Properties

Label 75.22.a.j.1.4
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 1.4
Root \(-144.571\) 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)
 
\(f(q)\) \(=\) \(q+594.286 q^{2} -59049.0 q^{3} -1.74398e6 q^{4} -3.50920e7 q^{6} -9.49356e8 q^{7} -2.28273e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\) \(q+594.286 q^{2} -59049.0 q^{3} -1.74398e6 q^{4} -3.50920e7 q^{6} -9.49356e8 q^{7} -2.28273e9 q^{8} +3.48678e9 q^{9} -8.95529e10 q^{11} +1.02980e11 q^{12} -1.19043e11 q^{13} -5.64189e11 q^{14} +2.30079e12 q^{16} -7.66551e11 q^{17} +2.07215e12 q^{18} +1.00568e13 q^{19} +5.60585e13 q^{21} -5.32200e13 q^{22} +1.37650e14 q^{23} +1.34793e14 q^{24} -7.07455e13 q^{26} -2.05891e14 q^{27} +1.65565e15 q^{28} +5.32391e14 q^{29} -2.67291e15 q^{31} +6.15455e15 q^{32} +5.28801e15 q^{33} -4.55550e14 q^{34} -6.08087e15 q^{36} +4.68003e16 q^{37} +5.97663e15 q^{38} +7.02937e15 q^{39} +1.01069e16 q^{41} +3.33148e16 q^{42} +6.08624e16 q^{43} +1.56178e17 q^{44} +8.18037e16 q^{46} -2.05864e17 q^{47} -1.35859e17 q^{48} +3.42731e17 q^{49} +4.52641e16 q^{51} +2.07608e17 q^{52} -1.16730e18 q^{53} -1.22358e17 q^{54} +2.16712e18 q^{56} -5.93846e17 q^{57} +3.16392e17 q^{58} -3.09436e18 q^{59} +4.71926e18 q^{61} -1.58847e18 q^{62} -3.31020e18 q^{63} -1.16755e18 q^{64} +3.14259e18 q^{66} +2.22543e19 q^{67} +1.33685e18 q^{68} -8.12812e18 q^{69} +3.77185e19 q^{71} -7.95938e18 q^{72} +2.43541e19 q^{73} +2.78127e19 q^{74} -1.75389e19 q^{76} +8.50176e19 q^{77} +4.17745e18 q^{78} +8.41111e19 q^{79} +1.21577e19 q^{81} +6.00636e18 q^{82} -5.17094e19 q^{83} -9.77648e19 q^{84} +3.61696e19 q^{86} -3.14372e19 q^{87} +2.04425e20 q^{88} -4.29525e20 q^{89} +1.13014e20 q^{91} -2.40059e20 q^{92} +1.57832e20 q^{93} -1.22342e20 q^{94} -3.63420e20 q^{96} -1.15264e21 q^{97} +2.03680e20 q^{98} -3.12252e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 92 q^{2} - 354294 q^{3} + 4390448 q^{4} - 5432508 q^{6} + 1327143454 q^{7} - 4252271424 q^{8} + 20920706406 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 594.286 0.410374 0.205187 0.978723i \(-0.434220\pi\)
0.205187 + 0.978723i \(0.434220\pi\)
\(3\) −59049.0 −0.577350
\(4\) −1.74398e6 −0.831593
\(5\) 0 0
\(6\) −3.50920e7 −0.236930
\(7\) −9.49356e8 −1.27028 −0.635140 0.772397i \(-0.719058\pi\)
−0.635140 + 0.772397i \(0.719058\pi\)
\(8\) −2.28273e9 −0.751639
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) −8.95529e10 −1.04101 −0.520507 0.853858i \(-0.674257\pi\)
−0.520507 + 0.853858i \(0.674257\pi\)
\(12\) 1.02980e11 0.480120
\(13\) −1.19043e11 −0.239496 −0.119748 0.992804i \(-0.538209\pi\)
−0.119748 + 0.992804i \(0.538209\pi\)
\(14\) −5.64189e11 −0.521291
\(15\) 0 0
\(16\) 2.30079e12 0.523140
\(17\) −7.66551e11 −0.0922205 −0.0461103 0.998936i \(-0.514683\pi\)
−0.0461103 + 0.998936i \(0.514683\pi\)
\(18\) 2.07215e12 0.136791
\(19\) 1.00568e13 0.376312 0.188156 0.982139i \(-0.439749\pi\)
0.188156 + 0.982139i \(0.439749\pi\)
\(20\) 0 0
\(21\) 5.60585e13 0.733397
\(22\) −5.32200e13 −0.427205
\(23\) 1.37650e14 0.692844 0.346422 0.938079i \(-0.387397\pi\)
0.346422 + 0.938079i \(0.387397\pi\)
\(24\) 1.34793e14 0.433959
\(25\) 0 0
\(26\) −7.07455e13 −0.0982831
\(27\) −2.05891e14 −0.192450
\(28\) 1.65565e15 1.05636
\(29\) 5.32391e14 0.234991 0.117496 0.993073i \(-0.462513\pi\)
0.117496 + 0.993073i \(0.462513\pi\)
\(30\) 0 0
\(31\) −2.67291e15 −0.585714 −0.292857 0.956156i \(-0.594606\pi\)
−0.292857 + 0.956156i \(0.594606\pi\)
\(32\) 6.15455e15 0.966322
\(33\) 5.28801e15 0.601029
\(34\) −4.55550e14 −0.0378449
\(35\) 0 0
\(36\) −6.08087e15 −0.277198
\(37\) 4.68003e16 1.60004 0.800019 0.599975i \(-0.204823\pi\)
0.800019 + 0.599975i \(0.204823\pi\)
\(38\) 5.97663e15 0.154429
\(39\) 7.02937e15 0.138273
\(40\) 0 0
\(41\) 1.01069e16 0.117594 0.0587971 0.998270i \(-0.481274\pi\)
0.0587971 + 0.998270i \(0.481274\pi\)
\(42\) 3.33148e16 0.300967
\(43\) 6.08624e16 0.429467 0.214734 0.976673i \(-0.431112\pi\)
0.214734 + 0.976673i \(0.431112\pi\)
\(44\) 1.56178e17 0.865699
\(45\) 0 0
\(46\) 8.18037e16 0.284325
\(47\) −2.05864e17 −0.570892 −0.285446 0.958395i \(-0.592142\pi\)
−0.285446 + 0.958395i \(0.592142\pi\)
\(48\) −1.35859e17 −0.302035
\(49\) 3.42731e17 0.613613
\(50\) 0 0
\(51\) 4.52641e16 0.0532435
\(52\) 2.07608e17 0.199163
\(53\) −1.16730e18 −0.916827 −0.458414 0.888739i \(-0.651582\pi\)
−0.458414 + 0.888739i \(0.651582\pi\)
\(54\) −1.22358e17 −0.0789766
\(55\) 0 0
\(56\) 2.16712e18 0.954792
\(57\) −5.93846e17 −0.217264
\(58\) 3.16392e17 0.0964345
\(59\) −3.09436e18 −0.788178 −0.394089 0.919072i \(-0.628940\pi\)
−0.394089 + 0.919072i \(0.628940\pi\)
\(60\) 0 0
\(61\) 4.71926e18 0.847053 0.423527 0.905884i \(-0.360792\pi\)
0.423527 + 0.905884i \(0.360792\pi\)
\(62\) −1.58847e18 −0.240362
\(63\) −3.31020e18 −0.423427
\(64\) −1.16755e18 −0.126586
\(65\) 0 0
\(66\) 3.14259e18 0.246647
\(67\) 2.22543e19 1.49152 0.745758 0.666217i \(-0.232088\pi\)
0.745758 + 0.666217i \(0.232088\pi\)
\(68\) 1.33685e18 0.0766899
\(69\) −8.12812e18 −0.400014
\(70\) 0 0
\(71\) 3.77185e19 1.37513 0.687563 0.726125i \(-0.258681\pi\)
