Properties

Label 1764.4.k.m.361.1
Level $1764$
Weight $4$
Character 1764.361
Analytic conductor $104.079$
Analytic rank $0$
Dimension $2$
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] = [1764,4,Mod(361,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.361");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.k (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: \(104.079369250\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1764.361
Dual form 1764.4.k.m.1549.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(4.00000 + 6.92820i) q^{5} +O(q^{10})\) \(q+(4.00000 + 6.92820i) q^{5} +(-20.0000 + 34.6410i) q^{11} +12.0000 q^{13} +(29.0000 - 50.2295i) q^{17} +(13.0000 + 22.5167i) q^{19} +(-32.0000 - 55.4256i) q^{23} +(30.5000 - 52.8275i) q^{25} +62.0000 q^{29} +(126.000 - 218.238i) q^{31} +(-13.0000 - 22.5167i) q^{37} +6.00000 q^{41} +416.000 q^{43} +(198.000 + 342.946i) q^{47} +(-225.000 + 389.711i) q^{53} -320.000 q^{55} +(-137.000 + 237.291i) q^{59} +(-288.000 - 498.831i) q^{61} +(48.0000 + 83.1384i) q^{65} +(238.000 - 412.228i) q^{67} +448.000 q^{71} +(-79.0000 + 136.832i) q^{73} +(468.000 + 810.600i) q^{79} +530.000 q^{83} +464.000 q^{85} +(195.000 + 337.750i) q^{89} +(-104.000 + 180.133i) q^{95} -214.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{5} - 40 q^{11} + 24 q^{13} + 58 q^{17} + 26 q^{19} - 64 q^{23} + 61 q^{25} + 124 q^{29} + 252 q^{31} - 26 q^{37} + 12 q^{41} + 832 q^{43} + 396 q^{47} - 450 q^{53} - 640 q^{55} - 274 q^{59} - 576 q^{61} + 96 q^{65} + 476 q^{67} + 896 q^{71} - 158 q^{73} + 936 q^{79} + 1060 q^{83} + 928 q^{85} + 390 q^{89} - 208 q^{95} - 428 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) 4.00000 + 6.92820i 0.357771 + 0.619677i 0.987588 0.157066i \(-0.0502036\pi\)
−0.629817 + 0.776743i \(0.716870\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −20.0000 + 34.6410i −0.548202 + 0.949514i 0.450195 + 0.892930i \(0.351354\pi\)
−0.998398 + 0.0565844i \(0.981979\pi\)
\(12\) 0 0
\(13\) 12.0000 0.256015 0.128008 0.991773i \(-0.459142\pi\)
0.128008 + 0.991773i \(0.459142\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 29.0000 50.2295i 0.413737 0.716614i −0.581558 0.813505i \(-0.697557\pi\)
0.995295 + 0.0968912i \(0.0308899\pi\)
\(18\) 0 0
\(19\) 13.0000 + 22.5167i 0.156969 + 0.271878i 0.933774 0.357863i \(-0.116495\pi\)
−0.776805 + 0.629741i \(0.783161\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −32.0000 55.4256i −0.290107 0.502480i 0.683728 0.729737i \(-0.260358\pi\)
−0.973835 + 0.227257i \(0.927024\pi\)
\(24\) 0 0
\(25\) 30.5000 52.8275i 0.244000 0.422620i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 62.0000 0.397004 0.198502 0.980101i \(-0.436392\pi\)
0.198502 + 0.980101i \(0.436392\pi\)
\(30\) 0 0
\(31\) 126.000 218.238i 0.730009 1.26441i −0.226870 0.973925i \(-0.572849\pi\)
0.956879 0.290487i \(-0.0938173\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −13.0000 22.5167i −0.0577618 0.100046i 0.835699 0.549188i \(-0.185063\pi\)
−0.893460 + 0.449142i \(0.851730\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.0228547 0.0114273 0.999935i \(-0.496362\pi\)
0.0114273 + 0.999935i \(0.496362\pi\)
\(42\) 0 0
\(43\) 416.000 1.47534 0.737668 0.675164i \(-0.235927\pi\)
0.737668 + 0.675164i \(0.235927\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 198.000 + 342.946i 0.614495 + 1.06434i 0.990473 + 0.137708i \(0.0439736\pi\)
−0.375978 + 0.926629i \(0.622693\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −225.000 + 389.711i −0.583134 + 1.01002i 0.411971 + 0.911197i \(0.364841\pi\)
−0.995105 + 0.0988214i \(0.968493\pi\)
\(54\) 0 0
\(55\) −320.000 −0.784523
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −137.000 + 237.291i −0.302303 + 0.523604i −0.976657 0.214804i \(-0.931089\pi\)
0.674354 + 0.738408i \(0.264422\pi\)
\(60\) 0 0
\(61\) −288.000 498.831i −0.604502 1.04703i −0.992130 0.125212i \(-0.960039\pi\)
0.387628 0.921816i \(-0.373295\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 48.0000 + 83.1384i 0.0915949 + 0.158647i
\(66\) 0 0
\(67\) 238.000 412.228i 0.433975 0.751667i −0.563236 0.826296i \(-0.690444\pi\)
0.997211 + 0.0746290i \(0.0237773\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 448.000 0.748843 0.374421 0.927259i \(-0.377841\pi\)
0.374421 + 0.927259i \(0.377841\pi\)
\(72\) 0 0
\(73\) −79.0000 + 136.832i −0.126661 + 0.219383i −0.922381 0.386281i \(-0.873759\pi\)
