Properties

Label 75.8.a.a.1.1
Level $75$
Weight $8$
Character 75.1
Self dual yes
Analytic conductor $23.429$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,8,Mod(1,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(23.4288769113\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 75.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-6.00000 q^{2} +27.0000 q^{3} -92.0000 q^{4} -162.000 q^{6} +64.0000 q^{7} +1320.00 q^{8} +729.000 q^{9} -948.000 q^{11} -2484.00 q^{12} +5098.00 q^{13} -384.000 q^{14} +3856.00 q^{16} -28386.0 q^{17} -4374.00 q^{18} -8620.00 q^{19} +1728.00 q^{21} +5688.00 q^{22} +15288.0 q^{23} +35640.0 q^{24} -30588.0 q^{26} +19683.0 q^{27} -5888.00 q^{28} +36510.0 q^{29} -276808. q^{31} -192096. q^{32} -25596.0 q^{33} +170316. q^{34} -67068.0 q^{36} -268526. q^{37} +51720.0 q^{38} +137646. q^{39} -629718. q^{41} -10368.0 q^{42} -685772. q^{43} +87216.0 q^{44} -91728.0 q^{46} -583296. q^{47} +104112. q^{48} -819447. q^{49} -766422. q^{51} -469016. q^{52} +428058. q^{53} -118098. q^{54} +84480.0 q^{56} -232740. q^{57} -219060. q^{58} +1.30638e6 q^{59} +300662. q^{61} +1.66085e6 q^{62} +46656.0 q^{63} +659008. q^{64} +153576. q^{66} +507244. q^{67} +2.61151e6 q^{68} +412776. q^{69} +5.56063e6 q^{71} +962280. q^{72} -1.36908e6 q^{73} +1.61116e6 q^{74} +793040. q^{76} -60672.0 q^{77} -825876. q^{78} -6.91372e6 q^{79} +531441. q^{81} +3.77831e6 q^{82} +4.37675e6 q^{83} -158976. q^{84} +4.11463e6 q^{86} +985770. q^{87} -1.25136e6 q^{88} -8.52831e6 q^{89} +326272. q^{91} -1.40650e6 q^{92} -7.47382e6 q^{93} +3.49978e6 q^{94} -5.18659e6 q^{96} +8.82681e6 q^{97} +4.91668e6 q^{98} -691092. q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.00000 −0.530330 −0.265165 0.964203i \(-0.585426\pi\)
−0.265165 + 0.964203i \(0.585426\pi\)
\(3\) 27.0000 0.577350
\(4\) −92.0000 −0.718750
\(5\) 0 0
\(6\) −162.000 −0.306186
\(7\) 64.0000 0.0705240 0.0352620 0.999378i \(-0.488773\pi\)
0.0352620 + 0.999378i \(0.488773\pi\)
\(8\) 1320.00 0.911505
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) −948.000 −0.214750 −0.107375 0.994219i \(-0.534245\pi\)
−0.107375 + 0.994219i \(0.534245\pi\)
\(12\) −2484.00 −0.414971
\(13\) 5098.00 0.643573 0.321787 0.946812i \(-0.395717\pi\)
0.321787 + 0.946812i \(0.395717\pi\)
\(14\) −384.000 −0.0374010
\(15\) 0 0
\(16\) 3856.00 0.235352
\(17\) −28386.0 −1.40131 −0.700653 0.713502i \(-0.747108\pi\)
−0.700653 + 0.713502i \(0.747108\pi\)
\(18\) −4374.00 −0.176777
\(19\) −8620.00 −0.288317 −0.144158 0.989555i \(-0.546047\pi\)
−0.144158 + 0.989555i \(0.546047\pi\)
\(20\) 0 0
\(21\) 1728.00 0.0407170
\(22\) 5688.00 0.113889
\(23\) 15288.0 0.262001 0.131001 0.991382i \(-0.458181\pi\)
0.131001 + 0.991382i \(0.458181\pi\)
\(24\) 35640.0 0.526258
\(25\) 0 0
\(26\) −30588.0 −0.341306
\(27\) 19683.0 0.192450
\(28\) −5888.00 −0.0506891
\(29\) 36510.0 0.277983 0.138992 0.990294i \(-0.455614\pi\)
0.138992 + 0.990294i \(0.455614\pi\)
\(30\) 0 0
\(31\) −276808. −1.66883 −0.834416 0.551135i \(-0.814195\pi\)
−0.834416 + 0.551135i \(0.814195\pi\)
\(32\) −192096. −1.03632
\(33\) −25596.0 −0.123986
\(34\) 170316. 0.743155
\(35\) 0 0
\(36\) −67068.0 −0.239583
\(37\) −268526. −0.871526 −0.435763 0.900061i \(-0.643521\pi\)
−0.435763 + 0.900061i \(0.643521\pi\)
\(38\) 51720.0 0.152903
\(39\) 137646. 0.371567
\(40\) 0 0
\(41\) −629718. −1.42693 −0.713465 0.700691i \(-0.752875\pi\)
−0.713465 + 0.700691i \(0.752875\pi\)
\(42\) −10368.0 −0.0215935
\(43\) −685772. −1.31535 −0.657673 0.753303i \(-0.728459\pi\)
−0.657673 + 0.753303i \(0.728459\pi\)
\(44\) 87216.0 0.154352
\(45\) 0 0
\(46\) −91728.0 −0.138947
\(47\) −583296. −0.819495 −0.409748 0.912199i \(-0.634383\pi\)
−0.409748 + 0.912199i \(0.634383\pi\)
\(48\) 104112. 0.135880
\(49\) −819447. −0.995026
\(50\) 0 0
\(51\) −766422. −0.809044
\(52\) −469016. −0.462568
\(53\) 428058. 0.394945 0.197473 0.980308i \(-0.436727\pi\)
0.197473 + 0.980308i \(0.436727\pi\)
\(54\) −118098. −0.102062
\(55\) 0 0
\(56\) 84480.0 0.0642830
\(57\) −232740. −0.166460
\(58\) −219060. −0.147423
\(59\) 1.30638e6 0.828109 0.414054 0.910252i \(-0.364112\pi\)
0.414054 + 0.910252i \(0.364112\pi\)
\(60\) 0 0
\(61\) 300662. 0.169599 0.0847997 0.996398i \(-0.472975\pi\)
0.0847997 + 0.996398i \(0.472975\pi\)
\(62\) 1.66085e6 0.885032
\(63\) 46656.0 0.0235080
\(64\) 659008. 0.314240
\(65\) 0 0
\(66\) 153576. 0.0657536
\(67\) 507244. 0.206042 0.103021 0.994679i \(-0.467149\pi\)
0.103021 + 0.994679i \(0.467149\pi\)
\(68\) 2.61151e6 1.00719
\(69\) 412776. 0.151266
\(70\) 0 0
\(71\) 5.56063e6 1.84383 0.921913 0.387397i \(-0.126626\pi\)
0.921913 + 0.387397i \(0.126626\pi\)
\(72\) 962280. 0.303835
\(73\) −1.36908e6 −0.411907 −0.205954 0.978562i \(-0.566030\pi\)
−0.205954 + 0.978562i \(0.566030\pi\)
\(74\) 1.61116e6 0.462196
\(75\) 0 0
\(76\) 793040. 0.207228
\(77\) −60672.0 −0.0151451
\(78\) −825876. −0.197053
\(79\) −6.91372e6 −1.57767 −0.788836 0.614603i \(-0.789316\pi\)
−0.788836 + 0.614603i \(0.789316\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 3.77831e6 0.756744
\(83\) 4.37675e6 0.840191 0.420096 0.907480i \(-0.361997\pi\)
