Properties

Label 10.24.a.c.1.2
Level $10$
Weight $24$
Character 10.1
Self dual yes
Analytic conductor $33.520$
Analytic rank $1$
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] = [10,24,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 24, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 24);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 10.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: \(33.5204037345\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1492261}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 373065 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5}\cdot 3\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-610.291\) of defining polynomial
Character \(\chi\) \(=\) 10.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2048.00 q^{2} +247238. q^{3} +4.19430e6 q^{4} -4.88281e7 q^{5} +5.06342e8 q^{6} -3.52608e9 q^{7} +8.58993e9 q^{8} -3.30168e10 q^{9} -1.00000e11 q^{10} -6.19384e11 q^{11} +1.03699e12 q^{12} -9.21814e12 q^{13} -7.22141e12 q^{14} -1.20721e13 q^{15} +1.75922e13 q^{16} +2.96454e13 q^{17} -6.76184e13 q^{18} -3.56432e14 q^{19} -2.04800e14 q^{20} -8.71779e14 q^{21} -1.26850e15 q^{22} +2.73214e15 q^{23} +2.12375e15 q^{24} +2.38419e15 q^{25} -1.88788e16 q^{26} -3.14387e16 q^{27} -1.47894e16 q^{28} +5.96584e16 q^{29} -2.47238e16 q^{30} +5.61319e16 q^{31} +3.60288e16 q^{32} -1.53135e17 q^{33} +6.07138e16 q^{34} +1.72172e17 q^{35} -1.38482e17 q^{36} -1.59225e18 q^{37} -7.29973e17 q^{38} -2.27907e18 q^{39} -4.19430e17 q^{40} -4.81098e18 q^{41} -1.78540e18 q^{42} -7.53412e18 q^{43} -2.59788e18 q^{44} +1.61215e18 q^{45} +5.59542e18 q^{46} +1.70653e18 q^{47} +4.34945e18 q^{48} -1.49355e19 q^{49} +4.88281e18 q^{50} +7.32946e18 q^{51} -3.86637e19 q^{52} +7.71411e19 q^{53} -6.43865e19 q^{54} +3.02434e19 q^{55} -3.02888e19 q^{56} -8.81234e19 q^{57} +1.22180e20 q^{58} +2.22753e20 q^{59} -5.06342e19 q^{60} +2.72118e20 q^{61} +1.14958e20 q^{62} +1.16420e20 q^{63} +7.37870e19 q^{64} +4.50105e20 q^{65} -3.13620e20 q^{66} +6.39125e20 q^{67} +1.24342e20 q^{68} +6.75487e20 q^{69} +3.52608e20 q^{70} -2.68803e21 q^{71} -2.83612e20 q^{72} +4.36224e21 q^{73} -3.26093e21 q^{74} +5.89460e20 q^{75} -1.49498e21 q^{76} +2.18400e21 q^{77} -4.66754e21 q^{78} -3.18360e21 q^{79} -8.58993e20 q^{80} -4.66452e21 q^{81} -9.85289e21 q^{82} -2.75933e21 q^{83} -3.65651e21 q^{84} -1.44753e21 q^{85} -1.54299e22 q^{86} +1.47498e22 q^{87} -5.32047e21 q^{88} -1.40148e22 q^{89} +3.30168e21 q^{90} +3.25039e22 q^{91} +1.14594e22 q^{92} +1.38779e22 q^{93} +3.49498e21 q^{94} +1.74039e22 q^{95} +8.90767e21 q^{96} +1.12387e23 q^{97} -3.05880e22 q^{98} +2.04501e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4096 q^{2} - 91884 q^{3} + 8388608 q^{4} - 97656250 q^{5} - 188178432 q^{6} + 2146058908 q^{7} + 17179869184 q^{8} - 12156555726 q^{9} - 200000000000 q^{10} - 692289571776 q^{11} - 385389428736 q^{12}+ \cdots + 18\!\cdots\!88 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2048.00 0.707107
\(3\) 247238. 0.805786 0.402893 0.915247i \(-0.368005\pi\)
0.402893 + 0.915247i \(0.368005\pi\)
\(4\) 4.19430e6 0.500000
\(5\) −4.88281e7 −0.447214
\(6\) 5.06342e8 0.569777
\(7\) −3.52608e9 −0.674007 −0.337004 0.941503i \(-0.609413\pi\)
−0.337004 + 0.941503i \(0.609413\pi\)
\(8\) 8.58993e9 0.353553
\(9\) −3.30168e10 −0.350708
\(10\) −1.00000e11 −0.316228
\(11\) −6.19384e11 −0.654552 −0.327276 0.944929i \(-0.606131\pi\)
−0.327276 + 0.944929i \(0.606131\pi\)
\(12\) 1.03699e12 0.402893
\(13\) −9.21814e12 −1.42658 −0.713288 0.700871i \(-0.752795\pi\)
−0.713288 + 0.700871i \(0.752795\pi\)
\(14\) −7.22141e12 −0.476595
\(15\) −1.20721e13 −0.360359
\(16\) 1.75922e13 0.250000
\(17\) 2.96454e13 0.209795 0.104897 0.994483i \(-0.466549\pi\)
0.104897 + 0.994483i \(0.466549\pi\)
\(18\) −6.76184e13 −0.247988
\(19\) −3.56432e14 −0.701957 −0.350978 0.936384i \(-0.614151\pi\)
−0.350978 + 0.936384i \(0.614151\pi\)
\(20\) −2.04800e14 −0.223607
\(21\) −8.71779e14 −0.543106
\(22\) −1.26850e15 −0.462838
\(23\) 2.73214e15 0.597905 0.298953 0.954268i \(-0.403363\pi\)
0.298953 + 0.954268i \(0.403363\pi\)
\(24\) 2.12375e15 0.284889
\(25\) 2.38419e15 0.200000
\(26\) −1.88788e16 −1.00874
\(27\) −3.14387e16 −1.08838
\(28\) −1.47894e16 −0.337004
\(29\) 5.96584e16 0.908018 0.454009 0.890997i \(-0.349993\pi\)
0.454009 + 0.890997i \(0.349993\pi\)
\(30\) −2.47238e16 −0.254812
\(31\) 5.61319e16 0.396780 0.198390 0.980123i \(-0.436429\pi\)
0.198390 + 0.980123i \(0.436429\pi\)
\(32\) 3.60288e16 0.176777
\(33\) −1.53135e17 −0.527429
\(34\) 6.07138e16 0.148347
\(35\) 1.72172e17 0.301425
\(36\) −1.38482e17 −0.175354
\(37\) −1.59225e18 −1.47127 −0.735634 0.677380i \(-0.763115\pi\)
−0.735634 + 0.677380i \(0.763115\pi\)
\(38\) −7.29973e17 −0.496359
\(39\) −2.27907e18 −1.14952
\(40\) −4.19430e17 −0.158114
\(41\) −4.81098e18 −1.36527 −0.682636 0.730759i \(-0.739167\pi\)
−0.682636 + 0.730759i \(0.739167\pi\)
\(42\) −1.78540e18 −0.384034
\(43\) −7.53412e18 −1.23636 −0.618180 0.786036i \(-0.712130\pi\)
−0.618180 + 0.786036i \(0.712130\pi\)
\(44\) −2.59788e18 −0.327276
\(45\) 1.61215e18 0.156841
\(46\) 5.59542e18 0.422783
\(47\) 1.70653e18 0.100691 0.0503454 0.998732i \(-0.483968\pi\)
0.0503454 + 0.998732i \(0.483968\pi\)
\(48\) 4.34945e18 0.201447
\(49\) −1.49355e19 −0.545715
\(50\) 4.88281e18 0.141421
\(51\) 7.32946e18 0.169050
\(52\) −3.86637e19 −0.713288
\(53\) 7.71411e19 1.14318 0.571589 0.820540i \(-0.306327\pi\)
0.571589 + 0.820540i \(0.306327\pi\)
\(54\) −6.43865e19 −0.769603
\(55\) 3.02434e19 0.292725
\(56\) −3.02888e19 −0.238297
\(57\) −8.81234e19 −0.565627
\(58\) 1.22180e20 0.642066