0.795720 + 0.605665i \(0.207093\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 468.000 + 810.600i 0.666508 + 1.15443i 0.978874 + 0.204464i \(0.0655450\pi\)
−0.312366 + 0.949962i \(0.601122\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 530.000 0.700904 0.350452 0.936581i \(-0.386028\pi\)
0.350452 + 0.936581i \(0.386028\pi\)
\(84\) 0 0
\(85\) 464.000 0.592093
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 195.000 + 337.750i 0.232247 + 0.402263i 0.958469 0.285197i \(-0.0920590\pi\)
−0.726222 + 0.687460i \(0.758726\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −104.000 + 180.133i −0.112318 + 0.194540i
\(96\) 0 0
\(97\) −214.000 −0.224004 −0.112002 0.993708i \(-0.535726\pi\)
−0.112002 + 0.993708i \(0.535726\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −716.000 + 1240.15i −0.705393 + 1.22178i 0.261157 + 0.965296i \(0.415896\pi\)
−0.966550 + 0.256480i \(0.917437\pi\)
\(102\) 0 0
\(103\) 382.000 + 661.643i 0.365433 + 0.632948i 0.988846 0.148945i \(-0.0475877\pi\)
−0.623413 + 0.781893i \(0.714254\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 162.000 + 280.592i 0.146366 + 0.253513i 0.929882 0.367859i \(-0.119909\pi\)
−0.783516 + 0.621372i \(0.786576\pi\)
\(108\) 0 0
\(109\) 667.000 1155.28i 0.586119 1.01519i −0.408615 0.912707i \(-0.633988\pi\)
0.994735 0.102482i \(-0.0326784\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1798.00 −1.49683 −0.748414 0.663232i \(-0.769184\pi\)
−0.748414 + 0.663232i \(0.769184\pi\)
\(114\) 0 0
\(115\) 256.000 443.405i 0.207584 0.359545i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −134.500 232.961i −0.101052 0.175027i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1488.00 1.06473
\(126\) 0 0
\(127\) −384.000 −0.268303 −0.134152 0.990961i \(-0.542831\pi\)
−0.134152 + 0.990961i \(0.542831\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 907.000 + 1570.97i 0.604923 + 1.04776i 0.992064 + 0.125737i \(0.0401296\pi\)
−0.387140 + 0.922021i \(0.626537\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 833.000 1442.80i 0.519474 0.899756i −0.480269 0.877121i \(-0.659461\pi\)
0.999744 0.0226350i \(-0.00720557\pi\)
\(138\) 0 0
\(139\) −1126.00 −0.687094 −0.343547 0.939135i \(-0.611628\pi\)
−0.343547 + 0.939135i \(0.611628\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −240.000 + 415.692i −0.140348 + 0.243090i
\(144\) 0 0
\(145\) 248.000 + 429.549i 0.142036 + 0.246014i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1347.00 + 2333.07i 0.740608 + 1.28277i 0.952219 + 0.305416i \(0.0987956\pi\)
−0.211611 + 0.977354i \(0.567871\pi\)
\(150\) 0 0
\(151\) 1324.00 2293.24i 0.713547 1.23590i −0.249970 0.968253i \(-0.580421\pi\)
0.963517 0.267646i \(-0.0862458\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2016.00 1.04470
\(156\) 0 0
\(157\) −278.000 + 481.510i −0.141317 + 0.244769i −0.927993 0.372598i \(-0.878467\pi\)
0.786676 + 0.617367i \(0.211800\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 164.000 + 284.056i 0.0788066 + 0.136497i 0.902735 0.430196i \(-0.141556\pi\)
−0.823929 + 0.566693i \(0.808222\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4268.00 −1.97765 −0.988826 0.149077i \(-0.952370\pi\)
−0.988826 + 0.149077i \(0.952370\pi\)
\(168\) 0 0
\(169\) −2053.00 −0.934456
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1738.00 + 3010.30i 0.763802 + 1.32294i 0.940878 + 0.338746i \(0.110003\pi\)
−0.177076 + 0.984197i \(0.556664\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1134.00 1964.15i 0.473515 0.820152i −0.526026 0.850469i \(-0.676318\pi\)
0.999540 + 0.0303171i \(0.00965173\pi\)
\(180\) 0 0
\(181\) 276.000 0.113342 0.0566710 0.998393i \(-0.481951\pi\)
0.0566710 + 0.998393i \(0.481951\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 104.000 180.133i 0.0413310 0.0715874i
\(186\) 0 0
\(187\) 1160.00 + 2009.18i 0.453624 + 0.785699i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1500.00 2598.08i −0.568252 0.984242i −0.996739 0.0806937i \(-0.974286\pi\)
0.428487 0.903548i \(-0.359047\pi\)
\(192\) 0 0
\(193\) −1639.00 + 2838.83i −0.611284 + 1.05877i 0.379740 + 0.925093i \(0.376013\pi\)
−0.991024 + 0.133682i \(0.957320\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2362.00 0.854241 0.427121 0.904195i \(-0.359528\pi\)
0.427121 + 0.904195i \(0.359528\pi\)
\(198\) 0 0
\(199\) −518.000 + 897.202i −0.184523 + 0.319603i −0.943416 0.331613i \(-0.892407\pi\)
