Properties

Label 25.24.a.a.1.1
Level $25$
Weight $24$
Character 25.1
Self dual yes
Analytic conductor $83.801$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [25,24,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, 24, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.1");
 
S:= CuspForms(chi, 24);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 24 \)
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: \(83.8010093363\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{144169}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 36042 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(190.348\) 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)
 
\(f(q)\) \(=\) \(q-5096.35 q^{2} +48964.9 q^{3} +1.75842e7 q^{4} -2.49542e8 q^{6} +5.17135e9 q^{7} -4.68639e10 q^{8} -9.17456e10 q^{9} +O(q^{10})\) \(q-5096.35 q^{2} +48964.9 q^{3} +1.75842e7 q^{4} -2.49542e8 q^{6} +5.17135e9 q^{7} -4.68639e10 q^{8} -9.17456e10 q^{9} +6.04602e11 q^{11} +8.61007e11 q^{12} -7.96710e12 q^{13} -2.63550e13 q^{14} +9.13280e13 q^{16} -1.98385e13 q^{17} +4.67568e14 q^{18} +6.27346e14 q^{19} +2.53215e14 q^{21} -3.08127e15 q^{22} +4.55685e15 q^{23} -2.29468e15 q^{24} +4.06032e16 q^{26} -9.10202e15 q^{27} +9.09340e16 q^{28} +4.14107e16 q^{29} +1.35683e15 q^{31} -7.23169e16 q^{32} +2.96043e16 q^{33} +1.01104e17 q^{34} -1.61327e18 q^{36} -3.41258e17 q^{37} -3.19718e18 q^{38} -3.90108e17 q^{39} -3.69518e18 q^{41} -1.29047e18 q^{42} +1.96955e18 q^{43} +1.06314e19 q^{44} -2.32233e19 q^{46} -2.44381e19 q^{47} +4.47186e18 q^{48} -6.25860e17 q^{49} -9.71391e17 q^{51} -1.40095e20 q^{52} +6.39437e19 q^{53} +4.63871e19 q^{54} -2.42350e20 q^{56} +3.07179e19 q^{57} -2.11044e20 q^{58} +2.81892e20 q^{59} -4.67790e20 q^{61} -6.91488e18 q^{62} -4.74449e20 q^{63} -3.97563e20 q^{64} -1.50874e20 q^{66} -2.77406e20 q^{67} -3.48844e20 q^{68} +2.23126e20 q^{69} +2.29069e21 q^{71} +4.29956e21 q^{72} +4.56908e21 q^{73} +1.73917e21 q^{74} +1.10314e22 q^{76} +3.12661e21 q^{77} +1.98813e21 q^{78} -3.99005e21 q^{79} +8.19155e21 q^{81} +1.88319e22 q^{82} -1.45920e22 q^{83} +4.45257e21 q^{84} -1.00375e22 q^{86} +2.02767e21 q^{87} -2.83340e22 q^{88} +1.80991e21 q^{89} -4.12007e22 q^{91} +8.01285e22 q^{92} +6.64369e19 q^{93} +1.24545e23 q^{94} -3.54099e21 q^{96} -8.25561e22 q^{97} +3.18960e21 q^{98} -5.54696e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1080 q^{2} - 339480 q^{3} + 25326656 q^{4} - 1809673056 q^{6} + 1359184400 q^{7} - 49459023360 q^{8} - 34999394166 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 1080 q^{2} - 339480 q^{3} + 25326656 q^{4} - 1809673056 q^{6} + 1359184400 q^{7} - 49459023360 q^{8} - 34999394166 q^{9} + 856801968264 q^{11} - 2146514952960 q^{12} - 4376109322060 q^{13} - 41666034529728 q^{14} + 15956586401792 q^{16} - 254028147597540 q^{17} + 695480683916520 q^{18} + 4260600979960 q^{19} + 17\!\cdots\!44 q^{21}+ \cdots - 41\!\cdots\!12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5096.35 −1.75960 −0.879801 0.475342i \(-0.842324\pi\)
−0.879801 + 0.475342i \(0.842324\pi\)
\(3\) 48964.9 0.159584 0.0797921 0.996812i \(-0.474574\pi\)
0.0797921 + 0.996812i \(0.474574\pi\)
\(4\) 1.75842e7 2.09620
\(5\) 0 0
\(6\) −2.49542e8 −0.280805
\(7\) 5.17135e9 0.988500 0.494250 0.869320i \(-0.335443\pi\)
0.494250 + 0.869320i \(0.335443\pi\)
\(8\) −4.68639e10 −1.92887
\(9\) −9.17456e10 −0.974533
\(10\) 0 0
\(11\) 6.04602e11 0.638931 0.319466 0.947598i \(-0.396497\pi\)
0.319466 + 0.947598i \(0.396497\pi\)
\(12\) 8.61007e11 0.334520
\(13\) −7.96710e12 −1.23297 −0.616484 0.787367i \(-0.711444\pi\)
−0.616484 + 0.787367i \(0.711444\pi\)
\(14\) −2.63550e13 −1.73937
\(15\) 0 0
\(16\) 9.13280e13 1.29785
\(17\) −1.98385e13 −0.140393 −0.0701967 0.997533i \(-0.522363\pi\)
−0.0701967 + 0.997533i \(0.522363\pi\)
\(18\) 4.67568e14 1.71479
\(19\) 6.27346e14 1.23550 0.617748 0.786377i \(-0.288045\pi\)
0.617748 + 0.786377i \(0.288045\pi\)
\(20\) 0 0
\(21\) 2.53215e14 0.157749
\(22\) −3.08127e15 −1.12426
\(23\) 4.55685e15 0.997229 0.498614 0.866824i \(-0.333842\pi\)
0.498614 + 0.866824i \(0.333842\pi\)
\(24\) −2.29468e15 −0.307818
\(25\) 0 0
\(26\) 4.06032e16 2.16953
\(27\) −9.10202e15 −0.315104
\(28\) 9.09340e16 2.07209
\(29\) 4.14107e16 0.630283 0.315142 0.949045i \(-0.397948\pi\)
0.315142 + 0.949045i \(0.397948\pi\)
\(30\) 0 0
\(31\) 1.35683e15 0.00959104 0.00479552 0.999989i \(-0.498474\pi\)
0.00479552 + 0.999989i \(0.498474\pi\)
\(32\) −7.23169e16 −0.354826
\(33\) 2.96043e16 0.101963
\(34\) 1.01104e17 0.247037
\(35\) 0 0
\(36\) −1.61327e18 −2.04281
\(37\) −3.41258e17 −0.315328 −0.157664 0.987493i \(-0.550396\pi\)
−0.157664 + 0.987493i \(0.550396\pi\)
\(38\) −3.19718e18 −2.17398
\(39\) −3.90108e17 −0.196762
\(40\) 0 0
\(41\) −3.69518e18 −1.04863 −0.524313 0.851525i \(-0.675678\pi\)
−0.524313 + 0.851525i \(0.675678\pi\)
\(42\) −1.29047e18 −0.277576
\(43\) 1.96955e18 0.323207 0.161603 0.986856i \(-0.448333\pi\)
0.161603 + 0.986856i \(0.448333\pi\)
\(44\) 1.06314e19 1.33933
\(45\) 0 0
\(46\) −2.32233e19 −1.75473
\(47\) −2.44381e19 −1.44192 −0.720962 0.692975i \(-0.756300\pi\)
−0.720962 + 0.692975i \(0.756300\pi\)
\(48\) 4.47186e18 0.207116
\(49\) −6.25860e17 −0.0228677
\(50\) 0 0
