Properties

Label 75.22.a.b.1.1
Level $75$
Weight $22$
Character 75.1
Self dual yes
Analytic conductor $209.608$
Analytic rank $1$
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] = [75,22,Mod(1,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.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: \(209.608008215\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 75.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-544.000 q^{2} -59049.0 q^{3} -1.80122e6 q^{4} +3.21227e7 q^{6} -1.27770e9 q^{7} +2.12071e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\) \(q-544.000 q^{2} -59049.0 q^{3} -1.80122e6 q^{4} +3.21227e7 q^{6} -1.27770e9 q^{7} +2.12071e9 q^{8} +3.48678e9 q^{9} -7.75859e10 q^{11} +1.06360e11 q^{12} +4.34111e11 q^{13} +6.95068e11 q^{14} +2.62376e12 q^{16} -1.28489e13 q^{17} -1.89681e12 q^{18} -2.86053e13 q^{19} +7.54468e13 q^{21} +4.22067e13 q^{22} -2.24022e14 q^{23} -1.25226e14 q^{24} -2.36156e14 q^{26} -2.05891e14 q^{27} +2.30141e15 q^{28} -5.16760e13 q^{29} +8.92111e15 q^{31} -5.87478e15 q^{32} +4.58137e15 q^{33} +6.98981e15 q^{34} -6.28045e15 q^{36} -4.39772e16 q^{37} +1.55613e16 q^{38} -2.56338e16 q^{39} +5.81681e16 q^{41} -4.10431e16 q^{42} +1.61438e17 q^{43} +1.39749e17 q^{44} +1.21868e17 q^{46} +1.60065e17 q^{47} -1.54930e17 q^{48} +1.07397e18 q^{49} +7.58716e17 q^{51} -7.81927e17 q^{52} -2.29953e18 q^{53} +1.12005e17 q^{54} -2.70963e18 q^{56} +1.68911e18 q^{57} +2.81118e16 q^{58} +5.15426e18 q^{59} +1.25169e18 q^{61} -4.85308e18 q^{62} -4.45506e18 q^{63} -2.30654e18 q^{64} -2.49227e18 q^{66} +5.40779e18 q^{67} +2.31437e19 q^{68} +1.32283e19 q^{69} -1.10432e19 q^{71} +7.39447e18 q^{72} +3.77012e19 q^{73} +2.39236e19 q^{74} +5.15242e19 q^{76} +9.91314e19 q^{77} +1.39448e19 q^{78} +6.31554e19 q^{79} +1.21577e19 q^{81} -3.16434e19 q^{82} +1.45158e20 q^{83} -1.35896e20 q^{84} -8.78222e19 q^{86} +3.05142e18 q^{87} -1.64537e20 q^{88} +1.37255e20 q^{89} -5.54663e20 q^{91} +4.03512e20 q^{92} -5.26783e20 q^{93} -8.70752e19 q^{94} +3.46900e20 q^{96} +3.24306e20 q^{97} -5.84238e20 q^{98} -2.70525e20 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −544.000 −0.375650 −0.187825 0.982202i \(-0.560144\pi\)
−0.187825 + 0.982202i \(0.560144\pi\)
\(3\) −59049.0 −0.577350
\(4\) −1.80122e6 −0.858887
\(5\) 0 0
\(6\) 3.21227e7 0.216882
\(7\) −1.27770e9 −1.70962 −0.854809 0.518943i \(-0.826326\pi\)
−0.854809 + 0.518943i \(0.826326\pi\)
\(8\) 2.12071e9 0.698292
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) −7.75859e10 −0.901903 −0.450951 0.892549i \(-0.648915\pi\)
−0.450951 + 0.892549i \(0.648915\pi\)
\(12\) 1.06360e11 0.495878
\(13\) 4.34111e11 0.873364 0.436682 0.899616i \(-0.356153\pi\)
0.436682 + 0.899616i \(0.356153\pi\)
\(14\) 6.95068e11 0.642219
\(15\) 0 0
\(16\) 2.62376e12 0.596573
\(17\) −1.28489e13 −1.54580 −0.772899 0.634529i \(-0.781194\pi\)
−0.772899 + 0.634529i \(0.781194\pi\)
\(18\) −1.89681e12 −0.125217
\(19\) −2.86053e13 −1.07037 −0.535184 0.844736i \(-0.679758\pi\)
−0.535184 + 0.844736i \(0.679758\pi\)
\(20\) 0 0
\(21\) 7.54468e13 0.987048
\(22\) 4.22067e13 0.338800
\(23\) −2.24022e14 −1.12758 −0.563792 0.825917i \(-0.690658\pi\)
−0.563792 + 0.825917i \(0.690658\pi\)
\(24\) −1.25226e14 −0.403159
\(25\) 0 0
\(26\) −2.36156e14 −0.328080
\(27\) −2.05891e14 −0.192450
\(28\) 2.30141e15 1.46837
\(29\) −5.16760e13 −0.0228092 −0.0114046 0.999935i \(-0.503630\pi\)
−0.0114046 + 0.999935i \(0.503630\pi\)
\(30\) 0 0
\(31\) 8.92111e15 1.95488 0.977442 0.211203i \(-0.0677382\pi\)
0.977442 + 0.211203i \(0.0677382\pi\)
\(32\) −5.87478e15 −0.922395
\(33\) 4.58137e15 0.520714
\(34\) 6.98981e15 0.580680
\(35\) 0 0
\(36\) −6.28045e15 −0.286296
\(37\) −4.39772e16 −1.50352 −0.751760 0.659437i \(-0.770795\pi\)
−0.751760 + 0.659437i \(0.770795\pi\)
\(38\) 1.55613e16 0.402084
\(39\) −2.56338e16 −0.504237
\(40\) 0 0
\(41\) 5.81681e16 0.676791 0.338395 0.941004i \(-0.390116\pi\)
0.338395 + 0.941004i \(0.390116\pi\)
\(42\) −4.10431e16 −0.370785
\(43\) 1.61438e17 1.13916 0.569582 0.821934i \(-0.307105\pi\)
0.569582 + 0.821934i \(0.307105\pi\)
\(44\) 1.39749e17 0.774632
\(45\) 0 0
\(46\) 1.21868e17 0.423577
\(47\) 1.60065e17 0.443883 0.221941 0.975060i \(-0.428761\pi\)
0.221941 + 0.975060i \(0.428761\pi\)
\(48\) −1.54930e17 −0.344432
\(49\) 1.07397e18 1.92279
\(50\) 0 0
\(51\) 7.58716e17 0.892467
\(52\) −7.81927e17 −0.750121
\(53\) −2.29953e18 −1.80610 −0.903050 0.429535i \(-0.858678\pi\)
−0.903050 + 0.429535i \(0.858678\pi\)
\(54\) 1.12005e17 0.0722940
\(55\) 0 0
\(56\) −2.70963e18 −1.19381
\(57\) 1.68911e18 0.617977
\(58\) 2.81118e16 0.00856829
\(59\) 5.15426e18 1.31286 0.656432 0.754385i \(-0.272065\pi\)
0.656432 + 0.754385i \(0.272065\pi\)
\(60\) 0 0
\(61\) 1.25169e18 0.224663 0.112332 0.993671i \(-0.464168\pi\)
0.112332 + 0.993671i \(0.464168\pi\)
\(62\) −4.85308e18 −0.734353
\(63\) −4.45506e18 −0.569872
\(64\) −2.30654e18 −0.250075
\(65\) 0 0
\(66\) −2.49227e18 −0.195606
\(67\) 5.40779e18 0.362438 0.181219 0.983443i \(-0.441996\pi\)
0.181219 + 0.983443i \(0.441996\pi\)
\(68\) 2.31437e19 1.32767
