Properties

Label 400.10.c.q.49.4
Level $400$
Weight $10$
Character 400.49
Analytic conductor $206.014$
Analytic rank $0$
Dimension $6$
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] = [400,10,Mod(49,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.49");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 400.c (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: \(206.014334466\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 1305x^{4} + 433104x^{2} + 16000000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 5^{8} \)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.4
Root \(6.48955i\) of defining polynomial
Character \(\chi\) \(=\) 400.49
Dual form 400.10.c.q.49.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+30.5073i q^{3} +4010.25i q^{7} +18752.3 q^{9} +O(q^{10})\) \(q+30.5073i q^{3} +4010.25i q^{7} +18752.3 q^{9} +42110.0 q^{11} -123743. i q^{13} +319945. i q^{17} +1.08733e6 q^{19} -122342. q^{21} +1.50672e6i q^{23} +1.17256e6i q^{27} +2.62160e6 q^{29} -3.27023e6 q^{31} +1.28466e6i q^{33} -2.51034e6i q^{37} +3.77508e6 q^{39} +2.95349e7 q^{41} -1.42413e7i q^{43} -1.35318e6i q^{47} +2.42715e7 q^{49} -9.76067e6 q^{51} -9.73342e7i q^{53} +3.31714e7i q^{57} -7.48924e6 q^{59} -9.11752e7 q^{61} +7.52015e7i q^{63} -2.94376e8i q^{67} -4.59659e7 q^{69} -1.56193e8 q^{71} +2.82539e8i q^{73} +1.68872e8i q^{77} -5.55294e8 q^{79} +3.33330e8 q^{81} +6.48378e6i q^{83} +7.99778e7i q^{87} +5.99001e8 q^{89} +4.96242e8 q^{91} -9.97660e7i q^{93} +9.25317e8i q^{97} +7.89660e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 116468 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 116468 q^{9} + 109398 q^{11} + 1637690 q^{19} - 4750788 q^{21} - 4350960 q^{29} - 8548132 q^{31} - 100828184 q^{39} + 11852622 q^{41} + 12907858 q^{49} + 51269398 q^{51} + 11341920 q^{59} + 250613852 q^{61} + 628549884 q^{69} - 595101192 q^{71} - 620050340 q^{79} + 2797694726 q^{81} + 2207720070 q^{89} - 2366375312 q^{91} - 1784469044 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 30.5073i 0.217449i 0.994072 + 0.108725i \(0.0346767\pi\)
−0.994072 + 0.108725i \(0.965323\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4010.25i 0.631292i 0.948877 + 0.315646i \(0.102221\pi\)
−0.948877 + 0.315646i \(0.897779\pi\)
\(8\) 0 0
\(9\) 18752.3 0.952716
\(10\) 0 0
\(11\) 42110.0 0.867198 0.433599 0.901106i \(-0.357243\pi\)
0.433599 + 0.901106i \(0.357243\pi\)
\(12\) 0 0
\(13\) − 123743.i − 1.20165i −0.799382 0.600824i \(-0.794839\pi\)
0.799382 0.600824i \(-0.205161\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 319945.i 0.929085i 0.885551 + 0.464543i \(0.153781\pi\)
−0.885551 + 0.464543i \(0.846219\pi\)
\(18\) 0 0
\(19\) 1.08733e6 1.91412 0.957059 0.289893i \(-0.0936197\pi\)
0.957059 + 0.289893i \(0.0936197\pi\)
\(20\) 0 0
\(21\) −122342. −0.137274
\(22\) 0 0
\(23\) 1.50672e6i 1.12268i 0.827585 + 0.561341i \(0.189714\pi\)
−0.827585 + 0.561341i \(0.810286\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.17256e6i 0.424617i
\(28\) 0 0
\(29\) 2.62160e6 0.688295 0.344148 0.938916i \(-0.388168\pi\)
0.344148 + 0.938916i \(0.388168\pi\)
\(30\) 0 0
\(31\) −3.27023e6 −0.635991 −0.317996 0.948092i \(-0.603010\pi\)
−0.317996 + 0.948092i \(0.603010\pi\)
\(32\) 0 0
\(33\) 1.28466e6i 0.188572i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.51034e6i − 0.220204i −0.993920 0.110102i \(-0.964882\pi\)
0.993920 0.110102i \(-0.0351177\pi\)
\(38\) 0 0
\(39\) 3.77508e6 0.261297
\(40\) 0 0
\(41\) 2.95349e7 1.63233 0.816165 0.577819i \(-0.196096\pi\)
0.816165 + 0.577819i \(0.196096\pi\)
\(42\) 0 0
\(43\) − 1.42413e7i − 0.635244i −0.948217 0.317622i \(-0.897116\pi\)
0.948217 0.317622i \(-0.102884\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 1.35318e6i − 0.0404496i −0.999795 0.0202248i \(-0.993562\pi\)
0.999795 0.0202248i \(-0.00643820\pi\)
\(48\) 0 0
\(49\) 2.42715e7 0.601470
\(50\) 0 0
\(51\) −9.76067e6 −0.202029
\(52\) 0 0
\(53\) − 9.73342e7i − 1.69443i −0.531249 0.847216i \(-0.678277\pi\)
0.531249 0.847216i \(-0.321723\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.31714e7i 0.416224i
\(58\) 0 0
\(59\) −7.48924e6 −0.0804644 −0.0402322 0.999190i \(-0.512810\pi\)
−0.0402322 + 0.999190i \(0.512810\pi\)
\(60\) 0 0
\(61\) −9.11752e7 −0.843126 −0.421563 0.906799i \(-0.638518\pi\)
−0.421563 + 0.906799i \(0.638518\pi\)
\(62\) 0 0
