Properties

Label 960.3.c.j.449.10
Level $960$
Weight $3$
Character 960.449
Analytic conductor $26.158$
Analytic rank $0$
Dimension $12$
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] = [960,3,Mod(449,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 960.c (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: \(26.1581053786\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 34x^{10} + 305x^{8} + 616x^{6} + 305x^{4} + 34x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.10
Root \(3.28615i\) of defining polynomial
Character \(\chi\) \(=\) 960.449
Dual form 960.3.c.j.449.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.49147 + 1.67109i) q^{3} +(-4.19906 + 2.71439i) q^{5} +12.7692i q^{7} +(3.41489 + 8.32698i) q^{9} +O(q^{10})\) \(q+(2.49147 + 1.67109i) q^{3} +(-4.19906 + 2.71439i) q^{5} +12.7692i q^{7} +(3.41489 + 8.32698i) q^{9} +12.6296i q^{11} -7.44085i q^{13} +(-14.9978 - 0.254179i) q^{15} +14.0550 q^{17} -31.0176 q^{19} +(-21.3386 + 31.8142i) q^{21} +7.50423 q^{23} +(10.2641 - 22.7958i) q^{25} +(-5.40707 + 26.4530i) q^{27} -15.7298i q^{29} +20.4893 q^{31} +(-21.1053 + 31.4663i) q^{33} +(-34.6607 - 53.6186i) q^{35} -12.9261i q^{37} +(12.4344 - 18.5387i) q^{39} -13.8451i q^{41} -30.0797i q^{43} +(-36.9420 - 25.6961i) q^{45} +20.2570 q^{47} -114.053 q^{49} +(35.0176 + 23.4872i) q^{51} +29.1185 q^{53} +(-34.2817 - 53.0324i) q^{55} +(-77.2795 - 51.8333i) q^{57} +47.6333i q^{59} -43.0176 q^{61} +(-106.329 + 43.6054i) q^{63} +(20.1974 + 31.2445i) q^{65} -0.630153i q^{67} +(18.6966 + 12.5403i) q^{69} +90.4047i q^{71} -46.2193i q^{73} +(63.6667 - 39.6427i) q^{75} -161.270 q^{77} -37.9610 q^{79} +(-57.6771 + 56.8713i) q^{81} +80.2267 q^{83} +(-59.0176 + 38.1507i) q^{85} +(26.2860 - 39.1904i) q^{87} +140.923i q^{89} +95.0138 q^{91} +(51.0486 + 34.2396i) q^{93} +(130.245 - 84.1939i) q^{95} +10.3429i q^{97} +(-105.166 + 43.1286i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 8 q^{9} - 16 q^{15} - 4 q^{21} + 36 q^{25} + 48 q^{31} + 128 q^{39} + 68 q^{45} - 252 q^{49} + 48 q^{51} + 48 q^{55} - 144 q^{61} - 268 q^{69} + 304 q^{75} - 432 q^{79} - 188 q^{81} - 336 q^{85} - 560 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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
\(3\) 2.49147 + 1.67109i 0.830491 + 0.557032i
\(4\) 0 0
\(5\) −4.19906 + 2.71439i −0.839811 + 0.542879i
\(6\) 0 0
\(7\) 12.7692i 1.82417i 0.409998 + 0.912086i \(0.365529\pi\)
−0.409998 + 0.912086i \(0.634471\pi\)
\(8\) 0 0
\(9\) 3.41489 + 8.32698i 0.379432 + 0.925220i
\(10\) 0 0
\(11\) 12.6296i 1.14815i 0.818804 + 0.574073i \(0.194637\pi\)
−0.818804 + 0.574073i \(0.805363\pi\)
\(12\) 0 0
\(13\) 7.44085i 0.572373i −0.958174 0.286187i \(-0.907612\pi\)
0.958174 0.286187i \(-0.0923877\pi\)
\(14\) 0 0
\(15\) −14.9978 0.254179i −0.999856 0.0169452i
\(16\) 0 0
\(17\) 14.0550 0.826763 0.413381 0.910558i \(-0.364348\pi\)
0.413381 + 0.910558i \(0.364348\pi\)
\(18\) 0 0
\(19\) −31.0176 −1.63250 −0.816252 0.577696i \(-0.803952\pi\)
−0.816252 + 0.577696i \(0.803952\pi\)
\(20\) 0 0
\(21\) −21.3386 + 31.8142i −1.01612 + 1.51496i
\(22\) 0 0
\(23\) 7.50423 0.326271 0.163135 0.986604i \(-0.447839\pi\)
0.163135 + 0.986604i \(0.447839\pi\)
\(24\) 0 0
\(25\) 10.2641 22.7958i 0.410565 0.911831i
\(26\) 0 0
\(27\) −5.40707 + 26.4530i −0.200262 + 0.979742i
\(28\) 0 0
\(29\) 15.7298i 0.542407i −0.962522 0.271204i \(-0.912578\pi\)
0.962522 0.271204i \(-0.0874217\pi\)
\(30\) 0 0
\(31\) 20.4893 0.660945 0.330473 0.943816i \(-0.392792\pi\)
0.330473 + 0.943816i \(0.392792\pi\)
\(32\) 0 0
\(33\) −21.1053 + 31.4663i −0.639553 + 0.953525i
\(34\) 0 0
\(35\) −34.6607 53.6186i −0.990305 1.53196i
\(36\) 0 0
\(37\) 12.9261i 0.349355i −0.984626 0.174677i \(-0.944112\pi\)
0.984626 0.174677i \(-0.0558883\pi\)
\(38\) 0 0
\(39\) 12.4344 18.5387i 0.318830 0.475351i
\(40\) 0 0
\(41\) 13.8451i 0.337685i −0.985643 0.168843i \(-0.945997\pi\)
0.985643 0.168843i \(-0.0540029\pi\)
\(42\) 0 0
\(43\) 30.0797i 0.699528i −0.936838 0.349764i \(-0.886262\pi\)
0.936838 0.349764i \(-0.113738\pi\)
\(44\) 0 0
\(45\) −36.9420 25.6961i −0.820933 0.571024i
\(46\) 0 0
\(47\) 20.2570 0.431000 0.215500 0.976504i \(-0.430862\pi\)
0.215500 + 0.976504i \(0.430862\pi\)
\(48\) 0 0
\(49\) −114.053 −2.32761
\(50\) 0 0
\(51\) 35.0176 + 23.4872i 0.686619 + 0.460533i
\(52\) 0 0
\(53\) 29.1185 0.549406 0.274703 0.961529i \(-0.411420\pi\)
0.274703 + 0.961529i \(0.411420\pi\)
\(54\) 0 0
\(55\) −34.2817 53.0324i −0.623304 0.964226i
\(56\) 0 0
\(57\) −77.2795 51.8333i −1.35578 0.909356i
\(58\) 0 0
\(59\) 47.6333i 0.807344i 0.914904 + 0.403672i \(0.132266\pi\)
−0.914904 + 0.403672i \(0.867734\pi\)
\(60\) 0 0
\(61\) −43.0176 −0.705206 −0.352603 0.935773i \(-0.614703\pi\)
−0.352603 + 0.935773i \(0.614703\pi\)
\(62\) 0 0
\(63\) −106.329 + 43.6054i −1.68776 + 0.692149i
\(64\) 0 0
\(65\) 20.1974 + 31.2445i 0.310729 + 0.480685i
\(66\) 0 0
\(67\) 0.630153i 0.00940527i −0.999989 0.00470264i \(-0.998503\pi\)
0.999989 0.00470264i \(-0.00149690\pi\)
\(68\) 0 0
\(69\) 18.6966 + 12.5403i 0.270965 + 0.181743i
