Properties

Label 1089.2.d.g.1088.1
Level $1089$
Weight $2$
Character 1089.1088
Analytic conductor $8.696$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(8.69570878012\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 2 x^{14} - 16 x^{12} - 72 x^{10} + 26 x^{8} + 360 x^{6} + 725 x^{4} + 1000 x^{2} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1088.1
Root \(0.0783900 - 1.17295i\) of defining polynomial
Character \(\chi\) \(=\) 1089.1088
Dual form 1089.2.d.g.1088.2

$q$-expansion

\(f(q)\) \(=\) \(q-2.43632 q^{2} +3.93565 q^{4} -3.79576i q^{5} -0.367791i q^{7} -4.71586 q^{8} +O(q^{10})\) \(q-2.43632 q^{2} +3.93565 q^{4} -3.79576i q^{5} -0.367791i q^{7} -4.71586 q^{8} +9.24768i q^{10} +0.948263i q^{13} +0.896057i q^{14} +3.61803 q^{16} -3.43470 q^{17} -4.26229i q^{19} -14.9388i q^{20} -4.96800i q^{23} -9.40778 q^{25} -2.31027i q^{26} -1.44750i q^{28} -2.48342 q^{29} +3.51391 q^{31} +0.617031 q^{32} +8.36801 q^{34} -1.39605 q^{35} -7.18234 q^{37} +10.3843i q^{38} +17.9003i q^{40} -2.71910 q^{41} +1.88749i q^{43} +12.1036i q^{46} -0.0206077i q^{47} +6.86473 q^{49} +22.9204 q^{50} +3.73203i q^{52} +5.54524i q^{53} +1.73445i q^{56} +6.05040 q^{58} +6.62419i q^{59} -9.75250i q^{61} -8.56101 q^{62} -8.73935 q^{64} +3.59938 q^{65} +4.46351 q^{67} -13.5178 q^{68} +3.40122 q^{70} +10.3266i q^{71} -4.39587i q^{73} +17.4985 q^{74} -16.7749i q^{76} -10.9371i q^{79} -13.7332i q^{80} +6.62460 q^{82} -9.18325 q^{83} +13.0373i q^{85} -4.59852i q^{86} +3.04837i q^{89} +0.348763 q^{91} -19.5523i q^{92} +0.0502070i q^{94} -16.1786 q^{95} -15.0868 q^{97} -16.7247 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4} + O(q^{10}) \) \( 16 q + 16 q^{4} + 40 q^{16} - 32 q^{25} + 16 q^{31} + 40 q^{34} + 8 q^{37} + 16 q^{49} + 32 q^{58} - 104 q^{64} + 96 q^{67} - 64 q^{70} + 88 q^{82} + 48 q^{91} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.43632 −1.72274 −0.861369 0.507980i \(-0.830392\pi\)
−0.861369 + 0.507980i \(0.830392\pi\)
\(3\) 0 0
\(4\) 3.93565 1.96782
\(5\) − 3.79576i − 1.69751i −0.528782 0.848757i \(-0.677351\pi\)
0.528782 0.848757i \(-0.322649\pi\)
\(6\) 0 0
\(7\) − 0.367791i − 0.139012i −0.997582 0.0695060i \(-0.977858\pi\)
0.997582 0.0695060i \(-0.0221423\pi\)
\(8\) −4.71586 −1.66731
\(9\) 0 0
\(10\) 9.24768i 2.92437i
\(11\) 0 0
\(12\) 0 0
\(13\) 0.948263i 0.263001i 0.991316 + 0.131500i \(0.0419795\pi\)
−0.991316 + 0.131500i \(0.958021\pi\)
\(14\) 0.896057i 0.239481i
\(15\) 0 0
\(16\) 3.61803 0.904508
\(17\) −3.43470 −0.833036 −0.416518 0.909127i \(-0.636750\pi\)
−0.416518 + 0.909127i \(0.636750\pi\)
\(18\) 0 0
\(19\) − 4.26229i − 0.977837i −0.872329 0.488919i \(-0.837391\pi\)
0.872329 0.488919i \(-0.162609\pi\)
\(20\) − 14.9388i − 3.34041i
\(21\) 0 0
\(22\) 0 0
\(23\) − 4.96800i − 1.03590i −0.855411 0.517950i \(-0.826695\pi\)
0.855411 0.517950i \(-0.173305\pi\)
\(24\) 0 0
\(25\) −9.40778 −1.88156
\(26\) − 2.31027i − 0.453081i
\(27\) 0 0
\(28\) − 1.44750i − 0.273551i
\(29\) −2.48342 −0.461159 −0.230580 0.973053i \(-0.574062\pi\)
−0.230580 + 0.973053i \(0.574062\pi\)
\(30\) 0 0
\(31\) 3.51391 0.631117 0.315559 0.948906i \(-0.397808\pi\)
0.315559 + 0.948906i \(0.397808\pi\)
\(32\) 0.617031 0.109077
\(33\) 0 0
\(34\) 8.36801 1.43510
\(35\) −1.39605 −0.235975
\(36\) 0 0
\(37\) −7.18234 −1.18077 −0.590385 0.807122i \(-0.701024\pi\)
−0.590385 + 0.807122i \(0.701024\pi\)
\(38\) 10.3843i 1.68456i
\(39\) 0 0
\(40\) 17.9003i 2.83028i
\(41\) −2.71910 −0.424653 −0.212326 0.977199i \(-0.568104\pi\)
−0.212326 + 0.977199i \(0.568104\pi\)
\(42\) 0 0
\(43\) 1.88749i 0.287839i 0.989589 + 0.143919i \(0.0459706\pi\)
−0.989589 + 0.143919i \(0.954029\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 12.1036i 1.78458i
\(47\) − 0.0206077i − 0.00300594i −0.999999 0.00150297i \(-0.999522\pi\)
0.999999 0.00150297i \(-0.000478411\pi\)
\(48\) 0 0
\(49\) 6.86473 0.980676
\(50\) 22.9204 3.24143
\(51\) 0 0
\(52\) 3.73203i 0.517539i
\(53\) 5.54524i 0.761697i 0.924637 + 0.380849i \(0.124368\pi\)
−0.924637 + 0.380849i \(0.875632\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.73445i 0.231776i
\(57\) 0 0
\(58\) 6.05040 0.794456
\(59\) 6.62419i 0.862396i 0.902257 + 0.431198i \(0.141909\pi\)
−0.902257 + 0.431198i \(0.858091\pi\)
\(60\) 0 0
\(61\) − 9.75250i − 1.24868i −0.781153 0.624340i \(-0.785368\pi\)
0.781153 0.624340i \(-0.214632\pi\)
\(62\) −8.56101 −1.08725
\(63\) 0 0
\(64\) −8.73935 −1.09242
\(65\) 3.59938 0.446448
\(66\) 0 0
\(67\) 4.46351 0.545305 0.272652 0.962113i \(-0.412099\pi\)
0.272652 + 0.962113i \(0.412099\pi\)
\(68\) −13.5178 −1.63927
\(69\) 0 0
\(70\) 3.40122 0.406523
\(71\) 10.3266i 1.22555i 0.790258 + 0.612774i \(0.209946\pi\)
