Properties

Label 1089.3.b.h.485.6
Level $1089$
Weight $3$
Character 1089.485
Analytic conductor $29.673$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3317760000.7
Defining polynomial: \( x^{8} + 12x^{6} + 23x^{4} + 12x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 485.6
Root \(-0.837556i\) of defining polynomial
Character \(\chi\) \(=\) 1089.485
Dual form 1089.3.b.h.485.3

$q$-expansion

\(f(q)\) \(=\) \(q+0.356394i q^{2} +3.87298 q^{4} +3.44572i q^{5} -3.96812 q^{7} +2.80588i q^{8} +O(q^{10})\) \(q+0.356394i q^{2} +3.87298 q^{4} +3.44572i q^{5} -3.96812 q^{7} +2.80588i q^{8} -1.22803 q^{10} +4.75615 q^{13} -1.41421i q^{14} +14.4919 q^{16} -29.1280i q^{17} +14.8644 q^{19} +13.3452i q^{20} +22.9639i q^{23} +13.1270 q^{25} +1.69506i q^{26} -15.3685 q^{28} -4.63312i q^{29} +27.4919 q^{31} +16.3884i q^{32} +10.3810 q^{34} -13.6730i q^{35} +36.8569 q^{37} +5.29760i q^{38} -9.66829 q^{40} +68.3199i q^{41} +26.8968 q^{43} -8.18418 q^{46} +42.9425i q^{47} -33.2540 q^{49} +4.67839i q^{50} +18.4205 q^{52} +78.2194i q^{53} -11.1341i q^{56} +1.65122 q^{58} +92.7992i q^{59} -40.5612 q^{61} +9.79796i q^{62} +52.1270 q^{64} +16.3884i q^{65} -54.3327 q^{67} -112.812i q^{68} +4.87298 q^{70} -91.0029i q^{71} +109.347 q^{73} +13.1356i q^{74} +57.5697 q^{76} -143.292 q^{79} +49.9351i q^{80} -24.3488 q^{82} -44.0003i q^{83} +100.367 q^{85} +9.58587i q^{86} -64.7730i q^{89} -18.8730 q^{91} +88.9386i q^{92} -15.3044 q^{94} +51.2187i q^{95} +16.8891 q^{97} -11.8515i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{16} + 136 q^{25} + 96 q^{31} + 176 q^{34} + 16 q^{37} - 328 q^{49} + 416 q^{58} + 448 q^{64} + 216 q^{67} + 8 q^{70} + 208 q^{82} - 120 q^{91} + 352 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.356394i 0.178197i 0.996023 + 0.0890985i \(0.0283986\pi\)
−0.996023 + 0.0890985i \(0.971601\pi\)
\(3\) 0 0
\(4\) 3.87298 0.968246
\(5\) 3.44572i 0.689144i 0.938760 + 0.344572i \(0.111976\pi\)
−0.938760 + 0.344572i \(0.888024\pi\)
\(6\) 0 0
\(7\) −3.96812 −0.566874 −0.283437 0.958991i \(-0.591475\pi\)
−0.283437 + 0.958991i \(0.591475\pi\)
\(8\) 2.80588i 0.350735i
\(9\) 0 0
\(10\) −1.22803 −0.122803
\(11\) 0 0
\(12\) 0 0
\(13\) 4.75615 0.365858 0.182929 0.983126i \(-0.441442\pi\)
0.182929 + 0.983126i \(0.441442\pi\)
\(14\) − 1.41421i − 0.101015i
\(15\) 0 0
\(16\) 14.4919 0.905746
\(17\) − 29.1280i − 1.71341i −0.515804 0.856706i \(-0.672507\pi\)
0.515804 0.856706i \(-0.327493\pi\)
\(18\) 0 0
\(19\) 14.8644 0.782339 0.391169 0.920319i \(-0.372071\pi\)
0.391169 + 0.920319i \(0.372071\pi\)
\(20\) 13.3452i 0.667261i
\(21\) 0 0
\(22\) 0 0
\(23\) 22.9639i 0.998429i 0.866479 + 0.499214i \(0.166378\pi\)
−0.866479 + 0.499214i \(0.833622\pi\)
\(24\) 0 0
\(25\) 13.1270 0.525081
\(26\) 1.69506i 0.0651948i
\(27\) 0 0
\(28\) −15.3685 −0.548873
\(29\) − 4.63312i − 0.159763i −0.996804 0.0798814i \(-0.974546\pi\)
0.996804 0.0798814i \(-0.0254542\pi\)
\(30\) 0 0
\(31\) 27.4919 0.886837 0.443418 0.896315i \(-0.353766\pi\)
0.443418 + 0.896315i \(0.353766\pi\)
\(32\) 16.3884i 0.512137i
\(33\) 0 0
\(34\) 10.3810 0.305325
\(35\) − 13.6730i − 0.390658i
\(36\) 0 0
\(37\) 36.8569 0.996131 0.498066 0.867139i \(-0.334044\pi\)
0.498066 + 0.867139i \(0.334044\pi\)
\(38\) 5.29760i 0.139410i
\(39\) 0 0
\(40\) −9.66829 −0.241707
\(41\) 68.3199i 1.66634i 0.553019 + 0.833169i \(0.313476\pi\)
−0.553019 + 0.833169i \(0.686524\pi\)
\(42\) 0 0
\(43\) 26.8968 0.625508 0.312754 0.949834i \(-0.398748\pi\)
0.312754 + 0.949834i \(0.398748\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −8.18418 −0.177917
\(47\) 42.9425i 0.913670i 0.889552 + 0.456835i \(0.151017\pi\)
−0.889552 + 0.456835i \(0.848983\pi\)
\(48\) 0 0
\(49\) −33.2540 −0.678654
\(50\) 4.67839i 0.0935678i
\(51\) 0 0
\(52\) 18.4205 0.354240
\(53\) 78.2194i 1.47584i 0.674889 + 0.737919i \(0.264191\pi\)
−0.674889 + 0.737919i \(0.735809\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 11.1341i − 0.198823i
\(57\) 0 0
\(58\) 1.65122 0.0284692
\(59\) 92.7992i 1.57287i 0.617674 + 0.786434i \(0.288075\pi\)
−0.617674 + 0.786434i \(0.711925\pi\)
\(60\) 0 0
\(61\) −40.5612 −0.664937 −0.332469 0.943114i \(-0.607882\pi\)
−0.332469 + 0.943114i \(0.607882\pi\)
\(62\) 9.79796i 0.158032i
\(63\) 0 0
\(64\) 52.1270 0.814485
\(65\) 16.3884i 0.252129i
\(66\) 0 0
\(67\) −54.3327 −0.810935 −0.405468 0.914109i \(-0.632891\pi\)
−0.405468 + 0.914109i \(0.632891\pi\)
\(68\) − 112.812i − 1.65900i
\(69\) 0 0
\(70\) 4.87298 0.0696140
\(71\) − 91.0029i − 1.28173i −0.767653 0.640866i \(-0.778576\pi\)
0.767653 0.640866i \(-0.221424\pi\)
