Properties

Label 75.14.a.a.1.1
Level $75$
Weight $14$
Character 75.1
Self dual yes
Analytic conductor $80.423$
Analytic rank $0$
Dimension $1$
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] = [75,14,Mod(1,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 75.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: \(80.4231967139\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 75.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+12.0000 q^{2} +729.000 q^{3} -8048.00 q^{4} +8748.00 q^{6} -235088. q^{7} -194880. q^{8} +531441. q^{9} -1.11829e7 q^{11} -5.86699e6 q^{12} -8.04961e6 q^{13} -2.82106e6 q^{14} +6.35907e7 q^{16} +1.17495e8 q^{17} +6.37729e6 q^{18} -2.14061e8 q^{19} -1.71379e8 q^{21} -1.34195e8 q^{22} -8.30556e8 q^{23} -1.42068e8 q^{24} -9.65954e7 q^{26} +3.87420e8 q^{27} +1.89199e9 q^{28} -1.25240e9 q^{29} +6.15935e9 q^{31} +2.35954e9 q^{32} -8.15234e9 q^{33} +1.40994e9 q^{34} -4.27704e9 q^{36} +5.49819e9 q^{37} -2.56874e9 q^{38} -5.86817e9 q^{39} -4.67869e9 q^{41} -2.05655e9 q^{42} -7.11501e9 q^{43} +9.00000e10 q^{44} -9.96667e9 q^{46} +2.95288e10 q^{47} +4.63576e10 q^{48} -4.16226e10 q^{49} +8.56536e10 q^{51} +6.47833e10 q^{52} +2.04125e11 q^{53} +4.64905e9 q^{54} +4.58139e10 q^{56} -1.56051e11 q^{57} -1.50288e10 q^{58} -2.99098e10 q^{59} -1.34392e11 q^{61} +7.39122e10 q^{62} -1.24935e11 q^{63} -4.92620e11 q^{64} -9.78281e10 q^{66} -3.48519e11 q^{67} -9.45597e11 q^{68} -6.05475e11 q^{69} +1.31434e12 q^{71} -1.03567e11 q^{72} +1.17888e12 q^{73} +6.59783e10 q^{74} +1.72277e12 q^{76} +2.62897e12 q^{77} -7.04180e10 q^{78} -1.07242e12 q^{79} +2.82430e11 q^{81} -5.61443e10 q^{82} -1.12403e12 q^{83} +1.37926e12 q^{84} -8.53802e10 q^{86} -9.13000e11 q^{87} +2.17933e12 q^{88} +2.23561e12 q^{89} +1.89237e12 q^{91} +6.68431e12 q^{92} +4.49017e12 q^{93} +3.54345e11 q^{94} +1.72011e12 q^{96} +1.42153e13 q^{97} -4.99472e11 q^{98} -5.94306e12 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 12.0000 0.132583 0.0662913 0.997800i \(-0.478883\pi\)
0.0662913 + 0.997800i \(0.478883\pi\)
\(3\) 729.000 0.577350
\(4\) −8048.00 −0.982422
\(5\) 0 0
\(6\) 8748.00 0.0765466
\(7\) −235088. −0.755254 −0.377627 0.925958i \(-0.623260\pi\)
−0.377627 + 0.925958i \(0.623260\pi\)
\(8\) −194880. −0.262834
\(9\) 531441. 0.333333
\(10\) 0 0
\(11\) −1.11829e7 −1.90328 −0.951639 0.307218i \(-0.900602\pi\)
−0.951639 + 0.307218i \(0.900602\pi\)
\(12\) −5.86699e6 −0.567202
\(13\) −8.04961e6 −0.462534 −0.231267 0.972890i \(-0.574287\pi\)
−0.231267 + 0.972890i \(0.574287\pi\)
\(14\) −2.82106e6 −0.100134
\(15\) 0 0
\(16\) 6.35907e7 0.947575
\(17\) 1.17495e8 1.18059 0.590296 0.807187i \(-0.299011\pi\)
0.590296 + 0.807187i \(0.299011\pi\)
\(18\) 6.37729e6 0.0441942
\(19\) −2.14061e8 −1.04385 −0.521927 0.852990i \(-0.674787\pi\)
−0.521927 + 0.852990i \(0.674787\pi\)
\(20\) 0 0
\(21\) −1.71379e8 −0.436046
\(22\) −1.34195e8 −0.252341
\(23\) −8.30556e8 −1.16987 −0.584935 0.811080i \(-0.698880\pi\)
−0.584935 + 0.811080i \(0.698880\pi\)
\(24\) −1.42068e8 −0.151748
\(25\) 0 0
\(26\) −9.65954e7 −0.0613239
\(27\) 3.87420e8 0.192450
\(28\) 1.89199e9 0.741978
\(29\) −1.25240e9 −0.390981 −0.195491 0.980706i \(-0.562630\pi\)
−0.195491 + 0.980706i \(0.562630\pi\)
\(30\) 0 0
\(31\) 6.15935e9 1.24648 0.623238 0.782032i \(-0.285817\pi\)
0.623238 + 0.782032i \(0.285817\pi\)
\(32\) 2.35954e9 0.388466
\(33\) −8.15234e9 −1.09886
\(34\) 1.40994e9 0.156526
\(35\) 0 0
\(36\) −4.27704e9 −0.327474
\(37\) 5.49819e9 0.352297 0.176148 0.984364i \(-0.443636\pi\)
0.176148 + 0.984364i \(0.443636\pi\)
\(38\) −2.56874e9 −0.138397
\(39\) −5.86817e9 −0.267044
\(40\) 0 0
\(41\) −4.67869e9 −0.153826 −0.0769129 0.997038i \(-0.524506\pi\)
−0.0769129 + 0.997038i \(0.524506\pi\)
\(42\) −2.05655e9 −0.0578121
\(43\) −7.11501e9 −0.171645 −0.0858224 0.996310i \(-0.527352\pi\)
−0.0858224 + 0.996310i \(0.527352\pi\)
\(44\) 9.00000e10 1.86982
\(45\) 0 0
\(46\) −9.96667e9 −0.155104
\(47\) 2.95288e10 0.399585 0.199793 0.979838i \(-0.435973\pi\)
0.199793 + 0.979838i \(0.435973\pi\)
\(48\) 4.63576e10 0.547082
\(49\) −4.16226e10 −0.429591
\(50\) 0 0
\(51\) 8.56536e10 0.681615
\(52\) 6.47833e10 0.454403
\(53\) 2.04125e11 1.26504 0.632518 0.774545i \(-0.282021\pi\)
0.632518 + 0.774545i \(0.282021\pi\)
\(54\) 4.64905e9 0.0255155
\(55\) 0 0
\(56\) 4.58139e10 0.198507
\(57\) −1.56051e11 −0.602670
\(58\) −1.50288e10 −0.0518373
\(59\) −2.99098e10 −0.0923157 −0.0461579 0.998934i \(-0.514698\pi\)
−0.0461579 + 0.998934i \(0.514698\pi\)
\(60\) 0 0
\(61\) −1.34392e11 −0.333987 −0.166993 0.985958i \(-0.553406\pi\)
−0.166993 + 0.985958i \(0.553406\pi\)
\(62\) 7.39122e10 0.165261
\(63\) −1.24935e11 −0.251751
\(64\) −4.92620e11 −0.896071
\(65\) 0 0
\(66\) −9.78281e10 −0.145689
\(67\) −3.48519e11 −0.470695 −0.235348 0.971911i \(-0.575623\pi\)
−0.235348 + 0.971911i \(0.575623\pi\)
\(68\) −9.45597e11 −1.15984
\(69\) −6.05475e11 −0.675425
\(70\) 0 0
\(71\) 1.31434e12 1.21766 0.608831 0.793300i \(-0.291639\pi\)
0.608831 + 0.793300i \(0.291639\pi\)
\(72\) −1.03567e11 −0.0876115
\(73\) 1.17888e12 0.911737 0.455868 0.890047i \(-0.349329\pi\)
