Properties

Label 400.10.c.q.49.6
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 49.6
Root \(-27.7229i\) of defining polynomial
Character \(\chi\) \(=\) 400.49
Dual form 400.10.c.q.49.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+268.664i q^{3} +637.237i q^{7} -52497.3 q^{9} +O(q^{10})\) \(q+268.664i q^{3} +637.237i q^{7} -52497.3 q^{9} +49042.6 q^{11} +72726.2i q^{13} -67319.3i q^{17} +341136. q^{19} -171203. q^{21} +134355. i q^{23} -8.81603e6i q^{27} -4.45784e6 q^{29} -456520. q^{31} +1.31760e7i q^{33} -1.30516e7i q^{37} -1.95389e7 q^{39} -2.56667e7 q^{41} -3.42710e6i q^{43} -3.39814e7i q^{47} +3.99475e7 q^{49} +1.80863e7 q^{51} -8.42456e7i q^{53} +9.16509e7i q^{57} -7.46358e7 q^{59} +1.78017e8 q^{61} -3.34533e7i q^{63} +6.94299e7i q^{67} -3.60963e7 q^{69} +2.07860e8 q^{71} -3.02516e8i q^{73} +3.12518e7i q^{77} +3.72244e8 q^{79} +1.33524e9 q^{81} -4.50079e8i q^{83} -1.19766e9i q^{87} -5.82741e7 q^{89} -4.63438e7 q^{91} -1.22650e8i q^{93} -7.85850e8i q^{97} -2.57461e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 116468 q^{9}+O(q^{10}) \) 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
\(3\) 268.664i 1.91498i 0.288470 + 0.957489i \(0.406854\pi\)
−0.288470 + 0.957489i \(0.593146\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 637.237i 0.100314i 0.998741 + 0.0501568i \(0.0159721\pi\)
−0.998741 + 0.0501568i \(0.984028\pi\)
\(8\) 0 0
\(9\) −52497.3 −2.66714
\(10\) 0 0
\(11\) 49042.6 1.00996 0.504982 0.863130i \(-0.331499\pi\)
0.504982 + 0.863130i \(0.331499\pi\)
\(12\) 0 0
\(13\) 72726.2i 0.706229i 0.935580 + 0.353115i \(0.114877\pi\)
−0.935580 + 0.353115i \(0.885123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 67319.3i − 0.195488i −0.995212 0.0977439i \(-0.968837\pi\)
0.995212 0.0977439i \(-0.0311626\pi\)
\(18\) 0 0
\(19\) 341136. 0.600532 0.300266 0.953855i \(-0.402925\pi\)
0.300266 + 0.953855i \(0.402925\pi\)
\(20\) 0 0
\(21\) −171203. −0.192098
\(22\) 0 0
\(23\) 134355.i 0.100110i 0.998746 + 0.0500550i \(0.0159397\pi\)
−0.998746 + 0.0500550i \(0.984060\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 8.81603e6i − 3.19254i
\(28\) 0 0
\(29\) −4.45784e6 −1.17040 −0.585199 0.810890i \(-0.698984\pi\)
−0.585199 + 0.810890i \(0.698984\pi\)
\(30\) 0 0
\(31\) −456520. −0.0887834 −0.0443917 0.999014i \(-0.514135\pi\)
−0.0443917 + 0.999014i \(0.514135\pi\)
\(32\) 0 0
\(33\) 1.31760e7i 1.93406i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 1.30516e7i − 1.14487i −0.819951 0.572433i \(-0.805999\pi\)
0.819951 0.572433i \(-0.194001\pi\)
\(38\) 0 0
\(39\) −1.95389e7 −1.35241
\(40\) 0 0
\(41\) −2.56667e7 −1.41854 −0.709272 0.704935i \(-0.750976\pi\)
−0.709272 + 0.704935i \(0.750976\pi\)
\(42\) 0 0
\(43\) − 3.42710e6i − 0.152869i −0.997075 0.0764344i \(-0.975646\pi\)
0.997075 0.0764344i \(-0.0243536\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 3.39814e7i − 1.01578i −0.861421 0.507891i \(-0.830425\pi\)
0.861421 0.507891i \(-0.169575\pi\)
\(48\) 0 0
\(49\) 3.99475e7 0.989937
\(50\) 0 0
\(51\) 1.80863e7 0.374355
\(52\) 0 0
\(53\) − 8.42456e7i − 1.46658i −0.679916 0.733290i \(-0.737984\pi\)
0.679916 0.733290i \(-0.262016\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 9.16509e7i 1.15001i
\(58\) 0 0
\(59\) −7.46358e7 −0.801887 −0.400944 0.916103i \(-0.631318\pi\)
−0.400944 + 0.916103i \(0.631318\pi\)
\(60\) 0 0
\(61\) 1.78017e8 1.64618 0.823088 0.567913i \(-0.192249\pi\)
0.823088 + 0.567913i \(0.192249\pi\)
\(62\) 0 0
\(63\) − 3.34533e7i − 0.267551i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.94299e7i 0.420930i 0.977601 + 0.210465i \(0.0674978\pi\)
−0.977601 + 0.210465i \(0.932502\pi\)
\(68\) 0 0
\(69\) −3.60963e7 −0.191708
\(70\) 0 0
\(71\) 2.07860e8 0.970753 0.485376 0.874305i \(-0.338683\pi\)
0.485376 + 0.874305i \(0.338683\pi\)
\(72\) 0 0
\(73\) − 3.02516e8i − 1.24680i −0.781905 0.623398i \(-0.785752\pi\)
0.781905 0.623398i \(-0.214248\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.12518e7i 0.101313i
\(78\) 0 0
\(79\) 3.72244e8 1.07524 0.537621 0.843187i \(-0.319323\pi\)
0.537621 + 0.843187i \(0.319323\pi\)
\(80\) 0 0
\(81\) 1.33524e9 3.44650
\(82\) 0 0
\(83\) − 4.50079e8i − 1.04097i −0.853872 0.520484i \(-0.825752\pi\)
0.853872 0.520484i \(-0.174248\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 1.19766e9i − 2.24129i
