Properties

Label 441.4.a.v.1.2
Level $441$
Weight $4$
Character 441.1
Self dual yes
Analytic conductor $26.020$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.0198423125\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.6257832.1
Defining polynomial: \(x^{4} - 19 x^{2} + 42\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.59805\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.59805 q^{2} -5.44622 q^{4} -18.2917 q^{5} +21.4878 q^{8} +O(q^{10})\) \(q-1.59805 q^{2} -5.44622 q^{4} -18.2917 q^{5} +21.4878 q^{8} +29.2311 q^{10} +61.2673 q^{11} -32.4462 q^{13} +9.23111 q^{16} +81.3289 q^{17} +20.9084 q^{19} +99.6206 q^{20} -97.9084 q^{22} -33.7310 q^{23} +209.586 q^{25} +51.8508 q^{26} +52.0227 q^{29} -193.924 q^{31} -186.654 q^{32} -129.968 q^{34} -267.156 q^{37} -33.4128 q^{38} -393.048 q^{40} -203.176 q^{41} -21.9520 q^{43} -333.675 q^{44} +53.9040 q^{46} +247.921 q^{47} -334.929 q^{50} +176.709 q^{52} +140.826 q^{53} -1120.68 q^{55} -83.1351 q^{58} +221.468 q^{59} -652.526 q^{61} +309.902 q^{62} +224.435 q^{64} +593.496 q^{65} +604.478 q^{67} -442.935 q^{68} +716.031 q^{71} -388.876 q^{73} +426.929 q^{74} -113.872 q^{76} -289.741 q^{79} -168.853 q^{80} +324.686 q^{82} +115.652 q^{83} -1487.64 q^{85} +35.0805 q^{86} +1316.50 q^{88} -939.363 q^{89} +183.707 q^{92} -396.192 q^{94} -382.451 q^{95} -120.394 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{4} + O(q^{10}) \) \( 4q + 6q^{4} - 22q^{10} - 102q^{13} - 102q^{16} - 222q^{19} - 86q^{22} + 366q^{25} - 220q^{31} - 1020q^{34} - 374q^{37} - 822q^{40} - 838q^{43} + 1716q^{46} + 40q^{52} - 2510q^{55} - 1694q^{58} - 1332q^{61} - 686q^{64} + 1890q^{67} - 1750q^{73} - 2456q^{76} + 8q^{79} - 2480q^{82} - 1116q^{85} + 2682q^{88} + 1416q^{94} - 3010q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.59805 −0.564998 −0.282499 0.959268i \(-0.591163\pi\)
−0.282499 + 0.959268i \(0.591163\pi\)
\(3\) 0 0
\(4\) −5.44622 −0.680778
\(5\) −18.2917 −1.63606 −0.818029 0.575177i \(-0.804933\pi\)
−0.818029 + 0.575177i \(0.804933\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 21.4878 0.949635
\(9\) 0 0
\(10\) 29.2311 0.924369
\(11\) 61.2673 1.67934 0.839672 0.543094i \(-0.182747\pi\)
0.839672 + 0.543094i \(0.182747\pi\)
\(12\) 0 0
\(13\) −32.4462 −0.692228 −0.346114 0.938192i \(-0.612499\pi\)
−0.346114 + 0.938192i \(0.612499\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 9.23111 0.144236
\(17\) 81.3289 1.16030 0.580152 0.814509i \(-0.302993\pi\)
0.580152 + 0.814509i \(0.302993\pi\)
\(18\) 0 0
\(19\) 20.9084 0.252459 0.126230 0.992001i \(-0.459712\pi\)
0.126230 + 0.992001i \(0.459712\pi\)
\(20\) 99.6206 1.11379
\(21\) 0 0
\(22\) −97.9084 −0.948825
\(23\) −33.7310 −0.305800 −0.152900 0.988242i \(-0.548861\pi\)
−0.152900 + 0.988242i \(0.548861\pi\)
\(24\) 0 0
\(25\) 209.586 1.67669
\(26\) 51.8508 0.391107
\(27\) 0 0
\(28\) 0 0
\(29\) 52.0227 0.333116 0.166558 0.986032i \(-0.446735\pi\)
0.166558 + 0.986032i \(0.446735\pi\)
\(30\) 0 0
\(31\) −193.924 −1.12354 −0.561772 0.827292i \(-0.689880\pi\)
−0.561772 + 0.827292i \(0.689880\pi\)
\(32\) −186.654 −1.03113
\(33\) 0 0
\(34\) −129.968 −0.655568
\(35\) 0 0
\(36\) 0 0
\(37\) −267.156 −1.18703 −0.593515 0.804823i \(-0.702260\pi\)
−0.593515 + 0.804823i \(0.702260\pi\)
\(38\) −33.4128 −0.142639
\(39\) 0 0
\(40\) −393.048 −1.55366
\(41\) −203.176 −0.773921 −0.386960 0.922096i \(-0.626475\pi\)
−0.386960 + 0.922096i \(0.626475\pi\)
\(42\) 0 0
\(43\) −21.9520 −0.0778523 −0.0389262 0.999242i \(-0.512394\pi\)
−0.0389262 + 0.999242i \(0.512394\pi\)
\(44\) −333.675 −1.14326
\(45\) 0 0
\(46\) 53.9040 0.172776
\(47\) 247.921 0.769427 0.384713 0.923036i \(-0.374300\pi\)
0.384713 + 0.923036i \(0.374300\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −334.929 −0.947324
\(51\) 0 0
\(52\) 176.709 0.471253
\(53\) 140.826 0.364981 0.182490 0.983208i \(-0.441584\pi\)
0.182490 + 0.983208i \(0.441584\pi\)
\(54\) 0 0
\(55\) −1120.68 −2.74750
\(56\) 0 0
\(57\) 0 0
\(58\) −83.1351 −0.188210
\(59\) 221.468 0.488689 0.244344 0.969689i \(-0.421427\pi\)
0.244344 + 0.969689i \(0.421427\pi\)
\(60\) 0 0
\(61\) −652.526 −1.36963 −0.684815 0.728717i \(-0.740117\pi\)
−0.684815 + 0.728717i \(0.740117\pi\)
\(62\) 309.902 0.634800
\(63\) 0 0
\(64\) 224.435 0.438349
\(65\) 593.496 1.13253
\(66\) 0 0
\(67\) 604.478 1.10222 0.551110 0.834432i \(-0.314204\pi\)
0.551110 + 0.834432i \(0.314204\pi\)
\(68\) −442.935 −0.789909
\(69\) 0 0
\(70\) 0 0
