Properties

Label 5.20.a.b.1.4
Level $5$
Weight $20$
Character 5.1
Self dual yes
Analytic conductor $11.441$
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] = [5,20,Mod(1,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 20, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5.1");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 5.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(11.4408348278\)
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: \( x^{4} - 2x^{3} - 323561x^{2} - 26738538x + 10870990650 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-267.923\) of defining polynomial
Character \(\chi\) \(=\) 5.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1360.76 q^{2} +10517.7 q^{3} +1.32738e6 q^{4} -1.95312e6 q^{5} +1.43121e7 q^{6} +1.23996e8 q^{7} +1.09282e9 q^{8} -1.05164e9 q^{9} +O(q^{10})\) \(q+1360.76 q^{2} +10517.7 q^{3} +1.32738e6 q^{4} -1.95312e6 q^{5} +1.43121e7 q^{6} +1.23996e8 q^{7} +1.09282e9 q^{8} -1.05164e9 q^{9} -2.65773e9 q^{10} +3.15053e9 q^{11} +1.39611e10 q^{12} -4.41439e10 q^{13} +1.68729e11 q^{14} -2.05425e10 q^{15} +7.91131e11 q^{16} +6.29241e11 q^{17} -1.43103e12 q^{18} -1.75434e12 q^{19} -2.59254e12 q^{20} +1.30416e12 q^{21} +4.28712e12 q^{22} -8.87113e12 q^{23} +1.14940e13 q^{24} +3.81470e12 q^{25} -6.00693e13 q^{26} -2.32852e13 q^{27} +1.64590e14 q^{28} -6.96019e13 q^{29} -2.79534e13 q^{30} -2.01459e14 q^{31} +5.03589e14 q^{32} +3.31365e13 q^{33} +8.56247e14 q^{34} -2.42179e14 q^{35} -1.39592e15 q^{36} +7.44252e14 q^{37} -2.38723e15 q^{38} -4.64295e14 q^{39} -2.13441e15 q^{40} +1.02353e15 q^{41} +1.77464e15 q^{42} +4.31907e15 q^{43} +4.18195e15 q^{44} +2.05398e15 q^{45} -1.20715e16 q^{46} +3.87185e15 q^{47} +8.32092e15 q^{48} +3.97607e15 q^{49} +5.19089e15 q^{50} +6.61820e15 q^{51} -5.85958e16 q^{52} +2.20534e16 q^{53} -3.16856e16 q^{54} -6.15338e15 q^{55} +1.35505e17 q^{56} -1.84517e16 q^{57} -9.47114e16 q^{58} +2.82151e16 q^{59} -2.72677e16 q^{60} +1.90704e16 q^{61} -2.74137e17 q^{62} -1.30399e17 q^{63} +2.70484e17 q^{64} +8.62186e16 q^{65} +4.50908e16 q^{66} +2.68524e17 q^{67} +8.35243e17 q^{68} -9.33043e16 q^{69} -3.29548e17 q^{70} -1.98938e17 q^{71} -1.14925e18 q^{72} +2.55551e17 q^{73} +1.01275e18 q^{74} +4.01220e16 q^{75} -2.32867e18 q^{76} +3.90653e17 q^{77} -6.31794e17 q^{78} +8.98937e17 q^{79} -1.54518e18 q^{80} +9.77370e17 q^{81} +1.39278e18 q^{82} -8.68645e17 q^{83} +1.73111e18 q^{84} -1.22899e18 q^{85} +5.87722e18 q^{86} -7.32055e17 q^{87} +3.44295e18 q^{88} -2.78159e18 q^{89} +2.79498e18 q^{90} -5.47366e18 q^{91} -1.17754e19 q^{92} -2.11889e18 q^{93} +5.26866e18 q^{94} +3.42644e18 q^{95} +5.29663e18 q^{96} -2.57839e18 q^{97} +5.41048e18 q^{98} -3.31322e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 420 q^{2} + 3080 q^{3} + 2084992 q^{4} - 7812500 q^{5} - 58071752 q^{6} + 214021400 q^{7} + 81737760 q^{8} + 154064548 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 420 q^{2} + 3080 q^{3} + 2084992 q^{4} - 7812500 q^{5} - 58071752 q^{6} + 214021400 q^{7} + 81737760 q^{8} + 154064548 q^{9} + 820312500 q^{10} + 11585712768 q^{11} + 105629359360 q^{12} + 14812333160 q^{13} + 185174155944 q^{14} - 6015625000 q^{15} + 1785780098944 q^{16} + 849033742440 q^{17} - 2113778999620 q^{18} + 1978167708560 q^{19} - 4072250000000 q^{20} - 1487020185552 q^{21} - 7953348762240 q^{22} - 26569906952760 q^{23} - 39774243472320 q^{24} + 15258789062500 q^{25} - 48695658207912 q^{26} - 7557605929360 q^{27} + 236612033519360 q^{28} + 116267174339640 q^{29} + 113421390625000 q^{30} + 251049672388688 q^{31} - 142495342974720 q^{32} + 359905680636160 q^{33} + 411849015040344 q^{34} - 418010546875000 q^{35} - 168308645735296 q^{36} + 53471657716520 q^{37} - 52\!\cdots\!60 q^{38}+ \cdots - 25\!\cdots\!84 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1360.76 1.87930 0.939651 0.342134i \(-0.111150\pi\)
0.939651 + 0.342134i \(0.111150\pi\)
\(3\) 10517.7 0.308511 0.154256 0.988031i \(-0.450702\pi\)
0.154256 + 0.988031i \(0.450702\pi\)
\(4\) 1.32738e6 2.53178
\(5\) −1.95312e6 −0.447214
\(6\) 1.43121e7 0.579786
\(7\) 1.23996e8 1.16138 0.580692 0.814123i \(-0.302782\pi\)
0.580692 + 0.814123i \(0.302782\pi\)
\(8\) 1.09282e9 2.87867
\(9\) −1.05164e9 −0.904821
\(10\) −2.65773e9 −0.840450
\(11\) 3.15053e9 0.402859 0.201430 0.979503i \(-0.435441\pi\)
0.201430 + 0.979503i \(0.435441\pi\)
\(12\) 1.39611e10 0.781081
\(13\) −4.41439e10 −1.15454 −0.577270 0.816553i \(-0.695882\pi\)
−0.577270 + 0.816553i \(0.695882\pi\)
\(14\) 1.68729e11 2.18259
\(15\) −2.05425e10 −0.137970
\(16\) 7.91131e11 2.87812
\(17\) 6.29241e11 1.28692 0.643461 0.765479i \(-0.277498\pi\)
0.643461 + 0.765479i \(0.277498\pi\)
\(18\) −1.43103e12 −1.70043
\(19\) −1.75434e12 −1.24725 −0.623625 0.781724i \(-0.714341\pi\)
−0.623625 + 0.781724i \(0.714341\pi\)
\(20\) −2.59254e12 −1.13225
\(21\) 1.30416e12 0.358300
\(22\) 4.28712e12 0.757094
\(23\) −8.87113e12 −1.02699 −0.513493 0.858094i \(-0.671649\pi\)
−0.513493 + 0.858094i \(0.671649\pi\)
\(24\) 1.14940e13 0.888102
\(25\) 3.81470e12 0.200000
\(26\) −6.00693e13 −2.16973
\(27\) −2.32852e13 −0.587659
\(28\) 1.64590e14 2.94036
\(29\) −6.96019e13 −0.890922 −0.445461 0.895301i \(-0.646960\pi\)
−0.445461 + 0.895301i \(0.646960\pi\)
\(30\) −2.79534e13 −0.259288
\(31\) −2.01459e14 −1.36852 −0.684258 0.729240i \(-0.739874\pi\)
−0.684258 + 0.729240i \(0.739874\pi\)
\(32\) 5.03589e14 2.53018
\(33\) 3.31365e13 0.124287
\(34\) 8.56247e14 2.41852
\(35\) −2.42179e14 −0.519387
\(36\) −1.39592e15 −2.29080
\(37\) 7.44252e14 0.941464 0.470732 0.882276i \(-0.343990\pi\)
0.470732 + 0.882276i \(0.343990\pi\)
\(38\) −2.38723e15 −2.34396
\(39\) −4.64295e14 −0.356189
\(40\) −2.13441e15 −1.28738
