Properties

Label 1840.4.a.m.1.2
Level $1840$
Weight $4$
Character 1840.1
Self dual yes
Analytic conductor $108.564$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(108.563514411\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 68 x^{2} - 111 x + 342\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.58997\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.589969 q^{3} -5.00000 q^{5} +18.5077 q^{7} -26.6519 q^{9} +O(q^{10})\) \(q-0.589969 q^{3} -5.00000 q^{5} +18.5077 q^{7} -26.6519 q^{9} -47.9296 q^{11} +42.3717 q^{13} +2.94985 q^{15} +1.70534 q^{17} -21.4208 q^{19} -10.9190 q^{21} +23.0000 q^{23} +25.0000 q^{25} +31.6530 q^{27} +57.6332 q^{29} -295.699 q^{31} +28.2770 q^{33} -92.5387 q^{35} -7.85184 q^{37} -24.9980 q^{39} +465.929 q^{41} -182.374 q^{43} +133.260 q^{45} -449.193 q^{47} -0.463605 q^{49} -1.00610 q^{51} -368.316 q^{53} +239.648 q^{55} +12.6376 q^{57} +377.032 q^{59} +849.042 q^{61} -493.267 q^{63} -211.858 q^{65} -92.3424 q^{67} -13.5693 q^{69} +626.854 q^{71} +439.227 q^{73} -14.7492 q^{75} -887.068 q^{77} -641.707 q^{79} +700.928 q^{81} +609.932 q^{83} -8.52672 q^{85} -34.0018 q^{87} +1122.87 q^{89} +784.204 q^{91} +174.453 q^{93} +107.104 q^{95} -1428.80 q^{97} +1277.42 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 20 q^{5} + q^{7} + 32 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{3} - 20 q^{5} + q^{7} + 32 q^{9} + 39 q^{11} - 20 q^{13} - 20 q^{15} - 23 q^{17} - 53 q^{19} + 300 q^{21} + 92 q^{23} + 100 q^{25} - 137 q^{27} + 161 q^{29} - 388 q^{31} + 87 q^{33} - 5 q^{35} + 466 q^{37} - 1047 q^{39} + 484 q^{41} - 894 q^{43} - 160 q^{45} + 265 q^{47} + 1643 q^{49} - 1825 q^{51} + 576 q^{53} - 195 q^{55} + 178 q^{57} + 94 q^{59} + 1153 q^{61} - 60 q^{63} + 100 q^{65} + 1472 q^{67} + 92 q^{69} - 200 q^{71} + 1147 q^{73} + 100 q^{75} - 2176 q^{77} + 908 q^{79} - 1056 q^{81} + 1048 q^{83} + 115 q^{85} + 2167 q^{87} - 1784 q^{89} - 2329 q^{91} + 1483 q^{93} + 265 q^{95} - 2047 q^{97} + 2665 q^{99} + O(q^{100}) \)

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
\(3\) −0.589969 −0.113540 −0.0567698 0.998387i \(-0.518080\pi\)
−0.0567698 + 0.998387i \(0.518080\pi\)
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 18.5077 0.999324 0.499662 0.866220i \(-0.333458\pi\)
0.499662 + 0.866220i \(0.333458\pi\)
\(8\) 0 0
\(9\) −26.6519 −0.987109
\(10\) 0 0
\(11\) −47.9296 −1.31376 −0.656878 0.753997i \(-0.728123\pi\)
−0.656878 + 0.753997i \(0.728123\pi\)
\(12\) 0 0
\(13\) 42.3717 0.903984 0.451992 0.892022i \(-0.350714\pi\)
0.451992 + 0.892022i \(0.350714\pi\)
\(14\) 0 0
\(15\) 2.94985 0.0507765
\(16\) 0 0
\(17\) 1.70534 0.0243298 0.0121649 0.999926i \(-0.496128\pi\)
0.0121649 + 0.999926i \(0.496128\pi\)
\(18\) 0 0
\(19\) −21.4208 −0.258645 −0.129323 0.991603i \(-0.541280\pi\)
−0.129323 + 0.991603i \(0.541280\pi\)
\(20\) 0 0
\(21\) −10.9190 −0.113463
\(22\) 0 0
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 31.6530 0.225616
\(28\) 0 0
\(29\) 57.6332 0.369042 0.184521 0.982829i \(-0.440927\pi\)
0.184521 + 0.982829i \(0.440927\pi\)
\(30\) 0 0
\(31\) −295.699 −1.71320 −0.856599 0.515983i \(-0.827427\pi\)
−0.856599 + 0.515983i \(0.827427\pi\)
\(32\) 0 0
\(33\) 28.2770 0.149163
\(34\) 0 0
\(35\) −92.5387 −0.446911
\(36\) 0 0
\(37\) −7.85184 −0.0348874 −0.0174437 0.999848i \(-0.505553\pi\)
−0.0174437 + 0.999848i \(0.505553\pi\)
\(38\) 0 0
\(39\) −24.9980 −0.102638
\(40\) 0 0
\(41\) 465.929 1.77478 0.887390 0.461020i \(-0.152516\pi\)
0.887390 + 0.461020i \(0.152516\pi\)
\(42\) 0 0
\(43\) −182.374 −0.646784 −0.323392 0.946265i \(-0.604823\pi\)
−0.323392 + 0.946265i \(0.604823\pi\)
\(44\) 0 0
\(45\) 133.260 0.441448
\(46\) 0 0
\(47\) −449.193 −1.39408 −0.697038 0.717035i \(-0.745499\pi\)
−0.697038 + 0.717035i \(0.745499\pi\)
\(48\) 0 0
\(49\) −0.463605 −0.00135162
\(50\) 0 0
\(51\) −1.00610 −0.00276240
\(52\) 0 0
\(53\) −368.316 −0.954567 −0.477283 0.878749i \(-0.658379\pi\)
−0.477283 + 0.878749i \(0.658379\pi\)
\(54\) 0 0
\(55\) 239.648 0.587529
\(56\) 0 0
\(57\) 12.6376 0.0293665
\(58\) 0 0
\(59\) 377.032 0.831955 0.415977 0.909375i \(-0.363440\pi\)
0.415977 + 0.909375i \(0.363440\pi\)
\(60\) 0 0
\(61\) 849.042 1.78211 0.891055 0.453896i \(-0.149966\pi\)
0.891055 + 0.453896i \(0.149966\pi\)
