Properties

Label 1521.4.a.bf.1.9
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 13^{2} \)
Twist minimal: no (minimal twist has level 507)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(5.39246\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.83750 q^{2} +15.4014 q^{4} -21.1983 q^{5} +16.2806 q^{7} +35.8043 q^{8} +O(q^{10})\) \(q+4.83750 q^{2} +15.4014 q^{4} -21.1983 q^{5} +16.2806 q^{7} +35.8043 q^{8} -102.547 q^{10} -30.7532 q^{11} +78.7575 q^{14} +49.9922 q^{16} -46.2371 q^{17} +144.865 q^{19} -326.483 q^{20} -148.769 q^{22} -8.38045 q^{23} +324.366 q^{25} +250.744 q^{28} +242.958 q^{29} +87.9353 q^{31} -44.5973 q^{32} -223.672 q^{34} -345.121 q^{35} -49.6950 q^{37} +700.783 q^{38} -758.990 q^{40} +107.947 q^{41} -35.4166 q^{43} -473.643 q^{44} -40.5404 q^{46} +374.815 q^{47} -77.9418 q^{49} +1569.12 q^{50} +348.583 q^{53} +651.915 q^{55} +582.916 q^{56} +1175.31 q^{58} +679.430 q^{59} -230.403 q^{61} +425.387 q^{62} -615.677 q^{64} +295.642 q^{67} -712.117 q^{68} -1669.52 q^{70} +329.215 q^{71} -48.9973 q^{73} -240.399 q^{74} +2231.12 q^{76} -500.681 q^{77} -107.942 q^{79} -1059.75 q^{80} +522.192 q^{82} +515.654 q^{83} +980.147 q^{85} -171.328 q^{86} -1101.10 q^{88} +984.453 q^{89} -129.071 q^{92} +1813.17 q^{94} -3070.88 q^{95} -487.072 q^{97} -377.043 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8} - 54 q^{10} - 85 q^{11} - 158 q^{14} + 216 q^{16} - 178 q^{17} + 352 q^{19} - 402 q^{20} - 630 q^{22} - 150 q^{23} - 20 q^{25} + 940 q^{28} + 97 q^{29} + 717 q^{31} - 707 q^{32} + 632 q^{34} + 418 q^{35} + 1108 q^{37} + 660 q^{38} - 1506 q^{40} - 334 q^{41} + 242 q^{43} + 307 q^{44} + 979 q^{46} + 184 q^{47} - 38 q^{49} + 2031 q^{50} + 151 q^{53} + 2064 q^{55} - 2276 q^{56} + 1161 q^{58} - 537 q^{59} - 1340 q^{61} - 347 q^{62} + 893 q^{64} + 2308 q^{67} - 2785 q^{68} - 1420 q^{70} - 96 q^{71} + 2505 q^{73} + 1191 q^{74} + 2409 q^{76} + 2142 q^{77} - 1591 q^{79} + 2671 q^{80} + 1517 q^{82} - 1539 q^{83} + 4296 q^{85} + 3763 q^{86} - 3716 q^{88} + 592 q^{89} - 515 q^{92} - 692 q^{94} - 4158 q^{95} + 1445 q^{97} - 1457 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.83750 1.71031 0.855157 0.518368i \(-0.173460\pi\)
0.855157 + 0.518368i \(0.173460\pi\)
\(3\) 0 0
\(4\) 15.4014 1.92518
\(5\) −21.1983 −1.89603 −0.948015 0.318225i \(-0.896913\pi\)
−0.948015 + 0.318225i \(0.896913\pi\)
\(6\) 0 0
\(7\) 16.2806 0.879070 0.439535 0.898225i \(-0.355143\pi\)
0.439535 + 0.898225i \(0.355143\pi\)
\(8\) 35.8043 1.58234
\(9\) 0 0
\(10\) −102.547 −3.24281
\(11\) −30.7532 −0.842950 −0.421475 0.906840i \(-0.638487\pi\)
−0.421475 + 0.906840i \(0.638487\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 78.7575 1.50349
\(15\) 0 0
\(16\) 49.9922 0.781128
\(17\) −46.2371 −0.659656 −0.329828 0.944041i \(-0.606991\pi\)
−0.329828 + 0.944041i \(0.606991\pi\)
\(18\) 0 0
\(19\) 144.865 1.74917 0.874585 0.484873i \(-0.161134\pi\)
0.874585 + 0.484873i \(0.161134\pi\)
\(20\) −326.483 −3.65019
\(21\) 0 0
\(22\) −148.769 −1.44171
\(23\) −8.38045 −0.0759758 −0.0379879 0.999278i \(-0.512095\pi\)
−0.0379879 + 0.999278i \(0.512095\pi\)
\(24\) 0 0
\(25\) 324.366 2.59493
\(26\) 0 0
\(27\) 0 0
\(28\) 250.744 1.69237
\(29\) 242.958 1.55573 0.777865 0.628431i \(-0.216303\pi\)
0.777865 + 0.628431i \(0.216303\pi\)
\(30\) 0 0
\(31\) 87.9353 0.509472 0.254736 0.967011i \(-0.418011\pi\)
0.254736 + 0.967011i \(0.418011\pi\)
\(32\) −44.5973 −0.246368
\(33\) 0 0
\(34\) −223.672 −1.12822
\(35\) −345.121 −1.66674
\(36\) 0 0
\(37\) −49.6950 −0.220806 −0.110403 0.993887i \(-0.535214\pi\)
−0.110403 + 0.993887i \(0.535214\pi\)
\(38\) 700.783 2.99163
\(39\) 0 0
\(40\) −758.990 −3.00017
\(41\) 107.947 0.411181 0.205591 0.978638i \(-0.434088\pi\)
0.205591 + 0.978638i \(0.434088\pi\)
\(42\) 0 0
\(43\) −35.4166 −0.125604 −0.0628021 0.998026i \(-0.520004\pi\)
−0.0628021 + 0.998026i \(0.520004\pi\)
\(44\) −473.643 −1.62283
\(45\) 0 0
\(46\) −40.5404 −0.129943
\(47\) 374.815 1.16324 0.581621 0.813460i \(-0.302419\pi\)
0.581621 + 0.813460i \(0.302419\pi\)
\(48\) 0 0
\(49\) −77.9418 −0.227236
\(50\) 1569.12 4.43815
\(51\) 0 0
\(52\) 0 0
\(53\) 348.583 0.903426 0.451713 0.892163i \(-0.350813\pi\)
0.451713 + 0.892163i \(0.350813\pi\)
\(54\) 0 0
\(55\) 651.915 1.59826
\(56\) 582.916 1.39099
\(57\) 0 0
\(58\) 1175.31 2.66079
\(59\) 679.430 1.49923 0.749613 0.661877i \(-0.230240\pi\)
0.749613 + 0.661877i \(0.230240\pi\)
\(60\) 0 0
\(61\) −230.403 −0.483608 −0.241804 0.970325i \(-0.577739\pi\)
