Properties

Label 400.10.a.u.1.3
Level $400$
Weight $10$
Character 400.1
Self dual yes
Analytic conductor $206.014$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,10,Mod(1,400)]
 
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, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 400.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(206.014334466\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 652x + 4000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 5 \)
Twist minimal: no (minimal twist has level 25)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(22.2334\) of defining polynomial
Character \(\chi\) \(=\) 400.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+210.171 q^{3} -9905.49 q^{7} +24489.0 q^{9} +O(q^{10})\) \(q+210.171 q^{3} -9905.49 q^{7} +24489.0 q^{9} -36453.6 q^{11} +164867. q^{13} +82357.1 q^{17} +609617. q^{19} -2.08185e6 q^{21} -1.88578e6 q^{23} +1.01008e6 q^{27} +339235. q^{29} -547314. q^{31} -7.66150e6 q^{33} +5.25687e6 q^{37} +3.46503e7 q^{39} +2.05812e6 q^{41} +6.76158e6 q^{43} -3.15241e7 q^{47} +5.77651e7 q^{49} +1.73091e7 q^{51} -4.89593e7 q^{53} +1.28124e8 q^{57} -8.77960e7 q^{59} +3.84654e7 q^{61} -2.42575e8 q^{63} +1.36116e8 q^{67} -3.96337e8 q^{69} -3.49218e8 q^{71} +1.61345e8 q^{73} +3.61091e8 q^{77} +1.26975e8 q^{79} -2.69727e8 q^{81} +2.87494e8 q^{83} +7.12976e7 q^{87} -5.63133e8 q^{89} -1.63309e9 q^{91} -1.15030e8 q^{93} -4.71704e8 q^{97} -8.92711e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 89 q^{3} - 5258 q^{7} + 58234 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 89 q^{3} - 5258 q^{7} + 58234 q^{9} + 54699 q^{11} + 215884 q^{13} + 334983 q^{17} - 818845 q^{19} - 2375394 q^{21} - 3526854 q^{23} - 6633395 q^{27} + 2175480 q^{29} - 4274066 q^{31} - 22122137 q^{33} - 10305042 q^{37} + 50414092 q^{39} + 5926311 q^{41} + 24429956 q^{43} - 66858708 q^{47} - 6453929 q^{49} + 25634699 q^{51} + 132620514 q^{53} + 252946415 q^{57} - 5670960 q^{59} + 125306926 q^{61} - 284323404 q^{63} - 88829483 q^{67} - 314274942 q^{69} - 297550596 q^{71} + 181321729 q^{73} + 561214086 q^{77} + 310025170 q^{79} + 1398847363 q^{81} + 731088801 q^{83} - 1046385560 q^{87} - 1103860035 q^{89} - 1183187656 q^{91} + 107386758 q^{93} - 332236842 q^{97} + 892234522 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 210.171 1.49805 0.749027 0.662539i \(-0.230521\pi\)
0.749027 + 0.662539i \(0.230521\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −9905.49 −1.55932 −0.779659 0.626204i \(-0.784608\pi\)
−0.779659 + 0.626204i \(0.784608\pi\)
\(8\) 0 0
\(9\) 24489.0 1.24417
\(10\) 0 0
\(11\) −36453.6 −0.750712 −0.375356 0.926881i \(-0.622480\pi\)
−0.375356 + 0.926881i \(0.622480\pi\)
\(12\) 0 0
\(13\) 164867. 1.60099 0.800494 0.599341i \(-0.204571\pi\)
0.800494 + 0.599341i \(0.204571\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 82357.1 0.239156 0.119578 0.992825i \(-0.461846\pi\)
0.119578 + 0.992825i \(0.461846\pi\)
\(18\) 0 0
\(19\) 609617. 1.07316 0.536582 0.843848i \(-0.319715\pi\)
0.536582 + 0.843848i \(0.319715\pi\)
\(20\) 0 0
\(21\) −2.08185e6 −2.33594
\(22\) 0 0
\(23\) −1.88578e6 −1.40513 −0.702564 0.711620i \(-0.747962\pi\)
−0.702564 + 0.711620i \(0.747962\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.01008e6 0.365778
\(28\) 0 0
\(29\) 339235. 0.0890657 0.0445328 0.999008i \(-0.485820\pi\)
0.0445328 + 0.999008i \(0.485820\pi\)
\(30\) 0 0
\(31\) −547314. −0.106441 −0.0532205 0.998583i \(-0.516949\pi\)
−0.0532205 + 0.998583i \(0.516949\pi\)
\(32\) 0 0
\(33\) −7.66150e6 −1.12461
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.25687e6 0.461126 0.230563 0.973057i \(-0.425943\pi\)
0.230563 + 0.973057i \(0.425943\pi\)
\(38\) 0 0
\(39\) 3.46503e7 2.39837
\(40\) 0 0
\(41\) 2.05812e6 0.113748 0.0568739 0.998381i \(-0.481887\pi\)
0.0568739 + 0.998381i \(0.481887\pi\)
\(42\) 0 0
\(43\) 6.76158e6 0.301606 0.150803 0.988564i \(-0.451814\pi\)
0.150803 + 0.988564i \(0.451814\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.15241e7 −0.942330 −0.471165 0.882045i \(-0.656166\pi\)
−0.471165 + 0.882045i \(0.656166\pi\)
\(48\) 0 0
\(49\) 5.77651e7 1.43147
\(50\) 0 0
\(51\) 1.73091e7 0.358268
\(52\) 0 0
\(53\) −4.89593e7 −0.852303 −0.426151 0.904652i \(-0.640131\pi\)
−0.426151 + 0.904652i \(0.640131\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.28124e8 1.60766