\(62\) 0 0
\(63\) −493.267 −0.986441
\(64\) 0 0
\(65\) −211.858 −0.404274
\(66\) 0 0
\(67\) −92.3424 −0.168379 −0.0841897 0.996450i \(-0.526830\pi\)
−0.0841897 + 0.996450i \(0.526830\pi\)
\(68\) 0 0
\(69\) −13.5693 −0.0236747
\(70\) 0 0
\(71\) 626.854 1.04780 0.523901 0.851779i \(-0.324476\pi\)
0.523901 + 0.851779i \(0.324476\pi\)
\(72\) 0 0
\(73\) 439.227 0.704214 0.352107 0.935960i \(-0.385465\pi\)
0.352107 + 0.935960i \(0.385465\pi\)
\(74\) 0 0
\(75\) −14.7492 −0.0227079
\(76\) 0 0
\(77\) −887.068 −1.31287
\(78\) 0 0
\(79\) −641.707 −0.913894 −0.456947 0.889494i \(-0.651057\pi\)
−0.456947 + 0.889494i \(0.651057\pi\)
\(80\) 0 0
\(81\) 700.928 0.961492
\(82\) 0 0
\(83\) 609.932 0.806611 0.403306 0.915065i \(-0.367861\pi\)
0.403306 + 0.915065i \(0.367861\pi\)
\(84\) 0 0
\(85\) −8.52672 −0.0108806
\(86\) 0 0
\(87\) −34.0018 −0.0419009
\(88\) 0 0
\(89\) 1122.87 1.33735 0.668673 0.743557i \(-0.266863\pi\)
0.668673 + 0.743557i \(0.266863\pi\)
\(90\) 0 0
\(91\) 784.204 0.903373
\(92\) 0 0
\(93\) 174.453 0.194516
\(94\) 0 0
\(95\) 107.104 0.115670
\(96\) 0 0
\(97\) −1428.80 −1.49559 −0.747795 0.663930i \(-0.768887\pi\)
−0.747795 + 0.663930i \(0.768887\pi\)
\(98\) 0 0
\(99\) 1277.42 1.29682
\(100\) 0 0
\(101\) −1512.15 −1.48975 −0.744875 0.667204i \(-0.767491\pi\)
−0.744875 + 0.667204i \(0.767491\pi\)
\(102\) 0 0
\(103\) −957.279 −0.915762 −0.457881 0.889013i \(-0.651391\pi\)
−0.457881 + 0.889013i \(0.651391\pi\)
\(104\) 0 0
\(105\) 54.5950 0.0507422
\(106\) 0 0
\(107\) 1742.16 1.57403 0.787013 0.616936i \(-0.211626\pi\)
0.787013 + 0.616936i \(0.211626\pi\)
\(108\) 0 0
\(109\) 1166.77 1.02529 0.512644 0.858601i \(-0.328666\pi\)
0.512644 + 0.858601i \(0.328666\pi\)
\(110\) 0 0
\(111\) 4.63234 0.00396111
\(112\) 0 0
\(113\) −393.287 −0.327410 −0.163705 0.986509i \(-0.552345\pi\)
−0.163705 + 0.986509i \(0.552345\pi\)
\(114\) 0 0
\(115\) −115.000 −0.0932505
\(116\) 0 0
\(117\) −1129.29 −0.892331
\(118\) 0 0
\(119\) 31.5621 0.0243134
\(120\) 0 0
\(121\) 966.245 0.725954
\(122\) 0 0
\(123\) −274.884 −0.201508
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 1067.87 0.746127 0.373063 0.927806i \(-0.378307\pi\)
0.373063 + 0.927806i \(0.378307\pi\)
\(128\) 0 0
\(129\) 107.595 0.0734357
\(130\) 0 0
\(131\) 175.497 0.117047 0.0585237 0.998286i \(-0.481361\pi\)
0.0585237 + 0.998286i \(0.481361\pi\)
\(132\) 0 0
\(133\) −396.450 −0.258471
\(134\) 0 0
\(135\) −158.265 −0.100898
\(136\) 0 0
\(137\) 475.898 0.296779 0.148389 0.988929i \(-0.452591\pi\)
0.148389 + 0.988929i \(0.452591\pi\)
\(138\) 0 0
\(139\) −153.167 −0.0934638 −0.0467319 0.998907i \(-0.514881\pi\)
−0.0467319 + 0.998907i \(0.514881\pi\)
\(140\) 0 0
\(141\) 265.010 0.158283
\(142\) 0 0
\(143\) −2030.86 −1.18761
\(144\) 0 0
\(145\) −288.166 −0.165041
\(146\) 0 0
\(147\) 0.273513 0.000153462 0
\(148\) 0 0
\(149\) 506.906 0.278707 0.139353 0.990243i \(-0.455498\pi\)
0.139353 + 0.990243i \(0.455498\pi\)
\(150\) 0 0
\(151\) −2437.84 −1.31383 −0.656916 0.753964i \(-0.728139\pi\)
−0.656916 + 0.753964i \(0.728139\pi\)
\(152\) 0 0
\(153\) −45.4507 −0.0240162
\(154\) 0 0
\(155\) 1478.50 0.766166
\(156\) 0 0
\(157\) 255.966 0.130117 0.0650584 0.997881i \(-0.479277\pi\)
0.0650584 + 0.997881i \(0.479277\pi\)
\(158\) 0 0
\(159\) 217.295 0.108381
\(160\) 0 0
\(161\) 425.678 0.208373
\(162\) 0 0
\(163\) −321.632 −0.154553 −0.0772767 0.997010i \(-0.524622\pi\)
−0.0772767 + 0.997010i \(0.524622\pi\)
\(164\) 0 0
\(165\) −141.385 −0.0667079
\(166\) 0 0
\(167\) 2926.22 1.35591 0.677957 0.735102i \(-0.262866\pi\)
0.677957 + 0.735102i \(0.262866\pi\)
\(168\) 0 0
\(169\) −401.640 −0.182813
\(170\) 0 0
\(171\) 570.905 0.255311
\(172\) 0 0
\(173\) 1811.84 0.796254 0.398127 0.917330i \(-0.369660\pi\)
0.398127 + 0.917330i \(0.369660\pi\)
\(174\) 0 0
\(175\) 462.693 0.199865
\(176\) 0 0
\(177\) −222.437 −0.0944599
\(178\) 0 0
\(179\) −912.664 −0.381093 −0.190547 0.981678i \(-0.561026\pi\)
−0.190547 + 0.981678i \(0.561026\pi\)
\(180\) 0 0
\(181\) 3670.55 1.50735 0.753673 0.657249i \(-0.228280\pi\)
0.753673 + 0.657249i \(0.228280\pi\)
\(182\) 0 0
\(183\) −500.909 −0.202340
\(184\) 0 0
\(185\) 39.2592 0.0156021
\(186\) 0 0
\(187\) −81.7364 −0.0319634
\(188\) 0 0
\(189\) 585.826 0.225463
\(190\) 0 0
