Properties

Label 1089.3.b.j.485.13
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 485.13
Root \(2.98036i\) of defining polynomial
Character \(\chi\) \(=\) 1089.485
Dual form 1089.3.b.j.485.4

$q$-expansion

\(f(q)\) \(=\) \(q+2.98036i q^{2} -4.88257 q^{4} -7.85680i q^{5} -2.80793 q^{7} -2.63038i q^{8} +O(q^{10})\) \(q+2.98036i q^{2} -4.88257 q^{4} -7.85680i q^{5} -2.80793 q^{7} -2.63038i q^{8} +23.4161 q^{10} +1.51796 q^{13} -8.36866i q^{14} -11.6908 q^{16} -26.3991i q^{17} +14.5410 q^{19} +38.3614i q^{20} +25.0726i q^{23} -36.7293 q^{25} +4.52407i q^{26} +13.7099 q^{28} +50.7104i q^{29} -51.6484 q^{31} -45.3643i q^{32} +78.6790 q^{34} +22.0614i q^{35} -19.5563 q^{37} +43.3375i q^{38} -20.6664 q^{40} +30.9353i q^{41} -2.11848 q^{43} -74.7256 q^{46} +40.2708i q^{47} -41.1155 q^{49} -109.467i q^{50} -7.41155 q^{52} +41.1436i q^{53} +7.38594i q^{56} -151.135 q^{58} -96.0644i q^{59} +2.89529 q^{61} -153.931i q^{62} +88.4391 q^{64} -11.9263i q^{65} -88.2869 q^{67} +128.896i q^{68} -65.7509 q^{70} -11.3261i q^{71} -104.776 q^{73} -58.2848i q^{74} -70.9975 q^{76} -16.7886 q^{79} +91.8522i q^{80} -92.1985 q^{82} -22.7716i q^{83} -207.413 q^{85} -6.31384i q^{86} +30.7523i q^{89} -4.26233 q^{91} -122.419i q^{92} -120.022 q^{94} -114.246i q^{95} -28.5285 q^{97} -122.539i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{4} + 8 q^{7} + O(q^{10}) \) \( 16 q - 32 q^{4} + 8 q^{7} - 24 q^{10} - 4 q^{13} + 28 q^{16} + 20 q^{19} - 44 q^{25} - 16 q^{28} + 28 q^{31} + 148 q^{34} - 148 q^{37} - 224 q^{40} - 272 q^{43} - 208 q^{46} + 348 q^{49} - 520 q^{52} - 44 q^{58} - 224 q^{61} + 436 q^{64} + 24 q^{67} - 664 q^{70} + 4 q^{73} - 1052 q^{76} - 216 q^{79} + 348 q^{82} - 416 q^{85} - 168 q^{91} - 1140 q^{94} - 44 q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.98036i 1.49018i 0.666963 + 0.745091i \(0.267594\pi\)
−0.666963 + 0.745091i \(0.732406\pi\)
\(3\) 0 0
\(4\) −4.88257 −1.22064
\(5\) − 7.85680i − 1.57136i −0.618633 0.785680i \(-0.712313\pi\)
0.618633 0.785680i \(-0.287687\pi\)
\(6\) 0 0
\(7\) −2.80793 −0.401133 −0.200567 0.979680i \(-0.564278\pi\)
−0.200567 + 0.979680i \(0.564278\pi\)
\(8\) − 2.63038i − 0.328798i
\(9\) 0 0
\(10\) 23.4161 2.34161
\(11\) 0 0
\(12\) 0 0
\(13\) 1.51796 0.116766 0.0583831 0.998294i \(-0.481406\pi\)
0.0583831 + 0.998294i \(0.481406\pi\)
\(14\) − 8.36866i − 0.597762i
\(15\) 0 0
\(16\) −11.6908 −0.730674
\(17\) − 26.3991i − 1.55289i −0.630185 0.776445i \(-0.717021\pi\)
0.630185 0.776445i \(-0.282979\pi\)
\(18\) 0 0
\(19\) 14.5410 0.765316 0.382658 0.923890i \(-0.375009\pi\)
0.382658 + 0.923890i \(0.375009\pi\)
\(20\) 38.3614i 1.91807i
\(21\) 0 0
\(22\) 0 0
\(23\) 25.0726i 1.09011i 0.838399 + 0.545057i \(0.183492\pi\)
−0.838399 + 0.545057i \(0.816508\pi\)
\(24\) 0 0
\(25\) −36.7293 −1.46917
\(26\) 4.52407i 0.174003i
\(27\) 0 0
\(28\) 13.7099 0.489640
\(29\) 50.7104i 1.74863i 0.485356 + 0.874317i \(0.338690\pi\)
−0.485356 + 0.874317i \(0.661310\pi\)
\(30\) 0 0
\(31\) −51.6484 −1.66608 −0.833038 0.553216i \(-0.813401\pi\)
−0.833038 + 0.553216i \(0.813401\pi\)
\(32\) − 45.3643i − 1.41764i
\(33\) 0 0
\(34\) 78.6790 2.31409
\(35\) 22.0614i 0.630325i
\(36\) 0 0
\(37\) −19.5563 −0.528548 −0.264274 0.964448i \(-0.585132\pi\)
−0.264274 + 0.964448i \(0.585132\pi\)
\(38\) 43.3375i 1.14046i
\(39\) 0 0
\(40\) −20.6664 −0.516660
\(41\) 30.9353i 0.754520i 0.926107 + 0.377260i \(0.123134\pi\)
−0.926107 + 0.377260i \(0.876866\pi\)
\(42\) 0 0
\(43\) −2.11848 −0.0492670 −0.0246335 0.999697i \(-0.507842\pi\)
−0.0246335 + 0.999697i \(0.507842\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −74.7256 −1.62447
\(47\) 40.2708i 0.856826i 0.903583 + 0.428413i \(0.140927\pi\)
−0.903583 + 0.428413i \(0.859073\pi\)
\(48\) 0 0
\(49\) −41.1155 −0.839092
\(50\) − 109.467i − 2.18934i
\(51\) 0 0
\(52\) −7.41155 −0.142530
\(53\) 41.1436i 0.776295i 0.921597 + 0.388147i \(0.126885\pi\)
−0.921597 + 0.388147i \(0.873115\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 7.38594i 0.131892i
\(57\) 0 0
\(58\) −151.135 −2.60578
\(59\) − 96.0644i − 1.62821i −0.580718 0.814105i \(-0.697228\pi\)
0.580718 0.814105i \(-0.302772\pi\)
\(60\) 0 0
\(61\) 2.89529 0.0474637 0.0237319 0.999718i \(-0.492445\pi\)
0.0237319 + 0.999718i \(0.492445\pi\)
\(62\) − 153.931i − 2.48276i
\(63\) 0 0
\(64\) 88.4391 1.38186
\(65\) − 11.9263i − 0.183482i
\(66\) 0 0
\(67\) −88.2869 −1.31772 −0.658858 0.752268i \(-0.728960\pi\)
−0.658858 + 0.752268i \(0.728960\pi\)
\(68\) 128.896i 1.89552i
\(69\) 0 0
\(70\) −65.7509 −0.939299
\(71\) − 11.3261i − 0.159523i −0.996814 0.0797614i \(-0.974584\pi\)
