Properties

Label 169.10.a.a.1.1
Level $169$
Weight $10$
Character 169.1
Self dual yes
Analytic conductor $87.041$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,10,Mod(1,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 169.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: \(87.0410563117\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 1602x^{2} + 1544x + 342272 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-36.8028\) of defining polynomial
Character \(\chi\) \(=\) 169.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-28.8028 q^{2} -204.594 q^{3} +317.603 q^{4} +258.914 q^{5} +5892.88 q^{6} +8862.28 q^{7} +5599.18 q^{8} +22175.6 q^{9} +O(q^{10})\) \(q-28.8028 q^{2} -204.594 q^{3} +317.603 q^{4} +258.914 q^{5} +5892.88 q^{6} +8862.28 q^{7} +5599.18 q^{8} +22175.6 q^{9} -7457.46 q^{10} -36087.9 q^{11} -64979.6 q^{12} -255259. q^{14} -52972.2 q^{15} -323885. q^{16} +327405. q^{17} -638720. q^{18} -265525. q^{19} +82231.9 q^{20} -1.81317e6 q^{21} +1.03943e6 q^{22} -2.42458e6 q^{23} -1.14556e6 q^{24} -1.88609e6 q^{25} -509973. q^{27} +2.81469e6 q^{28} +3.99178e6 q^{29} +1.52575e6 q^{30} +6.45220e6 q^{31} +6.46202e6 q^{32} +7.38336e6 q^{33} -9.43018e6 q^{34} +2.29457e6 q^{35} +7.04304e6 q^{36} +8.15498e6 q^{37} +7.64787e6 q^{38} +1.44971e6 q^{40} +720241. q^{41} +5.22244e7 q^{42} -4.13245e7 q^{43} -1.14616e7 q^{44} +5.74158e6 q^{45} +6.98348e7 q^{46} -3.67369e7 q^{47} +6.62649e7 q^{48} +3.81864e7 q^{49} +5.43247e7 q^{50} -6.69850e7 q^{51} +1.67334e7 q^{53} +1.46887e7 q^{54} -9.34367e6 q^{55} +4.96215e7 q^{56} +5.43248e7 q^{57} -1.14975e8 q^{58} +5.89865e7 q^{59} -1.68241e7 q^{60} +1.28785e8 q^{61} -1.85842e8 q^{62} +1.96527e8 q^{63} -2.02955e7 q^{64} -2.12662e8 q^{66} +1.91470e8 q^{67} +1.03985e8 q^{68} +4.96054e8 q^{69} -6.60901e7 q^{70} -3.40865e8 q^{71} +1.24165e8 q^{72} -3.19890e8 q^{73} -2.34887e8 q^{74} +3.85882e8 q^{75} -8.43316e7 q^{76} -3.19821e8 q^{77} +2.44328e8 q^{79} -8.38584e7 q^{80} -3.32145e8 q^{81} -2.07450e7 q^{82} +4.38630e8 q^{83} -5.75868e8 q^{84} +8.47697e7 q^{85} +1.19026e9 q^{86} -8.16693e8 q^{87} -2.02063e8 q^{88} +7.90342e8 q^{89} -1.65374e8 q^{90} -7.70054e8 q^{92} -1.32008e9 q^{93} +1.05813e9 q^{94} -6.87481e7 q^{95} -1.32209e9 q^{96} +1.65696e8 q^{97} -1.09988e9 q^{98} -8.00272e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 33 q^{2} - 163 q^{3} + 1429 q^{4} - 471 q^{5} + 4529 q^{6} + 11241 q^{7} + 45543 q^{8} - 29953 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 33 q^{2} - 163 q^{3} + 1429 q^{4} - 471 q^{5} + 4529 q^{6} + 11241 q^{7} + 45543 q^{8} - 29953 q^{9} - 67831 q^{10} + 40140 q^{11} - 155479 q^{12} - 277653 q^{14} - 83307 q^{15} + 726609 q^{16} + 78717 q^{17} - 1691026 q^{18} - 209664 q^{19} - 870843 q^{20} - 1138431 q^{21} + 1364090 q^{22} - 4257444 q^{23} - 3561573 q^{24} - 2900157 q^{25} - 2077801 q^{27} - 4035181 q^{28} - 1647936 q^{29} - 744143 q^{30} + 11366002 q^{31} + 29458959 q^{32} + 14413222 q^{33} - 26257659 q^{34} - 13789797 q^{35} - 11587714 q^{36} - 4636891 q^{37} + 25172466 q^{38} + 22536791 q^{40} - 13859538 q^{41} + 75564923 q^{42} - 33368081 q^{43} - 66489222 q^{44} + 17423928 q^{45} - 71369332 q^{46} + 3943005 q^{47} - 620787 q^{48} + 23294923 q^{49} + 4217748 q^{50} - 19664471 q^{51} - 171019326 q^{53} + 64946915 q^{54} - 121160538 q^{55} - 281552967 q^{56} + 47829030 q^{57} - 79964734 q^{58} + 63389388 q^{59} - 37708135 q^{60} + 77050190 q^{61} - 95878740 q^{62} + 155695476 q^{63} + 768962465 q^{64} - 42396374 q^{66} + 41174072 q^{67} - 717615423 q^{68} + 546642556 q^{69} - 409056389 q^{70} - 252460989 q^{71} - 562579254 q^{72} - 594415068 q^{73} - 957058539 q^{74} + 533318748 q^{75} + 326897170 q^{76} + 561950454 q^{77} + 115998984 q^{79} + 509107233 q^{80} + 437803700 q^{81} - 875148240 q^{82} + 79577862 q^{83} - 108899441 q^{84} - 549463469 q^{85} + 589924887 q^{86} - 1087526510 q^{87} - 2327564370 q^{88} + 1152240276 q^{89} + 877550038 q^{90} - 4213481460 q^{92} - 1618266556 q^{93} + 1859909503 q^{94} - 1273705170 q^{95} - 3171454029 q^{96} - 1049098084 q^{97} - 420532254 q^{98} - 2132181050 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −28.8028 −1.27292 −0.636459 0.771311i \(-0.719601\pi\)
−0.636459 + 0.771311i \(0.719601\pi\)
\(3\) −204.594 −1.45830 −0.729150 0.684354i \(-0.760084\pi\)
−0.729150 + 0.684354i \(0.760084\pi\)
\(4\) 317.603 0.620318
\(5\) 258.914 0.185264 0.0926319 0.995700i \(-0.470472\pi\)
0.0926319 + 0.995700i \(0.470472\pi\)
\(6\) 5892.88 1.85629
\(7\) 8862.28 1.39510 0.697548 0.716538i \(-0.254274\pi\)
0.697548 + 0.716538i \(0.254274\pi\)
\(8\) 5599.18 0.483303
\(9\) 22175.6 1.12664
\(10\) −7457.46 −0.235825
\(11\) −36087.9 −0.743181 −0.371591 0.928397i \(-0.621188\pi\)
−0.371591 + 0.928397i \(0.621188\pi\)
\(12\) −64979.6 −0.904610
\(13\) 0 0
\(14\) −255259. −1.77584
\(15\) −52972.2 −0.270170
\(16\) −323885. −1.23552
\(17\) 327405. 0.950747 0.475373 0.879784i \(-0.342313\pi\)
0.475373 + 0.879784i \(0.342313\pi\)
\(18\) −638720. −1.43412
\(19\) −265525. −0.467428 −0.233714 0.972305i \(-0.575088\pi\)
−0.233714 + 0.972305i \(0.575088\pi\)
\(20\) 82231.9 0.114923
\(21\) −1.81317e6 −2.03447
\(22\) 1.03943e6 0.946008
\(23\) −2.42458e6 −1.80660 −0.903299 0.429012i \(-0.858862\pi\)
−0.903299 + 0.429012i \(0.858862\pi\)
\(24\) −1.14556e6 −0.704801
\(25\) −1.88609e6 −0.965677
\(26\) 0 0
\(27\) −509973. −0.184676
\(28\) 2.81469e6 0.865404
\(29\) 3.99178e6 1.04803 0.524017 0.851708i \(-0.324433\pi\)
0.524017 + 0.851708i \(0.324433\pi\)
\(30\) 1.52575e6 0.343904
