Properties

Label 441.4.c.a.440.5
Level $441$
Weight $4$
Character 441.440
Analytic conductor $26.020$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 48 x^{14} + 1647 x^{12} - 27620 x^{10} + 336765 x^{8} - 1200006 x^{6} + 3242464 x^{4} - 1762200 x^{2} + 810000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{8}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 440.5
Root \(1.57646 + 0.910170i\) of defining polynomial
Character \(\chi\) \(=\) 441.440
Dual form 441.4.c.a.440.11

$q$-expansion

\(f(q)\) \(=\) \(q-1.82034i q^{2} +4.68636 q^{4} -15.0874 q^{5} -23.0935i q^{8} +O(q^{10})\) \(q-1.82034i q^{2} +4.68636 q^{4} -15.0874 q^{5} -23.0935i q^{8} +27.4643i q^{10} +9.89034i q^{11} +67.8891i q^{13} -4.54712 q^{16} +70.1373 q^{17} +61.4580i q^{19} -70.7052 q^{20} +18.0038 q^{22} -131.515i q^{23} +102.631 q^{25} +123.581 q^{26} +158.738i q^{29} -76.4814i q^{31} -176.471i q^{32} -127.674i q^{34} +348.682 q^{37} +111.874 q^{38} +348.422i q^{40} -138.909 q^{41} +539.651 q^{43} +46.3497i q^{44} -239.402 q^{46} +223.643 q^{47} -186.823i q^{50} +318.153i q^{52} +530.011i q^{53} -149.220i q^{55} +288.958 q^{58} +542.876 q^{59} +134.197i q^{61} -139.222 q^{62} -357.614 q^{64} -1024.27i q^{65} +320.580 q^{67} +328.689 q^{68} -416.958i q^{71} +545.607i q^{73} -634.719i q^{74} +288.014i q^{76} -322.737 q^{79} +68.6044 q^{80} +252.861i q^{82} -885.170 q^{83} -1058.19 q^{85} -982.349i q^{86} +228.403 q^{88} +1624.62 q^{89} -616.327i q^{92} -407.106i q^{94} -927.244i q^{95} -739.155i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 64q^{4} + O(q^{10}) \) \( 16q - 64q^{4} + 376q^{16} + 528q^{22} + 40q^{25} + 2392q^{37} + 328q^{43} + 2784q^{46} + 6744q^{58} + 5432q^{64} - 616q^{67} + 4352q^{79} - 4608q^{85} - 1416q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.82034i − 0.643587i −0.946810 0.321794i \(-0.895714\pi\)
0.946810 0.321794i \(-0.104286\pi\)
\(3\) 0 0
\(4\) 4.68636 0.585795
\(5\) −15.0874 −1.34946 −0.674731 0.738064i \(-0.735740\pi\)
−0.674731 + 0.738064i \(0.735740\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) − 23.0935i − 1.02060i
\(9\) 0 0
\(10\) 27.4643i 0.868497i
\(11\) 9.89034i 0.271096i 0.990771 + 0.135548i \(0.0432794\pi\)
−0.990771 + 0.135548i \(0.956721\pi\)
\(12\) 0 0
\(13\) 67.8891i 1.44839i 0.689596 + 0.724194i \(0.257788\pi\)
−0.689596 + 0.724194i \(0.742212\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.54712 −0.0710487
\(17\) 70.1373 1.00064 0.500318 0.865842i \(-0.333217\pi\)
0.500318 + 0.865842i \(0.333217\pi\)
\(18\) 0 0
\(19\) 61.4580i 0.742075i 0.928618 + 0.371038i \(0.120998\pi\)
−0.928618 + 0.371038i \(0.879002\pi\)
\(20\) −70.7052 −0.790508
\(21\) 0 0
\(22\) 18.0038 0.174474
\(23\) − 131.515i − 1.19229i −0.802875 0.596147i \(-0.796697\pi\)
0.802875 0.596147i \(-0.203303\pi\)
\(24\) 0 0
\(25\) 102.631 0.821047
\(26\) 123.581 0.932165
\(27\) 0 0
\(28\) 0 0
\(29\) 158.738i 1.01645i 0.861225 + 0.508223i \(0.169698\pi\)
−0.861225 + 0.508223i \(0.830302\pi\)
\(30\) 0 0
\(31\) − 76.4814i − 0.443112i −0.975148 0.221556i \(-0.928886\pi\)
0.975148 0.221556i \(-0.0711136\pi\)
\(32\) − 176.471i − 0.974872i
\(33\) 0 0
\(34\) − 127.674i − 0.643996i
\(35\) 0 0
\(36\) 0 0
\(37\) 348.682 1.54927 0.774634 0.632410i \(-0.217934\pi\)
0.774634 + 0.632410i \(0.217934\pi\)
\(38\) 111.874 0.477590
\(39\) 0 0
\(40\) 348.422i 1.37726i
\(41\) −138.909 −0.529120 −0.264560 0.964369i \(-0.585227\pi\)
−0.264560 + 0.964369i \(0.585227\pi\)
\(42\) 0 0
\(43\) 539.651 1.91386 0.956931 0.290316i \(-0.0937604\pi\)
0.956931 + 0.290316i \(0.0937604\pi\)
\(44\) 46.3497i 0.158806i
\(45\) 0 0
\(46\) −239.402 −0.767346
\(47\) 223.643 0.694078 0.347039 0.937851i \(-0.387187\pi\)
0.347039 + 0.937851i \(0.387187\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) − 186.823i − 0.528415i
\(51\) 0 0
\(52\) 318.153i 0.848459i
\(53\) 530.011i 1.37363i 0.726830 + 0.686817i \(0.240993\pi\)
−0.726830 + 0.686817i \(0.759007\pi\)
\(54\) 0 0
\(55\) − 149.220i − 0.365833i
\(56\) 0 0
\(57\) 0 0
\(58\) 288.958 0.654172
\(59\) 542.876 1.19791 0.598953 0.800784i \(-0.295583\pi\)
0.598953 + 0.800784i \(0.295583\pi\)
\(60\) 0 0
\(61\) 134.197i 0.281674i 0.990033 + 0.140837i \(0.0449793\pi\)
−0.990033 + 0.140837i \(0.955021\pi\)
\(62\) −139.222 −0.285181
\(63\) 0 0
\(64\) −357.614 −0.698464
\(65\) − 1024.27i − 1.95454i
\(66\) 0 0
\(67\) 320.580 0.584553 0.292276 0.956334i \(-0.405587\pi\)
0.292276 + 0.956334i \(0.405587\pi\)
\(68\) 328.689 0.586167
\(69\) 0 0
\(70\) 0 0
\(71\) − 416.958i − 0.696955i −0.937317 0.348478i \(-0.886699\pi\)
0.937317 0.348478i \(-0.113301\pi\)