\(63\) 7.52015e7i 0.601442i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 2.94376e8i − 1.78470i −0.451344 0.892350i \(-0.649055\pi\)
0.451344 0.892350i \(-0.350945\pi\)
\(68\) 0 0
\(69\) −4.59659e7 −0.244127
\(70\) 0 0
\(71\) −1.56193e8 −0.729455 −0.364728 0.931114i \(-0.618838\pi\)
−0.364728 + 0.931114i \(0.618838\pi\)
\(72\) 0 0
\(73\) 2.82539e8i 1.16446i 0.813023 + 0.582232i \(0.197820\pi\)
−0.813023 + 0.582232i \(0.802180\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.68872e8i 0.547455i
\(78\) 0 0
\(79\) −5.55294e8 −1.60399 −0.801994 0.597332i \(-0.796228\pi\)
−0.801994 + 0.597332i \(0.796228\pi\)
\(80\) 0 0
\(81\) 3.33330e8 0.860383
\(82\) 0 0
\(83\) 6.48378e6i 0.0149960i 0.999972 + 0.00749802i \(0.00238672\pi\)
−0.999972 + 0.00749802i \(0.997613\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 7.99778e7i 0.149669i
\(88\) 0 0
\(89\) 5.99001e8 1.01198 0.505990 0.862539i \(-0.331127\pi\)
0.505990 + 0.862539i \(0.331127\pi\)
\(90\) 0 0
\(91\) 4.96242e8 0.758591
\(92\) 0 0
\(93\) − 9.97660e7i − 0.138296i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.25317e8i 1.06125i 0.847606 + 0.530625i \(0.178043\pi\)
−0.847606 + 0.530625i \(0.821957\pi\)
\(98\) 0 0
\(99\) 7.89660e8 0.826193
\(100\) 0 0
\(101\) 9.58959e8 0.916967 0.458483 0.888703i \(-0.348393\pi\)
0.458483 + 0.888703i \(0.348393\pi\)
\(102\) 0 0
\(103\) 1.60441e8i 0.140458i 0.997531 + 0.0702292i \(0.0223731\pi\)
−0.997531 + 0.0702292i \(0.977627\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 9.60457e8i − 0.708355i −0.935178 0.354178i \(-0.884761\pi\)
0.935178 0.354178i \(-0.115239\pi\)
\(108\) 0 0
\(109\) −9.98912e8 −0.677810 −0.338905 0.940821i \(-0.610057\pi\)
−0.338905 + 0.940821i \(0.610057\pi\)
\(110\) 0 0
\(111\) 7.65836e7 0.0478831
\(112\) 0 0
\(113\) − 2.50705e9i − 1.44647i −0.690601 0.723236i \(-0.742654\pi\)
0.690601 0.723236i \(-0.257346\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 2.32047e9i − 1.14483i
\(118\) 0 0
\(119\) −1.28306e9 −0.586524
\(120\) 0 0
\(121\) −5.84695e8 −0.247968
\(122\) 0 0
\(123\) 9.01030e8i 0.354949i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2.47541e9i 0.844364i 0.906511 + 0.422182i \(0.138736\pi\)
−0.906511 + 0.422182i \(0.861264\pi\)
\(128\) 0 0
\(129\) 4.34463e8 0.138134
\(130\) 0 0
\(131\) −1.92402e9 −0.570808 −0.285404 0.958407i \(-0.592128\pi\)
−0.285404 + 0.958407i \(0.592128\pi\)
\(132\) 0 0
\(133\) 4.36045e9i 1.20837i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.48594e8i 0.108796i 0.998519 + 0.0543978i \(0.0173239\pi\)
−0.998519 + 0.0543978i \(0.982676\pi\)
\(138\) 0 0
\(139\) 4.48415e9 1.01886 0.509429 0.860513i \(-0.329857\pi\)
0.509429 + 0.860513i \(0.329857\pi\)
\(140\) 0 0
\(141\) 4.12818e7 0.00879575
\(142\) 0 0
\(143\) − 5.21084e9i − 1.04207i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 7.40458e8i 0.130789i
\(148\) 0 0
\(149\) 2.20480e9 0.366463 0.183232 0.983070i \(-0.441344\pi\)
0.183232 + 0.983070i \(0.441344\pi\)
\(150\) 0 0
\(151\) 3.21248e9 0.502857 0.251428 0.967876i \(-0.419100\pi\)
0.251428 + 0.967876i \(0.419100\pi\)
\(152\) 0 0
\(153\) 5.99971e9i 0.885154i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 1.08870e10i − 1.43007i −0.699088 0.715036i \(-0.746410\pi\)
0.699088 0.715036i \(-0.253590\pi\)
\(158\) 0 0
\(159\) 2.96940e9 0.368453
\(160\) 0 0
\(161\) −6.04232e9 −0.708740
\(162\) 0 0
\(163\) 1.19994e10i 1.33142i 0.746212 + 0.665708i \(0.231870\pi\)
−0.746212 + 0.665708i \(0.768130\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 9.68608e9i − 0.963660i −0.876265 0.481830i \(-0.839972\pi\)
0.876265 0.481830i \(-0.160028\pi\)
\(168\) 0 0
\(169\) −4.70793e9 −0.443956
\(170\) 0 0
\(171\) 2.03899e10 1.82361
\(172\) 0 0
\(173\) − 7.35665e9i − 0.624414i −0.950014 0.312207i \(-0.898932\pi\)
0.950014 0.312207i \(-0.101068\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 2.28477e8i − 0.0174969i
\(178\) 0 0
\(179\) −2.00351e9 −0.145866 −0.0729329 0.997337i \(-0.523236\pi\)
−0.0729329 + 0.997337i \(0.523236\pi\)
\(180\) 0 0
\(181\) 5.63414e9 0.390188 0.195094 0.980785i \(-0.437499\pi\)
0.195094 + 0.980785i \(0.437499\pi\)
\(182\) 0 0
\(183\) − 2.78151e9i − 0.183337i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.34729e10i 0.805701i
\(188\) 0 0
\(189\) −4.70225e9 −0.268057
\(190\) 0 0
\(191\) −9.16925e9 −0.498521 −0.249261 0.968436i \(-0.580188\pi\)
−0.249261 + 0.968436i \(0.580188\pi\)
\(192\) 0 0
\(193\) 3.16327e10i 1.64107i 0.571594 + 0.820536i \(0.306325\pi\)
