Properties

Label 10.26.a.c.1.1
Level $10$
Weight $26$
Character 10.1
Self dual yes
Analytic conductor $39.600$
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] = [10,26,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, 26, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 26);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 26 \)
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: \(39.5996779952\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 148387471 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-12181.4\) 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)
 
\(f(q)\) \(=\) \(q-4096.00 q^{2} -701909. q^{3} +1.67772e7 q^{4} -2.44141e8 q^{5} +2.87502e9 q^{6} -7.10199e8 q^{7} -6.87195e10 q^{8} -3.54612e11 q^{9} +1.00000e12 q^{10} +6.69047e12 q^{11} -1.17761e13 q^{12} -4.06505e12 q^{13} +2.90898e12 q^{14} +1.71365e14 q^{15} +2.81475e14 q^{16} +2.05623e15 q^{17} +1.45249e15 q^{18} +1.32410e16 q^{19} -4.09600e15 q^{20} +4.98495e14 q^{21} -2.74041e16 q^{22} -2.83334e16 q^{23} +4.82348e16 q^{24} +5.96046e16 q^{25} +1.66504e16 q^{26} +8.43625e17 q^{27} -1.19152e16 q^{28} +8.79820e17 q^{29} -7.01909e17 q^{30} -3.20276e17 q^{31} -1.15292e18 q^{32} -4.69610e18 q^{33} -8.42232e18 q^{34} +1.73388e17 q^{35} -5.94940e18 q^{36} -4.64680e19 q^{37} -5.42350e19 q^{38} +2.85329e18 q^{39} +1.67772e19 q^{40} -1.18387e20 q^{41} -2.04184e18 q^{42} -1.34535e20 q^{43} +1.12247e20 q^{44} +8.65752e19 q^{45} +1.16053e20 q^{46} -7.78716e20 q^{47} -1.97570e20 q^{48} -1.34056e21 q^{49} -2.44141e20 q^{50} -1.44329e21 q^{51} -6.82002e19 q^{52} -3.18129e21 q^{53} -3.45549e21 q^{54} -1.63341e21 q^{55} +4.88045e19 q^{56} -9.29396e21 q^{57} -3.60374e21 q^{58} -2.03629e21 q^{59} +2.87502e21 q^{60} +3.16043e22 q^{61} +1.31185e21 q^{62} +2.51845e20 q^{63} +4.72237e21 q^{64} +9.92443e20 q^{65} +1.92352e22 q^{66} +4.98952e22 q^{67} +3.44978e22 q^{68} +1.98875e22 q^{69} -7.10199e20 q^{70} +1.16862e23 q^{71} +2.43688e22 q^{72} -1.07622e23 q^{73} +1.90333e23 q^{74} -4.18370e22 q^{75} +2.22147e23 q^{76} -4.75156e21 q^{77} -1.16871e22 q^{78} +2.51396e23 q^{79} -6.87195e22 q^{80} -2.91689e23 q^{81} +4.84913e23 q^{82} -1.47962e24 q^{83} +8.36336e21 q^{84} -5.02010e23 q^{85} +5.51054e23 q^{86} -6.17554e23 q^{87} -4.59765e23 q^{88} -1.69758e24 q^{89} -3.54612e23 q^{90} +2.88699e21 q^{91} -4.75355e23 q^{92} +2.24804e23 q^{93} +3.18962e24 q^{94} -3.23266e24 q^{95} +8.09246e23 q^{96} -4.56953e24 q^{97} +5.49095e24 q^{98} -2.37252e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8192 q^{2} + 545212 q^{3} + 33554432 q^{4} - 488281250 q^{5} - 2233188352 q^{6} + 38567856964 q^{7} - 137438953472 q^{8} + 353410472386 q^{9} + 2000000000000 q^{10} - 8379169876416 q^{11} + 9147139489792 q^{12}+ \cdots - 13\!\cdots\!88 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4096.00 −0.707107
\(3\) −701909. −0.762545 −0.381272 0.924463i \(-0.624514\pi\)
−0.381272 + 0.924463i \(0.624514\pi\)
\(4\) 1.67772e7 0.500000
\(5\) −2.44141e8 −0.447214
\(6\) 2.87502e9 0.539200
\(7\) −7.10199e8 −0.0193934 −0.00969672 0.999953i \(-0.503087\pi\)
−0.00969672 + 0.999953i \(0.503087\pi\)
\(8\) −6.87195e10 −0.353553
\(9\) −3.54612e11 −0.418526
\(10\) 1.00000e12 0.316228
\(11\) 6.69047e12 0.642758 0.321379 0.946951i \(-0.395854\pi\)
0.321379 + 0.946951i \(0.395854\pi\)
\(12\) −1.17761e13 −0.381272
\(13\) −4.06505e12 −0.0483920 −0.0241960 0.999707i \(-0.507703\pi\)
−0.0241960 + 0.999707i \(0.507703\pi\)
\(14\) 2.90898e12 0.0137132
\(15\) 1.71365e14 0.341020
\(16\) 2.81475e14 0.250000
\(17\) 2.05623e15 0.855974 0.427987 0.903785i \(-0.359223\pi\)
0.427987 + 0.903785i \(0.359223\pi\)
\(18\) 1.45249e15 0.295942
\(19\) 1.32410e16 1.37246 0.686230 0.727384i \(-0.259264\pi\)
0.686230 + 0.727384i \(0.259264\pi\)
\(20\) −4.09600e15 −0.223607
\(21\) 4.98495e14 0.0147884
\(22\) −2.74041e16 −0.454499
\(23\) −2.83334e16 −0.269588 −0.134794 0.990874i \(-0.543037\pi\)
−0.134794 + 0.990874i \(0.543037\pi\)
\(24\) 4.82348e16 0.269600
\(25\) 5.96046e16 0.200000
\(26\) 1.66504e16 0.0342183
\(27\) 8.43625e17 1.08169
\(28\) −1.19152e16 −0.00969672
\(29\) 8.79820e17 0.461763 0.230881 0.972982i \(-0.425839\pi\)
0.230881 + 0.972982i \(0.425839\pi\)
\(30\) −7.01909e17 −0.241138
\(31\) −3.20276e17 −0.0730302 −0.0365151 0.999333i \(-0.511626\pi\)
−0.0365151 + 0.999333i \(0.511626\pi\)
\(32\) −1.15292e18 −0.176777
\(33\) −4.69610e18 −0.490132
\(34\) −8.42232e18 −0.605265
\(35\) 1.73388e17 0.00867301
\(36\) −5.94940e18 −0.209263
\(37\) −4.64680e19 −1.16047 −0.580233 0.814450i \(-0.697039\pi\)
−0.580233 + 0.814450i \(0.697039\pi\)
\(38\) −5.42350e19 −0.970476
\(39\) 2.85329e18 0.0369011
\(40\) 1.67772e19 0.158114
\(41\) −1.18387e20 −0.819418 −0.409709 0.912216i \(-0.634370\pi\)
−0.409709 + 0.912216i \(0.634370\pi\)
\(42\) −2.04184e18 −0.0104569
\(43\) −1.34535e20 −0.513426 −0.256713 0.966488i \(-0.582640\pi\)
−0.256713 + 0.966488i \(0.582640\pi\)
\(44\) 1.12247e20 0.321379
\(45\) 8.65752e19 0.187170
\(46\) 1.16053e20 0.190627
\(47\) −7.78716e20 −0.977587 −0.488794 0.872399i \(-0.662563\pi\)
−0.488794 + 0.872399i \(0.662563\pi\)
\(48\) −1.97570e20 −0.190636
\(49\) −1.34056e21 −0.999624
\(50\) −2.44141e20 −0.141421
\(51\) −1.44329e21 −0.652718
\(52\) −6.82002e19 −0.0241960
\(53\) −3.18129e21 −0.889517 −0.444759 0.895651i \(-0.646711\pi\)
−0.444759 + 0.895651i \(0.646711\pi\)
\(54\) −3.45549e21 −0.764870
\(55\) −1.63341e21 −0.287450
\(56\) 4.88045e19 0.00685661
\(57\) −9.29396e21 −1.04656
\(58\) −3.60374e21 −0.326516
\(59\) −2.03629e21 −0.149001 −0.0745005 0.997221i \(-0.523736\pi\)
−0.0745005 + 0.997221i \(0.523736\pi\)
\(60\) 2.87502e21 0.170510