0.420096 + 0.907480i \(0.361997\pi\)
\(84\) −158976. −0.0292654
\(85\) 0 0
\(86\) 4.11463e6 0.697568
\(87\) 985770. 0.160494
\(88\) −1.25136e6 −0.195746
\(89\) −8.52831e6 −1.28232 −0.641162 0.767405i \(-0.721547\pi\)
−0.641162 + 0.767405i \(0.721547\pi\)
\(90\) 0 0
\(91\) 326272. 0.0453874
\(92\) −1.40650e6 −0.188313
\(93\) −7.47382e6 −0.963501
\(94\) 3.49978e6 0.434603
\(95\) 0 0
\(96\) −5.18659e6 −0.598319
\(97\) 8.82681e6 0.981981 0.490990 0.871165i \(-0.336635\pi\)
0.490990 + 0.871165i \(0.336635\pi\)
\(98\) 4.91668e6 0.527692
\(99\) −691092. −0.0715835
\(100\) 0 0
\(101\) 1.19864e7 1.15762 0.578808 0.815464i \(-0.303518\pi\)
0.578808 + 0.815464i \(0.303518\pi\)
\(102\) 4.59853e6 0.429061
\(103\) −7.20939e6 −0.650082 −0.325041 0.945700i \(-0.605378\pi\)
−0.325041 + 0.945700i \(0.605378\pi\)
\(104\) 6.72936e6 0.586620
\(105\) 0 0
\(106\) −2.56835e6 −0.209451
\(107\) −1.14261e7 −0.901683 −0.450842 0.892604i \(-0.648876\pi\)
−0.450842 + 0.892604i \(0.648876\pi\)
\(108\) −1.81084e6 −0.138324
\(109\) 4.02095e6 0.297397 0.148698 0.988883i \(-0.452492\pi\)
0.148698 + 0.988883i \(0.452492\pi\)
\(110\) 0 0
\(111\) −7.25020e6 −0.503176
\(112\) 246784. 0.0165979
\(113\) 1.77063e7 1.15439 0.577197 0.816605i \(-0.304147\pi\)
0.577197 + 0.816605i \(0.304147\pi\)
\(114\) 1.39644e6 0.0882786
\(115\) 0 0
\(116\) −3.35892e6 −0.199801
\(117\) 3.71644e6 0.214524
\(118\) −7.83828e6 −0.439171
\(119\) −1.81670e6 −0.0988257
\(120\) 0 0
\(121\) −1.85885e7 −0.953882
\(122\) −1.80397e6 −0.0899436
\(123\) −1.70024e7 −0.823838
\(124\) 2.54663e7 1.19947
\(125\) 0 0
\(126\) −279936. −0.0124670
\(127\) −1.67883e7 −0.727267 −0.363633 0.931542i \(-0.618464\pi\)
−0.363633 + 0.931542i \(0.618464\pi\)
\(128\) 2.06342e7 0.869668
\(129\) −1.85158e7 −0.759416
\(130\) 0 0
\(131\) 1.68268e7 0.653960 0.326980 0.945031i \(-0.393969\pi\)
0.326980 + 0.945031i \(0.393969\pi\)
\(132\) 2.35483e6 0.0891151
\(133\) −551680. −0.0203332
\(134\) −3.04346e6 −0.109270
\(135\) 0 0
\(136\) −3.74695e7 −1.27730
\(137\) −2.80449e7 −0.931820 −0.465910 0.884832i \(-0.654273\pi\)
−0.465910 + 0.884832i \(0.654273\pi\)
\(138\) −2.47666e6 −0.0802212
\(139\) −1.18273e7 −0.373537 −0.186769 0.982404i \(-0.559801\pi\)
−0.186769 + 0.982404i \(0.559801\pi\)
\(140\) 0 0
\(141\) −1.57490e7 −0.473136
\(142\) −3.33638e7 −0.977836
\(143\) −4.83290e6 −0.138208
\(144\) 2.81102e6 0.0784505
\(145\) 0 0
\(146\) 8.21449e6 0.218447
\(147\) −2.21251e7 −0.574479
\(148\) 2.47044e7 0.626409
\(149\) 2.07846e7 0.514743 0.257371 0.966313i \(-0.417144\pi\)
0.257371 + 0.966313i \(0.417144\pi\)
\(150\) 0 0
\(151\) 76112.0 0.00179901 0.000899505 1.00000i \(-0.499714\pi\)
0.000899505 1.00000i \(0.499714\pi\)
\(152\) −1.13784e7 −0.262802
\(153\) −2.06934e7 −0.467102
\(154\) 364032. 0.00803188
\(155\) 0 0
\(156\) −1.26634e7 −0.267064
\(157\) 3.21825e7 0.663698 0.331849 0.943332i \(-0.392328\pi\)
0.331849 + 0.943332i \(0.392328\pi\)
\(158\) 4.14823e7 0.836687
\(159\) 1.15576e7 0.228022
\(160\) 0 0
\(161\) 978432. 0.0184774
\(162\) −3.18865e6 −0.0589256
\(163\) −5.83435e7 −1.05520 −0.527601 0.849492i \(-0.676908\pi\)
−0.527601 + 0.849492i \(0.676908\pi\)
\(164\) 5.79341e7 1.02561
\(165\) 0 0
\(166\) −2.62605e7 −0.445579
\(167\) 2.58365e7 0.429266 0.214633 0.976695i \(-0.431145\pi\)
0.214633 + 0.976695i \(0.431145\pi\)
\(168\) 2.28096e6 0.0371138
\(169\) −3.67589e7 −0.585813
\(170\) 0 0
\(171\) −6.28398e6 −0.0961055
\(172\) 6.30910e7 0.945405
\(173\) −6.35201e7 −0.932716 −0.466358 0.884596i \(-0.654434\pi\)
−0.466358 + 0.884596i \(0.654434\pi\)
\(174\) −5.91462e6 −0.0851147
\(175\) 0 0
\(176\) −3.65549e6 −0.0505418
\(177\) 3.52723e7 0.478109
\(178\) 5.11699e7 0.680055
\(179\) −8.09559e7 −1.05503 −0.527513 0.849547i \(-0.676875\pi\)
−0.527513 + 0.849547i \(0.676875\pi\)
\(180\) 0 0
\(181\) 6.45032e7 0.808549 0.404274 0.914638i \(-0.367524\pi\)
0.404274 + 0.914638i \(0.367524\pi\)
\(182\) −1.95763e6 −0.0240703
\(183\) 8.11787e6 0.0979182
\(184\) 2.01802e7 0.238815
\(185\) 0 0
\(186\) 4.48429e7 0.510973
\(187\) 2.69099e7 0.300931
\(188\) 5.36632e7 0.589012
\(189\) 1.25971e6 0.0135723
\(190\) 0 0
\(191\) 5.68274e7 0.590121 0.295060 0.955479i \(-0.404660\pi\)
0.295060 + 0.955479i \(0.404660\pi\)
\(192\) 1.77932e7 0.181426
\(193\) −1.16377e8 −1.16524 −0.582621 0.812744i \(-0.697973\pi\)
−0.582621 + 0.812744i \(0.697973\pi\)
\(194\) −5.29609e7 −0.520774
\(195\) 0 0
\(196\) 7.53891e7 0.715175
\(197\) 1.18816e8 1.10724 0.553622 0.832768i \(-0.313245\pi\)
0.553622 + 0.832768i \(0.313245\pi\)
\(198\) 4.14655e6 0.0379629
\(199\) −9.50106e7 −0.854646 −0.427323 0.904099i \(-0.640543\pi\)
−0.427323 + 0.904099i \(0.640543\pi\)
\(200\) 0 0
\(201\) 1.36956e7 0.118958
\(202\) −7.19185e7 −0.613919
\(203\) 2.33664e6 0.0196045
\(204\) 7.05108e7 0.581501
\(205\) 0 0
\(206\) 4.32564e7 0.344758
\(207\) 1.11450e7 0.0873337
\(208\) 1.96579e7 0.151466
\(209\) 8.17176e6 0.0619161
\(210\) 0 0
\(211\) 1.79246e8 1.31360 0.656798 0.754067i \(-0.271910\pi\)
0.656798 + 0.754067i \(0.271910\pi\)
\(212\) −3.93813e7 −0.283867
\(213\) 1.50137e8 1.06453