\(88\) 0 0
\(89\) −5.82741e7 −0.0984511 −0.0492255 0.998788i \(-0.515675\pi\)
−0.0492255 + 0.998788i \(0.515675\pi\)
\(90\) 0 0
\(91\) −4.63438e7 −0.0708444
\(92\) 0 0
\(93\) − 1.22650e8i − 0.170018i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 7.85850e8i − 0.901295i −0.892702 0.450647i \(-0.851193\pi\)
0.892702 0.450647i \(-0.148807\pi\)
\(98\) 0 0
\(99\) −2.57461e9 −2.69372
\(100\) 0 0
\(101\) −1.60460e8 −0.153433 −0.0767167 0.997053i \(-0.524444\pi\)
−0.0767167 + 0.997053i \(0.524444\pi\)
\(102\) 0 0
\(103\) 1.39454e9i 1.22085i 0.792074 + 0.610425i \(0.209001\pi\)
−0.792074 + 0.610425i \(0.790999\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 1.56463e9i − 1.15394i −0.816764 0.576972i \(-0.804234\pi\)
0.816764 0.576972i \(-0.195766\pi\)
\(108\) 0 0
\(109\) −1.24703e9 −0.846172 −0.423086 0.906089i \(-0.639053\pi\)
−0.423086 + 0.906089i \(0.639053\pi\)
\(110\) 0 0
\(111\) 3.50649e9 2.19240
\(112\) 0 0
\(113\) − 1.81056e9i − 1.04463i −0.852754 0.522313i \(-0.825069\pi\)
0.852754 0.522313i \(-0.174931\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 3.81793e9i − 1.88361i
\(118\) 0 0
\(119\) 4.28984e7 0.0196101
\(120\) 0 0
\(121\) 4.72275e7 0.0200291
\(122\) 0 0
\(123\) − 6.89572e9i − 2.71648i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 3.06491e9i − 1.04545i −0.852503 0.522723i \(-0.824916\pi\)
0.852503 0.522723i \(-0.175084\pi\)
\(128\) 0 0
\(129\) 9.20739e8 0.292740
\(130\) 0 0
\(131\) 1.83508e9 0.544421 0.272211 0.962238i \(-0.412245\pi\)
0.272211 + 0.962238i \(0.412245\pi\)
\(132\) 0 0
\(133\) 2.17385e8i 0.0602416i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.62426e9i 1.12150i 0.827984 + 0.560751i \(0.189488\pi\)
−0.827984 + 0.560751i \(0.810512\pi\)
\(138\) 0 0
\(139\) 2.57140e8 0.0584255 0.0292128 0.999573i \(-0.490700\pi\)
0.0292128 + 0.999573i \(0.490700\pi\)
\(140\) 0 0
\(141\) 9.12958e9 1.94520
\(142\) 0 0
\(143\) 3.56668e9i 0.713267i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.07325e10i 1.89571i
\(148\) 0 0
\(149\) −3.14809e9 −0.523250 −0.261625 0.965170i \(-0.584258\pi\)
−0.261625 + 0.965170i \(0.584258\pi\)
\(150\) 0 0
\(151\) 1.02772e10 1.60871 0.804356 0.594147i \(-0.202510\pi\)
0.804356 + 0.594147i \(0.202510\pi\)
\(152\) 0 0
\(153\) 3.53408e9i 0.521393i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 3.25077e9i − 0.427009i −0.976942 0.213505i \(-0.931512\pi\)
0.976942 0.213505i \(-0.0684878\pi\)
\(158\) 0 0
\(159\) 2.26338e10 2.80847
\(160\) 0 0
\(161\) −8.56158e7 −0.0100424
\(162\) 0 0
\(163\) − 3.40049e9i − 0.377309i −0.982043 0.188655i \(-0.939587\pi\)
0.982043 0.188655i \(-0.0604127\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.27785e9i 0.525089i 0.964920 + 0.262544i \(0.0845616\pi\)
−0.964920 + 0.262544i \(0.915438\pi\)
\(168\) 0 0
\(169\) 5.31540e9 0.501240
\(170\) 0 0
\(171\) −1.79087e10 −1.60170
\(172\) 0 0
\(173\) 6.58861e9i 0.559224i 0.960113 + 0.279612i \(0.0902059\pi\)
−0.960113 + 0.279612i \(0.909794\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 2.00520e10i − 1.53560i
\(178\) 0 0
\(179\) −1.23663e10 −0.900326 −0.450163 0.892946i \(-0.648634\pi\)
−0.450163 + 0.892946i \(0.648634\pi\)
\(180\) 0 0
\(181\) −2.59914e10 −1.80002 −0.900009 0.435872i \(-0.856440\pi\)
−0.900009 + 0.435872i \(0.856440\pi\)
\(182\) 0 0
\(183\) 4.78267e10i 3.15239i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 3.30151e9i − 0.197436i
\(188\) 0 0
\(189\) 5.61790e9 0.320255
\(190\) 0 0
\(191\) −1.50506e10 −0.818284 −0.409142 0.912471i \(-0.634172\pi\)
−0.409142 + 0.912471i \(0.634172\pi\)
\(192\) 0 0
\(193\) 1.40329e10i 0.728014i 0.931396 + 0.364007i \(0.118592\pi\)
−0.931396 + 0.364007i \(0.881408\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.57285e10i 1.21707i 0.793526 + 0.608536i \(0.208243\pi\)
−0.793526 + 0.608536i \(0.791757\pi\)
\(198\) 0 0
\(199\) −2.94367e10 −1.33061 −0.665303 0.746573i \(-0.731698\pi\)
−0.665303 + 0.746573i \(0.731698\pi\)
\(200\) 0 0
\(201\) −1.86533e10 −0.806072
\(202\) 0 0
\(203\) − 2.84070e9i − 0.117407i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 7.05326e9i − 0.267007i
\(208\) 0 0
\(209\) 1.67302e10 0.606516
\(210\) 0 0
\(211\) −1.17275e10 −0.407319 −0.203659 0.979042i \(-0.565284\pi\)
−0.203659 + 0.979042i \(0.565284\pi\)
