Properties

Label 23.10.a.a.1.5
Level $23$
Weight $10$
Character 23.1
Self dual yes
Analytic conductor $11.846$
Analytic rank $1$
Dimension $7$
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] = [23,10,Mod(1,23)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(23, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("23.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 23.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: \(11.8458242318\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 640x^{5} - 1455x^{4} + 114552x^{3} + 321544x^{2} - 5741296x - 13379024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(8.70295\) of defining polynomial
Character \(\chi\) \(=\) 23.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+17.4059 q^{2} +86.8015 q^{3} -209.035 q^{4} -1221.93 q^{5} +1510.86 q^{6} -4757.43 q^{7} -12550.3 q^{8} -12148.5 q^{9} +O(q^{10})\) \(q+17.4059 q^{2} +86.8015 q^{3} -209.035 q^{4} -1221.93 q^{5} +1510.86 q^{6} -4757.43 q^{7} -12550.3 q^{8} -12148.5 q^{9} -21268.8 q^{10} +27321.2 q^{11} -18144.5 q^{12} +38402.5 q^{13} -82807.4 q^{14} -106066. q^{15} -111423. q^{16} -179820. q^{17} -211455. q^{18} -248560. q^{19} +255426. q^{20} -412952. q^{21} +475550. q^{22} -279841. q^{23} -1.08938e6 q^{24} -460009. q^{25} +668429. q^{26} -2.76302e6 q^{27} +994469. q^{28} +1.71960e6 q^{29} -1.84617e6 q^{30} +2.85950e6 q^{31} +4.48632e6 q^{32} +2.37152e6 q^{33} -3.12993e6 q^{34} +5.81325e6 q^{35} +2.53946e6 q^{36} +3.16336e6 q^{37} -4.32641e6 q^{38} +3.33339e6 q^{39} +1.53356e7 q^{40} +2.07242e7 q^{41} -7.18781e6 q^{42} +1.29160e7 q^{43} -5.71108e6 q^{44} +1.48446e7 q^{45} -4.87088e6 q^{46} -5.01135e7 q^{47} -9.67166e6 q^{48} -1.77205e7 q^{49} -8.00686e6 q^{50} -1.56087e7 q^{51} -8.02745e6 q^{52} -9.57964e7 q^{53} -4.80929e7 q^{54} -3.33846e7 q^{55} +5.97070e7 q^{56} -2.15754e7 q^{57} +2.99311e7 q^{58} +2.30378e6 q^{59} +2.21714e7 q^{60} +3.45895e7 q^{61} +4.97721e7 q^{62} +5.77956e7 q^{63} +1.35137e8 q^{64} -4.69252e7 q^{65} +4.12785e7 q^{66} -2.87322e8 q^{67} +3.75887e7 q^{68} -2.42906e7 q^{69} +1.01185e8 q^{70} -6.74618e7 q^{71} +1.52467e8 q^{72} +2.71643e8 q^{73} +5.50610e7 q^{74} -3.99295e7 q^{75} +5.19577e7 q^{76} -1.29979e8 q^{77} +5.80207e7 q^{78} -8.11254e7 q^{79} +1.36151e8 q^{80} -715836. q^{81} +3.60723e8 q^{82} -7.39059e8 q^{83} +8.63214e7 q^{84} +2.19728e8 q^{85} +2.24815e8 q^{86} +1.49264e8 q^{87} -3.42888e8 q^{88} +9.38670e6 q^{89} +2.58384e8 q^{90} -1.82697e8 q^{91} +5.84965e7 q^{92} +2.48209e8 q^{93} -8.72271e8 q^{94} +3.03723e8 q^{95} +3.89420e8 q^{96} -8.96236e7 q^{97} -3.08440e8 q^{98} -3.31911e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 89 q^{3} + 1536 q^{4} - 2388 q^{5} - 12518 q^{6} - 9896 q^{7} + 34920 q^{8} + 33064 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 89 q^{3} + 1536 q^{4} - 2388 q^{5} - 12518 q^{6} - 9896 q^{7} + 34920 q^{8} + 33064 q^{9} - 60820 q^{10} - 78484 q^{11} - 343492 q^{12} - 296769 q^{13} - 711120 q^{14} - 237870 q^{15} - 253440 q^{16} - 1128820 q^{17} - 499874 q^{18} - 1301252 q^{19} - 3482704 q^{20} - 108908 q^{21} - 1562088 q^{22} - 1958887 q^{23} - 4606464 q^{24} - 1320899 q^{25} + 692230 q^{26} + 2977921 q^{27} - 8371144 q^{28} + 2813849 q^{29} + 25535196 q^{30} + 7334751 q^{31} + 26028800 q^{32} + 646330 q^{33} + 14981564 q^{34} + 23410104 q^{35} + 40211900 q^{36} - 13324320 q^{37} + 37578632 q^{38} + 6304533 q^{39} - 45307920 q^{40} - 15691573 q^{41} + 124523248 q^{42} - 46474818 q^{43} + 43428040 q^{44} - 72736710 q^{45} + 8232227 q^{47} - 163054384 q^{48} + 29219031 q^{49} + 50366304 q^{50} - 136344764 q^{51} - 100922292 q^{52} - 53545400 q^{53} - 26171642 q^{54} - 181608484 q^{55} - 420111696 q^{56} - 218913370 q^{57} - 39304854 q^{58} - 341275144 q^{59} + 420822384 q^{60} - 277157656 q^{61} + 464777594 q^{62} - 574619276 q^{63} + 340566208 q^{64} + 106659278 q^{65} + 258025876 q^{66} + 89654580 q^{67} + 62700400 q^{68} + 24905849 q^{69} + 1187910040 q^{70} - 286098961 q^{71} + 1446323640 q^{72} - 637495039 q^{73} + 189880036 q^{74} - 160733159 q^{75} + 228563936 q^{76} + 511682536 q^{77} + 1199383686 q^{78} + 274469546 q^{79} - 345318560 q^{80} - 237775217 q^{81} - 570256066 q^{82} + 1164579762 q^{83} + 3447171416 q^{84} - 18639492 q^{85} + 415245796 q^{86} - 595368433 q^{87} + 103329440 q^{88} - 504153000 q^{89} - 1414126968 q^{90} - 1692320156 q^{91} - 429835776 q^{92} - 2753858687 q^{93} - 2214048622 q^{94} + 162962164 q^{95} - 3332565856 q^{96} - 3519929016 q^{97} + 2474592568 q^{98} - 1883749262 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 17.4059 0.769239 0.384620 0.923075i \(-0.374333\pi\)
0.384620 + 0.923075i \(0.374333\pi\)
\(3\) 86.8015 0.618702 0.309351 0.950948i \(-0.399888\pi\)
0.309351 + 0.950948i \(0.399888\pi\)
\(4\) −209.035 −0.408271
\(5\) −1221.93 −0.874343 −0.437171 0.899378i \(-0.644020\pi\)
−0.437171 + 0.899378i \(0.644020\pi\)
\(6\) 1510.86 0.475930
\(7\) −4757.43 −0.748913 −0.374456 0.927244i \(-0.622171\pi\)
−0.374456 + 0.927244i \(0.622171\pi\)
\(8\) −12550.3 −1.08330
\(9\) −12148.5 −0.617207
\(10\) −21268.8 −0.672579
\(11\) 27321.2 0.562643 0.281321 0.959614i \(-0.409227\pi\)
0.281321 + 0.959614i \(0.409227\pi\)
\(12\) −18144.5 −0.252598
\(13\) 38402.5 0.372919 0.186459 0.982463i \(-0.440299\pi\)
0.186459 + 0.982463i \(0.440299\pi\)
\(14\) −82807.4 −0.576093
\(15\) −106066. −0.540958
\(16\) −111423. −0.425044
\(17\) −179820. −0.522178 −0.261089 0.965315i \(-0.584082\pi\)
−0.261089 + 0.965315i \(0.584082\pi\)
\(18\) −211455. −0.474780
\(19\) −248560. −0.437563 −0.218781 0.975774i \(-0.570208\pi\)
−0.218781 + 0.975774i \(0.570208\pi\)
\(20\) 255426. 0.356969
\(21\) −412952. −0.463354
\(22\) 475550. 0.432807
\(23\) −279841. −0.208514
\(24\) −1.08938e6 −0.670239
\(25\) −460009. −0.235524
