Properties

Label 45.22.a.g.1.2
Level $45$
Weight $22$
Character 45.1
Self dual yes
Analytic conductor $125.765$
Analytic rank $0$
Dimension $4$
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] = [45,22,Mod(1,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, 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("45.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 45.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: \(125.764804929\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 3512166x^{2} + 363520480x + 2321089280000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{4}\cdot 5^{2}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-849.272\) of defining polynomial
Character \(\chi\) \(=\) 45.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-625.272 q^{2} -1.70619e6 q^{4} -9.76562e6 q^{5} +3.78633e8 q^{7} +2.37812e9 q^{8} +O(q^{10})\) \(q-625.272 q^{2} -1.70619e6 q^{4} -9.76562e6 q^{5} +3.78633e8 q^{7} +2.37812e9 q^{8} +6.10617e9 q^{10} +3.73130e10 q^{11} -9.27472e11 q^{13} -2.36749e11 q^{14} +2.09116e12 q^{16} +1.40356e13 q^{17} +3.31068e13 q^{19} +1.66620e13 q^{20} -2.33308e13 q^{22} -3.30762e14 q^{23} +9.53674e13 q^{25} +5.79922e14 q^{26} -6.46019e14 q^{28} -2.35712e15 q^{29} +3.97094e15 q^{31} -6.29483e15 q^{32} -8.77604e15 q^{34} -3.69759e15 q^{35} -4.11796e16 q^{37} -2.07007e16 q^{38} -2.32238e16 q^{40} +5.23046e16 q^{41} +2.10603e17 q^{43} -6.36630e16 q^{44} +2.06816e17 q^{46} -5.77988e17 q^{47} -4.15183e17 q^{49} -5.96306e16 q^{50} +1.58244e18 q^{52} -1.16230e18 q^{53} -3.64385e17 q^{55} +9.00436e17 q^{56} +1.47384e18 q^{58} -4.17973e18 q^{59} +9.77562e18 q^{61} -2.48292e18 q^{62} -4.49508e17 q^{64} +9.05735e18 q^{65} -1.53703e19 q^{67} -2.39473e19 q^{68} +2.31200e18 q^{70} +6.75134e18 q^{71} -1.58050e19 q^{73} +2.57485e19 q^{74} -5.64864e19 q^{76} +1.41280e19 q^{77} -1.48815e19 q^{79} -2.04215e19 q^{80} -3.27046e19 q^{82} +6.30804e19 q^{83} -1.37066e20 q^{85} -1.31684e20 q^{86} +8.87349e19 q^{88} -1.22115e20 q^{89} -3.51172e20 q^{91} +5.64342e20 q^{92} +3.61400e20 q^{94} -3.23308e20 q^{95} +5.41393e20 q^{97} +2.59602e20 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 897 q^{2} - 1163123 q^{4} - 39062500 q^{5} - 234577504 q^{7} - 76855629 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 897 q^{2} - 1163123 q^{4} - 39062500 q^{5} - 234577504 q^{7} - 76855629 q^{8} - 8759765625 q^{10} - 31491830256 q^{11} - 27017977768 q^{13} + 2233308126672 q^{14} - 11324681311775 q^{16} + 2946095028888 q^{17} - 24270353300752 q^{19} + 11358623046875 q^{20} - 56303932793676 q^{22} - 10350924920928 q^{23} + 381469726562500 q^{25} - 474751622871378 q^{26} - 18\!\cdots\!68 q^{28}+ \cdots - 19\!\cdots\!63 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −625.272 −0.431771 −0.215886 0.976419i \(-0.569264\pi\)
−0.215886 + 0.976419i \(0.569264\pi\)
\(3\) 0 0
\(4\) −1.70619e6 −0.813573
\(5\) −9.76562e6 −0.447214
\(6\) 0 0
\(7\) 3.78633e8 0.506628 0.253314 0.967384i \(-0.418479\pi\)
0.253314 + 0.967384i \(0.418479\pi\)
\(8\) 2.37812e9 0.783049
\(9\) 0 0
\(10\) 6.10617e9 0.193094
\(11\) 3.73130e10 0.433748 0.216874 0.976200i \(-0.430414\pi\)
0.216874 + 0.976200i \(0.430414\pi\)
\(12\) 0 0
\(13\) −9.27472e11 −1.86593 −0.932966 0.359965i \(-0.882789\pi\)
−0.932966 + 0.359965i \(0.882789\pi\)
\(14\) −2.36749e11 −0.218748
\(15\) 0 0
\(16\) 2.09116e12 0.475475
\(17\) 1.40356e13 1.68856 0.844279 0.535904i \(-0.180029\pi\)
0.844279 + 0.535904i \(0.180029\pi\)
\(18\) 0 0
\(19\) 3.31068e13 1.23881 0.619404 0.785072i \(-0.287374\pi\)
0.619404 + 0.785072i \(0.287374\pi\)
\(20\) 1.66620e13 0.363841
\(21\) 0 0
\(22\) −2.33308e13 −0.187280
\(23\) −3.30762e14 −1.66484 −0.832422 0.554143i \(-0.813046\pi\)
−0.832422 + 0.554143i \(0.813046\pi\)
\(24\) 0 0
\(25\) 9.53674e13 0.200000
\(26\) 5.79922e14 0.805656
\(27\) 0 0
\(28\) −6.46019e14 −0.412179
\(29\) −2.35712e15 −1.04041 −0.520203 0.854043i \(-0.674144\pi\)
−0.520203 + 0.854043i \(0.674144\pi\)
\(30\) 0 0
\(31\) 3.97094e15 0.870154 0.435077 0.900393i \(-0.356721\pi\)
0.435077 + 0.900393i \(0.356721\pi\)
\(32\) −6.29483e15 −0.988346
\(33\) 0 0
\(34\) −8.77604e15 −0.729071
\(35\) −3.69759e15 −0.226571
\(36\) 0 0
\(37\) −4.11796e16 −1.40788 −0.703938 0.710262i \(-0.748577\pi\)
−0.703938 + 0.710262i \(0.748577\pi\)
\(38\) −2.07007e16 −0.534882
\(39\) 0 0
\(40\) −2.32238e16 −0.350190
\(41\) 5.23046e16 0.608569 0.304284 0.952581i \(-0.401583\pi\)
0.304284 + 0.952581i \(0.401583\pi\)
\(42\) 0 0
\(43\) 2.10603e17 1.48609 0.743046 0.669240i \(-0.233380\pi\)
0.743046 + 0.669240i \(0.233380\pi\)
\(44\) −6.36630e16 −0.352886
\(45\) 0 0
\(46\) 2.06816e17 0.718832
\(47\) −5.77988e17 −1.60284 −0.801422 0.598099i \(-0.795923\pi\)
−0.801422 + 0.598099i \(0.795923\pi\)
\(48\) 0 0
\(49\) −4.15183e17 −0.743328
\(50\) −5.96306e16 −0.0863543
\(51\) 0 0
\(52\) 1.58244e18 1.51807