\(191\) 1840.96 0.697419 0.348710 0.937231i \(-0.386620\pi\)
0.348710 + 0.937231i \(0.386620\pi\)
\(192\) 0 0
\(193\) −611.817 −0.228184 −0.114092 0.993470i \(-0.536396\pi\)
−0.114092 + 0.993470i \(0.536396\pi\)
\(194\) 0 0
\(195\) 124.990 0.0459011
\(196\) 0 0
\(197\) 2830.26 1.02359 0.511796 0.859107i \(-0.328980\pi\)
0.511796 + 0.859107i \(0.328980\pi\)
\(198\) 0 0
\(199\) 1162.74 0.414195 0.207097 0.978320i \(-0.433598\pi\)
0.207097 + 0.978320i \(0.433598\pi\)
\(200\) 0 0
\(201\) 54.4792 0.0191177
\(202\) 0 0
\(203\) 1066.66 0.368792
\(204\) 0 0
\(205\) −2329.65 −0.793705
\(206\) 0 0
\(207\) −612.995 −0.205826
\(208\) 0 0
\(209\) 1026.69 0.339797
\(210\) 0 0
\(211\) −1399.58 −0.456641 −0.228320 0.973586i \(-0.573323\pi\)
−0.228320 + 0.973586i \(0.573323\pi\)
\(212\) 0 0
\(213\) −369.825 −0.118967
\(214\) 0 0
\(215\) 911.869 0.289251
\(216\) 0 0
\(217\) −5472.72 −1.71204
\(218\) 0 0
\(219\) −259.130 −0.0799562
\(220\) 0 0
\(221\) 72.2583 0.0219938
\(222\) 0 0
\(223\) −4257.98 −1.27863 −0.639317 0.768943i \(-0.720783\pi\)
−0.639317 + 0.768943i \(0.720783\pi\)
\(224\) 0 0
\(225\) −666.298 −0.197422
\(226\) 0 0
\(227\) 4025.03 1.17688 0.588438 0.808542i \(-0.299743\pi\)
0.588438 + 0.808542i \(0.299743\pi\)
\(228\) 0 0
\(229\) 3623.04 1.04549 0.522745 0.852489i \(-0.324908\pi\)
0.522745 + 0.852489i \(0.324908\pi\)
\(230\) 0 0
\(231\) 523.343 0.149063
\(232\) 0 0
\(233\) 6502.81 1.82838 0.914192 0.405282i \(-0.132827\pi\)
0.914192 + 0.405282i \(0.132827\pi\)
\(234\) 0 0
\(235\) 2245.96 0.623449
\(236\) 0 0
\(237\) 378.587 0.103763
\(238\) 0 0
\(239\) 2690.07 0.728059 0.364030 0.931387i \(-0.381401\pi\)
0.364030 + 0.931387i \(0.381401\pi\)
\(240\) 0 0
\(241\) −44.6958 −0.0119465 −0.00597326 0.999982i \(-0.501901\pi\)
−0.00597326 + 0.999982i \(0.501901\pi\)
\(242\) 0 0
\(243\) −1268.16 −0.334783
\(244\) 0 0
\(245\) 2.31802 0.000604461 0
\(246\) 0 0
\(247\) −907.635 −0.233811
\(248\) 0 0
\(249\) −359.841 −0.0915824
\(250\) 0 0
\(251\) 6801.41 1.71036 0.855181 0.518329i \(-0.173446\pi\)
0.855181 + 0.518329i \(0.173446\pi\)
\(252\) 0 0
\(253\) −1102.38 −0.273937
\(254\) 0 0
\(255\) 5.03050 0.00123538
\(256\) 0 0
\(257\) 5576.00 1.35339 0.676695 0.736263i \(-0.263411\pi\)
0.676695 + 0.736263i \(0.263411\pi\)
\(258\) 0 0
\(259\) −145.320 −0.0348638
\(260\) 0 0
\(261\) −1536.04 −0.364285
\(262\) 0 0
\(263\) 5669.58 1.32928 0.664641 0.747163i \(-0.268585\pi\)
0.664641 + 0.747163i \(0.268585\pi\)
\(264\) 0 0
\(265\) 1841.58 0.426895
\(266\) 0 0
\(267\) −662.458 −0.151842
\(268\) 0 0
\(269\) 6040.21 1.36906 0.684532 0.728983i \(-0.260006\pi\)
0.684532 + 0.728983i \(0.260006\pi\)
\(270\) 0 0
\(271\) 6899.26 1.54650 0.773248 0.634104i \(-0.218631\pi\)
0.773248 + 0.634104i \(0.218631\pi\)
\(272\) 0 0
\(273\) −462.657 −0.102569
\(274\) 0 0
\(275\) −1198.24 −0.262751
\(276\) 0 0
\(277\) 5617.68 1.21853 0.609267 0.792965i \(-0.291464\pi\)
0.609267 + 0.792965i \(0.291464\pi\)
\(278\) 0 0
\(279\) 7880.96 1.69111
\(280\) 0 0
\(281\) 3069.89 0.651723 0.325861 0.945418i \(-0.394346\pi\)
0.325861 + 0.945418i \(0.394346\pi\)
\(282\) 0 0
\(283\) −1404.86 −0.295089 −0.147544 0.989055i \(-0.547137\pi\)
−0.147544 + 0.989055i \(0.547137\pi\)
\(284\) 0 0
\(285\) −63.1880 −0.0131331
\(286\) 0 0
\(287\) 8623.30 1.77358
\(288\) 0 0
\(289\) −4910.09 −0.999408
\(290\) 0 0
\(291\) 842.946 0.169809
\(292\) 0 0
\(293\) −2407.05 −0.479936 −0.239968 0.970781i \(-0.577137\pi\)
−0.239968 + 0.970781i \(0.577137\pi\)
\(294\) 0 0
\(295\) −1885.16 −0.372062
\(296\) 0 0
\(297\) −1517.12 −0.296404
\(298\) 0 0
\(299\) 974.549 0.188494
\(300\) 0 0
\(301\) −3375.33 −0.646347
\(302\) 0 0
\(303\) 892.124 0.169146
\(304\) 0 0
\(305\) −4245.21 −0.796984
\(306\) 0 0
\(307\) 459.743 0.0854689 0.0427344 0.999086i \(-0.486393\pi\)
0.0427344 + 0.999086i \(0.486393\pi\)
\(308\) 0 0
\(309\) 564.765 0.103975
\(310\) 0 0
\(311\) −4119.48 −0.751107 −0.375553 0.926801i \(-0.622547\pi\)
−0.375553 + 0.926801i \(0.622547\pi\)
\(312\) 0 0
\(313\) 1684.15 0.304133 0.152066 0.988370i \(-0.451407\pi\)
0.152066 + 0.988370i \(0.451407\pi\)
\(314\) 0 0
\(315\) 2466.34 0.441150
\(316\) 0 0
\(317\) 8686.47 1.53906 0.769528 0.638613i \(-0.220492\pi\)
0.769528 + 0.638613i \(0.220492\pi\)
\(318\) 0 0
\(319\) −2762.34 −0.484831
\(320\) 0 0
