Properties

Label 25.34.a.e.1.10
Level $25$
Weight $34$
Character 25.1
Self dual yes
Analytic conductor $172.457$
Analytic rank $0$
Dimension $11$
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] = [25,34,Mod(1,25)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(25, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 34, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.1");
 
S:= CuspForms(chi, 34);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 25.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: \(172.457072203\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - x^{10} - 71909733210 x^{9} - 392361971317440 x^{8} + \cdots + 13\!\cdots\!48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{47}\cdot 3^{12}\cdot 5^{30}\cdot 7^{2}\cdot 11^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-139697.\) of defining polynomial
Character \(\chi\) \(=\) 25.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+140551. q^{2} -1.29136e8 q^{3} +1.11647e10 q^{4} -1.81502e13 q^{6} -4.00085e13 q^{7} +3.61884e14 q^{8} +1.11170e16 q^{9} +O(q^{10})\) \(q+140551. q^{2} -1.29136e8 q^{3} +1.11647e10 q^{4} -1.81502e13 q^{6} -4.00085e13 q^{7} +3.61884e14 q^{8} +1.11170e16 q^{9} +2.48192e17 q^{11} -1.44176e18 q^{12} +3.61009e18 q^{13} -5.62325e18 q^{14} -4.50407e19 q^{16} -5.03453e19 q^{17} +1.56251e21 q^{18} -1.19790e21 q^{19} +5.16654e21 q^{21} +3.48837e22 q^{22} -2.35311e22 q^{23} -4.67322e22 q^{24} +5.07403e23 q^{26} -7.17728e23 q^{27} -4.46683e23 q^{28} -2.45131e24 q^{29} -2.88511e24 q^{31} -9.43908e24 q^{32} -3.20505e25 q^{33} -7.07609e24 q^{34} +1.24118e26 q^{36} +3.58819e25 q^{37} -1.68366e26 q^{38} -4.66192e26 q^{39} +6.61568e25 q^{41} +7.26162e26 q^{42} +1.70988e27 q^{43} +2.77099e27 q^{44} -3.30733e27 q^{46} +4.53211e27 q^{47} +5.81637e27 q^{48} -6.13031e27 q^{49} +6.50138e27 q^{51} +4.03056e28 q^{52} +1.11477e28 q^{53} -1.00877e29 q^{54} -1.44784e28 q^{56} +1.54691e29 q^{57} -3.44534e29 q^{58} +1.06018e29 q^{59} +9.42654e28 q^{61} -4.05506e29 q^{62} -4.44775e29 q^{63} -9.39777e29 q^{64} -4.50473e30 q^{66} +2.59250e30 q^{67} -5.62090e29 q^{68} +3.03871e30 q^{69} -4.27372e30 q^{71} +4.02306e30 q^{72} +2.23754e30 q^{73} +5.04324e30 q^{74} -1.33741e31 q^{76} -9.92980e30 q^{77} -6.55239e31 q^{78} -2.21899e31 q^{79} +3.08843e31 q^{81} +9.29841e30 q^{82} +2.60461e31 q^{83} +5.76827e31 q^{84} +2.40325e32 q^{86} +3.16552e32 q^{87} +8.98167e31 q^{88} +1.65644e32 q^{89} -1.44435e32 q^{91} -2.62718e32 q^{92} +3.72571e32 q^{93} +6.36993e32 q^{94} +1.21892e33 q^{96} -1.37226e32 q^{97} -8.61622e32 q^{98} +2.75915e33 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 9393 q^{2} + 80330849 q^{3} + 49338206677 q^{4} - 17783977676723 q^{6} + 21344025107658 q^{7} - 970199832420105 q^{8} + 24\!\cdots\!58 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q + 9393 q^{2} + 80330849 q^{3} + 49338206677 q^{4} - 17783977676723 q^{6} + 21344025107658 q^{7} - 970199832420105 q^{8} + 24\!\cdots\!58 q^{9}+ \cdots + 13\!\cdots\!06 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 140551. 1.51649 0.758245 0.651970i \(-0.226057\pi\)
0.758245 + 0.651970i \(0.226057\pi\)
\(3\) −1.29136e8 −1.73199 −0.865996 0.500051i \(-0.833315\pi\)
−0.865996 + 0.500051i \(0.833315\pi\)
\(4\) 1.11647e10 1.29974
\(5\) 0 0
\(6\) −1.81502e13 −2.62655
\(7\) −4.00085e13 −0.455025 −0.227512 0.973775i \(-0.573059\pi\)
−0.227512 + 0.973775i \(0.573059\pi\)
\(8\) 3.61884e14 0.454553
\(9\) 1.11170e16 1.99980
\(10\) 0 0
\(11\) 2.48192e17 1.62858 0.814289 0.580460i \(-0.197127\pi\)
0.814289 + 0.580460i \(0.197127\pi\)
\(12\) −1.44176e18 −2.25114
\(13\) 3.61009e18 1.50471 0.752356 0.658757i \(-0.228917\pi\)
0.752356 + 0.658757i \(0.228917\pi\)
\(14\) −5.62325e18 −0.690040
\(15\) 0 0
\(16\) −4.50407e19 −0.610416
\(17\) −5.03453e19 −0.250930 −0.125465 0.992098i \(-0.540042\pi\)
−0.125465 + 0.992098i \(0.540042\pi\)
\(18\) 1.56251e21 3.03267
\(19\) −1.19790e21 −0.952764 −0.476382 0.879238i \(-0.658052\pi\)
−0.476382 + 0.879238i \(0.658052\pi\)
\(20\) 0 0
\(21\) 5.16654e21 0.788099
\(22\) 3.48837e22 2.46972
\(23\) −2.35311e22 −0.800081 −0.400040 0.916498i \(-0.631004\pi\)
−0.400040 + 0.916498i \(0.631004\pi\)
\(24\) −4.67322e22 −0.787282
\(25\) 0 0
\(26\) 5.07403e23 2.28188
\(27\) −7.17728e23 −1.73164
\(28\) −4.46683e23 −0.591414
\(29\) −2.45131e24 −1.81899 −0.909497 0.415711i \(-0.863533\pi\)
−0.909497 + 0.415711i \(0.863533\pi\)
\(30\) 0 0
\(31\) −2.88511e24 −0.712352 −0.356176 0.934419i \(-0.615920\pi\)
−0.356176 + 0.934419i \(0.615920\pi\)
\(32\) −9.43908e24 −1.38024
\(33\) −3.20505e25 −2.82068
\(34\) −7.07609e24 −0.380532
\(35\) 0 0
\(36\) 1.24118e26 2.59922
\(37\) 3.58819e25 0.478131 0.239066 0.971003i \(-0.423159\pi\)
0.239066 + 0.971003i \(0.423159\pi\)
\(38\) −1.68366e26 −1.44486
\(39\) −4.66192e26 −2.60615
\(40\) 0 0
\(41\) 6.61568e25 0.162047 0.0810234 0.996712i \(-0.474181\pi\)
0.0810234 + 0.996712i \(0.474181\pi\)
\(42\) 7.26162e26 1.19514
\(43\) 1.70988e27 1.90869 0.954346 0.298702i \(-0.0965538\pi\)
0.954346 + 0.298702i \(0.0965538\pi\)
\(44\) 2.77099e27 2.11673
\(45\) 0 0
\(46\) −3.30733e27 −1.21331