\(71\) 716.031 1.19686 0.598431 0.801174i \(-0.295791\pi\)
0.598431 + 0.801174i \(0.295791\pi\)
\(72\) 0 0
\(73\) −388.876 −0.623487 −0.311743 0.950166i \(-0.600913\pi\)
−0.311743 + 0.950166i \(0.600913\pi\)
\(74\) 426.929 0.670669
\(75\) 0 0
\(76\) −113.872 −0.171869
\(77\) 0 0
\(78\) 0 0
\(79\) −289.741 −0.412639 −0.206319 0.978485i \(-0.566149\pi\)
−0.206319 + 0.978485i \(0.566149\pi\)
\(80\) −168.853 −0.235979
\(81\) 0 0
\(82\) 324.686 0.437263
\(83\) 115.652 0.152946 0.0764728 0.997072i \(-0.475634\pi\)
0.0764728 + 0.997072i \(0.475634\pi\)
\(84\) 0 0
\(85\) −1487.64 −1.89832
\(86\) 35.0805 0.0439864
\(87\) 0 0
\(88\) 1316.50 1.59476
\(89\) −939.363 −1.11879 −0.559395 0.828901i \(-0.688967\pi\)
−0.559395 + 0.828901i \(0.688967\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 183.707 0.208182
\(93\) 0 0
\(94\) −396.192 −0.434724
\(95\) −382.451 −0.413038
\(96\) 0 0
\(97\) −120.394 −0.126022 −0.0630110 0.998013i \(-0.520070\pi\)
−0.0630110 + 0.998013i \(0.520070\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1141.45 −1.14145
\(101\) −1281.00 −1.26203 −0.631013 0.775772i \(-0.717360\pi\)
−0.631013 + 0.775772i \(0.717360\pi\)
\(102\) 0 0
\(103\) −531.339 −0.508295 −0.254147 0.967166i \(-0.581795\pi\)
−0.254147 + 0.967166i \(0.581795\pi\)
\(104\) −697.198 −0.657364
\(105\) 0 0
\(106\) −225.048 −0.206213
\(107\) 133.352 0.120482 0.0602411 0.998184i \(-0.480813\pi\)
0.0602411 + 0.998184i \(0.480813\pi\)
\(108\) 0 0
\(109\) −217.769 −0.191362 −0.0956811 0.995412i \(-0.530503\pi\)
−0.0956811 + 0.995412i \(0.530503\pi\)
\(110\) 1790.91 1.55233
\(111\) 0 0
\(112\) 0 0
\(113\) −2006.09 −1.67006 −0.835031 0.550204i \(-0.814550\pi\)
−0.835031 + 0.550204i \(0.814550\pi\)
\(114\) 0 0
\(115\) 616.997 0.500307
\(116\) −283.327 −0.226778
\(117\) 0 0
\(118\) −353.917 −0.276108
\(119\) 0 0
\(120\) 0 0
\(121\) 2422.68 1.82019
\(122\) 1042.77 0.773838
\(123\) 0 0
\(124\) 1056.16 0.764884
\(125\) −1547.22 −1.10710
\(126\) 0 0
\(127\) 1638.92 1.14512 0.572562 0.819861i \(-0.305950\pi\)
0.572562 + 0.819861i \(0.305950\pi\)
\(128\) 1134.57 0.783462
\(129\) 0 0
\(130\) −948.439 −0.639874
\(131\) −91.7511 −0.0611933 −0.0305967 0.999532i \(-0.509741\pi\)
−0.0305967 + 0.999532i \(0.509741\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −965.989 −0.622752
\(135\) 0 0
\(136\) 1747.58 1.10186
\(137\) −1867.13 −1.16438 −0.582188 0.813054i \(-0.697803\pi\)
−0.582188 + 0.813054i \(0.697803\pi\)
\(138\) 0 0
\(139\) −639.778 −0.390397 −0.195199 0.980764i \(-0.562535\pi\)
−0.195199 + 0.980764i \(0.562535\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1144.26 −0.676224
\(143\) −1987.89 −1.16249
\(144\) 0 0
\(145\) −951.583 −0.544998
\(146\) 621.446 0.352269
\(147\) 0 0
\(148\) 1454.99 0.808103
\(149\) −3136.53 −1.72453 −0.862264 0.506460i \(-0.830954\pi\)
−0.862264 + 0.506460i \(0.830954\pi\)
\(150\) 0 0
\(151\) −2721.37 −1.46663 −0.733317 0.679887i \(-0.762029\pi\)
−0.733317 + 0.679887i \(0.762029\pi\)
\(152\) 449.276 0.239744
\(153\) 0 0
\(154\) 0 0
\(155\) 3547.20 1.83818
\(156\) 0 0
\(157\) −2879.75 −1.46388 −0.731939 0.681370i \(-0.761384\pi\)
−0.731939 + 0.681370i \(0.761384\pi\)
\(158\) 463.022 0.233140
\(159\) 0 0
\(160\) 3414.22 1.68699
\(161\) 0 0
\(162\) 0 0
\(163\) 646.142 0.310489 0.155245 0.987876i \(-0.450383\pi\)
0.155245 + 0.987876i \(0.450383\pi\)
\(164\) 1106.54 0.526868
\(165\) 0 0
\(166\) −184.819 −0.0864139
\(167\) 3765.03 1.74459 0.872296 0.488979i \(-0.162630\pi\)
0.872296 + 0.488979i \(0.162630\pi\)
\(168\) 0 0
\(169\) −1144.24 −0.520821
\(170\) 2377.33 1.07255
\(171\) 0 0
\(172\) 119.555 0.0530001
\(173\) −2308.98 −1.01473 −0.507366 0.861731i \(-0.669381\pi\)
−0.507366 + 0.861731i \(0.669381\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 565.565 0.242222
\(177\) 0 0
\(178\) 1501.15 0.632114
\(179\) 3033.76 1.26678 0.633390 0.773833i \(-0.281663\pi\)
0.633390 + 0.773833i \(0.281663\pi\)
\(180\) 0 0
\(181\) 4079.71 1.67537 0.837686 0.546152i \(-0.183908\pi\)
0.837686 + 0.546152i \(0.183908\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −724.805 −0.290399
\(185\) 4886.73 1.94205
\(186\) 0 0
\(187\) 4982.80 1.94855
\(188\) −1350.24 −0.523809
\(189\) 0 0
\(190\) 611.177 0.233365
\(191\) −877.109 −0.332279 −0.166140 0.986102i \(-0.553130\pi\)
−0.166140 + 0.986102i \(0.553130\pi\)
\(192\) 0 0
\(193\) −1458.71 −0.544043 −0.272022 0.962291i \(-0.587692\pi\)
−0.272022 + 0.962291i \(0.587692\pi\)
