Properties

Label 1143.3.b.a.890.8
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 890.8
Character \(\chi\) \(=\) 1143.890
Dual form 1143.3.b.a.890.77

$q$-expansion

\(f(q)\) \(=\) \(q-3.57620i q^{2} -8.78923 q^{4} +5.13671i q^{5} +4.24305 q^{7} +17.1272i q^{8} +O(q^{10})\) \(q-3.57620i q^{2} -8.78923 q^{4} +5.13671i q^{5} +4.24305 q^{7} +17.1272i q^{8} +18.3699 q^{10} +2.99209i q^{11} -1.37667 q^{13} -15.1740i q^{14} +26.0936 q^{16} -29.5895i q^{17} -26.2947 q^{19} -45.1477i q^{20} +10.7003 q^{22} +1.34689i q^{23} -1.38580 q^{25} +4.92326i q^{26} -37.2931 q^{28} -4.79007i q^{29} +40.7958 q^{31} -24.8070i q^{32} -105.818 q^{34} +21.7953i q^{35} -54.4024 q^{37} +94.0351i q^{38} -87.9777 q^{40} +59.8727i q^{41} -36.0993 q^{43} -26.2981i q^{44} +4.81677 q^{46} +13.5014i q^{47} -30.9965 q^{49} +4.95590i q^{50} +12.0999 q^{52} +57.2572i q^{53} -15.3695 q^{55} +72.6718i q^{56} -17.1303 q^{58} +94.2485i q^{59} -95.2655 q^{61} -145.894i q^{62} +15.6595 q^{64} -7.07156i q^{65} -4.97125 q^{67} +260.069i q^{68} +77.9445 q^{70} -54.4301i q^{71} -52.2497 q^{73} +194.554i q^{74} +231.110 q^{76} +12.6956i q^{77} -31.2067 q^{79} +134.035i q^{80} +214.117 q^{82} -5.29944i q^{83} +151.993 q^{85} +129.099i q^{86} -51.2462 q^{88} -6.08374i q^{89} -5.84129 q^{91} -11.8382i q^{92} +48.2836 q^{94} -135.068i q^{95} -129.209 q^{97} +110.850i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84q - 160q^{4} + O(q^{10}) \) \( 84q - 160q^{4} - 48q^{10} + 16q^{13} + 360q^{16} + 64q^{19} - 8q^{22} - 388q^{25} - 120q^{28} - 160q^{31} + 192q^{34} - 152q^{37} + 208q^{40} - 24q^{43} + 56q^{46} + 564q^{49} - 80q^{52} + 136q^{55} - 136q^{58} + 168q^{61} - 736q^{64} + 168q^{67} - 608q^{70} + 80q^{73} - 32q^{76} - 168q^{79} + 528q^{82} + 288q^{85} - 392q^{88} + 176q^{91} + 176q^{94} - 120q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.57620i − 1.78810i −0.447966 0.894051i \(-0.647851\pi\)
0.447966 0.894051i \(-0.352149\pi\)
\(3\) 0 0
\(4\) −8.78923 −2.19731
\(5\) 5.13671i 1.02734i 0.857987 + 0.513671i \(0.171715\pi\)
−0.857987 + 0.513671i \(0.828285\pi\)
\(6\) 0 0
\(7\) 4.24305 0.606150 0.303075 0.952967i \(-0.401987\pi\)
0.303075 + 0.952967i \(0.401987\pi\)
\(8\) 17.1272i 2.14091i
\(9\) 0 0
\(10\) 18.3699 1.83699
\(11\) 2.99209i 0.272008i 0.990708 + 0.136004i \(0.0434260\pi\)
−0.990708 + 0.136004i \(0.956574\pi\)
\(12\) 0 0
\(13\) −1.37667 −0.105898 −0.0529489 0.998597i \(-0.516862\pi\)
−0.0529489 + 0.998597i \(0.516862\pi\)
\(14\) − 15.1740i − 1.08386i
\(15\) 0 0
\(16\) 26.0936 1.63085
\(17\) − 29.5895i − 1.74056i −0.492560 0.870278i \(-0.663939\pi\)
0.492560 0.870278i \(-0.336061\pi\)
\(18\) 0 0
\(19\) −26.2947 −1.38393 −0.691965 0.721931i \(-0.743255\pi\)
−0.691965 + 0.721931i \(0.743255\pi\)
\(20\) − 45.1477i − 2.25739i
\(21\) 0 0
\(22\) 10.7003 0.486378
\(23\) 1.34689i 0.0585606i 0.999571 + 0.0292803i \(0.00932155\pi\)
−0.999571 + 0.0292803i \(0.990678\pi\)
\(24\) 0 0
\(25\) −1.38580 −0.0554320
\(26\) 4.92326i 0.189356i
\(27\) 0 0
\(28\) −37.2931 −1.33190
\(29\) − 4.79007i − 0.165175i −0.996584 0.0825874i \(-0.973682\pi\)
0.996584 0.0825874i \(-0.0263184\pi\)
\(30\) 0 0
\(31\) 40.7958 1.31599 0.657997 0.753021i \(-0.271404\pi\)
0.657997 + 0.753021i \(0.271404\pi\)
\(32\) − 24.8070i − 0.775219i
\(33\) 0 0
\(34\) −105.818 −3.11229
\(35\) 21.7953i 0.622724i
\(36\) 0 0
\(37\) −54.4024 −1.47034 −0.735168 0.677885i \(-0.762897\pi\)
−0.735168 + 0.677885i \(0.762897\pi\)
\(38\) 94.0351i 2.47461i
\(39\) 0 0
\(40\) −87.9777 −2.19944
\(41\) 59.8727i 1.46031i 0.683281 + 0.730155i \(0.260552\pi\)
−0.683281 + 0.730155i \(0.739448\pi\)
\(42\) 0 0
\(43\) −36.0993 −0.839519 −0.419760 0.907635i \(-0.637886\pi\)
−0.419760 + 0.907635i \(0.637886\pi\)
\(44\) − 26.2981i − 0.597685i
\(45\) 0 0
\(46\) 4.81677 0.104712
\(47\) 13.5014i 0.287263i 0.989631 + 0.143632i \(0.0458780\pi\)
−0.989631 + 0.143632i \(0.954122\pi\)
\(48\) 0 0
\(49\) −30.9965 −0.632582
\(50\) 4.95590i 0.0991180i
\(51\) 0 0
\(52\) 12.0999 0.232690
\(53\) 57.2572i 1.08032i 0.841561 + 0.540162i \(0.181637\pi\)
−0.841561 + 0.540162i \(0.818363\pi\)
\(54\) 0 0
\(55\) −15.3695 −0.279445
\(56\) 72.6718i 1.29771i
\(57\) 0 0
\(58\) −17.1303 −0.295349
\(59\) 94.2485i 1.59743i 0.601708 + 0.798716i \(0.294487\pi\)
−0.601708 + 0.798716i \(0.705513\pi\)
\(60\) 0 0
\(61\) −95.2655 −1.56173 −0.780864 0.624701i \(-0.785221\pi\)
−0.780864 + 0.624701i \(0.785221\pi\)
\(62\) − 145.894i − 2.35313i
\(63\) 0 0
\(64\) 15.6595 0.244679
\(65\) − 7.07156i − 0.108793i
\(66\) 0 0
\(67\) −4.97125 −0.0741978 −0.0370989 0.999312i \(-0.511812\pi\)
−0.0370989 + 0.999312i \(0.511812\pi\)
