Properties

Label 45.22.a.g.1.1
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 1.1
Root \(-1711.97\) 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)
 
\(f(q)\) \(=\) \(q-1487.97 q^{2} +116903. q^{4} -9.76562e6 q^{5} -1.26833e9 q^{7} +2.94655e9 q^{8} +O(q^{10})\) \(q-1487.97 q^{2} +116903. q^{4} -9.76562e6 q^{5} -1.26833e9 q^{7} +2.94655e9 q^{8} +1.45310e10 q^{10} -4.25851e10 q^{11} +7.14056e11 q^{13} +1.88724e12 q^{14} -4.62954e12 q^{16} -9.08430e12 q^{17} -2.62762e13 q^{19} -1.14163e12 q^{20} +6.33654e13 q^{22} +2.96171e13 q^{23} +9.53674e13 q^{25} -1.06249e15 q^{26} -1.48271e14 q^{28} -4.45551e15 q^{29} +4.93427e14 q^{31} +7.09254e14 q^{32} +1.35172e16 q^{34} +1.23860e16 q^{35} +8.82960e15 q^{37} +3.90982e16 q^{38} -2.87749e16 q^{40} -1.17100e17 q^{41} -4.77156e16 q^{43} -4.97831e15 q^{44} -4.40694e16 q^{46} -3.81704e17 q^{47} +1.05012e18 q^{49} -1.41904e17 q^{50} +8.34750e16 q^{52} -1.01485e18 q^{53} +4.15870e17 q^{55} -3.73720e18 q^{56} +6.62967e18 q^{58} +8.26140e17 q^{59} -6.60518e18 q^{61} -7.34204e17 q^{62} +8.65351e18 q^{64} -6.97320e18 q^{65} -2.57407e19 q^{67} -1.06198e18 q^{68} -1.84301e19 q^{70} -3.32319e19 q^{71} -2.66123e19 q^{73} -1.31382e19 q^{74} -3.07176e18 q^{76} +5.40120e19 q^{77} -7.85851e19 q^{79} +4.52104e19 q^{80} +1.74241e20 q^{82} -1.68959e20 q^{83} +8.87139e19 q^{85} +7.09994e19 q^{86} -1.25479e20 q^{88} -6.58830e19 q^{89} -9.05659e20 q^{91} +3.46232e18 q^{92} +5.67964e20 q^{94} +2.56604e20 q^{95} -1.13619e21 q^{97} -1.56254e21 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 897 q^{2} - 1163123 q^{4} - 39062500 q^{5} - 234577504 q^{7} - 76855629 q^{8}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1487.97 −1.02749 −0.513747 0.857942i \(-0.671743\pi\)
−0.513747 + 0.857942i \(0.671743\pi\)
\(3\) 0 0
\(4\) 116903. 0.0557435
\(5\) −9.76562e6 −0.447214
\(6\) 0 0
\(7\) −1.26833e9 −1.69708 −0.848541 0.529129i \(-0.822519\pi\)
−0.848541 + 0.529129i \(0.822519\pi\)
\(8\) 2.94655e9 0.970218
\(9\) 0 0
\(10\) 1.45310e10 0.459509
\(11\) −4.25851e10 −0.495033 −0.247517 0.968884i \(-0.579615\pi\)
−0.247517 + 0.968884i \(0.579615\pi\)
\(12\) 0 0
\(13\) 7.14056e11 1.43657 0.718285 0.695749i \(-0.244927\pi\)
0.718285 + 0.695749i \(0.244927\pi\)
\(14\) 1.88724e12 1.74374
\(15\) 0 0
\(16\) −4.62954e12 −1.05264
\(17\) −9.08430e12 −1.09289 −0.546447 0.837494i \(-0.684020\pi\)
−0.546447 + 0.837494i \(0.684020\pi\)
\(18\) 0 0
\(19\) −2.62762e13 −0.983219 −0.491609 0.870816i \(-0.663591\pi\)
−0.491609 + 0.870816i \(0.663591\pi\)
\(20\) −1.14163e12 −0.0249293
\(21\) 0 0
\(22\) 6.33654e13 0.508644
\(23\) 2.96171e13 0.149074 0.0745368 0.997218i \(-0.476252\pi\)
0.0745368 + 0.997218i \(0.476252\pi\)
\(24\) 0 0
\(25\) 9.53674e13 0.200000
\(26\) −1.06249e15 −1.47607
\(27\) 0 0
\(28\) −1.48271e14 −0.0946014
\(29\) −4.45551e15 −1.96661 −0.983305 0.181963i \(-0.941755\pi\)
−0.983305 + 0.181963i \(0.941755\pi\)
\(30\) 0 0
\(31\) 4.93427e14 0.108125 0.0540623 0.998538i \(-0.482783\pi\)
0.0540623 + 0.998538i \(0.482783\pi\)
\(32\) 7.09254e14 0.111359
\(33\) 0 0
\(34\) 1.35172e16 1.12294
\(35\) 1.23860e16 0.758958
\(36\) 0 0
\(37\) 8.82960e15 0.301872 0.150936 0.988544i \(-0.451771\pi\)
0.150936 + 0.988544i \(0.451771\pi\)
\(38\) 3.90982e16 1.01025
\(39\) 0 0
\(40\) −2.87749e16 −0.433895
\(41\) −1.17100e17 −1.36247 −0.681235 0.732065i \(-0.738557\pi\)
−0.681235 + 0.732065i \(0.738557\pi\)
\(42\) 0 0
\(43\) −4.77156e16 −0.336699 −0.168350 0.985727i \(-0.553844\pi\)
−0.168350 + 0.985727i \(0.553844\pi\)
\(44\) −4.97831e15 −0.0275949
\(45\) 0 0
\(46\) −4.40694e16 −0.153172
\(47\) −3.81704e17 −1.05852 −0.529260 0.848460i \(-0.677530\pi\)
−0.529260 + 0.848460i \(0.677530\pi\)
\(48\) 0 0
\(49\) 1.05012e18 1.88009
\(50\) −1.41904e17 −0.205499
\(51\) 0 0
\(52\) 8.34750e16 0.0800795
\(53\) −1.01485e18 −0.797086 −0.398543 0.917150i \(-0.630484\pi\)
−0.398543 + 0.917150i \(0.630484\pi\)
\(54\) 0 0
\(55\) 4.15870e17 0.221386
\(56\) −3.73720e18 −1.64654
\(57\) 0 0
\(58\) 6.62967e18 2.02068
\(59\) 8.26140e17 0.210430 0.105215 0.994450i \(-0.466447\pi\)
0.105215 + 0.994450i \(0.466447\pi\)
\(60\) 0 0
\(61\) −6.60518e18 −1.18555 −0.592777 0.805367i \(-0.701969\pi\)
−0.592777 + 0.805367i \(0.701969\pi\)
\(62\) −7.34204e17 −0.111097
\(63\) 0 0
\(64\) 8.65351e18 0.938215
\(65\) −6.97320e18 −0.642454
\(66\) 0 0
\(67\) −2.57407e19 −1.72518 −0.862590 0.505904i \(-0.831159\pi\)
−0.862590 + 0.505904i \(0.831159\pi\)
\(68\) −1.06198e18 −0.0609218
\(69\) 0 0
\(70\) −1.84301e19 −0.779825
\(71\) −3.32319e19 −1.21155 −0.605777 0.795635i \(-0.707137\pi\)
−0.605777 + 0.795635i \(0.707137\pi\)
\(72\) 0 0
\(73\) −2.66123e19 −0.724757 −0.362378 0.932031i \(-0.618035\pi\)
−0.362378 + 0.932031i \(0.618035\pi\)
