Properties

Label 210.12.a.p.1.2
Level $210$
Weight $12$
Character 210.1
Self dual yes
Analytic conductor $161.352$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [210,12,Mod(1,210)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(210, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("210.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 210.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: \(161.352067918\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2535805712x^{2} - 66934369575900x - 478525314115194389 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2}\cdot 5\cdot 7 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(61281.8\) of defining polynomial
Character \(\chi\) \(=\) 210.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+32.0000 q^{2} +243.000 q^{3} +1024.00 q^{4} +3125.00 q^{5} +7776.00 q^{6} +16807.0 q^{7} +32768.0 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q+32.0000 q^{2} +243.000 q^{3} +1024.00 q^{4} +3125.00 q^{5} +7776.00 q^{6} +16807.0 q^{7} +32768.0 q^{8} +59049.0 q^{9} +100000. q^{10} +66140.8 q^{11} +248832. q^{12} -1.66136e6 q^{13} +537824. q^{14} +759375. q^{15} +1.04858e6 q^{16} -2.72893e6 q^{17} +1.88957e6 q^{18} +170139. q^{19} +3.20000e6 q^{20} +4.08410e6 q^{21} +2.11651e6 q^{22} +2.02888e7 q^{23} +7.96262e6 q^{24} +9.76562e6 q^{25} -5.31635e7 q^{26} +1.43489e7 q^{27} +1.72104e7 q^{28} +1.73752e8 q^{29} +2.43000e7 q^{30} +9.43176e7 q^{31} +3.35544e7 q^{32} +1.60722e7 q^{33} -8.73259e7 q^{34} +5.25219e7 q^{35} +6.04662e7 q^{36} +5.05619e8 q^{37} +5.44446e6 q^{38} -4.03710e8 q^{39} +1.02400e8 q^{40} +9.81031e8 q^{41} +1.30691e8 q^{42} -6.55889e8 q^{43} +6.77282e7 q^{44} +1.84528e8 q^{45} +6.49241e8 q^{46} -1.20259e9 q^{47} +2.54804e8 q^{48} +2.82475e8 q^{49} +3.12500e8 q^{50} -6.63131e8 q^{51} -1.70123e9 q^{52} +4.49627e9 q^{53} +4.59165e8 q^{54} +2.06690e8 q^{55} +5.50732e8 q^{56} +4.13439e7 q^{57} +5.56008e9 q^{58} -3.76456e8 q^{59} +7.77600e8 q^{60} -1.37967e9 q^{61} +3.01816e9 q^{62} +9.92437e8 q^{63} +1.07374e9 q^{64} -5.19175e9 q^{65} +5.14311e8 q^{66} -4.62132e8 q^{67} -2.79443e9 q^{68} +4.93017e9 q^{69} +1.68070e9 q^{70} +2.61019e10 q^{71} +1.93492e9 q^{72} -1.98822e10 q^{73} +1.61798e10 q^{74} +2.37305e9 q^{75} +1.74223e8 q^{76} +1.11163e9 q^{77} -1.29187e10 q^{78} +4.05647e10 q^{79} +3.27680e9 q^{80} +3.48678e9 q^{81} +3.13930e10 q^{82} -3.73268e10 q^{83} +4.18212e9 q^{84} -8.52792e9 q^{85} -2.09884e10 q^{86} +4.22218e10 q^{87} +2.16730e9 q^{88} -3.35790e10 q^{89} +5.90490e9 q^{90} -2.79225e10 q^{91} +2.07757e10 q^{92} +2.29192e10 q^{93} -3.84827e10 q^{94} +5.31686e8 q^{95} +8.15373e9 q^{96} -3.04604e10 q^{97} +9.03921e9 q^{98} +3.90555e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 128 q^{2} + 972 q^{3} + 4096 q^{4} + 12500 q^{5} + 31104 q^{6} + 67228 q^{7} + 131072 q^{8} + 236196 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 128 q^{2} + 972 q^{3} + 4096 q^{4} + 12500 q^{5} + 31104 q^{6} + 67228 q^{7} + 131072 q^{8} + 236196 q^{9} + 400000 q^{10} + 458260 q^{11} + 995328 q^{12} + 1574316 q^{13} + 2151296 q^{14} + 3037500 q^{15} + 4194304 q^{16} + 8678072 q^{17} + 7558272 q^{18} + 12442004 q^{19} + 12800000 q^{20} + 16336404 q^{21} + 14664320 q^{22} + 513088 q^{23} + 31850496 q^{24} + 39062500 q^{25} + 50378112 q^{26} + 57395628 q^{27} + 68841472 q^{28} + 58476696 q^{29} + 97200000 q^{30} + 145189572 q^{31} + 134217728 q^{32} + 111357180 q^{33} + 277698304 q^{34} + 210087500 q^{35} + 241864704 q^{36} + 340912752 q^{37} + 398144128 q^{38} + 382558788 q^{39} + 409600000 q^{40} + 915147368 q^{41} + 522764928 q^{42} + 462244024 q^{43} + 469258240 q^{44} + 738112500 q^{45} + 16418816 q^{46} + 901710040 q^{47} + 1019215872 q^{48} + 1129900996 q^{49} + 1250000000 q^{50} + 2108771496 q^{51} + 1612099584 q^{52} - 157945788 q^{53} + 1836660096 q^{54} + 1432062500 q^{55} + 2202927104 q^{56} + 3023406972 q^{57} + 1871254272 q^{58} + 2706989128 q^{59} + 3110400000 q^{60} + 8740846920 q^{61} + 4646066304 q^{62} + 3969746172 q^{63} + 4294967296 q^{64} + 4919737500 q^{65} + 3563429760 q^{66} + 5883134368 q^{67} + 8886345728 q^{68} + 124680384 q^{69} + 6722800000 q^{70} + 344015372 q^{71} + 7739670528 q^{72} + 10549706244 q^{73} + 10909208064 q^{74} + 9492187500 q^{75} + 12740612096 q^{76} + 7701975820 q^{77} + 12241881216 q^{78} - 430177976 q^{79} + 13107200000 q^{80} + 13947137604 q^{81} + 29284715776 q^{82} + 28504941432 q^{83} + 16728477696 q^{84} + 27118975000 q^{85} + 14791808768 q^{86} + 14209837128 q^{87} + 15016263680 q^{88} + 26763786680 q^{89} + 23619600000 q^{90} + 26459529012 q^{91} + 525402112 q^{92} + 35281065996 q^{93} + 28854721280 q^{94} + 38881262500 q^{95} + 32614907904 q^{96} + 62389990476 q^{97} + 36156831872 q^{98} + 27059794740 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 32.0000 0.707107
\(3\) 243.000 0.577350
\(4\) 1024.00 0.500000
\(5\) 3125.00 0.447214
\(6\) 7776.00 0.408248
\(7\) 16807.0 0.377964
\(8\) 32768.0 0.353553
\(9\) 59049.0 0.333333
\(10\) 100000. 0.316228
\(11\) 66140.8 0.123825 0.0619127 0.998082i \(-0.480280\pi\)
0.0619127 + 0.998082i \(0.480280\pi\)
\(12\) 248832. 0.288675
\(13\) −1.66136e6 −1.24101 −0.620505 0.784203i \(-0.713072\pi\)
−0.620505 + 0.784203i \(0.713072\pi\)
\(14\) 537824. 0.267261
\(15\) 759375. 0.258199
\(16\) 1.04858e6 0.250000
\(17\) −2.72893e6 −0.466148 −0.233074 0.972459i \(-0.574878\pi\)
−0.233074 + 0.972459i \(0.574878\pi\)
\(18\) 1.88957e6 0.235702
\(19\) 170139. 0.0157638 0.00788189 0.999969i \(-0.497491\pi\)
0.00788189 + 0.999969i \(0.497491\pi\)
\(20\) 3.20000e6 0.223607
\(21\) 4.08410e6 0.218218
\(22\) 2.11651e6 0.0875577
\(23\) 2.02888e7 0.657283 0.328642 0.944455i \(-0.393409\pi\)
0.328642 + 0.944455i \(0.393409\pi\)
\(24\) 7.96262e6 0.204124
\(25\) 9.76562e6 0.200000
