Properties

Label 252.9.d.d.181.8
Level $252$
Weight $9$
Character 252.181
Analytic conductor $102.659$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,9,Mod(181,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.181");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 252.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(102.659409735\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 14312 x^{8} - 30343 x^{7} + 170123918 x^{6} - 875537263 x^{5} + 496509566533 x^{4} - 17126006720262 x^{3} + \cdots + 39\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{19}\cdot 3^{15}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 181.8
Root \(-32.7935 + 56.7999i\) of defining polynomial
Character \(\chi\) \(=\) 252.181
Dual form 252.9.d.d.181.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+429.816i q^{5} +(-1615.33 + 1776.38i) q^{7} +O(q^{10})\) \(q+429.816i q^{5} +(-1615.33 + 1776.38i) q^{7} -5985.37 q^{11} -10608.4i q^{13} -54416.4i q^{17} -183537. i q^{19} +355836. q^{23} +205883. q^{25} -709300. q^{29} -647270. i q^{31} +(-763517. - 694293. i) q^{35} +3.29917e6 q^{37} +837610. i q^{41} +884577. q^{43} +8.95952e6i q^{47} +(-546251. - 5.73886e6i) q^{49} -9.91683e6 q^{53} -2.57261e6i q^{55} +1.24157e7i q^{59} +2.50177e6i q^{61} +4.55965e6 q^{65} +2.92355e7 q^{67} -1.60661e7 q^{71} -4.50356e6i q^{73} +(9.66831e6 - 1.06323e7i) q^{77} -4.46076e7 q^{79} +1.75410e7i q^{83} +2.33891e7 q^{85} -5.24371e6i q^{89} +(1.88445e7 + 1.71360e7i) q^{91} +7.88873e7 q^{95} +6.69113e7i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2338 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2338 q^{7} + 37596 q^{11} + 22380 q^{23} - 303998 q^{25} + 308892 q^{29} + 1480584 q^{35} - 5471108 q^{37} + 1177324 q^{43} - 6064142 q^{49} + 129132 q^{53} + 106801008 q^{65} - 5722372 q^{67} - 26985540 q^{71} - 48770148 q^{77} - 181197556 q^{79} + 337759224 q^{85} + 67638816 q^{91} - 103302096 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 429.816i 0.687706i 0.939023 + 0.343853i \(0.111732\pi\)
−0.939023 + 0.343853i \(0.888268\pi\)
\(6\) 0 0
\(7\) −1615.33 + 1776.38i −0.672772 + 0.739850i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5985.37 −0.408809 −0.204404 0.978887i \(-0.565526\pi\)
−0.204404 + 0.978887i \(0.565526\pi\)
\(12\) 0 0
\(13\) 10608.4i 0.371429i −0.982604 0.185714i \(-0.940540\pi\)
0.982604 0.185714i \(-0.0594599\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 54416.4i 0.651530i −0.945451 0.325765i \(-0.894378\pi\)
0.945451 0.325765i \(-0.105622\pi\)
\(18\) 0 0
\(19\) 183537.i 1.40835i −0.710028 0.704173i \(-0.751318\pi\)
0.710028 0.704173i \(-0.248682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 355836. 1.27157 0.635783 0.771868i \(-0.280677\pi\)
0.635783 + 0.771868i \(0.280677\pi\)
\(24\) 0 0
\(25\) 205883. 0.527060
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −709300. −1.00285 −0.501427 0.865200i \(-0.667192\pi\)
−0.501427 + 0.865200i \(0.667192\pi\)
\(30\) 0 0
\(31\) 647270.i 0.700872i −0.936587 0.350436i \(-0.886033\pi\)
0.936587 0.350436i \(-0.113967\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −763517. 694293.i −0.508799 0.462669i
\(36\) 0 0
\(37\) 3.29917e6 1.76035 0.880173 0.474653i \(-0.157426\pi\)
0.880173 + 0.474653i \(0.157426\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 837610.i 0.296419i 0.988956 + 0.148210i \(0.0473510\pi\)
−0.988956 + 0.148210i \(0.952649\pi\)
\(42\) 0 0
\(43\) 884577. 0.258739 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.95952e6i 1.83609i 0.396480 + 0.918043i \(0.370231\pi\)
−0.396480 + 0.918043i \(0.629769\pi\)
\(48\) 0 0
\(49\) −546251. 5.73886e6i −0.0947563 0.995500i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.91683e6 −1.25681 −0.628404 0.777887i \(-0.716292\pi\)
−0.628404 + 0.777887i \(0.716292\pi\)
\(54\) 0 0
\(55\) 2.57261e6i 0.281140i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.24157e7i 1.02462i 0.858801 + 0.512310i \(0.171210\pi\)
−0.858801 + 0.512310i \(0.828790\pi\)
\(60\) 0 0
\(61\) 2.50177e6i 0.180687i 0.995911 + 0.0903437i \(0.0287966\pi\)
−0.995911 + 0.0903437i \(0.971203\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.55965e6 0.255434
\(66\) 0 0
\(67\) 2.92355e7 1.45081 0.725407 0.688320i \(-0.241651\pi\)