−0.571594 + 0.820536i \(0.693675\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.59858e10i 1.22924i 0.788822 + 0.614621i \(0.210691\pi\)
−0.788822 + 0.614621i \(0.789309\pi\)
\(198\) 0 0
\(199\) 1.05766e10 0.478088 0.239044 0.971009i \(-0.423166\pi\)
0.239044 + 0.971009i \(0.423166\pi\)
\(200\) 0 0
\(201\) 8.98061e9 0.388082
\(202\) 0 0
\(203\) 1.05133e10i 0.434515i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.82544e10i 1.06960i
\(208\) 0 0
\(209\) 4.57873e10 1.65992
\(210\) 0 0
\(211\) −1.80228e10 −0.625965 −0.312983 0.949759i \(-0.601328\pi\)
−0.312983 + 0.949759i \(0.601328\pi\)
\(212\) 0 0
\(213\) − 4.76502e9i − 0.158620i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 1.31145e10i − 0.401496i
\(218\) 0 0
\(219\) −8.61951e9 −0.253212
\(220\) 0 0
\(221\) 3.95911e10 1.11643
\(222\) 0 0
\(223\) 4.44522e10i 1.20371i 0.798606 + 0.601855i \(0.205571\pi\)
−0.798606 + 0.601855i \(0.794429\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 5.59677e10i − 1.39901i −0.714627 0.699505i \(-0.753404\pi\)
0.714627 0.699505i \(-0.246596\pi\)
\(228\) 0 0
\(229\) 1.47705e9 0.0354923 0.0177462 0.999843i \(-0.494351\pi\)
0.0177462 + 0.999843i \(0.494351\pi\)
\(230\) 0 0
\(231\) −5.15182e9 −0.119044
\(232\) 0 0
\(233\) 7.83279e9i 0.174106i 0.996204 + 0.0870532i \(0.0277450\pi\)
−0.996204 + 0.0870532i \(0.972255\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 1.69405e10i − 0.348786i
\(238\) 0 0
\(239\) 5.56371e10 1.10300 0.551498 0.834176i \(-0.314057\pi\)
0.551498 + 0.834176i \(0.314057\pi\)
\(240\) 0 0
\(241\) −1.16053e10 −0.221606 −0.110803 0.993842i \(-0.535342\pi\)
−0.110803 + 0.993842i \(0.535342\pi\)
\(242\) 0 0
\(243\) 3.32485e10i 0.611707i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 1.34549e11i − 2.30009i
\(248\) 0 0
\(249\) −1.97803e8 −0.00326088
\(250\) 0 0
\(251\) 3.45974e10 0.550189 0.275094 0.961417i \(-0.411291\pi\)
0.275094 + 0.961417i \(0.411291\pi\)
\(252\) 0 0
\(253\) 6.34479e10i 0.973587i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 3.67735e10i − 0.525818i −0.964821 0.262909i \(-0.915318\pi\)
0.964821 0.262909i \(-0.0846820\pi\)
\(258\) 0 0
\(259\) 1.00671e10 0.139013
\(260\) 0 0
\(261\) 4.91609e10 0.655750
\(262\) 0 0
\(263\) 1.33758e11i 1.72392i 0.506974 + 0.861962i \(0.330764\pi\)
−0.506974 + 0.861962i \(0.669236\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 1.82739e10i 0.220055i
\(268\) 0 0
\(269\) 1.42461e11 1.65886 0.829429 0.558612i \(-0.188666\pi\)
0.829429 + 0.558612i \(0.188666\pi\)
\(270\) 0 0
\(271\) 1.11046e11 1.25067 0.625333 0.780358i \(-0.284963\pi\)
0.625333 + 0.780358i \(0.284963\pi\)
\(272\) 0 0
\(273\) 1.51390e10i 0.164955i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.80726e10i 0.286500i 0.989687 + 0.143250i \(0.0457552\pi\)
−0.989687 + 0.143250i \(0.954245\pi\)
\(278\) 0 0
\(279\) −6.13244e10 −0.605919
\(280\) 0 0
\(281\) 5.47143e10 0.523507 0.261753 0.965135i \(-0.415699\pi\)
0.261753 + 0.965135i \(0.415699\pi\)
\(282\) 0 0
\(283\) − 1.09950e11i − 1.01895i −0.860484 0.509477i \(-0.829839\pi\)
0.860484 0.509477i \(-0.170161\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.18442e11i 1.03048i
\(288\) 0 0
\(289\) 1.62229e10 0.136801
\(290\) 0 0
\(291\) −2.82289e10 −0.230768
\(292\) 0 0
\(293\) 2.30453e11i 1.82675i 0.407124 + 0.913373i \(0.366532\pi\)
−0.407124 + 0.913373i \(0.633468\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.93764e10i 0.368227i
\(298\) 0 0
\(299\) 1.86446e11 1.34907
\(300\) 0 0
\(301\) 5.71111e10 0.401025
\(302\) 0 0
\(303\) 2.92552e10i 0.199394i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 6.85036e10i − 0.440140i −0.975484 0.220070i \(-0.929371\pi\)
0.975484 0.220070i \(-0.0706286\pi\)
\(308\) 0 0
\(309\) −4.89462e9 −0.0305426
\(310\) 0 0
\(311\) 1.98797e11 1.20500 0.602502 0.798118i \(-0.294171\pi\)
0.602502 + 0.798118i \(0.294171\pi\)
\(312\) 0 0
\(313\) 6.15202e10i 0.362300i 0.983455 + 0.181150i \(0.0579820\pi\)
−0.983455 + 0.181150i \(0.942018\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 2.25932e11i − 1.25664i −0.777955 0.628320i \(-0.783743\pi\)
0.777955 0.628320i \(-0.216257\pi\)
\(318\) 0 0
\(319\) 1.10395e11 0.596888
\(320\) 0 0
\(321\) 2.93010e10 0.154031
\(322\) 0 0
\(323\) 3.47885e11i 1.77838i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 3.04741e10i − 0.147389i
\(328\) 0 0
\(329\) 5.42658e9 0.0255355
