Properties

Label 531.4.a.c.1.3
Level $531$
Weight $4$
Character 531.1
Self dual yes
Analytic conductor $31.330$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.3300142130\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 41 x^{5} - 7 x^{4} + 484 x^{3} + 63 x^{2} - 1736 x - 44\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.74916\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.74916 q^{2} -0.442134 q^{4} -13.7745 q^{5} -35.6462 q^{7} +23.2088 q^{8} +O(q^{10})\) \(q-2.74916 q^{2} -0.442134 q^{4} -13.7745 q^{5} -35.6462 q^{7} +23.2088 q^{8} +37.8683 q^{10} +30.8537 q^{11} +14.6435 q^{13} +97.9970 q^{14} -60.2674 q^{16} +77.8827 q^{17} +98.6377 q^{19} +6.09019 q^{20} -84.8216 q^{22} +117.065 q^{23} +64.7375 q^{25} -40.2572 q^{26} +15.7604 q^{28} +54.2015 q^{29} -231.433 q^{31} -19.9854 q^{32} -214.112 q^{34} +491.009 q^{35} -258.311 q^{37} -271.171 q^{38} -319.690 q^{40} -326.921 q^{41} +250.920 q^{43} -13.6415 q^{44} -321.830 q^{46} -509.390 q^{47} +927.650 q^{49} -177.974 q^{50} -6.47437 q^{52} +313.730 q^{53} -424.994 q^{55} -827.304 q^{56} -149.008 q^{58} -59.0000 q^{59} +528.854 q^{61} +636.245 q^{62} +537.082 q^{64} -201.707 q^{65} -373.284 q^{67} -34.4346 q^{68} -1349.86 q^{70} +521.969 q^{71} +809.404 q^{73} +710.137 q^{74} -43.6111 q^{76} -1099.82 q^{77} -926.088 q^{79} +830.155 q^{80} +898.756 q^{82} +1311.77 q^{83} -1072.80 q^{85} -689.818 q^{86} +716.075 q^{88} -510.201 q^{89} -521.983 q^{91} -51.7585 q^{92} +1400.39 q^{94} -1358.69 q^{95} -856.449 q^{97} -2550.26 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 26q^{4} + 2q^{5} - 59q^{7} + 21q^{8} + O(q^{10}) \) \( 7q + 26q^{4} + 2q^{5} - 59q^{7} + 21q^{8} - 71q^{10} + 5q^{11} - 67q^{13} + 65q^{14} - 94q^{16} + 23q^{17} - 176q^{19} + 207q^{20} - 704q^{22} + 218q^{23} - 183q^{25} - 58q^{26} - 938q^{28} - 168q^{29} - 604q^{31} + 448q^{32} - 610q^{34} + 336q^{35} - 505q^{37} + 453q^{38} - 1080q^{40} + 265q^{41} - 493q^{43} - 504q^{44} + 381q^{46} + 244q^{47} + 770q^{49} - 1639q^{50} + 160q^{52} - 686q^{53} - 116q^{55} - 2190q^{56} + 1584q^{58} - 413q^{59} - 838q^{61} - 286q^{62} + 205q^{64} - 490q^{65} - 1504q^{67} - 3047q^{68} + 1530q^{70} + 1267q^{71} - 666q^{73} - 528q^{74} - 64q^{76} - 1109q^{77} - 2741q^{79} - 1213q^{80} + 953q^{82} + 2025q^{83} - 1274q^{85} - 4394q^{86} - 1639q^{88} - 616q^{89} - 2415q^{91} - 218q^{92} + 900q^{94} - 2554q^{95} - 1298q^{97} + 172q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.74916 −0.971974 −0.485987 0.873966i \(-0.661540\pi\)
−0.485987 + 0.873966i \(0.661540\pi\)
\(3\) 0 0
\(4\) −0.442134 −0.0552668
\(5\) −13.7745 −1.23203 −0.616015 0.787734i \(-0.711254\pi\)
−0.616015 + 0.787734i \(0.711254\pi\)
\(6\) 0 0
\(7\) −35.6462 −1.92471 −0.962357 0.271790i \(-0.912384\pi\)
−0.962357 + 0.271790i \(0.912384\pi\)
\(8\) 23.2088 1.02569
\(9\) 0 0
\(10\) 37.8683 1.19750
\(11\) 30.8537 0.845703 0.422851 0.906199i \(-0.361029\pi\)
0.422851 + 0.906199i \(0.361029\pi\)
\(12\) 0 0
\(13\) 14.6435 0.312413 0.156206 0.987724i \(-0.450074\pi\)
0.156206 + 0.987724i \(0.450074\pi\)
\(14\) 97.9970 1.87077
\(15\) 0 0
\(16\) −60.2674 −0.941679
\(17\) 77.8827 1.11114 0.555568 0.831471i \(-0.312501\pi\)
0.555568 + 0.831471i \(0.312501\pi\)
\(18\) 0 0
\(19\) 98.6377 1.19100 0.595501 0.803355i \(-0.296954\pi\)
0.595501 + 0.803355i \(0.296954\pi\)
\(20\) 6.09019 0.0680904
\(21\) 0 0
\(22\) −84.8216 −0.822001
\(23\) 117.065 1.06129 0.530647 0.847593i \(-0.321949\pi\)
0.530647 + 0.847593i \(0.321949\pi\)
\(24\) 0 0
\(25\) 64.7375 0.517900
\(26\) −40.2572 −0.303657
\(27\) 0 0
\(28\) 15.7604 0.106373
\(29\) 54.2015 0.347068 0.173534 0.984828i \(-0.444481\pi\)
0.173534 + 0.984828i \(0.444481\pi\)
\(30\) 0 0
\(31\) −231.433 −1.34086 −0.670428 0.741975i \(-0.733890\pi\)
−0.670428 + 0.741975i \(0.733890\pi\)
\(32\) −19.9854 −0.110405
\(33\) 0 0
\(34\) −214.112 −1.08000
\(35\) 491.009 2.37131
\(36\) 0 0
\(37\) −258.311 −1.14773 −0.573866 0.818950i \(-0.694557\pi\)
−0.573866 + 0.818950i \(0.694557\pi\)
\(38\) −271.171 −1.15762
\(39\) 0 0
\(40\) −319.690 −1.26368
\(41\) −326.921 −1.24528 −0.622639 0.782509i \(-0.713940\pi\)
−0.622639 + 0.782509i \(0.713940\pi\)
\(42\) 0 0
\(43\) 250.920 0.889882 0.444941 0.895560i \(-0.353225\pi\)
0.444941 + 0.895560i \(0.353225\pi\)
\(44\) −13.6415 −0.0467393
\(45\) 0 0
\(46\) −321.830 −1.03155
\(47\) −509.390 −1.58090 −0.790449 0.612528i \(-0.790153\pi\)
−0.790449 + 0.612528i \(0.790153\pi\)
\(48\) 0 0
\(49\) 927.650 2.70452
\(50\) −177.974 −0.503385
\(51\) 0 0
\(52\) −6.47437 −0.0172660
