Properties

Label 1150.4.a.s.1.3
Level $1150$
Weight $4$
Character 1150.1
Self dual yes
Analytic conductor $67.852$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1150,4,Mod(1,1150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1150, 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("1150.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.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: \(67.8521965066\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 38x^{3} + 38x^{2} + 202x + 101 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-5.76842\) of defining polynomial
Character \(\chi\) \(=\) 1150.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +1.80744 q^{3} +4.00000 q^{4} -3.61488 q^{6} +9.19471 q^{7} -8.00000 q^{8} -23.7332 q^{9} +43.1821 q^{11} +7.22976 q^{12} +25.6824 q^{13} -18.3894 q^{14} +16.0000 q^{16} +22.9669 q^{17} +47.4663 q^{18} +101.823 q^{19} +16.6189 q^{21} -86.3643 q^{22} -23.0000 q^{23} -14.4595 q^{24} -51.3647 q^{26} -91.6972 q^{27} +36.7788 q^{28} -146.682 q^{29} +23.9712 q^{31} -32.0000 q^{32} +78.0491 q^{33} -45.9339 q^{34} -94.9326 q^{36} +294.572 q^{37} -203.646 q^{38} +46.4193 q^{39} -127.823 q^{41} -33.2378 q^{42} -502.072 q^{43} +172.729 q^{44} +46.0000 q^{46} +607.879 q^{47} +28.9190 q^{48} -258.457 q^{49} +41.5114 q^{51} +102.729 q^{52} +674.254 q^{53} +183.394 q^{54} -73.5577 q^{56} +184.039 q^{57} +293.365 q^{58} +14.8056 q^{59} -478.549 q^{61} -47.9424 q^{62} -218.220 q^{63} +64.0000 q^{64} -156.098 q^{66} +458.422 q^{67} +91.8678 q^{68} -41.5711 q^{69} +1012.83 q^{71} +189.865 q^{72} -750.521 q^{73} -589.144 q^{74} +407.292 q^{76} +397.047 q^{77} -92.8387 q^{78} -641.648 q^{79} +475.058 q^{81} +255.645 q^{82} +900.187 q^{83} +66.4756 q^{84} +1004.14 q^{86} -265.120 q^{87} -345.457 q^{88} -923.770 q^{89} +236.142 q^{91} -92.0000 q^{92} +43.3265 q^{93} -1215.76 q^{94} -57.8381 q^{96} +168.230 q^{97} +516.915 q^{98} -1024.85 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 10 q^{2} + 12 q^{3} + 20 q^{4} - 24 q^{6} - 24 q^{7} - 40 q^{8} + 11 q^{9} - 54 q^{11} + 48 q^{12} + 36 q^{13} + 48 q^{14} + 80 q^{16} + 132 q^{17} - 22 q^{18} - 50 q^{19} - 158 q^{21} + 108 q^{22}+ \cdots - 1740 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 1.80744 0.347842 0.173921 0.984760i \(-0.444356\pi\)
0.173921 + 0.984760i \(0.444356\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) −3.61488 −0.245961
\(7\) 9.19471 0.496468 0.248234 0.968700i \(-0.420150\pi\)
0.248234 + 0.968700i \(0.420150\pi\)
\(8\) −8.00000 −0.353553
\(9\) −23.7332 −0.879006
\(10\) 0 0
\(11\) 43.1821 1.18363 0.591814 0.806075i \(-0.298412\pi\)
0.591814 + 0.806075i \(0.298412\pi\)
\(12\) 7.22976 0.173921
\(13\) 25.6824 0.547924 0.273962 0.961741i \(-0.411666\pi\)
0.273962 + 0.961741i \(0.411666\pi\)
\(14\) −18.3894 −0.351056
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 22.9669 0.327665 0.163832 0.986488i \(-0.447614\pi\)
0.163832 + 0.986488i \(0.447614\pi\)
\(18\) 47.4663 0.621551
\(19\) 101.823 1.22946 0.614732 0.788736i \(-0.289264\pi\)
0.614732 + 0.788736i \(0.289264\pi\)
\(20\) 0 0
\(21\) 16.6189 0.172692
\(22\) −86.3643 −0.836951
\(23\) −23.0000 −0.208514
\(24\) −14.4595 −0.122981
\(25\) 0 0
\(26\) −51.3647 −0.387440
\(27\) −91.6972 −0.653597
\(28\) 36.7788 0.248234
\(29\) −146.682 −0.939250 −0.469625 0.882866i \(-0.655611\pi\)
−0.469625 + 0.882866i \(0.655611\pi\)
\(30\) 0 0
\(31\) 23.9712 0.138882 0.0694412 0.997586i \(-0.477878\pi\)
0.0694412 + 0.997586i \(0.477878\pi\)
\(32\) −32.0000 −0.176777
\(33\) 78.0491 0.411715
\(34\) −45.9339 −0.231694
\(35\) 0 0
\(36\) −94.9326 −0.439503
\(37\) 294.572 1.30885 0.654423 0.756129i \(-0.272912\pi\)
0.654423 + 0.756129i \(0.272912\pi\)
\(38\) −203.646 −0.869362
\(39\) 46.4193 0.190591
\(40\) 0 0
\(41\) −127.823 −0.486891 −0.243445 0.969915i \(-0.578278\pi\)
−0.243445 + 0.969915i \(0.578278\pi\)
\(42\) −33.2378 −0.122112
\(43\) −502.072 −1.78059 −0.890293 0.455388i \(-0.849501\pi\)
−0.890293 + 0.455388i \(0.849501\pi\)
\(44\) 172.729 0.591814
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) 607.879 1.88656 0.943279 0.332002i \(-0.107724\pi\)
0.943279 + 0.332002i \(0.107724\pi\)
\(48\) 28.9190 0.0869605
\(49\) −258.457 −0.753520
\(50\) 0 0
\(51\) 41.5114 0.113976
\(52\) 102.729 0.273962
\(53\) 674.254 1.74747 0.873735 0.486401i \(-0.161691\pi\)
0.873735 + 0.486401i \(0.161691\pi\)
\(54\) 183.394 0.462163
\(55\) 0 0
\(56\) −73.5577 −0.175528
\(57\) 184.039 0.427659
\(58\) 293.365 0.664150
\(59\) 14.8056 0.0326700 0.0163350 0.999867i \(-0.494800\pi\)
0.0163350 + 0.999867i \(0.494800\pi\)
\(60\) 0 0
\(61\) −478.549 −1.00446 −0.502229 0.864735i \(-0.667487\pi\)
−0.502229 + 0.864735i \(0.667487\pi\)
\(62\) −47.9424 −0.0982047
\(63\) −218.220 −0.436398
