Properties

Label 531.5.b.a.296.52
Level $531$
Weight $5$
Character 531.296
Analytic conductor $54.889$
Analytic rank $0$
Dimension $76$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [531,5,Mod(296,531)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(531, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("531.296");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 531.b (of order \(2\), degree \(1\), 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: \(54.8894503975\)
Analytic rank: \(0\)
Dimension: \(76\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 296.52
Character \(\chi\) \(=\) 531.296
Dual form 531.5.b.a.296.25

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.23497i q^{2} +5.53498 q^{4} -9.83059i q^{5} -84.9856 q^{7} +69.6650i q^{8} +O(q^{10})\) \(q+3.23497i q^{2} +5.53498 q^{4} -9.83059i q^{5} -84.9856 q^{7} +69.6650i q^{8} +31.8017 q^{10} +4.98375i q^{11} +170.315 q^{13} -274.926i q^{14} -136.804 q^{16} -209.426i q^{17} -96.6592 q^{19} -54.4121i q^{20} -16.1223 q^{22} -778.979i q^{23} +528.359 q^{25} +550.964i q^{26} -470.393 q^{28} +357.930i q^{29} +74.8477 q^{31} +672.082i q^{32} +677.487 q^{34} +835.459i q^{35} -1846.48 q^{37} -312.690i q^{38} +684.848 q^{40} -627.641i q^{41} +2902.85 q^{43} +27.5849i q^{44} +2519.97 q^{46} +2788.16i q^{47} +4821.55 q^{49} +1709.23i q^{50} +942.691 q^{52} +1856.53i q^{53} +48.9932 q^{55} -5920.52i q^{56} -1157.89 q^{58} -453.188i q^{59} +3752.57 q^{61} +242.130i q^{62} -4363.03 q^{64} -1674.30i q^{65} +7055.01 q^{67} -1159.17i q^{68} -2702.68 q^{70} -8612.42i q^{71} +3578.48 q^{73} -5973.31i q^{74} -535.007 q^{76} -423.547i q^{77} +3518.38 q^{79} +1344.87i q^{80} +2030.40 q^{82} -7820.57i q^{83} -2058.78 q^{85} +9390.61i q^{86} -347.193 q^{88} +11047.1i q^{89} -14474.3 q^{91} -4311.63i q^{92} -9019.63 q^{94} +950.217i q^{95} +17797.5 q^{97} +15597.6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 76 q - 632 q^{4} - 48 q^{7} - 72 q^{10} + 448 q^{13} + 5696 q^{16} - 2240 q^{19} + 1008 q^{22} - 11180 q^{25} + 5376 q^{28} - 3440 q^{31} - 7112 q^{34} + 7016 q^{37} - 832 q^{40} - 608 q^{43} + 17728 q^{46} + 30628 q^{49} - 27056 q^{52} - 9312 q^{55} + 4616 q^{58} - 10056 q^{61} - 42864 q^{64} + 31160 q^{67} + 7544 q^{70} + 4488 q^{73} - 15632 q^{76} + 19208 q^{79} + 9768 q^{82} - 7568 q^{85} - 41736 q^{88} - 31240 q^{91} + 27360 q^{94} + 22216 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/531\mathbb{Z}\right)^\times\).

\(n\) \(119\) \(415\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.23497i 0.808742i 0.914595 + 0.404371i \(0.132510\pi\)
−0.914595 + 0.404371i \(0.867490\pi\)
\(3\) 0 0
\(4\) 5.53498 0.345936
\(5\) − 9.83059i − 0.393224i −0.980481 0.196612i \(-0.937006\pi\)
0.980481 0.196612i \(-0.0629939\pi\)
\(6\) 0 0
\(7\) −84.9856 −1.73440 −0.867200 0.497960i \(-0.834083\pi\)
−0.867200 + 0.497960i \(0.834083\pi\)
\(8\) 69.6650i 1.08852i
\(9\) 0 0
\(10\) 31.8017 0.318017
\(11\) 4.98375i 0.0411880i 0.999788 + 0.0205940i \(0.00655574\pi\)
−0.999788 + 0.0205940i \(0.993444\pi\)
\(12\) 0 0
\(13\) 170.315 1.00778 0.503891 0.863767i \(-0.331901\pi\)
0.503891 + 0.863767i \(0.331901\pi\)
\(14\) − 274.926i − 1.40268i
\(15\) 0 0
\(16\) −136.804 −0.534392
\(17\) − 209.426i − 0.724658i −0.932050 0.362329i \(-0.881982\pi\)
0.932050 0.362329i \(-0.118018\pi\)
\(18\) 0 0
\(19\) −96.6592 −0.267754 −0.133877 0.990998i \(-0.542743\pi\)
−0.133877 + 0.990998i \(0.542743\pi\)
\(20\) − 54.4121i − 0.136030i
\(21\) 0 0
\(22\) −16.1223 −0.0333105
\(23\) − 778.979i − 1.47255i −0.676682 0.736275i \(-0.736583\pi\)
0.676682 0.736275i \(-0.263417\pi\)
\(24\) 0 0
\(25\) 528.359 0.845375
\(26\) 550.964i 0.815036i
\(27\) 0 0
\(28\) −470.393 −0.599991
\(29\) 357.930i 0.425601i 0.977096 + 0.212801i \(0.0682585\pi\)
−0.977096 + 0.212801i \(0.931742\pi\)
\(30\) 0 0
\(31\) 74.8477 0.0778852 0.0389426 0.999241i \(-0.487601\pi\)
0.0389426 + 0.999241i \(0.487601\pi\)
\(32\) 672.082i 0.656330i
\(33\) 0 0
\(34\) 677.487 0.586062
\(35\) 835.459i 0.682007i
\(36\) 0 0
\(37\) −1846.48 −1.34878 −0.674391 0.738375i \(-0.735594\pi\)
−0.674391 + 0.738375i \(0.735594\pi\)
\(38\) − 312.690i − 0.216544i
\(39\) 0 0
\(40\) 684.848 0.428030
\(41\) − 627.641i − 0.373374i −0.982419 0.186687i \(-0.940225\pi\)
0.982419 0.186687i \(-0.0597750\pi\)
\(42\) 0 0
\(43\) 2902.85 1.56995 0.784977 0.619525i \(-0.212675\pi\)
0.784977 + 0.619525i \(0.212675\pi\)
\(44\) 27.5849i 0.0142484i
\(45\) 0 0
\(46\) 2519.97 1.19091
\(47\) 2788.16i 1.26218i 0.775708 + 0.631092i \(0.217393\pi\)
−0.775708 + 0.631092i \(0.782607\pi\)
\(48\) 0 0
\(49\) 4821.55 2.00814
\(50\) 1709.23i 0.683691i
\(51\) 0 0
\(52\) 942.691 0.348628
\(53\) 1856.53i 0.660921i 0.943820 + 0.330461i \(0.107204\pi\)