−0.790258 + 0.612774i \(0.790054\pi\)
\(72\) 0 0
\(73\) − 4.39587i − 0.514498i −0.966345 0.257249i \(-0.917184\pi\)
0.966345 0.257249i \(-0.0828160\pi\)
\(74\) 17.4985 2.03416
\(75\) 0 0
\(76\) − 16.7749i − 1.92421i
\(77\) 0 0
\(78\) 0 0
\(79\) − 10.9371i − 1.23052i −0.788324 0.615261i \(-0.789051\pi\)
0.788324 0.615261i \(-0.210949\pi\)
\(80\) − 13.7332i − 1.53542i
\(81\) 0 0
\(82\) 6.62460 0.731565
\(83\) −9.18325 −1.00799 −0.503996 0.863706i \(-0.668137\pi\)
−0.503996 + 0.863706i \(0.668137\pi\)
\(84\) 0 0
\(85\) 13.0373i 1.41409i
\(86\) − 4.59852i − 0.495871i
\(87\) 0 0
\(88\) 0 0
\(89\) 3.04837i 0.323127i 0.986862 + 0.161563i \(0.0516536\pi\)
−0.986862 + 0.161563i \(0.948346\pi\)
\(90\) 0 0
\(91\) 0.348763 0.0365603
\(92\) − 19.5523i − 2.03847i
\(93\) 0 0
\(94\) 0.0502070i 0.00517845i
\(95\) −16.1786 −1.65989
\(96\) 0 0
\(97\) −15.0868 −1.53184 −0.765919 0.642938i \(-0.777715\pi\)
−0.765919 + 0.642938i \(0.777715\pi\)
\(98\) −16.7247 −1.68945
\(99\) 0 0
\(100\) −37.0257 −3.70257
\(101\) 17.4598 1.73732 0.868658 0.495413i \(-0.164983\pi\)
0.868658 + 0.495413i \(0.164983\pi\)
\(102\) 0 0
\(103\) −9.31561 −0.917894 −0.458947 0.888464i \(-0.651773\pi\)
−0.458947 + 0.888464i \(0.651773\pi\)
\(104\) − 4.47187i − 0.438503i
\(105\) 0 0
\(106\) − 13.5100i − 1.31220i
\(107\) −15.4255 −1.49124 −0.745620 0.666371i \(-0.767847\pi\)
−0.745620 + 0.666371i \(0.767847\pi\)
\(108\) 0 0
\(109\) 10.6286i 1.01803i 0.860757 + 0.509016i \(0.169991\pi\)
−0.860757 + 0.509016i \(0.830009\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 1.33068i − 0.125738i
\(113\) 1.04702i 0.0984953i 0.998787 + 0.0492476i \(0.0156824\pi\)
−0.998787 + 0.0492476i \(0.984318\pi\)
\(114\) 0 0
\(115\) −18.8573 −1.75846
\(116\) −9.77386 −0.907480
\(117\) 0 0
\(118\) − 16.1386i − 1.48568i
\(119\) 1.26325i 0.115802i
\(120\) 0 0
\(121\) 0 0
\(122\) 23.7602i 2.15115i
\(123\) 0 0
\(124\) 13.8295 1.24193
\(125\) 16.7309i 1.49646i
\(126\) 0 0
\(127\) 17.5450i 1.55686i 0.627729 + 0.778432i \(0.283984\pi\)
−0.627729 + 0.778432i \(0.716016\pi\)
\(128\) 20.0578 1.77287
\(129\) 0 0
\(130\) −8.76923 −0.769112
\(131\) −13.8890 −1.21349 −0.606744 0.794897i \(-0.707525\pi\)
−0.606744 + 0.794897i \(0.707525\pi\)
\(132\) 0 0
\(133\) −1.56764 −0.135931
\(134\) −10.8745 −0.939417
\(135\) 0 0
\(136\) 16.1975 1.38893
\(137\) − 1.03773i − 0.0886593i −0.999017 0.0443296i \(-0.985885\pi\)
0.999017 0.0443296i \(-0.0141152\pi\)
\(138\) 0 0
\(139\) − 19.8351i − 1.68239i −0.540732 0.841195i \(-0.681853\pi\)
0.540732 0.841195i \(-0.318147\pi\)
\(140\) −5.49435 −0.464358
\(141\) 0 0
\(142\) − 25.1590i − 2.11130i
\(143\) 0 0
\(144\) 0 0
\(145\) 9.42646i 0.782825i
\(146\) 10.7097i 0.886345i
\(147\) 0 0
\(148\) −28.2672 −2.32355
\(149\) −8.03599 −0.658334 −0.329167 0.944272i \(-0.606768\pi\)
−0.329167 + 0.944272i \(0.606768\pi\)
\(150\) 0 0
\(151\) 10.0735i 0.819773i 0.912137 + 0.409886i \(0.134432\pi\)
−0.912137 + 0.409886i \(0.865568\pi\)
\(152\) 20.1004i 1.63036i
\(153\) 0 0
\(154\) 0 0
\(155\) − 13.3380i − 1.07133i
\(156\) 0 0
\(157\) −16.1823 −1.29149 −0.645746 0.763552i \(-0.723453\pi\)
−0.645746 + 0.763552i \(0.723453\pi\)
\(158\) 26.6463i 2.11986i
\(159\) 0 0
\(160\) − 2.34210i − 0.185159i
\(161\) −1.82719 −0.144003
\(162\) 0 0
\(163\) 4.27584 0.334910 0.167455 0.985880i \(-0.446445\pi\)
0.167455 + 0.985880i \(0.446445\pi\)
\(164\) −10.7014 −0.835642
\(165\) 0 0
\(166\) 22.3733 1.73651
\(167\) −18.3529 −1.42019 −0.710094 0.704107i \(-0.751347\pi\)
−0.710094 + 0.704107i \(0.751347\pi\)
\(168\) 0 0
\(169\) 12.1008 0.930831
\(170\) − 31.7630i − 2.43611i
\(171\) 0 0
\(172\) 7.42848i 0.566416i
\(173\) 6.18239 0.470038 0.235019 0.971991i \(-0.424485\pi\)
0.235019 + 0.971991i \(0.424485\pi\)
\(174\) 0 0
\(175\) 3.46010i 0.261559i
\(176\) 0 0
\(177\) 0 0
\(178\) − 7.42680i − 0.556662i
\(179\) − 9.22567i − 0.689559i −0.938684 0.344780i \(-0.887954\pi\)
0.938684 0.344780i \(-0.112046\pi\)
\(180\) 0 0
\(181\) −4.18767 −0.311267 −0.155634 0.987815i \(-0.549742\pi\)
−0.155634 + 0.987815i \(0.549742\pi\)
\(182\) −0.849698 −0.0629838
\(183\) 0 0
\(184\) 23.4284i 1.72716i
\(185\) 27.2624i 2.00437i
\(186\) 0 0
\(187\) 0 0
\(188\) − 0.0811047i − 0.00591517i
\(189\) 0 0
\(190\) 39.4163 2.85956
\(191\) − 20.1830i − 1.46039i −0.683240 0.730194i \(-0.739430\pi\)
0.683240 0.730194i \(-0.260570\pi\)
\(192\) 0 0
\(193\) 8.01396i 0.576857i 0.957501 + 0.288429i \(0.0931328\pi\)
−0.957501 + 0.288429i \(0.906867\pi\)
\(194\) 36.7564 2.63895
\(195\) 0 0
\(196\) 27.0172 1.92980
\(197\) 21.1710 1.50837 0.754187 0.656659i \(-0.228031\pi\)
0.754187 + 0.656659i \(0.228031\pi\)
\(198\) 0 0
\(199\) −10.5160 −0.745457 −0.372729 0.927940i \(-0.621578\pi\)