\(214\) 6.85565e7 0.478190
\(215\) 0 0
\(216\) 2.59816e7 0.175419
\(217\) −1.77157e7 −0.117693
\(218\) −2.41257e7 −0.157718
\(219\) −3.69652e7 −0.237815
\(220\) 0 0
\(221\) −1.44712e8 −0.901843
\(222\) 4.35012e7 0.266849
\(223\) 2.06537e8 1.24718 0.623592 0.781750i \(-0.285673\pi\)
0.623592 + 0.781750i \(0.285673\pi\)
\(224\) −1.22941e7 −0.0730853
\(225\) 0 0
\(226\) −1.06238e8 −0.612209
\(227\) −4.33954e7 −0.246237 −0.123118 0.992392i \(-0.539290\pi\)
−0.123118 + 0.992392i \(0.539290\pi\)
\(228\) 2.14121e7 0.119643
\(229\) −3.61931e7 −0.199160 −0.0995799 0.995030i \(-0.531750\pi\)
−0.0995799 + 0.995030i \(0.531750\pi\)
\(230\) 0 0
\(231\) −1.63814e6 −0.00874400
\(232\) 4.81932e7 0.253383
\(233\) −9.22347e7 −0.477693 −0.238846 0.971057i \(-0.576769\pi\)
−0.238846 + 0.971057i \(0.576769\pi\)
\(234\) −2.22987e7 −0.113769
\(235\) 0 0
\(236\) −1.20187e8 −0.595203
\(237\) −1.86670e8 −0.910870
\(238\) 1.09002e7 0.0524102
\(239\) 4.98468e7 0.236181 0.118090 0.993003i \(-0.462323\pi\)
0.118090 + 0.993003i \(0.462323\pi\)
\(240\) 0 0
\(241\) 1.99374e8 0.917506 0.458753 0.888564i \(-0.348296\pi\)
0.458753 + 0.888564i \(0.348296\pi\)
\(242\) 1.11531e8 0.505872
\(243\) 1.43489e7 0.0641500
\(244\) −2.76609e7 −0.121900
\(245\) 0 0
\(246\) 1.02014e8 0.436906
\(247\) −4.39448e7 −0.185553
\(248\) −3.65387e8 −1.52115
\(249\) 1.18172e8 0.485085
\(250\) 0 0
\(251\) −3.94678e8 −1.57538 −0.787689 0.616073i \(-0.788723\pi\)
−0.787689 + 0.616073i \(0.788723\pi\)
\(252\) −4.29235e6 −0.0168964
\(253\) −1.44930e7 −0.0562649
\(254\) 1.00730e8 0.385691
\(255\) 0 0
\(256\) −2.08158e8 −0.775451
\(257\) 1.42885e8 0.525076 0.262538 0.964922i \(-0.415441\pi\)
0.262538 + 0.964922i \(0.415441\pi\)
\(258\) 1.11095e8 0.402741
\(259\) −1.71857e7 −0.0614635
\(260\) 0 0
\(261\) 2.66158e7 0.0926611
\(262\) −1.00961e8 −0.346815
\(263\) −4.40241e8 −1.49226 −0.746131 0.665799i \(-0.768091\pi\)
−0.746131 + 0.665799i \(0.768091\pi\)
\(264\) −3.37867e7 −0.113014
\(265\) 0 0
\(266\) 3.31008e6 0.0107833
\(267\) −2.30264e8 −0.740350
\(268\) −4.66664e7 −0.148092
\(269\) 2.75405e8 0.862657 0.431329 0.902195i \(-0.358045\pi\)
0.431329 + 0.902195i \(0.358045\pi\)
\(270\) 0 0
\(271\) −4.24670e8 −1.29616 −0.648080 0.761572i \(-0.724428\pi\)
−0.648080 + 0.761572i \(0.724428\pi\)
\(272\) −1.09456e8 −0.329800
\(273\) 8.80934e6 0.0262044
\(274\) 1.68269e8 0.494172
\(275\) 0 0
\(276\) −3.79754e7 −0.108723
\(277\) −5.16158e8 −1.45916 −0.729581 0.683894i \(-0.760285\pi\)
−0.729581 + 0.683894i \(0.760285\pi\)
\(278\) 7.09638e7 0.198098
\(279\) −2.01793e8 −0.556277
\(280\) 0 0
\(281\) −3.11043e8 −0.836273 −0.418137 0.908384i \(-0.637317\pi\)
−0.418137 + 0.908384i \(0.637317\pi\)
\(282\) 9.44940e7 0.250918
\(283\) 5.94308e8 1.55869 0.779344 0.626596i \(-0.215552\pi\)
0.779344 + 0.626596i \(0.215552\pi\)
\(284\) −5.11578e8 −1.32525
\(285\) 0 0
\(286\) 2.89974e7 0.0732957
\(287\) −4.03020e7 −0.100633
\(288\) −1.40038e8 −0.345440
\(289\) 3.95426e8 0.963658
\(290\) 0 0
\(291\) 2.38324e8 0.566947
\(292\) 1.25956e8 0.296058
\(293\) −1.15515e8 −0.268288 −0.134144 0.990962i \(-0.542828\pi\)
−0.134144 + 0.990962i \(0.542828\pi\)
\(294\) 1.32750e8 0.304663
\(295\) 0 0
\(296\) −3.54454e8 −0.794400
\(297\) −1.86595e7 −0.0413287
\(298\) −1.24708e8 −0.272984
\(299\) 7.79382e7 0.168617
\(300\) 0 0
\(301\) −4.38894e7 −0.0927635
\(302\) −456672. −0.000954070 0
\(303\) 3.23633e8 0.668350
\(304\) −3.32387e7 −0.0678558
\(305\) 0 0
\(306\) 1.24160e8 0.247718
\(307\) 2.60600e8 0.514032 0.257016 0.966407i \(-0.417261\pi\)
0.257016 + 0.966407i \(0.417261\pi\)
\(308\) 5.58182e6 0.0108855
\(309\) −1.94654e8 −0.375325
\(310\) 0 0
\(311\) 5.76795e8 1.08733 0.543663 0.839303i \(-0.317037\pi\)
0.543663 + 0.839303i \(0.317037\pi\)
\(312\) 1.81693e8 0.338685
\(313\) 4.60074e8 0.848053 0.424026 0.905650i \(-0.360616\pi\)
0.424026 + 0.905650i \(0.360616\pi\)
\(314\) −1.93095e8 −0.351979
\(315\) 0 0
\(316\) 6.36062e8 1.13395
\(317\) −6.25561e7 −0.110297 −0.0551483 0.998478i \(-0.517563\pi\)
−0.0551483 + 0.998478i \(0.517563\pi\)
\(318\) −6.93454e7 −0.120927
\(319\) −3.46115e7 −0.0596970
\(320\) 0 0
\(321\) −3.08504e8 −0.520587
\(322\) −5.87059e6 −0.00979910
\(323\) 2.44687e8 0.404020
\(324\) −4.88926e7 −0.0798611
\(325\) 0 0
\(326\) 3.50061e8 0.559606
\(327\) 1.08566e8 0.171702
\(328\) −8.31228e8 −1.30065
\(329\) −3.73309e7 −0.0577941
\(330\) 0 0
\(331\) 6.84236e8 1.03707 0.518535 0.855057i \(-0.326478\pi\)
0.518535 + 0.855057i \(0.326478\pi\)
\(332\) −4.02661e8 −0.603888
\(333\) −1.95755e8 −0.290509
\(334\) −1.55019e8 −0.227652
\(335\) 0 0
\(336\) 6.66317e6 0.00958282
\(337\) 6.26313e8 0.891429 0.445714 0.895175i \(-0.352950\pi\)
0.445714 + 0.895175i \(0.352950\pi\)
\(338\) 2.20553e8 0.310674
\(339\) 4.78071e8 0.666489
\(340\) 0 0
\(341\) 2.62414e8 0.358382
\(342\) 3.77039e7 0.0509677
\(343\) −1.05151e8 −0.140697
\(344\) −9.05219e8 −1.19894
\(345\) 0 0
\(346\) 3.81120e8 0.494647
\(347\) 1.25340e9 1.61041 0.805203 0.593000i \(-0.202057\pi\)
0.805203 + 0.593000i \(0.202057\pi\)
\(348\) −9.06908e7 −0.115355
\(349\) 2.65350e8 0.334142 0.167071 0.985945i \(-0.446569\pi\)