\(31\) 6.45220e6 1.25482 0.627408 0.778691i \(-0.284116\pi\)
0.627408 + 0.778691i \(0.284116\pi\)
\(32\) 6.46202e6 1.08942
\(33\) 7.38336e6 1.08378
\(34\) −9.43018e6 −1.21022
\(35\) 2.29457e6 0.258461
\(36\) 7.04304e6 0.698874
\(37\) 8.15498e6 0.715344 0.357672 0.933847i \(-0.383571\pi\)
0.357672 + 0.933847i \(0.383571\pi\)
\(38\) 7.64787e6 0.594997
\(39\) 0 0
\(40\) 1.44971e6 0.0895386
\(41\) 720241. 0.0398062 0.0199031 0.999802i \(-0.493664\pi\)
0.0199031 + 0.999802i \(0.493664\pi\)
\(42\) 5.22244e7 2.58971
\(43\) −4.13245e7 −1.84332 −0.921658 0.388004i \(-0.873165\pi\)
−0.921658 + 0.388004i \(0.873165\pi\)
\(44\) −1.14616e7 −0.461009
\(45\) 5.74158e6 0.208725
\(46\) 6.98348e7 2.29965
\(47\) −3.67369e7 −1.09815 −0.549076 0.835772i \(-0.685020\pi\)
−0.549076 + 0.835772i \(0.685020\pi\)
\(48\) 6.62649e7 1.80176
\(49\) 3.81864e7 0.946295
\(50\) 5.43247e7 1.22923
\(51\) −6.69850e7 −1.38647
\(52\) 0 0
\(53\) 1.67334e7 0.291302 0.145651 0.989336i \(-0.453472\pi\)
0.145651 + 0.989336i \(0.453472\pi\)
\(54\) 1.46887e7 0.235077
\(55\) −9.34367e6 −0.137685
\(56\) 4.96215e7 0.674255
\(57\) 5.43248e7 0.681649
\(58\) −1.14975e8 −1.33406
\(59\) 5.89865e7 0.633751 0.316875 0.948467i \(-0.397366\pi\)
0.316875 + 0.948467i \(0.397366\pi\)
\(60\) −1.68241e7 −0.167591
\(61\) 1.28785e8 1.19091 0.595456 0.803388i \(-0.296971\pi\)
0.595456 + 0.803388i \(0.296971\pi\)
\(62\) −1.85842e8 −1.59728
\(63\) 1.96527e8 1.57177
\(64\) −2.02955e7 −0.151213
\(65\) 0 0
\(66\) −2.12662e8 −1.37956
\(67\) 1.91470e8 1.16082 0.580410 0.814324i \(-0.302892\pi\)
0.580410 + 0.814324i \(0.302892\pi\)
\(68\) 1.03985e8 0.589766
\(69\) 4.96054e8 2.63456
\(70\) −6.60901e7 −0.328999
\(71\) −3.40865e8 −1.59192 −0.795958 0.605351i \(-0.793033\pi\)
−0.795958 + 0.605351i \(0.793033\pi\)
\(72\) 1.24165e8 0.544508
\(73\) −3.19890e8 −1.31840 −0.659200 0.751968i \(-0.729105\pi\)
−0.659200 + 0.751968i \(0.729105\pi\)
\(74\) −2.34887e8 −0.910574
\(75\) 3.85882e8 1.40825
\(76\) −8.43316e7 −0.289954
\(77\) −3.19821e8 −1.03681
\(78\) 0 0
\(79\) 2.44328e8 0.705751 0.352876 0.935670i \(-0.385204\pi\)
0.352876 + 0.935670i \(0.385204\pi\)
\(80\) −8.38584e7 −0.228898
\(81\) −3.32145e8 −0.857325
\(82\) −2.07450e7 −0.0506700
\(83\) 4.38630e8 1.01449 0.507244 0.861802i \(-0.330664\pi\)
0.507244 + 0.861802i \(0.330664\pi\)
\(84\) −5.75868e8 −1.26202
\(85\) 8.47697e7 0.176139
\(86\) 1.19026e9 2.34639
\(87\) −8.16693e8 −1.52835
\(88\) −2.02063e8 −0.359182
\(89\) 7.90342e8 1.33524 0.667621 0.744501i \(-0.267312\pi\)
0.667621 + 0.744501i \(0.267312\pi\)
\(90\) −1.65374e8 −0.265690
\(91\) 0 0
\(92\) −7.70054e8 −1.12067
\(93\) −1.32008e9 −1.82990
\(94\) 1.05813e9 1.39786
\(95\) −6.87481e7 −0.0865974
\(96\) −1.32209e9 −1.58869
\(97\) 1.65696e8 0.190037 0.0950185 0.995476i \(-0.469709\pi\)
0.0950185 + 0.995476i \(0.469709\pi\)
\(98\) −1.09988e9 −1.20456
\(99\) −8.00272e8 −0.837296
\(100\) −5.99027e8 −0.599027
\(101\) −7.41636e8 −0.709161 −0.354580 0.935026i \(-0.615376\pi\)
−0.354580 + 0.935026i \(0.615376\pi\)
\(102\) 1.92936e9 1.76487
\(103\) 1.42175e9 1.24467 0.622336 0.782750i \(-0.286184\pi\)
0.622336 + 0.782750i \(0.286184\pi\)
\(104\) 0 0
\(105\) −4.69454e8 −0.376913
\(106\) −4.81970e8 −0.370803
\(107\) −1.44196e9 −1.06347 −0.531736 0.846910i \(-0.678460\pi\)
−0.531736 + 0.846910i \(0.678460\pi\)
\(108\) −1.61969e8 −0.114558
\(109\) 8.42758e7 0.0571852 0.0285926 0.999591i \(-0.490897\pi\)
0.0285926 + 0.999591i \(0.490897\pi\)
\(110\) 2.69124e8 0.175261
\(111\) −1.66846e9 −1.04319
\(112\) −2.87036e9 −1.72367
\(113\) 8.60241e8 0.496326 0.248163 0.968718i \(-0.420173\pi\)
0.248163 + 0.968718i \(0.420173\pi\)
\(114\) −1.56471e9 −0.867683
\(115\) −6.27758e8 −0.334697
\(116\) 1.26780e9 0.650115
\(117\) 0 0
\(118\) −1.69898e9 −0.806712
\(119\) 2.90155e9 1.32638
\(120\) −2.96601e8 −0.130574
\(121\) −1.05561e9 −0.447681
\(122\) −3.70936e9 −1.51593
\(123\) −1.47357e8 −0.0580494
\(124\) 2.04924e9 0.778386
\(125\) −9.94026e8 −0.364169
\(126\) −5.66052e9 −2.00073
\(127\) −1.44120e9 −0.491596 −0.245798 0.969321i \(-0.579050\pi\)
−0.245798 + 0.969321i \(0.579050\pi\)
\(128\) −2.72399e9 −0.896934
\(129\) 8.45474e9 2.68811
\(130\) 0 0
\(131\) −4.54928e9 −1.34965 −0.674827 0.737976i \(-0.735782\pi\)
−0.674827 + 0.737976i \(0.735782\pi\)
\(132\) 2.34498e9 0.672289
\(133\) −2.35316e9 −0.652107
\(134\) −5.51489e9 −1.47763
\(135\) −1.32039e8 −0.0342138
\(136\) 1.83320e9 0.459499
\(137\) 1.99742e9 0.484426 0.242213 0.970223i \(-0.422127\pi\)
0.242213 + 0.970223i \(0.422127\pi\)
\(138\) −1.42878e10 −3.35358
\(139\) 3.55325e8 0.0807345 0.0403672 0.999185i \(-0.487147\pi\)
0.0403672 + 0.999185i \(0.487147\pi\)
\(140\) 7.28762e8 0.160328
\(141\) 7.51615e9 1.60143
\(142\) 9.81789e9 2.02638
\(143\) 0 0
\(144\) −7.18235e9 −1.39199
\(145\) 1.03353e9 0.194163
\(146\) 9.21372e9 1.67821
\(147\) −7.81270e9 −1.37998
\(148\) 2.59005e9 0.443741
\(149\) −5.92546e9 −0.984881 −0.492440 0.870346i \(-0.663895\pi\)
−0.492440 + 0.870346i \(0.663895\pi\)
\(150\) −1.11145e10 −1.79258
\(151\) 1.07568e10 1.68378 0.841891 0.539647i \(-0.181442\pi\)
0.841891 + 0.539647i \(0.181442\pi\)
\(152\) −1.48672e9 −0.225909
\(153\) 7.26040e9 1.07115
\(154\) 9.21176e9 1.31977
\(155\) 1.67056e9 0.232472
\(156\) 0 0
\(157\) 8.87634e9 1.16597 0.582983 0.812485i \(-0.301886\pi\)
0.582983 + 0.812485i \(0.301886\pi\)
\(158\) −7.03734e9 −0.898363
\(159\) −3.42355e9 −0.424805
\(160\) 1.67311e9 0.201829
\(161\) −2.14873e10 −2.52038
\(162\) 9.56672e9 1.09130
\(163\) −2.75052e9 −0.305190 −0.152595 0.988289i \(-0.548763\pi\)
−0.152595 + 0.988289i \(0.548763\pi\)
