Properties

Label 225.8.a.i.1.1
Level $225$
Weight $8$
Character 225.1
Self dual yes
Analytic conductor $70.287$
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] = [225,8,Mod(1,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, 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("225.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 225.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: \(70.2866307339\)
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 \(\chi\) \(=\) 225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+6.00000 q^{2} -92.0000 q^{4} +64.0000 q^{7} -1320.00 q^{8} +O(q^{10})\) \(q+6.00000 q^{2} -92.0000 q^{4} +64.0000 q^{7} -1320.00 q^{8} +948.000 q^{11} +5098.00 q^{13} +384.000 q^{14} +3856.00 q^{16} +28386.0 q^{17} -8620.00 q^{19} +5688.00 q^{22} -15288.0 q^{23} +30588.0 q^{26} -5888.00 q^{28} -36510.0 q^{29} -276808. q^{31} +192096. q^{32} +170316. q^{34} -268526. q^{37} -51720.0 q^{38} +629718. q^{41} -685772. q^{43} -87216.0 q^{44} -91728.0 q^{46} +583296. q^{47} -819447. q^{49} -469016. q^{52} -428058. q^{53} -84480.0 q^{56} -219060. q^{58} -1.30638e6 q^{59} +300662. q^{61} -1.66085e6 q^{62} +659008. q^{64} +507244. q^{67} -2.61151e6 q^{68} -5.56063e6 q^{71} -1.36908e6 q^{73} -1.61116e6 q^{74} +793040. q^{76} +60672.0 q^{77} -6.91372e6 q^{79} +3.77831e6 q^{82} -4.37675e6 q^{83} -4.11463e6 q^{86} -1.25136e6 q^{88} +8.52831e6 q^{89} +326272. q^{91} +1.40650e6 q^{92} +3.49978e6 q^{94} +8.82681e6 q^{97} -4.91668e6 q^{98} +O(q^{100})\)

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 \(-0.414574\pi\)
0.265165 + 0.964203i \(0.414574\pi\)
\(3\) 0 0
\(4\) −92.0000 −0.718750
\(5\) 0 0
\(6\) 0 0
\(7\) 64.0000 0.0705240 0.0352620 0.999378i \(-0.488773\pi\)
0.0352620 + 0.999378i \(0.488773\pi\)
\(8\) −1320.00 −0.911505
\(9\) 0 0
\(10\) 0 0
\(11\) 948.000 0.214750 0.107375 0.994219i \(-0.465755\pi\)
0.107375 + 0.994219i \(0.465755\pi\)
\(12\) 0 0
\(13\) 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 \(-0.252892\pi\)
0.700653 + 0.713502i \(0.252892\pi\)
\(18\) 0 0
\(19\) −8620.00 −0.288317 −0.144158 0.989555i \(-0.546047\pi\)
−0.144158 + 0.989555i \(0.546047\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 5688.00 0.113889
\(23\) −15288.0 −0.262001 −0.131001 0.991382i \(-0.541819\pi\)
−0.131001 + 0.991382i \(0.541819\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 30588.0 0.341306
\(27\) 0 0
\(28\) −5888.00 −0.0506891
\(29\) −36510.0 −0.277983 −0.138992 0.990294i \(-0.544386\pi\)
−0.138992 + 0.990294i \(0.544386\pi\)
\(30\) 0 0
\(31\) −276808. −1.66883 −0.834416 0.551135i \(-0.814195\pi\)
−0.834416 + 0.551135i \(0.814195\pi\)
\(32\) 192096. 1.03632
\(33\) 0 0
\(34\) 170316. 0.743155
\(35\) 0 0
\(36\) 0 0
\(37\) −268526. −0.871526 −0.435763 0.900061i \(-0.643521\pi\)
−0.435763 + 0.900061i \(0.643521\pi\)
\(38\) −51720.0 −0.152903
\(39\) 0 0
\(40\) 0 0
\(41\) 629718. 1.42693 0.713465 0.700691i \(-0.247125\pi\)
0.713465 + 0.700691i \(0.247125\pi\)
\(42\) 0 0
\(43\) −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 \(-0.365617\pi\)
0.409748 + 0.912199i \(0.365617\pi\)
\(48\) 0 0
\(49\) −819447. −0.995026
\(50\) 0 0
\(51\) 0 0
\(52\) −469016. −0.462568
\(53\) −428058. −0.394945 −0.197473 0.980308i \(-0.563273\pi\)
−0.197473 + 0.980308i \(0.563273\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −84480.0 −0.0642830
\(57\) 0 0
\(58\) −219060. −0.147423
\(59\) −1.30638e6 −0.828109 −0.414054 0.910252i \(-0.635888\pi\)
−0.414054 + 0.910252i \(0.635888\pi\)
\(60\) 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
\(63\) 0 0
\(64\) 659008. 0.314240
\(65\) 0 0
\(66\) 0 0
\(67\) 507244. 0.206042 0.103021 0.994679i \(-0.467149\pi\)
0.103021 + 0.994679i \(0.467149\pi\)
\(68\) −2.61151e6 −1.00719
\(69\) 0 0
\(70\) 0 0
\(71\) −5.56063e6 −1.84383 −0.921913 0.387397i \(-0.873374\pi\)
−0.921913 + 0.387397i \(0.873374\pi\)
\(72\) 0 0