\(72\) 0 0
\(73\) 109.347 1.49791 0.748954 0.662622i \(-0.230556\pi\)
0.748954 + 0.662622i \(0.230556\pi\)
\(74\) 13.1356i 0.177508i
\(75\) 0 0
\(76\) 57.5697 0.757496
\(77\) 0 0
\(78\) 0 0
\(79\) −143.292 −1.81383 −0.906913 0.421318i \(-0.861568\pi\)
−0.906913 + 0.421318i \(0.861568\pi\)
\(80\) 49.9351i 0.624189i
\(81\) 0 0
\(82\) −24.3488 −0.296936
\(83\) − 44.0003i − 0.530124i −0.964231 0.265062i \(-0.914608\pi\)
0.964231 0.265062i \(-0.0853924\pi\)
\(84\) 0 0
\(85\) 100.367 1.18079
\(86\) 9.58587i 0.111464i
\(87\) 0 0
\(88\) 0 0
\(89\) − 64.7730i − 0.727786i −0.931441 0.363893i \(-0.881447\pi\)
0.931441 0.363893i \(-0.118553\pi\)
\(90\) 0 0
\(91\) −18.8730 −0.207395
\(92\) 88.9386i 0.966724i
\(93\) 0 0
\(94\) −15.3044 −0.162813
\(95\) 51.2187i 0.539144i
\(96\) 0 0
\(97\) 16.8891 0.174115 0.0870573 0.996203i \(-0.472254\pi\)
0.0870573 + 0.996203i \(0.472254\pi\)
\(98\) − 11.8515i − 0.120934i
\(99\) 0 0
\(100\) 50.8407 0.508407
\(101\) − 164.568i − 1.62939i −0.579889 0.814696i \(-0.696904\pi\)
0.579889 0.814696i \(-0.303096\pi\)
\(102\) 0 0
\(103\) 145.381 1.41147 0.705733 0.708478i \(-0.250618\pi\)
0.705733 + 0.708478i \(0.250618\pi\)
\(104\) 13.3452i 0.128319i
\(105\) 0 0
\(106\) −27.8769 −0.262990
\(107\) 189.414i 1.77022i 0.465377 + 0.885112i \(0.345919\pi\)
−0.465377 + 0.885112i \(0.654081\pi\)
\(108\) 0 0
\(109\) −57.5418 −0.527906 −0.263953 0.964536i \(-0.585026\pi\)
−0.263953 + 0.964536i \(0.585026\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −57.5057 −0.513444
\(113\) 29.7769i 0.263512i 0.991282 + 0.131756i \(0.0420616\pi\)
−0.991282 + 0.131756i \(0.957938\pi\)
\(114\) 0 0
\(115\) −79.1270 −0.688061
\(116\) − 17.9440i − 0.154690i
\(117\) 0 0
\(118\) −33.0731 −0.280280
\(119\) 115.583i 0.971289i
\(120\) 0 0
\(121\) 0 0
\(122\) − 14.4558i − 0.118490i
\(123\) 0 0
\(124\) 106.476 0.858676
\(125\) 131.375i 1.05100i
\(126\) 0 0
\(127\) −12.5923 −0.0991520 −0.0495760 0.998770i \(-0.515787\pi\)
−0.0495760 + 0.998770i \(0.515787\pi\)
\(128\) 84.1312i 0.657275i
\(129\) 0 0
\(130\) −5.84072 −0.0449286
\(131\) 20.3597i 0.155418i 0.996976 + 0.0777089i \(0.0247605\pi\)
−0.996976 + 0.0777089i \(0.975240\pi\)
\(132\) 0 0
\(133\) −58.9839 −0.443488
\(134\) − 19.3638i − 0.144506i
\(135\) 0 0
\(136\) 81.7298 0.600955
\(137\) − 35.9170i − 0.262168i −0.991371 0.131084i \(-0.958154\pi\)
0.991371 0.131084i \(-0.0418458\pi\)
\(138\) 0 0
\(139\) −101.091 −0.727273 −0.363637 0.931541i \(-0.618465\pi\)
−0.363637 + 0.931541i \(0.618465\pi\)
\(140\) − 52.9554i − 0.378253i
\(141\) 0 0
\(142\) 32.4329 0.228401
\(143\) 0 0
\(144\) 0 0
\(145\) 15.9644 0.110100
\(146\) 38.9707i 0.266923i
\(147\) 0 0
\(148\) 142.746 0.964500
\(149\) − 210.311i − 1.41148i −0.708469 0.705742i \(-0.750614\pi\)
0.708469 0.705742i \(-0.249386\pi\)
\(150\) 0 0
\(151\) 212.078 1.40449 0.702246 0.711934i \(-0.252180\pi\)
0.702246 + 0.711934i \(0.252180\pi\)
\(152\) 41.7079i 0.274394i
\(153\) 0 0
\(154\) 0 0
\(155\) 94.7295i 0.611158i
\(156\) 0 0
\(157\) 119.173 0.759066 0.379533 0.925178i \(-0.376085\pi\)
0.379533 + 0.925178i \(0.376085\pi\)
\(158\) − 51.0685i − 0.323218i
\(159\) 0 0
\(160\) −56.4697 −0.352936
\(161\) − 91.1233i − 0.565983i
\(162\) 0 0
\(163\) 266.268 1.63355 0.816773 0.576959i \(-0.195761\pi\)
0.816773 + 0.576959i \(0.195761\pi\)
\(164\) 264.602i 1.61342i
\(165\) 0 0
\(166\) 15.6814 0.0944665
\(167\) 113.090i 0.677184i 0.940933 + 0.338592i \(0.109951\pi\)
−0.940933 + 0.338592i \(0.890049\pi\)
\(168\) 0 0
\(169\) −146.379 −0.866148
\(170\) 35.7702i 0.210413i
\(171\) 0 0
\(172\) 104.171 0.605645
\(173\) 117.615i 0.679856i 0.940452 + 0.339928i \(0.110403\pi\)
−0.940452 + 0.339928i \(0.889597\pi\)
\(174\) 0 0
\(175\) −52.0896 −0.297655
\(176\) 0 0
\(177\) 0 0
\(178\) 23.0847 0.129689
\(179\) − 231.526i − 1.29344i −0.762727 0.646721i \(-0.776140\pi\)
0.762727 0.646721i \(-0.223860\pi\)
\(180\) 0 0
\(181\) −10.8105 −0.0597265 −0.0298633 0.999554i \(-0.509507\pi\)
−0.0298633 + 0.999554i \(0.509507\pi\)
\(182\) − 6.72622i − 0.0369572i
\(183\) 0 0
\(184\) −64.4339 −0.350184
\(185\) 126.998i 0.686478i
\(186\) 0 0
\(187\) 0 0
\(188\) 166.315i 0.884657i
\(189\) 0 0
\(190\) −18.2540 −0.0960739
\(191\) − 49.5203i − 0.259269i −0.991562 0.129634i \(-0.958620\pi\)
0.991562 0.129634i \(-0.0413803\pi\)
\(192\) 0 0
\(193\) 8.40418 0.0435450 0.0217725 0.999763i \(-0.493069\pi\)
0.0217725 + 0.999763i \(0.493069\pi\)
\(194\) 6.01918i 0.0310267i
\(195\) 0 0
\(196\) −128.792 −0.657104
\(197\) − 125.190i − 0.635481i −0.948178 0.317741i \(-0.897076\pi\)