−0.372729 + 0.927940i \(0.621578\pi\)
\(200\) 44.3658 3.13713
\(201\) 0 0
\(202\) −42.5376 −2.99294
\(203\) 0.913380i 0.0641067i
\(204\) 0 0
\(205\) 10.3211i 0.720854i
\(206\) 22.6958 1.58129
\(207\) 0 0
\(208\) 3.43085i 0.237886i
\(209\) 0 0
\(210\) 0 0
\(211\) 3.69822i 0.254596i 0.991864 + 0.127298i \(0.0406305\pi\)
−0.991864 + 0.127298i \(0.959369\pi\)
\(212\) 21.8241i 1.49889i
\(213\) 0 0
\(214\) 37.5815 2.56902
\(215\) 7.16444 0.488611
\(216\) 0 0
\(217\) − 1.29239i − 0.0877329i
\(218\) − 25.8946i − 1.75380i
\(219\) 0 0
\(220\) 0 0
\(221\) − 3.25699i − 0.219089i
\(222\) 0 0
\(223\) −17.8971 −1.19848 −0.599240 0.800570i \(-0.704530\pi\)
−0.599240 + 0.800570i \(0.704530\pi\)
\(224\) − 0.226939i − 0.0151630i
\(225\) 0 0
\(226\) − 2.55087i − 0.169681i
\(227\) −17.3784 −1.15344 −0.576722 0.816940i \(-0.695668\pi\)
−0.576722 + 0.816940i \(0.695668\pi\)
\(228\) 0 0
\(229\) 4.27666 0.282609 0.141305 0.989966i \(-0.454870\pi\)
0.141305 + 0.989966i \(0.454870\pi\)
\(230\) 45.9425 3.02936
\(231\) 0 0
\(232\) 11.7114 0.768894
\(233\) 23.3828 1.53186 0.765929 0.642925i \(-0.222279\pi\)
0.765929 + 0.642925i \(0.222279\pi\)
\(234\) 0 0
\(235\) −0.0782219 −0.00510263
\(236\) 26.0705i 1.69704i
\(237\) 0 0
\(238\) − 3.07768i − 0.199497i
\(239\) 19.9957 1.29342 0.646708 0.762738i \(-0.276145\pi\)
0.646708 + 0.762738i \(0.276145\pi\)
\(240\) 0 0
\(241\) 2.32570i 0.149811i 0.997191 + 0.0749057i \(0.0238656\pi\)
−0.997191 + 0.0749057i \(0.976134\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) − 38.3824i − 2.45718i
\(245\) − 26.0569i − 1.66471i
\(246\) 0 0
\(247\) 4.04178 0.257172
\(248\) −16.5711 −1.05227
\(249\) 0 0
\(250\) − 40.7618i − 2.57800i
\(251\) − 3.21004i − 0.202616i −0.994855 0.101308i \(-0.967697\pi\)
0.994855 0.101308i \(-0.0323027\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) − 42.7451i − 2.68207i
\(255\) 0 0
\(256\) −31.3885 −1.96178
\(257\) − 18.0428i − 1.12548i −0.826635 0.562739i \(-0.809748\pi\)
0.826635 0.562739i \(-0.190252\pi\)
\(258\) 0 0
\(259\) 2.64160i 0.164141i
\(260\) 14.1659 0.878531
\(261\) 0 0
\(262\) 33.8381 2.09052
\(263\) 2.05966 0.127004 0.0635019 0.997982i \(-0.479773\pi\)
0.0635019 + 0.997982i \(0.479773\pi\)
\(264\) 0 0
\(265\) 21.0484 1.29299
\(266\) 3.81926 0.234174
\(267\) 0 0
\(268\) 17.5668 1.07306
\(269\) 2.87997i 0.175595i 0.996138 + 0.0877974i \(0.0279828\pi\)
−0.996138 + 0.0877974i \(0.972017\pi\)
\(270\) 0 0
\(271\) 10.1705i 0.617814i 0.951092 + 0.308907i \(0.0999632\pi\)
−0.951092 + 0.308907i \(0.900037\pi\)
\(272\) −12.4268 −0.753488
\(273\) 0 0
\(274\) 2.52824i 0.152737i
\(275\) 0 0
\(276\) 0 0
\(277\) − 7.31072i − 0.439259i −0.975583 0.219629i \(-0.929515\pi\)
0.975583 0.219629i \(-0.0704848\pi\)
\(278\) 48.3246i 2.89832i
\(279\) 0 0
\(280\) 6.58356 0.393443
\(281\) 2.30543 0.137530 0.0687652 0.997633i \(-0.478094\pi\)
0.0687652 + 0.997633i \(0.478094\pi\)
\(282\) 0 0
\(283\) − 24.4328i − 1.45238i −0.687494 0.726190i \(-0.741289\pi\)
0.687494 0.726190i \(-0.258711\pi\)
\(284\) 40.6420i 2.41166i
\(285\) 0 0
\(286\) 0 0
\(287\) 1.00006i 0.0590318i
\(288\) 0 0
\(289\) −5.20286 −0.306051
\(290\) − 22.9659i − 1.34860i
\(291\) 0 0
\(292\) − 17.3006i − 1.01244i
\(293\) 10.6614 0.622846 0.311423 0.950271i \(-0.399194\pi\)
0.311423 + 0.950271i \(0.399194\pi\)
\(294\) 0 0
\(295\) 25.1438 1.46393
\(296\) 33.8709 1.96871
\(297\) 0 0
\(298\) 19.5782 1.13414
\(299\) 4.71097 0.272443
\(300\) 0 0
\(301\) 0.694201 0.0400131
\(302\) − 24.5423i − 1.41225i
\(303\) 0 0
\(304\) − 15.4211i − 0.884462i
\(305\) −37.0181 −2.11965
\(306\) 0 0
\(307\) 4.56848i 0.260737i 0.991466 + 0.130369i \(0.0416160\pi\)
−0.991466 + 0.130369i \(0.958384\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 32.4955i 1.84562i
\(311\) 9.38221i 0.532017i 0.963971 + 0.266008i \(0.0857049\pi\)
−0.963971 + 0.266008i \(0.914295\pi\)
\(312\) 0 0
\(313\) −5.01601 −0.283522 −0.141761 0.989901i \(-0.545276\pi\)
−0.141761 + 0.989901i \(0.545276\pi\)
\(314\) 39.4253 2.22490
\(315\) 0 0
\(316\) − 43.0446i − 2.42145i
\(317\) − 11.6762i − 0.655801i −0.944712 0.327900i \(-0.893659\pi\)
0.944712 0.327900i \(-0.106341\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 33.1725i 1.85440i
\(321\) 0 0
\(322\) 4.45162 0.248079
\(323\) 14.6397i 0.814574i
\(324\) 0 0
\(325\) − 8.92105i − 0.494851i
\(326\) −10.4173 −0.576961
\(327\) 0 0
\(328\) 12.8229 0.708026
\(329\) −0.00757934 −0.000417863 0
\(330\) 0 0
\(331\) −29.0516 −1.59682 −0.798411 0.602113i \(-0.794326\pi\)
−0.798411 + 0.602113i \(0.794326\pi\)
\(332\) −36.1420 −1.98355
\(333\) 0 0
\(334\) 44.7134 2.44661
\(335\) − 16.9424i − 0.925663i
\(336\) 0 0
\(337\) 28.8803i 1.57321i 0.617456 + 0.786605i \(0.288163\pi\)
