Properties

Label 900.3.f.f.199.10
Level $900$
Weight $3$
Character 900.199
Analytic conductor $24.523$
Analytic rank $0$
Dimension $16$
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] = [900,3,Mod(199,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("900.199");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.f (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: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 5x^{14} + 12x^{12} + 25x^{10} + 53x^{8} + 100x^{6} + 192x^{4} + 320x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{28} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.10
Root \(0.957636 - 1.04064i\) of defining polynomial
Character \(\chi\) \(=\) 900.199
Dual form 900.3.f.f.199.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.169449 + 1.99281i) q^{2} +(-3.94257 + 0.675358i) q^{4} -12.3959 q^{7} +(-2.01392 - 7.74236i) q^{8} +O(q^{10})\) \(q+(0.169449 + 1.99281i) q^{2} +(-3.94257 + 0.675358i) q^{4} -12.3959 q^{7} +(-2.01392 - 7.74236i) q^{8} -11.0403i q^{11} -2.82009i q^{13} +(-2.10047 - 24.7027i) q^{14} +(15.0878 - 5.32529i) q^{16} -6.52606i q^{17} +27.9928i q^{19} +(22.0012 - 1.87077i) q^{22} +7.90421 q^{23} +(5.61989 - 0.477860i) q^{26} +(48.8718 - 8.37167i) q^{28} +50.7169 q^{29} +36.3467i q^{31} +(13.1689 + 29.1647i) q^{32} +(13.0052 - 1.10583i) q^{34} -18.9279i q^{37} +(-55.7842 + 4.74333i) q^{38} -5.30410 q^{41} -45.5870 q^{43} +(7.45616 + 43.5273i) q^{44} +(1.33936 + 15.7516i) q^{46} +11.7246 q^{47} +104.658 q^{49} +(1.90457 + 11.1184i) q^{52} +41.1680i q^{53} +(24.9644 + 95.9735i) q^{56} +(8.59391 + 101.069i) q^{58} +10.7008i q^{59} +56.1297 q^{61} +(-72.4319 + 6.15889i) q^{62} +(-55.8882 + 31.1850i) q^{64} +16.1709 q^{67} +(4.40743 + 25.7295i) q^{68} -66.1617i q^{71} -15.6330i q^{73} +(37.7198 - 3.20731i) q^{74} +(-18.9051 - 110.363i) q^{76} +136.855i q^{77} -123.057i q^{79} +(-0.898773 - 10.5701i) q^{82} +99.6700 q^{83} +(-7.72465 - 90.8461i) q^{86} +(-85.4781 + 22.2343i) q^{88} +101.083 q^{89} +34.9575i q^{91} +(-31.1629 + 5.33817i) q^{92} +(1.98672 + 23.3649i) q^{94} +127.293i q^{97} +(17.7342 + 208.564i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 20 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 20 q^{4} - 40 q^{14} + 68 q^{16} + 72 q^{26} + 128 q^{29} + 184 q^{34} + 32 q^{41} + 344 q^{44} + 304 q^{46} + 112 q^{49} - 232 q^{56} - 352 q^{61} + 220 q^{64} + 264 q^{74} - 48 q^{76} + 400 q^{86} + 160 q^{89} + 192 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.169449 + 1.99281i 0.0847243 + 0.996404i
\(3\) 0 0
\(4\) −3.94257 + 0.675358i −0.985644 + 0.168839i
\(5\) 0 0
\(6\) 0 0
\(7\) −12.3959 −1.77084 −0.885422 0.464789i \(-0.846130\pi\)
−0.885422 + 0.464789i \(0.846130\pi\)
\(8\) −2.01392 7.74236i −0.251740 0.967795i
\(9\) 0 0
\(10\) 0 0
\(11\) 11.0403i 1.00366i −0.864965 0.501832i \(-0.832659\pi\)
0.864965 0.501832i \(-0.167341\pi\)
\(12\) 0 0
\(13\) 2.82009i 0.216930i −0.994100 0.108465i \(-0.965407\pi\)
0.994100 0.108465i \(-0.0345935\pi\)
\(14\) −2.10047 24.7027i −0.150034 1.76448i
\(15\) 0 0
\(16\) 15.0878 5.32529i 0.942987 0.332831i
\(17\) 6.52606i 0.383886i −0.981406 0.191943i \(-0.938521\pi\)
0.981406 0.191943i \(-0.0614789\pi\)
\(18\) 0 0
\(19\) 27.9928i 1.47330i 0.676273 + 0.736651i \(0.263594\pi\)
−0.676273 + 0.736651i \(0.736406\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 22.0012 1.87077i 1.00006 0.0850348i
\(23\) 7.90421 0.343661 0.171831 0.985126i \(-0.445032\pi\)
0.171831 + 0.985126i \(0.445032\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 5.61989 0.477860i 0.216150 0.0183792i
\(27\) 0 0
\(28\) 48.8718 8.37167i 1.74542 0.298988i
\(29\) 50.7169 1.74886 0.874429 0.485153i \(-0.161236\pi\)
0.874429 + 0.485153i \(0.161236\pi\)
\(30\) 0 0
\(31\) 36.3467i 1.17247i 0.810140 + 0.586236i \(0.199391\pi\)
−0.810140 + 0.586236i \(0.800609\pi\)
\(32\) 13.1689 + 29.1647i 0.411528 + 0.911397i
\(33\) 0 0
\(34\) 13.0052 1.10583i 0.382506 0.0325245i
\(35\) 0 0
\(36\) 0 0
\(37\) 18.9279i 0.511566i −0.966734 0.255783i \(-0.917667\pi\)
0.966734 0.255783i \(-0.0823332\pi\)
\(38\) −55.7842 + 4.74333i −1.46801 + 0.124825i
\(39\) 0 0
\(40\) 0 0
\(41\) −5.30410 −0.129368 −0.0646842 0.997906i \(-0.520604\pi\)
−0.0646842 + 0.997906i \(0.520604\pi\)
\(42\) 0 0
\(43\) −45.5870 −1.06016 −0.530081 0.847947i \(-0.677838\pi\)
−0.530081 + 0.847947i \(0.677838\pi\)
\(44\) 7.45616 + 43.5273i 0.169458 + 0.989256i
\(45\) 0 0
\(46\) 1.33936 + 15.7516i 0.0291165 + 0.342426i
\(47\) 11.7246 0.249460 0.124730 0.992191i \(-0.460194\pi\)
0.124730 + 0.992191i \(0.460194\pi\)
\(48\) 0 0
\(49\) 104.658 2.13589
\(50\) 0 0
\(51\) 0 0
\(52\) 1.90457 + 11.1184i 0.0366263 + 0.213815i
\(53\) 41.1680i 0.776755i 0.921500 + 0.388378i \(0.126964\pi\)
−0.921500 + 0.388378i \(0.873036\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 24.9644 + 95.9735i 0.445793 + 1.71381i
\(57\) 0 0
\(58\) 8.59391 + 101.069i 0.148171 + 1.74257i
\(59\) 10.7008i 0.181370i 0.995880 + 0.0906848i \(0.0289056\pi\)
−0.995880 + 0.0906848i \(0.971094\pi\)
\(60\) 0 0
\(61\) 56.1297 0.920159 0.460080 0.887878i \(-0.347821\pi\)
0.460080 + 0.887878i \(0.347821\pi\)
\(62\) −72.4319 + 6.15889i −1.16826 + 0.0993370i
\(63\) 0 0
\(64\) −55.8882 + 31.1850i −0.873254 + 0.487266i
\(65\) 0 0
\(66\) 0 0
