Properties

Label 95.9.d.d.94.1
Level $95$
Weight $9$
Character 95.94
Self dual yes
Analytic conductor $38.701$
Analytic rank $0$
Dimension $2$
CM discriminant -95
Inner twists $4$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [95,9,Mod(94,95)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(95, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("95.94");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 95.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(38.7009679558\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{95}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 95 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 94.1
Root \(-9.74679\) of defining polynomial
Character \(\chi\) \(=\) 95.94

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-29.2404 q^{2} +77.9744 q^{3} +599.000 q^{4} +625.000 q^{5} -2280.00 q^{6} -10029.5 q^{8} -481.000 q^{9} +O(q^{10})\) \(q-29.2404 q^{2} +77.9744 q^{3} +599.000 q^{4} +625.000 q^{5} -2280.00 q^{6} -10029.5 q^{8} -481.000 q^{9} -18275.2 q^{10} +25438.0 q^{11} +46706.6 q^{12} +22222.7 q^{13} +48734.0 q^{15} +139921. q^{16} +14064.6 q^{18} +130321. q^{19} +374375. q^{20} -743817. q^{22} -782040. q^{24} +390625. q^{25} -649800. q^{26} -549095. q^{27} -1.42500e6 q^{30} -1.52380e6 q^{32} +1.98351e6 q^{33} -288119. q^{36} -1.88940e6 q^{37} -3.81064e6 q^{38} +1.73280e6 q^{39} -6.26841e6 q^{40} +1.52374e7 q^{44} -300625. q^{45} +1.09102e7 q^{48} +5.76480e6 q^{49} -1.14220e7 q^{50} +1.33114e7 q^{52} -4.40267e6 q^{53} +1.60558e7 q^{54} +1.58988e7 q^{55} +1.01617e7 q^{57} +2.91916e7 q^{60} -2.32594e7 q^{61} +8.73684e6 q^{64} +1.38892e7 q^{65} -5.79986e7 q^{66} +4.01615e7 q^{67} +4.82417e6 q^{72} +5.52467e7 q^{74} +3.04587e7 q^{75} +7.80623e7 q^{76} -5.06677e7 q^{78} +8.74506e7 q^{80} -3.96595e7 q^{81} -2.55129e8 q^{88} +8.79039e6 q^{90} +8.14506e7 q^{95} -1.18818e8 q^{96} +1.66373e8 q^{97} -1.68565e8 q^{98} -1.22357e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 1198 q^{4} + 1250 q^{5} - 4560 q^{6} - 962 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 1198 q^{4} + 1250 q^{5} - 4560 q^{6} - 962 q^{9} + 50876 q^{11} + 279842 q^{16} + 260642 q^{19} + 748750 q^{20} - 1564080 q^{24} + 781250 q^{25} - 1299600 q^{26} - 2850000 q^{30} - 576238 q^{36} + 3465600 q^{39} + 30474724 q^{44} - 601250 q^{45} + 11529602 q^{49} + 32111520 q^{54} + 31797500 q^{55} - 46518724 q^{61} + 17473678 q^{64} - 115997280 q^{66} + 110493360 q^{74} + 156124558 q^{76} + 174901250 q^{80} - 79319038 q^{81} + 162901250 q^{95} - 237635280 q^{96} - 24471356 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −29.2404 −1.82752 −0.913762 0.406250i \(-0.866836\pi\)
−0.913762 + 0.406250i \(0.866836\pi\)
\(3\) 77.9744 0.962646 0.481323 0.876543i \(-0.340156\pi\)
0.481323 + 0.876543i \(0.340156\pi\)
\(4\) 599.000 2.33984
\(5\) 625.000 1.00000
\(6\) −2280.00 −1.75926
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −10029.5 −2.44860
\(9\) −481.000 −0.0733120
\(10\) −18275.2 −1.82752
\(11\) 25438.0 1.73745 0.868725 0.495295i \(-0.164940\pi\)
0.868725 + 0.495295i \(0.164940\pi\)
\(12\) 46706.6 2.25244
\(13\) 22222.7 0.778078 0.389039 0.921221i \(-0.372807\pi\)
0.389039 + 0.921221i \(0.372807\pi\)
\(14\) 0 0
\(15\) 48734.0 0.962646
\(16\) 139921. 2.13503
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 14064.6 0.133979
\(19\) 130321. 1.00000
\(20\) 374375. 2.33984
\(21\) 0 0
\(22\) −743817. −3.17523
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) −782040. −2.35713
\(25\) 390625. 1.00000
\(26\) −649800. −1.42196
\(27\) −549095. −1.03322
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) −1.42500e6 −1.75926
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −1.52380e6 −1.45321
\(33\) 1.98351e6 1.67255
