Properties

Label 1143.3.b.a.890.19
Level $1143$
Weight $3$
Character 1143.890
Analytic conductor $31.144$
Analytic rank $0$
Dimension $84$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1143 = 3^{2} \cdot 127 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1143.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.1444942164\)
Analytic rank: \(0\)
Dimension: \(84\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 890.19
Character \(\chi\) \(=\) 1143.890
Dual form 1143.3.b.a.890.66

$q$-expansion

\(f(q)\) \(=\) \(q-2.47320i q^{2} -2.11671 q^{4} -7.47527i q^{5} -3.57682 q^{7} -4.65775i q^{8} +O(q^{10})\) \(q-2.47320i q^{2} -2.11671 q^{4} -7.47527i q^{5} -3.57682 q^{7} -4.65775i q^{8} -18.4878 q^{10} +1.28153i q^{11} -14.4150 q^{13} +8.84618i q^{14} -19.9864 q^{16} +10.3912i q^{17} -3.18314 q^{19} +15.8230i q^{20} +3.16948 q^{22} +24.2785i q^{23} -30.8796 q^{25} +35.6512i q^{26} +7.57108 q^{28} -17.6691i q^{29} +32.6017 q^{31} +30.7993i q^{32} +25.6995 q^{34} +26.7377i q^{35} -49.9752 q^{37} +7.87253i q^{38} -34.8180 q^{40} +10.5982i q^{41} +38.4537 q^{43} -2.71262i q^{44} +60.0455 q^{46} +10.6464i q^{47} -36.2064 q^{49} +76.3715i q^{50} +30.5124 q^{52} -63.3509i q^{53} +9.57977 q^{55} +16.6599i q^{56} -43.6992 q^{58} +1.85123i q^{59} +18.8709 q^{61} -80.6304i q^{62} -3.77284 q^{64} +107.756i q^{65} -40.7181 q^{67} -21.9951i q^{68} +66.1275 q^{70} -5.52989i q^{71} -27.0768 q^{73} +123.599i q^{74} +6.73777 q^{76} -4.58380i q^{77} +40.6548 q^{79} +149.404i q^{80} +26.2115 q^{82} -45.3514i q^{83} +77.6770 q^{85} -95.1037i q^{86} +5.96905 q^{88} -77.0121i q^{89} +51.5599 q^{91} -51.3905i q^{92} +26.3305 q^{94} +23.7948i q^{95} -112.360 q^{97} +89.5455i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84q - 160q^{4} + O(q^{10}) \) \( 84q - 160q^{4} - 48q^{10} + 16q^{13} + 360q^{16} + 64q^{19} - 8q^{22} - 388q^{25} - 120q^{28} - 160q^{31} + 192q^{34} - 152q^{37} + 208q^{40} - 24q^{43} + 56q^{46} + 564q^{49} - 80q^{52} + 136q^{55} - 136q^{58} + 168q^{61} - 736q^{64} + 168q^{67} - 608q^{70} + 80q^{73} - 32q^{76} - 168q^{79} + 528q^{82} + 288q^{85} - 392q^{88} + 176q^{91} + 176q^{94} - 120q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1143\mathbb{Z}\right)^\times\).

\(n\) \(128\) \(892\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.47320i − 1.23660i −0.785943 0.618300i \(-0.787822\pi\)
0.785943 0.618300i \(-0.212178\pi\)
\(3\) 0 0
\(4\) −2.11671 −0.529177
\(5\) − 7.47527i − 1.49505i −0.664232 0.747527i \(-0.731241\pi\)
0.664232 0.747527i \(-0.268759\pi\)
\(6\) 0 0
\(7\) −3.57682 −0.510974 −0.255487 0.966813i \(-0.582236\pi\)
−0.255487 + 0.966813i \(0.582236\pi\)
\(8\) − 4.65775i − 0.582219i
\(9\) 0 0
\(10\) −18.4878 −1.84878
\(11\) 1.28153i 0.116503i 0.998302 + 0.0582513i \(0.0185525\pi\)
−0.998302 + 0.0582513i \(0.981448\pi\)
\(12\) 0 0
\(13\) −14.4150 −1.10885 −0.554424 0.832235i \(-0.687061\pi\)
−0.554424 + 0.832235i \(0.687061\pi\)
\(14\) 8.84618i 0.631870i
\(15\) 0 0
\(16\) −19.9864 −1.24915
\(17\) 10.3912i 0.611247i 0.952152 + 0.305623i \(0.0988648\pi\)
−0.952152 + 0.305623i \(0.901135\pi\)
\(18\) 0 0
\(19\) −3.18314 −0.167534 −0.0837668 0.996485i \(-0.526695\pi\)
−0.0837668 + 0.996485i \(0.526695\pi\)
\(20\) 15.8230i 0.791148i
\(21\) 0 0
\(22\) 3.16948 0.144067
\(23\) 24.2785i 1.05559i 0.849373 + 0.527794i \(0.176981\pi\)
−0.849373 + 0.527794i \(0.823019\pi\)
\(24\) 0 0
\(25\) −30.8796 −1.23519
\(26\) 35.6512i 1.37120i
\(27\) 0 0
\(28\) 7.57108 0.270396
\(29\) − 17.6691i − 0.609280i −0.952468 0.304640i \(-0.901464\pi\)
0.952468 0.304640i \(-0.0985361\pi\)
\(30\) 0 0
\(31\) 32.6017 1.05167 0.525833 0.850588i \(-0.323754\pi\)
0.525833 + 0.850588i \(0.323754\pi\)
\(32\) 30.7993i 0.962477i
\(33\) 0 0
\(34\) 25.6995 0.755867
\(35\) 26.7377i 0.763933i
\(36\) 0 0
\(37\) −49.9752 −1.35068 −0.675340 0.737506i \(-0.736003\pi\)
−0.675340 + 0.737506i \(0.736003\pi\)
\(38\) 7.87253i 0.207172i
\(39\) 0 0
\(40\) −34.8180 −0.870449
\(41\) 10.5982i 0.258493i 0.991613 + 0.129246i \(0.0412558\pi\)
−0.991613 + 0.129246i \(0.958744\pi\)
\(42\) 0 0
\(43\) 38.4537 0.894273 0.447136 0.894466i \(-0.352444\pi\)
0.447136 + 0.894466i \(0.352444\pi\)
\(44\) − 2.71262i − 0.0616505i
\(45\) 0 0
\(46\) 60.0455 1.30534
\(47\) 10.6464i 0.226518i 0.993565 + 0.113259i \(0.0361290\pi\)
−0.993565 + 0.113259i \(0.963871\pi\)
\(48\) 0 0
\(49\) −36.2064 −0.738906
\(50\) 76.3715i 1.52743i
\(51\) 0 0
\(52\) 30.5124 0.586777
\(53\) − 63.3509i − 1.19530i −0.801757 0.597650i \(-0.796101\pi\)
0.801757 0.597650i \(-0.203899\pi\)
\(54\) 0 0
\(55\) 9.57977 0.174178
\(56\) 16.6599i 0.297499i
\(57\) 0 0
\(58\) −43.6992 −0.753435
\(59\) 1.85123i 0.0313768i 0.999877 + 0.0156884i \(0.00499398\pi\)
