Properties

Label 91.10.a.c.1.7
Level $91$
Weight $10$
Character 91.1
Self dual yes
Analytic conductor $46.868$
Analytic rank $0$
Dimension $14$
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] = [91,10,Mod(1,91)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(91, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 91.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: \(46.8682610909\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} - 4752 x^{12} + 9346 x^{11} + 8576824 x^{10} - 26923636 x^{9} - 7450416552 x^{8} + 31594524240 x^{7} + 3232668379296 x^{6} + \cdots - 24\!\cdots\!40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(4.59170\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.59170 q^{2} +254.265 q^{3} -505.283 q^{4} -1903.58 q^{5} -658.979 q^{6} +2401.00 q^{7} +2636.49 q^{8} +44967.7 q^{9} +O(q^{10})\) \(q-2.59170 q^{2} +254.265 q^{3} -505.283 q^{4} -1903.58 q^{5} -658.979 q^{6} +2401.00 q^{7} +2636.49 q^{8} +44967.7 q^{9} +4933.51 q^{10} +18355.1 q^{11} -128476. q^{12} -28561.0 q^{13} -6222.67 q^{14} -484014. q^{15} +251872. q^{16} -373800. q^{17} -116543. q^{18} -246186. q^{19} +961846. q^{20} +610490. q^{21} -47571.0 q^{22} +1.88050e6 q^{23} +670368. q^{24} +1.67049e6 q^{25} +74021.5 q^{26} +6.42903e6 q^{27} -1.21318e6 q^{28} -280114. q^{29} +1.25442e6 q^{30} +4.45227e6 q^{31} -2.00266e6 q^{32} +4.66707e6 q^{33} +968777. q^{34} -4.57049e6 q^{35} -2.27214e7 q^{36} +1.76160e7 q^{37} +638041. q^{38} -7.26207e6 q^{39} -5.01877e6 q^{40} +3.32158e7 q^{41} -1.58221e6 q^{42} -71574.6 q^{43} -9.27453e6 q^{44} -8.55996e7 q^{45} -4.87368e6 q^{46} +2.72431e7 q^{47} +6.40422e7 q^{48} +5.76480e6 q^{49} -4.32941e6 q^{50} -9.50443e7 q^{51} +1.44314e7 q^{52} -3.68347e7 q^{53} -1.66621e7 q^{54} -3.49404e7 q^{55} +6.33022e6 q^{56} -6.25966e7 q^{57} +725972. q^{58} +1.71700e8 q^{59} +2.44564e8 q^{60} +7.90773e7 q^{61} -1.15390e7 q^{62} +1.07968e8 q^{63} -1.23768e8 q^{64} +5.43681e7 q^{65} -1.20956e7 q^{66} +5.56151e7 q^{67} +1.88875e8 q^{68} +4.78145e8 q^{69} +1.18453e7 q^{70} +1.06698e7 q^{71} +1.18557e8 q^{72} +9.59021e7 q^{73} -4.56553e7 q^{74} +4.24747e8 q^{75} +1.24394e8 q^{76} +4.40706e7 q^{77} +1.88211e7 q^{78} -1.92106e8 q^{79} -4.79458e8 q^{80} +7.49577e8 q^{81} -8.60854e7 q^{82} -5.38923e8 q^{83} -3.08471e8 q^{84} +7.11558e8 q^{85} +185500. q^{86} -7.12233e7 q^{87} +4.83931e7 q^{88} +8.19043e7 q^{89} +2.21849e8 q^{90} -6.85750e7 q^{91} -9.50183e8 q^{92} +1.13206e9 q^{93} -7.06060e7 q^{94} +4.68635e8 q^{95} -5.09207e8 q^{96} -7.56732e8 q^{97} -1.49406e7 q^{98} +8.25388e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 27 q^{2} + 163 q^{3} + 2389 q^{4} + 2964 q^{5} + 471 q^{6} + 33614 q^{7} + 43263 q^{8} + 144129 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 27 q^{2} + 163 q^{3} + 2389 q^{4} + 2964 q^{5} + 471 q^{6} + 33614 q^{7} + 43263 q^{8} + 144129 q^{9} + 126524 q^{10} + 81825 q^{11} + 157399 q^{12} - 399854 q^{13} + 64827 q^{14} + 163856 q^{15} + 166361 q^{16} - 44922 q^{17} - 826396 q^{18} + 171756 q^{19} + 3899724 q^{20} + 391363 q^{21} + 917579 q^{22} + 1930479 q^{23} + 2992373 q^{24} + 8222344 q^{25} - 771147 q^{26} + 4139125 q^{27} + 5735989 q^{28} - 3799608 q^{29} - 5918004 q^{30} - 4392203 q^{31} + 3135663 q^{32} + 17499977 q^{33} - 20071132 q^{34} + 7116564 q^{35} + 2121398 q^{36} + 29198909 q^{37} - 44208366 q^{38} - 4655443 q^{39} + 134932928 q^{40} + 48410973 q^{41} + 1130871 q^{42} + 52650242 q^{43} - 14827353 q^{44} + 99215088 q^{45} - 34410455 q^{46} + 160580841 q^{47} + 227620515 q^{48} + 80707214 q^{49} + 149462949 q^{50} + 57114360 q^{51} - 68232229 q^{52} + 80753796 q^{53} + 301368833 q^{54} + 328919412 q^{55} + 103874463 q^{56} + 151101102 q^{57} + 335044204 q^{58} + 442445502 q^{59} + 561078360 q^{60} + 270199089 q^{61} + 543824517 q^{62} + 346053729 q^{63} + 223643137 q^{64} - 84654804 q^{65} + 317483345 q^{66} + 92500909 q^{67} + 255771204 q^{68} + 292017029 q^{69} + 303784124 q^{70} + 84383796 q^{71} + 1456696818 q^{72} + 367274315 q^{73} + 1091659407 q^{74} + 1154152501 q^{75} + 674789222 q^{76} + 196461825 q^{77} - 13452231 q^{78} + 434861545 q^{79} + 2644363752 q^{80} + 644207518 q^{81} + 634104331 q^{82} + 1013603934 q^{83} + 377914999 q^{84} + 1103701048 q^{85} + 2514069096 q^{86} + 1039292304 q^{87} + 1071310221 q^{88} + 1069739706 q^{89} - 1271572324 q^{90} - 960049454 q^{91} + 2301673917 q^{92} - 933838861 q^{93} + 2025486277 q^{94} + 2504029998 q^{95} - 116199027 q^{96} + 2839636281 q^{97} + 155649627 q^{98} + 5063037274 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.59170 −0.114538 −0.0572690 0.998359i \(-0.518239\pi\)
−0.0572690 + 0.998359i \(0.518239\pi\)
\(3\) 254.265 1.81235 0.906173 0.422907i \(-0.138990\pi\)
0.906173 + 0.422907i \(0.138990\pi\)
\(4\) −505.283 −0.986881
\(5\) −1903.58 −1.36209 −0.681045 0.732241i \(-0.738474\pi\)
−0.681045 + 0.732241i \(0.738474\pi\)
\(6\) −658.979 −0.207583
\(7\) 2401.00 0.377964
\(8\) 2636.49 0.227573
\(9\) 44967.7 2.28460
\(10\) 4933.51 0.156011
\(11\) 18355.1 0.377999 0.188999 0.981977i \(-0.439476\pi\)
0.188999 + 0.981977i \(0.439476\pi\)
\(12\) −128476. −1.78857
\(13\) −28561.0 −0.277350
\(14\) −6222.67 −0.0432913
\(15\) −484014. −2.46858
\(16\) 251872. 0.960815
\(17\) −373800. −1.08547 −0.542737 0.839903i \(-0.682612\pi\)
−0.542737 + 0.839903i \(0.682612\pi\)
\(18\) −116543. −0.261673
\(19\) −246186. −0.433384 −0.216692 0.976240i \(-0.569527\pi\)
−0.216692 + 0.976240i \(0.569527\pi\)
\(20\) 961846. 1.34422
\(21\) 610490. 0.685002
\(22\) −47571.0 −0.0432952
\(23\) 1.88050e6 1.40119 0.700595 0.713559i \(-0.252918\pi\)
0.700595 + 0.713559i \(0.252918\pi\)
\(24\) 670368. 0.412442
\(25\) 1.67049e6 0.855290
\(26\) 74021.5 0.0317671
\(27\) 6.42903e6 2.32814