\(34\) 0 0
\(35\) 0 0
\(36\) −288119. −0.171539
\(37\) −1.88940e6 −1.00813 −0.504065 0.863666i \(-0.668163\pi\)
−0.504065 + 0.863666i \(0.668163\pi\)
\(38\) −3.81064e6 −1.82752
\(39\) 1.73280e6 0.749014
\(40\) −6.26841e6 −2.44860
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 1.52374e7 4.06536
\(45\) −300625. −0.0733120
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.09102e7 2.05527
\(49\) 5.76480e6 1.00000
\(50\) −1.14220e7 −1.82752
\(51\) 0 0
\(52\) 1.33114e7 1.82058
\(53\) −4.40267e6 −0.557972 −0.278986 0.960295i \(-0.589998\pi\)
−0.278986 + 0.960295i \(0.589998\pi\)
\(54\) 1.60558e7 1.88823
\(55\) 1.58988e7 1.73745
\(56\) 0 0
\(57\) 1.01617e7 0.962646
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 2.91916e7 2.25244
\(61\) −2.32594e7 −1.67988 −0.839940 0.542679i \(-0.817410\pi\)
−0.839940 + 0.542679i \(0.817410\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 8.73684e6 0.520756
\(65\) 1.38892e7 0.778078
\(66\) −5.79986e7 −3.05662
\(67\) 4.01615e7 1.99302 0.996509 0.0834847i \(-0.0266050\pi\)
0.996509 + 0.0834847i \(0.0266050\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 4.82417e6 0.179511
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 5.52467e7 1.84238
\(75\) 3.04587e7 0.962646
\(76\) 7.80623e7 2.33984
\(77\) 0 0
\(78\) −5.06677e7 −1.36884
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 8.74506e7 2.13503
\(81\) −3.96595e7 −0.921313
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −2.55129e8 −4.25431
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 8.79039e6 0.133979
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.14506e7 1.00000
\(96\) −1.18818e8 −1.39893
\(97\) 1.66373e8 1.87930 0.939651 0.342133i \(-0.111149\pi\)
0.939651 + 0.342133i \(0.111149\pi\)
\(98\) −1.68565e8 −1.82752
\(99\) −1.22357e7 −0.127376
\(100\) 2.33984e8 2.33984
\(101\) −1.95809e8 −1.88168 −0.940842 0.338845i \(-0.889964\pi\)
−0.940842 + 0.338845i \(0.889964\pi\)
\(102\) 0 0
\(103\) 2.24840e8 1.99767 0.998837 0.0482057i \(-0.0153503\pi\)
0.998837 + 0.0482057i \(0.0153503\pi\)
\(104\) −2.22881e8 −1.90520
\(105\) 0 0
\(106\) 1.28736e8 1.01971
\(107\) −2.59413e8 −1.97905 −0.989526 0.144353i \(-0.953890\pi\)
−0.989526 + 0.144353i \(0.953890\pi\)
\(108\) −3.28908e8 −2.41757
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) −4.64886e8 −3.17523
\(111\) −1.47324e8 −0.970472
\(112\) 0 0
\(113\) 2.16814e8 1.32976 0.664880 0.746951i \(-0.268483\pi\)
0.664880 + 0.746951i \(0.268483\pi\)
\(114\) −2.97132e8 −1.75926
\(115\) 0 0
\(116\) 0 0
\(117\) −1.06891e7 −0.0570425
\(118\) 0 0
\(119\) 0 0
\(120\) −4.88775e8 −2.35713
\(121\) 4.32733e8 2.01873
\(122\) 6.80113e8 3.07002
\(123\) 0 0
\(124\) 0 0
\(125\) 2.44141e8 1.00000
\(126\) 0 0
\(127\) 4.11470e8 1.58170 0.790849 0.612011i \(-0.209639\pi\)
0.790849 + 0.612011i \(0.209639\pi\)
\(128\) 1.34625e8 0.501518
\(129\) 0 0
\(130\) −4.06125e8 −1.42196
\(131\) −1.72921e8 −0.587170 −0.293585 0.955933i \(-0.594848\pi\)
−0.293585 + 0.955933i \(0.594848\pi\)
\(132\) 1.18812e9 3.91350
\(133\) 0 0
\(134\) −1.17434e9 −3.64229
\(135\) −3.43185e8 −1.03322
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 7.43150e8 1.99075 0.995376 0.0960536i \(-0.0306220\pi\)
0.995376 + 0.0960536i \(0.0306220\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.65301e8 1.35187
\(144\) −6.73020e7 −0.156523
\(145\) 0 0
\(146\) 0 0
\(147\) 4.49507e8 0.962646
\(148\) −1.13175e9 −2.35887
\(149\) 9.27735e8 1.88226 0.941128 0.338050i \(-0.109767\pi\)
0.941128 + 0.338050i \(0.109767\pi\)
\(150\) −8.90625e8 −1.75926
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) −1.30705e9 −2.44860
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 1.03795e9 1.75258
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) −3.43295e8 −0.537130
