Properties

Label 75.22.a.j.1.5
Level $75$
Weight $22$
Character 75.1
Self dual yes
Analytic conductor $209.608$
Analytic rank $1$
Dimension $6$
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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 530373x^{4} - 23850203x^{3} + 59940607490x^{2} + 2888057920800x - 684850637484000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{6}\cdot 5^{7}\cdot 7^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-388.716\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q+1570.86 q^{2} -59049.0 q^{3} +370456. q^{4} -9.27578e7 q^{6} +1.39580e9 q^{7} -2.71240e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\) \(q+1570.86 q^{2} -59049.0 q^{3} +370456. q^{4} -9.27578e7 q^{6} +1.39580e9 q^{7} -2.71240e9 q^{8} +3.48678e9 q^{9} -1.07847e11 q^{11} -2.18751e10 q^{12} +6.34039e11 q^{13} +2.19261e12 q^{14} -5.03771e12 q^{16} -9.95276e12 q^{17} +5.47726e12 q^{18} +2.82734e12 q^{19} -8.24206e13 q^{21} -1.69413e14 q^{22} +7.33909e13 q^{23} +1.60165e14 q^{24} +9.95988e14 q^{26} -2.05891e14 q^{27} +5.17083e14 q^{28} -1.24530e15 q^{29} +6.63391e15 q^{31} -2.22523e15 q^{32} +6.36827e15 q^{33} -1.56344e16 q^{34} +1.29170e15 q^{36} -5.11485e16 q^{37} +4.44137e15 q^{38} -3.74394e16 q^{39} +5.89040e16 q^{41} -1.29471e17 q^{42} -1.83716e17 q^{43} -3.99527e16 q^{44} +1.15287e17 q^{46} +4.57340e16 q^{47} +2.97472e17 q^{48} +1.38971e18 q^{49} +5.87700e17 q^{51} +2.34884e17 q^{52} +1.30917e18 q^{53} -3.23427e17 q^{54} -3.78597e18 q^{56} -1.66952e17 q^{57} -1.95620e18 q^{58} -5.67465e18 q^{59} -7.20648e18 q^{61} +1.04210e19 q^{62} +4.86685e18 q^{63} +7.06931e18 q^{64} +1.00037e19 q^{66} +2.10884e18 q^{67} -3.68706e18 q^{68} -4.33366e18 q^{69} +4.72038e19 q^{71} -9.45756e18 q^{72} +5.09819e19 q^{73} -8.03473e19 q^{74} +1.04741e18 q^{76} -1.50533e20 q^{77} -5.88121e19 q^{78} +7.07733e18 q^{79} +1.21577e19 q^{81} +9.25300e19 q^{82} -5.62246e19 q^{83} -3.05332e19 q^{84} -2.88592e20 q^{86} +7.35338e19 q^{87} +2.92525e20 q^{88} +1.35953e20 q^{89} +8.84991e20 q^{91} +2.71881e19 q^{92} -3.91726e20 q^{93} +7.18418e19 q^{94} +1.31398e20 q^{96} +6.40980e19 q^{97} +2.18304e21 q^{98} -3.76040e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 92 q^{2} - 354294 q^{3} + 4390448 q^{4} - 5432508 q^{6} + 1327143454 q^{7} - 4252271424 q^{8} + 20920706406 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 92 q^{2} - 354294 q^{3} + 4390448 q^{4} - 5432508 q^{6} + 1327143454 q^{7} - 4252271424 q^{8} + 20920706406 q^{9} - 78310112516 q^{11} - 259251563952 q^{12} + 116029746338 q^{13} - 313825730148 q^{14} + 1996701612800 q^{16} + 5382900513068 q^{17} + 320784164892 q^{18} + 6969638997622 q^{19} - 78366493815246 q^{21} - 19701817271864 q^{22} + 295754895311052 q^{23} + 251092375315776 q^{24} - 274314326263628 q^{26} - 12\!\cdots\!94 q^{27}+ \cdots - 27\!\cdots\!16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1570.86 1.08473 0.542367 0.840142i \(-0.317528\pi\)
0.542367 + 0.840142i \(0.317528\pi\)
\(3\) −59049.0 −0.577350
\(4\) 370456. 0.176647
\(5\) 0 0
\(6\) −9.27578e7 −0.626271
\(7\) 1.39580e9 1.86764 0.933821 0.357741i \(-0.116453\pi\)
0.933821 + 0.357741i \(0.116453\pi\)
\(8\) −2.71240e9 −0.893118
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) −1.07847e11 −1.25368 −0.626839 0.779149i \(-0.715652\pi\)
−0.626839 + 0.779149i \(0.715652\pi\)
\(12\) −2.18751e10 −0.101987
\(13\) 6.34039e11 1.27559 0.637794 0.770207i \(-0.279847\pi\)
0.637794 + 0.770207i \(0.279847\pi\)
\(14\) 2.19261e12 2.02589
\(15\) 0 0
\(16\) −5.03771e12 −1.14544
\(17\) −9.95276e12 −1.19737 −0.598687 0.800983i \(-0.704311\pi\)
−0.598687 + 0.800983i \(0.704311\pi\)
\(18\) 5.47726e12 0.361578
\(19\) 2.82734e12 0.105795 0.0528976 0.998600i \(-0.483154\pi\)
0.0528976 + 0.998600i \(0.483154\pi\)
\(20\) 0 0
\(21\) −8.24206e13 −1.07828
\(22\) −1.69413e14 −1.35991
\(23\) 7.33909e13 0.369403 0.184701 0.982795i \(-0.440868\pi\)
0.184701 + 0.982795i \(0.440868\pi\)
\(24\) 1.60165e14 0.515642
\(25\) 0 0
\(26\) 9.95988e14 1.38367
\(27\) −2.05891e14 −0.192450
\(28\) 5.17083e14 0.329914
\(29\) −1.24530e15 −0.549662 −0.274831 0.961493i \(-0.588622\pi\)
−0.274831 + 0.961493i \(0.588622\pi\)
\(30\) 0 0
\(31\) 6.63391e15 1.45369 0.726845 0.686802i \(-0.240986\pi\)
0.726845 + 0.686802i \(0.240986\pi\)
\(32\) −2.22523e15 −0.349382
\(33\) 6.36827e15 0.723811
\(34\) −1.56344e16 −1.29883
\(35\) 0 0
\(36\) 1.29170e15 0.0588824
\(37\) −5.11485e16 −1.74870 −0.874349 0.485297i \(-0.838712\pi\)
−0.874349 + 0.485297i \(0.838712\pi\)
\(38\) 4.44137e15 0.114760
\(39\) −3.74394e16 −0.736462
\(40\) 0 0
\(41\) 5.89040e16 0.685352 0.342676 0.939454i \(-0.388667\pi\)
0.342676 + 0.939454i \(0.388667\pi\)
\(42\) −1.29471e17 −1.16965
\(43\) −1.83716e17 −1.29637 −0.648183 0.761485i \(-0.724471\pi\)
−0.648183 + 0.761485i \(0.724471\pi\)
\(44\) −3.99527e16 −0.221459
\(45\) 0 0
\(46\) 1.15287e17 0.400703
\(47\) 4.57340e16 0.126827 0.0634135 0.997987i \(-0.479801\pi\)
0.0634135 + 0.997987i \(0.479801\pi\)
\(48\) 2.97472e17 0.661322
\(49\) 1.38971e18 2.48809
\(50\) 0 0
\(51\) 5.87700e17 0.691304
\(52\) 2.34884e17 0.225329
\(53\) 1.30917e18 1.02825 0.514126 0.857715i \(-0.328116\pi\)