\(47\) 4.53211e27 1.16597 0.582983 0.812485i \(-0.301886\pi\)
0.582983 + 0.812485i \(0.301886\pi\)
\(48\) 5.81637e27 1.05724
\(49\) −6.13031e27 −0.792952
\(50\) 0 0
\(51\) 6.50138e27 0.434608
\(52\) 4.03056e28 1.95573
\(53\) 1.11477e28 0.395031 0.197516 0.980300i \(-0.436713\pi\)
0.197516 + 0.980300i \(0.436713\pi\)
\(54\) −1.00877e29 −2.62601
\(55\) 0 0
\(56\) −1.44784e28 −0.206833
\(57\) 1.54691e29 1.65018
\(58\) −3.44534e29 −2.75848
\(59\) 1.06018e29 0.640210 0.320105 0.947382i \(-0.396282\pi\)
0.320105 + 0.947382i \(0.396282\pi\)
\(60\) 0 0
\(61\) 9.42654e28 0.328405 0.164203 0.986427i \(-0.447495\pi\)
0.164203 + 0.986427i \(0.447495\pi\)
\(62\) −4.05506e29 −1.08027
\(63\) −4.44775e29 −0.909957
\(64\) −9.39777e29 −1.48271
\(65\) 0 0
\(66\) −4.50473e30 −4.27754
\(67\) 2.59250e30 1.92081 0.960405 0.278607i \(-0.0898727\pi\)
0.960405 + 0.278607i \(0.0898727\pi\)
\(68\) −5.62090e29 −0.326143
\(69\) 3.03871e30 1.38573
\(70\) 0 0
\(71\) −4.27372e30 −1.21631 −0.608153 0.793820i \(-0.708089\pi\)
−0.608153 + 0.793820i \(0.708089\pi\)
\(72\) 4.02306e30 0.909013
\(73\) 2.23754e30 0.402664 0.201332 0.979523i \(-0.435473\pi\)
0.201332 + 0.979523i \(0.435473\pi\)
\(74\) 5.04324e30 0.725081
\(75\) 0 0
\(76\) −1.33741e31 −1.23835
\(77\) −9.92980e30 −0.741043
\(78\) −6.55239e31 −3.95220
\(79\) −2.21899e31 −1.08470 −0.542349 0.840153i \(-0.682465\pi\)
−0.542349 + 0.840153i \(0.682465\pi\)
\(80\) 0 0
\(81\) 3.08843e31 0.999390
\(82\) 9.29841e30 0.245742
\(83\) 2.60461e31 0.563579 0.281790 0.959476i \(-0.409072\pi\)
0.281790 + 0.959476i \(0.409072\pi\)
\(84\) 5.76827e31 1.02432
\(85\) 0 0
\(86\) 2.40325e32 2.89451
\(87\) 3.16552e32 3.15048
\(88\) 8.98167e31 0.740275
\(89\) 1.65644e32 1.13302 0.566512 0.824053i \(-0.308292\pi\)
0.566512 + 0.824053i \(0.308292\pi\)
\(90\) 0 0
\(91\) −1.44435e32 −0.684681
\(92\) −2.62718e32 −1.03990
\(93\) 3.72571e32 1.23379
\(94\) 6.36993e32 1.76817
\(95\) 0 0
\(96\) 1.21892e33 2.39057
\(97\) −1.37226e32 −0.226831 −0.113416 0.993548i \(-0.536179\pi\)
−0.113416 + 0.993548i \(0.536179\pi\)
\(98\) −8.61622e32 −1.20250
\(99\) 2.75915e33 3.25682
\(100\) 0 0
\(101\) 1.03548e33 0.878698 0.439349 0.898317i \(-0.355209\pi\)
0.439349 + 0.898317i \(0.355209\pi\)
\(102\) 9.13777e32 0.659079
\(103\) −2.45463e33 −1.50720 −0.753600 0.657333i \(-0.771684\pi\)
−0.753600 + 0.657333i \(0.771684\pi\)
\(104\) 1.30643e33 0.683971
\(105\) 0 0
\(106\) 1.56681e33 0.599061
\(107\) 1.11407e33 0.364822 0.182411 0.983222i \(-0.441610\pi\)
0.182411 + 0.983222i \(0.441610\pi\)
\(108\) −8.01320e33 −2.25068
\(109\) −2.70218e33 −0.651894 −0.325947 0.945388i \(-0.605683\pi\)
−0.325947 + 0.945388i \(0.605683\pi\)
\(110\) 0 0
\(111\) −4.63364e33 −0.828120
\(112\) 1.80201e33 0.277754
\(113\) −2.20215e33 −0.293124 −0.146562 0.989201i \(-0.546821\pi\)
−0.146562 + 0.989201i \(0.546821\pi\)
\(114\) 2.17421e34 2.50248
\(115\) 0 0
\(116\) −2.73681e34 −2.36422
\(117\) 4.01334e34 3.00912
\(118\) 1.49010e34 0.970872
\(119\) 2.01424e33 0.114179
\(120\) 0 0
\(121\) 3.83741e34 1.65227
\(122\) 1.32491e34 0.498023
\(123\) −8.54321e33 −0.280664
\(124\) −3.22114e34 −0.925872
\(125\) 0 0
\(126\) −6.25136e34 −1.37994
\(127\) 4.18545e34 0.810925 0.405463 0.914112i \(-0.367110\pi\)
0.405463 + 0.914112i \(0.367110\pi\)
\(128\) −5.10056e34 −0.868267
\(129\) −2.20807e35 −3.30584
\(130\) 0 0
\(131\) 1.65700e35 1.92463 0.962314 0.271941i \(-0.0876655\pi\)
0.962314 + 0.271941i \(0.0876655\pi\)
\(132\) −3.57833e35 −3.66616
\(133\) 4.79261e34 0.433531
\(134\) 3.64379e35 2.91289
\(135\) 0 0
\(136\) −1.82192e34 −0.114061
\(137\) −1.27396e35 −0.706751 −0.353375 0.935482i \(-0.614966\pi\)
−0.353375 + 0.935482i \(0.614966\pi\)
\(138\) 4.27094e35 2.10145
\(139\) −2.10212e35 −0.918148 −0.459074 0.888398i \(-0.651819\pi\)
−0.459074 + 0.888398i \(0.651819\pi\)
\(140\) 0 0
\(141\) −5.85257e35 −2.01944
\(142\) −6.00676e35 −1.84452
\(143\) 8.95997e35 2.45054
\(144\) −5.00717e35 −1.22071
\(145\) 0 0
\(146\) 3.14488e35 0.610636
\(147\) 7.91642e35 1.37339
\(148\) 4.00610e35 0.621446
\(149\) 3.92609e35 0.544987 0.272494 0.962158i \(-0.412152\pi\)
0.272494 + 0.962158i \(0.412152\pi\)
\(150\) 0 0
\(151\) 1.34206e36 1.49504 0.747521 0.664239i \(-0.231244\pi\)
0.747521 + 0.664239i \(0.231244\pi\)
\(152\) −4.33500e35 −0.433082
\(153\) −5.59689e35 −0.501808
\(154\) −1.39564e36 −1.12378
\(155\) 0 0
\(156\) −5.20489e36 −3.38732
\(157\) 1.18714e36 0.695279 0.347640 0.937628i \(-0.386983\pi\)
0.347640 + 0.937628i \(0.386983\pi\)
\(158\) −3.11882e36 −1.64493
\(159\) −1.43956e36 −0.684191
\(160\) 0 0
\(161\) 9.41447e35 0.364057
\(162\) 4.34082e36 1.51556
\(163\) 2.79919e36 0.882950 0.441475 0.897273i \(-0.354455\pi\)
0.441475 + 0.897273i \(0.354455\pi\)
\(164\) 7.38619e35 0.210619
\(165\) 0 0
\(166\) 3.66082e36 0.854662
\(167\) 1.00045e36 0.211531 0.105766 0.994391i \(-0.466271\pi\)
0.105766 + 0.994391i \(0.466271\pi\)
\(168\) 1.86969e36 0.358233
\(169\) 7.27665e36 1.26416
\(170\) 0 0
\(171\) −1.33170e37 −1.90533
\(172\) 1.90903e37 2.48080
\(173\) −3.30706e36 −0.390554 −0.195277 0.980748i \(-0.562561\pi\)
−0.195277 + 0.980748i \(0.562561\pi\)
\(174\) 4.44917e37 4.77767
\(175\) 0 0
\(176\) −1.11787e37 −0.994109
