Properties

Label 75.22.b.b.49.1
Level $75$
Weight $22$
Character 75.49
Analytic conductor $209.608$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,22,Mod(49,75)]
 
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, 1]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.49");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(209.608008215\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.22.b.b.49.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1728.00i q^{2} -59049.0i q^{3} -888832. q^{4} -1.02037e8 q^{6} -5.38430e8i q^{7} -2.08798e9i q^{8} -3.48678e9 q^{9} +O(q^{10})\) \(q-1728.00i q^{2} -59049.0i q^{3} -888832. q^{4} -1.02037e8 q^{6} -5.38430e8i q^{7} -2.08798e9i q^{8} -3.48678e9 q^{9} -6.41130e10 q^{11} +5.24846e10i q^{12} -1.30980e11i q^{13} -9.30407e11 q^{14} -5.47204e12 q^{16} -8.24203e12i q^{17} +6.02516e12i q^{18} -1.34921e13 q^{19} -3.17937e13 q^{21} +1.10787e14i q^{22} -2.33185e14i q^{23} -1.23293e14 q^{24} -2.26334e14 q^{26} +2.05891e14i q^{27} +4.78574e14i q^{28} +2.02456e15 q^{29} -6.86919e15 q^{31} +5.07688e15i q^{32} +3.78581e15i q^{33} -1.42422e16 q^{34} +3.09917e15 q^{36} -3.44400e15i q^{37} +2.33144e16i q^{38} -7.73424e15 q^{39} -2.18424e16 q^{41} +5.49396e16i q^{42} -7.17928e16i q^{43} +5.69857e16 q^{44} -4.02943e17 q^{46} -2.83545e17i q^{47} +3.23118e17i q^{48} +2.68639e17 q^{49} -4.86684e17 q^{51} +1.16419e17i q^{52} -2.17229e18i q^{53} +3.55780e17 q^{54} -1.12423e18 q^{56} +7.96695e17i q^{57} -3.49844e18i q^{58} -1.53483e18 q^{59} +4.31159e18 q^{61} +1.18700e19i q^{62} +1.87739e18i q^{63} -2.70285e18 q^{64} +6.54188e18 q^{66} -9.24391e18i q^{67} +7.32578e18i q^{68} -1.37693e19 q^{69} -2.03874e19 q^{71} +7.28033e18i q^{72} +1.66178e19i q^{73} -5.95123e18 q^{74} +1.19922e19 q^{76} +3.45204e19i q^{77} +1.33648e19i q^{78} -6.79403e19 q^{79} +1.21577e19 q^{81} +3.77437e19i q^{82} +3.95037e19i q^{83} +2.82593e19 q^{84} -1.24058e20 q^{86} -1.19548e20i q^{87} +1.33867e20i q^{88} -4.16117e19 q^{89} -7.05236e19 q^{91} +2.07262e20i q^{92} +4.05619e20i q^{93} -4.89965e20 q^{94} +2.99785e20 q^{96} -5.71815e19i q^{97} -4.64209e20i q^{98} +2.23548e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1777664 q^{4} - 204073344 q^{6} - 6973568802 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 1777664 q^{4} - 204073344 q^{6} - 6973568802 q^{9} - 128226080376 q^{11} - 1860813416448 q^{14} - 10944079986688 q^{16} - 26984203506040 q^{19} - 63587483465184 q^{21} - 246585903022080 q^{24} - 452667253199616 q^{26} + 40\!\cdots\!40 q^{29}+ \cdots + 44\!\cdots\!76 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1728.00i − 1.19324i −0.802523 0.596621i \(-0.796509\pi\)
0.802523 0.596621i \(-0.203491\pi\)
\(3\) − 59049.0i − 0.577350i
\(4\) −888832. −0.423828
\(5\) 0 0
\(6\) −1.02037e8 −0.688919
\(7\) − 5.38430e8i − 0.720443i −0.932867 0.360222i \(-0.882701\pi\)
0.932867 0.360222i \(-0.117299\pi\)
\(8\) − 2.08798e9i − 0.687513i
\(9\) −3.48678e9 −0.333333
\(10\) 0 0
\(11\) −6.41130e10 −0.745286 −0.372643 0.927975i \(-0.621548\pi\)
−0.372643 + 0.927975i \(0.621548\pi\)
\(12\) 5.24846e10i 0.244697i
\(13\) − 1.30980e11i − 0.263512i −0.991282 0.131756i \(-0.957938\pi\)
0.991282 0.131756i \(-0.0420615\pi\)
\(14\) −9.30407e11 −0.859663
\(15\) 0 0
\(16\) −5.47204e12 −1.24420
\(17\) − 8.24203e12i − 0.991563i −0.868447 0.495782i \(-0.834882\pi\)
0.868447 0.495782i \(-0.165118\pi\)
\(18\) 6.02516e12i 0.397748i
\(19\) −1.34921e13 −0.504855 −0.252428 0.967616i \(-0.581229\pi\)
−0.252428 + 0.967616i \(0.581229\pi\)
\(20\) 0 0
\(21\) −3.17937e13 −0.415948
\(22\) 1.10787e14i 0.889308i
\(23\) − 2.33185e14i − 1.17370i −0.809695 0.586851i \(-0.800367\pi\)
0.809695 0.586851i \(-0.199633\pi\)
\(24\) −1.23293e14 −0.396936
\(25\) 0 0
\(26\) −2.26334e14 −0.314434
\(27\) 2.05891e14i 0.192450i
\(28\) 4.78574e14i 0.305344i
\(29\) 2.02456e15 0.893618 0.446809 0.894629i \(-0.352560\pi\)
0.446809 + 0.894629i \(0.352560\pi\)
\(30\) 0 0
\(31\) −6.86919e15 −1.50525 −0.752624 0.658451i \(-0.771212\pi\)
−0.752624 + 0.658451i \(0.771212\pi\)
\(32\) 5.07688e15i 0.797117i
\(33\) 3.78581e15i 0.430291i
\(34\) −1.42422e16 −1.18318
\(35\) 0 0
\(36\) 3.09917e15 0.141276
\(37\) − 3.44400e15i − 0.117746i −0.998265 0.0588728i \(-0.981249\pi\)
0.998265 0.0588728i \(-0.0187506\pi\)
\(38\) 2.33144e16i 0.602415i
\(39\) −7.73424e15 −0.152139
\(40\) 0 0
\(41\) −2.18424e16 −0.254138 −0.127069 0.991894i \(-0.540557\pi\)
−0.127069 + 0.991894i \(0.540557\pi\)
\(42\) 5.49396e16i 0.496327i
\(43\) − 7.17928e16i − 0.506597i −0.967388 0.253298i \(-0.918485\pi\)
0.967388 0.253298i \(-0.0815154\pi\)
\(44\) 5.69857e16 0.315873
\(45\) 0 0
\(46\) −4.02943e17 −1.40051
\(47\) − 2.83545e17i − 0.786310i −0.919472 0.393155i \(-0.871383\pi\)
0.919472 0.393155i \(-0.128617\pi\)
\(48\) 3.23118e17i 0.718338i
\(49\) 2.68639e17 0.480962
\(50\) 0 0
\(51\) −4.86684e17 −0.572479
\(52\) 1.16419e17i 0.111684i
\(53\) − 2.17229e18i − 1.70616i −0.521779 0.853081i \(-0.674731\pi\)
0.521779 0.853081i \(-0.325269\pi\)
\(54\) 3.55780e17 0.229640
\(55\) 0 0
\(56\) −1.12423e18 −0.495314
\(57\) 7.96695e17i 0.291478i
\(58\) − 3.49844e18i − 1.06630i
\(59\) −1.53483e18 −0.390944 −0.195472 0.980709i \(-0.562624\pi\)
−0.195472 + 0.980709i \(0.562624\pi\)
\(60\) 0 0
