Properties

Label 475.3.c.f.151.1
Level $475$
Weight $3$
Character 475.151
Analytic conductor $12.943$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 42x^{6} + 771x^{4} - 7098x^{2} + 28561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 151.1
Root \(-3.32308 + 1.39897i\) of defining polynomial
Character \(\chi\) \(=\) 475.151
Dual form 475.3.c.f.151.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.79793i q^{2} +1.15894i q^{3} -3.82843 q^{4} +3.24264 q^{6} -6.64617 q^{7} -0.480049i q^{8} +7.65685 q^{9} +O(q^{10})\) \(q-2.79793i q^{2} +1.15894i q^{3} -3.82843 q^{4} +3.24264 q^{6} -6.64617 q^{7} -0.480049i q^{8} +7.65685 q^{9} -6.24264 q^{11} -4.43692i q^{12} +18.6254i q^{13} +18.5955i q^{14} -16.6569 q^{16} -29.3376 q^{17} -21.4234i q^{18} +(3.89949 - 18.5955i) q^{19} -7.70252i q^{21} +17.4665i q^{22} -19.9385 q^{23} +0.556349 q^{24} +52.1127 q^{26} +19.3043i q^{27} +25.4444 q^{28} +29.4885i q^{29} +26.2981i q^{31} +44.6846i q^{32} -7.23486i q^{33} +82.0847i q^{34} -29.3137 q^{36} -47.4825i q^{37} +(-52.0291 - 10.9105i) q^{38} -21.5858 q^{39} +41.7031i q^{41} -21.5511 q^{42} -14.9050 q^{43} +23.8995 q^{44} +55.7866i q^{46} -5.50587 q^{47} -19.3043i q^{48} -4.82843 q^{49} -34.0006i q^{51} -71.3061i q^{52} +55.4786i q^{53} +54.0122 q^{54} +3.19049i q^{56} +(21.5511 + 4.51929i) q^{57} +82.5070 q^{58} +86.5967i q^{59} +20.5858 q^{61} +73.5802 q^{62} -50.8888 q^{63} +58.3970 q^{64} -20.2426 q^{66} +0.363570i q^{67} +112.317 q^{68} -23.1076i q^{69} -100.680i q^{71} -3.67567i q^{72} -0.472329 q^{73} -132.853 q^{74} +(-14.9289 + 71.1917i) q^{76} +41.4896 q^{77} +60.3956i q^{78} -120.597i q^{79} +46.5391 q^{81} +116.682 q^{82} -42.1576 q^{83} +29.4885i q^{84} +41.7031i q^{86} -34.1755 q^{87} +2.99678i q^{88} +4.51203i q^{89} -123.788i q^{91} +76.3331 q^{92} -30.4779 q^{93} +15.4050i q^{94} -51.7868 q^{96} -11.5894i q^{97} +13.5096i q^{98} -47.7990 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} - 8 q^{6} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 8 q^{6} + 16 q^{9} - 16 q^{11} - 88 q^{16} - 48 q^{19} - 120 q^{24} + 168 q^{26} - 144 q^{36} - 184 q^{39} + 112 q^{44} - 16 q^{49} + 104 q^{54} + 176 q^{61} - 8 q^{64} - 128 q^{66} - 384 q^{74} - 176 q^{76} - 216 q^{81} - 584 q^{96} - 224 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.79793i 1.39897i −0.714649 0.699483i \(-0.753414\pi\)
0.714649 0.699483i \(-0.246586\pi\)
\(3\) 1.15894i 0.386314i 0.981168 + 0.193157i \(0.0618727\pi\)
−0.981168 + 0.193157i \(0.938127\pi\)
\(4\) −3.82843 −0.957107
\(5\) 0 0
\(6\) 3.24264 0.540440
\(7\) −6.64617 −0.949453 −0.474726 0.880133i \(-0.657453\pi\)
−0.474726 + 0.880133i \(0.657453\pi\)
\(8\) 0.480049i 0.0600062i
\(9\) 7.65685 0.850762
\(10\) 0 0
\(11\) −6.24264 −0.567513 −0.283756 0.958896i \(-0.591581\pi\)
−0.283756 + 0.958896i \(0.591581\pi\)
\(12\) 4.43692i 0.369744i
\(13\) 18.6254i 1.43273i 0.697728 + 0.716363i \(0.254194\pi\)
−0.697728 + 0.716363i \(0.745806\pi\)
\(14\) 18.5955i 1.32825i
\(15\) 0 0
\(16\) −16.6569 −1.04105
\(17\) −29.3376 −1.72574 −0.862871 0.505424i \(-0.831336\pi\)
−0.862871 + 0.505424i \(0.831336\pi\)
\(18\) 21.4234i 1.19019i
\(19\) 3.89949 18.5955i 0.205237 0.978712i
\(20\) 0 0
\(21\) 7.70252i 0.366787i
\(22\) 17.4665i 0.793931i
\(23\) −19.9385 −0.866892 −0.433446 0.901180i \(-0.642702\pi\)
−0.433446 + 0.901180i \(0.642702\pi\)
\(24\) 0.556349 0.0231812
\(25\) 0 0
\(26\) 52.1127 2.00433
\(27\) 19.3043i 0.714975i
\(28\) 25.4444 0.908728
\(29\) 29.4885i 1.01685i 0.861107 + 0.508423i \(0.169771\pi\)
−0.861107 + 0.508423i \(0.830229\pi\)
\(30\) 0 0
\(31\) 26.2981i 0.848324i 0.905586 + 0.424162i \(0.139431\pi\)
−0.905586 + 0.424162i \(0.860569\pi\)
\(32\) 44.6846i 1.39639i
\(33\) 7.23486i 0.219238i
\(34\) 82.0847i 2.41425i
\(35\) 0 0
\(36\) −29.3137 −0.814270
\(37\) 47.4825i 1.28331i −0.766993 0.641655i \(-0.778248\pi\)
0.766993 0.641655i \(-0.221752\pi\)
\(38\) −52.0291 10.9105i −1.36919 0.287119i
\(39\) −21.5858 −0.553482
\(40\) 0 0
\(41\) 41.7031i 1.01715i 0.861018 + 0.508574i \(0.169827\pi\)
−0.861018 + 0.508574i \(0.830173\pi\)
\(42\) −21.5511 −0.513122
\(43\) −14.9050 −0.346627 −0.173314 0.984867i \(-0.555447\pi\)
−0.173314 + 0.984867i \(0.555447\pi\)
\(44\) 23.8995 0.543170
\(45\) 0 0
\(46\) 55.7866i 1.21275i
\(47\) −5.50587 −0.117146 −0.0585731 0.998283i \(-0.518655\pi\)
−0.0585731 + 0.998283i \(0.518655\pi\)
\(48\) 19.3043i 0.402173i
\(49\) −4.82843 −0.0985393
\(50\) 0 0
\(51\) 34.0006i 0.666678i
\(52\) 71.3061i 1.37127i
\(53\) 55.4786i 1.04677i 0.852098 + 0.523383i \(0.175330\pi\)
−0.852098 + 0.523383i \(0.824670\pi\)
\(54\) 54.0122 1.00023
\(55\) 0 0
\(56\) 3.19049i 0.0569730i
\(57\) 21.5511 + 4.51929i 0.378090 + 0.0792857i
\(58\) 82.5070 1.42253
\(59\) 86.5967i 1.46774i 0.679290 + 0.733870i \(0.262288\pi\)
−0.679290 + 0.733870i \(0.737712\pi\)
\(60\) 0 0
\(61\) 20.5858 0.337472 0.168736 0.985661i \(-0.446031\pi\)
0.168736 + 0.985661i \(0.446031\pi\)
\(62\) 73.5802 1.18678
\(63\) −50.8888 −0.807758
\(64\) 58.3970 0.912453
\(65\) 0 0