\(72\) 0 0
\(73\) 545.607i 0.874774i 0.899273 + 0.437387i \(0.144096\pi\)
−0.899273 + 0.437387i \(0.855904\pi\)
\(74\) − 634.719i − 0.997089i
\(75\) 0 0
\(76\) 288.014i 0.434704i
\(77\) 0 0
\(78\) 0 0
\(79\) −322.737 −0.459630 −0.229815 0.973234i \(-0.573812\pi\)
−0.229815 + 0.973234i \(0.573812\pi\)
\(80\) 68.6044 0.0958776
\(81\) 0 0
\(82\) 252.861i 0.340535i
\(83\) −885.170 −1.17060 −0.585301 0.810816i \(-0.699024\pi\)
−0.585301 + 0.810816i \(0.699024\pi\)
\(84\) 0 0
\(85\) −1058.19 −1.35032
\(86\) − 982.349i − 1.23174i
\(87\) 0 0
\(88\) 228.403 0.276680
\(89\) 1624.62 1.93494 0.967471 0.252984i \(-0.0814119\pi\)
0.967471 + 0.252984i \(0.0814119\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 616.327i − 0.698441i
\(93\) 0 0
\(94\) − 407.106i − 0.446700i
\(95\) − 927.244i − 1.00140i
\(96\) 0 0
\(97\) − 739.155i − 0.773710i −0.922141 0.386855i \(-0.873561\pi\)
0.922141 0.386855i \(-0.126439\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 480.965 0.480965
\(101\) 239.516 0.235967 0.117984 0.993016i \(-0.462357\pi\)
0.117984 + 0.993016i \(0.462357\pi\)
\(102\) 0 0
\(103\) 51.1361i 0.0489184i 0.999701 + 0.0244592i \(0.00778637\pi\)
−0.999701 + 0.0244592i \(0.992214\pi\)
\(104\) 1567.80 1.47822
\(105\) 0 0
\(106\) 964.800 0.884054
\(107\) 1190.99i 1.07605i 0.842929 + 0.538025i \(0.180829\pi\)
−0.842929 + 0.538025i \(0.819171\pi\)
\(108\) 0 0
\(109\) 389.170 0.341979 0.170989 0.985273i \(-0.445304\pi\)
0.170989 + 0.985273i \(0.445304\pi\)
\(110\) −271.631 −0.235446
\(111\) 0 0
\(112\) 0 0
\(113\) 718.545i 0.598186i 0.954224 + 0.299093i \(0.0966841\pi\)
−0.954224 + 0.299093i \(0.903316\pi\)
\(114\) 0 0
\(115\) 1984.23i 1.60896i
\(116\) 743.905i 0.595430i
\(117\) 0 0
\(118\) − 988.220i − 0.770958i
\(119\) 0 0
\(120\) 0 0
\(121\) 1233.18 0.926507
\(122\) 244.283 0.181282
\(123\) 0 0
\(124\) − 358.420i − 0.259573i
\(125\) 337.493 0.241490
\(126\) 0 0
\(127\) −179.456 −0.125387 −0.0626934 0.998033i \(-0.519969\pi\)
−0.0626934 + 0.998033i \(0.519969\pi\)
\(128\) − 760.787i − 0.525349i
\(129\) 0 0
\(130\) −1864.53 −1.25792
\(131\) −2446.87 −1.63194 −0.815968 0.578096i \(-0.803796\pi\)
−0.815968 + 0.578096i \(0.803796\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) − 583.564i − 0.376211i
\(135\) 0 0
\(136\) − 1619.72i − 1.02125i
\(137\) 511.557i 0.319016i 0.987197 + 0.159508i \(0.0509908\pi\)
−0.987197 + 0.159508i \(0.949009\pi\)
\(138\) 0 0
\(139\) − 599.427i − 0.365775i −0.983134 0.182888i \(-0.941456\pi\)
0.983134 0.182888i \(-0.0585444\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −759.006 −0.448552
\(143\) −671.447 −0.392652
\(144\) 0 0
\(145\) − 2394.95i − 1.37166i
\(146\) 993.191 0.562994
\(147\) 0 0
\(148\) 1634.05 0.907554
\(149\) 2193.50i 1.20603i 0.797729 + 0.603016i \(0.206034\pi\)
−0.797729 + 0.603016i \(0.793966\pi\)
\(150\) 0 0
\(151\) −717.366 −0.386612 −0.193306 0.981139i \(-0.561921\pi\)
−0.193306 + 0.981139i \(0.561921\pi\)
\(152\) 1419.28 0.757360
\(153\) 0 0
\(154\) 0 0
\(155\) 1153.91i 0.597963i
\(156\) 0 0
\(157\) − 1802.94i − 0.916500i −0.888823 0.458250i \(-0.848476\pi\)
0.888823 0.458250i \(-0.151524\pi\)
\(158\) 587.492i 0.295812i
\(159\) 0 0
\(160\) 2662.49i 1.31555i
\(161\) 0 0
\(162\) 0 0
\(163\) −2907.81 −1.39728 −0.698642 0.715472i \(-0.746212\pi\)
−0.698642 + 0.715472i \(0.746212\pi\)
\(164\) −650.977 −0.309956
\(165\) 0 0
\(166\) 1611.31i 0.753385i
\(167\) −3491.37 −1.61779 −0.808893 0.587956i \(-0.799933\pi\)
−0.808893 + 0.587956i \(0.799933\pi\)
\(168\) 0 0
\(169\) −2411.93 −1.09783
\(170\) 1926.27i 0.869048i
\(171\) 0 0
\(172\) 2529.00 1.12113
\(173\) −1754.75 −0.771165 −0.385583 0.922673i \(-0.625999\pi\)
−0.385583 + 0.922673i \(0.625999\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) − 44.9726i − 0.0192610i
\(177\) 0 0
\(178\) − 2957.37i − 1.24530i
\(179\) 791.707i 0.330586i 0.986244 + 0.165293i \(0.0528571\pi\)
−0.986244 + 0.165293i \(0.947143\pi\)
\(180\) 0 0
\(181\) 2522.19i 1.03576i 0.855452 + 0.517882i \(0.173279\pi\)
−0.855452 + 0.517882i \(0.826721\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −3037.14 −1.21685
\(185\) −5260.71 −2.09068
\(186\) 0 0
\(187\) 693.682i 0.271268i
\(188\) 1048.07 0.406588
\(189\) 0 0
\(190\) −1687.90 −0.644490
\(191\) − 903.283i − 0.342195i −0.985254 0.171098i \(-0.945269\pi\)
0.985254 0.171098i \(-0.0547313\pi\)
\(192\) 0 0
\(193\) −198.875 −0.0741727 −0.0370863 0.999312i \(-0.511808\pi\)
−0.0370863 + 0.999312i \(0.511808\pi\)
\(194\) −1345.51 −0.497950
\(195\) 0 0
\(196\) 0 0
\(197\) 3220.69i 1.16480i 0.812904 + 0.582398i \(0.197886\pi\)
−0.812904 + 0.582398i \(0.802114\pi\)