0.996814 0.0797614i \(-0.0254158\pi\)
\(72\) 0 0
\(73\) −104.776 −1.43529 −0.717645 0.696409i \(-0.754780\pi\)
−0.717645 + 0.696409i \(0.754780\pi\)
\(74\) − 58.2848i − 0.787633i
\(75\) 0 0
\(76\) −70.9975 −0.934178
\(77\) 0 0
\(78\) 0 0
\(79\) −16.7886 −0.212514 −0.106257 0.994339i \(-0.533887\pi\)
−0.106257 + 0.994339i \(0.533887\pi\)
\(80\) 91.8522i 1.14815i
\(81\) 0 0
\(82\) −92.1985 −1.12437
\(83\) − 22.7716i − 0.274356i −0.990546 0.137178i \(-0.956197\pi\)
0.990546 0.137178i \(-0.0438033\pi\)
\(84\) 0 0
\(85\) −207.413 −2.44015
\(86\) − 6.31384i − 0.0734168i
\(87\) 0 0
\(88\) 0 0
\(89\) 30.7523i 0.345531i 0.984963 + 0.172766i \(0.0552703\pi\)
−0.984963 + 0.172766i \(0.944730\pi\)
\(90\) 0 0
\(91\) −4.26233 −0.0468388
\(92\) − 122.419i − 1.33064i
\(93\) 0 0
\(94\) −120.022 −1.27683
\(95\) − 114.246i − 1.20259i
\(96\) 0 0
\(97\) −28.5285 −0.294108 −0.147054 0.989128i \(-0.546979\pi\)
−0.147054 + 0.989128i \(0.546979\pi\)
\(98\) − 122.539i − 1.25040i
\(99\) 0 0
\(100\) 179.334 1.79334
\(101\) 3.39859i 0.0336494i 0.999858 + 0.0168247i \(0.00535573\pi\)
−0.999858 + 0.0168247i \(0.994644\pi\)
\(102\) 0 0
\(103\) 22.3116 0.216617 0.108309 0.994117i \(-0.465457\pi\)
0.108309 + 0.994117i \(0.465457\pi\)
\(104\) − 3.99282i − 0.0383925i
\(105\) 0 0
\(106\) −122.623 −1.15682
\(107\) 12.8713i 0.120292i 0.998190 + 0.0601461i \(0.0191567\pi\)
−0.998190 + 0.0601461i \(0.980843\pi\)
\(108\) 0 0
\(109\) −109.301 −1.00277 −0.501383 0.865226i \(-0.667175\pi\)
−0.501383 + 0.865226i \(0.667175\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 32.8269 0.293098
\(113\) − 92.0329i − 0.814451i −0.913328 0.407225i \(-0.866496\pi\)
0.913328 0.407225i \(-0.133504\pi\)
\(114\) 0 0
\(115\) 196.991 1.71296
\(116\) − 247.597i − 2.13446i
\(117\) 0 0
\(118\) 286.307 2.42633
\(119\) 74.1270i 0.622916i
\(120\) 0 0
\(121\) 0 0
\(122\) 8.62901i 0.0707296i
\(123\) 0 0
\(124\) 252.177 2.03368
\(125\) 92.1551i 0.737241i
\(126\) 0 0
\(127\) 8.28300 0.0652205 0.0326102 0.999468i \(-0.489618\pi\)
0.0326102 + 0.999468i \(0.489618\pi\)
\(128\) 82.1234i 0.641589i
\(129\) 0 0
\(130\) 35.5448 0.273421
\(131\) 1.63767i 0.0125013i 0.999980 + 0.00625065i \(0.00198966\pi\)
−0.999980 + 0.00625065i \(0.998010\pi\)
\(132\) 0 0
\(133\) −40.8302 −0.306994
\(134\) − 263.127i − 1.96364i
\(135\) 0 0
\(136\) −69.4398 −0.510587
\(137\) 198.118i 1.44612i 0.690785 + 0.723060i \(0.257265\pi\)
−0.690785 + 0.723060i \(0.742735\pi\)
\(138\) 0 0
\(139\) 187.446 1.34854 0.674268 0.738487i \(-0.264459\pi\)
0.674268 + 0.738487i \(0.264459\pi\)
\(140\) − 107.716i − 0.769402i
\(141\) 0 0
\(142\) 33.7560 0.237718
\(143\) 0 0
\(144\) 0 0
\(145\) 398.421 2.74773
\(146\) − 312.271i − 2.13884i
\(147\) 0 0
\(148\) 95.4849 0.645168
\(149\) 239.185i 1.60527i 0.596470 + 0.802636i \(0.296570\pi\)
−0.596470 + 0.802636i \(0.703430\pi\)
\(150\) 0 0
\(151\) −58.6757 −0.388581 −0.194290 0.980944i \(-0.562240\pi\)
−0.194290 + 0.980944i \(0.562240\pi\)
\(152\) − 38.2484i − 0.251634i
\(153\) 0 0
\(154\) 0 0
\(155\) 405.791i 2.61801i
\(156\) 0 0
\(157\) 20.4204 0.130066 0.0650330 0.997883i \(-0.479285\pi\)
0.0650330 + 0.997883i \(0.479285\pi\)
\(158\) − 50.0361i − 0.316684i
\(159\) 0 0
\(160\) −356.419 −2.22762
\(161\) − 70.4023i − 0.437281i
\(162\) 0 0
\(163\) 104.554 0.641434 0.320717 0.947175i \(-0.396076\pi\)
0.320717 + 0.947175i \(0.396076\pi\)
\(164\) − 151.044i − 0.920999i
\(165\) 0 0
\(166\) 67.8676 0.408841
\(167\) 39.2102i 0.234792i 0.993085 + 0.117396i \(0.0374546\pi\)
−0.993085 + 0.117396i \(0.962545\pi\)
\(168\) 0 0
\(169\) −166.696 −0.986366
\(170\) − 618.165i − 3.63627i
\(171\) 0 0
\(172\) 10.3436 0.0601374
\(173\) 91.4277i 0.528484i 0.964456 + 0.264242i \(0.0851217\pi\)
−0.964456 + 0.264242i \(0.914878\pi\)
\(174\) 0 0
\(175\) 103.134 0.589334
\(176\) 0 0
\(177\) 0 0
\(178\) −91.6530 −0.514904
\(179\) − 218.182i − 1.21890i −0.792826 0.609448i \(-0.791391\pi\)
0.792826 0.609448i \(-0.208609\pi\)
\(180\) 0 0
\(181\) −44.8209 −0.247629 −0.123815 0.992305i \(-0.539513\pi\)
−0.123815 + 0.992305i \(0.539513\pi\)
\(182\) − 12.7033i − 0.0697983i
\(183\) 0 0
\(184\) 65.9506 0.358427
\(185\) 153.650i 0.830540i
\(186\) 0 0
\(187\) 0 0
\(188\) − 196.625i − 1.04588i
\(189\) 0 0
\(190\) 340.494 1.79207
\(191\) − 148.973i − 0.779964i −0.920822 0.389982i \(-0.872481\pi\)
0.920822 0.389982i \(-0.127519\pi\)
\(192\) 0 0
\(193\) −193.706 −1.00366 −0.501830 0.864966i \(-0.667340\pi\)
−0.501830 + 0.864966i \(0.667340\pi\)
\(194\) − 85.0253i − 0.438275i
\(195\) 0 0
\(196\) 200.749 1.02423
