Properties

Label 1521.4.a.bm.1.1
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $18$
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: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \( x^{18} - 100 x^{16} + 4066 x^{14} - 86197 x^{12} + 1016064 x^{10} - 6594119 x^{8} + 22251817 x^{6} - 37096332 x^{4} + 27824160 x^{2} - 6889792 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 7\cdot 13^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.85886\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.85886 q^{2} +15.6085 q^{4} -14.8714 q^{5} -29.6598 q^{7} -36.9687 q^{8} +O(q^{10})\) \(q-4.85886 q^{2} +15.6085 q^{4} -14.8714 q^{5} -29.6598 q^{7} -36.9687 q^{8} +72.2582 q^{10} +60.0283 q^{11} +144.113 q^{14} +54.7574 q^{16} +49.0713 q^{17} -96.7335 q^{19} -232.121 q^{20} -291.669 q^{22} -42.4461 q^{23} +96.1597 q^{25} -462.946 q^{28} -176.826 q^{29} -269.559 q^{31} +29.6906 q^{32} -238.431 q^{34} +441.085 q^{35} -186.663 q^{37} +470.014 q^{38} +549.777 q^{40} +52.7630 q^{41} +527.456 q^{43} +936.952 q^{44} +206.240 q^{46} +61.3270 q^{47} +536.706 q^{49} -467.226 q^{50} +340.468 q^{53} -892.707 q^{55} +1096.48 q^{56} +859.172 q^{58} +595.111 q^{59} -415.671 q^{61} +1309.75 q^{62} -582.322 q^{64} +640.901 q^{67} +765.930 q^{68} -2143.17 q^{70} +551.454 q^{71} +431.954 q^{73} +906.968 q^{74} -1509.86 q^{76} -1780.43 q^{77} +998.743 q^{79} -814.322 q^{80} -256.368 q^{82} +461.462 q^{83} -729.762 q^{85} -2562.84 q^{86} -2219.17 q^{88} -1025.25 q^{89} -662.520 q^{92} -297.979 q^{94} +1438.57 q^{95} +261.585 q^{97} -2607.78 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 56 q^{4} - 94 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 56 q^{4} - 94 q^{7} + 156 q^{10} + 88 q^{16} - 448 q^{19} - 256 q^{22} - 16 q^{25} - 1300 q^{28} - 818 q^{31} - 900 q^{34} - 524 q^{37} + 800 q^{40} + 752 q^{43} - 1650 q^{46} + 2948 q^{49} - 464 q^{55} - 1626 q^{58} - 1784 q^{61} - 4274 q^{64} - 2872 q^{67} - 5772 q^{70} - 3078 q^{73} - 5726 q^{76} + 3326 q^{79} - 1038 q^{82} - 2340 q^{85} + 780 q^{88} + 1220 q^{94} - 4630 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.85886 −1.71787 −0.858933 0.512088i \(-0.828872\pi\)
−0.858933 + 0.512088i \(0.828872\pi\)
\(3\) 0 0
\(4\) 15.6085 1.95106
\(5\) −14.8714 −1.33014 −0.665071 0.746780i \(-0.731599\pi\)
−0.665071 + 0.746780i \(0.731599\pi\)
\(6\) 0 0
\(7\) −29.6598 −1.60148 −0.800741 0.599011i \(-0.795560\pi\)
−0.800741 + 0.599011i \(0.795560\pi\)
\(8\) −36.9687 −1.63380
\(9\) 0 0
\(10\) 72.2582 2.28501
\(11\) 60.0283 1.64538 0.822691 0.568488i \(-0.192471\pi\)
0.822691 + 0.568488i \(0.192471\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 144.113 2.75113
\(15\) 0 0
\(16\) 54.7574 0.855585
\(17\) 49.0713 0.700091 0.350046 0.936733i \(-0.386166\pi\)
0.350046 + 0.936733i \(0.386166\pi\)
\(18\) 0 0
\(19\) −96.7335 −1.16801 −0.584005 0.811750i \(-0.698515\pi\)
−0.584005 + 0.811750i \(0.698515\pi\)
\(20\) −232.121 −2.59519
\(21\) 0 0
\(22\) −291.669 −2.82655
\(23\) −42.4461 −0.384810 −0.192405 0.981316i \(-0.561629\pi\)
−0.192405 + 0.981316i \(0.561629\pi\)
\(24\) 0 0
\(25\) 96.1597 0.769278
\(26\) 0 0
\(27\) 0 0
\(28\) −462.946 −3.12459
\(29\) −176.826 −1.13227 −0.566133 0.824314i \(-0.691561\pi\)
−0.566133 + 0.824314i \(0.691561\pi\)
\(30\) 0 0
\(31\) −269.559 −1.56175 −0.780875 0.624688i \(-0.785226\pi\)
−0.780875 + 0.624688i \(0.785226\pi\)
\(32\) 29.6906 0.164019
\(33\) 0 0
\(34\) −238.431 −1.20266
\(35\) 441.085 2.13020
\(36\) 0 0
\(37\) −186.663 −0.829383 −0.414691 0.909962i \(-0.636110\pi\)
−0.414691 + 0.909962i \(0.636110\pi\)
\(38\) 470.014 2.00648
\(39\) 0 0
\(40\) 549.777 2.17319
\(41\) 52.7630 0.200981 0.100490 0.994938i \(-0.467959\pi\)
0.100490 + 0.994938i \(0.467959\pi\)
\(42\) 0 0
\(43\) 527.456 1.87061 0.935306 0.353839i \(-0.115124\pi\)
0.935306 + 0.353839i \(0.115124\pi\)
\(44\) 936.952 3.21025
\(45\) 0 0
\(46\) 206.240 0.661052
\(47\) 61.3270 0.190329 0.0951644 0.995462i \(-0.469662\pi\)
0.0951644 + 0.995462i \(0.469662\pi\)
\(48\) 0 0
\(49\) 536.706 1.56474
\(50\) −467.226 −1.32152
\(51\) 0 0
\(52\) 0 0
\(53\) 340.468 0.882394 0.441197 0.897410i \(-0.354554\pi\)
0.441197 + 0.897410i \(0.354554\pi\)
\(54\) 0 0
\(55\) −892.707 −2.18859
\(56\) 1096.48 2.61650
\(57\) 0 0
\(58\) 859.172 1.94508
\(59\) 595.111 1.31317 0.656583 0.754254i \(-0.272001\pi\)
0.656583 + 0.754254i \(0.272001\pi\)
\(60\) 0 0
\(61\) −415.671 −0.872480 −0.436240 0.899830i \(-0.643690\pi\)