\(68\) 260.069i 3.82454i
\(69\) 0 0
\(70\) 77.9445 1.11349
\(71\) − 54.4301i − 0.766621i −0.923620 0.383310i \(-0.874784\pi\)
0.923620 0.383310i \(-0.125216\pi\)
\(72\) 0 0
\(73\) −52.2497 −0.715750 −0.357875 0.933770i \(-0.616499\pi\)
−0.357875 + 0.933770i \(0.616499\pi\)
\(74\) 194.554i 2.62911i
\(75\) 0 0
\(76\) 231.110 3.04092
\(77\) 12.6956i 0.164878i
\(78\) 0 0
\(79\) −31.2067 −0.395022 −0.197511 0.980301i \(-0.563286\pi\)
−0.197511 + 0.980301i \(0.563286\pi\)
\(80\) 134.035i 1.67544i
\(81\) 0 0
\(82\) 214.117 2.61118
\(83\) − 5.29944i − 0.0638487i −0.999490 0.0319243i \(-0.989836\pi\)
0.999490 0.0319243i \(-0.0101636\pi\)
\(84\) 0 0
\(85\) 151.993 1.78815
\(86\) 129.099i 1.50115i
\(87\) 0 0
\(88\) −51.2462 −0.582343
\(89\) − 6.08374i − 0.0683567i −0.999416 0.0341783i \(-0.989119\pi\)
0.999416 0.0341783i \(-0.0108814\pi\)
\(90\) 0 0
\(91\) −5.84129 −0.0641899
\(92\) − 11.8382i − 0.128676i
\(93\) 0 0
\(94\) 48.2836 0.513656
\(95\) − 135.068i − 1.42177i
\(96\) 0 0
\(97\) −129.209 −1.33205 −0.666024 0.745930i \(-0.732005\pi\)
−0.666024 + 0.745930i \(0.732005\pi\)
\(98\) 110.850i 1.13112i
\(99\) 0 0
\(100\) 12.1801 0.121801
\(101\) − 61.6473i − 0.610369i −0.952293 0.305184i \(-0.901282\pi\)
0.952293 0.305184i \(-0.0987181\pi\)
\(102\) 0 0
\(103\) −11.2073 −0.108809 −0.0544046 0.998519i \(-0.517326\pi\)
−0.0544046 + 0.998519i \(0.517326\pi\)
\(104\) − 23.5786i − 0.226717i
\(105\) 0 0
\(106\) 204.763 1.93173
\(107\) 19.9500i 0.186448i 0.995645 + 0.0932241i \(0.0297173\pi\)
−0.995645 + 0.0932241i \(0.970283\pi\)
\(108\) 0 0
\(109\) −79.7628 −0.731769 −0.365884 0.930660i \(-0.619233\pi\)
−0.365884 + 0.930660i \(0.619233\pi\)
\(110\) 54.9644i 0.499676i
\(111\) 0 0
\(112\) 110.716 0.988540
\(113\) 76.9397i 0.680882i 0.940266 + 0.340441i \(0.110576\pi\)
−0.940266 + 0.340441i \(0.889424\pi\)
\(114\) 0 0
\(115\) −6.91861 −0.0601618
\(116\) 42.1010i 0.362940i
\(117\) 0 0
\(118\) 337.052 2.85637
\(119\) − 125.550i − 1.05504i
\(120\) 0 0
\(121\) 112.047 0.926012
\(122\) 340.689i 2.79253i
\(123\) 0 0
\(124\) −358.564 −2.89164
\(125\) 121.299i 0.970395i
\(126\) 0 0
\(127\) 11.2694 0.0887357
\(128\) − 155.229i − 1.21273i
\(129\) 0 0
\(130\) −25.2893 −0.194533
\(131\) 50.7426i 0.387348i 0.981066 + 0.193674i \(0.0620405\pi\)
−0.981066 + 0.193674i \(0.937960\pi\)
\(132\) 0 0
\(133\) −111.570 −0.838869
\(134\) 17.7782i 0.132673i
\(135\) 0 0
\(136\) 506.786 3.72637
\(137\) 24.8446i 0.181348i 0.995881 + 0.0906738i \(0.0289021\pi\)
−0.995881 + 0.0906738i \(0.971098\pi\)
\(138\) 0 0
\(139\) −193.959 −1.39539 −0.697695 0.716394i \(-0.745791\pi\)
−0.697695 + 0.716394i \(0.745791\pi\)
\(140\) − 191.564i − 1.36831i
\(141\) 0 0
\(142\) −194.653 −1.37080
\(143\) − 4.11912i − 0.0288050i
\(144\) 0 0
\(145\) 24.6052 0.169691
\(146\) 186.856i 1.27983i
\(147\) 0 0
\(148\) 478.155 3.23078
\(149\) − 91.0229i − 0.610892i −0.952209 0.305446i \(-0.901195\pi\)
0.952209 0.305446i \(-0.0988055\pi\)
\(150\) 0 0
\(151\) −292.154 −1.93479 −0.967397 0.253264i \(-0.918496\pi\)
−0.967397 + 0.253264i \(0.918496\pi\)
\(152\) − 450.355i − 2.96286i
\(153\) 0 0
\(154\) 45.4020 0.294818
\(155\) 209.556i 1.35198i
\(156\) 0 0
\(157\) 66.7254 0.425003 0.212501 0.977161i \(-0.431839\pi\)
0.212501 + 0.977161i \(0.431839\pi\)
\(158\) 111.602i 0.706339i
\(159\) 0 0
\(160\) 127.427 0.796416
\(161\) 5.71494i 0.0354965i
\(162\) 0 0
\(163\) 170.933 1.04867 0.524336 0.851512i \(-0.324314\pi\)
0.524336 + 0.851512i \(0.324314\pi\)
\(164\) − 526.235i − 3.20875i
\(165\) 0 0
\(166\) −18.9519 −0.114168
\(167\) − 108.587i − 0.650220i −0.945676 0.325110i \(-0.894599\pi\)
0.945676 0.325110i \(-0.105401\pi\)
\(168\) 0 0
\(169\) −167.105 −0.988786
\(170\) − 543.556i − 3.19739i
\(171\) 0 0
\(172\) 317.285 1.84468
\(173\) 114.070i 0.659362i 0.944092 + 0.329681i \(0.106941\pi\)
−0.944092 + 0.329681i \(0.893059\pi\)
\(174\) 0 0
\(175\) −5.88002 −0.0336001
\(176\) 78.0743i 0.443604i
\(177\) 0 0
\(178\) −21.7567 −0.122229
\(179\) − 152.335i − 0.851033i −0.904951 0.425517i \(-0.860092\pi\)
0.904951 0.425517i \(-0.139908\pi\)
\(180\) 0 0
\(181\) −67.4737 −0.372783 −0.186391 0.982476i \(-0.559679\pi\)
−0.186391 + 0.982476i \(0.559679\pi\)
\(182\) 20.8896i 0.114778i
\(183\) 0 0
\(184\) −23.0686 −0.125373
\(185\) − 279.450i − 1.51054i
\(186\) 0 0
\(187\) 88.5343 0.473445
\(188\) − 118.667i − 0.631205i
\(189\) 0 0
\(190\) −483.031 −2.54227
\(191\) 58.6665i 0.307155i 0.988137 + 0.153577i \(0.0490794\pi\)
−0.988137 + 0.153577i \(0.950921\pi\)
\(192\) 0 0
\(193\) 8.85349 0.0458730 0.0229365 0.999737i \(-0.492698\pi\)
0.0229365 + 0.999737i \(0.492698\pi\)
