Properties

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

$q$-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 \(0.285235\pi\)
−0.624665 + 0.780893i \(0.714765\pi\)
\(3\) − 7.84147i − 1.50909i −0.656248 0.754546i \(-0.727857\pi\)
0.656248 0.754546i \(-0.272143\pi\)
\(4\) −11.5134 −1.43917
\(5\) 0 0
\(6\) 34.6389 2.35688
\(7\) − 8.97260i − 0.484475i −0.970217 0.242237i \(-0.922119\pi\)
0.970217 0.242237i \(-0.0778813\pi\)
\(8\) − 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 \(-0.945102\pi\)
0.985164 0.171615i \(-0.0548985\pi\)
\(14\) 39.6355 0.756646
\(15\) 0 0
\(16\) −23.5490 −0.367954
\(17\) 25.1771i 0.359197i 0.983740 + 0.179599i \(0.0574799\pi\)
−0.983740 + 0.179599i \(0.942520\pi\)
\(18\) − 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 \(0.319595\pi\)
−0.536902 + 0.843645i \(0.680405\pi\)
\(38\) 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 \(0.252669\pi\)
−0.701153 + 0.713011i \(0.747331\pi\)
\(44\) 332.750 1.14009
\(45\) 0 0
\(46\) −101.600 −0.325655
\(47\) 110.070i 0.341603i 0.985305 + 0.170801i \(0.0546357\pi\)
−0.985305 + 0.170801i \(0.945364\pi\)
\(48\) 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 \(0.183829\pi\)
−0.837822 + 0.545943i \(0.816171\pi\)
\(54\) −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 \(0.0942290\pi\)
−0.956502 + 0.291724i \(0.905771\pi\)
\(68\) − 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 \(0.191228\pi\)
−0.824907 + 0.565269i \(0.808772\pi\)
\(74\) −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 \(0.0974896\pi\)
−0.953464 + 0.301507i \(0.902510\pi\)
\(84\) 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 \(0.183455\pi\)
−0.838463 + 0.544958i \(0.816545\pi\)
\(98\) 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 \(-0.966399\pi\)
0.994434 0.105365i \(-0.0336010\pi\)
\(104\) −249.685 −0.235419
\(105\) 0 0
\(106\) −1861.05 −1.70529
\(107\) 59.2679i 0.0535481i 0.999642 + 0.0267741i \(0.00852346\pi\)
−0.999642 + 0.0267741i \(0.991477\pi\)
\(108\) − 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 \(-0.726450\pi\)
0.652905 0.757439i \(-0.273550\pi\)
\(114\) 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 \(0.0941680\pi\)
−0.956558 + 0.291541i \(0.905832\pi\)
\(128\) 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 \(0.00111499\pi\)
−0.999994 + 0.00350284i \(0.998885\pi\)
\(138\) 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 \(0.0118626\pi\)
−0.999306 + 0.0372589i \(0.988137\pi\)
\(158\) − 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 \(0.288692\pi\)
−0.616149 + 0.787630i \(0.711308\pi\)
\(164\) −4749.44 −2.26139
\(165\) 0 0
\(166\) −2014.24 −0.941778
\(167\) − 1555.42i − 0.720732i −0.932811 0.360366i \(-0.882652\pi\)
0.932811 0.360366i \(-0.117348\pi\)
\(168\) 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 \(0.0331006\pi\)
−0.994598 + 0.103801i \(0.966899\pi\)
\(174\) −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 \(0.0875432\pi\)
−0.962418 + 0.271571i \(0.912457\pi\)
\(194\) −4599.57 −1.70222
\(195\) 0 0
\(196\) −3022.18 −1.10138
\(197\) 1071.99i 0.387695i 0.981032 + 0.193847i \(0.0620966\pi\)
−0.981032 + 0.193847i \(0.937903\pi\)
\(198\) 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 \(-0.738673\pi\)
0.681503 0.731815i \(-0.261327\pi\)
\(224\) −2047.42 −0.610709
\(225\) 0 0
\(226\) 8038.26 2.36592
\(227\) − 2742.09i − 0.801757i −0.916131 0.400879i \(-0.868705\pi\)
0.916131 0.400879i \(-0.131295\pi\)
\(228\) 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 \(-0.714897\pi\)
0.624991 0.780632i \(-0.285103\pi\)
\(234\) −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 \(0.0573293\pi\)
−0.983825 + 0.179133i \(0.942671\pi\)
\(258\) 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 \(-0.992388\pi\)
0.999714 0.0239130i \(-0.00761246\pi\)
\(264\) 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 \(0.0413359\pi\)