\(53\) −1.16230e18 −0.912894 −0.456447 0.889751i \(-0.650878\pi\)
−0.456447 + 0.889751i \(0.650878\pi\)
\(54\) 0 0
\(55\) −3.64385e17 −0.193978
\(56\) 9.00436e17 0.396715
\(57\) 0 0
\(58\) 1.47384e18 0.449217
\(59\) −4.17973e18 −1.06464 −0.532319 0.846544i \(-0.678679\pi\)
−0.532319 + 0.846544i \(0.678679\pi\)
\(60\) 0 0
\(61\) 9.77562e18 1.75461 0.877306 0.479931i \(-0.159338\pi\)
0.877306 + 0.479931i \(0.159338\pi\)
\(62\) −2.48292e18 −0.375707
\(63\) 0 0
\(64\) −4.49508e17 −0.0487358
\(65\) 9.05735e18 0.834470
\(66\) 0 0
\(67\) −1.53703e19 −1.03014 −0.515071 0.857147i \(-0.672235\pi\)
−0.515071 + 0.857147i \(0.672235\pi\)
\(68\) −2.39473e19 −1.37377
\(69\) 0 0
\(70\) 2.31200e18 0.0978269
\(71\) 6.75134e18 0.246137 0.123069 0.992398i \(-0.460726\pi\)
0.123069 + 0.992398i \(0.460726\pi\)
\(72\) 0 0
\(73\) −1.58050e19 −0.430431 −0.215216 0.976567i \(-0.569045\pi\)
−0.215216 + 0.976567i \(0.569045\pi\)
\(74\) 2.57485e19 0.607880
\(75\) 0 0
\(76\) −5.64864e19 −1.00786
\(77\) 1.41280e19 0.219749
\(78\) 0 0
\(79\) −1.48815e19 −0.176833 −0.0884164 0.996084i \(-0.528181\pi\)
−0.0884164 + 0.996084i \(0.528181\pi\)
\(80\) −2.04215e19 −0.212639
\(81\) 0 0
\(82\) −3.27046e19 −0.262763
\(83\) 6.30804e19 0.446246 0.223123 0.974790i \(-0.428375\pi\)
0.223123 + 0.974790i \(0.428375\pi\)
\(84\) 0 0
\(85\) −1.37066e20 −0.755146
\(86\) −1.31684e20 −0.641652
\(87\) 0 0
\(88\) 8.87349e19 0.339646
\(89\) −1.22115e20 −0.415119 −0.207559 0.978222i \(-0.566552\pi\)
−0.207559 + 0.978222i \(0.566552\pi\)
\(90\) 0 0
\(91\) −3.51172e20 −0.945334
\(92\) 5.64342e20 1.35447
\(93\) 0 0
\(94\) 3.61400e20 0.692063
\(95\) −3.23308e20 −0.554012
\(96\) 0 0
\(97\) 5.41393e20 0.745434 0.372717 0.927945i \(-0.378426\pi\)
0.372717 + 0.927945i \(0.378426\pi\)
\(98\) 2.59602e20 0.320948
\(99\) 0 0
\(100\) −1.62715e20 −0.162715
\(101\) −4.54105e20 −0.409055 −0.204527 0.978861i \(-0.565566\pi\)
−0.204527 + 0.978861i \(0.565566\pi\)
\(102\) 0 0
\(103\) 1.33612e21 0.979616 0.489808 0.871830i \(-0.337067\pi\)
0.489808 + 0.871830i \(0.337067\pi\)
\(104\) −2.20564e21 −1.46112
\(105\) 0 0
\(106\) 7.26751e20 0.394161
\(107\) 1.60585e21 0.789179 0.394590 0.918858i \(-0.370887\pi\)
0.394590 + 0.918858i \(0.370887\pi\)
\(108\) 0 0
\(109\) 2.31707e21 0.937476 0.468738 0.883337i \(-0.344709\pi\)
0.468738 + 0.883337i \(0.344709\pi\)
\(110\) 2.27840e20 0.0837542
\(111\) 0 0
\(112\) 7.91784e20 0.240889
\(113\) −1.71334e21 −0.474810 −0.237405 0.971411i \(-0.576297\pi\)
−0.237405 + 0.971411i \(0.576297\pi\)
\(114\) 0 0
\(115\) 3.23010e21 0.744541
\(116\) 4.02169e21 0.846447
\(117\) 0 0
\(118\) 2.61347e21 0.459680
\(119\) 5.31433e21 0.855471
\(120\) 0 0
\(121\) −6.00799e21 −0.811863
\(122\) −6.11242e21 −0.757591
\(123\) 0 0
\(124\) −6.77517e21 −0.707934
\(125\) −9.31323e20 −0.0894427
\(126\) 0 0
\(127\) 5.90533e21 0.480071 0.240035 0.970764i \(-0.422841\pi\)
0.240035 + 0.970764i \(0.422841\pi\)
\(128\) 1.34823e22 1.00939
\(129\) 0 0
\(130\) −5.66330e21 −0.360300
\(131\) 2.01878e22 1.18506 0.592530 0.805548i \(-0.298129\pi\)
0.592530 + 0.805548i \(0.298129\pi\)
\(132\) 0 0
\(133\) 1.25353e22 0.627615
\(134\) 9.61063e21 0.444786
\(135\) 0 0
\(136\) 3.33783e22 1.32222
\(137\) 2.23489e22 0.819768 0.409884 0.912138i \(-0.365569\pi\)
0.409884 + 0.912138i \(0.365569\pi\)
\(138\) 0 0
\(139\) −7.44374e21 −0.234496 −0.117248 0.993103i \(-0.537407\pi\)
−0.117248 + 0.993103i \(0.537407\pi\)
\(140\) 6.30878e21 0.184332
\(141\) 0 0
\(142\) −4.22142e21 −0.106275
\(143\) −3.46068e22 −0.809344
\(144\) 0 0
\(145\) 2.30188e22 0.465284
\(146\) 9.88241e21 0.185848
\(147\) 0 0
\(148\) 7.02601e22 1.14541
\(149\) 2.41970e21 0.0367541 0.0183770 0.999831i \(-0.494150\pi\)
0.0183770 + 0.999831i \(0.494150\pi\)
\(150\) 0 0
\(151\) 6.11367e22 0.807316 0.403658 0.914910i \(-0.367738\pi\)
0.403658 + 0.914910i \(0.367738\pi\)
\(152\) 7.87319e22 0.970048
\(153\) 0 0
\(154\) −8.83382e21 −0.0948813
\(155\) −3.87787e22 −0.389145
\(156\) 0 0
\(157\) 1.98289e23 1.73921 0.869604 0.493750i \(-0.164374\pi\)
0.869604 + 0.493750i \(0.164374\pi\)
\(158\) 9.30499e21 0.0763513
\(159\) 0 0
\(160\) 6.14729e22 0.442002
\(161\) −1.25238e23 −0.843457
\(162\) 0 0
\(163\) −8.73258e22 −0.516622 −0.258311 0.966062i \(-0.583166\pi\)
−0.258311 + 0.966062i \(0.583166\pi\)
\(164\) −8.92415e22 −0.495115
\(165\) 0 0
\(166\) −3.94424e22 −0.192676
\(167\) 1.33825e23 0.613782 0.306891 0.951745i \(-0.400711\pi\)
0.306891 + 0.951745i \(0.400711\pi\)
\(168\) 0 0
\(169\) 6.13140e23 2.48170
\(170\) 8.57035e22 0.326051
\(171\) 0 0
\(172\) −3.59328e23 −1.20905
\(173\) −2.26996e23 −0.718677 −0.359338 0.933207i \(-0.616998\pi\)
−0.359338 + 0.933207i \(0.616998\pi\)
\(174\) 0 0
\(175\) 3.61093e22 0.101326
\(176\) 7.80276e22 0.206236
\(177\) 0 0
\(178\) 7.63548e22 0.179236
\(179\) 5.38150e22 0.119110 0.0595548 0.998225i \(-0.481032\pi\)
0.0595548 + 0.998225i \(0.481032\pi\)