\(28\) −1.21318e6 −0.373006
\(29\) −280114. −0.0735435 −0.0367718 0.999324i \(-0.511707\pi\)
−0.0367718 + 0.999324i \(0.511707\pi\)
\(30\) 1.25442e6 0.282746
\(31\) 4.45227e6 0.865873 0.432937 0.901424i \(-0.357477\pi\)
0.432937 + 0.901424i \(0.357477\pi\)
\(32\) −2.00266e6 −0.337623
\(33\) 4.66707e6 0.685064
\(34\) 968777. 0.124328
\(35\) −4.57049e6 −0.514822
\(36\) −2.27214e7 −2.25463
\(37\) 1.76160e7 1.54525 0.772624 0.634863i \(-0.218944\pi\)
0.772624 + 0.634863i \(0.218944\pi\)
\(38\) 638041. 0.0496389
\(39\) −7.26207e6 −0.502654
\(40\) −5.01877e6 −0.309976
\(41\) 3.32158e7 1.83577 0.917883 0.396851i \(-0.129897\pi\)
0.917883 + 0.396851i \(0.129897\pi\)
\(42\) −1.58221e6 −0.0784588
\(43\) −71574.6 −0.00319265 −0.00159632 0.999999i \(-0.500508\pi\)
−0.00159632 + 0.999999i \(0.500508\pi\)
\(44\) −9.27453e6 −0.373040
\(45\) −8.55996e7 −3.11183
\(46\) −4.87368e6 −0.160490
\(47\) 2.72431e7 0.814360 0.407180 0.913348i \(-0.366512\pi\)
0.407180 + 0.913348i \(0.366512\pi\)
\(48\) 6.40422e7 1.74133
\(49\) 5.76480e6 0.142857
\(50\) −4.32941e6 −0.0979633
\(51\) −9.50443e7 −1.96725
\(52\) 1.44314e7 0.273712
\(53\) −3.68347e7 −0.641232 −0.320616 0.947209i \(-0.603890\pi\)
−0.320616 + 0.947209i \(0.603890\pi\)
\(54\) −1.66621e7 −0.266660
\(55\) −3.49404e7 −0.514868
\(56\) 6.33022e6 0.0860147
\(57\) −6.25966e7 −0.785441
\(58\) 725972. 0.00842353
\(59\) 1.71700e8 1.84475 0.922375 0.386296i \(-0.126246\pi\)
0.922375 + 0.386296i \(0.126246\pi\)
\(60\) 2.44564e8 2.43619
\(61\) 7.90773e7 0.731253 0.365627 0.930762i \(-0.380855\pi\)
0.365627 + 0.930762i \(0.380855\pi\)
\(62\) −1.15390e7 −0.0991754
\(63\) 1.07968e8 0.863497
\(64\) −1.23768e8 −0.922145
\(65\) 5.43681e7 0.377776
\(66\) −1.20956e7 −0.0784659
\(67\) 5.56151e7 0.337176 0.168588 0.985687i \(-0.446079\pi\)
0.168588 + 0.985687i \(0.446079\pi\)
\(68\) 1.88875e8 1.07123
\(69\) 4.78145e8 2.53944
\(70\) 1.18453e7 0.0589667
\(71\) 1.06698e7 0.0498304 0.0249152 0.999690i \(-0.492068\pi\)
0.0249152 + 0.999690i \(0.492068\pi\)
\(72\) 1.18557e8 0.519914
\(73\) 9.59021e7 0.395253 0.197627 0.980277i \(-0.436677\pi\)
0.197627 + 0.980277i \(0.436677\pi\)
\(74\) −4.56553e7 −0.176990
\(75\) 4.24747e8 1.55008
\(76\) 1.24394e8 0.427698
\(77\) 4.40706e7 0.142870
\(78\) 1.88211e7 0.0575730
\(79\) −1.92106e8 −0.554905 −0.277453 0.960739i \(-0.589490\pi\)
−0.277453 + 0.960739i \(0.589490\pi\)
\(80\) −4.79458e8 −1.30872
\(81\) 7.49577e8 1.93479
\(82\) −8.60854e7 −0.210265
\(83\) −5.38923e8 −1.24645 −0.623226 0.782042i \(-0.714178\pi\)
−0.623226 + 0.782042i \(0.714178\pi\)
\(84\) −3.08471e8 −0.676016
\(85\) 7.11558e8 1.47851
\(86\) 185500. 0.000365679 0
\(87\) −7.12233e7 −0.133286
\(88\) 4.83931e7 0.0860224
\(89\) 8.19043e7 0.138373 0.0691865 0.997604i \(-0.477960\pi\)
0.0691865 + 0.997604i \(0.477960\pi\)
\(90\) 2.21849e8 0.356423
\(91\) −6.85750e7 −0.104828
\(92\) −9.50183e8 −1.38281
\(93\) 1.13206e9 1.56926
\(94\) −7.06060e7 −0.0932752
\(95\) 4.68635e8 0.590308
\(96\) −5.09207e8 −0.611890
\(97\) −7.56732e8 −0.867899 −0.433949 0.900937i \(-0.642880\pi\)
−0.433949 + 0.900937i \(0.642880\pi\)
\(98\) −1.49406e7 −0.0163626
\(99\) 8.25388e8 0.863575
\(100\) −8.44070e8 −0.844070
\(101\) 1.68137e9 1.60774 0.803871 0.594803i \(-0.202770\pi\)
0.803871 + 0.594803i \(0.202770\pi\)
\(102\) 2.46326e8 0.225325
\(103\) −1.91743e9 −1.67862 −0.839310 0.543654i \(-0.817040\pi\)
−0.839310 + 0.543654i \(0.817040\pi\)
\(104\) −7.53009e7 −0.0631175
\(105\) −1.16212e9 −0.933035
\(106\) 9.54644e7 0.0734455
\(107\) −2.42552e9 −1.78887 −0.894435 0.447198i \(-0.852422\pi\)
−0.894435 + 0.447198i \(0.852422\pi\)
\(108\) −3.24848e9 −2.29759
\(109\) 3.42217e8 0.232211 0.116105 0.993237i \(-0.462959\pi\)
0.116105 + 0.993237i \(0.462959\pi\)
\(110\) 9.05551e7 0.0589720
\(111\) 4.47912e9 2.80053
\(112\) 6.04745e8 0.363154
\(113\) 1.67651e9 0.967284 0.483642 0.875266i \(-0.339314\pi\)
0.483642 + 0.875266i \(0.339314\pi\)
\(114\) 1.62232e8 0.0899629
\(115\) −3.57967e9 −1.90855
\(116\) 1.41537e8 0.0725787
\(117\) −1.28432e9 −0.633633
\(118\) −4.44996e8 −0.211294
\(119\) −8.97494e8 −0.410270
\(120\) −1.27610e9 −0.561783
\(121\) −2.02104e9 −0.857117
\(122\) −2.04945e8 −0.0837563
\(123\) 8.44562e9 3.32704
\(124\) −2.24966e9 −0.854514
\(125\) 5.38020e8 0.197108
\(126\) −2.79819e8 −0.0989032
\(127\) 2.63364e9 0.898337 0.449169 0.893447i \(-0.351720\pi\)
0.449169 + 0.893447i \(0.351720\pi\)
\(128\) 1.34613e9 0.443244
\(129\) −1.81989e7 −0.00578618
\(130\) −1.40906e8 −0.0432697
\(131\) 3.61555e9 1.07264 0.536319 0.844015i \(-0.319814\pi\)
0.536319 + 0.844015i \(0.319814\pi\)
\(132\) −2.35819e9 −0.676077
\(133\) −5.91093e8 −0.163804
\(134\) −1.44138e8 −0.0386194
\(135\) −1.22382e10 −3.17113
\(136\) −9.85521e8 −0.247025
\(137\) −3.29030e9 −0.797982 −0.398991 0.916955i \(-0.630640\pi\)
−0.398991 + 0.916955i \(0.630640\pi\)
\(138\) −1.23921e9 −0.290863
\(139\) 6.76805e9 1.53779 0.768895 0.639375i \(-0.220807\pi\)
0.768895 + 0.639375i \(0.220807\pi\)
\(140\) 2.30939e9 0.508068
\(141\) 6.92697e9 1.47590
\(142\) −2.76529e7 −0.00570747
\(143\) −5.24241e8 −0.104838
\(144\) 1.13261e10 2.19508
\(145\) 5.33220e8 0.100173
\(146\) −2.48549e8 −0.0452715
\(147\) 1.46579e9 0.258907
\(148\) −8.90104e9 −1.52498
\(149\) −1.16876e9 −0.194261 −0.0971305 0.995272i \(-0.530966\pi\)
−0.0971305 + 0.995272i \(0.530966\pi\)
\(150\) −1.10082e9 −0.177543
\(151\) 9.70697e9 1.51945 0.759727 0.650242i \(-0.225333\pi\)
0.759727 + 0.650242i \(0.225333\pi\)
\(152\) −6.49068e8 −0.0986266
\(153\) −1.68089e10 −2.47987
\(154\) −1.14218e8 −0.0163640
\(155\) −8.47526e9 −1.17940
\(156\) 3.66940e9 0.496060
\(157\) −1.39697e10 −1.83500 −0.917502 0.397731i \(-0.869798\pi\)
−0.917502 + 0.397731i \(0.869798\pi\)
\(158\) 4.97881e8 0.0635577
\(159\) −9.36577e9 −1.16213
