Properties

Label 3.18.a.b.1.1
Level $3$
Weight $18$
Character 3.1
Self dual yes
Analytic conductor $5.497$
Analytic rank $0$
Dimension $2$
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] = [3,18,Mod(1,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 3.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: \(5.49666262034\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14569}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3642 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(60.8511\) of defining polynomial
Character \(\chi\) \(=\) 3.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-65.1063 q^{2} +6561.00 q^{3} -126833. q^{4} +1.46604e6 q^{5} -427163. q^{6} +2.28730e7 q^{7} +1.67913e7 q^{8} +4.30467e7 q^{9} +O(q^{10})\) \(q-65.1063 q^{2} +6561.00 q^{3} -126833. q^{4} +1.46604e6 q^{5} -427163. q^{6} +2.28730e7 q^{7} +1.67913e7 q^{8} +4.30467e7 q^{9} -9.54488e7 q^{10} -5.40173e8 q^{11} -8.32152e8 q^{12} -3.45398e7 q^{13} -1.48918e9 q^{14} +9.61872e9 q^{15} +1.55311e10 q^{16} -9.43866e9 q^{17} -2.80261e9 q^{18} -2.25061e9 q^{19} -1.85943e11 q^{20} +1.50070e11 q^{21} +3.51687e10 q^{22} -3.45854e11 q^{23} +1.10167e11 q^{24} +1.38635e12 q^{25} +2.24876e9 q^{26} +2.82430e11 q^{27} -2.90106e12 q^{28} +5.11838e11 q^{29} -6.26239e11 q^{30} +1.14436e11 q^{31} -3.21203e12 q^{32} -3.54407e12 q^{33} +6.14516e11 q^{34} +3.35329e13 q^{35} -5.45975e12 q^{36} -1.56972e13 q^{37} +1.46529e11 q^{38} -2.26616e11 q^{39} +2.46167e13 q^{40} -8.00761e13 q^{41} -9.77050e12 q^{42} -3.66737e13 q^{43} +6.85118e13 q^{44} +6.31084e13 q^{45} +2.25173e13 q^{46} -1.17577e14 q^{47} +1.01899e14 q^{48} +2.90545e14 q^{49} -9.02599e13 q^{50} -6.19270e13 q^{51} +4.38080e12 q^{52} -3.90043e14 q^{53} -1.83880e13 q^{54} -7.91917e14 q^{55} +3.84067e14 q^{56} -1.47663e13 q^{57} -3.33239e13 q^{58} +1.82562e15 q^{59} -1.21997e15 q^{60} +1.41053e15 q^{61} -7.45051e12 q^{62} +9.84609e14 q^{63} -1.82656e15 q^{64} -5.06369e13 q^{65} +2.30742e14 q^{66} -1.47114e15 q^{67} +1.19713e15 q^{68} -2.26915e15 q^{69} -2.18320e15 q^{70} -7.31441e15 q^{71} +7.22809e14 q^{72} +1.34580e16 q^{73} +1.02199e15 q^{74} +9.09582e15 q^{75} +2.85452e14 q^{76} -1.23554e16 q^{77} +1.47541e13 q^{78} -8.37779e15 q^{79} +2.27692e16 q^{80} +1.85302e15 q^{81} +5.21346e15 q^{82} -2.55978e16 q^{83} -1.90338e16 q^{84} -1.38375e16 q^{85} +2.38769e15 q^{86} +3.35817e15 q^{87} -9.07018e15 q^{88} +4.47540e16 q^{89} -4.10876e15 q^{90} -7.90031e14 q^{91} +4.38657e16 q^{92} +7.50815e14 q^{93} +7.65502e15 q^{94} -3.29949e15 q^{95} -2.10742e16 q^{96} +7.41658e16 q^{97} -1.89163e16 q^{98} -2.32527e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 594 q^{2} + 13122 q^{3} + 176516 q^{4} + 382860 q^{5} + 3897234 q^{6} + 24471568 q^{7} + 130340232 q^{8} + 86093442 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 594 q^{2} + 13122 q^{3} + 176516 q^{4} + 382860 q^{5} + 3897234 q^{6} + 24471568 q^{7} + 130340232 q^{8} + 86093442 q^{9} - 809382420 q^{10} - 987553512 q^{11} + 1158121476 q^{12} - 2519398244 q^{13} - 435565296 q^{14} + 2511944460 q^{15} + 50611323920 q^{16} - 34313126364 q^{17} + 25569752274 q^{18} + 80053542184 q^{19} - 514526095080 q^{20} + 160557957648 q^{21} - 259702850088 q^{22} + 297228742704 q^{23} + 855162262152 q^{24} + 1796695260350 q^{25} - 1635537155076 q^{26} + 564859072962 q^{27} - 2416139220704 q^{28} - 470374069572 q^{29} - 5310358057620 q^{30} + 3400754454592 q^{31} + 5026499631648 q^{32} - 6479338592232 q^{33} - 15780403538748 q^{34} + 31801338154080 q^{35} + 7598435004036 q^{36} + 10652012180428 q^{37} + 54393717084264 q^{38} - 16529771878884 q^{39} - 98377731579600 q^{40} - 113376799448748 q^{41} - 2857743907056 q^{42} + 61637031489880 q^{43} - 67200775334352 q^{44} + 16480867602060 q^{45} + 446377123388592 q^{46} - 279645641926560 q^{47} + 332060896239120 q^{48} + 60469362475890 q^{49} + 180203550461550 q^{50} - 225128422074204 q^{51} - 749398920924104 q^{52} - 530964038611476 q^{53} + 167763144669714 q^{54} - 307321280077680 q^{55} + 565580346246720 q^{56} + 525231290269224 q^{57} - 680706445363812 q^{58} + 17\!\cdots\!56 q^{59}+ \cdots - 42\!\cdots\!52 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −65.1063 −0.179833 −0.0899163 0.995949i \(-0.528660\pi\)
−0.0899163 + 0.995949i \(0.528660\pi\)
\(3\) 6561.00 0.577350
\(4\) −126833. −0.967660
\(5\) 1.46604e6 1.67843 0.839213 0.543803i \(-0.183016\pi\)
0.839213 + 0.543803i \(0.183016\pi\)
\(6\) −427163. −0.103826
\(7\) 2.28730e7 1.49965 0.749825 0.661636i \(-0.230137\pi\)
0.749825 + 0.661636i \(0.230137\pi\)
\(8\) 1.67913e7 0.353849
\(9\) 4.30467e7 0.333333
\(10\) −9.54488e7 −0.301836
\(11\) −5.40173e8 −0.759792 −0.379896 0.925029i \(-0.624040\pi\)
−0.379896 + 0.925029i \(0.624040\pi\)
\(12\) −8.32152e8 −0.558679
\(13\) −3.45398e7 −0.0117436 −0.00587181 0.999983i \(-0.501869\pi\)
−0.00587181 + 0.999983i \(0.501869\pi\)
\(14\) −1.48918e9 −0.269686
\(15\) 9.61872e9 0.969039
\(16\) 1.55311e10 0.904027
\(17\) −9.43866e9 −0.328167 −0.164083 0.986446i \(-0.552467\pi\)
−0.164083 + 0.986446i \(0.552467\pi\)
\(18\) −2.80261e9 −0.0599442
\(19\) −2.25061e9 −0.0304015 −0.0152007 0.999884i \(-0.504839\pi\)
−0.0152007 + 0.999884i \(0.504839\pi\)
\(20\) −1.85943e11 −1.62415
\(21\) 1.50070e11 0.865824
\(22\) 3.51687e10 0.136635
\(23\) −3.45854e11 −0.920886 −0.460443 0.887689i \(-0.652310\pi\)
−0.460443 + 0.887689i \(0.652310\pi\)
\(24\) 1.10167e11 0.204295
\(25\) 1.38635e12 1.81711
\(26\) 2.24876e9 0.00211188
\(27\) 2.82430e11 0.192450
\(28\) −2.90106e12 −1.45115
\(29\) 5.11838e11 0.189998 0.0949992 0.995477i \(-0.469715\pi\)
0.0949992 + 0.995477i \(0.469715\pi\)
\(30\) −6.26239e11 −0.174265
\(31\) 1.14436e11 0.0240984 0.0120492 0.999927i \(-0.496165\pi\)
0.0120492 + 0.999927i \(0.496165\pi\)
\(32\) −3.21203e12 −0.516423
