Properties

Label 1617.4.a.be.1.7
Level $1617$
Weight $4$
Character 1617.1
Self dual yes
Analytic conductor $95.406$
Analytic rank $1$
Dimension $16$
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] = [1617,4,Mod(1,1617)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1617, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1617.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1617.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: \(95.4060884793\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 92 x^{14} + 346 x^{13} + 3385 x^{12} - 11756 x^{11} - 63875 x^{10} + 199850 x^{9} + \cdots + 5479424 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-1.36369\) of defining polynomial
Character \(\chi\) \(=\) 1617.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.36369 q^{2} -3.00000 q^{3} -6.14035 q^{4} +14.9279 q^{5} +4.09107 q^{6} +19.2830 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-1.36369 q^{2} -3.00000 q^{3} -6.14035 q^{4} +14.9279 q^{5} +4.09107 q^{6} +19.2830 q^{8} +9.00000 q^{9} -20.3570 q^{10} +11.0000 q^{11} +18.4211 q^{12} -36.0351 q^{13} -44.7838 q^{15} +22.8267 q^{16} +59.4444 q^{17} -12.2732 q^{18} -69.7279 q^{19} -91.6627 q^{20} -15.0006 q^{22} -29.1591 q^{23} -57.8491 q^{24} +97.8429 q^{25} +49.1407 q^{26} -27.0000 q^{27} +34.4863 q^{29} +61.0711 q^{30} +309.908 q^{31} -185.393 q^{32} -33.0000 q^{33} -81.0637 q^{34} -55.2632 q^{36} -347.598 q^{37} +95.0872 q^{38} +108.105 q^{39} +287.856 q^{40} -407.791 q^{41} -168.699 q^{43} -67.5439 q^{44} +134.351 q^{45} +39.7639 q^{46} -568.465 q^{47} -68.4802 q^{48} -133.427 q^{50} -178.333 q^{51} +221.268 q^{52} -146.442 q^{53} +36.8196 q^{54} +164.207 q^{55} +209.184 q^{57} -47.0286 q^{58} +458.470 q^{59} +274.988 q^{60} +638.295 q^{61} -422.618 q^{62} +70.2044 q^{64} -537.930 q^{65} +45.0017 q^{66} -179.242 q^{67} -365.009 q^{68} +87.4772 q^{69} -31.5981 q^{71} +173.547 q^{72} -187.321 q^{73} +474.015 q^{74} -293.529 q^{75} +428.154 q^{76} -147.422 q^{78} +939.768 q^{79} +340.756 q^{80} +81.0000 q^{81} +556.100 q^{82} +620.179 q^{83} +887.381 q^{85} +230.053 q^{86} -103.459 q^{87} +212.113 q^{88} +775.065 q^{89} -183.213 q^{90} +179.047 q^{92} -929.725 q^{93} +775.210 q^{94} -1040.89 q^{95} +556.179 q^{96} -297.150 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9} - 178 q^{10} + 176 q^{11} - 216 q^{12} - 104 q^{13} + 120 q^{15} + 220 q^{16} - 180 q^{17} + 36 q^{18} - 152 q^{19} - 298 q^{20} + 44 q^{22} + 4 q^{23} - 198 q^{24} + 588 q^{25} - 406 q^{26} - 432 q^{27} + 412 q^{29} + 534 q^{30} - 628 q^{31} + 592 q^{32} - 528 q^{33} + 88 q^{34} + 648 q^{36} + 148 q^{37} - 446 q^{38} + 312 q^{39} - 1376 q^{40} - 596 q^{41} - 260 q^{43} + 792 q^{44} - 360 q^{45} - 148 q^{46} - 2220 q^{47} - 660 q^{48} + 82 q^{50} + 540 q^{51} - 1046 q^{52} + 168 q^{53} - 108 q^{54} - 440 q^{55} + 456 q^{57} + 538 q^{58} + 48 q^{59} + 894 q^{60} - 1504 q^{61} - 1276 q^{62} + 630 q^{64} + 1224 q^{65} - 132 q^{66} + 116 q^{67} - 356 q^{68} - 12 q^{69} + 320 q^{71} + 594 q^{72} - 652 q^{73} + 1062 q^{74} - 1764 q^{75} + 594 q^{76} + 1218 q^{78} + 1136 q^{79} - 1970 q^{80} + 1296 q^{81} + 416 q^{82} - 3300 q^{83} - 1148 q^{85} + 1864 q^{86} - 1236 q^{87} + 726 q^{88} - 2416 q^{89} - 1602 q^{90} - 1064 q^{92} + 1884 q^{93} - 914 q^{94} + 1120 q^{95} - 1776 q^{96} - 3616 q^{97} + 1584 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.36369 −0.482137 −0.241068 0.970508i \(-0.577498\pi\)
−0.241068 + 0.970508i \(0.577498\pi\)
\(3\) −3.00000 −0.577350
\(4\) −6.14035 −0.767544
\(5\) 14.9279 1.33519 0.667597 0.744523i \(-0.267323\pi\)
0.667597 + 0.744523i \(0.267323\pi\)
\(6\) 4.09107 0.278362
\(7\) 0 0
\(8\) 19.2830 0.852198
\(9\) 9.00000 0.333333
\(10\) −20.3570 −0.643746
\(11\) 11.0000 0.301511
\(12\) 18.4211 0.443142
\(13\) −36.0351 −0.768796 −0.384398 0.923167i \(-0.625591\pi\)
−0.384398 + 0.923167i \(0.625591\pi\)
\(14\) 0 0
\(15\) −44.7838 −0.770875
\(16\) 22.8267 0.356668
\(17\) 59.4444 0.848081 0.424041 0.905643i \(-0.360611\pi\)
0.424041 + 0.905643i \(0.360611\pi\)
\(18\) −12.2732 −0.160712
\(19\) −69.7279 −0.841931 −0.420965 0.907077i \(-0.638309\pi\)
−0.420965 + 0.907077i \(0.638309\pi\)
\(20\) −91.6627 −1.02482
\(21\) 0 0
\(22\) −15.0006 −0.145370
\(23\) −29.1591 −0.264352 −0.132176 0.991226i \(-0.542196\pi\)
−0.132176 + 0.991226i \(0.542196\pi\)
\(24\) −57.8491 −0.492017
\(25\) 97.8429 0.782743
\(26\) 49.1407 0.370665
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 34.4863 0.220826 0.110413 0.993886i \(-0.464783\pi\)
0.110413 + 0.993886i \(0.464783\pi\)
\(30\) 61.0711 0.371667
\(31\) 309.908 1.79552 0.897761 0.440484i \(-0.145193\pi\)
0.897761 + 0.440484i \(0.145193\pi\)
\(32\) −185.393 −1.02416
\(33\) −33.0000 −0.174078
\(34\) −81.0637 −0.408891
\(35\) 0 0
\(36\) −55.2632 −0.255848
\(37\) −347.598 −1.54445 −0.772226 0.635348i \(-0.780857\pi\)
−0.772226 + 0.635348i \(0.780857\pi\)
\(38\) 95.0872 0.405926
\(39\) 108.105 0.443865
\(40\) 287.856 1.13785
\(41\) −407.791 −1.55332 −0.776661 0.629919i \(-0.783088\pi\)