\(26\) 668429. 0.286864
\(27\) −2.76302e6 −1.00057
\(28\) 994469. 0.305760
\(29\) 1.71960e6 0.451477 0.225739 0.974188i \(-0.427520\pi\)
0.225739 + 0.974188i \(0.427520\pi\)
\(30\) −1.84617e6 −0.416126
\(31\) 2.85950e6 0.556112 0.278056 0.960565i \(-0.410310\pi\)
0.278056 + 0.960565i \(0.410310\pi\)
\(32\) 4.48632e6 0.756337
\(33\) 2.37152e6 0.348108
\(34\) −3.12993e6 −0.401680
\(35\) 5.81325e6 0.654807
\(36\) 2.53946e6 0.251988
\(37\) 3.16336e6 0.277486 0.138743 0.990328i \(-0.455694\pi\)
0.138743 + 0.990328i \(0.455694\pi\)
\(38\) −4.32641e6 −0.336590
\(39\) 3.33339e6 0.230726
\(40\) 1.53356e7 0.947173
\(41\) 2.07242e7 1.14538 0.572691 0.819771i \(-0.305900\pi\)
0.572691 + 0.819771i \(0.305900\pi\)
\(42\) −7.18781e6 −0.356430
\(43\) 1.29160e7 0.576130 0.288065 0.957611i \(-0.406988\pi\)
0.288065 + 0.957611i \(0.406988\pi\)
\(44\) −5.71108e6 −0.229711
\(45\) 1.48446e7 0.539651
\(46\) −4.87088e6 −0.160397
\(47\) −5.01135e7 −1.49801 −0.749005 0.662565i \(-0.769468\pi\)
−0.749005 + 0.662565i \(0.769468\pi\)
\(48\) −9.67166e6 −0.262976
\(49\) −1.77205e7 −0.439129
\(50\) −8.00686e6 −0.181175
\(51\) −1.56087e7 −0.323073
\(52\) −8.02745e6 −0.152252
\(53\) −9.57964e7 −1.66766 −0.833830 0.552021i \(-0.813857\pi\)
−0.833830 + 0.552021i \(0.813857\pi\)
\(54\) −4.80929e7 −0.769678
\(55\) −3.33846e7 −0.491942
\(56\) 5.97070e7 0.811295
\(57\) −2.15754e7 −0.270721
\(58\) 2.99311e7 0.347294
\(59\) 2.30378e6 0.0247519 0.0123759 0.999923i \(-0.496061\pi\)
0.0123759 + 0.999923i \(0.496061\pi\)
\(60\) 2.21714e7 0.220858
\(61\) 3.45895e7 0.319860 0.159930 0.987128i \(-0.448873\pi\)
0.159930 + 0.987128i \(0.448873\pi\)
\(62\) 4.97721e7 0.427783
\(63\) 5.77956e7 0.462235
\(64\) 1.35137e8 1.00685
\(65\) −4.69252e7 −0.326059
\(66\) 4.12785e7 0.267779
\(67\) −2.87322e8 −1.74193 −0.870967 0.491342i \(-0.836507\pi\)
−0.870967 + 0.491342i \(0.836507\pi\)
\(68\) 3.75887e7 0.213190
\(69\) −2.42906e7 −0.129008
\(70\) 1.01185e8 0.503703
\(71\) −6.74618e7 −0.315061 −0.157531 0.987514i \(-0.550353\pi\)
−0.157531 + 0.987514i \(0.550353\pi\)
\(72\) 1.52467e8 0.668619
\(73\) 2.71643e8 1.11955 0.559777 0.828643i \(-0.310887\pi\)
0.559777 + 0.828643i \(0.310887\pi\)
\(74\) 5.50610e7 0.213453
\(75\) −3.99295e7 −0.145720
\(76\) 5.19577e7 0.178644
\(77\) −1.29979e8 −0.421370
\(78\) 5.80207e7 0.177483
\(79\) −8.11254e7 −0.234334 −0.117167 0.993112i \(-0.537381\pi\)
−0.117167 + 0.993112i \(0.537381\pi\)
\(80\) 1.36151e8 0.371634
\(81\) −715836. −0.00184770
\(82\) 3.60723e8 0.881073
\(83\) −7.39059e8 −1.70934 −0.854669 0.519173i \(-0.826240\pi\)
−0.854669 + 0.519173i \(0.826240\pi\)
\(84\) 8.63214e7 0.189174
\(85\) 2.19728e8 0.456563
\(86\) 2.24815e8 0.443182
\(87\) 1.49264e8 0.279330
\(88\) −3.42888e8 −0.609509
\(89\) 9.38670e6 0.0158583 0.00792917 0.999969i \(-0.497476\pi\)
0.00792917 + 0.999969i \(0.497476\pi\)
\(90\) 2.58384e8 0.415121
\(91\) −1.82697e8 −0.279284
\(92\) 5.84965e7 0.0851304
\(93\) 2.48209e8 0.344068
\(94\) −8.72271e8 −1.15233
\(95\) 3.03723e8 0.382580
\(96\) 3.89420e8 0.467948
\(97\) −8.96236e7 −0.102790 −0.0513948 0.998678i \(-0.516367\pi\)
−0.0513948 + 0.998678i \(0.516367\pi\)
\(98\) −3.08440e8 −0.337795
\(99\) −3.31911e8 −0.347267
\(100\) 9.61578e7 0.0961578
\(101\) −9.77148e7 −0.0934360 −0.0467180 0.998908i \(-0.514876\pi\)
−0.0467180 + 0.998908i \(0.514876\pi\)
\(102\) −2.71683e8 −0.248520
\(103\) −4.56398e8 −0.399555 −0.199777 0.979841i \(-0.564022\pi\)
−0.199777 + 0.979841i \(0.564022\pi\)
\(104\) −4.81961e8 −0.403982
\(105\) 5.04599e8 0.405131
\(106\) −1.66742e9 −1.28283
\(107\) 3.48422e8 0.256967 0.128484 0.991712i \(-0.458989\pi\)
0.128484 + 0.991712i \(0.458989\pi\)
\(108\) 5.77568e8 0.408504
\(109\) −2.27997e9 −1.54707 −0.773536 0.633752i \(-0.781514\pi\)
−0.773536 + 0.633752i \(0.781514\pi\)
\(110\) −5.81089e8 −0.378421
\(111\) 2.74584e8 0.171681
\(112\) 5.30086e8 0.318321
\(113\) 1.23565e9 0.712925 0.356463 0.934310i \(-0.383983\pi\)
0.356463 + 0.934310i \(0.383983\pi\)
\(114\) −3.75539e8 −0.208249
\(115\) 3.41946e8 0.182313
\(116\) −3.59456e8 −0.184325
\(117\) −4.66532e8 −0.230168
\(118\) 4.00994e7 0.0190401
\(119\) 8.55483e8 0.391066
\(120\) 1.33115e9 0.586018
\(121\) −1.61150e9 −0.683433
\(122\) 6.02061e8 0.246049
\(123\) 1.79889e9 0.708651
\(124\) −5.97735e8 −0.227044
\(125\) 2.94868e9 1.08027
\(126\) 1.00598e9 0.355569
\(127\) 4.71970e9 1.60990 0.804948 0.593345i \(-0.202193\pi\)
0.804948 + 0.593345i \(0.202193\pi\)
\(128\) 5.51813e7 0.0181697
\(129\) 1.12113e9 0.356453
\(130\) −8.16775e8 −0.250817
\(131\) 4.41996e8 0.131129 0.0655643 0.997848i \(-0.479115\pi\)
0.0655643 + 0.997848i \(0.479115\pi\)
\(132\) −4.95731e8 −0.142123
\(133\) 1.18251e9 0.327696
\(134\) −5.00109e9 −1.33996
\(135\) 3.37622e9 0.874841
\(136\) 2.25679e9 0.565674
\(137\) −1.47146e9 −0.356867 −0.178433 0.983952i \(-0.557103\pi\)
−0.178433 + 0.983952i \(0.557103\pi\)
\(138\) −4.22800e8 −0.0992383
\(139\) −8.12026e8 −0.184503 −0.0922515 0.995736i \(-0.529406\pi\)
−0.0922515 + 0.995736i \(0.529406\pi\)
\(140\) −1.21517e9 −0.267339
\(141\) −4.34993e9 −0.926822
\(142\) −1.17423e9 −0.242358
\(143\) 1.04920e9 0.209820
\(144\) 1.35362e9 0.262340
\(145\) −2.10123e9 −0.394746
\(146\) 4.72819e9 0.861205
\(147\) −1.53816e9 −0.271690
\(148\) −6.61251e8 −0.113289
\(149\) 7.31833e9 1.21639 0.608196 0.793787i \(-0.291893\pi\)
0.608196 + 0.793787i \(0.291893\pi\)
\(150\) −6.95008e8 −0.112093
\(151\) −7.84048e9 −1.22729 −0.613644 0.789583i \(-0.710297\pi\)
−0.613644 + 0.789583i \(0.710297\pi\)
\(152\) 3.11949e9 0.474010
\(153\) 2.18455e9 0.322292
\(154\) −2.26240e9 −0.324135
\(155\) −3.49411e9 −0.486232
\(156\) −6.96795e8 −0.0941986
\(157\) −1.00803e10 −1.32412 −0.662059 0.749452i \(-0.730317\pi\)
−0.662059 + 0.749452i \(0.730317\pi\)
\(158\) −1.41206e9 −0.180259
