Properties

Label 575.4.b.i.24.2
Level $575$
Weight $4$
Character 575.24
Analytic conductor $33.926$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [575,4,Mod(24,575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.24");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 575.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(33.9260982533\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 55x^{8} + 1079x^{6} + 8937x^{4} + 26936x^{2} + 8464 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 115)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.2
Root \(-3.41740i\) of defining polynomial
Character \(\chi\) \(=\) 575.24
Dual form 575.4.b.i.24.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.41740i q^{2} +7.84147i q^{3} -11.5134 q^{4} +34.6389 q^{6} +8.97260i q^{7} +15.5200i q^{8} -34.4886 q^{9} +O(q^{10})\) \(q-4.41740i q^{2} +7.84147i q^{3} -11.5134 q^{4} +34.6389 q^{6} +8.97260i q^{7} +15.5200i q^{8} -34.4886 q^{9} -28.9011 q^{11} -90.2818i q^{12} +16.0879i q^{13} +39.6355 q^{14} -23.5490 q^{16} -25.1771i q^{17} +152.350i q^{18} +35.6298 q^{19} -70.3583 q^{21} +127.668i q^{22} -23.0000i q^{23} -121.700 q^{24} +71.0668 q^{26} -58.7218i q^{27} -103.305i q^{28} -138.272 q^{29} +40.1277 q^{31} +228.186i q^{32} -226.627i q^{33} -111.217 q^{34} +397.081 q^{36} -379.745i q^{37} -157.391i q^{38} -126.153 q^{39} +412.514 q^{41} +310.801i q^{42} -402.095i q^{43} +332.750 q^{44} -101.600 q^{46} -110.070i q^{47} -184.659i q^{48} +262.492 q^{49} +197.426 q^{51} -185.227i q^{52} -421.300i q^{53} -259.397 q^{54} -139.255 q^{56} +279.390i q^{57} +610.802i q^{58} -755.913 q^{59} -307.032 q^{61} -177.260i q^{62} -309.453i q^{63} +819.594 q^{64} -1001.10 q^{66} -319.974i q^{67} +289.874i q^{68} +180.354 q^{69} -554.138 q^{71} -535.264i q^{72} -705.131i q^{73} -1677.48 q^{74} -410.219 q^{76} -259.318i q^{77} +557.268i q^{78} -1170.51 q^{79} -470.728 q^{81} -1822.24i q^{82} -455.978i q^{83} +810.063 q^{84} -1776.21 q^{86} -1084.26i q^{87} -448.546i q^{88} +1495.57 q^{89} -144.351 q^{91} +264.808i q^{92} +314.660i q^{93} -486.222 q^{94} -1789.31 q^{96} -1041.24i q^{97} -1159.53i q^{98} +996.760 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 44 q^{4} + 38 q^{6} - 154 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 44 q^{4} + 38 q^{6} - 154 q^{9} + 46 q^{11} - 186 q^{14} + 564 q^{16} + 322 q^{19} - 120 q^{21} - 210 q^{24} - 514 q^{26} - 802 q^{29} + 64 q^{31} + 1326 q^{34} - 1318 q^{36} - 670 q^{39} - 24 q^{41} - 94 q^{44} - 276 q^{46} + 1476 q^{49} - 1986 q^{51} + 16 q^{54} + 686 q^{56} - 2648 q^{59} - 3346 q^{61} - 4932 q^{64} - 5562 q^{66} + 184 q^{69} - 216 q^{71} - 2916 q^{74} - 6954 q^{76} - 1312 q^{79} - 638 q^{81} + 1436 q^{84} + 224 q^{86} - 1140 q^{89} - 3178 q^{91} + 1896 q^{94} - 11982 q^{96} - 4042 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/575\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 4.41740i − 1.56179i −0.624665 0.780893i \(-0.714765\pi\)
0.624665 0.780893i \(-0.285235\pi\)
\(3\) 7.84147i 1.50909i 0.656248 + 0.754546i \(0.272143\pi\)
−0.656248 + 0.754546i \(0.727857\pi\)
\(4\) −11.5134 −1.43917
\(5\) 0 0
\(6\) 34.6389 2.35688
\(7\) 8.97260i 0.484475i 0.970217 + 0.242237i \(0.0778813\pi\)
−0.970217 + 0.242237i \(0.922119\pi\)
\(8\) 15.5200i 0.685894i
\(9\) −34.4886 −1.27736
\(10\) 0 0
\(11\) −28.9011 −0.792183 −0.396092 0.918211i \(-0.629634\pi\)
−0.396092 + 0.918211i \(0.629634\pi\)
\(12\) − 90.2818i − 2.17184i
\(13\) 16.0879i 0.343230i 0.985164 + 0.171615i \(0.0548985\pi\)
−0.985164 + 0.171615i \(0.945102\pi\)
\(14\) 39.6355 0.756646
\(15\) 0 0
\(16\) −23.5490 −0.367954
\(17\) − 25.1771i − 0.359197i −0.983740 0.179599i \(-0.942520\pi\)
0.983740 0.179599i \(-0.0574799\pi\)
\(18\) 152.350i 1.99496i
\(19\) 35.6298 0.430212 0.215106 0.976591i \(-0.430990\pi\)
0.215106 + 0.976591i \(0.430990\pi\)
\(20\) 0 0
\(21\) −70.3583 −0.731117
\(22\) 127.668i 1.23722i
\(23\) − 23.0000i − 0.208514i
\(24\) −121.700 −1.03508
\(25\) 0 0
\(26\) 71.0668 0.536051
\(27\) − 58.7218i − 0.418556i
\(28\) − 103.305i − 0.697243i
\(29\) −138.272 −0.885395 −0.442698 0.896671i \(-0.645978\pi\)
−0.442698 + 0.896671i \(0.645978\pi\)
\(30\) 0 0
\(31\) 40.1277 0.232489 0.116244 0.993221i \(-0.462914\pi\)
0.116244 + 0.993221i \(0.462914\pi\)
\(32\) 228.186i 1.26056i
\(33\) − 226.627i − 1.19548i
\(34\) −111.217 −0.560989
\(35\) 0 0
\(36\) 397.081 1.83834
\(37\) − 379.745i − 1.68729i −0.536902 0.843645i \(-0.680405\pi\)
0.536902 0.843645i \(-0.319595\pi\)
\(38\) − 157.391i − 0.671899i
\(39\) −126.153 −0.517965
\(40\) 0 0
\(41\) 412.514 1.57132 0.785658 0.618662i \(-0.212325\pi\)
0.785658 + 0.618662i \(0.212325\pi\)
\(42\) 310.801i 1.14185i
\(43\) − 402.095i − 1.42602i −0.701153 0.713011i \(-0.747331\pi\)
0.701153 0.713011i \(-0.252669\pi\)
\(44\) 332.750 1.14009
\(45\) 0 0
\(46\) −101.600 −0.325655
\(47\) − 110.070i − 0.341603i −0.985305 0.170801i \(-0.945364\pi\)
0.985305 0.170801i \(-0.0546357\pi\)
\(48\) − 184.659i − 0.555276i
\(49\) 262.492 0.765284