0.758893 + 0.651216i \(0.225741\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 24.0000 + 41.5692i 0.00817674 + 0.0141625i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1040.00 −0.344202
\(210\) 0 0
\(211\) 3524.00 1.14977 0.574887 0.818233i \(-0.305046\pi\)
0.574887 + 0.818233i \(0.305046\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1664.00 + 2882.13i 0.527832 + 0.914232i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 348.000 602.754i 0.105923 0.183464i
\(222\) 0 0
\(223\) 1336.00 0.401189 0.200595 0.979674i \(-0.435713\pi\)
0.200595 + 0.979674i \(0.435713\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −645.000 + 1117.17i −0.188591 + 0.326649i −0.944781 0.327703i \(-0.893725\pi\)
0.756190 + 0.654352i \(0.227059\pi\)
\(228\) 0 0
\(229\) 2762.00 + 4783.92i 0.797022 + 1.38048i 0.921547 + 0.388267i \(0.126926\pi\)
−0.124525 + 0.992216i \(0.539741\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3157.00 + 5468.08i 0.887648 + 1.53745i 0.842648 + 0.538464i \(0.180995\pi\)
0.0449994 + 0.998987i \(0.485671\pi\)
\(234\) 0 0
\(235\) −1584.00 + 2743.57i −0.439697 + 0.761577i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3960.00 1.07176 0.535881 0.844294i \(-0.319980\pi\)
0.535881 + 0.844294i \(0.319980\pi\)
\(240\) 0 0
\(241\) −3509.00 + 6077.77i −0.937903 + 1.62450i −0.168529 + 0.985697i \(0.553902\pi\)
−0.769374 + 0.638798i \(0.779432\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 156.000 + 270.200i 0.0401864 + 0.0696049i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −2394.00 −0.602024 −0.301012 0.953620i \(-0.597324\pi\)
−0.301012 + 0.953620i \(0.597324\pi\)
\(252\) 0 0
\(253\) 2560.00 0.636149
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1383.00 + 2395.43i 0.335678 + 0.581411i 0.983615 0.180283i \(-0.0577014\pi\)
−0.647937 + 0.761694i \(0.724368\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3984.00 6900.49i 0.934084 1.61788i 0.157823 0.987467i \(-0.449552\pi\)
0.776260 0.630413i \(-0.217114\pi\)
\(264\) 0 0
\(265\) −3600.00 −0.834514
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1450.00 2511.47i 0.328654 0.569246i −0.653591 0.756848i \(-0.726738\pi\)
0.982245 + 0.187602i \(0.0600715\pi\)
\(270\) 0 0
\(271\) 1320.00 + 2286.31i 0.295883 + 0.512484i 0.975190 0.221370i \(-0.0710528\pi\)
−0.679307 + 0.733854i \(0.737719\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1220.00 + 2113.10i 0.267523 + 0.463363i
\(276\) 0 0
\(277\) 761.000 1318.09i 0.165069 0.285908i −0.771611 0.636095i \(-0.780549\pi\)
0.936680 + 0.350187i \(0.113882\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4534.00 0.962547 0.481274 0.876570i \(-0.340174\pi\)
0.481274 + 0.876570i \(0.340174\pi\)
\(282\) 0 0
\(283\) 2417.00 4186.37i 0.507688 0.879342i −0.492272 0.870441i \(-0.663834\pi\)
0.999960 0.00890034i \(-0.00283310\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 774.500 + 1341.47i 0.157643 + 0.273046i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −4656.00 −0.928350 −0.464175 0.885744i \(-0.653649\pi\)
−0.464175 + 0.885744i \(0.653649\pi\)
\(294\) 0 0
\(295\) −2192.00 −0.432621
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −384.000 665.108i −0.0742719 0.128643i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2304.00 3990.65i 0.432546 0.749192i
\(306\) 0 0
\(307\) 7238.00 1.34558 0.672792 0.739831i \(-0.265095\pi\)
0.672792 + 0.739831i \(0.265095\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −548.000 + 949.164i −0.0999171 + 0.173062i −0.911650 0.410967i \(-0.865191\pi\)
0.811733 + 0.584029i \(0.198524\pi\)
\(312\) 0 0
\(313\) −1909.00 3306.48i −0.344738 0.597104i 0.640568 0.767901i \(-0.278699\pi\)
−0.985306 + 0.170797i \(0.945366\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 999.000 + 1730.32i 0.177001 + 0.306575i 0.940852 0.338818i \(-0.110027\pi\)
−0.763851 + 0.645393i \(0.776694\pi\)
\(318\) 0 0
\(319\) −1240.00 + 2147.74i −0.217638 + 0.376961i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1508.00 0.259775
\(324\) 0 0
\(325\) 366.000 633.931i 0.0624678 0.108197i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 3968.00 + 6872.78i 0.658915 + 1.14127i 0.980897 + 0.194529i \(0.0623178\pi\)
−0.321981 + 0.946746i \(0.604349\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3808.00 0.621055
\(336\) 0 0
\(337\) 2766.00 0.447103 0.223551 0.974692i \(-0.428235\pi\)