\(73\) −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
\(78\) 0 0
\(79\) −6.91372e6 −1.57767 −0.788836 0.614603i \(-0.789316\pi\)
−0.788836 + 0.614603i \(0.789316\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 3.77831e6 0.756744
\(83\) −4.37675e6 −0.840191 −0.420096 0.907480i \(-0.638003\pi\)
−0.420096 + 0.907480i \(0.638003\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.11463e6 −0.697568
\(87\) 0 0
\(88\) −1.25136e6 −0.195746
\(89\) 8.52831e6 1.28232 0.641162 0.767405i \(-0.278453\pi\)
0.641162 + 0.767405i \(0.278453\pi\)
\(90\) 0 0
\(91\) 326272. 0.0453874
\(92\) 1.40650e6 0.188313
\(93\) 0 0
\(94\) 3.49978e6 0.434603
\(95\) 0 0
\(96\) 0 0
\(97\) 8.82681e6 0.981981 0.490990 0.871165i \(-0.336635\pi\)
0.490990 + 0.871165i \(0.336635\pi\)
\(98\) −4.91668e6 −0.527692
\(99\) 0 0
\(100\) 0 0
\(101\) −1.19864e7 −1.15762 −0.578808 0.815464i \(-0.696482\pi\)
−0.578808 + 0.815464i \(0.696482\pi\)
\(102\) 0 0
\(103\) −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 \(-0.351124\pi\)
0.450842 + 0.892604i \(0.351124\pi\)
\(108\) 0 0
\(109\) 4.02095e6 0.297397 0.148698 0.988883i \(-0.452492\pi\)
0.148698 + 0.988883i \(0.452492\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 246784. 0.0165979
\(113\) −1.77063e7 −1.15439 −0.577197 0.816605i \(-0.695853\pi\)
−0.577197 + 0.816605i \(0.695853\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 3.35892e6 0.199801
\(117\) 0 0
\(118\) −7.83828e6 −0.439171
\(119\) 1.81670e6 0.0988257
\(120\) 0 0
\(121\) −1.85885e7 −0.953882
\(122\) 1.80397e6 0.0899436
\(123\) 0 0
\(124\) 2.54663e7 1.19947
\(125\) 0 0
\(126\) 0 0
\(127\) −1.67883e7 −0.727267 −0.363633 0.931542i \(-0.618464\pi\)
−0.363633 + 0.931542i \(0.618464\pi\)
\(128\) −2.06342e7 −0.869668
\(129\) 0 0
\(130\) 0 0
\(131\) −1.68268e7 −0.653960 −0.326980 0.945031i \(-0.606031\pi\)
−0.326980 + 0.945031i \(0.606031\pi\)
\(132\) 0 0
\(133\) −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 \(-0.345727\pi\)
0.465910 + 0.884832i \(0.345727\pi\)
\(138\) 0 0
\(139\) −1.18273e7 −0.373537 −0.186769 0.982404i \(-0.559801\pi\)
−0.186769 + 0.982404i \(0.559801\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −3.33638e7 −0.977836
\(143\) 4.83290e6 0.138208
\(144\) 0 0
\(145\) 0 0
\(146\) −8.21449e6 −0.218447
\(147\) 0 0
\(148\) 2.47044e7 0.626409
\(149\) −2.07846e7 −0.514743 −0.257371 0.966313i \(-0.582856\pi\)
−0.257371 + 0.966313i \(0.582856\pi\)
\(150\) 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
\(153\) 0 0
\(154\) 364032. 0.00803188
\(155\) 0 0
\(156\) 0 0
\(157\) 3.21825e7 0.663698 0.331849 0.943332i \(-0.392328\pi\)
0.331849 + 0.943332i \(0.392328\pi\)
\(158\) −4.14823e7 −0.836687
\(159\) 0 0
\(160\) 0 0
\(161\) −978432. −0.0184774
\(162\) 0 0
\(163\) −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 \(-0.568855\pi\)
−0.214633 + 0.976695i \(0.568855\pi\)
\(168\) 0 0
\(169\) −3.67589e7 −0.585813
\(170\) 0 0
\(171\) 0 0
\(172\) 6.30910e7 0.945405
\(173\) 6.35201e7 0.932716 0.466358 0.884596i \(-0.345566\pi\)
0.466358 + 0.884596i \(0.345566\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.65549e6 0.0505418
\(177\) 0 0
\(178\) 5.11699e7 0.680055
\(179\) 8.09559e7 1.05503 0.527513 0.849547i \(-0.323125\pi\)
0.527513 + 0.849547i \(0.323125\pi\)
\(180\) 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
\(183\) 0 0
\(184\) 2.01802e7 0.238815
\(185\) 0 0
\(186\) 0 0
\(187\) 2.69099e7 0.300931
\(188\) −5.36632e7 −0.589012
\(189\) 0 0
\(190\) 0 0
\(191\) −5.68274e7 −0.590121 −0.295060 0.955479i \(-0.595340\pi\)
−0.295060 + 0.955479i \(0.595340\pi\)
\(192\) 0 0
\(193\) −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 \(-0.686755\pi\)
−0.553622 + 0.832768i \(0.686755\pi\)
\(198\) 0 0
\(199\) −9.50106e7 −0.854646 −0.427323 0.904099i \(-0.640543\pi\)
−0.427323 + 0.904099i \(0.640543\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −7.19185e7 −0.613919
\(203\) −2.33664e6 −0.0196045
\(204\) 0 0