\(53\) 313.730 0.813097 0.406549 0.913629i \(-0.366732\pi\)
0.406549 + 0.913629i \(0.366732\pi\)
\(54\) 0 0
\(55\) −424.994 −1.04193
\(56\) −827.304 −1.97416
\(57\) 0 0
\(58\) −149.008 −0.337341
\(59\) −59.0000 −0.130189
\(60\) 0 0
\(61\) 528.854 1.11005 0.555023 0.831835i \(-0.312709\pi\)
0.555023 + 0.831835i \(0.312709\pi\)
\(62\) 636.245 1.30328
\(63\) 0 0
\(64\) 537.082 1.04899
\(65\) −201.707 −0.384902
\(66\) 0 0
\(67\) −373.284 −0.680656 −0.340328 0.940307i \(-0.610538\pi\)
−0.340328 + 0.940307i \(0.610538\pi\)
\(68\) −34.4346 −0.0614090
\(69\) 0 0
\(70\) −1349.86 −2.30485
\(71\) 521.969 0.872483 0.436241 0.899830i \(-0.356309\pi\)
0.436241 + 0.899830i \(0.356309\pi\)
\(72\) 0 0
\(73\) 809.404 1.29772 0.648860 0.760908i \(-0.275246\pi\)
0.648860 + 0.760908i \(0.275246\pi\)
\(74\) 710.137 1.11556
\(75\) 0 0
\(76\) −43.6111 −0.0658229
\(77\) −1099.82 −1.62773
\(78\) 0 0
\(79\) −926.088 −1.31890 −0.659449 0.751749i \(-0.729211\pi\)
−0.659449 + 0.751749i \(0.729211\pi\)
\(80\) 830.155 1.16018
\(81\) 0 0
\(82\) 898.756 1.21038
\(83\) 1311.77 1.73476 0.867380 0.497646i \(-0.165802\pi\)
0.867380 + 0.497646i \(0.165802\pi\)
\(84\) 0 0
\(85\) −1072.80 −1.36895
\(86\) −689.818 −0.864942
\(87\) 0 0
\(88\) 716.075 0.867430
\(89\) −510.201 −0.607654 −0.303827 0.952727i \(-0.598265\pi\)
−0.303827 + 0.952727i \(0.598265\pi\)
\(90\) 0 0
\(91\) −521.983 −0.601305
\(92\) −51.7585 −0.0586543
\(93\) 0 0
\(94\) 1400.39 1.53659
\(95\) −1358.69 −1.46735
\(96\) 0 0
\(97\) −856.449 −0.896486 −0.448243 0.893912i \(-0.647950\pi\)
−0.448243 + 0.893912i \(0.647950\pi\)
\(98\) −2550.26 −2.62872
\(99\) 0 0
\(100\) −28.6227 −0.0286227
\(101\) −811.456 −0.799435 −0.399717 0.916638i \(-0.630892\pi\)
−0.399717 + 0.916638i \(0.630892\pi\)
\(102\) 0 0
\(103\) −419.340 −0.401153 −0.200577 0.979678i \(-0.564282\pi\)
−0.200577 + 0.979678i \(0.564282\pi\)
\(104\) 339.856 0.320439
\(105\) 0 0
\(106\) −862.494 −0.790309
\(107\) 1165.05 1.05261 0.526305 0.850296i \(-0.323577\pi\)
0.526305 + 0.850296i \(0.323577\pi\)
\(108\) 0 0
\(109\) −2016.17 −1.77169 −0.885844 0.463984i \(-0.846419\pi\)
−0.885844 + 0.463984i \(0.846419\pi\)
\(110\) 1168.38 1.01273
\(111\) 0 0
\(112\) 2148.30 1.81246
\(113\) 731.760 0.609188 0.304594 0.952482i \(-0.401479\pi\)
0.304594 + 0.952482i \(0.401479\pi\)
\(114\) 0 0
\(115\) −1612.51 −1.30755
\(116\) −23.9643 −0.0191813
\(117\) 0 0
\(118\) 162.200 0.126540
\(119\) −2776.22 −2.13862
\(120\) 0 0
\(121\) −379.051 −0.284787
\(122\) −1453.90 −1.07894
\(123\) 0 0
\(124\) 102.324 0.0741048
\(125\) 830.087 0.593962
\(126\) 0 0
\(127\) −1187.43 −0.829666 −0.414833 0.909898i \(-0.636160\pi\)
−0.414833 + 0.909898i \(0.636160\pi\)
\(128\) −1316.64 −0.909185
\(129\) 0 0
\(130\) 554.523 0.374115
\(131\) 250.996 0.167402 0.0837008 0.996491i \(-0.473326\pi\)
0.0837008 + 0.996491i \(0.473326\pi\)
\(132\) 0 0
\(133\) −3516.06 −2.29234
\(134\) 1026.22 0.661580
\(135\) 0 0
\(136\) 1807.56 1.13968
\(137\) −2474.51 −1.54315 −0.771576 0.636137i \(-0.780531\pi\)
−0.771576 + 0.636137i \(0.780531\pi\)
\(138\) 0 0
\(139\) 2067.65 1.26169 0.630847 0.775907i \(-0.282708\pi\)
0.630847 + 0.775907i \(0.282708\pi\)
\(140\) −217.092 −0.131054
\(141\) 0 0
\(142\) −1434.97 −0.848031
\(143\) 451.804 0.264208
\(144\) 0 0
\(145\) −746.600 −0.427598
\(146\) −2225.18 −1.26135
\(147\) 0 0
\(148\) 114.208 0.0634314
\(149\) −70.2921 −0.0386480 −0.0193240 0.999813i \(-0.506151\pi\)
−0.0193240 + 0.999813i \(0.506151\pi\)
\(150\) 0 0
\(151\) 2156.31 1.16211 0.581054 0.813865i \(-0.302641\pi\)
0.581054 + 0.813865i \(0.302641\pi\)
\(152\) 2289.26 1.22160
\(153\) 0 0
\(154\) 3023.57 1.58212
\(155\) 3187.88 1.65198
\(156\) 0 0
\(157\) 105.729 0.0537459 0.0268729 0.999639i \(-0.491445\pi\)
0.0268729 + 0.999639i \(0.491445\pi\)
\(158\) 2545.96 1.28194
\(159\) 0 0
\(160\) 275.289 0.136022
\(161\) −4172.92 −2.04268
\(162\) 0 0
\(163\) −1006.11 −0.483463 −0.241731 0.970343i \(-0.577715\pi\)
−0.241731 + 0.970343i \(0.577715\pi\)
\(164\) 144.543 0.0688226
\(165\) 0 0
\(166\) −3606.25 −1.68614
\(167\) 668.739 0.309872 0.154936 0.987925i \(-0.450483\pi\)
0.154936 + 0.987925i \(0.450483\pi\)
\(168\) 0 0
\(169\) −1982.57 −0.902398
\(170\) 2949.29 1.33059
\(171\) 0 0
\(172\) −110.940 −0.0491809
\(173\) 2879.96 1.26566 0.632831 0.774290i \(-0.281893\pi\)
0.632831 + 0.774290i \(0.281893\pi\)
\(174\) 0 0
\(175\) −2307.64 −0.996809
\(176\) −1859.47 −0.796380
\(177\) 0 0
\(178\) 1402.62 0.590624
\(179\) −1700.00 −0.709856 −0.354928 0.934894i \(-0.615495\pi\)
−0.354928 + 0.934894i \(0.615495\pi\)