0.514126 + 0.857715i \(0.328116\pi\)
\(54\) −3.23427e17 −0.208757
\(55\) 0 0
\(56\) −3.78597e18 −1.66803
\(57\) −1.66952e17 −0.0610809
\(58\) −1.95620e18 −0.596236
\(59\) −5.67465e18 −1.44542 −0.722708 0.691153i \(-0.757103\pi\)
−0.722708 + 0.691153i \(0.757103\pi\)
\(60\) 0 0
\(61\) −7.20648e18 −1.29348 −0.646741 0.762710i \(-0.723868\pi\)
−0.646741 + 0.762710i \(0.723868\pi\)
\(62\) 1.04210e19 1.57687
\(63\) 4.86685e18 0.622547
\(64\) 7.06931e18 0.766456
\(65\) 0 0
\(66\) 1.00037e19 0.785143
\(67\) 2.10884e18 0.141338 0.0706688 0.997500i \(-0.477487\pi\)
0.0706688 + 0.997500i \(0.477487\pi\)
\(68\) −3.68706e18 −0.211513
\(69\) −4.33366e18 −0.213275
\(70\) 0 0
\(71\) 4.72038e19 1.72093 0.860467 0.509506i \(-0.170172\pi\)
0.860467 + 0.509506i \(0.170172\pi\)
\(72\) −9.45756e18 −0.297706
\(73\) 5.09819e19 1.38844 0.694218 0.719765i \(-0.255750\pi\)
0.694218 + 0.719765i \(0.255750\pi\)
\(74\) −8.03473e19 −1.89687
\(75\) 0 0
\(76\) 1.04741e18 0.0186884
\(77\) −1.50533e20 −2.34142
\(78\) −5.88121e19 −0.798865
\(79\) 7.07733e18 0.0840979 0.0420490 0.999116i \(-0.486611\pi\)
0.0420490 + 0.999116i \(0.486611\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) 9.25300e19 0.743425
\(83\) −5.62246e19 −0.397747 −0.198873 0.980025i \(-0.563728\pi\)
−0.198873 + 0.980025i \(0.563728\pi\)
\(84\) −3.05332e19 −0.190476
\(85\) 0 0
\(86\) −2.88592e20 −1.40621
\(87\) 7.35338e19 0.317347
\(88\) 2.92525e20 1.11968
\(89\) 1.35953e20 0.462161 0.231080 0.972935i \(-0.425774\pi\)
0.231080 + 0.972935i \(0.425774\pi\)
\(90\) 0 0
\(91\) 8.84991e20 2.38234
\(92\) 2.71881e19 0.0652540
\(93\) −3.91726e20 −0.839288
\(94\) 7.18418e19 0.137574
\(95\) 0 0
\(96\) 1.31398e20 0.201716
\(97\) 6.40980e19 0.0882553 0.0441277 0.999026i \(-0.485949\pi\)
0.0441277 + 0.999026i \(0.485949\pi\)
\(98\) 2.18304e21 2.69891
\(99\) −3.76040e20 −0.417893
\(100\) 0 0
\(101\) −9.83553e20 −0.885979 −0.442989 0.896527i \(-0.646082\pi\)
−0.442989 + 0.896527i \(0.646082\pi\)
\(102\) 9.23196e20 0.749881
\(103\) −1.10209e21 −0.808027 −0.404014 0.914753i \(-0.632385\pi\)
−0.404014 + 0.914753i \(0.632385\pi\)
\(104\) −1.71977e21 −1.13925
\(105\) 0 0
\(106\) 2.05653e21 1.11538
\(107\) −1.92359e21 −0.945330 −0.472665 0.881242i \(-0.656708\pi\)
−0.472665 + 0.881242i \(0.656708\pi\)
\(108\) −7.62736e19 −0.0339958
\(109\) −4.10268e20 −0.165993 −0.0829964 0.996550i \(-0.526449\pi\)
−0.0829964 + 0.996550i \(0.526449\pi\)
\(110\) 0 0
\(111\) 3.02027e21 1.00961
\(112\) −7.03164e21 −2.13928
\(113\) −5.36495e21 −1.48676 −0.743382 0.668867i \(-0.766779\pi\)
−0.743382 + 0.668867i \(0.766779\pi\)
\(114\) −2.62258e20 −0.0662565
\(115\) 0 0
\(116\) −4.61329e20 −0.0970962
\(117\) 2.21076e21 0.425196
\(118\) −8.91409e21 −1.56789
\(119\) −1.38921e22 −2.23627
\(120\) 0 0
\(121\) 4.23079e21 0.571709
\(122\) −1.13204e22 −1.40308
\(123\) −3.47822e21 −0.395688
\(124\) 2.45757e21 0.256790
\(125\) 0 0
\(126\) 7.64515e21 0.675298
\(127\) −1.26019e22 −1.02447 −0.512233 0.858846i \(-0.671182\pi\)
−0.512233 + 0.858846i \(0.671182\pi\)
\(128\) 1.57716e22 1.18078
\(129\) 1.08482e22 0.748457
\(130\) 0 0
\(131\) −2.96143e22 −1.73841 −0.869206 0.494450i \(-0.835369\pi\)
−0.869206 + 0.494450i \(0.835369\pi\)
\(132\) 2.35917e21 0.127859
\(133\) 3.94640e21 0.197587
\(134\) 3.31269e21 0.153314
\(135\) 0 0
\(136\) 2.69959e22 1.06940
\(137\) −5.25367e22 −1.92707 −0.963534 0.267586i \(-0.913774\pi\)
−0.963534 + 0.267586i \(0.913774\pi\)
\(138\) −6.80758e21 −0.231346
\(139\) −4.58307e21 −0.144378 −0.0721889 0.997391i \(-0.522998\pi\)
−0.0721889 + 0.997391i \(0.522998\pi\)
\(140\) 0 0
\(141\) −2.70055e21 −0.0732236
\(142\) 7.41507e22 1.86676
\(143\) −6.83794e22 −1.59918
\(144\) −1.75654e22 −0.381814
\(145\) 0 0
\(146\) 8.00855e22 1.50608
\(147\) −8.20610e22 −1.43650
\(148\) −1.89483e22 −0.308903
\(149\) 2.00094e22 0.303934 0.151967 0.988386i \(-0.451439\pi\)
0.151967 + 0.988386i \(0.451439\pi\)
\(150\) 0 0
\(151\) −4.39241e22 −0.580022 −0.290011 0.957023i \(-0.593659\pi\)
−0.290011 + 0.957023i \(0.593659\pi\)
\(152\) −7.66889e21 −0.0944876
\(153\) −3.47031e22 −0.399125
\(154\) −2.36467e23 −2.53982
\(155\) 0 0
\(156\) −1.38696e22 −0.130094
\(157\) −1.86853e23 −1.63891 −0.819453 0.573146i \(-0.805723\pi\)
−0.819453 + 0.573146i \(0.805723\pi\)
\(158\) 1.11175e22 0.0912238
\(159\) −7.73052e22 −0.593662
\(160\) 0 0
\(161\) 1.02439e23 0.689912
\(162\) 1.90980e22 0.120526
\(163\) −1.28320e23 −0.759143 −0.379572 0.925162i \(-0.623929\pi\)
−0.379572 + 0.925162i \(0.623929\pi\)
\(164\) 2.18213e22 0.121066
\(165\) 0 0
\(166\) −8.83212e22 −0.431449
\(167\) 4.96700e22 0.227809 0.113905 0.993492i \(-0.463664\pi\)
0.113905 + 0.993492i \(0.463664\pi\)
\(168\) 2.23558e23 0.963035
\(169\) 1.54941e23 0.627127
\(170\) 0 0
\(171\) 9.85834e21 0.0352650
\(172\) −6.80586e22 −0.228999
\(173\) −2.12973e23 −0.674281 −0.337140 0.941454i \(-0.609460\pi\)
−0.337140 + 0.941454i \(0.609460\pi\)
\(174\) 1.15511e23 0.344237
\(175\) 0 0
\(176\) 5.43304e23 1.43602
\(177\) 3.35082e23 0.834511
\(178\) 2.13563e23 0.501322
\(179\) −1.81256e23 −0.401177 −0.200588 0.979676i \(-0.564285\pi\)
−0.200588 + 0.979676i \(0.564285\pi\)
\(180\) 0 0