\(160\) 3.81222e9 0.459873
\(161\) 4.51507e9 0.529600
\(162\) −1.94268e9 −0.221607
\(163\) −1.12027e10 −1.24302 −0.621509 0.783407i \(-0.713480\pi\)
−0.621509 + 0.783407i \(0.713480\pi\)
\(164\) −1.67834e10 −1.81168
\(165\) −8.88413e9 −0.933119
\(166\) 1.39673e9 0.142766
\(167\) 1.66802e9 0.165950 0.0829751 0.996552i \(-0.473558\pi\)
0.0829751 + 0.996552i \(0.473558\pi\)
\(168\) 1.60955e9 0.155888
\(169\) 8.15731e8 0.0769231
\(170\) −1.84414e9 −0.169346
\(171\) −1.10704e10 −0.990108
\(172\) 3.61654e7 0.00315076
\(173\) −8.64313e9 −0.733607 −0.366804 0.930298i \(-0.619548\pi\)
−0.366804 + 0.930298i \(0.619548\pi\)
\(174\) 1.84589e8 0.0152663
\(175\) 4.01084e9 0.323269
\(176\) 4.62314e9 0.363187
\(177\) 4.36574e10 3.34332
\(178\) −2.12271e8 −0.0158490
\(179\) −2.38067e10 −1.73325 −0.866623 0.498963i \(-0.833714\pi\)
−0.866623 + 0.498963i \(0.833714\pi\)
\(180\) 4.32521e10 3.07100
\(181\) 4.65133e9 0.322125 0.161062 0.986944i \(-0.448508\pi\)
0.161062 + 0.986944i \(0.448508\pi\)
\(182\) 1.77726e8 0.0120068
\(183\) 2.01066e10 1.32528
\(184\) 4.95791e9 0.318874
\(185\) −3.35334e10 −2.10477
\(186\) −2.93395e9 −0.179740
\(187\) −6.86114e9 −0.410307
\(188\) −1.37655e10 −0.803677
\(189\) 1.54361e10 0.879952
\(190\) −1.21456e9 −0.0676127
\(191\) 2.48256e9 0.134974 0.0674870 0.997720i \(-0.478502\pi\)
0.0674870 + 0.997720i \(0.478502\pi\)
\(192\) −3.14699e10 −1.67124
\(193\) −5.03343e9 −0.261130 −0.130565 0.991440i \(-0.541679\pi\)
−0.130565 + 0.991440i \(0.541679\pi\)
\(194\) 1.96122e9 0.0994074
\(195\) 1.38239e10 0.684661
\(196\) −2.91286e9 −0.140983
\(197\) 1.06453e10 0.503570 0.251785 0.967783i \(-0.418982\pi\)
0.251785 + 0.967783i \(0.418982\pi\)
\(198\) −2.13916e9 −0.0989121
\(199\) 2.20681e10 0.997530 0.498765 0.866737i \(-0.333787\pi\)
0.498765 + 0.866737i \(0.333787\pi\)
\(200\) 4.40423e9 0.194641
\(201\) 1.41410e10 0.611079
\(202\) −4.35760e9 −0.184148
\(203\) −6.72554e8 −0.0277968
\(204\) 4.80243e10 1.94144
\(205\) −6.32289e10 −2.50048
\(206\) 4.96940e9 0.192266
\(207\) 8.45617e10 3.20116
\(208\) −7.19371e9 −0.266482
\(209\) −4.51878e9 −0.163818
\(210\) 3.01186e9 0.106868
\(211\) 3.56797e10 1.23923 0.619613 0.784908i \(-0.287290\pi\)
0.619613 + 0.784908i \(0.287290\pi\)
\(212\) 1.86119e10 0.632820
\(213\) 2.71296e9 0.0903099
\(214\) 6.28623e9 0.204894
\(215\) 1.36248e8 0.00434867
\(216\) 1.69501e10 0.529822
\(217\) 1.06899e10 0.327269
\(218\) −8.86923e8 −0.0265969
\(219\) 2.43846e10 0.716335
\(220\) 1.76548e10 0.508114
\(221\) 1.06761e10 0.301056
\(222\) −1.16085e10 −0.320767
\(223\) 7.94326e9 0.215093 0.107547 0.994200i \(-0.465701\pi\)
0.107547 + 0.994200i \(0.465701\pi\)
\(224\) −4.80839e9 −0.127610
\(225\) 7.51181e10 1.95399
\(226\) −4.34502e9 −0.110791
\(227\) 7.84841e10 1.96185 0.980923 0.194394i \(-0.0622741\pi\)
0.980923 + 0.194394i \(0.0622741\pi\)
\(228\) 3.16290e10 0.775137
\(229\) −7.05521e10 −1.69532 −0.847658 0.530544i \(-0.821988\pi\)
−0.847658 + 0.530544i \(0.821988\pi\)
\(230\) 9.27744e9 0.218601
\(231\) 1.12056e10 0.258930
\(232\) −7.38519e8 −0.0167365
\(233\) −5.80112e10 −1.28947 −0.644733 0.764408i \(-0.723031\pi\)
−0.644733 + 0.764408i \(0.723031\pi\)
\(234\) 3.32858e9 0.0725751
\(235\) −5.18594e10 −1.10923
\(236\) −8.67574e10 −1.82055
\(237\) −4.88458e10 −1.00568
\(238\) 2.32603e9 0.0469915
\(239\) −4.01270e10 −0.795511 −0.397756 0.917491i \(-0.630211\pi\)
−0.397756 + 0.917491i \(0.630211\pi\)
\(240\) −1.21909e11 −2.37185
\(241\) −8.33869e9 −0.159229 −0.0796143 0.996826i \(-0.525369\pi\)
−0.0796143 + 0.996826i \(0.525369\pi\)
\(242\) 5.23792e9 0.0981725
\(243\) 6.40487e10 1.17837
\(244\) −3.99564e10 −0.721660
\(245\) −1.09738e10 −0.194584
\(246\) −2.18885e10 −0.381073
\(247\) 7.03133e9 0.120199
\(248\) 1.17384e10 0.197050
\(249\) −1.37029e11 −2.25900
\(250\) −1.39439e9 −0.0225763
\(251\) 3.04136e10 0.483655 0.241827 0.970319i \(-0.422253\pi\)
0.241827 + 0.970319i \(0.422253\pi\)
\(252\) −5.45542e10 −0.852169
\(253\) 3.45167e10 0.529648
\(254\) −6.82560e9 −0.102894
\(255\) 1.80924e11 2.67958
\(256\) 5.98805e10 0.871376
\(257\) 2.81976e10 0.403192 0.201596 0.979469i \(-0.435387\pi\)
0.201596 + 0.979469i \(0.435387\pi\)
\(258\) 4.71661e7 0.000662738 0
\(259\) 4.22959e10 0.584049
\(260\) −2.74713e10 −0.372820
\(261\) −1.25961e10 −0.168017
\(262\) −9.37042e9 −0.122858
\(263\) −1.07105e11 −1.38042 −0.690208 0.723611i \(-0.742481\pi\)
−0.690208 + 0.723611i \(0.742481\pi\)
\(264\) 1.23047e10 0.155902
\(265\) 7.01177e10 0.873416
\(266\) 1.53194e9 0.0187618
\(267\) 2.08254e10 0.250780
\(268\) −2.81014e10 −0.332752
\(269\) 2.99847e10 0.349152 0.174576 0.984644i \(-0.444145\pi\)
0.174576 + 0.984644i \(0.444145\pi\)
\(270\) 3.17176e10 0.363215
\(271\) 7.25971e10 0.817631 0.408815 0.912617i \(-0.365942\pi\)
0.408815 + 0.912617i \(0.365942\pi\)
\(272\) −9.41497e10 −1.04294
\(273\) −1.74362e10 −0.189985
\(274\) 8.52747e9 0.0913993
\(275\) 3.06620e10 0.323299
\(276\) −2.41598e11 −2.50613
\(277\) 4.16560e10 0.425127 0.212563 0.977147i \(-0.431819\pi\)
0.212563 + 0.977147i \(0.431819\pi\)
\(278\) −1.75408e10 −0.176135
\(279\) 2.00209e11 1.97817
\(280\) −1.20501e10 −0.117160
\(281\) 5.45536e10 0.521969 0.260985 0.965343i \(-0.415953\pi\)
0.260985 + 0.965343i \(0.415953\pi\)
\(282\) −1.79526e10 −0.169047
\(283\) 6.14842e10 0.569802 0.284901 0.958557i \(-0.408039\pi\)
0.284901 + 0.958557i \(0.408039\pi\)
\(284\) −5.39128e9 −0.0491767
\(285\) 1.19158e11 1.06984
\(286\) 1.35867e9 0.0120079
\(287\) 7.97511e10 0.693854
\(288\) −9.00551e10 −0.771333
\(289\) 2.11385e10 0.178252
\(290\) −1.38195e9 −0.0114736
\(291\) −1.92410e11 −1.57293
\(292\) −4.84577e10 −0.390068
\(293\) 2.80875e10 0.222643 0.111321 0.993784i \(-0.464492\pi\)
0.111321 + 0.993784i \(0.464492\pi\)
\(294\) −3.79888e9 −0.0296546