\(67\) 16.1709 0.241357 0.120679 0.992692i \(-0.461493\pi\)
0.120679 + 0.992692i \(0.461493\pi\)
\(68\) 4.40743 + 25.7295i 0.0648151 + 0.378375i
\(69\) 0 0
\(70\) 0 0
\(71\) 66.1617i 0.931855i −0.884823 0.465928i \(-0.845721\pi\)
0.884823 0.465928i \(-0.154279\pi\)
\(72\) 0 0
\(73\) 15.6330i 0.214150i −0.994251 0.107075i \(-0.965851\pi\)
0.994251 0.107075i \(-0.0341485\pi\)
\(74\) 37.7198 3.20731i 0.509727 0.0433421i
\(75\) 0 0
\(76\) −18.9051 110.363i −0.248752 1.45215i
\(77\) 136.855i 1.77733i
\(78\) 0 0
\(79\) 123.057i 1.55768i −0.627223 0.778840i \(-0.715809\pi\)
0.627223 0.778840i \(-0.284191\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −0.898773 10.5701i −0.0109606 0.128903i
\(83\) 99.6700 1.20084 0.600422 0.799684i \(-0.294999\pi\)
0.600422 + 0.799684i \(0.294999\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −7.72465 90.8461i −0.0898215 1.05635i
\(87\) 0 0
\(88\) −85.4781 + 22.2343i −0.971342 + 0.252663i
\(89\) 101.083 1.13576 0.567881 0.823110i \(-0.307763\pi\)
0.567881 + 0.823110i \(0.307763\pi\)
\(90\) 0 0
\(91\) 34.9575i 0.384148i
\(92\) −31.1629 + 5.33817i −0.338728 + 0.0580236i
\(93\) 0 0
\(94\) 1.98672 + 23.3649i 0.0211353 + 0.248563i
\(95\) 0 0
\(96\) 0 0
\(97\) 127.293i 1.31230i 0.754630 + 0.656151i \(0.227817\pi\)
−0.754630 + 0.656151i \(0.772183\pi\)
\(98\) 17.7342 + 208.564i 0.180962 + 2.12821i
\(99\) 0 0
\(100\) 0 0
\(101\) 94.3535 0.934193 0.467096 0.884206i \(-0.345300\pi\)
0.467096 + 0.884206i \(0.345300\pi\)
\(102\) 0 0
\(103\) −31.8455 −0.309180 −0.154590 0.987979i \(-0.549406\pi\)
−0.154590 + 0.987979i \(0.549406\pi\)
\(104\) −21.8341 + 5.67943i −0.209943 + 0.0546099i
\(105\) 0 0
\(106\) −82.0400 + 6.97587i −0.773962 + 0.0658101i
\(107\) 33.7912 0.315805 0.157903 0.987455i \(-0.449527\pi\)
0.157903 + 0.987455i \(0.449527\pi\)
\(108\) 0 0
\(109\) 83.4266 0.765382 0.382691 0.923876i \(-0.374997\pi\)
0.382691 + 0.923876i \(0.374997\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −187.027 + 66.0118i −1.66988 + 0.589391i
\(113\) 111.796i 0.989342i −0.869080 0.494671i \(-0.835289\pi\)
0.869080 0.494671i \(-0.164711\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −199.955 + 34.2520i −1.72375 + 0.295276i
\(117\) 0 0
\(118\) −21.3247 + 1.81324i −0.180717 + 0.0153664i
\(119\) 80.8964i 0.679802i
\(120\) 0 0
\(121\) −0.888544 −0.00734334
\(122\) 9.51110 + 111.856i 0.0779599 + 0.916851i
\(123\) 0 0
\(124\) −24.5470 143.299i −0.197960 1.15564i
\(125\) 0 0
\(126\) 0 0
\(127\) −16.6855 −0.131382 −0.0656909 0.997840i \(-0.520925\pi\)
−0.0656909 + 0.997840i \(0.520925\pi\)
\(128\) −71.6160 106.090i −0.559500 0.828831i
\(129\) 0 0
\(130\) 0 0
\(131\) 196.418i 1.49937i 0.661794 + 0.749686i \(0.269796\pi\)
−0.661794 + 0.749686i \(0.730204\pi\)
\(132\) 0 0
\(133\) 346.995i 2.60899i
\(134\) 2.74015 + 32.2256i 0.0204488 + 0.240490i
\(135\) 0 0
\(136\) −50.5271 + 13.1430i −0.371523 + 0.0966396i
\(137\) 117.127i 0.854942i 0.904029 + 0.427471i \(0.140595\pi\)
−0.904029 + 0.427471i \(0.859405\pi\)
\(138\) 0 0
\(139\) 187.238i 1.34704i 0.739170 + 0.673519i \(0.235218\pi\)
−0.739170 + 0.673519i \(0.764782\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 131.848 11.2110i 0.928505 0.0789508i
\(143\) −31.1346 −0.217725
\(144\) 0 0
\(145\) 0 0
\(146\) 31.1535 2.64899i 0.213380 0.0181437i
\(147\) 0 0
\(148\) 12.7831 + 74.6248i 0.0863725 + 0.504222i
\(149\) −50.2274 −0.337096 −0.168548 0.985693i \(-0.553908\pi\)
−0.168548 + 0.985693i \(0.553908\pi\)
\(150\) 0 0
\(151\) 213.160i 1.41166i −0.708382 0.705829i \(-0.750575\pi\)
0.708382 0.705829i \(-0.249425\pi\)
\(152\) 216.730 56.3752i 1.42585 0.370890i
\(153\) 0 0
\(154\) −272.725 + 23.1898i −1.77094 + 0.150583i
\(155\) 0 0
\(156\) 0 0
\(157\) 203.918i 1.29884i 0.760431 + 0.649419i \(0.224988\pi\)
−0.760431 + 0.649419i \(0.775012\pi\)
\(158\) 245.228 20.8518i 1.55208 0.131973i
\(159\) 0 0
\(160\) 0 0
\(161\) −97.9798 −0.608570
\(162\) 0 0
\(163\) 215.898 1.32452 0.662262 0.749272i \(-0.269596\pi\)
0.662262 + 0.749272i \(0.269596\pi\)
\(164\) 20.9118 3.58216i 0.127511 0.0218425i
\(165\) 0 0
\(166\) 16.8889 + 198.623i 0.101741 + 1.19653i
\(167\) −255.029 −1.52712 −0.763560 0.645737i \(-0.776550\pi\)
−0.763560 + 0.645737i \(0.776550\pi\)
\(168\) 0 0
\(169\) 161.047 0.952942
\(170\) 0 0
\(171\) 0 0
\(172\) 179.730 30.7875i 1.04494 0.178997i
\(173\) 235.426i 1.36084i −0.732822 0.680421i \(-0.761797\pi\)
0.732822 0.680421i \(-0.238203\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −58.7929 166.574i −0.334051 0.946443i
\(177\) 0 0
\(178\) 17.1284 + 201.439i 0.0962267 + 1.13168i
\(179\) 102.669i 0.573572i −0.957995 0.286786i \(-0.907413\pi\)
0.957995 0.286786i \(-0.0925869\pi\)
\(180\) 0 0
\(181\) −56.8222 −0.313935 −0.156967 0.987604i \(-0.550172\pi\)
−0.156967 + 0.987604i \(0.550172\pi\)
\(182\) −69.6636 + 5.92350i −0.382767 + 0.0325467i
\(183\) 0 0
\(184\) −15.9185 61.1972i −0.0865134 0.332594i
\(185\) 0 0
\(186\) 0 0
\(187\) −72.0498 −0.385293
\(188\) −46.2251 + 7.91830i −0.245878 + 0.0421186i
\(189\) 0 0
\(190\) 0 0
\(191\) 158.493i 0.829808i −0.909865 0.414904i \(-0.863815\pi\)
0.909865 0.414904i \(-0.136185\pi\)
\(192\) 0 0
\(193\) 156.732i 0.812084i 0.913854 + 0.406042i \(0.133091\pi\)
−0.913854 + 0.406042i \(0.866909\pi\)
\(194\) −253.671 + 21.5697i −1.30758 + 0.111184i
