Properties

Label 387.3.b.c.343.5
Level $387$
Weight $3$
Character 387.343
Analytic conductor $10.545$
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] = [387,3,Mod(343,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, 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("387.343");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 387.b (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: \(10.5449862307\)
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 343.5
Root \(2.58315i\) of defining polynomial
Character \(\chi\) \(=\) 387.343
Dual form 387.3.b.c.343.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.58315i q^{2} -2.67267 q^{4} -7.02284i q^{5} -0.845536i q^{7} +3.42869i q^{8} +O(q^{10})\) \(q+2.58315i q^{2} -2.67267 q^{4} -7.02284i q^{5} -0.845536i q^{7} +3.42869i q^{8} +18.1411 q^{10} -14.5475 q^{11} -15.5505 q^{13} +2.18415 q^{14} -19.5475 q^{16} -6.95691 q^{17} +30.8865i q^{19} +18.7698i q^{20} -37.5784i q^{22} -17.5074 q^{23} -24.3203 q^{25} -40.1692i q^{26} +2.25984i q^{28} -7.86076i q^{29} -57.7456 q^{31} -36.7794i q^{32} -17.9708i q^{34} -5.93806 q^{35} +32.5310i q^{37} -79.7846 q^{38} +24.0791 q^{40} +18.4425 q^{41} +(22.2772 - 36.7794i) q^{43} +38.8807 q^{44} -45.2242i q^{46} +25.7806 q^{47} +48.2851 q^{49} -62.8230i q^{50} +41.5613 q^{52} +79.9247 q^{53} +102.165i q^{55} +2.89908 q^{56} +20.3055 q^{58} -18.4243 q^{59} -76.1107i q^{61} -149.166i q^{62} +16.8168 q^{64} +109.208i q^{65} +9.00391 q^{67} +18.5935 q^{68} -15.3389i q^{70} +51.6630i q^{71} -77.4556i q^{73} -84.0326 q^{74} -82.5496i q^{76} +12.3004i q^{77} +7.04014 q^{79} +137.279i q^{80} +47.6397i q^{82} -83.8385 q^{83} +48.8573i q^{85} +(95.0069 + 57.5454i) q^{86} -49.8789i q^{88} -91.8322i q^{89} +13.1485i q^{91} +46.7915 q^{92} +66.5953i q^{94} +216.911 q^{95} -155.976 q^{97} +124.728i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 16 q^{4} - 2 q^{10} - 38 q^{11} + 30 q^{13} - 36 q^{14} - 68 q^{16} + 20 q^{17} + 80 q^{23} - 84 q^{25} - 112 q^{31} - 208 q^{35} - 170 q^{38} + 206 q^{40} + 172 q^{41} + 10 q^{43} + 36 q^{44} - 30 q^{47} - 6 q^{49} - 120 q^{52} + 110 q^{53} + 264 q^{56} + 430 q^{58} + 12 q^{59} + 100 q^{64} - 70 q^{67} + 50 q^{68} + 50 q^{74} + 178 q^{79} - 10 q^{83} + 372 q^{86} - 150 q^{92} + 130 q^{95} - 380 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/387\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(173\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.58315i 1.29158i 0.763517 + 0.645788i \(0.223471\pi\)
−0.763517 + 0.645788i \(0.776529\pi\)
\(3\) 0 0
\(4\) −2.67267 −0.668168
\(5\) 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 \(-0.0192362\pi\)
\(8\) 3.42869i 0.428586i
\(9\) 0 0
\(10\) 18.1411 1.81411
\(11\) −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 \(0.704075\pi\)
\(14\) 2.18415 0.156011
\(15\) 0 0
\(16\) −19.5475 −1.22172
\(17\) −6.95691 −0.409230 −0.204615 0.978843i \(-0.565594\pi\)
−0.204615 + 0.978843i \(0.565594\pi\)
\(18\) 0 0
\(19\) 30.8865i 1.62561i 0.582539 + 0.812803i \(0.302060\pi\)
−0.582539 + 0.812803i \(0.697940\pi\)
\(20\) 18.7698i 0.938488i
\(21\) 0 0
\(22\) 37.5784i 1.70811i
\(23\) −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
\(26\) 40.1692i 1.54497i
\(27\) 0 0
\(28\) 2.25984i 0.0807086i
\(29\) 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 \(-0.881394\pi\)
−0.931380 + 0.364048i \(0.881394\pi\)
\(32\) 36.7794i 1.14936i
\(33\) 0 0
\(34\) 17.9708i 0.528552i
\(35\) −5.93806 −0.169659
\(36\) 0 0
\(37\) 32.5310i 0.879217i 0.898189 + 0.439609i \(0.144883\pi\)
−0.898189 + 0.439609i \(0.855117\pi\)
\(38\) −79.7846 −2.09959
\(39\) 0 0
\(40\) 24.0791 0.601978
\(41\) 18.4425 0.449817 0.224908 0.974380i \(-0.427792\pi\)
0.224908 + 0.974380i \(0.427792\pi\)
\(42\) 0 0
\(43\) 22.2772 36.7794i 0.518074 0.855336i
\(44\) 38.8807 0.883653
\(45\) 0 0
\(46\) 45.2242i 0.983135i
\(47\) 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
\(50\) 62.8230i 1.25646i
\(51\) 0 0
\(52\) 41.5613 0.799256
\(53\) 79.9247 1.50801 0.754006 0.656867i \(-0.228119\pi\)
0.754006 + 0.656867i \(0.228119\pi\)
\(54\) 0 0
\(55\) 102.165i 1.85754i
\(56\) 2.89908 0.0517693
\(57\) 0 0
\(58\) 20.3055 0.350095