\(50\) 0 0
\(51\) 197.426 0.542061
\(52\) − 185.227i − 0.493967i
\(53\) − 421.300i − 1.09189i −0.837822 0.545943i \(-0.816171\pi\)
0.837822 0.545943i \(-0.183829\pi\)
\(54\) −259.397 −0.653695
\(55\) 0 0
\(56\) −139.255 −0.332298
\(57\) 279.390i 0.649229i
\(58\) 610.802i 1.38280i
\(59\) −755.913 −1.66799 −0.833996 0.551770i \(-0.813952\pi\)
−0.833996 + 0.551770i \(0.813952\pi\)
\(60\) 0 0
\(61\) −307.032 −0.644450 −0.322225 0.946663i \(-0.604431\pi\)
−0.322225 + 0.946663i \(0.604431\pi\)
\(62\) − 177.260i − 0.363098i
\(63\) − 309.453i − 0.618847i
\(64\) 819.594 1.60077
\(65\) 0 0
\(66\) −1001.10 −1.86708
\(67\) − 319.974i − 0.583448i −0.956502 0.291724i \(-0.905771\pi\)
0.956502 0.291724i \(-0.0942290\pi\)
\(68\) 289.874i 0.516947i
\(69\) 180.354 0.314667
\(70\) 0 0
\(71\) −554.138 −0.926254 −0.463127 0.886292i \(-0.653273\pi\)
−0.463127 + 0.886292i \(0.653273\pi\)
\(72\) − 535.264i − 0.876131i
\(73\) − 705.131i − 1.13054i −0.824907 0.565269i \(-0.808772\pi\)
0.824907 0.565269i \(-0.191228\pi\)
\(74\) −1677.48 −2.63518
\(75\) 0 0
\(76\) −410.219 −0.619149
\(77\) − 259.318i − 0.383793i
\(78\) 557.268i 0.808950i
\(79\) −1170.51 −1.66699 −0.833495 0.552526i \(-0.813664\pi\)
−0.833495 + 0.552526i \(0.813664\pi\)
\(80\) 0 0
\(81\) −470.728 −0.645717
\(82\) − 1822.24i − 2.45406i
\(83\) − 455.978i − 0.603014i −0.953464 0.301507i \(-0.902510\pi\)
0.953464 0.301507i \(-0.0974896\pi\)
\(84\) 810.063 1.05220
\(85\) 0 0
\(86\) −1776.21 −2.22714
\(87\) − 1084.26i − 1.33614i
\(88\) − 448.546i − 0.543354i
\(89\) 1495.57 1.78124 0.890621 0.454746i \(-0.150270\pi\)
0.890621 + 0.454746i \(0.150270\pi\)
\(90\) 0 0
\(91\) −144.351 −0.166286
\(92\) 264.808i 0.300088i
\(93\) 314.660i 0.350847i
\(94\) −486.222 −0.533510
\(95\) 0 0
\(96\) −1789.31 −1.90230
\(97\) − 1041.24i − 1.08992i −0.838463 0.544958i \(-0.816545\pi\)
0.838463 0.544958i \(-0.183455\pi\)
\(98\) − 1159.53i − 1.19521i
\(99\) 996.760 1.01190
\(100\) 0 0
\(101\) 1450.22 1.42873 0.714367 0.699771i \(-0.246715\pi\)
0.714367 + 0.699771i \(0.246715\pi\)
\(102\) − 872.108i − 0.846584i
\(103\) 220.283i 0.210729i 0.994434 + 0.105365i \(0.0336010\pi\)
−0.994434 + 0.105365i \(0.966399\pi\)
\(104\) −249.685 −0.235419
\(105\) 0 0
\(106\) −1861.05 −1.70529
\(107\) − 59.2679i − 0.0535481i −0.999642 0.0267741i \(-0.991477\pi\)
0.999642 0.0267741i \(-0.00852346\pi\)
\(108\) 676.087i 0.602375i
\(109\) −1966.29 −1.72786 −0.863930 0.503612i \(-0.832004\pi\)
−0.863930 + 0.503612i \(0.832004\pi\)
\(110\) 0 0
\(111\) 2977.76 2.54627
\(112\) − 211.296i − 0.178264i
\(113\) 1819.68i 1.51488i 0.652905 + 0.757439i \(0.273550\pi\)
−0.652905 + 0.757439i \(0.726450\pi\)
\(114\) 1234.17 1.01396
\(115\) 0 0
\(116\) 1591.98 1.27424
\(117\) − 554.851i − 0.438427i
\(118\) 3339.17i 2.60505i
\(119\) 225.904 0.174022
\(120\) 0 0
\(121\) −495.725 −0.372445
\(122\) 1356.28i 1.00649i
\(123\) 3234.72i 2.37126i
\(124\) −462.006 −0.334592
\(125\) 0 0
\(126\) −1366.97 −0.966506
\(127\) − 834.517i − 0.583082i −0.956558 0.291541i \(-0.905832\pi\)
0.956558 0.291541i \(-0.0941680\pi\)
\(128\) − 1794.98i − 1.23950i
\(129\) 3153.02 2.15200
\(130\) 0 0
\(131\) 510.293 0.340340 0.170170 0.985415i \(-0.445568\pi\)
0.170170 + 0.985415i \(0.445568\pi\)
\(132\) 2609.25i 1.72050i
\(133\) 319.692i 0.208427i
\(134\) −1413.45 −0.911221
\(135\) 0 0
\(136\) 390.750 0.246371
\(137\) − 11.2339i − 0.00700567i −0.999994 0.00350284i \(-0.998885\pi\)
0.999994 0.00350284i \(-0.00111499\pi\)
\(138\) − 796.694i − 0.491443i
\(139\) 1656.81 1.01100 0.505501 0.862826i \(-0.331308\pi\)
0.505501 + 0.862826i \(0.331308\pi\)
\(140\) 0 0
\(141\) 863.109 0.515510
\(142\) 2447.85i 1.44661i
\(143\) − 464.959i − 0.271901i
\(144\) 812.174 0.470008
\(145\) 0 0
\(146\) −3114.84 −1.76566
\(147\) 2058.33i 1.15488i
\(148\) 4372.15i 2.42830i
\(149\) 510.206 0.280522 0.140261 0.990115i \(-0.455206\pi\)
0.140261 + 0.990115i \(0.455206\pi\)
\(150\) 0 0
\(151\) −2337.38 −1.25969 −0.629846 0.776720i \(-0.716882\pi\)
−0.629846 + 0.776720i \(0.716882\pi\)
\(152\) 552.974i 0.295080i
\(153\) 868.325i 0.458823i
\(154\) −1145.51 −0.599402
\(155\) 0 0
\(156\) 1452.45 0.745442
\(157\) − 146.592i − 0.0745178i −0.999306 0.0372589i \(-0.988137\pi\)
0.999306 0.0372589i \(-0.0118626\pi\)
\(158\) 5170.59i 2.60348i
\(159\) 3303.61 1.64776
\(160\) 0 0
\(161\) 206.370 0.101020
\(162\) 2079.39i 1.00847i
\(163\) − 3278.19i − 1.57526i −0.616149 0.787630i \(-0.711308\pi\)
0.616149 0.787630i \(-0.288692\pi\)
\(164\) −4749.44 −2.26139
\(165\) 0 0
\(166\) −2014.24 −0.941778
\(167\) 1555.42i 0.720732i 0.932811 + 0.360366i \(0.117348\pi\)
−0.932811 + 0.360366i \(0.882652\pi\)
\(168\) − 1091.96i − 0.501469i
\(169\) 1938.18 0.882193
\(170\) 0 0
\(171\) −1228.82 −0.549534
\(172\) 4629.48i 2.05229i
\(173\) − 472.392i − 0.207603i −0.994598 0.103801i \(-0.966899\pi\)
0.994598 0.103801i \(-0.0331006\pi\)
\(174\) −4789.58 −2.08677
\(175\) 0 0
\(176\) 680.594 0.291487
\(177\) − 5927.47i − 2.51715i
\(178\) − 6606.54i − 2.78192i
