Properties

Label 1089.2.d.g.1088.4
Level $1089$
Weight $2$
Character 1089.1088
Analytic conductor $8.696$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1089.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.69570878012\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 2 x^{14} - 16 x^{12} - 72 x^{10} + 26 x^{8} + 360 x^{6} + 725 x^{4} + 1000 x^{2} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1088.4
Root \(0.556839 - 1.81878i\) of defining polynomial
Character \(\chi\) \(=\) 1089.1088
Dual form 1089.2.d.g.1088.3

$q$-expansion

\(f(q)\) \(=\) \(q-2.35080 q^{2} +3.52626 q^{4} +2.24814i q^{5} -4.05107i q^{7} -3.58792 q^{8} +O(q^{10})\) \(q-2.35080 q^{2} +3.52626 q^{4} +2.24814i q^{5} -4.05107i q^{7} -3.58792 q^{8} -5.28492i q^{10} +4.65266i q^{13} +9.52326i q^{14} +1.38197 q^{16} +0.0762914 q^{17} -2.39581i q^{19} +7.92750i q^{20} +3.22717i q^{23} -0.0541194 q^{25} -10.9375i q^{26} -14.2851i q^{28} +1.83922 q^{29} +1.67476 q^{31} +3.92711 q^{32} -0.179346 q^{34} +9.10737 q^{35} +7.26691 q^{37} +5.63207i q^{38} -8.06613i q^{40} -8.44210 q^{41} +4.28086i q^{43} -7.58644i q^{46} -6.21587i q^{47} -9.41120 q^{49} +0.127224 q^{50} +16.4065i q^{52} -1.22899i q^{53} +14.5349i q^{56} -4.32363 q^{58} +0.580290i q^{59} +3.78221i q^{61} -3.93702 q^{62} -11.9958 q^{64} -10.4598 q^{65} +12.9984 q^{67} +0.269023 q^{68} -21.4096 q^{70} +1.12047i q^{71} +13.3068i q^{73} -17.0831 q^{74} -8.44824i q^{76} +0.659861i q^{79} +3.10685i q^{80} +19.8457 q^{82} +10.2111 q^{83} +0.171513i q^{85} -10.0634i q^{86} -6.58983i q^{89} +18.8483 q^{91} +11.3798i q^{92} +14.6123i q^{94} +5.38611 q^{95} +16.4446 q^{97} +22.1238 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4} + O(q^{10}) \) \( 16 q + 16 q^{4} + 40 q^{16} - 32 q^{25} + 16 q^{31} + 40 q^{34} + 8 q^{37} + 16 q^{49} + 32 q^{58} - 104 q^{64} + 96 q^{67} - 64 q^{70} + 88 q^{82} + 48 q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.35080 −1.66227 −0.831133 0.556074i \(-0.812307\pi\)
−0.831133 + 0.556074i \(0.812307\pi\)
\(3\) 0 0
\(4\) 3.52626 1.76313
\(5\) 2.24814i 1.00540i 0.864462 + 0.502699i \(0.167659\pi\)
−0.864462 + 0.502699i \(0.832341\pi\)
\(6\) 0 0
\(7\) − 4.05107i − 1.53116i −0.643339 0.765581i \(-0.722452\pi\)
0.643339 0.765581i \(-0.277548\pi\)
\(8\) −3.58792 −1.26852
\(9\) 0 0
\(10\) − 5.28492i − 1.67124i
\(11\) 0 0
\(12\) 0 0
\(13\) 4.65266i 1.29042i 0.764007 + 0.645208i \(0.223229\pi\)
−0.764007 + 0.645208i \(0.776771\pi\)
\(14\) 9.52326i 2.54520i
\(15\) 0 0
\(16\) 1.38197 0.345492
\(17\) 0.0762914 0.0185034 0.00925169 0.999957i \(-0.497055\pi\)
0.00925169 + 0.999957i \(0.497055\pi\)
\(18\) 0 0
\(19\) − 2.39581i − 0.549636i −0.961496 0.274818i \(-0.911382\pi\)
0.961496 0.274818i \(-0.0886177\pi\)
\(20\) 7.92750i 1.77264i
\(21\) 0 0
\(22\) 0 0
\(23\) 3.22717i 0.672912i 0.941699 + 0.336456i \(0.109228\pi\)
−0.941699 + 0.336456i \(0.890772\pi\)
\(24\) 0 0
\(25\) −0.0541194 −0.0108239
\(26\) − 10.9375i − 2.14502i
\(27\) 0 0
\(28\) − 14.2851i − 2.69963i
\(29\) 1.83922 0.341534 0.170767 0.985311i \(-0.445375\pi\)
0.170767 + 0.985311i \(0.445375\pi\)
\(30\) 0 0
\(31\) 1.67476 0.300795 0.150398 0.988626i \(-0.451945\pi\)
0.150398 + 0.988626i \(0.451945\pi\)
\(32\) 3.92711 0.694222
\(33\) 0 0
\(34\) −0.179346 −0.0307575
\(35\) 9.10737 1.53943
\(36\) 0 0
\(37\) 7.26691 1.19467 0.597337 0.801991i \(-0.296226\pi\)
0.597337 + 0.801991i \(0.296226\pi\)
\(38\) 5.63207i 0.913642i
\(39\) 0 0
\(40\) − 8.06613i − 1.27537i
\(41\) −8.44210 −1.31843 −0.659217 0.751953i \(-0.729112\pi\)
−0.659217 + 0.751953i \(0.729112\pi\)
\(42\) 0 0
\(43\) 4.28086i 0.652825i 0.945227 + 0.326413i \(0.105840\pi\)
−0.945227 + 0.326413i \(0.894160\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) − 7.58644i − 1.11856i
\(47\) − 6.21587i − 0.906678i −0.891338 0.453339i \(-0.850233\pi\)
0.891338 0.453339i \(-0.149767\pi\)
\(48\) 0 0
\(49\) −9.41120 −1.34446
\(50\) 0.127224 0.0179922
\(51\) 0 0
\(52\) 16.4065i 2.27517i
\(53\) − 1.22899i − 0.168815i −0.996431 0.0844074i \(-0.973100\pi\)
0.996431 0.0844074i \(-0.0268997\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 14.5349i 1.94231i
\(57\) 0 0
\(58\) −4.32363 −0.567721
\(59\) 0.580290i 0.0755474i 0.999286 + 0.0377737i \(0.0120266\pi\)
−0.999286 + 0.0377737i \(0.987973\pi\)
\(60\) 0 0
\(61\) 3.78221i 0.484263i 0.970243 + 0.242131i \(0.0778465\pi\)
−0.970243 + 0.242131i \(0.922153\pi\)
\(62\) −3.93702 −0.500001
\(63\) 0 0
\(64\) −11.9958 −1.49947
\(65\) −10.4598 −1.29738
\(66\) 0 0
\(67\) 12.9984 1.58801 0.794003 0.607914i \(-0.207993\pi\)
0.794003 + 0.607914i \(0.207993\pi\)
\(68\) 0.269023 0.0326238
\(69\) 0 0
\(70\) −21.4096 −2.55894
\(71\) 1.12047i 0.132976i 0.997787 + 0.0664878i \(0.0211793\pi\)