\(197\) 331.307i 1.68176i 0.541220 + 0.840881i \(0.317963\pi\)
−0.541220 + 0.840881i \(0.682037\pi\)
\(198\) 0 0
\(199\) −294.829 −1.48155 −0.740775 0.671753i \(-0.765542\pi\)
−0.740775 + 0.671753i \(0.765542\pi\)
\(200\) 96.6123i 0.483061i
\(201\) 0 0
\(202\) −10.1290 −0.0501438
\(203\) − 142.391i − 0.701435i
\(204\) 0 0
\(205\) 243.053 1.18562
\(206\) 66.4966i 0.322799i
\(207\) 0 0
\(208\) −17.7461 −0.0853180
\(209\) 0 0
\(210\) 0 0
\(211\) 115.727 0.548470 0.274235 0.961663i \(-0.411575\pi\)
0.274235 + 0.961663i \(0.411575\pi\)
\(212\) − 200.887i − 0.947579i
\(213\) 0 0
\(214\) −38.3611 −0.179257
\(215\) 16.6445i 0.0774162i
\(216\) 0 0
\(217\) 145.025 0.668319
\(218\) − 325.758i − 1.49430i
\(219\) 0 0
\(220\) 0 0
\(221\) − 40.0728i − 0.181325i
\(222\) 0 0
\(223\) −244.092 −1.09458 −0.547291 0.836942i \(-0.684341\pi\)
−0.547291 + 0.836942i \(0.684341\pi\)
\(224\) 127.380i 0.568661i
\(225\) 0 0
\(226\) 274.292 1.21368
\(227\) 268.225i 1.18161i 0.806815 + 0.590804i \(0.201189\pi\)
−0.806815 + 0.590804i \(0.798811\pi\)
\(228\) 0 0
\(229\) 186.092 0.812628 0.406314 0.913734i \(-0.366814\pi\)
0.406314 + 0.913734i \(0.366814\pi\)
\(230\) 587.104i 2.55263i
\(231\) 0 0
\(232\) 133.388 0.574947
\(233\) − 162.181i − 0.696056i −0.937484 0.348028i \(-0.886851\pi\)
0.937484 0.348028i \(-0.113149\pi\)
\(234\) 0 0
\(235\) 316.400 1.34638
\(236\) 469.041i 1.98746i
\(237\) 0 0
\(238\) −220.925 −0.928258
\(239\) − 71.0937i − 0.297463i −0.988878 0.148732i \(-0.952481\pi\)
0.988878 0.148732i \(-0.0475190\pi\)
\(240\) 0 0
\(241\) −212.799 −0.882984 −0.441492 0.897265i \(-0.645551\pi\)
−0.441492 + 0.897265i \(0.645551\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −14.1364 −0.0579363
\(245\) 323.036i 1.31852i
\(246\) 0 0
\(247\) 22.0727 0.0893631
\(248\) 135.855i 0.547802i
\(249\) 0 0
\(250\) −274.656 −1.09862
\(251\) − 187.607i − 0.747437i −0.927542 0.373718i \(-0.878083\pi\)
0.927542 0.373718i \(-0.121917\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 24.6864i 0.0971904i
\(255\) 0 0
\(256\) 108.999 0.425776
\(257\) − 290.794i − 1.13149i −0.824579 0.565747i \(-0.808588\pi\)
0.824579 0.565747i \(-0.191412\pi\)
\(258\) 0 0
\(259\) 54.9127 0.212018
\(260\) 58.2311i 0.223966i
\(261\) 0 0
\(262\) −4.88086 −0.0186292
\(263\) − 458.564i − 1.74359i −0.489870 0.871795i \(-0.662956\pi\)
0.489870 0.871795i \(-0.337044\pi\)
\(264\) 0 0
\(265\) 323.257 1.21984
\(266\) − 121.689i − 0.457477i
\(267\) 0 0
\(268\) 431.067 1.60846
\(269\) 208.752i 0.776030i 0.921653 + 0.388015i \(0.126839\pi\)
−0.921653 + 0.388015i \(0.873161\pi\)
\(270\) 0 0
\(271\) 460.949 1.70092 0.850459 0.526042i \(-0.176324\pi\)
0.850459 + 0.526042i \(0.176324\pi\)
\(272\) 308.626i 1.13466i
\(273\) 0 0
\(274\) −590.465 −2.15498
\(275\) 0 0
\(276\) 0 0
\(277\) 105.652 0.381417 0.190708 0.981647i \(-0.438922\pi\)
0.190708 + 0.981647i \(0.438922\pi\)
\(278\) 558.659i 2.00956i
\(279\) 0 0
\(280\) 58.0299 0.207250
\(281\) 430.979i 1.53373i 0.641806 + 0.766867i \(0.278185\pi\)
−0.641806 + 0.766867i \(0.721815\pi\)
\(282\) 0 0
\(283\) −301.717 −1.06614 −0.533068 0.846072i \(-0.678961\pi\)
−0.533068 + 0.846072i \(0.678961\pi\)
\(284\) 55.3006i 0.194720i
\(285\) 0 0
\(286\) 0 0
\(287\) − 86.8643i − 0.302663i
\(288\) 0 0
\(289\) −407.914 −1.41147
\(290\) 1187.44i 4.09462i
\(291\) 0 0
\(292\) 511.577 1.75198
\(293\) − 46.3147i − 0.158071i −0.996872 0.0790354i \(-0.974816\pi\)
0.996872 0.0790354i \(-0.0251840\pi\)
\(294\) 0 0
\(295\) −754.759 −2.55851
\(296\) 51.4405i 0.173786i
\(297\) 0 0
\(298\) −712.860 −2.39215
\(299\) 38.0593i 0.127289i
\(300\) 0 0
\(301\) 5.94855 0.0197626
\(302\) − 174.875i − 0.579056i
\(303\) 0 0
\(304\) −169.996 −0.559197
\(305\) − 22.7477i − 0.0745826i
\(306\) 0 0
\(307\) 448.259 1.46013 0.730063 0.683379i \(-0.239491\pi\)
0.730063 + 0.683379i \(0.239491\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1209.40 −3.90131
\(311\) 334.674i 1.07612i 0.842906 + 0.538062i \(0.180843\pi\)
−0.842906 + 0.538062i \(0.819157\pi\)
\(312\) 0 0
\(313\) 102.316 0.326888 0.163444 0.986553i \(-0.447740\pi\)
0.163444 + 0.986553i \(0.447740\pi\)
\(314\) 60.8601i 0.193822i
\(315\) 0 0
\(316\) 81.9714 0.259403
\(317\) − 377.875i − 1.19204i −0.802971 0.596018i \(-0.796749\pi\)
0.802971 0.596018i \(-0.203251\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) − 694.848i − 2.17140i
\(321\) 0 0
\(322\) 209.824 0.651629
\(323\) − 383.870i − 1.18845i
\(324\) 0 0
\(325\) −55.7537 −0.171550
\(326\) 311.608i 0.955853i
\(327\) 0 0
\(328\) 81.3717 0.248084
\(329\) − 113.078i − 0.343701i
\(330\) 0 0