\(198\) 0 0
\(199\) 2849.92i 1.01520i 0.861592 + 0.507601i \(0.169468\pi\)
−0.861592 + 0.507601i \(0.830532\pi\)
\(200\) − 2370.11i − 0.837959i
\(201\) 0 0
\(202\) − 436.000i − 0.151866i
\(203\) 0 0
\(204\) 0 0
\(205\) 2095.78 0.714027
\(206\) 93.0851 0.0314832
\(207\) 0 0
\(208\) − 308.700i − 0.102906i
\(209\) −607.841 −0.201173
\(210\) 0 0
\(211\) 1204.50 0.392993 0.196496 0.980505i \(-0.437044\pi\)
0.196496 + 0.980505i \(0.437044\pi\)
\(212\) 2483.82i 0.804668i
\(213\) 0 0
\(214\) 2168.01 0.692532
\(215\) −8141.96 −2.58268
\(216\) 0 0
\(217\) 0 0
\(218\) − 708.421i − 0.220093i
\(219\) 0 0
\(220\) − 699.299i − 0.214303i
\(221\) 4761.56i 1.44931i
\(222\) 0 0
\(223\) 3377.73i 1.01430i 0.861857 + 0.507151i \(0.169301\pi\)
−0.861857 + 0.507151i \(0.830699\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1308.00 0.384985
\(227\) 4523.96 1.32276 0.661378 0.750052i \(-0.269972\pi\)
0.661378 + 0.750052i \(0.269972\pi\)
\(228\) 0 0
\(229\) − 3913.99i − 1.12945i −0.825280 0.564724i \(-0.808983\pi\)
0.825280 0.564724i \(-0.191017\pi\)
\(230\) 3611.96 1.03550
\(231\) 0 0
\(232\) 3665.82 1.03738
\(233\) 4369.16i 1.22847i 0.789124 + 0.614234i \(0.210535\pi\)
−0.789124 + 0.614234i \(0.789465\pi\)
\(234\) 0 0
\(235\) −3374.20 −0.936632
\(236\) 2544.12 0.701728
\(237\) 0 0
\(238\) 0 0
\(239\) 1945.23i 0.526471i 0.964732 + 0.263235i \(0.0847896\pi\)
−0.964732 + 0.263235i \(0.915210\pi\)
\(240\) 0 0
\(241\) − 4041.23i − 1.08016i −0.841614 0.540080i \(-0.818394\pi\)
0.841614 0.540080i \(-0.181606\pi\)
\(242\) − 2244.81i − 0.596288i
\(243\) 0 0
\(244\) 628.894i 0.165003i
\(245\) 0 0
\(246\) 0 0
\(247\) −4172.33 −1.07481
\(248\) −1766.22 −0.452239
\(249\) 0 0
\(250\) − 614.352i − 0.155420i
\(251\) 4415.70 1.11042 0.555212 0.831709i \(-0.312637\pi\)
0.555212 + 0.831709i \(0.312637\pi\)
\(252\) 0 0
\(253\) 1300.73 0.323226
\(254\) 326.671i 0.0806974i
\(255\) 0 0
\(256\) −4245.80 −1.03657
\(257\) −697.600 −0.169319 −0.0846597 0.996410i \(-0.526980\pi\)
−0.0846597 + 0.996410i \(0.526980\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) − 4800.11i − 1.14496i
\(261\) 0 0
\(262\) 4454.13i 1.05029i
\(263\) − 797.510i − 0.186983i −0.995620 0.0934915i \(-0.970197\pi\)
0.995620 0.0934915i \(-0.0298028\pi\)
\(264\) 0 0
\(265\) − 7996.51i − 1.85367i
\(266\) 0 0
\(267\) 0 0
\(268\) 1502.35 0.342428
\(269\) 410.703 0.0930892 0.0465446 0.998916i \(-0.485179\pi\)
0.0465446 + 0.998916i \(0.485179\pi\)
\(270\) 0 0
\(271\) − 3791.37i − 0.849850i −0.905228 0.424925i \(-0.860300\pi\)
0.905228 0.424925i \(-0.139700\pi\)
\(272\) −318.923 −0.0710939
\(273\) 0 0
\(274\) 931.207 0.205315
\(275\) 1015.05i 0.222582i
\(276\) 0 0
\(277\) 3246.62 0.704226 0.352113 0.935958i \(-0.385463\pi\)
0.352113 + 0.935958i \(0.385463\pi\)
\(278\) −1091.16 −0.235408
\(279\) 0 0
\(280\) 0 0
\(281\) 1599.58i 0.339583i 0.985480 + 0.169791i \(0.0543094\pi\)
−0.985480 + 0.169791i \(0.945691\pi\)
\(282\) 0 0
\(283\) − 4266.27i − 0.896125i −0.894002 0.448062i \(-0.852114\pi\)
0.894002 0.448062i \(-0.147886\pi\)
\(284\) − 1954.02i − 0.408273i
\(285\) 0 0
\(286\) 1222.26i 0.252706i
\(287\) 0 0
\(288\) 0 0
\(289\) 6.24157 0.00127042
\(290\) −4359.63 −0.882780
\(291\) 0 0
\(292\) 2556.91i 0.512438i
\(293\) −2926.77 −0.583562 −0.291781 0.956485i \(-0.594248\pi\)
−0.291781 + 0.956485i \(0.594248\pi\)
\(294\) 0 0
\(295\) −8190.62 −1.61653
\(296\) − 8052.28i − 1.58118i
\(297\) 0 0
\(298\) 3992.92 0.776187
\(299\) 8928.44 1.72691
\(300\) 0 0
\(301\) 0 0
\(302\) 1305.85i 0.248819i
\(303\) 0 0
\(304\) − 279.457i − 0.0527235i
\(305\) − 2024.68i − 0.380108i
\(306\) 0 0
\(307\) − 3571.36i − 0.663935i −0.943291 0.331968i \(-0.892288\pi\)
0.943291 0.331968i \(-0.107712\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 2100.51 0.384841
\(311\) 2573.41 0.469212 0.234606 0.972091i \(-0.424620\pi\)
0.234606 + 0.972091i \(0.424620\pi\)
\(312\) 0 0
\(313\) 1476.35i 0.266608i 0.991075 + 0.133304i \(0.0425587\pi\)
−0.991075 + 0.133304i \(0.957441\pi\)
\(314\) −3281.97 −0.589848
\(315\) 0 0
\(316\) −1512.46 −0.269249
\(317\) − 2526.90i − 0.447712i −0.974622 0.223856i \(-0.928136\pi\)
0.974622 0.223856i \(-0.0718645\pi\)
\(318\) 0 0
\(319\) −1569.98 −0.275554
\(320\) 5395.47 0.942550
\(321\) 0 0
\(322\) 0 0
\(323\) 4310.50i 0.742546i
\(324\) 0 0
\(325\) 6967.52i 1.18919i
\(326\) 5293.20i 0.899274i
\(327\) 0 0
\(328\) 3207.89i 0.540019i
\(329\) 0 0
\(330\) 0 0
\(331\) 1475.56 0.245027 0.122513 0.992467i \(-0.460905\pi\)
0.122513 + 0.992467i \(0.460905\pi\)
\(332\) −4148.23 −0.685733
\(333\) 0 0
\(334\) 6355.48i 1.04119i
\(335\) −4836.73 −0.788832