−0.997787 + 0.0664878i \(0.978821\pi\)
\(72\) 0 0
\(73\) 13.3068i 1.55744i 0.627371 + 0.778720i \(0.284131\pi\)
−0.627371 + 0.778720i \(0.715869\pi\)
\(74\) −17.0831 −1.98586
\(75\) 0 0
\(76\) − 8.44824i − 0.969079i
\(77\) 0 0
\(78\) 0 0
\(79\) 0.659861i 0.0742402i 0.999311 + 0.0371201i \(0.0118184\pi\)
−0.999311 + 0.0371201i \(0.988182\pi\)
\(80\) 3.10685i 0.347356i
\(81\) 0 0
\(82\) 19.8457 2.19159
\(83\) 10.2111 1.12082 0.560409 0.828216i \(-0.310644\pi\)
0.560409 + 0.828216i \(0.310644\pi\)
\(84\) 0 0
\(85\) 0.171513i 0.0186032i
\(86\) − 10.0634i − 1.08517i
\(87\) 0 0
\(88\) 0 0
\(89\) − 6.58983i − 0.698520i −0.937026 0.349260i \(-0.886433\pi\)
0.937026 0.349260i \(-0.113567\pi\)
\(90\) 0 0
\(91\) 18.8483 1.97584
\(92\) 11.3798i 1.18643i
\(93\) 0 0
\(94\) 14.6123i 1.50714i
\(95\) 5.38611 0.552603
\(96\) 0 0
\(97\) 16.4446 1.66970 0.834851 0.550477i \(-0.185554\pi\)
0.834851 + 0.550477i \(0.185554\pi\)
\(98\) 22.1238 2.23485
\(99\) 0 0
\(100\) −0.190839 −0.0190839
\(101\) −1.72190 −0.171335 −0.0856675 0.996324i \(-0.527302\pi\)
−0.0856675 + 0.996324i \(0.527302\pi\)
\(102\) 0 0
\(103\) −3.23086 −0.318346 −0.159173 0.987251i \(-0.550883\pi\)
−0.159173 + 0.987251i \(0.550883\pi\)
\(104\) − 16.6934i − 1.63692i
\(105\) 0 0
\(106\) 2.88911i 0.280615i
\(107\) 14.8184 1.43255 0.716276 0.697817i \(-0.245845\pi\)
0.716276 + 0.697817i \(0.245845\pi\)
\(108\) 0 0
\(109\) 14.9258i 1.42963i 0.699313 + 0.714815i \(0.253489\pi\)
−0.699313 + 0.714815i \(0.746511\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 5.59845i − 0.529004i
\(113\) 11.2465i 1.05799i 0.848626 + 0.528993i \(0.177430\pi\)
−0.848626 + 0.528993i \(0.822570\pi\)
\(114\) 0 0
\(115\) −7.25513 −0.676544
\(116\) 6.48556 0.602169
\(117\) 0 0
\(118\) − 1.36415i − 0.125580i
\(119\) − 0.309062i − 0.0283317i
\(120\) 0 0
\(121\) 0 0
\(122\) − 8.89122i − 0.804973i
\(123\) 0 0
\(124\) 5.90562 0.530340
\(125\) 11.1190i 0.994515i
\(126\) 0 0
\(127\) 6.06148i 0.537869i 0.963158 + 0.268935i \(0.0866715\pi\)
−0.963158 + 0.268935i \(0.913328\pi\)
\(128\) 20.3455 1.79830
\(129\) 0 0
\(130\) 24.5889 2.15659
\(131\) 7.15083 0.624771 0.312385 0.949955i \(-0.398872\pi\)
0.312385 + 0.949955i \(0.398872\pi\)
\(132\) 0 0
\(133\) −9.70560 −0.841582
\(134\) −30.5566 −2.63969
\(135\) 0 0
\(136\) −0.273727 −0.0234719
\(137\) 10.6225i 0.907538i 0.891119 + 0.453769i \(0.149921\pi\)
−0.891119 + 0.453769i \(0.850079\pi\)
\(138\) 0 0
\(139\) − 0.0530764i − 0.00450188i −0.999997 0.00225094i \(-0.999284\pi\)
0.999997 0.00225094i \(-0.000716497\pi\)
\(140\) 32.1149 2.71421
\(141\) 0 0
\(142\) − 2.63400i − 0.221041i
\(143\) 0 0
\(144\) 0 0
\(145\) 4.13482i 0.343378i
\(146\) − 31.2816i − 2.58888i
\(147\) 0 0
\(148\) 25.6250 2.10636
\(149\) 20.3949 1.67082 0.835409 0.549628i \(-0.185231\pi\)
0.835409 + 0.549628i \(0.185231\pi\)
\(150\) 0 0
\(151\) − 1.18918i − 0.0967737i −0.998829 0.0483869i \(-0.984592\pi\)
0.998829 0.0483869i \(-0.0154080\pi\)
\(152\) 8.59597i 0.697225i
\(153\) 0 0
\(154\) 0 0
\(155\) 3.76508i 0.302419i
\(156\) 0 0
\(157\) −1.73309 −0.138315 −0.0691577 0.997606i \(-0.522031\pi\)
−0.0691577 + 0.997606i \(0.522031\pi\)
\(158\) − 1.55120i − 0.123407i
\(159\) 0 0
\(160\) 8.82869i 0.697969i
\(161\) 13.0735 1.03034
\(162\) 0 0
\(163\) −1.00260 −0.0785299 −0.0392650 0.999229i \(-0.512502\pi\)
−0.0392650 + 0.999229i \(0.512502\pi\)
\(164\) −29.7690 −2.32457
\(165\) 0 0
\(166\) −24.0044 −1.86310
\(167\) 13.5182 1.04607 0.523033 0.852312i \(-0.324800\pi\)
0.523033 + 0.852312i \(0.324800\pi\)
\(168\) 0 0
\(169\) −8.64727 −0.665175
\(170\) − 0.403194i − 0.0309235i
\(171\) 0 0
\(172\) 15.0954i 1.15101i
\(173\) −7.52294 −0.571959 −0.285979 0.958236i \(-0.592319\pi\)
−0.285979 + 0.958236i \(0.592319\pi\)
\(174\) 0 0
\(175\) 0.219242i 0.0165731i
\(176\) 0 0
\(177\) 0 0
\(178\) 15.4914i 1.16113i
\(179\) 12.5003i 0.934317i 0.884173 + 0.467159i \(0.154722\pi\)
−0.884173 + 0.467159i \(0.845278\pi\)
\(180\) 0 0
\(181\) −18.0010 −1.33800 −0.669002 0.743261i \(-0.733278\pi\)
−0.669002 + 0.743261i \(0.733278\pi\)
\(182\) −44.3085 −3.28437
\(183\) 0 0
\(184\) − 11.5788i − 0.853604i
\(185\) 16.3370i 1.20112i
\(186\) 0 0
\(187\) 0 0
\(188\) − 21.9187i − 1.59859i
\(189\) 0 0
\(190\) −12.6617 −0.918573
\(191\) 21.6570i 1.56704i 0.621364 + 0.783522i \(0.286579\pi\)
−0.621364 + 0.783522i \(0.713421\pi\)
\(192\) 0 0
\(193\) 5.18693i 0.373363i 0.982420 + 0.186682i \(0.0597733\pi\)
−0.982420 + 0.186682i \(0.940227\pi\)
\(194\) −38.6581 −2.77549
\(195\) 0 0
\(196\) −33.1863 −2.37045
\(197\) 5.14679 0.366693 0.183347 0.983048i \(-0.441307\pi\)
0.183347 + 0.983048i \(0.441307\pi\)
\(198\) 0 0
\(199\) 24.2595 1.71971 0.859855 0.510539i \(-0.170554\pi\)