\(61\) 4.31159e18 0.773881 0.386940 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(62\) 1.18700e19i 1.79613i
\(63\) 1.87739e18i 0.240148i
\(64\) −2.70285e18 −0.293044
\(65\) 0 0
\(66\) 6.54188e18 0.513442
\(67\) − 9.24391e18i − 0.619541i −0.950811 0.309771i \(-0.899748\pi\)
0.950811 0.309771i \(-0.100252\pi\)
\(68\) 7.32578e18i 0.420252i
\(69\) −1.37693e19 −0.677637
\(70\) 0 0
\(71\) −2.03874e19 −0.743273 −0.371636 0.928378i \(-0.621203\pi\)
−0.371636 + 0.928378i \(0.621203\pi\)
\(72\) 7.28033e18i 0.229171i
\(73\) 1.66178e19i 0.452566i 0.974062 + 0.226283i \(0.0726575\pi\)
−0.974062 + 0.226283i \(0.927343\pi\)
\(74\) −5.95123e18 −0.140499
\(75\) 0 0
\(76\) 1.19922e19 0.213972
\(77\) 3.45204e19i 0.536936i
\(78\) 1.33648e19i 0.181538i
\(79\) −6.79403e19 −0.807315 −0.403658 0.914910i \(-0.632261\pi\)
−0.403658 + 0.914910i \(0.632261\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) 3.77437e19i 0.303249i
\(83\) 3.95037e19i 0.279459i 0.990190 + 0.139730i \(0.0446233\pi\)
−0.990190 + 0.139730i \(0.955377\pi\)
\(84\) 2.82593e19 0.176290
\(85\) 0 0
\(86\) −1.24058e20 −0.604493
\(87\) − 1.19548e20i − 0.515931i
\(88\) 1.33867e20i 0.512394i
\(89\) −4.16117e19 −0.141456 −0.0707278 0.997496i \(-0.522532\pi\)
−0.0707278 + 0.997496i \(0.522532\pi\)
\(90\) 0 0
\(91\) −7.05236e19 −0.189845
\(92\) 2.07262e20i 0.497448i
\(93\) 4.05619e20i 0.869055i
\(94\) −4.89965e20 −0.938259
\(95\) 0 0
\(96\) 2.99785e20 0.460216
\(97\) − 5.71815e19i − 0.0787322i −0.999225 0.0393661i \(-0.987466\pi\)
0.999225 0.0393661i \(-0.0125339\pi\)
\(98\) − 4.64209e20i − 0.573904i
\(99\) 2.23548e20 0.248429
\(100\) 0 0
\(101\) 4.32417e20 0.389518 0.194759 0.980851i \(-0.437607\pi\)
0.194759 + 0.980851i \(0.437607\pi\)
\(102\) 8.40989e20i 0.683107i
\(103\) 1.84123e21i 1.34995i 0.737841 + 0.674974i \(0.235845\pi\)
−0.737841 + 0.674974i \(0.764155\pi\)
\(104\) −2.73483e20 −0.181168
\(105\) 0 0
\(106\) −3.75371e21 −2.03587
\(107\) 2.43805e21i 1.19815i 0.800691 + 0.599077i \(0.204466\pi\)
−0.800691 + 0.599077i \(0.795534\pi\)
\(108\) − 1.83003e20i − 0.0815658i
\(109\) 4.13676e21 1.67372 0.836859 0.547418i \(-0.184389\pi\)
0.836859 + 0.547418i \(0.184389\pi\)
\(110\) 0 0
\(111\) −2.03365e20 −0.0679805
\(112\) 2.94631e21i 0.896374i
\(113\) 3.47910e21i 0.964146i 0.876131 + 0.482073i \(0.160116\pi\)
−0.876131 + 0.482073i \(0.839884\pi\)
\(114\) 1.37669e21 0.347804
\(115\) 0 0
\(116\) −1.79950e21 −0.378741
\(117\) 4.56699e20i 0.0878373i
\(118\) 2.65219e21i 0.466491i
\(119\) −4.43775e21 −0.714365
\(120\) 0 0
\(121\) −3.28977e21 −0.444548
\(122\) − 7.45043e21i − 0.923428i
\(123\) 1.28977e21i 0.146727i
\(124\) 6.10556e21 0.637966
\(125\) 0 0
\(126\) 3.24413e21 0.286554
\(127\) − 1.37141e21i − 0.111488i −0.998445 0.0557438i \(-0.982247\pi\)
0.998445 0.0557438i \(-0.0177530\pi\)
\(128\) 1.53175e22i 1.14679i
\(129\) −4.23929e21 −0.292484
\(130\) 0 0
\(131\) −2.45276e22 −1.43981 −0.719907 0.694071i \(-0.755815\pi\)
−0.719907 + 0.694071i \(0.755815\pi\)
\(132\) − 3.36495e21i − 0.182370i
\(133\) 7.26455e21i 0.363719i
\(134\) −1.59735e22 −0.739263
\(135\) 0 0
\(136\) −1.72092e22 −0.681713
\(137\) − 1.02835e22i − 0.377204i −0.982054 0.188602i \(-0.939604\pi\)
0.982054 0.188602i \(-0.0603955\pi\)
\(138\) 2.37934e22i 0.808586i
\(139\) −8.70692e21 −0.274289 −0.137145 0.990551i \(-0.543793\pi\)
−0.137145 + 0.990551i \(0.543793\pi\)
\(140\) 0 0
\(141\) −1.67430e22 −0.453977
\(142\) 3.52294e22i 0.886905i
\(143\) 8.39753e21i 0.196392i
\(144\) 1.90798e22 0.414733
\(145\) 0 0
\(146\) 2.87155e22 0.540022
\(147\) − 1.58629e22i − 0.277683i
\(148\) 3.06114e21i 0.0499039i
\(149\) 9.03997e22 1.37313 0.686564 0.727069i \(-0.259118\pi\)
0.686564 + 0.727069i \(0.259118\pi\)
\(150\) 0 0
\(151\) −4.75206e22 −0.627514 −0.313757 0.949503i \(-0.601588\pi\)
−0.313757 + 0.949503i \(0.601588\pi\)
\(152\) 2.81712e22i 0.347094i
\(153\) 2.87382e22i 0.330521i
\(154\) 5.96512e22 0.640695
\(155\) 0 0
\(156\) 6.87444e21 0.0644806
\(157\) 1.50901e23i 1.32356i 0.749697 + 0.661781i \(0.230199\pi\)
−0.749697 + 0.661781i \(0.769801\pi\)
\(158\) 1.17401e23i 0.963323i
\(159\) −1.28271e23 −0.985053
\(160\) 0 0
\(161\) −1.25554e23 −0.845586
\(162\) − 2.10084e22i − 0.132583i
\(163\) − 4.83503e22i − 0.286042i −0.989720 0.143021i \(-0.954318\pi\)
0.989720 0.143021i \(-0.0456816\pi\)
\(164\) 1.94142e22 0.107711
\(165\) 0 0
\(166\) 6.82624e22 0.333462
\(167\) − 4.78731e20i − 0.00219568i −0.999999 0.00109784i \(-0.999651\pi\)
0.999999 0.00109784i \(-0.000349453\pi\)
\(168\) 6.63846e22i 0.285970i
\(169\) 2.29909e23 0.930562
\(170\) 0 0
\(171\) 4.70440e22 0.168285
\(172\) 6.38118e22i 0.214710i
\(173\) − 1.61804e23i − 0.512277i −0.966640 0.256139i \(-0.917550\pi\)
0.966640 0.256139i \(-0.0824504\pi\)
\(174\) −2.06580e23 −0.615631
\(175\) 0 0
\(176\) 3.50829e23 0.927284
\(177\) 9.06303e22i 0.225712i
\(178\) 7.19050e22i 0.168791i
\(179\) 8.76377e22 0.193970 0.0969849 0.995286i \(-0.469080\pi\)
0.0969849 + 0.995286i \(0.469080\pi\)
\(180\) 0 0
\(181\) 9.36624e22 0.184476 0.0922381 0.995737i \(-0.470598\pi\)
0.0922381 + 0.995737i \(0.470598\pi\)
\(182\) 1.21865e23i 0.226532i
\(183\) − 2.54595e23i − 0.446800i
\(184\) −4.86885e23 −0.806936
\(185\) 0 0
\(186\) 7.00910e23 1.03699
\(187\) 5.28422e23i 0.738999i
\(188\) 2.52024e23i 0.333260i
\(189\) 1.10858e23 0.138649
\(190\) 0 0