\(180\) 0 0
\(181\) 5.67762e23 1.11826 0.559129 0.829081i \(-0.311136\pi\)
0.559129 + 0.829081i \(0.311136\pi\)
\(182\) 2.19578e23 0.408168
\(183\) 0 0
\(184\) −7.86592e23 −1.30365
\(185\) 4.02145e23 0.629621
\(186\) 0 0
\(187\) 5.23709e23 0.732409
\(188\) 9.86156e23 1.30403
\(189\) 0 0
\(190\) 2.02156e23 0.239206
\(191\) 1.06663e22 0.0119444 0.00597219 0.999982i \(-0.498099\pi\)
0.00597219 + 0.999982i \(0.498099\pi\)
\(192\) 0 0
\(193\) 6.54963e23 0.657453 0.328727 0.944425i \(-0.393380\pi\)
0.328727 + 0.944425i \(0.393380\pi\)
\(194\) −3.38518e23 −0.321857
\(195\) 0 0
\(196\) 7.08379e23 0.604752
\(197\) 2.10493e24 1.70350 0.851752 0.523945i \(-0.175540\pi\)
0.851752 + 0.523945i \(0.175540\pi\)
\(198\) 0 0
\(199\) −7.42491e23 −0.540423 −0.270211 0.962801i \(-0.587094\pi\)
−0.270211 + 0.962801i \(0.587094\pi\)
\(200\) 2.26795e23 0.156610
\(201\) 0 0
\(202\) 2.83939e23 0.176618
\(203\) −8.92484e23 −0.527099
\(204\) 0 0
\(205\) −5.10787e23 −0.272160
\(206\) −8.35441e23 −0.422970
\(207\) 0 0
\(208\) −1.93949e24 −0.887204
\(209\) 1.23531e24 0.537331
\(210\) 0 0
\(211\) −1.58957e23 −0.0625624 −0.0312812 0.999511i \(-0.509959\pi\)
−0.0312812 + 0.999511i \(0.509959\pi\)
\(212\) 1.98310e24 0.742706
\(213\) 0 0
\(214\) −1.00409e24 −0.340745
\(215\) −2.05667e24 −0.664601
\(216\) 0 0
\(217\) 1.50353e24 0.440844
\(218\) −1.44880e24 −0.404775
\(219\) 0 0
\(220\) 6.21709e23 0.157815
\(221\) −1.30176e25 −3.15073
\(222\) 0 0
\(223\) −3.69828e24 −0.814327 −0.407164 0.913355i \(-0.633482\pi\)
−0.407164 + 0.913355i \(0.633482\pi\)
\(224\) −2.38343e24 −0.500724
\(225\) 0 0
\(226\) 1.07130e24 0.205009
\(227\) 4.74876e23 0.0867578 0.0433789 0.999059i \(-0.486188\pi\)
0.0433789 + 0.999059i \(0.486188\pi\)
\(228\) 0 0
\(229\) −5.00087e23 −0.0833245 −0.0416623 0.999132i \(-0.513265\pi\)
−0.0416623 + 0.999132i \(0.513265\pi\)
\(230\) −2.01969e24 −0.321471
\(231\) 0 0
\(232\) −5.60552e24 −0.814689
\(233\) 1.84833e24 0.256769 0.128385 0.991724i \(-0.459021\pi\)
0.128385 + 0.991724i \(0.459021\pi\)
\(234\) 0 0
\(235\) 5.64442e24 0.716814
\(236\) 7.13140e24 0.866161
\(237\) 0 0
\(238\) −3.32290e24 −0.369368
\(239\) −6.80041e24 −0.723364 −0.361682 0.932301i \(-0.617797\pi\)
−0.361682 + 0.932301i \(0.617797\pi\)
\(240\) 0 0
\(241\) 1.15905e25 1.12960 0.564798 0.825229i \(-0.308954\pi\)
0.564798 + 0.825229i \(0.308954\pi\)
\(242\) 3.75662e24 0.350539
\(243\) 0 0
\(244\) −1.66790e25 −1.42751
\(245\) 4.05452e24 0.332426
\(246\) 0 0
\(247\) −3.07056e25 −2.31153
\(248\) 9.44338e24 0.681373
\(249\) 0 0
\(250\) 5.82330e23 0.0386188
\(251\) −7.75895e24 −0.493434 −0.246717 0.969088i \(-0.579352\pi\)
−0.246717 + 0.969088i \(0.579352\pi\)
\(252\) 0 0
\(253\) −1.23417e25 −0.722123
\(254\) −3.69244e24 −0.207281
\(255\) 0 0
\(256\) −7.48740e24 −0.387089
\(257\) 9.07103e24 0.450152 0.225076 0.974341i \(-0.427737\pi\)
0.225076 + 0.974341i \(0.427737\pi\)
\(258\) 0 0
\(259\) −1.55920e25 −0.713270
\(260\) −1.54535e25 −0.678903
\(261\) 0 0
\(262\) −1.26229e25 −0.511675
\(263\) −4.14573e25 −1.61460 −0.807301 0.590140i \(-0.799073\pi\)
−0.807301 + 0.590140i \(0.799073\pi\)
\(264\) 0 0
\(265\) 1.13506e25 0.408259
\(266\) −7.83799e24 −0.270986
\(267\) 0 0
\(268\) 2.62246e25 0.838097
\(269\) 4.62596e25 1.42168 0.710842 0.703352i \(-0.248314\pi\)
0.710842 + 0.703352i \(0.248314\pi\)
\(270\) 0 0
\(271\) 3.86928e25 1.10015 0.550075 0.835115i \(-0.314599\pi\)
0.550075 + 0.835115i \(0.314599\pi\)
\(272\) 2.93506e25 0.802868
\(273\) 0 0
\(274\) −1.39742e25 −0.353952
\(275\) 3.55845e24 0.0867496
\(276\) 0 0
\(277\) −7.06699e25 −1.59660 −0.798302 0.602257i \(-0.794268\pi\)
−0.798302 + 0.602257i \(0.794268\pi\)
\(278\) 4.65436e24 0.101249
\(279\) 0 0
\(280\) −8.79332e24 −0.177416
\(281\) −7.68544e25 −1.49366 −0.746831 0.665014i \(-0.768425\pi\)
−0.746831 + 0.665014i \(0.768425\pi\)
\(282\) 0 0
\(283\) −3.92878e24 −0.0708761 −0.0354381 0.999372i \(-0.511283\pi\)
−0.0354381 + 0.999372i \(0.511283\pi\)
\(284\) −1.15190e25 −0.200251
\(285\) 0 0
\(286\) 2.16387e25 0.349452
\(287\) 1.98043e25 0.308318
\(288\) 0 0
\(289\) 1.27905e26 1.85123
\(290\) −1.43930e25 −0.200896
\(291\) 0 0
\(292\) 2.69663e25 0.350188
\(293\) 3.51075e25 0.439835 0.219918 0.975518i \(-0.429421\pi\)
0.219918 + 0.975518i \(0.429421\pi\)
\(294\) 0 0
\(295\) 4.08177e25 0.476121
\(296\) −9.79301e25 −1.10244
\(297\) 0 0
\(298\) −1.51297e24 −0.0158694
\(299\) 3.06773e26 3.10648
\(300\) 0 0
\(301\) 7.97413e25 0.752896
\(302\) −3.82270e25 −0.348576
\(303\) 0 0
\(304\) 6.92316e25 0.589023
\(305\) −9.54651e25 −0.784686
\(306\) 0 0
\(307\) 1.03340e26 0.793081 0.396540 0.918017i \(-0.370211\pi\)
0.396540 + 0.918017i \(0.370211\pi\)
\(308\) −2.41050e25 −0.178782
\(309\) 0 0
\(310\) 2.42473e25 0.168021
\(311\) 5.45153e25 0.365203 0.182602 0.983187i \(-0.441548\pi\)
0.182602 + 0.983187i \(0.441548\pi\)
\(312\) 0 0
