Properties

Label 688.3.b.e.257.2
Level $688$
Weight $3$
Character 688.257
Analytic conductor $18.747$
Analytic rank $0$
Dimension $6$
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] = [688,3,Mod(257,688)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(688, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("688.257");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 688.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.7466421880\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 20x^{4} + 121x^{2} + 214 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 43)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 257.2
Root \(-2.58315i\) of defining polynomial
Character \(\chi\) \(=\) 688.257
Dual form 688.3.b.e.257.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.59415i q^{3} -7.02284i q^{5} -0.845536i q^{7} -3.91793 q^{9} +O(q^{10})\) \(q-3.59415i q^{3} -7.02284i q^{5} -0.845536i q^{7} -3.91793 q^{9} -14.5475 q^{11} -15.5505 q^{13} -25.2412 q^{15} +6.95691 q^{17} +30.8865i q^{19} -3.03899 q^{21} -17.5074 q^{23} -24.3203 q^{25} -18.2657i q^{27} -7.86076i q^{29} +57.7456 q^{31} +52.2860i q^{33} -5.93806 q^{35} -32.5310i q^{37} +55.8907i q^{39} -18.4425 q^{41} +(-22.2772 - 36.7794i) q^{43} +27.5150i q^{45} +25.7806 q^{47} +48.2851 q^{49} -25.0042i q^{51} -79.9247 q^{53} +102.165i q^{55} +111.011 q^{57} -18.4243 q^{59} +76.1107i q^{61} +3.31275i q^{63} +109.208i q^{65} -9.00391 q^{67} +62.9242i q^{69} -51.6630i q^{71} +77.4556i q^{73} +87.4108i q^{75} +12.3004i q^{77} -7.04014 q^{79} -100.911 q^{81} -83.8385 q^{83} -48.8573i q^{85} -28.2528 q^{87} -91.8322i q^{89} +13.1485i q^{91} -207.546i q^{93} +216.911 q^{95} -155.976 q^{97} +56.9961 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 36 q^{9} - 38 q^{11} + 30 q^{13} - 28 q^{15} - 20 q^{17} + 56 q^{21} + 80 q^{23} - 84 q^{25} + 112 q^{31} - 208 q^{35} - 172 q^{41} - 10 q^{43} - 30 q^{47} - 6 q^{49} - 110 q^{53} + 420 q^{57} + 12 q^{59} + 70 q^{67} - 178 q^{79} + 382 q^{81} - 10 q^{83} + 510 q^{87} + 130 q^{95} - 380 q^{97} + 466 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/688\mathbb{Z}\right)^\times\).

\(n\) \(431\) \(433\) \(517\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 0
\(3\) 3.59415i 1.19805i −0.800730 0.599025i \(-0.795555\pi\)
0.800730 0.599025i \(-0.204445\pi\)
\(4\) 0 0
\(5\) 7.02284i 1.40457i −0.711897 0.702284i \(-0.752164\pi\)
0.711897 0.702284i \(-0.247836\pi\)
\(6\) 0 0
\(7\) 0.845536i 0.120791i −0.998175 0.0603954i \(-0.980764\pi\)
0.998175 0.0603954i \(-0.0192362\pi\)
\(8\) 0 0
\(9\) −3.91793 −0.435325
\(10\) 0 0
\(11\) −14.5475 −1.32250 −0.661251 0.750165i \(-0.729974\pi\)
−0.661251 + 0.750165i \(0.729974\pi\)
\(12\) 0 0
\(13\) −15.5505 −1.19619 −0.598095 0.801425i \(-0.704075\pi\)
−0.598095 + 0.801425i \(0.704075\pi\)
\(14\) 0 0
\(15\) −25.2412 −1.68274
\(16\) 0 0
\(17\) 6.95691 0.409230 0.204615 0.978843i \(-0.434406\pi\)
0.204615 + 0.978843i \(0.434406\pi\)
\(18\) 0 0
\(19\) 30.8865i 1.62561i 0.582539 + 0.812803i \(0.302060\pi\)
−0.582539 + 0.812803i \(0.697940\pi\)
\(20\) 0 0
\(21\) −3.03899 −0.144714
\(22\) 0 0
\(23\) −17.5074 −0.761190 −0.380595 0.924742i \(-0.624281\pi\)
−0.380595 + 0.924742i \(0.624281\pi\)
\(24\) 0 0
\(25\) −24.3203 −0.972811
\(26\) 0 0
\(27\) 18.2657i 0.676509i
\(28\) 0 0
\(29\) 7.86076i 0.271061i −0.990773 0.135530i \(-0.956726\pi\)
0.990773 0.135530i \(-0.0432738\pi\)
\(30\) 0 0
\(31\) 57.7456 1.86276 0.931380 0.364048i \(-0.118606\pi\)
0.931380 + 0.364048i \(0.118606\pi\)
\(32\) 0 0
\(33\) 52.2860i 1.58442i
\(34\) 0 0
\(35\) −5.93806 −0.169659
\(36\) 0 0
\(37\) 32.5310i 0.879217i −0.898189 0.439609i \(-0.855117\pi\)
0.898189 0.439609i \(-0.144883\pi\)
\(38\) 0 0
\(39\) 55.8907i 1.43310i
\(40\) 0 0
\(41\) −18.4425 −0.449817 −0.224908 0.974380i \(-0.572208\pi\)
−0.224908 + 0.974380i \(0.572208\pi\)
\(42\) 0 0
\(43\) −22.2772 36.7794i −0.518074 0.855336i
\(44\) 0 0
\(45\) 27.5150i 0.611444i
\(46\) 0 0
\(47\) 25.7806 0.548524 0.274262 0.961655i \(-0.411566\pi\)
0.274262 + 0.961655i \(0.411566\pi\)
\(48\) 0 0
\(49\) 48.2851 0.985410
\(50\) 0 0
\(51\) 25.0042i 0.490278i
\(52\) 0 0
\(53\) −79.9247 −1.50801 −0.754006 0.656867i \(-0.771881\pi\)
−0.754006 + 0.656867i \(0.771881\pi\)
\(54\) 0 0
\(55\) 102.165i 1.85754i
\(56\) 0 0
\(57\) 111.011 1.94756
\(58\) 0 0