\(181\) −4.41216e23 −0.869012 −0.434506 0.900669i \(-0.643077\pi\)
−0.434506 + 0.900669i \(0.643077\pi\)
\(182\) 1.39020e24 2.58421
\(183\) 4.25536e23 0.746792
\(184\) −1.99066e23 −0.329920
\(185\) 0 0
\(186\) −6.15347e23 −0.910404
\(187\) 1.07338e24 1.50112
\(188\) 1.69424e22 0.0224036
\(189\) −2.87383e23 −0.359428
\(190\) 0 0
\(191\) −1.24018e24 −1.38878 −0.694391 0.719598i \(-0.744326\pi\)
−0.694391 + 0.719598i \(0.744326\pi\)
\(192\) −4.17436e23 −0.442514
\(193\) 1.62959e24 1.63579 0.817894 0.575369i \(-0.195142\pi\)
0.817894 + 0.575369i \(0.195142\pi\)
\(194\) 1.00689e23 0.0957335
\(195\) 0 0
\(196\) 5.14827e23 0.439514
\(197\) 4.64338e23 0.375784 0.187892 0.982190i \(-0.439834\pi\)
0.187892 + 0.982190i \(0.439834\pi\)
\(198\) −5.90707e23 −0.453302
\(199\) 6.70207e23 0.487811 0.243906 0.969799i \(-0.421571\pi\)
0.243906 + 0.969799i \(0.421571\pi\)
\(200\) 0 0
\(201\) −1.24525e23 −0.0816013
\(202\) −1.54503e24 −0.961051
\(203\) −1.73819e24 −1.02657
\(204\) 2.17717e23 0.122117
\(205\) 0 0
\(206\) −1.73123e24 −0.876494
\(207\) 2.55898e23 0.123134
\(208\) −3.19411e24 −1.46111
\(209\) −3.04921e23 −0.132633
\(210\) 0 0
\(211\) −2.27034e23 −0.0893562 −0.0446781 0.999001i \(-0.514226\pi\)
−0.0446781 + 0.999001i \(0.514226\pi\)
\(212\) 4.84991e23 0.181638
\(213\) −2.78734e24 −0.993582
\(214\) −3.02170e24 −1.02543
\(215\) 0 0
\(216\) 5.58459e23 0.171881
\(217\) 9.25961e24 2.71497
\(218\) −6.44474e23 −0.180058
\(219\) −3.01043e24 −0.801614
\(220\) 0 0
\(221\) −6.31044e24 −1.52736
\(222\) 4.74443e24 1.09516
\(223\) 3.62983e24 0.799255 0.399627 0.916678i \(-0.369140\pi\)
0.399627 + 0.916678i \(0.369140\pi\)
\(224\) −3.10598e24 −0.652521
\(225\) 0 0
\(226\) −8.42759e24 −1.61274
\(227\) −2.69270e24 −0.491944 −0.245972 0.969277i \(-0.579107\pi\)
−0.245972 + 0.969277i \(0.579107\pi\)
\(228\) −6.18483e22 −0.0107898
\(229\) 9.71974e23 0.161950 0.0809752 0.996716i \(-0.474197\pi\)
0.0809752 + 0.996716i \(0.474197\pi\)
\(230\) 0 0
\(231\) 8.88884e24 1.35182
\(232\) 3.37776e24 0.490913
\(233\) 7.37996e24 1.02522 0.512610 0.858622i \(-0.328679\pi\)
0.512610 + 0.858622i \(0.328679\pi\)
\(234\) 3.47279e24 0.461225
\(235\) 0 0
\(236\) −2.10221e24 −0.255329
\(237\) −4.17909e23 −0.0485539
\(238\) −2.18225e25 −2.42575
\(239\) −9.21658e24 −0.980374 −0.490187 0.871617i \(-0.663071\pi\)
−0.490187 + 0.871617i \(0.663071\pi\)
\(240\) 0 0
\(241\) −5.28386e23 −0.0514958 −0.0257479 0.999668i \(-0.508197\pi\)
−0.0257479 + 0.999668i \(0.508197\pi\)
\(242\) 6.64599e24 0.620152
\(243\) −7.17898e23 −0.0641500
\(244\) −2.66969e24 −0.228490
\(245\) 0 0
\(246\) −5.46380e24 −0.429217
\(247\) 1.79265e24 0.134951
\(248\) −1.79938e25 −1.29832
\(249\) 3.32001e24 0.229639
\(250\) 0 0
\(251\) 4.59287e24 0.292085 0.146043 0.989278i \(-0.453346\pi\)
0.146043 + 0.989278i \(0.453346\pi\)
\(252\) 1.80296e24 0.109971
\(253\) −7.91501e24 −0.463112
\(254\) −1.97959e25 −1.11127
\(255\) 0 0
\(256\) 9.94954e24 0.514379
\(257\) −1.06831e25 −0.530152 −0.265076 0.964227i \(-0.585397\pi\)
−0.265076 + 0.964227i \(0.585397\pi\)
\(258\) 1.70411e25 0.811876
\(259\) −7.13931e25 −3.26594
\(260\) 0 0
\(261\) −4.34210e24 −0.183221
\(262\) −4.65200e25 −1.88571
\(263\) −3.27052e25 −1.27374 −0.636872 0.770970i \(-0.719772\pi\)
−0.636872 + 0.770970i \(0.719772\pi\)
\(264\) −1.72733e25 −0.646449
\(265\) 0 0
\(266\) 6.19926e24 0.214330
\(267\) −8.02788e24 −0.266829
\(268\) 7.81232e23 0.0249669
\(269\) −5.84629e25 −1.79673 −0.898363 0.439255i \(-0.855243\pi\)
−0.898363 + 0.439255i \(0.855243\pi\)
\(270\) 0 0
\(271\) 1.83048e25 0.520459 0.260229 0.965547i \(-0.416202\pi\)
0.260229 + 0.965547i \(0.416202\pi\)
\(272\) 5.01391e25 1.37152
\(273\) −5.22579e25 −1.37545
\(274\) −8.25279e25 −2.09036
\(275\) 0 0
\(276\) −1.60543e24 −0.0376744
\(277\) 4.54193e25 1.02613 0.513066 0.858349i \(-0.328510\pi\)
0.513066 + 0.858349i \(0.328510\pi\)
\(278\) −7.19936e24 −0.156612
\(279\) 2.31310e25 0.484563
\(280\) 0 0
\(281\) 2.38319e25 0.463173 0.231586 0.972814i \(-0.425608\pi\)
0.231586 + 0.972814i \(0.425608\pi\)
\(282\) −4.24219e24 −0.0794281
\(283\) 4.26446e25 0.769319 0.384660 0.923058i \(-0.374319\pi\)
0.384660 + 0.923058i \(0.374319\pi\)
\(284\) 1.74869e25 0.303998
\(285\) 0 0
\(286\) −1.07415e26 −1.73468
\(287\) 8.22181e25 1.27999
\(288\) −7.75891e24 −0.116461
\(289\) 2.99655e25 0.433704
\(290\) 0 0
\(291\) −3.78492e24 −0.0509542
\(292\) 1.88866e25 0.245263
\(293\) −1.02681e26 −1.28641 −0.643207 0.765692i \(-0.722397\pi\)
−0.643207 + 0.765692i \(0.722397\pi\)
\(294\) −1.28907e26 −1.55822
\(295\) 0 0
\(296\) 1.38735e26 1.56180
\(297\) 2.22048e25 0.241270
\(298\) 3.14320e25 0.329687
\(299\) 4.65327e25 0.471206
\(300\) 0 0
\(301\) −2.56430e26 −2.42115
\(302\) −6.89987e25 −0.629170
\(303\) 5.80778e25 0.511520
\(304\) −1.42433e25 −0.121182
\(305\) 0 0
\(306\) −5.45138e25 −0.432944
\(307\) −1.11445e26 −0.855275 −0.427638 0.903950i \(-0.640654\pi\)
−0.427638 + 0.903950i \(0.640654\pi\)
\(308\) −5.57660e25 −0.413606
\(309\) 6.50773e25 0.466515
\(310\) 0 0
\(311\) −7.44229e24 −0.0498565 −0.0249283 0.999689i \(-0.507936\pi\)
−0.0249283 + 0.999689i \(0.507936\pi\)
\(312\) 1.01551e26 0.657747
