Properties

Label 1521.4.a.bk.1.3
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - 70x^{8} + 1645x^{6} - 14700x^{4} + 44100x^{2} - 27648 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3.27897\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.27897 q^{2} +2.75167 q^{4} +17.5414 q^{5} +26.6999 q^{7} +17.2091 q^{8} +O(q^{10})\) \(q-3.27897 q^{2} +2.75167 q^{4} +17.5414 q^{5} +26.6999 q^{7} +17.2091 q^{8} -57.5178 q^{10} -21.4026 q^{11} -87.5483 q^{14} -78.4417 q^{16} -83.9630 q^{17} +77.1142 q^{19} +48.2682 q^{20} +70.1785 q^{22} -142.119 q^{23} +182.701 q^{25} +73.4693 q^{28} -134.223 q^{29} -122.559 q^{31} +119.535 q^{32} +275.312 q^{34} +468.354 q^{35} +222.587 q^{37} -252.855 q^{38} +301.873 q^{40} +198.321 q^{41} +154.656 q^{43} -58.8928 q^{44} +466.006 q^{46} -78.7956 q^{47} +369.884 q^{49} -599.072 q^{50} +477.088 q^{53} -375.431 q^{55} +459.482 q^{56} +440.114 q^{58} -42.9282 q^{59} +496.539 q^{61} +401.869 q^{62} +235.581 q^{64} +484.659 q^{67} -231.038 q^{68} -1535.72 q^{70} +382.432 q^{71} -193.622 q^{73} -729.858 q^{74} +212.193 q^{76} -571.447 q^{77} +1049.60 q^{79} -1375.98 q^{80} -650.289 q^{82} +861.900 q^{83} -1472.83 q^{85} -507.112 q^{86} -368.320 q^{88} +967.645 q^{89} -391.065 q^{92} +258.369 q^{94} +1352.69 q^{95} -591.470 q^{97} -1212.84 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 60 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 60 q^{4} + 80 q^{10} + 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} + 960 q^{25} - 990 q^{29} + 120 q^{35} - 1380 q^{38} + 2000 q^{40} - 740 q^{43} + 1550 q^{49} - 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} + 3140 q^{64} - 1200 q^{68} + 4380 q^{74} - 4320 q^{77} + 1100 q^{79} - 4780 q^{82} + 6340 q^{88} + 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.27897 −1.15929 −0.579646 0.814868i \(-0.696809\pi\)
−0.579646 + 0.814868i \(0.696809\pi\)
\(3\) 0 0
\(4\) 2.75167 0.343959
\(5\) 17.5414 1.56895 0.784476 0.620160i \(-0.212932\pi\)
0.784476 + 0.620160i \(0.212932\pi\)
\(6\) 0 0
\(7\) 26.6999 1.44166 0.720829 0.693112i \(-0.243761\pi\)
0.720829 + 0.693112i \(0.243761\pi\)
\(8\) 17.2091 0.760544
\(9\) 0 0
\(10\) −57.5178 −1.81887
\(11\) −21.4026 −0.586647 −0.293324 0.956013i \(-0.594761\pi\)
−0.293324 + 0.956013i \(0.594761\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −87.5483 −1.67130
\(15\) 0 0
\(16\) −78.4417 −1.22565
\(17\) −83.9630 −1.19788 −0.598942 0.800793i \(-0.704412\pi\)
−0.598942 + 0.800793i \(0.704412\pi\)
\(18\) 0 0
\(19\) 77.1142 0.931116 0.465558 0.885017i \(-0.345854\pi\)
0.465558 + 0.885017i \(0.345854\pi\)
\(20\) 48.2682 0.539654
\(21\) 0 0
\(22\) 70.1785 0.680096
\(23\) −142.119 −1.28843 −0.644216 0.764844i \(-0.722816\pi\)
−0.644216 + 0.764844i \(0.722816\pi\)
\(24\) 0 0
\(25\) 182.701 1.46161
\(26\) 0 0
\(27\) 0 0
\(28\) 73.4693 0.495871
\(29\) −134.223 −0.859469 −0.429734 0.902955i \(-0.641393\pi\)
−0.429734 + 0.902955i \(0.641393\pi\)
\(30\) 0 0
\(31\) −122.559 −0.710074 −0.355037 0.934852i \(-0.615532\pi\)
−0.355037 + 0.934852i \(0.615532\pi\)
\(32\) 119.535 0.660344
\(33\) 0 0
\(34\) 275.312 1.38870
\(35\) 468.354 2.26189
\(36\) 0 0
\(37\) 222.587 0.989003 0.494502 0.869177i \(-0.335351\pi\)
0.494502 + 0.869177i \(0.335351\pi\)
\(38\) −252.855 −1.07944
\(39\) 0 0
\(40\) 301.873 1.19326
\(41\) 198.321 0.755427 0.377713 0.925923i \(-0.376710\pi\)
0.377713 + 0.925923i \(0.376710\pi\)
\(42\) 0 0
\(43\) 154.656 0.548483 0.274242 0.961661i \(-0.411573\pi\)
0.274242 + 0.961661i \(0.411573\pi\)
\(44\) −58.8928 −0.201782
\(45\) 0 0
\(46\) 466.006 1.49367
\(47\) −78.7956 −0.244543 −0.122271 0.992497i \(-0.539018\pi\)
−0.122271 + 0.992497i \(0.539018\pi\)
\(48\) 0 0
\(49\) 369.884 1.07838
\(50\) −599.072 −1.69443
\(51\) 0 0
\(52\) 0 0
\(53\) 477.088 1.23647 0.618237 0.785992i \(-0.287847\pi\)
0.618237 + 0.785992i \(0.287847\pi\)
\(54\) 0 0
\(55\) −375.431 −0.920421
\(56\) 459.482 1.09644
\(57\) 0 0
\(58\) 440.114 0.996376
\(59\) −42.9282 −0.0947249 −0.0473625 0.998878i \(-0.515082\pi\)
−0.0473625 + 0.998878i \(0.515082\pi\)
\(60\) 0 0
\(61\) 496.539 1.04222 0.521109 0.853490i \(-0.325519\pi\)
0.521109 + 0.853490i \(0.325519\pi\)
\(62\) 401.869 0.823183
\(63\) 0 0
\(64\) 235.581 0.460119
\(65\) 0 0
\(66\) 0 0
\(67\) 484.659 0.883739 0.441869 0.897079i \(-0.354315\pi\)
0.441869 + 0.897079i \(0.354315\pi\)
\(68\) −231.038 −0.412022
\(69\) 0 0
\(70\) −1535.72 −2.62219
\(71\) 382.432 0.639245 0.319622 0.947545i \(-0.396444\pi\)