−0.436240 + 0.899830i \(0.643690\pi\)
\(62\) 1309.75 2.68288
\(63\) 0 0
\(64\) −582.322 −1.13735
\(65\) 0 0
\(66\) 0 0
\(67\) 640.901 1.16864 0.584318 0.811525i \(-0.301362\pi\)
0.584318 + 0.811525i \(0.301362\pi\)
\(68\) 765.930 1.36592
\(69\) 0 0
\(70\) −2143.17 −3.65939
\(71\) 551.454 0.921768 0.460884 0.887461i \(-0.347533\pi\)
0.460884 + 0.887461i \(0.347533\pi\)
\(72\) 0 0
\(73\) 431.954 0.692553 0.346276 0.938133i \(-0.387446\pi\)
0.346276 + 0.938133i \(0.387446\pi\)
\(74\) 906.968 1.42477
\(75\) 0 0
\(76\) −1509.86 −2.27886
\(77\) −1780.43 −2.63505
\(78\) 0 0
\(79\) 998.743 1.42237 0.711186 0.703004i \(-0.248159\pi\)
0.711186 + 0.703004i \(0.248159\pi\)
\(80\) −814.322 −1.13805
\(81\) 0 0
\(82\) −256.368 −0.345258
\(83\) 461.462 0.610265 0.305133 0.952310i \(-0.401299\pi\)
0.305133 + 0.952310i \(0.401299\pi\)
\(84\) 0 0
\(85\) −729.762 −0.931221
\(86\) −2562.84 −3.21346
\(87\) 0 0
\(88\) −2219.17 −2.68823
\(89\) −1025.25 −1.22108 −0.610541 0.791985i \(-0.709048\pi\)
−0.610541 + 0.791985i \(0.709048\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −662.520 −0.750788
\(93\) 0 0
\(94\) −297.979 −0.326959
\(95\) 1438.57 1.55362
\(96\) 0 0
\(97\) 261.585 0.273813 0.136907 0.990584i \(-0.456284\pi\)
0.136907 + 0.990584i \(0.456284\pi\)
\(98\) −2607.78 −2.68802
\(99\) 0 0
\(100\) 1500.91 1.50091
\(101\) −290.237 −0.285937 −0.142969 0.989727i \(-0.545665\pi\)
−0.142969 + 0.989727i \(0.545665\pi\)
\(102\) 0 0
\(103\) −198.640 −0.190025 −0.0950123 0.995476i \(-0.530289\pi\)
−0.0950123 + 0.995476i \(0.530289\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −1654.29 −1.51583
\(107\) −684.251 −0.618216 −0.309108 0.951027i \(-0.600030\pi\)
−0.309108 + 0.951027i \(0.600030\pi\)
\(108\) 0 0
\(109\) 1725.33 1.51612 0.758059 0.652186i \(-0.226148\pi\)
0.758059 + 0.652186i \(0.226148\pi\)
\(110\) 4337.54 3.75971
\(111\) 0 0
\(112\) −1624.10 −1.37020
\(113\) 2032.55 1.69209 0.846045 0.533111i \(-0.178977\pi\)
0.846045 + 0.533111i \(0.178977\pi\)
\(114\) 0 0
\(115\) 631.235 0.511852
\(116\) −2759.99 −2.20912
\(117\) 0 0
\(118\) −2891.56 −2.25584
\(119\) −1455.45 −1.12118
\(120\) 0 0
\(121\) 2272.40 1.70728
\(122\) 2019.69 1.49880
\(123\) 0 0
\(124\) −4207.41 −3.04707
\(125\) 428.897 0.306894
\(126\) 0 0
\(127\) −1074.21 −0.750559 −0.375279 0.926912i \(-0.622453\pi\)
−0.375279 + 0.926912i \(0.622453\pi\)
\(128\) 2591.90 1.78979
\(129\) 0 0
\(130\) 0 0
\(131\) −1969.09 −1.31328 −0.656641 0.754203i \(-0.728023\pi\)
−0.656641 + 0.754203i \(0.728023\pi\)
\(132\) 0 0
\(133\) 2869.10 1.87054
\(134\) −3114.05 −2.00756
\(135\) 0 0
\(136\) −1814.10 −1.14381
\(137\) 0.346542 0.000216110 0 0.000108055 1.00000i \(-0.499966\pi\)
0.000108055 1.00000i \(0.499966\pi\)
\(138\) 0 0
\(139\) −1863.25 −1.13697 −0.568486 0.822693i \(-0.692471\pi\)
−0.568486 + 0.822693i \(0.692471\pi\)
\(140\) 6884.67 4.15615
\(141\) 0 0
\(142\) −2679.43 −1.58347
\(143\) 0 0
\(144\) 0 0
\(145\) 2629.65 1.50608
\(146\) −2098.80 −1.18971
\(147\) 0 0
\(148\) −2913.53 −1.61818
\(149\) −24.6640 −0.0135608 −0.00678038 0.999977i \(-0.502158\pi\)
−0.00678038 + 0.999977i \(0.502158\pi\)
\(150\) 0 0
\(151\) 447.901 0.241388 0.120694 0.992690i \(-0.461488\pi\)
0.120694 + 0.992690i \(0.461488\pi\)
\(152\) 3576.11 1.90829
\(153\) 0 0
\(154\) 8650.86 4.52666
\(155\) 4008.73 2.07735
\(156\) 0 0
\(157\) −547.454 −0.278290 −0.139145 0.990272i \(-0.544435\pi\)
−0.139145 + 0.990272i \(0.544435\pi\)
\(158\) −4852.75 −2.44344
\(159\) 0 0
\(160\) −441.542 −0.218169
\(161\) 1258.94 0.616266
\(162\) 0 0
\(163\) −190.029 −0.0913143 −0.0456572 0.998957i \(-0.514538\pi\)
−0.0456572 + 0.998957i \(0.514538\pi\)
\(164\) 823.552 0.392126
\(165\) 0 0
\(166\) −2242.18 −1.04835
\(167\) −851.170 −0.394404 −0.197202 0.980363i \(-0.563186\pi\)
−0.197202 + 0.980363i \(0.563186\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 3545.81 1.59971
\(171\) 0 0
\(172\) 8232.81 3.64968
\(173\) 2688.72 1.18162 0.590809 0.806811i \(-0.298809\pi\)
0.590809 + 0.806811i \(0.298809\pi\)
\(174\) 0 0
\(175\) −2852.08 −1.23198
\(176\) 3287.00 1.40776
\(177\) 0 0
\(178\) 4981.54 2.09765
\(179\) 991.338 0.413944 0.206972 0.978347i \(-0.433639\pi\)
0.206972 + 0.978347i \(0.433639\pi\)
\(180\) 0 0
\(181\) 225.981 0.0928011 0.0464006 0.998923i \(-0.485225\pi\)
0.0464006 + 0.998923i \(0.485225\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1569.18 0.628702
\(185\) 2775.94 1.10320
\(186\) 0 0
\(187\) 2945.67 1.15192