\(180\) 0 0
\(181\) −1750.62 −0.718910 −0.359455 0.933162i \(-0.617037\pi\)
−0.359455 + 0.933162i \(0.617037\pi\)
\(182\) 1435.01 0.584452
\(183\) 0 0
\(184\) 2716.93 1.08856
\(185\) 3558.11 1.41404
\(186\) 0 0
\(187\) 2402.97 0.939691
\(188\) 225.219 0.0873711
\(189\) 0 0
\(190\) 3735.24 1.42623
\(191\) −1504.78 −0.570062 −0.285031 0.958518i \(-0.592004\pi\)
−0.285031 + 0.958518i \(0.592004\pi\)
\(192\) 0 0
\(193\) −803.437 −0.299651 −0.149826 0.988712i \(-0.547871\pi\)
−0.149826 + 0.988712i \(0.547871\pi\)
\(194\) 2354.51 0.871361
\(195\) 0 0
\(196\) −410.146 −0.149470
\(197\) 1329.45 0.480808 0.240404 0.970673i \(-0.422720\pi\)
0.240404 + 0.970673i \(0.422720\pi\)
\(198\) 0 0
\(199\) −2857.91 −1.01805 −0.509025 0.860752i \(-0.669994\pi\)
−0.509025 + 0.860752i \(0.669994\pi\)
\(200\) 1502.48 0.531206
\(201\) 0 0
\(202\) 2230.82 0.777030
\(203\) −1932.08 −0.668006
\(204\) 0 0
\(205\) 4503.18 1.53422
\(206\) 1152.83 0.389911
\(207\) 0 0
\(208\) −882.524 −0.294192
\(209\) 3043.33 1.00723
\(210\) 0 0
\(211\) −3121.15 −1.01834 −0.509168 0.860667i \(-0.670047\pi\)
−0.509168 + 0.860667i \(0.670047\pi\)
\(212\) −138.711 −0.0449373
\(213\) 0 0
\(214\) −3202.89 −1.02311
\(215\) −3456.30 −1.09636
\(216\) 0 0
\(217\) 8249.69 2.58076
\(218\) 5542.76 1.72203
\(219\) 0 0
\(220\) 187.905 0.0575842
\(221\) 1140.47 0.347133
\(222\) 0 0
\(223\) −2100.41 −0.630734 −0.315367 0.948970i \(-0.602128\pi\)
−0.315367 + 0.948970i \(0.602128\pi\)
\(224\) 712.402 0.212497
\(225\) 0 0
\(226\) −2011.72 −0.592114
\(227\) −2601.37 −0.760613 −0.380306 0.924861i \(-0.624181\pi\)
−0.380306 + 0.924861i \(0.624181\pi\)
\(228\) 0 0
\(229\) −1502.73 −0.433637 −0.216819 0.976212i \(-0.569568\pi\)
−0.216819 + 0.976212i \(0.569568\pi\)
\(230\) 4433.06 1.27090
\(231\) 0 0
\(232\) 1257.95 0.355984
\(233\) 1672.77 0.470329 0.235164 0.971956i \(-0.424437\pi\)
0.235164 + 0.971956i \(0.424437\pi\)
\(234\) 0 0
\(235\) 7016.60 1.94771
\(236\) 26.0859 0.00719512
\(237\) 0 0
\(238\) 7632.27 2.07868
\(239\) −3489.49 −0.944420 −0.472210 0.881486i \(-0.656544\pi\)
−0.472210 + 0.881486i \(0.656544\pi\)
\(240\) 0 0
\(241\) −872.061 −0.233089 −0.116544 0.993185i \(-0.537182\pi\)
−0.116544 + 0.993185i \(0.537182\pi\)
\(242\) 1042.07 0.276806
\(243\) 0 0
\(244\) −233.825 −0.0613487
\(245\) −12777.9 −3.33205
\(246\) 0 0
\(247\) 1444.40 0.372084
\(248\) −5371.26 −1.37531
\(249\) 0 0
\(250\) −2282.04 −0.577316
\(251\) 463.531 0.116565 0.0582825 0.998300i \(-0.481438\pi\)
0.0582825 + 0.998300i \(0.481438\pi\)
\(252\) 0 0
\(253\) 3611.88 0.897538
\(254\) 3264.44 0.806413
\(255\) 0 0
\(256\) −677.006 −0.165285
\(257\) −2168.52 −0.526338 −0.263169 0.964750i \(-0.584768\pi\)
−0.263169 + 0.964750i \(0.584768\pi\)
\(258\) 0 0
\(259\) 9207.80 2.20905
\(260\) 89.1814 0.0212723
\(261\) 0 0
\(262\) −690.027 −0.162710
\(263\) −567.749 −0.133114 −0.0665569 0.997783i \(-0.521201\pi\)
−0.0665569 + 0.997783i \(0.521201\pi\)
\(264\) 0 0
\(265\) −4321.48 −1.00176
\(266\) 9666.20 2.22809
\(267\) 0 0
\(268\) 165.042 0.0376177
\(269\) 326.225 0.0739415 0.0369708 0.999316i \(-0.488229\pi\)
0.0369708 + 0.999316i \(0.488229\pi\)
\(270\) 0 0
\(271\) 1478.00 0.331300 0.165650 0.986185i \(-0.447028\pi\)
0.165650 + 0.986185i \(0.447028\pi\)
\(272\) −4693.79 −1.04633
\(273\) 0 0
\(274\) 6802.82 1.49990
\(275\) 1997.39 0.437989
\(276\) 0 0
\(277\) 8647.79 1.87580 0.937898 0.346911i \(-0.112769\pi\)
0.937898 + 0.346911i \(0.112769\pi\)
\(278\) −5684.29 −1.22633
\(279\) 0 0
\(280\) 11395.7 2.43223
\(281\) 4653.34 0.987882 0.493941 0.869495i \(-0.335556\pi\)
0.493941 + 0.869495i \(0.335556\pi\)
\(282\) 0 0
\(283\) −2594.93 −0.545062 −0.272531 0.962147i \(-0.587861\pi\)
−0.272531 + 0.962147i \(0.587861\pi\)
\(284\) −230.780 −0.0482193
\(285\) 0 0
\(286\) −1242.08 −0.256803
\(287\) 11653.5 2.39680
\(288\) 0 0
\(289\) 1152.71 0.234625
\(290\) 2052.52 0.415614
\(291\) 0 0
\(292\) −357.865 −0.0717208
\(293\) 7299.73 1.45548 0.727739 0.685854i \(-0.240571\pi\)
0.727739 + 0.685854i \(0.240571\pi\)
\(294\) 0 0
\(295\) 812.697 0.160397
\(296\) −5995.07 −1.17722
\(297\) 0 0
\(298\) 193.244 0.0375648
\(299\) 1714.24 0.331561
\(300\) 0 0
\(301\) −8944.34 −1.71277
\(302\) −5928.04 −1.12954
\(303\) 0 0
\(304\) −5944.64 −1.12154
\(305\) −7284.72 −1.36761
\(306\) 0 0
\(307\) 3677.41 0.683651 0.341826 0.939763i \(-0.388955\pi\)
0.341826 + 0.939763i \(0.388955\pi\)
\(308\) 486.266 0.0899597
\(309\) 0 0
\(310\) −8763.97 −1.60568
\(311\) −3389.46 −0.618002 −0.309001 0.951062i \(-0.599995\pi\)
−0.309001 + 0.951062i \(0.599995\pi\)