−0.617456 + 0.786605i \(0.711837\pi\)
\(338\) −29.4814 −1.60358
\(339\) 0 0
\(340\) 51.3101i 2.78268i
\(341\) 0 0
\(342\) 0 0
\(343\) − 5.09933i − 0.275338i
\(344\) − 8.90111i − 0.479916i
\(345\) 0 0
\(346\) −15.0623 −0.809752
\(347\) −22.1430 −1.18870 −0.594350 0.804207i \(-0.702590\pi\)
−0.594350 + 0.804207i \(0.702590\pi\)
\(348\) 0 0
\(349\) 15.6625i 0.838393i 0.907895 + 0.419197i \(0.137688\pi\)
−0.907895 + 0.419197i \(0.862312\pi\)
\(350\) − 8.42991i − 0.450598i
\(351\) 0 0
\(352\) 0 0
\(353\) 11.1636i 0.594180i 0.954850 + 0.297090i \(0.0960161\pi\)
−0.954850 + 0.297090i \(0.903984\pi\)
\(354\) 0 0
\(355\) 39.1975 2.08038
\(356\) 11.9973i 0.635856i
\(357\) 0 0
\(358\) 22.4767i 1.18793i
\(359\) 16.5643 0.874233 0.437116 0.899405i \(-0.356000\pi\)
0.437116 + 0.899405i \(0.356000\pi\)
\(360\) 0 0
\(361\) 0.832847 0.0438340
\(362\) 10.2025 0.536232
\(363\) 0 0
\(364\) 1.37261 0.0719442
\(365\) −16.6857 −0.873368
\(366\) 0 0
\(367\) 9.81027 0.512092 0.256046 0.966665i \(-0.417580\pi\)
0.256046 + 0.966665i \(0.417580\pi\)
\(368\) − 17.9744i − 0.936981i
\(369\) 0 0
\(370\) − 66.4200i − 3.45301i
\(371\) 2.03949 0.105885
\(372\) 0 0
\(373\) − 12.6350i − 0.654216i −0.944987 0.327108i \(-0.893926\pi\)
0.944987 0.327108i \(-0.106074\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0.0971830i 0.00501183i
\(377\) − 2.35493i − 0.121285i
\(378\) 0 0
\(379\) 35.1020 1.80307 0.901535 0.432706i \(-0.142441\pi\)
0.901535 + 0.432706i \(0.142441\pi\)
\(380\) −63.6735 −3.26638
\(381\) 0 0
\(382\) 49.1721i 2.51587i
\(383\) − 22.5865i − 1.15412i −0.816702 0.577059i \(-0.804200\pi\)
0.816702 0.577059i \(-0.195800\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 19.5246i − 0.993774i
\(387\) 0 0
\(388\) −59.3765 −3.01439
\(389\) 7.35884i 0.373108i 0.982445 + 0.186554i \(0.0597319\pi\)
−0.982445 + 0.186554i \(0.940268\pi\)
\(390\) 0 0
\(391\) 17.0636i 0.862942i
\(392\) −32.3731 −1.63509
\(393\) 0 0
\(394\) −51.5794 −2.59853
\(395\) −41.5146 −2.08883
\(396\) 0 0
\(397\) 16.7327 0.839790 0.419895 0.907573i \(-0.362067\pi\)
0.419895 + 0.907573i \(0.362067\pi\)
\(398\) 25.6202 1.28423
\(399\) 0 0
\(400\) −34.0377 −1.70188
\(401\) − 5.07343i − 0.253355i −0.991944 0.126677i \(-0.959569\pi\)
0.991944 0.126677i \(-0.0404313\pi\)
\(402\) 0 0
\(403\) 3.33211i 0.165984i
\(404\) 68.7157 3.41873
\(405\) 0 0
\(406\) − 2.22528i − 0.110439i
\(407\) 0 0
\(408\) 0 0
\(409\) − 32.3477i − 1.59949i −0.600340 0.799745i \(-0.704968\pi\)
0.600340 0.799745i \(-0.295032\pi\)
\(410\) − 25.1454i − 1.24184i
\(411\) 0 0
\(412\) −36.6630 −1.80625
\(413\) 2.43632 0.119883
\(414\) 0 0
\(415\) 34.8574i 1.71108i
\(416\) 0.585108i 0.0286873i
\(417\) 0 0
\(418\) 0 0
\(419\) − 27.9075i − 1.36337i −0.731645 0.681686i \(-0.761247\pi\)
0.731645 0.681686i \(-0.238753\pi\)
\(420\) 0 0
\(421\) 26.7286 1.30267 0.651337 0.758788i \(-0.274208\pi\)
0.651337 + 0.758788i \(0.274208\pi\)
\(422\) − 9.01005i − 0.438603i
\(423\) 0 0
\(424\) − 26.1506i − 1.26998i
\(425\) 32.3129 1.56740
\(426\) 0 0
\(427\) −3.58689 −0.173582
\(428\) −60.7094 −2.93450
\(429\) 0 0
\(430\) −17.4549 −0.841748
\(431\) 0.716622 0.0345185 0.0172592 0.999851i \(-0.494506\pi\)
0.0172592 + 0.999851i \(0.494506\pi\)
\(432\) 0 0
\(433\) 11.4324 0.549407 0.274703 0.961529i \(-0.411420\pi\)
0.274703 + 0.961529i \(0.411420\pi\)
\(434\) 3.14867i 0.151141i
\(435\) 0 0
\(436\) 41.8303i 2.00331i
\(437\) −21.1751 −1.01294
\(438\) 0 0
\(439\) 8.57525i 0.409274i 0.978838 + 0.204637i \(0.0656015\pi\)
−0.978838 + 0.204637i \(0.934399\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 7.93508i 0.377433i
\(443\) − 13.9769i − 0.664063i −0.943268 0.332031i \(-0.892266\pi\)
0.943268 0.332031i \(-0.107734\pi\)
\(444\) 0 0
\(445\) 11.5709 0.548512
\(446\) 43.6031 2.06467
\(447\) 0 0
\(448\) 3.21426i 0.151859i
\(449\) − 21.6079i − 1.01974i −0.860252 0.509870i \(-0.829694\pi\)
0.860252 0.509870i \(-0.170306\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 4.12070i 0.193821i
\(453\) 0 0
\(454\) 42.3393 1.98708
\(455\) − 1.32382i − 0.0620616i
\(456\) 0 0
\(457\) 9.80506i 0.458662i 0.973349 + 0.229331i \(0.0736537\pi\)
−0.973349 + 0.229331i \(0.926346\pi\)
\(458\) −10.4193 −0.486862
\(459\) 0 0
\(460\) −74.2159 −3.46033
\(461\) −12.3585 −0.575595 −0.287797 0.957691i \(-0.592923\pi\)
−0.287797 + 0.957691i \(0.592923\pi\)
\(462\) 0 0
\(463\) −33.8717 −1.57415 −0.787076 0.616856i \(-0.788406\pi\)
−0.787076 + 0.616856i \(0.788406\pi\)
\(464\) −8.98509 −0.417122
\(465\) 0 0
\(466\) −56.9680 −2.63899
\(467\) 15.7369i 0.728219i 0.931356 + 0.364109i \(0.118627\pi\)
−0.931356 + 0.364109i \(0.881373\pi\)
\(468\) 0 0
\(469\) − 1.64164i − 0.0758040i