\(70\) 0 0
\(71\) 90.4047i 1.27330i 0.771151 + 0.636652i \(0.219681\pi\)
−0.771151 + 0.636652i \(0.780319\pi\)
\(72\) 0 0
\(73\) 46.2193i 0.633140i −0.948569 0.316570i \(-0.897469\pi\)
0.948569 0.316570i \(-0.102531\pi\)
\(74\) 0 0
\(75\) 63.6667 39.6427i 0.848890 0.528570i
\(76\) 0 0
\(77\) −161.270 −2.09442
\(78\) 0 0
\(79\) −37.9610 −0.480519 −0.240260 0.970709i \(-0.577233\pi\)
−0.240260 + 0.970709i \(0.577233\pi\)
\(80\) 0 0
\(81\) −57.6771 + 56.8713i −0.712063 + 0.702115i
\(82\) 0 0
\(83\) 80.2267 0.966587 0.483294 0.875458i \(-0.339440\pi\)
0.483294 + 0.875458i \(0.339440\pi\)
\(84\) 0 0
\(85\) −59.0176 + 38.1507i −0.694324 + 0.448832i
\(86\) 0 0
\(87\) 26.2860 39.1904i 0.302138 0.450465i
\(88\) 0 0
\(89\) 140.923i 1.58341i 0.610907 + 0.791703i \(0.290805\pi\)
−0.610907 + 0.791703i \(0.709195\pi\)
\(90\) 0 0
\(91\) 95.0138 1.04411
\(92\) 0 0
\(93\) 51.0486 + 34.2396i 0.548909 + 0.368168i
\(94\) 0 0
\(95\) 130.245 84.1939i 1.37100 0.886252i
\(96\) 0 0
\(97\) 10.3429i 0.106628i 0.998578 + 0.0533138i \(0.0169784\pi\)
−0.998578 + 0.0533138i \(0.983022\pi\)
\(98\) 0 0
\(99\) −105.166 + 43.1286i −1.06229 + 0.435643i
\(100\) 0 0
\(101\) 19.2739i 0.190830i −0.995438 0.0954152i \(-0.969582\pi\)
0.995438 0.0954152i \(-0.0304179\pi\)
\(102\) 0 0
\(103\) 6.97008i 0.0676707i 0.999427 + 0.0338353i \(0.0107722\pi\)
−0.999427 + 0.0338353i \(0.989228\pi\)
\(104\) 0 0
\(105\) 3.24566 191.511i 0.0309110 1.82391i
\(106\) 0 0
\(107\) 73.7731 0.689468 0.344734 0.938700i \(-0.387969\pi\)
0.344734 + 0.938700i \(0.387969\pi\)
\(108\) 0 0
\(109\) −74.0314 −0.679187 −0.339593 0.940572i \(-0.610289\pi\)
−0.339593 + 0.940572i \(0.610289\pi\)
\(110\) 0 0
\(111\) 21.6008 32.2051i 0.194602 0.290136i
\(112\) 0 0
\(113\) −147.215 −1.30279 −0.651394 0.758739i \(-0.725816\pi\)
−0.651394 + 0.758739i \(0.725816\pi\)
\(114\) 0 0
\(115\) −31.5107 + 20.3694i −0.274006 + 0.177126i
\(116\) 0 0
\(117\) 61.9598 25.4096i 0.529571 0.217176i
\(118\) 0 0
\(119\) 179.471i 1.50816i
\(120\) 0 0
\(121\) −38.5069 −0.318239
\(122\) 0 0
\(123\) 23.1364 34.4947i 0.188101 0.280444i
\(124\) 0 0
\(125\) 18.7770 + 123.582i 0.150216 + 0.988653i
\(126\) 0 0
\(127\) 90.0171i 0.708796i −0.935095 0.354398i \(-0.884686\pi\)
0.935095 0.354398i \(-0.115314\pi\)
\(128\) 0 0
\(129\) 50.2660 74.9428i 0.389659 0.580952i
\(130\) 0 0
\(131\) 11.2911i 0.0861917i −0.999071 0.0430958i \(-0.986278\pi\)
0.999071 0.0430958i \(-0.0137221\pi\)
\(132\) 0 0
\(133\) 396.070i 2.97797i
\(134\) 0 0
\(135\) −49.0994 125.755i −0.363699 0.931516i
\(136\) 0 0
\(137\) 22.4905 0.164164 0.0820822 0.996626i \(-0.473843\pi\)
0.0820822 + 0.996626i \(0.473843\pi\)
\(138\) 0 0
\(139\) 91.0955 0.655363 0.327682 0.944788i \(-0.393733\pi\)
0.327682 + 0.944788i \(0.393733\pi\)
\(140\) 0 0
\(141\) 50.4698 + 33.8514i 0.357942 + 0.240081i
\(142\) 0 0
\(143\) 93.9750 0.657168
\(144\) 0 0
\(145\) 42.6969 + 66.0504i 0.294461 + 0.455520i
\(146\) 0 0
\(147\) −284.159 190.593i −1.93306 1.29655i
\(148\) 0 0
\(149\) 228.330i 1.53242i 0.642593 + 0.766208i \(0.277859\pi\)
−0.642593 + 0.766208i \(0.722141\pi\)
\(150\) 0 0
\(151\) 74.0390 0.490324 0.245162 0.969482i \(-0.421159\pi\)
0.245162 + 0.969482i \(0.421159\pi\)
\(152\) 0 0
\(153\) 47.9961 + 117.035i 0.313700 + 0.764937i
\(154\) 0 0
\(155\) −86.0358 + 55.6161i −0.555069 + 0.358813i
\(156\) 0 0
\(157\) 245.742i 1.56523i 0.622504 + 0.782616i \(0.286115\pi\)
−0.622504 + 0.782616i \(0.713885\pi\)
\(158\) 0 0
\(159\) 72.5481 + 48.6598i 0.456277 + 0.306037i
\(160\) 0 0
\(161\) 95.8231i 0.595175i
\(162\) 0 0
\(163\) 60.1570i 0.369061i −0.982827 0.184531i \(-0.940924\pi\)
0.982827 0.184531i \(-0.0590765\pi\)
\(164\) 0 0
\(165\) 3.21017 189.417i 0.0194556 1.14798i
\(166\) 0 0
\(167\) −81.5664 −0.488421 −0.244211 0.969722i \(-0.578529\pi\)
−0.244211 + 0.969722i \(0.578529\pi\)
\(168\) 0 0
\(169\) 113.634 0.672389
\(170\) 0 0
\(171\) −105.921 258.283i −0.619424 1.51043i
\(172\) 0 0
\(173\) 167.064 0.965687 0.482843 0.875707i \(-0.339604\pi\)
0.482843 + 0.875707i \(0.339604\pi\)
\(174\) 0 0
\(175\) 291.084 + 131.065i 1.66334 + 0.748942i
\(176\) 0 0
\(177\) −79.5997 + 118.677i −0.449716 + 0.670492i
\(178\) 0 0
\(179\) 270.104i 1.50896i 0.656322 + 0.754481i \(0.272111\pi\)
−0.656322 + 0.754481i \(0.727889\pi\)
\(180\) 0 0
\(181\) −86.9786 −0.480545 −0.240272 0.970705i \(-0.577237\pi\)
−0.240272 + 0.970705i \(0.577237\pi\)
\(182\) 0 0
\(183\) −107.177 71.8865i −0.585668 0.392822i
\(184\) 0 0
\(185\) 35.0866 + 54.2776i 0.189657 + 0.293392i
\(186\) 0 0
\(187\) 177.509i 0.949244i
\(188\) 0 0
\(189\) −337.785 69.0440i −1.78722 0.365312i
\(190\) 0 0
\(191\) 302.223i 1.58232i −0.611610 0.791159i \(-0.709478\pi\)
0.611610 0.791159i \(-0.290522\pi\)
\(192\) 0 0
\(193\) 306.780i 1.58953i −0.606916 0.794766i \(-0.707593\pi\)
0.606916 0.794766i \(-0.292407\pi\)
\(194\) 0 0
\(195\) −1.89130 + 111.597i −0.00969900 + 0.572291i
\(196\) 0 0
\(197\) −289.956 −1.47186 −0.735930 0.677058i \(-0.763255\pi\)
−0.735930 + 0.677058i \(0.763255\pi\)
\(198\) 0 0