\(74\) −1.31382e19 −0.310172
\(75\) 0 0
\(76\) −3.07176e18 −0.0548081
\(77\) 5.40120e19 0.840112
\(78\) 0 0
\(79\) −7.85851e19 −0.933804 −0.466902 0.884309i \(-0.654630\pi\)
−0.466902 + 0.884309i \(0.654630\pi\)
\(80\) 4.52104e19 0.470753
\(81\) 0 0
\(82\) 1.74241e20 1.39993
\(83\) −1.68959e20 −1.19526 −0.597630 0.801772i \(-0.703891\pi\)
−0.597630 + 0.801772i \(0.703891\pi\)
\(84\) 0 0
\(85\) 8.87139e19 0.488757
\(86\) 7.09994e19 0.345956
\(87\) 0 0
\(88\) −1.25479e20 −0.480290
\(89\) −6.58830e19 −0.223964 −0.111982 0.993710i \(-0.535720\pi\)
−0.111982 + 0.993710i \(0.535720\pi\)
\(90\) 0 0
\(91\) −9.05659e20 −2.43798
\(92\) 3.46232e18 0.00830989
\(93\) 0 0
\(94\) 5.67964e20 1.08762
\(95\) 2.56604e20 0.439709
\(96\) 0 0
\(97\) −1.13619e21 −1.56440 −0.782201 0.623026i \(-0.785903\pi\)
−0.782201 + 0.623026i \(0.785903\pi\)
\(98\) −1.56254e21 −1.93178
\(99\) 0 0
\(100\) 1.11487e19 0.0111487
\(101\) −6.08234e20 −0.547894 −0.273947 0.961745i \(-0.588329\pi\)
−0.273947 + 0.961745i \(0.588329\pi\)
\(102\) 0 0
\(103\) 1.93831e21 1.42113 0.710564 0.703633i \(-0.248440\pi\)
0.710564 + 0.703633i \(0.248440\pi\)
\(104\) 2.10400e21 1.39379
\(105\) 0 0
\(106\) 1.51007e21 0.819001
\(107\) 1.21945e20 0.0599286 0.0299643 0.999551i \(-0.490461\pi\)
0.0299643 + 0.999551i \(0.490461\pi\)
\(108\) 0 0
\(109\) 4.65871e21 1.88490 0.942448 0.334351i \(-0.108517\pi\)
0.942448 + 0.334351i \(0.108517\pi\)
\(110\) −6.18802e20 −0.227472
\(111\) 0 0
\(112\) 5.87179e21 1.78641
\(113\) −1.27385e21 −0.353017 −0.176509 0.984299i \(-0.556480\pi\)
−0.176509 + 0.984299i \(0.556480\pi\)
\(114\) 0 0
\(115\) −2.89230e20 −0.0666677
\(116\) −5.20861e20 −0.109626
\(117\) 0 0
\(118\) −1.22927e21 −0.216215
\(119\) 1.15219e22 1.85473
\(120\) 0 0
\(121\) −5.58676e21 −0.754942
\(122\) 9.82831e21 1.21815
\(123\) 0 0
\(124\) 5.76829e19 0.00602725
\(125\) −9.31323e20 −0.0894427
\(126\) 0 0
\(127\) −1.08103e22 −0.878821 −0.439411 0.898286i \(-0.644813\pi\)
−0.439411 + 0.898286i \(0.644813\pi\)
\(128\) −1.43636e22 −1.07537
\(129\) 0 0
\(130\) 1.03759e22 0.660118
\(131\) 2.63526e22 1.54694 0.773472 0.633830i \(-0.218518\pi\)
0.773472 + 0.633830i \(0.218518\pi\)
\(132\) 0 0
\(133\) 3.33269e22 1.66860
\(134\) 3.83013e22 1.77261
\(135\) 0 0
\(136\) −2.67674e22 −1.06034
\(137\) 1.96734e22 0.721629 0.360815 0.932638i \(-0.382499\pi\)
0.360815 + 0.932638i \(0.382499\pi\)
\(138\) 0 0
\(139\) 1.37361e22 0.432722 0.216361 0.976313i \(-0.430581\pi\)
0.216361 + 0.976313i \(0.430581\pi\)
\(140\) 1.44796e21 0.0423070
\(141\) 0 0
\(142\) 4.94481e22 1.24486
\(143\) −3.04081e22 −0.711151
\(144\) 0 0
\(145\) 4.35108e22 0.879495
\(146\) 3.95983e22 0.744683
\(147\) 0 0
\(148\) 1.03220e21 0.0168274
\(149\) −9.14642e22 −1.38930 −0.694649 0.719349i \(-0.744440\pi\)
−0.694649 + 0.719349i \(0.744440\pi\)
\(150\) 0 0
\(151\) 1.22303e23 1.61502 0.807511 0.589853i \(-0.200814\pi\)
0.807511 + 0.589853i \(0.200814\pi\)
\(152\) −7.74243e22 −0.953936
\(153\) 0 0
\(154\) −8.03682e22 −0.863210
\(155\) −4.81862e21 −0.0483548
\(156\) 0 0
\(157\) −1.10353e23 −0.967918 −0.483959 0.875091i \(-0.660802\pi\)
−0.483959 + 0.875091i \(0.660802\pi\)
\(158\) 1.16932e23 0.959477
\(159\) 0 0
\(160\) −6.92631e21 −0.0498015
\(161\) −3.75643e22 −0.252990
\(162\) 0 0
\(163\) 3.12086e23 1.84631 0.923155 0.384429i \(-0.125602\pi\)
0.923155 + 0.384429i \(0.125602\pi\)
\(164\) −1.36893e22 −0.0759488
\(165\) 0 0
\(166\) 2.51406e23 1.22812
\(167\) −5.58649e22 −0.256222 −0.128111 0.991760i \(-0.540891\pi\)
−0.128111 + 0.991760i \(0.540891\pi\)
\(168\) 0 0
\(169\) 2.62811e23 1.06374
\(170\) −1.32004e23 −0.502195
\(171\) 0 0
\(172\) −5.57809e21 −0.0187688
\(173\) 1.34399e23 0.425512 0.212756 0.977105i \(-0.431756\pi\)
0.212756 + 0.977105i \(0.431756\pi\)
\(174\) 0 0
\(175\) −1.20957e23 −0.339416
\(176\) 1.97150e23 0.521090
\(177\) 0 0
\(178\) 9.80319e22 0.230122
\(179\) −5.77287e23 −1.27772 −0.638859 0.769324i \(-0.720593\pi\)
−0.638859 + 0.769324i \(0.720593\pi\)
\(180\) 0 0
\(181\) 2.15236e23 0.423925 0.211963 0.977278i \(-0.432014\pi\)
0.211963 + 0.977278i \(0.432014\pi\)
\(182\) 1.34759e24 2.50501
\(183\) 0 0
\(184\) 8.72684e22 0.144634
\(185\) −8.62266e22 −0.135001
\(186\) 0 0
\(187\) 3.86856e23 0.541019
\(188\) −4.46222e22 −0.0590057
\(189\) 0 0
\(190\) −3.81819e23 −0.451798
\(191\) −4.41173e23 −0.494036 −0.247018 0.969011i \(-0.579451\pi\)
−0.247018 + 0.969011i \(0.579451\pi\)
\(192\) 0 0
\(193\) −7.07886e23 −0.710578 −0.355289 0.934757i \(-0.615617\pi\)
−0.355289 + 0.934757i \(0.615617\pi\)
\(194\) 1.69062e24 1.60741
\(195\) 0 0
\(196\) 1.22761e23 0.104803
\(197\) −2.25891e23 −0.182812 −0.0914058 0.995814i \(-0.529136\pi\)
−0.0914058 + 0.995814i \(0.529136\pi\)
\(198\) 0 0