\(195\) 0 0
\(196\) −412.624 + 70.6819i −2.10522 + 0.360622i
\(197\) 260.127i 1.32044i −0.751072 0.660221i \(-0.770463\pi\)
0.751072 0.660221i \(-0.229537\pi\)
\(198\) 0 0
\(199\) 14.0326i 0.0705157i 0.999378 + 0.0352579i \(0.0112253\pi\)
−0.999378 + 0.0352579i \(0.988775\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 15.9881 + 188.028i 0.0791489 + 0.930834i
\(203\) −628.682 −3.09696
\(204\) 0 0
\(205\) 0 0
\(206\) −5.39618 63.4620i −0.0261950 0.308068i
\(207\) 0 0
\(208\) −15.0178 42.5488i −0.0722009 0.204562i
\(209\) 309.049 1.47870
\(210\) 0 0
\(211\) 74.4941i 0.353052i 0.984296 + 0.176526i \(0.0564860\pi\)
−0.984296 + 0.176526i \(0.943514\pi\)
\(212\) −27.8031 162.308i −0.131147 0.765604i
\(213\) 0 0
\(214\) 5.72587 + 67.3393i 0.0267564 + 0.314670i
\(215\) 0 0
\(216\) 0 0
\(217\) 450.550i 2.07627i
\(218\) 14.1365 + 166.253i 0.0648465 + 0.762630i
\(219\) 0 0
\(220\) 0 0
\(221\) −18.4041 −0.0832763
\(222\) 0 0
\(223\) 159.996 0.717471 0.358736 0.933439i \(-0.383208\pi\)
0.358736 + 0.933439i \(0.383208\pi\)
\(224\) −163.240 361.523i −0.728752 1.61394i
\(225\) 0 0
\(226\) 222.787 18.9436i 0.985784 0.0838213i
\(227\) 175.978 0.775236 0.387618 0.921820i \(-0.373298\pi\)
0.387618 + 0.921820i \(0.373298\pi\)
\(228\) 0 0
\(229\) 114.170 0.498560 0.249280 0.968431i \(-0.419806\pi\)
0.249280 + 0.968431i \(0.419806\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −102.140 392.668i −0.440258 1.69254i
\(233\) 260.062i 1.11615i 0.829792 + 0.558073i \(0.188459\pi\)
−0.829792 + 0.558073i \(0.811541\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −7.22687 42.1887i −0.0306223 0.178766i
\(237\) 0 0
\(238\) −161.211 + 13.7078i −0.677358 + 0.0575958i
\(239\) 140.089i 0.586147i −0.956090 0.293073i \(-0.905322\pi\)
0.956090 0.293073i \(-0.0946780\pi\)
\(240\) 0 0
\(241\) 105.920 0.439503 0.219752 0.975556i \(-0.429475\pi\)
0.219752 + 0.975556i \(0.429475\pi\)
\(242\) −0.150563 1.77070i −0.000622159 0.00731693i
\(243\) 0 0
\(244\) −221.296 + 37.9076i −0.906949 + 0.155359i
\(245\) 0 0
\(246\) 0 0
\(247\) 78.9419 0.319603
\(248\) 281.409 73.1993i 1.13471 0.295159i
\(249\) 0 0
\(250\) 0 0
\(251\) 167.879i 0.668839i 0.942424 + 0.334420i \(0.108540\pi\)
−0.942424 + 0.334420i \(0.891460\pi\)
\(252\) 0 0
\(253\) 87.2650i 0.344921i
\(254\) −2.82733 33.2510i −0.0111312 0.130909i
\(255\) 0 0
\(256\) 199.282 160.694i 0.778447 0.627710i
\(257\) 198.849i 0.773732i −0.922136 0.386866i \(-0.873558\pi\)
0.922136 0.386866i \(-0.126442\pi\)
\(258\) 0 0
\(259\) 234.629i 0.905903i
\(260\) 0 0
\(261\) 0 0
\(262\) −391.423 + 33.2827i −1.49398 + 0.127033i
\(263\) 480.528 1.82710 0.913552 0.406722i \(-0.133328\pi\)
0.913552 + 0.406722i \(0.133328\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 691.496 58.7979i 2.59961 0.221045i
\(267\) 0 0
\(268\) −63.7552 + 10.9212i −0.237892 + 0.0407506i
\(269\) −291.496 −1.08363 −0.541815 0.840498i \(-0.682263\pi\)
−0.541815 + 0.840498i \(0.682263\pi\)
\(270\) 0 0
\(271\) 174.063i 0.642299i −0.947029 0.321150i \(-0.895931\pi\)
0.947029 0.321150i \(-0.104069\pi\)
\(272\) −34.7532 98.4638i −0.127769 0.361999i
\(273\) 0 0
\(274\) −233.412 + 19.8470i −0.851868 + 0.0724344i
\(275\) 0 0
\(276\) 0 0
\(277\) 50.5203i 0.182384i −0.995833 0.0911918i \(-0.970932\pi\)
0.995833 0.0911918i \(-0.0290676\pi\)
\(278\) −373.130 + 31.7273i −1.34219 + 0.114127i
\(279\) 0 0
\(280\) 0 0
\(281\) 66.0514 0.235058 0.117529 0.993069i \(-0.462503\pi\)
0.117529 + 0.993069i \(0.462503\pi\)
\(282\) 0 0
\(283\) 116.934 0.413196 0.206598 0.978426i \(-0.433761\pi\)
0.206598 + 0.978426i \(0.433761\pi\)
\(284\) 44.6828 + 260.848i 0.157334 + 0.918477i
\(285\) 0 0
\(286\) −5.27572 62.0454i −0.0184466 0.216942i
\(287\) 65.7491 0.229091
\(288\) 0 0
\(289\) 246.411 0.852631
\(290\) 0 0
\(291\) 0 0
\(292\) 10.5578 + 61.6341i 0.0361570 + 0.211076i
\(293\) 68.3732i 0.233356i −0.993170 0.116678i \(-0.962776\pi\)
0.993170 0.116678i \(-0.0372245\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −146.547 + 38.1194i −0.495091 + 0.128782i
\(297\) 0 0
\(298\) −8.51096 100.094i −0.0285603 0.335884i
\(299\) 22.2905i 0.0745503i
\(300\) 0 0
\(301\) 565.092 1.87738
\(302\) 424.788 36.1197i 1.40658 0.119602i
\(303\) 0 0
\(304\) 149.070 + 422.349i 0.490361 + 1.38930i
\(305\) 0 0
\(306\) 0 0
\(307\) 369.497 1.20357 0.601786 0.798657i \(-0.294456\pi\)
0.601786 + 0.798657i \(0.294456\pi\)
\(308\) −92.4258 539.560i −0.300084 1.75182i
\(309\) 0 0
\(310\) 0 0
\(311\) 303.446i 0.975712i 0.872924 + 0.487856i \(0.162221\pi\)
−0.872924 + 0.487856i \(0.837779\pi\)
\(312\) 0 0
\(313\) 297.693i 0.951097i −0.879689 0.475549i \(-0.842250\pi\)
0.879689 0.475549i \(-0.157750\pi\)
\(314\) −406.369 + 34.5536i −1.29417 + 0.110043i
\(315\) 0 0
\(316\) 83.1072 + 485.160i 0.262998 + 1.53532i
\(317\) 264.678i 0.834948i −0.908689 0.417474i \(-0.862916\pi\)
0.908689 0.417474i \(-0.137084\pi\)
\(318\) 0 0
\(319\) 559.931i 1.75527i
\(320\) 0 0
\(321\) 0 0
\(322\) −16.6026 195.255i −0.0515607 0.606382i
\(323\) 182.682 0.565580
\(324\) 0 0
\(325\) 0 0
\(326\) 36.5835 + 430.243i 0.112219 + 1.31976i
\(327\) 0 0
\(328\) 10.6820 + 41.0663i 0.0325672 + 0.125202i
\(329\) −145.337 −0.441754
\(330\) 0 0
\(331\) 473.426i 1.43029i 0.698976 + 0.715145i \(0.253639\pi\)
−0.698976 + 0.715145i \(0.746361\pi\)
