Properties

Label 2205.4.a.bx.1.6
Level $2205$
Weight $4$
Character 2205.1
Self dual yes
Analytic conductor $130.099$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(130.099211563\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 34x^{4} + 241x^{2} - 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(4.91092\) of defining polynomial
Character \(\chi\) \(=\) 2205.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.91092 q^{2} +16.1171 q^{4} -5.00000 q^{5} +39.8626 q^{8} +O(q^{10})\) \(q+4.91092 q^{2} +16.1171 q^{4} -5.00000 q^{5} +39.8626 q^{8} -24.5546 q^{10} +12.0023 q^{11} -46.9540 q^{13} +66.8251 q^{16} -44.7080 q^{17} +31.7414 q^{19} -80.5857 q^{20} +58.9423 q^{22} -39.4080 q^{23} +25.0000 q^{25} -230.587 q^{26} +87.2713 q^{29} -304.718 q^{31} +9.27181 q^{32} -219.557 q^{34} -151.061 q^{37} +155.879 q^{38} -199.313 q^{40} -282.927 q^{41} -143.177 q^{43} +193.443 q^{44} -193.530 q^{46} -88.8029 q^{47} +122.773 q^{50} -756.764 q^{52} +149.940 q^{53} -60.0114 q^{55} +428.582 q^{58} +712.934 q^{59} -65.0347 q^{61} -1496.45 q^{62} -489.068 q^{64} +234.770 q^{65} +698.725 q^{67} -720.565 q^{68} -291.400 q^{71} -476.241 q^{73} -741.849 q^{74} +511.580 q^{76} +256.947 q^{79} -334.126 q^{80} -1389.43 q^{82} -1177.61 q^{83} +223.540 q^{85} -703.129 q^{86} +478.443 q^{88} -1009.96 q^{89} -635.145 q^{92} -436.104 q^{94} -158.707 q^{95} +1149.16 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 20 q^{4} - 30 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 20 q^{4} - 30 q^{5} + 100 q^{16} - 44 q^{17} - 100 q^{20} - 24 q^{22} + 150 q^{25} - 168 q^{26} + 380 q^{37} - 56 q^{38} - 612 q^{41} - 328 q^{43} + 432 q^{46} - 120 q^{47} + 1120 q^{58} - 136 q^{59} - 2264 q^{62} - 1052 q^{64} + 1112 q^{67} - 264 q^{68} + 1400 q^{79} - 500 q^{80} - 2912 q^{83} + 220 q^{85} + 1384 q^{88} - 372 q^{89}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.91092 1.73627 0.868136 0.496326i \(-0.165318\pi\)
0.868136 + 0.496326i \(0.165318\pi\)
\(3\) 0 0
\(4\) 16.1171 2.01464
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 39.8626 1.76170
\(9\) 0 0
\(10\) −24.5546 −0.776485
\(11\) 12.0023 0.328984 0.164492 0.986378i \(-0.447402\pi\)
0.164492 + 0.986378i \(0.447402\pi\)
\(12\) 0 0
\(13\) −46.9540 −1.00175 −0.500873 0.865521i \(-0.666988\pi\)
−0.500873 + 0.865521i \(0.666988\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 66.8251 1.04414
\(17\) −44.7080 −0.637840 −0.318920 0.947782i \(-0.603320\pi\)
−0.318920 + 0.947782i \(0.603320\pi\)
\(18\) 0 0
\(19\) 31.7414 0.383262 0.191631 0.981467i \(-0.438622\pi\)
0.191631 + 0.981467i \(0.438622\pi\)
\(20\) −80.5857 −0.900976
\(21\) 0 0
\(22\) 58.9423 0.571206
\(23\) −39.4080 −0.357267 −0.178634 0.983916i \(-0.557168\pi\)
−0.178634 + 0.983916i \(0.557168\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −230.587 −1.73930
\(27\) 0 0
\(28\) 0 0
\(29\) 87.2713 0.558823 0.279412 0.960171i \(-0.409861\pi\)
0.279412 + 0.960171i \(0.409861\pi\)
\(30\) 0 0
\(31\) −304.718 −1.76545 −0.882726 0.469887i \(-0.844295\pi\)
−0.882726 + 0.469887i \(0.844295\pi\)
\(32\) 9.27181 0.0512200
\(33\) 0 0
\(34\) −219.557 −1.10746
\(35\) 0 0
\(36\) 0 0
\(37\) −151.061 −0.671197 −0.335599 0.942005i \(-0.608939\pi\)
−0.335599 + 0.942005i \(0.608939\pi\)
\(38\) 155.879 0.665447
\(39\) 0 0
\(40\) −199.313 −0.787855
\(41\) −282.927 −1.07770 −0.538851 0.842401i \(-0.681141\pi\)
−0.538851 + 0.842401i \(0.681141\pi\)
\(42\) 0 0
\(43\) −143.177 −0.507773 −0.253886 0.967234i \(-0.581709\pi\)
−0.253886 + 0.967234i \(0.581709\pi\)
\(44\) 193.443 0.662786
\(45\) 0 0
\(46\) −193.530 −0.620313
\(47\) −88.8029 −0.275601 −0.137800 0.990460i \(-0.544003\pi\)
−0.137800 + 0.990460i \(0.544003\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 122.773 0.347255
\(51\) 0 0
\(52\) −756.764 −2.01816
\(53\) 149.940 0.388601 0.194301 0.980942i \(-0.437756\pi\)
0.194301 + 0.980942i \(0.437756\pi\)
\(54\) 0 0
\(55\) −60.0114 −0.147126
\(56\) 0 0
\(57\) 0 0
\(58\) 428.582 0.970269
\(59\) 712.934 1.57315 0.786577 0.617493i \(-0.211851\pi\)
0.786577 + 0.617493i \(0.211851\pi\)
\(60\) 0 0
\(61\) −65.0347 −0.136506 −0.0682528 0.997668i \(-0.521742\pi\)
−0.0682528 + 0.997668i \(0.521742\pi\)
\(62\) −1496.45 −3.06531
\(63\) 0 0
\(64\) −489.068 −0.955211
\(65\) 234.770 0.447994
\(66\) 0 0
\(67\) 698.725 1.27407 0.637036 0.770834i \(-0.280160\pi\)
0.637036 + 0.770834i \(0.280160\pi\)