\(199\) −382.595 −1.92259 −0.961293 0.275527i \(-0.911148\pi\)
−0.961293 + 0.275527i \(0.911148\pi\)
\(200\) 0 0
\(201\) 1.05305 1.57001i 0.00523903 0.00781100i
\(202\) 0 0
\(203\) 200.857 0.989445
\(204\) 0 0
\(205\) 37.5810 + 58.1363i 0.183322 + 0.283592i
\(206\) 0 0
\(207\) 25.6261 + 62.4876i 0.123798 + 0.301872i
\(208\) 0 0
\(209\) 391.740i 1.87435i
\(210\) 0 0
\(211\) −321.115 −1.52187 −0.760937 0.648826i \(-0.775260\pi\)
−0.760937 + 0.648826i \(0.775260\pi\)
\(212\) 0 0
\(213\) −151.075 + 225.241i −0.709271 + 1.05747i
\(214\) 0 0
\(215\) 81.6482 + 126.306i 0.379759 + 0.587471i
\(216\) 0 0
\(217\) 261.632i 1.20568i
\(218\) 0 0
\(219\) 77.2368 115.154i 0.352679 0.525818i
\(220\) 0 0
\(221\) 104.581i 0.473217i
\(222\) 0 0
\(223\) 292.432i 1.31135i 0.755042 + 0.655676i \(0.227616\pi\)
−0.755042 + 0.655676i \(0.772384\pi\)
\(224\) 0 0
\(225\) 224.871 + 7.62426i 0.999426 + 0.0338856i
\(226\) 0 0
\(227\) 370.155 1.63064 0.815319 0.579013i \(-0.196562\pi\)
0.815319 + 0.579013i \(0.196562\pi\)
\(228\) 0 0
\(229\) 381.985 1.66806 0.834028 0.551722i \(-0.186029\pi\)
0.834028 + 0.551722i \(0.186029\pi\)
\(230\) 0 0
\(231\) −401.800 269.498i −1.73939 1.16666i
\(232\) 0 0
\(233\) 144.262 0.619150 0.309575 0.950875i \(-0.399813\pi\)
0.309575 + 0.950875i \(0.399813\pi\)
\(234\) 0 0
\(235\) −85.0603 + 54.9855i −0.361959 + 0.233981i
\(236\) 0 0
\(237\) −94.5789 63.4365i −0.399067 0.267665i
\(238\) 0 0
\(239\) 249.478i 1.04384i 0.852994 + 0.521921i \(0.174784\pi\)
−0.852994 + 0.521921i \(0.825216\pi\)
\(240\) 0 0
\(241\) 30.4541 0.126366 0.0631829 0.998002i \(-0.479875\pi\)
0.0631829 + 0.998002i \(0.479875\pi\)
\(242\) 0 0
\(243\) −238.738 + 45.3095i −0.982463 + 0.186459i
\(244\) 0 0
\(245\) 478.914 309.584i 1.95475 1.26361i
\(246\) 0 0
\(247\) 230.797i 0.934401i
\(248\) 0 0
\(249\) 199.883 + 134.066i 0.802742 + 0.538420i
\(250\) 0 0
\(251\) 68.9183i 0.274575i −0.990531 0.137287i \(-0.956162\pi\)
0.990531 0.137287i \(-0.0438384\pi\)
\(252\) 0 0
\(253\) 94.7755i 0.374607i
\(254\) 0 0
\(255\) −210.794 3.57247i −0.826644 0.0140097i
\(256\) 0 0
\(257\) −362.374 −1.41002 −0.705009 0.709199i \(-0.749057\pi\)
−0.705009 + 0.709199i \(0.749057\pi\)
\(258\) 0 0
\(259\) 165.057 0.637284
\(260\) 0 0
\(261\) 130.982 53.7155i 0.501846 0.205807i
\(262\) 0 0
\(263\) −23.4485 −0.0891580 −0.0445790 0.999006i \(-0.514195\pi\)
−0.0445790 + 0.999006i \(0.514195\pi\)
\(264\) 0 0
\(265\) −122.270 + 79.0391i −0.461397 + 0.298261i
\(266\) 0 0
\(267\) −235.496 + 351.106i −0.882007 + 1.31500i
\(268\) 0 0
\(269\) 396.738i 1.47486i −0.675421 0.737432i \(-0.736038\pi\)
0.675421 0.737432i \(-0.263962\pi\)
\(270\) 0 0
\(271\) 143.926 0.531092 0.265546 0.964098i \(-0.414448\pi\)
0.265546 + 0.964098i \(0.414448\pi\)
\(272\) 0 0
\(273\) 236.724 + 158.777i 0.867122 + 0.581601i
\(274\) 0 0
\(275\) 287.902 + 129.632i 1.04692 + 0.471389i
\(276\) 0 0
\(277\) 360.884i 1.30283i 0.758722 + 0.651415i \(0.225824\pi\)
−0.758722 + 0.651415i \(0.774176\pi\)
\(278\) 0 0
\(279\) 69.9686 + 170.614i 0.250784 + 0.611520i
\(280\) 0 0
\(281\) 531.655i 1.89201i −0.324149 0.946006i \(-0.605078\pi\)
0.324149 0.946006i \(-0.394922\pi\)
\(282\) 0 0
\(283\) 257.400i 0.909541i 0.890609 + 0.454770i \(0.150279\pi\)
−0.890609 + 0.454770i \(0.849721\pi\)
\(284\) 0 0
\(285\) 465.197 + 7.88400i 1.63227 + 0.0276632i
\(286\) 0 0
\(287\) 176.791 0.615996
\(288\) 0 0
\(289\) −91.4579 −0.316463
\(290\) 0 0
\(291\) −17.2839 + 25.7690i −0.0593949 + 0.0885533i
\(292\) 0 0
\(293\) 310.456 1.05958 0.529789 0.848130i \(-0.322271\pi\)
0.529789 + 0.848130i \(0.322271\pi\)
\(294\) 0 0
\(295\) −129.295 200.015i −0.438290 0.678016i
\(296\) 0 0
\(297\) −334.091 68.2892i −1.12489 0.229930i
\(298\) 0 0
\(299\) 55.8379i 0.186749i
\(300\) 0 0
\(301\) 384.094 1.27606
\(302\) 0 0
\(303\) 32.2085 48.0204i 0.106299 0.158483i
\(304\) 0 0
\(305\) 180.633 116.767i 0.592240 0.382841i
\(306\) 0 0
\(307\) 266.169i 0.866999i −0.901154 0.433499i \(-0.857279\pi\)
0.901154 0.433499i \(-0.142721\pi\)
\(308\) 0 0
\(309\) −11.6477 + 17.3658i −0.0376947 + 0.0561999i
\(310\) 0 0
\(311\) 186.580i 0.599934i 0.953950 + 0.299967i \(0.0969757\pi\)
−0.953950 + 0.299967i \(0.903024\pi\)
\(312\) 0 0
\(313\) 20.5021i 0.0655019i −0.999464 0.0327510i \(-0.989573\pi\)
0.999464 0.0327510i \(-0.0104268\pi\)
\(314\) 0 0
\(315\) 328.119 471.720i 1.04165 1.49752i
\(316\) 0 0
\(317\) 18.9792 0.0598711 0.0299356 0.999552i \(-0.490470\pi\)
0.0299356 + 0.999552i \(0.490470\pi\)
\(318\) 0 0
\(319\) 198.661 0.622763
\(320\) 0 0
\(321\) 183.804 + 123.282i 0.572597 + 0.384055i
\(322\) 0 0
\(323\) −435.951 −1.34969
\(324\) 0 0
\(325\) −169.620 76.3739i −0.521908 0.234997i
\(326\) 0 0
\(327\) −184.447 123.713i −0.564059 0.378328i
\(328\) 0 0
\(329\) 258.666i 0.786219i
\(330\) 0 0
\(331\) 413.193 1.24832 0.624159 0.781297i \(-0.285442\pi\)
0.624159 + 0.781297i \(0.285442\pi\)
\(332\) 0 0
\(333\) 107.636 44.1413i 0.323230 0.132556i
\(334\) 0 0
\(335\) 1.71048 + 2.64605i 0.00510592 + 0.00789865i
\(336\) 0 0
\(337\) 484.733i 1.43838i 0.694816 + 0.719188i \(0.255486\pi\)