\(58\) 0 0
\(59\) −8.77960e7 −0.943281 −0.471640 0.881791i \(-0.656338\pi\)
−0.471640 + 0.881791i \(0.656338\pi\)
\(60\) 0 0
\(61\) 3.84654e7 0.355701 0.177851 0.984057i \(-0.443086\pi\)
0.177851 + 0.984057i \(0.443086\pi\)
\(62\) 0 0
\(63\) −2.42575e8 −1.94005
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.36116e8 0.825227 0.412613 0.910906i \(-0.364616\pi\)
0.412613 + 0.910906i \(0.364616\pi\)
\(68\) 0 0
\(69\) −3.96337e8 −2.10496
\(70\) 0 0
\(71\) −3.49218e8 −1.63092 −0.815462 0.578810i \(-0.803517\pi\)
−0.815462 + 0.578810i \(0.803517\pi\)
\(72\) 0 0
\(73\) 1.61345e8 0.664971 0.332486 0.943108i \(-0.392113\pi\)
0.332486 + 0.943108i \(0.392113\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.61091e8 1.17060
\(78\) 0 0
\(79\) 1.26975e8 0.366772 0.183386 0.983041i \(-0.441294\pi\)
0.183386 + 0.983041i \(0.441294\pi\)
\(80\) 0 0
\(81\) −2.69727e8 −0.696213
\(82\) 0 0
\(83\) 2.87494e8 0.664931 0.332466 0.943115i \(-0.392119\pi\)
0.332466 + 0.943115i \(0.392119\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 7.12976e7 0.133425
\(88\) 0 0
\(89\) −5.63133e8 −0.951385 −0.475692 0.879612i \(-0.657802\pi\)
−0.475692 + 0.879612i \(0.657802\pi\)
\(90\) 0 0
\(91\) −1.63309e9 −2.49645
\(92\) 0 0
\(93\) −1.15030e8 −0.159454
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.71704e8 −0.541000 −0.270500 0.962720i \(-0.587189\pi\)
−0.270500 + 0.962720i \(0.587189\pi\)
\(98\) 0 0
\(99\) −8.92711e8 −0.934012
\(100\) 0 0
\(101\) −1.25673e9 −1.20170 −0.600851 0.799361i \(-0.705172\pi\)
−0.600851 + 0.799361i \(0.705172\pi\)
\(102\) 0 0
\(103\) −1.95398e9 −1.71062 −0.855309 0.518119i \(-0.826633\pi\)
−0.855309 + 0.518119i \(0.826633\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.12306e9 −1.56580 −0.782899 0.622148i \(-0.786260\pi\)
−0.782899 + 0.622148i \(0.786260\pi\)
\(108\) 0 0
\(109\) −1.18863e8 −0.0806544 −0.0403272 0.999187i \(-0.512840\pi\)
−0.0403272 + 0.999187i \(0.512840\pi\)
\(110\) 0 0
\(111\) 1.10484e9 0.690791
\(112\) 0 0
\(113\) −1.85455e9 −1.07001 −0.535003 0.844850i \(-0.679689\pi\)
−0.535003 + 0.844850i \(0.679689\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.03742e9 1.99190
\(118\) 0 0
\(119\) −8.15787e8 −0.372920
\(120\) 0 0
\(121\) −1.02908e9 −0.436432
\(122\) 0 0
\(123\) 4.32557e8 0.170400
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 5.39208e9 1.83924 0.919622 0.392805i \(-0.128495\pi\)
0.919622 + 0.392805i \(0.128495\pi\)
\(128\) 0 0
\(129\) 1.42109e9 0.451823
\(130\) 0 0
\(131\) −3.83690e9 −1.13831 −0.569154 0.822231i \(-0.692729\pi\)
−0.569154 + 0.822231i \(0.692729\pi\)
\(132\) 0 0
\(133\) −6.03856e9 −1.67340
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.55408e9 −0.619430 −0.309715 0.950829i \(-0.600234\pi\)
−0.309715 + 0.950829i \(0.600234\pi\)
\(138\) 0 0
\(139\) 4.09121e9 0.929577 0.464789 0.885422i \(-0.346130\pi\)
0.464789 + 0.885422i \(0.346130\pi\)
\(140\) 0 0
\(141\) −6.62547e9 −1.41166
\(142\) 0 0
\(143\) −6.00999e9 −1.20188
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.21406e10 2.14443
\(148\) 0 0
\(149\) −6.13643e9 −1.01995 −0.509974 0.860190i \(-0.670345\pi\)
−0.509974 + 0.860190i \(0.670345\pi\)
\(150\) 0 0
\(151\) 5.83754e9 0.913763 0.456882 0.889528i \(-0.348966\pi\)
0.456882 + 0.889528i \(0.348966\pi\)
\(152\) 0 0
\(153\) 2.01684e9 0.297550
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1.35031e9 −0.177371 −0.0886857 0.996060i \(-0.528267\pi\)
−0.0886857 + 0.996060i \(0.528267\pi\)
\(158\) 0 0
\(159\) −1.02898e10 −1.27680
\(160\) 0 0
\(161\) 1.86796e10 2.19104
\(162\) 0 0
\(163\) 4.50806e9 0.500202 0.250101 0.968220i \(-0.419536\pi\)
0.250101 + 0.968220i \(0.419536\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.81836e10 −1.80908 −0.904538 0.426393i \(-0.859784\pi\)
−0.904538 + 0.426393i \(0.859784\pi\)
\(168\) 0 0
\(169\) 1.65766e10 1.56316
\(170\) 0 0
\(171\) 1.49289e10 1.33520
\(172\) 0 0
\(173\) 9.77143e9 0.829375 0.414687 0.909964i \(-0.363891\pi\)
0.414687 + 0.909964i \(0.363891\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1.84522e10 −1.41309
\(178\) 0 0
\(179\) −6.65866e9 −0.484784 −0.242392 0.970178i \(-0.577932\pi\)
−0.242392 + 0.970178i \(0.577932\pi\)
\(180\) 0 0
\(181\) −7.32873e9 −0.507546 −0.253773 0.967264i \(-0.581672\pi\)
−0.253773 + 0.967264i \(0.581672\pi\)
\(182\) 0 0
\(183\) 8.08432e9 0.532860
