Properties

Label 25.34.a.f.1.14
Level $25$
Weight $34$
Character 25.1
Self dual yes
Analytic conductor $172.457$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 26286285043 x^{14} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: multiple of \( 2^{85}\cdot 3^{26}\cdot 5^{66}\cdot 7^{4}\cdot 11^{8} \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.14
Root \(78034.2\) 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)
 
\(f(q)\) \(=\) \(q+156068. q^{2} -6.96239e7 q^{3} +1.57674e10 q^{4} -1.08661e13 q^{6} -1.55193e14 q^{7} +1.12018e15 q^{8} -7.11573e14 q^{9} +O(q^{10})\) \(q+156068. q^{2} -6.96239e7 q^{3} +1.57674e10 q^{4} -1.08661e13 q^{6} -1.55193e14 q^{7} +1.12018e15 q^{8} -7.11573e14 q^{9} +4.13199e16 q^{11} -1.09779e18 q^{12} -2.38246e18 q^{13} -2.42207e19 q^{14} +3.93836e19 q^{16} -2.91772e20 q^{17} -1.11054e20 q^{18} -3.61361e20 q^{19} +1.08051e22 q^{21} +6.44874e21 q^{22} +3.81748e22 q^{23} -7.79913e22 q^{24} -3.71827e23 q^{26} +4.36586e23 q^{27} -2.44699e24 q^{28} +1.65188e24 q^{29} -2.76253e24 q^{31} -3.47573e24 q^{32} -2.87685e24 q^{33} -4.55364e25 q^{34} -1.12197e25 q^{36} -8.70723e24 q^{37} -5.63971e25 q^{38} +1.65876e26 q^{39} +2.69048e26 q^{41} +1.68634e27 q^{42} -1.33984e26 q^{43} +6.51509e26 q^{44} +5.95788e27 q^{46} -2.88434e27 q^{47} -2.74204e27 q^{48} +1.63539e28 q^{49} +2.03143e28 q^{51} -3.75653e28 q^{52} -2.92563e28 q^{53} +6.81373e28 q^{54} -1.73844e29 q^{56} +2.51594e28 q^{57} +2.57806e29 q^{58} +1.51134e29 q^{59} +2.88604e29 q^{61} -4.31144e29 q^{62} +1.10431e29 q^{63} -8.80755e29 q^{64} -4.48986e29 q^{66} -8.72314e29 q^{67} -4.60049e30 q^{68} -2.65788e30 q^{69} +3.08526e30 q^{71} -7.97090e29 q^{72} +2.41300e30 q^{73} -1.35892e30 q^{74} -5.69774e30 q^{76} -6.41256e30 q^{77} +2.58880e31 q^{78} -3.21352e31 q^{79} -2.64411e31 q^{81} +4.19898e31 q^{82} +7.79056e31 q^{83} +1.70369e32 q^{84} -2.09107e31 q^{86} -1.15010e32 q^{87} +4.62857e31 q^{88} +1.41115e32 q^{89} +3.69741e32 q^{91} +6.01918e32 q^{92} +1.92338e32 q^{93} -4.50154e32 q^{94} +2.41994e32 q^{96} +7.13656e32 q^{97} +2.55232e33 q^{98} -2.94021e31 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 72851326872 q^{4} + 11001777346872 q^{6} + 27\!\cdots\!68 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 72851326872 q^{4} + 11001777346872 q^{6} + 27\!\cdots\!68 q^{9}+ \cdots - 34\!\cdots\!24 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 156068. 1.68392 0.841958 0.539544i \(-0.181403\pi\)
0.841958 + 0.539544i \(0.181403\pi\)
\(3\) −6.96239e7 −0.933808 −0.466904 0.884308i \(-0.654631\pi\)
−0.466904 + 0.884308i \(0.654631\pi\)
\(4\) 1.57674e10 1.83557
\(5\) 0 0
\(6\) −1.08661e13 −1.57245
\(7\) −1.55193e14 −1.76504 −0.882520 0.470275i \(-0.844155\pi\)
−0.882520 + 0.470275i \(0.844155\pi\)
\(8\) 1.12018e15 1.40703
\(9\) −7.11573e14 −0.128002
\(10\) 0 0
\(11\) 4.13199e16 0.271132 0.135566 0.990768i \(-0.456715\pi\)
0.135566 + 0.990768i \(0.456715\pi\)
\(12\) −1.09779e18 −1.71407
\(13\) −2.38246e18 −0.993025 −0.496513 0.868029i \(-0.665386\pi\)
−0.496513 + 0.868029i \(0.665386\pi\)
\(14\) −2.42207e19 −2.97218
\(15\) 0 0
\(16\) 3.93836e19 0.533747
\(17\) −2.91772e20 −1.45424 −0.727120 0.686511i \(-0.759142\pi\)
−0.727120 + 0.686511i \(0.759142\pi\)
\(18\) −1.11054e20 −0.215545
\(19\) −3.61361e20 −0.287414 −0.143707 0.989620i \(-0.545902\pi\)
−0.143707 + 0.989620i \(0.545902\pi\)
\(20\) 0 0
\(21\) 1.08051e22 1.64821
\(22\) 6.44874e21 0.456563
\(23\) 3.81748e22 1.29798 0.648989 0.760798i \(-0.275192\pi\)
0.648989 + 0.760798i \(0.275192\pi\)
\(24\) −7.79913e22 −1.31390
\(25\) 0 0
\(26\) −3.71827e23 −1.67217
\(27\) 4.36586e23 1.05334
\(28\) −2.44699e24 −3.23985
\(29\) 1.65188e24 1.22577 0.612887 0.790170i \(-0.290008\pi\)
0.612887 + 0.790170i \(0.290008\pi\)
\(30\) 0 0
\(31\) −2.76253e24 −0.682086 −0.341043 0.940048i \(-0.610780\pi\)
−0.341043 + 0.940048i \(0.610780\pi\)
\(32\) −3.47573e24 −0.508244
\(33\) −2.87685e24 −0.253185
\(34\) −4.55364e25 −2.44882
\(35\) 0 0
\(36\) −1.12197e25 −0.234957
\(37\) −8.70723e24 −0.116025 −0.0580125 0.998316i \(-0.518476\pi\)
−0.0580125 + 0.998316i \(0.518476\pi\)
\(38\) −5.63971e25 −0.483980
\(39\) 1.65876e26 0.927295
\(40\) 0 0
\(41\) 2.69048e26 0.659015 0.329508 0.944153i \(-0.393117\pi\)
0.329508 + 0.944153i \(0.393117\pi\)
\(42\) 1.68634e27 2.77544
\(43\) −1.33984e26 −0.149563 −0.0747815 0.997200i \(-0.523826\pi\)
−0.0747815 + 0.997200i \(0.523826\pi\)
\(44\) 6.51509e26 0.497681
\(45\) 0 0
\(46\) 5.95788e27 2.18568
\(47\) −2.88434e27 −0.742047 −0.371024 0.928623i \(-0.620993\pi\)
−0.371024 + 0.928623i \(0.620993\pi\)
\(48\) −2.74204e27 −0.498418
\(49\) 1.63539e28 2.11537
\(50\) 0 0
\(51\) 2.03143e28 1.35798
\(52\) −3.75653e28 −1.82277
\(53\) −2.92563e28 −1.03673 −0.518367 0.855158i \(-0.673460\pi\)
−0.518367 + 0.855158i \(0.673460\pi\)
\(54\) 6.81373e28 1.77373
\(55\) 0 0
\(56\) −1.73844e29 −2.48346
\(57\) 2.51594e28 0.268389
\(58\) 2.57806e29 2.06410
\(59\) 1.51134e29 0.912650 0.456325 0.889813i \(-0.349166\pi\)
0.456325 + 0.889813i \(0.349166\pi\)
\(60\) 0 0
\(61\) 2.88604e29 1.00545 0.502725 0.864446i \(-0.332331\pi\)