\(51\) −9.71391e17 −0.0224046
\(52\) −1.40095e20 −2.58455
\(53\) 6.39437e19 0.947600 0.473800 0.880632i \(-0.342882\pi\)
0.473800 + 0.880632i \(0.342882\pi\)
\(54\) 4.63871e19 0.554458
\(55\) 0 0
\(56\) −2.42350e20 −1.90669
\(57\) 3.07179e19 0.197166
\(58\) −2.11044e20 −1.10905
\(59\) 2.81892e20 1.21698 0.608491 0.793561i \(-0.291775\pi\)
0.608491 + 0.793561i \(0.291775\pi\)
\(60\) 0 0
\(61\) −4.67790e20 −1.37644 −0.688221 0.725501i \(-0.741608\pi\)
−0.688221 + 0.725501i \(0.741608\pi\)
\(62\) −6.91488e18 −0.0168764
\(63\) −4.74449e20 −0.963326
\(64\) −3.97563e20 −0.673498
\(65\) 0 0
\(66\) −1.50874e20 −0.179415
\(67\) −2.77406e20 −0.277495 −0.138747 0.990328i \(-0.544308\pi\)
−0.138747 + 0.990328i \(0.544308\pi\)
\(68\) −3.48844e20 −0.294293
\(69\) 2.23126e20 0.159142
\(70\) 0 0
\(71\) 2.29069e21 1.17624 0.588119 0.808775i \(-0.299869\pi\)
0.588119 + 0.808775i \(0.299869\pi\)
\(72\) 4.29956e21 1.87975
\(73\) 4.56908e21 1.70457 0.852286 0.523075i \(-0.175215\pi\)
0.852286 + 0.523075i \(0.175215\pi\)
\(74\) 1.73917e21 0.554852
\(75\) 0 0
\(76\) 1.10314e22 2.58984
\(77\) 3.12661e21 0.631583
\(78\) 1.98813e21 0.346223
\(79\) −3.99005e21 −0.600160 −0.300080 0.953914i \(-0.597013\pi\)
−0.300080 + 0.953914i \(0.597013\pi\)
\(80\) 0 0
\(81\) 8.19155e21 0.924247
\(82\) 1.88319e22 1.84517
\(83\) −1.45920e22 −1.24370 −0.621850 0.783136i \(-0.713619\pi\)
−0.621850 + 0.783136i \(0.713619\pi\)
\(84\) 4.45257e21 0.330673
\(85\) 0 0
\(86\) −1.00375e22 −0.568715
\(87\) 2.02767e21 0.100583
\(88\) −2.83340e22 −1.23242
\(89\) 1.80991e21 0.0691310 0.0345655 0.999402i \(-0.488995\pi\)
0.0345655 + 0.999402i \(0.488995\pi\)
\(90\) 0 0
\(91\) −4.12007e22 −1.21879
\(92\) 8.01285e22 2.09039
\(93\) 6.64369e19 0.00153058
\(94\) 1.24545e23 2.53721
\(95\) 0 0
\(96\) −3.54099e21 −0.0566246
\(97\) −8.25561e22 −1.17186 −0.585928 0.810363i \(-0.699270\pi\)
−0.585928 + 0.810363i \(0.699270\pi\)
\(98\) 3.18960e21 0.0402380
\(99\) −5.54696e22 −0.622659
\(100\) 0 0
\(101\) 4.84234e22 0.431877 0.215938 0.976407i \(-0.430719\pi\)
0.215938 + 0.976407i \(0.430719\pi\)
\(102\) 4.95055e21 0.0394231
\(103\) 3.23687e22 0.230408 0.115204 0.993342i \(-0.463248\pi\)
0.115204 + 0.993342i \(0.463248\pi\)
\(104\) 3.73370e23 2.37824
\(105\) 0 0
\(106\) −3.25880e23 −1.66740
\(107\) 5.92757e22 0.272247 0.136124 0.990692i \(-0.456536\pi\)
0.136124 + 0.990692i \(0.456536\pi\)
\(108\) −1.60052e23 −0.660521
\(109\) 2.86887e23 1.06490 0.532448 0.846463i \(-0.321272\pi\)
0.532448 + 0.846463i \(0.321272\pi\)
\(110\) 0 0
\(111\) −1.67096e22 −0.0503214
\(112\) 4.72290e23 1.28292
\(113\) 4.87844e22 0.119641 0.0598204 0.998209i \(-0.480947\pi\)
0.0598204 + 0.998209i \(0.480947\pi\)
\(114\) −1.56549e23 −0.346933
\(115\) 0 0
\(116\) 7.28174e23 1.32120
\(117\) 7.30947e23 1.20157
\(118\) −1.43662e24 −2.14140
\(119\) −1.02592e23 −0.138779
\(120\) 0 0
\(121\) −5.29886e23 −0.591767
\(122\) 2.38402e24 2.42199
\(123\) −1.80934e23 −0.167344
\(124\) 2.38587e22 0.0201047
\(125\) 0 0
\(126\) 2.41796e24 1.69507
\(127\) −1.36871e24 −0.876131 −0.438065 0.898943i \(-0.644336\pi\)
−0.438065 + 0.898943i \(0.644336\pi\)
\(128\) 2.63276e24 1.53991
\(129\) 9.64389e22 0.0515787
\(130\) 0 0
\(131\) 2.63272e24 1.17974 0.589869 0.807499i \(-0.299180\pi\)
0.589869 + 0.807499i \(0.299180\pi\)
\(132\) 5.20567e23 0.213735
\(133\) 3.24423e24 1.22129
\(134\) 1.41376e24 0.488280
\(135\) 0 0
\(136\) 9.29711e23 0.270801
\(137\) 1.21837e24 0.326207 0.163103 0.986609i \(-0.447850\pi\)
0.163103 + 0.986609i \(0.447850\pi\)
\(138\) −1.13713e24 −0.280027
\(139\) −3.72223e24 −0.843593 −0.421797 0.906690i \(-0.638600\pi\)
−0.421797 + 0.906690i \(0.638600\pi\)
\(140\) 0 0
\(141\) −1.19661e24 −0.230108
\(142\) −1.16741e25 −2.06971
\(143\) −4.81693e24 −0.787782
\(144\) −8.37895e24 −1.26480
\(145\) 0 0
\(146\) −2.32856e25 −2.99937
\(147\) −3.06452e22 −0.00364932
\(148\) −6.00074e24 −0.660990
\(149\) 1.53044e25 1.56018 0.780088 0.625670i \(-0.215174\pi\)
0.780088 + 0.625670i \(0.215174\pi\)
\(150\) 0 0
\(151\) −1.25232e25 −1.09516 −0.547582 0.836752i \(-0.684452\pi\)
−0.547582 + 0.836752i \(0.684452\pi\)
\(152\) −2.93999e25 −2.38311
\(153\) 1.82010e24 0.136818
\(154\) −1.59343e25 −1.11134
\(155\) 0 0
\(156\) −6.85973e24 −0.412453
\(157\) −1.92166e23 −0.0107357 −0.00536785 0.999986i \(-0.501709\pi\)
−0.00536785 + 0.999986i \(0.501709\pi\)
\(158\) 2.03347e25 1.05604
\(159\) 3.13099e24 0.151222
\(160\) 0 0
\(161\) 2.35651e25 0.985761
\(162\) −4.17470e25 −1.62631
\(163\) 2.27639e25 0.826206 0.413103 0.910684i \(-0.364445\pi\)
0.413103 + 0.910684i \(0.364445\pi\)
\(164\) −6.49767e25 −2.19813
\(165\) 0 0
\(166\) 7.43657e25 2.18842
\(167\) 1.17984e25 0.324029 0.162015 0.986788i \(-0.448201\pi\)
0.162015 + 0.986788i \(0.448201\pi\)
\(168\) −1.18666e25 −0.304278
\(169\) 2.17208e25 0.520211
\(170\) 0 0
\(171\) −5.75563e25 −1.20403
\(172\) 3.46330e25 0.677506
\(173\) 4.75865e25 0.870871 0.435435 0.900220i \(-0.356594\pi\)
0.435435 + 0.900220i \(0.356594\pi\)
\(174\) −1.03337e25 −0.176987
\(175\) 0 0
\(176\) 5.52172e25 0.829236
\(177\) 1.38028e25 0.194211
\(178\) −9.22395e24 −0.121643
\(179\) 9.84804e25 1.21770 0.608850 0.793286i \(-0.291631\pi\)
0.608850 + 0.793286i \(0.291631\pi\)