\(159\) −8.31527e9 −1.03179
\(160\) −5.48198e9 −0.661298
\(161\) 1.33132e9 0.156159
\(162\) −1.24598e7 −0.00142132
\(163\) −3.70861e9 −0.411497 −0.205748 0.978605i \(-0.565963\pi\)
−0.205748 + 0.978605i \(0.565963\pi\)
\(164\) −4.33208e9 −0.467626
\(165\) −2.89784e9 −0.304366
\(166\) −1.28640e10 −1.31489
\(167\) 1.85468e10 1.84521 0.922604 0.385748i \(-0.126057\pi\)
0.922604 + 0.385748i \(0.126057\pi\)
\(168\) 5.18266e9 0.501950
\(169\) −9.12975e9 −0.860932
\(170\) 3.82457e9 0.351206
\(171\) 3.01963e9 0.270067
\(172\) −2.69990e9 −0.235217
\(173\) 2.30537e10 1.95674 0.978370 0.206862i \(-0.0663250\pi\)
0.978370 + 0.206862i \(0.0663250\pi\)
\(174\) 2.59807e9 0.214872
\(175\) 2.18846e9 0.176387
\(176\) −3.04420e9 −0.239148
\(177\) 1.99972e8 0.0153140
\(178\) 1.63384e8 0.0121989
\(179\) 1.30784e10 0.952175 0.476087 0.879398i \(-0.342055\pi\)
0.476087 + 0.879398i \(0.342055\pi\)
\(180\) −3.10304e9 −0.220324
\(181\) 1.02950e9 0.0712972 0.0356486 0.999364i \(-0.488650\pi\)
0.0356486 + 0.999364i \(0.488650\pi\)
\(182\) −3.18001e9 −0.214836
\(183\) 3.00242e9 0.197898
\(184\) 3.51208e9 0.225883
\(185\) −3.86540e9 −0.242617
\(186\) 4.32030e9 0.264670
\(187\) −4.91291e9 −0.293800
\(188\) 1.04755e10 0.611594
\(189\) 1.31449e10 0.749340
\(190\) 5.28658e9 0.294295
\(191\) 1.74223e10 0.947229 0.473614 0.880732i \(-0.342949\pi\)
0.473614 + 0.880732i \(0.342949\pi\)
\(192\) 1.17301e10 0.622939
\(193\) −9.23682e9 −0.479198 −0.239599 0.970872i \(-0.577016\pi\)
−0.239599 + 0.970872i \(0.577016\pi\)
\(194\) −1.55998e9 −0.0790699
\(195\) −4.07318e9 −0.201733
\(196\) 3.70419e9 0.179284
\(197\) −7.76986e9 −0.367549 −0.183774 0.982968i \(-0.558832\pi\)
−0.183774 + 0.982968i \(0.558832\pi\)
\(198\) −5.77721e9 −0.267131
\(199\) −2.26974e9 −0.102598 −0.0512989 0.998683i \(-0.516336\pi\)
−0.0512989 + 0.998683i \(0.516336\pi\)
\(200\) 5.77323e9 0.255143
\(201\) −2.49400e10 −1.07774
\(202\) −1.70081e9 −0.0718746
\(203\) −8.18087e9 −0.338117
\(204\) 3.26276e9 0.131901
\(205\) −2.53235e10 −1.00146
\(206\) −7.94402e9 −0.307353
\(207\) 3.39965e9 0.128697
\(208\) −4.27890e9 −0.158507
\(209\) −6.79096e9 −0.246191
\(210\) 8.78301e9 0.311642
\(211\) 1.46051e10 0.507264 0.253632 0.967301i \(-0.418375\pi\)
0.253632 + 0.967301i \(0.418375\pi\)
\(212\) 2.00248e10 0.680858
\(213\) −5.85579e9 −0.194929
\(214\) 6.06459e9 0.197669
\(215\) −1.57825e10 −0.503736
\(216\) 3.46766e10 1.08391
\(217\) −1.36039e10 −0.416479
\(218\) −3.96850e10 −1.19007
\(219\) 2.35790e10 0.692671
\(220\) 6.97855e9 0.200846
\(221\) −6.90555e9 −0.194730
\(222\) 4.77938e9 0.132064
\(223\) 3.29844e10 0.893176 0.446588 0.894740i \(-0.352639\pi\)
0.446588 + 0.894740i \(0.352639\pi\)
\(224\) −2.13434e10 −0.566431
\(225\) 5.58841e9 0.145367
\(226\) 2.15077e10 0.548410
\(227\) −5.49978e10 −1.37477 −0.687383 0.726295i \(-0.741241\pi\)
−0.687383 + 0.726295i \(0.741241\pi\)
\(228\) 4.51001e9 0.110528
\(229\) −5.66697e10 −1.36173 −0.680865 0.732408i \(-0.738396\pi\)
−0.680865 + 0.732408i \(0.738396\pi\)
\(230\) 5.95189e9 0.140242
\(231\) −1.12824e10 −0.260703
\(232\) −2.15814e10 −0.489084
\(233\) −2.20205e10 −0.489469 −0.244735 0.969590i \(-0.578701\pi\)
−0.244735 + 0.969590i \(0.578701\pi\)
\(234\) −8.12041e9 −0.177054
\(235\) 6.12353e10 1.30977
\(236\) −4.81571e8 −0.0101055
\(237\) −7.04181e9 −0.144983
\(238\) 1.48905e10 0.300823
\(239\) 2.61002e10 0.517432 0.258716 0.965953i \(-0.416701\pi\)
0.258716 + 0.965953i \(0.416701\pi\)
\(240\) 1.18181e10 0.229931
\(241\) −7.14912e10 −1.36514 −0.682568 0.730822i \(-0.739137\pi\)
−0.682568 + 0.730822i \(0.739137\pi\)
\(242\) −2.80496e10 −0.525724
\(243\) 5.43224e10 0.999427
\(244\) −7.23041e9 −0.130590
\(245\) 2.16532e10 0.383950
\(246\) 3.13113e10 0.545122
\(247\) −9.54532e9 −0.163175
\(248\) −3.58874e10 −0.602434
\(249\) −6.41515e10 −1.05757
\(250\) 5.13245e10 0.830988
\(251\) −4.35139e10 −0.691984 −0.345992 0.938238i \(-0.612458\pi\)
−0.345992 + 0.938238i \(0.612458\pi\)
\(252\) −1.20813e10 −0.188717
\(253\) −7.64559e9 −0.117319
\(254\) 8.21506e10 1.23840
\(255\) 1.90727e10 0.282477
\(256\) −6.82296e10 −0.992871
\(257\) −7.66199e10 −1.09558 −0.547788 0.836617i \(-0.684530\pi\)
−0.547788 + 0.836617i \(0.684530\pi\)
\(258\) 1.95143e10 0.274198
\(259\) −1.50494e10 −0.207813
\(260\) 9.80900e9 0.133120
\(261\) −2.08905e10 −0.278655
\(262\) 7.69334e9 0.100869
\(263\) 1.11739e11 1.44013 0.720067 0.693904i \(-0.244111\pi\)
0.720067 + 0.693904i \(0.244111\pi\)
\(264\) −2.97632e10 −0.377105
\(265\) 1.17057e11 1.45811
\(266\) 2.05826e10 0.252077
\(267\) 8.14780e8 0.00981160
\(268\) 6.00602e10 0.711181
\(269\) 8.03903e10 0.936092 0.468046 0.883704i \(-0.344958\pi\)
0.468046 + 0.883704i \(0.344958\pi\)
\(270\) 5.87662e10 0.672962
\(271\) 1.13706e11 1.28063 0.640314 0.768113i \(-0.278804\pi\)
0.640314 + 0.768113i \(0.278804\pi\)
\(272\) 2.00361e10 0.221949
\(273\) −1.58584e10 −0.172793
\(274\) −2.56121e10 −0.274516
\(275\) −1.25680e10 −0.132516
\(276\) 5.07759e9 0.0526704
\(277\) 1.21360e11 1.23856 0.619278 0.785172i \(-0.287425\pi\)
0.619278 + 0.785172i \(0.287425\pi\)
\(278\) −1.41340e10 −0.141927
\(279\) −3.47386e10 −0.343236
\(280\) −7.29578e10 −0.709350
\(281\) −5.81904e10 −0.556766 −0.278383 0.960470i \(-0.589798\pi\)
−0.278383 + 0.960470i \(0.589798\pi\)
\(282\) −7.57144e10 −0.712948
\(283\) 1.03826e11 0.962203 0.481102 0.876665i \(-0.340237\pi\)
0.481102 + 0.876665i \(0.340237\pi\)
\(284\) 1.41019e10 0.128630
\(285\) 2.63637e10 0.236703
\(286\) 1.82623e10 0.161402
\(287\) −9.85939e10 −0.857791
\(288\) −5.45020e10 −0.466817
\(289\) −8.62525e10 −0.727330
\(290\) −3.65738e10 −0.303654
\(291\) −7.77947e9 −0.0635962
\(292\) −5.67828e10 −0.457082
\(293\) −8.50392e9 −0.0674086 −0.0337043 0.999432i \(-0.510730\pi\)
−0.0337043 + 0.999432i \(0.510730\pi\)
\(294\) −2.67731e10 −0.208995
