Properties

Label 475.3.c.i.151.14
Level $475$
Weight $3$
Character 475.151
Analytic conductor $12.943$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,3,Mod(151,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.151");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 475.c (of order \(2\), degree \(1\), 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: \(12.9428125571\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 42x^{12} + 677x^{10} + 5313x^{8} + 21125x^{6} + 40138x^{4} + 30565x^{2} + 3675 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 151.14
Root \(3.82982i\) of defining polynomial
Character \(\chi\) \(=\) 475.151
Dual form 475.3.c.i.151.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.82982i q^{2} +0.102062i q^{3} -10.6675 q^{4} -0.390878 q^{6} +9.96826 q^{7} -25.5353i q^{8} +8.98958 q^{9} +O(q^{10})\) \(q+3.82982i q^{2} +0.102062i q^{3} -10.6675 q^{4} -0.390878 q^{6} +9.96826 q^{7} -25.5353i q^{8} +8.98958 q^{9} +5.29329 q^{11} -1.08875i q^{12} -13.5363i q^{13} +38.1766i q^{14} +55.1254 q^{16} -5.23391 q^{17} +34.4285i q^{18} +(11.7515 + 14.9299i) q^{19} +1.01738i q^{21} +20.2723i q^{22} +37.5555 q^{23} +2.60618 q^{24} +51.8414 q^{26} +1.83605i q^{27} -106.336 q^{28} -26.4485i q^{29} +29.3216i q^{31} +108.979i q^{32} +0.540244i q^{33} -20.0449i q^{34} -95.8963 q^{36} -38.9258i q^{37} +(-57.1787 + 45.0062i) q^{38} +1.38154 q^{39} -39.5253i q^{41} -3.89638 q^{42} -30.3917 q^{43} -56.4662 q^{44} +143.831i q^{46} -77.0113 q^{47} +5.62621i q^{48} +50.3662 q^{49} -0.534183i q^{51} +144.398i q^{52} +41.0262i q^{53} -7.03174 q^{54} -254.542i q^{56} +(-1.52377 + 1.19938i) q^{57} +101.293 q^{58} +50.0262i q^{59} +32.2395 q^{61} -112.296 q^{62} +89.6105 q^{63} -196.869 q^{64} -2.06903 q^{66} -100.135i q^{67} +55.8327 q^{68} +3.83299i q^{69} +90.8238i q^{71} -229.551i q^{72} -4.79733 q^{73} +149.079 q^{74} +(-125.359 - 159.265i) q^{76} +52.7649 q^{77} +5.29103i q^{78} +128.561i q^{79} +80.7189 q^{81} +151.374 q^{82} +100.182 q^{83} -10.8529i q^{84} -116.395i q^{86} +2.69939 q^{87} -135.166i q^{88} +127.458i q^{89} -134.933i q^{91} -400.623 q^{92} -2.99262 q^{93} -294.939i q^{94} -11.1226 q^{96} +14.5978i q^{97} +192.893i q^{98} +47.5845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 28 q^{4} - 4 q^{6} + 20 q^{7} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 28 q^{4} - 4 q^{6} + 20 q^{7} - 36 q^{9} - 4 q^{11} + 36 q^{16} - 22 q^{17} + 39 q^{19} - 12 q^{23} - 44 q^{24} + 30 q^{26} - 98 q^{28} + 4 q^{36} - 37 q^{38} - 32 q^{39} - 250 q^{42} - 90 q^{43} - 52 q^{44} - 148 q^{47} + 234 q^{49} + 98 q^{54} + 195 q^{57} + 274 q^{58} + 222 q^{61} - 518 q^{62} - 198 q^{63} - 218 q^{64} + 92 q^{66} - 80 q^{68} + 228 q^{73} - 92 q^{74} - 351 q^{76} + 260 q^{77} + 402 q^{81} - 58 q^{82} + 280 q^{83} + 282 q^{87} - 302 q^{92} + 358 q^{93} + 190 q^{96} - 180 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.82982i 1.91491i 0.288586 + 0.957454i \(0.406815\pi\)
−0.288586 + 0.957454i \(0.593185\pi\)
\(3\) 0.102062i 0.0340206i 0.999855 + 0.0170103i \(0.00541481\pi\)
−0.999855 + 0.0170103i \(0.994585\pi\)
\(4\) −10.6675 −2.66687
\(5\) 0 0
\(6\) −0.390878 −0.0651464
\(7\) 9.96826 1.42404 0.712019 0.702161i \(-0.247781\pi\)
0.712019 + 0.702161i \(0.247781\pi\)
\(8\) 25.5353i 3.19191i
\(9\) 8.98958 0.998843
\(10\) 0 0
\(11\) 5.29329 0.481208 0.240604 0.970623i \(-0.422654\pi\)
0.240604 + 0.970623i \(0.422654\pi\)
\(12\) 1.08875i 0.0907288i
\(13\) 13.5363i 1.04125i −0.853785 0.520625i \(-0.825699\pi\)
0.853785 0.520625i \(-0.174301\pi\)
\(14\) 38.1766i 2.72690i
\(15\) 0 0
\(16\) 55.1254 3.44534
\(17\) −5.23391 −0.307877 −0.153938 0.988080i \(-0.549196\pi\)
−0.153938 + 0.988080i \(0.549196\pi\)
\(18\) 34.4285i 1.91269i
\(19\) 11.7515 + 14.9299i 0.618501 + 0.785784i
\(20\) 0 0
\(21\) 1.01738i 0.0484467i
\(22\) 20.2723i 0.921470i
\(23\) 37.5555 1.63285 0.816425 0.577452i \(-0.195953\pi\)
0.816425 + 0.577452i \(0.195953\pi\)
\(24\) 2.60618 0.108591
\(25\) 0 0
\(26\) 51.8414 1.99390
\(27\) 1.83605i 0.0680019i
\(28\) −106.336 −3.79773
\(29\) 26.4485i 0.912018i −0.889975 0.456009i \(-0.849278\pi\)
0.889975 0.456009i \(-0.150722\pi\)
\(30\) 0 0
\(31\) 29.3216i 0.945859i 0.881100 + 0.472929i \(0.156803\pi\)
−0.881100 + 0.472929i \(0.843197\pi\)
\(32\) 108.979i 3.40560i
\(33\) 0.540244i 0.0163710i
\(34\) 20.0449i 0.589556i
\(35\) 0 0
\(36\) −95.8963 −2.66379
\(37\) 38.9258i 1.05205i −0.850470 0.526024i \(-0.823682\pi\)
0.850470 0.526024i \(-0.176318\pi\)
\(38\) −57.1787 + 45.0062i −1.50470 + 1.18437i
\(39\) 1.38154 0.0354240
\(40\) 0 0
\(41\) 39.5253i 0.964031i −0.876163 0.482015i \(-0.839905\pi\)
0.876163 0.482015i \(-0.160095\pi\)
\(42\) −3.89638 −0.0927709
\(43\) −30.3917 −0.706784 −0.353392 0.935475i \(-0.614972\pi\)
−0.353392 + 0.935475i \(0.614972\pi\)
\(44\) −56.4662 −1.28332
\(45\) 0 0
\(46\) 143.831i 3.12676i
\(47\) −77.0113 −1.63854 −0.819270 0.573409i \(-0.805621\pi\)
−0.819270 + 0.573409i \(0.805621\pi\)
\(48\) 5.62621i 0.117213i
\(49\) 50.3662 1.02788
\(50\) 0 0
\(51\) 0.534183i 0.0104742i