\(59\) 2.22753e20 0.961668 0.480834 0.876812i \(-0.340334\pi\)
0.480834 + 0.876812i \(0.340334\pi\)
\(60\) −5.06342e19 −0.180179
\(61\) 2.72118e20 0.800690 0.400345 0.916364i \(-0.368890\pi\)
0.400345 + 0.916364i \(0.368890\pi\)
\(62\) 1.14958e20 0.280566
\(63\) 1.16420e20 0.236380
\(64\) 7.37870e19 0.125000
\(65\) 4.50105e20 0.637984
\(66\) −3.13620e20 −0.372949
\(67\) 6.39125e20 0.639331 0.319665 0.947531i \(-0.396430\pi\)
0.319665 + 0.947531i \(0.396430\pi\)
\(68\) 1.24342e20 0.104897
\(69\) 6.75487e20 0.481784
\(70\) 3.52608e20 0.213140
\(71\) −2.68803e21 −1.38027 −0.690134 0.723681i \(-0.742448\pi\)
−0.690134 + 0.723681i \(0.742448\pi\)
\(72\) −2.83612e20 −0.123994
\(73\) 4.36224e21 1.62741 0.813704 0.581280i \(-0.197448\pi\)
0.813704 + 0.581280i \(0.197448\pi\)
\(74\) −3.26093e21 −1.04034
\(75\) 5.89460e20 0.161157
\(76\) −1.49498e21 −0.350978
\(77\) 2.18400e21 0.441173
\(78\) −4.66754e21 −0.812830
\(79\) −3.18360e21 −0.478859 −0.239429 0.970914i \(-0.576960\pi\)
−0.239429 + 0.970914i \(0.576960\pi\)
\(80\) −8.58993e20 −0.111803
\(81\) −4.66452e21 −0.526296
\(82\) −9.85289e21 −0.965393
\(83\) −2.75933e21 −0.235183 −0.117592 0.993062i \(-0.537517\pi\)
−0.117592 + 0.993062i \(0.537517\pi\)
\(84\) −3.65651e21 −0.271553
\(85\) −1.44753e21 −0.0938231
\(86\) −1.54299e22 −0.874239
\(87\) 1.47498e22 0.731668
\(88\) −5.32047e21 −0.231419
\(89\) −1.40148e22 −0.535304 −0.267652 0.963516i \(-0.586248\pi\)
−0.267652 + 0.963516i \(0.586248\pi\)
\(90\) 3.30168e21 0.110904
\(91\) 3.25039e22 0.961522
\(92\) 1.14594e22 0.298953
\(93\) 1.38779e22 0.319720
\(94\) 3.49498e21 0.0711991
\(95\) 1.74039e22 0.313925
\(96\) 8.90767e21 0.142444
\(97\) 1.12387e23 1.59529 0.797644 0.603128i \(-0.206079\pi\)
0.797644 + 0.603128i \(0.206079\pi\)
\(98\) −3.05880e22 −0.385878
\(99\) 2.04501e22 0.229557
\(100\) 1.00000e22 0.100000
\(101\) 1.15829e23 1.03305 0.516525 0.856272i \(-0.327225\pi\)
0.516525 + 0.856272i \(0.327225\pi\)
\(102\) 1.50107e22 0.119536
\(103\) −1.69254e23 −1.20479 −0.602394 0.798199i \(-0.705786\pi\)
−0.602394 + 0.798199i \(0.705786\pi\)
\(104\) −7.91832e22 −0.504371
\(105\) 4.25673e22 0.242884
\(106\) 1.57985e23 0.808348
\(107\) 1.60256e23 0.736041 0.368020 0.929818i \(-0.380036\pi\)
0.368020 + 0.929818i \(0.380036\pi\)
\(108\) −1.31864e23 −0.544191
\(109\) −1.24517e23 −0.462193 −0.231096 0.972931i \(-0.574231\pi\)
−0.231096 + 0.972931i \(0.574231\pi\)
\(110\) 6.19384e22 0.206987
\(111\) −3.93664e23 −1.18553
\(112\) −6.20314e22 −0.168502
\(113\) −8.30660e22 −0.203714 −0.101857 0.994799i \(-0.532478\pi\)
−0.101857 + 0.994799i \(0.532478\pi\)
\(114\) −1.80477e23 −0.399959
\(115\) −1.33405e23 −0.267391
\(116\) 2.50225e23 0.454009
\(117\) 3.04353e23 0.500312
\(118\) 4.56198e23 0.680002
\(119\) −1.04532e23 −0.141403
\(120\) −1.03699e23 −0.127406
\(121\) −5.11794e23 −0.571562
\(122\) 5.57298e23 0.566174
\(123\) −1.18945e24 −1.10012
\(124\) 2.35434e23 0.198390
\(125\) −1.16415e23 −0.0894427
\(126\) 2.38428e23 0.167146
\(127\) −1.35449e24 −0.867031 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(128\) 1.51116e23 0.0883883
\(129\) −1.86272e24 −0.996243
\(130\) 9.21814e23 0.451123
\(131\) −3.88681e24 −1.74170 −0.870850 0.491548i \(-0.836431\pi\)
−0.870850 + 0.491548i \(0.836431\pi\)
\(132\) −6.42295e23 −0.263715
\(133\) 1.25681e24 0.473124
\(134\) 1.30893e24 0.452075
\(135\) 1.53509e24 0.486739
\(136\) 2.54652e23 0.0741737
\(137\) 4.74583e24 1.27065 0.635324 0.772246i \(-0.280867\pi\)
0.635324 + 0.772246i \(0.280867\pi\)
\(138\) 1.38340e24 0.340673
\(139\) 8.52208e24 1.93141 0.965707 0.259633i \(-0.0836015\pi\)
0.965707 + 0.259633i \(0.0836015\pi\)
\(140\) 7.22141e23 0.150713
\(141\) 4.21919e23 0.0811352
\(142\) −5.50509e24 −0.975997
\(143\) 5.70957e24 0.933768
\(144\) −5.80837e23 −0.0876771
\(145\) −2.91301e24 −0.406078
\(146\) 8.93386e24 1.15075
\(147\) −3.69262e24 −0.439729
\(148\) −6.67838e24 −0.735634
\(149\) 4.97391e24 0.507056 0.253528 0.967328i \(-0.418409\pi\)
0.253528 + 0.967328i \(0.418409\pi\)
\(150\) 1.20721e24 0.113955
\(151\) 2.56571e24 0.224374 0.112187 0.993687i \(-0.464214\pi\)
0.112187 + 0.993687i \(0.464214\pi\)
\(152\) −3.06173e24 −0.248179
\(153\) −9.78796e23 −0.0735768
\(154\) 4.47282e24 0.311956
\(155\) −2.74082e24 −0.177446
\(156\) −9.55911e24 −0.574758
\(157\) −2.22180e25 −1.24125 −0.620623 0.784109i \(-0.713120\pi\)
−0.620623 + 0.784109i \(0.713120\pi\)
\(158\) −6.52002e24 −0.338604
\(159\) 1.90722e25 0.921157
\(160\) −1.75922e24 −0.0790569
\(161\) −9.63373e24 −0.402992
\(162\) −9.55295e24 −0.372147
\(163\) −2.96191e25 −1.07501 −0.537507 0.843259i \(-0.680634\pi\)
−0.537507 + 0.843259i \(0.680634\pi\)
\(164\) −2.01787e25 −0.682636
\(165\) 7.47729e24 0.235873
\(166\) −5.65112e24 −0.166300
\(167\) 1.84259e25 0.506044 0.253022 0.967460i \(-0.418575\pi\)
0.253022 + 0.967460i \(0.418575\pi\)
\(168\) −7.48852e24 −0.192017
\(169\) 4.32202e25 1.03512
\(170\) −2.96454e24 −0.0663430
\(171\) 1.17682e25 0.246182
\(172\) −3.16004e25 −0.618180
\(173\) −9.26470e25 −1.69551 −0.847757 0.530385i \(-0.822047\pi\)
−0.847757 + 0.530385i \(0.822047\pi\)
\(174\) 3.02076e25 0.517368
\(175\) −8.40682e24 −0.134801
\(176\) −1.08963e25 −0.163638
\(177\) 5.50728e25 0.774899
\(178\) −2.87022e25 −0.378517
\(179\) 1.33235e26 1.64744 0.823719 0.566998i \(-0.191895\pi\)
0.823719 + 0.566998i \(0.191895\pi\)
\(180\) 6.76184e24 0.0784207
\(181\) −3.72764e25 −0.405630 −0.202815 0.979217i \(-0.565009\pi\)
−0.202815 + 0.979217i \(0.565009\pi\)
\(182\) 6.65680e25 0.679899
\(183\) 6.72779e25 0.645185
\(184\) 2.34689e25 0.211391
\(185\) 7.77466e25 0.657971