0.725407 + 0.688320i \(0.241651\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.60661e7 −0.632233 −0.316117 0.948720i \(-0.602379\pi\)
−0.316117 + 0.948720i \(0.602379\pi\)
\(72\) 0 0
\(73\) 4.50356e6i 0.158586i −0.996851 0.0792929i \(-0.974734\pi\)
0.996851 0.0792929i \(-0.0252662\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 9.66831e6 1.06323e7i 0.275035 0.302457i
\(78\) 0 0
\(79\) −4.46076e7 −1.14525 −0.572625 0.819817i \(-0.694075\pi\)
−0.572625 + 0.819817i \(0.694075\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.75410e7i 0.369609i 0.982775 + 0.184805i \(0.0591652\pi\)
−0.982775 + 0.184805i \(0.940835\pi\)
\(84\) 0 0
\(85\) 2.33891e7 0.448061
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.24371e6i 0.0835755i −0.999127 0.0417877i \(-0.986695\pi\)
0.999127 0.0417877i \(-0.0133053\pi\)
\(90\) 0 0
\(91\) 1.88445e7 + 1.71360e7i 0.274801 + 0.249887i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.88873e7 0.968529
\(96\) 0 0
\(97\) 6.69113e7i 0.755810i 0.925844 + 0.377905i \(0.123355\pi\)
−0.925844 + 0.377905i \(0.876645\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.80485e7i 0.269540i 0.990877 + 0.134770i \(0.0430296\pi\)
−0.990877 + 0.134770i \(0.956970\pi\)
\(102\) 0 0
\(103\) 1.64415e7i 0.146081i 0.997329 + 0.0730405i \(0.0232702\pi\)
−0.997329 + 0.0730405i \(0.976730\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.32225e8 1.77163 0.885817 0.464034i \(-0.153598\pi\)
0.885817 + 0.464034i \(0.153598\pi\)
\(108\) 0 0
\(109\) −1.99604e7 −0.141404 −0.0707022 0.997497i \(-0.522524\pi\)
−0.0707022 + 0.997497i \(0.522524\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 7.62616e7 0.467727 0.233863 0.972269i \(-0.424863\pi\)
0.233863 + 0.972269i \(0.424863\pi\)
\(114\) 0 0
\(115\) 1.52944e8i 0.874464i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 9.66642e7 + 8.79002e7i 0.482034 + 0.438331i
\(120\) 0 0
\(121\) −1.78534e8 −0.832876
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2.56389e8i 1.05017i
\(126\) 0 0
\(127\) 5.66167e6 0.0217635 0.0108818 0.999941i \(-0.496536\pi\)
0.0108818 + 0.999941i \(0.496536\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.29714e8i 0.440454i −0.975449 0.220227i \(-0.929320\pi\)
0.975449 0.220227i \(-0.0706798\pi\)
\(132\) 0 0
\(133\) 3.26032e8 + 2.96472e8i 1.04197 + 0.947496i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.30172e8 0.937256 0.468628 0.883396i \(-0.344749\pi\)
0.468628 + 0.883396i \(0.344749\pi\)
\(138\) 0 0
\(139\) 2.04732e8i 0.548436i 0.961668 + 0.274218i \(0.0884190\pi\)
−0.961668 + 0.274218i \(0.911581\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.34950e7i 0.151843i
\(144\) 0 0
\(145\) 3.04869e8i 0.689669i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.62729e8 0.330157 0.165079 0.986280i \(-0.447212\pi\)
0.165079 + 0.986280i \(0.447212\pi\)
\(150\) 0 0
\(151\) −1.71122e8 −0.329152 −0.164576 0.986364i \(-0.552626\pi\)
−0.164576 + 0.986364i \(0.552626\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.78207e8 0.481994
\(156\) 0 0
\(157\) 2.38413e8i 0.392402i 0.980564 + 0.196201i \(0.0628605\pi\)
−0.980564 + 0.196201i \(0.937139\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −5.74791e8 + 6.32100e8i −0.855474 + 0.940768i
\(162\) 0 0
\(163\) 1.05318e9 1.49194 0.745971 0.665979i \(-0.231986\pi\)
0.745971 + 0.665979i \(0.231986\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.05696e9i 1.35891i 0.733715 + 0.679457i \(0.237785\pi\)
−0.733715 + 0.679457i \(0.762215\pi\)
\(168\) 0 0
\(169\) 7.03193e8 0.862041
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.68924e9i 1.88585i 0.333008 + 0.942924i \(0.391936\pi\)
−0.333008 + 0.942924i \(0.608064\pi\)
\(174\) 0 0
\(175\) −3.32568e8 + 3.65726e8i −0.354591 + 0.389946i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 7.98123e8 0.777424 0.388712 0.921359i \(-0.372920\pi\)
0.388712 + 0.921359i \(0.372920\pi\)
\(180\) 0 0
\(181\) 8.63968e8i 0.804977i −0.915425 0.402489i \(-0.868145\pi\)
0.915425 0.402489i \(-0.131855\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.41804e9i 1.21060i
\(186\) 0 0
\(187\) 3.25702e8i 0.266351i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.85383e8 0.665270 0.332635 0.943056i \(-0.392062\pi\)
0.332635 + 0.943056i \(0.392062\pi\)
\(192\) 0 0
\(193\) 1.84525e9 1.32992 0.664960 0.746879i \(-0.268449\pi\)
