Properties

Label 1620.4.i.m.541.1
Level $1620$
Weight $4$
Character 1620.541
Analytic conductor $95.583$
Analytic rank $0$
Dimension $4$
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] = [1620,4,Mod(541,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.541");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1620.i (of order \(3\), degree \(2\), not 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: \(95.5830942093\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{41})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 11x^{2} + 10x + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 540)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 541.1
Root \(-1.35078 + 2.33962i\) of defining polynomial
Character \(\chi\) \(=\) 1620.541
Dual form 1620.4.i.m.1081.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.50000 - 4.33013i) q^{5} +(-8.05234 + 13.9471i) q^{7} +O(q^{10})\) \(q+(-2.50000 - 4.33013i) q^{5} +(-8.05234 + 13.9471i) q^{7} +(-27.7617 + 48.0847i) q^{11} +(-21.7094 - 37.6017i) q^{13} -25.5234 q^{17} -103.361 q^{19} +(-2.23828 - 3.87682i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(-2.23828 + 3.87682i) q^{29} +(22.5523 + 39.0618i) q^{31} +80.5234 q^{35} +69.5703 q^{37} +(-241.570 - 418.412i) q^{41} +(-75.8664 + 131.404i) q^{43} +(33.8086 - 58.5582i) q^{47} +(41.8195 + 72.4336i) q^{49} +278.953 q^{53} +277.617 q^{55} +(-128.953 - 223.353i) q^{59} +(244.670 - 423.780i) q^{61} +(-108.547 + 188.009i) q^{65} +(386.298 + 669.087i) q^{67} +536.859 q^{71} -65.5703 q^{73} +(-447.094 - 774.389i) q^{77} +(-374.641 + 648.897i) q^{79} +(-630.234 + 1091.60i) q^{83} +(63.8086 + 110.520i) q^{85} +1326.28 q^{89} +699.245 q^{91} +(258.402 + 447.566i) q^{95} +(16.4820 - 28.5477i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{5} - 13 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{5} - 13 q^{7} - 15 q^{11} - 10 q^{13} + 90 q^{17} + 86 q^{19} - 105 q^{23} - 50 q^{25} - 105 q^{29} + 71 q^{31} + 130 q^{35} - 298 q^{37} - 390 q^{41} - 169 q^{43} - 345 q^{47} + 417 q^{49} + 1500 q^{53} + 150 q^{55} - 900 q^{59} - q^{61} - 50 q^{65} + 335 q^{67} + 3300 q^{71} + 314 q^{73} - 1020 q^{77} - 346 q^{79} - 600 q^{83} - 225 q^{85} + 3000 q^{89} + 1606 q^{91} - 215 q^{95} + 623 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) −2.50000 4.33013i −0.223607 0.387298i
\(6\) 0 0
\(7\) −8.05234 + 13.9471i −0.434786 + 0.753071i −0.997278 0.0737316i \(-0.976509\pi\)
0.562492 + 0.826802i \(0.309843\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −27.7617 + 48.0847i −0.760952 + 1.31801i 0.181408 + 0.983408i \(0.441934\pi\)
−0.942360 + 0.334600i \(0.891399\pi\)
\(12\) 0 0
\(13\) −21.7094 37.6017i −0.463161 0.802219i 0.535955 0.844246i \(-0.319952\pi\)
−0.999116 + 0.0420276i \(0.986618\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −25.5234 −0.364138 −0.182069 0.983286i \(-0.558279\pi\)
−0.182069 + 0.983286i \(0.558279\pi\)
\(18\) 0 0
\(19\) −103.361 −1.24803 −0.624016 0.781411i \(-0.714500\pi\)
−0.624016 + 0.781411i \(0.714500\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.23828 3.87682i −0.0202919 0.0351467i 0.855701 0.517470i \(-0.173126\pi\)
−0.875993 + 0.482324i \(0.839793\pi\)
\(24\) 0 0
\(25\) −12.5000 + 21.6506i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.23828 + 3.87682i −0.0143324 + 0.0248244i −0.873103 0.487536i \(-0.837896\pi\)
0.858770 + 0.512361i \(0.171229\pi\)
\(30\) 0 0
\(31\) 22.5523 + 39.0618i 0.130662 + 0.226313i 0.923932 0.382557i \(-0.124956\pi\)
−0.793270 + 0.608870i \(0.791623\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 80.5234 0.388884
\(36\) 0 0
\(37\) 69.5703 0.309116 0.154558 0.987984i \(-0.450605\pi\)
0.154558 + 0.987984i \(0.450605\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −241.570 418.412i −0.920169 1.59378i −0.799152 0.601130i \(-0.794718\pi\)
−0.121018 0.992650i \(-0.538616\pi\)
\(42\) 0 0
\(43\) −75.8664 + 131.404i −0.269059 + 0.466023i −0.968619 0.248550i \(-0.920046\pi\)
0.699560 + 0.714573i \(0.253379\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 33.8086 58.5582i 0.104925 0.181736i −0.808782 0.588108i \(-0.799873\pi\)
0.913708 + 0.406372i \(0.133206\pi\)
\(48\) 0 0
\(49\) 41.8195 + 72.4336i 0.121923 + 0.211177i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 278.953 0.722965 0.361483 0.932379i \(-0.382271\pi\)
0.361483 + 0.932379i \(0.382271\pi\)
\(54\) 0 0
\(55\) 277.617 0.680616
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −128.953 223.353i −0.284547 0.492850i 0.687952 0.725756i \(-0.258510\pi\)
−0.972499 + 0.232906i \(0.925176\pi\)
\(60\) 0 0
\(61\) 244.670 423.780i 0.513553 0.889500i −0.486324 0.873779i \(-0.661662\pi\)