\(188\) 957.222 0.371344
\(189\) 0 0
\(190\) −6989.79 −2.66891
\(191\) 2265.08 0.858090 0.429045 0.903283i \(-0.358850\pi\)
0.429045 + 0.903283i \(0.358850\pi\)
\(192\) 0 0
\(193\) −20.6018 −0.00768369 −0.00384184 0.999993i \(-0.501223\pi\)
−0.00384184 + 0.999993i \(0.501223\pi\)
\(194\) −1271.00 −0.470375
\(195\) 0 0
\(196\) 8377.19 3.05291
\(197\) −72.6929 −0.0262901 −0.0131451 0.999914i \(-0.504184\pi\)
−0.0131451 + 0.999914i \(0.504184\pi\)
\(198\) 0 0
\(199\) −3599.20 −1.28211 −0.641057 0.767494i \(-0.721504\pi\)
−0.641057 + 0.767494i \(0.721504\pi\)
\(200\) −3554.90 −1.25685
\(201\) 0 0
\(202\) 1410.22 0.491202
\(203\) 5244.63 1.81330
\(204\) 0 0
\(205\) −784.662 −0.267333
\(206\) 965.161 0.326437
\(207\) 0 0
\(208\) 0 0
\(209\) −5806.74 −1.92182
\(210\) 0 0
\(211\) −340.466 −0.111084 −0.0555419 0.998456i \(-0.517689\pi\)
−0.0555419 + 0.998456i \(0.517689\pi\)
\(212\) 5314.20 1.72161
\(213\) 0 0
\(214\) 3324.68 1.06201
\(215\) −7844.04 −2.48818
\(216\) 0 0
\(217\) 7995.08 2.50111
\(218\) −8383.14 −2.60449
\(219\) 0 0
\(220\) −13933.8 −4.27008
\(221\) 0 0
\(222\) 0 0
\(223\) −1963.17 −0.589522 −0.294761 0.955571i \(-0.595240\pi\)
−0.294761 + 0.955571i \(0.595240\pi\)
\(224\) −880.620 −0.262674
\(225\) 0 0
\(226\) −9875.87 −2.90678
\(227\) −610.759 −0.178579 −0.0892897 0.996006i \(-0.528460\pi\)
−0.0892897 + 0.996006i \(0.528460\pi\)
\(228\) 0 0
\(229\) −4357.13 −1.25732 −0.628662 0.777679i \(-0.716397\pi\)
−0.628662 + 0.777679i \(0.716397\pi\)
\(230\) −3067.08 −0.879292
\(231\) 0 0
\(232\) 6537.01 1.84990
\(233\) −2776.38 −0.780630 −0.390315 0.920681i \(-0.627634\pi\)
−0.390315 + 0.920681i \(0.627634\pi\)
\(234\) 0 0
\(235\) −912.020 −0.253164
\(236\) 9288.79 2.56207
\(237\) 0 0
\(238\) 7071.82 1.92604
\(239\) −2740.18 −0.741622 −0.370811 0.928708i \(-0.620920\pi\)
−0.370811 + 0.928708i \(0.620920\pi\)
\(240\) 0 0
\(241\) 1734.06 0.463489 0.231745 0.972777i \(-0.425557\pi\)
0.231745 + 0.972777i \(0.425557\pi\)
\(242\) −11041.3 −2.93289
\(243\) 0 0
\(244\) −6488.01 −1.70226
\(245\) −7981.60 −2.08133
\(246\) 0 0
\(247\) 0 0
\(248\) 9965.24 2.55159
\(249\) 0 0
\(250\) −2083.95 −0.527202
\(251\) −1183.57 −0.297635 −0.148817 0.988865i \(-0.547547\pi\)
−0.148817 + 0.988865i \(0.547547\pi\)
\(252\) 0 0
\(253\) −2547.97 −0.633159
\(254\) 5219.45 1.28936
\(255\) 0 0
\(256\) −7935.08 −1.93727
\(257\) −7585.34 −1.84109 −0.920546 0.390635i \(-0.872255\pi\)
−0.920546 + 0.390635i \(0.872255\pi\)
\(258\) 0 0
\(259\) 5536.39 1.32824
\(260\) 0 0
\(261\) 0 0
\(262\) 9567.52 2.25604
\(263\) 4698.49 1.10160 0.550801 0.834637i \(-0.314322\pi\)
0.550801 + 0.834637i \(0.314322\pi\)
\(264\) 0 0
\(265\) −5063.25 −1.17371
\(266\) −13940.5 −3.21335
\(267\) 0 0
\(268\) 10003.5 2.28008
\(269\) −7395.94 −1.67635 −0.838175 0.545401i \(-0.816378\pi\)
−0.838175 + 0.545401i \(0.816378\pi\)
\(270\) 0 0
\(271\) −1931.83 −0.433028 −0.216514 0.976280i \(-0.569469\pi\)
−0.216514 + 0.976280i \(0.569469\pi\)
\(272\) 2687.02 0.598987
\(273\) 0 0
\(274\) −1.68380 −0.000371248 0
\(275\) 5772.30 1.26576
\(276\) 0 0
\(277\) 1830.40 0.397032 0.198516 0.980098i \(-0.436388\pi\)
0.198516 + 0.980098i \(0.436388\pi\)
\(278\) 9053.29 1.95317
\(279\) 0 0
\(280\) −16306.3 −3.48031
\(281\) −273.145 −0.0579875 −0.0289938 0.999580i \(-0.509230\pi\)
−0.0289938 + 0.999580i \(0.509230\pi\)
\(282\) 0 0
\(283\) −1147.91 −0.241116 −0.120558 0.992706i \(-0.538468\pi\)
−0.120558 + 0.992706i \(0.538468\pi\)
\(284\) 8607.37 1.79843
\(285\) 0 0
\(286\) 0 0
\(287\) −1564.94 −0.321867
\(288\) 0 0
\(289\) −2505.00 −0.509872
\(290\) −12777.1 −2.58724
\(291\) 0 0
\(292\) 6742.15 1.35121
\(293\) 7106.25 1.41690 0.708450 0.705761i \(-0.249395\pi\)
0.708450 + 0.705761i \(0.249395\pi\)
\(294\) 0 0
\(295\) −8850.15 −1.74670
\(296\) 6900.67 1.35504
\(297\) 0 0
\(298\) 119.839 0.0232956
\(299\) 0 0
\(300\) 0 0
\(301\) −15644.3 −2.99575
\(302\) −2176.29 −0.414673
\(303\) 0 0
\(304\) −5296.88 −0.999331
\(305\) 6181.63 1.16052
\(306\) 0 0
\(307\) −7964.11 −1.48057 −0.740287 0.672291i \(-0.765310\pi\)
−0.740287 + 0.672291i \(0.765310\pi\)
\(308\) −27789.9 −5.14115
\(309\) 0 0
\(310\) −19477.9 −3.56861
\(311\) 7003.74 1.27700 0.638498 0.769623i \(-0.279556\pi\)
0.638498 + 0.769623i \(0.279556\pi\)
\(312\) 0 0
\(313\) −1848.21 −0.333761 −0.166881 0.985977i \(-0.553369\pi\)
−0.166881 + 0.985977i \(0.553369\pi\)
\(314\) 2660.00 0.478065
\(315\) 0 0
\(316\) 15588.9 2.77514