−0.991580 + 0.129496i \(0.958664\pi\)
\(278\) 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 \(0.109612\pi\)
−0.941293 + 0.337591i \(0.890388\pi\)
\(284\) 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 \(-0.812015\pi\)
0.830622 0.556837i \(-0.187985\pi\)
\(294\) 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 \(0.0853291\pi\)
−0.964284 + 0.264870i \(0.914671\pi\)
\(308\) − 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 \(0.191606\pi\)
−0.824233 + 0.566250i \(0.808394\pi\)
\(314\) −647.554 −0.116381
\(315\) 0 0
\(316\) 13476.5 2.39909
\(317\) 5650.73i 1.00119i 0.865682 + 0.500594i \(0.166885\pi\)
−0.865682 + 0.500594i \(0.833115\pi\)
\(318\) 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 \(0.108980\pi\)
−0.941962 + 0.335721i \(0.891020\pi\)
\(338\) 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 \(-0.923634\pi\)
0.971359 0.237617i \(-0.0763663\pi\)
\(348\) − 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 \(-0.793382\pi\)
0.796622 0.604478i \(-0.206618\pi\)
\(354\) −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 \(-0.896071\pi\)
0.947170 0.320733i \(-0.103929\pi\)
\(368\) − 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 \(0.0749703\pi\)
−0.972392 + 0.233355i \(0.925030\pi\)
\(374\) 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 \(-0.970804\pi\)
0.995797 0.0915919i \(-0.0291955\pi\)
\(384\) 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 \(0.0485275\pi\)
−0.988401 + 0.151864i \(0.951473\pi\)
\(398\) 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 \(-0.922925\pi\)
0.970827 0.239780i \(-0.0770753\pi\)
\(434\) 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 \(-0.769741\pi\)
0.749574 0.661921i \(-0.230259\pi\)
\(444\) −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 \(0.0863177\pi\)
−0.963457 + 0.267864i \(0.913682\pi\)
\(458\) − 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 \(-0.759903\pi\)
0.728760 0.684769i \(-0.240097\pi\)
\(464\) 3256.17 0.325784
\(465\) 0 0
\(466\) 24528.8 2.43836
\(467\) − 5246.31i − 0.519851i −0.965629 0.259925i \(-0.916302\pi\)
0.965629 0.259925i \(-0.0836979\pi\)
\(468\) − 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 \(-0.863373\pi\)
0.909288 0.416168i \(-0.136627\pi\)
\(488\) 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 \(-0.887864\pi\)
0.938586 0.345045i \(-0.112136\pi\)
\(504\) 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 \(-0.815729\pi\)
0.837063 0.547107i \(-0.184271\pi\)
\(524\) −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 \(0.0879668\pi\)
−0.962056 + 0.272852i \(0.912033\pi\)
\(548\) − 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 \(0.398453\pi\)
−0.313637 + 0.949543i \(0.601547\pi\)
\(558\) − 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 \(0.132726\pi\)
−0.914320 + 0.404993i \(0.867274\pi\)
\(564\) −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 \(0.353838\pi\)
−0.443215 + 0.896416i \(0.646162\pi\)
\(578\) 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 \(0.0985224\pi\)
−0.952481 + 0.304599i \(0.901478\pi\)
\(588\) 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 \(0.326138\pi\)
−0.519446 + 0.854503i \(0.673862\pi\)
\(594\) 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 \(-0.876438\pi\)
0.925599 0.378506i \(-0.123562\pi\)
\(608\) − 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 \(-0.992531\pi\)
0.999725 0.0234635i \(-0.00746935\pi\)
\(614\) −12587.4 −0.827341
\(615\) 0 0
\(616\) 4024.62 0.263241
\(617\) 3234.25i 0.211031i 0.994418 + 0.105515i \(0.0336492\pi\)
−0.994418 + 0.105515i \(0.966351\pi\)
\(618\) − 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 \(0.322894\pi\)
−0.528129 + 0.849164i \(0.677106\pi\)
\(644\) −2376.01 −0.145385
\(645\) 0 0
\(646\) −3962.65 −0.241344
\(647\) − 23560.3i − 1.43161i −0.698300 0.715805i \(-0.746060\pi\)
