Properties

Label 441.4.a.w.1.2
Level $441$
Weight $4$
Character 441.1
Self dual yes
Analytic conductor $26.020$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(0\)
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 \(-0.195067\pi\)
0.818029 + 0.575177i \(0.195067\pi\)
\(6\) 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 \(-0.387501\pi\)
0.346114 + 0.938192i \(0.387501\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 9.23111 0.144236
\(17\) −81.3289 −1.16030 −0.580152 0.814509i \(-0.697007\pi\)
−0.580152 + 0.814509i \(0.697007\pi\)
\(18\) 0 0
\(19\) −20.9084 −0.252459 −0.126230 0.992001i \(-0.540288\pi\)
−0.126230 + 0.992001i \(0.540288\pi\)
\(20\) −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 \(-0.310120\pi\)
0.561772 + 0.827292i \(0.310120\pi\)
\(32\) −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 \(-0.373525\pi\)
0.386960 + 0.922096i \(0.373525\pi\)
\(42\) 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 \(-0.625700\pi\)
−0.384713 + 0.923036i \(0.625700\pi\)
\(48\) 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 \(-0.578573\pi\)
−0.244344 + 0.969689i \(0.578573\pi\)
\(60\) 0 0
\(61\) 652.526 1.36963 0.684815 0.728717i \(-0.259883\pi\)
0.684815 + 0.728717i \(0.259883\pi\)
\(62\) −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 \(-0.399087\pi\)
0.311743 + 0.950166i \(0.399087\pi\)
\(74\) 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 \(-0.524366\pi\)
−0.0764728 + 0.997072i \(0.524366\pi\)
\(84\) 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 \(-0.311033\pi\)
0.559395 + 0.828901i \(0.311033\pi\)
\(90\) 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 \(-0.479930\pi\)
0.0630110 + 0.998013i \(0.479930\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1141.45 −1.14145
\(101\) 1281.00 1.26203 0.631013 0.775772i \(-0.282640\pi\)
0.631013 + 0.775772i \(0.282640\pi\)
\(102\) 0 0
\(103\) 531.339 0.508295 0.254147 0.967166i \(-0.418205\pi\)
0.254147 + 0.967166i \(0.418205\pi\)
\(104\) 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 \(-0.490259\pi\)
0.0305967 + 0.999532i \(0.490259\pi\)
\(132\) 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 \(-0.437465\pi\)
0.195199 + 0.980764i \(0.437465\pi\)
\(140\) 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 \(-0.238616\pi\)
0.731939 + 0.681370i \(0.238616\pi\)
\(158\) 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 \(-0.837370\pi\)
−0.872296 + 0.488979i \(0.837370\pi\)
\(168\) 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 \(-0.330619\pi\)
0.507366 + 0.861731i \(0.330619\pi\)
\(174\) 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 \(-0.816092\pi\)
−0.837686 + 0.546152i \(0.816092\pi\)
\(182\) 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 \(-0.297039\pi\)
0.595285 + 0.803515i \(0.297039\pi\)
\(200\) 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 \(-0.451549\pi\)
0.151626 + 0.988438i \(0.451549\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 3205.84 0.943580
\(227\) 2948.60 0.862140 0.431070 0.902319i \(-0.358136\pi\)
0.431070 + 0.902319i \(0.358136\pi\)
\(228\) 0 0
\(229\) −4038.85 −1.16548 −0.582739 0.812659i \(-0.698019\pi\)
−0.582739 + 0.812659i \(0.698019\pi\)
\(230\) 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 \(-0.667065\pi\)
−0.501083 + 0.865399i \(0.667065\pi\)
\(242\) −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 \(-0.389450\pi\)
0.340362 + 0.940295i \(0.389450\pi\)
\(252\) 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 \(-0.726211\pi\)
−0.652337 + 0.757929i \(0.726211\pi\)
\(258\) 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 \(-0.610934\pi\)
−0.341496 + 0.939883i \(0.610934\pi\)
\(270\) 0 0
\(271\) 6897.25 1.54604 0.773022 0.634379i \(-0.218744\pi\)
0.773022 + 0.634379i \(0.218744\pi\)