\(26\) −5.31635e7 −0.877526
\(27\) 1.43489e7 0.192450
\(28\) 1.72104e7 0.188982
\(29\) 1.73752e8 1.57305 0.786524 0.617560i \(-0.211879\pi\)
0.786524 + 0.617560i \(0.211879\pi\)
\(30\) 2.43000e7 0.182574
\(31\) 9.43176e7 0.591702 0.295851 0.955234i \(-0.404397\pi\)
0.295851 + 0.955234i \(0.404397\pi\)
\(32\) 3.35544e7 0.176777
\(33\) 1.60722e7 0.0714906
\(34\) −8.73259e7 −0.329616
\(35\) 5.25219e7 0.169031
\(36\) 6.04662e7 0.166667
\(37\) 5.05619e8 1.19871 0.599354 0.800484i \(-0.295424\pi\)
0.599354 + 0.800484i \(0.295424\pi\)
\(38\) 5.44446e6 0.0111467
\(39\) −4.03710e8 −0.716497
\(40\) 1.02400e8 0.158114
\(41\) 9.81031e8 1.32243 0.661213 0.750198i \(-0.270042\pi\)
0.661213 + 0.750198i \(0.270042\pi\)
\(42\) 1.30691e8 0.154303
\(43\) −6.55889e8 −0.680383 −0.340192 0.940356i \(-0.610492\pi\)
−0.340192 + 0.940356i \(0.610492\pi\)
\(44\) 6.77282e7 0.0619127
\(45\) 1.84528e8 0.149071
\(46\) 6.49241e8 0.464770
\(47\) −1.20259e9 −0.764853 −0.382426 0.923986i \(-0.624911\pi\)
−0.382426 + 0.923986i \(0.624911\pi\)
\(48\) 2.54804e8 0.144338
\(49\) 2.82475e8 0.142857
\(50\) 3.12500e8 0.141421
\(51\) −6.63131e8 −0.269131
\(52\) −1.70123e9 −0.620505
\(53\) 4.49627e9 1.47685 0.738423 0.674338i \(-0.235571\pi\)
0.738423 + 0.674338i \(0.235571\pi\)
\(54\) 4.59165e8 0.136083
\(55\) 2.06690e8 0.0553764
\(56\) 5.50732e8 0.133631
\(57\) 4.13439e7 0.00910122
\(58\) 5.56008e9 1.11231
\(59\) −3.76456e8 −0.0685533 −0.0342766 0.999412i \(-0.510913\pi\)
−0.0342766 + 0.999412i \(0.510913\pi\)
\(60\) 7.77600e8 0.129099
\(61\) −1.37967e9 −0.209151 −0.104575 0.994517i \(-0.533348\pi\)
−0.104575 + 0.994517i \(0.533348\pi\)
\(62\) 3.01816e9 0.418397
\(63\) 9.92437e8 0.125988
\(64\) 1.07374e9 0.125000
\(65\) −5.19175e9 −0.554996
\(66\) 5.14311e8 0.0505515
\(67\) −4.62132e8 −0.0418171 −0.0209086 0.999781i \(-0.506656\pi\)
−0.0209086 + 0.999781i \(0.506656\pi\)
\(68\) −2.79443e9 −0.233074
\(69\) 4.93017e9 0.379483
\(70\) 1.68070e9 0.119523
\(71\) 2.61019e10 1.71693 0.858463 0.512875i \(-0.171420\pi\)
0.858463 + 0.512875i \(0.171420\pi\)
\(72\) 1.93492e9 0.117851
\(73\) −1.98822e10 −1.12251 −0.561254 0.827644i \(-0.689681\pi\)
−0.561254 + 0.827644i \(0.689681\pi\)
\(74\) 1.61798e10 0.847615
\(75\) 2.37305e9 0.115470
\(76\) 1.74223e8 0.00788189
\(77\) 1.11163e9 0.0468016
\(78\) −1.29187e10 −0.506640
\(79\) 4.05647e10 1.48320 0.741599 0.670844i \(-0.234068\pi\)
0.741599 + 0.670844i \(0.234068\pi\)
\(80\) 3.27680e9 0.111803
\(81\) 3.48678e9 0.111111
\(82\) 3.13930e10 0.935097
\(83\) −3.73268e10 −1.04014 −0.520070 0.854124i \(-0.674094\pi\)
−0.520070 + 0.854124i \(0.674094\pi\)
\(84\) 4.18212e9 0.109109
\(85\) −8.52792e9 −0.208468
\(86\) −2.09884e10 −0.481104
\(87\) 4.22218e10 0.908200
\(88\) 2.16730e9 0.0437789
\(89\) −3.35790e10 −0.637415 −0.318707 0.947853i \(-0.603249\pi\)
−0.318707 + 0.947853i \(0.603249\pi\)
\(90\) 5.90490e9 0.105409
\(91\) −2.79225e10 −0.469057
\(92\) 2.07757e10 0.328642
\(93\) 2.29192e10 0.341619
\(94\) −3.84827e10 −0.540833
\(95\) 5.31686e8 0.00704977
\(96\) 8.15373e9 0.102062
\(97\) −3.04604e10 −0.360156 −0.180078 0.983652i \(-0.557635\pi\)
−0.180078 + 0.983652i \(0.557635\pi\)
\(98\) 9.03921e9 0.101015
\(99\) 3.90555e9 0.0412751
\(100\) 1.00000e10 0.100000
\(101\) 1.47836e11 1.39963 0.699813 0.714326i \(-0.253266\pi\)
0.699813 + 0.714326i \(0.253266\pi\)
\(102\) −2.12202e10 −0.190304
\(103\) 8.87420e10 0.754266 0.377133 0.926159i \(-0.376910\pi\)
0.377133 + 0.926159i \(0.376910\pi\)
\(104\) −5.44394e10 −0.438763
\(105\) 1.27628e10 0.0975900
\(106\) 1.43881e11 1.04429
\(107\) 3.28122e10 0.226165 0.113082 0.993586i \(-0.463928\pi\)
0.113082 + 0.993586i \(0.463928\pi\)
\(108\) 1.46933e10 0.0962250
\(109\) 1.54613e11 0.962497 0.481248 0.876584i \(-0.340183\pi\)
0.481248 + 0.876584i \(0.340183\pi\)
\(110\) 6.61408e9 0.0391570
\(111\) 1.22865e11 0.692075
\(112\) 1.76234e10 0.0944911
\(113\) −2.41500e11 −1.23306 −0.616531 0.787330i \(-0.711463\pi\)
−0.616531 + 0.787330i \(0.711463\pi\)
\(114\) 1.32300e9 0.00643553
\(115\) 6.34024e10 0.293946
\(116\) 1.77922e11 0.786524
\(117\) −9.81016e10 −0.413670
\(118\) −1.20466e10 −0.0484745
\(119\) −4.58652e10 −0.176187
\(120\) 2.48832e10 0.0912871
\(121\) −2.80937e11 −0.984667
\(122\) −4.41493e10 −0.147892
\(123\) 2.38390e11 0.763503
\(124\) 9.65812e10 0.295851
\(125\) 3.05176e10 0.0894427
\(126\) 3.17580e10 0.0890871
\(127\) −7.42829e10 −0.199512 −0.0997558 0.995012i \(-0.531806\pi\)
−0.0997558 + 0.995012i \(0.531806\pi\)
\(128\) 3.43597e10 0.0883883
\(129\) −1.59381e11 −0.392819
\(130\) −1.66136e11 −0.392442
\(131\) 2.17771e11 0.493183 0.246592 0.969119i \(-0.420689\pi\)
0.246592 + 0.969119i \(0.420689\pi\)
\(132\) 1.64579e10 0.0357453
\(133\) 2.85953e9 0.00595815
\(134\) −1.47882e10 −0.0295692
\(135\) 4.48403e10 0.0860663
\(136\) −8.94217e10 −0.164808
\(137\) 8.39855e11 1.48676 0.743381 0.668868i \(-0.233221\pi\)
0.743381 + 0.668868i \(0.233221\pi\)
\(138\) 1.57765e11 0.268335
\(139\) −2.47328e11 −0.404289 −0.202145 0.979356i \(-0.564791\pi\)
−0.202145 + 0.979356i \(0.564791\pi\)
\(140\) 5.37824e10 0.0845154
\(141\) −2.92228e11 −0.441588
\(142\) 8.35262e11 1.21405
\(143\) −1.09884e11 −0.153668
\(144\) 6.19174e10 0.0833333
\(145\) 5.42976e11 0.703488
\(146\) −6.36231e11 −0.793732
\(147\) 6.86415e10 0.0824786
\(148\) 5.17754e11 0.599354
\(149\) −1.41322e11 −0.157647 −0.0788236 0.996889i \(-0.525116\pi\)
−0.0788236 + 0.996889i \(0.525116\pi\)
\(150\) 7.59375e10 0.0816497
\(151\) 1.11781e11 0.115876 0.0579382 0.998320i \(-0.481547\pi\)
0.0579382 + 0.998320i \(0.481547\pi\)
\(152\) 5.57513e9 0.00557333
\(153\) −1.61141e11 −0.155383
\(154\) 3.55721e10 0.0330937
\(155\) 2.94742e11 0.264617
\(156\) −4.13399e11 −0.358249
\(157\) −2.14126e12 −1.79152 −0.895758 0.444541i \(-0.853367\pi\)
−0.895758 + 0.444541i \(0.853367\pi\)
