Properties

Label 225.8.b.f.199.1
Level $225$
Weight $8$
Character 225.199
Analytic conductor $70.287$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,8,Mod(199,225)]
 
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, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.199");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(70.2866307339\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 225.199
Dual form 225.8.b.f.199.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-6.00000i q^{2} +92.0000 q^{4} -64.0000i q^{7} -1320.00i q^{8} +948.000 q^{11} +5098.00i q^{13} -384.000 q^{14} +3856.00 q^{16} -28386.0i q^{17} +8620.00 q^{19} -5688.00i q^{22} -15288.0i q^{23} +30588.0 q^{26} -5888.00i q^{28} +36510.0 q^{29} -276808. q^{31} -192096. i q^{32} -170316. q^{34} +268526. i q^{37} -51720.0i q^{38} +629718. q^{41} -685772. i q^{43} +87216.0 q^{44} -91728.0 q^{46} -583296. i q^{47} +819447. q^{49} +469016. i q^{52} -428058. i q^{53} -84480.0 q^{56} -219060. i q^{58} +1.30638e6 q^{59} +300662. q^{61} +1.66085e6i q^{62} -659008. q^{64} -507244. i q^{67} -2.61151e6i q^{68} -5.56063e6 q^{71} -1.36908e6i q^{73} +1.61116e6 q^{74} +793040. q^{76} -60672.0i q^{77} +6.91372e6 q^{79} -3.77831e6i q^{82} -4.37675e6i q^{83} -4.11463e6 q^{86} -1.25136e6i q^{88} -8.52831e6 q^{89} +326272. q^{91} -1.40650e6i q^{92} -3.49978e6 q^{94} -8.82681e6i q^{97} -4.91668e6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 184 q^{4} + 1896 q^{11} - 768 q^{14} + 7712 q^{16} + 17240 q^{19} + 61176 q^{26} + 73020 q^{29} - 553616 q^{31} - 340632 q^{34} + 1259436 q^{41} + 174432 q^{44} - 183456 q^{46} + 1638894 q^{49} - 168960 q^{56}+ \cdots - 6999552 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 6.00000i − 0.530330i −0.964203 0.265165i \(-0.914574\pi\)
0.964203 0.265165i \(-0.0854264\pi\)
\(3\) 0 0
\(4\) 92.0000 0.718750
\(5\) 0 0
\(6\) 0 0
\(7\) − 64.0000i − 0.0705240i −0.999378 0.0352620i \(-0.988773\pi\)
0.999378 0.0352620i \(-0.0112266\pi\)
\(8\) − 1320.00i − 0.911505i
\(9\) 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
\(13\) 5098.00i 0.643573i 0.946812 + 0.321787i \(0.104283\pi\)
−0.946812 + 0.321787i \(0.895717\pi\)
\(14\) −384.000 −0.0374010
\(15\) 0 0
\(16\) 3856.00 0.235352
\(17\) − 28386.0i − 1.40131i −0.713502 0.700653i \(-0.752892\pi\)
0.713502 0.700653i \(-0.247108\pi\)
\(18\) 0 0
\(19\) 8620.00 0.288317 0.144158 0.989555i \(-0.453953\pi\)
0.144158 + 0.989555i \(0.453953\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 5688.00i − 0.113889i
\(23\) − 15288.0i − 0.262001i −0.991382 0.131001i \(-0.958181\pi\)
0.991382 0.131001i \(-0.0418190\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 30588.0 0.341306
\(27\) 0 0
\(28\) − 5888.00i − 0.0506891i
\(29\) 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.i − 1.03632i
\(33\) 0 0
\(34\) −170316. −0.743155
\(35\) 0 0
\(36\) 0 0
\(37\) 268526.i 0.871526i 0.900061 + 0.435763i \(0.143521\pi\)
−0.900061 + 0.435763i \(0.856479\pi\)
\(38\) − 51720.0i − 0.152903i
\(39\) 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.i − 1.31535i −0.753303 0.657673i \(-0.771541\pi\)
0.753303 0.657673i \(-0.228459\pi\)
\(44\) 87216.0 0.154352
\(45\) 0 0
\(46\) −91728.0 −0.138947
\(47\) − 583296.i − 0.819495i −0.912199 0.409748i \(-0.865617\pi\)
0.912199 0.409748i \(-0.134383\pi\)
\(48\) 0 0
\(49\) 819447. 0.995026
\(50\) 0 0
\(51\) 0 0
\(52\) 469016.i 0.462568i
\(53\) − 428058.i − 0.394945i −0.980308 0.197473i \(-0.936727\pi\)
0.980308 0.197473i \(-0.0632734\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −84480.0 −0.0642830
\(57\) 0 0
\(58\) − 219060.i − 0.147423i
\(59\) 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 \(0.472975\pi\)
\(62\) 1.66085e6i 0.885032i
\(63\) 0 0
\(64\) −659008. −0.314240
\(65\) 0 0
\(66\) 0 0
\(67\) − 507244.i − 0.206042i −0.994679 0.103021i \(-0.967149\pi\)
0.994679 0.103021i \(-0.0328508\pi\)
\(68\) − 2.61151e6i − 1.00719i
\(69\) 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
\(73\) − 1.36908e6i − 0.411907i −0.978562 0.205954i \(-0.933970\pi\)
0.978562 0.205954i \(-0.0660296\pi\)
\(74\) 1.61116e6 0.462196
\(75\) 0 0
\(76\) 793040. 0.207228
\(77\) − 60672.0i − 0.0151451i
\(78\) 0 0
\(79\) 6.91372e6 1.57767 0.788836 0.614603i \(-0.210684\pi\)