0.502725 + 0.864446i \(0.332331\pi\)
\(62\) −4.31144e29 −1.14857
\(63\) 1.10431e29 0.225929
\(64\) −8.80755e29 −1.38959
\(65\) 0 0
\(66\) −4.48986e29 −0.426342
\(67\) −8.72314e29 −0.646307 −0.323153 0.946347i \(-0.604743\pi\)
−0.323153 + 0.946347i \(0.604743\pi\)
\(68\) −4.60049e30 −2.66936
\(69\) −2.65788e30 −1.21206
\(70\) 0 0
\(71\) 3.08526e30 0.878069 0.439034 0.898470i \(-0.355321\pi\)
0.439034 + 0.898470i \(0.355321\pi\)
\(72\) −7.97090e29 −0.180103
\(73\) 2.41300e30 0.434240 0.217120 0.976145i \(-0.430334\pi\)
0.217120 + 0.976145i \(0.430334\pi\)
\(74\) −1.35892e30 −0.195376
\(75\) 0 0
\(76\) −5.69774e30 −0.527568
\(77\) −6.41256e30 −0.478558
\(78\) 2.58880e31 1.56149
\(79\) −3.21352e31 −1.57085 −0.785423 0.618960i \(-0.787554\pi\)
−0.785423 + 0.618960i \(0.787554\pi\)
\(80\) 0 0
\(81\) −2.64411e31 −0.855613
\(82\) 4.19898e31 1.10973
\(83\) 7.79056e31 1.68570 0.842850 0.538148i \(-0.180876\pi\)
0.842850 + 0.538148i \(0.180876\pi\)
\(84\) 1.70369e32 3.02540
\(85\) 0 0
\(86\) −2.09107e31 −0.251851
\(87\) −1.15010e32 −1.14464
\(88\) 4.62857e31 0.381490
\(89\) 1.41115e32 0.965245 0.482623 0.875828i \(-0.339684\pi\)
0.482623 + 0.875828i \(0.339684\pi\)
\(90\) 0 0
\(91\) 3.69741e32 1.75273
\(92\) 6.01918e32 2.38253
\(93\) 1.92338e32 0.636937
\(94\) −4.50154e32 −1.24954
\(95\) 0 0
\(96\) 2.41994e32 0.474602
\(97\) 7.13656e32 1.17966 0.589828 0.807529i \(-0.299196\pi\)
0.589828 + 0.807529i \(0.299196\pi\)
\(98\) 2.55232e33 3.56210
\(99\) −2.94021e31 −0.0347055
\(100\) 0 0
\(101\) −9.02530e32 −0.765877 −0.382939 0.923774i \(-0.625088\pi\)
−0.382939 + 0.923774i \(0.625088\pi\)
\(102\) 3.17042e33 2.28672
\(103\) 6.15871e32 0.378160 0.189080 0.981962i \(-0.439449\pi\)
0.189080 + 0.981962i \(0.439449\pi\)
\(104\) −2.66878e33 −1.39722
\(105\) 0 0
\(106\) −4.56598e33 −1.74577
\(107\) −4.51029e33 −1.47697 −0.738485 0.674270i \(-0.764459\pi\)
−0.738485 + 0.674270i \(0.764459\pi\)
\(108\) 6.88384e33 1.93348
\(109\) −1.07835e33 −0.260149 −0.130074 0.991504i \(-0.541522\pi\)
−0.130074 + 0.991504i \(0.541522\pi\)
\(110\) 0 0
\(111\) 6.06231e32 0.108345
\(112\) −6.11206e33 −0.942085
\(113\) 2.37121e31 0.00315629 0.00157814 0.999999i \(-0.499498\pi\)
0.00157814 + 0.999999i \(0.499498\pi\)
\(114\) 3.92659e33 0.451944
\(115\) 0 0
\(116\) 2.60458e34 2.24999
\(117\) 1.69529e33 0.127110
\(118\) 2.35873e34 1.53682
\(119\) 4.52809e34 2.56679
\(120\) 0 0
\(121\) −2.15178e34 −0.926488
\(122\) 4.50420e34 1.69309
\(123\) −1.87321e34 −0.615394
\(124\) −4.35580e34 −1.25202
\(125\) 0 0
\(126\) 1.72348e34 0.380446
\(127\) −6.78580e34 −1.31474 −0.657370 0.753568i \(-0.728331\pi\)
−0.657370 + 0.753568i \(0.728331\pi\)
\(128\) −1.07602e35 −1.83170
\(129\) 9.32850e33 0.139663
\(130\) 0 0
\(131\) 7.18678e34 0.834753 0.417377 0.908734i \(-0.362950\pi\)
0.417377 + 0.908734i \(0.362950\pi\)
\(132\) −4.53606e34 −0.464739
\(133\) 5.60808e34 0.507296
\(134\) −1.36141e35 −1.08833
\(135\) 0 0
\(136\) −3.26837e35 −2.04616
\(137\) −1.24185e34 −0.0688940 −0.0344470 0.999407i \(-0.510967\pi\)
−0.0344470 + 0.999407i \(0.510967\pi\)
\(138\) −4.14811e35 −2.04101
\(139\) 3.36802e34 0.147106 0.0735531 0.997291i \(-0.476566\pi\)
0.0735531 + 0.997291i \(0.476566\pi\)
\(140\) 0 0
\(141\) 2.00819e35 0.692930
\(142\) 4.81511e35 1.47859
\(143\) −9.84431e34 −0.269241
\(144\) −2.80243e34 −0.0683210
\(145\) 0 0
\(146\) 3.76593e35 0.731223
\(147\) −1.13862e36 −1.97535
\(148\) −1.37291e35 −0.212972
\(149\) 7.60967e35 1.05631 0.528156 0.849147i \(-0.322884\pi\)
0.528156 + 0.849147i \(0.322884\pi\)
\(150\) 0 0
\(151\) 3.06240e35 0.341147 0.170574 0.985345i \(-0.445438\pi\)
0.170574 + 0.985345i \(0.445438\pi\)
\(152\) −4.04790e35 −0.404399
\(153\) 2.07617e35 0.186146
\(154\) −1.00080e36 −0.805851
\(155\) 0 0
\(156\) 2.61544e36 1.70212
\(157\) 2.10956e36 1.23551 0.617755 0.786370i \(-0.288042\pi\)
0.617755 + 0.786370i \(0.288042\pi\)
\(158\) −5.01528e36 −2.64517
\(159\) 2.03694e36 0.968110
\(160\) 0 0
\(161\) −5.92446e36 −2.29098
\(162\) −4.12663e36 −1.44078
\(163\) 2.09987e36 0.662365 0.331182 0.943567i \(-0.392552\pi\)
0.331182 + 0.943567i \(0.392552\pi\)
\(164\) 4.24219e36 1.20967
\(165\) 0 0
\(166\) 1.21586e37 2.83858
\(167\) 4.97146e36 1.05114 0.525572 0.850749i \(-0.323851\pi\)
0.525572 + 0.850749i \(0.323851\pi\)
\(168\) 1.21037e37 2.31908
\(169\) −8.00138e34 −0.0139006
\(170\) 0 0
\(171\) 2.57135e35 0.0367896
\(172\) −2.11259e36 −0.274533
\(173\) 1.51427e37 1.78831 0.894157 0.447753i \(-0.147776\pi\)
0.894157 + 0.447753i \(0.147776\pi\)
\(174\) −1.79494e37 −1.92747
\(175\) 0 0
\(176\) 1.62733e36 0.144716
\(177\) −1.05225e37 −0.852240
\(178\) 2.20236e37 1.62539
\(179\) 9.15862e36 0.616247 0.308124 0.951346i \(-0.400299\pi\)
0.308124 + 0.951346i \(0.400299\pi\)
\(180\) 0 0
\(181\) 1.09431e37 0.612975 0.306488 0.951875i \(-0.400846\pi\)
0.306488 + 0.951875i \(0.400846\pi\)
\(182\) 5.77049e37 2.95145
\(183\) −2.00938e37 −0.938898
\(184\) 4.27626e37 1.82629
\(185\) 0 0
\(186\) 3.00179e37 1.07255
\(187\) −1.20560e37 −0.394290
\(188\) −4.54786e37 −1.36208
\(189\) −6.77551e37 −1.85918