0.455868 + 0.890047i \(0.349329\pi\)
\(74\) 6.59783e10 0.0467084
\(75\) 0 0
\(76\) 1.72277e12 1.02551
\(77\) 2.62897e12 1.43746
\(78\) −7.04180e10 −0.0354054
\(79\) −1.07242e12 −0.496351 −0.248176 0.968715i \(-0.579831\pi\)
−0.248176 + 0.968715i \(0.579831\pi\)
\(80\) 0 0
\(81\) 2.82430e11 0.111111
\(82\) −5.61443e10 −0.0203946
\(83\) −1.12403e12 −0.377371 −0.188685 0.982038i \(-0.560423\pi\)
−0.188685 + 0.982038i \(0.560423\pi\)
\(84\) 1.37926e12 0.428381
\(85\) 0 0
\(86\) −8.53802e10 −0.0227571
\(87\) −9.13000e11 −0.225733
\(88\) 2.17933e12 0.500247
\(89\) 2.23561e12 0.476827 0.238414 0.971164i \(-0.423373\pi\)
0.238414 + 0.971164i \(0.423373\pi\)
\(90\) 0 0
\(91\) 1.89237e12 0.349330
\(92\) 6.68431e12 1.14931
\(93\) 4.49017e12 0.719653
\(94\) 3.54345e11 0.0529780
\(95\) 0 0
\(96\) 1.72011e12 0.224281
\(97\) 1.42153e13 1.73276 0.866380 0.499385i \(-0.166441\pi\)
0.866380 + 0.499385i \(0.166441\pi\)
\(98\) −4.99472e11 −0.0569563
\(99\) −5.94306e12 −0.634426
\(100\) 0 0
\(101\) 1.70194e13 1.59535 0.797675 0.603088i \(-0.206063\pi\)
0.797675 + 0.603088i \(0.206063\pi\)
\(102\) 1.02784e12 0.0903703
\(103\) −1.09904e13 −0.906928 −0.453464 0.891275i \(-0.649812\pi\)
−0.453464 + 0.891275i \(0.649812\pi\)
\(104\) 1.56871e12 0.121570
\(105\) 0 0
\(106\) 2.44950e12 0.167722
\(107\) 1.96403e13 1.26519 0.632593 0.774485i \(-0.281991\pi\)
0.632593 + 0.774485i \(0.281991\pi\)
\(108\) −3.11796e12 −0.189067
\(109\) −9.82099e12 −0.560897 −0.280448 0.959869i \(-0.590483\pi\)
−0.280448 + 0.959869i \(0.590483\pi\)
\(110\) 0 0
\(111\) 4.00818e12 0.203399
\(112\) −1.49494e13 −0.715660
\(113\) 1.70267e13 0.769344 0.384672 0.923053i \(-0.374315\pi\)
0.384672 + 0.923053i \(0.374315\pi\)
\(114\) −1.87261e12 −0.0799035
\(115\) 0 0
\(116\) 1.00793e13 0.384109
\(117\) −4.27789e12 −0.154178
\(118\) −3.58918e11 −0.0122395
\(119\) −2.76216e13 −0.891648
\(120\) 0 0
\(121\) 9.05347e13 2.62247
\(122\) −1.61270e12 −0.0442808
\(123\) −3.41076e12 −0.0888113
\(124\) −4.95705e13 −1.22457
\(125\) 0 0
\(126\) −1.49922e12 −0.0333778
\(127\) 4.49347e13 0.950292 0.475146 0.879907i \(-0.342395\pi\)
0.475146 + 0.879907i \(0.342395\pi\)
\(128\) −2.52408e13 −0.507270
\(129\) −5.18685e12 −0.0990992
\(130\) 0 0
\(131\) −1.20182e12 −0.0207768 −0.0103884 0.999946i \(-0.503307\pi\)
−0.0103884 + 0.999946i \(0.503307\pi\)
\(132\) 6.56100e13 1.07954
\(133\) 5.03233e13 0.788375
\(134\) −4.18223e12 −0.0624060
\(135\) 0 0
\(136\) −2.28974e13 −0.310300
\(137\) −1.71562e13 −0.221685 −0.110842 0.993838i \(-0.535355\pi\)
−0.110842 + 0.993838i \(0.535355\pi\)
\(138\) −7.26570e12 −0.0895495
\(139\) 1.05644e14 1.24236 0.621182 0.783666i \(-0.286653\pi\)
0.621182 + 0.783666i \(0.286653\pi\)
\(140\) 0 0
\(141\) 2.15265e13 0.230701
\(142\) 1.57720e13 0.161441
\(143\) 9.00181e13 0.880330
\(144\) 3.37947e13 0.315858
\(145\) 0 0
\(146\) 1.41465e13 0.120880
\(147\) −3.03429e13 −0.248024
\(148\) −4.42494e13 −0.346104
\(149\) −8.53533e13 −0.639012 −0.319506 0.947584i \(-0.603517\pi\)
−0.319506 + 0.947584i \(0.603517\pi\)
\(150\) 0 0
\(151\) −6.16414e13 −0.423177 −0.211589 0.977359i \(-0.567864\pi\)
−0.211589 + 0.977359i \(0.567864\pi\)
\(152\) 4.17163e13 0.274361
\(153\) 6.24415e13 0.393531
\(154\) 3.15476e13 0.190582
\(155\) 0 0
\(156\) 4.72270e13 0.262350
\(157\) 1.18021e14 0.628942 0.314471 0.949267i \(-0.398173\pi\)
0.314471 + 0.949267i \(0.398173\pi\)
\(158\) −1.28690e13 −0.0658075
\(159\) 1.48807e14 0.730369
\(160\) 0 0
\(161\) 1.95254e14 0.883550
\(162\) 3.38915e12 0.0147314
\(163\) −1.54710e14 −0.646099 −0.323050 0.946382i \(-0.604708\pi\)
−0.323050 + 0.946382i \(0.604708\pi\)
\(164\) 3.76541e13 0.151122
\(165\) 0 0
\(166\) −1.34883e13 −0.0500328
\(167\) −3.76012e14 −1.34136 −0.670679 0.741748i \(-0.733997\pi\)
−0.670679 + 0.741748i \(0.733997\pi\)
\(168\) 3.33984e13 0.114608
\(169\) −2.38079e14 −0.786063
\(170\) 0 0
\(171\) −1.13761e14 −0.347951
\(172\) 5.72616e13 0.168628
\(173\) −3.73562e14 −1.05941 −0.529704 0.848182i \(-0.677697\pi\)
−0.529704 + 0.848182i \(0.677697\pi\)
\(174\) −1.09560e13 −0.0299283
\(175\) 0 0
\(176\) −7.11128e14 −1.80350
\(177\) −2.18043e13 −0.0532985
\(178\) 2.68273e13 0.0632190
\(179\) 4.23349e13 0.0961952 0.0480976 0.998843i \(-0.484684\pi\)
0.0480976 + 0.998843i \(0.484684\pi\)
\(180\) 0 0
\(181\) −3.10447e14 −0.656261 −0.328130 0.944632i \(-0.606419\pi\)
−0.328130 + 0.944632i \(0.606419\pi\)
\(182\) 2.27084e13 0.0463151
\(183\) −9.79718e13 −0.192827
\(184\) 1.61859e14 0.307482
\(185\) 0 0
\(186\) 5.38820e13 0.0954134
\(187\) −1.31393e15 −2.24700
\(188\) −2.37648e14 −0.392561
\(189\) −9.10779e13 −0.145349
\(190\) 0 0
\(191\) −8.62273e14 −1.28507 −0.642537 0.766255i \(-0.722118\pi\)
−0.642537 + 0.766255i \(0.722118\pi\)
\(192\) −3.59120e14 −0.517347
\(193\) 9.37837e14 1.30618 0.653092 0.757278i \(-0.273471\pi\)
0.653092 + 0.757278i \(0.273471\pi\)
\(194\) 1.70583e14 0.229734
\(195\) 0 0
\(196\) 3.34979e14 0.422040
\(197\) 6.71715e14 0.818756 0.409378 0.912365i \(-0.365746\pi\)
0.409378 + 0.912365i \(0.365746\pi\)
\(198\) −7.13167e13 −0.0841138
\(199\) −4.36451e13 −0.0498185 −0.0249093 0.999690i \(-0.507930\pi\)
