Properties

Label 9.20.a.c.1.1
Level $9$
Weight $20$
Character 9.1
Self dual yes
Analytic conductor $20.594$
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] = [9,20,Mod(1,9)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 20, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 9.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: \(20.5935026901\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{87481}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 21870 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(148.386\) of defining polynomial
Character \(\chi\) \(=\) 9.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1238.32 q^{2} +1.00914e6 q^{4} -6.69930e6 q^{5} +214621. q^{7} -6.00397e8 q^{8} +O(q^{10})\) \(q-1238.32 q^{2} +1.00914e6 q^{4} -6.69930e6 q^{5} +214621. q^{7} -6.00397e8 q^{8} +8.29585e9 q^{10} +1.30816e10 q^{11} +1.34409e10 q^{13} -2.65768e8 q^{14} +2.14402e11 q^{16} +3.48802e11 q^{17} -6.20630e11 q^{19} -6.76052e12 q^{20} -1.61991e13 q^{22} +2.65405e12 q^{23} +2.58072e13 q^{25} -1.66441e13 q^{26} +2.16582e11 q^{28} -1.27229e14 q^{29} +2.32290e12 q^{31} +4.92835e13 q^{32} -4.31927e14 q^{34} -1.43781e12 q^{35} -7.19096e14 q^{37} +7.68535e14 q^{38} +4.02224e15 q^{40} -1.72545e15 q^{41} +2.48385e15 q^{43} +1.32011e16 q^{44} -3.28655e15 q^{46} -1.39689e15 q^{47} -1.13988e16 q^{49} -3.19574e16 q^{50} +1.35637e16 q^{52} -1.05483e16 q^{53} -8.76375e16 q^{55} -1.28858e14 q^{56} +1.57550e17 q^{58} +2.80846e16 q^{59} -3.60392e16 q^{61} -2.87648e15 q^{62} -1.73437e17 q^{64} -9.00446e16 q^{65} +2.91862e17 q^{67} +3.51989e17 q^{68} +1.78046e15 q^{70} -3.20038e17 q^{71} -3.15819e17 q^{73} +8.90467e17 q^{74} -6.26301e17 q^{76} +2.80758e15 q^{77} +1.83029e18 q^{79} -1.43634e18 q^{80} +2.13665e18 q^{82} +1.64758e18 q^{83} -2.33673e18 q^{85} -3.07579e18 q^{86} -7.85414e18 q^{88} -5.75855e18 q^{89} +2.88469e15 q^{91} +2.67830e18 q^{92} +1.72979e18 q^{94} +4.15779e18 q^{95} +6.23309e18 q^{97} +1.41154e19 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 702 q^{2} + 772484 q^{4} - 6016140 q^{5} + 113892064 q^{7} - 1008501624 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 702 q^{2} + 772484 q^{4} - 6016140 q^{5} + 113892064 q^{7} - 1008501624 q^{8} + 8662242420 q^{10} + 6650071272 q^{11} - 44072356148 q^{13} + 60701219424 q^{14} + 119603620880 q^{16} - 336281471748 q^{17} + 602118925096 q^{19} - 6922191196440 q^{20} - 19648459326744 q^{22} - 2368252165968 q^{23} + 7200399078350 q^{25} - 47489309789364 q^{26} - 26685589805888 q^{28} - 280977251970492 q^{29} + 41610149253712 q^{31} + 212406109003296 q^{32} - 799347349171332 q^{34} + 76222408017600 q^{35} + 637994163989884 q^{37} + 14\!\cdots\!32 q^{38}+ \cdots + 14\!\cdots\!02 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1238.32 −1.71020 −0.855099 0.518465i \(-0.826504\pi\)
−0.855099 + 0.518465i \(0.826504\pi\)
\(3\) 0 0
\(4\) 1.00914e6 1.92478
\(5\) −6.69930e6 −1.53396 −0.766981 0.641670i \(-0.778242\pi\)
−0.766981 + 0.641670i \(0.778242\pi\)
\(6\) 0 0
\(7\) 214621. 0.00201021 0.00100510 0.999999i \(-0.499680\pi\)
0.00100510 + 0.999999i \(0.499680\pi\)
\(8\) −6.00397e8 −1.58155
\(9\) 0 0
\(10\) 8.29585e9 2.62338
\(11\) 1.30816e10 1.67275 0.836373 0.548161i \(-0.184672\pi\)
0.836373 + 0.548161i \(0.184672\pi\)
\(12\) 0 0
\(13\) 1.34409e10 0.351533 0.175766 0.984432i \(-0.443760\pi\)
0.175766 + 0.984432i \(0.443760\pi\)
\(14\) −2.65768e8 −0.00343785
\(15\) 0 0
\(16\) 2.14402e11 0.779990
\(17\) 3.48802e11 0.713368 0.356684 0.934225i \(-0.383907\pi\)
0.356684 + 0.934225i \(0.383907\pi\)
\(18\) 0 0
\(19\) −6.20630e11 −0.441238 −0.220619 0.975360i \(-0.570808\pi\)
−0.220619 + 0.975360i \(0.570808\pi\)
\(20\) −6.76052e12 −2.95253
\(21\) 0 0
\(22\) −1.61991e13 −2.86073
\(23\) 2.65405e12 0.307252 0.153626 0.988129i \(-0.450905\pi\)
0.153626 + 0.988129i \(0.450905\pi\)
\(24\) 0 0
\(25\) 2.58072e13 1.35304
\(26\) −1.66441e13 −0.601191
\(27\) 0 0
\(28\) 2.16582e11 0.00386920
\(29\) −1.27229e14 −1.62857 −0.814286 0.580464i \(-0.802871\pi\)
−0.814286 + 0.580464i \(0.802871\pi\)
\(30\) 0 0
\(31\) 2.32290e12 0.0157795 0.00788977 0.999969i \(-0.497489\pi\)
0.00788977 + 0.999969i \(0.497489\pi\)
\(32\) 4.92835e13 0.247615
\(33\) 0 0
\(34\) −4.31927e14 −1.22000
\(35\) −1.43781e12 −0.00308358
\(36\) 0 0
\(37\) −7.19096e14 −0.909642 −0.454821 0.890583i \(-0.650297\pi\)
−0.454821 + 0.890583i \(0.650297\pi\)
\(38\) 7.68535e14 0.754605
\(39\) 0 0
\(40\) 4.02224e15 2.42604
\(41\) −1.72545e15 −0.823105 −0.411553 0.911386i \(-0.635013\pi\)
−0.411553 + 0.911386i \(0.635013\pi\)
\(42\) 0 0
\(43\) 2.48385e15 0.753660 0.376830 0.926282i \(-0.377014\pi\)
0.376830 + 0.926282i \(0.377014\pi\)
\(44\) 1.32011e16 3.21966
\(45\) 0 0
\(46\) −3.28655e15 −0.525462
\(47\) −1.39689e15 −0.182067 −0.0910336 0.995848i \(-0.529017\pi\)
−0.0910336 + 0.995848i \(0.529017\pi\)
\(48\) 0 0
\(49\) −1.13988e16 −0.999996