\(191\) 1.20858e24 1.35340 0.676699 0.736260i \(-0.263410\pi\)
0.676699 + 0.736260i \(0.263410\pi\)
\(192\) 1.59601e23i 0.169189i
\(193\) − 1.78822e24i − 1.79502i −0.440997 0.897509i \(-0.645375\pi\)
0.440997 0.897509i \(-0.354625\pi\)
\(194\) −9.88096e22 −0.0939466
\(195\) 0 0
\(196\) −2.38775e23 −0.203845
\(197\) − 1.90963e24i − 1.54545i −0.634743 0.772723i \(-0.718894\pi\)
0.634743 0.772723i \(-0.281106\pi\)
\(198\) − 3.86292e23i − 0.296436i
\(199\) −1.44254e24 −1.04995 −0.524977 0.851116i \(-0.675926\pi\)
−0.524977 + 0.851116i \(0.675926\pi\)
\(200\) 0 0
\(201\) −5.45844e23 −0.357692
\(202\) − 7.47216e23i − 0.464790i
\(203\) − 1.09008e24i − 0.643801i
\(204\) 4.32580e23 0.242633
\(205\) 0 0
\(206\) 3.18165e24 1.61082
\(207\) 8.13065e23i 0.391234i
\(208\) 7.16728e23i 0.327861i
\(209\) 8.65020e23 0.376262
\(210\) 0 0
\(211\) 3.98848e24 1.56979 0.784895 0.619629i \(-0.212717\pi\)
0.784895 + 0.619629i \(0.212717\pi\)
\(212\) 1.93080e24i 0.723119i
\(213\) 1.20385e24i 0.429129i
\(214\) 4.21295e24 1.42969
\(215\) 0 0
\(216\) 4.29896e23 0.132312
\(217\) 3.69858e24i 1.08445i
\(218\) − 7.14833e24i − 1.99715i
\(219\) 9.81262e23 0.261289
\(220\) 0 0
\(221\) −1.07954e24 −0.261289
\(222\) 3.51414e23i 0.0811172i
\(223\) − 4.62963e24i − 1.01940i −0.860352 0.509700i \(-0.829756\pi\)
0.860352 0.509700i \(-0.170244\pi\)
\(224\) 2.73354e24 0.574278
\(225\) 0 0
\(226\) 6.01188e24 1.15046
\(227\) 3.43010e24i 0.626664i 0.949644 + 0.313332i \(0.101445\pi\)
−0.949644 + 0.313332i \(0.898555\pi\)
\(228\) − 7.08128e23i − 0.123537i
\(229\) −8.11792e23 −0.135261 −0.0676304 0.997710i \(-0.521544\pi\)
−0.0676304 + 0.997710i \(0.521544\pi\)
\(230\) 0 0
\(231\) 2.03839e24 0.310000
\(232\) − 4.22724e24i − 0.614374i
\(233\) 8.22188e23i 0.114218i 0.998368 + 0.0571089i \(0.0181882\pi\)
−0.998368 + 0.0571089i \(0.981812\pi\)
\(234\) 7.89177e23 0.104811
\(235\) 0 0
\(236\) 1.36421e24 0.165693
\(237\) 4.01181e24i 0.466104i
\(238\) 7.66844e24i 0.852411i
\(239\) 8.85525e24 0.941940 0.470970 0.882149i \(-0.343904\pi\)
0.470970 + 0.882149i \(0.343904\pi\)
\(240\) 0 0
\(241\) 7.46934e24 0.727953 0.363977 0.931408i \(-0.381419\pi\)
0.363977 + 0.931408i \(0.381419\pi\)
\(242\) 5.68472e24i 0.530454i
\(243\) − 7.17898e23i − 0.0641500i
\(244\) −3.83228e24 −0.327992
\(245\) 0 0
\(246\) 2.22873e24 0.175081
\(247\) 1.76720e24i 0.133035i
\(248\) 1.43427e25i 1.03488i
\(249\) 2.33266e24 0.161346
\(250\) 0 0
\(251\) 9.46474e23 0.0601914 0.0300957 0.999547i \(-0.490419\pi\)
0.0300957 + 0.999547i \(0.490419\pi\)
\(252\) − 1.66868e24i − 0.101781i
\(253\) 1.49502e25i 0.874744i
\(254\) −2.36979e24 −0.133032
\(255\) 0 0
\(256\) 2.08004e25 1.07535
\(257\) 1.91825e25i 0.951936i 0.879463 + 0.475968i \(0.157902\pi\)
−0.879463 + 0.475968i \(0.842098\pi\)
\(258\) 7.32550e24i 0.349004i
\(259\) −1.85435e24 −0.0848290
\(260\) 0 0
\(261\) −7.05921e24 −0.297873
\(262\) 4.23837e25i 1.71805i
\(263\) 8.88429e23i 0.0346009i 0.999850 + 0.0173004i \(0.00550718\pi\)
−0.999850 + 0.0173004i \(0.994493\pi\)
\(264\) 7.90469e24 0.295831
\(265\) 0 0
\(266\) 1.25531e25 0.434006
\(267\) 2.45713e24i 0.0816694i
\(268\) 8.21628e24i 0.262579i
\(269\) 2.13847e25 0.657211 0.328605 0.944467i \(-0.393421\pi\)
0.328605 + 0.944467i \(0.393421\pi\)
\(270\) 0 0
\(271\) −1.56435e25 −0.444791 −0.222395 0.974957i \(-0.571388\pi\)
−0.222395 + 0.974957i \(0.571388\pi\)
\(272\) 4.51007e25i 1.23370i
\(273\) 4.16435e24i 0.109607i
\(274\) −1.77699e25 −0.450095
\(275\) 0 0
\(276\) 1.22386e25 0.287202
\(277\) 8.04973e25i 1.81863i 0.416112 + 0.909313i \(0.363392\pi\)
−0.416112 + 0.909313i \(0.636608\pi\)
\(278\) 1.50456e25i 0.327294i
\(279\) 2.39514e25 0.501749
\(280\) 0 0
\(281\) 8.33171e25 1.61926 0.809632 0.586938i \(-0.199667\pi\)
0.809632 + 0.586938i \(0.199667\pi\)
\(282\) 2.89320e25i 0.541704i
\(283\) − 4.46130e24i − 0.0804829i −0.999190 0.0402415i \(-0.987187\pi\)
0.999190 0.0402415i \(-0.0128127\pi\)
\(284\) 1.81209e25 0.315020
\(285\) 0 0
\(286\) 1.45109e25 0.234343
\(287\) 1.17606e25i 0.183092i
\(288\) − 1.77020e25i − 0.265706i
\(289\) 1.16088e24 0.0168020
\(290\) 0 0
\(291\) −3.37651e24 −0.0454560
\(292\) − 1.47704e25i − 0.191810i
\(293\) 9.67128e25i 1.21164i 0.795602 + 0.605820i \(0.207155\pi\)
−0.795602 + 0.605820i \(0.792845\pi\)
\(294\) −2.74111e25 −0.331344
\(295\) 0 0
\(296\) −7.19099e24 −0.0809516
\(297\) − 1.32003e25i − 0.143430i
\(298\) − 1.56211e26i − 1.63848i
\(299\) −3.05426e25 −0.309284
\(300\) 0 0
\(301\) −3.86554e25 −0.364974
\(302\) 8.21155e25i 0.748777i
\(303\) − 2.55338e25i − 0.224889i
\(304\) 7.38293e25 0.628140
\(305\) 0 0
\(306\) 4.96596e25 0.394392
\(307\) − 1.68163e26i − 1.29056i −0.763948 0.645278i \(-0.776742\pi\)
0.763948 0.645278i \(-0.223258\pi\)
\(308\) − 3.06828e25i − 0.227569i
\(309\) 1.08723e26 0.779393
\(310\) 0 0
\(311\) 2.30370e26 1.54327 0.771636 0.636065i \(-0.219439\pi\)
0.771636 + 0.636065i \(0.219439\pi\)
\(312\) 1.61489e25i 0.104597i
\(313\) − 2.79658e26i − 1.75151i −0.482759 0.875753i \(-0.660365\pi\)
0.482759 0.875753i \(-0.339635\pi\)
\(314\) 2.60756e26 1.57933
\(315\) 0 0
\(316\) 6.03875e25 0.342163
\(317\) − 2.98501e25i − 0.163615i −0.996648 0.0818075i \(-0.973931\pi\)
0.996648 0.0818075i \(-0.0260693\pi\)
\(318\) 2.21653e26i 1.17541i
\(319\) −1.29801e26 −0.666002
\(320\) 0 0
\(321\) 1.43964e26 0.691755