−0.694816 + 0.719188i \(0.744514\pi\)
\(338\) 0 0
\(339\) −366.783 246.010i −1.08195 0.725694i
\(340\) 0 0
\(341\) 258.772i 0.758862i
\(342\) 0 0
\(343\) 830.672i 2.42178i
\(344\) 0 0
\(345\) −112.547 1.90741i −0.326224 0.00552874i
\(346\) 0 0
\(347\) 336.214 0.968915 0.484457 0.874815i \(-0.339017\pi\)
0.484457 + 0.874815i \(0.339017\pi\)
\(348\) 0 0
\(349\) 428.261 1.22711 0.613555 0.789652i \(-0.289739\pi\)
0.613555 + 0.789652i \(0.289739\pi\)
\(350\) 0 0
\(351\) 196.833 + 40.2332i 0.560778 + 0.114625i
\(352\) 0 0
\(353\) 558.817 1.58305 0.791525 0.611137i \(-0.209287\pi\)
0.791525 + 0.611137i \(0.209287\pi\)
\(354\) 0 0
\(355\) −245.394 379.614i −0.691250 1.06934i
\(356\) 0 0
\(357\) −299.913 + 447.147i −0.840092 + 1.25251i
\(358\) 0 0
\(359\) 206.915i 0.576365i −0.957576 0.288182i \(-0.906949\pi\)
0.957576 0.288182i \(-0.0930509\pi\)
\(360\) 0 0
\(361\) 601.090 1.66507
\(362\) 0 0
\(363\) −95.9389 64.3487i −0.264295 0.177269i
\(364\) 0 0
\(365\) 125.457 + 194.077i 0.343718 + 0.531718i
\(366\) 0 0
\(367\) 185.425i 0.505246i −0.967565 0.252623i \(-0.918707\pi\)
0.967565 0.252623i \(-0.0812932\pi\)
\(368\) 0 0
\(369\) 115.288 47.2794i 0.312433 0.128128i
\(370\) 0 0
\(371\) 371.821i 1.00221i
\(372\) 0 0
\(373\) 427.345i 1.14570i 0.819661 + 0.572848i \(0.194162\pi\)
−0.819661 + 0.572848i \(0.805838\pi\)
\(374\) 0 0
\(375\) −159.734 + 339.279i −0.425958 + 0.904743i
\(376\) 0 0
\(377\) −117.043 −0.310459
\(378\) 0 0
\(379\) 117.727 0.310626 0.155313 0.987865i \(-0.450361\pi\)
0.155313 + 0.987865i \(0.450361\pi\)
\(380\) 0 0
\(381\) 150.427 224.275i 0.394822 0.588649i
\(382\) 0 0
\(383\) 470.016 1.22720 0.613598 0.789619i \(-0.289722\pi\)
0.613598 + 0.789619i \(0.289722\pi\)
\(384\) 0 0
\(385\) 677.182 437.750i 1.75891 1.13701i
\(386\) 0 0
\(387\) 250.473 102.719i 0.647217 0.265423i
\(388\) 0 0
\(389\) 128.160i 0.329461i −0.986339 0.164730i \(-0.947325\pi\)
0.986339 0.164730i \(-0.0526754\pi\)
\(390\) 0 0
\(391\) 105.472 0.269749
\(392\) 0 0
\(393\) 18.8685 28.1315i 0.0480115 0.0715814i
\(394\) 0 0
\(395\) 159.401 103.041i 0.403546 0.260864i
\(396\) 0 0
\(397\) 276.316i 0.696011i −0.937493 0.348005i \(-0.886859\pi\)
0.937493 0.348005i \(-0.113141\pi\)
\(398\) 0 0
\(399\) 661.871 986.798i 1.65882 2.47318i
\(400\) 0 0
\(401\) 549.912i 1.37135i 0.727907 + 0.685675i \(0.240493\pi\)
−0.727907 + 0.685675i \(0.759507\pi\)
\(402\) 0 0
\(403\) 152.458i 0.378307i
\(404\) 0 0
\(405\) 87.8182 395.364i 0.216835 0.976208i
\(406\) 0 0
\(407\) 163.252 0.401110
\(408\) 0 0
\(409\) −219.166 −0.535858 −0.267929 0.963439i \(-0.586339\pi\)
−0.267929 + 0.963439i \(0.586339\pi\)
\(410\) 0 0
\(411\) 56.0346 + 37.5838i 0.136337 + 0.0914448i
\(412\) 0 0
\(413\) −608.240 −1.47273
\(414\) 0 0
\(415\) −336.877 + 217.767i −0.811751 + 0.524740i
\(416\) 0 0
\(417\) 226.962 + 152.229i 0.544274 + 0.365058i
\(418\) 0 0
\(419\) 204.522i 0.488119i 0.969760 + 0.244060i \(0.0784792\pi\)
−0.969760 + 0.244060i \(0.921521\pi\)
\(420\) 0 0
\(421\) −577.186 −1.37099 −0.685494 0.728078i \(-0.740414\pi\)
−0.685494 + 0.728078i \(0.740414\pi\)
\(422\) 0 0
\(423\) 69.1754 + 168.680i 0.163535 + 0.398770i
\(424\) 0 0
\(425\) 144.262 320.394i 0.339440 0.753868i
\(426\) 0 0
\(427\) 549.301i 1.28642i
\(428\) 0 0
\(429\) 234.136 + 157.041i 0.545772 + 0.366063i
\(430\) 0 0
\(431\) 663.363i 1.53913i 0.638571 + 0.769563i \(0.279526\pi\)
−0.638571 + 0.769563i \(0.720474\pi\)
\(432\) 0 0
\(433\) 226.876i 0.523964i 0.965073 + 0.261982i \(0.0843760\pi\)
−0.965073 + 0.261982i \(0.915624\pi\)
\(434\) 0 0
\(435\) −3.99818 + 235.913i −0.00919122 + 0.542329i
\(436\) 0 0
\(437\) −232.763 −0.532639
\(438\) 0 0
\(439\) −282.524 −0.643564 −0.321782 0.946814i \(-0.604282\pi\)
−0.321782 + 0.946814i \(0.604282\pi\)
\(440\) 0 0
\(441\) −389.477 949.715i −0.883168 2.15355i
\(442\) 0 0
\(443\) 455.605 1.02845 0.514227 0.857654i \(-0.328079\pi\)
0.514227 + 0.857654i \(0.328079\pi\)
\(444\) 0 0
\(445\) −382.521 591.744i −0.859597 1.32976i
\(446\) 0 0
\(447\) −381.561 + 568.878i −0.853604 + 1.27266i
\(448\) 0 0
\(449\) 157.206i 0.350124i −0.984557 0.175062i \(-0.943987\pi\)
0.984557 0.175062i \(-0.0560127\pi\)
\(450\) 0 0
\(451\) 174.858 0.387712
\(452\) 0 0
\(453\) 184.466 + 123.726i 0.407210 + 0.273126i
\(454\) 0 0
\(455\) −398.968 + 257.905i −0.876853 + 0.566824i
\(456\) 0 0
\(457\) 397.152i 0.869041i 0.900662 + 0.434520i \(0.143082\pi\)
−0.900662 + 0.434520i \(0.856918\pi\)
\(458\) 0 0
\(459\) −75.9962 + 371.797i −0.165569 + 0.810014i
\(460\) 0 0
\(461\) 350.730i 0.760803i −0.924821 0.380401i \(-0.875786\pi\)
0.924821 0.380401i \(-0.124214\pi\)
\(462\) 0 0
\(463\) 308.385i 0.666058i 0.942917 + 0.333029i \(0.108071\pi\)
−0.942917 + 0.333029i \(0.891929\pi\)
\(464\) 0 0
\(465\) −307.296 5.20794i −0.660851 0.0111999i
\(466\) 0 0
\(467\) −800.129 −1.71334 −0.856669 0.515866i \(-0.827470\pi\)
−0.856669 + 0.515866i \(0.827470\pi\)
\(468\) 0 0
\(469\) 8.04656 0.0171568
\(470\) 0 0
\(471\) −410.657 + 612.259i −0.871884 + 1.29991i
\(472\) 0 0
\(473\) 379.895 0.803160
\(474\) 0 0