0.859855 + 0.510539i \(0.170554\pi\)
\(200\) 0.194176 0.0137303
\(201\) 0 0
\(202\) 4.04783 0.284804
\(203\) − 7.45081i − 0.522945i
\(204\) 0 0
\(205\) − 18.9790i − 1.32555i
\(206\) 7.59511 0.529176
\(207\) 0 0
\(208\) 6.42982i 0.445828i
\(209\) 0 0
\(210\) 0 0
\(211\) − 18.7436i − 1.29036i −0.764029 0.645182i \(-0.776782\pi\)
0.764029 0.645182i \(-0.223218\pi\)
\(212\) − 4.33373i − 0.297642i
\(213\) 0 0
\(214\) −34.8352 −2.38128
\(215\) −9.62396 −0.656349
\(216\) 0 0
\(217\) − 6.78456i − 0.460566i
\(218\) − 35.0875i − 2.37643i
\(219\) 0 0
\(220\) 0 0
\(221\) 0.354958i 0.0238771i
\(222\) 0 0
\(223\) −5.48860 −0.367544 −0.183772 0.982969i \(-0.558831\pi\)
−0.183772 + 0.982969i \(0.558831\pi\)
\(224\) − 15.9090i − 1.06297i
\(225\) 0 0
\(226\) − 26.4383i − 1.75865i
\(227\) −13.7638 −0.913536 −0.456768 0.889586i \(-0.650993\pi\)
−0.456768 + 0.889586i \(0.650993\pi\)
\(228\) 0 0
\(229\) −21.1326 −1.39648 −0.698242 0.715861i \(-0.746034\pi\)
−0.698242 + 0.715861i \(0.746034\pi\)
\(230\) 17.0554 1.12460
\(231\) 0 0
\(232\) −6.59897 −0.433244
\(233\) 14.9655 0.980420 0.490210 0.871604i \(-0.336920\pi\)
0.490210 + 0.871604i \(0.336920\pi\)
\(234\) 0 0
\(235\) 13.9741 0.911571
\(236\) 2.04625i 0.133200i
\(237\) 0 0
\(238\) 0.726543i 0.0470948i
\(239\) 5.40500 0.349620 0.174810 0.984602i \(-0.444069\pi\)
0.174810 + 0.984602i \(0.444069\pi\)
\(240\) 0 0
\(241\) 11.7091i 0.754252i 0.926162 + 0.377126i \(0.123088\pi\)
−0.926162 + 0.377126i \(0.876912\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 13.3370i 0.853817i
\(245\) − 21.1577i − 1.35171i
\(246\) 0 0
\(247\) 11.1469 0.709260
\(248\) −6.00889 −0.381565
\(249\) 0 0
\(250\) − 26.1386i − 1.65315i
\(251\) − 22.5981i − 1.42638i −0.700972 0.713189i \(-0.747250\pi\)
0.700972 0.713189i \(-0.252750\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) − 14.2493i − 0.894081i
\(255\) 0 0
\(256\) −23.8365 −1.48978
\(257\) − 27.5164i − 1.71642i −0.513295 0.858212i \(-0.671575\pi\)
0.513295 0.858212i \(-0.328425\pi\)
\(258\) 0 0
\(259\) − 29.4388i − 1.82924i
\(260\) −36.8840 −2.28745
\(261\) 0 0
\(262\) −16.8102 −1.03853
\(263\) −32.0386 −1.97559 −0.987793 0.155773i \(-0.950213\pi\)
−0.987793 + 0.155773i \(0.950213\pi\)
\(264\) 0 0
\(265\) 2.76294 0.169726
\(266\) 22.8159 1.39893
\(267\) 0 0
\(268\) 45.8356 2.79986
\(269\) − 27.0412i − 1.64873i −0.566058 0.824365i \(-0.691532\pi\)
0.566058 0.824365i \(-0.308468\pi\)
\(270\) 0 0
\(271\) 14.8811i 0.903963i 0.892027 + 0.451982i \(0.149283\pi\)
−0.892027 + 0.451982i \(0.850717\pi\)
\(272\) 0.105432 0.00639276
\(273\) 0 0
\(274\) − 24.9713i − 1.50857i
\(275\) 0 0
\(276\) 0 0
\(277\) − 7.87985i − 0.473454i −0.971576 0.236727i \(-0.923925\pi\)
0.971576 0.236727i \(-0.0760747\pi\)
\(278\) 0.124772i 0.00748332i
\(279\) 0 0
\(280\) −32.6765 −1.95279
\(281\) −21.2297 −1.26646 −0.633229 0.773965i \(-0.718271\pi\)
−0.633229 + 0.773965i \(0.718271\pi\)
\(282\) 0 0
\(283\) 4.29511i 0.255318i 0.991818 + 0.127659i \(0.0407463\pi\)
−0.991818 + 0.127659i \(0.959254\pi\)
\(284\) 3.95107i 0.234453i
\(285\) 0 0
\(286\) 0 0
\(287\) 34.1996i 2.01874i
\(288\) 0 0
\(289\) −16.9942 −0.999658
\(290\) − 9.72012i − 0.570785i
\(291\) 0 0
\(292\) 46.9231i 2.74597i
\(293\) 19.0955 1.11557 0.557787 0.829984i \(-0.311651\pi\)
0.557787 + 0.829984i \(0.311651\pi\)
\(294\) 0 0
\(295\) −1.30457 −0.0759551
\(296\) −26.0731 −1.51547
\(297\) 0 0
\(298\) −47.9444 −2.77735
\(299\) −15.0150 −0.868337
\(300\) 0 0
\(301\) 17.3421 0.999581
\(302\) 2.79551i 0.160864i
\(303\) 0 0
\(304\) − 3.31093i − 0.189895i
\(305\) −8.50293 −0.486877
\(306\) 0 0
\(307\) 1.86240i 0.106293i 0.998587 + 0.0531463i \(0.0169250\pi\)
−0.998587 + 0.0531463i \(0.983075\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) − 8.85095i − 0.502700i
\(311\) 8.95461i 0.507769i 0.967235 + 0.253885i \(0.0817084\pi\)
−0.967235 + 0.253885i \(0.918292\pi\)
\(312\) 0 0
\(313\) 17.1369 0.968633 0.484316 0.874893i \(-0.339068\pi\)
0.484316 + 0.874893i \(0.339068\pi\)
\(314\) 4.07414 0.229917
\(315\) 0 0
\(316\) 2.32684i 0.130895i
\(317\) 27.3042i 1.53356i 0.641912 + 0.766778i \(0.278141\pi\)
−0.641912 + 0.766778i \(0.721859\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) − 26.9682i − 1.50757i
\(321\) 0 0
\(322\) −30.7332 −1.71270
\(323\) − 0.182780i − 0.0101701i
\(324\) 0 0
\(325\) − 0.251799i − 0.0139673i
\(326\) 2.35692 0.130538
\(327\) 0 0
\(328\) 30.2896 1.67246
\(329\) −25.1809 −1.38827
\(330\) 0 0
\(331\) −5.87217 −0.322764 −0.161382 0.986892i \(-0.551595\pi\)
−0.161382 + 0.986892i \(0.551595\pi\)
\(332\) 36.0071 1.97615
\(333\) 0 0
\(334\) −31.7785 −1.73884
\(335\) 29.2222i 1.59658i
\(336\) 0 0
\(337\) − 16.5708i − 0.902666i −0.892356 0.451333i \(-0.850949\pi\)