−0.0249093 + 0.999690i \(0.507930\pi\)
\(200\) 0 0
\(201\) −2.54070e14 −0.271756
\(202\) 2.04233e14 0.211515
\(203\) 2.94424e14 0.295290
\(204\) −6.89340e14 −0.669634
\(205\) 0 0
\(206\) −1.31885e14 −0.120243
\(207\) −4.41391e14 −0.389957
\(208\) −5.11880e14 −0.438285
\(209\) 2.39383e15 1.98675
\(210\) 0 0
\(211\) −1.62162e15 −1.26507 −0.632534 0.774533i \(-0.717985\pi\)
−0.632534 + 0.774533i \(0.717985\pi\)
\(212\) −1.64280e15 −1.24280
\(213\) 9.58151e14 0.703018
\(214\) 2.35684e14 0.167742
\(215\) 0 0
\(216\) −7.55005e13 −0.0505825
\(217\) −1.44799e15 −0.941406
\(218\) −1.17852e14 −0.0743651
\(219\) 8.59401e14 0.526392
\(220\) 0 0
\(221\) −9.45786e14 −0.546064
\(222\) 4.80982e13 0.0269671
\(223\) −1.47333e15 −0.802266 −0.401133 0.916020i \(-0.631383\pi\)
−0.401133 + 0.916020i \(0.631383\pi\)
\(224\) −5.54701e14 −0.293391
\(225\) 0 0
\(226\) 2.04320e14 0.102002
\(227\) 3.74889e15 1.81859 0.909294 0.416153i \(-0.136622\pi\)
0.909294 + 0.416153i \(0.136622\pi\)
\(228\) 1.25590e15 0.592076
\(229\) −1.47993e13 −0.00678126 −0.00339063 0.999994i \(-0.501079\pi\)
−0.00339063 + 0.999994i \(0.501079\pi\)
\(230\) 0 0
\(231\) 1.91652e15 0.829917
\(232\) 2.44068e14 0.102763
\(233\) −3.63053e15 −1.48647 −0.743236 0.669030i \(-0.766710\pi\)
−0.743236 + 0.669030i \(0.766710\pi\)
\(234\) −5.13347e13 −0.0204413
\(235\) 0 0
\(236\) 2.40714e14 0.0906930
\(237\) −7.81795e14 −0.286569
\(238\) −3.31459e14 −0.118217
\(239\) −4.33900e15 −1.50592 −0.752962 0.658063i \(-0.771376\pi\)
−0.752962 + 0.658063i \(0.771376\pi\)
\(240\) 0 0
\(241\) 3.02372e15 0.994103 0.497051 0.867721i \(-0.334416\pi\)
0.497051 + 0.867721i \(0.334416\pi\)
\(242\) 1.08642e15 0.347693
\(243\) 2.05891e14 0.0641500
\(244\) 1.08159e15 0.328116
\(245\) 0 0
\(246\) −4.09292e13 −0.0117748
\(247\) 1.72311e15 0.482818
\(248\) −1.20033e15 −0.327617
\(249\) −8.19414e14 −0.217875
\(250\) 0 0
\(251\) −1.75146e15 −0.442099 −0.221050 0.975263i \(-0.570948\pi\)
−0.221050 + 0.975263i \(0.570948\pi\)
\(252\) 1.00548e15 0.247326
\(253\) 9.28803e15 2.22659
\(254\) 5.39216e14 0.125992
\(255\) 0 0
\(256\) 3.73265e15 0.828816
\(257\) −4.87604e15 −1.05561 −0.527803 0.849367i \(-0.676984\pi\)
−0.527803 + 0.849367i \(0.676984\pi\)
\(258\) −6.22421e13 −0.0131388
\(259\) −1.29256e15 −0.266074
\(260\) 0 0
\(261\) −6.65577e14 −0.130327
\(262\) −1.44219e13 −0.00275463
\(263\) −4.67882e15 −0.871815 −0.435907 0.899992i \(-0.643572\pi\)
−0.435907 + 0.899992i \(0.643572\pi\)
\(264\) 1.58873e15 0.288818
\(265\) 0 0
\(266\) 6.03879e14 0.104525
\(267\) 1.62976e15 0.275296
\(268\) 2.80488e15 0.462422
\(269\) −1.80262e15 −0.290078 −0.145039 0.989426i \(-0.546331\pi\)
−0.145039 + 0.989426i \(0.546331\pi\)
\(270\) 0 0
\(271\) 6.10016e15 0.935494 0.467747 0.883862i \(-0.345066\pi\)
0.467747 + 0.883862i \(0.345066\pi\)
\(272\) 7.47156e15 1.11870
\(273\) 1.37954e15 0.201686
\(274\) −2.05874e14 −0.0293915
\(275\) 0 0
\(276\) 4.87286e15 0.663552
\(277\) 1.07023e16 1.42351 0.711754 0.702428i \(-0.247901\pi\)
0.711754 + 0.702428i \(0.247901\pi\)
\(278\) 1.26773e15 0.164716
\(279\) 3.27333e15 0.415492
\(280\) 0 0
\(281\) −2.45460e15 −0.297433 −0.148717 0.988880i \(-0.547514\pi\)
−0.148717 + 0.988880i \(0.547514\pi\)
\(282\) 2.58318e14 0.0305869
\(283\) −4.01155e15 −0.464195 −0.232098 0.972692i \(-0.574559\pi\)
−0.232098 + 0.972692i \(0.574559\pi\)
\(284\) −1.05778e16 −1.19626
\(285\) 0 0
\(286\) 1.08022e15 0.116716
\(287\) 1.09990e15 0.116178
\(288\) 1.25396e15 0.129489
\(289\) 3.90041e15 0.393799
\(290\) 0 0
\(291\) 1.03629e16 1.00041
\(292\) −9.48759e15 −0.895710
\(293\) −2.08187e15 −0.192227 −0.0961133 0.995370i \(-0.530641\pi\)
−0.0961133 + 0.995370i \(0.530641\pi\)
\(294\) −3.64115e14 −0.0328837
\(295\) 0 0
\(296\) −1.07149e15 −0.0925957
\(297\) −4.33249e15 −0.366286
\(298\) −1.02424e15 −0.0847219
\(299\) 6.68565e15 0.541104
\(300\) 0 0
\(301\) 1.67265e15 0.129636
\(302\) −7.39697e14 −0.0561059
\(303\) 1.24072e16 0.921075
\(304\) −1.36123e16 −0.989130
\(305\) 0 0
\(306\) 7.49298e14 0.0521753
\(307\) 1.32352e16 0.902260 0.451130 0.892458i \(-0.351021\pi\)
0.451130 + 0.892458i \(0.351021\pi\)
\(308\) −2.11579e16 −1.41219
\(309\) −8.01202e15 −0.523615
\(310\) 0 0
\(311\) −8.09301e15 −0.507187 −0.253593 0.967311i \(-0.581612\pi\)
−0.253593 + 0.967311i \(0.581612\pi\)
\(312\) 1.14359e15 0.0701883
\(313\) 1.48181e16 0.890748 0.445374 0.895345i \(-0.353071\pi\)
0.445374 + 0.895345i \(0.353071\pi\)
\(314\) 1.41625e15 0.0833868
\(315\) 0 0
\(316\) 8.63084e15 0.487626
\(317\) 2.43171e16 1.34594 0.672970 0.739670i \(-0.265018\pi\)
0.672970 + 0.739670i \(0.265018\pi\)
\(318\) 1.78569e15 0.0968342
\(319\) 1.40055e16 0.744147
\(320\) 0 0
\(321\) 1.43178e16 0.730455
\(322\) 2.34304e15 0.117143
\(323\) −2.51511e16 −1.23237
\(324\) −2.27299e15 −0.109158
\(325\) 0 0
\(326\) −1.85652e15 −0.0856615
\(327\) −7.15950e15 −0.323834
\(328\) 9.11783e14 0.0404307
\(329\) −6.94186e15 −0.301788
\(330\) 0 0
\(331\) 1.16232e16 0.485783 0.242892 0.970053i \(-0.421904\pi\)
0.242892 + 0.970053i \(0.421904\pi\)
\(332\) 9.04615e15 0.370737
\(333\) 2.92196e15 0.117432
