Properties

Label 1856.4.a.bg.1.4
Level $1856$
Weight $4$
Character 1856.1
Self dual yes
Analytic conductor $109.508$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.507544971\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - 4x^{8} - 138x^{7} + 394x^{6} + 5872x^{5} - 10822x^{4} - 85158x^{3} + 30654x^{2} + 439999x + 396802 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 928)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.95108\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.95108 q^{3} -18.0628 q^{5} -22.0180 q^{7} -23.1933 q^{9} +O(q^{10})\) \(q-1.95108 q^{3} -18.0628 q^{5} -22.0180 q^{7} -23.1933 q^{9} +11.1881 q^{11} +0.532875 q^{13} +35.2419 q^{15} +56.2274 q^{17} +26.5240 q^{19} +42.9588 q^{21} -66.9071 q^{23} +201.265 q^{25} +97.9310 q^{27} +29.0000 q^{29} -85.2968 q^{31} -21.8288 q^{33} +397.706 q^{35} +180.274 q^{37} -1.03968 q^{39} +46.4366 q^{41} +212.454 q^{43} +418.936 q^{45} -108.280 q^{47} +141.792 q^{49} -109.704 q^{51} +444.412 q^{53} -202.088 q^{55} -51.7503 q^{57} +43.3659 q^{59} -33.1252 q^{61} +510.670 q^{63} -9.62522 q^{65} +988.140 q^{67} +130.541 q^{69} +154.560 q^{71} -1030.98 q^{73} -392.683 q^{75} -246.339 q^{77} +631.568 q^{79} +435.148 q^{81} -145.157 q^{83} -1015.62 q^{85} -56.5812 q^{87} +169.477 q^{89} -11.7328 q^{91} +166.421 q^{93} -479.097 q^{95} -29.4573 q^{97} -259.489 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9} + 64 q^{11} - 70 q^{13} - 170 q^{15} - 66 q^{17} + 42 q^{19} - 76 q^{21} - 40 q^{23} + 111 q^{25} + 322 q^{27} + 261 q^{29} + 64 q^{31} - 52 q^{33} + 496 q^{35} + 54 q^{37} - 590 q^{39} - 378 q^{41} - 32 q^{43} - 1046 q^{45} - 1164 q^{47} - 351 q^{49} + 376 q^{51} - 278 q^{53} - 614 q^{55} + 28 q^{57} + 640 q^{59} - 1054 q^{61} - 1660 q^{63} - 708 q^{65} + 1184 q^{67} - 188 q^{69} - 1988 q^{71} - 750 q^{73} + 3126 q^{75} - 1260 q^{77} - 2916 q^{79} + 293 q^{81} + 2832 q^{83} - 56 q^{85} + 116 q^{87} - 370 q^{89} + 3016 q^{91} + 1696 q^{93} - 4412 q^{95} - 2234 q^{97} + 4118 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.95108 −0.375485 −0.187743 0.982218i \(-0.560117\pi\)
−0.187743 + 0.982218i \(0.560117\pi\)
\(4\) 0 0
\(5\) −18.0628 −1.61559 −0.807793 0.589466i \(-0.799338\pi\)
−0.807793 + 0.589466i \(0.799338\pi\)
\(6\) 0 0
\(7\) −22.0180 −1.18886 −0.594430 0.804148i \(-0.702622\pi\)
−0.594430 + 0.804148i \(0.702622\pi\)
\(8\) 0 0
\(9\) −23.1933 −0.859011
\(10\) 0 0
\(11\) 11.1881 0.306667 0.153334 0.988174i \(-0.450999\pi\)
0.153334 + 0.988174i \(0.450999\pi\)
\(12\) 0 0
\(13\) 0.532875 0.0113687 0.00568435 0.999984i \(-0.498191\pi\)
0.00568435 + 0.999984i \(0.498191\pi\)
\(14\) 0 0
\(15\) 35.2419 0.606628
\(16\) 0 0
\(17\) 56.2274 0.802185 0.401092 0.916038i \(-0.368631\pi\)
0.401092 + 0.916038i \(0.368631\pi\)
\(18\) 0 0
\(19\) 26.5240 0.320264 0.160132 0.987096i \(-0.448808\pi\)
0.160132 + 0.987096i \(0.448808\pi\)
\(20\) 0 0
\(21\) 42.9588 0.446399
\(22\) 0 0
\(23\) −66.9071 −0.606570 −0.303285 0.952900i \(-0.598083\pi\)
−0.303285 + 0.952900i \(0.598083\pi\)
\(24\) 0 0
\(25\) 201.265 1.61012
\(26\) 0 0
\(27\) 97.9310 0.698031
\(28\) 0 0
\(29\) 29.0000 0.185695
\(30\) 0 0
\(31\) −85.2968 −0.494186 −0.247093 0.968992i \(-0.579475\pi\)
−0.247093 + 0.968992i \(0.579475\pi\)
\(32\) 0 0
\(33\) −21.8288 −0.115149
\(34\) 0 0
\(35\) 397.706 1.92070
\(36\) 0 0
\(37\) 180.274 0.800995 0.400498 0.916298i \(-0.368837\pi\)
0.400498 + 0.916298i \(0.368837\pi\)
\(38\) 0 0
\(39\) −1.03968 −0.00426877
\(40\) 0 0
\(41\) 46.4366 0.176882 0.0884412 0.996081i \(-0.471811\pi\)
0.0884412 + 0.996081i \(0.471811\pi\)
\(42\) 0 0
\(43\) 212.454 0.753464 0.376732 0.926322i \(-0.377048\pi\)
0.376732 + 0.926322i \(0.377048\pi\)
\(44\) 0 0
\(45\) 418.936 1.38781
\(46\) 0 0
\(47\) −108.280 −0.336048 −0.168024 0.985783i \(-0.553739\pi\)
−0.168024 + 0.985783i \(0.553739\pi\)
\(48\) 0 0
\(49\) 141.792 0.413386
\(50\) 0 0
\(51\) −109.704 −0.301208
\(52\) 0 0
\(53\) 444.412 1.15179 0.575893 0.817525i \(-0.304655\pi\)
0.575893 + 0.817525i \(0.304655\pi\)
\(54\) 0 0
\(55\) −202.088 −0.495447
\(56\) 0 0
\(57\) −51.7503 −0.120254
\(58\) 0 0
\(59\) 43.3659 0.0956908 0.0478454 0.998855i \(-0.484765\pi\)
0.0478454 + 0.998855i \(0.484765\pi\)
\(60\) 0 0