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −3.00221e9 −0.179537
\(188\) 0 0
\(189\) −1.00053e10 −0.570364
\(190\) 0 0
\(191\) −8.06956e9 −0.438732 −0.219366 0.975643i \(-0.570399\pi\)
−0.219366 + 0.975643i \(0.570399\pi\)
\(192\) 0 0
\(193\) 3.12603e9 0.162176 0.0810878 0.996707i \(-0.474161\pi\)
0.0810878 + 0.996707i \(0.474161\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.40133e10 −1.13594 −0.567968 0.823051i \(-0.692270\pi\)
−0.567968 + 0.823051i \(0.692270\pi\)
\(198\) 0 0
\(199\) −9.73605e9 −0.440093 −0.220046 0.975489i \(-0.570621\pi\)
−0.220046 + 0.975489i \(0.570621\pi\)
\(200\) 0 0
\(201\) 2.86077e10 1.23623
\(202\) 0 0
\(203\) −3.36029e9 −0.138882
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −4.61808e10 −1.74822
\(208\) 0 0
\(209\) −2.22227e10 −0.805637
\(210\) 0 0
\(211\) 2.39305e10 0.831151 0.415576 0.909559i \(-0.363580\pi\)
0.415576 + 0.909559i \(0.363580\pi\)
\(212\) 0 0
\(213\) −7.33956e10 −2.44321
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 5.42141e9 0.165975
\(218\) 0 0
\(219\) 3.39101e10 0.996163
\(220\) 0 0
\(221\) 1.35779e10 0.382885
\(222\) 0 0
\(223\) −2.02289e10 −0.547773 −0.273886 0.961762i \(-0.588309\pi\)
−0.273886 + 0.961762i \(0.588309\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −4.42152e10 −1.10524 −0.552618 0.833434i \(-0.686371\pi\)
−0.552618 + 0.833434i \(0.686371\pi\)
\(228\) 0 0
\(229\) −4.89483e10 −1.17619 −0.588096 0.808791i \(-0.700122\pi\)
−0.588096 + 0.808791i \(0.700122\pi\)
\(230\) 0 0
\(231\) 7.58909e10 1.75362
\(232\) 0 0
\(233\) 6.68132e10 1.48512 0.742559 0.669781i \(-0.233612\pi\)
0.742559 + 0.669781i \(0.233612\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 2.66865e10 0.549444
\(238\) 0 0
\(239\) 2.21153e10 0.438431 0.219216 0.975676i \(-0.429650\pi\)
0.219216 + 0.975676i \(0.429650\pi\)
\(240\) 0 0
\(241\) 9.97637e10 1.90500 0.952502 0.304532i \(-0.0985002\pi\)
0.952502 + 0.304532i \(0.0985002\pi\)
\(242\) 0 0
\(243\) −7.65703e10 −1.40874
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.00506e11 1.71812
\(248\) 0 0
\(249\) 6.04229e10 0.996104
\(250\) 0 0
\(251\) 4.97276e10 0.790799 0.395399 0.918509i \(-0.370606\pi\)
0.395399 + 0.918509i \(0.370606\pi\)
\(252\) 0 0
\(253\) 6.87435e10 1.05485
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.81308e10 0.831204 0.415602 0.909547i \(-0.363571\pi\)
0.415602 + 0.909547i \(0.363571\pi\)
\(258\) 0 0
\(259\) −5.20718e10 −0.719041
\(260\) 0 0
\(261\) 8.30753e9 0.110813
\(262\) 0 0
\(263\) 3.23476e10 0.416909 0.208454 0.978032i \(-0.433157\pi\)
0.208454 + 0.978032i \(0.433157\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −1.18354e11 −1.42523
\(268\) 0 0
\(269\) 6.24220e9 0.0726863 0.0363431 0.999339i \(-0.488429\pi\)
0.0363431 + 0.999339i \(0.488429\pi\)
\(270\) 0 0
\(271\) −8.05816e10 −0.907557 −0.453779 0.891114i \(-0.649924\pi\)
−0.453779 + 0.891114i \(0.649924\pi\)
\(272\) 0 0
\(273\) −3.43228e11 −3.73982
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −7.07976e10 −0.722536 −0.361268 0.932462i \(-0.617656\pi\)
−0.361268 + 0.932462i \(0.617656\pi\)
\(278\) 0 0
\(279\) −1.34031e10 −0.132430
\(280\) 0 0
\(281\) 8.15619e10 0.780385 0.390193 0.920733i \(-0.372408\pi\)
0.390193 + 0.920733i \(0.372408\pi\)
\(282\) 0 0
\(283\) −1.89301e11 −1.75434 −0.877168 0.480184i \(-0.840570\pi\)
−0.877168 + 0.480184i \(0.840570\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.03867e10 −0.177369
\(288\) 0 0
\(289\) −1.11805e11 −0.942805
\(290\) 0 0
\(291\) −9.91387e10 −0.810447
\(292\) 0 0
\(293\) −5.39489e10 −0.427640 −0.213820 0.976873i \(-0.568591\pi\)
−0.213820 + 0.976873i \(0.568591\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −3.68209e10 −0.274594
\(298\) 0 0
\(299\) −3.10903e11 −2.24959
\(300\) 0 0
\(301\) −6.69768e10 −0.470300
\(302\) 0 0
\(303\) −2.64129e11 −1.80022
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.30037e11 0.835496 0.417748 0.908563i \(-0.362819\pi\)
0.417748 + 0.908563i \(0.362819\pi\)
\(308\) 0 0
\(309\) −4.10671e11 −2.56260
\(310\) 0 0
\(311\) −2.09580e9 −0.0127036 −0.00635182 0.999980i \(-0.502022\pi\)
−0.00635182 + 0.999980i \(0.502022\pi\)
\(312\) 0 0
\(313\) −2.43788e11 −1.43570 −0.717848 0.696200i \(-0.754873\pi\)
−0.717848 + 0.696200i \(0.754873\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.46483e11 −1.92715 −0.963573 0.267445i \(-0.913821\pi\)