\(179\) −2429.45 −1.01444 −0.507222 0.861815i \(-0.669328\pi\)
−0.507222 + 0.861815i \(0.669328\pi\)
\(180\) 0 0
\(181\) −982.359 −0.403415 −0.201708 0.979446i \(-0.564649\pi\)
−0.201708 + 0.979446i \(0.564649\pi\)
\(182\) 637.653i 0.259703i
\(183\) − 2407.58i − 0.972533i
\(184\) 356.960 0.143019
\(185\) 0 0
\(186\) 1389.98 0.547947
\(187\) 727.648i 0.284550i
\(188\) 1267.28i 0.491626i
\(189\) 526.887 0.202780
\(190\) 0 0
\(191\) 1361.14 0.515649 0.257824 0.966192i \(-0.416994\pi\)
0.257824 + 0.966192i \(0.416994\pi\)
\(192\) 6426.82i 2.41571i
\(193\) − 1456.29i − 0.543142i −0.962418 0.271571i \(-0.912457\pi\)
0.962418 0.271571i \(-0.0875432\pi\)
\(194\) −4599.57 −1.70222
\(195\) 0 0
\(196\) −3022.18 −1.10138
\(197\) − 1071.99i − 0.387695i −0.981032 0.193847i \(-0.937903\pi\)
0.981032 0.193847i \(-0.0620966\pi\)
\(198\) − 4403.08i − 1.58037i
\(199\) 2879.31 1.02567 0.512836 0.858486i \(-0.328595\pi\)
0.512836 + 0.858486i \(0.328595\pi\)
\(200\) 0 0
\(201\) 2509.07 0.880477
\(202\) − 6406.19i − 2.23138i
\(203\) − 1240.66i − 0.428952i
\(204\) −2273.04 −0.780120
\(205\) 0 0
\(206\) 973.077 0.329114
\(207\) 793.238i 0.266347i
\(208\) − 378.855i − 0.126293i
\(209\) −1029.74 −0.340807
\(210\) 0 0
\(211\) −4746.47 −1.54863 −0.774313 0.632802i \(-0.781905\pi\)
−0.774313 + 0.632802i \(0.781905\pi\)
\(212\) 4850.59i 1.57141i
\(213\) − 4345.25i − 1.39780i
\(214\) −261.810 −0.0836306
\(215\) 0 0
\(216\) 911.363 0.287085
\(217\) 360.050i 0.112635i
\(218\) 8685.90i 2.69855i
\(219\) 5529.26 1.70609
\(220\) 0 0
\(221\) 405.048 0.123287
\(222\) − 13153.9i − 3.97673i
\(223\) 4874.04i 1.46363i 0.681503 + 0.731815i \(0.261327\pi\)
−0.681503 + 0.731815i \(0.738673\pi\)
\(224\) −2047.42 −0.610709
\(225\) 0 0
\(226\) 8038.26 2.36592
\(227\) 2742.09i 0.801757i 0.916131 + 0.400879i \(0.131295\pi\)
−0.916131 + 0.400879i \(0.868705\pi\)
\(228\) − 3216.72i − 0.934353i
\(229\) −1528.38 −0.441041 −0.220520 0.975382i \(-0.570776\pi\)
−0.220520 + 0.975382i \(0.570776\pi\)
\(230\) 0 0
\(231\) 2033.44 0.579179
\(232\) − 2145.98i − 0.607287i
\(233\) 5552.78i 1.56126i 0.624991 + 0.780632i \(0.285103\pi\)
−0.624991 + 0.780632i \(0.714897\pi\)
\(234\) −2450.99 −0.684729
\(235\) 0 0
\(236\) 8703.12 2.40053
\(237\) − 9178.49i − 2.51564i
\(238\) − 997.909i − 0.271785i
\(239\) −2779.63 −0.752299 −0.376149 0.926559i \(-0.622752\pi\)
−0.376149 + 0.926559i \(0.622752\pi\)
\(240\) 0 0
\(241\) −5568.82 −1.48846 −0.744231 0.667922i \(-0.767184\pi\)
−0.744231 + 0.667922i \(0.767184\pi\)
\(242\) 2189.81i 0.581680i
\(243\) − 5276.68i − 1.39300i
\(244\) 3534.98 0.927475
\(245\) 0 0
\(246\) 14289.0 3.70340
\(247\) 573.209i 0.147662i
\(248\) 622.783i 0.159463i
\(249\) 3575.54 0.910003
\(250\) 0 0
\(251\) −387.065 −0.0973360 −0.0486680 0.998815i \(-0.515498\pi\)
−0.0486680 + 0.998815i \(0.515498\pi\)
\(252\) 3562.85i 0.890628i
\(253\) 664.726i 0.165182i
\(254\) −3686.39 −0.910649
\(255\) 0 0
\(256\) −1372.41 −0.335061
\(257\) − 1476.07i − 0.358267i −0.983825 0.179133i \(-0.942671\pi\)
0.983825 0.179133i \(-0.0573293\pi\)
\(258\) − 13928.1i − 3.36096i
\(259\) 3407.30 0.817449
\(260\) 0 0
\(261\) 4768.81 1.13097
\(262\) − 2254.17i − 0.531537i
\(263\) 203.984i 0.0478259i 0.999714 + 0.0239130i \(0.00761246\pi\)
−0.999714 + 0.0239130i \(0.992388\pi\)
\(264\) 3517.26 0.819971
\(265\) 0 0
\(266\) 1412.20 0.325518
\(267\) 11727.5i 2.68806i
\(268\) 3683.98i 0.839683i
\(269\) 6973.71 1.58065 0.790324 0.612689i \(-0.209912\pi\)
0.790324 + 0.612689i \(0.209912\pi\)
\(270\) 0 0
\(271\) −2164.97 −0.485287 −0.242643 0.970116i \(-0.578015\pi\)
−0.242643 + 0.970116i \(0.578015\pi\)
\(272\) 592.898i 0.132168i
\(273\) − 1131.92i − 0.250941i
\(274\) −49.6246 −0.0109414
\(275\) 0 0
\(276\) −2076.48 −0.452861
\(277\) − 1194.00i − 0.258992i −0.991580 0.129496i \(-0.958664\pi\)
0.991580 0.129496i \(-0.0413359\pi\)
\(278\) − 7318.81i − 1.57897i
\(279\) −1383.95 −0.296971
\(280\) 0 0
\(281\) 6485.98 1.37694 0.688472 0.725263i \(-0.258282\pi\)
0.688472 + 0.725263i \(0.258282\pi\)
\(282\) − 3812.69i − 0.805116i
\(283\) − 3214.41i − 0.675182i −0.941293 0.337591i \(-0.890388\pi\)
0.941293 0.337591i \(-0.109612\pi\)
\(284\) 6380.00 1.33304
\(285\) 0 0
\(286\) −2053.91 −0.424651
\(287\) 3701.33i 0.761263i
\(288\) − 7869.81i − 1.61018i
\(289\) 4279.11 0.870977
\(290\) 0 0
\(291\) 8164.85 1.64478
\(292\) 8118.44i 1.62704i
\(293\) 5585.47i 1.11367i 0.830622 + 0.556837i \(0.187985\pi\)
−0.830622 + 0.556837i \(0.812015\pi\)
\(294\) 9092.44 1.80368
\(295\) 0 0
\(296\) 5893.65 1.15730
\(297\) 1697.13i 0.331573i
\(298\) − 2253.78i − 0.438115i
\(299\) 370.022 0.0715684
\(300\) 0 0
\(301\) 3607.84 0.690872
\(302\) 10325.1i 1.96737i
\(303\) 11371.8i 2.15609i
\(304\) −839.047 −0.158298
\(305\) 0 0
\(306\) 3835.73 0.716583
\(307\) − 2849.51i − 0.529740i −0.964284 0.264870i \(-0.914671\pi\)
0.964284 0.264870i \(-0.0853291\pi\)
\(308\) 2985.63i 0.552344i
\(309\) −1727.34 −0.318010
\(310\) 0 0
\(311\) −6617.84 −1.20664 −0.603318 0.797501i \(-0.706155\pi\)
−0.603318 + 0.797501i \(0.706155\pi\)