−0.776661 + 0.629919i \(0.783088\pi\)
\(42\) 0 0
\(43\) −168.699 −0.598288 −0.299144 0.954208i \(-0.596701\pi\)
−0.299144 + 0.954208i \(0.596701\pi\)
\(44\) −67.5439 −0.231423
\(45\) 134.351 0.445065
\(46\) 39.7639 0.127454
\(47\) −568.465 −1.76424 −0.882119 0.471027i \(-0.843884\pi\)
−0.882119 + 0.471027i \(0.843884\pi\)
\(48\) −68.4802 −0.205922
\(49\) 0 0
\(50\) −133.427 −0.377389
\(51\) −178.333 −0.489640
\(52\) 221.268 0.590085
\(53\) −146.442 −0.379536 −0.189768 0.981829i \(-0.560774\pi\)
−0.189768 + 0.981829i \(0.560774\pi\)
\(54\) 36.8196 0.0927873
\(55\) 164.207 0.402576
\(56\) 0 0
\(57\) 209.184 0.486089
\(58\) −47.0286 −0.106468
\(59\) 458.470 1.01166 0.505828 0.862635i \(-0.331187\pi\)
0.505828 + 0.862635i \(0.331187\pi\)
\(60\) 274.988 0.591680
\(61\) 638.295 1.33976 0.669880 0.742469i \(-0.266346\pi\)
0.669880 + 0.742469i \(0.266346\pi\)
\(62\) −422.618 −0.865687
\(63\) 0 0
\(64\) 70.2044 0.137118
\(65\) −537.930 −1.02649
\(66\) 45.0017 0.0839293
\(67\) −179.242 −0.326834 −0.163417 0.986557i \(-0.552252\pi\)
−0.163417 + 0.986557i \(0.552252\pi\)
\(68\) −365.009 −0.650940
\(69\) 87.4772 0.152624
\(70\) 0 0
\(71\) −31.5981 −0.0528169 −0.0264085 0.999651i \(-0.508407\pi\)
−0.0264085 + 0.999651i \(0.508407\pi\)
\(72\) 173.547 0.284066
\(73\) −187.321 −0.300333 −0.150166 0.988661i \(-0.547981\pi\)
−0.150166 + 0.988661i \(0.547981\pi\)
\(74\) 474.015 0.744638
\(75\) −293.529 −0.451917
\(76\) 428.154 0.646219
\(77\) 0 0
\(78\) −147.422 −0.214003
\(79\) 939.768 1.33838 0.669191 0.743091i \(-0.266641\pi\)
0.669191 + 0.743091i \(0.266641\pi\)
\(80\) 340.756 0.476221
\(81\) 81.0000 0.111111
\(82\) 556.100 0.748914
\(83\) 620.179 0.820162 0.410081 0.912049i \(-0.365500\pi\)
0.410081 + 0.912049i \(0.365500\pi\)
\(84\) 0 0
\(85\) 887.381 1.13235
\(86\) 230.053 0.288457
\(87\) −103.459 −0.127494
\(88\) 212.113 0.256947
\(89\) 775.065 0.923109 0.461555 0.887112i \(-0.347292\pi\)
0.461555 + 0.887112i \(0.347292\pi\)
\(90\) −183.213 −0.214582
\(91\) 0 0
\(92\) 179.047 0.202902
\(93\) −929.725 −1.03664
\(94\) 775.210 0.850604
\(95\) −1040.89 −1.12414
\(96\) 556.179 0.591300
\(97\) −297.150 −0.311041 −0.155520 0.987833i \(-0.549705\pi\)
−0.155520 + 0.987833i \(0.549705\pi\)
\(98\) 0 0
\(99\) 99.0000 0.100504
\(100\) −600.790 −0.600790
\(101\) −64.1014 −0.0631518 −0.0315759 0.999501i \(-0.510053\pi\)
−0.0315759 + 0.999501i \(0.510053\pi\)
\(102\) 243.191 0.236073
\(103\) 239.724 0.229327 0.114664 0.993404i \(-0.463421\pi\)
0.114664 + 0.993404i \(0.463421\pi\)
\(104\) −694.867 −0.655167
\(105\) 0 0
\(106\) 199.702 0.182988
\(107\) −518.817 −0.468747 −0.234374 0.972147i \(-0.575304\pi\)
−0.234374 + 0.972147i \(0.575304\pi\)
\(108\) 165.790 0.147714
\(109\) 550.789 0.484001 0.242000 0.970276i \(-0.422197\pi\)
0.242000 + 0.970276i \(0.422197\pi\)
\(110\) −223.928 −0.194097
\(111\) 1042.79 0.891690
\(112\) 0 0
\(113\) 17.9046 0.0149055 0.00745274 0.999972i \(-0.497628\pi\)
0.00745274 + 0.999972i \(0.497628\pi\)
\(114\) −285.262 −0.234361
\(115\) −435.285 −0.352961
\(116\) −211.758 −0.169493
\(117\) −324.316 −0.256265
\(118\) −625.210 −0.487756
\(119\) 0 0
\(120\) −863.567 −0.656938
\(121\) 121.000 0.0909091
\(122\) −870.437 −0.645948
\(123\) 1223.37 0.896811
\(124\) −1902.95 −1.37814
\(125\) −405.399 −0.290080
\(126\) 0 0
\(127\) 1828.76 1.27777 0.638883 0.769304i \(-0.279397\pi\)
0.638883 + 0.769304i \(0.279397\pi\)
\(128\) 1387.41 0.958051
\(129\) 506.097 0.345422
\(130\) 733.569 0.494910
\(131\) −2891.36 −1.92839 −0.964197 0.265187i \(-0.914566\pi\)
−0.964197 + 0.265187i \(0.914566\pi\)
\(132\) 202.632 0.133612
\(133\) 0 0
\(134\) 244.430 0.157579
\(135\) −403.054 −0.256958
\(136\) 1146.27 0.722733
\(137\) −288.402 −0.179853 −0.0899263 0.995948i \(-0.528663\pi\)
−0.0899263 + 0.995948i \(0.528663\pi\)
\(138\) −119.292 −0.0735854
\(139\) −1334.10 −0.814075 −0.407038 0.913411i \(-0.633438\pi\)
−0.407038 + 0.913411i \(0.633438\pi\)
\(140\) 0 0
\(141\) 1705.40 1.01858
\(142\) 43.0900 0.0254650
\(143\) −396.386 −0.231801
\(144\) 205.441 0.118889
\(145\) 514.808 0.294845
\(146\) 255.448 0.144801
\(147\) 0 0
\(148\) 2134.37 1.18544
\(149\) 1942.90 1.06825 0.534124 0.845406i \(-0.320642\pi\)
0.534124 + 0.845406i \(0.320642\pi\)
\(150\) 400.282 0.217886
\(151\) 1082.91 0.583615 0.291807 0.956477i \(-0.405743\pi\)
0.291807 + 0.956477i \(0.405743\pi\)
\(152\) −1344.57 −0.717492
\(153\) 534.999 0.282694
\(154\) 0 0
\(155\) 4626.29 2.39737
\(156\) −663.805 −0.340686
\(157\) −1605.16 −0.815960 −0.407980 0.912991i \(-0.633767\pi\)
−0.407980 + 0.912991i \(0.633767\pi\)
\(158\) −1281.55 −0.645283
\(159\) 439.327 0.219125
\(160\) −2767.53 −1.36745
\(161\) 0 0
\(162\) −110.459 −0.0535708
\(163\) −3730.05 −1.79239 −0.896197 0.443657i \(-0.853681\pi\)
−0.896197 + 0.443657i \(0.853681\pi\)
\(164\) 2503.98 1.19224
\(165\) −492.621 −0.232427
\(166\) −845.731 −0.395430
\(167\) −2620.90 −1.21444 −0.607220 0.794534i \(-0.707715\pi\)
−0.607220 + 0.794534i \(0.707715\pi\)
