Properties

Label 1280.2.f.k.129.6
Level $1280$
Weight $2$
Character 1280.129
Analytic conductor $10.221$
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] = [1280,2,Mod(129,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.2208514587\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 640)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.6
Root \(1.45161 + 1.45161i\) of defining polynomial
Character \(\chi\) \(=\) 1280.129
Dual form 1280.2.f.k.129.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.90321 q^{3} +(2.21432 + 0.311108i) q^{5} +3.52543i q^{7} +5.42864 q^{9} +O(q^{10})\) \(q+2.90321 q^{3} +(2.21432 + 0.311108i) q^{5} +3.52543i q^{7} +5.42864 q^{9} +3.80642i q^{11} -2.62222 q^{13} +(6.42864 + 0.903212i) q^{15} -5.80642i q^{17} -5.05086i q^{19} +10.2351i q^{21} -0.474572i q^{23} +(4.80642 + 1.37778i) q^{25} +7.05086 q^{27} +2.00000i q^{29} -2.75557 q^{31} +11.0509i q^{33} +(-1.09679 + 7.80642i) q^{35} -7.18421 q^{37} -7.61285 q^{39} -5.18421 q^{41} +1.95407 q^{43} +(12.0207 + 1.68889i) q^{45} +5.33185i q^{47} -5.42864 q^{49} -16.8573i q^{51} +5.37778 q^{53} +(-1.18421 + 8.42864i) q^{55} -14.6637i q^{57} -5.05086i q^{59} -12.2351i q^{61} +19.1383i q^{63} +(-5.80642 - 0.815792i) q^{65} +7.76049 q^{67} -1.37778i q^{69} +4.85728 q^{71} +6.66370i q^{73} +(13.9541 + 4.00000i) q^{75} -13.4193 q^{77} +5.24443 q^{79} +4.18421 q^{81} +12.1476 q^{83} +(1.80642 - 12.8573i) q^{85} +5.80642i q^{87} -12.1017 q^{89} -9.24443i q^{91} -8.00000 q^{93} +(1.57136 - 11.1842i) q^{95} -13.8064i q^{97} +20.6637i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} + 6 q^{9} - 16 q^{13} + 12 q^{15} + 2 q^{25} + 16 q^{27} - 16 q^{31} - 20 q^{35} - 16 q^{37} + 8 q^{39} - 4 q^{41} - 28 q^{43} + 32 q^{45} - 6 q^{49} + 32 q^{53} + 20 q^{55} - 8 q^{65} - 20 q^{67} - 24 q^{71} + 44 q^{75} + 32 q^{79} - 2 q^{81} + 60 q^{83} - 16 q^{85} - 20 q^{89} - 48 q^{93} + 36 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.90321 1.67617 0.838085 0.545540i \(-0.183675\pi\)
0.838085 + 0.545540i \(0.183675\pi\)
\(4\) 0 0
\(5\) 2.21432 + 0.311108i 0.990274 + 0.139132i
\(6\) 0 0
\(7\) 3.52543i 1.33249i 0.745735 + 0.666243i \(0.232099\pi\)
−0.745735 + 0.666243i \(0.767901\pi\)
\(8\) 0 0
\(9\) 5.42864 1.80955
\(10\) 0 0
\(11\) 3.80642i 1.14768i 0.818967 + 0.573840i \(0.194547\pi\)
−0.818967 + 0.573840i \(0.805453\pi\)
\(12\) 0 0
\(13\) −2.62222 −0.727272 −0.363636 0.931541i \(-0.618465\pi\)
−0.363636 + 0.931541i \(0.618465\pi\)
\(14\) 0 0
\(15\) 6.42864 + 0.903212i 1.65987 + 0.233208i
\(16\) 0 0
\(17\) 5.80642i 1.40826i −0.710069 0.704132i \(-0.751336\pi\)
0.710069 0.704132i \(-0.248664\pi\)
\(18\) 0 0
\(19\) 5.05086i 1.15875i −0.815063 0.579373i \(-0.803298\pi\)
0.815063 0.579373i \(-0.196702\pi\)
\(20\) 0 0
\(21\) 10.2351i 2.23347i
\(22\) 0 0
\(23\) 0.474572i 0.0989552i −0.998775 0.0494776i \(-0.984244\pi\)
0.998775 0.0494776i \(-0.0157556\pi\)
\(24\) 0 0
\(25\) 4.80642 + 1.37778i 0.961285 + 0.275557i
\(26\) 0 0
\(27\) 7.05086 1.35694
\(28\) 0 0
\(29\) 2.00000i 0.371391i 0.982607 + 0.185695i \(0.0594537\pi\)
−0.982607 + 0.185695i \(0.940546\pi\)
\(30\) 0 0
\(31\) −2.75557 −0.494915 −0.247457 0.968899i \(-0.579595\pi\)
−0.247457 + 0.968899i \(0.579595\pi\)
\(32\) 0 0
\(33\) 11.0509i 1.92371i
\(34\) 0 0
\(35\) −1.09679 + 7.80642i −0.185391 + 1.31953i
\(36\) 0 0
\(37\) −7.18421 −1.18108 −0.590538 0.807010i \(-0.701085\pi\)
−0.590538 + 0.807010i \(0.701085\pi\)
\(38\) 0 0
\(39\) −7.61285 −1.21903
\(40\) 0 0
\(41\) −5.18421 −0.809637 −0.404819 0.914397i \(-0.632665\pi\)
−0.404819 + 0.914397i \(0.632665\pi\)
\(42\) 0 0
\(43\) 1.95407 0.297992 0.148996 0.988838i \(-0.452396\pi\)
0.148996 + 0.988838i \(0.452396\pi\)
\(44\) 0 0
\(45\) 12.0207 + 1.68889i 1.79195 + 0.251765i
\(46\) 0 0
\(47\) 5.33185i 0.777730i 0.921295 + 0.388865i \(0.127133\pi\)
−0.921295 + 0.388865i \(0.872867\pi\)
\(48\) 0 0
\(49\) −5.42864 −0.775520
\(50\) 0 0
\(51\) 16.8573i 2.36049i
\(52\) 0 0
\(53\) 5.37778 0.738695 0.369348 0.929291i \(-0.379581\pi\)
0.369348 + 0.929291i \(0.379581\pi\)
\(54\) 0 0
\(55\) −1.18421 + 8.42864i −0.159679 + 1.13652i
\(56\) 0 0
\(57\) 14.6637i 1.94225i
\(58\) 0 0
\(59\) 5.05086i 0.657565i −0.944406 0.328783i \(-0.893362\pi\)
0.944406 0.328783i \(-0.106638\pi\)
\(60\) 0 0
\(61\) 12.2351i 1.56654i −0.621682 0.783270i \(-0.713550\pi\)
0.621682 0.783270i \(-0.286450\pi\)
\(62\) 0 0
\(63\) 19.1383i 2.41120i
\(64\) 0 0
\(65\) −5.80642 0.815792i −0.720198 0.101187i
\(66\) 0 0
\(67\) 7.76049 0.948095 0.474047 0.880499i \(-0.342793\pi\)
0.474047 + 0.880499i \(0.342793\pi\)
\(68\) 0 0
\(69\) 1.37778i 0.165866i
\(70\) 0 0
\(71\) 4.85728 0.576453 0.288226 0.957562i \(-0.406934\pi\)