\(186\) 2.84220e25 0.226076
\(187\) −1.83619e25 −0.137322
\(188\) 7.15772e24 0.0503454
\(189\) 1.10855e26 0.733577
\(190\) 3.56432e25 0.221978
\(191\) −1.74153e26 −1.02105 −0.510525 0.859863i \(-0.670549\pi\)
−0.510525 + 0.859863i \(0.670549\pi\)
\(192\) 1.82429e25 0.100723
\(193\) −2.40877e26 −1.25281 −0.626407 0.779496i \(-0.715475\pi\)
−0.626407 + 0.779496i \(0.715475\pi\)
\(194\) 2.30168e26 1.12804
\(195\) 1.11283e26 0.514079
\(196\) −6.26441e25 −0.272857
\(197\) −2.32299e26 −0.954303 −0.477152 0.878821i \(-0.658331\pi\)
−0.477152 + 0.878821i \(0.658331\pi\)
\(198\) 4.18817e25 0.162321
\(199\) −3.32534e26 −1.21626 −0.608129 0.793838i \(-0.708080\pi\)
−0.608129 + 0.793838i \(0.708080\pi\)
\(200\) 2.04800e25 0.0707107
\(201\) 1.58016e26 0.515164
\(202\) 2.37218e26 0.730476
\(203\) −2.10360e26 −0.612010
\(204\) 3.07420e25 0.0845249
\(205\) 2.34911e26 0.610568
\(206\) −3.46632e26 −0.851913
\(207\) −9.02064e25 −0.209690
\(208\) −1.62167e26 −0.356644
\(209\) 2.20768e26 0.459467
\(210\) 8.71779e25 0.171745
\(211\) −4.88737e26 −0.911647 −0.455823 0.890070i \(-0.650655\pi\)
−0.455823 + 0.890070i \(0.650655\pi\)
\(212\) 3.23553e26 0.571589
\(213\) −6.64583e26 −1.11220
\(214\) 3.28205e26 0.520459
\(215\) 3.67877e26 0.552917
\(216\) −2.70057e26 −0.384801
\(217\) −1.97925e26 −0.267433
\(218\) −2.55010e26 −0.326820
\(219\) 1.07851e27 1.31134
\(220\) 1.26850e26 0.146362
\(221\) −2.73276e26 −0.299288
\(222\) −8.06224e26 −0.838294
\(223\) −1.80253e27 −1.77982 −0.889909 0.456138i \(-0.849232\pi\)
−0.889909 + 0.456138i \(0.849232\pi\)
\(224\) −1.27040e26 −0.119149
\(225\) −7.87182e25 −0.0701416
\(226\) −1.70119e26 −0.144048
\(227\) 7.37727e26 0.593743 0.296871 0.954917i \(-0.404057\pi\)
0.296871 + 0.954917i \(0.404057\pi\)
\(228\) −3.69616e26 −0.282814
\(229\) 1.94352e27 1.41410 0.707052 0.707162i \(-0.250025\pi\)
0.707052 + 0.707162i \(0.250025\pi\)
\(230\) −2.73214e26 −0.189074
\(231\) 5.39966e26 0.355491
\(232\) 5.12461e26 0.321033
\(233\) −2.43138e26 −0.144964 −0.0724820 0.997370i \(-0.523092\pi\)
−0.0724820 + 0.997370i \(0.523092\pi\)
\(234\) 6.23316e26 0.353774
\(235\) −8.33269e25 −0.0450303
\(236\) 9.34293e26 0.480834
\(237\) −7.87106e26 −0.385858
\(238\) −2.14082e26 −0.0999872
\(239\) 3.60889e27 1.60619 0.803097 0.595848i \(-0.203184\pi\)
0.803097 + 0.595848i \(0.203184\pi\)
\(240\) −2.12375e26 −0.0900897
\(241\) 2.00066e27 0.809056 0.404528 0.914526i \(-0.367436\pi\)
0.404528 + 0.914526i \(0.367436\pi\)
\(242\) −1.04815e27 −0.404155
\(243\) 1.80649e27 0.664301
\(244\) 1.14135e27 0.400345
\(245\) 7.29274e26 0.244051
\(246\) −2.43600e27 −0.777901
\(247\) 3.28564e27 1.00139
\(248\) 4.82169e26 0.140283
\(249\) −6.82211e26 −0.189507
\(250\) −2.38419e26 −0.0632456
\(251\) −1.85132e27 −0.469065 −0.234533 0.972108i \(-0.575356\pi\)
−0.234533 + 0.972108i \(0.575356\pi\)
\(252\) 4.88300e26 0.118190
\(253\) −1.69224e27 −0.391360
\(254\) −2.77400e27 −0.613083
\(255\) −3.57884e26 −0.0756014
\(256\) 3.09485e26 0.0625000
\(257\) −7.13742e27 −1.37819 −0.689097 0.724669i \(-0.741992\pi\)
−0.689097 + 0.724669i \(0.741992\pi\)
\(258\) −3.81484e27 −0.704450
\(259\) 5.61440e27 0.991644
\(260\) 1.88788e27 0.318992
\(261\) −1.96973e27 −0.318449
\(262\) −7.96019e27 −1.23157
\(263\) −5.02930e27 −0.744761 −0.372380 0.928080i \(-0.621458\pi\)
−0.372380 + 0.928080i \(0.621458\pi\)
\(264\) −1.31542e27 −0.186474
\(265\) −3.76666e27 −0.511244
\(266\) 2.57394e27 0.334549
\(267\) −3.46497e27 −0.431341
\(268\) 2.68068e27 0.319665
\(269\) −1.17031e28 −1.33705 −0.668527 0.743688i \(-0.733075\pi\)
−0.668527 + 0.743688i \(0.733075\pi\)
\(270\) 3.14387e27 0.344177
\(271\) −1.49314e27 −0.156659 −0.0783294 0.996928i \(-0.524959\pi\)
−0.0783294 + 0.996928i \(0.524959\pi\)
\(272\) 5.21528e26 0.0524487
\(273\) 8.03618e27 0.774781
\(274\) 9.71945e27 0.898484
\(275\) −1.47673e27 −0.130910
\(276\) 2.83320e27 0.240892
\(277\) 3.00946e27 0.245454 0.122727 0.992440i \(-0.460836\pi\)
0.122727 + 0.992440i \(0.460836\pi\)
\(278\) 1.74532e28 1.36572
\(279\) −1.85329e27 −0.139154
\(280\) 1.47894e27 0.106570
\(281\) −1.97459e28 −1.36569 −0.682847 0.730562i \(-0.739258\pi\)
−0.682847 + 0.730562i \(0.739258\pi\)
\(282\) 8.64091e26 0.0573713
\(283\) 2.06841e28 1.31854 0.659270 0.751906i \(-0.270866\pi\)
0.659270 + 0.751906i \(0.270866\pi\)
\(284\) −1.12744e28 −0.690134
\(285\) 4.30290e27 0.252956
\(286\) 1.16932e28 0.660274
\(287\) 1.69639e28 0.920203
\(288\) −1.18956e27 −0.0619970
\(289\) −1.90887e28 −0.955986
\(290\) −5.96584e27 −0.287140
\(291\) 2.77862e28 1.28546
\(292\) 1.82965e28 0.813704
\(293\) −2.59950e28 −1.11151 −0.555754 0.831347i \(-0.687570\pi\)
−0.555754 + 0.831347i \(0.687570\pi\)
\(294\) −7.56249e27 −0.310936
\(295\) −1.08766e28 −0.430071
\(296\) −1.36773e28 −0.520171
\(297\) 1.94726e28 0.712403
\(298\) 1.01866e28 0.358543
\(299\) −2.51852e28 −0.852957
\(300\) 2.47238e27 0.0805786
\(301\) 2.65659e28 0.833316
\(302\) 5.25458e27 0.158657
\(303\) 2.86372e28 0.832417
\(304\) −6.27042e27 −0.175489
\(305\) −1.32870e28 −0.358080
\(306\) −2.00457e27 −0.0520266
\(307\) 5.17015e28 1.29245 0.646223 0.763149i \(-0.276348\pi\)
0.646223 + 0.763149i \(0.276348\pi\)
\(308\) 9.16034e27 0.220586
\(309\) −4.18459e28 −0.970801
\(310\) −5.61319e27 −0.125473
\(311\) 2.19009e28 0.471756 0.235878 0.971783i \(-0.424203\pi\)
0.235878 + 0.971783i \(0.424203\pi\)
\(312\) −1.95771e28 −0.406415
\(313\) −3.44520e28 −0.689373 −0.344686 0.938718i \(-0.612015\pi\)
−0.344686 + 0.938718i \(0.612015\pi\)
\(314\) −4.55024e28 −0.877694
\(315\) −5.68456e27 −0.105712
\(316\) −1.33530e28 −0.239429
\(317\) −4.25122e28 −0.735075 −0.367538 0.930009i \(-0.619799\pi\)