0.664960 + 0.746879i \(0.268449\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4.23099e8 −0.280916 −0.140458 0.990087i \(-0.544858\pi\)
−0.140458 + 0.990087i \(0.544858\pi\)
\(198\) 0 0
\(199\) 5.11852e8i 0.326387i −0.986594 0.163193i \(-0.947821\pi\)
0.986594 0.163193i \(-0.0521794\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.14575e9 1.25999e9i 0.674692 0.741962i
\(204\) 0 0
\(205\) −3.60018e8 −0.203849
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.09854e9i 0.575744i
\(210\) 0 0
\(211\) 7.28727e8 0.367650 0.183825 0.982959i \(-0.441152\pi\)
0.183825 + 0.982959i \(0.441152\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.80205e8i 0.177936i
\(216\) 0 0
\(217\) 1.14980e9 + 1.04555e9i 0.518540 + 0.471527i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −5.77270e8 −0.241997
\(222\) 0 0
\(223\) 2.96999e9i 1.20098i 0.799632 + 0.600490i \(0.205028\pi\)
−0.799632 + 0.600490i \(0.794972\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.60001e9i 1.73243i −0.499674 0.866213i \(-0.666547\pi\)
0.499674 0.866213i \(-0.333453\pi\)
\(228\) 0 0
\(229\) 3.13006e9i 1.13818i −0.822275 0.569090i \(-0.807296\pi\)
0.822275 0.569090i \(-0.192704\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.12300e9 −0.381026 −0.190513 0.981685i \(-0.561015\pi\)
−0.190513 + 0.981685i \(0.561015\pi\)
\(234\) 0 0
\(235\) −3.85095e9 −1.26269
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.01010e9 1.84200 0.921002 0.389559i \(-0.127373\pi\)
0.921002 + 0.389559i \(0.127373\pi\)
\(240\) 0 0
\(241\) 3.33451e9i 0.988470i 0.869328 + 0.494235i \(0.164552\pi\)
−0.869328 + 0.494235i \(0.835448\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.46666e9 2.34788e8i 0.684612 0.0651645i
\(246\) 0 0
\(247\) −1.94703e9 −0.523100
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6.31335e9i 1.59061i 0.606206 + 0.795307i \(0.292691\pi\)
−0.606206 + 0.795307i \(0.707309\pi\)
\(252\) 0 0
\(253\) −2.12981e9 −0.519827
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.72277e9i 1.31182i −0.754839 0.655910i \(-0.772285\pi\)
0.754839 0.655910i \(-0.227715\pi\)
\(258\) 0 0
\(259\) −5.32924e9 + 5.86058e9i −1.18431 + 1.30239i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −2.52582e9 −0.527935 −0.263967 0.964532i \(-0.585031\pi\)
−0.263967 + 0.964532i \(0.585031\pi\)
\(264\) 0 0
\(265\) 4.26241e9i 0.864315i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.83838e9i 0.351096i −0.984471 0.175548i \(-0.943830\pi\)
0.984471 0.175548i \(-0.0561698\pi\)
\(270\) 0 0
\(271\) 1.32017e9i 0.244768i −0.992483 0.122384i \(-0.960946\pi\)
0.992483 0.122384i \(-0.0390539\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.23228e9 −0.215467
\(276\) 0 0
\(277\) 4.31304e9 0.732595 0.366298 0.930498i \(-0.380625\pi\)
0.366298 + 0.930498i \(0.380625\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6.61995e9 −1.06177 −0.530884 0.847444i \(-0.678140\pi\)
−0.530884 + 0.847444i \(0.678140\pi\)
\(282\) 0 0
\(283\) 4.71632e9i 0.735288i 0.929967 + 0.367644i \(0.119835\pi\)
−0.929967 + 0.367644i \(0.880165\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.48791e9 1.35301e9i −0.219306 0.199422i
\(288\) 0 0
\(289\) 4.01461e9 0.575509
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.03696e8i 0.0819121i 0.999161 + 0.0409561i \(0.0130404\pi\)
−0.999161 + 0.0409561i \(0.986960\pi\)
\(294\) 0 0
\(295\) −5.33646e9 −0.704637
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.77484e9i 0.472296i
\(300\) 0 0
\(301\) −1.42888e9 + 1.57134e9i −0.174072 + 0.191428i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.07530e9 −0.124260
\(306\) 0 0
\(307\) 1.43825e10i 1.61913i −0.587029 0.809566i \(-0.699703\pi\)
0.587029 0.809566i \(-0.300297\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.44388e9i 1.00951i −0.863264 0.504753i \(-0.831584\pi\)
0.863264 0.504753i \(-0.168416\pi\)
\(312\) 0 0
\(313\) 1.60438e10i 1.67159i −0.549039 0.835797i \(-0.685006\pi\)
0.549039 0.835797i \(-0.314994\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4.74781e9 −0.470171 −0.235086 0.971975i \(-0.575537\pi\)
−0.235086 + 0.971975i \(0.575537\pi\)
\(318\) 0 0
\(319\) 4.24542e9 0.409976
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −9.98744e9 −0.917580
\(324\) 0 0
\(325\) 2.18408e9i 0.195765i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1.59155e10 1.44725e10i −1.35843 1.23527i