\(59\) −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 \(-0.785564\pi\)
0.781537 0.623858i \(-0.214436\pi\)
\(62\) 149.166i 2.40590i
\(63\) 0 0
\(64\) 16.8168 0.262763
\(65\) 109.208i 1.68013i
\(66\) 0 0
\(67\) 9.00391 0.134387 0.0671934 0.997740i \(-0.478596\pi\)
0.0671934 + 0.997740i \(0.478596\pi\)
\(68\) 18.5935 0.273435
\(69\) 0 0
\(70\) 15.3389i 0.219127i
\(71\) 51.6630i 0.727648i 0.931468 + 0.363824i \(0.118529\pi\)
−0.931468 + 0.363824i \(0.881471\pi\)
\(72\) 0 0
\(73\) 77.4556i 1.06104i −0.847674 0.530518i \(-0.821997\pi\)
0.847674 0.530518i \(-0.178003\pi\)
\(74\) −84.0326 −1.13558
\(75\) 0 0
\(76\) 82.5496i 1.08618i
\(77\) 12.3004i 0.159746i
\(78\) 0 0
\(79\) 7.04014 0.0891157 0.0445578 0.999007i \(-0.485812\pi\)
0.0445578 + 0.999007i \(0.485812\pi\)
\(80\) 137.279i 1.71599i
\(81\) 0 0
\(82\) 47.6397i 0.580972i
\(83\) −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
\(86\) 95.0069 + 57.5454i 1.10473 + 0.669132i
\(87\) 0 0
\(88\) 49.8789i 0.566805i
\(89\) 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
\(92\) 46.7915 0.508603
\(93\) 0 0
\(94\) 66.5953i 0.708461i
\(95\) 216.911 2.28327
\(96\) 0 0
\(97\) −155.976 −1.60800 −0.803998 0.594632i \(-0.797298\pi\)
−0.803998 + 0.594632i \(0.797298\pi\)
\(98\) 124.728i 1.27273i
\(99\) 0 0
\(100\) 65.0001 0.650001
\(101\) −61.9489 −0.613355 −0.306678 0.951813i \(-0.599217\pi\)
−0.306678 + 0.951813i \(0.599217\pi\)
\(102\) 0 0
\(103\) −19.2194 −0.186596 −0.0932978 0.995638i \(-0.529741\pi\)
−0.0932978 + 0.995638i \(0.529741\pi\)
\(104\) 53.3177i 0.512670i
\(105\) 0 0
\(106\) 206.458i 1.94771i
\(107\) −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 \(0.435771\pi\)
\(110\) −263.907 −2.39916
\(111\) 0 0
\(112\) 16.5281i 0.147573i
\(113\) 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
\(116\) 21.0092i 0.181114i
\(117\) 0 0
\(118\) 47.5928i 0.403329i
\(119\) 5.88232i 0.0494313i
\(120\) 0 0
\(121\) 90.6301 0.749009
\(122\) 196.606 1.61152
\(123\) 0 0
\(124\) 154.335 1.24464
\(125\) 4.77360i 0.0381888i
\(126\) 0 0
\(127\) 19.3094 0.152043 0.0760214 0.997106i \(-0.475778\pi\)
0.0760214 + 0.997106i \(0.475778\pi\)
\(128\) 103.677i 0.809979i
\(129\) 0 0
\(130\) −282.102 −2.17001
\(131\) 19.3546i 0.147745i 0.997268 + 0.0738725i \(0.0235358\pi\)
−0.997268 + 0.0738725i \(0.976464\pi\)
\(132\) 0 0
\(133\) 26.1157 0.196358
\(134\) 23.2585i 0.173571i
\(135\) 0 0
\(136\) 23.8531i 0.175390i
\(137\) 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 \(-0.444392\pi\)
0.173811 + 0.984779i \(0.444392\pi\)
\(140\) 15.8705 0.113361
\(141\) 0 0
\(142\) −133.453 −0.939813
\(143\) 226.221 1.58196
\(144\) 0 0
\(145\) −55.2048 −0.380723
\(146\) 200.080 1.37041
\(147\) 0 0
\(148\) 86.9448i 0.587465i
\(149\) 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 \(-0.145615\pi\)
\(152\) −105.900 −0.696712
\(153\) 0 0
\(154\) −31.7739 −0.206324
\(155\) 405.538i 2.61637i
\(156\) 0 0
\(157\) 172.766i 1.10042i 0.835026 + 0.550210i \(0.185452\pi\)
−0.835026 + 0.550210i \(0.814548\pi\)
\(158\) 18.1857i 0.115100i
\(159\) 0 0
\(160\) −258.296 −1.61435
\(161\) 14.8031i 0.0919448i
\(162\) 0 0
\(163\) 243.720i 1.49521i 0.664142 + 0.747607i \(0.268797\pi\)
−0.664142 + 0.747607i \(0.731203\pi\)
\(164\) −49.2907 −0.300553
\(165\) 0 0
\(166\) 216.568i 1.30462i
\(167\) −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
\(170\) −126.206 −0.742387
\(171\) 0 0
\(172\) −59.5396 + 98.2994i −0.346161 + 0.571508i
\(173\) 137.397 0.794201 0.397100 0.917775i \(-0.370016\pi\)
0.397100 + 0.917775i \(0.370016\pi\)
\(174\) 0 0
\(175\) 20.5637i 0.117507i
\(176\) 284.368 1.61573
\(177\) 0 0
\(178\) 237.217 1.33268
\(179\) 131.174i 0.732816i 0.930454 + 0.366408i \(0.119413\pi\)
−0.930454 + 0.366408i \(0.880587\pi\)
\(180\) 0 0
\(181\) 229.555 1.26826 0.634129 0.773227i \(-0.281359\pi\)
0.634129 + 0.773227i \(0.281359\pi\)
\(182\) −33.9645 −0.186618
\(183\) 0 0
\(184\) 60.0273i 0.326235i
\(185\) 228.460 1.23492
\(186\) 0 0
\(187\) 101.206 0.541207
\(188\) −68.9032 −0.366507
\(189\) 0 0
\(190\) 560.314i 2.94902i
\(191\) 209.496i 1.09684i 0.836204 + 0.548419i \(0.184770\pi\)
−0.836204 + 0.548419i \(0.815230\pi\)
\(192\) 0 0
\(193\) −261.300 −1.35389 −0.676943 0.736035i \(-0.736696\pi\)