\(475\) −318.369 + 707.070i −0.670250 + 1.48857i
\(476\) 0 0
\(477\) 99.4364 + 242.469i 0.208462 + 0.508321i
\(478\) 0 0
\(479\) 630.135i 1.31552i −0.753227 0.657761i \(-0.771504\pi\)
0.753227 0.657761i \(-0.228496\pi\)
\(480\) 0 0
\(481\) −96.1814 −0.199961
\(482\) 0 0
\(483\) −160.130 + 238.741i −0.331531 + 0.494287i
\(484\) 0 0
\(485\) −28.0746 43.4303i −0.0578858 0.0895470i
\(486\) 0 0
\(487\) 103.952i 0.213454i −0.994288 0.106727i \(-0.965963\pi\)
0.994288 0.106727i \(-0.0340372\pi\)
\(488\) 0 0
\(489\) 100.528 149.879i 0.205579 0.306502i
\(490\) 0 0
\(491\) 286.049i 0.582585i 0.956634 + 0.291292i \(0.0940853\pi\)
−0.956634 + 0.291292i \(0.905915\pi\)
\(492\) 0 0
\(493\) 221.082i 0.448442i
\(494\) 0 0
\(495\) 324.532 466.563i 0.655619 0.942551i
\(496\) 0 0
\(497\) −1154.40 −2.32273
\(498\) 0 0
\(499\) −528.285 −1.05869 −0.529344 0.848407i \(-0.677562\pi\)
−0.529344 + 0.848407i \(0.677562\pi\)
\(500\) 0 0
\(501\) −203.220 136.305i −0.405630 0.272066i
\(502\) 0 0
\(503\) 733.418 1.45809 0.729044 0.684467i \(-0.239965\pi\)
0.729044 + 0.684467i \(0.239965\pi\)
\(504\) 0 0
\(505\) 52.3169 + 80.9321i 0.103598 + 0.160262i
\(506\) 0 0
\(507\) 283.116 + 189.893i 0.558413 + 0.374542i
\(508\) 0 0
\(509\) 312.619i 0.614183i 0.951680 + 0.307092i \(0.0993558\pi\)
−0.951680 + 0.307092i \(0.900644\pi\)
\(510\) 0 0
\(511\) 590.183 1.15496
\(512\) 0 0
\(513\) 167.714 820.509i 0.326928 1.59943i
\(514\) 0 0
\(515\) −18.9195 29.2677i −0.0367370 0.0568306i
\(516\) 0 0
\(517\) 255.838i 0.494851i
\(518\) 0 0
\(519\) 416.235 + 279.179i 0.801995 + 0.537918i
\(520\) 0 0
\(521\) 715.719i 1.37374i 0.726780 + 0.686870i \(0.241016\pi\)
−0.726780 + 0.686870i \(0.758984\pi\)
\(522\) 0 0
\(523\) 109.080i 0.208567i 0.994548 + 0.104283i \(0.0332549\pi\)
−0.994548 + 0.104283i \(0.966745\pi\)
\(524\) 0 0
\(525\) 506.207 + 812.974i 0.964203 + 1.54852i
\(526\) 0 0
\(527\) 287.977 0.546445
\(528\) 0 0
\(529\) −472.686 −0.893547
\(530\) 0 0
\(531\) −396.641 + 162.662i −0.746971 + 0.306332i
\(532\) 0 0
\(533\) −103.019 −0.193282
\(534\) 0 0
\(535\) −309.777 + 200.249i −0.579023 + 0.374297i
\(536\) 0 0
\(537\) −451.370 + 672.958i −0.840540 + 1.25318i
\(538\) 0 0
\(539\) 1440.44i 2.67243i
\(540\) 0 0
\(541\) 14.9710 0.0276729 0.0138364 0.999904i \(-0.495596\pi\)
0.0138364 + 0.999904i \(0.495596\pi\)
\(542\) 0 0
\(543\) −216.705 145.350i −0.399088 0.267679i
\(544\) 0 0
\(545\) 310.862 200.950i 0.570389 0.368716i
\(546\) 0 0
\(547\) 842.765i 1.54070i 0.637618 + 0.770352i \(0.279920\pi\)
−0.637618 + 0.770352i \(0.720080\pi\)
\(548\) 0 0
\(549\) −146.900 358.206i −0.267578 0.652471i
\(550\) 0 0
\(551\) 487.901i 0.885482i
\(552\) 0 0
\(553\) 484.733i 0.876551i
\(554\) 0 0
\(555\) −3.28555 + 193.864i −0.00591990 + 0.349305i
\(556\) 0 0
\(557\) −845.989 −1.51883 −0.759415 0.650606i \(-0.774515\pi\)
−0.759415 + 0.650606i \(0.774515\pi\)
\(558\) 0 0
\(559\) −223.819 −0.400391
\(560\) 0 0
\(561\) −296.634 + 442.258i −0.528759 + 0.788339i
\(562\) 0 0
\(563\) −336.509 −0.597707 −0.298853 0.954299i \(-0.596604\pi\)
−0.298853 + 0.954299i \(0.596604\pi\)
\(564\) 0 0
\(565\) 618.164 399.600i 1.09410 0.707256i
\(566\) 0 0
\(567\) −726.202 736.491i −1.28078 1.29893i
\(568\) 0 0
\(569\) 552.736i 0.971416i 0.874121 + 0.485708i \(0.161438\pi\)
−0.874121 + 0.485708i \(0.838562\pi\)
\(570\) 0 0
\(571\) 772.222 1.35240 0.676202 0.736717i \(-0.263625\pi\)
0.676202 + 0.736717i \(0.263625\pi\)
\(572\) 0 0
\(573\) 505.043 752.980i 0.881401 1.31410i
\(574\) 0 0
\(575\) 77.0245 171.065i 0.133956 0.297504i
\(576\) 0 0
\(577\) 848.557i 1.47064i −0.677722 0.735318i \(-0.737033\pi\)
0.677722 0.735318i \(-0.262967\pi\)
\(578\) 0 0
\(579\) 512.658 764.334i 0.885420 1.32009i
\(580\) 0 0
\(581\) 1024.43i 1.76322i
\(582\) 0 0
\(583\) 367.755i 0.630798i
\(584\) 0 0
\(585\) −191.201 + 274.880i −0.326839 + 0.469880i
\(586\) 0 0
\(587\) −575.536 −0.980470 −0.490235 0.871590i \(-0.663089\pi\)
−0.490235 + 0.871590i \(0.663089\pi\)
\(588\) 0 0
\(589\) −635.529 −1.07900
\(590\) 0 0
\(591\) −722.419 484.544i −1.22237 0.819872i
\(592\) 0 0
\(593\) −156.935 −0.264646 −0.132323 0.991207i \(-0.542244\pi\)
−0.132323 + 0.991207i \(0.542244\pi\)
\(594\) 0 0
\(595\) −487.154 753.608i −0.818747 1.26657i
\(596\) 0 0
\(597\) −953.225 639.352i −1.59669 1.07094i
\(598\) 0 0
\(599\) 517.564i 0.864047i −0.901862 0.432024i \(-0.857800\pi\)
0.901862 0.432024i \(-0.142200\pi\)
\(600\) 0 0
\(601\) −883.588 −1.47020 −0.735098 0.677961i \(-0.762864\pi\)
−0.735098 + 0.677961i \(0.762864\pi\)
\(602\) 0 0
\(603\) 5.24727 2.15190i 0.00870194 0.00356866i
\(604\) 0 0
\(605\) 161.693 104.523i 0.267260 0.172765i
\(606\) 0 0
\(607\) 242.929i 0.400212i −0.979774 0.200106i \(-0.935871\pi\)
0.979774 0.200106i \(-0.0641287\pi\)
\(608\) 0 0
\(609\) 500.431 + 335.652i 0.821725 + 0.551152i
\(610\) 0 0
\(611\) 150.729i 0.246693i
\(612\) 0 0
\(613\) 157.263i 0.256546i 0.991739 + 0.128273i \(0.0409434\pi\)
−0.991739 + 0.128273i \(0.959057\pi\)
\(614\) 0 0
\(615\) −3.51912 + 207.646i −0.00572215 + 0.337637i
\(616\) 0 0