\(332\) −392.956 + 67.3129i −1.18360 + 0.202750i
\(333\) 0 0
\(334\) −43.2143 508.224i −0.129384 1.52163i
\(335\) 0 0
\(336\) 0 0
\(337\) 29.7588i 0.0883051i 0.999025 + 0.0441526i \(0.0140588\pi\)
−0.999025 + 0.0441526i \(0.985941\pi\)
\(338\) 27.2892 + 320.936i 0.0807373 + 0.949515i
\(339\) 0 0
\(340\) 0 0
\(341\) 401.278 1.17677
\(342\) 0 0
\(343\) −689.937 −2.01148
\(344\) 91.8086 + 352.951i 0.266885 + 1.02602i
\(345\) 0 0
\(346\) 469.158 39.8925i 1.35595 0.115296i
\(347\) 306.190 0.882391 0.441195 0.897411i \(-0.354555\pi\)
0.441195 + 0.897411i \(0.354555\pi\)
\(348\) 0 0
\(349\) −649.149 −1.86002 −0.930012 0.367528i \(-0.880204\pi\)
−0.930012 + 0.367528i \(0.880204\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 321.988 145.389i 0.914737 0.413036i
\(353\) 275.547i 0.780587i −0.920691 0.390293i \(-0.872374\pi\)
0.920691 0.390293i \(-0.127626\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −398.527 + 68.2671i −1.11946 + 0.191761i
\(357\) 0 0
\(358\) 204.600 17.3972i 0.571510 0.0485955i
\(359\) 507.672i 1.41413i 0.707149 + 0.707065i \(0.249981\pi\)
−0.707149 + 0.707065i \(0.750019\pi\)
\(360\) 0 0
\(361\) −422.594 −1.17062
\(362\) −9.62845 113.236i −0.0265979 0.312806i
\(363\) 0 0
\(364\) −23.6088 137.823i −0.0648594 0.378633i
\(365\) 0 0
\(366\) 0 0
\(367\) 62.7671 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(368\) 119.257 42.0923i 0.324068 0.114381i
\(369\) 0 0
\(370\) 0 0
\(371\) 510.315i 1.37551i
\(372\) 0 0
\(373\) 272.776i 0.731302i 0.930752 + 0.365651i \(0.119154\pi\)
−0.930752 + 0.365651i \(0.880846\pi\)
\(374\) −12.2087 143.581i −0.0326437 0.383908i
\(375\) 0 0
\(376\) −23.6124 90.7761i −0.0627990 0.241426i
\(377\) 143.026i 0.379379i
\(378\) 0 0
\(379\) 376.828i 0.994270i −0.867673 0.497135i \(-0.834385\pi\)
0.867673 0.497135i \(-0.165615\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 315.847 26.8565i 0.826825 0.0703049i
\(383\) 412.206 1.07625 0.538127 0.842864i \(-0.319132\pi\)
0.538127 + 0.842864i \(0.319132\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −312.337 + 26.5581i −0.809164 + 0.0688032i
\(387\) 0 0
\(388\) −85.9685 501.863i −0.221568 1.29346i
\(389\) −161.289 −0.414623 −0.207312 0.978275i \(-0.566471\pi\)
−0.207312 + 0.978275i \(0.566471\pi\)
\(390\) 0 0
\(391\) 51.5834i 0.131927i
\(392\) −210.774 810.303i −0.537689 2.06710i
\(393\) 0 0
\(394\) 518.383 44.0782i 1.31569 0.111874i
\(395\) 0 0
\(396\) 0 0
\(397\) 186.505i 0.469785i −0.972021 0.234893i \(-0.924526\pi\)
0.972021 0.234893i \(-0.0754738\pi\)
\(398\) −27.9643 + 2.37781i −0.0702622 + 0.00597440i
\(399\) 0 0
\(400\) 0 0
\(401\) −239.061 −0.596162 −0.298081 0.954541i \(-0.596347\pi\)
−0.298081 + 0.954541i \(0.596347\pi\)
\(402\) 0 0
\(403\) 102.501 0.254344
\(404\) −371.996 + 63.7223i −0.920781 + 0.157729i
\(405\) 0 0
\(406\) −106.529 1252.84i −0.262387 3.08582i
\(407\) −208.970 −0.513441
\(408\) 0 0
\(409\) −47.8016 −0.116874 −0.0584372 0.998291i \(-0.518612\pi\)
−0.0584372 + 0.998291i \(0.518612\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 125.553 21.5071i 0.304741 0.0522017i
\(413\) 132.646i 0.321177i
\(414\) 0 0
\(415\) 0 0
\(416\) 82.2470 37.1374i 0.197709 0.0892726i
\(417\) 0 0
\(418\) 52.3679 + 615.875i 0.125282 + 1.47339i
\(419\) 239.009i 0.570428i −0.958464 0.285214i \(-0.907935\pi\)
0.958464 0.285214i \(-0.0920647\pi\)
\(420\) 0 0
\(421\) −257.592 −0.611857 −0.305929 0.952054i \(-0.598967\pi\)
−0.305929 + 0.952054i \(0.598967\pi\)
\(422\) −148.452 + 12.6229i −0.351783 + 0.0299121i
\(423\) 0 0
\(424\) 318.738 82.9092i 0.751740 0.195541i
\(425\) 0 0
\(426\) 0 0
\(427\) −695.779 −1.62946
\(428\) −133.224 + 22.8211i −0.311271 + 0.0533204i
\(429\) 0 0
\(430\) 0 0
\(431\) 343.164i 0.796205i −0.917341 0.398103i \(-0.869669\pi\)
0.917341 0.398103i \(-0.130331\pi\)
\(432\) 0 0
\(433\) 234.760i 0.542171i 0.962555 + 0.271085i \(0.0873826\pi\)
−0.962555 + 0.271085i \(0.912617\pi\)
\(434\) 897.859 76.3450i 2.06880 0.175910i
\(435\) 0 0
\(436\) −328.916 + 56.3428i −0.754394 + 0.129227i
\(437\) 221.261i 0.506317i
\(438\) 0 0
\(439\) 374.473i 0.853013i 0.904484 + 0.426507i \(0.140256\pi\)
−0.904484 + 0.426507i \(0.859744\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −3.11854 36.6758i −0.00705553 0.0829768i
\(443\) −108.557 −0.245050 −0.122525 0.992465i \(-0.539099\pi\)
−0.122525 + 0.992465i \(0.539099\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 27.1111 + 318.842i 0.0607873 + 0.714891i
\(447\) 0 0
\(448\) 692.785 386.566i 1.54640 0.862872i
\(449\) −431.511 −0.961050 −0.480525 0.876981i \(-0.659554\pi\)
−0.480525 + 0.876981i \(0.659554\pi\)
\(450\) 0 0
\(451\) 58.5589i 0.129842i
\(452\) 75.5020 + 440.762i 0.167040 + 0.975138i
\(453\) 0 0
\(454\) 29.8193 + 350.692i 0.0656813 + 0.772448i
\(455\) 0 0
\(456\) 0 0
\(457\) 219.747i 0.480847i 0.970668 + 0.240424i \(0.0772864\pi\)
−0.970668 + 0.240424i \(0.922714\pi\)
\(458\) 19.3460 + 227.520i 0.0422402 + 0.496768i
\(459\) 0 0
\(460\) 0 0
\(461\) −223.434 −0.484673 −0.242337 0.970192i \(-0.577914\pi\)
−0.242337 + 0.970192i \(0.577914\pi\)
\(462\) 0 0
\(463\) −740.855 −1.60012 −0.800059 0.599921i \(-0.795198\pi\)
−0.800059 + 0.599921i \(0.795198\pi\)
\(464\) 765.206 270.082i 1.64915 0.582074i
\(465\) 0 0
\(466\) −518.254 + 44.0672i −1.11213 + 0.0945648i