0.788836 + 0.614603i \(0.210684\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 3.77831e6i − 0.756744i
\(83\) − 4.37675e6i − 0.840191i −0.907480 0.420096i \(-0.861997\pi\)
0.907480 0.420096i \(-0.138003\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.11463e6 −0.697568
\(87\) 0 0
\(88\) − 1.25136e6i − 0.195746i
\(89\) −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
\(92\) − 1.40650e6i − 0.188313i
\(93\) 0 0
\(94\) −3.49978e6 −0.434603
\(95\) 0 0
\(96\) 0 0
\(97\) − 8.82681e6i − 0.981981i −0.871165 0.490990i \(-0.836635\pi\)
0.871165 0.490990i \(-0.163365\pi\)
\(98\) − 4.91668e6i − 0.527692i
\(99\) 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
\(103\) − 7.20939e6i − 0.650082i −0.945700 0.325041i \(-0.894622\pi\)
0.945700 0.325041i \(-0.105378\pi\)
\(104\) 6.72936e6 0.586620
\(105\) 0 0
\(106\) −2.56835e6 −0.209451
\(107\) − 1.14261e7i − 0.901683i −0.892604 0.450842i \(-0.851124\pi\)
0.892604 0.450842i \(-0.148876\pi\)
\(108\) 0 0
\(109\) −4.02095e6 −0.297397 −0.148698 0.988883i \(-0.547508\pi\)
−0.148698 + 0.988883i \(0.547508\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 246784.i − 0.0165979i
\(113\) − 1.77063e7i − 1.15439i −0.816605 0.577197i \(-0.804147\pi\)
0.816605 0.577197i \(-0.195853\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 3.35892e6 0.199801
\(117\) 0 0
\(118\) − 7.83828e6i − 0.439171i
\(119\) −1.81670e6 −0.0988257
\(120\) 0 0
\(121\) −1.85885e7 −0.953882
\(122\) − 1.80397e6i − 0.0899436i
\(123\) 0 0
\(124\) −2.54663e7 −1.19947
\(125\) 0 0
\(126\) 0 0
\(127\) 1.67883e7i 0.727267i 0.931542 + 0.363633i \(0.118464\pi\)
−0.931542 + 0.363633i \(0.881536\pi\)
\(128\) − 2.06342e7i − 0.869668i
\(129\) 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.i − 0.0203332i
\(134\) −3.04346e6 −0.109270
\(135\) 0 0
\(136\) −3.74695e7 −1.27730
\(137\) − 2.80449e7i − 0.931820i −0.884832 0.465910i \(-0.845727\pi\)
0.884832 0.465910i \(-0.154273\pi\)
\(138\) 0 0
\(139\) 1.18273e7 0.373537 0.186769 0.982404i \(-0.440199\pi\)
0.186769 + 0.982404i \(0.440199\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 3.33638e7i 0.977836i
\(143\) 4.83290e6i 0.138208i
\(144\) 0 0
\(145\) 0 0
\(146\) −8.21449e6 −0.218447
\(147\) 0 0
\(148\) 2.47044e7i 0.626409i
\(149\) 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 \(0.499714\pi\)
\(152\) − 1.13784e7i − 0.262802i
\(153\) 0 0
\(154\) −364032. −0.00803188
\(155\) 0 0
\(156\) 0 0
\(157\) − 3.21825e7i − 0.663698i −0.943332 0.331849i \(-0.892328\pi\)
0.943332 0.331849i \(-0.107672\pi\)
\(158\) − 4.14823e7i − 0.836687i
\(159\) 0 0
\(160\) 0 0
\(161\) −978432. −0.0184774
\(162\) 0 0
\(163\) − 5.83435e7i − 1.05520i −0.849492 0.527601i \(-0.823092\pi\)
0.849492 0.527601i \(-0.176908\pi\)
\(164\) 5.79341e7 1.02561
\(165\) 0 0
\(166\) −2.62605e7 −0.445579
\(167\) 2.58365e7i 0.429266i 0.976695 + 0.214633i \(0.0688555\pi\)
−0.976695 + 0.214633i \(0.931145\pi\)
\(168\) 0 0
\(169\) 3.67589e7 0.585813
\(170\) 0 0
\(171\) 0 0
\(172\) − 6.30910e7i − 0.945405i
\(173\) 6.35201e7i 0.932716i 0.884596 + 0.466358i \(0.154434\pi\)
−0.884596 + 0.466358i \(0.845566\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.65549e6 0.0505418
\(177\) 0 0
\(178\) 5.11699e7i 0.680055i
\(179\) −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 \(0.367524\pi\)
\(182\) − 1.95763e6i − 0.0240703i
\(183\) 0 0
\(184\) −2.01802e7 −0.238815
\(185\) 0 0
\(186\) 0 0
\(187\) − 2.69099e7i − 0.300931i
\(188\) − 5.36632e7i − 0.589012i
\(189\) 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
\(193\) − 1.16377e8i − 1.16524i −0.812744 0.582621i \(-0.802027\pi\)
0.812744 0.582621i \(-0.197973\pi\)
\(194\) −5.29609e7 −0.520774
\(195\) 0 0
\(196\) 7.53891e7 0.715175
\(197\) 1.18816e8i 1.10724i 0.832768 + 0.553622i \(0.186755\pi\)
−0.832768 + 0.553622i \(0.813245\pi\)
\(198\) 0 0
\(199\) 9.50106e7 0.854646 0.427323 0.904099i \(-0.359457\pi\)
0.427323 + 0.904099i \(0.359457\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 7.19185e7i 0.613919i
\(203\) − 2.33664e6i − 0.0196045i
\(204\) 0 0
\(205\) 0 0
\(206\) −4.32564e7 −0.344758
\(207\) 0 0
\(208\) 1.96579e7i 0.151466i
\(209\) 8.17176e6 0.0619161
\(210\) 0 0
\(211\) 1.79246e8 1.31360 0.656798 0.754067i \(-0.271910\pi\)
