
\(636\) 0 0
\(637\) −1.03628e6 598298.i −2.55388 1.47448i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −614771. + 354938.i −1.49623 + 0.863846i −0.999991 0.00434221i \(-0.998618\pi\)
−0.496235 + 0.868188i \(0.665284\pi\)
\(642\) 0 0
\(643\) 233098. + 403737.i 0.563788 + 0.976510i 0.997161 + 0.0752955i \(0.0239900\pi\)
−0.433373 + 0.901215i \(0.642677\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 564612. 1.34878 0.674391 0.738375i \(-0.264406\pi\)
0.674391 + 0.738375i \(0.264406\pi\)
\(648\) 0 0
\(649\) −393603. + 227247.i −0.934478 + 0.539521i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 270597. 0.634595 0.317298 0.948326i \(-0.397225\pi\)
0.317298 + 0.948326i \(0.397225\pi\)
\(654\) 0 0
\(655\) −385583. + 667850.i −0.898743 + 1.55667i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −226186. 130588.i −0.520828 0.300700i 0.216445 0.976295i \(-0.430554\pi\)
−0.737274 + 0.675594i \(0.763887\pi\)
\(660\) 0 0
\(661\) 155129. + 89564.0i 0.355051 + 0.204989i 0.666908 0.745140i \(-0.267617\pi\)
−0.311856 + 0.950129i \(0.600951\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −836734. 1.02246e6i −1.89210 2.31207i
\(666\) 0 0
\(667\) 228405. 131869.i 0.513397 0.296410i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1063.25 1841.60i 0.00236151 0.00409026i
\(672\) 0 0
\(673\) 42358.2i 0.0935207i −0.998906 0.0467603i \(-0.985110\pi\)
0.998906 0.0467603i \(-0.0148897\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 517746.i 1.12964i −0.825215 0.564819i \(-0.808946\pi\)
0.825215 0.564819i \(-0.191054\pi\)
\(678\) 0 0
\(679\) −846877. + 488945.i −1.83688 + 1.06052i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 342764.i 0.734775i −0.930068 0.367387i \(-0.880252\pi\)
0.930068 0.367387i \(-0.119748\pi\)
\(684\) 0 0
\(685\) 136053. 0.289953
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 125594. + 217535.i 0.264564 + 0.458238i
\(690\) 0 0
\(691\) 322926. 0.676311 0.338156 0.941090i \(-0.390197\pi\)
0.338156 + 0.941090i \(0.390197\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.30941e6 −2.71085
\(696\) 0 0
\(697\) −895286. 516893.i −1.84288 1.06398i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 359397. + 622494.i 0.731372 + 1.26677i 0.956297 + 0.292398i \(0.0944532\pi\)
−0.224924 + 0.974376i \(0.572214\pi\)
\(702\) 0 0
\(703\) −77835.9 475555.i −0.157496 0.962255i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −160927. + 278734.i −0.321951 + 0.557636i
\(708\) 0 0
\(709\) 394995. 684151.i 0.785776 1.36100i −0.142759 0.989758i \(-0.545597\pi\)
0.928535 0.371246i \(-0.121069\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 20375.7 + 11763.9i 0.0400805 + 0.0231405i
\(714\) 0 0
\(715\) 966301.i 1.89017i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 188294. + 326135.i 0.364233 + 0.630869i 0.988653 0.150219i \(-0.0479980\pi\)
−0.624420 + 0.781089i \(0.714665\pi\)
\(720\) 0 0
\(721\) 939311.i 1.80692i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.32441e6 764648.i 2.51968 1.45474i
\(726\) 0 0
\(727\) 310749. + 538233.i 0.587950 + 1.01836i 0.994501 + 0.104731i \(0.0333983\pi\)
−0.406550 + 0.913628i \(0.633268\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 717810. 1.24328e6i 1.34331 2.32667i
\(732\) 0 0
\(733\) −409694. −0.762521 −0.381261 0.924468i \(-0.624510\pi\)
−0.381261 + 0.924468i \(0.624510\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −268123. 154801.i −0.493627 0.284996i
\(738\) 0 0
\(739\) 6909.45 + 11967.5i 0.0126519 + 0.0219137i 0.872282 0.489003i \(-0.162639\pi\)
−0.859630 + 0.510917i \(0.829306\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −66341.6 + 38302.4i −0.120173 + 0.0693822i −0.558882 0.829247i \(-0.688769\pi\)
0.438708 + 0.898629i \(0.355436\pi\)
\(744\) 0 0
\(745\) 133329. 230933.i 0.240222 0.416077i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 710294.i 1.26612i
\(750\) 0 0
\(751\) 462729. + 267157.i 0.820440 + 0.473681i 0.850568 0.525865i \(-0.176258\pi\)
−0.0301283 + 0.999546i \(0.509592\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 671464. 387670.i 1.17795 0.680093i
\(756\) 0 0
\(757\) −386193. 668905.i −0.673926 1.16727i −0.976782 0.214237i \(-0.931273\pi\)