\(199\) −5.29674e22 −0.0385524 −0.0192762 0.999814i \(-0.506136\pi\)
−0.0192762 + 0.999814i \(0.506136\pi\)
\(200\) 2.81005e23 0.194044
\(201\) 0 0
\(202\) 9.05034e23 0.562957
\(203\) 5.65106e24 3.33750
\(204\) 0 0
\(205\) 1.14356e24 0.609315
\(206\) −2.88415e24 −1.46020
\(207\) 0 0
\(208\) −3.30575e24 −1.51219
\(209\) 1.11898e24 0.486726
\(210\) 0 0
\(211\) 2.91715e24 1.14813 0.574067 0.818808i \(-0.305365\pi\)
0.574067 + 0.818808i \(0.305365\pi\)
\(212\) −1.18639e23 −0.0444324
\(213\) 0 0
\(214\) −1.81450e23 −0.0615763
\(215\) 4.65973e23 0.150576
\(216\) 0 0
\(217\) −6.25828e23 −0.183496
\(218\) −6.93202e24 −1.93672
\(219\) 0 0
\(220\) 4.86163e22 0.0123408
\(221\) −6.48670e24 −1.57002
\(222\) 0 0
\(223\) 4.10996e24 0.904975 0.452487 0.891771i \(-0.350537\pi\)
0.452487 + 0.891771i \(0.350537\pi\)
\(224\) −8.99569e23 −0.188986
\(225\) 0 0
\(226\) 1.89546e24 0.362723
\(227\) −4.85462e24 −0.886919 −0.443460 0.896294i \(-0.646249\pi\)
−0.443460 + 0.896294i \(0.646249\pi\)
\(228\) 0 0
\(229\) −3.91845e24 −0.652893 −0.326446 0.945216i \(-0.605851\pi\)
−0.326446 + 0.945216i \(0.605851\pi\)
\(230\) 4.30365e23 0.0685007
\(231\) 0 0
\(232\) −1.31284e25 −1.90804
\(233\) −1.12801e25 −1.56702 −0.783511 0.621378i \(-0.786573\pi\)
−0.783511 + 0.621378i \(0.786573\pi\)
\(234\) 0 0
\(235\) 3.72758e24 0.473385
\(236\) 9.65779e22 0.0117301
\(237\) 0 0
\(238\) −1.71442e25 −1.90572
\(239\) 9.29369e24 0.988577 0.494288 0.869298i \(-0.335429\pi\)
0.494288 + 0.869298i \(0.335429\pi\)
\(240\) 0 0
\(241\) −2.88908e24 −0.281567 −0.140783 0.990040i \(-0.544962\pi\)
−0.140783 + 0.990040i \(0.544962\pi\)
\(242\) 8.31293e24 0.775698
\(243\) 0 0
\(244\) −7.72163e23 −0.0660870
\(245\) −1.02550e25 −0.840801
\(246\) 0 0
\(247\) −1.87627e25 −1.41246
\(248\) 1.45391e24 0.104904
\(249\) 0 0
\(250\) 1.38578e24 0.0919018
\(251\) −1.09500e25 −0.696370 −0.348185 0.937426i \(-0.613202\pi\)
−0.348185 + 0.937426i \(0.613202\pi\)
\(252\) 0 0
\(253\) −1.26125e24 −0.0737964
\(254\) 1.60855e25 0.902983
\(255\) 0 0
\(256\) 3.22485e24 0.166721
\(257\) 1.13821e25 0.564840 0.282420 0.959291i \(-0.408863\pi\)
0.282420 + 0.959291i \(0.408863\pi\)
\(258\) 0 0
\(259\) −1.11988e25 −0.512302
\(260\) −8.15186e23 −0.0358127
\(261\) 0 0
\(262\) −3.92119e25 −1.58948
\(263\) −3.72385e23 −0.0145030 −0.00725148 0.999974i \(-0.502308\pi\)
−0.00725148 + 0.999974i \(0.502308\pi\)
\(264\) 0 0
\(265\) 9.91065e24 0.356468
\(266\) −4.95895e25 −1.71448
\(267\) 0 0
\(268\) −3.00915e24 −0.0961676
\(269\) −3.72748e25 −1.14556 −0.572779 0.819710i \(-0.694135\pi\)
−0.572779 + 0.819710i \(0.694135\pi\)
\(270\) 0 0
\(271\) 2.21667e25 0.630267 0.315133 0.949047i \(-0.397951\pi\)
0.315133 + 0.949047i \(0.397951\pi\)
\(272\) 4.20562e25 1.15042
\(273\) 0 0
\(274\) −2.92735e25 −0.741470
\(275\) −4.06123e24 −0.0990067
\(276\) 0 0
\(277\) 1.52006e25 0.343419 0.171709 0.985148i \(-0.445071\pi\)
0.171709 + 0.985148i \(0.445071\pi\)
\(278\) −2.04389e25 −0.444619
\(279\) 0 0
\(280\) 3.64961e25 0.736355
\(281\) 4.82436e25 0.937612 0.468806 0.883301i \(-0.344684\pi\)
0.468806 + 0.883301i \(0.344684\pi\)
\(282\) 0 0
\(283\) 3.95762e25 0.713965 0.356983 0.934111i \(-0.383806\pi\)
0.356983 + 0.934111i \(0.383806\pi\)
\(284\) −3.88490e24 −0.0675362
\(285\) 0 0
\(286\) 4.52464e25 0.730703
\(287\) 1.48522e26 2.31222
\(288\) 0 0
\(289\) 1.34326e25 0.194416
\(290\) −6.47428e25 −0.903676
\(291\) 0 0
\(292\) −3.11105e24 −0.0404005
\(293\) 6.72372e25 0.842364 0.421182 0.906976i \(-0.361615\pi\)
0.421182 + 0.906976i \(0.361615\pi\)
\(294\) 0 0
\(295\) −8.06777e24 −0.0941071
\(296\) 2.60169e25 0.292882
\(297\) 0 0
\(298\) 1.36096e26 1.42749
\(299\) 2.11483e25 0.214155
\(300\) 0 0
\(301\) 6.05192e25 0.571406
\(302\) −1.81983e26 −1.65942
\(303\) 0 0
\(304\) 1.21647e26 1.03497
\(305\) 6.45037e25 0.530196
\(306\) 0 0
\(307\) −1.82823e25 −0.140307 −0.0701533 0.997536i \(-0.522349\pi\)
−0.0701533 + 0.997536i \(0.522349\pi\)
\(308\) 6.31414e24 0.0468308
\(309\) 0 0
\(310\) 7.16996e24 0.0496843
\(311\) −1.19682e26 −0.801762 −0.400881 0.916130i \(-0.631296\pi\)
−0.400881 + 0.916130i \(0.631296\pi\)
\(312\) 0 0
\(313\) −2.72575e26 −1.70715 −0.853573 0.520974i \(-0.825569\pi\)
−0.853573 + 0.520974i \(0.825569\pi\)
\(314\) 1.64202e26 0.994530
\(315\) 0 0
\(316\) −9.18680e24 −0.0520535
\(317\) 6.50489e25 0.356548 0.178274 0.983981i \(-0.442949\pi\)
0.178274 + 0.983981i \(0.442949\pi\)
\(318\) 0 0
\(319\) 1.89738e26 0.973538
\(320\) −8.45069e25 −0.419583
\(321\) 0 0
\(322\) 5.58945e25 0.259946
\(323\) 2.38701e26 1.07455
\(324\) 0 0
\(325\) 6.80977e25 0.287314
\(326\) −4.64375e26 −1.89707
\(327\) 0 0
\(328\) −3.45041e26 −1.32189
\(329\) 4.84127e26 1.79640
\(330\) 0 0
\(331\) 2.10948e26 0.734482 0.367241 0.930126i \(-0.380302\pi\)
0.367241 + 0.930126i \(0.380302\pi\)