\(158\) 1.29807e12 1.04878
\(159\) 1.09259e12 0.852657
\(160\) 1.04858e11 0.0790569
\(161\) 3.40993e11 0.248430
\(162\) 1.11577e11 0.0785674
\(163\) 1.09710e12 0.746814 0.373407 0.927668i \(-0.378189\pi\)
0.373407 + 0.927668i \(0.378189\pi\)
\(164\) 1.00458e12 0.661213
\(165\) 5.02257e10 0.0319716
\(166\) −1.19446e12 −0.735490
\(167\) 1.73886e12 1.03591 0.517957 0.855407i \(-0.326693\pi\)
0.517957 + 0.855407i \(0.326693\pi\)
\(168\) 1.33828e11 0.0771517
\(169\) 9.67953e11 0.540104
\(170\) −2.72893e11 −0.147409
\(171\) 1.00466e10 0.00525459
\(172\) −6.71630e11 −0.340192
\(173\) 5.88694e11 0.288826 0.144413 0.989518i \(-0.453871\pi\)
0.144413 + 0.989518i \(0.453871\pi\)
\(174\) 1.35110e12 0.642194
\(175\) 1.64131e11 0.0755929
\(176\) 6.93536e10 0.0309563
\(177\) −9.14788e10 −0.0395793
\(178\) −1.07453e12 −0.450720
\(179\) −3.59513e12 −1.46226 −0.731128 0.682240i \(-0.761006\pi\)
−0.731128 + 0.682240i \(0.761006\pi\)
\(180\) 1.88957e11 0.0745356
\(181\) 2.38944e12 0.914247 0.457123 0.889403i \(-0.348880\pi\)
0.457123 + 0.889403i \(0.348880\pi\)
\(182\) −8.93519e11 −0.331674
\(183\) −3.35259e11 −0.120753
\(184\) 6.64823e11 0.232385
\(185\) 1.58006e12 0.536079
\(186\) 7.33413e11 0.241561
\(187\) −1.80494e11 −0.0577209
\(188\) −1.23145e12 −0.382426
\(189\) 2.41162e11 0.0727393
\(190\) 1.70139e10 0.00498494
\(191\) −3.22923e10 −0.00919210 −0.00459605 0.999989i \(-0.501463\pi\)
−0.00459605 + 0.999989i \(0.501463\pi\)
\(192\) 2.60919e11 0.0721688
\(193\) 2.01541e11 0.0541750 0.0270875 0.999633i \(-0.491377\pi\)
0.0270875 + 0.999633i \(0.491377\pi\)
\(194\) −9.74731e11 −0.254669
\(195\) −1.26159e12 −0.320427
\(196\) 2.89255e11 0.0714286
\(197\) −6.83789e12 −1.64194 −0.820971 0.570970i \(-0.806567\pi\)
−0.820971 + 0.570970i \(0.806567\pi\)
\(198\) 1.24978e11 0.0291859
\(199\) 6.86478e12 1.55932 0.779659 0.626204i \(-0.215392\pi\)
0.779659 + 0.626204i \(0.215392\pi\)
\(200\) 3.20000e11 0.0707107
\(201\) −1.12298e11 −0.0241431
\(202\) 4.73075e12 0.989686
\(203\) 2.92026e12 0.594556
\(204\) −6.79046e11 −0.134565
\(205\) 3.06572e12 0.591407
\(206\) 2.83974e12 0.533346
\(207\) 1.19803e12 0.219094
\(208\) −1.74206e12 −0.310252
\(209\) 1.12532e10 0.00195195
\(210\) 4.08410e11 0.0690066
\(211\) 3.64997e12 0.600809 0.300404 0.953812i \(-0.402878\pi\)
0.300404 + 0.953812i \(0.402878\pi\)
\(212\) 4.60418e12 0.738423
\(213\) 6.34277e12 0.991268
\(214\) 1.04999e12 0.159923
\(215\) −2.04965e12 −0.304277
\(216\) 4.70185e11 0.0680414
\(217\) 1.58520e12 0.223642
\(218\) 4.94761e12 0.680588
\(219\) −4.83138e12 −0.648080
\(220\) 2.11651e11 0.0276882
\(221\) 4.53374e12 0.578494
\(222\) 3.93169e12 0.489371
\(223\) −1.18816e13 −1.44277 −0.721386 0.692533i \(-0.756495\pi\)
−0.721386 + 0.692533i \(0.756495\pi\)
\(224\) 5.63949e11 0.0668153
\(225\) 5.76650e11 0.0666667
\(226\) −7.72799e12 −0.871907
\(227\) −3.23541e12 −0.356277 −0.178138 0.984005i \(-0.557007\pi\)
−0.178138 + 0.984005i \(0.557007\pi\)
\(228\) 4.23361e10 0.00455061
\(229\) 4.37790e11 0.0459378 0.0229689 0.999736i \(-0.492688\pi\)
0.0229689 + 0.999736i \(0.492688\pi\)
\(230\) 2.02888e12 0.207851
\(231\) 2.70126e11 0.0270209
\(232\) 5.69352e12 0.556156
\(233\) 2.27567e12 0.217096 0.108548 0.994091i \(-0.465380\pi\)
0.108548 + 0.994091i \(0.465380\pi\)
\(234\) −3.13925e12 −0.292509
\(235\) −3.75808e12 −0.342053
\(236\) −3.85491e11 −0.0342766
\(237\) 9.85722e12 0.856325
\(238\) −1.46769e12 −0.124583
\(239\) 1.18415e13 0.982239 0.491119 0.871092i \(-0.336588\pi\)
0.491119 + 0.871092i \(0.336588\pi\)
\(240\) 7.96262e11 0.0645497
\(241\) −6.95309e12 −0.550914 −0.275457 0.961313i \(-0.588829\pi\)
−0.275457 + 0.961313i \(0.588829\pi\)
\(242\) −8.98999e12 −0.696265
\(243\) 8.47289e11 0.0641500
\(244\) −1.41278e12 −0.104575
\(245\) 8.82735e11 0.0638877
\(246\) 7.62850e12 0.539878
\(247\) −2.82663e11 −0.0195630
\(248\) 3.09060e12 0.209198
\(249\) −9.07042e12 −0.600525
\(250\) 9.76562e11 0.0632456
\(251\) 1.80184e13 1.14159 0.570797 0.821091i \(-0.306634\pi\)
0.570797 + 0.821091i \(0.306634\pi\)
\(252\) 1.01626e12 0.0629941
\(253\) 1.34192e12 0.0813884
\(254\) −2.37705e12 −0.141076
\(255\) −2.07228e12 −0.120359
\(256\) 1.09951e12 0.0625000
\(257\) 1.36485e13 0.759369 0.379685 0.925116i \(-0.376033\pi\)
0.379685 + 0.925116i \(0.376033\pi\)
\(258\) −5.10019e12 −0.277765
\(259\) 8.49793e12 0.453069
\(260\) −5.31635e12 −0.277498
\(261\) 1.02599e13 0.524349
\(262\) 6.96868e12 0.348733
\(263\) −5.43786e11 −0.0266484 −0.0133242 0.999911i \(-0.504241\pi\)
−0.0133242 + 0.999911i \(0.504241\pi\)
\(264\) 5.26654e11 0.0252757
\(265\) 1.40508e13 0.660465
\(266\) 9.15051e10 0.00421304
\(267\) −8.15969e12 −0.368012
\(268\) −4.73223e11 −0.0209086
\(269\) 1.28824e13 0.557645 0.278822 0.960343i \(-0.410056\pi\)
0.278822 + 0.960343i \(0.410056\pi\)
\(270\) 1.43489e12 0.0608581
\(271\) 1.37500e13 0.571440 0.285720 0.958313i \(-0.407767\pi\)
0.285720 + 0.958313i \(0.407767\pi\)
\(272\) −2.86149e12 −0.116537
\(273\) −6.78516e12 −0.270810
\(274\) 2.68754e13 1.05130
\(275\) 6.45906e11 0.0247651
\(276\) 5.04850e12 0.189741
\(277\) 3.09874e12 0.114168 0.0570842 0.998369i \(-0.481820\pi\)
0.0570842 + 0.998369i \(0.481820\pi\)
\(278\) −7.91450e12 −0.285876
\(279\) 5.56936e12 0.197234
\(280\) 1.72104e12 0.0597614
\(281\) 4.07746e12 0.138837 0.0694185 0.997588i \(-0.477886\pi\)
0.0694185 + 0.997588i \(0.477886\pi\)
\(282\) −9.35131e12 −0.312250
\(283\) 4.41972e12 0.144733 0.0723667 0.997378i \(-0.476945\pi\)
0.0723667 + 0.997378i \(0.476945\pi\)
\(284\) 2.67284e13 0.858463
\(285\) 1.29200e11 0.00407019
\(286\) −3.51627e12 −0.108660
\(287\) 1.64882e13 0.499830
\(288\) 1.98136e12 0.0589256
\(289\) −2.68248e13 −0.782706
\(290\) 1.73752e13 0.497441
\(291\) −7.40187e12 −0.207936
\(292\) −2.03594e13 −0.561254