0.319622 + 0.947545i \(0.396444\pi\)
\(72\) 0 0
\(73\) −193.622 −0.310435 −0.155217 0.987880i \(-0.549608\pi\)
−0.155217 + 0.987880i \(0.549608\pi\)
\(74\) −729.858 −1.14654
\(75\) 0 0
\(76\) 212.193 0.320265
\(77\) −571.447 −0.845745
\(78\) 0 0
\(79\) 1049.60 1.49480 0.747399 0.664375i \(-0.231302\pi\)
0.747399 + 0.664375i \(0.231302\pi\)
\(80\) −1375.98 −1.92299
\(81\) 0 0
\(82\) −650.289 −0.875761
\(83\) 861.900 1.13983 0.569914 0.821704i \(-0.306976\pi\)
0.569914 + 0.821704i \(0.306976\pi\)
\(84\) 0 0
\(85\) −1472.83 −1.87942
\(86\) −507.112 −0.635853
\(87\) 0 0
\(88\) −368.320 −0.446171
\(89\) 967.645 1.15247 0.576237 0.817283i \(-0.304521\pi\)
0.576237 + 0.817283i \(0.304521\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −391.065 −0.443167
\(93\) 0 0
\(94\) 258.369 0.283497
\(95\) 1352.69 1.46088
\(96\) 0 0
\(97\) −591.470 −0.619120 −0.309560 0.950880i \(-0.600182\pi\)
−0.309560 + 0.950880i \(0.600182\pi\)
\(98\) −1212.84 −1.25016
\(99\) 0 0
\(100\) 502.733 0.502733
\(101\) −255.110 −0.251331 −0.125665 0.992073i \(-0.540107\pi\)
−0.125665 + 0.992073i \(0.540107\pi\)
\(102\) 0 0
\(103\) −247.355 −0.236627 −0.118313 0.992976i \(-0.537749\pi\)
−0.118313 + 0.992976i \(0.537749\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −1564.36 −1.43343
\(107\) −683.484 −0.617522 −0.308761 0.951140i \(-0.599914\pi\)
−0.308761 + 0.951140i \(0.599914\pi\)
\(108\) 0 0
\(109\) −1697.76 −1.49189 −0.745946 0.666006i \(-0.768002\pi\)
−0.745946 + 0.666006i \(0.768002\pi\)
\(110\) 1231.03 1.06704
\(111\) 0 0
\(112\) −2094.38 −1.76697
\(113\) 380.709 0.316939 0.158469 0.987364i \(-0.449344\pi\)
0.158469 + 0.987364i \(0.449344\pi\)
\(114\) 0 0
\(115\) −2492.97 −2.02149
\(116\) −369.337 −0.295622
\(117\) 0 0
\(118\) 140.760 0.109814
\(119\) −2241.80 −1.72694
\(120\) 0 0
\(121\) −872.930 −0.655845
\(122\) −1628.14 −1.20824
\(123\) 0 0
\(124\) −337.243 −0.244236
\(125\) 1012.16 0.724241
\(126\) 0 0
\(127\) 123.231 0.0861025 0.0430513 0.999073i \(-0.486292\pi\)
0.0430513 + 0.999073i \(0.486292\pi\)
\(128\) −1728.74 −1.19376
\(129\) 0 0
\(130\) 0 0
\(131\) 1218.41 0.812616 0.406308 0.913736i \(-0.366816\pi\)
0.406308 + 0.913736i \(0.366816\pi\)
\(132\) 0 0
\(133\) 2058.94 1.34235
\(134\) −1589.18 −1.02451
\(135\) 0 0
\(136\) −1444.93 −0.911042
\(137\) 2728.83 1.70175 0.850875 0.525369i \(-0.176073\pi\)
0.850875 + 0.525369i \(0.176073\pi\)
\(138\) 0 0
\(139\) 3112.78 1.89944 0.949722 0.313094i \(-0.101366\pi\)
0.949722 + 0.313094i \(0.101366\pi\)
\(140\) 1288.75 0.777998
\(141\) 0 0
\(142\) −1253.99 −0.741071
\(143\) 0 0
\(144\) 0 0
\(145\) −2354.46 −1.34846
\(146\) 634.881 0.359885
\(147\) 0 0
\(148\) 612.487 0.340176
\(149\) 1370.57 0.753567 0.376784 0.926301i \(-0.377030\pi\)
0.376784 + 0.926301i \(0.377030\pi\)
\(150\) 0 0
\(151\) 2847.56 1.53464 0.767321 0.641263i \(-0.221589\pi\)
0.767321 + 0.641263i \(0.221589\pi\)
\(152\) 1327.07 0.708154
\(153\) 0 0
\(154\) 1873.76 0.980466
\(155\) −2149.86 −1.11407
\(156\) 0 0
\(157\) 3354.00 1.70496 0.852479 0.522761i \(-0.175098\pi\)
0.852479 + 0.522761i \(0.175098\pi\)
\(158\) −3441.61 −1.73291
\(159\) 0 0
\(160\) 2096.81 1.03605
\(161\) −3794.57 −1.85748
\(162\) 0 0
\(163\) 2196.18 1.05533 0.527664 0.849453i \(-0.323068\pi\)
0.527664 + 0.849453i \(0.323068\pi\)
\(164\) 545.713 0.259836
\(165\) 0 0
\(166\) −2826.15 −1.32139
\(167\) 912.535 0.422839 0.211419 0.977395i \(-0.432191\pi\)
0.211419 + 0.977395i \(0.432191\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 4829.37 2.17880
\(171\) 0 0
\(172\) 425.562 0.188656
\(173\) −899.636 −0.395364 −0.197682 0.980266i \(-0.563341\pi\)
−0.197682 + 0.980266i \(0.563341\pi\)
\(174\) 0 0
\(175\) 4878.10 2.10714
\(176\) 1678.85 0.719025
\(177\) 0 0
\(178\) −3172.88 −1.33605
\(179\) −313.278 −0.130813 −0.0654064 0.997859i \(-0.520834\pi\)
−0.0654064 + 0.997859i \(0.520834\pi\)
\(180\) 0 0
\(181\) −2745.06 −1.12728 −0.563642 0.826019i \(-0.690600\pi\)
−0.563642 + 0.826019i \(0.690600\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −2445.75 −0.979909
\(185\) 3904.50 1.55170
\(186\) 0 0
\(187\) 1797.02 0.702735
\(188\) −216.819 −0.0841126
\(189\) 0 0
\(190\) −4435.44 −1.69358
\(191\) −89.8679 −0.0340451 −0.0170226 0.999855i \(-0.505419\pi\)
−0.0170226 + 0.999855i \(0.505419\pi\)
\(192\) 0 0
\(193\) −848.954 −0.316627 −0.158314 0.987389i \(-0.550606\pi\)
−0.158314 + 0.987389i \(0.550606\pi\)
\(194\) 1939.41 0.717741
\(195\) 0 0
\(196\) 1017.80 0.370918