\(332\) −1.97518e25 −0.0666280
\(333\) 0 0
\(334\) 8.31254e25 0.263267
\(335\) 2.51374e26 0.771524
\(336\) 0 0
\(337\) 1.58978e26 0.458377 0.229189 0.973382i \(-0.426393\pi\)
0.229189 + 0.973382i \(0.426393\pi\)
\(338\) −3.91056e26 −1.09298
\(339\) 0 0
\(340\) 1.03709e25 0.0272450
\(341\) −2.10126e25 −0.0535253
\(342\) 0 0
\(343\) −6.23473e26 −1.49358
\(344\) −1.40597e26 −0.326671
\(345\) 0 0
\(346\) −1.99982e26 −0.437211
\(347\) 3.27291e26 0.694184 0.347092 0.937831i \(-0.387169\pi\)
0.347092 + 0.937831i \(0.387169\pi\)
\(348\) 0 0
\(349\) 6.28483e26 1.25495 0.627475 0.778637i \(-0.284088\pi\)
0.627475 + 0.778637i \(0.284088\pi\)
\(350\) 1.79981e26 0.348748
\(351\) 0 0
\(352\) −3.02037e25 −0.0551267
\(353\) −5.78370e26 −1.02464 −0.512320 0.858795i \(-0.671214\pi\)
−0.512320 + 0.858795i \(0.671214\pi\)
\(354\) 0 0
\(355\) 3.24530e26 0.541823
\(356\) −7.70189e24 −0.0124845
\(357\) 0 0
\(358\) 8.58985e26 1.31285
\(359\) 5.85883e25 0.0869599 0.0434799 0.999054i \(-0.486156\pi\)
0.0434799 + 0.999054i \(0.486156\pi\)
\(360\) 0 0
\(361\) −2.37694e25 −0.0332807
\(362\) −3.20264e26 −0.435581
\(363\) 0 0
\(364\) −1.05874e26 −0.135902
\(365\) 2.59886e26 0.324121
\(366\) 0 0
\(367\) −6.27704e25 −0.0739198 −0.0369599 0.999317i \(-0.511767\pi\)
−0.0369599 + 0.999317i \(0.511767\pi\)
\(368\) −1.37114e26 −0.156920
\(369\) 0 0
\(370\) 1.28303e26 0.138713
\(371\) 1.28717e27 1.35272
\(372\) 0 0
\(373\) −1.70679e27 −1.69526 −0.847631 0.530587i \(-0.821972\pi\)
−0.847631 + 0.530587i \(0.821972\pi\)
\(374\) −5.75630e26 −0.555893
\(375\) 0 0
\(376\) −1.12471e27 −1.02700
\(377\) −3.18148e27 −2.82518
\(378\) 0 0
\(379\) 2.35215e27 1.97585 0.987923 0.154946i \(-0.0495204\pi\)
0.987923 + 0.154946i \(0.0495204\pi\)
\(380\) 2.99977e25 0.0245109
\(381\) 0 0
\(382\) 6.56452e26 0.507619
\(383\) −9.18944e26 −0.691356 −0.345678 0.938353i \(-0.612351\pi\)
−0.345678 + 0.938353i \(0.612351\pi\)
\(384\) 0 0
\(385\) −5.27461e26 −0.375710
\(386\) 1.05331e27 0.730114
\(387\) 0 0
\(388\) −1.32824e26 −0.0872053
\(389\) 1.24941e27 0.798428 0.399214 0.916858i \(-0.369283\pi\)
0.399214 + 0.916858i \(0.369283\pi\)
\(390\) 0 0
\(391\) −2.69051e26 −0.162922
\(392\) 3.09422e27 1.82409
\(393\) 0 0
\(394\) 3.36119e26 0.187838
\(395\) 7.67432e26 0.417610
\(396\) 0 0
\(397\) −3.17780e27 −1.63993 −0.819967 0.572410i \(-0.806008\pi\)
−0.819967 + 0.572410i \(0.806008\pi\)
\(398\) 7.88139e25 0.0396123
\(399\) 0 0
\(400\) −4.41508e26 −0.210527
\(401\) 2.00476e27 0.931208 0.465604 0.884993i \(-0.345837\pi\)
0.465604 + 0.884993i \(0.345837\pi\)
\(402\) 0 0
\(403\) 3.52334e26 0.155329
\(404\) −7.11042e25 −0.0305415
\(405\) 0 0
\(406\) −8.40860e27 −3.42926
\(407\) −3.76009e26 −0.149437
\(408\) 0 0
\(409\) 1.65077e27 0.623150 0.311575 0.950222i \(-0.399144\pi\)
0.311575 + 0.950222i \(0.399144\pi\)
\(410\) −1.70158e27 −0.626067
\(411\) 0 0
\(412\) 2.26594e26 0.0792187
\(413\) −1.04782e27 −0.357117
\(414\) 0 0
\(415\) 1.64999e27 0.534536
\(416\) 5.06447e26 0.159976
\(417\) 0 0
\(418\) −1.66500e27 −0.500108
\(419\) −2.41867e27 −0.708484 −0.354242 0.935154i \(-0.615261\pi\)
−0.354242 + 0.935154i \(0.615261\pi\)
\(420\) 0 0
\(421\) −5.33902e27 −1.48764 −0.743822 0.668377i \(-0.766989\pi\)
−0.743822 + 0.668377i \(0.766989\pi\)
\(422\) −4.34063e27 −1.17970
\(423\) 0 0
\(424\) −2.99031e27 −0.773347
\(425\) −8.66347e26 −0.218579
\(426\) 0 0
\(427\) 8.37755e27 2.01198
\(428\) 1.42557e25 0.00334063
\(429\) 0 0
\(430\) −6.93354e26 −0.154716
\(431\) −1.47687e27 −0.321610 −0.160805 0.986986i \(-0.551409\pi\)
−0.160805 + 0.986986i \(0.551409\pi\)
\(432\) 0 0
\(433\) 1.41696e27 0.293924 0.146962 0.989142i \(-0.453051\pi\)
0.146962 + 0.989142i \(0.453051\pi\)
\(434\) 9.31213e26 0.188541
\(435\) 0 0
\(436\) 5.44616e26 0.105071
\(437\) −7.78226e26 −0.146572
\(438\) 0 0
\(439\) 1.76504e27 0.316867 0.158433 0.987370i \(-0.449356\pi\)
0.158433 + 0.987370i \(0.449356\pi\)
\(440\) 1.22538e27 0.214792
\(441\) 0 0
\(442\) 9.65202e27 1.61318
\(443\) −1.08549e28 −1.77169 −0.885843 0.463984i \(-0.846419\pi\)
−0.885843 + 0.463984i \(0.846419\pi\)
\(444\) 0 0
\(445\) 6.43388e26 0.100160
\(446\) −6.11550e27 −0.929856
\(447\) 0 0
\(448\) −1.09755e28 −1.59223
\(449\) −7.34487e27 −1.04087 −0.520436 0.853900i \(-0.674231\pi\)
−0.520436 + 0.853900i \(0.674231\pi\)
\(450\) 0 0
\(451\) 4.98672e27 0.674468
\(452\) −1.48917e26 −0.0196784
\(453\) 0 0
\(454\) 7.22353e27 0.911304
\(455\) 8.84432e27 1.09030
\(456\) 0 0
\(457\) −1.52263e28 −1.79256 −0.896282 0.443485i \(-0.853742\pi\)
−0.896282 + 0.443485i \(0.853742\pi\)
\(458\) 5.83054e27 0.670843
\(459\) 0 0
\(460\) −3.38117e25 −0.00371629
\(461\) 1.36988e28 1.47171 0.735855 0.677140i \(-0.236781\pi\)
0.735855 + 0.677140i \(0.236781\pi\)