\(68\) −720.565 −1.28502
\(69\) 0 0
\(70\) 0 0
\(71\) −291.400 −0.487082 −0.243541 0.969891i \(-0.578309\pi\)
−0.243541 + 0.969891i \(0.578309\pi\)
\(72\) 0 0
\(73\) −476.241 −0.763560 −0.381780 0.924253i \(-0.624689\pi\)
−0.381780 + 0.924253i \(0.624689\pi\)
\(74\) −741.849 −1.16538
\(75\) 0 0
\(76\) 511.580 0.772135
\(77\) 0 0
\(78\) 0 0
\(79\) 256.947 0.365935 0.182967 0.983119i \(-0.441430\pi\)
0.182967 + 0.983119i \(0.441430\pi\)
\(80\) −334.126 −0.466955
\(81\) 0 0
\(82\) −1389.43 −1.87118
\(83\) −1177.61 −1.55735 −0.778673 0.627430i \(-0.784107\pi\)
−0.778673 + 0.627430i \(0.784107\pi\)
\(84\) 0 0
\(85\) 223.540 0.285251
\(86\) −703.129 −0.881632
\(87\) 0 0
\(88\) 478.443 0.579570
\(89\) −1009.96 −1.20287 −0.601433 0.798923i \(-0.705403\pi\)
−0.601433 + 0.798923i \(0.705403\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −635.145 −0.719765
\(93\) 0 0
\(94\) −436.104 −0.478518
\(95\) −158.707 −0.171400
\(96\) 0 0
\(97\) 1149.16 1.20288 0.601441 0.798917i \(-0.294593\pi\)
0.601441 + 0.798917i \(0.294593\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 402.929 0.402929
\(101\) −288.883 −0.284604 −0.142302 0.989823i \(-0.545450\pi\)
−0.142302 + 0.989823i \(0.545450\pi\)
\(102\) 0 0
\(103\) −1113.21 −1.06493 −0.532464 0.846453i \(-0.678734\pi\)
−0.532464 + 0.846453i \(0.678734\pi\)
\(104\) −1871.71 −1.76477
\(105\) 0 0
\(106\) 736.345 0.674718
\(107\) 263.840 0.238377 0.119189 0.992872i \(-0.461971\pi\)
0.119189 + 0.992872i \(0.461971\pi\)
\(108\) 0 0
\(109\) −1246.96 −1.09575 −0.547877 0.836559i \(-0.684564\pi\)
−0.547877 + 0.836559i \(0.684564\pi\)
\(110\) −294.711 −0.255451
\(111\) 0 0
\(112\) 0 0
\(113\) −2034.45 −1.69367 −0.846835 0.531855i \(-0.821495\pi\)
−0.846835 + 0.531855i \(0.821495\pi\)
\(114\) 0 0
\(115\) 197.040 0.159775
\(116\) 1406.56 1.12583
\(117\) 0 0
\(118\) 3501.16 2.73142
\(119\) 0 0
\(120\) 0 0
\(121\) −1186.95 −0.891769
\(122\) −319.380 −0.237011
\(123\) 0 0
\(124\) −4911.19 −3.55676
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 1143.07 0.798670 0.399335 0.916805i \(-0.369241\pi\)
0.399335 + 0.916805i \(0.369241\pi\)
\(128\) −2475.95 −1.70973
\(129\) 0 0
\(130\) 1152.94 0.777841
\(131\) −1544.70 −1.03024 −0.515118 0.857119i \(-0.672252\pi\)
−0.515118 + 0.857119i \(0.672252\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 3431.38 2.21214
\(135\) 0 0
\(136\) −1782.18 −1.12368
\(137\) 243.730 0.151995 0.0759974 0.997108i \(-0.475786\pi\)
0.0759974 + 0.997108i \(0.475786\pi\)
\(138\) 0 0
\(139\) 624.857 0.381293 0.190646 0.981659i \(-0.438942\pi\)
0.190646 + 0.981659i \(0.438942\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1431.04 −0.845707
\(143\) −563.555 −0.329559
\(144\) 0 0
\(145\) −436.356 −0.249913
\(146\) −2338.78 −1.32575
\(147\) 0 0
\(148\) −2434.67 −1.35222
\(149\) 3082.32 1.69472 0.847359 0.531020i \(-0.178191\pi\)
0.847359 + 0.531020i \(0.178191\pi\)
\(150\) 0 0
\(151\) −1840.16 −0.991720 −0.495860 0.868402i \(-0.665147\pi\)
−0.495860 + 0.868402i \(0.665147\pi\)
\(152\) 1265.30 0.675191
\(153\) 0 0
\(154\) 0 0
\(155\) 1523.59 0.789535
\(156\) 0 0
\(157\) −129.576 −0.0658679 −0.0329339 0.999458i \(-0.510485\pi\)
−0.0329339 + 0.999458i \(0.510485\pi\)
\(158\) 1261.85 0.635362
\(159\) 0 0
\(160\) −46.3591 −0.0229063
\(161\) 0 0
\(162\) 0 0
\(163\) 3575.75 1.71825 0.859124 0.511768i \(-0.171009\pi\)
0.859124 + 0.511768i \(0.171009\pi\)
\(164\) −4559.97 −2.17118
\(165\) 0 0
\(166\) −5783.16 −2.70398
\(167\) 2054.63 0.952049 0.476025 0.879432i \(-0.342077\pi\)
0.476025 + 0.879432i \(0.342077\pi\)
\(168\) 0 0
\(169\) 7.67889 0.00349517
\(170\) 1097.79 0.495273
\(171\) 0 0
\(172\) −2307.60 −1.02298
\(173\) 3279.38 1.44120 0.720598 0.693353i \(-0.243867\pi\)
0.720598 + 0.693353i \(0.243867\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 802.054 0.343506
\(177\) 0 0
\(178\) −4959.81 −2.08850
\(179\) −3287.59 −1.37277 −0.686385 0.727239i \(-0.740803\pi\)
−0.686385 + 0.727239i \(0.740803\pi\)
\(180\) 0 0
\(181\) 2709.27 1.11259 0.556293 0.830986i \(-0.312223\pi\)
0.556293 + 0.830986i \(0.312223\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1570.91 −0.629396
\(185\) 755.306 0.300169
\(186\) 0 0
\(187\) −536.598 −0.209839
\(188\) −1431.25 −0.555237
\(189\) 0 0
\(190\) −779.397 −0.297597
\(191\) −2106.69 −0.798087 −0.399044 0.916932i \(-0.630658\pi\)
−0.399044 + 0.916932i \(0.630658\pi\)