\(334\) −4.51214e15 −0.177841
\(335\) 0 0
\(336\) −1.08981e16 −0.413186
\(337\) −4.62652e16 −1.72052 −0.860262 0.509853i \(-0.829700\pi\)
−0.860262 + 0.509853i \(0.829700\pi\)
\(338\) −2.85695e15 −0.104218
\(339\) 1.24125e16 0.444181
\(340\) 0 0
\(341\) −6.88795e16 −2.37239
\(342\) −1.36513e15 −0.0461323
\(343\) 3.25624e16 1.07970
\(344\) 1.38657e15 0.0451142
\(345\) 0 0
\(346\) −4.48275e15 −0.140459
\(347\) −4.79404e15 −0.147421 −0.0737106 0.997280i \(-0.523484\pi\)
−0.0737106 + 0.997280i \(0.523484\pi\)
\(348\) 7.34782e15 0.221765
\(349\) 3.76900e16 1.11651 0.558253 0.829671i \(-0.311472\pi\)
0.558253 + 0.829671i \(0.311472\pi\)
\(350\) 0 0
\(351\) −3.11859e15 −0.0890146
\(352\) −2.63866e16 −0.739360
\(353\) −4.80179e16 −1.32089 −0.660446 0.750873i \(-0.729633\pi\)
−0.660446 + 0.750873i \(0.729633\pi\)
\(354\) −2.61651e14 −0.00706645
\(355\) 0 0
\(356\) −1.79922e16 −0.468446
\(357\) −2.01361e16 −0.514793
\(358\) 5.08018e14 0.0127538
\(359\) 4.06616e16 1.00247 0.501234 0.865312i \(-0.332880\pi\)
0.501234 + 0.865312i \(0.332880\pi\)
\(360\) 0 0
\(361\) 3.76929e15 0.0896320
\(362\) −3.72536e15 −0.0870087
\(363\) 6.59998e16 1.51408
\(364\) −1.52298e16 −0.343190
\(365\) 0 0
\(366\) −1.17566e15 −0.0255656
\(367\) −2.96733e16 −0.633923 −0.316961 0.948438i \(-0.602663\pi\)
−0.316961 + 0.948438i \(0.602663\pi\)
\(368\) −5.28156e16 −1.10854
\(369\) −2.48645e15 −0.0512752
\(370\) 0 0
\(371\) −4.79873e16 −0.955424
\(372\) −3.61369e16 −0.707003
\(373\) 9.01346e16 1.73294 0.866471 0.499227i \(-0.166383\pi\)
0.866471 + 0.499227i \(0.166383\pi\)
\(374\) −1.57672e16 −0.297912
\(375\) 0 0
\(376\) −5.75457e15 −0.105025
\(377\) 1.00813e16 0.180842
\(378\) −1.09293e15 −0.0192707
\(379\) −1.54841e16 −0.268369 −0.134184 0.990956i \(-0.542841\pi\)
−0.134184 + 0.990956i \(0.542841\pi\)
\(380\) 0 0
\(381\) 3.27574e16 0.548652
\(382\) −1.03473e16 −0.170378
\(383\) −9.37088e15 −0.151701 −0.0758505 0.997119i \(-0.524167\pi\)
−0.0758505 + 0.997119i \(0.524167\pi\)
\(384\) −1.84006e16 −0.292872
\(385\) 0 0
\(386\) 1.12540e16 0.173177
\(387\) −3.78121e15 −0.0572150
\(388\) −1.14404e17 −1.70230
\(389\) 2.95806e16 0.432847 0.216423 0.976300i \(-0.430561\pi\)
0.216423 + 0.976300i \(0.430561\pi\)
\(390\) 0 0
\(391\) −9.75858e16 −1.38114
\(392\) 8.11142e15 0.112911
\(393\) −8.76130e14 −0.0119955
\(394\) 8.06058e15 0.108553
\(395\) 0 0
\(396\) 4.78297e16 0.623274
\(397\) −1.80617e16 −0.231538 −0.115769 0.993276i \(-0.536933\pi\)
−0.115769 + 0.993276i \(0.536933\pi\)
\(398\) −5.23742e14 −0.00660507
\(399\) 3.66857e16 0.455169
\(400\) 0 0
\(401\) −1.20412e17 −1.44621 −0.723107 0.690736i \(-0.757287\pi\)
−0.723107 + 0.690736i \(0.757287\pi\)
\(402\) −3.04884e15 −0.0360301
\(403\) −4.95804e16 −0.576537
\(404\) −1.36972e17 −1.56731
\(405\) 0 0
\(406\) 3.53309e15 0.0391503
\(407\) −6.14858e16 −0.670519
\(408\) −1.66922e16 −0.179152
\(409\) −1.77522e16 −0.187521 −0.0937606 0.995595i \(-0.529889\pi\)
−0.0937606 + 0.995595i \(0.529889\pi\)
\(410\) 0 0
\(411\) −1.25068e16 −0.127990
\(412\) 8.84510e16 0.890986
\(413\) 7.03144e15 0.0697219
\(414\) −5.29670e15 −0.0517015
\(415\) 0 0
\(416\) −1.89934e16 −0.179679
\(417\) 7.70145e16 0.717279
\(418\) 2.87259e16 0.263408
\(419\) 1.75670e17 1.58602 0.793008 0.609212i \(-0.208514\pi\)
0.793008 + 0.609212i \(0.208514\pi\)
\(420\) 0 0
\(421\) 1.84473e17 1.61473 0.807365 0.590052i \(-0.200893\pi\)
0.807365 + 0.590052i \(0.200893\pi\)
\(422\) −1.94595e16 −0.167726
\(423\) 1.56928e16 0.133195
\(424\) −3.97799e16 −0.332495
\(425\) 0 0
\(426\) 1.14978e16 0.0932079
\(427\) 3.15939e16 0.252245
\(428\) −1.58065e17 −1.24295
\(429\) 6.56232e16 0.508259
\(430\) 0 0
\(431\) 8.05532e16 0.605314 0.302657 0.953100i \(-0.402126\pi\)
0.302657 + 0.953100i \(0.402126\pi\)
\(432\) 2.46363e16 0.182361
\(433\) 1.97092e17 1.43714 0.718568 0.695457i \(-0.244798\pi\)
0.718568 + 0.695457i \(0.244798\pi\)
\(434\) −1.73759e16 −0.124814
\(435\) 0 0
\(436\) 7.90393e16 0.551037
\(437\) 1.77790e17 1.22117
\(438\) 1.03128e16 0.0697903
\(439\) 9.89007e16 0.659447 0.329724 0.944078i \(-0.393044\pi\)
0.329724 + 0.944078i \(0.393044\pi\)
\(440\) 0 0
\(441\) −2.21200e16 −0.143197
\(442\) −1.13494e16 −0.0723985
\(443\) 1.25104e17 0.786404 0.393202 0.919452i \(-0.371367\pi\)
0.393202 + 0.919452i \(0.371367\pi\)
\(444\) −3.22578e16 −0.199823
\(445\) 0 0
\(446\) −1.76800e16 −0.106366
\(447\) −6.22225e16 −0.368934
\(448\) 1.15809e17 0.676761
\(449\) −1.80095e17 −1.03729 −0.518645 0.854990i \(-0.673563\pi\)
−0.518645 + 0.854990i \(0.673563\pi\)
\(450\) 0 0
\(451\) 5.23213e16 0.292773
\(452\) −1.37031e17 −0.755820
\(453\) −4.49366e16 −0.244322
\(454\) 4.49867e16 0.241113
\(455\) 0 0
\(456\) 3.04112e16 0.158402
\(457\) 9.43597e16 0.484542 0.242271 0.970209i \(-0.422108\pi\)
0.242271 + 0.970209i \(0.422108\pi\)
\(458\) −1.77592e14 −0.000899076 0
\(459\) 4.55198e16 0.227205
\(460\) 0 0
\(461\) 8.00500e16 0.388423 0.194212 0.980960i \(-0.437785\pi\)
0.194212 + 0.980960i \(0.437785\pi\)
\(462\) 2.29982e16 0.110033
\(463\) −2.14174e17 −1.01039 −0.505196 0.863004i \(-0.668580\pi\)
−0.505196 + 0.863004i \(0.668580\pi\)