\(164\) 2.28751e8 0.0246925
\(165\) 1.91166e9 0.200785
\(166\) −1.26338e10 −1.29136
\(167\) 9.17849e9 0.913160 0.456580 0.889682i \(-0.349074\pi\)
0.456580 + 0.889682i \(0.349074\pi\)
\(168\) −1.01523e10 −0.983266
\(169\) 0 0
\(170\) −2.44161e9 −0.224210
\(171\) −5.88818e9 −0.526622
\(172\) −1.31248e10 −1.14344
\(173\) 9.75049e8 0.0827598 0.0413799 0.999143i \(-0.486825\pi\)
0.0413799 + 0.999143i \(0.486825\pi\)
\(174\) 2.35231e10 1.94546
\(175\) −1.67150e10 −1.34721
\(176\) 1.16883e10 0.918218
\(177\) −1.20683e10 −0.924198
\(178\) −2.27641e10 −1.69965
\(179\) −8.88356e9 −0.646768 −0.323384 0.946268i \(-0.604820\pi\)
−0.323384 + 0.946268i \(0.604820\pi\)
\(180\) 1.82354e9 0.129476
\(181\) −5.22791e9 −0.362055 −0.181028 0.983478i \(-0.557942\pi\)
−0.181028 + 0.983478i \(0.557942\pi\)
\(182\) 0 0
\(183\) −2.63485e10 −1.73671
\(184\) −1.35757e10 −0.873134
\(185\) 2.11144e9 0.132527
\(186\) 3.80220e10 2.32931
\(187\) −1.18154e10 −0.706577
\(188\) −1.16678e10 −0.681204
\(189\) −4.51952e9 −0.257641
\(190\) 1.98014e9 0.110231
\(191\) 9.23471e9 0.502080 0.251040 0.967977i \(-0.419227\pi\)
0.251040 + 0.967977i \(0.419227\pi\)
\(192\) 4.15232e9 0.220514
\(193\) −1.07065e10 −0.555441 −0.277720 0.960662i \(-0.589579\pi\)
−0.277720 + 0.960662i \(0.589579\pi\)
\(194\) −4.77250e9 −0.241901
\(195\) 0 0
\(196\) 1.21281e10 0.587004
\(197\) 2.77001e10 1.31034 0.655169 0.755482i \(-0.272597\pi\)
0.655169 + 0.755482i \(0.272597\pi\)
\(198\) 2.30501e10 1.06581
\(199\) 3.32299e10 1.50207 0.751035 0.660262i \(-0.229555\pi\)
0.751035 + 0.660262i \(0.229555\pi\)
\(200\) −1.05606e10 −0.466715
\(201\) −3.91737e10 −1.69282
\(202\) 2.13612e10 0.902703
\(203\) 3.53763e10 1.46211
\(204\) −2.12746e10 −0.860055
\(205\) 1.86481e8 0.00737465
\(206\) −4.09503e10 −1.58436
\(207\) −5.37666e10 −2.03538
\(208\) 0 0
\(209\) 9.58225e9 0.347383
\(210\) 1.35216e10 0.479780
\(211\) −1.49261e10 −0.518411 −0.259205 0.965822i \(-0.583461\pi\)
−0.259205 + 0.965822i \(0.583461\pi\)
\(212\) 5.31458e9 0.180700
\(213\) 6.97389e10 2.32149
\(214\) 4.15325e10 1.35371
\(215\) −1.06995e10 −0.341500
\(216\) −2.85543e9 −0.0892545
\(217\) 5.71812e10 1.75059
\(218\) −2.42738e9 −0.0727920
\(219\) 6.54474e10 1.92262
\(220\) −2.96758e9 −0.0854083
\(221\) 0 0
\(222\) 4.80563e10 1.32789
\(223\) −2.23373e10 −0.604865 −0.302433 0.953171i \(-0.597799\pi\)
−0.302433 + 0.953171i \(0.597799\pi\)
\(224\) 5.72683e10 1.51984
\(225\) −4.18252e10 −1.08797
\(226\) −2.47774e10 −0.631782
\(227\) −5.20726e9 −0.130165 −0.0650824 0.997880i \(-0.520731\pi\)
−0.0650824 + 0.997880i \(0.520731\pi\)
\(228\) 1.72537e10 0.422840
\(229\) −1.35573e10 −0.325773 −0.162887 0.986645i \(-0.552080\pi\)
−0.162887 + 0.986645i \(0.552080\pi\)
\(230\) 1.80812e10 0.426042
\(231\) 6.54335e10 1.51198
\(232\) 2.23507e10 0.506518
\(233\) 2.06224e8 0.00458393 0.00229196 0.999997i \(-0.499270\pi\)
0.00229196 + 0.999997i \(0.499270\pi\)
\(234\) 0 0
\(235\) −9.51170e9 −0.203448
\(236\) 1.87343e10 0.393127
\(237\) −4.99880e10 −1.02920
\(238\) −8.35729e10 −1.68838
\(239\) −1.89887e10 −0.376448 −0.188224 0.982126i \(-0.560273\pi\)
−0.188224 + 0.982126i \(0.560273\pi\)
\(240\) 1.71569e10 0.333801
\(241\) 6.11533e10 1.16773 0.583866 0.811850i \(-0.301539\pi\)
0.583866 + 0.811850i \(0.301539\pi\)
\(242\) 3.04045e10 0.569861
\(243\) 7.79927e10 1.43491
\(244\) 4.09024e10 0.738745
\(245\) 9.88700e9 0.175314
\(246\) 4.24430e9 0.0738920
\(247\) 0 0
\(248\) 3.61270e10 0.606457
\(249\) −8.97410e10 −1.47943
\(250\) 2.86308e10 0.463557
\(251\) −3.20685e9 −0.0509973 −0.0254986 0.999675i \(-0.508117\pi\)
−0.0254986 + 0.999675i \(0.508117\pi\)
\(252\) 6.24174e10 0.974997
\(253\) 8.74981e10 1.34263
\(254\) 4.15107e10 0.625761
\(255\) −1.73433e10 −0.256863
\(256\) 8.88499e10 1.29294
\(257\) −2.80963e10 −0.401744 −0.200872 0.979617i \(-0.564378\pi\)
−0.200872 + 0.979617i \(0.564378\pi\)
\(258\) −2.43520e11 −3.42174
\(259\) 7.22717e10 0.997975
\(260\) 0 0
\(261\) 8.85201e10 1.18076
\(262\) 1.31032e11 1.71800
\(263\) −1.25524e11 −1.61780 −0.808902 0.587944i \(-0.799938\pi\)
−0.808902 + 0.587944i \(0.799938\pi\)
\(264\) 4.13408e10 0.523795
\(265\) 4.33252e9 0.0539677
\(266\) 6.77776e10 0.830078
\(267\) −1.61699e11 −1.94718
\(268\) 6.08116e10 0.720078
\(269\) −1.78558e10 −0.207918 −0.103959 0.994582i \(-0.533151\pi\)
−0.103959 + 0.994582i \(0.533151\pi\)
\(270\) 3.80310e9 0.0435513
\(271\) −1.25204e11 −1.41013 −0.705063 0.709144i \(-0.749081\pi\)
−0.705063 + 0.709144i \(0.749081\pi\)
\(272\) −1.06041e11 −1.17467
\(273\) 0 0
\(274\) −5.75314e10 −0.616634
\(275\) 6.80650e10 0.717673
\(276\) 1.57548e11 1.63427
\(277\) −5.55594e10 −0.567020 −0.283510 0.958969i \(-0.591499\pi\)
−0.283510 + 0.958969i \(0.591499\pi\)
\(278\) −1.02344e10 −0.102768
\(279\) 1.43081e11 1.41372
\(280\) 1.28477e10 0.124915
\(281\) −1.01237e11 −0.968635 −0.484317 0.874892i \(-0.660932\pi\)
−0.484317 + 0.874892i \(0.660932\pi\)
\(282\) −2.16486e11 −2.03849
\(283\) −1.08041e11 −1.00127 −0.500635 0.865658i \(-0.666900\pi\)
−0.500635 + 0.865658i \(0.666900\pi\)
\(284\) −1.08260e11 −0.987495
\(285\) 1.40654e10 0.126285
\(286\) 0 0
\(287\) 6.38298e9 0.0555335
\(288\) 1.43299e11 1.22738
\(289\) −1.13940e10 −0.0960810
\(290\) −2.97685e10 −0.247153
\(291\) −3.39003e10 −0.277131
\(292\) −1.01598e11 −0.817828
\(293\) −3.53224e9 −0.0279992 −0.0139996 0.999902i \(-0.504456\pi\)
−0.0139996 + 0.999902i \(0.504456\pi\)
\(294\) 2.25028e11 1.75660
\(295\) 1.52724e10 0.117411
\(296\) 4.56612e10 0.345728
\(297\) 1.84039e10 0.137248
\(298\) 1.70670e11 1.25367
\(299\) 0 0
\(300\) 1.22557e11 0.873562