\(330\) 0 0
\(331\) 9.46168e9 0.788236 0.394118 0.919060i \(-0.371050\pi\)
0.394118 + 0.919060i \(0.371050\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.25659e10i 0.997734i
\(336\) 0 0
\(337\) 1.10076e8 0.00853438 0.00426719 0.999991i \(-0.498642\pi\)
0.00426719 + 0.999991i \(0.498642\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3.87415e9i 0.286522i
\(342\) 0 0
\(343\) 1.10768e10 + 8.29978e9i 0.800271 + 0.599639i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.97560e9 0.136264 0.0681320 0.997676i \(-0.478296\pi\)
0.0681320 + 0.997676i \(0.478296\pi\)
\(348\) 0 0
\(349\) 2.57684e10i 1.73694i −0.495741 0.868470i \(-0.665104\pi\)
0.495741 0.868470i \(-0.334896\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.67150e10i 1.07648i −0.842790 0.538242i \(-0.819088\pi\)
0.842790 0.538242i \(-0.180912\pi\)
\(354\) 0 0
\(355\) 6.90548e9i 0.434791i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2.99902e10 −1.80552 −0.902759 0.430146i \(-0.858462\pi\)
−0.902759 + 0.430146i \(0.858462\pi\)
\(360\) 0 0
\(361\) −1.67023e10 −0.983441
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.93570e9 0.109060
\(366\) 0 0
\(367\) 6.86910e9i 0.378648i 0.981915 + 0.189324i \(0.0606296\pi\)
−0.981915 + 0.189324i \(0.939370\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1.60189e10 1.76161e10i 0.845546 0.929850i
\(372\) 0 0
\(373\) −3.47622e10 −1.79586 −0.897929 0.440140i \(-0.854929\pi\)
−0.897929 + 0.440140i \(0.854929\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7.52452e9i 0.372489i
\(378\) 0 0
\(379\) 3.56196e9 0.172636 0.0863181 0.996268i \(-0.472490\pi\)
0.0863181 + 0.996268i \(0.472490\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2.68874e10i 1.24955i 0.780805 + 0.624774i \(0.214809\pi\)
−0.780805 + 0.624774i \(0.785191\pi\)
\(384\) 0 0
\(385\) 4.56993e9 + 4.15560e9i 0.208002 + 0.189143i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.38727e10 −0.605845 −0.302923 0.953015i \(-0.597962\pi\)
−0.302923 + 0.953015i \(0.597962\pi\)
\(390\) 0 0
\(391\) 1.93633e10i 0.828463i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.91731e10i 0.787595i
\(396\) 0 0
\(397\) 4.21587e10i 1.69717i 0.529061 + 0.848584i \(0.322544\pi\)
−0.529061 + 0.848584i \(0.677456\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.47182e10 0.955960 0.477980 0.878371i \(-0.341369\pi\)
0.477980 + 0.878371i \(0.341369\pi\)
\(402\) 0 0
\(403\) −6.86648e9 −0.260324
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.97468e10 −0.719645
\(408\) 0 0
\(409\) 4.86302e10i 1.73785i 0.494941 + 0.868926i \(0.335190\pi\)
−0.494941 + 0.868926i \(0.664810\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2.20550e10 2.00554e10i −0.758064 0.689335i
\(414\) 0 0
\(415\) −7.53942e9 −0.254182
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.62633e10i 0.527659i 0.964569 + 0.263830i \(0.0849856\pi\)
−0.964569 + 0.263830i \(0.915014\pi\)
\(420\) 0 0
\(421\) −3.58983e9 −0.114274 −0.0571368 0.998366i \(-0.518197\pi\)
−0.0571368 + 0.998366i \(0.518197\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.12034e10i 0.343395i
\(426\) 0 0
\(427\) −4.44409e9 4.04117e9i −0.133682 0.121561i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4.73438e10 1.37200 0.686000 0.727601i \(-0.259365\pi\)
0.686000 + 0.727601i \(0.259365\pi\)
\(432\) 0 0
\(433\) 2.17783e10i 0.619543i −0.950811 0.309772i \(-0.899747\pi\)
0.950811 0.309772i \(-0.100253\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6.53092e10i 1.79081i
\(438\) 0 0
\(439\) 1.08680e10i 0.292611i −0.989239 0.146305i \(-0.953262\pi\)
0.989239 0.146305i \(-0.0467382\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6.16237e10 1.60005 0.800024 0.599968i \(-0.204820\pi\)
0.800024 + 0.599968i \(0.204820\pi\)
\(444\) 0 0
\(445\) 2.25383e9 0.0574754
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 4.33484e10 1.06657 0.533283 0.845937i \(-0.320958\pi\)
0.533283 + 0.845937i \(0.320958\pi\)
\(450\) 0 0
\(451\) 5.01340e9i 0.121179i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −7.36532e9 + 8.09967e9i −0.171849 + 0.188983i
\(456\) 0 0
\(457\) −1.42235e10 −0.326094 −0.163047 0.986618i \(-0.552132\pi\)
−0.163047 + 0.986618i \(0.552132\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.44851e10i 0.763532i 0.924259 + 0.381766i \(0.124684\pi\)