\(212\) 0 0
\(213\) 5.58445e10i 1.85897i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 2.90911e8i − 0.00890619i
\(218\) 0 0
\(219\) 8.12751e10 2.38759
\(220\) 0 0
\(221\) 4.89588e9 0.138059
\(222\) 0 0
\(223\) − 2.58340e10i − 0.699550i −0.936834 0.349775i \(-0.886258\pi\)
0.936834 0.349775i \(-0.113742\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 2.50896e10i − 0.627159i −0.949562 0.313580i \(-0.898472\pi\)
0.949562 0.313580i \(-0.101528\pi\)
\(228\) 0 0
\(229\) 2.30463e10 0.553785 0.276892 0.960901i \(-0.410695\pi\)
0.276892 + 0.960901i \(0.410695\pi\)
\(230\) 0 0
\(231\) −8.39622e9 −0.194013
\(232\) 0 0
\(233\) 3.17197e10i 0.705062i 0.935800 + 0.352531i \(0.114679\pi\)
−0.935800 + 0.352531i \(0.885321\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.00009e11i 2.05906i
\(238\) 0 0
\(239\) 2.23794e10 0.443668 0.221834 0.975084i \(-0.428796\pi\)
0.221834 + 0.975084i \(0.428796\pi\)
\(240\) 0 0
\(241\) −2.10823e10 −0.402569 −0.201285 0.979533i \(-0.564512\pi\)
−0.201285 + 0.979533i \(0.564512\pi\)
\(242\) 0 0
\(243\) 1.85206e11i 3.40743i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.48095e10i 0.424113i
\(248\) 0 0
\(249\) 1.20920e11 1.99343
\(250\) 0 0
\(251\) 7.96509e10 1.26666 0.633329 0.773883i \(-0.281688\pi\)
0.633329 + 0.773883i \(0.281688\pi\)
\(252\) 0 0
\(253\) 6.58910e9i 0.101108i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.30815e10i 0.473027i 0.971628 + 0.236514i \(0.0760048\pi\)
−0.971628 + 0.236514i \(0.923995\pi\)
\(258\) 0 0
\(259\) 8.31695e9 0.114846
\(260\) 0 0
\(261\) 2.34025e11 3.12162
\(262\) 0 0
\(263\) − 6.05028e10i − 0.779784i −0.920861 0.389892i \(-0.872512\pi\)
0.920861 0.389892i \(-0.127488\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 1.56562e10i − 0.188532i
\(268\) 0 0
\(269\) −2.61933e10 −0.305004 −0.152502 0.988303i \(-0.548733\pi\)
−0.152502 + 0.988303i \(0.548733\pi\)
\(270\) 0 0
\(271\) −5.42857e10 −0.611398 −0.305699 0.952128i \(-0.598890\pi\)
−0.305699 + 0.952128i \(0.598890\pi\)
\(272\) 0 0
\(273\) − 1.24509e10i − 0.135666i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 1.76561e11i − 1.80192i −0.433900 0.900961i \(-0.642863\pi\)
0.433900 0.900961i \(-0.357137\pi\)
\(278\) 0 0
\(279\) 2.39661e10 0.236798
\(280\) 0 0
\(281\) −7.91126e9 −0.0756950 −0.0378475 0.999284i \(-0.512050\pi\)
−0.0378475 + 0.999284i \(0.512050\pi\)
\(282\) 0 0
\(283\) 1.34806e11i 1.24931i 0.780899 + 0.624657i \(0.214761\pi\)
−0.780899 + 0.624657i \(0.785239\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 1.63558e10i − 0.142299i
\(288\) 0 0
\(289\) 1.14056e11 0.961785
\(290\) 0 0
\(291\) 2.11130e11 1.72596
\(292\) 0 0
\(293\) 1.02784e11i 0.814741i 0.913263 + 0.407370i \(0.133554\pi\)
−0.913263 + 0.407370i \(0.866446\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 4.32361e11i − 3.22435i
\(298\) 0 0
\(299\) −9.77110e9 −0.0707006
\(300\) 0 0
\(301\) 2.18388e9 0.0153348
\(302\) 0 0
\(303\) − 4.31098e10i − 0.293822i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 2.61472e11i − 1.67998i −0.542606 0.839988i \(-0.682562\pi\)
0.542606 0.839988i \(-0.317438\pi\)
\(308\) 0 0
\(309\) −3.74662e11 −2.33790
\(310\) 0 0
\(311\) 8.24828e10 0.499967 0.249984 0.968250i \(-0.419575\pi\)
0.249984 + 0.968250i \(0.419575\pi\)
\(312\) 0 0
\(313\) − 1.35766e11i − 0.799540i −0.916615 0.399770i \(-0.869090\pi\)
0.916615 0.399770i \(-0.130910\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.06853e10i 0.504394i 0.967676 + 0.252197i \(0.0811531\pi\)
−0.967676 + 0.252197i \(0.918847\pi\)
\(318\) 0 0
\(319\) −2.18624e11 −1.18206
\(320\) 0 0
\(321\) 4.20360e11 2.20978
\(322\) 0 0
\(323\) − 2.29650e10i − 0.117397i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 3.35033e11i − 1.62040i
\(328\) 0 0
\(329\) 2.16542e10 0.101897
\(330\) 0 0
\(331\) 3.45169e11 1.58054 0.790271 0.612758i \(-0.209940\pi\)
0.790271 + 0.612758i \(0.209940\pi\)
\(332\) 0 0
\(333\) 6.85173e11i 3.05352i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 8.39418e10i − 0.354523i −0.984164 0.177261i \(-0.943276\pi\)
0.984164 0.177261i \(-0.0567238\pi\)
\(338\) 0 0
\(339\) 4.86433e11 2.00043
\(340\) 0 0
\(341\) −2.23889e10 −0.0896681
\(342\) 0 0
\(343\) 5.11709e10i 0.199618i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.56153e11i 0.578188i 0.957301 + 0.289094i \(0.0933540\pi\)