\(467\) −249.381 −0.534007 −0.267004 0.963696i \(-0.586034\pi\)
−0.267004 + 0.963696i \(0.586034\pi\)
\(468\) 0 0
\(469\) −200.454 −0.427406
\(470\) 0 0
\(471\) 0 0
\(472\) 82.8495 21.5506i 0.175529 0.0456580i
\(473\) 503.294i 1.06405i
\(474\) 0 0
\(475\) 0 0
\(476\) −54.6340 318.940i −0.114777 0.670043i
\(477\) 0 0
\(478\) 279.171 23.7379i 0.584039 0.0496609i
\(479\) 210.915i 0.440324i 0.975463 + 0.220162i \(0.0706587\pi\)
−0.975463 + 0.220162i \(0.929341\pi\)
\(480\) 0 0
\(481\) −53.3784 −0.110974
\(482\) 17.9480 + 211.079i 0.0372366 + 0.437923i
\(483\) 0 0
\(484\) 3.50315 0.600085i 0.00723791 0.00123984i
\(485\) 0 0
\(486\) 0 0
\(487\) −710.541 −1.45902 −0.729508 0.683972i \(-0.760251\pi\)
−0.729508 + 0.683972i \(0.760251\pi\)
\(488\) −113.041 434.576i −0.231641 0.890525i
\(489\) 0 0
\(490\) 0 0
\(491\) 697.876i 1.42134i −0.703528 0.710668i \(-0.748393\pi\)
0.703528 0.710668i \(-0.251607\pi\)
\(492\) 0 0
\(493\) 330.982i 0.671363i
\(494\) 13.3766 + 157.316i 0.0270781 + 0.318454i
\(495\) 0 0
\(496\) 193.557 + 548.390i 0.390235 + 1.10563i
\(497\) 820.135i 1.65017i
\(498\) 0 0
\(499\) 875.602i 1.75471i 0.479838 + 0.877357i \(0.340695\pi\)
−0.479838 + 0.877357i \(0.659305\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −334.550 + 28.4468i −0.666434 + 0.0566670i
\(503\) −142.849 −0.283995 −0.141997 0.989867i \(-0.545352\pi\)
−0.141997 + 0.989867i \(0.545352\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 173.902 14.7869i 0.343681 0.0292232i
\(507\) 0 0
\(508\) 65.7837 11.2687i 0.129496 0.0221824i
\(509\) −147.662 −0.290102 −0.145051 0.989424i \(-0.546335\pi\)
−0.145051 + 0.989424i \(0.546335\pi\)
\(510\) 0 0
\(511\) 193.785i 0.379227i
\(512\) 354.000 + 369.903i 0.691407 + 0.722466i
\(513\) 0 0
\(514\) 396.268 33.6947i 0.770950 0.0655539i
\(515\) 0 0
\(516\) 0 0
\(517\) 129.443i 0.250374i
\(518\) −467.571 + 39.7576i −0.902646 + 0.0767520i
\(519\) 0 0
\(520\) 0 0
\(521\) 348.592 0.669082 0.334541 0.942381i \(-0.391419\pi\)
0.334541 + 0.942381i \(0.391419\pi\)
\(522\) 0 0
\(523\) 370.317 0.708063 0.354032 0.935233i \(-0.384811\pi\)
0.354032 + 0.935233i \(0.384811\pi\)
\(524\) −132.652 774.392i −0.253153 1.47785i
\(525\) 0 0
\(526\) 81.4249 + 957.601i 0.154800 + 1.82053i
\(527\) 237.201 0.450096
\(528\) 0 0
\(529\) −466.523 −0.881897
\(530\) 0 0
\(531\) 0 0
\(532\) 234.346 + 1368.06i 0.440500 + 2.57153i
\(533\) 14.9580i 0.0280638i
\(534\) 0 0
\(535\) 0 0
\(536\) −32.5670 125.201i −0.0607594 0.233584i
\(537\) 0 0
\(538\) −49.3937 580.897i −0.0918098 1.07973i
\(539\) 1155.46i 2.14371i
\(540\) 0 0
\(541\) −279.719 −0.517041 −0.258520 0.966006i \(-0.583235\pi\)
−0.258520 + 0.966006i \(0.583235\pi\)
\(542\) 346.874 29.4947i 0.639990 0.0544183i
\(543\) 0 0
\(544\) 190.331 85.9411i 0.349873 0.157980i
\(545\) 0 0
\(546\) 0 0
\(547\) 387.716 0.708804 0.354402 0.935093i \(-0.384685\pi\)
0.354402 + 0.935093i \(0.384685\pi\)
\(548\) −79.1026 461.782i −0.144348 0.842668i
\(549\) 0 0
\(550\) 0 0
\(551\) 1419.71i 2.57660i
\(552\) 0 0
\(553\) 1525.40i 2.75841i
\(554\) 100.677 8.56059i 0.181728 0.0154523i
\(555\) 0 0
\(556\) −126.453 738.201i −0.227433 1.32770i
\(557\) 43.5564i 0.0781983i −0.999235 0.0390991i \(-0.987551\pi\)
0.999235 0.0390991i \(-0.0124488\pi\)
\(558\) 0 0
\(559\) 128.559i 0.229981i
\(560\) 0 0
\(561\) 0 0
\(562\) 11.1923 + 131.628i 0.0199152 + 0.234213i
\(563\) 361.646 0.642355 0.321178 0.947019i \(-0.395921\pi\)
0.321178 + 0.947019i \(0.395921\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 19.8144 + 233.028i 0.0350078 + 0.411710i
\(567\) 0 0
\(568\) −512.248 + 133.245i −0.901845 + 0.234586i
\(569\) 888.559 1.56161 0.780807 0.624772i \(-0.214808\pi\)
0.780807 + 0.624772i \(0.214808\pi\)
\(570\) 0 0
\(571\) 447.745i 0.784142i −0.919935 0.392071i \(-0.871759\pi\)
0.919935 0.392071i \(-0.128241\pi\)
\(572\) 122.751 21.0270i 0.214599 0.0367605i
\(573\) 0 0
\(574\) 11.1411 + 131.025i 0.0194096 + 0.228267i
\(575\) 0 0
\(576\) 0 0
\(577\) 1069.90i 1.85425i 0.374756 + 0.927124i \(0.377727\pi\)
−0.374756 + 0.927124i \(0.622273\pi\)
\(578\) 41.7539 + 491.049i 0.0722386 + 0.849566i
\(579\) 0 0
\(580\) 0 0
\(581\) −1235.50 −2.12651
\(582\) 0 0
\(583\) 454.508 0.779602
\(584\) −121.036 + 31.4836i −0.207253 + 0.0539102i
\(585\) 0 0
\(586\) 136.255 11.5857i 0.232517 0.0197709i
\(587\) −129.637 −0.220847 −0.110424 0.993885i \(-0.535221\pi\)
−0.110424 + 0.993885i \(0.535221\pi\)
\(588\) 0 0
\(589\) −1017.44 −1.72741
\(590\) 0 0
\(591\) 0 0
\(592\) −100.797 285.581i −0.170265 0.482400i
\(593\) 892.757i 1.50549i 0.658311 + 0.752746i \(0.271271\pi\)
−0.658311 + 0.752746i \(0.728729\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 198.025 33.9214i 0.332257 0.0569151i
\(597\) 0 0
\(598\) 44.4208 3.77710i 0.0742823 0.00631623i
\(599\) 1030.62i 1.72057i 0.509816 + 0.860284i \(0.329714\pi\)
−0.509816 + 0.860284i \(0.670286\pi\)
\(600\) 0 0
\(601\) −815.961 −1.35767 −0.678836 0.734289i \(-0.737515\pi\)
−0.678836 + 0.734289i \(0.737515\pi\)
\(602\) 95.7540 + 1126.12i 0.159060 + 1.87063i
\(603\) 0 0
\(604\) 143.959 + 840.401i 0.238343 + 1.39139i
\(605\) 0 0
\(606\) 0 0
\(607\) 842.678 1.38827 0.694133 0.719847i \(-0.255788\pi\)
0.694133 + 0.719847i \(0.255788\pi\)
\(608\) −816.400 + 368.634i −1.34276 + 0.606305i