\(295\) −2.81507e9 −0.0216416
\(296\) −3.97009e10 −0.300599
\(297\) −7.54891e10 −0.562963
\(298\) 1.27382e11 0.935697
\(299\) −1.07466e10 −0.0777589
\(300\) 8.34665e9 0.0594931
\(301\) −6.14471e10 −0.431472
\(302\) −1.36471e11 −0.944077
\(303\) −8.48180e9 −0.0578091
\(304\) 2.76952e10 0.185983
\(305\) −4.22660e10 −0.279667
\(306\) 3.80240e10 0.247920
\(307\) −1.78304e11 −1.14561 −0.572806 0.819691i \(-0.694145\pi\)
−0.572806 + 0.819691i \(0.694145\pi\)
\(308\) 2.71701e10 0.172033
\(309\) −3.96161e10 −0.247205
\(310\) −6.08181e10 −0.374029
\(311\) 1.47575e11 0.894522 0.447261 0.894403i \(-0.352400\pi\)
0.447261 + 0.894403i \(0.352400\pi\)
\(312\) −4.18349e10 −0.249944
\(313\) 2.46802e11 1.45344 0.726722 0.686931i \(-0.241043\pi\)
0.726722 + 0.686931i \(0.241043\pi\)
\(314\) −1.75457e11 −1.01856
\(315\) −7.06223e10 −0.404152
\(316\) 1.69580e10 0.0956718
\(317\) 4.21754e10 0.234581 0.117290 0.993098i \(-0.462579\pi\)
0.117290 + 0.993098i \(0.462579\pi\)
\(318\) −1.44735e11 −0.793690
\(319\) 4.69815e10 0.254020
\(320\) −1.65128e11 −0.880330
\(321\) 3.02435e10 0.158986
\(322\) 2.31729e10 0.120124
\(323\) 4.46962e10 0.228486
\(324\) 1.49635e8 0.000754362 0
\(325\) −1.76655e10 −0.0878315
\(326\) −6.45516e10 −0.316539
\(327\) −1.97905e11 −0.957177
\(328\) −2.60094e11 −1.24079
\(329\) 2.38412e11 1.12188
\(330\) −5.04394e10 −0.234130
\(331\) 3.27950e11 1.50170 0.750848 0.660475i \(-0.229645\pi\)
0.750848 + 0.660475i \(0.229645\pi\)
\(332\) 1.54489e11 0.697873
\(333\) −3.84300e10 −0.171266
\(334\) 3.22824e11 1.41941
\(335\) 3.51087e11 1.52305
\(336\) 4.60122e10 0.196946
\(337\) −3.96900e11 −1.67628 −0.838140 0.545455i \(-0.816357\pi\)
−0.838140 + 0.545455i \(0.816357\pi\)
\(338\) −1.58911e11 −0.662262
\(339\) 1.07257e11 0.441089
\(340\) −4.59308e10 −0.186401
\(341\) 7.81249e10 0.312892
\(342\) 5.25594e10 0.207746
\(343\) 2.76283e11 1.07778
\(344\) −1.62099e11 −0.624121
\(345\) 2.96815e10 0.112798
\(346\) 4.01270e11 1.50520
\(347\) 1.37404e11 0.508765 0.254383 0.967104i \(-0.418128\pi\)
0.254383 + 0.967104i \(0.418128\pi\)
\(348\) −3.12013e10 −0.114042
\(349\) 1.29416e11 0.466954 0.233477 0.972362i \(-0.424990\pi\)
0.233477 + 0.972362i \(0.424990\pi\)
\(350\) 3.80921e10 0.135684
\(351\) −1.06107e11 −0.373131
\(352\) 1.22572e11 0.425547
\(353\) −1.00183e11 −0.343405 −0.171703 0.985149i \(-0.554927\pi\)
−0.171703 + 0.985149i \(0.554927\pi\)
\(354\) 3.48069e9 0.0117802
\(355\) 8.24336e10 0.275472
\(356\) −1.96215e9 −0.00647450
\(357\) 7.42573e10 0.241954
\(358\) 2.27642e11 0.732450
\(359\) 5.68478e10 0.180630 0.0903148 0.995913i \(-0.471213\pi\)
0.0903148 + 0.995913i \(0.471213\pi\)
\(360\) −1.86304e11 −0.584602
\(361\) −2.60906e11 −0.808539
\(362\) 1.79194e10 0.0548446
\(363\) −1.39881e11 −0.422842
\(364\) 3.81901e10 0.114023
\(365\) −3.31929e11 −0.978875
\(366\) 5.22599e10 0.152231
\(367\) −1.76569e11 −0.508063 −0.254031 0.967196i \(-0.581757\pi\)
−0.254031 + 0.967196i \(0.581757\pi\)
\(368\) 3.11806e10 0.0886277
\(369\) −2.51768e11 −0.706938
\(370\) −6.72808e10 −0.186631
\(371\) 4.55745e11 1.24893
\(372\) −5.18843e10 −0.140473
\(373\) −6.26809e11 −1.67666 −0.838331 0.545161i \(-0.816468\pi\)
−0.838331 + 0.545161i \(0.816468\pi\)
\(374\) −8.55135e10 −0.226002
\(375\) 2.55950e11 0.668367
\(376\) 6.28937e11 1.62279
\(377\) 6.60368e10 0.168364
\(378\) 2.28799e11 0.576422
\(379\) −3.36046e11 −0.836608 −0.418304 0.908307i \(-0.637375\pi\)
−0.418304 + 0.908307i \(0.637375\pi\)
\(380\) −6.34888e10 −0.156196
\(381\) 4.09677e11 0.996046
\(382\) 3.03250e11 0.728645
\(383\) −7.94383e11 −1.88641 −0.943203 0.332216i \(-0.892204\pi\)
−0.943203 + 0.332216i \(0.892204\pi\)
\(384\) 4.78982e9 0.0112416
\(385\) 1.58825e11 0.368422
\(386\) −1.60775e11 −0.368618
\(387\) −1.56910e11 −0.355592
\(388\) 1.87344e10 0.0419661
\(389\) −6.84594e11 −1.51586 −0.757932 0.652334i \(-0.773790\pi\)
−0.757932 + 0.652334i \(0.773790\pi\)
\(390\) −7.08973e10 −0.155181
\(391\) 5.03211e10 0.108882
\(392\) 2.22396e11 0.475708
\(393\) 3.83659e10 0.0811296
\(394\) −1.35241e11 −0.282733
\(395\) 9.91297e10 0.204888
\(396\) 6.93810e10 0.141779
\(397\) −9.42340e11 −1.90393 −0.951964 0.306211i \(-0.900939\pi\)
−0.951964 + 0.306211i \(0.900939\pi\)
\(398\) −3.95069e10 −0.0789222
\(399\) 1.02643e11 0.202747
\(400\) 5.12554e10 0.100108
\(401\) 7.70083e11 1.48726 0.743632 0.668589i \(-0.233101\pi\)
0.743632 + 0.668589i \(0.233101\pi\)
\(402\) −4.34102e11 −0.829039
\(403\) 1.09812e11 0.207384
\(404\) 2.04258e10 0.0381472
\(405\) 8.74702e8 0.00161552
\(406\) −1.42395e11 −0.260093
\(407\) 8.64267e10 0.156125
\(408\) 1.95893e11 0.349984
\(409\) −8.77773e11 −1.55106 −0.775528 0.631313i \(-0.782516\pi\)
−0.775528 + 0.631313i \(0.782516\pi\)
\(410\) −4.40779e11 −0.770360
\(411\) −1.27725e11 −0.220794
\(412\) 9.54031e10 0.163127
\(413\) −1.09601e10 −0.0185370
\(414\) 5.91739e10 0.0989985
\(415\) 9.03080e11 1.49455
\(416\) 1.72286e11 0.282052
\(417\) −7.04851e10 −0.114152
\(418\) −1.18203e11 −0.189380
\(419\) −8.32714e11 −1.31988 −0.659938 0.751320i \(-0.729417\pi\)
−0.659938 + 0.751320i \(0.729417\pi\)
\(420\) −1.05479e11 −0.165403
\(421\) −2.38934e11 −0.370689 −0.185344 0.982674i \(-0.559340\pi\)
−0.185344 + 0.982674i \(0.559340\pi\)
\(422\) 2.54215e11 0.390207
\(423\) 6.08804e11 0.924582
\(424\) 1.20227e12 1.80657
\(425\) 8.27189e10 0.122986
\(426\) −1.01925e11 −0.149947
\(427\) −1.64557e11 −0.239547
\(428\) −7.28322e10 −0.104912
\(429\) 9.10723e10 0.129816
\(430\) −2.74709e11 −0.387493
\(431\) 1.24978e12 1.74456 0.872282 0.489003i \(-0.162639\pi\)
0.872282 + 0.489003i \(0.162639\pi\)
\(432\) 3.07863e11 0.425286
\(433\) −1.65520e11 −0.226284 −0.113142 0.993579i \(-0.536092\pi\)
−0.113142 + 0.993579i \(0.536092\pi\)
\(434\) −2.36787e11 −0.320372
\(435\) −1.82390e11 −0.244230