\(322\) 2.16957e26i 1.00899i
\(323\) 1.11202e26i 0.500596i
\(324\) −1.08061e25 −0.0470920
\(325\) 0 0
\(326\) −8.35494e25 −0.341317
\(327\) − 2.44272e26i − 0.966322i
\(328\) 4.56064e25i 0.174723i
\(329\) −1.52669e26 −0.566492
\(330\) 0 0
\(331\) 2.55594e26 0.889933 0.444967 0.895547i \(-0.353216\pi\)
0.444967 + 0.895547i \(0.353216\pi\)
\(332\) − 3.51122e25i − 0.118443i
\(333\) 1.20085e25i 0.0392485i
\(334\) −8.27248e23 −0.00261998
\(335\) 0 0
\(336\) 1.73977e26 0.517522
\(337\) 4.91931e25i 0.141837i 0.997482 + 0.0709187i \(0.0225931\pi\)
−0.997482 + 0.0709187i \(0.977407\pi\)
\(338\) − 3.97282e26i − 1.11039i
\(339\) 2.05437e26 0.556650
\(340\) 0 0
\(341\) 4.40405e26 1.12184
\(342\) − 8.12921e25i − 0.200805i
\(343\) − 4.45381e26i − 1.06695i
\(344\) −1.49902e26 −0.348292
\(345\) 0 0
\(346\) −2.79597e26 −0.611271
\(347\) − 2.98136e26i − 0.632345i −0.948702 0.316173i \(-0.897602\pi\)
0.948702 0.316173i \(-0.102398\pi\)
\(348\) 1.06258e26i 0.218666i
\(349\) −7.72834e26 −1.54319 −0.771595 0.636115i \(-0.780541\pi\)
−0.771595 + 0.636115i \(0.780541\pi\)
\(350\) 0 0
\(351\) 2.69676e25 0.0507129
\(352\) − 3.25494e26i − 0.594081i
\(353\) − 7.30755e26i − 1.29461i −0.762233 0.647303i \(-0.775897\pi\)
0.762233 0.647303i \(-0.224103\pi\)
\(354\) 1.56609e26 0.269329
\(355\) 0 0
\(356\) 3.69858e25 0.0599529
\(357\) 2.62045e26i 0.412439i
\(358\) − 1.51438e26i − 0.231453i
\(359\) 1.58936e25 0.0235901 0.0117951 0.999930i \(-0.496245\pi\)
0.0117951 + 0.999930i \(0.496245\pi\)
\(360\) 0 0
\(361\) −5.32173e26 −0.745121
\(362\) − 1.61849e26i − 0.220125i
\(363\) 1.94258e26i 0.256660i
\(364\) 6.26836e25 0.0804618
\(365\) 0 0
\(366\) −4.39940e26 −0.533141
\(367\) 1.40734e27i 1.65732i 0.559752 + 0.828660i \(0.310896\pi\)
−0.559752 + 0.828660i \(0.689104\pi\)
\(368\) 1.27600e27i 1.46032i
\(369\) 7.61598e25 0.0847127
\(370\) 0 0
\(371\) −1.16962e27 −1.22919
\(372\) − 3.60527e26i − 0.368330i
\(373\) − 9.30077e26i − 0.923797i −0.886933 0.461898i \(-0.847169\pi\)
0.886933 0.461898i \(-0.152831\pi\)
\(374\) 9.13112e26 0.881805
\(375\) 0 0
\(376\) −5.92035e26 −0.540599
\(377\) − 2.65177e26i − 0.235479i
\(378\) − 1.91562e26i − 0.165442i
\(379\) −2.18541e27 −1.83578 −0.917892 0.396830i \(-0.870110\pi\)
−0.917892 + 0.396830i \(0.870110\pi\)
\(380\) 0 0
\(381\) −8.09801e25 −0.0643674
\(382\) − 2.08843e27i − 1.61493i
\(383\) 2.10347e27i 1.58252i 0.611482 + 0.791258i \(0.290574\pi\)
−0.611482 + 0.791258i \(0.709426\pi\)
\(384\) 9.04484e26 0.662099
\(385\) 0 0
\(386\) −3.09004e27 −2.14189
\(387\) 2.50326e26i 0.168866i
\(388\) 5.08247e25i 0.0333689i
\(389\) 2.97815e26 0.190316 0.0951582 0.995462i \(-0.469664\pi\)
0.0951582 + 0.995462i \(0.469664\pi\)
\(390\) 0 0
\(391\) −1.92192e27 −1.16380
\(392\) − 5.60912e26i − 0.330667i
\(393\) 1.44833e27i 0.831277i
\(394\) −3.29984e27 −1.84409
\(395\) 0 0
\(396\) −1.98697e26 −0.105291
\(397\) − 6.36504e26i − 0.328474i −0.986421 0.164237i \(-0.947484\pi\)
0.986421 0.164237i \(-0.0525162\pi\)
\(398\) 2.49271e27i 1.25285i
\(399\) 4.28964e26 0.209993
\(400\) 0 0
\(401\) 2.43888e27 1.13286 0.566428 0.824111i \(-0.308325\pi\)
0.566428 + 0.824111i \(0.308325\pi\)
\(402\) 9.43218e26i 0.426814i
\(403\) 8.99728e26i 0.396651i
\(404\) −3.84346e26 −0.165089
\(405\) 0 0
\(406\) −1.88367e27 −0.768211
\(407\) 2.20805e26i 0.0877542i
\(408\) 1.01618e27i 0.393587i
\(409\) −5.48032e26 −0.206876 −0.103438 0.994636i \(-0.532984\pi\)
−0.103438 + 0.994636i \(0.532984\pi\)
\(410\) 0 0
\(411\) −6.07232e26 −0.217779
\(412\) − 1.63654e27i − 0.572146i
\(413\) 8.26399e26i 0.281653i
\(414\) 1.40498e27 0.466837
\(415\) 0 0
\(416\) 6.64970e26 0.210050
\(417\) 5.14135e26i 0.158361i
\(418\) − 1.49475e27i − 0.448971i
\(419\) 6.08246e27 1.78169 0.890844 0.454309i \(-0.150114\pi\)
0.890844 + 0.454309i \(0.150114\pi\)
\(420\) 0 0
\(421\) −4.05990e27 −1.13124 −0.565618 0.824667i \(-0.691362\pi\)
−0.565618 + 0.824667i \(0.691362\pi\)
\(422\) − 6.89209e27i − 1.87314i
\(423\) 9.88659e26i 0.262103i
\(424\) −4.53568e27 −1.17301
\(425\) 0 0
\(426\) 2.08026e27 0.512055
\(427\) − 2.32149e27i − 0.557537i
\(428\) − 2.16702e27i − 0.507811i
\(429\) 4.95866e26 0.113387
\(430\) 0 0
\(431\) −7.87214e27 −1.71428 −0.857140 0.515084i \(-0.827761\pi\)
−0.857140 + 0.515084i \(0.827761\pi\)
\(432\) − 1.12664e27i − 0.239446i
\(433\) − 1.73785e27i − 0.360486i −0.983622 0.180243i \(-0.942312\pi\)
0.983622 0.180243i \(-0.0576884\pi\)
\(434\) 6.39115e27 1.29401
\(435\) 0 0
\(436\) −3.67689e27 −0.709369
\(437\) 3.14615e27i 0.592550i
\(438\) − 1.69562e27i − 0.311782i
\(439\) −8.37416e27 −1.50336 −0.751681 0.659526i \(-0.770757\pi\)
−0.751681 + 0.659526i \(0.770757\pi\)
\(440\) 0 0
\(441\) −9.36687e26 −0.160321
\(442\) 1.86545e27i 0.311781i
\(443\) − 3.30286e25i − 0.00539077i −0.999996 0.00269539i \(-0.999142\pi\)
0.999996 0.00269539i \(-0.000857969\pi\)
\(444\) 1.80757e26 0.0288120
\(445\) 0 0
\(446\) −7.99999e27 −1.21639
\(447\) − 5.33801e27i − 0.792776i
\(448\) 1.45530e27i 0.211121i
\(449\) −5.21713e27 −0.739341 −0.369670 0.929163i \(-0.620529\pi\)
−0.369670 + 0.929163i \(0.620529\pi\)
\(450\) 0 0
\(451\) 1.40038e27 0.189406
\(452\) − 3.09233e27i − 0.408632i
\(453\) 2.80604e27i 0.362296i
\(454\) 5.92721e27 0.747763
\(455\) 0 0
\(456\) 1.66348e27 0.200395
\(457\) − 2.15211e26i − 0.0253363i −0.999920 0.0126682i \(-0.995967\pi\)