0.999876 0.0157208i \(-0.00500428\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −108.547 + 188.009i −0.207132 + 0.358763i
\(66\) 0 0
\(67\) 386.298 + 669.087i 0.704385 + 1.22003i 0.966913 + 0.255106i \(0.0821103\pi\)
−0.262529 + 0.964924i \(0.584556\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 536.859 0.897373 0.448687 0.893689i \(-0.351892\pi\)
0.448687 + 0.893689i \(0.351892\pi\)
\(72\) 0 0
\(73\) −65.5703 −0.105129 −0.0525645 0.998618i \(-0.516740\pi\)
−0.0525645 + 0.998618i \(0.516740\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −447.094 774.389i −0.661702 1.14610i
\(78\) 0 0
\(79\) −374.641 + 648.897i −0.533549 + 0.924134i 0.465683 + 0.884951i \(0.345809\pi\)
−0.999232 + 0.0391822i \(0.987525\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −630.234 + 1091.60i −0.833460 + 1.44360i 0.0618176 + 0.998087i \(0.480310\pi\)
−0.895278 + 0.445508i \(0.853023\pi\)
\(84\) 0 0
\(85\) 63.8086 + 110.520i 0.0814237 + 0.141030i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1326.28 1.57961 0.789806 0.613356i \(-0.210181\pi\)
0.789806 + 0.613356i \(0.210181\pi\)
\(90\) 0 0
\(91\) 699.245 0.805504
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 258.402 + 447.566i 0.279069 + 0.483361i
\(96\) 0 0
\(97\) 16.4820 28.5477i 0.0172526 0.0298823i −0.857270 0.514867i \(-0.827841\pi\)
0.874523 + 0.484984i \(0.161175\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 910.090 1576.32i 0.896607 1.55297i 0.0648041 0.997898i \(-0.479358\pi\)
0.831803 0.555071i \(-0.187309\pi\)
\(102\) 0 0
\(103\) −687.496 1190.78i −0.657680 1.13913i −0.981215 0.192918i \(-0.938205\pi\)
0.323535 0.946216i \(-0.395129\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1914.19 1.72945 0.864727 0.502243i \(-0.167492\pi\)
0.864727 + 0.502243i \(0.167492\pi\)
\(108\) 0 0
\(109\) −563.978 −0.495590 −0.247795 0.968812i \(-0.579706\pi\)
−0.247795 + 0.968812i \(0.579706\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 620.613 + 1074.93i 0.516658 + 0.894878i 0.999813 + 0.0193433i \(0.00615754\pi\)
−0.483155 + 0.875535i \(0.660509\pi\)
\(114\) 0 0
\(115\) −11.1914 + 19.3841i −0.00907483 + 0.0157181i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 205.523 355.977i 0.158322 0.274222i
\(120\) 0 0
\(121\) −875.926 1517.15i −0.658096 1.13986i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −1699.42 −1.18740 −0.593698 0.804688i \(-0.702333\pi\)
−0.593698 + 0.804688i \(0.702333\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −786.949 1363.04i −0.524855 0.909076i −0.999581 0.0289425i \(-0.990786\pi\)
0.474726 0.880134i \(-0.342547\pi\)
\(132\) 0 0
\(133\) 832.298 1441.58i 0.542627 0.939857i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 611.859 1059.77i 0.381567 0.660893i −0.609719 0.792617i \(-0.708718\pi\)
0.991286 + 0.131724i \(0.0420512\pi\)
\(138\) 0 0
\(139\) 1072.64 + 1857.86i 0.654531 + 1.13368i 0.982011 + 0.188823i \(0.0604673\pi\)
−0.327480 + 0.944858i \(0.606199\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2410.76 1.40977
\(144\) 0 0
\(145\) 22.3828 0.0128193
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −417.996 723.990i −0.229823 0.398064i 0.727933 0.685648i \(-0.240481\pi\)
−0.957755 + 0.287584i \(0.907148\pi\)
\(150\) 0 0
\(151\) −418.244 + 724.419i −0.225405 + 0.390413i −0.956441 0.291926i \(-0.905704\pi\)
0.731036 + 0.682339i \(0.239037\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 112.762 195.309i 0.0584338 0.101210i
\(156\) 0 0
\(157\) −705.487 1221.94i −0.358624 0.621155i 0.629107 0.777319i \(-0.283421\pi\)
−0.987731 + 0.156164i \(0.950087\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 72.0937 0.0352906
\(162\) 0 0
\(163\) −1926.56 −0.925765 −0.462883 0.886420i \(-0.653185\pi\)
−0.462883 + 0.886420i \(0.653185\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −312.094 540.562i −0.144614 0.250479i 0.784615 0.619983i \(-0.212861\pi\)
−0.929229 + 0.369505i \(0.879527\pi\)
\(168\) 0 0
\(169\) 155.906 270.038i 0.0709633 0.122912i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1795.52 3109.94i 0.789082 1.36673i −0.137449 0.990509i \(-0.543890\pi\)
0.926530 0.376221i \(-0.122776\pi\)
\(174\) 0 0
\(175\) −201.309 348.677i −0.0869571 0.150614i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −424.078 −0.177079 −0.0885393 0.996073i \(-0.528220\pi\)
−0.0885393 + 0.996073i \(0.528220\pi\)
\(180\) 0 0
\(181\) 3492.80 1.43435 0.717177 0.696891i \(-0.245434\pi\)
0.717177 + 0.696891i \(0.245434\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −173.926 301.248i −0.0691204 0.119720i
\(186\) 0 0
\(187\) 708.574 1227.29i 0.277091 0.479936i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 598.195 1036.10i 0.226617 0.392513i −0.730186 0.683248i \(-0.760567\pi\)