−0.241804 + 0.970325i \(0.577739\pi\)
\(62\) 425.387 0.871358
\(63\) 0 0
\(64\) −615.677 −1.20249
\(65\) 0 0
\(66\) 0 0
\(67\) 295.642 0.539082 0.269541 0.962989i \(-0.413128\pi\)
0.269541 + 0.962989i \(0.413128\pi\)
\(68\) −712.117 −1.26995
\(69\) 0 0
\(70\) −1669.52 −2.85066
\(71\) 329.215 0.550290 0.275145 0.961403i \(-0.411274\pi\)
0.275145 + 0.961403i \(0.411274\pi\)
\(72\) 0 0
\(73\) −48.9973 −0.0785575 −0.0392787 0.999228i \(-0.512506\pi\)
−0.0392787 + 0.999228i \(0.512506\pi\)
\(74\) −240.399 −0.377647
\(75\) 0 0
\(76\) 2231.12 3.36746
\(77\) −500.681 −0.741012
\(78\) 0 0
\(79\) −107.942 −0.153727 −0.0768636 0.997042i \(-0.524491\pi\)
−0.0768636 + 0.997042i \(0.524491\pi\)
\(80\) −1059.75 −1.48104
\(81\) 0 0
\(82\) 522.192 0.703250
\(83\) 515.654 0.681932 0.340966 0.940076i \(-0.389246\pi\)
0.340966 + 0.940076i \(0.389246\pi\)
\(84\) 0 0
\(85\) 980.147 1.25073
\(86\) −171.328 −0.214823
\(87\) 0 0
\(88\) −1101.10 −1.33384
\(89\) 984.453 1.17249 0.586246 0.810133i \(-0.300605\pi\)
0.586246 + 0.810133i \(0.300605\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −129.071 −0.146267
\(93\) 0 0
\(94\) 1813.17 1.98951
\(95\) −3070.88 −3.31648
\(96\) 0 0
\(97\) −487.072 −0.509842 −0.254921 0.966962i \(-0.582049\pi\)
−0.254921 + 0.966962i \(0.582049\pi\)
\(98\) −377.043 −0.388644
\(99\) 0 0
\(100\) 4995.70 4.99570
\(101\) −766.375 −0.755021 −0.377511 0.926005i \(-0.623220\pi\)
−0.377511 + 0.926005i \(0.623220\pi\)
\(102\) 0 0
\(103\) 1229.87 1.17653 0.588266 0.808668i \(-0.299811\pi\)
0.588266 + 0.808668i \(0.299811\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 1686.27 1.54514
\(107\) −76.8261 −0.0694117 −0.0347059 0.999398i \(-0.511049\pi\)
−0.0347059 + 0.999398i \(0.511049\pi\)
\(108\) 0 0
\(109\) 626.001 0.550092 0.275046 0.961431i \(-0.411307\pi\)
0.275046 + 0.961431i \(0.411307\pi\)
\(110\) 3153.64 2.73353
\(111\) 0 0
\(112\) 813.904 0.686667
\(113\) 1343.21 1.11822 0.559108 0.829095i \(-0.311144\pi\)
0.559108 + 0.829095i \(0.311144\pi\)
\(114\) 0 0
\(115\) 177.651 0.144052
\(116\) 3741.90 2.99506
\(117\) 0 0
\(118\) 3286.74 2.56415
\(119\) −752.769 −0.579884
\(120\) 0 0
\(121\) −385.238 −0.289435
\(122\) −1114.58 −0.827122
\(123\) 0 0
\(124\) 1354.33 0.980824
\(125\) −4226.22 −3.02404
\(126\) 0 0
\(127\) −2146.69 −1.49990 −0.749952 0.661492i \(-0.769923\pi\)
−0.749952 + 0.661492i \(0.769923\pi\)
\(128\) −2621.56 −1.81028
\(129\) 0 0
\(130\) 0 0
\(131\) 798.626 0.532644 0.266322 0.963884i \(-0.414192\pi\)
0.266322 + 0.963884i \(0.414192\pi\)
\(132\) 0 0
\(133\) 2358.48 1.53764
\(134\) 1430.17 0.921999
\(135\) 0 0
\(136\) −1655.49 −1.04380
\(137\) 601.153 0.374891 0.187445 0.982275i \(-0.439979\pi\)
0.187445 + 0.982275i \(0.439979\pi\)
\(138\) 0 0
\(139\) −2134.18 −1.30229 −0.651146 0.758953i \(-0.725711\pi\)
−0.651146 + 0.758953i \(0.725711\pi\)
\(140\) −5315.35 −3.20878
\(141\) 0 0
\(142\) 1592.58 0.941169
\(143\) 0 0
\(144\) 0 0
\(145\) −5150.29 −2.94971
\(146\) −237.024 −0.134358
\(147\) 0 0
\(148\) −765.373 −0.425090
\(149\) −3439.93 −1.89134 −0.945670 0.325127i \(-0.894593\pi\)
−0.945670 + 0.325127i \(0.894593\pi\)
\(150\) 0 0
\(151\) 2224.04 1.19861 0.599305 0.800521i \(-0.295444\pi\)
0.599305 + 0.800521i \(0.295444\pi\)
\(152\) 5186.78 2.76779
\(153\) 0 0
\(154\) −2422.05 −1.26736
\(155\) −1864.07 −0.965975
\(156\) 0 0
\(157\) 2465.32 1.25321 0.626604 0.779338i \(-0.284444\pi\)
0.626604 + 0.779338i \(0.284444\pi\)
\(158\) −522.171 −0.262922
\(159\) 0 0
\(160\) 945.386 0.467121
\(161\) −136.439 −0.0667881
\(162\) 0 0
\(163\) −243.565 −0.117040 −0.0585199 0.998286i \(-0.518638\pi\)
−0.0585199 + 0.998286i \(0.518638\pi\)
\(164\) 1662.53 0.791597
\(165\) 0 0
\(166\) 2494.47 1.16632
\(167\) −409.099 −0.189563 −0.0947814 0.995498i \(-0.530215\pi\)
−0.0947814 + 0.995498i \(0.530215\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 4741.46 2.13914
\(171\) 0 0
\(172\) −545.466 −0.241810
\(173\) 2618.97 1.15096 0.575481 0.817815i \(-0.304815\pi\)
0.575481 + 0.817815i \(0.304815\pi\)
\(174\) 0 0
\(175\) 5280.88 2.28113
\(176\) −1537.42 −0.658452
\(177\) 0 0
\(178\) 4762.29 2.00533
\(179\) −163.311 −0.0681925 −0.0340963 0.999419i \(-0.510855\pi\)
−0.0340963 + 0.999419i \(0.510855\pi\)
\(180\) 0 0
\(181\) −3313.07 −1.36054 −0.680272 0.732960i \(-0.738138\pi\)
−0.680272 + 0.732960i \(0.738138\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −300.056 −0.120220
\(185\) 1053.45 0.418654
\(186\) 0 0