\(313\) −6.17771e25 −0.386912 −0.193456 0.981109i \(-0.561970\pi\)
−0.193456 + 0.981109i \(0.561970\pi\)
\(314\) −1.23984e26 −0.750940
\(315\) 0 0
\(316\) 2.53907e25 0.143866
\(317\) 1.39085e26 0.762354 0.381177 0.924502i \(-0.375519\pi\)
0.381177 + 0.924502i \(0.375519\pi\)
\(318\) 0 0
\(319\) −8.79513e25 −0.451274
\(320\) 4.38973e24 0.0217953
\(321\) 0 0
\(322\) 7.83075e25 0.364181
\(323\) 4.64672e26 2.09180
\(324\) 0 0
\(325\) −8.84506e25 −0.373186
\(326\) 5.46024e25 0.223063
\(327\) 0 0
\(328\) 1.24387e26 0.476539
\(329\) −2.18846e26 −0.812047
\(330\) 0 0
\(331\) 8.96648e25 0.312197 0.156098 0.987742i \(-0.450108\pi\)
0.156098 + 0.987742i \(0.450108\pi\)
\(332\) −1.07627e26 −0.363054
\(333\) 0 0
\(334\) −8.36769e25 −0.265013
\(335\) 1.50101e26 0.460694
\(336\) 0 0
\(337\) −4.20032e26 −1.21107 −0.605534 0.795820i \(-0.707040\pi\)
−0.605534 + 0.795820i \(0.707040\pi\)
\(338\) −3.83379e26 −1.07153
\(339\) 0 0
\(340\) 2.33860e26 0.614367
\(341\) 1.48168e26 0.377427
\(342\) 0 0
\(343\) −3.68686e26 −0.883219
\(344\) 5.00839e26 1.16368
\(345\) 0 0
\(346\) 1.41934e26 0.310304
\(347\) −5.85880e26 −1.24265 −0.621326 0.783552i \(-0.713406\pi\)
−0.621326 + 0.783552i \(0.713406\pi\)
\(348\) 0 0
\(349\) 5.60384e26 1.11897 0.559485 0.828840i \(-0.310999\pi\)
0.559485 + 0.828840i \(0.310999\pi\)
\(350\) −2.25781e25 −0.0437495
\(351\) 0 0
\(352\) −2.34879e26 −0.428693
\(353\) 1.96463e26 0.348054 0.174027 0.984741i \(-0.444322\pi\)
0.174027 + 0.984741i \(0.444322\pi\)
\(354\) 0 0
\(355\) −6.59310e25 −0.110076
\(356\) 2.08350e26 0.337729
\(357\) 0 0
\(358\) −3.36490e25 −0.0514281
\(359\) −2.20403e26 −0.327135 −0.163567 0.986532i \(-0.552300\pi\)
−0.163567 + 0.986532i \(0.552300\pi\)
\(360\) 0 0
\(361\) 3.81849e26 0.534646
\(362\) −3.55006e26 −0.482831
\(363\) 0 0
\(364\) 5.99165e26 0.769098
\(365\) 1.54346e26 0.192495
\(366\) 0 0
\(367\) 1.39438e27 1.64205 0.821025 0.570892i \(-0.193402\pi\)
0.821025 + 0.570892i \(0.193402\pi\)
\(368\) −6.91677e26 −0.791592
\(369\) 0 0
\(370\) −2.51450e26 −0.271852
\(371\) −4.40084e26 −0.462498
\(372\) 0 0
\(373\) 8.14444e26 0.808944 0.404472 0.914550i \(-0.367455\pi\)
0.404472 + 0.914550i \(0.367455\pi\)
\(374\) −3.27461e26 −0.316233
\(375\) 0 0
\(376\) −1.37453e27 −1.25511
\(377\) 2.18616e27 1.94133
\(378\) 0 0
\(379\) −2.19248e26 −0.184172 −0.0920862 0.995751i \(-0.529354\pi\)
−0.0920862 + 0.995751i \(0.529354\pi\)
\(380\) 5.51625e26 0.450729
\(381\) 0 0
\(382\) −6.66934e24 −0.00515724
\(383\) −1.46274e27 −1.10048 −0.550239 0.835007i \(-0.685463\pi\)
−0.550239 + 0.835007i \(0.685463\pi\)
\(384\) 0 0
\(385\) −1.37968e26 −0.0982748
\(386\) −4.09530e26 −0.283870
\(387\) 0 0
\(388\) −9.23718e26 −0.606465
\(389\) −1.72831e27 −1.10446 −0.552231 0.833691i \(-0.686223\pi\)
−0.552231 + 0.833691i \(0.686223\pi\)
\(390\) 0 0
\(391\) −4.64243e27 −2.81119
\(392\) −9.87354e26 −0.582062
\(393\) 0 0
\(394\) −1.31616e27 −0.735524
\(395\) 1.45327e26 0.0790820
\(396\) 0 0
\(397\) 7.87988e26 0.406648 0.203324 0.979111i \(-0.434825\pi\)
0.203324 + 0.979111i \(0.434825\pi\)
\(398\) 4.64258e26 0.233339
\(399\) 0 0
\(400\) 1.99429e26 0.0950950
\(401\) 3.13221e27 1.45491 0.727453 0.686157i \(-0.240704\pi\)
0.727453 + 0.686157i \(0.240704\pi\)
\(402\) 0 0
\(403\) −3.68294e27 −1.62365
\(404\) 7.74788e26 0.332796
\(405\) 0 0
\(406\) 5.58045e26 0.227586
\(407\) −1.53654e27 −0.610663
\(408\) 0 0
\(409\) 2.95830e27 1.11673 0.558363 0.829596i \(-0.311429\pi\)
0.558363 + 0.829596i \(0.311429\pi\)
\(410\) 3.19381e26 0.117511
\(411\) 0 0
\(412\) −2.27968e27 −0.796990
\(413\) −1.58259e27 −0.539376
\(414\) 0 0
\(415\) −6.16020e26 −0.199567
\(416\) 5.83828e27 1.84419
\(417\) 0 0
\(418\) −7.72407e26 −0.232004
\(419\) −1.05350e27 −0.308595 −0.154298 0.988024i \(-0.549311\pi\)
−0.154298 + 0.988024i \(0.549311\pi\)
\(420\) 0 0
\(421\) −3.19984e27 −0.891592 −0.445796 0.895135i \(-0.647079\pi\)
−0.445796 + 0.895135i \(0.647079\pi\)
\(422\) 9.93913e25 0.0270127
\(423\) 0 0
\(424\) −2.76408e27 −0.714841
\(425\) 1.33854e27 0.337712
\(426\) 0 0
\(427\) 3.70138e27 0.888936
\(428\) −2.73988e27 −0.642055
\(429\) 0 0
\(430\) 1.28598e27 0.286956
\(431\) −6.57772e27 −1.43240 −0.716200 0.697895i \(-0.754120\pi\)
−0.716200 + 0.697895i \(0.754120\pi\)
\(432\) 0 0
\(433\) −9.38190e26 −0.194611 −0.0973056 0.995255i \(-0.531022\pi\)
−0.0973056 + 0.995255i \(0.531022\pi\)
\(434\) −9.40116e26 −0.190344
\(435\) 0 0
\(436\) −3.95335e27 −0.762706
\(437\) −1.09505e28 −2.06242
\(438\) 0 0
\(439\) 9.48686e27 1.70312 0.851559 0.524258i \(-0.175657\pi\)
0.851559 + 0.524258i \(0.175657\pi\)
\(440\) −8.66552e26 −0.151894
\(441\) 0 0
\(442\) 8.13953e27 1.36040
\(443\) −2.87822e27 −0.469770 −0.234885 0.972023i \(-0.575471\pi\)
−0.234885 + 0.972023i \(0.575471\pi\)
\(444\) 0 0
\(445\) 1.19252e27 0.185647
\(446\) 2.31243e27 0.351603
\(447\) 0 0
\(448\) −1.70199e26 −0.0246909
\(449\) 7.14238e27 1.01218 0.506089 0.862482i \(-0.331091\pi\)