\(50\) −3.19574e16 −2.31396
\(51\) 0 0
\(52\) 1.35637e16 0.676623
\(53\) −1.05483e16 −0.439099 −0.219550 0.975601i \(-0.570459\pi\)
−0.219550 + 0.975601i \(0.570459\pi\)
\(54\) 0 0
\(55\) −8.76375e16 −2.56593
\(56\) −1.28858e14 −0.00317925
\(57\) 0 0
\(58\) 1.57550e17 2.78518
\(59\) 2.80846e16 0.422060 0.211030 0.977480i \(-0.432318\pi\)
0.211030 + 0.977480i \(0.432318\pi\)
\(60\) 0 0
\(61\) −3.60392e16 −0.394586 −0.197293 0.980345i \(-0.563215\pi\)
−0.197293 + 0.980345i \(0.563215\pi\)
\(62\) −2.87648e15 −0.0269861
\(63\) 0 0
\(64\) −1.73437e17 −1.20346
\(65\) −9.00446e16 −0.539238
\(66\) 0 0
\(67\) 2.91862e17 1.31059 0.655296 0.755373i \(-0.272544\pi\)
0.655296 + 0.755373i \(0.272544\pi\)
\(68\) 3.51989e17 1.37308
\(69\) 0 0
\(70\) 1.78046e15 0.00527353
\(71\) −3.20038e17 −0.828414 −0.414207 0.910183i \(-0.635941\pi\)
−0.414207 + 0.910183i \(0.635941\pi\)
\(72\) 0 0
\(73\) −3.15819e17 −0.627873 −0.313936 0.949444i \(-0.601648\pi\)
−0.313936 + 0.949444i \(0.601648\pi\)
\(74\) 8.90467e17 1.55567
\(75\) 0 0
\(76\) −6.26301e17 −0.849286
\(77\) 2.80758e15 0.00336256
\(78\) 0 0
\(79\) 1.83029e18 1.71815 0.859077 0.511846i \(-0.171038\pi\)
0.859077 + 0.511846i \(0.171038\pi\)
\(80\) −1.43634e18 −1.19647
\(81\) 0 0
\(82\) 2.13665e18 1.40767
\(83\) 1.64758e18 0.967397 0.483698 0.875235i \(-0.339293\pi\)
0.483698 + 0.875235i \(0.339293\pi\)
\(84\) 0 0
\(85\) −2.33673e18 −1.09428
\(86\) −3.07579e18 −1.28891
\(87\) 0 0
\(88\) −7.85414e18 −2.64553
\(89\) −5.75855e18 −1.74224 −0.871120 0.491069i \(-0.836606\pi\)
−0.871120 + 0.491069i \(0.836606\pi\)
\(90\) 0 0
\(91\) 2.88469e15 0.000706654 0
\(92\) 2.67830e18 0.591392
\(93\) 0 0
\(94\) 1.72979e18 0.311371
\(95\) 4.15779e18 0.676843
\(96\) 0 0
\(97\) 6.23309e18 0.832477 0.416238 0.909256i \(-0.363348\pi\)
0.416238 + 0.909256i \(0.363348\pi\)
\(98\) 1.41154e19 1.71019
\(99\) 0 0
\(100\) 2.60430e19 2.60430
\(101\) −8.80411e17 −0.0801000 −0.0400500 0.999198i \(-0.512752\pi\)
−0.0400500 + 0.999198i \(0.512752\pi\)
\(102\) 0 0
\(103\) −1.96998e19 −1.48767 −0.743837 0.668361i \(-0.766996\pi\)
−0.743837 + 0.668361i \(0.766996\pi\)
\(104\) −8.06987e18 −0.555968
\(105\) 0 0
\(106\) 1.30622e19 0.750946
\(107\) −2.58893e19 −1.36136 −0.680682 0.732579i \(-0.738316\pi\)
−0.680682 + 0.732579i \(0.738316\pi\)
\(108\) 0 0
\(109\) −3.42533e19 −1.51060 −0.755302 0.655377i \(-0.772510\pi\)
−0.755302 + 0.655377i \(0.772510\pi\)
\(110\) 1.08523e20 4.38825
\(111\) 0 0
\(112\) 4.60151e16 0.00156794
\(113\) −5.49844e19 −1.72185 −0.860923 0.508735i \(-0.830113\pi\)
−0.860923 + 0.508735i \(0.830113\pi\)
\(114\) 0 0
\(115\) −1.77803e19 −0.471313
\(116\) −1.28392e20 −3.13464
\(117\) 0 0
\(118\) −3.47776e19 −0.721806
\(119\) 7.48601e16 0.00143402
\(120\) 0 0
\(121\) 1.09969e20 1.79808
\(122\) 4.46279e19 0.674821
\(123\) 0 0
\(124\) 2.34412e18 0.0303721
\(125\) −4.51110e19 −0.541549
\(126\) 0 0
\(127\) −8.76640e19 −0.905078 −0.452539 0.891745i \(-0.649482\pi\)
−0.452539 + 0.891745i \(0.649482\pi\)
\(128\) 1.88931e20 1.81054
\(129\) 0 0
\(130\) 1.11504e20 0.922204
\(131\) 1.23083e20 0.946497 0.473249 0.880929i \(-0.343081\pi\)
0.473249 + 0.880929i \(0.343081\pi\)
\(132\) 0 0
\(133\) −1.33200e17 −0.000886980 0
\(134\) −3.61418e20 −2.24137
\(135\) 0 0
\(136\) −2.09419e20 −1.12823
\(137\) 1.10729e20 0.556435 0.278218 0.960518i \(-0.410256\pi\)
0.278218 + 0.960518i \(0.410256\pi\)
\(138\) 0 0
\(139\) −2.36141e20 −1.03402 −0.517012 0.855978i \(-0.672956\pi\)
−0.517012 + 0.855978i \(0.672956\pi\)
\(140\) −1.45095e18 −0.00593520
\(141\) 0 0
\(142\) 3.96308e20 1.41675
\(143\) 1.75828e20 0.588025
\(144\) 0 0
\(145\) 8.52349e20 2.49817
\(146\) 3.91084e20 1.07379
\(147\) 0 0
\(148\) −7.25666e20 −1.75086
\(149\) −2.05810e20 −0.465796 −0.232898 0.972501i \(-0.574821\pi\)
−0.232898 + 0.972501i \(0.574821\pi\)
\(150\) 0 0
\(151\) −2.24449e20 −0.447546 −0.223773 0.974641i \(-0.571837\pi\)
−0.223773 + 0.974641i \(0.571837\pi\)
\(152\) 3.72624e20 0.697842
\(153\) 0 0
\(154\) −3.47667e18 −0.00575065
\(155\) −1.55618e19 −0.0242052
\(156\) 0 0
\(157\) 3.13212e20 0.431312 0.215656 0.976469i \(-0.430811\pi\)
0.215656 + 0.976469i \(0.430811\pi\)
\(158\) −2.26647e21 −2.93838
\(159\) 0 0
\(160\) −3.30165e20 −0.379832
\(161\) 5.69615e17 0.000617640 0
\(162\) 0 0
\(163\) 2.00591e20 0.193432 0.0967160 0.995312i \(-0.469166\pi\)
0.0967160 + 0.995312i \(0.469166\pi\)
\(164\) −1.74122e21 −1.58429
\(165\) 0 0
\(166\) −2.04022e21 −1.65444
\(167\) 2.56060e21 1.96126 0.980632 0.195860i \(-0.0627497\pi\)
0.980632 + 0.195860i \(0.0627497\pi\)
\(168\) 0 0
\(169\) −1.28126e21 −0.876425
\(170\) 2.89361e21 1.87144
\(171\) 0 0
\(172\) 2.50655e21 1.45063
\(173\) −2.86308e21 −1.56818 −0.784089 0.620649i \(-0.786869\pi\)
−0.784089 + 0.620649i \(0.786869\pi\)
\(174\) 0 0
\(175\) 5.53876e18 0.00271989
\(176\) 2.80472e21 1.30472
\(177\) 0 0
\(178\) 7.13090e21 2.97958