−0.999877 + 0.0156884i \(0.995006\pi\)
\(60\) 0 0
\(61\) 18.8709 0.309358 0.154679 0.987965i \(-0.450566\pi\)
0.154679 + 0.987965i \(0.450566\pi\)
\(62\) − 80.6304i − 1.30049i
\(63\) 0 0
\(64\) −3.77284 −0.0589506
\(65\) 107.756i 1.65779i
\(66\) 0 0
\(67\) −40.7181 −0.607733 −0.303867 0.952715i \(-0.598278\pi\)
−0.303867 + 0.952715i \(0.598278\pi\)
\(68\) − 21.9951i − 0.323458i
\(69\) 0 0
\(70\) 66.1275 0.944679
\(71\) − 5.52989i − 0.0778857i −0.999241 0.0389429i \(-0.987601\pi\)
0.999241 0.0389429i \(-0.0123990\pi\)
\(72\) 0 0
\(73\) −27.0768 −0.370915 −0.185458 0.982652i \(-0.559377\pi\)
−0.185458 + 0.982652i \(0.559377\pi\)
\(74\) 123.599i 1.67025i
\(75\) 0 0
\(76\) 6.73777 0.0886549
\(77\) − 4.58380i − 0.0595298i
\(78\) 0 0
\(79\) 40.6548 0.514618 0.257309 0.966329i \(-0.417164\pi\)
0.257309 + 0.966329i \(0.417164\pi\)
\(80\) 149.404i 1.86754i
\(81\) 0 0
\(82\) 26.2115 0.319652
\(83\) − 45.3514i − 0.546403i −0.961957 0.273201i \(-0.911917\pi\)
0.961957 0.273201i \(-0.0880826\pi\)
\(84\) 0 0
\(85\) 77.6770 0.913847
\(86\) − 95.1037i − 1.10586i
\(87\) 0 0
\(88\) 5.96905 0.0678301
\(89\) − 77.0121i − 0.865305i −0.901561 0.432652i \(-0.857578\pi\)
0.901561 0.432652i \(-0.142422\pi\)
\(90\) 0 0
\(91\) 51.5599 0.566592
\(92\) − 51.3905i − 0.558593i
\(93\) 0 0
\(94\) 26.3305 0.280112
\(95\) 23.7948i 0.250472i
\(96\) 0 0
\(97\) −112.360 −1.15835 −0.579174 0.815204i \(-0.696625\pi\)
−0.579174 + 0.815204i \(0.696625\pi\)
\(98\) 89.5455i 0.913730i
\(99\) 0 0
\(100\) 65.3632 0.653632
\(101\) 85.3813i 0.845359i 0.906279 + 0.422680i \(0.138910\pi\)
−0.906279 + 0.422680i \(0.861090\pi\)
\(102\) 0 0
\(103\) 32.7075 0.317549 0.158774 0.987315i \(-0.449246\pi\)
0.158774 + 0.987315i \(0.449246\pi\)
\(104\) 67.1416i 0.645592i
\(105\) 0 0
\(106\) −156.679 −1.47811
\(107\) 200.409i 1.87298i 0.350694 + 0.936490i \(0.385946\pi\)
−0.350694 + 0.936490i \(0.614054\pi\)
\(108\) 0 0
\(109\) −38.9278 −0.357136 −0.178568 0.983928i \(-0.557146\pi\)
−0.178568 + 0.983928i \(0.557146\pi\)
\(110\) − 23.6927i − 0.215388i
\(111\) 0 0
\(112\) 71.4876 0.638282
\(113\) − 24.1549i − 0.213760i −0.994272 0.106880i \(-0.965914\pi\)
0.994272 0.106880i \(-0.0340860\pi\)
\(114\) 0 0
\(115\) 181.488 1.57816
\(116\) 37.4004i 0.322417i
\(117\) 0 0
\(118\) 4.57846 0.0388005
\(119\) − 37.1674i − 0.312331i
\(120\) 0 0
\(121\) 119.358 0.986427
\(122\) − 46.6714i − 0.382552i
\(123\) 0 0
\(124\) −69.0082 −0.556518
\(125\) 43.9519i 0.351615i
\(126\) 0 0
\(127\) 11.2694 0.0887357
\(128\) 132.528i 1.03538i
\(129\) 0 0
\(130\) 266.502 2.05002
\(131\) − 26.7909i − 0.204511i −0.994758 0.102255i \(-0.967394\pi\)
0.994758 0.102255i \(-0.0326059\pi\)
\(132\) 0 0
\(133\) 11.3855 0.0856053
\(134\) 100.704i 0.751522i
\(135\) 0 0
\(136\) 48.3996 0.355880
\(137\) 205.936i 1.50318i 0.659629 + 0.751592i \(0.270714\pi\)
−0.659629 + 0.751592i \(0.729286\pi\)
\(138\) 0 0
\(139\) −159.715 −1.14903 −0.574513 0.818495i \(-0.694809\pi\)
−0.574513 + 0.818495i \(0.694809\pi\)
\(140\) − 56.5959i − 0.404256i
\(141\) 0 0
\(142\) −13.6765 −0.0963134
\(143\) − 18.4733i − 0.129184i
\(144\) 0 0
\(145\) −132.081 −0.910906
\(146\) 66.9663i 0.458674i
\(147\) 0 0
\(148\) 105.783 0.714749
\(149\) 43.3666i 0.291051i 0.989355 + 0.145526i \(0.0464873\pi\)
−0.989355 + 0.145526i \(0.953513\pi\)
\(150\) 0 0
\(151\) −227.463 −1.50638 −0.753189 0.657804i \(-0.771486\pi\)
−0.753189 + 0.657804i \(0.771486\pi\)
\(152\) 14.8263i 0.0975412i
\(153\) 0 0
\(154\) −11.3366 −0.0736145
\(155\) − 243.706i − 1.57230i
\(156\) 0 0
\(157\) 59.8855 0.381437 0.190718 0.981645i \(-0.438918\pi\)
0.190718 + 0.981645i \(0.438918\pi\)
\(158\) − 100.547i − 0.636376i
\(159\) 0 0
\(160\) 230.233 1.43895
\(161\) − 86.8398i − 0.539377i
\(162\) 0 0
\(163\) 143.932 0.883017 0.441508 0.897257i \(-0.354444\pi\)
0.441508 + 0.897257i \(0.354444\pi\)
\(164\) − 22.4333i − 0.136788i
\(165\) 0 0
\(166\) −112.163 −0.675681
\(167\) − 18.5191i − 0.110893i −0.998462 0.0554463i \(-0.982342\pi\)
0.998462 0.0554463i \(-0.0176582\pi\)
\(168\) 0 0
\(169\) 38.7927 0.229543
\(170\) − 192.111i − 1.13006i
\(171\) 0 0
\(172\) −81.3954 −0.473229
\(173\) 8.77175i 0.0507038i 0.999679 + 0.0253519i \(0.00807062\pi\)
−0.999679 + 0.0253519i \(0.991929\pi\)
\(174\) 0 0
\(175\) 110.451 0.631148
\(176\) − 25.6131i − 0.145529i
\(177\) 0 0
\(178\) −190.466 −1.07004
\(179\) − 82.8360i − 0.462771i −0.972862 0.231385i \(-0.925674\pi\)
0.972862 0.231385i \(-0.0743258\pi\)
\(180\) 0 0
\(181\) 219.331 1.21178 0.605888 0.795550i \(-0.292818\pi\)
0.605888 + 0.795550i \(0.292818\pi\)
\(182\) − 127.518i − 0.700647i
\(183\) 0 0
\(184\) 113.083 0.614583
\(185\) 373.578i 2.01934i
\(186\) 0 0