\(66\) −20.2426 −0.306707
\(67\) 0.363570i 0.00542642i 0.999996 + 0.00271321i \(0.000863642\pi\)
−0.999996 + 0.00271321i \(0.999136\pi\)
\(68\) 112.317 1.65172
\(69\) 23.1076i 0.334892i
\(70\) 0 0
\(71\) 100.680i 1.41803i −0.705193 0.709015i \(-0.749140\pi\)
0.705193 0.709015i \(-0.250860\pi\)
\(72\) 3.67567i 0.0510509i
\(73\) −0.472329 −0.00647026 −0.00323513 0.999995i \(-0.501030\pi\)
−0.00323513 + 0.999995i \(0.501030\pi\)
\(74\) −132.853 −1.79531
\(75\) 0 0
\(76\) −14.9289 + 71.1917i −0.196433 + 0.936732i
\(77\) 41.4896 0.538827
\(78\) 60.3956i 0.774302i
\(79\) 120.597i 1.52655i −0.646075 0.763274i \(-0.723591\pi\)
0.646075 0.763274i \(-0.276409\pi\)
\(80\) 0 0
\(81\) 46.5391 0.574557
\(82\) 116.682 1.42296
\(83\) −42.1576 −0.507923 −0.253962 0.967214i \(-0.581734\pi\)
−0.253962 + 0.967214i \(0.581734\pi\)
\(84\) 29.4885i 0.351054i
\(85\) 0 0
\(86\) 41.7031i 0.484920i
\(87\) −34.1755 −0.392822
\(88\) 2.99678i 0.0340543i
\(89\) 4.51203i 0.0506970i 0.999679 + 0.0253485i \(0.00806954\pi\)
−0.999679 + 0.0253485i \(0.991930\pi\)
\(90\) 0 0
\(91\) 123.788i 1.36031i
\(92\) 76.3331 0.829708
\(93\) −30.4779 −0.327720
\(94\) 15.4050i 0.163883i
\(95\) 0 0
\(96\) −51.7868 −0.539446
\(97\) 11.5894i 0.119479i −0.998214 0.0597393i \(-0.980973\pi\)
0.998214 0.0597393i \(-0.0190269\pi\)
\(98\) 13.5096i 0.137853i
\(99\) −47.7990 −0.482818
\(100\) 0 0
\(101\) −96.4924 −0.955371 −0.477685 0.878531i \(-0.658524\pi\)
−0.477685 + 0.878531i \(0.658524\pi\)
\(102\) −95.1313 −0.932660
\(103\) 131.702i 1.27866i −0.768934 0.639329i \(-0.779212\pi\)
0.768934 0.639329i \(-0.220788\pi\)
\(104\) 8.94113 0.0859724
\(105\) 0 0
\(106\) 155.225 1.46439
\(107\) 43.2785i 0.404472i 0.979337 + 0.202236i \(0.0648208\pi\)
−0.979337 + 0.202236i \(0.935179\pi\)
\(108\) 73.9052i 0.684307i
\(109\) 14.0835i 0.129206i −0.997911 0.0646032i \(-0.979422\pi\)
0.997911 0.0646032i \(-0.0205782\pi\)
\(110\) 0 0
\(111\) 55.0294 0.495761
\(112\) 110.704 0.988431
\(113\) 78.2256i 0.692262i −0.938186 0.346131i \(-0.887495\pi\)
0.938186 0.346131i \(-0.112505\pi\)
\(114\) 12.6447 60.2986i 0.110918 0.528935i
\(115\) 0 0
\(116\) 112.895i 0.973231i
\(117\) 142.612i 1.21891i
\(118\) 242.292 2.05332
\(119\) 194.983 1.63851
\(120\) 0 0
\(121\) −82.0294 −0.677929
\(122\) 57.5976i 0.472112i
\(123\) −48.3315 −0.392939
\(124\) 100.680i 0.811937i
\(125\) 0 0
\(126\) 142.383i 1.13003i
\(127\) 71.5532i 0.563411i −0.959501 0.281705i \(-0.909100\pi\)
0.959501 0.281705i \(-0.0909001\pi\)
\(128\) 15.3474i 0.119902i
\(129\) 17.2740i 0.133907i
\(130\) 0 0
\(131\) −154.593 −1.18010 −0.590049 0.807367i \(-0.700892\pi\)
−0.590049 + 0.807367i \(0.700892\pi\)
\(132\) 27.6981i 0.209834i
\(133\) −25.9167 + 123.589i −0.194862 + 0.929241i
\(134\) 1.01724 0.00759137
\(135\) 0 0
\(136\) 14.0835i 0.103555i
\(137\) −182.004 −1.32850 −0.664248 0.747513i \(-0.731248\pi\)
−0.664248 + 0.747513i \(0.731248\pi\)
\(138\) −64.6534 −0.468503
\(139\) −180.191 −1.29634 −0.648169 0.761497i \(-0.724465\pi\)
−0.648169 + 0.761497i \(0.724465\pi\)
\(140\) 0 0
\(141\) 6.38098i 0.0452552i
\(142\) −281.696 −1.98378
\(143\) 116.272i 0.813090i
\(144\) −127.539 −0.885688
\(145\) 0 0
\(146\) 1.32154i 0.00905167i
\(147\) 5.59587i 0.0380671i
\(148\) 181.783i 1.22827i
\(149\) 139.054 0.933247 0.466624 0.884456i \(-0.345470\pi\)
0.466624 + 0.884456i \(0.345470\pi\)
\(150\) 0 0
\(151\) 49.9530i 0.330815i −0.986225 0.165407i \(-0.947106\pi\)
0.986225 0.165407i \(-0.0528939\pi\)
\(152\) −8.92677 1.87195i −0.0587288 0.0123155i
\(153\) −224.634 −1.46819
\(154\) 116.085i 0.753800i
\(155\) 0 0
\(156\) 82.6396 0.529741
\(157\) 19.7429 0.125751 0.0628754 0.998021i \(-0.479973\pi\)
0.0628754 + 0.998021i \(0.479973\pi\)
\(158\) −337.423 −2.13559
\(159\) −64.2965 −0.404380
\(160\) 0 0
\(161\) 132.515 0.823073
\(162\) 130.213i 0.803786i
\(163\) 266.123 1.63266 0.816330 0.577586i \(-0.196005\pi\)
0.816330 + 0.577586i \(0.196005\pi\)
\(164\) 159.657i 0.973520i
\(165\) 0 0
\(166\) 117.954i 0.710567i
\(167\) 91.4058i 0.547340i 0.961824 + 0.273670i \(0.0882376\pi\)
−0.961824 + 0.273670i \(0.911762\pi\)
\(168\) −3.69759 −0.0220095
\(169\) −177.907 −1.05270
\(170\) 0 0
\(171\) 29.8579 142.383i 0.174607 0.832651i
\(172\) 57.0626 0.331759
\(173\) 136.221i 0.787404i 0.919238 + 0.393702i \(0.128806\pi\)
−0.919238 + 0.393702i \(0.871194\pi\)
\(174\) 95.6208i 0.549545i
\(175\) 0 0
\(176\) 103.983 0.590811
\(177\) −100.361 −0.567009
\(178\) 12.6244 0.0709234
\(179\) 328.339i 1.83429i −0.398549 0.917147i \(-0.630486\pi\)
0.398549 0.917147i \(-0.369514\pi\)
\(180\) 0 0
\(181\) 83.4062i 0.460808i −0.973095 0.230404i \(-0.925995\pi\)
0.973095 0.230404i \(-0.0740047\pi\)
\(182\) −346.350 −1.90302
\(183\) 23.8577i 0.130370i
\(184\) 9.57147i 0.0520189i
\(185\) 0 0
\(186\) 85.2752i 0.458469i
\(187\) 183.144 0.979381
\(188\) 21.0788 0.112121
\(189\) 128.300i 0.678835i
\(190\) 0 0
\(191\) −89.9878 −0.471140 −0.235570 0.971857i \(-0.575696\pi\)
−0.235570 + 0.971857i \(0.575696\pi\)
\(192\) 67.6787i 0.352493i