−0.957301 + 0.289094i \(0.906646\pi\)
\(348\) 0 0
\(349\) −6.31862e9 −0.0227986 −0.0113993 0.999935i \(-0.503629\pi\)
−0.0113993 + 0.999935i \(0.503629\pi\)
\(350\) 0 0
\(351\) 6.41156e11 2.25466
\(352\) 0 0
\(353\) − 2.83875e11i − 0.973064i −0.873663 0.486532i \(-0.838262\pi\)
0.873663 0.486532i \(-0.161738\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 1.15253e10i 0.0375529i
\(358\) 0 0
\(359\) 6.00733e11 1.90878 0.954391 0.298558i \(-0.0965058\pi\)
0.954391 + 0.298558i \(0.0965058\pi\)
\(360\) 0 0
\(361\) −2.06314e11 −0.639361
\(362\) 0 0
\(363\) 1.26883e10i 0.0383552i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 4.97260e11i 1.43082i 0.698702 + 0.715412i \(0.253761\pi\)
−0.698702 + 0.715412i \(0.746239\pi\)
\(368\) 0 0
\(369\) 1.34743e12 3.78346
\(370\) 0 0
\(371\) 5.36845e10 0.147118
\(372\) 0 0
\(373\) 1.39183e11i 0.372303i 0.982521 + 0.186151i \(0.0596015\pi\)
−0.982521 + 0.186151i \(0.940398\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 3.24202e11i − 0.826569i
\(378\) 0 0
\(379\) −5.65247e11 −1.40722 −0.703610 0.710587i \(-0.748430\pi\)
−0.703610 + 0.710587i \(0.748430\pi\)
\(380\) 0 0
\(381\) 8.23432e11 2.00201
\(382\) 0 0
\(383\) 5.38864e11i 1.27963i 0.768528 + 0.639816i \(0.220989\pi\)
−0.768528 + 0.639816i \(0.779011\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.79914e11i 0.407723i
\(388\) 0 0
\(389\) −7.67552e11 −1.69955 −0.849776 0.527144i \(-0.823263\pi\)
−0.849776 + 0.527144i \(0.823263\pi\)
\(390\) 0 0
\(391\) 9.04466e9 0.0195703
\(392\) 0 0
\(393\) 4.93021e11i 1.04255i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 6.25258e11i 1.26329i 0.775259 + 0.631643i \(0.217619\pi\)
−0.775259 + 0.631643i \(0.782381\pi\)
\(398\) 0 0
\(399\) −5.84034e10 −0.115361
\(400\) 0 0
\(401\) −1.61329e11 −0.311574 −0.155787 0.987791i \(-0.549791\pi\)
−0.155787 + 0.987791i \(0.549791\pi\)
\(402\) 0 0
\(403\) − 3.32009e10i − 0.0627014i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 6.40083e11i − 1.15628i
\(408\) 0 0
\(409\) 7.24004e11 1.27934 0.639670 0.768649i \(-0.279071\pi\)
0.639670 + 0.768649i \(0.279071\pi\)
\(410\) 0 0
\(411\) −1.24237e12 −2.14765
\(412\) 0 0
\(413\) − 4.75607e10i − 0.0804403i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 6.90842e10i 0.111884i
\(418\) 0 0
\(419\) 2.80040e11 0.443871 0.221936 0.975061i \(-0.428763\pi\)
0.221936 + 0.975061i \(0.428763\pi\)
\(420\) 0 0
\(421\) −6.55915e10 −0.101760 −0.0508801 0.998705i \(-0.516203\pi\)
−0.0508801 + 0.998705i \(0.516203\pi\)
\(422\) 0 0
\(423\) 1.78393e12i 2.70924i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.13439e11i 0.165134i
\(428\) 0 0
\(429\) −9.58239e11 −1.36589
\(430\) 0 0
\(431\) −1.00292e12 −1.39997 −0.699987 0.714156i \(-0.746811\pi\)
−0.699987 + 0.714156i \(0.746811\pi\)
\(432\) 0 0
\(433\) − 7.98482e11i − 1.09162i −0.837910 0.545808i \(-0.816223\pi\)
0.837910 0.545808i \(-0.183777\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.58332e10i 0.0601193i
\(438\) 0 0
\(439\) 7.98379e11 1.02593 0.512966 0.858409i \(-0.328547\pi\)
0.512966 + 0.858409i \(0.328547\pi\)
\(440\) 0 0
\(441\) −2.09714e12 −2.64030
\(442\) 0 0
\(443\) − 7.56642e11i − 0.933413i −0.884412 0.466707i \(-0.845440\pi\)
0.884412 0.466707i \(-0.154560\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 8.45779e11i − 1.00201i
\(448\) 0 0
\(449\) 6.39419e11 0.742467 0.371234 0.928540i \(-0.378935\pi\)
0.371234 + 0.928540i \(0.378935\pi\)
\(450\) 0 0
\(451\) −1.25876e12 −1.43268
\(452\) 0 0
\(453\) 2.76111e12i 3.08065i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 1.34276e12i − 1.44005i −0.693949 0.720024i \(-0.744131\pi\)
0.693949 0.720024i \(-0.255869\pi\)
\(458\) 0 0
\(459\) −5.93489e11 −0.624102
\(460\) 0 0
\(461\) 3.47112e11 0.357945 0.178972 0.983854i \(-0.442723\pi\)
0.178972 + 0.983854i \(0.442723\pi\)
\(462\) 0 0
\(463\) 7.20214e11i 0.728362i 0.931328 + 0.364181i \(0.118651\pi\)
−0.931328 + 0.364181i \(0.881349\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7.31553e10i 0.0711737i 0.999367 + 0.0355869i \(0.0113300\pi\)
−0.999367 + 0.0355869i \(0.988670\pi\)
\(468\) 0 0
\(469\) −4.42433e10 −0.0422250
\(470\) 0 0
\(471\) 8.73364e11 0.817713
\(472\) 0 0
\(473\) − 1.68074e11i − 0.154392i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 4.42267e12i 3.91158i
