Properties

Label 1089.3.b.i.485.8
Level $1089$
Weight $3$
Character 1089.485
Analytic conductor $29.673$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 48 x^{14} + 921 x^{12} + 8986 x^{10} + 46812 x^{8} + 125072 x^{6} + 152129 x^{4} + 65614 x^{2} + 5041\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 485.8
Root \(-0.311356i\) of defining polynomial
Character \(\chi\) \(=\) 1089.485
Dual form 1089.3.b.i.485.9

$q$-expansion

\(f(q)\) \(=\) \(q-0.311356i q^{2} +3.90306 q^{4} -5.94641i q^{5} +8.67101 q^{7} -2.46067i q^{8} +O(q^{10})\) \(q-0.311356i q^{2} +3.90306 q^{4} -5.94641i q^{5} +8.67101 q^{7} -2.46067i q^{8} -1.85145 q^{10} +18.6982 q^{13} -2.69977i q^{14} +14.8461 q^{16} -9.59948i q^{17} +5.76332 q^{19} -23.2092i q^{20} +41.5830i q^{23} -10.3598 q^{25} -5.82181i q^{26} +33.8434 q^{28} +17.3733i q^{29} -13.6771 q^{31} -14.4651i q^{32} -2.98886 q^{34} -51.5613i q^{35} -7.21391 q^{37} -1.79445i q^{38} -14.6321 q^{40} -53.1361i q^{41} +43.3682 q^{43} +12.9471 q^{46} +18.8439i q^{47} +26.1864 q^{49} +3.22558i q^{50} +72.9802 q^{52} +54.5516i q^{53} -21.3365i q^{56} +5.40928 q^{58} +35.3312i q^{59} -117.852 q^{61} +4.25847i q^{62} +54.8805 q^{64} -111.187i q^{65} -91.5111 q^{67} -37.4673i q^{68} -16.0540 q^{70} -115.816i q^{71} -52.9453 q^{73} +2.24610i q^{74} +22.4946 q^{76} +2.04912 q^{79} -88.2809i q^{80} -16.5443 q^{82} +28.8194i q^{83} -57.0824 q^{85} -13.5030i q^{86} +134.980i q^{89} +162.132 q^{91} +162.301i q^{92} +5.86717 q^{94} -34.2710i q^{95} -9.97627 q^{97} -8.15330i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{4} - 8 q^{7} + O(q^{10}) \) \( 16 q - 32 q^{4} - 8 q^{7} + 24 q^{10} + 4 q^{13} + 28 q^{16} - 20 q^{19} - 44 q^{25} + 16 q^{28} + 28 q^{31} + 148 q^{34} - 148 q^{37} + 224 q^{40} + 272 q^{43} + 208 q^{46} + 348 q^{49} + 520 q^{52} - 44 q^{58} + 224 q^{61} + 436 q^{64} + 24 q^{67} - 664 q^{70} - 4 q^{73} + 1052 q^{76} + 216 q^{79} + 348 q^{82} + 416 q^{85} - 168 q^{91} + 1140 q^{94} - 44 q^{97} + O(q^{100}) \)

Character values

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

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

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.311356i − 0.155678i −0.996966 0.0778391i \(-0.975198\pi\)
0.996966 0.0778391i \(-0.0248020\pi\)
\(3\) 0 0
\(4\) 3.90306 0.975764
\(5\) − 5.94641i − 1.18928i −0.803992 0.594641i \(-0.797294\pi\)
0.803992 0.594641i \(-0.202706\pi\)
\(6\) 0 0
\(7\) 8.67101 1.23872 0.619358 0.785109i \(-0.287393\pi\)
0.619358 + 0.785109i \(0.287393\pi\)
\(8\) − 2.46067i − 0.307583i
\(9\) 0 0
\(10\) −1.85145 −0.185145
\(11\) 0 0
\(12\) 0 0
\(13\) 18.6982 1.43832 0.719162 0.694842i \(-0.244526\pi\)
0.719162 + 0.694842i \(0.244526\pi\)
\(14\) − 2.69977i − 0.192841i
\(15\) 0 0
\(16\) 14.8461 0.927880
\(17\) − 9.59948i − 0.564675i −0.959315 0.282338i \(-0.908890\pi\)
0.959315 0.282338i \(-0.0911098\pi\)
\(18\) 0 0
\(19\) 5.76332 0.303333 0.151666 0.988432i \(-0.451536\pi\)
0.151666 + 0.988432i \(0.451536\pi\)
\(20\) − 23.2092i − 1.16046i
\(21\) 0 0
\(22\) 0 0
\(23\) 41.5830i 1.80796i 0.427578 + 0.903979i \(0.359367\pi\)
−0.427578 + 0.903979i \(0.640633\pi\)
\(24\) 0 0
\(25\) −10.3598 −0.414390
\(26\) − 5.82181i − 0.223916i
\(27\) 0 0
\(28\) 33.8434 1.20869
\(29\) 17.3733i 0.599078i 0.954084 + 0.299539i \(0.0968329\pi\)
−0.954084 + 0.299539i \(0.903167\pi\)
\(30\) 0 0
\(31\) −13.6771 −0.441198 −0.220599 0.975365i \(-0.570801\pi\)
−0.220599 + 0.975365i \(0.570801\pi\)
\(32\) − 14.4651i − 0.452034i
\(33\) 0 0
\(34\) −2.98886 −0.0879076
\(35\) − 51.5613i − 1.47318i
\(36\) 0 0
\(37\) −7.21391 −0.194970 −0.0974852 0.995237i \(-0.531080\pi\)
−0.0974852 + 0.995237i \(0.531080\pi\)
\(38\) − 1.79445i − 0.0472223i
\(39\) 0 0
\(40\) −14.6321 −0.365803
\(41\) − 53.1361i − 1.29600i −0.761640 0.648001i \(-0.775605\pi\)
0.761640 0.648001i \(-0.224395\pi\)
\(42\) 0 0
\(43\) 43.3682 1.00856 0.504282 0.863539i \(-0.331757\pi\)
0.504282 + 0.863539i \(0.331757\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 12.9471 0.281459
\(47\) 18.8439i 0.400934i 0.979700 + 0.200467i \(0.0642460\pi\)
−0.979700 + 0.200467i \(0.935754\pi\)
\(48\) 0 0
\(49\) 26.1864 0.534416
\(50\) 3.22558i 0.0645115i
\(51\) 0 0
\(52\) 72.9802 1.40347
\(53\) 54.5516i 1.02928i 0.857408 + 0.514638i \(0.172074\pi\)
−0.857408 + 0.514638i \(0.827926\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 21.3365i − 0.381008i
\(57\) 0 0
\(58\) 5.40928 0.0932634
\(59\) 35.3312i 0.598833i 0.954122 + 0.299417i \(0.0967920\pi\)
−0.954122 + 0.299417i \(0.903208\pi\)
\(60\) 0 0
\(61\) −117.852 −1.93201 −0.966004 0.258528i \(-0.916762\pi\)