\(179\) −3.79065e20 −0.150179 −0.0750896 0.997177i \(-0.523924\pi\)
−0.0750896 + 0.997177i \(0.523924\pi\)
\(180\) 0 0
\(181\) −3.15578e21 −1.12502 −0.562510 0.826790i \(-0.690164\pi\)
−0.562510 + 0.826790i \(0.690164\pi\)
\(182\) −3.57216e18 −0.00120852
\(183\) 0 0
\(184\) −1.59348e21 −0.485935
\(185\) 4.81744e21 1.39536
\(186\) 0 0
\(187\) 4.56288e21 1.19328
\(188\) −1.40965e21 −0.350439
\(189\) 0 0
\(190\) −5.14865e21 −1.15754
\(191\) −2.79667e21 −0.598170 −0.299085 0.954226i \(-0.596681\pi\)
−0.299085 + 0.954226i \(0.596681\pi\)
\(192\) 0 0
\(193\) −7.34293e20 −0.142257 −0.0711287 0.997467i \(-0.522660\pi\)
−0.0711287 + 0.997467i \(0.522660\pi\)
\(194\) −7.71854e21 −1.42370
\(195\) 0 0
\(196\) −1.15030e22 −1.92477
\(197\) 5.32518e21 0.848995 0.424498 0.905429i \(-0.360451\pi\)
0.424498 + 0.905429i \(0.360451\pi\)
\(198\) 0 0
\(199\) −2.54553e21 −0.368701 −0.184350 0.982861i \(-0.559018\pi\)
−0.184350 + 0.982861i \(0.559018\pi\)
\(200\) −1.54945e22 −2.13990
\(201\) 0 0
\(202\) 1.09023e21 0.136987
\(203\) −2.73061e19 −0.00327376
\(204\) 0 0
\(205\) 1.15593e22 1.26261
\(206\) 2.43945e22 2.54422
\(207\) 0 0
\(208\) 2.88175e21 0.274192
\(209\) −8.11882e21 −0.738080
\(210\) 0 0
\(211\) 7.77610e21 0.645771 0.322886 0.946438i \(-0.395347\pi\)
0.322886 + 0.946438i \(0.395347\pi\)
\(212\) −1.06447e22 −0.845168
\(213\) 0 0
\(214\) 3.20591e22 2.32820
\(215\) −1.66401e22 −1.15609
\(216\) 0 0
\(217\) 4.98543e17 3.17201e−5 0
\(218\) 4.24164e22 2.58343
\(219\) 0 0
\(220\) −8.84383e22 −4.93884
\(221\) 4.68820e21 0.250772
\(222\) 0 0
\(223\) −1.89249e22 −0.929259 −0.464629 0.885505i \(-0.653812\pi\)
−0.464629 + 0.885505i \(0.653812\pi\)
\(224\) 1.05773e19 0.000497757 0
\(225\) 0 0
\(226\) 6.80881e22 2.94470
\(227\) −2.42145e22 −1.00422 −0.502111 0.864803i \(-0.667443\pi\)
−0.502111 + 0.864803i \(0.667443\pi\)
\(228\) 0 0
\(229\) −3.29966e22 −1.25902 −0.629510 0.776992i \(-0.716744\pi\)
−0.629510 + 0.776992i \(0.716744\pi\)
\(230\) 2.20176e22 0.806039
\(231\) 0 0
\(232\) 7.63882e22 2.57567
\(233\) 8.99479e21 0.291145 0.145573 0.989348i \(-0.453498\pi\)
0.145573 + 0.989348i \(0.453498\pi\)
\(234\) 0 0
\(235\) 9.35817e21 0.279284
\(236\) 2.83412e22 0.812371
\(237\) 0 0
\(238\) −9.27005e19 −0.00245245
\(239\) −1.22851e22 −0.312319 −0.156159 0.987732i \(-0.549911\pi\)
−0.156159 + 0.987732i \(0.549911\pi\)
\(240\) 0 0
\(241\) −4.48313e22 −1.05298 −0.526489 0.850182i \(-0.676492\pi\)
−0.526489 + 0.850182i \(0.676492\pi\)
\(242\) −1.36176e23 −3.07507
\(243\) 0 0
\(244\) −3.63685e22 −0.759491
\(245\) 7.63643e22 1.53396
\(246\) 0 0
\(247\) −8.34181e21 −0.155110
\(248\) −1.39466e21 −0.0249562
\(249\) 0 0
\(250\) 5.58617e22 0.926155
\(251\) −3.85123e22 −0.614750 −0.307375 0.951588i \(-0.599451\pi\)
−0.307375 + 0.951588i \(0.599451\pi\)
\(252\) 0 0
\(253\) 3.47192e22 0.513955
\(254\) 1.08556e23 1.54786
\(255\) 0 0
\(256\) −1.43025e23 −1.89292
\(257\) −9.07321e22 −1.15717 −0.578584 0.815623i \(-0.696394\pi\)
−0.578584 + 0.815623i \(0.696394\pi\)
\(258\) 0 0
\(259\) −1.54333e20 −0.00182857
\(260\) −9.08674e22 −1.03791
\(261\) 0 0
\(262\) −1.52415e23 −1.61870
\(263\) 2.56086e21 0.0262305 0.0131152 0.999914i \(-0.495825\pi\)
0.0131152 + 0.999914i \(0.495825\pi\)
\(264\) 0 0
\(265\) 7.06664e22 0.673561
\(266\) 1.64944e20 0.00151691
\(267\) 0 0
\(268\) 2.94529e23 2.52260
\(269\) −3.82577e22 −0.316281 −0.158140 0.987417i \(-0.550550\pi\)
−0.158140 + 0.987417i \(0.550550\pi\)
\(270\) 0 0
\(271\) 3.64857e22 0.281135 0.140567 0.990071i \(-0.455107\pi\)
0.140567 + 0.990071i \(0.455107\pi\)
\(272\) 7.47838e22 0.556420
\(273\) 0 0
\(274\) −1.37117e23 −0.951615
\(275\) 3.37599e23 2.26329
\(276\) 0 0
\(277\) 1.78571e23 1.11752 0.558758 0.829331i \(-0.311278\pi\)
0.558758 + 0.829331i \(0.311278\pi\)
\(278\) 2.92417e23 1.76839
\(279\) 0 0
\(280\) 8.63257e20 0.00487684
\(281\) 2.28261e23 1.24659 0.623293 0.781989i \(-0.285795\pi\)
0.623293 + 0.781989i \(0.285795\pi\)
\(282\) 0 0
\(283\) 1.87538e23 0.957453 0.478727 0.877964i \(-0.341098\pi\)
0.478727 + 0.877964i \(0.341098\pi\)
\(284\) −3.22962e23 −1.59451
\(285\) 0 0
\(286\) −2.17731e23 −1.00564
\(287\) −3.70317e20 −0.00165461
\(288\) 0 0
\(289\) −1.17410e23 −0.491106
\(290\) −1.05548e24 −4.27236
\(291\) 0 0
\(292\) −3.18705e23 −1.20852
\(293\) −2.82617e23 −1.03742 −0.518711 0.854950i \(-0.673588\pi\)
−0.518711 + 0.854950i \(0.673588\pi\)
\(294\) 0 0
\(295\) −1.88147e23 −0.647424
\(296\) 4.31743e23 1.43865
\(297\) 0 0
\(298\) 2.54857e23 0.796604
\(299\) 3.56728e22 0.108009
\(300\) 0 0
\(301\) 5.33087e20 0.00151501
\(302\) 2.77939e23 0.765392
\(303\) 0 0
\(304\) −1.33064e23 −0.344162
\(305\) 2.41438e23 0.605280
\(306\) 0 0
\(307\) −2.47019e23 −0.581991 −0.290996 0.956724i \(-0.593986\pi\)
−0.290996 + 0.956724i \(0.593986\pi\)
\(308\) 2.83324e21 0.00647219
\(309\) 0 0
\(310\) 1.92704e22 0.0413957
\(311\) −3.27126e23 −0.681539 −0.340770 0.940147i \(-0.610688\pi\)