\(33\) −3.54407e12 −0.438666
\(34\) 6.14516e11 0.0590150
\(35\) 3.35329e13 2.51705
\(36\) −5.45975e12 −0.322553
\(37\) −1.56972e13 −0.734697 −0.367349 0.930083i \(-0.619734\pi\)
−0.367349 + 0.930083i \(0.619734\pi\)
\(38\) 1.46529e11 0.00546717
\(39\) −2.26616e11 −0.00678018
\(40\) 2.46167e13 0.593910
\(41\) −8.00761e13 −1.56617 −0.783087 0.621912i \(-0.786356\pi\)
−0.783087 + 0.621912i \(0.786356\pi\)
\(42\) −9.77050e12 −0.155703
\(43\) −3.66737e13 −0.478489 −0.239245 0.970959i \(-0.576900\pi\)
−0.239245 + 0.970959i \(0.576900\pi\)
\(44\) 6.85118e13 0.735221
\(45\) 6.31084e13 0.559475
\(46\) 2.25173e13 0.165605
\(47\) −1.17577e14 −0.720263 −0.360132 0.932901i \(-0.617268\pi\)
−0.360132 + 0.932901i \(0.617268\pi\)
\(48\) 1.01899e14 0.521940
\(49\) 2.90545e14 1.24895
\(50\) −9.02599e13 −0.326776
\(51\) −6.19270e13 −0.189467
\(52\) 4.38080e12 0.0113638
\(53\) −3.90043e14 −0.860534 −0.430267 0.902702i \(-0.641581\pi\)
−0.430267 + 0.902702i \(0.641581\pi\)
\(54\) −1.83880e13 −0.0346088
\(55\) −7.91917e14 −1.27525
\(56\) 3.84067e14 0.530651
\(57\) −1.47663e13 −0.0175523
\(58\) −3.33239e13 −0.0341679
\(59\) 1.82562e15 1.61870 0.809352 0.587323i \(-0.199818\pi\)
0.809352 + 0.587323i \(0.199818\pi\)
\(60\) −1.21997e15 −0.937701
\(61\) 1.41053e15 0.942059 0.471029 0.882118i \(-0.343883\pi\)
0.471029 + 0.882118i \(0.343883\pi\)
\(62\) −7.45051e12 −0.00433368
\(63\) 9.84609e14 0.499884
\(64\) −1.82656e15 −0.811157
\(65\) −5.06369e13 −0.0197108
\(66\) 2.30742e14 0.0788865
\(67\) −1.47114e15 −0.442607 −0.221304 0.975205i \(-0.571031\pi\)
−0.221304 + 0.975205i \(0.571031\pi\)
\(68\) 1.19713e15 0.317554
\(69\) −2.26915e15 −0.531674
\(70\) −2.18320e15 −0.452648
\(71\) −7.31441e15 −1.34426 −0.672130 0.740433i \(-0.734620\pi\)
−0.672130 + 0.740433i \(0.734620\pi\)
\(72\) 7.22809e14 0.117950
\(73\) 1.34580e16 1.95315 0.976577 0.215169i \(-0.0690302\pi\)
0.976577 + 0.215169i \(0.0690302\pi\)
\(74\) 1.02199e15 0.132122
\(75\) 9.09582e15 1.04911
\(76\) 2.85452e14 0.0294183
\(77\) −1.23554e16 −1.13942
\(78\) 1.47541e13 0.00121930
\(79\) −8.37779e15 −0.621297 −0.310648 0.950525i \(-0.600546\pi\)
−0.310648 + 0.950525i \(0.600546\pi\)
\(80\) 2.27692e16 1.51734
\(81\) 1.85302e15 0.111111
\(82\) 5.21346e15 0.281649
\(83\) −2.55978e16 −1.24750 −0.623748 0.781625i \(-0.714391\pi\)
−0.623748 + 0.781625i \(0.714391\pi\)
\(84\) −1.90338e16 −0.837823
\(85\) −1.38375e16 −0.550803
\(86\) 2.38769e15 0.0860480
\(87\) 3.35817e15 0.109696
\(88\) −9.07018e15 −0.268852
\(89\) 4.47540e16 1.20508 0.602541 0.798088i \(-0.294155\pi\)
0.602541 + 0.798088i \(0.294155\pi\)
\(90\) −4.10876e15 −0.100612
\(91\) −7.90031e14 −0.0176113
\(92\) 4.38657e16 0.891105
\(93\) 7.50815e14 0.0139132
\(94\) 7.65502e15 0.129527
\(95\) −3.29949e15 −0.0510266
\(96\) −2.10742e16 −0.298157
\(97\) 7.41658e16 0.960824 0.480412 0.877043i \(-0.340487\pi\)
0.480412 + 0.877043i \(0.340487\pi\)
\(98\) −1.89163e16 −0.224602
\(99\) −2.32527e16 −0.253264
\(100\) −1.75835e17 −1.75835
\(101\) −7.87706e16 −0.723823 −0.361912 0.932212i \(-0.617876\pi\)
−0.361912 + 0.932212i \(0.617876\pi\)
\(102\) 4.03184e15 0.0340723
\(103\) 6.67667e16 0.519330 0.259665 0.965699i \(-0.416388\pi\)
0.259665 + 0.965699i \(0.416388\pi\)
\(104\) −5.79968e14 −0.00415547
\(105\) 2.20009e17 1.45322
\(106\) 2.53943e16 0.154752
\(107\) −2.57732e17 −1.45013 −0.725064 0.688681i \(-0.758190\pi\)
−0.725064 + 0.688681i \(0.758190\pi\)
\(108\) −3.58214e16 −0.186226
\(109\) 1.67981e17 0.807488 0.403744 0.914872i \(-0.367709\pi\)
0.403744 + 0.914872i \(0.367709\pi\)
\(110\) 5.15588e16 0.229332
\(111\) −1.02990e17 −0.424178
\(112\) 3.55242e17 1.35572
\(113\) 3.01755e17 1.06779 0.533897 0.845550i \(-0.320727\pi\)
0.533897 + 0.845550i \(0.320727\pi\)
\(114\) 9.61376e14 0.00315647
\(115\) −5.07037e17 −1.54564
\(116\) −6.49181e16 −0.183854
\(117\) −1.48683e15 −0.00391454
\(118\) −1.18859e17 −0.291096
\(119\) −2.15891e17 −0.492135
\(120\) 1.61510e17 0.342894
\(121\) −2.13660e17 −0.422716
\(122\) −9.18343e16 −0.169413
\(123\) −5.25379e17 −0.904231
\(124\) −1.45143e16 −0.0233191
\(125\) 9.13942e17 1.37146
\(126\) −6.41043e16 −0.0898953
\(127\) 1.84942e17 0.242496 0.121248 0.992622i \(-0.461310\pi\)
0.121248 + 0.992622i \(0.461310\pi\)
\(128\) 5.39929e17 0.662295
\(129\) −2.40616e17 −0.276256
\(130\) 3.29679e15 0.00354464
\(131\) 9.78482e16 0.0985705 0.0492852 0.998785i \(-0.484306\pi\)
0.0492852 + 0.998785i \(0.484306\pi\)
\(132\) 4.49506e17 0.424480
\(133\) −5.14782e16 −0.0455916
\(134\) 9.57806e16 0.0795951
\(135\) 4.14054e17 0.323013
\(136\) −1.58487e17 −0.116122
\(137\) −3.72170e17 −0.256222 −0.128111 0.991760i \(-0.540891\pi\)
−0.128111 + 0.991760i \(0.540891\pi\)
\(138\) 1.47736e17 0.0956123
\(139\) 4.12327e17 0.250967 0.125483 0.992096i \(-0.459952\pi\)
0.125483 + 0.992096i \(0.459952\pi\)
\(140\) −4.25308e18 −2.43565
\(141\) −7.71424e17 −0.415844
\(142\) 4.76214e17 0.241742
\(143\) 1.86575e16 0.00892271
\(144\) 6.68561e17 0.301342
\(145\) 7.50378e17 0.318898
\(146\) −8.76202e17 −0.351241
\(147\) 1.90626e18 0.721083
\(148\) 1.99093e18 0.710937
\(149\) 4.80416e18 1.62007 0.810035 0.586382i \(-0.199448\pi\)
0.810035 + 0.586382i \(0.199448\pi\)
\(150\) −5.92195e17 −0.188664
\(151\) −4.91996e18 −1.48135 −0.740674 0.671864i \(-0.765494\pi\)
−0.740674 + 0.671864i \(0.765494\pi\)
\(152\) −3.77906e16 −0.0107575
\(153\) −4.06303e17 −0.109389
\(154\) 8.04414e17 0.204905
\(155\) 1.67768e17 0.0404474
\(156\) 2.87424e16 0.00656091
\(157\) 6.42241e18 1.38852 0.694258 0.719726i \(-0.255733\pi\)
0.694258 + 0.719726i \(0.255733\pi\)
\(158\) 5.45447e17 0.111729
\(159\) −2.55907e18 −0.496829
\(160\) −4.70898e18 −0.866777
\(161\) −7.91072e18 −1.38101
\(162\) −1.20643e17 −0.0199814
\(163\) −7.83455e18 −1.23146 −0.615729 0.787958i \(-0.711138\pi\)
−0.615729 + 0.787958i \(0.711138\pi\)