−0.367538 + 0.930009i \(0.619799\pi\)
\(318\) 3.90598e28 0.651356
\(319\) −3.69514e28 −0.594345
\(320\) −3.60288e27 −0.0559017
\(321\) 3.96214e28 0.593092
\(322\) −1.97299e28 −0.284959
\(323\) −1.05666e28 −0.147267
\(324\) −1.95644e28 −0.263148
\(325\) −2.19778e28 −0.285315
\(326\) −6.06599e28 −0.760150
\(327\) −3.07852e28 −0.372429
\(328\) −4.13260e28 −0.482697
\(329\) −6.01737e27 −0.0678663
\(330\) 1.53135e28 0.166788
\(331\) 1.43295e29 1.50733 0.753667 0.657256i \(-0.228283\pi\)
0.753667 + 0.657256i \(0.228283\pi\)
\(332\) −1.15735e28 −0.117592
\(333\) 5.25710e28 0.515985
\(334\) 3.77362e28 0.357827
\(335\) −3.12073e28 −0.285917
\(336\) −1.53365e28 −0.135776
\(337\) 1.66695e28 0.142619 0.0713095 0.997454i \(-0.477282\pi\)
0.0713095 + 0.997454i \(0.477282\pi\)
\(338\) 8.85150e28 0.731939
\(339\) −2.05370e28 −0.164150
\(340\) −6.07138e27 −0.0469116
\(341\) −3.47672e28 −0.259713
\(342\) 2.41014e28 0.174077
\(343\) 1.49168e29 1.04182
\(344\) −6.47176e28 −0.437120
\(345\) −3.29827e28 −0.215460
\(346\) −1.89741e29 −1.19891
\(347\) −7.87942e28 −0.481621 −0.240811 0.970572i \(-0.577413\pi\)
−0.240811 + 0.970572i \(0.577413\pi\)
\(348\) 6.18651e28 0.365834
\(349\) −1.45984e29 −0.835244 −0.417622 0.908621i \(-0.637136\pi\)
−0.417622 + 0.908621i \(0.637136\pi\)
\(350\) −1.72172e28 −0.0953190
\(351\) 2.89807e29 1.55266
\(352\) −2.23157e28 −0.115710
\(353\) 5.29769e28 0.265875 0.132937 0.991124i \(-0.457559\pi\)
0.132937 + 0.991124i \(0.457559\pi\)
\(354\) 1.12789e29 0.547936
\(355\) 1.31252e29 0.617275
\(356\) −5.87822e28 −0.267652
\(357\) −2.58442e28 −0.113941
\(358\) 2.72866e29 1.16492
\(359\) 2.82466e29 1.16783 0.583915 0.811815i \(-0.301520\pi\)
0.583915 + 0.811815i \(0.301520\pi\)
\(360\) 1.38482e28 0.0554518
\(361\) −1.30786e29 −0.507256
\(362\) −7.63420e28 −0.286824
\(363\) −1.26535e29 −0.460557
\(364\) 1.36331e29 0.480761
\(365\) −2.13000e29 −0.727799
\(366\) 1.37785e29 0.456215
\(367\) 2.59135e29 0.831509 0.415755 0.909477i \(-0.363517\pi\)
0.415755 + 0.909477i \(0.363517\pi\)
\(368\) 4.80643e28 0.149476
\(369\) 1.58843e29 0.478812
\(370\) 1.59225e29 0.465255
\(371\) −2.72006e29 −0.770509
\(372\) 5.82082e28 0.159860
\(373\) −6.81722e29 −1.81533 −0.907665 0.419695i \(-0.862137\pi\)
−0.907665 + 0.419695i \(0.862137\pi\)
\(374\) −3.76052e28 −0.0971011
\(375\) −2.87822e28 −0.0720717
\(376\) 1.46590e28 0.0355996
\(377\) −5.49939e29 −1.29536
\(378\) 2.27032e29 0.518718
\(379\) 8.03139e29 1.78008 0.890041 0.455881i \(-0.150676\pi\)
0.890041 + 0.455881i \(0.150676\pi\)
\(380\) 7.29973e28 0.156962
\(381\) −3.34882e29 −0.698642
\(382\) −3.56665e29 −0.721992
\(383\) −9.93629e29 −1.95181 −0.975907 0.218189i \(-0.929985\pi\)
−0.975907 + 0.218189i \(0.929985\pi\)
\(384\) 3.73615e28 0.0712221
\(385\) −1.06640e29 −0.197298
\(386\) −4.93317e29 −0.885874
\(387\) 2.48752e29 0.433602
\(388\) 4.71383e29 0.797644
\(389\) −8.33130e29 −1.36865 −0.684325 0.729177i \(-0.739903\pi\)
−0.684325 + 0.729177i \(0.739903\pi\)
\(390\) 2.27907e29 0.363509
\(391\) 8.09953e28 0.125437
\(392\) −1.28295e29 −0.192939
\(393\) −9.60966e29 −1.40344
\(394\) −4.75749e29 −0.674794
\(395\) 1.55449e29 0.214152
\(396\) 8.57738e28 0.114778
\(397\) 6.39815e29 0.831694 0.415847 0.909435i \(-0.363485\pi\)
0.415847 + 0.909435i \(0.363485\pi\)
\(398\) −6.81030e29 −0.860025
\(399\) 3.10730e29 0.381237
\(400\) 4.19430e28 0.0500000
\(401\) 1.37492e30 1.59264 0.796318 0.604879i \(-0.206778\pi\)
0.796318 + 0.604879i \(0.206778\pi\)
\(402\) 3.23616e29 0.364276
\(403\) −5.17432e29 −0.566037
\(404\) 4.85821e29 0.516525
\(405\) 2.27760e29 0.235367
\(406\) −4.30817e29 −0.432757
\(407\) 9.86214e29 0.963021
\(408\) 6.29596e28 0.0597682
\(409\) 2.04005e30 1.88288 0.941440 0.337181i \(-0.109473\pi\)
0.941440 + 0.337181i \(0.109473\pi\)
\(410\) 4.81098e29 0.431737
\(411\) 1.17335e30 1.02387
\(412\) −7.09902e29 −0.602394
\(413\) −7.85444e29 −0.648171
\(414\) −1.84743e29 −0.148273
\(415\) 1.34733e29 0.105177
\(416\) −3.32119e29 −0.252185
\(417\) 2.10698e30 1.55631
\(418\) 4.52134e29 0.324892
\(419\) −5.19038e29 −0.362859 −0.181430 0.983404i \(-0.558072\pi\)
−0.181430 + 0.983404i \(0.558072\pi\)
\(420\) 1.78540e29 0.121442
\(421\) −1.09510e30 −0.724787 −0.362394 0.932025i \(-0.618040\pi\)
−0.362394 + 0.932025i \(0.618040\pi\)
\(422\) −1.00093e30 −0.644632
\(423\) −5.63443e28 −0.0353131
\(424\) 6.62637e29 0.404174
\(425\) 7.06802e28 0.0419590
\(426\) −1.36106e30 −0.786445
\(427\) −9.59510e29 −0.539671
\(428\) 6.72164e29 0.368020
\(429\) 1.41162e30 0.752418
\(430\) 7.53412e29 0.390972
\(431\) −1.78506e30 −0.901911 −0.450956 0.892546i \(-0.648917\pi\)
−0.450956 + 0.892546i \(0.648917\pi\)
\(432\) −5.53076e29 −0.272096
\(433\) 1.83380e27 0.000878501 0 0.000439250 1.00000i \(-0.499860\pi\)
0.000439250 1.00000i \(0.499860\pi\)
\(434\) −4.05351e29 −0.189104
\(435\) −7.20204e29 −0.327212
\(436\) −5.22261e29 −0.231096
\(437\) −9.73821e29 −0.419704
\(438\) 2.20879e30 0.927260
\(439\) 2.76531e30 1.13084 0.565420 0.824803i \(-0.308714\pi\)
0.565420 + 0.824803i \(0.308714\pi\)
\(440\) 2.59788e29 0.103494
\(441\) 4.93123e29 0.191387
\(442\) −5.59668e29 −0.211629
\(443\) −4.05895e30 −1.49545 −0.747724 0.664010i \(-0.768853\pi\)
−0.747724 + 0.664010i \(0.768853\pi\)
\(444\) −1.65115e30 −0.592764
\(445\) 6.84314e29 0.239395
\(446\) −3.69157e30 −1.25852
\(447\) 1.22974e30 0.408579
\(448\) −2.60179e29 −0.0842509
\(449\) −2.95251e30 −0.931876 −0.465938 0.884817i \(-0.654283\pi\)
−0.465938 + 0.884817i \(0.654283\pi\)
\(450\) −1.61215e29 −0.0495976
\(451\) 2.97984e30 0.893641
\(452\) −3.48404e29 −0.101857