\(168\) 0 0
\(169\) −898.469 −0.408953
\(170\) −1210.11 −0.545949
\(171\) −627.552 −0.280644
\(172\) 1035.87 0.459212
\(173\) −3669.46 −1.61262 −0.806311 0.591491i \(-0.798539\pi\)
−0.806311 + 0.591491i \(0.798539\pi\)
\(174\) 141.086 0.0614694
\(175\) 0 0
\(176\) 251.094 0.107539
\(177\) −1375.41 −0.584079
\(178\) −1056.95 −0.445065
\(179\) 826.957 0.345305 0.172653 0.984983i \(-0.444766\pi\)
0.172653 + 0.984983i \(0.444766\pi\)
\(180\) −824.964 −0.341607
\(181\) 1884.81 0.774015 0.387007 0.922077i \(-0.373509\pi\)
0.387007 + 0.922077i \(0.373509\pi\)
\(182\) 0 0
\(183\) −1914.89 −0.773511
\(184\) −562.276 −0.225280
\(185\) −5188.92 −2.06214
\(186\) 1267.86 0.499805
\(187\) 653.888 0.255706
\(188\) 3490.58 1.35413
\(189\) 0 0
\(190\) 1419.46 0.541990
\(191\) −3516.26 −1.33208 −0.666040 0.745916i \(-0.732012\pi\)
−0.666040 + 0.745916i \(0.732012\pi\)
\(192\) −210.613 −0.0791651
\(193\) 2069.30 0.771771 0.385886 0.922547i \(-0.373896\pi\)
0.385886 + 0.922547i \(0.373896\pi\)
\(194\) 405.220 0.149964
\(195\) 1613.79 0.592645
\(196\) 0 0
\(197\) 2787.62 1.00817 0.504086 0.863653i \(-0.331829\pi\)
0.504086 + 0.863653i \(0.331829\pi\)
\(198\) −135.005 −0.0484566
\(199\) −2896.00 −1.03162 −0.515810 0.856703i \(-0.672509\pi\)
−0.515810 + 0.856703i \(0.672509\pi\)
\(200\) 1886.71 0.667052
\(201\) 537.726 0.188698
\(202\) 87.4144 0.0304478
\(203\) 0 0
\(204\) 1095.03 0.375820
\(205\) −6087.47 −2.07399
\(206\) −326.909 −0.110567
\(207\) −262.432 −0.0881172
\(208\) −822.565 −0.274205
\(209\) −767.007 −0.253852
\(210\) 0 0
\(211\) −4292.75 −1.40059 −0.700297 0.713852i \(-0.746949\pi\)
−0.700297 + 0.713852i \(0.746949\pi\)
\(212\) 899.208 0.291310
\(213\) 94.7943 0.0304939
\(214\) 707.505 0.226000
\(215\) −2518.33 −0.798830
\(216\) −520.642 −0.164006
\(217\) 0 0
\(218\) −751.106 −0.233355
\(219\) 561.963 0.173397
\(220\) −1008.29 −0.308995
\(221\) −2142.09 −0.652001
\(222\) −1422.05 −0.429917
\(223\) −5426.45 −1.62951 −0.814757 0.579802i \(-0.803130\pi\)
−0.814757 + 0.579802i \(0.803130\pi\)
\(224\) 0 0
\(225\) 880.586 0.260914
\(226\) −24.4163 −0.00718649
\(227\) −1359.43 −0.397484 −0.198742 0.980052i \(-0.563686\pi\)
−0.198742 + 0.980052i \(0.563686\pi\)
\(228\) −1284.46 −0.373095
\(229\) 6406.28 1.84864 0.924321 0.381615i \(-0.124632\pi\)
0.924321 + 0.381615i \(0.124632\pi\)
\(230\) 593.593 0.170175
\(231\) 0 0
\(232\) 665.000 0.188187
\(233\) −3453.40 −0.970985 −0.485493 0.874241i \(-0.661360\pi\)
−0.485493 + 0.874241i \(0.661360\pi\)
\(234\) 442.266 0.123555
\(235\) −8486.01 −2.35560
\(236\) −2815.16 −0.776490
\(237\) −2819.30 −0.772715
\(238\) 0 0
\(239\) −639.754 −0.173148 −0.0865738 0.996245i \(-0.527592\pi\)
−0.0865738 + 0.996245i \(0.527592\pi\)
\(240\) −1022.27 −0.274946
\(241\) −1994.68 −0.533148 −0.266574 0.963814i \(-0.585892\pi\)
−0.266574 + 0.963814i \(0.585892\pi\)
\(242\) −165.006 −0.0438306
\(243\) −243.000 −0.0641500
\(244\) −3919.36 −1.02832
\(245\) 0 0
\(246\) −1668.30 −0.432385
\(247\) 2512.66 0.647273
\(248\) 5975.97 1.53014
\(249\) −1860.54 −0.473521
\(250\) 552.838 0.139858
\(251\) −4945.08 −1.24355 −0.621775 0.783196i \(-0.713588\pi\)
−0.621775 + 0.783196i \(0.713588\pi\)
\(252\) 0 0
\(253\) −320.750 −0.0797050
\(254\) −2493.86 −0.616058
\(255\) −2662.14 −0.653764
\(256\) −2453.63 −0.599030
\(257\) −2416.27 −0.586470 −0.293235 0.956040i \(-0.594732\pi\)
−0.293235 + 0.956040i \(0.594732\pi\)
\(258\) −690.159 −0.166540
\(259\) 0 0
\(260\) 3303.08 0.787878
\(261\) 310.376 0.0736085
\(262\) 3942.92 0.929750
\(263\) −97.0388 −0.0227516 −0.0113758 0.999935i \(-0.503621\pi\)
−0.0113758 + 0.999935i \(0.503621\pi\)
\(264\) −636.340 −0.148349
\(265\) −2186.08 −0.506754
\(266\) 0 0
\(267\) −2325.20 −0.532957
\(268\) 1100.61 0.250860
\(269\) 2678.68 0.607145 0.303573 0.952808i \(-0.401820\pi\)
0.303573 + 0.952808i \(0.401820\pi\)
\(270\) 549.640 0.123889
\(271\) 250.523 0.0561556 0.0280778 0.999606i \(-0.491061\pi\)
0.0280778 + 0.999606i \(0.491061\pi\)
\(272\) 1356.92 0.302483
\(273\) 0 0
\(274\) 393.290 0.0867136
\(275\) 1076.27 0.236006
\(276\) −537.141 −0.117145
\(277\) 2187.39 0.474467 0.237234 0.971453i \(-0.423759\pi\)
0.237234 + 0.971453i \(0.423759\pi\)
\(278\) 1819.29 0.392496
\(279\) 2789.17 0.598507
\(280\) 0 0
\(281\) 1341.51 0.284797 0.142399 0.989809i \(-0.454518\pi\)
0.142399 + 0.989809i \(0.454518\pi\)
\(282\) −2325.63 −0.491097
\(283\) 5108.89 1.07312 0.536559 0.843863i \(-0.319724\pi\)
0.536559 + 0.843863i \(0.319724\pi\)
\(284\) 194.023 0.0405393
\(285\) 3122.68 0.649023
\(286\) 540.548 0.111760
\(287\) 0 0
\(288\) −1668.54 −0.341387
\(289\) −1379.37 −0.280758
\(290\) −702.039 −0.142156
\(291\) 891.449 0.179580
\(292\) 1150.22 0.230519
\(293\) −3363.87 −0.670714 −0.335357 0.942091i \(-0.608857\pi\)
−0.335357 + 0.942091i \(0.608857\pi\)
\(294\) 0 0
\(295\) 6844.00 1.35076
\(296\) −6702.75 −1.31618
\(297\) −297.000 −0.0580259
\(298\) −2649.52 −0.515042
\(299\) 1050.75 0.203233
\(300\) 1802.37 0.346866
\(301\) 0 0
\(302\) −1476.75 −0.281382