\(462\) 0 0
\(463\) −8.82461e27 −0.905931 −0.452965 0.891528i \(-0.649634\pi\)
−0.452965 + 0.891528i \(0.649634\pi\)
\(464\) 2.06270e28 2.07013
\(465\) 0 0
\(466\) 1.67844e28 1.61011
\(467\) −1.04648e28 −0.981532 −0.490766 0.871291i \(-0.663283\pi\)
−0.490766 + 0.871291i \(0.663283\pi\)
\(468\) 0 0
\(469\) 3.26477e28 2.92777
\(470\) −5.54652e27 −0.486400
\(471\) 0 0
\(472\) 2.43426e27 0.204163
\(473\) 2.03198e27 0.166677
\(474\) 0 0
\(475\) −2.50590e27 −0.196644
\(476\) 1.34694e27 0.103389
\(477\) 0 0
\(478\) −1.38287e28 −1.01576
\(479\) 9.99111e27 0.717944 0.358972 0.933348i \(-0.383127\pi\)
0.358972 + 0.933348i \(0.383127\pi\)
\(480\) 0 0
\(481\) 6.30483e27 0.433661
\(482\) 4.29887e27 0.289308
\(483\) 0 0
\(484\) −6.53107e26 −0.0420831
\(485\) 1.10956e28 0.699622
\(486\) 0 0
\(487\) −2.33380e28 −1.40932 −0.704661 0.709544i \(-0.748901\pi\)
−0.704661 + 0.709544i \(0.748901\pi\)
\(488\) −1.94625e28 −1.15025
\(489\) 0 0
\(490\) 1.52592e28 0.863918
\(491\) −2.29620e28 −1.27249 −0.636245 0.771487i \(-0.719513\pi\)
−0.636245 + 0.771487i \(0.719513\pi\)
\(492\) 0 0
\(493\) 4.04752e28 2.14930
\(494\) 2.79183e28 1.45130
\(495\) 0 0
\(496\) −2.28434e27 −0.113816
\(497\) 4.21490e28 2.05611
\(498\) 0 0
\(499\) −1.58930e28 −0.743276 −0.371638 0.928378i \(-0.621204\pi\)
−0.371638 + 0.928378i \(0.621204\pi\)
\(500\) −1.08874e26 −0.00498585
\(501\) 0 0
\(502\) 1.62933e28 0.715516
\(503\) 4.31616e28 1.85624 0.928120 0.372282i \(-0.121425\pi\)
0.928120 + 0.372282i \(0.121425\pi\)
\(504\) 0 0
\(505\) 5.93978e27 0.245025
\(506\) 1.87670e27 0.0758253
\(507\) 0 0
\(508\) −1.26376e27 −0.0489886
\(509\) −3.30343e28 −1.25438 −0.627189 0.778867i \(-0.715794\pi\)
−0.627189 + 0.778867i \(0.715794\pi\)
\(510\) 0 0
\(511\) 3.37532e28 1.22997
\(512\) 2.53241e28 0.904065
\(513\) 0 0
\(514\) −1.69362e28 −0.580369
\(515\) −1.89288e28 −0.635547
\(516\) 0 0
\(517\) 1.62549e28 0.524003
\(518\) 1.66635e28 0.526387
\(519\) 0 0
\(520\) −2.05469e28 −0.623320
\(521\) −4.42108e27 −0.131442 −0.0657208 0.997838i \(-0.520935\pi\)
−0.0657208 + 0.997838i \(0.520935\pi\)
\(522\) 0 0
\(523\) −6.75490e28 −1.92909 −0.964543 0.263927i \(-0.914982\pi\)
−0.964543 + 0.263927i \(0.914982\pi\)
\(524\) 3.08069e27 0.0862321
\(525\) 0 0
\(526\) 5.54097e26 0.0149017
\(527\) −4.48244e27 −0.118169
\(528\) 0 0
\(529\) −3.85944e28 −0.977777
\(530\) −1.47467e28 −0.366268
\(531\) 0 0
\(532\) 3.89601e27 0.0930138
\(533\) −8.36160e28 −1.95728
\(534\) 0 0
\(535\) −1.19087e27 −0.0268009
\(536\) −7.58462e28 −1.67380
\(537\) 0 0
\(538\) 5.54638e28 1.17705
\(539\) −4.47193e28 −0.930706
\(540\) 0 0
\(541\) 9.90502e27 0.198282 0.0991412 0.995073i \(-0.468390\pi\)
0.0991412 + 0.995073i \(0.468390\pi\)
\(542\) −3.29834e28 −0.647595
\(543\) 0 0
\(544\) −6.44308e27 −0.121704
\(545\) −4.54952e28 −0.842952
\(546\) 0 0
\(547\) −1.52806e28 −0.272442 −0.136221 0.990678i \(-0.543496\pi\)
−0.136221 + 0.990678i \(0.543496\pi\)
\(548\) 2.29988e27 0.0402262
\(549\) 0 0
\(550\) 6.04299e27 0.101729
\(551\) 1.17074e29 1.93361
\(552\) 0 0
\(553\) 9.96718e28 1.58474
\(554\) −2.26181e28 −0.352861
\(555\) 0 0
\(556\) 1.60579e27 0.0241214
\(557\) 6.67397e28 0.983795 0.491897 0.870653i \(-0.336304\pi\)
0.491897 + 0.870653i \(0.336304\pi\)
\(558\) 0 0
\(559\) −3.40716e28 −0.483692
\(560\) −5.73417e28 −0.798907
\(561\) 0 0
\(562\) −7.17850e28 −0.963390
\(563\) −4.32189e28 −0.569293 −0.284646 0.958633i \(-0.591876\pi\)
−0.284646 + 0.958633i \(0.591876\pi\)
\(564\) 0 0
\(565\) 1.24400e28 0.157874
\(566\) −5.88883e28 −0.733595
\(567\) 0 0
\(568\) −9.79195e28 −1.17547
\(569\) 2.28226e28 0.268960 0.134480 0.990916i \(-0.457064\pi\)
0.134480 + 0.990916i \(0.457064\pi\)
\(570\) 0 0
\(571\) 1.97421e28 0.224240 0.112120 0.993695i \(-0.464236\pi\)
0.112120 + 0.993695i \(0.464236\pi\)
\(572\) −3.55479e27 −0.0396420
\(573\) 0 0
\(574\) −2.20996e29 −2.37579
\(575\) 2.82451e27 0.0298147
\(576\) 0 0
\(577\) −1.13098e29 −1.15109 −0.575546 0.817770i \(-0.695210\pi\)
−0.575546 + 0.817770i \(0.695210\pi\)
\(578\) −1.99873e28 −0.199762
\(579\) 0 0
\(580\) 5.08653e27 0.0490262
\(581\) 2.14296e29 2.02845
\(582\) 0 0
\(583\) 4.32175e28 0.394584
\(584\) −7.84145e28 −0.703172
\(585\) 0 0
\(586\) −1.00047e29 −0.865523
\(587\) 6.65281e28 0.565334 0.282667 0.959218i \(-0.408781\pi\)
0.282667 + 0.959218i \(0.408781\pi\)
\(588\) 0 0
\(589\) −1.29654e28 −0.106310
\(590\) 1.20046e28 0.0966944
\(591\) 0 0
\(592\) −4.08770e28 −0.317761
\(593\) 3.42927e28 0.261895 0.130948 0.991389i \(-0.458198\pi\)
0.130948 + 0.991389i \(0.458198\pi\)
\(594\) 0 0
\(595\) −1.12518e29 −0.829461
\(596\) −1.06924e28 −0.0774443
\(597\) 0 0
\(598\) −3.14680e28 −0.220043
\(599\) 7.57014e28 0.520143 0.260071 0.965589i \(-0.416254\pi\)