0.302856 0.953036i \(-0.402060\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 134628. 0.232469 0.116235 0.993222i \(-0.462918\pi\)
0.116235 + 0.993222i \(0.462918\pi\)
\(762\) 0 0
\(763\) −604392. + 348946.i −1.03817 + 0.599390i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.76863e6 −3.00640
\(768\) 0 0
\(769\) 14900.5 25808.5i 0.0251970 0.0436425i −0.853152 0.521663i \(-0.825312\pi\)
0.878349 + 0.478020i \(0.158645\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −789192. 455640.i −1.32076 0.762541i −0.336910 0.941537i \(-0.609382\pi\)
−0.983850 + 0.178996i \(0.942715\pi\)
\(774\) 0 0
\(775\) 118149. + 68213.3i 0.196710 + 0.113571i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 566163. 463323.i 0.932968 0.763501i
\(780\) 0 0
\(781\) −214341. + 123750.i −0.351401 + 0.202881i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 255305. 442201.i 0.414305 0.717597i
\(786\) 0 0
\(787\) 398175.i 0.642872i −0.946931 0.321436i \(-0.895834\pi\)
0.946931 0.321436i \(-0.104166\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 870805.i 1.39177i
\(792\) 0 0
\(793\) 7166.48 4137.57i 0.0113962 0.00657959i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 85943.7i 0.135300i 0.997709 + 0.0676499i \(0.0215501\pi\)
−0.997709 + 0.0676499i \(0.978450\pi\)
\(798\) 0 0
\(799\) −1.43755e6 −2.25180
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 33874.0 + 58671.5i 0.0525334 + 0.0909905i
\(804\) 0 0
\(805\) −896093. −1.38281
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 593956. 0.907522 0.453761 0.891123i \(-0.350082\pi\)
0.453761 + 0.891123i \(0.350082\pi\)
\(810\) 0 0
\(811\) 801583. + 462794.i 1.21873 + 0.703633i 0.964646 0.263550i \(-0.0848933\pi\)
0.254082 + 0.967183i \(0.418227\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 166571. + 288509.i 0.250775 + 0.434355i
\(816\) 0 0
\(817\) 643417. + 786231.i 0.963937 + 1.17789i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −434476. + 752534.i −0.644584 + 1.11645i 0.339814 + 0.940493i \(0.389636\pi\)
−0.984398 + 0.175959i \(0.943697\pi\)
\(822\) 0 0
\(823\) 510698. 884555.i 0.753988 1.30595i −0.191888 0.981417i \(-0.561461\pi\)
0.945876 0.324529i \(-0.105206\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 799326. + 461491.i 1.16873 + 0.674765i 0.953380 0.301773i \(-0.0975785\pi\)
0.215347 + 0.976538i \(0.430912\pi\)
\(828\) 0 0
\(829\) 390005.i 0.567494i −0.958899 0.283747i \(-0.908422\pi\)
0.958899 0.283747i \(-0.0915776\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.05838e6 + 1.83317e6i 1.52529 + 2.64188i
\(834\) 0 0
\(835\) 610811.i 0.876060i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −361978. + 208988.i −0.514232 + 0.296892i −0.734571 0.678531i \(-0.762617\pi\)
0.220340 + 0.975423i \(0.429283\pi\)
\(840\) 0 0
\(841\) 226493. + 392297.i 0.320230 + 0.554655i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1.23440e6 + 2.13805e6i −1.72879 + 2.99436i
\(846\) 0 0
\(847\) −740526. −1.03222
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −283048. 163418.i −0.390842 0.225653i
\(852\) 0 0
\(853\) 203831. + 353045.i 0.280138 + 0.485213i 0.971419 0.237373i \(-0.0762865\pi\)
−0.691281 + 0.722586i \(0.742953\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 959729. 554100.i 1.30673 0.754443i 0.325184 0.945651i \(-0.394574\pi\)
0.981550 + 0.191208i \(0.0612405\pi\)
\(858\) 0 0
\(859\) 104151. 180394.i 0.141148 0.244476i −0.786781 0.617232i \(-0.788254\pi\)
0.927929 + 0.372756i \(0.121587\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 90149.7i 0.121044i −0.998167 0.0605219i \(-0.980723\pi\)
0.998167 0.0605219i \(-0.0192765\pi\)
\(864\) 0 0
\(865\) 154209. + 89032.6i 0.206100 + 0.118992i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −30674.7 + 17710.0i −0.0406200 + 0.0234520i
\(870\) 0 0
\(871\) −602398. 1.04338e6i −0.794049 1.37533i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2.90864e6 −3.79904
\(876\) 0 0
\(877\) −311539. + 179867.i −0.405055 + 0.233858i −0.688663 0.725082i \(-0.741802\pi\)
0.283608 + 0.958940i \(0.408469\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 671185. 0.864750 0.432375 0.901694i \(-0.357676\pi\)
0.432375 + 0.901694i \(0.357676\pi\)
\(882\) 0 0
\(883\) 347001. 601024.i 0.445051 0.770851i −0.553005 0.833178i \(-0.686519\pi\)