0.999920 0.0126682i \(-0.00403251\pi\)
\(458\) 1.40278e27i 0.161399i
\(459\) 1.69696e27 0.190826
\(460\) 0 0
\(461\) 1.68699e28 1.81239 0.906197 0.422855i \(-0.138972\pi\)
0.906197 + 0.422855i \(0.138972\pi\)
\(462\) − 3.52234e27i − 0.369906i
\(463\) − 1.90352e28i − 1.95415i −0.212898 0.977074i \(-0.568290\pi\)
0.212898 0.977074i \(-0.431710\pi\)
\(464\) −1.10785e28 −1.11184
\(465\) 0 0
\(466\) 1.42074e27 0.136290
\(467\) 1.21027e28i 1.13515i 0.823321 + 0.567576i \(0.192119\pi\)
−0.823321 + 0.567576i \(0.807881\pi\)
\(468\) − 4.05929e26i − 0.0372279i
\(469\) −4.97720e27 −0.446344
\(470\) 0 0
\(471\) 8.91053e27 0.764159
\(472\) 3.20469e27i 0.268779i
\(473\) 4.60286e27i 0.377559i
\(474\) 6.93240e27 0.556175
\(475\) 0 0
\(476\) 3.94442e27 0.302768
\(477\) 7.57429e27i 0.568721i
\(478\) − 1.53019e28i − 1.12396i
\(479\) −6.95253e27 −0.499597 −0.249798 0.968298i \(-0.580364\pi\)
−0.249798 + 0.968298i \(0.580364\pi\)
\(480\) 0 0
\(481\) −4.51095e26 −0.0310274
\(482\) − 1.29070e28i − 0.868625i
\(483\) 7.41382e27i 0.488199i
\(484\) 2.92405e27 0.188412
\(485\) 0 0
\(486\) −1.24053e27 −0.0765466
\(487\) 1.06412e28i 0.642596i 0.946978 + 0.321298i \(0.104119\pi\)
−0.946978 + 0.321298i \(0.895881\pi\)
\(488\) − 9.00250e27i − 0.532053i
\(489\) −2.85504e27 −0.165146
\(490\) 0 0
\(491\) 1.68064e28 0.931361 0.465681 0.884953i \(-0.345810\pi\)
0.465681 + 0.884953i \(0.345810\pi\)
\(492\) − 1.14639e27i − 0.0621869i
\(493\) − 1.66865e28i − 0.886079i
\(494\) 3.05372e27 0.158743
\(495\) 0 0
\(496\) 3.75885e28 1.87283
\(497\) 1.09772e28i 0.535486i
\(498\) − 4.03083e27i − 0.192525i
\(499\) 5.12285e27 0.239583 0.119792 0.992799i \(-0.461777\pi\)
0.119792 + 0.992799i \(0.461777\pi\)
\(500\) 0 0
\(501\) −2.82686e25 −0.00126768
\(502\) − 1.63551e27i − 0.0718229i
\(503\) 1.99606e28i 0.858442i 0.903200 + 0.429221i \(0.141212\pi\)
−0.903200 + 0.429221i \(0.858788\pi\)
\(504\) 3.91994e27 0.165105
\(505\) 0 0
\(506\) 2.58339e28 1.04378
\(507\) − 1.35759e28i − 0.537260i
\(508\) 1.21895e27i 0.0472516i
\(509\) 2.57966e27 0.0979550 0.0489775 0.998800i \(-0.484404\pi\)
0.0489775 + 0.998800i \(0.484404\pi\)
\(510\) 0 0
\(511\) 8.94749e27 0.326048
\(512\) − 3.81989e27i − 0.136369i
\(513\) − 2.77790e27i − 0.0971594i
\(514\) 3.31474e28 1.13589
\(515\) 0 0
\(516\) 3.76802e27 0.123963
\(517\) 1.81789e28i 0.586026i
\(518\) 3.20432e27i 0.101222i
\(519\) −9.55436e27 −0.295763
\(520\) 0 0
\(521\) −2.61230e28 −0.776652 −0.388326 0.921522i \(-0.626947\pi\)
−0.388326 + 0.921522i \(0.626947\pi\)
\(522\) 1.21983e28i 0.355435i
\(523\) 6.70750e28i 1.91555i 0.287523 + 0.957774i \(0.407168\pi\)
−0.287523 + 0.957774i \(0.592832\pi\)
\(524\) 2.18009e28 0.610233
\(525\) 0 0
\(526\) 1.53520e27 0.0412873
\(527\) 5.66161e28i 1.49255i
\(528\) − 2.07161e28i − 0.535367i
\(529\) −1.49036e28 −0.377577
\(530\) 0 0
\(531\) 5.35163e27 0.130315
\(532\) − 6.45696e27i − 0.154155i
\(533\) 2.86092e27i 0.0669684i
\(534\) 4.24592e27 0.0974514
\(535\) 0 0
\(536\) −1.93011e28 −0.425943
\(537\) − 5.17492e27i − 0.111988i
\(538\) − 3.69528e28i − 0.784212i
\(539\) −1.72233e28 −0.358454
\(540\) 0 0
\(541\) −2.15196e28 −0.430787 −0.215394 0.976527i \(-0.569103\pi\)
−0.215394 + 0.976527i \(0.569103\pi\)
\(542\) 2.70319e28i 0.530743i
\(543\) − 5.53067e27i − 0.106507i
\(544\) 4.18438e28 0.790392
\(545\) 0 0
\(546\) 7.19599e27 0.130788
\(547\) 7.46789e28i 1.33147i 0.746189 + 0.665734i \(0.231882\pi\)
−0.746189 + 0.665734i \(0.768118\pi\)
\(548\) 9.14032e27i 0.159870i
\(549\) −1.50336e28 −0.257960
\(550\) 0 0
\(551\) −2.73156e28 −0.451148
\(552\) 2.87500e28i 0.465884i
\(553\) 3.65811e28i 0.581625i
\(554\) 1.39099e29 2.17006
\(555\) 0 0
\(556\) 7.73899e27 0.116252
\(557\) 7.95166e28i 1.17214i 0.810262 + 0.586068i \(0.199325\pi\)
−0.810262 + 0.586068i \(0.800675\pi\)
\(558\) − 4.13880e28i − 0.598709i
\(559\) −9.40343e27 −0.133494
\(560\) 0 0
\(561\) 3.12028e28 0.426661
\(562\) − 1.43972e29i − 1.93217i
\(563\) 5.46305e28i 0.719609i 0.933028 + 0.359805i \(0.117157\pi\)
−0.933028 + 0.359805i \(0.882843\pi\)
\(564\) 1.48817e28 0.192408
\(565\) 0 0
\(566\) −7.70912e27 −0.0960356
\(567\) − 6.54605e27i − 0.0800492i
\(568\) 4.25683e28i 0.511010i
\(569\) −9.43478e28 −1.11187 −0.555933 0.831227i \(-0.687639\pi\)
−0.555933 + 0.831227i \(0.687639\pi\)
\(570\) 0 0
\(571\) 8.05027e28 0.914390 0.457195 0.889367i \(-0.348854\pi\)
0.457195 + 0.889367i \(0.348854\pi\)
\(572\) − 7.46400e27i − 0.0832364i
\(573\) − 7.13655e28i − 0.781384i
\(574\) 2.03223e28 0.218473
\(575\) 0 0
\(576\) 9.42426e27 0.0976812
\(577\) 1.67132e28i 0.170104i 0.996377 + 0.0850519i \(0.0271056\pi\)
−0.996377 + 0.0850519i \(0.972894\pi\)
\(578\) − 2.00600e27i − 0.0200488i
\(579\) −1.05592e29 −1.03635
\(580\) 0 0
\(581\) 2.12700e28 0.201334
\(582\) 5.83461e27i 0.0542401i
\(583\) 1.39272e29i 1.27158i
\(584\) 3.46975e28 0.311145
\(585\) 0 0
\(586\) 1.67120e29 1.44578
\(587\) − 3.15730e28i − 0.268297i −0.990961 0.134149i \(-0.957170\pi\)
0.990961 0.134149i \(-0.0428300\pi\)
\(588\) 1.40994e28i 0.117690i
\(589\) 9.26799e28 0.759932
\(590\) 0 0
\(591\) −1.12762e29 −0.892264
\(592\) 1.88457e28i 0.146499i
\(593\) − 5.48493e27i − 0.0418887i −0.999781 0.0209443i \(-0.993333\pi\)
0.999781 0.0209443i \(-0.00666728\pi\)
\(594\) −2.28101e28 −0.171147
\(595\) 0 0
\(596\) −8.03502e28 −0.581971