\(312\) 0 0
\(313\) −4847.47 −0.875384 −0.437692 0.899125i \(-0.644204\pi\)
−0.437692 + 0.899125i \(0.644204\pi\)
\(314\) −290.666 −0.0522396
\(315\) 0 0
\(316\) 409.455 0.0728913
\(317\) −10460.9 −1.85344 −0.926721 0.375750i \(-0.877385\pi\)
−0.926721 + 0.375750i \(0.877385\pi\)
\(318\) 0 0
\(319\) 1672.31 0.293516
\(320\) −7398.05 −1.29239
\(321\) 0 0
\(322\) 11472.0 1.98544
\(323\) 7682.17 1.32337
\(324\) 0 0
\(325\) 947.980 0.161798
\(326\) 2765.95 0.469913
\(327\) 0 0
\(328\) −7587.42 −1.27727
\(329\) 18157.8 3.04277
\(330\) 0 0
\(331\) −2859.84 −0.474898 −0.237449 0.971400i \(-0.576311\pi\)
−0.237449 + 0.971400i \(0.576311\pi\)
\(332\) −579.977 −0.0958747
\(333\) 0 0
\(334\) −1838.47 −0.301187
\(335\) 5141.82 0.838589
\(336\) 0 0
\(337\) 4699.94 0.759710 0.379855 0.925046i \(-0.375974\pi\)
0.379855 + 0.925046i \(0.375974\pi\)
\(338\) 5450.39 0.877108
\(339\) 0 0
\(340\) 474.320 0.0756577
\(341\) −7140.55 −1.13397
\(342\) 0 0
\(343\) −20840.6 −3.28071
\(344\) 5823.54 0.912745
\(345\) 0 0
\(346\) −7917.47 −1.23019
\(347\) −951.938 −0.147270 −0.0736350 0.997285i \(-0.523460\pi\)
−0.0736350 + 0.997285i \(0.523460\pi\)
\(348\) 0 0
\(349\) −10647.9 −1.63315 −0.816575 0.577239i \(-0.804130\pi\)
−0.816575 + 0.577239i \(0.804130\pi\)
\(350\) 6344.08 0.968872
\(351\) 0 0
\(352\) −616.622 −0.0933695
\(353\) −11818.1 −1.78191 −0.890957 0.454088i \(-0.849965\pi\)
−0.890957 + 0.454088i \(0.849965\pi\)
\(354\) 0 0
\(355\) −7189.87 −1.07493
\(356\) 225.578 0.0335831
\(357\) 0 0
\(358\) 4673.58 0.689962
\(359\) −8794.89 −1.29297 −0.646485 0.762926i \(-0.723762\pi\)
−0.646485 + 0.762926i \(0.723762\pi\)
\(360\) 0 0
\(361\) 2870.40 0.418486
\(362\) 4812.74 0.698762
\(363\) 0 0
\(364\) 230.787 0.0332322
\(365\) −11149.2 −1.59883
\(366\) 0 0
\(367\) 13284.1 1.88945 0.944723 0.327869i \(-0.106331\pi\)
0.944723 + 0.327869i \(0.106331\pi\)
\(368\) −7055.21 −0.999397
\(369\) 0 0
\(370\) −9781.80 −1.37441
\(371\) −11183.3 −1.56498
\(372\) 0 0
\(373\) −11897.3 −1.65153 −0.825763 0.564018i \(-0.809255\pi\)
−0.825763 + 0.564018i \(0.809255\pi\)
\(374\) −6606.13 −0.913355
\(375\) 0 0
\(376\) −11822.3 −1.62151
\(377\) 793.697 0.108428
\(378\) 0 0
\(379\) 1743.93 0.236357 0.118179 0.992992i \(-0.462294\pi\)
0.118179 + 0.992992i \(0.462294\pi\)
\(380\) 600.722 0.0810958
\(381\) 0 0
\(382\) 4136.87 0.554085
\(383\) −1572.59 −0.209806 −0.104903 0.994482i \(-0.533453\pi\)
−0.104903 + 0.994482i \(0.533453\pi\)
\(384\) 0 0
\(385\) 15149.4 2.00542
\(386\) 2208.77 0.291253
\(387\) 0 0
\(388\) 378.665 0.0495459
\(389\) 3545.77 0.462153 0.231077 0.972936i \(-0.425775\pi\)
0.231077 + 0.972936i \(0.425775\pi\)
\(390\) 0 0
\(391\) 9117.34 1.17924
\(392\) 21529.6 2.77400
\(393\) 0 0
\(394\) −3654.86 −0.467333
\(395\) 12756.4 1.62492
\(396\) 0 0
\(397\) 7176.14 0.907205 0.453602 0.891204i \(-0.350139\pi\)
0.453602 + 0.891204i \(0.350139\pi\)
\(398\) 7856.85 0.989518
\(399\) 0 0
\(400\) −3901.56 −0.487695
\(401\) −7431.68 −0.925487 −0.462744 0.886492i \(-0.653135\pi\)
−0.462744 + 0.886492i \(0.653135\pi\)
\(402\) 0 0
\(403\) −3388.97 −0.418900
\(404\) 358.773 0.0441822
\(405\) 0 0
\(406\) 5311.58 0.649284
\(407\) −7969.84 −0.970639
\(408\) 0 0
\(409\) −5038.83 −0.609179 −0.304589 0.952484i \(-0.598519\pi\)
−0.304589 + 0.952484i \(0.598519\pi\)
\(410\) −12379.9 −1.49122
\(411\) 0 0
\(412\) 185.405 0.0221705
\(413\) 2103.12 0.250576
\(414\) 0 0
\(415\) −18069.0 −2.13728
\(416\) −292.655 −0.0344918
\(417\) 0 0
\(418\) −8366.60 −0.979005
\(419\) 3543.16 0.413114 0.206557 0.978435i \(-0.433774\pi\)
0.206557 + 0.978435i \(0.433774\pi\)
\(420\) 0 0
\(421\) 13131.9 1.52022 0.760109 0.649796i \(-0.225145\pi\)
0.760109 + 0.649796i \(0.225145\pi\)
\(422\) 8580.54 0.989797
\(423\) 0 0
\(424\) 7281.29 0.833987
\(425\) 5041.93 0.575458
\(426\) 0 0
\(427\) −18851.6 −2.13652
\(428\) −515.107 −0.0581744
\(429\) 0 0
\(430\) 9501.92 1.06564
\(431\) 9117.45 1.01896 0.509480 0.860482i \(-0.329838\pi\)
0.509480 + 0.860482i \(0.329838\pi\)
\(432\) 0 0
\(433\) −2554.40 −0.283502 −0.141751 0.989902i \(-0.545273\pi\)
−0.141751 + 0.989902i \(0.545273\pi\)
\(434\) −22679.7 −2.50843
\(435\) 0 0
\(436\) 891.417 0.0979155
\(437\) 11547.0 1.26400
\(438\) 0 0
\(439\) 1523.94 0.165681 0.0828404 0.996563i \(-0.473601\pi\)
0.0828404 + 0.996563i \(0.473601\pi\)
\(440\) −9863.59 −1.06870
\(441\) 0 0
\(442\) −3135.34 −0.337404
\(443\) −10110.8 −1.08437 −0.542186 0.840258i \(-0.682403\pi\)
−0.542186 + 0.840258i \(0.682403\pi\)
\(444\) 0 0
\(445\) 7027.78 0.748649
\(446\) 5774.35 0.613057