\(197\) 4343.86 1.57100 0.785501 0.618860i \(-0.212405\pi\)
0.785501 + 0.618860i \(0.212405\pi\)
\(198\) 0 0
\(199\) −3328.41 −1.18565 −0.592825 0.805331i \(-0.701988\pi\)
−0.592825 + 0.805331i \(0.701988\pi\)
\(200\) 3144.13 1.11162
\(201\) 0 0
\(202\) 836.499 0.291366
\(203\) −3583.74 −1.23906
\(204\) 0 0
\(205\) 3478.83 1.18523
\(206\) 811.069 0.274320
\(207\) 0 0
\(208\) 0 0
\(209\) −1650.44 −0.546236
\(210\) 0 0
\(211\) −4599.88 −1.50080 −0.750401 0.660983i \(-0.770139\pi\)
−0.750401 + 0.660983i \(0.770139\pi\)
\(212\) 1312.79 0.425296
\(213\) 0 0
\(214\) 2241.13 0.715889
\(215\) 2712.88 0.860544
\(216\) 0 0
\(217\) −3272.32 −1.02368
\(218\) 5566.92 1.72954
\(219\) 0 0
\(220\) −1033.06 −0.316587
\(221\) 0 0
\(222\) 0 0
\(223\) 2529.58 0.759611 0.379806 0.925066i \(-0.375991\pi\)
0.379806 + 0.925066i \(0.375991\pi\)
\(224\) 3191.57 0.951991
\(225\) 0 0
\(226\) −1248.33 −0.367425
\(227\) 37.2670 0.0108965 0.00544823 0.999985i \(-0.498266\pi\)
0.00544823 + 0.999985i \(0.498266\pi\)
\(228\) 0 0
\(229\) 4094.45 1.18152 0.590762 0.806846i \(-0.298827\pi\)
0.590762 + 0.806846i \(0.298827\pi\)
\(230\) 8174.40 2.34349
\(231\) 0 0
\(232\) −2309.86 −0.653664
\(233\) −1466.04 −0.412205 −0.206103 0.978530i \(-0.566078\pi\)
−0.206103 + 0.978530i \(0.566078\pi\)
\(234\) 0 0
\(235\) −1382.19 −0.383676
\(236\) −118.124 −0.0325815
\(237\) 0 0
\(238\) 7350.81 2.00203
\(239\) −5520.53 −1.49412 −0.747058 0.664759i \(-0.768534\pi\)
−0.747058 + 0.664759i \(0.768534\pi\)
\(240\) 0 0
\(241\) −2665.63 −0.712483 −0.356241 0.934394i \(-0.615942\pi\)
−0.356241 + 0.934394i \(0.615942\pi\)
\(242\) 2862.31 0.760316
\(243\) 0 0
\(244\) 1366.31 0.358480
\(245\) 6488.30 1.69193
\(246\) 0 0
\(247\) 0 0
\(248\) −2109.14 −0.540042
\(249\) 0 0
\(250\) −3318.84 −0.839607
\(251\) −1579.21 −0.397127 −0.198564 0.980088i \(-0.563628\pi\)
−0.198564 + 0.980088i \(0.563628\pi\)
\(252\) 0 0
\(253\) 3041.72 0.755855
\(254\) −404.073 −0.0998180
\(255\) 0 0
\(256\) 3783.86 0.923794
\(257\) 2663.91 0.646577 0.323288 0.946300i \(-0.395212\pi\)
0.323288 + 0.946300i \(0.395212\pi\)
\(258\) 0 0
\(259\) 5943.06 1.42581
\(260\) 0 0
\(261\) 0 0
\(262\) −3995.12 −0.942059
\(263\) 2436.93 0.571360 0.285680 0.958325i \(-0.407781\pi\)
0.285680 + 0.958325i \(0.407781\pi\)
\(264\) 0 0
\(265\) 8368.80 1.93997
\(266\) −6751.21 −1.55618
\(267\) 0 0
\(268\) 1333.62 0.303970
\(269\) 2683.18 0.608165 0.304083 0.952646i \(-0.401650\pi\)
0.304083 + 0.952646i \(0.401650\pi\)
\(270\) 0 0
\(271\) 3857.14 0.864592 0.432296 0.901732i \(-0.357704\pi\)
0.432296 + 0.901732i \(0.357704\pi\)
\(272\) 6586.20 1.46819
\(273\) 0 0
\(274\) −8947.76 −1.97282
\(275\) −3910.27 −0.857448
\(276\) 0 0
\(277\) 1104.91 0.239666 0.119833 0.992794i \(-0.461764\pi\)
0.119833 + 0.992794i \(0.461764\pi\)
\(278\) −10206.7 −2.20201
\(279\) 0 0
\(280\) 8059.97 1.72027
\(281\) 4982.58 1.05778 0.528890 0.848691i \(-0.322609\pi\)
0.528890 + 0.848691i \(0.322609\pi\)
\(282\) 0 0
\(283\) 2584.86 0.542947 0.271473 0.962446i \(-0.412489\pi\)
0.271473 + 0.962446i \(0.412489\pi\)
\(284\) 1052.33 0.219874
\(285\) 0 0
\(286\) 0 0
\(287\) 5295.14 1.08907
\(288\) 0 0
\(289\) 2136.78 0.434924
\(290\) 7720.22 1.56326
\(291\) 0 0
\(292\) −532.784 −0.106777
\(293\) −92.5482 −0.0184530 −0.00922649 0.999957i \(-0.502937\pi\)
−0.00922649 + 0.999957i \(0.502937\pi\)
\(294\) 0 0
\(295\) −753.020 −0.148619
\(296\) 3830.54 0.752180
\(297\) 0 0
\(298\) −4494.07 −0.873605
\(299\) 0 0
\(300\) 0 0
\(301\) 4129.29 0.790726
\(302\) −9337.07 −1.77910
\(303\) 0 0
\(304\) −6048.96 −1.14122
\(305\) 8709.99 1.63519
\(306\) 0 0
\(307\) 3979.46 0.739803 0.369901 0.929071i \(-0.379391\pi\)
0.369901 + 0.929071i \(0.379391\pi\)
\(308\) −1572.43 −0.290901
\(309\) 0 0
\(310\) 7049.34 1.29153
\(311\) −3450.91 −0.629207 −0.314604 0.949223i \(-0.601872\pi\)
−0.314604 + 0.949223i \(0.601872\pi\)
\(312\) 0 0
\(313\) −6189.03 −1.11765 −0.558825 0.829285i \(-0.688748\pi\)
−0.558825 + 0.829285i \(0.688748\pi\)
\(314\) −10997.7 −1.97654
\(315\) 0 0
\(316\) 2888.15 0.514149
\(317\) 5437.78 0.963459 0.481729 0.876320i \(-0.340009\pi\)
0.481729 + 0.876320i \(0.340009\pi\)
\(318\) 0 0
\(319\) 2872.72 0.504205
\(320\) 4132.42 0.721904
\(321\) 0 0
\(322\) 12442.3 2.15336
\(323\) −6474.73 −1.11537
\(324\) 0 0
\(325\) 0 0
\(326\) −7201.23 −1.22343
\(327\) 0 0
\(328\) 3412.93 0.574535
\(329\) −2103.83 −0.352547
\(330\) 0 0
\(331\) −6626.64 −1.10040 −0.550201 0.835032i \(-0.685449\pi\)
−0.550201 + 0.835032i \(0.685449\pi\)