\(187\) −13.3166 −0.0712119
\(188\) − 22.5352i − 0.119868i
\(189\) 0 0
\(190\) 58.8493 0.309733
\(191\) − 44.5096i − 0.233034i −0.993189 0.116517i \(-0.962827\pi\)
0.993189 0.116517i \(-0.0371730\pi\)
\(192\) 0 0
\(193\) −73.5363 −0.381017 −0.190508 0.981686i \(-0.561014\pi\)
−0.190508 + 0.981686i \(0.561014\pi\)
\(194\) 277.888i 1.43241i
\(195\) 0 0
\(196\) 76.6384 0.391012
\(197\) − 267.608i − 1.35842i −0.733945 0.679209i \(-0.762323\pi\)
0.733945 0.679209i \(-0.237677\pi\)
\(198\) 0 0
\(199\) −173.678 −0.872753 −0.436376 0.899764i \(-0.643738\pi\)
−0.436376 + 0.899764i \(0.643738\pi\)
\(200\) 143.830i 0.719149i
\(201\) 0 0
\(202\) 211.165 1.04537
\(203\) 63.1992i 0.311326i
\(204\) 0 0
\(205\) 79.2244 0.386461
\(206\) − 80.8921i − 0.392680i
\(207\) 0 0
\(208\) 288.104 1.38512
\(209\) − 4.07928i − 0.0195181i
\(210\) 0 0
\(211\) −384.984 −1.82457 −0.912284 0.409559i \(-0.865683\pi\)
−0.912284 + 0.409559i \(0.865683\pi\)
\(212\) 134.095i 0.632526i
\(213\) 0 0
\(214\) 495.651 2.31613
\(215\) − 287.452i − 1.33699i
\(216\) 0 0
\(217\) −116.610 −0.537374
\(218\) 96.2762i 0.441634i
\(219\) 0 0
\(220\) −20.2776 −0.0921709
\(221\) − 149.789i − 0.677779i
\(222\) 0 0
\(223\) −391.016 −1.75344 −0.876719 0.481004i \(-0.840272\pi\)
−0.876719 + 0.481004i \(0.840272\pi\)
\(224\) − 110.163i − 0.491801i
\(225\) 0 0
\(226\) −59.7398 −0.264335
\(227\) 118.447i 0.521792i 0.965367 + 0.260896i \(0.0840180\pi\)
−0.965367 + 0.260896i \(0.915982\pi\)
\(228\) 0 0
\(229\) −13.0828 −0.0571299 −0.0285650 0.999592i \(-0.509094\pi\)
−0.0285650 + 0.999592i \(0.509094\pi\)
\(230\) − 448.857i − 1.95155i
\(231\) 0 0
\(232\) −82.2984 −0.354734
\(233\) 31.3816i 0.134685i 0.997730 + 0.0673425i \(0.0214520\pi\)
−0.997730 + 0.0673425i \(0.978548\pi\)
\(234\) 0 0
\(235\) 79.5844 0.338657
\(236\) − 3.91852i − 0.0166039i
\(237\) 0 0
\(238\) −91.9223 −0.386228
\(239\) − 151.225i − 0.632741i −0.948636 0.316370i \(-0.897536\pi\)
0.948636 0.316370i \(-0.102464\pi\)
\(240\) 0 0
\(241\) −138.477 −0.574593 −0.287297 0.957842i \(-0.592757\pi\)
−0.287297 + 0.957842i \(0.592757\pi\)
\(242\) − 295.195i − 1.21981i
\(243\) 0 0
\(244\) −39.9441 −0.163705
\(245\) 270.652i 1.10470i
\(246\) 0 0
\(247\) 45.8850 0.185769
\(248\) − 151.850i − 0.612300i
\(249\) 0 0
\(250\) 108.702 0.434807
\(251\) − 243.313i − 0.969376i −0.874687 0.484688i \(-0.838933\pi\)
0.874687 0.484688i \(-0.161067\pi\)
\(252\) 0 0
\(253\) −31.1136 −0.122979
\(254\) − 27.8715i − 0.109730i
\(255\) 0 0
\(256\) 312.677 1.22139
\(257\) − 158.311i − 0.615996i −0.951387 0.307998i \(-0.900341\pi\)
0.951387 0.307998i \(-0.0996590\pi\)
\(258\) 0 0
\(259\) 178.752 0.690163
\(260\) − 228.088i − 0.877263i
\(261\) 0 0
\(262\) −66.2593 −0.252898
\(263\) − 85.3123i − 0.324381i −0.986759 0.162191i \(-0.948144\pi\)
0.986759 0.162191i \(-0.0518559\pi\)
\(264\) 0 0
\(265\) −473.565 −1.78704
\(266\) − 28.1586i − 0.105859i
\(267\) 0 0
\(268\) 86.1884 0.321598
\(269\) − 118.069i − 0.438920i −0.975622 0.219460i \(-0.929570\pi\)
0.975622 0.219460i \(-0.0704295\pi\)
\(270\) 0 0
\(271\) 334.758 1.23527 0.617634 0.786466i \(-0.288091\pi\)
0.617634 + 0.786466i \(0.288091\pi\)
\(272\) − 207.682i − 0.763538i
\(273\) 0 0
\(274\) 509.321 1.85884
\(275\) − 39.5732i − 0.143902i
\(276\) 0 0
\(277\) −30.6223 −0.110550 −0.0552750 0.998471i \(-0.517604\pi\)
−0.0552750 + 0.998471i \(0.517604\pi\)
\(278\) 395.006i 1.42089i
\(279\) 0 0
\(280\) 124.537 0.444777
\(281\) − 4.97564i − 0.0177069i −0.999961 0.00885345i \(-0.997182\pi\)
0.999961 0.00885345i \(-0.00281818\pi\)
\(282\) 0 0
\(283\) −230.905 −0.815918 −0.407959 0.913000i \(-0.633759\pi\)
−0.407959 + 0.913000i \(0.633759\pi\)
\(284\) 11.7052i 0.0412153i
\(285\) 0 0
\(286\) −45.6880 −0.159748
\(287\) − 37.9078i − 0.132083i
\(288\) 0 0
\(289\) 181.023 0.626377
\(290\) 326.663i 1.12643i
\(291\) 0 0
\(292\) 57.3137 0.196280
\(293\) 420.765i 1.43606i 0.696014 + 0.718028i \(0.254955\pi\)
−0.696014 + 0.718028i \(0.745045\pi\)
\(294\) 0 0
\(295\) 13.8384 0.0469100
\(296\) 232.772i 0.786392i
\(297\) 0 0
\(298\) 107.254 0.359913
\(299\) − 349.975i − 1.17049i
\(300\) 0 0
\(301\) −137.542 −0.456950
\(302\) 562.562i 1.86279i
\(303\) 0 0
\(304\) 63.6194 0.209274
\(305\) − 141.065i − 0.462507i
\(306\) 0 0
\(307\) 176.646 0.575393 0.287697 0.957722i \(-0.407111\pi\)
0.287697 + 0.957722i \(0.407111\pi\)
\(308\) 9.70256i 0.0315018i
\(309\) 0 0
\(310\) −602.734 −1.94430
\(311\) 10.4719i 0.0336717i 0.999858 + 0.0168359i \(0.00535927\pi\)
−0.999858 + 0.0168359i \(0.994641\pi\)
\(312\) 0 0
\(313\) −569.849 −1.82060 −0.910302 0.413946i \(-0.864150\pi\)
−0.910302 + 0.413946i \(0.864150\pi\)
\(314\) − 148.109i − 0.471684i
\(315\) 0 0