\(164\) 1.01563e19 1.51552
\(165\) −5.19577e18 −0.736269
\(166\) 1.66658e18 0.224341
\(167\) 1.03503e19 1.32392 0.661962 0.749537i \(-0.269724\pi\)
0.661962 + 0.749537i \(0.269724\pi\)
\(168\) 2.51986e18 0.306371
\(169\) −8.64922e18 −0.999862
\(170\) 9.00908e17 0.0990523
\(171\) −9.68814e16 −0.0101338
\(172\) 4.65144e18 0.463015
\(173\) −8.18965e18 −0.776021 −0.388011 0.921655i \(-0.626838\pi\)
−0.388011 + 0.921655i \(0.626838\pi\)
\(174\) −2.18638e17 −0.0197268
\(175\) 3.17099e19 2.72503
\(176\) −8.38945e18 −0.686872
\(177\) 1.19779e19 0.934560
\(178\) −2.91377e18 −0.216713
\(179\) −2.91944e18 −0.207038 −0.103519 0.994627i \(-0.533010\pi\)
−0.103519 + 0.994627i \(0.533010\pi\)
\(180\) −8.00424e18 −0.541382
\(181\) −2.50385e19 −1.61562 −0.807812 0.589440i \(-0.799349\pi\)
−0.807812 + 0.589440i \(0.799349\pi\)
\(182\) 5.14360e16 0.00316709
\(183\) 9.25447e18 0.543898
\(184\) −5.80732e18 −0.325855
\(185\) −2.30128e19 −1.23313
\(186\) −4.88828e16 −0.00250205
\(187\) 5.09851e18 0.249338
\(188\) 1.49127e19 0.696970
\(189\) 6.46002e18 0.288608
\(190\) 2.14818e17 0.00917624
\(191\) −2.01953e19 −0.825025 −0.412513 0.910952i \(-0.635349\pi\)
−0.412513 + 0.910952i \(0.635349\pi\)
\(192\) −1.19841e19 −0.468322
\(193\) 4.28113e19 1.60074 0.800372 0.599504i \(-0.204635\pi\)
0.800372 + 0.599504i \(0.204635\pi\)
\(194\) −4.82866e18 −0.172788
\(195\) −3.32229e17 −0.0113800
\(196\) −3.68507e19 −1.20856
\(197\) 4.01977e19 1.26252 0.631261 0.775571i \(-0.282538\pi\)
0.631261 + 0.775571i \(0.282538\pi\)
\(198\) 1.51390e18 0.0455451
\(199\) 4.29123e19 1.23689 0.618444 0.785829i \(-0.287763\pi\)
0.618444 + 0.785829i \(0.287763\pi\)
\(200\) 2.32785e19 0.642984
\(201\) −9.65215e18 −0.255539
\(202\) 5.12846e18 0.130167
\(203\) 1.17073e19 0.284931
\(204\) 7.85440e18 0.183340
\(205\) −1.17395e20 −2.62871
\(206\) −4.34694e18 −0.0933925
\(207\) −1.48879e19 −0.306962
\(208\) −5.36440e17 −0.0106165
\(209\) 1.21572e18 0.0230988
\(210\) −1.43240e19 −0.261336
\(211\) 1.34631e19 0.235909 0.117955 0.993019i \(-0.462366\pi\)
0.117955 + 0.993019i \(0.462366\pi\)
\(212\) 4.94704e19 0.832704
\(213\) −4.79898e19 −0.776109
\(214\) 1.67800e19 0.260780
\(215\) −5.37652e19 −0.803109
\(216\) 4.74235e18 0.0680983
\(217\) 2.61750e18 0.0361392
\(218\) −1.09367e19 −0.145213
\(219\) 8.82980e19 1.12765
\(220\) 1.00441e20 1.23401
\(221\) 3.26010e17 0.00385386
\(222\) 6.70527e18 0.0762809
\(223\) −5.53968e19 −0.606587 −0.303293 0.952897i \(-0.598086\pi\)
−0.303293 + 0.952897i \(0.598086\pi\)
\(224\) −7.34689e19 −0.774454
\(225\) 5.96777e19 0.605704
\(226\) −1.96461e19 −0.192024
\(227\) 1.02921e19 0.0968915 0.0484457 0.998826i \(-0.484573\pi\)
0.0484457 + 0.998826i \(0.484573\pi\)
\(228\) 1.87285e18 0.0169847
\(229\) 1.55018e20 1.35451 0.677254 0.735750i \(-0.263170\pi\)
0.677254 + 0.735750i \(0.263170\pi\)
\(230\) 3.30113e19 0.277956
\(231\) −8.10637e19 −0.657846
\(232\) 8.59441e18 0.0672308
\(233\) −1.64581e19 −0.124124 −0.0620618 0.998072i \(-0.519768\pi\)
−0.0620618 + 0.998072i \(0.519768\pi\)
\(234\) 9.68019e16 0.000703962 0
\(235\) −1.72373e20 −1.20891
\(236\) −2.31549e20 −1.56636
\(237\) −5.49667e19 −0.358706
\(238\) 1.40558e19 0.0885020
\(239\) −7.36122e19 −0.447268 −0.223634 0.974673i \(-0.571792\pi\)
−0.223634 + 0.974673i \(0.571792\pi\)
\(240\) 1.49389e20 0.876037
\(241\) 1.00349e20 0.568028 0.284014 0.958820i \(-0.408334\pi\)
0.284014 + 0.958820i \(0.408334\pi\)
\(242\) 1.39107e19 0.0760181
\(243\) 1.21577e19 0.0641500
\(244\) −1.78902e20 −0.911593
\(245\) 4.25951e20 2.09627
\(246\) 3.42055e19 0.162610
\(247\) 7.77357e16 0.000357023 0
\(248\) 1.92153e18 0.00852721
\(249\) −1.67947e20 −0.720243
\(250\) −5.95034e19 −0.246633
\(251\) 3.34304e20 1.33942 0.669708 0.742624i \(-0.266419\pi\)
0.669708 + 0.742624i \(0.266419\pi\)
\(252\) −1.24881e20 −0.483718
\(253\) 1.86821e20 0.699682
\(254\) −1.20409e19 −0.0436086
\(255\) −9.07878e19 −0.318006
\(256\) 2.04259e20 0.692055
\(257\) −6.36862e19 −0.208744 −0.104372 0.994538i \(-0.533283\pi\)
−0.104372 + 0.994538i \(0.533283\pi\)
\(258\) 1.56656e19 0.0496798
\(259\) −3.59043e20 −1.10179
\(260\) 6.42244e18 0.0190733
\(261\) 2.20330e19 0.0633328
\(262\) −6.37054e18 −0.0177262
\(263\) −3.67784e20 −0.990761 −0.495381 0.868676i \(-0.664971\pi\)
−0.495381 + 0.868676i \(0.664971\pi\)
\(264\) −5.95095e19 −0.155222
\(265\) −5.71821e20 −1.44434
\(266\) 3.35156e18 0.00819885
\(267\) 2.93631e20 0.695754
\(268\) 1.86589e20 0.428293
\(269\) 1.44158e20 0.320586 0.160293 0.987069i \(-0.448756\pi\)
0.160293 + 0.987069i \(0.448756\pi\)
\(270\) −2.69576e19 −0.0580883
\(271\) 6.03451e20 1.26009 0.630047 0.776557i \(-0.283036\pi\)
0.630047 + 0.776557i \(0.283036\pi\)
\(272\) −1.46592e20 −0.296671
\(273\) −5.18339e18 −0.0101679
\(274\) 2.42306e19 0.0460770
\(275\) −7.48867e20 −1.38063
\(276\) 2.87803e20 0.514480
\(277\) 3.10539e20 0.538318 0.269159 0.963096i \(-0.413254\pi\)
0.269159 + 0.963096i \(0.413254\pi\)
\(278\) −2.68451e19 −0.0451320
\(279\) 4.92610e18 0.00803281
\(280\) 5.63059e20 0.890657
\(281\) 6.43730e20 0.987871 0.493935 0.869499i \(-0.335558\pi\)
0.493935 + 0.869499i \(0.335558\pi\)
\(282\) 5.02246e19 0.0747823
\(283\) −3.43859e20 −0.496817 −0.248408 0.968655i \(-0.579907\pi\)
−0.248408 + 0.968655i \(0.579907\pi\)
\(284\) 9.27709e20 1.30079
\(285\) −2.16480e19 −0.0294602
\(286\) −1.21472e18 −0.00160459
\(287\) −1.83158e21 −2.34872
\(288\) −1.38268e20 −0.172141
\(289\) −7.38152e20 −0.892307
\(290\) −4.88544e19 −0.0573483
\(291\) 4.86602e20 0.554732
\(292\) −1.70692e21 −1.88999
\(293\) 4.41631e20 0.474990 0.237495 0.971389i \(-0.423674\pi\)
0.237495 + 0.971389i \(0.423674\pi\)
\(294\) −1.24110e20 −0.129674
\(295\) 2.67644e21 2.71688
\(296\) −2.63576e20 −0.259972
\(297\) −1.52561e20 −0.146222
\(298\) −3.12781e20 −0.291341
\(299\) 1.19457e19 0.0108145