\(194\) 192.396 0.0712021
\(195\) 0 0
\(196\) 0 0
\(197\) −952.250 −0.344391 −0.172195 0.985063i \(-0.555086\pi\)
−0.172195 + 0.985063i \(0.555086\pi\)
\(198\) 0 0
\(199\) −3342.22 −1.19057 −0.595285 0.803515i \(-0.702961\pi\)
−0.595285 + 0.803515i \(0.702961\pi\)
\(200\) 4503.54 1.59224
\(201\) 0 0
\(202\) 2047.11 0.713041
\(203\) 0 0
\(204\) 0 0
\(205\) 3716.43 1.26618
\(206\) 849.108 0.287185
\(207\) 0 0
\(208\) −299.515 −0.0998442
\(209\) 1281.00 0.423966
\(210\) 0 0
\(211\) 1439.27 0.469589 0.234794 0.972045i \(-0.424558\pi\)
0.234794 + 0.972045i \(0.424558\pi\)
\(212\) −766.971 −0.248471
\(213\) 0 0
\(214\) −213.103 −0.0680721
\(215\) 401.539 0.127371
\(216\) 0 0
\(217\) 0 0
\(218\) 348.007 0.108119
\(219\) 0 0
\(220\) 6103.48 1.87044
\(221\) −2638.82 −0.803194
\(222\) 0 0
\(223\) −1009.86 −0.303253 −0.151626 0.988438i \(-0.548451\pi\)
−0.151626 + 0.988438i \(0.548451\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 3205.84 0.943580
\(227\) −2948.60 −0.862140 −0.431070 0.902319i \(-0.641864\pi\)
−0.431070 + 0.902319i \(0.641864\pi\)
\(228\) 0 0
\(229\) 4038.85 1.16548 0.582739 0.812659i \(-0.301981\pi\)
0.582739 + 0.812659i \(0.301981\pi\)
\(230\) −985.995 −0.282672
\(231\) 0 0
\(232\) 1117.85 0.316339
\(233\) −2995.80 −0.842323 −0.421162 0.906986i \(-0.638378\pi\)
−0.421162 + 0.906986i \(0.638378\pi\)
\(234\) 0 0
\(235\) −4534.90 −1.25883
\(236\) −1206.16 −0.332688
\(237\) 0 0
\(238\) 0 0
\(239\) 1810.28 0.489948 0.244974 0.969530i \(-0.421221\pi\)
0.244974 + 0.969530i \(0.421221\pi\)
\(240\) 0 0
\(241\) 3749.43 1.00217 0.501083 0.865399i \(-0.332935\pi\)
0.501083 + 0.865399i \(0.332935\pi\)
\(242\) −3871.57 −1.02841
\(243\) 0 0
\(244\) 3553.80 0.932414
\(245\) 0 0
\(246\) 0 0
\(247\) −678.400 −0.174759
\(248\) −4167.01 −1.06696
\(249\) 0 0
\(250\) 2472.54 0.625508
\(251\) −2706.96 −0.680724 −0.340362 0.940295i \(-0.610550\pi\)
−0.340362 + 0.940295i \(0.610550\pi\)
\(252\) 0 0
\(253\) −2066.61 −0.513544
\(254\) −2619.09 −0.646992
\(255\) 0 0
\(256\) −3608.59 −0.881003
\(257\) 5375.28 1.30467 0.652337 0.757929i \(-0.273789\pi\)
0.652337 + 0.757929i \(0.273789\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −3232.31 −0.770998
\(261\) 0 0
\(262\) 146.623 0.0345741
\(263\) −5246.56 −1.23010 −0.615051 0.788487i \(-0.710865\pi\)
−0.615051 + 0.788487i \(0.710865\pi\)
\(264\) 0 0
\(265\) −2575.95 −0.597129
\(266\) 0 0
\(267\) 0 0
\(268\) −3292.12 −0.750367
\(269\) 3013.32 0.682993 0.341496 0.939883i \(-0.389066\pi\)
0.341496 + 0.939883i \(0.389066\pi\)
\(270\) 0 0
\(271\) −6897.25 −1.54604 −0.773022 0.634379i \(-0.781256\pi\)
−0.773022 + 0.634379i \(0.781256\pi\)
\(272\) 750.756 0.167358
\(273\) 0 0
\(274\) 2983.77 0.657869
\(275\) 12840.7 2.81573
\(276\) 0 0
\(277\) 3318.61 0.719840 0.359920 0.932983i \(-0.382804\pi\)
0.359920 + 0.932983i \(0.382804\pi\)
\(278\) 1022.40 0.220574
\(279\) 0 0
\(280\) 0 0
\(281\) −6274.14 −1.33197 −0.665986 0.745964i \(-0.731989\pi\)
−0.665986 + 0.745964i \(0.731989\pi\)
\(282\) 0 0
\(283\) −7772.47 −1.63260 −0.816300 0.577629i \(-0.803978\pi\)
−0.816300 + 0.577629i \(0.803978\pi\)
\(284\) −3899.66 −0.814797
\(285\) 0 0
\(286\) 3176.76 0.656803
\(287\) 0 0
\(288\) 0 0
\(289\) 1701.39 0.346304
\(290\) 1520.68 0.307922
\(291\) 0 0
\(292\) 2117.91 0.424456
\(293\) −854.897 −0.170456 −0.0852280 0.996361i \(-0.527162\pi\)
−0.0852280 + 0.996361i \(0.527162\pi\)
\(294\) 0 0
\(295\) −4051.02 −0.799523
\(296\) −5740.58 −1.12725
\(297\) 0 0
\(298\) 5012.35 0.974354
\(299\) 1094.44 0.211683
\(300\) 0 0
\(301\) 0 0
\(302\) 4348.89 0.828645
\(303\) 0 0
\(304\) 193.008 0.0364137
\(305\) 11935.8 2.24079
\(306\) 0 0
\(307\) −2550.68 −0.474185 −0.237092 0.971487i \(-0.576194\pi\)
−0.237092 + 0.971487i \(0.576194\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −5668.63 −1.03857
\(311\) −7480.50 −1.36392 −0.681962 0.731388i \(-0.738873\pi\)
−0.681962 + 0.731388i \(0.738873\pi\)
\(312\) 0 0
\(313\) −6.24784 −0.00112827 −0.000564135 1.00000i \(-0.500180\pi\)
−0.000564135 1.00000i \(0.500180\pi\)
\(314\) 4601.99 0.827088
\(315\) 0 0
\(316\) 1578.00 0.280915
\(317\) 965.803 0.171120 0.0855598 0.996333i \(-0.472732\pi\)
0.0855598 + 0.996333i \(0.472732\pi\)
\(318\) 0 0
\(319\) 3187.29 0.559417
\(320\) −4105.29 −0.717164
\(321\) 0 0
\(322\) 0 0
\(323\) 1700.46 0.292929
\(324\) 0 0
\(325\) −6800.27 −1.16065
\(326\) −1032.57 −0.175426
\(327\) 0 0
\(328\) −4365.80 −0.734942
\(329\) 0 0
\(330\) 0 0