0.892356 0.451333i \(-0.149051\pi\)
\(338\) 20.3280 1.10570
\(339\) 0 0
\(340\) 0.604800i 0.0327999i
\(341\) 0 0
\(342\) 0 0
\(343\) 9.76796i 0.527420i
\(344\) − 15.3594i − 0.828122i
\(345\) 0 0
\(346\) 17.6849 0.950748
\(347\) −2.51817 −0.135183 −0.0675913 0.997713i \(-0.521531\pi\)
−0.0675913 + 0.997713i \(0.521531\pi\)
\(348\) 0 0
\(349\) − 8.72945i − 0.467277i −0.972324 0.233638i \(-0.924937\pi\)
0.972324 0.233638i \(-0.0750632\pi\)
\(350\) − 0.515393i − 0.0275489i
\(351\) 0 0
\(352\) 0 0
\(353\) 0.536500i 0.0285550i 0.999898 + 0.0142775i \(0.00454483\pi\)
−0.999898 + 0.0142775i \(0.995455\pi\)
\(354\) 0 0
\(355\) −2.51897 −0.133693
\(356\) − 23.2374i − 1.23158i
\(357\) 0 0
\(358\) − 29.3857i − 1.55308i
\(359\) −15.8058 −0.834197 −0.417099 0.908861i \(-0.636953\pi\)
−0.417099 + 0.908861i \(0.636953\pi\)
\(360\) 0 0
\(361\) 13.2601 0.697900
\(362\) 42.3167 2.22412
\(363\) 0 0
\(364\) 66.4639 3.48365
\(365\) −29.9155 −1.56585
\(366\) 0 0
\(367\) 28.9332 1.51030 0.755152 0.655550i \(-0.227563\pi\)
0.755152 + 0.655550i \(0.227563\pi\)
\(368\) 4.45985i 0.232486i
\(369\) 0 0
\(370\) − 38.4050i − 1.99658i
\(371\) −4.97873 −0.258483
\(372\) 0 0
\(373\) − 6.60935i − 0.342219i −0.985252 0.171109i \(-0.945265\pi\)
0.985252 0.171109i \(-0.0547352\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 22.3020i 1.15014i
\(377\) 8.55727i 0.440722i
\(378\) 0 0
\(379\) 1.54854 0.0795429 0.0397715 0.999209i \(-0.487337\pi\)
0.0397715 + 0.999209i \(0.487337\pi\)
\(380\) 18.9928 0.974310
\(381\) 0 0
\(382\) − 50.9112i − 2.60484i
\(383\) 13.5078i 0.690217i 0.938563 + 0.345109i \(0.112158\pi\)
−0.938563 + 0.345109i \(0.887842\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 12.1934i − 0.620629i
\(387\) 0 0
\(388\) 57.9880 2.94390
\(389\) 14.9795i 0.759489i 0.925091 + 0.379745i \(0.123988\pi\)
−0.925091 + 0.379745i \(0.876012\pi\)
\(390\) 0 0
\(391\) 0.246206i 0.0124512i
\(392\) 33.7666 1.70547
\(393\) 0 0
\(394\) −12.0991 −0.609542
\(395\) −1.48346 −0.0746409
\(396\) 0 0
\(397\) −20.7132 −1.03956 −0.519782 0.854299i \(-0.673987\pi\)
−0.519782 + 0.854299i \(0.673987\pi\)
\(398\) −57.0292 −2.85861
\(399\) 0 0
\(400\) −0.0747911 −0.00373956
\(401\) − 27.0448i − 1.35055i −0.737565 0.675276i \(-0.764025\pi\)
0.737565 0.675276i \(-0.235975\pi\)
\(402\) 0 0
\(403\) 7.79208i 0.388151i
\(404\) −6.07185 −0.302086
\(405\) 0 0
\(406\) 17.5154i 0.869273i
\(407\) 0 0
\(408\) 0 0
\(409\) − 6.10550i − 0.301898i −0.988542 0.150949i \(-0.951767\pi\)
0.988542 0.150949i \(-0.0482329\pi\)
\(410\) 44.6158i 2.20342i
\(411\) 0 0
\(412\) −11.3928 −0.561285
\(413\) 2.35080 0.115675
\(414\) 0 0
\(415\) 22.9561i 1.12687i
\(416\) 18.2715i 0.895836i
\(417\) 0 0
\(418\) 0 0
\(419\) − 35.6619i − 1.74220i −0.491110 0.871098i \(-0.663409\pi\)
0.491110 0.871098i \(-0.336591\pi\)
\(420\) 0 0
\(421\) 14.3076 0.697312 0.348656 0.937251i \(-0.386638\pi\)
0.348656 + 0.937251i \(0.386638\pi\)
\(422\) 44.0625i 2.14493i
\(423\) 0 0
\(424\) 4.40952i 0.214145i
\(425\) −0.00412884 −0.000200278 0
\(426\) 0 0
\(427\) 15.3220 0.741485
\(428\) 52.2536 2.52577
\(429\) 0 0
\(430\) 22.6240 1.09103
\(431\) 8.70321 0.419219 0.209609 0.977785i \(-0.432781\pi\)
0.209609 + 0.977785i \(0.432781\pi\)
\(432\) 0 0
\(433\) −37.5533 −1.80469 −0.902347 0.431009i \(-0.858158\pi\)
−0.902347 + 0.431009i \(0.858158\pi\)
\(434\) 15.9491i 0.765583i
\(435\) 0 0
\(436\) 52.6321i 2.52062i
\(437\) 7.73170 0.369857
\(438\) 0 0
\(439\) − 32.4669i − 1.54956i −0.632232 0.774779i \(-0.717861\pi\)
0.632232 0.774779i \(-0.282139\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 0.834435i − 0.0396900i
\(443\) 18.8215i 0.894235i 0.894475 + 0.447117i \(0.147549\pi\)
−0.894475 + 0.447117i \(0.852451\pi\)
\(444\) 0 0
\(445\) 14.8148 0.702290
\(446\) 12.9026 0.610955
\(447\) 0 0
\(448\) 48.5958i 2.29594i
\(449\) − 7.22187i − 0.340821i −0.985373 0.170410i \(-0.945491\pi\)
0.985373 0.170410i \(-0.0545094\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 39.6582i 1.86536i
\(453\) 0 0
\(454\) 32.3559 1.51854
\(455\) 42.3735i 1.98650i
\(456\) 0 0
\(457\) − 4.22219i − 0.197506i −0.995112 0.0987529i \(-0.968515\pi\)
0.995112 0.0987529i \(-0.0314853\pi\)
\(458\) 49.6786 2.32133
\(459\) 0 0
\(460\) −25.5834 −1.19283
\(461\) 14.1922 0.660997 0.330499 0.943806i \(-0.392783\pi\)
0.330499 + 0.943806i \(0.392783\pi\)
\(462\) 0 0
\(463\) −40.3784 −1.87654 −0.938271 0.345902i \(-0.887573\pi\)
−0.938271 + 0.345902i \(0.887573\pi\)
\(464\) 2.54174 0.117997
\(465\) 0 0
\(466\) −35.1808 −1.62972
\(467\) 11.3007i 0.522932i 0.965213 + 0.261466i \(0.0842060\pi\)
−0.965213 + 0.261466i \(0.915794\pi\)
\(468\) 0 0
\(469\) − 52.6574i − 2.43149i