\(41\) 1.02353e15 0.488265 0.244132 0.969742i \(-0.421497\pi\)
0.244132 + 0.969742i \(0.421497\pi\)
\(42\) 1.77464e15 0.673354
\(43\) 4.31907e15 1.31051 0.655255 0.755408i \(-0.272561\pi\)
0.655255 + 0.755408i \(0.272561\pi\)
\(44\) 4.18195e15 1.01995
\(45\) 2.05398e15 0.404648
\(46\) −1.20715e16 −1.93002
\(47\) 3.87185e15 0.504649 0.252324 0.967643i \(-0.418805\pi\)
0.252324 + 0.967643i \(0.418805\pi\)
\(48\) 8.32092e15 0.887931
\(49\) 3.97607e15 0.348812
\(50\) 5.19089e15 0.375860
\(51\) 6.61820e15 0.397030
\(52\) −5.85958e16 −2.92304
\(53\) 2.20534e16 0.918028 0.459014 0.888429i \(-0.348203\pi\)
0.459014 + 0.888429i \(0.348203\pi\)
\(54\) −3.16856e16 −1.10439
\(55\) −6.15338e15 −0.180164
\(56\) 1.35505e17 3.34324
\(57\) −1.84517e16 −0.384790
\(58\) −9.47114e16 −1.67431
\(59\) 2.82151e16 0.424020 0.212010 0.977267i \(-0.431999\pi\)
0.212010 + 0.977267i \(0.431999\pi\)
\(60\) −2.72677e16 −0.349310
\(61\) 1.90704e16 0.208798 0.104399 0.994536i \(-0.466708\pi\)
0.104399 + 0.994536i \(0.466708\pi\)
\(62\) −2.74137e17 −2.57185
\(63\) −1.30399e17 −1.05084
\(64\) 2.70484e17 1.87686
\(65\) 8.62186e16 0.516326
\(66\) 4.50908e16 0.233572
\(67\) 2.68524e17 1.20579 0.602897 0.797819i \(-0.294013\pi\)
0.602897 + 0.797819i \(0.294013\pi\)
\(68\) 8.35243e17 3.25820
\(69\) −9.33043e16 −0.316836
\(70\) −3.29548e17 −0.976084
\(71\) −1.98938e17 −0.514949 −0.257474 0.966285i \(-0.582890\pi\)
−0.257474 + 0.966285i \(0.582890\pi\)
\(72\) −1.14925e18 −2.60468
\(73\) 2.55551e17 0.508055 0.254028 0.967197i \(-0.418245\pi\)
0.254028 + 0.967197i \(0.418245\pi\)
\(74\) 1.01275e18 1.76929
\(75\) 4.01220e16 0.0617022
\(76\) −2.32867e18 −3.15776
\(77\) 3.90653e17 0.467874
\(78\) −6.31794e17 −0.669386
\(79\) 8.98937e17 0.843862 0.421931 0.906628i \(-0.361352\pi\)
0.421931 + 0.906628i \(0.361352\pi\)
\(80\) −1.54518e18 −1.28713
\(81\) 9.77370e17 0.723522
\(82\) 1.39278e18 0.917597
\(83\) −8.68645e17 −0.510035 −0.255018 0.966936i \(-0.582081\pi\)
−0.255018 + 0.966936i \(0.582081\pi\)
\(84\) 1.73111e18 0.907135
\(85\) −1.22899e18 −0.575529
\(86\) 5.87722e18 2.46284
\(87\) −7.32055e17 −0.274860
\(88\) 3.44295e18 1.15970
\(89\) −2.78159e18 −0.841567 −0.420783 0.907161i \(-0.638245\pi\)
−0.420783 + 0.907161i \(0.638245\pi\)
\(90\) 2.79498e18 0.760456
\(91\) −5.47366e18 −1.34086
\(92\) −1.17754e19 −2.60010
\(93\) −2.11889e18 −0.422202
\(94\) 5.26866e18 0.948387
\(95\) 3.42644e18 0.557787
\(96\) 5.29663e18 0.780589
\(97\) −2.57839e18 −0.344364 −0.172182 0.985065i \(-0.555082\pi\)
−0.172182 + 0.985065i \(0.555082\pi\)
\(98\) 5.41048e18 0.655523
\(99\) −3.31322e18 −0.364515
\(100\) 5.06355e18 0.506355
\(101\) −3.76803e18 −0.342816 −0.171408 0.985200i \(-0.554832\pi\)
−0.171408 + 0.985200i \(0.554832\pi\)
\(102\) 9.00579e18 0.746140
\(103\) −3.38045e18 −0.255283 −0.127641 0.991820i \(-0.540741\pi\)
−0.127641 + 0.991820i \(0.540741\pi\)
\(104\) −4.82412e19 −3.32354
\(105\) −2.54718e18 −0.160237
\(106\) 3.00095e19 1.72525
\(107\) −8.94715e18 −0.470477 −0.235239 0.971938i \(-0.575587\pi\)
−0.235239 + 0.971938i \(0.575587\pi\)
\(108\) −3.09084e19 −1.48782
\(109\) 4.27066e19 1.88340 0.941701 0.336452i \(-0.109227\pi\)
0.941701 + 0.336452i \(0.109227\pi\)
\(110\) −8.37328e18 −0.338583
\(111\) 7.82785e18 0.290452
\(112\) 9.80969e19 3.34260
\(113\) −1.68233e19 −0.526823 −0.263412 0.964684i \(-0.584848\pi\)
−0.263412 + 0.964684i \(0.584848\pi\)
\(114\) −2.51083e19 −0.723138
\(115\) 1.73264e19 0.459282
\(116\) −9.23881e19 −2.25562
\(117\) 4.64234e19 1.04465
\(118\) 3.83939e19 0.796862
\(119\) 7.80233e19 1.49461
\(120\) −2.24492e19 −0.397171
\(121\) −5.12332e19 −0.837704
\(122\) 2.59502e19 0.392394
\(123\) 1.07653e19 0.150635
\(124\) −2.67412e20 −3.46478
\(125\) −7.45058e18 −0.0894427
\(126\) −1.77441e20 −1.97485
\(127\) −2.55831e19 −0.264130 −0.132065 0.991241i \(-0.542161\pi\)
−0.132065 + 0.991241i \(0.542161\pi\)
\(128\) 1.04038e20 0.997001
\(129\) 4.54269e19 0.404307
\(130\) 1.17323e20 0.970333
\(131\) 3.95361e19 0.304030 0.152015 0.988378i \(-0.451424\pi\)
0.152015 + 0.988378i \(0.451424\pi\)
\(132\) 4.39847e19 0.314666
\(133\) −2.17530e20 −1.44854
\(134\) 3.65397e20 2.26605
\(135\) 4.54790e19 0.262809
\(136\) 6.87645e20 3.70463
\(137\) −1.82400e20 −0.916602 −0.458301 0.888797i \(-0.651542\pi\)
−0.458301 + 0.888797i \(0.651542\pi\)
\(138\) −1.26965e20 −0.595432
\(139\) 9.57423e18 0.0419240 0.0209620 0.999780i \(-0.493327\pi\)
0.0209620 + 0.999780i \(0.493327\pi\)
\(140\) −3.21464e20 −1.31497
\(141\) 4.07232e19 0.155690
\(142\) −2.70707e20 −0.967744
\(143\) −1.39077e20 −0.465117
\(144\) −8.31984e20 −2.60418
\(145\) 1.35941e20 0.398433
\(146\) 3.47744e20 0.954790
\(147\) 4.18193e19 0.107612
\(148\) 9.87905e20 2.38358
\(149\) 6.85635e20 1.55176 0.775878 0.630882i \(-0.217307\pi\)
0.775878 + 0.630882i \(0.217307\pi\)
\(150\) 5.45965e19 0.115957
\(151\) −3.55608e20 −0.709073 −0.354537 0.935042i \(-0.615361\pi\)
−0.354537 + 0.935042i \(0.615361\pi\)
\(152\) −1.91717e21 −3.59042
\(153\) −6.61734e20 −1.16443
\(154\) 5.31585e20 0.879277
\(155\) 3.93474e20 0.612019
\(156\) −6.16296e20 −0.901790
\(157\) 7.66927e20 1.05611 0.528053 0.849212i \(-0.322922\pi\)
0.528053 + 0.849212i \(0.322922\pi\)
\(158\) 1.22324e21 1.58587
\(159\) 2.31953e20 0.283222
\(160\) −9.83573e20 −1.13153
\(161\) −1.09998e21 −1.19272
\(162\) 1.32997e21 1.35972
\(163\) −3.87952e20 −0.374107 −0.187053 0.982350i \(-0.559894\pi\)
−0.187053 + 0.982350i \(0.559894\pi\)
\(164\) 1.35862e21 1.23618
\(165\) −6.47197e19 −0.0555826
\(166\) −1.18202e21 −0.958511
\(167\) −2.05376e21 −1.57306 −0.786528 0.617555i \(-0.788123\pi\)
−0.786528 + 0.617555i \(0.788123\pi\)
\(168\) 1.42520e21 1.03143
\(169\) 4.86766e20 0.332963
