Properties

Label 640.2.c.a.129.1
Level $640$
Weight $2$
Character 640.129
Analytic conductor $5.110$
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] = [640,2,Mod(129,640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(640, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("640.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 640.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.11042572936\)
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: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.1
Root \(1.45161 - 1.45161i\) of defining polynomial
Character \(\chi\) \(=\) 640.129
Dual form 640.2.c.a.129.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.90321i q^{3} +(-0.311108 - 2.21432i) q^{5} -3.52543i q^{7} -5.42864 q^{9} +O(q^{10})\) \(q-2.90321i q^{3} +(-0.311108 - 2.21432i) q^{5} -3.52543i q^{7} -5.42864 q^{9} +3.80642 q^{11} -2.62222i q^{13} +(-6.42864 + 0.903212i) q^{15} +5.80642i q^{17} +5.05086 q^{19} -10.2351 q^{21} +0.474572i q^{23} +(-4.80642 + 1.37778i) q^{25} +7.05086i q^{27} +2.00000 q^{29} +2.75557 q^{31} -11.0509i q^{33} +(-7.80642 + 1.09679i) q^{35} +7.18421i q^{37} -7.61285 q^{39} +5.18421 q^{41} +1.95407i q^{43} +(1.68889 + 12.0207i) q^{45} +5.33185i q^{47} -5.42864 q^{49} +16.8573 q^{51} -5.37778i q^{53} +(-1.18421 - 8.42864i) q^{55} -14.6637i q^{57} -5.05086 q^{59} -12.2351 q^{61} +19.1383i q^{63} +(-5.80642 + 0.815792i) q^{65} -7.76049i q^{67} +1.37778 q^{69} +4.85728 q^{71} +6.66370i q^{73} +(4.00000 + 13.9541i) q^{75} -13.4193i q^{77} -5.24443 q^{79} +4.18421 q^{81} -12.1476i q^{83} +(12.8573 - 1.80642i) q^{85} -5.80642i q^{87} +12.1017 q^{89} -9.24443 q^{91} -8.00000i q^{93} +(-1.57136 - 11.1842i) q^{95} +13.8064i q^{97} -20.6637 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{5} - 6 q^{9} - 4 q^{11} - 12 q^{15} + 4 q^{19} - 8 q^{21} - 2 q^{25} + 12 q^{29} + 16 q^{31} - 20 q^{35} + 8 q^{39} + 4 q^{41} + 10 q^{45} - 6 q^{49} + 48 q^{51} + 20 q^{55} - 4 q^{59} - 20 q^{61} - 8 q^{65} + 8 q^{69} - 24 q^{71} + 24 q^{75} - 32 q^{79} - 2 q^{81} + 24 q^{85} + 20 q^{89} - 56 q^{91} - 36 q^{95} - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/640\mathbb{Z}\right)^\times\).

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

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
\(3\) 2.90321i 1.67617i −0.545540 0.838085i \(-0.683675\pi\)
0.545540 0.838085i \(-0.316325\pi\)
\(4\) 0 0
\(5\) −0.311108 2.21432i −0.139132 0.990274i
\(6\) 0 0
\(7\) 3.52543i 1.33249i −0.745735 0.666243i \(-0.767901\pi\)
0.745735 0.666243i \(-0.232099\pi\)
\(8\) 0 0
\(9\) −5.42864 −1.80955
\(10\) 0 0
\(11\) 3.80642 1.14768 0.573840 0.818967i \(-0.305453\pi\)
0.573840 + 0.818967i \(0.305453\pi\)
\(12\) 0 0
\(13\) 2.62222i 0.727272i −0.931541 0.363636i \(-0.881535\pi\)
0.931541 0.363636i \(-0.118465\pi\)
\(14\) 0 0
\(15\) −6.42864 + 0.903212i −1.65987 + 0.233208i
\(16\) 0 0
\(17\) 5.80642i 1.40826i 0.710069 + 0.704132i \(0.248664\pi\)
−0.710069 + 0.704132i \(0.751336\pi\)
\(18\) 0 0
\(19\) 5.05086 1.15875 0.579373 0.815063i \(-0.303298\pi\)
0.579373 + 0.815063i \(0.303298\pi\)
\(20\) 0 0
\(21\) −10.2351 −2.23347
\(22\) 0 0
\(23\) 0.474572i 0.0989552i 0.998775 + 0.0494776i \(0.0157556\pi\)
−0.998775 + 0.0494776i \(0.984244\pi\)
\(24\) 0 0
\(25\) −4.80642 + 1.37778i −0.961285 + 0.275557i
\(26\) 0 0
\(27\) 7.05086i 1.35694i
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 2.75557 0.494915 0.247457 0.968899i \(-0.420405\pi\)
0.247457 + 0.968899i \(0.420405\pi\)
\(32\) 0 0
\(33\) 11.0509i 1.92371i
\(34\) 0 0
\(35\) −7.80642 + 1.09679i −1.31953 + 0.185391i
\(36\) 0 0
\(37\) 7.18421i 1.18108i 0.807010 + 0.590538i \(0.201085\pi\)
−0.807010 + 0.590538i \(0.798915\pi\)
\(38\) 0 0
\(39\) −7.61285 −1.21903
\(40\) 0 0
\(41\) 5.18421 0.809637 0.404819 0.914397i \(-0.367335\pi\)
0.404819 + 0.914397i \(0.367335\pi\)
\(42\) 0 0
\(43\) 1.95407i 0.297992i 0.988838 + 0.148996i \(0.0476042\pi\)
−0.988838 + 0.148996i \(0.952396\pi\)
\(44\) 0 0
\(45\) 1.68889 + 12.0207i 0.251765 + 1.79195i
\(46\) 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
\(51\) 16.8573 2.36049
\(52\) 0 0
\(53\) 5.37778i 0.738695i −0.929291 0.369348i \(-0.879581\pi\)
0.929291 0.369348i \(-0.120419\pi\)
\(54\) 0 0