−0.924259 + 0.381766i \(0.875316\pi\)
\(462\) 0 0
\(463\) 4.18685e10 0.911095 0.455548 0.890211i \(-0.349444\pi\)
0.455548 + 0.890211i \(0.349444\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.41461e10i 0.297420i −0.988881 0.148710i \(-0.952488\pi\)
0.988881 0.148710i \(-0.0475120\pi\)
\(468\) 0 0
\(469\) −4.72249e10 + 5.19334e10i −0.976067 + 1.07339i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −5.29451e9 −0.105775
\(474\) 0 0
\(475\) 3.77872e10i 0.742284i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 7.79025e10i 1.47982i 0.672705 + 0.739910i \(0.265132\pi\)
−0.672705 + 0.739910i \(0.734868\pi\)
\(480\) 0 0
\(481\) 3.49988e10i 0.653843i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.87596e10 −0.519775
\(486\) 0 0
\(487\) 8.13869e10 1.44690 0.723450 0.690376i \(-0.242555\pi\)
0.723450 + 0.690376i \(0.242555\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.21353e10 0.208796 0.104398 0.994536i \(-0.466708\pi\)
0.104398 + 0.994536i \(0.466708\pi\)
\(492\) 0 0
\(493\) 3.85976e10i 0.653390i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.59520e10 2.85395e10i 0.425349 0.467758i
\(498\) 0 0
\(499\) 6.31179e10 1.01801 0.509003 0.860765i \(-0.330014\pi\)
0.509003 + 0.860765i \(0.330014\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.26002e11i 1.96836i 0.177160 + 0.984182i \(0.443309\pi\)
−0.177160 + 0.984182i \(0.556691\pi\)
\(504\) 0 0
\(505\) −1.20557e10 −0.185364
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 9.14104e10i 1.36183i 0.732360 + 0.680917i \(0.238419\pi\)
−0.732360 + 0.680917i \(0.761581\pi\)
\(510\) 0 0
\(511\) 8.00003e9 + 7.27471e9i 0.117330 + 0.106692i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −7.06684e9 −0.100461
\(516\) 0 0
\(517\) 5.36260e10i 0.750608i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3.69946e9i 0.0502097i −0.999685 0.0251048i \(-0.992008\pi\)
0.999685 0.0251048i \(-0.00799196\pi\)
\(522\) 0 0
\(523\) 9.10568e10i 1.21704i 0.793538 + 0.608521i \(0.208237\pi\)
−0.793538 + 0.608521i \(0.791763\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3.52221e10 −0.456639
\(528\) 0 0
\(529\) 4.83085e10 0.616880
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.88567e9 0.110099
\(534\) 0 0
\(535\) 9.98142e10i 1.21836i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3.26951e9 + 3.43492e10i 0.0387372 + 0.406969i
\(540\) 0 0
\(541\) 8.28845e9 0.0967575 0.0483787 0.998829i \(-0.484595\pi\)
0.0483787 + 0.998829i \(0.484595\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 8.57930e9i 0.0972447i
\(546\) 0 0
\(547\) 6.46698e10 0.722358 0.361179 0.932497i \(-0.382374\pi\)
0.361179 + 0.932497i \(0.382374\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1.30183e11i 1.41237i
\(552\) 0 0
\(553\) 7.20557e10 7.92400e10i 0.770492 0.847313i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.02550e11 1.06541 0.532705 0.846301i \(-0.321175\pi\)
0.532705 + 0.846301i \(0.321175\pi\)
\(558\) 0 0
\(559\) 9.38392e9i 0.0961030i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.46724e11i 1.46038i 0.683244 + 0.730191i \(0.260569\pi\)
−0.683244 + 0.730191i \(0.739431\pi\)
\(564\) 0 0
\(565\) 3.27785e10i 0.321659i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.15957e11 −1.10623 −0.553116 0.833104i \(-0.686561\pi\)
−0.553116 + 0.833104i \(0.686561\pi\)
\(570\) 0 0
\(571\) −1.40905e11 −1.32551 −0.662753 0.748838i \(-0.730612\pi\)
−0.662753 + 0.748838i \(0.730612\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 7.32606e10 0.670192
\(576\) 0 0
\(577\) 1.35313e11i 1.22078i −0.792103 0.610388i \(-0.791014\pi\)
0.792103 0.610388i \(-0.208986\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −3.11595e10 2.83345e10i −0.273455 0.248663i
\(582\) 0 0
\(583\) 5.93558e10 0.513794
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5.72362e10i 0.482079i −0.970515 0.241039i \(-0.922512\pi\)
0.970515 0.241039i \(-0.0774883\pi\)
\(588\) 0 0
\(589\) −1.18798e11 −0.987071
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.06242e11i 0.859167i −0.903027 0.429584i \(-0.858660\pi\)
0.903027 0.429584i \(-0.141340\pi\)
\(594\) 0 0
\(595\) −3.77809e10 + 4.15479e10i −0.301443 + 0.331498i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1.89779e11 1.47415 0.737073 0.675813i \(-0.236207\pi\)
0.737073 + 0.675813i \(0.236207\pi\)