\(336\) 0 0
\(337\) −6727.28 −1.08741 −0.543706 0.839275i \(-0.682979\pi\)
−0.543706 + 0.839275i \(0.682979\pi\)
\(338\) 4390.54i 0.706549i
\(339\) 0 0
\(340\) −4959.07 −0.791010
\(341\) 756.428 0.120126
\(342\) 0 0
\(343\) 0 0
\(344\) − 12462.4i − 1.95328i
\(345\) 0 0
\(346\) 3194.25i 0.496312i
\(347\) − 538.160i − 0.0832563i −0.999133 0.0416281i \(-0.986746\pi\)
0.999133 0.0416281i \(-0.0132545\pi\)
\(348\) 0 0
\(349\) − 6975.93i − 1.06995i −0.844867 0.534976i \(-0.820321\pi\)
0.844867 0.534976i \(-0.179679\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1745.36 0.264283
\(353\) 8876.80 1.33843 0.669213 0.743071i \(-0.266631\pi\)
0.669213 + 0.743071i \(0.266631\pi\)
\(354\) 0 0
\(355\) 6290.83i 0.940514i
\(356\) 7613.57 1.13348
\(357\) 0 0
\(358\) 1441.18 0.212761
\(359\) − 11045.8i − 1.62389i −0.583735 0.811944i \(-0.698409\pi\)
0.583735 0.811944i \(-0.301591\pi\)
\(360\) 0 0
\(361\) 3081.92 0.449324
\(362\) 4591.25 0.666604
\(363\) 0 0
\(364\) 0 0
\(365\) − 8231.82i − 1.18047i
\(366\) 0 0
\(367\) 8326.07i 1.18424i 0.805848 + 0.592122i \(0.201710\pi\)
−0.805848 + 0.592122i \(0.798290\pi\)
\(368\) 598.015i 0.0847110i
\(369\) 0 0
\(370\) 9576.29i 1.34553i
\(371\) 0 0
\(372\) 0 0
\(373\) −4545.32 −0.630959 −0.315479 0.948932i \(-0.602165\pi\)
−0.315479 + 0.948932i \(0.602165\pi\)
\(374\) 1262.74 0.174584
\(375\) 0 0
\(376\) − 5164.70i − 0.708375i
\(377\) −10776.6 −1.47221
\(378\) 0 0
\(379\) 11527.2 1.56230 0.781151 0.624343i \(-0.214633\pi\)
0.781151 + 0.624343i \(0.214633\pi\)
\(380\) − 4345.40i − 0.586617i
\(381\) 0 0
\(382\) −1644.28 −0.220233
\(383\) −3920.46 −0.523044 −0.261522 0.965197i \(-0.584224\pi\)
−0.261522 + 0.965197i \(0.584224\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 362.020i 0.0477366i
\(387\) 0 0
\(388\) − 3463.95i − 0.453235i
\(389\) 796.437i 0.103807i 0.998652 + 0.0519035i \(0.0165288\pi\)
−0.998652 + 0.0519035i \(0.983471\pi\)
\(390\) 0 0
\(391\) − 9224.11i − 1.19305i
\(392\) 0 0
\(393\) 0 0
\(394\) 5862.76 0.749648
\(395\) 4869.28 0.620254
\(396\) 0 0
\(397\) − 3855.07i − 0.487357i −0.969856 0.243678i \(-0.921646\pi\)
0.969856 0.243678i \(-0.0783541\pi\)
\(398\) 5187.82 0.653371
\(399\) 0 0
\(400\) −466.675 −0.0583344
\(401\) − 4655.35i − 0.579744i −0.957065 0.289872i \(-0.906387\pi\)
0.957065 0.289872i \(-0.0936127\pi\)
\(402\) 0 0
\(403\) 5192.26 0.641798
\(404\) 1122.46 0.138229
\(405\) 0 0
\(406\) 0 0
\(407\) 3448.58i 0.420000i
\(408\) 0 0
\(409\) − 9790.50i − 1.18364i −0.806070 0.591821i \(-0.798409\pi\)
0.806070 0.591821i \(-0.201591\pi\)
\(410\) − 3815.03i − 0.459539i
\(411\) 0 0
\(412\) 239.642i 0.0286561i
\(413\) 0 0
\(414\) 0 0
\(415\) 13354.9 1.57968
\(416\) 11980.4 1.41199
\(417\) 0 0
\(418\) 1106.48i 0.129473i
\(419\) 3007.46 0.350654 0.175327 0.984510i \(-0.443902\pi\)
0.175327 + 0.984510i \(0.443902\pi\)
\(420\) 0 0
\(421\) 7646.06 0.885145 0.442573 0.896733i \(-0.354066\pi\)
0.442573 + 0.896733i \(0.354066\pi\)
\(422\) − 2192.61i − 0.252925i
\(423\) 0 0
\(424\) 12239.8 1.40193
\(425\) 7198.25 0.821568
\(426\) 0 0
\(427\) 0 0
\(428\) 5581.41i 0.630345i
\(429\) 0 0
\(430\) 14821.1i 1.66218i
\(431\) − 14991.6i − 1.67545i −0.546090 0.837727i \(-0.683884\pi\)
0.546090 0.837727i \(-0.316116\pi\)
\(432\) 0 0
\(433\) 5666.63i 0.628916i 0.949271 + 0.314458i \(0.101823\pi\)
−0.949271 + 0.314458i \(0.898177\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1823.79 0.200330
\(437\) 8082.65 0.884772
\(438\) 0 0
\(439\) 5531.10i 0.601332i 0.953729 + 0.300666i \(0.0972090\pi\)
−0.953729 + 0.300666i \(0.902791\pi\)
\(440\) −3446.01 −0.373368
\(441\) 0 0
\(442\) 8667.66 0.932757
\(443\) 403.222i 0.0432452i 0.999766 + 0.0216226i \(0.00688323\pi\)
−0.999766 + 0.0216226i \(0.993117\pi\)
\(444\) 0 0
\(445\) −24511.4 −2.61113
\(446\) 6148.61 0.652792
\(447\) 0 0
\(448\) 0 0
\(449\) 8429.03i 0.885948i 0.896534 + 0.442974i \(0.146077\pi\)
−0.896534 + 0.442974i \(0.853923\pi\)
\(450\) 0 0
\(451\) − 1373.86i − 0.143442i
\(452\) 3367.36i 0.350415i
\(453\) 0 0
\(454\) − 8235.15i − 0.851310i
\(455\) 0 0
\(456\) 0 0
\(457\) 685.661 0.0701835 0.0350917 0.999384i \(-0.488828\pi\)
0.0350917 + 0.999384i \(0.488828\pi\)
\(458\) −7124.79 −0.726899
\(459\) 0 0
\(460\) 9298.80i 0.942519i
\(461\) −4864.48 −0.491456 −0.245728 0.969339i \(-0.579027\pi\)
−0.245728 + 0.969339i \(0.579027\pi\)
\(462\) 0 0
\(463\) −8354.23 −0.838562 −0.419281 0.907857i \(-0.637718\pi\)
−0.419281 + 0.907857i \(0.637718\pi\)
\(464\) − 721.802i − 0.0722173i
\(465\) 0 0
\(466\) 7953.36 0.790627
\(467\) 1002.94 0.0993800 0.0496900 0.998765i \(-0.484177\pi\)