\(301\) −3.66229e11 −2.57160
\(302\) −3.09826e11 −2.14332
\(303\) 1.51734e11 1.03417
\(304\) 8.59996e10 0.577518
\(305\) 3.33441e10 0.220633
\(306\) −2.09120e11 −1.36348
\(307\) 1.82031e11 1.16956 0.584781 0.811191i \(-0.301180\pi\)
0.584781 + 0.811191i \(0.301180\pi\)
\(308\) −1.01576e11 −0.643152
\(309\) −2.90881e11 −1.81510
\(310\) −4.81170e10 −0.295918
\(311\) −1.51907e11 −0.920781 −0.460390 0.887717i \(-0.652291\pi\)
−0.460390 + 0.887717i \(0.652291\pi\)
\(312\) 0 0
\(313\) 6.10194e10 0.359351 0.179675 0.983726i \(-0.442495\pi\)
0.179675 + 0.983726i \(0.442495\pi\)
\(314\) −2.55664e11 −1.48418
\(315\) 5.08835e10 0.291192
\(316\) 7.75994e10 0.437790
\(317\) −6.89227e10 −0.383350 −0.191675 0.981458i \(-0.561392\pi\)
−0.191675 + 0.981458i \(0.561392\pi\)
\(318\) 9.86080e10 0.540742
\(319\) −1.44055e11 −0.778880
\(320\) −5.25478e9 −0.0280143
\(321\) 2.95016e11 1.55086
\(322\) 6.18896e11 3.20823
\(323\) −8.69341e10 −0.444405
\(324\) −1.05490e11 −0.531814
\(325\) 0 0
\(326\) 7.92227e10 0.388481
\(327\) −1.72423e10 −0.0833931
\(328\) 4.03276e9 0.0192385
\(329\) −3.25573e11 −1.53203
\(330\) −5.50611e10 −0.255583
\(331\) −4.70422e10 −0.215408 −0.107704 0.994183i \(-0.534350\pi\)
−0.107704 + 0.994183i \(0.534350\pi\)
\(332\) 1.39310e11 0.629306
\(333\) 1.80842e11 0.805934
\(334\) −2.64366e11 −1.16238
\(335\) 4.95744e10 0.215058
\(336\) 5.87258e11 2.51363
\(337\) −3.76711e11 −1.59101 −0.795507 0.605944i \(-0.792795\pi\)
−0.795507 + 0.605944i \(0.792795\pi\)
\(338\) 0 0
\(339\) −1.76000e11 −0.723792
\(340\) 2.69231e10 0.109262
\(341\) −2.32846e11 −0.932556
\(342\) 1.69596e11 0.670346
\(343\) −1.92062e10 −0.0749235
\(344\) −2.31383e11 −0.890880
\(345\) 1.28435e11 0.488089
\(346\) −2.80842e10 −0.105346
\(347\) 1.68331e11 0.623278 0.311639 0.950201i \(-0.399122\pi\)
0.311639 + 0.950201i \(0.399122\pi\)
\(348\) −2.59384e11 −0.948062
\(349\) 9.36126e10 0.337769 0.168884 0.985636i \(-0.445984\pi\)
0.168884 + 0.985636i \(0.445984\pi\)
\(350\) 4.81441e11 1.71489
\(351\) 0 0
\(352\) −2.33201e11 −0.809634
\(353\) 3.49043e11 1.19644 0.598222 0.801330i \(-0.295874\pi\)
0.598222 + 0.801330i \(0.295874\pi\)
\(354\) 3.47600e11 1.17643
\(355\) −8.82548e10 −0.294924
\(356\) 2.51015e11 0.828276
\(357\) −5.93640e11 −1.93426
\(358\) 2.55872e11 0.823282
\(359\) −5.83173e11 −1.85299 −0.926493 0.376312i \(-0.877192\pi\)
−0.926493 + 0.376312i \(0.877192\pi\)
\(360\) 3.21481e10 0.100878
\(361\) −2.52184e11 −0.781512
\(362\) 1.50579e11 0.460866
\(363\) 2.15971e11 0.652854
\(364\) 0 0
\(365\) −8.28239e10 −0.244252
\(366\) 7.58912e11 2.21068
\(367\) 5.28604e11 1.52101 0.760507 0.649329i \(-0.224950\pi\)
0.760507 + 0.649329i \(0.224950\pi\)
\(368\) 7.85286e11 2.23209
\(369\) 1.59718e10 0.0448472
\(370\) −6.08154e10 −0.168696
\(371\) 1.48296e11 0.406394
\(372\) −4.19261e11 −1.13512
\(373\) −4.19405e11 −1.12187 −0.560937 0.827858i \(-0.689559\pi\)
−0.560937 + 0.827858i \(0.689559\pi\)
\(374\) 3.40316e11 0.899414
\(375\) 2.03372e11 0.531067
\(376\) −2.05697e11 −0.530740
\(377\) 0 0
\(378\) 1.30175e11 0.327955
\(379\) −4.24128e11 −1.05590 −0.527948 0.849277i \(-0.677038\pi\)
−0.527948 + 0.849277i \(0.677038\pi\)
\(380\) −2.18346e10 −0.0537180
\(381\) 2.94861e11 0.716894
\(382\) −2.65986e11 −0.639107
\(383\) −6.52696e10 −0.154995 −0.0774973 0.996993i \(-0.524693\pi\)
−0.0774973 + 0.996993i \(0.524693\pi\)
\(384\) 5.57311e11 1.30800
\(385\) −8.28062e10 −0.192083
\(386\) 3.08376e11 0.707030
\(387\) −9.16396e11 −2.07675
\(388\) 5.26254e10 0.117883
\(389\) −5.91803e11 −1.31040 −0.655200 0.755455i \(-0.727416\pi\)
−0.655200 + 0.755455i \(0.727416\pi\)
\(390\) 0 0
\(391\) −7.93819e11 −1.71762
\(392\) 2.13813e11 0.457347
\(393\) 9.30755e11 1.96820
\(394\) −7.97842e11 −1.66795
\(395\) 6.32600e10 0.130750
\(396\) −2.54169e11 −0.519390
\(397\) 3.05131e11 0.616495 0.308248 0.951306i \(-0.400257\pi\)
0.308248 + 0.951306i \(0.400257\pi\)
\(398\) −9.57115e11 −1.91201
\(399\) 4.81441e11 0.950967
\(400\) 6.10876e11 1.19312
\(401\) −2.56722e11 −0.495808 −0.247904 0.968785i \(-0.579742\pi\)
−0.247904 + 0.968785i \(0.579742\pi\)
\(402\) 1.12831e12 2.15483
\(403\) 0 0
\(404\) −2.35546e11 −0.439906
\(405\) −8.59971e10 −0.158831
\(406\) −1.01894e12 −1.86114
\(407\) −2.94296e11 −0.531631
\(408\) −3.75061e11 −0.670087
\(409\) 4.57003e11 0.807540 0.403770 0.914860i \(-0.367700\pi\)
0.403770 + 0.914860i \(0.367700\pi\)
\(410\) −5.37117e9 −0.00938732
\(411\) −4.08660e11 −0.706438
\(412\) 4.51551e11 0.772093
\(413\) 5.22755e11 0.884144
\(414\) 1.54863e12 2.59087
\(415\) 1.13567e11 0.187948
\(416\) 0 0
\(417\) −7.26973e10 −0.117735
\(418\) −2.75996e11 −0.442190
\(419\) −1.65068e11 −0.261637 −0.130818 0.991406i \(-0.541760\pi\)
−0.130818 + 0.991406i \(0.541760\pi\)
\(420\) −1.49100e11 −0.233806
\(421\) 9.10255e10 0.141219 0.0706096 0.997504i \(-0.477506\pi\)
0.0706096 + 0.997504i \(0.477506\pi\)
\(422\) 4.29913e11 0.659894
\(423\) −8.14664e11 −1.23722
\(424\) 9.36935e10 0.140787
\(425\) −6.17514e11 −0.918114
\(426\) −2.00868e12 −2.95507
\(427\) 1.14133e12 1.66144
\(428\) −4.57971e11 −0.659691
\(429\) 0 0
\(430\) 3.08176e11 0.434701
\(431\) −7.94605e11 −1.10918 −0.554592 0.832122i \(-0.687126\pi\)
−0.554592 + 0.832122i \(0.687126\pi\)
\(432\) 1.65173e11 0.228171
\(433\) −8.26325e11 −1.12968 −0.564840 0.825200i \(-0.691062\pi\)
−0.564840 + 0.825200i \(0.691062\pi\)
\(434\) −1.64698e12 −2.22836
\(435\) −2.11453e11 −0.283148
\(436\) 2.67662e10 0.0354730
\(437\) 6.43787e11 0.844453
\(438\) −1.88507e12 −2.44734
\(439\) 1.21562e12 1.56210 0.781050 0.624469i \(-0.214684\pi\)