\(316\) −86.0543 −0.272324
\(317\) − 68.0365i − 0.214626i −0.994225 0.107313i \(-0.965775\pi\)
0.994225 0.107313i \(-0.0342247\pi\)
\(318\) 0 0
\(319\) 22.6435 0.0709827
\(320\) 28.2030i 0.0881343i
\(321\) 0 0
\(322\) −214.772 −0.666994
\(323\) − 33.0766i − 0.102404i
\(324\) 0 0
\(325\) 445.131 1.36963
\(326\) − 355.972i − 1.09194i
\(327\) 0 0
\(328\) 49.3638 0.150499
\(329\) − 38.0801i − 0.115745i
\(330\) 0 0
\(331\) 80.7152 0.243853 0.121926 0.992539i \(-0.461093\pi\)
0.121926 + 0.992539i \(0.461093\pi\)
\(332\) 95.9958i 0.289144i
\(333\) 0 0
\(334\) −45.8013 −0.137130
\(335\) 304.379i 0.908594i
\(336\) 0 0
\(337\) −652.457 −1.93607 −0.968037 0.250806i \(-0.919304\pi\)
−0.968037 + 0.250806i \(0.919304\pi\)
\(338\) − 95.9421i − 0.283852i
\(339\) 0 0
\(340\) −164.420 −0.483587
\(341\) 41.7800i 0.122522i
\(342\) 0 0
\(343\) 304.768 0.888535
\(344\) − 179.108i − 0.520663i
\(345\) 0 0
\(346\) 21.6943 0.0627002
\(347\) − 409.886i − 1.18123i −0.806955 0.590613i \(-0.798886\pi\)
0.806955 0.590613i \(-0.201114\pi\)
\(348\) 0 0
\(349\) −415.021 −1.18917 −0.594586 0.804032i \(-0.702684\pi\)
−0.594586 + 0.804032i \(0.702684\pi\)
\(350\) − 273.167i − 0.780476i
\(351\) 0 0
\(352\) −39.4702 −0.112131
\(353\) 145.012i 0.410799i 0.978678 + 0.205400i \(0.0658494\pi\)
−0.978678 + 0.205400i \(0.934151\pi\)
\(354\) 0 0
\(355\) −41.3374 −0.116443
\(356\) 163.012i 0.457899i
\(357\) 0 0
\(358\) −204.870 −0.572262
\(359\) − 188.574i − 0.525277i −0.964894 0.262638i \(-0.915407\pi\)
0.964894 0.262638i \(-0.0845926\pi\)
\(360\) 0 0
\(361\) −350.868 −0.971933
\(362\) − 542.450i − 1.49848i
\(363\) 0 0
\(364\) −109.137 −0.299828
\(365\) 202.406i 0.554538i
\(366\) 0 0
\(367\) −693.878 −1.89068 −0.945338 0.326093i \(-0.894268\pi\)
−0.945338 + 0.326093i \(0.894268\pi\)
\(368\) − 485.239i − 1.31859i
\(369\) 0 0
\(370\) 923.932 2.49711
\(371\) 226.595i 0.610767i
\(372\) 0 0
\(373\) −698.882 −1.87368 −0.936839 0.349760i \(-0.886263\pi\)
−0.936839 + 0.349760i \(0.886263\pi\)
\(374\) 32.9346i 0.0880605i
\(375\) 0 0
\(376\) 49.5881 0.131883
\(377\) 254.701i 0.675598i
\(378\) 0 0
\(379\) −117.384 −0.309721 −0.154861 0.987936i \(-0.549493\pi\)
−0.154861 + 0.987936i \(0.549493\pi\)
\(380\) − 50.3667i − 0.132544i
\(381\) 0 0
\(382\) −110.081 −0.288170
\(383\) 565.781i 1.47724i 0.674124 + 0.738618i \(0.264521\pi\)
−0.674124 + 0.738618i \(0.735479\pi\)
\(384\) 0 0
\(385\) −34.2651 −0.0890003
\(386\) 181.870i 0.471165i
\(387\) 0 0
\(388\) 237.833 0.612971
\(389\) − 293.432i − 0.754325i −0.926147 0.377163i \(-0.876900\pi\)
0.926147 0.377163i \(-0.123100\pi\)
\(390\) 0 0
\(391\) −252.283 −0.645224
\(392\) 168.640i 0.430205i
\(393\) 0 0
\(394\) −661.848 −1.67982
\(395\) − 303.905i − 0.769381i
\(396\) 0 0
\(397\) −156.374 −0.393890 −0.196945 0.980415i \(-0.563102\pi\)
−0.196945 + 0.980415i \(0.563102\pi\)
\(398\) 429.539i 1.07924i
\(399\) 0 0
\(400\) 617.172 1.54293
\(401\) 547.372i 1.36502i 0.730878 + 0.682508i \(0.239111\pi\)
−0.730878 + 0.682508i \(0.760889\pi\)
\(402\) 0 0
\(403\) −469.953 −1.16614
\(404\) − 180.727i − 0.447345i
\(405\) 0 0
\(406\) 156.304 0.384985
\(407\) − 64.0447i − 0.157358i
\(408\) 0 0
\(409\) −256.309 −0.626674 −0.313337 0.949642i \(-0.601447\pi\)
−0.313337 + 0.949642i \(0.601447\pi\)
\(410\) − 195.938i − 0.477897i
\(411\) 0 0
\(412\) −69.2323 −0.168039
\(413\) − 6.62152i − 0.0160327i
\(414\) 0 0
\(415\) −339.014 −0.816902
\(416\) − 443.972i − 1.06724i
\(417\) 0 0
\(418\) −10.0889 −0.0241361
\(419\) − 6.95844i − 0.0166072i −0.999966 0.00830362i \(-0.997357\pi\)
0.999966 0.00830362i \(-0.00264316\pi\)
\(420\) 0 0
\(421\) 164.509 0.390757 0.195378 0.980728i \(-0.437407\pi\)
0.195378 + 0.980728i \(0.437407\pi\)
\(422\) 952.141i 2.25626i
\(423\) 0 0
\(424\) −295.073 −0.695927
\(425\) − 320.876i − 0.755003i
\(426\) 0 0
\(427\) −67.4976 −0.158074
\(428\) − 424.207i − 0.991138i
\(429\) 0 0
\(430\) −710.926 −1.65332
\(431\) − 436.670i − 1.01316i −0.862194 0.506578i \(-0.830910\pi\)
0.862194 0.506578i \(-0.169090\pi\)
\(432\) 0 0
\(433\) 64.1886 0.148242 0.0741208 0.997249i \(-0.476385\pi\)
0.0741208 + 0.997249i \(0.476385\pi\)
\(434\) 288.400i 0.664516i
\(435\) 0 0
\(436\) 82.3989 0.188988
\(437\) − 77.2818i − 0.176846i
\(438\) 0 0
\(439\) −258.306 −0.588397 −0.294199 0.955744i \(-0.595053\pi\)
−0.294199 + 0.955744i \(0.595053\pi\)
\(440\) − 44.6202i − 0.101410i
\(441\) 0 0
\(442\) −370.459 −0.838141
\(443\) 276.205i 0.623487i 0.950166 + 0.311744i \(0.100913\pi\)
−0.950166 + 0.311744i \(0.899087\pi\)
\(444\) 0 0
\(445\) −575.686 −1.29368
\(446\) 967.061i 2.16830i
\(447\) 0 0
\(448\) 13.4948 0.0301222
\(449\) − 555.627i − 1.23748i −0.785597 0.618738i \(-0.787644\pi\)