\(303\) 192.304 0.0364607
\(304\) −1591.66 −0.300290
\(305\) 9528.43 1.78884
\(306\) −729.573 −0.136297
\(307\) −9002.53 −1.67362 −0.836810 0.547493i \(-0.815582\pi\)
−0.836810 + 0.547493i \(0.815582\pi\)
\(308\) 0 0
\(309\) −719.173 −0.132402
\(310\) −6308.82 −1.15586
\(311\) −7726.85 −1.40884 −0.704420 0.709783i \(-0.748793\pi\)
−0.704420 + 0.709783i \(0.748793\pi\)
\(312\) 2084.60 0.378261
\(313\) −416.130 −0.0751472 −0.0375736 0.999294i \(-0.511963\pi\)
−0.0375736 + 0.999294i \(0.511963\pi\)
\(314\) 2188.94 0.393404
\(315\) 0 0
\(316\) −5770.50 −1.02727
\(317\) 9868.54 1.74849 0.874247 0.485481i \(-0.161356\pi\)
0.874247 + 0.485481i \(0.161356\pi\)
\(318\) −599.105 −0.105648
\(319\) 379.349 0.0665814
\(320\) 1048.01 0.183079
\(321\) 1556.45 0.270631
\(322\) 0 0
\(323\) −4144.93 −0.714026
\(324\) −497.369 −0.0852827
\(325\) −3525.78 −0.601770
\(326\) 5086.63 0.864179
\(327\) −1652.37 −0.279438
\(328\) −7863.44 −1.32374
\(329\) 0 0
\(330\) 671.783 0.112062
\(331\) −8652.30 −1.43678 −0.718388 0.695642i \(-0.755120\pi\)
−0.718388 + 0.695642i \(0.755120\pi\)
\(332\) −3808.12 −0.629511
\(333\) −3128.38 −0.514818
\(334\) 3574.10 0.585527
\(335\) −2675.71 −0.436387
\(336\) 0 0
\(337\) −205.761 −0.0332596 −0.0166298 0.999862i \(-0.505294\pi\)
−0.0166298 + 0.999862i \(0.505294\pi\)
\(338\) 1225.23 0.197171
\(339\) −53.7137 −0.00860569
\(340\) −5448.83 −0.869131
\(341\) 3408.99 0.541370
\(342\) 855.785 0.135309
\(343\) 0 0
\(344\) −3253.03 −0.509860
\(345\) 1305.85 0.203782
\(346\) 5004.00 0.777505
\(347\) 7646.27 1.18292 0.591460 0.806334i \(-0.298552\pi\)
0.591460 + 0.806334i \(0.298552\pi\)
\(348\) 635.274 0.0978570
\(349\) −3364.07 −0.515973 −0.257987 0.966148i \(-0.583059\pi\)
−0.257987 + 0.966148i \(0.583059\pi\)
\(350\) 0 0
\(351\) 972.949 0.147955
\(352\) −2039.32 −0.308796
\(353\) −679.645 −0.102475 −0.0512377 0.998686i \(-0.516317\pi\)
−0.0512377 + 0.998686i \(0.516317\pi\)
\(354\) 1875.63 0.281606
\(355\) −471.694 −0.0705209
\(356\) −4759.17 −0.708527
\(357\) 0 0
\(358\) −1127.71 −0.166484
\(359\) −7506.64 −1.10358 −0.551790 0.833983i \(-0.686055\pi\)
−0.551790 + 0.833983i \(0.686055\pi\)
\(360\) 2590.70 0.379283
\(361\) −1997.01 −0.291152
\(362\) −2570.29 −0.373181
\(363\) −363.000 −0.0524864
\(364\) 0 0
\(365\) −2796.32 −0.401002
\(366\) 2611.31 0.372938
\(367\) 3508.84 0.499073 0.249537 0.968365i \(-0.419722\pi\)
0.249537 + 0.968365i \(0.419722\pi\)
\(368\) −665.607 −0.0942857
\(369\) −3670.11 −0.517774
\(370\) 7076.07 0.994236
\(371\) 0 0
\(372\) 5708.84 0.795670
\(373\) 743.711 0.103238 0.0516192 0.998667i \(-0.483562\pi\)
0.0516192 + 0.998667i \(0.483562\pi\)
\(374\) −891.700 −0.123285
\(375\) 1216.20 0.167478
\(376\) −10961.7 −1.50348
\(377\) −1242.72 −0.169770
\(378\) 0 0
\(379\) −4002.42 −0.542455 −0.271227 0.962515i \(-0.587430\pi\)
−0.271227 + 0.962515i \(0.587430\pi\)
\(380\) 6391.45 0.862828
\(381\) −5486.28 −0.737718
\(382\) 4795.08 0.642245
\(383\) −6347.65 −0.846867 −0.423433 0.905927i \(-0.639175\pi\)
−0.423433 + 0.905927i \(0.639175\pi\)
\(384\) −4162.22 −0.553131
\(385\) 0 0
\(386\) −2821.89 −0.372099
\(387\) −1518.29 −0.199429
\(388\) 1824.60 0.238738
\(389\) 12464.6 1.62463 0.812317 0.583216i \(-0.198206\pi\)
0.812317 + 0.583216i \(0.198206\pi\)
\(390\) −2200.71 −0.285736
\(391\) −1733.34 −0.224192
\(392\) 0 0
\(393\) 8674.09 1.11336
\(394\) −3801.45 −0.486077
\(395\) 14028.8 1.78700
\(396\) −607.895 −0.0771411
\(397\) 5647.60 0.713967 0.356983 0.934111i \(-0.383805\pi\)
0.356983 + 0.934111i \(0.383805\pi\)
\(398\) 3949.25 0.497382
\(399\) 0 0
\(400\) 2233.43 0.279179
\(401\) 2200.12 0.273986 0.136993 0.990572i \(-0.456256\pi\)
0.136993 + 0.990572i \(0.456256\pi\)
\(402\) −733.291 −0.0909782
\(403\) −11167.6 −1.38039
\(404\) 393.605 0.0484718
\(405\) 1209.16 0.148355
\(406\) 0 0
\(407\) −3823.58 −0.465670
\(408\) −3438.81 −0.417270
\(409\) −14938.4 −1.80600 −0.903000 0.429641i \(-0.858640\pi\)
−0.903000 + 0.429641i \(0.858640\pi\)
\(410\) 8301.41 0.999945
\(411\) 865.205 0.103838
\(412\) −1471.99 −0.176019
\(413\) 0 0
\(414\) 357.875 0.0424846
\(415\) 9257.98 1.09508
\(416\) 6680.66 0.787371
\(417\) 4002.29 0.470007
\(418\) 1045.96 0.122391
\(419\) 11426.1 1.33222 0.666112 0.745851i \(-0.267957\pi\)
0.666112 + 0.745851i \(0.267957\pi\)
\(420\) 0 0
\(421\) −6672.95 −0.772493 −0.386246 0.922396i \(-0.626229\pi\)
−0.386246 + 0.922396i \(0.626229\pi\)
\(422\) 5853.98 0.675278
\(423\) −5116.19 −0.588079
\(424\) −2823.85 −0.323440
\(425\) 5816.21 0.663830
\(426\) −129.270 −0.0147022
\(427\) 0 0
\(428\) 3185.72 0.359784
\(429\) 1189.16 0.133830
\(430\) 3434.22 0.385145
\(431\) 13249.3 1.48074 0.740369 0.672201i \(-0.234651\pi\)
0.740369 + 0.672201i \(0.234651\pi\)
\(432\) −616.322 −0.0686408
\(433\) 1900.88 0.210971 0.105485 0.994421i \(-0.466360\pi\)
0.105485 + 0.994421i \(0.466360\pi\)
\(434\) 0 0
\(435\) −1544.43 −0.170229
\(436\) −3382.04 −0.371492
\(437\) 2033.20 0.222566
\(438\) −766.343 −0.0836012
\(439\) −16372.1 −1.77995 −0.889975 0.456010i \(-0.849278\pi\)
