Properties

Label 1520.2.g.g.1519.2
Level $1520$
Weight $2$
Character 1520.1519
Analytic conductor $12.137$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1520,2,Mod(1519,1520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("1520.1519");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.g (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: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 20x^{14} + 271x^{12} - 2000x^{10} + 10645x^{8} - 29570x^{6} + 58816x^{4} - 56840x^{2} + 38416 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1519.2
Root \(-2.56586 - 1.48140i\) of defining polynomial
Character \(\chi\) \(=\) 1520.1519
Dual form 1520.2.g.g.1519.16

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.96281i q^{3} +(-0.602114 + 2.15348i) q^{5} +0.562696 q^{7} -5.77822 q^{9} +O(q^{10})\) \(q-2.96281i q^{3} +(-0.602114 + 2.15348i) q^{5} +0.562696 q^{7} -5.77822 q^{9} -4.19004i q^{11} +0.500297 q^{13} +(6.38033 + 1.78395i) q^{15} -2.59328i q^{17} +(3.27492 - 2.87662i) q^{19} -1.66716i q^{21} -5.09416 q^{23} +(-4.27492 - 2.59328i) q^{25} +8.23131i q^{27} +6.71292i q^{29} +4.79577 q^{31} -12.4143 q^{33} +(-0.338807 + 1.21175i) q^{35} -3.89142 q^{37} -1.48228i q^{39} -11.6408i q^{41} -8.46045 q^{43} +(3.47915 - 12.4432i) q^{45} -10.2635 q^{47} -6.68337 q^{49} -7.68337 q^{51} -5.73420 q^{53} +(9.02315 + 2.52288i) q^{55} +(-8.52285 - 9.70294i) q^{57} +6.83135 q^{59} +7.25736 q^{61} -3.25138 q^{63} +(-0.301236 + 1.07738i) q^{65} -4.43443i q^{67} +15.0930i q^{69} -4.79577 q^{71} +10.4420i q^{73} +(-7.68337 + 12.6657i) q^{75} -2.35772i q^{77} -7.96489 q^{79} +7.05313 q^{81} -2.12599 q^{83} +(5.58456 + 1.56145i) q^{85} +19.8891 q^{87} -8.67627i q^{89} +0.281515 q^{91} -14.2089i q^{93} +(4.22285 + 8.78451i) q^{95} +14.1549 q^{97} +24.2110i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{9} + 32 q^{15} - 8 q^{19} - 8 q^{25} + 96 q^{31} - 24 q^{45} - 8 q^{49} - 24 q^{51} - 72 q^{59} - 24 q^{61} - 96 q^{71} - 24 q^{75} + 32 q^{79} - 8 q^{81} - 56 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.96281i 1.71058i −0.518152 0.855288i \(-0.673380\pi\)
0.518152 0.855288i \(-0.326620\pi\)
\(4\) 0 0
\(5\) −0.602114 + 2.15348i −0.269274 + 0.963064i
\(6\) 0 0
\(7\) 0.562696 0.212679 0.106339 0.994330i \(-0.466087\pi\)
0.106339 + 0.994330i \(0.466087\pi\)
\(8\) 0 0
\(9\) −5.77822 −1.92607
\(10\) 0 0
\(11\) 4.19004i 1.26334i −0.775236 0.631672i \(-0.782369\pi\)
0.775236 0.631672i \(-0.217631\pi\)
\(12\) 0 0
\(13\) 0.500297 0.138757 0.0693787 0.997590i \(-0.477898\pi\)
0.0693787 + 0.997590i \(0.477898\pi\)
\(14\) 0 0
\(15\) 6.38033 + 1.78395i 1.64739 + 0.460613i