\(52\) 144.398i 2.77688i
\(53\) 41.0262i 0.774080i 0.922063 + 0.387040i \(0.126502\pi\)
−0.922063 + 0.387040i \(0.873498\pi\)
\(54\) −7.03174 −0.130217
\(55\) 0 0
\(56\) 254.542i 4.54540i
\(57\) −1.52377 + 1.19938i −0.0267329 + 0.0210418i
\(58\) 101.293 1.74643
\(59\) 50.0262i 0.847902i 0.905685 + 0.423951i \(0.139357\pi\)
−0.905685 + 0.423951i \(0.860643\pi\)
\(60\) 0 0
\(61\) 32.2395 0.528517 0.264258 0.964452i \(-0.414873\pi\)
0.264258 + 0.964452i \(0.414873\pi\)
\(62\) −112.296 −1.81123
\(63\) 89.6105 1.42239
\(64\) −196.869 −3.07607
\(65\) 0 0
\(66\) −2.06903 −0.0313490
\(67\) 100.135i 1.49455i −0.664516 0.747274i \(-0.731362\pi\)
0.664516 0.747274i \(-0.268638\pi\)
\(68\) 55.8327 0.821069
\(69\) 3.83299i 0.0555506i
\(70\) 0 0
\(71\) 90.8238i 1.27921i 0.768704 + 0.639604i \(0.220902\pi\)
−0.768704 + 0.639604i \(0.779098\pi\)
\(72\) 229.551i 3.18822i
\(73\) −4.79733 −0.0657169 −0.0328584 0.999460i \(-0.510461\pi\)
−0.0328584 + 0.999460i \(0.510461\pi\)
\(74\) 149.079 2.01457
\(75\) 0 0
\(76\) −125.359 159.265i −1.64946 2.09559i
\(77\) 52.7649 0.685259
\(78\) 5.29103i 0.0678337i
\(79\) 128.561i 1.62736i 0.581314 + 0.813679i \(0.302539\pi\)
−0.581314 + 0.813679i \(0.697461\pi\)
\(80\) 0 0
\(81\) 80.7189 0.996529
\(82\) 151.374 1.84603
\(83\) 100.182 1.20701 0.603503 0.797360i \(-0.293771\pi\)
0.603503 + 0.797360i \(0.293771\pi\)
\(84\) 10.8529i 0.129201i
\(85\) 0 0
\(86\) 116.395i 1.35343i
\(87\) 2.69939 0.0310274
\(88\) 135.166i 1.53597i
\(89\) 127.458i 1.43211i 0.698043 + 0.716056i \(0.254054\pi\)
−0.698043 + 0.716056i \(0.745946\pi\)
\(90\) 0 0
\(91\) 134.933i 1.48278i
\(92\) −400.623 −4.35460
\(93\) −2.99262 −0.0321787
\(94\) 294.939i 3.13765i
\(95\) 0 0
\(96\) −11.1226 −0.115861
\(97\) 14.5978i 0.150493i 0.997165 + 0.0752464i \(0.0239743\pi\)
−0.997165 + 0.0752464i \(0.976026\pi\)
\(98\) 192.893i 1.96830i
\(99\) 47.5845 0.480651
\(100\) 0 0
\(101\) −18.3954 −0.182132 −0.0910662 0.995845i \(-0.529028\pi\)
−0.0910662 + 0.995845i \(0.529028\pi\)
\(102\) 2.04582 0.0200571
\(103\) 184.511i 1.79137i −0.444692 0.895684i \(-0.646687\pi\)
0.444692 0.895684i \(-0.353313\pi\)
\(104\) −345.652 −3.32358
\(105\) 0 0
\(106\) −157.123 −1.48229
\(107\) 131.063i 1.22489i 0.790514 + 0.612444i \(0.209814\pi\)
−0.790514 + 0.612444i \(0.790186\pi\)
\(108\) 19.5861i 0.181352i
\(109\) 23.3256i 0.213996i 0.994259 + 0.106998i \(0.0341239\pi\)
−0.994259 + 0.106998i \(0.965876\pi\)
\(110\) 0 0
\(111\) 3.97284 0.0357913
\(112\) 549.505 4.90629
\(113\) 130.214i 1.15234i 0.817330 + 0.576169i \(0.195453\pi\)
−0.817330 + 0.576169i \(0.804547\pi\)
\(114\) −4.59342 5.83577i −0.0402931 0.0511910i
\(115\) 0 0
\(116\) 282.139i 2.43224i
\(117\) 121.685i 1.04004i
\(118\) −191.591 −1.62366
\(119\) −52.1730 −0.438428
\(120\) 0 0
\(121\) −92.9811 −0.768439
\(122\) 123.471i 1.01206i
\(123\) 4.03402 0.0327969
\(124\) 312.788i 2.52249i
\(125\) 0 0
\(126\) 343.192i 2.72374i
\(127\) 99.8187i 0.785974i −0.919544 0.392987i \(-0.871442\pi\)
0.919544 0.392987i \(-0.128558\pi\)
\(128\) 318.054i 2.48480i
\(129\) 3.10184i 0.0240452i
\(130\) 0 0
\(131\) 46.5614 0.355430 0.177715 0.984082i \(-0.443129\pi\)
0.177715 + 0.984082i \(0.443129\pi\)
\(132\) 5.76305i 0.0436594i
\(133\) 117.142 + 148.825i 0.880769 + 1.11899i
\(134\) 383.498 2.86192
\(135\) 0 0
\(136\) 133.649i 0.982715i
\(137\) −28.3675 −0.207062 −0.103531 0.994626i \(-0.533014\pi\)
−0.103531 + 0.994626i \(0.533014\pi\)
\(138\) −14.6797 −0.106374
\(139\) 155.882 1.12146 0.560728 0.828000i \(-0.310521\pi\)
0.560728 + 0.828000i \(0.310521\pi\)
\(140\) 0 0
\(141\) 7.85993i 0.0557442i
\(142\) −347.838 −2.44957
\(143\) 71.6513i 0.501058i
\(144\) 495.555 3.44135
\(145\) 0 0
\(146\) 18.3729i 0.125842i
\(147\) 5.14047i 0.0349692i
\(148\) 415.240i 2.80568i
\(149\) −42.8941 −0.287880 −0.143940 0.989586i \(-0.545977\pi\)
−0.143940 + 0.989586i \(0.545977\pi\)
\(150\) 0 0
\(151\) 35.9816i 0.238289i 0.992877 + 0.119144i \(0.0380151\pi\)
−0.992877 + 0.119144i \(0.961985\pi\)
\(152\) 381.239 300.078i 2.50815 1.97420i
\(153\) −47.0506 −0.307521
\(154\) 202.080i 1.31221i
\(155\) 0 0
\(156\) −14.7375 −0.0944713
\(157\) −202.251 −1.28823 −0.644113 0.764931i \(-0.722773\pi\)
−0.644113 + 0.764931i \(0.722773\pi\)
\(158\) −492.366 −3.11624
\(159\) −4.18722 −0.0263347
\(160\) 0 0
\(161\) 374.363 2.32524
\(162\) 309.138i 1.90826i
\(163\) 99.4595 0.610181 0.305091 0.952323i \(-0.401313\pi\)
0.305091 + 0.952323i \(0.401313\pi\)
\(164\) 421.635i 2.57095i
\(165\) 0 0
\(166\) 383.677i 2.31131i
\(167\) 95.2905i 0.570602i −0.958438 0.285301i \(-0.907906\pi\)
0.958438 0.285301i \(-0.0920935\pi\)
\(168\) 25.9791 0.154637
\(169\) −14.2301 −0.0842016
\(170\) 0 0
\(171\) 105.641 + 134.214i 0.617785 + 0.784874i
\(172\) 324.203 1.88490
\(173\) 245.293i 1.41788i −0.705269 0.708940i \(-0.749174\pi\)
0.705269 0.708940i \(-0.250826\pi\)
\(174\) 10.3382i 0.0594147i
\(175\) 0 0
\(176\) 291.795 1.65793
\(177\) −5.10578 −0.0288462
\(178\) −488.140 −2.74236
\(179\) 81.3066i 0.454227i −0.973868 0.227113i \(-0.927071\pi\)
0.973868 0.227113i \(-0.0729288\pi\)