\(61\) −33.1252 −0.0695286 −0.0347643 0.999396i \(-0.511068\pi\)
−0.0347643 + 0.999396i \(0.511068\pi\)
\(62\) 0 0
\(63\) 510.670 1.02124
\(64\) 0 0
\(65\) −9.62522 −0.0183671
\(66\) 0 0
\(67\) 988.140 1.80180 0.900899 0.434028i \(-0.142908\pi\)
0.900899 + 0.434028i \(0.142908\pi\)
\(68\) 0 0
\(69\) 130.541 0.227758
\(70\) 0 0
\(71\) 154.560 0.258350 0.129175 0.991622i \(-0.458767\pi\)
0.129175 + 0.991622i \(0.458767\pi\)
\(72\) 0 0
\(73\) −1030.98 −1.65297 −0.826484 0.562961i \(-0.809662\pi\)
−0.826484 + 0.562961i \(0.809662\pi\)
\(74\) 0 0
\(75\) −392.683 −0.604575
\(76\) 0 0
\(77\) −246.339 −0.364584
\(78\) 0 0
\(79\) 631.568 0.899455 0.449728 0.893166i \(-0.351521\pi\)
0.449728 + 0.893166i \(0.351521\pi\)
\(80\) 0 0
\(81\) 435.148 0.596911
\(82\) 0 0
\(83\) −145.157 −0.191965 −0.0959824 0.995383i \(-0.530599\pi\)
−0.0959824 + 0.995383i \(0.530599\pi\)
\(84\) 0 0
\(85\) −1015.62 −1.29600
\(86\) 0 0
\(87\) −56.5812 −0.0697258
\(88\) 0 0
\(89\) 169.477 0.201849 0.100924 0.994894i \(-0.467820\pi\)
0.100924 + 0.994894i \(0.467820\pi\)
\(90\) 0 0
\(91\) −11.7328 −0.0135158
\(92\) 0 0
\(93\) 166.421 0.185559
\(94\) 0 0
\(95\) −479.097 −0.517414
\(96\) 0 0
\(97\) −29.4573 −0.0308344 −0.0154172 0.999881i \(-0.504908\pi\)
−0.0154172 + 0.999881i \(0.504908\pi\)
\(98\) 0 0
\(99\) −259.489 −0.263430
\(100\) 0 0
\(101\) −887.027 −0.873886 −0.436943 0.899489i \(-0.643939\pi\)
−0.436943 + 0.899489i \(0.643939\pi\)
\(102\) 0 0
\(103\) −778.724 −0.744951 −0.372475 0.928042i \(-0.621491\pi\)
−0.372475 + 0.928042i \(0.621491\pi\)
\(104\) 0 0
\(105\) −775.956 −0.721196
\(106\) 0 0
\(107\) −1837.16 −1.65986 −0.829930 0.557868i \(-0.811620\pi\)
−0.829930 + 0.557868i \(0.811620\pi\)
\(108\) 0 0
\(109\) −232.609 −0.204403 −0.102201 0.994764i \(-0.532589\pi\)
−0.102201 + 0.994764i \(0.532589\pi\)
\(110\) 0 0
\(111\) −351.728 −0.300762
\(112\) 0 0
\(113\) 1873.74 1.55988 0.779939 0.625855i \(-0.215250\pi\)
0.779939 + 0.625855i \(0.215250\pi\)
\(114\) 0 0
\(115\) 1208.53 0.979966
\(116\) 0 0
\(117\) −12.3591 −0.00976583
\(118\) 0 0
\(119\) −1238.01 −0.953685
\(120\) 0 0
\(121\) −1205.83 −0.905955
\(122\) 0 0
\(123\) −90.6014 −0.0664167
\(124\) 0 0
\(125\) −1377.56 −0.985699
\(126\) 0 0
\(127\) −1672.06 −1.16828 −0.584141 0.811652i \(-0.698568\pi\)
−0.584141 + 0.811652i \(0.698568\pi\)
\(128\) 0 0
\(129\) −414.515 −0.282915
\(130\) 0 0
\(131\) 2611.20 1.74154 0.870769 0.491693i \(-0.163622\pi\)
0.870769 + 0.491693i \(0.163622\pi\)
\(132\) 0 0
\(133\) −584.004 −0.380749
\(134\) 0 0
\(135\) −1768.91 −1.12773
\(136\) 0 0
\(137\) −1506.03 −0.939186 −0.469593 0.882883i \(-0.655599\pi\)
−0.469593 + 0.882883i \(0.655599\pi\)
\(138\) 0 0
\(139\) 2818.87 1.72009 0.860047 0.510214i \(-0.170434\pi\)
0.860047 + 0.510214i \(0.170434\pi\)
\(140\) 0 0
\(141\) 211.263 0.126181
\(142\) 0 0
\(143\) 5.96186 0.00348640
\(144\) 0 0
\(145\) −523.821 −0.300007
\(146\) 0 0
\(147\) −276.646 −0.155220
\(148\) 0 0
\(149\) −1672.65 −0.919658 −0.459829 0.888007i \(-0.652089\pi\)
−0.459829 + 0.888007i \(0.652089\pi\)
\(150\) 0 0
\(151\) −1065.84 −0.574417 −0.287209 0.957868i \(-0.592727\pi\)
−0.287209 + 0.957868i \(0.592727\pi\)
\(152\) 0 0
\(153\) −1304.10 −0.689085
\(154\) 0 0
\(155\) 1540.70 0.798400
\(156\) 0 0
\(157\) −2554.74 −1.29866 −0.649332 0.760505i \(-0.724952\pi\)
−0.649332 + 0.760505i \(0.724952\pi\)
\(158\) 0 0
\(159\) −867.081 −0.432478
\(160\) 0 0
\(161\) 1473.16 0.721126
\(162\) 0 0
\(163\) 1345.04 0.646327 0.323164 0.946343i \(-0.395254\pi\)
0.323164 + 0.946343i \(0.395254\pi\)
\(164\) 0 0
\(165\) 394.290 0.186033
\(166\) 0 0
\(167\) 801.061 0.371185 0.185593 0.982627i \(-0.440579\pi\)
0.185593 + 0.982627i \(0.440579\pi\)
\(168\) 0 0
\(169\) −2196.72 −0.999871
\(170\) 0 0
\(171\) −615.178 −0.275110
\(172\) 0 0
\(173\) −1999.30 −0.878635 −0.439318 0.898332i \(-0.644780\pi\)
−0.439318 + 0.898332i \(0.644780\pi\)
\(174\) 0 0
\(175\) −4431.45 −1.91420
\(176\) 0 0
\(177\) −84.6102 −0.0359305
\(178\) 0 0
\(179\) 1183.93 0.494362 0.247181 0.968969i \(-0.420496\pi\)
0.247181 + 0.968969i \(0.420496\pi\)
\(180\) 0 0
\(181\) 1164.04 0.478025 0.239013 0.971016i \(-0.423176\pi\)
0.239013 + 0.971016i \(0.423176\pi\)
\(182\) 0 0
\(183\) 64.6297 0.0261069
\(184\) 0 0