−0.966004 + 0.258528i \(0.916762\pi\)
\(62\) 4.25847i 0.0686849i
\(63\) 0 0
\(64\) 54.8805 0.857508
\(65\) − 111.187i − 1.71057i
\(66\) 0 0
\(67\) −91.5111 −1.36584 −0.682919 0.730494i \(-0.739290\pi\)
−0.682919 + 0.730494i \(0.739290\pi\)
\(68\) − 37.4673i − 0.550990i
\(69\) 0 0
\(70\) −16.0540 −0.229342
\(71\) − 115.816i − 1.63120i −0.578613 0.815602i \(-0.696406\pi\)
0.578613 0.815602i \(-0.303594\pi\)
\(72\) 0 0
\(73\) −52.9453 −0.725278 −0.362639 0.931930i \(-0.618124\pi\)
−0.362639 + 0.931930i \(0.618124\pi\)
\(74\) 2.24610i 0.0303526i
\(75\) 0 0
\(76\) 22.4946 0.295981
\(77\) 0 0
\(78\) 0 0
\(79\) 2.04912 0.0259383 0.0129691 0.999916i \(-0.495872\pi\)
0.0129691 + 0.999916i \(0.495872\pi\)
\(80\) − 88.2809i − 1.10351i
\(81\) 0 0
\(82\) −16.5443 −0.201759
\(83\) 28.8194i 0.347222i 0.984814 + 0.173611i \(0.0555435\pi\)
−0.984814 + 0.173611i \(0.944457\pi\)
\(84\) 0 0
\(85\) −57.0824 −0.671558
\(86\) − 13.5030i − 0.157011i
\(87\) 0 0
\(88\) 0 0
\(89\) 134.980i 1.51663i 0.651891 + 0.758313i \(0.273976\pi\)
−0.651891 + 0.758313i \(0.726024\pi\)
\(90\) 0 0
\(91\) 162.132 1.78168
\(92\) 162.301i 1.76414i
\(93\) 0 0
\(94\) 5.86717 0.0624168
\(95\) − 34.2710i − 0.360748i
\(96\) 0 0
\(97\) −9.97627 −0.102848 −0.0514241 0.998677i \(-0.516376\pi\)
−0.0514241 + 0.998677i \(0.516376\pi\)
\(98\) − 8.15330i − 0.0831970i
\(99\) 0 0
\(100\) −40.4347 −0.404347
\(101\) − 172.426i − 1.70719i −0.520939 0.853594i \(-0.674418\pi\)
0.520939 0.853594i \(-0.325582\pi\)
\(102\) 0 0
\(103\) −113.639 −1.10329 −0.551643 0.834080i \(-0.685999\pi\)
−0.551643 + 0.834080i \(0.685999\pi\)
\(104\) − 46.0101i − 0.442405i
\(105\) 0 0
\(106\) 16.9850 0.160236
\(107\) 83.1724i 0.777312i 0.921383 + 0.388656i \(0.127061\pi\)
−0.921383 + 0.388656i \(0.872939\pi\)
\(108\) 0 0
\(109\) 125.432 1.15075 0.575377 0.817888i \(-0.304855\pi\)
0.575377 + 0.817888i \(0.304855\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 128.731 1.14938
\(113\) − 101.186i − 0.895451i −0.894171 0.447726i \(-0.852234\pi\)
0.894171 0.447726i \(-0.147766\pi\)
\(114\) 0 0
\(115\) 247.270 2.15017
\(116\) 67.8089i 0.584559i
\(117\) 0 0
\(118\) 11.0006 0.0932252
\(119\) − 83.2372i − 0.699472i
\(120\) 0 0
\(121\) 0 0
\(122\) 36.6941i 0.300771i
\(123\) 0 0
\(124\) −53.3827 −0.430505
\(125\) − 87.0569i − 0.696455i
\(126\) 0 0
\(127\) 123.996 0.976346 0.488173 0.872747i \(-0.337663\pi\)
0.488173 + 0.872747i \(0.337663\pi\)
\(128\) − 74.9478i − 0.585529i
\(129\) 0 0
\(130\) −34.6189 −0.266299
\(131\) − 52.5607i − 0.401227i −0.979670 0.200614i \(-0.935706\pi\)
0.979670 0.200614i \(-0.0642935\pi\)
\(132\) 0 0
\(133\) 49.9738 0.375743
\(134\) 28.4926i 0.212631i
\(135\) 0 0
\(136\) −23.6211 −0.173685
\(137\) − 125.783i − 0.918121i −0.888405 0.459060i \(-0.848186\pi\)
0.888405 0.459060i \(-0.151814\pi\)
\(138\) 0 0
\(139\) 53.6460 0.385943 0.192971 0.981204i \(-0.438188\pi\)
0.192971 + 0.981204i \(0.438188\pi\)
\(140\) − 201.247i − 1.43748i
\(141\) 0 0
\(142\) −36.0599 −0.253943
\(143\) 0 0
\(144\) 0 0
\(145\) 103.309 0.712473
\(146\) 16.4848i 0.112910i
\(147\) 0 0
\(148\) −28.1563 −0.190245
\(149\) 4.23732i 0.0284384i 0.999899 + 0.0142192i \(0.00452626\pi\)
−0.999899 + 0.0142192i \(0.995474\pi\)
\(150\) 0 0
\(151\) 94.3874 0.625082 0.312541 0.949904i \(-0.398820\pi\)
0.312541 + 0.949904i \(0.398820\pi\)
\(152\) − 14.1816i − 0.0933001i
\(153\) 0 0
\(154\) 0 0
\(155\) 81.3299i 0.524709i
\(156\) 0 0
\(157\) −148.683 −0.947028 −0.473514 0.880786i \(-0.657015\pi\)
−0.473514 + 0.880786i \(0.657015\pi\)
\(158\) − 0.638008i − 0.00403802i
\(159\) 0 0
\(160\) −86.0153 −0.537596
\(161\) 360.567i 2.23954i
\(162\) 0 0
\(163\) −91.7512 −0.562891 −0.281445 0.959577i \(-0.590814\pi\)
−0.281445 + 0.959577i \(0.590814\pi\)
\(164\) − 207.393i − 1.26459i
\(165\) 0 0
\(166\) 8.97310 0.0540548
\(167\) − 14.2677i − 0.0854351i −0.999087 0.0427175i \(-0.986398\pi\)
0.999087 0.0427175i \(-0.0136016\pi\)
\(168\) 0 0
\(169\) 180.623 1.06878
\(170\) 17.7730i 0.104547i
\(171\) 0 0
\(172\) 169.269 0.984120
\(173\) − 232.280i − 1.34266i −0.741158 0.671331i \(-0.765723\pi\)
0.741158 0.671331i \(-0.234277\pi\)
\(174\) 0 0
\(175\) −89.8295 −0.513312
\(176\) 0 0
\(177\) 0 0
\(178\) 42.0268 0.236105
\(179\) 143.658i 0.802560i 0.915956 + 0.401280i \(0.131434\pi\)
−0.915956 + 0.401280i \(0.868566\pi\)
\(180\) 0 0
\(181\) −239.706 −1.32434 −0.662171 0.749353i \(-0.730365\pi\)
−0.662171 + 0.749353i \(0.730365\pi\)
\(182\) − 50.4810i − 0.277368i
\(183\) 0 0
\(184\) 102.322 0.556098
\(185\) 42.8968i 0.231875i
\(186\) 0 0
\(187\) 0 0
\(188\) 73.5489i 0.391218i
\(189\) 0 0
\(190\) −10.6705 −0.0561606