\(180\) 0 0
\(181\) 4.76456e25 0.518465 0.259233 0.965815i \(-0.416530\pi\)
0.259233 + 0.965815i \(0.416530\pi\)
\(182\) 2.09973e26 2.14458
\(183\) −2.29053e25 −0.219659
\(184\) −2.13552e26 −1.92353
\(185\) 0 0
\(186\) −3.38586e23 −0.00269321
\(187\) −1.19944e25 −0.0897017
\(188\) −4.29724e26 −3.02256
\(189\) −4.70697e25 −0.311481
\(190\) 0 0
\(191\) −8.54117e25 −0.500765 −0.250382 0.968147i \(-0.580556\pi\)
−0.250382 + 0.968147i \(0.580556\pi\)
\(192\) −1.94666e25 −0.107480
\(193\) 2.66797e26 1.38762 0.693812 0.720156i \(-0.255930\pi\)
0.693812 + 0.720156i \(0.255930\pi\)
\(194\) 4.20735e26 2.06200
\(195\) 0 0
\(196\) −1.10052e25 −0.0479352
\(197\) 1.00070e26 0.411095 0.205547 0.978647i \(-0.434103\pi\)
0.205547 + 0.978647i \(0.434103\pi\)
\(198\) 2.82693e26 1.09563
\(199\) 4.46897e26 1.63454 0.817272 0.576251i \(-0.195485\pi\)
0.817272 + 0.576251i \(0.195485\pi\)
\(200\) 0 0
\(201\) −1.35831e25 −0.0442838
\(202\) −2.46783e26 −0.759931
\(203\) 2.14149e26 0.623035
\(204\) −1.70811e25 −0.0469645
\(205\) 0 0
\(206\) −1.64962e26 −0.405426
\(207\) −4.18071e26 −0.971832
\(208\) −7.27620e26 −1.60021
\(209\) 3.79295e26 0.789396
\(210\) 0 0
\(211\) −7.96987e26 −1.48663 −0.743315 0.668941i \(-0.766748\pi\)
−0.743315 + 0.668941i \(0.766748\pi\)
\(212\) 1.12440e27 1.98636
\(213\) 1.12163e26 0.187709
\(214\) −3.02090e26 −0.479047
\(215\) 0 0
\(216\) 4.26556e26 0.607796
\(217\) 7.01664e24 0.00948074
\(218\) −1.46208e27 −1.87379
\(219\) 2.23724e26 0.272023
\(220\) 0 0
\(221\) 1.58056e26 0.173101
\(222\) 8.51582e25 0.0885456
\(223\) 5.69004e26 0.561836 0.280918 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(224\) −3.73976e26 −0.350745
\(225\) 0 0
\(226\) −2.48623e26 −0.210520
\(227\) 1.82165e27 1.46611 0.733056 0.680168i \(-0.238093\pi\)
0.733056 + 0.680168i \(0.238093\pi\)
\(228\) 5.40150e26 0.413298
\(229\) 1.11330e27 0.810032 0.405016 0.914310i \(-0.367266\pi\)
0.405016 + 0.914310i \(0.367266\pi\)
\(230\) 0 0
\(231\) 1.53094e26 0.100791
\(232\) −1.94067e27 −1.21574
\(233\) 1.63214e27 0.973118 0.486559 0.873648i \(-0.338252\pi\)
0.486559 + 0.873648i \(0.338252\pi\)
\(234\) −3.72516e27 −2.11428
\(235\) 0 0
\(236\) 4.95683e27 2.55104
\(237\) −1.95372e26 −0.0957761
\(238\) 5.22845e26 0.244196
\(239\) 2.58550e27 1.15072 0.575358 0.817902i \(-0.304863\pi\)
0.575358 + 0.817902i \(0.304863\pi\)
\(240\) 0 0
\(241\) 1.86784e27 0.755340 0.377670 0.925940i \(-0.376725\pi\)
0.377670 + 0.925940i \(0.376725\pi\)
\(242\) 2.70049e27 1.04127
\(243\) 1.25799e27 0.462600
\(244\) −8.22571e27 −2.88530
\(245\) 0 0
\(246\) 9.22102e26 0.294459
\(247\) −4.99813e27 −1.52333
\(248\) −6.35863e25 −0.0184999
\(249\) −7.14493e26 −0.198475
\(250\) 0 0
\(251\) −4.11170e27 −1.04177 −0.520887 0.853626i \(-0.674399\pi\)
−0.520887 + 0.853626i \(0.674399\pi\)
\(252\) −8.34280e27 −2.01932
\(253\) 2.75508e27 0.637160
\(254\) 6.97543e27 1.54164
\(255\) 0 0
\(256\) −1.00825e28 −2.03614
\(257\) 5.10426e27 0.985603 0.492802 0.870142i \(-0.335973\pi\)
0.492802 + 0.870142i \(0.335973\pi\)
\(258\) −4.91487e26 −0.0907580
\(259\) −1.76476e27 −0.311702
\(260\) 0 0
\(261\) −3.79925e27 −0.614232
\(262\) −1.34173e28 −2.07587
\(263\) −9.95433e27 −1.47408 −0.737040 0.675849i \(-0.763777\pi\)
−0.737040 + 0.675849i \(0.763777\pi\)
\(264\) −1.38737e27 −0.196674
\(265\) 0 0
\(266\) −1.65337e28 −2.14898
\(267\) 8.86222e25 0.0110322
\(268\) −4.87795e27 −0.581684
\(269\) 6.42524e26 0.0734071 0.0367036 0.999326i \(-0.488314\pi\)
0.0367036 + 0.999326i \(0.488314\pi\)
\(270\) 0 0
\(271\) 1.19246e28 1.25111 0.625557 0.780178i \(-0.284872\pi\)
0.625557 + 0.780178i \(0.284872\pi\)
\(272\) −1.81181e27 −0.182210
\(273\) −2.01739e27 −0.194500
\(274\) −6.20925e27 −0.573994
\(275\) 0 0
\(276\) 3.92348e27 0.333593
\(277\) 5.29510e27 0.431874 0.215937 0.976407i \(-0.430719\pi\)
0.215937 + 0.976407i \(0.430719\pi\)
\(278\) 1.89698e28 1.48439
\(279\) −1.24483e26 −0.00934678
\(280\) 0 0
\(281\) 9.67433e27 0.669111 0.334556 0.942376i \(-0.391414\pi\)
0.334556 + 0.942376i \(0.391414\pi\)
\(282\) 6.09834e27 0.404899
\(283\) 2.62436e28 1.67294 0.836468 0.548016i \(-0.184617\pi\)
0.836468 + 0.548016i \(0.184617\pi\)
\(284\) 4.02799e28 2.46563
\(285\) 0 0
\(286\) 2.45488e28 1.38618
\(287\) −1.91091e28 −1.03657
\(288\) 6.63476e27 0.345789
\(289\) −1.95740e28 −0.980290
\(290\) 0 0
\(291\) −4.04235e27 −0.187010
\(292\) 8.03435e28 3.57312
\(293\) −3.20151e28 −1.36892 −0.684458 0.729052i \(-0.739961\pi\)
−0.684458 + 0.729052i \(0.739961\pi\)
\(294\) 1.56178e26 0.00642136
\(295\) 0 0
\(296\) 1.59927e28 0.608228
\(297\) −5.50310e27 −0.201330
\(298\) −7.79965e28 −2.74529
\(299\) −3.63049e28 −1.22955
\(300\) 0 0
\(301\) 1.01853e28 0.319490
\(302\) 6.38225e28 1.92705
\(303\) 2.37105e27 0.0689207
\(304\) 5.72943e28 1.60349
\(305\) 0 0
\(306\) −9.27586e27 −0.240745
\(307\) 1.15407e28 0.288496 0.144248 0.989542i \(-0.453924\pi\)
0.144248 + 0.989542i \(0.453924\pi\)
\(308\) 5.49789e28 1.32392
\(309\) 1.58493e27 0.0367695
\(310\) 0 0
\(311\) 3.62817e28 0.781525 0.390763 0.920491i \(-0.372211\pi\)
0.390763 + 0.920491i \(0.372211\pi\)
\(312\) 1.82820e28 0.379530
\(313\) −2.56117e28 −0.512481 −0.256241 0.966613i \(-0.582484\pi\)
−0.256241 + 0.966613i \(0.582484\pi\)
