Properties

Label 1521.4.a.bn.1.1
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1521,4,Mod(1,1521)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1521, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1521.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 100 x^{16} + 4066 x^{14} - 86197 x^{12} + 1016064 x^{10} - 6594119 x^{8} + 22251817 x^{6} + \cdots - 6889792 \) 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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.85886 q^{2} +15.6085 q^{4} -14.8714 q^{5} +29.6598 q^{7} -36.9687 q^{8} +O(q^{10})\) \(q-4.85886 q^{2} +15.6085 q^{4} -14.8714 q^{5} +29.6598 q^{7} -36.9687 q^{8} +72.2582 q^{10} +60.0283 q^{11} -144.113 q^{14} +54.7574 q^{16} -49.0713 q^{17} +96.7335 q^{19} -232.121 q^{20} -291.669 q^{22} +42.4461 q^{23} +96.1597 q^{25} +462.946 q^{28} +176.826 q^{29} +269.559 q^{31} +29.6906 q^{32} +238.431 q^{34} -441.085 q^{35} +186.663 q^{37} -470.014 q^{38} +549.777 q^{40} +52.7630 q^{41} +527.456 q^{43} +936.952 q^{44} -206.240 q^{46} +61.3270 q^{47} +536.706 q^{49} -467.226 q^{50} -340.468 q^{53} -892.707 q^{55} -1096.48 q^{56} -859.172 q^{58} +595.111 q^{59} -415.671 q^{61} -1309.75 q^{62} -582.322 q^{64} -640.901 q^{67} -765.930 q^{68} +2143.17 q^{70} +551.454 q^{71} -431.954 q^{73} -906.968 q^{74} +1509.86 q^{76} +1780.43 q^{77} +998.743 q^{79} -814.322 q^{80} -256.368 q^{82} +461.462 q^{83} +729.762 q^{85} -2562.84 q^{86} -2219.17 q^{88} -1025.25 q^{89} +662.520 q^{92} -297.979 q^{94} -1438.57 q^{95} -261.585 q^{97} -2607.78 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 56 q^{4} + 94 q^{7}+O(q^{10}) \) 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 \(-0.204440\pi\)
0.800741 + 0.599011i \(0.204440\pi\)
\(8\) −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 \(-0.613834\pi\)
−0.350046 + 0.936733i \(0.613834\pi\)
\(18\) 0 0
\(19\) 96.7335 1.16801 0.584005 0.811750i \(-0.301485\pi\)
0.584005 + 0.811750i \(0.301485\pi\)
\(20\) −232.121 −2.59519
\(21\) 0 0
\(22\) −291.669 −2.82655
\(23\) 42.4461 0.384810 0.192405 0.981316i \(-0.438371\pi\)
0.192405 + 0.981316i \(0.438371\pi\)
\(24\) 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 \(-0.308439\pi\)
0.566133 + 0.824314i \(0.308439\pi\)
\(30\) 0 0
\(31\) 269.559 1.56175 0.780875 0.624688i \(-0.214774\pi\)
0.780875 + 0.624688i \(0.214774\pi\)
\(32\) 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 \(-0.363890\pi\)
0.414691 + 0.909962i \(0.363890\pi\)
\(38\) −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 \(-0.645446\pi\)
−0.441197 + 0.897410i \(0.645446\pi\)
\(54\) 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 \(-0.698638\pi\)
−0.584318 + 0.811525i \(0.698638\pi\)
\(68\) −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 \(-0.612554\pi\)
−0.346276 + 0.938133i \(0.612554\pi\)
\(74\) −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 \(-0.543716\pi\)
−0.136907 + 0.990584i \(0.543716\pi\)
\(98\) −2607.78 −2.68802
\(99\) 0 0
\(100\) 1500.91 1.50091
\(101\) 290.237 0.285937 0.142969 0.989727i \(-0.454335\pi\)
0.142969 + 0.989727i \(0.454335\pi\)
\(102\) 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 \(-0.399970\pi\)
0.309108 + 0.951027i \(0.399970\pi\)
\(108\) 0 0
\(109\) −1725.33 −1.51612 −0.758059 0.652186i \(-0.773852\pi\)
−0.758059 + 0.652186i \(0.773852\pi\)
\(110\) 4337.54 3.75971
\(111\) 0 0
\(112\) 1624.10 1.37020
\(113\) −2032.55 −1.69209 −0.846045 0.533111i \(-0.821023\pi\)
−0.846045 + 0.533111i \(0.821023\pi\)
\(114\) 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 \(-0.271977\pi\)
0.656641 + 0.754203i \(0.271977\pi\)
\(132\) 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 \(-0.538512\pi\)
−0.120694 + 0.992690i \(0.538512\pi\)
\(152\) −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 \(-0.485462\pi\)
0.0456572 + 0.998957i \(0.485462\pi\)
\(164\) 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 \(-0.701191\pi\)
−0.590809 + 0.806811i \(0.701191\pi\)
\(174\) 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 \(-0.566361\pi\)
−0.206972 + 0.978347i \(0.566361\pi\)
\(180\) 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 \(-0.641150\pi\)
−0.429045 + 0.903283i \(0.641150\pi\)
\(192\) 0 0