0.223551 + 0.974692i \(0.428235\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5040.00 + 8729.54i 0.800385 + 1.38631i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4176.00 7233.04i 0.646050 1.11899i −0.338008 0.941143i \(-0.609753\pi\)
0.984058 0.177848i \(-0.0569137\pi\)
\(348\) 0 0
\(349\) 5924.00 0.908609 0.454304 0.890847i \(-0.349888\pi\)
0.454304 + 0.890847i \(0.349888\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1113.00 + 1927.77i −0.167816 + 0.290666i −0.937652 0.347576i \(-0.887005\pi\)
0.769836 + 0.638242i \(0.220338\pi\)
\(354\) 0 0
\(355\) 1792.00 + 3103.84i 0.267914 + 0.464041i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1940.00 + 3360.18i 0.285207 + 0.493993i 0.972659 0.232237i \(-0.0746043\pi\)
−0.687452 + 0.726229i \(0.741271\pi\)
\(360\) 0 0
\(361\) 3091.50 5354.64i 0.450722 0.780673i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1264.00 −0.181262
\(366\) 0 0
\(367\) −1292.00 + 2237.81i −0.183765 + 0.318291i −0.943160 0.332340i \(-0.892162\pi\)
0.759394 + 0.650630i \(0.225495\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −5267.00 9122.71i −0.731139 1.26637i −0.956397 0.292071i \(-0.905656\pi\)
0.225257 0.974299i \(-0.427678\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 744.000 0.101639
\(378\) 0 0
\(379\) −4472.00 −0.606098 −0.303049 0.952975i \(-0.598005\pi\)
−0.303049 + 0.952975i \(0.598005\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1234.00 2137.35i −0.164633 0.285153i 0.771892 0.635754i \(-0.219311\pi\)
−0.936525 + 0.350601i \(0.885977\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −523.000 + 905.863i −0.0681675 + 0.118070i −0.898095 0.439802i \(-0.855049\pi\)
0.829927 + 0.557872i \(0.188382\pi\)
\(390\) 0 0
\(391\) −3712.00 −0.480112
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3744.00 + 6484.80i −0.476914 + 0.826040i
\(396\) 0 0
\(397\) 1062.00 + 1839.44i 0.134258 + 0.232541i 0.925314 0.379203i \(-0.123802\pi\)
−0.791056 + 0.611744i \(0.790468\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 5799.00 + 10044.2i 0.722165 + 1.25083i 0.960130 + 0.279553i \(0.0901863\pi\)
−0.237965 + 0.971274i \(0.576480\pi\)
\(402\) 0 0
\(403\) 1512.00 2618.86i 0.186894 0.323709i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1040.00 0.126661
\(408\) 0 0
\(409\) −1385.00 + 2398.89i −0.167442 + 0.290018i −0.937520 0.347932i \(-0.886884\pi\)
0.770078 + 0.637950i \(0.220217\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 2120.00 + 3671.95i 0.250763 + 0.434335i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 9438.00 1.10042 0.550211 0.835026i \(-0.314547\pi\)
0.550211 + 0.835026i \(0.314547\pi\)
\(420\) 0 0
\(421\) 5550.00 0.642495 0.321248 0.946995i \(-0.395898\pi\)
0.321248 + 0.946995i \(0.395898\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1769.00 3064.00i −0.201904 0.349708i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1500.00 + 2598.08i −0.167639 + 0.290359i −0.937589 0.347744i \(-0.886948\pi\)
0.769950 + 0.638104i \(0.220281\pi\)
\(432\) 0 0
\(433\) −12926.0 −1.43460 −0.717302 0.696762i \(-0.754623\pi\)
−0.717302 + 0.696762i \(0.754623\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 832.000 1441.07i 0.0910754 0.157747i
\(438\) 0 0
\(439\) −204.000 353.338i −0.0221786 0.0384144i 0.854723 0.519084i \(-0.173727\pi\)
−0.876902 + 0.480670i \(0.840394\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −7226.00 12515.8i −0.774983 1.34231i −0.934804 0.355164i \(-0.884425\pi\)
0.159821 0.987146i \(-0.448908\pi\)
\(444\) 0 0
\(445\) −1560.00 + 2702.00i −0.166182 + 0.287836i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −10258.0 −1.07818 −0.539092 0.842247i \(-0.681233\pi\)
−0.539092 + 0.842247i \(0.681233\pi\)
\(450\) 0 0
\(451\) −120.000 + 207.846i −0.0125290 + 0.0217009i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2749.00 + 4761.41i 0.281385 + 0.487373i 0.971726 0.236111i \(-0.0758730\pi\)
−0.690341 + 0.723484i \(0.742540\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −16316.0 −1.64840 −0.824199 0.566300i \(-0.808375\pi\)
−0.824199 + 0.566300i \(0.808375\pi\)
\(462\) 0 0
\(463\) 8944.00 0.897760 0.448880 0.893592i \(-0.351823\pi\)
0.448880 + 0.893592i \(0.351823\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4711.00 + 8159.69i 0.466807 + 0.808534i 0.999281 0.0379124i \(-0.0120708\pi\)
−0.532474 + 0.846447i \(0.678737\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8320.00 + 14410.7i −0.808782 + 1.40085i
\(474\) 0 0