\(170\) −1.67236e21 −1.08159
\(171\) 1.84493e21 1.12854
\(172\) 5.73305e21 3.31792
\(173\) −2.48811e21 −1.36280 −0.681399 0.731912i \(-0.738628\pi\)
−0.681399 + 0.731912i \(0.738628\pi\)
\(174\) −9.96151e20 −0.516544
\(175\) 4.73007e20 0.232277
\(176\) 2.49248e21 1.15948
\(177\) 2.96759e20 0.130815
\(178\) −3.78508e21 −1.58156
\(179\) 3.14092e21 1.24438 0.622191 0.782866i \(-0.286243\pi\)
0.622191 + 0.782866i \(0.286243\pi\)
\(180\) 2.72641e21 1.02448
\(181\) −3.35426e21 −1.19578 −0.597889 0.801579i \(-0.703994\pi\)
−0.597889 + 0.801579i \(0.703994\pi\)
\(182\) −7.44834e21 −2.51989
\(183\) 2.00577e20 0.0644165
\(184\) −9.69451e21 −2.95635
\(185\) −1.45362e21 −0.421035
\(186\) −2.88330e21 −0.793446
\(187\) 1.98244e21 0.518449
\(188\) 5.13942e21 1.27766
\(189\) −2.88727e21 −0.682497
\(190\) 4.66256e21 1.04825
\(191\) 6.93292e21 1.48286 0.741428 0.671033i \(-0.234149\pi\)
0.741428 + 0.671033i \(0.234149\pi\)
\(192\) 2.84488e21 0.579032
\(193\) 8.38013e21 1.62351 0.811757 0.583995i \(-0.198511\pi\)
0.811757 + 0.583995i \(0.198511\pi\)
\(194\) −3.50858e21 −0.647164
\(195\) 9.06826e20 0.159292
\(196\) 5.27776e21 0.883114
\(197\) 2.41151e21 0.384467 0.192234 0.981349i \(-0.438427\pi\)
0.192234 + 0.981349i \(0.438427\pi\)
\(198\) −4.50850e21 −0.685035
\(199\) 1.85924e20 0.0269296 0.0134648 0.999909i \(-0.495714\pi\)
0.0134648 + 0.999909i \(0.495714\pi\)
\(200\) 4.16876e21 0.575734
\(201\) 2.82427e21 0.372001
\(202\) −5.12738e21 −0.644255
\(203\) −8.63034e21 −1.03470
\(204\) 8.78487e21 1.00519
\(205\) −1.99909e21 −0.218359
\(206\) −4.59999e21 −0.479753
\(207\) 9.32922e21 0.929238
\(208\) −3.49236e22 −3.32290
\(209\) −5.52709e21 −0.502466
\(210\) −3.46610e21 −0.301133
\(211\) 9.50589e21 0.789422 0.394711 0.918805i \(-0.370845\pi\)
0.394711 + 0.918805i \(0.370845\pi\)
\(212\) 2.92733e22 2.32424
\(213\) −2.09238e21 −0.158867
\(214\) −1.21749e22 −0.884169
\(215\) −8.43569e21 −0.586077
\(216\) −2.54465e22 −1.69168
\(217\) −2.49800e22 −1.58937
\(218\) 5.81134e22 3.53948
\(219\) 2.68783e21 0.156741
\(220\) −8.16788e21 −0.456135
\(221\) −2.77772e22 −1.48580
\(222\) 1.06518e22 0.545847
\(223\) 2.74715e22 1.34892 0.674459 0.738312i \(-0.264377\pi\)
0.674459 + 0.738312i \(0.264377\pi\)
\(224\) 6.24430e22 2.93851
\(225\) −4.01168e21 −0.180964
\(226\) −2.28924e22 −0.990060
\(227\) 1.41286e22 0.585942 0.292971 0.956121i \(-0.405356\pi\)
0.292971 + 0.956121i \(0.405356\pi\)
\(228\) −2.44924e22 −0.974204
\(229\) −3.83709e22 −1.46408 −0.732041 0.681261i \(-0.761432\pi\)
−0.732041 + 0.681261i \(0.761432\pi\)
\(230\) 2.35771e22 0.863129
\(231\) 4.10879e21 0.144344
\(232\) −7.60620e22 −2.56467
\(233\) −4.40483e22 −1.42576 −0.712882 0.701284i \(-0.752611\pi\)
−0.712882 + 0.701284i \(0.752611\pi\)
\(234\) 6.31712e22 1.96322
\(235\) −7.56221e21 −0.225686
\(236\) 3.74521e22 1.07353
\(237\) 9.45479e21 0.260341
\(238\) 1.06171e23 2.80883
\(239\) −5.68968e22 −1.44646 −0.723232 0.690605i \(-0.757344\pi\)
−0.723232 + 0.690605i \(0.757344\pi\)
\(240\) −1.62518e22 −0.397095
\(241\) 8.31189e22 1.95226 0.976131 0.217182i \(-0.0696866\pi\)
0.976131 + 0.217182i \(0.0696866\pi\)
\(242\) −6.97162e22 −1.57430
\(243\) 3.73433e22 0.810873
\(244\) 2.53136e22 0.528629
\(245\) −7.76576e21 −0.155993
\(246\) 1.46490e22 0.283089
\(247\) 7.74432e22 1.44000
\(248\) −2.20157e23 −3.93951
\(249\) −9.13619e21 −0.157352
\(250\) −1.01385e22 −0.168090
\(251\) 5.17728e22 0.826421 0.413211 0.910635i \(-0.364407\pi\)
0.413211 + 0.910635i \(0.364407\pi\)
\(252\) −1.73089e23 −2.66050
\(253\) −2.79488e22 −0.413731
\(254\) −3.48124e22 −0.496380
\(255\) −1.29262e22 −0.177557
\(256\) −2.41073e20 −0.00319058
\(257\) −3.72920e22 −0.475610 −0.237805 0.971313i \(-0.576428\pi\)
−0.237805 + 0.971313i \(0.576428\pi\)
\(258\) 6.18151e22 0.759814
\(259\) 9.22841e22 1.09340
\(260\) 1.14445e23 1.30722
\(261\) 7.31960e22 0.806125
\(262\) 5.37992e22 0.571365
\(263\) 3.03845e21 0.0311223 0.0155612 0.999879i \(-0.495047\pi\)
0.0155612 + 0.999879i \(0.495047\pi\)
\(264\) 3.62121e22 0.357780
\(265\) −4.30731e22 −0.410554
\(266\) −2.96007e23 −2.72224
\(267\) −2.92561e22 −0.259633
\(268\) 3.56434e23 3.05280
\(269\) −1.00175e23 −0.828159 −0.414079 0.910241i \(-0.635896\pi\)
−0.414079 + 0.910241i \(0.635896\pi\)
\(270\) 6.18860e22 0.493897
\(271\) 3.74901e22 0.288874 0.144437 0.989514i \(-0.453863\pi\)
0.144437 + 0.989514i \(0.453863\pi\)
\(272\) 4.97812e23 3.70392
\(273\) −5.75706e22 −0.413672
\(274\) −2.48203e23 −1.72257
\(275\) 1.20183e22 0.0805718
\(276\) −1.23850e23 −0.802159
\(277\) 2.51054e22 0.157112 0.0785559 0.996910i \(-0.474969\pi\)
0.0785559 + 0.996910i \(0.474969\pi\)
\(278\) 1.30282e22 0.0787879
\(279\) 2.11862e23 1.23826
\(280\) −2.64657e23 −1.49514
\(281\) −1.50547e23 −0.822173 −0.411086 0.911596i \(-0.634851\pi\)
−0.411086 + 0.911596i \(0.634851\pi\)
\(282\) 5.54145e22 0.292588
\(283\) −2.94312e23 −1.50258 −0.751289 0.659973i \(-0.770568\pi\)
−0.751289 + 0.659973i \(0.770568\pi\)
\(284\) −2.64067e23 −1.30373
\(285\) 3.60384e22 0.172084
\(286\) −1.89250e23 −0.874096
\(287\) 1.26914e23 0.567063
\(288\) −5.29594e23 −2.28936
\(289\) 1.56872e23 0.656171
\(290\) 1.84983e23 0.748775
\(291\) −2.71189e22 −0.106240
\(292\) 3.39214e23 1.28628
\(293\) −4.17390e23 −1.53215 −0.766073 0.642754i \(-0.777792\pi\)
−0.766073 + 0.642754i \(0.777792\pi\)
\(294\) 5.69060e22 0.202236
\(295\) −5.51075e22 −0.189628
\(296\) 8.13330e23 2.71017
\(297\) −7.33609e22 −0.236744
\(298\) 9.32985e23 2.91622
\(299\) 3.91606e23 1.18570
\(300\) 5.32572e22 0.156216
\(301\) 5.35547e23 1.52200
\(302\) −4.83898e23 −1.33256
\(303\) −3.96312e22 −0.105763
\(304\) −1.38791e24 −3.58973
\(305\) −3.72468e22 −0.0933772
\(306\) −9.00462e23 −2.18832