0.956803 + 0.290736i \(0.0939000\pi\)
\(192\) 0 0
\(193\) 791.546 + 1371.00i 0.295216 + 0.511329i 0.975035 0.222051i \(-0.0712750\pi\)
−0.679819 + 0.733380i \(0.737942\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2803.03 −1.01374 −0.506872 0.862021i \(-0.669198\pi\)
−0.506872 + 0.862021i \(0.669198\pi\)
\(198\) 0 0
\(199\) 2568.59 0.914989 0.457494 0.889213i \(-0.348747\pi\)
0.457494 + 0.889213i \(0.348747\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −36.0469 62.4350i −0.0124630 0.0215866i
\(204\) 0 0
\(205\) −1207.85 + 2092.06i −0.411512 + 0.712760i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2869.48 4970.08i 0.949693 1.64492i
\(210\) 0 0
\(211\) −1168.60 2024.07i −0.381278 0.660392i 0.609968 0.792426i \(-0.291182\pi\)
−0.991245 + 0.132034i \(0.957849\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 758.664 0.240653
\(216\) 0 0
\(217\) −726.397 −0.227240
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 554.098 + 959.725i 0.168654 + 0.292118i
\(222\) 0 0
\(223\) 1413.28 2447.88i 0.424397 0.735077i −0.571967 0.820277i \(-0.693819\pi\)
0.996364 + 0.0851996i \(0.0271528\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3194.13 5532.40i 0.933930 1.61761i 0.157398 0.987535i \(-0.449689\pi\)
0.776532 0.630078i \(-0.216977\pi\)
\(228\) 0 0
\(229\) 2380.57 + 4123.27i 0.686955 + 1.18984i 0.972818 + 0.231571i \(0.0743864\pi\)
−0.285863 + 0.958270i \(0.592280\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1580.58 −0.444408 −0.222204 0.975000i \(-0.571325\pi\)
−0.222204 + 0.975000i \(0.571325\pi\)
\(234\) 0 0
\(235\) −338.086 −0.0938480
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 900.234 + 1559.25i 0.243646 + 0.422007i 0.961750 0.273929i \(-0.0883233\pi\)
−0.718104 + 0.695936i \(0.754990\pi\)
\(240\) 0 0
\(241\) −2621.83 + 4541.14i −0.700775 + 1.21378i 0.267420 + 0.963580i \(0.413829\pi\)
−0.968195 + 0.250198i \(0.919504\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 209.098 362.168i 0.0545256 0.0944410i
\(246\) 0 0
\(247\) 2243.90 + 3886.55i 0.578040 + 1.00120i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4934.37 1.24085 0.620427 0.784264i \(-0.286959\pi\)
0.620427 + 0.784264i \(0.286959\pi\)
\(252\) 0 0
\(253\) 248.554 0.0617648
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2758.46 + 4777.80i 0.669526 + 1.15965i 0.978037 + 0.208433i \(0.0668363\pi\)
−0.308510 + 0.951221i \(0.599830\pi\)
\(258\) 0 0
\(259\) −560.204 + 970.302i −0.134399 + 0.232786i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2598.37 4500.52i 0.609212 1.05519i −0.382159 0.924097i \(-0.624819\pi\)
0.991371 0.131089i \(-0.0418473\pi\)
\(264\) 0 0
\(265\) −697.383 1207.90i −0.161660 0.280003i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −115.055 −0.0260781 −0.0130391 0.999915i \(-0.504151\pi\)
−0.0130391 + 0.999915i \(0.504151\pi\)
\(270\) 0 0
\(271\) 2804.16 0.628564 0.314282 0.949330i \(-0.398236\pi\)
0.314282 + 0.949330i \(0.398236\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −694.043 1202.12i −0.152190 0.263602i
\(276\) 0 0
\(277\) 3973.19 6881.77i 0.861827 1.49273i −0.00833675 0.999965i \(-0.502654\pi\)
0.870164 0.492763i \(-0.164013\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2806.10 4860.31i 0.595722 1.03182i −0.397722 0.917506i \(-0.630199\pi\)
0.993444 0.114316i \(-0.0364675\pi\)
\(282\) 0 0
\(283\) −3773.25 6535.45i −0.792566 1.37276i −0.924373 0.381489i \(-0.875411\pi\)
0.131807 0.991275i \(-0.457922\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7780.83 1.60031
\(288\) 0 0
\(289\) −4261.55 −0.867404
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1506.98 2610.17i −0.300474 0.520437i 0.675769 0.737113i \(-0.263812\pi\)
−0.976243 + 0.216677i \(0.930478\pi\)
\(294\) 0 0
\(295\) −644.766 + 1116.77i −0.127253 + 0.220409i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −97.1835 + 168.327i −0.0187969 + 0.0325571i
\(300\) 0 0
\(301\) −1221.80 2116.23i −0.233966 0.405240i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2446.70 −0.459336
\(306\) 0 0
\(307\) −4316.35 −0.802434 −0.401217 0.915983i \(-0.631413\pi\)
−0.401217 + 0.915983i \(0.631413\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 3815.02 + 6607.80i 0.695594 + 1.20480i 0.969980 + 0.243184i \(0.0781920\pi\)
−0.274386 + 0.961620i \(0.588475\pi\)
\(312\) 0 0
\(313\) −3351.41 + 5804.81i −0.605217 + 1.04827i 0.386800 + 0.922164i \(0.373580\pi\)
−0.992017 + 0.126103i \(0.959753\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −892.453 + 1545.77i −0.158123 + 0.273878i −0.934192 0.356771i \(-0.883878\pi\)
0.776069 + 0.630649i \(0.217211\pi\)
\(318\) 0 0
\(319\) −124.277 215.254i −0.0218125 0.0377804i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2638.13 0.454456