0.781050 + 0.624469i \(0.214684\pi\)
\(440\) −5.23169e10 −0.0665434
\(441\) 8.46807e11 1.06613
\(442\) 0 0
\(443\) −8.73136e11 −1.07712 −0.538561 0.842586i \(-0.681032\pi\)
−0.538561 + 0.842586i \(0.681032\pi\)
\(444\) −5.29907e11 −0.647108
\(445\) 2.04631e11 0.247372
\(446\) 6.43377e11 0.769943
\(447\) 1.21231e12 1.43625
\(448\) −1.79864e11 −0.210957
\(449\) −1.22107e12 −1.41785 −0.708927 0.705282i \(-0.750821\pi\)
−0.708927 + 0.705282i \(0.750821\pi\)
\(450\) 1.20468e12 1.38489
\(451\) −2.59920e10 −0.0295832
\(452\) 2.73215e11 0.307880
\(453\) −2.20077e12 −2.45546
\(454\) 1.49984e11 0.165689
\(455\) 0 0
\(456\) 3.04174e11 0.329443
\(457\) −6.32643e11 −0.678478 −0.339239 0.940700i \(-0.610170\pi\)
−0.339239 + 0.940700i \(0.610170\pi\)
\(458\) 3.90490e11 0.414682
\(459\) −1.66968e11 −0.175580
\(460\) −1.99378e11 −0.207619
\(461\) −8.09117e11 −0.834367 −0.417184 0.908822i \(-0.636983\pi\)
−0.417184 + 0.908822i \(0.636983\pi\)
\(462\) −1.88467e12 −1.92463
\(463\) −7.69685e11 −0.778393 −0.389196 0.921155i \(-0.627247\pi\)
−0.389196 + 0.921155i \(0.627247\pi\)
\(464\) −1.29288e12 −1.29487
\(465\) −3.41787e11 −0.339014
\(466\) −5.93984e9 −0.00583496
\(467\) 1.59708e12 1.55382 0.776909 0.629613i \(-0.216787\pi\)
0.776909 + 0.629613i \(0.216787\pi\)
\(468\) 0 0
\(469\) 1.69686e12 1.61946
\(470\) 2.73964e11 0.258972
\(471\) −1.81604e12 −1.70033
\(472\) 3.30276e11 0.306294
\(473\) 1.49132e12 1.36992
\(474\) 1.43980e12 1.31008
\(475\) 5.00804e11 0.451384
\(476\) 9.21542e11 0.822780
\(477\) 3.71074e11 0.328192
\(478\) 5.46929e11 0.479187
\(479\) −1.61308e12 −1.40006 −0.700028 0.714115i \(-0.746829\pi\)
−0.700028 + 0.714115i \(0.746829\pi\)
\(480\) −3.42308e11 −0.294328
\(481\) 0 0
\(482\) −1.76139e12 −1.48643
\(483\) 4.39617e12 3.67547
\(484\) −3.35265e11 −0.277705
\(485\) 4.29009e10 0.0352070
\(486\) −2.24641e12 −1.82653
\(487\) −9.29058e11 −0.748449 −0.374225 0.927338i \(-0.622091\pi\)
−0.374225 + 0.927338i \(0.622091\pi\)
\(488\) 7.21089e11 0.575572
\(489\) 5.62739e11 0.445058
\(490\) −2.84774e11 −0.223160
\(491\) −2.30662e12 −1.79106 −0.895528 0.445005i \(-0.853202\pi\)
−0.895528 + 0.445005i \(0.853202\pi\)
\(492\) −4.68010e10 −0.0360091
\(493\) 1.30693e12 0.996415
\(494\) 0 0
\(495\) −2.07202e11 −0.155121
\(496\) −2.08977e12 −1.55036
\(497\) −3.02084e12 −2.22088
\(498\) 2.58480e12 1.88319
\(499\) 1.31410e11 0.0948800 0.0474400 0.998874i \(-0.484894\pi\)
0.0474400 + 0.998874i \(0.484894\pi\)
\(500\) −3.15706e11 −0.225901
\(501\) −1.87786e12 −1.33166
\(502\) 9.23664e10 0.0649153
\(503\) −1.88419e12 −1.31241 −0.656203 0.754584i \(-0.727838\pi\)
−0.656203 + 0.754584i \(0.727838\pi\)
\(504\) 1.10039e12 0.759641
\(505\) −1.92020e11 −0.131382
\(506\) −2.52019e12 −1.70906
\(507\) 0 0
\(508\) −4.57730e11 −0.304946
\(509\) 1.00390e11 0.0662919 0.0331459 0.999451i \(-0.489447\pi\)
0.0331459 + 0.999451i \(0.489447\pi\)
\(510\) 4.99537e11 0.326966
\(511\) −2.83495e12 −1.83930
\(512\) −1.16445e12 −0.748866
\(513\) 1.35411e11 0.0863226
\(514\) 8.09252e11 0.511387
\(515\) 3.68110e11 0.230593
\(516\) 2.68525e12 1.66748
\(517\) 1.32576e12 0.816126
\(518\) −2.08163e12 −1.27034
\(519\) −1.99489e11 −0.120689
\(520\) 0 0
\(521\) −1.04516e11 −0.0621462 −0.0310731 0.999517i \(-0.509892\pi\)
−0.0310731 + 0.999517i \(0.509892\pi\)
\(522\) −2.54963e12 −1.50300
\(523\) 1.17694e12 0.687855 0.343927 0.938996i \(-0.388243\pi\)
0.343927 + 0.938996i \(0.388243\pi\)
\(524\) −1.44487e12 −0.837215
\(525\) 3.41979e12 1.96464
\(526\) 3.61544e12 2.05933
\(527\) 2.11248e12 1.19301
\(528\) −2.39136e12 −1.33904
\(529\) 4.07744e12 2.26379
\(530\) −1.24789e11 −0.0686964
\(531\) 1.30806e12 0.714008
\(532\) −7.47370e11 −0.404514
\(533\) 0 0
\(534\) 4.65739e12 2.47860
\(535\) −3.73343e11 −0.197023
\(536\) 1.07208e12 0.561028
\(537\) 1.81752e12 0.943181
\(538\) 5.14296e11 0.264663
\(539\) −1.37807e12 −0.703269
\(540\) −4.19360e10 −0.0212234
\(541\) 2.98502e12 1.49817 0.749083 0.662476i \(-0.230494\pi\)
0.749083 + 0.662476i \(0.230494\pi\)
\(542\) 3.60624e12 1.79497
\(543\) 1.06960e12 0.527985
\(544\) 2.11570e12 1.03576
\(545\) 2.18202e10 0.0105943
\(546\) 0 0
\(547\) 3.78628e12 1.80830 0.904148 0.427219i \(-0.140507\pi\)
0.904148 + 0.427219i \(0.140507\pi\)
\(548\) 6.34387e11 0.300498
\(549\) 2.85588e12 1.34173
\(550\) −1.96047e12 −0.913539
\(551\) −1.05992e12 −0.489880
\(552\) 2.77750e12 1.27329
\(553\) 2.16530e12 0.984591
\(554\) 1.60027e12 0.721770
\(555\) −4.31987e11 −0.193265
\(556\) 1.12852e11 0.0500811
\(557\) −1.00551e12 −0.442627 −0.221313 0.975203i \(-0.571034\pi\)
−0.221313 + 0.975203i \(0.571034\pi\)
\(558\) −4.12115e12 −1.79955
\(559\) 0 0
\(560\) −7.43177e11 −0.319335
\(561\) 2.41735e12 1.03040
\(562\) 2.91591e12 1.23299
\(563\) 2.76515e11 0.115993 0.0579964 0.998317i \(-0.481529\pi\)
0.0579964 + 0.998317i \(0.481529\pi\)
\(564\) 2.38715e12 0.993399
\(565\) 2.22728e11 0.0919512
\(566\) 3.11190e12 1.27453
\(567\) −2.94356e12 −1.19605
\(568\) −1.90857e12 −0.769378
\(569\) 8.39220e11 0.335638 0.167819 0.985818i \(-0.446328\pi\)
0.167819 + 0.985818i \(0.446328\pi\)
\(570\) −4.05125e11 −0.160750
\(571\) −1.02285e11 −0.0402672 −0.0201336 0.999797i \(-0.506409\pi\)
−0.0201336 + 0.999797i \(0.506409\pi\)
\(572\) 0 0
\(573\) −1.88936e12 −0.732184
\(574\) −1.83848e11 −0.0706896
\(575\) 4.57297e12 1.74459
\(576\) −4.50064e11 −0.170362
\(577\) 1.44021e12 0.540921 0.270461 0.962731i \(-0.412824\pi\)
0.270461 + 0.962731i \(0.412824\pi\)
\(578\) 3.28181e11 0.122303
\(579\) 2.19047e12 0.809999
\(580\) 3.28251e11 0.120443
\(581\) 3.88726e12 1.41531