\(272\) −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 \(-0.196022\pi\)
0.816300 + 0.577629i \(0.196022\pi\)
\(284\) −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 \(-0.472838\pi\)
0.0852280 + 0.996361i \(0.472838\pi\)
\(294\) 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 \(-0.423806\pi\)
0.237092 + 0.971487i \(0.423806\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −5668.63 −1.03857
\(311\) 7480.50 1.36392 0.681962 0.731388i \(-0.261127\pi\)
0.681962 + 0.731388i \(0.261127\pi\)
\(312\) 0 0
\(313\) 6.24784 0.00112827 0.000564135 1.00000i \(-0.499820\pi\)
0.000564135 1.00000i \(0.499820\pi\)
\(314\) −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 \(-0.207378\pi\)
0.795176 + 0.606378i \(0.207378\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −11435.8 −1.73162
\(353\) −5881.76 −0.886840 −0.443420 0.896314i \(-0.646235\pi\)
−0.443420 + 0.896314i \(0.646235\pi\)
\(354\) 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 \(-0.581211\pi\)
−0.252373 + 0.967630i \(0.581211\pi\)
\(368\) −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 \(-0.749387\pi\)
−0.705744 + 0.708467i \(0.749387\pi\)
\(384\) 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 \(-0.464147\pi\)
0.112398 + 0.993663i \(0.464147\pi\)
\(398\) −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 \(-0.541681\pi\)
−0.130572 + 0.991439i \(0.541681\pi\)
\(410\) −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 \(-0.788103\pi\)
−0.786488 + 0.617605i \(0.788103\pi\)
\(420\) 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 \(-0.171716\pi\)
0.857986 + 0.513673i \(0.171716\pi\)
\(434\) 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 \(-0.530886\pi\)
−0.0968780 + 0.995296i \(0.530886\pi\)
\(440\) 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 \(-0.636370\pi\)
−0.415432 + 0.909624i \(0.636370\pi\)
\(462\) 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 \(-0.319880\pi\)
0.536146 + 0.844125i \(0.319880\pi\)
\(468\) 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 \(-0.627664\pi\)
−0.390402 + 0.920644i \(0.627664\pi\)
\(480\) 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 \(-0.705890\pi\)
−0.602654 + 0.798003i \(0.705890\pi\)
\(504\) 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 \(-0.396152\pi\)
0.320491 + 0.947252i \(0.396152\pi\)
\(510\) 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 \(-0.302987\pi\)
0.580169 + 0.814496i \(0.302987\pi\)
\(522\) 0 0
\(523\) 18847.2 1.57577 0.787886 0.615821i \(-0.211175\pi\)
0.787886 + 0.615821i \(0.211175\pi\)
\(524\) −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 \(-0.858777\pi\)
−0.903185 + 0.429251i \(0.858777\pi\)
\(564\) 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 \(-0.835382\pi\)
−0.869225 + 0.494417i \(0.835382\pi\)
\(578\) −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 \(-0.368260\pi\)
0.402157 + 0.915571i \(0.368260\pi\)
\(588\) 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 \(-0.546170\pi\)
−0.144539 + 0.989499i \(0.546170\pi\)
\(594\) 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 \(-0.702930\pi\)
−0.595208 + 0.803572i \(0.702930\pi\)
\(602\) 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 \(-0.463891\pi\)
0.113198 + 0.993572i \(0.463891\pi\)
\(608\) 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 \(-0.519585\pi\)
−0.0614893 + 0.998108i \(0.519585\pi\)
\(620\) −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 \(-0.692316\pi\)
−0.568087 + 0.822969i \(0.692316\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −2717.43 −0.165504
\(647\) −8022.39 −0.487470 −0.243735 0.969842i \(-0.578373\pi\)
−0.243735 + 0.969842i \(0.578373\pi\)
\(648\) 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 \(-0.493940\pi\)
0.0190377 + 0.999819i \(0.493940\pi\)