\(59\) −18.4243 −0.312277 −0.156138 0.987735i \(-0.549905\pi\)
−0.156138 + 0.987735i \(0.549905\pi\)
\(60\) 0 0
\(61\) 76.1107i 1.24772i 0.781537 + 0.623858i \(0.214436\pi\)
−0.781537 + 0.623858i \(0.785564\pi\)
\(62\) 0 0
\(63\) 3.31275i 0.0525833i
\(64\) 0 0
\(65\) 109.208i 1.68013i
\(66\) 0 0
\(67\) −9.00391 −0.134387 −0.0671934 0.997740i \(-0.521404\pi\)
−0.0671934 + 0.997740i \(0.521404\pi\)
\(68\) 0 0
\(69\) 62.9242i 0.911944i
\(70\) 0 0
\(71\) 51.6630i 0.727648i −0.931468 0.363824i \(-0.881471\pi\)
0.931468 0.363824i \(-0.118529\pi\)
\(72\) 0 0
\(73\) 77.4556i 1.06104i 0.847674 + 0.530518i \(0.178003\pi\)
−0.847674 + 0.530518i \(0.821997\pi\)
\(74\) 0 0
\(75\) 87.4108i 1.16548i
\(76\) 0 0
\(77\) 12.3004i 0.159746i
\(78\) 0 0
\(79\) −7.04014 −0.0891157 −0.0445578 0.999007i \(-0.514188\pi\)
−0.0445578 + 0.999007i \(0.514188\pi\)
\(80\) 0 0
\(81\) −100.911 −1.24582
\(82\) 0 0
\(83\) −83.8385 −1.01010 −0.505051 0.863089i \(-0.668526\pi\)
−0.505051 + 0.863089i \(0.668526\pi\)
\(84\) 0 0
\(85\) 48.8573i 0.574791i
\(86\) 0 0
\(87\) −28.2528 −0.324744
\(88\) 0 0
\(89\) 91.8322i 1.03182i −0.856642 0.515911i \(-0.827453\pi\)
0.856642 0.515911i \(-0.172547\pi\)
\(90\) 0 0
\(91\) 13.1485i 0.144489i
\(92\) 0 0
\(93\) 207.546i 2.23168i
\(94\) 0 0
\(95\) 216.911 2.28327
\(96\) 0 0
\(97\) −155.976 −1.60800 −0.803998 0.594632i \(-0.797298\pi\)
−0.803998 + 0.594632i \(0.797298\pi\)
\(98\) 0 0
\(99\) 56.9961 0.575718
\(100\) 0 0
\(101\) 61.9489 0.613355 0.306678 0.951813i \(-0.400783\pi\)
0.306678 + 0.951813i \(0.400783\pi\)
\(102\) 0 0
\(103\) 19.2194 0.186596 0.0932978 0.995638i \(-0.470259\pi\)
0.0932978 + 0.995638i \(0.470259\pi\)
\(104\) 0 0
\(105\) 21.3423i 0.203260i
\(106\) 0 0
\(107\) −151.561 −1.41646 −0.708230 0.705981i \(-0.750506\pi\)
−0.708230 + 0.705981i \(0.750506\pi\)
\(108\) 0 0
\(109\) 43.6904 0.400829 0.200415 0.979711i \(-0.435771\pi\)
0.200415 + 0.979711i \(0.435771\pi\)
\(110\) 0 0
\(111\) −116.922 −1.05335
\(112\) 0 0
\(113\) 30.9047i 0.273493i 0.990606 + 0.136747i \(0.0436646\pi\)
−0.990606 + 0.136747i \(0.956335\pi\)
\(114\) 0 0
\(115\) 122.951i 1.06914i
\(116\) 0 0
\(117\) 60.9256 0.520731
\(118\) 0 0
\(119\) 5.88232i 0.0494313i
\(120\) 0 0
\(121\) 90.6301 0.749009
\(122\) 0 0
\(123\) 66.2851i 0.538903i
\(124\) 0 0
\(125\) 4.77360i 0.0381888i
\(126\) 0 0
\(127\) −19.3094 −0.152043 −0.0760214 0.997106i \(-0.524222\pi\)
−0.0760214 + 0.997106i \(0.524222\pi\)
\(128\) 0 0
\(129\) −132.191 + 80.0676i −1.02474 + 0.620679i
\(130\) 0 0
\(131\) 19.3546i 0.147745i −0.997268 0.0738725i \(-0.976464\pi\)
0.997268 0.0738725i \(-0.0235358\pi\)
\(132\) 0 0
\(133\) 26.1157 0.196358
\(134\) 0 0
\(135\) −128.277 −0.950203
\(136\) 0 0
\(137\) 74.8826i 0.546588i 0.961931 + 0.273294i \(0.0881132\pi\)
−0.961931 + 0.273294i \(0.911887\pi\)
\(138\) 0 0
\(139\) −48.3195 −0.347622 −0.173811 0.984779i \(-0.555608\pi\)
−0.173811 + 0.984779i \(0.555608\pi\)
\(140\) 0 0
\(141\) 92.6596i 0.657160i
\(142\) 0 0
\(143\) 226.221 1.58196
\(144\) 0 0
\(145\) −55.2048 −0.380723
\(146\) 0 0
\(147\) 173.544i 1.18057i
\(148\) 0 0
\(149\) 56.9508i 0.382220i −0.981569 0.191110i \(-0.938791\pi\)
0.981569 0.191110i \(-0.0612087\pi\)
\(150\) 0 0
\(151\) 133.385i 0.883346i −0.897176 0.441673i \(-0.854385\pi\)
0.897176 0.441673i \(-0.145615\pi\)
\(152\) 0 0
\(153\) −27.2567 −0.178148
\(154\) 0 0
\(155\) 405.538i 2.61637i
\(156\) 0 0
\(157\) 172.766i 1.10042i −0.835026 0.550210i \(-0.814548\pi\)
0.835026 0.550210i \(-0.185452\pi\)
\(158\) 0 0
\(159\) 287.261i 1.80668i
\(160\) 0 0
\(161\) 14.8031i 0.0919448i
\(162\) 0 0
\(163\) 243.720i 1.49521i 0.664142 + 0.747607i \(0.268797\pi\)
−0.664142 + 0.747607i \(0.731203\pi\)
\(164\) 0 0
\(165\) 367.196 2.22543
\(166\) 0 0
\(167\) −246.040 −1.47330 −0.736648 0.676277i \(-0.763592\pi\)
−0.736648 + 0.676277i \(0.763592\pi\)
\(168\) 0 0
\(169\) 72.8168 0.430869
\(170\) 0 0
\(171\) 121.011i 0.707667i
\(172\) 0 0
\(173\) −137.397 −0.794201 −0.397100 0.917775i \(-0.629984\pi\)
−0.397100 + 0.917775i \(0.629984\pi\)
\(174\) 0 0
\(175\) 20.5637i 0.117507i
\(176\) 0 0
\(177\) 66.2199i 0.374123i
\(178\) 0 0
\(179\) 131.174i 0.732816i −0.930454 0.366408i \(-0.880587\pi\)
0.930454 0.366408i \(-0.119413\pi\)
\(180\) 0 0
\(181\) 229.555 1.26826 0.634129 0.773227i \(-0.281359\pi\)
0.634129 + 0.773227i \(0.281359\pi\)