\(330\) 0 0
\(331\) 8.38825e10 0.384101 0.192050 0.981385i \(-0.438486\pi\)
0.192050 + 0.981385i \(0.438486\pi\)
\(332\) 0 0
\(333\) − 4.70746e10i − 0.209791i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 2.19457e11i − 0.926862i −0.886133 0.463431i \(-0.846618\pi\)
0.886133 0.463431i \(-0.153382\pi\)
\(338\) 0 0
\(339\) 7.64834e10 0.314535
\(340\) 0 0
\(341\) −1.37710e11 −0.551530
\(342\) 0 0
\(343\) 2.59163e11i 1.01100i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.85960e11i 0.688551i 0.938869 + 0.344276i \(0.111875\pi\)
−0.938869 + 0.344276i \(0.888125\pi\)
\(348\) 0 0
\(349\) −2.73237e11 −0.985881 −0.492940 0.870063i \(-0.664078\pi\)
−0.492940 + 0.870063i \(0.664078\pi\)
\(350\) 0 0
\(351\) 1.45096e11 0.510240
\(352\) 0 0
\(353\) 3.02861e11i 1.03814i 0.854731 + 0.519072i \(0.173722\pi\)
−0.854731 + 0.519072i \(0.826278\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 3.91427e10i − 0.127539i
\(358\) 0 0
\(359\) 2.47660e11 0.786919 0.393460 0.919342i \(-0.371278\pi\)
0.393460 + 0.919342i \(0.371278\pi\)
\(360\) 0 0
\(361\) 8.59591e11 2.66385
\(362\) 0 0
\(363\) − 1.78375e10i − 0.0539205i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 2.66354e11i − 0.766412i −0.923663 0.383206i \(-0.874820\pi\)
0.923663 0.383206i \(-0.125180\pi\)
\(368\) 0 0
\(369\) 5.53847e11 1.55515
\(370\) 0 0
\(371\) 3.90335e11 1.06968
\(372\) 0 0
\(373\) 5.20850e11i 1.39323i 0.717445 + 0.696616i \(0.245312\pi\)
−0.717445 + 0.696616i \(0.754688\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 3.24405e11i − 0.827088i
\(378\) 0 0
\(379\) 3.28308e10 0.0817344 0.0408672 0.999165i \(-0.486988\pi\)
0.0408672 + 0.999165i \(0.486988\pi\)
\(380\) 0 0
\(381\) −7.55180e10 −0.183607
\(382\) 0 0
\(383\) − 7.15293e10i − 0.169859i −0.996387 0.0849297i \(-0.972933\pi\)
0.996387 0.0849297i \(-0.0270666\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 2.67057e11i − 0.605207i
\(388\) 0 0
\(389\) 2.02681e11 0.448787 0.224394 0.974499i \(-0.427960\pi\)
0.224394 + 0.974499i \(0.427960\pi\)
\(390\) 0 0
\(391\) −4.82067e11 −1.04307
\(392\) 0 0
\(393\) − 5.86968e10i − 0.124122i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1.63266e10i 0.0329867i 0.999864 + 0.0164934i \(0.00525024\pi\)
−0.999864 + 0.0164934i \(0.994750\pi\)
\(398\) 0 0
\(399\) −1.33026e11 −0.262759
\(400\) 0 0
\(401\) −8.20766e11 −1.58515 −0.792574 0.609776i \(-0.791259\pi\)
−0.792574 + 0.609776i \(0.791259\pi\)
\(402\) 0 0
\(403\) 4.04670e11i 0.764237i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 1.05710e11i − 0.190960i
\(408\) 0 0
\(409\) 4.24628e11 0.750332 0.375166 0.926958i \(-0.377586\pi\)
0.375166 + 0.926958i \(0.377586\pi\)
\(410\) 0 0
\(411\) −1.36854e10 −0.0236575
\(412\) 0 0
\(413\) − 3.00337e10i − 0.0507966i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.36799e11i 0.221550i
\(418\) 0 0
\(419\) −1.26375e10 −0.0200308 −0.0100154 0.999950i \(-0.503188\pi\)
−0.0100154 + 0.999950i \(0.503188\pi\)
\(420\) 0 0
\(421\) −3.04545e10 −0.0472479 −0.0236240 0.999721i \(-0.507520\pi\)
−0.0236240 + 0.999721i \(0.507520\pi\)
\(422\) 0 0
\(423\) − 2.53752e10i − 0.0385370i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 3.65635e11i − 0.532259i
\(428\) 0 0
\(429\) 1.58969e11 0.226597
\(430\) 0 0
\(431\) −4.05830e11 −0.566495 −0.283248 0.959047i \(-0.591412\pi\)
−0.283248 + 0.959047i \(0.591412\pi\)
\(432\) 0 0
\(433\) − 1.36978e11i − 0.187265i −0.995607 0.0936324i \(-0.970152\pi\)
0.995607 0.0936324i \(-0.0298478\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.63829e12i 2.14895i
\(438\) 0 0
\(439\) 7.84981e11 1.00872 0.504358 0.863495i \(-0.331729\pi\)
0.504358 + 0.863495i \(0.331729\pi\)
\(440\) 0 0
\(441\) 4.55146e11 0.573030
\(442\) 0 0
\(443\) 8.87799e11i 1.09521i 0.836737 + 0.547605i \(0.184460\pi\)
−0.836737 + 0.547605i \(0.815540\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 6.72624e10i 0.0796872i
\(448\) 0 0
\(449\) −8.35477e11 −0.970122 −0.485061 0.874480i \(-0.661203\pi\)
−0.485061 + 0.874480i \(0.661203\pi\)
\(450\) 0 0
\(451\) 1.24371e12 1.41555
\(452\) 0 0
\(453\) 9.80041e10i 0.109346i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 6.01172e11i 0.644727i 0.946616 + 0.322364i \(0.104477\pi\)
−0.946616 + 0.322364i \(0.895523\pi\)
\(458\) 0 0
\(459\) −3.75154e11 −0.394505
\(460\) 0 0
\(461\) 1.52807e12 1.57576 0.787879 0.615829i \(-0.211179\pi\)
0.787879 + 0.615829i \(0.211179\pi\)
\(462\) 0 0