\(478\) 0 0
\(479\) −4.80307e10 −0.0416878 −0.0208439 0.999783i \(-0.506635\pi\)
−0.0208439 + 0.999783i \(0.506635\pi\)
\(480\) 0 0
\(481\) 9.49191e11 0.808539
\(482\) 0 0
\(483\) − 2.30019e10i − 0.0192310i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1.24384e12i 1.00204i 0.865435 + 0.501021i \(0.167042\pi\)
−0.865435 + 0.501021i \(0.832958\pi\)
\(488\) 0 0
\(489\) 9.13590e11 0.722539
\(490\) 0 0
\(491\) 2.63145e11 0.204328 0.102164 0.994768i \(-0.467423\pi\)
0.102164 + 0.994768i \(0.467423\pi\)
\(492\) 0 0
\(493\) 3.00099e11i 0.228798i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.32456e11i 0.0973798i
\(498\) 0 0
\(499\) 3.75452e10 0.0271083 0.0135542 0.999908i \(-0.495685\pi\)
0.0135542 + 0.999908i \(0.495685\pi\)
\(500\) 0 0
\(501\) −1.41797e12 −1.00553
\(502\) 0 0
\(503\) − 2.86189e11i − 0.199341i −0.995020 0.0996707i \(-0.968221\pi\)
0.995020 0.0996707i \(-0.0317789\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.42806e12i 0.959864i
\(508\) 0 0
\(509\) 1.48505e12 0.980646 0.490323 0.871541i \(-0.336879\pi\)
0.490323 + 0.871541i \(0.336879\pi\)
\(510\) 0 0
\(511\) 1.92774e11 0.125071
\(512\) 0 0
\(513\) − 3.00746e12i − 1.91722i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 1.66654e12i − 1.02590i
\(518\) 0 0
\(519\) −1.77012e12 −1.07090
\(520\) 0 0
\(521\) −1.77521e12 −1.05556 −0.527778 0.849383i \(-0.676975\pi\)
−0.527778 + 0.849383i \(0.676975\pi\)
\(522\) 0 0
\(523\) − 1.20614e12i − 0.704922i −0.935826 0.352461i \(-0.885345\pi\)
0.935826 0.352461i \(-0.114655\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.07326e10i 0.0173561i
\(528\) 0 0
\(529\) 1.78310e12 0.989978
\(530\) 0 0
\(531\) 3.91818e12 2.13875
\(532\) 0 0
\(533\) − 1.86664e12i − 1.00182i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 3.32237e12i − 1.72410i
\(538\) 0 0
\(539\) 1.95913e12 0.999802
\(540\) 0 0
\(541\) 3.24195e12 1.62711 0.813557 0.581485i \(-0.197528\pi\)
0.813557 + 0.581485i \(0.197528\pi\)
\(542\) 0 0
\(543\) − 6.98296e12i − 3.44699i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 9.33463e11i 0.445814i 0.974840 + 0.222907i \(0.0715547\pi\)
−0.974840 + 0.222907i \(0.928445\pi\)
\(548\) 0 0
\(549\) −9.34540e12 −4.39059
\(550\) 0 0
\(551\) −1.52073e12 −0.702861
\(552\) 0 0
\(553\) 2.37208e11i 0.107861i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.31760e12i 0.580012i 0.957025 + 0.290006i \(0.0936572\pi\)
−0.957025 + 0.290006i \(0.906343\pi\)
\(558\) 0 0
\(559\) 2.49240e11 0.107960
\(560\) 0 0
\(561\) 8.86998e11 0.378085
\(562\) 0 0
\(563\) 4.33363e12i 1.81788i 0.416931 + 0.908938i \(0.363106\pi\)
−0.416931 + 0.908938i \(0.636894\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 8.50868e11i 0.345731i
\(568\) 0 0
\(569\) −3.08940e12 −1.23558 −0.617788 0.786345i \(-0.711971\pi\)
−0.617788 + 0.786345i \(0.711971\pi\)
\(570\) 0 0
\(571\) 7.43095e11 0.292538 0.146269 0.989245i \(-0.453274\pi\)
0.146269 + 0.989245i \(0.453274\pi\)
\(572\) 0 0
\(573\) − 4.04356e12i − 1.56700i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2.25852e12i 0.848269i 0.905599 + 0.424134i \(0.139422\pi\)
−0.905599 + 0.424134i \(0.860578\pi\)
\(578\) 0 0
\(579\) −3.77013e12 −1.39413
\(580\) 0 0
\(581\) 2.86807e11 0.104423
\(582\) 0 0
\(583\) − 4.13162e12i − 1.48119i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 4.75794e12i − 1.65405i −0.562169 0.827023i \(-0.690033\pi\)
0.562169 0.827023i \(-0.309967\pi\)
\(588\) 0 0
\(589\) −1.55735e11 −0.0533173
\(590\) 0 0
\(591\) −6.91232e12 −2.33067
\(592\) 0 0
\(593\) − 2.01077e12i − 0.667755i −0.942616 0.333878i \(-0.891643\pi\)
0.942616 0.333878i \(-0.108357\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 7.90857e12i − 2.54808i
\(598\) 0 0
\(599\) −3.29427e12 −1.04554 −0.522768 0.852475i \(-0.675101\pi\)
−0.522768 + 0.852475i \(0.675101\pi\)
\(600\) 0 0
\(601\) −1.98887e12 −0.621830 −0.310915 0.950438i \(-0.600635\pi\)
−0.310915 + 0.950438i \(0.600635\pi\)
\(602\) 0 0
\(603\) − 3.64488e12i − 1.12268i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 4.11631e12i − 1.23072i −0.788246 0.615360i \(-0.789011\pi\)
0.788246 0.615360i \(-0.210989\pi\)
\(608\) 0 0
\(609\) 7.63194e11 0.224832
\(610\) 0 0
\(611\) 2.47134e12 0.717376
\(612\) 0 0