0.506089 + 0.862482i \(0.331091\pi\)
\(450\) 0 0
\(451\) 1.95164e27 0.263965
\(452\) 2.92328e27 0.386293
\(453\) 0 0
\(454\) −2.96926e26 −0.0374595
\(455\) 3.42941e27 0.422766
\(456\) 0 0
\(457\) −4.86346e26 −0.0572566 −0.0286283 0.999590i \(-0.509114\pi\)
−0.0286283 + 0.999590i \(0.509114\pi\)
\(458\) 3.12690e26 0.0359772
\(459\) 0 0
\(460\) −5.51115e27 −0.605739
\(461\) −2.16847e26 −0.0232966 −0.0116483 0.999932i \(-0.503708\pi\)
−0.0116483 + 0.999932i \(0.503708\pi\)
\(462\) 0 0
\(463\) 1.29183e28 1.32619 0.663094 0.748536i \(-0.269243\pi\)
0.663094 + 0.748536i \(0.269243\pi\)
\(464\) −4.92912e27 −0.494687
\(465\) 0 0
\(466\) −1.15571e27 −0.110866
\(467\) 2.06226e28 1.93426 0.967131 0.254277i \(-0.0818376\pi\)
0.967131 + 0.254277i \(0.0818376\pi\)
\(468\) 0 0
\(469\) −5.81972e27 −0.521900
\(470\) −3.52929e27 −0.309500
\(471\) 0 0
\(472\) −9.93990e27 −0.833664
\(473\) 7.85824e27 0.644590
\(474\) 0 0
\(475\) 3.15731e27 0.247762
\(476\) −9.06724e27 −0.695989
\(477\) 0 0
\(478\) 4.25210e27 0.312328
\(479\) −3.66438e27 −0.263317 −0.131658 0.991295i \(-0.542030\pi\)
−0.131658 + 0.991295i \(0.542030\pi\)
\(480\) 0 0
\(481\) 3.81930e28 2.62700
\(482\) −7.24721e27 −0.487727
\(483\) 0 0
\(484\) 1.02507e28 0.660510
\(485\) −5.28704e27 −0.333368
\(486\) 0 0
\(487\) −6.79839e27 −0.410537 −0.205268 0.978706i \(-0.565807\pi\)
−0.205268 + 0.978706i \(0.565807\pi\)
\(488\) 2.32476e28 1.37395
\(489\) 0 0
\(490\) −2.53518e27 −0.143532
\(491\) 1.83806e28 1.01860 0.509301 0.860588i \(-0.329904\pi\)
0.509301 + 0.860588i \(0.329904\pi\)
\(492\) 0 0
\(493\) −3.30835e28 −1.75679
\(494\) 1.91994e28 0.998053
\(495\) 0 0
\(496\) 8.30389e27 0.413736
\(497\) 2.55628e27 0.124700
\(498\) 0 0
\(499\) 7.03152e27 0.328847 0.164423 0.986390i \(-0.447424\pi\)
0.164423 + 0.986390i \(0.447424\pi\)
\(500\) 1.58901e27 0.0727682
\(501\) 0 0
\(502\) 4.85145e27 0.213051
\(503\) −1.71742e28 −0.738607 −0.369303 0.929309i \(-0.620404\pi\)
−0.369303 + 0.929309i \(0.620404\pi\)
\(504\) 0 0
\(505\) 4.43462e27 0.182935
\(506\) 7.71694e27 0.311792
\(507\) 0 0
\(508\) −1.00756e28 −0.390573
\(509\) 4.25735e28 1.61660 0.808300 0.588771i \(-0.200388\pi\)
0.808300 + 0.588771i \(0.200388\pi\)
\(510\) 0 0
\(511\) −5.98429e27 −0.218069
\(512\) −2.35927e28 −0.842254
\(513\) 0 0
\(514\) −5.67186e27 −0.194363
\(515\) −1.30481e28 −0.438098
\(516\) 0 0
\(517\) −2.15665e28 −0.695231
\(518\) 9.74922e27 0.307969
\(519\) 0 0
\(520\) 2.15395e28 0.653431
\(521\) −4.64258e27 −0.138027 −0.0690133 0.997616i \(-0.521985\pi\)
−0.0690133 + 0.997616i \(0.521985\pi\)
\(522\) 0 0
\(523\) 5.19658e28 1.48406 0.742028 0.670369i \(-0.233864\pi\)
0.742028 + 0.670369i \(0.233864\pi\)
\(524\) −3.44442e28 −0.964134
\(525\) 0 0
\(526\) 2.59221e28 0.697139
\(527\) 5.57344e28 1.46931
\(528\) 0 0
\(529\) 6.99320e28 1.77170
\(530\) −7.09718e27 −0.176274
\(531\) 0 0
\(532\) −2.13876e28 −0.510611
\(533\) −4.85111e28 −1.13555
\(534\) 0 0
\(535\) −1.56821e28 −0.352932
\(536\) −3.65525e28 −0.806653
\(537\) 0 0
\(538\) −2.89248e28 −0.613843
\(539\) −1.54917e28 −0.322417
\(540\) 0 0
\(541\) 3.03293e27 0.0607144 0.0303572 0.999539i \(-0.490336\pi\)
0.0303572 + 0.999539i \(0.490336\pi\)
\(542\) −2.41935e28 −0.475014
\(543\) 0 0
\(544\) −8.83514e28 −1.66888
\(545\) −2.26276e28 −0.419252
\(546\) 0 0
\(547\) −3.74354e27 −0.0667445 −0.0333722 0.999443i \(-0.510625\pi\)
−0.0333722 + 0.999443i \(0.510625\pi\)
\(548\) −3.81315e28 −0.666942
\(549\) 0 0
\(550\) −2.22500e27 −0.0374560
\(551\) −7.80367e28 −1.28886
\(552\) 0 0
\(553\) −5.63464e27 −0.0895885
\(554\) 4.41879e28 0.689368
\(555\) 0 0
\(556\) 1.27004e28 0.190780
\(557\) 1.24990e28 0.184245 0.0921225 0.995748i \(-0.470635\pi\)
0.0921225 + 0.995748i \(0.470635\pi\)
\(558\) 0 0
\(559\) −1.95328e29 −2.77295
\(560\) −7.73226e27 −0.107729
\(561\) 0 0
\(562\) 4.80549e28 0.644920
\(563\) 1.37259e27 0.0180802 0.00904008 0.999959i \(-0.497122\pi\)
0.00904008 + 0.999959i \(0.497122\pi\)
\(564\) 0 0
\(565\) 1.67318e28 0.212342
\(566\) 2.45655e27 0.0306023
\(567\) 0 0
\(568\) 1.60555e28 0.192738
\(569\) 4.80572e28 0.566343 0.283172 0.959069i \(-0.408613\pi\)
0.283172 + 0.959069i \(0.408613\pi\)
\(570\) 0 0
\(571\) −4.42408e28 −0.502508 −0.251254 0.967921i \(-0.580843\pi\)
−0.251254 + 0.967921i \(0.580843\pi\)
\(572\) 5.90457e28 0.658461
\(573\) 0 0
\(574\) −1.23831e28 −0.133123
\(575\) −3.15439e28 −0.332969
\(576\) 0 0
\(577\) −9.25254e28 −0.941705 −0.470852 0.882212i \(-0.656054\pi\)
−0.470852 + 0.882212i \(0.656054\pi\)
\(578\) −7.99754e28 −0.799308
\(579\) 0 0
\(580\) −3.92743e28 −0.378542
\(581\) 2.38843e28 0.226081
\(582\) 0 0
\(583\) −4.33688e28 −0.395966
\(584\) −3.75862e28 −0.337049
\(585\) 0 0
\(586\) −2.19517e28 −0.189908
\(587\) 1.94963e29 1.65673 0.828364 0.560190i \(-0.189272\pi\)
0.828364 + 0.560190i \(0.189272\pi\)
\(588\) 0 0
\(589\) 1.31465e29 1.07795
