Properties

Label 440.2.y.b.361.3
Level $440$
Weight $2$
Character 440.361
Analytic conductor $3.513$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [440,2,Mod(81,440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(440, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("440.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 440.y (of order \(5\), degree \(4\), 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: \(3.51341768894\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 15 x^{10} - 22 x^{9} + 89 x^{8} - 118 x^{7} + 205 x^{6} - 68 x^{5} + 1061 x^{4} + \cdots + 400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 361.3
Root \(-1.51700 - 1.10216i\) of defining polynomial
Character \(\chi\) \(=\) 440.361
Dual form 440.2.y.b.401.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.45455 - 1.78334i) q^{3} +(-0.309017 - 0.951057i) q^{5} +(-0.700550 - 0.508979i) q^{7} +(1.91748 - 5.90141i) q^{9} +(-1.21410 - 3.08642i) q^{11} +(-1.37052 + 4.21802i) q^{13} +(-2.45455 - 1.78334i) q^{15} +(-0.169969 - 0.523110i) q^{17} +(0.209505 - 0.152214i) q^{19} -2.62722 q^{21} +7.92701 q^{23} +(-0.809017 + 0.587785i) q^{25} +(-3.00497 - 9.24833i) q^{27} +(7.57410 + 5.50291i) q^{29} +(-1.64270 + 5.05570i) q^{31} +(-8.48418 - 5.41063i) q^{33} +(-0.267586 + 0.823546i) q^{35} +(-5.16751 - 3.75441i) q^{37} +(4.15814 + 12.7974i) q^{39} +(-7.07258 + 5.13853i) q^{41} +6.38622 q^{43} -6.20511 q^{45} +(-0.665989 + 0.483870i) q^{47} +(-1.93141 - 5.94426i) q^{49} +(-1.35008 - 0.980890i) q^{51} +(0.741981 - 2.28358i) q^{53} +(-2.56018 + 2.10843i) q^{55} +(0.242791 - 0.747235i) q^{57} +(10.9337 + 7.94382i) q^{59} +(-2.04186 - 6.28421i) q^{61} +(-4.34699 + 3.15827i) q^{63} +4.43509 q^{65} +7.17725 q^{67} +(19.4573 - 14.1365i) q^{69} +(0.864970 + 2.66210i) q^{71} +(2.33866 + 1.69913i) q^{73} +(-0.937555 + 2.88550i) q^{75} +(-0.720387 + 2.78014i) q^{77} +(-1.82769 + 5.62505i) q^{79} +(-8.80862 - 6.39983i) q^{81} +(-2.19347 - 6.75080i) q^{83} +(-0.444984 + 0.323300i) q^{85} +28.4046 q^{87} -16.6470 q^{89} +(3.10700 - 2.25737i) q^{91} +(4.98393 + 15.3390i) q^{93} +(-0.209505 - 0.152214i) q^{95} +(-2.42082 + 7.45053i) q^{97} +(-20.5422 + 1.24672i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{3} + 3 q^{5} - 8 q^{7} + 10 q^{9} - 4 q^{11} - 7 q^{13} + q^{15} + 7 q^{17} + 3 q^{19} + 4 q^{21} + 36 q^{23} - 3 q^{25} + 8 q^{27} + 13 q^{29} + 2 q^{31} - 19 q^{33} - 2 q^{35} - 22 q^{37}+ \cdots - 79 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/440\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(221\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{3}{5}\right)\)

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
\(3\) 2.45455 1.78334i 1.41714 1.02961i 0.424900 0.905240i \(-0.360309\pi\)
0.992236 0.124369i \(-0.0396907\pi\)
\(4\) 0 0
\(5\) −0.309017 0.951057i −0.138197 0.425325i
\(6\) 0 0
\(7\) −0.700550 0.508979i −0.264783 0.192376i 0.447470 0.894299i \(-0.352325\pi\)
−0.712253 + 0.701923i \(0.752325\pi\)
\(8\) 0 0
\(9\) 1.91748 5.90141i 0.639161 1.96714i
\(10\) 0 0
\(11\) −1.21410 3.08642i −0.366064 0.930590i
\(12\) 0 0
\(13\) −1.37052 + 4.21802i −0.380114 + 1.16987i 0.559850 + 0.828594i \(0.310859\pi\)
−0.939963 + 0.341275i \(0.889141\pi\)
\(14\) 0 0
\(15\) −2.45455 1.78334i −0.633762 0.460455i
\(16\) 0 0
\(17\) −0.169969 0.523110i −0.0412235 0.126873i 0.928327 0.371765i \(-0.121247\pi\)
−0.969550 + 0.244892i \(0.921247\pi\)
\(18\) 0 0
\(19\) 0.209505 0.152214i 0.0480637 0.0349203i −0.563494 0.826120i \(-0.690543\pi\)
0.611558 + 0.791200i \(0.290543\pi\)
\(20\) 0 0
\(21\) −2.62722 −0.573306
\(22\) 0 0
\(23\) 7.92701 1.65290 0.826448 0.563013i \(-0.190358\pi\)
0.826448 + 0.563013i \(0.190358\pi\)
\(24\) 0 0
\(25\) −0.809017 + 0.587785i −0.161803 + 0.117557i
\(26\) 0 0
\(27\) −3.00497 9.24833i −0.578306 1.77984i
\(28\) 0 0
\(29\) 7.57410 + 5.50291i 1.40648 + 1.02186i 0.993823 + 0.110976i \(0.0353975\pi\)
0.412652 + 0.910889i \(0.364602\pi\)
\(30\) 0 0
\(31\) −1.64270 + 5.05570i −0.295037 + 0.908031i 0.688172 + 0.725548i \(0.258413\pi\)
−0.983209 + 0.182483i \(0.941587\pi\)
\(32\) 0 0
\(33\) −8.48418 5.41063i −1.47691 0.941869i
\(34\) 0 0
\(35\) −0.267586 + 0.823546i −0.0452303 + 0.139205i
\(36\) 0 0
\(37\) −5.16751 3.75441i −0.849533 0.617222i 0.0754844 0.997147i \(-0.475950\pi\)
−0.925017 + 0.379925i \(0.875950\pi\)
\(38\) 0 0
\(39\) 4.15814 + 12.7974i 0.665836 + 2.04923i
\(40\) 0 0
\(41\) −7.07258 + 5.13853i −1.10455 + 0.802503i −0.981797 0.189934i \(-0.939173\pi\)
−0.122754 + 0.992437i \(0.539173\pi\)
\(42\) 0 0
\(43\) 6.38622 0.973890 0.486945 0.873433i \(-0.338111\pi\)
0.486945 + 0.873433i \(0.338111\pi\)
\(44\) 0 0
\(45\) −6.20511 −0.925003
\(46\) 0 0
\(47\) −0.665989 + 0.483870i −0.0971445 + 0.0705796i −0.635297 0.772268i \(-0.719122\pi\)
0.538153 + 0.842847i \(0.319122\pi\)
\(48\) 0 0
\(49\) −1.93141 5.94426i −0.275916 0.849181i
\(50\) 0 0
\(51\) −1.35008 0.980890i −0.189049 0.137352i
\(52\) 0 0
\(53\) 0.741981 2.28358i 0.101919 0.313674i −0.887076 0.461623i \(-0.847267\pi\)
0.988995 + 0.147949i \(0.0472672\pi\)
\(54\) 0 0
\(55\) −2.56018 + 2.10843i −0.345215 + 0.284301i
\(56\) 0 0
\(57\) 0.242791 0.747235i 0.0321585 0.0989737i
\(58\) 0 0
\(59\) 10.9337 + 7.94382i 1.42345 + 1.03420i 0.991190 + 0.132449i \(0.0422840\pi\)
0.432261 + 0.901749i \(0.357716\pi\)
\(60\) 0 0
\(61\) −2.04186 6.28421i −0.261434 0.804611i −0.992494 0.122297i \(-0.960974\pi\)
0.731060 0.682313i \(-0.239026\pi\)