\(185\) −3256.25 −1.29408
\(186\) 0 0
\(187\) 629.077 0.246004
\(188\) 0 0
\(189\) −2156.24 −0.829860
\(190\) 0 0
\(191\) 1613.60 0.611288 0.305644 0.952146i \(-0.401128\pi\)
0.305644 + 0.952146i \(0.401128\pi\)
\(192\) 0 0
\(193\) −1048.74 −0.391139 −0.195570 0.980690i \(-0.562656\pi\)
−0.195570 + 0.980690i \(0.562656\pi\)
\(194\) 0 0
\(195\) 18.7795 0.00689657
\(196\) 0 0
\(197\) 1487.86 0.538099 0.269050 0.963126i \(-0.413290\pi\)
0.269050 + 0.963126i \(0.413290\pi\)
\(198\) 0 0
\(199\) 297.754 0.106067 0.0530333 0.998593i \(-0.483111\pi\)
0.0530333 + 0.998593i \(0.483111\pi\)
\(200\) 0 0
\(201\) −1927.94 −0.676549
\(202\) 0 0
\(203\) −638.521 −0.220766
\(204\) 0 0
\(205\) −838.775 −0.285769
\(206\) 0 0
\(207\) 1551.80 0.521050
\(208\) 0 0
\(209\) 296.753 0.0982145
\(210\) 0 0
\(211\) 3947.96 1.28810 0.644049 0.764984i \(-0.277253\pi\)
0.644049 + 0.764984i \(0.277253\pi\)
\(212\) 0 0
\(213\) −301.558 −0.0970065
\(214\) 0 0
\(215\) −3837.52 −1.21729
\(216\) 0 0
\(217\) 1878.06 0.587517
\(218\) 0 0
\(219\) 2011.51 0.620664
\(220\) 0 0
\(221\) 29.9622 0.00911979
\(222\) 0 0
\(223\) 3790.90 1.13837 0.569187 0.822208i \(-0.307258\pi\)
0.569187 + 0.822208i \(0.307258\pi\)
\(224\) 0 0
\(225\) −4668.00 −1.38311
\(226\) 0 0
\(227\) −1952.76 −0.570966 −0.285483 0.958384i \(-0.592154\pi\)
−0.285483 + 0.958384i \(0.592154\pi\)
\(228\) 0 0
\(229\) 878.288 0.253445 0.126723 0.991938i \(-0.459554\pi\)
0.126723 + 0.991938i \(0.459554\pi\)
\(230\) 0 0
\(231\) 480.627 0.136896
\(232\) 0 0
\(233\) −2556.16 −0.718711 −0.359355 0.933201i \(-0.617003\pi\)
−0.359355 + 0.933201i \(0.617003\pi\)
\(234\) 0 0
\(235\) 1955.84 0.542914
\(236\) 0 0
\(237\) −1232.24 −0.337732
\(238\) 0 0
\(239\) 13.0160 0.00352274 0.00176137 0.999998i \(-0.499439\pi\)
0.00176137 + 0.999998i \(0.499439\pi\)
\(240\) 0 0
\(241\) 1987.42 0.531207 0.265603 0.964082i \(-0.414429\pi\)
0.265603 + 0.964082i \(0.414429\pi\)
\(242\) 0 0
\(243\) −3493.14 −0.922162
\(244\) 0 0
\(245\) −2561.15 −0.667861
\(246\) 0 0
\(247\) 14.1340 0.00364098
\(248\) 0 0
\(249\) 283.213 0.0720799
\(250\) 0 0
\(251\) −5553.49 −1.39655 −0.698273 0.715831i \(-0.746048\pi\)
−0.698273 + 0.715831i \(0.746048\pi\)
\(252\) 0 0
\(253\) −748.564 −0.186015
\(254\) 0 0
\(255\) 1981.56 0.486628
\(256\) 0 0
\(257\) −1089.82 −0.264518 −0.132259 0.991215i \(-0.542223\pi\)
−0.132259 + 0.991215i \(0.542223\pi\)
\(258\) 0 0
\(259\) −3969.27 −0.952271
\(260\) 0 0
\(261\) −672.606 −0.159514
\(262\) 0 0
\(263\) 6808.83 1.59639 0.798195 0.602399i \(-0.205788\pi\)
0.798195 + 0.602399i \(0.205788\pi\)
\(264\) 0 0
\(265\) −8027.32 −1.86081
\(266\) 0 0
\(267\) −330.663 −0.0757912
\(268\) 0 0
\(269\) −4978.43 −1.12840 −0.564201 0.825637i \(-0.690816\pi\)
−0.564201 + 0.825637i \(0.690816\pi\)
\(270\) 0 0
\(271\) −231.445 −0.0518793 −0.0259397 0.999664i \(-0.508258\pi\)
−0.0259397 + 0.999664i \(0.508258\pi\)
\(272\) 0 0
\(273\) 22.8917 0.00507497
\(274\) 0 0
\(275\) 2251.77 0.493771
\(276\) 0 0
\(277\) −2395.60 −0.519630 −0.259815 0.965658i \(-0.583662\pi\)
−0.259815 + 0.965658i \(0.583662\pi\)
\(278\) 0 0
\(279\) 1978.31 0.424511
\(280\) 0 0
\(281\) 7597.72 1.61296 0.806480 0.591261i \(-0.201370\pi\)
0.806480 + 0.591261i \(0.201370\pi\)
\(282\) 0 0
\(283\) −5203.18 −1.09292 −0.546461 0.837484i \(-0.684026\pi\)
−0.546461 + 0.837484i \(0.684026\pi\)
\(284\) 0 0
\(285\) 934.756 0.194281
\(286\) 0 0
\(287\) −1022.44 −0.210288
\(288\) 0 0
\(289\) −1751.48 −0.356500
\(290\) 0 0
\(291\) 57.4735 0.0115779
\(292\) 0 0
\(293\) 6628.25 1.32159 0.660796 0.750566i \(-0.270219\pi\)
0.660796 + 0.750566i \(0.270219\pi\)
\(294\) 0 0
\(295\) −783.309 −0.154597
\(296\) 0 0
\(297\) 1095.66 0.214063
\(298\) 0 0
\(299\) −35.6532 −0.00689590
\(300\) 0 0
\(301\) −4677.81 −0.895763
\(302\) 0 0
\(303\) 1730.66 0.328131
\(304\) 0 0
\(305\) 598.333 0.112329
\(306\) 0 0
\(307\) 8142.39 1.51371 0.756857 0.653580i \(-0.226734\pi\)
0.756857 + 0.653580i \(0.226734\pi\)
\(308\) 0 0
\(309\) 1519.35 0.279718
\(310\) 0 0
\(311\) 1020.13 0.186000 0.0930000 0.995666i \(-0.470354\pi\)
0.0930000 + 0.995666i \(0.470354\pi\)
\(312\) 0 0
\(313\) −6853.77 −1.23769 −0.618846 0.785512i \(-0.712400\pi\)
−0.618846 + 0.785512i \(0.712400\pi\)
\(314\) 0 0