\(613\) 2.06864e12i 0.591715i 0.955232 + 0.295858i \(0.0956054\pi\)
−0.955232 + 0.295858i \(0.904395\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 5.42058e12i − 1.50578i −0.658145 0.752891i \(-0.728659\pi\)
0.658145 0.752891i \(-0.271341\pi\)
\(618\) 0 0
\(619\) 3.87240e12 1.06016 0.530081 0.847947i \(-0.322162\pi\)
0.530081 + 0.847947i \(0.322162\pi\)
\(620\) 0 0
\(621\) 1.18447e12 0.319605
\(622\) 0 0
\(623\) − 3.71344e10i − 0.00987599i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 4.49480e12i 1.16147i
\(628\) 0 0
\(629\) −8.78623e11 −0.223807
\(630\) 0 0
\(631\) 3.22937e12 0.810933 0.405467 0.914110i \(-0.367109\pi\)
0.405467 + 0.914110i \(0.367109\pi\)
\(632\) 0 0
\(633\) − 3.15076e12i − 0.780006i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.90523e12i 0.699123i
\(638\) 0 0
\(639\) −1.09121e13 −2.58913
\(640\) 0 0
\(641\) 1.10935e12 0.259543 0.129771 0.991544i \(-0.458576\pi\)
0.129771 + 0.991544i \(0.458576\pi\)
\(642\) 0 0
\(643\) − 2.83993e12i − 0.655177i −0.944821 0.327588i \(-0.893764\pi\)
0.944821 0.327588i \(-0.106236\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.66962e12i 0.598935i 0.954107 + 0.299467i \(0.0968090\pi\)
−0.954107 + 0.299467i \(0.903191\pi\)
\(648\) 0 0
\(649\) −3.66033e12 −0.809878
\(650\) 0 0
\(651\) 7.81574e10 0.0170552
\(652\) 0 0
\(653\) − 4.72156e12i − 1.01619i −0.861300 0.508097i \(-0.830349\pi\)
0.861300 0.508097i \(-0.169651\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.58813e13i 3.32538i
\(658\) 0 0
\(659\) 4.01448e12 0.829173 0.414586 0.910010i \(-0.363926\pi\)
0.414586 + 0.910010i \(0.363926\pi\)
\(660\) 0 0
\(661\) −8.01591e12 −1.63323 −0.816613 0.577186i \(-0.804151\pi\)
−0.816613 + 0.577186i \(0.804151\pi\)
\(662\) 0 0
\(663\) 1.31535e12i 0.264380i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 5.98931e11i − 0.117168i
\(668\) 0 0
\(669\) 6.94065e12 1.33962
\(670\) 0 0
\(671\) 8.73040e12 1.66258
\(672\) 0 0
\(673\) − 3.55147e12i − 0.667330i −0.942692 0.333665i \(-0.891714\pi\)
0.942692 0.333665i \(-0.108286\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.06872e12i 1.11032i 0.831744 + 0.555160i \(0.187343\pi\)
−0.831744 + 0.555160i \(0.812657\pi\)
\(678\) 0 0
\(679\) 5.00773e11 0.0904122
\(680\) 0 0
\(681\) 6.74068e12 1.20100
\(682\) 0 0
\(683\) − 9.96514e11i − 0.175223i −0.996155 0.0876113i \(-0.972077\pi\)
0.996155 0.0876113i \(-0.0279233\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 6.19170e12i 1.06049i
\(688\) 0 0
\(689\) 6.12686e12 1.03574
\(690\) 0 0
\(691\) −1.02702e13 −1.71367 −0.856835 0.515591i \(-0.827572\pi\)
−0.856835 + 0.515591i \(0.827572\pi\)
\(692\) 0 0
\(693\) − 1.64063e12i − 0.270217i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.72786e12i 0.277308i
\(698\) 0 0
\(699\) −8.52194e12 −1.35018
\(700\) 0 0
\(701\) −8.26127e12 −1.29216 −0.646079 0.763270i \(-0.723592\pi\)
−0.646079 + 0.763270i \(0.723592\pi\)
\(702\) 0 0
\(703\) − 4.45236e12i − 0.687529i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 1.02251e11i − 0.0153915i
\(708\) 0 0
\(709\) 8.11428e12 1.20598 0.602992 0.797747i \(-0.293975\pi\)
0.602992 + 0.797747i \(0.293975\pi\)
\(710\) 0 0
\(711\) −1.95418e13 −2.86782
\(712\) 0 0
\(713\) − 6.13355e10i − 0.00888810i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.01254e12i 0.849614i
\(718\) 0 0
\(719\) 1.06399e13 1.48476 0.742380 0.669979i \(-0.233697\pi\)
0.742380 + 0.669979i \(0.233697\pi\)
\(720\) 0 0
\(721\) −8.88651e11 −0.122468
\(722\) 0 0
\(723\) − 5.66405e12i − 0.770911i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 1.35040e13i − 1.79290i −0.443145 0.896450i \(-0.646137\pi\)
0.443145 0.896450i \(-0.353863\pi\)
\(728\) 0 0
\(729\) −2.34766e13 −3.07866
\(730\) 0 0
\(731\) −2.30710e11 −0.0298840
\(732\) 0 0
\(733\) 3.03764e12i 0.388659i 0.980936 + 0.194329i \(0.0622530\pi\)
−0.980936 + 0.194329i \(0.937747\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.40502e12i 0.425124i
\(738\) 0 0
\(739\) 7.29483e12 0.899736 0.449868 0.893095i \(-0.351471\pi\)
0.449868 + 0.893095i \(0.351471\pi\)
\(740\) 0 0
\(741\) −6.66542e12 −0.812168
\(742\) 0 0
\(743\) 1.18034e13i 1.42088i 0.703756 + 0.710442i \(0.251505\pi\)
−0.703756 + 0.710442i \(0.748495\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 2.36279e13i 2.77641i
\(748\) 0 0
\(749\) 9.97041e11 0.115756