\(590\) −2.55221e28 −0.205575
\(591\) 0 0
\(592\) −8.61133e28 −0.669410
\(593\) 7.23704e28 0.552696 0.276348 0.961058i \(-0.410876\pi\)
0.276348 + 0.961058i \(0.410876\pi\)
\(594\) 0 0
\(595\) −5.18978e28 −0.382578
\(596\) −4.12846e27 −0.0299022
\(597\) 0 0
\(598\) −1.91816e29 −1.34129
\(599\) −1.37531e29 −0.944970 −0.472485 0.881339i \(-0.656643\pi\)
−0.472485 + 0.881339i \(0.656643\pi\)
\(600\) 0 0
\(601\) 9.83934e28 0.652806 0.326403 0.945231i \(-0.394163\pi\)
0.326403 + 0.945231i \(0.394163\pi\)
\(602\) −4.98600e28 −0.325079
\(603\) 0 0
\(604\) −1.04311e29 −0.656811
\(605\) 5.86717e28 0.363076
\(606\) 0 0
\(607\) 1.43248e29 0.856262 0.428131 0.903717i \(-0.359172\pi\)
0.428131 + 0.903717i \(0.359172\pi\)
\(608\) −2.08401e29 −1.22437
\(609\) 0 0
\(610\) 5.96916e28 0.338805
\(611\) 5.36068e29 2.99080
\(612\) 0 0
\(613\) 2.28269e29 1.23058 0.615292 0.788299i \(-0.289038\pi\)
0.615292 + 0.788299i \(0.289038\pi\)
\(614\) −6.46159e28 −0.342430
\(615\) 0 0
\(616\) 3.35980e28 0.172074
\(617\) −2.30467e28 −0.116042 −0.0580210 0.998315i \(-0.518479\pi\)
−0.0580210 + 0.998315i \(0.518479\pi\)
\(618\) 0 0
\(619\) 1.54693e29 0.752867 0.376433 0.926444i \(-0.377150\pi\)
0.376433 + 0.926444i \(0.377150\pi\)
\(620\) 6.61638e28 0.316598
\(621\) 0 0
\(622\) −3.40869e28 −0.157684
\(623\) −4.62366e28 −0.210311
\(624\) 0 0
\(625\) 9.09495e27 0.0400000
\(626\) 3.86275e28 0.167057
\(627\) 0 0
\(628\) −3.38318e29 −1.41497
\(629\) −5.77979e29 −2.37728
\(630\) 0 0
\(631\) 3.94122e29 1.56792 0.783958 0.620814i \(-0.213198\pi\)
0.783958 + 0.620814i \(0.213198\pi\)
\(632\) −3.53900e28 −0.138469
\(633\) 0 0
\(634\) −8.69657e28 −0.329163
\(635\) −5.76693e28 −0.214694
\(636\) 0 0
\(637\) 3.85070e29 1.38700
\(638\) 5.49935e28 0.194847
\(639\) 0 0
\(640\) −1.31663e29 −0.451412
\(641\) 1.02757e29 0.346578 0.173289 0.984871i \(-0.444561\pi\)
0.173289 + 0.984871i \(0.444561\pi\)
\(642\) 0 0
\(643\) 3.14849e29 1.02775 0.513875 0.857865i \(-0.328210\pi\)
0.513875 + 0.857865i \(0.328210\pi\)
\(644\) 2.13679e29 0.686214
\(645\) 0 0
\(646\) −2.90546e29 −0.903179
\(647\) −1.46297e29 −0.447446 −0.223723 0.974653i \(-0.571821\pi\)
−0.223723 + 0.974653i \(0.571821\pi\)
\(648\) 0 0
\(649\) −1.55958e29 −0.461785
\(650\) 5.53057e28 0.161131
\(651\) 0 0
\(652\) 1.48994e29 0.420310
\(653\) −7.78420e28 −0.216086 −0.108043 0.994146i \(-0.534458\pi\)
−0.108043 + 0.994146i \(0.534458\pi\)
\(654\) 0 0
\(655\) −1.97147e29 −0.529975
\(656\) 1.09377e29 0.289359
\(657\) 0 0
\(658\) 1.36838e29 0.350618
\(659\) 2.75206e29 0.694003 0.347001 0.937865i \(-0.387200\pi\)
0.347001 + 0.937865i \(0.387200\pi\)
\(660\) 0 0
\(661\) 4.14436e29 1.01238 0.506188 0.862423i \(-0.331054\pi\)
0.506188 + 0.862423i \(0.331054\pi\)
\(662\) −5.60649e28 −0.134798
\(663\) 0 0
\(664\) 1.50013e29 0.349433
\(665\) −1.22415e29 −0.280678
\(666\) 0 0
\(667\) 7.79646e29 1.73211
\(668\) −2.28330e29 −0.499356
\(669\) 0 0
\(670\) −9.38538e28 −0.198914
\(671\) 3.64758e29 0.761060
\(672\) 0 0
\(673\) −4.45773e28 −0.0901478 −0.0450739 0.998984i \(-0.514352\pi\)
−0.0450739 + 0.998984i \(0.514352\pi\)
\(674\) 2.62634e29 0.522904
\(675\) 0 0
\(676\) −1.04613e30 −2.01905
\(677\) −4.10130e27 −0.00779365 −0.00389682 0.999992i \(-0.501240\pi\)
−0.00389682 + 0.999992i \(0.501240\pi\)
\(678\) 0 0
\(679\) 2.04989e29 0.377658
\(680\) −3.25960e29 −0.591317
\(681\) 0 0
\(682\) −9.26453e28 −0.162962
\(683\) −2.97788e29 −0.515810 −0.257905 0.966170i \(-0.583032\pi\)
−0.257905 + 0.966170i \(0.583032\pi\)
\(684\) 0 0
\(685\) −2.18251e29 −0.366611
\(686\) 2.30529e29 0.381349
\(687\) 0 0
\(688\) 4.40405e29 0.706600
\(689\) 1.07800e30 1.70340
\(690\) 0 0
\(691\) −7.07918e28 −0.108508 −0.0542542 0.998527i \(-0.517278\pi\)
−0.0542542 + 0.998527i \(0.517278\pi\)
\(692\) 3.87297e29 0.584696
\(693\) 0 0
\(694\) 3.66334e29 0.536541
\(695\) 7.26927e28 0.104870
\(696\) 0 0
\(697\) 7.34125e29 1.02760
\(698\) −3.50392e29 −0.483140
\(699\) 0 0
\(700\) −6.16092e28 −0.0824359
\(701\) 1.23613e30 1.62939 0.814696 0.579889i \(-0.196904\pi\)
0.814696 + 0.579889i \(0.196904\pi\)
\(702\) 0 0
\(703\) −1.36332e30 −1.74409
\(704\) −1.67725e28 −0.0211391
\(705\) 0 0
\(706\) −1.22843e29 −0.150280
\(707\) −1.71939e29 −0.207239
\(708\) 0 0
\(709\) 1.35420e30 1.58452 0.792260 0.610184i \(-0.208904\pi\)
0.792260 + 0.610184i \(0.208904\pi\)
\(710\) 4.12248e28 0.0475276
\(711\) 0 0
\(712\) −2.90403e29 −0.325058
\(713\) −1.31344e30 −1.44867
\(714\) 0 0
\(715\) 3.37957e29 0.361950
\(716\) −9.18185e28 −0.0969044
\(717\) 0 0
\(718\) 1.37812e29 0.141247
\(719\) −6.73356e29 −0.680129 −0.340065 0.940402i \(-0.610449\pi\)
−0.340065 + 0.940402i \(0.610449\pi\)
\(720\) 0 0
\(721\) 5.05901e29 0.496301
\(722\) −2.38759e29 −0.230845
\(723\) 0 0
\(724\) −9.68709e29 −0.909784
\(725\) −2.24793e29 −0.208081
\(726\) 0 0
\(727\) −1.55564e30 −1.39893 −0.699467 0.714665i \(-0.746579\pi\)