\(61\) 3.16043e22 1.52448 0.762242 0.647292i \(-0.224099\pi\)
0.762242 + 0.647292i \(0.224099\pi\)
\(62\) 1.31185e21 0.0516402
\(63\) 2.51845e20 0.00811665
\(64\) 4.72237e21 0.125000
\(65\) 9.92443e20 0.0216416
\(66\) 1.92352e22 0.346576
\(67\) 4.98952e22 0.744945 0.372472 0.928043i \(-0.378510\pi\)
0.372472 + 0.928043i \(0.378510\pi\)
\(68\) 3.44978e22 0.427987
\(69\) 1.98875e22 0.205573
\(70\) −7.10199e20 −0.00613274
\(71\) 1.16862e23 0.845169 0.422585 0.906324i \(-0.361123\pi\)
0.422585 + 0.906324i \(0.361123\pi\)
\(72\) 2.43688e22 0.147971
\(73\) −1.07622e23 −0.550001 −0.275001 0.961444i \(-0.588678\pi\)
−0.275001 + 0.961444i \(0.588678\pi\)
\(74\) 1.90333e23 0.820574
\(75\) −4.18370e22 −0.152509
\(76\) 2.22147e23 0.686230
\(77\) −4.75156e21 −0.0124653
\(78\) −1.16871e22 −0.0260930
\(79\) 2.51396e23 0.478651 0.239326 0.970939i \(-0.423074\pi\)
0.239326 + 0.970939i \(0.423074\pi\)
\(80\) −6.87195e22 −0.111803
\(81\) −2.91689e23 −0.406310
\(82\) 4.84913e23 0.579416
\(83\) −1.47962e24 −1.51941 −0.759703 0.650270i \(-0.774656\pi\)
−0.759703 + 0.650270i \(0.774656\pi\)
\(84\) 8.36336e21 0.00739418
\(85\) −5.02010e23 −0.382803
\(86\) 5.51054e23 0.363047
\(87\) −6.17554e23 −0.352115
\(88\) −4.59765e23 −0.227249
\(89\) −1.69758e24 −0.728544 −0.364272 0.931293i \(-0.618682\pi\)
−0.364272 + 0.931293i \(0.618682\pi\)
\(90\) −3.54612e23 −0.132349
\(91\) 2.88699e21 0.000938487 0
\(92\) −4.75355e23 −0.134794
\(93\) 2.24804e23 0.0556888
\(94\) 3.18962e24 0.691259
\(95\) −3.23266e24 −0.613783
\(96\) 8.09246e23 0.134800
\(97\) −4.56953e24 −0.668689 −0.334345 0.942451i \(-0.608515\pi\)
−0.334345 + 0.942451i \(0.608515\pi\)
\(98\) 5.49095e24 0.706841
\(99\) −2.37252e24 −0.269011
\(100\) 1.00000e24 0.100000
\(101\) −1.34474e25 −1.18747 −0.593735 0.804661i \(-0.702347\pi\)
−0.593735 + 0.804661i \(0.702347\pi\)
\(102\) 5.91170e24 0.461541
\(103\) −1.45443e25 −1.00514 −0.502571 0.864536i \(-0.667612\pi\)
−0.502571 + 0.864536i \(0.667612\pi\)
\(104\) 2.79348e23 0.0171092
\(105\) −1.21703e23 −0.00661355
\(106\) 1.30305e25 0.628984
\(107\) −3.33812e25 −1.43286 −0.716432 0.697657i \(-0.754226\pi\)
−0.716432 + 0.697657i \(0.754226\pi\)
\(108\) 1.41537e25 0.540845
\(109\) 4.20504e25 1.43199 0.715993 0.698107i \(-0.245974\pi\)
0.715993 + 0.698107i \(0.245974\pi\)
\(110\) 6.69047e24 0.203258
\(111\) 3.26163e25 0.884908
\(112\) −1.99903e23 −0.00484836
\(113\) 7.61141e25 1.65190 0.825952 0.563741i \(-0.190638\pi\)
0.825952 + 0.563741i \(0.190638\pi\)
\(114\) 3.80681e25 0.740031
\(115\) 6.91733e24 0.120563
\(116\) 1.47609e25 0.230881
\(117\) 1.44152e24 0.0202533
\(118\) 8.34063e24 0.105360
\(119\) −1.46033e24 −0.0166003
\(120\) −1.17761e25 −0.120569
\(121\) −6.35847e25 −0.586862
\(122\) −1.29451e26 −1.07797
\(123\) 8.30968e25 0.624843
\(124\) −5.37333e24 −0.0365151
\(125\) −1.45519e25 −0.0894427
\(126\) −1.03156e24 −0.00573934
\(127\) −2.50269e26 −1.26142 −0.630710 0.776018i \(-0.717236\pi\)
−0.630710 + 0.776018i \(0.717236\pi\)
\(128\) −1.93428e25 −0.0883883
\(129\) 9.44310e25 0.391510
\(130\) −4.06505e24 −0.0153029
\(131\) −3.42885e26 −1.17289 −0.586445 0.809989i \(-0.699473\pi\)
−0.586445 + 0.809989i \(0.699473\pi\)
\(132\) −7.87875e25 −0.245066
\(133\) −9.40373e24 −0.0266167
\(134\) −2.04371e26 −0.526755
\(135\) −2.05963e26 −0.483746
\(136\) −1.41303e26 −0.302632
\(137\) 1.44437e26 0.282274 0.141137 0.989990i \(-0.454924\pi\)
0.141137 + 0.989990i \(0.454924\pi\)
\(138\) −8.14590e25 −0.145362
\(139\) 4.93300e26 0.804314 0.402157 0.915571i \(-0.368261\pi\)
0.402157 + 0.915571i \(0.368261\pi\)
\(140\) 2.90898e24 0.00433650
\(141\) 5.46588e26 0.745454
\(142\) −4.78666e26 −0.597625
\(143\) −2.71971e25 −0.0311044
\(144\) −9.98144e25 −0.104631
\(145\) −2.14800e26 −0.206507
\(146\) 4.40818e26 0.388910
\(147\) 9.40954e26 0.762258
\(148\) −7.79604e26 −0.580233
\(149\) 1.50844e27 1.03205 0.516023 0.856575i \(-0.327412\pi\)
0.516023 + 0.856575i \(0.327412\pi\)
\(150\) 1.71365e26 0.107840
\(151\) 1.70083e26 0.0985027 0.0492513 0.998786i \(-0.484316\pi\)
0.0492513 + 0.998786i \(0.484316\pi\)
\(152\) −9.09913e26 −0.485238
\(153\) −7.29164e26 −0.358247
\(154\) 1.94624e25 0.00881429
\(155\) 7.81923e25 0.0326601
\(156\) 4.78703e25 0.0184505
\(157\) 3.29198e27 1.17142 0.585708 0.810522i \(-0.300816\pi\)
0.585708 + 0.810522i \(0.300816\pi\)
\(158\) −1.02972e27 −0.338458
\(159\) 2.23297e27 0.678296
\(160\) 2.81475e26 0.0790569
\(161\) 2.01223e25 0.00522823
\(162\) 1.19476e27 0.287305
\(163\) 5.07392e26 0.112979 0.0564896 0.998403i \(-0.482009\pi\)
0.0564896 + 0.998403i \(0.482009\pi\)
\(164\) −1.98620e27 −0.409709
\(165\) 1.14651e27 0.219194
\(166\) 6.06052e27 1.07438
\(167\) −8.44897e26 −0.138947 −0.0694733 0.997584i \(-0.522132\pi\)
−0.0694733 + 0.997584i \(0.522132\pi\)
\(168\) −3.42563e25 −0.00522847
\(169\) −7.03989e27 −0.997658
\(170\) 2.05623e27 0.270683
\(171\) −4.69541e27 −0.574410
\(172\) −2.25712e27 −0.256713
\(173\) −7.54461e27 −0.798106 −0.399053 0.916928i \(-0.630661\pi\)
−0.399053 + 0.916928i \(0.630661\pi\)
\(174\) 2.52950e27 0.248983
\(175\) −4.23312e25 −0.00387869
\(176\) 1.88320e27 0.160690
\(177\) 1.42929e27 0.113620
\(178\) 6.95330e27 0.515159
\(179\) −5.46396e27 −0.377437 −0.188718 0.982031i \(-0.560433\pi\)
−0.188718 + 0.982031i \(0.560433\pi\)
\(180\) 1.45249e27 0.0935852
\(181\) −2.74350e28 −1.64939 −0.824693 0.565581i \(-0.808652\pi\)
−0.824693 + 0.565581i \(0.808652\pi\)
\(182\) −1.18251e25 −0.000663611 0
\(183\) −2.21833e28 −1.16249
\(184\) 1.94705e27 0.0953137
\(185\) 1.13447e28 0.518976
\(186\) −9.20799e26 −0.0393779
\(187\) 1.37571e28 0.550184
\(188\) −1.30647e28 −0.488794