\(464\) −7.96410e16 −0.370484
\(465\) 0 0
\(466\) −4.35663e16 −0.197080
\(467\) 1.80681e17 0.806031 0.403015 0.915193i \(-0.367962\pi\)
0.403015 + 0.915193i \(0.367962\pi\)
\(468\) 3.44285e16 0.151468
\(469\) 8.19326e16 0.355495
\(470\) 0 0
\(471\) 8.60372e16 0.363120
\(472\) 5.82883e15 0.0242638
\(473\) 7.95665e16 0.326688
\(474\) −9.38154e15 −0.0379940
\(475\) 0 0
\(476\) 2.22298e17 0.875974
\(477\) 1.08480e17 0.421679
\(478\) −5.20680e16 −0.199659
\(479\) −2.66712e17 −1.00893 −0.504466 0.863431i \(-0.668311\pi\)
−0.504466 + 0.863431i \(0.668311\pi\)
\(480\) 0 0
\(481\) −4.42583e16 −0.162949
\(482\) 3.62847e16 0.131801
\(483\) 1.42340e17 0.510118
\(484\) −7.28623e17 −2.57637
\(485\) 0 0
\(486\) 2.47069e15 0.00850517
\(487\) 2.63552e17 0.895216 0.447608 0.894230i \(-0.352276\pi\)
0.447608 + 0.894230i \(0.352276\pi\)
\(488\) 2.61903e16 0.0877833
\(489\) −1.12784e17 −0.373026
\(490\) 0 0
\(491\) 4.11733e17 1.32613 0.663065 0.748562i \(-0.269255\pi\)
0.663065 + 0.748562i \(0.269255\pi\)
\(492\) 2.74498e16 0.0872502
\(493\) −1.47150e17 −0.461590
\(494\) 2.06773e16 0.0640132
\(495\) 0 0
\(496\) 3.91677e17 1.18113
\(497\) −3.08984e17 −0.919645
\(498\) −9.83297e15 −0.0288864
\(499\) 3.99658e17 1.15887 0.579435 0.815018i \(-0.303273\pi\)
0.579435 + 0.815018i \(0.303273\pi\)
\(500\) 0 0
\(501\) −2.74113e17 −0.774433
\(502\) −2.10175e16 −0.0586146
\(503\) 2.83581e17 0.780702 0.390351 0.920666i \(-0.372354\pi\)
0.390351 + 0.920666i \(0.372354\pi\)
\(504\) 2.43474e16 0.0661690
\(505\) 0 0
\(506\) 1.11456e17 0.295207
\(507\) −1.73559e17 −0.453834
\(508\) −3.61634e17 −0.933588
\(509\) 6.40327e17 1.63206 0.816030 0.578009i \(-0.196170\pi\)
0.816030 + 0.578009i \(0.196170\pi\)
\(510\) 0 0
\(511\) −2.77140e17 −0.688593
\(512\) 2.51565e17 0.617156
\(513\) −8.29318e16 −0.200890
\(514\) −5.85124e16 −0.139955
\(515\) 0 0
\(516\) 4.17437e16 0.0973572
\(517\) −3.30218e17 −0.760522
\(518\) −1.55107e16 −0.0352767
\(519\) −2.72327e17 −0.611650
\(520\) 0 0
\(521\) −4.01348e17 −0.879175 −0.439588 0.898200i \(-0.644875\pi\)
−0.439588 + 0.898200i \(0.644875\pi\)
\(522\) −7.98692e15 −0.0172791
\(523\) 5.05985e17 1.08113 0.540564 0.841303i \(-0.318211\pi\)
0.540564 + 0.841303i \(0.318211\pi\)
\(524\) 9.67228e15 0.0204115
\(525\) 0 0
\(526\) −5.61459e16 −0.115587
\(527\) 7.23691e17 1.47158
\(528\) −5.18413e17 −1.04125
\(529\) 1.85786e17 0.368597
\(530\) 0 0
\(531\) −1.58953e16 −0.0307719
\(532\) −4.05002e17 −0.774517
\(533\) 3.76616e16 0.0711496
\(534\) 1.95571e16 0.0364995
\(535\) 0 0
\(536\) 6.79193e16 0.123715
\(537\) 3.08621e16 0.0555383
\(538\) −2.16314e16 −0.0384592
\(539\) 4.65462e17 0.817631
\(540\) 0 0
\(541\) −1.69124e17 −0.290017 −0.145009 0.989430i \(-0.546321\pi\)
−0.145009 + 0.989430i \(0.546321\pi\)
\(542\) 7.32020e16 0.124030
\(543\) −2.26316e17 −0.378892
\(544\) 2.77234e17 0.458620
\(545\) 0 0
\(546\) 1.65544e16 0.0267400
\(547\) 4.32104e17 0.689717 0.344858 0.938655i \(-0.387927\pi\)
0.344858 + 0.938655i \(0.387927\pi\)
\(548\) 1.38073e17 0.217788
\(549\) −7.14214e16 −0.111329
\(550\) 0 0
\(551\) 2.68091e17 0.408128
\(552\) 1.17995e17 0.177525
\(553\) 2.52113e17 0.374871
\(554\) 1.28428e17 0.188732
\(555\) 0 0
\(556\) −8.50224e17 −1.22053
\(557\) −1.36804e18 −1.94107 −0.970534 0.240966i \(-0.922536\pi\)
−0.970534 + 0.240966i \(0.922536\pi\)
\(558\) 3.92800e16 0.0550870
\(559\) 5.72731e16 0.0793915
\(560\) 0 0
\(561\) −9.57856e17 −1.29730
\(562\) −2.94552e16 −0.0394345
\(563\) 9.52405e17 1.26043 0.630213 0.776422i \(-0.282968\pi\)
0.630213 + 0.776422i \(0.282968\pi\)
\(564\) −1.73245e17 −0.226645
\(565\) 0 0
\(566\) −4.81386e16 −0.0615442
\(567\) −6.63958e16 −0.0839171
\(568\) −2.56138e17 −0.320044
\(569\) 1.53632e17 0.189780 0.0948902 0.995488i \(-0.469750\pi\)
0.0948902 + 0.995488i \(0.469750\pi\)
\(570\) 0 0
\(571\) −1.27956e18 −1.54500 −0.772498 0.635017i \(-0.780993\pi\)
−0.772498 + 0.635017i \(0.780993\pi\)
\(572\) −7.24466e17 −0.864856
\(573\) −6.28597e17 −0.741938
\(574\) 1.31988e16 0.0154031
\(575\) 0 0
\(576\) −2.61799e17 −0.298690
\(577\) −3.56770e17 −0.402481 −0.201241 0.979542i \(-0.564497\pi\)
−0.201241 + 0.979542i \(0.564497\pi\)
\(578\) 4.68049e16 0.0522108
\(579\) 6.83683e17 0.754126
\(580\) 0 0
\(581\) 2.64245e17 0.285011
\(582\) 1.24355e17 0.132637
\(583\) −2.28271e18 −2.40772
\(584\) −2.29739e17 −0.239636
\(585\) 0 0
\(586\) −2.49824e16 −0.0254859
\(587\) −1.28968e18 −1.30118 −0.650588 0.759431i \(-0.725477\pi\)
−0.650588 + 0.759431i \(0.725477\pi\)
\(588\) 2.44200e17 0.243665
\(589\) −1.31848e18 −1.30114
\(590\) 0 0
\(591\) 4.89680e17 0.472709
\(592\) 3.49634e17 0.333827
\(593\) 1.88640e18 1.78147 0.890735 0.454523i \(-0.150190\pi\)
0.890735 + 0.454523i \(0.150190\pi\)
\(594\) −5.19899e16 −0.0485631
\(595\) 0 0
\(596\) 6.86923e17 0.627780
\(597\) −3.18173e16 −0.0287627
\(598\) 8.02278e16 0.0717410
\(599\) −1.44668e18 −1.27967 −0.639834 0.768513i \(-0.720997\pi\)
−0.639834 + 0.768513i \(0.720997\pi\)
\(600\) 0 0
\(601\) −4.44358e16 −0.0384635 −0.0192317 0.999815i \(-0.506122\pi\)
−0.0192317 + 0.999815i \(0.506122\pi\)
\(602\) 2.00719e16 0.0171874