\(194\) 462.076i 2.38184i
\(195\) 0 0
\(196\) 272.435 1.38998
\(197\) 177.858i 0.902835i 0.892313 + 0.451417i \(0.149081\pi\)
−0.892313 + 0.451417i \(0.850919\pi\)
\(198\) 0 0
\(199\) 40.7549 0.204798 0.102399 0.994743i \(-0.467348\pi\)
0.102399 + 0.994743i \(0.467348\pi\)
\(200\) − 23.7349i − 0.118675i
\(201\) 0 0
\(202\) −220.463 −1.09140
\(203\) − 20.3245i − 0.100121i
\(204\) 0 0
\(205\) −307.549 −1.50024
\(206\) 40.0797i 0.194562i
\(207\) 0 0
\(208\) −35.9223 −0.172703
\(209\) − 78.6759i − 0.376440i
\(210\) 0 0
\(211\) −38.7869 −0.183824 −0.0919122 0.995767i \(-0.529298\pi\)
−0.0919122 + 0.995767i \(0.529298\pi\)
\(212\) − 503.246i − 2.37380i
\(213\) 0 0
\(214\) 71.3451 0.333388
\(215\) − 185.432i − 0.862474i
\(216\) 0 0
\(217\) 173.099 0.797690
\(218\) 285.248i 1.30848i
\(219\) 0 0
\(220\) 135.086 0.614027
\(221\) 40.7350i 0.184321i
\(222\) 0 0
\(223\) −227.007 −1.01797 −0.508985 0.860775i \(-0.669979\pi\)
−0.508985 + 0.860775i \(0.669979\pi\)
\(224\) − 105.257i − 0.469899i
\(225\) 0 0
\(226\) 275.152 1.21749
\(227\) 130.052i 0.572918i 0.958093 + 0.286459i \(0.0924782\pi\)
−0.958093 + 0.286459i \(0.907522\pi\)
\(228\) 0 0
\(229\) 271.372 1.18503 0.592516 0.805559i \(-0.298135\pi\)
0.592516 + 0.805559i \(0.298135\pi\)
\(230\) 24.7424i 0.107575i
\(231\) 0 0
\(232\) 82.0407 0.353624
\(233\) − 230.603i − 0.989712i −0.868975 0.494856i \(-0.835221\pi\)
0.868975 0.494856i \(-0.164779\pi\)
\(234\) 0 0
\(235\) −69.3526 −0.295118
\(236\) − 828.371i − 3.51005i
\(237\) 0 0
\(238\) −448.991 −1.88652
\(239\) 120.743i 0.505200i 0.967571 + 0.252600i \(0.0812857\pi\)
−0.967571 + 0.252600i \(0.918714\pi\)
\(240\) 0 0
\(241\) 323.560 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(242\) − 400.704i − 1.65580i
\(243\) 0 0
\(244\) 837.310 3.43160
\(245\) − 159.220i − 0.649878i
\(246\) 0 0
\(247\) 36.1991 0.146555
\(248\) 698.720i 2.81742i
\(249\) 0 0
\(250\) 433.791 1.73516
\(251\) 103.429i 0.412069i 0.978545 + 0.206035i \(0.0660559\pi\)
−0.978545 + 0.206035i \(0.933944\pi\)
\(252\) 0 0
\(253\) −4.03003 −0.0159290
\(254\) − 40.3018i − 0.158668i
\(255\) 0 0
\(256\) −492.494 −1.92381
\(257\) − 436.399i − 1.69805i −0.528352 0.849025i \(-0.677190\pi\)
0.528352 0.849025i \(-0.322810\pi\)
\(258\) 0 0
\(259\) −230.832 −0.891244
\(260\) 62.1536i 0.239052i
\(261\) 0 0
\(262\) 181.466 0.692618
\(263\) − 195.280i − 0.742510i −0.928531 0.371255i \(-0.878928\pi\)
0.928531 0.371255i \(-0.121072\pi\)
\(264\) 0 0
\(265\) −294.114 −1.10986
\(266\) 398.996i 1.49998i
\(267\) 0 0
\(268\) 43.6934 0.163035
\(269\) 178.865i 0.664927i 0.943116 + 0.332464i \(0.107880\pi\)
−0.943116 + 0.332464i \(0.892120\pi\)
\(270\) 0 0
\(271\) −430.708 −1.58933 −0.794664 0.607050i \(-0.792353\pi\)
−0.794664 + 0.607050i \(0.792353\pi\)
\(272\) − 772.096i − 2.83859i
\(273\) 0 0
\(274\) 88.8494 0.324268
\(275\) − 4.14643i − 0.0150779i
\(276\) 0 0
\(277\) 84.5848 0.305360 0.152680 0.988276i \(-0.451210\pi\)
0.152680 + 0.988276i \(0.451210\pi\)
\(278\) 693.638i 2.49510i
\(279\) 0 0
\(280\) −373.294 −1.33319
\(281\) 364.237i 1.29622i 0.761548 + 0.648109i \(0.224440\pi\)
−0.761548 + 0.648109i \(0.775560\pi\)
\(282\) 0 0
\(283\) 463.975 1.63949 0.819744 0.572730i \(-0.194116\pi\)
0.819744 + 0.572730i \(0.194116\pi\)
\(284\) 478.398i 1.68450i
\(285\) 0 0
\(286\) −14.7308 −0.0515063
\(287\) 254.043i 0.885167i
\(288\) 0 0
\(289\) −586.537 −2.02954
\(290\) − 87.9932i − 0.303425i
\(291\) 0 0
\(292\) 459.235 1.57272
\(293\) − 132.578i − 0.452484i −0.974071 0.226242i \(-0.927356\pi\)
0.974071 0.226242i \(-0.0726440\pi\)
\(294\) 0 0
\(295\) −484.127 −1.64111
\(296\) − 931.764i − 3.14785i
\(297\) 0 0
\(298\) −325.516 −1.09234
\(299\) − 1.85423i − 0.00620144i
\(300\) 0 0
\(301\) −153.171 −0.508875
\(302\) 1044.80i 3.45961i
\(303\) 0 0
\(304\) −686.123 −2.25698
\(305\) − 489.351i − 1.60443i
\(306\) 0 0
\(307\) −147.861 −0.481631 −0.240815 0.970571i \(-0.577415\pi\)
−0.240815 + 0.970571i \(0.577415\pi\)
\(308\) − 111.584i − 0.362287i
\(309\) 0 0
\(310\) 749.416 2.41747
\(311\) 95.6322i 0.307499i 0.988110 + 0.153749i \(0.0491349\pi\)
−0.988110 + 0.153749i \(0.950865\pi\)
\(312\) 0 0
\(313\) 544.450 1.73946 0.869728 0.493532i \(-0.164294\pi\)
0.869728 + 0.493532i \(0.164294\pi\)
\(314\) − 238.624i − 0.759948i
\(315\) 0 0
\(316\) 274.283 0.867984
\(317\) − 64.0473i − 0.202042i −0.994884 0.101021i \(-0.967789\pi\)
0.994884 0.101021i \(-0.0322109\pi\)
\(318\) 0 0
\(319\) 14.3323 0.0449289
\(320\) 80.4381i 0.251369i
\(321\) 0 0
\(322\) 20.4378 0.0634714
\(323\) 778.045i 2.40881i
\(324\) 0 0
\(325\) 1.90779 0.00587013