\(193\) 311.613i 1.61458i 0.590158 + 0.807288i \(0.299065\pi\)
−0.590158 + 0.807288i \(0.700935\pi\)
\(194\) −32.4264 −0.167146
\(195\) 0 0
\(196\) 18.4853 0.0943127
\(197\) 14.9050 0.0756598 0.0378299 0.999284i \(-0.487956\pi\)
0.0378299 + 0.999284i \(0.487956\pi\)
\(198\) 133.738i 0.675446i
\(199\) 293.149 1.47311 0.736556 0.676377i \(-0.236451\pi\)
0.736556 + 0.676377i \(0.236451\pi\)
\(200\) 0 0
\(201\) −0.421356 −0.00209630
\(202\) 269.979i 1.33653i
\(203\) 195.986i 0.965448i
\(204\) 130.169i 0.638082i
\(205\) 0 0
\(206\) −368.492 −1.78880
\(207\) −152.666 −0.737518
\(208\) 310.241i 1.49154i
\(209\) −24.3431 + 116.085i −0.116474 + 0.555432i
\(210\) 0 0
\(211\) 36.6437i 0.173667i 0.996223 + 0.0868333i \(0.0276748\pi\)
−0.996223 + 0.0868333i \(0.972325\pi\)
\(212\) 212.396i 1.00187i
\(213\) 116.682 0.547805
\(214\) 121.090 0.565843
\(215\) 0 0
\(216\) 9.26703 0.0429029
\(217\) 174.781i 0.805444i
\(218\) −39.4047 −0.180755
\(219\) 0.547401i 0.00249955i
\(220\) 0 0
\(221\) 546.426i 2.47251i
\(222\) 153.969i 0.693553i
\(223\) 122.780i 0.550581i −0.961361 0.275291i \(-0.911226\pi\)
0.961361 0.275291i \(-0.0887741\pi\)
\(224\) 296.981i 1.32581i
\(225\) 0 0
\(226\) −218.870 −0.968452
\(227\) 108.742i 0.479038i 0.970892 + 0.239519i \(0.0769898\pi\)
−0.970892 + 0.239519i \(0.923010\pi\)
\(228\) −82.5070 17.3018i −0.361873 0.0758849i
\(229\) −266.627 −1.16431 −0.582156 0.813077i \(-0.697791\pi\)
−0.582156 + 0.813077i \(0.697791\pi\)
\(230\) 0 0
\(231\) 48.0841i 0.208156i
\(232\) 14.1560 0.0610171
\(233\) −164.623 −0.706535 −0.353267 0.935522i \(-0.614929\pi\)
−0.353267 + 0.935522i \(0.614929\pi\)
\(234\) 399.019 1.70521
\(235\) 0 0
\(236\) 331.529i 1.40478i
\(237\) 139.765 0.589727
\(238\) 545.549i 2.29222i
\(239\) −277.825 −1.16245 −0.581225 0.813743i \(-0.697426\pi\)
−0.581225 + 0.813743i \(0.697426\pi\)
\(240\) 0 0
\(241\) 102.549i 0.425515i 0.977105 + 0.212758i \(0.0682445\pi\)
−0.977105 + 0.212758i \(0.931756\pi\)
\(242\) 229.513i 0.948400i
\(243\) 227.675i 0.936934i
\(244\) −78.8112 −0.322997
\(245\) 0 0
\(246\) 135.228i 0.549708i
\(247\) 346.350 + 72.6298i 1.40223 + 0.294048i
\(248\) 12.6244 0.0509047
\(249\) 48.8582i 0.196218i
\(250\) 0 0
\(251\) 260.184 1.03659 0.518294 0.855202i \(-0.326567\pi\)
0.518294 + 0.855202i \(0.326567\pi\)
\(252\) 194.824 0.773111
\(253\) 124.469 0.491972
\(254\) −200.201 −0.788193
\(255\) 0 0
\(256\) 276.529 1.08019
\(257\) 373.585i 1.45364i −0.686829 0.726819i \(-0.740998\pi\)
0.686829 0.726819i \(-0.259002\pi\)
\(258\) −48.3315 −0.187331
\(259\) 315.577i 1.21844i
\(260\) 0 0
\(261\) 225.790i 0.865094i
\(262\) 432.541i 1.65092i
\(263\) 222.353 0.845450 0.422725 0.906258i \(-0.361074\pi\)
0.422725 + 0.906258i \(0.361074\pi\)
\(264\) −3.47309 −0.0131556
\(265\) 0 0
\(266\) 345.794 + 72.5132i 1.29998 + 0.272606i
\(267\) −5.22918 −0.0195850
\(268\) 1.39190i 0.00519366i
\(269\) 216.765i 0.805820i 0.915240 + 0.402910i \(0.132001\pi\)
−0.915240 + 0.402910i \(0.867999\pi\)
\(270\) 0 0
\(271\) 7.01724 0.0258939 0.0129469 0.999916i \(-0.495879\pi\)
0.0129469 + 0.999916i \(0.495879\pi\)
\(272\) 488.672 1.79659
\(273\) 143.463 0.525505
\(274\) 509.235i 1.85852i
\(275\) 0 0
\(276\) 88.4656i 0.320528i
\(277\) 247.993 0.895282 0.447641 0.894213i \(-0.352264\pi\)
0.447641 + 0.894213i \(0.352264\pi\)
\(278\) 504.162i 1.81353i
\(279\) 201.360i 0.721722i
\(280\) 0 0
\(281\) 350.899i 1.24875i 0.781125 + 0.624375i \(0.214646\pi\)
−0.781125 + 0.624375i \(0.785354\pi\)
\(282\) −17.8535 −0.0633105
\(283\) 222.353 0.785700 0.392850 0.919602i \(-0.371489\pi\)
0.392850 + 0.919602i \(0.371489\pi\)
\(284\) 385.447i 1.35721i
\(285\) 0 0
\(286\) −325.321 −1.13749
\(287\) 277.166i 0.965735i
\(288\) 342.143i 1.18800i
\(289\) 571.696 1.97819
\(290\) 0 0
\(291\) 13.4315 0.0461562
\(292\) 1.80828 0.00619273
\(293\) 432.788i 1.47709i −0.674204 0.738546i \(-0.735513\pi\)
0.674204 0.738546i \(-0.264487\pi\)
\(294\) −15.6569 −0.0532546
\(295\) 0 0
\(296\) −22.7939 −0.0770066
\(297\) 120.510i 0.405757i
\(298\) 389.063i 1.30558i
\(299\) 371.363i 1.24202i
\(300\) 0 0
\(301\) 99.0610 0.329106
\(302\) −139.765 −0.462799
\(303\) 111.829i 0.369073i
\(304\) −64.9533 + 309.743i −0.213662 + 1.01889i
\(305\) 0 0
\(306\) 628.510i 2.05396i
\(307\) 185.788i 0.605174i 0.953122 + 0.302587i \(0.0978503\pi\)
−0.953122 + 0.302587i \(0.902150\pi\)
\(308\) −158.840 −0.515715
\(309\) 152.635 0.493963
\(310\) 0 0
\(311\) −288.345 −0.927155 −0.463578 0.886056i \(-0.653434\pi\)
−0.463578 + 0.886056i \(0.653434\pi\)
\(312\) 10.3622i 0.0332123i
\(313\) 561.976 1.79545 0.897725 0.440556i \(-0.145219\pi\)
0.897725 + 0.440556i \(0.145219\pi\)
\(314\) 55.2392i 0.175921i
\(315\) 0 0
\(316\) 461.698i 1.46107i
\(317\) 18.3642i 0.0579313i 0.999580 + 0.0289656i \(0.00922134\pi\)
−0.999580 + 0.0289656i \(0.990779\pi\)
\(318\) 179.897i 0.565714i
\(319\) 184.086i 0.577073i
\(320\) 0 0
\(321\) −50.1573 −0.156253
\(322\) 370.767i 1.15145i
\(323\) −114.402 + 545.549i −0.354185 + 1.68901i
\(324\) −178.172 −0.549912
\(325\) 0 0
\(326\) 744.596i 2.28404i