\(600\) 0 0
\(601\) 9.88010e9i 0.0757292i 0.999283 + 0.0378646i \(0.0120556\pi\)
−0.999283 + 0.0378646i \(0.987944\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 7.67370e10i 0.572774i
\(606\) 0 0
\(607\) 4.22612e10i 0.311306i −0.987812 0.155653i \(-0.950252\pi\)
0.987812 0.155653i \(-0.0497481\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 9.50459e10 0.681975
\(612\) 0 0
\(613\) 1.42522e11 1.00935 0.504675 0.863310i \(-0.331612\pi\)
0.504675 + 0.863310i \(0.331612\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.43687e10 0.0991463 0.0495731 0.998770i \(-0.484214\pi\)
0.0495731 + 0.998770i \(0.484214\pi\)
\(618\) 0 0
\(619\) 1.28847e11i 0.877632i 0.898577 + 0.438816i \(0.144602\pi\)
−0.898577 + 0.438816i \(0.855398\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 9.31483e9 + 8.47030e9i 0.0618333 + 0.0562272i
\(624\) 0 0
\(625\) −2.97771e10 −0.195147
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.79529e11i 1.14692i
\(630\) 0 0
\(631\) 6.12007e10 0.386046 0.193023 0.981194i \(-0.438171\pi\)
0.193023 + 0.981194i \(0.438171\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.43348e9i 0.0149669i
\(636\) 0 0
\(637\) −6.08800e10 + 5.79484e9i −0.369757 + 0.0351952i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −4.27354e10 −0.253137 −0.126569 0.991958i \(-0.540396\pi\)
−0.126569 + 0.991958i \(0.540396\pi\)
\(642\) 0 0
\(643\) 3.10685e11i 1.81751i 0.417332 + 0.908754i \(0.362965\pi\)
−0.417332 + 0.908754i \(0.637035\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.02430e11i 0.584532i −0.956337 0.292266i \(-0.905591\pi\)
0.956337 0.292266i \(-0.0944092\pi\)
\(648\) 0 0
\(649\) 7.43124e10i 0.418873i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −2.30695e11 −1.26878 −0.634389 0.773014i \(-0.718748\pi\)
−0.634389 + 0.773014i \(0.718748\pi\)
\(654\) 0 0
\(655\) 5.57531e10 0.302903
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 2.41191e11 1.27885 0.639425 0.768853i \(-0.279172\pi\)
0.639425 + 0.768853i \(0.279172\pi\)
\(660\) 0 0
\(661\) 1.04104e11i 0.545333i −0.962109 0.272666i \(-0.912094\pi\)
0.962109 0.272666i \(-0.0879055\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.27429e11 + 1.40134e11i −0.651599 + 0.716566i
\(666\) 0 0
\(667\) −2.52395e11 −1.27520
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1.49740e10i 0.0738666i
\(672\) 0 0
\(673\) −1.69473e11 −0.826116 −0.413058 0.910705i \(-0.635539\pi\)
−0.413058 + 0.910705i \(0.635539\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.96573e11i 0.935770i −0.883789 0.467885i \(-0.845016\pi\)
0.883789 0.467885i \(-0.154984\pi\)
\(678\) 0 0
\(679\) −1.18860e11 1.08084e11i −0.559186 0.508488i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 2.25604e11 1.03673 0.518363 0.855161i \(-0.326542\pi\)
0.518363 + 0.855161i \(0.326542\pi\)
\(684\) 0 0
\(685\) 1.41913e11i 0.644557i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.05201e11i 0.466815i
\(690\) 0 0
\(691\) 2.41724e11i 1.06025i 0.847920 + 0.530125i \(0.177855\pi\)
−0.847920 + 0.530125i \(0.822145\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8.79971e10 −0.377163
\(696\) 0 0
\(697\) 4.55797e10 0.193126
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −2.33717e11 −0.967875 −0.483937 0.875103i \(-0.660794\pi\)
−0.483937 + 0.875103i \(0.660794\pi\)
\(702\) 0 0
\(703\) 6.05521e11i 2.47918i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4.98247e10 4.53074e10i −0.199419 0.181339i
\(708\) 0 0
\(709\) 6.42622e9 0.0254314 0.0127157 0.999919i \(-0.495952\pi\)
0.0127157 + 0.999919i \(0.495952\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.30322e11i 0.891205i
\(714\) 0 0
\(715\) −2.72912e10 −0.104423
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 2.71193e11i 1.01476i −0.861722 0.507380i \(-0.830614\pi\)
0.861722 0.507380i \(-0.169386\pi\)
\(720\) 0 0
\(721\) −2.92064e10 2.65584e10i −0.108078 0.0982792i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.46033e11 −0.528565
\(726\) 0 0
\(727\) 3.67678e11i 1.31622i 0.752920 + 0.658111i \(0.228644\pi\)
−0.752920 + 0.658111i \(0.771356\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 4.81355e10i 0.168576i
\(732\) 0 0
\(733\) 5.48520e11i 1.90010i −0.312099 0.950050i \(-0.601032\pi\)
0.312099 0.950050i \(-0.398968\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.74985e11 −0.593106