\(177\) −1.36908e37 −1.10884
\(178\) 2.32814e37 1.71822
\(179\) 7.68220e36 0.516905 0.258452 0.966024i \(-0.416787\pi\)
0.258452 + 0.966024i \(0.416787\pi\)
\(180\) 0 0
\(181\) 1.70691e37 0.956123 0.478062 0.878326i \(-0.341340\pi\)
0.478062 + 0.878326i \(0.341340\pi\)
\(182\) −2.03005e37 −1.03831
\(183\) −1.21730e37 −0.568795
\(184\) −8.51554e36 −0.363679
\(185\) 0 0
\(186\) 5.23653e37 1.87103
\(187\) −1.24953e37 −0.408658
\(188\) 5.05995e37 1.51545
\(189\) 2.87152e37 0.787939
\(190\) 0 0
\(191\) 3.12199e37 0.720080 0.360040 0.932937i \(-0.382763\pi\)
0.360040 + 0.932937i \(0.382763\pi\)
\(192\) 1.21359e38 2.56804
\(193\) −3.37722e37 −0.655940 −0.327970 0.944688i \(-0.606365\pi\)
−0.327970 + 0.944688i \(0.606365\pi\)
\(194\) −1.92873e37 −0.343987
\(195\) 0 0
\(196\) −6.84430e37 −1.03063
\(197\) −1.81875e37 −0.251814 −0.125907 0.992042i \(-0.540184\pi\)
−0.125907 + 0.992042i \(0.540184\pi\)
\(198\) 3.87801e38 4.93894
\(199\) 1.94624e37 0.228098 0.114049 0.993475i \(-0.463618\pi\)
0.114049 + 0.993475i \(0.463618\pi\)
\(200\) 0 0
\(201\) −3.34785e38 −3.32683
\(202\) 1.45538e38 1.33254
\(203\) 9.80733e37 0.827687
\(204\) 7.25859e37 0.564878
\(205\) 0 0
\(206\) −3.45000e38 −2.28565
\(207\) −2.61595e38 −1.60000
\(208\) −1.62601e38 −0.918499
\(209\) −2.97309e38 −1.55165
\(210\) 0 0
\(211\) −4.61601e37 −0.205876 −0.102938 0.994688i \(-0.532824\pi\)
−0.102938 + 0.994688i \(0.532824\pi\)
\(212\) 1.24460e38 0.513438
\(213\) 5.51890e38 2.10663
\(214\) 1.56584e38 0.553249
\(215\) 0 0
\(216\) −2.59734e38 −0.787122
\(217\) 1.15429e38 0.324138
\(218\) −3.79795e38 −0.988591
\(219\) −2.88946e38 −0.697411
\(220\) 0 0
\(221\) −1.81751e38 −0.377577
\(222\) −6.51263e38 −1.25583
\(223\) 3.65945e38 0.655218 0.327609 0.944813i \(-0.393757\pi\)
0.327609 + 0.944813i \(0.393757\pi\)
\(224\) 3.77644e38 0.628044
\(225\) 0 0
\(226\) −3.09514e38 −0.444520
\(227\) 8.68482e38 1.15967 0.579834 0.814734i \(-0.303117\pi\)
0.579834 + 0.814734i \(0.303117\pi\)
\(228\) 1.72708e39 2.14481
\(229\) 3.80140e38 0.439195 0.219598 0.975591i \(-0.429526\pi\)
0.219598 + 0.975591i \(0.429526\pi\)
\(230\) 0 0
\(231\) 1.28229e39 1.28348
\(232\) −8.87089e38 −0.826829
\(233\) 7.33719e38 0.637026 0.318513 0.947918i \(-0.396817\pi\)
0.318513 + 0.947918i \(0.396817\pi\)
\(234\) 5.64079e39 4.56329
\(235\) 0 0
\(236\) 1.18366e39 0.832107
\(237\) 2.86551e39 1.87869
\(238\) 2.83104e38 0.173152
\(239\) −2.06579e39 −1.17902 −0.589509 0.807762i \(-0.700679\pi\)
−0.589509 + 0.807762i \(0.700679\pi\)
\(240\) 0 0
\(241\) 1.68438e39 0.837833 0.418917 0.908025i \(-0.362410\pi\)
0.418917 + 0.908025i \(0.362410\pi\)
\(242\) 5.39353e39 2.50564
\(243\) 1.62204e36 0.000703975 0
\(244\) 1.05244e39 0.426842
\(245\) 0 0
\(246\) −1.20076e39 −0.425624
\(247\) −4.32452e39 −1.43364
\(248\) −1.04408e39 −0.323802
\(249\) −3.36349e39 −0.976115
\(250\) 0 0
\(251\) 3.13801e39 0.798065 0.399032 0.916937i \(-0.369346\pi\)
0.399032 + 0.916937i \(0.369346\pi\)
\(252\) −4.96577e39 −1.18271
\(253\) −5.84024e39 −1.30299
\(254\) 5.88270e39 1.22976
\(255\) 0 0
\(256\) 9.03730e38 0.165989
\(257\) −6.46986e39 −1.11429 −0.557145 0.830415i \(-0.688103\pi\)
−0.557145 + 0.830415i \(0.688103\pi\)
\(258\) −3.10346e40 −5.01327
\(259\) −1.43558e39 −0.217562
\(260\) 0 0
\(261\) −2.72512e40 −3.63762
\(262\) 2.32894e40 2.91868
\(263\) 9.40764e39 1.10716 0.553581 0.832796i \(-0.313261\pi\)
0.553581 + 0.832796i \(0.313261\pi\)
\(264\) −1.15985e40 −1.28215
\(265\) 0 0
\(266\) 6.73607e39 0.657446
\(267\) −2.13905e40 −1.96239
\(268\) 2.89444e40 2.49655
\(269\) −1.06054e40 −0.860229 −0.430115 0.902774i \(-0.641527\pi\)
−0.430115 + 0.902774i \(0.641527\pi\)
\(270\) 0 0
\(271\) 5.90847e39 0.424113 0.212057 0.977257i \(-0.431984\pi\)
0.212057 + 0.977257i \(0.431984\pi\)
\(272\) 2.26759e39 0.153171
\(273\) 1.86517e40 1.18586
\(274\) −1.79056e40 −1.07178
\(275\) 0 0
\(276\) 3.39263e40 1.80109
\(277\) 4.69522e39 0.234822 0.117411 0.993083i \(-0.462540\pi\)
0.117411 + 0.993083i \(0.462540\pi\)
\(278\) −2.95455e40 −1.39236
\(279\) −3.20738e40 −1.42456
\(280\) 0 0
\(281\) −2.77236e40 −1.09445 −0.547225 0.836986i \(-0.684316\pi\)
−0.547225 + 0.836986i \(0.684316\pi\)
\(282\) −8.22586e40 −3.06246
\(283\) −1.78562e40 −0.627064 −0.313532 0.949578i \(-0.601512\pi\)
−0.313532 + 0.949578i \(0.601512\pi\)
\(284\) −4.77147e40 −1.58088
\(285\) 0 0
\(286\) 1.25933e41 3.71622
\(287\) −2.64684e39 −0.0737353
\(288\) −1.04934e41 −2.76020
\(289\) −3.77198e40 −0.937034
\(290\) 0 0
\(291\) 1.77208e40 0.392870
\(292\) 2.49814e40 0.523359
\(293\) 7.90450e40 1.56516 0.782580 0.622551i \(-0.213904\pi\)
0.782580 + 0.622551i \(0.213904\pi\)
\(294\) 1.11266e41 2.08273
\(295\) 0 0
\(296\) 1.29851e40 0.217336
\(297\) −1.78134e41 −2.82011
\(298\) 5.51816e40 0.826468
\(299\) −8.49496e40 −1.20389
\(300\) 0 0
\(301\) −6.84098e40 −0.868503
\(302\) 1.88628e41 2.26721
\(303\) −1.33718e41 −1.52190
\(304\) 5.39542e40 0.581582
\(305\) 0 0
\(306\) −7.86649e40 −0.760987
\(307\) 1.77046e41 1.62294 0.811471 0.584393i \(-0.198667\pi\)
0.811471 + 0.584393i \(0.198667\pi\)
\(308\) −1.10863e41 −0.963164
\(309\) 3.16980e41 2.61046
\(310\) 0 0