\(160\) −9.52378e8 −1.45321
\(161\) 0 0
\(162\) 1.15966e9 1.68372
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 1.23969e9 1.67255
\(166\) 0 0
\(167\) 6.99256e8 0.899022 0.449511 0.893275i \(-0.351598\pi\)
0.449511 + 0.893275i \(0.351598\pi\)
\(168\) 0 0
\(169\) −3.21883e8 −0.394594
\(170\) 0 0
\(171\) −6.26844e7 −0.0733120
\(172\) 0 0
\(173\) −1.54318e9 −1.72279 −0.861394 0.507937i \(-0.830408\pi\)
−0.861394 + 0.507937i \(0.830408\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.55931e9 3.70950
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) −1.80074e8 −0.171539
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) −1.81363e9 −1.61713
\(184\) 0 0
\(185\) −1.18087e9 −1.00813
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) −2.38165e9 −1.82752
\(191\) −2.32896e9 −1.74996 −0.874979 0.484160i \(-0.839125\pi\)
−0.874979 + 0.484160i \(0.839125\pi\)
\(192\) 6.81249e8 0.501304
\(193\) −1.78121e9 −1.28377 −0.641884 0.766802i \(-0.721847\pi\)
−0.641884 + 0.766802i \(0.721847\pi\)
\(194\) −4.86482e9 −3.43447
\(195\) 1.08300e9 0.749014
\(196\) 3.45312e9 2.33984
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 3.57776e8 0.232783
\(199\) −2.53711e9 −1.61781 −0.808904 0.587941i \(-0.799939\pi\)
−0.808904 + 0.587941i \(0.799939\pi\)
\(200\) −3.91775e9 −2.44860
\(201\) 3.13157e9 1.91857
\(202\) 5.72552e9 3.43882
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) −6.57441e9 −3.65080
\(207\) 0 0
\(208\) 3.10942e9 1.66122
\(209\) 3.31511e9 1.73745
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −2.63720e9 −1.30557
\(213\) 0 0
\(214\) 7.58535e9 3.61677
\(215\) 0 0
\(216\) 5.50713e9 2.52994
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 9.52335e9 4.06536
\(221\) 0 0
\(222\) 4.30782e9 1.77356
\(223\) 4.28518e9 1.73280 0.866402 0.499347i \(-0.166427\pi\)
0.866402 + 0.499347i \(0.166427\pi\)
\(224\) 0 0
\(225\) −1.87891e8 −0.0733120
\(226\) −6.33972e9 −2.43017
\(227\) −4.22128e9 −1.58979 −0.794897 0.606745i \(-0.792475\pi\)
−0.794897 + 0.606745i \(0.792475\pi\)
\(228\) 6.08686e9 2.25244
\(229\) 8.62779e8 0.313731 0.156866 0.987620i \(-0.449861\pi\)
0.156866 + 0.987620i \(0.449861\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 3.12554e8 0.104246
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4.42349e9 1.35573 0.677866 0.735186i \(-0.262905\pi\)
0.677866 + 0.735186i \(0.262905\pi\)
\(240\) 6.81891e9 2.05527
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −1.26533e10 −3.68928
\(243\) 5.10190e8 0.146321
\(244\) −1.39324e10 −3.93066
\(245\) 3.60300e9 1.00000
\(246\) 0 0
\(247\) 2.89608e9 0.778078
\(248\) 0 0
\(249\) 0 0
\(250\) −7.13877e9 −1.82752
\(251\) 6.88071e8 0.173356 0.0866779 0.996236i \(-0.472375\pi\)
0.0866779 + 0.996236i \(0.472375\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −1.20316e10 −2.89059
\(255\) 0 0
\(256\) −6.17313e9 −1.43729
\(257\) −6.10159e9 −1.39865 −0.699327 0.714802i \(-0.746517\pi\)
−0.699327 + 0.714802i \(0.746517\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 8.31962e9 1.82058
\(261\) 0 0
\(262\) 5.05629e9 1.07307
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) −1.98935e10 −4.09540
\(265\) −2.75167e9 −0.557972
\(266\) 0 0
\(267\) 0 0
\(268\) 2.40568e10 4.66335
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 1.00348e10 1.88823
\(271\) −1.03139e10 −1.91225 −0.956126 0.292955i \(-0.905361\pi\)
−0.956126 + 0.292955i \(0.905361\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 9.93672e9 1.73745
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) −2.17300e10 −3.63815
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 6.35106e9 0.962646
\(286\) −1.65296e10 −2.47058
\(287\) 0 0
\(288\) 7.32950e8 0.106538
\(289\) 6.97576e9 1.00000
