Properties

Label 225.10.a.a.1.1
Level $225$
Weight $10$
Character 225.1
Self dual yes
Analytic conductor $115.883$
Analytic rank $0$
Dimension $1$
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] = [225,10,Mod(1,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 225.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: \(115.883063137\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-36.0000 q^{2} +784.000 q^{4} +4480.00 q^{7} -9792.00 q^{8} +O(q^{10})\) \(q-36.0000 q^{2} +784.000 q^{4} +4480.00 q^{7} -9792.00 q^{8} -1476.00 q^{11} +151522. q^{13} -161280. q^{14} -48896.0 q^{16} +108162. q^{17} +593084. q^{19} +53136.0 q^{22} -969480. q^{23} -5.45479e6 q^{26} +3.51232e6 q^{28} +6.64252e6 q^{29} +7.07060e6 q^{31} +6.77376e6 q^{32} -3.89383e6 q^{34} +7.47241e6 q^{37} -2.13510e7 q^{38} +4.35015e6 q^{41} +4.35872e6 q^{43} -1.15718e6 q^{44} +3.49013e7 q^{46} +2.83092e7 q^{47} -2.02832e7 q^{49} +1.18793e8 q^{52} +1.61117e7 q^{53} -4.38682e7 q^{56} -2.39131e8 q^{58} +8.60760e7 q^{59} +3.22139e7 q^{61} -2.54542e8 q^{62} -2.18821e8 q^{64} -9.95315e7 q^{67} +8.47990e7 q^{68} +4.41705e7 q^{71} +2.35606e7 q^{73} -2.69007e8 q^{74} +4.64978e8 q^{76} -6.61248e6 q^{77} -4.01755e8 q^{79} -1.56605e8 q^{82} -7.44529e8 q^{83} -1.56914e8 q^{86} +1.44530e7 q^{88} -7.69871e8 q^{89} +6.78819e8 q^{91} -7.60072e8 q^{92} -1.01913e9 q^{94} -9.07131e8 q^{97} +7.30195e8 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −36.0000 −1.59099 −0.795495 0.605960i \(-0.792789\pi\)
−0.795495 + 0.605960i \(0.792789\pi\)
\(3\) 0 0
\(4\) 784.000 1.53125
\(5\) 0 0
\(6\) 0 0
\(7\) 4480.00 0.705240 0.352620 0.935767i \(-0.385291\pi\)
0.352620 + 0.935767i \(0.385291\pi\)
\(8\) −9792.00 −0.845214
\(9\) 0 0
\(10\) 0 0
\(11\) −1476.00 −0.0303962 −0.0151981 0.999885i \(-0.504838\pi\)
−0.0151981 + 0.999885i \(0.504838\pi\)
\(12\) 0 0
\(13\) 151522. 1.47140 0.735700 0.677308i \(-0.236853\pi\)
0.735700 + 0.677308i \(0.236853\pi\)
\(14\) −161280. −1.12203
\(15\) 0 0
\(16\) −48896.0 −0.186523
\(17\) 108162. 0.314090 0.157045 0.987591i \(-0.449803\pi\)
0.157045 + 0.987591i \(0.449803\pi\)
\(18\) 0 0
\(19\) 593084. 1.04406 0.522029 0.852927i \(-0.325175\pi\)
0.522029 + 0.852927i \(0.325175\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 53136.0 0.0483601
\(23\) −969480. −0.722376 −0.361188 0.932493i \(-0.617629\pi\)
−0.361188 + 0.932493i \(0.617629\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −5.45479e6 −2.34098
\(27\) 0 0
\(28\) 3.51232e6 1.07990
\(29\) 6.64252e6 1.74398 0.871991 0.489522i \(-0.162829\pi\)
0.871991 + 0.489522i \(0.162829\pi\)
\(30\) 0 0
\(31\) 7.07060e6 1.37508 0.687541 0.726145i \(-0.258690\pi\)
0.687541 + 0.726145i \(0.258690\pi\)
\(32\) 6.77376e6 1.14197
\(33\) 0 0
\(34\) −3.89383e6 −0.499715
\(35\) 0 0
\(36\) 0 0
\(37\) 7.47241e6 0.655470 0.327735 0.944770i \(-0.393715\pi\)
0.327735 + 0.944770i \(0.393715\pi\)
\(38\) −2.13510e7 −1.66109
\(39\) 0 0
\(40\) 0 0
\(41\) 4.35015e6 0.240423 0.120212 0.992748i \(-0.461643\pi\)
0.120212 + 0.992748i \(0.461643\pi\)
\(42\) 0 0
\(43\) 4.35872e6 0.194424 0.0972121 0.995264i \(-0.469007\pi\)
0.0972121 + 0.995264i \(0.469007\pi\)
\(44\) −1.15718e6 −0.0465442
\(45\) 0 0
\(46\) 3.49013e7 1.14929
\(47\) 2.83092e7 0.846229 0.423115 0.906076i \(-0.360937\pi\)
0.423115 + 0.906076i \(0.360937\pi\)
\(48\) 0 0
\(49\) −2.02832e7 −0.502637
\(50\) 0 0
\(51\) 0 0
\(52\) 1.18793e8 2.25308
\(53\) 1.61117e7 0.280479 0.140239 0.990118i \(-0.455213\pi\)
0.140239 + 0.990118i \(0.455213\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −4.38682e7 −0.596078
\(57\) 0 0
\(58\) −2.39131e8 −2.77466
\(59\) 8.60760e7 0.924800 0.462400 0.886671i \(-0.346988\pi\)
0.462400 + 0.886671i \(0.346988\pi\)
\(60\) 0 0
\(61\) 3.22139e7 0.297892 0.148946 0.988845i \(-0.452412\pi\)
0.148946 + 0.988845i \(0.452412\pi\)
\(62\) −2.54542e8 −2.18774
\(63\) 0 0
\(64\) −2.18821e8 −1.63034
\(65\) 0 0
\(66\) 0 0
\(67\) −9.95315e7 −0.603426 −0.301713 0.953399i \(-0.597558\pi\)
−0.301713 + 0.953399i \(0.597558\pi\)
\(68\) 8.47990e7 0.480951
\(69\) 0 0
\(70\) 0 0
\(71\) 4.41705e7 0.206286 0.103143 0.994667i \(-0.467110\pi\)
0.103143 + 0.994667i \(0.467110\pi\)
\(72\) 0 0
\(73\) 2.35606e7 0.0971033 0.0485517 0.998821i \(-0.484539\pi\)
0.0485517 + 0.998821i \(0.484539\pi\)
\(74\) −2.69007e8 −1.04285
\(75\) 0 0
\(76\) 4.64978e8 1.59872
\(77\) −6.61248e6 −0.0214366
\(78\) 0 0
\(79\) −4.01755e8 −1.16048 −0.580242 0.814444i \(-0.697042\pi\)
−0.580242 + 0.814444i \(0.697042\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1.56605e8 −0.382511