\(436\) 4.76594e11 0.631625
\(437\) 6.95573e10 0.0912381
\(438\) 4.10414e11 0.532830
\(439\) −1.47477e12 −1.89510 −0.947552 0.319601i \(-0.896451\pi\)
−0.947552 + 0.319601i \(0.896451\pi\)
\(440\) 4.18986e11 0.532920
\(441\) 2.15277e11 0.271034
\(442\) −1.20197e11 −0.149794
\(443\) −1.24460e12 −1.53537 −0.767687 0.640825i \(-0.778592\pi\)
−0.767687 + 0.640825i \(0.778592\pi\)
\(444\) −5.73976e10 −0.0700924
\(445\) −1.14699e10 −0.0138656
\(446\) 5.74124e11 0.687066
\(447\) 6.35242e11 0.752585
\(448\) −6.42904e11 −0.754041
\(449\) 1.26508e12 1.46896 0.734479 0.678631i \(-0.237426\pi\)
0.734479 + 0.678631i \(0.237426\pi\)
\(450\) 9.72713e10 0.111822
\(451\) 5.66210e11 0.644441
\(452\) −2.58295e11 −0.291067
\(453\) −6.80565e11 −0.759325
\(454\) −9.57286e11 −1.05752
\(455\) 2.23243e11 0.244190
\(456\) 2.70777e11 0.293271
\(457\) 2.54937e10 0.0273407 0.0136703 0.999907i \(-0.495648\pi\)
0.0136703 + 0.999907i \(0.495648\pi\)
\(458\) −9.86387e11 −1.04750
\(459\) 4.96848e11 0.522476
\(460\) −7.14787e10 −0.0744332
\(461\) 1.25077e12 1.28981 0.644903 0.764264i \(-0.276898\pi\)
0.644903 + 0.764264i \(0.276898\pi\)
\(462\) −1.96379e11 −0.200543
\(463\) 7.17086e11 0.725199 0.362599 0.931945i \(-0.381889\pi\)
0.362599 + 0.931945i \(0.381889\pi\)
\(464\) −1.91602e11 −0.191898
\(465\) −3.03294e11 −0.300833
\(466\) −3.83286e11 −0.376519
\(467\) −4.76349e11 −0.463446 −0.231723 0.972782i \(-0.574436\pi\)
−0.231723 + 0.972782i \(0.574436\pi\)
\(468\) 9.75215e10 0.0939710
\(469\) 1.36691e12 1.30456
\(470\) 1.06585e12 1.00753
\(471\) −8.74989e11 −0.819235
\(472\) −2.89131e10 −0.0268136
\(473\) 3.52881e11 0.324156
\(474\) −1.22569e11 −0.111527
\(475\) 1.14340e11 0.103057
\(476\) −1.78826e11 −0.159661
\(477\) 1.16378e12 1.02929
\(478\) 4.54298e11 0.398029
\(479\) 2.64981e11 0.229988 0.114994 0.993366i \(-0.463315\pi\)
0.114994 + 0.993366i \(0.463315\pi\)
\(480\) −4.75844e11 −0.409147
\(481\) 1.21481e11 0.103480
\(482\) −1.24437e12 −1.05012
\(483\) 1.15561e11 0.0966160
\(484\) 3.36860e11 0.279026
\(485\) 1.09514e11 0.0898734
\(486\) 9.45531e11 0.768798
\(487\) −1.44171e12 −1.16144 −0.580720 0.814103i \(-0.697229\pi\)
−0.580720 + 0.814103i \(0.697229\pi\)
\(488\) −4.34107e11 −0.346504
\(489\) −3.21913e11 −0.254594
\(490\) 3.76893e11 0.295349
\(491\) 4.61624e10 0.0358444 0.0179222 0.999839i \(-0.494295\pi\)
0.0179222 + 0.999839i \(0.494295\pi\)
\(492\) −3.76031e11 −0.289322
\(493\) −3.09219e11 −0.235752
\(494\) −1.66145e11 −0.125521
\(495\) 4.05573e11 0.303631
\(496\) −3.18613e11 −0.236372
\(497\) 3.20945e11 0.235954
\(498\) −1.11661e12 −0.813525
\(499\) 3.30266e11 0.238458 0.119229 0.992867i \(-0.461958\pi\)
0.119229 + 0.992867i \(0.461958\pi\)
\(500\) −6.16378e11 −0.441044
\(501\) 1.60989e12 1.14163
\(502\) −7.57398e11 −0.532301
\(503\) −8.35422e11 −0.581902 −0.290951 0.956738i \(-0.593972\pi\)
−0.290951 + 0.956738i \(0.593972\pi\)
\(504\) −7.25350e11 −0.500738
\(505\) 1.19401e11 0.0816951
\(506\) −1.33078e11 −0.0902464
\(507\) −7.92476e11 −0.532660
\(508\) −9.86582e11 −0.657274
\(509\) −2.24802e12 −1.48446 −0.742232 0.670144i \(-0.766233\pi\)
−0.742232 + 0.670144i \(0.766233\pi\)
\(510\) 3.31978e11 0.217292
\(511\) −1.29232e12 −0.838449
\(512\) −1.21585e12 −0.781925
\(513\) 6.86777e11 0.437812
\(514\) −1.33364e12 −0.842760
\(515\) 5.57687e11 0.349348
\(516\) −2.34355e11 −0.145530
\(517\) −1.36916e12 −0.842844
\(518\) −2.61949e11 −0.159858
\(519\) 2.00110e12 1.21064
\(520\) 5.88923e11 0.353219
\(521\) −1.16406e12 −0.692158 −0.346079 0.938205i \(-0.612487\pi\)
−0.346079 + 0.938205i \(0.612487\pi\)
\(522\) −3.63618e11 −0.214352
\(523\) −1.70100e12 −0.994136 −0.497068 0.867712i \(-0.665590\pi\)
−0.497068 + 0.867712i \(0.665590\pi\)
\(524\) −9.23925e10 −0.0535360
\(525\) 1.89962e11 0.109131
\(526\) 1.94491e12 1.10781
\(527\) −5.14196e11 −0.290389
\(528\) −2.64241e11 −0.147961
\(529\) 7.83110e10 0.0434783
\(530\) 2.03748e12 1.12163
\(531\) −2.79875e10 −0.0152770
\(532\) −2.47185e11 −0.133789
\(533\) 7.95860e11 0.427134
\(534\) 1.41820e10 0.00754746
\(535\) −4.25747e11 −0.224678
\(536\) 3.60596e12 1.88703
\(537\) 1.13523e12 0.589113
\(538\) 1.39927e12 0.720079
\(539\) −4.84144e11 −0.247073
\(540\) −7.05748e11 −0.357172
\(541\) 2.74399e12 1.37719 0.688596 0.725145i \(-0.258227\pi\)
0.688596 + 0.725145i \(0.258227\pi\)
\(542\) 1.97916e12 0.985109
\(543\) 8.93621e10 0.0441117
\(544\) −8.06732e11 −0.394943
\(545\) 2.78597e12 1.35267
\(546\) −2.76030e11 −0.132919
\(547\) −2.82754e12 −1.35041 −0.675205 0.737630i \(-0.735945\pi\)
−0.675205 + 0.737630i \(0.735945\pi\)
\(548\) 3.07587e11 0.145698
\(549\) −4.20210e11 −0.197420
\(550\) −2.18757e11 −0.101937
\(551\) −4.27424e11 −0.197550
\(552\) 3.04854e11 0.139754
\(553\) 3.85949e11 0.175496
\(554\) 2.11237e12 0.952746
\(555\) −3.35523e11 −0.150108
\(556\) 1.69742e11 0.0753272
\(557\) 2.18194e12 0.960495 0.480248 0.877133i \(-0.340547\pi\)
0.480248 + 0.877133i \(0.340547\pi\)
\(558\) −6.04656e11 −0.264031
\(559\) 4.96007e11 0.214850
\(560\) −6.47728e11 −0.278321
\(561\) −4.26448e11 −0.181775
\(562\) −1.01286e12 −0.428286
\(563\) −4.18295e11 −0.175467 −0.0877334 0.996144i \(-0.527962\pi\)
−0.0877334 + 0.996144i \(0.527962\pi\)
\(564\) 9.09287e11 0.378395
\(565\) −1.50988e12 −0.623341
\(566\) 1.80718e12 0.740164
\(567\) 3.40554e9 0.00138376
\(568\) 8.46662e11 0.341305
\(569\) 3.01046e12 1.20400 0.602001 0.798495i \(-0.294370\pi\)
0.602001 + 0.798495i \(0.294370\pi\)
\(570\) 4.58883e11 0.182081
\(571\) −1.96565e11 −0.0773825 −0.0386913 0.999251i \(-0.512319\pi\)
−0.0386913 + 0.999251i \(0.512319\pi\)
\(572\) −2.19320e11 −0.0856634
\(573\) 1.51228e12 0.586053
\(574\) −1.71612e12 −0.659847
\(575\) 1.28729e11 0.0491102
\(576\) −1.64171e12 −0.621434
\(577\) 1.24381e12 0.467157 0.233578 0.972338i \(-0.424956\pi\)