\(290\) 0 0
\(291\) 1.29729e10 1.80910
\(292\) 0 0
\(293\) −5.95616e8 −0.0808157 −0.0404079 0.999183i \(-0.512866\pi\)
−0.0404079 + 0.999183i \(0.512866\pi\)
\(294\) −1.31437e10 −1.75926
\(295\) 0 0
\(296\) 1.89496e10 2.46850
\(297\) −1.39679e10 −1.79517
\(298\) −2.71273e10 −3.43987
\(299\) 0 0
\(300\) 1.82448e10 2.25244
\(301\) 0 0
\(302\) 0 0
\(303\) −1.52681e10 −1.81140
\(304\) 1.82346e10 2.13503
\(305\) −1.45371e10 −1.67988
\(306\) 0 0
\(307\) 5.39419e9 0.607257 0.303628 0.952791i \(-0.401802\pi\)
0.303628 + 0.952791i \(0.401802\pi\)
\(308\) 0 0
\(309\) 1.75318e10 1.92305
\(310\) 0 0
\(311\) 1.48383e10 1.58615 0.793073 0.609126i \(-0.208480\pi\)
0.793073 + 0.609126i \(0.208480\pi\)
\(312\) −1.73790e10 −1.83403
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.24159e10 −1.22954 −0.614769 0.788707i \(-0.710751\pi\)
−0.614769 + 0.788707i \(0.710751\pi\)
\(318\) 1.00381e10 0.981617
\(319\) 0 0
\(320\) 5.46052e9 0.520756
\(321\) −2.02276e10 −1.90513
\(322\) 0 0
\(323\) 0 0
\(324\) −2.37561e10 −2.15573
\(325\) 8.68074e9 0.778078
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) −3.62492e10 −3.05662
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 9.08800e8 0.0739080
\(334\) −2.04465e10 −1.64298
\(335\) 2.51010e10 1.99302
\(336\) 0 0
\(337\) 9.04476e9 0.701257 0.350629 0.936515i \(-0.385968\pi\)
0.350629 + 0.936515i \(0.385968\pi\)
\(338\) 9.41197e9 0.721131
\(339\) 1.69059e10 1.28009
\(340\) 0 0
\(341\) 0 0
\(342\) 1.83292e9 0.133979
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 4.51232e10 3.14844
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −2.90598e10 −1.95880 −0.979402 0.201922i \(-0.935281\pi\)
−0.979402 + 0.201922i \(0.935281\pi\)
\(350\) 0 0
\(351\) −1.22024e10 −0.803926
\(352\) −3.87625e10 −2.52488
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −3.09210e10 −1.86156 −0.930778 0.365586i \(-0.880869\pi\)
−0.930778 + 0.365586i \(0.880869\pi\)
\(360\) 3.01510e9 0.179511
\(361\) 1.69836e10 1.00000
\(362\) 0 0
\(363\) 3.37421e10 1.94332
\(364\) 0 0
\(365\) 0 0
\(366\) 5.30313e10 2.95535
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 3.45292e10 1.84238
\(371\) 0 0
\(372\) 0 0
\(373\) −3.70714e10 −1.91516 −0.957578 0.288175i \(-0.906952\pi\)
−0.957578 + 0.288175i \(0.906952\pi\)
\(374\) 0 0
\(375\) 1.90367e10 0.962646
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 4.87889e10 2.33984
\(381\) 3.20841e10 1.52262
\(382\) 6.80996e10 3.19809
\(383\) 2.70081e10 1.25516 0.627579 0.778553i \(-0.284046\pi\)
0.627579 + 0.778553i \(0.284046\pi\)
\(384\) 1.04973e10 0.482785
\(385\) 0 0
\(386\) 5.20833e10 2.34612
\(387\) 0 0
\(388\) 9.96576e10 4.39728
\(389\) 1.56692e10 0.684304 0.342152 0.939645i \(-0.388844\pi\)
0.342152 + 0.939645i \(0.388844\pi\)
\(390\) −3.16673e10 −1.36884
\(391\) 0 0
\(392\) −5.78178e10 −2.44860
\(393\) −1.34834e10 −0.565237
\(394\) 0 0
\(395\) 0 0
\(396\) −7.32917e9 −0.298040
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 7.41861e10 2.95658
\(399\) 0 0
\(400\) 5.46566e10 2.13503
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) −9.15683e10 −3.50624
\(403\) 0 0
\(404\) −1.17289e11 −4.40285
\(405\) −2.47872e10 −0.921313
\(406\) 0 0
\(407\) −4.80625e10 −1.75157
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.34679e11 4.67425
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −3.38630e10 −1.13071
\(417\) 5.79466e10 1.91639
\(418\) −9.69350e10 −3.17523
\(419\) −4.16612e10 −1.35169 −0.675844 0.737045i \(-0.736220\pi\)
−0.675844 + 0.737045i \(0.736220\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 4.41563e10 1.36625
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −1.55389e11 −4.63067
\(429\) 4.40790e10 1.30137
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −7.68300e10 −2.20595