\(324\) 0 0
\(325\) 1085.47 0.185265
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 544.477 + 943.061i 0.0912400 + 0.158032i
\(330\) 0 0
\(331\) −3788.49 + 6561.85i −0.629106 + 1.08964i 0.358625 + 0.933482i \(0.383246\pi\)
−0.987731 + 0.156162i \(0.950088\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1931.49 3345.44i 0.315010 0.545614i
\(336\) 0 0
\(337\) 5390.92 + 9337.35i 0.871401 + 1.50931i 0.860548 + 0.509370i \(0.170121\pi\)
0.0108534 + 0.999941i \(0.496545\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2504.37 −0.397710
\(342\) 0 0
\(343\) −6870.89 −1.08161
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2420.41 4192.28i −0.374451 0.648569i 0.615793 0.787908i \(-0.288836\pi\)
−0.990245 + 0.139339i \(0.955502\pi\)
\(348\) 0 0
\(349\) 4254.21 7368.50i 0.652500 1.13016i −0.330014 0.943976i \(-0.607054\pi\)
0.982514 0.186187i \(-0.0596131\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4719.69 8174.74i 0.711626 1.23257i −0.252621 0.967565i \(-0.581293\pi\)
0.964247 0.265006i \(-0.0853740\pi\)
\(354\) 0 0
\(355\) −1342.15 2324.67i −0.200659 0.347551i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10696.6 1.57255 0.786277 0.617874i \(-0.212006\pi\)
0.786277 + 0.617874i \(0.212006\pi\)
\(360\) 0 0
\(361\) 3824.48 0.557586
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 163.926 + 283.928i 0.0235076 + 0.0407163i
\(366\) 0 0
\(367\) 3129.69 5420.79i 0.445146 0.771016i −0.552916 0.833237i \(-0.686485\pi\)
0.998062 + 0.0622208i \(0.0198183\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2246.23 + 3890.58i −0.314335 + 0.544444i
\(372\) 0 0
\(373\) −305.050 528.363i −0.0423456 0.0733447i 0.844076 0.536224i \(-0.180150\pi\)
−0.886421 + 0.462879i \(0.846816\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 194.367 0.0265528
\(378\) 0 0
\(379\) −2025.74 −0.274552 −0.137276 0.990533i \(-0.543835\pi\)
−0.137276 + 0.990533i \(0.543835\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −4562.07 7901.74i −0.608645 1.05420i −0.991464 0.130381i \(-0.958380\pi\)
0.382819 0.923823i \(-0.374953\pi\)
\(384\) 0 0
\(385\) −2235.47 + 3871.95i −0.295922 + 0.512552i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2547.60 4412.57i 0.332052 0.575131i −0.650862 0.759196i \(-0.725592\pi\)
0.982914 + 0.184065i \(0.0589256\pi\)
\(390\) 0 0
\(391\) 57.1287 + 98.9498i 0.00738906 + 0.0127982i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3746.41 0.477221
\(396\) 0 0
\(397\) 6140.93 0.776334 0.388167 0.921589i \(-0.373108\pi\)
0.388167 + 0.921589i \(0.373108\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1090.99 + 1889.65i 0.135864 + 0.235324i 0.925927 0.377702i \(-0.123286\pi\)
−0.790063 + 0.613026i \(0.789952\pi\)
\(402\) 0 0
\(403\) 979.194 1696.01i 0.121035 0.209639i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1931.39 + 3345.27i −0.235222 + 0.407417i
\(408\) 0 0
\(409\) −3546.37 6142.49i −0.428745 0.742607i 0.568017 0.823017i \(-0.307711\pi\)
−0.996762 + 0.0804092i \(0.974377\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4153.50 0.494868
\(414\) 0 0
\(415\) 6302.34 0.745470
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2883.81 4994.90i −0.336237 0.582379i 0.647485 0.762078i \(-0.275821\pi\)
−0.983722 + 0.179699i \(0.942488\pi\)
\(420\) 0 0
\(421\) −2233.95 + 3869.31i −0.258613 + 0.447931i −0.965871 0.259025i \(-0.916599\pi\)
0.707258 + 0.706956i \(0.249932\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 319.043 552.599i 0.0364138 0.0630705i
\(426\) 0 0
\(427\) 3940.33 + 6824.84i 0.446571 + 0.773483i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4560.33 0.509660 0.254830 0.966986i \(-0.417981\pi\)
0.254830 + 0.966986i \(0.417981\pi\)
\(432\) 0 0
\(433\) −1412.82 −0.156803 −0.0784016 0.996922i \(-0.524982\pi\)
−0.0784016 + 0.996922i \(0.524982\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 231.351 + 400.712i 0.0253250 + 0.0438642i
\(438\) 0 0
\(439\) 976.231 1690.88i 0.106134 0.183830i −0.808067 0.589091i \(-0.799486\pi\)
0.914201 + 0.405261i \(0.132819\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2078.95 + 3600.85i −0.222966 + 0.386189i −0.955707 0.294319i \(-0.904907\pi\)
0.732741 + 0.680508i \(0.238241\pi\)
\(444\) 0 0
\(445\) −3315.70 5742.97i −0.353212 0.611781i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −4352.20 −0.457446 −0.228723 0.973492i \(-0.573455\pi\)
−0.228723 + 0.973492i \(0.573455\pi\)
\(450\) 0 0
\(451\) 26825.6 2.80082
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1748.11 3027.82i −0.180116 0.311970i
\(456\) 0 0
\(457\) 1910.16 3308.50i 0.195522 0.338655i −0.751549 0.659677i \(-0.770693\pi\)