\(603\) −1.85217e17 −0.156898
\(604\) 4.96090e17 0.415739
\(605\) 0 0
\(606\) 1.48886e17 0.122118
\(607\) −2.98050e16 −0.0241860 −0.0120930 0.999927i \(-0.503849\pi\)
−0.0120930 + 0.999927i \(0.503849\pi\)
\(608\) −5.05087e17 −0.405502
\(609\) 2.14635e17 0.170486
\(610\) 0 0
\(611\) −2.37695e17 −0.184822
\(612\) −5.02529e17 −0.386613
\(613\) 8.84082e17 0.672976 0.336488 0.941688i \(-0.390761\pi\)
0.336488 + 0.941688i \(0.390761\pi\)
\(614\) 1.58823e17 0.119624
\(615\) 0 0
\(616\) −5.12333e17 −0.377814
\(617\) −1.43684e18 −1.04846 −0.524232 0.851575i \(-0.675648\pi\)
−0.524232 + 0.851575i \(0.675648\pi\)
\(618\) −9.61443e16 −0.0694222
\(619\) 1.68862e18 1.20654 0.603272 0.797535i \(-0.293863\pi\)
0.603272 + 0.797535i \(0.293863\pi\)
\(620\) 0 0
\(621\) −3.21774e17 −0.225142
\(622\) −9.71162e16 −0.0672441
\(623\) −5.25565e17 −0.360126
\(624\) −3.73161e17 −0.253044
\(625\) 0 0
\(626\) 1.77817e17 0.118098
\(627\) 1.74510e18 1.14705
\(628\) −9.49831e17 −0.617887
\(629\) 6.46008e17 0.415919
\(630\) 0 0
\(631\) −3.53490e17 −0.222939 −0.111470 0.993768i \(-0.535556\pi\)
−0.111470 + 0.993768i \(0.535556\pi\)
\(632\) 2.08993e17 0.130458
\(633\) −1.18216e18 −0.730387
\(634\) 2.91805e17 0.178448
\(635\) 0 0
\(636\) −1.19760e18 −0.717531
\(637\) 3.35046e17 0.198700
\(638\) 1.68066e17 0.0986608
\(639\) 6.98492e17 0.405888
\(640\) 0 0
\(641\) 1.61802e18 0.921313 0.460656 0.887579i \(-0.347614\pi\)
0.460656 + 0.887579i \(0.347614\pi\)
\(642\) 1.71814e17 0.0968456
\(643\) 1.96065e18 1.09403 0.547015 0.837123i \(-0.315764\pi\)
0.547015 + 0.837123i \(0.315764\pi\)
\(644\) −1.57140e18 −0.868018
\(645\) 0 0
\(646\) −3.01813e17 −0.163390
\(647\) 5.96114e17 0.319486 0.159743 0.987159i \(-0.448933\pi\)
0.159743 + 0.987159i \(0.448933\pi\)
\(648\) −5.50399e16 −0.0292038
\(649\) 3.34479e17 0.175703
\(650\) 0 0
\(651\) −1.05558e18 −0.543521
\(652\) 1.24511e18 0.634742
\(653\) 2.58318e18 1.30382 0.651912 0.758295i \(-0.273967\pi\)
0.651912 + 0.758295i \(0.273967\pi\)
\(654\) −8.59140e16 −0.0429347
\(655\) 0 0
\(656\) −2.97521e17 −0.145761
\(657\) 6.26503e17 0.303912
\(658\) −8.33023e16 −0.0400119
\(659\) 2.64137e18 1.25624 0.628121 0.778116i \(-0.283824\pi\)
0.628121 + 0.778116i \(0.283824\pi\)
\(660\) 0 0
\(661\) 4.12451e18 1.92337 0.961685 0.274156i \(-0.0883983\pi\)
0.961685 + 0.274156i \(0.0883983\pi\)
\(662\) 1.39478e17 0.0644064
\(663\) −6.89478e17 −0.315270
\(664\) 2.19050e17 0.0991861
\(665\) 0 0
\(666\) 3.50636e16 0.0155695
\(667\) 1.04019e18 0.457398
\(668\) 3.02614e18 1.31778
\(669\) −1.07406e18 −0.463189
\(670\) 0 0
\(671\) 1.50289e18 0.635670
\(672\) −4.04377e17 −0.169389
\(673\) −2.79726e18 −1.16047 −0.580236 0.814449i \(-0.697039\pi\)
−0.580236 + 0.814449i \(0.697039\pi\)
\(674\) −5.55183e17 −0.228111
\(675\) 0 0
\(676\) 1.91606e18 0.772245
\(677\) 4.25553e18 1.69874 0.849372 0.527795i \(-0.176981\pi\)
0.849372 + 0.527795i \(0.176981\pi\)
\(678\) 1.48950e17 0.0588906
\(679\) −3.34184e18 −1.30867
\(680\) 0 0
\(681\) 2.73294e18 1.04996
\(682\) −8.26553e17 −0.314538
\(683\) −1.60893e18 −0.606461 −0.303230 0.952917i \(-0.598065\pi\)
−0.303230 + 0.952917i \(0.598065\pi\)
\(684\) 9.15548e17 0.341835
\(685\) 0 0
\(686\) 3.90749e17 0.143150
\(687\) −1.07887e16 −0.00391516
\(688\) −4.52448e17 −0.162646
\(689\) −1.64313e18 −0.585122
\(690\) 0 0
\(691\) −3.06331e18 −1.07049 −0.535247 0.844696i \(-0.679782\pi\)
−0.535247 + 0.844696i \(0.679782\pi\)
\(692\) 3.00643e18 1.04079
\(693\) 1.39714e18 0.479153
\(694\) −5.75285e16 −0.0195455
\(695\) 0 0
\(696\) 1.77925e17 0.0593305
\(697\) −5.49721e17 −0.181606
\(698\) 4.52280e17 0.148029
\(699\) −2.64666e18 −0.858214
\(700\) 0 0
\(701\) 2.99144e18 0.952166 0.476083 0.879400i \(-0.342056\pi\)
0.476083 + 0.879400i \(0.342056\pi\)
\(702\) −3.74230e16 −0.0118018
\(703\) −1.17695e18 −0.367747
\(704\) 5.50893e18 1.70547
\(705\) 0 0
\(706\) −5.76215e17 −0.175127
\(707\) −4.00106e18 −1.20489
\(708\) 1.75481e17 0.0523616
\(709\) −5.31694e18 −1.57203 −0.786015 0.618207i \(-0.787859\pi\)
−0.786015 + 0.618207i \(0.787859\pi\)
\(710\) 0 0
\(711\) −5.69928e17 −0.165450
\(712\) −4.35676e17 −0.125327
\(713\) −5.11568e18 −1.45822
\(714\) −2.41634e17 −0.0682525
\(715\) 0 0
\(716\) −3.40711e17 −0.0945043
\(717\) −3.16313e18 −0.869446
\(718\) 4.87939e17 0.132910
\(719\) −4.03153e18 −1.08826 −0.544129 0.839001i \(-0.683140\pi\)
−0.544129 + 0.839001i \(0.683140\pi\)
\(720\) 0 0
\(721\) 2.58372e18 0.684961
\(722\) 4.52315e16 0.0118836
\(723\) 2.20429e18 0.573945
\(724\) 2.49848e18 0.644725
\(725\) 0 0
\(726\) 7.91998e17 0.200741
\(727\) 4.77643e18 1.19986 0.599928 0.800054i \(-0.295196\pi\)
0.599928 + 0.800054i \(0.295196\pi\)
\(728\) −3.68785e17 −0.0918161
\(729\) 1.50095e17 0.0370370
\(730\) 0 0
\(731\) −8.35976e17 −0.202643
\(732\) 7.88477e17 0.189438
\(733\) −1.71668e18 −0.408803 −0.204401 0.978887i \(-0.565525\pi\)
−0.204401 + 0.978887i \(0.565525\pi\)
\(734\) −3.56080e17 −0.0840471
\(735\) 0 0
\(736\) −1.95973e18 −0.454455
\(737\) 3.89745e18 0.895864
\(738\) −2.98374e16 −0.00679820
\(739\) −8.69723e17 −0.196423 −0.0982114 0.995166i \(-0.531312\pi\)