0.687563 + 0.726125i \(0.258681\pi\)
\(72\) −7.95938e18 −0.250546
\(73\) 2.43541e19 0.663257 0.331629 0.943410i \(-0.392402\pi\)
0.331629 + 0.943410i \(0.392402\pi\)
\(74\) 2.78127e19 0.656615
\(75\) 0 0
\(76\) −1.75389e19 −0.312939
\(77\) 8.50176e19 1.32238
\(78\) 4.17745e18 0.0567438
\(79\) 8.41111e19 0.999468 0.499734 0.866179i \(-0.333431\pi\)
0.499734 + 0.866179i \(0.333431\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) 6.00636e18 0.0482576
\(83\) −5.17094e19 −0.365805 −0.182902 0.983131i \(-0.558549\pi\)
−0.182902 + 0.983131i \(0.558549\pi\)
\(84\) −9.77648e19 −0.609888
\(85\) 0 0
\(86\) 3.61696e19 0.176242
\(87\) −3.14372e19 −0.135672
\(88\) 2.04425e20 0.782466
\(89\) −4.29525e20 −1.46014 −0.730069 0.683374i \(-0.760512\pi\)
−0.730069 + 0.683374i \(0.760512\pi\)
\(90\) 0 0
\(91\) 1.13014e20 0.304227
\(92\) −2.40059e20 −0.576164
\(93\) 1.57832e20 0.338162
\(94\) −1.22342e20 −0.234279
\(95\) 0 0
\(96\) −3.63420e20 −0.557906
\(97\) −1.15264e21 −1.58705 −0.793523 0.608540i \(-0.791755\pi\)
−0.793523 + 0.608540i \(0.791755\pi\)
\(98\) 2.03680e20 0.251811
\(99\) −3.12252e20 −0.347004
\(100\) 0 0
\(101\) 1.58871e21 1.43110 0.715549 0.698563i \(-0.246177\pi\)
0.715549 + 0.698563i \(0.246177\pi\)
\(102\) 2.68998e19 0.0218498
\(103\) 1.50388e21 1.10261 0.551304 0.834305i \(-0.314131\pi\)
0.551304 + 0.834305i \(0.314131\pi\)
\(104\) 2.71743e20 0.180015
\(105\) 0 0
\(106\) −6.93712e20 −0.376242
\(107\) 2.43784e21 1.19805 0.599025 0.800730i \(-0.295555\pi\)
0.599025 + 0.800730i \(0.295555\pi\)
\(108\) 3.59069e20 0.160040
\(109\) −1.83664e21 −0.743097 −0.371548 0.928414i \(-0.621173\pi\)
−0.371548 + 0.928414i \(0.621173\pi\)
\(110\) 0 0
\(111\) −2.76351e21 −0.923782
\(112\) −2.18427e21 −0.664534
\(113\) −2.87660e21 −0.797178 −0.398589 0.917130i \(-0.630500\pi\)
−0.398589 + 0.917130i \(0.630500\pi\)
\(114\) −3.52914e20 −0.0891596
\(115\) 0 0
\(116\) −9.28478e20 −0.195417
\(117\) −4.15077e20 −0.0798320
\(118\) −1.83893e21 −0.323448
\(119\) 7.27730e20 0.117146
\(120\) 0 0
\(121\) 6.19468e20 0.0837091
\(122\) 2.80459e21 0.347609
\(123\) −5.96800e20 −0.0678930
\(124\) 4.66149e21 0.487076
\(125\) 0 0
\(126\) −1.96720e21 −0.173764
\(127\) −1.13467e22 −0.922421 −0.461210 0.887291i \(-0.652585\pi\)
−0.461210 + 0.887291i \(0.652585\pi\)
\(128\) −1.36009e22 −1.01827
\(129\) −3.59386e21 −0.247953
\(130\) 0 0
\(131\) 3.19824e22 1.87742 0.938712 0.344703i \(-0.112020\pi\)
0.938712 + 0.344703i \(0.112020\pi\)
\(132\) −9.22216e21 −0.499812
\(133\) −9.54752e21 −0.478022
\(134\) 1.32254e22 0.612080
\(135\) 0 0
\(136\) 1.74983e21 0.0693165
\(137\) −1.70156e22 −0.624141 −0.312070 0.950059i \(-0.601022\pi\)
−0.312070 + 0.950059i \(0.601022\pi\)
\(138\) −4.83043e21 −0.164155
\(139\) −2.23908e22 −0.705365 −0.352683 0.935743i \(-0.614731\pi\)
−0.352683 + 0.935743i \(0.614731\pi\)
\(140\) 0 0
\(141\) 1.21561e22 0.329604
\(142\) 2.24156e22 0.564316
\(143\) 1.06606e22 0.249319
\(144\) 8.02237e21 0.174380
\(145\) 0 0
\(146\) 1.44733e22 0.272184
\(147\) −2.02379e22 −0.354270
\(148\) −8.16186e22 −1.33058
\(149\) −9.05514e22 −1.37543 −0.687717 0.725979i \(-0.741387\pi\)
−0.687717 + 0.725979i \(0.741387\pi\)
\(150\) 0 0
\(151\) −6.35537e22 −0.839234 −0.419617 0.907701i \(-0.637836\pi\)
−0.419617 + 0.907701i \(0.637836\pi\)
\(152\) −2.29570e22 −0.282851
\(153\) −2.67280e21 −0.0307402
\(154\) 5.05247e22 0.542671
\(155\) 0 0
\(156\) −1.22591e22 −0.114987
\(157\) 5.32189e22 0.466787 0.233394 0.972382i \(-0.425017\pi\)
0.233394 + 0.972382i \(0.425017\pi\)
\(158\) 4.99860e22 0.410156
\(159\) 6.89282e22 0.529330
\(160\) 0 0
\(161\) −1.30679e23 −0.880106
\(162\) 7.22513e21 0.0455972
\(163\) 9.96667e22 0.589631 0.294815 0.955554i \(-0.404742\pi\)
0.294815 + 0.955554i \(0.404742\pi\)
\(164\) −1.76261e22 −0.0977904
\(165\) 0 0
\(166\) −3.07301e22 −0.150117
\(167\) 6.26848e20 0.00287501 0.00143750 0.999999i \(-0.499542\pi\)
0.00143750 + 0.999999i \(0.499542\pi\)
\(168\) −1.27966e23 −0.551250
\(169\) −2.32893e23 −0.942642
\(170\) 0 0
\(171\) 3.50660e22 0.125437
\(172\) −1.06143e23 −0.357142
\(173\) 5.71132e23 1.80822 0.904112 0.427295i \(-0.140534\pi\)
0.904112 + 0.427295i \(0.140534\pi\)
\(174\) −1.86827e22 −0.0556765
\(175\) 0 0
\(176\) −2.06043e23 −0.544595
\(177\) 1.82719e23 0.455055
\(178\) −2.55261e23 −0.599203
\(179\) −5.62914e23 −1.24591 −0.622953 0.782260i \(-0.714067\pi\)
−0.622953 + 0.782260i \(0.714067\pi\)
\(180\) 0 0
\(181\) 6.00250e23 1.18224 0.591122 0.806582i \(-0.298685\pi\)
0.591122 + 0.806582i \(0.298685\pi\)
\(182\) 6.71627e22 0.124847
\(183\) −2.78668e23 −0.489046
\(184\) −3.14218e23 −0.520768
\(185\) 0 0
\(186\) 9.37976e22 0.138773
\(187\) 6.86469e22 0.0960028
\(188\) 3.59023e23 0.474749
\(189\) 1.95464e23 0.244466
\(190\) 0 0
\(191\) 3.23094e23 0.361809 0.180904 0.983501i \(-0.442098\pi\)
0.180904 + 0.983501i \(0.442098\pi\)
\(192\) 6.89425e22 0.0730843
\(193\) −7.50257e23 −0.753110 −0.376555 0.926394i \(-0.622891\pi\)
−0.376555 + 0.926394i \(0.622891\pi\)
\(194\) −6.84996e23 −0.651283
\(195\) 0 0
\(196\) −5.97715e23 −0.510276