\(475\) 1586.00 0.153201
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −6910.00 + 11968.5i −0.659136 + 1.14166i 0.321704 + 0.946840i \(0.395744\pi\)
−0.980840 + 0.194816i \(0.937589\pi\)
\(480\) 0 0
\(481\) −156.000 270.200i −0.0147879 0.0256134i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −856.000 1482.64i −0.0801422 0.138810i
\(486\) 0 0
\(487\) 6632.00 11487.0i 0.617094 1.06884i −0.372920 0.927864i \(-0.621643\pi\)
0.990013 0.140974i \(-0.0450234\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 5940.00 0.545964 0.272982 0.962019i \(-0.411990\pi\)
0.272982 + 0.962019i \(0.411990\pi\)
\(492\) 0 0
\(493\) 1798.00 3114.23i 0.164255 0.284498i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 4126.00 + 7146.44i 0.370151 + 0.641120i 0.989588 0.143926i \(-0.0459728\pi\)
−0.619438 + 0.785046i \(0.712639\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 4704.00 0.416980 0.208490 0.978024i \(-0.433145\pi\)
0.208490 + 0.978024i \(0.433145\pi\)
\(504\) 0 0
\(505\) −11456.0 −1.00948
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5394.00 + 9342.68i 0.469715 + 0.813570i 0.999400 0.0346241i \(-0.0110234\pi\)
−0.529686 + 0.848194i \(0.677690\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −3056.00 + 5293.15i −0.261482 + 0.452901i
\(516\) 0 0
\(517\) −15840.0 −1.34747
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7293.00 12631.8i 0.613267 1.06221i −0.377419 0.926043i \(-0.623188\pi\)
0.990686 0.136167i \(-0.0434784\pi\)
\(522\) 0 0
\(523\) 13.0000 + 22.5167i 0.00108690 + 0.00188257i 0.866568 0.499058i \(-0.166321\pi\)
−0.865481 + 0.500941i \(0.832987\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −7308.00 12657.8i −0.604064 1.04627i
\(528\) 0 0
\(529\) 4035.50 6989.69i 0.331676 0.574479i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 72.0000 0.00585116
\(534\) 0 0
\(535\) −1296.00 + 2244.74i −0.104731 + 0.181399i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −5607.00 9711.61i −0.445589 0.771783i 0.552504 0.833510i \(-0.313672\pi\)
−0.998093 + 0.0617272i \(0.980339\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 10672.0 0.838786
\(546\) 0 0
\(547\) −5424.00 −0.423973 −0.211987 0.977273i \(-0.567993\pi\)
−0.211987 + 0.977273i \(0.567993\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 806.000 + 1396.03i 0.0623172 + 0.107936i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −8809.00 + 15257.6i −0.670106 + 1.16066i 0.307767 + 0.951462i \(0.400418\pi\)
−0.977874 + 0.209197i \(0.932915\pi\)
\(558\) 0 0
\(559\) 4992.00 0.377709
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1781.00 3084.78i 0.133322 0.230920i −0.791633 0.610997i \(-0.790769\pi\)
0.924955 + 0.380076i \(0.124102\pi\)
\(564\) 0 0
\(565\) −7192.00 12456.9i −0.535522 0.927551i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1419.00 + 2457.78i 0.104548 + 0.181082i 0.913553 0.406719i \(-0.133327\pi\)
−0.809006 + 0.587801i \(0.799994\pi\)
\(570\) 0 0
\(571\) 180.000 311.769i 0.0131922 0.0228496i −0.859354 0.511381i \(-0.829134\pi\)
0.872546 + 0.488532i \(0.162467\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3904.00 −0.283144
\(576\) 0 0
\(577\) 11009.0 19068.1i 0.794299 1.37577i −0.128984 0.991647i \(-0.541172\pi\)
0.923283 0.384120i \(-0.125495\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −9000.00 15588.5i −0.639351 1.10739i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1454.00 0.102237 0.0511184 0.998693i \(-0.483721\pi\)
0.0511184 + 0.998693i \(0.483721\pi\)
\(588\) 0 0
\(589\) 6552.00 0.458354
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −6909.00 11966.7i −0.478446 0.828693i 0.521248 0.853405i \(-0.325467\pi\)
−0.999695 + 0.0247118i \(0.992133\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 3348.00 5798.91i 0.228373 0.395554i −0.728953 0.684564i \(-0.759993\pi\)
0.957326 + 0.289010i \(0.0933260\pi\)
\(600\) 0 0
\(601\) −10010.0 −0.679395 −0.339698 0.940535i \(-0.610325\pi\)
−0.339698 + 0.940535i \(0.610325\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1076.00 1863.69i 0.0723068 0.125239i
\(606\) 0 0
\(607\) 1440.00 + 2494.15i 0.0962896 + 0.166779i 0.910146 0.414287i \(-0.135969\pi\)
−0.813856 + 0.581066i \(0.802636\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2376.00 + 4115.35i 0.157320 + 0.272487i
\(612\) 0 0
\(613\) −3261.00 + 5648.22i −0.214862 + 0.372152i −0.953230 0.302246i \(-0.902264\pi\)