\(317\) −2266.75 −0.401619 −0.200810 0.979630i \(-0.564357\pi\)
−0.200810 + 0.979630i \(0.564357\pi\)
\(318\) 0 0
\(319\) −10614.6 −1.86301
\(320\) 8659.97 1.51283
\(321\) 0 0
\(322\) −6117.03 −1.05866
\(323\) −4746.84 −0.817713
\(324\) 0 0
\(325\) 0 0
\(326\) 923.325 0.156866
\(327\) 0 0
\(328\) −1950.58 −0.328362
\(329\) −1818.95 −0.304808
\(330\) 0 0
\(331\) 8471.48 1.40675 0.703376 0.710818i \(-0.251675\pi\)
0.703376 + 0.710818i \(0.251675\pi\)
\(332\) 7202.73 1.19067
\(333\) 0 0
\(334\) 4135.71 0.677533
\(335\) −9531.13 −1.55445
\(336\) 0 0
\(337\) 642.992 0.103935 0.0519674 0.998649i \(-0.483451\pi\)
0.0519674 + 0.998649i \(0.483451\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −11390.5 −1.81687
\(341\) −16181.2 −2.56968
\(342\) 0 0
\(343\) −5745.30 −0.904423
\(344\) −19499.4 −3.05621
\(345\) 0 0
\(346\) −13064.1 −2.02986
\(347\) 11357.5 1.75706 0.878531 0.477686i \(-0.158524\pi\)
0.878531 + 0.477686i \(0.158524\pi\)
\(348\) 0 0
\(349\) −7539.82 −1.15644 −0.578219 0.815881i \(-0.696252\pi\)
−0.578219 + 0.815881i \(0.696252\pi\)
\(350\) 13857.9 2.11638
\(351\) 0 0
\(352\) 1782.28 0.269874
\(353\) 11978.3 1.80607 0.903033 0.429572i \(-0.141336\pi\)
0.903033 + 0.429572i \(0.141336\pi\)
\(354\) 0 0
\(355\) −8200.91 −1.22608
\(356\) −16002.6 −2.38241
\(357\) 0 0
\(358\) −4816.77 −0.711101
\(359\) 8585.68 1.26221 0.631107 0.775696i \(-0.282601\pi\)
0.631107 + 0.775696i \(0.282601\pi\)
\(360\) 0 0
\(361\) 2498.36 0.364246
\(362\) −1098.01 −0.159420
\(363\) 0 0
\(364\) 0 0
\(365\) −6423.77 −0.921193
\(366\) 0 0
\(367\) 2302.65 0.327514 0.163757 0.986501i \(-0.447639\pi\)
0.163757 + 0.986501i \(0.447639\pi\)
\(368\) −2324.24 −0.329237
\(369\) 0 0
\(370\) −13487.9 −1.89514
\(371\) −10098.2 −1.41314
\(372\) 0 0
\(373\) −8961.58 −1.24400 −0.622002 0.783016i \(-0.713680\pi\)
−0.622002 + 0.783016i \(0.713680\pi\)
\(374\) −14312.6 −1.97884
\(375\) 0 0
\(376\) −2267.18 −0.310959
\(377\) 0 0
\(378\) 0 0
\(379\) −4627.11 −0.627120 −0.313560 0.949568i \(-0.601522\pi\)
−0.313560 + 0.949568i \(0.601522\pi\)
\(380\) 22453.9 3.03121
\(381\) 0 0
\(382\) −11005.7 −1.47408
\(383\) 2130.00 0.284172 0.142086 0.989854i \(-0.454619\pi\)
0.142086 + 0.989854i \(0.454619\pi\)
\(384\) 0 0
\(385\) 26477.6 3.50499
\(386\) 100.101 0.0131995
\(387\) 0 0
\(388\) 4082.94 0.534227
\(389\) −5935.17 −0.773586 −0.386793 0.922167i \(-0.626417\pi\)
−0.386793 + 0.922167i \(0.626417\pi\)
\(390\) 0 0
\(391\) −2082.89 −0.269402
\(392\) −19841.3 −2.55647
\(393\) 0 0
\(394\) 353.204 0.0451629
\(395\) −14852.7 −1.89196
\(396\) 0 0
\(397\) 11708.6 1.48019 0.740095 0.672502i \(-0.234781\pi\)
0.740095 + 0.672502i \(0.234781\pi\)
\(398\) 17488.0 2.20250
\(399\) 0 0
\(400\) 5265.46 0.658182
\(401\) 4600.01 0.572852 0.286426 0.958102i \(-0.407533\pi\)
0.286426 + 0.958102i \(0.407533\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −4530.16 −0.557881
\(405\) 0 0
\(406\) −25482.9 −3.11501
\(407\) −11205.0 −1.36465
\(408\) 0 0
\(409\) −544.220 −0.0657945 −0.0328972 0.999459i \(-0.510473\pi\)
−0.0328972 + 0.999459i \(0.510473\pi\)
\(410\) 3812.56 0.459242
\(411\) 0 0
\(412\) −3100.47 −0.370750
\(413\) −17650.9 −2.10301
\(414\) 0 0
\(415\) −6862.60 −0.811739
\(416\) 0 0
\(417\) 0 0
\(418\) 28214.2 3.30143
\(419\) −10617.4 −1.23793 −0.618965 0.785418i \(-0.712448\pi\)
−0.618965 + 0.785418i \(0.712448\pi\)
\(420\) 0 0
\(421\) 1404.23 0.162560 0.0812801 0.996691i \(-0.474099\pi\)
0.0812801 + 0.996691i \(0.474099\pi\)
\(422\) 1654.28 0.190827
\(423\) 0 0
\(424\) −12586.6 −1.44165
\(425\) 4718.69 0.538564
\(426\) 0 0
\(427\) 12328.8 1.39726
\(428\) −10680.1 −1.20618
\(429\) 0 0
\(430\) 38113.1 4.27436
\(431\) 1352.99 0.151209 0.0756046 0.997138i \(-0.475911\pi\)
0.0756046 + 0.997138i \(0.475911\pi\)
\(432\) 0 0
\(433\) 12465.1 1.38346 0.691728 0.722158i \(-0.256850\pi\)
0.691728 + 0.722158i \(0.256850\pi\)
\(434\) −38847.0 −4.29658
\(435\) 0 0
\(436\) 26929.8 2.95804
\(437\) 4105.96 0.449461
\(438\) 0 0
\(439\) 4202.96 0.456939 0.228470 0.973551i \(-0.426628\pi\)
0.228470 + 0.973551i \(0.426628\pi\)
\(440\) 33002.2 3.57572
\(441\) 0 0
\(442\) 0 0
\(443\) 8198.85 0.879321 0.439661 0.898164i \(-0.355099\pi\)
0.439661 + 0.898164i \(0.355099\pi\)
\(444\) 0 0
\(445\) 15246.9 1.62421
\(446\) 9538.75 1.01272
\(447\) 0 0
\(448\) 17271.6 1.82144
\(449\) −642.438 −0.0675245 −0.0337623 0.999430i \(-0.510749\pi\)
−0.0337623 + 0.999430i \(0.510749\pi\)
\(450\) 0 0
\(451\) 3167.27 0.330690
\(452\) 31725.1 3.30138