\(191\) − 37.5317i − 0.196501i −0.995162 0.0982506i \(-0.968675\pi\)
0.995162 0.0982506i \(-0.0313247\pi\)
\(192\) 0 0
\(193\) 203.572 1.05478 0.527389 0.849624i \(-0.323171\pi\)
0.527389 + 0.849624i \(0.323171\pi\)
\(194\) 3.10617i 0.0160112i
\(195\) 0 0
\(196\) 102.207 0.521464
\(197\) 208.477i 1.05826i 0.848541 + 0.529129i \(0.177481\pi\)
−0.848541 + 0.529129i \(0.822519\pi\)
\(198\) 0 0
\(199\) −249.874 −1.25565 −0.627825 0.778355i \(-0.716055\pi\)
−0.627825 + 0.778355i \(0.716055\pi\)
\(200\) 25.4919i 0.127460i
\(201\) 0 0
\(202\) −53.6859 −0.265772
\(203\) 150.644i 0.742088i
\(204\) 0 0
\(205\) −315.969 −1.54131
\(206\) 35.3821i 0.171758i
\(207\) 0 0
\(208\) 277.595 1.33459
\(209\) 0 0
\(210\) 0 0
\(211\) 121.553 0.576081 0.288040 0.957618i \(-0.406996\pi\)
0.288040 + 0.957618i \(0.406996\pi\)
\(212\) 212.918i 1.00433i
\(213\) 0 0
\(214\) 25.8963 0.121011
\(215\) − 257.885i − 1.19947i
\(216\) 0 0
\(217\) −118.595 −0.546519
\(218\) − 39.0541i − 0.179147i
\(219\) 0 0
\(220\) 0 0
\(221\) − 179.493i − 0.812186i
\(222\) 0 0
\(223\) −71.8121 −0.322027 −0.161014 0.986952i \(-0.551476\pi\)
−0.161014 + 0.986952i \(0.551476\pi\)
\(224\) − 125.427i − 0.559942i
\(225\) 0 0
\(226\) −31.5049 −0.139402
\(227\) − 234.045i − 1.03104i −0.856879 0.515518i \(-0.827600\pi\)
0.856879 0.515518i \(-0.172400\pi\)
\(228\) 0 0
\(229\) 68.1310 0.297515 0.148758 0.988874i \(-0.452473\pi\)
0.148758 + 0.988874i \(0.452473\pi\)
\(230\) − 76.9889i − 0.334735i
\(231\) 0 0
\(232\) 42.7498 0.184267
\(233\) − 106.735i − 0.458090i −0.973416 0.229045i \(-0.926440\pi\)
0.973416 0.229045i \(-0.0735603\pi\)
\(234\) 0 0
\(235\) 112.054 0.476824
\(236\) 137.900i 0.584320i
\(237\) 0 0
\(238\) −25.9164 −0.108893
\(239\) 348.588i 1.45853i 0.684232 + 0.729265i \(0.260138\pi\)
−0.684232 + 0.729265i \(0.739862\pi\)
\(240\) 0 0
\(241\) −377.429 −1.56609 −0.783047 0.621962i \(-0.786336\pi\)
−0.783047 + 0.621962i \(0.786336\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −459.985 −1.88518
\(245\) − 155.715i − 0.635571i
\(246\) 0 0
\(247\) 107.764 0.436291
\(248\) 33.6549i 0.135705i
\(249\) 0 0
\(250\) −27.1057 −0.108423
\(251\) 301.660i 1.20183i 0.799311 + 0.600917i \(0.205198\pi\)
−0.799311 + 0.600917i \(0.794802\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) − 38.6069i − 0.151996i
\(255\) 0 0
\(256\) 196.187 0.766354
\(257\) − 14.9765i − 0.0582743i −0.999575 0.0291372i \(-0.990724\pi\)
0.999575 0.0291372i \(-0.00927596\pi\)
\(258\) 0 0
\(259\) −62.5519 −0.241513
\(260\) − 433.970i − 1.66912i
\(261\) 0 0
\(262\) −16.3651 −0.0624623
\(263\) 470.671i 1.78962i 0.446443 + 0.894812i \(0.352691\pi\)
−0.446443 + 0.894812i \(0.647309\pi\)
\(264\) 0 0
\(265\) 324.386 1.22410
\(266\) − 15.5597i − 0.0584950i
\(267\) 0 0
\(268\) −357.173 −1.33274
\(269\) − 199.826i − 0.742848i −0.928463 0.371424i \(-0.878870\pi\)
0.928463 0.371424i \(-0.121130\pi\)
\(270\) 0 0
\(271\) −87.2697 −0.322028 −0.161014 0.986952i \(-0.551476\pi\)
−0.161014 + 0.986952i \(0.551476\pi\)
\(272\) − 142.515i − 0.523951i
\(273\) 0 0
\(274\) −39.1632 −0.142931
\(275\) 0 0
\(276\) 0 0
\(277\) −299.219 −1.08021 −0.540106 0.841597i \(-0.681616\pi\)
−0.540106 + 0.841597i \(0.681616\pi\)
\(278\) − 16.7030i − 0.0600828i
\(279\) 0 0
\(280\) −126.875 −0.453126
\(281\) 298.705i 1.06301i 0.847056 + 0.531504i \(0.178373\pi\)
−0.847056 + 0.531504i \(0.821627\pi\)
\(282\) 0 0
\(283\) −204.517 −0.722674 −0.361337 0.932435i \(-0.617680\pi\)
−0.361337 + 0.932435i \(0.617680\pi\)
\(284\) − 452.035i − 1.59167i
\(285\) 0 0
\(286\) 0 0
\(287\) − 460.743i − 1.60538i
\(288\) 0 0
\(289\) 196.850 0.681142
\(290\) − 32.1658i − 0.110916i
\(291\) 0 0
\(292\) −206.648 −0.707700
\(293\) 102.843i 0.351001i 0.984479 + 0.175500i \(0.0561543\pi\)
−0.984479 + 0.175500i \(0.943846\pi\)
\(294\) 0 0
\(295\) 210.093 0.712181
\(296\) 17.7510i 0.0599697i
\(297\) 0 0
\(298\) 1.31932 0.00442724
\(299\) 777.528i 2.60043i
\(300\) 0 0
\(301\) 376.046 1.24932
\(302\) − 29.3881i − 0.0973116i
\(303\) 0 0
\(304\) 85.5627 0.281456
\(305\) 700.799i 2.29770i
\(306\) 0 0
\(307\) 39.5643 0.128874 0.0644369 0.997922i \(-0.479475\pi\)
0.0644369 + 0.997922i \(0.479475\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 25.3226 0.0816857
\(311\) − 139.455i − 0.448407i −0.974542 0.224203i \(-0.928022\pi\)
0.974542 0.224203i \(-0.0719780\pi\)
\(312\) 0 0
\(313\) −226.078 −0.722295 −0.361148 0.932509i \(-0.617615\pi\)
−0.361148 + 0.932509i \(0.617615\pi\)
\(314\) 46.2935i 0.147432i
\(315\) 0 0
\(316\) 7.99784 0.0253096
\(317\) 352.182i 1.11098i 0.831522 + 0.555492i \(0.187470\pi\)
−0.831522 + 0.555492i \(0.812530\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) − 326.342i − 1.01982i