\(433\) −3.16978e10 −0.901733 −0.450866 0.892591i \(-0.648885\pi\)
−0.450866 + 0.892591i \(0.648885\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) −1.59456e11 −4.25431
\(441\) −2.77287e9 −0.0733120
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) −8.82474e10 −2.27075
\(445\) 0 0
\(446\) −1.25300e11 −3.16674
\(447\) 7.23395e10 1.81195
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 5.49399e9 0.133979
\(451\) 0 0
\(452\) 1.29871e11 3.11143
\(453\) 0 0
\(454\) 1.23432e11 2.90538
\(455\) 0 0
\(456\) −1.01916e11 −2.35713
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) −2.52280e10 −0.573351
\(459\) 0 0
\(460\) 0 0
\(461\) −8.57825e10 −1.89931 −0.949653 0.313304i \(-0.898564\pi\)
−0.949653 + 0.313304i \(0.898564\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −6.40278e9 −0.133470
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 5.09066e10 1.00000
\(476\) 0 0
\(477\) 2.11768e9 0.0409060
\(478\) −1.29345e11 −2.47763
\(479\) −1.66110e10 −0.315539 −0.157770 0.987476i \(-0.550430\pi\)
−0.157770 + 0.987476i \(0.550430\pi\)
\(480\) −7.42610e10 −1.39893
\(481\) −4.19875e10 −0.784403
\(482\) 0 0
\(483\) 0 0
\(484\) 2.59207e11 4.72352
\(485\) 1.03983e11 1.87930
\(486\) −1.49181e10 −0.267405
\(487\) −2.98415e10 −0.530524 −0.265262 0.964176i \(-0.585458\pi\)
−0.265262 + 0.964176i \(0.585458\pi\)
\(488\) 2.33279e11 4.11335
\(489\) 0 0
\(490\) −1.05353e11 −1.82752
\(491\) 3.84904e10 0.662257 0.331129 0.943586i \(-0.392571\pi\)
0.331129 + 0.943586i \(0.392571\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −8.46826e10 −1.42196
\(495\) −7.64730e9 −0.127376
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −8.87190e10 −1.43092 −0.715459 0.698655i \(-0.753782\pi\)
−0.715459 + 0.698655i \(0.753782\pi\)
\(500\) 1.46240e11 2.33984
\(501\) 5.45240e10 0.865440
\(502\) −2.01195e10 −0.316812
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) −1.22381e11 −1.88168
\(506\) 0 0
\(507\) −2.50986e10 −0.379855
\(508\) 2.46471e11 3.70093
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.46041e11 2.12517
\(513\) −7.15587e10 −1.03322
\(514\) 1.78413e11 2.55607
\(515\) 1.40525e11 1.99767
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −1.20328e11 −1.65844
\(520\) −1.39301e11 −1.90520
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −1.18966e11 −1.59007 −0.795034 0.606565i \(-0.792547\pi\)
−0.795034 + 0.606565i \(0.792547\pi\)
\(524\) −1.03580e11 −1.37389
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 2.77535e11 3.57094
\(529\) 7.83110e10 1.00000
\(530\) 8.04598e10 1.01971
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −1.62133e11 −1.97905
\(536\) −4.02798e11 −4.88010
\(537\) 0 0
\(538\) 0 0
\(539\) 1.46645e11 1.73745
\(540\) −2.05568e11 −2.41757
\(541\) −5.22741e10 −0.610236 −0.305118 0.952315i \(-0.598696\pi\)
−0.305118 + 0.952315i \(0.598696\pi\)
\(542\) 3.01582e11 3.49469
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 7.61783e10 0.850907 0.425453 0.904980i \(-0.360115\pi\)
0.425453 + 0.904980i \(0.360115\pi\)
\(548\) 0 0
\(549\) 1.11878e10 0.123155
\(550\) −2.90553e11 −3.17523
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −9.20778e10 −0.970472
\(556\) 4.45147e11 4.65805
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.76163e11 −1.75340 −0.876702 0.481034i \(-0.840261\pi\)
−0.876702 + 0.481034i \(0.840261\pi\)
\(564\) 0 0
\(565\) 1.35509e11 1.32976
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) −1.85707e11 −1.75926
\(571\) −1.69333e10 −0.159293 −0.0796464 0.996823i \(-0.525379\pi\)
−0.0796464 + 0.996823i \(0.525379\pi\)
\(572\) 3.38615e11 3.16317
\(573\) −1.81599e11 −1.68459
\(574\) 0 0
\(575\) 0 0
\(576\) −4.20242e9 −0.0381777
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −2.03974e11 −1.82752