\(55\) −1.18421 8.42864i −0.159679 1.13652i
\(56\) 0 0
\(57\) 14.6637i 1.94225i
\(58\) 0 0
\(59\) −5.05086 −0.657565 −0.328783 0.944406i \(-0.606638\pi\)
−0.328783 + 0.944406i \(0.606638\pi\)
\(60\) 0 0
\(61\) −12.2351 −1.56654 −0.783270 0.621682i \(-0.786450\pi\)
−0.783270 + 0.621682i \(0.786450\pi\)
\(62\) 0 0
\(63\) 19.1383i 2.41120i
\(64\) 0 0
\(65\) −5.80642 + 0.815792i −0.720198 + 0.101187i
\(66\) 0 0
\(67\) 7.76049i 0.948095i −0.880499 0.474047i \(-0.842793\pi\)
0.880499 0.474047i \(-0.157207\pi\)
\(68\) 0 0
\(69\) 1.37778 0.165866
\(70\) 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
\(75\) 4.00000 + 13.9541i 0.461880 + 1.61128i
\(76\) 0 0
\(77\) 13.4193i 1.52927i
\(78\) 0 0
\(79\) −5.24443 −0.590045 −0.295022 0.955490i \(-0.595327\pi\)
−0.295022 + 0.955490i \(0.595327\pi\)
\(80\) 0 0
\(81\) 4.18421 0.464912
\(82\) 0 0
\(83\) 12.1476i 1.33338i −0.745336 0.666689i \(-0.767711\pi\)
0.745336 0.666689i \(-0.232289\pi\)
\(84\) 0 0
\(85\) 12.8573 1.80642i 1.39457 0.195934i
\(86\) 0 0
\(87\) 5.80642i 0.622514i
\(88\) 0 0
\(89\) 12.1017 1.28278 0.641389 0.767216i \(-0.278358\pi\)
0.641389 + 0.767216i \(0.278358\pi\)
\(90\) 0 0
\(91\) −9.24443 −0.969080
\(92\) 0 0
\(93\) 8.00000i 0.829561i
\(94\) 0 0
\(95\) −1.57136 11.1842i −0.161218 1.14748i
\(96\) 0 0
\(97\) 13.8064i 1.40183i 0.713245 + 0.700915i \(0.247225\pi\)
−0.713245 + 0.700915i \(0.752775\pi\)
\(98\) 0 0
\(99\) −20.6637 −2.07678
\(100\) 0 0
\(101\) 19.7146 1.96167 0.980836 0.194836i \(-0.0624173\pi\)
0.980836 + 0.194836i \(0.0624173\pi\)
\(102\) 0 0
\(103\) 3.13828i 0.309223i −0.987975 0.154612i \(-0.950587\pi\)
0.987975 0.154612i \(-0.0494127\pi\)
\(104\) 0 0
\(105\) 3.18421 + 22.6637i 0.310747 + 2.21175i
\(106\) 0 0
\(107\) 5.39207i 0.521272i −0.965437 0.260636i \(-0.916068\pi\)
0.965437 0.260636i \(-0.0839322\pi\)
\(108\) 0 0
\(109\) −8.62222 −0.825858 −0.412929 0.910763i \(-0.635494\pi\)
−0.412929 + 0.910763i \(0.635494\pi\)
\(110\) 0 0
\(111\) 20.8573 1.97969
\(112\) 0 0
\(113\) 5.51114i 0.518444i −0.965818 0.259222i \(-0.916534\pi\)
0.965818 0.259222i \(-0.0834662\pi\)
\(114\) 0 0
\(115\) 1.05086 0.147643i 0.0979927 0.0137678i
\(116\) 0 0
\(117\) 14.2351i 1.31603i
\(118\) 0 0
\(119\) 20.4701 1.87649
\(120\) 0 0
\(121\) 3.48886 0.317169
\(122\) 0 0
\(123\) 15.0509i 1.35709i
\(124\) 0 0
\(125\) 4.54617 + 10.2143i 0.406622 + 0.913597i
\(126\) 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
\(131\) −4.66370 −0.407470 −0.203735 0.979026i \(-0.565308\pi\)
−0.203735 + 0.979026i \(0.565308\pi\)
\(132\) 0 0
\(133\) 17.8064i 1.54401i
\(134\) 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
\(139\) 11.8064 1.00141 0.500704 0.865619i \(-0.333075\pi\)
0.500704 + 0.865619i \(0.333075\pi\)
\(140\) 0 0
\(141\) 15.4795 1.30361
\(142\) 0 0
\(143\) 9.98126i 0.834675i
\(144\) 0 0
\(145\) −0.622216 4.42864i −0.0516722 0.367778i
\(146\) 0 0
\(147\) 15.7605i 1.29990i
\(148\) 0 0
\(149\) 5.47949 0.448898 0.224449 0.974486i \(-0.427942\pi\)
0.224449 + 0.974486i \(0.427942\pi\)
\(150\) 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
\(155\) −0.857279 6.10171i −0.0688583 0.490101i
\(156\) 0 0
\(157\) 0.815792i 0.0651073i 0.999470 + 0.0325536i \(0.0103640\pi\)
−0.999470 + 0.0325536i \(0.989636\pi\)
\(158\) 0 0
\(159\) −15.6128 −1.23818
\(160\) 0 0
\(161\) 1.67307 0.131856
\(162\) 0 0
\(163\) 10.9032i 0.854005i −0.904250 0.427003i \(-0.859569\pi\)
0.904250 0.427003i \(-0.140431\pi\)
\(164\) 0 0
\(165\) −24.4701 + 3.43801i −1.90500 + 0.267649i
\(166\) 0 0
\(167\) 6.57628i 0.508888i 0.967088 + 0.254444i \(0.0818925\pi\)
−0.967088 + 0.254444i \(0.918108\pi\)
\(168\) 0 0
\(169\) 6.12399 0.471076
\(170\) 0 0
\(171\) −27.4193 −2.09680
\(172\) 0 0
\(173\) 10.5303i 0.800608i 0.916382 + 0.400304i \(0.131095\pi\)
−0.916382 + 0.400304i \(0.868905\pi\)
\(174\) 0 0
\(175\) 4.85728 + 16.9447i 0.367176 + 1.28090i
\(176\) 0 0
\(177\) 14.6637i 1.10219i
\(178\) 0 0
\(179\) −6.29529 −0.470532 −0.235266 0.971931i \(-0.575596\pi\)
−0.235266 + 0.971931i \(0.575596\pi\)
\(180\) 0 0
\(181\) −0.488863 −0.0363369 −0.0181684 0.999835i \(-0.505784\pi\)
−0.0181684 + 0.999835i \(0.505784\pi\)
\(182\) 0 0
\(183\) 35.5210i 2.62579i
\(184\) 0 0
\(185\) 15.9081 2.23506i 1.16959 0.164325i
\(186\) 0 0
\(187\) 22.1017i 1.61624i