\(189\) −5.99142e26 −0.0209777
\(190\) 1.32410e28 0.434010
\(191\) −2.53537e28 −0.778258 −0.389129 0.921183i \(-0.627224\pi\)
−0.389129 + 0.921183i \(0.627224\pi\)
\(192\) −3.31467e27 −0.0953181
\(193\) 4.66476e27 0.125708 0.0628540 0.998023i \(-0.479980\pi\)
0.0628540 + 0.998023i \(0.479980\pi\)
\(194\) 1.87168e28 0.472835
\(195\) −6.96605e26 −0.0165027
\(196\) −2.24909e28 −0.499812
\(197\) −1.14775e28 −0.239343 −0.119672 0.992814i \(-0.538184\pi\)
−0.119672 + 0.992814i \(0.538184\pi\)
\(198\) 9.71784e27 0.190219
\(199\) 1.02205e29 1.87849 0.939243 0.343254i \(-0.111529\pi\)
0.939243 + 0.343254i \(0.111529\pi\)
\(200\) −4.09600e27 −0.0707107
\(201\) −3.50219e28 −0.568053
\(202\) 5.50807e28 0.839668
\(203\) −6.24848e26 −0.00895517
\(204\) −2.42143e28 −0.326359
\(205\) 2.89030e28 0.366455
\(206\) 5.95734e28 0.710742
\(207\) 1.00474e28 0.112829
\(208\) −1.14421e27 −0.0120980
\(209\) 8.85883e28 0.882161
\(210\) 4.98495e26 0.00467649
\(211\) 1.45021e29 1.28204 0.641020 0.767524i \(-0.278512\pi\)
0.641020 + 0.767524i \(0.278512\pi\)
\(212\) −5.33731e28 −0.444759
\(213\) −8.20264e28 −0.644479
\(214\) 1.36729e29 1.01319
\(215\) 3.28453e28 0.229611
\(216\) −5.79735e28 −0.382435
\(217\) 2.27460e26 0.00141631
\(218\) −1.72238e29 −1.01257
\(219\) 7.55406e28 0.419400
\(220\) −2.74041e28 −0.143725
\(221\) −8.35868e27 −0.0414223
\(222\) −1.33597e29 −0.625724
\(223\) −3.26058e29 −1.44372 −0.721861 0.692038i \(-0.756713\pi\)
−0.721861 + 0.692038i \(0.756713\pi\)
\(224\) 8.18804e26 0.00342831
\(225\) −2.11365e28 −0.0837052
\(226\) −3.11763e29 −1.16807
\(227\) −4.25210e29 −1.50758 −0.753791 0.657115i \(-0.771777\pi\)
−0.753791 + 0.657115i \(0.771777\pi\)
\(228\) −1.55927e29 −0.523281
\(229\) 5.46098e29 1.73511 0.867555 0.497342i \(-0.165690\pi\)
0.867555 + 0.497342i \(0.165690\pi\)
\(230\) −2.83334e28 −0.0852512
\(231\) 3.33517e27 0.00950534
\(232\) −6.04608e28 −0.163258
\(233\) −2.50604e29 −0.641268 −0.320634 0.947203i \(-0.603896\pi\)
−0.320634 + 0.947203i \(0.603896\pi\)
\(234\) −5.90445e27 −0.0143213
\(235\) 1.90116e29 0.437190
\(236\) −3.41632e28 −0.0745005
\(237\) −1.76457e29 −0.364993
\(238\) 5.98153e27 0.0117382
\(239\) 6.64864e29 1.23811 0.619055 0.785348i \(-0.287516\pi\)
0.619055 + 0.785348i \(0.287516\pi\)
\(240\) 4.82348e28 0.0852551
\(241\) 2.77555e29 0.465731 0.232866 0.972509i \(-0.425190\pi\)
0.232866 + 0.972509i \(0.425190\pi\)
\(242\) 2.60443e29 0.414974
\(243\) −5.10055e29 −0.771859
\(244\) 5.30232e29 0.762242
\(245\) 3.27286e29 0.447045
\(246\) −3.40365e29 −0.441830
\(247\) −5.38252e28 −0.0664161
\(248\) 2.20092e28 0.0258201
\(249\) 1.03856e30 1.15862
\(250\) 5.96046e28 0.0632456
\(251\) −1.49015e30 −1.50421 −0.752105 0.659043i \(-0.770962\pi\)
−0.752105 + 0.659043i \(0.770962\pi\)
\(252\) 4.22526e27 0.00405833
\(253\) −1.89563e29 −0.173280
\(254\) 1.02510e30 0.891959
\(255\) 3.52365e29 0.291904
\(256\) 7.92282e28 0.0625000
\(257\) −7.29296e29 −0.547948 −0.273974 0.961737i \(-0.588338\pi\)
−0.273974 + 0.961737i \(0.588338\pi\)
\(258\) −3.86790e29 −0.276840
\(259\) 3.30016e28 0.0225054
\(260\) 1.66504e28 0.0108208
\(261\) −3.11995e29 −0.193260
\(262\) 1.40446e30 0.829358
\(263\) −5.88360e29 −0.331281 −0.165640 0.986186i \(-0.552969\pi\)
−0.165640 + 0.986186i \(0.552969\pi\)
\(264\) 3.22713e29 0.173288
\(265\) 7.76681e29 0.397804
\(266\) 3.85177e28 0.0188209
\(267\) 1.19155e30 0.555548
\(268\) 8.37103e29 0.372472
\(269\) 1.22413e30 0.519904 0.259952 0.965622i \(-0.416293\pi\)
0.259952 + 0.965622i \(0.416293\pi\)
\(270\) 8.43625e29 0.342060
\(271\) −1.53605e30 −0.594688 −0.297344 0.954770i \(-0.596101\pi\)
−0.297344 + 0.954770i \(0.596101\pi\)
\(272\) 5.78778e29 0.213993
\(273\) −2.02641e27 −0.000715638 0
\(274\) −5.91613e29 −0.199598
\(275\) 3.98783e29 0.128552
\(276\) 3.33656e29 0.102786
\(277\) −4.32632e29 −0.127386 −0.0636929 0.997970i \(-0.520288\pi\)
−0.0636929 + 0.997970i \(0.520288\pi\)
\(278\) −2.02056e30 −0.568736
\(279\) 1.13574e29 0.0305650
\(280\) −1.19152e28 −0.00306637
\(281\) −3.63667e30 −0.895107 −0.447554 0.894257i \(-0.647705\pi\)
−0.447554 + 0.894257i \(0.647705\pi\)
\(282\) −2.23882e30 −0.527116
\(283\) −8.38888e30 −1.88962 −0.944808 0.327624i \(-0.893752\pi\)
−0.944808 + 0.327624i \(0.893752\pi\)
\(284\) 1.96062e30 0.422585
\(285\) 2.26903e30 0.468037
\(286\) 1.11399e29 0.0219941
\(287\) 8.40783e28 0.0158913
\(288\) 4.08840e29 0.0739856
\(289\) −1.54254e30 −0.267309
\(290\) 8.79820e29 0.146022
\(291\) 3.20739e30 0.509906
\(292\) −1.80559e30 −0.275001
\(293\) −8.70672e30 −1.27060 −0.635301 0.772265i \(-0.719124\pi\)
−0.635301 + 0.772265i \(0.719124\pi\)
\(294\) −3.85415e30 −0.538998
\(295\) 4.97141e29 0.0666352
\(296\) 3.19326e30 0.410287
\(297\) 5.64425e30 0.695265
\(298\) −6.17856e30 −0.729767
\(299\) 1.15177e29 0.0130459
\(300\) −7.01909e29 −0.0762545
\(301\) 9.55463e28 0.00995710
\(302\) −6.96658e29 −0.0696519
\(303\) 9.43888e30 0.905498
\(304\) 3.72700e30 0.343115
\(305\) −7.71589e30 −0.681770
\(306\) 2.98666e30 0.253319
\(307\) 3.24336e30 0.264098 0.132049 0.991243i \(-0.457844\pi\)
0.132049 + 0.991243i \(0.457844\pi\)
\(308\) −7.97180e28 −0.00623265
\(309\) 1.02088e31 0.766465
\(310\) −3.20276e29 −0.0230942
\(311\) −1.24495e30 −0.0862281 −0.0431140 0.999070i \(-0.513728\pi\)
−0.0431140 + 0.999070i \(0.513728\pi\)
\(312\) −1.96077e29 −0.0130465
\(313\) 2.33420e31 1.49222 0.746112 0.665820i \(-0.231918\pi\)
0.746112 + 0.665820i \(0.231918\pi\)
\(314\) −1.34839e31 −0.828316
\(315\) −6.14857e28 −0.00362988
\(316\) 4.21772e30 0.239326
\(317\) 2.11447e31 1.15335 0.576674 0.816974i \(-0.304350\pi\)
0.576674 + 0.816974i \(0.304350\pi\)
\(318\) −9.14626e30 −0.479628