\(617\) −471.040 −0.763436 −0.381718 0.924279i \(-0.624667\pi\)
−0.381718 + 0.924279i \(0.624667\pi\)
\(618\) 0 0
\(619\) 550.320 0.889047 0.444524 0.895767i \(-0.353373\pi\)
0.444524 + 0.895767i \(0.353373\pi\)
\(620\) 0 0
\(621\) −40.5759 + 198.510i −0.0653396 + 0.319662i
\(622\) 0 0
\(623\) −1799.48 −2.88841
\(624\) 0 0
\(625\) −414.295 467.958i −0.662872 0.748733i
\(626\) 0 0
\(627\) 654.634 976.009i 1.04407 1.55663i
\(628\) 0 0
\(629\) 181.676i 0.288834i
\(630\) 0 0
\(631\) 347.362 0.550495 0.275248 0.961373i \(-0.411240\pi\)
0.275248 + 0.961373i \(0.411240\pi\)
\(632\) 0 0
\(633\) −800.051 536.614i −1.26390 0.847732i
\(634\) 0 0
\(635\) 244.342 + 377.987i 0.384790 + 0.595255i
\(636\) 0 0
\(637\) 848.649i 1.33226i
\(638\) 0 0
\(639\) −752.798 + 308.721i −1.17809 + 0.483132i
\(640\) 0 0
\(641\) 333.802i 0.520752i −0.965507 0.260376i \(-0.916153\pi\)
0.965507 0.260376i \(-0.0838465\pi\)
\(642\) 0 0
\(643\) 495.512i 0.770625i 0.922786 + 0.385313i \(0.125906\pi\)
−0.922786 + 0.385313i \(0.874094\pi\)
\(644\) 0 0
\(645\) −7.64561 + 451.131i −0.0118537 + 0.699428i
\(646\) 0 0
\(647\) −149.493 −0.231056 −0.115528 0.993304i \(-0.536856\pi\)
−0.115528 + 0.993304i \(0.536856\pi\)
\(648\) 0 0
\(649\) −601.590 −0.926948
\(650\) 0 0
\(651\) −437.212 + 651.850i −0.671601 + 1.00131i
\(652\) 0 0
\(653\) 261.846 0.400990 0.200495 0.979695i \(-0.435745\pi\)
0.200495 + 0.979695i \(0.435745\pi\)
\(654\) 0 0
\(655\) 30.6485 + 47.4120i 0.0467916 + 0.0723847i
\(656\) 0 0
\(657\) 384.867 157.833i 0.585794 0.240234i
\(658\) 0 0
\(659\) 148.718i 0.225672i 0.993614 + 0.112836i \(0.0359935\pi\)
−0.993614 + 0.112836i \(0.964006\pi\)
\(660\) 0 0
\(661\) 535.548 0.810209 0.405104 0.914270i \(-0.367235\pi\)
0.405104 + 0.914270i \(0.367235\pi\)
\(662\) 0 0
\(663\) 174.765 260.561i 0.263597 0.393002i
\(664\) 0 0
\(665\) 1075.09 + 1663.12i 1.61668 + 2.50093i
\(666\) 0 0
\(667\) 118.040i 0.176972i
\(668\) 0 0
\(669\) −488.681 + 728.586i −0.730465 + 1.08907i
\(670\) 0 0
\(671\) 543.295i 0.809680i
\(672\) 0 0
\(673\) 678.388i 1.00801i 0.863702 + 0.504003i \(0.168140\pi\)
−0.863702 + 0.504003i \(0.831860\pi\)
\(674\) 0 0
\(675\) 547.519 + 394.776i 0.811139 + 0.584853i
\(676\) 0 0
\(677\) 809.743 1.19608 0.598038 0.801468i \(-0.295947\pi\)
0.598038 + 0.801468i \(0.295947\pi\)
\(678\) 0 0
\(679\) −132.070 −0.194507
\(680\) 0 0
\(681\) 922.231 + 618.563i 1.35423 + 0.908316i
\(682\) 0 0
\(683\) 150.099 0.219764 0.109882 0.993945i \(-0.464953\pi\)
0.109882 + 0.993945i \(0.464953\pi\)
\(684\) 0 0
\(685\) −94.4390 + 61.0481i −0.137867 + 0.0891214i
\(686\) 0 0
\(687\) 951.705 + 638.333i 1.38531 + 0.929160i
\(688\) 0 0
\(689\) 216.667i 0.314465i
\(690\) 0 0
\(691\) −334.001 −0.483359 −0.241679 0.970356i \(-0.577698\pi\)
−0.241679 + 0.970356i \(0.577698\pi\)
\(692\) 0 0
\(693\) −550.719 1342.89i −0.794688 1.93780i
\(694\) 0 0
\(695\) −382.515 + 247.269i −0.550381 + 0.355783i
\(696\) 0 0
\(697\) 194.592i 0.279185i
\(698\) 0 0
\(699\) 359.425 + 241.076i 0.514199 + 0.344886i
\(700\) 0 0
\(701\) 924.471i 1.31879i −0.751797 0.659394i \(-0.770813\pi\)
0.751797 0.659394i \(-0.229187\pi\)
\(702\) 0 0
\(703\) 400.937i 0.570324i
\(704\) 0 0
\(705\) −303.812 5.14890i −0.430938 0.00730340i
\(706\) 0 0
\(707\) 246.112 0.348108
\(708\) 0 0
\(709\) −797.459 −1.12477 −0.562383 0.826877i \(-0.690115\pi\)
−0.562383 + 0.826877i \(0.690115\pi\)
\(710\) 0 0
\(711\) −129.633 316.101i −0.182324 0.444586i
\(712\) 0 0
\(713\) 153.757 0.215647
\(714\) 0 0
\(715\) −394.606 + 255.085i −0.551897 + 0.356762i
\(716\) 0 0
\(717\) −416.902 + 621.568i −0.581453 + 0.866902i
\(718\) 0 0
\(719\) 907.966i 1.26282i 0.775450 + 0.631409i \(0.217523\pi\)
−0.775450 + 0.631409i \(0.782477\pi\)
\(720\) 0 0
\(721\) −89.0024 −0.123443
\(722\) 0 0
\(723\) 75.8757 + 50.8918i 0.104946 + 0.0703897i
\(724\) 0 0
\(725\) −358.573 161.453i −0.494584 0.222694i
\(726\) 0 0
\(727\) 1038.16i 1.42801i 0.700140 + 0.714005i \(0.253121\pi\)
−0.700140 + 0.714005i \(0.746879\pi\)
\(728\) 0 0
\(729\) −670.527 286.067i −0.919790 0.392410i
\(730\) 0 0
\(731\) 422.769i 0.578344i
\(732\) 0 0
\(733\) 399.107i 0.544485i −0.962229 0.272242i \(-0.912235\pi\)
0.962229 0.272242i \(-0.0877652\pi\)
\(734\) 0 0
\(735\) 1710.55 + 28.9898i 2.32727 + 0.0394418i
\(736\) 0 0
\(737\) 7.95858 0.0107986
\(738\) 0 0
\(739\) 452.504 0.612319 0.306159 0.951980i \(-0.400956\pi\)
0.306159 + 0.951980i \(0.400956\pi\)
\(740\) 0 0
\(741\) −385.684 + 575.025i −0.520491 + 0.776012i
\(742\) 0 0
\(743\) 486.909 0.655328 0.327664 0.944794i \(-0.393739\pi\)
0.327664 + 0.944794i \(0.393739\pi\)
\(744\) 0 0
\(745\) −619.777 958.770i −0.831916 1.28694i
\(746\) 0 0
\(747\) 273.965 + 668.046i 0.366754 + 0.894306i
\(748\) 0 0
\(749\) 942.024i 1.25771i
\(750\) 0 0
\(751\) 470.899 0.627029 0.313515 0.949583i \(-0.398494\pi\)
0.313515 + 0.949583i \(0.398494\pi\)
\(752\) 0 0
\(753\) 115.169 171.708i 0.152947 0.228032i
\(754\) 0 0
\(755\) −310.894 + 200.971i −0.411780 + 0.266187i
\(756\) 0 0
\(757\) 886.264i 1.17076i −0.810760 0.585379i \(-0.800946\pi\)