0.167071 + 0.985945i \(0.446569\pi\)
\(350\) 0 0
\(351\) 1.00344e8 0.123856
\(352\) 1.82107e8 0.222550
\(353\) 5.69636e8 0.689264 0.344632 0.938738i \(-0.388004\pi\)
0.344632 + 0.938738i \(0.388004\pi\)
\(354\) −2.11634e8 −0.253556
\(355\) 0 0
\(356\) 7.84605e8 0.921671
\(357\) −4.90510e7 −0.0570570
\(358\) 4.85735e8 0.559512
\(359\) 9.32541e8 1.06374 0.531872 0.846825i \(-0.321489\pi\)
0.531872 + 0.846825i \(0.321489\pi\)
\(360\) 0 0
\(361\) −8.19567e8 −0.916874
\(362\) −3.87019e8 −0.428798
\(363\) −5.01889e8 −0.550724
\(364\) −3.00170e7 −0.0326222
\(365\) 0 0
\(366\) −4.87072e7 −0.0519290
\(367\) 8.52565e8 0.900318 0.450159 0.892948i \(-0.351367\pi\)
0.450159 + 0.892948i \(0.351367\pi\)
\(368\) 5.89505e7 0.0616624
\(369\) −4.59064e8 −0.475643
\(370\) 0 0
\(371\) 2.73957e7 0.0278531
\(372\) 6.87591e8 0.692516
\(373\) −3.81183e8 −0.380323 −0.190162 0.981753i \(-0.560901\pi\)
−0.190162 + 0.981753i \(0.560901\pi\)
\(374\) −1.61460e8 −0.159593
\(375\) 0 0
\(376\) −7.69951e8 −0.746974
\(377\) 1.86128e8 0.178903
\(378\) −7.55827e6 −0.00719782
\(379\) −1.48353e9 −1.39978 −0.699889 0.714251i \(-0.746767\pi\)
−0.699889 + 0.714251i \(0.746767\pi\)
\(380\) 0 0
\(381\) −4.53284e8 −0.419888
\(382\) −3.40964e8 −0.312959
\(383\) 7.61930e8 0.692978 0.346489 0.938054i \(-0.387374\pi\)
0.346489 + 0.938054i \(0.387374\pi\)
\(384\) 5.57124e8 0.502103
\(385\) 0 0
\(386\) 6.98262e8 0.617963
\(387\) −4.99928e8 −0.438449
\(388\) −8.12067e8 −0.705799
\(389\) 1.60902e9 1.38592 0.692959 0.720977i \(-0.256307\pi\)
0.692959 + 0.720977i \(0.256307\pi\)
\(390\) 0 0
\(391\) −4.33965e8 −0.367144
\(392\) −1.08167e9 −0.906971
\(393\) 4.54323e8 0.377564
\(394\) −7.12896e8 −0.587205
\(395\) 0 0
\(396\) 6.35805e7 0.0514506
\(397\) −1.88016e9 −1.50809 −0.754046 0.656822i \(-0.771900\pi\)
−0.754046 + 0.656822i \(0.771900\pi\)
\(398\) 5.70064e8 0.453245
\(399\) −1.48954e7 −0.0117394
\(400\) 0 0
\(401\) 2.68592e8 0.208012 0.104006 0.994577i \(-0.466834\pi\)
0.104006 + 0.994577i \(0.466834\pi\)
\(402\) −8.21735e7 −0.0630871
\(403\) −1.41117e9 −1.07402
\(404\) −1.10275e9 −0.832037
\(405\) 0 0
\(406\) −1.40198e7 −0.0103969
\(407\) 2.54563e8 0.187161
\(408\) −1.01168e9 −0.737448
\(409\) 8.99478e7 0.0650069 0.0325034 0.999472i \(-0.489652\pi\)
0.0325034 + 0.999472i \(0.489652\pi\)
\(410\) 0 0
\(411\) −7.57212e8 −0.537986
\(412\) 6.63264e8 0.467247
\(413\) 8.36083e7 0.0584015
\(414\) −6.68697e7 −0.0463157
\(415\) 0 0
\(416\) −9.79305e8 −0.666947
\(417\) −3.19337e8 −0.215662
\(418\) −4.90306e7 −0.0328360
\(419\) 1.69054e9 1.12273 0.561367 0.827567i \(-0.310276\pi\)
0.561367 + 0.827567i \(0.310276\pi\)
\(420\) 0 0
\(421\) −1.13333e9 −0.740232 −0.370116 0.928985i \(-0.620682\pi\)
−0.370116 + 0.928985i \(0.620682\pi\)
\(422\) −1.07548e9 −0.696639
\(423\) −4.25223e8 −0.273165
\(424\) 5.65037e8 0.359995
\(425\) 0 0
\(426\) −9.00822e8 −0.564554
\(427\) 1.92424e7 0.0119608
\(428\) 1.05120e9 0.648085
\(429\) −1.30488e8 −0.0797942
\(430\) 0 0
\(431\) 2.19943e9 1.32324 0.661621 0.749839i \(-0.269869\pi\)
0.661621 + 0.749839i \(0.269869\pi\)
\(432\) 7.58976e7 0.0452934
\(433\) 1.51738e8 0.0898227 0.0449114 0.998991i \(-0.485699\pi\)
0.0449114 + 0.998991i \(0.485699\pi\)
\(434\) 1.06294e8 0.0624160
\(435\) 0 0
\(436\) −3.69927e8 −0.213754
\(437\) −1.31783e8 −0.0755393
\(438\) 2.21791e8 0.126120
\(439\) 9.90763e8 0.558912 0.279456 0.960158i \(-0.409846\pi\)
0.279456 + 0.960158i \(0.409846\pi\)
\(440\) 0 0
\(441\) −5.97377e8 −0.331675
\(442\) 8.68271e8 0.478275
\(443\) 1.77376e9 0.969351 0.484675 0.874694i \(-0.338938\pi\)
0.484675 + 0.874694i \(0.338938\pi\)
\(444\) 6.67019e8 0.361658
\(445\) 0 0
\(446\) −1.23922e9 −0.661419
\(447\) 5.61185e8 0.297187
\(448\) 4.21765e7 0.0221614
\(449\) −2.77010e8 −0.144422 −0.0722110 0.997389i \(-0.523006\pi\)
−0.0722110 + 0.997389i \(0.523006\pi\)
\(450\) 0 0
\(451\) 5.96973e8 0.306434
\(452\) −1.62898e9 −0.829720
\(453\) 2.05502e6 0.00103866
\(454\) 2.60372e8 0.130587
\(455\) 0 0
\(456\) −3.07217e8 −0.151729
\(457\) −2.94758e9 −1.44464 −0.722320 0.691559i \(-0.756924\pi\)
−0.722320 + 0.691559i \(0.756924\pi\)
\(458\) 2.17159e8 0.105620
\(459\) −5.58722e8 −0.269681
\(460\) 0 0
\(461\) −2.76687e9 −1.31533 −0.657667 0.753309i \(-0.728457\pi\)
−0.657667 + 0.753309i \(0.728457\pi\)
\(462\) 9.82886e6 0.00463721
\(463\) −4.63553e8 −0.217053 −0.108527 0.994094i \(-0.534613\pi\)
−0.108527 + 0.994094i \(0.534613\pi\)
\(464\) 1.40783e8 0.0654238
\(465\) 0 0
\(466\) 5.53408e8 0.253335
\(467\) 4.17922e8 0.189883 0.0949415 0.995483i \(-0.469734\pi\)
0.0949415 + 0.995483i \(0.469734\pi\)
\(468\) −3.41913e8 −0.154189
\(469\) 3.24636e7 0.0145309
\(470\) 0 0
\(471\) 8.68927e8 0.383186
\(472\) 1.72442e9 0.754825
\(473\) 6.50112e8 0.282471
\(474\) 1.12002e9 0.483062
\(475\) 0 0
\(476\) 1.67137e8 0.0710310
\(477\) 3.12054e8 0.131648
\(478\) −2.99081e8 −0.125254
\(479\) −1.50973e9 −0.627660 −0.313830 0.949479i \(-0.601612\pi\)
−0.313830 + 0.949479i \(0.601612\pi\)
\(480\) 0 0
\(481\) −1.36895e9 −0.560891
\(482\) −1.19624e9 −0.486581
\(483\) 2.64177e7 0.0106679