\(582\) 9.76424e11 0.352765
\(583\) −6.03874e11 −0.216490
\(584\) −1.79112e12 −0.637187
\(585\) 0 0
\(586\) 1.01738e11 0.0356407
\(587\) 2.13185e11 0.0741116 0.0370558 0.999313i \(-0.488202\pi\)
0.0370558 + 0.999313i \(0.488202\pi\)
\(588\) −2.48134e12 −0.856028
\(589\) −1.71322e12 −0.586536
\(590\) −4.39889e11 −0.149455
\(591\) −5.66727e12 −1.91087
\(592\) −2.64128e12 −0.883825
\(593\) 1.76049e12 0.584640 0.292320 0.956321i \(-0.405573\pi\)
0.292320 + 0.956321i \(0.405573\pi\)
\(594\) −5.30084e11 −0.174705
\(595\) 7.51253e11 0.245731
\(596\) −1.88194e12 −0.610940
\(597\) −6.79863e12 −2.19047
\(598\) 0 0
\(599\) −2.06753e12 −0.656191 −0.328096 0.944644i \(-0.606407\pi\)
−0.328096 + 0.944644i \(0.606407\pi\)
\(600\) 2.16062e12 0.680610
\(601\) −1.24403e12 −0.388951 −0.194476 0.980907i \(-0.562301\pi\)
−0.194476 + 0.980907i \(0.562301\pi\)
\(602\) 1.05484e13 3.27344
\(603\) 4.24597e12 1.30782
\(604\) 3.41639e12 1.04448
\(605\) −2.73312e11 −0.0829391
\(606\) −4.37037e12 −1.31641
\(607\) −1.31511e12 −0.393200 −0.196600 0.980484i \(-0.562990\pi\)
−0.196600 + 0.980484i \(0.562990\pi\)
\(608\) −1.71583e12 −0.509223
\(609\) −7.23776e12 −2.13219
\(610\) −9.60406e11 −0.280847
\(611\) 0 0
\(612\) 2.30593e12 0.664452
\(613\) 3.08252e12 0.881726 0.440863 0.897574i \(-0.354672\pi\)
0.440863 + 0.897574i \(0.354672\pi\)
\(614\) −5.24301e12 −1.48876
\(615\) −3.81528e10 −0.0107544
\(616\) −1.79074e12 −0.501094
\(617\) 2.29075e10 0.00636348 0.00318174 0.999995i \(-0.498987\pi\)
0.00318174 + 0.999995i \(0.498987\pi\)
\(618\) 8.37818e12 2.31048
\(619\) 4.45350e12 1.21925 0.609626 0.792689i \(-0.291320\pi\)
0.609626 + 0.792689i \(0.291320\pi\)
\(620\) 5.30576e11 0.144207
\(621\) 1.23647e12 0.333635
\(622\) 4.37535e12 1.17208
\(623\) 7.00424e12 1.86279
\(624\) 0 0
\(625\) 3.42640e12 0.898210
\(626\) −1.75753e12 −0.457423
\(627\) −1.96047e12 −0.506589
\(628\) 2.81915e12 0.723270
\(629\) 2.66998e12 0.680111
\(630\) −1.46559e12 −0.370663
\(631\) −3.83831e12 −0.963847 −0.481923 0.876213i \(-0.660062\pi\)
−0.481923 + 0.876213i \(0.660062\pi\)
\(632\) 1.36804e12 0.341092
\(633\) 3.05378e12 0.755998
\(634\) 1.98517e12 0.487973
\(635\) −3.73148e11 −0.0910749
\(636\) −1.08733e12 −0.263515
\(637\) 0 0
\(638\) 4.14919e12 0.991449
\(639\) −7.55890e12 −1.79351
\(640\) −7.05279e11 −0.166169
\(641\) 9.09238e11 0.212724 0.106362 0.994327i \(-0.466080\pi\)
0.106362 + 0.994327i \(0.466080\pi\)
\(642\) −8.49729e12 −1.97412
\(643\) −8.23839e12 −1.90061 −0.950305 0.311321i \(-0.899229\pi\)
−0.950305 + 0.311321i \(0.899229\pi\)
\(644\) −6.82444e12 −1.56344
\(645\) 2.18905e12 0.498009
\(646\) 2.50395e12 0.565691
\(647\) −8.73559e12 −1.95985 −0.979925 0.199365i \(-0.936112\pi\)
−0.979925 + 0.199365i \(0.936112\pi\)
\(648\) −1.85974e12 −0.414348
\(649\) −2.12870e12 −0.470992
\(650\) 0 0
\(651\) −1.16989e13 −2.55289
\(652\) −8.73573e11 −0.189315
\(653\) −6.13232e11 −0.131982 −0.0659911 0.997820i \(-0.521021\pi\)
−0.0659911 + 0.997820i \(0.521021\pi\)
\(654\) 4.96627e11 0.106153
\(655\) −1.17787e12 −0.250042
\(656\) −2.33275e11 −0.0491815
\(657\) −7.09375e12 −1.48536
\(658\) 9.37742e12 1.95015
\(659\) −8.83233e12 −1.82428 −0.912139 0.409882i \(-0.865570\pi\)
−0.912139 + 0.409882i \(0.865570\pi\)
\(660\) 6.07148e11 0.124551
\(661\) 5.61801e12 1.14466 0.572329 0.820024i \(-0.306040\pi\)
0.572329 + 0.820024i \(0.306040\pi\)
\(662\) 1.35495e12 0.274196
\(663\) 0 0
\(664\) 2.45597e12 0.490306
\(665\) −6.09265e11 −0.120812
\(666\) −5.20875e12 −1.02589
\(667\) −9.67839e12 −1.89338
\(668\) 2.91511e12 0.566450
\(669\) 4.57007e12 0.882075
\(670\) −1.42788e12 −0.273751
\(671\) −4.64757e12 −0.885064
\(672\) −1.17167e13 −2.21638
\(673\) 4.50151e12 0.845844 0.422922 0.906166i \(-0.361004\pi\)
0.422922 + 0.906166i \(0.361004\pi\)
\(674\) 1.08503e13 2.02523
\(675\) 9.61854e11 0.178337
\(676\) 0 0
\(677\) 4.33098e12 0.792387 0.396193 0.918167i \(-0.370331\pi\)
0.396193 + 0.918167i \(0.370331\pi\)
\(678\) 5.06929e12 0.921327
\(679\) 1.46844e12 0.265120
\(680\) 4.74641e11 0.0851285
\(681\) 1.06537e12 0.189819
\(682\) 6.70664e12 1.18707
\(683\) −6.25107e12 −1.09916 −0.549580 0.835441i \(-0.685212\pi\)
−0.549580 + 0.835441i \(0.685212\pi\)
\(684\) −1.87010e12 −0.326673
\(685\) 5.17161e11 0.0897466
\(686\) 5.53193e11 0.0953715
\(687\) 2.77375e12 0.475075
\(688\) 1.33844e13 2.27746
\(689\) 0 0
\(690\) −3.69930e12 −0.621296
\(691\) −7.34082e12 −1.22488 −0.612440 0.790517i \(-0.709812\pi\)
−0.612440 + 0.790517i \(0.709812\pi\)
\(692\) 3.09679e11 0.0513374
\(693\) −7.09223e12 −1.16811
\(694\) −4.84841e12 −0.793381
\(695\) 9.19986e10 0.0149572
\(696\) −4.57281e12 −0.738656
\(697\) 2.35810e11 0.0378456
\(698\) −2.69631e12 −0.429952
\(699\) −4.21922e10 −0.00668474
\(700\) −5.30875e12 −0.835701
\(701\) −1.71509e11 −0.0268259 −0.0134130 0.999910i \(-0.504270\pi\)
−0.0134130 + 0.999910i \(0.504270\pi\)
\(702\) 0 0
\(703\) −2.16535e12 −0.334372
\(704\) 7.32421e11 0.112379
\(705\) 1.94604e12 0.296688
\(706\) −1.00534e13 −1.52297
\(707\) −6.57259e12 −0.989348
\(708\) −3.83292e12 −0.573297
\(709\) 1.00896e13 1.49956 0.749780 0.661687i \(-0.230159\pi\)
0.749780 + 0.661687i \(0.230159\pi\)
\(710\) 2.54199e12 0.375414
\(711\) 5.41813e12 0.795126
\(712\) 4.42527e12 0.645327
\(713\) −1.56439e13 −2.26695
\(714\) 1.70985e13 2.46216
\(715\) 0 0
\(716\) −2.82145e12 −0.401202
\(717\) 3.88497e12 0.548974
\(718\) 1.67970e13 2.35870
\(719\) 8.18068e12 1.14159 0.570794 0.821093i \(-0.306635\pi\)
0.570794 + 0.821093i \(0.306635\pi\)
\(720\) −1.85961e12 −0.257885
\(721\) 1.25999e13 1.73644