−0.340770 + 0.940147i \(0.610688\pi\)
\(312\) 0 0
\(313\) 3.73412e23 0.732011 0.366005 0.930613i \(-0.380725\pi\)
0.366005 + 0.930613i \(0.380725\pi\)
\(314\) −3.87855e23 −0.737628
\(315\) 0 0
\(316\) 1.84701e24 3.30706
\(317\) 5.83518e22 0.101389 0.0506945 0.998714i \(-0.483857\pi\)
0.0506945 + 0.998714i \(0.483857\pi\)
\(318\) 0 0
\(319\) −1.66436e24 −2.72419
\(320\) 1.16191e24 1.84606
\(321\) 0 0
\(322\) −7.05363e20 −0.00105629
\(323\) −2.16477e23 −0.314766
\(324\) 0 0
\(325\) 3.46871e23 0.475638
\(326\) −2.48394e23 −0.330807
\(327\) 0 0
\(328\) 1.03595e24 1.30178
\(329\) −2.99801e20 −0.000365993 0
\(330\) 0 0
\(331\) 3.84253e23 0.442845 0.221422 0.975178i \(-0.428930\pi\)
0.221422 + 0.975178i \(0.428930\pi\)
\(332\) 1.66263e24 1.86202
\(333\) 0 0
\(334\) −3.17084e24 −3.35415
\(335\) −1.95527e24 −2.01040
\(336\) 0 0
\(337\) 1.30128e24 1.26440 0.632202 0.774803i \(-0.282151\pi\)
0.632202 + 0.774803i \(0.282151\pi\)
\(338\) 1.58661e24 1.49886
\(339\) 0 0
\(340\) −2.35808e24 −2.10624
\(341\) 3.03872e22 0.0263952
\(342\) 0 0
\(343\) −4.89287e21 −0.00402040
\(344\) −1.49130e24 −1.19195
\(345\) 0 0
\(346\) 3.54539e24 2.68189
\(347\) 8.56826e22 0.0630613 0.0315306 0.999503i \(-0.489962\pi\)
0.0315306 + 0.999503i \(0.489962\pi\)
\(348\) 0 0
\(349\) −1.11294e24 −0.775586 −0.387793 0.921746i \(-0.626763\pi\)
−0.387793 + 0.921746i \(0.626763\pi\)
\(350\) −6.85873e21 −0.00465155
\(351\) 0 0
\(352\) 6.44706e23 0.414197
\(353\) 2.23748e24 1.39926 0.699631 0.714505i \(-0.253348\pi\)
0.699631 + 0.714505i \(0.253348\pi\)
\(354\) 0 0
\(355\) 2.14403e24 1.27076
\(356\) −5.81117e24 −3.35343
\(357\) 0 0
\(358\) 4.69402e23 0.256836
\(359\) −2.02884e24 −1.08106 −0.540531 0.841324i \(-0.681777\pi\)
−0.540531 + 0.841324i \(0.681777\pi\)
\(360\) 0 0
\(361\) −1.59324e24 −0.805309
\(362\) 3.90785e24 1.92401
\(363\) 0 0
\(364\) 2.91105e21 0.00136015
\(365\) 2.11577e24 0.963133
\(366\) 0 0
\(367\) −2.03395e24 −0.879049 −0.439525 0.898230i \(-0.644853\pi\)
−0.439525 + 0.898230i \(0.644853\pi\)
\(368\) 5.69034e23 0.239654
\(369\) 0 0
\(370\) −5.96551e24 −2.38633
\(371\) −2.26389e21 −0.000882680 0
\(372\) 0 0
\(373\) −7.78333e23 −0.288358 −0.144179 0.989552i \(-0.546054\pi\)
−0.144179 + 0.989552i \(0.546054\pi\)
\(374\) −5.65029e24 −2.04075
\(375\) 0 0
\(376\) 8.38687e23 0.287949
\(377\) −1.71008e24 −0.572497
\(378\) 0 0
\(379\) 3.15677e24 1.00501 0.502505 0.864575i \(-0.332412\pi\)
0.502505 + 0.864575i \(0.332412\pi\)
\(380\) 4.19578e24 1.30277
\(381\) 0 0
\(382\) 3.46316e24 1.02299
\(383\) 3.09305e24 0.891249 0.445625 0.895220i \(-0.352982\pi\)
0.445625 + 0.895220i \(0.352982\pi\)
\(384\) 0 0
\(385\) −1.88088e22 −0.00515804
\(386\) 9.09286e23 0.243288
\(387\) 0 0
\(388\) 6.29005e24 1.60233
\(389\) 4.58505e24 1.13978 0.569892 0.821719i \(-0.306985\pi\)
0.569892 + 0.821719i \(0.306985\pi\)
\(390\) 0 0
\(391\) 9.25738e23 0.219184
\(392\) 6.84383e24 1.58155
\(393\) 0 0
\(394\) −6.59426e24 −1.45195
\(395\) −1.22617e25 −2.63558
\(396\) 0 0
\(397\) 5.62169e24 1.15175 0.575873 0.817539i \(-0.304662\pi\)
0.575873 + 0.817539i \(0.304662\pi\)
\(398\) 3.15217e24 0.630551
\(399\) 0 0
\(400\) 5.53311e24 1.05536
\(401\) 8.34521e24 1.55441 0.777206 0.629247i \(-0.216636\pi\)
0.777206 + 0.629247i \(0.216636\pi\)
\(402\) 0 0
\(403\) 3.12218e22 0.00554703
\(404\) −8.88456e23 −0.154175
\(405\) 0 0
\(406\) 3.38136e22 0.00559878
\(407\) −9.40691e24 −1.52160
\(408\) 0 0
\(409\) −1.09120e25 −1.68474 −0.842370 0.538900i \(-0.818840\pi\)
−0.842370 + 0.538900i \(0.818840\pi\)
\(410\) −1.43141e25 −2.15932
\(411\) 0 0
\(412\) −1.98798e25 −2.86344
\(413\) 6.02754e21 0.000848427 0
\(414\) 0 0
\(415\) −1.10376e25 −1.48395
\(416\) 6.62414e23 0.0870448
\(417\) 0 0
\(418\) 1.00537e25 1.26226
\(419\) 1.35175e25 1.65906 0.829529 0.558463i \(-0.188609\pi\)
0.829529 + 0.558463i \(0.188609\pi\)
\(420\) 0 0
\(421\) 9.91185e24 1.16272 0.581360 0.813647i \(-0.302521\pi\)
0.581360 + 0.813647i \(0.302521\pi\)
\(422\) −9.62927e24 −1.10440
\(423\) 0 0
\(424\) 6.33318e24 0.694458
\(425\) 9.00159e24 0.965215
\(426\) 0 0
\(427\) −7.73477e21 −0.000793200 0
\(428\) −2.61259e25 −2.62032
\(429\) 0 0
\(430\) 2.06057e25 1.97714
\(431\) −1.91272e24 −0.179522 −0.0897610 0.995963i \(-0.528610\pi\)
−0.0897610 + 0.995963i \(0.528610\pi\)
\(432\) 0 0
\(433\) 5.77174e24 0.518408 0.259204 0.965823i \(-0.416540\pi\)
0.259204 + 0.965823i \(0.416540\pi\)
\(434\) −6.17353e20 −5.42477e−5 0
\(435\) 0 0
\(436\) −3.45663e25 −2.90758
\(437\) −1.64718e24 −0.135571
\(438\) 0 0
\(439\) 8.11473e21 0.000639530 0 0.000319765 1.00000i \(-0.499898\pi\)
0.000319765 1.00000i \(0.499898\pi\)
\(440\) 5.26173e25 4.05815
\(441\) 0 0
\(442\) −5.80548e24 −0.428871
\(443\) −2.51728e24 −0.182010 −0.0910051 0.995850i \(-0.529008\pi\)
−0.0910051 + 0.995850i \(0.529008\pi\)
\(444\) 0 0
\(445\) 3.85783e25 2.67253
\(446\) 2.34350e25 1.58922
\(447\) 0 0
\(448\) −3.72232e22 −0.00241920