\(331\) 6710.20 1.11428 0.557139 0.830419i \(-0.311899\pi\)
0.557139 + 0.830419i \(0.311899\pi\)
\(332\) −629.868 −0.104122
\(333\) 0 0
\(334\) −6016.72 −0.985690
\(335\) −11056.9 −1.80330
\(336\) 0 0
\(337\) −605.546 −0.0978819 −0.0489409 0.998802i \(-0.515585\pi\)
−0.0489409 + 0.998802i \(0.515585\pi\)
\(338\) 1828.56 0.294262
\(339\) 0 0
\(340\) 8102.03 1.29234
\(341\) −11881.2 −1.88682
\(342\) 0 0
\(343\) 0 0
\(344\) −471.700 −0.0739313
\(345\) 0 0
\(346\) 3689.88 0.573321
\(347\) 6938.16 1.07337 0.536686 0.843782i \(-0.319676\pi\)
0.536686 + 0.843782i \(0.319676\pi\)
\(348\) 0 0
\(349\) −10368.9 −1.59035 −0.795176 0.606378i \(-0.792622\pi\)
−0.795176 + 0.606378i \(0.792622\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −11435.8 −1.73162
\(353\) 5881.76 0.886840 0.443420 0.896314i \(-0.353765\pi\)
0.443420 + 0.896314i \(0.353765\pi\)
\(354\) 0 0
\(355\) −13097.4 −1.95814
\(356\) 5115.98 0.761647
\(357\) 0 0
\(358\) −4848.11 −0.715728
\(359\) 589.269 0.0866307 0.0433153 0.999061i \(-0.486208\pi\)
0.0433153 + 0.999061i \(0.486208\pi\)
\(360\) 0 0
\(361\) −6421.84 −0.936264
\(362\) −6519.60 −0.946581
\(363\) 0 0
\(364\) 0 0
\(365\) 7113.21 1.02006
\(366\) 0 0
\(367\) 3548.72 0.504745 0.252373 0.967630i \(-0.418789\pi\)
0.252373 + 0.967630i \(0.418789\pi\)
\(368\) −311.375 −0.0441074
\(369\) 0 0
\(370\) −7809.25 −1.09725
\(371\) 0 0
\(372\) 0 0
\(373\) −1581.33 −0.219513 −0.109756 0.993959i \(-0.535007\pi\)
−0.109756 + 0.993959i \(0.535007\pi\)
\(374\) −7962.79 −1.10092
\(375\) 0 0
\(376\) 5327.29 0.730675
\(377\) −1687.94 −0.230592
\(378\) 0 0
\(379\) 3057.01 0.414322 0.207161 0.978307i \(-0.433578\pi\)
0.207161 + 0.978307i \(0.433578\pi\)
\(380\) 2082.91 0.281187
\(381\) 0 0
\(382\) 1401.67 0.187737
\(383\) 10579.7 1.41149 0.705744 0.708467i \(-0.250613\pi\)
0.705744 + 0.708467i \(0.250613\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2331.10 0.307383
\(387\) 0 0
\(388\) 655.691 0.0857930
\(389\) −7393.29 −0.963637 −0.481819 0.876271i \(-0.660024\pi\)
−0.481819 + 0.876271i \(0.660024\pi\)
\(390\) 0 0
\(391\) −2743.31 −0.354821
\(392\) 0 0
\(393\) 0 0
\(394\) 1521.75 0.194580
\(395\) 5299.86 0.675101
\(396\) 0 0
\(397\) −1778.17 −0.224796 −0.112398 0.993663i \(-0.535853\pi\)
−0.112398 + 0.993663i \(0.535853\pi\)
\(398\) 5341.04 0.672669
\(399\) 0 0
\(400\) 1934.71 0.241839
\(401\) −147.606 −0.0183818 −0.00919090 0.999958i \(-0.502926\pi\)
−0.00919090 + 0.999958i \(0.502926\pi\)
\(402\) 0 0
\(403\) 6292.12 0.777748
\(404\) 6976.63 0.859159
\(405\) 0 0
\(406\) 0 0
\(407\) −16367.9 −1.99343
\(408\) 0 0
\(409\) 2160.05 0.261143 0.130572 0.991439i \(-0.458319\pi\)
0.130572 + 0.991439i \(0.458319\pi\)
\(410\) −5939.06 −0.715388
\(411\) 0 0
\(412\) 2893.79 0.346036
\(413\) 0 0
\(414\) 0 0
\(415\) −2115.47 −0.250228
\(416\) 6056.22 0.713776
\(417\) 0 0
\(418\) −2047.11 −0.239540
\(419\) 13491.0 1.57298 0.786488 0.617605i \(-0.211897\pi\)
0.786488 + 0.617605i \(0.211897\pi\)
\(420\) 0 0
\(421\) −14146.7 −1.63769 −0.818847 0.574012i \(-0.805386\pi\)
−0.818847 + 0.574012i \(0.805386\pi\)
\(422\) −2300.03 −0.265317
\(423\) 0 0
\(424\) 3026.05 0.346598
\(425\) 17045.4 1.94546
\(426\) 0 0
\(427\) 0 0
\(428\) −726.262 −0.0820215
\(429\) 0 0
\(430\) −641.682 −0.0719643
\(431\) −9088.52 −1.01573 −0.507864 0.861437i \(-0.669565\pi\)
−0.507864 + 0.861437i \(0.669565\pi\)
\(432\) 0 0
\(433\) −15461.2 −1.71597 −0.857986 0.513673i \(-0.828284\pi\)
−0.857986 + 0.513673i \(0.828284\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1186.02 0.130275
\(437\) −705.263 −0.0772021
\(438\) 0 0
\(439\) 1782.18 0.193756 0.0968780 0.995296i \(-0.469114\pi\)
0.0968780 + 0.995296i \(0.469114\pi\)
\(440\) −24081.0 −2.60913
\(441\) 0 0
\(442\) 4216.97 0.453803
\(443\) −13424.7 −1.43979 −0.719896 0.694082i \(-0.755810\pi\)
−0.719896 + 0.694082i \(0.755810\pi\)
\(444\) 0 0
\(445\) 17182.5 1.83041
\(446\) 1613.81 0.171337
\(447\) 0 0
\(448\) 0 0
\(449\) −418.639 −0.0440018 −0.0220009 0.999758i \(-0.507004\pi\)
−0.0220009 + 0.999758i \(0.507004\pi\)
\(450\) 0 0
\(451\) −12448.0 −1.29968
\(452\) 10925.6 1.13694
\(453\) 0 0
\(454\) 4712.03 0.487107
\(455\) 0 0
\(456\) 0 0
\(457\) −708.410 −0.0725121 −0.0362560 0.999343i \(-0.511543\pi\)
−0.0362560 + 0.999343i \(0.511543\pi\)
\(458\) −6454.30 −0.658493
\(459\) 0 0
\(460\) −3360.30 −0.340598
\(461\) 8223.97 0.830865 0.415432 0.909624i \(-0.363630\pi\)
0.415432 + 0.909624i \(0.363630\pi\)
\(462\) 0 0