\(69\) 1.32283e19 0.651011
\(70\) 0 0
\(71\) −1.10432e19 −0.402609 −0.201305 0.979529i \(-0.564518\pi\)
−0.201305 + 0.979529i \(0.564518\pi\)
\(72\) 7.39447e18 0.232764
\(73\) 3.77012e19 1.02675 0.513375 0.858164i \(-0.328395\pi\)
0.513375 + 0.858164i \(0.328395\pi\)
\(74\) 2.39236e19 0.564798
\(75\) 0 0
\(76\) 5.15242e19 0.919325
\(77\) 9.91314e19 1.54191
\(78\) 1.39448e19 0.189417
\(79\) 6.31554e19 0.750457 0.375229 0.926932i \(-0.377564\pi\)
0.375229 + 0.926932i \(0.377564\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) −3.16434e19 −0.254237
\(83\) 1.45158e20 1.02688 0.513442 0.858124i \(-0.328370\pi\)
0.513442 + 0.858124i \(0.328370\pi\)
\(84\) −1.35896e20 −0.847762
\(85\) 0 0
\(86\) −8.78222e19 −0.427928
\(87\) 3.05142e18 0.0131689
\(88\) −1.64537e20 −0.629791
\(89\) 1.37255e20 0.466588 0.233294 0.972406i \(-0.425050\pi\)
0.233294 + 0.972406i \(0.425050\pi\)
\(90\) 0 0
\(91\) −5.54663e20 −1.49312
\(92\) 4.03512e20 0.968467
\(93\) −5.26783e20 −1.12865
\(94\) −8.70752e19 −0.166745
\(95\) 0 0
\(96\) 3.46900e20 0.532545
\(97\) 3.24306e20 0.446532 0.223266 0.974758i \(-0.428328\pi\)
0.223266 + 0.974758i \(0.428328\pi\)
\(98\) −5.84238e20 −0.722298
\(99\) −2.70525e20 −0.300634
\(100\) 0 0
\(101\) 1.14557e21 1.03192 0.515962 0.856611i \(-0.327435\pi\)
0.515962 + 0.856611i \(0.327435\pi\)
\(102\) −4.12741e20 −0.335256
\(103\) −8.95787e20 −0.656771 −0.328385 0.944544i \(-0.606504\pi\)
−0.328385 + 0.944544i \(0.606504\pi\)
\(104\) 9.20624e20 0.609863
\(105\) 0 0
\(106\) 1.25094e21 0.678463
\(107\) −1.25783e21 −0.618146 −0.309073 0.951038i \(-0.600019\pi\)
−0.309073 + 0.951038i \(0.600019\pi\)
\(108\) 3.70854e20 0.165293
\(109\) −1.66725e21 −0.674562 −0.337281 0.941404i \(-0.609507\pi\)
−0.337281 + 0.941404i \(0.609507\pi\)
\(110\) 0 0
\(111\) 2.59681e21 0.868057
\(112\) −3.35237e21 −1.01991
\(113\) 6.51198e21 1.80463 0.902317 0.431073i \(-0.141865\pi\)
0.902317 + 0.431073i \(0.141865\pi\)
\(114\) −9.18877e20 −0.232143
\(115\) 0 0
\(116\) 9.30797e19 0.0195905
\(117\) 1.51365e21 0.291121
\(118\) −2.80392e21 −0.493178
\(119\) 1.64170e22 2.64272
\(120\) 0 0
\(121\) −1.38067e21 −0.186571
\(122\) −6.80917e20 −0.0843949
\(123\) −3.43477e21 −0.390745
\(124\) −1.60688e22 −1.67902
\(125\) 0 0
\(126\) 2.42355e21 0.214073
\(127\) 7.99589e21 0.650022 0.325011 0.945710i \(-0.394632\pi\)
0.325011 + 0.945710i \(0.394632\pi\)
\(128\) 1.35751e22 1.01634
\(129\) −9.53274e21 −0.657697
\(130\) 0 0
\(131\) −5.53406e21 −0.324859 −0.162430 0.986720i \(-0.551933\pi\)
−0.162430 + 0.986720i \(0.551933\pi\)
\(132\) −8.25204e21 −0.447234
\(133\) 3.65489e22 1.82992
\(134\) −2.94184e21 −0.136150
\(135\) 0 0
\(136\) −2.72489e22 −1.07942
\(137\) −5.02831e21 −0.184440 −0.0922201 0.995739i \(-0.529396\pi\)
−0.0922201 + 0.995739i \(0.529396\pi\)
\(138\) −7.19619e21 −0.244552
\(139\) 3.06514e21 0.0965594 0.0482797 0.998834i \(-0.484626\pi\)
0.0482797 + 0.998834i \(0.484626\pi\)
\(140\) 0 0
\(141\) −9.45166e21 −0.256276
\(142\) 6.00752e21 0.151240
\(143\) −3.36809e22 −0.787690
\(144\) 9.14847e21 0.198858
\(145\) 0 0
\(146\) −2.05094e22 −0.385699
\(147\) −6.34167e22 −1.11012
\(148\) 7.92124e22 1.29135
\(149\) 3.18365e22 0.483581 0.241791 0.970328i \(-0.422265\pi\)
0.241791 + 0.970328i \(0.422265\pi\)
\(150\) 0 0
\(151\) 6.56248e22 0.866583 0.433292 0.901254i \(-0.357352\pi\)
0.433292 + 0.901254i \(0.357352\pi\)
\(152\) −6.06635e22 −0.747429
\(153\) −4.48014e22 −0.515266
\(154\) −5.39275e22 −0.579219
\(155\) 0 0
\(156\) 4.61720e22 0.433083
\(157\) 6.59284e22 0.578264 0.289132 0.957289i \(-0.406633\pi\)
0.289132 + 0.957289i \(0.406633\pi\)
\(158\) −3.43565e22 −0.281910
\(159\) 1.35785e23 1.04275
\(160\) 0 0
\(161\) 2.86233e23 1.92774
\(162\) −6.61377e21 −0.0417389
\(163\) 8.79055e21 0.0520052 0.0260026 0.999662i \(-0.491722\pi\)
0.0260026 + 0.999662i \(0.491722\pi\)
\(164\) −1.04773e23 −0.581286
\(165\) 0 0
\(166\) −7.89661e22 −0.385750
\(167\) 1.51028e23 0.692683 0.346341 0.938109i \(-0.387424\pi\)
0.346341 + 0.938109i \(0.387424\pi\)
\(168\) 1.60001e23 0.689247
\(169\) −5.86123e22 −0.237235
\(170\) 0 0
\(171\) −9.97404e22 −0.356789
\(172\) −2.90784e23 −0.978414
\(173\) −6.14868e23 −1.94669 −0.973347 0.229338i \(-0.926344\pi\)
−0.973347 + 0.229338i \(0.926344\pi\)
\(174\) −1.65997e21 −0.00494690
\(175\) 0 0
\(176\) −2.03567e23 −0.538051
\(177\) −3.04354e23 −0.757983
\(178\) −7.46667e22 −0.175274
\(179\) 4.83508e23 1.07016 0.535078 0.844803i \(-0.320282\pi\)
0.535078 + 0.844803i \(0.320282\pi\)
\(180\) 0 0
\(181\) −9.28497e23 −1.82876 −0.914378 0.404862i \(-0.867320\pi\)
−0.914378 + 0.404862i \(0.867320\pi\)
\(182\) 3.01737e23 0.560891
\(183\) −7.39108e22 −0.129709
\(184\) −4.75087e23 −0.787382
\(185\) 0 0
\(186\) 2.86570e23 0.423979
\(187\) 9.96895e23 1.39416
\(188\) −2.88311e23 −0.381245
\(189\) 2.63067e23 0.329016
\(190\) 0 0
\(191\) −5.96634e23 −0.668125 −0.334062 0.942551i \(-0.608420\pi\)
−0.334062 + 0.942551i \(0.608420\pi\)
\(192\) 1.36199e23 0.144381
\(193\) 8.56373e23 0.859629 0.429815 0.902917i \(-0.358579\pi\)
0.429815 + 0.902917i \(0.358579\pi\)
\(194\) −1.76423e23 −0.167740