\(453\) 0 0
\(454\) 2967.59 0.306775
\(455\) 0 0
\(456\) 0 0
\(457\) 1925.81 0.197124 0.0985620 0.995131i \(-0.468576\pi\)
0.0985620 + 0.995131i \(0.468576\pi\)
\(458\) 21170.7 2.15991
\(459\) 0 0
\(460\) 9852.63 0.998655
\(461\) −9658.81 −0.975826 −0.487913 0.872892i \(-0.662242\pi\)
−0.487913 + 0.872892i \(0.662242\pi\)
\(462\) 0 0
\(463\) −12025.9 −1.20711 −0.603556 0.797320i \(-0.706250\pi\)
−0.603556 + 0.797320i \(0.706250\pi\)
\(464\) −9682.53 −0.968750
\(465\) 0 0
\(466\) 13490.0 1.34102
\(467\) 2470.93 0.244842 0.122421 0.992478i \(-0.460934\pi\)
0.122421 + 0.992478i \(0.460934\pi\)
\(468\) 0 0
\(469\) −19009.0 −1.87155
\(470\) 4431.38 0.434903
\(471\) 0 0
\(472\) −22000.4 −2.14545
\(473\) 31662.3 3.07787
\(474\) 0 0
\(475\) −9301.86 −0.898523
\(476\) −22717.4 −2.18750
\(477\) 0 0
\(478\) 13314.2 1.27401
\(479\) −5606.05 −0.534754 −0.267377 0.963592i \(-0.586157\pi\)
−0.267377 + 0.963592i \(0.586157\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −8425.58 −0.796213
\(483\) 0 0
\(484\) 35468.7 3.33102
\(485\) −3890.14 −0.364211
\(486\) 0 0
\(487\) −778.566 −0.0724440 −0.0362220 0.999344i \(-0.511532\pi\)
−0.0362220 + 0.999344i \(0.511532\pi\)
\(488\) 15366.8 1.42546
\(489\) 0 0
\(490\) 38781.5 3.57544
\(491\) −9336.64 −0.858160 −0.429080 0.903266i \(-0.641162\pi\)
−0.429080 + 0.903266i \(0.641162\pi\)
\(492\) 0 0
\(493\) −8677.08 −0.792690
\(494\) 0 0
\(495\) 0 0
\(496\) −14760.4 −1.33621
\(497\) −16356.0 −1.47619
\(498\) 0 0
\(499\) 16615.2 1.49057 0.745287 0.666744i \(-0.232312\pi\)
0.745287 + 0.666744i \(0.232312\pi\)
\(500\) 6694.44 0.598769
\(501\) 0 0
\(502\) 5750.80 0.511296
\(503\) −7494.12 −0.664307 −0.332153 0.943225i \(-0.607775\pi\)
−0.332153 + 0.943225i \(0.607775\pi\)
\(504\) 0 0
\(505\) 4316.24 0.380337
\(506\) 12380.2 1.08768
\(507\) 0 0
\(508\) −16766.9 −1.46439
\(509\) −11882.9 −1.03478 −0.517388 0.855751i \(-0.673096\pi\)
−0.517388 + 0.855751i \(0.673096\pi\)
\(510\) 0 0
\(511\) −12811.7 −1.10911
\(512\) 17820.3 1.53819
\(513\) 0 0
\(514\) 36856.1 3.16275
\(515\) 2954.06 0.252760
\(516\) 0 0
\(517\) 3681.35 0.313164
\(518\) −26900.5 −2.28174
\(519\) 0 0
\(520\) 0 0
\(521\) −20244.7 −1.70237 −0.851186 0.524864i \(-0.824116\pi\)
−0.851186 + 0.524864i \(0.824116\pi\)
\(522\) 0 0
\(523\) −15231.8 −1.27350 −0.636748 0.771072i \(-0.719721\pi\)
−0.636748 + 0.771072i \(0.719721\pi\)
\(524\) −30734.5 −2.56230
\(525\) 0 0
\(526\) −22829.3 −1.89240
\(527\) −13227.6 −1.09337
\(528\) 0 0
\(529\) −10365.3 −0.851921
\(530\) 24601.6 2.01628
\(531\) 0 0
\(532\) 44782.4 3.64955
\(533\) 0 0
\(534\) 0 0
\(535\) 10175.8 0.822315
\(536\) −23693.3 −1.90932
\(537\) 0 0
\(538\) 35935.8 2.87975
\(539\) 32217.6 2.57460
\(540\) 0 0
\(541\) −3599.15 −0.286025 −0.143013 0.989721i \(-0.545679\pi\)
−0.143013 + 0.989721i \(0.545679\pi\)
\(542\) 9386.51 0.743884
\(543\) 0 0
\(544\) 1456.96 0.114828
\(545\) −25658.2 −2.01665
\(546\) 0 0
\(547\) −9212.26 −0.720087 −0.360044 0.932936i \(-0.617238\pi\)
−0.360044 + 0.932936i \(0.617238\pi\)
\(548\) 5.40900 0.000421644 0
\(549\) 0 0
\(550\) −28046.8 −2.17440
\(551\) 17105.0 1.32250
\(552\) 0 0
\(553\) −29622.6 −2.27790
\(554\) −8893.64 −0.682048
\(555\) 0 0
\(556\) −29082.6 −2.21830
\(557\) −25245.0 −1.92040 −0.960202 0.279306i \(-0.909896\pi\)
−0.960202 + 0.279306i \(0.909896\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 24152.7 1.82256
\(561\) 0 0
\(562\) 1327.18 0.0996148
\(563\) 1694.84 0.126872 0.0634359 0.997986i \(-0.479794\pi\)
0.0634359 + 0.997986i \(0.479794\pi\)
\(564\) 0 0
\(565\) −30226.9 −2.25072
\(566\) 5577.51 0.414206
\(567\) 0 0
\(568\) −20386.5 −1.50598
\(569\) −494.893 −0.0364622 −0.0182311 0.999834i \(-0.505803\pi\)
−0.0182311 + 0.999834i \(0.505803\pi\)
\(570\) 0 0
\(571\) −1750.91 −0.128325 −0.0641623 0.997939i \(-0.520438\pi\)
−0.0641623 + 0.997939i \(0.520438\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 7603.84 0.552924
\(575\) −4081.60 −0.296025
\(576\) 0 0
\(577\) −8535.56 −0.615841 −0.307920 0.951412i \(-0.599633\pi\)
−0.307920 + 0.951412i \(0.599633\pi\)
\(578\) 12171.5 0.875892
\(579\) 0 0
\(580\) 41045.0 2.93845
\(581\) −13686.9 −0.977328
\(582\) 0 0
\(583\) 20437.7 1.45188
\(584\) −15968.7 −1.13149
\(585\) 0 0
\(586\) −34528.3 −2.43404
\(587\) 19499.0 1.37105 0.685527 0.728047i \(-0.259572\pi\)
0.685527 + 0.728047i \(0.259572\pi\)
\(588\) 0 0
\(589\) 26075.4 1.82414
\(590\) 43001.6 3.00059
\(591\) 0 0
\(592\) −10221.2 −0.709607