\(182\) 0 0
\(183\) 273.553 1.49483
\(184\) 0 0
\(185\) −228.460 −1.23492
\(186\) 0 0
\(187\) −101.206 −0.541207
\(188\) 0 0
\(189\) −15.4443 −0.0817161
\(190\) 0 0
\(191\) 209.496i 1.09684i −0.836204 0.548419i \(-0.815230\pi\)
0.836204 0.548419i \(-0.184770\pi\)
\(192\) 0 0
\(193\) −261.300 −1.35389 −0.676943 0.736035i \(-0.736696\pi\)
−0.676943 + 0.736035i \(0.736696\pi\)
\(194\) 0 0
\(195\) 392.512 2.01288
\(196\) 0 0
\(197\) 94.0645 0.477485 0.238742 0.971083i \(-0.423265\pi\)
0.238742 + 0.971083i \(0.423265\pi\)
\(198\) 0 0
\(199\) 28.5992i 0.143715i 0.997415 + 0.0718573i \(0.0228926\pi\)
−0.997415 + 0.0718573i \(0.977107\pi\)
\(200\) 0 0
\(201\) 32.3614i 0.161002i
\(202\) 0 0
\(203\) −6.64655 −0.0327416
\(204\) 0 0
\(205\) 129.519i 0.631798i
\(206\) 0 0
\(207\) 68.5926 0.331365
\(208\) 0 0
\(209\) 449.322i 2.14987i
\(210\) 0 0
\(211\) 22.6639i 0.107412i −0.998557 0.0537059i \(-0.982897\pi\)
0.998557 0.0537059i \(-0.0171033\pi\)
\(212\) 0 0
\(213\) −185.685 −0.871760
\(214\) 0 0
\(215\) −258.296 + 156.449i −1.20138 + 0.727670i
\(216\) 0 0
\(217\) 48.8260i 0.225004i
\(218\) 0 0
\(219\) 278.387 1.27117
\(220\) 0 0
\(221\) −108.183 −0.489517
\(222\) 0 0
\(223\) 263.106i 1.17985i −0.807460 0.589923i \(-0.799158\pi\)
0.807460 0.589923i \(-0.200842\pi\)
\(224\) 0 0
\(225\) 95.2851 0.423489
\(226\) 0 0
\(227\) 54.4887i 0.240038i 0.992772 + 0.120019i \(0.0382956\pi\)
−0.992772 + 0.120019i \(0.961704\pi\)
\(228\) 0 0
\(229\) 189.993 0.829664 0.414832 0.909898i \(-0.363840\pi\)
0.414832 + 0.909898i \(0.363840\pi\)
\(230\) 0 0
\(231\) 44.2097 0.191384
\(232\) 0 0
\(233\) 209.186i 0.897795i 0.893583 + 0.448898i \(0.148183\pi\)
−0.893583 + 0.448898i \(0.851817\pi\)
\(234\) 0 0
\(235\) 181.053i 0.770440i
\(236\) 0 0
\(237\) 25.3033i 0.106765i
\(238\) 0 0
\(239\) 68.1912 0.285319 0.142659 0.989772i \(-0.454435\pi\)
0.142659 + 0.989772i \(0.454435\pi\)
\(240\) 0 0
\(241\) 380.930i 1.58062i −0.612705 0.790312i \(-0.709918\pi\)
0.612705 0.790312i \(-0.290082\pi\)
\(242\) 0 0
\(243\) 198.298i 0.816043i
\(244\) 0 0
\(245\) 339.098i 1.38407i
\(246\) 0 0
\(247\) 480.300i 1.94453i
\(248\) 0 0
\(249\) 301.328i 1.21015i
\(250\) 0 0
\(251\) 175.752 0.700208 0.350104 0.936711i \(-0.386146\pi\)
0.350104 + 0.936711i \(0.386146\pi\)
\(252\) 0 0
\(253\) 254.689 1.00667
\(254\) 0 0
\(255\) −175.600 −0.688629
\(256\) 0 0
\(257\) 162.749i 0.633263i −0.948549 0.316632i \(-0.897448\pi\)
0.948549 0.316632i \(-0.102552\pi\)
\(258\) 0 0
\(259\) −27.5062 −0.106201
\(260\) 0 0
\(261\) 30.7979i 0.118000i
\(262\) 0 0
\(263\) 78.4612i 0.298331i −0.988812 0.149166i \(-0.952341\pi\)
0.988812 0.149166i \(-0.0476588\pi\)
\(264\) 0 0
\(265\) 561.298i 2.11811i
\(266\) 0 0
\(267\) −330.059 −1.23618
\(268\) 0 0
\(269\) −48.1852 −0.179127 −0.0895636 0.995981i \(-0.528547\pi\)
−0.0895636 + 0.995981i \(0.528547\pi\)
\(270\) 0 0
\(271\) −81.9474 −0.302389 −0.151194 0.988504i \(-0.548312\pi\)
−0.151194 + 0.988504i \(0.548312\pi\)
\(272\) 0 0
\(273\) 47.2576 0.173105
\(274\) 0 0
\(275\) 353.799 1.28654
\(276\) 0 0
\(277\) 1.40205i 0.00506156i 0.999997 + 0.00253078i \(0.000805573\pi\)
−0.999997 + 0.00253078i \(0.999194\pi\)
\(278\) 0 0
\(279\) −226.243 −0.810907
\(280\) 0 0
\(281\) 248.005 0.882580 0.441290 0.897364i \(-0.354521\pi\)
0.441290 + 0.897364i \(0.354521\pi\)
\(282\) 0 0
\(283\) 138.544 0.489553 0.244777 0.969580i \(-0.421285\pi\)
0.244777 + 0.969580i \(0.421285\pi\)
\(284\) 0 0
\(285\) 779.611i 2.73548i
\(286\) 0 0
\(287\) 15.5938i 0.0543338i
\(288\) 0 0
\(289\) −240.601 −0.832531
\(290\) 0 0
\(291\) 560.600i 1.92646i
\(292\) 0 0
\(293\) 493.442 1.68410 0.842051 0.539398i \(-0.181348\pi\)
0.842051 + 0.539398i \(0.181348\pi\)
\(294\) 0 0
\(295\) 129.391i 0.438614i
\(296\) 0 0
\(297\) 265.721i 0.894684i
\(298\) 0 0
\(299\) 272.248 0.910527
\(300\) 0 0
\(301\) −31.0983 + 18.8362i −0.103317 + 0.0625786i
\(302\) 0 0
\(303\) 222.654i 0.734831i
\(304\) 0 0
\(305\) 534.513 1.75250
\(306\) 0 0
\(307\) −74.5297 −0.242768 −0.121384 0.992606i \(-0.538733\pi\)
−0.121384 + 0.992606i \(0.538733\pi\)
\(308\) 0 0
\(309\) 69.0773i 0.223551i
\(310\) 0 0
\(311\) −21.3831 −0.0687560 −0.0343780 0.999409i \(-0.510945\pi\)
−0.0343780 + 0.999409i \(0.510945\pi\)
\(312\) 0 0
\(313\) 231.173i 0.738571i −0.929316 0.369285i \(-0.879602\pi\)
0.929316 0.369285i \(-0.120398\pi\)