0.656798 + 0.754067i \(0.271910\pi\)
\(212\) − 3.93813e7i − 0.283867i
\(213\) 0 0
\(214\) −6.85565e7 −0.478190
\(215\) 0 0
\(216\) 0 0
\(217\) 1.77157e7i 0.117693i
\(218\) 2.41257e7i 0.157718i
\(219\) 0 0
\(220\) 0 0
\(221\) 1.44712e8 0.901843
\(222\) 0 0
\(223\) 2.06537e8i 1.24718i 0.781750 + 0.623592i \(0.214327\pi\)
−0.781750 + 0.623592i \(0.785673\pi\)
\(224\) −1.22941e7 −0.0730853
\(225\) 0 0
\(226\) −1.06238e8 −0.612209
\(227\) − 4.33954e7i − 0.246237i −0.992392 0.123118i \(-0.960710\pi\)
0.992392 0.123118i \(-0.0392895\pi\)
\(228\) 0 0
\(229\) 3.61931e7 0.199160 0.0995799 0.995030i \(-0.468250\pi\)
0.0995799 + 0.995030i \(0.468250\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 4.81932e7i − 0.253383i
\(233\) 9.22347e7i 0.477693i 0.971057 + 0.238846i \(0.0767692\pi\)
−0.971057 + 0.238846i \(0.923231\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.20187e8 0.595203
\(237\) 0 0
\(238\) 1.09002e7i 0.0524102i
\(239\) 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 \(0.348296\pi\)
\(242\) 1.11531e8i 0.505872i
\(243\) 0 0
\(244\) 2.76609e7 0.121900
\(245\) 0 0
\(246\) 0 0
\(247\) 4.39448e7i 0.185553i
\(248\) 3.65387e8i 1.52115i
\(249\) 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
\(253\) − 1.44930e7i − 0.0562649i
\(254\) 1.00730e8 0.385691
\(255\) 0 0
\(256\) −2.08158e8 −0.775451
\(257\) 1.42885e8i 0.525076i 0.964922 + 0.262538i \(0.0845594\pi\)
−0.964922 + 0.262538i \(0.915441\pi\)
\(258\) 0 0
\(259\) 1.71857e7 0.0614635
\(260\) 0 0
\(261\) 0 0
\(262\) 1.00961e8i 0.346815i
\(263\) 4.40241e8i 1.49226i 0.665799 + 0.746131i \(0.268091\pi\)
−0.665799 + 0.746131i \(0.731909\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −3.31008e6 −0.0107833
\(267\) 0 0
\(268\) − 4.66664e7i − 0.148092i
\(269\) 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 \(0.724428\pi\)
\(272\) − 1.09456e8i − 0.329800i
\(273\) 0 0
\(274\) −1.68269e8 −0.494172
\(275\) 0 0
\(276\) 0 0
\(277\) 5.16158e8i 1.45916i 0.683894 + 0.729581i \(0.260285\pi\)
−0.683894 + 0.729581i \(0.739715\pi\)
\(278\) − 7.09638e7i − 0.198098i
\(279\) 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
\(283\) 5.94308e8i 1.55869i 0.626596 + 0.779344i \(0.284448\pi\)
−0.626596 + 0.779344i \(0.715552\pi\)
\(284\) −5.11578e8 −1.32525
\(285\) 0 0
\(286\) 2.89974e7 0.0732957
\(287\) − 4.03020e7i − 0.100633i
\(288\) 0 0
\(289\) −3.95426e8 −0.963658
\(290\) 0 0
\(291\) 0 0
\(292\) − 1.25956e8i − 0.296058i
\(293\) 1.15515e8i 0.268288i 0.990962 + 0.134144i \(0.0428284\pi\)
−0.990962 + 0.134144i \(0.957172\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3.54454e8 0.794400
\(297\) 0 0
\(298\) − 1.24708e8i − 0.272984i
\(299\) 7.79382e7 0.168617
\(300\) 0 0
\(301\) −4.38894e7 −0.0927635
\(302\) − 456672.i 0 0.000954070i
\(303\) 0 0
\(304\) 3.32387e7 0.0678558
\(305\) 0 0
\(306\) 0 0
\(307\) − 2.60600e8i − 0.514032i −0.966407 0.257016i \(-0.917261\pi\)
0.966407 0.257016i \(-0.0827392\pi\)
\(308\) − 5.58182e6i − 0.0108855i
\(309\) 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
\(313\) 4.60074e8i 0.848053i 0.905650 + 0.424026i \(0.139384\pi\)
−0.905650 + 0.424026i \(0.860616\pi\)
\(314\) −1.93095e8 −0.351979
\(315\) 0 0
\(316\) 6.36062e8 1.13395
\(317\) − 6.25561e7i − 0.110297i −0.998478 0.0551483i \(-0.982437\pi\)
0.998478 0.0551483i \(-0.0175632\pi\)
\(318\) 0 0
\(319\) 3.46115e7 0.0596970
\(320\) 0 0
\(321\) 0 0
\(322\) 5.87059e6i 0.00979910i
\(323\) − 2.44687e8i − 0.404020i
\(324\) 0 0
\(325\) 0 0
\(326\) −3.50061e8 −0.559606
\(327\) 0 0
\(328\) − 8.31228e8i − 1.30065i
\(329\) −3.73309e7 −0.0577941
\(330\) 0 0
\(331\) 6.84236e8 1.03707 0.518535 0.855057i \(-0.326478\pi\)
0.518535 + 0.855057i \(0.326478\pi\)
\(332\) − 4.02661e8i − 0.603888i
\(333\) 0 0
\(334\) 1.55019e8 0.227652
\(335\) 0 0
\(336\) 0 0
\(337\) − 6.26313e8i − 0.891429i −0.895175 0.445714i \(-0.852950\pi\)
0.895175 0.445714i \(-0.147050\pi\)
\(338\) − 2.20553e8i − 0.310674i
\(339\) 0 0
\(340\) 0 0
\(341\) −2.62414e8 −0.358382
\(342\) 0 0
\(343\) − 1.05151e8i − 0.140697i
\(344\) −9.05219e8 −1.19894
\(345\) 0 0
\(346\) 3.81120e8 0.494647
\(347\) 1.25340e9i 1.61041i 0.593000 + 0.805203i \(0.297943\pi\)
−0.593000 + 0.805203i \(0.702057\pi\)
\(348\) 0 0
\(349\) −2.65350e8 −0.334142 −0.167071 0.985945i \(-0.553431\pi\)