\(449\) 1.07451e25 0.683706 0.341853 0.939753i \(-0.388945\pi\)
0.341853 + 0.939753i \(0.388945\pi\)
\(450\) 0 0
\(451\) −2.25716e25 −1.37685
\(452\) −5.54869e25 −3.31417
\(453\) 0 0
\(454\) 2.99852e25 1.71742
\(455\) −1.93254e22 −0.00108398
\(456\) 0 0
\(457\) 8.09140e24 0.435331 0.217665 0.976023i \(-0.430156\pi\)
0.217665 + 0.976023i \(0.430156\pi\)
\(458\) 4.08603e25 2.15317
\(459\) 0 0
\(460\) −1.79428e25 −0.907173
\(461\) −1.48075e25 −0.733368 −0.366684 0.930346i \(-0.619507\pi\)
−0.366684 + 0.930346i \(0.619507\pi\)
\(462\) 0 0
\(463\) −2.65292e25 −1.26097 −0.630485 0.776202i \(-0.717144\pi\)
−0.630485 + 0.776202i \(0.717144\pi\)
\(464\) −2.72783e25 −1.27027
\(465\) 0 0
\(466\) −1.11384e25 −0.497916
\(467\) −3.57468e25 −1.56576 −0.782882 0.622170i \(-0.786251\pi\)
−0.782882 + 0.622170i \(0.786251\pi\)
\(468\) 0 0
\(469\) 6.26397e22 0.00263456
\(470\) −1.15884e25 −0.477631
\(471\) 0 0
\(472\) −1.68619e25 −0.667510
\(473\) 3.24927e25 1.26068
\(474\) 0 0
\(475\) −1.60167e25 −0.597013
\(476\) 7.55442e22 0.00276016
\(477\) 0 0
\(478\) 1.52128e25 0.534127
\(479\) 2.88220e24 0.0992056 0.0496028 0.998769i \(-0.484204\pi\)
0.0496028 + 0.998769i \(0.484204\pi\)
\(480\) 0 0
\(481\) −9.66528e24 −0.319769
\(482\) 5.55152e25 1.80080
\(483\) 0 0
\(484\) 1.10974e26 3.46090
\(485\) −4.17574e25 −1.27699
\(486\) 0 0
\(487\) 6.03233e25 1.77403 0.887014 0.461743i \(-0.152776\pi\)
0.887014 + 0.461743i \(0.152776\pi\)
\(488\) 2.16378e25 0.624059
\(489\) 0 0
\(490\) −9.45632e25 −2.62337
\(491\) −4.20599e25 −1.14444 −0.572222 0.820099i \(-0.693918\pi\)
−0.572222 + 0.820099i \(0.693918\pi\)
\(492\) 0 0
\(493\) −4.43779e25 −1.16177
\(494\) 1.03298e25 0.265269
\(495\) 0 0
\(496\) 4.98034e23 0.0123079
\(497\) −6.86869e22 −0.00166528
\(498\) 0 0
\(499\) −3.96358e25 −0.924979 −0.462490 0.886625i \(-0.653044\pi\)
−0.462490 + 0.886625i \(0.653044\pi\)
\(500\) −4.55232e25 −1.04236
\(501\) 0 0
\(502\) 4.76904e25 1.05134
\(503\) −5.09940e25 −1.10312 −0.551561 0.834135i \(-0.685968\pi\)
−0.551561 + 0.834135i \(0.685968\pi\)
\(504\) 0 0
\(505\) 5.89814e24 0.122870
\(506\) −4.29933e25 −0.878965
\(507\) 0 0
\(508\) −8.84651e25 −1.74207
\(509\) 4.45056e25 0.860193 0.430096 0.902783i \(-0.358480\pi\)
0.430096 + 0.902783i \(0.358480\pi\)
\(510\) 0 0
\(511\) −6.77814e22 −0.00126215
\(512\) 7.80561e25 1.42673
\(513\) 0 0
\(514\) 1.12355e26 1.97899
\(515\) 1.31975e26 2.28204
\(516\) 0 0
\(517\) −1.82735e25 −0.304552
\(518\) 1.91113e23 0.00312721
\(519\) 0 0
\(520\) 5.40625e25 0.852833
\(521\) 1.65819e24 0.0256847 0.0128424 0.999918i \(-0.495912\pi\)
0.0128424 + 0.999918i \(0.495912\pi\)
\(522\) 0 0
\(523\) −4.69898e25 −0.701839 −0.350920 0.936406i \(-0.614131\pi\)
−0.350920 + 0.936406i \(0.614131\pi\)
\(524\) 1.24207e26 1.82180
\(525\) 0 0
\(526\) −3.17115e24 −0.0448593
\(527\) 8.10231e23 0.0112566
\(528\) 0 0
\(529\) −6.75715e25 −0.905596
\(530\) −8.75073e25 −1.15192
\(531\) 0 0
\(532\) −1.34417e23 −0.00170724
\(533\) −2.31916e25 −0.289349
\(534\) 0 0
\(535\) 1.73440e26 2.08828
\(536\) −1.75233e26 −2.07277
\(537\) 0 0
\(538\) 4.73752e25 0.540903
\(539\) −1.49115e26 −1.67274
\(540\) 0 0
\(541\) 6.20242e25 0.671718 0.335859 0.941912i \(-0.390973\pi\)
0.335859 + 0.941912i \(0.390973\pi\)
\(542\) −4.51808e25 −0.480796
\(543\) 0 0
\(544\) 1.71902e25 0.176641
\(545\) 2.29473e26 2.31721
\(546\) 0 0
\(547\) 7.76624e25 0.757410 0.378705 0.925517i \(-0.376370\pi\)
0.378705 + 0.925517i \(0.376370\pi\)
\(548\) 1.11740e26 1.07101
\(549\) 0 0
\(550\) −4.18054e26 −3.87068
\(551\) 7.89624e25 0.718588
\(552\) 0 0
\(553\) 3.92818e23 0.00345384
\(554\) −2.21128e26 −1.91117
\(555\) 0 0
\(556\) −2.38299e26 −1.99027
\(557\) −3.67420e25 −0.301675 −0.150837 0.988559i \(-0.548197\pi\)
−0.150837 + 0.988559i \(0.548197\pi\)
\(558\) 0 0
\(559\) 3.33852e25 0.264936
\(560\) −3.08269e23 −0.00240516
\(561\) 0 0
\(562\) −2.82659e26 −2.13191
\(563\) −1.24531e26 −0.923526 −0.461763 0.887003i \(-0.652783\pi\)
−0.461763 + 0.887003i \(0.652783\pi\)
\(564\) 0 0
\(565\) 3.68357e26 2.64125
\(566\) −2.32231e26 −1.63743
\(567\) 0 0
\(568\) 1.92150e26 1.31018
\(569\) 5.63513e25 0.377866 0.188933 0.981990i \(-0.439497\pi\)
0.188933 + 0.981990i \(0.439497\pi\)
\(570\) 0 0
\(571\) −1.23813e26 −0.803012 −0.401506 0.915856i \(-0.631513\pi\)
−0.401506 + 0.915856i \(0.631513\pi\)
\(572\) 1.77435e26 1.13182
\(573\) 0 0
\(574\) 4.58570e23 0.00282971
\(575\) 6.84936e25 0.415724
\(576\) 0 0
\(577\) −3.01722e26 −1.77189 −0.885945 0.463790i \(-0.846489\pi\)
−0.885945 + 0.463790i \(0.846489\pi\)
\(578\) 1.45390e26 0.839888
\(579\) 0 0
\(580\) 8.60137e26 4.80841
\(581\) 3.53605e23 0.00194467
\(582\) 0 0
\(583\) −1.37989e26 −0.734501
\(584\) 1.89617e26 0.993014
\(585\) 0 0
\(586\) 3.49968e26 1.77420
\(587\) 6.00743e25 0.299659 0.149829 0.988712i \(-0.452128\pi\)
0.149829 + 0.988712i \(0.452128\pi\)
\(588\) 0 0
\(589\) −1.44166e24 −0.00696254