0.0496900 + 0.998765i \(0.484177\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 6142.19i 0.602804i
\(471\) 0 0
\(472\) − 12536.9i − 1.22258i
\(473\) 5337.34i 0.518839i
\(474\) 0 0
\(475\) 6307.49i 0.609279i
\(476\) 0 0
\(477\) 0 0
\(478\) 3540.98 0.338830
\(479\) −6052.76 −0.577364 −0.288682 0.957425i \(-0.593217\pi\)
−0.288682 + 0.957425i \(0.593217\pi\)
\(480\) 0 0
\(481\) 23671.7i 2.24394i
\(482\) −7356.41 −0.695177
\(483\) 0 0
\(484\) 5779.13 0.542743
\(485\) 11152.0i 1.04409i
\(486\) 0 0
\(487\) 15309.4 1.42451 0.712255 0.701920i \(-0.247674\pi\)
0.712255 + 0.701920i \(0.247674\pi\)
\(488\) 3099.07 0.287476
\(489\) 0 0
\(490\) 0 0
\(491\) − 4291.01i − 0.394400i −0.980363 0.197200i \(-0.936815\pi\)
0.980363 0.197200i \(-0.0631848\pi\)
\(492\) 0 0
\(493\) 11133.5i 1.01709i
\(494\) 7595.06i 0.691736i
\(495\) 0 0
\(496\) 347.770i 0.0314826i
\(497\) 0 0
\(498\) 0 0
\(499\) 6891.53 0.618251 0.309126 0.951021i \(-0.399964\pi\)
0.309126 + 0.951021i \(0.399964\pi\)
\(500\) 1581.61 0.141464
\(501\) 0 0
\(502\) − 8038.07i − 0.714655i
\(503\) 13534.6 1.19975 0.599877 0.800092i \(-0.295216\pi\)
0.599877 + 0.800092i \(0.295216\pi\)
\(504\) 0 0
\(505\) −3613.68 −0.318429
\(506\) − 2367.77i − 0.208024i
\(507\) 0 0
\(508\) −840.995 −0.0734510
\(509\) −12087.8 −1.05262 −0.526310 0.850293i \(-0.676425\pi\)
−0.526310 + 0.850293i \(0.676425\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1642.50i 0.141776i
\(513\) 0 0
\(514\) 1269.87i 0.108972i
\(515\) − 771.513i − 0.0660134i
\(516\) 0 0
\(517\) 2211.91i 0.188161i
\(518\) 0 0
\(519\) 0 0
\(520\) −23654.0 −1.99480
\(521\) −7625.87 −0.641258 −0.320629 0.947205i \(-0.603894\pi\)
−0.320629 + 0.947205i \(0.603894\pi\)
\(522\) 0 0
\(523\) − 15390.7i − 1.28678i −0.765537 0.643392i \(-0.777527\pi\)
0.765537 0.643392i \(-0.222473\pi\)
\(524\) −11466.9 −0.955981
\(525\) 0 0
\(526\) −1451.74 −0.120340
\(527\) − 5364.20i − 0.443393i
\(528\) 0 0
\(529\) −5129.20 −0.421567
\(530\) −14556.4 −1.19300
\(531\) 0 0
\(532\) 0 0
\(533\) − 9430.40i − 0.766371i
\(534\) 0 0
\(535\) − 17969.0i − 1.45209i
\(536\) − 7403.30i − 0.596593i
\(537\) 0 0
\(538\) − 747.619i − 0.0599110i
\(539\) 0 0
\(540\) 0 0
\(541\) −13700.9 −1.08881 −0.544406 0.838822i \(-0.683245\pi\)
−0.544406 + 0.838822i \(0.683245\pi\)
\(542\) −6901.59 −0.546953
\(543\) 0 0
\(544\) − 12377.2i − 0.975491i
\(545\) −5871.58 −0.461487
\(546\) 0 0
\(547\) −6139.00 −0.479863 −0.239931 0.970790i \(-0.577125\pi\)
−0.239931 + 0.970790i \(0.577125\pi\)
\(548\) 2397.34i 0.186878i
\(549\) 0 0
\(550\) 1847.74 0.143251
\(551\) −9755.73 −0.754280
\(552\) 0 0
\(553\) 0 0
\(554\) − 5909.95i − 0.453231i
\(555\) 0 0
\(556\) − 2809.13i − 0.214269i
\(557\) 22732.7i 1.72929i 0.502382 + 0.864646i \(0.332457\pi\)
−0.502382 + 0.864646i \(0.667543\pi\)
\(558\) 0 0
\(559\) 36636.4i 2.77202i
\(560\) 0 0
\(561\) 0 0
\(562\) 2911.78 0.218551
\(563\) −9917.61 −0.742411 −0.371206 0.928551i \(-0.621055\pi\)
−0.371206 + 0.928551i \(0.621055\pi\)
\(564\) 0 0
\(565\) − 10841.0i − 0.807230i
\(566\) −7766.06 −0.576735
\(567\) 0 0
\(568\) −9629.02 −0.711311
\(569\) 5137.02i 0.378480i 0.981931 + 0.189240i \(0.0606024\pi\)
−0.981931 + 0.189240i \(0.939398\pi\)
\(570\) 0 0
\(571\) −18186.0 −1.33286 −0.666429 0.745568i \(-0.732178\pi\)
−0.666429 + 0.745568i \(0.732178\pi\)
\(572\) −3146.64 −0.230014
\(573\) 0 0
\(574\) 0 0
\(575\) − 13497.5i − 0.978930i
\(576\) 0 0
\(577\) 12398.6i 0.894562i 0.894394 + 0.447281i \(0.147608\pi\)
−0.894394 + 0.447281i \(0.852392\pi\)
\(578\) − 11.3618i 0 0.000817626i
\(579\) 0 0
\(580\) − 11223.6i − 0.803509i
\(581\) 0 0
\(582\) 0 0
\(583\) −5241.99 −0.372386
\(584\) 12600.0 0.892793
\(585\) 0 0
\(586\) 5327.72i 0.375573i
\(587\) 18977.6 1.33439 0.667195 0.744883i \(-0.267495\pi\)
0.667195 + 0.744883i \(0.267495\pi\)
\(588\) 0 0
\(589\) 4700.40 0.328822
\(590\) 14909.7i 1.04038i
\(591\) 0 0
\(592\) −1585.50 −0.110074
\(593\) 10728.9 0.742972 0.371486 0.928439i \(-0.378848\pi\)
0.371486 + 0.928439i \(0.378848\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10279.5i 0.706488i
\(597\) 0 0
\(598\) − 16252.8i − 1.11142i
\(599\) 1821.37i 0.124239i 0.998069 + 0.0621196i \(0.0197860\pi\)
−0.998069 + 0.0621196i \(0.980214\pi\)
\(600\) 0 0
\(601\) − 18933.3i − 1.28503i −0.766273 0.642516i \(-0.777891\pi\)
0.766273 0.642516i \(-0.222109\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −3361.84 −0.226475
\(605\) −18605.5 −1.25029
\(606\) 0 0
\(607\) − 15384.4i − 1.02872i −0.857574 0.514360i \(-0.828029\pi\)
0.857574 0.514360i \(-0.171971\pi\)
\(608\) 10845.5 0.723428
\(609\) 0 0
\(610\) −3685.61 −0.244633