\(750\) 0 0
\(751\) −1.20855e13 −1.38639 −0.693197 0.720748i \(-0.743798\pi\)
−0.693197 + 0.720748i \(0.743798\pi\)
\(752\) 0 0
\(753\) 2.13993e13i 2.42562i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 1.48077e13i − 1.63891i −0.573144 0.819455i \(-0.694276\pi\)
0.573144 0.819455i \(-0.305724\pi\)
\(758\) 0 0
\(759\) −1.77025e12 −0.193619
\(760\) 0 0
\(761\) −1.75185e13 −1.89350 −0.946749 0.321972i \(-0.895654\pi\)
−0.946749 + 0.321972i \(0.895654\pi\)
\(762\) 0 0
\(763\) − 7.94656e11i − 0.0848826i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 5.42798e12i − 0.566316i
\(768\) 0 0
\(769\) −4.16729e12 −0.429720 −0.214860 0.976645i \(-0.568930\pi\)
−0.214860 + 0.976645i \(0.568930\pi\)
\(770\) 0 0
\(771\) −8.88781e12 −0.905837
\(772\) 0 0
\(773\) − 1.05896e13i − 1.06677i −0.845872 0.533386i \(-0.820919\pi\)
0.845872 0.533386i \(-0.179081\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 2.23446e12i 0.219927i
\(778\) 0 0
\(779\) −8.75583e12 −0.851881
\(780\) 0 0
\(781\) 1.01940e13 0.980426
\(782\) 0 0
\(783\) 3.93004e13i 3.73654i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 2.50861e12i 0.233102i 0.993185 + 0.116551i \(0.0371839\pi\)
−0.993185 + 0.116551i \(0.962816\pi\)
\(788\) 0 0
\(789\) 1.62549e13 1.49327
\(790\) 0 0
\(791\) 1.15376e12 0.104790
\(792\) 0 0
\(793\) 1.29465e13i 1.16258i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 1.85822e13i − 1.63131i −0.578541 0.815653i \(-0.696378\pi\)
0.578541 0.815653i \(-0.303622\pi\)
\(798\) 0 0
\(799\) −2.28760e12 −0.198573
\(800\) 0 0
\(801\) 3.05923e12 0.262583
\(802\) 0 0
\(803\) − 1.48362e13i − 1.25922i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 7.03720e12i − 0.584075i
\(808\) 0 0
\(809\) 4.47333e12 0.367166 0.183583 0.983004i \(-0.441230\pi\)
0.183583 + 0.983004i \(0.441230\pi\)
\(810\) 0 0
\(811\) 3.49582e12 0.283763 0.141881 0.989884i \(-0.454685\pi\)
0.141881 + 0.989884i \(0.454685\pi\)
\(812\) 0 0
\(813\) − 1.45846e13i − 1.17081i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 1.16911e12i − 0.0918026i
\(818\) 0 0
\(819\) 2.43293e12 0.188952
\(820\) 0 0
\(821\) −6.84197e12 −0.525578 −0.262789 0.964853i \(-0.584642\pi\)
−0.262789 + 0.964853i \(0.584642\pi\)
\(822\) 0 0
\(823\) − 8.11058e12i − 0.616244i −0.951347 0.308122i \(-0.900300\pi\)
0.951347 0.308122i \(-0.0997005\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 5.06349e10i − 0.00376422i −0.999998 0.00188211i \(-0.999401\pi\)
0.999998 0.00188211i \(-0.000599095\pi\)
\(828\) 0 0
\(829\) 1.50474e13 1.10654 0.553269 0.833002i \(-0.313380\pi\)
0.553269 + 0.833002i \(0.313380\pi\)
\(830\) 0 0
\(831\) 4.74356e13 3.45064
\(832\) 0 0
\(833\) − 2.68924e12i − 0.193521i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 4.02469e12i 0.283444i
\(838\) 0 0
\(839\) 2.68352e13 1.86972 0.934859 0.355020i \(-0.115526\pi\)
0.934859 + 0.355020i \(0.115526\pi\)
\(840\) 0 0
\(841\) 5.36519e12 0.369831
\(842\) 0 0
\(843\) − 2.12547e12i − 0.144954i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 3.00951e10i 0.00200919i
\(848\) 0 0
\(849\) −3.62176e13 −2.39241
\(850\) 0 0
\(851\) 1.75354e12 0.114613
\(852\) 0 0
\(853\) 4.22107e12i 0.272993i 0.990641 + 0.136497i \(0.0435843\pi\)
−0.990641 + 0.136497i \(0.956416\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 7.57740e12i 0.479851i 0.970791 + 0.239926i \(0.0771230\pi\)
−0.970791 + 0.239926i \(0.922877\pi\)
\(858\) 0 0
\(859\) 1.41463e13 0.886486 0.443243 0.896401i \(-0.353828\pi\)
0.443243 + 0.896401i \(0.353828\pi\)
\(860\) 0 0
\(861\) 4.39421e12 0.272500
\(862\) 0 0
\(863\) − 1.52498e13i − 0.935868i −0.883763 0.467934i \(-0.844998\pi\)
0.883763 0.467934i \(-0.155002\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 3.06427e13i 1.84180i
\(868\) 0 0
\(869\) 1.82558e13 1.08596
\(870\) 0 0
\(871\) −5.04937e12 −0.297273
\(872\) 0 0
\(873\) 4.12550e13i 2.40388i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 3.30350e12i − 0.188572i −0.995545 0.0942858i \(-0.969943\pi\)
0.995545 0.0942858i \(-0.0300568\pi\)
\(878\) 0 0
\(879\) −2.76142e13 −1.56021
\(880\) 0 0
\(881\) 3.14728e13 1.76013 0.880063 0.474856i \(-0.157500\pi\)
0.880063 + 0.474856i \(0.157500\pi\)
\(882\) 0 0
\(883\) − 1.10669e13i − 0.612637i −0.951929 0.306318i \(-0.900903\pi\)