−0.963573 + 0.267445i \(0.913821\pi\)
\(318\) 0 0
\(319\) −1.23664e10 −0.0668626
\(320\) 0 0
\(321\) −4.46207e11 −2.34565
\(322\) 0 0
\(323\) 5.02063e10 0.256653
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −2.49816e10 −0.120825
\(328\) 0 0
\(329\) 3.12262e11 1.46939
\(330\) 0 0
\(331\) 1.43303e11 0.656189 0.328094 0.944645i \(-0.393594\pi\)
0.328094 + 0.944645i \(0.393594\pi\)
\(332\) 0 0
\(333\) 1.28735e11 0.573718
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 3.34820e11 1.41409 0.707044 0.707170i \(-0.250028\pi\)
0.707044 + 0.707170i \(0.250028\pi\)
\(338\) 0 0
\(339\) −3.89773e11 −1.60293
\(340\) 0 0
\(341\) 1.99515e10 0.0799064
\(342\) 0 0
\(343\) −1.72469e11 −0.672804
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9.22101e10 0.341425 0.170713 0.985321i \(-0.445393\pi\)
0.170713 + 0.985321i \(0.445393\pi\)
\(348\) 0 0
\(349\) −4.71738e11 −1.70211 −0.851053 0.525080i \(-0.824035\pi\)
−0.851053 + 0.525080i \(0.824035\pi\)
\(350\) 0 0
\(351\) 1.66528e11 0.585606
\(352\) 0 0
\(353\) −4.01029e11 −1.37464 −0.687322 0.726353i \(-0.741214\pi\)
−0.687322 + 0.726353i \(0.741214\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −1.71455e11 −0.558654
\(358\) 0 0
\(359\) 2.36110e10 0.0750220 0.0375110 0.999296i \(-0.488057\pi\)
0.0375110 + 0.999296i \(0.488057\pi\)
\(360\) 0 0
\(361\) 4.89457e10 0.151681
\(362\) 0 0
\(363\) −2.16284e11 −0.653799
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.23035e11 −0.354022 −0.177011 0.984209i \(-0.556643\pi\)
−0.177011 + 0.984209i \(0.556643\pi\)
\(368\) 0 0
\(369\) 5.04012e10 0.141521
\(370\) 0 0
\(371\) 4.84966e11 1.32901
\(372\) 0 0
\(373\) −2.39248e11 −0.639968 −0.319984 0.947423i \(-0.603678\pi\)
−0.319984 + 0.947423i \(0.603678\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.59287e10 0.142593
\(378\) 0 0
\(379\) 5.90629e11 1.47041 0.735205 0.677845i \(-0.237086\pi\)
0.735205 + 0.677845i \(0.237086\pi\)
\(380\) 0 0
\(381\) 1.13326e12 2.75529
\(382\) 0 0
\(383\) 5.30525e10 0.125983 0.0629914 0.998014i \(-0.479936\pi\)
0.0629914 + 0.998014i \(0.479936\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.65584e11 0.375249
\(388\) 0 0
\(389\) 3.10395e11 0.687293 0.343646 0.939099i \(-0.388338\pi\)
0.343646 + 0.939099i \(0.388338\pi\)
\(390\) 0 0
\(391\) −1.55307e11 −0.336045
\(392\) 0 0
\(393\) −8.06407e11 −1.70525
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −2.09351e11 −0.422979 −0.211489 0.977380i \(-0.567831\pi\)
−0.211489 + 0.977380i \(0.567831\pi\)
\(398\) 0 0
\(399\) −1.26913e12 −2.50685
\(400\) 0 0
\(401\) 4.43868e11 0.857244 0.428622 0.903484i \(-0.358999\pi\)
0.428622 + 0.903484i \(0.358999\pi\)
\(402\) 0 0
\(403\) −9.02338e10 −0.170411
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.91632e11 −0.346172
\(408\) 0 0
\(409\) 4.56031e11 0.805823 0.402912 0.915239i \(-0.367998\pi\)
0.402912 + 0.915239i \(0.367998\pi\)
\(410\) 0 0
\(411\) −5.36794e11 −0.927940
\(412\) 0 0
\(413\) 8.69663e11 1.47087
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 8.59856e11 1.39256
\(418\) 0 0
\(419\) 1.12392e12 1.78144 0.890720 0.454552i \(-0.150201\pi\)
0.890720 + 0.454552i \(0.150201\pi\)
\(420\) 0 0
\(421\) −1.02822e12 −1.59521 −0.797605 0.603180i \(-0.793900\pi\)
−0.797605 + 0.603180i \(0.793900\pi\)
\(422\) 0 0
\(423\) −7.71994e11 −1.17242
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −3.81018e11 −0.554652
\(428\) 0 0
\(429\) −1.26313e12 −1.80048
\(430\) 0 0
\(431\) 1.37714e12 1.92234 0.961171 0.275953i \(-0.0889934\pi\)
0.961171 + 0.275953i \(0.0889934\pi\)
\(432\) 0 0
\(433\) 1.94790e11 0.266299 0.133150 0.991096i \(-0.457491\pi\)
0.133150 + 0.991096i \(0.457491\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.14961e12 −1.50793
\(438\) 0 0
\(439\) 5.32533e11 0.684315 0.342157 0.939643i \(-0.388842\pi\)
0.342157 + 0.939643i \(0.388842\pi\)
\(440\) 0 0
\(441\) 1.41461e12 1.78099
\(442\) 0 0
\(443\) −7.26633e11 −0.896393 −0.448196 0.893935i \(-0.647933\pi\)
−0.448196 + 0.893935i \(0.647933\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −1.28970e12 −1.52794
\(448\) 0 0
\(449\) 9.34830e11 1.08549 0.542743 0.839899i \(-0.317386\pi\)
0.542743 + 0.839899i \(0.317386\pi\)
\(450\) 0 0
\(451\) −7.50258e10 −0.0853918
\(452\) 0 0
\(453\) 1.22688e12 1.36887
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.09190e12 −1.17101 −0.585503 0.810670i \(-0.699103\pi\)