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) −156.098 −0.291127
\(67\) 458.422 0.835898 0.417949 0.908471i \(-0.362749\pi\)
0.417949 + 0.908471i \(0.362749\pi\)
\(68\) 91.8678 0.163832
\(69\) −41.5711 −0.0725301
\(70\) 0 0
\(71\) 1012.83 1.69296 0.846481 0.532419i \(-0.178717\pi\)
0.846481 + 0.532419i \(0.178717\pi\)
\(72\) 189.865 0.310776
\(73\) −750.521 −1.20331 −0.601657 0.798755i \(-0.705492\pi\)
−0.601657 + 0.798755i \(0.705492\pi\)
\(74\) −589.144 −0.925494
\(75\) 0 0
\(76\) 407.292 0.614732
\(77\) 397.047 0.587633
\(78\) −92.8387 −0.134768
\(79\) −641.648 −0.913811 −0.456905 0.889515i \(-0.651042\pi\)
−0.456905 + 0.889515i \(0.651042\pi\)
\(80\) 0 0
\(81\) 475.058 0.651657
\(82\) 255.645 0.344284
\(83\) 900.187 1.19046 0.595231 0.803554i \(-0.297060\pi\)
0.595231 + 0.803554i \(0.297060\pi\)
\(84\) 66.4756 0.0863462
\(85\) 0 0
\(86\) 1004.14 1.25906
\(87\) −265.120 −0.326711
\(88\) −345.457 −0.418476
\(89\) −923.770 −1.10022 −0.550109 0.835093i \(-0.685414\pi\)
−0.550109 + 0.835093i \(0.685414\pi\)
\(90\) 0 0
\(91\) 236.142 0.272026
\(92\) −92.0000 −0.104257
\(93\) 43.3265 0.0483091
\(94\) −1215.76 −1.33400
\(95\) 0 0
\(96\) −57.8381 −0.0614904
\(97\) 168.230 0.176095 0.0880473 0.996116i \(-0.471937\pi\)
0.0880473 + 0.996116i \(0.471937\pi\)
\(98\) 516.915 0.532819
\(99\) −1024.85 −1.04042
\(100\) 0 0
\(101\) −633.410 −0.624026 −0.312013 0.950078i \(-0.601003\pi\)
−0.312013 + 0.950078i \(0.601003\pi\)
\(102\) −83.0228 −0.0805929
\(103\) 1300.63 1.24422 0.622111 0.782929i \(-0.286275\pi\)
0.622111 + 0.782929i \(0.286275\pi\)
\(104\) −205.459 −0.193720
\(105\) 0 0
\(106\) −1348.51 −1.23565
\(107\) −495.620 −0.447789 −0.223894 0.974613i \(-0.571877\pi\)
−0.223894 + 0.974613i \(0.571877\pi\)
\(108\) −366.789 −0.326799
\(109\) −198.210 −0.174175 −0.0870874 0.996201i \(-0.527756\pi\)
−0.0870874 + 0.996201i \(0.527756\pi\)
\(110\) 0 0
\(111\) 532.421 0.455272
\(112\) 147.115 0.124117
\(113\) 868.555 0.723069 0.361534 0.932359i \(-0.382253\pi\)
0.361534 + 0.932359i \(0.382253\pi\)
\(114\) −368.078 −0.302401
\(115\) 0 0
\(116\) −586.730 −0.469625
\(117\) −609.524 −0.481628
\(118\) −29.6113 −0.0231012
\(119\) 211.174 0.162675
\(120\) 0 0
\(121\) 533.696 0.400974
\(122\) 957.098 0.710259
\(123\) −231.032 −0.169361
\(124\) 95.8848 0.0694412
\(125\) 0 0
\(126\) 436.439 0.308580
\(127\) 2152.32 1.50384 0.751918 0.659257i \(-0.229129\pi\)
0.751918 + 0.659257i \(0.229129\pi\)
\(128\) −128.000 −0.0883883
\(129\) −907.464 −0.619363
\(130\) 0 0
\(131\) 1726.29 1.15135 0.575673 0.817680i \(-0.304740\pi\)
0.575673 + 0.817680i \(0.304740\pi\)
\(132\) 312.196 0.205858
\(133\) 936.233 0.610389
\(134\) −916.844 −0.591069
\(135\) 0 0
\(136\) −183.736 −0.115847
\(137\) −1771.09 −1.10448 −0.552242 0.833684i \(-0.686227\pi\)
−0.552242 + 0.833684i \(0.686227\pi\)
\(138\) 83.1423 0.0512865
\(139\) 2789.08 1.70192 0.850959 0.525232i \(-0.176021\pi\)
0.850959 + 0.525232i \(0.176021\pi\)
\(140\) 0 0
\(141\) 1098.70 0.656224
\(142\) −2025.65 −1.19710
\(143\) 1109.02 0.648537
\(144\) −379.731 −0.219751
\(145\) 0 0
\(146\) 1501.04 0.850871
\(147\) −467.146 −0.262106
\(148\) 1178.29 0.654423
\(149\) −2190.05 −1.20413 −0.602067 0.798445i \(-0.705656\pi\)
−0.602067 + 0.798445i \(0.705656\pi\)
\(150\) 0 0
\(151\) −1569.59 −0.845902 −0.422951 0.906153i \(-0.639006\pi\)
−0.422951 + 0.906153i \(0.639006\pi\)
\(152\) −814.584 −0.434681
\(153\) −545.078 −0.288019
\(154\) −794.094 −0.415519
\(155\) 0 0
\(156\) 185.677 0.0952954
\(157\) 2633.86 1.33888 0.669442 0.742864i \(-0.266533\pi\)
0.669442 + 0.742864i \(0.266533\pi\)
\(158\) 1283.30 0.646162
\(159\) 1218.67 0.607844
\(160\) 0 0
\(161\) −211.478 −0.103521
\(162\) −950.116 −0.460791
\(163\) −287.657 −0.138227 −0.0691135 0.997609i \(-0.522017\pi\)
−0.0691135 + 0.997609i \(0.522017\pi\)
\(164\) −511.290 −0.243445
\(165\) 0 0
\(166\) −1800.37 −0.841784
\(167\) 1295.54 0.600310 0.300155 0.953890i \(-0.402962\pi\)
0.300155 + 0.953890i \(0.402962\pi\)
\(168\) −132.951 −0.0610560
\(169\) −1537.42 −0.699780
\(170\) 0 0
\(171\) −2416.58 −1.08071
\(172\) −2008.29 −0.890293
\(173\) −1313.46 −0.577230 −0.288615 0.957445i \(-0.593195\pi\)
−0.288615 + 0.957445i \(0.593195\pi\)
\(174\) 530.240 0.231019
\(175\) 0 0
\(176\) 690.914 0.295907
\(177\) 26.7603 0.0113640
\(178\) 1847.54 0.777972
\(179\) 1916.37 0.800204 0.400102 0.916471i \(-0.368975\pi\)
0.400102 + 0.916471i \(0.368975\pi\)
\(180\) 0 0
\(181\) 4443.32 1.82469 0.912345 0.409421i \(-0.134269\pi\)
0.912345 + 0.409421i \(0.134269\pi\)
\(182\) −472.284 −0.192352
\(183\) −864.949 −0.349393
\(184\) 184.000 0.0737210
\(185\) 0 0
\(186\) −86.6530 −0.0341597
\(187\) 991.762 0.387833
\(188\) 2431.51 0.943279
\(189\) −843.129 −0.324490
\(190\) 0 0
\(191\) 1903.54 0.721129 0.360564 0.932734i \(-0.382584\pi\)