\(293\) 2.07901e13 0.562451 0.281226 0.959642i \(-0.409259\pi\)
0.281226 + 0.959642i \(0.409259\pi\)
\(294\) 2.19653e12 0.0583212
\(295\) −1.17643e12 −0.0306580
\(296\) 1.65681e13 0.423807
\(297\) 9.49048e11 0.0238302
\(298\) −4.52231e12 −0.111473
\(299\) −3.37069e13 −0.815695
\(300\) 2.43000e12 0.0577350
\(301\) −1.10235e13 −0.257161
\(302\) 3.57699e12 0.0819369
\(303\) 3.59241e13 0.808075
\(304\) 1.78404e11 0.00394094
\(305\) −4.31146e12 −0.0935352
\(306\) −5.15650e12 −0.109872
\(307\) −2.48567e13 −0.520213 −0.260107 0.965580i \(-0.583758\pi\)
−0.260107 + 0.965580i \(0.583758\pi\)
\(308\) 1.13831e12 0.0234008
\(309\) 2.15643e13 0.435475
\(310\) 9.43176e12 0.187113
\(311\) −4.76556e12 −0.0928821 −0.0464411 0.998921i \(-0.514788\pi\)
−0.0464411 + 0.998921i \(0.514788\pi\)
\(312\) −1.32288e13 −0.253320
\(313\) 5.20906e13 0.980089 0.490045 0.871697i \(-0.336980\pi\)
0.490045 + 0.871697i \(0.336980\pi\)
\(314\) −6.85203e13 −1.26679
\(315\) 3.10136e12 0.0563436
\(316\) 4.15382e13 0.741599
\(317\) 1.32321e13 0.232168 0.116084 0.993239i \(-0.462966\pi\)
0.116084 + 0.993239i \(0.462966\pi\)
\(318\) 3.49630e13 0.602920
\(319\) 1.14921e13 0.194783
\(320\) 3.35544e12 0.0559017
\(321\) 7.97337e12 0.130576
\(322\) 1.09118e13 0.175666
\(323\) −4.64299e11 −0.00734825
\(324\) 3.57047e12 0.0555556
\(325\) −1.62242e13 −0.248202
\(326\) 3.51071e13 0.528078
\(327\) 3.75709e13 0.555698
\(328\) 3.21464e13 0.467548
\(329\) −2.02119e13 −0.289087
\(330\) 1.60722e12 0.0226073
\(331\) −7.72344e13 −1.06846 −0.534228 0.845341i \(-0.679398\pi\)
−0.534228 + 0.845341i \(0.679398\pi\)
\(332\) −3.82227e13 −0.520070
\(333\) 2.98563e13 0.399570
\(334\) 5.56435e13 0.732502
\(335\) −1.44416e12 −0.0187012
\(336\) 4.28249e12 0.0545545
\(337\) −1.09778e14 −1.37579 −0.687895 0.725810i \(-0.741465\pi\)
−0.687895 + 0.725810i \(0.741465\pi\)
\(338\) 3.09745e13 0.381911
\(339\) −5.86844e13 −0.711909
\(340\) −8.73259e12 −0.104234
\(341\) 6.23824e12 0.0732677
\(342\) 3.21490e11 0.00371556
\(343\) 4.74756e12 0.0539949
\(344\) −2.14922e13 −0.240552
\(345\) 1.54068e13 0.169710
\(346\) 1.88382e13 0.204231
\(347\) 9.79105e13 1.04476 0.522381 0.852712i \(-0.325044\pi\)
0.522381 + 0.852712i \(0.325044\pi\)
\(348\) 4.32352e13 0.454100
\(349\) −3.78226e13 −0.391032 −0.195516 0.980701i \(-0.562638\pi\)
−0.195516 + 0.980701i \(0.562638\pi\)
\(350\) 5.25219e12 0.0534522
\(351\) −2.38387e13 −0.238832
\(352\) 2.21932e12 0.0218894
\(353\) 1.88359e14 1.82905 0.914523 0.404533i \(-0.132566\pi\)
0.914523 + 0.404533i \(0.132566\pi\)
\(354\) −2.92732e12 −0.0279868
\(355\) 8.15685e13 0.767833
\(356\) −3.43848e13 −0.318707
\(357\) −1.11452e13 −0.101722
\(358\) −1.15044e14 −1.03397
\(359\) 1.12973e14 0.999894 0.499947 0.866056i \(-0.333353\pi\)
0.499947 + 0.866056i \(0.333353\pi\)
\(360\) 6.04662e12 0.0527046
\(361\) −1.16461e14 −0.999752
\(362\) 7.64620e13 0.646470
\(363\) −6.82677e13 −0.568498
\(364\) −2.85926e13 −0.234529
\(365\) −6.21319e13 −0.502000
\(366\) −1.07283e13 −0.0853855
\(367\) −1.22823e14 −0.962976 −0.481488 0.876453i \(-0.659904\pi\)
−0.481488 + 0.876453i \(0.659904\pi\)
\(368\) 2.12743e13 0.164321
\(369\) 5.79289e13 0.440809
\(370\) 5.05619e13 0.379065
\(371\) 7.55688e13 0.558195
\(372\) 2.34692e13 0.170810
\(373\) 2.45034e12 0.0175723 0.00878614 0.999961i \(-0.497203\pi\)
0.00878614 + 0.999961i \(0.497203\pi\)
\(374\) −5.77580e12 −0.0408149
\(375\) 7.41577e12 0.0516398
\(376\) −3.94063e13 −0.270416
\(377\) −2.88665e14 −1.95217
\(378\) 7.71719e12 0.0514344
\(379\) 2.46438e12 0.0161880 0.00809398 0.999967i \(-0.497424\pi\)
0.00809398 + 0.999967i \(0.497424\pi\)
\(380\) 5.44446e11 0.00352489
\(381\) −1.80507e13 −0.115188
\(382\) −1.03335e12 −0.00649980
\(383\) −2.49930e14 −1.54962 −0.774810 0.632194i \(-0.782155\pi\)
−0.774810 + 0.632194i \(0.782155\pi\)
\(384\) 8.34942e12 0.0510310
\(385\) 3.47384e12 0.0209303
\(386\) 6.44932e12 0.0383075
\(387\) −3.87296e13 −0.226794
\(388\) −3.11914e13 −0.180078
\(389\) −2.61748e14 −1.48991 −0.744955 0.667115i \(-0.767529\pi\)
−0.744955 + 0.667115i \(0.767529\pi\)
\(390\) −4.03710e13 −0.226576
\(391\) −5.53667e13 −0.306391
\(392\) 9.25615e12 0.0505076
\(393\) 5.29184e13 0.284740
\(394\) −2.18812e14 −1.16103
\(395\) 1.26765e14 0.663306
\(396\) 3.99928e12 0.0206376
\(397\) 8.24283e13 0.419496 0.209748 0.977755i \(-0.432736\pi\)
0.209748 + 0.977755i \(0.432736\pi\)
\(398\) 2.19673e14 1.10260
\(399\) 6.94867e11 0.00343994
\(400\) 1.02400e13 0.0500000
\(401\) −1.23889e14 −0.596675 −0.298338 0.954460i \(-0.596432\pi\)
−0.298338 + 0.954460i \(0.596432\pi\)
\(402\) −3.59354e12 −0.0170718
\(403\) −1.56695e14 −0.734308
\(404\) 1.51384e14 0.699813
\(405\) 1.08962e13 0.0496904
\(406\) 9.34482e13 0.420415
\(407\) 3.34420e13 0.148431
\(408\) −2.17295e13 −0.0951521
\(409\) −3.30706e14 −1.42878 −0.714389 0.699749i \(-0.753295\pi\)
−0.714389 + 0.699749i \(0.753295\pi\)
\(410\) 9.81031e13 0.418188
\(411\) 2.04085e14 0.858382
\(412\) 9.08718e13 0.377133
\(413\) −6.32710e12 −0.0259107
\(414\) 3.83370e13 0.154923
\(415\) −1.16646e14 −0.465164
\(416\) −5.57460e13 −0.219382
\(417\) −6.01007e13 −0.233417
\(418\) 3.60101e11 0.00138024
\(419\) −3.05793e14 −1.15678 −0.578389 0.815761i \(-0.696318\pi\)
−0.578389 + 0.815761i \(0.696318\pi\)
\(420\) 1.30691e13 0.0487950
\(421\) −4.78033e14 −1.76160 −0.880798 0.473492i \(-0.842993\pi\)
−0.880798 + 0.473492i \(0.842993\pi\)
\(422\) 1.16799e14 0.424836
\(423\) −7.10115e13 −0.254951
\(424\) 1.47334e14 0.522144
\(425\) −2.66497e13 −0.0932296
\(426\) 2.02969e14 0.700932
\(427\) −2.31880e13 −0.0790516
\(428\) 3.35997e13 0.113082
\(429\) −2.67017e13 −0.0887205
\(430\) −6.55889e13 −0.215156
\(431\) −2.72466e14 −0.882445 −0.441222 0.897398i \(-0.645455\pi\)
−0.441222 + 0.897398i \(0.645455\pi\)
\(432\) 1.50459e13 0.0481125
\(433\) −3.61376e14 −1.14097 −0.570487 0.821307i \(-0.693245\pi\)