\(321\) 0 0
\(322\) 112.265 0.348648
\(323\) − 55.3249i − 0.171284i
\(324\) 0 0
\(325\) −193.709 −0.596028
\(326\) 28.5673i 0.0876298i
\(327\) 0 0
\(328\) −130.750 −0.398629
\(329\) 163.396i 0.496644i
\(330\) 0 0
\(331\) −84.8580 −0.256369 −0.128184 0.991750i \(-0.540915\pi\)
−0.128184 + 0.991750i \(0.540915\pi\)
\(332\) 112.484i 0.338806i
\(333\) 0 0
\(334\) −4.44233 −0.0133004
\(335\) 544.162i 1.62437i
\(336\) 0 0
\(337\) 527.174 1.56432 0.782158 0.623080i \(-0.214119\pi\)
0.782158 + 0.623080i \(0.214119\pi\)
\(338\) − 56.2383i − 0.166385i
\(339\) 0 0
\(340\) −222.796 −0.655282
\(341\) 0 0
\(342\) 0 0
\(343\) −197.817 −0.576726
\(344\) − 106.715i − 0.310217i
\(345\) 0 0
\(346\) −72.3220 −0.209023
\(347\) − 126.318i − 0.364029i −0.983296 0.182015i \(-0.941738\pi\)
0.983296 0.182015i \(-0.0582618\pi\)
\(348\) 0 0
\(349\) −451.455 −1.29357 −0.646784 0.762673i \(-0.723887\pi\)
−0.646784 + 0.762673i \(0.723887\pi\)
\(350\) 27.9690i 0.0799114i
\(351\) 0 0
\(352\) 0 0
\(353\) 145.710i 0.412775i 0.978470 + 0.206388i \(0.0661708\pi\)
−0.978470 + 0.206388i \(0.933829\pi\)
\(354\) 0 0
\(355\) −688.686 −1.93996
\(356\) 526.833i 1.47987i
\(357\) 0 0
\(358\) 44.7289 0.124941
\(359\) 429.208i 1.19557i 0.801658 + 0.597783i \(0.203952\pi\)
−0.801658 + 0.597783i \(0.796048\pi\)
\(360\) 0 0
\(361\) −327.784 −0.907989
\(362\) 74.6339i 0.206171i
\(363\) 0 0
\(364\) 632.812 1.73850
\(365\) 314.834i 0.862559i
\(366\) 0 0
\(367\) 132.361 0.360658 0.180329 0.983606i \(-0.442284\pi\)
0.180329 + 0.983606i \(0.442284\pi\)
\(368\) 617.345i 1.67757i
\(369\) 0 0
\(370\) 13.3562 0.0360978
\(371\) 473.017i 1.27498i
\(372\) 0 0
\(373\) −478.180 −1.28198 −0.640992 0.767547i \(-0.721477\pi\)
−0.640992 + 0.767547i \(0.721477\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 46.3686 0.123321
\(377\) 324.849i 0.861669i
\(378\) 0 0
\(379\) −264.206 −0.697113 −0.348557 0.937288i \(-0.613328\pi\)
−0.348557 + 0.937288i \(0.613328\pi\)
\(380\) − 133.762i − 0.352005i
\(381\) 0 0
\(382\) −11.6857 −0.0305909
\(383\) 236.431i 0.617313i 0.951174 + 0.308657i \(0.0998794\pi\)
−0.951174 + 0.308657i \(0.900121\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 63.3834i − 0.164206i
\(387\) 0 0
\(388\) −38.9379 −0.100356
\(389\) − 511.217i − 1.31418i −0.753811 0.657091i \(-0.771787\pi\)
0.753811 0.657091i \(-0.228213\pi\)
\(390\) 0 0
\(391\) 399.175 1.02091
\(392\) − 64.4360i − 0.164378i
\(393\) 0 0
\(394\) 64.9106 0.164748
\(395\) − 12.1849i − 0.0308479i
\(396\) 0 0
\(397\) 389.224 0.980413 0.490207 0.871606i \(-0.336921\pi\)
0.490207 + 0.871606i \(0.336921\pi\)
\(398\) 77.7999i 0.195477i
\(399\) 0 0
\(400\) −153.802 −0.384504
\(401\) 226.050i 0.563715i 0.959456 + 0.281858i \(0.0909506\pi\)
−0.959456 + 0.281858i \(0.909049\pi\)
\(402\) 0 0
\(403\) −255.738 −0.634586
\(404\) − 672.988i − 1.66581i
\(405\) 0 0
\(406\) 46.9039 0.115527
\(407\) 0 0
\(408\) 0 0
\(409\) −181.766 −0.444415 −0.222208 0.974999i \(-0.571326\pi\)
−0.222208 + 0.974999i \(0.571326\pi\)
\(410\) 98.3789i 0.239948i
\(411\) 0 0
\(412\) −443.538 −1.07655
\(413\) 306.357i 0.741784i
\(414\) 0 0
\(415\) 171.372 0.412944
\(416\) − 270.471i − 0.650172i
\(417\) 0 0
\(418\) 0 0
\(419\) 171.909i 0.410284i 0.978732 + 0.205142i \(0.0657656\pi\)
−0.978732 + 0.205142i \(0.934234\pi\)
\(420\) 0 0
\(421\) −120.846 −0.287046 −0.143523 0.989647i \(-0.545843\pi\)
−0.143523 + 0.989647i \(0.545843\pi\)
\(422\) − 37.8463i − 0.0896832i
\(423\) 0 0
\(424\) 134.233 0.316588
\(425\) 99.4482i 0.233996i
\(426\) 0 0
\(427\) −1021.90 −2.39321
\(428\) 324.627i 0.758473i
\(429\) 0 0
\(430\) −80.2942 −0.186731
\(431\) 474.296i 1.10045i 0.835015 + 0.550227i \(0.185459\pi\)
−0.835015 + 0.550227i \(0.814541\pi\)
\(432\) 0 0
\(433\) 254.226 0.587127 0.293564 0.955940i \(-0.405159\pi\)
0.293564 + 0.955940i \(0.405159\pi\)
\(434\) 36.9252i 0.0850811i
\(435\) 0 0
\(436\) 489.569 1.12286
\(437\) 239.656i 0.548412i
\(438\) 0 0
\(439\) 515.246 1.17368 0.586841 0.809703i \(-0.300372\pi\)
0.586841 + 0.809703i \(0.300372\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −55.8863 −0.126440
\(443\) 526.041i 1.18745i 0.804668 + 0.593726i \(0.202344\pi\)
−0.804668 + 0.593726i \(0.797656\pi\)
\(444\) 0 0
\(445\) 802.644 1.80369
\(446\) 22.3592i 0.0501326i
\(447\) 0 0
\(448\) 475.870 1.06221
\(449\) − 541.858i − 1.20681i −0.797435 0.603405i \(-0.793810\pi\)
0.797435 0.603405i \(-0.206190\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 394.935i − 0.873749i
\(453\) 0 0
\(454\) −72.8714 −0.160510
\(455\) − 964.106i − 2.11891i
\(456\) 0 0
\(457\) 122.430 0.267899 0.133950 0.990988i \(-0.457234\pi\)