\(188\) 0 0
\(189\) 24.8573 1.80810
\(190\) 0 0
\(191\) 10.4889 0.758947 0.379474 0.925203i \(-0.376105\pi\)
0.379474 + 0.925203i \(0.376105\pi\)
\(192\) 0 0
\(193\) 13.8064i 0.993808i −0.867805 0.496904i \(-0.834470\pi\)
0.867805 0.496904i \(-0.165530\pi\)
\(194\) 0 0
\(195\) 2.36842 + 16.8573i 0.169606 + 1.20717i
\(196\) 0 0
\(197\) 16.7239i 1.19153i 0.803159 + 0.595765i \(0.203151\pi\)
−0.803159 + 0.595765i \(0.796849\pi\)
\(198\) 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
\(203\) 7.05086i 0.494873i
\(204\) 0 0
\(205\) −1.61285 11.4795i −0.112646 0.801763i
\(206\) 0 0
\(207\) 2.57628i 0.179064i
\(208\) 0 0
\(209\) 19.2257 1.32987
\(210\) 0 0
\(211\) −4.66370 −0.321063 −0.160531 0.987031i \(-0.551321\pi\)
−0.160531 + 0.987031i \(0.551321\pi\)
\(212\) 0 0
\(213\) 14.1017i 0.966233i
\(214\) 0 0
\(215\) 4.32693 0.607926i 0.295094 0.0414602i
\(216\) 0 0
\(217\) 9.71456i 0.659467i
\(218\) 0 0
\(219\) 19.3461 1.30729
\(220\) 0 0
\(221\) 15.2257 1.02419
\(222\) 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
\(227\) 12.3225i 0.817872i 0.912563 + 0.408936i \(0.134100\pi\)
−0.912563 + 0.408936i \(0.865900\pi\)
\(228\) 0 0
\(229\) −13.2257 −0.873979 −0.436989 0.899467i \(-0.643955\pi\)
−0.436989 + 0.899467i \(0.643955\pi\)
\(230\) 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
\(235\) 11.8064 1.65878i 0.770166 0.108207i
\(236\) 0 0
\(237\) 15.2257i 0.989015i
\(238\) 0 0
\(239\) 22.9590 1.48509 0.742547 0.669794i \(-0.233618\pi\)
0.742547 + 0.669794i \(0.233618\pi\)
\(240\) 0 0
\(241\) −14.0415 −0.904492 −0.452246 0.891893i \(-0.649377\pi\)
−0.452246 + 0.891893i \(0.649377\pi\)
\(242\) 0 0
\(243\) 9.00492i 0.577666i
\(244\) 0 0
\(245\) 1.68889 + 12.0207i 0.107899 + 0.767977i
\(246\) 0 0
\(247\) 13.2444i 0.842723i
\(248\) 0 0
\(249\) −35.2672 −2.23497
\(250\) 0 0
\(251\) −24.9304 −1.57359 −0.786797 0.617212i \(-0.788262\pi\)
−0.786797 + 0.617212i \(0.788262\pi\)
\(252\) 0 0
\(253\) 1.80642i 0.113569i
\(254\) 0 0
\(255\) −5.24443 37.3274i −0.328419 2.33753i
\(256\) 0 0
\(257\) 25.7146i 1.60403i −0.597304 0.802015i \(-0.703761\pi\)
0.597304 0.802015i \(-0.296239\pi\)
\(258\) 0 0
\(259\) 25.3274 1.57377
\(260\) 0 0
\(261\) −10.8573 −0.672049
\(262\) 0 0
\(263\) 2.57628i 0.158860i 0.996840 + 0.0794302i \(0.0253101\pi\)
−0.996840 + 0.0794302i \(0.974690\pi\)
\(264\) 0 0
\(265\) −11.9081 + 1.67307i −0.731511 + 0.102776i
\(266\) 0 0
\(267\) 35.1338i 2.15016i
\(268\) 0 0
\(269\) −25.7462 −1.56977 −0.784887 0.619639i \(-0.787279\pi\)
−0.784887 + 0.619639i \(0.787279\pi\)
\(270\) 0 0
\(271\) −30.1847 −1.83359 −0.916795 0.399359i \(-0.869233\pi\)
−0.916795 + 0.399359i \(0.869233\pi\)
\(272\) 0 0
\(273\) 26.8385i 1.62434i
\(274\) 0 0
\(275\) −18.2953 + 5.24443i −1.10325 + 0.316251i
\(276\) 0 0
\(277\) 10.5303i 0.632707i −0.948641 0.316354i \(-0.897541\pi\)
0.948641 0.316354i \(-0.102459\pi\)
\(278\) 0 0
\(279\) −14.9590 −0.895571
\(280\) 0 0
\(281\) −7.93978 −0.473647 −0.236824 0.971553i \(-0.576106\pi\)
−0.236824 + 0.971553i \(0.576106\pi\)
\(282\) 0 0
\(283\) 10.9032i 0.648129i −0.946035 0.324064i \(-0.894951\pi\)
0.946035 0.324064i \(-0.105049\pi\)
\(284\) 0 0
\(285\) −32.4701 + 4.56199i −1.92336 + 0.270229i
\(286\) 0 0
\(287\) 18.2766i 1.07883i
\(288\) 0 0
\(289\) −16.7146 −0.983209
\(290\) 0 0
\(291\) 40.0830 2.34971
\(292\) 0 0
\(293\) 4.42864i 0.258724i −0.991597 0.129362i \(-0.958707\pi\)
0.991597 0.129362i \(-0.0412929\pi\)
\(294\) 0 0
\(295\) 1.57136 + 11.1842i 0.0914881 + 0.651170i
\(296\) 0 0
\(297\) 26.8385i 1.55733i
\(298\) 0 0
\(299\) 1.24443 0.0719673
\(300\) 0 0
\(301\) 6.88892 0.397071
\(302\) 0 0
\(303\) 57.2355i 3.28810i
\(304\) 0 0
\(305\) 3.80642 + 27.0923i 0.217955 + 1.55130i
\(306\) 0 0
\(307\) 5.27163i 0.300868i −0.988620 0.150434i \(-0.951933\pi\)
0.988620 0.150434i \(-0.0480671\pi\)
\(308\) 0 0
\(309\) −9.11108 −0.518311
\(310\) 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
\(315\) 42.3783 5.95407i 2.38774 0.335474i
\(316\) 0 0
\(317\) 16.7239i 0.939309i −0.882850 0.469655i \(-0.844378\pi\)
0.882850 0.469655i \(-0.155622\pi\)
\(318\) 0 0
\(319\) 7.61285 0.426238
\(320\) 0 0
\(321\) −15.6543 −0.873740
\(322\) 0 0
\(323\) 29.3274i 1.63182i
\(324\) 0 0
\(325\) 3.61285 + 12.6035i 0.200405 + 0.699115i