−0.167071 + 0.985945i \(0.553431\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 1.82107e8i − 0.222550i
\(353\) − 5.69636e8i − 0.689264i −0.938738 0.344632i \(-0.888004\pi\)
0.938738 0.344632i \(-0.111996\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −7.84605e8 −0.921671
\(357\) 0 0
\(358\) 4.85735e8i 0.559512i
\(359\) 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
\(362\) − 3.87019e8i − 0.428798i
\(363\) 0 0
\(364\) 3.00170e7 0.0326222
\(365\) 0 0
\(366\) 0 0
\(367\) − 8.52565e8i − 0.900318i −0.892948 0.450159i \(-0.851367\pi\)
0.892948 0.450159i \(-0.148633\pi\)
\(368\) − 5.89505e7i − 0.0616624i
\(369\) 0 0
\(370\) 0 0
\(371\) −2.73957e7 −0.0278531
\(372\) 0 0
\(373\) − 3.81183e8i − 0.380323i −0.981753 0.190162i \(-0.939099\pi\)
0.981753 0.190162i \(-0.0609012\pi\)
\(374\) −1.61460e8 −0.159593
\(375\) 0 0
\(376\) −7.69951e8 −0.746974
\(377\) 1.86128e8i 0.178903i
\(378\) 0 0
\(379\) 1.48353e9 1.39978 0.699889 0.714251i \(-0.253233\pi\)
0.699889 + 0.714251i \(0.253233\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 3.40964e8i 0.312959i
\(383\) − 7.61930e8i − 0.692978i −0.938054 0.346489i \(-0.887374\pi\)
0.938054 0.346489i \(-0.112626\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −6.98262e8 −0.617963
\(387\) 0 0
\(388\) − 8.12067e8i − 0.705799i
\(389\) 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
\(392\) − 1.08167e9i − 0.906971i
\(393\) 0 0
\(394\) 7.12896e8 0.587205
\(395\) 0 0
\(396\) 0 0
\(397\) 1.88016e9i 1.50809i 0.656822 + 0.754046i \(0.271900\pi\)
−0.656822 + 0.754046i \(0.728100\pi\)
\(398\) − 5.70064e8i − 0.453245i
\(399\) 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
\(403\) − 1.41117e9i − 1.07402i
\(404\) −1.10275e9 −0.832037
\(405\) 0 0
\(406\) −1.40198e7 −0.0103969
\(407\) 2.54563e8i 0.187161i
\(408\) 0 0
\(409\) −8.99478e7 −0.0650069 −0.0325034 0.999472i \(-0.510348\pi\)
−0.0325034 + 0.999472i \(0.510348\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 6.63264e8i − 0.467247i
\(413\) − 8.36083e7i − 0.0584015i
\(414\) 0 0
\(415\) 0 0
\(416\) 9.79305e8 0.666947
\(417\) 0 0
\(418\) − 4.90306e7i − 0.0328360i
\(419\) 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 \(0.620682\pi\)
\(422\) − 1.07548e9i − 0.696639i
\(423\) 0 0
\(424\) −5.65037e8 −0.359995
\(425\) 0 0
\(426\) 0 0
\(427\) − 1.92424e7i − 0.0119608i
\(428\) − 1.05120e9i − 0.648085i
\(429\) 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
\(433\) 1.51738e8i 0.0898227i 0.998991 + 0.0449114i \(0.0143005\pi\)
−0.998991 + 0.0449114i \(0.985699\pi\)
\(434\) 1.06294e8 0.0624160
\(435\) 0 0
\(436\) −3.69927e8 −0.213754
\(437\) − 1.31783e8i − 0.0755393i
\(438\) 0 0
\(439\) −9.90763e8 −0.558912 −0.279456 0.960158i \(-0.590154\pi\)
−0.279456 + 0.960158i \(0.590154\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 8.68271e8i − 0.478275i
\(443\) − 1.77376e9i − 0.969351i −0.874694 0.484675i \(-0.838938\pi\)
0.874694 0.484675i \(-0.161062\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 1.23922e9 0.661419
\(447\) 0 0
\(448\) 4.21765e7i 0.0221614i
\(449\) −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
\(452\) − 1.62898e9i − 0.829720i
\(453\) 0 0
\(454\) −2.60372e8 −0.130587
\(455\) 0 0
\(456\) 0 0
\(457\) 2.94758e9i 1.44464i 0.691559 + 0.722320i \(0.256924\pi\)
−0.691559 + 0.722320i \(0.743076\pi\)
\(458\) − 2.17159e8i − 0.105620i
\(459\) 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
\(463\) − 4.63553e8i − 0.217053i −0.994094 0.108527i \(-0.965387\pi\)
0.994094 0.108527i \(-0.0346132\pi\)
\(464\) 1.40783e8 0.0654238
\(465\) 0 0
\(466\) 5.53408e8 0.253335
\(467\) 4.17922e8i 0.189883i 0.995483 + 0.0949415i \(0.0302664\pi\)
−0.995483 + 0.0949415i \(0.969734\pi\)
\(468\) 0 0
\(469\) −3.24636e7 −0.0145309
\(470\) 0 0
\(471\) 0 0
\(472\) − 1.72442e9i − 0.754825i
\(473\) − 6.50112e8i − 0.282471i
\(474\) 0 0
\(475\) 0 0
\(476\) −1.67137e8 −0.0710310
\(477\) 0 0
\(478\) − 2.99081e8i − 0.125254i
\(479\) −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
\(482\) − 1.19624e9i − 0.486581i
\(483\) 0 0
\(484\) −1.71014e9 −0.685603
\(485\) 0 0
\(486\) 0 0
\(487\) 9.29460e8i 0.364653i 0.983238 + 0.182326i \(0.0583627\pi\)
−0.983238 + 0.182326i \(0.941637\pi\)