0.288226 + 0.957562i \(0.406934\pi\)
\(72\) 0 0
\(73\) 6.66370i 0.779927i 0.920830 + 0.389964i \(0.127512\pi\)
−0.920830 + 0.389964i \(0.872488\pi\)
\(74\) 0 0
\(75\) 13.9541 + 4.00000i 1.61128 + 0.461880i
\(76\) 0 0
\(77\) −13.4193 −1.52927
\(78\) 0 0
\(79\) 5.24443 0.590045 0.295022 0.955490i \(-0.404673\pi\)
0.295022 + 0.955490i \(0.404673\pi\)
\(80\) 0 0
\(81\) 4.18421 0.464912
\(82\) 0 0
\(83\) 12.1476 1.33338 0.666689 0.745336i \(-0.267711\pi\)
0.666689 + 0.745336i \(0.267711\pi\)
\(84\) 0 0
\(85\) 1.80642 12.8573i 0.195934 1.39457i
\(86\) 0 0
\(87\) 5.80642i 0.622514i
\(88\) 0 0
\(89\) −12.1017 −1.28278 −0.641389 0.767216i \(-0.721642\pi\)
−0.641389 + 0.767216i \(0.721642\pi\)
\(90\) 0 0
\(91\) 9.24443i 0.969080i
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) 1.57136 11.1842i 0.161218 1.14748i
\(96\) 0 0
\(97\) 13.8064i 1.40183i −0.713245 0.700915i \(-0.752775\pi\)
0.713245 0.700915i \(-0.247225\pi\)
\(98\) 0 0
\(99\) 20.6637i 2.07678i
\(100\) 0 0
\(101\) 19.7146i 1.96167i −0.194836 0.980836i \(-0.562417\pi\)
0.194836 0.980836i \(-0.437583\pi\)
\(102\) 0 0
\(103\) 3.13828i 0.309223i 0.987975 + 0.154612i \(0.0494127\pi\)
−0.987975 + 0.154612i \(0.950587\pi\)
\(104\) 0 0
\(105\) −3.18421 + 22.6637i −0.310747 + 2.21175i
\(106\) 0 0
\(107\) −5.39207 −0.521272 −0.260636 0.965437i \(-0.583932\pi\)
−0.260636 + 0.965437i \(0.583932\pi\)
\(108\) 0 0
\(109\) 8.62222i 0.825858i −0.910763 0.412929i \(-0.864506\pi\)
0.910763 0.412929i \(-0.135494\pi\)
\(110\) 0 0
\(111\) −20.8573 −1.97969
\(112\) 0 0
\(113\) 5.51114i 0.518444i 0.965818 + 0.259222i \(0.0834662\pi\)
−0.965818 + 0.259222i \(0.916534\pi\)
\(114\) 0 0
\(115\) 0.147643 1.05086i 0.0137678 0.0979927i
\(116\) 0 0
\(117\) −14.2351 −1.31603
\(118\) 0 0
\(119\) 20.4701 1.87649
\(120\) 0 0
\(121\) −3.48886 −0.317169
\(122\) 0 0
\(123\) −15.0509 −1.35709
\(124\) 0 0
\(125\) 10.2143 + 4.54617i 0.913597 + 0.406622i
\(126\) 0 0
\(127\) 10.2810i 0.912291i −0.889905 0.456145i \(-0.849230\pi\)
0.889905 0.456145i \(-0.150770\pi\)
\(128\) 0 0
\(129\) 5.67307 0.499486
\(130\) 0 0
\(131\) 4.66370i 0.407470i 0.979026 + 0.203735i \(0.0653080\pi\)
−0.979026 + 0.203735i \(0.934692\pi\)
\(132\) 0 0
\(133\) 17.8064 1.54401
\(134\) 0 0
\(135\) 15.6128 + 2.19358i 1.34374 + 0.188793i
\(136\) 0 0
\(137\) 11.3461i 0.969366i 0.874690 + 0.484683i \(0.161065\pi\)
−0.874690 + 0.484683i \(0.838935\pi\)
\(138\) 0 0
\(139\) 11.8064i 1.00141i 0.865619 + 0.500704i \(0.166925\pi\)
−0.865619 + 0.500704i \(0.833075\pi\)
\(140\) 0 0
\(141\) 15.4795i 1.30361i
\(142\) 0 0
\(143\) 9.98126i 0.834675i
\(144\) 0 0
\(145\) −0.622216 + 4.42864i −0.0516722 + 0.367778i
\(146\) 0 0
\(147\) −15.7605 −1.29990
\(148\) 0 0
\(149\) 5.47949i 0.448898i −0.974486 0.224449i \(-0.927942\pi\)
0.974486 0.224449i \(-0.0720582\pi\)
\(150\) 0 0
\(151\) −23.6128 −1.92159 −0.960793 0.277266i \(-0.910572\pi\)
−0.960793 + 0.277266i \(0.910572\pi\)
\(152\) 0 0
\(153\) 31.5210i 2.54832i
\(154\) 0 0
\(155\) −6.10171 0.857279i −0.490101 0.0688583i
\(156\) 0 0
\(157\) 0.815792 0.0651073 0.0325536 0.999470i \(-0.489636\pi\)
0.0325536 + 0.999470i \(0.489636\pi\)
\(158\) 0 0
\(159\) 15.6128 1.23818
\(160\) 0 0
\(161\) 1.67307 0.131856
\(162\) 0 0
\(163\) 10.9032 0.854005 0.427003 0.904250i \(-0.359569\pi\)
0.427003 + 0.904250i \(0.359569\pi\)
\(164\) 0 0
\(165\) −3.43801 + 24.4701i −0.267649 + 1.90500i
\(166\) 0 0
\(167\) 6.57628i 0.508888i −0.967088 0.254444i \(-0.918108\pi\)
0.967088 0.254444i \(-0.0818925\pi\)
\(168\) 0 0
\(169\) −6.12399 −0.471076
\(170\) 0 0
\(171\) 27.4193i 2.09680i
\(172\) 0 0
\(173\) 10.5303 0.800608 0.400304 0.916382i \(-0.368905\pi\)
0.400304 + 0.916382i \(0.368905\pi\)
\(174\) 0 0
\(175\) −4.85728 + 16.9447i −0.367176 + 1.28090i
\(176\) 0 0
\(177\) 14.6637i 1.10219i
\(178\) 0 0
\(179\) 6.29529i 0.470532i 0.971931 + 0.235266i \(0.0755961\pi\)
−0.971931 + 0.235266i \(0.924404\pi\)
\(180\) 0 0
\(181\) 0.488863i 0.0363369i 0.999835 + 0.0181684i \(0.00578351\pi\)
−0.999835 + 0.0181684i \(0.994216\pi\)
\(182\) 0 0
\(183\) 35.5210i 2.62579i
\(184\) 0 0
\(185\) −15.9081 2.23506i −1.16959 0.164325i
\(186\) 0 0
\(187\) 22.1017 1.61624
\(188\) 0 0
\(189\) 24.8573i 1.80810i
\(190\) 0 0
\(191\) −10.4889 −0.758947 −0.379474 0.925203i \(-0.623895\pi\)
−0.379474 + 0.925203i \(0.623895\pi\)
\(192\) 0 0
\(193\) 13.8064i 0.993808i 0.867805 + 0.496904i \(0.165530\pi\)
−0.867805 + 0.496904i \(0.834470\pi\)
\(194\) 0 0
\(195\) −16.8573 2.36842i −1.20717 0.169606i
\(196\) 0 0
\(197\) −16.7239 −1.19153 −0.595765 0.803159i \(-0.703151\pi\)
−0.595765 + 0.803159i \(0.703151\pi\)
\(198\) 0 0
\(199\) 20.8573 1.47853 0.739267 0.673413i \(-0.235172\pi\)