0.133950 + 0.990988i \(0.457234\pi\)
\(458\) − 21.2130i − 0.0463166i
\(459\) 0 0
\(460\) 965.107 2.09806
\(461\) 308.300i 0.668764i 0.942438 + 0.334382i \(0.108528\pi\)
−0.942438 + 0.334382i \(0.891472\pi\)
\(462\) 0 0
\(463\) 369.654 0.798389 0.399195 0.916866i \(-0.369290\pi\)
0.399195 + 0.916866i \(0.369290\pi\)
\(464\) 257.925i 0.555873i
\(465\) 0 0
\(466\) −33.2326 −0.0713146
\(467\) 108.850i 0.233084i 0.993186 + 0.116542i \(0.0371809\pi\)
−0.993186 + 0.116542i \(0.962819\pi\)
\(468\) 0 0
\(469\) −793.494 −1.69188
\(470\) − 34.8886i − 0.0742311i
\(471\) 0 0
\(472\) 86.9382 0.184191
\(473\) 0 0
\(474\) 0 0
\(475\) −59.7066 −0.125698
\(476\) − 324.879i − 0.682520i
\(477\) 0 0
\(478\) 108.535 0.227061
\(479\) − 302.678i − 0.631897i −0.948776 0.315948i \(-0.897677\pi\)
0.948776 0.315948i \(-0.102323\pi\)
\(480\) 0 0
\(481\) −134.887 −0.280431
\(482\) 117.515i 0.243807i
\(483\) 0 0
\(484\) 0 0
\(485\) 59.3229i 0.122315i
\(486\) 0 0
\(487\) 464.193 0.953169 0.476584 0.879129i \(-0.341875\pi\)
0.476584 + 0.879129i \(0.341875\pi\)
\(488\) 289.996i 0.594253i
\(489\) 0 0
\(490\) −48.4829 −0.0989446
\(491\) − 347.094i − 0.706913i −0.935451 0.353456i \(-0.885006\pi\)
0.935451 0.353456i \(-0.114994\pi\)
\(492\) 0 0
\(493\) 166.774 0.338285
\(494\) − 33.5530i − 0.0679210i
\(495\) 0 0
\(496\) −203.052 −0.409379
\(497\) − 1004.24i − 2.02060i
\(498\) 0 0
\(499\) 454.952 0.911728 0.455864 0.890049i \(-0.349330\pi\)
0.455864 + 0.890049i \(0.349330\pi\)
\(500\) − 339.788i − 0.679576i
\(501\) 0 0
\(502\) 93.9239 0.187099
\(503\) − 414.477i − 0.824010i −0.911182 0.412005i \(-0.864829\pi\)
0.911182 0.412005i \(-0.135171\pi\)
\(504\) 0 0
\(505\) −1025.31 −2.03033
\(506\) 0 0
\(507\) 0 0
\(508\) 483.963 0.952684
\(509\) 150.010i 0.294714i 0.989083 + 0.147357i \(0.0470767\pi\)
−0.989083 + 0.147357i \(0.952923\pi\)
\(510\) 0 0
\(511\) −459.089 −0.898413
\(512\) − 360.875i − 0.704834i
\(513\) 0 0
\(514\) −4.66303 −0.00907204
\(515\) 675.741i 1.31212i
\(516\) 0 0
\(517\) 0 0
\(518\) 19.4759i 0.0375983i
\(519\) 0 0
\(520\) −273.595 −0.526144
\(521\) 437.080i 0.838924i 0.907773 + 0.419462i \(0.137781\pi\)
−0.907773 + 0.419462i \(0.862219\pi\)
\(522\) 0 0
\(523\) −410.273 −0.784461 −0.392231 0.919867i \(-0.628297\pi\)
−0.392231 + 0.919867i \(0.628297\pi\)
\(524\) − 205.148i − 0.391503i
\(525\) 0 0
\(526\) 146.546 0.278605
\(527\) 131.293i 0.249134i
\(528\) 0 0
\(529\) −1200.15 −2.26871
\(530\) − 101.000i − 0.190565i
\(531\) 0 0
\(532\) 195.051 0.366636
\(533\) − 993.550i − 1.86407i
\(534\) 0 0
\(535\) 494.577 0.924443
\(536\) 225.178i 0.420109i
\(537\) 0 0
\(538\) −62.2171 −0.115645
\(539\) 0 0
\(540\) 0 0
\(541\) −499.682 −0.923626 −0.461813 0.886977i \(-0.652801\pi\)
−0.461813 + 0.886977i \(0.652801\pi\)
\(542\) 27.1720i 0.0501328i
\(543\) 0 0
\(544\) −138.857 −0.255252
\(545\) − 745.871i − 1.36857i
\(546\) 0 0
\(547\) −695.564 −1.27160 −0.635798 0.771855i \(-0.719329\pi\)
−0.635798 + 0.771855i \(0.719329\pi\)
\(548\) − 490.936i − 0.895869i
\(549\) 0 0
\(550\) 0 0
\(551\) 100.128i 0.181720i
\(552\) 0 0
\(553\) 17.7680 0.0321301
\(554\) 93.1637i 0.168165i
\(555\) 0 0
\(556\) 209.383 0.376589
\(557\) − 433.734i − 0.778697i −0.921091 0.389348i \(-0.872700\pi\)
0.921091 0.389348i \(-0.127300\pi\)
\(558\) 0 0
\(559\) 810.909 1.45064
\(560\) − 765.484i − 1.36694i
\(561\) 0 0
\(562\) 93.0037 0.165487
\(563\) − 672.668i − 1.19479i −0.801946 0.597396i \(-0.796202\pi\)
0.801946 0.597396i \(-0.203798\pi\)
\(564\) 0 0
\(565\) −601.693 −1.06494
\(566\) 63.6776i 0.112505i
\(567\) 0 0
\(568\) −284.984 −0.501732
\(569\) 543.176i 0.954614i 0.878737 + 0.477307i \(0.158387\pi\)
−0.878737 + 0.477307i \(0.841613\pi\)
\(570\) 0 0
\(571\) 607.861 1.06456 0.532278 0.846570i \(-0.321336\pi\)
0.532278 + 0.846570i \(0.321336\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −143.455 −0.249922
\(575\) − 430.790i − 0.749200i
\(576\) 0 0
\(577\) 131.051 0.227125 0.113562 0.993531i \(-0.463774\pi\)
0.113562 + 0.993531i \(0.463774\pi\)
\(578\) − 61.2905i − 0.106039i
\(579\) 0 0
\(580\) 403.219 0.695205
\(581\) 249.893i 0.430109i
\(582\) 0 0
\(583\) 0 0
\(584\) 130.281i 0.223083i
\(585\) 0 0
\(586\) 32.0209 0.0546431
\(587\) − 133.668i − 0.227713i −0.993497 0.113857i \(-0.963680\pi\)
0.993497 0.113857i \(-0.0363204\pi\)
\(588\) 0 0
\(589\) −78.8257 −0.133830
\(590\) − 65.4139i − 0.110871i
\(591\) 0 0
\(592\) −107.098 −0.180909
\(593\) 164.004i 0.276567i 0.990393 + 0.138283i \(0.0441585\pi\)
−0.990393 + 0.138283i \(0.955841\pi\)
\(594\) 0 0
\(595\) −494.962 −0.831869
\(596\) 16.5385i 0.0277492i
\(597\) 0 0
\(598\) 242.088 0.404830