\(180\) 0 0
\(181\) 257.547i 1.42291i −0.702730 0.711457i \(-0.748036\pi\)
0.702730 0.711457i \(-0.251964\pi\)
\(182\) 516.768 2.83939
\(183\) 3.29043i 0.0179805i
\(184\) 958.991i 5.21191i
\(185\) 0 0
\(186\) 11.4612i 0.0616193i
\(187\) −27.7046 −0.148153
\(188\) 821.518 4.36978
\(189\) 18.3022i 0.0968373i
\(190\) 0 0
\(191\) −236.760 −1.23958 −0.619791 0.784767i \(-0.712783\pi\)
−0.619791 + 0.784767i \(0.712783\pi\)
\(192\) 20.0928i 0.104650i
\(193\) 283.058i 1.46662i −0.679894 0.733310i \(-0.737974\pi\)
0.679894 0.733310i \(-0.262026\pi\)
\(194\) −55.9069 −0.288180
\(195\) 0 0
\(196\) −537.281 −2.74123
\(197\) 28.6273 0.145316 0.0726581 0.997357i \(-0.476852\pi\)
0.0726581 + 0.997357i \(0.476852\pi\)
\(198\) 182.240i 0.920403i
\(199\) −335.738 −1.68713 −0.843564 0.537029i \(-0.819547\pi\)
−0.843564 + 0.537029i \(0.819547\pi\)
\(200\) 0 0
\(201\) 10.2199 0.0508455
\(202\) 70.4509i 0.348767i
\(203\) 263.646i 1.29875i
\(204\) 5.69839i 0.0279333i
\(205\) 0 0
\(206\) 706.643 3.43030
\(207\) 337.609 1.63096
\(208\) 746.192i 3.58746i
\(209\) 62.2042 + 79.0283i 0.297628 + 0.378126i
\(210\) 0 0
\(211\) 116.161i 0.550528i −0.961369 0.275264i \(-0.911235\pi\)
0.961369 0.275264i \(-0.0887652\pi\)
\(212\) 437.647i 2.06437i
\(213\) −9.26965 −0.0435195
\(214\) −501.947 −2.34555
\(215\) 0 0
\(216\) 46.8841 0.217056
\(217\) 292.286i 1.34694i
\(218\) −89.3327 −0.409783
\(219\) 0.489625i 0.00223573i
\(220\) 0 0
\(221\) 70.8475i 0.320577i
\(222\) 15.2152i 0.0685371i
\(223\) 275.634i 1.23603i 0.786168 + 0.618013i \(0.212062\pi\)
−0.786168 + 0.618013i \(0.787938\pi\)
\(224\) 1086.33i 4.84970i
\(225\) 0 0
\(226\) −498.697 −2.20662
\(227\) 176.419i 0.777174i 0.921412 + 0.388587i \(0.127037\pi\)
−0.921412 + 0.388587i \(0.872963\pi\)
\(228\) 16.2548 12.7944i 0.0712932 0.0561158i
\(229\) −89.7773 −0.392041 −0.196020 0.980600i \(-0.562802\pi\)
−0.196020 + 0.980600i \(0.562802\pi\)
\(230\) 0 0
\(231\) 5.38529i 0.0233129i
\(232\) −675.370 −2.91108
\(233\) 57.1941 0.245468 0.122734 0.992440i \(-0.460834\pi\)
0.122734 + 0.992440i \(0.460834\pi\)
\(234\) 466.032 1.99159
\(235\) 0 0
\(236\) 533.655i 2.26125i
\(237\) −13.1212 −0.0553638
\(238\) 199.813i 0.839550i
\(239\) −30.8457 −0.129062 −0.0645308 0.997916i \(-0.520555\pi\)
−0.0645308 + 0.997916i \(0.520555\pi\)
\(240\) 0 0
\(241\) 307.041i 1.27403i −0.770852 0.637015i \(-0.780169\pi\)
0.770852 0.637015i \(-0.219831\pi\)
\(242\) 356.100i 1.47149i
\(243\) 24.7628i 0.101904i
\(244\) −343.915 −1.40949
\(245\) 0 0
\(246\) 15.4496i 0.0628031i
\(247\) 202.095 159.072i 0.818197 0.644014i
\(248\) 748.736 3.01910
\(249\) 10.2247i 0.0410631i
\(250\) 0 0
\(251\) −252.181 −1.00470 −0.502352 0.864663i \(-0.667532\pi\)
−0.502352 + 0.864663i \(0.667532\pi\)
\(252\) −955.919 −3.79333
\(253\) 198.792 0.785741
\(254\) 382.287 1.50507
\(255\) 0 0
\(256\) 430.613 1.68208
\(257\) 349.189i 1.35871i 0.733808 + 0.679356i \(0.237741\pi\)
−0.733808 + 0.679356i \(0.762259\pi\)
\(258\) 11.8795 0.0460444
\(259\) 388.022i 1.49815i
\(260\) 0 0
\(261\) 237.761i 0.910962i
\(262\) 178.321i 0.680616i
\(263\) −230.049 −0.874713 −0.437356 0.899288i \(-0.644085\pi\)
−0.437356 + 0.899288i \(0.644085\pi\)
\(264\) 13.7953 0.0522548
\(265\) 0 0
\(266\) −569.973 + 448.633i −2.14275 + 1.68659i
\(267\) −13.0086 −0.0487213
\(268\) 1068.19i 3.98577i
\(269\) 282.058i 1.04854i 0.851551 + 0.524272i \(0.175663\pi\)
−0.851551 + 0.524272i \(0.824337\pi\)
\(270\) 0 0
\(271\) 202.339 0.746638 0.373319 0.927703i \(-0.378220\pi\)
0.373319 + 0.927703i \(0.378220\pi\)
\(272\) −288.521 −1.06074
\(273\) 13.7715 0.0504451
\(274\) 108.642i 0.396505i
\(275\) 0 0
\(276\) 40.8884i 0.148146i
\(277\) 108.878 0.393063 0.196531 0.980498i \(-0.437032\pi\)
0.196531 + 0.980498i \(0.437032\pi\)
\(278\) 597.001i 2.14748i
\(279\) 263.589i 0.944764i
\(280\) 0 0
\(281\) 221.302i 0.787551i 0.919207 + 0.393776i \(0.128831\pi\)
−0.919207 + 0.393776i \(0.871169\pi\)
\(282\) 30.1021 0.106745
\(283\) −148.837 −0.525927 −0.262964 0.964806i \(-0.584700\pi\)
−0.262964 + 0.964806i \(0.584700\pi\)
\(284\) 968.862i 3.41149i
\(285\) 0 0
\(286\) 274.411 0.959481
\(287\) 393.998i 1.37282i
\(288\) 979.678i 3.40166i
\(289\) −261.606 −0.905212
\(290\) 0 0
\(291\) −1.48988 −0.00511986
\(292\) 51.1755 0.175259
\(293\) 68.3191i 0.233171i −0.993181 0.116586i \(-0.962805\pi\)
0.993181 0.116586i \(-0.0371949\pi\)
\(294\) −19.6871 −0.0669628
\(295\) 0 0
\(296\) −993.980 −3.35804
\(297\) 9.71876i 0.0327231i
\(298\) 164.276i 0.551263i
\(299\) 508.361i 1.70020i
\(300\) 0 0
\(301\) −302.952 −1.00649
\(302\) −137.803 −0.456301
\(303\) 1.87747i 0.00619626i
\(304\) 647.808 + 823.017i 2.13095 + 2.70729i
\(305\) 0 0
\(306\) 180.195i 0.588874i
\(307\) 145.822i 0.474991i −0.971389 0.237495i \(-0.923674\pi\)
0.971389 0.237495i \(-0.0763264\pi\)
\(308\) −562.869 −1.82750
\(309\) 18.8315 0.0609435
\(310\) 0 0
\(311\) −43.9351 −0.141270 −0.0706351 0.997502i \(-0.522503\pi\)
−0.0706351 + 0.997502i \(0.522503\pi\)
\(312\) 35.2779i 0.113070i
\(313\) −204.710 −0.654026 −0.327013 0.945020i \(-0.606042\pi\)