\(326\) − 611.293i − 1.87513i
\(327\) 0 0
\(328\) −1025.45 −3.12639
\(329\) 57.2870i 0.174125i
\(330\) 0 0
\(331\) −5.52777 −0.0167002 −0.00835010 0.999965i \(-0.502658\pi\)
−0.00835010 + 0.999965i \(0.502658\pi\)
\(332\) 46.5780i 0.140295i
\(333\) 0 0
\(334\) −388.328 −1.16266
\(335\) − 25.5359i − 0.0762265i
\(336\) 0 0
\(337\) 496.451 1.47315 0.736574 0.676357i \(-0.236442\pi\)
0.736574 + 0.676357i \(0.236442\pi\)
\(338\) 597.601i 1.76805i
\(339\) 0 0
\(340\) −1335.90 −3.92911
\(341\) 122.065i 0.357961i
\(342\) 0 0
\(343\) −339.429 −0.989590
\(344\) − 618.282i − 1.79733i
\(345\) 0 0
\(346\) 407.936 1.17901
\(347\) 675.910i 1.94787i 0.226834 + 0.973933i \(0.427162\pi\)
−0.226834 + 0.973933i \(0.572838\pi\)
\(348\) 0 0
\(349\) −82.2486 −0.235669 −0.117835 0.993033i \(-0.537595\pi\)
−0.117835 + 0.993033i \(0.537595\pi\)
\(350\) 21.0281i 0.0600804i
\(351\) 0 0
\(352\) 74.2248 0.210866
\(353\) − 418.136i − 1.18452i −0.805747 0.592260i \(-0.798236\pi\)
0.805747 0.592260i \(-0.201764\pi\)
\(354\) 0 0
\(355\) 279.592 0.787582
\(356\) 53.4714i 0.150201i
\(357\) 0 0
\(358\) −544.781 −1.52173
\(359\) 162.868i 0.453671i 0.973933 + 0.226836i \(0.0728381\pi\)
−0.973933 + 0.226836i \(0.927162\pi\)
\(360\) 0 0
\(361\) 330.410 0.915262
\(362\) 241.300i 0.666573i
\(363\) 0 0
\(364\) 51.3404 0.141045
\(365\) − 268.392i − 0.735320i
\(366\) 0 0
\(367\) 333.984 0.910038 0.455019 0.890482i \(-0.349632\pi\)
0.455019 + 0.890482i \(0.349632\pi\)
\(368\) 35.1453i 0.0955036i
\(369\) 0 0
\(370\) −999.369 −2.70100
\(371\) 242.945i 0.654839i
\(372\) 0 0
\(373\) −338.457 −0.907390 −0.453695 0.891157i \(-0.649895\pi\)
−0.453695 + 0.891157i \(0.649895\pi\)
\(374\) − 316.616i − 0.846568i
\(375\) 0 0
\(376\) −231.241 −0.615003
\(377\) 6.59435i 0.0174916i
\(378\) 0 0
\(379\) −326.685 −0.861965 −0.430982 0.902360i \(-0.641833\pi\)
−0.430982 + 0.902360i \(0.641833\pi\)
\(380\) 1187.14i 3.12406i
\(381\) 0 0
\(382\) 209.803 0.549224
\(383\) 300.534i 0.784684i 0.919819 + 0.392342i \(0.128335\pi\)
−0.919819 + 0.392342i \(0.871665\pi\)
\(384\) 0 0
\(385\) −65.2135 −0.169386
\(386\) − 31.6619i − 0.0820255i
\(387\) 0 0
\(388\) 1135.64 2.92692
\(389\) 659.134i 1.69443i 0.531249 + 0.847216i \(0.321723\pi\)
−0.531249 + 0.847216i \(0.678277\pi\)
\(390\) 0 0
\(391\) 39.8539 0.101928
\(392\) − 530.885i − 1.35430i
\(393\) 0 0
\(394\) 636.058 1.61436
\(395\) − 160.300i − 0.405822i
\(396\) 0 0
\(397\) 677.485 1.70651 0.853256 0.521492i \(-0.174624\pi\)
0.853256 + 0.521492i \(0.174624\pi\)
\(398\) − 145.748i − 0.366200i
\(399\) 0 0
\(400\) −36.1605 −0.0904013
\(401\) 96.4243i 0.240460i 0.992746 + 0.120230i \(0.0383631\pi\)
−0.992746 + 0.120230i \(0.961637\pi\)
\(402\) 0 0
\(403\) −56.1624 −0.139361
\(404\) 541.832i 1.34117i
\(405\) 0 0
\(406\) −72.6846 −0.179026
\(407\) − 162.777i − 0.399943i
\(408\) 0 0
\(409\) −473.994 −1.15891 −0.579454 0.815005i \(-0.696734\pi\)
−0.579454 + 0.815005i \(0.696734\pi\)
\(410\) 1099.86i 2.68258i
\(411\) 0 0
\(412\) 98.5039 0.239087
\(413\) 399.901i 0.968283i
\(414\) 0 0
\(415\) 27.2217 0.0655944
\(416\) 34.1511i 0.0820940i
\(417\) 0 0
\(418\) −281.361 −0.673113
\(419\) − 615.306i − 1.46851i −0.678873 0.734256i \(-0.737531\pi\)
0.678873 0.734256i \(-0.262469\pi\)
\(420\) 0 0
\(421\) −196.033 −0.465637 −0.232819 0.972520i \(-0.574795\pi\)
−0.232819 + 0.972520i \(0.574795\pi\)
\(422\) 138.710i 0.328697i
\(423\) 0 0
\(424\) −980.658 −2.31287
\(425\) 41.0051i 0.0964826i
\(426\) 0 0
\(427\) −404.216 −0.946642
\(428\) − 175.345i − 0.409684i
\(429\) 0 0
\(430\) −663.142 −1.54219
\(431\) − 687.036i − 1.59405i −0.603946 0.797025i \(-0.706406\pi\)
0.603946 0.797025i \(-0.293594\pi\)
\(432\) 0 0
\(433\) 229.945 0.531052 0.265526 0.964104i \(-0.414454\pi\)
0.265526 + 0.964104i \(0.414454\pi\)
\(434\) − 619.036i − 1.42635i
\(435\) 0 0
\(436\) 701.053 1.60792
\(437\) − 35.4162i − 0.0810438i
\(438\) 0 0
\(439\) −417.590 −0.951229 −0.475615 0.879654i \(-0.657774\pi\)
−0.475615 + 0.879654i \(0.657774\pi\)
\(440\) − 263.237i − 0.598266i
\(441\) 0 0
\(442\) 145.676 0.329585
\(443\) − 220.764i − 0.498339i −0.968460 0.249169i \(-0.919842\pi\)
0.968460 0.249169i \(-0.0801576\pi\)
\(444\) 0 0
\(445\) 31.2504 0.0702257
\(446\) 811.824i 1.82023i
\(447\) 0 0
\(448\) 66.4439 0.148312
\(449\) − 123.950i − 0.276059i −0.990428 0.138029i \(-0.955923\pi\)
0.990428 0.138029i \(-0.0440768\pi\)
\(450\) 0 0
\(451\) −179.144 −0.397216
\(452\) − 676.241i − 1.49611i
\(453\) 0 0
\(454\) 465.093 1.02443
\(455\) − 30.0050i − 0.0659450i
\(456\) 0 0
\(457\) −217.255 −0.475394 −0.237697 0.971339i \(-0.576393\pi\)
−0.237697 + 0.971339i \(0.576393\pi\)