\(738\) 0 0
\(739\) 2.70662e10 0.0907504 0.0453752 0.998970i \(-0.485552\pi\)
0.0453752 + 0.998970i \(0.485552\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −5.45102e11 −1.78864 −0.894319 0.447429i \(-0.852340\pi\)
−0.894319 + 0.447429i \(0.852340\pi\)
\(744\) 0 0
\(745\) 6.99438e10i 0.227051i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −3.75119e11 + 4.12520e11i −1.19191 + 1.31074i
\(750\) 0 0
\(751\) −2.15475e11 −0.677388 −0.338694 0.940897i \(-0.609985\pi\)
−0.338694 + 0.940897i \(0.609985\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 7.35508e10i 0.226360i
\(756\) 0 0
\(757\) 2.21637e11 0.674930 0.337465 0.941338i \(-0.390431\pi\)
0.337465 + 0.941338i \(0.390431\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.70134e11i 1.10362i −0.833970 0.551810i \(-0.813937\pi\)
0.833970 0.551810i \(-0.186063\pi\)
\(762\) 0 0
\(763\) 3.22425e10 3.54572e10i 0.0951329 0.104618i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.31710e11 0.380573
\(768\) 0 0
\(769\) 2.51630e11i 0.719543i −0.933040 0.359772i \(-0.882855\pi\)
0.933040 0.359772i \(-0.117145\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 8.76043e10i 0.245362i 0.992446 + 0.122681i \(0.0391492\pi\)
−0.992446 + 0.122681i \(0.960851\pi\)
\(774\) 0 0
\(775\) 1.33262e11i 0.369402i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.53733e11 0.417461
\(780\) 0 0
\(781\) 9.61615e10 0.258462
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1.02474e11 −0.269857
\(786\) 0 0
\(787\) 4.92202e11i 1.28305i −0.767101 0.641526i \(-0.778302\pi\)
0.767101 0.641526i \(-0.221698\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.23187e11 + 1.35470e11i −0.314673 + 0.346048i
\(792\) 0 0
\(793\) 2.65397e10 0.0671125
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 7.26288e11i 1.80001i −0.435877 0.900006i \(-0.643562\pi\)
0.435877 0.900006i \(-0.356438\pi\)
\(798\) 0 0
\(799\) 4.87545e11 1.19627
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 2.69554e10i 0.0648312i
\(804\) 0 0
\(805\) −2.71687e11 2.47055e11i −0.646972 0.588315i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −3.13737e11 −0.732439 −0.366219 0.930529i \(-0.619348\pi\)
−0.366219 + 0.930529i \(0.619348\pi\)
\(810\) 0 0
\(811\) 5.18509e11i 1.19860i 0.800526 + 0.599298i \(0.204553\pi\)
−0.800526 + 0.599298i \(0.795447\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 4.52674e11i 1.02602i
\(816\) 0 0
\(817\) 1.62353e11i 0.364394i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −5.87453e11 −1.29300 −0.646502 0.762912i \(-0.723769\pi\)
−0.646502 + 0.762912i \(0.723769\pi\)
\(822\) 0 0
\(823\) 5.98317e11 1.30416 0.652082 0.758149i \(-0.273896\pi\)
0.652082 + 0.758149i \(0.273896\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 5.41942e11 1.15859 0.579296 0.815117i \(-0.303328\pi\)
0.579296 + 0.815117i \(0.303328\pi\)
\(828\) 0 0
\(829\) 3.02398e11i 0.640266i 0.947373 + 0.320133i \(0.103728\pi\)
−0.947373 + 0.320133i \(0.896272\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.12288e11 + 2.97250e10i −0.648598 + 0.0617366i
\(834\) 0 0
\(835\) −4.54298e11 −0.934533
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.67906e11i 0.338859i −0.985542 0.169430i \(-0.945807\pi\)
0.985542 0.169430i \(-0.0541925\pi\)
\(840\) 0 0
\(841\) 2.86008e9 0.00571734
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3.02244e11i 0.592831i
\(846\) 0 0
\(847\) 2.88391e11 3.17145e11i 0.560335 0.616203i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.17397e12 2.23840
\(852\) 0 0
\(853\) 1.12501e10i 0.0212501i −0.999944 0.0106250i \(-0.996618\pi\)
0.999944 0.0106250i \(-0.00338212\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.60609e11i 0.483132i 0.970384 + 0.241566i \(0.0776611\pi\)
−0.970384 + 0.241566i \(0.922339\pi\)
\(858\) 0 0
\(859\) 5.95207e11i 1.09319i 0.837397 + 0.546595i \(0.184076\pi\)
−0.837397 + 0.546595i \(0.815924\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −7.81026e11 −1.40806 −0.704032 0.710168i \(-0.748619\pi\)
−0.704032 + 0.710168i \(0.748619\pi\)
\(864\) 0 0
\(865\) −7.26063e11 −1.29691
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2.66993e11 0.468188
\(870\) 0 0
\(871\) 3.10141e11i 0.538874i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −4.55444e11 4.14151e11i −0.776967 0.706524i