\(331\) −170.972 −0.516533 −0.258266 0.966074i \(-0.583151\pi\)
−0.258266 + 0.966074i \(0.583151\pi\)
\(332\) 111.184i 0.334891i
\(333\) 0 0
\(334\) −116.861 −0.349882
\(335\) 693.653i 2.07061i
\(336\) 0 0
\(337\) −191.936 −0.569543 −0.284772 0.958595i \(-0.591918\pi\)
−0.284772 + 0.958595i \(0.591918\pi\)
\(338\) − 496.814i − 1.46986i
\(339\) 0 0
\(340\) 1012.71 2.97855
\(341\) 0 0
\(342\) 0 0
\(343\) 253.038 0.737721
\(344\) 5.57241i 0.0161989i
\(345\) 0 0
\(346\) −272.488 −0.787537
\(347\) − 185.859i − 0.535616i −0.963472 0.267808i \(-0.913701\pi\)
0.963472 0.267808i \(-0.0862992\pi\)
\(348\) 0 0
\(349\) 176.390 0.505416 0.252708 0.967543i \(-0.418679\pi\)
0.252708 + 0.967543i \(0.418679\pi\)
\(350\) 307.375i 0.878216i
\(351\) 0 0
\(352\) 0 0
\(353\) − 232.825i − 0.659561i −0.944058 0.329780i \(-0.893025\pi\)
0.944058 0.329780i \(-0.106975\pi\)
\(354\) 0 0
\(355\) −88.9871 −0.250668
\(356\) − 150.150i − 0.421770i
\(357\) 0 0
\(358\) 650.263 1.81638
\(359\) − 403.124i − 1.12291i −0.827508 0.561454i \(-0.810242\pi\)
0.827508 0.561454i \(-0.189758\pi\)
\(360\) 0 0
\(361\) −149.559 −0.414291
\(362\) − 133.583i − 0.369012i
\(363\) 0 0
\(364\) 20.8111 0.0571734
\(365\) 823.205i 2.25536i
\(366\) 0 0
\(367\) −541.513 −1.47551 −0.737757 0.675067i \(-0.764115\pi\)
−0.737757 + 0.675067i \(0.764115\pi\)
\(368\) − 293.119i − 0.796518i
\(369\) 0 0
\(370\) −457.932 −1.23766
\(371\) − 115.529i − 0.311398i
\(372\) 0 0
\(373\) 160.025 0.429021 0.214510 0.976722i \(-0.431184\pi\)
0.214510 + 0.976722i \(0.431184\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 105.928 0.281723
\(377\) 76.9763i 0.204181i
\(378\) 0 0
\(379\) −378.620 −0.998998 −0.499499 0.866314i \(-0.666483\pi\)
−0.499499 + 0.866314i \(0.666483\pi\)
\(380\) 557.813i 1.46793i
\(381\) 0 0
\(382\) 443.994 1.16229
\(383\) 239.567i 0.625502i 0.949835 + 0.312751i \(0.101250\pi\)
−0.949835 + 0.312751i \(0.898750\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 577.315i − 1.49564i
\(387\) 0 0
\(388\) 139.292 0.359001
\(389\) 502.620i 1.29208i 0.763302 + 0.646042i \(0.223577\pi\)
−0.763302 + 0.646042i \(0.776423\pi\)
\(390\) 0 0
\(391\) 661.896 1.69283
\(392\) 108.150i 0.275892i
\(393\) 0 0
\(394\) −987.416 −2.50613
\(395\) 131.905i 0.333935i
\(396\) 0 0
\(397\) 97.1039 0.244594 0.122297 0.992494i \(-0.460974\pi\)
0.122297 + 0.992494i \(0.460974\pi\)
\(398\) − 878.696i − 2.20778i
\(399\) 0 0
\(400\) 429.395 1.07349
\(401\) − 657.148i − 1.63877i −0.573241 0.819387i \(-0.694314\pi\)
0.573241 0.819387i \(-0.305686\pi\)
\(402\) 0 0
\(403\) −78.4002 −0.194541
\(404\) − 16.5939i − 0.0410739i
\(405\) 0 0
\(406\) 424.378 1.04527
\(407\) 0 0
\(408\) 0 0
\(409\) −101.555 −0.248302 −0.124151 0.992263i \(-0.539621\pi\)
−0.124151 + 0.992263i \(0.539621\pi\)
\(410\) 724.385i 1.76679i
\(411\) 0 0
\(412\) −108.938 −0.264412
\(413\) 269.742i 0.653129i
\(414\) 0 0
\(415\) −178.912 −0.431113
\(416\) − 68.8613i − 0.165532i
\(417\) 0 0
\(418\) 0 0
\(419\) 0.439819i 0.00104969i 1.00000 0.000524844i \(0.000167063\pi\)
−1.00000 0.000524844i \(0.999833\pi\)
\(420\) 0 0
\(421\) −418.561 −0.994208 −0.497104 0.867691i \(-0.665603\pi\)
−0.497104 + 0.867691i \(0.665603\pi\)
\(422\) 344.909i 0.817320i
\(423\) 0 0
\(424\) 108.224 0.255244
\(425\) 969.622i 2.28146i
\(426\) 0 0
\(427\) −8.12977 −0.0190393
\(428\) − 62.8449i − 0.146834i
\(429\) 0 0
\(430\) −49.6066 −0.115364
\(431\) 252.069i 0.584846i 0.956289 + 0.292423i \(0.0944615\pi\)
−0.956289 + 0.292423i \(0.905538\pi\)
\(432\) 0 0
\(433\) −124.450 −0.287412 −0.143706 0.989620i \(-0.545902\pi\)
−0.143706 + 0.989620i \(0.545902\pi\)
\(434\) 432.228i 0.995916i
\(435\) 0 0
\(436\) 533.672 1.22402
\(437\) 364.581i 0.834283i
\(438\) 0 0
\(439\) 73.6933 0.167866 0.0839332 0.996471i \(-0.473252\pi\)
0.0839332 + 0.996471i \(0.473252\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 119.432 0.270207
\(443\) − 369.937i − 0.835071i −0.908661 0.417536i \(-0.862894\pi\)
0.908661 0.417536i \(-0.137106\pi\)
\(444\) 0 0
\(445\) 241.614 0.542954
\(446\) − 727.483i − 1.63113i
\(447\) 0 0
\(448\) −248.331 −0.554310
\(449\) 726.549i 1.61815i 0.587705 + 0.809075i \(0.300031\pi\)
−0.587705 + 0.809075i \(0.699969\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 449.357i 0.994153i
\(453\) 0 0
\(454\) −799.409 −1.76081
\(455\) 33.4883i 0.0736006i
\(456\) 0 0
\(457\) 365.520 0.799825 0.399912 0.916553i \(-0.369041\pi\)
0.399912 + 0.916553i \(0.369041\pi\)
\(458\) 554.621i 1.21096i
\(459\) 0 0
\(460\) −961.821 −2.09092
\(461\) − 153.670i − 0.333340i −0.986013 0.166670i \(-0.946699\pi\)