\(314\) 0 0
\(315\) 23.2649 0.0738568
\(316\) 0 0
\(317\) −160.636 −0.506737 −0.253369 0.967370i \(-0.581539\pi\)
−0.253369 + 0.967370i \(0.581539\pi\)
\(318\) 0 0
\(319\) 114.354i 0.358478i
\(320\) 0 0
\(321\) 544.734i 1.69699i
\(322\) 0 0
\(323\) 214.875i 0.665247i
\(324\) 0 0
\(325\) 378.191 1.16367
\(326\) 0 0
\(327\) 157.030i 0.480213i
\(328\) 0 0
\(329\) 21.7985i 0.0662567i
\(330\) 0 0
\(331\) 409.618i 1.23752i 0.785581 + 0.618759i \(0.212364\pi\)
−0.785581 + 0.618759i \(0.787636\pi\)
\(332\) 0 0
\(333\) 127.454i 0.382746i
\(334\) 0 0
\(335\) 63.2330i 0.188755i
\(336\) 0 0
\(337\) −476.795 −1.41482 −0.707411 0.706803i \(-0.750137\pi\)
−0.707411 + 0.706803i \(0.750137\pi\)
\(338\) 0 0
\(339\) 111.076 0.327659
\(340\) 0 0
\(341\) −840.054 −2.46350
\(342\) 0 0
\(343\) 82.2580i 0.239819i
\(344\) 0 0
\(345\) 441.906 1.28089
\(346\) 0 0
\(347\) 180.778i 0.520973i −0.965477 0.260487i \(-0.916117\pi\)
0.965477 0.260487i \(-0.0838830\pi\)
\(348\) 0 0
\(349\) 280.319i 0.803206i −0.915814 0.401603i \(-0.868453\pi\)
0.915814 0.401603i \(-0.131547\pi\)
\(350\) 0 0
\(351\) 284.041i 0.809233i
\(352\) 0 0
\(353\) −117.502 −0.332866 −0.166433 0.986053i \(-0.553225\pi\)
−0.166433 + 0.986053i \(0.553225\pi\)
\(354\) 0 0
\(355\) −362.821 −1.02203
\(356\) 0 0
\(357\) −21.1420 −0.0592212
\(358\) 0 0
\(359\) 208.289 0.580193 0.290096 0.956997i \(-0.406313\pi\)
0.290096 + 0.956997i \(0.406313\pi\)
\(360\) 0 0
\(361\) −592.977 −1.64260
\(362\) 0 0
\(363\) 325.738i 0.897351i
\(364\) 0 0
\(365\) 543.958 1.49030
\(366\) 0 0
\(367\) 444.860 1.21215 0.606076 0.795407i \(-0.292743\pi\)
0.606076 + 0.795407i \(0.292743\pi\)
\(368\) 0 0
\(369\) 72.2563 0.195817
\(370\) 0 0
\(371\) 67.5792i 0.182154i
\(372\) 0 0
\(373\) 71.8648i 0.192667i −0.995349 0.0963335i \(-0.969288\pi\)
0.995349 0.0963335i \(-0.0307115\pi\)
\(374\) 0 0
\(375\) −17.1570 −0.0457521
\(376\) 0 0
\(377\) 122.238i 0.324240i
\(378\) 0 0
\(379\) −146.399 −0.386276 −0.193138 0.981172i \(-0.561867\pi\)
−0.193138 + 0.981172i \(0.561867\pi\)
\(380\) 0 0
\(381\) 69.4011i 0.182155i
\(382\) 0 0
\(383\) 698.948i 1.82493i −0.409155 0.912465i \(-0.634176\pi\)
0.409155 0.912465i \(-0.365824\pi\)
\(384\) 0 0
\(385\) 86.3841 0.224374
\(386\) 0 0
\(387\) 87.2804 + 144.099i 0.225531 + 0.372349i
\(388\) 0 0
\(389\) 500.337i 1.28621i −0.765776 0.643107i \(-0.777645\pi\)
0.765776 0.643107i \(-0.222355\pi\)
\(390\) 0 0
\(391\) −121.797 −0.311502
\(392\) 0 0
\(393\) −69.5633 −0.177006
\(394\) 0 0
\(395\) 49.4418i 0.125169i
\(396\) 0 0
\(397\) −546.245 −1.37593 −0.687966 0.725743i \(-0.741496\pi\)
−0.687966 + 0.725743i \(0.741496\pi\)
\(398\) 0 0
\(399\) 93.8637i 0.235247i
\(400\) 0 0
\(401\) 393.142 0.980405 0.490203 0.871609i \(-0.336923\pi\)
0.490203 + 0.871609i \(0.336923\pi\)
\(402\) 0 0
\(403\) −897.970 −2.22821
\(404\) 0 0
\(405\) 708.683i 1.74983i
\(406\) 0 0
\(407\) 473.246i 1.16277i
\(408\) 0 0
\(409\) 788.894i 1.92884i 0.264382 + 0.964418i \(0.414832\pi\)
−0.264382 + 0.964418i \(0.585168\pi\)
\(410\) 0 0
\(411\) 269.139 0.654840
\(412\) 0 0
\(413\) 15.5784i 0.0377202i
\(414\) 0 0
\(415\) 588.784i 1.41876i
\(416\) 0 0
\(417\) 173.668i 0.416469i
\(418\) 0 0
\(419\) 221.714i 0.529149i −0.964365 0.264575i \(-0.914768\pi\)
0.964365 0.264575i \(-0.0852316\pi\)
\(420\) 0 0
\(421\) 253.802i 0.602855i 0.953489 + 0.301427i \(0.0974631\pi\)
−0.953489 + 0.301427i \(0.902537\pi\)
\(422\) 0 0
\(423\) −101.007 −0.238786
\(424\) 0 0
\(425\) −169.194 −0.398104
\(426\) 0 0
\(427\) 64.3544 0.150713
\(428\) 0 0
\(429\) 813.071i 1.89527i
\(430\) 0 0
\(431\) −455.340 −1.05647 −0.528237 0.849097i \(-0.677147\pi\)
−0.528237 + 0.849097i \(0.677147\pi\)
\(432\) 0 0
\(433\) 611.084i 1.41128i 0.708571 + 0.705640i \(0.249340\pi\)
−0.708571 + 0.705640i \(0.750660\pi\)
\(434\) 0 0
\(435\) 198.415i 0.456125i
\(436\) 0 0
\(437\) 540.742i 1.23740i
\(438\) 0 0
\(439\) −598.936 −1.36432 −0.682159 0.731204i \(-0.738959\pi\)
−0.682159 + 0.731204i \(0.738959\pi\)
\(440\) 0 0
\(441\) −189.177 −0.428974
\(442\) 0 0
\(443\) 379.068 0.855683 0.427842 0.903854i \(-0.359274\pi\)
0.427842 + 0.903854i \(0.359274\pi\)
\(444\) 0 0
\(445\) −644.923 −1.44927
\(446\) 0 0
\(447\) −204.690 −0.457919
\(448\) 0 0
\(449\) 260.317i 0.579770i −0.957061 0.289885i \(-0.906383\pi\)
0.957061 0.289885i \(-0.0936171\pi\)
\(450\) 0 0