\(599\) 42.8102i 0.0714694i 0.999361 + 0.0357347i \(0.0113771\pi\)
−0.999361 + 0.0357347i \(0.988623\pi\)
\(600\) 0 0
\(601\) −311.489 −0.518284 −0.259142 0.965839i \(-0.583440\pi\)
−0.259142 + 0.965839i \(0.583440\pi\)
\(602\) − 117.084i − 0.194492i
\(603\) 0 0
\(604\) 368.399 0.609933
\(605\) 0 0
\(606\) 0 0
\(607\) 1126.14 1.85526 0.927631 0.373498i \(-0.121842\pi\)
0.927631 + 0.373498i \(0.121842\pi\)
\(608\) − 83.3669i − 0.137117i
\(609\) 0 0
\(610\) 218.198 0.357702
\(611\) 352.348i 0.576674i
\(612\) 0 0
\(613\) 466.854 0.761589 0.380795 0.924660i \(-0.375650\pi\)
0.380795 + 0.924660i \(0.375650\pi\)
\(614\) − 12.3186i − 0.0200628i
\(615\) 0 0
\(616\) 0 0
\(617\) 130.650i 0.211750i 0.994379 + 0.105875i \(0.0337644\pi\)
−0.994379 + 0.105875i \(0.966236\pi\)
\(618\) 0 0
\(619\) 1096.95 1.77214 0.886068 0.463555i \(-0.153426\pi\)
0.886068 + 0.463555i \(0.153426\pi\)
\(620\) 317.435i 0.511992i
\(621\) 0 0
\(622\) −43.4200 −0.0698072
\(623\) 1170.41i 1.87867i
\(624\) 0 0
\(625\) −776.669 −1.24267
\(626\) 70.3909i 0.112446i
\(627\) 0 0
\(628\) −580.320 −0.924076
\(629\) 69.2497i 0.110095i
\(630\) 0 0
\(631\) −1161.69 −1.84102 −0.920511 0.390716i \(-0.872227\pi\)
−0.920511 + 0.390716i \(0.872227\pi\)
\(632\) − 5.04221i − 0.00797818i
\(633\) 0 0
\(634\) 109.654 0.172956
\(635\) − 737.331i − 1.16115i
\(636\) 0 0
\(637\) 489.639 0.768664
\(638\) 0 0
\(639\) 0 0
\(640\) −445.670 −0.696359
\(641\) 387.958i 0.605238i 0.953112 + 0.302619i \(0.0978610\pi\)
−0.953112 + 0.302619i \(0.902139\pi\)
\(642\) 0 0
\(643\) −115.275 −0.179277 −0.0896383 0.995974i \(-0.528571\pi\)
−0.0896383 + 0.995974i \(0.528571\pi\)
\(644\) 1407.31i 2.18527i
\(645\) 0 0
\(646\) −17.2257 −0.0266652
\(647\) 182.585i 0.282203i 0.989995 + 0.141101i \(0.0450643\pi\)
−0.989995 + 0.141101i \(0.954936\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 60.3125i 0.0927885i
\(651\) 0 0
\(652\) −358.110 −0.549249
\(653\) − 10.8119i − 0.0165573i −0.999966 0.00827865i \(-0.997365\pi\)
0.999966 0.00827865i \(-0.00263521\pi\)
\(654\) 0 0
\(655\) −312.548 −0.477172
\(656\) − 788.863i − 1.20253i
\(657\) 0 0
\(658\) 50.8743 0.0773166
\(659\) − 875.394i − 1.32837i −0.747569 0.664184i \(-0.768779\pi\)
0.747569 0.664184i \(-0.231221\pi\)
\(660\) 0 0
\(661\) 1015.92 1.53695 0.768475 0.639880i \(-0.221016\pi\)
0.768475 + 0.639880i \(0.221016\pi\)
\(662\) 26.4211i 0.0399110i
\(663\) 0 0
\(664\) 70.9149 0.106800
\(665\) − 297.165i − 0.446864i
\(666\) 0 0
\(667\) −722.433 −1.08311
\(668\) − 55.6875i − 0.0833645i
\(669\) 0 0
\(670\) 169.428 0.252878
\(671\) 0 0
\(672\) 0 0
\(673\) −113.798 −0.169091 −0.0845454 0.996420i \(-0.526944\pi\)
−0.0845454 + 0.996420i \(0.526944\pi\)
\(674\) − 164.139i − 0.243530i
\(675\) 0 0
\(676\) 704.984 1.04288
\(677\) 853.302i 1.26042i 0.776426 + 0.630208i \(0.217031\pi\)
−0.776426 + 0.630208i \(0.782969\pi\)
\(678\) 0 0
\(679\) −86.5043 −0.127400
\(680\) 140.461i 0.206560i
\(681\) 0 0
\(682\) 0 0
\(683\) 193.740i 0.283661i 0.989891 + 0.141830i \(0.0452987\pi\)
−0.989891 + 0.141830i \(0.954701\pi\)
\(684\) 0 0
\(685\) −747.954 −1.09190
\(686\) 61.5916i 0.0897836i
\(687\) 0 0
\(688\) 643.848 0.935826
\(689\) 1020.02i 1.48043i
\(690\) 0 0
\(691\) −253.670 −0.367106 −0.183553 0.983010i \(-0.558760\pi\)
−0.183553 + 0.983010i \(0.558760\pi\)
\(692\) − 906.604i − 1.31012i
\(693\) 0 0
\(694\) −39.3300 −0.0566714
\(695\) − 319.001i − 0.458994i
\(696\) 0 0
\(697\) −510.078 −0.731820
\(698\) 140.563i 0.201380i
\(699\) 0 0
\(700\) −350.610 −0.500871
\(701\) − 1139.33i − 1.62529i −0.582760 0.812644i \(-0.698027\pi\)
0.582760 0.812644i \(-0.301973\pi\)
\(702\) 0 0
\(703\) −41.5761 −0.0591409
\(704\) 0 0
\(705\) 0 0
\(706\) 45.3676 0.0642601
\(707\) − 1495.11i − 2.11472i
\(708\) 0 0
\(709\) 374.159 0.527728 0.263864 0.964560i \(-0.415003\pi\)
0.263864 + 0.964560i \(0.415003\pi\)
\(710\) 214.427i 0.302010i
\(711\) 0 0
\(712\) 332.140 0.466489
\(713\) − 568.737i − 0.797667i
\(714\) 0 0
\(715\) 0 0
\(716\) 560.706i 0.783109i
\(717\) 0 0
\(718\) 133.637 0.186124
\(719\) 757.515i 1.05357i 0.849999 + 0.526784i \(0.176602\pi\)
−0.849999 + 0.526784i \(0.823398\pi\)
\(720\) 0 0
\(721\) −985.361 −1.36666
\(722\) 102.058i 0.141354i
\(723\) 0 0
\(724\) −935.585 −1.29224
\(725\) − 179.983i − 0.248252i
\(726\) 0 0
\(727\) 845.080 1.16242 0.581211 0.813753i \(-0.302579\pi\)
0.581211 + 0.813753i \(0.302579\pi\)
\(728\) − 398.954i − 0.548014i
\(729\) 0 0
\(730\) 98.0256 0.134282
\(731\) − 416.312i − 0.569511i
\(732\) 0 0
\(733\) 1358.08 1.85278 0.926388 0.376570i \(-0.122897\pi\)
0.926388 + 0.376570i \(0.122897\pi\)
\(734\) − 41.2115i − 0.0561465i