\(312\) − 1957.90i − 0.355269i
\(313\) − 6271.26i − 1.13250i −0.824233 0.566250i \(-0.808394\pi\)
0.824233 0.566250i \(-0.191606\pi\)
\(314\) −647.554 −0.116381
\(315\) 0 0
\(316\) 13476.5 2.39909
\(317\) − 5650.73i − 1.00119i −0.865682 0.500594i \(-0.833115\pi\)
0.865682 0.500594i \(-0.166885\pi\)
\(318\) − 14593.3i − 2.57344i
\(319\) 3996.22 0.701395
\(320\) 0 0
\(321\) 464.748 0.0808090
\(322\) − 911.617i − 0.157772i
\(323\) − 897.055i − 0.154531i
\(324\) 5419.67 0.929299
\(325\) 0 0
\(326\) −14481.0 −2.46022
\(327\) − 15418.6i − 2.60750i
\(328\) 6402.23i 1.07776i
\(329\) 987.612 0.165498
\(330\) 0 0
\(331\) −6391.40 −1.06134 −0.530669 0.847579i \(-0.678059\pi\)
−0.530669 + 0.847579i \(0.678059\pi\)
\(332\) 5249.86i 0.867841i
\(333\) 13096.9i 2.15527i
\(334\) 6870.92 1.12563
\(335\) 0 0
\(336\) 1656.87 0.269017
\(337\) − 4153.87i − 0.671441i −0.941962 0.335721i \(-0.891020\pi\)
0.941962 0.335721i \(-0.108980\pi\)
\(338\) − 8561.70i − 1.37780i
\(339\) −14269.0 −2.28609
\(340\) 0 0
\(341\) −1159.74 −0.184174
\(342\) 5428.19i 0.858254i
\(343\) 5432.84i 0.855236i
\(344\) 6240.52 0.978100
\(345\) 0 0
\(346\) −2086.74 −0.324231
\(347\) 3071.86i 0.475234i 0.971359 + 0.237617i \(0.0763663\pi\)
−0.971359 + 0.237617i \(0.923634\pi\)
\(348\) 12483.4i 1.92294i
\(349\) 8358.91 1.28207 0.641035 0.767512i \(-0.278505\pi\)
0.641035 + 0.767512i \(0.278505\pi\)
\(350\) 0 0
\(351\) 944.712 0.143661
\(352\) − 6594.82i − 0.998594i
\(353\) 8018.12i 1.20896i 0.796622 + 0.604478i \(0.206618\pi\)
−0.796622 + 0.604478i \(0.793382\pi\)
\(354\) −26184.0 −3.93125
\(355\) 0 0
\(356\) −17219.1 −2.56352
\(357\) 1771.42i 0.262615i
\(358\) 10731.8i 1.58434i
\(359\) −2138.46 −0.314384 −0.157192 0.987568i \(-0.550244\pi\)
−0.157192 + 0.987568i \(0.550244\pi\)
\(360\) 0 0
\(361\) −5589.52 −0.814918
\(362\) 4339.47i 0.630048i
\(363\) − 3887.21i − 0.562054i
\(364\) 1661.96 0.239315
\(365\) 0 0
\(366\) −10635.2 −1.51889
\(367\) 4509.96i 0.641467i 0.947170 + 0.320733i \(0.103929\pi\)
−0.947170 + 0.320733i \(0.896071\pi\)
\(368\) 541.628i 0.0767237i
\(369\) −14227.1 −2.00713
\(370\) 0 0
\(371\) 3780.15 0.528991
\(372\) − 3622.80i − 0.504929i
\(373\) − 3362.09i − 0.466709i −0.972392 0.233355i \(-0.925030\pi\)
0.972392 0.233355i \(-0.0749703\pi\)
\(374\) 3214.31 0.444406
\(375\) 0 0
\(376\) 1708.29 0.234303
\(377\) − 2224.51i − 0.303894i
\(378\) − 2327.47i − 0.316699i
\(379\) −2107.16 −0.285587 −0.142793 0.989753i \(-0.545608\pi\)
−0.142793 + 0.989753i \(0.545608\pi\)
\(380\) 0 0
\(381\) 6543.84 0.879924
\(382\) − 6012.71i − 0.805333i
\(383\) 1373.05i 0.183184i 0.995797 + 0.0915919i \(0.0291955\pi\)
−0.995797 + 0.0915919i \(0.970804\pi\)
\(384\) 14075.3 1.87052
\(385\) 0 0
\(386\) −6433.03 −0.848271
\(387\) 13867.7i 1.82154i
\(388\) 11988.2i 1.56858i
\(389\) −7008.05 −0.913425 −0.456713 0.889614i \(-0.650973\pi\)
−0.456713 + 0.889614i \(0.650973\pi\)
\(390\) 0 0
\(391\) −579.074 −0.0748978
\(392\) 4073.89i 0.524904i
\(393\) 4001.44i 0.513604i
\(394\) −4735.39 −0.605496
\(395\) 0 0
\(396\) −11476.1 −1.45630
\(397\) − 2402.53i − 0.303727i −0.988401 0.151864i \(-0.951473\pi\)
0.988401 0.151864i \(-0.0485275\pi\)
\(398\) − 12719.0i − 1.60188i
\(399\) −2506.85 −0.314535
\(400\) 0 0
\(401\) −11902.0 −1.48218 −0.741091 0.671404i \(-0.765691\pi\)
−0.741091 + 0.671404i \(0.765691\pi\)
\(402\) − 11083.5i − 1.37512i
\(403\) 645.572i 0.0797971i
\(404\) −16696.9 −2.05620
\(405\) 0 0
\(406\) −5480.48 −0.669930
\(407\) 10975.1i 1.33664i
\(408\) 3064.05i 0.371797i
\(409\) −5277.96 −0.638089 −0.319044 0.947740i \(-0.603362\pi\)
−0.319044 + 0.947740i \(0.603362\pi\)
\(410\) 0 0
\(411\) 88.0903 0.0105722
\(412\) − 2536.20i − 0.303276i
\(413\) − 6782.51i − 0.808100i
\(414\) 3504.05 0.415977
\(415\) 0 0
\(416\) −3671.03 −0.432662
\(417\) 12991.9i 1.52569i
\(418\) 4548.77i 0.532267i
\(419\) 11196.4 1.30545 0.652723 0.757597i \(-0.273627\pi\)
0.652723 + 0.757597i \(0.273627\pi\)
\(420\) 0 0
\(421\) 5176.82 0.599293 0.299647 0.954050i \(-0.403131\pi\)
0.299647 + 0.954050i \(0.403131\pi\)
\(422\) 20967.0i 2.41862i
\(423\) 3796.16i 0.436349i
\(424\) 6538.58 0.748918
\(425\) 0 0
\(426\) −19194.7 −2.18307
\(427\) − 2754.88i − 0.312220i
\(428\) 682.375i 0.0770650i
\(429\) 3645.96 0.410324
\(430\) 0 0
\(431\) 9348.93 1.04483 0.522415 0.852691i \(-0.325031\pi\)
0.522415 + 0.852691i \(0.325031\pi\)
\(432\) 1382.84i 0.154009i
\(433\) 4320.91i 0.479560i 0.970827 + 0.239780i \(0.0770753\pi\)
−0.970827 + 0.239780i \(0.922925\pi\)
\(434\) 1590.48 0.175912
\(435\) 0 0
\(436\) 22638.7 2.48669
\(437\) − 819.484i − 0.0897054i
\(438\) − 24424.9i − 2.66454i
\(439\) 7016.03 0.762772 0.381386 0.924416i \(-0.375447\pi\)
0.381386 + 0.924416i \(0.375447\pi\)
\(440\) 0 0
\(441\) −9053.00 −0.977541
\(442\) − 1789.26i − 0.192548i
\(443\) 12343.6i 1.32384i 0.749574 + 0.661921i \(0.230259\pi\)
−0.749574 + 0.661921i \(0.769741\pi\)
\(444\) −34284.1 −3.66453
\(445\) 0 0
\(446\) 21530.5 2.28588
\(447\) 4000.77i 0.423333i
\(448\) 7353.88i 0.775532i