\(463\) −9414.17 −0.944954 −0.472477 0.881343i \(-0.656640\pi\)
−0.472477 + 0.881343i \(0.656640\pi\)
\(464\) 480.227 0.0480474
\(465\) 0 0
\(466\) 4787.45 0.475911
\(467\) −10821.5 −1.07229 −0.536146 0.844125i \(-0.680120\pi\)
−0.536146 + 0.844125i \(0.680120\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 7247.02 0.711234
\(471\) 0 0
\(472\) 4758.85 0.464076
\(473\) −1344.94 −0.130741
\(474\) 0 0
\(475\) 4382.11 0.423295
\(476\) 0 0
\(477\) 0 0
\(478\) −2892.93 −0.276819
\(479\) 8185.50 0.780804 0.390402 0.920644i \(-0.372336\pi\)
0.390402 + 0.920644i \(0.372336\pi\)
\(480\) 0 0
\(481\) 8668.19 0.821695
\(482\) −5991.79 −0.566221
\(483\) 0 0
\(484\) −13194.4 −1.23915
\(485\) 2202.21 0.206179
\(486\) 0 0
\(487\) −4003.15 −0.372485 −0.186242 0.982504i \(-0.559631\pi\)
−0.186242 + 0.982504i \(0.559631\pi\)
\(488\) −14021.3 −1.30065
\(489\) 0 0
\(490\) 0 0
\(491\) 11180.8 1.02766 0.513831 0.857891i \(-0.328226\pi\)
0.513831 + 0.857891i \(0.328226\pi\)
\(492\) 0 0
\(493\) 4230.95 0.386516
\(494\) 1084.12 0.0987386
\(495\) 0 0
\(496\) −1790.14 −0.162056
\(497\) 0 0
\(498\) 0 0
\(499\) −3771.08 −0.338310 −0.169155 0.985589i \(-0.554104\pi\)
−0.169155 + 0.985589i \(0.554104\pi\)
\(500\) 8426.48 0.753688
\(501\) 0 0
\(502\) 4325.87 0.384607
\(503\) 13597.2 1.20531 0.602654 0.798003i \(-0.294110\pi\)
0.602654 + 0.798003i \(0.294110\pi\)
\(504\) 0 0
\(505\) 23431.7 2.06475
\(506\) 3302.55 0.290151
\(507\) 0 0
\(508\) −8925.93 −0.779575
\(509\) −7360.75 −0.640982 −0.320491 0.947252i \(-0.603848\pi\)
−0.320491 + 0.947252i \(0.603848\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −3309.87 −0.285698
\(513\) 0 0
\(514\) −8590.00 −0.737137
\(515\) 9719.08 0.831600
\(516\) 0 0
\(517\) 15189.5 1.29213
\(518\) 0 0
\(519\) 0 0
\(520\) 12752.9 1.07549
\(521\) −13798.8 −1.16034 −0.580169 0.814496i \(-0.697013\pi\)
−0.580169 + 0.814496i \(0.697013\pi\)
\(522\) 0 0
\(523\) −18847.2 −1.57577 −0.787886 0.615821i \(-0.788825\pi\)
−0.787886 + 0.615821i \(0.788825\pi\)
\(524\) 499.697 0.0416591
\(525\) 0 0
\(526\) 8384.29 0.695005
\(527\) −15771.7 −1.30365
\(528\) 0 0
\(529\) −11029.2 −0.906486
\(530\) 4116.51 0.337377
\(531\) 0 0
\(532\) 0 0
\(533\) 6592.29 0.535730
\(534\) 0 0
\(535\) −2439.23 −0.197116
\(536\) 12988.9 1.04671
\(537\) 0 0
\(538\) −4815.44 −0.385889
\(539\) 0 0
\(540\) 0 0
\(541\) −14469.5 −1.14990 −0.574948 0.818190i \(-0.694978\pi\)
−0.574948 + 0.818190i \(0.694978\pi\)
\(542\) 11022.2 0.873511
\(543\) 0 0
\(544\) −15180.4 −1.19642
\(545\) 3983.36 0.313080
\(546\) 0 0
\(547\) 5749.63 0.449427 0.224713 0.974425i \(-0.427855\pi\)
0.224713 + 0.974425i \(0.427855\pi\)
\(548\) 10168.8 0.792681
\(549\) 0 0
\(550\) −20520.2 −1.59088
\(551\) 1087.71 0.0840983
\(552\) 0 0
\(553\) 0 0
\(554\) −5303.31 −0.406708
\(555\) 0 0
\(556\) 3484.37 0.265774
\(557\) 5430.77 0.413122 0.206561 0.978434i \(-0.433773\pi\)
0.206561 + 0.978434i \(0.433773\pi\)
\(558\) 0 0
\(559\) 712.260 0.0538915
\(560\) 0 0
\(561\) 0 0
\(562\) 10026.4 0.752561
\(563\) 24130.7 1.80637 0.903185 0.429251i \(-0.141223\pi\)
0.903185 + 0.429251i \(0.141223\pi\)
\(564\) 0 0
\(565\) 36694.7 2.73232
\(566\) 12420.8 0.922415
\(567\) 0 0
\(568\) 15385.9 1.13658
\(569\) 18048.5 1.32976 0.664880 0.746950i \(-0.268482\pi\)
0.664880 + 0.746950i \(0.268482\pi\)
\(570\) 0 0
\(571\) −11274.8 −0.826336 −0.413168 0.910655i \(-0.635578\pi\)
−0.413168 + 0.910655i \(0.635578\pi\)
\(572\) 10826.5 0.791396
\(573\) 0 0
\(574\) 0 0
\(575\) −7069.54 −0.512731
\(576\) 0 0
\(577\) 24094.9 1.73845 0.869225 0.494417i \(-0.164618\pi\)
0.869225 + 0.494417i \(0.164618\pi\)
\(578\) −2718.91 −0.195661
\(579\) 0 0
\(580\) 5182.53 0.371022
\(581\) 0 0
\(582\) 0 0
\(583\) 8628.04 0.612928
\(584\) −8356.10 −0.592085
\(585\) 0 0
\(586\) 1366.17 0.0963072
\(587\) −11438.9 −0.804315 −0.402157 0.915571i \(-0.631740\pi\)
−0.402157 + 0.915571i \(0.631740\pi\)
\(588\) 0 0
\(589\) −4054.66 −0.283649
\(590\) 6473.74 0.451729
\(591\) 0 0
\(592\) −2466.14 −0.171213
\(593\) 4174.44 0.289079 0.144539 0.989499i \(-0.453830\pi\)
0.144539 + 0.989499i \(0.453830\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 17082.2 1.17402
\(597\) 0 0
\(598\) −1748.98 −0.119601
\(599\) −11455.5 −0.781403 −0.390702 0.920517i \(-0.627768\pi\)
−0.390702 + 0.920517i \(0.627768\pi\)
\(600\) 0 0
\(601\) 17539.2 1.19042 0.595208 0.803572i \(-0.297070\pi\)
0.595208 + 0.803572i \(0.297070\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 14821.2 0.998452