\(332\) 2371.66 0.392054
\(333\) 0 0
\(334\) −2992.18 −0.490194
\(335\) 8501.60 1.38654
\(336\) 0 0
\(337\) −5538.63 −0.895277 −0.447638 0.894215i \(-0.647735\pi\)
−0.447638 + 0.894215i \(0.647735\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −4052.74 −0.646443
\(341\) 2623.08 0.416563
\(342\) 0 0
\(343\) 717.813 0.112998
\(344\) 2661.49 0.417146
\(345\) 0 0
\(346\) 2949.88 0.458343
\(347\) −9398.94 −1.45407 −0.727034 0.686602i \(-0.759102\pi\)
−0.727034 + 0.686602i \(0.759102\pi\)
\(348\) 0 0
\(349\) −5757.33 −0.883045 −0.441522 0.897250i \(-0.645561\pi\)
−0.441522 + 0.897250i \(0.645561\pi\)
\(350\) −15995.2 −2.44279
\(351\) 0 0
\(352\) −2558.36 −0.387389
\(353\) 3457.82 0.521364 0.260682 0.965425i \(-0.416053\pi\)
0.260682 + 0.965425i \(0.416053\pi\)
\(354\) 0 0
\(355\) 6708.40 1.00294
\(356\) 2662.64 0.396403
\(357\) 0 0
\(358\) 1027.23 0.151650
\(359\) −7168.96 −1.05394 −0.526968 0.849885i \(-0.676671\pi\)
−0.526968 + 0.849885i \(0.676671\pi\)
\(360\) 0 0
\(361\) −912.407 −0.133023
\(362\) 9000.97 1.30685
\(363\) 0 0
\(364\) 0 0
\(365\) −3396.40 −0.487057
\(366\) 0 0
\(367\) 3910.11 0.556147 0.278073 0.960560i \(-0.410304\pi\)
0.278073 + 0.960560i \(0.410304\pi\)
\(368\) 11148.1 1.57917
\(369\) 0 0
\(370\) −12802.7 −1.79887
\(371\) 12738.2 1.78257
\(372\) 0 0
\(373\) 11377.6 1.57938 0.789691 0.613505i \(-0.210241\pi\)
0.789691 + 0.613505i \(0.210241\pi\)
\(374\) −5892.39 −0.814675
\(375\) 0 0
\(376\) −1356.00 −0.185986
\(377\) 0 0
\(378\) 0 0
\(379\) 4032.22 0.546494 0.273247 0.961944i \(-0.411902\pi\)
0.273247 + 0.961944i \(0.411902\pi\)
\(380\) 3722.16 0.502481
\(381\) 0 0
\(382\) 294.675 0.0394682
\(383\) −1990.96 −0.265622 −0.132811 0.991141i \(-0.542400\pi\)
−0.132811 + 0.991141i \(0.542400\pi\)
\(384\) 0 0
\(385\) −10024.0 −1.32693
\(386\) 2783.70 0.367063
\(387\) 0 0
\(388\) −1627.53 −0.212952
\(389\) 11122.4 1.44969 0.724846 0.688911i \(-0.241911\pi\)
0.724846 + 0.688911i \(0.241911\pi\)
\(390\) 0 0
\(391\) 11932.8 1.54339
\(392\) 6365.39 0.820155
\(393\) 0 0
\(394\) −14243.4 −1.82125
\(395\) 18411.4 2.34527
\(396\) 0 0
\(397\) −10778.1 −1.36256 −0.681282 0.732021i \(-0.738577\pi\)
−0.681282 + 0.732021i \(0.738577\pi\)
\(398\) 10913.8 1.37452
\(399\) 0 0
\(400\) −14331.4 −1.79142
\(401\) −4711.58 −0.586746 −0.293373 0.955998i \(-0.594778\pi\)
−0.293373 + 0.955998i \(0.594778\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −701.978 −0.0864473
\(405\) 0 0
\(406\) 11751.0 1.43643
\(407\) −4763.94 −0.580196
\(408\) 0 0
\(409\) −1184.78 −0.143236 −0.0716179 0.997432i \(-0.522816\pi\)
−0.0716179 + 0.997432i \(0.522816\pi\)
\(410\) −11407.0 −1.37403
\(411\) 0 0
\(412\) −680.638 −0.0813899
\(413\) −1146.18 −0.136561
\(414\) 0 0
\(415\) 15118.9 1.78834
\(416\) 0 0
\(417\) 0 0
\(418\) 5411.75 0.633248
\(419\) 6168.30 0.719191 0.359596 0.933108i \(-0.382915\pi\)
0.359596 + 0.933108i \(0.382915\pi\)
\(420\) 0 0
\(421\) 10328.8 1.19571 0.597857 0.801603i \(-0.296019\pi\)
0.597857 + 0.801603i \(0.296019\pi\)
\(422\) 15082.9 1.73987
\(423\) 0 0
\(424\) 8210.27 0.940392
\(425\) −15340.1 −1.75084
\(426\) 0 0
\(427\) 13257.5 1.50252
\(428\) −1880.72 −0.212402
\(429\) 0 0
\(430\) −8895.46 −0.997622
\(431\) −11312.5 −1.26427 −0.632137 0.774857i \(-0.717822\pi\)
−0.632137 + 0.774857i \(0.717822\pi\)
\(432\) 0 0
\(433\) −10475.7 −1.16266 −0.581331 0.813667i \(-0.697468\pi\)
−0.581331 + 0.813667i \(0.697468\pi\)
\(434\) 10729.8 1.18675
\(435\) 0 0
\(436\) −4671.68 −0.513149
\(437\) −10959.4 −1.19968
\(438\) 0 0
\(439\) −2040.62 −0.221853 −0.110926 0.993829i \(-0.535382\pi\)
−0.110926 + 0.993829i \(0.535382\pi\)
\(440\) −6460.85 −0.700020
\(441\) 0 0
\(442\) 0 0
\(443\) 4089.28 0.438572 0.219286 0.975661i \(-0.429627\pi\)
0.219286 + 0.975661i \(0.429627\pi\)
\(444\) 0 0
\(445\) 16973.9 1.80818
\(446\) −8294.43 −0.880611
\(447\) 0 0
\(448\) 6289.99 0.663335
\(449\) −15217.9 −1.59951 −0.799753 0.600330i \(-0.795036\pi\)
−0.799753 + 0.600330i \(0.795036\pi\)
\(450\) 0 0
\(451\) −4244.58 −0.443169
\(452\) 1047.58 0.109014
\(453\) 0 0
\(454\) −122.197 −0.0126322
\(455\) 0 0
\(456\) 0 0
\(457\) −876.316 −0.0896988 −0.0448494 0.998994i \(-0.514281\pi\)
−0.0448494 + 0.998994i \(0.514281\pi\)
\(458\) −13425.6 −1.36973
\(459\) 0 0
\(460\) −6859.84 −0.695308
\(461\) −16293.4 −1.64611 −0.823057 0.567959i \(-0.807733\pi\)
−0.823057 + 0.567959i \(0.807733\pi\)
\(462\) 0 0
\(463\) −11704.8 −1.17488 −0.587438 0.809269i \(-0.699863\pi\)