0.948178 0.317741i \(-0.102924\pi\)
\(198\) 0 0
\(199\) 255.728 1.28506 0.642532 0.766259i \(-0.277884\pi\)
0.642532 + 0.766259i \(0.277884\pi\)
\(200\) 36.8329i 0.184164i
\(201\) 0 0
\(202\) 58.6512 0.290353
\(203\) 18.3848i 0.0905654i
\(204\) 0 0
\(205\) −235.411 −1.14835
\(206\) 51.8129i 0.251519i
\(207\) 0 0
\(208\) 68.9259 0.331374
\(209\) 0 0
\(210\) 0 0
\(211\) 149.332 0.707736 0.353868 0.935295i \(-0.384866\pi\)
0.353868 + 0.935295i \(0.384866\pi\)
\(212\) 302.942i 1.42897i
\(213\) 0 0
\(214\) −67.5060 −0.315449
\(215\) 92.6789i 0.431065i
\(216\) 0 0
\(217\) −109.091 −0.502725
\(218\) − 20.5075i − 0.0940713i
\(219\) 0 0
\(220\) 0 0
\(221\) − 138.537i − 0.626866i
\(222\) 0 0
\(223\) −211.270 −0.947400 −0.473700 0.880686i \(-0.657082\pi\)
−0.473700 + 0.880686i \(0.657082\pi\)
\(224\) − 65.0310i − 0.290317i
\(225\) 0 0
\(226\) −10.6123 −0.0469571
\(227\) − 175.091i − 0.771325i −0.922640 0.385662i \(-0.873973\pi\)
0.922640 0.385662i \(-0.126027\pi\)
\(228\) 0 0
\(229\) −142.460 −0.622095 −0.311047 0.950394i \(-0.600680\pi\)
−0.311047 + 0.950394i \(0.600680\pi\)
\(230\) − 28.2004i − 0.122610i
\(231\) 0 0
\(232\) 13.0000 0.0560345
\(233\) 264.240i 1.13408i 0.823692 + 0.567038i \(0.191911\pi\)
−0.823692 + 0.567038i \(0.808089\pi\)
\(234\) 0 0
\(235\) −147.968 −0.629650
\(236\) 359.410i 1.52292i
\(237\) 0 0
\(238\) −41.1932 −0.173081
\(239\) − 176.997i − 0.740573i −0.928918 0.370287i \(-0.879259\pi\)
0.928918 0.370287i \(-0.120741\pi\)
\(240\) 0 0
\(241\) −14.6084 −0.0606156 −0.0303078 0.999541i \(-0.509649\pi\)
−0.0303078 + 0.999541i \(0.509649\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −157.093 −0.643823
\(245\) − 114.584i − 0.467690i
\(246\) 0 0
\(247\) 70.6976 0.286225
\(248\) 77.1392i 0.311045i
\(249\) 0 0
\(250\) −46.8213 −0.187285
\(251\) − 217.159i − 0.865174i −0.901592 0.432587i \(-0.857601\pi\)
0.901592 0.432587i \(-0.142399\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) − 4.48782i − 0.0176686i
\(255\) 0 0
\(256\) 178.524 0.697360
\(257\) − 33.4151i − 0.130020i −0.997885 0.0650099i \(-0.979292\pi\)
0.997885 0.0650099i \(-0.0207079\pi\)
\(258\) 0 0
\(259\) −146.252 −0.564681
\(260\) 63.4719i 0.244123i
\(261\) 0 0
\(262\) −7.25608 −0.0276950
\(263\) − 335.360i − 1.27513i −0.770396 0.637566i \(-0.779941\pi\)
0.770396 0.637566i \(-0.220059\pi\)
\(264\) 0 0
\(265\) −269.522 −1.01706
\(266\) − 21.0215i − 0.0790282i
\(267\) 0 0
\(268\) −210.429 −0.785185
\(269\) − 337.759i − 1.25561i −0.778371 0.627805i \(-0.783954\pi\)
0.778371 0.627805i \(-0.216046\pi\)
\(270\) 0 0
\(271\) 436.501 1.61071 0.805353 0.592796i \(-0.201976\pi\)
0.805353 + 0.592796i \(0.201976\pi\)
\(272\) − 422.121i − 1.55192i
\(273\) 0 0
\(274\) 12.8006 0.0467176
\(275\) 0 0
\(276\) 0 0
\(277\) 199.130 0.718881 0.359440 0.933168i \(-0.382968\pi\)
0.359440 + 0.933168i \(0.382968\pi\)
\(278\) − 36.0282i − 0.129598i
\(279\) 0 0
\(280\) 38.3649 0.137018
\(281\) 250.442i 0.891253i 0.895219 + 0.445627i \(0.147019\pi\)
−0.895219 + 0.445627i \(0.852981\pi\)
\(282\) 0 0
\(283\) 43.0813 0.152231 0.0761153 0.997099i \(-0.475748\pi\)
0.0761153 + 0.997099i \(0.475748\pi\)
\(284\) − 352.453i − 1.24103i
\(285\) 0 0
\(286\) 0 0
\(287\) − 271.101i − 0.944604i
\(288\) 0 0
\(289\) −559.441 −1.93578
\(290\) 5.68963i 0.0196194i
\(291\) 0 0
\(292\) 423.500 1.45034
\(293\) − 7.51229i − 0.0256392i −0.999918 0.0128196i \(-0.995919\pi\)
0.999918 0.0128196i \(-0.00408072\pi\)
\(294\) 0 0
\(295\) −319.760 −1.08393
\(296\) 103.416i 0.349378i
\(297\) 0 0
\(298\) 74.9536 0.251522
\(299\) 109.220i 0.365283i
\(300\) 0 0
\(301\) −106.730 −0.354584
\(302\) 75.5835i 0.250276i
\(303\) 0 0
\(304\) 215.414 0.708600
\(305\) − 139.762i − 0.458238i
\(306\) 0 0
\(307\) −360.275 −1.17353 −0.586767 0.809756i \(-0.699599\pi\)
−0.586767 + 0.809756i \(0.699599\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −33.7610 −0.108907
\(311\) − 415.622i − 1.33641i −0.743979 0.668203i \(-0.767064\pi\)
0.743979 0.668203i \(-0.232936\pi\)
\(312\) 0 0
\(313\) −63.2863 −0.202193 −0.101096 0.994877i \(-0.532235\pi\)
−0.101096 + 0.994877i \(0.532235\pi\)
\(314\) 42.4727i 0.135263i
\(315\) 0 0
\(316\) −554.969 −1.75623
\(317\) − 422.491i − 1.33278i −0.745604 0.666389i \(-0.767839\pi\)
0.745604 0.666389i \(-0.232161\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 179.615i 0.561297i
\(321\) 0 0
\(322\) 32.4758 0.100857
\(323\) − 432.972i − 1.34047i
\(324\) 0 0
\(325\) 62.4341 0.192105
\(326\) 94.8963i 0.291093i
\(327\) 0 0
\(328\) −191.698 −0.584444
\(329\) − 170.401i − 0.517936i