0.947072 + 0.321022i \(0.104026\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −5932.85 + 10276.0i −0.599394 + 1.03818i 0.393517 + 0.919317i \(0.371258\pi\)
−0.992911 + 0.118863i \(0.962075\pi\)
\(462\) 0 0
\(463\) 2653.61 + 4596.19i 0.266358 + 0.461346i 0.967919 0.251264i \(-0.0808463\pi\)
−0.701560 + 0.712610i \(0.747513\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2810.91 0.278529 0.139265 0.990255i \(-0.455526\pi\)
0.139265 + 0.990255i \(0.455526\pi\)
\(468\) 0 0
\(469\) −12442.4 −1.22503
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −4212.36 7296.03i −0.409481 0.709242i
\(474\) 0 0
\(475\) 1292.01 2237.83i 0.124803 0.216166i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −9370.19 + 16229.7i −0.893810 + 1.54812i −0.0585401 + 0.998285i \(0.518645\pi\)
−0.835270 + 0.549840i \(0.814689\pi\)
\(480\) 0 0
\(481\) −1510.33 2615.96i −0.143170 0.247979i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −164.820 −0.0154312
\(486\) 0 0
\(487\) −8260.64 −0.768635 −0.384318 0.923201i \(-0.625563\pi\)
−0.384318 + 0.923201i \(0.625563\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −7462.29 12925.1i −0.685883 1.18798i −0.973159 0.230136i \(-0.926083\pi\)
0.287276 0.957848i \(-0.407250\pi\)
\(492\) 0 0
\(493\) 57.1287 98.9498i 0.00521896 0.00903950i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4322.98 + 7487.61i −0.390165 + 0.675786i
\(498\) 0 0
\(499\) 2967.06 + 5139.10i 0.266180 + 0.461037i 0.967872 0.251443i \(-0.0809051\pi\)
−0.701692 + 0.712480i \(0.747572\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −2704.62 −0.239747 −0.119874 0.992789i \(-0.538249\pi\)
−0.119874 + 0.992789i \(0.538249\pi\)
\(504\) 0 0
\(505\) −9100.90 −0.801950
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1383.25 + 2395.85i 0.120454 + 0.208633i 0.919947 0.392043i \(-0.128232\pi\)
−0.799493 + 0.600676i \(0.794898\pi\)
\(510\) 0 0
\(511\) 527.995 914.513i 0.0457086 0.0791696i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −3437.48 + 5953.89i −0.294123 + 0.509436i
\(516\) 0 0
\(517\) 1877.17 + 3251.35i 0.159686 + 0.276585i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1502.48 0.126344 0.0631718 0.998003i \(-0.479878\pi\)
0.0631718 + 0.998003i \(0.479878\pi\)
\(522\) 0 0
\(523\) −23380.0 −1.95475 −0.977377 0.211504i \(-0.932164\pi\)
−0.977377 + 0.211504i \(0.932164\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −575.613 996.991i −0.0475789 0.0824092i
\(528\) 0 0
\(529\) 6073.48 10519.6i 0.499176 0.864599i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −10488.7 + 18166.9i −0.852374 + 1.47635i
\(534\) 0 0
\(535\) −4785.47 8288.67i −0.386717 0.669814i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4643.93 −0.371110
\(540\) 0 0
\(541\) 18343.1 1.45773 0.728865 0.684657i \(-0.240048\pi\)
0.728865 + 0.684657i \(0.240048\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1409.95 + 2442.10i 0.110817 + 0.191941i
\(546\) 0 0
\(547\) −2856.60 + 4947.78i −0.223290 + 0.386749i −0.955805 0.294002i \(-0.905013\pi\)
0.732515 + 0.680751i \(0.238346\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 231.351 400.712i 0.0178873 0.0309817i
\(552\) 0 0
\(553\) −6033.47 10450.3i −0.463959 0.803600i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −20519.3 −1.56092 −0.780460 0.625206i \(-0.785015\pi\)
−0.780460 + 0.625206i \(0.785015\pi\)
\(558\) 0 0
\(559\) 6588.05 0.498470
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12722.9 + 22036.7i 0.952408 + 1.64962i 0.740192 + 0.672396i \(0.234735\pi\)
0.212216 + 0.977223i \(0.431932\pi\)
\(564\) 0 0
\(565\) 3103.07 5374.67i 0.231057 0.400202i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3657.47 6334.92i 0.269471 0.466738i −0.699254 0.714873i \(-0.746484\pi\)
0.968725 + 0.248135i \(0.0798178\pi\)
\(570\) 0 0
\(571\) −3796.52 6575.76i −0.278247 0.481939i 0.692702 0.721224i \(-0.256420\pi\)
−0.970949 + 0.239285i \(0.923087\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 111.914 0.00811677
\(576\) 0 0
\(577\) −19291.2 −1.39186 −0.695929 0.718110i \(-0.745007\pi\)
−0.695929 + 0.718110i \(0.745007\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −10149.7 17579.8i −0.724753 1.25531i
\(582\) 0 0
\(583\) −7744.22 + 13413.4i −0.550142 + 0.952874i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 10151.2 17582.4i 0.713773 1.23629i −0.249657 0.968334i \(-0.580318\pi\)
0.963431 0.267957i \(-0.0863487\pi\)
\(588\) 0 0
\(589\) −2331.03 4037.46i −0.163070 0.282446i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −11878.2 −0.822562 −0.411281 0.911509i \(-0.634918\pi\)
−0.411281 + 0.911509i \(0.634918\pi\)
\(594\) 0 0
\(595\) −2055.23 −0.141607
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −804.093 1392.73i −0.0548487 0.0950007i 0.837297 0.546748i \(-0.184134\pi\)