\(463\) − 7.80402e11i − 0.789231i −0.918846 0.394615i \(-0.870878\pi\)
0.918846 0.394615i \(-0.129122\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.04751e11i 0.393788i 0.980425 + 0.196894i \(0.0630855\pi\)
−0.980425 + 0.196894i \(0.936915\pi\)
\(468\) 0 0
\(469\) 1.18052e12 1.12667
\(470\) 0 0
\(471\) 3.32132e11 0.310968
\(472\) 0 0
\(473\) − 5.99700e11i − 0.550883i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 1.82524e12i − 1.61431i
\(478\) 0 0
\(479\) −2.06972e12 −1.79640 −0.898199 0.439588i \(-0.855124\pi\)
−0.898199 + 0.439588i \(0.855124\pi\)
\(480\) 0 0
\(481\) −3.10638e11 −0.264607
\(482\) 0 0
\(483\) − 1.84335e11i − 0.154115i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 2.41184e11i 0.194298i 0.995270 + 0.0971490i \(0.0309723\pi\)
−0.995270 + 0.0971490i \(0.969028\pi\)
\(488\) 0 0
\(489\) −3.66068e11 −0.289516
\(490\) 0 0
\(491\) −2.27883e12 −1.76948 −0.884739 0.466088i \(-0.845663\pi\)
−0.884739 + 0.466088i \(0.845663\pi\)
\(492\) 0 0
\(493\) 8.38767e11i 0.639485i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 6.26373e11i − 0.460499i
\(498\) 0 0
\(499\) 9.88752e11 0.713896 0.356948 0.934124i \(-0.383817\pi\)
0.356948 + 0.934124i \(0.383817\pi\)
\(500\) 0 0
\(501\) 2.95496e11 0.209547
\(502\) 0 0
\(503\) 1.22385e12i 0.852455i 0.904616 + 0.426228i \(0.140158\pi\)
−0.904616 + 0.426228i \(0.859842\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 1.43626e11i − 0.0965380i
\(508\) 0 0
\(509\) −1.58447e12 −1.04629 −0.523146 0.852243i \(-0.675242\pi\)
−0.523146 + 0.852243i \(0.675242\pi\)
\(510\) 0 0
\(511\) −1.13305e12 −0.735117
\(512\) 0 0
\(513\) 1.27495e12i 0.812767i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 5.69823e10i − 0.0350778i
\(518\) 0 0
\(519\) 2.24432e11 0.135778
\(520\) 0 0
\(521\) −2.71561e12 −1.61472 −0.807362 0.590057i \(-0.799105\pi\)
−0.807362 + 0.590057i \(0.799105\pi\)
\(522\) 0 0
\(523\) − 2.16171e12i − 1.26340i −0.775214 0.631698i \(-0.782358\pi\)
0.775214 0.631698i \(-0.217642\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 1.04630e12i − 0.590890i
\(528\) 0 0
\(529\) −4.69046e11 −0.260415
\(530\) 0 0
\(531\) −1.40441e11 −0.0766597
\(532\) 0 0
\(533\) − 3.65475e12i − 1.96149i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 6.11217e10i − 0.0317184i
\(538\) 0 0
\(539\) 1.02207e12 0.521594
\(540\) 0 0
\(541\) −2.29090e12 −1.14979 −0.574895 0.818227i \(-0.694957\pi\)
−0.574895 + 0.818227i \(0.694957\pi\)
\(542\) 0 0
\(543\) 1.71882e11i 0.0848462i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3.39447e12i 1.62117i 0.585620 + 0.810586i \(0.300851\pi\)
−0.585620 + 0.810586i \(0.699149\pi\)
\(548\) 0 0
\(549\) −1.70974e12 −0.803259
\(550\) 0 0
\(551\) 2.85053e12 1.31748
\(552\) 0 0
\(553\) − 2.22687e12i − 1.01259i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.74706e12i 0.769060i 0.923113 + 0.384530i \(0.125636\pi\)
−0.923113 + 0.384530i \(0.874364\pi\)
\(558\) 0 0
\(559\) −1.76226e12 −0.763340
\(560\) 0 0
\(561\) −4.11022e11 −0.175199
\(562\) 0 0
\(563\) − 2.58864e12i − 1.08588i −0.839770 0.542942i \(-0.817310\pi\)
0.839770 0.542942i \(-0.182690\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.33674e12i 0.543153i
\(568\) 0 0
\(569\) 1.99294e12 0.797055 0.398527 0.917156i \(-0.369521\pi\)
0.398527 + 0.917156i \(0.369521\pi\)
\(570\) 0 0
\(571\) −3.50761e10 −0.0138086 −0.00690428 0.999976i \(-0.502198\pi\)
−0.00690428 + 0.999976i \(0.502198\pi\)
\(572\) 0 0
\(573\) − 2.79729e11i − 0.108403i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 7.75900e11i − 0.291417i −0.989328 0.145708i \(-0.953454\pi\)
0.989328 0.145708i \(-0.0465461\pi\)
\(578\) 0 0
\(579\) −9.65027e11 −0.356850
\(580\) 0 0
\(581\) −2.60016e10 −0.00946689
\(582\) 0 0
\(583\) − 4.09874e12i − 1.46941i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 1.24477e12i − 0.432730i −0.976313 0.216365i \(-0.930580\pi\)
0.976313 0.216365i \(-0.0694201\pi\)
\(588\) 0 0
\(589\) −3.55581e12 −1.21736
\(590\) 0 0
\(591\) −7.92756e11 −0.267298
\(592\) 0 0
\(593\) 2.57794e11i 0.0856103i 0.999083 + 0.0428052i \(0.0136295\pi\)
−0.999083 + 0.0428052i \(0.986371\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 3.22664e11i 0.103960i
\(598\) 0 0
\(599\) 2.32089e12 0.736604 0.368302 0.929706i \(-0.379939\pi\)
0.368302 + 0.929706i \(0.379939\pi\)
\(600\) 0 0
\(601\) 1.78665e12 0.558605 0.279302 0.960203i \(-0.409897\pi\)
0.279302 + 0.960203i \(0.409897\pi\)