\(447\) 0 0
\(448\) −19144.9 −2.01900
\(449\) −17927.7 −1.88432 −0.942162 0.335158i \(-0.891210\pi\)
−0.942162 + 0.335158i \(0.891210\pi\)
\(450\) 0 0
\(451\) −10086.7 −1.05314
\(452\) −323.536 −0.0336678
\(453\) 0 0
\(454\) 7151.58 0.739296
\(455\) 7190.07 0.740826
\(456\) 0 0
\(457\) −8911.04 −0.912125 −0.456062 0.889948i \(-0.650741\pi\)
−0.456062 + 0.889948i \(0.650741\pi\)
\(458\) 4131.23 0.421484
\(459\) 0 0
\(460\) 712.948 0.0722639
\(461\) −16724.3 −1.68965 −0.844825 0.535042i \(-0.820296\pi\)
−0.844825 + 0.535042i \(0.820296\pi\)
\(462\) 0 0
\(463\) −14050.4 −1.41031 −0.705157 0.709051i \(-0.749123\pi\)
−0.705157 + 0.709051i \(0.749123\pi\)
\(464\) −3266.58 −0.326826
\(465\) 0 0
\(466\) −4598.70 −0.457147
\(467\) −10526.3 −1.04304 −0.521520 0.853239i \(-0.674635\pi\)
−0.521520 + 0.853239i \(0.674635\pi\)
\(468\) 0 0
\(469\) 13306.2 1.31007
\(470\) −19289.7 −1.89313
\(471\) 0 0
\(472\) −1369.32 −0.133534
\(473\) 7741.80 0.752576
\(474\) 0 0
\(475\) 6385.56 0.616820
\(476\) 1227.46 0.118195
\(477\) 0 0
\(478\) 9593.16 0.917952
\(479\) 6393.78 0.609894 0.304947 0.952369i \(-0.401361\pi\)
0.304947 + 0.952369i \(0.401361\pi\)
\(480\) 0 0
\(481\) −3782.56 −0.358566
\(482\) 2397.43 0.226556
\(483\) 0 0
\(484\) 167.592 0.0157393
\(485\) 11797.2 1.10450
\(486\) 0 0
\(487\) 14196.4 1.32095 0.660473 0.750850i \(-0.270356\pi\)
0.660473 + 0.750850i \(0.270356\pi\)
\(488\) 12274.1 1.13857
\(489\) 0 0
\(490\) 35128.6 3.23867
\(491\) 5867.49 0.539299 0.269650 0.962958i \(-0.413092\pi\)
0.269650 + 0.962958i \(0.413092\pi\)
\(492\) 0 0
\(493\) 4221.36 0.385640
\(494\) −3970.87 −0.361656
\(495\) 0 0
\(496\) 13947.9 1.26266
\(497\) −18606.2 −1.67928
\(498\) 0 0
\(499\) −8345.43 −0.748683 −0.374341 0.927291i \(-0.622131\pi\)
−0.374341 + 0.927291i \(0.622131\pi\)
\(500\) −367.010 −0.0328264
\(501\) 0 0
\(502\) −1274.32 −0.113298
\(503\) 1607.53 0.142498 0.0712488 0.997459i \(-0.477302\pi\)
0.0712488 + 0.997459i \(0.477302\pi\)
\(504\) 0 0
\(505\) 11177.4 0.984929
\(506\) −9929.64 −0.872384
\(507\) 0 0
\(508\) 525.004 0.0458530
\(509\) 9106.68 0.793018 0.396509 0.918031i \(-0.370221\pi\)
0.396509 + 0.918031i \(0.370221\pi\)
\(510\) 0 0
\(511\) −28852.2 −2.49774
\(512\) 12394.3 1.06984
\(513\) 0 0
\(514\) 5961.61 0.511586
\(515\) 5776.21 0.494233
\(516\) 0 0
\(517\) −15716.5 −1.33697
\(518\) −25313.7 −2.14714
\(519\) 0 0
\(520\) −4681.36 −0.394791
\(521\) 7301.03 0.613942 0.306971 0.951719i \(-0.400684\pi\)
0.306971 + 0.951719i \(0.400684\pi\)
\(522\) 0 0
\(523\) 4874.89 0.407579 0.203790 0.979015i \(-0.434674\pi\)
0.203790 + 0.979015i \(0.434674\pi\)
\(524\) −110.974 −0.00925175
\(525\) 0 0
\(526\) 1560.83 0.129383
\(527\) −18024.6 −1.48987
\(528\) 0 0
\(529\) 1537.22 0.126343
\(530\) 11880.4 0.973685
\(531\) 0 0
\(532\) 1554.57 0.126690
\(533\) −4787.25 −0.389041
\(534\) 0 0
\(535\) −16047.9 −1.29685
\(536\) −8663.47 −0.698143
\(537\) 0 0
\(538\) −896.843 −0.0718692
\(539\) 28621.4 2.28722
\(540\) 0 0
\(541\) −1382.28 −0.109850 −0.0549249 0.998490i \(-0.517492\pi\)
−0.0549249 + 0.998490i \(0.517492\pi\)
\(542\) −4063.26 −0.322015
\(543\) 0 0
\(544\) −1556.51 −0.122675
\(545\) 27771.8 2.18277
\(546\) 0 0
\(547\) −20117.5 −1.57251 −0.786255 0.617902i \(-0.787983\pi\)
−0.786255 + 0.617902i \(0.787983\pi\)
\(548\) 1094.07 0.0852851
\(549\) 0 0
\(550\) −5491.14 −0.425714
\(551\) 5346.31 0.413358
\(552\) 0 0
\(553\) 33011.5 2.53850
\(554\) −23774.1 −1.82322
\(555\) 0 0
\(556\) −914.178 −0.0697298
\(557\) 12926.3 0.983311 0.491655 0.870790i \(-0.336392\pi\)
0.491655 + 0.870790i \(0.336392\pi\)
\(558\) 0 0
\(559\) 3674.33 0.278010
\(560\) −29591.9 −2.23301
\(561\) 0 0
\(562\) −12792.8 −0.960195
\(563\) −1483.56 −0.111056 −0.0555279 0.998457i \(-0.517684\pi\)
−0.0555279 + 0.998457i \(0.517684\pi\)
\(564\) 0 0
\(565\) −10079.6 −0.750538
\(566\) 7133.87 0.529786
\(567\) 0 0
\(568\) 12114.2 0.894899
\(569\) −535.139 −0.0394274 −0.0197137 0.999806i \(-0.506275\pi\)
−0.0197137 + 0.999806i \(0.506275\pi\)
\(570\) 0 0
\(571\) 11303.3 0.828420 0.414210 0.910181i \(-0.364058\pi\)
0.414210 + 0.910181i \(0.364058\pi\)
\(572\) −199.758 −0.0146019
\(573\) 0 0
\(574\) −32037.2 −2.32963
\(575\) 7578.50 0.549644
\(576\) 0 0
\(577\) −22867.6 −1.64990 −0.824950 0.565206i \(-0.808797\pi\)
−0.824950 + 0.565206i \(0.808797\pi\)
\(578\) −3168.99 −0.228049
\(579\) 0 0
\(580\) 330.097 0.0236320
\(581\) −46759.5 −3.33892
\(582\) 0 0
\(583\) 9679.73 0.687639
\(584\) 18785.3 1.33106
\(585\) 0 0
\(586\) −20068.1 −1.41469
\(587\) 17589.4 1.23679 0.618393 0.785869i \(-0.287784\pi\)