−0.892146 + 0.451747i \(0.850801\pi\)
\(600\) 0 0
\(601\) −9517.75 + 16485.2i −0.645985 + 1.11888i 0.338088 + 0.941115i \(0.390220\pi\)
−0.984073 + 0.177765i \(0.943113\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −4379.63 + 7585.74i −0.294309 + 0.509759i
\(606\) 0 0
\(607\) 3082.24 + 5338.59i 0.206102 + 0.356980i 0.950483 0.310775i \(-0.100589\pi\)
−0.744381 + 0.667755i \(0.767255\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2935.85 −0.194389
\(612\) 0 0
\(613\) −10601.8 −0.698538 −0.349269 0.937023i \(-0.613570\pi\)
−0.349269 + 0.937023i \(0.613570\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −7086.92 12274.9i −0.462413 0.800923i 0.536668 0.843794i \(-0.319683\pi\)
−0.999081 + 0.0428709i \(0.986350\pi\)
\(618\) 0 0
\(619\) 1111.12 1924.51i 0.0721479 0.124964i −0.827695 0.561179i \(-0.810348\pi\)
0.899842 + 0.436215i \(0.143681\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −10679.7 + 18497.7i −0.686793 + 1.18956i
\(624\) 0 0
\(625\) −312.500 541.266i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1775.67 −0.112561
\(630\) 0 0
\(631\) 20385.1 1.28608 0.643041 0.765832i \(-0.277672\pi\)
0.643041 + 0.765832i \(0.277672\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4248.55 + 7358.71i 0.265510 + 0.459877i
\(636\) 0 0
\(637\) 1815.75 3144.97i 0.112940 0.195618i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 6379.38 11049.4i 0.393090 0.680851i −0.599766 0.800176i \(-0.704740\pi\)
0.992855 + 0.119324i \(0.0380729\pi\)
\(642\) 0 0
\(643\) 6652.21 + 11522.0i 0.407990 + 0.706660i 0.994664 0.103163i \(-0.0328964\pi\)
−0.586674 + 0.809823i \(0.699563\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24988.8 1.51841 0.759206 0.650851i \(-0.225588\pi\)
0.759206 + 0.650851i \(0.225588\pi\)
\(648\) 0 0
\(649\) 14319.8 0.866106
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7394.11 + 12807.0i 0.443115 + 0.767497i 0.997919 0.0644840i \(-0.0205402\pi\)
−0.554804 + 0.831981i \(0.687207\pi\)
\(654\) 0 0
\(655\) −3934.75 + 6815.18i −0.234723 + 0.406551i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1500.94 + 2599.70i −0.0887226 + 0.153672i −0.906971 0.421192i \(-0.861612\pi\)
0.818249 + 0.574864i \(0.194945\pi\)
\(660\) 0 0
\(661\) 9242.46 + 16008.4i 0.543858 + 0.941989i 0.998678 + 0.0514058i \(0.0163702\pi\)
−0.454820 + 0.890583i \(0.650296\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8322.98 −0.485340
\(666\) 0 0
\(667\) 20.0397 0.00116333
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 13584.9 + 23529.7i 0.781578 + 1.35373i
\(672\) 0 0
\(673\) −463.411 + 802.651i −0.0265426 + 0.0459732i −0.878992 0.476837i \(-0.841783\pi\)
0.852449 + 0.522810i \(0.175116\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −16802.3 + 29102.5i −0.953865 + 1.65214i −0.216920 + 0.976189i \(0.569601\pi\)
−0.736945 + 0.675953i \(0.763732\pi\)
\(678\) 0 0
\(679\) 265.438 + 459.752i 0.0150023 + 0.0259848i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 6468.33 0.362377 0.181189 0.983448i \(-0.442006\pi\)
0.181189 + 0.983448i \(0.442006\pi\)
\(684\) 0 0
\(685\) −6118.59 −0.341284
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −6055.90 10489.1i −0.334850 0.579976i
\(690\) 0 0
\(691\) 15760.2 27297.4i 0.867649 1.50281i 0.00325629 0.999995i \(-0.498963\pi\)
0.864393 0.502817i \(-0.167703\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5363.18 9289.31i 0.292715 0.506998i
\(696\) 0 0
\(697\) 6165.70 + 10679.3i 0.335068 + 0.580355i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −32088.9 −1.72893 −0.864465 0.502693i \(-0.832343\pi\)
−0.864465 + 0.502693i \(0.832343\pi\)
\(702\) 0 0
\(703\) −7190.85 −0.385787
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 14656.7 + 25386.2i 0.779664 + 1.35042i
\(708\) 0 0
\(709\) −4338.98 + 7515.34i −0.229836 + 0.398088i −0.957759 0.287571i \(-0.907152\pi\)
0.727923 + 0.685659i \(0.240486\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 100.957 174.863i 0.00530277 0.00918466i
\(714\) 0 0
\(715\) −6026.89 10438.9i −0.315235 0.546003i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −22564.2 −1.17038 −0.585190 0.810896i \(-0.698980\pi\)
−0.585190 + 0.810896i \(0.698980\pi\)
\(720\) 0 0
\(721\) 22143.8 1.14380
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −55.9571 96.9205i −0.00286648 0.00496488i
\(726\) 0 0
\(727\) 423.322 733.215i 0.0215958 0.0374050i −0.855026 0.518586i \(-0.826459\pi\)
0.876621 + 0.481181i \(0.159792\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1936.37 3353.89i 0.0979744 0.169697i
\(732\) 0 0
\(733\) −14590.7 25271.8i −0.735225 1.27345i −0.954625 0.297812i \(-0.903743\pi\)
0.219400 0.975635i \(-0.429590\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −42897.1 −2.14401