\(602\) 0 0
\(603\) − 5.52022e12i − 1.70031i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 1.45787e12i 0.435883i 0.975962 + 0.217941i \(0.0699342\pi\)
−0.975962 + 0.217941i \(0.930066\pi\)
\(608\) 0 0
\(609\) −3.20731e11 −0.0944851
\(610\) 0 0
\(611\) −1.67447e11 −0.0486062
\(612\) 0 0
\(613\) − 1.42075e12i − 0.406394i −0.979138 0.203197i \(-0.934867\pi\)
0.979138 0.203197i \(-0.0651331\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.20441e12i 0.334573i 0.985908 + 0.167286i \(0.0535004\pi\)
−0.985908 + 0.167286i \(0.946500\pi\)
\(618\) 0 0
\(619\) −4.91349e12 −1.34519 −0.672593 0.740013i \(-0.734819\pi\)
−0.672593 + 0.740013i \(0.734819\pi\)
\(620\) 0 0
\(621\) −1.76671e12 −0.476710
\(622\) 0 0
\(623\) 2.40214e12i 0.638856i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 1.39685e12i 0.360948i
\(628\) 0 0
\(629\) 8.03171e11 0.204588
\(630\) 0 0
\(631\) −4.58663e12 −1.15176 −0.575879 0.817535i \(-0.695340\pi\)
−0.575879 + 0.817535i \(0.695340\pi\)
\(632\) 0 0
\(633\) − 5.49826e11i − 0.136116i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 3.00344e12i − 0.722755i
\(638\) 0 0
\(639\) −2.92898e12 −0.694963
\(640\) 0 0
\(641\) −2.30636e12 −0.539593 −0.269796 0.962917i \(-0.586956\pi\)
−0.269796 + 0.962917i \(0.586956\pi\)
\(642\) 0 0
\(643\) 2.59493e12i 0.598654i 0.954151 + 0.299327i \(0.0967622\pi\)
−0.954151 + 0.299327i \(0.903238\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 5.14811e12i − 1.15499i −0.816394 0.577495i \(-0.804030\pi\)
0.816394 0.577495i \(-0.195970\pi\)
\(648\) 0 0
\(649\) −3.15372e11 −0.0697786
\(650\) 0 0
\(651\) 4.00087e11 0.0873051
\(652\) 0 0
\(653\) − 4.42559e12i − 0.952492i −0.879312 0.476246i \(-0.841997\pi\)
0.879312 0.476246i \(-0.158003\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 5.29826e12i 1.10940i
\(658\) 0 0
\(659\) 1.09827e12 0.226842 0.113421 0.993547i \(-0.463819\pi\)
0.113421 + 0.993547i \(0.463819\pi\)
\(660\) 0 0
\(661\) 7.99232e11 0.162842 0.0814209 0.996680i \(-0.474054\pi\)
0.0814209 + 0.996680i \(0.474054\pi\)
\(662\) 0 0
\(663\) 1.20782e12i 0.242768i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.95000e12i 0.772736i
\(668\) 0 0
\(669\) −1.35612e12 −0.261746
\(670\) 0 0
\(671\) −3.83939e12 −0.731157
\(672\) 0 0
\(673\) − 7.68445e12i − 1.44393i −0.691931 0.721963i \(-0.743240\pi\)
0.691931 0.721963i \(-0.256760\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.28854e12i 0.235749i 0.993029 + 0.117874i \(0.0376080\pi\)
−0.993029 + 0.117874i \(0.962392\pi\)
\(678\) 0 0
\(679\) −3.71076e12 −0.669959
\(680\) 0 0
\(681\) 1.70742e12 0.304214
\(682\) 0 0
\(683\) − 8.11444e11i − 0.142681i −0.997452 0.0713404i \(-0.977272\pi\)
0.997452 0.0713404i \(-0.0227277\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 4.50607e10i 0.00771778i
\(688\) 0 0
\(689\) −1.20445e13 −2.03611
\(690\) 0 0
\(691\) −2.26741e12 −0.378338 −0.189169 0.981945i \(-0.560579\pi\)
−0.189169 + 0.981945i \(0.560579\pi\)
\(692\) 0 0
\(693\) 3.16673e12i 0.521569i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 9.44955e12i 1.51657i
\(698\) 0 0
\(699\) −2.38957e11 −0.0378594
\(700\) 0 0
\(701\) −4.92113e12 −0.769721 −0.384861 0.922975i \(-0.625751\pi\)
−0.384861 + 0.922975i \(0.625751\pi\)
\(702\) 0 0
\(703\) − 2.72956e12i − 0.421496i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3.84566e12i 0.578874i
\(708\) 0 0
\(709\) 6.12354e12 0.910112 0.455056 0.890463i \(-0.349619\pi\)
0.455056 + 0.890463i \(0.349619\pi\)
\(710\) 0 0
\(711\) −1.04130e13 −1.52815
\(712\) 0 0
\(713\) − 4.92732e12i − 0.714016i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 1.69734e12i 0.239846i
\(718\) 0 0
\(719\) −2.30376e11 −0.0321483 −0.0160741 0.999871i \(-0.505117\pi\)
−0.0160741 + 0.999871i \(0.505117\pi\)
\(720\) 0 0
\(721\) −6.43408e11 −0.0886702
\(722\) 0 0
\(723\) − 3.54048e11i − 0.0481880i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 9.25894e12i − 1.22930i −0.788801 0.614648i \(-0.789298\pi\)
0.788801 0.614648i \(-0.210702\pi\)
\(728\) 0 0
\(729\) 5.54661e12 0.727368
\(730\) 0 0
\(731\) 4.55643e12 0.590196
\(732\) 0 0
\(733\) 7.22531e12i 0.924461i 0.886760 + 0.462231i \(0.152951\pi\)
−0.886760 + 0.462231i \(0.847049\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 1.23962e13i − 1.54769i
\(738\) 0 0
\(739\) −1.24725e13 −1.53834 −0.769170 0.639044i \(-0.779330\pi\)
−0.769170 + 0.639044i \(0.779330\pi\)