\(449\) −18146.9 −1.90737 −0.953683 0.300815i \(-0.902741\pi\)
−0.953683 + 0.300815i \(0.902741\pi\)
\(450\) 0 0
\(451\) −11922.1 −1.24477
\(452\) − 20950.7i − 2.18017i
\(453\) − 18328.5i − 1.90099i
\(454\) 12112.9 1.25217
\(455\) 0 0
\(456\) −4336.13 −0.445302
\(457\) − 5233.82i − 0.535728i −0.963457 0.267864i \(-0.913682\pi\)
0.963457 0.267864i \(-0.0863177\pi\)
\(458\) 6751.47i 0.688811i
\(459\) −1478.45 −0.150344
\(460\) 0 0
\(461\) −2335.01 −0.235905 −0.117952 0.993019i \(-0.537633\pi\)
−0.117952 + 0.993019i \(0.537633\pi\)
\(462\) − 8982.49i − 0.904553i
\(463\) 13644.1i 1.36954i 0.728760 + 0.684769i \(0.240097\pi\)
−0.728760 + 0.684769i \(0.759903\pi\)
\(464\) 3256.17 0.325784
\(465\) 0 0
\(466\) 24528.8 2.43836
\(467\) 5246.31i 0.519851i 0.965629 + 0.259925i \(0.0836979\pi\)
−0.965629 + 0.259925i \(0.916302\pi\)
\(468\) 6388.21i 0.630972i
\(469\) 2871.00 0.282666
\(470\) 0 0
\(471\) 1149.50 0.112454
\(472\) − 11731.8i − 1.14407i
\(473\) 11621.0i 1.12967i
\(474\) −40545.0 −3.92889
\(475\) 0 0
\(476\) −2600.92 −0.250448
\(477\) 14530.0i 1.39473i
\(478\) 12278.7i 1.17493i
\(479\) −12278.3 −1.17121 −0.585603 0.810598i \(-0.699142\pi\)
−0.585603 + 0.810598i \(0.699142\pi\)
\(480\) 0 0
\(481\) 6109.31 0.579128
\(482\) 24599.7i 2.32466i
\(483\) 1618.24i 0.152448i
\(484\) 5707.47 0.536013
\(485\) 0 0
\(486\) −23309.2 −2.17557
\(487\) 8945.24i 0.832336i 0.909288 + 0.416168i \(0.136627\pi\)
−0.909288 + 0.416168i \(0.863373\pi\)
\(488\) − 4765.14i − 0.442024i
\(489\) 25705.8 2.37721
\(490\) 0 0
\(491\) −7792.94 −0.716274 −0.358137 0.933669i \(-0.616588\pi\)
−0.358137 + 0.933669i \(0.616588\pi\)
\(492\) − 37242.6i − 3.41265i
\(493\) 3481.29i 0.318031i
\(494\) 2532.09 0.230616
\(495\) 0 0
\(496\) −944.970 −0.0855451
\(497\) − 4972.06i − 0.448747i
\(498\) − 15794.6i − 1.42123i
\(499\) 8577.71 0.769521 0.384761 0.923016i \(-0.374284\pi\)
0.384761 + 0.923016i \(0.374284\pi\)
\(500\) 0 0
\(501\) −12196.8 −1.08765
\(502\) 1709.82i 0.152018i
\(503\) 7784.99i 0.690091i 0.938586 + 0.345045i \(0.112136\pi\)
−0.938586 + 0.345045i \(0.887864\pi\)
\(504\) 4802.71 0.424464
\(505\) 0 0
\(506\) 2936.36 0.257978
\(507\) 15198.2i 1.33131i
\(508\) 9608.12i 0.839156i
\(509\) 11390.4 0.991891 0.495946 0.868354i \(-0.334822\pi\)
0.495946 + 0.868354i \(0.334822\pi\)
\(510\) 0 0
\(511\) 6326.85 0.547717
\(512\) − 8297.40i − 0.716205i
\(513\) − 2092.24i − 0.180068i
\(514\) −6520.37 −0.559535
\(515\) 0 0
\(516\) −36301.9 −3.09710
\(517\) 3181.14i 0.270612i
\(518\) − 15051.4i − 1.27668i
\(519\) 3704.24 0.313292
\(520\) 0 0
\(521\) −12824.5 −1.07841 −0.539206 0.842174i \(-0.681276\pi\)
−0.539206 + 0.842174i \(0.681276\pi\)
\(522\) − 21065.7i − 1.76632i
\(523\) 13087.4i 1.09421i 0.837063 + 0.547107i \(0.184271\pi\)
−0.837063 + 0.547107i \(0.815729\pi\)
\(524\) −5875.20 −0.489808
\(525\) 0 0
\(526\) 901.080 0.0746938
\(527\) − 1010.30i − 0.0835093i
\(528\) 5336.86i 0.439880i
\(529\) −529.000 −0.0434783
\(530\) 0 0
\(531\) 26070.4 2.13062
\(532\) − 3680.73i − 0.299962i
\(533\) 6636.50i 0.539322i
\(534\) 51805.0 4.19817
\(535\) 0 0
\(536\) 4966.00 0.400184
\(537\) − 19050.4i − 1.53089i
\(538\) − 30805.6i − 2.46863i
\(539\) −7586.33 −0.606245
\(540\) 0 0
\(541\) 15464.3 1.22895 0.614473 0.788938i \(-0.289369\pi\)
0.614473 + 0.788938i \(0.289369\pi\)
\(542\) 9563.54i 0.757914i
\(543\) − 7703.14i − 0.608791i
\(544\) 5745.06 0.452789
\(545\) 0 0
\(546\) −5000.14 −0.391916
\(547\) − 6981.32i − 0.545703i −0.962056 0.272852i \(-0.912033\pi\)
0.962056 0.272852i \(-0.0879668\pi\)
\(548\) 129.340i 0.0100824i
\(549\) 10589.1 0.823192
\(550\) 0 0
\(551\) −4926.60 −0.380908
\(552\) 2799.09i 0.215828i
\(553\) − 10502.5i − 0.807615i
\(554\) −5274.38 −0.404489
\(555\) 0 0
\(556\) −19075.5 −1.45501
\(557\) − 24964.8i − 1.89909i −0.313637 0.949543i \(-0.601547\pi\)
0.313637 0.949543i \(-0.398453\pi\)
\(558\) 6113.45i 0.463805i
\(559\) 6468.88 0.489454
\(560\) 0 0
\(561\) −5705.83 −0.429412
\(562\) − 28651.2i − 2.15049i
\(563\) − 10820.3i − 0.809986i −0.914320 0.404993i \(-0.867274\pi\)
0.914320 0.404993i \(-0.132726\pi\)
\(564\) −9937.31 −0.741908
\(565\) 0 0
\(566\) −14199.3 −1.05449
\(567\) − 4223.65i − 0.312834i
\(568\) − 8600.23i − 0.635312i
\(569\) −8438.73 −0.621740 −0.310870 0.950453i \(-0.600620\pi\)
−0.310870 + 0.950453i \(0.600620\pi\)
\(570\) 0 0
\(571\) 23107.6 1.69356 0.846781 0.531942i \(-0.178538\pi\)
0.846781 + 0.531942i \(0.178538\pi\)
\(572\) 5353.26i 0.391313i
\(573\) 10673.4i 0.778161i
\(574\) 16350.2 1.18893
\(575\) 0 0
\(576\) −28266.7 −2.04475
\(577\) − 24848.7i − 1.79283i −0.443215 0.896416i \(-0.646162\pi\)
0.443215 0.896416i \(-0.353838\pi\)
\(578\) − 18902.5i − 1.36028i
\(579\) 11419.5 0.819651
\(580\) 0 0
\(581\) 4091.31 0.292145
\(582\) − 36067.4i − 2.56880i
\(583\) 12176.0i 0.864974i
\(584\) 10943.6 0.775430
\(585\) 0 0
\(586\) 24673.2 1.73932
\(587\) − 8663.94i − 0.609198i −0.952481 0.304599i \(-0.901478\pi\)
0.952481 0.304599i \(-0.0985224\pi\)
\(588\) − 23698.3i − 1.66208i
\(589\) 1429.74 0.100019