\(326\) 0 0
\(327\) 25.0321i 1.38428i
\(328\) 0 0
\(329\) 18.7971 1.03632
\(330\) 0 0
\(331\) −3.33630 −0.183379 −0.0916897 0.995788i \(-0.529227\pi\)
−0.0916897 + 0.995788i \(0.529227\pi\)
\(332\) 0 0
\(333\) 39.0005i 2.13721i
\(334\) 0 0
\(335\) −17.1842 + 2.41435i −0.938874 + 0.131910i
\(336\) 0 0
\(337\) 3.61285i 0.196804i −0.995147 0.0984022i \(-0.968627\pi\)
0.995147 0.0984022i \(-0.0313732\pi\)
\(338\) 0 0
\(339\) −16.0000 −0.869001
\(340\) 0 0
\(341\) 10.4889 0.568004
\(342\) 0 0
\(343\) 5.53972i 0.299117i
\(344\) 0 0
\(345\) −0.428639 3.05086i −0.0230772 0.164253i
\(346\) 0 0
\(347\) 15.7605i 0.846067i −0.906114 0.423034i \(-0.860965\pi\)
0.906114 0.423034i \(-0.139035\pi\)
\(348\) 0 0
\(349\) 0.285442 0.0152794 0.00763968 0.999971i \(-0.497568\pi\)
0.00763968 + 0.999971i \(0.497568\pi\)
\(350\) 0 0
\(351\) 18.4889 0.986862
\(352\) 0 0
\(353\) 4.38715i 0.233505i 0.993161 + 0.116752i \(0.0372484\pi\)
−0.993161 + 0.116752i \(0.962752\pi\)
\(354\) 0 0
\(355\) −1.51114 10.7556i −0.0802028 0.570846i
\(356\) 0 0
\(357\) 59.4291i 3.14532i
\(358\) 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
\(363\) 10.1289i 0.531630i
\(364\) 0 0
\(365\) 14.7556 2.07313i 0.772342 0.108513i
\(366\) 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
\(371\) −18.9590 −0.984302
\(372\) 0 0
\(373\) 24.8988i 1.28921i 0.764516 + 0.644605i \(0.222978\pi\)
−0.764516 + 0.644605i \(0.777022\pi\)
\(374\) 0 0
\(375\) 29.6543 13.1985i 1.53134 0.681568i
\(376\) 0 0
\(377\) 5.24443i 0.270102i
\(378\) 0 0
\(379\) 9.31756 0.478611 0.239305 0.970944i \(-0.423080\pi\)
0.239305 + 0.970944i \(0.423080\pi\)
\(380\) 0 0
\(381\) −29.8479 −1.52915
\(382\) 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
\(387\) 10.6079i 0.539231i
\(388\) 0 0
\(389\) 18.9906 0.962863 0.481432 0.876484i \(-0.340117\pi\)
0.481432 + 0.876484i \(0.340117\pi\)
\(390\) 0 0
\(391\) −2.75557 −0.139355
\(392\) 0 0
\(393\) 13.5397i 0.682988i
\(394\) 0 0
\(395\) 1.63158 + 11.6128i 0.0820939 + 0.584306i
\(396\) 0 0
\(397\) 32.9906i 1.65575i 0.560911 + 0.827876i \(0.310451\pi\)
−0.560911 + 0.827876i \(0.689549\pi\)
\(398\) 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
\(403\) 7.22570i 0.359938i
\(404\) 0 0
\(405\) −1.30174 9.26517i −0.0646840 0.460390i
\(406\) 0 0
\(407\) 27.3461i 1.35550i
\(408\) 0 0
\(409\) 33.3876 1.65091 0.825456 0.564466i \(-0.190918\pi\)
0.825456 + 0.564466i \(0.190918\pi\)
\(410\) 0 0
\(411\) 32.9403 1.62482
\(412\) 0 0
\(413\) 17.8064i 0.876197i
\(414\) 0 0
\(415\) −26.8988 + 3.77923i −1.32041 + 0.185515i
\(416\) 0 0
\(417\) 34.2766i 1.67853i
\(418\) 0 0
\(419\) 27.4193 1.33952 0.669760 0.742578i \(-0.266397\pi\)
0.669760 + 0.742578i \(0.266397\pi\)
\(420\) 0 0
\(421\) −12.2351 −0.596301 −0.298150 0.954519i \(-0.596370\pi\)
−0.298150 + 0.954519i \(0.596370\pi\)
\(422\) 0 0
\(423\) 28.9447i 1.40734i
\(424\) 0 0
\(425\) −8.00000 27.9081i −0.388057 1.35374i
\(426\) 0 0
\(427\) 43.1338i 2.08739i
\(428\) 0 0
\(429\) −28.9777 −1.39906
\(430\) 0 0
\(431\) 28.4701 1.37136 0.685679 0.727904i \(-0.259505\pi\)
0.685679 + 0.727904i \(0.259505\pi\)
\(432\) 0 0
\(433\) 34.0098i 1.63441i −0.576348 0.817204i \(-0.695523\pi\)
0.576348 0.817204i \(-0.304477\pi\)
\(434\) 0 0
\(435\) −12.8573 + 1.80642i −0.616459 + 0.0866114i
\(436\) 0 0
\(437\) 2.39700i 0.114664i
\(438\) 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
\(443\) 13.0968i 0.622247i 0.950369 + 0.311124i \(0.100705\pi\)
−0.950369 + 0.311124i \(0.899295\pi\)
\(444\) 0 0
\(445\) −3.76494 26.7971i −0.178475 1.27030i
\(446\) 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
\(451\) 19.7333 0.929205
\(452\) 0 0
\(453\) 68.5531i 3.22091i
\(454\) 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
\(459\) −40.9403 −1.91093
\(460\) 0 0
\(461\) −10.7368 −0.500064 −0.250032 0.968238i \(-0.580441\pi\)
−0.250032 + 0.968238i \(0.580441\pi\)
\(462\) 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
\(467\) 8.70964i 0.403034i 0.979485 + 0.201517i \(0.0645871\pi\)
−0.979485 + 0.201517i \(0.935413\pi\)
\(468\) 0 0
\(469\) −27.3590 −1.26332
\(470\) 0 0
\(471\) 2.36842 0.109131
\(472\) 0 0
\(473\) 7.43801i 0.342000i
\(474\) 0 0
\(475\) −24.2766 + 6.95899i −1.11388 + 0.319300i
\(476\) 0 0
\(477\) 29.1941i 1.33670i