\(458\) − 970.482i − 2.11896i
\(459\) 0 0
\(460\) 60.8092 0.132194
\(461\) − 762.750i − 1.65456i −0.561793 0.827278i \(-0.689888\pi\)
0.561793 0.827278i \(-0.310112\pi\)
\(462\) 0 0
\(463\) 384.073 0.829532 0.414766 0.909928i \(-0.363863\pi\)
0.414766 + 0.909928i \(0.363863\pi\)
\(464\) − 124.990i − 0.269375i
\(465\) 0 0
\(466\) −824.682 −1.76970
\(467\) − 456.988i − 0.978562i −0.872126 0.489281i \(-0.837259\pi\)
0.872126 0.489281i \(-0.162741\pi\)
\(468\) 0 0
\(469\) −21.0933 −0.0449750
\(470\) 248.019i 0.527700i
\(471\) 0 0
\(472\) −1614.22 −3.41995
\(473\) − 108.012i − 0.228356i
\(474\) 0 0
\(475\) 36.4392 0.0767140
\(476\) 1103.48i 2.31824i
\(477\) 0 0
\(478\) 431.801 0.903349
\(479\) − 804.934i − 1.68045i −0.542241 0.840223i \(-0.682424\pi\)
0.542241 0.840223i \(-0.317576\pi\)
\(480\) 0 0
\(481\) 74.8943 0.155705
\(482\) − 1157.12i − 2.40066i
\(483\) 0 0
\(484\) −984.810 −2.03473
\(485\) − 663.708i − 1.36847i
\(486\) 0 0
\(487\) −563.230 −1.15653 −0.578265 0.815849i \(-0.696270\pi\)
−0.578265 + 0.815849i \(0.696270\pi\)
\(488\) − 1631.63i − 3.34351i
\(489\) 0 0
\(490\) −569.404 −1.16205
\(491\) 122.399i 0.249285i 0.992202 + 0.124642i \(0.0397783\pi\)
−0.992202 + 0.124642i \(0.960222\pi\)
\(492\) 0 0
\(493\) −141.736 −0.287496
\(494\) − 129.455i − 0.262055i
\(495\) 0 0
\(496\) 1064.51 2.14619
\(497\) − 230.950i − 0.464687i
\(498\) 0 0
\(499\) 127.102 0.254713 0.127356 0.991857i \(-0.459351\pi\)
0.127356 + 0.991857i \(0.459351\pi\)
\(500\) − 1066.13i − 2.13225i
\(501\) 0 0
\(502\) 369.884 0.736822
\(503\) 628.301i 1.24911i 0.780982 + 0.624554i \(0.214719\pi\)
−0.780982 + 0.624554i \(0.785281\pi\)
\(504\) 0 0
\(505\) 316.664 0.627058
\(506\) 14.4122i 0.0284826i
\(507\) 0 0
\(508\) −99.0496 −0.194979
\(509\) − 567.425i − 1.11478i −0.830249 0.557392i \(-0.811802\pi\)
0.830249 0.557392i \(-0.188198\pi\)
\(510\) 0 0
\(511\) −221.698 −0.433852
\(512\) 1140.34i 2.22723i
\(513\) 0 0
\(514\) −1560.65 −3.03629
\(515\) − 57.5689i − 0.111784i
\(516\) 0 0
\(517\) −40.3973 −0.0781378
\(518\) 825.503i 1.59364i
\(519\) 0 0
\(520\) 121.116 0.232916
\(521\) − 373.194i − 0.716303i −0.933664 0.358151i \(-0.883407\pi\)
0.933664 0.358151i \(-0.116593\pi\)
\(522\) 0 0
\(523\) −323.518 −0.618581 −0.309290 0.950968i \(-0.600091\pi\)
−0.309290 + 0.950968i \(0.600091\pi\)
\(524\) − 445.988i − 0.851123i
\(525\) 0 0
\(526\) −698.361 −1.32768
\(527\) − 1207.13i − 2.29056i
\(528\) 0 0
\(529\) 527.186 0.996571
\(530\) 1051.81i 1.98455i
\(531\) 0 0
\(532\) 980.611 1.84325
\(533\) − 82.4250i − 0.154644i
\(534\) 0 0
\(535\) −102.477 −0.191546
\(536\) − 85.1438i − 0.158850i
\(537\) 0 0
\(538\) 639.659 1.18896
\(539\) − 92.7443i − 0.172067i
\(540\) 0 0
\(541\) −765.040 −1.41412 −0.707061 0.707153i \(-0.749979\pi\)
−0.707061 + 0.707153i \(0.749979\pi\)
\(542\) 1540.30i 2.84188i
\(543\) 0 0
\(544\) −734.027 −1.34931
\(545\) − 409.719i − 0.751777i
\(546\) 0 0
\(547\) 285.623 0.522162 0.261081 0.965317i \(-0.415921\pi\)
0.261081 + 0.965317i \(0.415921\pi\)
\(548\) − 218.365i − 0.398476i
\(549\) 0 0
\(550\) −14.8285 −0.0269609
\(551\) 125.953i 0.228590i
\(552\) 0 0
\(553\) −132.412 −0.239442
\(554\) − 302.493i − 0.546015i
\(555\) 0 0
\(556\) 1704.75 3.06610
\(557\) − 197.446i − 0.354481i −0.984168 0.177240i \(-0.943283\pi\)
0.984168 0.177240i \(-0.0567170\pi\)
\(558\) 0 0
\(559\) 49.6969 0.0889032
\(560\) 568.718i 1.01557i
\(561\) 0 0
\(562\) 1302.59 2.31777
\(563\) 282.539i 0.501845i 0.968007 + 0.250923i \(0.0807340\pi\)
−0.968007 + 0.250923i \(0.919266\pi\)
\(564\) 0 0
\(565\) −395.217 −0.699499
\(566\) − 1659.27i − 2.93157i
\(567\) 0 0
\(568\) 932.237 1.64126
\(569\) 83.5021i 0.146752i 0.997304 + 0.0733762i \(0.0233774\pi\)
−0.997304 + 0.0733762i \(0.976623\pi\)
\(570\) 0 0
\(571\) 352.279 0.616951 0.308476 0.951232i \(-0.400181\pi\)
0.308476 + 0.951232i \(0.400181\pi\)
\(572\) 36.2039i 0.0632935i
\(573\) 0 0
\(574\) 908.509 1.58277
\(575\) − 1.86653i − 0.00324613i
\(576\) 0 0
\(577\) 1028.99 1.78334 0.891668 0.452689i \(-0.149535\pi\)
0.891668 + 0.452689i \(0.149535\pi\)
\(578\) 2097.57i 3.62902i
\(579\) 0 0
\(580\) −216.261 −0.372863
\(581\) − 22.4858i − 0.0387019i
\(582\) 0 0
\(583\) −171.318 −0.293857
\(584\) − 894.894i − 1.53235i
\(585\) 0 0
\(586\) −474.125 −0.809088
\(587\) − 913.820i − 1.55676i −0.627791 0.778382i \(-0.716041\pi\)
0.627791 0.778382i \(-0.283959\pi\)
\(588\) 0 0
\(589\) −1072.71 −1.82124
\(590\) 1731.34i 2.93447i
\(591\) 0 0
\(592\) −1419.56 −2.39790
\(593\) 767.567i 1.29438i 0.762329 + 0.647190i \(0.224056\pi\)
−0.762329 + 0.647190i \(0.775944\pi\)
\(594\) 0 0