\(190\) 0 0
\(191\) −2.18522e37 −0.504016 −0.252008 0.967725i \(-0.581091\pi\)
−0.252008 + 0.967725i \(0.581091\pi\)
\(192\) 6.13216e37 1.29761
\(193\) 7.41572e37 1.44032 0.720158 0.693810i \(-0.244069\pi\)
0.720158 + 0.693810i \(0.244069\pi\)
\(194\) 1.11379e38 1.98644
\(195\) 0 0
\(196\) 2.57859e38 3.88290
\(197\) −9.94695e37 −1.37720 −0.688601 0.725141i \(-0.741775\pi\)
−0.688601 + 0.725141i \(0.741775\pi\)
\(198\) −4.58875e36 −0.0584411
\(199\) 1.85341e37 0.217217 0.108609 0.994085i \(-0.465360\pi\)
0.108609 + 0.994085i \(0.465360\pi\)
\(200\) 0 0
\(201\) 6.07339e37 0.603526
\(202\) −1.40857e38 −1.28967
\(203\) −2.56360e38 −2.16354
\(204\) 3.20304e38 2.49267
\(205\) 0 0
\(206\) 9.61180e37 0.636789
\(207\) −2.71642e37 −0.166144
\(208\) −9.38299e37 −0.530025
\(209\) −1.49314e37 −0.0779269
\(210\) 0 0
\(211\) −2.61345e38 −1.16561 −0.582805 0.812612i \(-0.698045\pi\)
−0.582805 + 0.812612i \(0.698045\pi\)
\(212\) −4.61296e38 −1.90300
\(213\) −2.14808e38 −0.819948
\(214\) −7.03914e38 −2.48709
\(215\) 0 0
\(216\) 4.89055e38 1.48208
\(217\) 4.28725e38 1.20391
\(218\) −1.68296e38 −0.438068
\(219\) −1.68002e38 −0.405496
\(220\) 0 0
\(221\) 6.95134e38 1.44410
\(222\) 9.46135e37 0.182444
\(223\) −2.67594e38 −0.479122 −0.239561 0.970881i \(-0.577003\pi\)
−0.239561 + 0.970881i \(0.577003\pi\)
\(224\) 5.39410e38 0.897070
\(225\) 0 0
\(226\) 3.70072e36 0.00531492
\(227\) −3.53241e38 −0.471677 −0.235838 0.971792i \(-0.575784\pi\)
−0.235838 + 0.971792i \(0.575784\pi\)
\(228\) 3.96699e38 0.492647
\(229\) 1.17673e39 1.35954 0.679769 0.733427i \(-0.262080\pi\)
0.679769 + 0.733427i \(0.262080\pi\)
\(230\) 0 0
\(231\) 4.46468e38 0.446881
\(232\) 1.85040e39 1.72470
\(233\) 1.12975e38 0.0980870 0.0490435 0.998797i \(-0.484383\pi\)
0.0490435 + 0.998797i \(0.484383\pi\)
\(234\) 2.64582e38 0.214042
\(235\) 0 0
\(236\) 2.38299e39 1.67523
\(237\) 2.23738e39 1.46687
\(238\) 7.06692e39 4.32226
\(239\) 9.43852e38 0.538690 0.269345 0.963044i \(-0.413193\pi\)
0.269345 + 0.963044i \(0.413193\pi\)
\(240\) 0 0
\(241\) 7.67029e38 0.381532 0.190766 0.981636i \(-0.438903\pi\)
0.190766 + 0.981636i \(0.438903\pi\)
\(242\) −3.35825e39 −1.56013
\(243\) −5.86073e38 −0.254359
\(244\) 4.55055e39 1.84558
\(245\) 0 0
\(246\) −2.92350e39 −1.03627
\(247\) 8.60929e38 0.285409
\(248\) −3.09453e39 −0.959714
\(249\) −5.42409e39 −1.57412
\(250\) 0 0
\(251\) −2.55001e39 −0.648524 −0.324262 0.945967i \(-0.605116\pi\)
−0.324262 + 0.945967i \(0.605116\pi\)
\(252\) 1.74122e39 0.414709
\(253\) 1.57738e39 0.351923
\(254\) −1.05905e40 −2.21391
\(255\) 0 0
\(256\) −9.22761e39 −1.69484
\(257\) −7.67124e39 −1.32120 −0.660601 0.750737i \(-0.729698\pi\)
−0.660601 + 0.750737i \(0.729698\pi\)
\(258\) 1.45588e39 0.235181
\(259\) 1.35130e39 0.204789
\(260\) 0 0
\(261\) −1.17543e39 −0.156902
\(262\) 1.12163e40 1.40565
\(263\) −1.76090e39 −0.207236 −0.103618 0.994617i \(-0.533042\pi\)
−0.103618 + 0.994617i \(0.533042\pi\)
\(264\) −3.22259e39 −0.356238
\(265\) 0 0
\(266\) 8.75244e39 0.854244
\(267\) −9.82497e39 −0.901354
\(268\) −1.37541e40 −1.18634
\(269\) 1.37145e38 0.0111241 0.00556206 0.999985i \(-0.498230\pi\)
0.00556206 + 0.999985i \(0.498230\pi\)
\(270\) 0 0
\(271\) 1.20503e40 0.864978 0.432489 0.901639i \(-0.357635\pi\)
0.432489 + 0.901639i \(0.357635\pi\)
\(272\) −1.14910e40 −0.776197
\(273\) −2.57428e40 −1.63671
\(274\) −1.93814e39 −0.116012
\(275\) 0 0
\(276\) −4.19079e40 −2.22483
\(277\) −1.15971e40 −0.580005 −0.290002 0.957026i \(-0.593656\pi\)
−0.290002 + 0.957026i \(0.593656\pi\)
\(278\) 5.25642e39 0.247714
\(279\) 1.96574e39 0.0873086
\(280\) 0 0
\(281\) −3.63872e40 −1.43646 −0.718232 0.695804i \(-0.755048\pi\)
−0.718232 + 0.695804i \(0.755048\pi\)
\(282\) 3.13415e40 1.16684
\(283\) −1.87432e40 −0.658213 −0.329107 0.944293i \(-0.606748\pi\)
−0.329107 + 0.944293i \(0.606748\pi\)
\(284\) 4.86466e40 1.61176
\(285\) 0 0
\(286\) −1.53639e40 −0.453378
\(287\) −4.17543e40 −1.16319
\(288\) 2.47324e39 0.0650564
\(289\) 4.48762e40 1.11481
\(290\) 0 0
\(291\) −4.96875e40 −1.10157
\(292\) 3.80467e40 0.797077
\(293\) −2.34908e40 −0.465139 −0.232569 0.972580i \(-0.574713\pi\)
−0.232569 + 0.972580i \(0.574713\pi\)
\(294\) −1.77703e41 −3.32631
\(295\) 0 0
\(296\) −9.75366e39 −0.163250
\(297\) 1.80397e40 0.285593
\(298\) 1.18763e41 1.77874
\(299\) −9.09499e40 −1.28893
\(300\) 0 0
\(301\) 2.07934e40 0.263985
\(302\) 4.77944e40 0.574463
\(303\) 6.28377e40 0.715182
\(304\) −1.42317e40 −0.153406
\(305\) 0 0
\(306\) 3.24024e40 0.313454
\(307\) −2.12921e40 −0.195180 −0.0975901 0.995227i \(-0.531113\pi\)
−0.0975901 + 0.995227i \(0.531113\pi\)
\(308\) −1.01110e41 −0.878427
\(309\) −4.28793e40 −0.353129
\(310\) 0 0
\(311\) 9.65601e40 0.714910 0.357455 0.933930i \(-0.383645\pi\)
0.357455 + 0.933930i \(0.383645\pi\)
\(312\) 1.85811e41 1.30473
\(313\) −2.25901e40 −0.150465 −0.0752327 0.997166i \(-0.523970\pi\)
−0.0752327 + 0.997166i \(0.523970\pi\)
\(314\) 3.29235e41 2.08050
\(315\) 0 0
\(316\) −5.06689e41 −2.88340
\(317\) −1.90392e40 −0.102842 −0.0514209 0.998677i \(-0.516375\pi\)
−0.0514209 + 0.998677i \(0.516375\pi\)