\(83\) −7.44529e8 −1.72199 −0.860994 0.508615i \(-0.830158\pi\)
−0.860994 + 0.508615i \(0.830158\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.56914e8 −0.309327
\(87\) 0 0
\(88\) 1.44530e7 0.0256913
\(89\) −7.69871e8 −1.30066 −0.650329 0.759653i \(-0.725369\pi\)
−0.650329 + 0.759653i \(0.725369\pi\)
\(90\) 0 0
\(91\) 6.78819e8 1.03769
\(92\) −7.60072e8 −1.10614
\(93\) 0 0
\(94\) −1.01913e9 −1.34634
\(95\) 0 0
\(96\) 0 0
\(97\) −9.07131e8 −1.04039 −0.520196 0.854047i \(-0.674141\pi\)
−0.520196 + 0.854047i \(0.674141\pi\)
\(98\) 7.30195e8 0.799690
\(99\) 0 0
\(100\) 0 0
\(101\) 4.21902e8 0.403427 0.201714 0.979445i \(-0.435349\pi\)
0.201714 + 0.979445i \(0.435349\pi\)
\(102\) 0 0
\(103\) −5.79043e8 −0.506924 −0.253462 0.967345i \(-0.581569\pi\)
−0.253462 + 0.967345i \(0.581569\pi\)
\(104\) −1.48370e9 −1.24365
\(105\) 0 0
\(106\) −5.80022e8 −0.446239
\(107\) 1.39714e9 1.03042 0.515208 0.857065i \(-0.327715\pi\)
0.515208 + 0.857065i \(0.327715\pi\)
\(108\) 0 0
\(109\) −2.68530e9 −1.82211 −0.911054 0.412286i \(-0.864730\pi\)
−0.911054 + 0.412286i \(0.864730\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.19054e8 −0.131544
\(113\) 4.54749e8 0.262373 0.131186 0.991358i \(-0.458121\pi\)
0.131186 + 0.991358i \(0.458121\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 5.20774e9 2.67047
\(117\) 0 0
\(118\) −3.09873e9 −1.47135
\(119\) 4.84566e8 0.221509
\(120\) 0 0
\(121\) −2.35577e9 −0.999076
\(122\) −1.15970e9 −0.473944
\(123\) 0 0
\(124\) 5.54335e9 2.10559
\(125\) 0 0
\(126\) 0 0
\(127\) 8.38650e7 0.0286064 0.0143032 0.999898i \(-0.495447\pi\)
0.0143032 + 0.999898i \(0.495447\pi\)
\(128\) 4.40938e9 1.45189
\(129\) 0 0
\(130\) 0 0
\(131\) 3.73601e9 1.10838 0.554188 0.832391i \(-0.313029\pi\)
0.554188 + 0.832391i \(0.313029\pi\)
\(132\) 0 0
\(133\) 2.65702e9 0.736312
\(134\) 3.58313e9 0.960044
\(135\) 0 0
\(136\) −1.05912e9 −0.265473
\(137\) 6.43598e9 1.56089 0.780444 0.625225i \(-0.214993\pi\)
0.780444 + 0.625225i \(0.214993\pi\)
\(138\) 0 0
\(139\) 1.81833e9 0.413148 0.206574 0.978431i \(-0.433769\pi\)
0.206574 + 0.978431i \(0.433769\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1.59014e9 −0.328199
\(143\) −2.23646e8 −0.0447250
\(144\) 0 0
\(145\) 0 0
\(146\) −8.48183e8 −0.154490
\(147\) 0 0
\(148\) 5.85837e9 1.00369
\(149\) 8.30199e9 1.37989 0.689944 0.723863i \(-0.257635\pi\)
0.689944 + 0.723863i \(0.257635\pi\)
\(150\) 0 0
\(151\) 3.84057e9 0.601173 0.300587 0.953755i \(-0.402818\pi\)
0.300587 + 0.953755i \(0.402818\pi\)
\(152\) −5.80748e9 −0.882453
\(153\) 0 0
\(154\) 2.38049e8 0.0341054
\(155\) 0 0
\(156\) 0 0
\(157\) 2.17912e9 0.286242 0.143121 0.989705i \(-0.454286\pi\)
0.143121 + 0.989705i \(0.454286\pi\)
\(158\) 1.44632e10 1.84632
\(159\) 0 0
\(160\) 0 0
\(161\) −4.34327e9 −0.509449
\(162\) 0 0
\(163\) 1.54147e10 1.71038 0.855188 0.518317i \(-0.173441\pi\)
0.855188 + 0.518317i \(0.173441\pi\)
\(164\) 3.41052e9 0.368148
\(165\) 0 0
\(166\) 2.68030e10 2.73967
\(167\) −5.65506e9 −0.562617 −0.281309 0.959617i \(-0.590768\pi\)
−0.281309 + 0.959617i \(0.590768\pi\)
\(168\) 0 0
\(169\) 1.23544e10 1.16502
\(170\) 0 0
\(171\) 0 0
\(172\) 3.41723e9 0.297712
\(173\) 7.69892e7 0.00653465 0.00326733 0.999995i \(-0.498960\pi\)
0.00326733 + 0.999995i \(0.498960\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 7.21705e7 0.00566960
\(177\) 0 0
\(178\) 2.77154e10 2.06933
\(179\) −2.32247e10 −1.69087 −0.845436 0.534077i \(-0.820659\pi\)
−0.845436 + 0.534077i \(0.820659\pi\)
\(180\) 0 0
\(181\) −1.23532e10 −0.855513 −0.427756 0.903894i \(-0.640696\pi\)
−0.427756 + 0.903894i \(0.640696\pi\)
\(182\) −2.44375e10 −1.65095
\(183\) 0 0
\(184\) 9.49315e9 0.610562
\(185\) 0 0
\(186\) 0 0
\(187\) −1.59647e8 −0.00954715
\(188\) 2.21945e10 1.29579
\(189\) 0 0
\(190\) 0 0
\(191\) 4.20433e9 0.228584 0.114292 0.993447i \(-0.463540\pi\)
0.114292 + 0.993447i \(0.463540\pi\)
\(192\) 0 0
\(193\) 4.38611e9 0.227547 0.113774 0.993507i \(-0.463706\pi\)
0.113774 + 0.993507i \(0.463706\pi\)
\(194\) 3.26567e10 1.65525
\(195\) 0 0
\(196\) −1.59020e10 −0.769663
\(197\) −3.36694e10 −1.59271 −0.796356 0.604828i \(-0.793242\pi\)
−0.796356 + 0.604828i \(0.793242\pi\)
\(198\) 0 0
\(199\) 1.02732e10 0.464374 0.232187 0.972671i \(-0.425412\pi\)
0.232187 + 0.972671i \(0.425412\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −1.51885e10 −0.641849
\(203\) 2.97585e10 1.22993
\(204\) 0 0
\(205\) 0 0
\(206\) 2.08455e10 0.806512
\(207\) 0 0
\(208\) −7.40882e9 −0.274450
\(209\) −8.75392e8 −0.0317354
\(210\) 0 0
\(211\) 7.96696e9 0.276708 0.138354 0.990383i \(-0.455819\pi\)