\(205\) 0 0
\(206\) −4.32564e7 −0.344758
\(207\) 0 0
\(208\) 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
\(213\) 0 0
\(214\) 6.85565e7 0.478190
\(215\) 0 0
\(216\) 0 0
\(217\) −1.77157e7 −0.117693
\(218\) 2.41257e7 0.157718
\(219\) 0 0
\(220\) 0 0
\(221\) 1.44712e8 0.901843
\(222\) 0 0
\(223\) 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 \(-0.460710\pi\)
0.123118 + 0.992392i \(0.460710\pi\)
\(228\) 0 0
\(229\) −3.61931e7 −0.199160 −0.0995799 0.995030i \(-0.531750\pi\)
−0.0995799 + 0.995030i \(0.531750\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 4.81932e7 0.253383
\(233\) 9.22347e7 0.477693 0.238846 0.971057i \(-0.423231\pi\)
0.238846 + 0.971057i \(0.423231\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.20187e8 0.595203
\(237\) 0 0
\(238\) 1.09002e7 0.0524102
\(239\) −4.98468e7 −0.236181 −0.118090 0.993003i \(-0.537677\pi\)
−0.118090 + 0.993003i \(0.537677\pi\)
\(240\) 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
\(243\) 0 0
\(244\) −2.76609e7 −0.121900
\(245\) 0 0
\(246\) 0 0
\(247\) −4.39448e7 −0.185553
\(248\) 3.65387e8 1.52115
\(249\) 0 0
\(250\) 0 0
\(251\) 3.94678e8 1.57538 0.787689 0.616073i \(-0.211277\pi\)
0.787689 + 0.616073i \(0.211277\pi\)
\(252\) 0 0
\(253\) −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 \(-0.584559\pi\)
−0.262538 + 0.964922i \(0.584559\pi\)
\(258\) 0 0
\(259\) −1.71857e7 −0.0614635
\(260\) 0 0
\(261\) 0 0
\(262\) −1.00961e8 −0.346815
\(263\) 4.40241e8 1.49226 0.746131 0.665799i \(-0.231909\pi\)
0.746131 + 0.665799i \(0.231909\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −3.31008e6 −0.0107833
\(267\) 0 0
\(268\) −4.66664e7 −0.148092
\(269\) −2.75405e8 −0.862657 −0.431329 0.902195i \(-0.641955\pi\)
−0.431329 + 0.902195i \(0.641955\pi\)
\(270\) 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
\(273\) 0 0
\(274\) 1.68269e8 0.494172
\(275\) 0 0
\(276\) 0 0
\(277\) −5.16158e8 −1.45916 −0.729581 0.683894i \(-0.760285\pi\)
−0.729581 + 0.683894i \(0.760285\pi\)
\(278\) −7.09638e7 −0.198098
\(279\) 0 0
\(280\) 0 0
\(281\) 3.11043e8 0.836273 0.418137 0.908384i \(-0.362683\pi\)
0.418137 + 0.908384i \(0.362683\pi\)
\(282\) 0 0
\(283\) 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
\(288\) 0 0
\(289\) 3.95426e8 0.963658
\(290\) 0 0
\(291\) 0 0
\(292\) 1.25956e8 0.296058
\(293\) 1.15515e8 0.268288 0.134144 0.990962i \(-0.457172\pi\)
0.134144 + 0.990962i \(0.457172\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3.54454e8 0.794400
\(297\) 0 0
\(298\) −1.24708e8 −0.272984
\(299\) −7.79382e7 −0.168617
\(300\) 0 0
\(301\) −4.38894e7 −0.0927635
\(302\) 456672. 0.000954070 0
\(303\) 0 0
\(304\) −3.32387e7 −0.0678558
\(305\) 0 0
\(306\) 0 0
\(307\) 2.60600e8 0.514032 0.257016 0.966407i \(-0.417261\pi\)
0.257016 + 0.966407i \(0.417261\pi\)
\(308\) −5.58182e6 −0.0108855
\(309\) 0 0
\(310\) 0 0
\(311\) −5.76795e8 −1.08733 −0.543663 0.839303i \(-0.682963\pi\)
−0.543663 + 0.839303i \(0.682963\pi\)
\(312\) 0 0
\(313\) 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 \(-0.482437\pi\)
0.0551483 + 0.998478i \(0.482437\pi\)
\(318\) 0 0
\(319\) −3.46115e7 −0.0596970
\(320\) 0 0
\(321\) 0 0
\(322\) −5.87059e6 −0.00979910
\(323\) −2.44687e8 −0.404020
\(324\) 0 0
\(325\) 0 0
\(326\) −3.50061e8 −0.559606
\(327\) 0 0
\(328\) −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
\(333\) 0 0
\(334\) −1.55019e8 −0.227652
\(335\) 0 0
\(336\) 0 0
\(337\) 6.26313e8 0.891429 0.445714 0.895175i \(-0.352950\pi\)
0.445714 + 0.895175i \(0.352950\pi\)
\(338\) −2.20553e8 −0.310674
\(339\) 0 0
\(340\) 0 0
\(341\) −2.62414e8 −0.358382
\(342\) 0 0
\(343\) −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 \(-0.797943\pi\)
−0.805203 + 0.593000i \(0.797943\pi\)
\(348\) 0 0
\(349\) 2.65350e8 0.334142 0.167071 0.985945i \(-0.446569\pi\)
0.167071 + 0.985945i \(0.446569\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.82107e8 0.222550