\(484\) 1.71014e9 0.685603
\(485\) 0 0
\(486\) −8.60934e7 −0.0340207
\(487\) −9.29460e8 −0.364653 −0.182326 0.983238i \(-0.558363\pi\)
−0.182326 + 0.983238i \(0.558363\pi\)
\(488\) 3.96874e8 0.154591
\(489\) −1.57527e9 −0.609221
\(490\) 0 0
\(491\) 5.12803e9 1.95508 0.977541 0.210743i \(-0.0675885\pi\)
0.977541 + 0.210743i \(0.0675885\pi\)
\(492\) 1.56422e9 0.592134
\(493\) −1.03637e9 −0.389540
\(494\) 2.63669e8 0.0984043
\(495\) 0 0
\(496\) −1.06737e9 −0.392762
\(497\) 3.55880e8 0.130034
\(498\) −7.09033e8 −0.257255
\(499\) −4.10649e8 −0.147951 −0.0739757 0.997260i \(-0.523569\pi\)
−0.0739757 + 0.997260i \(0.523569\pi\)
\(500\) 0 0
\(501\) 6.97586e8 0.247837
\(502\) 2.36807e9 0.835470
\(503\) −5.02041e9 −1.75894 −0.879470 0.475954i \(-0.842103\pi\)
−0.879470 + 0.475954i \(0.842103\pi\)
\(504\) 6.15859e7 0.0214277
\(505\) 0 0
\(506\) 8.69581e7 0.0298389
\(507\) −9.92491e8 −0.338219
\(508\) 1.54452e9 0.522723
\(509\) −3.24926e9 −1.09212 −0.546062 0.837745i \(-0.683874\pi\)
−0.546062 + 0.837745i \(0.683874\pi\)
\(510\) 0 0
\(511\) −8.76212e7 −0.0290493
\(512\) −1.39223e9 −0.458423
\(513\) −1.69667e8 −0.0554866
\(514\) −8.57312e8 −0.278463
\(515\) 0 0
\(516\) 1.70346e9 0.545830
\(517\) 5.52965e8 0.175987
\(518\) 1.03114e8 0.0325959
\(519\) −1.71504e9 −0.538504
\(520\) 0 0
\(521\) −2.10950e9 −0.653503 −0.326752 0.945110i \(-0.605954\pi\)
−0.326752 + 0.945110i \(0.605954\pi\)
\(522\) −1.59695e8 −0.0491410
\(523\) 5.28911e9 1.61669 0.808345 0.588709i \(-0.200364\pi\)
0.808345 + 0.588709i \(0.200364\pi\)
\(524\) −1.54806e9 −0.470034
\(525\) 0 0
\(526\) 2.64144e9 0.791391
\(527\) 7.85747e9 2.33854
\(528\) −9.86982e7 −0.0291803
\(529\) −3.17110e9 −0.931355
\(530\) 0 0
\(531\) 9.52351e8 0.276036
\(532\) 5.07546e7 0.0146145
\(533\) −3.21030e9 −0.918334
\(534\) 1.38159e9 0.392630
\(535\) 0 0
\(536\) 6.69562e8 0.187808
\(537\) −2.18581e9 −0.609119
\(538\) −1.65243e9 −0.457493
\(539\) 7.76836e8 0.213682
\(540\) 0 0
\(541\) 3.04614e9 0.827101 0.413551 0.910481i \(-0.364288\pi\)
0.413551 + 0.910481i \(0.364288\pi\)
\(542\) 2.54802e9 0.687393
\(543\) 1.74159e9 0.466816
\(544\) 5.45284e9 1.45220
\(545\) 0 0
\(546\) −5.28561e7 −0.0138970
\(547\) 4.85537e9 1.26843 0.634215 0.773157i \(-0.281323\pi\)
0.634215 + 0.773157i \(0.281323\pi\)
\(548\) 2.58013e9 0.669746
\(549\) 2.19183e8 0.0565331
\(550\) 0 0
\(551\) −3.14716e8 −0.0801472
\(552\) 5.44864e8 0.137880
\(553\) −4.42478e8 −0.111264
\(554\) 3.09695e9 0.773838
\(555\) 0 0
\(556\) 1.08811e9 0.268480
\(557\) −1.27762e9 −0.313263 −0.156631 0.987657i \(-0.550063\pi\)
−0.156631 + 0.987657i \(0.550063\pi\)
\(558\) 1.21076e9 0.295011
\(559\) −3.49607e9 −0.846522
\(560\) 0 0
\(561\) 7.26568e8 0.173743
\(562\) 1.86626e9 0.443501
\(563\) −4.71265e9 −1.11297 −0.556487 0.830856i \(-0.687851\pi\)
−0.556487 + 0.830856i \(0.687851\pi\)
\(564\) 1.44891e9 0.340066
\(565\) 0 0
\(566\) −3.56585e9 −0.826619
\(567\) 3.40122e7 0.00783600
\(568\) 7.34003e9 1.68066
\(569\) 4.57800e9 1.04180 0.520898 0.853619i \(-0.325597\pi\)
0.520898 + 0.853619i \(0.325597\pi\)
\(570\) 0 0
\(571\) 4.95119e9 1.11297 0.556485 0.830858i \(-0.312150\pi\)
0.556485 + 0.830858i \(0.312150\pi\)
\(572\) 4.44627e8 0.0993367
\(573\) 1.53434e9 0.340706
\(574\) 2.41812e8 0.0533686
\(575\) 0 0
\(576\) 4.80417e8 0.104747
\(577\) −8.51847e9 −1.84606 −0.923031 0.384725i \(-0.874296\pi\)
−0.923031 + 0.384725i \(0.874296\pi\)
\(578\) −2.37256e9 −0.511057
\(579\) −3.14218e9 −0.672753
\(580\) 0 0
\(581\) 2.80112e8 0.0592536
\(582\) −1.42994e9 −0.300669
\(583\) −4.05799e8 −0.0848147
\(584\) −1.80719e9 −0.375455
\(585\) 0 0
\(586\) 6.93088e8 0.142281
\(587\) 5.62247e8 0.114735 0.0573673 0.998353i \(-0.481729\pi\)
0.0573673 + 0.998353i \(0.481729\pi\)
\(588\) 2.03551e9 0.412907
\(589\) 2.38608e9 0.481152
\(590\) 0 0
\(591\) 3.20803e9 0.639268
\(592\) −1.03544e9 −0.205115
\(593\) −3.62110e9 −0.713099 −0.356549 0.934277i \(-0.616047\pi\)
−0.356549 + 0.934277i \(0.616047\pi\)
\(594\) 1.11957e8 0.0219179
\(595\) 0 0
\(596\) −1.91219e9 −0.369971
\(597\) −2.56529e9 −0.493430
\(598\) −4.67629e8 −0.0894227
\(599\) −7.48104e9 −1.42222 −0.711112 0.703079i \(-0.751808\pi\)
−0.711112 + 0.703079i \(0.751808\pi\)
\(600\) 0 0
\(601\) −5.81270e9 −1.09224 −0.546119 0.837707i \(-0.683895\pi\)
−0.546119 + 0.837707i \(0.683895\pi\)
\(602\) 2.63336e8 0.0491953
\(603\) 3.69781e8 0.0686806
\(604\) −7.00230e6 −0.00129304
\(605\) 0 0
\(606\) −1.94180e9 −0.354446
\(607\) −3.84051e9 −0.696993 −0.348497 0.937310i \(-0.613308\pi\)
−0.348497 + 0.937310i \(0.613308\pi\)
\(608\) 1.65587e9 0.298788
\(609\) 6.30893e7 0.0113187
\(610\) 0 0
\(611\) −2.97364e9 −0.527405
\(612\) 1.90379e9 0.335730
\(613\) −1.70484e9 −0.298932 −0.149466 0.988767i \(-0.547755\pi\)
−0.149466 + 0.988767i \(0.547755\pi\)
\(614\) −1.56360e9 −0.272606
\(615\) 0 0
\(616\) −8.00870e7 −0.0138048
\(617\) 2.80809e9 0.481297 0.240649 0.970612i \(-0.422640\pi\)
0.240649 + 0.970612i \(0.422640\pi\)
\(618\) 1.16792e9 0.199046
\(619\) −2.54365e9 −0.431063 −0.215532 0.976497i \(-0.569148\pi\)
−0.215532 + 0.976497i \(0.569148\pi\)
\(620\) 0 0
\(621\) 3.00914e8 0.0504222