0.738368 + 0.674398i \(0.235597\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −6614.00 −0.431555 −0.215778 0.976443i \(-0.569229\pi\)
−0.215778 + 0.976443i \(0.569229\pi\)
\(618\) 0 0
\(619\) 2633.00 4560.49i 0.170968 0.296125i −0.767791 0.640701i \(-0.778644\pi\)
0.938759 + 0.344576i \(0.111977\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2139.50 + 3705.72i 0.136928 + 0.237166i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1508.00 −0.0955928
\(630\) 0 0
\(631\) 3344.00 0.210971 0.105485 0.994421i \(-0.466360\pi\)
0.105485 + 0.994421i \(0.466360\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1536.00 2660.43i −0.0959910 0.166261i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −2441.00 + 4227.94i −0.150411 + 0.260520i −0.931379 0.364052i \(-0.881393\pi\)
0.780967 + 0.624572i \(0.214726\pi\)
\(642\) 0 0
\(643\) 15898.0 0.975048 0.487524 0.873110i \(-0.337900\pi\)
0.487524 + 0.873110i \(0.337900\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −3066.00 + 5310.47i −0.186301 + 0.322683i −0.944014 0.329905i \(-0.892983\pi\)
0.757713 + 0.652588i \(0.226317\pi\)
\(648\) 0 0
\(649\) −5480.00 9491.64i −0.331447 0.574082i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −12099.0 20956.1i −0.725070 1.25586i −0.958945 0.283591i \(-0.908474\pi\)
0.233876 0.972266i \(-0.424859\pi\)
\(654\) 0 0
\(655\) −7256.00 + 12567.8i −0.432848 + 0.749715i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −17456.0 −1.03185 −0.515925 0.856634i \(-0.672552\pi\)
−0.515925 + 0.856634i \(0.672552\pi\)
\(660\) 0 0
\(661\) −328.000 + 568.113i −0.0193006 + 0.0334297i −0.875514 0.483192i \(-0.839477\pi\)
0.856214 + 0.516622i \(0.172811\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1984.00 3436.39i −0.115174 0.199487i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 23040.0 1.32556
\(672\) 0 0
\(673\) −18214.0 −1.04324 −0.521618 0.853179i \(-0.674671\pi\)
−0.521618 + 0.853179i \(0.674671\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −15126.0 26199.0i −0.858699 1.48731i −0.873170 0.487415i \(-0.837940\pi\)
0.0144709 0.999895i \(-0.495394\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −5418.00 + 9384.25i −0.303534 + 0.525737i −0.976934 0.213542i \(-0.931500\pi\)
0.673400 + 0.739279i \(0.264833\pi\)
\(684\) 0 0
\(685\) 13328.0 0.743411
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −2700.00 + 4676.54i −0.149291 + 0.258580i
\(690\) 0 0
\(691\) 4789.00 + 8294.79i 0.263650 + 0.456655i 0.967209 0.253982i \(-0.0817403\pi\)
−0.703559 + 0.710637i \(0.748407\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4504.00 7801.16i −0.245822 0.425777i
\(696\) 0 0
\(697\) 174.000 301.377i 0.00945584 0.0163780i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −12442.0 −0.670368 −0.335184 0.942153i \(-0.608798\pi\)
−0.335184 + 0.942153i \(0.608798\pi\)
\(702\) 0 0
\(703\) 338.000 585.433i 0.0181336 0.0314083i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 12587.0 + 21801.3i 0.666734 + 1.15482i 0.978812 + 0.204761i \(0.0656418\pi\)
−0.312078 + 0.950057i \(0.601025\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −16128.0 −0.847123
\(714\) 0 0
\(715\) −3840.00 −0.200850
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −17094.0 29607.7i −0.886646 1.53572i −0.843815 0.536634i \(-0.819696\pi\)
−0.0428311 0.999082i \(-0.513638\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1891.00 3275.31i 0.0968689 0.167782i
\(726\) 0 0
\(727\) 5204.00 0.265482 0.132741 0.991151i \(-0.457622\pi\)
0.132741 + 0.991151i \(0.457622\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 12064.0 20895.5i 0.610401 1.05725i
\(732\) 0 0
\(733\) −16440.0 28474.9i −0.828411 1.43485i −0.899284 0.437365i \(-0.855912\pi\)
0.0708733 0.997485i \(-0.477421\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9520.00 + 16489.1i 0.475812 + 0.824131i
\(738\) 0 0
\(739\) 1956.00 3387.89i 0.0973648 0.168641i −0.813228 0.581945i \(-0.802292\pi\)
0.910593 + 0.413304i \(0.135625\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 16008.0 0.790413 0.395206 0.918592i \(-0.370673\pi\)
0.395206 + 0.918592i \(0.370673\pi\)
\(744\) 0 0
\(745\) −10776.0 + 18664.6i −0.529936 + 0.917876i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −4980.00 8625.61i −0.241974 0.419112i 0.719302 0.694697i \(-0.244462\pi\)
−0.961277 + 0.275585i \(0.911128\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 21184.0 1.02115
\(756\) 0 0