\(321\) −1027.82 −0.178714
\(322\) 0 0
\(323\) −36.5298 −0.00629279
\(324\) 0 0
\(325\) 1059.29 0.180797
\(326\) 0 0
\(327\) −688.360 −0.116411
\(328\) 0 0
\(329\) −8313.55 −1.39313
\(330\) 0 0
\(331\) −4307.91 −0.715359 −0.357680 0.933844i \(-0.616432\pi\)
−0.357680 + 0.933844i \(0.616432\pi\)
\(332\) 0 0
\(333\) 209.267 0.0344377
\(334\) 0 0
\(335\) 461.712 0.0753015
\(336\) 0 0
\(337\) −290.156 −0.0469015 −0.0234507 0.999725i \(-0.507465\pi\)
−0.0234507 + 0.999725i \(0.507465\pi\)
\(338\) 0 0
\(339\) 232.027 0.0371740
\(340\) 0 0
\(341\) 14172.7 2.25072
\(342\) 0 0
\(343\) −6356.73 −1.00067
\(344\) 0 0
\(345\) 67.8465 0.0105876
\(346\) 0 0
\(347\) −1042.94 −0.161349 −0.0806743 0.996741i \(-0.525707\pi\)
−0.0806743 + 0.996741i \(0.525707\pi\)
\(348\) 0 0
\(349\) −1819.56 −0.279080 −0.139540 0.990216i \(-0.544562\pi\)
−0.139540 + 0.990216i \(0.544562\pi\)
\(350\) 0 0
\(351\) 1341.19 0.203953
\(352\) 0 0
\(353\) −4514.29 −0.680656 −0.340328 0.940307i \(-0.610538\pi\)
−0.340328 + 0.940307i \(0.610538\pi\)
\(354\) 0 0
\(355\) −3134.27 −0.468591
\(356\) 0 0
\(357\) −18.6207 −0.00276053
\(358\) 0 0
\(359\) −11527.9 −1.69476 −0.847379 0.530988i \(-0.821821\pi\)
−0.847379 + 0.530988i \(0.821821\pi\)
\(360\) 0 0
\(361\) −6400.15 −0.933103
\(362\) 0 0
\(363\) −570.055 −0.0824246
\(364\) 0 0
\(365\) −2196.13 −0.314934
\(366\) 0 0
\(367\) −6894.09 −0.980568 −0.490284 0.871563i \(-0.663107\pi\)
−0.490284 + 0.871563i \(0.663107\pi\)
\(368\) 0 0
\(369\) −12417.9 −1.75190
\(370\) 0 0
\(371\) −6816.69 −0.953922
\(372\) 0 0
\(373\) 7733.25 1.07349 0.536746 0.843744i \(-0.319653\pi\)
0.536746 + 0.843744i \(0.319653\pi\)
\(374\) 0 0
\(375\) 73.7462 0.0101553
\(376\) 0 0
\(377\) 2442.02 0.333608
\(378\) 0 0
\(379\) 9495.72 1.28697 0.643486 0.765458i \(-0.277488\pi\)
0.643486 + 0.765458i \(0.277488\pi\)
\(380\) 0 0
\(381\) −630.011 −0.0847150
\(382\) 0 0
\(383\) 12877.0 1.71797 0.858987 0.511998i \(-0.171094\pi\)
0.858987 + 0.511998i \(0.171094\pi\)
\(384\) 0 0
\(385\) 4435.34 0.587132
\(386\) 0 0
\(387\) 4860.61 0.638447
\(388\) 0 0
\(389\) −8200.40 −1.06884 −0.534418 0.845221i \(-0.679469\pi\)
−0.534418 + 0.845221i \(0.679469\pi\)
\(390\) 0 0
\(391\) 39.2229 0.00507312
\(392\) 0 0
\(393\) −103.538 −0.0132895
\(394\) 0 0
\(395\) 3208.53 0.408706
\(396\) 0 0
\(397\) 7842.68 0.991468 0.495734 0.868475i \(-0.334899\pi\)
0.495734 + 0.868475i \(0.334899\pi\)
\(398\) 0 0
\(399\) 233.893 0.0293467
\(400\) 0 0
\(401\) −14534.2 −1.80999 −0.904993 0.425427i \(-0.860124\pi\)
−0.904993 + 0.425427i \(0.860124\pi\)
\(402\) 0 0
\(403\) −12529.3 −1.54870
\(404\) 0 0
\(405\) −3504.64 −0.429992
\(406\) 0 0
\(407\) 376.335 0.0458335
\(408\) 0 0
\(409\) 13388.0 1.61857 0.809283 0.587419i \(-0.199856\pi\)
0.809283 + 0.587419i \(0.199856\pi\)
\(410\) 0 0
\(411\) −280.765 −0.0336962
\(412\) 0 0
\(413\) 6978.00 0.831393
\(414\) 0 0
\(415\) −3049.66 −0.360727
\(416\) 0 0
\(417\) 90.3640 0.0106119
\(418\) 0 0
\(419\) 3103.10 0.361805 0.180903 0.983501i \(-0.442098\pi\)
0.180903 + 0.983501i \(0.442098\pi\)
\(420\) 0 0
\(421\) −6075.82 −0.703367 −0.351684 0.936119i \(-0.614391\pi\)
−0.351684 + 0.936119i \(0.614391\pi\)
\(422\) 0 0
\(423\) 11971.9 1.37610
\(424\) 0 0
\(425\) 42.6336 0.00486596
\(426\) 0 0
\(427\) 15713.8 1.78090
\(428\) 0 0
\(429\) 1198.14 0.134841
\(430\) 0 0
\(431\) −14120.3 −1.57808 −0.789040 0.614341i \(-0.789422\pi\)
−0.789040 + 0.614341i \(0.789422\pi\)
\(432\) 0 0
\(433\) 8555.96 0.949592 0.474796 0.880096i \(-0.342522\pi\)
0.474796 + 0.880096i \(0.342522\pi\)
\(434\) 0 0
\(435\) 170.009 0.0187387
\(436\) 0 0
\(437\) −492.678 −0.0539313
\(438\) 0 0
\(439\) 10894.1 1.18439 0.592194 0.805796i \(-0.298262\pi\)
0.592194 + 0.805796i \(0.298262\pi\)
\(440\) 0 0
\(441\) 12.3560 0.00133419
\(442\) 0 0
\(443\) −16120.5 −1.72891 −0.864456 0.502708i \(-0.832337\pi\)
−0.864456 + 0.502708i \(0.832337\pi\)
\(444\) 0 0
\(445\) −5614.34 −0.598079
\(446\) 0 0
\(447\) −299.059 −0.0316443
\(448\) 0 0
\(449\) 4811.29 0.505699 0.252849 0.967506i \(-0.418632\pi\)
0.252849 + 0.967506i \(0.418632\pi\)
\(450\) 0 0
\(451\) −22331.8 −2.33163
\(452\) 0 0
\(453\) 1438.25 0.149172
\(454\) 0 0
\(455\) −3921.02 −0.404001
\(456\) 0 0
\(457\) −226.329 −0.0231667 −0.0115834 0.999933i \(-0.503687\pi\)