0.138354 + 0.990383i \(0.455819\pi\)
\(212\) 1.26316e10 0.429483
\(213\) 0 0
\(214\) −5.02970e10 −1.63938
\(215\) 0 0
\(216\) 0 0
\(217\) 3.16763e10 0.969763
\(218\) 9.66710e10 2.89896
\(219\) 0 0
\(220\) 0 0
\(221\) 1.63889e10 0.462152
\(222\) 0 0
\(223\) −6.96581e9 −0.188625 −0.0943126 0.995543i \(-0.530065\pi\)
−0.0943126 + 0.995543i \(0.530065\pi\)
\(224\) 3.03464e10 0.805363
\(225\) 0 0
\(226\) −1.63709e10 −0.417432
\(227\) 3.35697e10 0.839133 0.419567 0.907725i \(-0.362182\pi\)
0.419567 + 0.907725i \(0.362182\pi\)
\(228\) 0 0
\(229\) 2.93198e10 0.704534 0.352267 0.935900i \(-0.385411\pi\)
0.352267 + 0.935900i \(0.385411\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.50436e10 −1.47404
\(233\) −8.20079e10 −1.82286 −0.911431 0.411453i \(-0.865022\pi\)
−0.911431 + 0.411453i \(0.865022\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 6.74836e10 1.41610
\(237\) 0 0
\(238\) −1.74444e10 −0.352419
\(239\) −6.26609e10 −1.24224 −0.621121 0.783715i \(-0.713323\pi\)
−0.621121 + 0.783715i \(0.713323\pi\)
\(240\) 0 0
\(241\) 7.75548e10 1.48092 0.740460 0.672100i \(-0.234608\pi\)
0.740460 + 0.672100i \(0.234608\pi\)
\(242\) 8.48077e10 1.58952
\(243\) 0 0
\(244\) 2.52557e10 0.456148
\(245\) 0 0
\(246\) 0 0
\(247\) 8.98653e10 1.53623
\(248\) −6.92353e10 −1.16224
\(249\) 0 0
\(250\) 0 0
\(251\) 5.81901e10 0.925374 0.462687 0.886522i \(-0.346885\pi\)
0.462687 + 0.886522i \(0.346885\pi\)
\(252\) 0 0
\(253\) 1.43095e9 0.0219575
\(254\) −3.01914e9 −0.0455126
\(255\) 0 0
\(256\) −4.67014e10 −0.679595
\(257\) −7.41485e9 −0.106024 −0.0530119 0.998594i \(-0.516882\pi\)
−0.0530119 + 0.998594i \(0.516882\pi\)
\(258\) 0 0
\(259\) 3.34764e10 0.462264
\(260\) 0 0
\(261\) 0 0
\(262\) −1.34496e11 −1.76342
\(263\) −1.05271e11 −1.35677 −0.678387 0.734705i \(-0.737321\pi\)
−0.678387 + 0.734705i \(0.737321\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −9.56526e10 −1.17147
\(267\) 0 0
\(268\) −7.80327e10 −0.923995
\(269\) −4.67239e10 −0.544069 −0.272034 0.962288i \(-0.587696\pi\)
−0.272034 + 0.962288i \(0.587696\pi\)
\(270\) 0 0
\(271\) 2.86868e10 0.323087 0.161544 0.986866i \(-0.448353\pi\)
0.161544 + 0.986866i \(0.448353\pi\)
\(272\) −5.28869e9 −0.0585852
\(273\) 0 0
\(274\) −2.31695e11 −2.48336
\(275\) 0 0
\(276\) 0 0
\(277\) −8.50676e10 −0.868171 −0.434085 0.900872i \(-0.642928\pi\)
−0.434085 + 0.900872i \(0.642928\pi\)
\(278\) −6.54598e10 −0.657314
\(279\) 0 0
\(280\) 0 0
\(281\) 7.87257e8 0.00753248 0.00376624 0.999993i \(-0.498801\pi\)
0.00376624 + 0.999993i \(0.498801\pi\)
\(282\) 0 0
\(283\) −2.48961e10 −0.230724 −0.115362 0.993324i \(-0.536803\pi\)
−0.115362 + 0.993324i \(0.536803\pi\)
\(284\) 3.46297e10 0.315875
\(285\) 0 0
\(286\) 8.05127e9 0.0711570
\(287\) 1.94887e10 0.169556
\(288\) 0 0
\(289\) −1.06889e11 −0.901347
\(290\) 0 0
\(291\) 0 0
\(292\) 1.84715e10 0.148689
\(293\) 1.57074e11 1.24509 0.622543 0.782586i \(-0.286100\pi\)
0.622543 + 0.782586i \(0.286100\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −7.31698e10 −0.554012
\(297\) 0 0
\(298\) −2.98871e11 −2.19539
\(299\) −1.46898e11 −1.06290
\(300\) 0 0
\(301\) 1.95270e10 0.137116
\(302\) −1.38261e11 −0.956461
\(303\) 0 0
\(304\) −2.89994e10 −0.194741
\(305\) 0 0
\(306\) 0 0
\(307\) 2.45737e11 1.57887 0.789437 0.613831i \(-0.210372\pi\)
0.789437 + 0.613831i \(0.210372\pi\)
\(308\) −5.18418e9 −0.0328248
\(309\) 0 0
\(310\) 0 0
\(311\) 1.61050e11 0.976201 0.488101 0.872787i \(-0.337690\pi\)
0.488101 + 0.872787i \(0.337690\pi\)
\(312\) 0 0
\(313\) 2.44646e11 1.44075 0.720374 0.693586i \(-0.243970\pi\)
0.720374 + 0.693586i \(0.243970\pi\)
\(314\) −7.84484e10 −0.455408
\(315\) 0 0
\(316\) −3.14976e11 −1.77699
\(317\) 1.12832e11 0.627575 0.313787 0.949493i \(-0.398402\pi\)
0.313787 + 0.949493i \(0.398402\pi\)
\(318\) 0 0
\(319\) −9.80436e9 −0.0530104
\(320\) 0 0
\(321\) 0 0
\(322\) 1.56358e11 0.810528
\(323\) 6.41492e10 0.327929
\(324\) 0 0
\(325\) 0 0
\(326\) −5.54930e11 −2.72119
\(327\) 0 0
\(328\) −4.25967e10 −0.203209
\(329\) 1.26825e11 0.596794
\(330\) 0 0
\(331\) 2.87348e11 1.31578 0.657889 0.753115i \(-0.271450\pi\)
0.657889 + 0.753115i \(0.271450\pi\)
\(332\) −5.83711e11 −2.63679
\(333\) 0 0
\(334\) 2.03582e11 0.895119
\(335\) 0 0
\(336\) 0 0
\(337\) 2.52635e10 0.106699 0.0533494 0.998576i \(-0.483010\pi\)
0.0533494 + 0.998576i \(0.483010\pi\)
\(338\) −4.44759e11 −1.85353
\(339\) 0 0
\(340\) 0 0
\(341\) −1.04362e10 −0.0417973
\(342\) 0 0
\(343\) −2.71653e11 −1.05972
\(344\) −4.26805e10 −0.164330
\(345\) 0 0
\(346\) −2.77161e9 −0.0103966