\(453\) 6.34340e29 0.180798
\(454\) 1.51086e30 0.419840
\(455\) −1.58710e30 −0.430006
\(456\) −7.56974e29 −0.199979
\(457\) −7.87636e29 −0.202903 −0.101452 0.994840i \(-0.532349\pi\)
−0.101452 + 0.994840i \(0.532349\pi\)
\(458\) 3.98033e30 0.999923
\(459\) −9.32014e29 −0.228337
\(460\) −5.59542e29 −0.133696
\(461\) 5.04802e30 1.17641 0.588207 0.808710i \(-0.299834\pi\)
0.588207 + 0.808710i \(0.299834\pi\)
\(462\) 1.10585e30 0.251370
\(463\) −3.07277e30 −0.681316 −0.340658 0.940187i \(-0.610650\pi\)
−0.340658 + 0.940187i \(0.610650\pi\)
\(464\) 1.04952e30 0.227004
\(465\) −6.77632e29 −0.142983
\(466\) −4.97947e29 −0.102505
\(467\) −7.91191e30 −1.58905 −0.794524 0.607233i \(-0.792279\pi\)
−0.794524 + 0.607233i \(0.792279\pi\)
\(468\) 1.27655e30 0.250156
\(469\) −2.25360e30 −0.430913
\(470\) −1.70653e29 −0.0318412
\(471\) −5.49311e30 −1.00018
\(472\) 1.91343e30 0.340001
\(473\) 4.66651e30 0.809262
\(474\) −1.61199e30 −0.272843
\(475\) −8.49801e29 −0.140391
\(476\) −4.38439e29 −0.0707016
\(477\) −2.54695e30 −0.400922
\(478\) 7.39101e30 1.13575
\(479\) −1.14342e30 −0.171533 −0.0857664 0.996315i \(-0.527334\pi\)
−0.0857664 + 0.996315i \(0.527334\pi\)
\(480\) −4.34945e29 −0.0637030
\(481\) 1.46776e31 2.09887
\(482\) 4.09736e30 0.572089
\(483\) −2.38182e30 −0.324726
\(484\) −2.14662e30 −0.285781
\(485\) −5.48762e30 −0.713435
\(486\) 3.69970e30 0.469731
\(487\) 3.69536e30 0.458220 0.229110 0.973401i \(-0.426419\pi\)
0.229110 + 0.973401i \(0.426419\pi\)
\(488\) 2.33748e30 0.283087
\(489\) −7.32295e30 −0.866232
\(490\) 1.49355e30 0.172570
\(491\) 9.29094e30 1.04863 0.524315 0.851524i \(-0.324321\pi\)
0.524315 + 0.851524i \(0.324321\pi\)
\(492\) −4.98894e30 −0.550059
\(493\) 1.76860e30 0.190497
\(494\) 6.72900e30 0.708093
\(495\) −9.98539e29 −0.102661
\(496\) 9.87483e29 0.0991951
\(497\) 9.47821e30 0.930311
\(498\) −1.39717e30 −0.134002
\(499\) 3.27584e29 0.0307019 0.0153510 0.999882i \(-0.495113\pi\)
0.0153510 + 0.999882i \(0.495113\pi\)
\(500\) −4.88281e29 −0.0447214
\(501\) 4.55557e30 0.407764
\(502\) −3.79150e30 −0.331679
\(503\) 1.69287e30 0.144741 0.0723705 0.997378i \(-0.476944\pi\)
0.0723705 + 0.997378i \(0.476944\pi\)
\(504\) 1.00004e30 0.0835729
\(505\) −5.65571e30 −0.461994
\(506\) −3.46571e30 −0.276733
\(507\) 1.06857e31 0.834084
\(508\) −5.68116e30 −0.433515
\(509\) −7.45718e30 −0.556315 −0.278157 0.960536i \(-0.589724\pi\)
−0.278157 + 0.960536i \(0.589724\pi\)
\(510\) −7.32946e29 −0.0534583
\(511\) −1.53816e31 −1.09688
\(512\) 6.33825e29 0.0441942
\(513\) 1.12058e31 0.763998
\(514\) −1.46174e31 −0.974530
\(515\) 8.26435e30 0.538797
\(516\) −7.81280e30 −0.498121
\(517\) −1.05700e30 −0.0659073
\(518\) 1.14983e31 0.701198
\(519\) −2.29058e31 −1.36622
\(520\) 3.86637e30 0.225561
\(521\) 9.54796e30 0.544850 0.272425 0.962177i \(-0.412174\pi\)
0.272425 + 0.962177i \(0.412174\pi\)
\(522\) −4.03400e30 −0.225178
\(523\) −1.33111e31 −0.726851 −0.363425 0.931623i \(-0.618393\pi\)
−0.363425 + 0.931623i \(0.618393\pi\)
\(524\) −1.63025e31 −0.870850
\(525\) −2.07848e30 −0.108621
\(526\) −1.03000e31 −0.526625
\(527\) 1.66405e30 0.0832425
\(528\) −2.69398e30 −0.131857
\(529\) −1.34159e31 −0.642509
\(530\) −7.71411e30 −0.361504
\(531\) −7.35458e30 −0.337265
\(532\) 5.27143e30 0.236562
\(533\) 4.43483e31 1.94766
\(534\) −7.09627e30 −0.305004
\(535\) −7.82502e30 −0.329167
\(536\) 5.49004e30 0.226038
\(537\) 3.29408e31 1.32748
\(538\) −2.39679e31 −0.945440
\(539\) 9.25082e30 0.357198
\(540\) 6.43865e30 0.243370
\(541\) −2.43916e31 −0.902551 −0.451275 0.892385i \(-0.649031\pi\)
−0.451275 + 0.892385i \(0.649031\pi\)
\(542\) −3.05796e30 −0.110775
\(543\) −9.21611e30 −0.326851
\(544\) 1.06809e30 0.0370868
\(545\) 6.07992e30 0.206699
\(546\) 1.64581e31 0.547853
\(547\) −3.76722e31 −1.22791 −0.613955 0.789341i \(-0.710422\pi\)
−0.613955 + 0.789341i \(0.710422\pi\)
\(548\) 1.99054e31 0.635324
\(549\) −8.98447e30 −0.280809
\(550\) −3.02434e30 −0.0925676
\(551\) −2.12642e31 −0.637389
\(552\) 5.80239e30 0.170336
\(553\) 1.12256e31 0.322754
\(554\) 6.16337e30 0.173563
\(555\) 1.92219e31 0.530184
\(556\) 3.57442e31 0.965707
\(557\) 6.10400e30 0.161540 0.0807700 0.996733i \(-0.474262\pi\)
0.0807700 + 0.996733i \(0.474262\pi\)
\(558\) −3.79555e30 −0.0983968
\(559\) 6.94506e31 1.76376
\(560\) 3.02888e30 0.0753563
\(561\) −4.53975e30 −0.110652
\(562\) −4.04395e31 −0.965691
\(563\) 1.68208e31 0.393549 0.196775 0.980449i \(-0.436953\pi\)
0.196775 + 0.980449i \(0.436953\pi\)
\(564\) 1.76966e30 0.0405676
\(565\) 4.05596e30 0.0911038
\(566\) 4.23611e31 0.932348
\(567\) 1.64475e31 0.354727
\(568\) −2.30900e31 −0.487999
\(569\) 6.94377e31 1.43815 0.719076 0.694932i \(-0.244566\pi\)
0.719076 + 0.694932i \(0.244566\pi\)
\(570\) 8.81234e30 0.178867
\(571\) 2.16544e31 0.430756 0.215378 0.976531i \(-0.430902\pi\)
0.215378 + 0.976531i \(0.430902\pi\)
\(572\) 2.39477e31 0.466884
\(573\) −4.30572e31 −0.822749
\(574\) 3.47420e31 0.650682
\(575\) 6.51392e30 0.119581
\(576\) −2.43621e30 −0.0438385
\(577\) −2.82605e30 −0.0498492 −0.0249246 0.999689i \(-0.507935\pi\)
−0.0249246 + 0.999689i \(0.507935\pi\)
\(578\) −3.90937e31 −0.675984
\(579\) −5.95539e31 −1.00950
\(580\) −1.22180e31 −0.203039
\(581\) 9.72963e30 0.158515
\(582\) 5.69061e31 0.908959
\(583\) −4.77800e31 −0.748269
\(584\) 3.74713e31 0.575376
\(585\) −1.48610e31 −0.223746
\(586\) −5.32378e31 −0.785955
\(587\) −8.73921e31 −1.26513 −0.632563 0.774509i \(-0.717997\pi\)
−0.632563 + 0.774509i \(0.717997\pi\)
\(588\) −1.54880e31 −0.219865
\(589\) −2.00072e31 −0.278523
\(590\) −2.22753e31 −0.304106
\(591\) −5.74331e31 −0.768965
\(592\) −2.80112e31 −0.367817