0.360564 + 0.932734i \(0.382584\pi\)
\(192\) 115.676 0.0434803
\(193\) 3417.60 1.27463 0.637317 0.770602i \(-0.280044\pi\)
0.637317 + 0.770602i \(0.280044\pi\)
\(194\) −336.460 −0.124518
\(195\) 0 0
\(196\) −1033.83 −0.376760
\(197\) 1609.57 0.582117 0.291059 0.956705i \(-0.405993\pi\)
0.291059 + 0.956705i \(0.405993\pi\)
\(198\) 2049.70 0.735685
\(199\) −4518.91 −1.60973 −0.804866 0.593456i \(-0.797763\pi\)
−0.804866 + 0.593456i \(0.797763\pi\)
\(200\) 0 0
\(201\) 828.570 0.290760
\(202\) 1266.82 0.441253
\(203\) −1348.70 −0.466307
\(204\) 166.046 0.0569878
\(205\) 0 0
\(206\) −2601.26 −0.879798
\(207\) 545.863 0.183285
\(208\) 410.918 0.136981
\(209\) 4396.94 1.45523
\(210\) 0 0
\(211\) 5408.64 1.76467 0.882337 0.470618i \(-0.155969\pi\)
0.882337 + 0.470618i \(0.155969\pi\)
\(212\) 2697.02 0.873735
\(213\) 1830.62 0.588883
\(214\) 991.240 0.316634
\(215\) 0 0
\(216\) 733.577 0.231082
\(217\) 220.408 0.0689506
\(218\) 396.420 0.123160
\(219\) −1356.52 −0.418563
\(220\) 0 0
\(221\) 589.846 0.179535
\(222\) −1064.84 −0.321926
\(223\) 5812.74 1.74552 0.872758 0.488154i \(-0.162329\pi\)
0.872758 + 0.488154i \(0.162329\pi\)
\(224\) −294.231 −0.0877639
\(225\) 0 0
\(226\) −1737.11 −0.511287
\(227\) 2033.30 0.594514 0.297257 0.954798i \(-0.403928\pi\)
0.297257 + 0.954798i \(0.403928\pi\)
\(228\) 736.156 0.213830
\(229\) −435.903 −0.125787 −0.0628936 0.998020i \(-0.520033\pi\)
−0.0628936 + 0.998020i \(0.520033\pi\)
\(230\) 0 0
\(231\) 717.639 0.204403
\(232\) 1173.46 0.332075
\(233\) −1582.05 −0.444822 −0.222411 0.974953i \(-0.571393\pi\)
−0.222411 + 0.974953i \(0.571393\pi\)
\(234\) 1219.05 0.340562
\(235\) 0 0
\(236\) 59.2226 0.0163350
\(237\) −1159.74 −0.317862
\(238\) −422.349 −0.115029
\(239\) −1732.60 −0.468923 −0.234461 0.972125i \(-0.575333\pi\)
−0.234461 + 0.972125i \(0.575333\pi\)
\(240\) 0 0
\(241\) −4441.73 −1.18721 −0.593604 0.804757i \(-0.702296\pi\)
−0.593604 + 0.804757i \(0.702296\pi\)
\(242\) −1067.39 −0.283531
\(243\) 3334.46 0.880271
\(244\) −1914.20 −0.502229
\(245\) 0 0
\(246\) 462.063 0.119756
\(247\) 2615.06 0.673652
\(248\) −191.770 −0.0491023
\(249\) 1627.03 0.414093
\(250\) 0 0
\(251\) 4714.54 1.18557 0.592787 0.805359i \(-0.298028\pi\)
0.592787 + 0.805359i \(0.298028\pi\)
\(252\) −872.878 −0.218199
\(253\) −993.189 −0.246803
\(254\) −4304.63 −1.06337
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 4229.41 1.02655 0.513275 0.858224i \(-0.328432\pi\)
0.513275 + 0.858224i \(0.328432\pi\)
\(258\) 1814.93 0.437956
\(259\) 2708.50 0.649800
\(260\) 0 0
\(261\) 3481.24 0.825606
\(262\) −3452.57 −0.814125
\(263\) 4848.80 1.13684 0.568422 0.822737i \(-0.307554\pi\)
0.568422 + 0.822737i \(0.307554\pi\)
\(264\) −624.393 −0.145563
\(265\) 0 0
\(266\) −1872.47 −0.431610
\(267\) −1669.66 −0.382702
\(268\) 1833.69 0.417949
\(269\) 4603.64 1.04345 0.521726 0.853113i \(-0.325288\pi\)
0.521726 + 0.853113i \(0.325288\pi\)
\(270\) 0 0
\(271\) −1521.31 −0.341008 −0.170504 0.985357i \(-0.554540\pi\)
−0.170504 + 0.985357i \(0.554540\pi\)
\(272\) 367.471 0.0819162
\(273\) 426.812 0.0946222
\(274\) 3542.18 0.780988
\(275\) 0 0
\(276\) −166.285 −0.0362650
\(277\) −3293.74 −0.714446 −0.357223 0.934019i \(-0.616276\pi\)
−0.357223 + 0.934019i \(0.616276\pi\)
\(278\) −5578.16 −1.20344
\(279\) −568.912 −0.122078
\(280\) 0 0
\(281\) 8857.86 1.88048 0.940242 0.340508i \(-0.110599\pi\)
0.940242 + 0.340508i \(0.110599\pi\)
\(282\) −2197.41 −0.464020
\(283\) −5470.44 −1.14906 −0.574530 0.818483i \(-0.694815\pi\)
−0.574530 + 0.818483i \(0.694815\pi\)
\(284\) 4051.30 0.846481
\(285\) 0 0
\(286\) −2218.04 −0.458585
\(287\) −1175.29 −0.241726
\(288\) 759.461 0.155388
\(289\) −4385.52 −0.892636
\(290\) 0 0
\(291\) 304.066 0.0612531
\(292\) −3002.09 −0.601657
\(293\) 629.599 0.125534 0.0627672 0.998028i \(-0.480007\pi\)
0.0627672 + 0.998028i \(0.480007\pi\)
\(294\) 934.292 0.185337
\(295\) 0 0
\(296\) −2356.57 −0.462747
\(297\) −3959.68 −0.773616
\(298\) 4380.10 0.851452
\(299\) −590.694 −0.114250
\(300\) 0 0
\(301\) −4616.40 −0.884004
\(302\) 3139.17 0.598143
\(303\) −1144.85 −0.217062
\(304\) 1629.17 0.307366
\(305\) 0 0
\(306\) 1090.16 0.203660
\(307\) −791.929 −0.147224 −0.0736120 0.997287i \(-0.523453\pi\)
−0.0736120 + 0.997287i \(0.523453\pi\)
\(308\) 1588.19 0.293816
\(309\) 2350.81 0.432793
\(310\) 0 0
\(311\) 2109.26 0.384582 0.192291 0.981338i \(-0.438408\pi\)
0.192291 + 0.981338i \(0.438408\pi\)
\(312\) −371.355 −0.0673840
\(313\) −207.600 −0.0374896 −0.0187448 0.999824i \(-0.505967\pi\)
−0.0187448 + 0.999824i \(0.505967\pi\)
\(314\) −5267.72 −0.946734
\(315\) 0 0
\(316\) −2566.59 −0.456905
\(317\) −4013.96 −0.711188 −0.355594 0.934640i \(-0.615721\pi\)
−0.355594 + 0.934640i \(0.615721\pi\)
\(318\) −2437.35 −0.429811
\(319\) −6334.06 −1.11172