\(590\) 0 0
\(591\) 8405.94 0.585066
\(592\) 8942.64i 0.620845i
\(593\) − 24678.9i − 1.70901i −0.519446 0.854503i \(-0.673862\pi\)
0.519446 0.854503i \(-0.326138\pi\)
\(594\) 7496.88 0.517846
\(595\) 0 0
\(596\) −5874.20 −0.403719
\(597\) 22578.0i 1.54783i
\(598\) − 1634.54i − 0.111774i
\(599\) −19698.4 −1.34367 −0.671833 0.740702i \(-0.734493\pi\)
−0.671833 + 0.740702i \(0.734493\pi\)
\(600\) 0 0
\(601\) −18449.1 −1.25217 −0.626086 0.779754i \(-0.715344\pi\)
−0.626086 + 0.779754i \(0.715344\pi\)
\(602\) − 15937.3i − 1.07899i
\(603\) 11035.5i 0.745272i
\(604\) 26911.2 1.81291
\(605\) 0 0
\(606\) 50233.9 3.36735
\(607\) 11321.0i 0.757012i 0.925599 + 0.378506i \(0.123562\pi\)
−0.925599 + 0.378506i \(0.876438\pi\)
\(608\) 8130.20i 0.542308i
\(609\) 9728.59 0.647327
\(610\) 0 0
\(611\) 1770.80 0.117248
\(612\) − 9997.36i − 0.660326i
\(613\) 712.219i 0.0469270i 0.999725 + 0.0234635i \(0.00746935\pi\)
−0.999725 + 0.0234635i \(0.992531\pi\)
\(614\) −12587.4 −0.827341
\(615\) 0 0
\(616\) 4024.62 0.263241
\(617\) − 3234.25i − 0.211031i −0.994418 0.105515i \(-0.966351\pi\)
0.994418 0.105515i \(-0.0336492\pi\)
\(618\) 7630.36i 0.496663i
\(619\) −26905.1 −1.74703 −0.873513 0.486801i \(-0.838164\pi\)
−0.873513 + 0.486801i \(0.838164\pi\)
\(620\) 0 0
\(621\) −1350.60 −0.0872750
\(622\) 29233.6i 1.88451i
\(623\) 13419.2i 0.862967i
\(624\) 2970.78 0.190587
\(625\) 0 0
\(626\) −27702.6 −1.76872
\(627\) − 8074.68i − 0.514309i
\(628\) 1687.77i 0.107244i
\(629\) −9560.90 −0.606070
\(630\) 0 0
\(631\) 16199.5 1.02202 0.511008 0.859576i \(-0.329272\pi\)
0.511008 + 0.859576i \(0.329272\pi\)
\(632\) − 18166.3i − 1.14338i
\(633\) − 37219.3i − 2.33702i
\(634\) −24961.5 −1.56364
\(635\) 0 0
\(636\) −38035.7 −2.37141
\(637\) 4222.96i 0.262668i
\(638\) − 17652.9i − 1.09543i
\(639\) 19111.5 1.18316
\(640\) 0 0
\(641\) 19943.5 1.22889 0.614446 0.788959i \(-0.289380\pi\)
0.614446 + 0.788959i \(0.289380\pi\)
\(642\) − 2052.97i − 0.126206i
\(643\) − 27691.0i − 1.69833i −0.528129 0.849164i \(-0.677106\pi\)
0.528129 0.849164i \(-0.322894\pi\)
\(644\) −2376.01 −0.145385
\(645\) 0 0
\(646\) −3962.65 −0.241344
\(647\) 23560.3i 1.43161i 0.698300 + 0.715805i \(0.253940\pi\)
−0.698300 + 0.715805i \(0.746060\pi\)
\(648\) − 7305.70i − 0.442893i
\(649\) 21846.7 1.32136
\(650\) 0 0
\(651\) −2823.32 −0.169976
\(652\) 37743.0i 2.26707i
\(653\) 5571.58i 0.333894i 0.985966 + 0.166947i \(0.0533909\pi\)
−0.985966 + 0.166947i \(0.946609\pi\)
\(654\) −68110.2 −4.07235
\(655\) 0 0
\(656\) −9714.32 −0.578171
\(657\) 24319.0i 1.44410i
\(658\) − 4362.68i − 0.258472i
\(659\) −10177.4 −0.601602 −0.300801 0.953687i \(-0.597254\pi\)
−0.300801 + 0.953687i \(0.597254\pi\)
\(660\) 0 0
\(661\) −27413.8 −1.61312 −0.806561 0.591151i \(-0.798674\pi\)
−0.806561 + 0.591151i \(0.798674\pi\)
\(662\) 28233.3i 1.65758i
\(663\) 3176.17i 0.186052i
\(664\) 7076.79 0.413604
\(665\) 0 0
\(666\) 57854.1 3.36607
\(667\) 3180.25i 0.184618i
\(668\) − 17908.2i − 1.03726i
\(669\) −38219.6 −2.20875
\(670\) 0 0
\(671\) 8873.57 0.510522
\(672\) − 16054.8i − 0.921616i
\(673\) 12767.5i 0.731277i 0.930757 + 0.365639i \(0.119149\pi\)
−0.930757 + 0.365639i \(0.880851\pi\)
\(674\) −18349.3 −1.04865
\(675\) 0 0
\(676\) −22315.0 −1.26963
\(677\) − 17036.9i − 0.967180i −0.875295 0.483590i \(-0.839333\pi\)
0.875295 0.483590i \(-0.160667\pi\)
\(678\) 63031.7i 3.57038i
\(679\) 9342.63 0.528037
\(680\) 0 0
\(681\) −21502.0 −1.20993
\(682\) 5123.02i 0.287640i
\(683\) − 510.213i − 0.0285838i −0.999898 0.0142919i \(-0.995451\pi\)
0.999898 0.0142919i \(-0.00454941\pi\)
\(684\) 14147.9 0.790875
\(685\) 0 0
\(686\) 23999.0 1.33569
\(687\) − 11984.8i − 0.665570i
\(688\) 9468.96i 0.524710i
\(689\) 6777.84 0.374768
\(690\) 0 0
\(691\) 22793.1 1.25483 0.627416 0.778684i \(-0.284112\pi\)
0.627416 + 0.778684i \(0.284112\pi\)
\(692\) 5438.83i 0.298776i
\(693\) 8943.53i 0.490240i
\(694\) 13569.6 0.742213
\(695\) 0 0
\(696\) 16827.7 0.916452
\(697\) − 10385.9i − 0.564412i
\(698\) − 36924.6i − 2.00232i
\(699\) −43541.9 −2.35609
\(700\) 0 0
\(701\) 26324.3 1.41834 0.709168 0.705039i \(-0.249071\pi\)
0.709168 + 0.705039i \(0.249071\pi\)
\(702\) − 4173.17i − 0.224368i
\(703\) − 13530.2i − 0.725892i
\(704\) −23687.2 −1.26810
\(705\) 0 0
\(706\) 35419.2 1.88813
\(707\) 13012.2i 0.692185i
\(708\) 68245.2i 3.62262i
\(709\) 10961.3 0.580620 0.290310 0.956933i \(-0.406242\pi\)
0.290310 + 0.956933i \(0.406242\pi\)
\(710\) 0 0
\(711\) 40369.2 2.12934
\(712\) 23211.3i 1.22174i
\(713\) − 922.938i − 0.0484773i
\(714\) 7825.07 0.410148
\(715\) 0 0
\(716\) 27971.2 1.45996
\(717\) − 21796.4i − 1.13529i
\(718\) 9446.44i 0.491000i
\(719\) 1304.68 0.0676723 0.0338362 0.999427i \(-0.489228\pi\)
0.0338362 + 0.999427i \(0.489228\pi\)
\(720\) 0 0
\(721\) −1976.51 −0.102093
\(722\) 24691.1i 1.27273i
\(723\) − 43667.7i − 2.24622i
\(724\) 11310.3 0.580585
\(725\) 0 0
\(726\) −17171.3 −0.877808
\(727\) − 1583.59i − 0.0807869i −0.999184 0.0403934i \(-0.987139\pi\)
0.999184 0.0403934i \(-0.0128611\pi\)