\(622\) −3.46077e9 −0.576642
\(623\) −5.45812e8 −0.0904346
\(624\) 5.30763e8 0.0874489
\(625\) 0 0
\(626\) −2.76045e9 −0.449748
\(627\) 2.20638e8 0.0357473
\(628\) −2.96079e9 −0.477033
\(629\) 7.62238e9 1.22127
\(630\) 0 0
\(631\) −1.51146e8 −0.0239494 −0.0119747 0.999928i \(-0.503812\pi\)
−0.0119747 + 0.999928i \(0.503812\pi\)
\(632\) −9.12611e9 −1.43806
\(633\) 4.83965e9 0.758405
\(634\) 3.75337e8 0.0584936
\(635\) 0 0
\(636\) −1.06330e9 −0.163891
\(637\) −4.17754e9 −0.640373
\(638\) 2.07669e8 0.0316591
\(639\) 4.05370e9 0.614609
\(640\) 0 0
\(641\) −1.23625e10 −1.85397 −0.926987 0.375094i \(-0.877610\pi\)
−0.926987 + 0.375094i \(0.877610\pi\)
\(642\) 1.85102e9 0.276083
\(643\) −2.86744e9 −0.425359 −0.212680 0.977122i \(-0.568219\pi\)
−0.212680 + 0.977122i \(0.568219\pi\)
\(644\) −9.00157e7 −0.0132806
\(645\) 0 0
\(646\) −1.46812e9 −0.214264
\(647\) 4.10640e9 0.596068 0.298034 0.954555i \(-0.403669\pi\)
0.298034 + 0.954555i \(0.403669\pi\)
\(648\) 7.01502e8 0.101278
\(649\) −1.23845e9 −0.177837
\(650\) 0 0
\(651\) −4.78324e8 −0.0679499
\(652\) 5.36760e9 0.758427
\(653\) −6.91100e9 −0.971280 −0.485640 0.874159i \(-0.661413\pi\)
−0.485640 + 0.874159i \(0.661413\pi\)
\(654\) −6.51394e8 −0.0910587
\(655\) 0 0
\(656\) −2.42819e9 −0.335830
\(657\) −9.98061e8 −0.137302
\(658\) 2.23986e8 0.0306499
\(659\) 3.42444e9 0.466112 0.233056 0.972463i \(-0.425127\pi\)
0.233056 + 0.972463i \(0.425127\pi\)
\(660\) 0 0
\(661\) −6.76437e9 −0.911008 −0.455504 0.890234i \(-0.650541\pi\)
−0.455504 + 0.890234i \(0.650541\pi\)
\(662\) −4.10541e9 −0.549989
\(663\) −3.90722e9 −0.520679
\(664\) 5.77731e9 0.765839
\(665\) 0 0
\(666\) 1.17453e9 0.154065
\(667\) 5.58165e8 0.0728320
\(668\) −2.37696e9 −0.308535
\(669\) 5.57650e9 0.720062
\(670\) 0 0
\(671\) −2.85028e8 −0.0364215
\(672\) −3.31942e8 −0.0421958
\(673\) 1.74959e9 0.221250 0.110625 0.993862i \(-0.464715\pi\)
0.110625 + 0.993862i \(0.464715\pi\)
\(674\) −3.75788e9 −0.472752
\(675\) 0 0
\(676\) 3.38182e9 0.421053
\(677\) −8.30011e9 −1.02807 −0.514036 0.857769i \(-0.671850\pi\)
−0.514036 + 0.857769i \(0.671850\pi\)
\(678\) −2.86842e9 −0.353459
\(679\) 5.64916e8 0.0692532
\(680\) 0 0
\(681\) −1.17168e9 −0.142165
\(682\) −1.57448e9 −0.190061
\(683\) 1.21232e10 1.45594 0.727969 0.685610i \(-0.240464\pi\)
0.727969 + 0.685610i \(0.240464\pi\)
\(684\) 5.78126e8 0.0690759
\(685\) 0 0
\(686\) 6.30908e8 0.0746160
\(687\) −9.77213e8 −0.114985
\(688\) −2.64434e9 −0.309569
\(689\) 2.18224e9 0.254176
\(690\) 0 0
\(691\) 8.21846e9 0.947583 0.473791 0.880637i \(-0.342885\pi\)
0.473791 + 0.880637i \(0.342885\pi\)
\(692\) 5.84385e9 0.670390
\(693\) −4.42299e7 −0.00504835
\(694\) −7.52038e9 −0.854046
\(695\) 0 0
\(696\) 1.30122e9 0.146291
\(697\) 1.78752e10 1.99957
\(698\) −1.59210e9 −0.177205
\(699\) −2.49034e9 −0.275796
\(700\) 0 0
\(701\) 4.72231e9 0.517775 0.258888 0.965907i \(-0.416644\pi\)
0.258888 + 0.965907i \(0.416644\pi\)
\(702\) −6.02064e8 −0.0656844
\(703\) 2.31469e9 0.251275
\(704\) −6.24740e8 −0.0674831
\(705\) 0 0
\(706\) −3.41781e9 −0.365537
\(707\) 7.67131e8 0.0816397
\(708\) −3.24505e9 −0.343641
\(709\) 2.78975e9 0.293970 0.146985 0.989139i \(-0.453043\pi\)
0.146985 + 0.989139i \(0.453043\pi\)
\(710\) 0 0
\(711\) −5.04010e9 −0.525891
\(712\) −1.12574e10 −1.16885
\(713\) −4.23184e9 −0.437236
\(714\) 2.94306e8 0.0302591
\(715\) 0 0
\(716\) 7.44794e9 0.758299
\(717\) 1.34586e9 0.136359
\(718\) −5.59524e9 −0.564136
\(719\) 1.51985e9 0.152493 0.0762463 0.997089i \(-0.475706\pi\)
0.0762463 + 0.997089i \(0.475706\pi\)
\(720\) 0 0
\(721\) −4.61401e8 −0.0458464
\(722\) 4.91740e9 0.486246
\(723\) 5.38310e9 0.529722
\(724\) −5.93429e9 −0.581144
\(725\) 0 0
\(726\) 3.01133e9 0.292066
\(727\) 8.11761e9 0.783534 0.391767 0.920065i \(-0.371864\pi\)
0.391767 + 0.920065i \(0.371864\pi\)
\(728\) 4.30679e8 0.0413708
\(729\) 3.87420e8 0.0370370
\(730\) 0 0
\(731\) 1.94663e10 1.84320
\(732\) −7.46844e8 −0.0703787
\(733\) 1.03241e10 0.968249 0.484124 0.874999i \(-0.339138\pi\)
0.484124 + 0.874999i \(0.339138\pi\)
\(734\) −5.11539e9 −0.477466
\(735\) 0 0
\(736\) −2.93676e9 −0.271517
\(737\) −4.80867e8 −0.0442475
\(738\) 2.75439e9 0.252248
\(739\) −1.35365e10 −1.23382 −0.616908 0.787035i \(-0.711615\pi\)
−0.616908 + 0.787035i \(0.711615\pi\)
\(740\) 0 0
\(741\) −1.18651e9 −0.107129
\(742\) −1.64374e8 −0.0147713
\(743\) 1.71936e10 1.53782 0.768910 0.639356i \(-0.220799\pi\)
0.768910 + 0.639356i \(0.220799\pi\)
\(744\) −9.86544e9 −0.878236
\(745\) 0 0
\(746\) 2.28710e9 0.201697
\(747\) 3.19065e9 0.280064
\(748\) −2.47571e9 −0.216294
\(749\) −7.31269e8 −0.0635903
\(750\) 0 0
\(751\) 1.12478e10 0.969013 0.484506 0.874788i \(-0.338999\pi\)
0.484506 + 0.874788i \(0.338999\pi\)
\(752\) −2.24919e9 −0.192870
\(753\) −1.06563e10 −0.909545
\(754\) −1.11677e9 −0.0948775
\(755\) 0 0
\(756\) −1.15894e8 −0.00975512
\(757\) −1.63068e10 −1.36626 −0.683131 0.730296i \(-0.739382\pi\)
−0.683131 + 0.730296i \(0.739382\pi\)
\(758\) 8.90118e9 0.742345
\(759\) −3.91312e8 −0.0324845
\(760\) 0 0
\(761\) 6.14069e9 0.505093 0.252546 0.967585i \(-0.418732\pi\)