\(590\) 2.32986e26 1.10722
\(591\) 0 0
\(592\) −1.54176e26 −0.709511
\(593\) 9.94196e25 0.450248 0.225124 0.974330i \(-0.427721\pi\)
0.225124 + 0.974330i \(0.427721\pi\)
\(594\) 0 0
\(595\) −5.01511e23 −0.00219973
\(596\) −2.07690e26 −0.896554
\(597\) 0 0
\(598\) −4.41742e25 −0.184717
\(599\) 1.95677e26 0.805350 0.402675 0.915343i \(-0.368080\pi\)
0.402675 + 0.915343i \(0.368080\pi\)
\(600\) 0 0
\(601\) −4.73445e26 −1.88782 −0.943912 0.330197i \(-0.892885\pi\)
−0.943912 + 0.330197i \(0.892885\pi\)
\(602\) −6.60129e23 −0.00259097
\(603\) 0 0
\(604\) −2.26500e26 −0.861426
\(605\) −7.36715e26 −2.75818
\(606\) 0 0
\(607\) −1.22707e26 −0.445222 −0.222611 0.974907i \(-0.571458\pi\)
−0.222611 + 0.974907i \(0.571458\pi\)
\(608\) −3.05868e25 −0.109257
\(609\) 0 0
\(610\) −2.98976e26 −1.03515
\(611\) −1.87754e25 −0.0640026
\(612\) 0 0
\(613\) 3.20354e26 1.05866 0.529329 0.848417i \(-0.322444\pi\)
0.529329 + 0.848417i \(0.322444\pi\)
\(614\) 3.05888e26 0.995320
\(615\) 0 0
\(616\) −1.68566e24 −0.00531807
\(617\) −9.79781e25 −0.304383 −0.152191 0.988351i \(-0.548633\pi\)
−0.152191 + 0.988351i \(0.548633\pi\)
\(618\) 0 0
\(619\) −1.22713e25 −0.0369682 −0.0184841 0.999829i \(-0.505884\pi\)
−0.0184841 + 0.999829i \(0.505884\pi\)
\(620\) −1.57040e25 −0.0465896
\(621\) 0 0
\(622\) 4.05085e26 1.16557
\(623\) −1.23591e24 −0.00350226
\(624\) 0 0
\(625\) −1.90020e26 −0.522324
\(626\) −4.62402e26 −1.25188
\(627\) 0 0
\(628\) 3.16074e26 0.830179
\(629\) −2.50822e26 −0.648909
\(630\) 0 0
\(631\) 4.30803e26 1.08143 0.540716 0.841205i \(-0.318153\pi\)
0.540716 + 0.841205i \(0.318153\pi\)
\(632\) −1.09890e27 −2.71735
\(633\) 0 0
\(634\) −7.22579e25 −0.173395
\(635\) 5.87288e26 1.38836
\(636\) 0 0
\(637\) −1.53211e26 −0.351532
\(638\) 2.06101e27 4.65890
\(639\) 0 0
\(640\) −1.26571e27 −2.77730
\(641\) −5.54923e26 −1.19972 −0.599862 0.800104i \(-0.704778\pi\)
−0.599862 + 0.800104i \(0.704778\pi\)
\(642\) 0 0
\(643\) 6.72911e26 1.41239 0.706193 0.708019i \(-0.250411\pi\)
0.706193 + 0.708019i \(0.250411\pi\)
\(644\) 5.74820e23 0.00118882
\(645\) 0 0
\(646\) 2.68066e26 0.538311
\(647\) 3.57860e26 0.708146 0.354073 0.935218i \(-0.384796\pi\)
0.354073 + 0.935218i \(0.384796\pi\)
\(648\) 0 0
\(649\) 3.67391e26 0.705999
\(650\) −4.29536e26 −0.813435
\(651\) 0 0
\(652\) 2.02423e26 0.372313
\(653\) −5.10990e26 −0.926269 −0.463135 0.886288i \(-0.653275\pi\)
−0.463135 + 0.886288i \(0.653275\pi\)
\(654\) 0 0
\(655\) −8.24568e26 −1.45189
\(656\) −3.69940e26 −0.642014
\(657\) 0 0
\(658\) 3.71249e23 0.000625920 0
\(659\) 6.67000e26 1.10844 0.554222 0.832369i \(-0.313016\pi\)
0.554222 + 0.832369i \(0.313016\pi\)
\(660\) 0 0
\(661\) 2.27178e26 0.366819 0.183410 0.983037i \(-0.441287\pi\)
0.183410 + 0.983037i \(0.441287\pi\)
\(662\) −4.75827e26 −0.757352
\(663\) 0 0
\(664\) −9.89202e26 −1.52999
\(665\) 8.92348e23 0.00136059
\(666\) 0 0
\(667\) −3.37674e26 −0.500382
\(668\) 2.58400e27 3.77500
\(669\) 0 0
\(670\) 2.42125e27 3.43818
\(671\) −4.71450e26 −0.660043
\(672\) 0 0
\(673\) 2.72017e26 0.370214 0.185107 0.982718i \(-0.440737\pi\)
0.185107 + 0.982718i \(0.440737\pi\)
\(674\) −1.61139e27 −2.16238
\(675\) 0 0
\(676\) −1.29297e27 −1.68692
\(677\) −4.27674e26 −0.550200 −0.275100 0.961416i \(-0.588711\pi\)
−0.275100 + 0.961416i \(0.588711\pi\)
\(678\) 0 0
\(679\) 1.33775e24 0.00167345
\(680\) 1.40296e27 1.73066
\(681\) 0 0
\(682\) −3.76290e25 −0.0451409
\(683\) 1.11866e26 0.132344 0.0661719 0.997808i \(-0.478921\pi\)
0.0661719 + 0.997808i \(0.478921\pi\)
\(684\) 0 0
\(685\) −7.41805e26 −0.853551
\(686\) 6.05892e24 0.00687569
\(687\) 0 0
\(688\) 5.32543e26 0.587847
\(689\) −1.41779e26 −0.154358
\(690\) 0 0
\(691\) 1.82536e27 1.93333 0.966667 0.256037i \(-0.0824171\pi\)
0.966667 + 0.256037i \(0.0824171\pi\)
\(692\) −2.88924e27 −3.01839
\(693\) 0 0
\(694\) −1.06102e26 −0.107847
\(695\) 1.58198e27 1.58615
\(696\) 0 0
\(697\) −6.01839e26 −0.587177
\(698\) 1.37817e27 1.32641
\(699\) 0 0
\(700\) 5.58937e24 0.00523518
\(701\) 2.03227e27 1.87785 0.938924 0.344125i \(-0.111824\pi\)
0.938924 + 0.344125i \(0.111824\pi\)
\(702\) 0 0
\(703\) 4.46292e26 0.401369
\(704\) −2.26883e27 −2.01308
\(705\) 0 0
\(706\) −2.77070e27 −2.39301
\(707\) −1.88955e23 −0.000161017 0
\(708\) 0 0
\(709\) −1.86039e27 −1.54335 −0.771675 0.636017i \(-0.780581\pi\)
−0.771675 + 0.636017i \(0.780581\pi\)
\(710\) −2.65499e27 −2.17324
\(711\) 0 0
\(712\) 3.45742e27 2.75544
\(713\) 6.16510e24 0.00484830
\(714\) 0 0
\(715\) −1.17793e27 −0.902009
\(716\) −3.82529e26 −0.289062
\(717\) 0 0
\(718\) 2.51234e27 1.84883
\(719\) −1.92780e27 −1.40003 −0.700014 0.714129i \(-0.746823\pi\)
−0.700014 + 0.714129i \(0.746823\pi\)
\(720\) 0 0
\(721\) −4.22798e24 −0.00299053
\(722\) 1.97293e27 1.37724
\(723\) 0 0
\(724\) −3.18462e27 −2.16541
\(725\) −3.28343e27 −2.20352
\(726\) 0 0
\(727\) −1.11559e27 −0.729339 −0.364670 0.931137i \(-0.618818\pi\)