0.618393 + 0.785869i \(0.287784\pi\)
\(588\) 0 0
\(589\) −22828.0 −1.59696
\(590\) −2234.23 −0.155901
\(591\) 0 0
\(592\) 15567.7 1.08079
\(593\) 16110.5 1.11565 0.557825 0.829959i \(-0.311636\pi\)
0.557825 + 0.829959i \(0.311636\pi\)
\(594\) 0 0
\(595\) 38241.1 2.63485
\(596\) 31.0785 0.00213595
\(597\) 0 0
\(598\) −4712.70 −0.322269
\(599\) −22231.7 −1.51646 −0.758231 0.651986i \(-0.773936\pi\)
−0.758231 + 0.651986i \(0.773936\pi\)
\(600\) 0 0
\(601\) −16668.9 −1.13135 −0.565673 0.824629i \(-0.691384\pi\)
−0.565673 + 0.824629i \(0.691384\pi\)
\(602\) 24589.4 1.66477
\(603\) 0 0
\(604\) −953.380 −0.0642259
\(605\) 5221.25 0.350866
\(606\) 0 0
\(607\) 11293.3 0.755155 0.377577 0.925978i \(-0.376757\pi\)
0.377577 + 0.925978i \(0.376757\pi\)
\(608\) −1971.31 −0.131492
\(609\) 0 0
\(610\) 20026.8 1.32928
\(611\) −7459.23 −0.493892
\(612\) 0 0
\(613\) −9187.06 −0.605321 −0.302661 0.953098i \(-0.597875\pi\)
−0.302661 + 0.953098i \(0.597875\pi\)
\(614\) −10109.8 −0.664491
\(615\) 0 0
\(616\) −25525.3 −1.66955
\(617\) 5405.24 0.352685 0.176342 0.984329i \(-0.443573\pi\)
0.176342 + 0.984329i \(0.443573\pi\)
\(618\) 0 0
\(619\) 26890.8 1.74610 0.873049 0.487633i \(-0.162140\pi\)
0.873049 + 0.487633i \(0.162140\pi\)
\(620\) −1409.47 −0.0912994
\(621\) 0 0
\(622\) 9318.16 0.600682
\(623\) 18186.7 1.16956
\(624\) 0 0
\(625\) −19526.2 −1.24968
\(626\) 13326.5 0.850850
\(627\) 0 0
\(628\) −46.7465 −0.00297036
\(629\) −20117.9 −1.27529
\(630\) 0 0
\(631\) 13088.1 0.825718 0.412859 0.910795i \(-0.364530\pi\)
0.412859 + 0.910795i \(0.364530\pi\)
\(632\) −21493.3 −1.35278
\(633\) 0 0
\(634\) 28758.6 1.80150
\(635\) 16356.3 1.02217
\(636\) 0 0
\(637\) 13584.0 0.844926
\(638\) −4597.46 −0.285290
\(639\) 0 0
\(640\) 18136.1 1.12014
\(641\) −11372.4 −0.700754 −0.350377 0.936609i \(-0.613947\pi\)
−0.350377 + 0.936609i \(0.613947\pi\)
\(642\) 0 0
\(643\) 28882.8 1.77142 0.885712 0.464236i \(-0.153671\pi\)
0.885712 + 0.464236i \(0.153671\pi\)
\(644\) 1844.99 0.112893
\(645\) 0 0
\(646\) −21119.5 −1.28628
\(647\) −3509.03 −0.213222 −0.106611 0.994301i \(-0.534000\pi\)
−0.106611 + 0.994301i \(0.534000\pi\)
\(648\) 0 0
\(649\) −1820.37 −0.110101
\(650\) −2606.15 −0.157264
\(651\) 0 0
\(652\) 444.835 0.0267194
\(653\) 12551.5 0.752185 0.376092 0.926582i \(-0.377268\pi\)
0.376092 + 0.926582i \(0.377268\pi\)
\(654\) 0 0
\(655\) −3457.35 −0.206244
\(656\) 19702.7 1.17265
\(657\) 0 0
\(658\) −49918.7 −2.95750
\(659\) 26042.1 1.53938 0.769692 0.638415i \(-0.220410\pi\)
0.769692 + 0.638415i \(0.220410\pi\)
\(660\) 0 0
\(661\) 8841.43 0.520260 0.260130 0.965574i \(-0.416235\pi\)
0.260130 + 0.965574i \(0.416235\pi\)
\(662\) 7862.16 0.461588
\(663\) 0 0
\(664\) 30444.5 1.77933
\(665\) 48432.0 2.82423
\(666\) 0 0
\(667\) 6345.10 0.368341
\(668\) −295.673 −0.0171256
\(669\) 0 0
\(670\) −14135.7 −0.815087
\(671\) 16317.1 0.938770
\(672\) 0 0
\(673\) 5539.99 0.317312 0.158656 0.987334i \(-0.449284\pi\)
0.158656 + 0.987334i \(0.449284\pi\)
\(674\) −12920.9 −0.738418
\(675\) 0 0
\(676\) 876.562 0.0498727
\(677\) −6105.80 −0.346625 −0.173312 0.984867i \(-0.555447\pi\)
−0.173312 + 0.984867i \(0.555447\pi\)
\(678\) 0 0
\(679\) 30529.1 1.72548
\(680\) −24898.3 −1.40413
\(681\) 0 0
\(682\) 19630.5 1.10218
\(683\) −6169.93 −0.345660 −0.172830 0.984952i \(-0.555291\pi\)
−0.172830 + 0.984952i \(0.555291\pi\)
\(684\) 0 0
\(685\) 34085.2 1.90121
\(686\) 57294.0 3.18877
\(687\) 0 0
\(688\) −15122.3 −0.837983
\(689\) 4594.09 0.254022
\(690\) 0 0
\(691\) −24137.5 −1.32885 −0.664423 0.747357i \(-0.731323\pi\)
−0.664423 + 0.747357i \(0.731323\pi\)
\(692\) −1273.33 −0.0699491
\(693\) 0 0
\(694\) 2617.03 0.143143
\(695\) −28480.8 −1.55445
\(696\) 0 0
\(697\) −25461.5 −1.38368
\(698\) 29272.8 1.58738
\(699\) 0 0
\(700\) 1020.29 0.0550904
\(701\) −1381.52 −0.0744354 −0.0372177 0.999307i \(-0.511849\pi\)
−0.0372177 + 0.999307i \(0.511849\pi\)
\(702\) 0 0
\(703\) −25479.2 −1.36695
\(704\) 16571.0 0.887133
\(705\) 0 0
\(706\) 32489.9 1.73197
\(707\) 28925.3 1.53868
\(708\) 0 0
\(709\) 11440.3 0.605995 0.302998 0.952991i \(-0.402013\pi\)
0.302998 + 0.952991i \(0.402013\pi\)
\(710\) 19766.1 1.04480
\(711\) 0 0
\(712\) −11841.1 −0.623266
\(713\) −27092.7 −1.42304
\(714\) 0 0
\(715\) −6223.39 −0.325513
\(716\) 751.630 0.0392315
\(717\) 0 0
\(718\) 24178.5 1.25673
\(719\) −23773.0 −1.23308 −0.616539 0.787324i \(-0.711466\pi\)
−0.616539 + 0.787324i \(0.711466\pi\)
\(720\) 0 0
\(721\) 14947.9 0.772105
\(722\) −7891.17 −0.406757
\(723\) 0 0
\(724\) 774.010 0.0397319