\(593\) −7599.31 −0.526250 −0.263125 0.964762i \(-0.584753\pi\)
−0.263125 + 0.964762i \(0.584753\pi\)
\(594\) 0 0
\(595\) 21644.6 1.49133
\(596\) −384.968 −0.0264579
\(597\) 0 0
\(598\) 0 0
\(599\) −26240.9 −1.78994 −0.894971 0.446124i \(-0.852804\pi\)
−0.894971 + 0.446124i \(0.852804\pi\)
\(600\) 0 0
\(601\) 20319.6 1.37912 0.689561 0.724227i \(-0.257803\pi\)
0.689561 + 0.724227i \(0.257803\pi\)
\(602\) 76013.3 5.14630
\(603\) 0 0
\(604\) 6991.06 0.470964
\(605\) −33793.8 −2.27093
\(606\) 0 0
\(607\) 1361.39 0.0910333 0.0455166 0.998964i \(-0.485507\pi\)
0.0455166 + 0.998964i \(0.485507\pi\)
\(608\) −2872.08 −0.191576
\(609\) 0 0
\(610\) −30035.7 −1.99362
\(611\) 0 0
\(612\) 0 0
\(613\) 13883.5 0.914764 0.457382 0.889270i \(-0.348787\pi\)
0.457382 + 0.889270i \(0.348787\pi\)
\(614\) 38696.5 2.54343
\(615\) 0 0
\(616\) 65820.1 4.30514
\(617\) −15316.8 −0.999403 −0.499701 0.866198i \(-0.666557\pi\)
−0.499701 + 0.866198i \(0.666557\pi\)
\(618\) 0 0
\(619\) 10325.0 0.670432 0.335216 0.942141i \(-0.391191\pi\)
0.335216 + 0.942141i \(0.391191\pi\)
\(620\) 62570.3 4.05304
\(621\) 0 0
\(622\) −34030.2 −2.19371
\(623\) 30408.7 1.95554
\(624\) 0 0
\(625\) −18398.3 −1.17749
\(626\) 8980.21 0.573357
\(627\) 0 0
\(628\) −8544.93 −0.542962
\(629\) −9159.79 −0.580643
\(630\) 0 0
\(631\) −20967.7 −1.32284 −0.661420 0.750016i \(-0.730046\pi\)
−0.661420 + 0.750016i \(0.730046\pi\)
\(632\) −36922.2 −2.32387
\(633\) 0 0
\(634\) 11013.8 0.689928
\(635\) 15975.1 0.998349
\(636\) 0 0
\(637\) 0 0
\(638\) 51574.6 3.20041
\(639\) 0 0
\(640\) −38545.2 −2.38068
\(641\) 20292.1 1.25037 0.625187 0.780475i \(-0.285023\pi\)
0.625187 + 0.780475i \(0.285023\pi\)
\(642\) 0 0
\(643\) −9106.14 −0.558493 −0.279247 0.960219i \(-0.590085\pi\)
−0.279247 + 0.960219i \(0.590085\pi\)
\(644\) 19650.2 1.20237
\(645\) 0 0
\(646\) 23064.2 1.40472
\(647\) 15251.4 0.926732 0.463366 0.886167i \(-0.346641\pi\)
0.463366 + 0.886167i \(0.346641\pi\)
\(648\) 0 0
\(649\) 35723.5 2.16066
\(650\) 0 0
\(651\) 0 0
\(652\) −2966.07 −0.178160
\(653\) −14421.0 −0.864220 −0.432110 0.901821i \(-0.642231\pi\)
−0.432110 + 0.901821i \(0.642231\pi\)
\(654\) 0 0
\(655\) 29283.2 1.74685
\(656\) 2889.17 0.171956
\(657\) 0 0
\(658\) 8838.01 0.523619
\(659\) 11890.9 0.702892 0.351446 0.936208i \(-0.385690\pi\)
0.351446 + 0.936208i \(0.385690\pi\)
\(660\) 0 0
\(661\) 3696.05 0.217488 0.108744 0.994070i \(-0.465317\pi\)
0.108744 + 0.994070i \(0.465317\pi\)
\(662\) −41161.7 −2.41661
\(663\) 0 0
\(664\) −17059.6 −0.997051
\(665\) −42667.6 −2.48809
\(666\) 0 0
\(667\) 7505.57 0.435707
\(668\) −13285.5 −0.769507
\(669\) 0 0
\(670\) 46310.4 2.67034
\(671\) −24952.0 −1.43556
\(672\) 0 0
\(673\) −9587.03 −0.549112 −0.274556 0.961571i \(-0.588531\pi\)
−0.274556 + 0.961571i \(0.588531\pi\)
\(674\) −3124.21 −0.178546
\(675\) 0 0
\(676\) 0 0
\(677\) 12413.6 0.704718 0.352359 0.935865i \(-0.385380\pi\)
0.352359 + 0.935865i \(0.385380\pi\)
\(678\) 0 0
\(679\) −7758.56 −0.438507
\(680\) 26978.3 1.52143
\(681\) 0 0
\(682\) 78622.0 4.41436
\(683\) −23012.6 −1.28924 −0.644622 0.764502i \(-0.722985\pi\)
−0.644622 + 0.764502i \(0.722985\pi\)
\(684\) 0 0
\(685\) −5.15357 −0.000287457 0
\(686\) 27915.6 1.55368
\(687\) 0 0
\(688\) 28882.2 1.60047
\(689\) 0 0
\(690\) 0 0
\(691\) −30604.2 −1.68486 −0.842431 0.538804i \(-0.818876\pi\)
−0.842431 + 0.538804i \(0.818876\pi\)
\(692\) 41967.0 2.30541
\(693\) 0 0
\(694\) −55184.3 −3.01840
\(695\) 27709.3 1.51233
\(696\) 0 0
\(697\) 2589.15 0.140705
\(698\) 36634.9 1.98661
\(699\) 0 0
\(700\) −44516.7 −2.40368
\(701\) 4042.16 0.217789 0.108895 0.994053i \(-0.465269\pi\)
0.108895 + 0.994053i \(0.465269\pi\)
\(702\) 0 0
\(703\) 18056.5 0.968727
\(704\) −34955.8 −1.87137
\(705\) 0 0
\(706\) −58200.9 −3.10258
\(707\) 8608.38 0.457923
\(708\) 0 0
\(709\) 33867.3 1.79395 0.896977 0.442077i \(-0.145758\pi\)
0.896977 + 0.442077i \(0.145758\pi\)
\(710\) 39847.1 2.10624
\(711\) 0 0
\(712\) 37902.1 1.99500
\(713\) 11441.7 0.600976
\(714\) 0 0
\(715\) 0 0
\(716\) 15473.3 0.807632
\(717\) 0 0
\(718\) −41716.6 −2.16831
\(719\) −7082.45 −0.367358 −0.183679 0.982986i \(-0.558801\pi\)
−0.183679 + 0.982986i \(0.558801\pi\)
\(720\) 0 0
\(721\) 5891.62 0.304321
\(722\) −12139.2 −0.625725
\(723\) 0 0
\(724\) 3527.22 0.181061
\(725\) −17003.5 −0.871027
\(726\) 0 0
\(727\) 29511.0 1.50551 0.752754 0.658302i \(-0.228725\pi\)
0.752754 + 0.658302i \(0.228725\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 31212.2 1.58249