\(605\) −44314.9 −2.97794
\(606\) 0 0
\(607\) −3385.72 −0.226396 −0.113198 0.993572i \(-0.536109\pi\)
−0.113198 + 0.993572i \(0.536109\pi\)
\(608\) −3902.65 −0.260318
\(609\) 0 0
\(610\) −19074.1 −1.26604
\(611\) −8044.11 −0.532619
\(612\) 0 0
\(613\) 4271.88 0.281468 0.140734 0.990047i \(-0.455054\pi\)
0.140734 + 0.990047i \(0.455054\pi\)
\(614\) 4076.12 0.267913
\(615\) 0 0
\(616\) 0 0
\(617\) −13123.9 −0.856321 −0.428160 0.903703i \(-0.640838\pi\)
−0.428160 + 0.903703i \(0.640838\pi\)
\(618\) 0 0
\(619\) 1893.94 0.122979 0.0614893 0.998108i \(-0.480415\pi\)
0.0614893 + 0.998108i \(0.480415\pi\)
\(620\) −19318.9 −1.25139
\(621\) 0 0
\(622\) 11954.3 0.770614
\(623\) 0 0
\(624\) 0 0
\(625\) 2102.97 0.134590
\(626\) 9.98439 0.000637470 0
\(627\) 0 0
\(628\) 15683.7 0.996576
\(629\) −21727.5 −1.37731
\(630\) 0 0
\(631\) 20443.8 1.28979 0.644894 0.764272i \(-0.276902\pi\)
0.644894 + 0.764272i \(0.276902\pi\)
\(632\) −6225.90 −0.391856
\(633\) 0 0
\(634\) −1543.41 −0.0966821
\(635\) −29978.6 −1.87349
\(636\) 0 0
\(637\) 0 0
\(638\) −5093.46 −0.316069
\(639\) 0 0
\(640\) −20753.3 −1.28179
\(641\) −19228.5 −1.18484 −0.592419 0.805630i \(-0.701827\pi\)
−0.592419 + 0.805630i \(0.701827\pi\)
\(642\) 0 0
\(643\) 18525.1 1.13617 0.568087 0.822969i \(-0.307684\pi\)
0.568087 + 0.822969i \(0.307684\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −2717.43 −0.165504
\(647\) 8022.39 0.487470 0.243735 0.969842i \(-0.421627\pi\)
0.243735 + 0.969842i \(0.421627\pi\)
\(648\) 0 0
\(649\) 13568.7 0.820676
\(650\) 10867.2 0.655764
\(651\) 0 0
\(652\) −3519.03 −0.211374
\(653\) −2051.69 −0.122954 −0.0614769 0.998109i \(-0.519581\pi\)
−0.0614769 + 0.998109i \(0.519581\pi\)
\(654\) 0 0
\(655\) 1678.28 0.100116
\(656\) −1875.54 −0.111627
\(657\) 0 0
\(658\) 0 0
\(659\) −14765.2 −0.872792 −0.436396 0.899755i \(-0.643745\pi\)
−0.436396 + 0.899755i \(0.643745\pi\)
\(660\) 0 0
\(661\) −647.064 −0.0380755 −0.0190377 0.999819i \(-0.506060\pi\)
−0.0190377 + 0.999819i \(0.506060\pi\)
\(662\) −10723.3 −0.629564
\(663\) 0 0
\(664\) 2485.11 0.145243
\(665\) 0 0
\(666\) 0 0
\(667\) −1754.78 −0.101867
\(668\) −20505.2 −1.18768
\(669\) 0 0
\(670\) 17669.6 1.01886
\(671\) −39978.5 −2.30008
\(672\) 0 0
\(673\) −22596.6 −1.29426 −0.647130 0.762380i \(-0.724031\pi\)
−0.647130 + 0.762380i \(0.724031\pi\)
\(674\) 967.695 0.0553030
\(675\) 0 0
\(676\) 6231.80 0.354563
\(677\) 25204.3 1.43084 0.715420 0.698695i \(-0.246235\pi\)
0.715420 + 0.698695i \(0.246235\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −31966.2 −1.80272
\(681\) 0 0
\(682\) 18986.8 1.06605
\(683\) 17020.5 0.953547 0.476774 0.879026i \(-0.341806\pi\)
0.476774 + 0.879026i \(0.341806\pi\)
\(684\) 0 0
\(685\) 34152.9 1.90499
\(686\) 0 0
\(687\) 0 0
\(688\) −202.641 −0.0112291
\(689\) −4569.28 −0.252650
\(690\) 0 0
\(691\) 19290.5 1.06201 0.531003 0.847370i \(-0.321815\pi\)
0.531003 + 0.847370i \(0.321815\pi\)
\(692\) 12575.2 0.690806
\(693\) 0 0
\(694\) −11087.6 −0.606452
\(695\) 11702.6 0.638713
\(696\) 0 0
\(697\) −16524.1 −0.897983
\(698\) 16570.0 0.898545
\(699\) 0 0
\(700\) 0 0
\(701\) 28511.4 1.53618 0.768088 0.640345i \(-0.221208\pi\)
0.768088 + 0.640345i \(0.221208\pi\)
\(702\) 0 0
\(703\) −5585.81 −0.299677
\(704\) 13750.5 0.736138
\(705\) 0 0
\(706\) −9399.37 −0.501062
\(707\) 0 0
\(708\) 0 0
\(709\) 28426.5 1.50575 0.752877 0.658161i \(-0.228665\pi\)
0.752877 + 0.658161i \(0.228665\pi\)
\(710\) 20930.4 1.10634
\(711\) 0 0
\(712\) −20184.8 −1.06244
\(713\) 6541.27 0.343580
\(714\) 0 0
\(715\) 36361.9 1.90190
\(716\) −16522.5 −0.862396
\(717\) 0 0
\(718\) −941.683 −0.0489461
\(719\) 21763.3 1.12884 0.564420 0.825488i \(-0.309100\pi\)
0.564420 + 0.825488i \(0.309100\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 10262.4 0.528987
\(723\) 0 0
\(724\) −22219.0 −1.14056
\(725\) 10903.2 0.558532
\(726\) 0 0
\(727\) 13422.8 0.684763 0.342382 0.939561i \(-0.388766\pi\)
0.342382 + 0.939561i \(0.388766\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −11367.3 −0.576332
\(731\) −1785.33 −0.0903323
\(732\) 0 0
\(733\) −4559.52 −0.229754 −0.114877 0.993380i \(-0.536647\pi\)
−0.114877 + 0.993380i \(0.536647\pi\)
\(734\) −5671.04 −0.285180
\(735\) 0 0
\(736\) 6296.04 0.315319
\(737\) 37034.7 1.85101
\(738\) 0 0
\(739\) −36665.4 −1.82511 −0.912557 0.408950i \(-0.865895\pi\)
−0.912557 + 0.408950i \(0.865895\pi\)
\(740\) −26614.2 −1.32210
\(741\) 0 0