0.785597 0.618738i \(-0.212356\pi\)
\(450\) 0 0
\(451\) −13.5819 −0.0301151
\(452\) 51.1288i 0.113117i
\(453\) 0 0
\(454\) 292.943 0.645248
\(455\) − 385.424i − 0.847086i
\(456\) 0 0
\(457\) 67.3557 0.147387 0.0736934 0.997281i \(-0.476521\pi\)
0.0736934 + 0.997281i \(0.476521\pi\)
\(458\) 32.3562i 0.0706468i
\(459\) 0 0
\(460\) −384.158 −0.835126
\(461\) 799.216i 1.73366i 0.498605 + 0.866829i \(0.333846\pi\)
−0.498605 + 0.866829i \(0.666154\pi\)
\(462\) 0 0
\(463\) −469.674 −1.01441 −0.507207 0.861824i \(-0.669322\pi\)
−0.507207 + 0.861824i \(0.669322\pi\)
\(464\) 353.142i 0.761081i
\(465\) 0 0
\(466\) 77.6130 0.166551
\(467\) − 584.567i − 1.25175i −0.779923 0.625875i \(-0.784742\pi\)
0.779923 0.625875i \(-0.215258\pi\)
\(468\) 0 0
\(469\) 145.641 0.310536
\(470\) − 196.828i − 0.418783i
\(471\) 0 0
\(472\) 8.62258 0.0182682
\(473\) 49.2796i 0.104185i
\(474\) 0 0
\(475\) 98.2941 0.206935
\(476\) 78.6726i 0.165278i
\(477\) 0 0
\(478\) −374.010 −0.782447
\(479\) − 442.649i − 0.924111i −0.886851 0.462055i \(-0.847112\pi\)
0.886851 0.462055i \(-0.152888\pi\)
\(480\) 0 0
\(481\) 720.393 1.49770
\(482\) 342.481i 0.710541i
\(483\) 0 0
\(484\) −252.645 −0.521995
\(485\) 839.919i 1.73179i
\(486\) 0 0
\(487\) 730.873 1.50077 0.750383 0.661004i \(-0.229869\pi\)
0.750383 + 0.661004i \(0.229869\pi\)
\(488\) − 87.8958i − 0.180114i
\(489\) 0 0
\(490\) 669.377 1.36608
\(491\) − 275.276i − 0.560644i −0.959906 0.280322i \(-0.909559\pi\)
0.959906 0.280322i \(-0.0904412\pi\)
\(492\) 0 0
\(493\) 183.603 0.372420
\(494\) − 113.483i − 0.229722i
\(495\) 0 0
\(496\) −651.589 −1.31369
\(497\) 19.7794i 0.0397976i
\(498\) 0 0
\(499\) −185.214 −0.371170 −0.185585 0.982628i \(-0.559418\pi\)
−0.185585 + 0.982628i \(0.559418\pi\)
\(500\) − 93.0333i − 0.186067i
\(501\) 0 0
\(502\) −601.762 −1.19873
\(503\) − 634.099i − 1.26064i −0.776337 0.630318i \(-0.782925\pi\)
0.776337 0.630318i \(-0.217075\pi\)
\(504\) 0 0
\(505\) 638.248 1.26386
\(506\) 76.9501i 0.152075i
\(507\) 0 0
\(508\) −23.8541 −0.0469569
\(509\) 492.444i 0.967473i 0.875214 + 0.483736i \(0.160721\pi\)
−0.875214 + 0.483736i \(0.839279\pi\)
\(510\) 0 0
\(511\) 96.8488 0.189528
\(512\) − 243.199i − 0.474999i
\(513\) 0 0
\(514\) −391.534 −0.761740
\(515\) − 244.497i − 0.474752i
\(516\) 0 0
\(517\) −13.6436 −0.0263900
\(518\) − 442.089i − 0.853454i
\(519\) 0 0
\(520\) 501.901 0.965195
\(521\) − 861.089i − 1.65276i −0.563111 0.826381i \(-0.690396\pi\)
0.563111 0.826381i \(-0.309604\pi\)
\(522\) 0 0
\(523\) 58.5594 0.111968 0.0559842 0.998432i \(-0.482170\pi\)
0.0559842 + 0.998432i \(0.482170\pi\)
\(524\) 56.7086i 0.108223i
\(525\) 0 0
\(526\) −210.994 −0.401129
\(527\) 338.770i 0.642828i
\(528\) 0 0
\(529\) −60.4458 −0.114264
\(530\) 1171.22i 2.20985i
\(531\) 0 0
\(532\) −24.0998 −0.0453003
\(533\) − 152.773i − 0.286629i
\(534\) 0 0
\(535\) 1498.11 2.80021
\(536\) 189.655i 0.353834i
\(537\) 0 0
\(538\) −292.009 −0.542768
\(539\) − 46.3995i − 0.0860845i
\(540\) 0 0
\(541\) 794.836 1.46920 0.734599 0.678501i \(-0.237370\pi\)
0.734599 + 0.678501i \(0.237370\pi\)
\(542\) − 827.922i − 1.52753i
\(543\) 0 0
\(544\) −320.041 −0.588311
\(545\) 290.996i 0.533938i
\(546\) 0 0
\(547\) −164.077 −0.299957 −0.149979 0.988689i \(-0.547921\pi\)
−0.149979 + 0.988689i \(0.547921\pi\)
\(548\) − 435.907i − 0.795450i
\(549\) 0 0
\(550\) −97.8723 −0.177950
\(551\) 56.2432i 0.102075i
\(552\) 0 0
\(553\) −145.415 −0.262956
\(554\) 75.7351i 0.136706i
\(555\) 0 0
\(556\) 338.069 0.608039
\(557\) − 806.590i − 1.44810i −0.689749 0.724048i \(-0.742279\pi\)
0.689749 0.724048i \(-0.257721\pi\)
\(558\) 0 0
\(559\) −554.311 −0.991612
\(560\) − 534.389i − 0.954266i
\(561\) 0 0
\(562\) −12.3057 −0.0218963
\(563\) − 136.258i − 0.242022i −0.992651 0.121011i \(-0.961386\pi\)
0.992651 0.121011i \(-0.0386136\pi\)
\(564\) 0 0
\(565\) −180.564 −0.319582
\(566\) 571.073i 1.00896i
\(567\) 0 0
\(568\) −25.7568 −0.0453465
\(569\) − 438.619i − 0.770860i −0.922737 0.385430i \(-0.874053\pi\)
0.922737 0.385430i \(-0.125947\pi\)
\(570\) 0 0
\(571\) 279.001 0.488618 0.244309 0.969697i \(-0.421439\pi\)
0.244309 + 0.969697i \(0.421439\pi\)
\(572\) 39.1025i 0.0683610i
\(573\) 0 0
\(574\) −93.7536 −0.163334
\(575\) − 749.711i − 1.30385i
\(576\) 0 0
\(577\) −193.520 −0.335391 −0.167695 0.985839i \(-0.553633\pi\)
−0.167695 + 0.985839i \(0.553633\pi\)
\(578\) − 447.706i − 0.774578i
\(579\) 0 0
\(580\) 279.578 0.482031
\(581\) 162.214i 0.279198i
\(582\) 0 0
\(583\) 81.1861 0.139256
\(584\) 126.117i 0.215954i
\(585\) 0 0
\(586\) 1040.63 1.77583
\(587\) 159.092i 0.271025i 0.990776 + 0.135512i \(0.0432681\pi\)
−0.990776 + 0.135512i \(0.956732\pi\)
\(588\) 0 0
\(589\) −103.776 −0.176189