\(595\) 644.912 1.08389
\(596\) 800.021i 1.34232i
\(597\) 0 0
\(598\) −6.63111 −0.0110888
\(599\) 289.204i 0.482811i 0.970424 + 0.241405i \(0.0776084\pi\)
−0.970424 + 0.241405i \(0.922392\pi\)
\(600\) 0 0
\(601\) 904.393 1.50481 0.752407 0.658699i \(-0.228893\pi\)
0.752407 + 0.658699i \(0.228893\pi\)
\(602\) 547.772i 0.909920i
\(603\) 0 0
\(604\) 2567.81 4.25134
\(605\) 575.555i 0.951331i
\(606\) 0 0
\(607\) 656.394 1.08137 0.540687 0.841224i \(-0.318164\pi\)
0.540687 + 0.841224i \(0.318164\pi\)
\(608\) 652.292i 1.07285i
\(609\) 0 0
\(610\) −1750.02 −2.86888
\(611\) − 18.5869i − 0.0304205i
\(612\) 0 0
\(613\) −175.804 −0.286793 −0.143397 0.989665i \(-0.545802\pi\)
−0.143397 + 0.989665i \(0.545802\pi\)
\(614\) 528.780i 0.861205i
\(615\) 0 0
\(616\) −217.440 −0.352987
\(617\) 399.481i 0.647457i 0.946150 + 0.323729i \(0.104936\pi\)
−0.946150 + 0.323729i \(0.895064\pi\)
\(618\) 0 0
\(619\) −946.567 −1.52919 −0.764593 0.644513i \(-0.777060\pi\)
−0.764593 + 0.644513i \(0.777060\pi\)
\(620\) − 1841.84i − 2.97071i
\(621\) 0 0
\(622\) 342.000 0.549839
\(623\) − 25.8136i − 0.0414344i
\(624\) 0 0
\(625\) −657.725 −1.05236
\(626\) − 1947.06i − 3.11032i
\(627\) 0 0
\(628\) −586.465 −0.933861
\(629\) 1609.74i 2.55920i
\(630\) 0 0
\(631\) −761.002 −1.20603 −0.603013 0.797731i \(-0.706033\pi\)
−0.603013 + 0.797731i \(0.706033\pi\)
\(632\) − 534.485i − 0.845704i
\(633\) 0 0
\(634\) −229.046 −0.361272
\(635\) 57.8878i 0.0911619i
\(636\) 0 0
\(637\) 42.6720 0.0669890
\(638\) − 51.2552i − 0.0803374i
\(639\) 0 0
\(640\) 797.369 1.24589
\(641\) 917.816i 1.43185i 0.698177 + 0.715925i \(0.253995\pi\)
−0.698177 + 0.715925i \(0.746005\pi\)
\(642\) 0 0
\(643\) −1204.48 −1.87322 −0.936608 0.350379i \(-0.886053\pi\)
−0.936608 + 0.350379i \(0.886053\pi\)
\(644\) − 50.2299i − 0.0779968i
\(645\) 0 0
\(646\) 2782.45 4.30719
\(647\) − 889.738i − 1.37517i −0.726102 0.687587i \(-0.758670\pi\)
0.726102 0.687587i \(-0.241330\pi\)
\(648\) 0 0
\(649\) −282.000 −0.434514
\(650\) − 6.82265i − 0.0104964i
\(651\) 0 0
\(652\) −1502.37 −2.30425
\(653\) 695.115i 1.06449i 0.846589 + 0.532247i \(0.178652\pi\)
−0.846589 + 0.532247i \(0.821348\pi\)
\(654\) 0 0
\(655\) −260.650 −0.397939
\(656\) 1562.29i 2.38155i
\(657\) 0 0
\(658\) 204.870 0.311352
\(659\) − 361.925i − 0.549203i −0.961558 0.274602i \(-0.911454\pi\)
0.961558 0.274602i \(-0.0885460\pi\)
\(660\) 0 0
\(661\) 560.721 0.848292 0.424146 0.905594i \(-0.360574\pi\)
0.424146 + 0.905594i \(0.360574\pi\)
\(662\) 19.7684i 0.0298617i
\(663\) 0 0
\(664\) 90.7648 0.136694
\(665\) − 573.101i − 0.861806i
\(666\) 0 0
\(667\) 6.45172 0.00967275
\(668\) 954.394i 1.42873i
\(669\) 0 0
\(670\) −91.3215 −0.136301
\(671\) − 285.043i − 0.424803i
\(672\) 0 0
\(673\) 981.245 1.45802 0.729008 0.684505i \(-0.239982\pi\)
0.729008 + 0.684505i \(0.239982\pi\)
\(674\) − 1775.41i − 2.63414i
\(675\) 0 0
\(676\) 1468.72 2.17267
\(677\) 148.636i 0.219551i 0.993956 + 0.109776i \(0.0350133\pi\)
−0.993956 + 0.109776i \(0.964987\pi\)
\(678\) 0 0
\(679\) −548.239 −0.807421
\(680\) 2603.21i 3.82826i
\(681\) 0 0
\(682\) 436.528 0.640070
\(683\) − 615.508i − 0.901183i −0.892730 0.450592i \(-0.851213\pi\)
0.892730 0.450592i \(-0.148787\pi\)
\(684\) 0 0
\(685\) −127.620 −0.186306
\(686\) 1213.87i 1.76949i
\(687\) 0 0
\(688\) −941.962 −1.36913
\(689\) − 78.8243i − 0.114404i
\(690\) 0 0
\(691\) 805.875 1.16624 0.583122 0.812385i \(-0.301831\pi\)
0.583122 + 0.812385i \(0.301831\pi\)
\(692\) − 1002.58i − 1.44882i
\(693\) 0 0
\(694\) 2417.19 3.48298
\(695\) − 996.313i − 1.43354i
\(696\) 0 0
\(697\) 1771.60 2.54175
\(698\) 294.138i 0.421401i
\(699\) 0 0
\(700\) 51.6808 0.0738298
\(701\) 188.156i 0.268411i 0.990954 + 0.134205i \(0.0428482\pi\)
−0.990954 + 0.134205i \(0.957152\pi\)
\(702\) 0 0
\(703\) 1430.49 2.03484
\(704\) 46.8545i 0.0665546i
\(705\) 0 0
\(706\) −1495.34 −2.11804
\(707\) − 261.572i − 0.369975i
\(708\) 0 0
\(709\) 840.891 1.18602 0.593012 0.805193i \(-0.297939\pi\)
0.593012 + 0.805193i \(0.297939\pi\)
\(710\) − 999.876i − 1.40828i
\(711\) 0 0
\(712\) 104.198 0.146345
\(713\) 54.9477i 0.0770655i
\(714\) 0 0
\(715\) 21.1587 0.0295926
\(716\) 1338.91i 1.86998i
\(717\) 0 0
\(718\) 582.449 0.811211
\(719\) − 789.346i − 1.09784i −0.835875 0.548919i \(-0.815039\pi\)
0.835875 0.548919i \(-0.184961\pi\)
\(720\) 0 0
\(721\) −47.5533 −0.0659547
\(722\) − 1181.61i − 1.63658i
\(723\) 0 0
\(724\) 593.042 0.819118
\(725\) 6.63808i 0.00915597i
\(726\) 0 0
\(727\) −431.536 −0.593584 −0.296792 0.954942i \(-0.595917\pi\)
−0.296792 + 0.954942i \(0.595917\pi\)
\(728\) − 100.045i − 0.137425i
\(729\) 0 0