\(742\) 0 0
\(743\) −10321.3 −0.509625 −0.254813 0.966990i \(-0.582014\pi\)
−0.254813 + 0.966990i \(0.582014\pi\)
\(744\) 0 0
\(745\) 57372.4 2.82143
\(746\) 2527.06 0.124024
\(747\) 0 0
\(748\) −27137.4 −1.32653
\(749\) 0 0
\(750\) 0 0
\(751\) −26678.1 −1.29627 −0.648134 0.761526i \(-0.724450\pi\)
−0.648134 + 0.761526i \(0.724450\pi\)
\(752\) 2288.59 0.110979
\(753\) 0 0
\(754\) 2697.42 0.130284
\(755\) 49778.4 2.39950
\(756\) 0 0
\(757\) −11630.8 −0.558425 −0.279212 0.960229i \(-0.590073\pi\)
−0.279212 + 0.960229i \(0.590073\pi\)
\(758\) −4885.27 −0.234091
\(759\) 0 0
\(760\) −8218.02 −0.392235
\(761\) −36091.7 −1.71921 −0.859607 0.510956i \(-0.829291\pi\)
−0.859607 + 0.510956i \(0.829291\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 4776.93 0.226208
\(765\) 0 0
\(766\) −16907.0 −0.797487
\(767\) −7185.79 −0.338284
\(768\) 0 0
\(769\) −33089.3 −1.55167 −0.775833 0.630938i \(-0.782670\pi\)
−0.775833 + 0.630938i \(0.782670\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 7944.47 0.370373
\(773\) 30990.8 1.44199 0.720997 0.692938i \(-0.243684\pi\)
0.720997 + 0.692938i \(0.243684\pi\)
\(774\) 0 0
\(775\) −40643.8 −1.88383
\(776\) −2587.00 −0.119675
\(777\) 0 0
\(778\) 11814.9 0.544453
\(779\) −4248.09 −0.195383
\(780\) 0 0
\(781\) 43869.2 2.00994
\(782\) 4383.95 0.200473
\(783\) 0 0
\(784\) 0 0
\(785\) 52675.4 2.39499
\(786\) 0 0
\(787\) 25421.2 1.15142 0.575711 0.817653i \(-0.304725\pi\)
0.575711 + 0.817653i \(0.304725\pi\)
\(788\) 5186.17 0.234454
\(789\) 0 0
\(790\) −8469.46 −0.381430
\(791\) 0 0
\(792\) 0 0
\(793\) 21172.0 0.948096
\(794\) 2841.62 0.127009
\(795\) 0 0
\(796\) 18202.4 0.810513
\(797\) 20283.3 0.901470 0.450735 0.892658i \(-0.351162\pi\)
0.450735 + 0.892658i \(0.351162\pi\)
\(798\) 0 0
\(799\) 20163.2 0.892768
\(800\) −39120.1 −1.72888
\(801\) 0 0
\(802\) 235.883 0.0103857
\(803\) −23825.4 −1.04705
\(804\) 0 0
\(805\) 0 0
\(806\) −10055.1 −0.439426
\(807\) 0 0
\(808\) −27525.9 −1.19846
\(809\) 16431.3 0.714086 0.357043 0.934088i \(-0.383785\pi\)
0.357043 + 0.934088i \(0.383785\pi\)
\(810\) 0 0
\(811\) −6371.81 −0.275887 −0.137944 0.990440i \(-0.544049\pi\)
−0.137944 + 0.990440i \(0.544049\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 26156.8 1.12628
\(815\) −11819.0 −0.507979
\(816\) 0 0
\(817\) −458.982 −0.0196545
\(818\) −3451.88 −0.147545
\(819\) 0 0
\(820\) −20240.5 −0.861987
\(821\) −5224.48 −0.222090 −0.111045 0.993815i \(-0.535420\pi\)
−0.111045 + 0.993815i \(0.535420\pi\)
\(822\) 0 0
\(823\) 3713.10 0.157267 0.0786333 0.996904i \(-0.474944\pi\)
0.0786333 + 0.996904i \(0.474944\pi\)
\(824\) −11417.3 −0.482695
\(825\) 0 0
\(826\) 0 0
\(827\) 10202.6 0.428996 0.214498 0.976724i \(-0.431188\pi\)
0.214498 + 0.976724i \(0.431188\pi\)
\(828\) 0 0
\(829\) −24995.2 −1.04719 −0.523594 0.851968i \(-0.675409\pi\)
−0.523594 + 0.851968i \(0.675409\pi\)
\(830\) 3380.64 0.141378
\(831\) 0 0
\(832\) −7282.06 −0.303437
\(833\) 0 0
\(834\) 0 0
\(835\) −68868.8 −2.85425
\(836\) −6976.63 −0.288626
\(837\) 0 0
\(838\) −21559.3 −0.888728
\(839\) −31173.6 −1.28275 −0.641377 0.767226i \(-0.721637\pi\)
−0.641377 + 0.767226i \(0.721637\pi\)
\(840\) 0 0
\(841\) −21682.6 −0.889033
\(842\) 22607.2 0.925293
\(843\) 0 0
\(844\) −7838.57 −0.319686
\(845\) 20930.1 0.852093
\(846\) 0 0
\(847\) 0 0
\(848\) 1299.98 0.0526434
\(849\) 0 0
\(850\) −27239.4 −1.09918
\(851\) 9011.43 0.362994
\(852\) 0 0
\(853\) 25280.7 1.01477 0.507383 0.861721i \(-0.330613\pi\)
0.507383 + 0.861721i \(0.330613\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 2865.43 0.114414
\(857\) 27410.7 1.09257 0.546285 0.837599i \(-0.316041\pi\)
0.546285 + 0.837599i \(0.316041\pi\)
\(858\) 0 0
\(859\) −24558.8 −0.975480 −0.487740 0.872989i \(-0.662179\pi\)
−0.487740 + 0.872989i \(0.662179\pi\)
\(860\) −2186.87 −0.0867113
\(861\) 0 0
\(862\) 14523.9 0.573883
\(863\) −7148.35 −0.281962 −0.140981 0.990012i \(-0.545026\pi\)
−0.140981 + 0.990012i \(0.545026\pi\)
\(864\) 0 0
\(865\) 42235.1 1.66016
\(866\) 24707.8 0.969520
\(867\) 0 0
\(868\) 0 0
\(869\) −17751.7 −0.692962
\(870\) 0 0
\(871\) −19613.0 −0.762988
\(872\) −4679.37 −0.181724
\(873\) 0 0
\(874\) 1127.05 0.0436190
\(875\) 0 0
\(876\) 0 0
\(877\) −28218.5 −1.08651 −0.543256 0.839567i \(-0.682809\pi\)
−0.543256 + 0.839567i \(0.682809\pi\)
\(878\) −2848.02 −0.109472
\(879\) 0 0
\(880\) −10345.1 −0.396289
\(881\) −7431.50 −0.284192 −0.142096 0.989853i \(-0.545384\pi\)