\(311\) 6.58911e40 0.487843 0.243922 0.969795i \(-0.421566\pi\)
0.243922 + 0.969795i \(0.421566\pi\)
\(312\) −1.68707e41 −1.18463
\(313\) −1.43570e41 −0.956276 −0.478138 0.878285i \(-0.658688\pi\)
−0.478138 + 0.878285i \(0.658688\pi\)
\(314\) 1.66854e41 1.05438
\(315\) 0 0
\(316\) −2.47744e41 −1.40983
\(317\) 1.92102e41 1.03766 0.518830 0.854877i \(-0.326368\pi\)
0.518830 + 0.854877i \(0.326368\pi\)
\(318\) −2.02332e41 −1.03757
\(319\) −6.08395e41 −2.96237
\(320\) 0 0
\(321\) −1.43867e41 −0.631869
\(322\) 1.32321e41 0.552088
\(323\) 6.03086e40 0.239077
\(324\) 3.44814e41 1.29895
\(325\) 0 0
\(326\) 3.93429e41 1.33898
\(327\) 3.48948e41 1.12908
\(328\) 2.39411e40 0.0736589
\(329\) −1.81323e41 −0.530543
\(330\) 0 0
\(331\) 6.51390e41 1.72456 0.862282 0.506429i \(-0.169035\pi\)
0.862282 + 0.506429i \(0.169035\pi\)
\(332\) 2.90797e41 0.732507
\(333\) 3.98899e41 0.956165
\(334\) 1.40615e41 0.320785
\(335\) 0 0
\(336\) −2.32704e41 −0.481068
\(337\) 1.11640e41 0.219749 0.109874 0.993945i \(-0.464955\pi\)
0.109874 + 0.993945i \(0.464955\pi\)
\(338\) 1.02274e42 1.91708
\(339\) 2.84376e41 0.507689
\(340\) 0 0
\(341\) −7.16062e41 −1.16012
\(342\) −1.87172e42 −2.88942
\(343\) 5.54571e41 0.815838
\(344\) 6.18778e41 0.867602
\(345\) 0 0
\(346\) −4.64810e41 −0.592271
\(347\) 4.51461e41 0.548509 0.274255 0.961657i \(-0.411569\pi\)
0.274255 + 0.961657i \(0.411569\pi\)
\(348\) 3.53420e42 4.09481
\(349\) 1.08192e42 1.19557 0.597783 0.801658i \(-0.296048\pi\)
0.597783 + 0.801658i \(0.296048\pi\)
\(350\) 0 0
\(351\) −2.59107e42 −2.60562
\(352\) −2.34271e42 −2.24783
\(353\) −1.02423e41 −0.0937809 −0.0468904 0.998900i \(-0.514931\pi\)
−0.0468904 + 0.998900i \(0.514931\pi\)
\(354\) −1.92425e42 −1.68154
\(355\) 0 0
\(356\) 1.84936e42 1.47264
\(357\) −2.60111e41 −0.197758
\(358\) 1.07974e42 0.783881
\(359\) −6.14289e40 −0.0425907 −0.0212953 0.999773i \(-0.506779\pi\)
−0.0212953 + 0.999773i \(0.506779\pi\)
\(360\) 0 0
\(361\) −1.45811e41 −0.0922407
\(362\) 2.39908e42 1.44995
\(363\) −4.95547e42 −2.86171
\(364\) −1.61257e42 −0.889908
\(365\) 0 0
\(366\) −1.71093e42 −0.862572
\(367\) 2.92870e42 1.41151 0.705757 0.708454i \(-0.250607\pi\)
0.705757 + 0.708454i \(0.250607\pi\)
\(368\) 1.05986e42 0.488382
\(369\) 7.35464e41 0.324061
\(370\) 0 0
\(371\) −4.46001e41 −0.179749
\(372\) 4.15964e42 1.60360
\(373\) −2.72869e42 −1.00637 −0.503186 0.864178i \(-0.667839\pi\)
−0.503186 + 0.864178i \(0.667839\pi\)
\(374\) −1.75623e42 −0.619726
\(375\) 0 0
\(376\) 1.64010e42 0.529993
\(377\) −8.84946e42 −2.73706
\(378\) 4.03596e42 1.19490
\(379\) 2.51319e42 0.712327 0.356163 0.934424i \(-0.384085\pi\)
0.356163 + 0.934424i \(0.384085\pi\)
\(380\) 0 0
\(381\) −5.40492e42 −1.40452
\(382\) 4.38799e42 1.09199
\(383\) 5.53799e42 1.31999 0.659997 0.751268i \(-0.270557\pi\)
0.659997 + 0.751268i \(0.270557\pi\)
\(384\) 6.58665e42 1.50383
\(385\) 0 0
\(386\) −4.74672e42 −0.994726
\(387\) 1.90087e43 3.81700
\(388\) −1.53209e42 −0.294822
\(389\) 6.43527e42 1.18686 0.593428 0.804887i \(-0.297774\pi\)
0.593428 + 0.804887i \(0.297774\pi\)
\(390\) 0 0
\(391\) 1.18468e42 0.200764
\(392\) −2.21846e42 −0.360439
\(393\) −2.13978e43 −3.33344
\(394\) −2.55627e42 −0.381873
\(395\) 0 0
\(396\) 3.08050e43 4.23303
\(397\) 2.59182e41 0.0341634 0.0170817 0.999854i \(-0.494562\pi\)
0.0170817 + 0.999854i \(0.494562\pi\)
\(398\) 2.73547e42 0.345908
\(399\) −6.18898e42 −0.750873
\(400\) 0 0
\(401\) −2.15733e41 −0.0241010 −0.0120505 0.999927i \(-0.503836\pi\)
−0.0120505 + 0.999927i \(0.503836\pi\)
\(402\) −4.70543e43 −5.04510
\(403\) −1.04155e43 −1.07188
\(404\) 1.15608e43 1.14208
\(405\) 0 0
\(406\) 1.37843e43 1.25518
\(407\) 8.90561e42 0.778674
\(408\) 2.35275e42 0.197552
\(409\) 3.07671e42 0.248114 0.124057 0.992275i \(-0.460409\pi\)
0.124057 + 0.992275i \(0.460409\pi\)
\(410\) 0 0
\(411\) 1.64514e43 1.22409
\(412\) −2.74051e43 −1.95897
\(413\) −4.24164e42 −0.291312
\(414\) −3.67675e43 −2.42638
\(415\) 0 0
\(416\) −3.40760e43 −2.07687
\(417\) 2.71458e43 1.59022
\(418\) −4.17871e43 −2.35306
\(419\) 8.31146e42 0.449931 0.224966 0.974367i \(-0.427773\pi\)
0.224966 + 0.974367i \(0.427773\pi\)
\(420\) 0 0
\(421\) −2.48499e43 −1.24357 −0.621786 0.783187i \(-0.713593\pi\)
−0.621786 + 0.783187i \(0.713593\pi\)
\(422\) −6.48786e42 −0.312209
\(423\) 5.03834e43 2.33169
\(424\) 4.03415e42 0.179563
\(425\) 0 0
\(426\) 7.75687e43 3.19469
\(427\) −3.77142e42 −0.149433
\(428\) 1.24383e43 0.474174
\(429\) −1.15705e44 −4.24432
\(430\) 0 0
\(431\) −5.49619e43 −1.86718 −0.933591 0.358340i \(-0.883343\pi\)
−0.933591 + 0.358340i \(0.883343\pi\)
\(432\) 3.23270e43 1.05702
\(433\) 1.92976e43 0.607369 0.303685 0.952773i \(-0.401783\pi\)
0.303685 + 0.952773i \(0.401783\pi\)
\(434\) 1.62237e43 0.491552
\(435\) 0 0
\(436\) −3.01690e43 −0.847293
\(437\) 2.81879e43 0.762288
\(438\) −4.06117e43 −1.05762
\(439\) −4.45489e43 −1.11731 −0.558653 0.829401i \(-0.688682\pi\)
−0.558653 + 0.829401i \(0.688682\pi\)
\(440\) 0 0
\(441\) −6.81506e43 −1.58574
\(442\) −2.55454e43 −0.572591
\(443\) −4.76827e43 −1.02967 −0.514837 0.857288i \(-0.672147\pi\)
−0.514837 + 0.857288i \(0.672147\pi\)
\(444\) −5.17331e43 −1.07634
\(445\) 0 0
\(446\) 5.14340e43 0.993631