−0.327013 + 0.945020i \(0.606042\pi\)
\(314\) 774.586i 2.46683i
\(315\) 0 0
\(316\) 1371.43i 4.33996i
\(317\) 88.8488i 0.280280i −0.990132 0.140140i \(-0.955245\pi\)
0.990132 0.140140i \(-0.0447553\pi\)
\(318\) 16.0363i 0.0504285i
\(319\) 140.000i 0.438871i
\(320\) 0 0
\(321\) −13.3765 −0.0416715
\(322\) 1433.74i 4.45262i
\(323\) −61.5064 78.1417i −0.190422 0.241925i
\(324\) −861.068 −2.65762
\(325\) 0 0
\(326\) 380.912i 1.16844i
\(327\) −2.38065 −0.00728029
\(328\) −1009.29 −3.07710
\(329\) −767.669 −2.33334
\(330\) 0 0
\(331\) 121.098i 0.365854i −0.983127 0.182927i \(-0.941443\pi\)
0.983127 0.182927i \(-0.0585572\pi\)
\(332\) −1068.69 −3.21893
\(333\) 349.926i 1.05083i
\(334\) 364.945 1.09265
\(335\) 0 0
\(336\) 56.0835i 0.166915i
\(337\) 51.4397i 0.152640i 0.997083 + 0.0763200i \(0.0243171\pi\)
−0.997083 + 0.0763200i \(0.975683\pi\)
\(338\) 54.4986i 0.161238i
\(339\) −13.2899 −0.0392033
\(340\) 0 0
\(341\) 155.208i 0.455155i
\(342\) −514.013 + 404.587i −1.50296 + 1.18300i
\(343\) 13.6188 0.0397049
\(344\) 776.061i 2.25599i
\(345\) 0 0
\(346\) 939.428 2.71511
\(347\) 500.847 1.44336 0.721682 0.692225i \(-0.243369\pi\)
0.721682 + 0.692225i \(0.243369\pi\)
\(348\) −28.7957 −0.0827463
\(349\) −585.972 −1.67900 −0.839501 0.543357i \(-0.817153\pi\)
−0.839501 + 0.543357i \(0.817153\pi\)
\(350\) 0 0
\(351\) 24.8533 0.0708070
\(352\) 576.859i 1.63880i
\(353\) −307.915 −0.872281 −0.436140 0.899879i \(-0.643655\pi\)
−0.436140 + 0.899879i \(0.643655\pi\)
\(354\) 19.5542i 0.0552378i
\(355\) 0 0
\(356\) 1359.66i 3.81926i
\(357\) 5.32487i 0.0149156i
\(358\) 311.389 0.869803
\(359\) 348.313 0.970232 0.485116 0.874450i \(-0.338777\pi\)
0.485116 + 0.874450i \(0.338777\pi\)
\(360\) 0 0
\(361\) −84.8034 + 350.898i −0.234912 + 0.972017i
\(362\) 986.359 2.72475
\(363\) 9.48983i 0.0261428i
\(364\) 1439.40i 3.95438i
\(365\) 0 0
\(366\) −12.6017 −0.0344310
\(367\) 220.044 0.599575 0.299788 0.954006i \(-0.403084\pi\)
0.299788 + 0.954006i \(0.403084\pi\)
\(368\) 2070.27 5.62572
\(369\) 355.316i 0.962915i
\(370\) 0 0
\(371\) 408.960i 1.10232i
\(372\) 31.9238 0.0858166
\(373\) 208.409i 0.558737i −0.960184 0.279369i \(-0.909875\pi\)
0.960184 0.279369i \(-0.0901252\pi\)
\(374\) 106.104i 0.283699i
\(375\) 0 0
\(376\) 1966.51i 5.23007i
\(377\) −358.014 −0.949639
\(378\) −70.0942 −0.185434
\(379\) 233.504i 0.616104i −0.951369 0.308052i \(-0.900323\pi\)
0.951369 0.308052i \(-0.0996771\pi\)
\(380\) 0 0
\(381\) 10.1877 0.0267393
\(382\) 906.748i 2.37368i
\(383\) 19.2030i 0.0501385i −0.999686 0.0250693i \(-0.992019\pi\)
0.999686 0.0250693i \(-0.00798063\pi\)
\(384\) 32.4612 0.0845343
\(385\) 0 0
\(386\) 1084.06 2.80844
\(387\) −273.209 −0.705966
\(388\) 155.722i 0.401345i
\(389\) 385.371 0.990672 0.495336 0.868702i \(-0.335045\pi\)
0.495336 + 0.868702i \(0.335045\pi\)
\(390\) 0 0
\(391\) −196.562 −0.502717
\(392\) 1286.12i 3.28091i
\(393\) 4.75214i 0.0120920i
\(394\) 109.637i 0.278267i
\(395\) 0 0
\(396\) −507.607 −1.28184
\(397\) 221.566 0.558102 0.279051 0.960276i \(-0.409980\pi\)
0.279051 + 0.960276i \(0.409980\pi\)
\(398\) 1285.82i 3.23069i
\(399\) −15.1894 + 11.9558i −0.0380686 + 0.0299643i
\(400\) 0 0
\(401\) 684.193i 1.70622i −0.521734 0.853108i \(-0.674715\pi\)
0.521734 0.853108i \(-0.325285\pi\)
\(402\) 39.1405i 0.0973644i
\(403\) 396.905 0.984875
\(404\) 196.233 0.485724
\(405\) 0 0
\(406\) 1009.71 2.48698
\(407\) 206.045i 0.506254i
\(408\) −13.6405 −0.0334326
\(409\) 371.270i 0.907750i 0.891065 + 0.453875i \(0.149959\pi\)
−0.891065 + 0.453875i \(0.850041\pi\)
\(410\) 0 0
\(411\) 2.89525i 0.00704439i
\(412\) 1968.27i 4.77735i
\(413\) 498.675i 1.20744i
\(414\) 1292.98i 3.12314i
\(415\) 0 0
\(416\) 1475.17 3.54608
\(417\) 15.9097i 0.0381526i
\(418\) −302.664 + 238.231i −0.724076 + 0.569930i
\(419\) 473.130 1.12919 0.564595 0.825368i \(-0.309033\pi\)
0.564595 + 0.825368i \(0.309033\pi\)
\(420\) 0 0
\(421\) 253.042i 0.601049i 0.953774 + 0.300524i \(0.0971617\pi\)
−0.953774 + 0.300524i \(0.902838\pi\)
\(422\) 444.877 1.05421
\(423\) −692.300 −1.63664
\(424\) 1047.62 2.47079
\(425\) 0 0
\(426\) 35.5011i 0.0833358i
\(427\) 321.372 0.752628
\(428\) 1398.11i 3.26662i
\(429\) 7.31287 0.0170463
\(430\) 0 0
\(431\) 128.719i 0.298653i −0.988788 0.149326i \(-0.952290\pi\)
0.988788 0.149326i \(-0.0477105\pi\)
\(432\) 101.213i 0.234290i
\(433\) 161.686i 0.373409i 0.982416 + 0.186705i \(0.0597807\pi\)
−0.982416 + 0.186705i \(0.940219\pi\)
\(434\) −1119.40 −2.57926
\(435\) 0 0
\(436\) 248.825i 0.570701i
\(437\) 441.335 + 560.700i 1.00992 + 1.28307i
\(438\) 1.87517 0.00428122
\(439\) 144.836i 0.329923i −0.986300 0.164962i \(-0.947250\pi\)
0.986300 0.164962i \(-0.0527501\pi\)
\(440\) 0 0
\(441\) 452.771 1.02669
\(442\) −271.333 −0.613875
\(443\) 236.138 0.533042 0.266521 0.963829i \(-0.414126\pi\)
0.266521 + 0.963829i \(0.414126\pi\)
\(444\) −42.3802 −0.0954510
\(445\) 0 0
\(446\) −1055.63 −2.36688
\(447\) 4.37785i 0.00979385i
\(448\) −1962.44 −4.38044
\(449\) 392.043i 0.873147i 0.899669 + 0.436574i \(0.143808\pi\)
−0.899669 + 0.436574i \(0.856192\pi\)