\(330\) 0 0
\(331\) −233.077 −0.704159 −0.352079 0.935970i \(-0.614525\pi\)
−0.352079 + 0.935970i \(0.614525\pi\)
\(332\) − 170.412i − 0.513290i
\(333\) 0 0
\(334\) −40.3045 −0.120672
\(335\) − 187.215i − 0.558851i
\(336\) 0 0
\(337\) −622.254 −1.84645 −0.923226 0.384257i \(-0.874458\pi\)
−0.923226 + 0.384257i \(0.874458\pi\)
\(338\) − 52.1686i − 0.154345i
\(339\) 0 0
\(340\) 388.720 1.14329
\(341\) 0 0
\(342\) 0 0
\(343\) 326.394 0.951585
\(344\) 75.4694i 0.219388i
\(345\) 0 0
\(346\) −41.9173 −0.121148
\(347\) 648.204i 1.86802i 0.357241 + 0.934012i \(0.383717\pi\)
−0.357241 + 0.934012i \(0.616283\pi\)
\(348\) 0 0
\(349\) −304.125 −0.871419 −0.435709 0.900087i \(-0.643502\pi\)
−0.435709 + 0.900087i \(0.643502\pi\)
\(350\) − 18.5644i − 0.0530412i
\(351\) 0 0
\(352\) 0 0
\(353\) − 250.753i − 0.710350i −0.934800 0.355175i \(-0.884421\pi\)
0.934800 0.355175i \(-0.115579\pi\)
\(354\) 0 0
\(355\) 313.571 0.883297
\(356\) − 250.865i − 0.704676i
\(357\) 0 0
\(358\) 82.5145 0.230487
\(359\) 114.425i 0.318732i 0.987220 + 0.159366i \(0.0509450\pi\)
−0.987220 + 0.159366i \(0.949055\pi\)
\(360\) 0 0
\(361\) −140.048 −0.387946
\(362\) − 3.85280i − 0.0106431i
\(363\) 0 0
\(364\) −73.0948 −0.200810
\(365\) 376.780i 1.03227i
\(366\) 0 0
\(367\) 39.9818 0.108942 0.0544711 0.998515i \(-0.482653\pi\)
0.0544711 + 0.998515i \(0.482653\pi\)
\(368\) 332.791i 0.904323i
\(369\) 0 0
\(370\) −45.2615 −0.122328
\(371\) − 310.384i − 0.836614i
\(372\) 0 0
\(373\) −442.965 −1.18757 −0.593787 0.804622i \(-0.702368\pi\)
−0.593787 + 0.804622i \(0.702368\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −120.492 −0.320456
\(377\) − 22.0358i − 0.0584505i
\(378\) 0 0
\(379\) −225.413 −0.594758 −0.297379 0.954760i \(-0.596113\pi\)
−0.297379 + 0.954760i \(0.596113\pi\)
\(380\) 198.369i 0.522024i
\(381\) 0 0
\(382\) 17.6487 0.0462009
\(383\) − 182.924i − 0.477608i −0.971068 0.238804i \(-0.923245\pi\)
0.971068 0.238804i \(-0.0767554\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.99520i 0.00775958i
\(387\) 0 0
\(388\) 65.4113 0.168586
\(389\) − 619.589i − 1.59277i −0.604787 0.796387i \(-0.706742\pi\)
0.604787 0.796387i \(-0.293258\pi\)
\(390\) 0 0
\(391\) 668.892 1.71072
\(392\) − 93.3070i − 0.238028i
\(393\) 0 0
\(394\) 44.6169 0.113241
\(395\) − 493.745i − 1.24999i
\(396\) 0 0
\(397\) −45.3649 −0.114269 −0.0571347 0.998366i \(-0.518196\pi\)
−0.0571347 + 0.998366i \(0.518196\pi\)
\(398\) 91.1398i 0.228995i
\(399\) 0 0
\(400\) 190.236 0.475590
\(401\) − 361.330i − 0.901073i −0.892758 0.450536i \(-0.851233\pi\)
0.892758 0.450536i \(-0.148767\pi\)
\(402\) 0 0
\(403\) 130.756 0.324456
\(404\) − 637.371i − 1.57765i
\(405\) 0 0
\(406\) −6.55222 −0.0161385
\(407\) 0 0
\(408\) 0 0
\(409\) 683.744 1.67175 0.835873 0.548922i \(-0.184962\pi\)
0.835873 + 0.548922i \(0.184962\pi\)
\(410\) − 83.8991i − 0.204632i
\(411\) 0 0
\(412\) 563.058 1.36665
\(413\) − 368.238i − 0.891618i
\(414\) 0 0
\(415\) 151.613 0.365332
\(416\) 77.9456i 0.187369i
\(417\) 0 0
\(418\) 0 0
\(419\) 413.509i 0.986895i 0.869775 + 0.493448i \(0.164264\pi\)
−0.869775 + 0.493448i \(0.835736\pi\)
\(420\) 0 0
\(421\) −328.460 −0.780189 −0.390095 0.920775i \(-0.627558\pi\)
−0.390095 + 0.920775i \(0.627558\pi\)
\(422\) 53.2211i 0.126116i
\(423\) 0 0
\(424\) −219.475 −0.517629
\(425\) − 382.364i − 0.899680i
\(426\) 0 0
\(427\) 160.952 0.376936
\(428\) 733.597i 1.71401i
\(429\) 0 0
\(430\) −33.0302 −0.0768145
\(431\) 542.164i 1.25792i 0.777438 + 0.628960i \(0.216519\pi\)
−0.777438 + 0.628960i \(0.783481\pi\)
\(432\) 0 0
\(433\) −716.903 −1.65567 −0.827833 0.560975i \(-0.810426\pi\)
−0.827833 + 0.560975i \(0.810426\pi\)
\(434\) − 38.8795i − 0.0895840i
\(435\) 0 0
\(436\) −222.858 −0.511143
\(437\) 341.345i 0.781110i
\(438\) 0 0
\(439\) 196.790 0.448269 0.224135 0.974558i \(-0.428044\pi\)
0.224135 + 0.974558i \(0.428044\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 49.3739 0.111706
\(443\) − 524.354i − 1.18364i −0.806069 0.591821i \(-0.798409\pi\)
0.806069 0.591821i \(-0.201591\pi\)
\(444\) 0 0
\(445\) 223.190 0.501549
\(446\) − 75.2954i − 0.168824i
\(447\) 0 0
\(448\) −206.846 −0.461710
\(449\) − 313.024i − 0.697159i −0.937279 0.348580i \(-0.886664\pi\)
0.937279 0.348580i \(-0.113336\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 115.325i 0.255145i
\(453\) 0 0
\(454\) 62.4013 0.137448
\(455\) − 65.0310i − 0.142925i
\(456\) 0 0
\(457\) 4.06821 0.00890200 0.00445100 0.999990i \(-0.498583\pi\)
0.00445100 + 0.999990i \(0.498583\pi\)
\(458\) − 50.7718i − 0.110855i
\(459\) 0 0