\(327\) 16.3220 0.0499142
\(328\) 20.0195 0.0610352
\(329\) 36.5929 0.111225
\(330\) 0 0
\(331\) 319.862i 0.966350i 0.875524 + 0.483175i \(0.160517\pi\)
−0.875524 + 0.483175i \(0.839483\pi\)
\(332\) 161.397 0.486137
\(333\) 363.567i 1.09179i
\(334\) 255.747 0.765710
\(335\) 0 0
\(336\) 128.300i 0.381845i
\(337\) 72.9992i 0.216615i −0.994117 0.108307i \(-0.965457\pi\)
0.994117 0.108307i \(-0.0345431\pi\)
\(338\) 497.771i 1.47269i
\(339\) 90.6589 0.267431
\(340\) 0 0
\(341\) 164.169i 0.481435i
\(342\) −398.379 83.5403i −1.16485 0.244270i
\(343\) 357.753 1.04301
\(344\) 7.15512i 0.0207998i
\(345\) 0 0
\(346\) 381.137 1.10155
\(347\) 370.850 1.06873 0.534365 0.845254i \(-0.320551\pi\)
0.534365 + 0.845254i \(0.320551\pi\)
\(348\) 130.838 0.375973
\(349\) −41.5980 −0.119192 −0.0595960 0.998223i \(-0.518981\pi\)
−0.0595960 + 0.998223i \(0.518981\pi\)
\(350\) 0 0
\(351\) −359.551 −1.02436
\(352\) 278.950i 0.792471i
\(353\) −402.468 −1.14014 −0.570068 0.821598i \(-0.693083\pi\)
−0.570068 + 0.821598i \(0.693083\pi\)
\(354\) 280.802i 0.793226i
\(355\) 0 0
\(356\) 17.2740i 0.0485224i
\(357\) 225.974i 0.632979i
\(358\) −918.669 −2.56612
\(359\) −386.556 −1.07676 −0.538379 0.842703i \(-0.680963\pi\)
−0.538379 + 0.842703i \(0.680963\pi\)
\(360\) 0 0
\(361\) −330.588 145.026i −0.915756 0.401735i
\(362\) −233.365 −0.644655
\(363\) 95.0673i 0.261893i
\(364\) 473.912i 1.30196i
\(365\) 0 0
\(366\) 66.7523 0.182383
\(367\) −404.829 −1.10308 −0.551539 0.834149i \(-0.685959\pi\)
−0.551539 + 0.834149i \(0.685959\pi\)
\(368\) 332.113 0.902481
\(369\) 319.315i 0.865351i
\(370\) 0 0
\(371\) 368.720i 0.993855i
\(372\) 116.682 0.313663
\(373\) 22.0399i 0.0590881i −0.999563 0.0295441i \(-0.990594\pi\)
0.999563 0.0295441i \(-0.00940554\pi\)
\(374\) 512.425i 1.37012i
\(375\) 0 0
\(376\) 2.64309i 0.00702949i
\(377\) −549.237 −1.45686
\(378\) −358.974 −0.949667
\(379\) 296.981i 0.783591i −0.920052 0.391796i \(-0.871854\pi\)
0.920052 0.391796i \(-0.128146\pi\)
\(380\) 0 0
\(381\) 82.9260 0.217653
\(382\) 251.780i 0.659109i
\(383\) 291.198i 0.760308i 0.924923 + 0.380154i \(0.124129\pi\)
−0.924923 + 0.380154i \(0.875871\pi\)
\(384\) −17.7868 −0.0463198
\(385\) 0 0
\(386\) 871.872 2.25874
\(387\) −114.125 −0.294897
\(388\) 44.3692i 0.114354i
\(389\) −292.243 −0.751266 −0.375633 0.926768i \(-0.622575\pi\)
−0.375633 + 0.926768i \(0.622575\pi\)
\(390\) 0 0
\(391\) 584.948 1.49603
\(392\) 2.31788i 0.00591297i
\(393\) 179.164i 0.455889i
\(394\) 41.7031i 0.105845i
\(395\) 0 0
\(396\) 182.995 0.462108
\(397\) 235.092 0.592172 0.296086 0.955161i \(-0.404319\pi\)
0.296086 + 0.955161i \(0.404319\pi\)
\(398\) 820.212i 2.06083i
\(399\) −143.233 30.0359i −0.358979 0.0752781i
\(400\) 0 0
\(401\) 254.731i 0.635239i −0.948218 0.317619i \(-0.897117\pi\)
0.948218 0.317619i \(-0.102883\pi\)
\(402\) 1.17893i 0.00293265i
\(403\) −489.813 −1.21542
\(404\) 369.414 0.914392
\(405\) 0 0
\(406\) −548.355 −1.35063
\(407\) 296.416i 0.728295i
\(408\) −16.3220 −0.0400048
\(409\) 671.308i 1.64134i 0.571402 + 0.820670i \(0.306400\pi\)
−0.571402 + 0.820670i \(0.693600\pi\)
\(410\) 0 0
\(411\) 210.932i 0.513216i
\(412\) 504.210i 1.22381i
\(413\) 575.536i 1.39355i
\(414\) 427.150i 1.03176i
\(415\) 0 0
\(416\) −832.269 −2.00065
\(417\) 208.831i 0.500793i
\(418\) 324.799 + 68.1105i 0.777030 + 0.162944i
\(419\) 524.492 1.25177 0.625886 0.779915i \(-0.284738\pi\)
0.625886 + 0.779915i \(0.284738\pi\)
\(420\) 0 0
\(421\) 599.022i 1.42285i 0.702760 + 0.711427i \(0.251951\pi\)
−0.702760 + 0.711427i \(0.748049\pi\)
\(422\) 102.527 0.242954
\(423\) −42.1576 −0.0996634
\(424\) 26.6325 0.0628124
\(425\) 0 0
\(426\) 326.470i 0.766361i
\(427\) −136.817 −0.320414
\(428\) 165.689i 0.387123i
\(429\) 134.752 0.314108
\(430\) 0 0
\(431\) 50.7272i 0.117696i 0.998267 + 0.0588482i \(0.0187428\pi\)
−0.998267 + 0.0588482i \(0.981257\pi\)
\(432\) 321.549i 0.744327i
\(433\) 651.322i 1.50421i 0.659045 + 0.752104i \(0.270961\pi\)
−0.659045 + 0.752104i \(0.729039\pi\)
\(434\) −489.026 −1.12679
\(435\) 0 0
\(436\) 53.9177i 0.123664i
\(437\) −77.7501 + 370.767i −0.177918 + 0.848438i
\(438\) −1.53159 −0.00349679
\(439\) 373.780i 0.851434i −0.904856 0.425717i \(-0.860022\pi\)
0.904856 0.425717i \(-0.139978\pi\)
\(440\) 0 0
\(441\) −36.9706 −0.0838335
\(442\) −1528.86 −3.45896
\(443\) 457.331 1.03235 0.516175 0.856483i \(-0.327356\pi\)
0.516175 + 0.856483i \(0.327356\pi\)
\(444\) −210.676 −0.474496
\(445\) 0 0
\(446\) −343.529 −0.770244
\(447\) 161.155i 0.360526i
\(448\) −388.116 −0.866331
\(449\) 309.196i 0.688632i 0.938854 + 0.344316i \(0.111889\pi\)
−0.938854 + 0.344316i \(0.888111\pi\)
\(450\) 0 0
\(451\) 260.337i 0.577245i
\(452\) 299.481i 0.662569i
\(453\) 57.8926 0.127798
\(454\) 304.252 0.670158
\(455\) 0 0
\(456\) 2.16948 10.3456i 0.00475763 0.0226877i
\(457\) 64.6534 0.141474 0.0707368 0.997495i \(-0.477465\pi\)
0.0707368 + 0.997495i \(0.477465\pi\)
\(458\) 746.006i 1.62883i
\(459\) 566.343i 1.23386i
\(460\) 0 0
\(461\) −328.531 −0.712649 −0.356324 0.934362i \(-0.615970\pi\)