\(470\) −32.8504 −1.51527
\(471\) 0 0
\(472\) − 2.08203i − 0.0958334i
\(473\) 0 0
\(474\) 0 0
\(475\) 0.129660i 0.00594920i
\(476\) − 1.08983i − 0.0499523i
\(477\) 0 0
\(478\) −12.7061 −0.581162
\(479\) 22.8983 1.04625 0.523126 0.852256i \(-0.324766\pi\)
0.523126 + 0.852256i \(0.324766\pi\)
\(480\) 0 0
\(481\) 33.8105i 1.54163i
\(482\) − 27.5258i − 1.25377i
\(483\) 0 0
\(484\) 0 0
\(485\) 36.9698i 1.67871i
\(486\) 0 0
\(487\) −7.54210 −0.341765 −0.170883 0.985291i \(-0.554662\pi\)
−0.170883 + 0.985291i \(0.554662\pi\)
\(488\) − 13.5703i − 0.614298i
\(489\) 0 0
\(490\) 49.7374i 2.24691i
\(491\) −38.1902 −1.72350 −0.861750 0.507334i \(-0.830631\pi\)
−0.861750 + 0.507334i \(0.830631\pi\)
\(492\) 0 0
\(493\) 0.140317 0.00631954
\(494\) −26.2041 −1.17898
\(495\) 0 0
\(496\) 2.31446 0.103922
\(497\) 4.53911 0.203607
\(498\) 0 0
\(499\) −8.05909 −0.360774 −0.180387 0.983596i \(-0.557735\pi\)
−0.180387 + 0.983596i \(0.557735\pi\)
\(500\) 39.2085i 1.75346i
\(501\) 0 0
\(502\) 53.1235i 2.37102i
\(503\) 26.0620 1.16205 0.581023 0.813887i \(-0.302653\pi\)
0.581023 + 0.813887i \(0.302653\pi\)
\(504\) 0 0
\(505\) − 3.87106i − 0.172260i
\(506\) 0 0
\(507\) 0 0
\(508\) 21.3743i 0.948332i
\(509\) − 27.9633i − 1.23945i −0.784819 0.619726i \(-0.787244\pi\)
0.784819 0.619726i \(-0.212756\pi\)
\(510\) 0 0
\(511\) 53.9068 2.38469
\(512\) 15.3439 0.678111
\(513\) 0 0
\(514\) 64.6855i 2.85315i
\(515\) − 7.26342i − 0.320065i
\(516\) 0 0
\(517\) 0 0
\(518\) 69.2047i 3.04068i
\(519\) 0 0
\(520\) 37.5290 1.64576
\(521\) − 32.2913i − 1.41471i −0.706859 0.707355i \(-0.749888\pi\)
0.706859 0.707355i \(-0.250112\pi\)
\(522\) 0 0
\(523\) − 17.3600i − 0.759101i −0.925171 0.379550i \(-0.876079\pi\)
0.925171 0.379550i \(-0.123921\pi\)
\(524\) 25.2156 1.10155
\(525\) 0 0
\(526\) 75.3163 3.28395
\(527\) 0.127769 0.00556572
\(528\) 0 0
\(529\) 12.5853 0.547189
\(530\) −6.49511 −0.282130
\(531\) 0 0
\(532\) −34.2244 −1.48382
\(533\) − 39.2782i − 1.70133i
\(534\) 0 0
\(535\) 33.3139i 1.44029i
\(536\) −46.6372 −2.01442
\(537\) 0 0
\(538\) 63.5684i 2.74063i
\(539\) 0 0
\(540\) 0 0
\(541\) − 10.0763i − 0.433214i −0.976259 0.216607i \(-0.930501\pi\)
0.976259 0.216607i \(-0.0694990\pi\)
\(542\) − 34.9825i − 1.50263i
\(543\) 0 0
\(544\) 0.299605 0.0128455
\(545\) −33.5552 −1.43735
\(546\) 0 0
\(547\) − 23.8597i − 1.02017i −0.860125 0.510084i \(-0.829614\pi\)
0.860125 0.510084i \(-0.170386\pi\)
\(548\) 37.4575i 1.60010i
\(549\) 0 0
\(550\) 0 0
\(551\) − 4.40642i − 0.187720i
\(552\) 0 0
\(553\) 2.67315 0.113674
\(554\) 18.5239i 0.787007i
\(555\) 0 0
\(556\) − 0.187161i − 0.00793739i
\(557\) 39.1556 1.65908 0.829538 0.558450i \(-0.188604\pi\)
0.829538 + 0.558450i \(0.188604\pi\)
\(558\) 0 0
\(559\) −19.9174 −0.842416
\(560\) 12.5861 0.531859
\(561\) 0 0
\(562\) 49.9068 2.10519
\(563\) 5.02067 0.211596 0.105798 0.994388i \(-0.466260\pi\)
0.105798 + 0.994388i \(0.466260\pi\)
\(564\) 0 0
\(565\) −25.2838 −1.06370
\(566\) − 10.0969i − 0.424406i
\(567\) 0 0
\(568\) − 4.02016i − 0.168682i
\(569\) −9.50917 −0.398645 −0.199323 0.979934i \(-0.563874\pi\)
−0.199323 + 0.979934i \(0.563874\pi\)
\(570\) 0 0
\(571\) 39.8291i 1.66680i 0.552673 + 0.833398i \(0.313608\pi\)
−0.552673 + 0.833398i \(0.686392\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) − 80.3963i − 3.35568i
\(575\) − 0.174653i − 0.00728352i
\(576\) 0 0
\(577\) −13.0577 −0.543600 −0.271800 0.962354i \(-0.587619\pi\)
−0.271800 + 0.962354i \(0.587619\pi\)
\(578\) 39.9499 1.66170
\(579\) 0 0
\(580\) 14.5804i 0.605419i
\(581\) − 41.3661i − 1.71616i
\(582\) 0 0
\(583\) 0 0
\(584\) − 47.7437i − 1.97565i
\(585\) 0 0
\(586\) −44.8898 −1.85438
\(587\) − 23.7515i − 0.980328i −0.871630 0.490164i \(-0.836937\pi\)
0.871630 0.490164i \(-0.163063\pi\)
\(588\) 0 0
\(589\) − 4.01240i − 0.165328i
\(590\) 3.06679 0.126258
\(591\) 0 0
\(592\) 10.0426 0.412749
\(593\) 37.1489 1.52552 0.762761 0.646680i \(-0.223843\pi\)
0.762761 + 0.646680i \(0.223843\pi\)
\(594\) 0 0
\(595\) 0.694814 0.0284846
\(596\) 71.9178 2.94587
\(597\) 0 0
\(598\) 35.2971 1.44341
\(599\) − 2.38579i − 0.0974805i −0.998811 0.0487403i \(-0.984479\pi\)
0.998811 0.0487403i \(-0.0155207\pi\)
\(600\) 0 0
\(601\) 6.79386i 0.277127i 0.990354 + 0.138564i \(0.0442485\pi\)
−0.990354 + 0.138564i \(0.955751\pi\)
\(602\) −40.7678 −1.66157
\(603\) 0 0
\(604\) − 4.19334i − 0.170624i
\(605\) 0 0
\(606\) 0 0
\(607\) − 40.4963i − 1.64370i −0.569707 0.821848i \(-0.692943\pi\)
0.569707 0.821848i \(-0.307057\pi\)
\(608\) − 9.40862i − 0.381570i
\(609\) 0 0
\(610\) 19.9887 0.809318
\(611\) 28.9203 1.16999
\(612\) 0 0
\(613\) − 0.202577i − 0.00818201i −0.999992 0.00409100i \(-0.998698\pi\)
0.999992 0.00409100i \(-0.00130221\pi\)