\(611\) 15182.9i 1.00529i
\(612\) 0 0
\(613\) −5507.20 −0.362861 −0.181431 0.983404i \(-0.558073\pi\)
−0.181431 + 0.983404i \(0.558073\pi\)
\(614\) −6501.08 −0.427300
\(615\) 0 0
\(616\) 0 0
\(617\) 18134.0i 1.18322i 0.806224 + 0.591610i \(0.201507\pi\)
−0.806224 + 0.591610i \(0.798493\pi\)
\(618\) 0 0
\(619\) 3635.84i 0.236085i 0.993009 + 0.118043i \(0.0376619\pi\)
−0.993009 + 0.118043i \(0.962338\pi\)
\(620\) 5407.64i 0.350284i
\(621\) 0 0
\(622\) − 4684.49i − 0.301979i
\(623\) 0 0
\(624\) 0 0
\(625\) −17920.8 −1.14693
\(626\) 2687.46 0.171586
\(627\) 0 0
\(628\) − 8449.24i − 0.536881i
\(629\) 24455.6 1.55025
\(630\) 0 0
\(631\) −5912.59 −0.373021 −0.186511 0.982453i \(-0.559718\pi\)
−0.186511 + 0.982453i \(0.559718\pi\)
\(632\) 7453.13i 0.469098i
\(633\) 0 0
\(634\) −4599.81 −0.288142
\(635\) 2707.53 0.169205
\(636\) 0 0
\(637\) 0 0
\(638\) 2857.89i 0.177343i
\(639\) 0 0
\(640\) 11478.3i 0.708939i
\(641\) − 27466.6i − 1.69245i −0.532822 0.846227i \(-0.678868\pi\)
0.532822 0.846227i \(-0.321132\pi\)
\(642\) 0 0
\(643\) − 28474.0i − 1.74635i −0.487403 0.873177i \(-0.662056\pi\)
0.487403 0.873177i \(-0.337944\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 7846.57 0.477894
\(647\) 1323.36 0.0804122 0.0402061 0.999191i \(-0.487199\pi\)
0.0402061 + 0.999191i \(0.487199\pi\)
\(648\) 0 0
\(649\) 5369.24i 0.324747i
\(650\) 12683.3 0.765351
\(651\) 0 0
\(652\) −13627.0 −0.818522
\(653\) − 3846.75i − 0.230528i −0.993335 0.115264i \(-0.963229\pi\)
0.993335 0.115264i \(-0.0367714\pi\)
\(654\) 0 0
\(655\) 36917.0 2.20224
\(656\) 631.635 0.0375933
\(657\) 0 0
\(658\) 0 0
\(659\) 6796.84i 0.401771i 0.979615 + 0.200886i \(0.0643819\pi\)
−0.979615 + 0.200886i \(0.935618\pi\)
\(660\) 0 0
\(661\) − 31064.4i − 1.82793i −0.405790 0.913966i \(-0.633004\pi\)
0.405790 0.913966i \(-0.366996\pi\)
\(662\) − 2686.02i − 0.157696i
\(663\) 0 0
\(664\) 20441.7i 1.19471i
\(665\) 0 0
\(666\) 0 0
\(667\) 20876.5 1.21190
\(668\) −16361.8 −0.947691
\(669\) 0 0
\(670\) 8804.49i 0.507682i
\(671\) −1327.25 −0.0763605
\(672\) 0 0
\(673\) 15508.2 0.888259 0.444129 0.895963i \(-0.353513\pi\)
0.444129 + 0.895963i \(0.353513\pi\)
\(674\) 12245.9i 0.699845i
\(675\) 0 0
\(676\) −11303.2 −0.643103
\(677\) −30674.8 −1.74140 −0.870701 0.491813i \(-0.836334\pi\)
−0.870701 + 0.491813i \(0.836334\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 24437.4i 1.37813i
\(681\) 0 0
\(682\) − 1376.96i − 0.0773114i
\(683\) − 17658.7i − 0.989296i −0.869093 0.494648i \(-0.835297\pi\)
0.869093 0.494648i \(-0.164703\pi\)
\(684\) 0 0
\(685\) − 7718.08i − 0.430500i
\(686\) 0 0
\(687\) 0 0
\(688\) −2453.86 −0.135977
\(689\) −35982.0 −1.98956
\(690\) 0 0
\(691\) − 13166.2i − 0.724843i −0.932014 0.362422i \(-0.881950\pi\)
0.932014 0.362422i \(-0.118050\pi\)
\(692\) −8223.42 −0.451745
\(693\) 0 0
\(694\) −979.634 −0.0535827
\(695\) 9043.82i 0.493599i
\(696\) 0 0
\(697\) −9742.69 −0.529456
\(698\) −12698.6 −0.688607
\(699\) 0 0
\(700\) 0 0
\(701\) − 25910.0i − 1.39602i −0.716090 0.698008i \(-0.754070\pi\)
0.716090 0.698008i \(-0.245930\pi\)
\(702\) 0 0
\(703\) 21429.3i 1.14967i
\(704\) − 3536.92i − 0.189350i
\(705\) 0 0
\(706\) − 16158.8i − 0.861394i
\(707\) 0 0
\(708\) 0 0
\(709\) 6208.49 0.328864 0.164432 0.986388i \(-0.447421\pi\)
0.164432 + 0.986388i \(0.447421\pi\)
\(710\) 11451.5 0.605303
\(711\) 0 0
\(712\) − 37518.2i − 1.97480i
\(713\) −10058.5 −0.528320
\(714\) 0 0
\(715\) 10130.4 0.529868
\(716\) 3710.23i 0.193656i
\(717\) 0 0
\(718\) −20107.1 −1.04511
\(719\) 28758.8 1.49169 0.745843 0.666122i \(-0.232047\pi\)
0.745843 + 0.666122i \(0.232047\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 5610.13i − 0.289180i
\(723\) 0 0
\(724\) 11819.9i 0.606745i
\(725\) 16291.4i 0.834550i
\(726\) 0 0
\(727\) 35275.7i 1.79959i 0.436312 + 0.899795i \(0.356284\pi\)
−0.436312 + 0.899795i \(0.643716\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −14984.7 −0.759738
\(731\) 37849.7 1.91508
\(732\) 0 0
\(733\) 7950.28i 0.400615i 0.979733 + 0.200307i \(0.0641940\pi\)
−0.979733 + 0.200307i \(0.935806\pi\)
\(734\) 15156.3 0.762164
\(735\) 0 0
\(736\) −23208.5 −1.16233
\(737\) 3170.64i 0.158470i
\(738\) 0 0
\(739\) 33353.6 1.66026 0.830130 0.557570i \(-0.188266\pi\)
0.830130 + 0.557570i \(0.188266\pi\)
\(740\) −24653.6 −1.22471
\(741\) 0 0
\(742\) 0 0
\(743\) − 32933.6i − 1.62613i −0.582171 0.813066i \(-0.697797\pi\)
0.582171 0.813066i \(-0.302203\pi\)
\(744\) 0 0
\(745\) − 33094.3i − 1.62749i
\(746\) 8274.02i 0.406077i
\(747\) 0 0
\(748\) 3250.85i 0.158907i
\(749\) 0 0
\(750\) 0 0
\(751\) −39636.6 −1.92591 −0.962956 0.269660i \(-0.913089\pi\)