\(470\) 0.190573 0.00879050
\(471\) 0 0
\(472\) − 31.2387i − 1.43788i
\(473\) 0 0
\(474\) 0 0
\(475\) 40.0987i 1.83986i
\(476\) 4.97171i 0.227878i
\(477\) 0 0
\(478\) −48.7159 −2.22821
\(479\) −12.4691 −0.569726 −0.284863 0.958568i \(-0.591948\pi\)
−0.284863 + 0.958568i \(0.591948\pi\)
\(480\) 0 0
\(481\) − 6.81075i − 0.310544i
\(482\) − 5.66614i − 0.258086i
\(483\) 0 0
\(484\) 0 0
\(485\) 57.2660i 2.60032i
\(486\) 0 0
\(487\) −6.95609 −0.315210 −0.157605 0.987502i \(-0.550377\pi\)
−0.157605 + 0.987502i \(0.550377\pi\)
\(488\) 45.9914i 2.08193i
\(489\) 0 0
\(490\) 63.4828i 2.86786i
\(491\) 0.301920 0.0136255 0.00681273 0.999977i \(-0.497831\pi\)
0.00681273 + 0.999977i \(0.497831\pi\)
\(492\) 0 0
\(493\) 8.52979 0.384162
\(494\) −9.84705 −0.443040
\(495\) 0 0
\(496\) 12.7135 0.570851
\(497\) 3.79805 0.170366
\(498\) 0 0
\(499\) 6.61671 0.296205 0.148102 0.988972i \(-0.452683\pi\)
0.148102 + 0.988972i \(0.452683\pi\)
\(500\) 65.8469i 2.94476i
\(501\) 0 0
\(502\) 7.82068i 0.349054i
\(503\) −10.3292 −0.460554 −0.230277 0.973125i \(-0.573963\pi\)
−0.230277 + 0.973125i \(0.573963\pi\)
\(504\) 0 0
\(505\) − 66.2732i − 2.94912i
\(506\) 0 0
\(507\) 0 0
\(508\) 69.0508i 3.06363i
\(509\) − 20.3250i − 0.900891i −0.892804 0.450445i \(-0.851265\pi\)
0.892804 0.450445i \(-0.148735\pi\)
\(510\) 0 0
\(511\) −1.61676 −0.0715214
\(512\) 36.3567 1.60675
\(513\) 0 0
\(514\) 43.9580i 1.93890i
\(515\) 35.3598i 1.55814i
\(516\) 0 0
\(517\) 0 0
\(518\) − 6.43579i − 0.282772i
\(519\) 0 0
\(520\) −16.9742 −0.744366
\(521\) 11.9982i 0.525650i 0.964843 + 0.262825i \(0.0846542\pi\)
−0.964843 + 0.262825i \(0.915346\pi\)
\(522\) 0 0
\(523\) − 37.8361i − 1.65446i −0.561865 0.827229i \(-0.689916\pi\)
0.561865 0.827229i \(-0.310084\pi\)
\(524\) −54.6623 −2.38793
\(525\) 0 0
\(526\) −5.01798 −0.218794
\(527\) −12.0692 −0.525743
\(528\) 0 0
\(529\) −1.68106 −0.0730898
\(530\) −51.2806 −2.22749
\(531\) 0 0
\(532\) −6.16966 −0.267489
\(533\) − 2.57842i − 0.111684i
\(534\) 0 0
\(535\) 58.5515i 2.53140i
\(536\) −21.0493 −0.909191
\(537\) 0 0
\(538\) − 7.01652i − 0.302504i
\(539\) 0 0
\(540\) 0 0
\(541\) − 36.8317i − 1.58352i −0.610834 0.791759i \(-0.709166\pi\)
0.610834 0.791759i \(-0.290834\pi\)
\(542\) − 24.7786i − 1.06433i
\(543\) 0 0
\(544\) −2.11931 −0.0908648
\(545\) 40.3435 1.72812
\(546\) 0 0
\(547\) − 36.6744i − 1.56809i −0.620706 0.784043i \(-0.713154\pi\)
0.620706 0.784043i \(-0.286846\pi\)
\(548\) − 4.08414i − 0.174466i
\(549\) 0 0
\(550\) 0 0
\(551\) 10.5851i 0.450939i
\(552\) 0 0
\(553\) −4.02258 −0.171057
\(554\) 17.8113i 0.756728i
\(555\) 0 0
\(556\) − 78.0639i − 3.31065i
\(557\) −7.46772 −0.316418 −0.158209 0.987406i \(-0.550572\pi\)
−0.158209 + 0.987406i \(0.550572\pi\)
\(558\) 0 0
\(559\) −1.78983 −0.0757018
\(560\) −5.05095 −0.213441
\(561\) 0 0
\(562\) −5.61676 −0.236929
\(563\) −3.91481 −0.164990 −0.0824948 0.996591i \(-0.526289\pi\)
−0.0824948 + 0.996591i \(0.526289\pi\)
\(564\) 0 0
\(565\) 3.97423 0.167197
\(566\) 59.5261i 2.50207i
\(567\) 0 0
\(568\) − 48.6990i − 2.04336i
\(569\) 37.6712 1.57926 0.789628 0.613585i \(-0.210273\pi\)
0.789628 + 0.613585i \(0.210273\pi\)
\(570\) 0 0
\(571\) − 30.3411i − 1.26974i −0.772621 0.634868i \(-0.781054\pi\)
0.772621 0.634868i \(-0.218946\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) − 2.43647i − 0.101696i
\(575\) 46.7379i 1.94911i
\(576\) 0 0
\(577\) −3.31962 −0.138198 −0.0690988 0.997610i \(-0.522012\pi\)
−0.0690988 + 0.997610i \(0.522012\pi\)
\(578\) 12.6758 0.527245
\(579\) 0 0
\(580\) 37.0992i 1.54046i
\(581\) 3.37752i 0.140123i
\(582\) 0 0
\(583\) 0 0
\(584\) 20.7303i 0.857826i
\(585\) 0 0
\(586\) −25.9746 −1.07300
\(587\) − 40.9075i − 1.68843i −0.536003 0.844216i \(-0.680067\pi\)
0.536003 0.844216i \(-0.319933\pi\)
\(588\) 0 0
\(589\) − 14.9773i − 0.617130i
\(590\) −61.2583 −2.52197
\(591\) 0 0
\(592\) −25.9860 −1.06802
\(593\) 4.62924 0.190100 0.0950500 0.995472i \(-0.469699\pi\)
0.0950500 + 0.995472i \(0.469699\pi\)
\(594\) 0 0
\(595\) 4.79500 0.196576
\(596\) −31.6268 −1.29549
\(597\) 0 0
\(598\) −11.4774 −0.469347
\(599\) − 30.5789i − 1.24942i −0.780858 0.624709i \(-0.785218\pi\)
0.780858 0.624709i \(-0.214782\pi\)
\(600\) 0 0
\(601\) − 31.2300i − 1.27390i −0.770905 0.636950i \(-0.780196\pi\)
0.770905 0.636950i \(-0.219804\pi\)
\(602\) −1.69129 −0.0689320
\(603\) 0 0
\(604\) 39.6459i 1.61317i
\(605\) 0 0
\(606\) 0 0
\(607\) 30.7667i 1.24878i 0.781112 + 0.624391i \(0.214653\pi\)
−0.781112 + 0.624391i \(0.785347\pi\)
\(608\) − 2.62997i − 0.106659i
\(609\) 0 0
\(610\) 90.1880 3.65160
\(611\) 0.0195415 0.000790566 0
\(612\) 0 0
\(613\) − 25.4710i − 1.02876i −0.857561 0.514382i \(-0.828021\pi\)
0.857561 0.514382i \(-0.171979\pi\)