−0.676943 + 0.736035i \(0.736696\pi\)
\(194\) 402.909i 2.07685i
\(195\) 0 0
\(196\) −129.050 −0.658419
\(197\) −94.0645 −0.477485 −0.238742 0.971083i \(-0.576735\pi\)
−0.238742 + 0.971083i \(0.576735\pi\)
\(198\) 0 0
\(199\) 28.5992i 0.143715i 0.997415 + 0.0718573i \(0.0228926\pi\)
−0.997415 + 0.0718573i \(0.977107\pi\)
\(200\) 83.3866i 0.416933i
\(201\) 0 0
\(202\) 160.023i 0.792195i
\(203\) −6.64655 −0.0327416
\(204\) 0 0
\(205\) 129.519i 0.631798i
\(206\) 49.6465i 0.241002i
\(207\) 0 0
\(208\) 303.973 1.46141
\(209\) 449.322i 2.14987i
\(210\) 0 0
\(211\) 22.6639i 0.107412i −0.998557 0.0537059i \(-0.982897\pi\)
0.998557 0.0537059i \(-0.0171033\pi\)
\(212\) −213.612 −1.00761
\(213\) 0 0
\(214\) 391.506i 1.82947i
\(215\) −258.296 156.449i −1.20138 0.727670i
\(216\) 0 0
\(217\) 48.8260i 0.225004i
\(218\) 112.859i 0.517701i
\(219\) 0 0
\(220\) 273.053i 1.24115i
\(221\) 108.183 0.489517
\(222\) 0 0
\(223\) 263.106i 1.17985i −0.807460 0.589923i \(-0.799158\pi\)
0.807460 0.589923i \(-0.200842\pi\)
\(224\) −31.0983 −0.138832
\(225\) 0 0
\(226\) −79.8316 −0.353237
\(227\) 54.4887i 0.240038i −0.992772 0.120019i \(-0.961704\pi\)
0.992772 0.120019i \(-0.0382956\pi\)
\(228\) 0 0
\(229\) 189.993 0.829664 0.414832 0.909898i \(-0.363840\pi\)
0.414832 + 0.909898i \(0.363840\pi\)
\(230\) −317.602 −1.38088
\(231\) 0 0
\(232\) 26.9521 0.116173
\(233\) 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
\(236\) 49.2422 0.208653
\(237\) 0 0
\(238\) −15.1949 −0.0638442
\(239\) 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 \(0.290082\pi\)
−0.612705 + 0.790312i \(0.709918\pi\)
\(242\) 234.111i 0.967402i
\(243\) 0 0
\(244\) 203.419i 0.833685i
\(245\) 339.098i 1.38407i
\(246\) 0 0
\(247\) 480.300i 1.94453i
\(248\) 197.992i 0.798353i
\(249\) 0 0
\(250\) 12.3309 0.0493237
\(251\) 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
\(254\) 49.8792i 0.196375i
\(255\) 0 0
\(256\) 335.082 1.30891
\(257\) 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
\(260\) 291.878i 1.12261i
\(261\) 0 0
\(262\) −49.9959 −0.190824
\(263\) 78.4612i 0.298331i 0.988812 + 0.149166i \(0.0476588\pi\)
−0.988812 + 0.149166i \(0.952341\pi\)
\(264\) 0 0
\(265\) 561.298i 2.11811i
\(266\) 67.4607i 0.253612i
\(267\) 0 0
\(268\) −24.0645 −0.0897930
\(269\) 48.1852 0.179127 0.0895636 0.995981i \(-0.471453\pi\)
0.0895636 + 0.995981i \(0.471453\pi\)
\(270\) 0 0
\(271\) 81.9474 0.302389 0.151194 0.988504i \(-0.451688\pi\)
0.151194 + 0.988504i \(0.451688\pi\)
\(272\) 135.990 0.499964
\(273\) 0 0
\(274\) −193.433 −0.705960
\(275\) 353.799 1.28654
\(276\) 0 0
\(277\) 1.40205i 0.00506156i −0.999997 0.00253078i \(-0.999194\pi\)
0.999997 0.00253078i \(-0.000805573\pi\)
\(278\) 124.817i 0.448981i
\(279\) 0 0
\(280\) 20.3598i 0.0727135i
\(281\) −248.005 −0.882580 −0.441290 0.897364i \(-0.645479\pi\)
−0.441290 + 0.897364i \(0.645479\pi\)
\(282\) 0 0
\(283\) −138.544 −0.489553 −0.244777 0.969580i \(-0.578715\pi\)
−0.244777 + 0.969580i \(0.578715\pi\)
\(284\) 138.078i 0.486191i
\(285\) 0 0
\(286\) 584.362i 2.04322i
\(287\) 15.5938i 0.0543338i
\(288\) 0 0
\(289\) −240.601 −0.832531
\(290\) 142.602i 0.491733i
\(291\) 0 0
\(292\) 207.014i 0.708950i
\(293\) −493.442 −1.68410 −0.842051 0.539398i \(-0.818652\pi\)
−0.842051 + 0.539398i \(0.818652\pi\)
\(294\) 0 0
\(295\) 129.391i 0.438614i
\(296\) −111.539 −0.376820
\(297\) 0 0
\(298\) 147.112 0.493666
\(299\) 272.248 0.910527
\(300\) 0 0
\(301\) −31.0983 18.8362i −0.103317 0.0625786i
\(302\) 344.554 1.14091
\(303\) 0 0
\(304\) 603.755i 1.98603i
\(305\) −534.513 −1.75250
\(306\) 0 0
\(307\) 74.5297 0.242768 0.121384 0.992606i \(-0.461267\pi\)
0.121384 + 0.992606i \(0.461267\pi\)
\(308\) 32.8751i 0.106737i
\(309\) 0 0
\(310\) −1047.57 −3.37924
\(311\) −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 \(0.120398\pi\)
−0.929316 + 0.369285i \(0.879602\pi\)
\(314\) −446.281 −1.42128
\(315\) 0 0
\(316\) −18.8160 −0.0595443
\(317\) 160.636 0.506737 0.253369 0.967370i \(-0.418461\pi\)
0.253369 + 0.967370i \(0.418461\pi\)
\(318\) 0 0
\(319\) 114.354i 0.358478i
\(320\) 118.102i 0.369068i
\(321\) 0 0
\(322\) −38.2387 −0.118754
\(323\) 214.875i 0.665247i
\(324\) 0 0
\(325\) 378.191 1.16367