\(300\) −1.15365e21 −1.01518
\(301\) −8.38837e20 −0.717567
\(302\) 3.20320e20 0.266395
\(303\) −5.16814e20 −0.417899
\(304\) −3.49544e19 −0.0274837
\(305\) 2.06790e21 1.58118
\(306\) 2.64529e19 0.0196717
\(307\) −2.55581e20 −0.184864 −0.0924319 0.995719i \(-0.529464\pi\)
−0.0924319 + 0.995719i \(0.529464\pi\)
\(308\) 1.56707e21 1.10257
\(309\) 4.38056e20 0.299835
\(310\) −1.09228e19 −0.00727376
\(311\) 1.33035e21 0.861988 0.430994 0.902355i \(-0.358163\pi\)
0.430994 + 0.902355i \(0.358163\pi\)
\(312\) −3.80517e18 −0.00239916
\(313\) −8.09712e20 −0.496825 −0.248413 0.968654i \(-0.579909\pi\)
−0.248413 + 0.968654i \(0.579909\pi\)
\(314\) −4.18140e20 −0.249701
\(315\) 1.44348e21 0.839017
\(316\) 1.06258e21 0.601204
\(317\) −2.88959e21 −1.59159 −0.795797 0.605564i \(-0.792948\pi\)
−0.795797 + 0.605564i \(0.792948\pi\)
\(318\) 1.66612e20 0.0893461
\(319\) −2.76481e20 −0.144359
\(320\) −2.67782e21 −1.36147
\(321\) −1.69098e21 −0.837232
\(322\) 5.15038e20 0.248350
\(323\) 2.12427e19 0.00997675
\(324\) −2.35024e20 −0.107518
\(325\) −4.78842e19 −0.0213395
\(326\) 5.10079e20 0.221456
\(327\) 1.10213e21 0.466203
\(328\) −1.34458e21 −0.554190
\(329\) −2.68935e21 −1.08014
\(330\) 3.38277e20 0.132405
\(331\) 2.93645e21 1.12017 0.560085 0.828435i \(-0.310768\pi\)
0.560085 + 0.828435i \(0.310768\pi\)
\(332\) 3.24665e21 1.20715
\(333\) −6.75714e20 −0.244899
\(334\) −6.73871e20 −0.238085
\(335\) −2.15676e21 −0.742883
\(336\) 2.33074e21 0.782728
\(337\) −3.01200e20 −0.0986282 −0.0493141 0.998783i \(-0.515704\pi\)
−0.0493141 + 0.998783i \(0.515704\pi\)
\(338\) 5.63119e20 0.179808
\(339\) 1.97981e21 0.616491
\(340\) 1.75505e21 0.532990
\(341\) −6.18152e19 −0.0183098
\(342\) 6.30759e18 0.00182239
\(343\) 1.32467e21 0.373342
\(344\) −6.15797e20 −0.169313
\(345\) −3.32667e21 −0.892375
\(346\) 5.33198e20 0.139554
\(347\) 1.24669e21 0.318390 0.159195 0.987247i \(-0.449110\pi\)
0.159195 + 0.987247i \(0.449110\pi\)
\(348\) −4.25928e20 −0.106148
\(349\) −9.93535e18 −0.00241639 −0.00120820 0.999999i \(-0.500385\pi\)
−0.00120820 + 0.999999i \(0.500385\pi\)
\(350\) −2.06452e21 −0.490050
\(351\) −9.75507e18 −0.00226006
\(352\) 1.73505e21 0.392374
\(353\) 6.30999e21 1.39298 0.696488 0.717569i \(-0.254745\pi\)
0.696488 + 0.717569i \(0.254745\pi\)
\(354\) −7.79836e20 −0.168064
\(355\) −1.07232e22 −2.25624
\(356\) −5.67629e21 −1.16611
\(357\) −1.41646e21 −0.284134
\(358\) 1.90074e20 0.0372321
\(359\) 1.07474e21 0.205589 0.102795 0.994703i \(-0.467222\pi\)
0.102795 + 0.994703i \(0.467222\pi\)
\(360\) 1.05967e21 0.197970
\(361\) −5.47532e21 −0.999076
\(362\) 1.63017e21 0.290542
\(363\) −1.40183e21 −0.244055
\(364\) 1.00202e20 0.0170418
\(365\) 1.97300e22 3.27822
\(366\) −6.02525e20 −0.0978105
\(367\) 4.61598e21 0.732153 0.366077 0.930585i \(-0.380701\pi\)
0.366077 + 0.930585i \(0.380701\pi\)
\(368\) −5.37148e21 −0.832506
\(369\) −3.44701e21 −0.522058
\(370\) 1.49828e21 0.221758
\(371\) −8.92147e21 −1.29050
\(372\) −9.52283e19 −0.0134633
\(373\) 6.59694e21 0.911628 0.455814 0.890075i \(-0.349348\pi\)
0.455814 + 0.890075i \(0.349348\pi\)
\(374\) −3.31945e20 −0.0448392
\(375\) 5.99638e21 0.791814
\(376\) −1.97427e21 −0.254865
\(377\) −1.76788e19 −0.00223127
\(378\) −4.20588e20 −0.0519011
\(379\) 1.41137e22 1.70297 0.851485 0.524379i \(-0.175702\pi\)
0.851485 + 0.524379i \(0.175702\pi\)
\(380\) 4.18485e20 0.0493764
\(381\) 1.21340e21 0.140005
\(382\) 1.31484e21 0.148366
\(383\) −1.26426e22 −1.39523 −0.697617 0.716471i \(-0.745756\pi\)
−0.697617 + 0.716471i \(0.745756\pi\)
\(384\) 3.54247e21 0.382376
\(385\) −1.81135e22 −1.91244
\(386\) −2.78729e21 −0.287866
\(387\) −1.57868e21 −0.159496
\(388\) −9.40669e21 −0.929752
\(389\) 5.65745e21 0.547078 0.273539 0.961861i \(-0.411806\pi\)
0.273539 + 0.961861i \(0.411806\pi\)
\(390\) 2.16302e19 0.00204650
\(391\) 3.26440e21 0.302204
\(392\) 4.87861e21 0.441941
\(393\) 6.41982e20 0.0569097
\(394\) −2.61713e21 −0.227042
\(395\) −1.22822e22 −1.04280
\(396\) 2.94921e21 0.245074
\(397\) 5.61679e21 0.456845 0.228422 0.973562i \(-0.426643\pi\)
0.228422 + 0.973562i \(0.426643\pi\)
\(398\) −2.79386e21 −0.222433
\(399\) −3.37749e20 −0.0263223
\(400\) 2.15314e22 1.64272
\(401\) −1.90221e22 −1.42079 −0.710396 0.703802i \(-0.751484\pi\)
−0.710396 + 0.703802i \(0.751484\pi\)
\(402\) 6.28416e20 0.0459543
\(403\) −3.95260e18 −0.000283003 0
\(404\) 9.99072e21 0.700415
\(405\) 2.71661e21 0.186492
\(406\) −7.62219e20 −0.0512399
\(407\) 8.47922e21 0.558217
\(408\) −1.03983e21 −0.0670428
\(409\) 1.22737e22 0.775045 0.387522 0.921860i \(-0.373331\pi\)
0.387522 + 0.921860i \(0.373331\pi\)
\(410\) 7.64317e21 0.472727
\(411\) −2.44181e21 −0.147930
\(412\) −8.46823e21 −0.502535
\(413\) 4.17574e22 2.42749
\(414\) 9.69295e20 0.0552018
\(415\) −3.75276e22 −2.09383
\(416\) 1.10943e20 0.00606467
\(417\) 2.70528e21 0.144896
\(418\) −7.91509e19 −0.00415392
\(419\) −2.12528e22 −1.09294 −0.546470 0.837479i \(-0.684029\pi\)
−0.546470 + 0.837479i \(0.684029\pi\)
\(420\) −2.79045e22 −1.40622
\(421\) −6.94281e21 −0.342876 −0.171438 0.985195i \(-0.554841\pi\)
−0.171438 + 0.985195i \(0.554841\pi\)
\(422\) −8.76535e20 −0.0424242
\(423\) −5.06131e21 −0.240088
\(424\) −6.54932e21 −0.304499
\(425\) −1.30852e22 −0.596316
\(426\) 3.12444e21 0.139570
\(427\) 3.22630e22 1.41276
\(428\) 3.26890e22 1.40323
\(429\) 1.22412e20 0.00515153
\(430\) 3.50046e21 0.144425
\(431\) −2.09760e22 −0.848526 −0.424263 0.905539i \(-0.639467\pi\)
−0.424263 + 0.905539i \(0.639467\pi\)
\(432\) 4.38643e21 0.173980
\(433\) −7.77111e21 −0.302229 −0.151114 0.988516i \(-0.548286\pi\)
−0.151114 + 0.988516i \(0.548286\pi\)
\(434\) −1.70416e20 −0.00649901
\(435\) 4.92323e21 0.184116
\(436\) −2.13056e22 −0.781374
\(437\) 7.78382e20 0.0279963
\(438\) −5.74876e21 −0.202789
\(439\) −3.62044e21 −0.125260 −0.0626301 0.998037i \(-0.519949\pi\)