\(738\) 0 0
\(739\) 5544.59 0.275996 0.137998 0.990432i \(-0.455933\pi\)
0.137998 + 0.990432i \(0.455933\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 10048.0 + 17403.6i 0.496131 + 0.859324i 0.999990 0.00446196i \(-0.00142029\pi\)
−0.503859 + 0.863786i \(0.668087\pi\)
\(744\) 0 0
\(745\) −2089.98 + 3619.95i −0.102780 + 0.178020i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −15413.7 + 26697.3i −0.751941 + 1.30240i
\(750\) 0 0
\(751\) 19589.0 + 33929.2i 0.951817 + 1.64860i 0.741490 + 0.670964i \(0.234120\pi\)
0.210327 + 0.977631i \(0.432547\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 4182.44 0.201609
\(756\) 0 0
\(757\) 37739.2 1.81196 0.905980 0.423322i \(-0.139136\pi\)
0.905980 + 0.423322i \(0.139136\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4556.30 + 7891.74i 0.217038 + 0.375920i 0.953901 0.300122i \(-0.0970272\pi\)
−0.736863 + 0.676042i \(0.763694\pi\)
\(762\) 0 0
\(763\) 4541.35 7865.84i 0.215475 0.373214i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −5598.98 + 9697.72i −0.263582 + 0.456538i
\(768\) 0 0
\(769\) 15504.9 + 26855.3i 0.727076 + 1.25933i 0.958114 + 0.286388i \(0.0924548\pi\)
−0.231037 + 0.972945i \(0.574212\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −13117.9 −0.610372 −0.305186 0.952293i \(-0.598719\pi\)
−0.305186 + 0.952293i \(0.598719\pi\)
\(774\) 0 0
\(775\) −1127.62 −0.0522648
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 24968.9 + 43247.5i 1.14840 + 1.98909i
\(780\) 0 0
\(781\) −14904.1 + 25814.7i −0.682858 + 1.18274i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3527.43 + 6109.69i −0.160382 + 0.277789i
\(786\) 0 0
\(787\) −4334.37 7507.35i −0.196320 0.340036i 0.751013 0.660288i \(-0.229566\pi\)
−0.947332 + 0.320252i \(0.896232\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −19989.6 −0.898542
\(792\) 0 0
\(793\) −21246.5 −0.951431
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −12589.3 21805.3i −0.559519 0.969115i −0.997537 0.0701484i \(-0.977653\pi\)
0.438018 0.898966i \(-0.355681\pi\)
\(798\) 0 0
\(799\) −862.911 + 1494.61i −0.0382073 + 0.0661769i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1820.34 3152.93i 0.0799982 0.138561i
\(804\) 0 0
\(805\) −180.234 312.175i −0.00789121 0.0136680i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −21442.3 −0.931856 −0.465928 0.884823i \(-0.654279\pi\)
−0.465928 + 0.884823i \(0.654279\pi\)
\(810\) 0 0
\(811\) −23957.5 −1.03731 −0.518656 0.854983i \(-0.673568\pi\)
−0.518656 + 0.854983i \(0.673568\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 4816.40 + 8342.25i 0.207007 + 0.358547i
\(816\) 0 0
\(817\) 7841.62 13582.1i 0.335794 0.581612i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −10332.2 + 17895.8i −0.439214 + 0.760742i −0.997629 0.0688203i \(-0.978076\pi\)
0.558415 + 0.829562i \(0.311410\pi\)
\(822\) 0 0
\(823\) −22979.2 39801.2i −0.973276 1.68576i −0.685512 0.728062i \(-0.740421\pi\)
−0.287764 0.957701i \(-0.592912\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −22841.3 −0.960423 −0.480211 0.877153i \(-0.659440\pi\)
−0.480211 + 0.877153i \(0.659440\pi\)
\(828\) 0 0
\(829\) −28412.5 −1.19036 −0.595179 0.803594i \(-0.702919\pi\)
−0.595179 + 0.803594i \(0.702919\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1067.38 1848.75i −0.0443967 0.0768974i
\(834\) 0 0
\(835\) −1560.47 + 2702.81i −0.0646733 + 0.112018i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 10331.0 17893.9i 0.425110 0.736312i −0.571321 0.820727i \(-0.693569\pi\)
0.996431 + 0.0844151i \(0.0269022\pi\)
\(840\) 0 0
\(841\) 12184.5 + 21104.1i 0.499589 + 0.865314i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1559.06 −0.0634715
\(846\) 0 0
\(847\) 28213.0 1.14452
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −155.718 269.712i −0.00627256 0.0108644i
\(852\) 0 0
\(853\) −11777.1 + 20398.6i −0.472733 + 0.818798i −0.999513 0.0312037i \(-0.990066\pi\)
0.526780 + 0.850002i \(0.323399\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 19445.5 33680.6i 0.775084 1.34248i −0.159664 0.987171i \(-0.551041\pi\)
0.934747 0.355313i \(-0.115626\pi\)
\(858\) 0 0
\(859\) −14971.0 25930.5i −0.594649 1.02996i −0.993596 0.112988i \(-0.963958\pi\)
0.398947 0.916974i \(-0.369376\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −11036.3 −0.435318 −0.217659 0.976025i \(-0.569842\pi\)
−0.217659 + 0.976025i \(0.569842\pi\)
\(864\) 0 0
\(865\) −17955.2 −0.705776
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −20801.3 36029.0i −0.812010 1.40644i
\(870\) 0 0
\(871\) 16772.6 29050.9i 0.652487 1.13014i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1006.54 + 1743.38i −0.0388884 + 0.0673567i
\(876\) 0 0