\(614\) − 4.37812i − 0.176686i
\(615\) 0 0
\(616\) 0 0
\(617\) − 8.08426i − 0.325460i −0.986671 0.162730i \(-0.947970\pi\)
0.986671 0.162730i \(-0.0520299\pi\)
\(618\) 0 0
\(619\) 20.9866 0.843523 0.421762 0.906707i \(-0.361412\pi\)
0.421762 + 0.906707i \(0.361412\pi\)
\(620\) 13.2766i 0.533203i
\(621\) 0 0
\(622\) − 21.0505i − 0.844048i
\(623\) −26.6959 −1.06955
\(624\) 0 0
\(625\) −25.2677 −1.01071
\(626\) −40.2853 −1.61012
\(627\) 0 0
\(628\) −6.11131 −0.243868
\(629\) 0.554403 0.0221055
\(630\) 0 0
\(631\) −24.2601 −0.965779 −0.482890 0.875681i \(-0.660413\pi\)
−0.482890 + 0.875681i \(0.660413\pi\)
\(632\) − 2.36753i − 0.0941753i
\(633\) 0 0
\(634\) − 64.1867i − 2.54918i
\(635\) −13.6270 −0.540772
\(636\) 0 0
\(637\) − 43.7871i − 1.73491i
\(638\) 0 0
\(639\) 0 0
\(640\) 45.7394i 1.80801i
\(641\) 16.1391i 0.637457i 0.947846 + 0.318728i \(0.103256\pi\)
−0.947846 + 0.318728i \(0.896744\pi\)
\(642\) 0 0
\(643\) 12.5840 0.496263 0.248132 0.968726i \(-0.420183\pi\)
0.248132 + 0.968726i \(0.420183\pi\)
\(644\) 46.1006 1.81662
\(645\) 0 0
\(646\) 0.429678i 0.0169055i
\(647\) 30.2507i 1.18928i 0.803993 + 0.594638i \(0.202705\pi\)
−0.803993 + 0.594638i \(0.797295\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0.591929i 0.0232174i
\(651\) 0 0
\(652\) −3.53543 −0.138458
\(653\) − 19.5926i − 0.766716i −0.923600 0.383358i \(-0.874768\pi\)
0.923600 0.383358i \(-0.125232\pi\)
\(654\) 0 0
\(655\) 16.0760i 0.628143i
\(656\) −11.6667 −0.455508
\(657\) 0 0
\(658\) 59.1953 2.30767
\(659\) 1.34943 0.0525664 0.0262832 0.999655i \(-0.491633\pi\)
0.0262832 + 0.999655i \(0.491633\pi\)
\(660\) 0 0
\(661\) 28.7859 1.11964 0.559821 0.828614i \(-0.310870\pi\)
0.559821 + 0.828614i \(0.310870\pi\)
\(662\) 13.8043 0.536519
\(663\) 0 0
\(664\) −36.6368 −1.42178
\(665\) − 21.8195i − 0.846125i
\(666\) 0 0
\(667\) 5.93548i 0.229823i
\(668\) 47.6685 1.84435
\(669\) 0 0
\(670\) − 68.6954i − 2.65394i
\(671\) 0 0
\(672\) 0 0
\(673\) 21.7509i 0.838436i 0.907886 + 0.419218i \(0.137696\pi\)
−0.907886 + 0.419218i \(0.862304\pi\)
\(674\) 38.9545i 1.50047i
\(675\) 0 0
\(676\) −30.4925 −1.17279
\(677\) −18.9054 −0.726592 −0.363296 0.931674i \(-0.618349\pi\)
−0.363296 + 0.931674i \(0.618349\pi\)
\(678\) 0 0
\(679\) − 66.6185i − 2.55658i
\(680\) − 0.615376i − 0.0235986i
\(681\) 0 0
\(682\) 0 0
\(683\) 33.7466i 1.29128i 0.763642 + 0.645640i \(0.223409\pi\)
−0.763642 + 0.645640i \(0.776591\pi\)
\(684\) 0 0
\(685\) −23.8807 −0.912436
\(686\) − 22.9625i − 0.876713i
\(687\) 0 0
\(688\) 5.91600i 0.225546i
\(689\) 5.71808 0.217841
\(690\) 0 0
\(691\) −27.5329 −1.04740 −0.523701 0.851902i \(-0.675449\pi\)
−0.523701 + 0.851902i \(0.675449\pi\)
\(692\) −26.5278 −1.00844
\(693\) 0 0
\(694\) 5.91971 0.224709
\(695\) 0.119323 0.00452618
\(696\) 0 0
\(697\) −0.644059 −0.0243955
\(698\) 20.5212i 0.776738i
\(699\) 0 0
\(700\) 0.773102i 0.0292205i
\(701\) 10.4287 0.393886 0.196943 0.980415i \(-0.436899\pi\)
0.196943 + 0.980415i \(0.436899\pi\)
\(702\) 0 0
\(703\) − 17.4101i − 0.656636i
\(704\) 0 0
\(705\) 0 0
\(706\) − 1.26120i − 0.0474661i
\(707\) 6.97553i 0.262342i
\(708\) 0 0
\(709\) −20.0467 −0.752869 −0.376435 0.926443i \(-0.622850\pi\)
−0.376435 + 0.926443i \(0.622850\pi\)
\(710\) 5.92160 0.222234
\(711\) 0 0
\(712\) 23.6438i 0.886088i
\(713\) 5.40473i 0.202409i
\(714\) 0 0
\(715\) 0 0
\(716\) 44.0793i 1.64732i
\(717\) 0 0
\(718\) 37.1562 1.38666
\(719\) − 10.2376i − 0.381799i −0.981610 0.190899i \(-0.938860\pi\)
0.981610 0.190899i \(-0.0611404\pi\)
\(720\) 0 0
\(721\) 13.0885i 0.487440i
\(722\) −31.1718 −1.16010
\(723\) 0 0
\(724\) −63.4761 −2.35907
\(725\) −0.0995374 −0.00369673
\(726\) 0 0
\(727\) −14.4456 −0.535759 −0.267880 0.963452i \(-0.586323\pi\)
−0.267880 + 0.963452i \(0.586323\pi\)
\(728\) −67.6261 −2.50639
\(729\) 0 0
\(730\) 70.3252 2.60285
\(731\) 0.326593i 0.0120795i
\(732\) 0 0
\(733\) 7.30917i 0.269970i 0.990848 + 0.134985i \(0.0430987\pi\)
−0.990848 + 0.134985i \(0.956901\pi\)
\(734\) −68.0162 −2.51053
\(735\) 0 0
\(736\) 12.6735i 0.467151i
\(737\) 0 0
\(738\) 0 0
\(739\) − 13.1337i − 0.483131i −0.970385 0.241565i \(-0.922339\pi\)
0.970385 0.241565i \(-0.0776609\pi\)
\(740\) 57.6085i 2.11773i
\(741\) 0 0
\(742\) 11.7040 0.429667
\(743\) −35.3724 −1.29769 −0.648844 0.760921i \(-0.724747\pi\)
−0.648844 + 0.760921i \(0.724747\pi\)
\(744\) 0 0
\(745\) 45.8506i 1.67984i
\(746\) 15.5372i 0.568859i
\(747\) 0 0
\(748\) 0 0
\(749\) − 60.0306i − 2.19347i
\(750\) 0 0
\(751\) −23.0686 −0.841785 −0.420892 0.907111i \(-0.638283\pi\)
−0.420892 + 0.907111i \(0.638283\pi\)
\(752\) − 8.59012i − 0.313249i
\(753\) 0 0
\(754\) − 20.1164i − 0.732597i
\(755\) 2.67343 0.0972960
\(756\) 0 0
\(757\) 9.79363 0.355956 0.177978 0.984034i \(-0.443044\pi\)