−0.699467 + 0.714665i \(0.746579\pi\)
\(728\) −8.35129e29 −0.740243
\(729\) 0 0
\(730\) −9.65079e28 −0.0831138
\(731\) 2.95593e30 2.50935
\(732\) 0 0
\(733\) −1.13297e30 −0.934599 −0.467299 0.884099i \(-0.654773\pi\)
−0.467299 + 0.884099i \(0.654773\pi\)
\(734\) −8.71865e29 −0.708990
\(735\) 0 0
\(736\) 2.08209e30 1.64544
\(737\) −5.73513e29 −0.446822
\(738\) 0 0
\(739\) −5.13263e28 −0.0388663 −0.0194331 0.999811i \(-0.506186\pi\)
−0.0194331 + 0.999811i \(0.506186\pi\)
\(740\) −6.86134e29 −0.512243
\(741\) 0 0
\(742\) 2.75172e29 0.199693
\(743\) −1.03039e30 −0.737255 −0.368627 0.929577i \(-0.620172\pi\)
−0.368627 + 0.929577i \(0.620172\pi\)
\(744\) 0 0
\(745\) −2.36299e28 −0.0164369
\(746\) −5.09249e29 −0.349279
\(747\) 0 0
\(748\) −8.93546e29 −0.595868
\(749\) 6.08029e29 0.399820
\(750\) 0 0
\(751\) −6.58297e29 −0.420923 −0.210461 0.977602i \(-0.567497\pi\)
−0.210461 + 0.977602i \(0.567497\pi\)
\(752\) −1.20867e30 −0.762113
\(753\) 0 0
\(754\) −1.36695e30 −0.838209
\(755\) −5.97038e29 −0.361043
\(756\) 0 0
\(757\) −1.14028e30 −0.670663 −0.335332 0.942100i \(-0.608848\pi\)
−0.335332 + 0.942100i \(0.608848\pi\)
\(758\) 1.37090e29 0.0795204
\(759\) 0 0
\(760\) −7.68866e29 −0.433819
\(761\) −2.10124e30 −1.16933 −0.584664 0.811276i \(-0.698774\pi\)
−0.584664 + 0.811276i \(0.698774\pi\)
\(762\) 0 0
\(763\) 8.77319e29 0.474952
\(764\) −1.81987e28 −0.00971763
\(765\) 0 0
\(766\) 9.14612e29 0.475155
\(767\) 3.87658e30 1.98654
\(768\) 0 0
\(769\) 3.41637e30 1.70349 0.851743 0.523961i \(-0.175546\pi\)
0.851743 + 0.523961i \(0.175546\pi\)
\(770\) 8.62677e28 0.0424322
\(771\) 0 0
\(772\) −1.11749e30 −0.534887
\(773\) −1.82343e30 −0.861001 −0.430500 0.902590i \(-0.641663\pi\)
−0.430500 + 0.902590i \(0.641663\pi\)
\(774\) 0 0
\(775\) 3.78699e29 0.174031
\(776\) 1.28750e30 0.583712
\(777\) 0 0
\(778\) 1.08066e30 0.476875
\(779\) 1.73164e30 0.753900
\(780\) 0 0
\(781\) 2.51913e29 0.106762
\(782\) 2.90278e30 1.21379
\(783\) 0 0
\(784\) −8.68214e29 −0.353434
\(785\) −1.93641e30 −0.777797
\(786\) 0 0
\(787\) −6.96799e29 −0.272504 −0.136252 0.990674i \(-0.543506\pi\)
−0.136252 + 0.990674i \(0.543506\pi\)
\(788\) −3.59141e30 −1.38593
\(789\) 0 0
\(790\) −9.08691e28 −0.0341454
\(791\) −6.48728e29 −0.240552
\(792\) 0 0
\(793\) −9.06662e30 −3.27399
\(794\) −4.92706e29 −0.175579
\(795\) 0 0
\(796\) 1.26683e30 0.439674
\(797\) 1.48389e30 0.508263 0.254132 0.967170i \(-0.418210\pi\)
0.254132 + 0.967170i \(0.418210\pi\)
\(798\) 0 0
\(799\) −8.11239e30 −2.70650
\(800\) −6.00321e29 −0.197669
\(801\) 0 0
\(802\) −1.95848e30 −0.628187
\(803\) −5.89732e29 −0.186699
\(804\) 0 0
\(805\) 1.22302e30 0.377205
\(806\) 2.30284e30 0.701044
\(807\) 0 0
\(808\) −1.07992e30 −0.320310
\(809\) 3.52590e30 1.03231 0.516155 0.856495i \(-0.327363\pi\)
0.516155 + 0.856495i \(0.327363\pi\)
\(810\) 0 0
\(811\) 3.08640e30 0.880508 0.440254 0.897873i \(-0.354888\pi\)
0.440254 + 0.897873i \(0.354888\pi\)
\(812\) 1.52275e30 0.428834
\(813\) 0 0
\(814\) 9.60753e29 0.263667
\(815\) 8.52791e29 0.231040
\(816\) 0 0
\(817\) 6.97239e30 1.84098
\(818\) −1.84974e30 −0.482171
\(819\) 0 0
\(820\) 8.71499e29 0.221422
\(821\) −5.43721e30 −1.36387 −0.681934 0.731413i \(-0.738861\pi\)
−0.681934 + 0.731413i \(0.738861\pi\)
\(822\) 0 0
\(823\) −3.97429e30 −0.971766 −0.485883 0.874024i \(-0.661502\pi\)
−0.485883 + 0.874024i \(0.661502\pi\)
\(824\) 3.17746e30 0.767087
\(825\) 0 0
\(826\) 9.89546e29 0.232887
\(827\) 1.38664e30 0.322223 0.161111 0.986936i \(-0.448492\pi\)
0.161111 + 0.986936i \(0.448492\pi\)
\(828\) 0 0
\(829\) 7.15876e30 1.62187 0.810933 0.585139i \(-0.198960\pi\)
0.810933 + 0.585139i \(0.198960\pi\)
\(830\) 3.85180e29 0.0861675
\(831\) 0 0
\(832\) 4.16907e29 0.0909377
\(833\) −5.82732e30 −1.25515
\(834\) 0 0
\(835\) −1.30688e30 −0.274491
\(836\) −2.10768e30 −0.437158
\(837\) 0 0
\(838\) 6.58727e29 0.133243
\(839\) −8.11621e30 −1.62126 −0.810630 0.585558i \(-0.800875\pi\)
−0.810630 + 0.585558i \(0.800875\pi\)
\(840\) 0 0
\(841\) 4.23174e29 0.0824443
\(842\) 2.00077e30 0.384964
\(843\) 0 0
\(844\) 2.71210e29 0.0508991
\(845\) −5.98770e30 −1.10985
\(846\) 0 0
\(847\) −2.27482e30 −0.411313
\(848\) −2.43055e30 −0.434058
\(849\) 0 0
\(850\) −8.36948e29 −0.145814
\(851\) 1.36207e31 2.34389
\(852\) 0 0
\(853\) 5.85669e30 0.983302 0.491651 0.870792i \(-0.336394\pi\)
0.491651 + 0.870792i \(0.336394\pi\)
\(854\) −2.31437e30 −0.383817
\(855\) 0 0
\(856\) 3.81891e30 0.617966
\(857\) 2.04604e30 0.327051 0.163526 0.986539i \(-0.447713\pi\)
0.163526 + 0.986539i \(0.447713\pi\)
\(858\) 0 0
\(859\) 2.53649e30 0.395644 0.197822 0.980238i \(-0.436613\pi\)
0.197822 + 0.980238i \(0.436613\pi\)
\(860\) 3.50906e30 0.540701
\(861\) 0 0
\(862\) 4.11286e30 0.618469
\(863\) 5.67025e30 0.842343 0.421172 0.906981i \(-0.361619\pi\)
0.421172 + 0.906981i \(0.361619\pi\)
\(864\) 0 0
\(865\) 2.21676e30 0.321402
\(866\) 5.86624e29 0.0840275