−0.356324 + 0.934362i \(0.615970\pi\)
\(462\) 134.536 0.291204
\(463\) −619.396 −1.33779 −0.668894 0.743357i \(-0.733232\pi\)
−0.668894 + 0.743357i \(0.733232\pi\)
\(464\) 491.186i 1.05859i
\(465\) 0 0
\(466\) 460.603i 0.988419i
\(467\) −303.167 −0.649179 −0.324589 0.945855i \(-0.605226\pi\)
−0.324589 + 0.945855i \(0.605226\pi\)
\(468\) 545.980i 1.16662i
\(469\) 2.41635i 0.00515213i
\(470\) 0 0
\(471\) 22.8808i 0.0485793i
\(472\) 41.5707 0.0880735
\(473\) 93.0464 0.196715
\(474\) 391.054i 0.825008i
\(475\) 0 0
\(476\) −746.477 −1.56823
\(477\) 424.792i 0.890548i
\(478\) 777.337i 1.62623i
\(479\) 313.314 0.654100 0.327050 0.945007i \(-0.393946\pi\)
0.327050 + 0.945007i \(0.393946\pi\)
\(480\) 0 0
\(481\) 884.382 1.83863
\(482\) 286.926 0.595281
\(483\) 153.577i 0.317964i
\(484\) 314.044 0.648851
\(485\) 0 0
\(486\) 637.019 1.31074
\(487\) 19.7644i 0.0405840i 0.999794 + 0.0202920i \(0.00645958\pi\)
−0.999794 + 0.0202920i \(0.993540\pi\)
\(488\) 9.88219i 0.0202504i
\(489\) 308.422i 0.630719i
\(490\) 0 0
\(491\) 232.222 0.472958 0.236479 0.971637i \(-0.424007\pi\)
0.236479 + 0.971637i \(0.424007\pi\)
\(492\) 185.033 0.376084
\(493\) 865.124i 1.75481i
\(494\) 203.213 969.064i 0.411363 1.96167i
\(495\) 0 0
\(496\) 438.043i 0.883151i
\(497\) 669.138i 1.34635i
\(498\) −136.702 −0.274502
\(499\) −824.237 −1.65178 −0.825889 0.563833i \(-0.809326\pi\)
−0.825889 + 0.563833i \(0.809326\pi\)
\(500\) 0 0
\(501\) −105.934 −0.211445
\(502\) 727.977i 1.45015i
\(503\) −330.777 −0.657608 −0.328804 0.944398i \(-0.606646\pi\)
−0.328804 + 0.944398i \(0.606646\pi\)
\(504\) 24.4291i 0.0484705i
\(505\) 0 0
\(506\) 348.256i 0.688252i
\(507\) 206.183i 0.406673i
\(508\) 273.936i 0.539244i
\(509\) 375.649i 0.738013i 0.929427 + 0.369006i \(0.120302\pi\)
−0.929427 + 0.369006i \(0.879698\pi\)
\(510\) 0 0
\(511\) 3.13918 0.00614320
\(512\) 712.320i 1.39125i
\(513\) 358.974 + 75.2771i 0.699755 + 0.146739i
\(514\) −1045.27 −2.03359
\(515\) 0 0
\(516\) 66.1322i 0.128163i
\(517\) 34.3712 0.0664819
\(518\) 882.962 1.70456
\(519\) −157.872 −0.304185
\(520\) 0 0
\(521\) 769.799i 1.47754i 0.673957 + 0.738770i \(0.264593\pi\)
−0.673957 + 0.738770i \(0.735407\pi\)
\(522\) 631.744 1.21024
\(523\) 228.999i 0.437856i −0.975741 0.218928i \(-0.929744\pi\)
0.975741 0.218928i \(-0.0702560\pi\)
\(524\) 591.848 1.12948
\(525\) 0 0
\(526\) 622.129i 1.18276i
\(527\) 771.522i 1.46399i
\(528\) 120.510i 0.228239i
\(529\) −131.456 −0.248499
\(530\) 0 0
\(531\) 663.058i 1.24870i
\(532\) 99.2202 473.152i 0.186504 0.889383i
\(533\) −776.738 −1.45730
\(534\) 14.6309i 0.0273987i
\(535\) 0 0
\(536\) 0.174531 0.000325618
\(537\) 380.525 0.708613
\(538\) 606.495 1.12731
\(539\) 30.1421 0.0559223
\(540\) 0 0
\(541\) −67.4487 −0.124674 −0.0623371 0.998055i \(-0.519855\pi\)
−0.0623371 + 0.998055i \(0.519855\pi\)
\(542\) 19.6338i 0.0362247i
\(543\) 96.6629 0.178016
\(544\) 1310.94i 2.40981i
\(545\) 0 0
\(546\) 401.399i 0.735163i
\(547\) 788.659i 1.44179i −0.693044 0.720895i \(-0.743731\pi\)
0.693044 0.720895i \(-0.256269\pi\)
\(548\) 696.789 1.27151
\(549\) 157.622 0.287108
\(550\) 0 0
\(551\) 548.355 + 114.990i 0.995200 + 0.208694i
\(552\) −11.0928 −0.0200956
\(553\) 801.510i 1.44939i
\(554\) 693.868i 1.25247i
\(555\) 0 0
\(556\) 689.848 1.24073
\(557\) −676.735 −1.21496 −0.607482 0.794333i \(-0.707820\pi\)
−0.607482 + 0.794333i \(0.707820\pi\)
\(558\) 563.393 1.00966
\(559\) 277.611i 0.496622i
\(560\) 0 0
\(561\) 212.253i 0.378348i
\(562\) 981.791 1.74696
\(563\) 822.729i 1.46133i 0.682736 + 0.730665i \(0.260790\pi\)
−0.682736 + 0.730665i \(0.739210\pi\)
\(564\) 24.4291i 0.0433140i
\(565\) 0 0
\(566\) 622.129i 1.09917i
\(567\) −309.307 −0.545515
\(568\) −48.3315 −0.0850906
\(569\) 414.388i 0.728274i 0.931345 + 0.364137i \(0.118636\pi\)
−0.931345 + 0.364137i \(0.881364\pi\)
\(570\) 0 0
\(571\) −1105.17 −1.93550 −0.967748 0.251919i \(-0.918938\pi\)
−0.967748 + 0.251919i \(0.918938\pi\)
\(572\) 445.138i 0.778214i
\(573\) 104.291i 0.182008i
\(574\) −775.492 −1.35103
\(575\) 0 0
\(576\) 447.137 0.776280
\(577\) 768.918 1.33261 0.666307 0.745678i \(-0.267874\pi\)
0.666307 + 0.745678i \(0.267874\pi\)
\(578\) 1599.57i 2.76741i
\(579\) −361.141 −0.623733
\(580\) 0 0
\(581\) 280.187 0.482249
\(582\) 37.5803i 0.0645710i
\(583\) 346.333i 0.594053i
\(584\) 0.226741i 0.000388255i
\(585\) 0 0
\(586\) −1210.91 −2.06640
\(587\) 285.313 0.486053 0.243026 0.970020i \(-0.421860\pi\)
0.243026 + 0.970020i \(0.421860\pi\)
\(588\) 21.4234i 0.0364343i
\(589\) 489.026 + 102.549i 0.830266 + 0.174107i
\(590\) 0 0
\(591\) 17.2740i 0.0292284i
\(592\) 790.909i 1.33599i
\(593\) 114.516 0.193114 0.0965569 0.995327i \(-0.469217\pi\)
0.0965569 + 0.995327i \(0.469217\pi\)
\(594\) −337.179 −0.567641
\(595\) 0 0
\(596\) −532.357 −0.893217
\(597\) 339.743i 0.569084i
\(598\) −1039.05 −1.73754
\(599\) 841.859i 1.40544i −0.711467 0.702720i \(-0.751969\pi\)
0.711467 0.702720i \(-0.248031\pi\)
\(600\) 0 0
\(601\) 1048.51i 1.74460i 0.488970 + 0.872300i \(0.337373\pi\)
−0.488970 + 0.872300i \(0.662627\pi\)