\(320\) 0 0
\(321\) −895.803 −0.155760
\(322\) 422.957 0.0732002
\(323\) 2338.56 0.402852
\(324\) 1900.23 0.325829
\(325\) 0 0
\(326\) 575.313 0.0977413
\(327\) −358.252 −0.0605853
\(328\) 1022.58 0.172142
\(329\) 5589.27 0.936615
\(330\) 0 0
\(331\) 1196.79 0.198736 0.0993678 0.995051i \(-0.468318\pi\)
0.0993678 + 0.995051i \(0.468318\pi\)
\(332\) 3600.75 0.595231
\(333\) −6991.12 −1.15048
\(334\) −2591.08 −0.424484
\(335\) 0 0
\(336\) 265.902 0.0431731
\(337\) 3245.42 0.524597 0.262299 0.964987i \(-0.415519\pi\)
0.262299 + 0.964987i \(0.415519\pi\)
\(338\) 3074.83 0.494819
\(339\) 1569.86 0.251514
\(340\) 0 0
\(341\) 1035.13 0.164385
\(342\) 4833.16 0.764174
\(343\) −5530.23 −0.870566
\(344\) 4016.57 0.629532
\(345\) 0 0
\(346\) 2626.93 0.408163
\(347\) 10510.0 1.62596 0.812981 0.582290i \(-0.197843\pi\)
0.812981 + 0.582290i \(0.197843\pi\)
\(348\) −1060.48 −0.163355
\(349\) 6150.02 0.943276 0.471638 0.881792i \(-0.343663\pi\)
0.471638 + 0.881792i \(0.343663\pi\)
\(350\) 0 0
\(351\) −2355.00 −0.358121
\(352\) −1381.83 −0.209238
\(353\) 1123.35 0.169376 0.0846882 0.996408i \(-0.473011\pi\)
0.0846882 + 0.996408i \(0.473011\pi\)
\(354\) −53.5206 −0.00803556
\(355\) 0 0
\(356\) −3695.08 −0.550109
\(357\) 381.685 0.0565852
\(358\) −3832.75 −0.565829
\(359\) 2209.46 0.324821 0.162411 0.986723i \(-0.448073\pi\)
0.162411 + 0.986723i \(0.448073\pi\)
\(360\) 0 0
\(361\) 3508.93 0.511581
\(362\) −8886.63 −1.29025
\(363\) 964.624 0.139476
\(364\) 944.568 0.136013
\(365\) 0 0
\(366\) 1729.90 0.247058
\(367\) −9017.63 −1.28261 −0.641303 0.767287i \(-0.721606\pi\)
−0.641303 + 0.767287i \(0.721606\pi\)
\(368\) −368.000 −0.0521286
\(369\) 3033.63 0.427980
\(370\) 0 0
\(371\) 6199.57 0.867563
\(372\) 173.306 0.0241546
\(373\) 2000.52 0.277702 0.138851 0.990313i \(-0.455659\pi\)
0.138851 + 0.990313i \(0.455659\pi\)
\(374\) −1983.52 −0.274239
\(375\) 0 0
\(376\) −4863.03 −0.666999
\(377\) −3767.15 −0.514637
\(378\) 1686.26 0.229449
\(379\) 5404.84 0.732528 0.366264 0.930511i \(-0.380637\pi\)
0.366264 + 0.930511i \(0.380637\pi\)
\(380\) 0 0
\(381\) 3890.18 0.523097
\(382\) −3807.09 −0.509915
\(383\) −9355.83 −1.24820 −0.624100 0.781345i \(-0.714534\pi\)
−0.624100 + 0.781345i \(0.714534\pi\)
\(384\) −231.352 −0.0307452
\(385\) 0 0
\(386\) −6835.20 −0.901302
\(387\) 11915.7 1.56515
\(388\) 672.920 0.0880473
\(389\) 7771.25 1.01290 0.506450 0.862269i \(-0.330958\pi\)
0.506450 + 0.862269i \(0.330958\pi\)
\(390\) 0 0
\(391\) −528.240 −0.0683229
\(392\) 2067.66 0.266409
\(393\) 3120.16 0.400487
\(394\) −3219.14 −0.411619
\(395\) 0 0
\(396\) −4099.39 −0.520208
\(397\) −2509.29 −0.317223 −0.158612 0.987341i \(-0.550702\pi\)
−0.158612 + 0.987341i \(0.550702\pi\)
\(398\) 9037.81 1.13825
\(399\) 1692.19 0.212319
\(400\) 0 0
\(401\) 1502.71 0.187136 0.0935681 0.995613i \(-0.470173\pi\)
0.0935681 + 0.995613i \(0.470173\pi\)
\(402\) −1657.14 −0.205599
\(403\) 615.637 0.0760969
\(404\) −2533.64 −0.312013
\(405\) 0 0
\(406\) 2697.41 0.329729
\(407\) 12720.2 1.54919
\(408\) −332.091 −0.0402965
\(409\) 6281.54 0.759419 0.379709 0.925106i \(-0.376024\pi\)
0.379709 + 0.925106i \(0.376024\pi\)
\(410\) 0 0
\(411\) −3201.14 −0.384186
\(412\) 5202.52 0.622111
\(413\) 136.134 0.0162196
\(414\) −1091.73 −0.129602
\(415\) 0 0
\(416\) −821.836 −0.0968601
\(417\) 5041.09 0.591999
\(418\) −8793.87 −1.02900
\(419\) −5437.48 −0.633982 −0.316991 0.948429i \(-0.602672\pi\)
−0.316991 + 0.948429i \(0.602672\pi\)
\(420\) 0 0
\(421\) 8424.12 0.975217 0.487608 0.873062i \(-0.337869\pi\)
0.487608 + 0.873062i \(0.337869\pi\)
\(422\) −10817.3 −1.24781
\(423\) −14426.9 −1.65830
\(424\) −5394.03 −0.617824
\(425\) 0 0
\(426\) −3661.24 −0.416403
\(427\) −4400.12 −0.498681
\(428\) −1982.48 −0.223894
\(429\) 2004.49 0.225589
\(430\) 0 0
\(431\) −2280.19 −0.254832 −0.127416 0.991849i \(-0.540668\pi\)
−0.127416 + 0.991849i \(0.540668\pi\)
\(432\) −1467.15 −0.163399
\(433\) −9676.90 −1.07400 −0.537000 0.843582i \(-0.680443\pi\)
−0.537000 + 0.843582i \(0.680443\pi\)
\(434\) −440.816 −0.0487554
\(435\) 0 0
\(436\) −792.839 −0.0870874
\(437\) −2341.93 −0.256361
\(438\) 2713.04 0.295969
\(439\) −10267.6 −1.11628 −0.558138 0.829748i \(-0.688484\pi\)
−0.558138 + 0.829748i \(0.688484\pi\)
\(440\) 0 0
\(441\) 6134.01 0.662348
\(442\) −1179.69 −0.126951
\(443\) 2947.70 0.316139 0.158069 0.987428i \(-0.449473\pi\)
0.158069 + 0.987428i \(0.449473\pi\)
\(444\) 2129.68 0.227636
\(445\) 0 0
\(446\) −11625.5 −1.23427
\(447\) −3958.39 −0.418849
\(448\) 588.462 0.0620585
\(449\) −14322.2 −1.50536 −0.752682 0.658385i \(-0.771240\pi\)
−0.752682 + 0.658385i \(0.771240\pi\)
\(450\) 0 0
\(451\) −5519.65 −0.576297
\(452\) 3474.22 0.361534
\(453\) −2836.94 −0.294240
\(454\) −4066.59 −0.420385
\(455\) 0 0
\(456\) −1472.31 −0.151200