\(597\) 8.51805e28i 0.606191i
\(598\) 5.27776e28i 0.369051i
\(599\) −1.25621e29 −0.863136 −0.431568 0.902080i \(-0.642040\pi\)
−0.431568 + 0.902080i \(0.642040\pi\)
\(600\) 0 0
\(601\) 3.99325e28 0.264938 0.132469 0.991187i \(-0.457709\pi\)
0.132469 + 0.991187i \(0.457709\pi\)
\(602\) 6.67965e28i 0.435503i
\(603\) 3.22315e28i 0.206514i
\(604\) 4.22378e28 0.265958
\(605\) 0 0
\(606\) −4.41223e28 −0.268347
\(607\) 2.46990e29i 1.47638i 0.674594 + 0.738189i \(0.264319\pi\)
−0.674594 + 0.738189i \(0.735681\pi\)
\(608\) − 6.84978e28i − 0.402429i
\(609\) −6.43684e28 −0.371699
\(610\) 0 0
\(611\) −3.71387e28 −0.207202
\(612\) − 2.55434e28i − 0.140084i
\(613\) 2.63911e28i 0.142273i 0.997467 + 0.0711364i \(0.0226626\pi\)
−0.997467 + 0.0711364i \(0.977337\pi\)
\(614\) −2.90585e29 −1.53995
\(615\) 0 0
\(616\) 7.20777e28 0.369151
\(617\) − 3.09820e29i − 1.55997i −0.625800 0.779984i \(-0.715227\pi\)
0.625800 0.779984i \(-0.284773\pi\)
\(618\) − 1.87873e29i − 0.930005i
\(619\) 2.50758e29 1.22040 0.610202 0.792246i \(-0.291088\pi\)
0.610202 + 0.792246i \(0.291088\pi\)
\(620\) 0 0
\(621\) 4.80107e28 0.225879
\(622\) − 3.98080e29i − 1.84150i
\(623\) 2.24050e28i 0.101911i
\(624\) 4.23221e28 0.189291
\(625\) 0 0
\(626\) −4.83249e29 −2.08997
\(627\) − 5.10785e28i − 0.217235i
\(628\) − 1.34125e29i − 0.560963i
\(629\) −2.83855e28 −0.116752
\(630\) 0 0
\(631\) −4.32770e28 −0.172167 −0.0860833 0.996288i \(-0.527435\pi\)
−0.0860833 + 0.996288i \(0.527435\pi\)
\(632\) 1.41858e29i 0.555039i
\(633\) − 2.35516e29i − 0.906318i
\(634\) −5.15809e28 −0.195232
\(635\) 0 0
\(636\) 1.14012e29 0.417493
\(637\) − 3.51864e28i − 0.126739i
\(638\) 2.24296e29i 0.794702i
\(639\) 7.10863e28 0.247758
\(640\) 0 0
\(641\) −8.73381e28 −0.294574 −0.147287 0.989094i \(-0.547054\pi\)
−0.147287 + 0.989094i \(0.547054\pi\)
\(642\) − 2.48770e29i − 0.825431i
\(643\) − 4.72013e29i − 1.54077i −0.637578 0.770386i \(-0.720063\pi\)
0.637578 0.770386i \(-0.279937\pi\)
\(644\) 1.11596e29 0.358383
\(645\) 0 0
\(646\) 1.92158e29 0.597332
\(647\) − 1.26799e28i − 0.0387812i −0.999812 0.0193906i \(-0.993827\pi\)
0.999812 0.0193906i \(-0.00617261\pi\)
\(648\) − 2.53849e28i − 0.0763903i
\(649\) 9.84027e28 0.291365
\(650\) 0 0
\(651\) 2.18397e29 0.626105
\(652\) 4.29753e28i 0.121233i
\(653\) − 2.76226e29i − 0.766790i −0.923584 0.383395i \(-0.874755\pi\)
0.923584 0.383395i \(-0.125245\pi\)
\(654\) −4.22102e29 −1.15306
\(655\) 0 0
\(656\) 1.19523e29 0.316198
\(657\) − 5.79425e28i − 0.150855i
\(658\) 2.63812e29i 0.675962i
\(659\) −6.46511e28 −0.163034 −0.0815172 0.996672i \(-0.525977\pi\)
−0.0815172 + 0.996672i \(0.525977\pi\)
\(660\) 0 0
\(661\) −2.27730e29 −0.556295 −0.278147 0.960538i \(-0.589720\pi\)
−0.278147 + 0.960538i \(0.589720\pi\)
\(662\) − 4.41667e29i − 1.06191i
\(663\) 6.37459e28i 0.150855i
\(664\) 8.24829e28 0.192132
\(665\) 0 0
\(666\) 2.07507e28 0.0468330
\(667\) − 4.72097e29i − 1.04884i
\(668\) 4.25512e26i 0 0.000930591i
\(669\) −2.73375e29 −0.588551
\(670\) 0 0
\(671\) −2.76429e29 −0.576763
\(672\) − 1.61413e29i − 0.331559i
\(673\) − 3.79243e29i − 0.766936i −0.923554 0.383468i \(-0.874730\pi\)
0.923554 0.383468i \(-0.125270\pi\)
\(674\) 8.50057e28 0.169246
\(675\) 0 0
\(676\) −2.04350e29 −0.394398
\(677\) 3.39717e29i 0.645559i 0.946474 + 0.322780i \(0.104617\pi\)
−0.946474 + 0.322780i \(0.895383\pi\)
\(678\) − 3.54995e29i − 0.664218i
\(679\) −3.07882e28 −0.0567220
\(680\) 0 0
\(681\) 2.02544e29 0.361805
\(682\) − 7.61020e29i − 1.33863i
\(683\) 4.54742e29i 0.787677i 0.919180 + 0.393838i \(0.128853\pi\)
−0.919180 + 0.393838i \(0.871147\pi\)
\(684\) −4.18143e28 −0.0713239
\(685\) 0 0
\(686\) −7.69619e29 −1.27313
\(687\) 4.79355e28i 0.0780929i
\(688\) 3.92853e29i 0.630306i
\(689\) −2.84526e29 −0.449594
\(690\) 0 0
\(691\) −3.71838e29 −0.569947 −0.284974 0.958535i \(-0.591985\pi\)
−0.284974 + 0.958535i \(0.591985\pi\)
\(692\) 1.43817e29i 0.217118i
\(693\) − 1.20365e29i − 0.178979i
\(694\) −5.15178e29 −0.754542
\(695\) 0 0
\(696\) −2.49614e29 −0.354709
\(697\) 1.80026e29i 0.251994i
\(698\) 1.33546e30i 1.84140i
\(699\) 4.85494e28 0.0659437
\(700\) 0 0
\(701\) −9.37969e29 −1.23637 −0.618186 0.786032i \(-0.712132\pi\)
−0.618186 + 0.786032i \(0.712132\pi\)
\(702\) − 4.66001e28i − 0.0605128i
\(703\) 4.64668e28i 0.0594445i
\(704\) 1.73288e29 0.218401
\(705\) 0 0
\(706\) −1.26275e30 −1.54478
\(707\) − 2.32826e29i − 0.280626i
\(708\) − 8.05551e28i − 0.0956629i
\(709\) 7.54578e29 0.882914 0.441457 0.897282i \(-0.354462\pi\)
0.441457 + 0.897282i \(0.354462\pi\)
\(710\) 0 0
\(711\) 2.36893e29 0.269105
\(712\) 8.68842e28i 0.0972525i
\(713\) 1.60179e30i 1.76671i
\(714\) 4.52814e29 0.492140
\(715\) 0 0
\(716\) −7.78952e28 −0.0822098
\(717\) − 5.22894e29i − 0.543829i
\(718\) − 2.74641e28i − 0.0281488i
\(719\) −1.22754e30 −1.23989 −0.619944 0.784646i \(-0.712845\pi\)
−0.619944 + 0.784646i \(0.712845\pi\)
\(720\) 0 0
\(721\) 9.91374e29 0.972561
\(722\) 9.19594e29i 0.889111i
\(723\) − 4.41057e29i − 0.420284i
\(724\) −8.32502e28 −0.0781862
\(725\) 0 0
\(726\) 3.35677e29 0.306258
\(727\) − 9.04407e29i − 0.813303i −0.913583 0.406652i \(-0.866696\pi\)
0.913583 0.406652i \(-0.133304\pi\)
\(728\) 1.47252e29i 0.130521i
\(729\) −4.23912e28 −0.0370370
\(730\) 0 0
\(731\) −5.91719e29 −0.502323
\(732\) 2.26292e29i 0.189367i