−0.943820 + 0.330461i \(0.892796\pi\)
\(54\) 0 0
\(55\) 48.9932 0.0161961
\(56\) − 5920.52i − 1.88792i
\(57\) 0 0
\(58\) −1157.89 −0.344202
\(59\) − 453.188i − 0.130189i
\(60\) 0 0
\(61\) 3752.57 1.00848 0.504242 0.863562i \(-0.331772\pi\)
0.504242 + 0.863562i \(0.331772\pi\)
\(62\) 242.130i 0.0629890i
\(63\) 0 0
\(64\) −4363.03 −1.06519
\(65\) − 1674.30i − 0.396284i
\(66\) 0 0
\(67\) 7055.01 1.57162 0.785811 0.618467i \(-0.212246\pi\)
0.785811 + 0.618467i \(0.212246\pi\)
\(68\) − 1159.17i − 0.250685i
\(69\) 0 0
\(70\) −2702.68 −0.551568
\(71\) − 8612.42i − 1.70847i −0.519883 0.854237i \(-0.674025\pi\)
0.519883 0.854237i \(-0.325975\pi\)
\(72\) 0 0
\(73\) 3578.48 0.671510 0.335755 0.941949i \(-0.391009\pi\)
0.335755 + 0.941949i \(0.391009\pi\)
\(74\) − 5973.31i − 1.09082i
\(75\) 0 0
\(76\) −535.007 −0.0926258
\(77\) − 423.547i − 0.0714365i
\(78\) 0 0
\(79\) 3518.38 0.563752 0.281876 0.959451i \(-0.409043\pi\)
0.281876 + 0.959451i \(0.409043\pi\)
\(80\) 1344.87i 0.210136i
\(81\) 0 0
\(82\) 2030.40 0.301963
\(83\) − 7820.57i − 1.13523i −0.823296 0.567613i \(-0.807867\pi\)
0.823296 0.567613i \(-0.192133\pi\)
\(84\) 0 0
\(85\) −2058.78 −0.284953
\(86\) 9390.61i 1.26969i
\(87\) 0 0
\(88\) −347.193 −0.0448338
\(89\) 11047.1i 1.39466i 0.716752 + 0.697328i \(0.245628\pi\)
−0.716752 + 0.697328i \(0.754372\pi\)
\(90\) 0 0
\(91\) −14474.3 −1.74790
\(92\) − 4311.63i − 0.509408i
\(93\) 0 0
\(94\) −9019.63 −1.02078
\(95\) 950.217i 0.105287i
\(96\) 0 0
\(97\) 17797.5 1.89154 0.945772 0.324831i \(-0.105307\pi\)
0.945772 + 0.324831i \(0.105307\pi\)
\(98\) 15597.6i 1.62407i
\(99\) 0 0
\(100\) 2924.46 0.292446
\(101\) − 10178.1i − 0.997757i −0.866672 0.498879i \(-0.833745\pi\)
0.866672 0.498879i \(-0.166255\pi\)
\(102\) 0 0
\(103\) 12961.9 1.22179 0.610893 0.791713i \(-0.290811\pi\)
0.610893 + 0.791713i \(0.290811\pi\)
\(104\) 11865.0i 1.09699i
\(105\) 0 0
\(106\) −6005.81 −0.534515
\(107\) 8348.89i 0.729224i 0.931159 + 0.364612i \(0.118798\pi\)
−0.931159 + 0.364612i \(0.881202\pi\)
\(108\) 0 0
\(109\) 18152.6 1.52787 0.763935 0.645294i \(-0.223265\pi\)
0.763935 + 0.645294i \(0.223265\pi\)
\(110\) 158.491i 0.0130985i
\(111\) 0 0
\(112\) 11626.4 0.926850
\(113\) 13979.2i 1.09478i 0.836879 + 0.547389i \(0.184378\pi\)
−0.836879 + 0.547389i \(0.815622\pi\)
\(114\) 0 0
\(115\) −7657.83 −0.579042
\(116\) 1981.14i 0.147231i
\(117\) 0 0
\(118\) 1466.05 0.105289
\(119\) 17798.2i 1.25685i
\(120\) 0 0
\(121\) 14616.2 0.998304
\(122\) 12139.5i 0.815604i
\(123\) 0 0
\(124\) 414.280 0.0269433
\(125\) − 11338.2i − 0.725645i
\(126\) 0 0
\(127\) −23300.1 −1.44461 −0.722304 0.691576i \(-0.756917\pi\)
−0.722304 + 0.691576i \(0.756917\pi\)
\(128\) − 3360.97i − 0.205137i
\(129\) 0 0
\(130\) 5416.31 0.320492
\(131\) − 558.954i − 0.0325712i −0.999867 0.0162856i \(-0.994816\pi\)
0.999867 0.0162856i \(-0.00518409\pi\)
\(132\) 0 0
\(133\) 8214.64 0.464393
\(134\) 22822.7i 1.27104i
\(135\) 0 0
\(136\) 14589.7 0.788802
\(137\) − 5606.38i − 0.298704i −0.988784 0.149352i \(-0.952281\pi\)
0.988784 0.149352i \(-0.0477188\pi\)
\(138\) 0 0
\(139\) 21532.7 1.11447 0.557235 0.830355i \(-0.311862\pi\)
0.557235 + 0.830355i \(0.311862\pi\)
\(140\) 4624.24i 0.235931i
\(141\) 0 0
\(142\) 27860.9 1.38172
\(143\) 848.808i 0.0415086i
\(144\) 0 0
\(145\) 3518.67 0.167356
\(146\) 11576.3i 0.543079i
\(147\) 0 0
\(148\) −10220.2 −0.466592
\(149\) 26085.3i 1.17496i 0.809238 + 0.587481i \(0.199880\pi\)
−0.809238 + 0.587481i \(0.800120\pi\)
\(150\) 0 0
\(151\) −17430.2 −0.764448 −0.382224 0.924070i \(-0.624842\pi\)
−0.382224 + 0.924070i \(0.624842\pi\)
\(152\) − 6733.76i − 0.291454i
\(153\) 0 0
\(154\) 1370.16 0.0577737
\(155\) − 735.797i − 0.0306263i
\(156\) 0 0
\(157\) −16653.9 −0.675640 −0.337820 0.941211i \(-0.609690\pi\)
−0.337820 + 0.941211i \(0.609690\pi\)
\(158\) 11381.8i 0.455930i
\(159\) 0 0
\(160\) 6606.96 0.258084
\(161\) 66202.0i 2.55399i
\(162\) 0 0
\(163\) 811.917 0.0305588 0.0152794 0.999883i \(-0.495136\pi\)
0.0152794 + 0.999883i \(0.495136\pi\)
\(164\) − 3473.98i − 0.129163i
\(165\) 0 0
\(166\) 25299.3 0.918105
\(167\) − 34611.4i − 1.24104i −0.784191 0.620520i \(-0.786922\pi\)
0.784191 0.620520i \(-0.213078\pi\)
\(168\) 0 0
\(169\) 446.280 0.0156255
\(170\) − 6660.10i − 0.230453i
\(171\) 0 0
\(172\) 16067.2 0.543104
\(173\) 8414.35i 0.281144i 0.990070 + 0.140572i \(0.0448941\pi\)
−0.990070 + 0.140572i \(0.955106\pi\)
\(174\) 0 0
\(175\) −44902.9 −1.46622
\(176\) − 681.799i − 0.0220105i
\(177\) 0 0
\(178\) −35736.9 −1.12792
\(179\) − 47015.1i − 1.46734i −0.679506 0.733670i \(-0.737806\pi\)
0.679506 0.733670i \(-0.262194\pi\)
\(180\) 0 0
\(181\) −12837.7 −0.391860 −0.195930 0.980618i \(-0.562772\pi\)