0.739267 + 0.673413i \(0.235172\pi\)
\(200\) 0 0
\(201\) 22.5303 1.58917
\(202\) 0 0
\(203\) −7.05086 −0.494873
\(204\) 0 0
\(205\) −11.4795 1.61285i −0.801763 0.112646i
\(206\) 0 0
\(207\) 2.57628i 0.179064i
\(208\) 0 0
\(209\) 19.2257 1.32987
\(210\) 0 0
\(211\) 4.66370i 0.321063i 0.987031 + 0.160531i \(0.0513207\pi\)
−0.987031 + 0.160531i \(0.948679\pi\)
\(212\) 0 0
\(213\) 14.1017 0.966233
\(214\) 0 0
\(215\) 4.32693 + 0.607926i 0.295094 + 0.0414602i
\(216\) 0 0
\(217\) 9.71456i 0.659467i
\(218\) 0 0
\(219\) 19.3461i 1.30729i
\(220\) 0 0
\(221\) 15.2257i 1.02419i
\(222\) 0 0
\(223\) 26.4558i 1.77161i 0.464054 + 0.885807i \(0.346394\pi\)
−0.464054 + 0.885807i \(0.653606\pi\)
\(224\) 0 0
\(225\) 26.0923 + 7.47949i 1.73949 + 0.498633i
\(226\) 0 0
\(227\) −12.3225 −0.817872 −0.408936 0.912563i \(-0.634100\pi\)
−0.408936 + 0.912563i \(0.634100\pi\)
\(228\) 0 0
\(229\) 13.2257i 0.873979i 0.899467 + 0.436989i \(0.143955\pi\)
−0.899467 + 0.436989i \(0.856045\pi\)
\(230\) 0 0
\(231\) −38.9590 −2.56331
\(232\) 0 0
\(233\) 6.66370i 0.436554i 0.975887 + 0.218277i \(0.0700436\pi\)
−0.975887 + 0.218277i \(0.929956\pi\)
\(234\) 0 0
\(235\) −1.65878 + 11.8064i −0.108207 + 0.770166i
\(236\) 0 0
\(237\) 15.2257 0.989015
\(238\) 0 0
\(239\) −22.9590 −1.48509 −0.742547 0.669794i \(-0.766382\pi\)
−0.742547 + 0.669794i \(0.766382\pi\)
\(240\) 0 0
\(241\) −14.0415 −0.904492 −0.452246 0.891893i \(-0.649377\pi\)
−0.452246 + 0.891893i \(0.649377\pi\)
\(242\) 0 0
\(243\) −9.00492 −0.577666
\(244\) 0 0
\(245\) −12.0207 1.68889i −0.767977 0.107899i
\(246\) 0 0
\(247\) 13.2444i 0.842723i
\(248\) 0 0
\(249\) 35.2672 2.23497
\(250\) 0 0
\(251\) 24.9304i 1.57359i −0.617212 0.786797i \(-0.711738\pi\)
0.617212 0.786797i \(-0.288262\pi\)
\(252\) 0 0
\(253\) 1.80642 0.113569
\(254\) 0 0
\(255\) 5.24443 37.3274i 0.328419 2.33753i
\(256\) 0 0
\(257\) 25.7146i 1.60403i 0.597304 + 0.802015i \(0.296239\pi\)
−0.597304 + 0.802015i \(0.703761\pi\)
\(258\) 0 0
\(259\) 25.3274i 1.57377i
\(260\) 0 0
\(261\) 10.8573i 0.672049i
\(262\) 0 0
\(263\) 2.57628i 0.158860i −0.996840 0.0794302i \(-0.974690\pi\)
0.996840 0.0794302i \(-0.0253101\pi\)
\(264\) 0 0
\(265\) 11.9081 + 1.67307i 0.731511 + 0.102776i
\(266\) 0 0
\(267\) −35.1338 −2.15016
\(268\) 0 0
\(269\) 25.7462i 1.56977i −0.619639 0.784887i \(-0.712721\pi\)
0.619639 0.784887i \(-0.287279\pi\)
\(270\) 0 0
\(271\) 30.1847 1.83359 0.916795 0.399359i \(-0.130767\pi\)
0.916795 + 0.399359i \(0.130767\pi\)
\(272\) 0 0
\(273\) 26.8385i 1.62434i
\(274\) 0 0
\(275\) −5.24443 + 18.2953i −0.316251 + 1.10325i
\(276\) 0 0
\(277\) 10.5303 0.632707 0.316354 0.948641i \(-0.397541\pi\)
0.316354 + 0.948641i \(0.397541\pi\)
\(278\) 0 0
\(279\) −14.9590 −0.895571
\(280\) 0 0
\(281\) 7.93978 0.473647 0.236824 0.971553i \(-0.423894\pi\)
0.236824 + 0.971553i \(0.423894\pi\)
\(282\) 0 0
\(283\) −10.9032 −0.648129 −0.324064 0.946035i \(-0.605049\pi\)
−0.324064 + 0.946035i \(0.605049\pi\)
\(284\) 0 0
\(285\) 4.56199 32.4701i 0.270229 1.92336i
\(286\) 0 0
\(287\) 18.2766i 1.07883i
\(288\) 0 0
\(289\) −16.7146 −0.983209
\(290\) 0 0
\(291\) 40.0830i 2.34971i
\(292\) 0 0
\(293\) 4.42864 0.258724 0.129362 0.991597i \(-0.458707\pi\)
0.129362 + 0.991597i \(0.458707\pi\)
\(294\) 0 0
\(295\) 1.57136 11.1842i 0.0914881 0.651170i
\(296\) 0 0
\(297\) 26.8385i 1.55733i
\(298\) 0 0
\(299\) 1.24443i 0.0719673i
\(300\) 0 0
\(301\) 6.88892i 0.397071i
\(302\) 0 0
\(303\) 57.2355i 3.28810i
\(304\) 0 0
\(305\) 3.80642 27.0923i 0.217955 1.55130i
\(306\) 0 0
\(307\) 5.27163 0.300868 0.150434 0.988620i \(-0.451933\pi\)
0.150434 + 0.988620i \(0.451933\pi\)
\(308\) 0 0
\(309\) 9.11108i 0.518311i
\(310\) 0 0
\(311\) −0.387152 −0.0219534 −0.0109767 0.999940i \(-0.503494\pi\)
−0.0109767 + 0.999940i \(0.503494\pi\)
\(312\) 0 0
\(313\) 11.3461i 0.641322i −0.947194 0.320661i \(-0.896095\pi\)
0.947194 0.320661i \(-0.103905\pi\)
\(314\) 0 0
\(315\) −5.95407 + 42.3783i −0.335474 + 2.38774i
\(316\) 0 0
\(317\) −16.7239 −0.939309 −0.469655 0.882850i \(-0.655622\pi\)
−0.469655 + 0.882850i \(0.655622\pi\)
\(318\) 0 0
\(319\) −7.61285 −0.426238
\(320\) 0 0
\(321\) −15.6543 −0.873740
\(322\) 0 0
\(323\) −29.3274 −1.63182
\(324\) 0 0
\(325\) −12.6035 3.61285i −0.699115 0.200405i
\(326\) 0 0
\(327\) 25.0321i 1.38428i
\(328\) 0 0
\(329\) −18.7971 −1.03632
\(330\) 0 0
\(331\) 3.33630i 0.183379i −0.995788 0.0916897i \(-0.970773\pi\)
0.995788 0.0916897i \(-0.0292268\pi\)
\(332\) 0 0
\(333\) −39.0005 −2.13721
\(334\) 0 0
\(335\) 17.1842 + 2.41435i 0.938874 + 0.131910i
\(336\) 0 0
\(337\) 3.61285i 0.196804i 0.995147 + 0.0984022i \(0.0313732\pi\)
−0.995147 + 0.0984022i \(0.968627\pi\)