\(307\) −9.44483e22 −0.222525 −0.111263 0.993791i \(-0.535489\pi\)
−0.111263 + 0.993791i \(0.535489\pi\)
\(308\) 5.18545e23 1.18455
\(309\) −3.55548e22 −0.0787576
\(310\) 5.35423e23 1.15017
\(311\) −2.94596e23 −0.613766 −0.306883 0.951747i \(-0.599286\pi\)
−0.306883 + 0.951747i \(0.599286\pi\)
\(312\) −5.07389e23 −1.02535
\(313\) −1.70785e23 −0.334795 −0.167398 0.985889i \(-0.553536\pi\)
−0.167398 + 0.985889i \(0.553536\pi\)
\(314\) 1.04360e24 1.98474
\(315\) 2.54685e23 0.469952
\(316\) 1.19323e24 2.13647
\(317\) 9.64787e21 0.0167636 0.00838182 0.999965i \(-0.497332\pi\)
0.00838182 + 0.999965i \(0.497332\pi\)
\(318\) 3.15632e23 0.532259
\(319\) −2.19283e23 −0.358916
\(320\) −5.28288e23 −0.839356
\(321\) −9.41039e22 −0.145148
\(322\) −1.49681e24 −2.24149
\(323\) −1.10390e24 −1.60511
\(324\) 1.29734e24 1.83180
\(325\) −1.68396e23 −0.230908
\(326\) −5.27909e23 −0.703059
\(327\) 4.49177e23 0.581050
\(328\) 1.11853e24 1.40555
\(329\) 4.80094e23 0.586091
\(330\) −8.80680e22 −0.104457
\(331\) 5.66277e23 0.652624 0.326312 0.945262i \(-0.394194\pi\)
0.326312 + 0.945262i \(0.394194\pi\)
\(332\) −1.15302e24 −1.29130
\(333\) −7.82684e23 −0.851856
\(334\) −2.79468e24 −2.95625
\(335\) −5.24462e23 −0.539248
\(336\) 1.03176e24 1.03123
\(337\) 1.18640e24 1.15278 0.576392 0.817174i \(-0.304460\pi\)
0.576392 + 0.817174i \(0.304460\pi\)
\(338\) 6.62372e23 0.625739
\(339\) −1.76943e23 −0.162531
\(340\) −1.63133e24 −1.45711
\(341\) −6.34702e23 −0.551319
\(342\) 2.51050e24 2.12086
\(343\) −9.20399e23 −0.756279
\(344\) 4.71995e24 3.77253
\(345\) 1.82235e23 0.141694
\(346\) −3.38572e24 −2.56111
\(347\) −1.39897e24 −1.02962 −0.514812 0.857303i \(-0.672138\pi\)
−0.514812 + 0.857303i \(0.672138\pi\)
\(348\) −9.71715e23 −0.695883
\(349\) 1.55030e24 1.08037 0.540186 0.841546i \(-0.318354\pi\)
0.540186 + 0.841546i \(0.318354\pi\)
\(350\) 6.43648e23 0.436518
\(351\) 1.02790e24 0.678475
\(352\) 1.58657e24 1.01931
\(353\) 2.69377e24 1.68461 0.842307 0.538997i \(-0.181197\pi\)
0.842307 + 0.538997i \(0.181197\pi\)
\(354\) 4.03818e23 0.245841
\(355\) 3.88551e23 0.230292
\(356\) −3.69223e24 −2.13066
\(357\) 8.20630e23 0.461104
\(358\) 4.27404e24 2.33857
\(359\) −2.68511e23 −0.143075 −0.0715376 0.997438i \(-0.522791\pi\)
−0.0715376 + 0.997438i \(0.522791\pi\)
\(360\) 2.24462e24 1.16485
\(361\) 1.09927e24 0.555632
\(362\) −4.56435e24 −2.24723
\(363\) −5.38858e23 −0.258441
\(364\) −7.26563e24 −3.39477
\(365\) −4.99124e23 −0.227209
\(366\) 2.72938e23 0.121058
\(367\) 4.64559e22 0.0200777 0.0100388 0.999950i \(-0.496804\pi\)
0.0100388 + 0.999950i \(0.496804\pi\)
\(368\) −7.01822e24 −2.95579
\(369\) −1.07639e24 −0.441792
\(370\) −1.97802e24 −0.791253
\(371\) 2.73454e24 1.06618
\(372\) −2.81257e24 −1.06892
\(373\) 2.90680e24 1.07691 0.538457 0.842653i \(-0.319007\pi\)
0.538457 + 0.842653i \(0.319007\pi\)
\(374\) 2.69763e24 0.974322
\(375\) −7.83633e22 −0.0275941
\(376\) 4.23122e24 1.45272
\(377\) 3.07250e24 1.02861
\(378\) −3.92889e24 −1.28262
\(379\) −5.17963e24 −1.64902 −0.824510 0.565848i \(-0.808549\pi\)
−0.824510 + 0.565848i \(0.808549\pi\)
\(380\) 4.54818e24 1.41219
\(381\) −2.69077e23 −0.0814870
\(382\) 9.43404e24 2.78673
\(383\) −1.72726e24 −0.497703 −0.248852 0.968542i \(-0.580053\pi\)
−0.248852 + 0.968542i \(0.580053\pi\)
\(384\) 1.09424e24 0.307586
\(385\) −7.62994e23 −0.209240
\(386\) 1.14033e25 3.05107
\(387\) −4.54210e24 −1.18578
\(388\) −3.42251e24 −0.871853
\(389\) −1.27175e24 −0.316141 −0.158071 0.987428i \(-0.550527\pi\)
−0.158071 + 0.987428i \(0.550527\pi\)
\(390\) 1.23397e24 0.299359
\(391\) −5.58208e24 −1.32165
\(392\) 4.34511e24 1.00411
\(393\) 4.15831e23 0.0937967
\(394\) 3.28148e24 0.722530
\(395\) −1.75574e24 −0.377387
\(396\) −4.39790e24 −0.922872
\(397\) −5.93194e24 −1.21531 −0.607655 0.794201i \(-0.707890\pi\)
−0.607655 + 0.794201i \(0.707890\pi\)
\(398\) 2.52998e23 0.0506089
\(399\) −2.28793e24 −0.446889
\(400\) 3.01793e24 0.575624
\(401\) 3.19633e24 0.595361 0.297680 0.954666i \(-0.403787\pi\)
0.297680 + 0.954666i \(0.403787\pi\)
\(402\) 3.84316e24 0.699102
\(403\) 8.89317e24 1.58001
\(404\) −5.00160e24 −0.867933
\(405\) −1.90893e24 −0.323569
\(406\) −1.17438e25 −1.94452
\(407\) 2.34479e24 0.379277
\(408\) 7.23248e24 1.14292
\(409\) −1.00284e25 −1.54831 −0.774156 0.632995i \(-0.781825\pi\)
−0.774156 + 0.632995i \(0.781825\pi\)
\(410\) −2.72028e24 −0.410362
\(411\) −1.91844e24 −0.282782
\(412\) −4.48715e24 −0.646319
\(413\) 3.49855e24 0.492450
\(414\) 1.26948e25 1.74632
\(415\) 1.69657e24 0.228095
\(416\) −2.22304e25 −2.92120
\(417\) 1.00699e23 0.0129340
\(418\) −7.52104e24 −0.944285
\(419\) 1.18441e25 1.45368 0.726840 0.686807i \(-0.240988\pi\)
0.726840 + 0.686807i \(0.240988\pi\)
\(420\) −3.38108e24 −0.405683
\(421\) 1.03966e25 1.21958 0.609792 0.792562i \(-0.291253\pi\)
0.609792 + 0.792562i \(0.291253\pi\)
\(422\) 1.29352e25 1.48356
\(423\) −4.07179e24 −0.456617
\(424\) 2.41004e25 2.64270
\(425\) 2.40037e24 0.257385
\(426\) −2.84723e24 −0.298560
\(427\) 2.36465e24 0.242494
\(428\) −1.18763e25 −1.19114
\(429\) −1.46277e24 −0.143494
\(430\) −1.14789e25 −1.10142
\(431\) −2.03143e25 −1.90663 −0.953316 0.301975i \(-0.902354\pi\)
−0.953316 + 0.301975i \(0.902354\pi\)
\(432\) −1.84217e25 −1.69135
\(433\) −2.89050e21 −0.000259620 0 −0.000129810 1.00000i \(-0.500041\pi\)
−0.000129810 1.00000i \(0.500041\pi\)
\(434\) −3.39918e25 −2.98691
\(435\) 1.42979e24 0.122921
\(436\) 5.66878e25 4.76835
\(437\) 1.55629e25 1.28091
\(438\) 3.65749e24 0.294563
\(439\) 1.26912e25 1.00020 0.500102 0.865967i \(-0.333296\pi\)
0.500102 + 0.865967i \(0.333296\pi\)
\(440\) −6.72451e24 −0.518633
\(441\) −4.18139e24 −0.315612
\(442\) −3.77981e25 −2.79228