\(192\) 0 0
\(193\) 3956.67 1.47569 0.737844 0.674972i \(-0.235844\pi\)
0.737844 + 0.674972i \(0.235844\pi\)
\(194\) 5643.44 2.08853
\(195\) 0 0
\(196\) 0 0
\(197\) −1892.16 −0.684319 −0.342160 0.939642i \(-0.611158\pi\)
−0.342160 + 0.939642i \(0.611158\pi\)
\(198\) 0 0
\(199\) −4422.23 −1.57529 −0.787646 0.616128i \(-0.788701\pi\)
−0.787646 + 0.616128i \(0.788701\pi\)
\(200\) 996.566 0.352339
\(201\) 0 0
\(202\) −1418.68 −0.494150
\(203\) 0 0
\(204\) 0 0
\(205\) 1414.63 0.481963
\(206\) −5466.87 −1.84901
\(207\) 0 0
\(208\) −3137.71 −1.04597
\(209\) 380.969 0.126087
\(210\) 0 0
\(211\) 1402.47 0.457584 0.228792 0.973475i \(-0.426522\pi\)
0.228792 + 0.973475i \(0.426522\pi\)
\(212\) 2416.61 0.782893
\(213\) 0 0
\(214\) 1295.70 0.413888
\(215\) 715.883 0.227083
\(216\) 0 0
\(217\) 0 0
\(218\) −6123.73 −1.90253
\(219\) 0 0
\(220\) −967.213 −0.296407
\(221\) 2099.22 0.638954
\(222\) 0 0
\(223\) −1111.09 −0.333651 −0.166826 0.985986i \(-0.553352\pi\)
−0.166826 + 0.985986i \(0.553352\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −9991.01 −2.94067
\(227\) −1715.41 −0.501567 −0.250784 0.968043i \(-0.580688\pi\)
−0.250784 + 0.968043i \(0.580688\pi\)
\(228\) 0 0
\(229\) −934.139 −0.269562 −0.134781 0.990875i \(-0.543033\pi\)
−0.134781 + 0.990875i \(0.543033\pi\)
\(230\) 967.648 0.277412
\(231\) 0 0
\(232\) 3478.86 0.984477
\(233\) −3585.12 −1.00802 −0.504010 0.863698i \(-0.668143\pi\)
−0.504010 + 0.863698i \(0.668143\pi\)
\(234\) 0 0
\(235\) 444.014 0.123252
\(236\) 11490.5 3.16934
\(237\) 0 0
\(238\) 0 0
\(239\) 5944.92 1.60897 0.804487 0.593970i \(-0.202440\pi\)
0.804487 + 0.593970i \(0.202440\pi\)
\(240\) 0 0
\(241\) −5606.87 −1.49863 −0.749316 0.662212i \(-0.769618\pi\)
−0.749316 + 0.662212i \(0.769618\pi\)
\(242\) −5828.99 −1.54835
\(243\) 0 0
\(244\) −1048.17 −0.275010
\(245\) 0 0
\(246\) 0 0
\(247\) −1490.39 −0.383931
\(248\) −12146.9 −3.11019
\(249\) 0 0
\(250\) −613.865 −0.155297
\(251\) 5182.08 1.30315 0.651574 0.758585i \(-0.274109\pi\)
0.651574 + 0.758585i \(0.274109\pi\)
\(252\) 0 0
\(253\) −472.986 −0.117535
\(254\) 5613.52 1.38671
\(255\) 0 0
\(256\) −8246.64 −2.01334
\(257\) −37.7581 −0.00916453 −0.00458226 0.999990i \(-0.501459\pi\)
−0.00458226 + 0.999990i \(0.501459\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 3783.82 0.902549
\(261\) 0 0
\(262\) −7585.90 −1.78877
\(263\) 1147.11 0.268950 0.134475 0.990917i \(-0.457065\pi\)
0.134475 + 0.990917i \(0.457065\pi\)
\(264\) 0 0
\(265\) −749.701 −0.173788
\(266\) 0 0
\(267\) 0 0
\(268\) 11261.5 2.56680
\(269\) −5417.07 −1.22782 −0.613912 0.789374i \(-0.710405\pi\)
−0.613912 + 0.789374i \(0.710405\pi\)
\(270\) 0 0
\(271\) 6108.76 1.36930 0.684651 0.728871i \(-0.259955\pi\)
0.684651 + 0.728871i \(0.259955\pi\)
\(272\) −2987.62 −0.665996
\(273\) 0 0
\(274\) 1196.94 0.263904
\(275\) 300.057 0.0657968
\(276\) 0 0
\(277\) −7247.04 −1.57196 −0.785979 0.618253i \(-0.787841\pi\)
−0.785979 + 0.618253i \(0.787841\pi\)
\(278\) 3068.62 0.662028
\(279\) 0 0
\(280\) 0 0
\(281\) −563.644 −0.119659 −0.0598295 0.998209i \(-0.519056\pi\)
−0.0598295 + 0.998209i \(0.519056\pi\)
\(282\) 0 0
\(283\) 49.6198 0.0104226 0.00521129 0.999986i \(-0.498341\pi\)
0.00521129 + 0.999986i \(0.498341\pi\)
\(284\) −4696.54 −0.981296
\(285\) 0 0
\(286\) −2767.58 −0.572204
\(287\) 0 0
\(288\) 0 0
\(289\) −2914.20 −0.593160
\(290\) −2142.91 −0.433918
\(291\) 0 0
\(292\) −7675.65 −1.53830
\(293\) 5681.57 1.13284 0.566418 0.824118i \(-0.308329\pi\)
0.566418 + 0.824118i \(0.308329\pi\)
\(294\) 0 0
\(295\) −3564.67 −0.703536
\(296\) −6021.70 −1.18245
\(297\) 0 0
\(298\) 15137.0 2.94249
\(299\) 1850.36 0.357891
\(300\) 0 0
\(301\) 0 0
\(302\) −9036.86 −1.72190
\(303\) 0 0
\(304\) 2121.12 0.400180
\(305\) 325.173 0.0610471
\(306\) 0 0
\(307\) 9112.83 1.69413 0.847063 0.531493i \(-0.178369\pi\)
0.847063 + 0.531493i \(0.178369\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 7482.24 1.37085
\(311\) −2520.08 −0.459487 −0.229743 0.973251i \(-0.573789\pi\)
−0.229743 + 0.973251i \(0.573789\pi\)
\(312\) 0 0
\(313\) 1041.28 0.188040 0.0940200 0.995570i \(-0.470028\pi\)
0.0940200 + 0.995570i \(0.470028\pi\)
\(314\) −636.335 −0.114365
\(315\) 0 0
\(316\) 4141.26 0.737228
\(317\) −1294.04 −0.229277 −0.114638 0.993407i \(-0.536571\pi\)
−0.114638 + 0.993407i \(0.536571\pi\)
\(318\) 0 0
\(319\) 1047.45 0.183844
\(320\) 2445.34 0.427183