\(728\) − 2240.32i − 0.114055i
\(729\) 28667.3 1.45645
\(730\) 0 0
\(731\) −10123.6 −0.512223
\(732\) 27719.4i 1.39964i
\(733\) − 34351.2i − 1.73095i −0.500948 0.865477i \(-0.667015\pi\)
0.500948 0.865477i \(-0.332985\pi\)
\(734\) 19922.3 1.00183
\(735\) 0 0
\(736\) 5248.27 0.262845
\(737\) 9247.61i 0.462198i
\(738\) 62846.5i 3.13471i
\(739\) −14996.6 −0.746494 −0.373247 0.927732i \(-0.621756\pi\)
−0.373247 + 0.927732i \(0.621756\pi\)
\(740\) 0 0
\(741\) −4494.80 −0.222835
\(742\) − 16698.4i − 0.826171i
\(743\) − 32781.7i − 1.61863i −0.587372 0.809317i \(-0.699838\pi\)
0.587372 0.809317i \(-0.300162\pi\)
\(744\) −4883.53 −0.240644
\(745\) 0 0
\(746\) −14851.7 −0.728899
\(747\) 15726.1i 0.770263i
\(748\) − 8377.69i − 0.409517i
\(749\) 531.787 0.0259427
\(750\) 0 0
\(751\) −24044.4 −1.16830 −0.584149 0.811647i \(-0.698571\pi\)
−0.584149 + 0.811647i \(0.698571\pi\)
\(752\) 2592.04i 0.125694i
\(753\) − 3035.16i − 0.146889i
\(754\) −9826.54 −0.474617
\(755\) 0 0
\(756\) −6066.26 −0.291835
\(757\) 21680.9i 1.04096i 0.853874 + 0.520479i \(0.174247\pi\)
−0.853874 + 0.520479i \(0.825753\pi\)
\(758\) 9308.15i 0.446025i
\(759\) −5212.43 −0.249274
\(760\) 0 0
\(761\) −10299.1 −0.490594 −0.245297 0.969448i \(-0.578885\pi\)
−0.245297 + 0.969448i \(0.578885\pi\)
\(762\) − 28906.7i − 1.37425i
\(763\) − 17642.8i − 0.837105i
\(764\) −15671.4 −0.742108
\(765\) 0 0
\(766\) 6065.29 0.286094
\(767\) − 12161.1i − 0.572505i
\(768\) − 10761.7i − 0.505637i
\(769\) −28377.9 −1.33073 −0.665365 0.746518i \(-0.731724\pi\)
−0.665365 + 0.746518i \(0.731724\pi\)
\(770\) 0 0
\(771\) 11574.5 0.540657
\(772\) 16766.9i 0.781675i
\(773\) 35409.5i 1.64759i 0.566885 + 0.823797i \(0.308148\pi\)
−0.566885 + 0.823797i \(0.691852\pi\)
\(774\) 61259.2 2.84485
\(775\) 0 0
\(776\) 16160.1 0.747568
\(777\) 26718.2i 1.23361i
\(778\) 30957.3i 1.42657i
\(779\) 14697.8 0.675999
\(780\) 0 0
\(781\) 16015.2 0.733763
\(782\) 2558.00i 0.116974i
\(783\) 8119.58i 0.370588i
\(784\) −6181.45 −0.281589
\(785\) 0 0
\(786\) 17676.0 0.802138
\(787\) − 6117.56i − 0.277087i −0.990356 0.138544i \(-0.955758\pi\)
0.990356 0.138544i \(-0.0442421\pi\)
\(788\) 12342.2i 0.557960i
\(789\) −1599.54 −0.0721737
\(790\) 0 0
\(791\) −16327.3 −0.733921
\(792\) 15469.7i 0.694057i
\(793\) − 4939.51i − 0.221194i
\(794\) −10612.9 −0.474357
\(795\) 0 0
\(796\) −33150.6 −1.47612
\(797\) − 4099.46i − 0.182196i −0.995842 0.0910980i \(-0.970962\pi\)
0.995842 0.0910980i \(-0.0290377\pi\)
\(798\) 11073.8i 0.491236i
\(799\) −2771.24 −0.122703
\(800\) 0 0
\(801\) −51580.3 −2.27528
\(802\) 52575.6i 2.31485i
\(803\) 20379.1i 0.895594i
\(804\) −28887.8 −1.26716
\(805\) 0 0
\(806\) 2851.75 0.124626
\(807\) 54684.1i 2.38534i
\(808\) 22507.4i 0.979960i
\(809\) −21358.6 −0.928216 −0.464108 0.885779i \(-0.653625\pi\)
−0.464108 + 0.885779i \(0.653625\pi\)
\(810\) 0 0
\(811\) 13967.7 0.604776 0.302388 0.953185i \(-0.402216\pi\)
0.302388 + 0.953185i \(0.402216\pi\)
\(812\) 14284.2i 0.617336i
\(813\) − 16976.6i − 0.732342i
\(814\) 48481.2 2.08755
\(815\) 0 0
\(816\) −4649.19 −0.199454
\(817\) − 14326.6i − 0.613492i
\(818\) 23314.8i 0.996557i
\(819\) 4978.45 0.212407
\(820\) 0 0
\(821\) −22387.6 −0.951684 −0.475842 0.879531i \(-0.657857\pi\)
−0.475842 + 0.879531i \(0.657857\pi\)
\(822\) − 389.130i − 0.0165115i
\(823\) − 22615.7i − 0.957877i −0.877848 0.478939i \(-0.841022\pi\)
0.877848 0.478939i \(-0.158978\pi\)
\(824\) −3418.80 −0.144538
\(825\) 0 0
\(826\) −29961.0 −1.26208
\(827\) − 10878.0i − 0.457394i −0.973498 0.228697i \(-0.926554\pi\)
0.973498 0.228697i \(-0.0734464\pi\)
\(828\) − 9132.86i − 0.383320i
\(829\) −27382.3 −1.14720 −0.573600 0.819136i \(-0.694453\pi\)
−0.573600 + 0.819136i \(0.694453\pi\)
\(830\) 0 0
\(831\) 9362.74 0.390842
\(832\) 13185.6i 0.549432i
\(833\) − 6608.81i − 0.274888i
\(834\) 57390.2 2.38281
\(835\) 0 0
\(836\) 11855.8 0.490480
\(837\) − 2356.37i − 0.0973096i
\(838\) − 49459.1i − 2.03883i
\(839\) 31799.7 1.30852 0.654260 0.756270i \(-0.272980\pi\)
0.654260 + 0.756270i \(0.272980\pi\)
\(840\) 0 0
\(841\) −5269.87 −0.216076
\(842\) − 22868.0i − 0.935968i
\(843\) 50859.6i 2.07793i
\(844\) 54647.9 2.22874
\(845\) 0 0
\(846\) 16769.1 0.681483
\(847\) − 4447.94i − 0.180440i
\(848\) 9921.21i 0.401764i
\(849\) 25205.7 1.01891
\(850\) 0 0
\(851\) −8734.14 −0.351824
\(852\) 50028.6i 2.01168i
\(853\) 32016.4i 1.28514i 0.766229 + 0.642568i \(0.222131\pi\)
−0.766229 + 0.642568i \(0.777869\pi\)
\(854\) −12169.4 −0.487620
\(855\) 0 0
\(856\) 919.839 0.0367283
\(857\) − 25280.1i − 1.00764i −0.863807 0.503822i \(-0.831927\pi\)
0.863807 0.503822i \(-0.168073\pi\)
\(858\) − 16105.7i − 0.640837i
\(859\) 22313.5 0.886296 0.443148 0.896448i \(-0.353862\pi\)
0.443148 + 0.896448i \(0.353862\pi\)
\(860\) 0 0
\(861\) −29023.8 −1.14881
\(862\) − 41297.9i − 1.63180i
\(863\) − 1478.28i − 0.0583096i −0.999575 0.0291548i \(-0.990718\pi\)
0.999575 0.0291548i \(-0.00928158\pi\)
\(864\) 13399.5 0.527615
\(865\) 0 0
\(866\) 19087.2 0.748970
\(867\) 33554.5i 1.31438i