\(353\) −5.69636e8 −0.689264 −0.344632 0.938738i \(-0.611996\pi\)
−0.344632 + 0.938738i \(0.611996\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −7.84605e8 −0.921671
\(357\) 0 0
\(358\) 4.85735e8 0.559512
\(359\) −9.32541e8 −1.06374 −0.531872 0.846825i \(-0.678511\pi\)
−0.531872 + 0.846825i \(0.678511\pi\)
\(360\) 0 0
\(361\) −8.19567e8 −0.916874
\(362\) 3.87019e8 0.428798
\(363\) 0 0
\(364\) −3.00170e7 −0.0326222
\(365\) 0 0
\(366\) 0 0
\(367\) 8.52565e8 0.900318 0.450159 0.892948i \(-0.351367\pi\)
0.450159 + 0.892948i \(0.351367\pi\)
\(368\) −5.89505e7 −0.0616624
\(369\) 0 0
\(370\) 0 0
\(371\) −2.73957e7 −0.0278531
\(372\) 0 0
\(373\) −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
\(378\) 0 0
\(379\) −1.48353e9 −1.39978 −0.699889 0.714251i \(-0.746767\pi\)
−0.699889 + 0.714251i \(0.746767\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −3.40964e8 −0.312959
\(383\) −7.61930e8 −0.692978 −0.346489 0.938054i \(-0.612626\pi\)
−0.346489 + 0.938054i \(0.612626\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −6.98262e8 −0.617963
\(387\) 0 0
\(388\) −8.12067e8 −0.705799
\(389\) −1.60902e9 −1.38592 −0.692959 0.720977i \(-0.743693\pi\)
−0.692959 + 0.720977i \(0.743693\pi\)
\(390\) 0 0
\(391\) −4.33965e8 −0.367144
\(392\) 1.08167e9 0.906971
\(393\) 0 0
\(394\) −7.12896e8 −0.587205
\(395\) 0 0
\(396\) 0 0
\(397\) −1.88016e9 −1.50809 −0.754046 0.656822i \(-0.771900\pi\)
−0.754046 + 0.656822i \(0.771900\pi\)
\(398\) −5.70064e8 −0.453245
\(399\) 0 0
\(400\) 0 0
\(401\) −2.68592e8 −0.208012 −0.104006 0.994577i \(-0.533166\pi\)
−0.104006 + 0.994577i \(0.533166\pi\)
\(402\) 0 0
\(403\) −1.41117e9 −1.07402
\(404\) 1.10275e9 0.832037
\(405\) 0 0
\(406\) −1.40198e7 −0.0103969
\(407\) −2.54563e8 −0.187161
\(408\) 0 0
\(409\) 8.99478e7 0.0650069 0.0325034 0.999472i \(-0.489652\pi\)
0.0325034 + 0.999472i \(0.489652\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 6.63264e8 0.467247
\(413\) −8.36083e7 −0.0584015
\(414\) 0 0
\(415\) 0 0
\(416\) 9.79305e8 0.666947
\(417\) 0 0
\(418\) −4.90306e7 −0.0328360
\(419\) −1.69054e9 −1.12273 −0.561367 0.827567i \(-0.689724\pi\)
−0.561367 + 0.827567i \(0.689724\pi\)
\(420\) 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
\(423\) 0 0
\(424\) 5.65037e8 0.359995
\(425\) 0 0
\(426\) 0 0
\(427\) 1.92424e7 0.0119608
\(428\) −1.05120e9 −0.648085
\(429\) 0 0
\(430\) 0 0
\(431\) −2.19943e9 −1.32324 −0.661621 0.749839i \(-0.730131\pi\)
−0.661621 + 0.749839i \(0.730131\pi\)
\(432\) 0 0
\(433\) 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
\(438\) 0 0
\(439\) 9.90763e8 0.558912 0.279456 0.960158i \(-0.409846\pi\)
0.279456 + 0.960158i \(0.409846\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 8.68271e8 0.478275
\(443\) −1.77376e9 −0.969351 −0.484675 0.874694i \(-0.661062\pi\)
−0.484675 + 0.874694i \(0.661062\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 1.23922e9 0.661419
\(447\) 0 0
\(448\) 4.21765e7 0.0221614
\(449\) 2.77010e8 0.144422 0.0722110 0.997389i \(-0.476994\pi\)
0.0722110 + 0.997389i \(0.476994\pi\)
\(450\) 0 0
\(451\) 5.96973e8 0.306434
\(452\) 1.62898e9 0.829720
\(453\) 0 0
\(454\) 2.60372e8 0.130587
\(455\) 0 0
\(456\) 0 0
\(457\) −2.94758e9 −1.44464 −0.722320 0.691559i \(-0.756924\pi\)
−0.722320 + 0.691559i \(0.756924\pi\)
\(458\) −2.17159e8 −0.105620
\(459\) 0 0
\(460\) 0 0
\(461\) 2.76687e9 1.31533 0.657667 0.753309i \(-0.271543\pi\)
0.657667 + 0.753309i \(0.271543\pi\)
\(462\) 0 0
\(463\) −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 \(-0.530266\pi\)
−0.0949415 + 0.995483i \(0.530266\pi\)
\(468\) 0 0
\(469\) 3.24636e7 0.0145309
\(470\) 0 0
\(471\) 0 0
\(472\) 1.72442e9 0.754825
\(473\) −6.50112e8 −0.282471
\(474\) 0 0
\(475\) 0 0
\(476\) −1.67137e8 −0.0710310
\(477\) 0 0
\(478\) −2.99081e8 −0.125254
\(479\) 1.50973e9 0.627660 0.313830 0.949479i \(-0.398388\pi\)
0.313830 + 0.949479i \(0.398388\pi\)
\(480\) 0 0
\(481\) −1.36895e9 −0.560891