0.698300 0.715805i \(-0.253940\pi\)
\(648\) 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 \(-0.946609\pi\)
0.985966 0.166947i \(-0.0533909\pi\)
\(654\) −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 \(-0.880851\pi\)
0.930757 0.365639i \(-0.119149\pi\)
\(674\) −18349.3 −1.04865
\(675\) 0 0
\(676\) −22315.0 −1.26963
\(677\) 17036.9i 0.967180i 0.875295 + 0.483590i \(0.160667\pi\)
−0.875295 + 0.483590i \(0.839333\pi\)
\(678\) − 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 \(0.00454941\pi\)
−0.999898 + 0.0142919i \(0.995451\pi\)
\(684\) 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 \(0.0128611\pi\)
−0.999184 + 0.0403934i \(0.987139\pi\)
\(728\) 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 \(0.332985\pi\)
−0.500948 + 0.865477i \(0.667015\pi\)
\(734\) 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 \(0.300162\pi\)
−0.587372 + 0.809317i \(0.699838\pi\)
\(744\) −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 \(-0.825753\pi\)
0.853874 0.520479i \(-0.174247\pi\)
\(758\) − 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 \(-0.691852\pi\)
0.566885 0.823797i \(-0.308148\pi\)
\(774\) 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 \(0.0442421\pi\)
−0.990356 + 0.138544i \(0.955758\pi\)
\(788\) − 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 \(0.0290377\pi\)
−0.995842 + 0.0910980i \(0.970962\pi\)
\(798\) − 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 \(0.158978\pi\)
−0.877848 + 0.478939i \(0.841022\pi\)
\(824\) −3418.80 −0.144538
\(825\) 0 0
\(826\) −29961.0 −1.26208
\(827\) 10878.0i 0.457394i 0.973498 + 0.228697i \(0.0734464\pi\)
−0.973498 + 0.228697i \(0.926554\pi\)
\(828\) 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 \(-0.777869\pi\)
0.766229 0.642568i \(-0.222131\pi\)
\(854\) −12169.4 −0.487620
\(855\) 0 0
\(856\) 919.839 0.0367283
\(857\) 25280.1i 1.00764i 0.863807 + 0.503822i \(0.168073\pi\)
−0.863807 + 0.503822i \(0.831927\pi\)
\(858\) 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 \(0.00928158\pi\)
−0.999575 + 0.0291548i \(0.990718\pi\)
\(864\) 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 \(0.218925\pi\)
−0.772661 + 0.634819i \(0.781075\pi\)
\(878\) 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 \(0.318990\pi\)
−0.538503 + 0.842623i \(0.681010\pi\)
\(884\) −4663.47 −0.177432
\(885\) 0 0
\(886\) 54526.6 2.06756
\(887\) 27374.8i 1.03625i 0.855304 + 0.518126i \(0.173370\pi\)
−0.855304 + 0.518126i \(0.826630\pi\)
\(888\) − 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 \(0.0458097\pi\)
−0.989662 + 0.143419i \(0.954190\pi\)
\(908\) 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 \(-0.791267\pi\)
0.792589 0.609756i \(-0.208733\pi\)
\(938\) 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 \(0.147937\pi\)
−0.893930 + 0.448207i \(0.852063\pi\)
\(948\) − 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 \(0.122816\pi\)
−0.926483 + 0.376336i \(0.877184\pi\)
\(954\) 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 \(0.268066\pi\)
−0.665858 + 0.746078i \(0.731934\pi\)
\(968\) 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 \(0.209253\pi\)
−0.791590 + 0.611052i \(0.790747\pi\)
\(978\) 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 \(0.0899594\pi\)
−0.960329 + 0.278869i \(0.910041\pi\)
\(984\) −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 \(0.266755\pi\)
−0.668923 + 0.743331i \(0.733245\pi\)
\(998\) 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.24.9 10
5.2 odd 4 575.4.a.j.1.2 5
5.3 odd 4 115.4.a.e.1.4 5
5.4 even 2 inner 575.4.b.i.24.2 10
15.8 even 4 1035.4.a.k.1.2 5
20.3 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 5.3 odd 4
575.4.a.j.1.2 5 5.2 odd 4
575.4.b.i.24.2 10 5.4 even 2 inner
575.4.b.i.24.9 10 1.1 even 1 trivial
1035.4.a.k.1.2 5 15.8 even 4
1840.4.a.n.1.2 5 20.3 even 4