\(457\) −13404.4 −1.37206 −0.686029 0.727574i \(-0.740648\pi\)
−0.686029 + 0.727574i \(0.740648\pi\)
\(458\) 871.805 0.0889449
\(459\) −2106.00 −0.214161
\(460\) 0 0
\(461\) −16928.6 −1.71029 −0.855144 0.518390i \(-0.826532\pi\)
−0.855144 + 0.518390i \(0.826532\pi\)
\(462\) −1435.28 −0.144535
\(463\) −13109.3 −1.31585 −0.657925 0.753083i \(-0.728566\pi\)
−0.657925 + 0.753083i \(0.728566\pi\)
\(464\) −2346.92 −0.234812
\(465\) 0 0
\(466\) 3164.10 0.314537
\(467\) 4676.02 0.463341 0.231671 0.972794i \(-0.425581\pi\)
0.231671 + 0.972794i \(0.425581\pi\)
\(468\) −2438.09 −0.240814
\(469\) 4215.06 0.414996
\(470\) 0 0
\(471\) 4760.54 0.465720
\(472\) −118.445 −0.0115506
\(473\) −21680.5 −2.10755
\(474\) 2319.48 0.224762
\(475\) 0 0
\(476\) 844.698 0.0813375
\(477\) −16002.2 −1.53604
\(478\) 3465.20 0.331579
\(479\) −13518.0 −1.28947 −0.644734 0.764407i \(-0.723032\pi\)
−0.644734 + 0.764407i \(0.723032\pi\)
\(480\) 0 0
\(481\) 7565.30 0.717148
\(482\) 8883.47 0.839483
\(483\) −382.234 −0.0360088
\(484\) 2134.79 0.200487
\(485\) 0 0
\(486\) −6668.92 −0.622446
\(487\) −14914.7 −1.38778 −0.693889 0.720082i \(-0.744104\pi\)
−0.693889 + 0.720082i \(0.744104\pi\)
\(488\) 3828.39 0.355129
\(489\) −519.922 −0.0480812
\(490\) 0 0
\(491\) −3706.99 −0.340722 −0.170361 0.985382i \(-0.554493\pi\)
−0.170361 + 0.985382i \(0.554493\pi\)
\(492\) −924.126 −0.0846806
\(493\) −3368.85 −0.307759
\(494\) −5230.11 −0.476344
\(495\) 0 0
\(496\) 383.539 0.0347206
\(497\) 9312.64 0.840501
\(498\) −3254.07 −0.292808
\(499\) 13037.0 1.16957 0.584784 0.811189i \(-0.301179\pi\)
0.584784 + 0.811189i \(0.301179\pi\)
\(500\) 0 0
\(501\) 2341.61 0.208813
\(502\) −9429.08 −0.838328
\(503\) −2976.34 −0.263834 −0.131917 0.991261i \(-0.542113\pi\)
−0.131917 + 0.991261i \(0.542113\pi\)
\(504\) 1745.76 0.154290
\(505\) 0 0
\(506\) 1986.38 0.174516
\(507\) −2778.79 −0.243413
\(508\) 8609.26 0.751918
\(509\) 20818.7 1.81292 0.906458 0.422297i \(-0.138776\pi\)
0.906458 + 0.422297i \(0.138776\pi\)
\(510\) 0 0
\(511\) −6900.83 −0.597406
\(512\) −512.000 −0.0441942
\(513\) −9336.88 −0.803574
\(514\) −8458.82 −0.725881
\(515\) 0 0
\(516\) −3629.86 −0.309681
\(517\) 26249.5 2.23298
\(518\) −5417.00 −0.459478
\(519\) −2374.01 −0.200785
\(520\) 0 0
\(521\) −10439.5 −0.877855 −0.438927 0.898522i \(-0.644641\pi\)
−0.438927 + 0.898522i \(0.644641\pi\)
\(522\) −6962.48 −0.583792
\(523\) −22364.9 −1.86988 −0.934942 0.354801i \(-0.884549\pi\)
−0.934942 + 0.354801i \(0.884549\pi\)
\(524\) 6905.15 0.575673
\(525\) 0 0
\(526\) −9697.60 −0.803870
\(527\) 550.545 0.0455069
\(528\) 1248.79 0.102929
\(529\) 529.000 0.0434783
\(530\) 0 0
\(531\) −351.385 −0.0287171
\(532\) 3744.93 0.305194
\(533\) −3282.78 −0.266779
\(534\) 3339.32 0.270611
\(535\) 0 0
\(536\) −3667.38 −0.295535
\(537\) 3463.73 0.278344
\(538\) −9207.27 −0.737832
\(539\) −11160.7 −0.891887
\(540\) 0 0
\(541\) −22462.5 −1.78510 −0.892550 0.450949i \(-0.851086\pi\)
−0.892550 + 0.450949i \(0.851086\pi\)
\(542\) 3042.63 0.241129
\(543\) 8031.03 0.634704
\(544\) −734.942 −0.0579235
\(545\) 0 0
\(546\) −853.625 −0.0669080
\(547\) −4367.35 −0.341379 −0.170690 0.985325i \(-0.554600\pi\)
−0.170690 + 0.985325i \(0.554600\pi\)
\(548\) −7084.35 −0.552242
\(549\) 11357.5 0.882924
\(550\) 0 0
\(551\) −14935.7 −1.15477
\(552\) 332.569 0.0256433
\(553\) −5899.77 −0.453677
\(554\) 6587.47 0.505189
\(555\) 0 0
\(556\) 11156.3 0.850959
\(557\) 13805.9 1.05022 0.525110 0.851034i \(-0.324024\pi\)
0.525110 + 0.851034i \(0.324024\pi\)
\(558\) 1137.82 0.0863225
\(559\) −12894.4 −0.975625
\(560\) 0 0
\(561\) 1792.55 0.134905
\(562\) −17715.7 −1.32970
\(563\) −8149.77 −0.610075 −0.305037 0.952340i \(-0.598669\pi\)
−0.305037 + 0.952340i \(0.598669\pi\)
\(564\) 4394.82 0.328112
\(565\) 0 0
\(566\) 10940.9 0.812508
\(567\) 4368.02 0.323527
\(568\) −8102.60 −0.598552
\(569\) −11539.9 −0.850226 −0.425113 0.905140i \(-0.639766\pi\)
−0.425113 + 0.905140i \(0.639766\pi\)
\(570\) 0 0
\(571\) −2436.53 −0.178573 −0.0892867 0.996006i \(-0.528459\pi\)
−0.0892867 + 0.996006i \(0.528459\pi\)
\(572\) 4436.08 0.324269
\(573\) 3440.54 0.250839
\(574\) 2350.58 0.170926
\(575\) 0 0
\(576\) −1518.92 −0.109876
\(577\) 6965.59 0.502567 0.251284 0.967913i \(-0.419147\pi\)
0.251284 + 0.967913i \(0.419147\pi\)
\(578\) 8771.04 0.631189
\(579\) 6177.11 0.443371
\(580\) 0 0
\(581\) 8276.96 0.591026
\(582\) −608.132 −0.0433125
\(583\) 29115.7 2.06835
\(584\) 6004.17 0.425435
\(585\) 0 0
\(586\) −1259.20 −0.0887662
\(587\) −3080.79 −0.216623 −0.108311 0.994117i \(-0.534544\pi\)
−0.108311 + 0.994117i \(0.534544\pi\)
\(588\) −1868.58 −0.131053
\(589\) 2440.82 0.170751
\(590\) 0 0
\(591\) 2909.20 0.202485
\(592\) 4713.15 0.327211
\(593\) 13741.0 0.951563 0.475782 0.879564i \(-0.342165\pi\)