\(62\) 0 0
\(63\) −4.34699 + 3.15827i −0.547669 + 0.397905i
\(64\) 0 0
\(65\) 4.43509 0.550105
\(66\) 0 0
\(67\) 7.17725 0.876840 0.438420 0.898770i \(-0.355538\pi\)
0.438420 + 0.898770i \(0.355538\pi\)
\(68\) 0 0
\(69\) 19.4573 14.1365i 2.34238 1.70184i
\(70\) 0 0
\(71\) 0.864970 + 2.66210i 0.102653 + 0.315934i 0.989172 0.146758i \(-0.0468838\pi\)
−0.886519 + 0.462691i \(0.846884\pi\)
\(72\) 0 0
\(73\) 2.33866 + 1.69913i 0.273719 + 0.198868i 0.716173 0.697923i \(-0.245892\pi\)
−0.442454 + 0.896791i \(0.645892\pi\)
\(74\) 0 0
\(75\) −0.937555 + 2.88550i −0.108260 + 0.333189i
\(76\) 0 0
\(77\) −0.720387 + 2.78014i −0.0820958 + 0.316826i
\(78\) 0 0
\(79\) −1.82769 + 5.62505i −0.205631 + 0.632867i 0.794056 + 0.607845i \(0.207966\pi\)
−0.999687 + 0.0250225i \(0.992034\pi\)
\(80\) 0 0
\(81\) −8.80862 6.39983i −0.978735 0.711093i
\(82\) 0 0
\(83\) −2.19347 6.75080i −0.240764 0.740996i −0.996304 0.0858936i \(-0.972625\pi\)
0.755540 0.655102i \(-0.227375\pi\)
\(84\) 0 0
\(85\) −0.444984 + 0.323300i −0.0482653 + 0.0350668i
\(86\) 0 0
\(87\) 28.4046 3.04529
\(88\) 0 0
\(89\) −16.6470 −1.76458 −0.882291 0.470704i \(-0.844000\pi\)
−0.882291 + 0.470704i \(0.844000\pi\)
\(90\) 0 0
\(91\) 3.10700 2.25737i 0.325702 0.236637i
\(92\) 0 0
\(93\) 4.98393 + 15.3390i 0.516809 + 1.59058i
\(94\) 0 0
\(95\) −0.209505 0.152214i −0.0214948 0.0156169i
\(96\) 0 0
\(97\) −2.42082 + 7.45053i −0.245797 + 0.756486i 0.749707 + 0.661770i \(0.230194\pi\)
−0.995504 + 0.0947164i \(0.969806\pi\)
\(98\) 0 0
\(99\) −20.5422 + 1.24672i −2.06457 + 0.125300i
\(100\) 0 0
\(101\) −5.62378 + 17.3082i −0.559587 + 1.72223i 0.123923 + 0.992292i \(0.460452\pi\)
−0.683511 + 0.729941i \(0.739548\pi\)
\(102\) 0 0
\(103\) −9.54845 6.93735i −0.940837 0.683558i 0.00778529 0.999970i \(-0.497522\pi\)
−0.948622 + 0.316412i \(0.897522\pi\)
\(104\) 0 0
\(105\) 0.811855 + 2.49863i 0.0792289 + 0.243841i
\(106\) 0 0
\(107\) 0.421469 0.306215i 0.0407450 0.0296030i −0.567226 0.823562i \(-0.691984\pi\)
0.607971 + 0.793959i \(0.291984\pi\)
\(108\) 0 0
\(109\) −6.22545 −0.596290 −0.298145 0.954521i \(-0.596368\pi\)
−0.298145 + 0.954521i \(0.596368\pi\)
\(110\) 0 0
\(111\) −19.3793 −1.83940
\(112\) 0 0
\(113\) 7.98968 5.80484i 0.751605 0.546073i −0.144719 0.989473i \(-0.546228\pi\)
0.896324 + 0.443400i \(0.146228\pi\)
\(114\) 0 0
\(115\) −2.44958 7.53904i −0.228425 0.703019i
\(116\) 0 0
\(117\) 22.2643 + 16.1760i 2.05834 + 1.49547i
\(118\) 0 0
\(119\) −0.147181 + 0.452976i −0.0134920 + 0.0415242i
\(120\) 0 0
\(121\) −8.05194 + 7.49442i −0.731995 + 0.681310i
\(122\) 0 0
\(123\) −8.19628 + 25.2256i −0.739034 + 2.27451i
\(124\) 0 0
\(125\) 0.809017 + 0.587785i 0.0723607 + 0.0525731i
\(126\) 0 0
\(127\) 1.14934 + 3.53730i 0.101987 + 0.313885i 0.989012 0.147838i \(-0.0472315\pi\)
−0.887024 + 0.461723i \(0.847231\pi\)
\(128\) 0 0
\(129\) 15.6753 11.3888i 1.38013 1.00273i
\(130\) 0 0
\(131\) 20.4259 1.78462 0.892308 0.451426i \(-0.149085\pi\)
0.892308 + 0.451426i \(0.149085\pi\)
\(132\) 0 0
\(133\) −0.224243 −0.0194443
\(134\) 0 0
\(135\) −7.86710 + 5.71579i −0.677092 + 0.491936i
\(136\) 0 0
\(137\) 4.14751 + 12.7647i 0.354346 + 1.09056i 0.956388 + 0.292100i \(0.0943540\pi\)
−0.602042 + 0.798465i \(0.705646\pi\)
\(138\) 0 0
\(139\) −11.3955 8.27933i −0.966555 0.702243i −0.0118911 0.999929i \(-0.503785\pi\)
−0.954664 + 0.297686i \(0.903785\pi\)
\(140\) 0 0
\(141\) −0.771803 + 2.37537i −0.0649975 + 0.200042i
\(142\) 0 0
\(143\) 14.6825 0.891094i 1.22781 0.0745170i
\(144\) 0 0
\(145\) 2.89305 8.90389i 0.240255 0.739428i
\(146\) 0 0
\(147\) −15.3414 11.1462i −1.26533 0.919319i
\(148\) 0 0
\(149\) 0.553674 + 1.70403i 0.0453588 + 0.139600i 0.971171 0.238384i \(-0.0766176\pi\)
−0.925812 + 0.377984i \(0.876618\pi\)
\(150\) 0 0
\(151\) −15.4418 + 11.2191i −1.25663 + 0.912996i −0.998587 0.0531366i \(-0.983078\pi\)
−0.258045 + 0.966133i \(0.583078\pi\)
\(152\) 0 0
\(153\) −3.41300 −0.275925
\(154\) 0 0
\(155\) 5.31588 0.426982
\(156\) 0 0
\(157\) −7.91828 + 5.75296i −0.631947 + 0.459137i −0.857074 0.515193i \(-0.827720\pi\)
0.225127 + 0.974329i \(0.427720\pi\)
\(158\) 0 0
\(159\) −2.25116 6.92837i −0.178529 0.549455i
\(160\) 0 0
\(161\) −5.55327 4.03469i −0.437659 0.317978i
\(162\) 0 0
\(163\) 0.122631 0.377420i 0.00960522 0.0295618i −0.946139 0.323761i \(-0.895053\pi\)
0.955744 + 0.294199i \(0.0950528\pi\)
\(164\) 0 0
\(165\) −2.52406 + 9.74091i −0.196498 + 0.758329i
\(166\) 0 0
\(167\) 4.29967 13.2330i 0.332718 1.02400i −0.635117 0.772416i \(-0.719048\pi\)
0.967835 0.251585i \(-0.0809518\pi\)
\(168\) 0 0
\(169\) −5.39618 3.92055i −0.415091 0.301581i
\(170\) 0 0
\(171\) −0.496556 1.52824i −0.0379726 0.116868i
\(172\) 0 0
\(173\) −4.31522 + 3.13519i −0.328080 + 0.238364i −0.739615 0.673030i \(-0.764993\pi\)
0.411535 + 0.911394i \(0.364993\pi\)
\(174\) 0 0
\(175\) 0.865927 0.0654580
\(176\) 0 0
\(177\) 41.0039 3.08204
\(178\) 0 0
\(179\) −6.17667 + 4.48761i −0.461666 + 0.335420i −0.794184 0.607677i \(-0.792102\pi\)
0.332519 + 0.943097i \(0.392102\pi\)
\(180\) 0 0
\(181\) 0.403121 + 1.24068i 0.0299638 + 0.0922189i 0.964920 0.262544i \(-0.0845615\pi\)
−0.934956 + 0.354763i \(0.884562\pi\)
\(182\) 0 0
\(183\) −16.2187 11.7836i −1.19892 0.871068i
\(184\) 0 0
\(185\) −1.97381 + 6.07477i −0.145118 + 0.446626i
\(186\) 0 0
\(187\) −1.40818 + 1.15970i −0.102976 + 0.0848058i
\(188\) 0 0
\(189\) −2.60208 + 8.00839i −0.189274 + 0.582524i
\(190\) 0 0