\(488\) − 3.96874e8i − 0.154591i
\(489\) 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
\(493\) − 1.03637e9i − 0.389540i
\(494\) 2.63669e8 0.0984043
\(495\) 0 0
\(496\) −1.06737e9 −0.392762
\(497\) 3.55880e8i 0.130034i
\(498\) 0 0
\(499\) 4.10649e8 0.147951 0.0739757 0.997260i \(-0.476431\pi\)
0.0739757 + 0.997260i \(0.476431\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 2.36807e9i − 0.835470i
\(503\) 5.02041e9i 1.75894i 0.475954 + 0.879470i \(0.342103\pi\)
−0.475954 + 0.879470i \(0.657897\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −8.69581e7 −0.0298389
\(507\) 0 0
\(508\) 1.54452e9i 0.522723i
\(509\) −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
\(512\) − 1.39223e9i − 0.458423i
\(513\) 0 0
\(514\) 8.57312e8 0.278463
\(515\) 0 0
\(516\) 0 0
\(517\) − 5.52965e8i − 0.175987i
\(518\) − 1.03114e8i − 0.0325959i
\(519\) 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
\(523\) 5.28911e9i 1.61669i 0.588709 + 0.808345i \(0.299636\pi\)
−0.588709 + 0.808345i \(0.700364\pi\)
\(524\) −1.54806e9 −0.470034
\(525\) 0 0
\(526\) 2.64144e9 0.791391
\(527\) 7.85747e9i 2.33854i
\(528\) 0 0
\(529\) 3.17110e9 0.931355
\(530\) 0 0
\(531\) 0 0
\(532\) − 5.07546e7i − 0.0146145i
\(533\) 3.21030e9i 0.918334i
\(534\) 0 0
\(535\) 0 0
\(536\) −6.69562e8 −0.187808
\(537\) 0 0
\(538\) − 1.65243e9i − 0.457493i
\(539\) 7.76836e8 0.213682
\(540\) 0 0
\(541\) 3.04614e9 0.827101 0.413551 0.910481i \(-0.364288\pi\)
0.413551 + 0.910481i \(0.364288\pi\)
\(542\) 2.54802e9i 0.687393i
\(543\) 0 0
\(544\) −5.45284e9 −1.45220
\(545\) 0 0
\(546\) 0 0
\(547\) − 4.85537e9i − 1.26843i −0.773157 0.634215i \(-0.781323\pi\)
0.773157 0.634215i \(-0.218677\pi\)
\(548\) − 2.58013e9i − 0.669746i
\(549\) 0 0
\(550\) 0 0
\(551\) 3.14716e8 0.0801472
\(552\) 0 0
\(553\) − 4.42478e8i − 0.111264i
\(554\) 3.09695e9 0.773838
\(555\) 0 0
\(556\) 1.08811e9 0.268480
\(557\) − 1.27762e9i − 0.313263i −0.987657 0.156631i \(-0.949937\pi\)
0.987657 0.156631i \(-0.0500635\pi\)
\(558\) 0 0
\(559\) 3.49607e9 0.846522
\(560\) 0 0
\(561\) 0 0
\(562\) − 1.86626e9i − 0.443501i
\(563\) 4.71265e9i 1.11297i 0.830856 + 0.556487i \(0.187851\pi\)
−0.830856 + 0.556487i \(0.812149\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 3.56585e9 0.826619
\(567\) 0 0
\(568\) 7.34003e9i 1.68066i
\(569\) 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 \(0.312150\pi\)
\(572\) 4.44627e8i 0.0993367i
\(573\) 0 0
\(574\) −2.41812e8 −0.0533686
\(575\) 0 0
\(576\) 0 0
\(577\) 8.51847e9i 1.84606i 0.384725 + 0.923031i \(0.374296\pi\)
−0.384725 + 0.923031i \(0.625704\pi\)
\(578\) 2.37256e9i 0.511057i
\(579\) 0 0
\(580\) 0 0
\(581\) −2.80112e8 −0.0592536
\(582\) 0 0
\(583\) − 4.05799e8i − 0.0848147i
\(584\) −1.80719e9 −0.375455
\(585\) 0 0
\(586\) 6.93088e8 0.142281
\(587\) 5.62247e8i 0.114735i 0.998353 + 0.0573673i \(0.0182706\pi\)
−0.998353 + 0.0573673i \(0.981729\pi\)
\(588\) 0 0
\(589\) −2.38608e9 −0.481152
\(590\) 0 0
\(591\) 0 0
\(592\) 1.03544e9i 0.205115i
\(593\) 3.62110e9i 0.713099i 0.934277 + 0.356549i \(0.116047\pi\)
−0.934277 + 0.356549i \(0.883953\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.91219e9 0.369971
\(597\) 0 0
\(598\) − 4.67629e8i − 0.0894227i
\(599\) −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 \(0.683895\pi\)
\(602\) 2.63336e8i 0.0491953i
\(603\) 0 0
\(604\) 7.00230e6 0.00129304
\(605\) 0 0
\(606\) 0 0
\(607\) 3.84051e9i 0.696993i 0.937310 + 0.348497i \(0.113308\pi\)
−0.937310 + 0.348497i \(0.886692\pi\)
\(608\) − 1.65587e9i − 0.298788i
\(609\) 0 0
\(610\) 0 0
\(611\) 2.97364e9 0.527405
\(612\) 0 0
\(613\) − 1.70484e9i − 0.298932i −0.988767 0.149466i \(-0.952245\pi\)
0.988767 0.149466i \(-0.0477555\pi\)
\(614\) −1.56360e9 −0.272606
\(615\) 0 0
\(616\) −8.00870e7 −0.0138048
\(617\) 2.80809e9i 0.481297i 0.970612 + 0.240649i \(0.0773601\pi\)
−0.970612 + 0.240649i \(0.922640\pi\)
\(618\) 0 0
\(619\) 2.54365e9 0.431063 0.215532 0.976497i \(-0.430852\pi\)
0.215532 + 0.976497i \(0.430852\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 3.46077e9i 0.576642i
\(623\) 5.45812e8i 0.0904346i
\(624\) 0 0
\(625\) 0 0
\(626\) 2.76045e9 0.449748
\(627\) 0 0
\(628\) − 2.96079e9i − 0.477033i
\(629\) 7.62238e9 1.22127
\(630\) 0 0