\(314\) 9.79346e26 0.0188906
\(315\) 0 0
\(316\) −7.01618e28 −1.25805
\(317\) 2.91424e27 0.0503899 0.0251950 0.999683i \(-0.491979\pi\)
0.0251950 + 0.999683i \(0.491979\pi\)
\(318\) −1.59566e28 −0.266091
\(319\) 2.50370e28 0.402708
\(320\) 0 0
\(321\) 2.90243e27 0.0434464
\(322\) −1.20096e29 −1.73455
\(323\) −1.24456e28 −0.173455
\(324\) 1.44042e29 1.93741
\(325\) 0 0
\(326\) −1.16013e29 −1.45379
\(327\) 1.40474e28 0.169940
\(328\) 1.73170e29 2.02267
\(329\) −1.26378e29 −1.42534
\(330\) 0 0
\(331\) 4.00537e28 0.421328 0.210664 0.977558i \(-0.432437\pi\)
0.210664 + 0.977558i \(0.432437\pi\)
\(332\) −2.56588e29 −2.60704
\(333\) 3.13089e28 0.307298
\(334\) −6.01289e28 −0.570163
\(335\) 0 0
\(336\) 2.31256e28 0.204735
\(337\) −1.31131e29 −1.12192 −0.560959 0.827843i \(-0.689568\pi\)
−0.560959 + 0.827843i \(0.689568\pi\)
\(338\) −1.10697e29 −0.915364
\(339\) 2.38872e27 0.0190928
\(340\) 0 0
\(341\) 8.20342e26 0.00612801
\(342\) 2.93327e29 2.11861
\(343\) −1.44770e29 −1.01110
\(344\) −9.23010e28 −0.623425
\(345\) 0 0
\(346\) −2.42518e29 −1.53239
\(347\) −1.14051e29 −0.697126 −0.348563 0.937285i \(-0.613330\pi\)
−0.348563 + 0.937285i \(0.613330\pi\)
\(348\) 3.56549e28 0.210843
\(349\) 7.28518e28 0.416820 0.208410 0.978042i \(-0.433171\pi\)
0.208410 + 0.978042i \(0.433171\pi\)
\(350\) 0 0
\(351\) 7.25167e28 0.388514
\(352\) −4.37230e28 −0.226709
\(353\) 3.68425e29 1.84902 0.924508 0.381164i \(-0.124477\pi\)
0.924508 + 0.381164i \(0.124477\pi\)
\(354\) −7.03438e28 −0.341734
\(355\) 0 0
\(356\) 3.18259e28 0.144912
\(357\) −5.02340e27 −0.0221469
\(358\) −5.01891e29 −2.14267
\(359\) −2.45802e29 −1.01625 −0.508123 0.861284i \(-0.669661\pi\)
−0.508123 + 0.861284i \(0.669661\pi\)
\(360\) 0 0
\(361\) 1.35734e29 0.526448
\(362\) −2.42819e29 −0.912292
\(363\) −2.59458e28 −0.0944367
\(364\) −7.24481e29 −2.55482
\(365\) 0 0
\(366\) 1.16733e29 0.386512
\(367\) 4.73025e29 1.51783 0.758917 0.651187i \(-0.225729\pi\)
0.758917 + 0.651187i \(0.225729\pi\)
\(368\) 4.16168e29 1.29425
\(369\) 3.39016e29 1.02192
\(370\) 0 0
\(371\) 3.30675e29 0.936703
\(372\) 1.16824e27 0.00320840
\(373\) 2.90241e29 0.772872 0.386436 0.922316i \(-0.373706\pi\)
0.386436 + 0.922316i \(0.373706\pi\)
\(374\) 6.11278e28 0.157839
\(375\) 0 0
\(376\) 1.14526e30 2.78129
\(377\) −3.29924e29 −0.777120
\(378\) 2.39884e29 0.548082
\(379\) 2.72591e29 0.604172 0.302086 0.953281i \(-0.402317\pi\)
0.302086 + 0.953281i \(0.402317\pi\)
\(380\) 0 0
\(381\) −6.70187e28 −0.139817
\(382\) 4.35288e29 0.881147
\(383\) 9.91721e29 1.94806 0.974032 0.226408i \(-0.0726984\pi\)
0.974032 + 0.226408i \(0.0726984\pi\)
\(384\) 1.28913e29 0.245746
\(385\) 0 0
\(386\) −1.35969e30 −2.44166
\(387\) −1.80698e29 −0.314976
\(388\) −1.45168e30 −2.45644
\(389\) 8.39840e29 1.37967 0.689836 0.723966i \(-0.257683\pi\)
0.689836 + 0.723966i \(0.257683\pi\)
\(390\) 0 0
\(391\) −9.04012e28 −0.140004
\(392\) 2.93303e28 0.0441089
\(393\) 1.28911e29 0.188268
\(394\) −5.09992e29 −0.723363
\(395\) 0 0
\(396\) −9.75388e29 −1.30522
\(397\) −9.94808e27 −0.0129315 −0.00646574 0.999979i \(-0.502058\pi\)
−0.00646574 + 0.999979i \(0.502058\pi\)
\(398\) −2.27754e30 −2.87615
\(399\) 1.58853e29 0.194898
\(400\) 0 0
\(401\) −4.81480e29 −0.557723 −0.278861 0.960331i \(-0.589957\pi\)
−0.278861 + 0.960331i \(0.589957\pi\)
\(402\) 6.92244e28 0.0779219
\(403\) −1.08100e28 −0.0118254
\(404\) 8.51487e29 0.905299
\(405\) 0 0
\(406\) −1.09138e30 −1.09629
\(407\) −2.06325e29 −0.201473
\(408\) 4.55232e28 0.0432156
\(409\) −8.22531e29 −0.759162 −0.379581 0.925159i \(-0.623932\pi\)
−0.379581 + 0.925159i \(0.623932\pi\)
\(410\) 0 0
\(411\) 5.96573e28 0.0520575
\(412\) 5.69178e29 0.482981
\(413\) 1.45776e30 1.20299
\(414\) 2.13064e30 1.71004
\(415\) 0 0
\(416\) 5.76156e29 0.437489
\(417\) −1.82259e29 −0.134624
\(418\) −1.93302e30 −1.38902
\(419\) 2.72055e30 1.90194 0.950969 0.309287i \(-0.100090\pi\)
0.950969 + 0.309287i \(0.100090\pi\)
\(420\) 0 0
\(421\) 1.73534e30 1.14852 0.574261 0.818672i \(-0.305289\pi\)
0.574261 + 0.818672i \(0.305289\pi\)
\(422\) 4.06173e30 2.61588
\(423\) 2.24209e30 1.40520
\(424\) −2.99665e30 −1.82780
\(425\) 0 0
\(426\) −5.71623e29 −0.330293
\(427\) −2.41911e30 −1.36061
\(428\) 1.04232e30 0.570684
\(429\) −2.35860e29 −0.125718
\(430\) 0 0
\(431\) 2.30954e30 1.16691 0.583455 0.812146i \(-0.301701\pi\)
0.583455 + 0.812146i \(0.301701\pi\)
\(432\) −8.31270e29 −0.408958
\(433\) 4.49441e29 0.215309 0.107655 0.994188i \(-0.465666\pi\)
0.107655 + 0.994188i \(0.465666\pi\)
\(434\) −3.57593e28 −0.0166823
\(435\) 0 0
\(436\) 5.04468e30 2.23223
\(437\) 2.85872e30 1.23207
\(438\) −1.14018e30 −0.478652
\(439\) −1.98640e30 −0.812315 −0.406157 0.913803i \(-0.633132\pi\)
−0.406157 + 0.913803i \(0.633132\pi\)
\(440\) 0 0
\(441\) 5.74199e28 0.0222853
\(442\) −8.05507e29 −0.304588
\(443\) 2.09074e30 0.770294 0.385147 0.922855i \(-0.374151\pi\)
0.385147 + 0.922855i \(0.374151\pi\)
\(444\) −2.93825e29 −0.105484
\(445\) 0 0
\(446\) −2.89984e30 −0.988607
\(447\) 7.49376e29 0.248979
\(448\) −2.05594e30 −0.665753
\(449\) 2.38796e30 0.753694 0.376847 0.926276i \(-0.377008\pi\)
0.376847 + 0.926276i \(0.377008\pi\)
\(450\) 0 0
\(451\) −2.23411e30 −0.670000