\(614\) − 11.1303i − 0.449181i
\(615\) 0 0
\(616\) 0 0
\(617\) − 4.20930i − 0.169460i −0.996404 0.0847300i \(-0.972997\pi\)
0.996404 0.0847300i \(-0.0270028\pi\)
\(618\) 0 0
\(619\) 38.5032 1.54757 0.773787 0.633446i \(-0.218360\pi\)
0.773787 + 0.633446i \(0.218360\pi\)
\(620\) − 52.4935i − 2.10819i
\(621\) 0 0
\(622\) − 22.8581i − 0.916525i
\(623\) 1.12116 0.0449185
\(624\) 0 0
\(625\) 16.4675 0.658700
\(626\) 12.2206 0.488433
\(627\) 0 0
\(628\) −63.6880 −2.54143
\(629\) 24.6692 0.983624
\(630\) 0 0
\(631\) −11.8328 −0.471058 −0.235529 0.971867i \(-0.575682\pi\)
−0.235529 + 0.971867i \(0.575682\pi\)
\(632\) 51.5779i 2.05166i
\(633\) 0 0
\(634\) 28.4469i 1.12977i
\(635\) 66.5964 2.64280
\(636\) 0 0
\(637\) 6.50957i 0.257918i
\(638\) 0 0
\(639\) 0 0
\(640\) − 76.1345i − 3.00948i
\(641\) − 35.9401i − 1.41955i −0.704430 0.709774i \(-0.748797\pi\)
0.704430 0.709774i \(-0.251203\pi\)
\(642\) 0 0
\(643\) 3.25527 0.128375 0.0641876 0.997938i \(-0.479554\pi\)
0.0641876 + 0.997938i \(0.479554\pi\)
\(644\) −7.19118 −0.283372
\(645\) 0 0
\(646\) − 35.6669i − 1.40330i
\(647\) − 12.1550i − 0.477863i −0.971036 0.238931i \(-0.923203\pi\)
0.971036 0.238931i \(-0.0767971\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 21.7345i 0.852498i
\(651\) 0 0
\(652\) 16.8282 0.659043
\(653\) − 25.5204i − 0.998690i −0.866403 0.499345i \(-0.833574\pi\)
0.866403 0.499345i \(-0.166426\pi\)
\(654\) 0 0
\(655\) 52.7193i 2.05992i
\(656\) −9.83781 −0.384102
\(657\) 0 0
\(658\) 0.0184657 0.000719867 0
\(659\) 27.9441 1.08855 0.544273 0.838908i \(-0.316806\pi\)
0.544273 + 0.838908i \(0.316806\pi\)
\(660\) 0 0
\(661\) −27.3798 −1.06495 −0.532475 0.846446i \(-0.678738\pi\)
−0.532475 + 0.846446i \(0.678738\pi\)
\(662\) 70.7790 2.75091
\(663\) 0 0
\(664\) 43.3069 1.68063
\(665\) 5.95037i 0.230745i
\(666\) 0 0
\(667\) 12.3376i 0.477715i
\(668\) −72.2304 −2.79468
\(669\) 0 0
\(670\) 41.2771i 1.59467i
\(671\) 0 0
\(672\) 0 0
\(673\) 13.5488i 0.522267i 0.965303 + 0.261134i \(0.0840963\pi\)
−0.965303 + 0.261134i \(0.915904\pi\)
\(674\) − 70.3616i − 2.71023i
\(675\) 0 0
\(676\) 47.6245 1.83171
\(677\) 36.6970 1.41038 0.705190 0.709018i \(-0.250862\pi\)
0.705190 + 0.709018i \(0.250862\pi\)
\(678\) 0 0
\(679\) 5.54881i 0.212944i
\(680\) − 61.4819i − 2.35772i
\(681\) 0 0
\(682\) 0 0
\(683\) − 30.8347i − 1.17986i −0.807455 0.589929i \(-0.799156\pi\)
0.807455 0.589929i \(-0.200844\pi\)
\(684\) 0 0
\(685\) −3.93897 −0.150500
\(686\) 12.4236i 0.474335i
\(687\) 0 0
\(688\) 6.82899i 0.260353i
\(689\) −5.25835 −0.200327
\(690\) 0 0
\(691\) −47.7199 −1.81535 −0.907676 0.419672i \(-0.862145\pi\)
−0.907676 + 0.419672i \(0.862145\pi\)
\(692\) 24.3317 0.924953
\(693\) 0 0
\(694\) 53.9474 2.04782
\(695\) −75.2892 −2.85588
\(696\) 0 0
\(697\) 9.33929 0.353751
\(698\) − 38.1588i − 1.44433i
\(699\) 0 0
\(700\) 13.6177i 0.514703i
\(701\) −14.0665 −0.531284 −0.265642 0.964072i \(-0.585584\pi\)
−0.265642 + 0.964072i \(0.585584\pi\)
\(702\) 0 0
\(703\) 30.6133i 1.15460i
\(704\) 0 0
\(705\) 0 0
\(706\) − 27.1981i − 1.02362i
\(707\) − 6.42157i − 0.241508i
\(708\) 0 0
\(709\) −9.07416 −0.340787 −0.170394 0.985376i \(-0.554504\pi\)
−0.170394 + 0.985376i \(0.554504\pi\)
\(710\) −95.4975 −3.58396
\(711\) 0 0
\(712\) − 14.3757i − 0.538751i
\(713\) − 17.4571i − 0.653775i
\(714\) 0 0
\(715\) 0 0
\(716\) − 36.3090i − 1.35693i
\(717\) 0 0
\(718\) −40.3560 −1.50607
\(719\) − 33.9415i − 1.26580i −0.774232 0.632901i \(-0.781864\pi\)
0.774232 0.632901i \(-0.218136\pi\)
\(720\) 0 0
\(721\) 3.42620i 0.127598i
\(722\) −2.02908 −0.0755146
\(723\) 0 0
\(724\) −16.4812 −0.612519
\(725\) 23.3635 0.867697
\(726\) 0 0
\(727\) 30.8992 1.14599 0.572993 0.819560i \(-0.305782\pi\)
0.572993 + 0.819560i \(0.305782\pi\)
\(728\) −1.64472 −0.0609572
\(729\) 0 0
\(730\) 40.6516 1.50458
\(731\) − 6.48294i − 0.239780i
\(732\) 0 0
\(733\) − 41.6009i − 1.53657i −0.640110 0.768283i \(-0.721111\pi\)
0.640110 0.768283i \(-0.278889\pi\)
\(734\) −23.9009 −0.882200
\(735\) 0 0
\(736\) − 3.06541i − 0.112993i
\(737\) 0 0
\(738\) 0 0
\(739\) 14.3707i 0.528634i 0.964436 + 0.264317i \(0.0851465\pi\)
−0.964436 + 0.264317i \(0.914854\pi\)
\(740\) 107.295i 3.94426i
\(741\) 0 0
\(742\) −4.96885 −0.182412
\(743\) −13.4298 −0.492692 −0.246346 0.969182i \(-0.579230\pi\)
−0.246346 + 0.969182i \(0.579230\pi\)
\(744\) 0 0
\(745\) 30.5027i 1.11753i
\(746\) 30.7829i 1.12704i
\(747\) 0 0
\(748\) 0 0
\(749\) 5.67337i 0.207301i
\(750\) 0 0
\(751\) −36.3646 −1.32696 −0.663481 0.748193i \(-0.730922\pi\)
−0.663481 + 0.748193i \(0.730922\pi\)
\(752\) − 0.0745594i − 0.00271890i
\(753\) 0 0
\(754\) 5.73737i 0.208943i
\(755\) 38.2367 1.39158
\(756\) 0 0