0.475782 + 0.879564i \(0.342165\pi\)
\(594\) 7919.36 0.547029
\(595\) 0 0
\(596\) −8760.21 −0.602067
\(597\) −8167.65 −0.559933
\(598\) 1181.39 0.0807869
\(599\) 24131.0 1.64602 0.823011 0.568026i \(-0.192293\pi\)
0.823011 + 0.568026i \(0.192293\pi\)
\(600\) 0 0
\(601\) −18108.3 −1.22904 −0.614520 0.788901i \(-0.710650\pi\)
−0.614520 + 0.788901i \(0.710650\pi\)
\(602\) 9232.81 0.625085
\(603\) −10879.8 −0.734759
\(604\) −6278.35 −0.422951
\(605\) 0 0
\(606\) 2289.70 0.153486
\(607\) −10257.1 −0.685873 −0.342936 0.939359i \(-0.611422\pi\)
−0.342936 + 0.939359i \(0.611422\pi\)
\(608\) −3258.34 −0.217340
\(609\) −2437.70 −0.162201
\(610\) 0 0
\(611\) 15611.8 1.03369
\(612\) −2180.31 −0.144010
\(613\) 2710.49 0.178590 0.0892951 0.996005i \(-0.471539\pi\)
0.0892951 + 0.996005i \(0.471539\pi\)
\(614\) 1583.86 0.104103
\(615\) 0 0
\(616\) −3176.38 −0.207760
\(617\) −20517.7 −1.33875 −0.669377 0.742923i \(-0.733439\pi\)
−0.669377 + 0.742923i \(0.733439\pi\)
\(618\) −4701.62 −0.306031
\(619\) −5220.67 −0.338992 −0.169496 0.985531i \(-0.554214\pi\)
−0.169496 + 0.985531i \(0.554214\pi\)
\(620\) 0 0
\(621\) 2109.03 0.136284
\(622\) −4218.52 −0.271941
\(623\) −8493.80 −0.546223
\(624\) 742.710 0.0476477
\(625\) 0 0
\(626\) 415.200 0.0265092
\(627\) 7947.20 0.506189
\(628\) 10535.4 0.669442
\(629\) 6765.41 0.428863
\(630\) 0 0
\(631\) −19100.5 −1.20504 −0.602520 0.798103i \(-0.705837\pi\)
−0.602520 + 0.798103i \(0.705837\pi\)
\(632\) 5133.18 0.323081
\(633\) 9775.80 0.613828
\(634\) 8027.93 0.502886
\(635\) 0 0
\(636\) 4874.70 0.303922
\(637\) −6637.79 −0.412871
\(638\) 12668.1 0.786106
\(639\) −24037.5 −1.48812
\(640\) 0 0
\(641\) 15088.0 0.929701 0.464851 0.885389i \(-0.346108\pi\)
0.464851 + 0.885389i \(0.346108\pi\)
\(642\) 1791.61 0.110139
\(643\) −1568.01 −0.0961687 −0.0480843 0.998843i \(-0.515312\pi\)
−0.0480843 + 0.998843i \(0.515312\pi\)
\(644\) −845.913 −0.0517603
\(645\) 0 0
\(646\) −4677.13 −0.284859
\(647\) 15035.4 0.913603 0.456801 0.889569i \(-0.348995\pi\)
0.456801 + 0.889569i \(0.348995\pi\)
\(648\) −3800.47 −0.230396
\(649\) 639.339 0.0386691
\(650\) 0 0
\(651\) 398.375 0.0239839
\(652\) −1150.63 −0.0691135
\(653\) 22403.7 1.34261 0.671307 0.741180i \(-0.265733\pi\)
0.671307 + 0.741180i \(0.265733\pi\)
\(654\) 716.505 0.0428403
\(655\) 0 0
\(656\) −2045.16 −0.121723
\(657\) 17812.2 1.05772
\(658\) −11178.5 −0.662287
\(659\) 890.918 0.0526635 0.0263317 0.999653i \(-0.491617\pi\)
0.0263317 + 0.999653i \(0.491617\pi\)
\(660\) 0 0
\(661\) −28636.0 −1.68504 −0.842519 0.538667i \(-0.818928\pi\)
−0.842519 + 0.538667i \(0.818928\pi\)
\(662\) −2393.58 −0.140527
\(663\) 1066.11 0.0624499
\(664\) −7201.50 −0.420892
\(665\) 0 0
\(666\) 13982.2 0.813515
\(667\) 3373.70 0.195847
\(668\) 5182.16 0.300155
\(669\) 10506.2 0.607164
\(670\) 0 0
\(671\) −20664.8 −1.18890
\(672\) −531.805 −0.0305280
\(673\) −15056.8 −0.862405 −0.431203 0.902255i \(-0.641911\pi\)
−0.431203 + 0.902255i \(0.641911\pi\)
\(674\) −6490.84 −0.370946
\(675\) 0 0
\(676\) −6149.66 −0.349890
\(677\) 1612.72 0.0915539 0.0457770 0.998952i \(-0.485424\pi\)
0.0457770 + 0.998952i \(0.485424\pi\)
\(678\) −3139.72 −0.177847
\(679\) 1546.83 0.0874253
\(680\) 0 0
\(681\) 3675.06 0.206797
\(682\) −2070.25 −0.116238
\(683\) −20264.3 −1.13527 −0.567637 0.823279i \(-0.692142\pi\)
−0.567637 + 0.823279i \(0.692142\pi\)
\(684\) −9666.33 −0.540353
\(685\) 0 0
\(686\) 11060.5 0.615583
\(687\) −787.868 −0.0437540
\(688\) −8033.15 −0.445147
\(689\) 17316.4 0.957481
\(690\) 0 0
\(691\) −7628.56 −0.419977 −0.209988 0.977704i \(-0.567343\pi\)
−0.209988 + 0.977704i \(0.567343\pi\)
\(692\) −5253.85 −0.288615
\(693\) −9423.19 −0.516533
\(694\) −21020.1 −1.14973
\(695\) 0 0
\(696\) 2120.96 0.115510
\(697\) −2935.69 −0.159537
\(698\) −12300.0 −0.666997
\(699\) −2859.46 −0.154728
\(700\) 0 0
\(701\) 18206.5 0.980953 0.490477 0.871454i \(-0.336823\pi\)
0.490477 + 0.871454i \(0.336823\pi\)
\(702\) 4710.00 0.253230
\(703\) 29994.2 1.60918
\(704\) 2763.66 0.147953
\(705\) 0 0
\(706\) −2246.70 −0.119767
\(707\) −5824.02 −0.309809
\(708\) 107.041 0.00568200
\(709\) 14793.4 0.783607 0.391804 0.920049i \(-0.371851\pi\)
0.391804 + 0.920049i \(0.371851\pi\)
\(710\) 0 0
\(711\) 15228.3 0.803245
\(712\) 7390.16 0.388986
\(713\) −551.337 −0.0289590
\(714\) −763.370 −0.0400118
\(715\) 0 0
\(716\) 7665.49 0.400102
\(717\) −3131.57 −0.163111
\(718\) −4418.92 −0.229683
\(719\) 32215.9 1.67100 0.835500 0.549490i \(-0.185178\pi\)
0.835500 + 0.549490i \(0.185178\pi\)
\(720\) 0 0
\(721\) 11958.9 0.617716
\(722\) −7017.86 −0.361742
\(723\) −8028.17 −0.412961
\(724\) 17773.3 0.912345
\(725\) 0 0
\(726\) −1929.25 −0.0986242
\(727\) 1853.37 0.0945497 0.0472748 0.998882i \(-0.484946\pi\)
0.0472748 + 0.998882i \(0.484946\pi\)
\(728\) −1889.14 −0.0961759