\(451\) 268.292 0.594883
\(452\) 0 0
\(453\) −479.407 −1.05829
\(454\) 0 0
\(455\) 92.3396 0.202944
\(456\) 0 0
\(457\) 670.232i 1.46659i −0.679910 0.733296i \(-0.737981\pi\)
0.679910 0.733296i \(-0.262019\pi\)
\(458\) 0 0
\(459\) 127.073i 0.276848i
\(460\) 0 0
\(461\) −557.387 −1.20908 −0.604541 0.796574i \(-0.706644\pi\)
−0.604541 + 0.796574i \(0.706644\pi\)
\(462\) 0 0
\(463\) 116.501i 0.251622i 0.992054 + 0.125811i \(0.0401533\pi\)
−0.992054 + 0.125811i \(0.959847\pi\)
\(464\) 0 0
\(465\) −1457.56 −3.13455
\(466\) 0 0
\(467\) 465.465i 0.996712i −0.866972 0.498356i \(-0.833937\pi\)
0.866972 0.498356i \(-0.166063\pi\)
\(468\) 0 0
\(469\) 7.61313i 0.0162327i
\(470\) 0 0
\(471\) −620.947 −1.31836
\(472\) 0 0
\(473\) 324.078 + 535.049i 0.685154 + 1.13118i
\(474\) 0 0
\(475\) 751.169i 1.58141i
\(476\) 0 0
\(477\) 313.139 0.656476
\(478\) 0 0
\(479\) −24.7759 −0.0517241 −0.0258621 0.999666i \(-0.508233\pi\)
−0.0258621 + 0.999666i \(0.508233\pi\)
\(480\) 0 0
\(481\) 505.873i 1.05171i
\(482\) 0 0
\(483\) 53.2046 0.110155
\(484\) 0 0
\(485\) 1095.39i 2.25854i
\(486\) 0 0
\(487\) −356.159 −0.731333 −0.365667 0.930746i \(-0.619159\pi\)
−0.365667 + 0.930746i \(0.619159\pi\)
\(488\) 0 0
\(489\) 875.966 1.79134
\(490\) 0 0
\(491\) 498.114i 1.01449i 0.861802 + 0.507244i \(0.169336\pi\)
−0.861802 + 0.507244i \(0.830664\pi\)
\(492\) 0 0
\(493\) 54.6866i 0.110926i
\(494\) 0 0
\(495\) 400.274i 0.808635i
\(496\) 0 0
\(497\) −43.6830 −0.0878933
\(498\) 0 0
\(499\) 0.670777i 0.00134424i −1.00000 0.000672122i \(-0.999786\pi\)
1.00000 0.000672122i \(-0.000213943\pi\)
\(500\) 0 0
\(501\) 884.306i 1.76508i
\(502\) 0 0
\(503\) 775.171i 1.54109i 0.637383 + 0.770547i \(0.280017\pi\)
−0.637383 + 0.770547i \(0.719983\pi\)
\(504\) 0 0
\(505\) 435.057i 0.861499i
\(506\) 0 0
\(507\) 261.715i 0.516203i
\(508\) 0 0
\(509\) 825.181 1.62118 0.810591 0.585613i \(-0.199146\pi\)
0.810591 + 0.585613i \(0.199146\pi\)
\(510\) 0 0
\(511\) 65.4915 0.128163
\(512\) 0 0
\(513\) 564.165 1.09974
\(514\) 0 0
\(515\) 134.974i 0.262086i
\(516\) 0 0
\(517\) −375.044 −0.725424
\(518\) 0 0
\(519\) 493.825i 0.951493i
\(520\) 0 0
\(521\) 801.382i 1.53816i −0.639151 0.769081i \(-0.720714\pi\)
0.639151 0.769081i \(-0.279286\pi\)
\(522\) 0 0
\(523\) 274.667i 0.525176i −0.964908 0.262588i \(-0.915424\pi\)
0.964908 0.262588i \(-0.0845760\pi\)
\(524\) 0 0
\(525\) 73.9090 0.140779
\(526\) 0 0
\(527\) 401.731 0.762298
\(528\) 0 0
\(529\) −222.492 −0.420590
\(530\) 0 0
\(531\) 72.1852 0.135942
\(532\) 0 0
\(533\) 286.789 0.538066
\(534\) 0 0
\(535\) 1064.39i 1.98952i
\(536\) 0 0
\(537\) −471.460 −0.877951
\(538\) 0 0
\(539\) −702.428 −1.30321
\(540\) 0 0
\(541\) 361.835 0.668826 0.334413 0.942427i \(-0.391462\pi\)
0.334413 + 0.942427i \(0.391462\pi\)
\(542\) 0 0
\(543\) 825.055i 1.51944i
\(544\) 0 0
\(545\) 306.830i 0.562992i
\(546\) 0 0
\(547\) 281.855 0.515273 0.257637 0.966242i \(-0.417056\pi\)
0.257637 + 0.966242i \(0.417056\pi\)
\(548\) 0 0
\(549\) 298.196i 0.543163i
\(550\) 0 0
\(551\) 242.791 0.440638
\(552\) 0 0
\(553\) 5.95269i 0.0107644i
\(554\) 0 0
\(555\) 821.121i 1.47950i
\(556\) 0 0
\(557\) 227.271 0.408026 0.204013 0.978968i \(-0.434601\pi\)
0.204013 + 0.978968i \(0.434601\pi\)
\(558\) 0 0
\(559\) 346.421 + 571.937i 0.619715 + 1.02314i
\(560\) 0 0
\(561\) 363.749i 0.648394i
\(562\) 0 0
\(563\) −789.251 −1.40187 −0.700934 0.713226i \(-0.747233\pi\)
−0.700934 + 0.713226i \(0.747233\pi\)
\(564\) 0 0
\(565\) 217.039 0.384140
\(566\) 0 0
\(567\) 85.3241i 0.150483i
\(568\) 0 0
\(569\) 581.996 1.02284 0.511420 0.859331i \(-0.329120\pi\)
0.511420 + 0.859331i \(0.329120\pi\)
\(570\) 0 0
\(571\) 761.473i 1.33358i 0.745246 + 0.666789i \(0.232332\pi\)
−0.745246 + 0.666789i \(0.767668\pi\)
\(572\) 0 0
\(573\) −752.960 −1.31407
\(574\) 0 0
\(575\) 425.784 0.740494
\(576\) 0 0
\(577\) 570.663i 0.989017i −0.869173 0.494509i \(-0.835348\pi\)
0.869173 0.494509i \(-0.164652\pi\)
\(578\) 0 0
\(579\) 939.152i 1.62202i
\(580\) 0 0
\(581\) 70.8885i 0.122011i
\(582\) 0 0
\(583\) 1162.70 1.99435
\(584\) 0 0
\(585\) 427.870i 0.731403i
\(586\) 0 0
\(587\) 188.120i 0.320477i −0.987078 0.160238i \(-0.948774\pi\)
0.987078 0.160238i \(-0.0512263\pi\)
\(588\) 0 0
\(589\) 1783.56i 3.02811i
\(590\) 0 0
\(591\) 338.082i 0.572051i
\(592\) 0 0
\(593\) 527.123i 0.888909i 0.895801 + 0.444455i \(0.146602\pi\)