\(315\) −9224.12 −1.64991
\(316\) 0 0
\(317\) −7542.08 −1.33630 −0.668148 0.744029i \(-0.732913\pi\)
−0.668148 + 0.744029i \(0.732913\pi\)
\(318\) 0 0
\(319\) 324.455 0.0569467
\(320\) 0 0
\(321\) 3584.44 0.623252
\(322\) 0 0
\(323\) 1491.37 0.256911
\(324\) 0 0
\(325\) 107.249 0.0183049
\(326\) 0 0
\(327\) 453.838 0.0767501
\(328\) 0 0
\(329\) 2384.11 0.399514
\(330\) 0 0
\(331\) −849.318 −0.141035 −0.0705177 0.997511i \(-0.522465\pi\)
−0.0705177 + 0.997511i \(0.522465\pi\)
\(332\) 0 0
\(333\) −4181.14 −0.688064
\(334\) 0 0
\(335\) −17848.6 −2.91096
\(336\) 0 0
\(337\) 8174.38 1.32133 0.660663 0.750682i \(-0.270275\pi\)
0.660663 + 0.750682i \(0.270275\pi\)
\(338\) 0 0
\(339\) −3655.81 −0.585711
\(340\) 0 0
\(341\) −954.309 −0.151551
\(342\) 0 0
\(343\) 4430.20 0.697401
\(344\) 0 0
\(345\) −2357.94 −0.367962
\(346\) 0 0
\(347\) 2726.57 0.421816 0.210908 0.977506i \(-0.432358\pi\)
0.210908 + 0.977506i \(0.432358\pi\)
\(348\) 0 0
\(349\) 7409.06 1.13638 0.568191 0.822896i \(-0.307643\pi\)
0.568191 + 0.822896i \(0.307643\pi\)
\(350\) 0 0
\(351\) 52.1850 0.00793570
\(352\) 0 0
\(353\) −5535.82 −0.834679 −0.417340 0.908751i \(-0.637037\pi\)
−0.417340 + 0.908751i \(0.637037\pi\)
\(354\) 0 0
\(355\) −2791.78 −0.417387
\(356\) 0 0
\(357\) 2415.46 0.358094
\(358\) 0 0
\(359\) −3735.97 −0.549239 −0.274619 0.961553i \(-0.588552\pi\)
−0.274619 + 0.961553i \(0.588552\pi\)
\(360\) 0 0
\(361\) −6155.48 −0.897431
\(362\) 0 0
\(363\) 2352.66 0.340173
\(364\) 0 0
\(365\) 18622.3 2.67051
\(366\) 0 0
\(367\) 11094.8 1.57805 0.789027 0.614359i \(-0.210585\pi\)
0.789027 + 0.614359i \(0.210585\pi\)
\(368\) 0 0
\(369\) −1077.02 −0.151944
\(370\) 0 0
\(371\) −9785.05 −1.36931
\(372\) 0 0
\(373\) −4753.81 −0.659901 −0.329951 0.943998i \(-0.607032\pi\)
−0.329951 + 0.943998i \(0.607032\pi\)
\(374\) 0 0
\(375\) 2687.72 0.370115
\(376\) 0 0
\(377\) 15.4534 0.00211111
\(378\) 0 0
\(379\) 6164.88 0.835537 0.417769 0.908553i \(-0.362812\pi\)
0.417769 + 0.908553i \(0.362812\pi\)
\(380\) 0 0
\(381\) 3262.33 0.438672
\(382\) 0 0
\(383\) 2734.38 0.364805 0.182402 0.983224i \(-0.441613\pi\)
0.182402 + 0.983224i \(0.441613\pi\)
\(384\) 0 0
\(385\) 4449.58 0.589017
\(386\) 0 0
\(387\) −4927.51 −0.647234
\(388\) 0 0
\(389\) 2349.93 0.306289 0.153144 0.988204i \(-0.451060\pi\)
0.153144 + 0.988204i \(0.451060\pi\)
\(390\) 0 0
\(391\) −3762.01 −0.486581
\(392\) 0 0
\(393\) −5094.65 −0.653921
\(394\) 0 0
\(395\) −11407.9 −1.45315
\(396\) 0 0
\(397\) −5065.74 −0.640409 −0.320204 0.947348i \(-0.603752\pi\)
−0.320204 + 0.947348i \(0.603752\pi\)
\(398\) 0 0
\(399\) 1139.44 0.142965
\(400\) 0 0
\(401\) 5031.88 0.626634 0.313317 0.949649i \(-0.398560\pi\)
0.313317 + 0.949649i \(0.398560\pi\)
\(402\) 0 0
\(403\) −45.4526 −0.00561825
\(404\) 0 0
\(405\) −7859.99 −0.964361
\(406\) 0 0
\(407\) 2016.92 0.245639
\(408\) 0 0
\(409\) −8107.90 −0.980220 −0.490110 0.871661i \(-0.663043\pi\)
−0.490110 + 0.871661i \(0.663043\pi\)
\(410\) 0 0
\(411\) 2938.37 0.352650
\(412\) 0 0
\(413\) −954.829 −0.113763
\(414\) 0 0
\(415\) 2621.95 0.310136
\(416\) 0 0
\(417\) −5499.83 −0.645870
\(418\) 0 0
\(419\) −12818.4 −1.49456 −0.747279 0.664510i \(-0.768640\pi\)
−0.747279 + 0.664510i \(0.768640\pi\)
\(420\) 0 0
\(421\) −6405.87 −0.741575 −0.370788 0.928718i \(-0.620912\pi\)
−0.370788 + 0.928718i \(0.620912\pi\)
\(422\) 0 0
\(423\) 2511.37 0.288669
\(424\) 0 0
\(425\) 11316.6 1.29161
\(426\) 0 0
\(427\) 729.349 0.0826597
\(428\) 0 0
\(429\) −11.6321 −0.00130909
\(430\) 0 0
\(431\) 1873.09 0.209336 0.104668 0.994507i \(-0.466622\pi\)
0.104668 + 0.994507i \(0.466622\pi\)
\(432\) 0 0
\(433\) 492.001 0.0546052 0.0273026 0.999627i \(-0.491308\pi\)
0.0273026 + 0.999627i \(0.491308\pi\)
\(434\) 0 0
\(435\) 1022.02 0.112648
\(436\) 0 0
\(437\) −1774.64 −0.194262
\(438\) 0 0
\(439\) −9655.65 −1.04975 −0.524874 0.851180i \(-0.675887\pi\)
−0.524874 + 0.851180i \(0.675887\pi\)
\(440\) 0 0
\(441\) −3288.61 −0.355104
\(442\) 0 0
\(443\) 3021.05 0.324005 0.162003 0.986790i \(-0.448205\pi\)
0.162003 + 0.986790i \(0.448205\pi\)
\(444\) 0 0
\(445\) −3061.23 −0.326104
\(446\) 0 0
\(447\) 3263.47 0.345318
\(448\) 0 0
\(449\) 1335.44 0.140364 0.0701820 0.997534i \(-0.477642\pi\)
0.0701820 + 0.997534i \(0.477642\pi\)