−0.195930 + 0.980618i \(0.562772\pi\)
\(182\) − 46824.0i − 1.41360i
\(183\) 0 0
\(184\) 54267.6 1.60289
\(185\) 18152.0i 0.530373i
\(186\) 0 0
\(187\) 1043.73 0.0298472
\(188\) 15432.4i 0.436635i
\(189\) 0 0
\(190\) −3073.92 −0.0851502
\(191\) − 60775.0i − 1.66594i −0.553321 0.832968i \(-0.686640\pi\)
0.553321 0.832968i \(-0.313360\pi\)
\(192\) 0 0
\(193\) −22371.4 −0.600590 −0.300295 0.953846i \(-0.597085\pi\)
−0.300295 + 0.953846i \(0.597085\pi\)
\(194\) 57574.5i 1.52977i
\(195\) 0 0
\(196\) 26687.2 0.694689
\(197\) − 31702.4i − 0.816882i −0.912785 0.408441i \(-0.866073\pi\)
0.912785 0.408441i \(-0.133927\pi\)
\(198\) 0 0
\(199\) −52481.1 −1.32525 −0.662624 0.748953i \(-0.730557\pi\)
−0.662624 + 0.748953i \(0.730557\pi\)
\(200\) 36808.2i 0.920204i
\(201\) 0 0
\(202\) 32925.9 0.806928
\(203\) − 30418.9i − 0.738162i
\(204\) 0 0
\(205\) −6170.08 −0.146819
\(206\) 41931.4i 0.988110i
\(207\) 0 0
\(208\) −23299.9 −0.538551
\(209\) − 481.725i − 0.0110283i
\(210\) 0 0
\(211\) 8850.73 0.198799 0.0993994 0.995048i \(-0.468308\pi\)
0.0993994 + 0.995048i \(0.468308\pi\)
\(212\) 10275.8i 0.228637i
\(213\) 0 0
\(214\) −27008.4 −0.589755
\(215\) − 28536.7i − 0.617343i
\(216\) 0 0
\(217\) −6360.97 −0.135084
\(218\) 58723.1i 1.23565i
\(219\) 0 0
\(220\) 271.176 0.00560281
\(221\) − 35668.5i − 0.730298i
\(222\) 0 0
\(223\) −17012.2 −0.342097 −0.171049 0.985263i \(-0.554716\pi\)
−0.171049 + 0.985263i \(0.554716\pi\)
\(224\) − 57117.3i − 1.13834i
\(225\) 0 0
\(226\) −45222.3 −0.885393
\(227\) 50453.0i 0.979118i 0.871970 + 0.489559i \(0.162842\pi\)
−0.871970 + 0.489559i \(0.837158\pi\)
\(228\) 0 0
\(229\) 23301.6 0.444340 0.222170 0.975008i \(-0.428686\pi\)
0.222170 + 0.975008i \(0.428686\pi\)
\(230\) − 24772.8i − 0.468295i
\(231\) 0 0
\(232\) −24935.2 −0.463273
\(233\) 31693.0i 0.583783i 0.956451 + 0.291892i \(0.0942846\pi\)
−0.956451 + 0.291892i \(0.905715\pi\)
\(234\) 0 0
\(235\) 27409.3 0.496321
\(236\) − 2508.38i − 0.0450370i
\(237\) 0 0
\(238\) −57576.7 −1.01647
\(239\) 26482.9i 0.463628i 0.972760 + 0.231814i \(0.0744661\pi\)
−0.972760 + 0.231814i \(0.925534\pi\)
\(240\) 0 0
\(241\) −24512.4 −0.422038 −0.211019 0.977482i \(-0.567678\pi\)
−0.211019 + 0.977482i \(0.567678\pi\)
\(242\) 47282.8i 0.807370i
\(243\) 0 0
\(244\) 20770.4 0.348871
\(245\) − 47398.7i − 0.789649i
\(246\) 0 0
\(247\) −16462.5 −0.269838
\(248\) 5214.26i 0.0847792i
\(249\) 0 0
\(250\) 36678.7 0.586860
\(251\) − 67859.2i − 1.07711i −0.842589 0.538557i \(-0.818970\pi\)
0.842589 0.538557i \(-0.181030\pi\)
\(252\) 0 0
\(253\) 3882.24 0.0606514
\(254\) − 75375.0i − 1.16831i
\(255\) 0 0
\(256\) −58935.9 −0.899291
\(257\) 63829.7i 0.966399i 0.875510 + 0.483200i \(0.160525\pi\)
−0.875510 + 0.483200i \(0.839475\pi\)
\(258\) 0 0
\(259\) 156924. 2.33933
\(260\) − 9267.21i − 0.137089i
\(261\) 0 0
\(262\) 1808.20 0.0263417
\(263\) − 45119.3i − 0.652305i −0.945317 0.326153i \(-0.894248\pi\)
0.945317 0.326153i \(-0.105752\pi\)
\(264\) 0 0
\(265\) 18250.8 0.259890
\(266\) 26574.1i 0.375574i
\(267\) 0 0
\(268\) 39049.3 0.543681
\(269\) − 48160.9i − 0.665565i −0.943004 0.332782i \(-0.892013\pi\)
0.943004 0.332782i \(-0.107987\pi\)
\(270\) 0 0
\(271\) 48564.2 0.661269 0.330634 0.943759i \(-0.392737\pi\)
0.330634 + 0.943759i \(0.392737\pi\)
\(272\) 28650.4i 0.387252i
\(273\) 0 0
\(274\) 18136.5 0.241575
\(275\) 2633.21i 0.0348193i
\(276\) 0 0
\(277\) 92108.4 1.20044 0.600219 0.799836i \(-0.295080\pi\)
0.600219 + 0.799836i \(0.295080\pi\)
\(278\) 69657.6i 0.901319i
\(279\) 0 0
\(280\) −58202.2 −0.742375
\(281\) − 79561.8i − 1.00761i −0.863818 0.503804i \(-0.831933\pi\)
0.863818 0.503804i \(-0.168067\pi\)
\(282\) 0 0
\(283\) −124898. −1.55948 −0.779742 0.626101i \(-0.784650\pi\)
−0.779742 + 0.626101i \(0.784650\pi\)
\(284\) − 47669.5i − 0.591023i
\(285\) 0 0
\(286\) −2745.87 −0.0335697
\(287\) 53340.5i 0.647579i
\(288\) 0 0
\(289\) 39661.6 0.474870
\(290\) 11382.8i 0.135348i
\(291\) 0 0
\(292\) 19806.8 0.232300
\(293\) − 38846.3i − 0.452496i −0.974070 0.226248i \(-0.927354\pi\)
0.974070 0.226248i \(-0.0726459\pi\)
\(294\) 0 0
\(295\) −4455.10 −0.0511934
\(296\) − 128635.i − 1.46817i
\(297\) 0 0
\(298\) −84385.2 −0.950241
\(299\) − 132672.i − 1.48401i
\(300\) 0 0
\(301\) −246700. −2.72293
\(302\) − 56386.1i − 0.618242i
\(303\) 0 0
\(304\) 13223.4 0.143086
\(305\) − 36890.0i − 0.396560i
\(306\) 0 0
\(307\) −53031.7 −0.562677 −0.281338 0.959609i \(-0.590778\pi\)
−0.281338 + 0.959609i \(0.590778\pi\)
\(308\) − 2344.32i − 0.0247125i
\(309\) 0 0
\(310\) 2380.28 0.0247688
\(311\) − 127407.i − 1.31726i −0.752465 0.658632i \(-0.771135\pi\)
0.752465 0.658632i \(-0.228865\pi\)
\(312\) 0 0
\(313\) 34530.6 0.352464 0.176232 0.984349i \(-0.443609\pi\)