\(730\) −959.823 −1.31483
\(731\) 1068.16i 1.46123i
\(732\) 0 0
\(733\) −86.6077 −0.118155 −0.0590775 0.998253i \(-0.518816\pi\)
−0.0590775 + 0.998253i \(0.518816\pi\)
\(734\) − 1194.39i − 1.62724i
\(735\) 0 0
\(736\) 33.4124 0.0453973
\(737\) − 14.8744i − 0.0201824i
\(738\) 0 0
\(739\) 810.019 1.09610 0.548051 0.836445i \(-0.315370\pi\)
0.548051 + 0.836445i \(0.315370\pi\)
\(740\) 2456.15i 3.31912i
\(741\) 0 0
\(742\) 868.821 1.17092
\(743\) 1176.59i 1.58356i 0.610804 + 0.791782i \(0.290846\pi\)
−0.610804 + 0.791782i \(0.709154\pi\)
\(744\) 0 0
\(745\) 467.558 0.627595
\(746\) 1210.39i 1.62251i
\(747\) 0 0
\(748\) −778.148 −1.04030
\(749\) 84.6487i 0.113016i
\(750\) 0 0
\(751\) 505.244 0.672761 0.336381 0.941726i \(-0.390797\pi\)
0.336381 + 0.941726i \(0.390797\pi\)
\(752\) 352.299i 0.468483i
\(753\) 0 0
\(754\) 23.5827 0.0312768
\(755\) − 1500.71i − 1.98770i
\(756\) 0 0
\(757\) 369.709 0.488387 0.244193 0.969727i \(-0.421477\pi\)
0.244193 + 0.969727i \(0.421477\pi\)
\(758\) 1168.29i 1.54128i
\(759\) 0 0
\(760\) 2313.34 3.04387
\(761\) − 706.092i − 0.927848i −0.885875 0.463924i \(-0.846441\pi\)
0.885875 0.463924i \(-0.153559\pi\)
\(762\) 0 0
\(763\) −338.438 −0.443562
\(764\) − 515.633i − 0.674913i
\(765\) 0 0
\(766\) 1074.77 1.40309
\(767\) − 129.749i − 0.169164i
\(768\) 0 0
\(769\) −165.103 −0.214698 −0.107349 0.994221i \(-0.534236\pi\)
−0.107349 + 0.994221i \(0.534236\pi\)
\(770\) 233.217i 0.302879i
\(771\) 0 0
\(772\) −77.8153 −0.100797
\(773\) 902.380i 1.16737i 0.811979 + 0.583687i \(0.198390\pi\)
−0.811979 + 0.583687i \(0.801610\pi\)
\(774\) 0 0
\(775\) −56.5348 −0.0729482
\(776\) − 2212.99i − 2.85179i
\(777\) 0 0
\(778\) 2357.20 3.02982
\(779\) − 1574.33i − 2.02097i
\(780\) 0 0
\(781\) 162.859 0.208527
\(782\) − 142.526i − 0.182258i
\(783\) 0 0
\(784\) −808.811 −1.03165
\(785\) 342.749i 0.436623i
\(786\) 0 0
\(787\) −1151.83 −1.46357 −0.731787 0.681533i \(-0.761313\pi\)
−0.731787 + 0.681533i \(0.761313\pi\)
\(788\) − 1563.24i − 1.98380i
\(789\) 0 0
\(790\) −573.265 −0.725652
\(791\) 326.459i 0.412717i
\(792\) 0 0
\(793\) 131.149 0.165384
\(794\) − 2422.82i − 3.05142i
\(795\) 0 0
\(796\) −358.204 −0.450005
\(797\) 1178.11i 1.47819i 0.673603 + 0.739093i \(0.264746\pi\)
−0.673603 + 0.739093i \(0.735254\pi\)
\(798\) 0 0
\(799\) 399.498 0.499998
\(800\) 34.3776i 0.0429720i
\(801\) 0 0
\(802\) 344.833 0.429966
\(803\) − 156.336i − 0.194690i
\(804\) 0 0
\(805\) −29.3560 −0.0364671
\(806\) 200.848i 0.249191i
\(807\) 0 0
\(808\) 1055.85 1.30674
\(809\) − 1162.01i − 1.43636i −0.695859 0.718178i \(-0.744976\pi\)
0.695859 0.718178i \(-0.255024\pi\)
\(810\) 0 0
\(811\) −1217.85 −1.50167 −0.750835 0.660490i \(-0.770349\pi\)
−0.750835 + 0.660490i \(0.770349\pi\)
\(812\) 178.637i 0.219996i
\(813\) 0 0
\(814\) −582.123 −0.715139
\(815\) 878.036i 1.07734i
\(816\) 0 0
\(817\) 949.220 1.16184
\(818\) 1695.10i 2.07225i
\(819\) 0 0
\(820\) 2703.12 3.29648
\(821\) 299.911i 0.365299i 0.983178 + 0.182650i \(0.0584674\pi\)
−0.983178 + 0.182650i \(0.941533\pi\)
\(822\) 0 0
\(823\) 642.499 0.780679 0.390340 0.920671i \(-0.372358\pi\)
0.390340 + 0.920671i \(0.372358\pi\)
\(824\) − 191.951i − 0.232950i
\(825\) 0 0
\(826\) 1430.13 1.73139
\(827\) 17.8512i 0.0215855i 0.999942 + 0.0107928i \(0.00343551\pi\)
−0.999942 + 0.0107928i \(0.996564\pi\)
\(828\) 0 0
\(829\) 1360.14 1.64070 0.820352 0.571859i \(-0.193777\pi\)
0.820352 + 0.571859i \(0.193777\pi\)
\(830\) − 97.3503i − 0.117289i
\(831\) 0 0
\(832\) −21.5579 −0.0259110
\(833\) 917.171i 1.10105i
\(834\) 0 0
\(835\) 557.779 0.667999
\(836\) 691.501i 0.827154i
\(837\) 0 0
\(838\) −2200.46 −2.62585
\(839\) − 479.083i − 0.571017i −0.958376 0.285509i \(-0.907838\pi\)
0.958376 0.285509i \(-0.0921625\pi\)
\(840\) 0 0
\(841\) 818.055 0.972717
\(842\) 701.055i 0.832607i
\(843\) 0 0
\(844\) 340.907 0.403918
\(845\) − 858.369i − 1.01582i
\(846\) 0 0
\(847\) 475.423 0.561302
\(848\) 1494.05i 1.76185i
\(849\) 0 0
\(850\) 146.643 0.172521
\(851\) − 73.2744i − 0.0861038i
\(852\) 0 0
\(853\) −1483.28 −1.73890 −0.869452 0.494018i \(-0.835528\pi\)
−0.869452 + 0.494018i \(0.835528\pi\)
\(854\) 1445.56i 1.69269i
\(855\) 0 0
\(856\) −341.688 −0.399168
\(857\) 1382.14i 1.61277i 0.591393 + 0.806383i \(0.298578\pi\)
−0.591393 + 0.806383i \(0.701422\pi\)
\(858\) 0 0
\(859\) 54.9906 0.0640170 0.0320085 0.999488i \(-0.489810\pi\)
0.0320085 + 0.999488i \(0.489810\pi\)
\(860\) 1629.80i 1.89512i
\(861\) 0 0
\(862\) −2456.98 −2.85032
\(863\) − 115.645i − 0.134004i −0.997753 0.0670020i \(-0.978657\pi\)
0.997753 0.0670020i \(-0.0213434\pi\)
\(864\) 0 0
\(865\) −585.943 −0.677391