\(729\) −6799.73 −0.345462
\(730\) 0 0
\(731\) −11531.1 −0.583436
\(732\) −3459.80 −0.174696
\(733\) 6958.47 0.350637 0.175319 0.984512i \(-0.443904\pi\)
0.175319 + 0.984512i \(0.443904\pi\)
\(734\) 18035.3 0.906940
\(735\) 0 0
\(736\) 736.000 0.0368605
\(737\) 19795.6 0.989392
\(738\) −6067.26 −0.302628
\(739\) 6469.54 0.322038 0.161019 0.986951i \(-0.448522\pi\)
0.161019 + 0.986951i \(0.448522\pi\)
\(740\) 0 0
\(741\) 4726.56 0.234325
\(742\) −12399.1 −0.613460
\(743\) 6992.55 0.345265 0.172632 0.984986i \(-0.444773\pi\)
0.172632 + 0.984986i \(0.444773\pi\)
\(744\) −346.612 −0.0170799
\(745\) 0 0
\(746\) −4001.04 −0.196365
\(747\) −21364.3 −1.04642
\(748\) 3967.05 0.193917
\(749\) −4557.08 −0.222313
\(750\) 0 0
\(751\) 24595.2 1.19506 0.597531 0.801846i \(-0.296148\pi\)
0.597531 + 0.801846i \(0.296148\pi\)
\(752\) 9726.06 0.471639
\(753\) 8521.25 0.412393
\(754\) 7534.30 0.363903
\(755\) 0 0
\(756\) −3372.52 −0.162245
\(757\) −35778.4 −1.71782 −0.858909 0.512128i \(-0.828857\pi\)
−0.858909 + 0.512128i \(0.828857\pi\)
\(758\) −10809.7 −0.517975
\(759\) −1795.13 −0.0858486
\(760\) 0 0
\(761\) −39181.2 −1.86638 −0.933191 0.359380i \(-0.882988\pi\)
−0.933191 + 0.359380i \(0.882988\pi\)
\(762\) −7780.37 −0.369886
\(763\) −1822.48 −0.0864722
\(764\) 7614.18 0.360564
\(765\) 0 0
\(766\) 18711.7 0.882610
\(767\) 380.244 0.0179007
\(768\) 462.705 0.0217401
\(769\) 9455.39 0.443394 0.221697 0.975116i \(-0.428840\pi\)
0.221697 + 0.975116i \(0.428840\pi\)
\(770\) 0 0
\(771\) 7644.41 0.357077
\(772\) 13670.4 0.637317
\(773\) 5683.75 0.264464 0.132232 0.991219i \(-0.457786\pi\)
0.132232 + 0.991219i \(0.457786\pi\)
\(774\) −23831.5 −1.10673
\(775\) 0 0
\(776\) −1345.84 −0.0622588
\(777\) 4895.46 0.226028
\(778\) −15542.5 −0.716228
\(779\) −13015.3 −0.598615
\(780\) 0 0
\(781\) 43736.0 2.00384
\(782\) 1056.48 0.0483116
\(783\) 13450.4 0.613891
\(784\) −4135.32 −0.188380
\(785\) 0 0
\(786\) −6240.32 −0.283187
\(787\) 18500.7 0.837966 0.418983 0.907994i \(-0.362387\pi\)
0.418983 + 0.907994i \(0.362387\pi\)
\(788\) 6438.28 0.291059
\(789\) 8763.91 0.395442
\(790\) 0 0
\(791\) 7986.11 0.358980
\(792\) 8198.79 0.367842
\(793\) −12290.3 −0.550366
\(794\) 5018.58 0.224311
\(795\) 0 0
\(796\) −18075.6 −0.804866
\(797\) −30464.4 −1.35396 −0.676979 0.736002i \(-0.736711\pi\)
−0.676979 + 0.736002i \(0.736711\pi\)
\(798\) −3384.37 −0.150132
\(799\) 13961.1 0.618159
\(800\) 0 0
\(801\) 21924.0 0.967099
\(802\) −3005.42 −0.132325
\(803\) −32409.1 −1.42427
\(804\) 3314.28 0.145380
\(805\) 0 0
\(806\) −1231.27 −0.0538087
\(807\) 8320.80 0.362957
\(808\) 5067.28 0.220626
\(809\) −20168.5 −0.876500 −0.438250 0.898853i \(-0.644401\pi\)
−0.438250 + 0.898853i \(0.644401\pi\)
\(810\) 0 0
\(811\) 3472.00 0.150331 0.0751655 0.997171i \(-0.476051\pi\)
0.0751655 + 0.997171i \(0.476051\pi\)
\(812\) −5394.81 −0.233154
\(813\) −2749.68 −0.118617
\(814\) −25440.5 −1.09544
\(815\) 0 0
\(816\) 664.182 0.0284939
\(817\) −51122.5 −2.18917
\(818\) −12563.1 −0.536990
\(819\) −5604.39 −0.239113
\(820\) 0 0
\(821\) 7377.27 0.313603 0.156802 0.987630i \(-0.449882\pi\)
0.156802 + 0.987630i \(0.449882\pi\)
\(822\) 6402.27 0.271660
\(823\) 19960.3 0.845411 0.422705 0.906267i \(-0.361081\pi\)
0.422705 + 0.906267i \(0.361081\pi\)
\(824\) −10405.0 −0.439899
\(825\) 0 0
\(826\) −272.267 −0.0114690
\(827\) 3105.37 0.130574 0.0652868 0.997867i \(-0.479204\pi\)
0.0652868 + 0.997867i \(0.479204\pi\)
\(828\) 2183.45 0.0916427
\(829\) −10780.3 −0.451646 −0.225823 0.974168i \(-0.572507\pi\)
−0.225823 + 0.974168i \(0.572507\pi\)
\(830\) 0 0
\(831\) −5953.23 −0.248514
\(832\) 1643.67 0.0684905
\(833\) −5935.98 −0.246902
\(834\) −10082.2 −0.418606
\(835\) 0 0
\(836\) 17587.7 0.727613
\(837\) −2198.09 −0.0907731
\(838\) 10875.0 0.448293
\(839\) 2884.27 0.118684 0.0593422 0.998238i \(-0.481100\pi\)
0.0593422 + 0.998238i \(0.481100\pi\)
\(840\) 0 0
\(841\) −2873.26 −0.117810
\(842\) −16848.2 −0.689582
\(843\) 16010.1 0.654111
\(844\) 21634.6 0.882337
\(845\) 0 0
\(846\) 28853.8 1.17259
\(847\) 4907.18 0.199071
\(848\) 10788.1 0.436868
\(849\) −9887.50 −0.399691
\(850\) 0 0
\(851\) −6775.15 −0.272913
\(852\) 7322.49 0.294442
\(853\) 13694.1 0.549681 0.274841 0.961490i \(-0.411375\pi\)
0.274841 + 0.961490i \(0.411375\pi\)
\(854\) 8800.24 0.352621
\(855\) 0 0
\(856\) 3964.96 0.158317
\(857\) 31610.2 1.25996 0.629978 0.776613i \(-0.283064\pi\)
0.629978 + 0.776613i \(0.283064\pi\)
\(858\) −4008.97 −0.159515
\(859\) −20316.3 −0.806964 −0.403482 0.914988i \(-0.632200\pi\)
−0.403482 + 0.914988i \(0.632200\pi\)
\(860\) 0 0
\(861\) −2124.27 −0.0840823
\(862\) 4560.37 0.180194
\(863\) 9902.42 0.390593 0.195297 0.980744i \(-0.437433\pi\)
0.195297 + 0.980744i \(0.437433\pi\)
\(864\) 2934.31 0.115541
\(865\) 0 0
\(866\) 19353.8 0.759433