\(318\) 3.17901e41 1.63021
\(319\) 6.82554e40 0.332346
\(320\) 0 0
\(321\) 3.14024e41 1.37921
\(322\) −9.24621e41 −3.85782
\(323\) 1.05435e41 0.417968
\(324\) −4.16909e41 −1.57054
\(325\) 0 0
\(326\) 3.27724e41 1.11537
\(327\) 7.50788e40 0.242929
\(328\) 3.01382e41 0.927253
\(329\) 4.47629e41 1.30974
\(330\) 0 0
\(331\) 8.34163e39 0.0220846 0.0110423 0.999939i \(-0.496485\pi\)
0.0110423 + 0.999939i \(0.496485\pi\)
\(332\) 1.22837e42 3.09422
\(333\) 6.19583e39 0.0148515
\(334\) 7.75889e41 1.77004
\(335\) 0 0
\(336\) 4.25546e41 0.879727
\(337\) 1.27475e41 0.250918 0.125459 0.992099i \(-0.459960\pi\)
0.125459 + 0.992099i \(0.459960\pi\)
\(338\) −1.24876e40 −0.0234075
\(339\) −1.65093e39 −0.00294737
\(340\) 0 0
\(341\) −1.14147e41 −0.184935
\(342\) 4.01307e40 0.0619506
\(343\) −1.33821e42 −1.96866
\(344\) −1.50086e41 −0.210439
\(345\) 0 0
\(346\) 2.36330e42 3.01137
\(347\) −2.90496e41 −0.352943 −0.176471 0.984306i \(-0.556468\pi\)
−0.176471 + 0.984306i \(0.556468\pi\)
\(348\) −1.81341e42 −2.10106
\(349\) −1.05476e42 −1.16556 −0.582779 0.812631i \(-0.698035\pi\)
−0.582779 + 0.812631i \(0.698035\pi\)
\(350\) 0 0
\(351\) −1.04015e42 −1.04599
\(352\) −1.43617e41 −0.137801
\(353\) −1.42033e42 −1.30049 −0.650246 0.759724i \(-0.725334\pi\)
−0.650246 + 0.759724i \(0.725334\pi\)
\(354\) −1.64224e42 −1.43510
\(355\) 0 0
\(356\) 2.22502e42 1.77177
\(357\) −3.15263e42 −2.39689
\(358\) 1.42937e42 1.03771
\(359\) −3.15371e41 −0.218657 −0.109329 0.994006i \(-0.534870\pi\)
−0.109329 + 0.994006i \(0.534870\pi\)
\(360\) 0 0
\(361\) −1.45019e42 −0.917393
\(362\) 1.70787e42 1.03220
\(363\) 1.49815e42 0.865162
\(364\) 5.82987e42 3.21726
\(365\) 0 0
\(366\) −3.13600e42 −1.58102
\(367\) −3.83973e41 −0.185059 −0.0925297 0.995710i \(-0.529495\pi\)
−0.0925297 + 0.995710i \(0.529495\pi\)
\(368\) 1.50346e42 0.692792
\(369\) −1.91447e41 −0.0843555
\(370\) 0 0
\(371\) 4.54037e42 1.82988
\(372\) 3.03268e42 1.16914
\(373\) 2.58713e42 0.954164 0.477082 0.878859i \(-0.341695\pi\)
0.477082 + 0.878859i \(0.341695\pi\)
\(374\) −1.88156e42 −0.663951
\(375\) 0 0
\(376\) −3.23098e42 −1.04408
\(377\) −3.93553e42 −1.21723
\(378\) −1.05744e43 −3.13071
\(379\) 1.90182e42 0.539042 0.269521 0.962994i \(-0.413135\pi\)
0.269521 + 0.962994i \(0.413135\pi\)
\(380\) 0 0
\(381\) 4.72454e42 1.22771
\(382\) −3.41044e42 −0.848720
\(383\) 6.82259e42 1.62618 0.813092 0.582135i \(-0.197783\pi\)
0.813092 + 0.582135i \(0.197783\pi\)
\(384\) 7.49165e42 1.71046
\(385\) 0 0
\(386\) 1.15736e43 2.42537
\(387\) 9.53396e40 0.0191444
\(388\) 1.12525e43 2.16534
\(389\) −2.03241e42 −0.374837 −0.187419 0.982280i \(-0.560012\pi\)
−0.187419 + 0.982280i \(0.560012\pi\)
\(390\) 0 0
\(391\) −1.11383e43 −1.88757
\(392\) 1.83193e43 2.97638
\(393\) −5.00372e42 −0.779499
\(394\) −1.55241e43 −2.31909
\(395\) 0 0
\(396\) −4.63596e41 −0.0637044
\(397\) −7.63461e42 −1.00634 −0.503168 0.864188i \(-0.667832\pi\)
−0.503168 + 0.864188i \(0.667832\pi\)
\(398\) 2.89258e42 0.365776
\(399\) −3.90456e42 −0.473717
\(400\) 0 0
\(401\) −1.09608e43 −1.22450 −0.612250 0.790664i \(-0.709736\pi\)
−0.612250 + 0.790664i \(0.709736\pi\)
\(402\) 9.47864e42 1.01629
\(403\) 6.58162e42 0.677328
\(404\) −1.42306e43 −1.40582
\(405\) 0 0
\(406\) −4.00097e43 −3.64322
\(407\) −3.59782e41 −0.0314580
\(408\) 2.27557e43 1.91072
\(409\) 1.27736e43 1.03010 0.515050 0.857160i \(-0.327773\pi\)
0.515050 + 0.857160i \(0.327773\pi\)
\(410\) 0 0
\(411\) 8.64628e41 0.0643338
\(412\) 9.71070e42 0.694139
\(413\) −2.34549e43 −1.61086
\(414\) −4.23947e42 −0.279773
\(415\) 0 0
\(416\) 8.28080e42 0.504699
\(417\) −2.34495e42 −0.137369
\(418\) −2.33032e42 −0.131222
\(419\) 1.81304e43 0.981467 0.490734 0.871310i \(-0.336729\pi\)
0.490734 + 0.871310i \(0.336729\pi\)
\(420\) 0 0
\(421\) 1.35628e43 0.678724 0.339362 0.940656i \(-0.389789\pi\)
0.339362 + 0.940656i \(0.389789\pi\)
\(422\) −4.07877e43 −1.96279
\(423\) 2.05242e42 0.0949839
\(424\) −3.27723e43 −1.45871
\(425\) 0 0
\(426\) −3.35247e43 −1.38072
\(427\) −4.47894e43 −1.77466
\(428\) −7.11156e43 −2.71108
\(429\) 6.85399e42 0.251419
\(430\) 0 0
\(431\) 4.12465e43 1.40124 0.700620 0.713535i \(-0.252907\pi\)
0.700620 + 0.713535i \(0.252907\pi\)
\(432\) 1.71943e43 0.562216
\(433\) −5.05620e43 −1.59138 −0.795690 0.605704i \(-0.792892\pi\)
−0.795690 + 0.605704i \(0.792892\pi\)
\(434\) 6.69105e43 2.02728
\(435\) 0 0
\(436\) −1.70028e43 −0.477521
\(437\) −1.37949e43 −0.373056
\(438\) −2.62198e43 −0.682822
\(439\) −2.66416e43 −0.668182 −0.334091 0.942541i \(-0.608429\pi\)
−0.334091 + 0.942541i \(0.608429\pi\)
\(440\) 0 0
\(441\) −1.16370e43 −0.270772
\(442\) 1.08489e44 2.43174
\(443\) −1.04497e43 −0.225653 −0.112826 0.993615i \(-0.535990\pi\)
−0.112826 + 0.993615i \(0.535990\pi\)
\(444\) 9.55870e42 0.198875
\(445\) 0 0
\(446\) −4.17630e43 −0.806800
\(447\) −5.29815e43 −0.986393
\(448\) 1.36687e44 2.45268
\(449\) −5.66906e43 −0.980497 −0.490248 0.871583i \(-0.663094\pi\)
−0.490248 + 0.871583i \(0.663094\pi\)
\(450\) 0 0
\(451\) 1.11170e43 0.178680
\(452\) 3.73879e41 0.00579358
\(453\) −2.13216e43 −0.318566
\(454\) −5.51298e43 −0.794264