−0.585503 + 0.810670i \(0.699103\pi\)
\(458\) 0 0
\(459\) 8.31870e10 0.0874779
\(460\) 0 0
\(461\) 9.93683e11 1.02469 0.512347 0.858779i \(-0.328776\pi\)
0.512347 + 0.858779i \(0.328776\pi\)
\(462\) 0 0
\(463\) 6.95734e11 0.703605 0.351802 0.936074i \(-0.385569\pi\)
0.351802 + 0.936074i \(0.385569\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.87957e12 −1.82866 −0.914328 0.404975i \(-0.867280\pi\)
−0.914328 + 0.404975i \(0.867280\pi\)
\(468\) 0 0
\(469\) −1.34830e12 −1.28679
\(470\) 0 0
\(471\) −2.83795e11 −0.265712
\(472\) 0 0
\(473\) −2.46484e11 −0.226419
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −1.19896e12 −1.06041
\(478\) 0 0
\(479\) 2.29011e11 0.198768 0.0993838 0.995049i \(-0.468313\pi\)
0.0993838 + 0.995049i \(0.468313\pi\)
\(480\) 0 0
\(481\) 8.66683e11 0.738256
\(482\) 0 0
\(483\) 3.92591e12 3.28230
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 4.81410e11 0.387824 0.193912 0.981019i \(-0.437882\pi\)
0.193912 + 0.981019i \(0.437882\pi\)
\(488\) 0 0
\(489\) 9.47465e11 0.749330
\(490\) 0 0
\(491\) 1.70320e12 1.32251 0.661257 0.750160i \(-0.270023\pi\)
0.661257 + 0.750160i \(0.270023\pi\)
\(492\) 0 0
\(493\) 2.79384e10 0.0213006
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.45917e12 2.54313
\(498\) 0 0
\(499\) −2.06075e12 −1.48789 −0.743947 0.668238i \(-0.767049\pi\)
−0.743947 + 0.668238i \(0.767049\pi\)
\(500\) 0 0
\(501\) −3.82168e12 −2.71010
\(502\) 0 0
\(503\) 5.97493e11 0.416176 0.208088 0.978110i \(-0.433276\pi\)
0.208088 + 0.978110i \(0.433276\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 3.48392e12 2.34170
\(508\) 0 0
\(509\) −8.46938e11 −0.559270 −0.279635 0.960106i \(-0.590213\pi\)
−0.279635 + 0.960106i \(0.590213\pi\)
\(510\) 0 0
\(511\) −1.59820e12 −1.03690
\(512\) 0 0
\(513\) 6.15760e11 0.392540
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 1.14917e12 0.707418
\(518\) 0 0
\(519\) 2.05367e12 1.24245
\(520\) 0 0
\(521\) 7.31013e11 0.434666 0.217333 0.976098i \(-0.430264\pi\)
0.217333 + 0.976098i \(0.430264\pi\)
\(522\) 0 0
\(523\) −1.04453e12 −0.610467 −0.305233 0.952278i \(-0.598734\pi\)
−0.305233 + 0.952278i \(0.598734\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.50751e10 −0.0254559
\(528\) 0 0
\(529\) 1.75502e12 0.974386
\(530\) 0 0
\(531\) −2.15003e12 −1.17360
\(532\) 0 0
\(533\) 3.39315e11 0.182109
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −1.39946e12 −0.726233
\(538\) 0 0
\(539\) −2.10575e12 −1.07462
\(540\) 0 0
\(541\) −1.88034e12 −0.943733 −0.471866 0.881670i \(-0.656420\pi\)
−0.471866 + 0.881670i \(0.656420\pi\)
\(542\) 0 0
\(543\) −1.54029e12 −0.760331
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.08624e12 0.518780 0.259390 0.965773i \(-0.416479\pi\)
0.259390 + 0.965773i \(0.416479\pi\)
\(548\) 0 0
\(549\) 9.41977e11 0.442553
\(550\) 0 0
\(551\) 2.06804e11 0.0955821
\(552\) 0 0
\(553\) −1.25775e12 −0.571914
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.11965e12 0.933072 0.466536 0.884502i \(-0.345502\pi\)
0.466536 + 0.884502i \(0.345502\pi\)
\(558\) 0 0
\(559\) 1.11476e12 0.482868
\(560\) 0 0
\(561\) −6.30978e11 −0.268956
\(562\) 0 0
\(563\) −3.04489e12 −1.27727 −0.638636 0.769509i \(-0.720501\pi\)
−0.638636 + 0.769509i \(0.720501\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.67178e12 1.08562
\(568\) 0 0
\(569\) 3.25275e12 1.30090 0.650452 0.759548i \(-0.274580\pi\)
0.650452 + 0.759548i \(0.274580\pi\)
\(570\) 0 0
\(571\) 1.35916e12 0.535066 0.267533 0.963549i \(-0.413792\pi\)
0.267533 + 0.963549i \(0.413792\pi\)
\(572\) 0 0
\(573\) −1.69599e12 −0.657245
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −2.35782e12 −0.885564 −0.442782 0.896629i \(-0.646008\pi\)
−0.442782 + 0.896629i \(0.646008\pi\)
\(578\) 0 0
\(579\) 6.57002e11 0.242948
\(580\) 0 0
\(581\) −2.84777e12 −1.03684
\(582\) 0 0
\(583\) 1.78474e12 0.639834
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.37065e12 0.824130 0.412065 0.911154i \(-0.364808\pi\)
0.412065 + 0.911154i \(0.364808\pi\)
\(588\) 0 0
\(589\) −3.33652e11 −0.114229
\(590\) 0 0
\(591\) −5.04690e12 −1.70169
\(592\) 0 0
\(593\) 4.56707e12 1.51667 0.758336 0.651864i \(-0.226013\pi\)
0.758336 + 0.651864i \(0.226013\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −2.04624e12 −0.659283
\(598\) 0 0
\(599\) 5.30493e12 1.68368 0.841839 0.539729i \(-0.181473\pi\)
0.841839 + 0.539729i \(0.181473\pi\)