\(460\) −306.458 −0.666212
\(461\) − 99.7105i − 0.216292i −0.994135 0.108146i \(-0.965509\pi\)
0.994135 0.108146i \(-0.0344914\pi\)
\(462\) 0 0
\(463\) 326.079 0.704273 0.352137 0.935949i \(-0.385455\pi\)
0.352137 + 0.935949i \(0.385455\pi\)
\(464\) − 67.1429i − 0.144704i
\(465\) 0 0
\(466\) −94.1734 −0.202089
\(467\) 422.171i 0.904007i 0.892016 + 0.452003i \(0.149291\pi\)
−0.892016 + 0.452003i \(0.850709\pi\)
\(468\) 0 0
\(469\) 215.598 0.459698
\(470\) − 52.7348i − 0.112202i
\(471\) 0 0
\(472\) −260.384 −0.551661
\(473\) 0 0
\(474\) 0 0
\(475\) 195.126 0.410791
\(476\) 447.653i 0.940447i
\(477\) 0 0
\(478\) 63.0807 0.131968
\(479\) − 467.797i − 0.976611i −0.872673 0.488305i \(-0.837615\pi\)
0.872673 0.488305i \(-0.162385\pi\)
\(480\) 0 0
\(481\) 175.297 0.364443
\(482\) − 5.20633i − 0.0108015i
\(483\) 0 0
\(484\) 0 0
\(485\) 58.1952i 0.119990i
\(486\) 0 0
\(487\) −414.175 −0.850463 −0.425231 0.905085i \(-0.639807\pi\)
−0.425231 + 0.905085i \(0.639807\pi\)
\(488\) − 113.810i − 0.233217i
\(489\) 0 0
\(490\) 40.8371 0.0833410
\(491\) 176.284i 0.359031i 0.983755 + 0.179516i \(0.0574530\pi\)
−0.983755 + 0.179516i \(0.942547\pi\)
\(492\) 0 0
\(493\) −134.954 −0.273740
\(494\) 25.1962i 0.0510044i
\(495\) 0 0
\(496\) 398.411 0.803249
\(497\) 361.110i 0.726580i
\(498\) 0 0
\(499\) 605.317 1.21306 0.606530 0.795061i \(-0.292561\pi\)
0.606530 + 0.795061i \(0.292561\pi\)
\(500\) 508.813i 1.01763i
\(501\) 0 0
\(502\) 77.3941 0.154171
\(503\) 180.872i 0.359587i 0.983704 + 0.179793i \(0.0575429\pi\)
−0.983704 + 0.179793i \(0.942457\pi\)
\(504\) 0 0
\(505\) 567.057 1.12288
\(506\) 0 0
\(507\) 0 0
\(508\) −48.7698 −0.0960035
\(509\) − 23.8193i − 0.0467962i −0.999726 0.0233981i \(-0.992551\pi\)
0.999726 0.0233981i \(-0.00744853\pi\)
\(510\) 0 0
\(511\) −433.903 −0.849126
\(512\) 400.150i 0.781543i
\(513\) 0 0
\(514\) 11.9089 0.0231691
\(515\) 500.942i 0.972704i
\(516\) 0 0
\(517\) 0 0
\(518\) − 52.1235i − 0.100624i
\(519\) 0 0
\(520\) −45.9839 −0.0884305
\(521\) 198.640i 0.381267i 0.981661 + 0.190633i \(0.0610542\pi\)
−0.981661 + 0.190633i \(0.938946\pi\)
\(522\) 0 0
\(523\) −1000.04 −1.91212 −0.956059 0.293173i \(-0.905289\pi\)
−0.956059 + 0.293173i \(0.905289\pi\)
\(524\) 78.8529i 0.150483i
\(525\) 0 0
\(526\) 119.520 0.227225
\(527\) − 800.786i − 1.51952i
\(528\) 0 0
\(529\) 1.66120 0.00314027
\(530\) − 96.0561i − 0.181238i
\(531\) 0 0
\(532\) −228.444 −0.429405
\(533\) 324.940i 0.609643i
\(534\) 0 0
\(535\) −652.668 −1.21994
\(536\) − 152.451i − 0.284424i
\(537\) 0 0
\(538\) 120.375 0.223746
\(539\) 0 0
\(540\) 0 0
\(541\) −491.351 −0.908227 −0.454113 0.890944i \(-0.650044\pi\)
−0.454113 + 0.890944i \(0.650044\pi\)
\(542\) 155.566i 0.287023i
\(543\) 0 0
\(544\) 477.361 0.877502
\(545\) − 198.273i − 0.363803i
\(546\) 0 0
\(547\) −195.142 −0.356750 −0.178375 0.983963i \(-0.557084\pi\)
−0.178375 + 0.983963i \(0.557084\pi\)
\(548\) − 139.106i − 0.253843i
\(549\) 0 0
\(550\) 0 0
\(551\) − 68.8688i − 0.124989i
\(552\) 0 0
\(553\) 568.601 1.02821
\(554\) 70.9687i 0.128102i
\(555\) 0 0
\(556\) −391.524 −0.704179
\(557\) − 615.014i − 1.10416i −0.833793 0.552078i \(-0.813835\pi\)
0.833793 0.552078i \(-0.186165\pi\)
\(558\) 0 0
\(559\) 127.925 0.228847
\(560\) − 198.149i − 0.353837i
\(561\) 0 0
\(562\) −89.2561 −0.158819
\(563\) 328.204i 0.582955i 0.956578 + 0.291477i \(0.0941468\pi\)
−0.956578 + 0.291477i \(0.905853\pi\)
\(564\) 0 0
\(565\) −102.603 −0.181598
\(566\) 15.3539i 0.0271270i
\(567\) 0 0
\(568\) 255.344 0.449549
\(569\) 830.739i 1.46000i 0.683448 + 0.729999i \(0.260480\pi\)
−0.683448 + 0.729999i \(0.739520\pi\)
\(570\) 0 0
\(571\) −1048.66 −1.83652 −0.918262 0.395973i \(-0.870407\pi\)
−0.918262 + 0.395973i \(0.870407\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 96.6189 0.168326
\(575\) 301.447i 0.524256i
\(576\) 0 0
\(577\) −518.202 −0.898096 −0.449048 0.893508i \(-0.648237\pi\)
−0.449048 + 0.893508i \(0.648237\pi\)
\(578\) − 199.382i − 0.344951i
\(579\) 0 0
\(580\) 61.8300 0.106603
\(581\) 174.598i 0.300513i
\(582\) 0 0
\(583\) 0 0
\(584\) 306.816i 0.525370i
\(585\) 0 0
\(586\) 2.67734 0.00456883
\(587\) 1016.52i 1.73173i 0.500282 + 0.865863i \(0.333230\pi\)
−0.500282 + 0.865863i \(0.666770\pi\)
\(588\) 0 0
\(589\) 408.652 0.693807
\(590\) − 113.961i − 0.193153i
\(591\) 0 0
\(592\) 534.127 0.902242
\(593\) 1073.37i 1.81007i 0.425341 + 0.905033i \(0.360154\pi\)
−0.425341 + 0.905033i \(0.639846\pi\)
\(594\) 0 0
\(595\) −398.268 −0.669358
\(596\) − 814.532i − 1.36666i
\(597\) 0 0
\(598\) −38.9252 −0.0650923