\(455\) 0 0
\(456\) 2.81830e43 0.377631
\(457\) −1.11812e44 −1.44501 −0.722505 0.691366i \(-0.757009\pi\)
−0.722505 + 0.691366i \(0.757009\pi\)
\(458\) 1.83650e44 2.28935
\(459\) −1.27383e44 −1.53181
\(460\) 0 0
\(461\) −2.91705e43 −0.326496 −0.163248 0.986585i \(-0.552197\pi\)
−0.163248 + 0.986585i \(0.552197\pi\)
\(462\) 6.96795e43 0.752510
\(463\) −1.20018e44 −1.25072 −0.625362 0.780335i \(-0.715048\pi\)
−0.625362 + 0.780335i \(0.715048\pi\)
\(464\) 6.50569e43 0.654254
\(465\) 0 0
\(466\) 1.76319e43 0.165170
\(467\) 2.20756e43 0.199611 0.0998054 0.995007i \(-0.468178\pi\)
0.0998054 + 0.995007i \(0.468178\pi\)
\(468\) 2.67304e43 0.233319
\(469\) 1.35377e44 1.14076
\(470\) 0 0
\(471\) −1.46875e44 −1.15373
\(472\) 1.69297e44 1.28412
\(473\) −5.53622e42 −0.0405512
\(474\) 3.49184e44 2.47008
\(475\) 0 0
\(476\) 7.13964e44 4.71152
\(477\) 2.08180e43 0.132704
\(478\) 1.47306e44 0.907108
\(479\) 7.80915e43 0.464588 0.232294 0.972646i \(-0.425377\pi\)
0.232294 + 0.972646i \(0.425377\pi\)
\(480\) 0 0
\(481\) 2.07446e43 0.115216
\(482\) 1.19709e44 0.642467
\(483\) 4.12484e44 2.13934
\(484\) −3.39281e44 −1.70063
\(485\) 0 0
\(486\) −9.14674e43 −0.428320
\(487\) 4.19385e44 1.89839 0.949194 0.314693i \(-0.101901\pi\)
0.949194 + 0.314693i \(0.101901\pi\)
\(488\) 3.23289e44 1.41470
\(489\) −1.46201e44 −0.618522
\(490\) 0 0
\(491\) 4.24412e44 1.67859 0.839293 0.543680i \(-0.182969\pi\)
0.839293 + 0.543680i \(0.182969\pi\)
\(492\) −2.95358e44 −1.12960
\(493\) −4.81971e44 −1.78257
\(494\) 1.34364e44 0.480604
\(495\) 0 0
\(496\) −1.08798e44 −0.364061
\(497\) −4.78810e44 −1.54983
\(498\) −8.46530e44 −2.65069
\(499\) 6.09953e44 1.84772 0.923862 0.382726i \(-0.125015\pi\)
0.923862 + 0.382726i \(0.125015\pi\)
\(500\) 0 0
\(501\) −3.46133e44 −0.981567
\(502\) −3.97976e44 −1.09206
\(503\) 3.00762e44 0.798639 0.399320 0.916812i \(-0.369246\pi\)
0.399320 + 0.916812i \(0.369246\pi\)
\(504\) 1.23703e44 0.317889
\(505\) 0 0
\(506\) 2.46179e44 0.592608
\(507\) 5.57087e42 0.0129805
\(508\) −1.06995e45 −2.41330
\(509\) 1.45444e44 0.317578 0.158789 0.987313i \(-0.449241\pi\)
0.158789 + 0.987313i \(0.449241\pi\)
\(510\) 0 0
\(511\) −3.74480e44 −0.766450
\(512\) −5.15847e44 −1.02227
\(513\) −1.57765e44 −0.302744
\(514\) −1.19724e45 −2.22479
\(515\) 0 0
\(516\) 1.47086e44 0.256361
\(517\) −1.19181e44 −0.201192
\(518\) 2.10895e44 0.344847
\(519\) −1.05430e45 −1.66994
\(520\) 0 0
\(521\) 3.35096e44 0.498135 0.249067 0.968486i \(-0.419876\pi\)
0.249067 + 0.968486i \(0.419876\pi\)
\(522\) −1.83448e44 −0.264210
\(523\) 7.95243e44 1.10974 0.554871 0.831937i \(-0.312768\pi\)
0.554871 + 0.831937i \(0.312768\pi\)
\(524\) 1.13317e45 1.53225
\(525\) 0 0
\(526\) −2.74821e44 −0.348967
\(527\) 8.06028e44 0.991916
\(528\) −1.13301e44 −0.135137
\(529\) 5.92309e44 0.684747
\(530\) 0 0
\(531\) −1.07543e44 −0.116821
\(532\) 8.84249e44 0.931178
\(533\) −6.40995e44 −0.654419
\(534\) −1.53337e45 −1.51780
\(535\) 0 0
\(536\) −9.77149e44 −0.909372
\(537\) −6.37659e44 −0.575457
\(538\) 2.14039e43 0.0187321
\(539\) 6.75741e44 0.573542
\(540\) 0 0
\(541\) 1.21161e45 0.967402 0.483701 0.875233i \(-0.339292\pi\)
0.483701 + 0.875233i \(0.339292\pi\)
\(542\) 1.88067e45 1.45655
\(543\) −7.61900e44 −0.572401
\(544\) 1.01412e45 0.739108
\(545\) 0 0
\(546\) −4.01764e45 −2.75609
\(547\) 1.99236e45 1.32610 0.663052 0.748573i \(-0.269261\pi\)
0.663052 + 0.748573i \(0.269261\pi\)
\(548\) −1.95808e44 −0.126460
\(549\) −2.05363e44 −0.128700
\(550\) 0 0
\(551\) −5.96924e44 −0.352304
\(552\) −2.97730e45 −1.70541
\(553\) 4.98715e45 2.77260
\(554\) −1.80993e45 −0.976679
\(555\) 0 0
\(556\) 5.31051e44 0.270024
\(557\) 1.74444e45 0.861082 0.430541 0.902571i \(-0.358323\pi\)
0.430541 + 0.902571i \(0.358323\pi\)
\(558\) 3.06790e44 0.147020
\(559\) 3.19212e44 0.148520
\(560\) 0 0
\(561\) 8.39384e44 0.368191
\(562\) −5.67890e45 −2.41888
\(563\) −1.30371e45 −0.539251 −0.269626 0.962965i \(-0.586900\pi\)
−0.269626 + 0.962965i \(0.586900\pi\)
\(564\) 3.16640e45 1.27192
\(565\) 0 0
\(566\) −2.92522e45 −1.10838
\(567\) 4.10348e45 1.51019
\(568\) 3.45604e45 1.23547
\(569\) 8.99757e43 0.0312445 0.0156222 0.999878i \(-0.495027\pi\)
0.0156222 + 0.999878i \(0.495027\pi\)
\(570\) 0 0
\(571\) 2.85011e45 0.934039 0.467020 0.884247i \(-0.345328\pi\)
0.467020 + 0.884247i \(0.345328\pi\)
\(572\) −1.55219e45 −0.494210
\(573\) 1.52143e45 0.470654
\(574\) −6.51653e45 −1.95871
\(575\) 0 0
\(576\) 6.26722e44 0.177870
\(577\) 6.49472e45 1.79126 0.895632 0.444797i \(-0.146724\pi\)
0.895632 + 0.444797i \(0.146724\pi\)
\(578\) 7.00376e45 1.87725
\(579\) −5.16312e45 −1.34498
\(580\) 0 0
\(581\) −1.20904e46 −2.97533
\(582\) −7.75465e45 −1.85495
\(583\) −1.20887e45 −0.281091
\(584\) 2.70299e45 0.610988
\(585\) 0 0
\(586\) −3.66618e45 −0.783254
\(587\) −5.04614e44 −0.104817 −0.0524084 0.998626i \(-0.516690\pi\)
−0.0524084 + 0.998626i \(0.516690\pi\)
\(588\) −1.79531e46 −3.62588
\(589\) 9.98271e44 0.196041
\(590\) 0 0
\(591\) 6.92546e45 1.28604
\(592\) −3.42922e44 −0.0619280
\(593\) 1.38106e45 0.242555 0.121277 0.992619i \(-0.461301\pi\)