\(195\) 0 0
\(196\) −1.93445e24 −1.65146
\(197\) −1.10520e24 −0.894430 −0.447215 0.894427i \(-0.647584\pi\)
−0.447215 + 0.894427i \(0.647584\pi\)
\(198\) 1.47166e23 0.112933
\(199\) 1.53187e24 1.11497 0.557487 0.830186i \(-0.311766\pi\)
0.557487 + 0.830186i \(0.311766\pi\)
\(200\) 0 0
\(201\) −3.19324e23 −0.209254
\(202\) −6.23191e23 −0.387643
\(203\) 6.60264e22 0.0389950
\(204\) −1.36661e24 −0.766528
\(205\) 0 0
\(206\) 4.87308e23 0.246716
\(207\) −7.81117e23 −0.375861
\(208\) 1.13900e24 0.521026
\(209\) 2.21937e24 0.965368
\(210\) 0 0
\(211\) 2.15918e24 0.849811 0.424906 0.905238i \(-0.360307\pi\)
0.424906 + 0.905238i \(0.360307\pi\)
\(212\) 4.14195e24 1.55124
\(213\) 6.52092e23 0.232446
\(214\) 6.84258e23 0.232207
\(215\) 0 0
\(216\) −4.36636e23 −0.134386
\(217\) −1.13985e25 −3.34210
\(218\) 9.06983e23 0.253400
\(219\) −2.22622e24 −0.592795
\(220\) 0 0
\(221\) −5.57785e24 −1.35005
\(222\) −1.41266e24 −0.326086
\(223\) −7.86309e24 −1.73138 −0.865689 0.500582i \(-0.833119\pi\)
−0.865689 + 0.500582i \(0.833119\pi\)
\(224\) 7.50620e24 1.57694
\(225\) 0 0
\(226\) −3.54252e24 −0.677912
\(227\) −5.06979e24 −0.926229 −0.463115 0.886298i \(-0.653268\pi\)
−0.463115 + 0.886298i \(0.653268\pi\)
\(228\) −3.04246e24 −0.530772
\(229\) −6.10932e24 −1.01793 −0.508967 0.860786i \(-0.669973\pi\)
−0.508967 + 0.860786i \(0.669973\pi\)
\(230\) 0 0
\(231\) −5.85361e24 −0.890221
\(232\) −1.09590e23 −0.0159275
\(233\) −2.40865e24 −0.334609 −0.167304 0.985905i \(-0.553506\pi\)
−0.167304 + 0.985905i \(0.553506\pi\)
\(234\) −8.23426e23 −0.109360
\(235\) 0 0
\(236\) −9.28393e24 −1.12760
\(237\) −3.72926e24 −0.433277
\(238\) −8.93087e24 −0.992740
\(239\) −8.12956e23 −0.0864748 −0.0432374 0.999065i \(-0.513767\pi\)
−0.0432374 + 0.999065i \(0.513767\pi\)
\(240\) 0 0
\(241\) 1.98089e24 0.193055 0.0965277 0.995330i \(-0.469226\pi\)
0.0965277 + 0.995330i \(0.469226\pi\)
\(242\) 7.51087e23 0.0700856
\(243\) −7.17898e23 −0.0641500
\(244\) −2.25456e24 −0.192960
\(245\) 0 0
\(246\) 1.86851e24 0.146784
\(247\) −1.24179e25 −0.934821
\(248\) 1.89191e25 1.36508
\(249\) −8.57145e24 −0.592872
\(250\) 0 0
\(251\) −1.71966e25 −1.09363 −0.546813 0.837255i \(-0.684159\pi\)
−0.546813 + 0.837255i \(0.684159\pi\)
\(252\) 8.02452e24 0.489456
\(253\) 1.73810e25 1.01697
\(254\) −4.34976e24 −0.244181
\(255\) 0 0
\(256\) −2.54768e24 −0.131712
\(257\) −1.42771e25 −0.708503 −0.354251 0.935150i \(-0.615264\pi\)
−0.354251 + 0.935150i \(0.615264\pi\)
\(258\) 5.18581e24 0.247064
\(259\) 5.61895e25 2.57044
\(260\) 0 0
\(261\) −1.80183e23 −0.00760307
\(262\) 3.01053e24 0.122034
\(263\) −1.23443e25 −0.480765 −0.240382 0.970678i \(-0.577273\pi\)
−0.240382 + 0.970678i \(0.577273\pi\)
\(264\) 9.71577e24 0.363610
\(265\) 0 0
\(266\) −1.98826e25 −0.687410
\(267\) −8.10477e24 −0.269384
\(268\) −9.74059e24 −0.311293
\(269\) −4.23127e24 −0.130039 −0.0650193 0.997884i \(-0.520711\pi\)
−0.0650193 + 0.997884i \(0.520711\pi\)
\(270\) 0 0
\(271\) −5.48006e25 −1.55815 −0.779073 0.626933i \(-0.784310\pi\)
−0.779073 + 0.626933i \(0.784310\pi\)
\(272\) −3.37124e25 −0.922182
\(273\) 3.27523e25 0.862053
\(274\) 2.73540e24 0.0692851
\(275\) 0 0
\(276\) −2.38270e25 −0.559144
\(277\) −1.74992e25 −0.395349 −0.197675 0.980268i \(-0.563339\pi\)
−0.197675 + 0.980268i \(0.563339\pi\)
\(278\) −1.66743e24 −0.0362726
\(279\) 3.11060e25 0.651628
\(280\) 0 0
\(281\) 2.52311e25 0.490365 0.245183 0.969477i \(-0.421152\pi\)
0.245183 + 0.969477i \(0.421152\pi\)
\(282\) 5.14171e24 0.0962701
\(283\) 1.19713e25 0.215966 0.107983 0.994153i \(-0.465561\pi\)
0.107983 + 0.994153i \(0.465561\pi\)
\(284\) 1.98912e25 0.345796
\(285\) 0 0
\(286\) 1.83224e25 0.295896
\(287\) −7.43213e25 −1.15705
\(288\) −2.04841e25 −0.307465
\(289\) 9.60027e25 1.38949
\(290\) 0 0
\(291\) −1.91500e25 −0.257805
\(292\) −6.79080e25 −0.881863
\(293\) −6.89938e25 −0.864371 −0.432186 0.901785i \(-0.642257\pi\)
−0.432186 + 0.901785i \(0.642257\pi\)
\(294\) 3.44987e25 0.417019
\(295\) 0 0
\(296\) −9.32629e25 −1.04990
\(297\) 1.59743e25 0.173571
\(298\) −1.73191e25 −0.181657
\(299\) −9.72505e25 −0.984791
\(300\) 0 0
\(301\) −2.06269e26 −1.94754
\(302\) −3.56999e25 −0.325532
\(303\) −6.76449e25 −0.595782
\(304\) −7.50532e25 −0.638553
\(305\) 0 0
\(306\) 2.43720e25 0.193560
\(307\) 6.78268e25 0.520533 0.260267 0.965537i \(-0.416190\pi\)
0.260267 + 0.965537i \(0.416190\pi\)
\(308\) −1.78557e26 −1.32432
\(309\) 5.28953e25 0.379187
\(310\) 0 0
\(311\) 1.00215e26 0.671347 0.335673 0.941978i \(-0.391036\pi\)
0.335673 + 0.941978i \(0.391036\pi\)
\(312\) −5.43619e25 −0.352105
\(313\) 2.29792e26 1.43920 0.719599 0.694390i \(-0.244326\pi\)
0.719599 + 0.694390i \(0.244326\pi\)
\(314\) −3.58651e25 −0.217225
\(315\) 0 0
\(316\) −1.13756e26 −0.644558
\(317\) −2.06457e26 −1.13164 −0.565818 0.824530i \(-0.691439\pi\)
−0.565818 + 0.824530i \(0.691439\pi\)
\(318\) −7.38669e25 −0.391711
\(319\) 4.00933e24 0.0205717
\(320\) 0 0
\(321\) 7.42734e25 0.356887
\(322\) −1.55711e26 −0.724155
\(323\) 3.67547e26 1.65457
\(324\) −2.18986e25 −0.0954319
\(325\) 0 0