\(295\) −3.26845e11 −2.51272
\(296\) 4.64443e10 0.351658
\(297\) 1.18006e11 0.880032
\(298\) 3.02906e9 0.0222503
\(299\) −5.37089e10 −0.388620
\(300\) −2.14617e11 −1.52975
\(301\) −1.71851e8 −0.00120671
\(302\) −2.51576e10 −0.174035
\(303\) 4.27513e11 2.91379
\(304\) −6.20074e10 −0.416402
\(305\) −1.50530e11 −0.996033
\(306\) 4.35637e10 0.284039
\(307\) 2.54042e11 1.63224 0.816119 0.577884i \(-0.196122\pi\)
0.816119 + 0.577884i \(0.196122\pi\)
\(308\) −2.22681e10 −0.140996
\(309\) −4.87536e11 −3.04224
\(310\) 2.19653e10 0.135086
\(311\) 2.66154e11 1.61329 0.806643 0.591040i \(-0.201282\pi\)
0.806643 + 0.591040i \(0.201282\pi\)
\(312\) −1.91464e10 −0.114391
\(313\) −3.34837e11 −1.97190 −0.985949 0.167047i \(-0.946577\pi\)
−0.985949 + 0.167047i \(0.946577\pi\)
\(314\) 3.62051e10 0.210178
\(315\) −2.05525e11 −1.17616
\(316\) 9.70678e10 0.547625
\(317\) −8.40674e10 −0.467586 −0.233793 0.972286i \(-0.575114\pi\)
−0.233793 + 0.972286i \(0.575114\pi\)
\(318\) 2.42733e10 0.133109
\(319\) −5.14153e9 −0.0277993
\(320\) 2.35602e11 1.25604
\(321\) −6.16726e11 −3.24205
\(322\) −1.17017e10 −0.0606594
\(323\) 9.20244e10 0.470427
\(324\) −3.78748e11 −1.90941
\(325\) −4.77108e10 −0.237215
\(326\) 2.90339e10 0.142373
\(327\) 8.70137e10 0.420846
\(328\) 8.75732e10 0.417772
\(329\) 6.54107e10 0.307799
\(330\) 2.30250e10 0.106878
\(331\) −9.02322e10 −0.413177 −0.206588 0.978428i \(-0.566236\pi\)
−0.206588 + 0.978428i \(0.566236\pi\)
\(332\) 2.72309e11 1.23010
\(333\) 7.92150e11 3.53027
\(334\) −4.32301e9 −0.0190076
\(335\) −1.05868e11 −0.459264
\(336\) 1.53765e11 0.658161
\(337\) −3.71614e11 −1.56949 −0.784743 0.619821i \(-0.787205\pi\)
−0.784743 + 0.619821i \(0.787205\pi\)
\(338\) −2.11413e9 −0.00881062
\(339\) 4.26279e11 1.75305
\(340\) −3.59538e11 −1.45912
\(341\) 8.17220e10 0.327299
\(342\) 2.86913e10 0.113405
\(343\) 1.38413e10 0.0539949
\(344\) −1.88706e8 −0.000726562 0
\(345\) −9.10186e11 −3.45895
\(346\) 2.24004e10 0.0840260
\(347\) 1.37583e11 0.509428 0.254714 0.967016i \(-0.418019\pi\)
0.254714 + 0.967016i \(0.418019\pi\)
\(348\) 3.59879e10 0.131538
\(349\) 1.26667e11 0.457034 0.228517 0.973540i \(-0.426612\pi\)
0.228517 + 0.973540i \(0.426612\pi\)
\(350\) −1.03949e10 −0.0370266
\(351\) −1.83619e11 −0.645708
\(352\) −3.67591e10 −0.127621
\(353\) 4.56985e11 1.56645 0.783223 0.621741i \(-0.213574\pi\)
0.783223 + 0.621741i \(0.213574\pi\)
\(354\) −1.13147e11 −0.382938
\(355\) −2.03108e10 −0.0678735
\(356\) −4.13848e10 −0.136558
\(357\) −2.28201e11 −0.743552
\(358\) 6.16998e10 0.198523
\(359\) 9.39943e10 0.298660 0.149330 0.988787i \(-0.452288\pi\)
0.149330 + 0.988787i \(0.452288\pi\)
\(360\) −2.25683e11 −0.708169
\(361\) −2.62080e11 −0.812178
\(362\) −1.20549e10 −0.0368955
\(363\) −5.13879e11 −1.55339
\(364\) 3.46498e10 0.103453
\(365\) −1.82557e11 −0.538371
\(366\) −5.21103e10 −0.151795
\(367\) 3.48049e10 0.100148 0.0500741 0.998746i \(-0.484054\pi\)
0.0500741 + 0.998746i \(0.484054\pi\)
\(368\) 4.73644e11 1.34629
\(369\) 1.49364e12 4.19399
\(370\) 8.69084e10 0.241076
\(371\) −8.84400e10 −0.242363
\(372\) −5.72010e11 −1.54867
\(373\) 5.00571e11 1.33899 0.669493 0.742819i \(-0.266512\pi\)
0.669493 + 0.742819i \(0.266512\pi\)
\(374\) 1.77820e10 0.0469958
\(375\) 1.36800e11 0.357227
\(376\) 7.18263e10 0.185327
\(377\) 8.00034e9 0.0203973
\(378\) −4.00057e10 −0.100788
\(379\) −1.07548e11 −0.267749 −0.133875 0.990998i \(-0.542742\pi\)
−0.133875 + 0.990998i \(0.542742\pi\)
\(380\) −2.36793e11 −0.582564
\(381\) 6.69642e11 1.62810
\(382\) −6.43405e9 −0.0154596
\(383\) 6.05542e11 1.43797 0.718985 0.695026i \(-0.244607\pi\)
0.718985 + 0.695026i \(0.244607\pi\)
\(384\) 3.42274e11 0.803311
\(385\) −8.38920e10 −0.194602
\(386\) 1.30451e10 0.0299093
\(387\) −3.21855e9 −0.00729391
\(388\) 3.82364e11 0.856513
\(389\) 1.31728e11 0.291678 0.145839 0.989308i \(-0.453412\pi\)
0.145839 + 0.989308i \(0.453412\pi\)
\(390\) −3.58274e10 −0.0784197
\(391\) −7.02929e11 −1.52095
\(392\) 1.51989e10 0.0325105
\(393\) 9.19308e11 1.94399
\(394\) −2.75894e10 −0.0576779
\(395\) 3.65689e11 0.755831
\(396\) −4.17055e11 −0.852245
\(397\) −6.44186e11 −1.30153 −0.650765 0.759279i \(-0.725552\pi\)
−0.650765 + 0.759279i \(0.725552\pi\)
\(398\) −5.71939e10 −0.114255
\(399\) −1.50294e11 −0.296869
\(400\) 4.20749e11 0.821776
\(401\) 4.16565e11 0.804513 0.402256 0.915527i \(-0.368226\pi\)
0.402256 + 0.915527i \(0.368226\pi\)
\(402\) −3.66492e10 −0.0699918
\(403\) −1.27161e11 −0.240150
\(404\) −8.49567e11 −1.58665
\(405\) −1.42688e12 −2.63536
\(406\) 1.74306e9 0.00318379
\(407\) 3.23343e11 0.584102
\(408\) −2.50583e11 −0.447694
\(409\) −8.23424e11 −1.45502 −0.727510 0.686097i \(-0.759322\pi\)
−0.727510 + 0.686097i \(0.759322\pi\)
\(410\) 1.63870e11 0.286400
\(411\) −8.36609e11 −1.44622
\(412\) 9.68845e11 1.65660
\(413\) 4.12253e11 0.697250
\(414\) −2.19158e11 −0.366654
\(415\) 1.02588e12 1.69778
\(416\) 5.71980e10 0.0936399
\(417\) 1.72088e12 2.78701
\(418\) 1.17113e10 0.0187634
\(419\) −6.66183e11 −1.05592 −0.527959 0.849270i \(-0.677043\pi\)
−0.527959 + 0.849270i \(0.677043\pi\)
\(420\) 5.87198e11 0.920795
\(421\) 6.72240e11 1.04293 0.521465 0.853273i \(-0.325386\pi\)
0.521465 + 0.853273i \(0.325386\pi\)
\(422\) −9.24711e10 −0.141938
\(423\) 1.22506e12 1.86049
\(424\) −9.71143e10 −0.145927
\(425\) −6.24429e11 −0.928395
\(426\) −7.03118e9 −0.0103439
\(427\) 1.89865e11 0.276388
\(428\) 1.22558e12 1.76540
\(429\) −1.33296e11 −0.190003
\(430\) −3.53114e8 −0.000498089 0
\(431\) 1.06235e12 1.48293 0.741465 0.670991i \(-0.234131\pi\)
0.741465 + 0.670991i \(0.234131\pi\)
\(432\) 1.61929e12 2.23691
\(433\) −1.17404e12 −1.60504 −0.802521 0.596624i \(-0.796508\pi\)
−0.802521 + 0.596624i \(0.796508\pi\)
\(434\) −2.77050e10 −0.0374848
\(435\) 1.35579e11 0.181548