\(757\) 39.9016 1.45025 0.725124 0.688618i \(-0.241782\pi\)
0.725124 + 0.688618i \(0.241782\pi\)
\(758\) −85.5197 −3.10622
\(759\) 0 0
\(760\) 76.2962 2.76755
\(761\) −3.41015 −0.123618 −0.0618089 0.998088i \(-0.519687\pi\)
−0.0618089 + 0.998088i \(0.519687\pi\)
\(762\) 0 0
\(763\) 3.90910 0.141519
\(764\) − 79.4331i − 2.87379i
\(765\) 0 0
\(766\) 55.0280i 1.98824i
\(767\) −6.28147 −0.226811
\(768\) 0 0
\(769\) 21.4154i 0.772260i 0.922444 + 0.386130i \(0.126188\pi\)
−0.922444 + 0.386130i \(0.873812\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 31.5401i 1.13515i
\(773\) 35.8866i 1.29075i 0.763866 + 0.645375i \(0.223299\pi\)
−0.763866 + 0.645375i \(0.776701\pi\)
\(774\) 0 0
\(775\) −33.0581 −1.18748
\(776\) 71.1474 2.55404
\(777\) 0 0
\(778\) − 17.9285i − 0.642767i
\(779\) 11.5896i 0.415241i
\(780\) 0 0
\(781\) 0 0
\(782\) − 41.5723i − 1.48662i
\(783\) 0 0
\(784\) 24.8368 0.887029
\(785\) 61.4243i 2.19233i
\(786\) 0 0
\(787\) − 48.0635i − 1.71328i −0.515917 0.856639i \(-0.672549\pi\)
0.515917 0.856639i \(-0.327451\pi\)
\(788\) 83.3218 2.96822
\(789\) 0 0
\(790\) 101.143 3.59850
\(791\) 0.385085 0.0136920
\(792\) 0 0
\(793\) 9.24793 0.328404
\(794\) −40.7662 −1.44674
\(795\) 0 0
\(796\) −41.3872 −1.46693
\(797\) 6.04580i 0.214153i 0.994251 + 0.107077i \(0.0341490\pi\)
−0.994251 + 0.107077i \(0.965851\pi\)
\(798\) 0 0
\(799\) 0.0707812i 0.00250406i
\(800\) −5.80489 −0.205234
\(801\) 0 0
\(802\) 12.3605i 0.436464i
\(803\) 0 0
\(804\) 0 0
\(805\) 6.93557i 0.244447i
\(806\) − 8.11809i − 0.285947i
\(807\) 0 0
\(808\) −82.3379 −2.89664
\(809\) 7.43425 0.261374 0.130687 0.991424i \(-0.458282\pi\)
0.130687 + 0.991424i \(0.458282\pi\)
\(810\) 0 0
\(811\) − 29.1631i − 1.02405i −0.858970 0.512027i \(-0.828895\pi\)
0.858970 0.512027i \(-0.171105\pi\)
\(812\) 3.59474i 0.126151i
\(813\) 0 0
\(814\) 0 0
\(815\) − 16.2301i − 0.568514i
\(816\) 0 0
\(817\) 8.04502 0.281460
\(818\) 78.8093i 2.75550i
\(819\) 0 0
\(820\) 40.6201i 1.41851i
\(821\) −1.07240 −0.0374270 −0.0187135 0.999825i \(-0.505957\pi\)
−0.0187135 + 0.999825i \(0.505957\pi\)
\(822\) 0 0
\(823\) −8.94619 −0.311845 −0.155922 0.987769i \(-0.549835\pi\)
−0.155922 + 0.987769i \(0.549835\pi\)
\(824\) 43.9311 1.53041
\(825\) 0 0
\(826\) −5.93565 −0.206528
\(827\) 35.3031 1.22761 0.613805 0.789458i \(-0.289638\pi\)
0.613805 + 0.789458i \(0.289638\pi\)
\(828\) 0 0
\(829\) 27.0734 0.940298 0.470149 0.882587i \(-0.344200\pi\)
0.470149 + 0.882587i \(0.344200\pi\)
\(830\) − 84.9237i − 2.94774i
\(831\) 0 0
\(832\) − 8.28720i − 0.287307i
\(833\) −23.5783 −0.816938
\(834\) 0 0
\(835\) 69.6630i 2.41079i
\(836\) 0 0
\(837\) 0 0
\(838\) 67.9916i 2.34873i
\(839\) 37.2329i 1.28542i 0.766108 + 0.642712i \(0.222191\pi\)
−0.766108 + 0.642712i \(0.777809\pi\)
\(840\) 0 0
\(841\) −22.8326 −0.787332
\(842\) −65.1195 −2.24417
\(843\) 0 0
\(844\) 14.5549i 0.501001i
\(845\) − 45.9317i − 1.58010i
\(846\) 0 0
\(847\) 0 0
\(848\) 20.0629i 0.688962i
\(849\) 0 0
\(850\) −78.7245 −2.70023
\(851\) 35.6819i 1.22316i
\(852\) 0 0
\(853\) 35.1190i 1.20245i 0.799080 + 0.601225i \(0.205320\pi\)
−0.799080 + 0.601225i \(0.794680\pi\)
\(854\) 8.73880 0.299035
\(855\) 0 0
\(856\) 72.7445 2.48636
\(857\) 29.9637 1.02354 0.511770 0.859122i \(-0.328990\pi\)
0.511770 + 0.859122i \(0.328990\pi\)
\(858\) 0 0
\(859\) 17.2684 0.589189 0.294595 0.955622i \(-0.404815\pi\)
0.294595 + 0.955622i \(0.404815\pi\)
\(860\) 28.1967 0.961500
\(861\) 0 0
\(862\) −1.74592 −0.0594663
\(863\) 34.8150i 1.18512i 0.805528 + 0.592558i \(0.201882\pi\)
−0.805528 + 0.592558i \(0.798118\pi\)
\(864\) 0 0
\(865\) − 23.4668i − 0.797897i
\(866\) −27.8530 −0.946484
\(867\) 0 0
\(868\) − 5.08638i − 0.172643i
\(869\) 0 0
\(870\) 0 0
\(871\) 4.23258i 0.143416i
\(872\) − 50.1228i − 1.69737i
\(873\) 0 0
\(874\) 51.5893 1.74503
\(875\) 6.15348 0.208025
\(876\) 0 0
\(877\) 1.98289i 0.0669576i 0.999439 + 0.0334788i \(0.0106586\pi\)
−0.999439 + 0.0334788i \(0.989341\pi\)
\(878\) − 20.8920i − 0.705072i
\(879\) 0 0
\(880\) 0 0
\(881\) 48.9571i 1.64941i 0.565566 + 0.824703i \(0.308658\pi\)
−0.565566 + 0.824703i \(0.691342\pi\)
\(882\) 0 0
\(883\) 34.7553 1.16961 0.584804 0.811175i \(-0.301171\pi\)
0.584804 + 0.811175i \(0.301171\pi\)
\(884\) − 12.8184i − 0.431129i
\(885\) 0 0
\(886\) 34.0522i 1.14401i
\(887\) 14.4931 0.486629 0.243315 0.969947i \(-0.421765\pi\)
0.243315 + 0.969947i \(0.421765\pi\)
\(888\) 0 0
\(889\) 6.45289 0.216423
\(890\) −28.1903 −0.944943
\(891\) 0 0
\(892\) −70.4368 −2.35840
\(893\) −0.0878361 −0.00293932
\(894\) 0 0
\(895\) −35.0184 −1.17054
\(896\) − 7.37708i − 0.246451i
\(897\) 0 0
\(898\) 52.6437i 1.75674i
\(899\) −8.72651 −0.291045
\(900\) 0 0
\(901\) − 19.0462i − 0.634521i