\(731\) 25883.0 1.30960
\(732\) 0 0
\(733\) −19353.4 −0.975217 −0.487608 0.873062i \(-0.662131\pi\)
−0.487608 + 0.873062i \(0.662131\pi\)
\(734\) −11188.3 −0.562625
\(735\) 0 0
\(736\) −1260.25 −0.0631162
\(737\) 38472.2 1.92285
\(738\) 0 0
\(739\) −35578.3 −1.77100 −0.885500 0.464639i \(-0.846184\pi\)
−0.885500 + 0.464639i \(0.846184\pi\)
\(740\) 43328.3 2.15241
\(741\) 0 0
\(742\) 49065.9 2.42758
\(743\) 12875.1 0.635722 0.317861 0.948137i \(-0.397036\pi\)
0.317861 + 0.948137i \(0.397036\pi\)
\(744\) 0 0
\(745\) 366.789 0.0180377
\(746\) 43543.1 2.13703
\(747\) 0 0
\(748\) 45977.5 2.24747
\(749\) 20294.8 0.990061
\(750\) 0 0
\(751\) −37817.4 −1.83752 −0.918758 0.394821i \(-0.870807\pi\)
−0.918758 + 0.394821i \(0.870807\pi\)
\(752\) 3358.11 0.162843
\(753\) 0 0
\(754\) 0 0
\(755\) −6660.93 −0.321081
\(756\) 0 0
\(757\) 6185.73 0.296994 0.148497 0.988913i \(-0.452557\pi\)
0.148497 + 0.988913i \(0.452557\pi\)
\(758\) 22482.4 1.07731
\(759\) 0 0
\(760\) −53181.8 −2.53830
\(761\) −15728.3 −0.749213 −0.374607 0.927184i \(-0.622222\pi\)
−0.374607 + 0.927184i \(0.622222\pi\)
\(762\) 0 0
\(763\) −51173.1 −2.42803
\(764\) 35354.5 1.67419
\(765\) 0 0
\(766\) −10349.4 −0.488169
\(767\) 0 0
\(768\) 0 0
\(769\) −8203.97 −0.384711 −0.192356 0.981325i \(-0.561613\pi\)
−0.192356 + 0.981325i \(0.561613\pi\)
\(770\) −128651. −6.02110
\(771\) 0 0
\(772\) −321.564 −0.0149914
\(773\) −3418.73 −0.159073 −0.0795363 0.996832i \(-0.525344\pi\)
−0.0795363 + 0.996832i \(0.525344\pi\)
\(774\) 0 0
\(775\) −25920.7 −1.20142
\(776\) −9670.43 −0.447356
\(777\) 0 0
\(778\) 28838.1 1.32892
\(779\) −5103.95 −0.234747
\(780\) 0 0
\(781\) 33102.8 1.51666
\(782\) 10120.5 0.462796
\(783\) 0 0
\(784\) 29388.7 1.33877
\(785\) 8141.42 0.370165
\(786\) 0 0
\(787\) −26092.8 −1.18184 −0.590921 0.806730i \(-0.701235\pi\)
−0.590921 + 0.806730i \(0.701235\pi\)
\(788\) −1134.63 −0.0512937
\(789\) 0 0
\(790\) 72167.4 3.25013
\(791\) −60285.1 −2.70985
\(792\) 0 0
\(793\) 0 0
\(794\) −56890.2 −2.54277
\(795\) 0 0
\(796\) −56178.1 −2.50148
\(797\) −18636.4 −0.828275 −0.414138 0.910214i \(-0.635917\pi\)
−0.414138 + 0.910214i \(0.635917\pi\)
\(798\) 0 0
\(799\) 3009.40 0.133248
\(800\) 2855.04 0.126176
\(801\) 0 0
\(802\) −22350.8 −0.984083
\(803\) 25929.4 1.13951
\(804\) 0 0
\(805\) −18722.3 −0.819721
\(806\) 0 0
\(807\) 0 0
\(808\) 10729.7 0.467164
\(809\) −9074.08 −0.394348 −0.197174 0.980368i \(-0.563176\pi\)
−0.197174 + 0.980368i \(0.563176\pi\)
\(810\) 0 0
\(811\) 20112.2 0.870818 0.435409 0.900233i \(-0.356604\pi\)
0.435409 + 0.900233i \(0.356604\pi\)
\(812\) 81860.8 3.53787
\(813\) 0 0
\(814\) 54443.7 2.34429
\(815\) 2826.01 0.121461
\(816\) 0 0
\(817\) −51022.7 −2.18489
\(818\) 2644.29 0.113026
\(819\) 0 0
\(820\) −12247.4 −0.521583
\(821\) 29190.8 1.24089 0.620443 0.784252i \(-0.286953\pi\)
0.620443 + 0.784252i \(0.286953\pi\)
\(822\) 0 0
\(823\) 27556.8 1.16716 0.583579 0.812056i \(-0.301652\pi\)
0.583579 + 0.812056i \(0.301652\pi\)
\(824\) 7343.44 0.310462
\(825\) 0 0
\(826\) 85763.2 3.61269
\(827\) 16596.0 0.697822 0.348911 0.937156i \(-0.386552\pi\)
0.348911 + 0.937156i \(0.386552\pi\)
\(828\) 0 0
\(829\) 20877.6 0.874681 0.437340 0.899296i \(-0.355920\pi\)
0.437340 + 0.899296i \(0.355920\pi\)
\(830\) 33344.4 1.39446
\(831\) 0 0
\(832\) 0 0
\(833\) 26336.9 1.09546
\(834\) 0 0
\(835\) 12658.1 0.524613
\(836\) −90634.6 −3.74960
\(837\) 0 0
\(838\) 51588.3 2.12660
\(839\) 31026.1 1.27669 0.638343 0.769752i \(-0.279620\pi\)
0.638343 + 0.769752i \(0.279620\pi\)
\(840\) 0 0
\(841\) 6878.38 0.282028
\(842\) −6822.94 −0.279257
\(843\) 0 0
\(844\) −5314.17 −0.216731
\(845\) 0 0
\(846\) 0 0
\(847\) −67398.9 −2.73418
\(848\) 18643.2 0.754963
\(849\) 0 0
\(850\) −22927.4 −0.925181
\(851\) 7923.10 0.319154
\(852\) 0 0
\(853\) 24134.3 0.968748 0.484374 0.874861i \(-0.339047\pi\)
0.484374 + 0.874861i \(0.339047\pi\)
\(854\) −59903.7 −2.40031
\(855\) 0 0
\(856\) 25295.9 1.01004
\(857\) 4872.72 0.194223 0.0971114 0.995274i \(-0.469040\pi\)
0.0971114 + 0.995274i \(0.469040\pi\)
\(858\) 0 0
\(859\) 17690.2 0.702658 0.351329 0.936252i \(-0.385730\pi\)
0.351329 + 0.936252i \(0.385730\pi\)
\(860\) −122434. −4.85460
\(861\) 0 0
\(862\) −6573.98 −0.259757
\(863\) −32855.1 −1.29595 −0.647973 0.761664i \(-0.724383\pi\)
−0.647973 + 0.761664i \(0.724383\pi\)
\(864\) 0 0
\(865\) −39985.2 −1.57172
\(866\) −60566.4 −2.37659
\(867\) 0 0
\(868\) 124791. 4.87983
\(869\) 59952.8 2.34035