0.810760 0.585379i \(-0.199054\pi\)
\(758\) 0 0
\(759\) −158.379 + 236.131i −0.208668 + 0.311108i
\(760\) 0 0
\(761\) 1022.30i 1.34336i −0.740841 0.671680i \(-0.765573\pi\)
0.740841 0.671680i \(-0.234427\pi\)
\(762\) 0 0
\(763\) 945.322i 1.23895i
\(764\) 0 0
\(765\) −519.218 361.158i −0.678717 0.472102i
\(766\) 0 0
\(767\) 354.432 0.462102
\(768\) 0 0
\(769\) 1051.96 1.36796 0.683982 0.729499i \(-0.260247\pi\)
0.683982 + 0.729499i \(0.260247\pi\)
\(770\) 0 0
\(771\) −902.847 605.562i −1.17101 0.785424i
\(772\) 0 0
\(773\) 983.554 1.27238 0.636192 0.771530i \(-0.280508\pi\)
0.636192 + 0.771530i \(0.280508\pi\)
\(774\) 0 0
\(775\) 210.305 467.070i 0.271361 0.602671i
\(776\) 0 0
\(777\) 411.234 + 275.825i 0.529259 + 0.354987i
\(778\) 0 0
\(779\) 429.441i 0.551272i
\(780\) 0 0
\(781\) −1141.77 −1.46194
\(782\) 0 0
\(783\) 416.101 + 85.0522i 0.531419 + 0.108624i
\(784\) 0 0
\(785\) −667.039 1031.88i −0.849731 1.31450i
\(786\) 0 0
\(787\) 984.726i 1.25124i 0.780128 + 0.625620i \(0.215154\pi\)
−0.780128 + 0.625620i \(0.784846\pi\)
\(788\) 0 0
\(789\) −58.4214 39.1847i −0.0740449 0.0496638i
\(790\) 0 0
\(791\) 1879.82i 2.37651i
\(792\) 0 0
\(793\) 320.087i 0.403641i
\(794\) 0 0
\(795\) −436.715 7.40130i −0.549327 0.00930982i
\(796\) 0 0
\(797\) −359.710 −0.451330 −0.225665 0.974205i \(-0.572456\pi\)
−0.225665 + 0.974205i \(0.572456\pi\)
\(798\) 0 0
\(799\) 284.712 0.356335
\(800\) 0 0
\(801\) −1173.46 + 481.236i −1.46500 + 0.600794i
\(802\) 0 0
\(803\) 583.731 0.726938
\(804\) 0 0
\(805\) −260.102 402.367i −0.323108 0.499834i
\(806\) 0 0
\(807\) 662.987 988.463i 0.821546 1.22486i
\(808\) 0 0
\(809\) 1174.75i 1.45211i −0.687638 0.726053i \(-0.741352\pi\)
0.687638 0.726053i \(-0.258648\pi\)
\(810\) 0 0
\(811\) −1000.55 −1.23373 −0.616863 0.787071i \(-0.711597\pi\)
−0.616863 + 0.787071i \(0.711597\pi\)
\(812\) 0 0
\(813\) 358.588 + 240.514i 0.441067 + 0.295835i
\(814\) 0 0
\(815\) 163.290 + 252.602i 0.200355 + 0.309942i
\(816\) 0 0
\(817\) 933.000i 1.14198i
\(818\) 0 0
\(819\) 324.461 + 791.178i 0.396167 + 0.966029i
\(820\) 0 0
\(821\) 536.992i 0.654071i 0.945012 + 0.327036i \(0.106050\pi\)
−0.945012 + 0.327036i \(0.893950\pi\)
\(822\) 0 0
\(823\) 563.712i 0.684948i −0.939527 0.342474i \(-0.888735\pi\)
0.939527 0.342474i \(-0.111265\pi\)
\(824\) 0 0
\(825\) 500.672 + 804.086i 0.606875 + 0.974649i
\(826\) 0 0
\(827\) −976.487 −1.18076 −0.590379 0.807126i \(-0.701022\pi\)
−0.590379 + 0.807126i \(0.701022\pi\)
\(828\) 0 0
\(829\) 358.250 0.432147 0.216074 0.976377i \(-0.430675\pi\)
0.216074 + 0.976377i \(0.430675\pi\)
\(830\) 0 0
\(831\) −603.071 + 899.133i −0.725718 + 1.08199i
\(832\) 0 0
\(833\) −1603.01 −1.92438
\(834\) 0 0
\(835\) 342.502 221.403i 0.410182 0.265154i
\(836\) 0 0
\(837\) −110.787 + 542.005i −0.132362 + 0.647556i
\(838\) 0 0
\(839\) 98.9138i 0.117895i −0.998261 0.0589474i \(-0.981226\pi\)
0.998261 0.0589474i \(-0.0187744\pi\)
\(840\) 0 0
\(841\) 593.573 0.705794
\(842\) 0 0
\(843\) 888.447 1324.61i 1.05391 1.57130i
\(844\) 0 0
\(845\) −477.154 + 308.447i −0.564680 + 0.365026i
\(846\) 0 0
\(847\) 491.703i 0.580523i
\(848\) 0 0
\(849\) −430.140 + 641.305i −0.506643 + 0.755366i
\(850\) 0 0
\(851\) 97.0007i 0.113984i
\(852\) 0 0
\(853\) 262.197i 0.307383i 0.988119 + 0.153691i \(0.0491161\pi\)
−0.988119 + 0.153691i \(0.950884\pi\)
\(854\) 0 0
\(855\) 1145.85 + 797.031i 1.34018 + 0.932200i
\(856\) 0 0
\(857\) 41.1542 0.0480212 0.0240106 0.999712i \(-0.492356\pi\)
0.0240106 + 0.999712i \(0.492356\pi\)
\(858\) 0 0
\(859\) −414.736 −0.482813 −0.241406 0.970424i \(-0.577609\pi\)
−0.241406 + 0.970424i \(0.577609\pi\)
\(860\) 0 0
\(861\) 440.470 + 295.434i 0.511579 + 0.343129i
\(862\) 0 0
\(863\) −1253.21 −1.45216 −0.726078 0.687613i \(-0.758659\pi\)
−0.726078 + 0.687613i \(0.758659\pi\)
\(864\) 0 0
\(865\) −701.510 + 453.477i −0.810995 + 0.524251i
\(866\) 0 0
\(867\) −227.865 152.835i −0.262820 0.176280i
\(868\) 0 0
\(869\) 479.433i 0.551706i
\(870\) 0 0
\(871\) −4.68888 −0.00538332
\(872\) 0 0
\(873\) −86.1249 + 35.3197i −0.0986539 + 0.0404579i
\(874\) 0 0
\(875\) −1578.04 + 239.768i −1.80347 + 0.274021i
\(876\) 0 0
\(877\) 861.957i 0.982847i −0.870921 0.491424i \(-0.836477\pi\)
0.870921 0.491424i \(-0.163523\pi\)
\(878\) 0 0
\(879\) 773.494 + 518.802i 0.879970 + 0.590218i
\(880\) 0 0
\(881\) 87.0009i 0.0987524i −0.998780 0.0493762i \(-0.984277\pi\)
0.998780 0.0493762i \(-0.0157233\pi\)
\(882\) 0 0
\(883\) 1022.42i 1.15789i −0.815366 0.578945i \(-0.803464\pi\)
0.815366 0.578945i \(-0.196536\pi\)
\(884\) 0 0
\(885\) 12.1074 714.397i 0.0136806 0.807228i
\(886\) 0 0
\(887\) −669.924 −0.755269 −0.377634 0.925955i \(-0.623262\pi\)
−0.377634 + 0.925955i \(0.623262\pi\)
\(888\) 0 0
\(889\) 1149.45 1.29297
\(890\) 0 0
\(891\) −718.263 728.439i −0.806131 0.817552i
\(892\) 0 0
\(893\) −628.324 −0.703610
\(894\) 0 0
\(895\) −733.169 1134.18i −0.819183 1.26724i
\(896\) 0 0
\(897\) 93.3104 139.119i 0.104025 0.155093i
\(898\) 0 0
\(899\) 322.293i 0.358502i
\(900\) 0 0
\(901\) 409.260 0.454229
\(902\) 0 0
\(903\) 956.960 + 641.858i 1.05976 + 0.710806i