\(187\) 1421.94 0.556057
\(188\) 5772.68 2.23945
\(189\) 0 0
\(190\) −14855.4 −5.67222
\(191\) 4281.90 1.62213 0.811066 0.584954i \(-0.198888\pi\)
0.811066 + 0.584954i \(0.198888\pi\)
\(192\) 0 0
\(193\) 1877.33 0.700171 0.350086 0.936718i \(-0.386152\pi\)
0.350086 + 0.936718i \(0.386152\pi\)
\(194\) −2356.21 −0.871990
\(195\) 0 0
\(196\) −1200.41 −0.437468
\(197\) 1991.26 0.720158 0.360079 0.932922i \(-0.382750\pi\)
0.360079 + 0.932922i \(0.382750\pi\)
\(198\) 0 0
\(199\) 1345.05 0.479137 0.239568 0.970879i \(-0.422994\pi\)
0.239568 + 0.970879i \(0.422994\pi\)
\(200\) 11613.7 4.10607
\(201\) 0 0
\(202\) −3707.34 −1.29132
\(203\) 3955.51 1.36760
\(204\) 0 0
\(205\) −2288.28 −0.779612
\(206\) 5949.50 2.01224
\(207\) 0 0
\(208\) 0 0
\(209\) −4455.05 −1.47446
\(210\) 0 0
\(211\) −288.763 −0.0942147 −0.0471073 0.998890i \(-0.515000\pi\)
−0.0471073 + 0.998890i \(0.515000\pi\)
\(212\) 5368.67 1.73925
\(213\) 0 0
\(214\) −371.646 −0.118716
\(215\) 750.771 0.238150
\(216\) 0 0
\(217\) 1431.64 0.447862
\(218\) 3028.28 0.940830
\(219\) 0 0
\(220\) 10040.4 3.07693
\(221\) 0 0
\(222\) 0 0
\(223\) −1798.41 −0.540046 −0.270023 0.962854i \(-0.587031\pi\)
−0.270023 + 0.962854i \(0.587031\pi\)
\(224\) −726.072 −0.216575
\(225\) 0 0
\(226\) 6497.77 1.91250
\(227\) −2486.10 −0.726909 −0.363454 0.931612i \(-0.618403\pi\)
−0.363454 + 0.931612i \(0.618403\pi\)
\(228\) 0 0
\(229\) 6074.71 1.75296 0.876480 0.481438i \(-0.159885\pi\)
0.876480 + 0.481438i \(0.159885\pi\)
\(230\) 859.386 0.246375
\(231\) 0 0
\(232\) 8698.95 2.46170
\(233\) −6367.16 −1.79024 −0.895121 0.445822i \(-0.852911\pi\)
−0.895121 + 0.445822i \(0.852911\pi\)
\(234\) 0 0
\(235\) −7945.42 −2.20554
\(236\) 10464.2 2.88627
\(237\) 0 0
\(238\) −3641.52 −0.991784
\(239\) 1886.43 0.510556 0.255278 0.966868i \(-0.417833\pi\)
0.255278 + 0.966868i \(0.417833\pi\)
\(240\) 0 0
\(241\) 5847.91 1.56306 0.781529 0.623869i \(-0.214440\pi\)
0.781529 + 0.623869i \(0.214440\pi\)
\(242\) −1863.59 −0.495025
\(243\) 0 0
\(244\) −3548.53 −0.931031
\(245\) 1652.23 0.430845
\(246\) 0 0
\(247\) 0 0
\(248\) 3148.46 0.806160
\(249\) 0 0
\(250\) −20444.3 −5.17205
\(251\) 2388.65 0.600678 0.300339 0.953832i \(-0.402900\pi\)
0.300339 + 0.953832i \(0.402900\pi\)
\(252\) 0 0
\(253\) 257.726 0.0640438
\(254\) −10384.6 −2.56531
\(255\) 0 0
\(256\) −7756.38 −1.89365
\(257\) −4447.21 −1.07941 −0.539707 0.841853i \(-0.681465\pi\)
−0.539707 + 0.841853i \(0.681465\pi\)
\(258\) 0 0
\(259\) −809.064 −0.194104
\(260\) 0 0
\(261\) 0 0
\(262\) 3863.36 0.910988
\(263\) −7590.87 −1.77975 −0.889873 0.456208i \(-0.849207\pi\)
−0.889873 + 0.456208i \(0.849207\pi\)
\(264\) 0 0
\(265\) −7389.36 −1.71292
\(266\) 11409.2 2.62985
\(267\) 0 0
\(268\) 4553.31 1.03783
\(269\) 557.911 0.126455 0.0632275 0.997999i \(-0.479861\pi\)
0.0632275 + 0.997999i \(0.479861\pi\)
\(270\) 0 0
\(271\) −2707.41 −0.606877 −0.303439 0.952851i \(-0.598135\pi\)
−0.303439 + 0.952851i \(0.598135\pi\)
\(272\) −2311.50 −0.515276
\(273\) 0 0
\(274\) 2908.08 0.641181
\(275\) −9975.31 −2.18740
\(276\) 0 0
\(277\) −1231.35 −0.267092 −0.133546 0.991043i \(-0.542636\pi\)
−0.133546 + 0.991043i \(0.542636\pi\)
\(278\) −10324.1 −2.22733
\(279\) 0 0
\(280\) −12356.8 −2.63736
\(281\) 4601.44 0.976864 0.488432 0.872602i \(-0.337569\pi\)
0.488432 + 0.872602i \(0.337569\pi\)
\(282\) 0 0
\(283\) −54.5677 −0.0114619 −0.00573094 0.999984i \(-0.501824\pi\)
−0.00573094 + 0.999984i \(0.501824\pi\)
\(284\) 5070.37 1.05941
\(285\) 0 0
\(286\) 0 0
\(287\) 1757.44 0.361457
\(288\) 0 0
\(289\) −2775.13 −0.564854
\(290\) −24914.5 −5.04494
\(291\) 0 0
\(292\) −754.627 −0.151237
\(293\) −4745.24 −0.946144 −0.473072 0.881024i \(-0.656855\pi\)
−0.473072 + 0.881024i \(0.656855\pi\)
\(294\) 0 0
\(295\) −14402.7 −2.84258
\(296\) −1779.30 −0.349390
\(297\) 0 0
\(298\) −16640.6 −3.23479
\(299\) 0 0
\(300\) 0 0
\(301\) −576.604 −0.110415
\(302\) 10758.8 2.05000
\(303\) 0 0
\(304\) 7242.10 1.36633
\(305\) 4884.15 0.916936
\(306\) 0 0
\(307\) 1028.82 0.191264 0.0956320 0.995417i \(-0.469513\pi\)
0.0956320 + 0.995417i \(0.469513\pi\)
\(308\) −7711.20 −1.42658
\(309\) 0 0
\(310\) −9017.46 −1.65212
\(311\) 2437.84 0.444492 0.222246 0.974991i \(-0.428661\pi\)
0.222246 + 0.974991i \(0.428661\pi\)
\(312\) 0 0
\(313\) 7934.20 1.43280 0.716402 0.697688i \(-0.245788\pi\)
0.716402 + 0.697688i \(0.245788\pi\)
\(314\) 11926.0 2.14338
\(315\) 0 0
\(316\) −1662.46 −0.295952
\(317\) −6942.72 −1.23010 −0.615050 0.788488i \(-0.710864\pi\)