\(313\) −1.84619e26 −1.15627 −0.578136 0.815940i \(-0.696220\pi\)
−0.578136 + 0.815940i \(0.696220\pi\)
\(314\) −2.93521e26 −1.77778
\(315\) 0 0
\(316\) 2.62184e24 0.0148557
\(317\) 2.29713e26 1.25911 0.629555 0.776956i \(-0.283237\pi\)
0.629555 + 0.776956i \(0.283237\pi\)
\(318\) −1.21436e26 −0.643965
\(319\) 1.34302e26 0.689099
\(320\) 0 0
\(321\) 1.13586e26 0.545787
\(322\) 1.60918e26 0.748370
\(323\) −2.81399e25 −0.126676
\(324\) 4.50388e24 0.0196275
\(325\) 0 0
\(326\) −2.01573e26 −0.823468
\(327\) 2.42259e25 0.0958360
\(328\) −1.59771e26 −0.612101
\(329\) 6.38355e25 0.236867
\(330\) 0 0
\(331\) 4.36673e25 0.152042 0.0760208 0.997106i \(-0.475778\pi\)
0.0760208 + 0.997106i \(0.475778\pi\)
\(332\) −2.08288e25 −0.0702609
\(333\) −1.78344e26 −0.582900
\(334\) 7.80248e25 0.247113
\(335\) 0 0
\(336\) 4.15211e26 1.23511
\(337\) −1.35323e25 −0.0390172 −0.0195086 0.999810i \(-0.506210\pi\)
−0.0195086 + 0.999810i \(0.506210\pi\)
\(338\) 2.43391e26 0.680266
\(339\) 3.16795e26 0.858383
\(340\) 0 0
\(341\) −7.15449e26 −1.82246
\(342\) 1.54861e25 0.0382532
\(343\) 1.16014e27 2.77921
\(344\) 4.98311e26 1.15781
\(345\) 0 0
\(346\) −3.34551e26 −0.731415
\(347\) 7.03159e26 1.49140 0.745700 0.666282i \(-0.232115\pi\)
0.745700 + 0.666282i \(0.232115\pi\)
\(348\) 2.72410e25 0.0560585
\(349\) −9.22111e26 −1.84126 −0.920632 0.390432i \(-0.872326\pi\)
−0.920632 + 0.390432i \(0.872326\pi\)
\(350\) 0 0
\(351\) −1.30543e26 −0.245487
\(352\) 2.39985e26 0.438013
\(353\) 4.12270e26 0.730378 0.365189 0.930933i \(-0.381004\pi\)
0.365189 + 0.930933i \(0.381004\pi\)
\(354\) 5.26368e26 0.905223
\(355\) 0 0
\(356\) 5.03646e25 0.0816395
\(357\) 8.20312e26 1.29111
\(358\) −2.84728e26 −0.435170
\(359\) 1.18737e26 0.176236 0.0881181 0.996110i \(-0.471915\pi\)
0.0881181 + 0.996110i \(0.471915\pi\)
\(360\) 0 0
\(361\) −7.06216e26 −0.988807
\(362\) −6.93089e26 −0.942647
\(363\) −2.49824e26 −0.330076
\(364\) 3.27851e26 0.420834
\(365\) 0 0
\(366\) 6.68458e26 0.810070
\(367\) −1.00595e27 −1.18463 −0.592313 0.805708i \(-0.701785\pi\)
−0.592313 + 0.805708i \(0.701785\pi\)
\(368\) −3.69722e26 −0.423130
\(369\) 2.05385e26 0.228451
\(370\) 0 0
\(371\) 1.82734e27 1.92041
\(372\) −1.45117e26 −0.148258
\(373\) 1.11174e27 1.10423 0.552117 0.833766i \(-0.313820\pi\)
0.552117 + 0.833766i \(0.313820\pi\)
\(374\) 1.68613e27 1.62832
\(375\) 0 0
\(376\) −1.24049e26 −0.113272
\(377\) −7.89569e26 −0.701142
\(378\) −4.51439e26 −0.389884
\(379\) 1.87213e24 0.00157263 0.000786313 1.00000i \(-0.499750\pi\)
0.000786313 1.00000i \(0.499750\pi\)
\(380\) 0 0
\(381\) 7.44131e26 0.591476
\(382\) −1.94815e27 −1.50646
\(383\) 3.86177e26 0.290535 0.145268 0.989392i \(-0.453596\pi\)
0.145268 + 0.989392i \(0.453596\pi\)
\(384\) −9.31295e26 −0.681726
\(385\) 0 0
\(386\) 2.55986e27 1.77439
\(387\) −6.40577e26 −0.432122
\(388\) 2.37455e25 0.0155901
\(389\) −6.45820e26 −0.412706 −0.206353 0.978478i \(-0.566160\pi\)
−0.206353 + 0.978478i \(0.566160\pi\)
\(390\) 0 0
\(391\) −7.30442e26 −0.442313
\(392\) −3.76945e27 −2.22216
\(393\) 1.74869e27 1.00367
\(394\) 7.29410e26 0.407626
\(395\) 0 0
\(396\) −1.39306e26 −0.0738196
\(397\) −2.30956e27 −1.19187 −0.595935 0.803033i \(-0.703218\pi\)
−0.595935 + 0.803033i \(0.703218\pi\)
\(398\) 1.05280e27 0.529145
\(399\) −2.33031e26 −0.114077
\(400\) 0 0
\(401\) 3.67069e27 1.70503 0.852516 0.522702i \(-0.175076\pi\)
0.852516 + 0.522702i \(0.175076\pi\)
\(402\) −1.95611e26 −0.0885157
\(403\) 4.20616e27 1.85431
\(404\) −3.64363e26 −0.156506
\(405\) 0 0
\(406\) −2.73046e27 −1.11356
\(407\) 5.51623e27 2.19231
\(408\) −1.59408e27 −0.617417
\(409\) 3.42126e27 1.29149 0.645745 0.763553i \(-0.276547\pi\)
0.645745 + 0.763553i \(0.276547\pi\)
\(410\) 0 0
\(411\) 3.10224e27 1.11259
\(412\) −4.08276e26 −0.142736
\(413\) −7.92067e27 −2.69952
\(414\) 4.01981e26 0.133568
\(415\) 0 0
\(416\) −1.41088e27 −0.445668
\(417\) 2.70625e26 0.0833566
\(418\) −4.78989e26 −0.143872
\(419\) 2.50977e27 0.735169 0.367585 0.929990i \(-0.380185\pi\)
0.367585 + 0.929990i \(0.380185\pi\)
\(420\) 0 0
\(421\) −1.62467e27 −0.452692 −0.226346 0.974047i \(-0.572678\pi\)
−0.226346 + 0.974047i \(0.572678\pi\)
\(422\) −3.56639e26 −0.0969276
\(423\) 1.59465e26 0.0422757
\(424\) −3.55100e27 −0.918351
\(425\) 0 0
\(426\) −4.37852e27 −1.07777
\(427\) −1.00588e28 −2.41576
\(428\) −7.12607e26 −0.166990
\(429\) 4.03773e27 0.923286
\(430\) 0 0
\(431\) 3.42615e27 0.746098 0.373049 0.927812i \(-0.378312\pi\)
0.373049 + 0.927812i \(0.378312\pi\)
\(432\) 1.03722e27 0.220441
\(433\) 8.30800e27 1.72335 0.861675 0.507460i \(-0.169416\pi\)
0.861675 + 0.507460i \(0.169416\pi\)
\(434\) 1.45456e28 2.94502
\(435\) 0 0
\(436\) −1.51986e26 −0.0293222
\(437\) 2.07501e26 0.0390810
\(438\) −4.72897e27 −0.869538
\(439\) 2.35594e27 0.422948 0.211474 0.977384i \(-0.432174\pi\)
0.211474 + 0.977384i \(0.432174\pi\)
\(440\) 0 0
\(441\) 4.84562e27 0.829362
\(442\) −9.91283e27 −1.65678
\(443\) 2.67071e27 0.435900 0.217950 0.975960i \(-0.430063\pi\)
0.217950 + 0.975960i \(0.430063\pi\)
\(444\) 1.11888e27 0.178345
\(445\) 0 0
\(446\) 5.70196e27 0.866979
\(447\) −1.18154e27 −0.175476
\(448\) 9.86734e27 1.43147
\(449\) −1.35877e28 −1.92556 −0.962782 0.270278i \(-0.912884\pi\)