\(867\) 0 0
\(868\) −2.56531e30 −0.358659
\(869\) −5.55275e29 −0.0767009
\(870\) 0 0
\(871\) 1.42555e31 1.92218
\(872\) 5.51026e30 0.734090
\(873\) 0 0
\(874\) 6.84702e30 0.890495
\(875\) −3.52630e29 −0.0453142
\(876\) 0 0
\(877\) 5.52103e30 0.692667 0.346333 0.938112i \(-0.387427\pi\)
0.346333 + 0.938112i \(0.387427\pi\)
\(878\) −5.93186e30 −0.735358
\(879\) 0 0
\(880\) −7.61989e29 −0.0922317
\(881\) 7.48461e30 0.895204 0.447602 0.894233i \(-0.352278\pi\)
0.447602 + 0.894233i \(0.352278\pi\)
\(882\) 0 0
\(883\) −3.41506e30 −0.398851 −0.199426 0.979913i \(-0.563908\pi\)
−0.199426 + 0.979913i \(0.563908\pi\)
\(884\) 2.22104e31 2.56335
\(885\) 0 0
\(886\) 1.79967e30 0.202833
\(887\) 3.07207e30 0.342164 0.171082 0.985257i \(-0.445274\pi\)
0.171082 + 0.985257i \(0.445274\pi\)
\(888\) 0 0
\(889\) 2.23596e30 0.243217
\(890\) −7.45652e29 −0.0801569
\(891\) 0 0
\(892\) 6.30996e30 0.662515
\(893\) −1.91353e31 −1.98562
\(894\) 0 0
\(895\) −5.25537e29 −0.0532674
\(896\) 5.10484e30 0.511385
\(897\) 0 0
\(898\) −4.46593e30 −0.437029
\(899\) −9.35999e30 −0.905313
\(900\) 0 0
\(901\) −1.63135e31 −1.54147
\(902\) −1.22031e30 −0.113973
\(903\) 0 0
\(904\) −4.07453e30 −0.371800
\(905\) −5.54456e30 −0.500100
\(906\) 0 0
\(907\) −1.67239e31 −1.47388 −0.736940 0.675958i \(-0.763730\pi\)
−0.736940 + 0.675958i \(0.763730\pi\)
\(908\) −8.10227e29 −0.0705839
\(909\) 0 0
\(910\) −2.14432e30 −0.182538
\(911\) 2.31676e31 1.94957 0.974783 0.223153i \(-0.0716349\pi\)
0.974783 + 0.223153i \(0.0716349\pi\)
\(912\) 0 0
\(913\) 2.35372e30 0.193558
\(914\) 3.04099e29 0.0247218
\(915\) 0 0
\(916\) 8.53242e29 0.0677906
\(917\) 7.64378e30 0.600385
\(918\) 0 0
\(919\) 9.52890e29 0.0731526 0.0365763 0.999331i \(-0.488355\pi\)
0.0365763 + 0.999331i \(0.488355\pi\)
\(920\) 7.68156e30 0.583012
\(921\) 0 0
\(922\) 1.35588e29 0.0100588
\(923\) −6.26168e30 −0.459275
\(924\) 0 0
\(925\) −3.92719e30 −0.281575
\(926\) −8.07746e30 −0.572610
\(927\) 0 0
\(928\) 1.48377e31 1.02828
\(929\) −2.43263e30 −0.166691 −0.0833453 0.996521i \(-0.526560\pi\)
−0.0833453 + 0.996521i \(0.526560\pi\)
\(930\) 0 0
\(931\) −1.37454e31 −0.920841
\(932\) −3.15360e30 −0.208900
\(933\) 0 0
\(934\) −1.28947e31 −0.835159
\(935\) −5.11435e30 −0.327543
\(936\) 0 0
\(937\) 1.14297e31 0.715763 0.357881 0.933767i \(-0.383499\pi\)
0.357881 + 0.933767i \(0.383499\pi\)
\(938\) 3.63890e30 0.225341
\(939\) 0 0
\(940\) −9.63043e30 −0.583181
\(941\) −7.76249e30 −0.464847 −0.232423 0.972615i \(-0.574666\pi\)
−0.232423 + 0.972615i \(0.574666\pi\)
\(942\) 0 0
\(943\) −1.73004e31 −1.01317
\(944\) −8.74049e30 −0.506209
\(945\) 0 0
\(946\) −4.91354e30 −0.278315
\(947\) −3.25786e31 −1.82498 −0.912488 0.409103i \(-0.865842\pi\)
−0.912488 + 0.409103i \(0.865842\pi\)
\(948\) 0 0
\(949\) 1.46587e31 0.803156
\(950\) −1.97418e30 −0.106976
\(951\) 0 0
\(952\) 1.26381e31 0.669876
\(953\) −3.62605e29 −0.0190090 −0.00950448 0.999955i \(-0.503025\pi\)
−0.00950448 + 0.999955i \(0.503025\pi\)
\(954\) 0 0
\(955\) −1.04163e29 −0.00534169
\(956\) 1.16028e31 0.588510
\(957\) 0 0
\(958\) 2.29124e30 0.113693
\(959\) 8.46205e30 0.415318
\(960\) 0 0
\(961\) −5.05711e30 −0.242833
\(962\) −2.38810e31 −1.13426
\(963\) 0 0
\(964\) −1.97756e31 −0.919010
\(965\) −6.39612e30 −0.294022
\(966\) 0 0
\(967\) 2.60517e30 0.117181 0.0585906 0.998282i \(-0.481339\pi\)
0.0585906 + 0.998282i \(0.481339\pi\)
\(968\) −1.42877e31 −0.635728
\(969\) 0 0
\(970\) 3.30584e30 0.143939
\(971\) 3.70602e31 1.59627 0.798135 0.602479i \(-0.205820\pi\)
0.798135 + 0.602479i \(0.205820\pi\)
\(972\) 0 0
\(973\) −2.81845e30 −0.118802
\(974\) 4.25084e30 0.177258
\(975\) 0 0
\(976\) 2.04424e31 0.834275
\(977\) 1.56254e30 0.0630866 0.0315433 0.999502i \(-0.489958\pi\)
0.0315433 + 0.999502i \(0.489958\pi\)
\(978\) 0 0
\(979\) −4.55646e30 −0.180057
\(980\) −6.91777e30 −0.270453
\(981\) 0 0
\(982\) −1.14929e31 −0.439804
\(983\) 1.61908e31 0.612996 0.306498 0.951871i \(-0.400843\pi\)
0.306498 + 0.951871i \(0.400843\pi\)
\(984\) 0 0
\(985\) −2.05560e31 −0.761830
\(986\) 2.06862e31 0.758530
\(987\) 0 0
\(988\) 5.23895e31 1.88060
\(989\) −6.96595e31 −2.47411
\(990\) 0 0
\(991\) −1.36425e31 −0.474374 −0.237187 0.971464i \(-0.576225\pi\)
−0.237187 + 0.971464i \(0.576225\pi\)
\(992\) −2.49964e31 −0.860013
\(993\) 0 0
\(994\) −1.59837e30 −0.0538419
\(995\) 7.25088e30 0.241684
\(996\) 0 0
\(997\) −1.27367e31 −0.415677 −0.207839 0.978163i \(-0.566643\pi\)
−0.207839 + 0.978163i \(0.566643\pi\)
\(998\) −4.39661e30 −0.141987
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 45.22.a.g.1.2 4
3.2 odd 2 15.22.a.e.1.3 4
15.2 even 4 75.22.b.h.49.5 8
15.8 even 4 75.22.b.h.49.4 8
15.14 odd 2 75.22.a.h.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.22.a.e.1.3 4 3.2 odd 2
45.22.a.g.1.2 4 1.1 even 1 trivial
75.22.a.h.1.2 4 15.14 odd 2
75.22.b.h.49.4 8 15.8 even 4
75.22.b.h.49.5 8 15.2 even 4