\(593\) −7.84285e31 −1.01005 −0.505027 0.863103i \(-0.668518\pi\)
−0.505027 + 0.863103i \(0.668518\pi\)
\(594\) 3.98800e31 0.503745
\(595\) 5.10410e30 0.0632374
\(596\) 2.08621e31 0.253528
\(597\) −8.22149e31 −0.980045
\(598\) −5.15793e31 −0.603132
\(599\) 1.13954e32 1.30714 0.653570 0.756866i \(-0.273271\pi\)
0.653570 + 0.756866i \(0.273271\pi\)
\(600\) 5.06342e30 0.0569777
\(601\) −1.12055e32 −1.23701 −0.618506 0.785780i \(-0.712262\pi\)
−0.618506 + 0.785780i \(0.712262\pi\)
\(602\) 5.44069e31 0.589243
\(603\) −2.11019e31 −0.224219
\(604\) 1.07614e31 0.112187
\(605\) 2.49899e31 0.255610
\(606\) 5.86491e31 0.588608
\(607\) −2.47121e31 −0.243355 −0.121677 0.992570i \(-0.538827\pi\)
−0.121677 + 0.992570i \(0.538827\pi\)
\(608\) −1.28418e31 −0.124090
\(609\) −5.20089e31 −0.493150
\(610\) −2.72118e31 −0.253201
\(611\) −1.57311e31 −0.143643
\(612\) −4.10537e30 −0.0367884
\(613\) 2.06753e32 1.81826 0.909131 0.416511i \(-0.136747\pi\)
0.909131 + 0.416511i \(0.136747\pi\)
\(614\) 1.05885e32 0.913897
\(615\) 5.80788e31 0.491988
\(616\) 1.87604e31 0.155978
\(617\) −1.21753e32 −0.993572 −0.496786 0.867873i \(-0.665487\pi\)
−0.496786 + 0.867873i \(0.665487\pi\)
\(618\) −8.57004e31 −0.686460
\(619\) 2.23981e32 1.76104 0.880518 0.474012i \(-0.157195\pi\)
0.880518 + 0.474012i \(0.157195\pi\)
\(620\) −1.14958e31 −0.0887228
\(621\) −8.58949e31 −0.650749
\(622\) 4.48530e31 0.333582
\(623\) 4.94171e31 0.360799
\(624\) −4.00938e31 −0.287379
\(625\) 5.68434e30 0.0400000
\(626\) −7.05576e31 −0.487460
\(627\) 5.45822e31 0.370233
\(628\) −9.31889e31 −0.620623
\(629\) −4.72029e31 −0.308664
\(630\) −1.16420e31 −0.0747499
\(631\) 7.64725e31 0.482134 0.241067 0.970508i \(-0.422503\pi\)
0.241067 + 0.970508i \(0.422503\pi\)
\(632\) −2.73469e31 −0.169302
\(633\) −1.20834e32 −0.734593
\(634\) −8.70649e31 −0.519777
\(635\) 6.61374e31 0.387748
\(636\) 7.99945e31 0.460578
\(637\) 1.37678e32 0.778503
\(638\) −7.56765e31 −0.420265
\(639\) 8.87502e31 0.484071
\(640\) −7.37870e30 −0.0395285
\(641\) −5.00356e31 −0.263276 −0.131638 0.991298i \(-0.542024\pi\)
−0.131638 + 0.991298i \(0.542024\pi\)
\(642\) 8.11446e31 0.419379
\(643\) 1.82713e32 0.927565 0.463783 0.885949i \(-0.346492\pi\)
0.463783 + 0.885949i \(0.346492\pi\)
\(644\) −4.04068e31 −0.201496
\(645\) 9.09530e31 0.445533
\(646\) −2.16404e31 −0.104133
\(647\) 1.03583e32 0.489653 0.244826 0.969567i \(-0.421269\pi\)
0.244826 + 0.969567i \(0.421269\pi\)
\(648\) −4.00680e31 −0.186074
\(649\) −1.37970e32 −0.629461
\(650\) −4.50105e31 −0.201748
\(651\) −4.89346e31 −0.215494
\(652\) −1.24231e32 −0.537507
\(653\) 2.79465e32 1.18802 0.594012 0.804456i \(-0.297543\pi\)
0.594012 + 0.804456i \(0.297543\pi\)
\(654\) −6.30481e31 −0.263347
\(655\) 1.89786e32 0.778912
\(656\) −8.46357e31 −0.341318
\(657\) −1.44027e32 −0.570745
\(658\) −1.23236e31 −0.0479887
\(659\) 1.29327e32 0.494886 0.247443 0.968902i \(-0.420410\pi\)
0.247443 + 0.968902i \(0.420410\pi\)
\(660\) 3.13620e31 0.117937
\(661\) 1.04268e32 0.385330 0.192665 0.981265i \(-0.438287\pi\)
0.192665 + 0.981265i \(0.438287\pi\)
\(662\) 2.93468e32 1.06585
\(663\) −6.75640e31 −0.241162
\(664\) −2.37025e31 −0.0831498
\(665\) −6.13676e31 −0.211587
\(666\) 1.07665e32 0.364857
\(667\) 1.62995e32 0.542909
\(668\) 7.72837e31 0.253022
\(669\) −4.45652e32 −1.43415
\(670\) −6.39125e31 −0.202174
\(671\) −1.68546e32 −0.524093
\(672\) −3.14091e31 −0.0960084
\(673\) 9.42317e31 0.283155 0.141577 0.989927i \(-0.454783\pi\)
0.141577 + 0.989927i \(0.454783\pi\)
\(674\) 3.41390e31 0.100847
\(675\) −7.49557e31 −0.217676
\(676\) 1.81279e32 0.517559
\(677\) −2.82722e32 −0.793577 −0.396789 0.917910i \(-0.629875\pi\)
−0.396789 + 0.917910i \(0.629875\pi\)
\(678\) −4.20599e31 −0.116072
\(679\) −3.96284e32 −1.07524
\(680\) −1.24342e31 −0.0331715
\(681\) 1.82394e32 0.478430
\(682\) −7.12032e31 −0.183645
\(683\) −6.52604e31 −0.165505 −0.0827525 0.996570i \(-0.526371\pi\)
−0.0827525 + 0.996570i \(0.526371\pi\)
\(684\) 4.93596e31 0.123091
\(685\) −2.31730e32 −0.568251
\(686\) 3.05496e32 0.736680
\(687\) 4.80512e32 1.13947
\(688\) −1.32542e32 −0.309090
\(689\) −7.11098e32 −1.63083
\(690\) −6.75487e31 −0.152353
\(691\) −5.18137e31 −0.114934 −0.0574668 0.998347i \(-0.518302\pi\)
−0.0574668 + 0.998347i \(0.518302\pi\)
\(692\) −3.88590e32 −0.847757
\(693\) −7.21085e31 −0.154723
\(694\) −1.61371e32 −0.340558
\(695\) −4.16117e32 −0.863755
\(696\) 1.26700e32 0.258684
\(697\) −1.42623e32 −0.286427
\(698\) −2.98976e32 −0.590607
\(699\) −6.01129e31 −0.116810
\(700\) −3.52608e31 −0.0674007
\(701\) −6.09165e32 −1.14545 −0.572727 0.819746i \(-0.694114\pi\)
−0.572727 + 0.819746i \(0.694114\pi\)
\(702\) 5.93524e32 1.09790
\(703\) 5.67529e32 1.03277
\(704\) −4.57025e31 −0.0818190
\(705\) −2.06015e31 −0.0362848
\(706\) 1.08497e32 0.188002
\(707\) −4.08422e32 −0.696282
\(708\) 2.30992e32 0.387449
\(709\) 8.97201e32 1.48067 0.740334 0.672239i \(-0.234667\pi\)
0.740334 + 0.672239i \(0.234667\pi\)
\(710\) 2.68803e32 0.436479
\(711\) 1.05112e32 0.167940
\(712\) −1.20386e32 −0.189259
\(713\) 1.53360e32 0.237237
\(714\) −5.29290e31 −0.0805683
\(715\) −2.78788e32 −0.417594
\(716\) 5.58829e32 0.823719
\(717\) 8.92254e32 1.29425
\(718\) 5.78490e32 0.825780
\(719\) −7.87343e32 −1.10607 −0.553034 0.833159i \(-0.686530\pi\)
−0.553034 + 0.833159i \(0.686530\pi\)
\(720\) 2.83612e31 0.0392104
\(721\) 5.96802e32 0.812035
\(722\) −2.67849e32 −0.358684
\(723\) 4.94639e32 0.651926
\(724\) −1.56348e32 −0.202815
\(725\) 1.42237e32 0.181604
\(726\) −2.59143e32 −0.325663
\(727\) −3.15970e32 −0.390841 −0.195421 0.980720i \(-0.562607\pi\)
−0.195421 + 0.980720i \(0.562607\pi\)