\(326\) −4.78206e24 −0.0195358
\(327\) 9.84493e25 0.389459
\(328\) 1.23358e26 0.472597
\(329\) −2.04515e26 −0.758870
\(330\) 0 0
\(331\) −1.09215e26 −0.380268 −0.190134 0.981758i \(-0.560892\pi\)
−0.190134 + 0.981758i \(0.560892\pi\)
\(332\) −2.61461e26 −0.881978
\(333\) −1.53339e26 −0.501173
\(334\) −8.21592e25 −0.260207
\(335\) 0 0
\(336\) 1.97954e26 0.588846
\(337\) 4.76493e26 1.37386 0.686930 0.726724i \(-0.258958\pi\)
0.686930 + 0.726724i \(0.258958\pi\)
\(338\) 3.18851e25 0.0891173
\(339\) −3.84526e26 −1.04191
\(340\) 0 0
\(341\) −6.92152e26 −1.76312
\(342\) 5.42588e25 0.134028
\(343\) −6.58553e26 −1.57762
\(344\) 3.42363e26 0.795469
\(345\) 0 0
\(346\) 3.34488e26 0.731276
\(347\) 5.67729e26 1.20415 0.602077 0.798438i \(-0.294340\pi\)
0.602077 + 0.798438i \(0.294340\pi\)
\(348\) −5.49626e24 −0.0113106
\(349\) −4.85816e26 −0.970075 −0.485037 0.874493i \(-0.661194\pi\)
−0.485037 + 0.874493i \(0.661194\pi\)
\(350\) 0 0
\(351\) −8.93796e25 −0.168079
\(352\) 4.55800e26 0.831910
\(353\) −1.71788e26 −0.304340 −0.152170 0.988354i \(-0.548626\pi\)
−0.152170 + 0.988354i \(0.548626\pi\)
\(354\) 1.65568e26 0.284737
\(355\) 0 0
\(356\) −2.47226e26 −0.400746
\(357\) −9.69410e26 −1.52578
\(358\) −2.63028e26 −0.402004
\(359\) 1.33028e27 1.97448 0.987239 0.159247i \(-0.0509068\pi\)
0.987239 + 0.159247i \(0.0509068\pi\)
\(360\) 0 0
\(361\) 1.04051e26 0.145687
\(362\) 5.05103e26 0.686973
\(363\) 8.15275e25 0.107717
\(364\) 9.99067e26 1.28242
\(365\) 0 0
\(366\) 4.02075e25 0.0487254
\(367\) −3.40629e26 −0.401132 −0.200566 0.979680i \(-0.564278\pi\)
−0.200566 + 0.979680i \(0.564278\pi\)
\(368\) −5.87780e26 −0.672686
\(369\) 2.02820e26 0.225597
\(370\) 0 0
\(371\) 2.93810e27 3.08774
\(372\) 9.48849e26 0.969385
\(373\) 1.62912e27 1.61812 0.809058 0.587729i \(-0.199978\pi\)
0.809058 + 0.587729i \(0.199978\pi\)
\(374\) −5.42311e26 −0.523717
\(375\) 0 0
\(376\) 3.39451e26 0.309960
\(377\) −2.24331e25 −0.0199207
\(378\) −1.43108e26 −0.123595
\(379\) 3.25550e26 0.273467 0.136734 0.990608i \(-0.456340\pi\)
0.136734 + 0.990608i \(0.456340\pi\)
\(380\) 0 0
\(381\) −4.72149e26 −0.375290
\(382\) 3.24569e26 0.250981
\(383\) 1.61833e27 1.21753 0.608765 0.793350i \(-0.291665\pi\)
0.608765 + 0.793350i \(0.291665\pi\)
\(384\) −8.01594e26 −0.586782
\(385\) 0 0
\(386\) −4.65867e26 −0.322920
\(387\) 5.62899e26 0.379722
\(388\) −5.84146e26 −0.383520
\(389\) −2.02736e27 −1.29557 −0.647784 0.761824i \(-0.724304\pi\)
−0.647784 + 0.761824i \(0.724304\pi\)
\(390\) 0 0
\(391\) 2.87844e27 1.74302
\(392\) 2.27758e27 1.34267
\(393\) 3.26781e26 0.187558
\(394\) 6.01230e26 0.335993
\(395\) 0 0
\(396\) 4.87275e26 0.258211
\(397\) −1.57647e27 −0.813551 −0.406775 0.913528i \(-0.633347\pi\)
−0.406775 + 0.913528i \(0.633347\pi\)
\(398\) −8.33337e26 −0.418840
\(399\) −2.15818e27 −1.05650
\(400\) 0 0
\(401\) 2.38019e27 1.10559 0.552797 0.833316i \(-0.313561\pi\)
0.552797 + 0.833316i \(0.313561\pi\)
\(402\) 1.73712e26 0.0786063
\(403\) 3.87275e27 1.70733
\(404\) −2.06342e27 −0.886306
\(405\) 0 0
\(406\) −3.59184e25 −0.0146485
\(407\) 3.41201e27 1.35603
\(408\) 1.60902e27 0.623202
\(409\) 1.28701e27 0.485832 0.242916 0.970047i \(-0.421896\pi\)
0.242916 + 0.970047i \(0.421896\pi\)
\(410\) 0 0
\(411\) 2.96916e26 0.106487
\(412\) 1.61351e27 0.564092
\(413\) −6.58558e27 −2.24450
\(414\) 4.24928e26 0.141192
\(415\) 0 0
\(416\) −2.55031e27 −0.805587
\(417\) −1.80993e26 −0.0557486
\(418\) −1.20733e27 −0.362641
\(419\) −6.36510e27 −1.86448 −0.932241 0.361839i \(-0.882149\pi\)
−0.932241 + 0.361839i \(0.882149\pi\)
\(420\) 0 0
\(421\) 1.16839e27 0.325556 0.162778 0.986663i \(-0.447955\pi\)
0.162778 + 0.986663i \(0.447955\pi\)
\(422\) −1.17459e27 −0.319232
\(423\) 5.58111e26 0.147961
\(424\) −4.87664e27 −1.26119
\(425\) 0 0
\(426\) −3.54738e26 −0.0873186
\(427\) −1.59928e27 −0.384088
\(428\) 2.26562e27 0.530917
\(429\) 1.98882e27 0.454773
\(430\) 0 0
\(431\) 4.87470e27 1.06154 0.530770 0.847516i \(-0.321903\pi\)
0.530770 + 0.847516i \(0.321903\pi\)
\(432\) −5.40208e26 −0.114811
\(433\) −3.42952e27 −0.711395 −0.355698 0.934601i \(-0.615757\pi\)
−0.355698 + 0.934601i \(0.615757\pi\)
\(434\) 6.20078e27 1.25546
\(435\) 0 0
\(436\) 3.00307e27 0.579372
\(437\) 6.40821e27 1.20693
\(438\) 1.21106e27 0.222684
\(439\) 6.10587e27 1.09615 0.548075 0.836429i \(-0.315361\pi\)
0.548075 + 0.836429i \(0.315361\pi\)
\(440\) 0 0
\(441\) 3.74469e27 0.640931
\(442\) 3.03435e27 0.507145
\(443\) 6.52956e27 1.06572 0.532862 0.846202i \(-0.321116\pi\)
0.532862 + 0.846202i \(0.321116\pi\)
\(444\) −4.67741e27 −0.745563
\(445\) 0 0
\(446\) 4.27752e27 0.650393
\(447\) −1.87991e27 −0.279196
\(448\) 2.94706e27 0.427533
\(449\) 3.56364e27 0.505018 0.252509 0.967595i \(-0.418744\pi\)
0.252509 + 0.967595i \(0.418744\pi\)
\(450\) 0 0
\(451\) −4.51302e27 −0.610399
\(452\) −1.17295e28 −1.54998
\(453\) −3.87508e27 −0.500322
\(454\) 2.75797e27 0.347939
\(455\) 0 0
\(456\) 3.58212e27 0.431528
\(457\) 5.64234e25 0.00664262 0.00332131 0.999994i \(-0.498943\pi\)
0.00332131 + 0.999994i \(0.498943\pi\)
\(458\) 3.32347e27 0.382388
\(459\) 2.64548e27 0.297489