−0.962956 + 0.269660i \(0.913089\pi\)
\(752\) −1016.93 −0.0493134
\(753\) 0 0
\(754\) 19617.1i 0.947496i
\(755\) 10823.2 0.521718
\(756\) 0 0
\(757\) −3996.51 −0.191883 −0.0959417 0.995387i \(-0.530586\pi\)
−0.0959417 + 0.995387i \(0.530586\pi\)
\(758\) − 20983.4i − 1.00548i
\(759\) 0 0
\(760\) −21413.3 −1.02203
\(761\) 26235.7 1.24973 0.624863 0.780734i \(-0.285155\pi\)
0.624863 + 0.780734i \(0.285155\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) − 4233.11i − 0.200456i
\(765\) 0 0
\(766\) 7136.57i 0.336625i
\(767\) 36855.4i 1.73503i
\(768\) 0 0
\(769\) 36456.9i 1.70958i 0.518971 + 0.854792i \(0.326315\pi\)
−0.518971 + 0.854792i \(0.673685\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −932.000 −0.0434500
\(773\) −9465.47 −0.440426 −0.220213 0.975452i \(-0.570675\pi\)
−0.220213 + 0.975452i \(0.570675\pi\)
\(774\) 0 0
\(775\) − 7849.36i − 0.363816i
\(776\) −17069.7 −0.789646
\(777\) 0 0
\(778\) 1449.79 0.0668089
\(779\) − 8537.06i − 0.392647i
\(780\) 0 0
\(781\) 4123.86 0.188941
\(782\) −16791.0 −0.767833
\(783\) 0 0
\(784\) 0 0
\(785\) 27201.8i 1.23678i
\(786\) 0 0
\(787\) 25017.6i 1.13314i 0.824013 + 0.566571i \(0.191730\pi\)
−0.824013 + 0.566571i \(0.808270\pi\)
\(788\) 15093.3i 0.682332i
\(789\) 0 0
\(790\) − 8863.75i − 0.399187i
\(791\) 0 0
\(792\) 0 0
\(793\) −9110.48 −0.407973
\(794\) −7017.54 −0.313657
\(795\) 0 0
\(796\) 13355.7i 0.594701i
\(797\) −38893.5 −1.72858 −0.864290 0.502994i \(-0.832232\pi\)
−0.864290 + 0.502994i \(0.832232\pi\)
\(798\) 0 0
\(799\) 15685.7 0.694519
\(800\) − 18111.3i − 0.800415i
\(801\) 0 0
\(802\) −8474.32 −0.373116
\(803\) −5396.24 −0.237147
\(804\) 0 0
\(805\) 0 0
\(806\) − 9451.67i − 0.413053i
\(807\) 0 0
\(808\) − 5531.26i − 0.240828i
\(809\) 7011.64i 0.304717i 0.988325 + 0.152359i \(0.0486869\pi\)
−0.988325 + 0.152359i \(0.951313\pi\)
\(810\) 0 0
\(811\) − 5013.82i − 0.217089i −0.994092 0.108544i \(-0.965381\pi\)
0.994092 0.108544i \(-0.0346189\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 6277.59 0.270306
\(815\) 43871.4 1.88558
\(816\) 0 0
\(817\) 33165.9i 1.42023i
\(818\) −17822.0 −0.761776
\(819\) 0 0
\(820\) 9821.58 0.418274
\(821\) − 696.609i − 0.0296124i −0.999890 0.0148062i \(-0.995287\pi\)
0.999890 0.0148062i \(-0.00471314\pi\)
\(822\) 0 0
\(823\) 6413.15 0.271626 0.135813 0.990734i \(-0.456635\pi\)
0.135813 + 0.990734i \(0.456635\pi\)
\(824\) 1180.91 0.0499260
\(825\) 0 0
\(826\) 0 0
\(827\) 16718.9i 0.702989i 0.936190 + 0.351494i \(0.114326\pi\)
−0.936190 + 0.351494i \(0.885674\pi\)
\(828\) 0 0
\(829\) − 14557.2i − 0.609882i −0.952371 0.304941i \(-0.901363\pi\)
0.952371 0.304941i \(-0.0986367\pi\)
\(830\) − 24310.5i − 1.01666i
\(831\) 0 0
\(832\) − 24278.1i − 1.01165i
\(833\) 0 0
\(834\) 0 0
\(835\) 52675.8 2.18314
\(836\) −2848.56 −0.117846
\(837\) 0 0
\(838\) − 5474.61i − 0.225677i
\(839\) 19467.0 0.801045 0.400523 0.916287i \(-0.368829\pi\)
0.400523 + 0.916287i \(0.368829\pi\)
\(840\) 0 0
\(841\) −808.829 −0.0331637
\(842\) − 13918.4i − 0.569668i
\(843\) 0 0
\(844\) 5644.74 0.230213
\(845\) 36389.9 1.48148
\(846\) 0 0
\(847\) 0 0
\(848\) − 2410.02i − 0.0975950i
\(849\) 0 0
\(850\) − 13103.3i − 0.528751i
\(851\) − 45856.9i − 1.84718i
\(852\) 0 0
\(853\) 22345.3i 0.896938i 0.893798 + 0.448469i \(0.148031\pi\)
−0.893798 + 0.448469i \(0.851969\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 27504.1 1.09821
\(857\) −48079.7 −1.91642 −0.958209 0.286068i \(-0.907652\pi\)
−0.958209 + 0.286068i \(0.907652\pi\)
\(858\) 0 0
\(859\) 26110.7i 1.03712i 0.855041 + 0.518560i \(0.173532\pi\)
−0.855041 + 0.518560i \(0.826468\pi\)
\(860\) −38156.2 −1.51292
\(861\) 0 0
\(862\) −27289.8 −1.07830
\(863\) 2928.55i 0.115514i 0.998331 + 0.0577572i \(0.0183949\pi\)
−0.998331 + 0.0577572i \(0.981605\pi\)
\(864\) 0 0
\(865\) 26474.8 1.04066
\(866\) 10315.2 0.404763
\(867\) 0 0
\(868\) 0 0
\(869\) − 3191.98i − 0.124604i
\(870\) 0 0
\(871\) 21763.9i 0.846660i
\(872\) − 8987.29i − 0.349023i
\(873\) 0 0
\(874\) − 14713.2i − 0.569428i
\(875\) 0 0
\(876\) 0 0
\(877\) 13090.6 0.504033 0.252016 0.967723i \(-0.418906\pi\)
0.252016 + 0.967723i \(0.418906\pi\)
\(878\) 10068.5 0.387010
\(879\) 0 0
\(880\) 678.521i 0.0259920i
\(881\) 7888.79 0.301680 0.150840 0.988558i \(-0.451802\pi\)
0.150840 + 0.988558i \(0.451802\pi\)
\(882\) 0 0
\(883\) 45061.9 1.71739 0.858694 0.512489i \(-0.171276\pi\)
0.858694 + 0.512489i \(0.171276\pi\)
\(884\) 22314.4i 0.848998i
\(885\) 0 0
\(886\) 734.000 0.0278321
\(887\) 3546.58 0.134253 0.0671266 0.997744i \(-0.478617\pi\)