\(590\) − 34.2252i − 0.0580089i
\(591\) 0 0
\(592\) 998.823 1.68720
\(593\) 502.148i 0.846793i 0.905945 + 0.423396i \(0.139162\pi\)
−0.905945 + 0.423396i \(0.860838\pi\)
\(594\) 0 0
\(595\) −277.836 −0.466952
\(596\) − 91.7945i − 0.154018i
\(597\) 0 0
\(598\) −865.558 −1.44742
\(599\) − 190.124i − 0.317403i −0.987327 0.158701i \(-0.949269\pi\)
0.987327 0.158701i \(-0.0507307\pi\)
\(600\) 0 0
\(601\) 199.419 0.331811 0.165906 0.986142i \(-0.446945\pi\)
0.165906 + 0.986142i \(0.446945\pi\)
\(602\) 340.169i 0.565064i
\(603\) 0 0
\(604\) 481.473 0.797141
\(605\) − 892.231i − 1.47476i
\(606\) 0 0
\(607\) 748.425 1.23299 0.616495 0.787359i \(-0.288552\pi\)
0.616495 + 0.787359i \(0.288552\pi\)
\(608\) − 98.0383i − 0.161247i
\(609\) 0 0
\(610\) −348.881 −0.571936
\(611\) − 153.467i − 0.251174i
\(612\) 0 0
\(613\) 458.107 0.747319 0.373660 0.927566i \(-0.378103\pi\)
0.373660 + 0.927566i \(0.378103\pi\)
\(614\) − 436.880i − 0.711530i
\(615\) 0 0
\(616\) −21.3502 −0.0346594
\(617\) 783.814i 1.27036i 0.772363 + 0.635182i \(0.219075\pi\)
−0.772363 + 0.635182i \(0.780925\pi\)
\(618\) 0 0
\(619\) 1123.39 1.81484 0.907421 0.420223i \(-0.138048\pi\)
0.907421 + 0.420223i \(0.138048\pi\)
\(620\) 515.855i 0.832024i
\(621\) 0 0
\(622\) 25.8991 0.0416384
\(623\) 275.458i 0.442148i
\(624\) 0 0
\(625\) −443.439 −0.709502
\(626\) 1409.35i 2.25136i
\(627\) 0 0
\(628\) −126.760 −0.201848
\(629\) − 519.302i − 0.825599i
\(630\) 0 0
\(631\) 367.570 0.582520 0.291260 0.956644i \(-0.405925\pi\)
0.291260 + 0.956644i \(0.405925\pi\)
\(632\) − 189.360i − 0.299620i
\(633\) 0 0
\(634\) −168.268 −0.265407
\(635\) − 84.2420i − 0.132665i
\(636\) 0 0
\(637\) 521.916 0.819334
\(638\) − 56.0018i − 0.0877771i
\(639\) 0 0
\(640\) 990.683 1.54794
\(641\) − 165.990i − 0.258955i −0.991582 0.129477i \(-0.958670\pi\)
0.991582 0.129477i \(-0.0413300\pi\)
\(642\) 0 0
\(643\) −225.814 −0.351188 −0.175594 0.984463i \(-0.556185\pi\)
−0.175594 + 0.984463i \(0.556185\pi\)
\(644\) 183.814i 0.285426i
\(645\) 0 0
\(646\) −81.8050 −0.126633
\(647\) − 727.421i − 1.12430i −0.827036 0.562149i \(-0.809975\pi\)
0.827036 0.562149i \(-0.190025\pi\)
\(648\) 0 0
\(649\) −2.37241 −0.00365548
\(650\) − 1100.90i − 1.69369i
\(651\) 0 0
\(652\) −304.662 −0.467272
\(653\) − 445.966i − 0.682949i −0.939891 0.341475i \(-0.889074\pi\)
0.939891 0.341475i \(-0.110926\pi\)
\(654\) 0 0
\(655\) −200.269 −0.305755
\(656\) − 211.820i − 0.322896i
\(657\) 0 0
\(658\) −94.1796 −0.143130
\(659\) − 1167.70i − 1.77192i −0.463760 0.885961i \(-0.653500\pi\)
0.463760 0.885961i \(-0.346500\pi\)
\(660\) 0 0
\(661\) 1018.80 1.54130 0.770652 0.637256i \(-0.219931\pi\)
0.770652 + 0.637256i \(0.219931\pi\)
\(662\) − 199.625i − 0.301548i
\(663\) 0 0
\(664\) −211.236 −0.318126
\(665\) − 85.1097i − 0.127984i
\(666\) 0 0
\(667\) 428.980 0.643148
\(668\) 39.1995i 0.0586818i
\(669\) 0 0
\(670\) 752.789 1.12357
\(671\) 24.1836i 0.0360411i
\(672\) 0 0
\(673\) 547.102 0.812930 0.406465 0.913666i \(-0.366761\pi\)
0.406465 + 0.913666i \(0.366761\pi\)
\(674\) 1613.66i 2.39415i
\(675\) 0 0
\(676\) −82.1129 −0.121469
\(677\) 619.136i 0.914529i 0.889331 + 0.457264i \(0.151171\pi\)
−0.889331 + 0.457264i \(0.848829\pi\)
\(678\) 0 0
\(679\) 401.890 0.591885
\(680\) − 361.800i − 0.532059i
\(681\) 0 0
\(682\) 103.330 0.151510
\(683\) − 667.806i − 0.977754i −0.872353 0.488877i \(-0.837407\pi\)
0.872353 0.488877i \(-0.162593\pi\)
\(684\) 0 0
\(685\) 1539.43 2.24734
\(686\) − 753.751i − 1.09876i
\(687\) 0 0
\(688\) −768.551 −1.11708
\(689\) 913.205i 1.32541i
\(690\) 0 0
\(691\) 624.648 0.903977 0.451988 0.892024i \(-0.350715\pi\)
0.451988 + 0.892024i \(0.350715\pi\)
\(692\) − 18.5672i − 0.0268313i
\(693\) 0 0
\(694\) −1013.73 −1.46070
\(695\) 1193.91i 1.71786i
\(696\) 0 0
\(697\) −110.128 −0.158003
\(698\) 1026.43i 1.47053i
\(699\) 0 0
\(700\) −233.792 −0.333989
\(701\) − 714.318i − 1.01900i −0.860471 0.509499i \(-0.829831\pi\)
0.860471 0.509499i \(-0.170169\pi\)
\(702\) 0 0
\(703\) 159.078 0.226284
\(704\) − 4.83500i − 0.00686790i
\(705\) 0 0
\(706\) 358.644 0.507994
\(707\) − 305.393i − 0.431956i
\(708\) 0 0
\(709\) 483.391 0.681792 0.340896 0.940101i \(-0.389270\pi\)
0.340896 + 0.940101i \(0.389270\pi\)
\(710\) 102.236i 0.143994i
\(711\) 0 0
\(712\) −358.703 −0.503797
\(713\) 791.519i 1.11013i
\(714\) 0 0
\(715\) −138.093 −0.193137
\(716\) 175.340i 0.244888i
\(717\) 0 0
\(718\) −466.382 −0.649556
\(719\) − 1146.34i − 1.59436i −0.603745 0.797178i \(-0.706325\pi\)
0.603745 0.797178i \(-0.293675\pi\)
\(720\) 0 0
\(721\) −116.989 −0.162259
\(722\) 867.765i 1.20189i
\(723\) 0 0
\(724\) −464.260 −0.641244
\(725\) 545.616i 0.752574i
\(726\) 0 0
\(727\) −396.069 −0.544799 −0.272400 0.962184i \(-0.587817\pi\)