\(191\) −20.2928 14.7436i −1.46834 1.06681i −0.981091 0.193547i \(-0.938001\pi\)
−0.487248 0.873264i \(-0.661999\pi\)
\(192\) 0 0
\(193\) −6.16676 18.9793i −0.443893 1.36616i −0.883693 0.468067i \(-0.844950\pi\)
0.439799 0.898096i \(-0.355050\pi\)
\(194\) 0 0
\(195\) 10.8862 7.90926i 0.779574 0.566394i
\(196\) 0 0
\(197\) −1.31088 −0.0933960 −0.0466980 0.998909i \(-0.514870\pi\)
−0.0466980 + 0.998909i \(0.514870\pi\)
\(198\) 0 0
\(199\) −4.99115 −0.353813 −0.176907 0.984228i \(-0.556609\pi\)
−0.176907 + 0.984228i \(0.556609\pi\)
\(200\) 0 0
\(201\) 17.6169 12.7994i 1.24260 0.902803i
\(202\) 0 0
\(203\) −2.50517 7.71012i −0.175829 0.541145i
\(204\) 0 0
\(205\) 7.07258 + 5.13853i 0.493970 + 0.358890i
\(206\) 0 0
\(207\) 15.1999 46.7805i 1.05647 3.25147i
\(208\) 0 0
\(209\) −0.724156 0.461817i −0.0500909 0.0319445i
\(210\) 0 0
\(211\) −1.98113 + 6.09730i −0.136387 + 0.419755i −0.995803 0.0915210i \(-0.970827\pi\)
0.859416 + 0.511276i \(0.170827\pi\)
\(212\) 0 0
\(213\) 6.87054 + 4.99174i 0.470762 + 0.342028i
\(214\) 0 0
\(215\) −1.97345 6.07366i −0.134588 0.414220i
\(216\) 0 0
\(217\) 3.72404 2.70567i 0.252804 0.183673i
\(218\) 0 0
\(219\) 8.77047 0.592654
\(220\) 0 0
\(221\) 2.43944 0.164094
\(222\) 0 0
\(223\) 12.0126 8.72766i 0.804423 0.584447i −0.107786 0.994174i \(-0.534376\pi\)
0.912208 + 0.409727i \(0.134376\pi\)
\(224\) 0 0
\(225\) 1.91748 + 5.90141i 0.127832 + 0.393427i
\(226\) 0 0
\(227\) 18.9925 + 13.7989i 1.26058 + 0.915865i 0.998786 0.0492591i \(-0.0156860\pi\)
0.261794 + 0.965124i \(0.415686\pi\)
\(228\) 0 0
\(229\) 6.06694 18.6721i 0.400914 1.23389i −0.523344 0.852121i \(-0.675316\pi\)
0.924259 0.381766i \(-0.124684\pi\)
\(230\) 0 0
\(231\) 3.18969 + 8.10869i 0.209866 + 0.533512i
\(232\) 0 0
\(233\) 6.48770 19.9671i 0.425024 1.30809i −0.477948 0.878388i \(-0.658619\pi\)
0.902972 0.429700i \(-0.141381\pi\)
\(234\) 0 0
\(235\) 0.665989 + 0.483870i 0.0434444 + 0.0315642i
\(236\) 0 0
\(237\) 5.54519 + 17.0664i 0.360199 + 1.10858i
\(238\) 0 0
\(239\) 15.2527 11.0817i 0.986614 0.716817i 0.0274367 0.999624i \(-0.491266\pi\)
0.959177 + 0.282807i \(0.0912655\pi\)
\(240\) 0 0
\(241\) −13.8318 −0.890987 −0.445494 0.895285i \(-0.646972\pi\)
−0.445494 + 0.895285i \(0.646972\pi\)
\(242\) 0 0
\(243\) −3.86143 −0.247711
\(244\) 0 0
\(245\) −5.05649 + 3.67376i −0.323048 + 0.234708i
\(246\) 0 0
\(247\) 0.354913 + 1.09231i 0.0225826 + 0.0695020i
\(248\) 0 0
\(249\) −17.4229 12.6585i −1.10413 0.802199i
\(250\) 0 0
\(251\) 4.71929 14.5245i 0.297879 0.916777i −0.684360 0.729144i \(-0.739919\pi\)
0.982239 0.187633i \(-0.0600815\pi\)
\(252\) 0 0
\(253\) −9.62416 24.4661i −0.605066 1.53817i
\(254\) 0 0
\(255\) −0.515684 + 1.58711i −0.0322934 + 0.0993889i
\(256\) 0 0
\(257\) −23.2900 16.9212i −1.45279 1.05551i −0.985170 0.171584i \(-0.945112\pi\)
−0.467620 0.883930i \(-0.654888\pi\)
\(258\) 0 0
\(259\) 1.70918 + 5.26031i 0.106203 + 0.326860i
\(260\) 0 0
\(261\) 46.9981 34.1461i 2.90911 2.11359i
\(262\) 0 0
\(263\) −9.23218 −0.569281 −0.284640 0.958634i \(-0.591874\pi\)
−0.284640 + 0.958634i \(0.591874\pi\)
\(264\) 0 0
\(265\) −2.40110 −0.147498
\(266\) 0 0
\(267\) −40.8610 + 29.6873i −2.50065 + 1.81683i
\(268\) 0 0
\(269\) 8.29165 + 25.5191i 0.505551 + 1.55593i 0.799842 + 0.600210i \(0.204916\pi\)
−0.294291 + 0.955716i \(0.595084\pi\)
\(270\) 0 0
\(271\) −19.0614 13.8489i −1.15790 0.841261i −0.168386 0.985721i \(-0.553856\pi\)
−0.989511 + 0.144460i \(0.953856\pi\)
\(272\) 0 0
\(273\) 3.60065 11.0817i 0.217921 0.670693i
\(274\) 0 0
\(275\) 2.79638 + 1.78334i 0.168628 + 0.107539i
\(276\) 0 0
\(277\) 3.08782 9.50334i 0.185529 0.571001i −0.814428 0.580265i \(-0.802949\pi\)
0.999957 + 0.00926435i \(0.00294898\pi\)
\(278\) 0 0
\(279\) 26.6859 + 19.3884i 1.59764 + 1.16076i
\(280\) 0 0
\(281\) 1.26401 + 3.89022i 0.0754044 + 0.232071i 0.981654 0.190673i \(-0.0610670\pi\)
−0.906249 + 0.422744i \(0.861067\pi\)
\(282\) 0 0
\(283\) −3.86118 + 2.80531i −0.229523 + 0.166758i −0.696603 0.717457i \(-0.745306\pi\)
0.467080 + 0.884215i \(0.345306\pi\)
\(284\) 0 0
\(285\) −0.785690 −0.0465402
\(286\) 0 0
\(287\) 7.57010 0.446849
\(288\) 0 0
\(289\) 13.5085 9.81452i 0.794620 0.577325i
\(290\) 0 0
\(291\) 7.34476 + 22.6048i 0.430557 + 1.32512i
\(292\) 0 0
\(293\) −0.116508 0.0846481i −0.00680648 0.00494520i 0.584377 0.811482i \(-0.301339\pi\)
−0.591183 + 0.806537i \(0.701339\pi\)
\(294\) 0 0
\(295\) 4.17631 12.8534i 0.243154 0.748352i
\(296\) 0 0
\(297\) −24.8959 + 20.5029i −1.44461 + 1.18970i
\(298\) 0 0
\(299\) −10.8641 + 33.4363i −0.628288 + 1.93367i
\(300\) 0 0
\(301\) −4.47387 3.25046i −0.257869 0.187353i
\(302\) 0 0
\(303\) 17.0625 + 52.5130i 0.980216 + 3.01679i
\(304\) 0 0
\(305\) −5.34567 + 3.88385i −0.306092 + 0.222389i
\(306\) 0 0
\(307\) −25.7576 −1.47006 −0.735031 0.678034i \(-0.762832\pi\)
−0.735031 + 0.678034i \(0.762832\pi\)
\(308\) 0 0
\(309\) −35.8088 −2.03709
\(310\) 0 0
\(311\) −13.2610 + 9.63467i −0.751961 + 0.546332i −0.896434 0.443177i \(-0.853851\pi\)
0.144473 + 0.989509i \(0.453851\pi\)
\(312\) 0 0
\(313\) 8.19629 + 25.2256i 0.463282 + 1.42583i 0.861131 + 0.508384i \(0.169757\pi\)
−0.397849 + 0.917451i \(0.630243\pi\)
\(314\) 0 0
\(315\) 4.34699 + 3.15827i 0.244925 + 0.177948i
\(316\) 0 0
\(317\) 5.16546 15.8977i 0.290121 0.892902i −0.694695 0.719304i \(-0.744461\pi\)
0.984817 0.173598i \(-0.0555392\pi\)
\(318\) 0 0
\(319\) 7.78858 30.0579i 0.436077 1.68292i
\(320\) 0 0
\(321\) 0.488433 1.50324i 0.0272617 0.0839028i
\(322\) 0 0