\(478\) 0 0
\(479\) 23.2257 1.06121 0.530605 0.847619i \(-0.321965\pi\)
0.530605 + 0.847619i \(0.321965\pi\)
\(480\) 0 0
\(481\) 18.8385 0.858964
\(482\) 0 0
\(483\) 4.85728i 0.221014i
\(484\) 0 0
\(485\) 30.5718 4.29529i 1.38820 0.195039i
\(486\) 0 0
\(487\) 32.8528i 1.48870i −0.667787 0.744352i \(-0.732759\pi\)
0.667787 0.744352i \(-0.267241\pi\)
\(488\) 0 0
\(489\) −31.6543 −1.43146
\(490\) 0 0
\(491\) 2.94914 0.133093 0.0665465 0.997783i \(-0.478802\pi\)
0.0665465 + 0.997783i \(0.478802\pi\)
\(492\) 0 0
\(493\) 11.6128i 0.523016i
\(494\) 0 0
\(495\) 6.42864 + 45.7560i 0.288946 + 2.05658i
\(496\) 0 0
\(497\) 17.1240i 0.768116i
\(498\) 0 0
\(499\) −35.0321 −1.56825 −0.784127 0.620601i \(-0.786889\pi\)
−0.784127 + 0.620601i \(0.786889\pi\)
\(500\) 0 0
\(501\) 19.0923 0.852983
\(502\) 0 0
\(503\) 16.2908i 0.726373i 0.931717 + 0.363186i \(0.118311\pi\)
−0.931717 + 0.363186i \(0.881689\pi\)
\(504\) 0 0
\(505\) −6.13335 43.6543i −0.272931 1.94259i
\(506\) 0 0
\(507\) 17.7792i 0.789603i
\(508\) 0 0
\(509\) 26.0000 1.15243 0.576215 0.817298i \(-0.304529\pi\)
0.576215 + 0.817298i \(0.304529\pi\)
\(510\) 0 0
\(511\) 23.4924 1.03924
\(512\) 0 0
\(513\) 35.6128i 1.57235i
\(514\) 0 0
\(515\) −6.94914 + 0.976342i −0.306216 + 0.0430228i
\(516\) 0 0
\(517\) 20.2953i 0.892586i
\(518\) 0 0
\(519\) 30.5718 1.34195
\(520\) 0 0
\(521\) 11.7146 0.513224 0.256612 0.966514i \(-0.417394\pi\)
0.256612 + 0.966514i \(0.417394\pi\)
\(522\) 0 0
\(523\) 38.0370i 1.66324i 0.555342 + 0.831622i \(0.312587\pi\)
−0.555342 + 0.831622i \(0.687413\pi\)
\(524\) 0 0
\(525\) 49.1941 14.1017i 2.14700 0.615449i
\(526\) 0 0
\(527\) 16.0000i 0.696971i
\(528\) 0 0
\(529\) 22.7748 0.990208
\(530\) 0 0
\(531\) 27.4193 1.18990
\(532\) 0 0
\(533\) 13.5941i 0.588826i
\(534\) 0 0
\(535\) −11.9398 + 1.67752i −0.516202 + 0.0725254i
\(536\) 0 0
\(537\) 18.2766i 0.788691i
\(538\) 0 0
\(539\) −20.6637 −0.890049
\(540\) 0 0
\(541\) −11.5111 −0.494902 −0.247451 0.968900i \(-0.579593\pi\)
−0.247451 + 0.968900i \(0.579593\pi\)
\(542\) 0 0
\(543\) 1.41927i 0.0609068i
\(544\) 0 0
\(545\) 2.68244 + 19.0923i 0.114903 + 0.817826i
\(546\) 0 0
\(547\) 30.2208i 1.29215i 0.763275 + 0.646073i \(0.223590\pi\)
−0.763275 + 0.646073i \(0.776410\pi\)
\(548\) 0 0
\(549\) 66.4197 2.83473
\(550\) 0 0
\(551\) 10.1017 0.430347
\(552\) 0 0
\(553\) 18.4889i 0.786226i
\(554\) 0 0
\(555\) −6.48886 46.1847i −0.275437 1.96043i
\(556\) 0 0
\(557\) 9.75605i 0.413377i 0.978407 + 0.206688i \(0.0662687\pi\)
−0.978407 + 0.206688i \(0.933731\pi\)
\(558\) 0 0
\(559\) 5.12399 0.216721
\(560\) 0 0
\(561\) 64.1659 2.70909
\(562\) 0 0
\(563\) 4.50622i 0.189914i 0.995481 + 0.0949572i \(0.0302714\pi\)
−0.995481 + 0.0949572i \(0.969729\pi\)
\(564\) 0 0
\(565\) −12.2034 + 1.71456i −0.513402 + 0.0721320i
\(566\) 0 0
\(567\) 14.7511i 0.619489i
\(568\) 0 0
\(569\) −5.30465 −0.222383 −0.111191 0.993799i \(-0.535467\pi\)
−0.111191 + 0.993799i \(0.535467\pi\)
\(570\) 0 0
\(571\) −16.3970 −0.686193 −0.343096 0.939300i \(-0.611476\pi\)
−0.343096 + 0.939300i \(0.611476\pi\)
\(572\) 0 0
\(573\) 30.4514i 1.27213i
\(574\) 0 0
\(575\) −0.653858 2.28100i −0.0272678 0.0951241i
\(576\) 0 0
\(577\) 39.8163i 1.65757i 0.559565 + 0.828786i \(0.310968\pi\)
−0.559565 + 0.828786i \(0.689032\pi\)
\(578\) 0 0
\(579\) −40.0830 −1.66579
\(580\) 0 0
\(581\) −42.8256 −1.77671
\(582\) 0 0
\(583\) 20.4701i 0.847786i
\(584\) 0 0
\(585\) 31.5210 4.42864i 1.30323 0.183102i
\(586\) 0 0
\(587\) 15.1699i 0.626130i −0.949732 0.313065i \(-0.898644\pi\)
0.949732 0.313065i \(-0.101356\pi\)
\(588\) 0 0
\(589\) 13.9180 0.573480
\(590\) 0 0
\(591\) 48.5531 1.99721
\(592\) 0 0
\(593\) 8.00000i 0.328521i 0.986417 + 0.164260i \(0.0525237\pi\)
−0.986417 + 0.164260i \(0.947476\pi\)
\(594\) 0 0
\(595\) −6.36842 45.3274i −0.261080 1.85824i
\(596\) 0 0
\(597\) 60.5531i 2.47827i
\(598\) 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 \(-0.763832\pi\)
−0.737156 + 0.675723i \(0.763832\pi\)
\(602\) 0 0
\(603\) 42.1289i 1.71562i
\(604\) 0 0
\(605\) −1.08541 7.72546i −0.0441283 0.314084i
\(606\) 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
\(611\) 13.9813 0.565621
\(612\) 0 0
\(613\) 33.9309i 1.37046i −0.728329 0.685228i \(-0.759703\pi\)
0.728329 0.685228i \(-0.240297\pi\)