\(452\) 8.57834e29 0.250791
\(453\) −6.13195e29 −0.174771
\(454\) −9.28376e30 −2.57977
\(455\) 0 0
\(456\) −1.43956e30 −0.380307
\(457\) −1.54921e30 −0.399093 −0.199547 0.979888i \(-0.563947\pi\)
−0.199547 + 0.979888i \(0.563947\pi\)
\(458\) −5.67374e30 −1.42533
\(459\) 1.80571e29 0.0442386
\(460\) 0 0
\(461\) 4.06665e30 0.947711 0.473855 0.880603i \(-0.342862\pi\)
0.473855 + 0.880603i \(0.342862\pi\)
\(462\) −7.80221e29 −0.177352
\(463\) 6.37975e29 0.141456 0.0707282 0.997496i \(-0.477468\pi\)
0.0707282 + 0.997496i \(0.477468\pi\)
\(464\) 3.78196e30 0.818013
\(465\) 0 0
\(466\) −8.31798e30 −1.71230
\(467\) −1.15365e30 −0.231701 −0.115851 0.993267i \(-0.536959\pi\)
−0.115851 + 0.993267i \(0.536959\pi\)
\(468\) 1.28531e31 2.51873
\(469\) −1.43456e30 −0.274304
\(470\) 0 0
\(471\) −9.40938e27 −0.00171325
\(472\) −1.32105e31 −2.34740
\(473\) 1.19080e30 0.206507
\(474\) 9.95686e29 0.168528
\(475\) 0 0
\(476\) −1.80400e30 −0.290908
\(477\) −5.86655e30 −0.923468
\(478\) −1.31766e31 −2.02480
\(479\) −2.12936e30 −0.319441 −0.159721 0.987162i \(-0.551059\pi\)
−0.159721 + 0.987162i \(0.551059\pi\)
\(480\) 0 0
\(481\) 2.71884e30 0.388790
\(482\) −9.51914e30 −1.32910
\(483\) 1.15386e30 0.157312
\(484\) −9.31762e30 −1.24046
\(485\) 0 0
\(486\) −6.41116e30 −0.813991
\(487\) −4.84200e30 −0.600401 −0.300201 0.953876i \(-0.597054\pi\)
−0.300201 + 0.953876i \(0.597054\pi\)
\(488\) 2.19225e31 2.65498
\(489\) 1.11463e30 0.131849
\(490\) 0 0
\(491\) −3.44338e29 −0.0388640 −0.0194320 0.999811i \(-0.506186\pi\)
−0.0194320 + 0.999811i \(0.506186\pi\)
\(492\) −3.18157e30 −0.350787
\(493\) −8.21528e29 −0.0884877
\(494\) 2.54722e31 2.68045
\(495\) 0 0
\(496\) 1.23917e29 0.0124477
\(497\) 1.18460e31 1.16271
\(498\) 3.64131e30 0.349237
\(499\) −1.27549e31 −1.19542 −0.597711 0.801711i \(-0.703923\pi\)
−0.597711 + 0.801711i \(0.703923\pi\)
\(500\) 0 0
\(501\) 5.77708e29 0.0517100
\(502\) 2.09546e31 1.83311
\(503\) −1.67840e31 −1.43504 −0.717520 0.696538i \(-0.754723\pi\)
−0.717520 + 0.696538i \(0.754723\pi\)
\(504\) 2.22345e31 1.85813
\(505\) 0 0
\(506\) −1.40409e31 −1.12115
\(507\) 1.06356e30 0.0830175
\(508\) −2.40677e31 −1.83654
\(509\) −2.11935e31 −1.58106 −0.790529 0.612425i \(-0.790194\pi\)
−0.790529 + 0.612425i \(0.790194\pi\)
\(510\) 0 0
\(511\) 2.36283e31 1.68497
\(512\) 2.92986e31 2.04288
\(513\) −5.71012e30 −0.389310
\(514\) −2.60131e31 −1.73427
\(515\) 0 0
\(516\) 1.69580e30 0.108119
\(517\) −1.47753e31 −0.921290
\(518\) 8.99386e30 0.548471
\(519\) 2.33007e30 0.138977
\(520\) 0 0
\(521\) −1.22820e30 −0.0700869 −0.0350434 0.999386i \(-0.511157\pi\)
−0.0350434 + 0.999386i \(0.511157\pi\)
\(522\) 1.93623e31 1.08080
\(523\) −7.24558e30 −0.395643 −0.197821 0.980238i \(-0.563387\pi\)
−0.197821 + 0.980238i \(0.563387\pi\)
\(524\) 4.62943e31 2.47296
\(525\) 0 0
\(526\) 5.07308e31 2.59379
\(527\) −2.69175e28 −0.00134652
\(528\) 2.70370e30 0.132333
\(529\) −1.15570e29 −0.00553483
\(530\) 0 0
\(531\) −2.58623e31 −1.18599
\(532\) 5.70471e31 2.56006
\(533\) 2.94399e31 1.29292
\(534\) −4.51650e29 −0.0194123
\(535\) 0 0
\(536\) 1.30003e31 0.535252
\(537\) 4.82208e30 0.194326
\(538\) −3.27453e30 −0.129167
\(539\) −3.78397e29 −0.0146109
\(540\) 0 0
\(541\) 9.30820e30 0.344427 0.172214 0.985060i \(-0.444908\pi\)
0.172214 + 0.985060i \(0.444908\pi\)
\(542\) −6.07719e31 −2.20146
\(543\) 2.33296e30 0.0827389
\(544\) 1.43466e30 0.0498152
\(545\) 0 0
\(546\) 1.02813e31 0.342242
\(547\) 2.67466e31 0.871796 0.435898 0.899996i \(-0.356431\pi\)
0.435898 + 0.899996i \(0.356431\pi\)
\(548\) 2.14241e31 0.683794
\(549\) 4.29177e31 1.34139
\(550\) 0 0
\(551\) 2.59789e31 0.778712
\(552\) −1.04565e31 −0.306965
\(553\) −2.06340e31 −0.593258
\(554\) −2.69857e31 −0.759926
\(555\) 0 0
\(556\) −6.54524e31 −1.76834
\(557\) −2.40510e31 −0.636501 −0.318250 0.948007i \(-0.603095\pi\)
−0.318250 + 0.948007i \(0.603095\pi\)
\(558\) 6.34410e29 0.0164466
\(559\) −1.56916e31 −0.398504
\(560\) 0 0
\(561\) −5.87305e29 −0.0143150
\(562\) −4.93038e31 −1.17737
\(563\) −2.15437e31 −0.504049 −0.252025 0.967721i \(-0.581096\pi\)
−0.252025 + 0.967721i \(0.581096\pi\)
\(564\) −2.10414e31 −0.482353
\(565\) 0 0
\(566\) −1.33747e32 −2.94370
\(567\) 4.23614e31 0.913618
\(568\) −1.07351e32 −2.26881
\(569\) 1.70978e31 0.354120 0.177060 0.984200i \(-0.443341\pi\)
0.177060 + 0.984200i \(0.443341\pi\)
\(570\) 0 0
\(571\) −8.69138e31 −1.72891 −0.864457 0.502707i \(-0.832337\pi\)
−0.864457 + 0.502707i \(0.832337\pi\)
\(572\) −8.47018e31 −1.65135
\(573\) −4.18217e30 −0.0799142
\(574\) 9.73865e31 1.82395
\(575\) 0 0
\(576\) 3.64747e31 0.656346
\(577\) −1.59167e31 −0.280757 −0.140378 0.990098i \(-0.544832\pi\)
−0.140378 + 0.990098i \(0.544832\pi\)
\(578\) 9.97560e31 1.72492
\(579\) 1.30637e31 0.221443
\(580\) 0 0
\(581\) −7.54602e31 −1.22940
\(582\) 2.06012e31 0.329063
\(583\) 3.86605e31 0.605451
\(584\) −2.14125e32 −3.28790
\(585\) 0 0
\(586\) 1.63160e32 2.40875
\(587\) −5.71080e31 −0.826721 −0.413361 0.910567i \(-0.635645\pi\)
−0.413361 + 0.910567i \(0.635645\pi\)
\(588\) −5.38870e29 −0.00764971
\(589\) 8.51202e29 0.0118497
\(590\) 0 0
\(591\) 4.89991e30 0.0656043
\(592\) −3.11664e31 −0.409248
\(593\) −2.68653e31 −0.345989 −0.172995 0.984923i \(-0.555344\pi\)