\(460\) 0 0
\(461\) 3.68168e27 0.395537 0.197768 0.980249i \(-0.436631\pi\)
0.197768 + 0.980249i \(0.436631\pi\)
\(462\) 3.18436e27 0.334412
\(463\) 6.25388e27 0.642020 0.321010 0.947076i \(-0.395978\pi\)
0.321010 + 0.947076i \(0.395978\pi\)
\(464\) −1.35585e26 −0.0136074
\(465\) 0 0
\(466\) 1.31031e27 0.125696
\(467\) 1.81487e26 0.0170223 0.00851114 0.999964i \(-0.497291\pi\)
0.00851114 + 0.999964i \(0.497291\pi\)
\(468\) −2.72641e27 −0.250040
\(469\) −6.90952e27 −0.619631
\(470\) 0 0
\(471\) −3.89301e27 −0.333861
\(472\) 1.09307e28 0.916762
\(473\) −1.25253e28 −1.02742
\(474\) 2.02872e27 0.162761
\(475\) 0 0
\(476\) −2.95706e28 −2.26980
\(477\) −8.01796e27 −0.602034
\(478\) 4.42248e26 0.0324843
\(479\) 4.83395e27 0.347360 0.173680 0.984802i \(-0.444434\pi\)
0.173680 + 0.984802i \(0.444434\pi\)
\(480\) 0 0
\(481\) −1.90910e28 −1.31312
\(482\) −1.07761e27 −0.0725214
\(483\) −1.69018e28 −1.11298
\(484\) 2.48689e27 0.160244
\(485\) 0 0
\(486\) 3.90537e26 0.0240980
\(487\) 6.30377e27 0.380668 0.190334 0.981719i \(-0.439043\pi\)
0.190334 + 0.981719i \(0.439043\pi\)
\(488\) 2.65447e27 0.156881
\(489\) −5.19073e26 −0.0300252
\(490\) 0 0
\(491\) 3.67372e27 0.203587 0.101794 0.994806i \(-0.467542\pi\)
0.101794 + 0.994806i \(0.467542\pi\)
\(492\) 6.18676e27 0.335606
\(493\) 6.63981e26 0.0352584
\(494\) 6.75531e27 0.351166
\(495\) 0 0
\(496\) 2.34068e28 1.16623
\(497\) 1.41099e28 0.688307
\(498\) 4.66287e27 0.222713
\(499\) −2.00088e28 −0.935761 −0.467881 0.883792i \(-0.654982\pi\)
−0.467881 + 0.883792i \(0.654982\pi\)
\(500\) 0 0
\(501\) −8.91805e27 −0.399921
\(502\) 9.35496e27 0.410821
\(503\) −1.40332e28 −0.603521 −0.301761 0.953384i \(-0.597574\pi\)
−0.301761 + 0.953384i \(0.597574\pi\)
\(504\) −9.44790e27 −0.397937
\(505\) 0 0
\(506\) −9.45525e27 −0.382026
\(507\) 3.46100e27 0.136967
\(508\) −1.44023e28 −0.558295
\(509\) −6.58158e27 −0.249916 −0.124958 0.992162i \(-0.539880\pi\)
−0.124958 + 0.992162i \(0.539880\pi\)
\(510\) 0 0
\(511\) −4.81708e28 −1.75535
\(512\) −2.70830e28 −0.966858
\(513\) 5.88957e27 0.205992
\(514\) 7.76673e27 0.266149
\(515\) 0 0
\(516\) 1.71705e28 0.564887
\(517\) −1.24188e28 −0.400339
\(518\) −3.05671e28 −0.965588
\(519\) 3.63073e28 1.12392
\(520\) 0 0
\(521\) −4.65750e28 −1.38470 −0.692351 0.721561i \(-0.743425\pi\)
−0.692351 + 0.721561i \(0.743425\pi\)
\(522\) 9.80196e25 0.00285610
\(523\) 3.89578e28 1.11257 0.556284 0.830992i \(-0.312227\pi\)
0.556284 + 0.830992i \(0.312227\pi\)
\(524\) 9.96804e27 0.279017
\(525\) 0 0
\(526\) 6.71532e27 0.180600
\(527\) −1.14627e29 −3.02186
\(528\) 1.20204e28 0.310644
\(529\) 1.07144e28 0.271445
\(530\) 0 0
\(531\) 1.79718e28 0.437621
\(532\) −6.58324e28 −1.57169
\(533\) 2.52514e28 0.591085
\(534\) 4.40900e27 0.101194
\(535\) 0 0
\(536\) 1.14684e28 0.253088
\(537\) −2.85507e28 −0.617854
\(538\) 2.30181e27 0.0488491
\(539\) −8.33247e28 −1.73417
\(540\) 0 0
\(541\) 7.65629e27 0.153267 0.0766333 0.997059i \(-0.475583\pi\)
0.0766333 + 0.997059i \(0.475583\pi\)
\(542\) 2.98116e28 0.585318
\(543\) 5.48268e28 1.05583
\(544\) 7.54846e28 1.42584
\(545\) 0 0
\(546\) −1.78172e28 −0.323830
\(547\) −6.77366e28 −1.20769 −0.603847 0.797100i \(-0.706366\pi\)
−0.603847 + 0.797100i \(0.706366\pi\)
\(548\) 9.05707e27 0.158413
\(549\) 4.36436e27 0.0748878
\(550\) 0 0
\(551\) 1.47821e27 0.0244142
\(552\) 2.80534e28 0.454595
\(553\) −8.06935e28 −1.28299
\(554\) 9.51957e27 0.148513
\(555\) 0 0
\(556\) −5.52097e27 −0.0829336
\(557\) 5.64828e28 0.832600 0.416300 0.909227i \(-0.363327\pi\)
0.416300 + 0.909227i \(0.363327\pi\)
\(558\) −1.69217e28 −0.244784
\(559\) 7.00819e28 0.994906
\(560\) 0 0
\(561\) −5.88657e28 −0.804919
\(562\) −1.37257e28 −0.184206
\(563\) −1.32120e28 −0.174033 −0.0870166 0.996207i \(-0.527733\pi\)
−0.0870166 + 0.996207i \(0.527733\pi\)
\(564\) 1.70245e28 0.220112
\(565\) 0 0
\(566\) −6.51240e27 −0.0811275
\(567\) −1.55338e28 −0.189957
\(568\) −2.34195e28 −0.281139
\(569\) 3.77888e27 0.0445332 0.0222666 0.999752i \(-0.492912\pi\)
0.0222666 + 0.999752i \(0.492912\pi\)
\(570\) 0 0
\(571\) −9.70592e28 −1.10245 −0.551223 0.834358i \(-0.685839\pi\)
−0.551223 + 0.834358i \(0.685839\pi\)
\(572\) 6.06666e28 0.676536
\(573\) 3.52306e28 0.385742
\(574\) 4.04308e28 0.434647
\(575\) 0 0
\(576\) −8.04239e27 −0.0833584
\(577\) 1.40057e27 0.0142547 0.00712737 0.999975i \(-0.497731\pi\)
0.00712737 + 0.999975i \(0.497731\pi\)
\(578\) −5.22255e28 −0.521964
\(579\) −5.05680e28 −0.496307
\(580\) 0 0
\(581\) −1.85468e29 −1.75558
\(582\) 1.04176e28 0.0968447
\(583\) 1.78411e29 1.62893
\(584\) 7.99534e28 0.716972
\(585\) 0 0
\(586\) 3.75326e28 0.324701
\(587\) 9.16040e28 0.778421 0.389211 0.921149i \(-0.372748\pi\)
0.389211 + 0.921149i \(0.372748\pi\)
\(588\) 1.14227e29 0.953471
\(589\) −2.55191e29 −2.09245
\(590\) 0 0
\(591\) 6.52610e28 0.516399
\(592\) −1.15385e29 −0.896959
\(593\) −1.47495e29 −1.12643 −0.563214 0.826311i \(-0.690435\pi\)
−0.563214 + 0.826311i \(0.690435\pi\)
\(594\) −8.68999e27 −0.0652021
\(595\) 0 0
\(596\) −5.73444e28 −0.415341
\(597\) −9.04554e28 −0.643730