\(347\) 9.04803e10 0.335020 0.167510 0.985870i \(-0.446427\pi\)
0.167510 + 0.985870i \(0.446427\pi\)
\(348\) 0 0
\(349\) −1.53822e10 −0.0555016 −0.0277508 0.999615i \(-0.508834\pi\)
−0.0277508 + 0.999615i \(0.508834\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −9.99807e9 −0.0347116
\(353\) −1.46875e11 −0.503457 −0.251728 0.967798i \(-0.580999\pi\)
−0.251728 + 0.967798i \(0.580999\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.03579e11 −1.99163
\(357\) 0 0
\(358\) 8.36088e11 2.69016
\(359\) 4.42246e11 1.40520 0.702602 0.711583i \(-0.252021\pi\)
0.702602 + 0.711583i \(0.252021\pi\)
\(360\) 0 0
\(361\) 2.90609e10 0.0900590
\(362\) 4.44716e11 1.36111
\(363\) 0 0
\(364\) 5.32194e11 1.58896
\(365\) 0 0
\(366\) 0 0
\(367\) −1.48110e11 −0.426175 −0.213088 0.977033i \(-0.568352\pi\)
−0.213088 + 0.977033i \(0.568352\pi\)
\(368\) 4.74037e10 0.134740
\(369\) 0 0
\(370\) 0 0
\(371\) 7.21805e10 0.197805
\(372\) 0 0
\(373\) −7.63489e10 −0.204227 −0.102114 0.994773i \(-0.532560\pi\)
−0.102114 + 0.994773i \(0.532560\pi\)
\(374\) 5.74730e9 0.0151894
\(375\) 0 0
\(376\) −2.77204e11 −0.715244
\(377\) 1.00649e12 2.56609
\(378\) 0 0
\(379\) −2.70192e11 −0.672660 −0.336330 0.941744i \(-0.609186\pi\)
−0.336330 + 0.941744i \(0.609186\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −1.51356e11 −0.363676
\(383\) −6.61033e11 −1.56974 −0.784872 0.619658i \(-0.787271\pi\)
−0.784872 + 0.619658i \(0.787271\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.57900e11 −0.362026
\(387\) 0 0
\(388\) −7.11191e11 −1.59310
\(389\) 3.09861e11 0.686109 0.343054 0.939316i \(-0.388538\pi\)
0.343054 + 0.939316i \(0.388538\pi\)
\(390\) 0 0
\(391\) −1.04861e11 −0.226891
\(392\) 1.98613e11 0.424835
\(393\) 0 0
\(394\) 1.21210e12 2.53399
\(395\) 0 0
\(396\) 0 0
\(397\) 6.50589e11 1.31447 0.657233 0.753688i \(-0.271727\pi\)
0.657233 + 0.753688i \(0.271727\pi\)
\(398\) −3.69836e11 −0.738814
\(399\) 0 0
\(400\) 0 0
\(401\) −2.76701e10 −0.0534393 −0.0267196 0.999643i \(-0.508506\pi\)
−0.0267196 + 0.999643i \(0.508506\pi\)
\(402\) 0 0
\(403\) 1.07135e12 2.02330
\(404\) 3.30771e11 0.617748
\(405\) 0 0
\(406\) −1.07131e12 −1.95680
\(407\) −1.10293e10 −0.0199238
\(408\) 0 0
\(409\) 2.08505e11 0.368436 0.184218 0.982885i \(-0.441025\pi\)
0.184218 + 0.982885i \(0.441025\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −4.53969e11 −0.776228
\(413\) 3.85620e11 0.652206
\(414\) 0 0
\(415\) 0 0
\(416\) 1.02637e12 1.68029
\(417\) 0 0
\(418\) 3.15141e10 0.0504907
\(419\) −4.50465e11 −0.714000 −0.357000 0.934104i \(-0.616200\pi\)
−0.357000 + 0.934104i \(0.616200\pi\)
\(420\) 0 0
\(421\) 8.60883e11 1.33559 0.667797 0.744343i \(-0.267237\pi\)
0.667797 + 0.744343i \(0.267237\pi\)
\(422\) −2.86810e11 −0.440239
\(423\) 0 0
\(424\) −1.57766e11 −0.237065
\(425\) 0 0
\(426\) 0 0
\(427\) 1.44318e11 0.210086
\(428\) 1.09536e12 1.57782
\(429\) 0 0
\(430\) 0 0
\(431\) 3.02405e10 0.0422125 0.0211063 0.999777i \(-0.493281\pi\)
0.0211063 + 0.999777i \(0.493281\pi\)
\(432\) 0 0
\(433\) −1.03636e12 −1.41682 −0.708410 0.705802i \(-0.750587\pi\)
−0.708410 + 0.705802i \(0.750587\pi\)
\(434\) −1.14035e12 −1.54288
\(435\) 0 0
\(436\) −2.10528e12 −2.79010
\(437\) −5.74983e11 −0.754204
\(438\) 0 0
\(439\) −5.90670e11 −0.759022 −0.379511 0.925187i \(-0.623908\pi\)
−0.379511 + 0.925187i \(0.623908\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −5.90001e11 −0.735280
\(443\) 1.27097e12 1.56790 0.783948 0.620827i \(-0.213203\pi\)
0.783948 + 0.620827i \(0.213203\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 2.50769e11 0.300101
\(447\) 0 0
\(448\) −9.80316e11 −1.14978
\(449\) 9.34644e11 1.08527 0.542635 0.839969i \(-0.317427\pi\)
0.542635 + 0.839969i \(0.317427\pi\)
\(450\) 0 0
\(451\) −6.42082e9 −0.00730796
\(452\) 3.56523e11 0.401758
\(453\) 0 0
\(454\) −1.20851e12 −1.33505
\(455\) 0 0
\(456\) 0 0
\(457\) 4.52481e11 0.485263 0.242632 0.970119i \(-0.421989\pi\)
0.242632 + 0.970119i \(0.421989\pi\)
\(458\) −1.05551e12 −1.12091
\(459\) 0 0
\(460\) 0 0
\(461\) 8.56467e11 0.883195 0.441597 0.897213i \(-0.354412\pi\)
0.441597 + 0.897213i \(0.354412\pi\)
\(462\) 0 0
\(463\) −9.21380e11 −0.931803 −0.465902 0.884836i \(-0.654270\pi\)
−0.465902 + 0.884836i \(0.654270\pi\)
\(464\) −3.24793e11 −0.325294
\(465\) 0 0
\(466\) 2.95228e12 2.90016
\(467\) 8.65382e10 0.0841941 0.0420971 0.999114i \(-0.486596\pi\)
0.0420971 + 0.999114i \(0.486596\pi\)
\(468\) 0 0
\(469\) −4.45901e11 −0.425560
\(470\) 0 0
\(471\) 0 0
\(472\) −8.42856e11 −0.781654
\(473\) −6.43346e9 −0.00590976
\(474\) 0 0
\(475\) 0 0
\(476\) 3.79900e11 0.339186
\(477\) 0 0