−0.889975 + 0.456010i \(0.849278\pi\)
\(440\) 3166.41 0.343075
\(441\) 0 0
\(442\) 2921.14 0.314354
\(443\) −15555.3 −1.66829 −0.834146 0.551543i \(-0.814039\pi\)
−0.834146 + 0.551543i \(0.814039\pi\)
\(444\) −6403.12 −0.684411
\(445\) 11570.1 1.23253
\(446\) 7399.98 0.785649
\(447\) −5828.71 −0.616753
\(448\) 0 0
\(449\) 3813.17 0.400790 0.200395 0.979715i \(-0.435777\pi\)
0.200395 + 0.979715i \(0.435777\pi\)
\(450\) −1200.85 −0.125796
\(451\) −4485.70 −0.468344
\(452\) −109.940 −0.0114406
\(453\) −3248.72 −0.336950
\(454\) 1853.85 0.191642
\(455\) 0 0
\(456\) 4033.70 0.414244
\(457\) 16706.2 1.71003 0.855015 0.518603i \(-0.173548\pi\)
0.855015 + 0.518603i \(0.173548\pi\)
\(458\) −8736.18 −0.891299
\(459\) −1605.00 −0.163213
\(460\) 2672.80 0.270913
\(461\) −8916.91 −0.900871 −0.450436 0.892809i \(-0.648731\pi\)
−0.450436 + 0.892809i \(0.648731\pi\)
\(462\) 0 0
\(463\) −11942.2 −1.19871 −0.599355 0.800483i \(-0.704576\pi\)
−0.599355 + 0.800483i \(0.704576\pi\)
\(464\) 787.209 0.0787614
\(465\) −13878.9 −1.38412
\(466\) 4709.36 0.468148
\(467\) −4457.05 −0.441644 −0.220822 0.975314i \(-0.570874\pi\)
−0.220822 + 0.975314i \(0.570874\pi\)
\(468\) 1991.42 0.196695
\(469\) 0 0
\(470\) 11572.3 1.13572
\(471\) 4815.48 0.471095
\(472\) 8840.69 0.862131
\(473\) −1855.69 −0.180391
\(474\) 3844.65 0.372554
\(475\) −6822.39 −0.659016
\(476\) 0 0
\(477\) −1317.98 −0.126512
\(478\) 872.426 0.0834808
\(479\) 12055.7 1.14998 0.574988 0.818162i \(-0.305007\pi\)
0.574988 + 0.818162i \(0.305007\pi\)
\(480\) 8302.59 0.789500
\(481\) 12525.7 1.18737
\(482\) 2720.13 0.257050
\(483\) 0 0
\(484\) −742.983 −0.0697767
\(485\) −4435.83 −0.415300
\(486\) 331.376 0.0309291
\(487\) −9926.82 −0.923670 −0.461835 0.886966i \(-0.652809\pi\)
−0.461835 + 0.886966i \(0.652809\pi\)
\(488\) 12308.3 1.14174
\(489\) 11190.2 1.03484
\(490\) 0 0
\(491\) 14220.0 1.30700 0.653501 0.756926i \(-0.273300\pi\)
0.653501 + 0.756926i \(0.273300\pi\)
\(492\) −7511.93 −0.688342
\(493\) 2050.02 0.187278
\(494\) −3426.48 −0.312074
\(495\) 1477.86 0.134192
\(496\) 7074.19 0.640405
\(497\) 0 0
\(498\) 2537.19 0.228302
\(499\) 12714.6 1.14065 0.570323 0.821420i \(-0.306818\pi\)
0.570323 + 0.821420i \(0.306818\pi\)
\(500\) 2489.29 0.222649
\(501\) 7862.71 0.701158
\(502\) 6743.56 0.599561
\(503\) −11421.1 −1.01241 −0.506204 0.862414i \(-0.668952\pi\)
−0.506204 + 0.862414i \(0.668952\pi\)
\(504\) 0 0
\(505\) −956.901 −0.0843199
\(506\) 437.403 0.0384287
\(507\) 2695.41 0.236109
\(508\) −11229.2 −0.980741
\(509\) −17644.2 −1.53648 −0.768238 0.640164i \(-0.778866\pi\)
−0.768238 + 0.640164i \(0.778866\pi\)
\(510\) 3630.34 0.315204
\(511\) 0 0
\(512\) −7753.27 −0.669237
\(513\) 1882.65 0.162030
\(514\) 3295.04 0.282759
\(515\) 3578.58 0.306197
\(516\) −3107.62 −0.265126
\(517\) −6253.12 −0.531938
\(518\) 0 0
\(519\) 11008.4 0.931048
\(520\) −10372.9 −0.874775
\(521\) −13697.3 −1.15180 −0.575902 0.817519i \(-0.695349\pi\)
−0.575902 + 0.817519i \(0.695349\pi\)
\(522\) −423.257 −0.0354894
\(523\) −15896.1 −1.32904 −0.664521 0.747270i \(-0.731364\pi\)
−0.664521 + 0.747270i \(0.731364\pi\)
\(524\) 17754.0 1.48013
\(525\) 0 0
\(526\) 132.331 0.0109694
\(527\) 18422.3 1.52275
\(528\) −753.282 −0.0620879
\(529\) −11316.7 −0.930118
\(530\) 2981.13 0.244325
\(531\) 4126.23 0.337218
\(532\) 0 0
\(533\) 14694.8 1.19419
\(534\) 3170.84 0.256958
\(535\) −7744.87 −0.625869
\(536\) −3456.33 −0.278528
\(537\) −2480.87 −0.199362
\(538\) −3652.89 −0.292727
\(539\) 0 0
\(540\) 2474.89 0.197227
\(541\) 6768.57 0.537900 0.268950 0.963154i \(-0.413323\pi\)
0.268950 + 0.963154i \(0.413323\pi\)
\(542\) −341.635 −0.0270747
\(543\) −5654.43 −0.446878
\(544\) −11020.6 −0.868572
\(545\) 8222.14 0.646235
\(546\) 0 0
\(547\) −6596.06 −0.515589 −0.257795 0.966200i \(-0.582996\pi\)
−0.257795 + 0.966200i \(0.582996\pi\)
\(548\) 1770.89 0.138045
\(549\) 5744.66 0.446587
\(550\) −1467.70 −0.113787
\(551\) −2404.66 −0.185920
\(552\) 1686.83 0.130065
\(553\) 0 0
\(554\) −2982.92 −0.228758
\(555\) 15566.7 1.19058
\(556\) 8191.81 0.624839
\(557\) −4740.49 −0.360612 −0.180306 0.983611i \(-0.557709\pi\)
−0.180306 + 0.983611i \(0.557709\pi\)
\(558\) −3803.57 −0.288562
\(559\) 6079.09 0.459961
\(560\) 0 0
\(561\) −1961.66 −0.147632
\(562\) −1829.41 −0.137311
\(563\) 15733.7 1.17779 0.588896 0.808208i \(-0.299563\pi\)
0.588896 + 0.808208i \(0.299563\pi\)
\(564\) −10471.7 −0.781808
\(565\) 267.278 0.0199017
\(566\) −6966.94 −0.517389
\(567\) 0 0
\(568\) −609.307 −0.0450105
\(569\) 5880.47 0.433255 0.216627 0.976254i \(-0.430494\pi\)
0.216627 + 0.976254i \(0.430494\pi\)
\(570\) −4258.37 −0.312918
\(571\) 6212.40 0.455308 0.227654 0.973742i \(-0.426895\pi\)
0.227654 + 0.973742i \(0.426895\pi\)
\(572\) 2433.95 0.177917
\(573\) 10548.8 0.769077
\(574\) 0 0
\(575\) −2853.01 −0.206920
\(576\) 631.839 0.0457060
\(577\) 13649.8 0.984831 0.492416 0.870360i \(-0.336114\pi\)
0.492416 + 0.870360i \(0.336114\pi\)
\(578\) 1881.03 0.135364
\(579\) −6207.91 −0.445582