\(609\) 0 0
\(610\) 0 0
\(611\) 33.0644i 0.0541152i
\(612\) 0 0
\(613\) 731.088i 1.19264i 0.802747 + 0.596320i \(0.203371\pi\)
−0.802747 + 0.596320i \(0.796629\pi\)
\(614\) 62.6107 + 736.336i 0.101972 + 1.19924i
\(615\) 0 0
\(616\) 1059.58 275.615i 1.72009 0.447426i
\(617\) 919.609i 1.49045i 0.666812 + 0.745226i \(0.267658\pi\)
−0.666812 + 0.745226i \(0.732342\pi\)
\(618\) 0 0
\(619\) 688.974i 1.11304i −0.830833 0.556522i \(-0.812135\pi\)
0.830833 0.556522i \(-0.187865\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −604.711 + 51.4186i −0.972203 + 0.0826665i
\(623\) −1253.01 −2.01126
\(624\) 0 0
\(625\) 0 0
\(626\) 593.246 50.4438i 0.947678 0.0805811i
\(627\) 0 0
\(628\) −137.717 803.960i −0.219295 1.28019i
\(629\) −123.525 −0.196383
\(630\) 0 0
\(631\) 418.968i 0.663975i 0.943284 + 0.331987i \(0.107719\pi\)
−0.943284 + 0.331987i \(0.892281\pi\)
\(632\) −952.749 + 247.827i −1.50751 + 0.392131i
\(633\) 0 0
\(634\) 527.454 44.8494i 0.831946 0.0707404i
\(635\) 0 0
\(636\) 0 0
\(637\) 295.146i 0.463337i
\(638\) 1115.83 94.8795i 1.74896 0.148714i
\(639\) 0 0
\(640\) 0 0
\(641\) 47.2426 0.0737014 0.0368507 0.999321i \(-0.488267\pi\)
0.0368507 + 0.999321i \(0.488267\pi\)
\(642\) 0 0
\(643\) 710.880 1.10557 0.552784 0.833325i \(-0.313565\pi\)
0.552784 + 0.833325i \(0.313565\pi\)
\(644\) 386.293 66.1714i 0.599834 0.102751i
\(645\) 0 0
\(646\) 30.9553 + 364.051i 0.0479184 + 0.563547i
\(647\) 468.195 0.723641 0.361820 0.932248i \(-0.382155\pi\)
0.361820 + 0.932248i \(0.382155\pi\)
\(648\) 0 0
\(649\) 118.140 0.182034
\(650\) 0 0
\(651\) 0 0
\(652\) −851.192 + 145.808i −1.30551 + 0.223632i
\(653\) 551.066i 0.843900i 0.906619 + 0.421950i \(0.138654\pi\)
−0.906619 + 0.421950i \(0.861346\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −80.0271 + 28.2459i −0.121993 + 0.0430578i
\(657\) 0 0
\(658\) −24.6272 289.629i −0.0374273 0.440165i
\(659\) 158.259i 0.240151i −0.992765 0.120075i \(-0.961686\pi\)
0.992765 0.120075i \(-0.0383136\pi\)
\(660\) 0 0
\(661\) 92.4953 0.139932 0.0699662 0.997549i \(-0.477711\pi\)
0.0699662 + 0.997549i \(0.477711\pi\)
\(662\) −943.447 + 80.2214i −1.42515 + 0.121180i
\(663\) 0 0
\(664\) −200.728 771.681i −0.302301 1.16217i
\(665\) 0 0
\(666\) 0 0
\(667\) 400.877 0.601015
\(668\) 1005.47 172.236i 1.50520 0.257838i
\(669\) 0 0
\(670\) 0 0
\(671\) 619.690i 0.923532i
\(672\) 0 0
\(673\) 956.062i 1.42060i −0.703900 0.710299i \(-0.748560\pi\)
0.703900 0.710299i \(-0.251440\pi\)
\(674\) −59.3037 + 5.04259i −0.0879876 + 0.00748159i
\(675\) 0 0
\(676\) −634.940 + 108.764i −0.939261 + 0.160894i
\(677\) 1116.67i 1.64944i −0.565543 0.824719i \(-0.691333\pi\)
0.565543 0.824719i \(-0.308667\pi\)
\(678\) 0 0
\(679\) 1577.92i 2.32388i
\(680\) 0 0
\(681\) 0 0
\(682\) 67.9961 + 799.671i 0.0997010 + 1.17254i
\(683\) −826.776 −1.21051 −0.605254 0.796033i \(-0.706928\pi\)
−0.605254 + 0.796033i \(0.706928\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −116.909 1374.91i −0.170421 2.00424i
\(687\) 0 0
\(688\) −687.806 + 242.764i −0.999718 + 0.352855i
\(689\) 116.097 0.168501
\(690\) 0 0
\(691\) 965.432i 1.39715i −0.715536 0.698576i \(-0.753818\pi\)
0.715536 0.698576i \(-0.246182\pi\)
\(692\) 158.996 + 928.183i 0.229764 + 1.34130i
\(693\) 0 0
\(694\) 51.8834 + 610.177i 0.0747600 + 0.879218i
\(695\) 0 0
\(696\) 0 0
\(697\) 34.6149i 0.0496627i
\(698\) −109.997 1293.63i −0.157589 1.85334i
\(699\) 0 0
\(700\) 0 0
\(701\) 1109.94 1.58337 0.791686 0.610928i \(-0.209203\pi\)
0.791686 + 0.610928i \(0.209203\pi\)
\(702\) 0 0
\(703\) 529.845 0.753691
\(704\) 344.292 + 617.024i 0.489052 + 0.876454i
\(705\) 0 0
\(706\) 549.113 46.6911i 0.777780 0.0661347i
\(707\) −1169.60 −1.65431
\(708\) 0 0
\(709\) 964.244 1.36001 0.680003 0.733210i \(-0.261979\pi\)
0.680003 + 0.733210i \(0.261979\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −203.573 782.620i −0.285917 1.09919i
\(713\) 287.292i 0.402934i
\(714\) 0 0
\(715\) 0 0
\(716\) 69.3385 + 404.782i 0.0968415 + 0.565338i
\(717\) 0 0
\(718\) −1011.69 + 86.0244i −1.40904 + 0.119811i
\(719\) 190.820i 0.265396i −0.991157 0.132698i \(-0.957636\pi\)
0.991157 0.132698i \(-0.0423641\pi\)
\(720\) 0 0
\(721\) 394.754 0.547509
\(722\) −71.6080 842.149i −0.0991801 1.16641i
\(723\) 0 0
\(724\) 224.026 38.3753i 0.309428 0.0530046i
\(725\) 0 0
\(726\) 0 0
\(727\) 202.134 0.278039 0.139019 0.990290i \(-0.455605\pi\)
0.139019 + 0.990290i \(0.455605\pi\)
\(728\) 270.654 70.4017i 0.371777 0.0967056i
\(729\) 0 0
\(730\) 0 0
\(731\) 297.503i 0.406981i
\(732\) 0 0
\(733\) 962.435i 1.31301i 0.754322 + 0.656504i \(0.227966\pi\)
−0.754322 + 0.656504i \(0.772034\pi\)
\(734\) 10.6358 + 125.083i 0.0144902 + 0.170413i
\(735\) 0 0
\(736\) 104.090 + 230.524i 0.141426 + 0.313212i
\(737\) 178.532i 0.242242i
\(738\) 0 0
\(739\) 932.112i 1.26132i 0.776061 + 0.630658i \(0.217215\pi\)
−0.776061 + 0.630658i \(0.782785\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 1016.96 86.4722i 1.37057 0.116539i
\(743\) 1153.70 1.55276 0.776379 0.630266i \(-0.217054\pi\)
0.776379 + 0.630266i \(0.217054\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −543.590 + 46.2215i −0.728672 + 0.0619591i
\(747\) 0 0
\(748\) 284.062 48.6594i 0.379762 0.0650526i
\(749\) −418.872 −0.559242
\(750\) 0 0
\(751\) 204.359i 0.272116i −0.990701 0.136058i \(-0.956557\pi\)