\(447\) −5.06998e43 −0.943914
\(448\) 3.75991e43 0.674668
\(449\) 7.03065e43 1.21599 0.607996 0.793940i \(-0.291974\pi\)
0.607996 + 0.793940i \(0.291974\pi\)
\(450\) 0 0
\(451\) 1.64196e43 0.263906
\(452\) −2.45863e43 −0.380986
\(453\) −1.73308e44 −2.58940
\(454\) 1.22066e44 1.75863
\(455\) 0 0
\(456\) 5.59803e43 0.750094
\(457\) −1.29751e44 −1.67685 −0.838424 0.545018i \(-0.816523\pi\)
−0.838424 + 0.545018i \(0.816523\pi\)
\(458\) 5.34291e43 0.666035
\(459\) 3.61342e43 0.434520
\(460\) 0 0
\(461\) −1.60383e43 −0.179512 −0.0897561 0.995964i \(-0.528609\pi\)
−0.0897561 + 0.995964i \(0.528609\pi\)
\(462\) 1.80228e44 1.94639
\(463\) −8.21430e43 −0.856020 −0.428010 0.903774i \(-0.640785\pi\)
−0.428010 + 0.903774i \(0.640785\pi\)
\(464\) 1.10409e44 1.11034
\(465\) 0 0
\(466\) 1.03125e44 0.966044
\(467\) 2.45616e43 0.222089 0.111045 0.993815i \(-0.464580\pi\)
0.111045 + 0.993815i \(0.464580\pi\)
\(468\) 4.48077e44 3.91107
\(469\) −1.03722e44 −0.874016
\(470\) 0 0
\(471\) −1.53303e44 −1.20422
\(472\) 3.83663e43 0.291010
\(473\) 4.24378e44 3.10845
\(474\) 4.02751e44 2.84901
\(475\) 0 0
\(476\) 2.24884e43 0.148403
\(477\) 1.23928e44 0.789982
\(478\) −2.90349e44 −1.78797
\(479\) 2.62104e44 1.55933 0.779664 0.626199i \(-0.215390\pi\)
0.779664 + 0.626199i \(0.215390\pi\)
\(480\) 0 0
\(481\) 1.29537e44 0.719450
\(482\) 2.36741e44 1.27057
\(483\) −1.21574e44 −0.630543
\(484\) 4.28435e44 2.14752
\(485\) 0 0
\(486\) 2.27979e41 0.00106757
\(487\) −1.10024e43 −0.0498037 −0.0249018 0.999690i \(-0.507927\pi\)
−0.0249018 + 0.999690i \(0.507927\pi\)
\(488\) 3.41131e43 0.149278
\(489\) −3.61475e44 −1.52926
\(490\) 0 0
\(491\) −2.25640e44 −0.892426 −0.446213 0.894927i \(-0.647228\pi\)
−0.446213 + 0.894927i \(0.647228\pi\)
\(492\) −9.53822e43 −0.364790
\(493\) 1.23412e44 0.456439
\(494\) −6.07817e44 −2.17409
\(495\) 0 0
\(496\) 1.29947e44 0.434831
\(497\) 1.70985e44 0.553450
\(498\) −4.72742e44 −1.48027
\(499\) −4.29566e44 −1.30128 −0.650641 0.759386i \(-0.725500\pi\)
−0.650641 + 0.759386i \(0.725500\pi\)
\(500\) 0 0
\(501\) −1.29194e44 −0.366371
\(502\) 4.41051e44 1.21026
\(503\) 2.02877e44 0.538717 0.269359 0.963040i \(-0.413188\pi\)
0.269359 + 0.963040i \(0.413188\pi\)
\(504\) −1.60957e44 −0.413624
\(505\) 0 0
\(506\) −8.20853e44 −1.97598
\(507\) −9.39676e44 −2.18951
\(508\) 4.67292e44 1.05399
\(509\) 4.74147e44 1.03531 0.517654 0.855590i \(-0.326806\pi\)
0.517654 + 0.855590i \(0.326806\pi\)
\(510\) 0 0
\(511\) −8.95206e43 −0.183222
\(512\) 5.65155e44 1.11999
\(513\) 8.59764e44 1.64984
\(514\) −9.09346e44 −1.68981
\(515\) 0 0
\(516\) −2.46524e45 −4.29673
\(517\) 1.12483e45 1.89886
\(518\) −2.01773e44 −0.329930
\(519\) 4.27059e44 0.676436
\(520\) 0 0
\(521\) 1.04032e45 1.54648 0.773240 0.634113i \(-0.218635\pi\)
0.773240 + 0.634113i \(0.218635\pi\)
\(522\) −3.83018e45 −5.51641
\(523\) −7.60705e44 −1.06154 −0.530772 0.847514i \(-0.678098\pi\)
−0.530772 + 0.847514i \(0.678098\pi\)
\(524\) 1.84999e45 2.50152
\(525\) 0 0
\(526\) 1.32225e45 1.67900
\(527\) 1.45252e44 0.178750
\(528\) 1.44358e45 1.72179
\(529\) −3.11290e44 −0.359871
\(530\) 0 0
\(531\) 1.17860e45 1.28029
\(532\) 5.35080e44 0.563478
\(533\) 2.38832e44 0.243834
\(534\) −3.00646e45 −2.97594
\(535\) 0 0
\(536\) 9.38184e44 0.873110
\(537\) −9.92047e44 −0.895275
\(538\) −1.49060e45 −1.30453
\(539\) −1.52149e45 −1.29138
\(540\) 0 0
\(541\) 9.20989e44 0.735359 0.367680 0.929953i \(-0.380152\pi\)
0.367680 + 0.929953i \(0.380152\pi\)
\(542\) 8.30442e44 0.643163
\(543\) −2.20423e45 −1.65600
\(544\) 4.75214e44 0.346344
\(545\) 0 0
\(546\) 2.62151e45 1.79835
\(547\) 2.34454e45 1.56051 0.780254 0.625463i \(-0.215090\pi\)
0.780254 + 0.625463i \(0.215090\pi\)
\(548\) −1.42234e45 −0.918593
\(549\) 1.04795e45 0.656744
\(550\) 0 0
\(551\) 2.93642e45 1.73307
\(552\) 1.09966e45 0.629889
\(553\) 8.87787e44 0.493565
\(554\) 6.59918e44 0.356106
\(555\) 0 0
\(556\) −2.34695e45 −1.19335
\(557\) −9.31697e44 −0.459900 −0.229950 0.973202i \(-0.573856\pi\)
−0.229950 + 0.973202i \(0.573856\pi\)
\(558\) −4.50800e45 −2.16033
\(559\) 6.17283e45 2.87203
\(560\) 0 0
\(561\) 1.61359e45 0.707793
\(562\) −3.89659e45 −1.65972
\(563\) 1.76320e45 0.729312 0.364656 0.931142i \(-0.381187\pi\)
0.364656 + 0.931142i \(0.381187\pi\)
\(564\) −6.53421e45 −2.62475
\(565\) 0 0
\(566\) −2.50970e45 −0.950936
\(567\) −1.23564e45 −0.454747
\(568\) −1.54659e45 −0.552875
\(569\) 1.99581e45 0.693054 0.346527 0.938040i \(-0.387361\pi\)
0.346527 + 0.938040i \(0.387361\pi\)
\(570\) 0 0
\(571\) −1.67320e45 −0.548342 −0.274171 0.961681i \(-0.588404\pi\)
−0.274171 + 0.961681i \(0.588404\pi\)
\(572\) 1.00035e46 3.18507
\(573\) −4.03161e45 −1.24717
\(574\) −3.72016e44 −0.111819
\(575\) 0 0
\(576\) −1.04475e46 −2.96511
\(577\) −6.81345e44 −0.187917 −0.0939586 0.995576i \(-0.529952\pi\)
−0.0939586 + 0.995576i \(0.529952\pi\)
\(578\) −5.30157e45 −1.42100
\(579\) 4.36120e45 1.13608
\(580\) 0 0
\(581\) −1.04207e45 −0.256443
\(582\) 2.49068e45 0.595783
\(583\) 2.76676e45 0.643339
\(584\) 8.09729e44 0.183032
\(585\) 0 0
\(586\) 1.11099e46 2.37355
\(587\) −7.97410e45 −1.65635 −0.828176 0.560467i \(-0.810622\pi\)
−0.828176 + 0.560467i \(0.810622\pi\)