0.260071 + 0.965589i \(0.416254\pi\)
\(600\) 0 0
\(601\) 2.14743e29 1.42475 0.712374 0.701800i \(-0.247620\pi\)
0.712374 + 0.701800i \(0.247620\pi\)
\(602\) −9.00507e28 −0.587116
\(603\) 0 0
\(604\) 1.42975e28 0.0900270
\(605\) 5.45582e28 0.337620
\(606\) 0 0
\(607\) 3.33523e28 0.199363 0.0996815 0.995019i \(-0.468218\pi\)
0.0996815 + 0.995019i \(0.468218\pi\)
\(608\) −1.86365e28 −0.109491
\(609\) 0 0
\(610\) −9.59796e28 −0.544773
\(611\) −2.72558e29 −1.52064
\(612\) 0 0
\(613\) 6.67007e28 0.359580 0.179790 0.983705i \(-0.442458\pi\)
0.179790 + 0.983705i \(0.442458\pi\)
\(614\) 2.72035e28 0.144164
\(615\) 0 0
\(616\) 1.59149e29 0.815092
\(617\) 2.22057e29 1.11807 0.559037 0.829143i \(-0.311171\pi\)
0.559037 + 0.829143i \(0.311171\pi\)
\(618\) 0 0
\(619\) 1.99692e29 0.971869 0.485935 0.873995i \(-0.338479\pi\)
0.485935 + 0.873995i \(0.338479\pi\)
\(620\) −5.63309e26 −0.00269547
\(621\) 0 0
\(622\) 1.78084e29 0.823806
\(623\) 8.35614e28 0.380085
\(624\) 0 0
\(625\) 9.09495e27 0.0400000
\(626\) 4.05583e29 1.75408
\(627\) 0 0
\(628\) −1.29006e28 −0.0539552
\(629\) −8.02107e28 −0.329914
\(630\) 0 0
\(631\) 1.07700e29 0.428456 0.214228 0.976784i \(-0.431276\pi\)
0.214228 + 0.976784i \(0.431276\pi\)
\(632\) −2.31555e29 −0.905993
\(633\) 0 0
\(634\) −9.67908e28 −0.366351
\(635\) 1.05570e29 0.393021
\(636\) 0 0
\(637\) 7.49841e29 2.70088
\(638\) −2.82325e29 −1.00030
\(639\) 0 0
\(640\) 1.40269e29 0.480920
\(641\) 4.82362e28 0.162691 0.0813455 0.996686i \(-0.474078\pi\)
0.0813455 + 0.996686i \(0.474078\pi\)
\(642\) 0 0
\(643\) 1.12338e29 0.366699 0.183350 0.983048i \(-0.441306\pi\)
0.183350 + 0.983048i \(0.441306\pi\)
\(644\) −4.39136e27 −0.0141026
\(645\) 0 0
\(646\) −3.55180e29 −1.10410
\(647\) 2.58352e28 0.0790162 0.0395081 0.999219i \(-0.487421\pi\)
0.0395081 + 0.999219i \(0.487421\pi\)
\(648\) 0 0
\(649\) −3.51812e28 −0.104170
\(650\) −1.01327e29 −0.295214
\(651\) 0 0
\(652\) 3.64837e28 0.102920
\(653\) −5.85458e29 −1.62520 −0.812601 0.582820i \(-0.801949\pi\)
−0.812601 + 0.582820i \(0.801949\pi\)
\(654\) 0 0
\(655\) −2.57350e29 −0.691815
\(656\) 5.42120e29 1.43418
\(657\) 0 0
\(658\) −7.20366e29 −1.84579
\(659\) 1.22801e29 0.309673 0.154837 0.987940i \(-0.450515\pi\)
0.154837 + 0.987940i \(0.450515\pi\)
\(660\) 0 0
\(661\) 4.50136e29 1.09958 0.549792 0.835301i \(-0.314707\pi\)
0.549792 + 0.835301i \(0.314707\pi\)
\(662\) −3.13884e29 −0.754676
\(663\) 0 0
\(664\) −4.97848e29 −1.15966
\(665\) −3.25458e29 −0.746222
\(666\) 0 0
\(667\) −1.31959e29 −0.293170
\(668\) −6.53076e27 −0.0142827
\(669\) 0 0
\(670\) −3.74036e29 −0.792736
\(671\) 2.81282e29 0.586889
\(672\) 0 0
\(673\) 6.19299e29 1.25240 0.626199 0.779663i \(-0.284610\pi\)
0.626199 + 0.779663i \(0.284610\pi\)
\(674\) −2.36555e29 −0.470980
\(675\) 0 0
\(676\) 3.07234e28 0.0592964
\(677\) −9.15907e28 −0.174049 −0.0870243 0.996206i \(-0.527736\pi\)
−0.0870243 + 0.996206i \(0.527736\pi\)
\(678\) 0 0
\(679\) 1.44107e30 2.65492
\(680\) 2.61400e29 0.474201
\(681\) 0 0
\(682\) 3.12661e28 0.0549969
\(683\) 1.08753e30 1.88375 0.941876 0.335961i \(-0.109061\pi\)
0.941876 + 0.335961i \(0.109061\pi\)
\(684\) 0 0
\(685\) −1.92123e29 −0.322722
\(686\) 9.27708e29 1.53465
\(687\) 0 0
\(688\) 2.20902e29 0.354422
\(689\) −7.24660e29 −1.14507
\(690\) 0 0
\(691\) −4.53419e29 −0.694992 −0.347496 0.937681i \(-0.612968\pi\)
−0.347496 + 0.937681i \(0.612968\pi\)
\(692\) 1.57116e28 0.0237196
\(693\) 0 0
\(694\) −4.86999e29 −0.713269
\(695\) −1.34142e29 −0.193519
\(696\) 0 0
\(697\) 1.06377e30 1.48903
\(698\) −9.35163e29 −1.28945
\(699\) 0 0
\(700\) −1.41402e28 −0.0189203
\(701\) 9.03929e29 1.19150 0.595752 0.803169i \(-0.296854\pi\)
0.595752 + 0.803169i \(0.296854\pi\)
\(702\) 0 0
\(703\) −2.32009e29 −0.296806
\(704\) −3.68510e29 −0.464448
\(705\) 0 0
\(706\) 8.60597e29 1.05281
\(707\) 7.71441e29 0.929821
\(708\) 0 0
\(709\) −9.52975e29 −1.11505 −0.557527 0.830159i \(-0.688250\pi\)
−0.557527 + 0.830159i \(0.688250\pi\)
\(710\) −4.82891e29 −0.556720
\(711\) 0 0
\(712\) −1.94128e29 −0.217294
\(713\) 1.46139e28 0.0161185
\(714\) 0 0
\(715\) 2.96955e29 0.318036
\(716\) −6.74864e28 −0.0712245
\(717\) 0 0
\(718\) −8.71776e28 −0.0893507
\(719\) 1.28953e30 1.30250 0.651249 0.758864i \(-0.274245\pi\)
0.651249 + 0.758864i \(0.274245\pi\)
\(720\) 0 0
\(721\) −2.45842e30 −2.41177
\(722\) 3.53681e28 0.0341957
\(723\) 0 0
\(724\) 2.51616e28 0.0236311
\(725\) −4.24911e29 −0.393322
\(726\) 0 0
\(727\) 1.09976e30 0.988981 0.494490 0.869183i \(-0.335355\pi\)
0.494490 + 0.869183i \(0.335355\pi\)
\(728\) −2.66857e30 −2.36537
\(729\) 0 0
\(730\) −3.86702e29 −0.333032
\(731\) 4.33463e29 0.367976
\(732\) 0 0
\(733\) 5.44126e29 0.448856 0.224428 0.974491i \(-0.427949\pi\)
0.224428 + 0.974491i \(0.427949\pi\)