\(482\) 1.19624e9 0.486581
\(483\) 0 0
\(484\) 1.71014e9 0.685603
\(485\) 0 0
\(486\) 0 0
\(487\) −9.29460e8 −0.364653 −0.182326 0.983238i \(-0.558363\pi\)
−0.182326 + 0.983238i \(0.558363\pi\)
\(488\) −3.96874e8 −0.154591
\(489\) 0 0
\(490\) 0 0
\(491\) −5.12803e9 −1.95508 −0.977541 0.210743i \(-0.932412\pi\)
−0.977541 + 0.210743i \(0.932412\pi\)
\(492\) 0 0
\(493\) −1.03637e9 −0.389540
\(494\) −2.63669e8 −0.0984043
\(495\) 0 0
\(496\) −1.06737e9 −0.392762
\(497\) −3.55880e8 −0.130034
\(498\) 0 0
\(499\) −4.10649e8 −0.147951 −0.0739757 0.997260i \(-0.523569\pi\)
−0.0739757 + 0.997260i \(0.523569\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 2.36807e9 0.835470
\(503\) 5.02041e9 1.75894 0.879470 0.475954i \(-0.157897\pi\)
0.879470 + 0.475954i \(0.157897\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −8.69581e7 −0.0298389
\(507\) 0 0
\(508\) 1.54452e9 0.522723
\(509\) 3.24926e9 1.09212 0.546062 0.837745i \(-0.316126\pi\)
0.546062 + 0.837745i \(0.316126\pi\)
\(510\) 0 0
\(511\) −8.76212e7 −0.0290493
\(512\) 1.39223e9 0.458423
\(513\) 0 0
\(514\) −8.57312e8 −0.278463
\(515\) 0 0
\(516\) 0 0
\(517\) 5.52965e8 0.175987
\(518\) −1.03114e8 −0.0325959
\(519\) 0 0
\(520\) 0 0
\(521\) 2.10950e9 0.653503 0.326752 0.945110i \(-0.394046\pi\)
0.326752 + 0.945110i \(0.394046\pi\)
\(522\) 0 0
\(523\) 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
\(528\) 0 0
\(529\) −3.17110e9 −0.931355
\(530\) 0 0
\(531\) 0 0
\(532\) 5.07546e7 0.0146145
\(533\) 3.21030e9 0.918334
\(534\) 0 0
\(535\) 0 0
\(536\) −6.69562e8 −0.187808
\(537\) 0 0
\(538\) −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
\(543\) 0 0
\(544\) 5.45284e9 1.45220
\(545\) 0 0
\(546\) 0 0
\(547\) 4.85537e9 1.26843 0.634215 0.773157i \(-0.281323\pi\)
0.634215 + 0.773157i \(0.281323\pi\)
\(548\) −2.58013e9 −0.669746
\(549\) 0 0
\(550\) 0 0
\(551\) 3.14716e8 0.0801472
\(552\) 0 0
\(553\) −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 \(-0.449937\pi\)
0.156631 + 0.987657i \(0.449937\pi\)
\(558\) 0 0
\(559\) −3.49607e9 −0.846522
\(560\) 0 0
\(561\) 0 0
\(562\) 1.86626e9 0.443501
\(563\) 4.71265e9 1.11297 0.556487 0.830856i \(-0.312149\pi\)
0.556487 + 0.830856i \(0.312149\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 3.56585e9 0.826619
\(567\) 0 0
\(568\) 7.34003e9 1.68066
\(569\) −4.57800e9 −1.04180 −0.520898 0.853619i \(-0.674403\pi\)
−0.520898 + 0.853619i \(0.674403\pi\)
\(570\) 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
\(573\) 0 0
\(574\) 2.41812e8 0.0533686
\(575\) 0 0
\(576\) 0 0
\(577\) −8.51847e9 −1.84606 −0.923031 0.384725i \(-0.874296\pi\)
−0.923031 + 0.384725i \(0.874296\pi\)
\(578\) 2.37256e9 0.511057
\(579\) 0 0
\(580\) 0 0
\(581\) −2.80112e8 −0.0592536
\(582\) 0 0
\(583\) −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 \(-0.518271\pi\)
−0.0573673 + 0.998353i \(0.518271\pi\)
\(588\) 0 0
\(589\) 2.38608e9 0.481152
\(590\) 0 0
\(591\) 0 0
\(592\) −1.03544e9 −0.205115
\(593\) 3.62110e9 0.713099 0.356549 0.934277i \(-0.383953\pi\)
0.356549 + 0.934277i \(0.383953\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.91219e9 0.369971
\(597\) 0 0
\(598\) −4.67629e8 −0.0894227
\(599\) 7.48104e9 1.42222 0.711112 0.703079i \(-0.248192\pi\)
0.711112 + 0.703079i \(0.248192\pi\)
\(600\) 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
\(603\) 0 0
\(604\) −7.00230e6 −0.00129304
\(605\) 0 0
\(606\) 0 0
\(607\) −3.84051e9 −0.696993 −0.348497 0.937310i \(-0.613308\pi\)
−0.348497 + 0.937310i \(0.613308\pi\)
\(608\) −1.65587e9 −0.298788
\(609\) 0 0
\(610\) 0 0
\(611\) 2.97364e9 0.527405
\(612\) 0 0
\(613\) −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 \(-0.577360\pi\)
−0.240649 + 0.970612i \(0.577360\pi\)
\(618\) 0 0
\(619\) −2.54365e9 −0.431063 −0.215532 0.976497i \(-0.569148\pi\)
−0.215532 + 0.976497i \(0.569148\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −3.46077e9 −0.576642
\(623\) 5.45812e8 0.0904346