\(323\) −0.115234 0.0837225i −0.00641180 0.00465845i
\(324\) 0 0
\(325\) −1.37052 4.21802i −0.0760227 0.233974i
\(326\) 0 0
\(327\) −15.2807 + 11.1021i −0.845025 + 0.613946i
\(328\) 0 0
\(329\) 0.712839 0.0393001
\(330\) 0 0
\(331\) 4.39584 0.241617 0.120809 0.992676i \(-0.461451\pi\)
0.120809 + 0.992676i \(0.461451\pi\)
\(332\) 0 0
\(333\) −32.0649 + 23.2965i −1.75715 + 1.27664i
\(334\) 0 0
\(335\) −2.21789 6.82597i −0.121176 0.372942i
\(336\) 0 0
\(337\) −10.1965 7.40817i −0.555437 0.403549i 0.274349 0.961630i \(-0.411538\pi\)
−0.829786 + 0.558082i \(0.811538\pi\)
\(338\) 0 0
\(339\) 9.25909 28.4966i 0.502885 1.54772i
\(340\) 0 0
\(341\) 17.5984 1.06806i 0.953006 0.0578387i
\(342\) 0 0
\(343\) −3.54556 + 10.9121i −0.191442 + 0.589199i
\(344\) 0 0
\(345\) −19.4573 14.1365i −1.04754 0.761085i
\(346\) 0 0
\(347\) −7.41108 22.8089i −0.397847 1.22445i −0.926722 0.375748i \(-0.877386\pi\)
0.528874 0.848700i \(-0.322614\pi\)
\(348\) 0 0
\(349\) −19.0848 + 13.8659i −1.02159 + 0.742227i −0.966608 0.256261i \(-0.917509\pi\)
−0.0549798 + 0.998487i \(0.517509\pi\)
\(350\) 0 0
\(351\) 43.1281 2.30201
\(352\) 0 0
\(353\) 6.46751 0.344231 0.172115 0.985077i \(-0.444940\pi\)
0.172115 + 0.985077i \(0.444940\pi\)
\(354\) 0 0
\(355\) 2.26452 1.64527i 0.120188 0.0873219i
\(356\) 0 0
\(357\) 0.446545 + 1.37432i 0.0236337 + 0.0727370i
\(358\) 0 0
\(359\) −9.56591 6.95004i −0.504869 0.366809i 0.306005 0.952030i \(-0.401008\pi\)
−0.810874 + 0.585221i \(0.801008\pi\)
\(360\) 0 0
\(361\) −5.85060 + 18.0063i −0.307926 + 0.947700i
\(362\) 0 0
\(363\) −6.39884 + 32.7547i −0.335852 + 1.71918i
\(364\) 0 0
\(365\) 0.893287 2.74925i 0.0467568 0.143903i
\(366\) 0 0
\(367\) −6.78533 4.92983i −0.354191 0.257335i 0.396434 0.918063i \(-0.370248\pi\)
−0.750625 + 0.660728i \(0.770248\pi\)
\(368\) 0 0
\(369\) 16.7630 + 51.5912i 0.872647 + 2.68573i
\(370\) 0 0
\(371\) −1.68209 + 1.22211i −0.0873298 + 0.0634488i
\(372\) 0 0
\(373\) 34.6263 1.79288 0.896441 0.443163i \(-0.146144\pi\)
0.896441 + 0.443163i \(0.146144\pi\)
\(374\) 0 0
\(375\) 3.03399 0.156675
\(376\) 0 0
\(377\) −33.5918 + 24.4059i −1.73007 + 1.25697i
\(378\) 0 0
\(379\) 4.96669 + 15.2859i 0.255122 + 0.785185i 0.993806 + 0.111132i \(0.0354478\pi\)
−0.738684 + 0.674052i \(0.764552\pi\)
\(380\) 0 0
\(381\) 9.12931 + 6.63283i 0.467709 + 0.339810i
\(382\) 0 0
\(383\) 2.60059 8.00380i 0.132884 0.408975i −0.862371 0.506277i \(-0.831021\pi\)
0.995255 + 0.0973020i \(0.0310213\pi\)
\(384\) 0 0
\(385\) 2.86668 0.173981i 0.146100 0.00886690i
\(386\) 0 0
\(387\) 12.2455 37.6877i 0.622473 1.91577i
\(388\) 0 0
\(389\) −15.0410 10.9279i −0.762608 0.554067i 0.137101 0.990557i \(-0.456222\pi\)
−0.899709 + 0.436490i \(0.856222\pi\)
\(390\) 0 0
\(391\) −1.34735 4.14670i −0.0681382 0.209708i
\(392\) 0 0
\(393\) 50.1364 36.4262i 2.52904 1.83746i
\(394\) 0 0
\(395\) 5.91452 0.297592
\(396\) 0 0
\(397\) −11.9897 −0.601743 −0.300872 0.953665i \(-0.597278\pi\)
−0.300872 + 0.953665i \(0.597278\pi\)
\(398\) 0 0
\(399\) −0.550415 + 0.399900i −0.0275552 + 0.0200200i
\(400\) 0 0
\(401\) 5.80924 + 17.8790i 0.290099 + 0.892834i 0.984824 + 0.173558i \(0.0555264\pi\)
−0.694724 + 0.719276i \(0.744474\pi\)
\(402\) 0 0
\(403\) −19.0737 13.8579i −0.950129 0.690309i
\(404\) 0 0
\(405\) −3.36459 + 10.3551i −0.167188 + 0.514551i
\(406\) 0 0
\(407\) −5.31383 + 20.5073i −0.263397 + 1.01651i
\(408\) 0 0
\(409\) −2.93699 + 9.03912i −0.145225 + 0.446956i −0.997040 0.0768869i \(-0.975502\pi\)
0.851815 + 0.523842i \(0.175502\pi\)
\(410\) 0 0
\(411\) 32.9441 + 23.9353i 1.62501 + 1.18064i
\(412\) 0 0
\(413\) −3.61638 11.1301i −0.177951 0.547676i
\(414\) 0 0
\(415\) −5.74257 + 4.17222i −0.281892 + 0.204806i
\(416\) 0 0
\(417\) −42.7357 −2.09278
\(418\) 0 0
\(419\) 32.9152 1.60801 0.804007 0.594620i \(-0.202698\pi\)
0.804007 + 0.594620i \(0.202698\pi\)
\(420\) 0 0
\(421\) 0.837812 0.608706i 0.0408325 0.0296665i −0.567182 0.823593i \(-0.691966\pi\)
0.608014 + 0.793926i \(0.291966\pi\)
\(422\) 0 0
\(423\) 1.57849 + 4.85809i 0.0767487 + 0.236208i
\(424\) 0 0
\(425\) 0.444984 + 0.323300i 0.0215849 + 0.0156824i
\(426\) 0 0
\(427\) −1.76811 + 5.44167i −0.0855646 + 0.263341i
\(428\) 0 0
\(429\) 34.4499 28.3711i 1.66326 1.36977i
\(430\) 0 0
\(431\) 7.96068 24.5005i 0.383452 1.18015i −0.554144 0.832421i \(-0.686954\pi\)
0.937597 0.347725i \(-0.113046\pi\)
\(432\) 0 0
\(433\) 14.9188 + 10.8391i 0.716951 + 0.520896i 0.885409 0.464813i \(-0.153879\pi\)
−0.168457 + 0.985709i \(0.553879\pi\)
\(434\) 0 0
\(435\) −8.77749 27.0143i −0.420848 1.29524i
\(436\) 0 0
\(437\) 1.66075 1.20660i 0.0794444 0.0577197i
\(438\) 0 0
\(439\) −4.44724 −0.212255 −0.106128 0.994353i \(-0.533845\pi\)
−0.106128 + 0.994353i \(0.533845\pi\)
\(440\) 0 0
\(441\) −38.7830 −1.84681
\(442\) 0 0
\(443\) 20.2851 14.7380i 0.963775 0.700224i 0.00975065 0.999952i \(-0.496896\pi\)
0.954024 + 0.299729i \(0.0968962\pi\)
\(444\) 0 0
\(445\) 5.14422 + 15.8323i 0.243859 + 0.750522i
\(446\) 0 0
\(447\) 4.39789 + 3.19525i 0.208013 + 0.151130i
\(448\) 0 0
\(449\) −5.77075 + 17.7605i −0.272338 + 0.838172i 0.717573 + 0.696483i \(0.245253\pi\)
−0.989911 + 0.141688i \(0.954747\pi\)
\(450\) 0 0
\(451\) 24.4464 + 15.5903i 1.15114 + 0.734116i
\(452\) 0 0
\(453\) −17.8952 + 55.0757i −0.840788 + 2.58768i
\(454\) 0 0
\(455\) −3.10700 2.25737i −0.145659 0.105827i
\(456\) 0 0
\(457\) −4.49832 13.8444i −0.210422 0.647614i −0.999447 0.0332521i \(-0.989414\pi\)
0.789025 0.614362i \(-0.210586\pi\)
\(458\) 0 0