−0.587438 + 0.809269i \(0.699863\pi\)
\(464\) 10528.7 1.05341
\(465\) 0 0
\(466\) 4807.12 0.477866
\(467\) 15616.1 1.54738 0.773688 0.633567i \(-0.218410\pi\)
0.773688 + 0.633567i \(0.218410\pi\)
\(468\) 0 0
\(469\) 12940.3 1.27405
\(470\) 4532.15 0.444792
\(471\) 0 0
\(472\) −738.757 −0.0720424
\(473\) −3310.03 −0.321766
\(474\) 0 0
\(475\) 14088.8 1.36093
\(476\) −6168.70 −0.593996
\(477\) 0 0
\(478\) 18101.7 1.73212
\(479\) 10376.0 0.989755 0.494877 0.868963i \(-0.335213\pi\)
0.494877 + 0.868963i \(0.335213\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 8740.53 0.825976
\(483\) 0 0
\(484\) −2402.01 −0.225584
\(485\) −10375.2 −0.971369
\(486\) 0 0
\(487\) 3994.27 0.371658 0.185829 0.982582i \(-0.440503\pi\)
0.185829 + 0.982582i \(0.440503\pi\)
\(488\) 8545.01 0.792652
\(489\) 0 0
\(490\) −21275.0 −1.96144
\(491\) −12267.3 −1.12753 −0.563763 0.825936i \(-0.690647\pi\)
−0.563763 + 0.825936i \(0.690647\pi\)
\(492\) 0 0
\(493\) 11269.8 1.02954
\(494\) 0 0
\(495\) 0 0
\(496\) 9613.75 0.870303
\(497\) 10210.9 0.921573
\(498\) 0 0
\(499\) 3578.76 0.321056 0.160528 0.987031i \(-0.448680\pi\)
0.160528 + 0.987031i \(0.448680\pi\)
\(500\) 2785.12 0.249109
\(501\) 0 0
\(502\) 5178.19 0.460386
\(503\) 1296.82 0.114955 0.0574774 0.998347i \(-0.481694\pi\)
0.0574774 + 0.998347i \(0.481694\pi\)
\(504\) 0 0
\(505\) −4474.99 −0.394325
\(506\) −9973.72 −0.876257
\(507\) 0 0
\(508\) 339.092 0.0296157
\(509\) 4728.60 0.411771 0.205886 0.978576i \(-0.433993\pi\)
0.205886 + 0.978576i \(0.433993\pi\)
\(510\) 0 0
\(511\) −5169.69 −0.447541
\(512\) 1422.78 0.122810
\(513\) 0 0
\(514\) −8734.90 −0.749571
\(515\) −4338.95 −0.371256
\(516\) 0 0
\(517\) 1686.43 0.143460
\(518\) −19487.1 −1.65293
\(519\) 0 0
\(520\) 0 0
\(521\) 9220.74 0.775370 0.387685 0.921792i \(-0.373275\pi\)
0.387685 + 0.921792i \(0.373275\pi\)
\(522\) 0 0
\(523\) −12102.7 −1.01188 −0.505939 0.862569i \(-0.668854\pi\)
−0.505939 + 0.862569i \(0.668854\pi\)
\(524\) 3352.65 0.279506
\(525\) 0 0
\(526\) −7990.64 −0.662373
\(527\) 10290.4 0.850585
\(528\) 0 0
\(529\) 8030.91 0.660057
\(530\) −27441.1 −2.24899
\(531\) 0 0
\(532\) 5665.52 0.461713
\(533\) 0 0
\(534\) 0 0
\(535\) −11989.3 −0.968862
\(536\) 8340.56 0.672122
\(537\) 0 0
\(538\) −8798.08 −0.705041
\(539\) −7916.48 −0.632629
\(540\) 0 0
\(541\) 12801.3 1.01732 0.508659 0.860968i \(-0.330141\pi\)
0.508659 + 0.860968i \(0.330141\pi\)
\(542\) −12647.5 −1.00231
\(543\) 0 0
\(544\) −10036.5 −0.791015
\(545\) −29781.2 −2.34071
\(546\) 0 0
\(547\) 400.693 0.0313207 0.0156603 0.999877i \(-0.495015\pi\)
0.0156603 + 0.999877i \(0.495015\pi\)
\(548\) 7508.84 0.585331
\(549\) 0 0
\(550\) 12821.7 0.994033
\(551\) −10350.5 −0.800265
\(552\) 0 0
\(553\) 28024.2 2.15499
\(554\) −3622.96 −0.277843
\(555\) 0 0
\(556\) 8565.35 0.653330
\(557\) −14475.5 −1.10116 −0.550582 0.834781i \(-0.685594\pi\)
−0.550582 + 0.834781i \(0.685594\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −36738.5 −2.77229
\(561\) 0 0
\(562\) −16337.8 −1.22628
\(563\) 14776.4 1.10613 0.553063 0.833139i \(-0.313459\pi\)
0.553063 + 0.833139i \(0.313459\pi\)
\(564\) 0 0
\(565\) 6678.17 0.497261
\(566\) −8475.68 −0.629434
\(567\) 0 0
\(568\) 6581.33 0.486173
\(569\) 6868.88 0.506078 0.253039 0.967456i \(-0.418570\pi\)
0.253039 + 0.967456i \(0.418570\pi\)
\(570\) 0 0
\(571\) 3011.00 0.220677 0.110338 0.993894i \(-0.464807\pi\)
0.110338 + 0.993894i \(0.464807\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −17362.6 −1.26255
\(575\) −25965.4 −1.88318
\(576\) 0 0
\(577\) −23106.6 −1.66714 −0.833572 0.552411i \(-0.813708\pi\)
−0.833572 + 0.552411i \(0.813708\pi\)
\(578\) −7006.45 −0.504204
\(579\) 0 0
\(580\) −6478.70 −0.463816
\(581\) 23012.6 1.64324
\(582\) 0 0
\(583\) −10210.9 −0.725374
\(584\) −3332.07 −0.236099
\(585\) 0 0
\(586\) 303.463 0.0213924
\(587\) −3024.81 −0.212687 −0.106343 0.994329i \(-0.533914\pi\)
−0.106343 + 0.994329i \(0.533914\pi\)
\(588\) 0 0
\(589\) −9451.05 −0.661161
\(590\) 2469.13 0.172293
\(591\) 0 0
\(592\) −17460.1 −1.21217
\(593\) 6396.07 0.442926 0.221463 0.975169i \(-0.428917\pi\)
0.221463 + 0.975169i \(0.428917\pi\)
\(594\) 0 0
\(595\) −39324.4 −2.70948
\(596\) 3771.36 0.259196
\(597\) 0 0
\(598\) 0 0
\(599\) −12095.9 −0.825084 −0.412542 0.910939i \(-0.635359\pi\)
−0.412542 + 0.910939i \(0.635359\pi\)
\(600\) 0 0
\(601\) −11816.5 −0.802005 −0.401003 0.916077i \(-0.631338\pi\)
−0.401003 + 0.916077i \(0.631338\pi\)
\(602\) −13539.8 −0.916682
\(603\) 0 0