0.986013 0.166670i \(-0.0533014\pi\)
\(462\) 0 0
\(463\) 325.357 0.702714 0.351357 0.936242i \(-0.385720\pi\)
0.351357 + 0.936242i \(0.385720\pi\)
\(464\) − 592.844i − 1.27768i
\(465\) 0 0
\(466\) 483.359 1.03725
\(467\) 835.965i 1.79007i 0.445991 + 0.895037i \(0.352851\pi\)
−0.445991 + 0.895037i \(0.647149\pi\)
\(468\) 0 0
\(469\) 247.904 0.528580
\(470\) 942.987i 2.00635i
\(471\) 0 0
\(472\) −252.686 −0.535352
\(473\) 0 0
\(474\) 0 0
\(475\) −534.082 −1.12438
\(476\) − 361.930i − 0.760357i
\(477\) 0 0
\(478\) 211.885 0.443274
\(479\) − 335.742i − 0.700922i −0.936577 0.350461i \(-0.886025\pi\)
0.936577 0.350461i \(-0.113975\pi\)
\(480\) 0 0
\(481\) −29.6857 −0.0617165
\(482\) − 634.219i − 1.31581i
\(483\) 0 0
\(484\) 0 0
\(485\) 224.143i 0.462150i
\(486\) 0 0
\(487\) 585.442 1.20214 0.601070 0.799196i \(-0.294741\pi\)
0.601070 + 0.799196i \(0.294741\pi\)
\(488\) − 7.61572i − 0.0156060i
\(489\) 0 0
\(490\) −962.766 −1.96483
\(491\) 325.583i 0.663102i 0.943437 + 0.331551i \(0.107572\pi\)
−0.943437 + 0.331551i \(0.892428\pi\)
\(492\) 0 0
\(493\) 1338.71 2.71543
\(494\) 65.7846i 0.133167i
\(495\) 0 0
\(496\) 603.810 1.21736
\(497\) 31.8030i 0.0639899i
\(498\) 0 0
\(499\) 403.323 0.808262 0.404131 0.914701i \(-0.367574\pi\)
0.404131 + 0.914701i \(0.367574\pi\)
\(500\) − 449.954i − 0.899908i
\(501\) 0 0
\(502\) 559.136 1.11382
\(503\) − 676.545i − 1.34502i −0.740088 0.672510i \(-0.765216\pi\)
0.740088 0.672510i \(-0.234784\pi\)
\(504\) 0 0
\(505\) 26.7021 0.0528754
\(506\) 0 0
\(507\) 0 0
\(508\) −40.4423 −0.0796109
\(509\) − 672.614i − 1.32144i −0.750632 0.660721i \(-0.770251\pi\)
0.750632 0.660721i \(-0.229749\pi\)
\(510\) 0 0
\(511\) 294.204 0.575742
\(512\) 653.350i 1.27607i
\(513\) 0 0
\(514\) 866.672 1.68613
\(515\) − 175.297i − 0.340383i
\(516\) 0 0
\(517\) 0 0
\(518\) 163.660i 0.315946i
\(519\) 0 0
\(520\) −31.3708 −0.0603284
\(521\) − 73.6126i − 0.141291i −0.997501 0.0706455i \(-0.977494\pi\)
0.997501 0.0706455i \(-0.0225059\pi\)
\(522\) 0 0
\(523\) 534.934 1.02282 0.511409 0.859337i \(-0.329124\pi\)
0.511409 + 0.859337i \(0.329124\pi\)
\(524\) − 7.99604i − 0.0152596i
\(525\) 0 0
\(526\) 1366.69 2.59827
\(527\) 1363.47i 2.58723i
\(528\) 0 0
\(529\) −99.6371 −0.188350
\(530\) 963.425i 1.81778i
\(531\) 0 0
\(532\) 199.356 0.374730
\(533\) 46.9586i 0.0881024i
\(534\) 0 0
\(535\) 101.127 0.189023
\(536\) 232.228i 0.433262i
\(537\) 0 0
\(538\) −622.158 −1.15643
\(539\) 0 0
\(540\) 0 0
\(541\) −15.3834 −0.0284351 −0.0142176 0.999899i \(-0.504526\pi\)
−0.0142176 + 0.999899i \(0.504526\pi\)
\(542\) 1373.79i 2.53468i
\(543\) 0 0
\(544\) −1197.58 −2.20143
\(545\) 858.760i 1.57571i
\(546\) 0 0
\(547\) 182.142 0.332984 0.166492 0.986043i \(-0.446756\pi\)
0.166492 + 0.986043i \(0.446756\pi\)
\(548\) − 967.328i − 1.76520i
\(549\) 0 0
\(550\) 0 0
\(551\) 737.380i 1.33826i
\(552\) 0 0
\(553\) 47.1412 0.0852463
\(554\) 314.883i 0.568380i
\(555\) 0 0
\(556\) −915.221 −1.64608
\(557\) − 268.975i − 0.482899i −0.970413 0.241450i \(-0.922377\pi\)
0.970413 0.241450i \(-0.0776229\pi\)
\(558\) 0 0
\(559\) −3.21577 −0.00575272
\(560\) − 257.915i − 0.460562i
\(561\) 0 0
\(562\) −1284.47 −2.28554
\(563\) 400.943i 0.712155i 0.934456 + 0.356078i \(0.115886\pi\)
−0.934456 + 0.356078i \(0.884114\pi\)
\(564\) 0 0
\(565\) −723.085 −1.27980
\(566\) − 899.225i − 1.58874i
\(567\) 0 0
\(568\) −29.7920 −0.0524508
\(569\) − 183.406i − 0.322330i −0.986927 0.161165i \(-0.948475\pi\)
0.986927 0.161165i \(-0.0515252\pi\)
\(570\) 0 0
\(571\) −693.995 −1.21540 −0.607701 0.794166i \(-0.707908\pi\)
−0.607701 + 0.794166i \(0.707908\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 258.887 0.451023
\(575\) − 920.902i − 1.60157i
\(576\) 0 0
\(577\) 531.359 0.920900 0.460450 0.887686i \(-0.347688\pi\)
0.460450 + 0.887686i \(0.347688\pi\)
\(578\) − 1215.73i − 2.10334i
\(579\) 0 0
\(580\) −1945.32 −3.35400
\(581\) 63.9411i 0.110053i
\(582\) 0 0
\(583\) 0 0
\(584\) 275.601i 0.471920i
\(585\) 0 0
\(586\) 138.035 0.235554
\(587\) − 536.946i − 0.914728i −0.889279 0.457364i \(-0.848794\pi\)
0.889279 0.457364i \(-0.151206\pi\)
\(588\) 0 0
\(589\) −751.019 −1.27508
\(590\) − 2249.46i − 3.81264i
\(591\) 0 0
\(592\) 228.628 0.386196
\(593\) 228.398i 0.385156i 0.981282 + 0.192578i \(0.0616849\pi\)
−0.981282 + 0.192578i \(0.938315\pi\)
\(594\) 0 0
\(595\) 582.401 0.978825
\(596\) − 1167.84i − 1.95946i
\(597\) 0 0
\(598\) −113.430 −0.189683
\(599\) − 595.595i − 0.994315i −0.867660 0.497158i \(-0.834377\pi\)
0.867660 0.497158i \(-0.165623\pi\)
\(600\) 0 0
\(601\) −792.909 −1.31932 −0.659658 0.751566i \(-0.729299\pi\)