0.177978 + 0.984034i \(0.443044\pi\)
\(758\) −3.64030 −0.132222
\(759\) 0 0
\(760\) −19.3249 −0.700988
\(761\) −48.4251 −1.75541 −0.877705 0.479201i \(-0.840926\pi\)
−0.877705 + 0.479201i \(0.840926\pi\)
\(762\) 0 0
\(763\) 60.4655 2.18900
\(764\) 76.3680i 2.76290i
\(765\) 0 0
\(766\) − 31.7542i − 1.14732i
\(767\) −2.69989 −0.0974876
\(768\) 0 0
\(769\) − 33.4223i − 1.20524i −0.798029 0.602619i \(-0.794124\pi\)
0.798029 0.602619i \(-0.205876\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 18.2904i 0.658287i
\(773\) − 6.51470i − 0.234317i −0.993113 0.117159i \(-0.962621\pi\)
0.993113 0.117159i \(-0.0373786\pi\)
\(774\) 0 0
\(775\) −0.0906368 −0.00325577
\(776\) −59.0021 −2.11805
\(777\) 0 0
\(778\) − 35.2137i − 1.26247i
\(779\) 20.2257i 0.724659i
\(780\) 0 0
\(781\) 0 0
\(782\) − 0.578780i − 0.0206971i
\(783\) 0 0
\(784\) −13.0060 −0.464499
\(785\) − 3.89622i − 0.139062i
\(786\) 0 0
\(787\) − 28.9940i − 1.03352i −0.856129 0.516761i \(-0.827137\pi\)
0.856129 0.516761i \(-0.172863\pi\)
\(788\) 18.1489 0.646527
\(789\) 0 0
\(790\) 3.48731 0.124073
\(791\) 45.5606 1.61995
\(792\) 0 0
\(793\) −17.5974 −0.624901
\(794\) 48.6925 1.72803
\(795\) 0 0
\(796\) 85.5451 3.03207
\(797\) 27.0503i 0.958170i 0.877768 + 0.479085i \(0.159032\pi\)
−0.877768 + 0.479085i \(0.840968\pi\)
\(798\) 0 0
\(799\) − 0.474217i − 0.0167766i
\(800\) −0.212533 −0.00751418
\(801\) 0 0
\(802\) 63.5768i 2.24498i
\(803\) 0 0
\(804\) 0 0
\(805\) 29.3911i 1.03590i
\(806\) − 18.3176i − 0.645210i
\(807\) 0 0
\(808\) 6.17803 0.217342
\(809\) −41.8697 −1.47206 −0.736031 0.676948i \(-0.763302\pi\)
−0.736031 + 0.676948i \(0.763302\pi\)
\(810\) 0 0
\(811\) 12.7916i 0.449174i 0.974454 + 0.224587i \(0.0721033\pi\)
−0.974454 + 0.224587i \(0.927897\pi\)
\(812\) − 26.2735i − 0.922018i
\(813\) 0 0
\(814\) 0 0
\(815\) − 2.25399i − 0.0789538i
\(816\) 0 0
\(817\) 10.2561 0.358816
\(818\) 14.3528i 0.501834i
\(819\) 0 0
\(820\) − 66.9248i − 2.33711i
\(821\) 11.8623 0.413998 0.206999 0.978341i \(-0.433630\pi\)
0.206999 + 0.978341i \(0.433630\pi\)
\(822\) 0 0
\(823\) 26.2761 0.915927 0.457963 0.888971i \(-0.348579\pi\)
0.457963 + 0.888971i \(0.348579\pi\)
\(824\) 11.5921 0.403829
\(825\) 0 0
\(826\) −5.52626 −0.192283
\(827\) −25.8883 −0.900224 −0.450112 0.892972i \(-0.648616\pi\)
−0.450112 + 0.892972i \(0.648616\pi\)
\(828\) 0 0
\(829\) 17.3291 0.601863 0.300932 0.953646i \(-0.402702\pi\)
0.300932 + 0.953646i \(0.402702\pi\)
\(830\) − 53.9651i − 1.87315i
\(831\) 0 0
\(832\) − 55.8124i − 1.93495i
\(833\) −0.717993 −0.0248770
\(834\) 0 0
\(835\) 30.3907i 1.05171i
\(836\) 0 0
\(837\) 0 0
\(838\) 83.8338i 2.89599i
\(839\) 31.0509i 1.07200i 0.844219 + 0.535999i \(0.180065\pi\)
−0.844219 + 0.535999i \(0.819935\pi\)
\(840\) 0 0
\(841\) −25.6173 −0.883354
\(842\) −33.6344 −1.15912
\(843\) 0 0
\(844\) − 66.0948i − 2.27508i
\(845\) − 19.4402i − 0.668765i
\(846\) 0 0
\(847\) 0 0
\(848\) − 1.69842i − 0.0583241i
\(849\) 0 0
\(850\) 0.00970608 0.000332916 0
\(851\) 23.4516i 0.803910i
\(852\) 0 0
\(853\) − 0.319368i − 0.0109350i −0.999985 0.00546748i \(-0.998260\pi\)
0.999985 0.00546748i \(-0.00174036\pi\)
\(854\) −36.0190 −1.23254
\(855\) 0 0
\(856\) −53.1674 −1.81722
\(857\) −13.2476 −0.452528 −0.226264 0.974066i \(-0.572651\pi\)
−0.226264 + 0.974066i \(0.572651\pi\)
\(858\) 0 0
\(859\) −8.58152 −0.292798 −0.146399 0.989226i \(-0.546768\pi\)
−0.146399 + 0.989226i \(0.546768\pi\)
\(860\) −33.9365 −1.15723
\(861\) 0 0
\(862\) −20.4595 −0.696853
\(863\) 53.7180i 1.82858i 0.405057 + 0.914292i \(0.367252\pi\)
−0.405057 + 0.914292i \(0.632748\pi\)
\(864\) 0 0
\(865\) − 16.9126i − 0.575046i
\(866\) 88.2802 2.99988
\(867\) 0 0
\(868\) − 23.9241i − 0.812037i
\(869\) 0 0
\(870\) 0 0
\(871\) 60.4771i 2.04919i
\(872\) − 53.5525i − 1.81352i
\(873\) 0 0
\(874\) −18.1757 −0.614801
\(875\) 45.0440 1.52276
\(876\) 0 0
\(877\) 44.0154i 1.48630i 0.669127 + 0.743148i \(0.266668\pi\)
−0.669127 + 0.743148i \(0.733332\pi\)
\(878\) 76.3231i 2.57578i
\(879\) 0 0
\(880\) 0 0
\(881\) 7.44194i 0.250725i 0.992111 + 0.125363i \(0.0400095\pi\)
−0.992111 + 0.125363i \(0.959991\pi\)
\(882\) 0 0
\(883\) −9.38632 −0.315875 −0.157937 0.987449i \(-0.550484\pi\)
−0.157937 + 0.987449i \(0.550484\pi\)
\(884\) 1.25167i 0.0420983i
\(885\) 0 0
\(886\) − 44.2455i − 1.48646i
\(887\) −24.1847 −0.812043 −0.406022 0.913864i \(-0.633084\pi\)
−0.406022 + 0.913864i \(0.633084\pi\)
\(888\) 0 0
\(889\) 24.5555 0.823565
\(890\) −34.8267 −1.16739
\(891\) 0 0
\(892\) −19.3542 −0.648026
\(893\) −14.8920 −0.498343
\(894\) 0 0
\(895\) −28.1024 −0.939360
\(896\) − 82.4210i − 2.75349i
\(897\) 0 0
\(898\) 16.9772i 0.566535i
\(899\) 3.08024 0.102732
\(900\) 0 0