\(319\) 5.88641e30 0.296802
\(320\) −1.15292e30 −0.0559017
\(321\) 2.34306e31 1.09262
\(322\) −8.24211e28 −0.00369692
\(323\) 2.72265e31 1.17479
\(324\) −4.89374e30 −0.203155
\(325\) −2.42296e29 −0.00967840
\(326\) −2.07828e30 −0.0798883
\(327\) −2.95156e31 −1.09195
\(328\) 8.13549e30 0.289708
\(329\) 5.53043e29 0.0189588
\(330\) −4.69610e30 −0.154993
\(331\) −4.35367e31 −1.38359 −0.691793 0.722096i \(-0.743179\pi\)
−0.691793 + 0.722096i \(0.743179\pi\)
\(332\) −2.48239e31 −0.759703
\(333\) 1.64781e31 0.485685
\(334\) 3.46070e30 0.0982501
\(335\) −1.21815e31 −0.333149
\(336\) 1.40314e29 0.00369709
\(337\) −5.97252e30 −0.151630 −0.0758148 0.997122i \(-0.524156\pi\)
−0.0758148 + 0.997122i \(0.524156\pi\)
\(338\) 2.88354e31 0.705451
\(339\) −5.34252e31 −1.25965
\(340\) −8.42232e30 −0.191402
\(341\) −2.14279e30 −0.0469408
\(342\) 1.92324e31 0.406169
\(343\) 1.90449e30 0.0387796
\(344\) 9.24514e30 0.181524
\(345\) −4.85534e30 −0.0919349
\(346\) 3.09027e31 0.564346
\(347\) −9.71425e31 −1.71116 −0.855580 0.517670i \(-0.826799\pi\)
−0.855580 + 0.517670i \(0.826799\pi\)
\(348\) −1.03608e31 −0.176057
\(349\) −7.20784e31 −1.18165 −0.590823 0.806801i \(-0.701197\pi\)
−0.590823 + 0.806801i \(0.701197\pi\)
\(350\) 1.73388e29 0.00274265
\(351\) −3.42938e30 −0.0523451
\(352\) −7.71358e30 −0.113625
\(353\) 6.98207e31 0.992660 0.496330 0.868134i \(-0.334681\pi\)
0.496330 + 0.868134i \(0.334681\pi\)
\(354\) −5.85437e30 −0.0803414
\(355\) −2.85307e31 −0.377971
\(356\) −2.84807e31 −0.364272
\(357\) 1.02502e30 0.0126584
\(358\) 2.23804e31 0.266888
\(359\) −1.33870e32 −1.54171 −0.770857 0.637009i \(-0.780171\pi\)
−0.770857 + 0.637009i \(0.780171\pi\)
\(360\) −5.94940e30 −0.0661747
\(361\) 8.22469e31 0.883649
\(362\) 1.12374e32 1.16629
\(363\) 4.46307e31 0.447508
\(364\) 4.84357e28 0.000469244 0
\(365\) 2.62748e31 0.245968
\(366\) 9.08629e31 0.822003
\(367\) 6.63971e31 0.580528 0.290264 0.956947i \(-0.406257\pi\)
0.290264 + 0.956947i \(0.406257\pi\)
\(368\) −7.97514e30 −0.0673970
\(369\) 4.19814e31 0.342947
\(370\) −4.64680e31 −0.366972
\(371\) 2.25935e30 0.0172508
\(372\) 3.77159e30 0.0278444
\(373\) 2.18553e32 1.56026 0.780129 0.625619i \(-0.215154\pi\)
0.780129 + 0.625619i \(0.215154\pi\)
\(374\) −5.63493e31 −0.389039
\(375\) 1.02141e31 0.0682041
\(376\) 5.35129e31 0.345629
\(377\) −3.57651e30 −0.0223456
\(378\) 2.45409e30 0.0148334
\(379\) −1.48811e32 −0.870253 −0.435126 0.900369i \(-0.643296\pi\)
−0.435126 + 0.900369i \(0.643296\pi\)
\(380\) −5.42350e31 −0.306892
\(381\) 1.75666e32 0.961889
\(382\) 1.03849e32 0.550311
\(383\) −2.09320e32 −1.07356 −0.536779 0.843723i \(-0.680359\pi\)
−0.536779 + 0.843723i \(0.680359\pi\)
\(384\) 1.35769e31 0.0674001
\(385\) 1.16005e30 0.00557465
\(386\) −1.91068e31 −0.0888889
\(387\) 4.77076e31 0.214882
\(388\) −7.66639e31 −0.334345
\(389\) −2.78926e32 −1.17793 −0.588965 0.808158i \(-0.700465\pi\)
−0.588965 + 0.808158i \(0.700465\pi\)
\(390\) 2.85329e30 0.0116691
\(391\) −5.82600e31 −0.230760
\(392\) 9.21229e31 0.353420
\(393\) 2.40674e32 0.894381
\(394\) 4.70120e31 0.169241
\(395\) −6.13759e31 −0.214059
\(396\) −3.98043e31 −0.134505
\(397\) 4.22963e32 1.38491 0.692455 0.721461i \(-0.256529\pi\)
0.692455 + 0.721461i \(0.256529\pi\)
\(398\) −4.18631e32 −1.32829
\(399\) 6.60056e30 0.0202964
\(400\) 1.67772e31 0.0500000
\(401\) −2.59389e32 −0.749284 −0.374642 0.927169i \(-0.622234\pi\)
−0.374642 + 0.927169i \(0.622234\pi\)
\(402\) 1.43450e32 0.401674
\(403\) 1.30194e30 0.00353408
\(404\) −2.25611e32 −0.593735
\(405\) 7.12132e31 0.181708
\(406\) 2.55938e30 0.00633226
\(407\) −3.10893e32 −0.745900
\(408\) 9.91820e31 0.230771
\(409\) 7.31347e32 1.65037 0.825186 0.564861i \(-0.191070\pi\)
0.825186 + 0.564861i \(0.191070\pi\)
\(410\) −1.18387e32 −0.259123
\(411\) −1.01381e32 −0.215246
\(412\) −2.44013e32 −0.502571
\(413\) 1.44617e30 0.00288964
\(414\) −4.11540e31 −0.0797825
\(415\) 3.61235e32 0.679499
\(416\) 4.68668e30 0.00855458
\(417\) −3.46252e32 −0.613325
\(418\) −3.62858e32 −0.623782
\(419\) −6.24972e32 −1.04276 −0.521381 0.853324i \(-0.674583\pi\)
−0.521381 + 0.853324i \(0.674583\pi\)
\(420\) −2.04184e30 −0.00330678
\(421\) −2.51168e32 −0.394855 −0.197427 0.980318i \(-0.563259\pi\)
−0.197427 + 0.980318i \(0.563259\pi\)
\(422\) −5.94008e32 −0.906539
\(423\) 2.76142e32 0.409146
\(424\) 2.18616e32 0.314492
\(425\) 1.22561e32 0.171195
\(426\) 3.35980e32 0.455716
\(427\) −2.24453e31 −0.0295650
\(428\) −5.60044e32 −0.716432
\(429\) 1.90899e31 0.0237185
\(430\) −1.34535e32 −0.162360
\(431\) −1.55381e33 −1.82151 −0.910754 0.412948i \(-0.864499\pi\)
−0.910754 + 0.412948i \(0.864499\pi\)
\(432\) 2.37459e32 0.270422
\(433\) 1.08349e33 1.19874 0.599370 0.800472i \(-0.295418\pi\)
0.599370 + 0.800472i \(0.295418\pi\)
\(434\) −9.31674e29 −0.00100148
\(435\) 1.50770e32 0.157471
\(436\) 7.05489e32 0.715993
\(437\) −3.75161e32 −0.369999
\(438\) −3.09414e32 −0.296561
\(439\) 1.66407e33 1.55012 0.775058 0.631890i \(-0.217721\pi\)
0.775058 + 0.631890i \(0.217721\pi\)
\(440\) 1.12247e32 0.101629
\(441\) 4.75380e32 0.418368
\(442\) 3.42371e31 0.0292900
\(443\) 3.77416e32 0.313887 0.156944 0.987608i \(-0.449836\pi\)
0.156944 + 0.987608i \(0.449836\pi\)
\(444\) 5.47211e32 0.442454
\(445\) 4.14449e32 0.325815
\(446\) 1.33553e33 1.02087
\(447\) −1.05879e33 −0.786981
\(448\) −3.35382e30 −0.00242418
\(449\) 2.15921e33 1.51781 0.758903 0.651204i \(-0.225736\pi\)
0.758903 + 0.651204i \(0.225736\pi\)
\(450\) 8.65752e31 0.0591885
\(451\) −7.92063e32 −0.526688
\(452\) 1.27698e33 0.825952
\(453\) −1.19383e32 −0.0751127
\(454\) 1.74166e33 1.06602
\(455\) −7.04832e29 −0.000419704 0