0.233578 + 0.972338i \(0.424956\pi\)
\(578\) −1.50130e12 −0.559491
\(579\) −8.01770e11 −0.296481
\(580\) 4.39230e11 0.161163
\(581\) 3.51602e12 1.28015
\(582\) −1.35409e11 −0.0489207
\(583\) −2.61727e12 −0.938297
\(584\) −3.40919e12 −1.21281
\(585\) 5.70070e11 0.201246
\(586\) −1.48018e11 −0.0518533
\(587\) 2.36780e11 0.0823139 0.0411570 0.999153i \(-0.486896\pi\)
0.0411570 + 0.999153i \(0.486896\pi\)
\(588\) 3.21529e11 0.110923
\(589\) −7.10757e11 −0.243334
\(590\) −4.89987e10 −0.0166476
\(591\) −6.74436e11 −0.227403
\(592\) −3.52469e11 −0.117943
\(593\) −1.88281e12 −0.625259 −0.312630 0.949875i \(-0.601210\pi\)
−0.312630 + 0.949875i \(0.601210\pi\)
\(594\) −1.31395e12 −0.433053
\(595\) −1.04534e12 −0.341926
\(596\) −1.52979e12 −0.496618
\(597\) −1.97017e11 −0.0634775
\(598\) −1.87054e11 −0.0598152
\(599\) −9.75007e11 −0.309447 −0.154724 0.987958i \(-0.549449\pi\)
−0.154724 + 0.987958i \(0.549449\pi\)
\(600\) 5.01125e11 0.157858
\(601\) −5.03518e12 −1.57427 −0.787135 0.616780i \(-0.788437\pi\)
−0.787135 + 0.616780i \(0.788437\pi\)
\(602\) −1.06954e12 −0.331905
\(603\) 3.49052e12 1.07513
\(604\) 1.63893e12 0.501066
\(605\) 1.96914e12 0.597555
\(606\) −1.47633e11 −0.0444690
\(607\) 9.31056e11 0.278373 0.139186 0.990266i \(-0.455551\pi\)
0.139186 + 0.990266i \(0.455551\pi\)
\(608\) −1.11512e12 −0.330945
\(609\) −7.10112e11 −0.209194
\(610\) −7.35678e11 −0.215131
\(611\) −1.92448e12 −0.558636
\(612\) −4.56646e11 −0.131583
\(613\) 1.84008e12 0.526337 0.263168 0.964750i \(-0.415233\pi\)
0.263168 + 0.964750i \(0.415233\pi\)
\(614\) −3.10353e12 −0.881249
\(615\) −2.19812e12 −0.619604
\(616\) 1.63127e12 0.456469
\(617\) −4.14604e12 −1.15173 −0.575865 0.817545i \(-0.695335\pi\)
−0.575865 + 0.817545i \(0.695335\pi\)
\(618\) −6.89553e11 −0.190160
\(619\) 5.89090e11 0.161277 0.0806387 0.996743i \(-0.474304\pi\)
0.0806387 + 0.996743i \(0.474304\pi\)
\(620\) 7.30391e11 0.198515
\(621\) 7.73207e11 0.208633
\(622\) 2.56867e12 0.688101
\(623\) −4.46566e10 −0.0118765
\(624\) −3.71416e11 −0.0980685
\(625\) −2.70463e12 −0.709004
\(626\) 4.29580e12 1.11805
\(627\) −5.89466e11 −0.152319
\(628\) 2.10714e12 0.540599
\(629\) −5.68836e11 −0.144897
\(630\) −1.22924e12 −0.310889
\(631\) 4.58081e12 1.15030 0.575149 0.818049i \(-0.304944\pi\)
0.575149 + 0.818049i \(0.304944\pi\)
\(632\) 1.01815e12 0.253853
\(633\) 1.26775e12 0.313845
\(634\) 7.34101e11 0.180449
\(635\) −5.76715e12 −1.40760
\(636\) 1.73818e12 0.421248
\(637\) −6.80509e11 −0.163760
\(638\) 8.17755e11 0.195402
\(639\) 8.19559e11 0.194458
\(640\) −6.74277e10 −0.0158865
\(641\) −1.39671e12 −0.326772 −0.163386 0.986562i \(-0.552242\pi\)
−0.163386 + 0.986562i \(0.552242\pi\)
\(642\) 5.26416e11 0.122299
\(643\) 3.37971e12 0.779704 0.389852 0.920877i \(-0.372526\pi\)
0.389852 + 0.920877i \(0.372526\pi\)
\(644\) −2.78293e11 −0.0637553
\(645\) −1.36995e12 −0.311662
\(646\) 7.77977e11 0.175760
\(647\) −5.91132e12 −1.32622 −0.663110 0.748522i \(-0.730764\pi\)
−0.663110 + 0.748522i \(0.730764\pi\)
\(648\) 8.98393e9 0.00200161
\(649\) 6.29421e10 0.0139264
\(650\) −3.07483e11 −0.0675634
\(651\) −1.18084e12 −0.257677
\(652\) 7.75228e11 0.168002
\(653\) 3.76996e12 0.811385 0.405692 0.914010i \(-0.367030\pi\)
0.405692 + 0.914010i \(0.367030\pi\)
\(654\) −3.44472e12 −0.736298
\(655\) −5.40089e11 −0.114651
\(656\) −2.30914e12 −0.486837
\(657\) −3.30005e12 −0.690997
\(658\) 4.14977e12 0.862993
\(659\) 3.45188e12 0.712969 0.356485 0.934301i \(-0.383975\pi\)
0.356485 + 0.934301i \(0.383975\pi\)
\(660\) 6.05749e11 0.124264
\(661\) −5.09909e12 −1.03893 −0.519465 0.854492i \(-0.673869\pi\)
−0.519465 + 0.854492i \(0.673869\pi\)
\(662\) 5.70827e12 1.15516
\(663\) −5.99412e11 −0.120480
\(664\) 9.27538e12 1.85172
\(665\) −1.44494e12 −0.286519
\(666\) −6.68909e11 −0.131745
\(667\) −4.81214e11 −0.0941395
\(668\) −3.87693e12 −0.753345
\(669\) 2.86310e12 0.552610
\(670\) 6.11099e12 1.17159
\(671\) 9.45027e11 0.179967
\(672\) −1.85264e12 −0.350452
\(673\) 9.60092e12 1.80403 0.902017 0.431700i \(-0.142086\pi\)
0.902017 + 0.431700i \(0.142086\pi\)
\(674\) −6.90840e12 −1.28946
\(675\) 1.27101e12 0.235659
\(676\) 1.90844e12 0.351494
\(677\) −1.75260e12 −0.320652 −0.160326 0.987064i \(-0.551254\pi\)
−0.160326 + 0.987064i \(0.551254\pi\)
\(678\) 1.86690e12 0.339303
\(679\) 4.26378e11 0.0769805
\(680\) −2.75764e12 −0.494593
\(681\) −4.77389e12 −0.850571
\(682\) 1.35983e12 0.240689
\(683\) 8.15393e12 1.43375 0.716876 0.697201i \(-0.245571\pi\)
0.716876 + 0.697201i \(0.245571\pi\)
\(684\) −6.31208e11 −0.110260
\(685\) 1.79802e12 0.312024
\(686\) 4.80896e12 0.829073
\(687\) −4.91902e12 −0.842506
\(688\) −1.43914e12 −0.244881
\(689\) −3.67882e12 −0.621902
\(690\) 5.16633e11 0.0867683
\(691\) −1.49514e12 −0.249477 −0.124738 0.992190i \(-0.539809\pi\)
−0.124738 + 0.992190i \(0.539809\pi\)
\(692\) −4.81903e12 −0.798881
\(693\) 1.57905e12 0.260073
\(694\) 2.39164e12 0.391362
\(695\) 9.92240e11 0.161319
\(696\) −1.87330e12 −0.302598
\(697\) −3.72663e12 −0.598094
\(698\) 2.25260e12 0.359199
\(699\) −1.91141e12 −0.302836
\(700\) −4.57464e11 −0.0720139
\(701\) 3.86276e12 0.604181 0.302090 0.953279i \(-0.402316\pi\)
0.302090 + 0.953279i \(0.402316\pi\)
\(702\) −1.84689e12 −0.287027
\(703\) −7.86284e11 −0.121417
\(704\) 3.69210e12 0.566495
\(705\) 5.31532e12 0.810360
\(706\) −1.74377e12 −0.264161
\(707\) 4.64872e11 0.0699754
\(708\) −4.18011e10 −0.00625228
\(709\) 3.87552e12 0.575999 0.287999 0.957631i \(-0.407010\pi\)
0.287999 + 0.957631i \(0.407010\pi\)
\(710\) 1.43483e12 0.211904
\(711\) 9.85552e11 0.144633
\(712\) −1.17805e11 −0.0171793
\(713\) −8.00205e11 −0.115957
\(714\) 1.29251e12 0.186120
\(715\) −1.28205e12 −0.183455
\(716\) −2.73385e12 −0.388746
\(717\) 2.26554e12 0.320137
\(718\) 9.89487e11 0.138947
\(719\) −1.06531e12 −0.148661 −0.0743303 0.997234i \(-0.523682\pi\)