\(598\) 5.29043e28 0.369937
\(599\) 1.19182e29 0.818899 0.409450 0.912333i \(-0.365721\pi\)
0.409450 + 0.912333i \(0.365721\pi\)
\(600\) 0 0
\(601\) 2.64125e29 1.75238 0.876190 0.481966i \(-0.160077\pi\)
0.876190 + 0.481966i \(0.160077\pi\)
\(602\) 1.12210e29 0.731593
\(603\) 1.88558e28 0.120813
\(604\) −1.18205e29 −0.744297
\(605\) 0 0
\(606\) 3.67988e28 0.223806
\(607\) 8.82263e28 0.527371 0.263686 0.964609i \(-0.415062\pi\)
0.263686 + 0.964609i \(0.415062\pi\)
\(608\) 1.68050e29 0.987302
\(609\) −3.89879e27 −0.0225138
\(610\) 0 0
\(611\) 6.94859e28 0.387671
\(612\) 8.06970e28 0.442555
\(613\) −3.27173e29 −1.76377 −0.881885 0.471465i \(-0.843725\pi\)
−0.881885 + 0.471465i \(0.843725\pi\)
\(614\) −3.68978e28 −0.195538
\(615\) 0 0
\(616\) 2.10229e29 1.07670
\(617\) −7.46428e28 −0.375832 −0.187916 0.982185i \(-0.560173\pi\)
−0.187916 + 0.982185i \(0.560173\pi\)
\(618\) −2.87751e28 −0.142442
\(619\) −8.32478e28 −0.405155 −0.202577 0.979266i \(-0.564932\pi\)
−0.202577 + 0.979266i \(0.564932\pi\)
\(620\) 0 0
\(621\) 4.61242e28 0.217004
\(622\) −5.45167e28 −0.252192
\(623\) −1.75371e29 −0.797686
\(624\) −6.72569e28 −0.300814
\(625\) 0 0
\(626\) −1.25007e29 −0.540635
\(627\) −1.31051e29 −0.557355
\(628\) −1.18751e29 −0.496663
\(629\) 5.65059e29 2.32414
\(630\) 0 0
\(631\) 3.63173e29 1.44479 0.722396 0.691479i \(-0.243041\pi\)
0.722396 + 0.691479i \(0.243041\pi\)
\(632\) 1.33934e29 0.524038
\(633\) −1.27497e29 −0.490639
\(634\) 1.12312e29 0.425099
\(635\) 0 0
\(636\) −2.44578e29 −0.895606
\(637\) 4.66221e29 1.67930
\(638\) −2.18108e27 −0.00772776
\(639\) −3.85054e28 −0.134203
\(640\) 0 0
\(641\) −4.09788e28 −0.138213 −0.0691067 0.997609i \(-0.522015\pi\)
−0.0691067 + 0.997609i \(0.522015\pi\)
\(642\) −4.04047e28 −0.134065
\(643\) 3.44481e29 1.12448 0.562238 0.826976i \(-0.309941\pi\)
0.562238 + 0.826976i \(0.309941\pi\)
\(644\) −5.15567e29 −1.65571
\(645\) 0 0
\(646\) −1.99945e29 −0.621541
\(647\) 2.99052e29 0.914645 0.457322 0.889301i \(-0.348809\pi\)
0.457322 + 0.889301i \(0.348809\pi\)
\(648\) 2.57829e28 0.0775880
\(649\) −3.99898e29 −1.18408
\(650\) 0 0
\(651\) 6.73069e29 1.92956
\(652\) −1.58337e28 −0.0446665
\(653\) −6.43185e29 −1.78545 −0.892724 0.450603i \(-0.851209\pi\)
−0.892724 + 0.450603i \(0.851209\pi\)
\(654\) −5.35564e28 −0.146300
\(655\) 0 0
\(656\) 1.52619e29 0.403755
\(657\) 1.31456e29 0.342250
\(658\) 1.11256e29 0.285070
\(659\) 1.00447e29 0.253303 0.126652 0.991947i \(-0.459577\pi\)
0.126652 + 0.991947i \(0.459577\pi\)
\(660\) 0 0
\(661\) −4.70181e29 −1.14855 −0.574274 0.818663i \(-0.694716\pi\)
−0.574274 + 0.818663i \(0.694716\pi\)
\(662\) 5.94131e28 0.142848
\(663\) 3.29367e29 0.779449
\(664\) 3.07839e29 0.717065
\(665\) 0 0
\(666\) 8.34163e28 0.188266
\(667\) 1.15766e28 0.0257193
\(668\) −2.72034e29 −0.594936
\(669\) 4.64307e29 0.999612
\(670\) 0 0
\(671\) −9.71132e28 −0.202624
\(672\) −4.43233e29 −0.910448
\(673\) 3.92345e29 0.793432 0.396716 0.917941i \(-0.370150\pi\)
0.396716 + 0.917941i \(0.370150\pi\)
\(674\) −2.59212e29 −0.516091
\(675\) 0 0
\(676\) 1.05573e29 0.203758
\(677\) 1.52718e29 0.290207 0.145104 0.989416i \(-0.453648\pi\)
0.145104 + 0.989416i \(0.453648\pi\)
\(678\) 2.09182e29 0.391392
\(679\) −4.14366e29 −0.763399
\(680\) 0 0
\(681\) 2.99366e29 0.534759
\(682\) 3.76531e29 0.662315
\(683\) 4.36102e29 0.755389 0.377695 0.925930i \(-0.376717\pi\)
0.377695 + 0.925930i \(0.376717\pi\)
\(684\) 1.79654e29 0.306442
\(685\) 0 0
\(686\) 3.58253e29 0.592634
\(687\) 3.60749e29 0.587705
\(688\) 4.23574e29 0.679595
\(689\) −9.98250e29 −1.57738
\(690\) 0 0
\(691\) 5.96732e29 0.914660 0.457330 0.889297i \(-0.348806\pi\)
0.457330 + 0.889297i \(0.348806\pi\)
\(692\) 1.10751e30 1.67199
\(693\) 3.45650e29 0.513970
\(694\) −3.08845e29 −0.452341
\(695\) 0 0
\(696\) 6.47118e27 0.00919573
\(697\) −7.47397e29 −1.04618
\(698\) 2.64284e29 0.364409
\(699\) 1.42229e29 0.193186
\(700\) 0 0
\(701\) 5.37209e29 0.708116 0.354058 0.935223i \(-0.384802\pi\)
0.354058 + 0.935223i \(0.384802\pi\)
\(702\) 4.86225e28 0.0631390
\(703\) 1.25798e30 1.60932
\(704\) 1.78955e29 0.225543
\(705\) 0 0
\(706\) 9.34527e28 0.114325
\(707\) −1.46370e30 −1.76420
\(708\) 5.48207e29 0.651021
\(709\) −1.27317e29 −0.148971 −0.0744854 0.997222i \(-0.523731\pi\)
−0.0744854 + 0.997222i \(0.523731\pi\)
\(710\) 0 0
\(711\) 2.20209e29 0.250152
\(712\) 2.91078e29 0.325814
\(713\) −1.99853e30 −2.20430
\(714\) 5.27359e29 0.573159
\(715\) 0 0
\(716\) −8.70902e29 −0.919142
\(717\) 4.80043e28 0.0499262
\(718\) −7.23674e29 −0.741713
\(719\) −1.86676e30 −1.88553 −0.942767 0.333451i \(-0.891787\pi\)
−0.942767 + 0.333451i \(0.891787\pi\)
\(720\) 0 0
\(721\) 1.14455e30 1.12283
\(722\) −5.66038e28 −0.0547275
\(723\) −1.16970e29 −0.111461
\(724\) 1.67242e30 1.57069
\(725\) 0 0
\(726\) −4.43509e28 −0.0404640
\(727\) 1.86570e30 1.67776 0.838882 0.544314i \(-0.183210\pi\)
0.838882 + 0.544314i \(0.183210\pi\)
\(728\) −1.17628e30 −1.04263
\(729\) 4.23912e28 0.0370370
\(730\) 0 0
\(731\) −2.07430e30 −1.76092
\(732\) 1.33129e29 0.111406
\(733\) 4.88128e29 0.402663 0.201331 0.979523i \(-0.435473\pi\)