\(902\) 0 0
\(903\) 0 0
\(904\) − 4.93759i − 0.164222i
\(905\) 15.8954i 0.528381i
\(906\) 0 0
\(907\) 22.7691 0.756037 0.378019 0.925798i \(-0.376606\pi\)
0.378019 + 0.925798i \(0.376606\pi\)
\(908\) −68.3952 −2.26978
\(909\) 0 0
\(910\) 3.22525i 0.106916i
\(911\) 22.6944i 0.751900i 0.926640 + 0.375950i \(0.122684\pi\)
−0.926640 + 0.375950i \(0.877316\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) − 23.8883i − 0.790153i
\(915\) 0 0
\(916\) 16.8314 0.556126
\(917\) 5.10826i 0.168690i
\(918\) 0 0
\(919\) − 1.25513i − 0.0414029i −0.999786 0.0207014i \(-0.993410\pi\)
0.999786 0.0207014i \(-0.00658994\pi\)
\(920\) 88.9286 2.93189
\(921\) 0 0
\(922\) 30.1093 0.991598
\(923\) −9.79237 −0.322320
\(924\) 0 0
\(925\) 67.5699 2.22169
\(926\) 82.5223 2.71185
\(927\) 0 0
\(928\) −1.53235 −0.0503017
\(929\) − 2.87512i − 0.0943296i −0.998887 0.0471648i \(-0.984981\pi\)
0.998887 0.0471648i \(-0.0150186\pi\)
\(930\) 0 0
\(931\) − 29.2595i − 0.958941i
\(932\) 92.0265 3.01443
\(933\) 0 0
\(934\) − 38.3402i − 1.25453i
\(935\) 0 0
\(936\) 0 0
\(937\) − 30.8451i − 1.00767i −0.863801 0.503833i \(-0.831923\pi\)
0.863801 0.503833i \(-0.168077\pi\)
\(938\) 3.99956i 0.130590i
\(939\) 0 0
\(940\) −0.307854 −0.0100411
\(941\) −6.54086 −0.213226 −0.106613 0.994301i \(-0.534001\pi\)
−0.106613 + 0.994301i \(0.534001\pi\)
\(942\) 0 0
\(943\) 13.5085i 0.439898i
\(944\) 23.9665i 0.780044i
\(945\) 0 0
\(946\) 0 0
\(947\) − 33.2579i − 1.08074i −0.841429 0.540368i \(-0.818285\pi\)
0.841429 0.540368i \(-0.181715\pi\)
\(948\) 0 0
\(949\) 4.16844 0.135313
\(950\) − 97.6933i − 3.16959i
\(951\) 0 0
\(952\) − 5.95731i − 0.193078i
\(953\) −54.9548 −1.78016 −0.890081 0.455803i \(-0.849352\pi\)
−0.890081 + 0.455803i \(0.849352\pi\)
\(954\) 0 0
\(955\) −76.6097 −2.47903
\(956\) 78.6961 2.54521
\(957\) 0 0
\(958\) 30.3786 0.981488
\(959\) −0.381668 −0.0123247
\(960\) 0 0
\(961\) −18.6524 −0.601691
\(962\) 16.5932i 0.534985i
\(963\) 0 0
\(964\) 9.15313i 0.294803i
\(965\) 30.4190 0.979224
\(966\) 0 0
\(967\) 34.0395i 1.09464i 0.836925 + 0.547318i \(0.184351\pi\)
−0.836925 + 0.547318i \(0.815649\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) − 139.518i − 4.47966i
\(971\) 38.0638i 1.22153i 0.791814 + 0.610763i \(0.209137\pi\)
−0.791814 + 0.610763i \(0.790863\pi\)
\(972\) 0 0
\(973\) −7.29517 −0.233872
\(974\) 16.9472 0.543025
\(975\) 0 0
\(976\) − 35.2849i − 1.12944i
\(977\) − 32.1005i − 1.02699i −0.858094 0.513493i \(-0.828351\pi\)
0.858094 0.513493i \(-0.171649\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) − 102.551i − 3.27586i
\(981\) 0 0
\(982\) −0.735573 −0.0234731
\(983\) − 21.4305i − 0.683526i −0.939786 0.341763i \(-0.888976\pi\)
0.939786 0.341763i \(-0.111024\pi\)
\(984\) 0 0
\(985\) − 80.3602i − 2.56049i
\(986\) −20.7813 −0.661811
\(987\) 0 0
\(988\) 15.9070 0.506069
\(989\) 9.37704 0.298172
\(990\) 0 0
\(991\) −29.3068 −0.930962 −0.465481 0.885058i \(-0.654119\pi\)
−0.465481 + 0.885058i \(0.654119\pi\)
\(992\) 2.16819 0.0688402
\(993\) 0 0
\(994\) −9.25326 −0.293496
\(995\) 39.9161i 1.26542i
\(996\) 0 0
\(997\) − 32.7622i − 1.03759i −0.854899 0.518795i \(-0.826381\pi\)
0.854899 0.518795i \(-0.173619\pi\)
\(998\) −16.1204 −0.510283
\(999\) 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 1089.2.d.g.1088.1 16
3.2 odd 2 inner 1089.2.d.g.1088.16 16
11.2 odd 10 99.2.j.a.62.1 yes 16
11.5 even 5 99.2.j.a.8.4 yes 16
11.10 odd 2 inner 1089.2.d.g.1088.15 16
33.2 even 10 99.2.j.a.62.4 yes 16
33.5 odd 10 99.2.j.a.8.1 16
33.32 even 2 inner 1089.2.d.g.1088.2 16
44.27 odd 10 1584.2.cd.c.305.1 16
44.35 even 10 1584.2.cd.c.161.4 16
99.2 even 30 891.2.u.c.458.4 32
99.5 odd 30 891.2.u.c.107.1 32
99.13 odd 30 891.2.u.c.755.4 32
99.16 even 15 891.2.u.c.701.1 32
99.38 odd 30 891.2.u.c.701.4 32
99.49 even 15 891.2.u.c.107.4 32
99.68 even 30 891.2.u.c.755.1 32
99.79 odd 30 891.2.u.c.458.1 32
132.35 odd 10 1584.2.cd.c.161.1 16
132.71 even 10 1584.2.cd.c.305.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.j.a.8.1 16 33.5 odd 10
99.2.j.a.8.4 yes 16 11.5 even 5
99.2.j.a.62.1 yes 16 11.2 odd 10
99.2.j.a.62.4 yes 16 33.2 even 10
891.2.u.c.107.1 32 99.5 odd 30
891.2.u.c.107.4 32 99.49 even 15
891.2.u.c.458.1 32 99.79 odd 30
891.2.u.c.458.4 32 99.2 even 30
891.2.u.c.701.1 32 99.16 even 15
891.2.u.c.701.4 32 99.38 odd 30
891.2.u.c.755.1 32 99.68 even 30
891.2.u.c.755.4 32 99.13 odd 30
1089.2.d.g.1088.1 16 1.1 even 1 trivial
1089.2.d.g.1088.2 16 33.32 even 2 inner
1089.2.d.g.1088.15 16 11.10 odd 2 inner
1089.2.d.g.1088.16 16 3.2 odd 2 inner
1584.2.cd.c.161.1 16 132.35 odd 10
1584.2.cd.c.161.4 16 44.35 even 10
1584.2.cd.c.305.1 16 44.27 odd 10
1584.2.cd.c.305.4 16 132.71 even 10