\(321\) 0 0
\(322\) 0 0
\(323\) −1419.09 −0.244460
\(324\) 0 0
\(325\) −1173.85 −0.200349
\(326\) 17560.2 2.98335
\(327\) 0 0
\(328\) −11278.2 −1.89858
\(329\) 0 0
\(330\) 0 0
\(331\) −7053.87 −1.17135 −0.585674 0.810547i \(-0.699170\pi\)
−0.585674 + 0.810547i \(0.699170\pi\)
\(332\) −18979.7 −3.13750
\(333\) 0 0
\(334\) 10090.1 1.65302
\(335\) −3493.63 −0.569783
\(336\) 0 0
\(337\) −1616.01 −0.261216 −0.130608 0.991434i \(-0.541693\pi\)
−0.130608 + 0.991434i \(0.541693\pi\)
\(338\) 37.7104 0.00606857
\(339\) 0 0
\(340\) 3602.83 0.574678
\(341\) −3657.32 −0.580806
\(342\) 0 0
\(343\) 0 0
\(344\) −5707.40 −0.894541
\(345\) 0 0
\(346\) 16104.8 2.50231
\(347\) −2320.46 −0.358987 −0.179494 0.983759i \(-0.557446\pi\)
−0.179494 + 0.983759i \(0.557446\pi\)
\(348\) 0 0
\(349\) 8883.26 1.36249 0.681247 0.732054i \(-0.261438\pi\)
0.681247 + 0.732054i \(0.261438\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 111.283 0.0168506
\(353\) 7213.82 1.08769 0.543843 0.839187i \(-0.316969\pi\)
0.543843 + 0.839187i \(0.316969\pi\)
\(354\) 0 0
\(355\) 1457.00 0.217830
\(356\) −16277.6 −2.42334
\(357\) 0 0
\(358\) −16145.1 −2.38350
\(359\) 571.681 0.0840451 0.0420225 0.999117i \(-0.486620\pi\)
0.0420225 + 0.999117i \(0.486620\pi\)
\(360\) 0 0
\(361\) −5851.48 −0.853110
\(362\) 13305.0 1.93175
\(363\) 0 0
\(364\) 0 0
\(365\) 2381.21 0.341474
\(366\) 0 0
\(367\) 7595.02 1.08026 0.540132 0.841580i \(-0.318374\pi\)
0.540132 + 0.841580i \(0.318374\pi\)
\(368\) −2633.45 −0.373038
\(369\) 0 0
\(370\) 3709.25 0.521174
\(371\) 0 0
\(372\) 0 0
\(373\) 12739.0 1.76837 0.884184 0.467139i \(-0.154715\pi\)
0.884184 + 0.467139i \(0.154715\pi\)
\(374\) −2635.19 −0.364338
\(375\) 0 0
\(376\) −3539.92 −0.485525
\(377\) −4097.74 −0.559799
\(378\) 0 0
\(379\) 2726.44 0.369519 0.184759 0.982784i \(-0.440849\pi\)
0.184759 + 0.982784i \(0.440849\pi\)
\(380\) −2557.90 −0.345309
\(381\) 0 0
\(382\) −10345.8 −1.38570
\(383\) 44.4732 0.00593336 0.00296668 0.999996i \(-0.499056\pi\)
0.00296668 + 0.999996i \(0.499056\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 19430.9 2.56220
\(387\) 0 0
\(388\) 18521.2 2.42338
\(389\) −2160.33 −0.281576 −0.140788 0.990040i \(-0.544964\pi\)
−0.140788 + 0.990040i \(0.544964\pi\)
\(390\) 0 0
\(391\) 1761.85 0.227879
\(392\) 0 0
\(393\) 0 0
\(394\) −9292.25 −1.18816
\(395\) −1284.74 −0.163651
\(396\) 0 0
\(397\) 10915.6 1.37994 0.689970 0.723838i \(-0.257624\pi\)
0.689970 + 0.723838i \(0.257624\pi\)
\(398\) −21717.2 −2.73514
\(399\) 0 0
\(400\) 1670.63 0.208829
\(401\) −7119.55 −0.886617 −0.443308 0.896369i \(-0.646195\pi\)
−0.443308 + 0.896369i \(0.646195\pi\)
\(402\) 0 0
\(403\) 14307.8 1.76854
\(404\) −4655.97 −0.573375
\(405\) 0 0
\(406\) 0 0
\(407\) −1813.08 −0.220813
\(408\) 0 0
\(409\) 10435.9 1.26166 0.630832 0.775920i \(-0.282714\pi\)
0.630832 + 0.775920i \(0.282714\pi\)
\(410\) 6947.16 0.836819
\(411\) 0 0
\(412\) −17941.7 −2.14545
\(413\) 0 0
\(414\) 0 0
\(415\) 5888.06 0.696467
\(416\) −435.349 −0.0513094
\(417\) 0 0
\(418\) 1870.91 0.218921
\(419\) −6763.41 −0.788579 −0.394289 0.918986i \(-0.629009\pi\)
−0.394289 + 0.918986i \(0.629009\pi\)
\(420\) 0 0
\(421\) 12708.5 1.47119 0.735597 0.677419i \(-0.236902\pi\)
0.735597 + 0.677419i \(0.236902\pi\)
\(422\) 6887.44 0.794491
\(423\) 0 0
\(424\) 5977.01 0.684598
\(425\) −1117.70 −0.127568
\(426\) 0 0
\(427\) 0 0
\(428\) 4252.34 0.480245
\(429\) 0 0
\(430\) 3515.64 0.394278
\(431\) 8478.80 0.947585 0.473793 0.880636i \(-0.342885\pi\)
0.473793 + 0.880636i \(0.342885\pi\)
\(432\) 0 0
\(433\) 13442.6 1.49194 0.745968 0.665982i \(-0.231987\pi\)
0.745968 + 0.665982i \(0.231987\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −20097.4 −2.20755
\(437\) −1250.87 −0.136927
\(438\) 0 0
\(439\) 11890.4 1.29270 0.646352 0.763040i \(-0.276294\pi\)
0.646352 + 0.763040i \(0.276294\pi\)
\(440\) −2392.21 −0.259192
\(441\) 0 0
\(442\) 10309.1 1.10940
\(443\) 11118.4 1.19245 0.596223 0.802819i \(-0.296667\pi\)
0.596223 + 0.802819i \(0.296667\pi\)
\(444\) 0 0
\(445\) 5049.78 0.537938
\(446\) −5456.48 −0.579309
\(447\) 0 0
\(448\) 0 0
\(449\) −14077.5 −1.47964 −0.739818 0.672807i \(-0.765088\pi\)
−0.739818 + 0.672807i \(0.765088\pi\)
\(450\) 0 0
\(451\) −3395.77 −0.354547
\(452\) −32789.5 −3.41214
\(453\) 0 0
\(454\) −8424.24 −0.870858
\(455\) 0 0
\(456\) 0 0
\(457\) −6557.10 −0.671178 −0.335589 0.942009i \(-0.608935\pi\)