\(602\) 277.166i 0.460408i
\(603\) 2.78380i 0.00461659i
\(604\) 191.242i 0.316625i
\(605\) 0 0
\(606\) −312.890 −0.516321
\(607\) 226.715i 0.373501i 0.982407 + 0.186750i \(0.0597955\pi\)
−0.982407 + 0.186750i \(0.940204\pi\)
\(608\) 830.933 + 174.247i 1.36667 + 0.286591i
\(609\) 227.136 0.372966
\(610\) 0 0
\(611\) 102.549i 0.167838i
\(612\) 859.994 1.40522
\(613\) −406.833 −0.663676 −0.331838 0.943336i \(-0.607669\pi\)
−0.331838 + 0.943336i \(0.607669\pi\)
\(614\) 519.823 0.846618
\(615\) 0 0
\(616\) 19.9171i 0.0323329i
\(617\) 215.903 0.349923 0.174962 0.984575i \(-0.444020\pi\)
0.174962 + 0.984575i \(0.444020\pi\)
\(618\) 427.061i 0.691038i
\(619\) 838.586 1.35474 0.677371 0.735641i \(-0.263119\pi\)
0.677371 + 0.735641i \(0.263119\pi\)
\(620\) 0 0
\(621\) 384.899i 0.619806i
\(622\) 806.771i 1.29706i
\(623\) 29.9877i 0.0481344i
\(624\) 359.551 0.576204
\(625\) 0 0
\(626\) 1572.37i 2.51177i
\(627\) −134.536 28.2123i −0.214571 0.0449957i
\(628\) −75.5841 −0.120357
\(629\) 1393.02i 2.21466i
\(630\) 0 0
\(631\) −267.661 −0.424185 −0.212093 0.977250i \(-0.568028\pi\)
−0.212093 + 0.977250i \(0.568028\pi\)
\(632\) −57.8926 −0.0916023
\(633\) −42.4679 −0.0670898
\(634\) 51.3818 0.0810439
\(635\) 0 0
\(636\) 246.154 0.387035
\(637\) 89.9315i 0.141180i
\(638\) −515.061 −0.807306
\(639\) 770.894i 1.20641i
\(640\) 0 0
\(641\) 449.257i 0.700868i 0.936587 + 0.350434i \(0.113966\pi\)
−0.936587 + 0.350434i \(0.886034\pi\)
\(642\) 140.337i 0.218593i
\(643\) 368.569 0.573202 0.286601 0.958050i \(-0.407475\pi\)
0.286601 + 0.958050i \(0.407475\pi\)
\(644\) −507.323 −0.787769
\(645\) 0 0
\(646\) 1526.41 + 320.089i 2.36286 + 0.495493i
\(647\) 371.599 0.574341 0.287170 0.957879i \(-0.407285\pi\)
0.287170 + 0.957879i \(0.407285\pi\)
\(648\) 22.3411i 0.0344770i
\(649\) 540.592i 0.832962i
\(650\) 0 0
\(651\) 202.561 0.311154
\(652\) −1018.83 −1.56263
\(653\) −257.669 −0.394593 −0.197296 0.980344i \(-0.563216\pi\)
−0.197296 + 0.980344i \(0.563216\pi\)
\(654\) 45.6677i 0.0698283i
\(655\) 0 0
\(656\) 694.643i 1.05891i
\(657\) −3.61655 −0.00550465
\(658\) 102.385i 0.155600i
\(659\) 32.4523i 0.0492448i 0.999697 + 0.0246224i \(0.00783834\pi\)
−0.999697 + 0.0246224i \(0.992162\pi\)
\(660\) 0 0
\(661\) 992.719i 1.50184i −0.660391 0.750922i \(-0.729610\pi\)
0.660391 0.750922i \(-0.270390\pi\)
\(662\) 894.952 1.35189
\(663\) 633.275 0.955167
\(664\) 20.2377i 0.0304785i
\(665\) 0 0
\(666\) −1017.23 −1.52738
\(667\) 587.958i 0.881496i
\(668\) 349.940i 0.523863i
\(669\) 142.294 0.212697
\(670\) 0 0
\(671\) −128.510 −0.191520
\(672\) 344.184 0.512178
\(673\) 649.462i 0.965025i 0.875889 + 0.482512i \(0.160276\pi\)
−0.875889 + 0.482512i \(0.839724\pi\)
\(674\) −204.247 −0.303037
\(675\) 0 0
\(676\) 681.103 1.00755
\(677\) 1219.86i 1.80186i 0.433969 + 0.900928i \(0.357113\pi\)
−0.433969 + 0.900928i \(0.642887\pi\)
\(678\) 253.658i 0.374126i
\(679\) 77.0252i 0.113439i
\(680\) 0 0
\(681\) −126.025 −0.185059
\(682\) −459.335 −0.673511
\(683\) 97.8453i 0.143258i −0.997431 0.0716290i \(-0.977180\pi\)
0.997431 0.0716290i \(-0.0228198\pi\)
\(684\) −114.309 + 545.104i −0.167118 + 0.796936i
\(685\) 0 0
\(686\) 1000.97i 1.45914i
\(687\) 309.006i 0.449790i
\(688\) 248.270 0.360857
\(689\) −1033.31 −1.49973
\(690\) 0 0
\(691\) −255.951 −0.370407 −0.185203 0.982700i \(-0.559294\pi\)
−0.185203 + 0.982700i \(0.559294\pi\)
\(692\) 521.512i 0.753630i
\(693\) 317.680 0.458413
\(694\) 1037.61i 1.49512i
\(695\) 0 0
\(696\) 16.4059i 0.0235717i
\(697\) 1223.47i 1.75534i
\(698\) 116.388i 0.166745i
\(699\) 190.788i 0.272944i
\(700\) 0 0
\(701\) −290.329 −0.414164 −0.207082 0.978324i \(-0.566397\pi\)
−0.207082 + 0.978324i \(0.566397\pi\)
\(702\) 1006.00i 1.43305i
\(703\) −882.962 185.158i −1.25599 0.263382i
\(704\) −364.551 −0.517829
\(705\) 0 0
\(706\) 1126.08i 1.59501i
\(707\) 641.305 0.907079
\(708\) 384.223 0.542688
\(709\) 248.916 0.351080 0.175540 0.984472i \(-0.443833\pi\)
0.175540 + 0.984472i \(0.443833\pi\)
\(710\) 0 0
\(711\) 923.396i 1.29873i
\(712\) 2.16600 0.00304213
\(713\) 524.344i 0.735405i
\(714\) 632.259 0.885517
\(715\) 0 0
\(716\) 1257.02i 1.75562i
\(717\) 321.984i 0.449070i
\(718\) 1081.56i 1.50635i
\(719\) −1233.16 −1.71511 −0.857555 0.514393i \(-0.828017\pi\)
−0.857555 + 0.514393i \(0.828017\pi\)
\(720\) 0 0
\(721\) 875.312i 1.21402i
\(722\) −405.774 + 924.963i −0.562014 + 1.28111i
\(723\) −118.848 −0.164382
\(724\) 319.315i 0.441042i
\(725\) 0 0
\(726\) −265.992 −0.366380
\(727\) 176.613 0.242933 0.121467 0.992596i \(-0.461240\pi\)
0.121467 + 0.992596i \(0.461240\pi\)
\(728\) −59.4242 −0.0816267
\(729\) 154.990 0.212606
\(730\) 0 0
\(731\) 437.276 0.598189
\(732\) 91.3376i 0.124778i
\(733\) −1135.13 −1.54860 −0.774301 0.632817i \(-0.781898\pi\)
−0.774301 + 0.632817i \(0.781898\pi\)
\(734\) 1132.69i 1.54317i
\(735\) 0 0
\(736\) 890.943i 1.21052i
\(737\) 2.26964i 0.00307956i
\(738\) 893.421 1.21060
\(739\) 556.673 0.753279 0.376640 0.926360i \(-0.377080\pi\)
0.376640 + 0.926360i \(0.377080\pi\)
\(740\) 0 0