\(443\) 8.29689e24 0.599901 0.299951 0.953955i \(-0.403030\pi\)
0.299951 + 0.953955i \(0.403030\pi\)
\(444\) 1.03905e25 0.735360
\(445\) 5.43280e24 0.376360
\(446\) 3.73821e25 2.53502
\(447\) 7.21134e24 0.478734
\(448\) 3.35388e25 2.17975
\(449\) 4.92023e23 0.0313073 0.0156536 0.999877i \(-0.495017\pi\)
0.0156536 + 0.999877i \(0.495017\pi\)
\(450\) −5.45894e24 −0.340086
\(451\) 3.22467e24 0.196702
\(452\) −2.23309e25 −1.33380
\(453\) −3.74020e24 −0.218757
\(454\) 1.92257e25 1.10116
\(455\) 1.06907e25 0.599653
\(456\) −2.01643e25 −1.10769
\(457\) −2.95954e25 −1.59228 −0.796141 0.605111i \(-0.793129\pi\)
−0.796141 + 0.605111i \(0.793129\pi\)
\(458\) −5.22136e25 −2.75145
\(459\) −1.46520e25 −0.756271
\(460\) 2.29987e25 1.16280
\(461\) −2.39904e25 −1.18817 −0.594084 0.804403i \(-0.702485\pi\)
−0.594084 + 0.804403i \(0.702485\pi\)
\(462\) 5.59107e24 0.271267
\(463\) 1.01589e25 0.482869 0.241435 0.970417i \(-0.422382\pi\)
0.241435 + 0.970417i \(0.422382\pi\)
\(464\) −5.50642e25 −2.56418
\(465\) 4.13846e24 0.188815
\(466\) −5.99391e25 −2.67944
\(467\) −2.35296e25 −1.03063 −0.515317 0.857000i \(-0.672326\pi\)
−0.515317 + 0.857000i \(0.672326\pi\)
\(468\) 6.16216e25 2.64483
\(469\) 3.32959e25 1.40039
\(470\) −1.02904e25 −0.424132
\(471\) 8.06634e24 0.325820
\(472\) 3.08339e25 1.22062
\(473\) 1.36074e25 0.527951
\(474\) 1.28657e25 0.489259
\(475\) −6.69226e24 −0.249450
\(476\) 1.03567e26 3.78402
\(477\) −2.31923e25 −0.830651
\(478\) −7.74229e25 −2.71834
\(479\) 3.50521e24 0.120650 0.0603249 0.998179i \(-0.480786\pi\)
0.0603249 + 0.998179i \(0.480786\pi\)
\(480\) −1.03450e25 −0.349090
\(481\) −3.28542e25 −1.08696
\(482\) 1.13105e26 3.66889
\(483\) −1.15693e25 −0.367969
\(484\) −6.80060e25 −2.12088
\(485\) 5.03593e24 0.154004
\(486\) 5.08152e25 1.52388
\(487\) 5.43728e25 1.59903 0.799515 0.600647i \(-0.205090\pi\)
0.799515 + 0.600647i \(0.205090\pi\)
\(488\) 2.08404e25 0.601060
\(489\) −4.08038e24 −0.115416
\(490\) −1.05673e25 −0.293159
\(491\) −5.76237e24 −0.156793 −0.0783966 0.996922i \(-0.524980\pi\)
−0.0783966 + 0.996922i \(0.524980\pi\)
\(492\) 1.42896e25 0.381375
\(493\) −4.37964e25 −1.14655
\(494\) 1.05382e26 2.70620
\(495\) 6.47113e24 0.163016
\(496\) −1.59380e26 −3.93875
\(497\) −2.46675e25 −0.598053
\(498\) −1.24322e25 −0.295711
\(499\) 7.67371e25 1.79081 0.895406 0.445251i \(-0.146885\pi\)
0.895406 + 0.445251i \(0.146885\pi\)
\(500\) −9.88975e24 −0.226449
\(501\) −2.16010e25 −0.485305
\(502\) 7.04504e25 1.55310
\(503\) 3.38074e24 0.0731334 0.0365667 0.999331i \(-0.488358\pi\)
0.0365667 + 0.999331i \(0.488358\pi\)
\(504\) −1.42502e26 −3.02504
\(505\) 7.35943e24 0.153312
\(506\) −3.80316e25 −0.777525
\(507\) 5.11968e24 0.102723
\(508\) −3.39585e25 −0.668718
\(509\) −5.88807e25 −1.13803 −0.569015 0.822327i \(-0.692675\pi\)
−0.569015 + 0.822327i \(0.692675\pi\)
\(510\) −1.75894e25 −0.333684
\(511\) 3.16873e25 0.590047
\(512\) −5.48737e25 −1.00300
\(513\) 4.08501e25 0.732957
\(514\) −5.07455e25 −0.893815
\(515\) 6.60245e24 0.114166
\(516\) 6.02988e25 1.02361
\(517\) 1.21984e25 0.203302
\(518\) 1.25577e26 2.05483
\(519\) −2.61693e25 −0.420438
\(520\) 9.42211e25 1.48633
\(521\) −1.35023e25 −0.209145 −0.104573 0.994517i \(-0.533347\pi\)
−0.104573 + 0.994517i \(0.533347\pi\)
\(522\) 9.96022e25 1.51495
\(523\) 2.74955e25 0.410673 0.205336 0.978691i \(-0.434171\pi\)
0.205336 + 0.978691i \(0.434171\pi\)
\(524\) 5.24795e25 0.769737
\(525\) 4.97496e24 0.0716600
\(526\) 4.13460e24 0.0584883
\(527\) −1.26766e26 −1.76117
\(528\) 2.62153e25 0.357711
\(529\) 4.08140e24 0.0546991
\(530\) −5.86122e25 −0.771556
\(531\) −2.96720e25 −0.383662
\(532\) −2.88745e26 −3.66737
\(533\) −4.51828e25 −0.563722
\(534\) −3.98105e25 −0.487928
\(535\) 1.74749e25 0.210404
\(536\) 2.93448e26 3.47109
\(537\) 3.30354e25 0.383906
\(538\) −1.36314e26 −1.55636
\(539\) 1.25267e25 0.140522
\(540\) 6.03679e25 0.665373
\(541\) 5.00313e25 0.541836 0.270918 0.962602i \(-0.412673\pi\)
0.270918 + 0.962602i \(0.412673\pi\)
\(542\) 5.10151e25 0.542882
\(543\) −3.52793e25 −0.368911
\(544\) 3.16879e26 3.25615
\(545\) −8.34113e25 −0.842283
\(546\) −7.83398e25 −0.777414
\(547\) 7.15124e25 0.697431 0.348716 0.937229i \(-0.386618\pi\)
0.348716 + 0.937229i \(0.386618\pi\)
\(548\) −2.42115e26 −2.32063
\(549\) −2.00551e25 −0.188925
\(550\) 1.63541e25 0.151419
\(551\) 1.22105e26 1.11120
\(552\) −1.01964e26 −0.912068
\(553\) 1.11464e26 0.980048
\(554\) 3.41624e25 0.295261
\(555\) −1.52888e25 −0.129894
\(556\) 1.27086e25 0.106142
\(557\) 5.59071e25 0.459031 0.229516 0.973305i \(-0.426286\pi\)
0.229516 + 0.973305i \(0.426286\pi\)
\(558\) 2.88293e26 2.32707
\(559\) −1.90661e26 −1.51304
\(560\) −1.91596e26 −1.49486
\(561\) 2.08509e25 0.159947
\(562\) −2.04859e26 −1.54511
\(563\) 1.34263e26 0.995696 0.497848 0.867264i \(-0.334124\pi\)
0.497848 + 0.867264i \(0.334124\pi\)
\(564\) 5.40551e25 0.394172
\(565\) 3.28579e25 0.235602
\(566\) −4.00488e26 −2.82380
\(567\) 1.21190e26 0.840286
\(568\) −2.17403e26 −1.48237
\(569\) −1.78969e26 −1.20009 −0.600043 0.799968i \(-0.704850\pi\)
−0.600043 + 0.799968i \(0.704850\pi\)
\(570\) 4.90396e25 0.323397
\(571\) 9.68510e25 0.628147 0.314073 0.949399i \(-0.398306\pi\)
0.314073 + 0.949399i \(0.398306\pi\)
\(572\) −1.84608e26 −1.17757
\(573\) 7.29187e25 0.457478
\(574\) 1.72699e26 1.06568
\(575\) −3.38407e25 −0.205397
\(576\) −2.84451e26 −1.69822
\(577\) 1.80741e26 1.06142 0.530708 0.847555i \(-0.321926\pi\)
0.530708 + 0.847555i \(0.321926\pi\)
\(578\) 2.13466e26 1.23314
\(579\) 8.81401e25 0.500872
\(580\) 1.80446e26 1.00874
\(581\) −1.07708e26 −0.592347
\(582\) −3.69023e25 −0.199657
\(583\) 6.94801e25 0.369836
\(584\) 2.79271e26 1.46252
\(585\) −9.06708e25 −0.467183
\(586\) −5.67968e26 −2.87937