−0.570487 + 0.821307i \(0.693245\pi\)
\(434\) 5.07262e13 0.158139
\(435\) 1.31943e14 0.406159
\(436\) 1.58323e14 0.481248
\(437\) 3.45192e12 0.0103613
\(438\) −1.54604e14 −0.458262
\(439\) −4.05139e14 −1.18590 −0.592951 0.805238i \(-0.702037\pi\)
−0.592951 + 0.805238i \(0.702037\pi\)
\(440\) 6.77282e12 0.0195785
\(441\) 1.66799e13 0.0476190
\(442\) 1.45080e14 0.409057
\(443\) −2.89899e14 −0.807284 −0.403642 0.914917i \(-0.632256\pi\)
−0.403642 + 0.914917i \(0.632256\pi\)
\(444\) 1.25814e14 0.346037
\(445\) −1.04934e14 −0.285061
\(446\) −3.80211e14 −1.02019
\(447\) −3.43413e13 −0.0910176
\(448\) 1.80464e13 0.0472456
\(449\) −6.42021e14 −1.66033 −0.830165 0.557518i \(-0.811754\pi\)
−0.830165 + 0.557518i \(0.811754\pi\)
\(450\) 1.84528e13 0.0471405
\(451\) 6.48861e13 0.163750
\(452\) −2.47296e14 −0.616531
\(453\) 2.71628e13 0.0669012
\(454\) −1.03533e14 −0.251926
\(455\) −8.72577e13 −0.209769
\(456\) 1.35476e12 0.00321777
\(457\) 7.48555e14 1.75665 0.878324 0.478066i \(-0.158662\pi\)
0.878324 + 0.478066i \(0.158662\pi\)
\(458\) 1.40093e13 0.0324830
\(459\) −3.91572e13 −0.0897102
\(460\) 6.49241e13 0.146973
\(461\) −1.97382e14 −0.441523 −0.220761 0.975328i \(-0.570854\pi\)
−0.220761 + 0.975328i \(0.570854\pi\)
\(462\) 8.64402e12 0.0191067
\(463\) 1.52003e14 0.332013 0.166007 0.986125i \(-0.446913\pi\)
0.166007 + 0.986125i \(0.446913\pi\)
\(464\) 1.82193e14 0.393262
\(465\) 7.16224e13 0.152777
\(466\) 7.28216e13 0.153510
\(467\) −5.83558e14 −1.21574 −0.607870 0.794036i \(-0.707976\pi\)
−0.607870 + 0.794036i \(0.707976\pi\)
\(468\) −1.00456e14 −0.206835
\(469\) −7.76705e12 −0.0158054
\(470\) −1.20259e14 −0.241868
\(471\) −5.20326e14 −1.03433
\(472\) −1.23357e13 −0.0242372
\(473\) −4.33810e13 −0.0842487
\(474\) 3.15431e14 0.605513
\(475\) 1.66152e12 0.00315275
\(476\) −4.69659e13 −0.0880937
\(477\) 2.65500e14 0.492282
\(478\) 3.78927e14 0.694548
\(479\) −1.00235e15 −1.81624 −0.908121 0.418708i \(-0.862483\pi\)
−0.908121 + 0.418708i \(0.862483\pi\)
\(480\) 2.54804e13 0.0456435
\(481\) −8.40014e14 −1.48761
\(482\) −2.22499e14 −0.389555
\(483\) 8.28614e13 0.143431
\(484\) −2.87680e14 −0.492334
\(485\) −9.51886e13 −0.161067
\(486\) 2.71132e13 0.0453609
\(487\) −1.92144e14 −0.317847 −0.158924 0.987291i \(-0.550802\pi\)
−0.158924 + 0.987291i \(0.550802\pi\)
\(488\) −4.52089e13 −0.0739460
\(489\) 2.66594e14 0.431174
\(490\) 2.82475e13 0.0451754
\(491\) −1.03762e15 −1.64093 −0.820466 0.571695i \(-0.806286\pi\)
−0.820466 + 0.571695i \(0.806286\pi\)
\(492\) 2.44112e14 0.381752
\(493\) −4.74159e14 −0.733273
\(494\) −9.04520e12 −0.0138331
\(495\) 1.22048e13 0.0184588
\(496\) 9.88991e13 0.147926
\(497\) 4.38695e14 0.648937
\(498\) −2.90253e14 −0.424635
\(499\) 9.42193e14 1.36329 0.681643 0.731685i \(-0.261266\pi\)
0.681643 + 0.731685i \(0.261266\pi\)
\(500\) 3.12500e13 0.0447214
\(501\) 4.22543e14 0.598085
\(502\) 5.76590e14 0.807229
\(503\) −3.71104e14 −0.513891 −0.256946 0.966426i \(-0.582716\pi\)
−0.256946 + 0.966426i \(0.582716\pi\)
\(504\) 3.25202e13 0.0445435
\(505\) 4.61987e14 0.625932
\(506\) 4.29413e13 0.0575503
\(507\) 2.35213e14 0.311829
\(508\) −7.60656e13 −0.0997558
\(509\) 6.34591e14 0.823278 0.411639 0.911347i \(-0.364957\pi\)
0.411639 + 0.911347i \(0.364957\pi\)
\(510\) −6.63131e13 −0.0851066
\(511\) −3.34160e14 −0.424268
\(512\) 3.51844e13 0.0441942
\(513\) 2.44131e12 0.00303374
\(514\) 4.36752e14 0.536955
\(515\) 2.77319e14 0.337318
\(516\) −1.63206e14 −0.196410
\(517\) −7.95400e13 −0.0947082
\(518\) 2.71934e14 0.320368
\(519\) 1.43053e14 0.166754
\(520\) −1.70123e14 −0.196221
\(521\) −5.41628e13 −0.0618149 −0.0309075 0.999522i \(-0.509840\pi\)
−0.0309075 + 0.999522i \(0.509840\pi\)
\(522\) 3.28317e14 0.370771
\(523\) 7.28576e14 0.814170 0.407085 0.913390i \(-0.366545\pi\)
0.407085 + 0.913390i \(0.366545\pi\)
\(524\) 2.22998e14 0.246592
\(525\) 3.98838e13 0.0436436
\(526\) −1.74012e13 −0.0188433
\(527\) −2.57386e14 −0.275821
\(528\) 1.68529e13 0.0178726
\(529\) −5.41175e14 −0.567978
\(530\) 4.49627e14 0.467019
\(531\) −2.22294e13 −0.0228511
\(532\) 2.92816e12 0.00297907
\(533\) −1.62984e15 −1.64114
\(534\) −2.61110e14 −0.260223
\(535\) 1.02538e14 0.101144
\(536\) −1.51431e13 −0.0147846
\(537\) −8.73618e14 −0.844234
\(538\) 4.12235e14 0.394315
\(539\) 1.86831e13 0.0176893
\(540\) 4.59165e13 0.0430331
\(541\) 2.21419e14 0.205414 0.102707 0.994712i \(-0.467250\pi\)
0.102707 + 0.994712i \(0.467250\pi\)
\(542\) 4.39999e14 0.404069
\(543\) 5.80633e14 0.527841
\(544\) −9.15678e13 −0.0824041
\(545\) 4.83165e14 0.430442
\(546\) −2.17125e14 −0.191492
\(547\) 9.82674e14 0.857984 0.428992 0.903308i \(-0.358869\pi\)
0.428992 + 0.903308i \(0.358869\pi\)
\(548\) 8.60012e14 0.743381
\(549\) −8.14679e13 −0.0697170
\(550\) 2.06690e13 0.0175115
\(551\) 2.95621e13 0.0247972
\(552\) 1.61552e14 0.134167
\(553\) 6.81771e14 0.560596
\(554\) 9.91596e13 0.0807293
\(555\) 3.83954e14 0.309505
\(556\) −2.53264e14 −0.202145
\(557\) 1.30649e15 1.03253 0.516265 0.856429i \(-0.327322\pi\)
0.516265 + 0.856429i \(0.327322\pi\)
\(558\) 1.78219e14 0.139466
\(559\) 1.08967e15 0.844362
\(560\) 5.50732e13 0.0422577
\(561\) −4.38600e13 −0.0333252
\(562\) 1.30479e14 0.0981726
\(563\) −1.38691e15 −1.03336 −0.516679 0.856179i \(-0.672832\pi\)
−0.516679 + 0.856179i \(0.672832\pi\)
\(564\) −2.99242e14 −0.220794
\(565\) −7.54686e14 −0.551442
\(566\) 1.41431e14 0.102342
\(567\) 5.86024e13 0.0419961
\(568\) 8.55308e14 0.607025
\(569\) −3.74583e14 −0.263288 −0.131644 0.991297i \(-0.542026\pi\)
−0.131644 + 0.991297i \(0.542026\pi\)
\(570\) 4.13439e12 0.00287806
\(571\) 2.71362e15 1.87090 0.935450 0.353458i \(-0.114994\pi\)
0.935450 + 0.353458i \(0.114994\pi\)
\(572\) −1.12521e14 −0.0768342
\(573\) −7.84702e12 −0.00530706
\(574\) 5.27622e14 0.353433