\(631\) −1.51146e8 −0.0239494 −0.0119747 0.999928i \(-0.503812\pi\)
−0.0119747 + 0.999928i \(0.503812\pi\)
\(632\) − 9.12611e9i − 1.43806i
\(633\) 0 0
\(634\) −3.75337e8 −0.0584936
\(635\) 0 0
\(636\) 0 0
\(637\) 4.17754e9i 0.640373i
\(638\) − 2.07669e8i − 0.0316591i
\(639\) 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
\(643\) − 2.86744e9i − 0.425359i −0.977122 0.212680i \(-0.931781\pi\)
0.977122 0.212680i \(-0.0682191\pi\)
\(644\) −9.00157e7 −0.0132806
\(645\) 0 0
\(646\) −1.46812e9 −0.214264
\(647\) 4.10640e9i 0.596068i 0.954555 + 0.298034i \(0.0963309\pi\)
−0.954555 + 0.298034i \(0.903669\pi\)
\(648\) 0 0
\(649\) 1.23845e9 0.177837
\(650\) 0 0
\(651\) 0 0
\(652\) − 5.36760e9i − 0.758427i
\(653\) 6.91100e9i 0.971280i 0.874159 + 0.485640i \(0.161413\pi\)
−0.874159 + 0.485640i \(0.838587\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2.42819e9 0.335830
\(657\) 0 0
\(658\) 2.23986e8i 0.0306499i
\(659\) 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 \(0.650541\pi\)
\(662\) − 4.10541e9i − 0.549989i
\(663\) 0 0
\(664\) −5.77731e9 −0.765839
\(665\) 0 0
\(666\) 0 0
\(667\) − 5.58165e8i − 0.0728320i
\(668\) 2.37696e9i 0.308535i
\(669\) 0 0
\(670\) 0 0
\(671\) 2.85028e8 0.0364215
\(672\) 0 0
\(673\) 1.74959e9i 0.221250i 0.993862 + 0.110625i \(0.0352853\pi\)
−0.993862 + 0.110625i \(0.964715\pi\)
\(674\) −3.75788e9 −0.472752
\(675\) 0 0
\(676\) 3.38182e9 0.421053
\(677\) − 8.30011e9i − 1.02807i −0.857769 0.514036i \(-0.828150\pi\)
0.857769 0.514036i \(-0.171850\pi\)
\(678\) 0 0
\(679\) −5.64916e8 −0.0692532
\(680\) 0 0
\(681\) 0 0
\(682\) 1.57448e9i 0.190061i
\(683\) − 1.21232e10i − 1.45594i −0.685610 0.727969i \(-0.740464\pi\)
0.685610 0.727969i \(-0.259536\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −6.30908e8 −0.0746160
\(687\) 0 0
\(688\) − 2.64434e9i − 0.309569i
\(689\) 2.18224e9 0.254176
\(690\) 0 0
\(691\) 8.21846e9 0.947583 0.473791 0.880637i \(-0.342885\pi\)
0.473791 + 0.880637i \(0.342885\pi\)
\(692\) 5.84385e9i 0.670390i
\(693\) 0 0
\(694\) 7.52038e9 0.854046
\(695\) 0 0
\(696\) 0 0
\(697\) − 1.78752e10i − 1.99957i
\(698\) 1.59210e9i 0.177205i
\(699\) 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
\(703\) 2.31469e9i 0.251275i
\(704\) −6.24740e8 −0.0674831
\(705\) 0 0
\(706\) −3.41781e9 −0.365537
\(707\) 7.67131e8i 0.0816397i
\(708\) 0 0
\(709\) −2.78975e9 −0.293970 −0.146985 0.989139i \(-0.546957\pi\)
−0.146985 + 0.989139i \(0.546957\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 1.12574e10i 1.16885i
\(713\) 4.23184e9i 0.437236i
\(714\) 0 0
\(715\) 0 0
\(716\) −7.44794e9 −0.758299
\(717\) 0 0
\(718\) − 5.59524e9i − 0.564136i
\(719\) 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
\(722\) 4.91740e9i 0.486246i
\(723\) 0 0
\(724\) 5.93429e9 0.581144
\(725\) 0 0
\(726\) 0 0
\(727\) − 8.11761e9i − 0.783534i −0.920065 0.391767i \(-0.871864\pi\)
0.920065 0.391767i \(-0.128136\pi\)
\(728\) − 4.30679e8i − 0.0413708i
\(729\) 0 0
\(730\) 0 0
\(731\) −1.94663e10 −1.84320
\(732\) 0 0
\(733\) 1.03241e10i 0.968249i 0.874999 + 0.484124i \(0.160862\pi\)
−0.874999 + 0.484124i \(0.839138\pi\)
\(734\) −5.11539e9 −0.477466
\(735\) 0 0
\(736\) −2.93676e9 −0.271517
\(737\) − 4.80867e8i − 0.0442475i
\(738\) 0 0
\(739\) 1.35365e10 1.23382 0.616908 0.787035i \(-0.288385\pi\)
0.616908 + 0.787035i \(0.288385\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 1.64374e8i 0.0147713i
\(743\) − 1.71936e10i − 1.53782i −0.639356 0.768910i \(-0.720799\pi\)
0.639356 0.768910i \(-0.279201\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −2.28710e9 −0.201697
\(747\) 0 0
\(748\) − 2.47571e9i − 0.216294i
\(749\) −7.31269e8 −0.0635903
\(750\) 0 0
\(751\) 1.12478e10 0.969013 0.484506 0.874788i \(-0.338999\pi\)
0.484506 + 0.874788i \(0.338999\pi\)
\(752\) − 2.24919e9i − 0.192870i
\(753\) 0 0
\(754\) 1.11677e9 0.0948775
\(755\) 0 0
\(756\) 0 0
\(757\) 1.63068e10i 1.36626i 0.730296 + 0.683131i \(0.239382\pi\)
−0.730296 + 0.683131i \(0.760618\pi\)
\(758\) − 8.90118e9i − 0.742345i
\(759\) 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
\(763\) 2.57341e8i 0.0209736i
\(764\) −5.22812e9 −0.424149
\(765\) 0 0
\(766\) −4.57158e9 −0.367507
\(767\) 6.65993e9i 0.532949i
\(768\) 0 0
\(769\) −2.45069e10 −1.94333 −0.971664 0.236368i \(-0.924043\pi\)