\(580\) −3161.11 −0.226306
\(581\) 0 0
\(582\) −1215.66 −0.0865819
\(583\) −1610.87 −0.114434
\(584\) −3612.12 −0.255943
\(585\) −4841.37 −0.342164
\(586\) 4587.27 0.323376
\(587\) 11833.8 0.832081 0.416041 0.909346i \(-0.363417\pi\)
0.416041 + 0.909346i \(0.363417\pi\)
\(588\) 0 0
\(589\) −21609.3 −1.51170
\(590\) −9333.09 −0.651249
\(591\) −8362.87 −0.582068
\(592\) −7934.53 −0.550857
\(593\) −1165.52 −0.0807119 −0.0403560 0.999185i \(-0.512849\pi\)
−0.0403560 + 0.999185i \(0.512849\pi\)
\(594\) 405.016 0.0279764
\(595\) 0 0
\(596\) −11930.1 −0.819927
\(597\) 8688.01 0.595606
\(598\) −1432.90 −0.0979859
\(599\) 5705.63 0.389191 0.194596 0.980884i \(-0.437661\pi\)
0.194596 + 0.980884i \(0.437661\pi\)
\(600\) −5660.13 −0.385123
\(601\) −6498.36 −0.441054 −0.220527 0.975381i \(-0.570778\pi\)
−0.220527 + 0.975381i \(0.570778\pi\)
\(602\) 0 0
\(603\) −1613.18 −0.108945
\(604\) −6649.44 −0.447950
\(605\) 1806.28 0.121381
\(606\) −262.243 −0.0175791
\(607\) 7509.07 0.502115 0.251057 0.967972i \(-0.419222\pi\)
0.251057 + 0.967972i \(0.419222\pi\)
\(608\) 12927.1 0.862273
\(609\) 0 0
\(610\) −12993.8 −0.862466
\(611\) 20484.7 1.35634
\(612\) −3285.09 −0.216980
\(613\) −9262.10 −0.610266 −0.305133 0.952310i \(-0.598701\pi\)
−0.305133 + 0.952310i \(0.598701\pi\)
\(614\) 12276.6 0.806914
\(615\) 18262.4 1.19742
\(616\) 0 0
\(617\) −14827.9 −0.967501 −0.483750 0.875206i \(-0.660726\pi\)
−0.483750 + 0.875206i \(0.660726\pi\)
\(618\) 980.728 0.0638360
\(619\) 11518.3 0.747917 0.373958 0.927445i \(-0.378000\pi\)
0.373958 + 0.927445i \(0.378000\pi\)
\(620\) −28407.0 −1.84009
\(621\) 787.295 0.0508745
\(622\) 10537.0 0.679254
\(623\) 0 0
\(624\) 2467.69 0.158312
\(625\) −18282.1 −1.17006
\(626\) 567.472 0.0362312
\(627\) 2301.02 0.146561
\(628\) 9856.25 0.626285
\(629\) −20662.7 −1.30982
\(630\) 0 0
\(631\) −12799.5 −0.807515 −0.403757 0.914866i \(-0.632296\pi\)
−0.403757 + 0.914866i \(0.632296\pi\)
\(632\) 18121.6 1.14057
\(633\) 12878.3 0.808633
\(634\) −13457.6 −0.843014
\(635\) 27299.6 1.70606
\(636\) −2697.62 −0.168188
\(637\) 0 0
\(638\) −517.314 −0.0321013
\(639\) −284.383 −0.0176056
\(640\) 20711.1 1.27918
\(641\) −11200.4 −0.690156 −0.345078 0.938574i \(-0.612147\pi\)
−0.345078 + 0.938574i \(0.612147\pi\)
\(642\) −2122.52 −0.130481
\(643\) −23688.2 −1.45283 −0.726417 0.687255i \(-0.758816\pi\)
−0.726417 + 0.687255i \(0.758816\pi\)
\(644\) 0 0
\(645\) 7554.98 0.461205
\(646\) 5652.40 0.344258
\(647\) −13624.6 −0.827881 −0.413940 0.910304i \(-0.635848\pi\)
−0.413940 + 0.910304i \(0.635848\pi\)
\(648\) 1561.93 0.0946887
\(649\) 5043.16 0.305025
\(650\) 4808.07 0.290135
\(651\) 0 0
\(652\) 22903.8 1.37574
\(653\) 26268.2 1.57420 0.787100 0.616825i \(-0.211581\pi\)
0.787100 + 0.616825i \(0.211581\pi\)
\(654\) 2253.32 0.134727
\(655\) −43162.1 −2.57478
\(656\) −9308.53 −0.554020
\(657\) −1685.89 −0.100111
\(658\) 0 0
\(659\) −13003.0 −0.768627 −0.384313 0.923203i \(-0.625562\pi\)
−0.384313 + 0.923203i \(0.625562\pi\)
\(660\) 3024.87 0.178398
\(661\) −22342.7 −1.31472 −0.657361 0.753576i \(-0.728327\pi\)
−0.657361 + 0.753576i \(0.728327\pi\)
\(662\) 11799.0 0.692723
\(663\) 6426.26 0.376433
\(664\) 11958.9 0.698941
\(665\) 0 0
\(666\) 4266.14 0.248213
\(667\) −1005.59 −0.0583756
\(668\) 16093.3 0.932137
\(669\) 16279.3 0.940800
\(670\) 3648.84 0.210398
\(671\) 7021.25 0.403953
\(672\) 0 0
\(673\) −17286.2 −0.990097 −0.495049 0.868865i \(-0.664850\pi\)
−0.495049 + 0.868865i \(0.664850\pi\)
\(674\) 280.593 0.0160357
\(675\) −2641.76 −0.150639
\(676\) 5516.92 0.313889
\(677\) −13111.0 −0.744311 −0.372155 0.928170i \(-0.621381\pi\)
−0.372155 + 0.928170i \(0.621381\pi\)
\(678\) 73.2488 0.00414912
\(679\) 0 0
\(680\) 17111.4 0.964989
\(681\) 4078.30 0.229487
\(682\) −4648.80 −0.261014
\(683\) 407.524 0.0228309 0.0114154 0.999935i \(-0.496366\pi\)
0.0114154 + 0.999935i \(0.496366\pi\)
\(684\) 3853.39 0.215406
\(685\) −4305.24 −0.240138
\(686\) 0 0
\(687\) −19218.8 −1.06731
\(688\) −3850.85 −0.213390
\(689\) 5277.07 0.291786
\(690\) −1780.78 −0.0982508
\(691\) −13252.8 −0.729611 −0.364806 0.931084i \(-0.618865\pi\)
−0.364806 + 0.931084i \(0.618865\pi\)
\(692\) 22531.8 1.23776
\(693\) 0 0
\(694\) −10427.1 −0.570330
\(695\) −19915.3 −1.08695
\(696\) −1995.00 −0.108650
\(697\) −24240.9 −1.31734
\(698\) 4587.55 0.248770
\(699\) 10360.2 0.560599
\(700\) 0 0
\(701\) 23142.2 1.24689 0.623443 0.781869i \(-0.285733\pi\)
0.623443 + 0.781869i \(0.285733\pi\)
\(702\) −1326.80 −0.0713345
\(703\) 24237.3 1.30032
\(704\) 772.248 0.0413426
\(705\) 25458.0 1.36001
\(706\) 926.824 0.0494072
\(707\) 0 0
\(708\) 8445.49 0.448307
\(709\) 22170.3 1.17436 0.587182 0.809455i \(-0.300237\pi\)
0.587182 + 0.809455i \(0.300237\pi\)
\(710\) 643.244 0.0340007
\(711\) 8457.91 0.446127
\(712\) 14945.6 0.786672
\(713\) −9036.64 −0.474649
\(714\) 0 0
\(715\) −5917.23 −0.309499
\(716\) −5077.81 −0.265037
\(717\) 1919.26 0.0999668
\(718\) 10236.7 0.532077
\(719\) −23361.8 −1.21175 −0.605875 0.795560i \(-0.707177\pi\)