\(197\) 1.99932e24 1.61804 0.809018 0.587784i \(-0.200000\pi\)
0.809018 + 0.587784i \(0.200000\pi\)
\(198\) −1.85567e23 −0.142402
\(199\) 1.47747e24 1.07538 0.537688 0.843144i \(-0.319298\pi\)
0.537688 + 0.843144i \(0.319298\pi\)
\(200\) 0 0
\(201\) −1.31409e24 −0.861127
\(202\) 9.44146e23 0.587286
\(203\) −5.05429e23 −0.298505
\(204\) −7.89395e22 −0.0442770
\(205\) 0 0
\(206\) 8.93731e23 0.452482
\(207\) 4.79957e23 0.230948
\(208\) −2.73893e23 −0.125290
\(209\) −9.00619e23 −0.391746
\(210\) 0 0
\(211\) −3.93789e24 −1.54988 −0.774939 0.632036i \(-0.782220\pi\)
−0.774939 + 0.632036i \(0.782220\pi\)
\(212\) 2.03575e24 0.762427
\(213\) −2.22724e24 −0.793929
\(214\) 1.44877e24 0.491649
\(215\) 0 0
\(216\) 4.69993e23 0.144653
\(217\) 2.53754e24 0.744022
\(218\) −1.09149e24 −0.304948
\(219\) −1.43809e24 −0.382932
\(220\) 0 0
\(221\) 9.12525e22 0.0220865
\(222\) −1.64231e24 −0.379097
\(223\) −4.06392e24 −0.894838 −0.447419 0.894324i \(-0.647657\pi\)
−0.447419 + 0.894324i \(0.647657\pi\)
\(224\) −5.84286e24 −1.22750
\(225\) 0 0
\(226\) −1.70952e24 −0.327142
\(227\) 3.21738e24 0.587801 0.293901 0.955836i \(-0.405047\pi\)
0.293901 + 0.955836i \(0.405047\pi\)
\(228\) 1.03565e24 0.180675
\(229\) −1.16796e25 −1.94605 −0.973026 0.230695i \(-0.925900\pi\)
−0.973026 + 0.230695i \(0.925900\pi\)
\(230\) 0 0
\(231\) −5.02020e24 −0.763476
\(232\) −1.21530e24 −0.176629
\(233\) −7.69497e24 −1.06898 −0.534490 0.845174i \(-0.679496\pi\)
−0.534490 + 0.845174i \(0.679496\pi\)
\(234\) −2.46674e23 −0.0327610
\(235\) 0 0
\(236\) 5.39648e24 0.655443
\(237\) −4.96668e24 −0.577043
\(238\) 4.32480e23 0.0480737
\(239\) −3.70261e24 −0.393849 −0.196924 0.980419i \(-0.563095\pi\)
−0.196924 + 0.980419i \(0.563095\pi\)
\(240\) 0 0
\(241\) −1.12205e25 −1.09354 −0.546768 0.837284i \(-0.684142\pi\)
−0.546768 + 0.837284i \(0.684142\pi\)
\(242\) 3.68141e23 0.0343521
\(243\) −7.17898e23 −0.0641500
\(244\) −8.23028e24 −0.704403
\(245\) 0 0
\(246\) −3.54670e23 −0.0278615
\(247\) −1.19720e24 −0.0901254
\(248\) 6.10152e24 0.440246
\(249\) 3.05339e24 0.211197
\(250\) 0 0
\(251\) −1.24318e25 −0.790608 −0.395304 0.918550i \(-0.629361\pi\)
−0.395304 + 0.918550i \(0.629361\pi\)
\(252\) 5.77291e24 0.352119
\(253\) −1.23270e25 −0.721260
\(254\) −6.74316e24 −0.378538
\(255\) 0 0
\(256\) −5.63429e24 −0.291286
\(257\) 2.23944e25 1.11133 0.555663 0.831408i \(-0.312465\pi\)
0.555663 + 0.831408i \(0.312465\pi\)
\(258\) −2.13578e24 −0.101754
\(259\) −4.44301e25 −2.03250
\(260\) 0 0
\(261\) 1.85633e24 0.0783305
\(262\) 1.90067e25 0.770446
\(263\) −2.05340e25 −0.799719 −0.399859 0.916577i \(-0.630941\pi\)
−0.399859 + 0.916577i \(0.630941\pi\)
\(264\) −1.20711e25 −0.451757
\(265\) 0 0
\(266\) −5.67395e24 −0.196168
\(267\) 2.53630e25 0.843011
\(268\) −3.88109e25 −1.24033
\(269\) −4.35651e25 −1.33887 −0.669437 0.742869i \(-0.733465\pi\)
−0.669437 + 0.742869i \(0.733465\pi\)
\(270\) 0 0
\(271\) 3.57994e25 1.01788 0.508942 0.860801i \(-0.330037\pi\)
0.508942 + 0.860801i \(0.330037\pi\)
\(272\) −1.76368e24 −0.0482442
\(273\) −6.67337e24 −0.175646
\(274\) −1.01122e25 −0.256131
\(275\) 0 0
\(276\) 1.41753e25 0.332648
\(277\) 4.54757e25 1.02741 0.513703 0.857968i \(-0.328273\pi\)
0.513703 + 0.857968i \(0.328273\pi\)
\(278\) −1.33065e25 −0.289464
\(279\) −9.31985e24 −0.195238
\(280\) 0 0
\(281\) −3.65534e24 −0.0710415 −0.0355207 0.999369i \(-0.511309\pi\)
−0.0355207 + 0.999369i \(0.511309\pi\)
\(282\) 7.22419e24 0.135261
\(283\) 3.26857e25 0.589659 0.294829 0.955550i \(-0.404737\pi\)
0.294829 + 0.955550i \(0.404737\pi\)
\(284\) −6.57803e25 −1.14354
\(285\) 0 0
\(286\) 6.33546e24 0.102314
\(287\) −9.59501e24 −0.149378
\(288\) 2.14596e25 0.322107
\(289\) −6.85043e25 −0.991495
\(290\) 0 0
\(291\) 6.80621e25 0.916282
\(292\) −4.24730e25 −0.551560
\(293\) −9.81757e25 −1.22997 −0.614984 0.788539i \(-0.710838\pi\)
−0.614984 + 0.788539i \(0.710838\pi\)
\(294\) −1.20271e25 −0.145383
\(295\) 0 0
\(296\) −1.06832e26 −1.20265
\(297\) 1.84381e25 0.200343
\(298\) −5.38134e25 −0.564443
\(299\) −1.63863e25 −0.165933
\(300\) 0 0
\(301\) −5.77801e25 −0.545544
\(302\) −3.77690e25 −0.344400
\(303\) −9.38116e25 −0.826245
\(304\) 2.31387e25 0.196864
\(305\) 0 0
\(306\) −1.58841e24 −0.0126150
\(307\) 1.52004e26 1.16655 0.583274 0.812276i \(-0.301772\pi\)
0.583274 + 0.812276i \(0.301772\pi\)
\(308\) −1.48269e26 −1.09968
\(309\) −8.88023e25 −0.636591
\(310\) 0 0
\(311\) −1.70761e26 −1.14394 −0.571971 0.820274i \(-0.693821\pi\)
−0.571971 + 0.820274i \(0.693821\pi\)
\(312\) −1.60461e25 −0.103931
\(313\) 1.54966e26 0.970554 0.485277 0.874360i \(-0.338719\pi\)
0.485277 + 0.874360i \(0.338719\pi\)
\(314\) 3.16272e25 0.191558
\(315\) 0 0
\(316\) −1.46688e26 −0.831150
\(317\) 1.62343e26 0.889840 0.444920 0.895570i \(-0.353232\pi\)
0.444920 + 0.895570i \(0.353232\pi\)
\(318\) 4.09630e25 0.217224
\(319\) −4.76772e25 −0.244629
\(320\) 0 0
\(321\) −1.43952e26 −0.691695
\(322\) −7.76608e25 −0.361173
\(323\) −7.70908e24 −0.0347037
\(324\) −2.12027e25 −0.0923992
\(325\) 0 0
\(326\) 5.92305e25 0.241969
\(327\) 1.08452e26 0.429027