\(614\) 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
\(619\) 41.4005 1.66403 0.832014 0.554755i \(-0.187188\pi\)
0.832014 + 0.554755i \(0.187188\pi\)
\(620\) 0 0
\(621\) −3.34614 −0.134276
\(622\) 0 0
\(623\) 42.6637i 1.70929i
\(624\) 0 0
\(625\) 21.2034 13.2444i 0.848137 0.529777i
\(626\) 0 0
\(627\) 55.8163i 2.22909i
\(628\) 0 0
\(629\) −41.7146 −1.66327
\(630\) 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
\(635\) −22.7654 + 3.19850i −0.903418 + 0.126929i
\(636\) 0 0
\(637\) 14.2351i 0.564014i
\(638\) 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
\(643\) 27.9639i 1.10279i −0.834245 0.551395i \(-0.814096\pi\)
0.834245 0.551395i \(-0.185904\pi\)
\(644\) 0 0
\(645\) −1.76494 12.5620i −0.0694943 0.494628i
\(646\) 0 0
\(647\) 7.13828i 0.280635i −0.990107 0.140317i \(-0.955188\pi\)
0.990107 0.140317i \(-0.0448123\pi\)
\(648\) 0 0
\(649\) −19.2257 −0.754675
\(650\) 0 0
\(651\) −28.2034 −1.10538
\(652\) 0 0
\(653\) 17.7649i 0.695196i 0.937644 + 0.347598i \(0.113003\pi\)
−0.937644 + 0.347598i \(0.886997\pi\)
\(654\) 0 0
\(655\) 1.45091 + 10.3269i 0.0566919 + 0.403507i
\(656\) 0 0
\(657\) 36.1748i 1.41131i
\(658\) 0 0
\(659\) 20.1936 0.786630 0.393315 0.919404i \(-0.371328\pi\)
0.393315 + 0.919404i \(0.371328\pi\)
\(660\) 0 0
\(661\) 22.0701 0.858426 0.429213 0.903203i \(-0.358791\pi\)
0.429213 + 0.903203i \(0.358791\pi\)
\(662\) 0 0
\(663\) 44.2034i 1.71672i
\(664\) 0 0
\(665\) −39.4291 + 5.53972i −1.52900 + 0.214821i
\(666\) 0 0
\(667\) 0.949145i 0.0367510i
\(668\) 0 0
\(669\) 76.8069 2.96953
\(670\) 0 0
\(671\) −46.5718 −1.79789
\(672\) 0 0
\(673\) 26.0098i 1.00261i 0.865272 + 0.501303i \(0.167146\pi\)
−0.865272 + 0.501303i \(0.832854\pi\)
\(674\) 0 0
\(675\) −9.71456 33.8894i −0.373914 1.30440i
\(676\) 0 0
\(677\) 7.86665i 0.302340i −0.988508 0.151170i \(-0.951696\pi\)
0.988508 0.151170i \(-0.0483041\pi\)
\(678\) 0 0
\(679\) 48.6735 1.86792
\(680\) 0 0
\(681\) 35.7748 1.37089
\(682\) 0 0
\(683\) 18.2494i 0.698292i −0.937068 0.349146i \(-0.886472\pi\)
0.937068 0.349146i \(-0.113528\pi\)
\(684\) 0 0
\(685\) 25.1240 3.52987i 0.959938 0.134870i
\(686\) 0 0
\(687\) 38.3970i 1.46494i
\(688\) 0 0
\(689\) −14.1017 −0.537232
\(690\) 0 0
\(691\) 25.2543 0.960718 0.480359 0.877072i \(-0.340506\pi\)
0.480359 + 0.877072i \(0.340506\pi\)
\(692\) 0 0
\(693\) 72.8484i 2.76728i
\(694\) 0 0
\(695\) −3.67307 26.1432i −0.139328 0.991668i
\(696\) 0 0
\(697\) 30.1017i 1.14018i
\(698\) 0 0
\(699\) 19.3461 0.731738
\(700\) 0 0
\(701\) 34.8069 1.31464 0.657319 0.753612i \(-0.271690\pi\)
0.657319 + 0.753612i \(0.271690\pi\)
\(702\) 0 0
\(703\) 36.2864i 1.36857i
\(704\) 0 0
\(705\) −4.81579 34.2766i −0.181373 1.29093i
\(706\) 0 0
\(707\) 69.5022i 2.61390i
\(708\) 0 0
\(709\) 7.51114 0.282087 0.141043 0.990003i \(-0.454954\pi\)
0.141043 + 0.990003i \(0.454954\pi\)
\(710\) 0 0
\(711\) 28.4701 1.06771
\(712\) 0 0
\(713\) 1.30772i 0.0489744i
\(714\) 0 0
\(715\) −22.1017 + 3.10525i −0.826557 + 0.116130i
\(716\) 0 0
\(717\) 66.6548i 2.48927i
\(718\) 0 0
\(719\) 26.7556 0.997814 0.498907 0.866655i \(-0.333735\pi\)
0.498907 + 0.866655i \(0.333735\pi\)
\(720\) 0 0
\(721\) −11.0638 −0.412036
\(722\) 0 0
\(723\) 40.7654i 1.51608i
\(724\) 0 0
\(725\) −9.61285 + 2.75557i −0.357012 + 0.102339i
\(726\) 0 0
\(727\) 25.6271i 0.950458i −0.879862 0.475229i \(-0.842365\pi\)
0.879862 0.475229i \(-0.157635\pi\)
\(728\) 0 0
\(729\) 38.6958 1.43318
\(730\) 0 0
\(731\) −11.3461 −0.419652
\(732\) 0 0
\(733\) 19.6543i 0.725949i 0.931799 + 0.362975i \(0.118239\pi\)
−0.931799 + 0.362975i \(0.881761\pi\)
\(734\) 0 0
\(735\) 34.8988 4.90321i 1.28726 0.180858i
\(736\) 0 0
\(737\) 29.5397i 1.08811i
\(738\) 0 0
\(739\) −32.5433 −1.19712 −0.598562 0.801077i \(-0.704261\pi\)
−0.598562 + 0.801077i \(0.704261\pi\)
\(740\) 0 0
\(741\) −38.4514 −1.41255
\(742\) 0 0
\(743\) 23.1383i 0.848861i −0.905461 0.424430i \(-0.860474\pi\)
0.905461 0.424430i \(-0.139526\pi\)
\(744\) 0 0
\(745\) −1.70471 12.1334i −0.0624559 0.444532i
\(746\) 0 0
\(747\) 65.9452i 2.41281i
\(748\) 0 0
\(749\) −19.0094 −0.694587
\(750\) 0 0
\(751\) 2.48886 0.0908199 0.0454099 0.998968i \(-0.485541\pi\)
0.0454099 + 0.998968i \(0.485541\pi\)
\(752\) 0 0
\(753\) 72.3783i 2.63761i
\(754\) 0 0