−0.605875 + 0.795560i \(0.707177\pi\)
\(720\) 3066.80 0.158740
\(721\) 0 0
\(722\) 2723.31 0.140375
\(723\) 5984.04 0.307813
\(724\) −11573.4 −0.594091
\(725\) 3374.24 0.172850
\(726\) 495.019 0.0253056
\(727\) 9556.05 0.487502 0.243751 0.969838i \(-0.421622\pi\)
0.243751 + 0.969838i \(0.421622\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 3813.31 0.193338
\(731\) −10028.2 −0.507397
\(732\) 11758.1 0.593704
\(733\) 18488.7 0.931645 0.465822 0.884878i \(-0.345759\pi\)
0.465822 + 0.884878i \(0.345759\pi\)
\(734\) −4784.97 −0.240622
\(735\) 0 0
\(736\) 5405.89 0.270739
\(737\) −1971.66 −0.0985443
\(738\) 5004.90 0.249638
\(739\) −10124.6 −0.503980 −0.251990 0.967730i \(-0.581085\pi\)
−0.251990 + 0.967730i \(0.581085\pi\)
\(740\) 31861.8 1.58279
\(741\) −7537.97 −0.373703
\(742\) 0 0
\(743\) −8393.20 −0.414424 −0.207212 0.978296i \(-0.566439\pi\)
−0.207212 + 0.978296i \(0.566439\pi\)
\(744\) −17927.9 −0.883427
\(745\) 29003.5 1.42632
\(746\) −1014.19 −0.0497750
\(747\) 5581.61 0.273387
\(748\) −4015.10 −0.196266
\(749\) 0 0
\(750\) −1658.51 −0.0807472
\(751\) 16004.1 0.777629 0.388814 0.921316i \(-0.372885\pi\)
0.388814 + 0.921316i \(0.372885\pi\)
\(752\) −12976.2 −0.629247
\(753\) 14835.3 0.717964
\(754\) 1694.68 0.0818523
\(755\) 16165.6 0.779239
\(756\) 0 0
\(757\) 16914.5 0.812110 0.406055 0.913849i \(-0.366904\pi\)
0.406055 + 0.913849i \(0.366904\pi\)
\(758\) 5458.05 0.261537
\(759\) 962.250 0.0460177
\(760\) −20071.6 −0.957991
\(761\) −2002.58 −0.0953920 −0.0476960 0.998862i \(-0.515188\pi\)
−0.0476960 + 0.998862i \(0.515188\pi\)
\(762\) 7481.58 0.355681
\(763\) 0 0
\(764\) 21591.1 1.02243
\(765\) 7986.43 0.377451
\(766\) 8656.23 0.408306
\(767\) −16521.0 −0.777756
\(768\) 7360.88 0.345850
\(769\) −21995.2 −1.03143 −0.515714 0.856761i \(-0.672473\pi\)
−0.515714 + 0.856761i \(0.672473\pi\)
\(770\) 0 0
\(771\) 7248.82 0.338599
\(772\) −12706.3 −0.592368
\(773\) −15395.6 −0.716352 −0.358176 0.933654i \(-0.616601\pi\)
−0.358176 + 0.933654i \(0.616601\pi\)
\(774\) 2070.48 0.0961522
\(775\) 30322.3 1.40543
\(776\) −5729.95 −0.265069
\(777\) 0 0
\(778\) −16997.9 −0.783296
\(779\) 28434.4 1.30779
\(780\) −9909.23 −0.454881
\(781\) −347.579 −0.0159249
\(782\) 2363.74 0.108091
\(783\) −931.129 −0.0424979
\(784\) 0 0
\(785\) −23961.7 −1.08946
\(786\) −11828.8 −0.536791
\(787\) 8616.65 0.390280 0.195140 0.980775i \(-0.437484\pi\)
0.195140 + 0.980775i \(0.437484\pi\)
\(788\) −17117.0 −0.773816
\(789\) 291.117 0.0131356
\(790\) −19130.9 −0.861578
\(791\) 0 0
\(792\) 1909.02 0.0856491
\(793\) −23001.1 −1.03000
\(794\) −7701.57 −0.344230
\(795\) 6558.24 0.292575
\(796\) 17782.5 0.791813
\(797\) 8226.45 0.365616 0.182808 0.983149i \(-0.441481\pi\)
0.182808 + 0.983149i \(0.441481\pi\)
\(798\) 0 0
\(799\) −33792.1 −1.49622
\(800\) −18139.4 −0.801655
\(801\) 6975.59 0.307703
\(802\) −3000.27 −0.132099
\(803\) −2060.53 −0.0905537
\(804\) −3301.83 −0.144834
\(805\) 0 0
\(806\) 15229.1 0.665537
\(807\) −8036.04 −0.350536
\(808\) −1236.07 −0.0538178
\(809\) −4112.03 −0.178704 −0.0893519 0.996000i \(-0.528480\pi\)
−0.0893519 + 0.996000i \(0.528480\pi\)
\(810\) −1648.92 −0.0715274
\(811\) −3267.94 −0.141495 −0.0707477 0.997494i \(-0.522539\pi\)
−0.0707477 + 0.997494i \(0.522539\pi\)
\(812\) 0 0
\(813\) −751.569 −0.0324215
\(814\) 5214.17 0.224517
\(815\) −55681.9 −2.39319
\(816\) −4070.76 −0.174639
\(817\) 11763.0 0.503717
\(818\) 20371.3 0.870739
\(819\) 0 0
\(820\) 37379.2 1.59188
\(821\) −17572.3 −0.746989 −0.373494 0.927632i \(-0.621840\pi\)
−0.373494 + 0.927632i \(0.621840\pi\)
\(822\) −1179.87 −0.0500641
\(823\) 17826.6 0.755037 0.377518 0.926002i \(-0.376778\pi\)
0.377518 + 0.926002i \(0.376778\pi\)
\(824\) 4622.61 0.195432
\(825\) −3228.82 −0.136258
\(826\) 0 0
\(827\) −10897.6 −0.458221 −0.229110 0.973400i \(-0.573582\pi\)
−0.229110 + 0.973400i \(0.573582\pi\)
\(828\) 1611.42 0.0676338
\(829\) 34462.5 1.44383 0.721914 0.691983i \(-0.243262\pi\)
0.721914 + 0.691983i \(0.243262\pi\)
\(830\) −12625.0 −0.527976
\(831\) −6562.17 −0.273934
\(832\) −2529.82 −0.105416
\(833\) 0 0
\(834\) −5457.87 −0.226608
\(835\) −39124.6 −1.62151
\(836\) 4709.70 0.194842
\(837\) −8367.52 −0.345548
\(838\) −15581.7 −0.642315
\(839\) 47930.2 1.97227 0.986135 0.165943i \(-0.0530669\pi\)
0.986135 + 0.165943i \(0.0530669\pi\)
\(840\) 0 0
\(841\) −23199.7 −0.951236
\(842\) 9099.82 0.372447
\(843\) −4024.54 −0.164428
\(844\) 26359.0 1.07502
\(845\) −13412.3 −0.546031
\(846\) 6976.89 0.283535
\(847\) 0 0
\(848\) −3342.80 −0.135368
\(849\) −15326.7 −0.619565
\(850\) −7931.50 −0.320057
\(851\) 10135.6 0.408279
\(852\) −582.070 −0.0234054
\(853\) 30569.4 1.22705 0.613526 0.789675i \(-0.289751\pi\)
0.613526 + 0.789675i \(0.289751\pi\)
\(854\) 0 0
\(855\) −9368.04 −0.374714
\(856\) −10004.4 −0.399466
\(857\) −27977.8 −1.11517 −0.557586 0.830119i \(-0.688272\pi\)
−0.557586 + 0.830119i \(0.688272\pi\)
\(858\) −1621.64 −0.0645245
\(859\) 32650.8 1.29689 0.648446 0.761260i \(-0.275419\pi\)
0.648446 + 0.761260i \(0.275419\pi\)