0.252546 + 0.967585i \(0.418732\pi\)
\(762\) 2.71970e9 0.222679
\(763\) 2.57341e8 0.0209736
\(764\) −5.22812e9 −0.424149
\(765\) 0 0
\(766\) −4.57158e9 −0.367507
\(767\) 6.65993e9 0.532949
\(768\) −5.62028e9 −0.447707
\(769\) 2.45069e10 1.94333 0.971664 0.236368i \(-0.0759569\pi\)
0.971664 + 0.236368i \(0.0759569\pi\)
\(770\) 0 0
\(771\) 3.85791e9 0.303153
\(772\) 1.07067e10 0.837518
\(773\) 1.01722e10 0.792110 0.396055 0.918227i \(-0.370379\pi\)
0.396055 + 0.918227i \(0.370379\pi\)
\(774\) 2.99957e9 0.232523
\(775\) 0 0
\(776\) 1.16514e10 0.895080
\(777\) −4.64013e8 −0.0354860
\(778\) −9.65411e9 −0.734994
\(779\) 5.42817e9 0.411408
\(780\) 0 0
\(781\) −5.27148e9 −0.395962
\(782\) 2.60379e9 0.194707
\(783\) 7.18626e8 0.0534979
\(784\) −3.15979e9 −0.234181
\(785\) 0 0
\(786\) −2.72594e9 −0.200234
\(787\) 9.79135e9 0.716030 0.358015 0.933716i \(-0.383454\pi\)
0.358015 + 0.933716i \(0.383454\pi\)
\(788\) −1.09311e10 −0.795832
\(789\) −1.18865e10 −0.861558
\(790\) 0 0
\(791\) 1.13320e9 0.0814124
\(792\) −9.12241e8 −0.0652487
\(793\) 1.53277e9 0.109150
\(794\) 1.12809e10 0.799786
\(795\) 0 0
\(796\) 8.74098e9 0.614277
\(797\) 9.75782e9 0.682729 0.341365 0.939931i \(-0.389111\pi\)
0.341365 + 0.939931i \(0.389111\pi\)
\(798\) 8.93722e7 0.00622576
\(799\) 1.65574e10 1.14836
\(800\) 0 0
\(801\) −6.21714e9 −0.427442
\(802\) −1.61155e9 −0.110315
\(803\) 1.29789e9 0.0884572
\(804\) −1.25999e9 −0.0855012
\(805\) 0 0
\(806\) 8.46700e9 0.569583
\(807\) 7.43592e9 0.498055
\(808\) 1.58221e10 1.05517
\(809\) −2.78706e9 −0.185066 −0.0925330 0.995710i \(-0.529496\pi\)
−0.0925330 + 0.995710i \(0.529496\pi\)
\(810\) 0 0
\(811\) −7.99983e9 −0.526633 −0.263316 0.964710i \(-0.584816\pi\)
−0.263316 + 0.964710i \(0.584816\pi\)
\(812\) −2.14971e8 −0.0140907
\(813\) −1.14661e10 −0.748339
\(814\) −1.52738e9 −0.0992569
\(815\) 0 0
\(816\) −2.95532e9 −0.190410
\(817\) 5.91135e9 0.379236
\(818\) −5.39687e8 −0.0344751
\(819\) 2.37852e8 0.0151291
\(820\) 0 0
\(821\) −1.02402e10 −0.645813 −0.322906 0.946431i \(-0.604660\pi\)
−0.322906 + 0.946431i \(0.604660\pi\)
\(822\) 4.54327e9 0.285310
\(823\) −2.78682e10 −1.74265 −0.871324 0.490707i \(-0.836738\pi\)
−0.871324 + 0.490707i \(0.836738\pi\)
\(824\) −9.51640e9 −0.592553
\(825\) 0 0
\(826\) −5.01650e8 −0.0309721
\(827\) −2.35125e10 −1.44554 −0.722769 0.691090i \(-0.757131\pi\)
−0.722769 + 0.691090i \(0.757131\pi\)
\(828\) −1.02534e9 −0.0627711
\(829\) −1.28598e10 −0.783960 −0.391980 0.919974i \(-0.628210\pi\)
−0.391980 + 0.919974i \(0.628210\pi\)
\(830\) 0 0
\(831\) −1.39363e10 −0.842448
\(832\) 3.35962e9 0.202236
\(833\) 2.32608e10 1.39434
\(834\) 1.91602e9 0.114372
\(835\) 0 0
\(836\) −7.51802e8 −0.0445022
\(837\) −5.44841e9 −0.321167
\(838\) −1.01433e10 −0.595420
\(839\) −7.99832e9 −0.467554 −0.233777 0.972290i \(-0.575109\pi\)
−0.233777 + 0.972290i \(0.575109\pi\)
\(840\) 0 0
\(841\) −1.59169e10 −0.922725
\(842\) 6.79996e9 0.392567
\(843\) −8.39816e9 −0.482822
\(844\) −1.64907e10 −0.944147
\(845\) 0 0
\(846\) 2.55134e9 0.144868
\(847\) −1.18966e9 −0.0672716
\(848\) 1.65059e9 0.0929510
\(849\) 1.60463e10 0.899909
\(850\) 0 0
\(851\) −4.10523e9 −0.228341
\(852\) −1.38126e10 −0.765133
\(853\) −4.20827e9 −0.232157 −0.116079 0.993240i \(-0.537032\pi\)
−0.116079 + 0.993240i \(0.537032\pi\)
\(854\) −1.15454e8 −0.00634318
\(855\) 0 0
\(856\) −1.50824e10 −0.821888
\(857\) −3.19307e10 −1.73291 −0.866453 0.499259i \(-0.833606\pi\)
−0.866453 + 0.499259i \(0.833606\pi\)
\(858\) 7.82930e8 0.0423173
\(859\) 2.18002e10 1.17350 0.586752 0.809767i \(-0.300406\pi\)
0.586752 + 0.809767i \(0.300406\pi\)
\(860\) 0 0
\(861\) −1.08815e9 −0.0581004
\(862\) −1.31966e10 −0.701755
\(863\) −1.04728e10 −0.554657 −0.277329 0.960775i \(-0.589449\pi\)
−0.277329 + 0.960775i \(0.589449\pi\)
\(864\) −3.78103e9 −0.199440
\(865\) 0 0
\(866\) −9.10427e8 −0.0476357
\(867\) 1.06765e10 0.556368
\(868\) 1.62985e9 0.0845916
\(869\) 6.55421e9 0.338806
\(870\) 0 0
\(871\) 2.58593e9 0.132603
\(872\) 5.30765e9 0.271078
\(873\) 6.43475e9 0.327327
\(874\) 7.90695e8 0.0400608
\(875\) 0 0
\(876\) 3.40080e9 0.170929
\(877\) 1.77787e10 0.890024 0.445012 0.895525i \(-0.353199\pi\)
0.445012 + 0.895525i \(0.353199\pi\)
\(878\) −5.94458e9 −0.296408
\(879\) −3.11890e9 −0.154896
\(880\) 0 0
\(881\) −7.64253e9 −0.376549 −0.188274 0.982116i \(-0.560289\pi\)
−0.188274 + 0.982116i \(0.560289\pi\)
\(882\) 3.58426e9 0.175897
\(883\) 2.76375e10 1.35094 0.675472 0.737386i \(-0.263940\pi\)
0.675472 + 0.737386i \(0.263940\pi\)
\(884\) 1.33135e10 0.648200
\(885\) 0 0
\(886\) −1.06425e10 −0.514076
\(887\) −3.23087e10 −1.55449 −0.777243 0.629200i \(-0.783383\pi\)
−0.777243 + 0.629200i \(0.783383\pi\)
\(888\) −9.57027e9 −0.458647
\(889\) −1.07445e9 −0.0512897
\(890\) 0 0
\(891\) −5.03806e8 −0.0238612
\(892\) −1.90014e10 −0.896414
\(893\) 5.02801e9 0.236274
\(894\) −3.36711e9 −0.157607
\(895\) 0 0
\(896\) 1.32059e9 0.0613325
\(897\) 2.10433e9 0.0973511
\(898\) 1.66206e9 0.0765914
\(899\) −1.01063e10 −0.463908
\(900\) 0 0
\(901\) −1.21509e10 −0.553439
\(902\) −3.58184e9 −0.162511
\(903\) −1.18501e9 −0.0535570