−0.364670 + 0.931137i \(0.618818\pi\)
\(728\) −1.73196e24 −0.00111761
\(729\) 0 0
\(730\) −2.61999e27 −1.64715
\(731\) 8.66372e26 0.537637
\(732\) 0 0
\(733\) 2.04762e27 1.23812 0.619059 0.785344i \(-0.287514\pi\)
0.619059 + 0.785344i \(0.287514\pi\)
\(734\) 2.51868e27 1.50335
\(735\) 0 0
\(736\) 1.30801e26 0.0760802
\(737\) 3.81802e27 2.19229
\(738\) 0 0
\(739\) 1.50373e27 0.841486 0.420743 0.907180i \(-0.361769\pi\)
0.420743 + 0.907180i \(0.361769\pi\)
\(740\) 4.86146e27 2.68575
\(741\) 0 0
\(742\) 2.80341e24 0.00150956
\(743\) −1.88836e27 −1.00390 −0.501950 0.864897i \(-0.667384\pi\)
−0.501950 + 0.864897i \(0.667384\pi\)
\(744\) 0 0
\(745\) 1.37878e27 0.714514
\(746\) 9.63821e26 0.493149
\(747\) 0 0
\(748\) 4.60457e27 2.29681
\(749\) −5.55638e24 −0.00273662
\(750\) 0 0
\(751\) 8.19244e26 0.393400 0.196700 0.980464i \(-0.436978\pi\)
0.196700 + 0.980464i \(0.436978\pi\)
\(752\) −2.99495e26 −0.142011
\(753\) 0 0
\(754\) 2.11762e27 0.979082
\(755\) 1.50365e27 0.686518
\(756\) 0 0
\(757\) 1.94096e27 0.864184 0.432092 0.901829i \(-0.357776\pi\)
0.432092 + 0.901829i \(0.357776\pi\)
\(758\) −3.90907e27 −1.71876
\(759\) 0 0
\(760\) −2.49632e27 −1.07046
\(761\) −2.40135e27 −1.01695 −0.508477 0.861076i \(-0.669791\pi\)
−0.508477 + 0.861076i \(0.669791\pi\)
\(762\) 0 0
\(763\) −7.35147e24 −0.00303663
\(764\) −2.82223e27 −1.15134
\(765\) 0 0
\(766\) −3.83018e27 −1.52421
\(767\) 3.77482e26 0.148368
\(768\) 0 0
\(769\) −3.49979e27 −1.34197 −0.670984 0.741472i \(-0.734128\pi\)
−0.670984 + 0.741472i \(0.734128\pi\)
\(770\) 2.32913e25 0.00882128
\(771\) 0 0
\(772\) −7.41002e26 −0.273814
\(773\) 3.84483e27 1.40337 0.701685 0.712487i \(-0.252431\pi\)
0.701685 + 0.712487i \(0.252431\pi\)
\(774\) 0 0
\(775\) 5.99475e25 0.0213503
\(776\) −3.74233e27 −1.31661
\(777\) 0 0
\(778\) −5.67774e27 −1.94926
\(779\) 1.07086e27 0.363186
\(780\) 0 0
\(781\) −4.18661e27 −1.38573
\(782\) −1.14636e27 −0.374848
\(783\) 0 0
\(784\) −2.44394e27 −0.779987
\(785\) −2.09830e27 −0.661615
\(786\) 0 0
\(787\) 4.22212e27 1.29948 0.649741 0.760155i \(-0.274877\pi\)
0.649741 + 0.760155i \(0.274877\pi\)
\(788\) 5.37384e27 1.63413
\(789\) 0 0
\(790\) 1.51838e28 4.50737
\(791\) −1.18008e25 −0.00346127
\(792\) 0 0
\(793\) −4.84399e26 −0.138710
\(794\) −6.96142e27 −1.96971
\(795\) 0 0
\(796\) −2.56879e27 −0.709667
\(797\) 2.10706e27 0.575205 0.287602 0.957750i \(-0.407142\pi\)
0.287602 + 0.957750i \(0.407142\pi\)
\(798\) 0 0
\(799\) −4.87237e26 −0.129881
\(800\) 1.27187e27 0.335033
\(801\) 0 0
\(802\) −1.03340e28 −2.65835
\(803\) −4.13142e27 −1.05027
\(804\) 0 0
\(805\) −3.81602e24 −0.000947436 0
\(806\) −3.86625e25 −0.00948652
\(807\) 0 0
\(808\) 5.28596e26 0.126682
\(809\) −2.31386e27 −0.548056 −0.274028 0.961722i \(-0.588356\pi\)
−0.274028 + 0.961722i \(0.588356\pi\)
\(810\) 0 0
\(811\) 2.32427e27 0.537760 0.268880 0.963174i \(-0.413347\pi\)
0.268880 + 0.963174i \(0.413347\pi\)
\(812\) −2.75556e25 −0.00630127
\(813\) 0 0
\(814\) 1.16487e28 2.60224
\(815\) −1.34382e27 −0.296717
\(816\) 0 0
\(817\) −1.54155e27 −0.332544
\(818\) 1.35125e28 2.88124
\(819\) 0 0
\(820\) 1.16649e28 2.43025
\(821\) −6.08231e27 −1.25259 −0.626295 0.779587i \(-0.715429\pi\)
−0.626295 + 0.779587i \(0.715429\pi\)
\(822\) 0 0
\(823\) −3.84390e27 −0.773523 −0.386762 0.922180i \(-0.626406\pi\)
−0.386762 + 0.922180i \(0.626406\pi\)
\(824\) 1.18277e28 2.35283
\(825\) 0 0
\(826\) −7.46400e24 −0.00145098
\(827\) −5.47093e27 −1.05138 −0.525689 0.850677i \(-0.676193\pi\)
−0.525689 + 0.850677i \(0.676193\pi\)
\(828\) 0 0
\(829\) 8.53655e26 0.160330 0.0801649 0.996782i \(-0.474455\pi\)
0.0801649 + 0.996782i \(0.474455\pi\)
\(830\) 1.36681e28 2.53785
\(831\) 0 0
\(832\) −2.33115e27 −0.423056
\(833\) −3.97594e27 −0.713365
\(834\) 0 0
\(835\) −1.71543e28 −3.00850
\(836\) −8.19301e27 −1.42064
\(837\) 0 0
\(838\) −1.67389e28 −2.83732
\(839\) 8.76504e27 1.46898 0.734489 0.678621i \(-0.237422\pi\)
0.734489 + 0.678621i \(0.237422\pi\)
\(840\) 0 0
\(841\) 1.00841e28 1.65224
\(842\) −1.22740e28 −1.98848
\(843\) 0 0
\(844\) 7.84716e27 1.24297
\(845\) 8.58357e27 1.34440
\(846\) 0 0
\(847\) 2.36016e25 0.00361451
\(848\) −2.26158e27 −0.342493
\(849\) 0 0
\(850\) −1.11468e28 −1.65071
\(851\) −1.90852e27 −0.279489
\(852\) 0 0
\(853\) 4.55754e27 0.652702 0.326351 0.945249i \(-0.394181\pi\)
0.326351 + 0.945249i \(0.394181\pi\)
\(854\) 9.57809e24 0.00135653
\(855\) 0 0
\(856\) 1.55438e28 2.15307
\(857\) −8.72531e27 −1.19526 −0.597631 0.801771i \(-0.703891\pi\)
−0.597631 + 0.801771i \(0.703891\pi\)
\(858\) 0 0
\(859\) −6.82608e27 −0.914610 −0.457305 0.889310i \(-0.651185\pi\)
−0.457305 + 0.889310i \(0.651185\pi\)
\(860\) −1.67921e28 −2.22521
\(861\) 0 0
\(862\) 2.36855e27 0.307018
\(863\) −1.70088e27 −0.218058 −0.109029 0.994039i \(-0.534774\pi\)
−0.109029 + 0.994039i \(0.534774\pi\)
\(864\) 0 0
\(865\) 1.91806e28 2.40552
\(866\) −7.14723e27 −0.886580
\(867\) 0 0