\(575\) 1.98133e14 0.131457
\(576\) 6.34034e13 0.0416667
\(577\) 1.15722e15 0.753268 0.376634 0.926362i \(-0.377081\pi\)
0.376634 + 0.926362i \(0.377081\pi\)
\(578\) −8.58394e14 −0.553457
\(579\) 4.89745e13 0.0312779
\(580\) 5.56008e14 0.351744
\(581\) −6.27352e14 −0.393136
\(582\) −2.36860e14 −0.147033
\(583\) 2.97387e14 0.182871
\(584\) −6.51501e14 −0.396866
\(585\) −3.06567e14 −0.184999
\(586\) 6.65283e14 0.397713
\(587\) 6.58289e12 0.00389859 0.00194929 0.999998i \(-0.499380\pi\)
0.00194929 + 0.999998i \(0.499380\pi\)
\(588\) 7.02889e13 0.0412393
\(589\) 1.60471e13 0.00932746
\(590\) −3.76456e13 −0.0216785
\(591\) −1.66161e15 −0.947976
\(592\) 5.30180e14 0.299677
\(593\) −6.61462e14 −0.370428 −0.185214 0.982698i \(-0.559298\pi\)
−0.185214 + 0.982698i \(0.559298\pi\)
\(594\) 3.03695e13 0.0168505
\(595\) −1.43329e14 −0.0787934
\(596\) −1.44714e14 −0.0788236
\(597\) 1.66814e15 0.900273
\(598\) −1.07862e15 −0.576783
\(599\) 1.91973e15 1.01717 0.508583 0.861013i \(-0.330170\pi\)
0.508583 + 0.861013i \(0.330170\pi\)
\(600\) 7.77600e13 0.0408248
\(601\) 2.36512e15 1.23039 0.615196 0.788374i \(-0.289077\pi\)
0.615196 + 0.788374i \(0.289077\pi\)
\(602\) −3.52753e14 −0.181840
\(603\) −2.72884e13 −0.0139390
\(604\) 1.14464e14 0.0579382
\(605\) −8.77928e14 −0.440357
\(606\) 1.14957e15 0.571395
\(607\) 3.33310e15 1.64176 0.820882 0.571098i \(-0.193483\pi\)
0.820882 + 0.571098i \(0.193483\pi\)
\(608\) 5.70893e12 0.00278667
\(609\) 7.09622e14 0.343267
\(610\) −1.37967e14 −0.0661393
\(611\) 1.99793e15 0.949190
\(612\) −1.65008e14 −0.0776913
\(613\) 1.94940e15 0.909637 0.454819 0.890584i \(-0.349704\pi\)
0.454819 + 0.890584i \(0.349704\pi\)
\(614\) −7.95413e14 −0.367846
\(615\) 7.44970e14 0.341449
\(616\) 3.64258e13 0.0165469
\(617\) 1.81576e15 0.817503 0.408751 0.912646i \(-0.365964\pi\)
0.408751 + 0.912646i \(0.365964\pi\)
\(618\) 6.90058e14 0.307928
\(619\) 1.78336e15 0.788753 0.394376 0.918949i \(-0.370961\pi\)
0.394376 + 0.918949i \(0.370961\pi\)
\(620\) 3.01816e14 0.132309
\(621\) 2.91122e14 0.126494
\(622\) −1.52498e14 −0.0656776
\(623\) −5.64361e14 −0.240920
\(624\) −4.23321e14 −0.179124
\(625\) 9.53674e13 0.0400000
\(626\) 1.66690e15 0.693028
\(627\) 2.73452e12 0.00112696
\(628\) −2.19265e15 −0.895758
\(629\) −1.37980e15 −0.558776
\(630\) 9.92437e13 0.0398410
\(631\) −4.41303e15 −1.75621 −0.878103 0.478471i \(-0.841191\pi\)
−0.878103 + 0.478471i \(0.841191\pi\)
\(632\) 1.32922e15 0.524390
\(633\) 8.86943e14 0.346877
\(634\) 4.23426e14 0.164167
\(635\) −2.32134e14 −0.0892243
\(636\) 1.11882e15 0.426328
\(637\) −4.69293e14 −0.177287
\(638\) 3.67748e14 0.137733
\(639\) 1.54129e15 0.572309
\(640\) 1.07374e14 0.0395285
\(641\) −2.49924e15 −0.912198 −0.456099 0.889929i \(-0.650754\pi\)
−0.456099 + 0.889929i \(0.650754\pi\)
\(642\) 2.55148e14 0.0923313
\(643\) −2.04153e15 −0.732479 −0.366239 0.930521i \(-0.619355\pi\)
−0.366239 + 0.930521i \(0.619355\pi\)
\(644\) 3.49177e14 0.124215
\(645\) −4.98065e14 −0.175674
\(646\) −1.48576e13 −0.00519600
\(647\) −1.64063e15 −0.568902 −0.284451 0.958691i \(-0.591811\pi\)
−0.284451 + 0.958691i \(0.591811\pi\)
\(648\) 1.14255e14 0.0392837
\(649\) −2.48991e13 −0.00848864
\(650\) −5.19175e14 −0.175505
\(651\) 3.85202e14 0.129120
\(652\) 1.12343e15 0.373407
\(653\) −1.06369e15 −0.350585 −0.175293 0.984516i \(-0.556087\pi\)
−0.175293 + 0.984516i \(0.556087\pi\)
\(654\) 1.20227e15 0.392938
\(655\) 6.80535e14 0.220558
\(656\) 1.02869e15 0.330607
\(657\) −1.17403e15 −0.374169
\(658\) −6.46780e14 −0.204416
\(659\) 1.39069e15 0.435874 0.217937 0.975963i \(-0.430067\pi\)
0.217937 + 0.975963i \(0.430067\pi\)
\(660\) 5.14311e13 0.0159858
\(661\) 2.83166e15 0.872838 0.436419 0.899743i \(-0.356247\pi\)
0.436419 + 0.899743i \(0.356247\pi\)
\(662\) −2.47150e15 −0.755512
\(663\) 1.10170e15 0.333994
\(664\) −1.22313e15 −0.367745
\(665\) 8.93604e12 0.00266456
\(666\) 9.55401e14 0.282538
\(667\) 3.52522e15 1.03394
\(668\) 1.78059e15 0.517957
\(669\) −2.88723e15 −0.832985
\(670\) −4.62132e13 −0.0132237
\(671\) −9.12522e13 −0.0258982
\(672\) 1.37040e14 0.0385758
\(673\) −2.97905e15 −0.831754 −0.415877 0.909421i \(-0.636525\pi\)
−0.415877 + 0.909421i \(0.636525\pi\)
\(674\) −3.51291e15 −0.972831
\(675\) 1.40126e14 0.0384900
\(676\) 9.91184e14 0.270052
\(677\) −7.08685e14 −0.191521 −0.0957603 0.995404i \(-0.530528\pi\)
−0.0957603 + 0.995404i \(0.530528\pi\)
\(678\) −1.87790e15 −0.503396
\(679\) −5.11947e14 −0.136126
\(680\) −2.79443e14 −0.0737045
\(681\) −7.86205e14 −0.205696
\(682\) 1.99624e14 0.0518081
\(683\) −5.19665e15 −1.33786 −0.668928 0.743327i \(-0.733247\pi\)
−0.668928 + 0.743327i \(0.733247\pi\)
\(684\) 1.02877e13 0.00262730
\(685\) 2.62455e15 0.664900
\(686\) 1.51922e14 0.0381802
\(687\) 1.06383e14 0.0265222
\(688\) −6.87749e14 −0.170096
\(689\) −7.46992e15 −1.83278
\(690\) 4.93017e14 0.120003
\(691\) −1.46449e15 −0.353637 −0.176819 0.984243i \(-0.556581\pi\)
−0.176819 + 0.984243i \(0.556581\pi\)
\(692\) 6.02823e14 0.144413
\(693\) 6.56405e13 0.0156005
\(694\) 3.13314e15 0.738758
\(695\) −7.72901e14 −0.180804
\(696\) 1.38353e15 0.321097
\(697\) −2.67717e15 −0.616446
\(698\) −1.21032e15 −0.276501
\(699\) 5.52989e14 0.125341
\(700\) 1.68070e14 0.0377964
\(701\) −6.77879e15 −1.51253 −0.756263 0.654267i \(-0.772977\pi\)
−0.756263 + 0.654267i \(0.772977\pi\)
\(702\) −7.62838e14 −0.168880
\(703\) 8.60257e13 0.0188962
\(704\) 7.10181e13 0.0154782
\(705\) −9.13214e14 −0.197484
\(706\) 6.02748e15 1.29333
\(707\) 2.48468e15 0.529009
\(708\) −9.36743e13 −0.0197896
\(709\) 4.86070e15 1.01893 0.509466 0.860491i \(-0.329843\pi\)
0.509466 + 0.860491i \(0.329843\pi\)
\(710\) 2.61019e15 0.542940
\(711\) 2.39530e15 0.494399
\(712\) −1.10032e15 −0.225360
\(713\) 1.91359e15 0.388916
\(714\) −3.56648e14 −0.0719282
\(715\) −3.43386e14 −0.0687226