0.176232 + 0.984349i \(0.443609\pi\)
\(314\) − 53874.7i − 0.546419i
\(315\) 0 0
\(316\) 19474.1 0.195022
\(317\) 15699.4i 0.156230i 0.996944 + 0.0781150i \(0.0248901\pi\)
−0.996944 + 0.0781150i \(0.975110\pi\)
\(318\) 0 0
\(319\) −1783.84 −0.0175297
\(320\) 42891.2i 0.418859i
\(321\) 0 0
\(322\) −214161. −2.06552
\(323\) 20243.0i 0.194030i
\(324\) 0 0
\(325\) 89987.7 0.851954
\(326\) 2626.53i 0.0247142i
\(327\) 0 0
\(328\) 43724.6 0.406423
\(329\) − 236954.i − 2.18913i
\(330\) 0 0
\(331\) −141162. −1.28843 −0.644216 0.764844i \(-0.722816\pi\)
−0.644216 + 0.764844i \(0.722816\pi\)
\(332\) − 43286.7i − 0.392715i
\(333\) 0 0
\(334\) 111967. 1.00368
\(335\) − 69354.9i − 0.617999i
\(336\) 0 0
\(337\) 38060.4 0.335130 0.167565 0.985861i \(-0.446410\pi\)
0.167565 + 0.985861i \(0.446410\pi\)
\(338\) 1443.70i 0.0126370i
\(339\) 0 0
\(340\) −11395.3 −0.0985755
\(341\) 373.022i 0.00320794i
\(342\) 0 0
\(343\) −205712. −1.74852
\(344\) 202227.i 1.70892i
\(345\) 0 0
\(346\) −27220.2 −0.227373
\(347\) − 139688.i − 1.16011i −0.814577 0.580055i \(-0.803031\pi\)
0.814577 0.580055i \(-0.196969\pi\)
\(348\) 0 0
\(349\) 169472. 1.39138 0.695691 0.718341i \(-0.255098\pi\)
0.695691 + 0.718341i \(0.255098\pi\)
\(350\) − 145260.i − 1.18579i
\(351\) 0 0
\(352\) −3349.49 −0.0270329
\(353\) 145139.i 1.16475i 0.812919 + 0.582377i \(0.197877\pi\)
−0.812919 + 0.582377i \(0.802123\pi\)
\(354\) 0 0
\(355\) −84665.2 −0.671813
\(356\) 61145.3i 0.482462i
\(357\) 0 0
\(358\) 152092. 1.18670
\(359\) 200226.i 1.55358i 0.629762 + 0.776788i \(0.283152\pi\)
−0.629762 + 0.776788i \(0.716848\pi\)
\(360\) 0 0
\(361\) −120978. −0.928308
\(362\) − 41529.6i − 0.316913i
\(363\) 0 0
\(364\) −80115.1 −0.604661
\(365\) − 35178.6i − 0.264054i
\(366\) 0 0
\(367\) 38293.8 0.284313 0.142157 0.989844i \(-0.454596\pi\)
0.142157 + 0.989844i \(0.454596\pi\)
\(368\) 106568.i 0.786919i
\(369\) 0 0
\(370\) −58721.2 −0.428935
\(371\) − 157778.i − 1.14630i
\(372\) 0 0
\(373\) −150209. −1.07964 −0.539818 0.841782i \(-0.681507\pi\)
−0.539818 + 0.841782i \(0.681507\pi\)
\(374\) 3376.43i 0.0241387i
\(375\) 0 0
\(376\) −194237. −1.37391
\(377\) 60961.0i 0.428913i
\(378\) 0 0
\(379\) −47965.7 −0.333928 −0.166964 0.985963i \(-0.553396\pi\)
−0.166964 + 0.985963i \(0.553396\pi\)
\(380\) 5259.43i 0.0364227i
\(381\) 0 0
\(382\) 196605. 1.34731
\(383\) 151408.i 1.03217i 0.856537 + 0.516086i \(0.172612\pi\)
−0.856537 + 0.516086i \(0.827388\pi\)
\(384\) 0 0
\(385\) −4163.72 −0.0280905
\(386\) − 72370.7i − 0.485723i
\(387\) 0 0
\(388\) 98509.0 0.654353
\(389\) 261549.i 1.72844i 0.503117 + 0.864218i \(0.332186\pi\)
−0.503117 + 0.864218i \(0.667814\pi\)
\(390\) 0 0
\(391\) −163139. −1.06710
\(392\) 335893.i 2.18589i
\(393\) 0 0
\(394\) 102556. 0.660647
\(395\) − 34587.7i − 0.221681i
\(396\) 0 0
\(397\) 269644. 1.71084 0.855421 0.517933i \(-0.173298\pi\)
0.855421 + 0.517933i \(0.173298\pi\)
\(398\) − 169775.i − 1.07178i
\(399\) 0 0
\(400\) −72281.9 −0.451762
\(401\) − 95186.4i − 0.591952i −0.955195 0.295976i \(-0.904355\pi\)
0.955195 0.295976i \(-0.0956448\pi\)
\(402\) 0 0
\(403\) 12747.7 0.0784913
\(404\) − 56335.7i − 0.345160i
\(405\) 0 0
\(406\) 98404.3 0.596983
\(407\) − 9202.40i − 0.0555536i
\(408\) 0 0
\(409\) 69543.9 0.415731 0.207866 0.978157i \(-0.433348\pi\)
0.207866 + 0.978157i \(0.433348\pi\)
\(410\) − 19960.0i − 0.118739i
\(411\) 0 0
\(412\) 71744.0 0.422660
\(413\) 38514.4i 0.225800i
\(414\) 0 0
\(415\) −76880.8 −0.446398
\(416\) 114466.i 0.661438i
\(417\) 0 0
\(418\) 1558.37 0.00891902
\(419\) 19338.6i 0.110153i 0.998482 + 0.0550766i \(0.0175403\pi\)
−0.998482 + 0.0550766i \(0.982460\pi\)
\(420\) 0 0
\(421\) −25467.9 −0.143691 −0.0718454 0.997416i \(-0.522889\pi\)
−0.0718454 + 0.997416i \(0.522889\pi\)
\(422\) 28631.8i 0.160777i
\(423\) 0 0
\(424\) −129335. −0.719423
\(425\) − 110652.i − 0.612608i
\(426\) 0 0
\(427\) −318915. −1.74912
\(428\) 46210.9i 0.252265i
\(429\) 0 0
\(430\) 92315.3 0.499271
\(431\) − 177704.i − 0.956627i −0.878189 0.478314i \(-0.841248\pi\)
0.878189 0.478314i \(-0.158752\pi\)
\(432\) 0 0
\(433\) −118232. −0.630606 −0.315303 0.948991i \(-0.602106\pi\)
−0.315303 + 0.948991i \(0.602106\pi\)
\(434\) − 20577.6i − 0.109248i
\(435\) 0 0
\(436\) 100474. 0.528545
\(437\) 75295.5i 0.394281i
\(438\) 0 0
\(439\) 148789. 0.772045 0.386022 0.922489i \(-0.373849\pi\)
0.386022 + 0.922489i \(0.373849\pi\)
\(440\) 3413.11i 0.0176297i
\(441\) 0 0
\(442\) 115386. 0.590623
\(443\) − 317446.i − 1.61757i −0.588105 0.808784i \(-0.700126\pi\)
0.588105 0.808784i \(-0.299874\pi\)
\(444\) 0 0
\(445\) 108599. 0.548412
\(446\) − 55033.8i − 0.276668i
\(447\) 0 0
\(448\) 370795. 1.84747
\(449\) − 267391.i − 1.32634i −0.748469 0.663169i \(-0.769211\pi\)