\(860\) 15463.4 0.613137
\(861\) 0 0
\(862\) −18068.0 −0.713918
\(863\) −1861.99 −0.0734447 −0.0367223 0.999326i \(-0.511692\pi\)
−0.0367223 + 0.999326i \(0.511692\pi\)
\(864\) 5005.61 0.197100
\(865\) −54777.4 −2.15316
\(866\) −2592.21 −0.101717
\(867\) 4138.10 0.162096
\(868\) 0 0
\(869\) 10337.4 0.403537
\(870\) 2106.12 0.0820736
\(871\) 6459.01 0.251269
\(872\) 10620.9 0.412464
\(873\) −2674.35 −0.103680
\(874\) −2772.66 −0.107307
\(875\) 0 0
\(876\) −3450.65 −0.133090
\(877\) 36169.3 1.39264 0.696322 0.717729i \(-0.254818\pi\)
0.696322 + 0.717729i \(0.254818\pi\)
\(878\) 22326.5 0.858179
\(879\) 10091.6 0.387237
\(880\) 3748.31 0.143586
\(881\) 34235.6 1.30922 0.654612 0.755965i \(-0.272832\pi\)
0.654612 + 0.755965i \(0.272832\pi\)
\(882\) 0 0
\(883\) 44521.5 1.69679 0.848396 0.529363i \(-0.177569\pi\)
0.848396 + 0.529363i \(0.177569\pi\)
\(884\) 13153.2 0.500440
\(885\) −20532.0 −0.779859
\(886\) 21212.6 0.804345
\(887\) −38230.4 −1.44718 −0.723592 0.690228i \(-0.757510\pi\)
−0.723592 + 0.690228i \(0.757510\pi\)
\(888\) 20108.2 0.759897
\(889\) 0 0
\(890\) −15778.0 −0.594248
\(891\) 891.000 0.0335013
\(892\) 33320.3 1.25072
\(893\) 39637.9 1.48537
\(894\) 7948.55 0.297359
\(895\) 12344.7 0.461050
\(896\) 0 0
\(897\) −3152.25 −0.117336
\(898\) −5199.98 −0.193236
\(899\) 10687.6 0.396497
\(900\) −5407.11 −0.200263
\(901\) −8705.17 −0.321877
\(902\) 6117.09 0.225806
\(903\) 0 0
\(904\) 345.255 0.0127024
\(905\) 28136.3 1.03346
\(906\) 4430.25 0.162456
\(907\) −11633.8 −0.425902 −0.212951 0.977063i \(-0.568308\pi\)
−0.212951 + 0.977063i \(0.568308\pi\)
\(908\) 8347.41 0.305086
\(909\) −576.913 −0.0210506
\(910\) 0 0
\(911\) −16159.9 −0.587708 −0.293854 0.955850i \(-0.594938\pi\)
−0.293854 + 0.955850i \(0.594938\pi\)
\(912\) 4774.99 0.173372
\(913\) 6821.97 0.247288
\(914\) −22782.1 −0.824469
\(915\) −28585.3 −1.03279
\(916\) −39336.8 −1.41891
\(917\) 0 0
\(918\) 2188.72 0.0786912
\(919\) −42542.8 −1.52705 −0.763525 0.645779i \(-0.776533\pi\)
−0.763525 + 0.645779i \(0.776533\pi\)
\(920\) −8393.61 −0.300793
\(921\) 27007.6 0.966265
\(922\) 12159.9 0.434343
\(923\) 1138.64 0.0406055
\(924\) 0 0
\(925\) −34010.0 −1.20891
\(926\) 16285.5 0.577942
\(927\) 2157.52 0.0764425
\(928\) −6393.51 −0.226161
\(929\) 13583.5 0.479720 0.239860 0.970807i \(-0.422898\pi\)
0.239860 + 0.970807i \(0.422898\pi\)
\(930\) 18926.4 0.667336
\(931\) 0 0
\(932\) 21205.1 0.745274
\(933\) 23180.5 0.813394
\(934\) 6078.03 0.212933
\(935\) 9761.19 0.341417
\(936\) −6253.80 −0.218389
\(937\) 9344.02 0.325780 0.162890 0.986644i \(-0.447918\pi\)
0.162890 + 0.986644i \(0.447918\pi\)
\(938\) 0 0
\(939\) 1248.39 0.0433862
\(940\) 52107.1 1.80803
\(941\) 40791.8 1.41315 0.706575 0.707638i \(-0.250239\pi\)
0.706575 + 0.707638i \(0.250239\pi\)
\(942\) −6566.82 −0.227132
\(943\) 11890.8 0.410623
\(944\) 10465.4 0.360825
\(945\) 0 0
\(946\) 2530.58 0.0869729
\(947\) 2524.31 0.0866198 0.0433099 0.999062i \(-0.486210\pi\)
0.0433099 + 0.999062i \(0.486210\pi\)
\(948\) 17311.5 0.593093
\(949\) 6750.14 0.230895
\(950\) 9303.61 0.317736
\(951\) −29605.6 −1.00949
\(952\) 0 0
\(953\) 27102.3 0.921227 0.460613 0.887601i \(-0.347630\pi\)
0.460613 + 0.887601i \(0.347630\pi\)
\(954\) 1797.32 0.0609961
\(955\) −52490.4 −1.77859
\(956\) 3928.32 0.132898
\(957\) −1138.05 −0.0384408
\(958\) −16440.2 −0.554446
\(959\) 0 0
\(960\) −3144.02 −0.105701
\(961\) 66252.1 2.22390
\(962\) −17081.2 −0.572474
\(963\) −4669.36 −0.156249
\(964\) 12248.0 0.409215
\(965\) 30890.4 1.03046
\(966\) 0 0
\(967\) −30010.1 −0.997995 −0.498997 0.866603i \(-0.666298\pi\)
−0.498997 + 0.866603i \(0.666298\pi\)
\(968\) 2333.25 0.0774726
\(969\) 12434.8 0.412243
\(970\) 6049.09 0.200231
\(971\) 27393.7 0.905362 0.452681 0.891672i \(-0.350468\pi\)
0.452681 + 0.891672i \(0.350468\pi\)
\(972\) 1492.11 0.0492380
\(973\) 0 0
\(974\) 13537.1 0.445335
\(975\) 10577.3 0.347432
\(976\) 14570.2 0.477849
\(977\) −4793.91 −0.156981 −0.0784907 0.996915i \(-0.525010\pi\)
−0.0784907 + 0.996915i \(0.525010\pi\)
\(978\) −15259.9 −0.498934
\(979\) 8525.72 0.278328
\(980\) 0 0
\(981\) 4957.11 0.161334
\(982\) −19391.6 −0.630154
\(983\) −34777.3 −1.12841 −0.564203 0.825636i \(-0.690816\pi\)
−0.564203 + 0.825636i \(0.690816\pi\)
\(984\) 23590.3 0.764260
\(985\) 41613.4 1.34611
\(986\) −2795.58 −0.0902936
\(987\) 0 0
\(988\) −15428.6 −0.496811
\(989\) 4919.11 0.158158
\(990\) −2015.35 −0.0646989
\(991\) 49474.6 1.58588 0.792942 0.609296i \(-0.208548\pi\)
0.792942 + 0.609296i \(0.208548\pi\)
\(992\) −57454.8 −1.83890
\(993\) 25956.9 0.829524
\(994\) 0 0
\(995\) −43231.3 −1.37741
\(996\) 11424.3 0.363448
\(997\) −40547.9 −1.28803 −0.644014 0.765014i \(-0.722732\pi\)
−0.644014 + 0.765014i \(0.722732\pi\)
\(998\) −17338.7 −0.549948
\(999\) 9385.14 0.297230
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 1617.4.a.be.1.7 16
7.6 odd 2 1617.4.a.bf.1.7 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1617.4.a.be.1.7 16 1.1 even 1 trivial
1617.4.a.bf.1.7 yes 16 7.6 odd 2