−0.335589 + 0.942009i \(0.608935\pi\)
\(458\) −4587.48 −0.468033
\(459\) 0 0
\(460\) 3175.72 0.321889
\(461\) −5157.65 −0.521075 −0.260538 0.965464i \(-0.583900\pi\)
−0.260538 + 0.965464i \(0.583900\pi\)
\(462\) 0 0
\(463\) −6069.13 −0.609193 −0.304596 0.952482i \(-0.598522\pi\)
−0.304596 + 0.952482i \(0.598522\pi\)
\(464\) 5831.91 0.583491
\(465\) 0 0
\(466\) −17606.2 −1.75020
\(467\) −16351.0 −1.62020 −0.810101 0.586290i \(-0.800588\pi\)
−0.810101 + 0.586290i \(0.800588\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 2180.52 0.214000
\(471\) 0 0
\(472\) 28419.4 2.77142
\(473\) −1718.45 −0.167049
\(474\) 0 0
\(475\) 793.535 0.0766523
\(476\) 0 0
\(477\) 0 0
\(478\) 29195.0 2.79362
\(479\) 9345.33 0.891439 0.445719 0.895173i \(-0.352948\pi\)
0.445719 + 0.895173i \(0.352948\pi\)
\(480\) 0 0
\(481\) 7092.93 0.672369
\(482\) −27534.9 −2.60204
\(483\) 0 0
\(484\) −19130.2 −1.79660
\(485\) −5745.81 −0.537946
\(486\) 0 0
\(487\) −648.574 −0.0603485 −0.0301742 0.999545i \(-0.509606\pi\)
−0.0301742 + 0.999545i \(0.509606\pi\)
\(488\) −2592.45 −0.240481
\(489\) 0 0
\(490\) 0 0
\(491\) 16109.9 1.48071 0.740354 0.672217i \(-0.234658\pi\)
0.740354 + 0.672217i \(0.234658\pi\)
\(492\) 0 0
\(493\) −3901.72 −0.356440
\(494\) −7319.16 −0.666609
\(495\) 0 0
\(496\) −20362.8 −1.84338
\(497\) 0 0
\(498\) 0 0
\(499\) 6357.21 0.570316 0.285158 0.958481i \(-0.407954\pi\)
0.285158 + 0.958481i \(0.407954\pi\)
\(500\) −2014.64 −0.180195
\(501\) 0 0
\(502\) 25448.8 2.26262
\(503\) −3100.43 −0.274833 −0.137417 0.990513i \(-0.543880\pi\)
−0.137417 + 0.990513i \(0.543880\pi\)
\(504\) 0 0
\(505\) 1444.42 0.127279
\(506\) −2322.80 −0.204073
\(507\) 0 0
\(508\) 18423.0 1.60903
\(509\) −16682.7 −1.45275 −0.726374 0.687300i \(-0.758796\pi\)
−0.726374 + 0.687300i \(0.758796\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −20691.0 −1.78598
\(513\) 0 0
\(514\) −185.427 −0.0159121
\(515\) 5566.04 0.476250
\(516\) 0 0
\(517\) −1065.84 −0.0906682
\(518\) 0 0
\(519\) 0 0
\(520\) 9358.55 0.789230
\(521\) −11268.9 −0.947597 −0.473798 0.880633i \(-0.657117\pi\)
−0.473798 + 0.880633i \(0.657117\pi\)
\(522\) 0 0
\(523\) 21181.0 1.77090 0.885448 0.464738i \(-0.153852\pi\)
0.885448 + 0.464738i \(0.153852\pi\)
\(524\) −24896.1 −2.07556
\(525\) 0 0
\(526\) 5633.37 0.466970
\(527\) 13623.3 1.12608
\(528\) 0 0
\(529\) −10614.0 −0.872360
\(530\) −3681.72 −0.301743
\(531\) 0 0
\(532\) 0 0
\(533\) 13284.5 1.07958
\(534\) 0 0
\(535\) −1319.20 −0.106606
\(536\) 27853.0 2.24453
\(537\) 0 0
\(538\) −26602.8 −2.13184
\(539\) 0 0
\(540\) 0 0
\(541\) −13392.8 −1.06433 −0.532163 0.846642i \(-0.678621\pi\)
−0.532163 + 0.846642i \(0.678621\pi\)
\(542\) 29999.7 2.37748
\(543\) 0 0
\(544\) −414.524 −0.0326702
\(545\) 6234.81 0.490036
\(546\) 0 0
\(547\) 14270.9 1.11550 0.557752 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(548\) 3928.23 0.306215
\(549\) 0 0
\(550\) 1473.56 0.114241
\(551\) 2770.11 0.214175
\(552\) 0 0
\(553\) 0 0
\(554\) −35589.6 −2.72935
\(555\) 0 0
\(556\) 10070.9 0.768169
\(557\) −12713.6 −0.967130 −0.483565 0.875309i \(-0.660658\pi\)
−0.483565 + 0.875309i \(0.660658\pi\)
\(558\) 0 0
\(559\) 6722.71 0.508659
\(560\) 0 0
\(561\) 0 0
\(562\) −2768.01 −0.207761
\(563\) −16455.5 −1.23183 −0.615913 0.787814i \(-0.711213\pi\)
−0.615913 + 0.787814i \(0.711213\pi\)
\(564\) 0 0
\(565\) 10172.2 0.757433
\(566\) 243.679 0.0180964
\(567\) 0 0
\(568\) −11616.0 −0.858091
\(569\) −22665.7 −1.66994 −0.834968 0.550298i \(-0.814514\pi\)
−0.834968 + 0.550298i \(0.814514\pi\)
\(570\) 0 0
\(571\) −12436.3 −0.911460 −0.455730 0.890118i \(-0.650622\pi\)
−0.455730 + 0.890118i \(0.650622\pi\)
\(572\) −9082.90 −0.663943
\(573\) 0 0
\(574\) 0 0
\(575\) −985.201 −0.0714534
\(576\) 0 0
\(577\) −15784.4 −1.13884 −0.569421 0.822046i \(-0.692832\pi\)
−0.569421 + 0.822046i \(0.692832\pi\)
\(578\) −14311.4 −1.02989
\(579\) 0 0
\(580\) −7032.82 −0.503486
\(581\) 0 0
\(582\) 0 0
\(583\) 1799.63 0.127844
\(584\) −18984.2 −1.34516
\(585\) 0 0
\(586\) 27901.7 1.96691
\(587\) 5373.31 0.377820 0.188910 0.981994i \(-0.439505\pi\)
0.188910 + 0.981994i \(0.439505\pi\)
\(588\) 0 0
\(589\) −9672.18 −0.676631
\(590\) −17505.8 −1.22153
\(591\) 0 0
\(592\) −10094.7 −0.700826
\(593\) −19885.4 −1.37706 −0.688530 0.725208i \(-0.741744\pi\)
−0.688530 + 0.725208i \(0.741744\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 49678.1 3.41425