\(587\) −1.87816e26 −0.936851 −0.468425 0.883503i \(-0.655179\pi\)
−0.468425 + 0.883503i \(0.655179\pi\)
\(588\) 5.55101e25 0.272450
\(589\) 3.53426e26 1.70688
\(590\) −7.49881e25 −0.356368
\(591\) 2.53636e25 0.118612
\(592\) 5.88801e26 2.70964
\(593\) −2.23387e26 −1.01167 −0.505835 0.862630i \(-0.668815\pi\)
−0.505835 + 0.862630i \(0.668815\pi\)
\(594\) −9.98266e25 −0.444913
\(595\) −1.52389e26 −0.668410
\(596\) 9.10099e26 3.92870
\(597\) 1.95550e24 0.00830810
\(598\) 5.32882e26 2.22828
\(599\) −4.10335e26 −1.68882 −0.844410 0.535697i \(-0.820049\pi\)
−0.844410 + 0.535697i \(0.820049\pi\)
\(600\) 4.38460e25 0.177620
\(601\) 2.50922e26 1.00053 0.500267 0.865871i \(-0.333235\pi\)
0.500267 + 0.865871i \(0.333235\pi\)
\(602\) 7.28751e26 2.86030
\(603\) −2.82391e26 −1.09103
\(604\) −4.72028e26 −1.79522
\(605\) 1.00065e26 0.374633
\(606\) −5.39285e25 −0.198760
\(607\) −1.61528e26 −0.586077 −0.293039 0.956101i \(-0.594666\pi\)
−0.293039 + 0.956101i \(0.594666\pi\)
\(608\) −8.83464e26 −3.15577
\(609\) −9.07718e25 −0.319217
\(610\) −5.06840e25 −0.175484
\(611\) −1.70919e26 −0.582637
\(612\) −8.78373e26 −2.94809
\(613\) −3.72600e26 −1.23131 −0.615657 0.788014i \(-0.711109\pi\)
−0.615657 + 0.788014i \(0.711109\pi\)
\(614\) −1.28521e26 −0.418192
\(615\) −2.10259e25 −0.0673661
\(616\) 4.26912e26 1.34686
\(617\) 1.16664e26 0.362432 0.181216 0.983443i \(-0.441997\pi\)
0.181216 + 0.983443i \(0.441997\pi\)
\(618\) −4.83815e25 −0.148009
\(619\) −4.75023e25 −0.143104 −0.0715522 0.997437i \(-0.522795\pi\)
−0.0715522 + 0.997437i \(0.522795\pi\)
\(620\) 5.22289e26 1.54949
\(621\) 2.06566e26 0.603517
\(622\) −4.00874e26 −1.15345
\(623\) −3.44906e26 −0.977382
\(624\) −3.67318e26 −1.02515
\(625\) 1.45519e25 0.0400000
\(626\) −2.32398e26 −0.629181
\(627\) −5.81325e25 −0.155016
\(628\) 1.01800e27 2.67382
\(629\) 4.68314e26 1.21159
\(630\) 3.46565e26 0.883181
\(631\) −2.59055e26 −0.650300 −0.325150 0.945662i \(-0.605415\pi\)
−0.325150 + 0.945662i \(0.605415\pi\)
\(632\) 9.82372e26 2.42920
\(633\) 9.99805e25 0.243546
\(634\) 1.31284e25 0.0315040
\(635\) 4.99670e25 0.118122
\(636\) 3.07889e26 0.717054
\(637\) −1.75519e26 −0.402717
\(638\) −2.98391e26 −0.674512
\(639\) 2.09211e26 0.465936
\(640\) −2.03199e26 −0.445873
\(641\) 6.38588e26 1.38061 0.690303 0.723521i \(-0.257477\pi\)
0.690303 + 0.723521i \(0.257477\pi\)
\(642\) −1.28053e26 −0.272776
\(643\) 2.77288e26 0.582006 0.291003 0.956722i \(-0.406011\pi\)
0.291003 + 0.956722i \(0.406011\pi\)
\(644\) −1.46010e27 −3.01971
\(645\) −8.87244e25 −0.180811
\(646\) −1.50214e27 −3.01649
\(647\) 3.56860e26 0.706166 0.353083 0.935592i \(-0.385133\pi\)
0.353083 + 0.935592i \(0.385133\pi\)
\(648\) 1.06809e27 2.08278
\(649\) 8.88924e25 0.170820
\(650\) −2.29146e26 −0.433946
\(651\) −2.62734e26 −0.490339
\(652\) −5.14959e26 −0.947154
\(653\) −1.33562e26 −0.242106 −0.121053 0.992646i \(-0.538627\pi\)
−0.121053 + 0.992646i \(0.538627\pi\)
\(654\) 6.11222e26 1.09197
\(655\) −7.72190e25 −0.135966
\(656\) 8.09749e26 1.40528
\(657\) −2.68748e26 −0.459699
\(658\) 6.53292e26 1.10144
\(659\) 1.09829e27 1.82518 0.912592 0.408872i \(-0.134078\pi\)
0.912592 + 0.408872i \(0.134078\pi\)
\(660\) −8.59077e25 −0.140723
\(661\) −1.98489e25 −0.0320496 −0.0160248 0.999872i \(-0.505101\pi\)
−0.0160248 + 0.999872i \(0.505101\pi\)
\(662\) 7.70567e26 1.22648
\(663\) −2.92153e26 −0.458387
\(664\) −9.49269e26 −1.46822
\(665\) 4.24864e26 0.647805
\(666\) −1.06505e27 −1.60089
\(667\) 6.17447e26 0.914964
\(668\) −2.72613e27 −3.98263
\(669\) 2.88938e26 0.416156
\(670\) −7.13667e26 −1.01341
\(671\) 6.00818e25 0.0841161
\(672\) 6.56759e26 0.906563
\(673\) −1.27707e26 −0.173809 −0.0869045 0.996217i \(-0.527698\pi\)
−0.0869045 + 0.996217i \(0.527698\pi\)
\(674\) 1.61441e27 2.16643
\(675\) −8.88262e25 −0.117532
\(676\) 6.46123e26 0.842989
\(677\) 1.05640e26 0.135905 0.0679525 0.997689i \(-0.478353\pi\)
0.0679525 + 0.997689i \(0.478353\pi\)
\(678\) −2.40777e26 −0.305445
\(679\) −3.19710e26 −0.399939
\(680\) −1.34306e27 −1.65676
\(681\) 1.48601e26 0.180770
\(682\) −8.63677e26 −1.03610
\(683\) −5.53869e26 −0.655256 −0.327628 0.944807i \(-0.606249\pi\)
−0.327628 + 0.944807i \(0.606249\pi\)
\(684\) 2.44892e27 2.85721
\(685\) 3.56251e26 0.409917
\(686\) −1.25244e27 −1.42128
\(687\) −4.03576e26 −0.451686
\(688\) 3.41695e27 3.77180
\(689\) −9.73526e26 −1.05990
\(690\) 2.47978e26 0.266285
\(691\) 8.40567e26 0.890289 0.445144 0.895459i \(-0.353152\pi\)
0.445144 + 0.895459i \(0.353152\pi\)
\(692\) −3.30267e27 −3.45030
\(693\) −4.10825e26 −0.423342
\(694\) −1.90366e27 −1.93497
\(695\) −1.86997e25 −0.0187490
\(696\) −8.00001e26 −0.791230
\(697\) 6.44050e26 0.628359
\(698\) 2.10958e27 2.03035
\(699\) −4.63289e26 −0.439864
\(700\) 6.27860e26 0.588073
\(701\) −2.09875e27 −1.93927 −0.969637 0.244551i \(-0.921360\pi\)
−0.969637 + 0.244551i \(0.921360\pi\)
\(702\) 1.39873e27 1.27506
\(703\) −1.30567e27 −1.17424
\(704\) 8.52167e26 0.756109
\(705\) −7.95374e25 −0.0696266
\(706\) 3.66557e27 3.16590
\(707\) −4.67220e26 −0.398141
\(708\) 3.93912e26 0.331194
\(709\) 5.69598e26 0.472530 0.236265 0.971689i \(-0.424077\pi\)
0.236265 + 0.971689i \(0.424077\pi\)
\(710\) 5.28725e26 0.432788
\(711\) −9.45356e26 −0.763544
\(712\) −3.03977e27 −2.42259
\(713\) 1.78716e27 1.40545
\(714\) 1.11668e27 0.866554
\(715\) 2.71634e26 0.208007
\(716\) 4.16920e27 3.15050
\(717\) −5.98426e26 −0.446250
\(718\) −3.65379e26 −0.268882
\(719\) 2.23197e27 1.62093 0.810464 0.585789i \(-0.199215\pi\)
0.810464 + 0.585789i \(0.199215\pi\)
\(720\) 1.62497e27 1.16463
\(721\) −4.19162e26 −0.296481
\(722\) 1.49585e27 1.04420
\(723\) 8.74224e26 0.602295
\(724\) −4.45238e27 −3.02744
\(725\) −2.65510e26 −0.178184