\(588\) 8.83844e45 1.78505
\(589\) 3.45607e45 0.678703
\(590\) 0 0
\(591\) 2.34865e45 0.436139
\(592\) −1.61615e45 −0.291859
\(593\) −9.65663e45 −1.69599 −0.847994 0.530005i \(-0.822190\pi\)
−0.847994 + 0.530005i \(0.822190\pi\)
\(594\) −2.50370e46 −4.27667
\(595\) 0 0
\(596\) 4.38335e45 0.708342
\(597\) −2.51330e45 −0.395064
\(598\) −1.19398e46 −1.82569
\(599\) −8.55789e45 −1.27299 −0.636494 0.771282i \(-0.719616\pi\)
−0.636494 + 0.771282i \(0.719616\pi\)
\(600\) 0 0
\(601\) −3.47394e45 −0.489095 −0.244548 0.969637i \(-0.578639\pi\)
−0.244548 + 0.969637i \(0.578639\pi\)
\(602\) −9.61507e45 −1.31707
\(603\) 2.88208e46 3.84123
\(604\) 1.49837e46 1.94317
\(605\) 0 0
\(606\) −1.87942e46 −2.30794
\(607\) 3.78540e45 0.452375 0.226188 0.974084i \(-0.427374\pi\)
0.226188 + 0.974084i \(0.427374\pi\)
\(608\) 1.13071e46 1.31504
\(609\) −1.26648e46 −1.43355
\(610\) 0 0
\(611\) 1.63613e46 1.75444
\(612\) −6.24875e45 −0.652220
\(613\) 9.19782e45 0.934517 0.467258 0.884121i \(-0.345242\pi\)
0.467258 + 0.884121i \(0.345242\pi\)
\(614\) 2.48840e46 2.46117
\(615\) 0 0
\(616\) −3.59344e45 −0.336843
\(617\) 3.00505e45 0.274250 0.137125 0.990554i \(-0.456214\pi\)
0.137125 + 0.990554i \(0.456214\pi\)
\(618\) 4.45519e46 3.95873
\(619\) −8.07148e45 −0.698325 −0.349162 0.937062i \(-0.613534\pi\)
−0.349162 + 0.937062i \(0.613534\pi\)
\(620\) 0 0
\(621\) 1.68890e46 1.38545
\(622\) 9.26106e45 0.739809
\(623\) −6.62716e45 −0.515554
\(624\) 2.09976e46 1.59083
\(625\) 0 0
\(626\) −2.01790e46 −1.45018
\(627\) 3.83932e46 2.68745
\(628\) 1.32541e46 0.903682
\(629\) −1.80649e45 −0.119977
\(630\) 0 0
\(631\) −1.34115e46 −0.845264 −0.422632 0.906301i \(-0.638894\pi\)
−0.422632 + 0.906301i \(0.638894\pi\)
\(632\) −8.03018e45 −0.493053
\(633\) 5.96092e45 0.356576
\(634\) 2.70002e46 1.57360
\(635\) 0 0
\(636\) −1.60722e46 −0.889271
\(637\) −2.21310e46 −1.19316
\(638\) −8.55107e46 −4.49241
\(639\) −4.75109e46 −2.43237
\(640\) 0 0
\(641\) −1.68745e46 −0.820489 −0.410244 0.911976i \(-0.634557\pi\)
−0.410244 + 0.911976i \(0.634557\pi\)
\(642\) −2.02206e46 −0.958222
\(643\) −3.29909e45 −0.152375 −0.0761874 0.997094i \(-0.524275\pi\)
−0.0761874 + 0.997094i \(0.524275\pi\)
\(644\) 1.05110e46 0.473179
\(645\) 0 0
\(646\) 8.47644e45 0.362557
\(647\) −1.31901e46 −0.549953 −0.274977 0.961451i \(-0.588670\pi\)
−0.274977 + 0.961451i \(0.588670\pi\)
\(648\) 1.11765e46 0.454276
\(649\) 2.63129e46 1.04263
\(650\) 0 0
\(651\) −1.49060e46 −0.561404
\(652\) 3.12520e46 1.14761
\(653\) −3.26301e46 −1.16829 −0.584146 0.811649i \(-0.698570\pi\)
−0.584146 + 0.811649i \(0.698570\pi\)
\(654\) 4.90451e46 1.71223
\(655\) 0 0
\(656\) −2.97975e45 −0.0989159
\(657\) 2.48747e46 0.805247
\(658\) −2.54852e46 −0.804563
\(659\) 1.87303e46 0.576682 0.288341 0.957528i \(-0.406896\pi\)
0.288341 + 0.957528i \(0.406896\pi\)
\(660\) 0 0
\(661\) −1.63902e46 −0.480021 −0.240011 0.970770i \(-0.577151\pi\)
−0.240011 + 0.970770i \(0.577151\pi\)
\(662\) 9.15536e46 2.61528
\(663\) 2.34706e46 0.653960
\(664\) 9.42568e45 0.256177
\(665\) 0 0
\(666\) 5.60657e46 1.45001
\(667\) 5.76821e46 1.45534
\(668\) 1.11697e46 0.274936
\(669\) −4.72566e46 −1.13483
\(670\) 0 0
\(671\) 2.33959e46 0.534834
\(672\) −4.87674e46 −1.08777
\(673\) 8.65953e46 1.88472 0.942358 0.334606i \(-0.108603\pi\)
0.942358 + 0.334606i \(0.108603\pi\)
\(674\) 1.56911e46 0.333246
\(675\) 0 0
\(676\) 8.12415e46 1.64308
\(677\) −4.79310e45 −0.0946027 −0.0473014 0.998881i \(-0.515062\pi\)
−0.0473014 + 0.998881i \(0.515062\pi\)
\(678\) 3.99694e46 0.769905
\(679\) 5.49021e45 0.103214
\(680\) 0 0
\(681\) −1.12152e47 −2.00854
\(682\) −1.00643e47 −1.75931
\(683\) −6.64508e46 −1.13386 −0.566929 0.823767i \(-0.691869\pi\)
−0.566929 + 0.823767i \(0.691869\pi\)
\(684\) −1.48680e47 −2.47644
\(685\) 0 0
\(686\) 7.79455e46 1.23721
\(687\) −4.90896e46 −0.760683
\(688\) −7.70142e46 −1.16510
\(689\) 4.02441e46 0.594408
\(690\) 0 0
\(691\) 4.37178e46 0.615560 0.307780 0.951458i \(-0.400414\pi\)
0.307780 + 0.951458i \(0.400414\pi\)
\(692\) −3.69222e46 −0.507619
\(693\) −1.10390e47 −1.48194
\(694\) 6.34534e46 0.831809
\(695\) 0 0
\(696\) 1.14555e47 1.43206
\(697\) −3.33068e45 −0.0406624
\(698\) 1.52064e47 1.81306
\(699\) −9.47494e46 −1.10332
\(700\) 0 0
\(701\) 9.99204e46 1.10996 0.554979 0.831864i \(-0.312726\pi\)
0.554979 + 0.831864i \(0.312726\pi\)
\(702\) −3.64177e47 −3.95139
\(703\) −4.29829e46 −0.455546
\(704\) −2.33245e47 −2.41470
\(705\) 0 0
\(706\) −1.43956e46 −0.142218
\(707\) −4.14281e46 −0.399829
\(708\) −1.52853e47 −1.44120
\(709\) −3.22308e46 −0.296898 −0.148449 0.988920i \(-0.547428\pi\)
−0.148449 + 0.988920i \(0.547428\pi\)
\(710\) 0 0
\(711\) −2.46685e47 −2.16918
\(712\) 5.99437e46 0.515020
\(713\) 6.78900e46 0.569939
\(714\) −3.65589e46 −0.299897
\(715\) 0 0
\(716\) 8.57693e46 0.671842
\(717\) 2.66767e47 2.04205
\(718\) −8.63391e45 −0.0645883
\(719\) −2.25121e47 −1.64585 −0.822923 0.568153i \(-0.807658\pi\)
−0.822923 + 0.568153i \(0.807658\pi\)
\(720\) 0 0
\(721\) 9.82060e46 0.685814
\(722\) −2.04940e46 −0.139882
\(723\) −2.17513e47 −1.45112
\(724\) 1.90571e47 1.24271
\(725\) 0 0
\(726\) −6.96497e47 −4.33976