\(338\) 0 0
\(339\) 16.0000i 0.869001i
\(340\) 0 0
\(341\) 10.4889i 0.568004i
\(342\) 0 0
\(343\) 5.53972i 0.299117i
\(344\) 0 0
\(345\) 0.428639 3.05086i 0.0230772 0.164253i
\(346\) 0 0
\(347\) −15.7605 −0.846067 −0.423034 0.906114i \(-0.639035\pi\)
−0.423034 + 0.906114i \(0.639035\pi\)
\(348\) 0 0
\(349\) 0.285442i 0.0152794i 0.999971 + 0.00763968i \(0.00243181\pi\)
−0.999971 + 0.00763968i \(0.997568\pi\)
\(350\) 0 0
\(351\) −18.4889 −0.986862
\(352\) 0 0
\(353\) 4.38715i 0.233505i −0.993161 0.116752i \(-0.962752\pi\)
0.993161 0.116752i \(-0.0372484\pi\)
\(354\) 0 0
\(355\) 10.7556 + 1.51114i 0.570846 + 0.0802028i
\(356\) 0 0
\(357\) 59.4291 3.14532
\(358\) 0 0
\(359\) −20.5906 −1.08673 −0.543364 0.839497i \(-0.682850\pi\)
−0.543364 + 0.839497i \(0.682850\pi\)
\(360\) 0 0
\(361\) −6.51114 −0.342691
\(362\) 0 0
\(363\) −10.1289 −0.531630
\(364\) 0 0
\(365\) −2.07313 + 14.7556i −0.108513 + 0.772342i
\(366\) 0 0
\(367\) 16.5575i 0.864297i 0.901802 + 0.432148i \(0.142244\pi\)
−0.901802 + 0.432148i \(0.857756\pi\)
\(368\) 0 0
\(369\) −28.1432 −1.46508
\(370\) 0 0
\(371\) 18.9590i 0.984302i
\(372\) 0 0
\(373\) −24.8988 −1.28921 −0.644605 0.764516i \(-0.722978\pi\)
−0.644605 + 0.764516i \(0.722978\pi\)
\(374\) 0 0
\(375\) 29.6543 + 13.1985i 1.53134 + 0.681568i
\(376\) 0 0
\(377\) 5.24443i 0.270102i
\(378\) 0 0
\(379\) 9.31756i 0.478611i 0.970944 + 0.239305i \(0.0769197\pi\)
−0.970944 + 0.239305i \(0.923080\pi\)
\(380\) 0 0
\(381\) 29.8479i 1.52915i
\(382\) 0 0
\(383\) 32.5575i 1.66361i 0.555066 + 0.831806i \(0.312693\pi\)
−0.555066 + 0.831806i \(0.687307\pi\)
\(384\) 0 0
\(385\) −29.7146 4.17484i −1.51439 0.212770i
\(386\) 0 0
\(387\) 10.6079 0.539231
\(388\) 0 0
\(389\) 18.9906i 0.962863i −0.876484 0.481432i \(-0.840117\pi\)
0.876484 0.481432i \(-0.159883\pi\)
\(390\) 0 0
\(391\) −2.75557 −0.139355
\(392\) 0 0
\(393\) 13.5397i 0.682988i
\(394\) 0 0
\(395\) 11.6128 + 1.63158i 0.584306 + 0.0820939i
\(396\) 0 0
\(397\) 32.9906 1.65575 0.827876 0.560911i \(-0.189549\pi\)
0.827876 + 0.560911i \(0.189549\pi\)
\(398\) 0 0
\(399\) 51.6958 2.58803
\(400\) 0 0
\(401\) −16.1017 −0.804081 −0.402041 0.915622i \(-0.631699\pi\)
−0.402041 + 0.915622i \(0.631699\pi\)
\(402\) 0 0
\(403\) 7.22570 0.359938
\(404\) 0 0
\(405\) 9.26517 + 1.30174i 0.460390 + 0.0646840i
\(406\) 0 0
\(407\) 27.3461i 1.35550i
\(408\) 0 0
\(409\) −33.3876 −1.65091 −0.825456 0.564466i \(-0.809082\pi\)
−0.825456 + 0.564466i \(0.809082\pi\)
\(410\) 0 0
\(411\) 32.9403i 1.62482i
\(412\) 0 0
\(413\) 17.8064 0.876197
\(414\) 0 0
\(415\) 26.8988 + 3.77923i 1.32041 + 0.185515i
\(416\) 0 0
\(417\) 34.2766i 1.67853i
\(418\) 0 0
\(419\) 27.4193i 1.33952i −0.742578 0.669760i \(-0.766397\pi\)
0.742578 0.669760i \(-0.233603\pi\)
\(420\) 0 0
\(421\) 12.2351i 0.596301i 0.954519 + 0.298150i \(0.0963696\pi\)
−0.954519 + 0.298150i \(0.903630\pi\)
\(422\) 0 0
\(423\) 28.9447i 1.40734i
\(424\) 0 0
\(425\) 8.00000 27.9081i 0.388057 1.35374i
\(426\) 0 0
\(427\) 43.1338 2.08739
\(428\) 0 0
\(429\) 28.9777i 1.39906i
\(430\) 0 0
\(431\) −28.4701 −1.37136 −0.685679 0.727904i \(-0.740495\pi\)
−0.685679 + 0.727904i \(0.740495\pi\)
\(432\) 0 0
\(433\) 34.0098i 1.63441i 0.576348 + 0.817204i \(0.304477\pi\)
−0.576348 + 0.817204i \(0.695523\pi\)
\(434\) 0 0
\(435\) −1.80642 + 12.8573i −0.0866114 + 0.616459i
\(436\) 0 0
\(437\) −2.39700 −0.114664
\(438\) 0 0
\(439\) −14.8385 −0.708205 −0.354103 0.935207i \(-0.615214\pi\)
−0.354103 + 0.935207i \(0.615214\pi\)
\(440\) 0 0
\(441\) −29.4701 −1.40334
\(442\) 0 0
\(443\) 13.0968 0.622247 0.311124 0.950369i \(-0.399295\pi\)
0.311124 + 0.950369i \(0.399295\pi\)
\(444\) 0 0
\(445\) −26.7971 3.76494i −1.27030 0.178475i
\(446\) 0 0
\(447\) 15.9081i 0.752429i
\(448\) 0 0
\(449\) 21.3876 1.00934 0.504672 0.863311i \(-0.331613\pi\)
0.504672 + 0.863311i \(0.331613\pi\)
\(450\) 0 0
\(451\) 19.7333i 0.929205i
\(452\) 0 0
\(453\) −68.5531 −3.22091
\(454\) 0 0
\(455\) 2.87601 20.4701i 0.134830 0.959654i
\(456\) 0 0
\(457\) 17.9813i 0.841128i 0.907263 + 0.420564i \(0.138168\pi\)
−0.907263 + 0.420564i \(0.861832\pi\)
\(458\) 0 0
\(459\) 40.9403i 1.91093i
\(460\) 0 0
\(461\) 10.7368i 0.500064i −0.968238 0.250032i \(-0.919559\pi\)
0.968238 0.250032i \(-0.0804412\pi\)
\(462\) 0 0
\(463\) 9.30327i 0.432360i −0.976354 0.216180i \(-0.930640\pi\)
0.976354 0.216180i \(-0.0693598\pi\)
\(464\) 0 0
\(465\) −17.7146 2.48886i −0.821493 0.115418i
\(466\) 0 0
\(467\) −8.70964 −0.403034 −0.201517 0.979485i \(-0.564587\pi\)
−0.201517 + 0.979485i \(0.564587\pi\)
\(468\) 0 0
\(469\) 27.3590i 1.26332i
\(470\) 0 0
\(471\) 2.36842 0.109131
\(472\) 0 0
\(473\) 7.43801i 0.342000i
\(474\) 0 0