\(734\) 9.34004e28 0.0759522
\(735\) 0 0
\(736\) 2.10061e28 0.0166008
\(737\) 1.09617e30 0.854021
\(738\) 0 0
\(739\) −3.50055e29 −0.265076 −0.132538 0.991178i \(-0.542313\pi\)
−0.132538 + 0.991178i \(0.542313\pi\)
\(740\) −1.00801e28 −0.00752545
\(741\) 0 0
\(742\) −1.91526e30 −1.38991
\(743\) −1.31687e30 −0.942239 −0.471120 0.882069i \(-0.656150\pi\)
−0.471120 + 0.882069i \(0.656150\pi\)
\(744\) 0 0
\(745\) 8.93205e29 0.621313
\(746\) 2.53965e30 1.74187
\(747\) 0 0
\(748\) 4.52245e28 0.0301583
\(749\) −1.54666e29 −0.101704
\(750\) 0 0
\(751\) 2.67518e30 1.71054 0.855271 0.518181i \(-0.173391\pi\)
0.855271 + 0.518181i \(0.173391\pi\)
\(752\) 1.76712e30 1.11424
\(753\) 0 0
\(754\) 4.73395e30 2.90285
\(755\) −1.19436e30 −0.722260
\(756\) 0 0
\(757\) −1.57579e30 −0.926812 −0.463406 0.886146i \(-0.653373\pi\)
−0.463406 + 0.886146i \(0.653373\pi\)
\(758\) −3.49993e30 −2.03017
\(759\) 0 0
\(760\) 7.56096e29 0.426613
\(761\) −1.14447e30 −0.636893 −0.318447 0.947941i \(-0.603161\pi\)
−0.318447 + 0.947941i \(0.603161\pi\)
\(762\) 0 0
\(763\) −5.90878e30 −3.19883
\(764\) −5.15743e28 −0.0275393
\(765\) 0 0
\(766\) 1.36736e30 0.710364
\(767\) 5.89910e29 0.302297
\(768\) 0 0
\(769\) −1.55372e29 −0.0774724 −0.0387362 0.999249i \(-0.512333\pi\)
−0.0387362 + 0.999249i \(0.512333\pi\)
\(770\) 7.84846e29 0.386039
\(771\) 0 0
\(772\) −8.27538e28 −0.0396101
\(773\) −2.20959e30 −1.04334 −0.521671 0.853147i \(-0.674691\pi\)
−0.521671 + 0.853147i \(0.674691\pi\)
\(774\) 0 0
\(775\) 4.70568e28 0.0216249
\(776\) −3.34785e30 −1.51781
\(777\) 0 0
\(778\) −1.85909e30 −0.820380
\(779\) 3.07695e30 1.33961
\(780\) 0 0
\(781\) 1.41518e30 0.599759
\(782\) 4.00340e29 0.167401
\(783\) 0 0
\(784\) −4.86155e30 −1.97905
\(785\) 1.07767e30 0.432866
\(786\) 0 0
\(787\) −2.75620e30 −1.07789 −0.538947 0.842340i \(-0.681178\pi\)
−0.538947 + 0.842340i \(0.681178\pi\)
\(788\) −2.64073e28 −0.0101906
\(789\) 0 0
\(790\) −1.14192e30 −0.429091
\(791\) 1.61567e30 0.599099
\(792\) 0 0
\(793\) −4.71647e30 −1.70313
\(794\) 4.72847e30 1.68502
\(795\) 0 0
\(796\) −6.19203e27 −0.00214905
\(797\) −4.23039e30 −1.44900 −0.724500 0.689275i \(-0.757929\pi\)
−0.724500 + 0.689275i \(0.757929\pi\)
\(798\) 0 0
\(799\) 3.46751e30 1.15685
\(800\) 6.76398e28 0.0222719
\(801\) 0 0
\(802\) −2.98302e30 −0.956810
\(803\) 1.13329e30 0.358779
\(804\) 0 0
\(805\) 3.66839e29 0.113141
\(806\) −5.24263e29 −0.159599
\(807\) 0 0
\(808\) −1.79219e30 −0.531576
\(809\) −3.93165e30 −1.15111 −0.575553 0.817764i \(-0.695213\pi\)
−0.575553 + 0.817764i \(0.695213\pi\)
\(810\) 0 0
\(811\) −1.50931e30 −0.430585 −0.215293 0.976550i \(-0.569071\pi\)
−0.215293 + 0.976550i \(0.569071\pi\)
\(812\) 6.60624e29 0.186044
\(813\) 0 0
\(814\) 5.59491e29 0.153545
\(815\) −3.04771e30 −0.825695
\(816\) 0 0
\(817\) 1.25379e30 0.331049
\(818\) −2.45630e30 −0.640282
\(819\) 0 0
\(820\) 1.33685e29 0.0339654
\(821\) 1.87901e29 0.0471332 0.0235666 0.999722i \(-0.492498\pi\)
0.0235666 + 0.999722i \(0.492498\pi\)
\(822\) 0 0
\(823\) −5.28445e30 −1.29212 −0.646058 0.763289i \(-0.723583\pi\)
−0.646058 + 0.763289i \(0.723583\pi\)
\(824\) 5.71134e30 1.37880
\(825\) 0 0
\(826\) 1.55912e30 0.366935
\(827\) 2.20857e30 0.513220 0.256610 0.966515i \(-0.417394\pi\)
0.256610 + 0.966515i \(0.417394\pi\)
\(828\) 0 0
\(829\) 5.77035e30 1.30731 0.653657 0.756791i \(-0.273234\pi\)
0.653657 + 0.756791i \(0.273234\pi\)
\(830\) −2.45514e30 −0.549233
\(831\) 0 0
\(832\) 6.17909e30 1.34781
\(833\) −9.53957e30 −2.05474
\(834\) 0 0
\(835\) 5.45556e29 0.114586
\(836\) 1.30811e29 0.0271318
\(837\) 0 0
\(838\) 3.59892e30 0.727963
\(839\) 4.79329e30 0.957488 0.478744 0.877955i \(-0.341092\pi\)
0.478744 + 0.877955i \(0.341092\pi\)
\(840\) 0 0
\(841\) 1.47187e31 2.86756
\(842\) 7.94430e30 1.52855
\(843\) 0 0
\(844\) 3.41022e29 0.0640010
\(845\) −2.56652e30 −0.475717
\(846\) 0 0
\(847\) 7.08586e30 1.28120
\(848\) 4.69829e30 0.839042
\(849\) 0 0
\(850\) 1.28910e30 0.224588
\(851\) 2.61507e29 0.0450011
\(852\) 0 0
\(853\) −2.24317e30 −0.376615 −0.188308 0.982110i \(-0.560300\pi\)
−0.188308 + 0.982110i \(0.560300\pi\)
\(854\) −1.24655e31 −2.06730
\(855\) 0 0
\(856\) 3.59317e29 0.0581438
\(857\) 1.67730e30 0.268109 0.134055 0.990974i \(-0.457200\pi\)
0.134055 + 0.990974i \(0.457200\pi\)
\(858\) 0 0
\(859\) −2.75580e30 −0.429853 −0.214926 0.976630i \(-0.568951\pi\)
−0.214926 + 0.976630i \(0.568951\pi\)
\(860\) 5.44735e28 0.00839366
\(861\) 0 0
\(862\) 2.19753e30 0.330453
\(863\) −4.97847e30 −0.739576 −0.369788 0.929116i \(-0.620570\pi\)
−0.369788 + 0.929116i \(0.620570\pi\)
\(864\) 0 0
\(865\) −1.31249e30 −0.190295
\(866\) −2.10839e30 −0.302005
\(867\) 0 0
\(868\) −7.31609e28 −0.0102287
\(869\) 3.34655e30 0.462264
\(870\) 0 0