−0.0982114 + 0.995166i \(0.531312\pi\)
\(740\) 0 0
\(741\) 1.25615e18 0.278755
\(742\) −5.75848e17 −0.126673
\(743\) −2.40272e18 −0.523933 −0.261966 0.965077i \(-0.584371\pi\)
−0.261966 + 0.965077i \(0.584371\pi\)
\(744\) −8.75044e17 −0.189150
\(745\) 0 0
\(746\) 1.08161e18 0.229758
\(747\) −5.97353e17 −0.125790
\(748\) 1.05745e19 2.20750
\(749\) −4.61721e18 −0.955537
\(750\) 0 0
\(751\) 9.37175e18 1.90617 0.953084 0.302706i \(-0.0978899\pi\)
0.953084 + 0.302706i \(0.0978899\pi\)
\(752\) 1.87775e18 0.378637
\(753\) −1.27681e18 −0.255246
\(754\) 1.20976e17 0.0239765
\(755\) 0 0
\(756\) 7.32995e17 0.142794
\(757\) −3.09120e18 −0.597040 −0.298520 0.954403i \(-0.596493\pi\)
−0.298520 + 0.954403i \(0.596493\pi\)
\(758\) −1.85810e17 −0.0355810
\(759\) 6.77097e18 1.28552
\(760\) 0 0
\(761\) −7.97787e18 −1.48897 −0.744486 0.667638i \(-0.767306\pi\)
−0.744486 + 0.667638i \(0.767306\pi\)
\(762\) 3.93088e17 0.0727416
\(763\) 2.30880e18 0.423620
\(764\) 6.93957e18 1.26248
\(765\) 0 0
\(766\) −1.12451e17 −0.0201129
\(767\) 2.40763e17 0.0426991
\(768\) 2.72110e18 0.478517
\(769\) 7.37344e18 1.28573 0.642863 0.765981i \(-0.277746\pi\)
0.642863 + 0.765981i \(0.277746\pi\)
\(770\) 0 0
\(771\) −3.55463e18 −0.609454
\(772\) −7.54771e18 −1.28322
\(773\) 1.67335e18 0.282111 0.141056 0.990002i \(-0.454950\pi\)
0.141056 + 0.990002i \(0.454950\pi\)
\(774\) −4.53745e16 −0.00758570
\(775\) 0 0
\(776\) −2.77027e18 −0.455429
\(777\) −9.42275e17 −0.153618
\(778\) 3.54967e17 0.0573879
\(779\) 1.00153e18 0.160572
\(780\) 0 0
\(781\) −1.46981e19 −2.31755
\(782\) −1.17103e18 −0.183115
\(783\) −4.85206e17 −0.0752444
\(784\) −2.64681e18 −0.407069
\(785\) 0 0
\(786\) −1.05136e16 −0.00159039
\(787\) 3.75359e18 0.563133 0.281566 0.959542i \(-0.409146\pi\)
0.281566 + 0.959542i \(0.409146\pi\)
\(788\) −5.40596e18 −0.804363
\(789\) −3.41086e18 −0.503342
\(790\) 0 0
\(791\) −4.00277e18 −0.581050
\(792\) 1.15818e18 0.166749
\(793\) 1.08180e18 0.154480
\(794\) −2.16741e17 −0.0306978
\(795\) 0 0
\(796\) 3.51256e17 0.0489428
\(797\) −3.38853e18 −0.468309 −0.234154 0.972199i \(-0.575232\pi\)
−0.234154 + 0.972199i \(0.575232\pi\)
\(798\) 4.40228e17 0.0603474
\(799\) 3.46947e18 0.471747
\(800\) 0 0
\(801\) 1.18810e18 0.158942
\(802\) −1.44495e18 −0.191743
\(803\) −1.31833e19 −1.73529
\(804\) 2.04476e18 0.266979
\(805\) 0 0
\(806\) −5.94965e17 −0.0764387
\(807\) −1.31411e18 −0.167476
\(808\) −3.31674e18 −0.419313
\(809\) −3.13119e18 −0.392685 −0.196343 0.980535i \(-0.562906\pi\)
−0.196343 + 0.980535i \(0.562906\pi\)
\(810\) 0 0
\(811\) 1.04731e19 1.29253 0.646264 0.763114i \(-0.276331\pi\)
0.646264 + 0.763114i \(0.276331\pi\)
\(812\) −2.36953e18 −0.290100
\(813\) 4.44702e18 0.540108
\(814\) −7.37829e17 −0.0888991
\(815\) 0 0
\(816\) 5.44677e18 0.645881
\(817\) 1.52305e18 0.179172
\(818\) −2.13026e17 −0.0248620
\(819\) 1.00568e18 0.116443
\(820\) 0 0
\(821\) −6.85162e18 −0.780841 −0.390421 0.920637i \(-0.627670\pi\)
−0.390421 + 0.920637i \(0.627670\pi\)
\(822\) −1.50082e17 −0.0169692
\(823\) −3.06934e17 −0.0344308 −0.0172154 0.999852i \(-0.505480\pi\)
−0.0172154 + 0.999852i \(0.505480\pi\)
\(824\) 2.14181e18 0.238372
\(825\) 0 0
\(826\) 8.43773e16 0.00924390
\(827\) −7.75365e18 −0.842792 −0.421396 0.906877i \(-0.638460\pi\)
−0.421396 + 0.906877i \(0.638460\pi\)
\(828\) 3.55232e18 0.383102
\(829\) 2.34336e18 0.250747 0.125373 0.992110i \(-0.459987\pi\)
0.125373 + 0.992110i \(0.459987\pi\)
\(830\) 0 0
\(831\) 7.80200e18 0.821863
\(832\) 3.96540e18 0.414463
\(833\) −4.89044e18 −0.507172
\(834\) 9.24174e17 0.0950987
\(835\) 0 0
\(836\) −1.92655e19 −1.95182
\(837\) 2.38626e18 0.239884
\(838\) 2.10804e18 0.210278
\(839\) −1.63297e19 −1.61632 −0.808158 0.588965i \(-0.799535\pi\)
−0.808158 + 0.588965i \(0.799535\pi\)
\(840\) 0 0
\(841\) −8.69212e18 −0.847134
\(842\) 2.21368e18 0.214085
\(843\) −1.78940e18 −0.171723
\(844\) 1.30508e19 1.24283
\(845\) 0 0
\(846\) 1.88314e17 0.0176593
\(847\) −2.12836e19 −1.98063
\(848\) 1.29804e19 1.19872
\(849\) −2.92442e18 −0.268003
\(850\) 0 0
\(851\) −4.56655e18 −0.412142
\(852\) −7.71120e18 −0.690660
\(853\) −1.93794e19 −1.72255 −0.861276 0.508137i \(-0.830334\pi\)
−0.861276 + 0.508137i \(0.830334\pi\)
\(854\) 3.79127e17 0.0334433
\(855\) 0 0
\(856\) −3.82751e18 −0.332534
\(857\) −1.20537e19 −1.03931 −0.519656 0.854376i \(-0.673940\pi\)
−0.519656 + 0.854376i \(0.673940\pi\)
\(858\) 7.87478e17 0.0673862
\(859\) −1.00612e19 −0.854465 −0.427232 0.904142i \(-0.640511\pi\)
−0.427232 + 0.904142i \(0.640511\pi\)
\(860\) 0 0
\(861\) 8.01830e17 0.0670751
\(862\) 9.66639e17 0.0802541
\(863\) −8.01407e18 −0.660363 −0.330182 0.943917i \(-0.607110\pi\)
−0.330182 + 0.943917i \(0.607110\pi\)
\(864\) 9.14136e17 0.0747604
\(865\) 0 0
\(866\) 2.36510e18 0.190539
\(867\) 2.84340e18 0.227360
\(868\) 1.16534e19 0.924858
\(869\) 1.19928e19 0.944694
\(870\) 0 0
\(871\) 2.80544e18 0.217712
\(872\) 1.91391e18 0.147423
\(873\) 7.55457e18 0.577587
\(874\) 2.13348e18 0.161906
\(875\) 0 0
\(876\) −6.91646e18 −0.517139
\(877\) −8.87791e17 −0.0658891 −0.0329445 0.999457i \(-0.510488\pi\)
−0.0329445 + 0.999457i \(0.510488\pi\)