−0.172995 + 0.984923i \(0.555344\pi\)
\(594\) 2.80457e31 0.354261
\(595\) 0 0
\(596\) 2.69115e32 3.27044
\(597\) 2.18822e31 0.260848
\(598\) 1.85023e32 2.16352
\(599\) 1.35452e30 0.0155373 0.00776867 0.999970i \(-0.497527\pi\)
0.00776867 + 0.999970i \(0.497527\pi\)
\(600\) 0 0
\(601\) −7.37959e31 −0.814659 −0.407329 0.913281i \(-0.633540\pi\)
−0.407329 + 0.913281i \(0.633540\pi\)
\(602\) −5.19077e31 −0.562175
\(603\) 2.54507e31 0.270428
\(604\) −2.20210e32 −2.29568
\(605\) 0 0
\(606\) −1.20837e31 −0.121273
\(607\) 3.23323e31 0.318395 0.159198 0.987247i \(-0.449109\pi\)
0.159198 + 0.987247i \(0.449109\pi\)
\(608\) −4.53677e31 −0.438385
\(609\) 1.04858e31 0.0994266
\(610\) 0 0
\(611\) 1.94701e32 1.77785
\(612\) 3.20049e31 0.286798
\(613\) −4.35574e31 −0.383059 −0.191530 0.981487i \(-0.561345\pi\)
−0.191530 + 0.981487i \(0.561345\pi\)
\(614\) −5.88154e31 −0.507639
\(615\) 0 0
\(616\) −1.46525e32 −1.21824
\(617\) 2.29127e32 1.86981 0.934904 0.354902i \(-0.115486\pi\)
0.934904 + 0.354902i \(0.115486\pi\)
\(618\) −8.07736e30 −0.0646997
\(619\) 2.32366e32 1.82697 0.913484 0.406875i \(-0.133382\pi\)
0.913484 + 0.406875i \(0.133382\pi\)
\(620\) 0 0
\(621\) −4.14765e31 −0.314231
\(622\) −1.84904e32 −1.37517
\(623\) 9.35970e30 0.0683360
\(624\) −3.56278e31 −0.255368
\(625\) 0 0
\(626\) 1.30526e32 0.901763
\(627\) 1.85721e31 0.125975
\(628\) −3.37908e30 −0.0225042
\(629\) 6.77005e30 0.0442700
\(630\) 0 0
\(631\) −2.22944e32 −1.40559 −0.702793 0.711394i \(-0.748064\pi\)
−0.702793 + 0.711394i \(0.748064\pi\)
\(632\) 1.86989e32 1.15763
\(633\) −3.90244e31 −0.237243
\(634\) −1.48520e31 −0.0886662
\(635\) 0 0
\(636\) 5.50560e31 0.316992
\(637\) 4.98629e30 0.0281951
\(638\) −1.27597e32 −0.708605
\(639\) −2.10161e32 −1.14628
\(640\) 0 0
\(641\) −2.75021e32 −1.44710 −0.723550 0.690272i \(-0.757491\pi\)
−0.723550 + 0.690272i \(0.757491\pi\)
\(642\) −1.47918e31 −0.0764483
\(643\) −1.47614e32 −0.749377 −0.374689 0.927151i \(-0.622250\pi\)
−0.374689 + 0.927151i \(0.622250\pi\)
\(644\) 4.14373e32 2.06635
\(645\) 0 0
\(646\) 6.34273e31 0.305212
\(647\) −1.82358e32 −0.862036 −0.431018 0.902343i \(-0.641846\pi\)
−0.431018 + 0.902343i \(0.641846\pi\)
\(648\) −3.83888e32 −1.78276
\(649\) 1.70432e32 0.777567
\(650\) 0 0
\(651\) 3.43569e29 0.00151298
\(652\) 4.00284e32 1.73189
\(653\) 4.35899e31 0.185304 0.0926519 0.995699i \(-0.470466\pi\)
0.0926519 + 0.995699i \(0.470466\pi\)
\(654\) −7.15905e31 −0.299028
\(655\) 0 0
\(656\) −3.37473e32 −1.36096
\(657\) −4.19193e32 −1.66116
\(658\) 6.44067e32 2.50803
\(659\) −3.58880e31 −0.137331 −0.0686653 0.997640i \(-0.521874\pi\)
−0.0686653 + 0.997640i \(0.521874\pi\)
\(660\) 0 0
\(661\) 2.78803e32 1.03034 0.515171 0.857088i \(-0.327729\pi\)
0.515171 + 0.857088i \(0.327729\pi\)
\(662\) −2.04128e32 −0.741370
\(663\) 7.73917e30 0.0276241
\(664\) 6.83836e32 2.39894
\(665\) 0 0
\(666\) −1.59561e32 −0.540721
\(667\) 1.88703e32 0.628537
\(668\) 2.07466e32 0.679230
\(669\) 2.78612e31 0.0896601
\(670\) 0 0
\(671\) −2.82827e32 −0.879452
\(672\) −1.83117e31 −0.0559734
\(673\) −1.90131e31 −0.0571320 −0.0285660 0.999592i \(-0.509094\pi\)
−0.0285660 + 0.999592i \(0.509094\pi\)
\(674\) 6.68289e32 1.97413
\(675\) 0 0
\(676\) 3.81943e32 1.09047
\(677\) 3.87840e32 1.08864 0.544318 0.838879i \(-0.316788\pi\)
0.544318 + 0.838879i \(0.316788\pi\)
\(678\) −1.21738e31 −0.0335957
\(679\) −4.26927e32 −1.15838
\(680\) 0 0
\(681\) 8.91968e31 0.233969
\(682\) −4.18075e30 −0.0107829
\(683\) −6.51681e32 −1.65271 −0.826356 0.563148i \(-0.809590\pi\)
−0.826356 + 0.563148i \(0.809590\pi\)
\(684\) −1.01208e33 −2.52389
\(685\) 0 0
\(686\) 7.37799e32 1.77914
\(687\) 5.45123e31 0.129268
\(688\) 1.79876e32 0.419474
\(689\) −5.09446e32 −1.16836
\(690\) 0 0
\(691\) −6.18205e31 −0.137131 −0.0685654 0.997647i \(-0.521842\pi\)
−0.0685654 + 0.997647i \(0.521842\pi\)
\(692\) 8.36770e32 1.82552
\(693\) −2.86853e32 −0.615499
\(694\) 5.81246e32 1.22666
\(695\) 0 0
\(696\) −9.50245e31 −0.194012
\(697\) 7.33069e31 0.147220
\(698\) −3.71278e32 −0.733436
\(699\) 7.99177e31 0.155294
\(700\) 0 0
\(701\) 5.26483e32 0.989982 0.494991 0.868898i \(-0.335171\pi\)
0.494991 + 0.868898i \(0.335171\pi\)
\(702\) −3.69571e32 −0.683629
\(703\) −2.14087e32 −0.389586
\(704\) −2.40367e32 −0.430319
\(705\) 0 0
\(706\) −1.87763e33 −3.25353
\(707\) 2.50415e32 0.426910
\(708\) 2.42711e32 0.407105
\(709\) −7.09958e32 −1.17166 −0.585829 0.810435i \(-0.699231\pi\)
−0.585829 + 0.810435i \(0.699231\pi\)
\(710\) 0 0
\(711\) 3.66070e32 0.584876
\(712\) −8.48196e31 −0.133345
\(713\) 6.18287e30 0.00956446
\(714\) 2.56010e31 0.0389698
\(715\) 0 0
\(716\) 1.73170e33 2.55254
\(717\) 1.26598e32 0.183636
\(718\) 1.25269e33 1.78819
\(719\) 4.58875e32 0.644633 0.322316 0.946632i \(-0.395539\pi\)
0.322316 + 0.946632i \(0.395539\pi\)
\(720\) 0 0
\(721\) 1.67390e32 0.227758
\(722\) −6.91748e32 −0.926339
\(723\) 9.14583e31 0.120540
\(724\) 8.37810e32 1.08681
\(725\) 0 0
\(726\) 1.32229e32 0.166171
\(727\) −1.25284e33 −1.54970 −0.774851 0.632144i \(-0.782175\pi\)
−0.774851 + 0.632144i \(0.782175\pi\)
\(728\) 1.93083e33 2.35089
\(729\) −7.09581e32 −0.850424
\(730\) 0 0
\(731\) −3.90730e31 −0.0453761
\(732\) −4.02771e32 −0.460448