\(741\) −84.1737 + 401.399i −0.113595 + 0.541699i
\(742\) −1031.65 −1.39037
\(743\) 126.242i 0.169909i 0.996385 + 0.0849544i \(0.0270745\pi\)
−0.996385 + 0.0849544i \(0.972926\pi\)
\(744\) 14.6309i 0.0196652i
\(745\) 0 0
\(746\) −61.6661 −0.0826623
\(747\) −322.795 −0.432122
\(748\) −701.154 −0.937372
\(749\) 287.636i 0.384027i
\(750\) 0 0
\(751\) 550.711i 0.733304i 0.930358 + 0.366652i \(0.119496\pi\)
−0.930358 + 0.366652i \(0.880504\pi\)
\(752\) 91.7104 0.121955
\(753\) 301.538i 0.400449i
\(754\) 1536.73i 2.03810i
\(755\) 0 0
\(756\) 491.186i 0.649718i
\(757\) −965.388 −1.27528 −0.637641 0.770334i \(-0.720090\pi\)
−0.637641 + 0.770334i \(0.720090\pi\)
\(758\) −830.933 −1.09622
\(759\) 144.252i 0.190056i
\(760\) 0 0
\(761\) 682.013 0.896207 0.448103 0.893982i \(-0.352100\pi\)
0.448103 + 0.893982i \(0.352100\pi\)
\(762\) 232.021i 0.304490i
\(763\) 93.6013i 0.122675i
\(764\) 344.512 0.450932
\(765\) 0 0
\(766\) 814.752 1.06365
\(767\) −1612.90 −2.10287
\(768\) 320.481i 0.417293i
\(769\) −279.579 −0.363561 −0.181781 0.983339i \(-0.558186\pi\)
−0.181781 + 0.983339i \(0.558186\pi\)
\(770\) 0 0
\(771\) 432.963 0.561561
\(772\) 1192.99i 1.54532i
\(773\) 527.781i 0.682770i −0.939924 0.341385i \(-0.889104\pi\)
0.939924 0.341385i \(-0.110896\pi\)
\(774\) 319.315i 0.412551i
\(775\) 0 0
\(776\) −5.56349 −0.00716945
\(777\) −365.735 −0.470701
\(778\) 817.675i 1.05100i
\(779\) 775.492 + 162.621i 0.995496 + 0.208756i
\(780\) 0 0
\(781\) 628.510i 0.804751i
\(782\) 1636.65i 2.09290i
\(783\) −569.256 −0.727020
\(784\) 80.4264 0.102585
\(785\) 0 0
\(786\) −501.289 −0.637773
\(787\) 761.947i 0.968167i 0.875022 + 0.484083i \(0.160847\pi\)
−0.875022 + 0.484083i \(0.839153\pi\)
\(788\) −57.0626 −0.0724145
\(789\) 257.694i 0.326609i
\(790\) 0 0
\(791\) 519.901i 0.657270i
\(792\) 22.9459i 0.0289721i
\(793\) 383.419i 0.483505i
\(794\) 657.772i 0.828428i
\(795\) 0 0
\(796\) −1122.30 −1.40993
\(797\) 989.767i 1.24187i −0.783864 0.620933i \(-0.786754\pi\)
0.783864 0.620933i \(-0.213246\pi\)
\(798\) −84.0386 + 400.755i −0.105311 + 0.502199i
\(799\) 161.529 0.202164
\(800\) 0 0
\(801\) 34.5480i 0.0431311i
\(802\) −712.719 −0.888677
\(803\) 2.94858 0.00367195
\(804\) 1.61313 0.00200638
\(805\) 0 0
\(806\) 1370.46i 1.70033i
\(807\) −251.219 −0.311299
\(808\) 46.3211i 0.0573281i
\(809\) 1148.89 1.42013 0.710066 0.704135i \(-0.248665\pi\)
0.710066 + 0.704135i \(0.248665\pi\)
\(810\) 0 0
\(811\) 660.963i 0.814997i −0.913206 0.407499i \(-0.866401\pi\)
0.913206 0.407499i \(-0.133599\pi\)
\(812\) 750.318i 0.924037i
\(813\) 8.13258i 0.0100032i
\(814\) 829.352 1.01886
\(815\) 0 0
\(816\) 566.343i 0.694047i
\(817\) −58.1219 + 277.166i −0.0711406 + 0.339248i
\(818\) 1878.28 2.29618
\(819\) 947.825i 1.15730i
\(820\) 0 0
\(821\) 1048.30 1.27686 0.638430 0.769680i \(-0.279584\pi\)
0.638430 + 0.769680i \(0.279584\pi\)
\(822\) −590.173 −0.717972
\(823\) −140.352 −0.170537 −0.0852686 0.996358i \(-0.527175\pi\)
−0.0852686 + 0.996358i \(0.527175\pi\)
\(824\) −63.2233 −0.0767273
\(825\) 0 0
\(826\) −1610.31 −1.94953
\(827\) 41.2618i 0.0498934i 0.999689 + 0.0249467i \(0.00794160\pi\)
−0.999689 + 0.0249467i \(0.992058\pi\)
\(828\) 584.472 0.705884
\(829\) 315.350i 0.380398i 0.981746 + 0.190199i \(0.0609133\pi\)
−0.981746 + 0.190199i \(0.939087\pi\)
\(830\) 0 0
\(831\) 287.410i 0.345860i
\(832\) 1087.67i 1.30729i
\(833\) 141.655 0.170053
\(834\) −584.294 −0.700593
\(835\) 0 0
\(836\) 93.1960 444.424i 0.111478 0.531608i
\(837\) −507.666 −0.606531
\(838\) 1467.49i 1.75119i
\(839\) 1541.79i 1.83765i 0.394667 + 0.918824i \(0.370860\pi\)
−0.394667 + 0.918824i \(0.629140\pi\)
\(840\) 0 0
\(841\) −28.5745 −0.0339768
\(842\) 1676.02 1.99053
\(843\) −406.671 −0.482410
\(844\) 140.288i 0.166218i
\(845\) 0 0
\(846\) 117.954i 0.139426i
\(847\) 545.182 0.643662
\(848\) 924.099i 1.08974i
\(849\) 257.694i 0.303527i
\(850\) 0 0
\(851\) 946.730i 1.11249i
\(852\) −446.710 −0.524308
\(853\) 633.242 0.742370 0.371185 0.928559i \(-0.378952\pi\)
0.371185 + 0.928559i \(0.378952\pi\)
\(854\) 382.804i 0.448248i
\(855\) 0 0
\(856\) 20.7758 0.0242708
\(857\) 365.640i 0.426651i −0.976981 0.213325i \(-0.931571\pi\)
0.976981 0.213325i \(-0.0684294\pi\)
\(858\) 377.028i 0.439426i
\(859\) −1141.98 −1.32943 −0.664713 0.747099i \(-0.731446\pi\)
−0.664713 + 0.747099i \(0.731446\pi\)
\(860\) 0 0
\(861\) 321.219 0.373077
\(862\) 141.931 0.164653
\(863\) 805.728i 0.933636i 0.884353 + 0.466818i \(0.154600\pi\)
−0.884353 + 0.466818i \(0.845400\pi\)
\(864\) −862.605 −0.998386
\(865\) 0 0
\(866\) 1822.35 2.10434
\(867\) 662.562i 0.764200i
\(868\) 669.138i 0.770896i
\(869\) 752.845i 0.866335i
\(870\) 0 0
\(871\) −6.77164 −0.00777456
\(872\) −6.76078 −0.00775318
\(873\) 88.7385i 0.101648i
\(874\) 1037.38 + 217.540i 1.18694 + 0.248901i
\(875\) 0 0
\(876\) 2.09569i 0.00239234i
\(877\) 884.064i 1.00806i 0.863688 + 0.504028i \(0.168149\pi\)
−0.863688 + 0.504028i \(0.831851\pi\)
\(878\) −1045.81 −1.19113
\(879\) 501.576 0.570621
\(880\) 0 0
\(881\) −1348.89 −1.53109 −0.765543 0.643384i \(-0.777530\pi\)