\(450\) 0 0
\(451\) 519.537 0.0542440
\(452\) 0 0
\(453\) 2079.54 0.215685
\(454\) 0 0
\(455\) 211.928 0.0218359
\(456\) 0 0
\(457\) −8984.92 −0.919687 −0.459843 0.888000i \(-0.652094\pi\)
−0.459843 + 0.888000i \(0.652094\pi\)
\(458\) 0 0
\(459\) 5506.40 0.559950
\(460\) 0 0
\(461\) 8468.56 0.855575 0.427788 0.903879i \(-0.359293\pi\)
0.427788 + 0.903879i \(0.359293\pi\)
\(462\) 0 0
\(463\) −13946.0 −1.39983 −0.699917 0.714224i \(-0.746780\pi\)
−0.699917 + 0.714224i \(0.746780\pi\)
\(464\) 0 0
\(465\) −3006.02 −0.299787
\(466\) 0 0
\(467\) −1242.51 −0.123119 −0.0615594 0.998103i \(-0.519607\pi\)
−0.0615594 + 0.998103i \(0.519607\pi\)
\(468\) 0 0
\(469\) −21756.9 −2.14209
\(470\) 0 0
\(471\) 4984.49 0.487629
\(472\) 0 0
\(473\) 2376.96 0.231063
\(474\) 0 0
\(475\) 5338.34 0.515663
\(476\) 0 0
\(477\) −10307.4 −0.989396
\(478\) 0 0
\(479\) −1054.91 −0.100626 −0.0503131 0.998733i \(-0.516022\pi\)
−0.0503131 + 0.998733i \(0.516022\pi\)
\(480\) 0 0
\(481\) 96.0634 0.00910627
\(482\) 0 0
\(483\) −2874.25 −0.270772
\(484\) 0 0
\(485\) 532.082 0.0498156
\(486\) 0 0
\(487\) 8140.23 0.757432 0.378716 0.925513i \(-0.376366\pi\)
0.378716 + 0.925513i \(0.376366\pi\)
\(488\) 0 0
\(489\) −2624.27 −0.242686
\(490\) 0 0
\(491\) −19696.4 −1.81035 −0.905177 0.425034i \(-0.860262\pi\)
−0.905177 + 0.425034i \(0.860262\pi\)
\(492\) 0 0
\(493\) 1630.59 0.148962
\(494\) 0 0
\(495\) 4687.10 0.425595
\(496\) 0 0
\(497\) −3403.09 −0.307142
\(498\) 0 0
\(499\) −3554.81 −0.318908 −0.159454 0.987205i \(-0.550973\pi\)
−0.159454 + 0.987205i \(0.550973\pi\)
\(500\) 0 0
\(501\) −1562.93 −0.139375
\(502\) 0 0
\(503\) 21333.8 1.89111 0.945555 0.325463i \(-0.105520\pi\)
0.945555 + 0.325463i \(0.105520\pi\)
\(504\) 0 0
\(505\) 16022.2 1.41184
\(506\) 0 0
\(507\) 4285.96 0.375436
\(508\) 0 0
\(509\) −11816.7 −1.02901 −0.514506 0.857487i \(-0.672025\pi\)
−0.514506 + 0.857487i \(0.672025\pi\)
\(510\) 0 0
\(511\) 22700.0 1.96515
\(512\) 0 0
\(513\) 2597.52 0.223554
\(514\) 0 0
\(515\) 14065.9 1.20353
\(516\) 0 0
\(517\) −1211.45 −0.103055
\(518\) 0 0
\(519\) 3900.79 0.329914
\(520\) 0 0
\(521\) 12945.2 1.08856 0.544278 0.838905i \(-0.316804\pi\)
0.544278 + 0.838905i \(0.316804\pi\)
\(522\) 0 0
\(523\) 2842.92 0.237690 0.118845 0.992913i \(-0.462081\pi\)
0.118845 + 0.992913i \(0.462081\pi\)
\(524\) 0 0
\(525\) 8646.09 0.718755
\(526\) 0 0
\(527\) −4796.01 −0.396428
\(528\) 0 0
\(529\) −7690.44 −0.632073
\(530\) 0 0
\(531\) −1005.80 −0.0821994
\(532\) 0 0
\(533\) 24.7449 0.00201092
\(534\) 0 0
\(535\) 33184.3 2.68165
\(536\) 0 0
\(537\) −2309.93 −0.185625
\(538\) 0 0
\(539\) 1586.38 0.126772
\(540\) 0 0
\(541\) −564.312 −0.0448459 −0.0224230 0.999749i \(-0.507138\pi\)
−0.0224230 + 0.999749i \(0.507138\pi\)
\(542\) 0 0
\(543\) −2271.14 −0.179491
\(544\) 0 0
\(545\) 4201.57 0.330230
\(546\) 0 0
\(547\) −21063.9 −1.64648 −0.823241 0.567692i \(-0.807836\pi\)
−0.823241 + 0.567692i \(0.807836\pi\)
\(548\) 0 0
\(549\) 768.282 0.0597258
\(550\) 0 0
\(551\) 769.195 0.0594715
\(552\) 0 0
\(553\) −13905.9 −1.06933
\(554\) 0 0
\(555\) 6353.20 0.485907
\(556\) 0 0
\(557\) 23042.5 1.75286 0.876430 0.481530i \(-0.159919\pi\)
0.876430 + 0.481530i \(0.159919\pi\)
\(558\) 0 0
\(559\) 113.212 0.00856590
\(560\) 0 0
\(561\) −1227.38 −0.0923707
\(562\) 0 0
\(563\) 9753.28 0.730109 0.365055 0.930986i \(-0.381050\pi\)
0.365055 + 0.930986i \(0.381050\pi\)
\(564\) 0 0
\(565\) −33844.9 −2.52012
\(566\) 0 0
\(567\) −9581.08 −0.709643
\(568\) 0 0
\(569\) −11380.6 −0.838485 −0.419242 0.907874i \(-0.637704\pi\)
−0.419242 + 0.907874i \(0.637704\pi\)
\(570\) 0 0
\(571\) −12977.3 −0.951109 −0.475555 0.879686i \(-0.657753\pi\)
−0.475555 + 0.879686i \(0.657753\pi\)
\(572\) 0 0
\(573\) −3148.26 −0.229529
\(574\) 0 0
\(575\) −13466.1 −0.976649
\(576\) 0 0
\(577\) 12229.5 0.882359 0.441180 0.897419i \(-0.354560\pi\)
0.441180 + 0.897419i \(0.354560\pi\)
\(578\) 0 0
\(579\) 2046.17 0.146867
\(580\) 0 0
\(581\) 3196.07 0.228219
\(582\) 0 0
\(583\) 4972.12 0.353215
\(584\) 0 0
\(585\) 223.241 0.0157775
\(586\) 0 0
\(587\) 13884.3 0.976263 0.488132 0.872770i \(-0.337679\pi\)
0.488132 + 0.872770i \(0.337679\pi\)
\(588\) 0 0
\(589\) −2262.41 −0.158270
\(590\) 0 0
\(591\) −2902.93 −0.202048