0.998056 + 0.0623271i \(0.0198522\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 520884. + 300732.i 0.662054 + 0.382237i 0.793059 0.609145i \(-0.208487\pi\)
−0.131005 + 0.991382i \(0.541820\pi\)
\(888\) 0 0
\(889\) 1.41098e6 + 814627.i 1.78532 + 1.03075i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 359670. 951609.i 0.451026 1.19332i
\(894\) 0 0
\(895\) −2.15177e6 + 1.24233e6i −2.68628 + 1.55092i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −51753.0 + 89638.8i −0.0640348 + 0.110912i
\(900\) 0 0
\(901\) 444349.i 0.547362i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 319308.i 0.389864i
\(906\) 0 0
\(907\) 1.25753e6 726034.i 1.52863 0.882556i 0.529212 0.848490i \(-0.322488\pi\)
0.999420 0.0340662i \(-0.0108457\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 405570.i 0.488685i 0.969689 + 0.244343i \(0.0785721\pi\)
−0.969689 + 0.244343i \(0.921428\pi\)
\(912\) 0 0
\(913\) 80036.8 0.0960169
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 690137. + 1.19535e6i 0.820723 + 1.42153i
\(918\) 0 0
\(919\) −567985. −0.672521 −0.336260 0.941769i \(-0.609162\pi\)
−0.336260 + 0.941769i \(0.609162\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −963130. −1.13053
\(924\) 0 0
\(925\) −1.64126e6 947581.i −1.91820 1.10747i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 548586. + 950180.i 0.635643 + 1.10097i 0.986378 + 0.164492i \(0.0525985\pi\)
−0.350735 + 0.936475i \(0.614068\pi\)
\(930\) 0 0
\(931\) −1.47830e6 + 241959.i −1.70555 + 0.279154i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 854688. 1.48036e6i 0.977652 1.69334i
\(936\) 0 0
\(937\) 243059. 420990.i 0.276842 0.479505i −0.693756 0.720210i \(-0.744045\pi\)
0.970598 + 0.240705i \(0.0773787\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −594241. 343085.i −0.671094 0.387456i 0.125397 0.992107i \(-0.459980\pi\)
−0.796491 + 0.604650i \(0.793313\pi\)
\(942\) 0 0
\(943\) 496193.i 0.557991i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 144312. + 249956.i 0.160918 + 0.278717i 0.935198 0.354125i \(-0.115221\pi\)
−0.774280 + 0.632843i \(0.781888\pi\)
\(948\) 0 0
\(949\) 263637.i 0.292735i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.39296e6 + 804224.i −1.53374 + 0.885505i −0.534556 + 0.845133i \(0.679521\pi\)
−0.999185 + 0.0403720i \(0.987146\pi\)
\(954\) 0 0
\(955\) 582300. + 1.00857e6i 0.638469 + 1.10586i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 121758. 210890.i 0.132391 0.229308i
\(960\) 0 0
\(961\) 914287. 0.990002
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 672460. + 388245.i 0.722124 + 0.416919i
\(966\) 0 0
\(967\) 84060.1 + 145596.i 0.0898953 + 0.155703i 0.907467 0.420124i \(-0.138013\pi\)
−0.817571 + 0.575827i \(0.804680\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −745165. + 430221.i −0.790340 + 0.456303i −0.840082 0.542459i \(-0.817493\pi\)
0.0497423 + 0.998762i \(0.484160\pi\)
\(972\) 0 0
\(973\) −1.17182e6 + 2.02966e6i −1.23776 + 2.14386i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 612468.i 0.641644i −0.947140 0.320822i \(-0.896041\pi\)
0.947140 0.320822i \(-0.103959\pi\)
\(978\) 0 0
\(979\) −778280. 449340.i −0.812027 0.468824i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −335085. + 193461.i −0.346775 + 0.200211i −0.663264 0.748386i \(-0.730829\pi\)
0.316489 + 0.948596i \(0.397496\pi\)
\(984\) 0 0
\(985\) 501315. + 868303.i 0.516700 + 0.894950i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 689063. 0.704476
\(990\) 0 0
\(991\) 22518.6 13001.1i 0.0229295 0.0132384i −0.488491 0.872569i \(-0.662453\pi\)
0.511421 + 0.859330i \(0.329119\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −504144. −0.509224
\(996\) 0 0
\(997\) −40472.2 + 70100.0i −0.0407162 + 0.0705225i −0.885665 0.464324i \(-0.846297\pi\)
0.844949 + 0.534847i \(0.179631\pi\)
\(998\) 0 0
\(999\) 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 684.5.y.f.145.12 yes 24
3.2 odd 2 inner 684.5.y.f.145.1 24
19.8 odd 6 inner 684.5.y.f.217.12 yes 24
57.8 even 6 inner 684.5.y.f.217.1 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
684.5.y.f.145.1 24 3.2 odd 2 inner
684.5.y.f.145.12 yes 24 1.1 even 1 trivial
684.5.y.f.217.1 yes 24 57.8 even 6 inner
684.5.y.f.217.12 yes 24 19.8 odd 6 inner