−0.971664 + 0.236368i \(0.924043\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 1.07067e10i − 0.837518i
\(773\) − 1.01722e10i − 0.792110i −0.918227 0.396055i \(-0.870379\pi\)
0.918227 0.396055i \(-0.129621\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −1.16514e10 −0.895080
\(777\) 0 0
\(778\) − 9.65411e9i − 0.734994i
\(779\) 5.42817e9 0.411408
\(780\) 0 0
\(781\) −5.27148e9 −0.395962
\(782\) 2.60379e9i 0.194707i
\(783\) 0 0
\(784\) 3.15979e9 0.234181
\(785\) 0 0
\(786\) 0 0
\(787\) − 9.79135e9i − 0.716030i −0.933716 0.358015i \(-0.883454\pi\)
0.933716 0.358015i \(-0.116546\pi\)
\(788\) 1.09311e10i 0.795832i
\(789\) 0 0
\(790\) 0 0
\(791\) −1.13320e9 −0.0814124
\(792\) 0 0
\(793\) 1.53277e9i 0.109150i
\(794\) 1.12809e10 0.799786
\(795\) 0 0
\(796\) 8.74098e9 0.614277
\(797\) 9.75782e9i 0.682729i 0.939931 + 0.341365i \(0.110889\pi\)
−0.939931 + 0.341365i \(0.889111\pi\)
\(798\) 0 0
\(799\) −1.65574e10 −1.14836
\(800\) 0 0
\(801\) 0 0
\(802\) 1.61155e9i 0.110315i
\(803\) − 1.29789e9i − 0.0884572i
\(804\) 0 0
\(805\) 0 0
\(806\) −8.46700e9 −0.569583
\(807\) 0 0
\(808\) 1.58221e10i 1.05517i
\(809\) −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 \(0.584816\pi\)
\(812\) − 2.14971e8i − 0.0140907i
\(813\) 0 0
\(814\) 1.52738e9 0.0992569
\(815\) 0 0
\(816\) 0 0
\(817\) − 5.91135e9i − 0.379236i
\(818\) 5.39687e8i 0.0344751i
\(819\) 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
\(823\) − 2.78682e10i − 1.74265i −0.490707 0.871324i \(-0.663262\pi\)
0.490707 0.871324i \(-0.336738\pi\)
\(824\) −9.51640e9 −0.592553
\(825\) 0 0
\(826\) −5.01650e8 −0.0309721
\(827\) − 2.35125e10i − 1.44554i −0.691090 0.722769i \(-0.742869\pi\)
0.691090 0.722769i \(-0.257131\pi\)
\(828\) 0 0
\(829\) 1.28598e10 0.783960 0.391980 0.919974i \(-0.371790\pi\)
0.391980 + 0.919974i \(0.371790\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 3.35962e9i − 0.202236i
\(833\) − 2.32608e10i − 1.39434i
\(834\) 0 0
\(835\) 0 0
\(836\) 7.51802e8 0.0445022
\(837\) 0 0
\(838\) − 1.01433e10i − 0.595420i
\(839\) −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
\(842\) 6.79996e9i 0.392567i
\(843\) 0 0
\(844\) 1.64907e10 0.944147
\(845\) 0 0
\(846\) 0 0
\(847\) 1.18966e9i 0.0672716i
\(848\) − 1.65059e9i − 0.0929510i
\(849\) 0 0
\(850\) 0 0
\(851\) 4.10523e9 0.228341
\(852\) 0 0
\(853\) − 4.20827e9i − 0.232157i −0.993240 0.116079i \(-0.962968\pi\)
0.993240 0.116079i \(-0.0370324\pi\)
\(854\) −1.15454e8 −0.00634318
\(855\) 0 0
\(856\) −1.50824e10 −0.821888
\(857\) − 3.19307e10i − 1.73291i −0.499259 0.866453i \(-0.666394\pi\)
0.499259 0.866453i \(-0.333606\pi\)
\(858\) 0 0
\(859\) −2.18002e10 −1.17350 −0.586752 0.809767i \(-0.699594\pi\)
−0.586752 + 0.809767i \(0.699594\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 1.31966e10i 0.701755i
\(863\) 1.04728e10i 0.554657i 0.960775 + 0.277329i \(0.0894490\pi\)
−0.960775 + 0.277329i \(0.910551\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 9.10427e8 0.0476357
\(867\) 0 0
\(868\) 1.62985e9i 0.0845916i
\(869\) 6.55421e9 0.338806
\(870\) 0 0
\(871\) 2.58593e9 0.132603
\(872\) 5.30765e9i 0.271078i
\(873\) 0 0
\(874\) −7.90695e8 −0.0400608
\(875\) 0 0
\(876\) 0 0
\(877\) − 1.77787e10i − 0.890024i −0.895525 0.445012i \(-0.853199\pi\)
0.895525 0.445012i \(-0.146801\pi\)
\(878\) 5.94458e9i 0.296408i
\(879\) 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
\(883\) 2.76375e10i 1.35094i 0.737386 + 0.675472i \(0.236060\pi\)
−0.737386 + 0.675472i \(0.763940\pi\)
\(884\) 1.33135e10 0.648200
\(885\) 0 0
\(886\) −1.06425e10 −0.514076
\(887\) − 3.23087e10i − 1.55449i −0.629200 0.777243i \(-0.716617\pi\)
0.629200 0.777243i \(-0.283383\pi\)
\(888\) 0 0
\(889\) 1.07445e9 0.0512897
\(890\) 0 0
\(891\) 0 0
\(892\) 1.90014e10i 0.896414i
\(893\) − 5.02801e9i − 0.236274i
\(894\) 0 0
\(895\) 0 0
\(896\) −1.32059e9 −0.0613325
\(897\) 0 0
\(898\) 1.66206e9i 0.0765914i
\(899\) −1.01063e10 −0.463908
\(900\) 0 0
\(901\) −1.21509e10 −0.553439
\(902\) − 3.58184e9i − 0.162511i
\(903\) 0 0
\(904\) −2.33723e10 −1.05223
\(905\) 0 0
\(906\) 0 0
\(907\) 2.27142e10i 1.01082i 0.862880 + 0.505409i \(0.168658\pi\)
−0.862880 + 0.505409i \(0.831342\pi\)
\(908\) − 3.99238e9i − 0.176983i