\(871\) −1.83803e31 −2.47834
\(872\) 1.37271e31 1.82876
\(873\) 0 0
\(874\) 1.15798e30 0.150602
\(875\) 1.18122e30 0.151792
\(876\) 0 0
\(877\) −5.93482e29 −0.0744581 −0.0372291 0.999307i \(-0.511853\pi\)
−0.0372291 + 0.999307i \(0.511853\pi\)
\(878\) −2.62632e30 −0.325579
\(879\) 0 0
\(880\) −1.92529e30 −0.233039
\(881\) 1.27954e31 1.53040 0.765202 0.643791i \(-0.222639\pi\)
0.765202 + 0.643791i \(0.222639\pi\)
\(882\) 0 0
\(883\) 1.12787e30 0.131726 0.0658632 0.997829i \(-0.479020\pi\)
0.0658632 + 0.997829i \(0.479020\pi\)
\(884\) −7.58313e29 −0.0875184
\(885\) 0 0
\(886\) 1.61518e31 1.82040
\(887\) 6.34061e30 0.706209 0.353104 0.935584i \(-0.385126\pi\)
0.353104 + 0.935584i \(0.385126\pi\)
\(888\) 0 0
\(889\) 1.37111e31 1.49143
\(890\) −9.57343e29 −0.102913
\(891\) 0 0
\(892\) 4.80465e29 0.0504465
\(893\) 1.00297e31 1.04076
\(894\) 0 0
\(895\) 5.63757e30 0.571413
\(896\) 1.82178e31 1.82499
\(897\) 0 0
\(898\) 1.09289e31 1.06949
\(899\) −2.19847e30 −0.212639
\(900\) 0 0
\(901\) 9.21921e30 0.871130
\(902\) −7.42009e30 −0.693011
\(903\) 0 0
\(904\) −3.75347e30 −0.342503
\(905\) −2.10191e30 −0.189585
\(906\) 0 0
\(907\) −4.56592e30 −0.402394 −0.201197 0.979551i \(-0.564483\pi\)
−0.201197 + 0.979551i \(0.564483\pi\)
\(908\) −5.67518e29 −0.0494400
\(909\) 0 0
\(910\) −1.31601e31 −1.12027
\(911\) 2.06100e31 1.73434 0.867172 0.498009i \(-0.165935\pi\)
0.867172 + 0.498009i \(0.165935\pi\)
\(912\) 0 0
\(913\) 7.19515e30 0.591694
\(914\) 2.26563e31 1.84185
\(915\) 0 0
\(916\) −4.58077e29 −0.0363945
\(917\) −3.34238e31 −2.62529
\(918\) 0 0
\(919\) −1.86231e31 −1.42968 −0.714839 0.699289i \(-0.753500\pi\)
−0.714839 + 0.699289i \(0.753500\pi\)
\(920\) −8.52230e29 −0.0646822
\(921\) 0 0
\(922\) −2.03834e31 −1.51217
\(923\) −2.37294e31 −1.74048
\(924\) 0 0
\(925\) 8.42056e29 0.0603744
\(926\) 1.31308e31 0.930838
\(927\) 0 0
\(928\) −3.16009e30 −0.219001
\(929\) −2.63294e31 −1.80417 −0.902083 0.431562i \(-0.857963\pi\)
−0.902083 + 0.431562i \(0.857963\pi\)
\(930\) 0 0
\(931\) −2.75931e31 −1.84854
\(932\) −1.31867e30 −0.0873513
\(933\) 0 0
\(934\) 1.55713e31 1.00852
\(935\) −3.77789e30 −0.241951
\(936\) 0 0
\(937\) 4.07588e30 0.255244 0.127622 0.991823i \(-0.459266\pi\)
0.127622 + 0.991823i \(0.459266\pi\)
\(938\) −4.85787e31 −3.00827
\(939\) 0 0
\(940\) 4.35764e29 0.0263881
\(941\) 3.48852e30 0.208906 0.104453 0.994530i \(-0.466691\pi\)
0.104453 + 0.994530i \(0.466691\pi\)
\(942\) 0 0
\(943\) −3.46817e30 −0.203108
\(944\) −3.82465e30 −0.221506
\(945\) 0 0
\(946\) −3.02352e30 −0.171260
\(947\) 3.11529e31 1.74511 0.872557 0.488513i \(-0.162460\pi\)
0.872557 + 0.488513i \(0.162460\pi\)
\(948\) 0 0
\(949\) −1.90027e31 −1.04116
\(950\) 3.72870e30 0.202050
\(951\) 0 0
\(952\) 3.39499e31 1.79949
\(953\) 1.98167e31 1.03886 0.519429 0.854514i \(-0.326145\pi\)
0.519429 + 0.854514i \(0.326145\pi\)
\(954\) 0 0
\(955\) 4.30833e30 0.220940
\(956\) 1.08646e30 0.0551068
\(957\) 0 0
\(958\) −1.48665e31 −0.737683
\(959\) −2.49524e31 −1.22466
\(960\) 0 0
\(961\) −2.05820e31 −0.988309
\(962\) −9.38140e30 −0.445584
\(963\) 0 0
\(964\) −3.37742e29 −0.0156955
\(965\) 6.91295e30 0.317780
\(966\) 0 0
\(967\) −8.26115e30 −0.371589 −0.185794 0.982589i \(-0.559486\pi\)
−0.185794 + 0.982589i \(0.559486\pi\)
\(968\) −1.64617e31 −0.732458
\(969\) 0 0
\(970\) −1.65099e31 −0.718857
\(971\) −5.95729e30 −0.256594 −0.128297 0.991736i \(-0.540951\pi\)
−0.128297 + 0.991736i \(0.540951\pi\)
\(972\) 0 0
\(973\) −1.74219e31 −0.734365
\(974\) 3.47263e31 1.44807
\(975\) 0 0
\(976\) 3.05790e31 1.24796
\(977\) −3.99242e31 −1.61192 −0.805960 0.591970i \(-0.798351\pi\)
−0.805960 + 0.591970i \(0.798351\pi\)
\(978\) 0 0
\(979\) 2.80563e30 0.110870
\(980\) −1.19884e30 −0.0468692
\(981\) 0 0
\(982\) 3.41668e31 1.30747
\(983\) −2.59366e30 −0.0981977 −0.0490988 0.998794i \(-0.515635\pi\)
−0.0490988 + 0.998794i \(0.515635\pi\)
\(984\) 0 0
\(985\) 2.20597e30 0.0817558
\(986\) −6.02259e31 −2.20839
\(987\) 0 0
\(988\) −2.19341e30 −0.0787357
\(989\) −1.41320e30 −0.0501929
\(990\) 0 0
\(991\) 6.46664e30 0.224856 0.112428 0.993660i \(-0.464137\pi\)
0.112428 + 0.993660i \(0.464137\pi\)
\(992\) 3.49965e29 0.0120407
\(993\) 0 0
\(994\) −6.27165e31 −2.11264
\(995\) 5.17260e29 0.0172411
\(996\) 0 0
\(997\) 3.67859e31 1.20056 0.600278 0.799791i \(-0.295057\pi\)
0.600278 + 0.799791i \(0.295057\pi\)
\(998\) 2.36483e31 0.763712
\(999\) 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.1.1 4
3.2 odd 2 15.22.a.e.1.4 4
15.2 even 4 75.22.b.h.49.7 8
15.8 even 4 75.22.b.h.49.2 8
15.14 odd 2 75.22.a.h.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.22.a.e.1.4 4 3.2 odd 2
45.22.a.g.1.1 4 1.1 even 1 trivial
75.22.a.h.1.1 4 15.14 odd 2
75.22.b.h.49.2 8 15.8 even 4
75.22.b.h.49.7 8 15.2 even 4