\(592\) 0 0
\(593\) −22356.1 −1.54815 −0.774077 0.633092i \(-0.781786\pi\)
−0.774077 + 0.633092i \(0.781786\pi\)
\(594\) 0 0
\(595\) 22362.0 1.54076
\(596\) 0 0
\(597\) −580.942 −0.0398264
\(598\) 0 0
\(599\) 3966.27 0.270546 0.135273 0.990808i \(-0.456809\pi\)
0.135273 + 0.990808i \(0.456809\pi\)
\(600\) 0 0
\(601\) −17343.0 −1.17710 −0.588550 0.808461i \(-0.700301\pi\)
−0.588550 + 0.808461i \(0.700301\pi\)
\(602\) 0 0
\(603\) −22918.2 −1.54777
\(604\) 0 0
\(605\) 21780.6 1.46365
\(606\) 0 0
\(607\) 17468.7 1.16810 0.584048 0.811719i \(-0.301468\pi\)
0.584048 + 0.811719i \(0.301468\pi\)
\(608\) 0 0
\(609\) 1245.80 0.0828942
\(610\) 0 0
\(611\) −57.6997 −0.00382042
\(612\) 0 0
\(613\) 5605.90 0.369364 0.184682 0.982798i \(-0.440875\pi\)
0.184682 + 0.982798i \(0.440875\pi\)
\(614\) 0 0
\(615\) 1636.52 0.107302
\(616\) 0 0
\(617\) 12413.8 0.809987 0.404993 0.914320i \(-0.367274\pi\)
0.404993 + 0.914320i \(0.367274\pi\)
\(618\) 0 0
\(619\) −24588.8 −1.59662 −0.798310 0.602247i \(-0.794272\pi\)
−0.798310 + 0.602247i \(0.794272\pi\)
\(620\) 0 0
\(621\) −6552.28 −0.423404
\(622\) 0 0
\(623\) −3731.54 −0.239970
\(624\) 0 0
\(625\) −275.570 −0.0176365
\(626\) 0 0
\(627\) −578.988 −0.0368781
\(628\) 0 0
\(629\) 10136.3 0.642546
\(630\) 0 0
\(631\) 15144.9 0.955481 0.477740 0.878501i \(-0.341456\pi\)
0.477740 + 0.878501i \(0.341456\pi\)
\(632\) 0 0
\(633\) −7702.78 −0.483662
\(634\) 0 0
\(635\) 30202.2 1.88746
\(636\) 0 0
\(637\) 75.5572 0.00469966
\(638\) 0 0
\(639\) −3584.75 −0.221925
\(640\) 0 0
\(641\) −1087.26 −0.0669957 −0.0334979 0.999439i \(-0.510665\pi\)
−0.0334979 + 0.999439i \(0.510665\pi\)
\(642\) 0 0
\(643\) −14014.7 −0.859541 −0.429771 0.902938i \(-0.641406\pi\)
−0.429771 + 0.902938i \(0.641406\pi\)
\(644\) 0 0
\(645\) 7487.30 0.457073
\(646\) 0 0
\(647\) −9390.59 −0.570606 −0.285303 0.958437i \(-0.592094\pi\)
−0.285303 + 0.958437i \(0.592094\pi\)
\(648\) 0 0
\(649\) 485.182 0.0293452
\(650\) 0 0
\(651\) −3664.25 −0.220604
\(652\) 0 0
\(653\) 8357.09 0.500824 0.250412 0.968139i \(-0.419434\pi\)
0.250412 + 0.968139i \(0.419434\pi\)
\(654\) 0 0
\(655\) −47165.5 −2.81360
\(656\) 0 0
\(657\) 23911.7 1.41992
\(658\) 0 0
\(659\) 13965.0 0.825489 0.412745 0.910847i \(-0.364570\pi\)
0.412745 + 0.910847i \(0.364570\pi\)
\(660\) 0 0
\(661\) −10501.3 −0.617934 −0.308967 0.951073i \(-0.599983\pi\)
−0.308967 + 0.951073i \(0.599983\pi\)
\(662\) 0 0
\(663\) −58.4585 −0.00342434
\(664\) 0 0
\(665\) 10548.8 0.615133
\(666\) 0 0
\(667\) −1940.31 −0.112637
\(668\) 0 0
\(669\) −7396.34 −0.427443
\(670\) 0 0
\(671\) −370.608 −0.0213221
\(672\) 0 0
\(673\) −30084.5 −1.72314 −0.861569 0.507641i \(-0.830518\pi\)
−0.861569 + 0.507641i \(0.830518\pi\)
\(674\) 0 0
\(675\) 19710.1 1.12391
\(676\) 0 0
\(677\) −14825.2 −0.841621 −0.420810 0.907149i \(-0.638254\pi\)
−0.420810 + 0.907149i \(0.638254\pi\)
\(678\) 0 0
\(679\) 648.591 0.0366578
\(680\) 0 0
\(681\) 3809.99 0.214389
\(682\) 0 0
\(683\) 30850.4 1.72834 0.864171 0.503199i \(-0.167844\pi\)
0.864171 + 0.503199i \(0.167844\pi\)
\(684\) 0 0
\(685\) 27203.1 1.51734
\(686\) 0 0
\(687\) −1713.61 −0.0951648
\(688\) 0 0
\(689\) 236.816 0.0130943
\(690\) 0 0
\(691\) −27669.0 −1.52327 −0.761635 0.648006i \(-0.775603\pi\)
−0.761635 + 0.648006i \(0.775603\pi\)
\(692\) 0 0
\(693\) 5713.42 0.313182
\(694\) 0 0
\(695\) −50916.6 −2.77896
\(696\) 0 0
\(697\) 2611.01 0.141892
\(698\) 0 0
\(699\) 4987.26 0.269865
\(700\) 0 0
\(701\) 18155.9 0.978229 0.489115 0.872220i \(-0.337320\pi\)
0.489115 + 0.872220i \(0.337320\pi\)
\(702\) 0 0
\(703\) 4781.58 0.256530
\(704\) 0 0
\(705\) −3815.99 −0.203856
\(706\) 0 0
\(707\) 19530.5 1.03893
\(708\) 0 0
\(709\) 12971.8 0.687118 0.343559 0.939131i \(-0.388367\pi\)
0.343559 + 0.939131i \(0.388367\pi\)
\(710\) 0 0
\(711\) −14648.1 −0.772642
\(712\) 0 0
\(713\) 5706.97 0.299758
\(714\) 0 0
\(715\) −107.688 −0.00563259
\(716\) 0 0
\(717\) −25.3952 −0.00132274
\(718\) 0 0
\(719\) 11163.6 0.579044 0.289522 0.957171i \(-0.406504\pi\)
0.289522 + 0.957171i \(0.406504\pi\)
\(720\) 0 0
\(721\) 17145.9 0.885642
\(722\) 0 0
\(723\) −3877.60 −0.199460
\(724\) 0 0
\(725\) 5836.68 0.298992
\(726\) 0 0
\(727\) 23149.5 1.18097 0.590487 0.807047i \(-0.298936\pi\)