\(326\) −629.565 −1.93118
\(327\) 0 0
\(328\) 63.2335i 0.192785i
\(329\) 21.7985i 0.0662567i
\(330\) 0 0
\(331\) 409.618i 1.23752i 0.785581 + 0.618759i \(0.212364\pi\)
−0.785581 + 0.618759i \(0.787636\pi\)
\(332\) 224.073 0.674918
\(333\) 0 0
\(334\) 635.559i 1.90287i
\(335\) 63.2330i 0.188755i
\(336\) 0 0
\(337\) −476.795 −1.41482 −0.707411 0.706803i \(-0.750137\pi\)
−0.707411 + 0.706803i \(0.750137\pi\)
\(338\) 188.097i 0.556500i
\(339\) 0 0
\(340\) 130.580i 0.384057i
\(341\) 840.054 2.46350
\(342\) 0 0
\(343\) 82.2580i 0.239819i
\(344\) 126.105 + 76.3815i 0.366585 + 0.222039i
\(345\) 0 0
\(346\) 354.917i 1.02577i
\(347\) 180.778i 0.520973i 0.965477 + 0.260487i \(0.0838830\pi\)
−0.965477 + 0.260487i \(0.916117\pi\)
\(348\) 0 0
\(349\) 280.319i 0.803206i 0.915814 + 0.401603i \(0.131547\pi\)
−0.915814 + 0.401603i \(0.868453\pi\)
\(350\) −53.1191 −0.151769
\(351\) 0 0
\(352\) 535.049i 1.52003i
\(353\) 117.502 0.332866 0.166433 0.986053i \(-0.446775\pi\)
0.166433 + 0.986053i \(0.446775\pi\)
\(354\) 0 0
\(355\) 362.821 1.02203
\(356\) 245.437i 0.689431i
\(357\) 0 0
\(358\) −338.843 −0.946488
\(359\) 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
\(362\) 592.975i 1.63805i
\(363\) 0 0
\(364\) 35.1416i 0.0965428i
\(365\) −543.958 −1.49030
\(366\) 0 0
\(367\) −444.860 −1.21215 −0.606076 0.795407i \(-0.707257\pi\)
−0.606076 + 0.795407i \(0.707257\pi\)
\(368\) 342.226 0.929961
\(369\) 0 0
\(370\) 590.148i 1.59499i
\(371\) 67.5792i 0.182154i
\(372\) 0 0
\(373\) 71.8648i 0.192667i 0.995349 + 0.0963335i \(0.0307115\pi\)
−0.995349 + 0.0963335i \(0.969288\pi\)
\(374\) 261.430i 0.699010i
\(375\) 0 0
\(376\) 88.3938i 0.235090i
\(377\) 122.238i 0.324240i
\(378\) 0 0
\(379\) 146.399 0.386276 0.193138 0.981172i \(-0.438133\pi\)
0.193138 + 0.981172i \(0.438133\pi\)
\(380\) −579.732 −1.52561
\(381\) 0 0
\(382\) −541.160 −1.41665
\(383\) 698.948i 1.82493i 0.409155 + 0.912465i \(0.365824\pi\)
−0.409155 + 0.912465i \(0.634176\pi\)
\(384\) 0 0
\(385\) 86.3841 0.224374
\(386\) 674.978i 1.74865i
\(387\) 0 0
\(388\) 416.872 1.07441
\(389\) 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
\(392\) 165.554i 0.422333i
\(393\) 0 0
\(394\) 242.983i 0.616708i
\(395\) 49.4418i 0.125169i
\(396\) 0 0
\(397\) −546.245 −1.37593 −0.687966 0.725743i \(-0.741496\pi\)
−0.687966 + 0.725743i \(0.741496\pi\)
\(398\) −73.8760 −0.185618
\(399\) 0 0
\(400\) 475.401 1.18850
\(401\) −393.142 −0.980405 −0.490203 0.871609i \(-0.663077\pi\)
−0.490203 + 0.871609i \(0.663077\pi\)
\(402\) 0 0
\(403\) 897.970 2.22821
\(404\) 165.569 0.409825
\(405\) 0 0
\(406\) 17.1691i 0.0422883i
\(407\) 473.246i 1.16277i
\(408\) 0 0
\(409\) 788.894i 1.92884i −0.264382 0.964418i \(-0.585168\pi\)
0.264382 0.964418i \(-0.414832\pi\)
\(410\) 334.566 0.816015
\(411\) 0 0
\(412\) 51.3670 0.124677
\(413\) 15.5784i 0.0377202i
\(414\) 0 0
\(415\) 588.784i 1.41876i
\(416\) 571.937i 1.37485i
\(417\) 0 0
\(418\) 1160.67 2.77671
\(419\) 221.714i 0.529149i 0.964365 + 0.264575i \(0.0852316\pi\)
−0.964365 + 0.264575i \(0.914768\pi\)
\(420\) 0 0
\(421\) 253.802i 0.602855i −0.953489 0.301427i \(-0.902537\pi\)
0.953489 0.301427i \(-0.0974631\pi\)
\(422\) 58.5442 0.138730
\(423\) 0 0
\(424\) 274.037i 0.646313i
\(425\) 169.194 0.398104
\(426\) 0 0
\(427\) −64.3544 −0.150713
\(428\) 405.074 0.946434
\(429\) 0 0
\(430\) 404.132 667.218i 0.939841 1.55167i
\(431\) −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 \(-0.750660\pi\)
0.708571 0.705640i \(-0.249340\pi\)
\(434\) −126.125 −0.290610
\(435\) 0 0
\(436\) −116.770 −0.267821
\(437\) 540.742i 1.23740i
\(438\) 0 0
\(439\) 598.936 1.36432 0.682159 0.731204i \(-0.261041\pi\)
0.682159 + 0.731204i \(0.261041\pi\)
\(440\) −350.291 −0.796117
\(441\) 0 0
\(442\) 279.454i 0.632248i
\(443\) 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
\(446\) 679.642 1.52386
\(447\) 0 0
\(448\) 14.2192i 0.0317393i
\(449\) 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
\(452\) 82.5982i 0.182739i
\(453\) 0 0
\(454\) 140.752 0.310027
\(455\) 92.3396 0.202944
\(456\) 0 0
\(457\) 670.232i 1.46659i 0.679910 + 0.733296i \(0.262019\pi\)
−0.679910 + 0.733296i \(0.737981\pi\)
\(458\) 490.781i 1.07157i
\(459\) 0 0