0.201331 + 0.979523i \(0.435473\pi\)
\(734\) 1.85302e29 0.150686
\(735\) 0 0
\(736\) 1.31608e30 1.04008
\(737\) −4.19568e29 −0.326884
\(738\) −1.10334e29 −0.0847456
\(739\) 1.67069e30 1.26511 0.632556 0.774515i \(-0.282006\pi\)
0.632556 + 0.774515i \(0.282006\pi\)
\(740\) 0 0
\(741\) 7.33262e29 0.539719
\(742\) −1.59833e30 −1.15991
\(743\) −1.62834e30 −1.16509 −0.582547 0.812797i \(-0.697944\pi\)
−0.582547 + 0.812797i \(0.697944\pi\)
\(744\) −1.11715e30 −0.788129
\(745\) 0 0
\(746\) −8.86240e29 −0.607846
\(747\) 5.06136e29 0.342295
\(748\) −1.79562e30 −1.19743
\(749\) 1.60712e30 1.05679
\(750\) 0 0
\(751\) 5.83073e29 0.372824 0.186412 0.982472i \(-0.440314\pi\)
0.186412 + 0.982472i \(0.440314\pi\)
\(752\) 4.19971e29 0.264809
\(753\) 1.01544e30 0.631406
\(754\) 1.22036e28 0.00748324
\(755\) 0 0
\(756\) −4.73840e29 −0.282587
\(757\) −1.39333e30 −0.819494 −0.409747 0.912199i \(-0.634383\pi\)
−0.409747 + 0.912199i \(0.634383\pi\)
\(758\) −1.77099e29 −0.102728
\(759\) −1.02633e30 −0.587148
\(760\) 0 0
\(761\) −9.17379e29 −0.510517 −0.255258 0.966873i \(-0.582161\pi\)
−0.255258 + 0.966873i \(0.582161\pi\)
\(762\) 2.56849e29 0.140978
\(763\) 2.13024e30 1.15324
\(764\) 1.07467e30 0.573843
\(765\) 0 0
\(766\) −8.80371e29 −0.457366
\(767\) 2.23752e30 1.14661
\(768\) 1.50438e29 0.0760438
\(769\) −1.13366e30 −0.565273 −0.282636 0.959227i \(-0.591209\pi\)
−0.282636 + 0.959227i \(0.591209\pi\)
\(770\) 0 0
\(771\) 8.43047e29 0.409054
\(772\) −1.54251e30 −0.738324
\(773\) 1.81526e29 0.0857143 0.0428572 0.999081i \(-0.486354\pi\)
0.0428572 + 0.999081i \(0.486354\pi\)
\(774\) −3.06217e29 −0.142643
\(775\) 0 0
\(776\) 6.87761e29 0.311809
\(777\) −3.31794e30 −1.48405
\(778\) 1.10289e30 0.486681
\(779\) −1.66391e30 −0.724415
\(780\) 0 0
\(781\) 8.56799e29 0.363114
\(782\) −1.56587e30 −0.654765
\(783\) 1.06396e28 0.00438963
\(784\) 2.81783e30 1.14709
\(785\) 0 0
\(786\) −1.77769e29 −0.0704561
\(787\) −7.88071e29 −0.308199 −0.154099 0.988055i \(-0.549248\pi\)
−0.154099 + 0.988055i \(0.549248\pi\)
\(788\) 1.99071e30 0.768214
\(789\) 7.28921e29 0.277570
\(790\) 0 0
\(791\) −8.32034e30 −3.08523
\(792\) −5.73706e29 −0.209930
\(793\) 5.43371e29 0.196213
\(794\) 8.57598e29 0.305611
\(795\) 0 0
\(796\) −2.75923e30 −0.957636
\(797\) 3.61162e30 1.23706 0.618528 0.785763i \(-0.287729\pi\)
0.618528 + 0.785763i \(0.287729\pi\)
\(798\) 1.17405e30 0.396876
\(799\) −2.05666e30 −0.686153
\(800\) 0 0
\(801\) 4.78579e29 0.155529
\(802\) −1.29482e30 −0.415317
\(803\) −2.92508e30 −0.926029
\(804\) 5.75172e29 0.179725
\(805\) 0 0
\(806\) −2.10678e30 −0.641358
\(807\) 2.49853e29 0.0750778
\(808\) 2.42943e30 0.720584
\(809\) 6.55296e30 1.91857 0.959286 0.282436i \(-0.0911424\pi\)
0.959286 + 0.282436i \(0.0911424\pi\)
\(810\) 0 0
\(811\) −2.95834e30 −0.843975 −0.421988 0.906602i \(-0.638667\pi\)
−0.421988 + 0.906602i \(0.638667\pi\)
\(812\) −1.18928e29 −0.0334923
\(813\) 3.23592e30 0.899596
\(814\) −1.85613e30 −0.509393
\(815\) 0 0
\(816\) 1.99069e30 0.532422
\(817\) −4.61797e30 −1.21933
\(818\) −7.00132e29 −0.182503
\(819\) −1.93399e30 −0.497706
\(820\) 0 0
\(821\) −6.60712e30 −1.65733 −0.828665 0.559744i \(-0.810899\pi\)
−0.828665 + 0.559744i \(0.810899\pi\)
\(822\) −1.61523e29 −0.0400018
\(823\) −4.27938e30 −1.04636 −0.523182 0.852221i \(-0.675255\pi\)
−0.523182 + 0.852221i \(0.675255\pi\)
\(824\) −1.89971e30 −0.458618
\(825\) 0 0
\(826\) 3.58256e30 0.843146
\(827\) −8.08422e29 −0.187858 −0.0939291 0.995579i \(-0.529943\pi\)
−0.0939291 + 0.995579i \(0.529943\pi\)
\(828\) 1.40696e30 0.322822
\(829\) 6.58152e30 1.49109 0.745544 0.666456i \(-0.232190\pi\)
0.745544 + 0.666456i \(0.232190\pi\)
\(830\) 0 0
\(831\) 1.03331e30 0.228255
\(832\) −1.00129e30 −0.218407
\(833\) −1.37993e31 −2.97225
\(834\) 9.84603e28 0.0209420
\(835\) 0 0
\(836\) −3.99756e30 −0.829141
\(837\) −1.83678e30 −0.376218
\(838\) 3.46262e30 0.700393
\(839\) −8.11970e30 −1.62196 −0.810979 0.585075i \(-0.801065\pi\)
−0.810979 + 0.585075i \(0.801065\pi\)
\(840\) 0 0
\(841\) −5.13017e30 −0.999480
\(842\) −6.35604e29 −0.122295
\(843\) −1.48987e30 −0.283113
\(844\) −3.88914e30 −0.729892
\(845\) 0 0
\(846\) −3.03613e29 −0.0555816
\(847\) 1.76409e30 0.318966
\(848\) −6.03340e30 −1.07747
\(849\) −7.06894e29 −0.124688
\(850\) 0 0
\(851\) 9.85186e30 1.69534
\(852\) −1.17456e30 −0.199645
\(853\) −1.29820e30 −0.217960 −0.108980 0.994044i \(-0.534759\pi\)
−0.108980 + 0.994044i \(0.534759\pi\)
\(854\) 8.70007e29 0.144283
\(855\) 0 0
\(856\) −2.66749e30 −0.431646
\(857\) 1.05399e30 0.168476 0.0842380 0.996446i \(-0.473154\pi\)
0.0842380 + 0.996446i \(0.473154\pi\)
\(858\) −1.08192e30 −0.170836
\(859\) 6.98517e30 1.08955 0.544777 0.838581i \(-0.316614\pi\)
0.544777 + 0.838581i \(0.316614\pi\)
\(860\) 0 0
\(861\) 4.38860e30 0.668025
\(862\) −2.65184e30 −0.398768
\(863\) −1.03987e31 −1.54478 −0.772392 0.635146i \(-0.780940\pi\)
−0.772392 + 0.635146i \(0.780940\pi\)
\(864\) 1.20956e30 0.177515
\(865\) 0 0
\(866\) 1.86566e30 0.267236
\(867\) −5.66887e30 −0.802224
\(868\) 2.05311e31 2.87049
\(869\) −4.89997e30 −0.676839