\(624\) 0 0
\(625\) 0 0
\(626\) 2.76045e9 0.449748
\(627\) 0 0
\(628\) −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
\(633\) 0 0
\(634\) 3.75337e8 0.0584936
\(635\) 0 0
\(636\) 0 0
\(637\) −4.17754e9 −0.640373
\(638\) −2.07669e8 −0.0316591
\(639\) 0 0
\(640\) 0 0
\(641\) 1.23625e10 1.85397 0.926987 0.375094i \(-0.122390\pi\)
0.926987 + 0.375094i \(0.122390\pi\)
\(642\) 0 0
\(643\) −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 \(-0.596331\pi\)
−0.298034 + 0.954555i \(0.596331\pi\)
\(648\) 0 0
\(649\) −1.23845e9 −0.177837
\(650\) 0 0
\(651\) 0 0
\(652\) 5.36760e9 0.758427
\(653\) 6.91100e9 0.971280 0.485640 0.874159i \(-0.338587\pi\)
0.485640 + 0.874159i \(0.338587\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2.42819e9 0.335830
\(657\) 0 0
\(658\) 2.23986e8 0.0306499
\(659\) −3.42444e9 −0.466112 −0.233056 0.972463i \(-0.574873\pi\)
−0.233056 + 0.972463i \(0.574873\pi\)
\(660\) 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
\(663\) 0 0
\(664\) 5.77731e9 0.765839
\(665\) 0 0
\(666\) 0 0
\(667\) 5.58165e8 0.0728320
\(668\) 2.37696e9 0.308535
\(669\) 0 0
\(670\) 0 0
\(671\) 2.85028e8 0.0364215
\(672\) 0 0
\(673\) 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 \(-0.328150\pi\)
0.514036 + 0.857769i \(0.328150\pi\)
\(678\) 0 0
\(679\) 5.64916e8 0.0692532
\(680\) 0 0
\(681\) 0 0
\(682\) −1.57448e9 −0.190061
\(683\) −1.21232e10 −1.45594 −0.727969 0.685610i \(-0.759536\pi\)
−0.727969 + 0.685610i \(0.759536\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −6.30908e8 −0.0746160
\(687\) 0 0
\(688\) −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
\(693\) 0 0
\(694\) −7.52038e9 −0.854046
\(695\) 0 0
\(696\) 0 0
\(697\) 1.78752e10 1.99957
\(698\) 1.59210e9 0.177205
\(699\) 0 0
\(700\) 0 0
\(701\) −4.72231e9 −0.517775 −0.258888 0.965907i \(-0.583356\pi\)
−0.258888 + 0.965907i \(0.583356\pi\)
\(702\) 0 0
\(703\) 2.31469e9 0.251275
\(704\) 6.24740e8 0.0674831
\(705\) 0 0
\(706\) −3.41781e9 −0.365537
\(707\) −7.67131e8 −0.0816397
\(708\) 0 0
\(709\) 2.78975e9 0.293970 0.146985 0.989139i \(-0.453043\pi\)
0.146985 + 0.989139i \(0.453043\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1.12574e10 −1.16885
\(713\) 4.23184e9 0.437236
\(714\) 0 0
\(715\) 0 0
\(716\) −7.44794e9 −0.758299
\(717\) 0 0
\(718\) −5.59524e9 −0.564136
\(719\) −1.51985e9 −0.152493 −0.0762463 0.997089i \(-0.524294\pi\)
−0.0762463 + 0.997089i \(0.524294\pi\)
\(720\) 0 0
\(721\) −4.61401e8 −0.0458464
\(722\) −4.91740e9 −0.486246
\(723\) 0 0
\(724\) −5.93429e9 −0.581144
\(725\) 0 0
\(726\) 0 0
\(727\) 8.11761e9 0.783534 0.391767 0.920065i \(-0.371864\pi\)
0.391767 + 0.920065i \(0.371864\pi\)
\(728\) −4.30679e8 −0.0413708
\(729\) 0 0
\(730\) 0 0
\(731\) −1.94663e10 −1.84320
\(732\) 0 0
\(733\) 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
\(738\) 0 0
\(739\) −1.35365e10 −1.23382 −0.616908 0.787035i \(-0.711615\pi\)
−0.616908 + 0.787035i \(0.711615\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −1.64374e8 −0.0147713
\(743\) −1.71936e10 −1.53782 −0.768910 0.639356i \(-0.779201\pi\)
−0.768910 + 0.639356i \(0.779201\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −2.28710e9 −0.201697
\(747\) 0 0
\(748\) −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
\(753\) 0 0
\(754\) −1.11677e9 −0.0948775
\(755\) 0 0
\(756\) 0 0
\(757\) −1.63068e10 −1.36626 −0.683131 0.730296i \(-0.739382\pi\)
−0.683131 + 0.730296i \(0.739382\pi\)
\(758\) −8.90118e9 −0.742345
\(759\) 0 0
\(760\) 0 0
\(761\) −6.14069e9 −0.505093 −0.252546 0.967585i \(-0.581268\pi\)
−0.252546 + 0.967585i \(0.581268\pi\)
\(762\) 0 0
\(763\) 2.57341e8 0.0209736
\(764\) 5.22812e9 0.424149
\(765\) 0 0
\(766\) −4.57158e9 −0.367507
\(767\) −6.65993e9 −0.532949
\(768\) 0 0
\(769\) 2.45069e10 1.94333 0.971664 0.236368i \(-0.0759569\pi\)
0.971664 + 0.236368i \(0.0759569\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.07067e10 0.837518