\(450\) 0 0
\(451\) 209.219i 0.463900i
\(452\) 1389.06i 3.07314i
\(453\) −3.67235 −0.00810673
\(454\) −675.651 −1.48822
\(455\) 0 0
\(456\) 30.6266 + 38.9100i 0.0671636 + 0.0853289i
\(457\) −599.374 −1.31154 −0.655770 0.754961i \(-0.727656\pi\)
−0.655770 + 0.754961i \(0.727656\pi\)
\(458\) 343.831i 0.750722i
\(459\) 9.60972i 0.0209362i
\(460\) 0 0
\(461\) −304.936 −0.661467 −0.330734 0.943724i \(-0.607296\pi\)
−0.330734 + 0.943724i \(0.607296\pi\)
\(462\) −20.6247 −0.0446421
\(463\) 457.460 0.988035 0.494018 0.869452i \(-0.335528\pi\)
0.494018 + 0.869452i \(0.335528\pi\)
\(464\) 1457.99i 3.14221i
\(465\) 0 0
\(466\) 219.043i 0.470049i
\(467\) −211.620 −0.453147 −0.226574 0.973994i \(-0.572752\pi\)
−0.226574 + 0.973994i \(0.572752\pi\)
\(468\) 1298.08i 2.77367i
\(469\) 998.169i 2.12829i
\(470\) 0 0
\(471\) 20.6422i 0.0438263i
\(472\) 1277.43 2.70643
\(473\) −160.872 −0.340110
\(474\) 50.2519i 0.106017i
\(475\) 0 0
\(476\) 556.555 1.16923
\(477\) 368.809i 0.773184i
\(478\) 118.133i 0.247141i
\(479\) −116.631 −0.243488 −0.121744 0.992562i \(-0.538849\pi\)
−0.121744 + 0.992562i \(0.538849\pi\)
\(480\) 0 0
\(481\) −526.909 −1.09544
\(482\) 1175.91 2.43965
\(483\) 38.2082i 0.0791061i
\(484\) 991.875 2.04933
\(485\) 0 0
\(486\) −94.8369 −0.195138
\(487\) 86.5090i 0.177637i −0.996048 0.0888183i \(-0.971691\pi\)
0.996048 0.0888183i \(-0.0283090\pi\)
\(488\) 823.245i 1.68698i
\(489\) 10.1510i 0.0207588i
\(490\) 0 0
\(491\) 114.292 0.232773 0.116387 0.993204i \(-0.462869\pi\)
0.116387 + 0.993204i \(0.462869\pi\)
\(492\) −43.0329 −0.0874653
\(493\) 138.429i 0.280789i
\(494\) 609.215 + 773.986i 1.23323 + 1.56677i
\(495\) 0 0
\(496\) 1616.37i 3.25881i
\(497\) 905.355i 1.82164i
\(498\) −39.1588 −0.0786322
\(499\) 293.641 0.588459 0.294230 0.955735i \(-0.404937\pi\)
0.294230 + 0.955735i \(0.404937\pi\)
\(500\) 0 0
\(501\) 9.72553 0.0194122
\(502\) 965.805i 1.92391i
\(503\) −67.3735 −0.133943 −0.0669717 0.997755i \(-0.521334\pi\)
−0.0669717 + 0.997755i \(0.521334\pi\)
\(504\) 2288.23i 4.54014i
\(505\) 0 0
\(506\) 761.338i 1.50462i
\(507\) 1.45235i 0.00286459i
\(508\) 1064.82i 2.09609i
\(509\) 513.534i 1.00891i 0.863439 + 0.504454i \(0.168306\pi\)
−0.863439 + 0.504454i \(0.831694\pi\)
\(510\) 0 0
\(511\) −47.8210 −0.0935833
\(512\) 376.954i 0.736239i
\(513\) −27.4121 + 21.5764i −0.0534348 + 0.0420593i
\(514\) −1337.33 −2.60181
\(515\) 0 0
\(516\) 33.0888i 0.0641256i
\(517\) −407.643 −0.788479
\(518\) 1486.05 2.86883
\(519\) 25.0351 0.0482372
\(520\) 0 0
\(521\) 294.021i 0.564340i 0.959364 + 0.282170i \(0.0910542\pi\)
−0.959364 + 0.282170i \(0.908946\pi\)
\(522\) 910.582 1.74441
\(523\) 39.6545i 0.0758211i −0.999281 0.0379106i \(-0.987930\pi\)
0.999281 0.0379106i \(-0.0120702\pi\)
\(524\) −496.693 −0.947888
\(525\) 0 0
\(526\) 881.047i 1.67499i
\(527\) 153.467i 0.291208i
\(528\) 29.7812i 0.0564037i
\(529\) 881.418 1.66620
\(530\) 0 0
\(531\) 449.715i 0.846921i
\(532\) −1249.61 1587.59i −2.34890 2.98419i
\(533\) −535.024 −1.00380
\(534\) 49.8205i 0.0932969i
\(535\) 0 0
\(536\) −2556.97 −4.77046
\(537\) 8.29831 0.0154531
\(538\) −1080.23 −2.00787
\(539\) 266.603 0.494625
\(540\) 0 0
\(541\) 200.431 0.370482 0.185241 0.982693i \(-0.440693\pi\)
0.185241 + 0.982693i \(0.440693\pi\)
\(542\) 774.921i 1.42974i
\(543\) 26.2858 0.0484084
\(544\) 570.387i 1.04851i
\(545\) 0 0
\(546\) 52.7424i 0.0965977i
\(547\) 460.716i 0.842260i −0.907000 0.421130i \(-0.861634\pi\)
0.907000 0.421130i \(-0.138366\pi\)
\(548\) 302.611 0.552209
\(549\) 289.820 0.527905
\(550\) 0 0
\(551\) 394.874 310.810i 0.716649 0.564084i
\(552\) 97.8765 0.177312
\(553\) 1281.53i 2.31742i
\(554\) 416.984i 0.752679i
\(555\) 0 0
\(556\) −1662.87 −2.99078
\(557\) −920.981 −1.65347 −0.826733 0.562594i \(-0.809803\pi\)
−0.826733 + 0.562594i \(0.809803\pi\)
\(558\) −1009.50 −1.80914
\(559\) 411.390i 0.735939i
\(560\) 0 0
\(561\) 2.82759i 0.00504026i
\(562\) −847.546 −1.50809
\(563\) 90.0804i 0.160001i 0.996795 + 0.0800004i \(0.0254921\pi\)
−0.996795 + 0.0800004i \(0.974508\pi\)
\(564\) 83.8457i 0.148663i
\(565\) 0 0
\(566\) 570.020i 1.00710i
\(567\) 804.627 1.41909
\(568\) 2319.21 4.08312
\(569\) 199.974i 0.351448i −0.984440 0.175724i \(-0.943773\pi\)
0.984440 0.175724i \(-0.0562266\pi\)
\(570\) 0 0
\(571\) 14.6282 0.0256186 0.0128093 0.999918i \(-0.495923\pi\)
0.0128093 + 0.999918i \(0.495923\pi\)
\(572\) 764.340i 1.33626i
\(573\) 24.1642i 0.0421714i
\(574\) 1508.94 2.62882
\(575\) 0 0
\(576\) −1769.77 −3.07251
\(577\) −1029.44 −1.78412 −0.892062 0.451912i \(-0.850742\pi\)
−0.892062 + 0.451912i \(0.850742\pi\)
\(578\) 1001.90i 1.73340i
\(579\) 28.8894 0.0498954
\(580\) 0 0
\(581\) 998.636 1.71882
\(582\) 5.70597i 0.00980406i
\(583\) 217.164i 0.372494i
\(584\) 122.501i 0.209762i
\(585\) 0 0
\(586\) 261.650 0.446501
\(587\) −904.645 −1.54113 −0.770566 0.637360i \(-0.780026\pi\)
−0.770566 + 0.637360i \(0.780026\pi\)
\(588\) 54.8360i 0.0932584i
\(589\) −437.769 + 344.574i −0.743240 + 0.585015i
\(590\) 0 0
\(591\) 2.92176i 0.00494375i
\(592\) 2145.80i 3.62466i