\(604\) 7835.54 0.527853
\(605\) −15312.4 −1.02899
\(606\) 0 0
\(607\) −25164.0 −1.68266 −0.841329 0.540523i \(-0.818226\pi\)
−0.841329 + 0.540523i \(0.818226\pi\)
\(608\) 9217.85 0.614857
\(609\) 0 0
\(610\) −28559.8 −1.89566
\(611\) 0 0
\(612\) 0 0
\(613\) 19583.0 1.29030 0.645148 0.764058i \(-0.276796\pi\)
0.645148 + 0.764058i \(0.276796\pi\)
\(614\) −13048.5 −0.857648
\(615\) 0 0
\(616\) −9834.10 −0.643226
\(617\) −19677.1 −1.28390 −0.641952 0.766745i \(-0.721875\pi\)
−0.641952 + 0.766745i \(0.721875\pi\)
\(618\) 0 0
\(619\) −4394.05 −0.285318 −0.142659 0.989772i \(-0.545565\pi\)
−0.142659 + 0.989772i \(0.545565\pi\)
\(620\) −5915.71 −0.383194
\(621\) 0 0
\(622\) 11315.5 0.729435
\(623\) 25836.0 1.66147
\(624\) 0 0
\(625\) −5082.96 −0.325310
\(626\) 20293.7 1.29568
\(627\) 0 0
\(628\) 9229.10 0.586435
\(629\) −18689.1 −1.18471
\(630\) 0 0
\(631\) −23887.7 −1.50706 −0.753529 0.657415i \(-0.771650\pi\)
−0.753529 + 0.657415i \(0.771650\pi\)
\(632\) 18062.7 1.13686
\(633\) 0 0
\(634\) −17830.4 −1.11693
\(635\) 2161.65 0.135091
\(636\) 0 0
\(637\) 0 0
\(638\) −9419.57 −0.584521
\(639\) 0 0
\(640\) −30324.6 −1.87295
\(641\) 5443.62 0.335429 0.167714 0.985836i \(-0.446361\pi\)
0.167714 + 0.985836i \(0.446361\pi\)
\(642\) 0 0
\(643\) 5839.00 0.358115 0.179057 0.983839i \(-0.442695\pi\)
0.179057 + 0.983839i \(0.442695\pi\)
\(644\) −10441.4 −0.638896
\(645\) 0 0
\(646\) 21230.5 1.29304
\(647\) −8708.86 −0.529182 −0.264591 0.964361i \(-0.585237\pi\)
−0.264591 + 0.964361i \(0.585237\pi\)
\(648\) 0 0
\(649\) 918.773 0.0555701
\(650\) 0 0
\(651\) 0 0
\(652\) 6043.17 0.362989
\(653\) −2794.93 −0.167495 −0.0837475 0.996487i \(-0.526689\pi\)
−0.0837475 + 0.996487i \(0.526689\pi\)
\(654\) 0 0
\(655\) 21372.6 1.27495
\(656\) −15556.6 −0.925890
\(657\) 0 0
\(658\) 6898.41 0.408705
\(659\) −31389.9 −1.85550 −0.927752 0.373197i \(-0.878262\pi\)
−0.927752 + 0.373197i \(0.878262\pi\)
\(660\) 0 0
\(661\) −20597.2 −1.21201 −0.606005 0.795461i \(-0.707229\pi\)
−0.606005 + 0.795461i \(0.707229\pi\)
\(662\) 21728.6 1.27569
\(663\) 0 0
\(664\) 14832.5 0.866889
\(665\) 36116.7 2.10608
\(666\) 0 0
\(667\) 19075.7 1.10737
\(668\) 2511.00 0.145439
\(669\) 0 0
\(670\) −27876.5 −1.60741
\(671\) −10627.2 −0.611414
\(672\) 0 0
\(673\) −17935.8 −1.02730 −0.513651 0.857999i \(-0.671707\pi\)
−0.513651 + 0.857999i \(0.671707\pi\)
\(674\) 18161.0 1.03789
\(675\) 0 0
\(676\) 0 0
\(677\) 24104.0 1.36838 0.684188 0.729305i \(-0.260157\pi\)
0.684188 + 0.729305i \(0.260157\pi\)
\(678\) 0 0
\(679\) −15792.2 −0.892560
\(680\) −25346.1 −1.42938
\(681\) 0 0
\(682\) −8601.02 −0.482918
\(683\) −6744.25 −0.377835 −0.188918 0.981993i \(-0.560498\pi\)
−0.188918 + 0.981993i \(0.560498\pi\)
\(684\) 0 0
\(685\) 47867.5 2.66996
\(686\) −2353.69 −0.130998
\(687\) 0 0
\(688\) −12131.5 −0.672249
\(689\) 0 0
\(690\) 0 0
\(691\) 30844.4 1.69809 0.849043 0.528323i \(-0.177179\pi\)
0.849043 + 0.528323i \(0.177179\pi\)
\(692\) −2475.50 −0.135989
\(693\) 0 0
\(694\) 30818.9 1.68569
\(695\) 54602.6 2.98014
\(696\) 0 0
\(697\) −16651.6 −0.904913
\(698\) 18878.1 1.02371
\(699\) 0 0
\(700\) 13422.9 0.724769
\(701\) 21007.6 1.13188 0.565940 0.824447i \(-0.308514\pi\)
0.565940 + 0.824447i \(0.308514\pi\)
\(702\) 0 0
\(703\) 17164.6 0.920877
\(704\) −5042.04 −0.269928
\(705\) 0 0
\(706\) −11338.1 −0.604413
\(707\) −6811.41 −0.362333
\(708\) 0 0
\(709\) 14763.4 0.782017 0.391008 0.920387i \(-0.372126\pi\)
0.391008 + 0.920387i \(0.372126\pi\)
\(710\) −21996.7 −1.16271
\(711\) 0 0
\(712\) 16652.3 0.876507
\(713\) 17418.0 0.914882
\(714\) 0 0
\(715\) 0 0
\(716\) −862.037 −0.0449942
\(717\) 0 0
\(718\) 23506.8 1.22182
\(719\) 26186.3 1.35825 0.679126 0.734022i \(-0.262359\pi\)
0.679126 + 0.734022i \(0.262359\pi\)
\(720\) 0 0
\(721\) −6604.34 −0.341135
\(722\) 2991.76 0.154213
\(723\) 0 0
\(724\) −7553.49 −0.387739
\(725\) −24522.7 −1.25621
\(726\) 0 0
\(727\) 20044.0 1.02254 0.511272 0.859419i \(-0.329175\pi\)
0.511272 + 0.859419i \(0.329175\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 11136.7 0.564641
\(731\) −12985.4 −0.657019
\(732\) 0 0
\(733\) −25555.5 −1.28774 −0.643871 0.765134i \(-0.722673\pi\)
−0.643871 + 0.765134i \(0.722673\pi\)
\(734\) −12821.1 −0.644737
\(735\) 0 0
\(736\) −16988.2 −0.850809
\(737\) −10372.9 −0.518443
\(738\) 0 0
\(739\) −11108.5 −0.552953 −0.276477 0.961021i \(-0.589167\pi\)
−0.276477 + 0.961021i \(0.589167\pi\)
\(740\) 10743.9 0.533720
\(741\) 0 0
\(742\) −41768.2 −2.06652