\(436\) −1.72916e11 −0.229164
\(437\) −4.62953e11 −0.607253
\(438\) −6.31975e10 −0.0820476
\(439\) 3.13450e11 0.402790 0.201395 0.979510i \(-0.435453\pi\)
0.201395 + 0.979510i \(0.435453\pi\)
\(440\) −9.21202e10 −0.117170
\(441\) 2.59230e11 0.326371
\(442\) −2.76692e10 −0.0344824
\(443\) 5.05500e10 0.0623597 0.0311799 0.999514i \(-0.490074\pi\)
0.0311799 + 0.999514i \(0.490074\pi\)
\(444\) −2.26322e12 −2.76379
\(445\) −1.55911e11 −0.188477
\(446\) −2.05865e10 −0.0246364
\(447\) −2.97174e11 −0.352068
\(448\) −2.97167e11 −0.348538
\(449\) 8.38208e10 0.0973292 0.0486646 0.998815i \(-0.484503\pi\)
0.0486646 + 0.998815i \(0.484503\pi\)
\(450\) −1.94684e11 −0.223807
\(451\) 6.09680e11 0.693917
\(452\) −8.47114e11 −0.954594
\(453\) 2.46814e12 2.75378
\(454\) −2.03407e11 −0.224706
\(455\) 1.30538e11 0.142786
\(456\) −1.65035e11 −0.178746
\(457\) −1.47870e12 −1.58583 −0.792913 0.609334i \(-0.791437\pi\)
−0.792913 + 0.609334i \(0.791437\pi\)
\(458\) 1.82850e11 0.194178
\(459\) −2.40317e12 −2.52713
\(460\) 1.80875e12 1.88351
\(461\) 1.45505e12 1.50046 0.750228 0.661179i \(-0.229943\pi\)
0.750228 + 0.661179i \(0.229943\pi\)
\(462\) −2.90416e10 −0.0296573
\(463\) −1.20315e12 −1.21676 −0.608382 0.793644i \(-0.708181\pi\)
−0.608382 + 0.793644i \(0.708181\pi\)
\(464\) −7.05529e10 −0.0706617
\(465\) −2.15496e12 −2.13748
\(466\) 1.50348e11 0.147693
\(467\) −1.29013e12 −1.25518 −0.627592 0.778542i \(-0.715959\pi\)
−0.627592 + 0.778542i \(0.715959\pi\)
\(468\) 6.48947e11 0.625321
\(469\) 1.33532e11 0.127440
\(470\) 1.34404e11 0.127049
\(471\) −3.55199e12 −3.32566
\(472\) 4.52687e11 0.419816
\(473\) −1.31376e9 −0.00120682
\(474\) 1.26594e11 0.115189
\(475\) −4.11252e11 −0.370669
\(476\) 4.53488e11 0.404888
\(477\) −1.65637e12 −1.46496
\(478\) 1.03997e11 0.0911163
\(479\) −7.45278e11 −0.646857 −0.323429 0.946253i \(-0.604836\pi\)
−0.323429 + 0.946253i \(0.604836\pi\)
\(480\) 9.69315e11 0.833450
\(481\) −5.03129e11 −0.428575
\(482\) 2.16114e10 0.0182377
\(483\) 1.14803e12 0.959819
\(484\) 1.02120e12 0.845873
\(485\) 1.44050e12 1.18216
\(486\) −1.65995e11 −0.134968
\(487\) 5.96435e11 0.480488 0.240244 0.970712i \(-0.422772\pi\)
0.240244 + 0.970712i \(0.422772\pi\)
\(488\) 2.08487e11 0.166414
\(489\) −2.84845e12 −2.25278
\(490\) 2.84407e10 0.0222873
\(491\) −5.32558e11 −0.413523 −0.206762 0.978391i \(-0.566292\pi\)
−0.206762 + 0.978391i \(0.566292\pi\)
\(492\) −4.26743e12 −3.28340
\(493\) 1.04707e11 0.0798295
\(494\) −1.82231e10 −0.0137674
\(495\) −1.57119e12 −1.17627
\(496\) 1.12140e12 0.831944
\(497\) 2.56182e10 0.0188341
\(498\) 3.55139e11 0.258741
\(499\) −9.55707e11 −0.690037 −0.345019 0.938596i \(-0.612127\pi\)
−0.345019 + 0.938596i \(0.612127\pi\)
\(500\) −2.71852e11 −0.194522
\(501\) 4.24120e11 0.300759
\(502\) −7.88228e10 −0.0553969
\(503\) 1.79300e12 1.24889 0.624444 0.781070i \(-0.285326\pi\)
0.624444 + 0.781070i \(0.285326\pi\)
\(504\) 2.84656e11 0.196509
\(505\) −3.20062e12 −2.18989
\(506\) −8.94570e10 −0.0606648
\(507\) 2.07412e11 0.139411
\(508\) −1.33073e12 −0.886552
\(509\) 2.18143e12 1.44050 0.720248 0.693717i \(-0.244028\pi\)
0.720248 + 0.693717i \(0.244028\pi\)
\(510\) −4.68901e11 −0.306913
\(511\) 2.30261e11 0.149392
\(512\) −8.44412e11 −0.543050
\(513\) −1.58274e12 −1.00898
\(514\) −7.30796e10 −0.0461809
\(515\) 3.64998e12 2.28643
\(516\) 9.19561e9 0.00571027
\(517\) 5.00051e11 0.307827
\(518\) −1.09618e11 −0.0668958
\(519\) −2.19765e12 −1.32955
\(520\) 1.43341e11 0.0859718
\(521\) 1.96104e11 0.116605 0.0583025 0.998299i \(-0.481431\pi\)
0.0583025 + 0.998299i \(0.481431\pi\)
\(522\) 3.26453e10 0.0192444
\(523\) 2.98958e12 1.74724 0.873620 0.486609i \(-0.161766\pi\)
0.873620 + 0.486609i \(0.161766\pi\)
\(524\) −1.82688e12 −1.05857
\(525\) 1.01982e12 0.585876
\(526\) 2.77585e11 0.158110
\(527\) −1.66426e12 −0.939882
\(528\) 1.17550e12 0.658220
\(529\) 1.73511e12 0.963336
\(530\) −1.81724e11 −0.100039
\(531\) 7.72098e12 4.21451
\(532\) 2.98669e11 0.161655
\(533\) −9.48677e11 −0.509150
\(534\) −5.39732e10 −0.0287238
\(535\) 4.61718e12 2.43660
\(536\) 1.46629e11 0.0767322
\(537\) −6.05321e12 −3.14124
\(538\) −7.77113e10 −0.0399911
\(539\) 1.05814e11 0.0539998
\(540\) 6.18373e12 3.12953
\(541\) 2.82256e11 0.141663 0.0708314 0.997488i \(-0.477435\pi\)
0.0708314 + 0.997488i \(0.477435\pi\)
\(542\) −1.88150e11 −0.0936498
\(543\) 1.18267e12 0.583801
\(544\) 7.48594e11 0.366481
\(545\) −6.51436e11 −0.316292
\(546\) 4.51894e10 0.0217606
\(547\) 2.45530e12 1.17263 0.586315 0.810083i \(-0.300578\pi\)
0.586315 + 0.810083i \(0.300578\pi\)
\(548\) 1.66253e12 0.787513
\(549\) 3.55593e12 1.67062
\(550\) −7.94668e10 −0.0370300
\(551\) 6.89603e10 0.0318726
\(552\) 1.26062e12 0.577910
\(553\) −4.61246e11 −0.209734
\(554\) −1.07960e11 −0.0486932
\(555\) −8.52636e12 −3.81457
\(556\) −3.41978e12 −1.51762
\(557\) −2.04361e11 −0.0899602 −0.0449801 0.998988i \(-0.514322\pi\)
−0.0449801 + 0.998988i \(0.514322\pi\)
\(558\) −5.18881e11 −0.226576
\(559\) 2.04424e9 0.000885481 0
\(560\) −1.15118e12 −0.494649
\(561\) −1.74455e12 −0.743619
\(562\) −1.41386e11 −0.0597853
\(563\) −3.84312e12 −1.61212 −0.806058 0.591837i \(-0.798403\pi\)
−0.806058 + 0.591837i \(0.798403\pi\)
\(564\) −3.50008e12 −1.45654
\(565\) −3.19137e12 −1.31753
\(566\) −1.59348e11 −0.0652641
\(567\) 1.79973e12 0.731281
\(568\) 2.81309e10 0.0113401
\(569\) 2.72425e12 1.08954 0.544768 0.838587i \(-0.316618\pi\)
0.544768 + 0.838587i \(0.316618\pi\)
\(570\) −3.08821e11 −0.122538
\(571\) −6.05815e11 −0.238494 −0.119247 0.992865i \(-0.538048\pi\)
−0.119247 + 0.992865i \(0.538048\pi\)
\(572\) 2.64890e11 0.103463
\(573\) 6.31229e11 0.244619
\(574\) −2.06691e11 −0.0794727
\(575\) 3.14135e12 1.19843
\(576\) −5.56557e12 −2.10673
\(577\) −4.47921e12 −1.68233 −0.841164 0.540781i \(-0.818129\pi\)