\(733\) − 1.38874e30i − 1.14559i −0.819699 0.572795i \(-0.805859\pi\)
0.819699 0.572795i \(-0.194141\pi\)
\(734\) 2.43189e30 1.97759
\(735\) 0 0
\(736\) 1.18385e30 0.935578
\(737\) 5.92655e29i 0.461736i
\(738\) − 1.31604e29i − 0.101083i
\(739\) −2.11506e30 −1.60161 −0.800804 0.598927i \(-0.795594\pi\)
−0.800804 + 0.598927i \(0.795594\pi\)
\(740\) 0 0
\(741\) 1.04351e29 0.0768080
\(742\) 2.02111e30i 1.46673i
\(743\) 2.26805e30i 1.62282i 0.584476 + 0.811411i \(0.301300\pi\)
−0.584476 + 0.811411i \(0.698700\pi\)
\(744\) 8.46923e29 0.597487
\(745\) 0 0
\(746\) −1.60717e30 −1.10231
\(747\) − 1.37741e29i − 0.0931530i
\(748\) − 4.69678e29i − 0.313208i
\(749\) 1.31272e30 0.863202
\(750\) 0 0
\(751\) −6.98349e29 −0.446533 −0.223266 0.974757i \(-0.571672\pi\)
−0.223266 + 0.974757i \(0.571672\pi\)
\(752\) 1.55157e30i 0.978326i
\(753\) − 5.58883e28i − 0.0347515i
\(754\) −4.58226e29 −0.280984
\(755\) 0 0
\(756\) −9.85341e28 −0.0587635
\(757\) 3.79778e29i 0.223369i 0.993744 + 0.111685i \(0.0356247\pi\)
−0.993744 + 0.111685i \(0.964375\pi\)
\(758\) 3.77639e30i 2.19054i
\(759\) 8.82794e29 0.505034
\(760\) 0 0
\(761\) −7.50371e29 −0.417577 −0.208789 0.977961i \(-0.566952\pi\)
−0.208789 + 0.977961i \(0.566952\pi\)
\(762\) 1.39934e29i 0.0768060i
\(763\) − 2.22736e30i − 1.20582i
\(764\) −1.07423e30 −0.573608
\(765\) 0 0
\(766\) 3.63479e30 1.88833
\(767\) 2.01032e29i 0.103018i
\(768\) − 1.22824e30i − 0.620856i
\(769\) −2.16884e29 −0.108144 −0.0540719 0.998537i \(-0.517220\pi\)
−0.0540719 + 0.998537i \(0.517220\pi\)
\(770\) 0 0
\(771\) 1.13271e30 0.549600
\(772\) 1.58943e30i 0.760779i
\(773\) − 3.08128e30i − 1.45495i −0.686136 0.727473i \(-0.740695\pi\)
0.686136 0.727473i \(-0.259305\pi\)
\(774\) 4.32563e29 0.201498
\(775\) 0 0
\(776\) −1.19394e29 −0.0541294
\(777\) 1.09498e29i 0.0489761i
\(778\) − 5.14625e29i − 0.227094i
\(779\) 2.94700e29 0.128303
\(780\) 0 0
\(781\) 1.30710e30 0.553951
\(782\) 3.32107e30i 1.38870i
\(783\) 4.16839e29i 0.171977i
\(784\) −1.47000e30 −0.598412
\(785\) 0 0
\(786\) 2.50271e30 0.991915
\(787\) − 1.46460e30i − 0.572775i −0.958114 0.286387i \(-0.907546\pi\)
0.958114 0.286387i \(-0.0924544\pi\)
\(788\) 1.69734e30i 0.655004i
\(789\) 5.24608e28 0.0199768
\(790\) 0 0
\(791\) 1.87325e30 0.694612
\(792\) − 4.66764e29i − 0.170798i
\(793\) − 5.64732e29i − 0.203927i
\(794\) −1.09988e30 −0.391949
\(795\) 0 0
\(796\) 1.28217e30 0.445000
\(797\) − 1.27731e30i − 0.437505i −0.975780 0.218752i \(-0.929801\pi\)
0.975780 0.218752i \(-0.0701987\pi\)
\(798\) − 7.41250e29i − 0.250573i
\(799\) −2.33698e30 −0.779677
\(800\) 0 0
\(801\) 1.45091e29 0.0471519
\(802\) − 4.21439e30i − 1.35177i
\(803\) − 1.06541e30i − 0.337292i
\(804\) 4.85163e29 0.151600
\(805\) 0 0
\(806\) 1.55473e30 0.473301
\(807\) − 1.26275e30i − 0.379441i
\(808\) − 9.02876e29i − 0.267799i
\(809\) 4.24975e30 1.24424 0.622120 0.782922i \(-0.286272\pi\)
0.622120 + 0.782922i \(0.286272\pi\)
\(810\) 0 0
\(811\) 2.05863e30 0.587298 0.293649 0.955913i \(-0.405130\pi\)
0.293649 + 0.955913i \(0.405130\pi\)
\(812\) 9.68902e29i 0.272861i
\(813\) 9.23732e29i 0.256800i
\(814\) 3.81551e29 0.104712
\(815\) 0 0
\(816\) 2.66315e30 0.712278
\(817\) 9.68636e29i 0.255758i
\(818\) 9.46999e29i 0.246854i
\(819\) 2.45901e29 0.0632818
\(820\) 0 0
\(821\) −1.80717e29 −0.0453309 −0.0226655 0.999743i \(-0.507215\pi\)
−0.0226655 + 0.999743i \(0.507215\pi\)
\(822\) 1.04930e30i 0.259863i
\(823\) − 6.23532e30i − 1.52462i −0.647214 0.762308i \(-0.724066\pi\)
0.647214 0.762308i \(-0.275934\pi\)
\(824\) 3.84445e30 0.928107
\(825\) 0 0
\(826\) 1.42802e30 0.336080
\(827\) − 3.37179e30i − 0.783524i −0.920067 0.391762i \(-0.871866\pi\)
0.920067 0.391762i \(-0.128134\pi\)
\(828\) − 7.22678e29i − 0.165816i
\(829\) 2.70711e29 0.0613315 0.0306658 0.999530i \(-0.490237\pi\)
0.0306658 + 0.999530i \(0.490237\pi\)
\(830\) 0 0
\(831\) 4.75328e30 1.04998
\(832\) 3.54020e29i 0.0772205i
\(833\) − 2.21413e30i − 0.476904i
\(834\) 8.88425e29 0.188963
\(835\) 0 0
\(836\) −7.68857e29 −0.159470
\(837\) − 1.41431e30i − 0.289685i
\(838\) − 1.05105e31i − 2.12599i
\(839\) 7.45457e30 1.48909 0.744547 0.667570i \(-0.232666\pi\)
0.744547 + 0.667570i \(0.232666\pi\)
\(840\) 0 0
\(841\) −1.03399e30 −0.201446
\(842\) 7.01551e30i 1.34984i
\(843\) − 4.91979e30i − 0.934882i
\(844\) −3.54509e30 −0.665321
\(845\) 0 0
\(846\) 1.70840e30 0.312753
\(847\) 1.77131e30i 0.320272i
\(848\) 1.18868e31i 2.12280i
\(849\) −2.63435e29 −0.0464668
\(850\) 0 0
\(851\) −8.03088e29 −0.138198
\(852\) − 1.07002e30i − 0.181877i
\(853\) − 6.10653e30i − 1.02525i −0.858613 0.512625i \(-0.828673\pi\)
0.858613 0.512625i \(-0.171327\pi\)
\(854\) −4.01153e30 −0.665277
\(855\) 0 0
\(856\) 5.09059e30 0.823746
\(857\) 4.08307e30i 0.652662i 0.945256 + 0.326331i \(0.105812\pi\)
−0.945256 + 0.326331i \(0.894188\pi\)
\(858\) − 8.56856e29i − 0.135298i
\(859\) −5.27189e30 −0.822316 −0.411158 0.911564i \(-0.634875\pi\)
−0.411158 + 0.911564i \(0.634875\pi\)
\(860\) 0 0
\(861\) 6.94452e29 0.105708
\(862\) 1.36031e31i 2.04555i
\(863\) 1.05537e31i 1.56780i 0.620884 + 0.783902i \(0.286774\pi\)
−0.620884 + 0.783902i \(0.713226\pi\)
\(864\) −1.04528e30 −0.153405
\(865\) 0 0
\(866\) −3.00300e30 −0.430147
\(867\) − 6.85488e28i − 0.00970062i
\(868\) − 3.28742e30i − 0.459619i
\(869\) 4.35586e30 0.601681
\(870\) 0 0