−0.0743303 + 0.997234i \(0.523682\pi\)
\(720\) −1.65403e12 −0.229375
\(721\) 2.17128e12 0.299232
\(722\) −4.54130e12 −0.621960
\(723\) −6.20554e12 −0.844613
\(724\) −2.15201e11 −0.0291086
\(725\) −7.91030e11 −0.106334
\(726\) −2.43475e12 −0.325267
\(727\) −1.00157e13 −1.32977 −0.664885 0.746946i \(-0.731520\pi\)
−0.664885 + 0.746946i \(0.731520\pi\)
\(728\) 2.29290e12 0.302547
\(729\) 4.72936e12 0.620195
\(730\) −5.77752e12 −0.752989
\(731\) −2.32256e12 −0.300843
\(732\) −6.27611e11 −0.0807961
\(733\) 9.32137e11 0.119265 0.0596323 0.998220i \(-0.481007\pi\)
0.0596323 + 0.998220i \(0.481007\pi\)
\(734\) −3.07334e12 −0.390822
\(735\) 1.87953e12 0.237551
\(736\) −1.25546e12 −0.157707
\(737\) −7.84997e12 −0.980086
\(738\) −4.38224e12 −0.543805
\(739\) 1.25556e13 1.54860 0.774298 0.632821i \(-0.218103\pi\)
0.774298 + 0.632821i \(0.218103\pi\)
\(740\) 8.08004e11 0.0990537
\(741\) −8.28549e11 −0.100957
\(742\) 7.93265e12 0.960728
\(743\) −1.07581e12 −0.129504 −0.0647522 0.997901i \(-0.520626\pi\)
−0.0647522 + 0.997901i \(0.520626\pi\)
\(744\) −3.11508e12 −0.372728
\(745\) −8.94250e12 −1.06354
\(746\) −1.09102e13 −1.28975
\(747\) 8.97846e12 1.05502
\(748\) 1.02697e12 0.119950
\(749\) −1.65759e12 −0.192446
\(750\) 4.45504e12 0.514134
\(751\) −7.04814e11 −0.0808527 −0.0404264 0.999183i \(-0.512872\pi\)
−0.0404264 + 0.999183i \(0.512872\pi\)
\(752\) 5.58378e12 0.636719
\(753\) −3.77707e12 −0.428132
\(754\) 1.14943e12 0.129512
\(755\) 9.58052e12 1.07307
\(756\) −2.74774e12 −0.305934
\(757\) 1.39364e13 1.54248 0.771239 0.636545i \(-0.219637\pi\)
0.771239 + 0.636545i \(0.219637\pi\)
\(758\) −5.84918e12 −0.643552
\(759\) −6.63649e11 −0.0725856
\(760\) −3.81181e12 −0.414448
\(761\) −1.23010e13 −1.32957 −0.664784 0.747036i \(-0.731476\pi\)
−0.664784 + 0.747036i \(0.731476\pi\)
\(762\) 7.13080e12 0.766198
\(763\) 1.08468e13 1.15862
\(764\) −3.64186e12 −0.386726
\(765\) −2.66937e12 −0.281794
\(766\) −1.38269e13 −1.45110
\(767\) 8.84710e10 0.00923043
\(768\) −5.92243e12 −0.614292
\(769\) −3.26322e12 −0.336494 −0.168247 0.985745i \(-0.553811\pi\)
−0.168247 + 0.985745i \(0.553811\pi\)
\(770\) 2.76449e12 0.283405
\(771\) −6.65072e12 −0.677835
\(772\) 1.93082e12 0.195643
\(773\) 1.21642e12 0.122539 0.0612697 0.998121i \(-0.480485\pi\)
0.0612697 + 0.998121i \(0.480485\pi\)
\(774\) −2.73116e12 −0.273535
\(775\) −1.31539e12 −0.130978
\(776\) 1.12480e12 0.111352
\(777\) −1.30632e12 −0.128574
\(778\) −1.19160e13 −1.16606
\(779\) −5.15121e12 −0.501176
\(780\) 8.51436e11 0.0823619
\(781\) −1.84314e12 −0.177267
\(782\) 8.75884e11 0.0837561
\(783\) −4.75129e12 −0.451735
\(784\) 1.97446e12 0.186649
\(785\) 1.23175e13 1.15773
\(786\) 6.67793e11 0.0624081
\(787\) −2.95303e12 −0.274398 −0.137199 0.990544i \(-0.543810\pi\)
−0.137199 + 0.990544i \(0.543810\pi\)
\(788\) 1.62417e12 0.150060
\(789\) 9.69910e12 0.891015
\(790\) 1.72544e12 0.157608
\(791\) −5.87854e12 −0.533919
\(792\) 4.16557e12 0.376194
\(793\) 1.32832e12 0.119282
\(794\) −1.64023e13 −1.46458
\(795\) 1.01607e13 0.902135
\(796\) 4.74455e11 0.0418877
\(797\) 1.70002e13 1.49242 0.746211 0.665710i \(-0.231871\pi\)
0.746211 + 0.665710i \(0.231871\pi\)
\(798\) 1.78660e12 0.155961
\(799\) 9.01143e12 0.782228
\(800\) −2.06375e12 −0.178136
\(801\) −1.14034e11 −0.00978789
\(802\) 1.34040e13 1.14406
\(803\) 7.42160e12 0.629909
\(804\) 5.21332e12 0.440009
\(805\) −1.62679e12 −0.136537
\(806\) 1.91137e12 0.159528
\(807\) 6.97800e12 0.579162
\(808\) 1.22635e12 0.101219
\(809\) 2.92170e12 0.239810 0.119905 0.992785i \(-0.461741\pi\)
0.119905 + 0.992785i \(0.461741\pi\)
\(810\) 1.52250e10 0.00124272
\(811\) 1.76690e13 1.43423 0.717115 0.696955i \(-0.245462\pi\)
0.717115 + 0.696955i \(0.245462\pi\)
\(812\) 1.71009e12 0.138044
\(813\) 9.86989e12 0.792327
\(814\) 1.50433e12 0.120098
\(815\) 4.53166e12 0.359789
\(816\) 1.73916e12 0.137320
\(817\) −3.21041e12 −0.252093
\(818\) −1.52784e13 −1.19313
\(819\) 2.21949e12 0.172376
\(820\) 5.29350e12 0.408866
\(821\) −2.28315e13 −1.75384 −0.876922 0.480633i \(-0.840407\pi\)
−0.876922 + 0.480633i \(0.840407\pi\)
\(822\) −2.22317e12 −0.169844
\(823\) −1.72245e13 −1.30872 −0.654362 0.756181i \(-0.727063\pi\)
−0.654362 + 0.756181i \(0.727063\pi\)
\(824\) 5.72791e12 0.432837
\(825\) −1.09092e12 −0.0819880
\(826\) −1.90770e11 −0.0142594
\(827\) 2.12680e13 1.58107 0.790535 0.612416i \(-0.209802\pi\)
0.790535 + 0.612416i \(0.209802\pi\)
\(828\) −7.10644e11 −0.0525431
\(829\) −2.26060e13 −1.66237 −0.831185 0.555996i \(-0.812337\pi\)
−0.831185 + 0.555996i \(0.812337\pi\)
\(830\) 1.57189e13 1.14966
\(831\) 1.05342e13 0.766297
\(832\) 5.18959e12 0.375472
\(833\) 3.18650e12 0.229304
\(834\) −1.22686e12 −0.0878105
\(835\) −2.26629e13 −1.61334
\(836\) 1.41955e12 0.100513
\(837\) −7.90086e12 −0.556429
\(838\) −1.44941e13 −1.01530
\(839\) 2.52979e13 1.76261 0.881305 0.472548i \(-0.156666\pi\)
0.881305 + 0.472548i \(0.156666\pi\)
\(840\) −6.33285e12 −0.438877
\(841\) −1.15501e13 −0.796168
\(842\) −4.15887e12 −0.285148
\(843\) −5.05101e12 −0.344472
\(844\) −3.05298e12 −0.207101
\(845\) 1.11559e13 0.752750
\(846\) 1.05968e13 0.711225
\(847\) 7.66660e12 0.511832
\(848\) 1.06739e13 0.708829
\(849\) 9.01225e12 0.595317
\(850\) 1.43980e12 0.0946055
\(851\) −8.85237e11 −0.0578597
\(852\) 1.22406e12 0.0795840
\(853\) 1.73877e13 1.12453 0.562266 0.826957i \(-0.309930\pi\)
0.562266 + 0.826957i \(0.309930\pi\)
\(854\) −2.86427e12 −0.184269
\(855\) −3.68978e12 −0.236131
\(856\) −4.37278e12 −0.278372
\(857\) −5.92067e12 −0.374936 −0.187468 0.982271i \(-0.560028\pi\)
−0.187468 + 0.982271i \(0.560028\pi\)
\(858\) 1.58519e12 0.0998596
\(859\) −1.22656e13 −0.768633 −0.384316 0.923201i \(-0.625563\pi\)
−0.384316 + 0.923201i \(0.625563\pi\)
\(860\) 3.29909e12 0.205661