\(735\) 0 0
\(736\) 601.502 0.817258
\(737\) 0 0
\(738\) 0 0
\(739\) 801.192 1.08416 0.542078 0.840328i \(-0.317638\pi\)
0.542078 + 0.840328i \(0.317638\pi\)
\(740\) 167.429i 0.226255i
\(741\) 0 0
\(742\) 147.277 0.198487
\(743\) − 893.596i − 1.20269i −0.798991 0.601343i \(-0.794632\pi\)
0.798991 0.601343i \(-0.205368\pi\)
\(744\) 0 0
\(745\) 25.1968 0.0338213
\(746\) 148.884i 0.199577i
\(747\) 0 0
\(748\) 0 0
\(749\) 721.189i 0.962868i
\(750\) 0 0
\(751\) 397.035 0.528676 0.264338 0.964430i \(-0.414847\pi\)
0.264338 + 0.964430i \(0.414847\pi\)
\(752\) 279.758i 0.372019i
\(753\) 0 0
\(754\) 101.144 0.134143
\(755\) − 561.266i − 0.743398i
\(756\) 0 0
\(757\) 593.110 0.783500 0.391750 0.920072i \(-0.371870\pi\)
0.391750 + 0.920072i \(0.371870\pi\)
\(758\) 82.2622i 0.108525i
\(759\) 0 0
\(760\) −84.3296 −0.110960
\(761\) 409.540i 0.538160i 0.963118 + 0.269080i \(0.0867196\pi\)
−0.963118 + 0.269080i \(0.913280\pi\)
\(762\) 0 0
\(763\) 1087.62 1.42546
\(764\) − 146.488i − 0.191739i
\(765\) 0 0
\(766\) 73.6143 0.0961022
\(767\) 660.630i 0.861316i
\(768\) 0 0
\(769\) 119.029 0.154784 0.0773921 0.997001i \(-0.475341\pi\)
0.0773921 + 0.997001i \(0.475341\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 794.553 1.02921
\(773\) 701.535i 0.907548i 0.891117 + 0.453774i \(0.149923\pi\)
−0.891117 + 0.453774i \(0.850077\pi\)
\(774\) 0 0
\(775\) 141.692 0.182828
\(776\) 24.5483i 0.0316344i
\(777\) 0 0
\(778\) −159.171 −0.204590
\(779\) − 306.240i − 0.393120i
\(780\) 0 0
\(781\) 0 0
\(782\) − 124.286i − 0.158933i
\(783\) 0 0
\(784\) 388.766 0.495874
\(785\) 884.132i 1.12628i
\(786\) 0 0
\(787\) 1539.13 1.95569 0.977844 0.209336i \(-0.0671303\pi\)
0.977844 + 0.209336i \(0.0671303\pi\)
\(788\) 813.697i 1.03261i
\(789\) 0 0
\(790\) −3.79385 −0.00480234
\(791\) − 877.384i − 1.10921i
\(792\) 0 0
\(793\) −2203.63 −2.77885
\(794\) − 121.187i − 0.152629i
\(795\) 0 0
\(796\) −975.273 −1.22522
\(797\) − 989.760i − 1.24186i −0.783867 0.620928i \(-0.786756\pi\)
0.783867 0.620928i \(-0.213244\pi\)
\(798\) 0 0
\(799\) 180.892 0.226398
\(800\) 149.855i 0.187319i
\(801\) 0 0
\(802\) 70.3821 0.0877582
\(803\) 0 0
\(804\) 0 0
\(805\) 2144.08 2.66345
\(806\) 79.6257i 0.0987912i
\(807\) 0 0
\(808\) −424.283 −0.525103
\(809\) − 94.1076i − 0.116326i −0.998307 0.0581629i \(-0.981476\pi\)
0.998307 0.0581629i \(-0.0185243\pi\)
\(810\) 0 0
\(811\) −370.312 −0.456612 −0.228306 0.973589i \(-0.573319\pi\)
−0.228306 + 0.973589i \(0.573319\pi\)
\(812\) 587.971i 0.724103i
\(813\) 0 0
\(814\) 0 0
\(815\) 545.590i 0.669436i
\(816\) 0 0
\(817\) 249.945 0.305930
\(818\) 56.5940i 0.0691858i
\(819\) 0 0
\(820\) −1233.24 −1.50396
\(821\) 213.696i 0.260288i 0.991495 + 0.130144i \(0.0415439\pi\)
−0.991495 + 0.130144i \(0.958456\pi\)
\(822\) 0 0
\(823\) 104.216 0.126630 0.0633148 0.997994i \(-0.479833\pi\)
0.0633148 + 0.997994i \(0.479833\pi\)
\(824\) 279.627i 0.339353i
\(825\) 0 0
\(826\) 95.3861 0.115480
\(827\) − 1329.26i − 1.60732i −0.595086 0.803662i \(-0.702882\pi\)
0.595086 0.803662i \(-0.297118\pi\)
\(828\) 0 0
\(829\) −350.092 −0.422306 −0.211153 0.977453i \(-0.567722\pi\)
−0.211153 + 0.977453i \(0.567722\pi\)
\(830\) − 53.3577i − 0.0642864i
\(831\) 0 0
\(832\) 1026.17 1.23338
\(833\) − 251.376i − 0.301772i
\(834\) 0 0
\(835\) −84.8413 −0.101606
\(836\) 0 0
\(837\) 0 0
\(838\) 53.5250 0.0638723
\(839\) − 1542.17i − 1.83811i −0.394132 0.919054i \(-0.628955\pi\)
0.394132 0.919054i \(-0.371045\pi\)
\(840\) 0 0
\(841\) 539.169 0.641105
\(842\) 37.6263i 0.0446868i
\(843\) 0 0
\(844\) 474.428 0.562119
\(845\) − 1074.06i − 1.27108i
\(846\) 0 0
\(847\) 0 0
\(848\) 809.878i 0.955044i
\(849\) 0 0
\(850\) 30.9638 0.0364280
\(851\) − 299.976i − 0.352498i
\(852\) 0 0
\(853\) 53.1329 0.0622895 0.0311448 0.999515i \(-0.490085\pi\)
0.0311448 + 0.999515i \(0.490085\pi\)
\(854\) 318.175i 0.372570i
\(855\) 0 0
\(856\) 204.660 0.239088
\(857\) 668.807i 0.780405i 0.920729 + 0.390202i \(0.127595\pi\)
−0.920729 + 0.390202i \(0.872405\pi\)
\(858\) 0 0
\(859\) 192.579 0.224189 0.112095 0.993698i \(-0.464244\pi\)
0.112095 + 0.993698i \(0.464244\pi\)
\(860\) − 1006.54i − 1.17040i
\(861\) 0 0
\(862\) 147.675 0.171317
\(863\) − 851.072i − 0.986179i −0.869979 0.493089i \(-0.835868\pi\)
0.869979 0.493089i \(-0.164132\pi\)
\(864\) 0 0
\(865\) −1381.23 −1.59680
\(866\) − 79.1549i − 0.0914029i
\(867\) 0 0
\(868\) −462.882 −0.533274
\(869\) 0 0
\(870\) 0 0
\(871\) −1711.10 −1.96452
\(872\) − 308.647i − 0.353953i
\(873\) 0 0
\(874\) 74.6185 0.0853758
\(875\) − 754.871i − 0.862709i
\(876\) 0 0
\(877\) 440.628 0.502427 0.251213 0.967932i \(-0.419170\pi\)