\(456\) 6.38676e32 0.370016
\(457\) 8.65127e30 0.00487671 0.00243836 0.999997i \(-0.499224\pi\)
0.00243836 + 0.999997i \(0.499224\pi\)
\(458\) −2.23682e33 −1.22691
\(459\) 1.73469e33 0.925897
\(460\) 1.16053e32 0.0602817
\(461\) −3.15151e33 −1.59315 −0.796575 0.604539i \(-0.793357\pi\)
−0.796575 + 0.604539i \(0.793357\pi\)
\(462\) −1.36608e31 −0.00672129
\(463\) −2.93719e33 −1.40660 −0.703298 0.710895i \(-0.748290\pi\)
−0.703298 + 0.710895i \(0.748290\pi\)
\(464\) 2.47647e32 0.115441
\(465\) −5.48839e31 −0.0249048
\(466\) 1.02648e33 0.453445
\(467\) −2.28078e33 −0.980895 −0.490448 0.871471i \(-0.663167\pi\)
−0.490448 + 0.871471i \(0.663167\pi\)
\(468\) 2.41846e31 0.0101267
\(469\) −3.54356e31 −0.0144470
\(470\) −7.78716e32 −0.309140
\(471\) −2.31067e33 −0.893257
\(472\) 1.39933e32 0.0526798
\(473\) −9.00099e32 −0.330009
\(474\) 7.22768e32 0.258089
\(475\) 7.89224e32 0.274492
\(476\) −2.45003e31 −0.00830013
\(477\) 1.12812e33 0.372286
\(478\) −2.72328e33 −0.875476
\(479\) −7.97173e32 −0.249666 −0.124833 0.992178i \(-0.539839\pi\)
−0.124833 + 0.992178i \(0.539839\pi\)
\(480\) −1.97570e32 −0.0602844
\(481\) 1.88895e32 0.0561573
\(482\) −1.13686e33 −0.329322
\(483\) −1.41241e31 −0.00398676
\(484\) −1.06677e33 −0.293431
\(485\) 1.11561e33 0.299047
\(486\) 2.08918e33 0.545787
\(487\) −7.32515e32 −0.186511 −0.0932555 0.995642i \(-0.529727\pi\)
−0.0932555 + 0.995642i \(0.529727\pi\)
\(488\) −2.17183e33 −0.538987
\(489\) −3.56143e32 −0.0861516
\(490\) −1.34056e33 −0.316109
\(491\) −2.35651e33 −0.541690 −0.270845 0.962623i \(-0.587303\pi\)
−0.270845 + 0.962623i \(0.587303\pi\)
\(492\) 1.39413e33 0.312421
\(493\) 1.80911e33 0.395257
\(494\) 2.20468e32 0.0469633
\(495\) 5.79229e32 0.120305
\(496\) −9.01496e31 −0.0182576
\(497\) −8.29952e31 −0.0163907
\(498\) −4.25394e33 −0.819265
\(499\) 1.11697e33 0.209789 0.104895 0.994483i \(-0.466549\pi\)
0.104895 + 0.994483i \(0.466549\pi\)
\(500\) −2.44141e32 −0.0447214
\(501\) 5.93041e32 0.105953
\(502\) 6.10366e33 1.06364
\(503\) 3.79303e33 0.644743 0.322372 0.946613i \(-0.395520\pi\)
0.322372 + 0.946613i \(0.395520\pi\)
\(504\) −1.73067e31 −0.00286967
\(505\) 3.28307e33 0.531052
\(506\) 7.76452e32 0.122527
\(507\) 4.94136e33 0.760759
\(508\) −4.19881e33 −0.630710
\(509\) −7.62370e33 −1.11736 −0.558681 0.829383i \(-0.688692\pi\)
−0.558681 + 0.829383i \(0.688692\pi\)
\(510\) −1.44329e33 −0.206408
\(511\) 7.64328e31 0.0106664
\(512\) −3.24519e32 −0.0441942
\(513\) 1.11704e34 1.48458
\(514\) 2.98720e33 0.387458
\(515\) 3.55085e33 0.449513
\(516\) 1.58429e33 0.195755
\(517\) −5.20997e33 −0.628353
\(518\) −1.35174e32 −0.0159137
\(519\) 5.29563e33 0.608592
\(520\) −6.82002e31 −0.00765145
\(521\) 5.33662e33 0.584514 0.292257 0.956340i \(-0.405594\pi\)
0.292257 + 0.956340i \(0.405594\pi\)
\(522\) 1.27793e33 0.136655
\(523\) 1.14894e34 1.19957 0.599786 0.800160i \(-0.295252\pi\)
0.599786 + 0.800160i \(0.295252\pi\)
\(524\) −5.75265e33 −0.586445
\(525\) 2.97126e31 0.00295767
\(526\) 2.40992e33 0.234251
\(527\) −6.58561e32 −0.0625120
\(528\) −1.32183e33 −0.122533
\(529\) −1.02430e34 −0.927322
\(530\) −3.18129e33 −0.281290
\(531\) 7.22092e32 0.0623607
\(532\) −1.57768e32 −0.0133084
\(533\) 4.81248e32 0.0396533
\(534\) −4.88059e33 −0.392831
\(535\) 8.14971e33 0.640796
\(536\) −3.42877e33 −0.263378
\(537\) 3.83520e33 0.287812
\(538\) −5.01403e33 −0.367628
\(539\) −8.96900e33 −0.642517
\(540\) −3.45549e33 −0.241873
\(541\) −2.08461e34 −1.42580 −0.712901 0.701265i \(-0.752619\pi\)
−0.712901 + 0.701265i \(0.752619\pi\)
\(542\) 6.29167e33 0.420508
\(543\) 1.92569e34 1.25773
\(544\) −2.37067e33 −0.151316
\(545\) −1.02662e34 −0.640404
\(546\) 8.30016e30 0.000506033 0
\(547\) −5.82664e33 −0.347197 −0.173599 0.984816i \(-0.555540\pi\)
−0.173599 + 0.984816i \(0.555540\pi\)
\(548\) 2.42324e33 0.141137
\(549\) −1.12073e34 −0.638036
\(550\) −1.63341e33 −0.0908998
\(551\) 1.16497e34 0.633751
\(552\) −1.36666e33 −0.0726809
\(553\) −1.78541e32 −0.00928270
\(554\) 1.77206e33 0.0900754
\(555\) −7.96297e33 −0.395743
\(556\) 8.27619e33 0.402157
\(557\) −2.41168e34 −1.14586 −0.572928 0.819606i \(-0.694193\pi\)
−0.572928 + 0.819606i \(0.694193\pi\)
\(558\) −4.65198e32 −0.0216127
\(559\) 5.46889e32 0.0248457
\(560\) 4.88045e31 0.00216825
\(561\) −9.65626e33 −0.419540
\(562\) 1.48958e34 0.632937
\(563\) 1.42755e34 0.593248 0.296624 0.954994i \(-0.404139\pi\)
0.296624 + 0.954994i \(0.404139\pi\)
\(564\) 9.17022e33 0.372727
\(565\) −1.85826e34 −0.738754
\(566\) 3.43609e34 1.33616
\(567\) 2.07158e32 0.00787975
\(568\) −8.03069e33 −0.298812
\(569\) −4.47698e34 −1.62960 −0.814802 0.579739i \(-0.803154\pi\)
−0.814802 + 0.579739i \(0.803154\pi\)
\(570\) −9.29396e33 −0.330952
\(571\) −1.37949e33 −0.0480581 −0.0240291 0.999711i \(-0.507649\pi\)
−0.0240291 + 0.999711i \(0.507649\pi\)
\(572\) −4.56291e32 −0.0155522
\(573\) 1.77960e34 0.593456
\(574\) −3.44385e32 −0.0112369
\(575\) −1.68880e33 −0.0539176
\(576\) −1.67461e33 −0.0523157
\(577\) −2.32241e33 −0.0709973 −0.0354986 0.999370i \(-0.511302\pi\)
−0.0354986 + 0.999370i \(0.511302\pi\)
\(578\) 6.31825e33 0.189016
\(579\) −3.27424e33 −0.0958579
\(580\) −3.60374e33 −0.103253
\(581\) 1.05082e33 0.0294665
\(582\) −1.31375e34 −0.360558
\(583\) −2.12843e34 −0.571745
\(584\) 7.39570e33 0.194455
\(585\) −3.51932e32 −0.00905755
\(586\) 3.56627e34 0.898451
\(587\) 4.35047e34 1.07290 0.536451 0.843931i \(-0.319765\pi\)
0.536451 + 0.843931i \(0.319765\pi\)
\(588\) 1.57866e34 0.381129
\(589\) −4.24076e33 −0.100231
\(590\) −2.03629e33 −0.0471182
\(591\) 8.05619e33 0.182510
\(592\) −1.30796e34 −0.290117
\(593\) 5.31713e34 1.15476 0.577382 0.816474i \(-0.304074\pi\)