\(773\) −1.01722e10 −0.792110 −0.396055 0.918227i \(-0.629621\pi\)
−0.396055 + 0.918227i \(0.629621\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −1.16514e10 −0.895080
\(777\) 0 0
\(778\) −9.65411e9 −0.734994
\(779\) −5.42817e9 −0.411408
\(780\) 0 0
\(781\) −5.27148e9 −0.395962
\(782\) −2.60379e9 −0.194707
\(783\) 0 0
\(784\) −3.15979e9 −0.234181
\(785\) 0 0
\(786\) 0 0
\(787\) 9.79135e9 0.716030 0.358015 0.933716i \(-0.383454\pi\)
0.358015 + 0.933716i \(0.383454\pi\)
\(788\) 1.09311e10 0.795832
\(789\) 0 0
\(790\) 0 0
\(791\) −1.13320e9 −0.0814124
\(792\) 0 0
\(793\) 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 \(-0.610889\pi\)
−0.341365 + 0.939931i \(0.610889\pi\)
\(798\) 0 0
\(799\) 1.65574e10 1.14836
\(800\) 0 0
\(801\) 0 0
\(802\) −1.61155e9 −0.110315
\(803\) −1.29789e9 −0.0884572
\(804\) 0 0
\(805\) 0 0
\(806\) −8.46700e9 −0.569583
\(807\) 0 0
\(808\) 1.58221e10 1.05517
\(809\) 2.78706e9 0.185066 0.0925330 0.995710i \(-0.470504\pi\)
0.0925330 + 0.995710i \(0.470504\pi\)
\(810\) 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
\(813\) 0 0
\(814\) −1.52738e9 −0.0992569
\(815\) 0 0
\(816\) 0 0
\(817\) 5.91135e9 0.379236
\(818\) 5.39687e8 0.0344751
\(819\) 0 0
\(820\) 0 0
\(821\) 1.02402e10 0.645813 0.322906 0.946431i \(-0.395340\pi\)
0.322906 + 0.946431i \(0.395340\pi\)
\(822\) 0 0
\(823\) −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 \(-0.242869\pi\)
0.722769 + 0.691090i \(0.242869\pi\)
\(828\) 0 0
\(829\) −1.28598e10 −0.783960 −0.391980 0.919974i \(-0.628210\pi\)
−0.391980 + 0.919974i \(0.628210\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 3.35962e9 0.202236
\(833\) −2.32608e10 −1.39434
\(834\) 0 0
\(835\) 0 0
\(836\) 7.51802e8 0.0445022
\(837\) 0 0
\(838\) −1.01433e10 −0.595420
\(839\) 7.99832e9 0.467554 0.233777 0.972290i \(-0.424891\pi\)
0.233777 + 0.972290i \(0.424891\pi\)
\(840\) 0 0
\(841\) −1.59169e10 −0.922725
\(842\) −6.79996e9 −0.392567
\(843\) 0 0
\(844\) −1.64907e10 −0.944147
\(845\) 0 0
\(846\) 0 0
\(847\) −1.18966e9 −0.0672716
\(848\) −1.65059e9 −0.0929510
\(849\) 0 0
\(850\) 0 0
\(851\) 4.10523e9 0.228341
\(852\) 0 0
\(853\) −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 \(-0.166394\pi\)
0.866453 + 0.499259i \(0.166394\pi\)
\(858\) 0 0
\(859\) 2.18002e10 1.17350 0.586752 0.809767i \(-0.300406\pi\)
0.586752 + 0.809767i \(0.300406\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −1.31966e10 −0.701755
\(863\) 1.04728e10 0.554657 0.277329 0.960775i \(-0.410551\pi\)
0.277329 + 0.960775i \(0.410551\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 9.10427e8 0.0476357
\(867\) 0 0
\(868\) 1.62985e9 0.0845916
\(869\) −6.55421e9 −0.338806
\(870\) 0 0
\(871\) 2.58593e9 0.132603
\(872\) −5.30765e9 −0.271078
\(873\) 0 0
\(874\) 7.90695e8 0.0400608
\(875\) 0 0
\(876\) 0 0
\(877\) 1.77787e10 0.890024 0.445012 0.895525i \(-0.353199\pi\)
0.445012 + 0.895525i \(0.353199\pi\)
\(878\) 5.94458e9 0.296408
\(879\) 0 0
\(880\) 0 0
\(881\) 7.64253e9 0.376549 0.188274 0.982116i \(-0.439711\pi\)
0.188274 + 0.982116i \(0.439711\pi\)
\(882\) 0 0
\(883\) 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 \(-0.216617\pi\)
0.777243 + 0.629200i \(0.216617\pi\)
\(888\) 0 0
\(889\) −1.07445e9 −0.0512897
\(890\) 0 0
\(891\) 0 0
\(892\) −1.90014e10 −0.896414
\(893\) −5.02801e9 −0.236274
\(894\) 0 0
\(895\) 0 0
\(896\) −1.32059e9 −0.0613325
\(897\) 0 0
\(898\) 1.66206e9 0.0765914
\(899\) 1.01063e10 0.463908
\(900\) 0 0
\(901\) −1.21509e10 −0.553439
\(902\) 3.58184e9 0.162511
\(903\) 0 0
\(904\) 2.33723e10 1.05223
\(905\) 0 0
\(906\) 0 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
\(909\) 0 0
\(910\) 0 0
\(911\) −7.50925e9 −0.329065 −0.164533 0.986372i \(-0.552612\pi\)
−0.164533 + 0.986372i \(0.552612\pi\)
\(912\) 0 0
\(913\) −4.14916e9 −0.180431
\(914\) −1.76855e10 −0.766136
\(915\) 0 0
\(916\) 3.32976e9 0.143146
\(917\) −1.07691e9 −0.0461199
\(918\) 0 0