\(757\) 12378.0 0.594301 0.297151 0.954831i \(-0.403964\pi\)
0.297151 + 0.954831i \(0.403964\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −17335.0 30025.1i −0.825747 1.43024i −0.901347 0.433097i \(-0.857421\pi\)
0.0756005 0.997138i \(-0.475913\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1644.00 + 2847.49i −0.0773943 + 0.134051i
\(768\) 0 0
\(769\) 10898.0 0.511043 0.255521 0.966803i \(-0.417753\pi\)
0.255521 + 0.966803i \(0.417753\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 12904.0 22350.4i 0.600420 1.03996i −0.392337 0.919821i \(-0.628333\pi\)
0.992757 0.120136i \(-0.0383332\pi\)
\(774\) 0 0
\(775\) −7686.00 13312.5i −0.356244 0.617033i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 78.0000 + 135.100i 0.00358747 + 0.00621368i
\(780\) 0 0
\(781\) −8960.00 + 15519.2i −0.410517 + 0.711037i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −4448.00 −0.202237
\(786\) 0 0
\(787\) −10527.0 + 18233.3i −0.476807 + 0.825854i −0.999647 0.0265772i \(-0.991539\pi\)
0.522840 + 0.852431i \(0.324873\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −3456.00 5985.97i −0.154762 0.268055i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −24276.0 −1.07892 −0.539461 0.842011i \(-0.681372\pi\)
−0.539461 + 0.842011i \(0.681372\pi\)
\(798\) 0 0
\(799\) 22968.0 1.01696
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −3160.00 5473.28i −0.138872 0.240533i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 10763.0 18642.1i 0.467747 0.810161i −0.531574 0.847012i \(-0.678399\pi\)
0.999321 + 0.0368510i \(0.0117327\pi\)
\(810\) 0 0
\(811\) 12806.0 0.554475 0.277238 0.960801i \(-0.410581\pi\)
0.277238 + 0.960801i \(0.410581\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −1312.00 + 2272.45i −0.0563894 + 0.0976693i
\(816\) 0 0
\(817\) 5408.00 + 9366.93i 0.231581 + 0.401111i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 6607.00 + 11443.7i 0.280860 + 0.486463i 0.971597 0.236643i \(-0.0760471\pi\)
−0.690737 + 0.723106i \(0.742714\pi\)
\(822\) 0 0
\(823\) −16124.0 + 27927.6i −0.682925 + 1.18286i 0.291159 + 0.956675i \(0.405959\pi\)
−0.974084 + 0.226186i \(0.927374\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 14316.0 0.601954 0.300977 0.953631i \(-0.402687\pi\)
0.300977 + 0.953631i \(0.402687\pi\)
\(828\) 0 0
\(829\) 12584.0 21796.1i 0.527214 0.913161i −0.472283 0.881447i \(-0.656570\pi\)
0.999497 0.0317144i \(-0.0100967\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −17072.0 29569.6i −0.707546 1.22551i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 9356.00 0.384988 0.192494 0.981298i \(-0.438342\pi\)
0.192494 + 0.981298i \(0.438342\pi\)
\(840\) 0 0
\(841\) −20545.0 −0.842388
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −8212.00 14223.6i −0.334321 0.579061i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −832.000 + 1441.07i −0.0335142 + 0.0580483i
\(852\) 0 0
\(853\) −2372.00 −0.0952119 −0.0476059 0.998866i \(-0.515159\pi\)
−0.0476059 + 0.998866i \(0.515159\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −5847.00 + 10127.3i −0.233057 + 0.403666i −0.958706 0.284398i \(-0.908206\pi\)
0.725649 + 0.688065i \(0.241539\pi\)
\(858\) 0 0
\(859\) −10253.0 17758.7i −0.407250 0.705378i 0.587330 0.809347i \(-0.300179\pi\)
−0.994580 + 0.103969i \(0.966846\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −14068.0 24366.5i −0.554902 0.961118i −0.997911 0.0646012i \(-0.979422\pi\)
0.443009 0.896517i \(-0.353911\pi\)
\(864\) 0 0
\(865\) −13904.0 + 24082.4i −0.546532 + 0.946621i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −37440.0 −1.46152
\(870\) 0 0
\(871\) 2856.00 4946.74i 0.111104 0.192438i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 18535.0 + 32103.6i 0.713663 + 1.23610i 0.963473 + 0.267806i \(0.0862985\pi\)
−0.249810 + 0.968295i \(0.580368\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −6198.00 −0.237021 −0.118511 0.992953i \(-0.537812\pi\)
−0.118511 + 0.992953i \(0.537812\pi\)
\(882\) 0 0
\(883\) −31876.0 −1.21485 −0.607425 0.794377i \(-0.707798\pi\)
−0.607425 + 0.794377i \(0.707798\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 66.0000 + 114.315i 0.00249838 + 0.00432732i 0.867272 0.497835i \(-0.165871\pi\)
−0.864774 + 0.502162i \(0.832538\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −5148.00 + 8916.60i −0.192913 + 0.334135i
\(894\) 0 0
\(895\) 18144.0 0.677639
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 7812.00 13530.8i 0.289816 0.501976i