\(725\) 3508.87 0.179746
\(726\) 0 0
\(727\) −38328.7 −1.95534 −0.977670 0.210146i \(-0.932606\pi\)
−0.977670 + 0.210146i \(0.932606\pi\)
\(728\) −12114.6 −0.616753
\(729\) 0 0
\(730\) 30650.8 1.55402
\(731\) 19542.3 0.988781
\(732\) 0 0
\(733\) −23440.6 −1.18117 −0.590586 0.806975i \(-0.701103\pi\)
−0.590586 + 0.806975i \(0.701103\pi\)
\(734\) −36520.2 −1.83649
\(735\) 0 0
\(736\) −2339.59 −0.117172
\(737\) −11517.2 −0.575633
\(738\) 0 0
\(739\) −20289.0 −1.00994 −0.504968 0.863138i \(-0.668496\pi\)
−0.504968 + 0.863138i \(0.668496\pi\)
\(740\) −1573.16 −0.0781495
\(741\) 0 0
\(742\) 30744.6 1.52112
\(743\) 21978.5 1.08521 0.542607 0.839987i \(-0.317437\pi\)
0.542607 + 0.839987i \(0.317437\pi\)
\(744\) 0 0
\(745\) 968.240 0.0476155
\(746\) 32707.6 1.60524
\(747\) 0 0
\(748\) −1062.43 −0.0519337
\(749\) −41529.4 −2.02597
\(750\) 0 0
\(751\) −11321.9 −0.550122 −0.275061 0.961427i \(-0.588698\pi\)
−0.275061 + 0.961427i \(0.588698\pi\)
\(752\) 30699.6 1.48870
\(753\) 0 0
\(754\) −2182.00 −0.105389
\(755\) −29702.2 −1.43175
\(756\) 0 0
\(757\) −29868.4 −1.43406 −0.717031 0.697042i \(-0.754499\pi\)
−0.717031 + 0.697042i \(0.754499\pi\)
\(758\) −4794.33 −0.229733
\(759\) 0 0
\(760\) −31533.4 −1.50505
\(761\) −25711.9 −1.22478 −0.612388 0.790557i \(-0.709791\pi\)
−0.612388 + 0.790557i \(0.709791\pi\)
\(762\) 0 0
\(763\) 71868.7 3.40999
\(764\) 665.313 0.0315055
\(765\) 0 0
\(766\) 4323.31 0.203926
\(767\) −863.964 −0.0406727
\(768\) 0 0
\(769\) −16360.5 −0.767199 −0.383600 0.923500i \(-0.625316\pi\)
−0.383600 + 0.923500i \(0.625316\pi\)
\(770\) −41648.2 −1.94922
\(771\) 0 0
\(772\) 355.227 0.0165608
\(773\) 8952.62 0.416563 0.208282 0.978069i \(-0.433213\pi\)
0.208282 + 0.978069i \(0.433213\pi\)
\(774\) 0 0
\(775\) −14982.4 −0.694429
\(776\) −19877.1 −0.919519
\(777\) 0 0
\(778\) −9747.88 −0.449201
\(779\) −32246.7 −1.48313
\(780\) 0 0
\(781\) 16104.6 0.737861
\(782\) −25065.0 −1.14619
\(783\) 0 0
\(784\) −55907.1 −2.54679
\(785\) −1456.37 −0.0662166
\(786\) 0 0
\(787\) −1427.86 −0.0646731 −0.0323365 0.999477i \(-0.510295\pi\)
−0.0323365 + 0.999477i \(0.510295\pi\)
\(788\) −587.794 −0.0265727
\(789\) 0 0
\(790\) −35069.4 −1.57938
\(791\) −26084.5 −1.17251
\(792\) 0 0
\(793\) 7744.25 0.346793
\(794\) −19728.3 −0.881779
\(795\) 0 0
\(796\) 1263.58 0.0562644
\(797\) −33475.6 −1.48779 −0.743894 0.668297i \(-0.767023\pi\)
−0.743894 + 0.668297i \(0.767023\pi\)
\(798\) 0 0
\(799\) −39672.7 −1.75659
\(800\) −1293.80 −0.0571785
\(801\) 0 0
\(802\) 20430.9 0.899549
\(803\) 24973.1 1.09749
\(804\) 0 0
\(805\) 57480.0 2.51665
\(806\) 9316.82 0.407160
\(807\) 0 0
\(808\) −18832.9 −0.819974
\(809\) −18354.9 −0.797679 −0.398839 0.917021i \(-0.630587\pi\)
−0.398839 + 0.917021i \(0.630587\pi\)
\(810\) 0 0
\(811\) 18590.6 0.804937 0.402469 0.915434i \(-0.368152\pi\)
0.402469 + 0.915434i \(0.368152\pi\)
\(812\) 854.237 0.0369185
\(813\) 0 0
\(814\) 21910.3 0.943436
\(815\) 13858.7 0.595641
\(816\) 0 0
\(817\) 24750.2 1.05985
\(818\) 13852.5 0.592106
\(819\) 0 0
\(820\) −1991.01 −0.0847915
\(821\) −11083.4 −0.471151 −0.235575 0.971856i \(-0.575697\pi\)
−0.235575 + 0.971856i \(0.575697\pi\)
\(822\) 0 0
\(823\) −11533.7 −0.488506 −0.244253 0.969712i \(-0.578543\pi\)
−0.244253 + 0.969712i \(0.578543\pi\)
\(824\) −9732.36 −0.411460
\(825\) 0 0
\(826\) −5781.82 −0.243554
\(827\) 25859.7 1.08734 0.543670 0.839299i \(-0.317034\pi\)
0.543670 + 0.839299i \(0.317034\pi\)
\(828\) 0 0
\(829\) −20734.3 −0.868674 −0.434337 0.900750i \(-0.643017\pi\)
−0.434337 + 0.900750i \(0.643017\pi\)
\(830\) 49674.4 2.07738
\(831\) 0 0
\(832\) 7864.74 0.327717
\(833\) 72247.9 3.00509
\(834\) 0 0
\(835\) −9211.56 −0.381772
\(836\) −1345.56 −0.0556666
\(837\) 0 0
\(838\) −9740.71 −0.401536
\(839\) −8282.44 −0.340812 −0.170406 0.985374i \(-0.554508\pi\)
−0.170406 + 0.985374i \(0.554508\pi\)
\(840\) 0 0
\(841\) −21451.2 −0.879544
\(842\) −36101.8 −1.47761
\(843\) 0 0
\(844\) 1379.97 0.0562802
\(845\) 27308.9 1.11178
\(846\) 0 0
\(847\) 13511.7 0.548133
\(848\) −18907.7 −0.765676
\(849\) 0 0
\(850\) −13861.1 −0.559330
\(851\) −30239.2 −1.21808
\(852\) 0 0
\(853\) 27976.9 1.12299 0.561495 0.827480i \(-0.310226\pi\)
0.561495 + 0.827480i \(0.310226\pi\)
\(854\) 51826.1 2.07664
\(855\) 0 0
\(856\) 27039.3 1.07965
\(857\) 30334.1 1.20910 0.604548 0.796569i \(-0.293354\pi\)
0.604548 + 0.796569i \(0.293354\pi\)
\(858\) 0 0
\(859\) 35079.4 1.39336 0.696679 0.717383i \(-0.254660\pi\)
0.696679 + 0.717383i \(0.254660\pi\)
\(860\) 1528.15 0.0605924
\(861\) 0 0
\(862\) −25065.3 −0.990403