\(475\) 6.95899 24.2766i 0.319300 1.11388i
\(476\) 0 0
\(477\) 29.1941 1.33670
\(478\) 0 0
\(479\) −23.2257 −1.06121 −0.530605 0.847619i \(-0.678035\pi\)
−0.530605 + 0.847619i \(0.678035\pi\)
\(480\) 0 0
\(481\) 18.8385 0.858964
\(482\) 0 0
\(483\) 4.85728 0.221014
\(484\) 0 0
\(485\) 4.29529 30.5718i 0.195039 1.38820i
\(486\) 0 0
\(487\) 32.8528i 1.48870i 0.667787 + 0.744352i \(0.267241\pi\)
−0.667787 + 0.744352i \(0.732759\pi\)
\(488\) 0 0
\(489\) 31.6543 1.43146
\(490\) 0 0
\(491\) 2.94914i 0.133093i 0.997783 + 0.0665465i \(0.0211981\pi\)
−0.997783 + 0.0665465i \(0.978802\pi\)
\(492\) 0 0
\(493\) 11.6128 0.523016
\(494\) 0 0
\(495\) −6.42864 + 45.7560i −0.288946 + 2.05658i
\(496\) 0 0
\(497\) 17.1240i 0.768116i
\(498\) 0 0
\(499\) 35.0321i 1.56825i 0.620601 + 0.784127i \(0.286889\pi\)
−0.620601 + 0.784127i \(0.713111\pi\)
\(500\) 0 0
\(501\) 19.0923i 0.852983i
\(502\) 0 0
\(503\) 16.2908i 0.726373i −0.931717 0.363186i \(-0.881689\pi\)
0.931717 0.363186i \(-0.118311\pi\)
\(504\) 0 0
\(505\) 6.13335 43.6543i 0.272931 1.94259i
\(506\) 0 0
\(507\) −17.7792 −0.789603
\(508\) 0 0
\(509\) 26.0000i 1.15243i 0.817298 + 0.576215i \(0.195471\pi\)
−0.817298 + 0.576215i \(0.804529\pi\)
\(510\) 0 0
\(511\) −23.4924 −1.03924
\(512\) 0 0
\(513\) 35.6128i 1.57235i
\(514\) 0 0
\(515\) −0.976342 + 6.94914i −0.0430228 + 0.306216i
\(516\) 0 0
\(517\) −20.2953 −0.892586
\(518\) 0 0
\(519\) 30.5718 1.34195
\(520\) 0 0
\(521\) −11.7146 −0.513224 −0.256612 0.966514i \(-0.582606\pi\)
−0.256612 + 0.966514i \(0.582606\pi\)
\(522\) 0 0
\(523\) 38.0370 1.66324 0.831622 0.555342i \(-0.187413\pi\)
0.831622 + 0.555342i \(0.187413\pi\)
\(524\) 0 0
\(525\) −14.1017 + 49.1941i −0.615449 + 2.14700i
\(526\) 0 0
\(527\) 16.0000i 0.696971i
\(528\) 0 0
\(529\) 22.7748 0.990208
\(530\) 0 0
\(531\) 27.4193i 1.18990i
\(532\) 0 0
\(533\) 13.5941 0.588826
\(534\) 0 0
\(535\) −11.9398 1.67752i −0.516202 0.0725254i
\(536\) 0 0
\(537\) 18.2766i 0.788691i
\(538\) 0 0
\(539\) 20.6637i 0.890049i
\(540\) 0 0
\(541\) 11.5111i 0.494902i −0.968900 0.247451i \(-0.920407\pi\)
0.968900 0.247451i \(-0.0795930\pi\)
\(542\) 0 0
\(543\) 1.41927i 0.0609068i
\(544\) 0 0
\(545\) 2.68244 19.0923i 0.114903 0.817826i
\(546\) 0 0
\(547\) −30.2208 −1.29215 −0.646073 0.763275i \(-0.723590\pi\)
−0.646073 + 0.763275i \(0.723590\pi\)
\(548\) 0 0
\(549\) 66.4197i 2.83473i
\(550\) 0 0
\(551\) 10.1017 0.430347
\(552\) 0 0
\(553\) 18.4889i 0.786226i
\(554\) 0 0
\(555\) −46.1847 6.48886i −1.96043 0.275437i
\(556\) 0 0
\(557\) 9.75605 0.413377 0.206688 0.978407i \(-0.433731\pi\)
0.206688 + 0.978407i \(0.433731\pi\)
\(558\) 0 0
\(559\) −5.12399 −0.216721
\(560\) 0 0
\(561\) 64.1659 2.70909
\(562\) 0 0
\(563\) −4.50622 −0.189914 −0.0949572 0.995481i \(-0.530271\pi\)
−0.0949572 + 0.995481i \(0.530271\pi\)
\(564\) 0 0
\(565\) −1.71456 + 12.2034i −0.0721320 + 0.513402i
\(566\) 0 0
\(567\) 14.7511i 0.619489i
\(568\) 0 0
\(569\) 5.30465 0.222383 0.111191 0.993799i \(-0.464533\pi\)
0.111191 + 0.993799i \(0.464533\pi\)
\(570\) 0 0
\(571\) 16.3970i 0.686193i −0.939300 0.343096i \(-0.888524\pi\)
0.939300 0.343096i \(-0.111476\pi\)
\(572\) 0 0
\(573\) −30.4514 −1.27213
\(574\) 0 0
\(575\) 0.653858 2.28100i 0.0272678 0.0951241i
\(576\) 0 0
\(577\) 39.8163i 1.65757i −0.559565 0.828786i \(-0.689032\pi\)
0.559565 0.828786i \(-0.310968\pi\)
\(578\) 0 0
\(579\) 40.0830i 1.66579i
\(580\) 0 0
\(581\) 42.8256i 1.77671i
\(582\) 0 0
\(583\) 20.4701i 0.847786i
\(584\) 0 0
\(585\) −31.5210 4.42864i −1.30323 0.183102i
\(586\) 0 0
\(587\) −15.1699 −0.626130 −0.313065 0.949732i \(-0.601356\pi\)
−0.313065 + 0.949732i \(0.601356\pi\)
\(588\) 0 0
\(589\) 13.9180i 0.573480i
\(590\) 0 0
\(591\) −48.5531 −1.99721
\(592\) 0 0
\(593\) 8.00000i 0.328521i −0.986417 0.164260i \(-0.947476\pi\)
0.986417 0.164260i \(-0.0525237\pi\)
\(594\) 0 0
\(595\) 45.3274 + 6.36842i 1.85824 + 0.261080i
\(596\) 0 0
\(597\) 60.5531 2.47827
\(598\) 0 0
\(599\) 1.83500 0.0749762 0.0374881 0.999297i \(-0.488064\pi\)
0.0374881 + 0.999297i \(0.488064\pi\)
\(600\) 0 0
\(601\) 36.1432 1.47431 0.737156 0.675723i \(-0.236168\pi\)
0.737156 + 0.675723i \(0.236168\pi\)
\(602\) 0 0
\(603\) 42.1289 1.71562
\(604\) 0 0
\(605\) −7.72546 1.08541i −0.314084 0.0441283i
\(606\) 0 0
\(607\) 17.5353i 0.711735i 0.934536 + 0.355867i \(0.115815\pi\)
−0.934536 + 0.355867i \(0.884185\pi\)
\(608\) 0 0
\(609\) −20.4701 −0.829491
\(610\) 0 0
\(611\) 13.9813i 0.565621i
\(612\) 0 0
\(613\) 33.9309 1.37046 0.685228 0.728329i \(-0.259703\pi\)
0.685228 + 0.728329i \(0.259703\pi\)
\(614\) 0 0
\(615\) −33.3274 4.68244i −1.34389 0.188814i
\(616\) 0 0
\(617\) 9.68598i 0.389943i −0.980809 0.194971i \(-0.937539\pi\)