−0.841164 + 0.540781i \(0.818129\pi\)
\(578\) −5.47847e10 −0.0204166
\(579\) −1.27983e12 −0.473257
\(580\) −2.69427e11 −0.0988587
\(581\) −1.29395e12 −0.471114
\(582\) 4.98670e11 0.180161
\(583\) −6.76105e11 −0.242385
\(584\) 2.52845e11 0.0899491
\(585\) 2.44481e12 0.863066
\(586\) −7.27942e10 −0.0255010
\(587\) 5.14513e12 1.78865 0.894325 0.447418i \(-0.147656\pi\)
0.894325 + 0.447418i \(0.147656\pi\)
\(588\) −7.40638e11 −0.255510
\(589\) −1.09609e12 −0.375255
\(590\) 8.47085e11 0.287802
\(591\) 2.70673e12 0.912643
\(592\) 4.43696e12 1.48470
\(593\) −4.16705e12 −1.38383 −0.691915 0.721979i \(-0.743233\pi\)
−0.691915 + 0.721979i \(0.743233\pi\)
\(594\) −3.05835e11 −0.100797
\(595\) 1.70845e12 0.558825
\(596\) 5.90552e11 0.191712
\(597\) 5.61115e12 1.80787
\(598\) 1.39197e11 0.0445118
\(599\) −1.65167e12 −0.524207 −0.262103 0.965040i \(-0.584416\pi\)
−0.262103 + 0.965040i \(0.584416\pi\)
\(600\) 1.11984e12 0.352757
\(601\) −5.18671e12 −1.62165 −0.810824 0.585289i \(-0.800981\pi\)
−0.810824 + 0.585289i \(0.800981\pi\)
\(602\) 4.45385e8 0.000138214 0
\(603\) 2.50089e12 0.770311
\(604\) −4.90477e12 −1.49952
\(605\) 3.84720e12 1.16747
\(606\) −1.10799e12 −0.333739
\(607\) −6.39931e11 −0.191330 −0.0956652 0.995414i \(-0.530498\pi\)
−0.0956652 + 0.995414i \(0.530498\pi\)
\(608\) 4.93028e11 0.146320
\(609\) −1.71007e11 −0.0503775
\(610\) 3.90128e11 0.114084
\(611\) −7.78091e11 −0.225863
\(612\) 8.49327e12 2.44734
\(613\) −4.23624e11 −0.121174 −0.0605868 0.998163i \(-0.519297\pi\)
−0.0605868 + 0.998163i \(0.519297\pi\)
\(614\) −6.58401e11 −0.186953
\(615\) −1.60769e13 −4.53173
\(616\) 1.16192e11 0.0325134
\(617\) −2.26957e12 −0.630465 −0.315233 0.949014i \(-0.602083\pi\)
−0.315233 + 0.949014i \(0.602083\pi\)
\(618\) 1.26355e12 0.348452
\(619\) −2.03001e12 −0.555763 −0.277882 0.960615i \(-0.589632\pi\)
−0.277882 + 0.960615i \(0.589632\pi\)
\(620\) 4.28240e12 1.16393
\(621\) 1.20898e13 3.26216
\(622\) −6.89791e11 −0.184782
\(623\) 1.96652e11 0.0523001
\(624\) −1.82911e12 −0.482958
\(625\) −4.28684e12 −1.12377
\(626\) 8.67798e11 0.225857
\(627\) −1.14897e12 −0.296896
\(628\) 7.05863e12 1.81093
\(629\) −6.58484e12 −1.67733
\(630\) 5.32658e11 0.134715
\(631\) −1.89988e12 −0.477083 −0.238542 0.971132i \(-0.576669\pi\)
−0.238542 + 0.971132i \(0.576669\pi\)
\(632\) −5.06486e11 −0.126282
\(633\) 9.07210e12 2.24590
\(634\) 2.17878e11 0.0535563
\(635\) −5.01334e12 −1.22362
\(636\) 4.73236e12 1.14689
\(637\) −1.64648e11 −0.0396214
\(638\) 1.33253e10 0.00318408
\(639\) 4.79797e11 0.113842
\(640\) −2.56247e12 −0.603738
\(641\) 3.58899e12 0.839675 0.419837 0.907599i \(-0.362087\pi\)
0.419837 + 0.907599i \(0.362087\pi\)
\(642\) 1.59837e12 0.371338
\(643\) −1.24458e12 −0.287127 −0.143564 0.989641i \(-0.545856\pi\)
−0.143564 + 0.989641i \(0.545856\pi\)
\(644\) −2.28139e12 −0.522653
\(645\) 3.46431e10 0.00788130
\(646\) −2.38500e11 −0.0538817
\(647\) 6.58112e12 1.47649 0.738246 0.674532i \(-0.235655\pi\)
0.738246 + 0.674532i \(0.235655\pi\)
\(648\) 1.97625e12 0.440306
\(649\) 3.15158e12 0.697313
\(650\) 1.23652e11 0.0271701
\(651\) 2.71807e12 0.593125
\(652\) 5.66052e12 1.22671
\(653\) 2.73810e12 0.589304 0.294652 0.955605i \(-0.404796\pi\)
0.294652 + 0.955605i \(0.404796\pi\)
\(654\) −2.25513e11 −0.0482028
\(655\) −6.88248e12 −1.46103
\(656\) 8.36613e12 1.76383
\(657\) 4.31250e12 0.902994
\(658\) −1.69525e11 −0.0352547
\(659\) −1.55865e12 −0.321933 −0.160966 0.986960i \(-0.551461\pi\)
−0.160966 + 0.986960i \(0.551461\pi\)
\(660\) 4.48900e12 0.920878
\(661\) −4.31825e12 −0.879836 −0.439918 0.898038i \(-0.644992\pi\)
−0.439918 + 0.898038i \(0.644992\pi\)
\(662\) 2.33855e11 0.0473244
\(663\) 2.71456e12 0.545618
\(664\) −1.42087e12 −0.283659
\(665\) 1.12519e12 0.223115
\(666\) −2.05301e12 −0.404350
\(667\) −5.26754e11 −0.103048
\(668\) −8.42823e11 −0.163773
\(669\) 2.01969e12 0.389823
\(670\) 2.74378e11 0.0526032
\(671\) 1.45147e12 0.276413
\(672\) −1.22261e12 −0.231273
\(673\) −3.11290e12 −0.584921 −0.292461 0.956278i \(-0.594474\pi\)
−0.292461 + 0.956278i \(0.594474\pi\)
\(674\) 9.63112e11 0.179766
\(675\) 1.07396e13 1.99123
\(676\) −4.12175e11 −0.0759139
\(677\) 5.23656e12 0.958071 0.479035 0.877796i \(-0.340987\pi\)
0.479035 + 0.877796i \(0.340987\pi\)
\(678\) −1.10479e12 −0.200791
\(679\) −1.81691e12 −0.328035
\(680\) 1.87602e12 0.336470
\(681\) 1.99558e13 3.55555
\(682\) −2.11799e11 −0.0374882
\(683\) 1.03001e13 1.81112 0.905560 0.424218i \(-0.139451\pi\)
0.905560 + 0.424218i \(0.139451\pi\)
\(684\) 5.59371e12 0.977118
\(685\) 6.26335e12 1.08692
\(686\) −3.58725e10 −0.00618447
\(687\) −1.79389e13 −3.07250
\(688\) −1.80276e10 −0.00306754
\(689\) 1.05203e12 0.177846
\(690\) 2.35893e12 0.396181
\(691\) −2.86194e12 −0.477540 −0.238770 0.971076i \(-0.576744\pi\)
−0.238770 + 0.971076i \(0.576744\pi\)
\(692\) 4.36723e12 0.723983
\(693\) 1.98176e12 0.326400
\(694\) −3.56575e11 −0.0583489
\(695\) −1.28835e13 −2.09461
\(696\) −1.87780e11 −0.0303324
\(697\) −1.24161e13 −1.99267
\(698\) −3.28282e11 −0.0523478
\(699\) −1.47502e13 −2.33696
\(700\) −2.02661e12 −0.319028
\(701\) 4.39926e12 0.688095 0.344048 0.938952i \(-0.388202\pi\)
0.344048 + 0.938952i \(0.388202\pi\)
\(702\) 4.75886e11 0.0739582
\(703\) −4.33681e12 −0.669686
\(704\) −2.27178e12 −0.348569
\(705\) −1.31860e13 −2.01031
\(706\) −1.18437e12 −0.179418
\(707\) 4.03696e12 0.607670
\(708\) −2.20594e13 −3.29946
\(709\) −5.62114e12 −0.835442 −0.417721 0.908575i \(-0.637171\pi\)
−0.417721 + 0.908575i \(0.637171\pi\)
\(710\) 5.26396e10 0.00777409
\(711\) −8.63857e12 −1.26773
\(712\) 2.15940e11 0.0314900
\(713\) 8.37249e12 1.21325
\(714\) 5.91429e11 0.0851649
\(715\) 9.97933e11 0.142799
\(716\) 1.20291e13 1.71051
\(717\) −1.02029e13 −1.44174