\(459\) −4.32715 + 3.14386i −0.201974 + 0.146743i
\(460\) 0 0
\(461\) 20.6525 0.961884 0.480942 0.876752i \(-0.340295\pi\)
0.480942 + 0.876752i \(0.340295\pi\)
\(462\) 0 0
\(463\) −21.8605 −1.01594 −0.507971 0.861374i \(-0.669604\pi\)
−0.507971 + 0.861374i \(0.669604\pi\)
\(464\) 0 0
\(465\) 13.0481 9.48000i 0.605091 0.439624i
\(466\) 0 0
\(467\) 6.27254 + 19.3049i 0.290258 + 0.893324i 0.984773 + 0.173845i \(0.0556192\pi\)
−0.694515 + 0.719479i \(0.744381\pi\)
\(468\) 0 0
\(469\) −5.02802 3.65307i −0.232172 0.168683i
\(470\) 0 0
\(471\) −9.17635 + 28.2419i −0.422824 + 1.30132i
\(472\) 0 0
\(473\) −7.75349 19.7105i −0.356506 0.906292i
\(474\) 0 0
\(475\) −0.0800238 + 0.246288i −0.00367174 + 0.0113005i
\(476\) 0 0
\(477\) −12.0536 8.75746i −0.551897 0.400977i
\(478\) 0 0
\(479\) −3.05665 9.40740i −0.139662 0.429835i 0.856624 0.515941i \(-0.172558\pi\)
−0.996286 + 0.0861060i \(0.972558\pi\)
\(480\) 0 0
\(481\) 22.9184 16.6512i 1.04499 0.759228i
\(482\) 0 0
\(483\) −20.8260 −0.947615
\(484\) 0 0
\(485\) 7.83395 0.355721
\(486\) 0 0
\(487\) −4.80458 + 3.49073i −0.217716 + 0.158180i −0.691298 0.722570i \(-0.742961\pi\)
0.473582 + 0.880750i \(0.342961\pi\)
\(488\) 0 0
\(489\) −0.372062 1.14509i −0.0168252 0.0517828i
\(490\) 0 0
\(491\) −22.1143 16.0670i −0.998005 0.725093i −0.0363455 0.999339i \(-0.511572\pi\)
−0.961660 + 0.274246i \(0.911572\pi\)
\(492\) 0 0
\(493\) 1.59127 4.89741i 0.0716670 0.220568i
\(494\) 0 0
\(495\) 7.53360 + 19.1516i 0.338610 + 0.860798i
\(496\) 0 0
\(497\) 0.749001 2.30519i 0.0335973 0.103402i
\(498\) 0 0
\(499\) 9.84395 + 7.15205i 0.440676 + 0.320170i 0.785903 0.618349i \(-0.212198\pi\)
−0.345228 + 0.938519i \(0.612198\pi\)
\(500\) 0 0
\(501\) −13.0452 40.1488i −0.582814 1.79372i
\(502\) 0 0
\(503\) 7.67903 5.57914i 0.342391 0.248762i −0.403279 0.915077i \(-0.632130\pi\)
0.745670 + 0.666315i \(0.232130\pi\)
\(504\) 0 0
\(505\) 18.1989 0.809842
\(506\) 0 0
\(507\) −20.2369 −0.898750
\(508\) 0 0
\(509\) 0.774792 0.562919i 0.0343421 0.0249510i −0.570482 0.821310i \(-0.693244\pi\)
0.604824 + 0.796359i \(0.293244\pi\)
\(510\) 0 0
\(511\) −0.773521 2.38065i −0.0342186 0.105314i
\(512\) 0 0
\(513\) −2.03728 1.48017i −0.0899483 0.0653512i
\(514\) 0 0
\(515\) −3.64718 + 11.2249i −0.160714 + 0.494627i
\(516\) 0 0
\(517\) 2.30200 + 1.46806i 0.101242 + 0.0645651i
\(518\) 0 0
\(519\) −5.00083 + 15.3910i −0.219512 + 0.675589i
\(520\) 0 0
\(521\) 14.7577 + 10.7221i 0.646547 + 0.469744i 0.862093 0.506750i \(-0.169153\pi\)
−0.215546 + 0.976494i \(0.569153\pi\)
\(522\) 0 0
\(523\) 8.75364 + 26.9409i 0.382770 + 1.17804i 0.938085 + 0.346405i \(0.112598\pi\)
−0.555315 + 0.831640i \(0.687402\pi\)
\(524\) 0 0
\(525\) 2.12546 1.54424i 0.0927628 0.0673961i
\(526\) 0 0
\(527\) 2.92390 0.127367
\(528\) 0 0
\(529\) 39.8375 1.73207
\(530\) 0 0
\(531\) 67.8450 49.2923i 2.94422 2.13910i
\(532\) 0 0
\(533\) −11.9813 36.8747i −0.518969 1.59722i
\(534\) 0 0
\(535\) −0.421469 0.306215i −0.0182217 0.0132388i
\(536\) 0 0
\(537\) −7.15803 + 22.0301i −0.308892 + 0.950671i
\(538\) 0 0
\(539\) −16.0016 + 13.1780i −0.689236 + 0.567618i
\(540\) 0 0
\(541\) −9.81343 + 30.2026i −0.421912 + 1.29851i 0.484008 + 0.875064i \(0.339181\pi\)
−0.905920 + 0.423448i \(0.860819\pi\)
\(542\) 0 0
\(543\) 3.20203 + 2.32641i 0.137412 + 0.0998358i
\(544\) 0 0
\(545\) 1.92377 + 5.92076i 0.0824053 + 0.253617i
\(546\) 0 0
\(547\) −2.89975 + 2.10679i −0.123984 + 0.0900798i −0.648049 0.761598i \(-0.724415\pi\)
0.524065 + 0.851678i \(0.324415\pi\)
\(548\) 0 0
\(549\) −41.0009 −1.74988
\(550\) 0 0
\(551\) 2.42443 0.103284
\(552\) 0 0
\(553\) 4.14342 3.01037i 0.176196 0.128014i
\(554\) 0 0
\(555\) 5.98853 + 18.4308i 0.254199 + 0.782344i
\(556\) 0 0
\(557\) 13.4746 + 9.78990i 0.570939 + 0.414811i 0.835446 0.549572i \(-0.185209\pi\)
−0.264507 + 0.964384i \(0.585209\pi\)
\(558\) 0 0
\(559\) −8.75244 + 26.9372i −0.370189 + 1.13932i
\(560\) 0 0
\(561\) −1.38831 + 5.35780i −0.0586144 + 0.226207i
\(562\) 0 0
\(563\) 12.3291 37.9450i 0.519609 1.59919i −0.255127 0.966907i \(-0.582117\pi\)
0.774736 0.632285i \(-0.217883\pi\)
\(564\) 0 0
\(565\) −7.98968 5.80484i −0.336128 0.244211i
\(566\) 0 0
\(567\) 2.91349 + 8.96681i 0.122355 + 0.376570i
\(568\) 0 0
\(569\) −36.7814 + 26.7233i −1.54196 + 1.12030i −0.592866 + 0.805301i \(0.702004\pi\)
−0.949093 + 0.314997i \(0.897996\pi\)
\(570\) 0 0
\(571\) −15.8877 −0.664880 −0.332440 0.943124i \(-0.607872\pi\)
−0.332440 + 0.943124i \(0.607872\pi\)
\(572\) 0 0
\(573\) −76.1027 −3.17923
\(574\) 0 0
\(575\) −6.41309 + 4.65938i −0.267444 + 0.194310i
\(576\) 0 0
\(577\) −4.06088 12.4981i −0.169057 0.520302i 0.830256 0.557383i \(-0.188194\pi\)
−0.999312 + 0.0370801i \(0.988194\pi\)
\(578\) 0 0
\(579\) −48.9832 35.5884i −2.03567 1.47900i
\(580\) 0 0
\(581\) −1.89938 + 5.84570i −0.0787997 + 0.242520i
\(582\) 0 0
\(583\) −7.94892 + 0.482427i −0.329211 + 0.0199801i
\(584\) 0 0
\(585\) 8.50422 26.1733i 0.351606 1.08213i
\(586\) 0 0
\(587\) −12.1061 8.79560i −0.499672 0.363033i 0.309220 0.950991i \(-0.399932\pi\)
−0.808892 + 0.587958i \(0.799932\pi\)
\(588\) 0 0
\(589\) 0.425397 + 1.30924i 0.0175282 + 0.0539461i
\(590\) 0 0
\(591\) −3.21761 + 2.33773i −0.132355 + 0.0961615i
\(592\) 0 0
\(593\) −13.7967 −0.566564 −0.283282 0.959037i \(-0.591423\pi\)
−0.283282 + 0.959037i \(0.591423\pi\)
\(594\) 0 0
\(595\) 0.476287 0.0195259
\(596\) 0 0
\(597\) −12.2510 + 8.90090i −0.501401 + 0.364289i
\(598\) 0 0