0.251213 + 0.967932i \(0.419170\pi\)
\(878\) − 160.425i − 0.182717i
\(879\) 0 0
\(880\) 0 0
\(881\) 448.427i 0.508997i 0.967073 + 0.254499i \(0.0819104\pi\)
−0.967073 + 0.254499i \(0.918090\pi\)
\(882\) 0 0
\(883\) −310.831 −0.352017 −0.176009 0.984389i \(-0.556319\pi\)
−0.176009 + 0.984389i \(0.556319\pi\)
\(884\) − 700.572i − 0.792502i
\(885\) 0 0
\(886\) 163.786 0.184860
\(887\) − 1292.83i − 1.45754i −0.684761 0.728768i \(-0.740093\pi\)
0.684761 0.728768i \(-0.259907\pi\)
\(888\) 0 0
\(889\) 1075.17 1.20942
\(890\) − 249.908i − 0.280796i
\(891\) 0 0
\(892\) −280.287 −0.314223
\(893\) 108.604i 0.121617i
\(894\) 0 0
\(895\) 854.250 0.954469
\(896\) − 649.873i − 0.725304i
\(897\) 0 0
\(898\) −168.711 −0.187874
\(899\) − 237.617i − 0.264312i
\(900\) 0 0
\(901\) 523.667 0.581206
\(902\) 0 0
\(903\) 0 0
\(904\) −248.985 −0.275426
\(905\) 1425.39i 1.57501i
\(906\) 0 0
\(907\) 815.519 0.899139 0.449570 0.893245i \(-0.351577\pi\)
0.449570 + 0.893245i \(0.351577\pi\)
\(908\) − 913.491i − 1.00605i
\(909\) 0 0
\(910\) −300.180 −0.329869
\(911\) − 243.043i − 0.266787i −0.991063 0.133393i \(-0.957413\pi\)
0.991063 0.133393i \(-0.0425874\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) − 38.1193i − 0.0417061i
\(915\) 0 0
\(916\) 265.919 0.290305
\(917\) − 455.755i − 0.497006i
\(918\) 0 0
\(919\) −290.283 −0.315868 −0.157934 0.987450i \(-0.550483\pi\)
−0.157934 + 0.987450i \(0.550483\pi\)
\(920\) − 608.448i − 0.661357i
\(921\) 0 0
\(922\) 95.9913 0.104112
\(923\) − 2165.54i − 2.34620i
\(924\) 0 0
\(925\) 74.7343 0.0807938
\(926\) − 115.094i − 0.124292i
\(927\) 0 0
\(928\) 251.306 0.270804
\(929\) 295.075i 0.317627i 0.987309 + 0.158813i \(0.0507668\pi\)
−0.987309 + 0.158813i \(0.949233\pi\)
\(930\) 0 0
\(931\) 150.921 0.162106
\(932\) − 416.593i − 0.446988i
\(933\) 0 0
\(934\) 33.8912 0.0362861
\(935\) 0 0
\(936\) 0 0
\(937\) −918.143 −0.979875 −0.489938 0.871758i \(-0.662980\pi\)
−0.489938 + 0.871758i \(0.662980\pi\)
\(938\) 247.059i 0.263389i
\(939\) 0 0
\(940\) 437.352 0.465268
\(941\) 998.146i 1.06073i 0.847770 + 0.530365i \(0.177945\pi\)
−0.847770 + 0.530365i \(0.822055\pi\)
\(942\) 0 0
\(943\) 2209.56 2.34312
\(944\) 524.529i 0.555645i
\(945\) 0 0
\(946\) 0 0
\(947\) − 780.779i − 0.824476i −0.911076 0.412238i \(-0.864747\pi\)
0.911076 0.412238i \(-0.135253\pi\)
\(948\) 0 0
\(949\) −989.982 −1.04318
\(950\) 18.5900i 0.0195684i
\(951\) 0 0
\(952\) −204.819 −0.215146
\(953\) − 93.2267i − 0.0978245i −0.998803 0.0489122i \(-0.984425\pi\)
0.998803 0.0489122i \(-0.0155755\pi\)
\(954\) 0 0
\(955\) −223.179 −0.233695
\(956\) 1360.56i 1.42318i
\(957\) 0 0
\(958\) −94.2409 −0.0983725
\(959\) − 1090.66i − 1.13729i
\(960\) 0 0
\(961\) −773.936 −0.805344
\(962\) 41.9980i 0.0436570i
\(963\) 0 0
\(964\) −1473.13 −1.52814
\(965\) − 1210.52i − 1.25443i
\(966\) 0 0
\(967\) 365.949 0.378437 0.189219 0.981935i \(-0.439405\pi\)
0.189219 + 0.981935i \(0.439405\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 18.4706 0.0190418
\(971\) − 284.163i − 0.292650i −0.989237 0.146325i \(-0.953256\pi\)
0.989237 0.146325i \(-0.0467445\pi\)
\(972\) 0 0
\(973\) 465.165 0.478073
\(974\) − 144.529i − 0.148388i
\(975\) 0 0
\(976\) −1749.65 −1.79267
\(977\) 882.999i 0.903786i 0.892072 + 0.451893i \(0.149251\pi\)
−0.892072 + 0.451893i \(0.850749\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) − 607.765i − 0.620168i
\(981\) 0 0
\(982\) −108.070 −0.110051
\(983\) − 1364.76i − 1.38836i −0.719799 0.694182i \(-0.755766\pi\)
0.719799 0.694182i \(-0.244234\pi\)
\(984\) 0 0
\(985\) 1239.69 1.25857
\(986\) − 51.9262i − 0.0526635i
\(987\) 0 0
\(988\) 420.608 0.425717
\(989\) 1803.38i 1.82344i
\(990\) 0 0
\(991\) −842.288 −0.849938 −0.424969 0.905208i \(-0.639715\pi\)
−0.424969 + 0.905208i \(0.639715\pi\)
\(992\) 197.841i 0.199437i
\(993\) 0 0
\(994\) −312.676 −0.314563
\(995\) 1485.85i 1.49332i
\(996\) 0 0
\(997\) −366.417 −0.367520 −0.183760 0.982971i \(-0.558827\pi\)
−0.183760 + 0.982971i \(0.558827\pi\)
\(998\) − 141.652i − 0.141936i
\(999\) 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.i.485.8 16
3.2 odd 2 inner 1089.3.b.i.485.9 16
11.3 even 5 99.3.l.a.53.4 32
11.4 even 5 99.3.l.a.71.5 yes 32
11.10 odd 2 1089.3.b.j.485.9 16
33.14 odd 10 99.3.l.a.53.5 yes 32
33.26 odd 10 99.3.l.a.71.4 yes 32
33.32 even 2 1089.3.b.j.485.8 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.3.l.a.53.4 32 11.3 even 5
99.3.l.a.53.5 yes 32 33.14 odd 10
99.3.l.a.71.4 yes 32 33.26 odd 10
99.3.l.a.71.5 yes 32 11.4 even 5
1089.3.b.i.485.8 16 1.1 even 1 trivial
1089.3.b.i.485.9 16 3.2 odd 2 inner
1089.3.b.j.485.8 16 33.32 even 2
1089.3.b.j.485.9 16 11.10 odd 2