\(593\) −418.448 −0.705645 −0.352823 0.935690i \(-0.614778\pi\)
−0.352823 + 0.935690i \(0.614778\pi\)
\(594\) −37.2211 −0.0626617
\(595\) 0 0
\(596\) 457.572 0.767739
\(597\) 34.2661i 0.0573972i
\(598\) 1946.93 3.25574
\(599\) 616.270i 1.02883i −0.857541 0.514416i \(-0.828009\pi\)
0.857541 0.514416i \(-0.171991\pi\)
\(600\) 0 0
\(601\) 618.358i 1.02888i 0.857526 + 0.514441i \(0.172000\pi\)
−0.857526 + 0.514441i \(0.828000\pi\)
\(602\) 1160.25i 1.92733i
\(603\) 900.169i 1.49282i
\(604\) 383.833i 0.635485i
\(605\) 0 0
\(606\) 7.19036 0.0118653
\(607\) 204.176i 0.336369i −0.985756 0.168185i \(-0.946209\pi\)
0.985756 0.168185i \(-0.0537905\pi\)
\(608\) −1627.05 + 1280.67i −2.67607 + 2.10637i
\(609\) 26.9082 0.0441842
\(610\) 0 0
\(611\) 1042.44i 1.70613i
\(612\) 501.912 0.820118
\(613\) 916.966 1.49587 0.747933 0.663774i \(-0.231046\pi\)
0.747933 + 0.663774i \(0.231046\pi\)
\(614\) 558.472 0.909563
\(615\) 0 0
\(616\) 1347.37i 2.18728i
\(617\) 438.672 0.710975 0.355488 0.934681i \(-0.384315\pi\)
0.355488 + 0.934681i \(0.384315\pi\)
\(618\) 72.1213i 0.116701i
\(619\) −164.123 −0.265142 −0.132571 0.991174i \(-0.542323\pi\)
−0.132571 + 0.991174i \(0.542323\pi\)
\(620\) 0 0
\(621\) 68.9539i 0.111037i
\(622\) 168.263i 0.270520i
\(623\) 1270.53i 2.03938i
\(624\) 76.1578 0.122048
\(625\) 0 0
\(626\) 784.002i 1.25240i
\(627\) −8.06578 + 6.34869i −0.0128641 + 0.0101255i
\(628\) 2157.52 3.43553
\(629\) 203.734i 0.323901i
\(630\) 0 0
\(631\) −1182.36 −1.87378 −0.936890 0.349624i \(-0.886309\pi\)
−0.936890 + 0.349624i \(0.886309\pi\)
\(632\) 3282.85 5.19438
\(633\) 11.8556 0.0187293
\(634\) 340.275 0.536711
\(635\) 0 0
\(636\) 44.6671 0.0702313
\(637\) 681.770i 1.07028i
\(638\) 536.173 0.840397
\(639\) 816.468i 1.27773i
\(640\) 0 0
\(641\) 123.199i 0.192199i −0.995372 0.0960993i \(-0.969363\pi\)
0.995372 0.0960993i \(-0.0306366\pi\)
\(642\) 51.2297i 0.0797971i
\(643\) 107.069 0.166515 0.0832577 0.996528i \(-0.473468\pi\)
0.0832577 + 0.996528i \(0.473468\pi\)
\(644\) −3993.52 −6.20112
\(645\) 0 0
\(646\) 299.268 235.558i 0.463264 0.364641i
\(647\) −277.921 −0.429553 −0.214777 0.976663i \(-0.568902\pi\)
−0.214777 + 0.976663i \(0.568902\pi\)
\(648\) 2061.18i 3.18083i
\(649\) 264.804i 0.408018i
\(650\) 0 0
\(651\) −29.8312 −0.0458237
\(652\) −1060.98 −1.62728
\(653\) −500.160 −0.765942 −0.382971 0.923760i \(-0.625099\pi\)
−0.382971 + 0.923760i \(0.625099\pi\)
\(654\) 9.11747i 0.0139411i
\(655\) 0 0
\(656\) 2178.85i 3.32141i
\(657\) −43.1260 −0.0656408
\(658\) 2940.03i 4.46813i
\(659\) 154.079i 0.233807i −0.993143 0.116903i \(-0.962703\pi\)
0.993143 0.116903i \(-0.0372968\pi\)
\(660\) 0 0
\(661\) 1164.81i 1.76220i −0.472933 0.881099i \(-0.656805\pi\)
0.472933 0.881099i \(-0.343195\pi\)
\(662\) 463.781 0.700576
\(663\) −7.23083 −0.0109062
\(664\) 2558.16i 3.85266i
\(665\) 0 0
\(666\) 1340.15 2.01224
\(667\) 993.288i 1.48919i
\(668\) 1016.51i 1.52172i
\(669\) −28.1317 −0.0420504
\(670\) 0 0
\(671\) 170.653 0.254327
\(672\) −110.873 −0.164990
\(673\) 943.259i 1.40157i 0.713371 + 0.700786i \(0.247167\pi\)
−0.713371 + 0.700786i \(0.752833\pi\)
\(674\) −197.005 −0.292292
\(675\) 0 0
\(676\) 151.799 0.224555
\(677\) 675.089i 0.997178i 0.866839 + 0.498589i \(0.166148\pi\)
−0.866839 + 0.498589i \(0.833852\pi\)
\(678\) 50.8980i 0.0750707i
\(679\) 145.515i 0.214307i
\(680\) 0 0
\(681\) −18.0056 −0.0264400
\(682\) −594.418 −0.871580
\(683\) 652.171i 0.954863i −0.878669 0.477431i \(-0.841568\pi\)
0.878669 0.477431i \(-0.158432\pi\)
\(684\) −1126.93 1431.72i −1.64756 2.09316i
\(685\) 0 0
\(686\) 52.1574i 0.0760312i
\(687\) 9.16285i 0.0133375i
\(688\) −1675.36 −2.43511
\(689\) 555.341 0.806011
\(690\) 0 0
\(691\) −1182.56 −1.71138 −0.855688 0.517491i \(-0.826866\pi\)
−0.855688 + 0.517491i \(0.826866\pi\)
\(692\) 2616.66i 3.78131i
\(693\) 474.335 0.684465
\(694\) 1918.15i 2.76391i
\(695\) 0 0
\(696\) 68.9296i 0.0990368i
\(697\) 206.872i 0.296803i
\(698\) 2244.17i 3.21514i
\(699\) 5.83734i 0.00835098i
\(700\) 0 0
\(701\) 935.918 1.33512 0.667559 0.744557i \(-0.267339\pi\)
0.667559 + 0.744557i \(0.267339\pi\)
\(702\) 95.1834i 0.135589i
\(703\) 581.157 457.437i 0.826682 0.650693i
\(704\) −1042.08 −1.48023
\(705\) 0 0
\(706\) 1179.26i 1.67034i
\(707\) −183.370 −0.259363
\(708\) 54.4658 0.0769291
\(709\) 343.933 0.485096 0.242548 0.970139i \(-0.422017\pi\)
0.242548 + 0.970139i \(0.422017\pi\)
\(710\) 0 0
\(711\) 1155.71i 1.62548i
\(712\) 3254.67 4.57117
\(713\) 1101.19i 1.54444i
\(714\) 20.3933 0.0285620
\(715\) 0 0
\(716\) 867.338i 1.21137i
\(717\) 3.14817i 0.00439076i
\(718\) 1333.98i 1.85790i
\(719\) 312.031 0.433979 0.216989 0.976174i \(-0.430376\pi\)
0.216989 + 0.976174i \(0.430376\pi\)
\(720\) 0 0
\(721\) 1839.25i 2.55097i
\(722\) −1343.87 324.781i −1.86132 0.449836i
\(723\) 31.3372 0.0433433
\(724\) 2747.38i 3.79473i
\(725\) 0 0
\(726\) 36.3443 0.0500610
\(727\) 492.307 0.677177 0.338588 0.940935i \(-0.390051\pi\)
0.338588 + 0.940935i \(0.390051\pi\)
\(728\) −3445.55 −4.73290
\(729\) 723.942 0.993062
\(730\) 0 0
\(731\) 159.067 0.217602
\(732\) 35.1006i 0.0479517i