0.748469 0.663169i \(-0.230789\pi\)
\(450\) 0 0
\(451\) 3128.01 0.0153785
\(452\) 77374.6i 0.378723i
\(453\) 0 0
\(454\) −163214. −0.791854
\(455\) 142291.i 0.687315i
\(456\) 0 0
\(457\) 58479.0 0.280006 0.140003 0.990151i \(-0.455289\pi\)
0.140003 + 0.990151i \(0.455289\pi\)
\(458\) 75380.0i 0.359356i
\(459\) 0 0
\(460\) −42385.9 −0.200311
\(461\) 164100.i 0.772160i 0.922465 + 0.386080i \(0.126171\pi\)
−0.922465 + 0.386080i \(0.873829\pi\)
\(462\) 0 0
\(463\) −105301. −0.491216 −0.245608 0.969369i \(-0.578988\pi\)
−0.245608 + 0.969369i \(0.578988\pi\)
\(464\) − 48966.5i − 0.227438i
\(465\) 0 0
\(466\) −102526. −0.472130
\(467\) 385404.i 1.76718i 0.468257 + 0.883592i \(0.344882\pi\)
−0.468257 + 0.883592i \(0.655118\pi\)
\(468\) 0 0
\(469\) −599574. −2.72582
\(470\) 88668.3i 0.401395i
\(471\) 0 0
\(472\) 31571.3 0.141713
\(473\) 14467.1i 0.0646633i
\(474\) 0 0
\(475\) −51070.8 −0.226353
\(476\) 98512.7i 0.434789i
\(477\) 0 0
\(478\) −85671.4 −0.374956
\(479\) 230440.i 1.00435i 0.864765 + 0.502176i \(0.167467\pi\)
−0.864765 + 0.502176i \(0.832533\pi\)
\(480\) 0 0
\(481\) −314484. −1.35928
\(482\) − 79296.8i − 0.341320i
\(483\) 0 0
\(484\) 80900.1 0.345349
\(485\) − 174960.i − 0.743800i
\(486\) 0 0
\(487\) −48164.1 −0.203079 −0.101540 0.994832i \(-0.532377\pi\)
−0.101540 + 0.994832i \(0.532377\pi\)
\(488\) 261423.i 1.09775i
\(489\) 0 0
\(490\) 153333. 0.638623
\(491\) − 211675.i − 0.878022i −0.898482 0.439011i \(-0.855329\pi\)
0.898482 0.439011i \(-0.144671\pi\)
\(492\) 0 0
\(493\) 74960.0 0.308415
\(494\) − 53255.8i − 0.218229i
\(495\) 0 0
\(496\) −10239.5 −0.0416212
\(497\) 731932.i 2.96318i
\(498\) 0 0
\(499\) 375387. 1.50757 0.753786 0.657120i \(-0.228226\pi\)
0.753786 + 0.657120i \(0.228226\pi\)
\(500\) − 62756.7i − 0.251027i
\(501\) 0 0
\(502\) 219522. 0.871107
\(503\) 69030.8i 0.272839i 0.990651 + 0.136420i \(0.0435595\pi\)
−0.990651 + 0.136420i \(0.956440\pi\)
\(504\) 0 0
\(505\) −100057. −0.392342
\(506\) 12558.9i 0.0490514i
\(507\) 0 0
\(508\) −128965. −0.499742
\(509\) − 231705.i − 0.894332i −0.894451 0.447166i \(-0.852433\pi\)
0.894451 0.447166i \(-0.147567\pi\)
\(510\) 0 0
\(511\) −304119. −1.16467
\(512\) − 244431.i − 0.932432i
\(513\) 0 0
\(514\) −206487. −0.781568
\(515\) − 127423.i − 0.480435i
\(516\) 0 0
\(517\) −13895.5 −0.0519868
\(518\) 507645.i 1.89191i
\(519\) 0 0
\(520\) 116640. 0.431361
\(521\) − 63854.4i − 0.235242i −0.993059 0.117621i \(-0.962473\pi\)
0.993059 0.117621i \(-0.0375268\pi\)
\(522\) 0 0
\(523\) 364579. 1.33287 0.666435 0.745563i \(-0.267819\pi\)
0.666435 + 0.745563i \(0.267819\pi\)
\(524\) − 3093.80i − 0.0112675i
\(525\) 0 0
\(526\) 145960. 0.527547
\(527\) − 15675.1i − 0.0564402i
\(528\) 0 0
\(529\) −326968. −1.16841
\(530\) 59040.7i 0.210184i
\(531\) 0 0
\(532\) 45467.9 0.160650
\(533\) − 106897.i − 0.376279i
\(534\) 0 0
\(535\) 82074.5 0.286748
\(536\) 491487.i 1.71073i
\(537\) 0 0
\(538\) 155799. 0.538270
\(539\) 24029.4i 0.0827114i
\(540\) 0 0
\(541\) 287660. 0.982844 0.491422 0.870922i \(-0.336477\pi\)
0.491422 + 0.870922i \(0.336477\pi\)
\(542\) 157104.i 0.534796i
\(543\) 0 0
\(544\) 140752. 0.475615
\(545\) − 178451.i − 0.600794i
\(546\) 0 0
\(547\) 40922.4 0.136768 0.0683842 0.997659i \(-0.478216\pi\)
0.0683842 + 0.997659i \(0.478216\pi\)
\(548\) − 31031.2i − 0.103333i
\(549\) 0 0
\(550\) −8518.35 −0.0281599
\(551\) − 34597.3i − 0.113956i
\(552\) 0 0
\(553\) −299011. −0.977771
\(554\) 297968.i 0.970845i
\(555\) 0 0
\(556\) 119183. 0.385536
\(557\) 6420.57i 0.0206949i 0.999946 + 0.0103475i \(0.00329375\pi\)
−0.999946 + 0.0103475i \(0.996706\pi\)
\(558\) 0 0
\(559\) 494399. 1.58217
\(560\) − 114294.i − 0.364459i
\(561\) 0 0
\(562\) 257380. 0.814896
\(563\) − 33869.4i − 0.106854i −0.998572 0.0534269i \(-0.982986\pi\)
0.998572 0.0534269i \(-0.0170144\pi\)
\(564\) 0 0
\(565\) 137424. 0.430492
\(566\) − 404040.i − 1.26122i
\(567\) 0 0
\(568\) 599984. 1.85970
\(569\) − 549854.i − 1.69833i −0.528126 0.849166i \(-0.677105\pi\)
0.528126 0.849166i \(-0.322895\pi\)
\(570\) 0 0
\(571\) 629932. 1.93206 0.966032 0.258423i \(-0.0832029\pi\)
0.966032 + 0.258423i \(0.0832029\pi\)
\(572\) 4698.13i 0.0143593i
\(573\) 0 0
\(574\) −172555. −0.523725
\(575\) − 411581.i − 1.24486i
\(576\) 0 0
\(577\) −563046. −1.69119 −0.845594 0.533826i \(-0.820754\pi\)
−0.845594 + 0.533826i \(0.820754\pi\)
\(578\) 128304.i 0.384048i
\(579\) 0 0
\(580\) 19475.7 0.0578946
\(581\) 664636.i 1.96894i
\(582\) 0 0
\(583\) −9252.47 −0.0272220
\(584\) 249295.i 0.730949i
\(585\) 0 0
\(586\) 125667. 0.365952
\(587\) 520388.i 1.51026i 0.655576 + 0.755129i \(0.272426\pi\)
−0.655576 + 0.755129i \(0.727574\pi\)
\(588\) 0 0
\(589\) −7234.72 −0.0208541