−0.765543 + 0.643384i \(0.777530\pi\)
\(882\) 103.441i 0.117280i
\(883\) −990.084 −1.12127 −0.560636 0.828062i \(-0.689443\pi\)
−0.560636 + 0.828062i \(0.689443\pi\)
\(884\) 2091.95i 2.36646i
\(885\) 0 0
\(886\) 1279.58i 1.44422i
\(887\) 864.925i 0.975112i 0.873092 + 0.487556i \(0.162112\pi\)
−0.873092 + 0.487556i \(0.837888\pi\)
\(888\) 26.4168i 0.0297487i
\(889\) 475.555i 0.534932i
\(890\) 0 0
\(891\) −290.527 −0.326068
\(892\) 470.053i 0.526965i
\(893\) −21.4701 + 102.385i −0.0240427 + 0.114652i
\(894\) 450.902 0.504364
\(895\) 0 0
\(896\) 102.002i 0.113841i
\(897\) 430.388 0.479809
\(898\) 865.109 0.963373
\(899\) −775.492 −0.862616
\(900\) 0 0
\(901\) 1627.61i 1.80645i
\(902\) −728.407 −0.807546
\(903\) 114.806i 0.127138i
\(904\) −37.5522 −0.0415400
\(905\) 0 0
\(906\) 161.980i 0.178786i
\(907\) 1019.55i 1.12409i 0.827107 + 0.562045i \(0.189985\pi\)
−0.827107 + 0.562045i \(0.810015\pi\)
\(908\) 416.310i 0.458491i
\(909\) −738.828 −0.812793
\(910\) 0 0
\(911\) 1069.97i 1.17450i −0.809405 0.587251i \(-0.800210\pi\)
0.809405 0.587251i \(-0.199790\pi\)
\(912\) −358.974 75.2771i −0.393612 0.0825407i
\(913\) 263.175 0.288253
\(914\) 180.896i 0.197917i
\(915\) 0 0
\(916\) 1020.76 1.11437
\(917\) 1027.45 1.12045
\(918\) −1584.59 −1.72613
\(919\) −1166.06 −1.26883 −0.634415 0.772993i \(-0.718759\pi\)
−0.634415 + 0.772993i \(0.718759\pi\)
\(920\) 0 0
\(921\) −215.318 −0.233787
\(922\) 919.208i 0.996972i
\(923\) 1875.21 2.03165
\(924\) 184.086i 0.199228i
\(925\) 0 0
\(926\) 1733.03i 1.87152i
\(927\) 1008.42i 1.08783i
\(928\) −1317.68 −1.41992
\(929\) 183.514 0.197539 0.0987696 0.995110i \(-0.468509\pi\)
0.0987696 + 0.995110i \(0.468509\pi\)
\(930\) 0 0
\(931\) −18.8284 + 89.7872i −0.0202239 + 0.0964417i
\(932\) 630.246 0.676229
\(933\) 334.175i 0.358173i
\(934\) 848.240i 0.908179i
\(935\) 0 0
\(936\) 68.4609 0.0731420
\(937\) 273.714 0.292118 0.146059 0.989276i \(-0.453341\pi\)
0.146059 + 0.989276i \(0.453341\pi\)
\(938\) −6.76078 −0.00720765
\(939\) 651.297i 0.693607i
\(940\) 0 0
\(941\) 313.160i 0.332795i 0.986059 + 0.166398i \(0.0532135\pi\)
−0.986059 + 0.166398i \(0.946786\pi\)
\(942\) 64.0190 0.0679607
\(943\) 831.498i 0.881758i
\(944\) 1442.43i 1.52800i
\(945\) 0 0
\(946\) 260.337i 0.275198i
\(947\) 129.698 0.136957 0.0684784 0.997653i \(-0.478186\pi\)
0.0684784 + 0.997653i \(0.478186\pi\)
\(948\) −535.081 −0.564431
\(949\) 8.79733i 0.00927010i
\(950\) 0 0
\(951\) −21.2830 −0.0223796
\(952\) 93.6013i 0.0983207i
\(953\) 353.917i 0.371372i 0.982609 + 0.185686i \(0.0594507\pi\)
−0.982609 + 0.185686i \(0.940549\pi\)
\(954\) 1188.54 1.24585
\(955\) 0 0
\(956\) 1063.63 1.11259
\(957\) 213.345 0.222931
\(958\) 876.631i 0.915063i
\(959\) 1209.63 1.26134
\(960\) 0 0
\(961\) 269.412 0.280346
\(962\) 2474.44i 2.57218i
\(963\) 331.377i 0.344109i
\(964\) 392.602i 0.407263i
\(965\) 0 0
\(966\) 429.698 0.444822
\(967\) −383.913 −0.397014 −0.198507 0.980099i \(-0.563609\pi\)
−0.198507 + 0.980099i \(0.563609\pi\)
\(968\) 39.3782i 0.0406799i
\(969\) −632.259 132.585i −0.652486 0.136827i
\(970\) 0 0
\(971\) 307.647i 0.316836i 0.987372 + 0.158418i \(0.0506393\pi\)
−0.987372 + 0.158418i \(0.949361\pi\)
\(972\) 871.637i 0.896746i
\(973\) 1197.58 1.23081
\(974\) 55.2994 0.0567756
\(975\) 0 0
\(976\) −342.894 −0.351326
\(977\) 264.741i 0.270974i 0.990779 + 0.135487i \(0.0432598\pi\)
−0.990779 + 0.135487i \(0.956740\pi\)
\(978\) 862.943 0.882355
\(979\) 28.1670i 0.0287712i
\(980\) 0 0
\(981\) 107.835i 0.109924i
\(982\) 649.743i 0.661652i
\(983\) 383.178i 0.389805i −0.980823 0.194902i \(-0.937561\pi\)
0.980823 0.194902i \(-0.0624390\pi\)
\(984\) 23.2015i 0.0235787i
\(985\) 0 0
\(986\) −2420.56 −2.45493
\(987\) 42.4091i 0.0429676i
\(988\) −1325.98 278.058i −1.34208 0.281435i
\(989\) 297.183 0.300488
\(990\) 0 0
\(991\) 1692.42i 1.70779i −0.520445 0.853895i \(-0.674234\pi\)
0.520445 0.853895i \(-0.325766\pi\)
\(992\) −1175.12 −1.18459
\(993\) −370.701 −0.373315
\(994\) 1872.20 1.88350
\(995\) 0 0
\(996\) 187.050i 0.187801i
\(997\) 968.728 0.971643 0.485822 0.874058i \(-0.338520\pi\)
0.485822 + 0.874058i \(0.338520\pi\)
\(998\) 2306.16i 2.31078i
\(999\) 916.617 0.917535
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.f.151.1 8
5.2 odd 4 95.3.d.d.94.8 yes 8
5.3 odd 4 95.3.d.d.94.1 8
5.4 even 2 inner 475.3.c.f.151.8 8
15.2 even 4 855.3.g.g.379.1 8
15.8 even 4 855.3.g.g.379.8 8
19.18 odd 2 inner 475.3.c.f.151.7 8
95.18 even 4 95.3.d.d.94.7 yes 8
95.37 even 4 95.3.d.d.94.2 yes 8
95.94 odd 2 inner 475.3.c.f.151.2 8
285.113 odd 4 855.3.g.g.379.2 8
285.227 odd 4 855.3.g.g.379.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.3.d.d.94.1 8 5.3 odd 4
95.3.d.d.94.2 yes 8 95.37 even 4
95.3.d.d.94.7 yes 8 95.18 even 4
95.3.d.d.94.8 yes 8 5.2 odd 4
475.3.c.f.151.1 8 1.1 even 1 trivial
475.3.c.f.151.2 8 95.94 odd 2 inner
475.3.c.f.151.7 8 19.18 odd 2 inner
475.3.c.f.151.8 8 5.4 even 2 inner
855.3.g.g.379.1 8 15.2 even 4
855.3.g.g.379.2 8 285.113 odd 4
855.3.g.g.379.7 8 285.227 odd 4
855.3.g.g.379.8 8 15.8 even 4