\(878\) 1.18681e18 0.0874312
\(879\) −1.51768e18 −0.110982
\(880\) 0 0
\(881\) 3.78774e18 0.272921 0.136460 0.990646i \(-0.456427\pi\)
0.136460 + 0.990646i \(0.456427\pi\)
\(882\) −2.65440e17 −0.0189854
\(883\) 2.75428e19 1.95553 0.977763 0.209712i \(-0.0672526\pi\)
0.977763 + 0.209712i \(0.0672526\pi\)
\(884\) 7.61169e18 0.536465
\(885\) 0 0
\(886\) 1.50124e18 0.104263
\(887\) 2.84165e19 1.95914 0.979572 0.201092i \(-0.0644489\pi\)
0.979572 + 0.201092i \(0.0644489\pi\)
\(888\) −7.81114e17 −0.0534602
\(889\) −1.05636e19 −0.717712
\(890\) 0 0
\(891\) −3.15838e18 −0.211475
\(892\) 1.18574e19 0.788164
\(893\) −6.32097e18 −0.417109
\(894\) −7.46670e17 −0.0489142
\(895\) 0 0
\(896\) 5.93382e18 0.383118
\(897\) 4.87384e18 0.312407
\(898\) −2.16114e18 −0.137527
\(899\) −7.71397e18 −0.487349
\(900\) 0 0
\(901\) 2.39836e19 1.49349
\(902\) 6.27856e17 0.0388166
\(903\) 1.21937e18 0.0748451
\(904\) −3.31816e18 −0.202210
\(905\) 0 0
\(906\) −5.39239e17 −0.0323928
\(907\) −7.51023e18 −0.447926 −0.223963 0.974598i \(-0.571899\pi\)
−0.223963 + 0.974598i \(0.571899\pi\)
\(908\) −3.01711e19 −1.78662
\(909\) 9.04482e18 0.531783
\(910\) 0 0
\(911\) −2.34028e19 −1.35643 −0.678216 0.734863i \(-0.737247\pi\)
−0.678216 + 0.734863i \(0.737247\pi\)
\(912\) −9.92337e18 −0.571074
\(913\) 1.25699e19 0.718242
\(914\) 1.13232e18 0.0642418
\(915\) 0 0
\(916\) 1.19105e17 0.00666206
\(917\) 2.82535e17 0.0156917
\(918\) 5.46238e17 0.0301234
\(919\) 2.20881e19 1.20950 0.604752 0.796414i \(-0.293272\pi\)
0.604752 + 0.796414i \(0.293272\pi\)
\(920\) 0 0
\(921\) 9.64848e18 0.520920
\(922\) 9.60600e17 0.0514982
\(923\) −1.05799e19 −0.563210
\(924\) −1.54241e19 −0.815329
\(925\) 0 0
\(926\) −2.57009e18 −0.133960
\(927\) −5.84077e18 −0.302309
\(928\) −2.95509e18 −0.151883
\(929\) −1.05632e19 −0.539132 −0.269566 0.962982i \(-0.586880\pi\)
−0.269566 + 0.962982i \(0.586880\pi\)
\(930\) 0 0
\(931\) 8.90980e18 0.448430
\(932\) 2.92185e19 1.46034
\(933\) −5.89981e18 −0.292824
\(934\) 2.16817e18 0.106866
\(935\) 0 0
\(936\) 8.33676e17 0.0405233
\(937\) 1.72833e19 0.834294 0.417147 0.908839i \(-0.363030\pi\)
0.417147 + 0.908839i \(0.363030\pi\)
\(938\) 9.83191e17 0.0471324
\(939\) 1.08024e19 0.514273
\(940\) 0 0
\(941\) 2.14038e19 1.00498 0.502490 0.864583i \(-0.332417\pi\)
0.502490 + 0.864583i \(0.332417\pi\)
\(942\) 1.03245e18 0.0481434
\(943\) 3.88591e18 0.179956
\(944\) −1.90199e18 −0.0874760
\(945\) 0 0
\(946\) 9.54799e17 0.0433131
\(947\) 5.91308e17 0.0266403 0.0133202 0.999911i \(-0.495760\pi\)
0.0133202 + 0.999911i \(0.495760\pi\)
\(948\) 6.29188e18 0.281531
\(949\) −9.48950e18 −0.421709
\(950\) 0 0
\(951\) 1.77271e19 0.777079
\(952\) 5.38289e18 0.234356
\(953\) 2.39396e19 1.03517 0.517587 0.855631i \(-0.326830\pi\)
0.517587 + 0.855631i \(0.326830\pi\)
\(954\) 1.30177e18 0.0559073
\(955\) 0 0
\(956\) 3.49203e19 1.47945
\(957\) 1.02100e19 0.429633
\(958\) −3.20055e18 −0.133767
\(959\) 4.03321e18 0.167429
\(960\) 0 0
\(961\) 1.35201e19 0.553702
\(962\) −5.31100e17 −0.0216042
\(963\) 1.04377e19 0.421729
\(964\) −2.43349e19 −0.976628
\(965\) 0 0
\(966\) 1.70808e18 0.0676327
\(967\) −1.99045e18 −0.0782853 −0.0391427 0.999234i \(-0.512463\pi\)
−0.0391427 + 0.999234i \(0.512463\pi\)
\(968\) −1.76434e19 −0.689275
\(969\) −1.83351e19 −0.711507
\(970\) 0 0
\(971\) 4.12385e19 1.57898 0.789492 0.613761i \(-0.210344\pi\)
0.789492 + 0.613761i \(0.210344\pi\)
\(972\) −1.65701e18 −0.0630224
\(973\) −2.48357e19 −0.938301
\(974\) 3.16262e18 0.118690
\(975\) 0 0
\(976\) −8.54608e18 −0.316478
\(977\) 1.25374e19 0.461202 0.230601 0.973048i \(-0.425931\pi\)
0.230601 + 0.973048i \(0.425931\pi\)
\(978\) −1.35340e18 −0.0494567
\(979\) −2.50006e19 −0.907535
\(980\) 0 0
\(981\) −5.21928e18 −0.186966
\(982\) 4.94080e18 0.175822
\(983\) 1.83966e19 0.650340 0.325170 0.945656i \(-0.394578\pi\)
0.325170 + 0.945656i \(0.394578\pi\)
\(984\) 6.64690e17 0.0233427
\(985\) 0 0
\(986\) −1.76580e18 −0.0611987
\(987\) −5.06062e18 −0.174238
\(988\) −1.38676e19 −0.474331
\(989\) 5.90941e18 0.200802
\(990\) 0 0
\(991\) −3.43064e19 −1.15053 −0.575263 0.817969i \(-0.695100\pi\)
−0.575263 + 0.817969i \(0.695100\pi\)
\(992\) 1.45333e19 0.484214
\(993\) 8.47329e18 0.280467
\(994\) −3.70781e18 −0.121929
\(995\) 0 0
\(996\) 6.59465e18 0.214045
\(997\) 5.72458e19 1.84597 0.922985 0.384835i \(-0.125742\pi\)
0.922985 + 0.384835i \(0.125742\pi\)
\(998\) 4.79589e18 0.153646
\(999\) 2.13011e18 0.0677995
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 75.14.a.a.1.1 1
5.2 odd 4 75.14.b.b.49.2 2
5.3 odd 4 75.14.b.b.49.1 2
5.4 even 2 3.14.a.a.1.1 1
15.14 odd 2 9.14.a.a.1.1 1
20.19 odd 2 48.14.a.c.1.1 1
35.34 odd 2 147.14.a.a.1.1 1
40.19 odd 2 192.14.a.e.1.1 1
40.29 even 2 192.14.a.j.1.1 1
60.59 even 2 144.14.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.14.a.a.1.1 1 5.4 even 2
9.14.a.a.1.1 1 15.14 odd 2
48.14.a.c.1.1 1 20.19 odd 2
75.14.a.a.1.1 1 1.1 even 1 trivial
75.14.b.b.49.1 2 5.3 odd 4
75.14.b.b.49.2 2 5.2 odd 4
144.14.a.k.1.1 1 60.59 even 2
147.14.a.a.1.1 1 35.34 odd 2
192.14.a.e.1.1 1 40.19 odd 2
192.14.a.j.1.1 1 40.29 even 2