\(590\) − 14412.1i − 0.0414022i
\(591\) 0 0
\(592\) 252607. 0.720778
\(593\) 41999.5i 0.119436i 0.998215 + 0.0597179i \(0.0190201\pi\)
−0.998215 + 0.0597179i \(0.980980\pi\)
\(594\) 0 0
\(595\) 174967. 0.494222
\(596\) 144382.i 0.406461i
\(597\) 0 0
\(598\) 429190. 1.20018
\(599\) − 39343.2i − 0.109652i −0.998496 0.0548260i \(-0.982540\pi\)
0.998496 0.0548260i \(-0.0174604\pi\)
\(600\) 0 0
\(601\) −80632.4 −0.223234 −0.111617 0.993751i \(-0.535603\pi\)
−0.111617 + 0.993751i \(0.535603\pi\)
\(602\) − 798067.i − 2.20215i
\(603\) 0 0
\(604\) −96475.7 −0.264450
\(605\) − 143686.i − 0.392557i
\(606\) 0 0
\(607\) −424244. −1.15143 −0.575717 0.817649i \(-0.695277\pi\)
−0.575717 + 0.817649i \(0.695277\pi\)
\(608\) − 64962.9i − 0.175735i
\(609\) 0 0
\(610\) 119338. 0.320715
\(611\) 474867.i 1.27201i
\(612\) 0 0
\(613\) 283622. 0.754777 0.377388 0.926055i \(-0.376822\pi\)
0.377388 + 0.926055i \(0.376822\pi\)
\(614\) − 171556.i − 0.455060i
\(615\) 0 0
\(616\) 29506.4 0.0777597
\(617\) 491931.i 1.29221i 0.763247 + 0.646106i \(0.223604\pi\)
−0.763247 + 0.646106i \(0.776396\pi\)
\(618\) 0 0
\(619\) −312185. −0.814762 −0.407381 0.913258i \(-0.633558\pi\)
−0.407381 + 0.913258i \(0.633558\pi\)
\(620\) − 4072.62i − 0.0105947i
\(621\) 0 0
\(622\) 412158. 1.06533
\(623\) − 938842.i − 2.41889i
\(624\) 0 0
\(625\) 218763. 0.560034
\(626\) 111705.i 0.285053i
\(627\) 0 0
\(628\) −92178.7 −0.233728
\(629\) 386702.i 0.977405i
\(630\) 0 0
\(631\) 70824.4 0.177879 0.0889395 0.996037i \(-0.471652\pi\)
0.0889395 + 0.996037i \(0.471652\pi\)
\(632\) 245108.i 0.613653i
\(633\) 0 0
\(634\) −50787.1 −0.126350
\(635\) 229053.i 0.568054i
\(636\) 0 0
\(637\) 821184. 2.02377
\(638\) − 5770.65i − 0.0141770i
\(639\) 0 0
\(640\) −33040.3 −0.0806649
\(641\) − 486066.i − 1.18298i −0.806311 0.591492i \(-0.798539\pi\)
0.806311 0.591492i \(-0.201461\pi\)
\(642\) 0 0
\(643\) 472005. 1.14163 0.570814 0.821079i \(-0.306628\pi\)
0.570814 + 0.821079i \(0.306628\pi\)
\(644\) 366427.i 0.883518i
\(645\) 0 0
\(646\) −65485.4 −0.156920
\(647\) − 296340.i − 0.707917i −0.935261 0.353958i \(-0.884835\pi\)
0.935261 0.353958i \(-0.115165\pi\)
\(648\) 0 0
\(649\) 2258.57 0.00536222
\(650\) 291107.i 0.689011i
\(651\) 0 0
\(652\) 4493.94 0.0105714
\(653\) 242277.i 0.568180i 0.958798 + 0.284090i \(0.0916914\pi\)
−0.958798 + 0.284090i \(0.908309\pi\)
\(654\) 0 0
\(655\) −5494.85 −0.0128078
\(656\) 85864.1i 0.199528i
\(657\) 0 0
\(658\) 766538. 1.77044
\(659\) − 247436.i − 0.569760i −0.958563 0.284880i \(-0.908046\pi\)
0.958563 0.284880i \(-0.0919537\pi\)
\(660\) 0 0
\(661\) −427955. −0.979480 −0.489740 0.871869i \(-0.662908\pi\)
−0.489740 + 0.871869i \(0.662908\pi\)
\(662\) − 456654.i − 1.04201i
\(663\) 0 0
\(664\) 544820. 1.23571
\(665\) − 80754.8i − 0.182610i
\(666\) 0 0
\(667\) 278820. 0.626719
\(668\) − 191573.i − 0.429321i
\(669\) 0 0
\(670\) 224361. 0.499802
\(671\) 18701.9i 0.0415375i
\(672\) 0 0
\(673\) 517331. 1.14219 0.571095 0.820884i \(-0.306519\pi\)
0.571095 + 0.820884i \(0.306519\pi\)
\(674\) 123124.i 0.271034i
\(675\) 0 0
\(676\) 2470.15 0.00540543
\(677\) 280479.i 0.611959i 0.952038 + 0.305980i \(0.0989840\pi\)
−0.952038 + 0.305980i \(0.901016\pi\)
\(678\) 0 0
\(679\) −1.51253e6 −3.28069
\(680\) − 143425.i − 0.310175i
\(681\) 0 0
\(682\) −1206.71 −0.00259439
\(683\) 242148.i 0.519086i 0.965732 + 0.259543i \(0.0835720\pi\)
−0.965732 + 0.259543i \(0.916428\pi\)
\(684\) 0 0
\(685\) −55114.1 −0.117458
\(686\) − 665472.i − 1.41410i
\(687\) 0 0
\(688\) −397122. −0.838971
\(689\) 316195.i 0.666065i
\(690\) 0 0
\(691\) −41284.0 −0.0864621 −0.0432310 0.999065i \(-0.513765\pi\)
−0.0432310 + 0.999065i \(0.513765\pi\)
\(692\) 46573.3i 0.0972578i
\(693\) 0 0
\(694\) 451885. 0.938230
\(695\) − 211679.i − 0.438236i
\(696\) 0 0
\(697\) −131445. −0.270568
\(698\) 548236.i 1.12527i
\(699\) 0 0
\(700\) −248537. −0.507218
\(701\) − 897088.i − 1.82557i −0.408438 0.912786i \(-0.633926\pi\)
0.408438 0.912786i \(-0.366074\pi\)
\(702\) 0 0
\(703\) 178479. 0.361142
\(704\) − 21744.3i − 0.0438732i
\(705\) 0 0
\(706\) −469520. −0.941986
\(707\) 864994.i 1.73051i
\(708\) 0 0
\(709\) 304280. 0.605314 0.302657 0.953100i \(-0.402126\pi\)
0.302657 + 0.953100i \(0.402126\pi\)
\(710\) − 273889.i − 0.543323i
\(711\) 0 0
\(712\) −769594. −1.51810
\(713\) − 58304.8i − 0.114690i
\(714\) 0 0
\(715\) 8344.29 0.0163221
\(716\) − 260227.i − 0.507606i
\(717\) 0 0
\(718\) −647726. −1.25644
\(719\) 761954.i 1.47391i 0.675942 + 0.736955i \(0.263737\pi\)
−0.675942 + 0.736955i \(0.736263\pi\)
\(720\) 0 0
\(721\) −1.10158e6 −2.11907
\(722\) − 391360.i − 0.750762i
\(723\) 0 0
\(724\) −71056.4 −0.135558
\(725\) 189116.i 0.359793i
\(726\) 0 0
\(727\) −101145. −0.191370 −0.0956852 0.995412i \(-0.530504\pi\)