\(579\) −1.38889e11 −1.23581
\(580\) 0 0
\(581\) 0 0
\(582\) −3.79331e11 −3.30618
\(583\) −1.11995e11 −0.969448
\(584\) 0 0
\(585\) −6.68070e9 −0.0570425
\(586\) 1.74160e10 0.147693
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 2.69254e11 2.25244
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −2.64366e11 −2.15238
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 4.08426e11 3.28071
\(595\) 0 0
\(596\) 5.55713e11 4.40419
\(597\) −1.97830e11 −1.55738
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) −3.05484e11 −2.35713
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) −1.93177e10 −0.146112
\(604\) 0 0
\(605\) 2.70458e11 2.01873
\(606\) 4.46444e11 3.31037
\(607\) −2.06910e11 −1.52414 −0.762072 0.647492i \(-0.775818\pi\)
−0.762072 + 0.647492i \(0.775818\pi\)
\(608\) −1.98584e11 −1.45321
\(609\) 0 0
\(610\) 4.25070e11 3.07002
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) −1.57728e11 −1.10978
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) −5.12635e11 −3.51443
\(619\) −2.93349e11 −1.99812 −0.999061 0.0433359i \(-0.986201\pi\)
−0.999061 + 0.0433359i \(0.986201\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −4.33878e11 −2.89872
\(623\) 0 0
\(624\) 2.42455e11 1.59916
\(625\) 1.52588e11 1.00000
\(626\) 0 0
\(627\) 2.58493e11 1.67255
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 6.28689e10 0.396569 0.198284 0.980145i \(-0.436463\pi\)
0.198284 + 0.980145i \(0.436463\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 3.63046e11 2.24701
\(635\) 2.57169e11 1.58170
\(636\) −2.05634e11 −1.25680
\(637\) 1.28109e11 0.778078
\(638\) 0 0
\(639\) 0 0
\(640\) 8.41408e10 0.501518
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 5.91463e11 3.48167
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 3.97763e11 2.25592
\(649\) 0 0
\(650\) −2.53828e11 −1.42196
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) −1.08076e11 −0.587170
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 7.42577e11 3.91350
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −2.65737e10 −0.135069
\(667\) 0 0
\(668\) 4.18854e11 2.10357
\(669\) 3.34134e11 1.66808
\(670\) −7.33962e11 −3.64229
\(671\) −5.91672e11 −2.91871
\(672\) 0 0
\(673\) 3.92999e11 1.91572 0.957859 0.287240i \(-0.0927378\pi\)
0.957859 + 0.287240i \(0.0927378\pi\)
\(674\) −2.64472e11 −1.28156
\(675\) −2.14490e11 −1.03322
\(676\) −1.92808e11 −0.923289
\(677\) −3.97601e11 −1.89275 −0.946373 0.323075i \(-0.895283\pi\)
−0.946373 + 0.323075i \(0.895283\pi\)
\(678\) −4.94335e11 −2.33939
\(679\) 0 0
\(680\) 0 0
\(681\) −3.29151e11 −1.53041
\(682\) 0 0
\(683\) −4.34597e11 −1.99712 −0.998560 0.0536512i \(-0.982914\pi\)
−0.998560 + 0.0536512i \(0.982914\pi\)
\(684\) −3.75480e10 −0.171539
\(685\) 0 0
\(686\) 0 0
\(687\) 6.72747e10 0.302012
\(688\) 0 0
\(689\) −9.78391e10 −0.434146
\(690\) 0 0
\(691\) −4.55396e11 −1.99745 −0.998727 0.0504447i \(-0.983936\pi\)
−0.998727 + 0.0504447i \(0.983936\pi\)
\(692\) −9.24364e11 −4.03106
\(693\) 0 0
\(694\) 0 0
\(695\) 4.64469e11 1.99075
\(696\) 0 0
\(697\) 0 0
\(698\) 8.49719e11 3.57976
\(699\) 0 0
\(700\) 0 0
\(701\) 4.77261e11 1.97644 0.988221 0.153031i \(-0.0489034\pi\)
0.988221 + 0.153031i \(0.0489034\pi\)
\(702\) 3.56802e11 1.46919
\(703\) −2.46228e11 −1.00813
\(704\) 2.22248e11 0.904788
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 4.24996e11 1.68190 0.840949 0.541114i \(-0.181997\pi\)
0.840949 + 0.541114i \(0.181997\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 3.53313e11 1.35187
\(716\) 0 0
\(717\) 3.44919e11 1.30509
\(718\) 9.04142e11 3.40204
\(719\) 2.61075e11 0.976899 0.488450 0.872592i \(-0.337563\pi\)
0.488450 + 0.872592i \(0.337563\pi\)
\(720\) −4.20638e10 −0.156523