\(727\) −1.90515e46 −0.116041 −0.0580205 0.998315i \(-0.518479\pi\)
−0.0580205 + 0.998315i \(0.518479\pi\)
\(728\) −5.22686e46 −0.311224
\(729\) −1.71897e47 −1.00061
\(730\) 0 0
\(731\) −8.60844e46 −0.478948
\(732\) −1.35908e47 −0.739286
\(733\) 2.27821e47 1.21165 0.605826 0.795597i \(-0.292843\pi\)
0.605826 + 0.795597i \(0.292843\pi\)
\(734\) 4.11633e47 2.14055
\(735\) 0 0
\(736\) 2.22112e47 1.10430
\(737\) 6.43438e47 3.12819
\(738\) 1.03370e47 0.491435
\(739\) 3.27147e47 1.52093 0.760467 0.649377i \(-0.224970\pi\)
0.760467 + 0.649377i \(0.224970\pi\)
\(740\) 0 0
\(741\) 5.58451e47 2.48304
\(742\) −6.26860e46 −0.272588
\(743\) −1.23139e47 −0.523698 −0.261849 0.965109i \(-0.584332\pi\)
−0.261849 + 0.965109i \(0.584332\pi\)
\(744\) 1.34827e47 0.560822
\(745\) 0 0
\(746\) −3.83520e47 −1.52615
\(747\) 2.89555e47 1.12704
\(748\) −1.39506e47 −0.531150
\(749\) −4.45724e46 −0.166003
\(750\) 0 0
\(751\) 4.24124e47 1.51159 0.755795 0.654808i \(-0.227251\pi\)
0.755795 + 0.654808i \(0.227251\pi\)
\(752\) −2.04129e47 −0.711723
\(753\) −4.05230e47 −1.38224
\(754\) −1.24380e48 −4.15072
\(755\) 0 0
\(756\) 3.20597e47 1.02412
\(757\) 4.38259e47 1.36977 0.684887 0.728649i \(-0.259852\pi\)
0.684887 + 0.728649i \(0.259852\pi\)
\(758\) 3.53232e47 1.08024
\(759\) 7.54184e47 2.25677
\(760\) 0 0
\(761\) −6.43868e47 −1.84480 −0.922401 0.386232i \(-0.873776\pi\)
−0.922401 + 0.386232i \(0.873776\pi\)
\(762\) −7.59667e47 −2.12993
\(763\) 1.08110e47 0.296628
\(764\) 3.48560e47 0.935918
\(765\) 0 0
\(766\) 7.78370e47 2.00176
\(767\) 3.82736e47 0.963332
\(768\) −1.16704e47 −0.287491
\(769\) 1.83655e46 0.0442812 0.0221406 0.999755i \(-0.492952\pi\)
0.0221406 + 0.999755i \(0.492952\pi\)
\(770\) 0 0
\(771\) 8.35491e47 1.92994
\(772\) −3.77056e47 −0.852552
\(773\) 1.27125e47 0.281364 0.140682 0.990055i \(-0.455070\pi\)
0.140682 + 0.990055i \(0.455070\pi\)
\(774\) 2.67170e48 5.78844
\(775\) 0 0
\(776\) −4.96599e46 −0.103107
\(777\) 1.85385e47 0.376815
\(778\) 9.04484e47 1.79986
\(779\) −7.92490e46 −0.154392
\(780\) 0 0
\(781\) −1.06070e48 −1.98085
\(782\) 1.66509e47 0.304456
\(783\) 1.75937e48 3.14984
\(784\) 2.76114e47 0.484031
\(785\) 0 0
\(786\) −3.00749e48 −5.05513
\(787\) −1.07309e48 −1.76626 −0.883128 0.469133i \(-0.844567\pi\)
−0.883128 + 0.469133i \(0.844567\pi\)
\(788\) −2.03057e47 −0.327292
\(789\) −1.21486e48 −1.91759
\(790\) 0 0
\(791\) 8.81047e46 0.133379
\(792\) 9.98491e47 1.48040
\(793\) 3.40307e47 0.494155
\(794\) 3.64283e46 0.0518084
\(795\) 0 0
\(796\) 2.17292e47 0.296468
\(797\) −8.32944e47 −1.11315 −0.556575 0.830798i \(-0.687885\pi\)
−0.556575 + 0.830798i \(0.687885\pi\)
\(798\) −8.69868e47 −1.13869
\(799\) −2.28170e47 −0.292575
\(800\) 0 0
\(801\) 1.84146e48 2.26582
\(802\) −3.03216e46 −0.0365489
\(803\) 5.55339e47 0.655770
\(804\) −3.73776e48 −4.32401
\(805\) 0 0
\(806\) −1.46391e48 −1.62550
\(807\) 1.36954e48 1.48991
\(808\) 3.74724e47 0.399415
\(809\) −4.91569e47 −0.513374 −0.256687 0.966495i \(-0.582631\pi\)
−0.256687 + 0.966495i \(0.582631\pi\)
\(810\) 0 0
\(811\) 1.31202e48 1.31551 0.657757 0.753230i \(-0.271505\pi\)
0.657757 + 0.753230i \(0.271505\pi\)
\(812\) 1.09496e48 1.07578
\(813\) −7.62995e47 −0.734560
\(814\) 1.25169e48 1.18085
\(815\) 0 0
\(816\) −2.92827e47 −0.265292
\(817\) −2.04826e48 −1.81853
\(818\) 4.32435e47 0.376262
\(819\) −1.60568e48 −1.36922
\(820\) 0 0
\(821\) −1.18387e48 −0.969711 −0.484855 0.874594i \(-0.661128\pi\)
−0.484855 + 0.874594i \(0.661128\pi\)
\(822\) 2.31226e48 1.85631
\(823\) −1.23245e48 −0.969776 −0.484888 0.874576i \(-0.661140\pi\)
−0.484888 + 0.874576i \(0.661140\pi\)
\(824\) −8.88289e47 −0.685102
\(825\) 0 0
\(826\) −5.96167e47 −0.441771
\(827\) 4.69422e45 0.00340975 0.00170487 0.999999i \(-0.499457\pi\)
0.00170487 + 0.999999i \(0.499457\pi\)
\(828\) −2.92063e48 −2.07958
\(829\) −4.02684e47 −0.281070 −0.140535 0.990076i \(-0.544882\pi\)
−0.140535 + 0.990076i \(0.544882\pi\)
\(830\) 0 0
\(831\) −6.06321e47 −0.406711
\(832\) −3.39268e48 −2.23105
\(833\) 3.08632e47 0.198975
\(834\) 3.81538e48 2.41156
\(835\) 0 0
\(836\) −3.31936e48 −2.01674
\(837\) 2.07072e48 1.23354
\(838\) 1.16819e48 0.682316
\(839\) 5.27811e46 0.0302278 0.0151139 0.999886i \(-0.495189\pi\)
0.0151139 + 0.999886i \(0.495189\pi\)
\(840\) 0 0
\(841\) 4.19284e48 2.30874
\(842\) −3.49269e48 −1.88586
\(843\) 3.58011e48 1.89558
\(844\) −5.15363e47 −0.267586
\(845\) 0 0
\(846\) 7.08144e48 3.53599
\(847\) −1.53529e48 −0.751822
\(848\) −5.02098e47 −0.241133
\(849\) 2.30587e48 1.08607
\(850\) 0 0
\(851\) −8.44342e47 −0.382544
\(852\) 6.16167e48 2.73808
\(853\) 3.55498e48 1.54945 0.774725 0.632298i \(-0.217888\pi\)
0.774725 + 0.632298i \(0.217888\pi\)
\(854\) −5.30078e47 −0.226613
\(855\) 0 0
\(856\) 4.03165e47 0.165831
\(857\) −1.12728e48 −0.454827 −0.227414 0.973798i \(-0.573027\pi\)
−0.227414 + 0.973798i \(0.573027\pi\)
\(858\) −1.62625e49 −6.43646
\(859\) 1.80056e48 0.699070 0.349535 0.936923i \(-0.386340\pi\)
0.349535 + 0.936923i \(0.386340\pi\)
\(860\) 0 0
\(861\) 3.41801e47 0.127709
\(862\) −7.72496e48 −2.83156
\(863\) 2.38697e48 0.858359 0.429180 0.903219i \(-0.358803\pi\)
0.429180 + 0.903219i \(0.358803\pi\)