\(328\) −2.30712e25 −0.0883883
\(329\) 1.95439e26 0.725193
\(330\) 0 0
\(331\) −3.05873e26 −1.06500 −0.532498 0.846431i \(-0.678747\pi\)
−0.532498 + 0.846431i \(0.678747\pi\)
\(332\) 9.01799e25 0.304200
\(333\) 1.63182e26 0.533346
\(334\) 3.72527e23 0.00117983
\(335\) 0 0
\(336\) 1.28979e26 0.383669
\(337\) 2.13469e25 0.0615490 0.0307745 0.999526i \(-0.490203\pi\)
0.0307745 + 0.999526i \(0.490203\pi\)
\(338\) −1.38405e26 −0.386836
\(339\) 1.69860e26 0.460251
\(340\) 0 0
\(341\) 2.39366e26 0.609737
\(342\) 2.08392e25 0.0514763
\(343\) 2.04885e26 0.490820
\(344\) −1.38932e26 −0.322804
\(345\) 0 0
\(346\) 3.39415e26 0.742049
\(347\) −2.78601e25 −0.0590913 −0.0295457 0.999563i \(-0.509406\pi\)
−0.0295457 + 0.999563i \(0.509406\pi\)
\(348\) 5.48257e25 0.112824
\(349\) −4.67113e26 −0.932729 −0.466364 0.884593i \(-0.654436\pi\)
−0.466364 + 0.884593i \(0.654436\pi\)
\(350\) 0 0
\(351\) 2.45099e25 0.0460910
\(352\) −5.51158e26 −1.00595
\(353\) −5.62641e25 −0.0996774 −0.0498387 0.998757i \(-0.515871\pi\)
−0.0498387 + 0.998757i \(0.515871\pi\)
\(354\) 1.08587e26 0.186743
\(355\) 0 0
\(356\) 7.49082e26 1.21424
\(357\) −4.29717e25 −0.0676342
\(358\) −3.34531e26 −0.511288
\(359\) 9.76558e26 1.44946 0.724730 0.689033i \(-0.241964\pi\)
0.724730 + 0.689033i \(0.241964\pi\)
\(360\) 0 0
\(361\) −6.13070e26 −0.858389
\(362\) 3.56720e26 0.485162
\(363\) −3.65790e25 −0.0483295
\(364\) −1.97094e26 −0.252993
\(365\) 0 0
\(366\) −1.65608e26 −0.200692
\(367\) −9.97652e25 −0.117486 −0.0587429 0.998273i \(-0.518709\pi\)
−0.0587429 + 0.998273i \(0.518709\pi\)
\(368\) 3.16705e26 0.362454
\(369\) 3.52404e25 0.0391980
\(370\) 0 0
\(371\) 1.10819e27 1.16463
\(372\) −2.75256e26 −0.281213
\(373\) 5.88039e26 0.584068 0.292034 0.956408i \(-0.405668\pi\)
0.292034 + 0.956408i \(0.405668\pi\)
\(374\) 4.07959e25 0.0393971
\(375\) 0 0
\(376\) 4.69932e26 0.429104
\(377\) −6.33774e25 −0.0562795
\(378\) 1.16161e26 0.100322
\(379\) 2.15064e27 1.80658 0.903290 0.429032i \(-0.141145\pi\)
0.903290 + 0.429032i \(0.141145\pi\)
\(380\) 0 0
\(381\) 6.70009e26 0.532560
\(382\) 1.92010e26 0.148477
\(383\) 1.98164e27 1.49086 0.745430 0.666583i \(-0.232244\pi\)
0.745430 + 0.666583i \(0.232244\pi\)
\(384\) 8.03119e26 0.587898
\(385\) 0 0
\(386\) −4.45867e26 −0.309057
\(387\) 2.12214e26 0.143156
\(388\) 2.01017e27 1.31978
\(389\) 4.42277e26 0.282633 0.141317 0.989964i \(-0.454866\pi\)
0.141317 + 0.989964i \(0.454866\pi\)
\(390\) 0 0
\(391\) −1.05516e26 −0.0638944
\(392\) −7.82361e26 −0.461215
\(393\) −1.88853e27 −1.08393
\(394\) 1.18817e27 0.664000
\(395\) 0 0
\(396\) 5.44559e26 0.288566
\(397\) −1.84878e27 −0.954079 −0.477040 0.878882i \(-0.658290\pi\)
−0.477040 + 0.878882i \(0.658290\pi\)
\(398\) 8.78037e26 0.441307
\(399\) 5.63771e26 0.275986
\(400\) 0 0
\(401\) 5.92825e26 0.275366 0.137683 0.990476i \(-0.456034\pi\)
0.137683 + 0.990476i \(0.456034\pi\)
\(402\) −7.80946e26 −0.353384
\(403\) 3.18191e26 0.140276
\(404\) −2.77067e27 −1.19009
\(405\) 0 0
\(406\) −3.00369e26 −0.122499
\(407\) −4.19110e27 −1.66566
\(408\) −1.03326e26 −0.0400199
\(409\) 1.08923e26 0.0411175 0.0205587 0.999789i \(-0.493455\pi\)
0.0205587 + 0.999789i \(0.493455\pi\)
\(410\) 0 0
\(411\) 1.00476e27 0.360348
\(412\) −2.62272e27 −0.916920
\(413\) 2.93764e27 1.00121
\(414\) 2.85232e26 0.0947751
\(415\) 0 0
\(416\) −7.32656e26 −0.231430
\(417\) 1.32215e27 0.407243
\(418\) −5.35225e26 −0.160763
\(419\) 3.50075e27 1.02545 0.512724 0.858553i \(-0.328636\pi\)
0.512724 + 0.858553i \(0.328636\pi\)
\(420\) 0 0
\(421\) −2.43696e27 −0.679027 −0.339514 0.940601i \(-0.610262\pi\)
−0.339514 + 0.940601i \(0.610262\pi\)
\(422\) −2.34023e27 −0.636030
\(423\) −7.17805e26 −0.190297
\(424\) 2.66464e27 0.689123
\(425\) 0 0
\(426\) −1.32362e27 −0.325808
\(427\) −4.48026e27 −1.07600
\(428\) −4.25153e27 −0.996290
\(429\) −6.29500e26 −0.143944
\(430\) 0 0
\(431\) 4.22655e27 0.920396 0.460198 0.887816i \(-0.347778\pi\)
0.460198 + 0.887816i \(0.347778\pi\)
\(432\) −4.73713e26 −0.100678
\(433\) −4.97259e26 −0.103148 −0.0515739 0.998669i \(-0.516424\pi\)
−0.0515739 + 0.998669i \(0.516424\pi\)
\(434\) 1.50802e27 0.305327
\(435\) 0 0
\(436\) 3.20305e27 0.617954
\(437\) 1.38433e27 0.260726
\(438\) −8.54633e26 −0.157145
\(439\) 5.68534e27 1.02066 0.510328 0.859980i \(-0.329524\pi\)
0.510328 + 0.859980i \(0.329524\pi\)
\(440\) 0 0
\(441\) 1.19503e27 0.204538
\(442\) 5.42301e25 0.00906372
\(443\) 6.47154e27 1.05625 0.528127 0.849166i \(-0.322895\pi\)
0.528127 + 0.849166i \(0.322895\pi\)
\(444\) 4.81950e27 0.768211
\(445\) 0 0
\(446\) −2.41513e27 −0.367219
\(447\) 5.34697e27 0.794107
\(448\) 1.10842e27 0.160799
\(449\) 6.40973e27 0.908350 0.454175 0.890912i \(-0.349934\pi\)
0.454175 + 0.890912i \(0.349934\pi\)
\(450\) 0 0
\(451\) −9.05098e26 −0.122417
\(452\) 5.01672e27 0.662928
\(453\) 3.75278e27 0.484532
\(454\) 1.91204e27 0.241219
\(455\) 0 0
\(456\) 1.35559e27 0.163304
\(457\) −9.55569e26 −0.112497 −0.0562487 0.998417i \(-0.517914\pi\)
−0.0562487 + 0.998417i \(0.517914\pi\)
\(458\) −6.94100e27 −0.798610
\(459\) 1.57826e26 0.0177478
\(460\) 0 0