\(599\) 9.69984 + 29.8530i 0.396325 + 1.21976i 0.927925 + 0.372767i \(0.121591\pi\)
−0.531600 + 0.846995i \(0.678409\pi\)
\(600\) 0 0
\(601\) 2.06813 + 1.50259i 0.0843608 + 0.0612917i 0.629166 0.777271i \(-0.283396\pi\)
−0.544805 + 0.838563i \(0.683396\pi\)
\(602\) 0 0
\(603\) 13.7623 42.3559i 0.560442 1.72486i
\(604\) 0 0
\(605\) 9.61580 + 5.34195i 0.390938 + 0.217181i
\(606\) 0 0
\(607\) −11.6898 + 35.9774i −0.474473 + 1.46028i 0.372194 + 0.928155i \(0.378605\pi\)
−0.846667 + 0.532123i \(0.821395\pi\)
\(608\) 0 0
\(609\) −19.8988 14.4573i −0.806340 0.585841i
\(610\) 0 0
\(611\) −1.12822 3.47231i −0.0456430 0.140475i
\(612\) 0 0
\(613\) −9.43309 + 6.85354i −0.380999 + 0.276812i −0.761757 0.647863i \(-0.775663\pi\)
0.380758 + 0.924675i \(0.375663\pi\)
\(614\) 0 0
\(615\) 26.5237 1.06954
\(616\) 0 0
\(617\) 38.5910 1.55361 0.776807 0.629739i \(-0.216838\pi\)
0.776807 + 0.629739i \(0.216838\pi\)
\(618\) 0 0
\(619\) 6.60453 4.79847i 0.265458 0.192867i −0.447092 0.894488i \(-0.647540\pi\)
0.712550 + 0.701621i \(0.247540\pi\)
\(620\) 0 0
\(621\) −23.8204 73.3117i −0.955880 2.94190i
\(622\) 0 0
\(623\) 11.6621 + 8.47300i 0.467231 + 0.339463i
\(624\) 0 0
\(625\) 0.309017 0.951057i 0.0123607 0.0380423i
\(626\) 0 0
\(627\) −2.60105 + 0.157860i −0.103876 + 0.00630432i
\(628\) 0 0
\(629\) −1.08566 + 3.34131i −0.0432880 + 0.133227i
\(630\) 0 0
\(631\) 10.5830 + 7.68900i 0.421303 + 0.306094i 0.778162 0.628064i \(-0.216152\pi\)
−0.356859 + 0.934158i \(0.616152\pi\)
\(632\) 0 0
\(633\) 6.01074 + 18.4992i 0.238906 + 0.735276i
\(634\) 0 0
\(635\) 3.00901 2.18617i 0.119409 0.0867556i
\(636\) 0 0
\(637\) 27.7201 1.09831
\(638\) 0 0
\(639\) 17.3687 0.687096
\(640\) 0 0
\(641\) −28.8414 + 20.9545i −1.13917 + 0.827654i −0.987003 0.160702i \(-0.948624\pi\)
−0.152164 + 0.988355i \(0.548624\pi\)
\(642\) 0 0
\(643\) −2.36588 7.28142i −0.0933011 0.287151i 0.893506 0.449051i \(-0.148238\pi\)
−0.986807 + 0.161900i \(0.948238\pi\)
\(644\) 0 0
\(645\) −15.6753 11.3888i −0.617215 0.448433i
\(646\) 0 0
\(647\) 2.83754 8.73304i 0.111555 0.343331i −0.879658 0.475607i \(-0.842228\pi\)
0.991213 + 0.132276i \(0.0422285\pi\)
\(648\) 0 0
\(649\) 11.2433 43.3906i 0.441340 1.70323i
\(650\) 0 0
\(651\) 4.31572 13.2824i 0.169146 0.520579i
\(652\) 0 0
\(653\) 4.73503 + 3.44020i 0.185296 + 0.134626i 0.676566 0.736382i \(-0.263467\pi\)
−0.491270 + 0.871007i \(0.663467\pi\)
\(654\) 0 0
\(655\) −6.31194 19.4262i −0.246628 0.759043i
\(656\) 0 0
\(657\) 14.5116 10.5433i 0.566152 0.411333i
\(658\) 0 0
\(659\) −22.6530 −0.882434 −0.441217 0.897400i \(-0.645453\pi\)
−0.441217 + 0.897400i \(0.645453\pi\)
\(660\) 0 0
\(661\) 1.68482 0.0655320 0.0327660 0.999463i \(-0.489568\pi\)
0.0327660 + 0.999463i \(0.489568\pi\)
\(662\) 0 0
\(663\) 5.98772 4.35034i 0.232544 0.168953i
\(664\) 0 0
\(665\) 0.0692948 + 0.213267i 0.00268714 + 0.00827015i
\(666\) 0 0
\(667\) 60.0400 + 43.6216i 2.32476 + 1.68904i
\(668\) 0 0
\(669\) 13.9212 42.8450i 0.538224 1.65648i
\(670\) 0 0
\(671\) −16.9167 + 13.9317i −0.653061 + 0.537826i
\(672\) 0 0
\(673\) −2.93100 + 9.02070i −0.112982 + 0.347722i −0.991521 0.129948i \(-0.958519\pi\)
0.878539 + 0.477671i \(0.158519\pi\)
\(674\) 0 0
\(675\) 7.86710 + 5.71579i 0.302805 + 0.220001i
\(676\) 0 0
\(677\) −1.27300 3.91790i −0.0489255 0.150577i 0.923609 0.383336i \(-0.125225\pi\)
−0.972535 + 0.232759i \(0.925225\pi\)
\(678\) 0 0
\(679\) 5.48807 3.98732i 0.210613 0.153019i
\(680\) 0 0
\(681\) 71.2262 2.72940
\(682\) 0 0
\(683\) 29.0594 1.11193 0.555963 0.831207i \(-0.312349\pi\)
0.555963 + 0.831207i \(0.312349\pi\)
\(684\) 0 0
\(685\) 10.8583 7.88904i 0.414875 0.301425i
\(686\) 0 0
\(687\) −18.4070 56.6510i −0.702272 2.16137i
\(688\) 0 0
\(689\) 8.61530 + 6.25938i 0.328217 + 0.238464i
\(690\) 0 0
\(691\) 6.01448 18.5107i 0.228802 0.704179i −0.769082 0.639150i \(-0.779286\pi\)
0.997883 0.0650287i \(-0.0207139\pi\)
\(692\) 0 0
\(693\) 15.0254 + 9.58217i 0.570768 + 0.363997i
\(694\) 0 0
\(695\) −4.35270 + 13.3962i −0.165107 + 0.508148i
\(696\) 0 0
\(697\) 3.89013 + 2.82635i 0.147349 + 0.107056i
\(698\) 0 0
\(699\) −19.6836 60.5800i −0.744504 2.29135i
\(700\) 0 0
\(701\) 12.2452 8.89666i 0.462495 0.336022i −0.332014 0.943274i \(-0.607728\pi\)
0.794509 + 0.607252i \(0.207728\pi\)
\(702\) 0 0
\(703\) −1.65409 −0.0623853
\(704\) 0 0
\(705\) 2.49761 0.0940653
\(706\) 0 0
\(707\) 12.7493 9.26288i 0.479486 0.348367i
\(708\) 0 0
\(709\) −8.35512 25.7144i −0.313783 0.965725i −0.976252 0.216636i \(-0.930492\pi\)
0.662469 0.749089i \(-0.269508\pi\)
\(710\) 0 0
\(711\) 29.6911 + 21.5719i 1.11350 + 0.809009i
\(712\) 0 0
\(713\) −13.0217 + 40.0766i −0.487666 + 1.50088i
\(714\) 0 0
\(715\) −5.38463 13.6885i −0.201374 0.511923i
\(716\) 0 0
\(717\) 17.6760 54.4013i 0.660124 2.03165i
\(718\) 0 0
\(719\) −3.03354 2.20400i −0.113132 0.0821953i 0.529781 0.848135i \(-0.322274\pi\)
−0.642913 + 0.765939i \(0.722274\pi\)
\(720\) 0 0
\(721\) 3.15820 + 9.71993i 0.117617 + 0.361989i
\(722\) 0 0
\(723\) −33.9510 + 24.6668i −1.26265 + 0.917369i
\(724\) 0 0
\(725\) −9.36210 −0.347700
\(726\) 0 0
\(727\) 45.5685 1.69004 0.845022 0.534732i \(-0.179587\pi\)
0.845022 + 0.534732i \(0.179587\pi\)
\(728\) 0 0
\(729\) 16.9478 12.3133i 0.627695 0.456047i
\(730\) 0 0
\(731\) −1.08546 3.34070i −0.0401472 0.123560i
\(732\) 0 0
\(733\) 30.0787 + 21.8535i 1.11098 + 0.807176i 0.982818 0.184577i \(-0.0590915\pi\)
0.128164 + 0.991753i \(0.459091\pi\)
\(734\) 0 0
\(735\) −5.85988 + 18.0349i −0.216145 + 0.665226i
\(736\) 0 0