−0.272400 + 0.962184i \(0.587817\pi\)
\(728\) − 240.153i − 0.329881i
\(729\) 0 0
\(730\) 500.591 0.685742
\(731\) 399.580i 0.546621i
\(732\) 0 0
\(733\) 1143.30 1.55975 0.779876 0.625934i \(-0.215282\pi\)
0.779876 + 0.625934i \(0.215282\pi\)
\(734\) 1716.10i 2.33801i
\(735\) 0 0
\(736\) −747.760 −1.01598
\(737\) − 52.1815i − 0.0708025i
\(738\) 0 0
\(739\) −940.492 −1.27265 −0.636327 0.771419i \(-0.719547\pi\)
−0.636327 + 0.771419i \(0.719547\pi\)
\(740\) − 790.756i − 1.06859i
\(741\) 0 0
\(742\) 560.414 0.755274
\(743\) 1295.75i 1.74394i 0.489555 + 0.871972i \(0.337159\pi\)
−0.489555 + 0.871972i \(0.662841\pi\)
\(744\) 0 0
\(745\) 324.177 0.435137
\(746\) 1728.47i 2.31699i
\(747\) 0 0
\(748\) 28.1874 0.0376837
\(749\) − 716.826i − 0.957044i
\(750\) 0 0
\(751\) −245.217 −0.326521 −0.163260 0.986583i \(-0.552201\pi\)
−0.163260 + 0.986583i \(0.552201\pi\)
\(752\) − 212.782i − 0.282955i
\(753\) 0 0
\(754\) 629.925 0.835444
\(755\) 1700.35i 2.25212i
\(756\) 0 0
\(757\) −210.908 −0.278610 −0.139305 0.990250i \(-0.544487\pi\)
−0.139305 + 0.990250i \(0.544487\pi\)
\(758\) 290.315i 0.383001i
\(759\) 0 0
\(760\) 110.830 0.145829
\(761\) 237.328i 0.311863i 0.987768 + 0.155932i \(0.0498379\pi\)
−0.987768 + 0.155932i \(0.950162\pi\)
\(762\) 0 0
\(763\) 139.238 0.182487
\(764\) 94.2138i 0.123316i
\(765\) 0 0
\(766\) 1399.29 1.82675
\(767\) − 26.6855i − 0.0347921i
\(768\) 0 0
\(769\) −231.536 −0.301087 −0.150543 0.988603i \(-0.548102\pi\)
−0.150543 + 0.988603i \(0.548102\pi\)
\(770\) 84.7444i 0.110058i
\(771\) 0 0
\(772\) 155.655 0.201625
\(773\) 1365.28i 1.76621i 0.469171 + 0.883107i \(0.344553\pi\)
−0.469171 + 0.883107i \(0.655447\pi\)
\(774\) 0 0
\(775\) −1006.73 −1.29900
\(776\) 523.344i 0.674412i
\(777\) 0 0
\(778\) −725.717 −0.932798
\(779\) − 33.7355i − 0.0433062i
\(780\) 0 0
\(781\) 7.08671 0.00907389
\(782\) 623.945i 0.797884i
\(783\) 0 0
\(784\) 723.634 0.923003
\(785\) − 447.660i − 0.570268i
\(786\) 0 0
\(787\) 690.289 0.877114 0.438557 0.898703i \(-0.355490\pi\)
0.438557 + 0.898703i \(0.355490\pi\)
\(788\) 566.449i 0.718844i
\(789\) 0 0
\(790\) −751.618 −0.951416
\(791\) 86.3975i 0.109226i
\(792\) 0 0
\(793\) −272.024 −0.343031
\(794\) 386.744i 0.487083i
\(795\) 0 0
\(796\) 367.625 0.461841
\(797\) − 877.407i − 1.10089i −0.834873 0.550443i \(-0.814459\pi\)
0.834873 0.550443i \(-0.185541\pi\)
\(798\) 0 0
\(799\) −110.628 −0.138459
\(800\) − 951.070i − 1.18884i
\(801\) 0 0
\(802\) 1353.76 1.68798
\(803\) − 34.6997i − 0.0432126i
\(804\) 0 0
\(805\) −649.151 −0.806398
\(806\) 1162.29i 1.44204i
\(807\) 0 0
\(808\) 397.685 0.492184
\(809\) − 271.708i − 0.335857i −0.985799 0.167928i \(-0.946292\pi\)
0.985799 0.167928i \(-0.0537077\pi\)
\(810\) 0 0
\(811\) 590.185 0.727725 0.363862 0.931453i \(-0.381458\pi\)
0.363862 + 0.931453i \(0.381458\pi\)
\(812\) − 133.774i − 0.164747i
\(813\) 0 0
\(814\) −158.395 −0.194589
\(815\) − 1075.93i − 1.32016i
\(816\) 0 0
\(817\) −122.404 −0.149821
\(818\) 633.904i 0.774944i
\(819\) 0 0
\(820\) −167.695 −0.204506
\(821\) − 1133.04i − 1.38007i −0.723774 0.690037i \(-0.757594\pi\)
0.723774 0.690037i \(-0.242406\pi\)
\(822\) 0 0
\(823\) −839.497 −1.02004 −0.510022 0.860161i \(-0.670363\pi\)
−0.510022 + 0.860161i \(0.670363\pi\)
\(824\) − 152.343i − 0.184883i
\(825\) 0 0
\(826\) −16.3763 −0.0198261
\(827\) − 104.942i − 0.126895i −0.997985 0.0634477i \(-0.979790\pi\)
0.997985 0.0634477i \(-0.0202096\pi\)
\(828\) 0 0
\(829\) −361.576 −0.436159 −0.218079 0.975931i \(-0.569979\pi\)
−0.218079 + 0.975931i \(0.569979\pi\)
\(830\) 838.449i 1.01018i
\(831\) 0 0
\(832\) 54.3855 0.0653672
\(833\) − 376.228i − 0.451654i
\(834\) 0 0
\(835\) −138.435 −0.165790
\(836\) 8.63465i 0.0103285i
\(837\) 0 0
\(838\) −17.2096 −0.0205365
\(839\) 1033.20i 1.23146i 0.787957 + 0.615730i \(0.211139\pi\)
−0.787957 + 0.615730i \(0.788861\pi\)
\(840\) 0 0
\(841\) 528.802 0.628778
\(842\) − 406.862i − 0.483209i
\(843\) 0 0
\(844\) 814.898 0.965519
\(845\) − 289.986i − 0.343179i
\(846\) 0 0
\(847\) −426.921 −0.504038
\(848\) 1266.16i 1.49311i
\(849\) 0 0
\(850\) −793.591 −0.933636
\(851\) − 1213.32i − 1.42576i
\(852\) 0 0
\(853\) 1162.20 1.36249 0.681245 0.732056i \(-0.261439\pi\)
0.681245 + 0.732056i \(0.261439\pi\)
\(854\) 166.935i 0.195474i
\(855\) 0 0
\(856\) 933.455 1.09048
\(857\) 172.029i 0.200735i 0.994950 + 0.100367i \(0.0320018\pi\)
−0.994950 + 0.100367i \(0.967998\pi\)
\(858\) 0 0
\(859\) −1205.40 −1.40326 −0.701631 0.712540i \(-0.747545\pi\)
−0.701631 + 0.712540i \(0.747545\pi\)
\(860\) 608.452i 0.707503i
\(861\) 0 0
\(862\) −1079.97 −1.25287
\(863\) 1547.43i 1.79308i 0.442965 + 0.896539i \(0.353927\pi\)
−0.442965 + 0.896539i \(0.646073\pi\)