−0.142096 + 0.989853i \(0.545384\pi\)
\(882\) 0 0
\(883\) 4937.77 0.188187 0.0940936 0.995563i \(-0.470005\pi\)
0.0940936 + 0.995563i \(0.470005\pi\)
\(884\) 14371.6 0.546797
\(885\) 0 0
\(886\) 21453.4 0.813478
\(887\) 41973.7 1.58888 0.794441 0.607341i \(-0.207764\pi\)
0.794441 + 0.607341i \(0.207764\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −27458.6 −1.03417
\(891\) 0 0
\(892\) 5499.93 0.206448
\(893\) 5183.65 0.194249
\(894\) 0 0
\(895\) −55492.5 −2.07253
\(896\) 0 0
\(897\) 0 0
\(898\) 669.008 0.0248609
\(899\) −10088.5 −0.374271
\(900\) 0 0
\(901\) 11453.2 0.423488
\(902\) 19892.6 0.734315
\(903\) 0 0
\(904\) −43106.4 −1.58595
\(905\) −74624.7 −2.74101
\(906\) 0 0
\(907\) 41424.2 1.51650 0.758252 0.651961i \(-0.226054\pi\)
0.758252 + 0.651961i \(0.226054\pi\)
\(908\) 16058.7 0.586925
\(909\) 0 0
\(910\) 0 0
\(911\) −40072.0 −1.45735 −0.728675 0.684860i \(-0.759864\pi\)
−0.728675 + 0.684860i \(0.759864\pi\)
\(912\) 0 0
\(913\) 7085.70 0.256848
\(914\) 1132.08 0.0409691
\(915\) 0 0
\(916\) −21996.5 −0.793432
\(917\) 0 0
\(918\) 0 0
\(919\) −21817.5 −0.783124 −0.391562 0.920152i \(-0.628065\pi\)
−0.391562 + 0.920152i \(0.628065\pi\)
\(920\) 13257.9 0.475109
\(921\) 0 0
\(922\) −13142.4 −0.469437
\(923\) −23232.5 −0.828501
\(924\) 0 0
\(925\) −55992.0 −1.99028
\(926\) 15044.4 0.533897
\(927\) 0 0
\(928\) −9710.26 −0.343486
\(929\) −11977.0 −0.422986 −0.211493 0.977380i \(-0.567833\pi\)
−0.211493 + 0.977380i \(0.567833\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 16315.8 0.573435
\(933\) 0 0
\(934\) 17293.4 0.605842
\(935\) −91143.8 −3.18794
\(936\) 0 0
\(937\) 15155.2 0.528389 0.264194 0.964469i \(-0.414894\pi\)
0.264194 + 0.964469i \(0.414894\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 24698.1 0.856981
\(941\) 6954.40 0.240921 0.120461 0.992718i \(-0.461563\pi\)
0.120461 + 0.992718i \(0.461563\pi\)
\(942\) 0 0
\(943\) 6853.33 0.236665
\(944\) 2044.39 0.0704865
\(945\) 0 0
\(946\) 2149.29 0.0738682
\(947\) 9557.49 0.327959 0.163979 0.986464i \(-0.447567\pi\)
0.163979 + 0.986464i \(0.447567\pi\)
\(948\) 0 0
\(949\) 12617.6 0.431595
\(950\) −7002.85 −0.239161
\(951\) 0 0
\(952\) 0 0
\(953\) −8437.24 −0.286788 −0.143394 0.989666i \(-0.545802\pi\)
−0.143394 + 0.989666i \(0.545802\pi\)
\(954\) 0 0
\(955\) 16043.8 0.543628
\(956\) −9859.21 −0.333546
\(957\) 0 0
\(958\) −13080.9 −0.441153
\(959\) 0 0
\(960\) 0 0
\(961\) 7815.69 0.262351
\(962\) −13852.2 −0.464256
\(963\) 0 0
\(964\) −20420.2 −0.682252
\(965\) 26682.3 0.890087
\(966\) 0 0
\(967\) 52344.7 1.74074 0.870369 0.492401i \(-0.163881\pi\)
0.870369 + 0.492401i \(0.163881\pi\)
\(968\) 52058.0 1.72852
\(969\) 0 0
\(970\) −3519.24 −0.116491
\(971\) −36983.0 −1.22229 −0.611143 0.791520i \(-0.709290\pi\)
−0.611143 + 0.791520i \(0.709290\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 6397.25 0.210453
\(975\) 0 0
\(976\) −6023.54 −0.197550
\(977\) 17873.0 0.585270 0.292635 0.956224i \(-0.405468\pi\)
0.292635 + 0.956224i \(0.405468\pi\)
\(978\) 0 0
\(979\) −57552.2 −1.87883
\(980\) 0 0
\(981\) 0 0
\(982\) −17867.5 −0.580627
\(983\) −33744.0 −1.09488 −0.547440 0.836845i \(-0.684398\pi\)
−0.547440 + 0.836845i \(0.684398\pi\)
\(984\) 0 0
\(985\) 17418.3 0.563444
\(986\) −6761.29 −0.218381
\(987\) 0 0
\(988\) 3694.72 0.118972
\(989\) 740.464 0.0238073
\(990\) 0 0
\(991\) −14506.5 −0.464999 −0.232499 0.972597i \(-0.574690\pi\)
−0.232499 + 0.972597i \(0.574690\pi\)
\(992\) 36196.8 1.15852
\(993\) 0 0
\(994\) 0 0
\(995\) 61134.7 1.94784
\(996\) 0 0
\(997\) 42368.8 1.34587 0.672936 0.739701i \(-0.265033\pi\)
0.672936 + 0.739701i \(0.265033\pi\)
\(998\) 6026.40 0.191145
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 441.4.a.v.1.2 4
3.2 odd 2 inner 441.4.a.v.1.3 4
7.2 even 3 441.4.e.x.361.3 8
7.3 odd 6 63.4.e.d.37.3 yes 8
7.4 even 3 441.4.e.x.226.3 8
7.5 odd 6 63.4.e.d.46.3 yes 8
7.6 odd 2 441.4.a.w.1.2 4
21.2 odd 6 441.4.e.x.361.2 8
21.5 even 6 63.4.e.d.46.2 yes 8
21.11 odd 6 441.4.e.x.226.2 8
21.17 even 6 63.4.e.d.37.2 8
21.20 even 2 441.4.a.w.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.4.e.d.37.2 8 21.17 even 6
63.4.e.d.37.3 yes 8 7.3 odd 6
63.4.e.d.46.2 yes 8 21.5 even 6
63.4.e.d.46.3 yes 8 7.5 odd 6
441.4.a.v.1.2 4 1.1 even 1 trivial
441.4.a.v.1.3 4 3.2 odd 2 inner
441.4.a.w.1.2 4 7.6 odd 2
441.4.a.w.1.3 4 21.20 even 2
441.4.e.x.226.2 8 21.11 odd 6
441.4.e.x.226.3 8 7.4 even 3
441.4.e.x.361.2 8 21.2 odd 6
441.4.e.x.361.3 8 7.2 even 3