0.980809 0.194971i \(-0.0624614\pi\)
\(618\) 0 0
\(619\) 41.4005i 1.66403i 0.554755 + 0.832014i \(0.312812\pi\)
−0.554755 + 0.832014i \(0.687188\pi\)
\(620\) 0 0
\(621\) 3.34614i 0.134276i
\(622\) 0 0
\(623\) 42.6637i 1.70929i
\(624\) 0 0
\(625\) 21.2034 + 13.2444i 0.848137 + 0.529777i
\(626\) 0 0
\(627\) 55.8163 2.22909
\(628\) 0 0
\(629\) 41.7146i 1.66327i
\(630\) 0 0
\(631\) 27.8163 1.10735 0.553674 0.832733i \(-0.313225\pi\)
0.553674 + 0.832733i \(0.313225\pi\)
\(632\) 0 0
\(633\) 13.5397i 0.538155i
\(634\) 0 0
\(635\) 3.19850 22.7654i 0.126929 0.903418i
\(636\) 0 0
\(637\) 14.2351 0.564014
\(638\) 0 0
\(639\) 26.3684 1.04312
\(640\) 0 0
\(641\) 42.8988 1.69440 0.847200 0.531275i \(-0.178287\pi\)
0.847200 + 0.531275i \(0.178287\pi\)
\(642\) 0 0
\(643\) 27.9639 1.10279 0.551395 0.834245i \(-0.314096\pi\)
0.551395 + 0.834245i \(0.314096\pi\)
\(644\) 0 0
\(645\) 12.5620 + 1.76494i 0.494628 + 0.0694943i
\(646\) 0 0
\(647\) 7.13828i 0.280635i 0.990107 + 0.140317i \(0.0448123\pi\)
−0.990107 + 0.140317i \(0.955188\pi\)
\(648\) 0 0
\(649\) 19.2257 0.754675
\(650\) 0 0
\(651\) 28.2034i 1.10538i
\(652\) 0 0
\(653\) 17.7649 0.695196 0.347598 0.937644i \(-0.386997\pi\)
0.347598 + 0.937644i \(0.386997\pi\)
\(654\) 0 0
\(655\) −1.45091 + 10.3269i −0.0566919 + 0.403507i
\(656\) 0 0
\(657\) 36.1748i 1.41131i
\(658\) 0 0
\(659\) 20.1936i 0.786630i −0.919404 0.393315i \(-0.871328\pi\)
0.919404 0.393315i \(-0.128672\pi\)
\(660\) 0 0
\(661\) 22.0701i 0.858426i −0.903203 0.429213i \(-0.858791\pi\)
0.903203 0.429213i \(-0.141209\pi\)
\(662\) 0 0
\(663\) 44.2034i 1.71672i
\(664\) 0 0
\(665\) 39.4291 + 5.53972i 1.52900 + 0.214821i
\(666\) 0 0
\(667\) 0.949145 0.0367510
\(668\) 0 0
\(669\) 76.8069i 2.96953i
\(670\) 0 0
\(671\) 46.5718 1.79789
\(672\) 0 0
\(673\) 26.0098i 1.00261i −0.865272 0.501303i \(-0.832854\pi\)
0.865272 0.501303i \(-0.167146\pi\)
\(674\) 0 0
\(675\) 33.8894 + 9.71456i 1.30440 + 0.373914i
\(676\) 0 0
\(677\) 7.86665 0.302340 0.151170 0.988508i \(-0.451696\pi\)
0.151170 + 0.988508i \(0.451696\pi\)
\(678\) 0 0
\(679\) 48.6735 1.86792
\(680\) 0 0
\(681\) −35.7748 −1.37089
\(682\) 0 0
\(683\) −18.2494 −0.698292 −0.349146 0.937068i \(-0.613528\pi\)
−0.349146 + 0.937068i \(0.613528\pi\)
\(684\) 0 0
\(685\) −3.52987 + 25.1240i −0.134870 + 0.959938i
\(686\) 0 0
\(687\) 38.3970i 1.46494i
\(688\) 0 0
\(689\) −14.1017 −0.537232
\(690\) 0 0
\(691\) 25.2543i 0.960718i −0.877072 0.480359i \(-0.840506\pi\)
0.877072 0.480359i \(-0.159494\pi\)
\(692\) 0 0
\(693\) −72.8484 −2.76728
\(694\) 0 0
\(695\) −3.67307 + 26.1432i −0.139328 + 0.991668i
\(696\) 0 0
\(697\) 30.1017i 1.14018i
\(698\) 0 0
\(699\) 19.3461i 0.731738i
\(700\) 0 0
\(701\) 34.8069i 1.31464i 0.753612 + 0.657319i \(0.228310\pi\)
−0.753612 + 0.657319i \(0.771690\pi\)
\(702\) 0 0
\(703\) 36.2864i 1.36857i
\(704\) 0 0
\(705\) −4.81579 + 34.2766i −0.181373 + 1.29093i
\(706\) 0 0
\(707\) 69.5022 2.61390
\(708\) 0 0
\(709\) 7.51114i 0.282087i −0.990003 0.141043i \(-0.954954\pi\)
0.990003 0.141043i \(-0.0450457\pi\)
\(710\) 0 0
\(711\) 28.4701 1.06771
\(712\) 0 0
\(713\) 1.30772i 0.0489744i
\(714\) 0 0
\(715\) 3.10525 22.1017i 0.116130 0.826557i
\(716\) 0 0
\(717\) −66.6548 −2.48927
\(718\) 0 0
\(719\) −26.7556 −0.997814 −0.498907 0.866655i \(-0.666265\pi\)
−0.498907 + 0.866655i \(0.666265\pi\)
\(720\) 0 0
\(721\) −11.0638 −0.412036
\(722\) 0 0
\(723\) −40.7654 −1.51608
\(724\) 0 0
\(725\) −2.75557 + 9.61285i −0.102339 + 0.357012i
\(726\) 0 0
\(727\) 25.6271i 0.950458i 0.879862 + 0.475229i \(0.157635\pi\)
−0.879862 + 0.475229i \(0.842365\pi\)
\(728\) 0 0
\(729\) −38.6958 −1.43318
\(730\) 0 0
\(731\) 11.3461i 0.419652i
\(732\) 0 0
\(733\) 19.6543 0.725949 0.362975 0.931799i \(-0.381761\pi\)
0.362975 + 0.931799i \(0.381761\pi\)
\(734\) 0 0
\(735\) −34.8988 4.90321i −1.28726 0.180858i
\(736\) 0 0
\(737\) 29.5397i 1.08811i
\(738\) 0 0
\(739\) 32.5433i 1.19712i 0.801077 + 0.598562i \(0.204261\pi\)
−0.801077 + 0.598562i \(0.795739\pi\)
\(740\) 0 0
\(741\) 38.4514i 1.41255i
\(742\) 0 0
\(743\) 23.1383i 0.848861i 0.905461 + 0.424430i \(0.139526\pi\)
−0.905461 + 0.424430i \(0.860474\pi\)
\(744\) 0 0
\(745\) 1.70471 12.1334i 0.0624559 0.444532i
\(746\) 0 0
\(747\) 65.9452 2.41281
\(748\) 0 0
\(749\) 19.0094i 0.694587i
\(750\) 0 0
\(751\) −2.48886 −0.0908199 −0.0454099 0.998968i \(-0.514459\pi\)
−0.0454099 + 0.998968i \(0.514459\pi\)
\(752\) 0 0
\(753\) 72.3783i 2.63761i
\(754\) 0 0
\(755\) −52.2864 7.34614i −1.90290 0.267353i
\(756\) 0 0
\(757\) −21.2859 −0.773650 −0.386825 0.922153i \(-0.626428\pi\)
−0.386825 + 0.922153i \(0.626428\pi\)
\(758\) 0 0
\(759\) 5.24443 0.190361
\(760\) 0 0