\(900\) 0 0
\(901\) 13050.0 + 22603.3i 0.482529 + 0.835765i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1104.00 + 1912.18i 0.0405505 + 0.0702355i
\(906\) 0 0
\(907\) −19122.0 + 33120.3i −0.700039 + 1.21250i 0.268413 + 0.963304i \(0.413501\pi\)
−0.968452 + 0.249200i \(0.919832\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 7008.00 0.254869 0.127434 0.991847i \(-0.459326\pi\)
0.127434 + 0.991847i \(0.459326\pi\)
\(912\) 0 0
\(913\) −10600.0 + 18359.7i −0.384237 + 0.665519i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −18332.0 31752.0i −0.658016 1.13972i −0.981128 0.193357i \(-0.938062\pi\)
0.323112 0.946361i \(-0.395271\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 5376.00 0.191715
\(924\) 0 0
\(925\) −1586.00 −0.0563755
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −22755.0 39412.8i −0.803625 1.39192i −0.917216 0.398391i \(-0.869569\pi\)
0.113591 0.993528i \(-0.463765\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −9280.00 + 16073.4i −0.324587 + 0.562200i
\(936\) 0 0
\(937\) 3838.00 0.133812 0.0669061 0.997759i \(-0.478687\pi\)
0.0669061 + 0.997759i \(0.478687\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 8416.00 14576.9i 0.291556 0.504989i −0.682622 0.730771i \(-0.739161\pi\)
0.974178 + 0.225782i \(0.0724939\pi\)
\(942\) 0 0
\(943\) −192.000 332.554i −0.00663031 0.0114840i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −20464.0 35444.7i −0.702208 1.21626i −0.967690 0.252144i \(-0.918864\pi\)
0.265482 0.964116i \(-0.414469\pi\)
\(948\) 0 0
\(949\) −948.000 + 1641.98i −0.0324272 + 0.0561655i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 24070.0 0.818157 0.409079 0.912499i \(-0.365850\pi\)
0.409079 + 0.912499i \(0.365850\pi\)
\(954\) 0 0
\(955\) 12000.0 20784.6i 0.406608 0.704266i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −16856.5 29196.3i −0.565825 0.980038i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −26224.0 −0.874798
\(966\) 0 0
\(967\) −17152.0 −0.570394 −0.285197 0.958469i \(-0.592059\pi\)
−0.285197 + 0.958469i \(0.592059\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 16455.0 + 28500.9i 0.543837 + 0.941954i 0.998679 + 0.0513817i \(0.0163625\pi\)
−0.454842 + 0.890572i \(0.650304\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −3411.00 + 5908.03i −0.111697 + 0.193464i −0.916454 0.400139i \(-0.868962\pi\)
0.804758 + 0.593603i \(0.202295\pi\)
\(978\) 0 0
\(979\) −15600.0 −0.509273
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 24210.0 41933.0i 0.785533 1.36058i −0.143147 0.989701i \(-0.545722\pi\)
0.928680 0.370882i \(-0.120945\pi\)
\(984\) 0 0
\(985\) 9448.00 + 16364.4i 0.305623 + 0.529354i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −13312.0 23057.1i −0.428005 0.741326i
\(990\) 0 0
\(991\) 24608.0 42622.3i 0.788798 1.36624i −0.137906 0.990445i \(-0.544037\pi\)
0.926704 0.375793i \(-0.122630\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −8288.00 −0.264068
\(996\) 0 0
\(997\) 17632.0 30539.5i 0.560091 0.970107i −0.437397 0.899269i \(-0.644099\pi\)
0.997488 0.0708379i \(-0.0225673\pi\)
\(998\) 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 1764.4.k.m.361.1 2
3.2 odd 2 196.4.e.a.165.1 2
7.2 even 3 inner 1764.4.k.m.1549.1 2
7.3 odd 6 252.4.a.d.1.1 1
7.4 even 3 1764.4.a.c.1.1 1
7.5 odd 6 1764.4.k.d.1549.1 2
7.6 odd 2 1764.4.k.d.361.1 2
21.2 odd 6 196.4.e.a.177.1 2
21.5 even 6 196.4.e.f.177.1 2
21.11 odd 6 196.4.a.d.1.1 1
21.17 even 6 28.4.a.a.1.1 1
21.20 even 2 196.4.e.f.165.1 2
28.3 even 6 1008.4.a.o.1.1 1
84.11 even 6 784.4.a.a.1.1 1
84.59 odd 6 112.4.a.g.1.1 1
105.17 odd 12 700.4.e.a.449.2 2
105.38 odd 12 700.4.e.a.449.1 2
105.59 even 6 700.4.a.n.1.1 1
168.59 odd 6 448.4.a.a.1.1 1
168.101 even 6 448.4.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.4.a.a.1.1 1 21.17 even 6
112.4.a.g.1.1 1 84.59 odd 6
196.4.a.d.1.1 1 21.11 odd 6
196.4.e.a.165.1 2 3.2 odd 2
196.4.e.a.177.1 2 21.2 odd 6
196.4.e.f.165.1 2 21.20 even 2
196.4.e.f.177.1 2 21.5 even 6
252.4.a.d.1.1 1 7.3 odd 6
448.4.a.a.1.1 1 168.59 odd 6
448.4.a.p.1.1 1 168.101 even 6
700.4.a.n.1.1 1 105.59 even 6
700.4.e.a.449.1 2 105.38 odd 12
700.4.e.a.449.2 2 105.17 odd 12
784.4.a.a.1.1 1 84.11 even 6
1008.4.a.o.1.1 1 28.3 even 6
1764.4.a.c.1.1 1 7.4 even 3
1764.4.k.d.361.1 2 7.6 odd 2
1764.4.k.d.1549.1 2 7.5 odd 6
1764.4.k.m.361.1 2 1.1 even 1 trivial
1764.4.k.m.1549.1 2 7.2 even 3 inner