\(919\) −2.49374e10 −1.05986 −0.529928 0.848043i \(-0.677781\pi\)
−0.529928 + 0.848043i \(0.677781\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.66012e10 0.697561
\(923\) −2.83481e10 −1.18664
\(924\) 0 0
\(925\) 0 0
\(926\) −2.78132e9 −0.115110
\(927\) 0 0
\(928\) −7.01342e9 −0.288079
\(929\) 8.66205e9 0.354459 0.177229 0.984170i \(-0.443287\pi\)
0.177229 + 0.984170i \(0.443287\pi\)
\(930\) 0 0
\(931\) 7.06363e9 0.286883
\(932\) −8.48559e9 −0.343342
\(933\) 0 0
\(934\) −2.50753e9 −0.100701
\(935\) 0 0
\(936\) 0 0
\(937\) −2.82655e10 −1.12245 −0.561226 0.827663i \(-0.689670\pi\)
−0.561226 + 0.827663i \(0.689670\pi\)
\(938\) 1.94782e8 0.00770616
\(939\) 0 0
\(940\) 0 0
\(941\) 4.67082e10 1.82738 0.913691 0.406410i \(-0.133220\pi\)
0.913691 + 0.406410i \(0.133220\pi\)
\(942\) 0 0
\(943\) −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 \(-0.852243\pi\)
−0.894184 + 0.447701i \(0.852243\pi\)
\(948\) 0 0
\(949\) −6.97958e9 −0.265093
\(950\) 0 0
\(951\) 0 0
\(952\) −2.39805e9 −0.0900801
\(953\) 3.82420e10 1.43125 0.715625 0.698484i \(-0.246142\pi\)
0.715625 + 0.698484i \(0.246142\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 4.58591e9 0.169755
\(957\) 0 0
\(958\) 9.05837e9 0.332867
\(959\) 1.79487e9 0.0657157
\(960\) 0 0
\(961\) 4.91101e10 1.78500
\(962\) −8.21367e9 −0.297457
\(963\) 0 0
\(964\) −1.83424e10 −0.659457
\(965\) 0 0
\(966\) 0 0
\(967\) 4.90012e10 1.74267 0.871333 0.490692i \(-0.163256\pi\)
0.871333 + 0.490692i \(0.163256\pi\)
\(968\) 2.45368e10 0.869468
\(969\) 0 0
\(970\) 0 0
\(971\) −2.72929e10 −0.956713 −0.478357 0.878166i \(-0.658767\pi\)
−0.478357 + 0.878166i \(0.658767\pi\)
\(972\) 0 0
\(973\) −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 \(-0.478445\pi\)
0.0676653 + 0.997708i \(0.478445\pi\)
\(978\) 0 0
\(979\) 8.08484e9 0.275380
\(980\) 0 0
\(981\) 0 0
\(982\) −3.07682e10 −1.03684
\(983\) 4.74320e8 0.0159270 0.00796351 0.999968i \(-0.497465\pi\)
0.00796351 + 0.999968i \(0.497465\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −6.21824e9 −0.206585
\(987\) 0 0
\(988\) 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
\(993\) 0 0
\(994\) −2.13528e9 −0.0689609
\(995\) 0 0
\(996\) 0 0
\(997\) 3.60690e10 1.15266 0.576330 0.817217i \(-0.304484\pi\)
0.576330 + 0.817217i \(0.304484\pi\)
\(998\) −2.46390e9 −0.0784631
\(999\) 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 225.8.a.i.1.1 1
3.2 odd 2 75.8.a.a.1.1 1
5.2 odd 4 225.8.b.f.199.2 2
5.3 odd 4 225.8.b.f.199.1 2
5.4 even 2 9.8.a.a.1.1 1
15.2 even 4 75.8.b.c.49.1 2
15.8 even 4 75.8.b.c.49.2 2
15.14 odd 2 3.8.a.a.1.1 1
20.19 odd 2 144.8.a.b.1.1 1
35.34 odd 2 441.8.a.a.1.1 1
40.19 odd 2 576.8.a.x.1.1 1
40.29 even 2 576.8.a.w.1.1 1
45.4 even 6 81.8.c.c.55.1 2
45.14 odd 6 81.8.c.a.55.1 2
45.29 odd 6 81.8.c.a.28.1 2
45.34 even 6 81.8.c.c.28.1 2
60.59 even 2 48.8.a.g.1.1 1
105.44 odd 6 147.8.e.b.67.1 2
105.59 even 6 147.8.e.a.79.1 2
105.74 odd 6 147.8.e.b.79.1 2
105.89 even 6 147.8.e.a.67.1 2
105.104 even 2 147.8.a.b.1.1 1
120.29 odd 2 192.8.a.i.1.1 1
120.59 even 2 192.8.a.a.1.1 1
165.164 even 2 363.8.a.b.1.1 1
195.194 odd 2 507.8.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.8.a.a.1.1 1 15.14 odd 2
9.8.a.a.1.1 1 5.4 even 2
48.8.a.g.1.1 1 60.59 even 2
75.8.a.a.1.1 1 3.2 odd 2
75.8.b.c.49.1 2 15.2 even 4
75.8.b.c.49.2 2 15.8 even 4
81.8.c.a.28.1 2 45.29 odd 6
81.8.c.a.55.1 2 45.14 odd 6
81.8.c.c.28.1 2 45.34 even 6
81.8.c.c.55.1 2 45.4 even 6
144.8.a.b.1.1 1 20.19 odd 2
147.8.a.b.1.1 1 105.104 even 2
147.8.e.a.67.1 2 105.89 even 6
147.8.e.a.79.1 2 105.59 even 6
147.8.e.b.67.1 2 105.44 odd 6
147.8.e.b.79.1 2 105.74 odd 6
192.8.a.a.1.1 1 120.59 even 2
192.8.a.i.1.1 1 120.29 odd 2
225.8.a.i.1.1 1 1.1 even 1 trivial
225.8.b.f.199.1 2 5.3 odd 4
225.8.b.f.199.2 2 5.2 odd 4
363.8.a.b.1.1 1 165.164 even 2
441.8.a.a.1.1 1 35.34 odd 2
507.8.a.a.1.1 1 195.194 odd 2
576.8.a.w.1.1 1 40.29 even 2
576.8.a.x.1.1 1 40.19 odd 2