\(761\) −19.1240 −0.693244 −0.346622 0.938005i \(-0.612671\pi\)
−0.346622 + 0.938005i \(0.612671\pi\)
\(762\) 0 0
\(763\) 30.3970 1.10045
\(764\) 0 0
\(765\) 9.80642 69.7975i 0.354552 2.52354i
\(766\) 0 0
\(767\) 13.2444i 0.478229i
\(768\) 0 0
\(769\) 33.9625 1.22472 0.612360 0.790579i \(-0.290220\pi\)
0.612360 + 0.790579i \(0.290220\pi\)
\(770\) 0 0
\(771\) 74.6548i 2.68863i
\(772\) 0 0
\(773\) −0.133353 −0.00479638 −0.00239819 0.999997i \(-0.500763\pi\)
−0.00239819 + 0.999997i \(0.500763\pi\)
\(774\) 0 0
\(775\) −13.2444 3.79658i −0.475754 0.136377i
\(776\) 0 0
\(777\) 73.5308i 2.63790i
\(778\) 0 0
\(779\) 26.1847i 0.938164i
\(780\) 0 0
\(781\) 18.4889i 0.661584i
\(782\) 0 0
\(783\) 14.1017i 0.503954i
\(784\) 0 0
\(785\) 1.80642 + 0.253799i 0.0644740 + 0.00905848i
\(786\) 0 0
\(787\) 1.12537 0.0401150 0.0200575 0.999799i \(-0.493615\pi\)
0.0200575 + 0.999799i \(0.493615\pi\)
\(788\) 0 0
\(789\) 7.47949i 0.266277i
\(790\) 0 0
\(791\) −19.4291 −0.690820
\(792\) 0 0
\(793\) 32.0830i 1.13930i
\(794\) 0 0
\(795\) 34.5718 + 4.85728i 1.22614 + 0.172270i
\(796\) 0 0
\(797\) −5.11108 −0.181044 −0.0905218 0.995894i \(-0.528853\pi\)
−0.0905218 + 0.995894i \(0.528853\pi\)
\(798\) 0 0
\(799\) 30.9590 1.09525
\(800\) 0 0
\(801\) −65.6958 −2.32125
\(802\) 0 0
\(803\) −25.3649 −0.895107
\(804\) 0 0
\(805\) 3.70471 + 0.520505i 0.130574 + 0.0183454i
\(806\) 0 0
\(807\) 74.7467i 2.63121i
\(808\) 0 0
\(809\) 18.7368 0.658752 0.329376 0.944199i \(-0.393162\pi\)
0.329376 + 0.944199i \(0.393162\pi\)
\(810\) 0 0
\(811\) 37.5210i 1.31754i 0.752344 + 0.658770i \(0.228923\pi\)
−0.752344 + 0.658770i \(0.771077\pi\)
\(812\) 0 0
\(813\) 87.6325 3.07341
\(814\) 0 0
\(815\) 24.1432 + 3.39207i 0.845699 + 0.118819i
\(816\) 0 0
\(817\) 9.86971i 0.345297i
\(818\) 0 0
\(819\) 50.1847i 1.75359i
\(820\) 0 0
\(821\) 18.4001i 0.642166i −0.947051 0.321083i \(-0.895953\pi\)
0.947051 0.321083i \(-0.104047\pi\)
\(822\) 0 0
\(823\) 0.649413i 0.0226371i 0.999936 + 0.0113186i \(0.00360288\pi\)
−0.999936 + 0.0113186i \(0.996397\pi\)
\(824\) 0 0
\(825\) −15.2257 + 53.1151i −0.530091 + 1.84923i
\(826\) 0 0
\(827\) −29.8622 −1.03841 −0.519205 0.854650i \(-0.673772\pi\)
−0.519205 + 0.854650i \(0.673772\pi\)
\(828\) 0 0
\(829\) 21.1753i 0.735449i −0.929935 0.367725i \(-0.880137\pi\)
0.929935 0.367725i \(-0.119863\pi\)
\(830\) 0 0
\(831\) 30.5718 1.06053
\(832\) 0 0
\(833\) 31.5210i 1.09214i
\(834\) 0 0
\(835\) 2.04593 14.5620i 0.0708024 0.503939i
\(836\) 0 0
\(837\) −19.4291 −0.671568
\(838\) 0 0
\(839\) −43.8163 −1.51271 −0.756353 0.654164i \(-0.773021\pi\)
−0.756353 + 0.654164i \(0.773021\pi\)
\(840\) 0 0
\(841\) 25.0000 0.862069
\(842\) 0 0
\(843\) 23.0509 0.793914
\(844\) 0 0
\(845\) −13.5605 1.90522i −0.466494 0.0655415i
\(846\) 0 0
\(847\) 12.2997i 0.422624i
\(848\) 0 0
\(849\) −31.6543 −1.08637
\(850\) 0 0
\(851\) 3.40943i 0.116874i
\(852\) 0 0
\(853\) 37.7846 1.29372 0.646860 0.762608i \(-0.276082\pi\)
0.646860 + 0.762608i \(0.276082\pi\)
\(854\) 0 0
\(855\) 8.53035 60.7150i 0.291732 2.07641i
\(856\) 0 0
\(857\) 19.5299i 0.667128i −0.942727 0.333564i \(-0.891749\pi\)
0.942727 0.333564i \(-0.108251\pi\)
\(858\) 0 0
\(859\) 33.2543i 1.13462i −0.823504 0.567311i \(-0.807984\pi\)
0.823504 0.567311i \(-0.192016\pi\)
\(860\) 0 0
\(861\) 53.0607i 1.80830i
\(862\) 0 0
\(863\) 44.9733i 1.53091i −0.643490 0.765454i \(-0.722514\pi\)
0.643490 0.765454i \(-0.277486\pi\)
\(864\) 0 0
\(865\) 23.3176 + 3.27607i 0.792821 + 0.111390i
\(866\) 0 0
\(867\) −48.5259 −1.64803
\(868\) 0 0
\(869\) 19.9625i 0.677182i
\(870\) 0 0
\(871\) −20.3497 −0.689523
\(872\) 0 0
\(873\) 74.9501i 2.53668i
\(874\) 0 0
\(875\) −16.0272 + 36.0098i −0.541818 + 1.21735i
\(876\) 0 0
\(877\) −8.30819 −0.280548 −0.140274 0.990113i \(-0.544798\pi\)
−0.140274 + 0.990113i \(0.544798\pi\)
\(878\) 0 0
\(879\) 12.8573 0.433665
\(880\) 0 0
\(881\) −39.3689 −1.32637 −0.663186 0.748455i \(-0.730796\pi\)
−0.663186 + 0.748455i \(0.730796\pi\)
\(882\) 0 0
\(883\) 7.29036 0.245340 0.122670 0.992447i \(-0.460854\pi\)
0.122670 + 0.992447i \(0.460854\pi\)
\(884\) 0 0
\(885\) 4.56199 32.4701i 0.153350 1.09147i
\(886\) 0 0
\(887\) 11.1097i 0.373027i −0.982452 0.186514i \(-0.940281\pi\)
0.982452 0.186514i \(-0.0597188\pi\)
\(888\) 0 0
\(889\) 36.2449 1.21562
\(890\) 0 0
\(891\) 15.9269i 0.533570i
\(892\) 0 0
\(893\) 26.9304 0.901192
\(894\) 0 0
\(895\) −1.95851 + 13.9398i −0.0654659 + 0.465955i
\(896\) 0 0
\(897\) 3.61285i 0.120629i
\(898\) 0 0
\(899\) 5.51114i 0.183807i
\(900\) 0 0
\(901\) 31.2257i 1.04028i
\(902\) 0 0
\(903\) 20.0000i 0.665558i
\(904\) 0 0
\(905\) −0.152089 + 1.08250i −0.00505561 + 0.0359835i
\(906\) 0 0