−0.0626301 + 0.998037i \(0.519949\pi\)
\(440\) −1.32973e22 −0.451248
\(441\) 1.25070e22 0.416318
\(442\) −2.12253e19 −0.000693050 0
\(443\) 4.20160e22 1.34581 0.672905 0.739729i \(-0.265046\pi\)
0.672905 + 0.739729i \(0.265046\pi\)
\(444\) 1.30625e22 0.410460
\(445\) 6.56113e22 2.02264
\(446\) 3.60668e21 0.109084
\(447\) 3.15201e22 0.935348
\(448\) −4.17790e22 −1.21645
\(449\) −3.62047e22 −1.03436 −0.517180 0.855877i \(-0.673018\pi\)
−0.517180 + 0.855877i \(0.673018\pi\)
\(450\) −3.88539e21 −0.108925
\(451\) 4.32549e22 1.18997
\(452\) −3.82725e22 −1.03326
\(453\) −3.22798e22 −0.855257
\(454\) −6.70083e20 −0.0174242
\(455\) −1.15822e21 −0.0295593
\(456\) −2.47944e20 −0.00621087
\(457\) −3.85735e22 −0.948422 −0.474211 0.880411i \(-0.657267\pi\)
−0.474211 + 0.880411i \(0.657267\pi\)
\(458\) −1.00927e22 −0.243585
\(459\) −2.66576e21 −0.0631557
\(460\) 6.43091e22 1.49565
\(461\) −3.51617e22 −0.802809 −0.401405 0.915901i \(-0.631478\pi\)
−0.401405 + 0.915901i \(0.631478\pi\)
\(462\) 5.27776e21 0.118302
\(463\) −5.56328e22 −1.22431 −0.612157 0.790737i \(-0.709698\pi\)
−0.612157 + 0.790737i \(0.709698\pi\)
\(464\) 7.94939e21 0.171764
\(465\) 1.10073e21 0.0233523
\(466\) 1.07153e21 0.0223215
\(467\) −2.54999e22 −0.521610 −0.260805 0.965392i \(-0.583988\pi\)
−0.260805 + 0.965392i \(0.583988\pi\)
\(468\) 1.88579e20 0.00378794
\(469\) −3.36494e22 −0.663756
\(470\) 1.12226e22 0.217401
\(471\) 4.21374e22 0.801661
\(472\) 3.06544e22 0.572778
\(473\) 1.98101e22 0.363553
\(474\) 3.57868e21 0.0645070
\(475\) −3.12012e21 −0.0552429
\(476\) 2.73821e22 0.476220
\(477\) −1.67901e22 −0.286845
\(478\) 4.79262e21 0.0804333
\(479\) 7.71958e22 1.27275 0.636373 0.771381i \(-0.280434\pi\)
0.636373 + 0.771381i \(0.280434\pi\)
\(480\) −3.08957e22 −0.500434
\(481\) 5.42180e20 0.00862800
\(482\) −6.53338e21 −0.102150
\(483\) −5.19023e22 −0.797325
\(484\) 2.70992e22 0.409045
\(485\) 1.08730e23 1.61267
\(486\) −7.91541e20 −0.0115363
\(487\) −3.67556e22 −0.526413 −0.263207 0.964739i \(-0.584780\pi\)
−0.263207 + 0.964739i \(0.584780\pi\)
\(488\) 2.36845e22 0.333347
\(489\) −5.14025e22 −0.710982
\(490\) −2.77321e22 −0.376978
\(491\) 8.85815e21 0.118345 0.0591726 0.998248i \(-0.481154\pi\)
0.0591726 + 0.998248i \(0.481154\pi\)
\(492\) 6.66355e22 0.874989
\(493\) −4.83107e21 −0.0623511
\(494\) −5.06109e18 −6.42044e−5 0
\(495\) −3.40894e22 −0.425085
\(496\) 1.77731e21 0.0217856
\(497\) −1.67303e23 −2.01592
\(498\) 1.09344e22 0.129523
\(499\) −3.19438e22 −0.371991 −0.185995 0.982551i \(-0.559551\pi\)
−0.185995 + 0.982551i \(0.559551\pi\)
\(500\) −1.15918e23 −1.32711
\(501\) 6.79084e22 0.764368
\(502\) −2.17653e22 −0.240871
\(503\) 1.46736e23 1.59664 0.798322 0.602230i \(-0.205721\pi\)
0.798322 + 0.602230i \(0.205721\pi\)
\(504\) 1.65328e22 0.176884
\(505\) −1.15481e23 −1.21488
\(506\) −1.21632e22 −0.125826
\(507\) −5.67476e22 −0.577271
\(508\) −2.34568e22 −0.234653
\(509\) −1.22599e22 −0.120611 −0.0603054 0.998180i \(-0.519207\pi\)
−0.0603054 + 0.998180i \(0.519207\pi\)
\(510\) 5.91086e21 0.0571879
\(511\) 3.07825e23 2.92905
\(512\) −8.40680e22 −0.786749
\(513\) −6.35639e20 −0.00585077
\(514\) 4.14637e21 0.0375389
\(515\) 9.78830e22 0.871657
\(516\) 3.05181e22 0.267322
\(517\) 6.35120e22 0.547250
\(518\) 2.33760e22 0.198138
\(519\) −5.37323e22 −0.448036
\(520\) −8.50258e20 −0.00697465
\(521\) −1.70219e23 −1.37368 −0.686842 0.726807i \(-0.741004\pi\)
−0.686842 + 0.726807i \(0.741004\pi\)
\(522\) −1.43449e21 −0.0113893
\(523\) 7.34078e21 0.0573427 0.0286713 0.999589i \(-0.490872\pi\)
0.0286713 + 0.999589i \(0.490872\pi\)
\(524\) −1.24104e22 −0.0953828
\(525\) 2.08049e23 1.57330
\(526\) 2.39451e22 0.178171
\(527\) −1.08012e21 −0.00790830
\(528\) −5.50432e22 −0.396566
\(529\) −2.14351e22 −0.151968
\(530\) 3.72292e22 0.259740
\(531\) 7.85868e22 0.539568
\(532\) 6.52915e21 0.0441172
\(533\) 2.76582e21 0.0183926
\(534\) −1.91172e22 −0.125119
\(535\) −3.77847e23 −2.43393
\(536\) −2.47023e22 −0.156616
\(537\) −1.91545e22 −0.119533
\(538\) −9.38560e21 −0.0576518
\(539\) −1.56944e23 −0.948945
\(540\) −5.25158e22 −0.312567
\(541\) 2.69319e23 1.57794 0.788970 0.614431i \(-0.210614\pi\)
0.788970 + 0.614431i \(0.210614\pi\)
\(542\) −3.92885e22 −0.226606
\(543\) −1.64278e23 −0.932781
\(544\) 3.03173e22 0.169473
\(545\) 2.46268e23 1.35531
\(546\) 3.37472e20 0.00182852
\(547\) −5.71353e22 −0.304798 −0.152399 0.988319i \(-0.548700\pi\)
−0.152399 + 0.988319i \(0.548700\pi\)
\(548\) 4.72035e22 0.247936
\(549\) 6.07186e22 0.314020
\(550\) 4.87560e22 0.248282
\(551\) −1.15195e21 −0.00577623
\(552\) −3.81018e22 −0.188133
\(553\) −1.91625e23 −0.931728
\(554\) −2.02181e22 −0.0968070
\(555\) −1.50987e23 −0.711951
\(556\) −5.22968e22 −0.242850
\(557\) 8.13046e22 0.371831 0.185916 0.982566i \(-0.440475\pi\)
0.185916 + 0.982566i \(0.440475\pi\)
\(558\) −3.20720e20 −0.00144456
\(559\) 1.26670e21 0.00561920
\(560\) 5.20801e23 2.27548
\(561\) 3.34513e22 0.143956
\(562\) −4.19109e22 −0.177651
\(563\) 2.45069e23 1.02322 0.511608 0.859219i \(-0.329050\pi\)
0.511608 + 0.859219i \(0.329050\pi\)
\(564\) 9.78422e22 0.402396
\(565\) 4.42386e23 1.79221
\(566\) 2.23874e22 0.0893438
\(567\) 4.23842e22 0.166628
\(568\) −1.22818e23 −0.475665
\(569\) −4.48105e23 −1.70972 −0.854862 0.518855i \(-0.826358\pi\)
−0.854862 + 0.518855i \(0.826358\pi\)
\(570\) 1.40942e21 0.00529791
\(571\) −4.10855e23 −1.52153 −0.760767 0.649025i \(-0.775177\pi\)
−0.760767 + 0.649025i \(0.775177\pi\)
\(572\) −2.36639e21 −0.00863415
\(573\) −1.32502e23 −0.476329
\(574\) 1.19248e23 0.422375
\(575\) −4.79473e23 −1.67335
\(576\) −7.86276e22 −0.270386
\(577\) 1.87387e23 0.634959 0.317480 0.948265i \(-0.397163\pi\)
0.317480 + 0.948265i \(0.397163\pi\)
\(578\) 4.80584e22 0.160466
\(579\) 2.80885e23 0.924190
\(580\) −9.51728e22 −0.308585
\(581\) −5.85500e23 −1.87081