\(478\) 2.25579e12 1.97640
\(479\) −7.63707e11 −0.662852 −0.331426 0.943481i \(-0.607530\pi\)
−0.331426 + 0.943481i \(0.607530\pi\)
\(480\) 0 0
\(481\) 1.13223e12 0.964458
\(482\) −2.79197e12 −2.35613
\(483\) 0 0
\(484\) −1.84692e12 −1.52984
\(485\) 0 0
\(486\) 0 0
\(487\) 5.25531e11 0.423368 0.211684 0.977338i \(-0.432105\pi\)
0.211684 + 0.977338i \(0.432105\pi\)
\(488\) −3.15439e11 −0.251783
\(489\) 0 0
\(490\) 0 0
\(491\) −2.37265e12 −1.84233 −0.921163 0.389177i \(-0.872759\pi\)
−0.921163 + 0.389177i \(0.872759\pi\)
\(492\) 0 0
\(493\) 7.18468e11 0.547768
\(494\) −3.23515e12 −2.44412
\(495\) 0 0
\(496\) −3.45724e11 −0.256485
\(497\) 1.97884e11 0.145481
\(498\) 0 0
\(499\) −1.33387e11 −0.0963080 −0.0481540 0.998840i \(-0.515334\pi\)
−0.0481540 + 0.998840i \(0.515334\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −2.09484e12 −1.47226
\(503\) −6.58632e11 −0.458762 −0.229381 0.973337i \(-0.573670\pi\)
−0.229381 + 0.973337i \(0.573670\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −5.15143e10 −0.0349342
\(507\) 0 0
\(508\) 6.57501e10 0.0438036
\(509\) 1.01965e12 0.673322 0.336661 0.941626i \(-0.390702\pi\)
0.336661 + 0.941626i \(0.390702\pi\)
\(510\) 0 0
\(511\) 1.05552e11 0.0684811
\(512\) −5.76350e11 −0.370656
\(513\) 0 0
\(514\) 2.66935e11 0.168683
\(515\) 0 0
\(516\) 0 0
\(517\) −4.17845e10 −0.0257221
\(518\) −1.20515e12 −0.735457
\(519\) 0 0
\(520\) 0 0
\(521\) 5.57535e11 0.331514 0.165757 0.986167i \(-0.446993\pi\)
0.165757 + 0.986167i \(0.446993\pi\)
\(522\) 0 0
\(523\) −2.12050e12 −1.23931 −0.619657 0.784873i \(-0.712728\pi\)
−0.619657 + 0.784873i \(0.712728\pi\)
\(524\) 2.92903e12 1.69720
\(525\) 0 0
\(526\) 3.78975e12 2.15861
\(527\) 7.64770e11 0.431900
\(528\) 0 0
\(529\) −8.61261e11 −0.478172
\(530\) 0 0
\(531\) 0 0
\(532\) 2.08310e12 1.12748
\(533\) 6.59143e11 0.353759
\(534\) 0 0
\(535\) 0 0
\(536\) 9.74612e11 0.510024
\(537\) 0 0
\(538\) 1.68206e12 0.865608
\(539\) 2.99380e10 0.0152782
\(540\) 0 0
\(541\) 1.92746e12 0.967379 0.483690 0.875240i \(-0.339296\pi\)
0.483690 + 0.875240i \(0.339296\pi\)
\(542\) −1.03272e12 −0.514029
\(543\) 0 0
\(544\) 7.32663e11 0.358682
\(545\) 0 0
\(546\) 0 0
\(547\) 2.32751e12 1.11160 0.555799 0.831317i \(-0.312412\pi\)
0.555799 + 0.831317i \(0.312412\pi\)
\(548\) 5.04581e12 2.39011
\(549\) 0 0
\(550\) 0 0
\(551\) 3.93957e12 1.82082
\(552\) 0 0
\(553\) −1.79986e12 −0.818419
\(554\) 3.06243e12 1.38125
\(555\) 0 0
\(556\) 1.42557e12 0.632633
\(557\) 4.94739e11 0.217785 0.108892 0.994054i \(-0.465270\pi\)
0.108892 + 0.994054i \(0.465270\pi\)
\(558\) 0 0
\(559\) 6.60441e11 0.286076
\(560\) 0 0
\(561\) 0 0
\(562\) −2.83413e10 −0.0119841
\(563\) 1.14083e12 0.478557 0.239279 0.970951i \(-0.423089\pi\)
0.239279 + 0.970951i \(0.423089\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 8.96260e11 0.367079
\(567\) 0 0
\(568\) −4.32517e11 −0.174356
\(569\) 1.64398e10 0.00657495 0.00328747 0.999995i \(-0.498954\pi\)
0.00328747 + 0.999995i \(0.498954\pi\)
\(570\) 0 0
\(571\) −3.67652e12 −1.44735 −0.723676 0.690139i \(-0.757549\pi\)
−0.723676 + 0.690139i \(0.757549\pi\)
\(572\) −1.75339e11 −0.0684851
\(573\) 0 0
\(574\) −7.01592e11 −0.269762
\(575\) 0 0
\(576\) 0 0
\(577\) 2.29045e12 0.860260 0.430130 0.902767i \(-0.358468\pi\)
0.430130 + 0.902767i \(0.358468\pi\)
\(578\) 3.84800e12 1.43403
\(579\) 0 0
\(580\) 0 0
\(581\) −3.33549e12 −1.21441
\(582\) 0 0
\(583\) −2.37809e10 −0.00852549
\(584\) −2.30706e11 −0.0820730
\(585\) 0 0
\(586\) −5.65465e12 −1.98092
\(587\) 4.68750e12 1.62956 0.814780 0.579771i \(-0.196858\pi\)
0.814780 + 0.579771i \(0.196858\pi\)
\(588\) 0 0
\(589\) 4.19346e12 1.43567
\(590\) 0 0
\(591\) 0 0
\(592\) −3.65371e11 −0.122261
\(593\) −2.33770e12 −0.776323 −0.388162 0.921591i \(-0.626890\pi\)
−0.388162 + 0.921591i \(0.626890\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.50876e12 2.11295
\(597\) 0 0
\(598\) 5.28831e12 1.69107
\(599\) 4.66995e12 1.48215 0.741075 0.671423i \(-0.234316\pi\)
0.741075 + 0.671423i \(0.234316\pi\)
\(600\) 0 0
\(601\) −3.96517e12 −1.23973 −0.619864 0.784709i \(-0.712812\pi\)
−0.619864 + 0.784709i \(0.712812\pi\)
\(602\) −7.02974e11 −0.218150
\(603\) 0 0
\(604\) 3.01101e12 0.920546
\(605\) 0 0
\(606\) 0 0
\(607\) −6.24743e12 −1.86790 −0.933948 0.357409i \(-0.883660\pi\)
−0.933948 + 0.357409i \(0.883660\pi\)
\(608\) 4.01741e12 1.19228
\(609\) 0 0
\(610\) 0 0
\(611\) 4.28947e12 1.24514
\(612\) 0 0
\(613\) 3.73193e12 1.06748 0.533742 0.845647i \(-0.320785\pi\)
0.533742 + 0.845647i \(0.320785\pi\)
\(614\) −8.84653e12 −2.51197
\(615\) 0 0