\(597\) 0 0
\(598\) 9086.99 0.621396
\(599\) 5568.32 0.379825 0.189913 0.981801i \(-0.439180\pi\)
0.189913 + 0.981801i \(0.439180\pi\)
\(600\) 0 0
\(601\) −5796.37 −0.393409 −0.196704 0.980463i \(-0.563024\pi\)
−0.196704 + 0.980463i \(0.563024\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −29658.0 −1.99796
\(605\) 5934.73 0.398811
\(606\) 0 0
\(607\) 9808.47 0.655871 0.327935 0.944700i \(-0.393647\pi\)
0.327935 + 0.944700i \(0.393647\pi\)
\(608\) 294.300 0.0196307
\(609\) 0 0
\(610\) 1596.90 0.105994
\(611\) 4169.65 0.276082
\(612\) 0 0
\(613\) 2745.09 0.180870 0.0904350 0.995902i \(-0.471174\pi\)
0.0904350 + 0.995902i \(0.471174\pi\)
\(614\) 44752.4 2.94146
\(615\) 0 0
\(616\) 0 0
\(617\) −15727.9 −1.02623 −0.513114 0.858321i \(-0.671508\pi\)
−0.513114 + 0.858321i \(0.671508\pi\)
\(618\) 0 0
\(619\) −13005.2 −0.844462 −0.422231 0.906488i \(-0.638753\pi\)
−0.422231 + 0.906488i \(0.638753\pi\)
\(620\) 24555.9 1.59063
\(621\) 0 0
\(622\) −12375.9 −0.797794
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 5113.64 0.326489
\(627\) 0 0
\(628\) −2088.39 −0.132700
\(629\) 6753.64 0.428116
\(630\) 0 0
\(631\) −10019.2 −0.632102 −0.316051 0.948742i \(-0.602357\pi\)
−0.316051 + 0.948742i \(0.602357\pi\)
\(632\) 10242.6 0.644666
\(633\) 0 0
\(634\) −6354.95 −0.398087
\(635\) −5715.35 −0.357176
\(636\) 0 0
\(637\) 0 0
\(638\) 5143.97 0.319203
\(639\) 0 0
\(640\) 12379.7 0.764613
\(641\) 28621.9 1.76365 0.881824 0.471579i \(-0.156316\pi\)
0.881824 + 0.471579i \(0.156316\pi\)
\(642\) 0 0
\(643\) 21100.2 1.29411 0.647054 0.762444i \(-0.276001\pi\)
0.647054 + 0.762444i \(0.276001\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −6969.06 −0.424449
\(647\) −3060.93 −0.185993 −0.0929965 0.995666i \(-0.529645\pi\)
−0.0929965 + 0.995666i \(0.529645\pi\)
\(648\) 0 0
\(649\) 8556.83 0.517543
\(650\) −5764.69 −0.347861
\(651\) 0 0
\(652\) 57630.9 3.46165
\(653\) 23083.9 1.38337 0.691687 0.722197i \(-0.256868\pi\)
0.691687 + 0.722197i \(0.256868\pi\)
\(654\) 0 0
\(655\) 7723.50 0.460736
\(656\) −18906.6 −1.12527
\(657\) 0 0
\(658\) 0 0
\(659\) −19482.0 −1.15161 −0.575806 0.817586i \(-0.695312\pi\)
−0.575806 + 0.817586i \(0.695312\pi\)
\(660\) 0 0
\(661\) −2297.05 −0.135166 −0.0675831 0.997714i \(-0.521529\pi\)
−0.0675831 + 0.997714i \(0.521529\pi\)
\(662\) −34641.0 −2.03378
\(663\) 0 0
\(664\) −46942.7 −2.74357
\(665\) 0 0
\(666\) 0 0
\(667\) −3439.19 −0.199649
\(668\) 33114.8 1.91804
\(669\) 0 0
\(670\) −17156.9 −0.989298
\(671\) −780.565 −0.0449081
\(672\) 0 0
\(673\) −17610.4 −1.00866 −0.504332 0.863510i \(-0.668261\pi\)
−0.504332 + 0.863510i \(0.668261\pi\)
\(674\) −7936.10 −0.453542
\(675\) 0 0
\(676\) 123.762 0.00704152
\(677\) 25472.5 1.44607 0.723033 0.690814i \(-0.242748\pi\)
0.723033 + 0.690814i \(0.242748\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 8910.89 0.502525
\(681\) 0 0
\(682\) −17960.8 −1.00844
\(683\) −6821.30 −0.382152 −0.191076 0.981575i \(-0.561198\pi\)
−0.191076 + 0.981575i \(0.561198\pi\)
\(684\) 0 0
\(685\) −1218.65 −0.0679741
\(686\) 0 0
\(687\) 0 0
\(688\) −9567.79 −0.530187
\(689\) −7040.29 −0.389280
\(690\) 0 0
\(691\) 16765.6 0.923003 0.461502 0.887139i \(-0.347311\pi\)
0.461502 + 0.887139i \(0.347311\pi\)
\(692\) 52854.3 2.90350
\(693\) 0 0
\(694\) −11395.6 −0.623300
\(695\) −3124.29 −0.170519
\(696\) 0 0
\(697\) 12649.1 0.687401
\(698\) 43625.0 2.36566
\(699\) 0 0
\(700\) 0 0
\(701\) 14137.8 0.761736 0.380868 0.924629i \(-0.375625\pi\)
0.380868 + 0.924629i \(0.375625\pi\)
\(702\) 0 0
\(703\) −4794.89 −0.257244
\(704\) −5869.93 −0.314249
\(705\) 0 0
\(706\) 35426.5 1.88852
\(707\) 0 0
\(708\) 0 0
\(709\) 2483.11 0.131531 0.0657654 0.997835i \(-0.479051\pi\)
0.0657654 + 0.997835i \(0.479051\pi\)
\(710\) 7155.21 0.378212
\(711\) 0 0
\(712\) −40259.5 −2.11908
\(713\) 12008.4 0.630738
\(714\) 0 0
\(715\) 2817.78 0.147383
\(716\) −52986.5 −2.76564
\(717\) 0 0
\(718\) 2807.48 0.145925
\(719\) −9074.07 −0.470662 −0.235331 0.971915i \(-0.575617\pi\)
−0.235331 + 0.971915i \(0.575617\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −28736.2 −1.48123
\(723\) 0 0
\(724\) 43665.6 2.24146
\(725\) 2181.78 0.111765
\(726\) 0 0
\(727\) 27924.6 1.42457 0.712286 0.701889i \(-0.247660\pi\)
0.712286 + 0.701889i \(0.247660\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 11693.9 0.592892
\(731\) 6401.14 0.323878
\(732\) 0 0