0.0671266 + 0.997744i \(0.478617\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 44619.1i 1.68049i
\(891\) 0 0
\(892\) 15829.2i 0.594173i
\(893\) 13744.6i 0.515058i
\(894\) 0 0
\(895\) − 11944.8i − 0.446114i
\(896\) 0 0
\(897\) 0 0
\(898\) 15343.7 0.570185
\(899\) 12140.5 0.450400
\(900\) 0 0
\(901\) 37173.5i 1.37451i
\(902\) −2500.89 −0.0923175
\(903\) 0 0
\(904\) 16593.7 0.610508
\(905\) − 38053.4i − 1.39772i
\(906\) 0 0
\(907\) −14679.5 −0.537405 −0.268703 0.963223i \(-0.586595\pi\)
−0.268703 + 0.963223i \(0.586595\pi\)
\(908\) 21200.9 0.774864
\(909\) 0 0
\(910\) 0 0
\(911\) − 12355.2i − 0.449336i −0.974435 0.224668i \(-0.927870\pi\)
0.974435 0.224668i \(-0.0721297\pi\)
\(912\) 0 0
\(913\) − 8754.64i − 0.317345i
\(914\) − 1248.14i − 0.0451692i
\(915\) 0 0
\(916\) − 18342.4i − 0.661626i
\(917\) 0 0
\(918\) 0 0
\(919\) −3070.36 −0.110209 −0.0551044 0.998481i \(-0.517549\pi\)
−0.0551044 + 0.998481i \(0.517549\pi\)
\(920\) 45822.7 1.64210
\(921\) 0 0
\(922\) 8855.01i 0.316295i
\(923\) 28306.9 1.00946
\(924\) 0 0
\(925\) 35785.5 1.27202
\(926\) 15207.5i 0.539688i
\(927\) 0 0
\(928\) 28012.6 0.990905
\(929\) 30528.7 1.07816 0.539082 0.842254i \(-0.318771\pi\)
0.539082 + 0.842254i \(0.318771\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 20475.5i 0.719631i
\(933\) 0 0
\(934\) − 1825.69i − 0.0639597i
\(935\) − 10465.9i − 0.366065i
\(936\) 0 0
\(937\) − 18235.1i − 0.635769i −0.948129 0.317885i \(-0.897028\pi\)
0.948129 0.317885i \(-0.102972\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −15812.7 −0.548674
\(941\) 13372.3 0.463258 0.231629 0.972804i \(-0.425594\pi\)
0.231629 + 0.972804i \(0.425594\pi\)
\(942\) 0 0
\(943\) 18268.6i 0.630867i
\(944\) −2468.52 −0.0851098
\(945\) 0 0
\(946\) 9715.77 0.333919
\(947\) − 5117.10i − 0.175590i −0.996139 0.0877948i \(-0.972018\pi\)
0.996139 0.0877948i \(-0.0279820\pi\)
\(948\) 0 0
\(949\) −37040.8 −1.26701
\(950\) 11481.8 0.392124
\(951\) 0 0
\(952\) 0 0
\(953\) − 35456.7i − 1.20520i −0.798044 0.602599i \(-0.794132\pi\)
0.798044 0.602599i \(-0.205868\pi\)
\(954\) 0 0
\(955\) 13628.2i 0.461779i
\(956\) 9116.05i 0.308404i
\(957\) 0 0
\(958\) 11018.1i 0.371584i
\(959\) 0 0
\(960\) 0 0
\(961\) 23941.6 0.803652
\(962\) 43090.5 1.44417
\(963\) 0 0
\(964\) − 18938.7i − 0.632752i
\(965\) 3000.51 0.100093
\(966\) 0 0
\(967\) −2804.92 −0.0932784 −0.0466392 0.998912i \(-0.514851\pi\)
−0.0466392 + 0.998912i \(0.514851\pi\)
\(968\) − 28478.5i − 0.945591i
\(969\) 0 0
\(970\) 20300.4 0.671964
\(971\) −27367.5 −0.904496 −0.452248 0.891892i \(-0.649378\pi\)
−0.452248 + 0.891892i \(0.649378\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) − 27868.4i − 0.916797i
\(975\) 0 0
\(976\) − 610.208i − 0.0200126i
\(977\) 24096.8i 0.789075i 0.918880 + 0.394537i \(0.129095\pi\)
−0.918880 + 0.394537i \(0.870905\pi\)
\(978\) 0 0
\(979\) 16068.1i 0.524554i
\(980\) 0 0
\(981\) 0 0
\(982\) −7811.09 −0.253831
\(983\) 9310.72 0.302102 0.151051 0.988526i \(-0.451734\pi\)
0.151051 + 0.988526i \(0.451734\pi\)
\(984\) 0 0
\(985\) − 48592.0i − 1.57185i
\(986\) 20266.7 0.654588
\(987\) 0 0
\(988\) −19553.0 −0.629621
\(989\) − 70972.3i − 2.28189i
\(990\) 0 0
\(991\) −24190.5 −0.775414 −0.387707 0.921783i \(-0.626733\pi\)
−0.387707 + 0.921783i \(0.626733\pi\)
\(992\) −13496.7 −0.431977
\(993\) 0 0
\(994\) 0 0
\(995\) − 42998.0i − 1.36998i
\(996\) 0 0
\(997\) 48866.7i 1.55228i 0.630560 + 0.776140i \(0.282825\pi\)
−0.630560 + 0.776140i \(0.717175\pi\)
\(998\) − 12544.9i − 0.397899i
\(999\) 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 441.4.c.a.440.5 16
3.2 odd 2 inner 441.4.c.a.440.12 16
7.2 even 3 441.4.p.c.80.3 16
7.3 odd 6 441.4.p.c.215.6 16
7.4 even 3 63.4.p.a.26.6 yes 16
7.5 odd 6 63.4.p.a.17.3 16
7.6 odd 2 inner 441.4.c.a.440.6 16
21.2 odd 6 441.4.p.c.80.6 16
21.5 even 6 63.4.p.a.17.6 yes 16
21.11 odd 6 63.4.p.a.26.3 yes 16
21.17 even 6 441.4.p.c.215.3 16
21.20 even 2 inner 441.4.c.a.440.11 16
28.11 odd 6 1008.4.bt.a.593.8 16
28.19 even 6 1008.4.bt.a.17.1 16
84.11 even 6 1008.4.bt.a.593.1 16
84.47 odd 6 1008.4.bt.a.17.8 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.4.p.a.17.3 16 7.5 odd 6
63.4.p.a.17.6 yes 16 21.5 even 6
63.4.p.a.26.3 yes 16 21.11 odd 6
63.4.p.a.26.6 yes 16 7.4 even 3
441.4.c.a.440.5 16 1.1 even 1 trivial
441.4.c.a.440.6 16 7.6 odd 2 inner
441.4.c.a.440.11 16 21.20 even 2 inner
441.4.c.a.440.12 16 3.2 odd 2 inner
441.4.p.c.80.3 16 7.2 even 3
441.4.p.c.80.6 16 21.2 odd 6
441.4.p.c.215.3 16 21.17 even 6
441.4.p.c.215.6 16 7.3 odd 6
1008.4.bt.a.17.1 16 28.19 even 6
1008.4.bt.a.17.8 16 84.47 odd 6
1008.4.bt.a.593.1 16 84.11 even 6
1008.4.bt.a.593.8 16 28.11 odd 6