\(461\) 1.06753e28 1.14689 0.573443 0.819245i \(-0.305607\pi\)
0.573443 + 0.819245i \(0.305607\pi\)
\(462\) −2.98343e27 −0.313311
\(463\) 1.09436e28 1.12346 0.561732 0.827319i \(-0.310135\pi\)
0.561732 + 0.827319i \(0.310135\pi\)
\(464\) 1.22492e27 0.122933
\(465\) 0 0
\(466\) −4.57301e27 −0.438682
\(467\) 1.44925e28 1.35931 0.679653 0.733534i \(-0.262130\pi\)
0.679653 + 0.733534i \(0.262130\pi\)
\(468\) 7.23885e26 0.0663877
\(469\) −2.11272e28 −1.89464
\(470\) 0 0
\(471\) −3.14252e27 −0.269500
\(472\) 7.06357e27 0.592425
\(473\) −5.45040e27 −0.447081
\(474\) −2.95162e27 −0.236804
\(475\) 0 0
\(476\) −1.26914e27 −0.0974177
\(477\) −4.07014e27 −0.305609
\(478\) −2.20041e27 −0.161626
\(479\) −3.46837e27 −0.249231 −0.124615 0.992205i \(-0.539770\pi\)
−0.124615 + 0.992205i \(0.539770\pi\)
\(480\) 0 0
\(481\) −5.57124e27 −0.383203
\(482\) −6.66818e27 −0.448759
\(483\) 7.71648e27 0.508130
\(484\) −1.08034e27 −0.0696119
\(485\) 0 0
\(486\) −4.26636e26 −0.0263255
\(487\) 2.37739e28 1.43564 0.717822 0.696227i \(-0.245139\pi\)
0.717822 + 0.696227i \(0.245139\pi\)
\(488\) −1.07728e28 −0.636678
\(489\) −5.88522e27 −0.340424
\(490\) 0 0
\(491\) 3.78642e27 0.209833 0.104916 0.994481i \(-0.466543\pi\)
0.104916 + 0.994481i \(0.466543\pi\)
\(492\) 1.04080e27 0.0564593
\(493\) −4.08105e26 −0.0216710
\(494\) −7.11476e26 −0.0369851
\(495\) 0 0
\(496\) −6.14980e27 −0.306410
\(497\) −3.58083e28 −1.74680
\(498\) 1.81458e27 0.0866700
\(499\) −1.98246e28 −0.927149 −0.463575 0.886058i \(-0.653433\pi\)
−0.463575 + 0.886058i \(0.653433\pi\)
\(500\) 0 0
\(501\) −3.70147e25 −0.00165989
\(502\) −7.38806e27 −0.324445
\(503\) −1.60989e28 −0.692361 −0.346180 0.938168i \(-0.612521\pi\)
−0.346180 + 0.938168i \(0.612521\pi\)
\(504\) 7.55628e27 0.318264
\(505\) 0 0
\(506\) −7.32576e27 −0.295987
\(507\) 1.37521e28 0.544234
\(508\) 1.97883e28 0.767079
\(509\) −1.05342e28 −0.400004 −0.200002 0.979795i \(-0.564095\pi\)
−0.200002 + 0.979795i \(0.564095\pi\)
\(510\) 0 0
\(511\) −2.31207e28 −0.842523
\(512\) 2.51748e28 0.898733
\(513\) −2.07061e27 −0.0724214
\(514\) 1.33087e28 0.456059
\(515\) 0 0
\(516\) 6.26761e27 0.206196
\(517\) 1.84357e28 0.594306
\(518\) −2.64042e28 −0.834085
\(519\) −3.37248e28 −1.04398
\(520\) 0 0
\(521\) −2.68944e28 −0.799589 −0.399794 0.916605i \(-0.630918\pi\)
−0.399794 + 0.916605i \(0.630918\pi\)
\(522\) 1.10319e27 0.0321448
\(523\) −2.49861e28 −0.713561 −0.356781 0.934188i \(-0.616126\pi\)
−0.356781 + 0.934188i \(0.616126\pi\)
\(524\) −5.57765e28 −1.56125
\(525\) 0 0
\(526\) −1.22030e28 −0.328184
\(527\) 2.04892e27 0.0540149
\(528\) 1.21666e28 0.314422
\(529\) −2.05239e28 −0.519967
\(530\) 0 0
\(531\) −1.07894e28 −0.262726
\(532\) 1.66506e28 0.397520
\(533\) −1.20315e27 −0.0281633
\(534\) 1.50729e28 0.345950
\(535\) 0 0
\(536\) −5.08004e28 −1.12108
\(537\) 3.32395e28 0.719324
\(538\) −2.58901e28 −0.549440
\(539\) −3.06925e28 −0.638779
\(540\) 0 0
\(541\) 6.35823e28 1.27281 0.636407 0.771353i \(-0.280420\pi\)
0.636407 + 0.771353i \(0.280420\pi\)
\(542\) 2.12751e28 0.417713
\(543\) −3.54441e28 −0.682569
\(544\) −4.71778e27 −0.0891147
\(545\) 0 0
\(546\) −3.96589e27 −0.0720805
\(547\) 1.45910e28 0.260146 0.130073 0.991504i \(-0.458479\pi\)
0.130073 + 0.991504i \(0.458479\pi\)
\(548\) 2.96749e28 0.519031
\(549\) 1.64550e28 0.282351
\(550\) 0 0
\(551\) 5.35417e27 0.0884302
\(552\) 1.85543e28 0.300666
\(553\) −7.98514e28 −1.26960
\(554\) 2.70256e28 0.421621
\(555\) 0 0
\(556\) 3.90490e28 0.586577
\(557\) −1.08130e29 −1.59392 −0.796958 0.604035i \(-0.793559\pi\)
−0.796958 + 0.604035i \(0.793559\pi\)
\(558\) −5.53865e27 −0.0801207
\(559\) −7.24523e27 −0.102856
\(560\) 0 0
\(561\) −4.05353e27 −0.0554273
\(562\) −2.17232e27 −0.0291536
\(563\) −9.29087e28 −1.22382 −0.611911 0.790927i \(-0.709599\pi\)
−0.611911 + 0.790927i \(0.709599\pi\)
\(564\) −2.11999e28 −0.274097
\(565\) 0 0
\(566\) 1.94247e28 0.241981
\(567\) −1.15420e28 −0.141142
\(568\) −8.61012e28 −1.03360
\(569\) −1.56535e29 −1.84473 −0.922364 0.386322i \(-0.873745\pi\)
−0.922364 + 0.386322i \(0.873745\pi\)
\(570\) 0 0
\(571\) 1.80389e28 0.204895 0.102447 0.994738i \(-0.467333\pi\)
0.102447 + 0.994738i \(0.467333\pi\)
\(572\) −1.85919e28 −0.207332
\(573\) −1.90784e28 −0.208890
\(574\) −5.70217e27 −0.0613007
\(575\) 0 0
\(576\) −4.07099e27 −0.0421953
\(577\) −2.28665e28 −0.232731 −0.116365 0.993206i \(-0.537124\pi\)
−0.116365 + 0.993206i \(0.537124\pi\)
\(578\) −4.07111e28 −0.406884
\(579\) 4.43019e28 0.434808
\(580\) 0 0
\(581\) 4.90906e28 0.464674
\(582\) 4.04483e28 0.376019
\(583\) 1.04535e29 0.954430
\(584\) −5.55938e28 −0.498530
\(585\) 0 0
\(586\) −5.83444e28 −0.504748
\(587\) 6.12097e28 0.520140 0.260070 0.965590i \(-0.416254\pi\)
0.260070 + 0.965590i \(0.416254\pi\)
\(588\) 3.52945e28 0.294608
\(589\) −2.68810e28 −0.220412
\(590\) 0 0
\(591\) −1.18058e29 −0.934173
\(592\) 1.07678e29 0.837043
\(593\) 1.13456e29 0.866471 0.433235 0.901281i \(-0.357372\pi\)
0.433235 + 0.901281i \(0.357372\pi\)
\(594\) 1.09575e28 0.0822157
\(595\) 0 0
\(596\) 1.57920e29 1.14380
\(597\) −8.72429e28 −0.620868