\(733\) 11177.6 0.563238 0.281619 0.959526i \(-0.409129\pi\)
0.281619 + 0.959526i \(0.409129\pi\)
\(734\) 37298.5 1.87563
\(735\) 0 0
\(736\) −365.384 −0.0182992
\(737\) 8386.30 0.419150
\(738\) 0 0
\(739\) 12147.5 0.604671 0.302335 0.953202i \(-0.402234\pi\)
0.302335 + 0.953202i \(0.402234\pi\)
\(740\) 12173.4 0.604732
\(741\) 0 0
\(742\) 0 0
\(743\) 8190.00 0.404390 0.202195 0.979345i \(-0.435192\pi\)
0.202195 + 0.979345i \(0.435192\pi\)
\(744\) 0 0
\(745\) −15411.6 −0.757901
\(746\) 62560.3 3.07037
\(747\) 0 0
\(748\) −8648.43 −0.422751
\(749\) 0 0
\(750\) 0 0
\(751\) −11143.0 −0.541432 −0.270716 0.962659i \(-0.587260\pi\)
−0.270716 + 0.962659i \(0.587260\pi\)
\(752\) −5934.26 −0.287766
\(753\) 0 0
\(754\) −20123.7 −0.971963
\(755\) 9200.78 0.443511
\(756\) 0 0
\(757\) −20072.7 −0.963746 −0.481873 0.876241i \(-0.660043\pi\)
−0.481873 + 0.876241i \(0.660043\pi\)
\(758\) 13389.3 0.641585
\(759\) 0 0
\(760\) −6326.48 −0.301955
\(761\) 5593.19 0.266430 0.133215 0.991087i \(-0.457470\pi\)
0.133215 + 0.991087i \(0.457470\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −33953.8 −1.60786
\(765\) 0 0
\(766\) 218.405 0.0103019
\(767\) −33475.1 −1.57590
\(768\) 0 0
\(769\) 35021.4 1.64227 0.821133 0.570737i \(-0.193342\pi\)
0.821133 + 0.570737i \(0.193342\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 63770.3 2.97298
\(773\) −36389.0 −1.69317 −0.846585 0.532254i \(-0.821345\pi\)
−0.846585 + 0.532254i \(0.821345\pi\)
\(774\) 0 0
\(775\) −7617.96 −0.353091
\(776\) 45808.6 2.11911
\(777\) 0 0
\(778\) −10609.2 −0.488893
\(779\) −8980.49 −0.413042
\(780\) 0 0
\(781\) −3497.47 −0.160242
\(782\) 8652.32 0.395660
\(783\) 0 0
\(784\) 0 0
\(785\) 647.878 0.0294570
\(786\) 0 0
\(787\) 23542.2 1.06632 0.533158 0.846016i \(-0.321005\pi\)
0.533158 + 0.846016i \(0.321005\pi\)
\(788\) −30496.2 −1.37866
\(789\) 0 0
\(790\) −6309.24 −0.284143
\(791\) 0 0
\(792\) 0 0
\(793\) 3053.64 0.136744
\(794\) 53605.4 2.39595
\(795\) 0 0
\(796\) −71273.6 −3.17365
\(797\) 9662.12 0.429423 0.214711 0.976678i \(-0.431119\pi\)
0.214711 + 0.976678i \(0.431119\pi\)
\(798\) 0 0
\(799\) 3970.20 0.175789
\(800\) 231.795 0.0102440
\(801\) 0 0
\(802\) −34963.5 −1.53941
\(803\) −5715.99 −0.251199
\(804\) 0 0
\(805\) 0 0
\(806\) 70264.2 3.07066
\(807\) 0 0
\(808\) −11515.7 −0.501385
\(809\) −24010.3 −1.04346 −0.521728 0.853112i \(-0.674713\pi\)
−0.521728 + 0.853112i \(0.674713\pi\)
\(810\) 0 0
\(811\) 13412.7 0.580746 0.290373 0.956913i \(-0.406221\pi\)
0.290373 + 0.956913i \(0.406221\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −8903.89 −0.383392
\(815\) −17878.7 −0.768423
\(816\) 0 0
\(817\) −4544.62 −0.194610
\(818\) 51249.7 2.19059
\(819\) 0 0
\(820\) 22799.9 0.970983
\(821\) 17462.1 0.742304 0.371152 0.928572i \(-0.378963\pi\)
0.371152 + 0.928572i \(0.378963\pi\)
\(822\) 0 0
\(823\) −14905.4 −0.631311 −0.315655 0.948874i \(-0.602224\pi\)
−0.315655 + 0.948874i \(0.602224\pi\)
\(824\) −44375.4 −1.87608
\(825\) 0 0
\(826\) 0 0
\(827\) −29123.5 −1.22458 −0.612288 0.790635i \(-0.709751\pi\)
−0.612288 + 0.790635i \(0.709751\pi\)
\(828\) 0 0
\(829\) 9272.75 0.388487 0.194244 0.980953i \(-0.437775\pi\)
0.194244 + 0.980953i \(0.437775\pi\)
\(830\) 28915.8 1.20926
\(831\) 0 0
\(832\) 22963.7 0.956879
\(833\) 0 0
\(834\) 0 0
\(835\) −10273.2 −0.425769
\(836\) 6140.13 0.254020
\(837\) 0 0
\(838\) −33214.6 −1.36919
\(839\) 20794.4 0.855664 0.427832 0.903858i \(-0.359277\pi\)
0.427832 + 0.903858i \(0.359277\pi\)
\(840\) 0 0
\(841\) −16772.7 −0.687717
\(842\) 62410.3 2.55439
\(843\) 0 0
\(844\) 22603.9 0.921869
\(845\) −38.3944 −0.00156309
\(846\) 0 0
\(847\) 0 0
\(848\) 10019.8 0.405755
\(849\) 0 0
\(850\) −5488.94 −0.221493
\(851\) 5953.02 0.239797
\(852\) 0 0
\(853\) 31567.2 1.26711 0.633553 0.773699i \(-0.281596\pi\)
0.633553 + 0.773699i \(0.281596\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 10517.4 0.419948
\(857\) −21907.7 −0.873223 −0.436612 0.899650i \(-0.643822\pi\)
−0.436612 + 0.899650i \(0.643822\pi\)
\(858\) 0 0
\(859\) −12141.1 −0.482247 −0.241124 0.970494i \(-0.577516\pi\)
−0.241124 + 0.970494i \(0.577516\pi\)
\(860\) 11538.0 0.457491
\(861\) 0 0
\(862\) 41638.7 1.64527
\(863\) 7598.88 0.299732 0.149866 0.988706i \(-0.452116\pi\)
0.149866 + 0.988706i \(0.452116\pi\)
\(864\) 0 0
\(865\) −16396.9 −0.644523
\(866\) 66015.4 2.59041
\(867\) 0 0
\(868\) 0 0
\(869\) 3083.96 0.120387
\(870\) 0 0