\(728\) 2.79206e32 0.339949
\(729\) 8.85767e32 1.06158
\(730\) −4.36224e32 −0.514631
\(731\) −2.23352e32 −0.259382
\(732\) 2.82184e32 0.322593
\(733\) 9.47917e31 0.106678 0.0533389 0.998576i \(-0.483014\pi\)
0.0533389 + 0.998576i \(0.483014\pi\)
\(734\) 5.30709e32 0.587966
\(735\) 1.80304e32 0.196653
\(736\) 9.84356e31 0.105696
\(737\) −3.95864e32 −0.418475
\(738\) 3.25311e32 0.338571
\(739\) −4.13181e32 −0.423379 −0.211689 0.977337i \(-0.567897\pi\)
−0.211689 + 0.977337i \(0.567897\pi\)
\(740\) 3.26093e32 0.328985
\(741\) 8.12334e32 0.806910
\(742\) −5.57068e32 −0.544832
\(743\) 2.23848e32 0.215566 0.107783 0.994174i \(-0.465625\pi\)
0.107783 + 0.994174i \(0.465625\pi\)
\(744\) 1.19210e32 0.113038
\(745\) −2.42867e32 −0.226763
\(746\) −1.39617e33 −1.28363
\(747\) 9.11043e31 0.0824807
\(748\) −7.70154e31 −0.0686608
\(749\) −5.65076e32 −0.496097
\(750\) −5.89460e31 −0.0509624
\(751\) 1.65408e33 1.40831 0.704154 0.710047i \(-0.251326\pi\)
0.704154 + 0.710047i \(0.251326\pi\)
\(752\) 3.00217e31 0.0251727
\(753\) −4.57715e32 −0.377967
\(754\) −1.12628e33 −0.915955
\(755\) −1.25279e32 −0.100343
\(756\) 4.64961e32 0.366789
\(757\) −6.98662e32 −0.542831 −0.271415 0.962462i \(-0.587492\pi\)
−0.271415 + 0.962462i \(0.587492\pi\)
\(758\) 1.64483e33 1.25871
\(759\) −4.18386e32 −0.315353
\(760\) 1.49498e32 0.110989
\(761\) −9.28523e32 −0.678999 −0.339499 0.940606i \(-0.610258\pi\)
−0.339499 + 0.940606i \(0.610258\pi\)
\(762\) −6.85838e32 −0.494014
\(763\) 4.39056e32 0.311521
\(764\) −7.30451e32 −0.510525
\(765\) 4.77928e31 0.0329045
\(766\) −2.03495e33 −1.38014
\(767\) −2.05337e33 −1.37189
\(768\) 7.65163e31 0.0503617
\(769\) 1.31566e33 0.853081 0.426540 0.904469i \(-0.359732\pi\)
0.426540 + 0.904469i \(0.359732\pi\)
\(770\) −2.18400e32 −0.139511
\(771\) −1.76464e33 −1.11053
\(772\) −1.01031e33 −0.626407
\(773\) −1.01794e33 −0.621813 −0.310906 0.950441i \(-0.600633\pi\)
−0.310906 + 0.950441i \(0.600633\pi\)
\(774\) 5.09445e32 0.306603
\(775\) 1.33829e32 0.0793561
\(776\) 9.65393e32 0.564020
\(777\) 1.38809e33 0.799054
\(778\) −1.70625e33 −0.967781
\(779\) 1.71479e33 0.958362
\(780\) 4.66754e32 0.257039
\(781\) 1.66492e33 0.903457
\(782\) 1.65878e32 0.0886977
\(783\) −1.87558e33 −0.988271
\(784\) −2.62749e32 −0.136429
\(785\) 1.08486e33 0.555102
\(786\) −1.96806e33 −0.992381
\(787\) 3.95568e33 1.96567 0.982837 0.184475i \(-0.0590583\pi\)
0.982837 + 0.184475i \(0.0590583\pi\)
\(788\) −9.74334e32 −0.477152
\(789\) −1.24343e33 −0.600118
\(790\) 3.18360e32 0.151428
\(791\) 2.92897e32 0.137305
\(792\) 1.75665e32 0.0811606
\(793\) −2.50842e33 −1.14225
\(794\) 1.31034e33 0.588097
\(795\) −9.31259e32 −0.411954
\(796\) −1.39475e33 −0.608129
\(797\) 2.05790e33 0.884408 0.442204 0.896914i \(-0.354197\pi\)
0.442204 + 0.896914i \(0.354197\pi\)
\(798\) 6.36375e32 0.269575
\(799\) 5.05909e31 0.0211244
\(800\) 8.58993e31 0.0353553
\(801\) 4.62722e32 0.187736
\(802\) 2.81583e33 1.12616
\(803\) −2.70190e33 −1.06522
\(804\) 6.62766e32 0.257582
\(805\) 4.70397e32 0.180224
\(806\) −1.05970e33 −0.400249
\(807\) −2.89344e33 −1.07738
\(808\) 9.94962e32 0.365238
\(809\) 3.34033e33 1.20887 0.604437 0.796653i \(-0.293398\pi\)
0.604437 + 0.796653i \(0.293398\pi\)
\(810\) 4.66452e32 0.166429
\(811\) 4.70664e33 1.65566 0.827830 0.560980i \(-0.189575\pi\)
0.827830 + 0.560980i \(0.189575\pi\)
\(812\) −8.82314e32 −0.306005
\(813\) −3.69161e32 −0.126234
\(814\) 2.01977e33 0.680958
\(815\) 1.44624e33 0.480761
\(816\) 1.28941e32 0.0422625
\(817\) 2.68540e33 0.867872
\(818\) 4.17802e33 1.33140
\(819\) −1.07317e33 −0.337214
\(820\) 9.85289e32 0.305284
\(821\) −1.93945e33 −0.592561 −0.296281 0.955101i \(-0.595746\pi\)
−0.296281 + 0.955101i \(0.595746\pi\)
\(822\) 2.40301e33 0.723986
\(823\) −6.10374e33 −1.81342 −0.906709 0.421756i \(-0.861414\pi\)
−0.906709 + 0.421756i \(0.861414\pi\)
\(824\) −1.45388e33 −0.425957
\(825\) −3.65102e32 −0.105486
\(826\) −1.60859e33 −0.458326
\(827\) −2.51659e33 −0.707129 −0.353564 0.935410i \(-0.615030\pi\)
−0.353564 + 0.935410i \(0.615030\pi\)
\(828\) −3.78353e32 −0.104845
\(829\) 3.51989e33 0.961948 0.480974 0.876735i \(-0.340283\pi\)
0.480974 + 0.876735i \(0.340283\pi\)
\(830\) 2.75933e32 0.0743714
\(831\) 7.44051e32 0.197784
\(832\) −6.80179e32 −0.178322
\(833\) −4.42770e32 −0.114488
\(834\) 4.31509e33 1.10048
\(835\) −8.99701e32 −0.226310
\(836\) 9.25970e32 0.229734
\(837\) −1.76471e33 −0.431849
\(838\) −1.06299e33 −0.256580
\(839\) −2.21395e33 −0.527116 −0.263558 0.964644i \(-0.584896\pi\)
−0.263558 + 0.964644i \(0.584896\pi\)
\(840\) 3.65651e32 0.0858726
\(841\) −7.57601e32 −0.175504
\(842\) −2.24277e33 −0.512502
\(843\) −4.88192e33 −1.10046
\(844\) −2.04991e33 −0.455823
\(845\) −2.11036e33 −0.462919
\(846\) −1.15393e32 −0.0249701
\(847\) 1.80462e33 0.385237
\(848\) 1.35708e33 0.285794
\(849\) 5.11389e33 1.06246
\(850\) 1.44753e32 0.0296695
\(851\) −4.35025e33 −0.879678
\(852\) −2.78746e33 −0.556101
\(853\) −1.33116e33 −0.262008 −0.131004 0.991382i \(-0.541820\pi\)
−0.131004 + 0.991382i \(0.541820\pi\)
\(854\) −1.96508e33 −0.381605
\(855\) −5.74621e32 −0.110096
\(856\) 1.37659e33 0.260230
\(857\) 2.64308e33 0.492982 0.246491 0.969145i \(-0.420722\pi\)
0.246491 + 0.969145i \(0.420722\pi\)
\(858\) 2.89100e33 0.532040
\(859\) −8.59233e32 −0.156023 −0.0780117 0.996952i \(-0.524857\pi\)
−0.0780117 + 0.996952i \(0.524857\pi\)
\(860\) 1.54299e33 0.276459
\(861\) 4.19411e33 0.741487
\(862\) −3.65580e33 −0.637748
\(863\) 6.91408e33 1.19017 0.595087 0.803661i \(-0.297118\pi\)
0.595087 + 0.803661i \(0.297118\pi\)
\(864\) −1.13270e33 −0.192401
\(865\) 4.52378e33 0.758257