−0.659658 + 0.751566i \(0.729299\pi\)
\(602\) 17.7288i 0.0294499i
\(603\) 0 0
\(604\) 286.488 0.474318
\(605\) 0 0
\(606\) 0 0
\(607\) 873.253 1.43864 0.719318 0.694680i \(-0.244454\pi\)
0.719318 + 0.694680i \(0.244454\pi\)
\(608\) − 659.643i − 1.08494i
\(609\) 0 0
\(610\) 67.7964 0.111142
\(611\) 61.1295i 0.100048i
\(612\) 0 0
\(613\) −315.018 −0.513895 −0.256948 0.966425i \(-0.582717\pi\)
−0.256948 + 0.966425i \(0.582717\pi\)
\(614\) 1335.97i 2.17585i
\(615\) 0 0
\(616\) 0 0
\(617\) − 738.239i − 1.19650i −0.801311 0.598249i \(-0.795864\pi\)
0.801311 0.598249i \(-0.204136\pi\)
\(618\) 0 0
\(619\) 925.109 1.49452 0.747261 0.664531i \(-0.231369\pi\)
0.747261 + 0.664531i \(0.231369\pi\)
\(620\) − 1981.30i − 3.19565i
\(621\) 0 0
\(622\) −997.451 −1.60362
\(623\) − 86.3503i − 0.138604i
\(624\) 0 0
\(625\) −194.189 −0.310702
\(626\) 304.938i 0.487122i
\(627\) 0 0
\(628\) −99.7039 −0.158764
\(629\) 516.269i 0.820777i
\(630\) 0 0
\(631\) −352.401 −0.558481 −0.279240 0.960221i \(-0.590083\pi\)
−0.279240 + 0.960221i \(0.590083\pi\)
\(632\) 44.1604i 0.0698740i
\(633\) 0 0
\(634\) 1126.21 1.77635
\(635\) − 65.0779i − 0.102485i
\(636\) 0 0
\(637\) −62.4117 −0.0979776
\(638\) 0 0
\(639\) 0 0
\(640\) 645.227 1.00817
\(641\) − 247.035i − 0.385390i −0.981259 0.192695i \(-0.938277\pi\)
0.981259 0.192695i \(-0.0617227\pi\)
\(642\) 0 0
\(643\) 657.375 1.02236 0.511178 0.859475i \(-0.329209\pi\)
0.511178 + 0.859475i \(0.329209\pi\)
\(644\) 343.744i 0.533764i
\(645\) 0 0
\(646\) 1144.07 1.77101
\(647\) − 645.586i − 0.997814i −0.866655 0.498907i \(-0.833735\pi\)
0.866655 0.498907i \(-0.166265\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) − 166.166i − 0.255640i
\(651\) 0 0
\(652\) −510.491 −0.782961
\(653\) 35.5445i 0.0544326i 0.999630 + 0.0272163i \(0.00866429\pi\)
−0.999630 + 0.0272163i \(0.991336\pi\)
\(654\) 0 0
\(655\) 12.8669 0.0196441
\(656\) − 361.658i − 0.551308i
\(657\) 0 0
\(658\) 337.013 0.512178
\(659\) − 721.237i − 1.09444i −0.836988 0.547221i \(-0.815686\pi\)
0.836988 0.547221i \(-0.184314\pi\)
\(660\) 0 0
\(661\) −617.542 −0.934253 −0.467127 0.884190i \(-0.654711\pi\)
−0.467127 + 0.884190i \(0.654711\pi\)
\(662\) − 509.560i − 0.769728i
\(663\) 0 0
\(664\) −59.8980 −0.0902078
\(665\) 320.795i 0.482398i
\(666\) 0 0
\(667\) −1271.44 −1.90621
\(668\) − 191.447i − 0.286597i
\(669\) 0 0
\(670\) −2067.34 −3.08558
\(671\) 0 0
\(672\) 0 0
\(673\) 840.469 1.24884 0.624420 0.781089i \(-0.285335\pi\)
0.624420 + 0.781089i \(0.285335\pi\)
\(674\) − 572.039i − 0.848723i
\(675\) 0 0
\(676\) 813.904 1.20400
\(677\) − 386.555i − 0.570983i −0.958381 0.285491i \(-0.907843\pi\)
0.958381 0.285491i \(-0.0921568\pi\)
\(678\) 0 0
\(679\) 80.1061 0.117977
\(680\) 545.575i 0.802316i
\(681\) 0 0
\(682\) 0 0
\(683\) − 122.406i − 0.179219i −0.995977 0.0896094i \(-0.971438\pi\)
0.995977 0.0896094i \(-0.0285619\pi\)
\(684\) 0 0
\(685\) 1556.58 2.27238
\(686\) 754.146i 1.09934i
\(687\) 0 0
\(688\) 24.7667 0.0359981
\(689\) 62.4544i 0.0906450i
\(690\) 0 0
\(691\) 244.981 0.354532 0.177266 0.984163i \(-0.443275\pi\)
0.177266 + 0.984163i \(0.443275\pi\)
\(692\) − 446.402i − 0.645090i
\(693\) 0 0
\(694\) 553.926 0.798165
\(695\) − 1472.73i − 2.11904i
\(696\) 0 0
\(697\) 816.665 1.17169
\(698\) 525.707i 0.753162i
\(699\) 0 0
\(700\) −503.557 −0.719367
\(701\) − 197.380i − 0.281569i −0.990040 0.140785i \(-0.955037\pi\)
0.990040 0.140785i \(-0.0449625\pi\)
\(702\) 0 0
\(703\) −284.368 −0.404506
\(704\) 0 0
\(705\) 0 0
\(706\) 693.903 0.982865
\(707\) − 9.54302i − 0.0134979i
\(708\) 0 0
\(709\) −525.864 −0.741698 −0.370849 0.928693i \(-0.620933\pi\)
−0.370849 + 0.928693i \(0.620933\pi\)
\(710\) − 265.214i − 0.373541i
\(711\) 0 0
\(712\) 80.8903 0.113610
\(713\) − 1294.96i − 1.81621i
\(714\) 0 0
\(715\) 0 0
\(716\) 1065.29i 1.48784i
\(717\) 0 0
\(718\) 1201.46 1.67334
\(719\) 54.4556i 0.0757379i 0.999283 + 0.0378690i \(0.0120569\pi\)
−0.999283 + 0.0378690i \(0.987943\pi\)
\(720\) 0 0
\(721\) −62.6493 −0.0868923
\(722\) − 445.740i − 0.617369i
\(723\) 0 0
\(724\) 218.841 0.302267
\(725\) − 1862.56i − 2.56905i
\(726\) 0 0
\(727\) 1312.96 1.80600 0.902999 0.429642i \(-0.141360\pi\)
0.902999 + 0.429642i \(0.141360\pi\)
\(728\) 11.2116i 0.0154005i
\(729\) 0 0
\(730\) −2453.45 −3.36089
\(731\) 55.9260i 0.0765062i
\(732\) 0 0
\(733\) −395.593 −0.539690 −0.269845 0.962904i \(-0.586973\pi\)
−0.269845 + 0.962904i \(0.586973\pi\)
\(734\) − 1613.91i − 2.19878i
\(735\) 0 0
\(736\) 1137.40 1.54538
\(737\) 0 0
\(738\) 0 0
\(739\) −1425.40 −1.92883 −0.964413 0.264401i \(-0.914826\pi\)