\(907\) −17.5383 −0.582351 −0.291175 0.956670i \(-0.594046\pi\)
−0.291175 + 0.956670i \(0.594046\pi\)
\(908\) 0 0
\(909\) 107.023i 3.54974i
\(910\) 0 0
\(911\) 44.4701 1.47336 0.736681 0.676241i \(-0.236392\pi\)
0.736681 + 0.676241i \(0.236392\pi\)
\(912\) 0 0
\(913\) 46.2391i 1.53029i
\(914\) 0 0
\(915\) 11.0509 78.6548i 0.365330 2.60025i
\(916\) 0 0
\(917\) −16.4415 −0.542948
\(918\) 0 0
\(919\) 7.87955 0.259922 0.129961 0.991519i \(-0.458515\pi\)
0.129961 + 0.991519i \(0.458515\pi\)
\(920\) 0 0
\(921\) 15.3047 0.504306
\(922\) 0 0
\(923\) −12.7368 −0.419238
\(924\) 0 0
\(925\) −34.5303 9.89829i −1.13535 0.325454i
\(926\) 0 0
\(927\) 17.0366i 0.559554i
\(928\) 0 0
\(929\) −15.7560 −0.516939 −0.258470 0.966019i \(-0.583218\pi\)
−0.258470 + 0.966019i \(0.583218\pi\)
\(930\) 0 0
\(931\) 27.4193i 0.898630i
\(932\) 0 0
\(933\) −1.12399 −0.0367976
\(934\) 0 0
\(935\) 48.9403 + 6.87601i 1.60052 + 0.224870i
\(936\) 0 0
\(937\) 10.2766i 0.335720i −0.985811 0.167860i \(-0.946314\pi\)
0.985811 0.167860i \(-0.0536857\pi\)
\(938\) 0 0
\(939\) 32.9403i 1.07496i
\(940\) 0 0
\(941\) 2.53341i 0.0825869i 0.999147 + 0.0412934i \(0.0131478\pi\)
−0.999147 + 0.0412934i \(0.986852\pi\)
\(942\) 0 0
\(943\) 2.46028i 0.0801178i
\(944\) 0 0
\(945\) −7.73329 + 55.0420i −0.251564 + 1.79052i
\(946\) 0 0
\(947\) 1.06821 0.0347121 0.0173560 0.999849i \(-0.494475\pi\)
0.0173560 + 0.999849i \(0.494475\pi\)
\(948\) 0 0
\(949\) 17.4737i 0.567219i
\(950\) 0 0
\(951\) −48.5531 −1.57444
\(952\) 0 0
\(953\) 51.6958i 1.67459i 0.546750 + 0.837296i \(0.315865\pi\)
−0.546750 + 0.837296i \(0.684135\pi\)
\(954\) 0 0
\(955\) −23.2257 3.26317i −0.751566 0.105594i
\(956\) 0 0
\(957\) −22.1017 −0.714447
\(958\) 0 0
\(959\) −40.0000 −1.29167
\(960\) 0 0
\(961\) −23.4068 −0.755059
\(962\) 0 0
\(963\) −29.2716 −0.943265
\(964\) 0 0
\(965\) −4.29529 + 30.5718i −0.138270 + 0.984142i
\(966\) 0 0
\(967\) 52.2623i 1.68064i 0.542090 + 0.840320i \(0.317633\pi\)
−0.542090 + 0.840320i \(0.682367\pi\)
\(968\) 0 0
\(969\) −85.1437 −2.73521
\(970\) 0 0
\(971\) 59.3560i 1.90482i 0.304813 + 0.952412i \(0.401406\pi\)
−0.304813 + 0.952412i \(0.598594\pi\)
\(972\) 0 0
\(973\) −41.6227 −1.33436
\(974\) 0 0
\(975\) −36.5906 10.4889i −1.17184 0.335912i
\(976\) 0 0
\(977\) 47.8707i 1.53152i −0.643128 0.765759i \(-0.722363\pi\)
0.643128 0.765759i \(-0.277637\pi\)
\(978\) 0 0
\(979\) 46.0642i 1.47222i
\(980\) 0 0
\(981\) 46.8069i 1.49443i
\(982\) 0 0
\(983\) 6.01429i 0.191826i 0.995390 + 0.0959130i \(0.0305771\pi\)
−0.995390 + 0.0959130i \(0.969423\pi\)
\(984\) 0 0
\(985\) −37.0321 5.20294i −1.17994 0.165780i
\(986\) 0 0
\(987\) −54.5718 −1.73704
\(988\) 0 0
\(989\) 0.927346i 0.0294879i
\(990\) 0 0
\(991\) −19.4291 −0.617186 −0.308593 0.951194i \(-0.599858\pi\)
−0.308593 + 0.951194i \(0.599858\pi\)
\(992\) 0 0
\(993\) 9.68598i 0.307375i
\(994\) 0 0
\(995\) 46.1847 + 6.48886i 1.46415 + 0.205711i
\(996\) 0 0
\(997\) −32.1334 −1.01767 −0.508837 0.860863i \(-0.669924\pi\)
−0.508837 + 0.860863i \(0.669924\pi\)
\(998\) 0 0
\(999\) −50.6548 −1.60265
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 1280.2.f.k.129.6 6
4.3 odd 2 1280.2.f.i.129.2 6
5.4 even 2 1280.2.f.j.129.2 6
8.3 odd 2 1280.2.f.l.129.5 6
8.5 even 2 1280.2.f.j.129.1 6
16.3 odd 4 640.2.c.d.129.1 yes 6
16.5 even 4 640.2.c.b.129.1 yes 6
16.11 odd 4 640.2.c.a.129.6 yes 6
16.13 even 4 640.2.c.c.129.6 yes 6
20.19 odd 2 1280.2.f.l.129.6 6
40.19 odd 2 1280.2.f.i.129.1 6
40.29 even 2 inner 1280.2.f.k.129.5 6
80.3 even 4 3200.2.a.bv.1.3 3
80.13 odd 4 3200.2.a.bo.1.1 3
80.19 odd 4 640.2.c.d.129.6 yes 6
80.27 even 4 3200.2.a.bs.1.3 3
80.29 even 4 640.2.c.c.129.1 yes 6
80.37 odd 4 3200.2.a.br.1.1 3
80.43 even 4 3200.2.a.bq.1.1 3
80.53 odd 4 3200.2.a.bt.1.3 3
80.59 odd 4 640.2.c.a.129.1 6
80.67 even 4 3200.2.a.bp.1.1 3
80.69 even 4 640.2.c.b.129.6 yes 6
80.77 odd 4 3200.2.a.bu.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.c.a.129.1 6 80.59 odd 4
640.2.c.a.129.6 yes 6 16.11 odd 4
640.2.c.b.129.1 yes 6 16.5 even 4
640.2.c.b.129.6 yes 6 80.69 even 4
640.2.c.c.129.1 yes 6 80.29 even 4
640.2.c.c.129.6 yes 6 16.13 even 4
640.2.c.d.129.1 yes 6 16.3 odd 4
640.2.c.d.129.6 yes 6 80.19 odd 4
1280.2.f.i.129.1 6 40.19 odd 2
1280.2.f.i.129.2 6 4.3 odd 2
1280.2.f.j.129.1 6 8.5 even 2
1280.2.f.j.129.2 6 5.4 even 2
1280.2.f.k.129.5 6 40.29 even 2 inner
1280.2.f.k.129.6 6 1.1 even 1 trivial
1280.2.f.l.129.5 6 8.3 odd 2
1280.2.f.l.129.6 6 20.19 odd 2
3200.2.a.bo.1.1 3 80.13 odd 4
3200.2.a.bp.1.1 3 80.67 even 4
3200.2.a.bq.1.1 3 80.43 even 4
3200.2.a.br.1.1 3 80.37 odd 4
3200.2.a.bs.1.3 3 80.27 even 4
3200.2.a.bt.1.3 3 80.53 odd 4
3200.2.a.bu.1.3 3 80.77 odd 4
3200.2.a.bv.1.3 3 80.3 even 4