−0.615050 + 0.788488i \(0.710864\pi\)
\(318\) 0 0
\(319\) −7471.75 −1.31140
\(320\) 13051.3 2.27997
\(321\) 0 0
\(322\) −660.023 −0.114229
\(323\) −6698.12 −1.15385
\(324\) 0 0
\(325\) 0 0
\(326\) −1178.25 −0.200175
\(327\) 0 0
\(328\) 3864.96 0.650630
\(329\) 6102.21 1.02257
\(330\) 0 0
\(331\) −5439.27 −0.903230 −0.451615 0.892213i \(-0.649152\pi\)
−0.451615 + 0.892213i \(0.649152\pi\)
\(332\) 7941.79 1.31284
\(333\) 0 0
\(334\) −1979.01 −0.324212
\(335\) −6267.10 −1.02211
\(336\) 0 0
\(337\) −1663.87 −0.268952 −0.134476 0.990917i \(-0.542935\pi\)
−0.134476 + 0.990917i \(0.542935\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 15095.6 2.40787
\(341\) −2704.29 −0.429460
\(342\) 0 0
\(343\) −6853.19 −1.07883
\(344\) −1268.07 −0.198749
\(345\) 0 0
\(346\) 12669.3 1.96851
\(347\) −9268.10 −1.43383 −0.716913 0.697163i \(-0.754446\pi\)
−0.716913 + 0.697163i \(0.754446\pi\)
\(348\) 0 0
\(349\) 5239.45 0.803614 0.401807 0.915724i \(-0.368382\pi\)
0.401807 + 0.915724i \(0.368382\pi\)
\(350\) 25546.3 3.90144
\(351\) 0 0
\(352\) 1371.51 0.207676
\(353\) −1218.01 −0.183649 −0.0918244 0.995775i \(-0.529270\pi\)
−0.0918244 + 0.995775i \(0.529270\pi\)
\(354\) 0 0
\(355\) −6978.78 −1.04337
\(356\) 15162.0 2.25726
\(357\) 0 0
\(358\) −790.018 −0.116631
\(359\) −5316.06 −0.781534 −0.390767 0.920490i \(-0.627790\pi\)
−0.390767 + 0.920490i \(0.627790\pi\)
\(360\) 0 0
\(361\) 14126.7 2.05959
\(362\) −16027.0 −2.32696
\(363\) 0 0
\(364\) 0 0
\(365\) 1038.66 0.148947
\(366\) 0 0
\(367\) −289.101 −0.0411197 −0.0205599 0.999789i \(-0.506545\pi\)
−0.0205599 + 0.999789i \(0.506545\pi\)
\(368\) −418.957 −0.0593469
\(369\) 0 0
\(370\) 5096.05 0.716030
\(371\) 5675.15 0.794175
\(372\) 0 0
\(373\) 6163.98 0.855654 0.427827 0.903861i \(-0.359279\pi\)
0.427827 + 0.903861i \(0.359279\pi\)
\(374\) 6878.64 0.951032
\(375\) 0 0
\(376\) 13420.0 1.84065
\(377\) 0 0
\(378\) 0 0
\(379\) −2440.03 −0.330702 −0.165351 0.986235i \(-0.552876\pi\)
−0.165351 + 0.986235i \(0.552876\pi\)
\(380\) −47295.9 −6.38481
\(381\) 0 0
\(382\) 20713.7 2.77436
\(383\) 6577.28 0.877502 0.438751 0.898609i \(-0.355421\pi\)
0.438751 + 0.898609i \(0.355421\pi\)
\(384\) 0 0
\(385\) 10613.6 1.40498
\(386\) 9081.58 1.19751
\(387\) 0 0
\(388\) −7501.59 −0.981535
\(389\) −4964.94 −0.647127 −0.323563 0.946206i \(-0.604881\pi\)
−0.323563 + 0.946206i \(0.604881\pi\)
\(390\) 0 0
\(391\) 387.488 0.0501179
\(392\) −2790.65 −0.359565
\(393\) 0 0
\(394\) 9632.70 1.23170
\(395\) 2288.19 0.291471
\(396\) 0 0
\(397\) −12073.9 −1.52637 −0.763187 0.646178i \(-0.776366\pi\)
−0.763187 + 0.646178i \(0.776366\pi\)
\(398\) 6506.69 0.819475
\(399\) 0 0
\(400\) 16215.8 2.02697
\(401\) 4916.90 0.612315 0.306158 0.951981i \(-0.400957\pi\)
0.306158 + 0.951981i \(0.400957\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −11803.3 −1.45355
\(405\) 0 0
\(406\) 19134.8 2.33902
\(407\) 1528.28 0.186128
\(408\) 0 0
\(409\) 15350.9 1.85588 0.927939 0.372733i \(-0.121579\pi\)
0.927939 + 0.372733i \(0.121579\pi\)
\(410\) −11069.6 −1.33338
\(411\) 0 0
\(412\) 18941.7 2.26503
\(413\) 11061.5 1.31792
\(414\) 0 0
\(415\) −10931.0 −1.29296
\(416\) 0 0
\(417\) 0 0
\(418\) −21551.3 −2.52179
\(419\) 5488.58 0.639939 0.319970 0.947428i \(-0.396327\pi\)
0.319970 + 0.947428i \(0.396327\pi\)
\(420\) 0 0
\(421\) −927.681 −0.107393 −0.0536964 0.998557i \(-0.517100\pi\)
−0.0536964 + 0.998557i \(0.517100\pi\)
\(422\) −1396.89 −0.161137
\(423\) 0 0
\(424\) 12480.8 1.42953
\(425\) −14997.8 −1.71176
\(426\) 0 0
\(427\) −3751.10 −0.425126
\(428\) −1183.23 −0.133630
\(429\) 0 0
\(430\) 3631.85 0.407311
\(431\) −11002.3 −1.22961 −0.614806 0.788678i \(-0.710766\pi\)
−0.614806 + 0.788678i \(0.710766\pi\)
\(432\) 0 0
\(433\) −9596.76 −1.06511 −0.532553 0.846397i \(-0.678767\pi\)
−0.532553 + 0.846397i \(0.678767\pi\)
\(434\) 6925.56 0.765985
\(435\) 0 0
\(436\) 9641.30 1.05902
\(437\) −1214.03 −0.132895
\(438\) 0 0
\(439\) −10346.3 −1.12483 −0.562416 0.826854i \(-0.690128\pi\)
−0.562416 + 0.826854i \(0.690128\pi\)
\(440\) 23341.4 2.52899
\(441\) 0 0
\(442\) 0 0
\(443\) −5650.38 −0.606000 −0.303000 0.952991i \(-0.597988\pi\)
−0.303000 + 0.952991i \(0.597988\pi\)
\(444\) 0 0
\(445\) −20868.7 −2.22308
\(446\) −8699.80 −0.923649
\(447\) 0 0
\(448\) −10023.6 −1.05708
\(449\) 11987.8 1.25999 0.629997 0.776597i \(-0.283056\pi\)
0.629997 + 0.776597i \(0.283056\pi\)
\(450\) 0 0
\(451\) −3319.71 −0.346605
\(452\) 20687.3 2.15276
\(453\) 0 0