−0.895801 + 0.444455i \(0.853398\pi\)
\(594\) 0 0
\(595\) −41.3106 −0.0694296
\(596\) 0 0
\(597\) 102.790 0.172177
\(598\) 0 0
\(599\) 1156.58 1.93085 0.965427 0.260675i \(-0.0839451\pi\)
0.965427 + 0.260675i \(0.0839451\pi\)
\(600\) 0 0
\(601\) 119.619i 0.199033i 0.995036 + 0.0995166i \(0.0317297\pi\)
−0.995036 + 0.0995166i \(0.968270\pi\)
\(602\) 0 0
\(603\) 35.2767 0.0585020
\(604\) 0 0
\(605\) 636.481i 1.05203i
\(606\) 0 0
\(607\) 295.289i 0.486472i −0.969967 0.243236i \(-0.921791\pi\)
0.969967 0.243236i \(-0.0782090\pi\)
\(608\) 0 0
\(609\) 23.8887i 0.0392262i
\(610\) 0 0
\(611\) −400.901 −0.656139
\(612\) 0 0
\(613\) 846.563 1.38102 0.690508 0.723324i \(-0.257387\pi\)
0.690508 + 0.723324i \(0.257387\pi\)
\(614\) 0 0
\(615\) 465.510 0.756926
\(616\) 0 0
\(617\) 495.684 0.803377 0.401689 0.915776i \(-0.368423\pi\)
0.401689 + 0.915776i \(0.368423\pi\)
\(618\) 0 0
\(619\) −228.258 −0.368753 −0.184376 0.982856i \(-0.559027\pi\)
−0.184376 + 0.982856i \(0.559027\pi\)
\(620\) 0 0
\(621\) 319.785i 0.514952i
\(622\) 0 0
\(623\) −77.6475 −0.124635
\(624\) 0 0
\(625\) −641.531 −1.02645
\(626\) 0 0
\(627\) −1614.93 −2.57565
\(628\) 0 0
\(629\) 226.316i 0.359802i
\(630\) 0 0
\(631\) 702.132i 1.11273i 0.830939 + 0.556364i \(0.187804\pi\)
−0.830939 + 0.556364i \(0.812196\pi\)
\(632\) 0 0
\(633\) −81.4574 −0.128685
\(634\) 0 0
\(635\) 135.607i 0.213555i
\(636\) 0 0
\(637\) −750.855 −1.17874
\(638\) 0 0
\(639\) 202.412i 0.316764i
\(640\) 0 0
\(641\) 897.002i 1.39938i −0.714447 0.699689i \(-0.753322\pi\)
0.714447 0.699689i \(-0.246678\pi\)
\(642\) 0 0
\(643\) 1054.92 1.64062 0.820311 0.571918i \(-0.193800\pi\)
0.820311 + 0.571918i \(0.193800\pi\)
\(644\) 0 0
\(645\) 562.302 + 928.355i 0.871786 + 1.43931i
\(646\) 0 0
\(647\) 1230.50i 1.90185i −0.309414 0.950927i \(-0.600133\pi\)
0.309414 0.950927i \(-0.399867\pi\)
\(648\) 0 0
\(649\) 268.028 0.412986
\(650\) 0 0
\(651\) −175.488 −0.269567
\(652\) 0 0
\(653\) 234.774i 0.359532i 0.983709 + 0.179766i \(0.0575340\pi\)
−0.983709 + 0.179766i \(0.942466\pi\)
\(654\) 0 0
\(655\) −135.924 −0.207518
\(656\) 0 0
\(657\) 303.465i 0.461896i
\(658\) 0 0
\(659\) 493.253 0.748488 0.374244 0.927330i \(-0.377902\pi\)
0.374244 + 0.927330i \(0.377902\pi\)
\(660\) 0 0
\(661\) 429.533 0.649823 0.324911 0.945744i \(-0.394666\pi\)
0.324911 + 0.945744i \(0.394666\pi\)
\(662\) 0 0
\(663\) 388.827i 0.586466i
\(664\) 0 0
\(665\) 183.406i 0.275799i
\(666\) 0 0
\(667\) 137.621i 0.206329i
\(668\) 0 0
\(669\) −945.642 −1.41352
\(670\) 0 0
\(671\) 1107.22i 1.65011i
\(672\) 0 0
\(673\) 349.722i 0.519647i −0.965656 0.259823i \(-0.916336\pi\)
0.965656 0.259823i \(-0.0836644\pi\)
\(674\) 0 0
\(675\) 444.228i 0.658115i
\(676\) 0 0
\(677\) 788.649i 1.16492i −0.812861 0.582458i \(-0.802091\pi\)
0.812861 0.582458i \(-0.197909\pi\)
\(678\) 0 0
\(679\) 131.883i 0.194231i
\(680\) 0 0
\(681\) 195.841 0.287578
\(682\) 0 0
\(683\) 94.5265 0.138399 0.0691995 0.997603i \(-0.477956\pi\)
0.0691995 + 0.997603i \(0.477956\pi\)
\(684\) 0 0
\(685\) 525.888 0.767720
\(686\) 0 0
\(687\) 682.864i 0.993980i
\(688\) 0 0
\(689\) 1242.87 1.80387
\(690\) 0 0
\(691\) 942.228i 1.36357i −0.731552 0.681786i \(-0.761203\pi\)
0.731552 0.681786i \(-0.238797\pi\)
\(692\) 0 0
\(693\) 48.1922i 0.0695415i
\(694\) 0 0
\(695\) 339.340i 0.488259i
\(696\) 0 0
\(697\) −128.303 −0.184079
\(698\) 0 0
\(699\) 751.847 1.07560
\(700\) 0 0
\(701\) 568.235 0.810607 0.405303 0.914182i \(-0.367166\pi\)
0.405303 + 0.914182i \(0.367166\pi\)
\(702\) 0 0
\(703\) 1004.77 1.42926
\(704\) 0 0
\(705\) −650.733 −0.923026
\(706\) 0 0
\(707\) 52.3800i 0.0740877i
\(708\) 0 0
\(709\) 1211.96 1.70939 0.854696 0.519128i \(-0.173743\pi\)
0.854696 + 0.519128i \(0.173743\pi\)
\(710\) 0 0
\(711\) 27.5828 0.0387943
\(712\) 0 0
\(713\) −1010.97 −1.41791
\(714\) 0 0
\(715\) 1588.71i 2.22197i
\(716\) 0 0
\(717\) 245.090i 0.341826i
\(718\) 0 0
\(719\) −692.271 −0.962825 −0.481413 0.876494i \(-0.659876\pi\)
−0.481413 + 0.876494i \(0.659876\pi\)
\(720\) 0 0
\(721\) 16.2507i 0.0225391i
\(722\) 0 0
\(723\) −1369.12 −1.89367
\(724\) 0 0
\(725\) 191.176i 0.263691i
\(726\) 0 0
\(727\) 441.195i 0.606871i 0.952852 + 0.303435i \(0.0981337\pi\)
−0.952852 + 0.303435i \(0.901866\pi\)
\(728\) 0 0
\(729\) −195.486 −0.268156
\(730\) 0 0
\(731\) −154.980 255.871i −0.212012 0.350029i