\(582\) −3.16809e22 −0.0997589
\(583\) 2.10691e23 0.653827
\(584\) 2.25977e23 0.691122
\(585\) −2.17975e21 −0.00657026
\(586\) −2.87530e22 −0.0854186
\(587\) 3.17863e23 0.930714 0.465357 0.885123i \(-0.345926\pi\)
0.465357 + 0.885123i \(0.345926\pi\)
\(588\) −2.41777e23 −0.697764
\(589\) −2.57551e20 −0.000732627 0
\(590\) −1.74253e23 −0.488583
\(591\) 2.63737e23 0.728917
\(592\) −2.43795e23 −0.664186
\(593\) 5.64469e23 1.51592 0.757958 0.652304i \(-0.226197\pi\)
0.757958 + 0.652304i \(0.226197\pi\)
\(594\) 9.93267e21 0.0262955
\(595\) −3.16505e23 −0.826013
\(596\) −6.09326e23 −1.56768
\(597\) 2.81547e23 0.714117
\(598\) −7.77743e20 −0.00194481
\(599\) 2.56471e22 0.0632282 0.0316141 0.999500i \(-0.489935\pi\)
0.0316141 + 0.999500i \(0.489935\pi\)
\(600\) 1.52730e23 0.371227
\(601\) 5.16554e23 1.23789 0.618946 0.785433i \(-0.287560\pi\)
0.618946 + 0.785433i \(0.287560\pi\)
\(602\) 5.46136e22 0.129042
\(603\) −6.33278e22 −0.147536
\(604\) 6.24014e23 1.43344
\(605\) −3.13236e23 −0.709497
\(606\) 3.36479e22 0.0751519
\(607\) 6.45213e23 1.42102 0.710508 0.703689i \(-0.248465\pi\)
0.710508 + 0.703689i \(0.248465\pi\)
\(608\) 7.22904e21 0.0157000
\(609\) 7.68115e22 0.164505
\(610\) −1.34633e23 −0.284347
\(611\) 4.06110e21 0.00845850
\(612\) 5.15327e22 0.105851
\(613\) −5.11815e23 −1.03681 −0.518405 0.855135i \(-0.673474\pi\)
−0.518405 + 0.855135i \(0.673474\pi\)
\(614\) 1.66399e22 0.0332445
\(615\) −7.70229e23 −1.51768
\(616\) −2.07462e23 −0.403184
\(617\) −3.00782e22 −0.0576537 −0.0288269 0.999584i \(-0.509177\pi\)
−0.0288269 + 0.999584i \(0.509177\pi\)
\(618\) −2.85203e22 −0.0539202
\(619\) −6.97971e23 −1.30157 −0.650784 0.759263i \(-0.725560\pi\)
−0.650784 + 0.759263i \(0.725560\pi\)
\(620\) −2.12786e22 −0.0391393
\(621\) −9.76794e22 −0.177225
\(622\) −8.66140e22 −0.155014
\(623\) 1.02366e24 1.80720
\(624\) −3.51959e21 −0.00612946
\(625\) 2.82181e23 0.484784
\(626\) 5.27174e22 0.0893454
\(627\) 7.97633e21 0.0133361
\(628\) −8.14575e23 −1.34361
\(629\) 1.48161e23 0.241103
\(630\) −9.39797e22 −0.150883
\(631\) −5.56309e23 −0.881183 −0.440592 0.897708i \(-0.645231\pi\)
−0.440592 + 0.897708i \(0.645231\pi\)
\(632\) −1.40674e23 −0.219845
\(633\) 8.83316e22 0.136202
\(634\) 1.88130e23 0.286220
\(635\) 2.71133e23 0.407011
\(636\) 3.24576e23 0.480762
\(637\) −1.00354e22 −0.0146672
\(638\) 1.80007e22 0.0259605
\(639\) −3.14861e23 −0.448086
\(640\) 7.91559e23 1.11161
\(641\) −3.15692e23 −0.437492 −0.218746 0.975782i \(-0.570197\pi\)
−0.218746 + 0.975782i \(0.570197\pi\)
\(642\) 1.10094e23 0.150562
\(643\) 1.03616e24 1.39841 0.699203 0.714924i \(-0.253539\pi\)
0.699203 + 0.714924i \(0.253539\pi\)
\(644\) 1.00334e24 1.33635
\(645\) −3.52754e23 −0.463675
\(646\) −1.38304e21 −0.00179414
\(647\) −5.82417e23 −0.745671 −0.372836 0.927897i \(-0.621614\pi\)
−0.372836 + 0.927897i \(0.621614\pi\)
\(648\) 3.11145e22 0.0393166
\(649\) −9.86149e23 −1.22988
\(650\) 3.11756e21 0.00383753
\(651\) 1.71734e22 0.0208650
\(652\) 9.93680e23 1.19163
\(653\) 1.52832e23 0.180905 0.0904526 0.995901i \(-0.471169\pi\)
0.0904526 + 0.995901i \(0.471169\pi\)
\(654\) −7.17554e22 −0.0838385
\(655\) 1.43450e23 0.165443
\(656\) −1.24367e24 −1.41586
\(657\) 5.79323e23 0.651051
\(658\) 1.75093e23 0.194245
\(659\) 6.03187e23 0.660581 0.330290 0.943879i \(-0.392853\pi\)
0.330290 + 0.943879i \(0.392853\pi\)
\(660\) 6.58996e23 0.712458
\(661\) −1.09124e24 −1.16468 −0.582340 0.812945i \(-0.697863\pi\)
−0.582340 + 0.812945i \(0.697863\pi\)
\(662\) −1.91181e23 −0.201443
\(663\) 2.13895e21 0.00222503
\(664\) −4.29820e23 −0.441426
\(665\) −7.54694e22 −0.0765221
\(666\) 4.39933e22 0.0440408
\(667\) −1.77021e23 −0.174967
\(668\) −1.31276e24 −1.28111
\(669\) −3.63458e23 −0.350213
\(670\) 1.40419e23 0.133595
\(671\) −7.61928e23 −0.715769
\(672\) −4.82030e23 −0.447131
\(673\) −3.64938e23 −0.334265 −0.167133 0.985934i \(-0.553451\pi\)
−0.167133 + 0.985934i \(0.553451\pi\)
\(674\) 1.96100e22 0.0177366
\(675\) 3.91545e23 0.349703
\(676\) 1.09701e24 0.967527
\(677\) −3.91131e23 −0.340658 −0.170329 0.985387i \(-0.554483\pi\)
−0.170329 + 0.985387i \(0.554483\pi\)
\(678\) −1.28898e23 −0.110865
\(679\) 1.69640e24 1.44090
\(680\) −2.32349e23 −0.194901
\(681\) 6.75267e22 0.0559403
\(682\) 4.02456e21 0.00329270
\(683\) 8.44250e23 0.682174 0.341087 0.940032i \(-0.389205\pi\)
0.341087 + 0.940032i \(0.389205\pi\)
\(684\) 1.22878e22 0.00980610
\(685\) −5.45617e23 −0.430049
\(686\) −8.62443e22 −0.0671391
\(687\) 1.01708e24 0.782025
\(688\) −5.69581e23 −0.432567
\(689\) 1.34720e22 0.0101058
\(690\) 2.16587e23 0.160478
\(691\) 1.21416e24 0.888610 0.444305 0.895876i \(-0.353451\pi\)
0.444305 + 0.895876i \(0.353451\pi\)
\(692\) 1.03872e24 0.750925
\(693\) −5.31859e23 −0.379808
\(694\) −8.11677e22 −0.0572569
\(695\) 6.04490e23 0.421229
\(696\) 5.63879e22 0.0388157
\(697\) 7.55811e23 0.513966
\(698\) 6.46854e20 0.000434546 0
\(699\) −1.07982e23 −0.0716628
\(700\) −4.02187e24 −2.63691
\(701\) 2.69078e24 1.74291 0.871456 0.490473i \(-0.163176\pi\)
0.871456 + 0.490473i \(0.163176\pi\)
\(702\) 6.35117e20 0.000406432 0
\(703\) 3.53283e22 0.0223359
\(704\) 9.86660e23 0.616311
\(705\) −1.13094e24 −0.697964
\(706\) −4.10820e23 −0.250502
\(707\) −1.80172e24 −1.08548
\(708\) −1.51919e24 −0.904336
\(709\) −1.01451e24 −0.596712 −0.298356 0.954455i \(-0.596438\pi\)
−0.298356 + 0.954455i \(0.596438\pi\)
\(710\) 6.98151e23 0.405745
\(711\) −3.60636e23 −0.207099
\(712\) 7.51476e23 0.426417
\(713\) −3.95782e22 −0.0221919
\(714\) 9.22204e22 0.0510966
\(715\) 2.73527e22 0.0149761
\(716\) 3.70282e23 0.200342
\(717\) −4.82970e23 −0.258230
\(718\) −6.99723e22 −0.0369716
\(719\) −3.62460e24 −1.89263 −0.946313 0.323251i \(-0.895224\pi\)
−0.946313 + 0.323251i \(0.895224\pi\)
\(720\) 9.80140e23 0.505780
\(721\) 1.52716e24 0.778814
\(722\) 3.56478e23 0.179666