\(454\) −12026.5 −1.24324
\(455\) 0 0
\(456\) 0 0
\(457\) 16437.6 1.68253 0.841265 0.540623i \(-0.181811\pi\)
0.841265 + 0.540623i \(0.181811\pi\)
\(458\) 29386.4 2.99811
\(459\) 0 0
\(460\) 2736.08 0.277326
\(461\) 8847.70 0.893880 0.446940 0.894564i \(-0.352514\pi\)
0.446940 + 0.894564i \(0.352514\pi\)
\(462\) 0 0
\(463\) −10269.6 −1.03081 −0.515407 0.856945i \(-0.672359\pi\)
−0.515407 + 0.856945i \(0.672359\pi\)
\(464\) 12146.0 1.21523
\(465\) 0 0
\(466\) −30801.1 −3.06188
\(467\) −14730.4 −1.45962 −0.729811 0.683649i \(-0.760392\pi\)
−0.729811 + 0.683649i \(0.760392\pi\)
\(468\) 0 0
\(469\) 4813.24 0.473891
\(470\) −38436.0 −3.77217
\(471\) 0 0
\(472\) 24326.6 2.37229
\(473\) 1089.18 0.105878
\(474\) 0 0
\(475\) 46989.2 4.53897
\(476\) −11593.7 −1.11638
\(477\) 0 0
\(478\) 9125.59 0.873211
\(479\) 635.077 0.0605792 0.0302896 0.999541i \(-0.490357\pi\)
0.0302896 + 0.999541i \(0.490357\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 28289.3 2.67332
\(483\) 0 0
\(484\) −5933.22 −0.557214
\(485\) 10325.1 0.966675
\(486\) 0 0
\(487\) 16611.1 1.54563 0.772815 0.634632i \(-0.218848\pi\)
0.772815 + 0.634632i \(0.218848\pi\)
\(488\) −8249.43 −0.765234
\(489\) 0 0
\(490\) 7992.66 0.736881
\(491\) 2584.14 0.237516 0.118758 0.992923i \(-0.462109\pi\)
0.118758 + 0.992923i \(0.462109\pi\)
\(492\) 0 0
\(493\) −11233.7 −1.02625
\(494\) 0 0
\(495\) 0 0
\(496\) 4396.08 0.397963
\(497\) 5359.82 0.483744
\(498\) 0 0
\(499\) −2183.88 −0.195919 −0.0979597 0.995190i \(-0.531232\pi\)
−0.0979597 + 0.995190i \(0.531232\pi\)
\(500\) −65089.8 −5.82180
\(501\) 0 0
\(502\) 11555.1 1.02735
\(503\) 17214.9 1.52600 0.762998 0.646401i \(-0.223727\pi\)
0.762998 + 0.646401i \(0.223727\pi\)
\(504\) 0 0
\(505\) 16245.8 1.43154
\(506\) 1246.75 0.109535
\(507\) 0 0
\(508\) −33062.0 −2.88758
\(509\) 20260.6 1.76431 0.882157 0.470955i \(-0.156091\pi\)
0.882157 + 0.470955i \(0.156091\pi\)
\(510\) 0 0
\(511\) −797.705 −0.0690575
\(512\) −16549.0 −1.42846
\(513\) 0 0
\(514\) −21513.4 −1.84614
\(515\) −26071.1 −2.23074
\(516\) 0 0
\(517\) −11526.8 −0.980554
\(518\) −3913.85 −0.331978
\(519\) 0 0
\(520\) 0 0
\(521\) 20731.5 1.74331 0.871653 0.490124i \(-0.163048\pi\)
0.871653 + 0.490124i \(0.163048\pi\)
\(522\) 0 0
\(523\) −944.353 −0.0789554 −0.0394777 0.999220i \(-0.512569\pi\)
−0.0394777 + 0.999220i \(0.512569\pi\)
\(524\) 12300.0 1.02543
\(525\) 0 0
\(526\) −36720.9 −3.04393
\(527\) −4065.87 −0.336076
\(528\) 0 0
\(529\) −12096.8 −0.994228
\(530\) −35746.0 −2.92964
\(531\) 0 0
\(532\) 36324.0 2.96023
\(533\) 0 0
\(534\) 0 0
\(535\) 1628.58 0.131607
\(536\) 10585.3 0.853012
\(537\) 0 0
\(538\) 2698.89 0.216278
\(539\) 2396.96 0.191548
\(540\) 0 0
\(541\) −4883.06 −0.388058 −0.194029 0.980996i \(-0.562156\pi\)
−0.194029 + 0.980996i \(0.562156\pi\)
\(542\) −13097.1 −1.03795
\(543\) 0 0
\(544\) 2062.05 0.162518
\(545\) −13270.1 −1.04299
\(546\) 0 0
\(547\) −16269.1 −1.27169 −0.635847 0.771815i \(-0.719349\pi\)
−0.635847 + 0.771815i \(0.719349\pi\)
\(548\) 9258.61 0.721730
\(549\) 0 0
\(550\) −48255.6 −3.74114
\(551\) 35196.0 2.72124
\(552\) 0 0
\(553\) −1757.37 −0.135137
\(554\) −5956.63 −0.456811
\(555\) 0 0
\(556\) −32869.3 −2.50714
\(557\) −15661.0 −1.19134 −0.595670 0.803229i \(-0.703113\pi\)
−0.595670 + 0.803229i \(0.703113\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −17253.3 −1.30194
\(561\) 0 0
\(562\) 22259.5 1.67074
\(563\) 9915.18 0.742230 0.371115 0.928587i \(-0.378976\pi\)
0.371115 + 0.928587i \(0.378976\pi\)
\(564\) 0 0
\(565\) −28473.7 −2.12017
\(566\) −263.971 −0.0196034
\(567\) 0 0
\(568\) 11787.3 0.870748
\(569\) 11299.1 0.832482 0.416241 0.909254i \(-0.363347\pi\)
0.416241 + 0.909254i \(0.363347\pi\)
\(570\) 0 0
\(571\) −17619.6 −1.29134 −0.645672 0.763615i \(-0.723423\pi\)
−0.645672 + 0.763615i \(0.723423\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 8501.61 0.618206
\(575\) −2718.33 −0.197152
\(576\) 0 0
\(577\) −22153.5 −1.59837 −0.799186 0.601083i \(-0.794736\pi\)
−0.799186 + 0.601083i \(0.794736\pi\)
\(578\) −13424.7 −0.966078
\(579\) 0 0
\(580\) −79321.7 −5.67872
\(581\) 8395.15 0.599466
\(582\) 0 0
\(583\) −10720.1 −0.761543
\(584\) −1754.31 −0.124305
\(585\) 0 0
\(586\) −22955.1 −1.61820
\(587\) 9891.28 0.695497 0.347748 0.937588i \(-0.386946\pi\)
0.347748 + 0.937588i \(0.386946\pi\)
\(588\) 0 0
\(589\) 12738.7 0.891153
\(590\) −69673.3 −4.86170
\(591\) 0 0
\(592\) −2484.36 −0.172477
\(593\) 9746.23 0.674924 0.337462 0.941339i \(-0.390432\pi\)