\(732\) 0 0
\(733\) 251.039i 0.342481i −0.985229 0.171240i \(-0.945222\pi\)
0.985229 0.171240i \(-0.0547775\pi\)
\(734\) 0 0
\(735\) −1218.77 −1.65819
\(736\) 0 0
\(737\) 130.985 0.177727
\(738\) 0 0
\(739\) 635.071i 0.859365i −0.902980 0.429683i \(-0.858625\pi\)
0.902980 0.429683i \(-0.141375\pi\)
\(740\) 0 0
\(741\) −1726.27 −2.32965
\(742\) 0 0
\(743\) 899.136i 1.21014i 0.796171 + 0.605071i \(0.206855\pi\)
−0.796171 + 0.605071i \(0.793145\pi\)
\(744\) 0 0
\(745\) −399.956 −0.536854
\(746\) 0 0
\(747\) 328.473 0.439723
\(748\) 0 0
\(749\) 128.151i 0.171096i
\(750\) 0 0
\(751\) 597.422i 0.795502i −0.917493 0.397751i \(-0.869791\pi\)
0.917493 0.397751i \(-0.130209\pi\)
\(752\) 0 0
\(753\) 631.680i 0.838885i
\(754\) 0 0
\(755\) −936.743 −1.24072
\(756\) 0 0
\(757\) 392.757i 0.518834i −0.965765 0.259417i \(-0.916470\pi\)
0.965765 0.259417i \(-0.0835304\pi\)
\(758\) 0 0
\(759\) 915.390i 1.20605i
\(760\) 0 0
\(761\) 472.103i 0.620371i 0.950676 + 0.310186i \(0.100391\pi\)
−0.950676 + 0.310186i \(0.899609\pi\)
\(762\) 0 0
\(763\) 36.9418i 0.0484165i
\(764\) 0 0
\(765\) 191.419i 0.250221i
\(766\) 0 0
\(767\) 286.507 0.373542
\(768\) 0 0
\(769\) −591.892 −0.769690 −0.384845 0.922981i \(-0.625745\pi\)
−0.384845 + 0.922981i \(0.625745\pi\)
\(770\) 0 0
\(771\) −584.943 −0.758681
\(772\) 0 0
\(773\) 632.541i 0.818293i −0.912469 0.409147i \(-0.865826\pi\)
0.912469 0.409147i \(-0.134174\pi\)
\(774\) 0 0
\(775\) −1404.39 −1.81211
\(776\) 0 0
\(777\) 98.8614i 0.127235i
\(778\) 0 0
\(779\) 569.624i 0.731225i
\(780\) 0 0
\(781\) 751.569i 0.962316i
\(782\) 0 0
\(783\) −143.583 −0.183375
\(784\) 0 0
\(785\) −1213.31 −1.54562
\(786\) 0 0
\(787\) −26.7235 −0.0339561 −0.0169781 0.999856i \(-0.505405\pi\)
−0.0169781 + 0.999856i \(0.505405\pi\)
\(788\) 0 0
\(789\) −282.001 −0.357416
\(790\) 0 0
\(791\) 26.1311 0.0330355
\(792\) 0 0
\(793\) 1183.56i 1.49251i
\(794\) 0 0
\(795\) 2017.39 2.53760
\(796\) 0 0
\(797\) −638.409 −0.801015 −0.400508 0.916293i \(-0.631166\pi\)
−0.400508 + 0.916293i \(0.631166\pi\)
\(798\) 0 0
\(799\) 179.354 0.224473
\(800\) 0 0
\(801\) 359.792i 0.449178i
\(802\) 0 0
\(803\) 1126.79i 1.40322i
\(804\) 0 0
\(805\) 103.960 0.129143
\(806\) 0 0
\(807\) 173.185i 0.214603i
\(808\) 0 0
\(809\) 996.366 1.23160 0.615801 0.787902i \(-0.288833\pi\)
0.615801 + 0.787902i \(0.288833\pi\)
\(810\) 0 0
\(811\) 1348.22i 1.66242i −0.555959 0.831209i \(-0.687649\pi\)
0.555959 0.831209i \(-0.312351\pi\)
\(812\) 0 0
\(813\) 294.531i 0.362277i
\(814\) 0 0
\(815\) 1711.60 2.10013
\(816\) 0 0
\(817\) 1135.99 688.065i 1.39044 0.842184i
\(818\) 0 0
\(819\) 51.5148i 0.0628996i
\(820\) 0 0
\(821\) −1363.54 −1.66083 −0.830417 0.557142i \(-0.811898\pi\)
−0.830417 + 0.557142i \(0.811898\pi\)
\(822\) 0 0
\(823\) −590.955 −0.718050 −0.359025 0.933328i \(-0.616891\pi\)
−0.359025 + 0.933328i \(0.616891\pi\)
\(824\) 0 0
\(825\) 1271.61i 1.54134i
\(826\) 0 0
\(827\) 1422.73 1.72035 0.860175 0.509999i \(-0.170354\pi\)
0.860175 + 0.509999i \(0.170354\pi\)
\(828\) 0 0
\(829\) 873.419i 1.05358i 0.849995 + 0.526791i \(0.176605\pi\)
−0.849995 + 0.526791i \(0.823395\pi\)
\(830\) 0 0
\(831\) 5.03919 0.00606400
\(832\) 0 0
\(833\) 335.915 0.403259
\(834\) 0 0
\(835\) 1727.90i 2.06934i
\(836\) 0 0
\(837\) 1054.77i 1.26017i
\(838\) 0 0
\(839\) 1155.58i 1.37732i 0.725082 + 0.688662i \(0.241802\pi\)
−0.725082 + 0.688662i \(0.758198\pi\)
\(840\) 0 0
\(841\) 779.208 0.926526
\(842\) 0 0
\(843\) 891.368i 1.05738i
\(844\) 0 0
\(845\) 511.381i 0.605184i
\(846\) 0 0
\(847\) 76.6310i 0.0904734i
\(848\) 0 0
\(849\) 497.947i 0.586509i
\(850\) 0 0
\(851\) 569.533i 0.669252i
\(852\) 0 0
\(853\) 86.1596 0.101008 0.0505039 0.998724i \(-0.483917\pi\)
0.0505039 + 0.998724i \(0.483917\pi\)
\(854\) 0 0
\(855\) −849.842 −0.993967
\(856\) 0 0
\(857\) −813.017 −0.948678 −0.474339 0.880342i \(-0.657313\pi\)
−0.474339 + 0.880342i \(0.657313\pi\)
\(858\) 0 0
\(859\) 282.918i 0.329357i 0.986347 + 0.164678i \(0.0526587\pi\)
−0.986347 + 0.164678i \(0.947341\pi\)
\(860\) 0 0
\(861\) 56.0464 0.0650946
\(862\) 0 0
\(863\) 579.879i 0.671934i −0.941874 0.335967i \(-0.890937\pi\)
0.941874 0.335967i \(-0.109063\pi\)
\(864\) 0 0
\(865\) 964.915i 1.11551i
\(866\) 0 0
\(867\) 864.758i 0.997414i
\(868\) 0 0
\(869\) 102.417 0.117856
\(870\) 0 0
\(871\) 140.015 0.160752
\(872\) 0 0