\(723\) 6.58392e23 0.327951
\(724\) 3.17571e24 1.56338
\(725\) 7.09585e23 0.345248
\(726\) 9.12678e22 0.0438891
\(727\) −2.60528e24 −1.23826 −0.619129 0.785289i \(-0.712514\pi\)
−0.619129 + 0.785289i \(0.712514\pi\)
\(728\) −1.32656e22 −0.00623176
\(729\) 7.97664e22 0.0370370
\(730\) −1.28455e24 −0.589531
\(731\) 3.46150e23 0.157024
\(732\) −1.17377e24 −0.526308
\(733\) −2.82031e23 −0.125001 −0.0625004 0.998045i \(-0.519907\pi\)
−0.0625004 + 0.998045i \(0.519907\pi\)
\(734\) −3.00529e23 −0.131665
\(735\) 2.79467e24 1.21028
\(736\) 1.11089e24 0.475567
\(737\) 7.94670e23 0.336289
\(738\) 2.24422e23 0.0938831
\(739\) −2.13421e24 −0.882592 −0.441296 0.897362i \(-0.645481\pi\)
−0.441296 + 0.897362i \(0.645481\pi\)
\(740\) 2.91879e24 1.19326
\(741\) 5.10024e20 0.000206127 0
\(742\) 5.80844e23 0.232074
\(743\) 1.65375e24 0.653229 0.326614 0.945158i \(-0.394092\pi\)
0.326614 + 0.945158i \(0.394092\pi\)
\(744\) 1.26071e22 0.00492319
\(745\) 7.04311e24 2.71917
\(746\) −4.29503e23 −0.163940
\(747\) −1.10190e24 −0.415832
\(748\) −6.46660e23 −0.241275
\(749\) −5.89511e24 −2.17469
\(750\) −3.90402e23 −0.142394
\(751\) −1.59968e24 −0.576890 −0.288445 0.957496i \(-0.593138\pi\)
−0.288445 + 0.957496i \(0.593138\pi\)
\(752\) −1.82610e24 −0.651137
\(753\) 2.19337e24 0.773312
\(754\) 1.15100e21 0.000401255 0
\(755\) −7.21288e24 −2.48633
\(756\) −8.19344e23 −0.279274
\(757\) −3.10841e24 −1.04767 −0.523834 0.851820i \(-0.675499\pi\)
−0.523834 + 0.851820i \(0.675499\pi\)
\(758\) −9.18890e23 −0.306250
\(759\) 1.22573e24 0.403962
\(760\) −5.54027e22 −0.0180557
\(761\) 3.83430e24 1.23571 0.617856 0.786292i \(-0.288002\pi\)
0.617856 + 0.786292i \(0.288002\pi\)
\(762\) −7.90003e22 −0.0251774
\(763\) 3.84224e24 1.21095
\(764\) 2.56144e24 0.798344
\(765\) −5.95659e23 −0.183601
\(766\) 8.23113e23 0.250908
\(767\) −6.30565e22 −0.0190094
\(768\) 1.34014e24 0.399558
\(769\) −3.27383e24 −0.965343 −0.482672 0.875801i \(-0.660334\pi\)
−0.482672 + 0.875801i \(0.660334\pi\)
\(770\) 1.17931e24 0.343918
\(771\) −4.17845e23 −0.120518
\(772\) −5.42990e24 −1.54898
\(773\) −7.30757e23 −0.206180 −0.103090 0.994672i \(-0.532873\pi\)
−0.103090 + 0.994672i \(0.532873\pi\)
\(774\) 1.02782e23 0.0286827
\(775\) 1.58648e23 0.0437895
\(776\) 1.24534e24 0.339987
\(777\) −2.35568e24 −0.636118
\(778\) −3.68336e23 −0.0983824
\(779\) 1.80220e23 0.0476140
\(780\) 4.21377e22 0.0110120
\(781\) 3.95104e24 1.02136
\(782\) −2.12533e23 −0.0543461
\(783\) 1.44558e23 0.0365652
\(784\) 4.51246e24 1.12909
\(785\) 9.41554e24 2.33052
\(786\) −4.17971e22 −0.0102342
\(787\) 6.01570e23 0.145714 0.0728570 0.997342i \(-0.476788\pi\)
0.0728570 + 0.997342i \(0.476788\pi\)
\(788\) −5.09841e24 −1.22169
\(789\) −2.41303e24 −0.572016
\(790\) 7.99650e23 0.187529
\(791\) 6.90204e24 1.60132
\(792\) −3.90442e23 −0.0896173
\(793\) −4.87194e22 −0.0110632
\(794\) −3.65689e23 −0.0821556
\(795\) −3.75172e24 −0.833891
\(796\) −5.44270e24 −1.19689
\(797\) 5.16820e24 1.12446 0.562230 0.826981i \(-0.309944\pi\)
0.562230 + 0.826981i \(0.309944\pi\)
\(798\) 2.19896e22 0.00473361
\(799\) 1.10977e24 0.236366
\(800\) −4.45299e24 −0.938398
\(801\) 1.92651e24 0.401694
\(802\) 1.23846e24 0.255505
\(803\) −7.26965e24 −1.48399
\(804\) 1.22421e24 0.247275
\(805\) −1.15975e25 −2.31792
\(806\) 2.57340e20 5.08931e−5 0
\(807\) 9.45821e23 0.185090
\(808\) −1.32266e24 −0.256124
\(809\) 8.29099e24 1.58871 0.794354 0.607455i \(-0.207810\pi\)
0.794354 + 0.607455i \(0.207810\pi\)
\(810\) −1.76869e23 −0.0335373
\(811\) −6.74898e23 −0.126637 −0.0633185 0.997993i \(-0.520168\pi\)
−0.0633185 + 0.997993i \(0.520168\pi\)
\(812\) −1.48487e24 −0.275717
\(813\) 3.95924e24 0.727515
\(814\) −5.52051e23 −0.100386
\(815\) −1.14858e25 −2.06691
\(816\) −9.61792e23 −0.171283
\(817\) 8.25381e22 0.0145468
\(818\) −7.99094e23 −0.139378
\(819\) −3.40082e22 −0.00587044
\(820\) 1.48896e25 2.54370
\(821\) −3.52716e24 −0.596361 −0.298180 0.954510i \(-0.596380\pi\)
−0.298180 + 0.954510i \(0.596380\pi\)
\(822\) 1.58977e23 0.0266026
\(823\) 2.10509e24 0.348636 0.174318 0.984689i \(-0.444228\pi\)
0.174318 + 0.984689i \(0.444228\pi\)
\(824\) 1.12110e24 0.183765
\(825\) −4.91331e24 −0.797106
\(826\) −2.71867e24 −0.436542
\(827\) 7.78739e24 1.23764 0.618821 0.785532i \(-0.287610\pi\)
0.618821 + 0.785532i \(0.287610\pi\)
\(828\) 1.88828e24 0.297035
\(829\) −3.67550e24 −0.572273 −0.286137 0.958189i \(-0.592371\pi\)
−0.286137 + 0.958189i \(0.592371\pi\)
\(830\) 2.44328e24 0.376539
\(831\) 2.03745e24 0.310798
\(832\) 6.30892e22 0.00952592
\(833\) −2.74235e24 −0.409865
\(834\) −1.76131e23 −0.0260570
\(835\) 1.51740e25 2.22211
\(836\) −1.54193e23 −0.0223518
\(837\) 3.23201e22 0.00463774
\(838\) 1.38369e24 0.196546
\(839\) −7.12132e23 −0.100135 −0.0500673 0.998746i \(-0.515944\pi\)
−0.0500673 + 0.998746i \(0.515944\pi\)
\(840\) 3.69423e24 0.514221
\(841\) −6.99517e24 −0.963901
\(842\) 4.52021e23 0.0616603
\(843\) 4.22351e24 0.570348
\(844\) −1.70757e24 −0.228280
\(845\) −1.26801e25 −1.67819
\(846\) 3.29524e23 0.0431756
\(847\) −4.88706e24 −0.633926
\(848\) −6.05779e24 −0.777946
\(849\) −2.25606e24 −0.286837
\(850\) 8.51933e23 0.107237
\(851\) 5.42895e24 0.676573
\(852\) 6.08670e24 0.751009
\(853\) 9.94213e24 1.21454 0.607271 0.794495i \(-0.292264\pi\)
0.607271 + 0.794495i \(0.292264\pi\)
\(854\) −2.10053e24 −0.254060
\(855\) −1.42032e23 −0.0170089
\(856\) −4.32765e24 −0.513127
\(857\) −7.12623e23 −0.0836610 −0.0418305 0.999125i \(-0.513319\pi\)
−0.0418305 + 0.999125i \(0.513319\pi\)
\(858\) −7.96978e21 −0.000926413 0
\(859\) −6.42042e24 −0.738961 −0.369480 0.929239i \(-0.620464\pi\)
−0.369480 + 0.929239i \(0.620464\pi\)
\(860\) 6.81921e24 0.777136
\(861\) −1.20170e25 −1.35603
\(862\) 1.36567e24 0.152593
\(863\) −3.45207e24 −0.381934 −0.190967 0.981596i \(-0.561162\pi\)