\(870\) 0 0
\(871\) 2.34758e30 0.316541
\(872\) −3.53575e30 −0.471041
\(873\) 1.13079e30 0.148844
\(874\) −3.48607e30 −0.453384
\(875\) 0 0
\(876\) 4.00990e30 0.509144
\(877\) −5.66163e30 −0.710306 −0.355153 0.934808i \(-0.615571\pi\)
−0.355153 + 0.934808i \(0.615571\pi\)
\(878\) −3.32159e30 −0.411770
\(879\) 4.07402e30 0.499045
\(880\) 0 0
\(881\) 3.59833e30 0.430382 0.215191 0.976572i \(-0.430963\pi\)
0.215191 + 0.976572i \(0.430963\pi\)
\(882\) −2.03711e30 −0.240766
\(883\) 1.24249e31 1.45112 0.725562 0.688157i \(-0.241580\pi\)
0.725562 + 0.688157i \(0.241580\pi\)
\(884\) 1.00469e31 1.15954
\(885\) 0 0
\(886\) −3.55208e30 −0.400340
\(887\) 9.60349e30 1.06962 0.534812 0.844971i \(-0.320383\pi\)
0.534812 + 0.844971i \(0.320383\pi\)
\(888\) 5.50708e30 0.606157
\(889\) −1.02163e31 −1.11129
\(890\) 0 0
\(891\) −9.43264e29 −0.100211
\(892\) 1.41631e31 1.48706
\(893\) −4.57869e30 −0.475118
\(894\) 1.02267e30 0.104880
\(895\) 0 0
\(896\) −1.73448e31 −1.73754
\(897\) 5.74254e30 0.568570
\(898\) −1.93862e30 −0.189710
\(899\) −4.61007e29 −0.0445894
\(900\) 0 0
\(901\) 2.95464e31 2.79187
\(902\) 2.45509e30 0.229297
\(903\) 1.21800e31 1.12441
\(904\) 1.38100e31 1.26016
\(905\) 0 0
\(906\) 2.10804e30 0.187946
\(907\) 1.94141e31 1.71097 0.855483 0.517830i \(-0.173260\pi\)
0.855483 + 0.517830i \(0.173260\pi\)
\(908\) 9.13179e30 0.795526
\(909\) 3.99436e30 0.343975
\(910\) 0 0
\(911\) −4.71847e30 −0.397062 −0.198531 0.980095i \(-0.563617\pi\)
−0.198531 + 0.980095i \(0.563617\pi\)
\(912\) 4.43182e30 0.368669
\(913\) −1.12622e31 −0.926150
\(914\) −3.06943e28 −0.00249530
\(915\) 0 0
\(916\) 1.10042e31 0.874291
\(917\) 7.07086e30 0.555385
\(918\) −1.43914e30 −0.111752
\(919\) −1.97911e31 −1.51935 −0.759673 0.650305i \(-0.774641\pi\)
−0.759673 + 0.650305i \(0.774641\pi\)
\(920\) 0 0
\(921\) −4.00510e30 −0.300530
\(922\) −2.00284e30 −0.148584
\(923\) −4.79399e30 −0.351624
\(924\) 1.05436e31 0.764599
\(925\) 0 0
\(926\) −3.40211e30 −0.241175
\(927\) −3.12342e30 −0.218924
\(928\) 3.03585e29 0.0210391
\(929\) −2.30269e31 −1.57786 −0.788932 0.614480i \(-0.789366\pi\)
−0.788932 + 0.614480i \(0.789366\pi\)
\(930\) 0 0
\(931\) −3.07211e31 −2.05809
\(932\) 4.33851e30 0.287391
\(933\) −5.91757e30 −0.387602
\(934\) −9.87288e28 −0.00639443
\(935\) 0 0
\(936\) 3.21002e30 0.203288
\(937\) −2.58396e31 −1.61816 −0.809078 0.587701i \(-0.800033\pi\)
−0.809078 + 0.587701i \(0.800033\pi\)
\(938\) 3.75878e30 0.232765
\(939\) −1.35690e31 −0.830921
\(940\) 0 0
\(941\) 2.93678e31 1.75865 0.879326 0.476221i \(-0.157994\pi\)
0.879326 + 0.476221i \(0.157994\pi\)
\(942\) 2.11780e30 0.125415
\(943\) −1.30309e31 −0.763138
\(944\) 1.35235e31 0.783220
\(945\) 0 0
\(946\) 6.81377e30 0.385949
\(947\) 8.23900e29 0.0461530 0.0230765 0.999734i \(-0.492654\pi\)
0.0230765 + 0.999734i \(0.492654\pi\)
\(948\) 6.71721e30 0.372135
\(949\) 1.63665e31 0.896728
\(950\) 0 0
\(951\) 1.21911e31 0.653350
\(952\) 3.48158e31 1.84539
\(953\) 3.34266e31 1.75233 0.876167 0.482008i \(-0.160093\pi\)
0.876167 + 0.482008i \(0.160093\pi\)
\(954\) 4.36177e30 0.226154
\(955\) 0 0
\(956\) 1.46431e30 0.0742720
\(957\) −2.36747e29 −0.0118771
\(958\) −2.62967e30 −0.130486
\(959\) 6.42466e30 0.315322
\(960\) 0 0
\(961\) 5.87607e31 2.82157
\(962\) 1.03855e31 0.493274
\(963\) −4.38577e30 −0.206049
\(964\) −3.56802e30 −0.165813
\(965\) 0 0
\(966\) 9.19456e30 0.418091
\(967\) −2.28891e31 −1.02956 −0.514779 0.857323i \(-0.672126\pi\)
−0.514779 + 0.857323i \(0.672126\pi\)
\(968\) −2.92801e30 −0.130281
\(969\) −2.17033e31 −0.955268
\(970\) 0 0
\(971\) −4.60425e31 −1.98316 −0.991578 0.129508i \(-0.958660\pi\)
−0.991578 + 0.129508i \(0.958660\pi\)
\(972\) 1.29309e30 0.0550976
\(973\) −3.91632e30 −0.165080
\(974\) −3.42925e30 −0.142998
\(975\) 0 0
\(976\) 3.28412e30 0.134028
\(977\) −7.20727e30 −0.290990 −0.145495 0.989359i \(-0.546477\pi\)
−0.145495 + 0.989359i \(0.546477\pi\)
\(978\) 2.82376e29 0.0112790
\(979\) −1.06491e31 −0.420817
\(980\) 0 0
\(981\) −5.81333e30 −0.224854
\(982\) −1.99850e30 −0.0764776
\(983\) −1.58951e31 −0.601800 −0.300900 0.953656i \(-0.597287\pi\)
−0.300900 + 0.953656i \(0.597287\pi\)
\(984\) −7.28415e30 −0.272854
\(985\) 0 0
\(986\) −3.61206e29 −0.0132448
\(987\) 1.20764e31 0.438134
\(988\) 2.23672e31 0.802905
\(989\) −3.61657e31 −1.28450
\(990\) 0 0
\(991\) 4.62622e31 1.60862 0.804309 0.594211i \(-0.202536\pi\)
0.804309 + 0.594211i \(0.202536\pi\)
\(992\) −5.24095e31 −1.80317
\(993\) 6.44905e30 0.219548
\(994\) −7.67580e30 −0.258563
\(995\) 0 0
\(996\) 1.54390e31 0.509210
\(997\) 1.13237e31 0.369562 0.184781 0.982780i \(-0.440842\pi\)
0.184781 + 0.982780i \(0.440842\pi\)
\(998\) 1.08848e31 0.351519
\(999\) 9.05451e30 0.289352
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 75.22.a.b.1.1 1
5.2 odd 4 75.22.b.c.49.1 2
5.3 odd 4 75.22.b.c.49.2 2
5.4 even 2 15.22.a.a.1.1 1
15.14 odd 2 45.22.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.22.a.a.1.1 1 5.4 even 2
45.22.a.a.1.1 1 15.14 odd 2
75.22.a.b.1.1 1 1.1 even 1 trivial
75.22.b.c.49.1 2 5.2 odd 4
75.22.b.c.49.2 2 5.3 odd 4