0.951929 0.306318i \(-0.0990973\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 6.92035e12i − 0.375381i −0.982228 0.187690i \(-0.939900\pi\)
0.982228 0.187690i \(-0.0601001\pi\)
\(888\) 0 0
\(889\) 1.95308e12 0.104872
\(890\) 0 0
\(891\) 6.54838e13 3.48084
\(892\) 0 0
\(893\) − 1.15923e13i − 0.610010i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 2.62514e12i − 0.135390i
\(898\) 0 0
\(899\) 2.03509e12 0.103912
\(900\) 0 0
\(901\) −5.67136e12 −0.286699
\(902\) 0 0
\(903\) 5.86729e11i 0.0293659i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 2.00486e13i − 0.983676i −0.870687 0.491838i \(-0.836325\pi\)
0.870687 0.491838i \(-0.163675\pi\)
\(908\) 0 0
\(909\) 8.42371e12 0.409229
\(910\) 0 0
\(911\) −2.53798e13 −1.22083 −0.610416 0.792081i \(-0.708998\pi\)
−0.610416 + 0.792081i \(0.708998\pi\)
\(912\) 0 0
\(913\) − 2.20730e13i − 1.05134i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.16938e12i 0.0546129i
\(918\) 0 0
\(919\) −6.15926e12 −0.284845 −0.142423 0.989806i \(-0.545489\pi\)
−0.142423 + 0.989806i \(0.545489\pi\)
\(920\) 0 0
\(921\) 7.02482e13 3.21712
\(922\) 0 0
\(923\) 1.51169e13i 0.685574i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 7.32095e13i − 3.25618i
\(928\) 0 0
\(929\) −2.20000e13 −0.969061 −0.484531 0.874774i \(-0.661010\pi\)
−0.484531 + 0.874774i \(0.661010\pi\)
\(930\) 0 0
\(931\) 1.36275e13 0.594489
\(932\) 0 0
\(933\) 2.21601e13i 0.957426i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 2.49274e12i − 0.105645i −0.998604 0.0528225i \(-0.983178\pi\)
0.998604 0.0528225i \(-0.0168217\pi\)
\(938\) 0 0
\(939\) 3.64753e13 1.53110
\(940\) 0 0
\(941\) −1.26366e13 −0.525382 −0.262691 0.964880i \(-0.584610\pi\)
−0.262691 + 0.964880i \(0.584610\pi\)
\(942\) 0 0
\(943\) − 3.44844e12i − 0.142010i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 2.05809e13i − 0.831551i −0.909467 0.415775i \(-0.863510\pi\)
0.909467 0.415775i \(-0.136490\pi\)
\(948\) 0 0
\(949\) 2.20008e13 0.880524
\(950\) 0 0
\(951\) −2.43639e13 −0.965904
\(952\) 0 0
\(953\) − 2.97887e13i − 1.16986i −0.811084 0.584929i \(-0.801122\pi\)
0.811084 0.584929i \(-0.198878\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 5.87364e13i − 2.26362i
\(958\) 0 0
\(959\) −2.94675e12 −0.112502
\(960\) 0 0
\(961\) −2.62312e13 −0.992118
\(962\) 0 0
\(963\) 8.21389e13i 3.07773i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.74473e13i 0.641664i 0.947136 + 0.320832i \(0.103963\pi\)
−0.947136 + 0.320832i \(0.896037\pi\)
\(968\) 0 0
\(969\) 6.16988e12 0.224812
\(970\) 0 0
\(971\) 2.53844e13 0.916390 0.458195 0.888852i \(-0.348496\pi\)
0.458195 + 0.888852i \(0.348496\pi\)
\(972\) 0 0
\(973\) 1.63859e11i 0.00586088i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 3.23029e13i − 1.13427i −0.823625 0.567135i \(-0.808052\pi\)
0.823625 0.567135i \(-0.191948\pi\)
\(978\) 0 0
\(979\) −2.85791e12 −0.0994321
\(980\) 0 0
\(981\) 6.54659e13 2.25686
\(982\) 0 0
\(983\) 1.78014e13i 0.608083i 0.952659 + 0.304041i \(0.0983361\pi\)
−0.952659 + 0.304041i \(0.901664\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 5.81771e12i 0.195130i
\(988\) 0 0
\(989\) 4.60447e11 0.0153037
\(990\) 0 0
\(991\) 3.38638e12 0.111533 0.0557666 0.998444i \(-0.482240\pi\)
0.0557666 + 0.998444i \(0.482240\pi\)
\(992\) 0 0
\(993\) 9.27345e13i 3.02670i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 4.17766e13i − 1.33908i −0.742778 0.669538i \(-0.766492\pi\)
0.742778 0.669538i \(-0.233508\pi\)
\(998\) 0 0
\(999\) −1.15063e14 −3.65503
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.49.6 6
4.3 odd 2 25.10.b.c.24.6 6
5.2 odd 4 400.10.a.y.1.3 3
5.3 odd 4 400.10.a.u.1.1 3
5.4 even 2 inner 400.10.c.q.49.1 6
12.11 even 2 225.10.b.m.199.1 6
20.3 even 4 25.10.a.d.1.3 yes 3
20.7 even 4 25.10.a.c.1.1 3
20.19 odd 2 25.10.b.c.24.1 6
60.23 odd 4 225.10.a.m.1.1 3
60.47 odd 4 225.10.a.p.1.3 3
60.59 even 2 225.10.b.m.199.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.10.a.c.1.1 3 20.7 even 4
25.10.a.d.1.3 yes 3 20.3 even 4
25.10.b.c.24.1 6 20.19 odd 2
25.10.b.c.24.6 6 4.3 odd 2
225.10.a.m.1.1 3 60.23 odd 4
225.10.a.p.1.3 3 60.47 odd 4
225.10.b.m.199.1 6 12.11 even 2
225.10.b.m.199.6 6 60.59 even 2
400.10.a.u.1.1 3 5.3 odd 4
400.10.a.y.1.3 3 5.2 odd 4
400.10.c.q.49.1 6 5.4 even 2 inner
400.10.c.q.49.6 6 1.1 even 1 trivial