\(599\) − 767.105i − 1.28064i −0.768107 0.640322i \(-0.778801\pi\)
0.768107 0.640322i \(-0.221199\pi\)
\(600\) 0 0
\(601\) 752.890 1.25273 0.626365 0.779530i \(-0.284542\pi\)
0.626365 + 0.779530i \(0.284542\pi\)
\(602\) − 38.0379i − 0.0631858i
\(603\) 0 0
\(604\) 821.376 1.35989
\(605\) 0 0
\(606\) 0 0
\(607\) −503.535 −0.829548 −0.414774 0.909925i \(-0.636139\pi\)
−0.414774 + 0.909925i \(0.636139\pi\)
\(608\) 243.604i 0.400664i
\(609\) 0 0
\(610\) 49.8105 0.0816566
\(611\) 204.241i 0.334273i
\(612\) 0 0
\(613\) 748.338 1.22078 0.610390 0.792101i \(-0.291013\pi\)
0.610390 + 0.792101i \(0.291013\pi\)
\(614\) − 128.400i − 0.209120i
\(615\) 0 0
\(616\) 0 0
\(617\) − 634.767i − 1.02880i −0.857552 0.514398i \(-0.828016\pi\)
0.857552 0.514398i \(-0.171984\pi\)
\(618\) 0 0
\(619\) −684.329 −1.10554 −0.552769 0.833334i \(-0.686429\pi\)
−0.552769 + 0.833334i \(0.686429\pi\)
\(620\) 366.886i 0.591751i
\(621\) 0 0
\(622\) 148.125 0.238143
\(623\) 257.027i 0.412563i
\(624\) 0 0
\(625\) −124.506 −0.199210
\(626\) − 22.5549i − 0.0360301i
\(627\) 0 0
\(628\) 461.556 0.734963
\(629\) − 1073.57i − 1.70678i
\(630\) 0 0
\(631\) −266.046 −0.421627 −0.210813 0.977526i \(-0.567611\pi\)
−0.210813 + 0.977526i \(0.567611\pi\)
\(632\) − 402.061i − 0.636173i
\(633\) 0 0
\(634\) 150.573 0.237497
\(635\) − 43.3895i − 0.0683300i
\(636\) 0 0
\(637\) −158.161 −0.248291
\(638\) 0 0
\(639\) 0 0
\(640\) −289.893 −0.452957
\(641\) − 450.083i − 0.702158i −0.936346 0.351079i \(-0.885815\pi\)
0.936346 0.351079i \(-0.114185\pi\)
\(642\) 0 0
\(643\) 142.988 0.222376 0.111188 0.993799i \(-0.464534\pi\)
0.111188 + 0.993799i \(0.464534\pi\)
\(644\) − 352.919i − 0.548011i
\(645\) 0 0
\(646\) 154.309 0.238868
\(647\) 796.539i 1.23113i 0.788088 + 0.615563i \(0.211071\pi\)
−0.788088 + 0.615563i \(0.788929\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 22.2511i 0.0342325i
\(651\) 0 0
\(652\) 1031.25 1.58167
\(653\) 54.6582i 0.0837032i 0.999124 + 0.0418516i \(0.0133257\pi\)
−0.999124 + 0.0418516i \(0.986674\pi\)
\(654\) 0 0
\(655\) −70.1539 −0.107105
\(656\) 990.087i 1.50928i
\(657\) 0 0
\(658\) 60.7298 0.0922946
\(659\) 333.527i 0.506111i 0.967452 + 0.253056i \(0.0814356\pi\)
−0.967452 + 0.253056i \(0.918564\pi\)
\(660\) 0 0
\(661\) −489.397 −0.740389 −0.370195 0.928954i \(-0.620709\pi\)
−0.370195 + 0.928954i \(0.620709\pi\)
\(662\) − 83.0671i − 0.125479i
\(663\) 0 0
\(664\) 123.460 0.185933
\(665\) − 203.242i − 0.305627i
\(666\) 0 0
\(667\) 106.394 0.159512
\(668\) 437.994i 0.655680i
\(669\) 0 0
\(670\) 66.7223 0.0995856
\(671\) 0 0
\(672\) 0 0
\(673\) −699.341 −1.03914 −0.519570 0.854428i \(-0.673908\pi\)
−0.519570 + 0.854428i \(0.673908\pi\)
\(674\) − 221.768i − 0.329032i
\(675\) 0 0
\(676\) −566.923 −0.838644
\(677\) 389.391i 0.575171i 0.957755 + 0.287585i \(0.0928525\pi\)
−0.957755 + 0.287585i \(0.907148\pi\)
\(678\) 0 0
\(679\) −67.0180 −0.0987011
\(680\) 281.618i 0.414144i
\(681\) 0 0
\(682\) 0 0
\(683\) 332.412i 0.486693i 0.969939 + 0.243347i \(0.0782453\pi\)
−0.969939 + 0.243347i \(0.921755\pi\)
\(684\) 0 0
\(685\) 123.760 0.180672
\(686\) 116.325i 0.169570i
\(687\) 0 0
\(688\) 389.787 0.566551
\(689\) 372.024i 0.539947i
\(690\) 0 0
\(691\) 873.853 1.26462 0.632310 0.774715i \(-0.282107\pi\)
0.632310 + 0.774715i \(0.282107\pi\)
\(692\) 455.521i 0.658267i
\(693\) 0 0
\(694\) −231.016 −0.332876
\(695\) − 348.331i − 0.501196i
\(696\) 0 0
\(697\) 1990.02 2.85512
\(698\) − 108.388i − 0.155284i
\(699\) 0 0
\(700\) −201.742 −0.288203
\(701\) 630.096i 0.898853i 0.893317 + 0.449426i \(0.148372\pi\)
−0.893317 + 0.449426i \(0.851628\pi\)
\(702\) 0 0
\(703\) 547.856 0.779312
\(704\) 0 0
\(705\) 0 0
\(706\) 89.3670 0.126582
\(707\) 653.027i 0.923660i
\(708\) 0 0
\(709\) −1007.61 −1.42117 −0.710587 0.703609i \(-0.751571\pi\)
−0.710587 + 0.703609i \(0.751571\pi\)
\(710\) 111.755i 0.157401i
\(711\) 0 0
\(712\) 181.745 0.255260
\(713\) 631.321i 0.885443i
\(714\) 0 0
\(715\) 0 0
\(716\) − 896.697i − 1.25237i
\(717\) 0 0
\(718\) −40.7803 −0.0567971
\(719\) 170.989i 0.237815i 0.992905 + 0.118907i \(0.0379391\pi\)
−0.992905 + 0.118907i \(0.962061\pi\)
\(720\) 0 0
\(721\) −576.889 −0.800124
\(722\) − 49.9124i − 0.0691308i
\(723\) 0 0
\(724\) −41.8689 −0.0578300
\(725\) − 60.8191i − 0.0838884i
\(726\) 0 0
\(727\) 408.204 0.561490 0.280745 0.959782i \(-0.409418\pi\)
0.280745 + 0.959782i \(0.409418\pi\)
\(728\) − 52.9554i − 0.0727409i
\(729\) 0 0
\(730\) −134.282 −0.183948
\(731\) − 783.451i − 1.07175i
\(732\) 0 0
\(733\) −1178.02 −1.60713 −0.803563 0.595220i \(-0.797065\pi\)