\(866\) 3.75563e30 0.000621194 0
\(867\) −4.71945e33 −0.770321
\(868\) −8.30159e32 −0.133716
\(869\) 1.97187e33 0.313438
\(870\) −1.47498e33 −0.231374
\(871\) −5.89154e33 −0.912054
\(872\) −1.06959e33 −0.163410
\(873\) −3.71064e33 −0.559481
\(874\) −1.99439e33 −0.296775
\(875\) 4.10489e32 0.0602850
\(876\) 4.52359e33 0.655672
\(877\) −8.45783e33 −1.20994 −0.604969 0.796249i \(-0.706815\pi\)
−0.604969 + 0.796249i \(0.706815\pi\)
\(878\) 5.66335e33 0.799625
\(879\) −6.42694e33 −0.895638
\(880\) 5.32047e32 0.0731811
\(881\) −6.75908e33 −0.917623 −0.458812 0.888534i \(-0.651725\pi\)
−0.458812 + 0.888534i \(0.651725\pi\)
\(882\) 1.00992e33 0.135331
\(883\) 1.02125e34 1.35077 0.675386 0.737464i \(-0.263977\pi\)
0.675386 + 0.737464i \(0.263977\pi\)
\(884\) −1.14620e33 −0.149644
\(885\) −2.68910e33 −0.346545
\(886\) −8.31273e33 −1.05744
\(887\) 9.62143e33 1.20814 0.604071 0.796931i \(-0.293544\pi\)
0.604071 + 0.796931i \(0.293544\pi\)
\(888\) −3.38155e33 −0.419147
\(889\) 4.77605e33 0.584385
\(890\) 1.40148e33 0.169278
\(891\) 2.88913e33 0.344488
\(892\) −7.56034e33 −0.889909
\(893\) −6.08264e32 −0.0706806
\(894\) 2.51850e33 0.288909
\(895\) −6.50563e33 −0.736757
\(896\) −5.32846e32 −0.0595744
\(897\) −6.22673e33 −0.687301
\(898\) −6.04673e33 −0.658936
\(899\) 3.34874e33 0.360284
\(900\) −3.30168e32 −0.0350708
\(901\) 2.28688e33 0.239833
\(902\) 6.10272e33 0.631900
\(903\) 6.56808e33 0.671475
\(904\) −7.13532e32 −0.0720239
\(905\) 1.82013e33 0.181403
\(906\) 1.29913e33 0.127843
\(907\) 4.24872e33 0.412833 0.206416 0.978464i \(-0.433820\pi\)
0.206416 + 0.978464i \(0.433820\pi\)
\(908\) 3.09425e33 0.296871
\(909\) −3.82430e33 −0.362299
\(910\) −3.25039e33 −0.304060
\(911\) 1.36929e34 1.26483 0.632417 0.774628i \(-0.282063\pi\)
0.632417 + 0.774628i \(0.282063\pi\)
\(912\) −1.55028e33 −0.141407
\(913\) 1.70909e33 0.153940
\(914\) −1.61308e33 −0.143474
\(915\) −3.28505e33 −0.288536
\(916\) 8.15172e33 0.707052
\(917\) 1.37052e34 1.17392
\(918\) −1.90876e33 −0.161459
\(919\) 5.57801e33 0.465963 0.232981 0.972481i \(-0.425152\pi\)
0.232981 + 0.972481i \(0.425152\pi\)
\(920\) −1.14594e33 −0.0945371
\(921\) 1.27826e34 1.04143
\(922\) 1.03383e34 0.831851
\(923\) 2.47787e34 1.96906
\(924\) 2.26478e33 0.177745
\(925\) −3.79622e33 −0.294253
\(926\) −6.29304e33 −0.481763
\(927\) 5.58822e33 0.422529
\(928\) 2.14942e33 0.160516
\(929\) −1.60947e34 −1.18714 −0.593571 0.804782i \(-0.702282\pi\)
−0.593571 + 0.804782i \(0.702282\pi\)
\(930\) −1.38779e33 −0.101104
\(931\) 5.32350e33 0.383068
\(932\) −1.01980e33 −0.0724820
\(933\) 5.41473e33 0.380135
\(934\) −1.62036e34 −1.12363
\(935\) 8.96577e32 0.0614121
\(936\) 2.61438e33 0.176887
\(937\) 2.39248e33 0.159898 0.0799489 0.996799i \(-0.474524\pi\)
0.0799489 + 0.996799i \(0.474524\pi\)
\(938\) −4.61538e33 −0.304702
\(939\) −8.51782e33 −0.555487
\(940\) −3.49498e32 −0.0225151
\(941\) −1.12281e34 −0.714537 −0.357268 0.934002i \(-0.616292\pi\)
−0.357268 + 0.934002i \(0.616292\pi\)
\(942\) −1.12499e34 −0.707234
\(943\) −1.31443e34 −0.816303
\(944\) 3.91871e33 0.240417
\(945\) −5.41286e33 −0.328066
\(946\) 9.55702e33 0.572235
\(947\) −4.28978e33 −0.253752 −0.126876 0.991919i \(-0.540495\pi\)
−0.126876 + 0.991919i \(0.540495\pi\)
\(948\) −3.30136e33 −0.192929
\(949\) −4.02117e34 −2.32162
\(950\) −1.74039e33 −0.0992717
\(951\) −1.05106e34 −0.592314
\(952\) −8.97923e32 −0.0499936
\(953\) 3.92575e33 0.215951 0.107975 0.994154i \(-0.465563\pi\)
0.107975 + 0.994154i \(0.465563\pi\)
\(954\) −5.21616e33 −0.283494
\(955\) 8.50357e33 0.456628
\(956\) 1.51368e34 0.803097
\(957\) −9.13578e33 −0.478915
\(958\) −2.34172e33 −0.121292
\(959\) −1.67341e34 −0.856426
\(960\) −8.90767e32 −0.0450448
\(961\) −1.68625e34 −0.842565
\(962\) 3.00597e34 1.48413
\(963\) −5.29115e33 −0.258136
\(964\) 8.39140e33 0.404528
\(965\) 1.17616e34 0.560276
\(966\) −4.87796e33 −0.229616
\(967\) −2.44005e33 −0.113500 −0.0567498 0.998388i \(-0.518074\pi\)
−0.0567498 + 0.998388i \(0.518074\pi\)
\(968\) −4.39627e33 −0.202078
\(969\) −2.61245e33 −0.118666
\(970\) −1.12387e34 −0.504475
\(971\) 1.61207e34 0.715094 0.357547 0.933895i \(-0.383613\pi\)
0.357547 + 0.933895i \(0.383613\pi\)
\(972\) 7.57699e33 0.332150
\(973\) −3.00495e34 −1.30179
\(974\) 7.56810e33 0.324010
\(975\) −5.43373e33 −0.229903
\(976\) 4.78716e33 0.200173
\(977\) −3.11737e34 −1.28825 −0.644126 0.764920i \(-0.722779\pi\)
−0.644126 + 0.764920i \(0.722779\pi\)
\(978\) −1.49974e34 −0.612519
\(979\) 8.68052e33 0.350384
\(980\) 3.05880e33 0.122025
\(981\) 4.11114e33 0.162095
\(982\) 1.90278e34 0.741494
\(983\) 3.16423e34 1.21872 0.609358 0.792895i \(-0.291427\pi\)
0.609358 + 0.792895i \(0.291427\pi\)
\(984\) −1.02173e34 −0.388950
\(985\) 1.13427e34 0.426777
\(986\) 3.62209e33 0.134702
\(987\) −1.48772e33 −0.0546857
\(988\) 1.37810e34 0.500697
\(989\) −2.05842e34 −0.739227
\(990\) −2.04501e33 −0.0725922
\(991\) −2.83514e34 −0.994779 −0.497390 0.867527i \(-0.665708\pi\)
−0.497390 + 0.867527i \(0.665708\pi\)
\(992\) 2.02236e33 0.0701415
\(993\) 3.54279e34 1.21459
\(994\) 1.94114e34 0.657829
\(995\) 1.62370e34 0.543927
\(996\) −2.86140e33 −0.0947537
\(997\) 1.22327e34 0.400430 0.200215 0.979752i \(-0.435836\pi\)
0.200215 + 0.979752i \(0.435836\pi\)
\(998\) 6.70891e32 0.0217096
\(999\) 5.00583e34 1.60130
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 10.24.a.c.1.2 2
5.2 odd 4 50.24.b.c.49.3 4
5.3 odd 4 50.24.b.c.49.2 4
5.4 even 2 50.24.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.24.a.c.1.2 2 1.1 even 1 trivial
50.24.a.b.1.1 2 5.4 even 2
50.24.b.c.49.2 4 5.3 odd 4
50.24.b.c.49.3 4 5.2 odd 4