−0.0956852 + 0.995412i \(0.530504\pi\)
\(728\) − 1.00835e6i − 1.90261i
\(729\) 0 0
\(730\) 113802. 0.213551
\(731\) − 607932.i − 1.13768i
\(732\) 0 0
\(733\) 216085. 0.402176 0.201088 0.979573i \(-0.435552\pi\)
0.201088 + 0.979573i \(0.435552\pi\)
\(734\) 123879.i 0.229936i
\(735\) 0 0
\(736\) 523538. 0.966479
\(737\) 35160.4i 0.0647320i
\(738\) 0 0
\(739\) −633797. −1.16054 −0.580272 0.814423i \(-0.697054\pi\)
−0.580272 + 0.814423i \(0.697054\pi\)
\(740\) 100471.i 0.183475i
\(741\) 0 0
\(742\) 510407. 0.927063
\(743\) 282503.i 0.511735i 0.966712 + 0.255868i \(0.0823611\pi\)
−0.966712 + 0.255868i \(0.917639\pi\)
\(744\) 0 0
\(745\) 256434. 0.462023
\(746\) − 485920.i − 0.873146i
\(747\) 0 0
\(748\) 5777.01 0.0103252
\(749\) − 709535.i − 1.26477i
\(750\) 0 0
\(751\) 870392. 1.54325 0.771623 0.636080i \(-0.219445\pi\)
0.771623 + 0.636080i \(0.219445\pi\)
\(752\) − 381433.i − 0.674501i
\(753\) 0 0
\(754\) −197207. −0.346880
\(755\) 171349.i 0.300599i
\(756\) 0 0
\(757\) −709614. −1.23831 −0.619157 0.785267i \(-0.712525\pi\)
−0.619157 + 0.785267i \(0.712525\pi\)
\(758\) − 155168.i − 0.270061i
\(759\) 0 0
\(760\) −66196.9 −0.114607
\(761\) 345513.i 0.596615i 0.954470 + 0.298308i \(0.0964222\pi\)
−0.954470 + 0.298308i \(0.903578\pi\)
\(762\) 0 0
\(763\) −1.54271e6 −2.64994
\(764\) − 336388.i − 0.576308i
\(765\) 0 0
\(766\) −489801. −0.834762
\(767\) − 77184.8i − 0.131202i
\(768\) 0 0
\(769\) 96909.7 0.163876 0.0819379 0.996637i \(-0.473889\pi\)
0.0819379 + 0.996637i \(0.473889\pi\)
\(770\) − 13469.5i − 0.0227180i
\(771\) 0 0
\(772\) −123825. −0.207766
\(773\) 826181.i 1.38266i 0.722538 + 0.691331i \(0.242975\pi\)
−0.722538 + 0.691331i \(0.757025\pi\)
\(774\) 0 0
\(775\) 39546.5 0.0658422
\(776\) 1.23987e6i 2.05897i
\(777\) 0 0
\(778\) −846102. −1.39786
\(779\) 60667.3i 0.0999723i
\(780\) 0 0
\(781\) 42922.1 0.0703687
\(782\) − 527749.i − 0.863006i
\(783\) 0 0
\(784\) −659609. −1.07314
\(785\) 163717.i 0.265678i
\(786\) 0 0
\(787\) 794567. 1.28287 0.641433 0.767179i \(-0.278340\pi\)
0.641433 + 0.767179i \(0.278340\pi\)
\(788\) − 175472.i − 0.282589i
\(789\) 0 0
\(790\) 111890. 0.179282
\(791\) − 1.18803e6i − 1.89878i
\(792\) 0 0
\(793\) 639120. 1.01633
\(794\) 872290.i 1.38363i
\(795\) 0 0
\(796\) −290482. −0.458451
\(797\) 779274.i 1.22680i 0.789773 + 0.613400i \(0.210198\pi\)
−0.789773 + 0.613400i \(0.789802\pi\)
\(798\) 0 0
\(799\) 583915. 0.914652
\(800\) 355101.i 0.554845i
\(801\) 0 0
\(802\) 307925. 0.478736
\(803\) 17834.2i 0.0276582i
\(804\) 0 0
\(805\) 650805. 1.00429
\(806\) 41238.4i 0.0634793i
\(807\) 0 0
\(808\) 709059. 1.08607
\(809\) 169320.i 0.258709i 0.991598 + 0.129354i \(0.0412905\pi\)
−0.991598 + 0.129354i \(0.958710\pi\)
\(810\) 0 0
\(811\) 157589. 0.239598 0.119799 0.992798i \(-0.461775\pi\)
0.119799 + 0.992798i \(0.461775\pi\)
\(812\) − 168368.i − 0.255357i
\(813\) 0 0
\(814\) 29769.5 0.0449285
\(815\) − 7981.63i − 0.0120165i
\(816\) 0 0
\(817\) −280587. −0.420362
\(818\) 224972.i 0.336219i
\(819\) 0 0
\(820\) −34151.3 −0.0507901
\(821\) − 436652.i − 0.647812i −0.946089 0.323906i \(-0.895004\pi\)
0.946089 0.323906i \(-0.104996\pi\)
\(822\) 0 0
\(823\) −787170. −1.16217 −0.581084 0.813844i \(-0.697371\pi\)
−0.581084 + 0.813844i \(0.697371\pi\)
\(824\) 902992.i 1.32993i
\(825\) 0 0
\(826\) −124593. −0.182614
\(827\) 545449.i 0.797523i 0.917055 + 0.398762i \(0.130560\pi\)
−0.917055 + 0.398762i \(0.869440\pi\)
\(828\) 0 0
\(829\) −450871. −0.656060 −0.328030 0.944667i \(-0.606385\pi\)
−0.328030 + 0.944667i \(0.606385\pi\)
\(830\) − 248707.i − 0.361021i
\(831\) 0 0
\(832\) −743091. −1.07348
\(833\) − 1.00976e6i − 1.45522i
\(834\) 0 0
\(835\) −340250. −0.488006
\(836\) − 2666.34i − 0.00381507i
\(837\) 0 0
\(838\) −62559.8 −0.0890855
\(839\) − 555172.i − 0.788686i −0.918963 0.394343i \(-0.870972\pi\)
0.918963 0.394343i \(-0.129028\pi\)
\(840\) 0 0
\(841\) 579167. 0.818864
\(842\) − 82387.9i − 0.116209i
\(843\) 0 0
\(844\) 48988.6 0.0687717
\(845\) − 4387.20i − 0.00614432i
\(846\) 0 0
\(847\) −1.24216e6 −1.73146
\(848\) − 253981.i − 0.353191i
\(849\) 0 0
\(850\) 357957. 0.495442
\(851\) 1.43837e6i 1.98615i
\(852\) 0 0
\(853\) −1.20523e6 −1.65643 −0.828215 0.560410i \(-0.810644\pi\)
−0.828215 + 0.560410i \(0.810644\pi\)
\(854\) − 1.03168e6i − 1.41458i
\(855\) 0 0
\(856\) −581625. −0.793772
\(857\) − 972072.i − 1.32354i −0.749708 0.661769i \(-0.769806\pi\)
0.749708 0.661769i \(-0.230194\pi\)
\(858\) 0 0
\(859\) 854142. 1.15756 0.578780 0.815484i \(-0.303529\pi\)
0.578780 + 0.815484i \(0.303529\pi\)
\(860\) − 157950.i − 0.213561i
\(861\) 0 0
\(862\) 574867. 0.773665
\(863\) − 126040.i − 0.169233i −0.996414 0.0846166i \(-0.973033\pi\)