\(864\) 0 0
\(865\) 65.5712 0.0758049
\(866\) − 158.751i − 0.183315i
\(867\) 0 0
\(868\) 246.830 0.284366
\(869\) 52.1003i 0.0599543i
\(870\) 0 0
\(871\) 586.952 0.673883
\(872\) 181.316i 0.207931i
\(873\) 0 0
\(874\) −191.133 −0.218688
\(875\) − 157.208i − 0.179666i
\(876\) 0 0
\(877\) −1072.48 −1.22289 −0.611445 0.791287i \(-0.709412\pi\)
−0.611445 + 0.791287i \(0.709412\pi\)
\(878\) 638.843i 0.727611i
\(879\) 0 0
\(880\) −191.465 −0.217574
\(881\) − 363.670i − 0.412793i −0.978468 0.206396i \(-0.933826\pi\)
0.978468 0.206396i \(-0.0661736\pi\)
\(882\) 0 0
\(883\) −910.225 −1.03083 −0.515416 0.856940i \(-0.672363\pi\)
−0.515416 + 0.856940i \(0.672363\pi\)
\(884\) 317.060i 0.358665i
\(885\) 0 0
\(886\) 683.109 0.771004
\(887\) − 425.744i − 0.479982i −0.970775 0.239991i \(-0.922855\pi\)
0.970775 0.239991i \(-0.0771446\pi\)
\(888\) 0 0
\(889\) −40.3087 −0.0453416
\(890\) 1423.79i 1.59976i
\(891\) 0 0
\(892\) 827.668 0.927879
\(893\) − 33.8888i − 0.0379494i
\(894\) 0 0
\(895\) −619.221 −0.691867
\(896\) − 474.029i − 0.529050i
\(897\) 0 0
\(898\) −1374.17 −1.53026
\(899\) − 576.042i − 0.640759i
\(900\) 0 0
\(901\) 658.292 0.730624
\(902\) 33.5907i 0.0372403i
\(903\) 0 0
\(904\) −112.507 −0.124455
\(905\) − 1639.56i − 1.81167i
\(906\) 0 0
\(907\) −735.774 −0.811218 −0.405609 0.914047i \(-0.632940\pi\)
−0.405609 + 0.914047i \(0.632940\pi\)
\(908\) − 250.717i − 0.276121i
\(909\) 0 0
\(910\) −953.230 −1.04751
\(911\) − 791.514i − 0.868841i −0.900710 0.434420i \(-0.856953\pi\)
0.900710 0.434420i \(-0.143047\pi\)
\(912\) 0 0
\(913\) 58.1192 0.0636574
\(914\) − 166.584i − 0.182258i
\(915\) 0 0
\(916\) 27.6924 0.0302319
\(917\) 95.8263i 0.104500i
\(918\) 0 0
\(919\) −264.051 −0.287324 −0.143662 0.989627i \(-0.545888\pi\)
−0.143662 + 0.989627i \(0.545888\pi\)
\(920\) − 845.328i − 0.918835i
\(921\) 0 0
\(922\) 1976.62 2.14384
\(923\) 79.7134i 0.0863634i
\(924\) 0 0
\(925\) 1543.22 1.66834
\(926\) 1161.60i 1.25442i
\(927\) 0 0
\(928\) 544.196 0.586418
\(929\) − 197.077i − 0.212139i −0.994359 0.106070i \(-0.966173\pi\)
0.994359 0.106070i \(-0.0338266\pi\)
\(930\) 0 0
\(931\) 115.250 0.123791
\(932\) − 66.4257i − 0.0712722i
\(933\) 0 0
\(934\) −1445.75 −1.54791
\(935\) 99.5453i 0.106466i
\(936\) 0 0
\(937\) −1280.61 −1.36671 −0.683354 0.730087i \(-0.739479\pi\)
−0.683354 + 0.730087i \(0.739479\pi\)
\(938\) − 360.200i − 0.384008i
\(939\) 0 0
\(940\) −168.457 −0.179209
\(941\) − 571.596i − 0.607434i −0.952762 0.303717i \(-0.901772\pi\)
0.952762 0.303717i \(-0.0982278\pi\)
\(942\) 0 0
\(943\) −257.309 −0.272862
\(944\) − 36.9994i − 0.0391943i
\(945\) 0 0
\(946\) 121.878 0.128835
\(947\) − 135.984i − 0.143594i −0.997419 0.0717971i \(-0.977127\pi\)
0.997419 0.0717971i \(-0.0228734\pi\)
\(948\) 0 0
\(949\) 390.313 0.411289
\(950\) − 243.101i − 0.255896i
\(951\) 0 0
\(952\) −173.117 −0.181845
\(953\) − 360.322i − 0.378092i −0.981968 0.189046i \(-0.939460\pi\)
0.981968 0.189046i \(-0.0605395\pi\)
\(954\) 0 0
\(955\) −332.721 −0.348399
\(956\) 320.099i 0.334832i
\(957\) 0 0
\(958\) −1094.76 −1.14275
\(959\) − 736.596i − 0.768088i
\(960\) 0 0
\(961\) 101.868 0.106002
\(962\) − 1781.68i − 1.85205i
\(963\) 0 0
\(964\) 293.115 0.304062
\(965\) 549.703i 0.569641i
\(966\) 0 0
\(967\) 1222.22 1.26393 0.631963 0.774999i \(-0.282250\pi\)
0.631963 + 0.774999i \(0.282250\pi\)
\(968\) − 555.939i − 0.574317i
\(969\) 0 0
\(970\) 2077.29 2.14153
\(971\) 286.990i 0.295561i 0.989020 + 0.147781i \(0.0472130\pi\)
−0.989020 + 0.147781i \(0.952787\pi\)
\(972\) 0 0
\(973\) 571.270 0.587123
\(974\) − 1807.59i − 1.85585i
\(975\) 0 0
\(976\) −377.160 −0.386435
\(977\) 1714.15i 1.75451i 0.480028 + 0.877253i \(0.340627\pi\)
−0.480028 + 0.877253i \(0.659373\pi\)
\(978\) 0 0
\(979\) 98.6933 0.100810
\(980\) − 572.892i − 0.584584i
\(981\) 0 0
\(982\) −680.812 −0.693292
\(983\) 807.305i 0.821267i 0.911801 + 0.410633i \(0.134692\pi\)
−0.911801 + 0.410633i \(0.865308\pi\)
\(984\) 0 0
\(985\) −2000.44 −2.03091
\(986\) − 454.087i − 0.460535i
\(987\) 0 0
\(988\) −97.1251 −0.0983048
\(989\) 933.599i 0.943983i
\(990\) 0 0
\(991\) 167.284 0.168804 0.0844019 0.996432i \(-0.473102\pi\)
0.0844019 + 0.996432i \(0.473102\pi\)
\(992\) 1004.11i 1.01220i
\(993\) 0 0
\(994\) 48.9183 0.0492136
\(995\) 1298.29i 1.30481i
\(996\) 0 0
\(997\) 1733.30 1.73851 0.869257 0.494360i \(-0.164598\pi\)
0.869257 + 0.494360i \(0.164598\pi\)
\(998\) 458.071i 0.458989i
\(999\) 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 1143.3.b.a.890.19 84
3.2 odd 2 inner 1143.3.b.a.890.66 yes 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1143.3.b.a.890.19 84 1.1 even 1 trivial
1143.3.b.a.890.66 yes 84 3.2 odd 2 inner