\(873\) 611.101 0.700001
\(874\) 0 0
\(875\) −4.03625 −0.00461286
\(876\) 0 0
\(877\) 305.763 0.348647 0.174323 0.984688i \(-0.444226\pi\)
0.174323 + 0.984688i \(0.444226\pi\)
\(878\) 0 0
\(879\) 1773.51i 2.01764i
\(880\) 0 0
\(881\) 911.059 1.03412 0.517059 0.855950i \(-0.327027\pi\)
0.517059 + 0.855950i \(0.327027\pi\)
\(882\) 0 0
\(883\) −396.432 −0.448960 −0.224480 0.974479i \(-0.572068\pi\)
−0.224480 + 0.974479i \(0.572068\pi\)
\(884\) 0 0
\(885\) 465.051 0.525482
\(886\) 0 0
\(887\) 266.284i 0.300207i 0.988670 + 0.150104i \(0.0479607\pi\)
−0.988670 + 0.150104i \(0.952039\pi\)
\(888\) 0 0
\(889\) 16.3268i 0.0183654i
\(890\) 0 0
\(891\) 1468.01 1.64759
\(892\) 0 0
\(893\) 796.274i 0.891685i
\(894\) 0 0
\(895\) −921.215 −1.02929
\(896\) 0 0
\(897\) 978.500i 1.09086i
\(898\) 0 0
\(899\) 453.924i 0.504921i
\(900\) 0 0
\(901\) −556.029 −0.617124
\(902\) 0 0
\(903\) 67.7000 + 111.772i 0.0749724 + 0.123779i
\(904\) 0 0
\(905\) 1612.13i 1.78135i
\(906\) 0 0
\(907\) 1168.24 1.28802 0.644012 0.765015i \(-0.277269\pi\)
0.644012 + 0.765015i \(0.277269\pi\)
\(908\) 0 0
\(909\) −242.711 −0.267009
\(910\) 0 0
\(911\) 981.585i 1.07748i 0.842472 + 0.538740i \(0.181100\pi\)
−0.842472 + 0.538740i \(0.818900\pi\)
\(912\) 0 0
\(913\) 1219.64 1.33586
\(914\) 0 0
\(915\) 1921.12i 2.09959i
\(916\) 0 0
\(917\) −16.3650 −0.0178462
\(918\) 0 0
\(919\) 1148.89 1.25015 0.625075 0.780564i \(-0.285068\pi\)
0.625075 + 0.780564i \(0.285068\pi\)
\(920\) 0 0
\(921\) 267.871i 0.290848i
\(922\) 0 0
\(923\) 803.384i 0.870405i
\(924\) 0 0
\(925\) 791.164i 0.855312i
\(926\) 0 0
\(927\) −75.3000 −0.0812298
\(928\) 0 0
\(929\) 870.176i 0.936680i −0.883548 0.468340i \(-0.844852\pi\)
0.883548 0.468340i \(-0.155148\pi\)
\(930\) 0 0
\(931\) 1491.36i 1.60189i
\(932\) 0 0
\(933\) 76.8541i 0.0823731i
\(934\) 0 0
\(935\) 710.752i 0.760162i
\(936\) 0 0
\(937\) 1301.72i 1.38925i 0.719374 + 0.694623i \(0.244429\pi\)
−0.719374 + 0.694623i \(0.755571\pi\)
\(938\) 0 0
\(939\) −830.870 −0.884845
\(940\) 0 0
\(941\) −539.895 −0.573746 −0.286873 0.957969i \(-0.592616\pi\)
−0.286873 + 0.957969i \(0.592616\pi\)
\(942\) 0 0
\(943\) 322.879 0.342396
\(944\) 0 0
\(945\) 108.463i 0.114776i
\(946\) 0 0
\(947\) −80.2459 −0.0847369 −0.0423685 0.999102i \(-0.513490\pi\)
−0.0423685 + 0.999102i \(0.513490\pi\)
\(948\) 0 0
\(949\) 1204.47i 1.26920i
\(950\) 0 0
\(951\) 577.349i 0.607097i
\(952\) 0 0
\(953\) 127.129i 0.133399i 0.997773 + 0.0666996i \(0.0212469\pi\)
−0.997773 + 0.0666996i \(0.978753\pi\)
\(954\) 0 0
\(955\) −1471.26 −1.54058
\(956\) 0 0
\(957\) 411.007 0.429475
\(958\) 0 0
\(959\) 63.3159 0.0660229
\(960\) 0 0
\(961\) 2373.55 2.46988
\(962\) 0 0
\(963\) 593.806 0.616621
\(964\) 0 0
\(965\) 1835.07i 1.90162i
\(966\) 0 0
\(967\) 487.850 0.504498 0.252249 0.967662i \(-0.418830\pi\)
0.252249 + 0.967662i \(0.418830\pi\)
\(968\) 0 0
\(969\) 772.293 0.797000
\(970\) 0 0
\(971\) −706.443 −0.727542 −0.363771 0.931488i \(-0.618511\pi\)
−0.363771 + 0.931488i \(0.618511\pi\)
\(972\) 0 0
\(973\) 40.8559i 0.0419896i
\(974\) 0 0
\(975\) 1359.28i 1.39413i
\(976\) 0 0
\(977\) 1385.87 1.41849 0.709247 0.704961i \(-0.249035\pi\)
0.709247 + 0.704961i \(0.249035\pi\)
\(978\) 0 0
\(979\) 1335.93i 1.36459i
\(980\) 0 0
\(981\) −171.176 −0.174491
\(982\) 0 0
\(983\) 868.493i 0.883513i −0.897135 0.441756i \(-0.854356\pi\)
0.897135 0.441756i \(-0.145644\pi\)
\(984\) 0 0
\(985\) 660.600i 0.670660i
\(986\) 0 0
\(987\) −78.3470 −0.0793789
\(988\) 0 0
\(989\) 390.015 + 643.911i 0.394353 + 0.651073i
\(990\) 0 0
\(991\) 1678.05i 1.69329i 0.532162 + 0.846643i \(0.321380\pi\)
−0.532162 + 0.846643i \(0.678620\pi\)
\(992\) 0 0
\(993\) 1472.23 1.48261
\(994\) 0 0
\(995\) 200.848 0.201857
\(996\) 0 0
\(997\) 83.7411i 0.0839931i −0.999118 0.0419965i \(-0.986628\pi\)
0.999118 0.0419965i \(-0.0133718\pi\)
\(998\) 0 0
\(999\) −594.204 −0.594798
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 688.3.b.e.257.2 6
4.3 odd 2 43.3.b.b.42.5 yes 6
12.11 even 2 387.3.b.c.343.2 6
43.42 odd 2 inner 688.3.b.e.257.5 6
172.171 even 2 43.3.b.b.42.2 6
516.515 odd 2 387.3.b.c.343.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.3.b.b.42.2 6 172.171 even 2
43.3.b.b.42.5 yes 6 4.3 odd 2
387.3.b.c.343.2 6 12.11 even 2
387.3.b.c.343.5 6 516.515 odd 2
688.3.b.e.257.2 6 1.1 even 1 trivial
688.3.b.e.257.5 6 43.42 odd 2 inner