0.996414 0.0846166i \(-0.0269666\pi\)
\(864\) 0 0
\(865\) 82718.1 0.110552
\(866\) − 382476.i − 0.509998i
\(867\) 0 0
\(868\) −35207.8 −0.0467305
\(869\) 17534.7i 0.0232198i
\(870\) 0 0
\(871\) 1.20158e6 1.58385
\(872\) 1.26460e6i 1.66311i
\(873\) 0 0
\(874\) −243579. −0.318872
\(875\) 963584.i 1.25856i
\(876\) 0 0
\(877\) −167079. −0.217232 −0.108616 0.994084i \(-0.534642\pi\)
−0.108616 + 0.994084i \(0.534642\pi\)
\(878\) 481329.i 0.624385i
\(879\) 0 0
\(880\) −6702.48 −0.00865507
\(881\) − 501079.i − 0.645586i −0.946470 0.322793i \(-0.895378\pi\)
0.946470 0.322793i \(-0.104622\pi\)
\(882\) 0 0
\(883\) 219320. 0.281292 0.140646 0.990060i \(-0.455082\pi\)
0.140646 + 0.990060i \(0.455082\pi\)
\(884\) − 197424.i − 0.252636i
\(885\) 0 0
\(886\) 1.02693e6 1.30820
\(887\) − 733514.i − 0.932311i −0.884703 0.466156i \(-0.845639\pi\)
0.884703 0.466156i \(-0.154361\pi\)
\(888\) 0 0
\(889\) 1.98017e6 2.50553
\(890\) 351315.i 0.443524i
\(891\) 0 0
\(892\) −94161.9 −0.118344
\(893\) − 269502.i − 0.337955i
\(894\) 0 0
\(895\) −462186. −0.576993
\(896\) 285634.i 0.355790i
\(897\) 0 0
\(898\) 865002. 1.07267
\(899\) 26790.3i 0.0331480i
\(900\) 0 0
\(901\) 388806. 0.478942
\(902\) 10119.0i 0.0124373i
\(903\) 0 0
\(904\) −973862. −1.19168
\(905\) 126202.i 0.154088i
\(906\) 0 0
\(907\) −917159. −1.11489 −0.557443 0.830215i \(-0.688217\pi\)
−0.557443 + 0.830215i \(0.688217\pi\)
\(908\) 279256.i 0.338712i
\(909\) 0 0
\(910\) −460308. −0.555860
\(911\) − 148683.i − 0.179153i −0.995980 0.0895765i \(-0.971449\pi\)
0.995980 0.0895765i \(-0.0285513\pi\)
\(912\) 0 0
\(913\) 38975.8 0.0467577
\(914\) 189178.i 0.226453i
\(915\) 0 0
\(916\) 128974. 0.153713
\(917\) 47503.0i 0.0564914i
\(918\) 0 0
\(919\) −180189. −0.213353 −0.106676 0.994294i \(-0.534021\pi\)
−0.106676 + 0.994294i \(0.534021\pi\)
\(920\) − 533482.i − 0.630296i
\(921\) 0 0
\(922\) −530859. −0.624478
\(923\) − 1.46683e6i − 1.72177i
\(924\) 0 0
\(925\) −975606. −1.14023
\(926\) − 340647.i − 0.397267i
\(927\) 0 0
\(928\) −240559. −0.279335
\(929\) − 1.13809e6i − 1.31869i −0.751839 0.659347i \(-0.770833\pi\)
0.751839 0.659347i \(-0.229167\pi\)
\(930\) 0 0
\(931\) −466047. −0.537688
\(932\) 175420.i 0.201952i
\(933\) 0 0
\(934\) −1.24677e6 −1.42920
\(935\) − 10260.5i − 0.0117366i
\(936\) 0 0
\(937\) −308142. −0.350971 −0.175486 0.984482i \(-0.556150\pi\)
−0.175486 + 0.984482i \(0.556150\pi\)
\(938\) − 1.93960e6i − 2.20449i
\(939\) 0 0
\(940\) 151710. 0.171695
\(941\) 899640.i 1.01599i 0.861360 + 0.507995i \(0.169613\pi\)
−0.861360 + 0.507995i \(0.830387\pi\)
\(942\) 0 0
\(943\) −488919. −0.549812
\(944\) 61998.1i 0.0695719i
\(945\) 0 0
\(946\) −46800.5 −0.0522959
\(947\) 599889.i 0.668915i 0.942411 + 0.334457i \(0.108553\pi\)
−0.942411 + 0.334457i \(0.891447\pi\)
\(948\) 0 0
\(949\) 609469. 0.676736
\(950\) − 165213.i − 0.183061i
\(951\) 0 0
\(952\) −1.23991e6 −1.36810
\(953\) 1.14195e6i 1.25737i 0.777662 + 0.628683i \(0.216406\pi\)
−0.777662 + 0.628683i \(0.783594\pi\)
\(954\) 0 0
\(955\) −597455. −0.655086
\(956\) 146582.i 0.160386i
\(957\) 0 0
\(958\) −745465. −0.812262
\(959\) 476462.i 0.518073i
\(960\) 0 0
\(961\) −917919. −0.993934
\(962\) − 1.01735e6i − 1.09931i
\(963\) 0 0
\(964\) −135675. −0.145998
\(965\) 219924.i 0.236166i
\(966\) 0 0
\(967\) −1.07074e6 −1.14507 −0.572533 0.819882i \(-0.694039\pi\)
−0.572533 + 0.819882i \(0.694039\pi\)
\(968\) 1.01823e6i 1.08667i
\(969\) 0 0
\(970\) 565991. 0.601542
\(971\) 505487.i 0.536132i 0.963401 + 0.268066i \(0.0863846\pi\)
−0.963401 + 0.268066i \(0.913615\pi\)
\(972\) 0 0
\(973\) −1.82997e6 −1.93294
\(974\) − 155809.i − 0.164239i
\(975\) 0 0
\(976\) −513368. −0.538926
\(977\) 650673.i 0.681669i 0.940123 + 0.340835i \(0.110710\pi\)
−0.940123 + 0.340835i \(0.889290\pi\)
\(978\) 0 0
\(979\) −55055.8 −0.0574431
\(980\) − 262351.i − 0.273168i
\(981\) 0 0
\(982\) 684760. 0.710094
\(983\) 1.60271e6i 1.65863i 0.558784 + 0.829313i \(0.311268\pi\)
−0.558784 + 0.829313i \(0.688732\pi\)
\(984\) 0 0
\(985\) −311653. −0.321217
\(986\) 242493.i 0.249429i
\(987\) 0 0
\(988\) −91119.8 −0.0933467
\(989\) − 2.26126e6i − 2.31184i
\(990\) 0 0
\(991\) 41952.2 0.0427177 0.0213588 0.999772i \(-0.493201\pi\)
0.0213588 + 0.999772i \(0.493201\pi\)
\(992\) 50303.8i 0.0511184i
\(993\) 0 0
\(994\) −2.36778e6 −2.39645
\(995\) 515920.i 0.521119i
\(996\) 0 0
\(997\) 634831. 0.638657 0.319328 0.947644i \(-0.396543\pi\)
0.319328 + 0.947644i \(0.396543\pi\)
\(998\) 1.21436e6i 1.21924i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 531.5.b.a.296.52 yes 76
3.2 odd 2 inner 531.5.b.a.296.25 76
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
531.5.b.a.296.25 76 3.2 odd 2 inner
531.5.b.a.296.52 yes 76 1.1 even 1 trivial