\(863\) 8904.72 0.351240 0.175620 0.984458i \(-0.443807\pi\)
0.175620 + 0.984458i \(0.443807\pi\)
\(864\) 0 0
\(865\) −39670.1 −1.55934
\(866\) 7022.44 0.275557
\(867\) 0 0
\(868\) −3647.47 −0.142631
\(869\) −28573.2 −1.11540
\(870\) 0 0
\(871\) −5466.17 −0.212645
\(872\) −46792.8 −1.81720
\(873\) 0 0
\(874\) −31744.6 −1.22858
\(875\) −29589.4 −1.14321
\(876\) 0 0
\(877\) 5487.15 0.211275 0.105637 0.994405i \(-0.466312\pi\)
0.105637 + 0.994405i \(0.466312\pi\)
\(878\) −4189.56 −0.161037
\(879\) 0 0
\(880\) 25613.3 0.981165
\(881\) 12650.9 0.483792 0.241896 0.970302i \(-0.422231\pi\)
0.241896 + 0.970302i \(0.422231\pi\)
\(882\) 0 0
\(883\) 31973.4 1.21856 0.609282 0.792954i \(-0.291458\pi\)
0.609282 + 0.792954i \(0.291458\pi\)
\(884\) −504.242 −0.0191849
\(885\) 0 0
\(886\) 27796.1 1.05398
\(887\) 48252.3 1.82655 0.913277 0.407338i \(-0.133543\pi\)
0.913277 + 0.407338i \(0.133543\pi\)
\(888\) 0 0
\(889\) 42327.4 1.59687
\(890\) −19320.5 −0.727667
\(891\) 0 0
\(892\) 928.662 0.0348586
\(893\) −50245.1 −1.88285
\(894\) 0 0
\(895\) 23416.8 0.874565
\(896\) 46933.2 1.74992
\(897\) 0 0
\(898\) 49286.1 1.83151
\(899\) −12544.0 −0.465368
\(900\) 0 0
\(901\) 24434.1 0.903462
\(902\) 27729.9 1.02362
\(903\) 0 0
\(904\) 16983.2 0.624839
\(905\) 24114.0 0.885719
\(906\) 0 0
\(907\) 33861.7 1.23964 0.619822 0.784742i \(-0.287205\pi\)
0.619822 + 0.784742i \(0.287205\pi\)
\(908\) 1150.16 0.0420366
\(909\) 0 0
\(910\) −19766.6 −0.720063
\(911\) −45845.4 −1.66732 −0.833659 0.552280i \(-0.813758\pi\)
−0.833659 + 0.552280i \(0.813758\pi\)
\(912\) 0 0
\(913\) 40472.8 1.46709
\(914\) 24497.9 0.886561
\(915\) 0 0
\(916\) 664.407 0.0239657
\(917\) −8947.05 −0.322200
\(918\) 0 0
\(919\) −14126.6 −0.507066 −0.253533 0.967327i \(-0.581593\pi\)
−0.253533 + 0.967327i \(0.581593\pi\)
\(920\) −37424.5 −1.34114
\(921\) 0 0
\(922\) 45977.8 1.64230
\(923\) 7643.42 0.272575
\(924\) 0 0
\(925\) −16722.4 −0.594410
\(926\) 38626.6 1.37079
\(927\) 0 0
\(928\) −1083.24 −0.0383179
\(929\) −27790.5 −0.981461 −0.490730 0.871311i \(-0.663270\pi\)
−0.490730 + 0.871311i \(0.663270\pi\)
\(930\) 0 0
\(931\) 91501.3 3.22109
\(932\) −739.588 −0.0259936
\(933\) 0 0
\(934\) 28938.5 1.01381
\(935\) −33099.7 −1.15773
\(936\) 0 0
\(937\) −6472.88 −0.225677 −0.112839 0.993613i \(-0.535994\pi\)
−0.112839 + 0.993613i \(0.535994\pi\)
\(938\) −36580.7 −1.27335
\(939\) 0 0
\(940\) −3102.28 −0.107644
\(941\) 12312.7 0.426550 0.213275 0.976992i \(-0.431587\pi\)
0.213275 + 0.976992i \(0.431587\pi\)
\(942\) 0 0
\(943\) −38271.0 −1.32161
\(944\) 3555.78 0.122596
\(945\) 0 0
\(946\) −21283.4 −0.731484
\(947\) −4368.65 −0.149907 −0.0749536 0.997187i \(-0.523881\pi\)
−0.0749536 + 0.997187i \(0.523881\pi\)
\(948\) 0 0
\(949\) 11852.5 0.405424
\(950\) −17554.9 −0.599533
\(951\) 0 0
\(952\) −64432.6 −2.19356
\(953\) −2917.34 −0.0991627 −0.0495814 0.998770i \(-0.515789\pi\)
−0.0495814 + 0.998770i \(0.515789\pi\)
\(954\) 0 0
\(955\) 20727.6 0.702333
\(956\) 1542.82 0.0521951
\(957\) 0 0
\(958\) −17577.5 −0.592801
\(959\) 88206.9 2.97012
\(960\) 0 0
\(961\) 23770.1 0.797895
\(962\) 10398.9 0.348516
\(963\) 0 0
\(964\) 385.568 0.0128821
\(965\) 11067.0 0.369179
\(966\) 0 0
\(967\) 42945.5 1.42816 0.714082 0.700062i \(-0.246845\pi\)
0.714082 + 0.700062i \(0.246845\pi\)
\(968\) −8797.31 −0.292104
\(969\) 0 0
\(970\) −32432.3 −1.07354
\(971\) −43742.9 −1.44570 −0.722851 0.691004i \(-0.757169\pi\)
−0.722851 + 0.691004i \(0.757169\pi\)
\(972\) 0 0
\(973\) −73703.7 −2.42840
\(974\) −39028.1 −1.28392
\(975\) 0 0
\(976\) −31872.7 −1.04531
\(977\) 29917.5 0.979679 0.489839 0.871813i \(-0.337055\pi\)
0.489839 + 0.871813i \(0.337055\pi\)
\(978\) 0 0
\(979\) −15741.6 −0.513895
\(980\) 5649.57 0.184152
\(981\) 0 0
\(982\) −16130.6 −0.524185
\(983\) 48495.4 1.57351 0.786756 0.617264i \(-0.211759\pi\)
0.786756 + 0.617264i \(0.211759\pi\)
\(984\) 0 0
\(985\) −18312.5 −0.592370
\(986\) −11605.2 −0.374832
\(987\) 0 0
\(988\) −638.617 −0.0205639
\(989\) 29373.9 0.944426
\(990\) 0 0
\(991\) −14192.8 −0.454944 −0.227472 0.973785i \(-0.573046\pi\)
−0.227472 + 0.973785i \(0.573046\pi\)
\(992\) 4625.27 0.148037
\(993\) 0 0
\(994\) 51151.4 1.63222
\(995\) 39366.4 1.25427
\(996\) 0 0
\(997\) −49366.8 −1.56817 −0.784083 0.620656i \(-0.786866\pi\)
−0.784083 + 0.620656i \(0.786866\pi\)
\(998\) 22942.9 0.727700
\(999\) 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 531.4.a.c.1.3 7
3.2 odd 2 177.4.a.b.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.4.a.b.1.5 7 3.2 odd 2
531.4.a.c.1.3 7 1.1 even 1 trivial