0.337462 + 0.941339i \(0.390432\pi\)
\(594\) 0 0
\(595\) 15957.4 1.09948
\(596\) −52979.7 −3.64116
\(597\) 0 0
\(598\) 0 0
\(599\) 8460.53 0.577109 0.288554 0.957464i \(-0.406825\pi\)
0.288554 + 0.957464i \(0.406825\pi\)
\(600\) 0 0
\(601\) 6792.99 0.461052 0.230526 0.973066i \(-0.425955\pi\)
0.230526 + 0.973066i \(0.425955\pi\)
\(602\) −2789.32 −0.188844
\(603\) 0 0
\(604\) 34253.4 2.30754
\(605\) 8166.38 0.548778
\(606\) 0 0
\(607\) 19073.4 1.27539 0.637697 0.770287i \(-0.279887\pi\)
0.637697 + 0.770287i \(0.279887\pi\)
\(608\) −6460.58 −0.430939
\(609\) 0 0
\(610\) 23627.1 1.56825
\(611\) 0 0
\(612\) 0 0
\(613\) −14465.2 −0.953091 −0.476545 0.879150i \(-0.658111\pi\)
−0.476545 + 0.879150i \(0.658111\pi\)
\(614\) 4976.93 0.327122
\(615\) 0 0
\(616\) −17926.6 −1.17254
\(617\) −12897.3 −0.841536 −0.420768 0.907168i \(-0.638239\pi\)
−0.420768 + 0.907168i \(0.638239\pi\)
\(618\) 0 0
\(619\) −27735.8 −1.80096 −0.900481 0.434895i \(-0.856786\pi\)
−0.900481 + 0.434895i \(0.856786\pi\)
\(620\) −28709.4 −1.85967
\(621\) 0 0
\(622\) 11793.0 0.760222
\(623\) 16027.5 1.03070
\(624\) 0 0
\(625\) 49042.7 3.13873
\(626\) 38381.7 2.45055
\(627\) 0 0
\(628\) 37969.3 2.41265
\(629\) 2297.75 0.145656
\(630\) 0 0
\(631\) 28560.1 1.80184 0.900918 0.433990i \(-0.142895\pi\)
0.900918 + 0.433990i \(0.142895\pi\)
\(632\) −3864.80 −0.243249
\(633\) 0 0
\(634\) −33585.4 −2.10386
\(635\) 45506.1 2.84386
\(636\) 0 0
\(637\) 0 0
\(638\) −36144.6 −2.24291
\(639\) 0 0
\(640\) 55572.5 3.43234
\(641\) −1862.53 −0.114767 −0.0573833 0.998352i \(-0.518276\pi\)
−0.0573833 + 0.998352i \(0.518276\pi\)
\(642\) 0 0
\(643\) 29495.2 1.80899 0.904494 0.426487i \(-0.140249\pi\)
0.904494 + 0.426487i \(0.140249\pi\)
\(644\) −2101.35 −0.128579
\(645\) 0 0
\(646\) −32402.2 −1.97345
\(647\) −27780.7 −1.68805 −0.844027 0.536301i \(-0.819821\pi\)
−0.844027 + 0.536301i \(0.819821\pi\)
\(648\) 0 0
\(649\) −20894.7 −1.26377
\(650\) 0 0
\(651\) 0 0
\(652\) −3751.25 −0.225322
\(653\) −276.678 −0.0165808 −0.00829039 0.999966i \(-0.502639\pi\)
−0.00829039 + 0.999966i \(0.502639\pi\)
\(654\) 0 0
\(655\) −16929.5 −1.00991
\(656\) 5396.49 0.321185
\(657\) 0 0
\(658\) 29519.5 1.74892
\(659\) −18114.9 −1.07080 −0.535398 0.844600i \(-0.679838\pi\)
−0.535398 + 0.844600i \(0.679838\pi\)
\(660\) 0 0
\(661\) 27094.4 1.59433 0.797163 0.603764i \(-0.206333\pi\)
0.797163 + 0.603764i \(0.206333\pi\)
\(662\) −26312.5 −1.54481
\(663\) 0 0
\(664\) 18462.6 1.07905
\(665\) −49995.8 −2.91542
\(666\) 0 0
\(667\) −2036.10 −0.118198
\(668\) −6300.70 −0.364942
\(669\) 0 0
\(670\) −30317.1 −1.74814
\(671\) 7085.64 0.407657
\(672\) 0 0
\(673\) 2410.31 0.138054 0.0690272 0.997615i \(-0.478010\pi\)
0.0690272 + 0.997615i \(0.478010\pi\)
\(674\) −8048.99 −0.459993
\(675\) 0 0
\(676\) 0 0
\(677\) −12508.3 −0.710091 −0.355046 0.934849i \(-0.615535\pi\)
−0.355046 + 0.934849i \(0.615535\pi\)
\(678\) 0 0
\(679\) −7929.83 −0.448187
\(680\) 35093.5 1.97908
\(681\) 0 0
\(682\) −13082.0 −0.734511
\(683\) 16793.8 0.940844 0.470422 0.882442i \(-0.344102\pi\)
0.470422 + 0.882442i \(0.344102\pi\)
\(684\) 0 0
\(685\) −12743.4 −0.710804
\(686\) −33152.3 −1.84513
\(687\) 0 0
\(688\) −1770.56 −0.0981131
\(689\) 0 0
\(690\) 0 0
\(691\) −26026.9 −1.43286 −0.716432 0.697657i \(-0.754226\pi\)
−0.716432 + 0.697657i \(0.754226\pi\)
\(692\) 40335.8 2.21580
\(693\) 0 0
\(694\) −44834.4 −2.45229
\(695\) 45240.8 2.46918
\(696\) 0 0
\(697\) −4991.14 −0.271238
\(698\) 25345.8 1.37443
\(699\) 0 0
\(700\) 81333.0 4.39157
\(701\) −5079.08 −0.273658 −0.136829 0.990595i \(-0.543691\pi\)
−0.136829 + 0.990595i \(0.543691\pi\)
\(702\) 0 0
\(703\) −7199.04 −0.386226
\(704\) 18934.1 1.01364
\(705\) 0 0
\(706\) −5892.11 −0.314097
\(707\) −12477.1 −0.663717
\(708\) 0 0
\(709\) −2530.72 −0.134052 −0.0670262 0.997751i \(-0.521351\pi\)
−0.0670262 + 0.997751i \(0.521351\pi\)
\(710\) −33759.9 −1.78449
\(711\) 0 0
\(712\) 35247.7 1.85529
\(713\) −736.937 −0.0387076
\(714\) 0 0
\(715\) 0 0
\(716\) −2515.22 −0.131283
\(717\) 0 0
\(718\) −25716.4 −1.33667
\(719\) −9503.67 −0.492944 −0.246472 0.969150i \(-0.579271\pi\)
−0.246472 + 0.969150i \(0.579271\pi\)
\(720\) 0 0
\(721\) 20023.0 1.03425
\(722\) 68338.2 3.52255
\(723\) 0 0
\(724\) −51026.0 −2.61929
\(725\) 78807.4 4.03701
\(726\) 0 0
\(727\) 33343.1 1.70100 0.850500 0.525975i \(-0.176300\pi\)
0.850500 + 0.525975i \(0.176300\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 5024.50 0.254747
\(731\) 1637.56 0.0828556