\(721\) 0 0
\(722\) −4.96606e11 −1.82752
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) −9.86631e11 −3.55147
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 2.99988e11 1.06217
\(730\) 0 0
\(731\) 0 0
\(732\) −1.08637e12 −3.78383
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 2.80942e11 0.962646
\(736\) 0 0
\(737\) 1.02163e12 3.46277
\(738\) 0 0
\(739\) 1.66265e11 0.557472 0.278736 0.960368i \(-0.410085\pi\)
0.278736 + 0.960368i \(0.410085\pi\)
\(740\) −7.07343e11 −2.35887
\(741\) 2.25820e11 0.749014
\(742\) 0 0
\(743\) −1.44136e11 −0.472952 −0.236476 0.971637i \(-0.575992\pi\)
−0.236476 + 0.971637i \(0.575992\pi\)
\(744\) 0 0
\(745\) 5.79834e11 1.88226
\(746\) 1.08398e12 3.49999
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) −5.56641e11 −1.75926
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 5.36519e10 0.166880
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) −8.16905e11 −2.44860
\(761\) −1.43885e11 −0.429019 −0.214509 0.976722i \(-0.568815\pi\)
−0.214509 + 0.976722i \(0.568815\pi\)
\(762\) −9.38153e11 −2.78262
\(763\) 0 0
\(764\) −1.39504e12 −4.09463
\(765\) 0 0
\(766\) −7.89726e11 −2.29383
\(767\) 0 0
\(768\) −4.81346e11 −1.38360
\(769\) 1.88845e11 0.540009 0.270005 0.962859i \(-0.412975\pi\)
0.270005 + 0.962859i \(0.412975\pi\)
\(770\) 0 0
\(771\) −4.75767e11 −1.34641
\(772\) −1.06695e12 −3.00382
\(773\) 6.76787e11 1.89554 0.947772 0.318948i \(-0.103329\pi\)
0.947772 + 0.318948i \(0.103329\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −1.66863e12 −4.60165
\(777\) 0 0
\(778\) −4.58174e11 −1.25058
\(779\) 0 0
\(780\) 6.48717e11 1.75258
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 8.06617e11 2.13503
\(785\) 0 0
\(786\) 3.94261e11 1.03298
\(787\) −4.73234e11 −1.23361 −0.616803 0.787117i \(-0.711573\pi\)
−0.616803 + 0.787117i \(0.711573\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 1.22717e11 0.311892
\(793\) −5.16886e11 −1.30708
\(794\) 0 0
\(795\) −2.14559e11 −0.537130
\(796\) −1.51973e12 −3.78542
\(797\) 7.38977e11 1.83146 0.915730 0.401794i \(-0.131613\pi\)
0.915730 + 0.401794i \(0.131613\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −5.95236e11 −1.45321
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 1.87581e12 4.48916
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 1.96385e12 4.60749
\(809\) −8.53205e11 −1.99186 −0.995931 0.0901140i \(-0.971277\pi\)
−0.995931 + 0.0901140i \(0.971277\pi\)
\(810\) 7.24787e11 1.68372
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) −8.04219e11 −1.84082
\(814\) 1.40537e12 3.20104
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8.68754e11 1.91216 0.956080 0.293105i \(-0.0946885\pi\)
0.956080 + 0.293105i \(0.0946885\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) −2.25502e12 −4.89150
\(825\) 7.74809e11 1.67255
\(826\) 0 0
\(827\) 9.23631e11 1.97459 0.987294 0.158902i \(-0.0507953\pi\)
0.987294 + 0.158902i \(0.0507953\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.94156e11 0.405189
\(833\) 0 0
\(834\) −1.69438e12 −3.50225
\(835\) 4.37035e11 0.899022
\(836\) 1.98575e12 4.06536
\(837\) 0 0
\(838\) 1.21819e12 2.47024
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 5.00246e11 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −2.01177e11 −0.394594
\(846\) 0 0
\(847\) 0 0
\(848\) −6.16025e11 −1.19128
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) −3.91778e10 −0.0733120
\(856\) 2.60177e12 4.84590
\(857\) −1.06418e12 −1.97285 −0.986424 0.164221i \(-0.947489\pi\)
−0.986424 + 0.164221i \(0.947489\pi\)
\(858\) −1.28889e12 −2.37829
\(859\) 9.30424e11 1.70887 0.854433 0.519561i \(-0.173905\pi\)
0.854433 + 0.519561i \(0.173905\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −5.37567e11 −0.969146 −0.484573 0.874751i \(-0.661025\pi\)