\(716\) −3.68142e15 −0.731128
\(717\) 2.87748e15 0.567096
\(718\) 3.61513e15 0.707032
\(719\) −6.11407e14 −0.118665 −0.0593324 0.998238i \(-0.518897\pi\)
−0.0593324 + 0.998238i \(0.518897\pi\)
\(720\) 1.93492e14 0.0372678
\(721\) 1.49149e15 0.285086
\(722\) −3.72676e15 −0.706931
\(723\) −1.68960e15 −0.318070
\(724\) 2.44678e15 0.457123
\(725\) 1.69680e15 0.314610
\(726\) −2.18457e15 −0.401989
\(727\) 5.40192e15 0.986526 0.493263 0.869880i \(-0.335804\pi\)
0.493263 + 0.869880i \(0.335804\pi\)
\(728\) −9.14963e14 −0.165837
\(729\) 2.05891e14 0.0370370
\(730\) −1.98822e15 −0.354968
\(731\) 1.78988e15 0.317159
\(732\) −3.43305e14 −0.0603767
\(733\) −1.00849e16 −1.76035 −0.880176 0.474647i \(-0.842576\pi\)
−0.880176 + 0.474647i \(0.842576\pi\)
\(734\) −3.93033e15 −0.680927
\(735\) 2.14505e14 0.0368856
\(736\) 6.80778e14 0.116192
\(737\) −3.05657e13 −0.00517802
\(738\) 1.85372e15 0.311699
\(739\) 6.32820e15 1.05617 0.528087 0.849190i \(-0.322909\pi\)
0.528087 + 0.849190i \(0.322909\pi\)
\(740\) 1.61798e15 0.268039
\(741\) −6.86870e13 −0.0112947
\(742\) 2.41820e15 0.394703
\(743\) −1.59106e15 −0.257779 −0.128890 0.991659i \(-0.541141\pi\)
−0.128890 + 0.991659i \(0.541141\pi\)
\(744\) 7.51015e14 0.120781
\(745\) −4.41632e14 −0.0705020
\(746\) 7.84109e13 0.0124255
\(747\) −2.20411e15 −0.346713
\(748\) −1.84826e14 −0.0288605
\(749\) 5.51475e14 0.0854822
\(750\) 2.37305e14 0.0365148
\(751\) 2.80993e14 0.0429216 0.0214608 0.999770i \(-0.493168\pi\)
0.0214608 + 0.999770i \(0.493168\pi\)
\(752\) −1.26100e15 −0.191213
\(753\) 4.37848e15 0.659099
\(754\) −9.23728e15 −1.38039
\(755\) 3.49316e14 0.0518215
\(756\) 2.46950e14 0.0363696
\(757\) 1.75331e15 0.256349 0.128174 0.991752i \(-0.459088\pi\)
0.128174 + 0.991752i \(0.459088\pi\)
\(758\) 7.88602e13 0.0114466
\(759\) 3.26085e14 0.0469896
\(760\) 1.74223e13 0.00249247
\(761\) −7.72125e15 −1.09666 −0.548330 0.836262i \(-0.684736\pi\)
−0.548330 + 0.836262i \(0.684736\pi\)
\(762\) −5.77624e14 −0.0814503
\(763\) 2.59858e15 0.363790
\(764\) −3.30673e13 −0.00459605
\(765\) −5.03565e14 −0.0694892
\(766\) −7.99777e15 −1.09575
\(767\) 6.25429e14 0.0850753
\(768\) 2.67181e14 0.0360844
\(769\) 4.61889e15 0.619359 0.309679 0.950841i \(-0.399778\pi\)
0.309679 + 0.950841i \(0.399778\pi\)
\(770\) 1.11163e14 0.0148000
\(771\) 3.31659e15 0.438422
\(772\) 2.06378e14 0.0270875
\(773\) −3.76789e15 −0.491034 −0.245517 0.969392i \(-0.578958\pi\)
−0.245517 + 0.969392i \(0.578958\pi\)
\(774\) −1.23935e15 −0.160368
\(775\) 9.21070e14 0.118340
\(776\) −9.98125e14 −0.127334
\(777\) 2.06500e15 0.261580
\(778\) −8.37592e15 −1.05353
\(779\) 1.66912e14 0.0208464
\(780\) −1.29187e15 −0.160214
\(781\) 1.72640e15 0.212599
\(782\) −1.77173e15 −0.216651
\(783\) 2.49316e15 0.302733
\(784\) 2.96197e14 0.0357143
\(785\) −6.69143e15 −0.801191
\(786\) 1.69339e15 0.201341
\(787\) 6.50876e15 0.768488 0.384244 0.923231i \(-0.374462\pi\)
0.384244 + 0.923231i \(0.374462\pi\)
\(788\) −7.00200e15 −0.820971
\(789\) −1.32140e14 −0.0153855
\(790\) 4.05647e15 0.469028
\(791\) −4.05889e15 −0.466054
\(792\) 1.27977e14 0.0145930
\(793\) 2.29212e15 0.259558
\(794\) 2.63770e15 0.296629
\(795\) 3.41435e15 0.381320
\(796\) 7.02954e15 0.779659
\(797\) 2.41321e14 0.0265812 0.0132906 0.999912i \(-0.495769\pi\)
0.0132906 + 0.999912i \(0.495769\pi\)
\(798\) 2.22357e13 0.00243240
\(799\) 3.28178e15 0.356535
\(800\) 3.27680e14 0.0353553
\(801\) −1.98280e15 −0.212472
\(802\) −3.96444e15 −0.421913
\(803\) −1.31503e15 −0.138995
\(804\) −1.14993e14 −0.0120716
\(805\) 1.06560e15 0.111101
\(806\) −5.01425e15 −0.519234
\(807\) 3.13041e15 0.321956
\(808\) 4.84429e15 0.494843
\(809\) 7.97869e15 0.809496 0.404748 0.914428i \(-0.367359\pi\)
0.404748 + 0.914428i \(0.367359\pi\)
\(810\) 3.48678e14 0.0351364
\(811\) 7.66758e15 0.767439 0.383720 0.923450i \(-0.374643\pi\)
0.383720 + 0.923450i \(0.374643\pi\)
\(812\) 2.99034e15 0.297278
\(813\) 3.34124e15 0.329921
\(814\) 1.07014e15 0.104956
\(815\) 3.42842e15 0.333986
\(816\) −6.95343e14 −0.0672827
\(817\) −1.11592e14 −0.0107254
\(818\) −1.05826e16 −1.01030
\(819\) −1.64879e15 −0.156352
\(820\) 3.13930e15 0.295703
\(821\) −3.56124e15 −0.333206 −0.166603 0.986024i \(-0.553280\pi\)
−0.166603 + 0.986024i \(0.553280\pi\)
\(822\) 6.53072e15 0.606968
\(823\) −6.38018e15 −0.589025 −0.294512 0.955648i \(-0.595157\pi\)
−0.294512 + 0.955648i \(0.595157\pi\)
\(824\) 2.90790e15 0.266673
\(825\) 1.56955e14 0.0142981
\(826\) −2.02467e14 −0.0183216
\(827\) 2.94158e15 0.264424 0.132212 0.991221i \(-0.457792\pi\)
0.132212 + 0.991221i \(0.457792\pi\)
\(828\) 1.22678e15 0.109547
\(829\) 3.48147e15 0.308825 0.154412 0.988006i \(-0.450652\pi\)
0.154412 + 0.988006i \(0.450652\pi\)
\(830\) −3.73268e15 −0.328921
\(831\) 7.52993e14 0.0659152
\(832\) −1.78387e15 −0.155126
\(833\) −7.70856e14 −0.0665926
\(834\) −1.92322e15 −0.165050
\(835\) 5.43393e15 0.463275
\(836\) 1.15232e13 0.000975977 0
\(837\) 1.35335e15 0.113873
\(838\) −9.78537e15 −0.817966
\(839\) −7.97328e15 −0.662135 −0.331067 0.943607i \(-0.607409\pi\)
−0.331067 + 0.943607i \(0.607409\pi\)
\(840\) 4.18212e14 0.0345033
\(841\) 1.79894e16 1.47448
\(842\) −1.52971e16 −1.24564
\(843\) 9.90824e14 0.0801576
\(844\) 3.73757e15 0.300404
\(845\) 3.02485e15 0.241542
\(846\) −2.27237e15 −0.180278
\(847\) −4.72171e15 −0.372169
\(848\) 4.71468e15 0.369211
\(849\) 1.07399e15 0.0835619
\(850\) −8.52792e14 −0.0659233
\(851\) 1.02584e16 0.787891
\(852\) 6.49499e15 0.495634
\(853\) 4.80230e15 0.364108 0.182054 0.983289i \(-0.441725\pi\)
0.182054 + 0.983289i \(0.441725\pi\)
\(854\) −7.42017e14 −0.0558979
\(855\) 3.13955e13 0.00234992
\(856\) 1.07519e15 0.0799613
\(857\) −1.15145e16 −0.850847 −0.425423 0.904994i \(-0.639875\pi\)
−0.425423 + 0.904994i \(0.639875\pi\)
\(858\) −8.54455e14 −0.0627349