\(600\) 0 0
\(601\) −2.71307e12 −0.848255 −0.424128 0.905602i \(-0.639419\pi\)
−0.424128 + 0.905602i \(0.639419\pi\)
\(602\) 0 0
\(603\) 3.33335e12 1.02672
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 4.00734e12 1.19814 0.599069 0.800698i \(-0.295538\pi\)
0.599069 + 0.800698i \(0.295538\pi\)
\(608\) 0 0
\(609\) −7.06237e11 −0.208052
\(610\) 0 0
\(611\) −5.19728e12 −1.50866
\(612\) 0 0
\(613\) −2.24954e12 −0.643459 −0.321730 0.946832i \(-0.604264\pi\)
−0.321730 + 0.946832i \(0.604264\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.19664e12 0.332414 0.166207 0.986091i \(-0.446848\pi\)
0.166207 + 0.986091i \(0.446848\pi\)
\(618\) 0 0
\(619\) 4.36723e12 1.19563 0.597817 0.801633i \(-0.296035\pi\)
0.597817 + 0.801633i \(0.296035\pi\)
\(620\) 0 0
\(621\) −1.90478e12 −0.513965
\(622\) 0 0
\(623\) 5.57811e12 1.48351
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −4.67058e12 −1.20689
\(628\) 0 0
\(629\) 4.32940e11 0.110281
\(630\) 0 0
\(631\) −2.78790e12 −0.700077 −0.350039 0.936735i \(-0.613832\pi\)
−0.350039 + 0.936735i \(0.613832\pi\)
\(632\) 0 0
\(633\) 5.02950e12 1.24511
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 9.52355e12 2.29177
\(638\) 0 0
\(639\) −8.55199e12 −2.02915
\(640\) 0 0
\(641\) −2.28295e12 −0.534115 −0.267058 0.963681i \(-0.586051\pi\)
−0.267058 + 0.963681i \(0.586051\pi\)
\(642\) 0 0
\(643\) 2.29995e12 0.530603 0.265301 0.964166i \(-0.414529\pi\)
0.265301 + 0.964166i \(0.414529\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.85647e12 −0.416503 −0.208251 0.978075i \(-0.566777\pi\)
−0.208251 + 0.978075i \(0.566777\pi\)
\(648\) 0 0
\(649\) 3.20048e12 0.708132
\(650\) 0 0
\(651\) 1.13942e12 0.248640
\(652\) 0 0
\(653\) 7.75397e12 1.66884 0.834420 0.551129i \(-0.185803\pi\)
0.834420 + 0.551129i \(0.185803\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 3.95117e12 0.827336
\(658\) 0 0
\(659\) −7.42166e12 −1.53291 −0.766455 0.642298i \(-0.777981\pi\)
−0.766455 + 0.642298i \(0.777981\pi\)
\(660\) 0 0
\(661\) 5.79783e12 1.18130 0.590648 0.806929i \(-0.298872\pi\)
0.590648 + 0.806929i \(0.298872\pi\)
\(662\) 0 0
\(663\) 2.85369e12 0.573583
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −6.39724e11 −0.125149
\(668\) 0 0
\(669\) −4.25153e12 −0.820593
\(670\) 0 0
\(671\) −1.40220e12 −0.267029
\(672\) 0 0
\(673\) −6.32456e12 −1.18840 −0.594200 0.804318i \(-0.702531\pi\)
−0.594200 + 0.804318i \(0.702531\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −4.95066e12 −0.905762 −0.452881 0.891571i \(-0.649604\pi\)
−0.452881 + 0.891571i \(0.649604\pi\)
\(678\) 0 0
\(679\) 4.67246e12 0.843591
\(680\) 0 0
\(681\) −9.29277e12 −1.65571
\(682\) 0 0
\(683\) −7.32611e12 −1.28819 −0.644095 0.764945i \(-0.722766\pi\)
−0.644095 + 0.764945i \(0.722766\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −1.02875e13 −1.76200
\(688\) 0 0
\(689\) −8.07176e12 −1.36453
\(690\) 0 0
\(691\) 5.66806e12 0.945765 0.472882 0.881126i \(-0.343214\pi\)
0.472882 + 0.881126i \(0.343214\pi\)
\(692\) 0 0
\(693\) 8.84274e12 1.45642
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.69501e11 0.0272034
\(698\) 0 0
\(699\) 1.40422e13 2.22479
\(700\) 0 0
\(701\) −4.20985e12 −0.658470 −0.329235 0.944248i \(-0.606791\pi\)
−0.329235 + 0.944248i \(0.606791\pi\)
\(702\) 0 0
\(703\) 3.20468e12 0.494863
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.24486e13 1.87384
\(708\) 0 0
\(709\) 1.19654e13 1.77835 0.889176 0.457566i \(-0.151279\pi\)
0.889176 + 0.457566i \(0.151279\pi\)
\(710\) 0 0
\(711\) 3.10949e12 0.456326
\(712\) 0 0
\(713\) 1.03211e12 0.149563
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 4.64799e12 0.656794
\(718\) 0 0
\(719\) 4.07118e11 0.0568120 0.0284060 0.999596i \(-0.490957\pi\)
0.0284060 + 0.999596i \(0.490957\pi\)
\(720\) 0 0
\(721\) 1.93551e13 2.66740
\(722\) 0 0
\(723\) 2.09675e13 2.85380
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −6.44424e12 −0.855592 −0.427796 0.903875i \(-0.640710\pi\)
−0.427796 + 0.903875i \(0.640710\pi\)
\(728\) 0 0
\(729\) −1.07838e13 −1.41416
\(730\) 0 0
\(731\) 5.56864e11 0.0721308
\(732\) 0 0
\(733\) 5.46122e12 0.698749 0.349375 0.936983i \(-0.386394\pi\)
0.349375 + 0.936983i \(0.386394\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −4.96192e12 −0.619507
\(738\) 0 0
\(739\) 1.88201e12 0.232125 0.116062 0.993242i \(-0.462973\pi\)