−0.0115834 + 0.999933i \(0.503687\pi\)
\(458\) 0 0
\(459\) 53.9793 0.00548919
\(460\) 0 0
\(461\) −4349.17 −0.439395 −0.219697 0.975568i \(-0.570507\pi\)
−0.219697 + 0.975568i \(0.570507\pi\)
\(462\) 0 0
\(463\) 989.313 0.0993030 0.0496515 0.998767i \(-0.484189\pi\)
0.0496515 + 0.998767i \(0.484189\pi\)
\(464\) 0 0
\(465\) −872.267 −0.0869902
\(466\) 0 0
\(467\) −8512.91 −0.843535 −0.421767 0.906704i \(-0.638590\pi\)
−0.421767 + 0.906704i \(0.638590\pi\)
\(468\) 0 0
\(469\) −1709.05 −0.168266
\(470\) 0 0
\(471\) −151.012 −0.0147734
\(472\) 0 0
\(473\) 8741.10 0.849717
\(474\) 0 0
\(475\) −535.519 −0.0517291
\(476\) 0 0
\(477\) 9816.33 0.942261
\(478\) 0 0
\(479\) 12058.6 1.15025 0.575125 0.818065i \(-0.304953\pi\)
0.575125 + 0.818065i \(0.304953\pi\)
\(480\) 0 0
\(481\) −332.696 −0.0315377
\(482\) 0 0
\(483\) −251.137 −0.0236587
\(484\) 0 0
\(485\) 7143.98 0.668848
\(486\) 0 0
\(487\) −11605.3 −1.07985 −0.539924 0.841714i \(-0.681547\pi\)
−0.539924 + 0.841714i \(0.681547\pi\)
\(488\) 0 0
\(489\) 189.753 0.0175479
\(490\) 0 0
\(491\) 4651.88 0.427569 0.213785 0.976881i \(-0.431421\pi\)
0.213785 + 0.976881i \(0.431421\pi\)
\(492\) 0 0
\(493\) 98.2845 0.00897872
\(494\) 0 0
\(495\) −6387.08 −0.579955
\(496\) 0 0
\(497\) 11601.6 1.04709
\(498\) 0 0
\(499\) 7953.03 0.713480 0.356740 0.934204i \(-0.383888\pi\)
0.356740 + 0.934204i \(0.383888\pi\)
\(500\) 0 0
\(501\) −1726.38 −0.153950
\(502\) 0 0
\(503\) 11805.6 1.04649 0.523245 0.852182i \(-0.324721\pi\)
0.523245 + 0.852182i \(0.324721\pi\)
\(504\) 0 0
\(505\) 7560.76 0.666237
\(506\) 0 0
\(507\) 236.955 0.0207565
\(508\) 0 0
\(509\) −2732.89 −0.237983 −0.118991 0.992895i \(-0.537966\pi\)
−0.118991 + 0.992895i \(0.537966\pi\)
\(510\) 0 0
\(511\) 8129.09 0.703738
\(512\) 0 0
\(513\) −678.032 −0.0583545
\(514\) 0 0
\(515\) 4786.40 0.409541
\(516\) 0 0
\(517\) 21529.6 1.83147
\(518\) 0 0
\(519\) −1068.93 −0.0904065
\(520\) 0 0
\(521\) 14710.9 1.23704 0.618518 0.785771i \(-0.287733\pi\)
0.618518 + 0.785771i \(0.287733\pi\)
\(522\) 0 0
\(523\) 7034.35 0.588127 0.294064 0.955786i \(-0.404992\pi\)
0.294064 + 0.955786i \(0.404992\pi\)
\(524\) 0 0
\(525\) −272.975 −0.0226926
\(526\) 0 0
\(527\) −504.269 −0.0416818
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) 0 0
\(531\) −10048.6 −0.821230
\(532\) 0 0
\(533\) 19742.2 1.60437
\(534\) 0 0
\(535\) −8710.79 −0.703926
\(536\) 0 0
\(537\) 538.444 0.0432692
\(538\) 0 0
\(539\) 22.2204 0.00177569
\(540\) 0 0
\(541\) 1552.90 0.123409 0.0617045 0.998094i \(-0.480346\pi\)
0.0617045 + 0.998094i \(0.480346\pi\)
\(542\) 0 0
\(543\) −2165.51 −0.171144
\(544\) 0 0
\(545\) −5833.86 −0.458523
\(546\) 0 0
\(547\) −174.657 −0.0136523 −0.00682614 0.999977i \(-0.502173\pi\)
−0.00682614 + 0.999977i \(0.502173\pi\)
\(548\) 0 0
\(549\) −22628.6 −1.75914
\(550\) 0 0
\(551\) −1234.55 −0.0954510
\(552\) 0 0
\(553\) −11876.5 −0.913276
\(554\) 0 0
\(555\) −23.1617 −0.00177146
\(556\) 0 0
\(557\) −1990.63 −0.151429 −0.0757143 0.997130i \(-0.524124\pi\)
−0.0757143 + 0.997130i \(0.524124\pi\)
\(558\) 0 0
\(559\) −7727.48 −0.584683
\(560\) 0 0
\(561\) 48.2220 0.00362912
\(562\) 0 0
\(563\) −208.006 −0.0155709 −0.00778543 0.999970i \(-0.502478\pi\)
−0.00778543 + 0.999970i \(0.502478\pi\)
\(564\) 0 0
\(565\) 1966.44 0.146422
\(566\) 0 0
\(567\) 12972.6 0.960842
\(568\) 0 0
\(569\) −3003.05 −0.221256 −0.110628 0.993862i \(-0.535286\pi\)
−0.110628 + 0.993862i \(0.535286\pi\)
\(570\) 0 0
\(571\) 10796.1 0.791249 0.395624 0.918412i \(-0.370528\pi\)
0.395624 + 0.918412i \(0.370528\pi\)
\(572\) 0 0
\(573\) −1086.11 −0.0791847
\(574\) 0 0
\(575\) 575.000 0.0417029
\(576\) 0 0
\(577\) −26011.3 −1.87671 −0.938357 0.345667i \(-0.887653\pi\)
−0.938357 + 0.345667i \(0.887653\pi\)
\(578\) 0 0
\(579\) 360.953 0.0259079
\(580\) 0 0
\(581\) 11288.5 0.806066
\(582\) 0 0
\(583\) 17653.2 1.25407
\(584\) 0 0
\(585\) 5646.44 0.399062
\(586\) 0 0
\(587\) 2774.97 0.195120 0.0975598 0.995230i \(-0.468896\pi\)
0.0975598 + 0.995230i \(0.468896\pi\)
\(588\) 0 0
\(589\) 6334.11 0.443111
\(590\) 0 0
\(591\) −1669.77 −0.116218
\(592\) 0 0
\(593\) −2384.07 −0.165096 −0.0825481 0.996587i \(-0.526306\pi\)
−0.0825481 + 0.996587i \(0.526306\pi\)
\(594\) 0 0
\(595\) −157.810 −0.0108733
\(596\) 0 0