−0.962782 + 0.270278i \(0.912884\pi\)
\(450\) 0 0
\(451\) −6.35263e27 −0.859211
\(452\) −1.98748e27 −0.262633
\(453\) 2.59367e27 0.334876
\(454\) −4.22986e27 −0.533629
\(455\) 0 0
\(456\) 4.52840e26 0.0545524
\(457\) 1.47262e28 1.73369 0.866844 0.498580i \(-0.166145\pi\)
0.866844 + 0.498580i \(0.166145\pi\)
\(458\) 1.52684e27 0.175673
\(459\) 2.04918e27 0.230435
\(460\) 0 0
\(461\) −9.63926e27 −1.03558 −0.517790 0.855508i \(-0.673245\pi\)
−0.517790 + 0.855508i \(0.673245\pi\)
\(462\) 1.39631e28 1.46637
\(463\) −1.41941e28 −1.45716 −0.728582 0.684959i \(-0.759820\pi\)
−0.728582 + 0.684959i \(0.759820\pi\)
\(464\) 6.27347e27 0.629606
\(465\) 0 0
\(466\) 1.15929e28 1.11209
\(467\) −1.36824e28 −1.28332 −0.641659 0.766990i \(-0.721754\pi\)
−0.641659 + 0.766990i \(0.721754\pi\)
\(468\) 8.18989e26 0.0751098
\(469\) 2.94351e27 0.263968
\(470\) 0 0
\(471\) 1.10335e28 0.946223
\(472\) 1.53919e28 1.29093
\(473\) 1.98132e28 1.62522
\(474\) −6.56478e26 −0.0526681
\(475\) 0 0
\(476\) −5.14640e27 −0.395030
\(477\) 4.56480e27 0.342751
\(478\) −1.44780e28 −1.06344
\(479\) −1.06076e28 −0.762247 −0.381124 0.924524i \(-0.624463\pi\)
−0.381124 + 0.924524i \(0.624463\pi\)
\(480\) 0 0
\(481\) −3.24302e28 −2.23062
\(482\) −8.30021e26 −0.0558593
\(483\) −6.04892e27 −0.398321
\(484\) 1.56732e27 0.100991
\(485\) 0 0
\(486\) −1.12772e27 −0.0695857
\(487\) 1.90709e28 1.15164 0.575822 0.817575i \(-0.304682\pi\)
0.575822 + 0.817575i \(0.304682\pi\)
\(488\) 1.95469e28 1.15523
\(489\) 7.57715e27 0.438292
\(490\) 0 0
\(491\) −7.05431e27 −0.390930 −0.195465 0.980711i \(-0.562622\pi\)
−0.195465 + 0.980711i \(0.562622\pi\)
\(492\) −1.28853e27 −0.0698973
\(493\) 1.23942e28 0.658150
\(494\) 2.81600e27 0.146386
\(495\) 0 0
\(496\) −3.34197e28 −1.66512
\(497\) 6.58870e28 3.21409
\(498\) 5.21528e27 0.249097
\(499\) −3.42851e27 −0.160343 −0.0801713 0.996781i \(-0.525547\pi\)
−0.0801713 + 0.996781i \(0.525547\pi\)
\(500\) 0 0
\(501\) −2.93297e27 −0.131526
\(502\) 7.21476e27 0.316835
\(503\) −1.83167e28 −0.787742 −0.393871 0.919166i \(-0.628864\pi\)
−0.393871 + 0.919166i \(0.628864\pi\)
\(504\) −1.32009e28 −0.556009
\(505\) 0 0
\(506\) −1.24334e28 −0.502353
\(507\) −9.14910e27 −0.362072
\(508\) −4.66846e27 −0.180969
\(509\) −2.36804e28 −0.899191 −0.449595 0.893232i \(-0.648432\pi\)
−0.449595 + 0.893232i \(0.648432\pi\)
\(510\) 0 0
\(511\) 7.11605e28 2.59310
\(512\) −1.74460e28 −0.622819
\(513\) −5.82125e26 −0.0203603
\(514\) −1.67817e28 −0.575074
\(515\) 0 0
\(516\) 4.01879e27 0.132213
\(517\) −4.93229e27 −0.159000
\(518\) −1.12149e29 −3.54268
\(519\) 1.25759e28 0.389296
\(520\) 0 0
\(521\) 5.04379e28 1.49955 0.749776 0.661692i \(-0.230161\pi\)
0.749776 + 0.661692i \(0.230161\pi\)
\(522\) −6.82083e27 −0.198745
\(523\) 4.63144e28 1.32266 0.661330 0.750095i \(-0.269992\pi\)
0.661330 + 0.750095i \(0.269992\pi\)
\(524\) −1.09708e28 −0.307086
\(525\) 0 0
\(526\) −5.13754e28 −1.38167
\(527\) −6.60257e28 −1.74061
\(528\) −3.20815e28 −0.829085
\(529\) −3.40854e28 −0.863542
\(530\) 0 0
\(531\) −1.97863e28 −0.481805
\(532\) 1.46197e27 0.0349033
\(533\) 3.73474e28 0.874228
\(534\) −1.26107e28 −0.289438
\(535\) 0 0
\(536\) −5.72001e27 −0.126231
\(537\) 1.07030e28 0.231620
\(538\) −9.18372e28 −1.94897
\(539\) −1.49877e29 −3.11926
\(540\) 0 0
\(541\) 3.70309e28 0.741299 0.370649 0.928773i \(-0.379135\pi\)
0.370649 + 0.928773i \(0.379135\pi\)
\(542\) 2.87543e28 0.564559
\(543\) 2.60534e28 0.501725
\(544\) 2.21472e28 0.418341
\(545\) 0 0
\(546\) −8.20899e28 −1.49199
\(547\) −4.71889e28 −0.841342 −0.420671 0.907213i \(-0.638205\pi\)
−0.420671 + 0.907213i \(0.638205\pi\)
\(548\) −1.94626e28 −0.340411
\(549\) −2.51275e28 −0.431160
\(550\) 0 0
\(551\) −3.52089e27 −0.0581515
\(552\) 1.17546e28 0.190480
\(553\) 9.87854e27 0.157065
\(554\) 7.13475e28 1.11308
\(555\) 0 0
\(556\) −1.69782e27 −0.0255040
\(557\) −2.12737e28 −0.313591 −0.156796 0.987631i \(-0.550116\pi\)
−0.156796 + 0.987631i \(0.550116\pi\)
\(558\) 3.63356e28 0.525622
\(559\) −1.16483e29 −1.65363
\(560\) 0 0
\(561\) −6.33819e28 −0.866673
\(562\) 3.74367e28 0.502419
\(563\) −1.06098e29 −1.39756 −0.698780 0.715336i \(-0.746273\pi\)
−0.698780 + 0.715336i \(0.746273\pi\)
\(564\) −1.00043e27 −0.0129348
\(565\) 0 0
\(566\) 6.69888e28 0.834507
\(567\) 1.69697e28 0.207516
\(568\) −1.28036e29 −1.53700
\(569\) 5.57124e28 0.656558 0.328279 0.944581i \(-0.393531\pi\)
0.328279 + 0.944581i \(0.393531\pi\)
\(570\) 0 0
\(571\) 1.27465e29 1.44781 0.723903 0.689902i \(-0.242346\pi\)
0.723903 + 0.689902i \(0.242346\pi\)
\(572\) −2.53316e28 −0.282490
\(573\) 7.32313e28 0.801814
\(574\) 1.29153e29 1.38845
\(575\) 0 0
\(576\) 2.46492e28 0.255485
\(577\) 2.13598e28 0.217396 0.108698 0.994075i \(-0.465332\pi\)
0.108698 + 0.994075i \(0.465332\pi\)
\(578\) 4.70716e28 0.470454
\(579\) −9.62257e28 −0.944422
\(580\) 0 0
\(581\) −7.84783e28 −0.742849
\(582\) −5.94559e27 −0.0552718
\(583\) −1.41191e29 −1.28910
\(584\) −1.38283e29 −1.24004
\(585\) 0 0
\(586\) −1.61298e29 −1.39542
\(587\) 3.06281e28 0.260267 0.130134 0.991496i \(-0.458459\pi\)
0.130134 + 0.991496i \(0.458459\pi\)
\(588\) −3.04000e28 −0.253753