0.577382 + 0.816474i \(0.304074\pi\)
\(594\) −2.31188e34 −0.491626
\(595\) 3.56527e32 0.00742387
\(596\) 2.53074e34 0.516023
\(597\) −7.17385e34 −1.43243
\(598\) −4.71763e32 −0.00922484
\(599\) 7.13123e34 1.36562 0.682809 0.730597i \(-0.260758\pi\)
0.682809 + 0.730597i \(0.260758\pi\)
\(600\) 2.87502e33 0.0539200
\(601\) 3.07608e34 0.565023 0.282512 0.959264i \(-0.408832\pi\)
0.282512 + 0.959264i \(0.408832\pi\)
\(602\) −3.91358e32 −0.00704073
\(603\) −1.76935e34 −0.311779
\(604\) 2.85351e33 0.0492513
\(605\) 1.55236e34 0.262452
\(606\) −3.86617e34 −0.640284
\(607\) 2.76184e34 0.448064 0.224032 0.974582i \(-0.428078\pi\)
0.224032 + 0.974582i \(0.428078\pi\)
\(608\) −1.52658e34 −0.242619
\(609\) 4.38586e32 0.00682871
\(610\) 3.16043e34 0.482084
\(611\) 3.16552e33 0.0473074
\(612\) −1.22333e34 −0.179124
\(613\) 4.94974e34 0.710112 0.355056 0.934845i \(-0.384462\pi\)
0.355056 + 0.934845i \(0.384462\pi\)
\(614\) −1.32848e34 −0.186746
\(615\) −2.02873e34 −0.279438
\(616\) 3.26525e32 0.00440715
\(617\) 1.12825e35 1.49225 0.746126 0.665805i \(-0.231912\pi\)
0.746126 + 0.665805i \(0.231912\pi\)
\(618\) −4.18151e34 −0.541973
\(619\) 5.35184e34 0.679782 0.339891 0.940465i \(-0.389610\pi\)
0.339891 + 0.940465i \(0.389610\pi\)
\(620\) 1.31185e33 0.0163301
\(621\) −2.39027e34 −0.291610
\(622\) 5.09934e33 0.0609725
\(623\) 1.20562e33 0.0141290
\(624\) 8.03131e32 0.00922527
\(625\) 3.55271e33 0.0400000
\(626\) −9.56089e34 −1.05516
\(627\) −6.21809e34 −0.672687
\(628\) 5.52302e34 0.585708
\(629\) −9.55490e34 −0.993329
\(630\) 2.51845e32 0.00256671
\(631\) −9.73214e34 −0.972392 −0.486196 0.873850i \(-0.661616\pi\)
−0.486196 + 0.873850i \(0.661616\pi\)
\(632\) −1.72758e34 −0.169229
\(633\) −1.01792e35 −0.977612
\(634\) −8.66087e34 −0.815541
\(635\) 6.11007e34 0.564124
\(636\) 3.74631e34 0.339148
\(637\) 5.44946e33 0.0483738
\(638\) −2.41107e34 −0.209871
\(639\) −4.14406e34 −0.353725
\(640\) 4.72237e33 0.0395285
\(641\) −2.79912e34 −0.229772 −0.114886 0.993379i \(-0.536650\pi\)
−0.114886 + 0.993379i \(0.536650\pi\)
\(642\) −9.59717e34 −0.772601
\(643\) −1.80728e35 −1.42688 −0.713442 0.700714i \(-0.752865\pi\)
−0.713442 + 0.700714i \(0.752865\pi\)
\(644\) 3.37597e32 0.00261412
\(645\) −2.30545e34 −0.175089
\(646\) −1.11520e35 −0.830702
\(647\) 7.31933e34 0.534771 0.267385 0.963590i \(-0.413840\pi\)
0.267385 + 0.963590i \(0.413840\pi\)
\(648\) 2.00447e34 0.143652
\(649\) −1.36237e34 −0.0957716
\(650\) 9.92443e32 0.00684367
\(651\) −1.59656e32 −0.00108000
\(652\) 8.51262e33 0.0564896
\(653\) −7.32271e32 −0.00476713 −0.00238357 0.999997i \(-0.500759\pi\)
−0.00238357 + 0.999997i \(0.500759\pi\)
\(654\) 1.20896e35 0.772128
\(655\) 8.37121e34 0.524532
\(656\) −3.33229e34 −0.204854
\(657\) 3.81639e34 0.230190
\(658\) −2.26526e33 −0.0134059
\(659\) 3.37009e35 1.95692 0.978460 0.206434i \(-0.0661860\pi\)
0.978460 + 0.206434i \(0.0661860\pi\)
\(660\) 1.92352e34 0.109597
\(661\) 1.22749e34 0.0686275 0.0343138 0.999411i \(-0.489075\pi\)
0.0343138 + 0.999411i \(0.489075\pi\)
\(662\) 1.78326e35 0.978343
\(663\) 5.86703e33 0.0315863
\(664\) 1.01679e35 0.537191
\(665\) 2.29583e33 0.0119034
\(666\) −6.74944e34 −0.343431
\(667\) −2.49283e34 −0.124486
\(668\) −1.41750e34 −0.0694733
\(669\) 2.28863e35 1.10090
\(670\) 4.98952e34 0.235572
\(671\) 2.11447e35 0.979875
\(672\) −5.74726e32 −0.00261424
\(673\) −4.17789e35 −1.86538 −0.932692 0.360673i \(-0.882547\pi\)
−0.932692 + 0.360673i \(0.882547\pi\)
\(674\) 2.44634e34 0.107218
\(675\) 5.02840e34 0.216338
\(676\) −1.18110e35 −0.498829
\(677\) 2.87478e35 1.19192 0.595959 0.803014i \(-0.296772\pi\)
0.595959 + 0.803014i \(0.296772\pi\)
\(678\) 2.18830e35 0.890707
\(679\) 3.24527e33 0.0129682
\(680\) 3.44978e34 0.135341
\(681\) 2.98459e35 1.14960
\(682\) 8.77688e33 0.0331922
\(683\) −3.52227e35 −1.30787 −0.653935 0.756551i \(-0.726883\pi\)
−0.653935 + 0.756551i \(0.726883\pi\)
\(684\) −7.87759e34 −0.287205
\(685\) −3.52629e34 −0.126237
\(686\) −7.80081e33 −0.0274213
\(687\) −3.83311e35 −1.32310
\(688\) −3.78681e34 −0.128357
\(689\) 1.29321e34 0.0430455
\(690\) 1.98875e34 0.0650078
\(691\) −1.41869e35 −0.455421 −0.227711 0.973729i \(-0.573124\pi\)
−0.227711 + 0.973729i \(0.573124\pi\)
\(692\) −1.26578e35 −0.399053
\(693\) 1.68496e33 0.00521705
\(694\) 3.97896e35 1.20997
\(695\) −1.20434e35 −0.359700
\(696\) 4.24380e34 0.124491
\(697\) −2.43431e35 −0.701400
\(698\) 2.95233e35 0.835550
\(699\) 1.75901e35 0.488995
\(700\) −7.10199e32 −0.00193934
\(701\) 4.11949e35 1.10502 0.552508 0.833508i \(-0.313671\pi\)
0.552508 + 0.833508i \(0.313671\pi\)
\(702\) 1.40467e34 0.0370136
\(703\) −6.15282e35 −1.59270
\(704\) 3.15948e34 0.0803448
\(705\) −1.33444e35 −0.333377
\(706\) −2.85986e35 −0.701917
\(707\) 9.55036e33 0.0230291
\(708\) 2.39795e34 0.0568099
\(709\) −3.69515e35 −0.860110 −0.430055 0.902803i \(-0.641506\pi\)
−0.430055 + 0.902803i \(0.641506\pi\)
\(710\) 1.16862e35 0.267266
\(711\) −8.91480e34 −0.200328
\(712\) 1.16657e35 0.257579
\(713\) 9.07449e33 0.0196881
\(714\) −4.19849e33 −0.00895087
\(715\) 6.63991e33 0.0139103
\(716\) −9.16700e34 −0.188718
\(717\) −4.66674e35 −0.944114
\(718\) 5.48333e35 1.09016
\(719\) 2.40654e35 0.470200 0.235100 0.971971i \(-0.424458\pi\)
0.235100 + 0.971971i \(0.424458\pi\)
\(720\) 2.43688e34 0.0467926
\(721\) 1.03293e34 0.0194931
\(722\) −3.36883e35 −0.624834
\(723\) −1.94818e35 −0.355141
\(724\) −4.60282e35 −0.824693
\(725\) 5.24414e34 0.0923526
\(726\) −1.82807e35 −0.316436
\(727\) −2.82511e34 −0.0480680 −0.0240340 0.999711i \(-0.507651\pi\)
−0.0240340 + 0.999711i \(0.507651\pi\)
\(728\) −1.98393e32 −0.000331805 0