\(597\) −685.984 −0.0470276
\(598\) 0 0
\(599\) 1090.87 0.0744102 0.0372051 0.999308i \(-0.488155\pi\)
0.0372051 + 0.999308i \(0.488155\pi\)
\(600\) 0 0
\(601\) −5192.37 −0.352414 −0.176207 0.984353i \(-0.556383\pi\)
−0.176207 + 0.984353i \(0.556383\pi\)
\(602\) 0 0
\(603\) 2461.10 0.166209
\(604\) 0 0
\(605\) −4831.22 −0.324657
\(606\) 0 0
\(607\) −1205.44 −0.0806054 −0.0403027 0.999188i \(-0.512832\pi\)
−0.0403027 + 0.999188i \(0.512832\pi\)
\(608\) 0 0
\(609\) −629.297 −0.0418726
\(610\) 0 0
\(611\) −19033.1 −1.26022
\(612\) 0 0
\(613\) −14243.2 −0.938466 −0.469233 0.883075i \(-0.655469\pi\)
−0.469233 + 0.883075i \(0.655469\pi\)
\(614\) 0 0
\(615\) 1374.42 0.0901170
\(616\) 0 0
\(617\) −422.492 −0.0275671 −0.0137835 0.999905i \(-0.504388\pi\)
−0.0137835 + 0.999905i \(0.504388\pi\)
\(618\) 0 0
\(619\) 8826.35 0.573119 0.286560 0.958062i \(-0.407488\pi\)
0.286560 + 0.958062i \(0.407488\pi\)
\(620\) 0 0
\(621\) 728.019 0.0470441
\(622\) 0 0
\(623\) 20781.7 1.33644
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) −605.715 −0.0385804
\(628\) 0 0
\(629\) −13.3901 −0.000848804 0
\(630\) 0 0
\(631\) 19684.1 1.24186 0.620929 0.783867i \(-0.286756\pi\)
0.620929 + 0.783867i \(0.286756\pi\)
\(632\) 0 0
\(633\) 825.710 0.0518468
\(634\) 0 0
\(635\) −5339.35 −0.333678
\(636\) 0 0
\(637\) −19.6437 −0.00122184
\(638\) 0 0
\(639\) −16706.9 −1.03429
\(640\) 0 0
\(641\) 15113.1 0.931253 0.465626 0.884981i \(-0.345829\pi\)
0.465626 + 0.884981i \(0.345829\pi\)
\(642\) 0 0
\(643\) 16917.8 1.03760 0.518798 0.854897i \(-0.326380\pi\)
0.518798 + 0.854897i \(0.326380\pi\)
\(644\) 0 0
\(645\) −537.975 −0.0328414
\(646\) 0 0
\(647\) −19564.5 −1.18881 −0.594405 0.804166i \(-0.702612\pi\)
−0.594405 + 0.804166i \(0.702612\pi\)
\(648\) 0 0
\(649\) −18071.0 −1.09299
\(650\) 0 0
\(651\) 3228.74 0.194384
\(652\) 0 0
\(653\) −22735.0 −1.36247 −0.681233 0.732067i \(-0.738556\pi\)
−0.681233 + 0.732067i \(0.738556\pi\)
\(654\) 0 0
\(655\) −877.483 −0.0523452
\(656\) 0 0
\(657\) −11706.2 −0.695136
\(658\) 0 0
\(659\) 30730.7 1.81654 0.908269 0.418388i \(-0.137405\pi\)
0.908269 + 0.418388i \(0.137405\pi\)
\(660\) 0 0
\(661\) −29204.3 −1.71848 −0.859240 0.511573i \(-0.829063\pi\)
−0.859240 + 0.511573i \(0.829063\pi\)
\(662\) 0 0
\(663\) −42.6302 −0.00249716
\(664\) 0 0
\(665\) 1982.25 0.115592
\(666\) 0 0
\(667\) 1325.56 0.0769506
\(668\) 0 0
\(669\) 2512.08 0.145176
\(670\) 0 0
\(671\) −40694.2 −2.34126
\(672\) 0 0
\(673\) −17530.1 −1.00406 −0.502032 0.864849i \(-0.667414\pi\)
−0.502032 + 0.864849i \(0.667414\pi\)
\(674\) 0 0
\(675\) 791.325 0.0451231
\(676\) 0 0
\(677\) −20553.4 −1.16681 −0.583407 0.812180i \(-0.698281\pi\)
−0.583407 + 0.812180i \(0.698281\pi\)
\(678\) 0 0
\(679\) −26443.8 −1.49458
\(680\) 0 0
\(681\) −2374.65 −0.133622
\(682\) 0 0
\(683\) −13174.8 −0.738098 −0.369049 0.929410i \(-0.620316\pi\)
−0.369049 + 0.929410i \(0.620316\pi\)
\(684\) 0 0
\(685\) −2379.49 −0.132723
\(686\) 0 0
\(687\) −2137.48 −0.118705
\(688\) 0 0
\(689\) −15606.2 −0.862913
\(690\) 0 0
\(691\) 14800.8 0.814831 0.407415 0.913243i \(-0.366430\pi\)
0.407415 + 0.913243i \(0.366430\pi\)
\(692\) 0 0
\(693\) 23642.1 1.29594
\(694\) 0 0
\(695\) 765.836 0.0417983
\(696\) 0 0
\(697\) 794.570 0.0431800
\(698\) 0 0
\(699\) −3836.46 −0.207594
\(700\) 0 0
\(701\) −12311.5 −0.663336 −0.331668 0.943396i \(-0.607611\pi\)
−0.331668 + 0.943396i \(0.607611\pi\)
\(702\) 0 0
\(703\) 168.192 0.00902347
\(704\) 0 0
\(705\) −1325.05 −0.0707862
\(706\) 0 0
\(707\) −27986.5 −1.48874
\(708\) 0 0
\(709\) 3893.89 0.206260 0.103130 0.994668i \(-0.467114\pi\)
0.103130 + 0.994668i \(0.467114\pi\)
\(710\) 0 0
\(711\) 17102.7 0.902113
\(712\) 0 0
\(713\) −6801.08 −0.357227
\(714\) 0 0
\(715\) 10154.3 0.531117
\(716\) 0 0
\(717\) −1587.06 −0.0826636
\(718\) 0 0
\(719\) 19942.5 1.03440 0.517198 0.855866i \(-0.326975\pi\)
0.517198 + 0.855866i \(0.326975\pi\)
\(720\) 0 0
\(721\) −17717.1 −0.915143
\(722\) 0 0
\(723\) 26.3692 0.00135640
\(724\) 0 0
\(725\) 1440.83 0.0738084
\(726\) 0 0
\(727\) −36167.0 −1.84506 −0.922530 0.385926i \(-0.873882\pi\)
−0.922530 + 0.385926i \(0.873882\pi\)
\(728\) 0 0
\(729\) −18176.9 −0.923481
\(730\) 0 0
\(731\) −311.010 −0.0157361
\(732\) 0 0
\(733\) −11669.5 −0.588025 −0.294013 0.955802i \(-0.594991\pi\)