\(871\) −1.21077e30 −0.163256
\(872\) − 8.63747e30i − 1.15070i
\(873\) 1.99379e29i 0.0262441i
\(874\) 5.43655e30 0.707056
\(875\) 0 0
\(876\) −8.72177e29 −0.110742
\(877\) − 7.16069e30i − 0.898378i −0.893437 0.449189i \(-0.851713\pi\)
0.893437 0.449189i \(-0.148287\pi\)
\(878\) 1.44706e31i 1.79388i
\(879\) 5.71079e30 0.699541
\(880\) 0 0
\(881\) −1.05943e31 −1.26714 −0.633568 0.773687i \(-0.718410\pi\)
−0.633568 + 0.773687i \(0.718410\pi\)
\(882\) 1.61860e30i 0.191301i
\(883\) 6.60744e30i 0.771695i 0.922562 + 0.385848i \(0.126091\pi\)
−0.922562 + 0.385848i \(0.873909\pi\)
\(884\) 9.59531e29 0.110742
\(885\) 0 0
\(886\) −5.70734e28 −0.00643250
\(887\) 2.74467e30i 0.305698i 0.988250 + 0.152849i \(0.0488447\pi\)
−0.988250 + 0.152849i \(0.951155\pi\)
\(888\) 4.24621e29i 0.0467374i
\(889\) −7.38406e29 −0.0803205
\(890\) 0 0
\(891\) −7.79465e29 −0.0828096
\(892\) 4.11496e30i 0.432051i
\(893\) 3.82561e30i 0.396973i
\(894\) −9.22409e30 −0.945974
\(895\) 0 0
\(896\) 8.24741e30 0.826196
\(897\) 1.80351e30i 0.178565i
\(898\) 9.01519e30i 0.882213i
\(899\) −1.39071e31 −1.34512
\(900\) 0 0
\(901\) −1.79040e31 −1.69177
\(902\) − 2.41986e30i − 0.226007i
\(903\) 2.28256e30i 0.210718i
\(904\) 7.26427e30 0.662863
\(905\) 0 0
\(906\) 4.84884e30 0.432307
\(907\) − 5.45568e30i − 0.480809i −0.970673 0.240404i \(-0.922720\pi\)
0.970673 0.240404i \(-0.0772800\pi\)
\(908\) − 3.04878e30i − 0.265598i
\(909\) −1.50774e30 −0.129839
\(910\) 0 0
\(911\) −8.75739e30 −0.736940 −0.368470 0.929640i \(-0.620118\pi\)
−0.368470 + 0.929640i \(0.620118\pi\)
\(912\) − 4.35955e30i − 0.362657i
\(913\) − 2.53270e30i − 0.208277i
\(914\) −3.71884e29 −0.0302324
\(915\) 0 0
\(916\) 7.21547e29 0.0573274
\(917\) 1.32064e31i 1.03730i
\(918\) − 2.93235e30i − 0.227702i
\(919\) −1.24295e31 −0.954206 −0.477103 0.878847i \(-0.658313\pi\)
−0.477103 + 0.878847i \(0.658313\pi\)
\(920\) 0 0
\(921\) −9.92984e30 −0.745102
\(922\) − 2.91512e31i − 2.16263i
\(923\) 2.67034e30i 0.195861i
\(924\) −1.81179e30 −0.131387
\(925\) 0 0
\(926\) −3.28929e31 −2.33177
\(927\) − 6.41997e30i − 0.449983i
\(928\) 1.02785e31i 0.712319i
\(929\) −2.24310e31 −1.53703 −0.768517 0.639829i \(-0.779005\pi\)
−0.768517 + 0.639829i \(0.779005\pi\)
\(930\) 0 0
\(931\) −3.62451e30 −0.242816
\(932\) − 7.30787e29i − 0.0484087i
\(933\) − 1.36031e31i − 0.891008i
\(934\) 2.09134e31 1.35451
\(935\) 0 0
\(936\) 9.53578e29 0.0603893
\(937\) 1.10052e31i 0.689176i 0.938754 + 0.344588i \(0.111981\pi\)
−0.938754 + 0.344588i \(0.888019\pi\)
\(938\) 8.60060e30i 0.532597i
\(939\) −1.65135e31 −1.01123
\(940\) 0 0
\(941\) −2.38036e31 −1.42545 −0.712725 0.701444i \(-0.752539\pi\)
−0.712725 + 0.701444i \(0.752539\pi\)
\(942\) − 1.53974e31i − 0.911828i
\(943\) 5.09332e30i 0.298283i
\(944\) 8.39866e30 0.486412
\(945\) 0 0
\(946\) 7.95373e30 0.450520
\(947\) − 8.09762e30i − 0.453610i −0.973940 0.226805i \(-0.927172\pi\)
0.973940 0.226805i \(-0.0728280\pi\)
\(948\) − 3.56582e30i − 0.197548i
\(949\) 2.17660e30 0.119257
\(950\) 0 0
\(951\) −1.76262e30 −0.0944631
\(952\) 9.26593e30i 0.491135i
\(953\) − 3.42232e30i − 0.179410i −0.995968 0.0897048i \(-0.971408\pi\)
0.995968 0.0897048i \(-0.0285924\pi\)
\(954\) 1.30884e31 0.678622
\(955\) 0 0
\(956\) −7.87083e30 −0.399220
\(957\) 7.66461e30i 0.384516i
\(958\) 1.20140e31i 0.596140i
\(959\) −5.53695e30 −0.271754
\(960\) 0 0
\(961\) 2.63603e31 1.26577
\(962\) 7.79493e29i 0.0370232i
\(963\) − 8.50095e30i − 0.399385i
\(964\) −6.63899e30 −0.308527
\(965\) 0 0
\(966\) 1.28111e31 0.582540
\(967\) − 1.33121e31i − 0.598780i −0.954131 0.299390i \(-0.903217\pi\)
0.954131 0.299390i \(-0.0967831\pi\)
\(968\) 6.86896e30i 0.305633i
\(969\) 6.56638e30 0.289019
\(970\) 0 0
\(971\) 9.71774e30 0.418565 0.209283 0.977855i \(-0.432887\pi\)
0.209283 + 0.977855i \(0.432887\pi\)
\(972\) 6.38091e29i 0.0271886i
\(973\) 4.68806e30i 0.197610i
\(974\) 1.83881e31 0.766773
\(975\) 0 0
\(976\) −2.35932e31 −0.962861
\(977\) 1.40637e31i 0.567816i 0.958851 + 0.283908i \(0.0916311\pi\)
−0.958851 + 0.283908i \(0.908369\pi\)
\(978\) 4.93351e30i 0.197060i
\(979\) 2.66785e30 0.105425
\(980\) 0 0
\(981\) −1.44240e31 −0.557906
\(982\) − 2.90414e31i − 1.11134i
\(983\) − 4.87291e31i − 1.84492i −0.386098 0.922458i \(-0.626177\pi\)
0.386098 0.922458i \(-0.373823\pi\)
\(984\) 2.69301e30 0.100877
\(985\) 0 0
\(986\) −2.88343e31 −1.05731
\(987\) 9.01495e30i 0.327064i
\(988\) − 1.57074e30i − 0.0563841i
\(989\) −1.67410e31 −0.594594
\(990\) 0 0
\(991\) 1.17847e30 0.0409773 0.0204887 0.999790i \(-0.493478\pi\)
0.0204887 + 0.999790i \(0.493478\pi\)
\(992\) − 3.48741e31i − 1.19986i
\(993\) − 1.50926e31i − 0.513803i
\(994\) 1.89685e31 0.638965
\(995\) 0 0
\(996\) −2.07334e30 −0.0683829
\(997\) − 5.05438e31i − 1.64956i −0.565453 0.824781i \(-0.691299\pi\)
0.565453 0.824781i \(-0.308701\pi\)
\(998\) − 8.85229e30i − 0.285881i
\(999\) 7.09089e29 0.0226602
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 75.22.b.b.49.1 2
5.2 odd 4 3.22.a.b.1.1 1
5.3 odd 4 75.22.a.a.1.1 1
5.4 even 2 inner 75.22.b.b.49.2 2
15.2 even 4 9.22.a.a.1.1 1
20.7 even 4 48.22.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.b.1.1 1 5.2 odd 4
9.22.a.a.1.1 1 15.2 even 4
48.22.a.d.1.1 1 20.7 even 4
75.22.a.a.1.1 1 5.3 odd 4
75.22.b.b.49.1 2 1.1 even 1 trivial
75.22.b.b.49.2 2 5.4 even 2 inner