\(722\) 7.26362e12 0.994800
\(723\) −1.25116e13 −1.70290
\(724\) −1.66040e12 −0.224589
\(725\) −7.52885e12 −1.01206
\(726\) −6.22058e12 −0.831029
\(727\) −9.98421e11 −0.132559 −0.0662795 0.997801i \(-0.521113\pi\)
−0.0662795 + 0.997801i \(0.521113\pi\)
\(728\) 0 0
\(729\) −9.41920e12 −1.23521
\(730\) 2.38556e12 0.310912
\(731\) −1.35298e13 −1.75253
\(732\) −8.36837e12 −1.07731
\(733\) 2.10457e12 0.269274 0.134637 0.990895i \(-0.457013\pi\)
0.134637 + 0.990895i \(0.457013\pi\)
\(734\) −1.52253e13 −1.93613
\(735\) −2.02282e12 −0.255661
\(736\) −1.56677e13 −1.96814
\(737\) −6.90977e12 −0.862700
\(738\) −4.60033e11 −0.0570867
\(739\) 1.28931e13 1.59023 0.795113 0.606461i \(-0.207412\pi\)
0.795113 + 0.606461i \(0.207412\pi\)
\(740\) 6.70599e11 0.0822092
\(741\) 0 0
\(742\) −4.27135e12 −0.517306
\(743\) −2.72357e11 −0.0327861 −0.0163930 0.999866i \(-0.505218\pi\)
−0.0163930 + 0.999866i \(0.505218\pi\)
\(744\) −7.39137e12 −0.884396
\(745\) −1.53418e12 −0.182463
\(746\) 1.20801e13 1.42805
\(747\) 9.72689e12 1.14296
\(748\) −3.75259e12 −0.438303
\(749\) −1.27790e13 −1.48365
\(750\) −5.85768e12 −0.676005
\(751\) −1.40530e13 −1.61209 −0.806045 0.591855i \(-0.798396\pi\)
−0.806045 + 0.591855i \(0.798396\pi\)
\(752\) 1.18985e13 1.35679
\(753\) 6.56102e11 0.0743693
\(754\) 0 0
\(755\) 2.78508e12 0.311944
\(756\) −1.43541e12 −0.159819
\(757\) 8.53220e12 0.944342 0.472171 0.881507i \(-0.343470\pi\)
0.472171 + 0.881507i \(0.343470\pi\)
\(758\) 1.22161e13 1.34407
\(759\) −1.79016e13 −1.95796
\(760\) −3.84933e11 −0.0418528
\(761\) −1.13835e13 −1.23039 −0.615197 0.788373i \(-0.710924\pi\)
−0.615197 + 0.788373i \(0.710924\pi\)
\(762\) −8.49283e12 −0.912547
\(763\) 7.46876e11 0.0797789
\(764\) 2.93297e12 0.311450
\(765\) 1.87982e12 0.198445
\(766\) 1.87995e12 0.197295
\(767\) 0 0
\(768\) −1.81781e13 −1.88549
\(769\) −1.94858e12 −0.200933 −0.100466 0.994940i \(-0.532033\pi\)
−0.100466 + 0.994940i \(0.532033\pi\)
\(770\) 2.38505e12 0.244506
\(771\) 5.74832e12 0.585864
\(772\) −3.40040e12 −0.344550
\(773\) 1.09355e13 1.10161 0.550807 0.834633i \(-0.314320\pi\)
0.550807 + 0.834633i \(0.314320\pi\)
\(774\) 2.63948e13 2.64353
\(775\) −1.21694e13 −1.21175
\(776\) 9.27760e11 0.0918455
\(777\) −1.47863e13 −1.45535
\(778\) 1.70456e13 1.66803
\(779\) −1.91242e11 −0.0186065
\(780\) 0 0
\(781\) 1.23011e13 1.18308
\(782\) 2.28642e13 2.18638
\(783\) −2.03570e12 −0.193547
\(784\) −1.23680e13 −1.16917
\(785\) 2.29821e12 0.216011
\(786\) −2.68084e13 −2.50535
\(787\) −7.50366e12 −0.697247 −0.348623 0.937263i \(-0.613351\pi\)
−0.348623 + 0.937263i \(0.613351\pi\)
\(788\) 8.79764e12 0.812827
\(789\) 2.56814e13 2.35924
\(790\) −1.82207e12 −0.166434
\(791\) 7.62369e12 0.692423
\(792\) −4.48087e12 −0.404668
\(793\) 0 0
\(794\) −8.78865e12 −0.784747
\(795\) −8.86406e11 −0.0787011
\(796\) 1.05539e13 0.931762
\(797\) 1.65844e12 0.145592 0.0727962 0.997347i \(-0.476808\pi\)
0.0727962 + 0.997347i \(0.476808\pi\)
\(798\) −1.38669e13 −1.21050
\(799\) −1.20278e13 −1.04406
\(800\) −1.21880e13 −1.05202
\(801\) 1.75263e13 1.50434
\(802\) 7.39433e12 0.631123
\(803\) 1.15441e13 0.979810
\(804\) −1.24417e13 −1.05009
\(805\) −5.56337e12 −0.466935
\(806\) 0 0
\(807\) 3.65318e12 0.303207
\(808\) −4.15256e12 −0.342740
\(809\) −7.07921e12 −0.581054 −0.290527 0.956867i \(-0.593831\pi\)
−0.290527 + 0.956867i \(0.593831\pi\)
\(810\) 2.47696e12 0.202179
\(811\) 1.84596e13 1.49840 0.749201 0.662343i \(-0.230438\pi\)
0.749201 + 0.662343i \(0.230438\pi\)
\(812\) 1.12356e13 0.906973
\(813\) 2.56161e13 2.05639
\(814\) 8.47657e12 0.676722
\(815\) −7.12147e11 −0.0565406
\(816\) 2.16954e13 1.71302
\(817\) 1.09727e13 0.861616
\(818\) −1.31630e13 −1.02793
\(819\) 0 0
\(820\) 5.92268e10 0.00457463
\(821\) −2.40377e13 −1.84650 −0.923250 0.384200i \(-0.874477\pi\)
−0.923250 + 0.384200i \(0.874477\pi\)
\(822\) 1.17706e13 0.899237
\(823\) −1.26494e12 −0.0961104 −0.0480552 0.998845i \(-0.515302\pi\)
−0.0480552 + 0.998845i \(0.515302\pi\)
\(824\) 7.96062e12 0.601554
\(825\) −1.39257e13 −1.04658
\(826\) −1.50568e13 −1.12544
\(827\) 9.81229e12 0.729450 0.364725 0.931115i \(-0.381163\pi\)
0.364725 + 0.931115i \(0.381163\pi\)
\(828\) −1.70764e13 −1.26258
\(829\) 1.26128e13 0.927504 0.463752 0.885965i \(-0.346503\pi\)
0.463752 + 0.885965i \(0.346503\pi\)
\(830\) −3.27107e12 −0.239242
\(831\) 1.13671e13 0.826886
\(832\) 0 0
\(833\) 1.25024e13 0.899687
\(834\) 2.09389e12 0.149867
\(835\) 2.37644e12 0.169175
\(836\) 3.04335e12 0.215488
\(837\) −3.29045e12 −0.231734
\(838\) 4.75441e12 0.333042
\(839\) 8.53632e12 0.594760 0.297380 0.954759i \(-0.403887\pi\)
0.297380 + 0.954759i \(0.403887\pi\)
\(840\) −2.62856e12 −0.182163
\(841\) 1.42715e12 0.0983759
\(842\) −2.62179e12 −0.179760
\(843\) 2.07124e13 1.41256
\(844\) −4.74056e12 −0.321580
\(845\) 0 0
\(846\) 2.34646e13 1.57488
\(847\) −9.35511e12 −0.624559
\(848\) −5.41970e12 −0.359910
\(849\) 2.21046e13 1.46015
\(850\) 1.77862e13 1.16868
\(851\) −1.97724e13 −1.29234
\(852\) 2.21493e13 1.44006
\(853\) 1.12025e13 0.724507 0.362254 0.932080i \(-0.382007\pi\)
0.362254 + 0.932080i \(0.382007\pi\)
\(854\) −3.28734e13 −2.11487
\(855\) −1.52453e12 −0.0975639
\(856\) −8.07379e12 −0.513979
\(857\) −2.23241e13 −1.41371 −0.706856 0.707357i \(-0.749887\pi\)
−0.706856 + 0.707357i \(0.749887\pi\)
\(858\) 0 0
\(859\) −2.11323e13 −1.32428 −0.662138 0.749382i \(-0.730351\pi\)
−0.662138 + 0.749382i \(0.730351\pi\)
\(860\) −3.39819e12 −0.211838
\(861\) −1.30592e12 −0.0809845
\(862\) 2.28869e13 1.41190
\(863\) −1.17061e13 −0.718395 −0.359197 0.933262i \(-0.616950\pi\)