\(877\) 12433.7 + 21535.8i 0.478742 + 0.829205i 0.999703 0.0243754i \(-0.00775970\pi\)
−0.520961 + 0.853580i \(0.674426\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −10250.2 −0.391983 −0.195991 0.980606i \(-0.562792\pi\)
−0.195991 + 0.980606i \(0.562792\pi\)
\(882\) 0 0
\(883\) −39991.9 −1.52416 −0.762081 0.647481i \(-0.775822\pi\)
−0.762081 + 0.647481i \(0.775822\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 12121.9 + 20995.8i 0.458867 + 0.794780i 0.998901 0.0468624i \(-0.0149222\pi\)
−0.540035 + 0.841643i \(0.681589\pi\)
\(888\) 0 0
\(889\) 13684.3 23702.0i 0.516263 0.894193i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −3494.49 + 6052.63i −0.130950 + 0.226812i
\(894\) 0 0
\(895\) 1060.19 + 1836.31i 0.0395960 + 0.0685822i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −201.914 −0.00749079
\(900\) 0 0
\(901\) −7119.84 −0.263259
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −8732.01 15124.3i −0.320731 0.555523i
\(906\) 0 0
\(907\) −5503.94 + 9533.10i −0.201494 + 0.348998i −0.949010 0.315246i \(-0.897913\pi\)
0.747516 + 0.664244i \(0.231246\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −6056.53 + 10490.2i −0.220265 + 0.381511i −0.954889 0.296965i \(-0.904026\pi\)
0.734623 + 0.678475i \(0.237359\pi\)
\(912\) 0 0
\(913\) −34992.8 60609.3i −1.26845 2.19701i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 25347.1 0.912799
\(918\) 0 0
\(919\) 5343.49 0.191801 0.0959007 0.995391i \(-0.469427\pi\)
0.0959007 + 0.995391i \(0.469427\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −11654.9 20186.8i −0.415628 0.719890i
\(924\) 0 0
\(925\) −869.629 + 1506.24i −0.0309116 + 0.0535404i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −3995.65 + 6920.67i −0.141112 + 0.244413i −0.927916 0.372790i \(-0.878401\pi\)
0.786804 + 0.617203i \(0.211734\pi\)
\(930\) 0 0
\(931\) −4322.51 7486.80i −0.152164 0.263555i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −7085.74 −0.247838
\(936\) 0 0
\(937\) 33059.7 1.15263 0.576315 0.817227i \(-0.304490\pi\)
0.576315 + 0.817227i \(0.304490\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −11880.5 20577.6i −0.411576 0.712871i 0.583486 0.812123i \(-0.301688\pi\)
−0.995062 + 0.0992521i \(0.968355\pi\)
\(942\) 0 0
\(943\) −1081.41 + 1873.05i −0.0373440 + 0.0646818i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −17484.1 + 30283.3i −0.599953 + 1.03915i 0.392874 + 0.919592i \(0.371481\pi\)
−0.992827 + 0.119557i \(0.961852\pi\)
\(948\) 0 0
\(949\) 1423.49 + 2465.56i 0.0486917 + 0.0843365i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −50724.6 −1.72417 −0.862084 0.506766i \(-0.830841\pi\)
−0.862084 + 0.506766i \(0.830841\pi\)
\(954\) 0 0
\(955\) −5981.95 −0.202693
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 9853.80 + 17067.3i 0.331800 + 0.574694i
\(960\) 0 0
\(961\) 13878.3 24037.9i 0.465855 0.806884i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 3957.73 6854.99i 0.132025 0.228673i
\(966\) 0 0
\(967\) −7558.50 13091.7i −0.251360 0.435368i 0.712541 0.701631i \(-0.247544\pi\)
−0.963900 + 0.266263i \(0.914211\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 48459.4 1.60158 0.800792 0.598943i \(-0.204412\pi\)
0.800792 + 0.598943i \(0.204412\pi\)
\(972\) 0 0
\(973\) −34549.0 −1.13832
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 10306.0 + 17850.5i 0.337481 + 0.584534i 0.983958 0.178399i \(-0.0570919\pi\)
−0.646477 + 0.762933i \(0.723759\pi\)
\(978\) 0 0
\(979\) −36819.8 + 63773.8i −1.20201 + 2.08194i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 4545.20 7872.52i 0.147476 0.255437i −0.782818 0.622251i \(-0.786218\pi\)
0.930294 + 0.366814i \(0.119552\pi\)
\(984\) 0 0
\(985\) 7007.58 + 12137.5i 0.226680 + 0.392622i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 679.242 0.0218389
\(990\) 0 0
\(991\) 29334.8 0.940312 0.470156 0.882583i \(-0.344198\pi\)
0.470156 + 0.882583i \(0.344198\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −6421.48 11122.3i −0.204598 0.354374i
\(996\) 0 0
\(997\) 1141.47 1977.09i 0.0362595 0.0628034i −0.847326 0.531073i \(-0.821789\pi\)
0.883586 + 0.468270i \(0.155122\pi\)
\(998\) 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 1620.4.i.m.541.1 4
3.2 odd 2 1620.4.i.p.541.1 4
9.2 odd 6 540.4.a.g.1.2 2
9.4 even 3 inner 1620.4.i.m.1081.1 4
9.5 odd 6 1620.4.i.p.1081.1 4
9.7 even 3 540.4.a.j.1.2 yes 2
36.7 odd 6 2160.4.a.z.1.1 2
36.11 even 6 2160.4.a.u.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
540.4.a.g.1.2 2 9.2 odd 6
540.4.a.j.1.2 yes 2 9.7 even 3
1620.4.i.m.541.1 4 1.1 even 1 trivial
1620.4.i.m.1081.1 4 9.4 even 3 inner
1620.4.i.p.541.1 4 3.2 odd 2
1620.4.i.p.1081.1 4 9.5 odd 6
2160.4.a.u.1.1 2 36.11 even 6
2160.4.a.z.1.1 2 36.7 odd 6