0.990701 0.136058i \(-0.0434434\pi\)
\(752\) 176.898 62.4369i 0.235237 0.0830279i
\(753\) 0 0
\(754\) 285.023 24.2356i 0.378015 0.0321427i
\(755\) 0 0
\(756\) 0 0
\(757\) 216.739i 0.286314i −0.989700 0.143157i \(-0.954275\pi\)
0.989700 0.143157i \(-0.0457253\pi\)
\(758\) 750.947 63.8530i 0.990695 0.0842388i
\(759\) 0 0
\(760\) 0 0
\(761\) −1324.78 −1.74085 −0.870424 0.492303i \(-0.836155\pi\)
−0.870424 + 0.492303i \(0.836155\pi\)
\(762\) 0 0
\(763\) −1034.15 −1.35537
\(764\) 107.040 + 624.872i 0.140104 + 0.817895i
\(765\) 0 0
\(766\) 69.8477 + 821.447i 0.0911849 + 1.07238i
\(767\) 30.1772 0.0393444
\(768\) 0 0
\(769\) −444.088 −0.577488 −0.288744 0.957406i \(-0.593238\pi\)
−0.288744 + 0.957406i \(0.593238\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −105.850 617.928i −0.137112 0.800425i
\(773\) 751.987i 0.972817i −0.873732 0.486408i \(-0.838307\pi\)
0.873732 0.486408i \(-0.161693\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 985.550 256.359i 1.27004 0.330359i
\(777\) 0 0
\(778\) −27.3301 321.417i −0.0351287 0.413133i
\(779\) 148.476i 0.190599i
\(780\) 0 0
\(781\) −730.446 −0.935271
\(782\) 102.796 8.74073i 0.131452 0.0111774i
\(783\) 0 0
\(784\) 1579.06 557.337i 2.01411 0.710889i
\(785\) 0 0
\(786\) 0 0
\(787\) −442.296 −0.562002 −0.281001 0.959707i \(-0.590666\pi\)
−0.281001 + 0.959707i \(0.590666\pi\)
\(788\) 175.679 + 1025.57i 0.222943 + 1.30148i
\(789\) 0 0
\(790\) 0 0
\(791\) 1385.81i 1.75197i
\(792\) 0 0
\(793\) 158.291i 0.199610i
\(794\) 371.668 31.6030i 0.468096 0.0398022i
\(795\) 0 0
\(796\) −9.47704 55.3247i −0.0119058 0.0695034i
\(797\) 56.2072i 0.0705235i 0.999378 + 0.0352618i \(0.0112265\pi\)
−0.999378 + 0.0352618i \(0.988774\pi\)
\(798\) 0 0
\(799\) 76.5155i 0.0957641i
\(800\) 0 0
\(801\) 0 0
\(802\) −40.5086 476.403i −0.0505094 0.594019i
\(803\) −172.593 −0.214935
\(804\) 0 0
\(805\) 0 0
\(806\) 17.3686 + 204.264i 0.0215491 + 0.253430i
\(807\) 0 0
\(808\) −190.021 730.518i −0.235174 0.904107i
\(809\) 1522.16 1.88153 0.940765 0.339060i \(-0.110109\pi\)
0.940765 + 0.339060i \(0.110109\pi\)
\(810\) 0 0
\(811\) 930.734i 1.14764i −0.818982 0.573819i \(-0.805461\pi\)
0.818982 0.573819i \(-0.194539\pi\)
\(812\) 2478.63 424.585i 3.05249 0.522888i
\(813\) 0 0
\(814\) −35.4098 416.438i −0.0435009 0.511595i
\(815\) 0 0
\(816\) 0 0
\(817\) 1276.10i 1.56194i
\(818\) −8.09992 95.2595i −0.00990210 0.116454i
\(819\) 0 0
\(820\) 0 0
\(821\) −349.814 −0.426083 −0.213041 0.977043i \(-0.568337\pi\)
−0.213041 + 0.977043i \(0.568337\pi\)
\(822\) 0 0
\(823\) −61.2187 −0.0743849 −0.0371924 0.999308i \(-0.511841\pi\)
−0.0371924 + 0.999308i \(0.511841\pi\)
\(824\) 64.1344 + 246.559i 0.0778330 + 0.299222i
\(825\) 0 0
\(826\) 264.338 22.4767i 0.320022 0.0272115i
\(827\) −46.2063 −0.0558721 −0.0279361 0.999610i \(-0.508893\pi\)
−0.0279361 + 0.999610i \(0.508893\pi\)
\(828\) 0 0
\(829\) −223.832 −0.270002 −0.135001 0.990845i \(-0.543104\pi\)
−0.135001 + 0.990845i \(0.543104\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 87.9444 + 157.610i 0.105702 + 0.189435i
\(833\) 683.007i 0.819937i
\(834\) 0 0
\(835\) 0 0
\(836\) −1218.45 + 208.718i −1.45747 + 0.249663i
\(837\) 0 0
\(838\) 476.300 40.4998i 0.568377 0.0483291i
\(839\) 361.794i 0.431220i 0.976480 + 0.215610i \(0.0691740\pi\)
−0.976480 + 0.215610i \(0.930826\pi\)
\(840\) 0 0
\(841\) 1731.20 2.05851
\(842\) −43.6486 513.332i −0.0518392 0.609657i
\(843\) 0 0
\(844\) −50.3101 293.698i −0.0596092 0.347984i
\(845\) 0 0
\(846\) 0 0
\(847\) 11.0143 0.0130039
\(848\) 219.232 + 621.134i 0.258528 + 0.732470i
\(849\) 0 0
\(850\) 0 0
\(851\) 149.610i 0.175805i
\(852\) 0 0
\(853\) 844.503i 0.990039i 0.868882 + 0.495019i \(0.164839\pi\)
−0.868882 + 0.495019i \(0.835161\pi\)
\(854\) −117.899 1386.55i −0.138055 1.62360i
\(855\) 0 0
\(856\) −68.0528 261.623i −0.0795009 0.305635i
\(857\) 1389.51i 1.62137i 0.585482 + 0.810685i \(0.300905\pi\)
−0.585482 + 0.810685i \(0.699095\pi\)
\(858\) 0 0
\(859\) 1205.45i 1.40332i 0.712512 + 0.701660i \(0.247558\pi\)
−0.712512 + 0.701660i \(0.752442\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 683.861 58.1487i 0.793342 0.0674579i
\(863\) 258.868 0.299963 0.149981 0.988689i \(-0.452079\pi\)
0.149981 + 0.988689i \(0.452079\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −467.832 + 39.7798i −0.540222 + 0.0459351i
\(867\) 0 0
\(868\) 304.282 + 1776.33i 0.350555 + 2.04646i
\(869\) −1358.58 −1.56339
\(870\) 0 0
\(871\) 45.6035i 0.0523576i
\(872\) −168.015 645.919i −0.192677 0.740732i
\(873\) 0 0
\(874\) −440.930 + 37.4923i −0.504497 + 0.0428974i
\(875\) 0 0
\(876\) 0 0
\(877\) 156.268i 0.178185i −0.996023 0.0890926i \(-0.971603\pi\)
0.996023 0.0890926i \(-0.0283967\pi\)
\(878\) −746.253 + 63.4539i −0.849946 + 0.0722710i
\(879\) 0 0
\(880\) 0 0
\(881\) 1343.58 1.52507 0.762533 0.646950i \(-0.223956\pi\)
0.762533 + 0.646950i \(0.223956\pi\)
\(882\) 0 0
\(883\) −149.478 −0.169284 −0.0846420 0.996411i \(-0.526975\pi\)
−0.0846420 + 0.996411i \(0.526975\pi\)
\(884\) 72.5594 12.4293i 0.0820807 0.0140603i
\(885\) 0 0
\(886\) −18.3949 216.334i −0.0207617 0.244169i
\(887\) 1532.07 1.72725 0.863626 0.504134i \(-0.168188\pi\)
0.863626 + 0.504134i \(0.168188\pi\)
\(888\) 0 0
\(889\) 206.832 0.232656
\(890\) 0 0
\(891\) 0 0