\(460\) 328.609i 0.714367i
\(461\) 557.387 1.20908 0.604541 0.796574i \(-0.293356\pi\)
0.604541 + 0.796574i \(0.293356\pi\)
\(462\) 0 0
\(463\) 116.501i 0.251622i 0.992054 + 0.125811i \(0.0401533\pi\)
−0.992054 + 0.125811i \(0.959847\pi\)
\(464\) 153.658i 0.331160i
\(465\) 0 0
\(466\) −540.360 −1.15957
\(467\) 465.465i 0.996712i 0.866972 + 0.498356i \(0.166063\pi\)
−0.866972 + 0.498356i \(0.833937\pi\)
\(468\) 0 0
\(469\) 7.61313i 0.0162327i
\(470\) 467.688 0.995081
\(471\) 0 0
\(472\) 63.1713i 0.133837i
\(473\) −324.078 + 535.049i −0.685154 + 1.13118i
\(474\) 0 0
\(475\) 751.169i 1.58141i
\(476\) 15.7215i 0.0330284i
\(477\) 0 0
\(478\) 176.148i 0.368511i
\(479\) −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
\(482\) −984.001 −2.04150
\(483\) 0 0
\(484\) −242.225 −0.500464
\(485\) 1095.39i 2.25854i
\(486\) 0 0
\(487\) 356.159 0.731333 0.365667 0.930746i \(-0.380841\pi\)
0.365667 + 0.930746i \(0.380841\pi\)
\(488\) 260.960 0.534754
\(489\) 0 0
\(490\) 875.942 1.78764
\(491\) 498.114i 1.01449i −0.861802 0.507244i \(-0.830664\pi\)
0.861802 0.507244i \(-0.169336\pi\)
\(492\) 0 0
\(493\) 54.6866i 0.110926i
\(494\) 1240.69 2.51151
\(495\) 0 0
\(496\) 1128.78 2.27577
\(497\) 43.6830 0.0878933
\(498\) 0 0
\(499\) 0.670777i 0.00134424i −1.00000 0.000672122i \(-0.999786\pi\)
1.00000 0.000672122i \(-0.000213943\pi\)
\(500\) 12.7583i 0.0255165i
\(501\) 0 0
\(502\) 453.995i 0.904372i
\(503\) 775.171i 1.54109i −0.637383 0.770547i \(-0.719983\pi\)
0.637383 0.770547i \(-0.280017\pi\)
\(504\) 0 0
\(505\) 435.057i 0.861499i
\(506\) 657.900i 1.30020i
\(507\) 0 0
\(508\) −51.6078 −0.101590
\(509\) −825.181 −1.62118 −0.810591 0.585613i \(-0.800854\pi\)
−0.810591 + 0.585613i \(0.800854\pi\)
\(510\) 0 0
\(511\) −65.4915 −0.128163
\(512\) 450.857i 0.880580i
\(513\) 0 0
\(514\) 420.404 0.817908
\(515\) 134.974i 0.262086i
\(516\) 0 0
\(517\) −375.044 −0.725424
\(518\) 71.0526i 0.137167i
\(519\) 0 0
\(520\) −374.441 −0.720080
\(521\) 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 \(-0.0845760\pi\)
\(524\) 51.7285i 0.0987185i
\(525\) 0 0
\(526\) −202.677 −0.385318
\(527\) 401.731 0.762298
\(528\) 0 0
\(529\) −222.492 −0.420590
\(530\) 1449.92 2.73569
\(531\) 0 0
\(532\) −69.7986 −0.131200
\(533\) −286.789 −0.538066
\(534\) 0 0
\(535\) 1064.39i 1.98952i
\(536\) 30.8716i 0.0575963i
\(537\) 0 0
\(538\) 124.470i 0.231356i
\(539\) −702.428 −1.30321
\(540\) 0 0
\(541\) 361.835 0.668826 0.334413 0.942427i \(-0.391462\pi\)
0.334413 + 0.942427i \(0.391462\pi\)
\(542\) 211.682i 0.390558i
\(543\) 0 0
\(544\) 255.871i 0.470352i
\(545\) 306.830i 0.562992i
\(546\) 0 0
\(547\) −281.855 −0.515273 −0.257637 0.966242i \(-0.582944\pi\)
−0.257637 + 0.966242i \(0.582944\pi\)
\(548\) 200.137i 0.365213i
\(549\) 0 0
\(550\) 913.918i 1.66167i
\(551\) 242.791 0.440638
\(552\) 0 0
\(553\) 5.95269i 0.0107644i
\(554\) 3.62171 0.00653739
\(555\) 0 0
\(556\) −129.142 −0.232270
\(557\) −227.271 −0.408026 −0.204013 0.978968i \(-0.565399\pi\)
−0.204013 + 0.978968i \(0.565399\pi\)
\(558\) 0 0
\(559\) −346.421 + 571.937i −0.619715 + 1.02314i
\(560\) 116.074 0.207276
\(561\) 0 0
\(562\) 640.635i 1.13992i
\(563\) −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
\(566\) 357.879i 0.632295i
\(567\) 0 0
\(568\) −177.136 −0.311860
\(569\) −581.996 −1.02284 −0.511420 0.859331i \(-0.670880\pi\)
−0.511420 + 0.859331i \(0.670880\pi\)
\(570\) 0 0
\(571\) 761.473i 1.33358i 0.745246 + 0.666789i \(0.232332\pi\)
−0.745246 + 0.666789i \(0.767668\pi\)
\(572\) −604.613 −1.05702
\(573\) 0 0
\(574\) 40.2811 0.0701762
\(575\) 425.784 0.740494
\(576\) 0 0
\(577\) 570.663i 0.989017i 0.869173 + 0.494509i \(0.164652\pi\)
−0.869173 + 0.494509i \(0.835348\pi\)
\(578\) 621.510i 1.07528i
\(579\) 0 0
\(580\) 147.544 0.254387
\(581\) 70.8885i 0.122011i
\(582\) 0 0
\(583\) −1162.70 −1.99435
\(584\) 265.571 0.454745
\(585\) 0 0
\(586\) 1274.64i 2.17515i
\(587\) 188.120i 0.320477i 0.987078 + 0.160238i \(0.0512263\pi\)
−0.987078 + 0.160238i \(0.948774\pi\)
\(588\) 0 0
\(589\) 1783.56i 3.02811i
\(590\) −334.237 −0.566503
\(591\) 0 0
\(592\) 635.901i 1.07416i
\(593\) 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
\(596\) 152.211i 0.255387i
\(597\) 0 0