\(732\) 0 0
\(733\) 32288.1 1.62699 0.813497 0.581569i \(-0.197561\pi\)
0.813497 + 0.581569i \(0.197561\pi\)
\(734\) −1398.53 −0.0703277
\(735\) 0 0
\(736\) 373.746 0.0187180
\(737\) −9091.96 −0.454419
\(738\) 0 0
\(739\) −14168.8 −0.705288 −0.352644 0.935758i \(-0.614717\pi\)
−0.352644 + 0.935758i \(0.614717\pi\)
\(740\) 16224.6 0.805983
\(741\) 0 0
\(742\) 27453.5 1.35829
\(743\) −13248.6 −0.654165 −0.327083 0.944996i \(-0.606066\pi\)
−0.327083 + 0.944996i \(0.606066\pi\)
\(744\) 0 0
\(745\) 72920.5 3.58604
\(746\) 29818.3 1.46344
\(747\) 0 0
\(748\) 21899.9 1.07051
\(749\) −1250.78 −0.0610178
\(750\) 0 0
\(751\) 5113.98 0.248484 0.124242 0.992252i \(-0.460350\pi\)
0.124242 + 0.992252i \(0.460350\pi\)
\(752\) 18737.8 0.908641
\(753\) 0 0
\(754\) 0 0
\(755\) −47145.9 −2.27260
\(756\) 0 0
\(757\) 27380.4 1.31461 0.657303 0.753626i \(-0.271697\pi\)
0.657303 + 0.753626i \(0.271697\pi\)
\(758\) −11803.6 −0.565604
\(759\) 0 0
\(760\) −109951. −5.24781
\(761\) 10613.8 0.505586 0.252793 0.967520i \(-0.418651\pi\)
0.252793 + 0.967520i \(0.418651\pi\)
\(762\) 0 0
\(763\) 10191.7 0.483569
\(764\) 65947.3 3.12289
\(765\) 0 0
\(766\) 31817.6 1.50080
\(767\) 0 0
\(768\) 0 0
\(769\) 9172.22 0.430115 0.215058 0.976601i \(-0.431006\pi\)
0.215058 + 0.976601i \(0.431006\pi\)
\(770\) 51343.2 2.40296
\(771\) 0 0
\(772\) 28913.5 1.34795
\(773\) −14952.1 −0.695718 −0.347859 0.937547i \(-0.613091\pi\)
−0.347859 + 0.937547i \(0.613091\pi\)
\(774\) 0 0
\(775\) 28523.2 1.32205
\(776\) −17439.3 −0.806745
\(777\) 0 0
\(778\) −24017.9 −1.10679
\(779\) 15637.7 0.719226
\(780\) 0 0
\(781\) −10124.4 −0.463867
\(782\) 1874.47 0.0857174
\(783\) 0 0
\(784\) −3896.48 −0.177500
\(785\) −52260.4 −2.37612
\(786\) 0 0
\(787\) −2671.52 −0.121003 −0.0605015 0.998168i \(-0.519270\pi\)
−0.0605015 + 0.998168i \(0.519270\pi\)
\(788\) 30668.1 1.38643
\(789\) 0 0
\(790\) 11069.1 0.498508
\(791\) 21868.3 0.982991
\(792\) 0 0
\(793\) 0 0
\(794\) −58407.4 −2.61058
\(795\) 0 0
\(796\) 20715.7 0.922423
\(797\) 12949.7 0.575536 0.287768 0.957700i \(-0.407087\pi\)
0.287768 + 0.957700i \(0.407087\pi\)
\(798\) 0 0
\(799\) −17330.4 −0.767339
\(800\) −14465.9 −0.639307
\(801\) 0 0
\(802\) 23785.5 1.04725
\(803\) 1506.82 0.0662200
\(804\) 0 0
\(805\) 2892.27 0.126632
\(806\) 0 0
\(807\) 0 0
\(808\) −27439.6 −1.19470
\(809\) 11640.4 0.505876 0.252938 0.967482i \(-0.418603\pi\)
0.252938 + 0.967482i \(0.418603\pi\)
\(810\) 0 0
\(811\) 18135.3 0.785225 0.392613 0.919704i \(-0.371571\pi\)
0.392613 + 0.919704i \(0.371571\pi\)
\(812\) 60920.4 2.63286
\(813\) 0 0
\(814\) 7393.06 0.318337
\(815\) 5163.16 0.221911
\(816\) 0 0
\(817\) −5130.61 −0.219703
\(818\) 74260.0 3.17413
\(819\) 0 0
\(820\) −35242.8 −1.50089
\(821\) 10789.1 0.458640 0.229320 0.973351i \(-0.426350\pi\)
0.229320 + 0.973351i \(0.426350\pi\)
\(822\) 0 0
\(823\) −12284.7 −0.520312 −0.260156 0.965567i \(-0.583774\pi\)
−0.260156 + 0.965567i \(0.583774\pi\)
\(824\) 44034.7 1.86168
\(825\) 0 0
\(826\) 53510.2 2.25407
\(827\) 36077.2 1.51696 0.758480 0.651696i \(-0.225942\pi\)
0.758480 + 0.651696i \(0.225942\pi\)
\(828\) 0 0
\(829\) −8861.83 −0.371271 −0.185636 0.982619i \(-0.559434\pi\)
−0.185636 + 0.982619i \(0.559434\pi\)
\(830\) −52878.5 −2.21137
\(831\) 0 0
\(832\) 0 0
\(833\) 3603.80 0.149897
\(834\) 0 0
\(835\) 8672.18 0.359417
\(836\) −68614.1 −2.83860
\(837\) 0 0
\(838\) 26551.0 1.09450
\(839\) 5833.37 0.240036 0.120018 0.992772i \(-0.461705\pi\)
0.120018 + 0.992772i \(0.461705\pi\)
\(840\) 0 0
\(841\) 34639.6 1.42030
\(842\) −4487.66 −0.183676
\(843\) 0 0
\(844\) −4447.37 −0.181380
\(845\) 0 0
\(846\) 0 0
\(847\) −6271.92 −0.254434
\(848\) 17426.4 0.705692
\(849\) 0 0
\(850\) −72551.7 −2.92765
\(851\) 416.466 0.0167759
\(852\) 0 0
\(853\) 28649.5 1.14999 0.574995 0.818157i \(-0.305004\pi\)
0.574995 + 0.818157i \(0.305004\pi\)
\(854\) −18146.0 −0.727098
\(855\) 0 0
\(856\) −2750.71 −0.109833
\(857\) −32336.1 −1.28889 −0.644445 0.764651i \(-0.722911\pi\)
−0.644445 + 0.764651i \(0.722911\pi\)
\(858\) 0 0
\(859\) −13878.5 −0.551254 −0.275627 0.961265i \(-0.588885\pi\)
−0.275627 + 0.961265i \(0.588885\pi\)
\(860\) 11562.9 0.458480
\(861\) 0 0
\(862\) −53223.7 −2.10302
\(863\) −32934.7 −1.29908 −0.649542 0.760326i \(-0.725039\pi\)
−0.649542 + 0.760326i \(0.725039\pi\)
\(864\) 0 0
\(865\) −55517.5 −2.18226
\(866\) −46424.3 −1.82167
\(867\) 0 0
\(868\) 22049.3 0.862213
\(869\) 3319.57 0.129584
\(870\) 0 0
\(871\) 0 0
\(872\) 22413.5 0.870434