\(733\) 1.67809e30 0.00188852 0.000944258 1.00000i \(-0.499699\pi\)
0.000944258 1.00000i \(0.499699\pi\)
\(734\) −2.41070e33 −2.67078
\(735\) 0 0
\(736\) −3.29537e32 −0.353842
\(737\) −1.67720e32 −0.177300
\(738\) −1.72775e33 −1.79817
\(739\) 2.96573e32 0.303893 0.151946 0.988389i \(-0.451446\pi\)
0.151946 + 0.988389i \(0.451446\pi\)
\(740\) 0 0
\(741\) −2.44733e32 −0.243099
\(742\) −1.68524e33 −1.64822
\(743\) 4.48312e32 0.431727 0.215863 0.976424i \(-0.430743\pi\)
0.215863 + 0.976424i \(0.430743\pi\)
\(744\) −3.11349e30 −0.00295229
\(745\) 0 0
\(746\) −1.47917e33 −1.35995
\(747\) 1.33875e33 1.21203
\(748\) −2.10912e32 −0.188033
\(749\) 3.06536e32 0.269116
\(750\) 0 0
\(751\) −3.50465e32 −0.298391 −0.149195 0.988808i \(-0.547668\pi\)
−0.149195 + 0.988808i \(0.547668\pi\)
\(752\) −2.23188e33 −1.87140
\(753\) −2.01329e32 −0.166251
\(754\) 1.68141e33 1.36742
\(755\) 0 0
\(756\) −8.27683e32 −0.652925
\(757\) −4.37490e32 −0.339911 −0.169956 0.985452i \(-0.554362\pi\)
−0.169956 + 0.985452i \(0.554362\pi\)
\(758\) −1.38922e33 −1.06310
\(759\) 1.34902e32 0.101681
\(760\) 0 0
\(761\) −1.53089e31 −0.0111949 −0.00559744 0.999984i \(-0.501782\pi\)
−0.00559744 + 0.999984i \(0.501782\pi\)
\(762\) 3.41551e32 0.246022
\(763\) 1.48360e33 1.05265
\(764\) −1.50190e33 −1.04970
\(765\) 0 0
\(766\) −5.05416e33 −3.42782
\(767\) −2.24586e33 −1.50050
\(768\) −4.93686e32 −0.324935
\(769\) 2.67481e33 1.73436 0.867180 0.497995i \(-0.165930\pi\)
0.867180 + 0.497995i \(0.165930\pi\)
\(770\) 0 0
\(771\) 2.49929e32 0.157287
\(772\) 4.69141e33 2.90873
\(773\) −8.38808e31 −0.0512387 −0.0256194 0.999672i \(-0.508156\pi\)
−0.0256194 + 0.999672i \(0.508156\pi\)
\(774\) 9.20900e32 0.554232
\(775\) 0 0
\(776\) 3.86890e33 2.26036
\(777\) −8.64114e31 −0.0497427
\(778\) −4.28012e33 −2.42767
\(779\) −2.31816e33 −1.29557
\(780\) 0 0
\(781\) 1.38496e33 0.751534
\(782\) 4.60716e32 0.246352
\(783\) −3.76921e32 −0.198605
\(784\) −5.71586e31 −0.0296788
\(785\) 0 0
\(786\) −6.56975e32 −0.331276
\(787\) 3.07440e33 1.52775 0.763874 0.645366i \(-0.223295\pi\)
0.763874 + 0.645366i \(0.223295\pi\)
\(788\) 1.75965e33 0.861737
\(789\) −4.87412e32 −0.235240
\(790\) 0 0
\(791\) 2.52281e32 0.118265
\(792\) 2.59952e33 1.20103
\(793\) 3.72693e33 1.69711
\(794\) 5.06989e31 0.0227543
\(795\) 0 0
\(796\) 7.85832e33 3.42633
\(797\) −2.31415e33 −0.994535 −0.497268 0.867597i \(-0.665663\pi\)
−0.497268 + 0.867597i \(0.665663\pi\)
\(798\) −8.09572e32 −0.342943
\(799\) 4.84816e32 0.202437
\(800\) 0 0
\(801\) −1.66052e32 −0.0673704
\(802\) 2.45379e33 0.981370
\(803\) 2.76247e33 1.08910
\(804\) −2.38848e32 −0.0928276
\(805\) 0 0
\(806\) 5.50915e31 0.0208081
\(807\) 3.14611e31 0.0117146
\(808\) −2.26931e33 −0.833035
\(809\) 3.80662e33 1.37763 0.688814 0.724938i \(-0.258132\pi\)
0.688814 + 0.724938i \(0.258132\pi\)
\(810\) 0 0
\(811\) 4.24454e33 1.49311 0.746554 0.665325i \(-0.231707\pi\)
0.746554 + 0.665325i \(0.231707\pi\)
\(812\) 3.76564e33 1.30601
\(813\) 5.83886e32 0.199658
\(814\) 1.05151e33 0.354512
\(815\) 0 0
\(816\) −8.87152e31 −0.0290778
\(817\) 1.23559e33 0.399321
\(818\) 4.19191e33 1.33582
\(819\) 3.77998e33 1.18775
\(820\) 0 0
\(821\) −2.99736e33 −0.915784 −0.457892 0.889008i \(-0.651395\pi\)
−0.457892 + 0.889008i \(0.651395\pi\)
\(822\) −3.04035e32 −0.0916004
\(823\) 5.19319e32 0.154289 0.0771447 0.997020i \(-0.475420\pi\)
0.0771447 + 0.997020i \(0.475420\pi\)
\(824\) −1.51692e33 −0.444428
\(825\) 0 0
\(826\) −7.42926e33 −2.11678
\(827\) −4.81781e32 −0.135374 −0.0676872 0.997707i \(-0.521562\pi\)
−0.0676872 + 0.997707i \(0.521562\pi\)
\(828\) −7.35144e33 −2.03715
\(829\) −2.15627e33 −0.589287 −0.294644 0.955607i \(-0.595201\pi\)
−0.294644 + 0.955607i \(0.595201\pi\)
\(830\) 0 0
\(831\) 2.59274e32 0.0689202
\(832\) 3.16743e33 0.830401
\(833\) 1.24161e31 0.00321048
\(834\) 9.28853e32 0.236885
\(835\) 0 0
\(836\) 6.66960e33 1.65473
\(837\) −1.23499e31 −0.00302218
\(838\) −1.38649e34 −3.34665
\(839\) −2.38371e32 −0.0567532 −0.0283766 0.999597i \(-0.509034\pi\)
−0.0283766 + 0.999597i \(0.509034\pi\)
\(840\) 0 0
\(841\) −2.60187e33 −0.602743
\(842\) −8.84388e33 −2.02094
\(843\) 4.73702e32 0.106780
\(844\) −1.40144e34 −3.11627
\(845\) 0 0
\(846\) −1.14265e34 −2.47260
\(847\) −2.74023e33 −0.584962
\(848\) 5.83985e33 1.22984
\(849\) 1.28501e33 0.266974
\(850\) 0 0
\(851\) −1.55506e33 −0.314454
\(852\) 1.97230e33 0.393475
\(853\) 1.94872e32 0.0383563 0.0191781 0.999816i \(-0.493895\pi\)
0.0191781 + 0.999816i \(0.493895\pi\)
\(854\) 1.23286e34 2.39414
\(855\) 0 0
\(856\) −2.77789e33 −0.525130
\(857\) 1.87495e33 0.349711 0.174855 0.984594i \(-0.444054\pi\)
0.174855 + 0.984594i \(0.444054\pi\)
\(858\) 1.20203e33 0.221213
\(859\) 6.46779e33 1.17445 0.587225 0.809424i \(-0.300220\pi\)
0.587225 + 0.809424i \(0.300220\pi\)
\(860\) 0 0
\(861\) −9.35673e32 −0.165420
\(862\) −1.17702e34 −2.05330
\(863\) 1.04050e34 1.79110 0.895550 0.444961i \(-0.146782\pi\)
0.895550 + 0.444961i \(0.146782\pi\)
\(864\) 6.58230e32 0.111807
\(865\) 0 0
\(866\) −2.29051e33 −0.378858
\(867\) −9.58438e32 −0.156439
\(868\) 1.23382e32 0.0198735
\(869\) −2.41239e33 −0.383461
\(870\) 0 0
\(871\) 2.21012e33 0.342142
\(872\) −1.34447e34 −2.05405