\(861\) −8.55811e12 −0.530718
\(862\) 2.17536e13 1.34199
\(863\) −1.24413e13 −0.763512 −0.381756 0.924263i \(-0.624681\pi\)
−0.381756 + 0.924263i \(0.624681\pi\)
\(864\) −1.23958e13 −0.756768
\(865\) −2.81700e13 −1.71086
\(866\) −2.88102e12 −0.174067
\(867\) −7.48685e12 −0.450001
\(868\) 2.84368e12 0.170036
\(869\) −2.21644e12 −0.131846
\(870\) −3.17466e12 −0.187872
\(871\) −1.10339e13 −0.649599
\(872\) 2.86142e13 1.67594
\(873\) 1.08879e12 0.0634426
\(874\) 1.21071e12 0.0701839
\(875\) −1.40282e13 −0.809030
\(876\) −4.92883e12 −0.282798
\(877\) −2.81673e13 −1.60786 −0.803929 0.594726i \(-0.797261\pi\)
−0.803929 + 0.594726i \(0.797261\pi\)
\(878\) −2.56696e13 −1.45779
\(879\) −7.38154e11 −0.0417058
\(880\) 3.71980e12 0.209097
\(881\) −1.04783e13 −0.586003 −0.293001 0.956112i \(-0.594654\pi\)
−0.293001 + 0.956112i \(0.594654\pi\)
\(882\) 3.74709e12 0.208490
\(883\) −1.07155e13 −0.593183 −0.296591 0.955004i \(-0.595850\pi\)
−0.296591 + 0.955004i \(0.595850\pi\)
\(884\) 1.44350e12 0.0795026
\(885\) −2.44352e11 −0.0133897
\(886\) −2.16634e13 −1.18107
\(887\) 3.90674e12 0.211913 0.105957 0.994371i \(-0.466210\pi\)
0.105957 + 0.994371i \(0.466210\pi\)
\(888\) −3.44610e12 −0.185982
\(889\) −2.24537e13 −1.20567
\(890\) −1.99644e11 −0.0106660
\(891\) −1.95575e10 −0.00103959
\(892\) −6.89489e12 −0.364658
\(893\) 1.24562e13 0.655473
\(894\) 1.10570e13 0.578918
\(895\) −1.59809e13 −0.832527
\(896\) −2.62521e11 −0.0136075
\(897\) −9.32820e11 −0.0481096
\(898\) 2.20199e13 1.12998
\(899\) 4.91719e12 0.251072
\(900\) −1.16817e12 −0.0593493
\(901\) 1.72261e13 0.870816
\(902\) 9.85539e12 0.495729
\(903\) −5.33370e12 −0.266953
\(904\) −1.55078e13 −0.772310
\(905\) −1.25798e12 −0.0623382
\(906\) −1.18459e13 −0.584103
\(907\) 1.35226e13 0.663481 0.331741 0.943371i \(-0.392364\pi\)
0.331741 + 0.943371i \(0.392364\pi\)
\(908\) 1.14965e13 0.561277
\(909\) 1.18709e12 0.0576694
\(910\) 3.88575e12 0.187840
\(911\) 3.63077e13 1.74649 0.873246 0.487280i \(-0.162011\pi\)
0.873246 + 0.487280i \(0.162011\pi\)
\(912\) 2.40399e12 0.115068
\(913\) −2.01920e13 −0.961746
\(914\) 4.43740e11 0.0210315
\(915\) −3.66875e12 −0.173031
\(916\) 1.18459e13 0.555955
\(917\) −2.10277e12 −0.0982039
\(918\) 8.64808e12 0.401909
\(919\) 3.11569e13 1.44090 0.720450 0.693506i \(-0.243935\pi\)
0.720450 + 0.693506i \(0.243935\pi\)
\(920\) −4.29152e12 −0.197499
\(921\) −1.54770e13 −0.708792
\(922\) 2.17708e13 0.992169
\(923\) −2.59070e12 −0.117492
\(924\) 2.35840e12 0.106437
\(925\) −1.45517e12 −0.0653546
\(926\) 1.24815e13 0.557851
\(927\) 5.54455e12 0.246608
\(928\) 7.71467e12 0.341469
\(929\) 8.29345e12 0.365312 0.182656 0.983177i \(-0.441530\pi\)
0.182656 + 0.983177i \(0.441530\pi\)
\(930\) −5.27911e12 −0.231413
\(931\) 4.40460e12 0.192147
\(932\) 4.60305e12 0.199836
\(933\) 1.28097e13 0.553443
\(934\) −8.29128e12 −0.356501
\(935\) 6.00324e12 0.256882
\(936\) 5.85510e12 0.249341
\(937\) 2.53064e13 1.07251 0.536257 0.844055i \(-0.319838\pi\)
0.536257 + 0.844055i \(0.319838\pi\)
\(938\) 2.37923e13 1.00352
\(939\) 2.14228e13 0.899250
\(940\) −1.28003e13 −0.534743
\(941\) −8.22803e12 −0.342092 −0.171046 0.985263i \(-0.554715\pi\)
−0.171046 + 0.985263i \(0.554715\pi\)
\(942\) −1.52300e13 −0.630188
\(943\) −5.79948e12 −0.238829
\(944\) −2.56694e11 −0.0105206
\(945\) −1.60622e13 −0.655180
\(946\) 6.14221e12 0.249353
\(947\) −2.37452e12 −0.0959405 −0.0479702 0.998849i \(-0.515275\pi\)
−0.0479702 + 0.998849i \(0.515275\pi\)
\(948\) 1.47198e12 0.0591923
\(949\) 1.04318e13 0.417503
\(950\) 1.99019e12 0.0792753
\(951\) 3.66089e12 0.145136
\(952\) −1.07365e13 −0.423641
\(953\) −4.17617e13 −1.64006 −0.820031 0.572318i \(-0.806044\pi\)
−0.820031 + 0.572318i \(0.806044\pi\)
\(954\) 2.02567e13 0.791772
\(955\) −2.12888e13 −0.828203
\(956\) −5.45585e12 −0.211253
\(957\) 4.07806e12 0.157163
\(958\) 4.61223e12 0.176916
\(959\) 7.00038e12 0.267262
\(960\) −1.43334e13 −0.544662
\(961\) −1.82629e13 −0.690740
\(962\) 2.11448e12 0.0796005
\(963\) −4.23280e12 −0.158602
\(964\) 1.49441e13 0.557345
\(965\) 1.12868e13 0.418983
\(966\) 2.01144e12 0.0743208
\(967\) 3.67341e11 0.0135098 0.00675492 0.999977i \(-0.497850\pi\)
0.00675492 + 0.999977i \(0.497850\pi\)
\(968\) 2.02247e13 0.740362
\(969\) 3.87970e12 0.141365
\(970\) 1.90619e12 0.0691342
\(971\) −3.92627e12 −0.141741 −0.0708703 0.997486i \(-0.522578\pi\)
−0.0708703 + 0.997486i \(0.522578\pi\)
\(972\) −1.13553e13 −0.408037
\(973\) 3.86316e12 0.138177
\(974\) −2.50942e13 −0.893425
\(975\) −1.53339e12 −0.0543415
\(976\) −3.85405e12 −0.135955
\(977\) −3.19527e13 −1.12197 −0.560986 0.827826i \(-0.689578\pi\)
−0.560986 + 0.827826i \(0.689578\pi\)
\(978\) −5.60318e12 −0.195844
\(979\) 2.56456e11 0.00892258
\(980\) −4.52627e12 −0.156756
\(981\) 2.76982e13 0.954864
\(982\) 8.03497e11 0.0275729
\(983\) −2.07291e12 −0.0708091 −0.0354045 0.999373i \(-0.511272\pi\)
−0.0354045 + 0.999373i \(0.511272\pi\)
\(984\) −2.25766e13 −0.767679
\(985\) 9.49424e12 0.321364
\(986\) −5.38223e12 −0.181349
\(987\) 2.06945e13 0.694109
\(988\) 1.99530e12 0.0666197
\(989\) −3.61443e12 −0.120132
\(990\) 7.05936e12 0.233565
\(991\) −1.25112e13 −0.412066 −0.206033 0.978545i \(-0.566055\pi\)
−0.206033 + 0.978545i \(0.566055\pi\)
\(992\) 1.28286e13 0.420608
\(993\) 2.84666e13 0.929103
\(994\) 5.58633e12 0.181505
\(995\) 2.77347e12 0.0897056
\(996\) 1.34099e13 0.431776
\(997\) −1.53878e13 −0.493227 −0.246614 0.969114i \(-0.579318\pi\)
−0.246614 + 0.969114i \(0.579318\pi\)
\(998\) 5.74858e12 0.183431
\(999\) −8.74042e12 −0.277644
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 23.10.a.a.1.5 7
3.2 odd 2 207.10.a.b.1.3 7
4.3 odd 2 368.10.a.f.1.3 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
23.10.a.a.1.5 7 1.1 even 1 trivial
207.10.a.b.1.3 7 3.2 odd 2
368.10.a.f.1.3 7 4.3 odd 2