\(718\) −2.43605e11 −0.0342079
\(719\) 4.66981e11 0.0651657 0.0325828 0.999469i \(-0.489627\pi\)
0.0325828 + 0.999469i \(0.489627\pi\)
\(720\) −2.15601e13 −2.98989
\(721\) −4.60375e12 −0.634458
\(722\) 6.79233e11 0.0930253
\(723\) −2.12024e12 −0.288577
\(724\) −2.35024e12 −0.317899
\(725\) −4.67928e11 −0.0629011
\(726\) 1.33182e12 0.177923
\(727\) 3.43399e12 0.455926 0.227963 0.973670i \(-0.426793\pi\)
0.227963 + 0.973670i \(0.426793\pi\)
\(728\) −1.80797e11 −0.0238562
\(729\) 1.53143e12 0.200827
\(730\) 4.73134e11 0.0616639
\(731\) 2.67546e10 0.00346553
\(732\) −1.01595e13 −1.30790
\(733\) 1.03659e13 1.32630 0.663149 0.748488i \(-0.269220\pi\)
0.663149 + 0.748488i \(0.269220\pi\)
\(734\) −9.02038e10 −0.0114708
\(735\) −2.79024e12 −0.352654
\(736\) −3.76600e12 −0.473075
\(737\) 1.02082e12 0.127452
\(738\) −3.87106e12 −0.480371
\(739\) −1.30128e12 −0.160499 −0.0802495 0.996775i \(-0.525572\pi\)
−0.0802495 + 0.996775i \(0.525572\pi\)
\(740\) 1.69438e13 2.07716
\(741\) 1.78782e12 0.217842
\(742\) 2.29210e11 0.0277598
\(743\) −1.65678e12 −0.199441 −0.0997204 0.995015i \(-0.531795\pi\)
−0.0997204 + 0.995015i \(0.531795\pi\)
\(744\) 2.98466e12 0.357122
\(745\) 2.22482e12 0.264601
\(746\) −1.29733e12 −0.153365
\(747\) −2.42341e13 −2.84764
\(748\) 3.46682e12 0.404924
\(749\) −5.82369e12 −0.676129
\(750\) −3.54544e11 −0.0409161
\(751\) −1.21749e13 −1.39664 −0.698321 0.715785i \(-0.746069\pi\)
−0.698321 + 0.715785i \(0.746069\pi\)
\(752\) 6.86178e12 0.782450
\(753\) 7.73311e12 0.876550
\(754\) −2.07345e10 −0.00233627
\(755\) −1.84780e13 −2.06963
\(756\) −7.79959e12 −0.868408
\(757\) −5.32413e12 −0.589274 −0.294637 0.955609i \(-0.595199\pi\)
−0.294637 + 0.955609i \(0.595199\pi\)
\(758\) 2.78733e11 0.0306674
\(759\) 8.77640e12 0.959905
\(760\) 1.23555e12 0.134338
\(761\) −1.33415e13 −1.44202 −0.721012 0.692923i \(-0.756323\pi\)
−0.721012 + 0.692923i \(0.756323\pi\)
\(762\) −1.73551e12 −0.186479
\(763\) 8.21662e11 0.0877673
\(764\) −1.25440e12 −0.133203
\(765\) 3.19971e13 3.37781
\(766\) −1.56938e12 −0.164702
\(767\) −4.90394e12 −0.511642
\(768\) 1.52255e13 1.57924
\(769\) 1.20587e13 1.24346 0.621729 0.783233i \(-0.286431\pi\)
0.621729 + 0.783233i \(0.286431\pi\)
\(770\) 2.17423e11 0.0222893
\(771\) 7.16965e12 0.730724
\(772\) 2.54331e12 0.257704
\(773\) −2.60650e12 −0.262573 −0.131286 0.991344i \(-0.541911\pi\)
−0.131286 + 0.991344i \(0.541911\pi\)
\(774\) 8.34151e9 0.000835430 0
\(775\) 7.43748e12 0.740573
\(776\) −1.99512e12 −0.197511
\(777\) 1.07544e13 1.05850
\(778\) −3.41399e11 −0.0334083
\(779\) −8.17728e12 −0.795591
\(780\) −6.98499e12 −0.675679
\(781\) 1.95846e11 0.0188358
\(782\) 1.82178e12 0.174207
\(783\) −1.80086e12 −0.171219
\(784\) 1.45199e12 0.137259
\(785\) 2.65923e13 2.49944
\(786\) −2.38257e12 −0.222661
\(787\) −3.98596e12 −0.370379 −0.185190 0.982703i \(-0.559290\pi\)
−0.185190 + 0.982703i \(0.559290\pi\)
\(788\) −5.37889e12 −0.496964
\(789\) −2.72331e13 −2.50179
\(790\) −9.47755e11 −0.0865714
\(791\) 4.02531e12 0.365599
\(792\) 2.17613e12 0.196527
\(793\) −2.25853e12 −0.202813
\(794\) 1.66954e12 0.149075
\(795\) 1.78285e13 1.58293
\(796\) −1.11506e13 −0.984444
\(797\) −8.84099e12 −0.776137 −0.388069 0.921631i \(-0.626858\pi\)
−0.388069 + 0.921631i \(0.626858\pi\)
\(798\) 3.89518e11 0.0340028
\(799\) −1.01835e13 −0.883966
\(800\) −3.34542e12 −0.288766
\(801\) 3.68305e12 0.316127
\(802\) −1.07961e12 −0.0921473
\(803\) 1.76029e12 0.149405
\(804\) −7.14520e12 −0.603062
\(805\) −8.59480e12 −0.721364
\(806\) 3.29564e11 0.0275063
\(807\) 7.62405e12 0.632783
\(808\) 4.43291e12 0.365880
\(809\) −3.43704e12 −0.282109 −0.141054 0.990002i \(-0.545049\pi\)
−0.141054 + 0.990002i \(0.545049\pi\)
\(810\) 3.69804e12 0.301849
\(811\) −1.68173e13 −1.36509 −0.682545 0.730843i \(-0.739127\pi\)
−0.682545 + 0.730843i \(0.739127\pi\)
\(812\) 3.39830e11 0.0274322
\(813\) 1.84589e13 1.48183
\(814\) −8.38008e11 −0.0669019
\(815\) 2.13252e13 1.69310
\(816\) −2.39390e13 −1.89017
\(817\) 1.76207e10 0.00138364
\(818\) 2.13407e12 0.166655
\(819\) −3.08366e12 −0.239491
\(820\) 3.19485e13 2.46768
\(821\) −3.29338e12 −0.252987 −0.126494 0.991967i \(-0.540372\pi\)
−0.126494 + 0.991967i \(0.540372\pi\)
\(822\) 2.16824e12 0.165647
\(823\) −1.56498e13 −1.18907 −0.594536 0.804069i \(-0.702664\pi\)
−0.594536 + 0.804069i \(0.702664\pi\)
\(824\) −5.05529e12 −0.382009
\(825\) 7.79628e12 0.585929
\(826\) −1.06844e12 −0.0798616
\(827\) 1.49483e13 1.11126 0.555630 0.831430i \(-0.312477\pi\)
0.555630 + 0.831430i \(0.312477\pi\)
\(828\) −4.27276e13 −3.15916
\(829\) 6.90186e12 0.507540 0.253770 0.967265i \(-0.418329\pi\)
0.253770 + 0.967265i \(0.418329\pi\)
\(830\) −2.65878e12 −0.194460
\(831\) 1.05917e13 0.770477
\(832\) 3.53494e12 0.255757
\(833\) −2.15488e12 −0.155068
\(834\) −4.46000e12 −0.319218
\(835\) −3.17521e12 −0.226039
\(836\) 2.28326e12 0.161669
\(837\) 2.86238e13 2.01587
\(838\) 1.72655e12 0.120943
\(839\) 1.53766e13 1.07135 0.535673 0.844425i \(-0.320058\pi\)
0.535673 + 0.844425i \(0.320058\pi\)
\(840\) −3.06391e12 −0.212334
\(841\) −1.44287e13 −0.994591
\(842\) −1.74225e12 −0.119455
\(843\) 1.38711e13 0.945988
\(844\) −1.80284e13 −1.22297
\(845\) −1.55281e12 −0.104776
\(846\) −3.17499e12 −0.213096
\(847\) −4.85251e12 −0.323960
\(848\) −9.27762e12 −0.616106
\(849\) 1.56333e13 1.03268
\(850\) 1.61833e12 0.106336
\(851\) 3.31267e13 2.16519
\(852\) −1.37081e12 −0.0891251
\(853\) 2.50784e13 1.62192 0.810960 0.585102i \(-0.198945\pi\)
0.810960 + 0.585102i \(0.198945\pi\)
\(854\) −4.92072e11 −0.0316569
\(855\) 2.10735e13 1.34862
\(856\) −6.39488e12 −0.407099
\(857\) −1.08161e12 −0.0684945 −0.0342472 0.999413i \(-0.510903\pi\)
−0.0342472 + 0.999413i \(0.510903\pi\)
\(858\) 3.45463e11 0.0217625
\(859\) 6.50623e12 0.407718 0.203859 0.979000i \(-0.434652\pi\)