\(743\) 28188.2 1.39182 0.695911 0.718128i \(-0.255001\pi\)
0.695911 + 0.718128i \(0.255001\pi\)
\(744\) 0 0
\(745\) 24041.7 1.18231
\(746\) −37306.8 −1.83096
\(747\) 0 0
\(748\) 4944.82 0.241712
\(749\) −18248.9 −0.890256
\(750\) 0 0
\(751\) 17176.6 0.834596 0.417298 0.908770i \(-0.362977\pi\)
0.417298 + 0.908770i \(0.362977\pi\)
\(752\) 6180.86 0.299724
\(753\) 0 0
\(754\) 0 0
\(755\) 49950.2 2.40778
\(756\) 0 0
\(757\) −4409.96 −0.211734 −0.105867 0.994380i \(-0.533762\pi\)
−0.105867 + 0.994380i \(0.533762\pi\)
\(758\) −13221.5 −0.633546
\(759\) 0 0
\(760\) 23278.6 1.11106
\(761\) −32443.6 −1.54544 −0.772721 0.634746i \(-0.781105\pi\)
−0.772721 + 0.634746i \(0.781105\pi\)
\(762\) 0 0
\(763\) −45330.1 −2.15080
\(764\) −247.287 −0.0117101
\(765\) 0 0
\(766\) 6528.30 0.307934
\(767\) 0 0
\(768\) 0 0
\(769\) −31994.1 −1.50031 −0.750155 0.661263i \(-0.770021\pi\)
−0.750155 + 0.661263i \(0.770021\pi\)
\(770\) 32868.4 1.53830
\(771\) 0 0
\(772\) −2336.04 −0.108907
\(773\) 8979.39 0.417809 0.208904 0.977936i \(-0.433010\pi\)
0.208904 + 0.977936i \(0.433010\pi\)
\(774\) 0 0
\(775\) −22391.7 −1.03785
\(776\) −10178.7 −0.470868
\(777\) 0 0
\(778\) −36470.2 −1.68062
\(779\) 15293.3 0.703390
\(780\) 0 0
\(781\) −8185.04 −0.375011
\(782\) −39127.2 −1.78924
\(783\) 0 0
\(784\) −29014.4 −1.32172
\(785\) 58833.9 2.67500
\(786\) 0 0
\(787\) −12570.5 −0.569366 −0.284683 0.958622i \(-0.591888\pi\)
−0.284683 + 0.958622i \(0.591888\pi\)
\(788\) 11952.9 0.540360
\(789\) 0 0
\(790\) −60370.6 −2.71885
\(791\) 10164.9 0.456917
\(792\) 0 0
\(793\) 0 0
\(794\) 35341.2 1.57961
\(795\) 0 0
\(796\) −9158.67 −0.407815
\(797\) −37863.3 −1.68280 −0.841398 0.540416i \(-0.818267\pi\)
−0.841398 + 0.540416i \(0.818267\pi\)
\(798\) 0 0
\(799\) 6615.91 0.292934
\(800\) 21839.2 0.965165
\(801\) 0 0
\(802\) 15449.2 0.680210
\(803\) 4144.01 0.182116
\(804\) 0 0
\(805\) −66562.1 −2.91429
\(806\) 0 0
\(807\) 0 0
\(808\) −4390.22 −0.191148
\(809\) −2503.79 −0.108812 −0.0544058 0.998519i \(-0.517326\pi\)
−0.0544058 + 0.998519i \(0.517326\pi\)
\(810\) 0 0
\(811\) −5409.55 −0.234223 −0.117112 0.993119i \(-0.537364\pi\)
−0.117112 + 0.993119i \(0.537364\pi\)
\(812\) −9861.27 −0.426186
\(813\) 0 0
\(814\) 15620.8 0.672617
\(815\) 38524.2 1.65576
\(816\) 0 0
\(817\) 11926.1 0.510702
\(818\) 3884.85 0.166052
\(819\) 0 0
\(820\) 9572.58 0.407669
\(821\) 31381.6 1.33401 0.667007 0.745051i \(-0.267575\pi\)
0.667007 + 0.745051i \(0.267575\pi\)
\(822\) 0 0
\(823\) −33046.1 −1.39965 −0.699827 0.714312i \(-0.746740\pi\)
−0.699827 + 0.714312i \(0.746740\pi\)
\(824\) −4256.76 −0.179965
\(825\) 0 0
\(826\) 3758.29 0.158314
\(827\) 33653.7 1.41506 0.707529 0.706684i \(-0.249810\pi\)
0.707529 + 0.706684i \(0.249810\pi\)
\(828\) 0 0
\(829\) 12898.5 0.540390 0.270195 0.962806i \(-0.412912\pi\)
0.270195 + 0.962806i \(0.412912\pi\)
\(830\) −49574.6 −2.07320
\(831\) 0 0
\(832\) 0 0
\(833\) −31056.6 −1.29177
\(834\) 0 0
\(835\) 16007.2 0.663414
\(836\) −4541.47 −0.187883
\(837\) 0 0
\(838\) −20225.7 −0.833753
\(839\) 395.829 0.0162879 0.00814395 0.999967i \(-0.497408\pi\)
0.00814395 + 0.999967i \(0.497408\pi\)
\(840\) 0 0
\(841\) −6373.17 −0.261313
\(842\) −33867.9 −1.38618
\(843\) 0 0
\(844\) −12657.4 −0.516214
\(845\) 0 0
\(846\) 0 0
\(847\) −23307.1 −0.945505
\(848\) −37423.6 −1.51548
\(849\) 0 0
\(850\) 50299.9 2.02973
\(851\) −31634.0 −1.27426
\(852\) 0 0
\(853\) −21248.9 −0.852930 −0.426465 0.904504i \(-0.640241\pi\)
−0.426465 + 0.904504i \(0.640241\pi\)
\(854\) −43471.1 −1.74186
\(855\) 0 0
\(856\) −11762.2 −0.469653
\(857\) 9920.37 0.395418 0.197709 0.980261i \(-0.436650\pi\)
0.197709 + 0.980261i \(0.436650\pi\)
\(858\) 0 0
\(859\) 20946.3 0.831990 0.415995 0.909367i \(-0.363433\pi\)
0.415995 + 0.909367i \(0.363433\pi\)
\(860\) 7464.95 0.295991
\(861\) 0 0
\(862\) 37093.2 1.46566
\(863\) 11271.4 0.444594 0.222297 0.974979i \(-0.428645\pi\)
0.222297 + 0.974979i \(0.428645\pi\)
\(864\) 0 0
\(865\) −15780.9 −0.620308
\(866\) 34349.7 1.34786
\(867\) 0 0
\(868\) −9004.34 −0.352105
\(869\) −22464.1 −0.876920
\(870\) 0 0
\(871\) 0 0
\(872\) −29217.0 −1.13465
\(873\) 0 0
\(874\) 35935.6 1.39078
\(875\) 27024.5 1.04411
\(876\) 0 0
\(877\) −9436.36 −0.363333 −0.181667 0.983360i \(-0.558149\pi\)
−0.181667 + 0.983360i \(0.558149\pi\)
\(878\) 6691.13 0.257192
\(879\) 0 0
\(880\) 29449.5 1.12811
\(881\) 20343.0 0.777949 0.388974 0.921249i \(-0.372829\pi\)
0.388974 + 0.921249i \(0.372829\pi\)
\(882\) 0 0