\(909\) 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
\(913\) − 4.14916e9i − 0.180431i
\(914\) 1.76855e10 0.766136
\(915\) 0 0
\(916\) 3.32976e9 0.143146
\(917\) 1.07691e9i 0.0461199i
\(918\) 0 0
\(919\) 2.49374e10 1.05986 0.529928 0.848043i \(-0.322219\pi\)
0.529928 + 0.848043i \(0.322219\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 1.66012e10i − 0.697561i
\(923\) − 2.83481e10i − 1.18664i
\(924\) 0 0
\(925\) 0 0
\(926\) −2.78132e9 −0.115110
\(927\) 0 0
\(928\) − 7.01342e9i − 0.288079i
\(929\) −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
\(932\) 8.48559e9i 0.343342i
\(933\) 0 0
\(934\) 2.50753e9 0.100701
\(935\) 0 0
\(936\) 0 0
\(937\) 2.82655e10i 1.12245i 0.827663 + 0.561226i \(0.189670\pi\)
−0.827663 + 0.561226i \(0.810330\pi\)
\(938\) 1.94782e8i 0.00770616i
\(939\) 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
\(943\) − 9.62713e9i − 0.373857i
\(944\) 5.03740e9 0.194897
\(945\) 0 0
\(946\) −3.90067e9 −0.149803
\(947\) 4.67392e10i 1.78837i 0.447701 + 0.894184i \(0.352243\pi\)
−0.447701 + 0.894184i \(0.647757\pi\)
\(948\) 0 0
\(949\) 6.97958e9 0.265093
\(950\) 0 0
\(951\) 0 0
\(952\) 2.39805e9i 0.0900801i
\(953\) 3.82420e10i 1.43125i 0.698484 + 0.715625i \(0.253858\pi\)
−0.698484 + 0.715625i \(0.746142\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 4.58591e9 0.169755
\(957\) 0 0
\(958\) 9.05837e9i 0.332867i
\(959\) −1.79487e9 −0.0657157
\(960\) 0 0
\(961\) 4.91101e10 1.78500
\(962\) 8.21367e9i 0.297457i
\(963\) 0 0
\(964\) 1.83424e10 0.659457
\(965\) 0 0
\(966\) 0 0
\(967\) − 4.90012e10i − 1.74267i −0.490692 0.871333i \(-0.663256\pi\)
0.490692 0.871333i \(-0.336744\pi\)
\(968\) 2.45368e10i 0.869468i
\(969\) 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
\(973\) − 7.56947e8i − 0.0263433i
\(974\) 5.57676e9 0.193386
\(975\) 0 0
\(976\) 1.15935e9 0.0399155
\(977\) − 3.94482e9i − 0.135331i −0.997708 0.0676653i \(-0.978445\pi\)
0.997708 0.0676653i \(-0.0215550\pi\)
\(978\) 0 0
\(979\) −8.08484e9 −0.275380
\(980\) 0 0
\(981\) 0 0
\(982\) 3.07682e10i 1.03684i
\(983\) 4.74320e8i 0.0159270i 0.999968 + 0.00796351i \(0.00253489\pi\)
−0.999968 + 0.00796351i \(0.997465\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −6.21824e9 −0.206585
\(987\) 0 0
\(988\) 4.04292e9i 0.133366i
\(989\) −1.04841e10 −0.344622
\(990\) 0 0
\(991\) 1.22197e10 0.398843 0.199421 0.979914i \(-0.436094\pi\)
0.199421 + 0.979914i \(0.436094\pi\)
\(992\) 5.31737e10i 1.72944i
\(993\) 0 0
\(994\) 2.13528e9 0.0689609
\(995\) 0 0
\(996\) 0 0
\(997\) − 3.60690e10i − 1.15266i −0.817217 0.576330i \(-0.804484\pi\)
0.817217 0.576330i \(-0.195516\pi\)
\(998\) − 2.46390e9i − 0.0784631i
\(999\) 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.b.f.199.1 2
3.2 odd 2 75.8.b.c.49.2 2
5.2 odd 4 225.8.a.i.1.1 1
5.3 odd 4 9.8.a.a.1.1 1
5.4 even 2 inner 225.8.b.f.199.2 2
15.2 even 4 75.8.a.a.1.1 1
15.8 even 4 3.8.a.a.1.1 1
15.14 odd 2 75.8.b.c.49.1 2
20.3 even 4 144.8.a.b.1.1 1
35.13 even 4 441.8.a.a.1.1 1
40.3 even 4 576.8.a.x.1.1 1
40.13 odd 4 576.8.a.w.1.1 1
45.13 odd 12 81.8.c.c.55.1 2
45.23 even 12 81.8.c.a.55.1 2
45.38 even 12 81.8.c.a.28.1 2
45.43 odd 12 81.8.c.c.28.1 2
60.23 odd 4 48.8.a.g.1.1 1
105.23 even 12 147.8.e.b.67.1 2
105.38 odd 12 147.8.e.a.79.1 2
105.53 even 12 147.8.e.b.79.1 2
105.68 odd 12 147.8.e.a.67.1 2
105.83 odd 4 147.8.a.b.1.1 1
120.53 even 4 192.8.a.i.1.1 1
120.83 odd 4 192.8.a.a.1.1 1
165.98 odd 4 363.8.a.b.1.1 1
195.38 even 4 507.8.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.8.a.a.1.1 1 15.8 even 4
9.8.a.a.1.1 1 5.3 odd 4
48.8.a.g.1.1 1 60.23 odd 4
75.8.a.a.1.1 1 15.2 even 4
75.8.b.c.49.1 2 15.14 odd 2
75.8.b.c.49.2 2 3.2 odd 2
81.8.c.a.28.1 2 45.38 even 12
81.8.c.a.55.1 2 45.23 even 12
81.8.c.c.28.1 2 45.43 odd 12
81.8.c.c.55.1 2 45.13 odd 12
144.8.a.b.1.1 1 20.3 even 4
147.8.a.b.1.1 1 105.83 odd 4
147.8.e.a.67.1 2 105.68 odd 12
147.8.e.a.79.1 2 105.38 odd 12
147.8.e.b.67.1 2 105.23 even 12
147.8.e.b.79.1 2 105.53 even 12
192.8.a.a.1.1 1 120.83 odd 4
192.8.a.i.1.1 1 120.53 even 4
225.8.a.i.1.1 1 5.2 odd 4
225.8.b.f.199.1 2 1.1 even 1 trivial
225.8.b.f.199.2 2 5.4 even 2 inner
363.8.a.b.1.1 1 165.98 odd 4
441.8.a.a.1.1 1 35.13 even 4
507.8.a.a.1.1 1 195.38 even 4
576.8.a.w.1.1 1 40.13 odd 4
576.8.a.x.1.1 1 40.3 even 4