\(871\) −32807.9 −1.27630
\(872\) −49707.2 −1.93039
\(873\) 0 0
\(874\) −6142.90 −0.237742
\(875\) 0 0
\(876\) 0 0
\(877\) 5560.67 0.214105 0.107053 0.994253i \(-0.465859\pi\)
0.107053 + 0.994253i \(0.465859\pi\)
\(878\) 58392.7 2.24449
\(879\) 0 0
\(880\) −4010.27 −0.153621
\(881\) 37010.1 1.41532 0.707662 0.706551i \(-0.249750\pi\)
0.707662 + 0.706551i \(0.249750\pi\)
\(882\) 0 0
\(883\) 38430.3 1.46465 0.732323 0.680958i \(-0.238436\pi\)
0.732323 + 0.680958i \(0.238436\pi\)
\(884\) 33833.4 1.28726
\(885\) 0 0
\(886\) 54601.8 2.07041
\(887\) −21019.9 −0.795694 −0.397847 0.917452i \(-0.630242\pi\)
−0.397847 + 0.917452i \(0.630242\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 24799.1 0.934007
\(891\) 0 0
\(892\) −17907.6 −0.672188
\(893\) −2818.73 −0.105627
\(894\) 0 0
\(895\) 16437.9 0.613921
\(896\) 0 0
\(897\) 0 0
\(898\) −69133.3 −2.56905
\(899\) −26593.2 −0.986576
\(900\) 0 0
\(901\) −6703.53 −0.247866
\(902\) −16676.4 −0.615590
\(903\) 0 0
\(904\) −81098.5 −2.98373
\(905\) −13546.3 −0.497564
\(906\) 0 0
\(907\) −14566.9 −0.533281 −0.266641 0.963796i \(-0.585914\pi\)
−0.266641 + 0.963796i \(0.585914\pi\)
\(908\) −27647.5 −1.01048
\(909\) 0 0
\(910\) 0 0
\(911\) 329.028 0.0119662 0.00598309 0.999982i \(-0.498096\pi\)
0.00598309 + 0.999982i \(0.498096\pi\)
\(912\) 0 0
\(913\) −14134.0 −0.512342
\(914\) −32201.4 −1.16535
\(915\) 0 0
\(916\) −15055.6 −0.543070
\(917\) 0 0
\(918\) 0 0
\(919\) −26414.9 −0.948147 −0.474073 0.880485i \(-0.657217\pi\)
−0.474073 + 0.880485i \(0.657217\pi\)
\(920\) 7854.54 0.281474
\(921\) 0 0
\(922\) −25328.8 −0.904729
\(923\) 13682.4 0.487932
\(924\) 0 0
\(925\) −3776.53 −0.134239
\(926\) −29805.0 −1.05772
\(927\) 0 0
\(928\) 809.163 0.0286229
\(929\) 18244.8 0.644342 0.322171 0.946682i \(-0.395587\pi\)
0.322171 + 0.946682i \(0.395587\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −57781.8 −2.03080
\(933\) 0 0
\(934\) −80298.5 −2.81311
\(935\) 2682.99 0.0938430
\(936\) 0 0
\(937\) −140.951 −0.00491427 −0.00245713 0.999997i \(-0.500782\pi\)
−0.00245713 + 0.999997i \(0.500782\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 7156.24 0.248309
\(941\) 21674.8 0.750879 0.375440 0.926847i \(-0.377492\pi\)
0.375440 + 0.926847i \(0.377492\pi\)
\(942\) 0 0
\(943\) 11149.6 0.385027
\(944\) 47641.9 1.64260
\(945\) 0 0
\(946\) −8439.15 −0.290043
\(947\) −22150.2 −0.760069 −0.380035 0.924972i \(-0.624088\pi\)
−0.380035 + 0.924972i \(0.624088\pi\)
\(948\) 0 0
\(949\) 22361.4 0.764893
\(950\) 3896.99 0.133089
\(951\) 0 0
\(952\) 0 0
\(953\) −15980.9 −0.543203 −0.271602 0.962410i \(-0.587553\pi\)
−0.271602 + 0.962410i \(0.587553\pi\)
\(954\) 0 0
\(955\) 10533.4 0.356916
\(956\) 95815.1 3.24151
\(957\) 0 0
\(958\) 45894.2 1.54778
\(959\) 0 0
\(960\) 0 0
\(961\) 63062.3 2.11682
\(962\) 34832.8 1.16742
\(963\) 0 0
\(964\) −90366.8 −3.01921
\(965\) −19783.4 −0.659947
\(966\) 0 0
\(967\) −13804.2 −0.459064 −0.229532 0.973301i \(-0.573720\pi\)
−0.229532 + 0.973301i \(0.573720\pi\)
\(968\) −47314.8 −1.57103
\(969\) 0 0
\(970\) −28217.2 −0.934020
\(971\) −24806.1 −0.819841 −0.409921 0.912121i \(-0.634444\pi\)
−0.409921 + 0.912121i \(0.634444\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −3185.10 −0.104781
\(975\) 0 0
\(976\) −4345.95 −0.142531
\(977\) 2069.03 0.0677524 0.0338762 0.999426i \(-0.489215\pi\)
0.0338762 + 0.999426i \(0.489215\pi\)
\(978\) 0 0
\(979\) −12121.8 −0.395724
\(980\) 0 0
\(981\) 0 0
\(982\) 79114.2 2.57091
\(983\) 22474.1 0.729210 0.364605 0.931162i \(-0.381204\pi\)
0.364605 + 0.931162i \(0.381204\pi\)
\(984\) 0 0
\(985\) 9460.80 0.306037
\(986\) −19161.1 −0.618877
\(987\) 0 0
\(988\) −24020.7 −0.773484
\(989\) 5642.31 0.181410
\(990\) 0 0
\(991\) 40924.6 1.31182 0.655910 0.754839i \(-0.272285\pi\)
0.655910 + 0.754839i \(0.272285\pi\)
\(992\) −2825.29 −0.0904265
\(993\) 0 0
\(994\) 0 0
\(995\) 22111.1 0.704492
\(996\) 0 0
\(997\) −60402.8 −1.91873 −0.959366 0.282164i \(-0.908948\pi\)
−0.959366 + 0.282164i \(0.908948\pi\)
\(998\) 31219.7 0.990224
\(999\) 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 2205.4.a.bx.1.6 yes 6
3.2 odd 2 2205.4.a.by.1.1 yes 6
7.6 odd 2 2205.4.a.by.1.6 yes 6
21.20 even 2 inner 2205.4.a.bx.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2205.4.a.bx.1.1 6 21.20 even 2 inner
2205.4.a.bx.1.6 yes 6 1.1 even 1 trivial
2205.4.a.by.1.1 yes 6 3.2 odd 2
2205.4.a.by.1.6 yes 6 7.6 odd 2