−0.964413 + 0.264401i \(0.914826\pi\)
\(740\) − 750.206i − 1.01379i
\(741\) 0 0
\(742\) 344.317 0.464039
\(743\) − 648.328i − 0.872581i −0.899806 0.436291i \(-0.856292\pi\)
0.899806 0.436291i \(-0.143708\pi\)
\(744\) 0 0
\(745\) 1879.23 2.52246
\(746\) 476.932i 0.639319i
\(747\) 0 0
\(748\) 0 0
\(749\) − 36.1417i − 0.0482532i
\(750\) 0 0
\(751\) −520.551 −0.693144 −0.346572 0.938023i \(-0.612654\pi\)
−0.346572 + 0.938023i \(0.612654\pi\)
\(752\) − 470.797i − 0.626060i
\(753\) 0 0
\(754\) −229.418 −0.304267
\(755\) 461.003i 0.610600i
\(756\) 0 0
\(757\) −546.651 −0.722128 −0.361064 0.932541i \(-0.617586\pi\)
−0.361064 + 0.932541i \(0.617586\pi\)
\(758\) − 1128.43i − 1.48869i
\(759\) 0 0
\(760\) −300.510 −0.395408
\(761\) 825.999i 1.08541i 0.839922 + 0.542707i \(0.182600\pi\)
−0.839922 + 0.542707i \(0.817400\pi\)
\(762\) 0 0
\(763\) 306.911 0.402243
\(764\) 727.372i 0.952058i
\(765\) 0 0
\(766\) −713.997 −0.932111
\(767\) − 145.822i − 0.190120i
\(768\) 0 0
\(769\) −712.096 −0.926003 −0.463001 0.886358i \(-0.653228\pi\)
−0.463001 + 0.886358i \(0.653228\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 945.785 1.22511
\(773\) 276.314i 0.357456i 0.983899 + 0.178728i \(0.0571982\pi\)
−0.983899 + 0.178728i \(0.942802\pi\)
\(774\) 0 0
\(775\) 1897.01 2.44776
\(776\) 75.0409i 0.0967022i
\(777\) 0 0
\(778\) −1497.99 −1.92544
\(779\) 449.831i 0.577446i
\(780\) 0 0
\(781\) 0 0
\(782\) 1972.69i 2.52262i
\(783\) 0 0
\(784\) 480.673 0.613103
\(785\) − 160.439i − 0.204381i
\(786\) 0 0
\(787\) −955.911 −1.21463 −0.607313 0.794463i \(-0.707753\pi\)
−0.607313 + 0.794463i \(0.707753\pi\)
\(788\) − 1617.63i − 2.05283i
\(789\) 0 0
\(790\) −393.124 −0.497625
\(791\) 258.422i 0.326703i
\(792\) 0 0
\(793\) 4.39493 0.00554216
\(794\) 289.405i 0.364490i
\(795\) 0 0
\(796\) 1439.52 1.80844
\(797\) − 1060.39i − 1.33048i −0.746631 0.665238i \(-0.768330\pi\)
0.746631 0.665238i \(-0.231670\pi\)
\(798\) 0 0
\(799\) 1063.11 1.33056
\(800\) 1666.20i 2.08275i
\(801\) 0 0
\(802\) 1958.54 2.44207
\(803\) 0 0
\(804\) 0 0
\(805\) −553.137 −0.687126
\(806\) − 233.661i − 0.289902i
\(807\) 0 0
\(808\) 8.93960 0.0110639
\(809\) − 366.478i − 0.453001i −0.974011 0.226501i \(-0.927271\pi\)
0.974011 0.226501i \(-0.0727286\pi\)
\(810\) 0 0
\(811\) −600.466 −0.740402 −0.370201 0.928952i \(-0.620711\pi\)
−0.370201 + 0.928952i \(0.620711\pi\)
\(812\) 695.236i 0.856202i
\(813\) 0 0
\(814\) 0 0
\(815\) − 821.458i − 1.00792i
\(816\) 0 0
\(817\) −30.8048 −0.0377048
\(818\) − 302.672i − 0.370015i
\(819\) 0 0
\(820\) −1186.72 −1.44722
\(821\) 796.899i 0.970645i 0.874335 + 0.485322i \(0.161298\pi\)
−0.874335 + 0.485322i \(0.838702\pi\)
\(822\) 0 0
\(823\) 822.371 0.999235 0.499618 0.866246i \(-0.333474\pi\)
0.499618 + 0.866246i \(0.333474\pi\)
\(824\) − 58.6879i − 0.0712232i
\(825\) 0 0
\(826\) −803.931 −0.973282
\(827\) 913.876i 1.10505i 0.833497 + 0.552524i \(0.186335\pi\)
−0.833497 + 0.552524i \(0.813665\pi\)
\(828\) 0 0
\(829\) −1298.36 −1.56618 −0.783091 0.621908i \(-0.786358\pi\)
−0.783091 + 0.621908i \(0.786358\pi\)
\(830\) − 533.222i − 0.642437i
\(831\) 0 0
\(832\) 134.247 0.161355
\(833\) 1085.41i 1.30302i
\(834\) 0 0
\(835\) 308.067 0.368942
\(836\) 0 0
\(837\) 0 0
\(838\) −1.31082 −0.00156423
\(839\) 58.6568i 0.0699128i 0.999389 + 0.0349564i \(0.0111292\pi\)
−0.999389 + 0.0349564i \(0.988871\pi\)
\(840\) 0 0
\(841\) −1730.54 −2.05772
\(842\) − 1247.47i − 1.48155i
\(843\) 0 0
\(844\) −565.046 −0.669485
\(845\) 1309.70i 1.54994i
\(846\) 0 0
\(847\) 0 0
\(848\) − 481.001i − 0.567218i
\(849\) 0 0
\(850\) −2889.83 −3.39980
\(851\) − 490.327i − 0.576178i
\(852\) 0 0
\(853\) 1144.33 1.34154 0.670771 0.741665i \(-0.265964\pi\)
0.670771 + 0.741665i \(0.265964\pi\)
\(854\) − 24.2297i − 0.0283720i
\(855\) 0 0
\(856\) 33.8564 0.0395519
\(857\) 1380.38i 1.61071i 0.592790 + 0.805357i \(0.298027\pi\)
−0.592790 + 0.805357i \(0.701973\pi\)
\(858\) 0 0
\(859\) 210.196 0.244698 0.122349 0.992487i \(-0.460957\pi\)
0.122349 + 0.992487i \(0.460957\pi\)
\(860\) − 81.2679i − 0.0944975i
\(861\) 0 0
\(862\) −751.256 −0.871527
\(863\) − 201.367i − 0.233333i −0.993171 0.116667i \(-0.962779\pi\)
0.993171 0.116667i \(-0.0372209\pi\)
\(864\) 0 0
\(865\) 718.329 0.830439
\(866\) − 370.905i − 0.428297i
\(867\) 0 0
\(868\) −708.096 −0.815778
\(869\) 0 0
\(870\) 0 0
\(871\) −134.016 −0.153865
\(872\) 287.505i 0.329707i
\(873\) 0 0
\(874\) −1086.59 −1.24323
\(875\) − 258.765i − 0.295732i
\(876\) 0 0
\(877\) 1058.24 1.20666 0.603332 0.797490i \(-0.293840\pi\)
0.603332 + 0.797490i \(0.293840\pi\)