0.116062 + 0.993242i \(0.462973\pi\)
\(740\) 0 0
\(741\) 2.11234e13 2.57384
\(742\) 0 0
\(743\) −1.36704e13 −1.64563 −0.822813 0.568312i \(-0.807597\pi\)
−0.822813 + 0.568312i \(0.807597\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 7.04042e12 0.827287
\(748\) 0 0
\(749\) 2.10300e13 2.44158
\(750\) 0 0
\(751\) −8.52585e12 −0.978043 −0.489021 0.872272i \(-0.662646\pi\)
−0.489021 + 0.872272i \(0.662646\pi\)
\(752\) 0 0
\(753\) 1.04513e13 1.18466
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −7.50792e12 −0.830975 −0.415487 0.909599i \(-0.636389\pi\)
−0.415487 + 0.909599i \(0.636389\pi\)
\(758\) 0 0
\(759\) 1.44479e13 1.58022
\(760\) 0 0
\(761\) −1.61560e13 −1.74624 −0.873118 0.487509i \(-0.837906\pi\)
−0.873118 + 0.487509i \(0.837906\pi\)
\(762\) 0 0
\(763\) 1.17740e12 0.125766
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.44747e13 −1.51018
\(768\) 0 0
\(769\) 1.11830e12 0.115317 0.0576583 0.998336i \(-0.481637\pi\)
0.0576583 + 0.998336i \(0.481637\pi\)
\(770\) 0 0
\(771\) 1.22174e13 1.24519
\(772\) 0 0
\(773\) −1.73165e13 −1.74442 −0.872212 0.489129i \(-0.837315\pi\)
−0.872212 + 0.489129i \(0.837315\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −1.09440e13 −1.07716
\(778\) 0 0
\(779\) 1.25466e12 0.122070
\(780\) 0 0
\(781\) 1.27302e13 1.22435
\(782\) 0 0
\(783\) 3.42654e11 0.0325783
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 3.88515e11 0.0361012 0.0180506 0.999837i \(-0.494254\pi\)
0.0180506 + 0.999837i \(0.494254\pi\)
\(788\) 0 0
\(789\) 6.79853e12 0.624552
\(790\) 0 0
\(791\) 1.83702e13 1.66848
\(792\) 0 0
\(793\) 6.34166e12 0.569474
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.08432e12 −0.0951910 −0.0475955 0.998867i \(-0.515156\pi\)
−0.0475955 + 0.998867i \(0.515156\pi\)
\(798\) 0 0
\(799\) −2.59624e12 −0.225363
\(800\) 0 0
\(801\) −1.37906e13 −1.18368
\(802\) 0 0
\(803\) −5.88161e12 −0.499202
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 1.31193e12 0.108888
\(808\) 0 0
\(809\) 4.06803e12 0.333900 0.166950 0.985965i \(-0.446608\pi\)
0.166950 + 0.985965i \(0.446608\pi\)
\(810\) 0 0
\(811\) −4.87468e12 −0.395688 −0.197844 0.980234i \(-0.563394\pi\)
−0.197844 + 0.980234i \(0.563394\pi\)
\(812\) 0 0
\(813\) −1.69359e13 −1.35957
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 4.12198e12 0.323673
\(818\) 0 0
\(819\) −3.99926e13 −3.10600
\(820\) 0 0
\(821\) −1.03829e12 −0.0797582 −0.0398791 0.999205i \(-0.512697\pi\)
−0.0398791 + 0.999205i \(0.512697\pi\)
\(822\) 0 0
\(823\) −1.10923e13 −0.842794 −0.421397 0.906876i \(-0.638460\pi\)
−0.421397 + 0.906876i \(0.638460\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.45526e13 1.08184 0.540922 0.841073i \(-0.318075\pi\)
0.540922 + 0.841073i \(0.318075\pi\)
\(828\) 0 0
\(829\) 1.37985e13 1.01470 0.507348 0.861741i \(-0.330626\pi\)
0.507348 + 0.861741i \(0.330626\pi\)
\(830\) 0 0
\(831\) −1.48796e13 −1.08240
\(832\) 0 0
\(833\) 4.75736e12 0.342345
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −5.52829e11 −0.0389337
\(838\) 0 0
\(839\) −9.20234e11 −0.0641165 −0.0320582 0.999486i \(-0.510206\pi\)
−0.0320582 + 0.999486i \(0.510206\pi\)
\(840\) 0 0
\(841\) −1.43921e13 −0.992067
\(842\) 0 0
\(843\) 1.71420e13 1.16906
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1.01936e13 0.680536
\(848\) 0 0
\(849\) −3.97855e13 −2.62809
\(850\) 0 0
\(851\) −9.91330e12 −0.647941
\(852\) 0 0
\(853\) −7.46964e12 −0.483091 −0.241545 0.970390i \(-0.577654\pi\)
−0.241545 + 0.970390i \(0.577654\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.05568e13 1.30179 0.650897 0.759166i \(-0.274393\pi\)
0.650897 + 0.759166i \(0.274393\pi\)
\(858\) 0 0
\(859\) 1.05923e12 0.0663775 0.0331888 0.999449i \(-0.489434\pi\)
0.0331888 + 0.999449i \(0.489434\pi\)
\(860\) 0 0
\(861\) −4.28469e12 −0.265709
\(862\) 0 0
\(863\) −1.53023e13 −0.939092 −0.469546 0.882908i \(-0.655582\pi\)
−0.469546 + 0.882908i \(0.655582\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −2.34982e13 −1.41237
\(868\) 0 0
\(869\) −4.62869e12 −0.275340
\(870\) 0 0
\(871\) 2.24410e13 1.32118
\(872\) 0 0
\(873\) −1.15516e13 −0.673095
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.74558e11 −0.00996416 −0.00498208 0.999988i \(-0.501586\pi\)
−0.00498208 + 0.999988i \(0.501586\pi\)
\(878\) 0 0
\(879\) −1.13385e13 −0.640629
\(880\) 0 0
\(881\) 8.59039e12 0.480420 0.240210 0.970721i \(-0.422784\pi\)