\(616\) 6.47494e10 0.0181185
\(617\) −6.05181e12 −1.68113 −0.840567 0.541708i \(-0.817778\pi\)
−0.840567 + 0.541708i \(0.817778\pi\)
\(618\) 0 0
\(619\) 4.69849e12 1.28632 0.643162 0.765730i \(-0.277622\pi\)
0.643162 + 0.765730i \(0.277622\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −5.79780e12 −1.55313
\(623\) −3.44902e12 −0.917275
\(624\) 0 0
\(625\) 0 0
\(626\) −8.80724e12 −2.29222
\(627\) 0 0
\(628\) 1.70843e12 0.438308
\(629\) 8.08231e11 0.205877
\(630\) 0 0
\(631\) 2.16875e12 0.544600 0.272300 0.962212i \(-0.412216\pi\)
0.272300 + 0.962212i \(0.412216\pi\)
\(632\) 3.93398e12 0.980857
\(633\) 0 0
\(634\) −4.06195e12 −0.998466
\(635\) 0 0
\(636\) 0 0
\(637\) −3.07335e12 −0.739580
\(638\) 3.52957e11 0.0843391
\(639\) 0 0
\(640\) 0 0
\(641\) −3.56446e12 −0.833936 −0.416968 0.908921i \(-0.636907\pi\)
−0.416968 + 0.908921i \(0.636907\pi\)
\(642\) 0 0
\(643\) 5.37917e12 1.24098 0.620491 0.784213i \(-0.286933\pi\)
0.620491 + 0.784213i \(0.286933\pi\)
\(644\) −3.40512e12 −0.780093
\(645\) 0 0
\(646\) −2.30937e12 −0.521732
\(647\) 6.01827e12 1.35021 0.675107 0.737720i \(-0.264097\pi\)
0.675107 + 0.737720i \(0.264097\pi\)
\(648\) 0 0
\(649\) −1.27048e11 −0.0281104
\(650\) 0 0
\(651\) 0 0
\(652\) 1.20852e13 2.61901
\(653\) −3.68383e11 −0.0792849 −0.0396424 0.999214i \(-0.512622\pi\)
−0.0396424 + 0.999214i \(0.512622\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −2.12705e11 −0.0448446
\(657\) 0 0
\(658\) −4.56572e12 −0.949494
\(659\) 5.24810e12 1.08397 0.541985 0.840388i \(-0.317673\pi\)
0.541985 + 0.840388i \(0.317673\pi\)
\(660\) 0 0
\(661\) 1.55039e11 0.0315888 0.0157944 0.999875i \(-0.494972\pi\)
0.0157944 + 0.999875i \(0.494972\pi\)
\(662\) −1.03445e13 −2.09339
\(663\) 0 0
\(664\) 7.29043e12 1.45545
\(665\) 0 0
\(666\) 0 0
\(667\) −6.43979e12 −1.25981
\(668\) −4.43357e12 −0.861508
\(669\) 0 0
\(670\) 0 0
\(671\) −4.75477e10 −0.00905479
\(672\) 0 0
\(673\) −5.89588e11 −0.110785 −0.0553925 0.998465i \(-0.517641\pi\)
−0.0553925 + 0.998465i \(0.517641\pi\)
\(674\) −9.09487e11 −0.169757
\(675\) 0 0
\(676\) 9.68586e12 1.78393
\(677\) 3.32980e12 0.609214 0.304607 0.952478i \(-0.401475\pi\)
0.304607 + 0.952478i \(0.401475\pi\)
\(678\) 0 0
\(679\) −4.06395e12 −0.733726
\(680\) 0 0
\(681\) 0 0
\(682\) 3.75703e11 0.0664991
\(683\) 2.28928e12 0.402536 0.201268 0.979536i \(-0.435494\pi\)
0.201268 + 0.979536i \(0.435494\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 9.77951e12 1.68600
\(687\) 0 0
\(688\) −2.13124e11 −0.0362647
\(689\) 2.44128e12 0.412697
\(690\) 0 0
\(691\) −9.46129e12 −1.57870 −0.789349 0.613945i \(-0.789582\pi\)
−0.789349 + 0.613945i \(0.789582\pi\)
\(692\) 6.03596e10 0.0100062
\(693\) 0 0
\(694\) −3.25729e12 −0.533014
\(695\) 0 0
\(696\) 0 0
\(697\) 4.70521e11 0.0755147
\(698\) 5.53761e11 0.0883024
\(699\) 0 0
\(700\) 0 0
\(701\) −8.22209e12 −1.28603 −0.643015 0.765854i \(-0.722317\pi\)
−0.643015 + 0.765854i \(0.722317\pi\)
\(702\) 0 0
\(703\) 4.43177e12 0.684349
\(704\) 3.22979e11 0.0495562
\(705\) 0 0
\(706\) 5.28751e12 0.800995
\(707\) 1.89012e12 0.284513
\(708\) 0 0
\(709\) 7.61957e12 1.13246 0.566230 0.824247i \(-0.308401\pi\)
0.566230 + 0.824247i \(0.308401\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 7.53858e12 1.09933
\(713\) −6.85481e12 −0.993327
\(714\) 0 0
\(715\) 0 0
\(716\) −1.82081e13 −2.58915
\(717\) 0 0
\(718\) −1.59209e13 −2.23567
\(719\) −7.94823e12 −1.10915 −0.554575 0.832134i \(-0.687119\pi\)
−0.554575 + 0.832134i \(0.687119\pi\)
\(720\) 0 0
\(721\) −2.59411e12 −0.357503
\(722\) −1.04619e12 −0.143283
\(723\) 0 0
\(724\) −9.68492e12 −1.31000
\(725\) 0 0
\(726\) 0 0
\(727\) 5.75787e12 0.764464 0.382232 0.924066i \(-0.375156\pi\)
0.382232 + 0.924066i \(0.375156\pi\)
\(728\) −6.64699e12 −0.877069
\(729\) 0 0
\(730\) 0 0
\(731\) 4.71447e11 0.0610668
\(732\) 0 0
\(733\) −1.07808e13 −1.37938 −0.689690 0.724105i \(-0.742253\pi\)
−0.689690 + 0.724105i \(0.742253\pi\)
\(734\) 5.33198e12 0.678041
\(735\) 0 0
\(736\) −6.56702e12 −0.824933
\(737\) 1.46908e11 0.0183418
\(738\) 0 0
\(739\) 7.37756e12 0.909940 0.454970 0.890507i \(-0.349650\pi\)
0.454970 + 0.890507i \(0.349650\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2.59850e12 −0.314706
\(743\) −1.02899e13 −1.23869 −0.619346 0.785118i \(-0.712602\pi\)
−0.619346 + 0.785118i \(0.712602\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 2.74856e12 0.324923
\(747\) 0 0
\(748\) −1.25163e11 −0.0146191
\(749\) 6.25918e12 0.726690
\(750\) 0 0
\(751\) −7.56555e12 −0.867882 −0.433941 0.900941i \(-0.642877\pi\)
−0.433941 + 0.900941i \(0.642877\pi\)