\(598\) 703.257i 1.17602i
\(599\) 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 \(-0.968270\pi\)
0.995036 0.0995166i \(-0.0317297\pi\)
\(602\) 48.6567 80.3317i 0.0808250 0.133441i
\(603\) 0 0
\(604\) 356.495i 0.590224i
\(605\) 636.481i 1.05203i
\(606\) 0 0
\(607\) 295.289i 0.486472i −0.969967 0.243236i \(-0.921791\pi\)
0.969967 0.243236i \(-0.0782090\pi\)
\(608\) 1135.99 1.86840
\(609\) 0 0
\(610\) 1380.73i 2.26349i
\(611\) −400.901 −0.656139
\(612\) 0 0
\(613\) 846.563 1.38102 0.690508 0.723324i \(-0.257387\pi\)
0.690508 + 0.723324i \(0.257387\pi\)
\(614\) 192.522i 0.313553i
\(615\) 0 0
\(616\) −42.1744 −0.0684649
\(617\) −495.684 −0.803377 −0.401689 0.915776i \(-0.631577\pi\)
−0.401689 + 0.915776i \(0.631577\pi\)
\(618\) 0 0
\(619\) 228.258 0.368753 0.184376 0.982856i \(-0.440973\pi\)
0.184376 + 0.982856i \(0.440973\pi\)
\(620\) 1083.87i 1.74818i
\(621\) 0 0
\(622\) 55.2358i 0.0888035i
\(623\) −77.6475 −0.124635
\(624\) 0 0
\(625\) −641.531 −1.02645
\(626\) −597.154 −0.953920
\(627\) 0 0
\(628\) 461.747i 0.735266i
\(629\) 226.316i 0.359802i
\(630\) 0 0
\(631\) 702.132i 1.11273i 0.830939 + 0.556364i \(0.187804\pi\)
−0.830939 + 0.556364i \(0.812196\pi\)
\(632\) 24.1384i 0.0381937i
\(633\) 0 0
\(634\) 414.946i 0.654490i
\(635\) 135.607i 0.213555i
\(636\) 0 0
\(637\) −750.855 −1.17874
\(638\) −295.395 −0.463001
\(639\) 0 0
\(640\) −728.109 −1.13767
\(641\) 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 \(-0.806200\pi\)
−0.820311 + 0.571918i \(0.806200\pi\)
\(644\) 39.5639i 0.0614346i
\(645\) 0 0
\(646\) 555.054 0.859217
\(647\) 1230.50i 1.90185i 0.309414 + 0.950927i \(0.399867\pi\)
−0.309414 + 0.950927i \(0.600133\pi\)
\(648\) 0 0
\(649\) 268.028 0.412986
\(650\) 976.926i 1.50296i
\(651\) 0 0
\(652\) 651.383i 0.999054i
\(653\) 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
\(656\) −360.505 −0.549550
\(657\) 0 0
\(658\) 56.3087 0.0855756
\(659\) 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 \(0.394666\pi\)
\(662\) −1058.11 −1.59835
\(663\) 0 0
\(664\) 287.456i 0.432916i
\(665\) 183.406i 0.275799i
\(666\) 0 0
\(667\) 137.621i 0.206329i
\(668\) 657.585 0.984409
\(669\) 0 0
\(670\) 163.341 0.243792
\(671\) 1107.22i 1.65011i
\(672\) 0 0
\(673\) 349.722i 0.519647i 0.965656 + 0.259823i \(0.0836644\pi\)
−0.965656 + 0.259823i \(0.916336\pi\)
\(674\) 1231.63i 1.82735i
\(675\) 0 0
\(676\) −194.616 −0.287893
\(677\) 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
\(680\) −167.516 −0.246348
\(681\) 0 0
\(682\) 2169.99i 3.18180i
\(683\) 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
\(686\) 212.485 0.309745
\(687\) 0 0
\(688\) −435.464 + 718.946i −0.632941 + 1.04498i
\(689\) −1242.87 −1.80387
\(690\) 0 0
\(691\) 942.228i 1.36357i −0.731552 0.681786i \(-0.761203\pi\)
0.731552 0.681786i \(-0.238797\pi\)
\(692\) −367.217 −0.530660
\(693\) 0 0
\(694\) −466.976 −0.672877
\(695\) 339.340i 0.488259i
\(696\) 0 0
\(697\) −128.303 −0.184079
\(698\) −724.106 −1.03740
\(699\) 0 0
\(700\) 54.9600i 0.0785142i
\(701\) −568.235 −0.810607 −0.405303 0.914182i \(-0.632834\pi\)
−0.405303 + 0.914182i \(0.632834\pi\)
\(702\) 0 0
\(703\) −1004.77 −1.42926
\(704\) −244.643 −0.347504
\(705\) 0 0
\(706\) 303.524i 0.429921i
\(707\) 52.3800i 0.0740877i
\(708\) 0 0
\(709\) 1211.96 1.70939 0.854696 0.519128i \(-0.173743\pi\)
0.854696 + 0.519128i \(0.173743\pi\)
\(710\) 937.222i 1.32003i
\(711\) 0 0
\(712\) 314.864 0.442225
\(713\) 1010.97 1.41791
\(714\) 0 0
\(715\) 1588.71i 2.22197i
\(716\) 350.585i 0.489645i
\(717\) 0 0
\(718\) 538.043i 0.749363i
\(719\) −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
\(722\) 1531.75i 2.12154i
\(723\) 0 0
\(724\) −613.525 −0.847410
\(725\) 191.176i 0.263691i
\(726\) 0 0
\(727\) 441.195i 0.606871i 0.952852 + 0.303435i \(0.0981337\pi\)
−0.952852 + 0.303435i \(0.901866\pi\)
\(728\) −45.0820 −0.0619258
\(729\) 0 0
\(730\) 1405.13i 1.92483i
\(731\) −154.980 + 255.871i −0.212012 + 0.350029i
\(732\) 0 0
\(733\) 251.039i 0.342481i 0.985229 + 0.171240i \(0.0547775\pi\)
−0.985229 + 0.171240i \(0.945222\pi\)
\(734\) 1149.14i 1.56559i
\(735\) 0 0
\(736\) 643.911i 0.874879i
\(737\) −130.985 −0.177727
\(738\) 0 0
\(739\) 635.071i 0.859365i −0.902980 0.429683i \(-0.858625\pi\)