−0.294013 + 0.955802i \(0.594991\pi\)
\(734\) 0 0
\(735\) −1.36756 −6.86304e−5 0
\(736\) 0 0
\(737\) 4425.93 0.221209
\(738\) 0 0
\(739\) −75.0536 −0.00373598 −0.00186799 0.999998i \(-0.500595\pi\)
−0.00186799 + 0.999998i \(0.500595\pi\)
\(740\) 0 0
\(741\) 535.477 0.0265469
\(742\) 0 0
\(743\) −28279.8 −1.39635 −0.698173 0.715929i \(-0.746003\pi\)
−0.698173 + 0.715929i \(0.746003\pi\)
\(744\) 0 0
\(745\) −2534.53 −0.124641
\(746\) 0 0
\(747\) −16255.9 −0.796213
\(748\) 0 0
\(749\) 32243.4 1.57296
\(750\) 0 0
\(751\) 17268.1 0.839044 0.419522 0.907745i \(-0.362198\pi\)
0.419522 + 0.907745i \(0.362198\pi\)
\(752\) 0 0
\(753\) −4012.62 −0.194194
\(754\) 0 0
\(755\) 12189.2 0.587564
\(756\) 0 0
\(757\) −12525.8 −0.601400 −0.300700 0.953719i \(-0.597220\pi\)
−0.300700 + 0.953719i \(0.597220\pi\)
\(758\) 0 0
\(759\) 650.371 0.0311027
\(760\) 0 0
\(761\) −18670.1 −0.889343 −0.444672 0.895694i \(-0.646680\pi\)
−0.444672 + 0.895694i \(0.646680\pi\)
\(762\) 0 0
\(763\) 21594.3 1.02460
\(764\) 0 0
\(765\) 227.254 0.0107404
\(766\) 0 0
\(767\) 15975.5 0.752074
\(768\) 0 0
\(769\) −32969.6 −1.54605 −0.773027 0.634373i \(-0.781258\pi\)
−0.773027 + 0.634373i \(0.781258\pi\)
\(770\) 0 0
\(771\) −3289.67 −0.153664
\(772\) 0 0
\(773\) −23251.3 −1.08188 −0.540938 0.841062i \(-0.681931\pi\)
−0.540938 + 0.841062i \(0.681931\pi\)
\(774\) 0 0
\(775\) −7392.48 −0.342640
\(776\) 0 0
\(777\) 85.7342 0.00395843
\(778\) 0 0
\(779\) −9980.57 −0.459039
\(780\) 0 0
\(781\) −30044.8 −1.37655
\(782\) 0 0
\(783\) 1824.26 0.0832617
\(784\) 0 0
\(785\) −1279.83 −0.0581900
\(786\) 0 0
\(787\) 29782.8 1.34897 0.674486 0.738288i \(-0.264365\pi\)
0.674486 + 0.738288i \(0.264365\pi\)
\(788\) 0 0
\(789\) −3344.88 −0.150926
\(790\) 0 0
\(791\) −7278.86 −0.327189
\(792\) 0 0
\(793\) 35975.3 1.61100
\(794\) 0 0
\(795\) −1086.47 −0.0484696
\(796\) 0 0
\(797\) 35307.2 1.56919 0.784596 0.620008i \(-0.212871\pi\)
0.784596 + 0.620008i \(0.212871\pi\)
\(798\) 0 0
\(799\) −766.029 −0.0339176
\(800\) 0 0
\(801\) −29926.6 −1.32011
\(802\) 0 0
\(803\) −21052.0 −0.925165
\(804\) 0 0
\(805\) −2128.39 −0.0931874
\(806\) 0 0
\(807\) −3563.54 −0.155443
\(808\) 0 0
\(809\) 7752.46 0.336912 0.168456 0.985709i \(-0.446122\pi\)
0.168456 + 0.985709i \(0.446122\pi\)
\(810\) 0 0
\(811\) −23471.6 −1.01627 −0.508137 0.861276i \(-0.669666\pi\)
−0.508137 + 0.861276i \(0.669666\pi\)
\(812\) 0 0
\(813\) −4070.35 −0.175589
\(814\) 0 0
\(815\) 1608.16 0.0691184
\(816\) 0 0
\(817\) 3906.59 0.167288
\(818\) 0 0
\(819\) −20900.6 −0.891727
\(820\) 0 0
\(821\) 2467.55 0.104894 0.0524470 0.998624i \(-0.483298\pi\)
0.0524470 + 0.998624i \(0.483298\pi\)
\(822\) 0 0
\(823\) −21984.5 −0.931144 −0.465572 0.885010i \(-0.654151\pi\)
−0.465572 + 0.885010i \(0.654151\pi\)
\(824\) 0 0
\(825\) 706.925 0.0298327
\(826\) 0 0
\(827\) 587.293 0.0246943 0.0123472 0.999924i \(-0.496070\pi\)
0.0123472 + 0.999924i \(0.496070\pi\)
\(828\) 0 0
\(829\) 41822.8 1.75219 0.876096 0.482137i \(-0.160139\pi\)
0.876096 + 0.482137i \(0.160139\pi\)
\(830\) 0 0
\(831\) −3314.26 −0.138352
\(832\) 0 0
\(833\) −0.790605 −3.28846e−5 0
\(834\) 0 0
\(835\) −14631.1 −0.606383
\(836\) 0 0
\(837\) −9359.77 −0.386524
\(838\) 0 0
\(839\) 10126.5 0.416693 0.208346 0.978055i \(-0.433192\pi\)
0.208346 + 0.978055i \(0.433192\pi\)
\(840\) 0 0
\(841\) −21067.4 −0.863808
\(842\) 0 0
\(843\) −1811.14 −0.0739964
\(844\) 0 0
\(845\) 2008.20 0.0817564
\(846\) 0 0
\(847\) 17883.0 0.725463
\(848\) 0 0
\(849\) 828.822 0.0335043
\(850\) 0 0
\(851\) −180.592 −0.00727453
\(852\) 0 0
\(853\) 11584.4 0.464995 0.232498 0.972597i \(-0.425310\pi\)
0.232498 + 0.972597i \(0.425310\pi\)
\(854\) 0 0
\(855\) −2854.53 −0.114179
\(856\) 0 0
\(857\) −9221.47 −0.367561 −0.183780 0.982967i \(-0.558834\pi\)
−0.183780 + 0.982967i \(0.558834\pi\)
\(858\) 0 0
\(859\) −34654.2 −1.37647 −0.688235 0.725488i \(-0.741614\pi\)
−0.688235 + 0.725488i \(0.741614\pi\)
\(860\) 0 0
\(861\) −5087.48 −0.201372
\(862\) 0 0
\(863\) −42887.7 −1.69167 −0.845837 0.533441i \(-0.820899\pi\)
−0.845837 + 0.533441i \(0.820899\pi\)
\(864\) 0 0
\(865\) −9059.22 −0.356096
\(866\) 0 0
\(867\) 2896.80 0.113472
\(868\) 0 0
\(869\) 30756.7 1.20063
\(870\) 0 0
\(871\) −3912.70 −0.152212
\(872\) 0 0
\(873\) 38080.2 1.47631