\(589\) 1.87563e28 0.153793
\(590\) 0 0
\(591\) −2.74187e28 −0.216959
\(592\) 2.57672e29 2.00303
\(593\) 1.27833e29 0.976270 0.488135 0.872768i \(-0.337677\pi\)
0.488135 + 0.872768i \(0.337677\pi\)
\(594\) 3.48807e28 0.261714
\(595\) 0 0
\(596\) 7.41261e27 0.0536890
\(597\) −3.95751e28 −0.281638
\(598\) 7.30964e28 0.511133
\(599\) −1.72328e29 −1.18406 −0.592030 0.805916i \(-0.701673\pi\)
−0.592030 + 0.805916i \(0.701673\pi\)
\(600\) 0 0
\(601\) 8.64690e28 0.573692 0.286846 0.957977i \(-0.407393\pi\)
0.286846 + 0.957977i \(0.407393\pi\)
\(602\) −4.02817e29 −2.62630
\(603\) 7.35306e27 0.0471125
\(604\) −1.62719e28 −0.102459
\(605\) 0 0
\(606\) 9.12323e28 0.554863
\(607\) 7.36779e28 0.440409 0.220205 0.975454i \(-0.429328\pi\)
0.220205 + 0.975454i \(0.429328\pi\)
\(608\) −6.29150e27 −0.0369629
\(609\) 1.02638e29 0.592691
\(610\) 0 0
\(611\) 2.89971e28 0.161779
\(612\) −1.28560e28 −0.0705043
\(613\) 3.09550e29 1.66877 0.834384 0.551184i \(-0.185824\pi\)
0.834384 + 0.551184i \(0.185824\pi\)
\(614\) −1.75064e29 −0.927746
\(615\) 0 0
\(616\) 4.08306e29 2.09117
\(617\) −1.25449e29 −0.631645 −0.315823 0.948818i \(-0.602280\pi\)
−0.315823 + 0.948818i \(0.602280\pi\)
\(618\) 1.02227e29 0.506044
\(619\) 1.34488e29 0.654532 0.327266 0.944932i \(-0.393873\pi\)
0.327266 + 0.944932i \(0.393873\pi\)
\(620\) 0 0
\(621\) −1.51105e28 −0.0710915
\(622\) −1.16908e28 −0.0540811
\(623\) 1.89763e29 0.863151
\(624\) 1.88609e29 0.843575
\(625\) 0 0
\(626\) −2.90010e29 −1.25425
\(627\) 1.80053e28 0.0765757
\(628\) −6.92209e28 −0.289508
\(629\) 5.09069e29 2.09385
\(630\) 0 0
\(631\) 2.24248e29 0.892114 0.446057 0.895005i \(-0.352828\pi\)
0.446057 + 0.895005i \(0.352828\pi\)
\(632\) −1.91966e28 −0.0751094
\(633\) 1.34061e28 0.0515898
\(634\) 3.60848e29 1.36580
\(635\) 0 0
\(636\) −2.86382e28 −0.104869
\(637\) 8.81131e29 3.17378
\(638\) 2.10970e29 0.747489
\(639\) 1.64589e29 0.573645
\(640\) 0 0
\(641\) −1.36025e29 −0.458784 −0.229392 0.973334i \(-0.573674\pi\)
−0.229392 + 0.973334i \(0.573674\pi\)
\(642\) 1.78428e29 0.592033
\(643\) −4.01819e28 −0.131164 −0.0655821 0.997847i \(-0.520890\pi\)
−0.0655821 + 0.997847i \(0.520890\pi\)
\(644\) 3.79492e28 0.121871
\(645\) 0 0
\(646\) −4.42039e28 −0.137410
\(647\) −5.10662e29 −1.56185 −0.780923 0.624627i \(-0.785251\pi\)
−0.780923 + 0.624627i \(0.785251\pi\)
\(648\) −3.29765e28 −0.0992354
\(649\) 6.11996e29 1.81209
\(650\) 0 0
\(651\) −5.46770e29 −1.56749
\(652\) −4.75369e28 −0.134101
\(653\) −2.87215e29 −0.797293 −0.398647 0.917105i \(-0.630520\pi\)
−0.398647 + 0.917105i \(0.630520\pi\)
\(654\) 3.80556e28 0.103957
\(655\) 0 0
\(656\) −2.96741e29 −0.785032
\(657\) 1.77763e29 0.462812
\(658\) 1.00277e29 0.256938
\(659\) −2.21222e29 −0.557869 −0.278935 0.960310i \(-0.589981\pi\)
−0.278935 + 0.960310i \(0.589981\pi\)
\(660\) 0 0
\(661\) −4.38039e29 −1.07003 −0.535017 0.844841i \(-0.679695\pi\)
−0.535017 + 0.844841i \(0.679695\pi\)
\(662\) 6.85953e28 0.164925
\(663\) 3.72625e29 0.881820
\(664\) 1.52504e29 0.355235
\(665\) 0 0
\(666\) −2.80154e29 −0.632291
\(667\) −9.13937e28 −0.203046
\(668\) 1.84006e28 0.0402419
\(669\) −2.14338e29 −0.461450
\(670\) 0 0
\(671\) 7.77200e29 1.62161
\(672\) 1.83405e29 0.376733
\(673\) −5.31936e29 −1.07573 −0.537863 0.843032i \(-0.680768\pi\)
−0.537863 + 0.843032i \(0.680768\pi\)
\(674\) −2.12573e28 −0.0423233
\(675\) 0 0
\(676\) 5.73988e28 0.110780
\(677\) −4.61865e29 −0.877676 −0.438838 0.898566i \(-0.644610\pi\)
−0.438838 + 0.898566i \(0.644610\pi\)
\(678\) 4.97641e29 0.931117
\(679\) 8.94679e28 0.164829
\(680\) 0 0
\(681\) 1.59001e29 0.284024
\(682\) −1.12387e30 −1.97688
\(683\) 7.71063e29 1.33559 0.667794 0.744346i \(-0.267239\pi\)
0.667794 + 0.744346i \(0.267239\pi\)
\(684\) 3.65208e27 0.00622947
\(685\) 0 0
\(686\) 1.82242e30 3.01471
\(687\) −5.73941e28 −0.0935021
\(688\) 9.25507e29 1.48491
\(689\) 8.30065e29 1.31163
\(690\) 0 0
\(691\) −4.81732e28 −0.0738390 −0.0369195 0.999318i \(-0.511755\pi\)
−0.0369195 + 0.999318i \(0.511755\pi\)
\(692\) −7.88972e28 −0.119110
\(693\) −5.24877e29 −0.780474
\(694\) 1.10457e30 1.61777
\(695\) 0 0
\(696\) −1.99453e29 −0.283429
\(697\) −5.86257e29 −0.820623
\(698\) −1.44851e30 −1.99728
\(699\) −4.35779e29 −0.591911
\(700\) 0 0
\(701\) 9.14664e28 0.120565 0.0602827 0.998181i \(-0.480800\pi\)
0.0602827 + 0.998181i \(0.480800\pi\)
\(702\) −2.05065e29 −0.266288
\(703\) −1.44614e29 −0.185004
\(704\) −7.62406e29 −0.960890
\(705\) 0 0
\(706\) 6.47620e29 0.792266
\(707\) −1.37284e30 −1.65469
\(708\) 1.24133e29 0.147414
\(709\) −9.79770e29 −1.14641 −0.573203 0.819413i \(-0.694299\pi\)
−0.573203 + 0.819413i \(0.694299\pi\)
\(710\) 0 0
\(711\) 2.46771e28 0.0280326
\(712\) −3.68759e29 −0.412764
\(713\) 4.86868e29 0.536997
\(714\) 1.28860e30 1.40051
\(715\) 0 0
\(716\) −6.71475e28 −0.0708668
\(717\) 5.44230e29 0.566019
\(718\) 1.86520e29 0.191169
\(719\) −3.43627e29 −0.347084 −0.173542 0.984827i \(-0.555521\pi\)
−0.173542 + 0.984827i \(0.555521\pi\)
\(720\) 0 0
\(721\) −1.53830e30 −1.50911
\(722\) −1.10937e30 −1.07259
\(723\) 3.12007e28 0.0297311
\(724\) −1.63451e29 −0.153509
\(725\) 0 0
\(726\) −3.92439e29 −0.358045
\(727\) 8.08332e29 0.726907 0.363453 0.931612i \(-0.381598\pi\)