\(904\) 2.33723e10 1.05223
\(905\) 0 0
\(906\) −1.23301e7 −0.000550832 0
\(907\) −2.27142e10 −1.01082 −0.505409 0.862880i \(-0.668658\pi\)
−0.505409 + 0.862880i \(0.668658\pi\)
\(908\) 3.99238e9 0.176983
\(909\) 8.73810e9 0.385872
\(910\) 0 0
\(911\) 7.50925e9 0.329065 0.164533 0.986372i \(-0.447388\pi\)
0.164533 + 0.986372i \(0.447388\pi\)
\(912\) −8.97445e8 −0.0391765
\(913\) −4.14916e9 −0.180431
\(914\) 1.76855e10 0.766136
\(915\) 0 0
\(916\) 3.32976e9 0.143146
\(917\) 1.07691e9 0.0461199
\(918\) 3.35233e9 0.143020
\(919\) −2.49374e10 −1.05986 −0.529928 0.848043i \(-0.677781\pi\)
−0.529928 + 0.848043i \(0.677781\pi\)
\(920\) 0 0
\(921\) 7.03619e9 0.296776
\(922\) 1.66012e10 0.697561
\(923\) 2.83481e10 1.18664
\(924\) 1.50709e8 0.00628475
\(925\) 0 0
\(926\) 2.78132e9 0.115110
\(927\) −5.25565e9 −0.216694
\(928\) −7.01342e9 −0.288079
\(929\) −8.66205e9 −0.354459 −0.177229 0.984170i \(-0.556713\pi\)
−0.177229 + 0.984170i \(0.556713\pi\)
\(930\) 0 0
\(931\) 7.06363e9 0.286883
\(932\) 8.48559e9 0.343342
\(933\) 1.55735e10 0.627768
\(934\) −2.50753e9 −0.100701
\(935\) 0 0
\(936\) 4.90570e9 0.195540
\(937\) −2.82655e10 −1.12245 −0.561226 0.827663i \(-0.689670\pi\)
−0.561226 + 0.827663i \(0.689670\pi\)
\(938\) −1.94782e8 −0.00770616
\(939\) 1.24220e10 0.489623
\(940\) 0 0
\(941\) −4.67082e10 −1.82738 −0.913691 0.406410i \(-0.866780\pi\)
−0.913691 + 0.406410i \(0.866780\pi\)
\(942\) −5.21356e9 −0.203215
\(943\) −9.62713e9 −0.373857
\(944\) 5.03740e9 0.194897
\(945\) 0 0
\(946\) −3.90067e9 −0.149803
\(947\) 4.67392e10 1.78837 0.894184 0.447701i \(-0.147757\pi\)
0.894184 + 0.447701i \(0.147757\pi\)
\(948\) 1.71737e10 0.654688
\(949\) −6.97958e9 −0.265093
\(950\) 0 0
\(951\) −1.68902e9 −0.0636798
\(952\) −2.39805e9 −0.0900801
\(953\) −3.82420e10 −1.43125 −0.715625 0.698484i \(-0.753858\pi\)
−0.715625 + 0.698484i \(0.753858\pi\)
\(954\) −1.87233e9 −0.0698171
\(955\) 0 0
\(956\) −4.58591e9 −0.169755
\(957\) −9.34510e8 −0.0344661
\(958\) 9.05837e9 0.332867
\(959\) −1.79487e9 −0.0657157
\(960\) 0 0
\(961\) 4.91101e10 1.78500
\(962\) 8.21367e9 0.297457
\(963\) −8.32961e9 −0.300561
\(964\) −1.83424e10 −0.659457
\(965\) 0 0
\(966\) −1.58506e8 −0.00565752
\(967\) 4.90012e10 1.74267 0.871333 0.490692i \(-0.163256\pi\)
0.871333 + 0.490692i \(0.163256\pi\)
\(968\) −2.45368e10 −0.869468
\(969\) 6.60656e9 0.233261
\(970\) 0 0
\(971\) 2.72929e10 0.956713 0.478357 0.878166i \(-0.341233\pi\)
0.478357 + 0.878166i \(0.341233\pi\)
\(972\) −1.32010e9 −0.0461078
\(973\) −7.56947e8 −0.0263433
\(974\) 5.57676e9 0.193386
\(975\) 0 0
\(976\) 1.15935e9 0.0399155
\(977\) −3.94482e9 −0.135331 −0.0676653 0.997708i \(-0.521555\pi\)
−0.0676653 + 0.997708i \(0.521555\pi\)
\(978\) 9.45165e9 0.323088
\(979\) 8.08484e9 0.275380
\(980\) 0 0
\(981\) 2.93127e9 0.0991322
\(982\) −3.07682e10 −1.03684
\(983\) −4.74320e8 −0.0159270 −0.00796351 0.999968i \(-0.502535\pi\)
−0.00796351 + 0.999968i \(0.502535\pi\)
\(984\) −2.24431e10 −0.750933
\(985\) 0 0
\(986\) 6.21824e9 0.206585
\(987\) −1.00794e9 −0.0333674
\(988\) 4.04292e9 0.133366
\(989\) −1.04841e10 −0.344622
\(990\) 0 0
\(991\) 1.22197e10 0.398843 0.199421 0.979914i \(-0.436094\pi\)
0.199421 + 0.979914i \(0.436094\pi\)
\(992\) 5.31737e10 1.72944
\(993\) 1.84744e10 0.598752
\(994\) −2.13528e9 −0.0689609
\(995\) 0 0
\(996\) −1.08718e10 −0.348655
\(997\) 3.60690e10 1.15266 0.576330 0.817217i \(-0.304484\pi\)
0.576330 + 0.817217i \(0.304484\pi\)
\(998\) 2.46390e9 0.0784631
\(999\) −5.28540e9 −0.167725
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 75.8.a.a.1.1 1
3.2 odd 2 225.8.a.i.1.1 1
5.2 odd 4 75.8.b.c.49.1 2
5.3 odd 4 75.8.b.c.49.2 2
5.4 even 2 3.8.a.a.1.1 1
15.2 even 4 225.8.b.f.199.2 2
15.8 even 4 225.8.b.f.199.1 2
15.14 odd 2 9.8.a.a.1.1 1
20.19 odd 2 48.8.a.g.1.1 1
35.4 even 6 147.8.e.b.79.1 2
35.9 even 6 147.8.e.b.67.1 2
35.19 odd 6 147.8.e.a.67.1 2
35.24 odd 6 147.8.e.a.79.1 2
35.34 odd 2 147.8.a.b.1.1 1
40.19 odd 2 192.8.a.a.1.1 1
40.29 even 2 192.8.a.i.1.1 1
45.4 even 6 81.8.c.a.55.1 2
45.14 odd 6 81.8.c.c.55.1 2
45.29 odd 6 81.8.c.c.28.1 2
45.34 even 6 81.8.c.a.28.1 2
55.54 odd 2 363.8.a.b.1.1 1
60.59 even 2 144.8.a.b.1.1 1
65.64 even 2 507.8.a.a.1.1 1
105.104 even 2 441.8.a.a.1.1 1
120.29 odd 2 576.8.a.w.1.1 1
120.59 even 2 576.8.a.x.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.8.a.a.1.1 1 5.4 even 2
9.8.a.a.1.1 1 15.14 odd 2
48.8.a.g.1.1 1 20.19 odd 2
75.8.a.a.1.1 1 1.1 even 1 trivial
75.8.b.c.49.1 2 5.2 odd 4
75.8.b.c.49.2 2 5.3 odd 4
81.8.c.a.28.1 2 45.34 even 6
81.8.c.a.55.1 2 45.4 even 6
81.8.c.c.28.1 2 45.29 odd 6
81.8.c.c.55.1 2 45.14 odd 6
144.8.a.b.1.1 1 60.59 even 2
147.8.a.b.1.1 1 35.34 odd 2
147.8.e.a.67.1 2 35.19 odd 6
147.8.e.a.79.1 2 35.24 odd 6
147.8.e.b.67.1 2 35.9 even 6
147.8.e.b.79.1 2 35.4 even 6
192.8.a.a.1.1 1 40.19 odd 2
192.8.a.i.1.1 1 40.29 even 2
225.8.a.i.1.1 1 3.2 odd 2
225.8.b.f.199.1 2 15.8 even 4
225.8.b.f.199.2 2 15.2 even 4
363.8.a.b.1.1 1 55.54 odd 2
441.8.a.a.1.1 1 105.104 even 2
507.8.a.a.1.1 1 65.64 even 2
576.8.a.w.1.1 1 120.29 odd 2
576.8.a.x.1.1 1 120.59 even 2