\(874\) 0 0
\(875\) −2313.47 −0.0893823
\(876\) 0 0
\(877\) 45935.5 1.76868 0.884339 0.466846i \(-0.154610\pi\)
0.884339 + 0.466846i \(0.154610\pi\)
\(878\) 0 0
\(879\) 1420.08 0.0544917
\(880\) 0 0
\(881\) −71.9309 −0.00275075 −0.00137538 0.999999i \(-0.500438\pi\)
−0.00137538 + 0.999999i \(0.500438\pi\)
\(882\) 0 0
\(883\) −20003.8 −0.762379 −0.381189 0.924497i \(-0.624485\pi\)
−0.381189 + 0.924497i \(0.624485\pi\)
\(884\) 0 0
\(885\) 1112.19 0.0422437
\(886\) 0 0
\(887\) −26357.3 −0.997737 −0.498869 0.866678i \(-0.666251\pi\)
−0.498869 + 0.866678i \(0.666251\pi\)
\(888\) 0 0
\(889\) 19763.9 0.745623
\(890\) 0 0
\(891\) −33595.2 −1.26317
\(892\) 0 0
\(893\) 9622.06 0.360571
\(894\) 0 0
\(895\) 4563.32 0.170430
\(896\) 0 0
\(897\) −574.954 −0.0214015
\(898\) 0 0
\(899\) −17042.1 −0.632242
\(900\) 0 0
\(901\) −628.105 −0.0232244
\(902\) 0 0
\(903\) 1991.34 0.0733860
\(904\) 0 0
\(905\) −18352.7 −0.674106
\(906\) 0 0
\(907\) 39300.4 1.43875 0.719377 0.694620i \(-0.244428\pi\)
0.719377 + 0.694620i \(0.244428\pi\)
\(908\) 0 0
\(909\) 40301.8 1.47055
\(910\) 0 0
\(911\) −31099.2 −1.13103 −0.565513 0.824740i \(-0.691322\pi\)
−0.565513 + 0.824740i \(0.691322\pi\)
\(912\) 0 0
\(913\) −29233.8 −1.05969
\(914\) 0 0
\(915\) 2504.54 0.0904893
\(916\) 0 0
\(917\) 3248.05 0.116968
\(918\) 0 0
\(919\) −18935.8 −0.679690 −0.339845 0.940481i \(-0.610375\pi\)
−0.339845 + 0.940481i \(0.610375\pi\)
\(920\) 0 0
\(921\) −271.235 −0.00970411
\(922\) 0 0
\(923\) 26560.9 0.947195
\(924\) 0 0
\(925\) −196.296 −0.00697748
\(926\) 0 0
\(927\) 25513.3 0.903957
\(928\) 0 0
\(929\) 40156.1 1.41817 0.709085 0.705123i \(-0.249108\pi\)
0.709085 + 0.705123i \(0.249108\pi\)
\(930\) 0 0
\(931\) 9.93077 0.000349590 0
\(932\) 0 0
\(933\) 2430.37 0.0852804
\(934\) 0 0
\(935\) 408.682 0.0142945
\(936\) 0 0
\(937\) 126.898 0.00442432 0.00221216 0.999998i \(-0.499296\pi\)
0.00221216 + 0.999998i \(0.499296\pi\)
\(938\) 0 0
\(939\) −993.594 −0.0345311
\(940\) 0 0
\(941\) −19266.6 −0.667452 −0.333726 0.942670i \(-0.608306\pi\)
−0.333726 + 0.942670i \(0.608306\pi\)
\(942\) 0 0
\(943\) 10716.4 0.370067
\(944\) 0 0
\(945\) −2929.13 −0.100830
\(946\) 0 0
\(947\) 54560.4 1.87220 0.936101 0.351732i \(-0.114407\pi\)
0.936101 + 0.351732i \(0.114407\pi\)
\(948\) 0 0
\(949\) 18610.8 0.636598
\(950\) 0 0
\(951\) −5124.75 −0.174744
\(952\) 0 0
\(953\) −44976.2 −1.52877 −0.764387 0.644758i \(-0.776958\pi\)
−0.764387 + 0.644758i \(0.776958\pi\)
\(954\) 0 0
\(955\) −9204.79 −0.311895
\(956\) 0 0
\(957\) 1629.69 0.0550476
\(958\) 0 0
\(959\) 8807.79 0.296578
\(960\) 0 0
\(961\) 57647.0 1.93505
\(962\) 0 0
\(963\) −46431.9 −1.55374
\(964\) 0 0
\(965\) 3059.08 0.102047
\(966\) 0 0
\(967\) 39431.7 1.31131 0.655656 0.755059i \(-0.272392\pi\)
0.655656 + 0.755059i \(0.272392\pi\)
\(968\) 0 0
\(969\) 21.5515 0.000714482 0
\(970\) 0 0
\(971\) −20249.9 −0.669258 −0.334629 0.942350i \(-0.608611\pi\)
−0.334629 + 0.942350i \(0.608611\pi\)
\(972\) 0 0
\(973\) −2834.78 −0.0934006
\(974\) 0 0
\(975\) −624.950 −0.0205276
\(976\) 0 0
\(977\) −9673.55 −0.316770 −0.158385 0.987377i \(-0.550629\pi\)
−0.158385 + 0.987377i \(0.550629\pi\)
\(978\) 0 0
\(979\) −53818.6 −1.75695
\(980\) 0 0
\(981\) −31096.7 −1.01207
\(982\) 0 0
\(983\) 17264.3 0.560168 0.280084 0.959975i \(-0.409638\pi\)
0.280084 + 0.959975i \(0.409638\pi\)
\(984\) 0 0
\(985\) −14151.3 −0.457764
\(986\) 0 0
\(987\) 4904.74 0.158176
\(988\) 0 0
\(989\) −4194.60 −0.134864
\(990\) 0 0
\(991\) −23892.9 −0.765876 −0.382938 0.923774i \(-0.625088\pi\)
−0.382938 + 0.923774i \(0.625088\pi\)
\(992\) 0 0
\(993\) 2541.53 0.0812217
\(994\) 0 0
\(995\) −5813.72 −0.185234
\(996\) 0 0
\(997\) −35830.2 −1.13817 −0.569083 0.822280i \(-0.692702\pi\)
−0.569083 + 0.822280i \(0.692702\pi\)
\(998\) 0 0
\(999\) −248.534 −0.00787115
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 1840.4.a.m.1.2 4
4.3 odd 2 230.4.a.h.1.3 4
12.11 even 2 2070.4.a.bj.1.2 4
20.3 even 4 1150.4.b.n.599.7 8
20.7 even 4 1150.4.b.n.599.2 8
20.19 odd 2 1150.4.a.p.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.h.1.3 4 4.3 odd 2
1150.4.a.p.1.2 4 20.19 odd 2
1150.4.b.n.599.2 8 20.7 even 4
1150.4.b.n.599.7 8 20.3 even 4
1840.4.a.m.1.2 4 1.1 even 1 trivial
2070.4.a.bj.1.2 4 12.11 even 2