\(870\) 0 0
\(871\) 0 0
\(872\) −63783.2 −2.47703
\(873\) 0 0
\(874\) −19950.3 −0.772114
\(875\) −12721.0 −0.491484
\(876\) 0 0
\(877\) −10874.1 −0.418691 −0.209346 0.977842i \(-0.567133\pi\)
−0.209346 + 0.977842i \(0.567133\pi\)
\(878\) −20421.6 −0.784960
\(879\) 0 0
\(880\) −48882.4 −1.87253
\(881\) −7130.10 −0.272667 −0.136333 0.990663i \(-0.543532\pi\)
−0.136333 + 0.990663i \(0.543532\pi\)
\(882\) 0 0
\(883\) 12029.0 0.458446 0.229223 0.973374i \(-0.426381\pi\)
0.229223 + 0.973374i \(0.426381\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −39837.1 −1.51056
\(887\) 40837.9 1.54589 0.772944 0.634474i \(-0.218783\pi\)
0.772944 + 0.634474i \(0.218783\pi\)
\(888\) 0 0
\(889\) 31861.0 1.20201
\(890\) −74082.7 −2.79018
\(891\) 0 0
\(892\) −30642.1 −1.15019
\(893\) −5932.37 −0.222306
\(894\) 0 0
\(895\) −14742.6 −0.550605
\(896\) −76875.2 −2.86632
\(897\) 0 0
\(898\) 3121.52 0.115998
\(899\) 47665.0 1.76832
\(900\) 0 0
\(901\) 16707.2 0.617756
\(902\) −15389.3 −0.568081
\(903\) 0 0
\(904\) −75140.7 −2.76454
\(905\) −3360.66 −0.123439
\(906\) 0 0
\(907\) −19247.0 −0.704616 −0.352308 0.935884i \(-0.614603\pi\)
−0.352308 + 0.935884i \(0.614603\pi\)
\(908\) −9533.04 −0.348420
\(909\) 0 0
\(910\) 0 0
\(911\) 11860.9 0.431359 0.215680 0.976464i \(-0.430803\pi\)
0.215680 + 0.976464i \(0.430803\pi\)
\(912\) 0 0
\(913\) 27700.8 1.00412
\(914\) −9357.25 −0.338633
\(915\) 0 0
\(916\) −68008.3 −2.45312
\(917\) 58402.8 2.10320
\(918\) 0 0
\(919\) −21211.9 −0.761390 −0.380695 0.924701i \(-0.624315\pi\)
−0.380695 + 0.924701i \(0.624315\pi\)
\(920\) −23335.9 −0.836263
\(921\) 0 0
\(922\) 46930.8 1.67634
\(923\) 0 0
\(924\) 0 0
\(925\) −17949.4 −0.638025
\(926\) 58432.4 2.07366
\(927\) 0 0
\(928\) −5250.07 −0.185713
\(929\) 37450.9 1.32263 0.661316 0.750108i \(-0.269998\pi\)
0.661316 + 0.750108i \(0.269998\pi\)
\(930\) 0 0
\(931\) −51917.5 −1.82763
\(932\) −43335.2 −1.52306
\(933\) 0 0
\(934\) −12005.9 −0.420605
\(935\) −43806.3 −1.53221
\(936\) 0 0
\(937\) −45853.2 −1.59868 −0.799338 0.600882i \(-0.794816\pi\)
−0.799338 + 0.600882i \(0.794816\pi\)
\(938\) 92362.2 3.21507
\(939\) 0 0
\(940\) −14235.3 −0.493940
\(941\) −14680.3 −0.508569 −0.254285 0.967129i \(-0.581840\pi\)
−0.254285 + 0.967129i \(0.581840\pi\)
\(942\) 0 0
\(943\) −2239.59 −0.0773393
\(944\) 32586.7 1.12353
\(945\) 0 0
\(946\) −153843. −5.28738
\(947\) −34542.6 −1.18531 −0.592653 0.805458i \(-0.701919\pi\)
−0.592653 + 0.805458i \(0.701919\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 45196.4 1.54354
\(951\) 0 0
\(952\) 53806.0 1.83179
\(953\) 44993.2 1.52935 0.764677 0.644414i \(-0.222899\pi\)
0.764677 + 0.644414i \(0.222899\pi\)
\(954\) 0 0
\(955\) −33684.9 −1.14138
\(956\) −42770.1 −1.44695
\(957\) 0 0
\(958\) 27239.0 0.918636
\(959\) −10.2784 −0.000346096 0
\(960\) 0 0
\(961\) 42871.1 1.43906
\(962\) 0 0
\(963\) 0 0
\(964\) 27066.2 0.904297
\(965\) 306.379 0.0102204
\(966\) 0 0
\(967\) −38006.9 −1.26393 −0.631964 0.774997i \(-0.717751\pi\)
−0.631964 + 0.774997i \(0.717751\pi\)
\(968\) −84007.4 −2.78936
\(969\) 0 0
\(970\) 18901.6 0.625665
\(971\) −29954.2 −0.989987 −0.494993 0.868897i \(-0.664829\pi\)
−0.494993 + 0.868897i \(0.664829\pi\)
\(972\) 0 0
\(973\) 55263.8 1.82084
\(974\) 3782.94 0.124449
\(975\) 0 0
\(976\) −22761.1 −0.746481
\(977\) −43605.7 −1.42791 −0.713957 0.700190i \(-0.753099\pi\)
−0.713957 + 0.700190i \(0.753099\pi\)
\(978\) 0 0
\(979\) −61544.0 −2.00915
\(980\) −124581. −4.06080
\(981\) 0 0
\(982\) 45365.4 1.47420
\(983\) −45624.8 −1.48037 −0.740185 0.672403i \(-0.765262\pi\)
−0.740185 + 0.672403i \(0.765262\pi\)
\(984\) 0 0
\(985\) 1081.05 0.0349696
\(986\) 42160.7 1.36173
\(987\) 0 0
\(988\) 0 0
\(989\) −22388.5 −0.719830
\(990\) 0 0
\(991\) 34882.4 1.11814 0.559069 0.829121i \(-0.311159\pi\)
0.559069 + 0.829121i \(0.311159\pi\)
\(992\) −8003.38 −0.256157
\(993\) 0 0
\(994\) 79471.6 2.53590
\(995\) 53525.3 1.70539
\(996\) 0 0
\(997\) −18448.4 −0.586024 −0.293012 0.956109i \(-0.594658\pi\)
−0.293012 + 0.956109i \(0.594658\pi\)
\(998\) −80730.7 −2.56061
\(999\) 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.bm.1.1 18
3.2 odd 2 inner 1521.4.a.bm.1.18 yes 18
13.12 even 2 1521.4.a.bn.1.18 yes 18
39.38 odd 2 1521.4.a.bn.1.1 yes 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1521.4.a.bm.1.1 18 1.1 even 1 trivial
1521.4.a.bm.1.18 yes 18 3.2 odd 2 inner
1521.4.a.bn.1.1 yes 18 39.38 odd 2
1521.4.a.bn.1.18 yes 18 13.12 even 2