\(752\) −1.38421e12 −0.157842
\(753\) 0 0
\(754\) −3.62336e13 −4.08263
\(755\) 0 0
\(756\) 0 0
\(757\) 6.88713e12 0.762267 0.381133 0.924520i \(-0.375534\pi\)
0.381133 + 0.924520i \(0.375534\pi\)
\(758\) 9.72690e12 1.07020
\(759\) 0 0
\(760\) 0 0
\(761\) −4.30641e12 −0.465462 −0.232731 0.972541i \(-0.574766\pi\)
−0.232731 + 0.972541i \(0.574766\pi\)
\(762\) 0 0
\(763\) −1.20302e13 −1.28502
\(764\) 3.29619e12 0.350020
\(765\) 0 0
\(766\) 2.37972e13 2.49745
\(767\) 1.30424e13 1.36075
\(768\) 0 0
\(769\) 2.16296e12 0.223039 0.111519 0.993762i \(-0.464428\pi\)
0.111519 + 0.993762i \(0.464428\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 3.43871e12 0.348432
\(773\) −8.15447e12 −0.821463 −0.410731 0.911756i \(-0.634727\pi\)
−0.410731 + 0.911756i \(0.634727\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 8.88263e12 0.879354
\(777\) 0 0
\(778\) −1.11550e13 −1.09159
\(779\) 2.58000e12 0.251016
\(780\) 0 0
\(781\) −6.51956e10 −0.00627031
\(782\) 3.77499e12 0.360982
\(783\) 0 0
\(784\) 9.91768e11 0.0937535
\(785\) 0 0
\(786\) 0 0
\(787\) 2.25199e12 0.209257 0.104629 0.994511i \(-0.466635\pi\)
0.104629 + 0.994511i \(0.466635\pi\)
\(788\) −2.63968e13 −2.43884
\(789\) 0 0
\(790\) 0 0
\(791\) 2.03727e12 0.185036
\(792\) 0 0
\(793\) 4.88112e12 0.438319
\(794\) −2.34212e13 −2.09130
\(795\) 0 0
\(796\) 8.05420e12 0.711072
\(797\) −1.16128e13 −1.01947 −0.509737 0.860331i \(-0.670257\pi\)
−0.509737 + 0.860331i \(0.670257\pi\)
\(798\) 0 0
\(799\) 3.06198e12 0.265792
\(800\) 0 0
\(801\) 0 0
\(802\) 9.96122e11 0.0850214
\(803\) −3.47755e10 −0.00295157
\(804\) 0 0
\(805\) 0 0
\(806\) −3.85687e13 −3.21904
\(807\) 0 0
\(808\) −4.13126e12 −0.340982
\(809\) −2.31768e12 −0.190233 −0.0951165 0.995466i \(-0.530322\pi\)
−0.0951165 + 0.995466i \(0.530322\pi\)
\(810\) 0 0
\(811\) −9.18815e12 −0.745821 −0.372910 0.927867i \(-0.621640\pi\)
−0.372910 + 0.927867i \(0.621640\pi\)
\(812\) 2.33307e13 1.88332
\(813\) 0 0
\(814\) 3.97054e11 0.0316986
\(815\) 0 0
\(816\) 0 0
\(817\) 2.58508e12 0.202990
\(818\) −7.50620e12 −0.586179
\(819\) 0 0
\(820\) 0 0
\(821\) 1.92679e13 1.48010 0.740048 0.672554i \(-0.234803\pi\)
0.740048 + 0.672554i \(0.234803\pi\)
\(822\) 0 0
\(823\) 1.04990e13 0.797713 0.398857 0.917013i \(-0.369407\pi\)
0.398857 + 0.917013i \(0.369407\pi\)
\(824\) 5.66999e12 0.428459
\(825\) 0 0
\(826\) −1.38823e13 −1.03765
\(827\) −8.10053e12 −0.602197 −0.301098 0.953593i \(-0.597353\pi\)
−0.301098 + 0.953593i \(0.597353\pi\)
\(828\) 0 0
\(829\) −1.52452e13 −1.12108 −0.560542 0.828126i \(-0.689407\pi\)
−0.560542 + 0.828126i \(0.689407\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −3.31561e13 −2.39888
\(833\) −2.19387e12 −0.157873
\(834\) 0 0
\(835\) 0 0
\(836\) −6.86307e11 −0.0485949
\(837\) 0 0
\(838\) 1.62168e13 1.13597
\(839\) −2.72258e13 −1.89694 −0.948468 0.316874i \(-0.897367\pi\)
−0.948468 + 0.316874i \(0.897367\pi\)
\(840\) 0 0
\(841\) 2.96160e13 2.04147
\(842\) −3.09918e13 −2.12492
\(843\) 0 0
\(844\) 6.24609e12 0.423709
\(845\) 0 0
\(846\) 0 0
\(847\) −1.05538e13 −0.704588
\(848\) −7.87798e11 −0.0523159
\(849\) 0 0
\(850\) 0 0
\(851\) −7.24435e12 −0.473496
\(852\) 0 0
\(853\) −5.38234e12 −0.348097 −0.174048 0.984737i \(-0.555685\pi\)
−0.174048 + 0.984737i \(0.555685\pi\)
\(854\) −5.19546e12 −0.334244
\(855\) 0 0
\(856\) −1.36808e13 −0.870922
\(857\) 2.13114e12 0.134958 0.0674791 0.997721i \(-0.478504\pi\)
0.0674791 + 0.997721i \(0.478504\pi\)
\(858\) 0 0
\(859\) 1.07650e13 0.674598 0.337299 0.941398i \(-0.390487\pi\)
0.337299 + 0.941398i \(0.390487\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −1.08866e12 −0.0671597
\(863\) 2.36698e13 1.45260 0.726299 0.687379i \(-0.241239\pi\)
0.726299 + 0.687379i \(0.241239\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 3.73089e13 2.25415
\(867\) 0 0
\(868\) 2.48342e13 1.48495
\(869\) 5.92990e11 0.0352743
\(870\) 0 0
\(871\) −1.50812e13 −0.887880
\(872\) 2.62945e13 1.54007
\(873\) 0 0
\(874\) 2.06994e13 1.19993
\(875\) 0 0
\(876\) 0 0
\(877\) 9.91806e12 0.566146 0.283073 0.959098i \(-0.408646\pi\)
0.283073 + 0.959098i \(0.408646\pi\)
\(878\) 2.12641e13 1.20760
\(879\) 0 0
\(880\) 0 0
\(881\) −1.28992e13 −0.721391 −0.360696 0.932684i \(-0.617461\pi\)
−0.360696 + 0.932684i \(0.617461\pi\)
\(882\) 0 0
\(883\) −1.00956e13 −0.558866 −0.279433 0.960165i \(-0.590146\pi\)
−0.279433 + 0.960165i \(0.590146\pi\)
\(884\) 1.28489e13 0.707671
\(885\) 0 0
\(886\) −4.57548e13 −2.49451
\(887\) −2.22403e12 −0.120638 −0.0603190 0.998179i \(-0.519212\pi\)
−0.0603190 + 0.998179i \(0.519212\pi\)
\(888\) 0 0