0.363453 + 0.931612i \(0.381598\pi\)
\(728\) −2.40045e30 −2.12771
\(729\) 4.23912e28 0.0370370
\(730\) 0 0
\(731\) 1.82848e30 1.55223
\(732\) 1.57642e29 0.131919
\(733\) 2.11039e30 1.74088 0.870442 0.492270i \(-0.163833\pi\)
0.870442 + 0.492270i \(0.163833\pi\)
\(734\) −1.58020e30 −1.28500
\(735\) 0 0
\(736\) −1.63312e29 −0.129063
\(737\) −2.27432e29 −0.177192
\(738\) 3.22632e29 0.247808
\(739\) 5.63357e29 0.426597 0.213298 0.976987i \(-0.431579\pi\)
0.213298 + 0.976987i \(0.431579\pi\)
\(740\) 0 0
\(741\) −1.05854e29 −0.0779141
\(742\) 2.87050e30 2.08313
\(743\) −6.01588e29 −0.430444 −0.215222 0.976565i \(-0.569047\pi\)
−0.215222 + 0.976565i \(0.569047\pi\)
\(744\) 1.06252e30 0.749584
\(745\) 0 0
\(746\) 1.74639e30 1.19780
\(747\) −1.96043e29 −0.132582
\(748\) 3.97640e29 0.265169
\(749\) −2.68495e30 −1.76554
\(750\) 0 0
\(751\) −1.25188e30 −0.800466 −0.400233 0.916413i \(-0.631071\pi\)
−0.400233 + 0.916413i \(0.631071\pi\)
\(752\) −2.30395e29 −0.145273
\(753\) −2.71204e29 −0.168636
\(754\) −1.24030e30 −0.760553
\(755\) 0 0
\(756\) −1.06463e29 −0.0634920
\(757\) 1.18833e30 0.698923 0.349462 0.936951i \(-0.386364\pi\)
0.349462 + 0.936951i \(0.386364\pi\)
\(758\) 2.94087e27 0.00170588
\(759\) 4.67373e29 0.267378
\(760\) 0 0
\(761\) 7.45622e29 0.414935 0.207468 0.978242i \(-0.433478\pi\)
0.207468 + 0.978242i \(0.433478\pi\)
\(762\) 1.16893e30 0.641594
\(763\) −5.72652e29 −0.310015
\(764\) −4.59432e29 −0.245325
\(765\) 0 0
\(766\) 6.06631e29 0.315154
\(767\) −3.59795e30 −1.84376
\(768\) −5.87510e29 −0.296977
\(769\) 1.07483e30 0.535938 0.267969 0.963428i \(-0.413648\pi\)
0.267969 + 0.963428i \(0.413648\pi\)
\(770\) 0 0
\(771\) 6.30828e29 0.306084
\(772\) 6.03692e29 0.288957
\(773\) 1.63839e30 0.773629 0.386815 0.922157i \(-0.373575\pi\)
0.386815 + 0.922157i \(0.373575\pi\)
\(774\) −1.00626e30 −0.468737
\(775\) 0 0
\(776\) −1.73859e29 −0.0788225
\(777\) 4.21569e30 1.88559
\(778\) −1.01449e30 −0.447676
\(779\) 1.66542e29 0.0725070
\(780\) 0 0
\(781\) −5.09080e30 −2.15750
\(782\) −1.14742e30 −0.479792
\(783\) 2.56396e29 0.105782
\(784\) −7.00096e30 −2.84996
\(785\) 0 0
\(786\) 2.74696e30 1.08872
\(787\) −5.04924e30 −1.97465 −0.987327 0.158699i \(-0.949270\pi\)
−0.987327 + 0.158699i \(0.949270\pi\)
\(788\) 1.72017e29 0.0663812
\(789\) 1.93121e30 0.735396
\(790\) 0 0
\(791\) −7.48839e30 −2.77674
\(792\) 1.01997e30 0.373228
\(793\) −4.56919e30 −1.64995
\(794\) −3.62800e30 −1.29286
\(795\) 0 0
\(796\) 2.48282e29 0.0861705
\(797\) 7.08681e29 0.242738 0.121369 0.992607i \(-0.461272\pi\)
0.121369 + 0.992607i \(0.461272\pi\)
\(798\) −3.66060e29 −0.123743
\(799\) −4.55180e29 −0.151859
\(800\) 0 0
\(801\) 4.74038e29 0.154054
\(802\) 5.76616e30 1.84950
\(803\) −5.49826e30 −1.74065
\(804\) −4.61310e28 −0.0144146
\(805\) 0 0
\(806\) 6.60729e30 2.01143
\(807\) 3.45218e30 1.03734
\(808\) 2.66779e30 0.791284
\(809\) −1.78478e30 −0.522547 −0.261274 0.965265i \(-0.584142\pi\)
−0.261274 + 0.965265i \(0.584142\pi\)
\(810\) 0 0
\(811\) −2.51646e30 −0.717912 −0.358956 0.933355i \(-0.616867\pi\)
−0.358956 + 0.933355i \(0.616867\pi\)
\(812\) −6.43923e29 −0.181341
\(813\) −1.08088e30 −0.300487
\(814\) 8.66524e30 2.37807
\(815\) 0 0
\(816\) −2.96067e30 −0.791850
\(817\) −5.19427e29 −0.137149
\(818\) 5.37432e30 1.40092
\(819\) 3.08577e30 0.794115
\(820\) 0 0
\(821\) 3.78741e30 0.950033 0.475017 0.879977i \(-0.342442\pi\)
0.475017 + 0.879977i \(0.342442\pi\)
\(822\) 4.87319e30 1.20687
\(823\) 3.18521e30 0.778825 0.389412 0.921064i \(-0.372678\pi\)
0.389412 + 0.921064i \(0.372678\pi\)
\(824\) 2.98931e30 0.721664
\(825\) 0 0
\(826\) −1.24423e31 −2.92826
\(827\) 7.58906e30 1.76352 0.881760 0.471699i \(-0.156359\pi\)
0.881760 + 0.471699i \(0.156359\pi\)
\(828\) 9.47991e28 0.0217513
\(829\) −1.36317e30 −0.308835 −0.154417 0.988006i \(-0.549350\pi\)
−0.154417 + 0.988006i \(0.549350\pi\)
\(830\) 0 0
\(831\) −2.68197e30 −0.592438
\(832\) 4.48222e30 0.977683
\(833\) −1.38315e31 −2.97917
\(834\) 4.25115e29 0.0904197
\(835\) 0 0
\(836\) −1.12960e29 −0.0234293
\(837\) −1.36586e30 −0.279763
\(838\) 3.94251e30 0.797463
\(839\) 8.94192e29 0.178620 0.0893100 0.996004i \(-0.471534\pi\)
0.0893100 + 0.996004i \(0.471534\pi\)
\(840\) 0 0
\(841\) −3.58207e30 −0.697872
\(842\) −2.55213e30 −0.491050
\(843\) −1.40725e30 −0.267413
\(844\) −8.41060e28 −0.0157845
\(845\) 0 0
\(846\) 2.50497e29 0.0458578
\(847\) 5.90533e30 1.06775
\(848\) −6.59523e30 −1.17780
\(849\) −2.51812e30 −0.444167
\(850\) 0 0
\(851\) −3.75384e30 −0.645974
\(852\) −1.03259e30 −0.175514
\(853\) 1.06093e31 1.78124 0.890618 0.454753i \(-0.150272\pi\)
0.890618 + 0.454753i \(0.150272\pi\)
\(854\) −1.58010e31 −2.62046
\(855\) 0 0
\(856\) 5.21756e30 0.844292
\(857\) −1.20757e31 −1.93026 −0.965128 0.261779i \(-0.915691\pi\)
−0.965128 + 0.261779i \(0.915691\pi\)
\(858\) 6.34272e30 1.00152
\(859\) 9.43078e29 0.147102 0.0735511 0.997291i \(-0.476567\pi\)
0.0735511 + 0.997291i \(0.476567\pi\)
\(860\) 0 0
\(861\) −4.85490e30 −0.739004
\(862\) 5.38202e30 0.809317
\(863\) −7.20039e30 −1.06965 −0.534826 0.844962i \(-0.679623\pi\)
−0.534826 + 0.844962i \(0.679623\pi\)
\(864\) 4.58156e29 0.0672386