\(892\) −630.796 + 108.055i −0.707171 + 0.121137i
\(893\) 328.204i 0.367529i
\(894\) 0 0
\(895\) 0 0
\(896\) 887.745 + 1315.09i 0.990786 + 1.46773i
\(897\) 0 0
\(898\) −73.1190 859.919i −0.0814243 0.957594i
\(899\) 1843.39i 2.05049i
\(900\) 0 0
\(901\) 268.665 0.298186
\(902\) −116.697 + 9.92273i −0.129376 + 0.0110008i
\(903\) 0 0
\(904\) −865.562 + 225.148i −0.957480 + 0.249057i
\(905\) 0 0
\(906\) 0 0
\(907\) −1245.02 −1.37268 −0.686341 0.727280i \(-0.740784\pi\)
−0.686341 + 0.727280i \(0.740784\pi\)
\(908\) −693.808 + 118.848i −0.764106 + 0.130890i
\(909\) 0 0
\(910\) 0 0
\(911\) 173.681i 0.190649i −0.995446 0.0953245i \(-0.969611\pi\)
0.995446 0.0953245i \(-0.0303889\pi\)
\(912\) 0 0
\(913\) 1100.39i 1.20524i
\(914\) −437.914 + 37.2359i −0.479118 + 0.0407395i
\(915\) 0 0
\(916\) −450.125 + 77.1058i −0.491403 + 0.0841766i
\(917\) 2434.78i 2.65515i
\(918\) 0 0
\(919\) 874.426i 0.951498i 0.879581 + 0.475749i \(0.157823\pi\)
−0.879581 + 0.475749i \(0.842177\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −37.8607 445.262i −0.0410636 0.482931i
\(923\) −186.582 −0.202147
\(924\) 0 0
\(925\) 0 0
\(926\) −125.537 1476.38i −0.135569 1.59436i
\(927\) 0 0
\(928\) 667.886 + 1479.14i 0.719705 + 1.59390i
\(929\) −1564.05 −1.68358 −0.841792 0.539803i \(-0.818499\pi\)
−0.841792 + 0.539803i \(0.818499\pi\)
\(930\) 0 0
\(931\) 2929.68i 3.14681i
\(932\) −175.635 1025.31i −0.188450 1.10012i
\(933\) 0 0
\(934\) −42.2573 496.970i −0.0452434 0.532087i
\(935\) 0 0
\(936\) 0 0
\(937\) 958.621i 1.02308i 0.859261 + 0.511538i \(0.170924\pi\)
−0.859261 + 0.511538i \(0.829076\pi\)
\(938\) −33.9666 399.466i −0.0362117 0.425869i
\(939\) 0 0
\(940\) 0 0
\(941\) 752.357 0.799529 0.399765 0.916618i \(-0.369092\pi\)
0.399765 + 0.916618i \(0.369092\pi\)
\(942\) 0 0
\(943\) −41.9247 −0.0444589
\(944\) 56.9849 + 161.451i 0.0603654 + 0.171029i
\(945\) 0 0
\(946\) −1002.97 + 85.2825i −1.06022 + 0.0901507i
\(947\) −1013.16 −1.06986 −0.534932 0.844895i \(-0.679663\pi\)
−0.534932 + 0.844895i \(0.679663\pi\)
\(948\) 0 0
\(949\) −44.0863 −0.0464555
\(950\) 0 0
\(951\) 0 0
\(952\) 626.329 162.919i 0.657909 0.171134i
\(953\) 21.5482i 0.0226109i −0.999936 0.0113054i \(-0.996401\pi\)
0.999936 0.0113054i \(-0.00359871\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 94.6102 + 552.312i 0.0989646 + 0.577732i
\(957\) 0 0
\(958\) −420.314 + 35.7393i −0.438741 + 0.0373062i
\(959\) 1451.90i 1.51397i
\(960\) 0 0
\(961\) −360.079 −0.374692
\(962\) −9.04490 106.373i −0.00940218 0.110575i
\(963\) 0 0
\(964\) −417.599 + 71.5340i −0.433193 + 0.0742054i
\(965\) 0 0
\(966\) 0 0
\(967\) 303.965 0.314338 0.157169 0.987572i \(-0.449763\pi\)
0.157169 + 0.987572i \(0.449763\pi\)
\(968\) 1.78946 + 6.87942i 0.00184861 + 0.00710684i
\(969\) 0 0
\(970\) 0 0
\(971\) 356.162i 0.366799i −0.983038 0.183399i \(-0.941290\pi\)
0.983038 0.183399i \(-0.0587101\pi\)
\(972\) 0 0
\(973\) 2320.99i 2.38539i
\(974\) −120.400 1415.97i −0.123614 1.45377i
\(975\) 0 0
\(976\) 846.873 298.907i 0.867698 0.306257i
\(977\) 1845.09i 1.88852i 0.329194 + 0.944262i \(0.393223\pi\)
−0.329194 + 0.944262i \(0.606777\pi\)
\(978\) 0 0
\(979\) 1115.99i 1.13993i
\(980\) 0 0
\(981\) 0 0
\(982\) 1390.73 118.254i 1.41623 0.120422i
\(983\) −289.444 −0.294449 −0.147225 0.989103i \(-0.547034\pi\)
−0.147225 + 0.989103i \(0.547034\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 659.583 56.0844i 0.668949 0.0568807i
\(987\) 0 0
\(988\) −311.234 + 53.3140i −0.315015 + 0.0539616i
\(989\) −360.329 −0.364337
\(990\) 0 0
\(991\) 441.980i 0.445994i 0.974819 + 0.222997i \(0.0715840\pi\)
−0.974819 + 0.222997i \(0.928416\pi\)
\(992\) −1060.04 + 478.645i −1.06859 + 0.482505i
\(993\) 0 0
\(994\) −1634.37 + 138.971i −1.64424 + 0.139810i
\(995\) 0 0
\(996\) 0 0
\(997\) 1045.42i 1.04856i −0.851545 0.524282i \(-0.824334\pi\)
0.851545 0.524282i \(-0.175666\pi\)
\(998\) −1744.91 + 148.370i −1.74841 + 0.148667i
\(999\) 0 0
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 900.3.f.f.199.10 16
3.2 odd 2 300.3.f.b.199.7 16
4.3 odd 2 inner 900.3.f.f.199.8 16
5.2 odd 4 900.3.c.u.451.2 8
5.3 odd 4 180.3.c.b.91.7 8
5.4 even 2 inner 900.3.f.f.199.7 16
12.11 even 2 300.3.f.b.199.9 16
15.2 even 4 300.3.c.d.151.7 8
15.8 even 4 60.3.c.a.31.2 yes 8
15.14 odd 2 300.3.f.b.199.10 16
20.3 even 4 180.3.c.b.91.8 8
20.7 even 4 900.3.c.u.451.1 8
20.19 odd 2 inner 900.3.f.f.199.9 16
40.3 even 4 2880.3.e.j.2431.1 8
40.13 odd 4 2880.3.e.j.2431.4 8
60.23 odd 4 60.3.c.a.31.1 8
60.47 odd 4 300.3.c.d.151.8 8
60.59 even 2 300.3.f.b.199.8 16
120.53 even 4 960.3.e.c.511.4 8
120.83 odd 4 960.3.e.c.511.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.3.c.a.31.1 8 60.23 odd 4
60.3.c.a.31.2 yes 8 15.8 even 4
180.3.c.b.91.7 8 5.3 odd 4
180.3.c.b.91.8 8 20.3 even 4
300.3.c.d.151.7 8 15.2 even 4
300.3.c.d.151.8 8 60.47 odd 4
300.3.f.b.199.7 16 3.2 odd 2
300.3.f.b.199.8 16 60.59 even 2
300.3.f.b.199.9 16 12.11 even 2
300.3.f.b.199.10 16 15.14 odd 2
900.3.c.u.451.1 8 20.7 even 4
900.3.c.u.451.2 8 5.2 odd 4
900.3.f.f.199.7 16 5.4 even 2 inner
900.3.f.f.199.8 16 4.3 odd 2 inner
900.3.f.f.199.9 16 20.19 odd 2 inner
900.3.f.f.199.10 16 1.1 even 1 trivial
960.3.e.c.511.4 8 120.53 even 4
960.3.e.c.511.7 8 120.83 odd 4
2880.3.e.j.2431.1 8 40.3 even 4
2880.3.e.j.2431.4 8 40.13 odd 4