\(729\) 6.05157e35 0.994888
\(730\) −1.07622e35 −0.173926
\(731\) −2.76634e35 −0.439479
\(732\) −3.72175e35 −0.581244
\(733\) 1.13647e36 1.74486 0.872428 0.488743i \(-0.162544\pi\)
0.872428 + 0.488743i \(0.162544\pi\)
\(734\) −2.71963e35 −0.410495
\(735\) −2.29725e35 −0.340892
\(736\) 3.26662e34 0.0476568
\(737\) 3.33822e35 0.478819
\(738\) −1.71956e35 −0.242500
\(739\) −7.59548e35 −1.05317 −0.526587 0.850121i \(-0.676528\pi\)
−0.526587 + 0.850121i \(0.676528\pi\)
\(740\) 1.90333e35 0.259488
\(741\) 3.77804e34 0.0506453
\(742\) −9.25428e33 −0.0121982
\(743\) −2.06273e35 −0.267352 −0.133676 0.991025i \(-0.542678\pi\)
−0.133676 + 0.991025i \(0.542678\pi\)
\(744\) −1.54484e34 −0.0196890
\(745\) −3.68271e35 −0.461545
\(746\) −8.95193e35 −1.10327
\(747\) 5.24691e35 0.635911
\(748\) 2.30807e35 0.275092
\(749\) 2.37073e34 0.0277881
\(750\) −4.18370e34 −0.0482276
\(751\) −7.61805e34 −0.0863664 −0.0431832 0.999067i \(-0.513750\pi\)
−0.0431832 + 0.999067i \(0.513750\pi\)
\(752\) −2.19189e35 −0.244397
\(753\) 1.04595e36 1.14703
\(754\) 1.46494e34 0.0158008
\(755\) −4.15241e34 −0.0440517
\(756\) −1.00519e34 −0.0104888
\(757\) 1.93605e36 1.98709 0.993546 0.113428i \(-0.0361831\pi\)
0.993546 + 0.113428i \(0.0361831\pi\)
\(758\) 6.09530e35 0.615362
\(759\) 1.33056e35 0.132134
\(760\) 2.22147e35 0.217005
\(761\) 1.19339e36 1.14677 0.573383 0.819288i \(-0.305631\pi\)
0.573383 + 0.819288i \(0.305631\pi\)
\(762\) −7.19527e35 −0.680158
\(763\) −2.98642e34 −0.0277711
\(764\) −4.25364e35 −0.389129
\(765\) 1.78019e35 0.160213
\(766\) 8.57374e35 0.759121
\(767\) 8.27761e33 0.00721046
\(768\) −5.56110e34 −0.0476590
\(769\) −9.68810e35 −0.816882 −0.408441 0.912785i \(-0.633927\pi\)
−0.408441 + 0.912785i \(0.633927\pi\)
\(770\) −4.75156e33 −0.00394187
\(771\) 5.11899e35 0.417835
\(772\) 7.82616e34 0.0628540
\(773\) −8.25175e35 −0.652082 −0.326041 0.945356i \(-0.605715\pi\)
−0.326041 + 0.945356i \(0.605715\pi\)
\(774\) −1.95410e35 −0.151945
\(775\) −1.90899e34 −0.0146060
\(776\) 3.14015e35 0.236417
\(777\) −2.31641e34 −0.0171614
\(778\) 1.14248e36 0.832923
\(779\) −1.56756e36 −1.12462
\(780\) −1.16871e34 −0.00825133
\(781\) 7.81861e35 0.543240
\(782\) 2.38633e35 0.163172
\(783\) 7.42239e35 0.499484
\(784\) −3.77335e35 −0.249906
\(785\) −8.03706e35 −0.523873
\(786\) −9.85801e35 −0.632423
\(787\) 8.02076e35 0.506444 0.253222 0.967408i \(-0.418510\pi\)
0.253222 + 0.967408i \(0.418510\pi\)
\(788\) −1.92561e35 −0.119672
\(789\) 4.12975e35 0.252616
\(790\) 2.51396e35 0.151363
\(791\) −5.40562e34 −0.0320361
\(792\) 1.63038e35 0.0951097
\(793\) −1.28473e35 −0.0737729
\(794\) −1.73246e36 −0.979279
\(795\) −5.45160e35 −0.303343
\(796\) 1.71471e36 0.939243
\(797\) 8.65699e35 0.466807 0.233404 0.972380i \(-0.425014\pi\)
0.233404 + 0.972380i \(0.425014\pi\)
\(798\) −2.70359e34 −0.0143518
\(799\) −1.60122e36 −0.836789
\(800\) −6.87195e34 −0.0353553
\(801\) 6.01984e35 0.304915
\(802\) 1.06246e36 0.529824
\(803\) −7.20038e35 −0.353518
\(804\) −5.87570e35 −0.284027
\(805\) −4.91268e33 −0.00233814
\(806\) −5.33273e33 −0.00249897
\(807\) −8.59226e35 −0.396450
\(808\) 9.24101e35 0.419834
\(809\) −3.72457e36 −1.66617 −0.833086 0.553144i \(-0.813428\pi\)
−0.833086 + 0.553144i \(0.813428\pi\)
\(810\) −2.91689e35 −0.128487
\(811\) 6.42225e35 0.278565 0.139282 0.990253i \(-0.455520\pi\)
0.139282 + 0.990253i \(0.455520\pi\)
\(812\) −1.04832e34 −0.00447758
\(813\) 1.07817e36 0.453476
\(814\) 1.27342e36 0.527431
\(815\) −1.23875e35 −0.0505258
\(816\) −4.06249e35 −0.163180
\(817\) −1.78137e36 −0.704657
\(818\) −2.99560e36 −1.16699
\(819\) −1.02376e33 −0.000392781 0
\(820\) 4.84913e35 0.183227
\(821\) −2.68244e36 −0.998254 −0.499127 0.866529i \(-0.666346\pi\)
−0.499127 + 0.866529i \(0.666346\pi\)
\(822\) 4.15258e35 0.152202
\(823\) −8.11664e35 −0.293007 −0.146504 0.989210i \(-0.546802\pi\)
−0.146504 + 0.989210i \(0.546802\pi\)
\(824\) 9.99476e35 0.355371
\(825\) −2.79909e35 −0.0980264
\(826\) −5.92351e33 −0.00204328
\(827\) −1.42939e36 −0.485661 −0.242830 0.970069i \(-0.578076\pi\)
−0.242830 + 0.970069i \(0.578076\pi\)
\(828\) 1.68567e35 0.0564147
\(829\) −1.47702e36 −0.486915 −0.243458 0.969912i \(-0.578282\pi\)
−0.243458 + 0.969912i \(0.578282\pi\)
\(830\) −1.47962e36 −0.480478
\(831\) 3.03668e35 0.0971374
\(832\) −1.91966e34 −0.00604900
\(833\) −2.75651e36 −0.855652
\(834\) 1.41825e36 0.433687
\(835\) 2.06274e35 0.0621388
\(836\) 1.48626e36 0.441080
\(837\) −2.70193e35 −0.0789960
\(838\) 2.55989e36 0.737344
\(839\) 3.59874e36 1.02124 0.510618 0.859808i \(-0.329417\pi\)
0.510618 + 0.859808i \(0.329417\pi\)
\(840\) 8.36336e33 0.00233824
\(841\) −2.85628e36 −0.786775
\(842\) 1.02878e36 0.279204
\(843\) 2.55261e36 0.682559
\(844\) 2.43306e36 0.641020
\(845\) 1.71872e36 0.446166
\(846\) −1.13108e36 −0.289310
\(847\) 4.51578e34 0.0113813
\(848\) −8.95452e35 −0.222379
\(849\) 5.88823e36 1.44092
\(850\) −5.02010e35 −0.121053
\(851\) 1.31660e36 0.312848
\(852\) −1.37618e36 −0.322240
\(853\) 2.51059e36 0.579312 0.289656 0.957131i \(-0.406459\pi\)
0.289656 + 0.957131i \(0.406459\pi\)
\(854\) 9.19361e34 0.0209056
\(855\) 1.14634e36 0.256884
\(856\) 2.29394e36 0.506594
\(857\) −1.62785e36 −0.354286 −0.177143 0.984185i \(-0.556686\pi\)
−0.177143 + 0.984185i \(0.556686\pi\)
\(858\) −7.81921e34 −0.0167715
\(859\) 1.44920e36 0.306348 0.153174 0.988199i \(-0.451051\pi\)
0.153174 + 0.988199i \(0.451051\pi\)
\(860\) 5.51054e35 0.114806
\(861\) −5.90153e34 −0.0121178
\(862\) 6.36440e36 1.28800
\(863\) 5.15478e36 1.02819 0.514097 0.857732i \(-0.328127\pi\)
0.514097 + 0.857732i \(0.328127\pi\)
\(864\) −9.72634e35 −0.191217
\(865\) 1.84195e36 0.356924