\(873\) 0 0
\(874\) −5872.87 −0.227292
\(875\) −68805.4 −2.65834
\(876\) 0 0
\(877\) 18375.7 0.707531 0.353765 0.935334i \(-0.384901\pi\)
0.353765 + 0.935334i \(0.384901\pi\)
\(878\) −50050.2 −1.92382
\(879\) 0 0
\(880\) 32590.7 1.24845
\(881\) −46883.4 −1.79290 −0.896448 0.443150i \(-0.853861\pi\)
−0.896448 + 0.443150i \(0.853861\pi\)
\(882\) 0 0
\(883\) 1050.05 0.0400194 0.0200097 0.999800i \(-0.493630\pi\)
0.0200097 + 0.999800i \(0.493630\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −27333.7 −1.03645
\(887\) −27916.8 −1.05677 −0.528385 0.849005i \(-0.677202\pi\)
−0.528385 + 0.849005i \(0.677202\pi\)
\(888\) 0 0
\(889\) −34949.4 −1.31852
\(890\) −100952. −3.80217
\(891\) 0 0
\(892\) −27698.0 −1.03968
\(893\) 54297.4 2.03471
\(894\) 0 0
\(895\) 3461.92 0.129295
\(896\) −42680.6 −1.59136
\(897\) 0 0
\(898\) 57990.8 2.15499
\(899\) 21364.6 0.792601
\(900\) 0 0
\(901\) −16117.5 −0.595950
\(902\) −16059.1 −0.592804
\(903\) 0 0
\(904\) 48092.7 1.76940
\(905\) 70231.3 2.57963
\(906\) 0 0
\(907\) 23220.1 0.850066 0.425033 0.905178i \(-0.360263\pi\)
0.425033 + 0.905178i \(0.360263\pi\)
\(908\) −38289.5 −1.39943
\(909\) 0 0
\(910\) 0 0
\(911\) −16344.9 −0.594435 −0.297217 0.954810i \(-0.596059\pi\)
−0.297217 + 0.954810i \(0.596059\pi\)
\(912\) 0 0
\(913\) −15858.0 −0.574834
\(914\) 79516.7 2.87766
\(915\) 0 0
\(916\) 93559.0 3.37476
\(917\) 13002.1 0.468231
\(918\) 0 0
\(919\) −32931.7 −1.18206 −0.591032 0.806648i \(-0.701279\pi\)
−0.591032 + 0.806648i \(0.701279\pi\)
\(920\) 6360.67 0.227940
\(921\) 0 0
\(922\) 42800.8 1.52882
\(923\) 0 0
\(924\) 0 0
\(925\) −16119.4 −0.572975
\(926\) −49679.0 −1.76302
\(927\) 0 0
\(928\) −10835.3 −0.383282
\(929\) −56305.5 −1.98851 −0.994253 0.107056i \(-0.965858\pi\)
−0.994253 + 0.107056i \(0.965858\pi\)
\(930\) 0 0
\(931\) −11291.0 −0.397473
\(932\) −98063.3 −3.44653
\(933\) 0 0
\(934\) −71258.5 −2.49641
\(935\) −30142.7 −1.05430
\(936\) 0 0
\(937\) 7095.61 0.247389 0.123695 0.992320i \(-0.460526\pi\)
0.123695 + 0.992320i \(0.460526\pi\)
\(938\) 23284.0 0.810502
\(939\) 0 0
\(940\) −122371. −4.24606
\(941\) −5185.17 −0.179630 −0.0898150 0.995958i \(-0.528628\pi\)
−0.0898150 + 0.995958i \(0.528628\pi\)
\(942\) 0 0
\(943\) −904.641 −0.0312398
\(944\) 33966.2 1.17109
\(945\) 0 0
\(946\) 5268.89 0.181085
\(947\) 38451.7 1.31944 0.659722 0.751510i \(-0.270674\pi\)
0.659722 + 0.751510i \(0.270674\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 227310. 7.76307
\(951\) 0 0
\(952\) −26952.4 −0.917575
\(953\) 20930.5 0.711445 0.355722 0.934592i \(-0.384235\pi\)
0.355722 + 0.934592i \(0.384235\pi\)
\(954\) 0 0
\(955\) −90768.8 −3.07561
\(956\) 29053.7 0.982910
\(957\) 0 0
\(958\) 3072.19 0.103609
\(959\) 9787.14 0.329555
\(960\) 0 0
\(961\) −22058.4 −0.740438
\(962\) 0 0
\(963\) 0 0
\(964\) 90066.1 3.00916
\(965\) −39796.1 −1.32755
\(966\) 0 0
\(967\) 50788.1 1.68897 0.844486 0.535578i \(-0.179906\pi\)
0.844486 + 0.535578i \(0.179906\pi\)
\(968\) −13793.2 −0.457986
\(969\) 0 0
\(970\) 49947.6 1.65332
\(971\) −15277.5 −0.504922 −0.252461 0.967607i \(-0.581240\pi\)
−0.252461 + 0.967607i \(0.581240\pi\)
\(972\) 0 0
\(973\) −34745.7 −1.14481
\(974\) 80356.3 2.64351
\(975\) 0 0
\(976\) −11518.4 −0.377760
\(977\) 1293.19 0.0423469 0.0211734 0.999776i \(-0.493260\pi\)
0.0211734 + 0.999776i \(0.493260\pi\)
\(978\) 0 0
\(979\) −30275.1 −0.988353
\(980\) 25446.7 0.829453
\(981\) 0 0
\(982\) 12500.8 0.406227
\(983\) 8474.63 0.274973 0.137487 0.990504i \(-0.456098\pi\)
0.137487 + 0.990504i \(0.456098\pi\)
\(984\) 0 0
\(985\) −42211.1 −1.36544
\(986\) −54343.0 −1.75521
\(987\) 0 0
\(988\) 0 0
\(989\) 296.807 0.00954289
\(990\) 0 0
\(991\) −7080.71 −0.226969 −0.113484 0.993540i \(-0.536201\pi\)
−0.113484 + 0.993540i \(0.536201\pi\)
\(992\) −3921.68 −0.125518
\(993\) 0 0
\(994\) 25928.1 0.827354
\(995\) −28512.8 −0.908458
\(996\) 0 0
\(997\) 19423.1 0.616985 0.308493 0.951227i \(-0.400175\pi\)
0.308493 + 0.951227i \(0.400175\pi\)
\(998\) −10564.5 −0.335084
\(999\) 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 1521.4.a.bf.1.9 9
3.2 odd 2 507.4.a.p.1.1 yes 9
13.12 even 2 1521.4.a.bi.1.1 9
39.5 even 4 507.4.b.k.337.17 18
39.8 even 4 507.4.b.k.337.2 18
39.38 odd 2 507.4.a.o.1.9 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.9 9 39.38 odd 2
507.4.a.p.1.1 yes 9 3.2 odd 2
507.4.b.k.337.2 18 39.8 even 4
507.4.b.k.337.17 18 39.5 even 4
1521.4.a.bf.1.9 9 1.1 even 1 trivial
1521.4.a.bi.1.1 9 13.12 even 2