\(859\) −5.67712e15 −0.414158 −0.207079 0.978324i \(-0.566396\pi\)
−0.207079 + 0.978324i \(0.566396\pi\)
\(860\) −2.09884e15 −0.152138
\(861\) 4.00663e15 0.288577
\(862\) −8.71892e15 −0.623983
\(863\) −8.65956e15 −0.615796 −0.307898 0.951419i \(-0.599625\pi\)
−0.307898 + 0.951419i \(0.599625\pi\)
\(864\) 4.81469e14 0.0340207
\(865\) 1.83967e15 0.129167
\(866\) −1.15640e16 −0.806790
\(867\) −6.51843e15 −0.451896
\(868\) 1.62324e15 0.111821
\(869\) 2.68298e15 0.183657
\(870\) 4.22218e15 0.287198
\(871\) 7.67767e14 0.0518955
\(872\) 5.06635e15 0.340294
\(873\) −1.79865e15 −0.120052
\(874\) 1.10461e14 0.00732652
\(875\) 5.12909e14 0.0338062
\(876\) −4.94733e15 −0.324040
\(877\) −4.62452e15 −0.301002 −0.150501 0.988610i \(-0.548089\pi\)
−0.150501 + 0.988610i \(0.548089\pi\)
\(878\) −1.29644e16 −0.838560
\(879\) 5.05200e15 0.324731
\(880\) 2.16730e14 0.0138441
\(881\) 6.67369e15 0.423642 0.211821 0.977309i \(-0.432061\pi\)
0.211821 + 0.977309i \(0.432061\pi\)
\(882\) 5.33756e14 0.0336718
\(883\) −2.91359e16 −1.82660 −0.913302 0.407282i \(-0.866477\pi\)
−0.913302 + 0.407282i \(0.866477\pi\)
\(884\) 4.64255e15 0.289247
\(885\) −2.85871e14 −0.0177004
\(886\) −9.27677e15 −0.570836
\(887\) −3.07655e16 −1.88141 −0.940705 0.339224i \(-0.889835\pi\)
−0.940705 + 0.339224i \(0.889835\pi\)
\(888\) 4.02605e15 0.244685
\(889\) −1.24847e15 −0.0754083
\(890\) −3.35790e15 −0.201568
\(891\) 2.30619e14 0.0137584
\(892\) −1.21667e16 −0.721386
\(893\) −2.04607e14 −0.0120570
\(894\) −1.09892e15 −0.0643592
\(895\) −1.12348e16 −0.653941
\(896\) 5.77484e14 0.0334077
\(897\) −8.19079e15 −0.470942
\(898\) −2.05447e16 −1.17403
\(899\) 1.63879e16 0.930776
\(900\) 5.90490e14 0.0333333
\(901\) −1.22700e16 −0.688428
\(902\) 2.07636e15 0.115789
\(903\) −2.67872e15 −0.148472
\(904\) −7.91346e15 −0.435953
\(905\) 7.46699e15 0.408864
\(906\) 8.69209e14 0.0473063
\(907\) 1.61863e16 0.875606 0.437803 0.899071i \(-0.355757\pi\)
0.437803 + 0.899071i \(0.355757\pi\)
\(908\) −3.31306e15 −0.178138
\(909\) 8.72956e15 0.466542
\(910\) −2.79225e15 −0.148329
\(911\) −2.78417e16 −1.47009 −0.735046 0.678018i \(-0.762839\pi\)
−0.735046 + 0.678018i \(0.762839\pi\)
\(912\) 4.33522e13 0.00227530
\(913\) −2.46883e15 −0.128796
\(914\) 2.39538e16 1.24214
\(915\) −1.04768e15 −0.0540025
\(916\) 4.48297e14 0.0229689
\(917\) 3.66008e15 0.186406
\(918\) −1.25303e15 −0.0634347
\(919\) 5.36430e15 0.269946 0.134973 0.990849i \(-0.456905\pi\)
0.134973 + 0.990849i \(0.456905\pi\)
\(920\) 2.07757e15 0.103926
\(921\) −6.04017e15 −0.300345
\(922\) −6.31623e15 −0.312204
\(923\) −4.33647e16 −2.13072
\(924\) 2.76609e14 0.0135105
\(925\) 4.93768e15 0.239742
\(926\) 4.86409e15 0.234769
\(927\) 5.24012e15 0.251422
\(928\) 5.83016e15 0.278078
\(929\) 7.30043e15 0.346148 0.173074 0.984909i \(-0.444630\pi\)
0.173074 + 0.984909i \(0.444630\pi\)
\(930\) 2.29192e15 0.108030
\(931\) 4.80602e13 0.00225197
\(932\) 2.33029e15 0.108548
\(933\) −1.15803e15 −0.0536255
\(934\) −1.86738e16 −0.859658
\(935\) −5.64043e14 −0.0258136
\(936\) −3.21459e15 −0.146254
\(937\) −2.97740e16 −1.34670 −0.673348 0.739326i \(-0.735144\pi\)
−0.673348 + 0.739326i \(0.735144\pi\)
\(938\) −2.48545e14 −0.0111761
\(939\) 1.26580e16 0.565855
\(940\) −3.84827e15 −0.171026
\(941\) −1.16623e16 −0.515279 −0.257640 0.966241i \(-0.582945\pi\)
−0.257640 + 0.966241i \(0.582945\pi\)
\(942\) −1.66504e16 −0.731384
\(943\) 1.99039e16 0.869209
\(944\) −3.94743e14 −0.0171383
\(945\) 7.53631e14 0.0325300
\(946\) −1.38819e15 −0.0595728
\(947\) 6.31543e15 0.269450 0.134725 0.990883i \(-0.456985\pi\)
0.134725 + 0.990883i \(0.456985\pi\)
\(948\) 1.00938e16 0.428162
\(949\) 3.30315e16 1.39304
\(950\) 5.31686e13 0.00222933
\(951\) 3.21539e15 0.134042
\(952\) −1.50291e15 −0.0622916
\(953\) 4.74094e16 1.95368 0.976840 0.213969i \(-0.0686392\pi\)
0.976840 + 0.213969i \(0.0686392\pi\)
\(954\) 8.49601e15 0.348096
\(955\) −1.00913e14 −0.00411083
\(956\) 1.21257e16 0.491119
\(957\) 2.79259e15 0.112458
\(958\) −3.20752e16 −1.28428
\(959\) 1.41154e16 0.561943
\(960\) 8.15373e14 0.0322749
\(961\) −1.65127e16 −0.649888
\(962\) −2.68805e16 −1.05190
\(963\) 1.93753e15 0.0753882
\(964\) −7.11996e15 −0.275457
\(965\) 6.29816e14 0.0242278
\(966\) 2.65156e15 0.101421
\(967\) 1.60190e16 0.609244 0.304622 0.952473i \(-0.401470\pi\)
0.304622 + 0.952473i \(0.401470\pi\)
\(968\) −9.20575e15 −0.348132
\(969\) −1.12825e14 −0.00424251
\(970\) −3.04604e15 −0.113891
\(971\) −4.54980e16 −1.69156 −0.845778 0.533534i \(-0.820863\pi\)
−0.845778 + 0.533534i \(0.820863\pi\)
\(972\) 8.67624e14 0.0320750
\(973\) −4.15684e15 −0.152807
\(974\) −6.14862e15 −0.224752
\(975\) −3.94248e15 −0.143299
\(976\) −1.44668e15 −0.0522877
\(977\) 3.50910e16 1.26117 0.630587 0.776118i \(-0.282814\pi\)
0.630587 + 0.776118i \(0.282814\pi\)
\(978\) 8.53102e15 0.304886
\(979\) −2.22094e15 −0.0789281
\(980\) 9.03921e14 0.0319438
\(981\) 9.12972e15 0.320832
\(982\) −3.32039e16 −1.16031
\(983\) −4.45932e16 −1.54962 −0.774808 0.632197i \(-0.782153\pi\)
−0.774808 + 0.632197i \(0.782153\pi\)
\(984\) 7.81158e15 0.269939
\(985\) −2.13684e16 −0.734299
\(986\) −1.51731e16 −0.518502
\(987\) −4.91148e15 −0.166905
\(988\) −2.89447e14 −0.00978149
\(989\) −1.33072e16 −0.447205
\(990\) 3.90555e14 0.0130523
\(991\) 2.37035e16 0.787783 0.393892 0.919157i \(-0.371129\pi\)
0.393892 + 0.919157i \(0.371129\pi\)
\(992\) 3.16477e15 0.104599
\(993\) −1.87679e16 −0.616873
\(994\) 1.40382e16 0.458868
\(995\) 2.14524e16 0.697349
\(996\) −9.28811e15 −0.300262
\(997\) 4.45651e16 1.43275 0.716376 0.697714i \(-0.245799\pi\)
0.716376 + 0.697714i \(0.245799\pi\)
\(998\) 3.01502e16 0.963989
\(999\) 7.25508e15 0.230692
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 210.12.a.p.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.12.a.p.1.2 4 1.1 even 1 trivial