\(737\) −8.71387 22.1520i −0.320980 0.815979i
\(738\) 0 0
\(739\) −1.19398 + 3.67469i −0.0439213 + 0.135176i −0.970613 0.240648i \(-0.922640\pi\)
0.926691 + 0.375824i \(0.122640\pi\)
\(740\) 0 0
\(741\) 2.81911 + 2.04820i 0.103562 + 0.0752425i
\(742\) 0 0
\(743\) −2.39028 7.35652i −0.0876908 0.269885i 0.897589 0.440833i \(-0.145317\pi\)
−0.985280 + 0.170948i \(0.945317\pi\)
\(744\) 0 0
\(745\) 1.44954 1.05315i 0.0531070 0.0385845i
\(746\) 0 0
\(747\) −44.0451 −1.61153
\(748\) 0 0
\(749\) −0.451118 −0.0164835
\(750\) 0 0
\(751\) 15.9688 11.6020i 0.582711 0.423364i −0.256989 0.966414i \(-0.582731\pi\)
0.839700 + 0.543050i \(0.182731\pi\)
\(752\) 0 0
\(753\) −14.3183 44.0671i −0.521787 1.60590i
\(754\) 0 0
\(755\) 15.4418 + 11.2191i 0.561983 + 0.408304i
\(756\) 0 0
\(757\) 3.27026 10.0648i 0.118859 0.365812i −0.873873 0.486154i \(-0.838399\pi\)
0.992732 + 0.120343i \(0.0383993\pi\)
\(758\) 0 0
\(759\) −67.2542 42.8901i −2.44117 1.55681i
\(760\) 0 0
\(761\) 15.6578 48.1899i 0.567596 1.74688i −0.0925129 0.995711i \(-0.529490\pi\)
0.660109 0.751170i \(-0.270510\pi\)
\(762\) 0 0
\(763\) 4.36124 + 3.16863i 0.157888 + 0.114712i
\(764\) 0 0
\(765\) 1.05468 + 3.24596i 0.0381319 + 0.117358i
\(766\) 0 0
\(767\) −48.4921 + 35.2316i −1.75095 + 1.27214i
\(768\) 0 0
\(769\) −16.2588 −0.586306 −0.293153 0.956066i \(-0.594705\pi\)
−0.293153 + 0.956066i \(0.594705\pi\)
\(770\) 0 0
\(771\) −87.3426 −3.14557
\(772\) 0 0
\(773\) −13.3405 + 9.69243i −0.479824 + 0.348612i −0.801257 0.598320i \(-0.795835\pi\)
0.321434 + 0.946932i \(0.395835\pi\)
\(774\) 0 0
\(775\) −1.64270 5.05570i −0.0590074 0.181606i
\(776\) 0 0
\(777\) 13.5762 + 9.86366i 0.487042 + 0.353857i
\(778\) 0 0
\(779\) −0.699582 + 2.15309i −0.0250651 + 0.0771426i
\(780\) 0 0
\(781\) 7.16621 5.90171i 0.256427 0.211180i
\(782\) 0 0
\(783\) 28.1328 86.5839i 1.00538 3.09426i
\(784\) 0 0
\(785\) 7.91828 + 5.75296i 0.282615 + 0.205332i
\(786\) 0 0
\(787\) 7.94742 + 24.4597i 0.283295 + 0.871892i 0.986905 + 0.161306i \(0.0515705\pi\)
−0.703610 + 0.710587i \(0.748430\pi\)
\(788\) 0 0
\(789\) −22.6609 + 16.4641i −0.806748 + 0.586137i
\(790\) 0 0
\(791\) −8.55171 −0.304064
\(792\) 0 0
\(793\) 29.3054 1.04066
\(794\) 0 0
\(795\) −5.89362 + 4.28197i −0.209025 + 0.151866i
\(796\) 0 0
\(797\) 1.72973 + 5.32356i 0.0612701 + 0.188570i 0.977006 0.213210i \(-0.0683917\pi\)
−0.915736 + 0.401780i \(0.868392\pi\)
\(798\) 0 0
\(799\) 0.366315 + 0.266143i 0.0129593 + 0.00941547i
\(800\) 0 0
\(801\) −31.9204 + 98.2410i −1.12785 + 3.47117i
\(802\) 0 0
\(803\) 2.40488 9.28098i 0.0848663 0.327519i
\(804\) 0 0
\(805\) −2.12116 + 6.52826i −0.0747611 + 0.230091i
\(806\) 0 0
\(807\) 65.8614 + 47.8511i 2.31843 + 1.68444i
\(808\) 0 0
\(809\) 6.08110 + 18.7157i 0.213800 + 0.658009i 0.999237 + 0.0390667i \(0.0124385\pi\)
−0.785437 + 0.618942i \(0.787562\pi\)
\(810\) 0 0
\(811\) 5.99885 4.35842i 0.210648 0.153045i −0.477459 0.878654i \(-0.658442\pi\)
0.688107 + 0.725609i \(0.258442\pi\)
\(812\) 0 0
\(813\) −71.4844 −2.50707
\(814\) 0 0
\(815\) −0.396843 −0.0139008
\(816\) 0 0
\(817\) 1.33795 0.972074i 0.0468088 0.0340086i
\(818\) 0 0
\(819\) −7.36404 22.6642i −0.257320 0.791950i
\(820\) 0 0
\(821\) 4.29404 + 3.11981i 0.149863 + 0.108882i 0.660190 0.751098i \(-0.270476\pi\)
−0.510327 + 0.859980i \(0.670476\pi\)
\(822\) 0 0
\(823\) −5.00458 + 15.4025i −0.174449 + 0.536898i −0.999608 0.0280024i \(-0.991085\pi\)
0.825159 + 0.564900i \(0.191085\pi\)
\(824\) 0 0
\(825\) 10.0441 0.609587i 0.349692 0.0212231i
\(826\) 0 0
\(827\) −3.08205 + 9.48557i −0.107173 + 0.329846i −0.990234 0.139412i \(-0.955479\pi\)
0.883061 + 0.469258i \(0.155479\pi\)
\(828\) 0 0
\(829\) 21.4275 + 15.5680i 0.744209 + 0.540700i 0.894027 0.448014i \(-0.147869\pi\)
−0.149817 + 0.988714i \(0.547869\pi\)
\(830\) 0 0
\(831\) −9.36843 28.8331i −0.324987 1.00021i
\(832\) 0 0
\(833\) −2.78123 + 2.02068i −0.0963638 + 0.0700124i
\(834\) 0 0
\(835\) −13.9140 −0.481514
\(836\) 0 0
\(837\) 51.6931 1.78677
\(838\) 0 0
\(839\) −19.4642 + 14.1416i −0.671980 + 0.488222i −0.870687 0.491837i \(-0.836326\pi\)
0.198707 + 0.980059i \(0.436326\pi\)
\(840\) 0 0
\(841\) 18.1235 + 55.7785i 0.624950 + 1.92340i
\(842\) 0 0
\(843\) 10.0401 + 7.29458i 0.345801 + 0.251239i
\(844\) 0 0
\(845\) −2.06116 + 6.34359i −0.0709059 + 0.218226i
\(846\) 0 0
\(847\) 9.45529 1.15194i 0.324888 0.0395812i
\(848\) 0 0
\(849\) −4.47465 + 13.7716i −0.153570 + 0.472639i
\(850\) 0 0
\(851\) −40.9629 29.7613i −1.40419 1.02020i
\(852\) 0 0
\(853\) 6.96743 + 21.4436i 0.238560 + 0.734213i 0.996629 + 0.0820386i \(0.0261431\pi\)
−0.758069 + 0.652175i \(0.773857\pi\)
\(854\) 0 0
\(855\) −1.30000 + 0.944506i −0.0444591 + 0.0323014i
\(856\) 0 0
\(857\) 10.0453 0.343143 0.171571 0.985172i \(-0.445116\pi\)
0.171571 + 0.985172i \(0.445116\pi\)
\(858\) 0 0
\(859\) 15.7921 0.538819 0.269409 0.963026i \(-0.413172\pi\)
0.269409 + 0.963026i \(0.413172\pi\)
\(860\) 0 0
\(861\) 18.5812 13.5000i 0.633245 0.460080i
\(862\) 0 0
\(863\) −6.99653 21.5331i −0.238165 0.732995i −0.996686 0.0813469i \(-0.974078\pi\)
0.758521 0.651648i \(-0.225922\pi\)
\(864\) 0 0
\(865\) 4.31522 + 3.13519i 0.146722 + 0.106600i
\(866\) 0 0
\(867\) 15.6548 48.1805i 0.531665 1.63630i
\(868\) 0 0
\(869\) 19.5802 1.18834i 0.664214 0.0403117i
\(870\) 0 0
\(871\) −9.83655 + 30.2738i −0.333299 + 1.02579i
\(872\) 0 0
\(873\) 39.3267 + 28.5725i 1.33101 + 0.967034i
\(874\) 0 0
\(875\) −0.267586 0.823546i −0.00904607 0.0278409i