\(865\) 0 0
\(866\) 1.30507e31 1.86938
\(867\) −1.76943e30 −0.250399
\(868\) 3.43028e30 0.479592
\(869\) −7.63271e29 −0.105432
\(870\) 0 0
\(871\) 1.33708e30 0.180289
\(872\) 1.11281e30 0.148251
\(873\) 2.23496e29 0.0294184
\(874\) 3.25956e29 0.0423925
\(875\) 0 0
\(876\) −1.11523e30 −0.141603
\(877\) −1.24970e31 −1.56787 −0.783933 0.620845i \(-0.786790\pi\)
−0.783933 + 0.620845i \(0.786790\pi\)
\(878\) 3.70086e30 0.458786
\(879\) 6.06322e30 0.742712
\(880\) 0 0
\(881\) 1.31068e31 1.56765 0.783826 0.620981i \(-0.213266\pi\)
0.783826 + 0.620981i \(0.213266\pi\)
\(882\) 7.61180e30 0.899637
\(883\) −1.52572e29 −0.0178191 −0.00890956 0.999960i \(-0.502836\pi\)
−0.00890956 + 0.999960i \(0.502836\pi\)
\(884\) −2.33774e30 −0.269803
\(885\) 0 0
\(886\) 4.19531e30 0.472836
\(887\) −2.62786e30 −0.292688 −0.146344 0.989234i \(-0.546751\pi\)
−0.146344 + 0.989234i \(0.546751\pi\)
\(888\) −8.19218e30 −0.901703
\(889\) −1.75898e31 −1.91334
\(890\) 0 0
\(891\) −1.31117e30 −0.139298
\(892\) 1.34469e30 0.141186
\(893\) 1.29306e29 0.0134177
\(894\) −1.85603e30 −0.190345
\(895\) 0 0
\(896\) 2.20139e31 2.20528
\(897\) −2.74771e30 −0.272051
\(898\) −2.13443e31 −2.08872
\(899\) −8.26121e30 −0.799037
\(900\) 0 0
\(901\) −1.30299e31 −1.23120
\(902\) −9.97911e30 −0.932015
\(903\) 1.51419e31 1.39785
\(904\) 1.45519e31 1.32786
\(905\) 0 0
\(906\) 4.07430e30 0.363251
\(907\) −6.91978e29 −0.0609840 −0.0304920 0.999535i \(-0.509707\pi\)
−0.0304920 + 0.999535i \(0.509707\pi\)
\(908\) −9.97526e29 −0.0869006
\(909\) −3.42944e30 −0.295326
\(910\) 0 0
\(911\) −4.14924e30 −0.349160 −0.174580 0.984643i \(-0.555857\pi\)
−0.174580 + 0.984643i \(0.555857\pi\)
\(912\) 8.41055e29 0.0699646
\(913\) 6.06368e30 0.498647
\(914\) 2.31328e31 1.88059
\(915\) 0 0
\(916\) 3.60074e29 0.0286081
\(917\) −4.13356e31 −3.24673
\(918\) 3.21899e30 0.249960
\(919\) 2.28464e31 1.75390 0.876949 0.480583i \(-0.159575\pi\)
0.876949 + 0.480583i \(0.159575\pi\)
\(920\) 0 0
\(921\) 6.58069e30 0.493793
\(922\) −1.51419e31 −1.12333
\(923\) 2.99290e31 2.19520
\(924\) 3.29292e30 0.238795
\(925\) 0 0
\(926\) −2.22970e31 −1.58063
\(927\) −3.84275e30 −0.269342
\(928\) 2.77109e30 0.192042
\(929\) −6.47040e29 −0.0443370 −0.0221685 0.999754i \(-0.507057\pi\)
−0.0221685 + 0.999754i \(0.507057\pi\)
\(930\) 0 0
\(931\) 3.92919e30 0.263228
\(932\) 2.73395e30 0.181102
\(933\) 4.39459e29 0.0287847
\(934\) −2.14931e31 −1.39206
\(935\) 0 0
\(936\) −5.99646e30 −0.379751
\(937\) −2.63556e30 −0.165047 −0.0825234 0.996589i \(-0.526298\pi\)
−0.0825234 + 0.996589i \(0.526298\pi\)
\(938\) 4.62385e30 0.286335
\(939\) 1.09015e31 0.667574
\(940\) 0 0
\(941\) 1.31020e31 0.784598 0.392299 0.919838i \(-0.371680\pi\)
0.392299 + 0.919838i \(0.371680\pi\)
\(942\) 1.73321e31 1.02640
\(943\) 4.32301e30 0.253171
\(944\) 2.85872e31 1.65564
\(945\) 0 0
\(946\) 3.11239e31 1.76294
\(947\) 2.51677e31 1.40983 0.704917 0.709290i \(-0.250984\pi\)
0.704917 + 0.709290i \(0.250984\pi\)
\(948\) −1.54817e29 −0.00857692
\(949\) 3.23245e31 1.77107
\(950\) 0 0
\(951\) −1.35643e31 −0.726947
\(952\) 3.76808e31 1.99725
\(953\) 1.18988e31 0.623776 0.311888 0.950119i \(-0.399039\pi\)
0.311888 + 0.950119i \(0.399039\pi\)
\(954\) 7.17067e30 0.371793
\(955\) 0 0
\(956\) −3.41434e30 −0.173180
\(957\) −7.93042e30 −0.397851
\(958\) −1.66631e31 −0.826835
\(959\) −7.33307e31 −3.59907
\(960\) 0 0
\(961\) 2.31832e31 1.11321
\(962\) −5.09433e31 −2.41963
\(963\) −6.70716e30 −0.315110
\(964\) −1.95744e29 −0.00909660
\(965\) 0 0
\(966\) −9.50202e30 −0.432072
\(967\) 1.10804e30 0.0498401 0.0249201 0.999689i \(-0.492067\pi\)
0.0249201 + 0.999689i \(0.492067\pi\)
\(968\) −1.14756e31 −0.510604
\(969\) 1.66163e30 0.0731366
\(970\) 0 0
\(971\) −2.93255e31 −1.26312 −0.631559 0.775328i \(-0.717585\pi\)
−0.631559 + 0.775328i \(0.717585\pi\)
\(972\) −2.65950e29 −0.0113319
\(973\) −6.39704e30 −0.269646
\(974\) 2.99578e31 1.24923
\(975\) 0 0
\(976\) 3.63042e31 1.48161
\(977\) −3.78584e31 −1.52851 −0.764257 0.644912i \(-0.776894\pi\)
−0.764257 + 0.644912i \(0.776894\pi\)
\(978\) 1.19027e31 0.475430
\(979\) −1.46621e31 −0.579401
\(980\) 0 0
\(981\) −1.43052e30 −0.0553310
\(982\) −1.10814e31 −0.424055
\(983\) −1.52257e31 −0.576455 −0.288228 0.957562i \(-0.593066\pi\)
−0.288228 + 0.957562i \(0.593066\pi\)
\(984\) 9.43433e30 0.353397
\(985\) 0 0
\(986\) 1.94695e31 0.713918
\(987\) −3.76942e30 −0.136756
\(988\) 6.64097e29 0.0238387
\(989\) −1.34831e31 −0.478881
\(990\) 0 0
\(991\) 4.68145e30 0.162782 0.0813912 0.996682i \(-0.474064\pi\)
0.0813912 + 0.996682i \(0.474064\pi\)
\(992\) −1.47620e31 −0.507893
\(993\) −2.57851e30 −0.0877812
\(994\) 1.03499e32 3.48643
\(995\) 0 0
\(996\) 1.22992e30 0.0405651
\(997\) 4.40459e31 1.43749 0.718747 0.695272i \(-0.244716\pi\)
0.718747 + 0.695272i \(0.244716\pi\)
\(998\) −5.38571e30 −0.173929
\(999\) 1.05310e31 0.336537
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.j.1.5 yes 6
5.2 odd 4 75.22.b.i.49.10 12
5.3 odd 4 75.22.b.i.49.3 12
5.4 even 2 75.22.a.i.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.22.a.i.1.2 6 5.4 even 2
75.22.a.j.1.5 yes 6 1.1 even 1 trivial
75.22.b.i.49.3 12 5.3 odd 4
75.22.b.i.49.10 12 5.2 odd 4