\(867\) −7926.56 −0.310496
\(868\) 881.633 0.0344753
\(869\) −27707.7 −1.08161
\(870\) 0 0
\(871\) 11773.4 0.458008
\(872\) 1585.68 0.0615801
\(873\) −3992.63 −0.154788
\(874\) 4683.86 0.181275
\(875\) 0 0
\(876\) −5426.09 −0.209281
\(877\) 11027.7 0.424606 0.212303 0.977204i \(-0.431904\pi\)
0.212303 + 0.977204i \(0.431904\pi\)
\(878\) 20535.2 0.789326
\(879\) 1137.96 0.0436661
\(880\) 0 0
\(881\) −27892.8 −1.06667 −0.533333 0.845906i \(-0.679061\pi\)
−0.533333 + 0.845906i \(0.679061\pi\)
\(882\) −12268.0 −0.468351
\(883\) 1618.76 0.0616937 0.0308468 0.999524i \(-0.490180\pi\)
0.0308468 + 0.999524i \(0.490180\pi\)
\(884\) 2359.38 0.0897677
\(885\) 0 0
\(886\) −5895.40 −0.223544
\(887\) −9518.69 −0.360323 −0.180161 0.983637i \(-0.557662\pi\)
−0.180161 + 0.983637i \(0.557662\pi\)
\(888\) −4259.37 −0.160963
\(889\) 19789.9 0.746606
\(890\) 0 0
\(891\) 20514.0 0.771319
\(892\) 23251.0 0.872758
\(893\) 61896.0 2.31945
\(894\) 7916.78 0.296171
\(895\) 0 0
\(896\) −1176.92 −0.0438820
\(897\) −1067.64 −0.0397409
\(898\) 28644.5 1.06445
\(899\) −3516.15 −0.130445
\(900\) 0 0
\(901\) 15485.6 0.572585
\(902\) 11039.3 0.407504
\(903\) −8343.87 −0.307494
\(904\) −6948.44 −0.255643
\(905\) 0 0
\(906\) 5673.87 0.208059
\(907\) 9580.94 0.350750 0.175375 0.984502i \(-0.443886\pi\)
0.175375 + 0.984502i \(0.443886\pi\)
\(908\) 8133.18 0.297257
\(909\) 15032.8 0.548522
\(910\) 0 0
\(911\) −18016.6 −0.655232 −0.327616 0.944811i \(-0.606245\pi\)
−0.327616 + 0.944811i \(0.606245\pi\)
\(912\) 2944.62 0.106915
\(913\) 38872.0 1.40906
\(914\) 26808.8 0.970192
\(915\) 0 0
\(916\) −1743.61 −0.0628936
\(917\) 15872.7 0.571606
\(918\) 4212.01 0.151435
\(919\) −36712.5 −1.31777 −0.658887 0.752242i \(-0.728973\pi\)
−0.658887 + 0.752242i \(0.728973\pi\)
\(920\) 0 0
\(921\) −1431.36 −0.0512107
\(922\) 33857.2 1.20936
\(923\) 26011.8 0.927614
\(924\) 2870.56 0.102202
\(925\) 0 0
\(926\) 26218.5 0.930447
\(927\) −30868.1 −1.09368
\(928\) 4693.84 0.166037
\(929\) 4117.08 0.145400 0.0727002 0.997354i \(-0.476838\pi\)
0.0727002 + 0.997354i \(0.476838\pi\)
\(930\) 0 0
\(931\) −26316.9 −0.926425
\(932\) −6328.20 −0.222411
\(933\) 3812.36 0.133774
\(934\) −9352.04 −0.327632
\(935\) 0 0
\(936\) 4876.19 0.170281
\(937\) −11540.5 −0.402362 −0.201181 0.979554i \(-0.564478\pi\)
−0.201181 + 0.979554i \(0.564478\pi\)
\(938\) −8430.12 −0.293447
\(939\) −375.225 −0.0130405
\(940\) 0 0
\(941\) −8600.43 −0.297945 −0.148972 0.988841i \(-0.547597\pi\)
−0.148972 + 0.988841i \(0.547597\pi\)
\(942\) −9521.08 −0.329314
\(943\) 2939.92 0.101524
\(944\) 236.890 0.00816750
\(945\) 0 0
\(946\) 43361.0 1.49026
\(947\) 48782.8 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(948\) −4638.96 −0.158931
\(949\) −19275.2 −0.659324
\(950\) 0 0
\(951\) −7255.00 −0.247381
\(952\) −1689.40 −0.0575143
\(953\) 42550.4 1.44632 0.723160 0.690681i \(-0.242689\pi\)
0.723160 + 0.690681i \(0.242689\pi\)
\(954\) 32004.4 1.08614
\(955\) 0 0
\(956\) −6930.40 −0.234461
\(957\) −11448.4 −0.386704
\(958\) 27036.1 0.911791
\(959\) −16284.6 −0.548341
\(960\) 0 0
\(961\) −29216.4 −0.980712
\(962\) −15130.6 −0.507100
\(963\) 11762.6 0.393609
\(964\) −17766.9 −0.593604
\(965\) 0 0
\(966\) 764.469 0.0254621
\(967\) −25119.6 −0.835360 −0.417680 0.908594i \(-0.637157\pi\)
−0.417680 + 0.908594i \(0.637157\pi\)
\(968\) −4269.57 −0.141766
\(969\) 4226.82 0.140129
\(970\) 0 0
\(971\) 11624.9 0.384204 0.192102 0.981375i \(-0.438470\pi\)
0.192102 + 0.981375i \(0.438470\pi\)
\(972\) 13337.8 0.440136
\(973\) 25644.8 0.844947
\(974\) 29829.3 0.981308
\(975\) 0 0
\(976\) −7656.78 −0.251114
\(977\) −15427.4 −0.505187 −0.252593 0.967573i \(-0.581283\pi\)
−0.252593 + 0.967573i \(0.581283\pi\)
\(978\) 1039.84 0.0339985
\(979\) −39890.4 −1.30225
\(980\) 0 0
\(981\) 4704.14 0.153101
\(982\) 7413.99 0.240927
\(983\) −42012.4 −1.36316 −0.681580 0.731744i \(-0.738707\pi\)
−0.681580 + 0.731744i \(0.738707\pi\)
\(984\) 1848.25 0.0598782
\(985\) 0 0
\(986\) 6737.70 0.217619
\(987\) 10102.3 0.325794
\(988\) 10460.2 0.336826
\(989\) 11547.6 0.371278
\(990\) 0 0
\(991\) 30889.0 0.990132 0.495066 0.868855i \(-0.335144\pi\)
0.495066 + 0.868855i \(0.335144\pi\)
\(992\) −767.078 −0.0245512
\(993\) 2163.13 0.0691286
\(994\) −18625.3 −0.594324
\(995\) 0 0
\(996\) 6508.14 0.207046
\(997\) 10011.2 0.318011 0.159005 0.987278i \(-0.449171\pi\)
0.159005 + 0.987278i \(0.449171\pi\)
\(998\) −26073.9 −0.827009
\(999\) −27011.4 −0.855458
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 1150.4.a.s.1.3 5
5.2 odd 4 1150.4.b.p.599.3 10
5.3 odd 4 1150.4.b.p.599.8 10
5.4 even 2 1150.4.a.t.1.3 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.4.a.s.1.3 5 1.1 even 1 trivial
1150.4.a.t.1.3 yes 5 5.4 even 2
1150.4.b.p.599.3 10 5.2 odd 4
1150.4.b.p.599.8 10 5.3 odd 4