0.203859 + 0.979000i \(0.434652\pi\)
\(860\) −6.88438e10 −0.00429162
\(861\) 2.02779e13 1.25750
\(862\) −2.75330e12 −0.169852
\(863\) 1.96548e13 1.20620 0.603100 0.797666i \(-0.293932\pi\)
0.603100 + 0.797666i \(0.293932\pi\)
\(864\) −1.28752e13 −0.786033
\(865\) 1.64529e13 0.999240
\(866\) 3.04275e12 0.183838
\(867\) 5.37479e12 0.323054
\(868\) −5.40143e12 −0.322976
\(869\) −3.52613e12 −0.209753
\(870\) −3.51381e11 −0.0207941
\(871\) −1.58842e12 −0.0935157
\(872\) 9.02251e11 0.0528449
\(873\) −3.40285e13 −1.98280
\(874\) 1.19983e12 0.0695536
\(875\) 1.29179e12 0.0744997
\(876\) −1.23211e13 −0.706938
\(877\) 6.41797e12 0.366353 0.183176 0.983080i \(-0.441362\pi\)
0.183176 + 0.983080i \(0.441362\pi\)
\(878\) −8.12369e11 −0.0461348
\(879\) 7.14166e12 0.403505
\(880\) −8.80051e12 −0.494693
\(881\) 6.78986e12 0.379725 0.189862 0.981811i \(-0.439196\pi\)
0.189862 + 0.981811i \(0.439196\pi\)
\(882\) −6.71846e11 −0.0373819
\(883\) −3.84292e12 −0.212734 −0.106367 0.994327i \(-0.533922\pi\)
−0.106367 + 0.994327i \(0.533922\pi\)
\(884\) −5.39445e12 −0.297107
\(885\) −8.31054e13 −4.55391
\(886\) −1.31010e11 −0.00714256
\(887\) −1.12350e13 −0.609419 −0.304710 0.952445i \(-0.598559\pi\)
−0.304710 + 0.952445i \(0.598559\pi\)
\(888\) 1.18092e13 0.637325
\(889\) 6.32336e12 0.339540
\(890\) 4.04075e11 0.0215877
\(891\) 1.37586e13 0.731347
\(892\) −4.01359e12 −0.212271
\(893\) −6.70688e12 −0.352931
\(894\) 7.70185e11 0.0403252
\(895\) 4.53179e13 2.36084
\(896\) 3.23206e12 0.167530
\(897\) −1.36563e13 −0.704315
\(898\) −2.17238e11 −0.0111479
\(899\) −1.24715e12 −0.0636794
\(900\) −3.79559e13 −1.92836
\(901\) 1.37688e13 0.696040
\(902\) −1.58011e12 −0.0794799
\(903\) −4.36956e10 −0.00218697
\(904\) 4.42011e12 0.220128
\(905\) −8.85418e12 −0.438763
\(906\) −6.39669e12 −0.315412
\(907\) −1.22450e13 −0.600796 −0.300398 0.953814i \(-0.597119\pi\)
−0.300398 + 0.953814i \(0.597119\pi\)
\(908\) −3.96567e13 −1.93611
\(909\) 7.56073e13 3.67305
\(910\) −3.38315e11 −0.0163544
\(911\) −2.30193e13 −1.10728 −0.553641 0.832755i \(-0.686762\pi\)
−0.553641 + 0.832755i \(0.686762\pi\)
\(912\) −1.57663e13 −0.754664
\(913\) −9.89199e12 −0.471157
\(914\) 3.83233e12 0.181637
\(915\) −3.82745e13 −1.80516
\(916\) 3.56488e13 1.67307
\(917\) 8.68093e12 0.405419
\(918\) 6.22829e12 0.289452
\(919\) 2.26848e13 1.04909 0.524547 0.851381i \(-0.324235\pi\)
0.524547 + 0.851381i \(0.324235\pi\)
\(920\) −9.43778e12 −0.434335
\(921\) 6.45941e13 2.95818
\(922\) −3.77105e12 −0.171859
\(923\) −3.04740e11 −0.0138205
\(924\) −5.66201e12 −0.255533
\(925\) 2.94273e13 1.32164
\(926\) 3.11821e12 0.139366
\(927\) −8.62225e13 −3.83497
\(928\) 5.60974e11 0.0248300
\(929\) −3.10075e13 −1.36583 −0.682915 0.730498i \(-0.739288\pi\)
−0.682915 + 0.730498i \(0.739288\pi\)
\(930\) 5.58501e12 0.244822
\(931\) −1.41922e12 −0.0619120
\(932\) 2.93121e13 1.27255
\(933\) 6.76736e13 2.92383
\(934\) 3.34363e12 0.143766
\(935\) 1.30607e13 0.558876
\(936\) −3.38611e12 −0.144198
\(937\) 5.09355e12 0.215870 0.107935 0.994158i \(-0.465576\pi\)
0.107935 + 0.994158i \(0.465576\pi\)
\(938\) −3.46075e11 −0.0145968
\(939\) −8.51375e13 −3.57376
\(940\) 2.62037e13 1.09468
\(941\) 4.47730e12 0.186150 0.0930749 0.995659i \(-0.470330\pi\)
0.0930749 + 0.995659i \(0.470330\pi\)
\(942\) 9.20570e12 0.380915
\(943\) 6.24622e13 2.57226
\(944\) 4.32465e13 1.77246
\(945\) −2.93838e13 −1.19857
\(946\) 3.40487e9 0.000138226 0
\(947\) −3.14466e13 −1.27057 −0.635286 0.772277i \(-0.719118\pi\)
−0.635286 + 0.772277i \(0.719118\pi\)
\(948\) 2.46810e13 0.992487
\(949\) −2.73906e12 −0.109624
\(950\) 1.06584e12 0.0424557
\(951\) −2.13754e13 −0.847427
\(952\) −2.36624e12 −0.0933666
\(953\) 4.14945e13 1.62957 0.814783 0.579765i \(-0.196856\pi\)
0.814783 + 0.579765i \(0.196856\pi\)
\(954\) 4.29282e12 0.167793
\(955\) −4.72575e12 −0.183847
\(956\) 2.02755e13 0.785075
\(957\) −1.30731e12 −0.0503820
\(958\) 1.93154e12 0.0740897
\(959\) −7.90001e12 −0.301609
\(960\) 5.99055e13 2.27639
\(961\) −6.61687e12 −0.250263
\(962\) 1.30396e12 0.0490881
\(963\) −1.09070e14 −4.08685
\(964\) 4.21340e12 0.157140
\(965\) 9.58154e12 0.355682
\(966\) −2.97534e12 −0.109936
\(967\) 4.47611e13 1.64620 0.823098 0.567900i \(-0.192244\pi\)
0.823098 + 0.567900i \(0.192244\pi\)
\(968\) −5.32845e12 −0.195057
\(969\) 2.33986e13 0.852576
\(970\) −3.73334e12 −0.135402
\(971\) 1.13104e13 0.408311 0.204155 0.978939i \(-0.434555\pi\)
0.204155 + 0.978939i \(0.434555\pi\)
\(972\) −3.23627e13 −1.16291
\(973\) 1.62501e13 0.581230
\(974\) −1.54578e12 −0.0550342
\(975\) −1.21312e13 −0.429915
\(976\) 1.99174e13 0.702599
\(977\) −4.12140e13 −1.44717 −0.723585 0.690235i \(-0.757507\pi\)
−0.723585 + 0.690235i \(0.757507\pi\)
\(978\) 7.38232e12 0.258029
\(979\) 1.50336e12 0.0523048
\(980\) 5.54485e12 0.192032
\(981\) 1.53887e13 0.530508
\(982\) 1.38023e12 0.0473641
\(983\) 2.16993e12 0.0741233 0.0370617 0.999313i \(-0.488200\pi\)
0.0370617 + 0.999313i \(0.488200\pi\)
\(984\) 2.22668e13 0.757147
\(985\) −2.02642e13 −0.685908
\(986\) −2.71368e11 −0.00914351
\(987\) 1.66317e13 0.557839
\(988\) −3.55281e12 −0.118622
\(989\) −1.34596e11 −0.00447351
\(990\) 4.07206e12 0.134727
\(991\) 3.84824e13 1.26745 0.633725 0.773559i \(-0.281525\pi\)
0.633725 + 0.773559i \(0.281525\pi\)
\(992\) −8.91639e12 −0.292339
\(993\) −2.29429e13 −0.748819
\(994\) −6.63947e10 −0.00215722
\(995\) −4.20084e13 −1.35873
\(996\) 6.92386e13 2.22936
\(997\) −5.11699e13 −1.64016 −0.820080 0.572249i \(-0.806071\pi\)
−0.820080 + 0.572249i \(0.806071\pi\)
\(998\) 2.47691e12 0.0790355
\(999\) 1.13253e14 3.59755
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 91.10.a.c.1.7 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.10.a.c.1.7 14 1.1 even 1 trivial