\(598\) −9.73815e27 −0.0680948
\(599\) −4.73524e28 −0.325357 −0.162679 0.986679i \(-0.552013\pi\)
−0.162679 + 0.986679i \(0.552013\pi\)
\(600\) 0 0
\(601\) −3.55276e28 −0.235713 −0.117856 0.993031i \(-0.537602\pi\)
−0.117856 + 0.993031i \(0.537602\pi\)
\(602\) −3.43379e28 −0.223877
\(603\) 7.75958e28 0.497172
\(604\) 1.10836e29 0.697901
\(605\) 0 0
\(606\) −5.57509e28 −0.339070
\(607\) 2.81288e29 1.68140 0.840698 0.541504i \(-0.182145\pi\)
0.840698 + 0.541504i \(0.182145\pi\)
\(608\) 6.18953e28 0.363639
\(609\) 2.98451e28 0.172342
\(610\) 0 0
\(611\) 2.45067e28 0.136726
\(612\) 4.66130e27 0.0255633
\(613\) 2.09223e29 1.12791 0.563954 0.825806i \(-0.309280\pi\)
0.563954 + 0.825806i \(0.309280\pi\)
\(614\) 9.03339e28 0.478721
\(615\) 0 0
\(616\) −1.94072e29 −0.993952
\(617\) −2.27935e29 −1.14767 −0.573835 0.818971i \(-0.694545\pi\)
−0.573835 + 0.818971i \(0.694545\pi\)
\(618\) −5.27739e28 −0.261241
\(619\) 1.54704e29 0.752919 0.376460 0.926433i \(-0.377141\pi\)
0.376460 + 0.926433i \(0.377141\pi\)
\(620\) 0 0
\(621\) −2.83410e28 −0.133338
\(622\) −1.01481e29 −0.469445
\(623\) 4.07772e29 1.85478
\(624\) 1.61731e28 0.0723361
\(625\) 0 0
\(626\) 9.20939e28 0.398291
\(627\) 5.31806e28 0.226175
\(628\) −9.28124e28 −0.388177
\(629\) −3.58748e28 −0.147556
\(630\) 0 0
\(631\) −3.19204e28 −0.126987 −0.0634935 0.997982i \(-0.520224\pi\)
−0.0634935 + 0.997982i \(0.520224\pi\)
\(632\) −1.92003e29 −0.751239
\(633\) 2.32528e29 0.894823
\(634\) 9.64782e28 0.365167
\(635\) 0 0
\(636\) −1.20209e29 −0.440187
\(637\) −4.07997e28 −0.146958
\(638\) −2.83339e28 −0.100390
\(639\) 1.31516e29 0.458375
\(640\) 0 0
\(641\) 5.35140e29 1.80492 0.902461 0.430772i \(-0.141759\pi\)
0.902461 + 0.430772i \(0.141759\pi\)
\(642\) −8.55485e28 −0.283854
\(643\) −9.30767e28 −0.303827 −0.151913 0.988394i \(-0.548543\pi\)
−0.151913 + 0.988394i \(0.548543\pi\)
\(644\) 2.27902e29 0.731890
\(645\) 0 0
\(646\) −4.58140e27 −0.0142415
\(647\) −5.11942e28 −0.156576 −0.0782881 0.996931i \(-0.524945\pi\)
−0.0782881 + 0.996931i \(0.524945\pi\)
\(648\) −2.77526e28 −0.0835154
\(649\) 2.77108e29 0.820503
\(650\) 0 0
\(651\) −1.49839e29 −0.429561
\(652\) −1.73816e29 −0.490333
\(653\) −6.68051e29 −1.85448 −0.927238 0.374473i \(-0.877824\pi\)
−0.927238 + 0.374473i \(0.877824\pi\)
\(654\) 6.44513e28 0.176062
\(655\) 0 0
\(656\) 2.32538e28 0.0615181
\(657\) 8.49175e28 0.221086
\(658\) 1.16146e29 0.297600
\(659\) −5.22900e28 −0.131863 −0.0659313 0.997824i \(-0.521002\pi\)
−0.0659313 + 0.997824i \(0.521002\pi\)
\(660\) 0 0
\(661\) 4.62032e29 1.12864 0.564322 0.825555i \(-0.309138\pi\)
0.564322 + 0.825555i \(0.309138\pi\)
\(662\) −1.81776e29 −0.437047
\(663\) −5.38837e27 −0.0127516
\(664\) 1.18038e29 0.274953
\(665\) 0 0
\(666\) 9.69770e28 0.218872
\(667\) 7.32839e28 0.162812
\(668\) −1.09321e27 −0.00239084
\(669\) 2.39971e29 0.516635
\(670\) 0 0
\(671\) −4.22623e29 −0.881794
\(672\) 3.45015e29 0.708697
\(673\) 3.53790e29 0.715463 0.357731 0.933825i \(-0.383550\pi\)
0.357731 + 0.933825i \(0.383550\pi\)
\(674\) 1.26862e28 0.0252581
\(675\) 0 0
\(676\) 4.06160e29 0.783894
\(677\) −5.61821e29 −1.06762 −0.533811 0.845604i \(-0.679240\pi\)
−0.533811 + 0.845604i \(0.679240\pi\)
\(678\) 1.00945e29 0.188875
\(679\) 1.09426e30 2.01599
\(680\) 0 0
\(681\) −1.89983e29 −0.339367
\(682\) 1.42252e29 0.250220
\(683\) 3.41615e29 0.591725 0.295862 0.955231i \(-0.404393\pi\)
0.295862 + 0.955231i \(0.404393\pi\)
\(684\) −6.11543e28 −0.104313
\(685\) 0 0
\(686\) 1.21760e29 0.201420
\(687\) 6.89667e29 1.12355
\(688\) 1.40032e29 0.224671
\(689\) 1.38959e29 0.219577
\(690\) 0 0
\(691\) 8.73790e28 0.133933 0.0669665 0.997755i \(-0.478668\pi\)
0.0669665 + 0.997755i \(0.478668\pi\)
\(692\) −9.96041e29 −1.50371
\(693\) 2.96438e29 0.440793
\(694\) −1.65569e28 −0.0242496
\(695\) 0 0
\(696\) 7.17625e28 0.101977
\(697\) −7.74743e27 −0.0108446
\(698\) −2.77599e29 −0.382768
\(699\) 4.54380e29 0.617176
\(700\) 0 0
\(701\) −8.83377e28 −0.116441 −0.0582206 0.998304i \(-0.518543\pi\)
−0.0582206 + 0.998304i \(0.518543\pi\)
\(702\) 1.45659e28 0.0189146
\(703\) 4.70663e29 0.602114
\(704\) 1.04557e29 0.131777
\(705\) 0 0
\(706\) −3.34369e28 −0.0409051
\(707\) −1.50825e30 −1.81790
\(708\) −3.18657e29 −0.378420
\(709\) 7.67334e29 0.897840 0.448920 0.893572i \(-0.351809\pi\)
0.448920 + 0.893572i \(0.351809\pi\)
\(710\) 0 0
\(711\) 2.93277e29 0.333156
\(712\) 9.80489e29 1.09750
\(713\) −3.67927e29 −0.405809
\(714\) −2.55375e28 −0.0277554
\(715\) 0 0
\(716\) 9.81708e29 1.03609
\(717\) 2.18635e29 0.227389
\(718\) 5.80354e29 0.594821
\(719\) −1.26076e30 −1.27344 −0.636722 0.771094i \(-0.719710\pi\)
−0.636722 + 0.771094i \(0.719710\pi\)
\(720\) 0 0
\(721\) −1.42771e30 −1.40062
\(722\) −3.64338e29 −0.352261
\(723\) 6.62559e29 0.631353
\(724\) −1.04682e30 −0.983145
\(725\) 0 0
\(726\) −2.17384e28 −0.0198332
\(727\) −5.58424e29 −0.502173 −0.251086 0.967965i \(-0.580788\pi\)
−0.251086 + 0.967965i \(0.580788\pi\)
\(728\) −2.57980e29 −0.228669
\(729\) 4.23912e28 0.0370370
\(730\) 0 0
\(731\) −4.66541e28 −0.0396057
\(732\) 4.85990e29 0.406687