0.590487 + 0.807047i \(0.298936\pi\)
\(728\) 0 0
\(729\) −4933.60 −0.250653
\(730\) 0 0
\(731\) 11945.7 0.604418
\(732\) 0 0
\(733\) 30271.8 1.52539 0.762697 0.646756i \(-0.223875\pi\)
0.762697 + 0.646756i \(0.223875\pi\)
\(734\) 0 0
\(735\) 4997.01 0.250772
\(736\) 0 0
\(737\) 11055.4 0.552553
\(738\) 0 0
\(739\) 31154.1 1.55077 0.775387 0.631487i \(-0.217555\pi\)
0.775387 + 0.631487i \(0.217555\pi\)
\(740\) 0 0
\(741\) −27.5765 −0.00136713
\(742\) 0 0
\(743\) −32817.8 −1.62041 −0.810207 0.586144i \(-0.800645\pi\)
−0.810207 + 0.586144i \(0.800645\pi\)
\(744\) 0 0
\(745\) 30212.8 1.48579
\(746\) 0 0
\(747\) 3366.68 0.164900
\(748\) 0 0
\(749\) 40450.6 1.97334
\(750\) 0 0
\(751\) −26390.0 −1.28227 −0.641135 0.767428i \(-0.721536\pi\)
−0.641135 + 0.767428i \(0.721536\pi\)
\(752\) 0 0
\(753\) 10835.3 0.524382
\(754\) 0 0
\(755\) 19252.1 0.928021
\(756\) 0 0
\(757\) −2340.44 −0.112371 −0.0561853 0.998420i \(-0.517894\pi\)
−0.0561853 + 0.998420i \(0.517894\pi\)
\(758\) 0 0
\(759\) 1460.51 0.0698458
\(760\) 0 0
\(761\) 34510.4 1.64389 0.821946 0.569565i \(-0.192888\pi\)
0.821946 + 0.569565i \(0.192888\pi\)
\(762\) 0 0
\(763\) 5121.58 0.243006
\(764\) 0 0
\(765\) 23555.7 1.11328
\(766\) 0 0
\(767\) 23.1086 0.00108788
\(768\) 0 0
\(769\) 24619.2 1.15447 0.577237 0.816577i \(-0.304131\pi\)
0.577237 + 0.816577i \(0.304131\pi\)
\(770\) 0 0
\(771\) 2126.32 0.0993225
\(772\) 0 0
\(773\) −4506.63 −0.209692 −0.104846 0.994488i \(-0.533435\pi\)
−0.104846 + 0.994488i \(0.533435\pi\)
\(774\) 0 0
\(775\) −17167.2 −0.795698
\(776\) 0 0
\(777\) 7744.34 0.357563
\(778\) 0 0
\(779\) 1231.68 0.0566491
\(780\) 0 0
\(781\) 1729.23 0.0792274
\(782\) 0 0
\(783\) 2840.00 0.129621
\(784\) 0 0
\(785\) 46145.7 2.09810
\(786\) 0 0
\(787\) −12917.4 −0.585076 −0.292538 0.956254i \(-0.594500\pi\)
−0.292538 + 0.956254i \(0.594500\pi\)
\(788\) 0 0
\(789\) −13284.6 −0.599421
\(790\) 0 0
\(791\) −41255.9 −1.85448
\(792\) 0 0
\(793\) −17.6516 −0.000790449 0
\(794\) 0 0
\(795\) 15661.9 0.698706
\(796\) 0 0
\(797\) 14608.1 0.649243 0.324621 0.945844i \(-0.394763\pi\)
0.324621 + 0.945844i \(0.394763\pi\)
\(798\) 0 0
\(799\) −6088.29 −0.269572
\(800\) 0 0
\(801\) −3930.73 −0.173390
\(802\) 0 0
\(803\) −11534.7 −0.506911
\(804\) 0 0
\(805\) −26609.4 −1.16504
\(806\) 0 0
\(807\) 9713.31 0.423698
\(808\) 0 0
\(809\) 16506.3 0.717342 0.358671 0.933464i \(-0.383230\pi\)
0.358671 + 0.933464i \(0.383230\pi\)
\(810\) 0 0
\(811\) −43809.7 −1.89688 −0.948438 0.316964i \(-0.897337\pi\)
−0.948438 + 0.316964i \(0.897337\pi\)
\(812\) 0 0
\(813\) 451.568 0.0194799
\(814\) 0 0
\(815\) −24295.1 −1.04420
\(816\) 0 0
\(817\) 5635.13 0.241308
\(818\) 0 0
\(819\) 272.123 0.0116102
\(820\) 0 0
\(821\) −17285.6 −0.734802 −0.367401 0.930063i \(-0.619752\pi\)
−0.367401 + 0.930063i \(0.619752\pi\)
\(822\) 0 0
\(823\) −39612.1 −1.67775 −0.838876 0.544323i \(-0.816787\pi\)
−0.838876 + 0.544323i \(0.816787\pi\)
\(824\) 0 0
\(825\) −4393.38 −0.185403
\(826\) 0 0
\(827\) 23037.5 0.968674 0.484337 0.874882i \(-0.339061\pi\)
0.484337 + 0.874882i \(0.339061\pi\)
\(828\) 0 0
\(829\) 32090.7 1.34446 0.672228 0.740344i \(-0.265337\pi\)
0.672228 + 0.740344i \(0.265337\pi\)
\(830\) 0 0
\(831\) 4674.00 0.195113
\(832\) 0 0
\(833\) 7972.57 0.331612
\(834\) 0 0
\(835\) −14469.4 −0.599682
\(836\) 0 0
\(837\) −8353.20 −0.344957
\(838\) 0 0
\(839\) −19295.2 −0.793973 −0.396987 0.917824i \(-0.629944\pi\)
−0.396987 + 0.917824i \(0.629944\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) −14823.7 −0.605643
\(844\) 0 0
\(845\) 39678.8 1.61538
\(846\) 0 0
\(847\) 26549.9 1.07705
\(848\) 0 0
\(849\) 10151.8 0.410376
\(850\) 0 0
\(851\) −12061.6 −0.485860
\(852\) 0 0
\(853\) −19213.6 −0.771234 −0.385617 0.922659i \(-0.626011\pi\)
−0.385617 + 0.922659i \(0.626011\pi\)
\(854\) 0 0
\(855\) 11111.8 0.444464
\(856\) 0 0
\(857\) −38335.0 −1.52800 −0.764001 0.645214i \(-0.776768\pi\)
−0.764001 + 0.645214i \(0.776768\pi\)
\(858\) 0 0
\(859\) −34344.3 −1.36416 −0.682080 0.731278i \(-0.738924\pi\)
−0.682080 + 0.731278i \(0.738924\pi\)
\(860\) 0 0
\(861\) 1994.86 0.0789601
\(862\) 0 0
\(863\) 39162.2 1.54472 0.772362 0.635183i \(-0.219075\pi\)
0.772362 + 0.635183i \(0.219075\pi\)
\(864\) 0 0
\(865\) 36112.9 1.41951
\(866\) 0 0