\(868\) 5.03098e23 6.10542e−5 0
\(869\) 2.39431e28 2.87404
\(870\) 0 0
\(871\) 3.92289e27 0.460716
\(872\) 2.05656e28 2.38910
\(873\) 0 0
\(874\) 2.03973e27 0.231854
\(875\) −9.68176e24 −0.00108862
\(876\) 0 0
\(877\) −9.69466e27 −1.06669 −0.533343 0.845899i \(-0.679064\pi\)
−0.533343 + 0.845899i \(0.679064\pi\)
\(878\) −1.00486e25 −0.00109372
\(879\) 0 0
\(880\) −1.87897e28 −2.00140
\(881\) 2.20901e27 0.232769 0.116385 0.993204i \(-0.462869\pi\)
0.116385 + 0.993204i \(0.462869\pi\)
\(882\) 0 0
\(883\) −5.45945e27 −0.563018 −0.281509 0.959559i \(-0.590835\pi\)
−0.281509 + 0.959559i \(0.590835\pi\)
\(884\) 4.73104e27 0.482681
\(885\) 0 0
\(886\) 3.11718e27 0.311273
\(887\) −1.69664e28 −1.67616 −0.838081 0.545546i \(-0.816322\pi\)
−0.838081 + 0.545546i \(0.816322\pi\)
\(888\) 0 0
\(889\) −1.88145e25 −0.00181939
\(890\) −4.77721e28 −4.57056
\(891\) 0 0
\(892\) −1.90978e28 −1.78862
\(893\) 8.66950e26 0.0803351
\(894\) 0 0
\(895\) 2.53947e27 0.230369
\(896\) 4.05485e25 0.00363956
\(897\) 0 0
\(898\) −1.33058e28 −1.16927
\(899\) −2.95541e26 −0.0256981
\(900\) 0 0
\(901\) −3.67927e27 −0.313239
\(902\) 2.79508e28 2.35468
\(903\) 0 0
\(904\) 3.30125e28 2.72319
\(905\) 2.11415e28 1.72574
\(906\) 0 0
\(907\) 1.06963e28 0.854996 0.427498 0.904016i \(-0.359395\pi\)
0.427498 + 0.904016i \(0.359395\pi\)
\(908\) −2.44357e28 −1.93290
\(909\) 0 0
\(910\) 2.39310e25 0.00185382
\(911\) 2.36272e28 1.81129 0.905645 0.424036i \(-0.139387\pi\)
0.905645 + 0.424036i \(0.139387\pi\)
\(912\) 0 0
\(913\) 2.15530e28 1.61821
\(914\) −1.00197e28 −0.744502
\(915\) 0 0
\(916\) −3.32982e28 −2.42333
\(917\) 2.64161e25 0.00190265
\(918\) 0 0
\(919\) 4.79736e27 0.338458 0.169229 0.985577i \(-0.445872\pi\)
0.169229 + 0.985577i \(0.445872\pi\)
\(920\) 1.06752e28 0.745406
\(921\) 0 0
\(922\) 1.83363e28 1.25420
\(923\) −4.30160e27 −0.291215
\(924\) 0 0
\(925\) −1.85578e28 −1.23078
\(926\) 3.28515e28 2.15651
\(927\) 0 0
\(928\) −6.27031e27 −0.403258
\(929\) 4.77499e27 0.303965 0.151982 0.988383i \(-0.451434\pi\)
0.151982 + 0.988383i \(0.451434\pi\)
\(930\) 0 0
\(931\) 7.07446e27 0.441237
\(932\) 9.07699e27 0.560390
\(933\) 0 0
\(934\) 4.42658e28 2.67777
\(935\) −3.05681e28 −1.83045
\(936\) 0 0
\(937\) −3.08658e27 −0.181114 −0.0905568 0.995891i \(-0.528865\pi\)
−0.0905568 + 0.995891i \(0.528865\pi\)
\(938\) −7.75677e25 −0.00450562
\(939\) 0 0
\(940\) 9.44368e27 0.537560
\(941\) −2.51804e27 −0.141893 −0.0709466 0.997480i \(-0.522602\pi\)
−0.0709466 + 0.997480i \(0.522602\pi\)
\(942\) 0 0
\(943\) −4.57943e27 −0.252901
\(944\) 6.02139e27 0.329202
\(945\) 0 0
\(946\) −4.02363e28 −2.15602
\(947\) −8.85552e27 −0.469774 −0.234887 0.972023i \(-0.575472\pi\)
−0.234887 + 0.972023i \(0.575472\pi\)
\(948\) 0 0
\(949\) −4.24489e27 −0.220718
\(950\) 1.98337e28 1.02101
\(951\) 0 0
\(952\) −4.49458e25 −0.00226797
\(953\) −1.35516e28 −0.677030 −0.338515 0.940961i \(-0.609925\pi\)
−0.338515 + 0.940961i \(0.609925\pi\)
\(954\) 0 0
\(955\) 1.87358e28 0.917570
\(956\) −1.23973e28 −0.601144
\(957\) 0 0
\(958\) −3.56907e27 −0.169661
\(959\) 2.37647e25 0.00111855
\(960\) 0 0
\(961\) −2.16653e28 −0.999751
\(962\) 1.19687e28 0.546868
\(963\) 0 0
\(964\) −4.52409e28 −2.02675
\(965\) 4.91925e27 0.218217
\(966\) 0 0
\(967\) 5.51108e27 0.239710 0.119855 0.992791i \(-0.461757\pi\)
0.119855 + 0.992791i \(0.461757\pi\)
\(968\) −6.60250e28 −2.84376
\(969\) 0 0
\(970\) 5.17088e28 2.18390
\(971\) −2.37023e28 −0.991308 −0.495654 0.868520i \(-0.665072\pi\)
−0.495654 + 0.868520i \(0.665072\pi\)
\(972\) 0 0
\(973\) −5.06808e25 −0.00207860
\(974\) −7.46993e28 −3.03394
\(975\) 0 0
\(976\) −7.72688e27 −0.307773
\(977\) 6.97540e27 0.275151 0.137575 0.990491i \(-0.456069\pi\)
0.137575 + 0.990491i \(0.456069\pi\)
\(978\) 0 0
\(979\) −7.53310e28 −2.91433
\(980\) 7.70621e28 2.95252
\(981\) 0 0
\(982\) 5.20835e28 1.95722
\(983\) −1.96330e28 −0.730683 −0.365341 0.930874i \(-0.619048\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(984\) 0 0
\(985\) −3.56750e28 −1.30233
\(986\) 5.49538e28 1.98686
\(987\) 0 0
\(988\) −8.41804e27 −0.298552
\(989\) 6.59227e27 0.231564
\(990\) 0 0
\(991\) 4.20556e28 1.44918 0.724592 0.689178i \(-0.242028\pi\)
0.724592 + 0.689178i \(0.242028\pi\)
\(992\) 1.14481e26 0.00390725
\(993\) 0 0
\(994\) 8.50560e25 0.00284796
\(995\) 1.70533e28 0.565573
\(996\) 0 0
\(997\) −1.15418e28 −0.375551 −0.187776 0.982212i \(-0.560128\pi\)
−0.187776 + 0.982212i \(0.560128\pi\)
\(998\) 4.90816e28 1.58190
\(999\) 0 0
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 9.20.a.c.1.1 2
3.2 odd 2 3.20.a.b.1.2 2
12.11 even 2 48.20.a.j.1.2 2
15.2 even 4 75.20.b.b.49.4 4
15.8 even 4 75.20.b.b.49.1 4
15.14 odd 2 75.20.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.20.a.b.1.2 2 3.2 odd 2
9.20.a.c.1.1 2 1.1 even 1 trivial
48.20.a.j.1.2 2 12.11 even 2
75.20.a.b.1.1 2 15.14 odd 2
75.20.b.b.49.1 4 15.8 even 4
75.20.b.b.49.4 4 15.2 even 4