\(866\) − 822.331i − 0.949574i
\(867\) 0 0
\(868\) −1521.40 −1.75277
\(869\) − 93.3732i − 0.107449i
\(870\) 0 0
\(871\) 6.84378 0.00785738
\(872\) − 1366.12i − 1.56665i
\(873\) 0 0
\(874\) −126.655 −0.144915
\(875\) 514.679i 0.588205i
\(876\) 0 0
\(877\) −884.784 −1.00888 −0.504438 0.863448i \(-0.668300\pi\)
−0.504438 + 0.863448i \(0.668300\pi\)
\(878\) 1493.39i 1.70089i
\(879\) 0 0
\(880\) −401.045 −0.455733
\(881\) 431.615i 0.489915i 0.969534 + 0.244958i \(0.0787740\pi\)
−0.969534 + 0.244958i \(0.921226\pi\)
\(882\) 0 0
\(883\) −306.380 −0.346976 −0.173488 0.984836i \(-0.555504\pi\)
−0.173488 + 0.984836i \(0.555504\pi\)
\(884\) − 358.029i − 0.405010i
\(885\) 0 0
\(886\) −789.497 −0.891080
\(887\) 701.411i 0.790768i 0.918516 + 0.395384i \(0.129389\pi\)
−0.918516 + 0.395384i \(0.870611\pi\)
\(888\) 0 0
\(889\) 47.8168 0.0537871
\(890\) − 111.758i − 0.125571i
\(891\) 0 0
\(892\) 1995.22 2.23679
\(893\) − 355.014i − 0.397552i
\(894\) 0 0
\(895\) 782.501 0.874302
\(896\) − 658.647i − 0.735097i
\(897\) 0 0
\(898\) −443.271 −0.493621
\(899\) − 195.415i − 0.217369i
\(900\) 0 0
\(901\) 1694.21 1.88037
\(902\) 640.657i 0.710262i
\(903\) 0 0
\(904\) −1317.77 −1.45771
\(905\) − 346.593i − 0.382976i
\(906\) 0 0
\(907\) 993.097 1.09492 0.547462 0.836830i \(-0.315594\pi\)
0.547462 + 0.836830i \(0.315594\pi\)
\(908\) − 1143.06i − 1.25888i
\(909\) 0 0
\(910\) −107.304 −0.117916
\(911\) − 1663.25i − 1.82574i −0.408245 0.912872i \(-0.633859\pi\)
0.408245 0.912872i \(-0.366141\pi\)
\(912\) 0 0
\(913\) 15.8564 0.0173673
\(914\) 776.949i 0.850053i
\(915\) 0 0
\(916\) −2385.15 −2.60388
\(917\) 215.304i 0.234791i
\(918\) 0 0
\(919\) 1638.37 1.78277 0.891387 0.453244i \(-0.149733\pi\)
0.891387 + 0.453244i \(0.149733\pi\)
\(920\) − 118.497i − 0.128801i
\(921\) 0 0
\(922\) −2727.75 −2.95851
\(923\) 74.9323i 0.0811834i
\(924\) 0 0
\(925\) 75.3909 0.0815037
\(926\) − 1373.52i − 1.48329i
\(927\) 0 0
\(928\) −118.827 −0.128047
\(929\) 885.774i 0.953471i 0.879047 + 0.476735i \(0.158180\pi\)
−0.879047 + 0.476735i \(0.841820\pi\)
\(930\) 0 0
\(931\) 815.043 0.875449
\(932\) 2026.82i 2.17470i
\(933\) 0 0
\(934\) −1634.28 −1.74977
\(935\) 454.775i 0.486390i
\(936\) 0 0
\(937\) −1680.01 −1.79297 −0.896485 0.443074i \(-0.853888\pi\)
−0.896485 + 0.443074i \(0.853888\pi\)
\(938\) 75.4338i 0.0804198i
\(939\) 0 0
\(940\) 609.556 0.648464
\(941\) − 1128.85i − 1.19963i −0.800141 0.599813i \(-0.795242\pi\)
0.800141 0.599813i \(-0.204758\pi\)
\(942\) 0 0
\(943\) −80.6422 −0.0855167
\(944\) 2459.28i 2.60517i
\(945\) 0 0
\(946\) −386.274 −0.408324
\(947\) − 1236.52i − 1.30573i −0.757476 0.652863i \(-0.773568\pi\)
0.757476 0.652863i \(-0.226432\pi\)
\(948\) 0 0
\(949\) 71.9307 0.0757963
\(950\) − 130.314i − 0.137172i
\(951\) 0 0
\(952\) 2150.32 2.25874
\(953\) − 1235.07i − 1.29598i −0.761651 0.647988i \(-0.775611\pi\)
0.761651 0.647988i \(-0.224389\pi\)
\(954\) 0 0
\(955\) −301.353 −0.315553
\(956\) − 1061.24i − 1.11008i
\(957\) 0 0
\(958\) −2878.61 −3.00481
\(959\) 105.417i 0.109924i
\(960\) 0 0
\(961\) 703.299 0.731840
\(962\) − 267.837i − 0.278417i
\(963\) 0 0
\(964\) −2843.84 −2.95005
\(965\) 45.4778i 0.0471273i
\(966\) 0 0
\(967\) −1620.15 −1.67544 −0.837718 0.546103i \(-0.816111\pi\)
−0.837718 + 0.546103i \(0.816111\pi\)
\(968\) 1919.06i 1.98250i
\(969\) 0 0
\(970\) −2373.55 −2.44696
\(971\) 1545.85i 1.59202i 0.605286 + 0.796008i \(0.293059\pi\)
−0.605286 + 0.796008i \(0.706941\pi\)
\(972\) 0 0
\(973\) −822.979 −0.845816
\(974\) 2014.23i 2.06799i
\(975\) 0 0
\(976\) −2485.82 −2.54695
\(977\) − 1453.06i − 1.48726i −0.668589 0.743632i \(-0.733102\pi\)
0.668589 0.743632i \(-0.266898\pi\)
\(978\) 0 0
\(979\) 18.2031 0.0185936
\(980\) 1399.42i 1.42798i
\(981\) 0 0
\(982\) 437.723 0.445746
\(983\) − 678.816i − 0.690555i −0.938501 0.345278i \(-0.887785\pi\)
0.938501 0.345278i \(-0.112215\pi\)
\(984\) 0 0
\(985\) −913.607 −0.927520
\(986\) 506.875i 0.514072i
\(987\) 0 0
\(988\) −318.162 −0.322027
\(989\) − 48.6220i − 0.0491628i
\(990\) 0 0
\(991\) −269.445 −0.271892 −0.135946 0.990716i \(-0.543407\pi\)
−0.135946 + 0.990716i \(0.543407\pi\)
\(992\) − 1012.02i − 1.02018i
\(993\) 0 0
\(994\) −825.922 −0.830908
\(995\) 209.346i 0.210398i
\(996\) 0 0
\(997\) −880.896 −0.883547 −0.441774 0.897127i \(-0.645651\pi\)
−0.441774 + 0.897127i \(0.645651\pi\)
\(998\) − 454.541i − 0.455452i
\(999\) 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.890.8 84
3.2 odd 2 inner 1143.3.b.a.890.77 yes 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1143.3.b.a.890.8 84 1.1 even 1 trivial
1143.3.b.a.890.77 yes 84 3.2 odd 2 inner