0.902980 0.429683i \(-0.141375\pi\)
\(740\) −610.600 −0.825135
\(741\) 0 0
\(742\) 174.567 0.235266
\(743\) 899.136i 1.21014i −0.796171 0.605071i \(-0.793145\pi\)
0.796171 0.605071i \(-0.206855\pi\)
\(744\) 0 0
\(745\) −399.956 −0.536854
\(746\) −185.638 −0.248844
\(747\) 0 0
\(748\) −270.490 −0.361617
\(749\) 128.151i 0.171096i
\(750\) 0 0
\(751\) 597.422i 0.795502i −0.917493 0.397751i \(-0.869791\pi\)
0.917493 0.397751i \(-0.130209\pi\)
\(752\) −503.947 −0.670143
\(753\) 0 0
\(754\) −315.760 −0.418780
\(755\) −936.743 −1.24072
\(756\) 0 0
\(757\) 392.757i 0.518834i 0.965765 + 0.259417i \(0.0835304\pi\)
−0.965765 + 0.259417i \(0.916470\pi\)
\(758\) 378.170i 0.498905i
\(759\) 0 0
\(760\) 743.720i 0.978579i
\(761\) 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
\(764\) 559.914i 0.732872i
\(765\) 0 0
\(766\) −1805.49 −2.35704
\(767\) 286.507 0.373542
\(768\) 0 0
\(769\) −591.892 −0.769690 −0.384845 0.922981i \(-0.625745\pi\)
−0.384845 + 0.922981i \(0.625745\pi\)
\(770\) 223.143i 0.289796i
\(771\) 0 0
\(772\) 698.369 0.904624
\(773\) 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
\(776\) 534.792i 0.689164i
\(777\) 0 0
\(778\) 1292.45 1.66124
\(779\) 569.624i 0.731225i
\(780\) 0 0
\(781\) 751.569i 0.962316i
\(782\) 314.621i 0.402328i
\(783\) 0 0
\(784\) −943.853 −1.20389
\(785\) 1213.31 1.54562
\(786\) 0 0
\(787\) 26.7235 0.0339561 0.0169781 0.999856i \(-0.494595\pi\)
0.0169781 + 0.999856i \(0.494595\pi\)
\(788\) 251.404 0.319040
\(789\) 0 0
\(790\) 127.716 0.161665
\(791\) 26.1311 0.0330355
\(792\) 0 0
\(793\) 1183.56i 1.49251i
\(794\) 1411.03i 1.77712i
\(795\) 0 0
\(796\) 76.4363i 0.0960255i
\(797\) 638.409 0.801015 0.400508 0.916293i \(-0.368834\pi\)
0.400508 + 0.916293i \(0.368834\pi\)
\(798\) 0 0
\(799\) −179.354 −0.224473
\(800\) 894.486i 1.11811i
\(801\) 0 0
\(802\) 1015.55i 1.26627i
\(803\) 1126.79i 1.40322i
\(804\) 0 0
\(805\) 103.960 0.129143
\(806\) 2319.59i 2.87791i
\(807\) 0 0
\(808\) 212.403i 0.262876i
\(809\) −996.366 −1.23160 −0.615801 0.787902i \(-0.711167\pi\)
−0.615801 + 0.787902i \(0.711167\pi\)
\(810\) 0 0
\(811\) 1348.22i 1.66242i −0.555959 0.831209i \(-0.687649\pi\)
0.555959 0.831209i \(-0.312351\pi\)
\(812\) 17.7641 0.0218769
\(813\) 0 0
\(814\) 1222.47 1.50180
\(815\) 1711.60 2.10013
\(816\) 0 0
\(817\) 1135.99 + 688.065i 1.39044 + 0.842184i
\(818\) 2037.83 2.49124
\(819\) 0 0
\(820\) 346.161i 0.422147i
\(821\) 1363.54 1.66083 0.830417 0.557142i \(-0.188102\pi\)
0.830417 + 0.557142i \(0.188102\pi\)
\(822\) 0 0
\(823\) 590.955 0.718050 0.359025 0.933328i \(-0.383109\pi\)
0.359025 + 0.933328i \(0.383109\pi\)
\(824\) 65.8972i 0.0799723i
\(825\) 0 0
\(826\) −40.2415 −0.0487185
\(827\) 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 \(-0.823395\pi\)
0.849995 0.526791i \(-0.176605\pi\)
\(830\) −1520.92 −1.83243
\(831\) 0 0
\(832\) −261.509 −0.314314
\(833\) −335.915 −0.403259
\(834\) 0 0
\(835\) 1727.90i 2.06934i
\(836\) 1200.89i 1.43647i
\(837\) 0 0
\(838\) −572.720 −0.683437
\(839\) 1155.58i 1.37732i −0.725082 0.688662i \(-0.758198\pi\)
0.725082 0.688662i \(-0.241802\pi\)
\(840\) 0 0
\(841\) 779.208 0.926526
\(842\) 655.608 0.778632
\(843\) 0 0
\(844\) 60.5731i 0.0717691i
\(845\) 511.381i 0.605184i
\(846\) 0 0
\(847\) 76.6310i 0.0904734i
\(848\) −1562.33 −1.84237
\(849\) 0 0
\(850\) 437.054i 0.514181i
\(851\) 569.533i 0.669252i
\(852\) 0 0
\(853\) 86.1596 0.101008 0.0505039 0.998724i \(-0.483917\pi\)
0.0505039 + 0.998724i \(0.483917\pi\)
\(854\) 166.237i 0.194657i
\(855\) 0 0
\(856\) 519.656i 0.607075i
\(857\) 813.017 0.948678 0.474339 0.880342i \(-0.342687\pi\)
0.474339 + 0.880342i \(0.342687\pi\)
\(858\) 0 0
\(859\) 282.918i 0.329357i 0.986347 + 0.164678i \(0.0526587\pi\)
−0.986347 + 0.164678i \(0.947341\pi\)
\(860\) 690.341 + 418.137i 0.802722 + 0.486206i
\(861\) 0 0
\(862\) 1176.21i 1.36452i
\(863\) 579.879i 0.671934i 0.941874 + 0.335967i \(0.109063\pi\)
−0.941874 + 0.335967i \(0.890937\pi\)
\(864\) 0 0
\(865\) 964.915i 1.11551i
\(866\) 1578.52 1.82277
\(867\) 0 0
\(868\) 130.496i 0.150341i
\(869\) −102.417 −0.117856
\(870\) 0 0
\(871\) −140.015 −0.160752
\(872\) 149.801i 0.171790i
\(873\) 0 0
\(874\) 1396.82 1.59819