\(755\) 7.34614 + 52.2864i 0.267353 + 1.90290i
\(756\) 0 0
\(757\) 21.2859i 0.773650i 0.922153 + 0.386825i \(0.126428\pi\)
−0.922153 + 0.386825i \(0.873572\pi\)
\(758\) 0 0
\(759\) 5.24443 0.190361
\(760\) 0 0
\(761\) 19.1240 0.693244 0.346622 0.938005i \(-0.387329\pi\)
0.346622 + 0.938005i \(0.387329\pi\)
\(762\) 0 0
\(763\) 30.3970i 1.10045i
\(764\) 0 0
\(765\) −69.7975 + 9.80642i −2.52354 + 0.354552i
\(766\) 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
\(771\) −74.6548 −2.68863
\(772\) 0 0
\(773\) 0.133353i 0.00479638i 0.999997 + 0.00239819i \(0.000763368\pi\)
−0.999997 + 0.00239819i \(0.999237\pi\)
\(774\) 0 0
\(775\) −13.2444 + 3.79658i −0.475754 + 0.136377i
\(776\) 0 0
\(777\) 73.5308i 2.63790i
\(778\) 0 0
\(779\) 26.1847 0.938164
\(780\) 0 0
\(781\) 18.4889 0.661584
\(782\) 0 0
\(783\) 14.1017i 0.503954i
\(784\) 0 0
\(785\) 1.80642 0.253799i 0.0644740 0.00905848i
\(786\) 0 0
\(787\) 1.12537i 0.0401150i −0.999799 0.0200575i \(-0.993615\pi\)
0.999799 0.0200575i \(-0.00638494\pi\)
\(788\) 0 0
\(789\) 7.47949 0.266277
\(790\) 0 0
\(791\) −19.4291 −0.690820
\(792\) 0 0
\(793\) 32.0830i 1.13930i
\(794\) 0 0
\(795\) 4.85728 + 34.5718i 0.172270 + 1.22614i
\(796\) 0 0
\(797\) 5.11108i 0.181044i −0.995894 0.0905218i \(-0.971147\pi\)
0.995894 0.0905218i \(-0.0288535\pi\)
\(798\) 0 0
\(799\) −30.9590 −1.09525
\(800\) 0 0
\(801\) −65.6958 −2.32125
\(802\) 0 0
\(803\) 25.3649i 0.895107i
\(804\) 0 0
\(805\) −0.520505 3.70471i −0.0183454 0.130574i
\(806\) 0 0
\(807\) 74.7467i 2.63121i
\(808\) 0 0
\(809\) −18.7368 −0.658752 −0.329376 0.944199i \(-0.606838\pi\)
−0.329376 + 0.944199i \(0.606838\pi\)
\(810\) 0 0
\(811\) 37.5210 1.31754 0.658770 0.752344i \(-0.271077\pi\)
0.658770 + 0.752344i \(0.271077\pi\)
\(812\) 0 0
\(813\) 87.6325i 3.07341i
\(814\) 0 0
\(815\) −24.1432 + 3.39207i −0.845699 + 0.118819i
\(816\) 0 0
\(817\) 9.86971i 0.345297i
\(818\) 0 0
\(819\) 50.1847 1.75359
\(820\) 0 0
\(821\) 18.4001 0.642166 0.321083 0.947051i \(-0.395953\pi\)
0.321083 + 0.947051i \(0.395953\pi\)
\(822\) 0 0
\(823\) 0.649413i 0.0226371i −0.999936 0.0113186i \(-0.996397\pi\)
0.999936 0.0113186i \(-0.00360288\pi\)
\(824\) 0 0
\(825\) 15.2257 + 53.1151i 0.530091 + 1.84923i
\(826\) 0 0
\(827\) 29.8622i 1.03841i −0.854650 0.519205i \(-0.826228\pi\)
0.854650 0.519205i \(-0.173772\pi\)
\(828\) 0 0
\(829\) −21.1753 −0.735449 −0.367725 0.929935i \(-0.619863\pi\)
−0.367725 + 0.929935i \(0.619863\pi\)
\(830\) 0 0
\(831\) −30.5718 −1.06053
\(832\) 0 0
\(833\) 31.5210i 1.09214i
\(834\) 0 0
\(835\) 14.5620 2.04593i 0.503939 0.0708024i
\(836\) 0 0
\(837\) 19.4291i 0.671568i
\(838\) 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
\(843\) 23.0509i 0.793914i
\(844\) 0 0
\(845\) −1.90522 13.5605i −0.0655415 0.466494i
\(846\) 0 0
\(847\) 12.2997i 0.422624i
\(848\) 0 0
\(849\) −31.6543 −1.08637
\(850\) 0 0
\(851\) −3.40943 −0.116874
\(852\) 0 0
\(853\) 37.7846i 1.29372i −0.762608 0.646860i \(-0.776082\pi\)
0.762608 0.646860i \(-0.223918\pi\)
\(854\) 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
\(859\) −33.2543 −1.13462 −0.567311 0.823504i \(-0.692016\pi\)
−0.567311 + 0.823504i \(0.692016\pi\)
\(860\) 0 0
\(861\) −53.0607 −1.80830
\(862\) 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
\(867\) 48.5259i 1.64803i
\(868\) 0 0
\(869\) −19.9625 −0.677182
\(870\) 0 0
\(871\) −20.3497 −0.689523
\(872\) 0 0
\(873\) 74.9501i 2.53668i
\(874\) 0 0
\(875\) 36.0098 16.0272i 1.21735 0.541818i
\(876\) 0 0
\(877\) 8.30819i 0.280548i −0.990113 0.140274i \(-0.955202\pi\)
0.990113 0.140274i \(-0.0447983\pi\)
\(878\) 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
\(883\) 7.29036i 0.245340i −0.992447 0.122670i \(-0.960854\pi\)
0.992447 0.122670i \(-0.0391457\pi\)
\(884\) 0 0
\(885\) 32.4701 4.56199i 1.09147 0.153350i
\(886\) 0 0
\(887\) 11.1097i 0.373027i 0.982452 + 0.186514i \(0.0597188\pi\)
−0.982452 + 0.186514i \(0.940281\pi\)
\(888\) 0 0
\(889\) −36.2449 −1.21562
\(890\) 0 0
\(891\) 15.9269 0.533570
\(892\) 0 0
\(893\) 26.9304i 0.901192i
\(894\) 0 0
\(895\) 1.95851 + 13.9398i 0.0654659 + 0.465955i
\(896\) 0 0
\(897\) 3.61285i 0.120629i
\(898\) 0 0
\(899\) 5.51114 0.183807
\(900\) 0 0
\(901\) 31.2257 1.04028
\(902\) 0 0
\(903\) 20.0000i 0.665558i
\(904\) 0 0