\(733\) −4.20539e29 −0.346908 −0.173454 0.984842i \(-0.555493\pi\)
−0.173454 + 0.984842i \(0.555493\pi\)
\(734\) −5.92890e28 −0.0482132
\(735\) 0 0
\(736\) 8.47177e29 0.669510
\(737\) −1.99293e30 −1.55269
\(738\) 2.09429e28 0.0160859
\(739\) −3.26857e29 −0.247509 −0.123755 0.992313i \(-0.539494\pi\)
−0.123755 + 0.992313i \(0.539494\pi\)
\(740\) 0 0
\(741\) 7.06932e28 0.0520339
\(742\) 6.58580e29 0.477934
\(743\) −1.86969e30 −1.33779 −0.668895 0.743357i \(-0.733232\pi\)
−0.668895 + 0.743357i \(0.733232\pi\)
\(744\) −3.60288e29 −0.254176
\(745\) 0 0
\(746\) 3.49463e29 0.239687
\(747\) −1.80299e29 −0.121935
\(748\) −1.19719e29 −0.0798353
\(749\) −2.31438e30 −1.52186
\(750\) 0 0
\(751\) −2.50654e30 −1.60271 −0.801355 0.598189i \(-0.795887\pi\)
−0.801355 + 0.598189i \(0.795887\pi\)
\(752\) −4.73651e29 −0.298656
\(753\) 7.34088e29 0.456458
\(754\) −3.76643e28 −0.0230957
\(755\) 0 0
\(756\) −3.40885e29 −0.203296
\(757\) 8.92584e29 0.524979 0.262490 0.964935i \(-0.415456\pi\)
0.262490 + 0.964935i \(0.415456\pi\)
\(758\) 1.27810e30 0.741374
\(759\) 7.27897e29 0.416420
\(760\) 0 0
\(761\) 1.97087e30 1.09678 0.548391 0.836222i \(-0.315241\pi\)
0.548391 + 0.836222i \(0.315241\pi\)
\(762\) 3.98177e29 0.218549
\(763\) 1.74362e30 0.943942
\(764\) −5.63469e29 −0.300878
\(765\) 0 0
\(766\) 1.17766e30 0.611811
\(767\) 3.68361e29 0.188765
\(768\) 3.32699e29 0.168174
\(769\) −2.71675e30 −1.35464 −0.677319 0.735689i \(-0.736858\pi\)
−0.677319 + 0.735689i \(0.736858\pi\)
\(770\) 0 0
\(771\) −1.32237e30 −0.641624
\(772\) 1.30843e30 0.626281
\(773\) −2.27866e30 −1.07596 −0.537978 0.842959i \(-0.680812\pi\)
−0.537978 + 0.842959i \(0.680812\pi\)
\(774\) 1.26116e29 0.0587475
\(775\) 0 0
\(776\) 2.63116e30 1.19289
\(777\) 2.62355e30 1.17346
\(778\) 2.62839e29 0.115986
\(779\) 1.01643e29 0.0442521
\(780\) 0 0
\(781\) −3.37780e30 −1.43152
\(782\) −6.27067e28 −0.0262206
\(783\) −1.09615e29 −0.0452241
\(784\) 7.88553e29 0.321005
\(785\) 0 0
\(786\) −1.12232e30 −0.444817
\(787\) 2.91885e30 1.14150 0.570751 0.821123i \(-0.306652\pi\)
0.570751 + 0.821123i \(0.306652\pi\)
\(788\) −3.48678e30 −1.34555
\(789\) 1.21251e30 0.461718
\(790\) 0 0
\(791\) 2.73092e30 1.01264
\(792\) 7.12785e29 0.260822
\(793\) −5.61795e29 −0.202866
\(794\) −1.09870e30 −0.391530
\(795\) 0 0
\(796\) −2.57667e30 −0.894275
\(797\) 2.36129e30 0.808791 0.404396 0.914584i \(-0.367482\pi\)
0.404396 + 0.914584i \(0.367482\pi\)
\(798\) 3.35041e29 0.113258
\(799\) 1.57806e29 0.0526479
\(800\) 0 0
\(801\) −1.49766e30 −0.486712
\(802\) 3.52308e29 0.113003
\(803\) −2.18098e30 −0.690460
\(804\) 2.29175e30 0.716107
\(805\) 0 0
\(806\) 1.89096e29 0.0575658
\(807\) 2.57248e30 0.772999
\(808\) −3.62658e30 −1.07567
\(809\) −5.23291e30 −1.53209 −0.766045 0.642787i \(-0.777778\pi\)
−0.766045 + 0.642787i \(0.777778\pi\)
\(810\) 0 0
\(811\) −2.58712e30 −0.738070 −0.369035 0.929416i \(-0.620312\pi\)
−0.369035 + 0.929416i \(0.620312\pi\)
\(812\) 8.81456e29 0.248235
\(813\) −2.11392e30 −0.587675
\(814\) −2.49071e30 −0.683545
\(815\) 0 0
\(816\) 1.04143e29 0.0278538
\(817\) 6.12083e29 0.161614
\(818\) 6.47316e28 0.0168736
\(819\) 3.94056e29 0.101409
\(820\) 0 0
\(821\) −1.25380e30 −0.314503 −0.157252 0.987559i \(-0.550263\pi\)
−0.157252 + 0.987559i \(0.550263\pi\)
\(822\) 5.97112e29 0.147877
\(823\) −2.38455e30 −0.583053 −0.291526 0.956563i \(-0.594163\pi\)
−0.291526 + 0.956563i \(0.594163\pi\)
\(824\) −3.43294e30 −0.828763
\(825\) 0 0
\(826\) 1.74580e30 0.410870
\(827\) −7.49200e30 −1.74096 −0.870482 0.492201i \(-0.836193\pi\)
−0.870482 + 0.492201i \(0.836193\pi\)
\(828\) −8.37035e29 −0.192055
\(829\) 4.27738e30 0.969070 0.484535 0.874772i \(-0.338989\pi\)
0.484535 + 0.874772i \(0.338989\pi\)
\(830\) 0 0
\(831\) −2.68530e30 −0.593173
\(832\) 1.38988e29 0.0303168
\(833\) −2.62721e29 −0.0565877
\(834\) 7.85737e29 0.167122
\(835\) 0 0
\(836\) 1.57066e30 0.325773
\(837\) 5.50328e29 0.112721
\(838\) 2.08045e30 0.420818
\(839\) −6.06783e30 −1.21208 −0.606042 0.795432i \(-0.707244\pi\)
−0.606042 + 0.795432i \(0.707244\pi\)
\(840\) 0 0
\(841\) −4.84940e30 −0.944779
\(842\) −1.44825e30 −0.278655
\(843\) 2.15844e29 0.0410158
\(844\) 6.86759e30 1.28887
\(845\) 0 0
\(846\) −4.26581e29 −0.0780931
\(847\) −5.88096e29 −0.106334
\(848\) −2.68572e30 −0.479629
\(849\) −1.93006e30 −0.340440
\(850\) 0 0
\(851\) 6.44208e30 1.10858
\(852\) 3.88426e30 0.660226
\(853\) 7.56989e30 1.27094 0.635470 0.772126i \(-0.280806\pi\)
0.635470 + 0.772126i \(0.280806\pi\)
\(854\) −2.66255e30 −0.441561
\(855\) 0 0
\(856\) −5.56492e30 −0.900501
\(857\) −7.92913e30 −1.26744 −0.633720 0.773563i \(-0.718473\pi\)
−0.633720 + 0.773563i \(0.718473\pi\)
\(858\) −3.74103e29 −0.0590710
\(859\) −8.04328e30 −1.25460 −0.627299 0.778778i \(-0.715840\pi\)
−0.627299 + 0.778778i \(0.715840\pi\)
\(860\) 0 0
\(861\) 5.66575e29 0.0862432
\(862\) 2.51178e30 0.377707
\(863\) 6.93106e30 1.02964 0.514821 0.857298i \(-0.327858\pi\)
0.514821 + 0.857298i \(0.327858\pi\)
\(864\) −1.26717e30 −0.185969
\(865\) 0 0
\(866\) −2.95514e29 −0.0423292
\(867\) 4.04511e30 0.572440