−0.803563 + 0.595220i \(0.797065\pi\)
\(734\) 14.2493i 0.0194132i
\(735\) 0 0
\(736\) −376.340 −0.511332
\(737\) 0 0
\(738\) 0 0
\(739\) −938.068 −1.26938 −0.634688 0.772769i \(-0.718871\pi\)
−0.634688 + 0.772769i \(0.718871\pi\)
\(740\) 491.863i 0.664679i
\(741\) 0 0
\(742\) 110.619 0.149082
\(743\) − 1050.56i − 1.41394i −0.707242 0.706971i \(-0.750061\pi\)
0.707242 0.706971i \(-0.249939\pi\)
\(744\) 0 0
\(745\) 724.673 0.972716
\(746\) − 157.870i − 0.211622i
\(747\) 0 0
\(748\) 0 0
\(749\) − 751.617i − 1.00349i
\(750\) 0 0
\(751\) 1074.61 1.43091 0.715456 0.698658i \(-0.246219\pi\)
0.715456 + 0.698658i \(0.246219\pi\)
\(752\) 622.320i 0.827553i
\(753\) 0 0
\(754\) 7.85344 0.0104157
\(755\) 730.763i 0.967898i
\(756\) 0 0
\(757\) −518.488 −0.684924 −0.342462 0.939532i \(-0.611261\pi\)
−0.342462 + 0.939532i \(0.611261\pi\)
\(758\) − 80.3359i − 0.105984i
\(759\) 0 0
\(760\) −143.714 −0.189097
\(761\) − 1304.01i − 1.71355i −0.515690 0.856775i \(-0.672464\pi\)
0.515690 0.856775i \(-0.327536\pi\)
\(762\) 0 0
\(763\) 228.333 0.299256
\(764\) − 191.791i − 0.251036i
\(765\) 0 0
\(766\) 65.1930 0.0851083
\(767\) 441.367i 0.575446i
\(768\) 0 0
\(769\) −141.048 −0.183418 −0.0917088 0.995786i \(-0.529233\pi\)
−0.0917088 + 0.995786i \(0.529233\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 32.5492 0.0421622
\(773\) 234.469i 0.303323i 0.988432 + 0.151661i \(0.0484624\pi\)
−0.988432 + 0.151661i \(0.951538\pi\)
\(774\) 0 0
\(775\) 360.887 0.465661
\(776\) 47.3889i 0.0610682i
\(777\) 0 0
\(778\) 220.818 0.283828
\(779\) 1015.54i 1.30364i
\(780\) 0 0
\(781\) 0 0
\(782\) 238.389i 0.304845i
\(783\) 0 0
\(784\) −481.915 −0.614688
\(785\) 410.638i 0.523106i
\(786\) 0 0
\(787\) 1454.04 1.84757 0.923784 0.382915i \(-0.125080\pi\)
0.923784 + 0.382915i \(0.125080\pi\)
\(788\) − 484.858i − 0.615302i
\(789\) 0 0
\(790\) 175.968 0.222744
\(791\) − 118.158i − 0.149378i
\(792\) 0 0
\(793\) −192.915 −0.243273
\(794\) − 16.1678i − 0.0203624i
\(795\) 0 0
\(796\) 990.429 1.24426
\(797\) 325.332i 0.408195i 0.978951 + 0.204098i \(0.0654260\pi\)
−0.978951 + 0.204098i \(0.934574\pi\)
\(798\) 0 0
\(799\) 1250.83 1.56549
\(800\) 215.130i 0.268913i
\(801\) 0 0
\(802\) 128.776 0.160568
\(803\) 0 0
\(804\) 0 0
\(805\) 313.985 0.390044
\(806\) 46.6006i 0.0578171i
\(807\) 0 0
\(808\) 461.760 0.571485
\(809\) − 894.483i − 1.10567i −0.833292 0.552833i \(-0.813547\pi\)
0.833292 0.552833i \(-0.186453\pi\)
\(810\) 0 0
\(811\) −92.4668 −0.114016 −0.0570079 0.998374i \(-0.518156\pi\)
−0.0570079 + 0.998374i \(0.518156\pi\)
\(812\) 71.2039i 0.0876896i
\(813\) 0 0
\(814\) 0 0
\(815\) 917.485i 1.12575i
\(816\) 0 0
\(817\) 399.806 0.489359
\(818\) 243.682i 0.297900i
\(819\) 0 0
\(820\) −911.743 −1.11188
\(821\) − 616.078i − 0.750399i −0.926944 0.375200i \(-0.877574\pi\)
0.926944 0.375200i \(-0.122426\pi\)
\(822\) 0 0
\(823\) 1584.99 1.92587 0.962937 0.269727i \(-0.0869336\pi\)
0.962937 + 0.269727i \(0.0869336\pi\)
\(824\) 407.922i 0.495051i
\(825\) 0 0
\(826\) 131.238 0.158884
\(827\) − 776.862i − 0.939374i −0.882833 0.469687i \(-0.844367\pi\)
0.882833 0.469687i \(-0.155633\pi\)
\(828\) 0 0
\(829\) −1047.54 −1.26361 −0.631807 0.775126i \(-0.717687\pi\)
−0.631807 + 0.775126i \(0.717687\pi\)
\(830\) 54.0338i 0.0651010i
\(831\) 0 0
\(832\) 247.924 0.297986
\(833\) 968.624i 1.16281i
\(834\) 0 0
\(835\) −389.675 −0.466677
\(836\) 0 0
\(837\) 0 0
\(838\) −147.372 −0.175862
\(839\) − 197.206i − 0.235049i −0.993070 0.117524i \(-0.962504\pi\)
0.993070 0.117524i \(-0.0374958\pi\)
\(840\) 0 0
\(841\) 819.534 0.974476
\(842\) − 117.061i − 0.139027i
\(843\) 0 0
\(844\) 578.362 0.685263
\(845\) − 504.381i − 0.596901i
\(846\) 0 0
\(847\) 0 0
\(848\) 1133.55i 1.33673i
\(849\) 0 0
\(850\) 136.272 0.160320
\(851\) 846.375i 0.994566i
\(852\) 0 0
\(853\) −1389.41 −1.62885 −0.814423 0.580271i \(-0.802947\pi\)
−0.814423 + 0.580271i \(0.802947\pi\)
\(854\) 57.3622i 0.0671688i
\(855\) 0 0
\(856\) −531.474 −0.620881
\(857\) 218.943i 0.255476i 0.991808 + 0.127738i \(0.0407717\pi\)
−0.991808 + 0.127738i \(0.959228\pi\)
\(858\) 0 0
\(859\) 679.659 0.791221 0.395611 0.918418i \(-0.370533\pi\)
0.395611 + 0.918418i \(0.370533\pi\)
\(860\) 358.944i 0.417377i
\(861\) 0 0
\(862\) −193.224 −0.224158
\(863\) 719.743i 0.834002i 0.908906 + 0.417001i \(0.136919\pi\)
−0.908906 + 0.417001i \(0.863081\pi\)
\(864\) 0 0
\(865\) −405.268 −0.468518
\(866\) − 255.500i − 0.295035i
\(867\) 0 0
\(868\) −422.509 −0.486761
\(869\) 0 0
\(870\) 0 0
\(871\) −258.414 −0.296687
\(872\) − 161.456i − 0.185155i
\(873\) 0 0