\(733\) −599.784 −0.818259 −0.409130 0.912476i \(-0.634168\pi\)
−0.409130 + 0.912476i \(0.634168\pi\)
\(734\) 842.728i 1.14813i
\(735\) 0 0
\(736\) 4092.77i 5.56083i
\(737\) 530.042i 0.719189i
\(738\) 1360.79 1.84389
\(739\) 509.437 0.689360 0.344680 0.938720i \(-0.387987\pi\)
0.344680 + 0.938720i \(0.387987\pi\)
\(740\) 0 0
\(741\) 16.2352 + 20.6262i 0.0219098 + 0.0278356i
\(742\) −1566.24 −2.11084
\(743\) 547.250i 0.736541i 0.929719 + 0.368271i \(0.120050\pi\)
−0.929719 + 0.368271i \(0.879950\pi\)
\(744\) 76.4174i 0.102712i
\(745\) 0 0
\(746\) 798.168 1.06993
\(747\) 900.590 1.20561
\(748\) 295.539 0.395105
\(749\) 1306.47i 1.74429i
\(750\) 0 0
\(751\) 1150.89i 1.53248i 0.642554 + 0.766240i \(0.277875\pi\)
−0.642554 + 0.766240i \(0.722125\pi\)
\(752\) −4245.28 −5.64532
\(753\) 25.7380i 0.0341807i
\(754\) 1371.13i 1.81847i
\(755\) 0 0
\(756\) 195.239i 0.258253i
\(757\) −600.187 −0.792849 −0.396424 0.918067i \(-0.629749\pi\)
−0.396424 + 0.918067i \(0.629749\pi\)
\(758\) 894.276 1.17978
\(759\) 20.2891i 0.0267314i
\(760\) 0 0
\(761\) −918.373 −1.20680 −0.603399 0.797440i \(-0.706187\pi\)
−0.603399 + 0.797440i \(0.706187\pi\)
\(762\) 39.0170i 0.0512034i
\(763\) 232.515i 0.304738i
\(764\) 2525.64 3.30581
\(765\) 0 0
\(766\) 73.5441 0.0960106
\(767\) 677.168 0.882878
\(768\) 43.9492i 0.0572255i
\(769\) 573.628 0.745940 0.372970 0.927843i \(-0.378339\pi\)
0.372970 + 0.927843i \(0.378339\pi\)
\(770\) 0 0
\(771\) −35.6389 −0.0462243
\(772\) 3019.52i 3.91129i
\(773\) 845.816i 1.09420i 0.837068 + 0.547100i \(0.184268\pi\)
−0.837068 + 0.547100i \(0.815732\pi\)
\(774\) 1046.34i 1.35186i
\(775\) 0 0
\(776\) 372.759 0.480359
\(777\) 39.6023 0.0509682
\(778\) 1475.90i 1.89705i
\(779\) 590.108 464.482i 0.757520 0.596254i
\(780\) 0 0
\(781\) 480.757i 0.615566i
\(782\) 752.797i 0.962656i
\(783\) 48.5609 0.0620190
\(784\) 2776.46 3.54140
\(785\) 0 0
\(786\) −18.1998 −0.0231550
\(787\) 435.404i 0.553245i 0.960979 + 0.276623i \(0.0892152\pi\)
−0.960979 + 0.276623i \(0.910785\pi\)
\(788\) −305.382 −0.387540
\(789\) 23.4793i 0.0297583i
\(790\) 0 0
\(791\) 1298.01i 1.64097i
\(792\) 1215.08i 1.53420i
\(793\) 436.402i 0.550318i
\(794\) 848.559i 1.06871i
\(795\) 0 0
\(796\) 3581.49 4.49936
\(797\) 42.1998i 0.0529483i −0.999649 0.0264742i \(-0.991572\pi\)
0.999649 0.0264742i \(-0.00842797\pi\)
\(798\) −45.7884 58.1725i −0.0573789 0.0728979i
\(799\) 403.070 0.504468
\(800\) 0 0
\(801\) 1145.79i 1.43045i
\(802\) 2620.33 3.26725
\(803\) −25.3937 −0.0316235
\(804\) −109.021 −0.135598
\(805\) 0 0
\(806\) 1520.07i 1.88595i
\(807\) −28.7874 −0.0356722
\(808\) 469.731i 0.581350i
\(809\) −1054.72 −1.30374 −0.651868 0.758332i \(-0.726014\pi\)
−0.651868 + 0.758332i \(0.726014\pi\)
\(810\) 0 0
\(811\) 388.797i 0.479404i 0.970846 + 0.239702i \(0.0770498\pi\)
−0.970846 + 0.239702i \(0.922950\pi\)
\(812\) 2812.44i 3.46360i
\(813\) 20.6511i 0.0254011i
\(814\) 789.116 0.969430
\(815\) 0 0
\(816\) 29.4471i 0.0360871i
\(817\) −357.149 453.745i −0.437147 0.555379i
\(818\) −1421.89 −1.73826
\(819\) 1212.99i 1.48106i
\(820\) 0 0
\(821\) 934.546 1.13830 0.569151 0.822233i \(-0.307272\pi\)
0.569151 + 0.822233i \(0.307272\pi\)
\(822\) 11.0883 0.0134894
\(823\) 349.311 0.424436 0.212218 0.977222i \(-0.431931\pi\)
0.212218 + 0.977222i \(0.431931\pi\)
\(824\) −4711.53 −5.71788
\(825\) 0 0
\(826\) −1909.83 −2.31215
\(827\) 770.532i 0.931719i 0.884859 + 0.465860i \(0.154255\pi\)
−0.884859 + 0.465860i \(0.845745\pi\)
\(828\) −3601.44 −4.34956
\(829\) 61.0383i 0.0736288i 0.999322 + 0.0368144i \(0.0117210\pi\)
−0.999322 + 0.0368144i \(0.988279\pi\)
\(830\) 0 0
\(831\) 11.1123i 0.0133723i
\(832\) 2664.86i 3.20296i
\(833\) −263.612 −0.316461
\(834\) −60.9311 −0.0730588
\(835\) 0 0
\(836\) −663.563 843.034i −0.793736 1.00841i
\(837\) −53.8360 −0.0643202
\(838\) 1812.00i 2.16229i
\(839\) 1088.98i 1.29795i −0.760810 0.648975i \(-0.775198\pi\)
0.760810 0.648975i \(-0.224802\pi\)
\(840\) 0 0
\(841\) 141.476 0.168223
\(842\) −969.103 −1.15095
\(843\) −22.5865 −0.0267930
\(844\) 1239.15i 1.46819i
\(845\) 0 0
\(846\) 2651.38i 3.13402i
\(847\) −926.859 −1.09429
\(848\) 2261.59i 2.66697i
\(849\) 15.1906i 0.0178924i
\(850\) 0 0
\(851\) 1461.88i 1.71784i
\(852\) 98.8840 0.116061
\(853\) −1351.24 −1.58411 −0.792053 0.610452i \(-0.790988\pi\)
−0.792053 + 0.610452i \(0.790988\pi\)
\(854\) 1230.80i 1.44121i
\(855\) 0 0
\(856\) 3346.73 3.90973
\(857\) 175.280i 0.204527i 0.994757 + 0.102264i \(0.0326085\pi\)
−0.994757 + 0.102264i \(0.967391\pi\)
\(858\) 28.0070i 0.0326421i
\(859\) −567.473 −0.660620 −0.330310 0.943872i \(-0.607153\pi\)
−0.330310 + 0.943872i \(0.607153\pi\)
\(860\) 0 0
\(861\) 40.2122 0.0467041
\(862\) 492.971 0.571892
\(863\) 1339.71i 1.55239i −0.630492 0.776196i \(-0.717147\pi\)
0.630492 0.776196i \(-0.282853\pi\)
\(864\) −200.091 −0.231587
\(865\) 0 0
\(866\) −619.228 −0.715044
\(867\) 26.7000i 0.0307959i
\(868\) 3117.95i 3.59211i
\(869\) 680.513i 0.783099i
\(870\) 0 0
\(871\) −1355.45 −1.55620
\(872\) 595.625 0.683056
\(873\) 131.228i 0.150319i
\(874\) −2147.38 + 1690.23i −2.45695 + 1.93390i