\(864\) 6.77469e48 2.39008
\(865\) 0 0
\(866\) 2.71230e48 0.921069
\(867\) 4.87098e48 1.62294
\(868\) 1.28873e48 0.421295
\(869\) −5.50737e48 −1.76652
\(870\) 0 0
\(871\) 9.35917e48 2.89027
\(872\) −9.77876e47 −0.296320
\(873\) −1.52554e48 −0.453616
\(874\) 3.96184e48 1.15600
\(875\) 0 0
\(876\) −3.22599e48 −0.906454
\(877\) 1.44993e48 0.399811 0.199905 0.979815i \(-0.435936\pi\)
0.199905 + 0.979815i \(0.435936\pi\)
\(878\) −6.26140e48 −1.69438
\(879\) −1.02075e49 −2.71084
\(880\) 0 0
\(881\) −4.86343e48 −1.24406 −0.622029 0.782994i \(-0.713691\pi\)
−0.622029 + 0.782994i \(0.713691\pi\)
\(882\) −9.57864e48 −2.40476
\(883\) −2.25361e48 −0.555299 −0.277650 0.960682i \(-0.589555\pi\)
−0.277650 + 0.960682i \(0.589555\pi\)
\(884\) −2.02920e48 −0.490752
\(885\) 0 0
\(886\) −6.70186e48 −1.56149
\(887\) 7.86411e48 1.79850 0.899248 0.437439i \(-0.144114\pi\)
0.899248 + 0.437439i \(0.144114\pi\)
\(888\) −1.67684e48 −0.376424
\(889\) −1.67454e48 −0.368991
\(890\) 0 0
\(891\) 7.66524e48 1.62758
\(892\) 4.08566e48 0.851613
\(893\) −5.42900e48 −1.11089
\(894\) −7.12592e48 −1.43144
\(895\) 0 0
\(896\) 2.04066e48 0.395083
\(897\) 1.09700e49 2.08513
\(898\) 9.88165e48 1.84404
\(899\) 7.07230e48 1.29576
\(900\) 0 0
\(901\) −5.61232e47 −0.0991251
\(902\) 2.30779e48 0.400211
\(903\) 8.83415e48 1.50424
\(904\) −7.96922e47 −0.133241
\(905\) 0 0
\(906\) −2.43587e49 −3.92680
\(907\) 1.83168e48 0.289954 0.144977 0.989435i \(-0.453689\pi\)
0.144977 + 0.989435i \(0.453689\pi\)
\(908\) 9.69632e48 1.50727
\(909\) 1.15114e49 1.75722
\(910\) 0 0
\(911\) 4.44938e48 0.655007 0.327504 0.944850i \(-0.393793\pi\)
0.327504 + 0.944850i \(0.393793\pi\)
\(912\) −6.96742e48 −1.00730
\(913\) 6.46445e48 0.917833
\(914\) −1.82367e49 −2.54292
\(915\) 0 0
\(916\) 4.24414e48 0.570840
\(917\) −6.62943e48 −0.875754
\(918\) 5.07871e48 0.658945
\(919\) −1.19748e49 −1.52603 −0.763013 0.646383i \(-0.776281\pi\)
−0.763013 + 0.646383i \(0.776281\pi\)
\(920\) 0 0
\(921\) −2.28630e49 −2.81092
\(922\) −2.25421e48 −0.272228
\(923\) −1.54285e49 −1.83019
\(924\) 1.43164e49 1.66819
\(925\) 0 0
\(926\) −1.15453e49 −1.29815
\(927\) −2.72880e49 −3.01409
\(928\) 2.31381e49 2.51065
\(929\) −6.38779e48 −0.680912 −0.340456 0.940260i \(-0.610581\pi\)
−0.340456 + 0.940260i \(0.610581\pi\)
\(930\) 0 0
\(931\) 7.34348e48 0.755496
\(932\) 8.19175e48 0.827969
\(933\) −8.50889e48 −0.844940
\(934\) 3.45215e48 0.336796
\(935\) 0 0
\(936\) 1.45236e49 1.36780
\(937\) −2.98186e47 −0.0275921 −0.0137960 0.999905i \(-0.504392\pi\)
−0.0137960 + 0.999905i \(0.504392\pi\)
\(938\) −1.45783e49 −1.32544
\(939\) 1.85401e49 1.65626
\(940\) 0 0
\(941\) 6.14468e48 0.529993 0.264997 0.964249i \(-0.414629\pi\)
0.264997 + 0.964249i \(0.414629\pi\)
\(942\) −2.15469e49 −1.82618
\(943\) −1.55674e48 −0.129651
\(944\) −4.77514e48 −0.390794
\(945\) 0 0
\(946\) 5.96469e49 4.71394
\(947\) −4.27472e46 −0.00331996 −0.00165998 0.999999i \(-0.500528\pi\)
−0.00165998 + 0.999999i \(0.500528\pi\)
\(948\) 3.19926e49 2.44181
\(949\) 8.07772e48 0.605894
\(950\) 0 0
\(951\) −2.48073e49 −1.79722
\(952\) 7.28922e47 0.0519005
\(953\) −1.36139e49 −0.952684 −0.476342 0.879260i \(-0.658038\pi\)
−0.476342 + 0.879260i \(0.658038\pi\)
\(954\) 1.74183e49 1.19800
\(955\) 0 0
\(956\) −2.30639e49 −1.53242
\(957\) 7.85656e49 5.13081
\(958\) 3.68390e49 2.36470
\(959\) 5.09693e48 0.321589
\(960\) 0 0
\(961\) −8.07961e48 −0.492555
\(962\) 1.82066e49 1.09104
\(963\) 1.23851e49 0.729570
\(964\) 1.88055e49 1.08897
\(965\) 0 0
\(966\) −1.70874e49 −0.956212
\(967\) −1.39797e49 −0.769060 −0.384530 0.923113i \(-0.625636\pi\)
−0.384530 + 0.923113i \(0.625636\pi\)
\(968\) 1.38870e49 0.751042
\(969\) −7.78799e48 −0.414079
\(970\) 0 0
\(971\) 2.21572e49 1.13867 0.569334 0.822107i \(-0.307201\pi\)
0.569334 + 0.822107i \(0.307201\pi\)
\(972\) 1.81095e46 0.000914985 0
\(973\) 8.41026e48 0.417780
\(974\) −1.54641e48 −0.0755268
\(975\) 0 0
\(976\) −4.24578e48 −0.200464
\(977\) 1.12838e49 0.523835 0.261918 0.965090i \(-0.415645\pi\)
0.261918 + 0.965090i \(0.415645\pi\)
\(978\) −5.08057e49 −2.31911
\(979\) 4.11114e49 1.84522
\(980\) 0 0
\(981\) −3.00401e49 −1.30366
\(982\) −3.17140e49 −1.35335
\(983\) 2.75705e49 1.15694 0.578471 0.815703i \(-0.303650\pi\)
0.578471 + 0.815703i \(0.303650\pi\)
\(984\) −3.09165e48 −0.127577
\(985\) 0 0
\(986\) 1.73457e49 0.692185
\(987\) 2.34153e49 0.918896
\(988\) −4.82819e49 −1.86335
\(989\) −4.02354e49 −1.52711
\(990\) 0 0
\(991\) 2.93272e49 1.07660 0.538300 0.842753i \(-0.319067\pi\)
0.538300 + 0.842753i \(0.319067\pi\)
\(992\) 2.72328e49 0.983218
\(993\) −8.41178e49 −2.98693
\(994\) 2.40322e49 0.839300
\(995\) 0 0
\(996\) −3.75523e49 −1.26870
\(997\) 1.66255e49 0.552464 0.276232 0.961091i \(-0.410914\pi\)
0.276232 + 0.961091i \(0.410914\pi\)
\(998\) −6.03760e49 −1.97338
\(999\) −2.57534e49 −0.827951
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 25.34.a.e.1.10 yes 11
5.2 odd 4 25.34.b.d.24.19 22
5.3 odd 4 25.34.b.d.24.4 22
5.4 even 2 25.34.a.d.1.2 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.34.a.d.1.2 11 5.4 even 2
25.34.a.e.1.10 yes 11 1.1 even 1 trivial
25.34.b.d.24.4 22 5.3 odd 4
25.34.b.d.24.19 22 5.2 odd 4