\(726\) −7.33257e26 −0.485689
\(727\) −1.57138e27 −1.02731 −0.513657 0.857996i \(-0.671710\pi\)
−0.513657 + 0.857996i \(0.671710\pi\)
\(728\) −5.98171e27 −3.85991
\(729\) −7.43193e26 −0.473358
\(730\) −6.79188e26 −0.426995
\(731\) 2.71774e27 1.68652
\(732\) 2.66243e26 0.163088
\(733\) −1.63155e27 −0.986535 −0.493267 0.869878i \(-0.664198\pi\)
−0.493267 + 0.869878i \(0.664198\pi\)
\(734\) 6.32154e25 0.0377320
\(735\) −8.16783e25 −0.0481257
\(736\) −4.46740e27 −2.59846
\(737\) 8.45995e26 0.485765
\(738\) −1.46471e27 −0.830261
\(739\) −2.01351e27 −1.12676 −0.563380 0.826198i \(-0.690499\pi\)
−0.563380 + 0.826198i \(0.690499\pi\)
\(740\) −1.92950e27 −1.06597
\(741\) 8.14529e26 0.444256
\(742\) 3.72105e27 2.00368
\(743\) 3.00829e27 1.59928 0.799642 0.600477i \(-0.205023\pi\)
0.799642 + 0.600477i \(0.205023\pi\)
\(744\) −2.31556e27 −1.21538
\(745\) −1.33913e27 −0.693967
\(746\) 3.95545e27 2.02385
\(747\) 9.13500e26 0.461491
\(748\) 2.63146e27 1.31260
\(749\) −1.10941e27 −0.546405
\(750\) −1.06634e26 −0.0518576
\(751\) −1.39340e27 −0.669108 −0.334554 0.942377i \(-0.608586\pi\)
−0.334554 + 0.942377i \(0.608586\pi\)
\(752\) 3.06314e27 1.45244
\(753\) 5.44534e26 0.254960
\(754\) 4.18093e27 1.93306
\(755\) 6.94548e26 0.317107
\(756\) −3.83251e27 −1.72793
\(757\) −1.80766e26 −0.0804832 −0.0402416 0.999190i \(-0.512813\pi\)
−0.0402416 + 0.999190i \(0.512813\pi\)
\(758\) −7.04823e27 −3.09901
\(759\) −2.93958e26 −0.127640
\(760\) 3.74446e27 1.60569
\(761\) −2.60179e27 −1.10184 −0.550918 0.834559i \(-0.685722\pi\)
−0.550918 + 0.834559i \(0.685722\pi\)
\(762\) −3.66149e26 −0.153139
\(763\) 5.29544e27 2.18735
\(764\) 9.20262e27 3.75426
\(765\) 1.29245e27 0.520751
\(766\) −2.35039e27 −0.935335
\(767\) −1.24552e27 −0.489549
\(768\) −2.53555e24 −0.000984330 0
\(769\) −3.78836e27 −1.45262 −0.726308 0.687369i \(-0.758765\pi\)
−0.726308 + 0.687369i \(0.758765\pi\)
\(770\) −1.03825e27 −0.393225
\(771\) −3.92228e26 −0.146731
\(772\) 1.11236e28 4.11038
\(773\) 1.74556e27 0.637134 0.318567 0.947900i \(-0.396798\pi\)
0.318567 + 0.947900i \(0.396798\pi\)
\(774\) −6.18071e27 −2.22843
\(775\) −7.68504e26 −0.273703
\(776\) −2.81771e27 −0.991311
\(777\) 9.70621e26 0.337326
\(778\) −1.73055e27 −0.594125
\(779\) −1.79562e27 −0.608988
\(780\) 1.20370e27 0.403293
\(781\) −6.26761e26 −0.207452
\(782\) −7.59587e27 −2.48378
\(783\) 1.62070e27 0.523558
\(784\) 3.14559e27 1.00392
\(785\) −1.49790e27 −0.472305
\(786\) 5.65846e26 0.176272
\(787\) −9.37222e25 −0.0288458 −0.0144229 0.999896i \(-0.504591\pi\)
−0.0144229 + 0.999896i \(0.504591\pi\)
\(788\) 3.20099e27 0.973385
\(789\) 3.19576e25 0.00960158
\(790\) −2.38913e27 −0.709224
\(791\) −2.08601e27 −0.611844
\(792\) −3.62074e27 −1.04932
\(793\) −8.41841e26 −0.241065
\(794\) −8.07195e27 −2.28394
\(795\) −4.53033e26 −0.126661
\(796\) 2.46791e26 0.0681798
\(797\) 2.07539e27 0.566559 0.283280 0.959037i \(-0.408578\pi\)
0.283280 + 0.959037i \(0.408578\pi\)
\(798\) −3.11332e27 −0.839840
\(799\) 2.43633e27 0.649444
\(800\) 1.92104e27 0.506036
\(801\) 2.92523e27 0.761467
\(802\) 4.34944e27 1.11886
\(803\) 8.05122e26 0.204675
\(804\) 3.74888e27 0.941823
\(805\) 2.14840e27 0.533402
\(806\) 1.21015e28 2.96931
\(807\) −1.05362e27 −0.255496
\(808\) −4.11776e27 −0.986855
\(809\) 8.60373e26 0.203787 0.101893 0.994795i \(-0.467510\pi\)
0.101893 + 0.994795i \(0.467510\pi\)
\(810\) −2.59759e27 −0.608083
\(811\) 5.38167e27 1.24514 0.622572 0.782563i \(-0.286088\pi\)
0.622572 + 0.782563i \(0.286088\pi\)
\(812\) −1.14557e28 −2.61964
\(813\) 3.94312e26 0.0891209
\(814\) 3.19070e27 0.712777
\(815\) 7.57718e26 0.167306
\(816\) 5.23587e27 1.14270
\(817\) −7.57710e27 −1.63453
\(818\) −1.36462e28 −2.90974
\(819\) 5.75631e27 1.21324
\(820\) −2.65355e27 −0.552836
\(821\) −2.81042e27 −0.578778 −0.289389 0.957212i \(-0.593452\pi\)
−0.289389 + 0.957212i \(0.593452\pi\)
\(822\) −2.61054e27 −0.531433
\(823\) −1.70827e27 −0.343762 −0.171881 0.985118i \(-0.554984\pi\)
−0.171881 + 0.985118i \(0.554984\pi\)
\(824\) −3.69421e27 −0.734875
\(825\) 1.26406e26 0.0248573
\(826\) 4.76069e27 0.925463
\(827\) 2.86276e27 0.550152 0.275076 0.961423i \(-0.411297\pi\)
0.275076 + 0.961423i \(0.411297\pi\)
\(828\) 1.23834e28 2.35262
\(829\) −3.36069e27 −0.631191 −0.315595 0.948894i \(-0.602204\pi\)
−0.315595 + 0.948894i \(0.602204\pi\)
\(830\) 2.30863e27 0.428659
\(831\) 2.64052e26 0.0484708
\(832\) −1.19402e28 −2.16691
\(833\) 2.50191e27 0.448894
\(834\) 1.37028e26 0.0243070
\(835\) 4.01126e27 0.703492
\(836\) −7.33655e27 −1.27213
\(837\) 4.69101e27 0.804220
\(838\) 1.61170e28 2.73190
\(839\) −4.83778e27 −0.810788 −0.405394 0.914142i \(-0.632866\pi\)
−0.405394 + 0.914142i \(0.632866\pi\)
\(840\) −2.78360e27 −0.461268
\(841\) −1.25884e27 −0.206257
\(842\) 1.41473e28 2.29197
\(843\) −1.58342e27 −0.253649
\(844\) 1.26179e28 1.99864
\(845\) −9.50714e26 −0.148906
\(846\) −5.54073e27 −0.858121
\(847\) −6.35271e27 −0.972896
\(848\) 1.74472e28 2.64219
\(849\) −3.09550e27 −0.463562
\(850\) 3.26632e27 0.483703
\(851\) −6.60235e27 −0.966870
\(852\) −2.77739e27 −0.402217
\(853\) −1.95293e27 −0.279687 −0.139843 0.990174i \(-0.544660\pi\)
−0.139843 + 0.990174i \(0.544660\pi\)
\(854\) 3.21772e27 0.455720
\(855\) −3.60337e27 −0.504697
\(856\) −9.77759e27 −1.35435
\(857\) 5.90679e27 0.809159 0.404579 0.914503i \(-0.367418\pi\)
0.404579 + 0.914503i \(0.367418\pi\)
\(858\) −1.99049e27 −0.269668
\(859\) 1.34994e28 1.80876 0.904380 0.426729i \(-0.140334\pi\)
0.904380 + 0.426729i \(0.140334\pi\)
\(860\) −1.11974e28 −1.48382
\(861\) 1.33485e27 0.174945
\(862\) −2.76428e28 −3.58314
\(863\) −1.46014e28 −1.87194 −0.935971 0.352077i \(-0.885476\pi\)
−0.935971 + 0.352077i \(0.885476\pi\)
\(864\) −1.17262e28 −1.48688