\(875\) 0 0
\(876\) 5.22307i 0.00596241i
\(877\) 1155.71i 1.31780i 0.752228 + 0.658902i \(0.228979\pi\)
−0.752228 + 0.658902i \(0.771021\pi\)
\(878\) 554.697 0.631773
\(879\) 6.97278 0.00793263
\(880\) 0 0
\(881\) 1056.32 1.19900 0.599500 0.800374i \(-0.295366\pi\)
0.599500 + 0.800374i \(0.295366\pi\)
\(882\) 1734.03i 1.96602i
\(883\) −624.622 −0.707386 −0.353693 0.935362i \(-0.615074\pi\)
−0.353693 + 0.935362i \(0.615074\pi\)
\(884\) 755.765i 0.854938i
\(885\) 0 0
\(886\) 904.364i 1.02073i
\(887\) 820.850i 0.925423i 0.886509 + 0.462711i \(0.153123\pi\)
−0.886509 + 0.462711i \(0.846877\pi\)
\(888\) 101.448i 0.114243i
\(889\) 995.019i 1.11926i
\(890\) 0 0
\(891\) 427.268 0.479538
\(892\) 2940.32i 3.29633i
\(893\) −905.001 1149.77i −1.01344 1.28754i
\(894\) 16.7664 0.0187543
\(895\) 0 0
\(896\) 3170.44i 3.53844i
\(897\) 51.8843 0.0578421
\(898\) −1501.45 −1.67200
\(899\) 775.514 0.862640
\(900\) 0 0
\(901\) 214.727i 0.238321i
\(902\) 801.269 0.888325
\(903\) 30.9199i 0.0342413i
\(904\) 3325.06 3.67816
\(905\) 0 0
\(906\) 14.0644i 0.0155236i
\(907\) 1430.19i 1.57684i −0.615140 0.788418i \(-0.710901\pi\)
0.615140 0.788418i \(-0.289099\pi\)
\(908\) 1881.94i 2.07263i
\(909\) −165.367 −0.181922
\(910\) 0 0
\(911\) 436.452i 0.479091i −0.970885 0.239546i \(-0.923001\pi\)
0.970885 0.239546i \(-0.0769985\pi\)
\(912\) −83.9987 + 66.1165i −0.0921038 + 0.0724962i
\(913\) 530.290 0.580822
\(914\) 2295.49i 2.51148i
\(915\) 0 0
\(916\) 957.699 1.04552
\(917\) 464.136 0.506146
\(918\) 36.8035 0.0400909
\(919\) 110.671 0.120426 0.0602130 0.998186i \(-0.480822\pi\)
0.0602130 + 0.998186i \(0.480822\pi\)
\(920\) 0 0
\(921\) 14.8829 0.0161595
\(922\) 1167.85i 1.26665i
\(923\) 1229.41 1.33198
\(924\) 57.4475i 0.0621727i
\(925\) 0 0
\(926\) 1751.99i 1.89200i
\(927\) 1658.68i 1.78929i
\(928\) 2882.34 3.10597
\(929\) −1781.29 −1.91743 −0.958714 0.284371i \(-0.908215\pi\)
−0.958714 + 0.284371i \(0.908215\pi\)
\(930\) 0 0
\(931\) 591.880 + 751.962i 0.635746 + 0.807693i
\(932\) −610.117 −0.654632
\(933\) 4.48410i 0.00480611i
\(934\) 810.465i 0.867736i
\(935\) 0 0
\(936\) −3107.27 −3.31973
\(937\) 400.832 0.427782 0.213891 0.976858i \(-0.431386\pi\)
0.213891 + 0.976858i \(0.431386\pi\)
\(938\) 3822.80 4.07548
\(939\) 20.8931i 0.0222504i
\(940\) 0 0
\(941\) 1092.72i 1.16123i −0.814177 0.580617i \(-0.802812\pi\)
0.814177 0.580617i \(-0.197188\pi\)
\(942\) 79.0557 0.0839232
\(943\) 1484.39i 1.57412i
\(944\) 2757.72i 2.92131i
\(945\) 0 0
\(946\) 616.111i 0.651280i
\(947\) 1260.43 1.33097 0.665487 0.746410i \(-0.268224\pi\)
0.665487 + 0.746410i \(0.268224\pi\)
\(948\) 139.971 0.147648
\(949\) 64.9379i 0.0684277i
\(950\) 0 0
\(951\) 9.06808 0.00953531
\(952\) 1332.25i 1.39942i
\(953\) 381.589i 0.400408i −0.979754 0.200204i \(-0.935839\pi\)
0.979754 0.200204i \(-0.0641606\pi\)
\(954\) −1412.47 −1.48058
\(955\) 0 0
\(956\) 329.046 0.344191
\(957\) 14.2886 0.0149307
\(958\) 446.675i 0.466258i
\(959\) −282.775 −0.294864
\(960\) 0 0
\(961\) 101.243 0.105351
\(962\) 2017.96i 2.09768i
\(963\) 1178.20i 1.22347i
\(964\) 3275.36i 3.39768i
\(965\) 0 0
\(966\) −146.331 −0.151481
\(967\) −1649.39 −1.70568 −0.852840 0.522172i \(-0.825122\pi\)
−0.852840 + 0.522172i \(0.825122\pi\)
\(968\) 2374.30i 2.45279i
\(969\) 7.97529 6.27746i 0.00823043 0.00647829i
\(970\) 0 0
\(971\) 1246.99i 1.28423i −0.766607 0.642116i \(-0.778057\pi\)
0.766607 0.642116i \(-0.221943\pi\)
\(972\) 264.157i 0.271766i
\(973\) 1553.88 1.59699
\(974\) 331.313 0.340158
\(975\) 0 0
\(976\) 1777.22 1.82092
\(977\) 162.841i 0.166674i 0.996521 + 0.0833372i \(0.0265578\pi\)
−0.996521 + 0.0833372i \(0.973442\pi\)
\(978\) −38.8766 −0.0397511
\(979\) 674.672i 0.689144i
\(980\) 0 0
\(981\) 209.687i 0.213748i
\(982\) 437.717i 0.445740i
\(983\) 984.880i 1.00191i 0.865472 + 0.500956i \(0.167018\pi\)
−0.865472 + 0.500956i \(0.832982\pi\)
\(984\) 103.010i 0.104685i
\(985\) 0 0
\(986\) −530.158 −0.537686
\(987\) 78.3498i 0.0793817i
\(988\) −2155.84 + 1696.89i −2.18203 + 1.71750i
\(989\) −1141.38 −1.15407
\(990\) 0 0
\(991\) 44.4482i 0.0448518i 0.999749 + 0.0224259i \(0.00713899\pi\)
−0.999749 + 0.0224259i \(0.992861\pi\)
\(992\) −3195.45 −3.22122
\(993\) 12.3595 0.0124466
\(994\) −3467.34 −3.48827
\(995\) 0 0
\(996\) 109.072i 0.109510i
\(997\) −666.830 −0.668836 −0.334418 0.942425i \(-0.608540\pi\)
−0.334418 + 0.942425i \(0.608540\pi\)
\(998\) 1124.59i 1.12685i
\(999\) 71.4697 0.0715412
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 475.3.c.i.151.14 yes 14
5.2 odd 4 475.3.d.d.474.2 28
5.3 odd 4 475.3.d.d.474.27 28
5.4 even 2 475.3.c.h.151.1 14
19.18 odd 2 inner 475.3.c.i.151.1 yes 14
95.18 even 4 475.3.d.d.474.1 28
95.37 even 4 475.3.d.d.474.28 28
95.94 odd 2 475.3.c.h.151.14 yes 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.3.c.h.151.1 14 5.4 even 2
475.3.c.h.151.14 yes 14 95.94 odd 2
475.3.c.i.151.1 yes 14 19.18 odd 2 inner
475.3.c.i.151.14 yes 14 1.1 even 1 trivial
475.3.d.d.474.1 28 95.18 even 4
475.3.d.d.474.2 28 5.2 odd 4
475.3.d.d.474.27 28 5.3 odd 4
475.3.d.d.474.28 28 95.37 even 4