\(876\) 0 0
\(877\) −8.87976e11 −1.50108 −0.750539 0.660827i \(-0.770206\pi\)
−0.750539 + 0.660827i \(0.770206\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 7.50548e11i 1.24588i −0.782271 0.622938i \(-0.785939\pi\)
0.782271 0.622938i \(-0.214061\pi\)
\(882\) 0 0
\(883\) −4.39811e11 −0.723474 −0.361737 0.932280i \(-0.617816\pi\)
−0.361737 + 0.932280i \(0.617816\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.20334e11i 0.355948i 0.984035 + 0.177974i \(0.0569543\pi\)
−0.984035 + 0.177974i \(0.943046\pi\)
\(888\) 0 0
\(889\) −9.14543e9 + 1.00573e10i −0.0146419 + 0.0161017i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.64440e12 2.58585
\(894\) 0 0
\(895\) 3.43046e11i 0.534639i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 4.59109e11i 0.702873i
\(900\) 0 0
\(901\) 5.39638e11i 0.818848i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 3.71348e11 0.553588
\(906\) 0 0
\(907\) 5.62985e11 0.831894 0.415947 0.909389i \(-0.363450\pi\)
0.415947 + 0.909389i \(0.363450\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.03509e12 1.50281 0.751405 0.659841i \(-0.229377\pi\)
0.751405 + 0.659841i \(0.229377\pi\)
\(912\) 0 0
\(913\) 1.04989e11i 0.151099i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.30421e11 + 2.09530e11i 0.325870 + 0.296325i
\(918\) 0 0
\(919\) 8.60972e11 1.20706 0.603528 0.797342i \(-0.293761\pi\)
0.603528 + 0.797342i \(0.293761\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.70435e11i 0.234829i
\(924\) 0 0
\(925\) 6.79243e11 0.927808
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 5.60419e11i 0.752403i −0.926538 0.376201i \(-0.877230\pi\)
0.926538 0.376201i \(-0.122770\pi\)
\(930\) 0 0
\(931\) −1.05329e12 + 1.00257e11i −1.40201 + 0.133450i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.39992e11 −0.183171
\(936\) 0 0
\(937\) 1.18541e12i 1.53783i 0.639349 + 0.768916i \(0.279204\pi\)
−0.639349 + 0.768916i \(0.720796\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.11841e12i 1.42641i −0.700956 0.713204i \(-0.747243\pi\)
0.700956 0.713204i \(-0.252757\pi\)
\(942\) 0 0
\(943\) 2.98052e11i 0.376916i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −8.43072e11 −1.04825 −0.524125 0.851641i \(-0.675607\pi\)
−0.524125 + 0.851641i \(0.675607\pi\)
\(948\) 0 0
\(949\) −4.77754e10 −0.0589033
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.08227e11 −0.131209 −0.0656047 0.997846i \(-0.520898\pi\)
−0.0656047 + 0.997846i \(0.520898\pi\)
\(954\) 0 0
\(955\) 3.80552e11i 0.457510i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −5.33335e11 + 5.86511e11i −0.630559 + 0.693429i
\(960\) 0 0
\(961\) 4.33933e11 0.508779
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 7.93118e11i 0.914594i
\(966\) 0 0
\(967\) −1.22545e12 −1.40149 −0.700746 0.713410i \(-0.747150\pi\)
−0.700746 + 0.713410i \(0.747150\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 8.37376e10i 0.0941984i 0.998890 + 0.0470992i \(0.0149977\pi\)
−0.998890 + 0.0470992i \(0.985002\pi\)
\(972\) 0 0
\(973\) −3.63681e11 3.30708e11i −0.405761 0.368972i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.20619e12 1.32385 0.661923 0.749572i \(-0.269740\pi\)
0.661923 + 0.749572i \(0.269740\pi\)
\(978\) 0 0
\(979\) 3.13855e10i 0.0341664i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1.48704e12i 1.59260i 0.604900 + 0.796301i \(0.293213\pi\)
−0.604900 + 0.796301i \(0.706787\pi\)
\(984\) 0 0
\(985\) 1.81855e11i 0.193188i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3.14764e11 0.329003
\(990\) 0 0
\(991\) 1.38578e12 1.43682 0.718408 0.695622i \(-0.244871\pi\)
0.718408 + 0.695622i \(0.244871\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2.20002e11 0.224458
\(996\) 0 0
\(997\) 7.07167e11i 0.715717i 0.933776 + 0.357859i \(0.116493\pi\)
−0.933776 + 0.357859i \(0.883507\pi\)
\(998\) 0 0
\(999\) 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 252.9.d.d.181.8 10
3.2 odd 2 84.9.d.a.13.7 yes 10
7.6 odd 2 inner 252.9.d.d.181.3 10
12.11 even 2 336.9.f.a.97.2 10
21.20 even 2 84.9.d.a.13.4 10
84.83 odd 2 336.9.f.a.97.9 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.9.d.a.13.4 10 21.20 even 2
84.9.d.a.13.7 yes 10 3.2 odd 2
252.9.d.d.181.3 10 7.6 odd 2 inner
252.9.d.d.181.8 10 1.1 even 1 trivial
336.9.f.a.97.2 10 12.11 even 2
336.9.f.a.97.9 10 84.83 odd 2