\(865\) 4.85959e27 0.609462
\(866\) −3.93328e24 −0.000487904 0
\(867\) 1.64994e27 0.202436
\(868\) −3.31580e28 −4.02393
\(869\) 2.83213e27 0.339958
\(870\) 1.94561e27 0.231006
\(871\) −1.18537e28 −1.39214
\(872\) 4.66704e28 5.42169
\(873\) 2.71154e27 0.311588
\(874\) 2.11774e28 2.40721
\(875\) −9.23841e26 −0.103877
\(876\) 3.56777e27 0.396833
\(877\) 2.30842e27 0.253992 0.126996 0.991903i \(-0.459467\pi\)
0.126996 + 0.991903i \(0.459467\pi\)
\(878\) 1.72696e28 1.87968
\(879\) −4.39001e27 −0.472684
\(880\) −4.86813e27 −0.518533
\(881\) −2.21402e26 −0.0233298 −0.0116649 0.999932i \(-0.503713\pi\)
−0.0116649 + 0.999932i \(0.503713\pi\)
\(882\) −5.68987e27 −0.593131
\(883\) −6.85195e27 −0.706622 −0.353311 0.935506i \(-0.614944\pi\)
−0.353311 + 0.935506i \(0.614944\pi\)
\(884\) −3.68709e28 −3.76173
\(885\) −5.79607e26 −0.0585023
\(886\) 1.12901e28 1.12740
\(887\) −1.86889e28 −1.84633 −0.923164 0.384407i \(-0.874406\pi\)
−0.923164 + 0.384407i \(0.874406\pi\)
\(888\) 8.55441e27 0.836116
\(889\) −3.17220e27 −0.306756
\(890\) 7.39274e27 0.707294
\(891\) 3.07924e27 0.291477
\(892\) 3.64651e28 3.41516
\(893\) −6.79253e27 −0.629423
\(894\) 9.81291e27 0.899687
\(895\) −6.13461e27 −0.556504
\(896\) 1.29002e28 1.15790
\(897\) 4.11882e27 0.365800
\(898\) 6.69525e26 0.0588358
\(899\) 1.40219e28 1.21924
\(900\) −5.32503e27 −0.458161
\(901\) 1.38769e28 1.18143
\(902\) 4.38801e27 0.369663
\(903\) 5.63275e27 0.469555
\(904\) −1.83847e28 −1.51655
\(905\) 6.55129e27 0.534768
\(906\) −5.08952e27 −0.411111
\(907\) 2.16205e27 0.172821 0.0864107 0.996260i \(-0.472460\pi\)
0.0864107 + 0.996260i \(0.472460\pi\)
\(908\) 1.87541e28 1.48347
\(909\) 3.96260e27 0.310187
\(910\) 1.45475e28 1.12693
\(911\) 7.62832e27 0.584796 0.292398 0.956297i \(-0.405547\pi\)
0.292398 + 0.956297i \(0.405547\pi\)
\(912\) −1.45977e28 −1.10747
\(913\) −2.73669e27 −0.205472
\(914\) −4.02722e28 −2.99238
\(915\) −3.91753e26 −0.0288079
\(916\) −5.09328e28 −3.70673
\(917\) 4.90231e27 0.353096
\(918\) −1.99379e28 −1.42126
\(919\) 8.34991e27 0.589093 0.294547 0.955637i \(-0.404831\pi\)
0.294547 + 0.955637i \(0.404831\pi\)
\(920\) 1.89346e28 1.32212
\(921\) −9.93383e26 −0.0686516
\(922\) −3.26451e28 −2.23293
\(923\) 8.78191e27 0.594529
\(924\) 5.45392e27 0.365448
\(925\) 2.83910e27 0.188293
\(926\) 1.38239e28 0.907457
\(927\) 3.55501e27 0.230985
\(928\) −3.50507e28 −2.25420
\(929\) 1.90486e28 1.21259 0.606295 0.795240i \(-0.292655\pi\)
0.606295 + 0.795240i \(0.292655\pi\)
\(930\) 5.63145e27 0.354840
\(931\) −6.97536e27 −0.435055
\(932\) −5.84688e28 −3.60972
\(933\) −3.09848e27 −0.189354
\(934\) −3.20182e28 −1.93687
\(935\) −3.87196e27 −0.231857
\(936\) 5.07323e28 3.00721
\(937\) 1.61387e28 0.946986 0.473493 0.880797i \(-0.342993\pi\)
0.473493 + 0.880797i \(0.342993\pi\)
\(938\) 4.53077e28 2.63176
\(939\) −1.79628e27 −0.103288
\(940\) −1.00379e28 −0.571386
\(941\) 1.27180e28 0.716670 0.358335 0.933593i \(-0.383345\pi\)
0.358335 + 0.933593i \(0.383345\pi\)
\(942\) 1.09764e28 0.612315
\(943\) −9.07990e27 −0.501441
\(944\) 2.23218e28 1.22038
\(945\) 5.63921e27 0.305222
\(946\) 1.85164e28 0.992179
\(947\) 1.45271e28 0.770647 0.385323 0.922782i \(-0.374090\pi\)
0.385323 + 0.922782i \(0.374090\pi\)
\(948\) 1.25501e28 0.659125
\(949\) −1.12810e28 −0.586570
\(950\) −9.10656e27 −0.468792
\(951\) 1.01474e26 0.00517177
\(952\) 8.52651e28 4.30250
\(953\) −2.74411e28 −1.37094 −0.685471 0.728100i \(-0.740403\pi\)
−0.685471 + 0.728100i \(0.740403\pi\)
\(954\) −3.15591e28 −1.56104
\(955\) −1.35409e28 −0.663153
\(956\) −7.55237e28 −3.66212
\(957\) −2.30636e27 −0.110730
\(958\) 4.76975e27 0.226738
\(959\) −2.26169e28 −1.06453
\(960\) −5.55641e27 −0.258951
\(961\) 1.89149e28 0.872834
\(962\) −4.47067e28 −2.04272
\(963\) 9.40917e27 0.425698
\(964\) 1.10330e29 4.94269
\(965\) −1.63674e28 −0.726058
\(966\) −1.57431e28 −0.691524
\(967\) −4.38569e28 −1.90760 −0.953798 0.300448i \(-0.902864\pi\)
−0.953798 + 0.300448i \(0.902864\pi\)
\(968\) −5.59885e28 −2.41148
\(969\) −1.16105e28 −0.495196
\(970\) 6.85269e27 0.289421
\(971\) 3.19230e28 1.33512 0.667561 0.744555i \(-0.267338\pi\)
0.667561 + 0.744555i \(0.267338\pi\)
\(972\) 4.95687e28 2.05295
\(973\) 1.18716e27 0.0486899
\(974\) 7.39883e28 3.00506
\(975\) −1.77114e27 −0.0712377
\(976\) 1.50872e28 0.600945
\(977\) 2.21216e28 0.872605 0.436303 0.899800i \(-0.356288\pi\)
0.436303 + 0.899800i \(0.356288\pi\)
\(978\) −5.55242e27 −0.216902
\(979\) −8.76350e27 −0.339033
\(980\) −1.03081e28 −0.394940
\(981\) −4.49119e28 −1.70414
\(982\) −7.84121e27 −0.294662
\(983\) −3.34360e27 −0.124439 −0.0622195 0.998062i \(-0.519818\pi\)
−0.0622195 + 0.998062i \(0.519818\pi\)
\(984\) 1.17645e28 0.433629
\(985\) −4.70997e27 −0.171939
\(986\) −5.95964e28 −2.15471
\(987\) 5.04950e27 0.180816
\(988\) 1.02797e29 3.64576
\(989\) −3.83150e28 −1.34587
\(990\) 8.80566e27 0.306357
\(991\) 1.69711e28 0.584803 0.292401 0.956296i \(-0.405546\pi\)
0.292401 + 0.956296i \(0.405546\pi\)
\(992\) −1.01452e29 −3.46259
\(993\) 5.95596e27 0.201342
\(994\) −3.35666e28 −1.12392
\(995\) −3.63132e26 −0.0120433
\(996\) −1.21272e28 −0.398379
\(997\) 4.73263e28 1.53992 0.769960 0.638092i \(-0.220276\pi\)
0.769960 + 0.638092i \(0.220276\pi\)
\(998\) 1.04421e29 3.36548
\(999\) −1.73301e28 −0.553259
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 5.20.a.b.1.4 4
3.2 odd 2 45.20.a.f.1.1 4
4.3 odd 2 80.20.a.g.1.2 4
5.2 odd 4 25.20.b.c.24.7 8
5.3 odd 4 25.20.b.c.24.2 8
5.4 even 2 25.20.a.c.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.20.a.b.1.4 4 1.1 even 1 trivial
25.20.a.c.1.1 4 5.4 even 2
25.20.b.c.24.2 8 5.3 odd 4
25.20.b.c.24.7 8 5.2 odd 4
45.20.a.f.1.1 4 3.2 odd 2
80.20.a.g.1.2 4 4.3 odd 2