\(740\) 0 0
\(741\) 4.10474e12 0.500154
\(742\) 0 0
\(743\) − 1.15645e13i − 1.39212i −0.717986 0.696058i \(-0.754936\pi\)
0.717986 0.696058i \(-0.245064\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1.21586e11i 0.0142870i
\(748\) 0 0
\(749\) 3.85167e12 0.447179
\(750\) 0 0
\(751\) 3.70732e12 0.425285 0.212642 0.977130i \(-0.431793\pi\)
0.212642 + 0.977130i \(0.431793\pi\)
\(752\) 0 0
\(753\) 1.05547e12i 0.119638i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 4.86635e11i − 0.0538607i −0.999637 0.0269303i \(-0.991427\pi\)
0.999637 0.0269303i \(-0.00857323\pi\)
\(758\) 0 0
\(759\) −1.93562e12 −0.211706
\(760\) 0 0
\(761\) −4.28395e12 −0.463035 −0.231518 0.972831i \(-0.574369\pi\)
−0.231518 + 0.972831i \(0.574369\pi\)
\(762\) 0 0
\(763\) − 4.00589e12i − 0.427896i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.26745e11i 0.0966899i
\(768\) 0 0
\(769\) 4.03625e12 0.416207 0.208103 0.978107i \(-0.433271\pi\)
0.208103 + 0.978107i \(0.433271\pi\)
\(770\) 0 0
\(771\) 1.12186e12 0.114339
\(772\) 0 0
\(773\) − 1.21916e13i − 1.22815i −0.789247 0.614076i \(-0.789529\pi\)
0.789247 0.614076i \(-0.210471\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 3.07120e11i 0.0302282i
\(778\) 0 0
\(779\) 3.21141e13 3.12447
\(780\) 0 0
\(781\) −6.57728e12 −0.632582
\(782\) 0 0
\(783\) 3.07397e12i 0.292262i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1.20299e12i 0.111783i 0.998437 + 0.0558913i \(0.0178000\pi\)
−0.998437 + 0.0558913i \(0.982200\pi\)
\(788\) 0 0
\(789\) −4.08059e12 −0.374866
\(790\) 0 0
\(791\) 1.00539e13 0.913147
\(792\) 0 0
\(793\) 1.12823e13i 1.01314i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.72611e13i 1.51533i 0.652646 + 0.757663i \(0.273659\pi\)
−0.652646 + 0.757663i \(0.726341\pi\)
\(798\) 0 0
\(799\) 4.32943e11 0.0375811
\(800\) 0 0
\(801\) 1.12326e13 0.964130
\(802\) 0 0
\(803\) 1.18977e13i 1.00982i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 4.34609e12i 0.360718i
\(808\) 0 0
\(809\) −7.44408e12 −0.611002 −0.305501 0.952192i \(-0.598824\pi\)
−0.305501 + 0.952192i \(0.598824\pi\)
\(810\) 0 0
\(811\) 9.59955e12 0.779214 0.389607 0.920981i \(-0.372611\pi\)
0.389607 + 0.920981i \(0.372611\pi\)
\(812\) 0 0
\(813\) 3.38772e12i 0.271957i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 1.54849e13i − 1.21593i
\(818\) 0 0
\(819\) 9.30568e12 0.722721
\(820\) 0 0
\(821\) −5.04043e12 −0.387189 −0.193595 0.981082i \(-0.562015\pi\)
−0.193595 + 0.981082i \(0.562015\pi\)
\(822\) 0 0
\(823\) 1.62323e13i 1.23333i 0.787225 + 0.616666i \(0.211517\pi\)
−0.787225 + 0.616666i \(0.788483\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.74898e13i 1.30020i 0.759850 + 0.650099i \(0.225273\pi\)
−0.759850 + 0.650099i \(0.774727\pi\)
\(828\) 0 0
\(829\) 1.04102e13 0.765535 0.382768 0.923845i \(-0.374971\pi\)
0.382768 + 0.923845i \(0.374971\pi\)
\(830\) 0 0
\(831\) −8.56420e11 −0.0622992
\(832\) 0 0
\(833\) 7.76555e12i 0.558817i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 3.83454e12i − 0.270053i
\(838\) 0 0
\(839\) −2.66219e13 −1.85485 −0.927426 0.374006i \(-0.877984\pi\)
−0.927426 + 0.374006i \(0.877984\pi\)
\(840\) 0 0
\(841\) −7.63439e12 −0.526250
\(842\) 0 0
\(843\) 1.66919e12i 0.113836i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 2.34477e12i − 0.156540i
\(848\) 0 0
\(849\) 3.35427e12 0.221571
\(850\) 0 0
\(851\) 3.78237e12 0.247219
\(852\) 0 0
\(853\) 7.30064e12i 0.472161i 0.971734 + 0.236080i \(0.0758629\pi\)
−0.971734 + 0.236080i \(0.924137\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 1.18598e13i − 0.751041i −0.926814 0.375520i \(-0.877464\pi\)
0.926814 0.375520i \(-0.122536\pi\)
\(858\) 0 0
\(859\) 1.67692e13 1.05085 0.525427 0.850839i \(-0.323906\pi\)
0.525427 + 0.850839i \(0.323906\pi\)
\(860\) 0 0
\(861\) −3.61336e12 −0.224077
\(862\) 0 0
\(863\) 6.71596e12i 0.412154i 0.978536 + 0.206077i \(0.0660698\pi\)
−0.978536 + 0.206077i \(0.933930\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 4.94917e11i 0.0297472i
\(868\) 0 0
\(869\) −2.33834e13 −1.39098
\(870\) 0 0
\(871\) −3.64270e13 −2.14458
\(872\) 0 0
\(873\) 1.73518e13i 1.01107i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 2.79757e13i − 1.59692i −0.602051 0.798458i \(-0.705649\pi\)
0.602051 0.798458i \(-0.294351\pi\)
\(878\) 0 0
\(879\) −7.03050e12 −0.397225
\(880\) 0 0
\(881\) 2.06225e13 1.15332 0.576660 0.816984i \(-0.304356\pi\)
0.576660 + 0.816984i \(0.304356\pi\)