\(905\) 0.152089 + 1.08250i 0.00505561 + 0.0359835i
\(906\) 0 0
\(907\) 17.5383i 0.582351i −0.956670 0.291175i \(-0.905954\pi\)
0.956670 0.291175i \(-0.0940463\pi\)
\(908\) 0 0
\(909\) −107.023 −3.54974
\(910\) 0 0
\(911\) −44.4701 −1.47336 −0.736681 0.676241i \(-0.763608\pi\)
−0.736681 + 0.676241i \(0.763608\pi\)
\(912\) 0 0
\(913\) 46.2391i 1.53029i
\(914\) 0 0
\(915\) 78.6548 11.0509i 2.60025 0.365330i
\(916\) 0 0
\(917\) 16.4415i 0.542948i
\(918\) 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
\(923\) 12.7368i 0.419238i
\(924\) 0 0
\(925\) −9.89829 34.5303i −0.325454 1.13535i
\(926\) 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
\(931\) −27.4193 −0.898630
\(932\) 0 0
\(933\) 1.12399i 0.0367976i
\(934\) 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
\(939\) −32.9403 −1.07496
\(940\) 0 0
\(941\) 2.53341 0.0825869 0.0412934 0.999147i \(-0.486852\pi\)
0.0412934 + 0.999147i \(0.486852\pi\)
\(942\) 0 0
\(943\) 2.46028i 0.0801178i
\(944\) 0 0
\(945\) −7.73329 55.0420i −0.251564 1.79052i
\(946\) 0 0
\(947\) 1.06821i 0.0347121i −0.999849 0.0173560i \(-0.994475\pi\)
0.999849 0.0173560i \(-0.00552488\pi\)
\(948\) 0 0
\(949\) 17.4737 0.567219
\(950\) 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
\(955\) −3.26317 23.2257i −0.105594 0.751566i
\(956\) 0 0
\(957\) 22.1017i 0.714447i
\(958\) 0 0
\(959\) 40.0000 1.29167
\(960\) 0 0
\(961\) −23.4068 −0.755059
\(962\) 0 0
\(963\) 29.2716i 0.943265i
\(964\) 0 0
\(965\) −30.5718 + 4.29529i −0.984142 + 0.138270i
\(966\) 0 0
\(967\) 52.2623i 1.68064i −0.542090 0.840320i \(-0.682367\pi\)
0.542090 0.840320i \(-0.317633\pi\)
\(968\) 0 0
\(969\) 85.1437 2.73521
\(970\) 0 0
\(971\) 59.3560 1.90482 0.952412 0.304813i \(-0.0985941\pi\)
0.952412 + 0.304813i \(0.0985941\pi\)
\(972\) 0 0
\(973\) 41.6227i 1.33436i
\(974\) 0 0
\(975\) 36.5906 10.4889i 1.17184 0.335912i
\(976\) 0 0
\(977\) 47.8707i 1.53152i 0.643128 + 0.765759i \(0.277637\pi\)
−0.643128 + 0.765759i \(0.722363\pi\)
\(978\) 0 0
\(979\) 46.0642 1.47222
\(980\) 0 0
\(981\) 46.8069 1.49443
\(982\) 0 0
\(983\) 6.01429i 0.191826i −0.995390 0.0959130i \(-0.969423\pi\)
0.995390 0.0959130i \(-0.0305771\pi\)
\(984\) 0 0
\(985\) 37.0321 5.20294i 1.17994 0.165780i
\(986\) 0 0
\(987\) 54.5718i 1.73704i
\(988\) 0 0
\(989\) −0.927346 −0.0294879
\(990\) 0 0
\(991\) 19.4291 0.617186 0.308593 0.951194i \(-0.400142\pi\)
0.308593 + 0.951194i \(0.400142\pi\)
\(992\) 0 0
\(993\) 9.68598i 0.307375i
\(994\) 0 0
\(995\) −6.48886 46.1847i −0.205711 1.46415i
\(996\) 0 0
\(997\) 32.1334i 1.01767i 0.860863 + 0.508837i \(0.169924\pi\)
−0.860863 + 0.508837i \(0.830076\pi\)
\(998\) 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 640.2.c.a.129.1 6
4.3 odd 2 640.2.c.b.129.6 yes 6
5.2 odd 4 3200.2.a.bq.1.1 3
5.3 odd 4 3200.2.a.bs.1.3 3
5.4 even 2 inner 640.2.c.a.129.6 yes 6
8.3 odd 2 640.2.c.c.129.1 yes 6
8.5 even 2 640.2.c.d.129.6 yes 6
16.3 odd 4 1280.2.f.j.129.2 6
16.5 even 4 1280.2.f.i.129.1 6
16.11 odd 4 1280.2.f.k.129.5 6
16.13 even 4 1280.2.f.l.129.6 6
20.3 even 4 3200.2.a.br.1.1 3
20.7 even 4 3200.2.a.bt.1.3 3
20.19 odd 2 640.2.c.b.129.1 yes 6
40.3 even 4 3200.2.a.bu.1.3 3
40.13 odd 4 3200.2.a.bp.1.1 3
40.19 odd 2 640.2.c.c.129.6 yes 6
40.27 even 4 3200.2.a.bo.1.1 3
40.29 even 2 640.2.c.d.129.1 yes 6
40.37 odd 4 3200.2.a.bv.1.3 3
80.19 odd 4 1280.2.f.k.129.6 6
80.29 even 4 1280.2.f.i.129.2 6
80.59 odd 4 1280.2.f.j.129.1 6
80.69 even 4 1280.2.f.l.129.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.c.a.129.1 6 1.1 even 1 trivial
640.2.c.a.129.6 yes 6 5.4 even 2 inner
640.2.c.b.129.1 yes 6 20.19 odd 2
640.2.c.b.129.6 yes 6 4.3 odd 2
640.2.c.c.129.1 yes 6 8.3 odd 2
640.2.c.c.129.6 yes 6 40.19 odd 2
640.2.c.d.129.1 yes 6 40.29 even 2
640.2.c.d.129.6 yes 6 8.5 even 2
1280.2.f.i.129.1 6 16.5 even 4
1280.2.f.i.129.2 6 80.29 even 4
1280.2.f.j.129.1 6 80.59 odd 4
1280.2.f.j.129.2 6 16.3 odd 4
1280.2.f.k.129.5 6 16.11 odd 4
1280.2.f.k.129.6 6 80.19 odd 4
1280.2.f.l.129.5 6 80.69 even 4
1280.2.f.l.129.6 6 16.13 even 4
3200.2.a.bo.1.1 3 40.27 even 4
3200.2.a.bp.1.1 3 40.13 odd 4
3200.2.a.bq.1.1 3 5.2 odd 4
3200.2.a.br.1.1 3 20.3 even 4
3200.2.a.bs.1.3 3 5.3 odd 4
3200.2.a.bt.1.3 3 20.7 even 4
3200.2.a.bu.1.3 3 40.3 even 4
3200.2.a.bv.1.3 3 40.37 odd 4