−0.190967 + 0.981596i \(0.561162\pi\)
\(864\) −9.07173e23 −0.0993856
\(865\) −1.20064e25 −1.30249
\(866\) 5.05948e23 0.0543506
\(867\) −4.84302e24 −0.515173
\(868\) −3.31986e23 −0.0349705
\(869\) 4.52545e24 0.472056
\(870\) −3.20533e23 −0.0331100
\(871\) 5.08130e22 0.00519781
\(872\) 2.82062e24 0.285729
\(873\) 3.19260e24 0.320275
\(874\) −5.06776e22 −0.00503465
\(875\) 2.09046e25 2.05671
\(876\) −1.11991e25 −1.09119
\(877\) 1.98750e25 1.91783 0.958915 0.283692i \(-0.0915595\pi\)
0.958915 + 0.283692i \(0.0915595\pi\)
\(878\) 2.35714e23 0.0225259
\(879\) 2.89754e24 0.274236
\(880\) −1.22993e25 −1.15286
\(881\) −1.53861e24 −0.142835 −0.0714175 0.997447i \(-0.522752\pi\)
−0.0714175 + 0.997447i \(0.522752\pi\)
\(882\) −8.14284e23 −0.0748675
\(883\) −8.67401e24 −0.789867 −0.394934 0.918710i \(-0.629232\pi\)
−0.394934 + 0.918710i \(0.629232\pi\)
\(884\) −4.13489e22 −0.00372923
\(885\) 1.75601e25 1.56859
\(886\) −2.73551e24 −0.242020
\(887\) 7.74587e24 0.678765 0.339383 0.940648i \(-0.389782\pi\)
0.339383 + 0.940648i \(0.389782\pi\)
\(888\) −1.72932e24 −0.150095
\(889\) 4.23018e24 0.363659
\(890\) −4.27171e24 −0.363737
\(891\) −1.00095e24 −0.0844214
\(892\) 7.02615e24 0.586970
\(893\) 2.64620e23 0.0218971
\(894\) −2.05216e24 −0.168206
\(895\) −4.28003e24 −0.347497
\(896\) 1.23498e25 0.993212
\(897\) 7.83760e22 0.00624378
\(898\) 2.35716e24 0.186011
\(899\) 5.85728e22 0.00457866
\(900\) −7.56911e24 −0.586116
\(901\) 3.68149e24 0.282399
\(902\) −2.81617e24 −0.213995
\(903\) −5.50361e24 −0.414288
\(904\) 5.06684e24 0.377838
\(905\) −3.67076e25 −2.71171
\(906\) 2.10162e24 0.153803
\(907\) −9.51449e24 −0.689801 −0.344900 0.938639i \(-0.612087\pi\)
−0.344900 + 0.938639i \(0.612087\pi\)
\(908\) −1.30538e24 −0.0937580
\(909\) −3.39082e24 −0.241274
\(910\) 7.54075e22 0.00531572
\(911\) −5.87611e24 −0.410378 −0.205189 0.978722i \(-0.565781\pi\)
−0.205189 + 0.978722i \(0.565781\pi\)
\(912\) −2.29336e23 −0.0158677
\(913\) 1.38272e25 0.947838
\(914\) 2.51138e24 0.170557
\(915\) 1.35675e25 0.912892
\(916\) −1.96615e25 −1.31070
\(917\) 2.23808e24 0.147821
\(918\) 1.73558e23 0.0113574
\(919\) 1.95287e25 1.26617 0.633084 0.774083i \(-0.281789\pi\)
0.633084 + 0.774083i \(0.281789\pi\)
\(920\) −8.51379e24 −0.546923
\(921\) −1.67686e24 −0.106731
\(922\) 2.28925e24 0.144371
\(923\) 2.52638e23 0.0157865
\(924\) 1.02816e25 0.636572
\(925\) −2.17618e25 −1.33503
\(926\) 3.62205e24 0.220171
\(927\) 2.87409e24 0.173110
\(928\) −1.64404e24 −0.0981195
\(929\) −4.51245e24 −0.266857 −0.133428 0.991058i \(-0.542599\pi\)
−0.133428 + 0.991058i \(0.542599\pi\)
\(930\) −7.16644e22 −0.00419951
\(931\) −6.53902e23 −0.0379700
\(932\) 2.08743e24 0.120110
\(933\) 8.72840e24 0.497669
\(934\) 1.66021e24 0.0938024
\(935\) 7.47463e24 0.418496
\(936\) −2.49657e22 −0.00138516
\(937\) 9.95126e24 0.547131 0.273566 0.961853i \(-0.411797\pi\)
0.273566 + 0.961853i \(0.411797\pi\)
\(938\) 2.19079e24 0.119365
\(939\) −5.31252e24 −0.286842
\(940\) 2.18627e25 1.16981
\(941\) −1.60734e25 −0.852309 −0.426155 0.904650i \(-0.640132\pi\)
−0.426155 + 0.904650i \(0.640132\pi\)
\(942\) −2.74341e24 −0.144165
\(943\) 2.76946e25 1.44227
\(944\) 2.83538e25 1.46335
\(945\) 9.47067e24 0.484407
\(946\) −1.28976e24 −0.0653786
\(947\) −1.75270e25 −0.880505 −0.440253 0.897874i \(-0.645111\pi\)
−0.440253 + 0.897874i \(0.645111\pi\)
\(948\) 6.97160e24 0.347105
\(949\) −4.64838e23 −0.0229371
\(950\) 2.03140e23 0.00993447
\(951\) −1.89586e25 −0.918907
\(952\) −3.62508e24 −0.174142
\(953\) −2.36935e25 −1.12808 −0.564040 0.825747i \(-0.690754\pi\)
−0.564040 + 0.825747i \(0.690754\pi\)
\(954\) 1.09314e24 0.0515840
\(955\) −2.96072e25 −1.38474
\(956\) 9.33647e24 0.432803
\(957\) −1.81399e24 −0.0833459
\(958\) −5.02594e24 −0.228881
\(959\) −8.51265e24 −0.384243
\(960\) −1.75692e25 −0.786043
\(961\) −2.25370e25 −0.999419
\(962\) −3.52994e22 −0.00155160
\(963\) −1.10945e25 −0.483376
\(964\) −1.27276e25 −0.549658
\(965\) 6.27633e25 2.68673
\(966\) 3.37917e24 0.143385
\(967\) −2.02177e25 −0.850369 −0.425185 0.905107i \(-0.639791\pi\)
−0.425185 + 0.905107i \(0.639791\pi\)
\(968\) −3.58763e24 −0.149578
\(969\) 1.39374e23 0.00576008
\(970\) −7.07904e24 −0.290011
\(971\) 2.75403e25 1.11842 0.559210 0.829026i \(-0.311105\pi\)
0.559210 + 0.829026i \(0.311105\pi\)
\(972\) −1.54200e24 −0.0620754
\(973\) 9.43117e24 0.376362
\(974\) 2.39302e24 0.0946663
\(975\) −3.14168e23 −0.0123203
\(976\) 2.19070e25 0.851646
\(977\) 3.78814e25 1.45990 0.729948 0.683502i \(-0.239544\pi\)
0.729948 + 0.683502i \(0.239544\pi\)
\(978\) 3.34663e24 0.127858
\(979\) −2.41749e25 −0.915612
\(980\) −5.40247e25 −2.02848
\(981\) 7.23105e24 0.269163
\(982\) −5.76722e23 −0.0212823
\(983\) −5.25889e25 −1.92393 −0.961965 0.273171i \(-0.911927\pi\)
−0.961965 + 0.273171i \(0.911927\pi\)
\(984\) −8.82178e24 −0.319962
\(985\) 5.89317e25 2.11905
\(986\) 3.14533e23 0.0112128
\(987\) −1.76448e25 −0.623621
\(988\) −9.85947e21 −0.000345477 0
\(989\) 1.26837e25 0.440634
\(990\) 2.21944e24 0.0764441
\(991\) 6.05240e24 0.206682 0.103341 0.994646i \(-0.467047\pi\)
0.103341 + 0.994646i \(0.467047\pi\)
\(992\) −3.67573e23 −0.0124450
\(993\) 1.92660e25 0.646731
\(994\) 1.08925e25 0.362528
\(995\) 6.29113e25 2.07602
\(996\) 2.13013e25 0.696950
\(997\) −4.03816e24 −0.131001 −0.0655005 0.997853i \(-0.520864\pi\)
−0.0655005 + 0.997853i \(0.520864\pi\)
\(998\) 2.07974e24 0.0668960
\(999\) −4.43336e24 −0.141393
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 3.18.a.b.1.1 2
3.2 odd 2 9.18.a.c.1.2 2
4.3 odd 2 48.18.a.h.1.2 2
5.2 odd 4 75.18.b.c.49.2 4
5.3 odd 4 75.18.b.c.49.3 4
5.4 even 2 75.18.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.18.a.b.1.1 2 1.1 even 1 trivial
9.18.a.c.1.2 2 3.2 odd 2
48.18.a.h.1.2 2 4.3 odd 2
75.18.a.b.1.2 2 5.4 even 2
75.18.b.c.49.2 4 5.2 odd 4
75.18.b.c.49.3 4 5.3 odd 4