\(882\) 0 0
\(883\) − 2.02048e13i − 1.11849i −0.829003 0.559244i \(-0.811091\pi\)
0.829003 0.559244i \(-0.188909\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 3.19954e13i − 1.73553i −0.496978 0.867763i \(-0.665557\pi\)
0.496978 0.867763i \(-0.334443\pi\)
\(888\) 0 0
\(889\) −9.92701e12 −0.533041
\(890\) 0 0
\(891\) 1.40365e13 0.746122
\(892\) 0 0
\(893\) − 1.47135e12i − 0.0774254i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 5.68798e12i 0.293354i
\(898\) 0 0
\(899\) −8.57323e12 −0.437750
\(900\) 0 0
\(901\) 3.11416e13 1.57427
\(902\) 0 0
\(903\) 1.74231e12i 0.0872026i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 2.40537e12i − 0.118018i −0.998257 0.0590091i \(-0.981206\pi\)
0.998257 0.0590091i \(-0.0187941\pi\)
\(908\) 0 0
\(909\) 1.79827e13 0.873609
\(910\) 0 0
\(911\) 3.11773e13 1.49970 0.749852 0.661606i \(-0.230125\pi\)
0.749852 + 0.661606i \(0.230125\pi\)
\(912\) 0 0
\(913\) 2.73032e11i 0.0130045i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 7.71582e12i − 0.360346i
\(918\) 0 0
\(919\) 3.42677e13 1.58477 0.792383 0.610024i \(-0.208840\pi\)
0.792383 + 0.610024i \(0.208840\pi\)
\(920\) 0 0
\(921\) 2.08986e12 0.0957082
\(922\) 0 0
\(923\) 1.93278e13i 0.876548i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 3.00864e12i 0.133817i
\(928\) 0 0
\(929\) 2.52782e13 1.11346 0.556730 0.830693i \(-0.312056\pi\)
0.556730 + 0.830693i \(0.312056\pi\)
\(930\) 0 0
\(931\) 2.63910e13 1.15129
\(932\) 0 0
\(933\) 6.06476e12i 0.262027i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 2.72636e13i 1.15546i 0.816227 + 0.577731i \(0.196062\pi\)
−0.816227 + 0.577731i \(0.803938\pi\)
\(938\) 0 0
\(939\) −1.87682e12 −0.0787819
\(940\) 0 0
\(941\) 1.57383e13 0.654340 0.327170 0.944966i \(-0.393905\pi\)
0.327170 + 0.944966i \(0.393905\pi\)
\(942\) 0 0
\(943\) 4.45008e13i 1.83259i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 9.71128e12i 0.392375i 0.980566 + 0.196188i \(0.0628562\pi\)
−0.980566 + 0.196188i \(0.937144\pi\)
\(948\) 0 0
\(949\) 3.49624e13 1.39927
\(950\) 0 0
\(951\) 6.89257e12 0.273256
\(952\) 0 0
\(953\) − 4.15385e12i − 0.163130i −0.996668 0.0815648i \(-0.974008\pi\)
0.996668 0.0815648i \(-0.0259917\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 3.36787e12i 0.129793i
\(958\) 0 0
\(959\) −1.79897e12 −0.0686818
\(960\) 0 0
\(961\) −1.57452e13 −0.595515
\(962\) 0 0
\(963\) − 1.80108e13i − 0.674861i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 1.84544e13i − 0.678706i −0.940659 0.339353i \(-0.889792\pi\)
0.940659 0.339353i \(-0.110208\pi\)
\(968\) 0 0
\(969\) −1.06130e13 −0.386707
\(970\) 0 0
\(971\) 2.95973e12 0.106848 0.0534238 0.998572i \(-0.482987\pi\)
0.0534238 + 0.998572i \(0.482987\pi\)
\(972\) 0 0
\(973\) 1.79826e13i 0.643197i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 4.44178e13i 1.55967i 0.625988 + 0.779833i \(0.284696\pi\)
−0.625988 + 0.779833i \(0.715304\pi\)
\(978\) 0 0
\(979\) 2.52239e13 0.877588
\(980\) 0 0
\(981\) −1.87319e13 −0.645761
\(982\) 0 0
\(983\) 4.57558e13i 1.56299i 0.623914 + 0.781493i \(0.285541\pi\)
−0.623914 + 0.781493i \(0.714459\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 1.65550e11i 0.00555269i
\(988\) 0 0
\(989\) 2.14576e13 0.713177
\(990\) 0 0
\(991\) −1.30469e13 −0.429710 −0.214855 0.976646i \(-0.568928\pi\)
−0.214855 + 0.976646i \(0.568928\pi\)
\(992\) 0 0
\(993\) 2.55903e12i 0.0835225i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 5.07578e12i − 0.162695i −0.996686 0.0813476i \(-0.974078\pi\)
0.996686 0.0813476i \(-0.0259224\pi\)
\(998\) 0 0
\(999\) 2.94352e12 0.0935022
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 400.10.c.q.49.4 6
4.3 odd 2 25.10.b.c.24.3 6
5.2 odd 4 400.10.a.y.1.2 3
5.3 odd 4 400.10.a.u.1.2 3
5.4 even 2 inner 400.10.c.q.49.3 6
12.11 even 2 225.10.b.m.199.4 6
20.3 even 4 25.10.a.d.1.1 yes 3
20.7 even 4 25.10.a.c.1.3 3
20.19 odd 2 25.10.b.c.24.4 6
60.23 odd 4 225.10.a.m.1.3 3
60.47 odd 4 225.10.a.p.1.1 3
60.59 even 2 225.10.b.m.199.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.10.a.c.1.3 3 20.7 even 4
25.10.a.d.1.1 yes 3 20.3 even 4
25.10.b.c.24.3 6 4.3 odd 2
25.10.b.c.24.4 6 20.19 odd 2
225.10.a.m.1.3 3 60.23 odd 4
225.10.a.p.1.1 3 60.47 odd 4
225.10.b.m.199.3 6 60.59 even 2
225.10.b.m.199.4 6 12.11 even 2
400.10.a.u.1.2 3 5.3 odd 4
400.10.a.y.1.2 3 5.2 odd 4
400.10.c.q.49.3 6 5.4 even 2 inner
400.10.c.q.49.4 6 1.1 even 1 trivial