\(876\) 0 0
\(877\) −19.2215 + 13.9652i −0.649064 + 0.471573i −0.862952 0.505285i \(-0.831387\pi\)
0.213888 + 0.976858i \(0.431387\pi\)
\(878\) 0 0
\(879\) −0.436931 −0.0147373
\(880\) 0 0
\(881\) −21.6772 −0.730323 −0.365161 0.930944i \(-0.618986\pi\)
−0.365161 + 0.930944i \(0.618986\pi\)
\(882\) 0 0
\(883\) 32.8621 23.8757i 1.10590 0.803482i 0.123885 0.992297i \(-0.460465\pi\)
0.982013 + 0.188815i \(0.0604645\pi\)
\(884\) 0 0
\(885\) −12.6709 38.9970i −0.425928 1.31087i
\(886\) 0 0
\(887\) −27.6287 20.0734i −0.927680 0.673999i 0.0177436 0.999843i \(-0.494352\pi\)
−0.945424 + 0.325843i \(0.894352\pi\)
\(888\) 0 0
\(889\) 0.995244 3.06305i 0.0333794 0.102731i
\(890\) 0 0
\(891\) −9.05805 + 34.9571i −0.303456 + 1.17111i
\(892\) 0 0
\(893\) −0.0658762 + 0.202746i −0.00220446 + 0.00678464i
\(894\) 0 0
\(895\) 6.17667 + 4.48761i 0.206463 + 0.150004i
\(896\) 0 0
\(897\) 32.9617 + 101.446i 1.10056 + 3.38717i
\(898\) 0 0
\(899\) −40.2630 + 29.2528i −1.34285 + 0.975635i
\(900\) 0 0
\(901\) −1.32068 −0.0439982
\(902\) 0 0
\(903\) −16.7780 −0.558337
\(904\) 0 0
\(905\) 1.05538 0.766782i 0.0350822 0.0254887i
\(906\) 0 0
\(907\) 3.57041 + 10.9886i 0.118554 + 0.364870i 0.992672 0.120843i \(-0.0385598\pi\)
−0.874118 + 0.485714i \(0.838560\pi\)
\(908\) 0 0
\(909\) 91.3594 + 66.3765i 3.03020 + 2.20157i
\(910\) 0 0
\(911\) −4.45162 + 13.7007i −0.147489 + 0.453924i −0.997323 0.0731269i \(-0.976702\pi\)
0.849834 + 0.527051i \(0.176702\pi\)
\(912\) 0 0
\(913\) −18.1727 + 14.9661i −0.601428 + 0.495305i
\(914\) 0 0
\(915\) −6.19500 + 19.0662i −0.204800 + 0.630310i
\(916\) 0 0
\(917\) −14.3093 10.3963i −0.472536 0.343318i
\(918\) 0 0
\(919\) −15.1775 46.7116i −0.500660 1.54087i −0.807946 0.589256i \(-0.799421\pi\)
0.307286 0.951617i \(-0.400579\pi\)
\(920\) 0 0
\(921\) −63.2232 + 45.9344i −2.08328 + 1.51359i
\(922\) 0 0
\(923\) −12.4143 −0.408621
\(924\) 0 0
\(925\) 6.38739 0.210016
\(926\) 0 0
\(927\) −59.2492 + 43.0470i −1.94600 + 1.41385i
\(928\) 0 0
\(929\) 7.41296 + 22.8147i 0.243211 + 0.748527i 0.995925 + 0.0901799i \(0.0287442\pi\)
−0.752714 + 0.658347i \(0.771256\pi\)
\(930\) 0 0
\(931\) −1.30944 0.951365i −0.0429152 0.0311797i
\(932\) 0 0
\(933\) −15.3679 + 47.2976i −0.503123 + 1.54845i
\(934\) 0 0
\(935\) 1.53809 + 0.980890i 0.0503010 + 0.0320785i
\(936\) 0 0
\(937\) 4.04879 12.4609i 0.132268 0.407079i −0.862887 0.505397i \(-0.831346\pi\)
0.995155 + 0.0983175i \(0.0313461\pi\)
\(938\) 0 0
\(939\) 65.1039 + 47.3008i 2.12459 + 1.54360i
\(940\) 0 0
\(941\) −11.3355 34.8872i −0.369528 1.13729i −0.947097 0.320948i \(-0.895998\pi\)
0.577569 0.816342i \(-0.304002\pi\)
\(942\) 0 0
\(943\) −56.0644 + 40.7332i −1.82571 + 1.32645i
\(944\) 0 0
\(945\) 8.42052 0.273919
\(946\) 0 0
\(947\) 20.6112 0.669774 0.334887 0.942258i \(-0.391302\pi\)
0.334887 + 0.942258i \(0.391302\pi\)
\(948\) 0 0
\(949\) −10.3722 + 7.53581i −0.336694 + 0.244623i
\(950\) 0 0
\(951\) −15.6720 48.2334i −0.508199 1.56407i
\(952\) 0 0
\(953\) −14.3964 10.4596i −0.466344 0.338819i 0.329671 0.944096i \(-0.393062\pi\)
−0.796015 + 0.605277i \(0.793062\pi\)
\(954\) 0 0
\(955\) −7.75118 + 23.8557i −0.250822 + 0.771951i
\(956\) 0 0
\(957\) −34.4859 87.6683i −1.11477 2.83391i
\(958\) 0 0
\(959\) 3.59144 11.0533i 0.115974 0.356931i
\(960\) 0 0
\(961\) 2.21787 + 1.61138i 0.0715443 + 0.0519800i
\(962\) 0 0
\(963\) −0.998941 3.07443i −0.0321904 0.0990720i
\(964\) 0 0
\(965\) −16.1448 + 11.7299i −0.519719 + 0.377598i
\(966\) 0 0
\(967\) −0.828077 −0.0266292 −0.0133146 0.999911i \(-0.504238\pi\)
−0.0133146 + 0.999911i \(0.504238\pi\)
\(968\) 0 0
\(969\) −0.432154 −0.0138828
\(970\) 0 0
\(971\) 0.100751 0.0732001i 0.00323326 0.00234910i −0.586167 0.810190i \(-0.699364\pi\)
0.589401 + 0.807841i \(0.299364\pi\)
\(972\) 0 0
\(973\) 3.76912 + 11.6002i 0.120832 + 0.371884i
\(974\) 0 0
\(975\) −10.8862 7.90926i −0.348636 0.253299i
\(976\) 0 0
\(977\) −6.63347 + 20.4157i −0.212223 + 0.653157i 0.787116 + 0.616805i \(0.211573\pi\)
−0.999339 + 0.0363512i \(0.988427\pi\)
\(978\) 0 0
\(979\) 20.2111 + 51.3797i 0.645950 + 1.64210i
\(980\) 0 0
\(981\) −11.9372 + 36.7390i −0.381126 + 1.17298i
\(982\) 0 0
\(983\) −25.6917 18.6661i −0.819436 0.595355i 0.0971146 0.995273i \(-0.469039\pi\)
−0.916551 + 0.399918i \(0.869039\pi\)
\(984\) 0 0
\(985\) 0.405083 + 1.24672i 0.0129070 + 0.0397237i
\(986\) 0 0
\(987\) 1.74970 1.27123i 0.0556935 0.0404637i
\(988\) 0 0
\(989\) 50.6237 1.60974
\(990\) 0 0
\(991\) 46.7101 1.48379 0.741897 0.670514i \(-0.233926\pi\)
0.741897 + 0.670514i \(0.233926\pi\)
\(992\) 0 0
\(993\) 10.7898 7.83927i 0.342405 0.248772i
\(994\) 0 0
\(995\) 1.54235 + 4.74686i 0.0488958 + 0.150486i
\(996\) 0 0
\(997\) −41.7896 30.3619i −1.32349 0.961572i −0.999882 0.0153805i \(-0.995104\pi\)
−0.323608 0.946191i \(-0.604896\pi\)
\(998\) 0 0
\(999\) −19.1939 + 59.0727i −0.607268 + 1.86898i
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 440.2.y.b.361.3 12
4.3 odd 2 880.2.bo.j.801.1 12
11.4 even 5 4840.2.a.bf.1.1 6
11.5 even 5 inner 440.2.y.b.401.3 yes 12
11.7 odd 10 4840.2.a.be.1.1 6
44.7 even 10 9680.2.a.cy.1.6 6
44.15 odd 10 9680.2.a.cx.1.6 6
44.27 odd 10 880.2.bo.j.401.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.b.361.3 12 1.1 even 1 trivial
440.2.y.b.401.3 yes 12 11.5 even 5 inner
880.2.bo.j.401.1 12 44.27 odd 10
880.2.bo.j.801.1 12 4.3 odd 2
4840.2.a.be.1.1 6 11.7 odd 10
4840.2.a.bf.1.1 6 11.4 even 5
9680.2.a.cx.1.6 6 44.15 odd 10
9680.2.a.cy.1.6 6 44.7 even 10