−0.359197 + 0.933262i \(0.616950\pi\)
\(864\) −3.29546e12 −0.201189
\(865\) 2.52454e11 0.0153324
\(866\) 2.38005e13 1.43799
\(867\) 2.33115e12 0.140115
\(868\) 1.81609e13 1.08592
\(869\) −8.81730e12 −0.524501
\(870\) 6.09045e12 0.360423
\(871\) 0 0
\(872\) 4.71876e11 0.0276378
\(873\) 3.67440e12 0.214103
\(874\) −1.85429e13 −1.07492
\(875\) −8.80934e12 −0.508051
\(876\) 2.07863e13 1.19264
\(877\) −1.17836e13 −0.672638 −0.336319 0.941748i \(-0.609182\pi\)
−0.336319 + 0.941748i \(0.609182\pi\)
\(878\) −3.50134e13 −1.98842
\(879\) 7.22674e11 0.0408312
\(880\) 3.02627e12 0.170113
\(881\) −1.93741e13 −1.08350 −0.541751 0.840539i \(-0.682238\pi\)
−0.541751 + 0.840539i \(0.682238\pi\)
\(882\) −2.43904e13 −1.35710
\(883\) −1.50199e13 −0.831466 −0.415733 0.909487i \(-0.636475\pi\)
−0.415733 + 0.909487i \(0.636475\pi\)
\(884\) 0 0
\(885\) −3.12464e12 −0.171220
\(886\) 2.51488e13 1.37109
\(887\) 1.23661e13 0.670776 0.335388 0.942080i \(-0.391133\pi\)
0.335388 + 0.942080i \(0.391133\pi\)
\(888\) −9.34200e12 −0.504175
\(889\) −1.27723e13 −0.685824
\(890\) −5.89394e12 −0.314884
\(891\) 1.19864e13 0.637148
\(892\) −7.09439e12 −0.375209
\(893\) 9.75457e12 0.513306
\(894\) −3.49180e13 −1.82823
\(895\) −2.30008e12 −0.119823
\(896\) −2.41408e13 −1.25131
\(897\) 0 0
\(898\) 3.51702e13 1.80481
\(899\) 2.57558e13 1.31509
\(900\) −1.32838e13 −0.674887
\(901\) 5.47860e12 0.276954
\(902\) 7.48644e11 0.0376570
\(903\) 7.49283e13 3.75017
\(904\) 4.81664e12 0.239876
\(905\) −1.35358e12 −0.0670757
\(906\) 6.33884e13 3.12560
\(907\) 9.76708e12 0.479217 0.239608 0.970870i \(-0.422981\pi\)
0.239608 + 0.970870i \(0.422981\pi\)
\(908\) −1.65384e12 −0.0807436
\(909\) −1.64462e13 −0.798967
\(910\) 0 0
\(911\) −1.91556e13 −0.921430 −0.460715 0.887548i \(-0.652407\pi\)
−0.460715 + 0.887548i \(0.652407\pi\)
\(912\) −1.75950e13 −0.842194
\(913\) −1.58293e13 −0.753949
\(914\) 1.82219e13 0.863647
\(915\) −6.82200e12 −0.321749
\(916\) −4.30586e12 −0.202083
\(917\) −4.03170e13 −1.88290
\(918\) 4.80914e12 0.223499
\(919\) −2.82464e13 −1.30630 −0.653150 0.757229i \(-0.726553\pi\)
−0.653150 + 0.757229i \(0.726553\pi\)
\(920\) −3.51493e12 −0.161760
\(921\) −3.72425e13 −1.70557
\(922\) 2.33049e13 1.06208
\(923\) 0 0
\(924\) 2.07819e13 0.937909
\(925\) −1.53810e13 −0.690792
\(926\) 2.21691e13 0.990830
\(927\) 3.15281e13 1.40229
\(928\) 2.57950e13 1.14175
\(929\) −1.81017e13 −0.797348 −0.398674 0.917093i \(-0.630529\pi\)
−0.398674 + 0.917093i \(0.630529\pi\)
\(930\) 9.84444e12 0.431537
\(931\) −1.01395e13 −0.442324
\(932\) 6.54974e10 0.00284350
\(933\) 3.10792e13 1.34277
\(934\) −4.60004e13 −1.97788
\(935\) −3.05916e12 −0.130903
\(936\) 0 0
\(937\) 2.51202e13 1.06462 0.532310 0.846550i \(-0.321324\pi\)
0.532310 + 0.846550i \(0.321324\pi\)
\(938\) −4.88745e13 −2.06143
\(939\) −1.24842e13 −0.524041
\(940\) −3.02095e12 −0.126202
\(941\) −8.09156e11 −0.0336418 −0.0168209 0.999859i \(-0.505355\pi\)
−0.0168209 + 0.999859i \(0.505355\pi\)
\(942\) 5.23072e13 2.16438
\(943\) −1.74628e12 −0.0719138
\(944\) −1.91048e13 −0.783014
\(945\) −1.17017e12 −0.0477315
\(946\) −4.29541e13 −1.74379
\(947\) 1.25569e13 0.507349 0.253675 0.967290i \(-0.418361\pi\)
0.253675 + 0.967290i \(0.418361\pi\)
\(948\) −1.58763e13 −0.638430
\(949\) 0 0
\(950\) −1.44246e13 −0.574575
\(951\) 1.41012e13 0.559040
\(952\) 1.62463e13 0.641045
\(953\) −3.64201e13 −1.43029 −0.715144 0.698977i \(-0.753639\pi\)
−0.715144 + 0.698977i \(0.753639\pi\)
\(954\) −1.06880e13 −0.417761
\(955\) 2.39100e12 0.0930173
\(956\) −6.03087e12 −0.233518
\(957\) 2.94728e13 1.13584
\(958\) 4.64612e13 1.78216
\(959\) 1.77017e13 0.675821
\(960\) 1.07509e12 0.0408532
\(961\) 1.51913e13 0.574564
\(962\) 0 0
\(963\) −3.19763e13 −1.19815
\(964\) 1.94225e13 0.724366
\(965\) −2.77205e12 −0.102903
\(966\) −1.26622e14 −4.67857
\(967\) −2.96691e13 −1.09115 −0.545575 0.838062i \(-0.683689\pi\)
−0.545575 + 0.838062i \(0.683689\pi\)
\(968\) −5.91055e12 −0.216366
\(969\) 1.77862e13 0.648076
\(970\) −1.23567e12 −0.0448156
\(971\) −1.05417e13 −0.380563 −0.190281 0.981730i \(-0.560940\pi\)
−0.190281 + 0.981730i \(0.560940\pi\)
\(972\) 2.47707e13 0.890103
\(973\) 3.14899e12 0.112632
\(974\) 2.67595e13 0.952714
\(975\) 0 0
\(976\) −4.17114e13 −1.47140
\(977\) 2.40729e13 0.845285 0.422643 0.906296i \(-0.361103\pi\)
0.422643 + 0.906296i \(0.361103\pi\)
\(978\) −1.62085e13 −0.566522
\(979\) −2.85218e13 −0.992328
\(980\) 3.14014e12 0.108751
\(981\) 1.86887e12 0.0644270
\(982\) 6.64372e13 2.27987
\(983\) 3.57135e13 1.21995 0.609974 0.792421i \(-0.291180\pi\)
0.609974 + 0.792421i \(0.291180\pi\)
\(984\) −8.25078e11 −0.0280555
\(985\) 7.17195e12 0.242758
\(986\) −3.76432e13 −1.26835
\(987\) 6.66102e13 2.23416
\(988\) 0 0
\(989\) 1.00195e14 3.33013
\(990\) 5.96799e12 0.197456
\(991\) 2.48780e13 0.819376 0.409688 0.912226i \(-0.365638\pi\)
0.409688 + 0.912226i \(0.365638\pi\)
\(992\) 4.16943e13 1.36702
\(993\) 9.62454e12 0.314129
\(994\) 8.70089e13 2.82699
\(995\) 8.60369e12 0.278279
\(996\) −2.85020e13 −0.917717
\(997\) −3.31717e13 −1.06326 −0.531630 0.846977i \(-0.678420\pi\)
−0.531630 + 0.846977i \(0.678420\pi\)
\(998\) −3.78497e12 −0.120774
\(999\) −4.15882e12 −0.132107
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 169.10.a.a.1.1 4
13.12 even 2 13.10.a.a.1.4 4
39.38 odd 2 117.10.a.c.1.1 4
52.51 odd 2 208.10.a.g.1.4 4
65.64 even 2 325.10.a.a.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.10.a.a.1.4 4 13.12 even 2
117.10.a.c.1.1 4 39.38 odd 2
169.10.a.a.1.1 4 1.1 even 1 trivial
208.10.a.g.1.4 4 52.51 odd 2
325.10.a.a.1.1 4 65.64 even 2