0.121277 + 0.992619i \(0.461301\pi\)
\(594\) 2.81543e45 0.480915
\(595\) 0 0
\(596\) 1.19985e46 1.93894
\(597\) −1.29041e45 −0.202839
\(598\) −1.41944e46 −2.17044
\(599\) −7.75620e45 −1.15374 −0.576868 0.816837i \(-0.695725\pi\)
−0.576868 + 0.816837i \(0.695725\pi\)
\(600\) 0 0
\(601\) −1.97032e45 −0.277400 −0.138700 0.990334i \(-0.544292\pi\)
−0.138700 + 0.990334i \(0.544292\pi\)
\(602\) 3.24520e45 0.444528
\(603\) 6.20715e44 0.0827288
\(604\) 4.82861e45 0.626200
\(605\) 0 0
\(606\) 9.80698e45 1.20431
\(607\) −3.36784e45 −0.402474 −0.201237 0.979543i \(-0.564496\pi\)
−0.201237 + 0.979543i \(0.564496\pi\)
\(608\) 1.25600e45 0.146076
\(609\) 1.78488e46 2.02033
\(610\) 0 0
\(611\) 6.87182e45 0.736872
\(612\) 3.27358e45 0.341684
\(613\) 9.87239e45 1.00305 0.501527 0.865142i \(-0.332772\pi\)
0.501527 + 0.865142i \(0.332772\pi\)
\(614\) −3.32303e45 −0.328667
\(615\) 0 0
\(616\) −7.18322e45 −0.673345
\(617\) 3.29814e45 0.300998 0.150499 0.988610i \(-0.451912\pi\)
0.150499 + 0.988610i \(0.451912\pi\)
\(618\) −6.69211e45 −0.594639
\(619\) 1.65886e46 1.43521 0.717604 0.696451i \(-0.245239\pi\)
0.717604 + 0.696451i \(0.245239\pi\)
\(620\) 0 0
\(621\) 1.66666e46 1.36721
\(622\) 1.50700e46 1.20385
\(623\) −2.19000e46 −1.70370
\(624\) 6.53280e45 0.494941
\(625\) 0 0
\(626\) −3.52560e45 −0.253371
\(627\) 1.03958e45 0.0727688
\(628\) 3.32623e46 2.26787
\(629\) 2.54052e45 0.168728
\(630\) 0 0
\(631\) 5.39111e45 0.339777 0.169888 0.985463i \(-0.445659\pi\)
0.169888 + 0.985463i \(0.445659\pi\)
\(632\) −3.59972e46 −2.21023
\(633\) 1.81958e46 1.08846
\(634\) −2.97141e45 −0.173177
\(635\) 0 0
\(636\) 3.21172e46 1.77703
\(637\) −3.89625e46 −2.10061
\(638\) 1.06525e46 0.559643
\(639\) −2.19539e45 −0.112395
\(640\) 0 0
\(641\) −3.59612e46 −1.74854 −0.874271 0.485438i \(-0.838660\pi\)
−0.874271 + 0.485438i \(0.838660\pi\)
\(642\) 4.90092e46 2.32247
\(643\) −3.23556e46 −1.49441 −0.747203 0.664596i \(-0.768604\pi\)
−0.747203 + 0.664596i \(0.768604\pi\)
\(644\) −9.34135e46 −4.20526
\(645\) 0 0
\(646\) 1.64551e46 0.703823
\(647\) −5.30852e45 −0.221336 −0.110668 0.993857i \(-0.535299\pi\)
−0.110668 + 0.993857i \(0.535299\pi\)
\(648\) −2.96188e46 −1.20387
\(649\) 6.24485e45 0.247448
\(650\) 0 0
\(651\) −2.98495e46 −1.12422
\(652\) 3.31096e46 1.21582
\(653\) 2.38611e46 0.854326 0.427163 0.904175i \(-0.359513\pi\)
0.427163 + 0.904175i \(0.359513\pi\)
\(654\) 1.17174e46 0.409072
\(655\) 0 0
\(656\) 1.05961e46 0.351748
\(657\) −1.71702e45 −0.0555837
\(658\) 6.98608e46 2.20550
\(659\) 2.81962e46 0.868122 0.434061 0.900883i \(-0.357080\pi\)
0.434061 + 0.900883i \(0.357080\pi\)
\(660\) 0 0
\(661\) 3.38643e46 0.991786 0.495893 0.868384i \(-0.334841\pi\)
0.495893 + 0.868384i \(0.334841\pi\)
\(662\) 1.30187e45 0.0371886
\(663\) −4.83980e46 −1.34851
\(664\) 8.72683e46 2.37183
\(665\) 0 0
\(666\) 9.66973e44 0.0250086
\(667\) 6.30600e46 1.59103
\(668\) 7.83872e46 1.92945
\(669\) 1.86309e46 0.447408
\(670\) 0 0
\(671\) 1.19251e46 0.272609
\(672\) −3.75558e46 −0.837692
\(673\) 3.57808e46 0.778756 0.389378 0.921078i \(-0.372690\pi\)
0.389378 + 0.921078i \(0.372690\pi\)
\(674\) 1.98949e46 0.422525
\(675\) 0 0
\(676\) −1.26161e45 −0.0255156
\(677\) −6.06906e46 −1.19787 −0.598933 0.800799i \(-0.704409\pi\)
−0.598933 + 0.800799i \(0.704409\pi\)
\(678\) −2.57658e44 −0.00496311
\(679\) −1.10754e47 −2.08214
\(680\) 0 0
\(681\) 2.45940e46 0.440456
\(682\) −1.78148e46 −0.311415
\(683\) −6.22121e46 −1.06153 −0.530766 0.847518i \(-0.678096\pi\)
−0.530766 + 0.847518i \(0.678096\pi\)
\(684\) 4.05436e45 0.0675299
\(685\) 0 0
\(686\) −2.08853e47 −3.31506
\(687\) −8.19285e46 −1.26955
\(688\) −5.27678e45 −0.0798289
\(689\) 6.97019e46 1.02950
\(690\) 0 0
\(691\) 1.23978e47 1.74565 0.872825 0.488033i \(-0.162285\pi\)
0.872825 + 0.488033i \(0.162285\pi\)
\(692\) 2.38762e47 3.28258
\(693\) 4.56301e45 0.0612566
\(694\) −4.53373e46 −0.594326
\(695\) 0 0
\(696\) −1.28832e47 −1.61054
\(697\) −7.85005e46 −0.958366
\(698\) −1.64615e47 −1.96270
\(699\) −7.86579e45 −0.0915944
\(700\) 0 0
\(701\) −7.57725e46 −0.841713 −0.420857 0.907127i \(-0.638270\pi\)
−0.420857 + 0.907127i \(0.638270\pi\)
\(702\) −1.62334e47 −1.76136
\(703\) 3.14645e45 0.0333471
\(704\) −3.63927e46 −0.376761
\(705\) 0 0
\(706\) −2.21669e47 −2.18992
\(707\) 1.40066e47 1.35180
\(708\) −1.65913e47 −1.56435
\(709\) −9.15900e46 −0.843694 −0.421847 0.906667i \(-0.638618\pi\)
−0.421847 + 0.906667i \(0.638618\pi\)
\(710\) 0 0
\(711\) 2.28665e46 0.201072
\(712\) 1.58074e47 1.35813
\(713\) −1.05459e47 −0.885332
\(714\) −4.92027e47 −4.03616
\(715\) 0 0
\(716\) 1.44408e47 1.13117
\(717\) −6.57147e46 −0.503033
\(718\) −4.92195e46 −0.368200
\(719\) 1.43667e47 1.05034 0.525169 0.850998i \(-0.324002\pi\)
0.525169 + 0.850998i \(0.324002\pi\)
\(720\) 0 0
\(721\) −9.55788e46 −0.667467
\(722\) −2.26329e47 −1.54481
\(723\) −5.34036e46 −0.356277
\(724\) 1.72544e47 1.12516
\(725\) 0 0
\(726\) 2.33815e47 1.45686
\(727\) −9.62263e46 −0.586106 −0.293053 0.956096i \(-0.594671\pi\)
−0.293053 + 0.956096i \(0.594671\pi\)
\(728\) 4.14177e47 2.46614
\(729\) 1.87793e47 1.09314