−0.484573 + 0.874751i \(0.661025\pi\)
\(864\) 8.36714e11 1.50149
\(865\) −9.64487e11 −1.72279
\(866\) 9.26856e11 1.64794
\(867\) 5.43930e11 0.962646
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 8.92498e11 1.55072
\(872\) 0 0
\(873\) −8.00256e10 −0.137775
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.20027e11 0.202900 0.101450 0.994841i \(-0.467652\pi\)
0.101450 + 0.994841i \(0.467652\pi\)
\(878\) 0 0
\(879\) −4.64428e10 −0.0777970
\(880\) 2.22457e12 3.70950
\(881\) −3.83197e11 −0.636090 −0.318045 0.948076i \(-0.603026\pi\)
−0.318045 + 0.948076i \(0.603026\pi\)
\(882\) 8.10798e10 0.133979
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 9.27492e11 1.49836 0.749179 0.662368i \(-0.230448\pi\)
0.749179 + 0.662368i \(0.230448\pi\)
\(888\) 1.47758e12 2.37629
\(889\) 0 0
\(890\) 0 0
\(891\) −1.00886e12 −1.60074
\(892\) 2.56682e12 4.05449
\(893\) 0 0
\(894\) −2.11524e12 −3.31138
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −1.12546e11 −0.171539
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −2.17452e12 −3.25604
\(905\) 0 0
\(906\) 0 0
\(907\) −1.00891e12 −1.49081 −0.745406 0.666611i \(-0.767744\pi\)
−0.745406 + 0.666611i \(0.767744\pi\)
\(908\) −2.52855e12 −3.71987
\(909\) 9.41840e10 0.137950
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 1.42183e12 2.05527
\(913\) 0 0
\(914\) 0 0
\(915\) −1.13352e12 −1.61713
\(916\) 5.16805e11 0.734082
\(917\) 0 0
\(918\) 0 0
\(919\) −7.64618e11 −1.07197 −0.535985 0.844227i \(-0.680060\pi\)
−0.535985 + 0.844227i \(0.680060\pi\)
\(920\) 0 0
\(921\) 4.20608e11 0.584574
\(922\) 2.50831e12 3.47103
\(923\) 0 0
\(924\) 0 0
\(925\) −7.38046e11 −1.00813
\(926\) 0 0
\(927\) −1.08148e11 −0.146454
\(928\) 0 0
\(929\) −5.77276e11 −0.775034 −0.387517 0.921863i \(-0.626667\pi\)
−0.387517 + 0.921863i \(0.626667\pi\)
\(930\) 0 0
\(931\) 7.51275e11 1.00000
\(932\) 0 0
\(933\) 1.15701e12 1.52690
\(934\) 0 0
\(935\) 0 0
\(936\) 1.07206e11 0.139674
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −1.48853e12 −1.82752
\(951\) −9.68124e11 −1.18361
\(952\) 0 0
\(953\) −1.12800e12 −1.36754 −0.683768 0.729699i \(-0.739660\pi\)
−0.683768 + 0.729699i \(0.739660\pi\)
\(954\) −6.19218e10 −0.0747567
\(955\) −1.45560e12 −1.74996
\(956\) 2.64967e12 3.17220
\(957\) 0 0
\(958\) 4.85712e11 0.576656
\(959\) 0 0
\(960\) 4.25781e11 0.501304
\(961\) 8.52891e11 1.00000
\(962\) 1.22773e12 1.43352
\(963\) 1.24778e11 0.145088
\(964\) 0 0
\(965\) −1.11326e12 −1.28377
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −4.34007e12 −4.94306
\(969\) 0 0
\(970\) −3.04051e12 −3.43447
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 3.05604e11 0.342368
\(973\) 0 0
\(974\) 8.72577e11 0.969545
\(975\) 6.76875e11 0.749014
\(976\) −3.25447e12 −3.58659
\(977\) −1.52462e12 −1.67334 −0.836669 0.547709i \(-0.815500\pi\)
−0.836669 + 0.547709i \(0.815500\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 2.15820e12 2.33984
\(981\) 0 0
\(982\) −1.12547e12 −1.21029
\(983\) 4.58563e10 0.0491117 0.0245559 0.999698i \(-0.492183\pi\)
0.0245559 + 0.999698i \(0.492183\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 1.73475e12 1.82058
\(989\) 0 0
\(990\) 2.23610e11 0.232783
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1.58569e12 −1.61781
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 2.59418e12 2.61504
\(999\) 1.03746e12 1.04162
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 95.9.d.d.94.1 2
5.4 even 2 inner 95.9.d.d.94.2 yes 2
19.18 odd 2 inner 95.9.d.d.94.2 yes 2
95.94 odd 2 CM 95.9.d.d.94.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.9.d.d.94.1 2 1.1 even 1 trivial
95.9.d.d.94.1 2 95.94 odd 2 CM
95.9.d.d.94.2 yes 2 5.4 even 2 inner
95.9.d.d.94.2 yes 2 19.18 odd 2 inner