\(883\) 46521.9 1.77303 0.886515 0.462699i \(-0.153119\pi\)
0.886515 + 0.462699i \(0.153119\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −13408.6 −0.508433
\(887\) 19955.1 0.755384 0.377692 0.925931i \(-0.376718\pi\)
0.377692 + 0.925931i \(0.376718\pi\)
\(888\) 0 0
\(889\) 3290.27 0.124130
\(890\) −55656.8 −2.09620
\(891\) 0 0
\(892\) 6960.57 0.261275
\(893\) −6076.25 −0.227698
\(894\) 0 0
\(895\) −5495.33 −0.205239
\(896\) −46157.3 −1.72099
\(897\) 0 0
\(898\) 49899.1 1.85429
\(899\) 16450.3 0.610286
\(900\) 0 0
\(901\) −40057.7 −1.48115
\(902\) 13917.9 0.513762
\(903\) 0 0
\(904\) 6551.67 0.241046
\(905\) −48152.2 −1.76866
\(906\) 0 0
\(907\) −2653.95 −0.0971587 −0.0485793 0.998819i \(-0.515469\pi\)
−0.0485793 + 0.998819i \(0.515469\pi\)
\(908\) 102.546 0.00374793
\(909\) 0 0
\(910\) 0 0
\(911\) −1797.50 −0.0653720 −0.0326860 0.999466i \(-0.510406\pi\)
−0.0326860 + 0.999466i \(0.510406\pi\)
\(912\) 0 0
\(913\) −18446.9 −0.668677
\(914\) 2873.42 0.103987
\(915\) 0 0
\(916\) 11266.6 0.406395
\(917\) 32531.3 1.17151
\(918\) 0 0
\(919\) −48642.0 −1.74597 −0.872987 0.487743i \(-0.837820\pi\)
−0.872987 + 0.487743i \(0.837820\pi\)
\(920\) −42901.9 −1.53743
\(921\) 0 0
\(922\) 53425.6 1.90833
\(923\) 0 0
\(924\) 0 0
\(925\) 40666.9 1.44554
\(926\) 38379.7 1.36202
\(927\) 0 0
\(928\) −16044.4 −0.567545
\(929\) 37745.3 1.33303 0.666513 0.745493i \(-0.267786\pi\)
0.666513 + 0.745493i \(0.267786\pi\)
\(930\) 0 0
\(931\) 28523.3 1.00410
\(932\) −4034.07 −0.141782
\(933\) 0 0
\(934\) −51204.6 −1.79386
\(935\) 31522.3 1.10256
\(936\) 0 0
\(937\) 2705.50 0.0943273 0.0471637 0.998887i \(-0.484982\pi\)
0.0471637 + 0.998887i \(0.484982\pi\)
\(938\) −42431.0 −1.47700
\(939\) 0 0
\(940\) −3803.32 −0.131969
\(941\) 5189.27 0.179772 0.0898860 0.995952i \(-0.471350\pi\)
0.0898860 + 0.995952i \(0.471350\pi\)
\(942\) 0 0
\(943\) −28185.2 −0.973316
\(944\) 3367.36 0.116100
\(945\) 0 0
\(946\) 10853.5 0.373021
\(947\) −72.2711 −0.00247993 −0.00123997 0.999999i \(-0.500395\pi\)
−0.00123997 + 0.999999i \(0.500395\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −46196.9 −1.57771
\(951\) 0 0
\(952\) −38579.5 −1.31341
\(953\) −44695.1 −1.51922 −0.759609 0.650380i \(-0.774610\pi\)
−0.759609 + 0.650380i \(0.774610\pi\)
\(954\) 0 0
\(955\) −1576.41 −0.0534151
\(956\) −15190.7 −0.513914
\(957\) 0 0
\(958\) −34022.7 −1.14742
\(959\) 72859.5 2.45334
\(960\) 0 0
\(961\) −14770.2 −0.495795
\(962\) 0 0
\(963\) 0 0
\(964\) −7334.93 −0.245065
\(965\) −14891.8 −0.496772
\(966\) 0 0
\(967\) 17936.9 0.596496 0.298248 0.954488i \(-0.403598\pi\)
0.298248 + 0.954488i \(0.403598\pi\)
\(968\) −15022.4 −0.498799
\(969\) 0 0
\(970\) 34020.1 1.12610
\(971\) 40914.6 1.35223 0.676113 0.736798i \(-0.263663\pi\)
0.676113 + 0.736798i \(0.263663\pi\)
\(972\) 0 0
\(973\) 83111.0 2.73835
\(974\) −13097.1 −0.430860
\(975\) 0 0
\(976\) −38949.3 −1.27740
\(977\) 24118.1 0.789770 0.394885 0.918731i \(-0.370784\pi\)
0.394885 + 0.918731i \(0.370784\pi\)
\(978\) 0 0
\(979\) −20710.1 −0.676096
\(980\) 17853.6 0.581953
\(981\) 0 0
\(982\) 40224.2 1.30713
\(983\) −2928.61 −0.0950235 −0.0475118 0.998871i \(-0.515129\pi\)
−0.0475118 + 0.998871i \(0.515129\pi\)
\(984\) 0 0
\(985\) 76197.5 2.46483
\(986\) −36953.3 −1.19354
\(987\) 0 0
\(988\) 0 0
\(989\) −21979.6 −0.706683
\(990\) 0 0
\(991\) 49809.9 1.59663 0.798317 0.602238i \(-0.205724\pi\)
0.798317 + 0.602238i \(0.205724\pi\)
\(992\) −14650.1 −0.468893
\(993\) 0 0
\(994\) −33481.3 −1.06837
\(995\) −58384.9 −1.86023
\(996\) 0 0
\(997\) −41151.7 −1.30721 −0.653604 0.756837i \(-0.726744\pi\)
−0.653604 + 0.756837i \(0.726744\pi\)
\(998\) −11734.7 −0.372198
\(999\) 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.bk.1.3 10
3.2 odd 2 507.4.a.r.1.8 10
13.2 odd 12 117.4.q.e.82.2 10
13.7 odd 12 117.4.q.e.10.2 10
13.12 even 2 inner 1521.4.a.bk.1.8 10
39.2 even 12 39.4.j.c.4.4 10
39.5 even 4 507.4.b.i.337.3 10
39.8 even 4 507.4.b.i.337.8 10
39.20 even 12 39.4.j.c.10.4 yes 10
39.38 odd 2 507.4.a.r.1.3 10
156.59 odd 12 624.4.bv.h.49.1 10
156.119 odd 12 624.4.bv.h.433.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.4 10 39.2 even 12
39.4.j.c.10.4 yes 10 39.20 even 12
117.4.q.e.10.2 10 13.7 odd 12
117.4.q.e.82.2 10 13.2 odd 12
507.4.a.r.1.3 10 39.38 odd 2
507.4.a.r.1.8 10 3.2 odd 2
507.4.b.i.337.3 10 39.5 even 4
507.4.b.i.337.8 10 39.8 even 4
624.4.bv.h.49.1 10 156.59 odd 12
624.4.bv.h.433.5 10 156.119 odd 12
1521.4.a.bk.1.3 10 1.1 even 1 trivial
1521.4.a.bk.1.8 10 13.12 even 2 inner