\(662\) −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 \(-0.753765\pi\)
−0.715420 + 0.698695i \(0.753765\pi\)
\(678\) 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 \(-0.678185\pi\)
−0.531003 + 0.847370i \(0.678185\pi\)
\(692\) −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 \(-0.690900\pi\)
−0.564420 + 0.825488i \(0.690900\pi\)
\(720\) 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 \(-0.611234\pi\)
−0.342382 + 0.939561i \(0.611234\pi\)
\(728\) 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 \(-0.463353\pi\)
0.114877 + 0.993380i \(0.463353\pi\)
\(734\) 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 \(-0.170709\pi\)
0.859607 + 0.510956i \(0.170709\pi\)
\(762\) 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 \(-0.217330\pi\)
0.775833 + 0.630938i \(0.217330\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 7944.47 0.370373
\(773\) −30990.8 −1.44199 −0.720997 0.692938i \(-0.756316\pi\)
−0.720997 + 0.692938i \(0.756316\pi\)
\(774\) 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 \(-0.695275\pi\)
−0.575711 + 0.817653i \(0.695275\pi\)
\(788\) 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 \(-0.648838\pi\)
−0.450735 + 0.892658i \(0.648838\pi\)
\(798\) 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 \(-0.455951\pi\)
0.137944 + 0.990440i \(0.455951\pi\)
\(812\) 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 \(-0.324591\pi\)
0.523594 + 0.851968i \(0.324591\pi\)
\(830\) 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 \(-0.278363\pi\)
0.641377 + 0.767226i \(0.278363\pi\)
\(840\) 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 \(-0.669387\pi\)
−0.507383 + 0.861721i \(0.669387\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 2865.43 0.114414
\(857\) −27410.7 −1.09257 −0.546285 0.837599i \(-0.683959\pi\)
−0.546285 + 0.837599i \(0.683959\pi\)
\(858\) 0 0
\(859\) 24558.8 0.975480 0.487740 0.872989i \(-0.337821\pi\)
0.487740 + 0.872989i \(0.337821\pi\)
\(860\) 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 \(-0.454616\pi\)
0.142096 + 0.989853i \(0.454616\pi\)
\(882\) 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 \(-0.792236\pi\)
−0.794441 + 0.607341i \(0.792236\pi\)
\(888\) 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 \(-0.432167\pi\)
0.211493 + 0.977380i \(0.432167\pi\)
\(930\) 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 \(-0.585106\pi\)
−0.264194 + 0.964469i \(0.585106\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 24698.1 0.856981
\(941\) −6954.40 −0.240921 −0.120461 0.992718i \(-0.538437\pi\)
−0.120461 + 0.992718i \(0.538437\pi\)
\(942\) 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 \(-0.290710\pi\)
0.611143 + 0.791520i \(0.290710\pi\)
\(972\) 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 \(-0.315602\pi\)
0.547440 + 0.836845i \(0.315602\pi\)
\(984\) 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 \(-0.734967\pi\)
−0.672936 + 0.739701i \(0.734967\pi\)
\(998\) 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.w.1.2 4
3.2 odd 2 inner 441.4.a.w.1.3 4
7.2 even 3 63.4.e.d.46.3 yes 8
7.3 odd 6 441.4.e.x.226.3 8
7.4 even 3 63.4.e.d.37.3 yes 8
7.5 odd 6 441.4.e.x.361.3 8
7.6 odd 2 441.4.a.v.1.2 4
21.2 odd 6 63.4.e.d.46.2 yes 8
21.5 even 6 441.4.e.x.361.2 8
21.11 odd 6 63.4.e.d.37.2 8
21.17 even 6 441.4.e.x.226.2 8
21.20 even 2 441.4.a.v.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.4.e.d.37.2 8 21.11 odd 6
63.4.e.d.37.3 yes 8 7.4 even 3
63.4.e.d.46.2 yes 8 21.2 odd 6
63.4.e.d.46.3 yes 8 7.2 even 3
441.4.a.v.1.2 4 7.6 odd 2
441.4.a.v.1.3 4 21.20 even 2
441.4.a.w.1.2 4 1.1 even 1 trivial
441.4.a.w.1.3 4 3.2 odd 2 inner
441.4.e.x.226.2 8 21.17 even 6
441.4.e.x.226.3 8 7.3 odd 6
441.4.e.x.361.2 8 21.5 even 6
441.4.e.x.361.3 8 7.5 odd 6