\(866\) −4.43796e36 −0.847637
\(867\) 1.08272e36 0.203835
\(868\) 3.81614e33 0.000708154 0
\(869\) 1.68196e36 0.307657
\(870\) −6.17554e35 −0.111348
\(871\) −2.02826e35 −0.0360494
\(872\) −2.88968e36 −0.506284
\(873\) 1.62041e36 0.279864
\(874\) 1.53666e36 0.261629
\(875\) 1.03348e34 0.00173460
\(876\) 1.26736e36 0.209700
\(877\) −1.01431e37 −1.65453 −0.827266 0.561810i \(-0.810105\pi\)
−0.827266 + 0.561810i \(0.810105\pi\)
\(878\) −6.81602e36 −1.09610
\(879\) 6.11132e36 0.968890
\(880\) −4.59765e35 −0.0718626
\(881\) −3.63479e36 −0.560120 −0.280060 0.959983i \(-0.590354\pi\)
−0.280060 + 0.959983i \(0.590354\pi\)
\(882\) −1.94716e36 −0.295831
\(883\) 4.65240e36 0.696896 0.348448 0.937328i \(-0.386709\pi\)
0.348448 + 0.937328i \(0.386709\pi\)
\(884\) −1.40235e35 −0.0207111
\(885\) −3.48947e35 −0.0508123
\(886\) −1.54589e36 −0.221952
\(887\) 9.05035e35 0.128121 0.0640604 0.997946i \(-0.479595\pi\)
0.0640604 + 0.997946i \(0.479595\pi\)
\(888\) −2.24138e36 −0.312862
\(889\) 1.77741e35 0.0244633
\(890\) −1.69758e36 −0.230386
\(891\) −1.95154e36 −0.261159
\(892\) −5.47034e36 −0.721861
\(893\) −1.03110e37 −1.34170
\(894\) 4.33679e36 0.556480
\(895\) 1.33397e36 0.168795
\(896\) 1.37373e34 0.00171415
\(897\) −8.08434e34 −0.00994808
\(898\) −8.84413e36 −1.07325
\(899\) −2.81785e35 −0.0337227
\(900\) −3.54612e35 −0.0418526
\(901\) −6.54146e36 −0.761403
\(902\) 3.24429e36 0.372424
\(903\) −6.70648e34 −0.00759273
\(904\) −5.23052e36 −0.584036
\(905\) 6.69799e36 0.737628
\(906\) 4.88991e35 0.0531127
\(907\) −3.58423e36 −0.383977 −0.191988 0.981397i \(-0.561494\pi\)
−0.191988 + 0.981397i \(0.561494\pi\)
\(908\) −7.13385e36 −0.753791
\(909\) 4.76862e36 0.496986
\(910\) 2.88699e33 0.000296776 0
\(911\) 5.15388e36 0.522582 0.261291 0.965260i \(-0.415852\pi\)
0.261291 + 0.965260i \(0.415852\pi\)
\(912\) −2.61602e36 −0.261641
\(913\) −9.89934e36 −0.976611
\(914\) −3.54356e34 −0.00344836
\(915\) 5.41585e36 0.519880
\(916\) 9.16200e36 0.867555
\(917\) 2.43517e35 0.0227464
\(918\) −7.10528e36 −0.654708
\(919\) 1.45442e36 0.132205 0.0661023 0.997813i \(-0.478944\pi\)
0.0661023 + 0.997813i \(0.478944\pi\)
\(920\) −4.75355e35 −0.0426256
\(921\) −2.27655e36 −0.201387
\(922\) 1.29086e37 1.12653
\(923\) −4.75049e35 −0.0408994
\(924\) 5.59548e34 0.00475267
\(925\) −2.76971e36 −0.232093
\(926\) 1.20307e37 0.994614
\(927\) 5.15758e36 0.420678
\(928\) −1.01436e36 −0.0816289
\(929\) 1.48492e37 1.17898 0.589491 0.807775i \(-0.299328\pi\)
0.589491 + 0.807775i \(0.299328\pi\)
\(930\) 2.24804e35 0.0176103
\(931\) −1.77504e37 −1.37194
\(932\) −4.20444e36 −0.320634
\(933\) 8.73845e35 0.0657528
\(934\) 9.34208e36 0.693598
\(935\) −3.35868e36 −0.246050
\(936\) −9.90602e34 −0.00716063
\(937\) 4.13990e36 0.295288 0.147644 0.989041i \(-0.452831\pi\)
0.147644 + 0.989041i \(0.452831\pi\)
\(938\) 1.45144e35 0.0102156
\(939\) −1.63840e37 −1.13789
\(940\) 3.18962e36 0.218595
\(941\) 3.04574e36 0.205979 0.102989 0.994682i \(-0.467159\pi\)
0.102989 + 0.994682i \(0.467159\pi\)
\(942\) 9.46450e36 0.631628
\(943\) 3.35430e36 0.220905
\(944\) −5.73164e35 −0.0372502
\(945\) 1.46275e35 0.00938150
\(946\) 3.68680e36 0.233352
\(947\) 1.93836e37 1.21076 0.605382 0.795935i \(-0.293020\pi\)
0.605382 + 0.795935i \(0.293020\pi\)
\(948\) −2.96046e36 −0.182497
\(949\) 4.37487e35 0.0266157
\(950\) −3.23266e36 −0.194095
\(951\) −1.48417e37 −0.879480
\(952\) 1.00353e35 0.00586908
\(953\) −7.72509e35 −0.0445905 −0.0222952 0.999751i \(-0.507097\pi\)
−0.0222952 + 0.999751i \(0.507097\pi\)
\(954\) −4.62079e36 −0.263246
\(955\) 6.18986e36 0.348048
\(956\) 1.11546e37 0.619055
\(957\) −4.13172e36 −0.226325
\(958\) 3.26522e36 0.176540
\(959\) −1.02579e35 −0.00547425
\(960\) 8.09246e35 0.0426275
\(961\) −1.91302e37 −0.994667
\(962\) −7.73713e35 −0.0397092
\(963\) 1.18374e37 0.599690
\(964\) 4.65659e36 0.232866
\(965\) −1.13886e36 −0.0562183
\(966\) 5.78521e34 0.00281907
\(967\) −6.95372e36 −0.334492 −0.167246 0.985915i \(-0.553487\pi\)
−0.167246 + 0.985915i \(0.553487\pi\)
\(968\) 4.36951e36 0.207487
\(969\) −1.91105e37 −0.895830
\(970\) −4.56953e36 −0.211458
\(971\) 2.67539e37 1.22221 0.611106 0.791549i \(-0.290725\pi\)
0.611106 + 0.791549i \(0.290725\pi\)
\(972\) −8.55729e36 −0.385930
\(973\) −3.50341e35 −0.0155984
\(974\) 3.00038e36 0.131883
\(975\) 1.70070e35 0.00738021
\(976\) 8.89581e36 0.381121
\(977\) 2.82491e37 1.19488 0.597438 0.801915i \(-0.296185\pi\)
0.597438 + 0.801915i \(0.296185\pi\)
\(978\) 1.45876e36 0.0609184
\(979\) −1.13576e37 −0.468278
\(980\) 5.49095e36 0.223523
\(981\) −1.49116e37 −0.599323
\(982\) 9.65226e36 0.383032
\(983\) 3.52532e37 1.38127 0.690636 0.723203i \(-0.257331\pi\)
0.690636 + 0.723203i \(0.257331\pi\)
\(984\) −5.71037e36 −0.220915
\(985\) 2.80213e36 0.107037
\(986\) −7.41013e36 −0.279489
\(987\) −3.88186e35 −0.0144569
\(988\) −9.03037e35 −0.0332081
\(989\) 3.81182e36 0.138413
\(990\) −2.37252e36 −0.0850687
\(991\) −1.01549e36 −0.0359547 −0.0179774 0.999838i \(-0.505723\pi\)
−0.0179774 + 0.999838i \(0.505723\pi\)
\(992\) 3.69253e35 0.0129100
\(993\) 3.05588e37 1.05505
\(994\) 3.39948e35 0.0115900
\(995\) −2.49523e37 −0.840084
\(996\) 1.74241e37 0.579308
\(997\) −5.04007e37 −1.65481 −0.827403 0.561608i \(-0.810183\pi\)
−0.827403 + 0.561608i \(0.810183\pi\)
\(998\) −4.57509e36 −0.148343
\(999\) −3.92016e37 −1.25526
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 10.26.a.c.1.1 2
5.2 odd 4 50.26.b.c.49.2 4
5.3 odd 4 50.26.b.c.49.3 4
5.4 even 2 50.26.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.26.a.c.1.1 2 1.1 even 1 trivial
50.26.a.e.1.2 2 5.4 even 2
50.26.b.c.49.2 4 5.2 odd 4
50.26.b.c.49.3 4 5.3 odd 4