\(193\) 20.6018 0.00768369 0.00384184 0.999993i \(-0.498777\pi\)
0.00384184 + 0.999993i \(0.498777\pi\)
\(194\) 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 \(-0.404760\pi\)
0.294761 + 0.955571i \(0.404760\pi\)
\(224\) 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 \(-0.283603\pi\)
0.628662 + 0.777679i \(0.283603\pi\)
\(230\) 3067.08 0.879292
\(231\) 0 0
\(232\) −6537.01 −1.84990
\(233\) 2776.38 0.780630 0.390315 0.920681i \(-0.372366\pi\)
0.390315 + 0.920681i \(0.372366\pi\)
\(234\) 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 \(-0.574443\pi\)
−0.231745 + 0.972777i \(0.574443\pi\)
\(242\) −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 \(-0.452453\pi\)
0.148817 + 0.988865i \(0.452453\pi\)
\(252\) 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 \(-0.127745\pi\)
0.920546 + 0.390635i \(0.127745\pi\)
\(258\) 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 \(-0.685678\pi\)
−0.550801 + 0.834637i \(0.685678\pi\)
\(264\) 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 \(-0.183622\pi\)
0.838175 + 0.545401i \(0.183622\pi\)
\(270\) 0 0
\(271\) 1931.83 0.433028 0.216514 0.976280i \(-0.430531\pi\)
0.216514 + 0.976280i \(0.430531\pi\)
\(272\) −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 \(-0.234690\pi\)
0.740287 + 0.672291i \(0.234690\pi\)
\(308\) 27789.9 5.14115
\(309\) 0 0
\(310\) 19477.9 3.56861
\(311\) −7003.74 −1.27700 −0.638498 0.769623i \(-0.720444\pi\)
−0.638498 + 0.769623i \(0.720444\pi\)
\(312\) 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 \(-0.748325\pi\)
−0.703376 + 0.710818i \(0.748325\pi\)
\(332\) 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 \(-0.841476\pi\)
−0.878531 + 0.477686i \(0.841476\pi\)
\(348\) 0 0
\(349\) 7539.82 1.15644 0.578219 0.815881i \(-0.303748\pi\)
0.578219 + 0.815881i \(0.303748\pi\)
\(350\) −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 \(-0.398478\pi\)
0.313560 + 0.949568i \(0.398478\pi\)
\(380\) −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 \(-0.373583\pi\)
0.386793 + 0.922167i \(0.373583\pi\)
\(390\) 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 \(-0.765219\pi\)
−0.740095 + 0.672502i \(0.765219\pi\)
\(398\) 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 \(-0.489527\pi\)
0.0328972 + 0.999459i \(0.489527\pi\)
\(410\) 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 \(-0.287552\pi\)
0.618965 + 0.785418i \(0.287552\pi\)
\(420\) 0 0
\(421\) −1404.23 −0.162560 −0.0812801 0.996691i \(-0.525901\pi\)
−0.0812801 + 0.996691i \(0.525901\pi\)
\(422\) 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 \(-0.644901\pi\)
−0.439661 + 0.898164i \(0.644901\pi\)
\(444\) 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 \(-0.531424\pi\)
−0.0985620 + 0.995131i \(0.531424\pi\)
\(458\) −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 \(-0.293750\pi\)
0.603556 + 0.797320i \(0.293750\pi\)
\(464\) 9682.53 0.968750
\(465\) 0 0
\(466\) −13490.0 −1.34102
\(467\) −2470.93 −0.244842 −0.122421 0.992478i \(-0.539066\pi\)
−0.122421 + 0.992478i \(0.539066\pi\)
\(468\) 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 \(-0.488468\pi\)
0.0362220 + 0.999344i \(0.488468\pi\)
\(488\) 15366.8 1.42546
\(489\) 0 0
\(490\) 38781.5 3.57544
\(491\) 9336.64 0.858160 0.429080 0.903266i \(-0.358838\pi\)
0.429080 + 0.903266i \(0.358838\pi\)
\(492\) 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 \(-0.767688\pi\)
−0.745287 + 0.666744i \(0.767688\pi\)
\(500\) 6694.44 0.598769
\(501\) 0 0
\(502\) −5750.80 −0.511296
\(503\) 7494.12 0.664307 0.332153 0.943225i \(-0.392225\pi\)
0.332153 + 0.943225i \(0.392225\pi\)
\(504\) 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 \(-0.175884\pi\)
0.851186 + 0.524864i \(0.175884\pi\)
\(522\) 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 \(-0.454321\pi\)
0.143013 + 0.989721i \(0.454321\pi\)
\(542\) −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 \(-0.520206\pi\)
−0.0634359 + 0.997986i \(0.520206\pi\)
\(564\) 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 \(-0.494197\pi\)
0.0182311 + 0.999834i \(0.494197\pi\)
\(570\) 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 \(-0.400367\pi\)
0.307920 + 0.951412i \(0.400367\pi\)