\(867\) 3417.28 0.133860
\(868\) 0 0
\(869\) 7066.05 0.275833
\(870\) 0 0
\(871\) 526.555 0.0204841
\(872\) 0 0
\(873\) 683.212 0.0264871
\(874\) 0 0
\(875\) 30331.0 1.17186
\(876\) 0 0
\(877\) 15235.0 0.586601 0.293301 0.956020i \(-0.405246\pi\)
0.293301 + 0.956020i \(0.405246\pi\)
\(878\) 0 0
\(879\) −12932.2 −0.496238
\(880\) 0 0
\(881\) 29455.1 1.12641 0.563206 0.826317i \(-0.309568\pi\)
0.563206 + 0.826317i \(0.309568\pi\)
\(882\) 0 0
\(883\) −37608.0 −1.43331 −0.716653 0.697430i \(-0.754327\pi\)
−0.716653 + 0.697430i \(0.754327\pi\)
\(884\) 0 0
\(885\) 1528.30 0.0580487
\(886\) 0 0
\(887\) −20517.3 −0.776666 −0.388333 0.921519i \(-0.626949\pi\)
−0.388333 + 0.921519i \(0.626949\pi\)
\(888\) 0 0
\(889\) 36815.5 1.38892
\(890\) 0 0
\(891\) 4868.48 0.183053
\(892\) 0 0
\(893\) −2872.01 −0.107624
\(894\) 0 0
\(895\) −21385.0 −0.798684
\(896\) 0 0
\(897\) 69.5621 0.00258931
\(898\) 0 0
\(899\) −2473.61 −0.0917680
\(900\) 0 0
\(901\) 24988.1 0.923944
\(902\) 0 0
\(903\) 9126.78 0.336346
\(904\) 0 0
\(905\) −21025.9 −0.772291
\(906\) 0 0
\(907\) 49117.1 1.79813 0.899067 0.437811i \(-0.144246\pi\)
0.899067 + 0.437811i \(0.144246\pi\)
\(908\) 0 0
\(909\) 20573.1 0.750678
\(910\) 0 0
\(911\) 23792.4 0.865289 0.432645 0.901564i \(-0.357580\pi\)
0.432645 + 0.901564i \(0.357580\pi\)
\(912\) 0 0
\(913\) −1624.03 −0.0588693
\(914\) 0 0
\(915\) −1167.39 −0.0421780
\(916\) 0 0
\(917\) −57493.3 −2.07044
\(918\) 0 0
\(919\) 43361.1 1.55642 0.778211 0.628003i \(-0.216127\pi\)
0.778211 + 0.628003i \(0.216127\pi\)
\(920\) 0 0
\(921\) −15886.4 −0.568377
\(922\) 0 0
\(923\) 82.3610 0.00293710
\(924\) 0 0
\(925\) 36282.8 1.28970
\(926\) 0 0
\(927\) 18061.2 0.639921
\(928\) 0 0
\(929\) −24782.7 −0.875235 −0.437617 0.899161i \(-0.644178\pi\)
−0.437617 + 0.899161i \(0.644178\pi\)
\(930\) 0 0
\(931\) 3760.88 0.132393
\(932\) 0 0
\(933\) −1990.34 −0.0698402
\(934\) 0 0
\(935\) −11362.9 −0.397440
\(936\) 0 0
\(937\) −31441.0 −1.09619 −0.548097 0.836415i \(-0.684647\pi\)
−0.548097 + 0.836415i \(0.684647\pi\)
\(938\) 0 0
\(939\) 13372.2 0.464735
\(940\) 0 0
\(941\) 2094.24 0.0725509 0.0362754 0.999342i \(-0.488451\pi\)
0.0362754 + 0.999342i \(0.488451\pi\)
\(942\) 0 0
\(943\) −3106.94 −0.107292
\(944\) 0 0
\(945\) 38947.8 1.34071
\(946\) 0 0
\(947\) 29337.6 1.00670 0.503350 0.864083i \(-0.332101\pi\)
0.503350 + 0.864083i \(0.332101\pi\)
\(948\) 0 0
\(949\) −549.381 −0.0187921
\(950\) 0 0
\(951\) 14715.2 0.501759
\(952\) 0 0
\(953\) −13702.6 −0.465763 −0.232881 0.972505i \(-0.574815\pi\)
−0.232881 + 0.972505i \(0.574815\pi\)
\(954\) 0 0
\(955\) −29146.1 −0.987588
\(956\) 0 0
\(957\) −633.037 −0.0213826
\(958\) 0 0
\(959\) 33159.7 1.11656
\(960\) 0 0
\(961\) −22515.5 −0.755780
\(962\) 0 0
\(963\) 42609.8 1.42584
\(964\) 0 0
\(965\) 18943.2 0.631919
\(966\) 0 0
\(967\) 15477.4 0.514706 0.257353 0.966318i \(-0.417150\pi\)
0.257353 + 0.966318i \(0.417150\pi\)
\(968\) 0 0
\(969\) −2909.78 −0.0964662
\(970\) 0 0
\(971\) 943.820 0.0311932 0.0155966 0.999878i \(-0.495035\pi\)
0.0155966 + 0.999878i \(0.495035\pi\)
\(972\) 0 0
\(973\) −62065.8 −2.04495
\(974\) 0 0
\(975\) −209.251 −0.00687323
\(976\) 0 0
\(977\) 41392.6 1.35544 0.677720 0.735320i \(-0.262968\pi\)
0.677720 + 0.735320i \(0.262968\pi\)
\(978\) 0 0
\(979\) 1896.13 0.0619004
\(980\) 0 0
\(981\) 5394.96 0.175584
\(982\) 0 0
\(983\) 9580.92 0.310869 0.155434 0.987846i \(-0.450322\pi\)
0.155434 + 0.987846i \(0.450322\pi\)
\(984\) 0 0
\(985\) −26874.9 −0.869346
\(986\) 0 0
\(987\) −4651.57 −0.150011
\(988\) 0 0
\(989\) −14214.7 −0.457029
\(990\) 0 0
\(991\) −25374.3 −0.813361 −0.406680 0.913570i \(-0.633314\pi\)
−0.406680 + 0.913570i \(0.633314\pi\)
\(992\) 0 0
\(993\) 1657.08 0.0529567
\(994\) 0 0
\(995\) −5378.28 −0.171360
\(996\) 0 0
\(997\) 51564.5 1.63798 0.818989 0.573810i \(-0.194535\pi\)
0.818989 + 0.573810i \(0.194535\pi\)
\(998\) 0 0
\(999\) 17654.4 0.559119
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 1856.4.a.bg.1.4 9
4.3 odd 2 1856.4.a.bf.1.6 9
8.3 odd 2 928.4.a.e.1.4 yes 9
8.5 even 2 928.4.a.d.1.6 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
928.4.a.d.1.6 9 8.5 even 2
928.4.a.e.1.4 yes 9 8.3 odd 2
1856.4.a.bf.1.6 9 4.3 odd 2
1856.4.a.bg.1.4 9 1.1 even 1 trivial