\(874\) −121.653 −0.139191
\(875\) − 521.312i − 0.595785i
\(876\) 0 0
\(877\) 920.439 1.04953 0.524766 0.851247i \(-0.324153\pi\)
0.524766 + 0.851247i \(0.324153\pi\)
\(878\) 70.1349i 0.0798802i
\(879\) 0 0
\(880\) 0 0
\(881\) − 309.389i − 0.351180i −0.984463 0.175590i \(-0.943817\pi\)
0.984463 0.175590i \(-0.0561832\pi\)
\(882\) 0 0
\(883\) −856.028 −0.969454 −0.484727 0.874665i \(-0.661081\pi\)
−0.484727 + 0.874665i \(0.661081\pi\)
\(884\) − 536.553i − 0.606960i
\(885\) 0 0
\(886\) 186.877 0.210922
\(887\) 508.096i 0.572826i 0.958106 + 0.286413i \(0.0924629\pi\)
−0.958106 + 0.286413i \(0.907537\pi\)
\(888\) 0 0
\(889\) 49.9677 0.0562067
\(890\) 79.5434i 0.0893746i
\(891\) 0 0
\(892\) −818.246 −0.917316
\(893\) 638.316i 0.714799i
\(894\) 0 0
\(895\) 797.774 0.891368
\(896\) − 333.843i − 0.372592i
\(897\) 0 0
\(898\) 111.560 0.124232
\(899\) − 127.373i − 0.141684i
\(900\) 0 0
\(901\) 2278.38 2.52872
\(902\) 0 0
\(903\) 0 0
\(904\) −83.5505 −0.0924231
\(905\) − 37.2500i − 0.0411602i
\(906\) 0 0
\(907\) −509.667 −0.561927 −0.280963 0.959719i \(-0.590654\pi\)
−0.280963 + 0.959719i \(0.590654\pi\)
\(908\) − 678.124i − 0.746832i
\(909\) 0 0
\(910\) 23.1767 0.0254689
\(911\) 451.807i 0.495946i 0.968767 + 0.247973i \(0.0797645\pi\)
−0.968767 + 0.247973i \(0.920236\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 1.44989i 0.00158631i
\(915\) 0 0
\(916\) −551.744 −0.602341
\(917\) − 80.7898i − 0.0881023i
\(918\) 0 0
\(919\) −1645.29 −1.79030 −0.895152 0.445762i \(-0.852933\pi\)
−0.895152 + 0.445762i \(0.852933\pi\)
\(920\) − 222.021i − 0.241327i
\(921\) 0 0
\(922\) 35.5362 0.0385426
\(923\) − 432.824i − 0.468932i
\(924\) 0 0
\(925\) 483.820 0.523049
\(926\) 116.212i 0.125499i
\(927\) 0 0
\(928\) 75.9293 0.0818204
\(929\) 1005.87i 1.08274i 0.840783 + 0.541372i \(0.182095\pi\)
−0.840783 + 0.541372i \(0.817905\pi\)
\(930\) 0 0
\(931\) −494.303 −0.530937
\(932\) 1023.40i 1.09806i
\(933\) 0 0
\(934\) −150.459 −0.161091
\(935\) 0 0
\(936\) 0 0
\(937\) 209.683 0.223781 0.111891 0.993721i \(-0.464309\pi\)
0.111891 + 0.993721i \(0.464309\pi\)
\(938\) 76.8380i 0.0819168i
\(939\) 0 0
\(940\) −573.077 −0.609656
\(941\) 601.375i 0.639081i 0.947573 + 0.319541i \(0.103529\pi\)
−0.947573 + 0.319541i \(0.896471\pi\)
\(942\) 0 0
\(943\) −1568.89 −1.66372
\(944\) 1344.84i 1.42462i
\(945\) 0 0
\(946\) 0 0
\(947\) − 18.4760i − 0.0195101i −0.999952 0.00975504i \(-0.996895\pi\)
0.999952 0.00975504i \(-0.00310517\pi\)
\(948\) 0 0
\(949\) 520.073 0.548022
\(950\) 69.5416i 0.0732017i
\(951\) 0 0
\(952\) −324.314 −0.340666
\(953\) − 1285.82i − 1.34923i −0.738169 0.674616i \(-0.764309\pi\)
0.738169 0.674616i \(-0.235691\pi\)
\(954\) 0 0
\(955\) 170.633 0.178673
\(956\) − 685.506i − 0.717057i
\(957\) 0 0
\(958\) 166.720 0.174029
\(959\) 142.523i 0.148616i
\(960\) 0 0
\(961\) −205.194 −0.213521
\(962\) 62.4747i 0.0649426i
\(963\) 0 0
\(964\) −56.5780 −0.0586908
\(965\) 28.9584i 0.0300088i
\(966\) 0 0
\(967\) −401.260 −0.414953 −0.207477 0.978240i \(-0.566525\pi\)
−0.207477 + 0.978240i \(0.566525\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −20.7404 −0.0213819
\(971\) 1310.59i 1.34973i 0.737942 + 0.674864i \(0.235798\pi\)
−0.737942 + 0.674864i \(0.764202\pi\)
\(972\) 0 0
\(973\) 401.141 0.412272
\(974\) − 147.610i − 0.151550i
\(975\) 0 0
\(976\) −587.810 −0.602264
\(977\) − 1220.43i − 1.24916i −0.780960 0.624581i \(-0.785270\pi\)
0.780960 0.624581i \(-0.214730\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) − 443.782i − 0.452839i
\(981\) 0 0
\(982\) −62.8266 −0.0639782
\(983\) − 89.1639i − 0.0907059i −0.998971 0.0453530i \(-0.985559\pi\)
0.998971 0.0453530i \(-0.0144412\pi\)
\(984\) 0 0
\(985\) 431.369 0.437938
\(986\) − 48.0967i − 0.0487796i
\(987\) 0 0
\(988\) 273.810 0.277136
\(989\) 617.655i 0.624525i
\(990\) 0 0
\(991\) 48.9921 0.0494370 0.0247185 0.999694i \(-0.492131\pi\)
0.0247185 + 0.999694i \(0.492131\pi\)
\(992\) 450.548i 0.454181i
\(993\) 0 0
\(994\) −128.698 −0.129474
\(995\) 881.166i 0.885594i
\(996\) 0 0
\(997\) 209.770 0.210401 0.105201 0.994451i \(-0.466451\pi\)
0.105201 + 0.994451i \(0.466451\pi\)
\(998\) 215.731i 0.216163i
\(999\) 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 1089.3.b.h.485.6 yes 8
3.2 odd 2 inner 1089.3.b.h.485.3 8
11.10 odd 2 inner 1089.3.b.h.485.4 yes 8
33.32 even 2 inner 1089.3.b.h.485.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1089.3.b.h.485.3 8 3.2 odd 2 inner
1089.3.b.h.485.4 yes 8 11.10 odd 2 inner
1089.3.b.h.485.5 yes 8 33.32 even 2 inner
1089.3.b.h.485.6 yes 8 1.1 even 1 trivial