\(868\) −4.42541e30 −0.618723
\(869\) −7.53239e30 −1.04046
\(870\) 0 0
\(871\) −2.64921e30 −0.357212
\(872\) 4.19255e30 0.558541
\(873\) −4.01900e30 −0.529015
\(874\) 8.22686e29 0.106995
\(875\) 0 0
\(876\) 2.50799e30 0.318443
\(877\) 1.42323e31 1.78558 0.892792 0.450470i \(-0.148744\pi\)
0.892792 + 0.450470i \(0.148744\pi\)
\(878\) 3.37872e30 0.418851
\(879\) 5.79718e30 0.710123
\(880\) 0 0
\(881\) −1.28748e31 −1.53991 −0.769954 0.638100i \(-0.779721\pi\)
−0.769954 + 0.638100i \(0.779721\pi\)
\(882\) 7.10189e29 0.0839370
\(883\) 7.16145e29 0.0836400 0.0418200 0.999125i \(-0.486684\pi\)
0.0418200 + 0.999125i \(0.486684\pi\)
\(884\) −1.59142e29 −0.0183669
\(885\) 0 0
\(886\) 3.84594e30 0.433459
\(887\) −1.69060e31 −1.88297 −0.941484 0.337057i \(-0.890568\pi\)
−0.941484 + 0.337057i \(0.890568\pi\)
\(888\) 6.30834e30 0.694351
\(889\) 1.07720e31 1.17173
\(890\) 0 0
\(891\) −1.08875e30 −0.115668
\(892\) 7.08739e30 0.744141
\(893\) −2.07034e30 −0.214834
\(894\) 3.17763e30 0.325881
\(895\) 0 0
\(896\) 1.29121e31 1.29349
\(897\) 9.67596e29 0.0958017
\(898\) 3.80921e30 0.372764
\(899\) −1.42303e30 −0.137638
\(900\) 0 0
\(901\) 8.94799e29 0.0845503
\(902\) −5.37887e29 −0.0502368
\(903\) 3.41185e30 0.314970
\(904\) 6.56649e30 0.599190
\(905\) 0 0
\(906\) 2.23022e30 0.198839
\(907\) −1.73394e31 −1.52812 −0.764058 0.645147i \(-0.776796\pi\)
−0.764058 + 0.645147i \(0.776796\pi\)
\(908\) −5.61103e30 −0.488811
\(909\) 5.53948e30 0.477033
\(910\) 0 0
\(911\) 1.43753e31 1.20969 0.604844 0.796344i \(-0.293235\pi\)
0.604844 + 0.796344i \(0.293235\pi\)
\(912\) −1.36632e30 −0.113659
\(913\) 4.63072e30 0.380807
\(914\) −5.67881e29 −0.0461660
\(915\) 0 0
\(916\) 2.03689e31 1.61832
\(917\) −3.03627e31 −2.38485
\(918\) 9.37938e28 0.00728326
\(919\) −1.44583e31 −1.10995 −0.554976 0.831867i \(-0.687272\pi\)
−0.554976 + 0.831867i \(0.687272\pi\)
\(920\) 0 0
\(921\) −8.97569e30 −0.673506
\(922\) 6.34418e30 0.470653
\(923\) −4.49013e30 −0.329337
\(924\) 8.75512e30 0.634901
\(925\) 0 0
\(926\) 6.50361e30 0.461041
\(927\) 5.24369e30 0.367536
\(928\) 3.27663e30 0.227077
\(929\) 2.70280e31 1.85204 0.926018 0.377479i \(-0.123209\pi\)
0.926018 + 0.377479i \(0.123209\pi\)
\(930\) 0 0
\(931\) 3.44679e30 0.230910
\(932\) 1.34199e31 0.888957
\(933\) 1.00833e31 0.660455
\(934\) 8.61271e30 0.557824
\(935\) 0 0
\(936\) 9.47508e29 0.0600049
\(937\) −1.56150e31 −0.977861 −0.488930 0.872323i \(-0.662613\pi\)
−0.488930 + 0.872323i \(0.662613\pi\)
\(938\) −1.25556e31 −0.777513
\(939\) −9.15057e30 −0.560350
\(940\) 0 0
\(941\) 2.18700e31 1.30966 0.654830 0.755776i \(-0.272740\pi\)
0.654830 + 0.755776i \(0.272740\pi\)
\(942\) −1.86755e30 −0.110596
\(943\) 1.39121e30 0.0814744
\(944\) −7.11947e30 −0.412327
\(945\) 0 0
\(946\) −3.23909e30 −0.183471
\(947\) −1.90573e31 −1.06755 −0.533773 0.845628i \(-0.679226\pi\)
−0.533773 + 0.845628i \(0.679226\pi\)
\(948\) 8.66177e30 0.479865
\(949\) −2.89918e30 −0.158848
\(950\) 0 0
\(951\) −9.58620e30 −0.513749
\(952\) −1.66121e30 −0.0880514
\(953\) 5.74012e30 0.300916 0.150458 0.988616i \(-0.451925\pi\)
0.150458 + 0.988616i \(0.451925\pi\)
\(954\) −2.41883e30 −0.125414
\(955\) 0 0
\(956\) 6.45726e30 0.327522
\(957\) 2.81529e30 0.141237
\(958\) −2.06120e30 −0.102278
\(959\) 1.61539e31 0.792834
\(960\) 0 0
\(961\) −1.36811e31 −0.656939
\(962\) −3.31091e30 −0.157257
\(963\) 8.50022e30 0.399350
\(964\) 1.95683e31 0.909377
\(965\) 0 0
\(966\) 4.58579e30 0.208523
\(967\) 7.45029e30 0.335116 0.167558 0.985862i \(-0.446412\pi\)
0.167558 + 0.985862i \(0.446412\pi\)
\(968\) −1.41408e30 −0.0629190
\(969\) 4.55214e29 0.0200362
\(970\) 0 0
\(971\) −4.22388e31 −1.81932 −0.909661 0.415352i \(-0.863658\pi\)
−0.909661 + 0.415352i \(0.863658\pi\)
\(972\) 1.25200e30 0.0533467
\(973\) 2.12568e31 0.896012
\(974\) 1.41285e31 0.589152
\(975\) 0 0
\(976\) 1.08580e31 0.443127
\(977\) 9.32935e30 0.376668 0.188334 0.982105i \(-0.439691\pi\)
0.188334 + 0.982105i \(0.439691\pi\)
\(978\) −3.49750e30 −0.139701
\(979\) 3.84652e31 1.52002
\(980\) 0 0
\(981\) −6.40396e30 −0.247699
\(982\) 2.25022e30 0.0861100
\(983\) −7.57232e30 −0.286693 −0.143346 0.989673i \(-0.545786\pi\)
−0.143346 + 0.989673i \(0.545786\pi\)
\(984\) 1.36233e30 0.0510310
\(985\) 0 0
\(986\) −2.42531e29 −0.00889324
\(987\) −1.15405e31 −0.418690
\(988\) 2.08788e30 0.0749476
\(989\) 8.37773e30 0.297554
\(990\) 0 0
\(991\) 8.84327e30 0.307496 0.153748 0.988110i \(-0.450866\pi\)
0.153748 + 0.988110i \(0.450866\pi\)
\(992\) −1.64505e31 −0.565989
\(993\) 1.80615e31 0.614876
\(994\) −2.12804e31 −0.716840
\(995\) 0 0
\(996\) −5.32503e30 −0.175630
\(997\) 3.16102e30 0.103164 0.0515820 0.998669i \(-0.483574\pi\)
0.0515820 + 0.998669i \(0.483574\pi\)
\(998\) −1.17815e31 −0.380478
\(999\) −9.63576e30 −0.307927
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.1.4 yes 6
5.2 odd 4 75.22.b.i.49.8 12
5.3 odd 4 75.22.b.i.49.5 12
5.4 even 2 75.22.a.i.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.22.a.i.1.3 6 5.4 even 2
75.22.a.j.1.4 yes 6 1.1 even 1 trivial
75.22.b.i.49.5 12 5.3 odd 4
75.22.b.i.49.8 12 5.2 odd 4