\(873\) 7.57417e33 1.14201
\(874\) −1.45691e34 −2.16795
\(875\) 0 0
\(876\) 3.93401e33 0.570214
\(877\) −8.52438e33 −1.21946 −0.609730 0.792609i \(-0.708722\pi\)
−0.609730 + 0.792609i \(0.708722\pi\)
\(878\) 1.01234e34 1.42935
\(879\) −1.56761e33 −0.218457
\(880\) 0 0
\(881\) 1.20706e34 1.63872 0.819362 0.573276i \(-0.194328\pi\)
0.819362 + 0.573276i \(0.194328\pi\)
\(882\) −2.92632e32 −0.0392133
\(883\) −4.60310e33 −0.608839 −0.304420 0.952538i \(-0.598462\pi\)
−0.304420 + 0.952538i \(0.598462\pi\)
\(884\) 2.77928e33 0.362853
\(885\) 0 0
\(886\) −1.06551e34 −1.35541
\(887\) 7.25037e33 0.910413 0.455207 0.890386i \(-0.349565\pi\)
0.455207 + 0.890386i \(0.349565\pi\)
\(888\) 7.83079e32 0.0970636
\(889\) −7.07809e33 −0.866055
\(890\) 0 0
\(891\) 4.95263e33 0.590530
\(892\) 1.00055e34 1.17772
\(893\) −1.53312e34 −1.78149
\(894\) −3.81909e33 −0.438105
\(895\) 0 0
\(896\) 1.36149e34 1.52220
\(897\) −1.77766e33 −0.196217
\(898\) −1.21699e34 −1.32620
\(899\) 5.61873e31 0.00604507
\(900\) 0 0
\(901\) −1.26855e33 −0.133037
\(902\) 1.13858e34 1.17893
\(903\) 4.98720e32 0.0509856
\(904\) −2.28623e33 −0.230772
\(905\) 0 0
\(906\) 3.12506e33 0.307528
\(907\) 3.03876e33 0.295266 0.147633 0.989042i \(-0.452835\pi\)
0.147633 + 0.989042i \(0.452835\pi\)
\(908\) 3.20322e34 3.07326
\(909\) −4.44264e33 −0.420878
\(910\) 0 0
\(911\) −1.60065e34 −1.47855 −0.739273 0.673406i \(-0.764831\pi\)
−0.739273 + 0.673406i \(0.764831\pi\)
\(912\) 2.80541e33 0.255891
\(913\) −8.82233e33 −0.794638
\(914\) 7.89533e33 0.702245
\(915\) 0 0
\(916\) 1.95764e34 1.69799
\(917\) 1.36147e34 1.16617
\(918\) −9.20251e32 −0.0778423
\(919\) −1.54454e34 −1.29024 −0.645120 0.764081i \(-0.723193\pi\)
−0.645120 + 0.764081i \(0.723193\pi\)
\(920\) 0 0
\(921\) 5.65088e32 0.0460395
\(922\) −2.07251e34 −1.66759
\(923\) −1.82501e34 −1.45026
\(924\) 2.69204e33 0.211277
\(925\) 0 0
\(926\) −3.25135e33 −0.248907
\(927\) −2.96969e33 −0.224540
\(928\) −2.99469e33 −0.223641
\(929\) 1.19786e34 0.883535 0.441768 0.897129i \(-0.354352\pi\)
0.441768 + 0.897129i \(0.354352\pi\)
\(930\) 0 0
\(931\) −3.92631e32 −0.0282529
\(932\) 2.86999e34 2.03985
\(933\) 1.77653e33 0.124719
\(934\) 5.87939e33 0.407702
\(935\) 0 0
\(936\) −3.42550e34 −2.31767
\(937\) 2.83646e33 0.189570 0.0947852 0.995498i \(-0.469784\pi\)
0.0947852 + 0.995498i \(0.469784\pi\)
\(938\) 7.31103e33 0.482665
\(939\) −1.25407e33 −0.0817839
\(940\) 0 0
\(941\) 2.88589e34 1.83653 0.918267 0.395961i \(-0.129588\pi\)
0.918267 + 0.395961i \(0.129588\pi\)
\(942\) 4.79535e31 0.00301464
\(943\) −1.68384e34 −1.04572
\(944\) 2.57446e34 1.57946
\(945\) 0 0
\(946\) −6.06872e33 −0.363370
\(947\) −2.29990e34 −1.36045 −0.680226 0.733002i \(-0.738118\pi\)
−0.680226 + 0.733002i \(0.738118\pi\)
\(948\) −3.43546e33 −0.200766
\(949\) −3.64023e34 −2.10168
\(950\) 0 0
\(951\) 1.42695e32 0.00804144
\(952\) 4.80786e33 0.267687
\(953\) 2.39769e34 1.31894 0.659470 0.751731i \(-0.270781\pi\)
0.659470 + 0.751731i \(0.270781\pi\)
\(954\) 2.98980e34 1.62494
\(955\) 0 0
\(956\) 4.54638e34 2.41213
\(957\) 1.22593e33 0.0642658
\(958\) 1.08520e34 0.562090
\(959\) 6.30062e33 0.322455
\(960\) 0 0
\(961\) −2.00115e34 −0.999908
\(962\) −1.38561e34 −0.684115
\(963\) −5.43829e33 −0.265314
\(964\) 3.28444e34 1.58334
\(965\) 0 0
\(966\) −5.88048e33 −0.276806
\(967\) 6.66196e33 0.309883 0.154942 0.987924i \(-0.450481\pi\)
0.154942 + 0.987924i \(0.450481\pi\)
\(968\) 2.48325e34 1.14144
\(969\) −6.09398e32 −0.0276808
\(970\) 0 0
\(971\) −3.35172e34 −1.48678 −0.743390 0.668858i \(-0.766783\pi\)
−0.743390 + 0.668858i \(0.766783\pi\)
\(972\) 2.21207e34 0.969701
\(973\) −1.92490e34 −0.833892
\(974\) 2.46765e34 1.05647
\(975\) 0 0
\(976\) −4.27224e34 −1.78642
\(977\) −1.17263e34 −0.484590 −0.242295 0.970203i \(-0.577900\pi\)
−0.242295 + 0.970203i \(0.577900\pi\)
\(978\) −5.68054e33 −0.232003
\(979\) 1.09428e33 0.0441699
\(980\) 0 0
\(981\) −2.63207e34 −1.03778
\(982\) 1.75487e33 0.0683851
\(983\) −1.51439e34 −0.583274 −0.291637 0.956529i \(-0.594200\pi\)
−0.291637 + 0.956529i \(0.594200\pi\)
\(984\) 8.47927e33 0.322786
\(985\) 0 0
\(986\) 4.18679e33 0.155703
\(987\) −6.18808e33 −0.227462
\(988\) −8.78881e34 −3.19319
\(989\) 8.97497e33 0.322311
\(990\) 0 0
\(991\) 2.60260e34 0.913187 0.456593 0.889675i \(-0.349069\pi\)
0.456593 + 0.889675i \(0.349069\pi\)
\(992\) −9.81216e31 −0.00340315
\(993\) 1.96122e33 0.0672374
\(994\) −6.03711e34 −2.04591
\(995\) 0 0
\(996\) −1.25638e34 −0.416043
\(997\) 1.96863e34 0.644422 0.322211 0.946668i \(-0.395574\pi\)
0.322211 + 0.946668i \(0.395574\pi\)
\(998\) 6.50036e34 2.10347
\(999\) 3.10613e33 0.0993613
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 25.24.a.a.1.1 2
5.2 odd 4 25.24.b.a.24.1 4
5.3 odd 4 25.24.b.a.24.4 4
5.4 even 2 1.24.a.a.1.2 2
15.14 odd 2 9.24.a.b.1.1 2
20.19 odd 2 16.24.a.b.1.2 2
35.34 odd 2 49.24.a.b.1.2 2
40.19 odd 2 64.24.a.g.1.1 2
40.29 even 2 64.24.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.24.a.a.1.2 2 5.4 even 2
9.24.a.b.1.1 2 15.14 odd 2
16.24.a.b.1.2 2 20.19 odd 2
25.24.a.a.1.1 2 1.1 even 1 trivial
25.24.b.a.24.1 4 5.2 odd 4
25.24.b.a.24.4 4 5.3 odd 4
49.24.a.b.1.2 2 35.34 odd 2
64.24.a.d.1.2 2 40.29 even 2
64.24.a.g.1.1 2 40.19 odd 2