\(889\) 3.75715e11 0.0201744
\(890\) 0 0
\(891\) 0 0
\(892\) −5.46119e12 −0.288832
\(893\) 1.67898e13 0.883513
\(894\) 0 0
\(895\) 0 0
\(896\) 1.97540e13 1.02393
\(897\) 0 0
\(898\) −3.36472e13 −1.72665
\(899\) 4.69666e13 2.39812
\(900\) 0 0
\(901\) 1.74267e12 0.0880957
\(902\) 2.31150e11 0.0116269
\(903\) 0 0
\(904\) −4.45290e12 −0.221761
\(905\) 0 0
\(906\) 0 0
\(907\) 6.35542e12 0.311825 0.155913 0.987771i \(-0.450168\pi\)
0.155913 + 0.987771i \(0.450168\pi\)
\(908\) 2.63186e13 1.28492
\(909\) 0 0
\(910\) 0 0
\(911\) 1.91432e13 0.920837 0.460418 0.887702i \(-0.347699\pi\)
0.460418 + 0.887702i \(0.347699\pi\)
\(912\) 0 0
\(913\) 1.09892e12 0.0523419
\(914\) −1.62893e13 −0.772049
\(915\) 0 0
\(916\) 2.29867e13 1.07882
\(917\) 1.67373e13 0.781671
\(918\) 0 0
\(919\) 2.86053e13 1.32290 0.661449 0.749991i \(-0.269942\pi\)
0.661449 + 0.749991i \(0.269942\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −3.08328e13 −1.40515
\(923\) 6.69280e12 0.303529
\(924\) 0 0
\(925\) 0 0
\(926\) 3.31697e13 1.48249
\(927\) 0 0
\(928\) 4.49948e13 1.99158
\(929\) 1.18899e13 0.523728 0.261864 0.965105i \(-0.415663\pi\)
0.261864 + 0.965105i \(0.415663\pi\)
\(930\) 0 0
\(931\) −1.20296e13 −0.524782
\(932\) −6.42942e13 −2.79126
\(933\) 0 0
\(934\) −3.11537e12 −0.133952
\(935\) 0 0
\(936\) 0 0
\(937\) 3.45871e13 1.46584 0.732920 0.680315i \(-0.238157\pi\)
0.732920 + 0.680315i \(0.238157\pi\)
\(938\) 1.60524e13 0.677061
\(939\) 0 0
\(940\) 0 0
\(941\) −2.59742e13 −1.07991 −0.539956 0.841693i \(-0.681559\pi\)
−0.539956 + 0.841693i \(0.681559\pi\)
\(942\) 0 0
\(943\) −4.21738e12 −0.173676
\(944\) −4.20877e12 −0.172497
\(945\) 0 0
\(946\) 2.31605e11 0.00940237
\(947\) 2.43739e13 0.984804 0.492402 0.870368i \(-0.336119\pi\)
0.492402 + 0.870368i \(0.336119\pi\)
\(948\) 0 0
\(949\) 3.56995e12 0.142878
\(950\) 0 0
\(951\) 0 0
\(952\) −4.74487e12 −0.187222
\(953\) −2.34891e13 −0.922461 −0.461230 0.887280i \(-0.652592\pi\)
−0.461230 + 0.887280i \(0.652592\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −4.91262e13 −1.90218
\(957\) 0 0
\(958\) 2.74934e13 1.05459
\(959\) 2.88332e13 1.10080
\(960\) 0 0
\(961\) 2.35538e13 0.890851
\(962\) −4.07604e13 −1.53444
\(963\) 0 0
\(964\) 6.08029e13 2.26766
\(965\) 0 0
\(966\) 0 0
\(967\) −4.68639e12 −0.172353 −0.0861766 0.996280i \(-0.527465\pi\)
−0.0861766 + 0.996280i \(0.527465\pi\)
\(968\) 2.30677e13 0.844433
\(969\) 0 0
\(970\) 0 0
\(971\) 4.63936e13 1.67483 0.837416 0.546566i \(-0.184065\pi\)
0.837416 + 0.546566i \(0.184065\pi\)
\(972\) 0 0
\(973\) 8.14611e12 0.291368
\(974\) −1.89191e13 −0.673574
\(975\) 0 0
\(976\) −1.57513e12 −0.0555639
\(977\) 2.53890e13 0.891496 0.445748 0.895159i \(-0.352938\pi\)
0.445748 + 0.895159i \(0.352938\pi\)
\(978\) 0 0
\(979\) 1.13633e12 0.0395350
\(980\) 0 0
\(981\) 0 0
\(982\) 8.54153e13 2.93112
\(983\) −8.90780e12 −0.304284 −0.152142 0.988359i \(-0.548617\pi\)
−0.152142 + 0.988359i \(0.548617\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −2.58649e13 −0.871493
\(987\) 0 0
\(988\) 7.04544e13 2.35235
\(989\) −4.22569e12 −0.140448
\(990\) 0 0
\(991\) 3.90840e13 1.28727 0.643633 0.765335i \(-0.277427\pi\)
0.643633 + 0.765335i \(0.277427\pi\)
\(992\) 4.78945e13 1.57030
\(993\) 0 0
\(994\) −7.12382e12 −0.231459
\(995\) 0 0
\(996\) 0 0
\(997\) 1.94045e12 0.0621975 0.0310988 0.999516i \(-0.490099\pi\)
0.0310988 + 0.999516i \(0.490099\pi\)
\(998\) 4.80195e12 0.153225
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 225.10.a.a.1.1 1
3.2 odd 2 75.10.a.d.1.1 1
5.2 odd 4 225.10.b.a.199.1 2
5.3 odd 4 225.10.b.a.199.2 2
5.4 even 2 9.10.a.c.1.1 1
15.2 even 4 75.10.b.a.49.2 2
15.8 even 4 75.10.b.a.49.1 2
15.14 odd 2 3.10.a.a.1.1 1
20.19 odd 2 144.10.a.l.1.1 1
45.4 even 6 81.10.c.a.55.1 2
45.14 odd 6 81.10.c.e.55.1 2
45.29 odd 6 81.10.c.e.28.1 2
45.34 even 6 81.10.c.a.28.1 2
60.59 even 2 48.10.a.e.1.1 1
105.104 even 2 147.10.a.a.1.1 1
120.29 odd 2 192.10.a.m.1.1 1
120.59 even 2 192.10.a.f.1.1 1
165.164 even 2 363.10.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.10.a.a.1.1 1 15.14 odd 2
9.10.a.c.1.1 1 5.4 even 2
48.10.a.e.1.1 1 60.59 even 2
75.10.a.d.1.1 1 3.2 odd 2
75.10.b.a.49.1 2 15.8 even 4
75.10.b.a.49.2 2 15.2 even 4
81.10.c.a.28.1 2 45.34 even 6
81.10.c.a.55.1 2 45.4 even 6
81.10.c.e.28.1 2 45.29 odd 6
81.10.c.e.55.1 2 45.14 odd 6
144.10.a.l.1.1 1 20.19 odd 2
147.10.a.a.1.1 1 105.104 even 2
192.10.a.f.1.1 1 120.59 even 2
192.10.a.m.1.1 1 120.29 odd 2
225.10.a.a.1.1 1 1.1 even 1 trivial
225.10.b.a.199.1 2 5.2 odd 4
225.10.b.a.199.2 2 5.3 odd 4
363.10.a.b.1.1 1 165.164 even 2