\(878\) 219.633i 0.250151i
\(879\) 0 0
\(880\) 0 0
\(881\) 402.743i 0.457144i 0.973527 + 0.228572i \(0.0734056\pi\)
−0.973527 + 0.228572i \(0.926594\pi\)
\(882\) 0 0
\(883\) −1161.84 −1.31579 −0.657893 0.753111i \(-0.728552\pi\)
−0.657893 + 0.753111i \(0.728552\pi\)
\(884\) 195.658i 0.221333i
\(885\) 0 0
\(886\) 1102.55 1.24441
\(887\) 488.442i 0.550668i 0.961349 + 0.275334i \(0.0887884\pi\)
−0.961349 + 0.275334i \(0.911212\pi\)
\(888\) 0 0
\(889\) −23.2581 −0.0261621
\(890\) 720.099i 0.809100i
\(891\) 0 0
\(892\) 1191.80 1.33609
\(893\) 585.578i 0.655743i
\(894\) 0 0
\(895\) −1714.22 −1.91532
\(896\) − 230.597i − 0.257363i
\(897\) 0 0
\(898\) −2165.38 −2.41134
\(899\) − 2619.11i − 2.91336i
\(900\) 0 0
\(901\) 1086.16 1.20550
\(902\) 0 0
\(903\) 0 0
\(904\) −242.082 −0.267790
\(905\) 352.149i 0.389115i
\(906\) 0 0
\(907\) −1687.27 −1.86028 −0.930138 0.367210i \(-0.880313\pi\)
−0.930138 + 0.367210i \(0.880313\pi\)
\(908\) − 1309.63i − 1.44232i
\(909\) 0 0
\(910\) −99.8073 −0.109678
\(911\) − 1092.72i − 1.19947i −0.800199 0.599735i \(-0.795273\pi\)
0.800199 0.599735i \(-0.204727\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 1089.38i 1.19188i
\(915\) 0 0
\(916\) −908.606 −0.991928
\(917\) − 4.59847i − 0.00501469i
\(918\) 0 0
\(919\) −1204.42 −1.31058 −0.655288 0.755379i \(-0.727453\pi\)
−0.655288 + 0.755379i \(0.727453\pi\)
\(920\) − 518.161i − 0.563219i
\(921\) 0 0
\(922\) 457.992 0.496737
\(923\) − 17.1926i − 0.0186269i
\(924\) 0 0
\(925\) 718.289 0.776529
\(926\) 969.682i 1.04717i
\(927\) 0 0
\(928\) 2300.44 2.47892
\(929\) − 227.170i − 0.244531i −0.992497 0.122266i \(-0.960984\pi\)
0.992497 0.122266i \(-0.0390160\pi\)
\(930\) 0 0
\(931\) −597.861 −0.642171
\(932\) 791.861i 0.849636i
\(933\) 0 0
\(934\) −2491.48 −2.66754
\(935\) 0 0
\(936\) 0 0
\(937\) −355.723 −0.379640 −0.189820 0.981819i \(-0.560791\pi\)
−0.189820 + 0.981819i \(0.560791\pi\)
\(938\) 738.844i 0.787680i
\(939\) 0 0
\(940\) −1544.84 −1.64345
\(941\) − 1363.18i − 1.44866i −0.689456 0.724328i \(-0.742150\pi\)
0.689456 0.724328i \(-0.257850\pi\)
\(942\) 0 0
\(943\) −775.630 −0.822513
\(944\) 1123.07i 1.18969i
\(945\) 0 0
\(946\) 0 0
\(947\) 90.3298i 0.0953852i 0.998862 + 0.0476926i \(0.0151868\pi\)
−0.998862 + 0.0476926i \(0.984813\pi\)
\(948\) 0 0
\(949\) −159.046 −0.167593
\(950\) − 1591.76i − 1.67553i
\(951\) 0 0
\(952\) 194.982 0.204813
\(953\) − 612.654i − 0.642869i −0.946932 0.321434i \(-0.895835\pi\)
0.946932 0.321434i \(-0.104165\pi\)
\(954\) 0 0
\(955\) −1170.45 −1.22561
\(956\) 347.120i 0.363096i
\(957\) 0 0
\(958\) 1000.63 1.04450
\(959\) − 556.303i − 0.580087i
\(960\) 0 0
\(961\) 1706.55 1.77581
\(962\) − 88.4741i − 0.0919689i
\(963\) 0 0
\(964\) 1039.01 1.07781
\(965\) 1521.91i 1.57711i
\(966\) 0 0
\(967\) 446.224 0.461452 0.230726 0.973019i \(-0.425890\pi\)
0.230726 + 0.973019i \(0.425890\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −668.027 −0.688688
\(971\) 152.782i 0.157345i 0.996901 + 0.0786723i \(0.0250681\pi\)
−0.996901 + 0.0786723i \(0.974932\pi\)
\(972\) 0 0
\(973\) −526.337 −0.540943
\(974\) 1744.83i 1.79141i
\(975\) 0 0
\(976\) −33.8482 −0.0346805
\(977\) 564.811i 0.578107i 0.957313 + 0.289054i \(0.0933406\pi\)
−0.957313 + 0.289054i \(0.906659\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) − 1577.25i − 1.60944i
\(981\) 0 0
\(982\) −970.356 −0.988142
\(983\) 945.431i 0.961781i 0.876781 + 0.480890i \(0.159687\pi\)
−0.876781 + 0.480890i \(0.840313\pi\)
\(984\) 0 0
\(985\) 2603.01 2.64265
\(986\) 3989.84i 4.04649i
\(987\) 0 0
\(988\) −107.771 −0.109080
\(989\) − 53.1159i − 0.0537067i
\(990\) 0 0
\(991\) 1225.61 1.23674 0.618371 0.785886i \(-0.287793\pi\)
0.618371 + 0.785886i \(0.287793\pi\)
\(992\) 2342.99i 2.36189i
\(993\) 0 0
\(994\) −94.7845 −0.0953566
\(995\) 2316.41i 2.32805i
\(996\) 0 0
\(997\) 1276.52 1.28036 0.640180 0.768225i \(-0.278860\pi\)
0.640180 + 0.768225i \(0.278860\pi\)
\(998\) 1202.05i 1.20446i
\(999\) 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.j.485.13 16
3.2 odd 2 inner 1089.3.b.j.485.4 16
11.2 odd 10 99.3.l.a.26.2 32
11.6 odd 10 99.3.l.a.80.7 yes 32
11.10 odd 2 1089.3.b.i.485.4 16
33.2 even 10 99.3.l.a.26.7 yes 32
33.17 even 10 99.3.l.a.80.2 yes 32
33.32 even 2 1089.3.b.i.485.13 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.3.l.a.26.2 32 11.2 odd 10
99.3.l.a.26.7 yes 32 33.2 even 10
99.3.l.a.80.2 yes 32 33.17 even 10
99.3.l.a.80.7 yes 32 11.6 odd 10
1089.3.b.i.485.4 16 11.10 odd 2
1089.3.b.i.485.13 16 33.32 even 2
1089.3.b.j.485.4 16 3.2 odd 2 inner
1089.3.b.j.485.13 16 1.1 even 1 trivial