\(868\) − 4145.39i − 0.162101i
\(869\) 33829.0 1.32056
\(870\) 0 0
\(871\) 5147.72 0.200257
\(872\) − 30516.9i − 1.18513i
\(873\) 35910.9i 1.39221i
\(874\) −3619.99 −0.140101
\(875\) 0 0
\(876\) −63660.5 −2.45535
\(877\) − 32974.6i − 1.26964i −0.772661 0.634819i \(-0.781075\pi\)
0.772661 0.634819i \(-0.218925\pi\)
\(878\) − 30992.6i − 1.19129i
\(879\) −43798.3 −1.68064
\(880\) 0 0
\(881\) −32000.7 −1.22376 −0.611879 0.790951i \(-0.709586\pi\)
−0.611879 + 0.790951i \(0.709586\pi\)
\(882\) 39990.7i 1.52671i
\(883\) − 44218.6i − 1.68525i −0.538503 0.842623i \(-0.681010\pi\)
0.538503 0.842623i \(-0.318990\pi\)
\(884\) −4663.47 −0.177432
\(885\) 0 0
\(886\) 54526.6 2.06756
\(887\) − 27374.8i − 1.03625i −0.855304 0.518126i \(-0.826630\pi\)
0.855304 0.518126i \(-0.173370\pi\)
\(888\) 46214.9i 1.74647i
\(889\) 7487.79 0.282489
\(890\) 0 0
\(891\) 13604.6 0.511526
\(892\) − 56116.7i − 2.10642i
\(893\) − 3921.76i − 0.146962i
\(894\) 17673.0 0.661155
\(895\) 0 0
\(896\) 16105.7 0.600505
\(897\) 2901.52i 0.108003i
\(898\) 80162.2i 2.97889i
\(899\) −5548.54 −0.205844
\(900\) 0 0
\(901\) −10607.1 −0.392203
\(902\) 52664.8i 1.94406i
\(903\) 28290.8i 1.04259i
\(904\) −28241.5 −1.03905
\(905\) 0 0
\(906\) −80964.3 −2.96894
\(907\) − 7835.16i − 0.286838i −0.989662 0.143419i \(-0.954190\pi\)
0.989662 0.143419i \(-0.0458097\pi\)
\(908\) − 31570.7i − 1.15387i
\(909\) −50016.0 −1.82500
\(910\) 0 0
\(911\) −36528.6 −1.32848 −0.664241 0.747519i \(-0.731245\pi\)
−0.664241 + 0.747519i \(0.731245\pi\)
\(912\) − 6579.36i − 0.238886i
\(913\) 13178.3i 0.477698i
\(914\) −23119.8 −0.836692
\(915\) 0 0
\(916\) 17596.8 0.634734
\(917\) 4578.65i 0.164886i
\(918\) 6530.89i 0.234805i
\(919\) 15741.5 0.565030 0.282515 0.959263i \(-0.408831\pi\)
0.282515 + 0.959263i \(0.408831\pi\)
\(920\) 0 0
\(921\) 22344.4 0.799427
\(922\) 10314.7i 0.368433i
\(923\) − 8914.93i − 0.317918i
\(924\) −23411.7 −0.833538
\(925\) 0 0
\(926\) 60271.5 2.13892
\(927\) − 7597.26i − 0.269177i
\(928\) − 31551.7i − 1.11609i
\(929\) −47428.9 −1.67502 −0.837510 0.546422i \(-0.815989\pi\)
−0.837510 + 0.546422i \(0.815989\pi\)
\(930\) 0 0
\(931\) 9352.54 0.329234
\(932\) − 63931.3i − 2.24693i
\(933\) − 51893.6i − 1.82092i
\(934\) 23175.0 0.811895
\(935\) 0 0
\(936\) 8611.29 0.300714
\(937\) 34978.0i 1.21951i 0.792589 + 0.609756i \(0.208733\pi\)
−0.792589 + 0.609756i \(0.791267\pi\)
\(938\) − 12682.3i − 0.441464i
\(939\) 49175.9 1.70905
\(940\) 0 0
\(941\) 45144.0 1.56392 0.781962 0.623326i \(-0.214219\pi\)
0.781962 + 0.623326i \(0.214219\pi\)
\(942\) − 5077.78i − 0.175629i
\(943\) − 9487.83i − 0.327642i
\(944\) 17801.0 0.613744
\(945\) 0 0
\(946\) 51334.6 1.76430
\(947\) − 26123.6i − 0.896413i −0.893930 0.448207i \(-0.852063\pi\)
0.893930 0.448207i \(-0.147937\pi\)
\(948\) 105675.i 3.62044i
\(949\) 11344.1 0.388035
\(950\) 0 0
\(951\) 44310.0 1.51088
\(952\) 3506.04i 0.119361i
\(953\) − 22143.5i − 0.752673i −0.926483 0.376336i \(-0.877184\pi\)
0.926483 0.376336i \(-0.122816\pi\)
\(954\) 64185.0 2.17827
\(955\) 0 0
\(956\) 32003.0 1.08269
\(957\) 31336.2i 1.05847i
\(958\) 54237.9i 1.82917i
\(959\) 100.797 0.00339407
\(960\) 0 0
\(961\) −28180.8 −0.945949
\(962\) − 26987.3i − 0.904474i
\(963\) 2044.07i 0.0684000i
\(964\) 64116.0 2.14215
\(965\) 0 0
\(966\) 7148.42 0.238092
\(967\) − 44869.8i − 1.49216i −0.665858 0.746078i \(-0.731934\pi\)
0.665858 0.746078i \(-0.268066\pi\)
\(968\) − 7693.65i − 0.255458i
\(969\) 7034.23 0.233201
\(970\) 0 0
\(971\) −25048.3 −0.827845 −0.413923 0.910312i \(-0.635842\pi\)
−0.413923 + 0.910312i \(0.635842\pi\)
\(972\) 60752.5i 2.00477i
\(973\) 14865.9i 0.489805i
\(974\) 39514.7 1.29993
\(975\) 0 0
\(976\) 7230.31 0.237128
\(977\) − 37320.7i − 1.22210i −0.791590 0.611052i \(-0.790747\pi\)
0.791590 0.611052i \(-0.209253\pi\)
\(978\) − 113553.i − 3.71269i
\(979\) −43223.8 −1.41107
\(980\) 0 0
\(981\) 67814.7 2.20709
\(982\) 34424.5i 1.11867i
\(983\) − 17189.4i − 0.557737i −0.960329 0.278869i \(-0.910041\pi\)
0.960329 0.278869i \(-0.0899594\pi\)
\(984\) −50202.9 −1.62643
\(985\) 0 0
\(986\) 15378.2 0.496697
\(987\) 7744.33i 0.249752i
\(988\) − 6599.58i − 0.212511i
\(989\) −9248.19 −0.297346
\(990\) 0 0
\(991\) −57797.1 −1.85266 −0.926330 0.376712i \(-0.877055\pi\)
−0.926330 + 0.376712i \(0.877055\pi\)
\(992\) 9156.57i 0.293066i
\(993\) − 50117.9i − 1.60166i
\(994\) −21963.5 −0.700846
\(995\) 0 0
\(996\) −41166.6 −1.30965
\(997\) − 46801.0i − 1.48666i −0.668923 0.743331i \(-0.733245\pi\)
0.668923 0.743331i \(-0.266755\pi\)
\(998\) − 37891.2i − 1.20183i
\(999\) −22299.3 −0.706226
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 575.4.b.i.24.2 10
5.2 odd 4 115.4.a.e.1.4 5
5.3 odd 4 575.4.a.j.1.2 5
5.4 even 2 inner 575.4.b.i.24.9 10
15.2 even 4 1035.4.a.k.1.2 5
20.7 even 4 1840.4.a.n.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.e.1.4 5 5.2 odd 4
575.4.a.j.1.2 5 5.3 odd 4
575.4.b.i.24.2 10 1.1 even 1 trivial
575.4.b.i.24.9 10 5.4 even 2 inner
1035.4.a.k.1.2 5 15.2 even 4
1840.4.a.n.1.2 5 20.7 even 4