\(901\) − 0.0937613i − 0.00312364i
\(902\) 0 0
\(903\) 0 0
\(904\) − 40.3517i − 1.34208i
\(905\) − 40.4687i − 1.34523i
\(906\) 0 0
\(907\) −35.8342 −1.18985 −0.594927 0.803780i \(-0.702819\pi\)
−0.594927 + 0.803780i \(0.702819\pi\)
\(908\) −48.5347 −1.61068
\(909\) 0 0
\(910\) − 99.6116i − 3.30209i
\(911\) − 8.75932i − 0.290209i −0.989416 0.145105i \(-0.953648\pi\)
0.989416 0.145105i \(-0.0463519\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 9.92552i 0.328307i
\(915\) 0 0
\(916\) −74.5191 −2.46218
\(917\) − 28.9685i − 0.956625i
\(918\) 0 0
\(919\) 41.5197i 1.36961i 0.728726 + 0.684805i \(0.240113\pi\)
−0.728726 + 0.684805i \(0.759887\pi\)
\(920\) 26.0308 0.858211
\(921\) 0 0
\(922\) −33.3630 −1.09875
\(923\) −5.21318 −0.171594
\(924\) 0 0
\(925\) −0.393281 −0.0129310
\(926\) 94.9214 3.11931
\(927\) 0 0
\(928\) 7.22283 0.237101
\(929\) 54.1884i 1.77786i 0.458038 + 0.888932i \(0.348552\pi\)
−0.458038 + 0.888932i \(0.651448\pi\)
\(930\) 0 0
\(931\) 22.5474i 0.738963i
\(932\) 52.7721 1.72861
\(933\) 0 0
\(934\) − 26.5656i − 0.869253i
\(935\) 0 0
\(936\) 0 0
\(937\) − 17.3958i − 0.568295i −0.958781 0.284147i \(-0.908290\pi\)
0.958781 0.284147i \(-0.0917105\pi\)
\(938\) 123.787i 4.04179i
\(939\) 0 0
\(940\) 49.2763 1.60722
\(941\) 21.5104 0.701219 0.350610 0.936522i \(-0.385974\pi\)
0.350610 + 0.936522i \(0.385974\pi\)
\(942\) 0 0
\(943\) − 27.2441i − 0.887191i
\(944\) 0.801941i 0.0261010i
\(945\) 0 0
\(946\) 0 0
\(947\) − 44.7887i − 1.45544i −0.685875 0.727719i \(-0.740580\pi\)
0.685875 0.727719i \(-0.259420\pi\)
\(948\) 0 0
\(949\) −61.9120 −2.00975
\(950\) − 0.304804i − 0.00988914i
\(951\) 0 0
\(952\) 1.10889i 0.0359393i
\(953\) 10.0337 0.325023 0.162511 0.986707i \(-0.448041\pi\)
0.162511 + 0.986707i \(0.448041\pi\)
\(954\) 0 0
\(955\) −48.6879 −1.57550
\(956\) 19.0594 0.616425
\(957\) 0 0
\(958\) −53.8294 −1.73915
\(959\) 43.0324 1.38959
\(960\) 0 0
\(961\) −28.1952 −0.909522
\(962\) − 79.4817i − 2.56259i
\(963\) 0 0
\(964\) 41.2894i 1.32984i
\(965\) −11.6609 −0.375378
\(966\) 0 0
\(967\) − 4.32167i − 0.138976i −0.997583 0.0694878i \(-0.977863\pi\)
0.997583 0.0694878i \(-0.0221365\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) − 86.9086i − 2.79047i
\(971\) − 57.3923i − 1.84181i −0.389790 0.920904i \(-0.627452\pi\)
0.389790 0.920904i \(-0.372548\pi\)
\(972\) 0 0
\(973\) −0.215016 −0.00689311
\(974\) 17.7300 0.568105
\(975\) 0 0
\(976\) 5.22689i 0.167309i
\(977\) 3.45017i 0.110381i 0.998476 + 0.0551903i \(0.0175766\pi\)
−0.998476 + 0.0551903i \(0.982423\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) − 74.6073i − 2.38324i
\(981\) 0 0
\(982\) 89.7775 2.86491
\(983\) 1.03468i 0.0330010i 0.999864 + 0.0165005i \(0.00525252\pi\)
−0.999864 + 0.0165005i \(0.994747\pi\)
\(984\) 0 0
\(985\) 11.5707i 0.368673i
\(986\) −0.329856 −0.0105048
\(987\) 0 0
\(988\) 39.3068 1.25052
\(989\) −13.8151 −0.439294
\(990\) 0 0
\(991\) 9.78910 0.310961 0.155481 0.987839i \(-0.450307\pi\)
0.155481 + 0.987839i \(0.450307\pi\)
\(992\) 6.57696 0.208819
\(993\) 0 0
\(994\) −10.6705 −0.338449
\(995\) 54.5386i 1.72899i
\(996\) 0 0
\(997\) − 11.6146i − 0.367838i −0.982941 0.183919i \(-0.941122\pi\)
0.982941 0.183919i \(-0.0588785\pi\)
\(998\) 18.9453 0.599703
\(999\) 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 1089.2.d.g.1088.4 16
3.2 odd 2 inner 1089.2.d.g.1088.13 16
11.3 even 5 99.2.j.a.35.1 yes 16
11.7 odd 10 99.2.j.a.17.4 yes 16
11.10 odd 2 inner 1089.2.d.g.1088.14 16
33.14 odd 10 99.2.j.a.35.4 yes 16
33.29 even 10 99.2.j.a.17.1 16
33.32 even 2 inner 1089.2.d.g.1088.3 16
44.3 odd 10 1584.2.cd.c.1025.1 16
44.7 even 10 1584.2.cd.c.17.4 16
99.7 odd 30 891.2.u.c.215.4 32
99.14 odd 30 891.2.u.c.431.4 32
99.25 even 15 891.2.u.c.134.4 32
99.29 even 30 891.2.u.c.215.1 32
99.40 odd 30 891.2.u.c.512.1 32
99.47 odd 30 891.2.u.c.134.1 32
99.58 even 15 891.2.u.c.431.1 32
99.95 even 30 891.2.u.c.512.4 32
132.47 even 10 1584.2.cd.c.1025.4 16
132.95 odd 10 1584.2.cd.c.17.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.j.a.17.1 16 33.29 even 10
99.2.j.a.17.4 yes 16 11.7 odd 10
99.2.j.a.35.1 yes 16 11.3 even 5
99.2.j.a.35.4 yes 16 33.14 odd 10
891.2.u.c.134.1 32 99.47 odd 30
891.2.u.c.134.4 32 99.25 even 15
891.2.u.c.215.1 32 99.29 even 30
891.2.u.c.215.4 32 99.7 odd 30
891.2.u.c.431.1 32 99.58 even 15
891.2.u.c.431.4 32 99.14 odd 30
891.2.u.c.512.1 32 99.40 odd 30
891.2.u.c.512.4 32 99.95 even 30
1089.2.d.g.1088.3 16 33.32 even 2 inner
1089.2.d.g.1088.4 16 1.1 even 1 trivial
1089.2.d.g.1088.13 16 3.2 odd 2 inner
1089.2.d.g.1088.14 16 11.10 odd 2 inner
1584.2.cd.c.17.1 16 132.95 odd 10
1584.2.cd.c.17.4 16 44.7 even 10
1584.2.cd.c.1025.1 16 44.3 odd 10
1584.2.cd.c.1025.4 16 132.47 even 10