\(730\) 0 0
\(731\) 3.90928e46 0.217500
\(732\) −3.16827e47 −1.72341
\(733\) 1.83322e47 0.974988 0.487494 0.873126i \(-0.337911\pi\)
0.487494 + 0.873126i \(0.337911\pi\)
\(734\) −5.99261e46 −0.311624
\(735\) 0 0
\(736\) −1.32685e47 −0.659689
\(737\) −3.60439e46 −0.175234
\(738\) −2.98788e46 −0.142048
\(739\) 2.28000e47 1.05999 0.529996 0.848000i \(-0.322194\pi\)
0.529996 + 0.848000i \(0.322194\pi\)
\(740\) 0 0
\(741\) −5.99412e46 −0.266517
\(742\) 7.08608e47 3.08135
\(743\) 4.61345e46 0.196205 0.0981026 0.995176i \(-0.468723\pi\)
0.0981026 + 0.995176i \(0.468723\pi\)
\(744\) 2.15453e47 0.896189
\(745\) 0 0
\(746\) 4.03770e47 1.60673
\(747\) −5.54356e46 −0.215774
\(748\) −1.90092e47 −0.723747
\(749\) 6.99965e47 2.60691
\(750\) 0 0
\(751\) −3.50719e46 −0.124997 −0.0624986 0.998045i \(-0.519907\pi\)
−0.0624986 + 0.998045i \(0.519907\pi\)
\(752\) −1.13596e47 −0.396066
\(753\) 1.77542e47 0.605597
\(754\) −6.14212e47 −2.04970
\(755\) 0 0
\(756\) −1.06832e48 −3.41266
\(757\) −1.87145e47 −0.584921 −0.292460 0.956278i \(-0.594474\pi\)
−0.292460 + 0.956278i \(0.594474\pi\)
\(758\) 2.96814e47 0.907701
\(759\) −1.09823e47 −0.328628
\(760\) 0 0
\(761\) 2.20228e47 0.630994 0.315497 0.948927i \(-0.397829\pi\)
0.315497 + 0.948927i \(0.397829\pi\)
\(762\) 7.37352e47 2.06737
\(763\) 1.67352e47 0.459173
\(764\) −3.44553e47 −0.925157
\(765\) 0 0
\(766\) 1.06479e48 2.73836
\(767\) −3.60071e47 −0.906284
\(768\) 6.42462e47 1.58266
\(769\) −2.66239e47 −0.641928 −0.320964 0.947091i \(-0.604007\pi\)
−0.320964 + 0.947091i \(0.604007\pi\)
\(770\) 0 0
\(771\) 5.34102e47 1.23375
\(772\) 1.16927e48 2.64380
\(773\) 5.04292e45 0.0111614 0.00558072 0.999984i \(-0.498224\pi\)
0.00558072 + 0.999984i \(0.498224\pi\)
\(774\) 1.48795e46 0.0322376
\(775\) 0 0
\(776\) 7.99423e47 1.65981
\(777\) −9.40828e46 −0.191233
\(778\) −3.17195e47 −0.631194
\(779\) −9.72234e46 −0.189410
\(780\) 0 0
\(781\) 1.27483e47 0.238072
\(782\) −1.73834e48 −3.17851
\(783\) 7.21186e47 1.29115
\(784\) 6.44075e47 1.12907
\(785\) 0 0
\(786\) −7.80922e47 −1.31261
\(787\) 1.14740e48 1.88857 0.944283 0.329133i \(-0.106757\pi\)
0.944283 + 0.329133i \(0.106757\pi\)
\(788\) −1.56838e48 −2.52795
\(789\) 1.22601e47 0.193518
\(790\) 0 0
\(791\) −3.67996e45 −0.00557097
\(792\) −3.29357e46 −0.0488316
\(793\) −6.87589e47 −0.998438
\(794\) −1.19152e48 −1.69459
\(795\) 0 0
\(796\) 2.92234e47 0.398718
\(797\) 1.04385e48 1.39501 0.697503 0.716582i \(-0.254294\pi\)
0.697503 + 0.716582i \(0.254294\pi\)
\(798\) −6.09379e47 −0.797700
\(799\) 8.41568e47 1.07911
\(800\) 0 0
\(801\) −1.00414e47 −0.123554
\(802\) −1.71063e48 −2.06196
\(803\) 9.97048e46 0.117736
\(804\) 9.57617e47 1.10781
\(805\) 0 0
\(806\) 1.02718e48 1.14056
\(807\) −9.54854e45 −0.0103878
\(808\) −1.01100e48 −1.07761
\(809\) 1.19411e48 1.24707 0.623537 0.781794i \(-0.285695\pi\)
0.623537 + 0.781794i \(0.285695\pi\)
\(810\) 0 0
\(811\) −1.04748e48 −1.05027 −0.525134 0.851019i \(-0.675985\pi\)
−0.525134 + 0.851019i \(0.675985\pi\)
\(812\) −4.04213e48 −3.97133
\(813\) −8.38990e47 −0.807723
\(814\) −5.61506e46 −0.0529727
\(815\) 0 0
\(816\) 8.00050e47 0.724819
\(817\) 4.84167e46 0.0429864
\(818\) 1.99356e48 1.73460
\(819\) −2.63098e47 −0.224354
\(820\) 0 0
\(821\) 8.56741e47 0.701758 0.350879 0.936421i \(-0.385883\pi\)
0.350879 + 0.936421i \(0.385883\pi\)
\(822\) 1.34941e47 0.108333
\(823\) 4.60629e47 0.362455 0.181228 0.983441i \(-0.441993\pi\)
0.181228 + 0.983441i \(0.441993\pi\)
\(824\) 6.89886e47 0.532082
\(825\) 0 0
\(826\) −3.66058e48 −2.71256
\(827\) 2.09756e48 1.52361 0.761803 0.647808i \(-0.224314\pi\)
0.761803 + 0.647808i \(0.224314\pi\)
\(828\) −4.28309e47 −0.304970
\(829\) −2.34867e48 −1.63935 −0.819675 0.572828i \(-0.805846\pi\)
−0.819675 + 0.572828i \(0.805846\pi\)
\(830\) 0 0
\(831\) 8.07432e47 0.541613
\(832\) 2.09836e48 1.37989
\(833\) −4.77160e48 −3.07625
\(834\) −3.65973e47 −0.231318
\(835\) 0 0
\(836\) −2.35430e47 −0.143040
\(837\) −1.20608e48 −0.718467
\(838\) 2.82958e48 1.65271
\(839\) 3.05667e48 1.75056 0.875279 0.483618i \(-0.160678\pi\)
0.875279 + 0.483618i \(0.160678\pi\)
\(840\) 0 0
\(841\) 9.12620e47 0.502523
\(842\) 2.11672e48 1.14291
\(843\) 2.53342e48 1.34138
\(844\) −4.12074e48 −2.13956
\(845\) 0 0
\(846\) 3.20318e47 0.159945
\(847\) 3.33942e48 1.63529
\(848\) −1.15222e48 −0.553354
\(849\) 1.30497e48 0.614645
\(850\) 0 0
\(851\) −3.32397e47 −0.150598
\(852\) −3.38696e48 −1.50507
\(853\) 3.72611e48 1.62404 0.812021 0.583629i \(-0.198368\pi\)
0.812021 + 0.583629i \(0.198368\pi\)
\(854\) −6.99021e48 −2.98838
\(855\) 0 0
\(856\) −5.05233e48 −2.07814
\(857\) −6.05125e47 −0.244153 −0.122076 0.992521i \(-0.538955\pi\)
−0.122076 + 0.992521i \(0.538955\pi\)
\(858\) 1.06969e48 0.423368
\(859\) 2.95430e48 1.14701 0.573505 0.819202i \(-0.305583\pi\)
0.573505 + 0.819202i \(0.305583\pi\)
\(860\) 0 0
\(861\) 2.90710e48 1.08619
\(862\) 6.43728e48 2.35957
\(863\) 5.76096e47 0.207165 0.103583 0.994621i \(-0.466969\pi\)
0.103583 + 0.994621i \(0.466969\pi\)
\(864\) −1.51746e48 −0.535352
\(865\) 0 0
\(866\) −7.89113e48 −2.67975
\(867\) −3.12446e48 −1.04102