\(904\) 0 0
\(905\) 365.228 236.094i 0.403567 0.260878i
\(906\) 0 0
\(907\) 1296.52i 1.42946i −0.699402 0.714729i \(-0.746550\pi\)
0.699402 0.714729i \(-0.253450\pi\)
\(908\) 0 0
\(909\) 160.493 65.8181i 0.176560 0.0724071i
\(910\) 0 0
\(911\) 926.622i 1.01715i −0.861018 0.508574i \(-0.830173\pi\)
0.861018 0.508574i \(-0.169827\pi\)
\(912\) 0 0
\(913\) 1013.23i 1.10978i
\(914\) 0 0
\(915\) 645.171 + 10.9341i 0.705105 + 0.0119499i
\(916\) 0 0
\(917\) 144.179 0.157229
\(918\) 0 0
\(919\) 478.391 0.520556 0.260278 0.965534i \(-0.416186\pi\)
0.260278 + 0.965534i \(0.416186\pi\)
\(920\) 0 0
\(921\) 444.793 663.152i 0.482946 0.720035i
\(922\) 0 0
\(923\) 672.687 0.728805
\(924\) 0 0
\(925\) −294.661 132.676i −0.318553 0.143433i
\(926\) 0 0
\(927\) −58.0397 + 23.8020i −0.0626102 + 0.0256764i
\(928\) 0 0
\(929\) 1452.00i 1.56297i 0.623923 + 0.781486i \(0.285538\pi\)
−0.623923 + 0.781486i \(0.714462\pi\)
\(930\) 0 0
\(931\) 3537.64 3.79983
\(932\) 0 0
\(933\) −311.792 + 464.858i −0.334182 + 0.498240i
\(934\) 0 0
\(935\) −481.828 745.369i −0.515324 0.797186i
\(936\) 0 0
\(937\) 1535.86i 1.63912i −0.572991 0.819562i \(-0.694217\pi\)
0.572991 0.819562i \(-0.305783\pi\)
\(938\) 0 0
\(939\) 34.2610 51.0805i 0.0364866 0.0543988i
\(940\) 0 0
\(941\) 1237.50i 1.31509i −0.753415 0.657545i \(-0.771595\pi\)
0.753415 0.657545i \(-0.228405\pi\)
\(942\) 0 0
\(943\) 103.897i 0.110177i
\(944\) 0 0
\(945\) 1605.79 626.960i 1.69925 0.663450i
\(946\) 0 0
\(947\) 603.800 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(948\) 0 0
\(949\) −343.911 −0.362393
\(950\) 0 0
\(951\) 47.2861 + 31.7160i 0.0497225 + 0.0333501i
\(952\) 0 0
\(953\) 534.553 0.560916 0.280458 0.959866i \(-0.409514\pi\)
0.280458 + 0.959866i \(0.409514\pi\)
\(954\) 0 0
\(955\) 820.352 + 1269.05i 0.859007 + 1.32885i
\(956\) 0 0
\(957\) 494.959 + 331.982i 0.517199 + 0.346899i
\(958\) 0 0
\(959\) 287.186i 0.299464i
\(960\) 0 0
\(961\) −541.188 −0.563151
\(962\) 0 0
\(963\) 251.927 + 614.307i 0.261606 + 0.637909i
\(964\) 0 0
\(965\) 832.721 + 1288.19i 0.862923 + 1.33491i
\(966\) 0 0
\(967\) 271.601i 0.280870i 0.990090 + 0.140435i \(0.0448500\pi\)
−0.990090 + 0.140435i \(0.955150\pi\)
\(968\) 0 0
\(969\) −1086.16 728.515i −1.12091 0.751822i
\(970\) 0 0
\(971\) 912.176i 0.939419i −0.882821 0.469709i \(-0.844359\pi\)
0.882821 0.469709i \(-0.155641\pi\)
\(972\) 0 0
\(973\) 1163.22i 1.19550i
\(974\) 0 0
\(975\) −294.976 473.735i −0.302539 0.485882i
\(976\) 0 0
\(977\) 333.628 0.341482 0.170741 0.985316i \(-0.445384\pi\)
0.170741 + 0.985316i \(0.445384\pi\)
\(978\) 0 0
\(979\) −1779.80 −1.81798
\(980\) 0 0
\(981\) −252.809 616.458i −0.257705 0.628397i
\(982\) 0 0
\(983\) −1600.35 −1.62803 −0.814013 0.580847i \(-0.802722\pi\)
−0.814013 + 0.580847i \(0.802722\pi\)
\(984\) 0 0
\(985\) 1217.54 787.056i 1.23608 0.799041i
\(986\) 0 0
\(987\) −432.256 + 644.460i −0.437949 + 0.652948i
\(988\) 0 0
\(989\) 225.725i 0.228236i
\(990\) 0 0
\(991\) 69.3757 0.0700057 0.0350029 0.999387i \(-0.488856\pi\)
0.0350029 + 0.999387i \(0.488856\pi\)
\(992\) 0 0
\(993\) 1029.46 + 690.485i 1.03672 + 0.695353i
\(994\) 0 0
\(995\) 1606.54 1038.51i 1.61461 1.04373i
\(996\) 0 0
\(997\) 482.497i 0.483949i 0.970283 + 0.241974i \(0.0777950\pi\)
−0.970283 + 0.241974i \(0.922205\pi\)
\(998\) 0 0
\(999\) 341.936 + 69.8925i 0.342278 + 0.0699625i
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 960.3.c.j.449.10 12
3.2 odd 2 inner 960.3.c.j.449.4 12
4.3 odd 2 960.3.c.k.449.3 12
5.4 even 2 inner 960.3.c.j.449.3 12
8.3 odd 2 120.3.c.a.89.10 yes 12
8.5 even 2 240.3.c.e.209.3 12
12.11 even 2 960.3.c.k.449.9 12
15.14 odd 2 inner 960.3.c.j.449.9 12
20.19 odd 2 960.3.c.k.449.10 12
24.5 odd 2 240.3.c.e.209.9 12
24.11 even 2 120.3.c.a.89.4 yes 12
40.3 even 4 600.3.l.g.401.4 12
40.13 odd 4 1200.3.l.y.401.9 12
40.19 odd 2 120.3.c.a.89.3 12
40.27 even 4 600.3.l.g.401.9 12
40.29 even 2 240.3.c.e.209.10 12
40.37 odd 4 1200.3.l.y.401.4 12
60.59 even 2 960.3.c.k.449.4 12
120.29 odd 2 240.3.c.e.209.4 12
120.53 even 4 1200.3.l.y.401.10 12
120.59 even 2 120.3.c.a.89.9 yes 12
120.77 even 4 1200.3.l.y.401.3 12
120.83 odd 4 600.3.l.g.401.3 12
120.107 odd 4 600.3.l.g.401.10 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.3.c.a.89.3 12 40.19 odd 2
120.3.c.a.89.4 yes 12 24.11 even 2
120.3.c.a.89.9 yes 12 120.59 even 2
120.3.c.a.89.10 yes 12 8.3 odd 2
240.3.c.e.209.3 12 8.5 even 2
240.3.c.e.209.4 12 120.29 odd 2
240.3.c.e.209.9 12 24.5 odd 2
240.3.c.e.209.10 12 40.29 even 2
600.3.l.g.401.3 12 120.83 odd 4
600.3.l.g.401.4 12 40.3 even 4
600.3.l.g.401.9 12 40.27 even 4
600.3.l.g.401.10 12 120.107 odd 4
960.3.c.j.449.3 12 5.4 even 2 inner
960.3.c.j.449.4 12 3.2 odd 2 inner
960.3.c.j.449.9 12 15.14 odd 2 inner
960.3.c.j.449.10 12 1.1 even 1 trivial
960.3.c.k.449.3 12 4.3 odd 2
960.3.c.k.449.4 12 60.59 even 2
960.3.c.k.449.9 12 12.11 even 2
960.3.c.k.449.10 12 20.19 odd 2
1200.3.l.y.401.3 12 120.77 even 4
1200.3.l.y.401.4 12 40.37 odd 4
1200.3.l.y.401.9 12 40.13 odd 4
1200.3.l.y.401.10 12 120.53 even 4