\(875\) −4.03625 −0.00461286
\(876\) 0 0
\(877\) 305.763 0.348647 0.174323 0.984688i \(-0.444226\pi\)
0.174323 + 0.984688i \(0.444226\pi\)
\(878\) 1547.14i 1.76212i
\(879\) 0 0
\(880\) 1997.07i 2.26940i
\(881\) −911.059 −1.03412 −0.517059 0.855950i \(-0.672973\pi\)
−0.517059 + 0.855950i \(0.672973\pi\)
\(882\) 0 0
\(883\) 396.432 0.448960 0.224480 0.974479i \(-0.427932\pi\)
0.224480 + 0.974479i \(0.427932\pi\)
\(884\) −289.138 −0.327079
\(885\) 0 0
\(886\) 979.189i 1.10518i
\(887\) 266.284i 0.300207i −0.988670 0.150104i \(-0.952039\pi\)
0.988670 0.150104i \(-0.0479607\pi\)
\(888\) 0 0
\(889\) 16.3268i 0.0183654i
\(890\) 1665.93i 1.87184i
\(891\) 0 0
\(892\) 703.195i 0.788336i
\(893\) 796.274i 0.891685i
\(894\) 0 0
\(895\) 921.215 1.02929
\(896\) −87.6629 −0.0978381
\(897\) 0 0
\(898\) 672.438 0.748817
\(899\) 453.924i 0.504921i
\(900\) 0 0
\(901\) −556.029 −0.617124
\(902\) 693.040i 0.768337i
\(903\) 0 0
\(904\) −105.963 −0.117215
\(905\) 1612.13i 1.78135i
\(906\) 0 0
\(907\) −1168.24 −1.28802 −0.644012 0.765015i \(-0.722731\pi\)
−0.644012 + 0.765015i \(0.722731\pi\)
\(908\) 145.630i 0.160386i
\(909\) 0 0
\(910\) 238.527i 0.262118i
\(911\) 981.585i 1.07748i −0.842472 0.538740i \(-0.818900\pi\)
0.842472 0.538740i \(-0.181100\pi\)
\(912\) 0 0
\(913\) 1219.64 1.33586
\(914\) −1731.31 −1.89421
\(915\) 0 0
\(916\) −507.789 −0.554355
\(917\) 16.3650 0.0178462
\(918\) 0 0
\(919\) −1148.89 −1.25015 −0.625075 0.780564i \(-0.714932\pi\)
−0.625075 + 0.780564i \(0.714932\pi\)
\(920\) −421.562 −0.458220
\(921\) 0 0
\(922\) 1439.82i 1.56162i
\(923\) 803.384i 0.870405i
\(924\) 0 0
\(925\) 791.164i 0.855312i
\(926\) −300.940 −0.324989
\(927\) 0 0
\(928\) −289.114 −0.311546
\(929\) 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
\(932\) 559.086i 0.599878i
\(933\) 0 0
\(934\) −1202.37 −1.28733
\(935\) 710.752i 0.760162i
\(936\) 0 0
\(937\) 1301.72i 1.38925i −0.719374 0.694623i \(-0.755571\pi\)
0.719374 0.694623i \(-0.244429\pi\)
\(938\) 19.6659 0.0209658
\(939\) 0 0
\(940\) 483.896i 0.514783i
\(941\) 539.895 0.573746 0.286873 0.957969i \(-0.407384\pi\)
0.286873 + 0.957969i \(0.407384\pi\)
\(942\) 0 0
\(943\) −322.879 −0.342396
\(944\) 360.150 0.381515
\(945\) 0 0
\(946\) −1382.11 837.142i −1.46101 0.884928i
\(947\) −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
\(950\) 1940.38 2.04251
\(951\) 0 0
\(952\) −20.1686 −0.0211855
\(953\) 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
\(956\) −182.253 −0.190641
\(957\) 0 0
\(958\) 63.9998i 0.0668057i
\(959\) 63.3159 0.0660229
\(960\) 0 0
\(961\) 2373.55 2.46988
\(962\) 1306.75 1.35836
\(963\) 0 0
\(964\) 1018.10i 1.05612i
\(965\) 1835.07i 1.90162i
\(966\) 0 0
\(967\) −487.850 −0.504498 −0.252249 0.967662i \(-0.581170\pi\)
−0.252249 + 0.967662i \(0.581170\pi\)
\(968\) 310.742i 0.321015i
\(969\) 0 0
\(970\) −2829.56 −2.91707
\(971\) −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
\(974\) 920.013i 0.944572i
\(975\) 0 0
\(976\) 1487.78i 1.52436i
\(977\) −1385.87 −1.41849 −0.709247 0.704961i \(-0.750965\pi\)
−0.709247 + 0.704961i \(0.750965\pi\)
\(978\) 0 0
\(979\) 1335.93i 1.36459i
\(980\) 906.299i 0.924795i
\(981\) 0 0
\(982\) 1286.70 1.31029
\(983\) 868.493i 0.883513i 0.897135 + 0.441756i \(0.145644\pi\)
−0.897135 + 0.441756i \(0.854356\pi\)
\(984\) 0 0
\(985\) 660.600i 0.670660i
\(986\) −141.264 −0.143270
\(987\) 0 0
\(988\) 1283.68i 1.29927i
\(989\) −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 \(0.678620\pi\)
\(992\) 2123.85i 2.14098i
\(993\) 0 0
\(994\) 112.840i 0.113521i
\(995\) 200.848 0.201857
\(996\) 0 0
\(997\) 83.7411i 0.0839931i 0.999118 + 0.0419965i \(0.0133718\pi\)
−0.999118 + 0.0419965i \(0.986628\pi\)
\(998\) 1.73272 0.00173619
\(999\) 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 387.3.b.c.343.5 6
3.2 odd 2 43.3.b.b.42.2 6
12.11 even 2 688.3.b.e.257.5 6
43.42 odd 2 inner 387.3.b.c.343.2 6
129.128 even 2 43.3.b.b.42.5 yes 6
516.515 odd 2 688.3.b.e.257.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.3.b.b.42.2 6 3.2 odd 2
43.3.b.b.42.5 yes 6 129.128 even 2
387.3.b.c.343.2 6 43.42 odd 2 inner
387.3.b.c.343.5 6 1.1 even 1 trivial
688.3.b.e.257.2 6 516.515 odd 2
688.3.b.e.257.5 6 12.11 even 2