\(868\) 6.75989e48 2.20986
\(869\) −1.32782e48 −0.425906
\(870\) 0 0
\(871\) 2.07825e48 0.641799
\(872\) −1.20794e48 −0.366037
\(873\) −5.07818e47 −0.150999
\(874\) −2.15295e48 −0.628195
\(875\) 0 0
\(876\) −2.64896e48 −0.744317
\(877\) 1.47230e48 0.405980 0.202990 0.979181i \(-0.434934\pi\)
0.202990 + 0.979181i \(0.434934\pi\)
\(878\) −4.15791e48 −1.12516
\(879\) 1.63552e48 0.434350
\(880\) 0 0
\(881\) −6.01645e48 −1.53900 −0.769499 0.638648i \(-0.779494\pi\)
−0.769499 + 0.638648i \(0.779494\pi\)
\(882\) −1.81617e48 −0.455957
\(883\) 1.72340e48 0.424654 0.212327 0.977199i \(-0.431896\pi\)
0.212327 + 0.977199i \(0.431896\pi\)
\(884\) 1.09605e49 2.65074
\(885\) 0 0
\(886\) −1.63086e48 −0.379980
\(887\) −4.24957e48 −0.971863 −0.485931 0.873997i \(-0.661520\pi\)
−0.485931 + 0.873997i \(0.661520\pi\)
\(888\) 6.79088e47 0.152445
\(889\) 1.05311e49 2.32057
\(890\) 0 0
\(891\) −1.09255e48 −0.231984
\(892\) −4.21927e48 −0.879462
\(893\) 1.04229e48 0.213274
\(894\) −8.26874e48 −1.66100
\(895\) 0 0
\(896\) 1.66990e49 3.23303
\(897\) 6.33229e48 1.20361
\(898\) −8.84761e48 −1.65107
\(899\) −4.56336e48 −0.836083
\(900\) 0 0
\(901\) 8.53615e48 1.50766
\(902\) 1.73502e48 0.300882
\(903\) −1.44772e48 −0.246511
\(904\) 2.65619e46 0.00444099
\(905\) 0 0
\(906\) −3.32763e48 −0.536438
\(907\) −1.37448e48 −0.217579 −0.108789 0.994065i \(-0.534697\pi\)
−0.108789 + 0.994065i \(0.534697\pi\)
\(908\) −5.56970e48 −0.865796
\(909\) 6.42216e47 0.0980341
\(910\) 0 0
\(911\) −6.36073e48 −0.936383 −0.468192 0.883627i \(-0.655094\pi\)
−0.468192 + 0.883627i \(0.655094\pi\)
\(912\) 9.90867e47 0.143252
\(913\) 3.21905e48 0.457047
\(914\) −1.74503e49 −2.43327
\(915\) 0 0
\(916\) 1.85540e49 2.49553
\(917\) −1.11534e49 −1.47337
\(918\) −1.98805e49 −2.57943
\(919\) 1.02620e49 1.30775 0.653876 0.756601i \(-0.273142\pi\)
0.653876 + 0.756601i \(0.273142\pi\)
\(920\) 0 0
\(921\) 1.48244e48 0.182261
\(922\) −4.55259e48 −0.549792
\(923\) −7.35050e48 −0.871945
\(924\) 7.03965e48 0.820282
\(925\) 0 0
\(926\) −1.87311e49 −2.10611
\(927\) −4.38237e47 −0.0484054
\(928\) −5.74148e48 −0.622992
\(929\) −8.18502e48 −0.872490 −0.436245 0.899828i \(-0.643692\pi\)
−0.436245 + 0.899828i \(0.643692\pi\)
\(930\) 0 0
\(931\) −5.90966e48 −0.607985
\(932\) 1.78133e48 0.180046
\(933\) −6.72289e48 −0.667589
\(934\) 3.44530e48 0.336128
\(935\) 0 0
\(936\) 1.89904e48 0.178847
\(937\) 2.10406e49 1.94695 0.973474 0.228800i \(-0.0734800\pi\)
0.973474 + 0.228800i \(0.0734800\pi\)
\(938\) 2.11281e49 1.92094
\(939\) 1.57281e48 0.140506
\(940\) 0 0
\(941\) 1.12511e49 0.970434 0.485217 0.874394i \(-0.338741\pi\)
0.485217 + 0.874394i \(0.338741\pi\)
\(942\) −2.29226e49 −1.94278
\(943\) 1.02708e49 0.855387
\(944\) 5.95220e48 0.487124
\(945\) 0 0
\(946\) −8.64029e47 −0.0682849
\(947\) −1.47061e49 −1.14215 −0.571075 0.820898i \(-0.693474\pi\)
−0.571075 + 0.820898i \(0.693474\pi\)
\(948\) 3.52777e49 2.69254
\(949\) −5.74887e48 −0.431211
\(950\) 0 0
\(951\) 1.32558e48 0.0960346
\(952\) 5.07228e49 3.61155
\(953\) 6.20681e47 0.0434346 0.0217173 0.999764i \(-0.493087\pi\)
0.0217173 + 0.999764i \(0.493087\pi\)
\(954\) 3.24903e48 0.223463
\(955\) 0 0
\(956\) 1.48821e49 0.988803
\(957\) −4.75221e48 −0.310348
\(958\) 1.21876e49 0.782327
\(959\) 1.92727e48 0.121601
\(960\) 0 0
\(961\) −8.77191e48 −0.534759
\(962\) 3.23758e48 0.194013
\(963\) 3.20940e48 0.189056
\(964\) 1.20941e49 0.700328
\(965\) 0 0
\(966\) 6.43757e49 3.60246
\(967\) 1.22644e49 0.674699 0.337350 0.941379i \(-0.390470\pi\)
0.337350 + 0.941379i \(0.390470\pi\)
\(968\) −2.41038e49 −1.30360
\(969\) −7.34080e48 −0.390302
\(970\) 0 0
\(971\) 1.12879e49 0.580090 0.290045 0.957013i \(-0.406330\pi\)
0.290045 + 0.957013i \(0.406330\pi\)
\(972\) −9.24086e48 −0.466895
\(973\) −5.22694e48 −0.259648
\(974\) 6.54527e49 3.19672
\(975\) 0 0
\(976\) 1.13663e49 0.536657
\(977\) −4.11771e49 −1.91159 −0.955796 0.294029i \(-0.905004\pi\)
−0.955796 + 0.294029i \(0.905004\pi\)
\(978\) −2.28174e49 −1.04154
\(979\) 5.83085e48 0.261708
\(980\) 0 0
\(981\) 7.67323e47 0.0332996
\(982\) 6.62374e49 2.82660
\(983\) 1.02350e49 0.429494 0.214747 0.976670i \(-0.431107\pi\)
0.214747 + 0.976670i \(0.431107\pi\)
\(984\) −2.09834e49 −0.865877
\(985\) 0 0
\(986\) −7.52204e49 −3.00170
\(987\) −3.11657e49 −1.22305
\(988\) 1.35746e49 0.523888
\(989\) −5.11482e48 −0.194129
\(990\) 0 0
\(991\) 1.68749e49 0.619478 0.309739 0.950822i \(-0.399758\pi\)
0.309739 + 0.950822i \(0.399758\pi\)
\(992\) 9.60182e48 0.346666
\(993\) −5.80777e47 −0.0206228
\(994\) −7.47272e49 −2.60978
\(995\) 0 0
\(996\) −8.55240e49 −2.88941
\(997\) −1.90551e49 −0.633200 −0.316600 0.948559i \(-0.602541\pi\)
−0.316600 + 0.948559i \(0.602541\pi\)
\(998\) 9.51944e49 3.11141
\(999\) −3.80145e48 −0.122213
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.f.1.14 16
5.2 odd 4 5.34.b.a.4.14 yes 16
5.3 odd 4 5.34.b.a.4.3 16
5.4 even 2 inner 25.34.a.f.1.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.34.b.a.4.3 16 5.3 odd 4
5.34.b.a.4.14 yes 16 5.2 odd 4
25.34.a.f.1.3 16 5.4 even 2 inner
25.34.a.f.1.14 16 1.1 even 1 trivial