0.240210 + 0.970721i \(0.422784\pi\)
\(882\) 0 0
\(883\) 1.70227e13 0.942336 0.471168 0.882043i \(-0.343833\pi\)
0.471168 + 0.882043i \(0.343833\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.01246e13 1.09162 0.545810 0.837909i \(-0.316222\pi\)
0.545810 + 0.837909i \(0.316222\pi\)
\(888\) 0 0
\(889\) −5.34112e13 −2.86797
\(890\) 0 0
\(891\) 9.83253e12 0.522655
\(892\) 0 0
\(893\) −1.92177e13 −1.01127
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −6.53428e13 −3.37002
\(898\) 0 0
\(899\) −1.85668e11 −0.00948023
\(900\) 0 0
\(901\) −4.03214e12 −0.203833
\(902\) 0 0
\(903\) −1.40766e13 −0.704535
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 2.10130e13 1.03099 0.515495 0.856893i \(-0.327608\pi\)
0.515495 + 0.856893i \(0.327608\pi\)
\(908\) 0 0
\(909\) −3.07761e13 −1.49512
\(910\) 0 0
\(911\) −1.95345e12 −0.0939656 −0.0469828 0.998896i \(-0.514961\pi\)
−0.0469828 + 0.998896i \(0.514961\pi\)
\(912\) 0 0
\(913\) −1.04802e13 −0.499172
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 3.80064e13 1.77499
\(918\) 0 0
\(919\) 2.47656e12 0.114533 0.0572663 0.998359i \(-0.481762\pi\)
0.0572663 + 0.998359i \(0.481762\pi\)
\(920\) 0 0
\(921\) 2.73301e13 1.25162
\(922\) 0 0
\(923\) −5.75744e13 −2.61109
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −4.78510e13 −2.12830
\(928\) 0 0
\(929\) 4.40844e13 1.94184 0.970922 0.239395i \(-0.0769490\pi\)
0.970922 + 0.239395i \(0.0769490\pi\)
\(930\) 0 0
\(931\) 3.52146e13 1.53621
\(932\) 0 0
\(933\) −4.40477e11 −0.0190308
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 4.92250e12 0.208621 0.104310 0.994545i \(-0.466736\pi\)
0.104310 + 0.994545i \(0.466736\pi\)
\(938\) 0 0
\(939\) −5.12372e13 −2.15075
\(940\) 0 0
\(941\) −1.54680e13 −0.643103 −0.321552 0.946892i \(-0.604204\pi\)
−0.321552 + 0.946892i \(0.604204\pi\)
\(942\) 0 0
\(943\) −3.88116e12 −0.159830
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −5.49107e12 −0.221862 −0.110931 0.993828i \(-0.535383\pi\)
−0.110931 + 0.993828i \(0.535383\pi\)
\(948\) 0 0
\(949\) 2.66004e13 1.06461
\(950\) 0 0
\(951\) −7.28207e13 −2.88697
\(952\) 0 0
\(953\) −1.76932e13 −0.694845 −0.347422 0.937709i \(-0.612943\pi\)
−0.347422 + 0.937709i \(0.612943\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −2.59905e12 −0.100164
\(958\) 0 0
\(959\) 2.52994e13 0.965888
\(960\) 0 0
\(961\) −2.61401e13 −0.988670
\(962\) 0 0
\(963\) −5.19916e13 −1.94812
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −2.89231e13 −1.06372 −0.531858 0.846834i \(-0.678506\pi\)
−0.531858 + 0.846834i \(0.678506\pi\)
\(968\) 0 0
\(969\) 1.05519e13 0.384481
\(970\) 0 0
\(971\) −2.69482e13 −0.972843 −0.486422 0.873724i \(-0.661698\pi\)
−0.486422 + 0.873724i \(0.661698\pi\)
\(972\) 0 0
\(973\) −4.05255e13 −1.44951
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.90598e13 1.37153 0.685764 0.727824i \(-0.259468\pi\)
0.685764 + 0.727824i \(0.259468\pi\)
\(978\) 0 0
\(979\) 2.05282e13 0.714216
\(980\) 0 0
\(981\) −2.91084e12 −0.100348
\(982\) 0 0
\(983\) −9.03830e12 −0.308742 −0.154371 0.988013i \(-0.549335\pi\)
−0.154371 + 0.988013i \(0.549335\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 6.56285e13 2.20123
\(988\) 0 0
\(989\) −1.27509e13 −0.423795
\(990\) 0 0
\(991\) −4.89096e13 −1.61088 −0.805439 0.592678i \(-0.798071\pi\)
−0.805439 + 0.592678i \(0.798071\pi\)
\(992\) 0 0
\(993\) 3.01181e13 0.983007
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 4.01445e13 1.28676 0.643380 0.765547i \(-0.277532\pi\)
0.643380 + 0.765547i \(0.277532\pi\)
\(998\) 0 0
\(999\) 5.30984e12 0.168670
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.a.u.1.3 3
4.3 odd 2 25.10.a.d.1.2 yes 3
5.2 odd 4 400.10.c.q.49.2 6
5.3 odd 4 400.10.c.q.49.5 6
5.4 even 2 400.10.a.y.1.1 3
12.11 even 2 225.10.a.m.1.2 3
20.3 even 4 25.10.b.c.24.2 6
20.7 even 4 25.10.b.c.24.5 6
20.19 odd 2 25.10.a.c.1.2 3
60.23 odd 4 225.10.b.m.199.5 6
60.47 odd 4 225.10.b.m.199.2 6
60.59 even 2 225.10.a.p.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.10.a.c.1.2 3 20.19 odd 2
25.10.a.d.1.2 yes 3 4.3 odd 2
25.10.b.c.24.2 6 20.3 even 4
25.10.b.c.24.5 6 20.7 even 4
225.10.a.m.1.2 3 12.11 even 2
225.10.a.p.1.2 3 60.59 even 2
225.10.b.m.199.2 6 60.47 odd 4
225.10.b.m.199.5 6 60.23 odd 4
400.10.a.u.1.3 3 1.1 even 1 trivial
400.10.a.y.1.1 3 5.4 even 2
400.10.c.q.49.2 6 5.2 odd 4
400.10.c.q.49.5 6 5.3 odd 4