\(578\) 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 \(-0.147196\pi\)
0.894971 + 0.446124i \(0.147196\pi\)
\(600\) 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 \(-0.651213\pi\)
−0.457382 + 0.889270i \(0.651213\pi\)
\(614\) −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 \(-0.608809\pi\)
−0.335216 + 0.942141i \(0.608809\pi\)
\(620\) −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 \(-0.269954\pi\)
0.661420 + 0.750016i \(0.269954\pi\)
\(632\) −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 \(-0.714977\pi\)
−0.625187 + 0.780475i \(0.714977\pi\)
\(642\) 0 0
\(643\) 9106.14 0.558493 0.279247 0.960219i \(-0.409915\pi\)
0.279247 + 0.960219i \(0.409915\pi\)
\(644\) 19650.2 1.20237
\(645\) 0 0
\(646\) 23064.2 1.40472
\(647\) −15251.4 −0.926732 −0.463366 0.886167i \(-0.653359\pi\)
−0.463366 + 0.886167i \(0.653359\pi\)
\(648\) 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 \(-0.357769\pi\)
0.432110 + 0.901821i \(0.357769\pi\)
\(654\) 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 \(-0.614310\pi\)
−0.351446 + 0.936208i \(0.614310\pi\)
\(660\) 0 0
\(661\) −3696.05 −0.217488 −0.108744 0.994070i \(-0.534683\pi\)
−0.108744 + 0.994070i \(0.534683\pi\)
\(662\) 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 \(-0.614620\pi\)
−0.352359 + 0.935865i \(0.614620\pi\)
\(678\) 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 \(-0.181124\pi\)
0.842431 + 0.538804i \(0.181124\pi\)
\(692\) −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 \(-0.534731\pi\)
−0.108895 + 0.994053i \(0.534731\pi\)
\(702\) 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 \(-0.854242\pi\)
−0.896977 + 0.442077i \(0.854242\pi\)
\(710\) 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 \(-0.441199\pi\)
0.183679 + 0.982986i \(0.441199\pi\)
\(720\) 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 \(-0.337869\pi\)
0.487608 + 0.873062i \(0.337869\pi\)
\(734\) −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 \(-0.153816\pi\)
0.885500 + 0.464639i \(0.153816\pi\)
\(740\) −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 \(-0.438387\pi\)
0.192356 + 0.981325i \(0.438387\pi\)
\(770\) 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 \(-0.298765\pi\)
0.590921 + 0.806730i \(0.298765\pi\)
\(788\) −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 \(-0.364083\pi\)
0.414138 + 0.910214i \(0.364083\pi\)
\(798\) 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 \(-0.436824\pi\)
0.197174 + 0.980368i \(0.436824\pi\)
\(810\) 0 0
\(811\) −20112.2 −0.870818 −0.435409 0.900233i \(-0.643396\pi\)
−0.435409 + 0.900233i \(0.643396\pi\)
\(812\) 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 \(-0.660953\pi\)
−0.484374 + 0.874861i \(0.660953\pi\)
\(854\) 59903.7 2.40031
\(855\) 0 0
\(856\) −25295.9 −1.01004
\(857\) −4872.72 −0.194223 −0.0971114 0.995274i \(-0.530960\pi\)
−0.0971114 + 0.995274i \(0.530960\pi\)
\(858\) 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 \(-0.432867\pi\)
0.209346 + 0.977842i \(0.432867\pi\)
\(878\) −20421.6 −0.784960
\(879\) 0 0
\(880\) −48882.4 −1.87253
\(881\) 7130.10 0.272667 0.136333 0.990663i \(-0.456468\pi\)
0.136333 + 0.990663i \(0.456468\pi\)
\(882\) 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 \(-0.781217\pi\)
−0.772944 + 0.634474i \(0.781217\pi\)
\(888\) 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 \(-0.569197\pi\)
−0.215680 + 0.976464i \(0.569197\pi\)
\(912\) 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 \(-0.777101\pi\)
−0.764677 + 0.644414i \(0.777101\pi\)
\(954\) 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 \(-0.282249\pi\)
0.631964 + 0.774997i \(0.282249\pi\)
\(968\) −84007.4 −2.78936
\(969\) 0 0
\(970\) −18901.6 −0.625665
\(971\) 29954.2 0.989987 0.494993 0.868897i \(-0.335171\pi\)
0.494993 + 0.868897i \(0.335171\pi\)
\(972\) 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.bn.1.1 yes 18
3.2 odd 2 inner 1521.4.a.bn.1.18 yes 18
13.12 even 2 1521.4.a.bm.1.18 yes 18
39.38 odd 2 1521.4.a.bm.1.1 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1521.4.a.bm.1.1 18 39.38 odd 2
1521.4.a.bm.1.18 yes 18 13.12 even 2
1521.4.a.bn.1.1 yes 18 1.1 even 1 trivial
1521.4.a.bn.1.18 yes 18 3.2 odd 2 inner