\(16\) 0 0
\(17\) 2.59328i 0.628962i −0.949264 0.314481i \(-0.898170\pi\)
0.949264 0.314481i \(-0.101830\pi\)
\(18\) 0 0
\(19\) 3.27492 2.87662i 0.751318 0.659941i
\(20\) 0 0
\(21\) 1.66716i 0.363804i
\(22\) 0 0
\(23\) −5.09416 −1.06221 −0.531103 0.847307i \(-0.678222\pi\)
−0.531103 + 0.847307i \(0.678222\pi\)
\(24\) 0 0
\(25\) −4.27492 2.59328i −0.854983 0.518655i
\(26\) 0 0
\(27\) 8.23131i 1.58412i
\(28\) 0 0
\(29\) 6.71292i 1.24656i 0.781999 + 0.623279i \(0.214200\pi\)
−0.781999 + 0.623279i \(0.785800\pi\)
\(30\) 0 0
\(31\) 4.79577 0.861346 0.430673 0.902508i \(-0.358276\pi\)
0.430673 + 0.902508i \(0.358276\pi\)
\(32\) 0 0
\(33\) −12.4143 −2.16105
\(34\) 0 0
\(35\) −0.338807 + 1.21175i −0.0572688 + 0.204823i
\(36\) 0 0
\(37\) −3.89142 −0.639746 −0.319873 0.947461i \(-0.603640\pi\)
−0.319873 + 0.947461i \(0.603640\pi\)
\(38\) 0 0
\(39\) 1.48228i 0.237355i
\(40\) 0 0
\(41\) 11.6408i 1.81799i −0.416804 0.908997i \(-0.636850\pi\)
0.416804 0.908997i \(-0.363150\pi\)
\(42\) 0 0
\(43\) −8.46045 −1.29021 −0.645104 0.764095i \(-0.723186\pi\)
−0.645104 + 0.764095i \(0.723186\pi\)
\(44\) 0 0
\(45\) 3.47915 12.4432i 0.518640 1.85493i
\(46\) 0 0
\(47\) −10.2635 −1.49708 −0.748540 0.663090i \(-0.769245\pi\)
−0.748540 + 0.663090i \(0.769245\pi\)
\(48\) 0 0
\(49\) −6.68337 −0.954768
\(50\) 0 0
\(51\) −7.68337 −1.07589
\(52\) 0 0
\(53\) −5.73420 −0.787653 −0.393827 0.919185i \(-0.628849\pi\)
−0.393827 + 0.919185i \(0.628849\pi\)
\(54\) 0 0
\(55\) 9.02315 + 2.52288i 1.21668 + 0.340185i
\(56\) 0 0
\(57\) −8.52285 9.70294i −1.12888 1.28519i
\(58\) 0 0
\(59\) 6.83135 0.889366 0.444683 0.895688i \(-0.353316\pi\)
0.444683 + 0.895688i \(0.353316\pi\)
\(60\) 0 0
\(61\) 7.25736 0.929210 0.464605 0.885518i \(-0.346196\pi\)
0.464605 + 0.885518i \(0.346196\pi\)
\(62\) 0 0
\(63\) −3.25138 −0.409635
\(64\) 0 0
\(65\) −0.301236 + 1.07738i −0.0373637 + 0.133632i
\(66\) 0 0
\(67\) 4.43443i 0.541752i −0.962614 0.270876i \(-0.912687\pi\)
0.962614 0.270876i \(-0.0873134\pi\)
\(68\) 0 0
\(69\) 15.0930i 1.81698i
\(70\) 0 0
\(71\) −4.79577 −0.569153 −0.284577 0.958653i \(-0.591853\pi\)
−0.284577 + 0.958653i \(0.591853\pi\)
\(72\) 0 0
\(73\) 10.4420i 1.22214i 0.791575 + 0.611072i \(0.209261\pi\)
−0.791575 + 0.611072i \(0.790739\pi\)
\(74\) 0 0
\(75\) −7.68337 + 12.6657i −0.887200 + 1.46251i
\(76\) 0 0
\(77\) 2.35772i 0.268687i
\(78\) 0 0
\(79\) −7.96489 −0.896120 −0.448060 0.894003i \(-0.647885\pi\)
−0.448060 + 0.894003i \(0.647885\pi\)
\(80\) 0 0
\(81\) 7.05313 0.783681
\(82\) 0 0
\(83\) −2.12599 −0.233357 −0.116679 0.993170i \(-0.537225\pi\)
−0.116679 + 0.993170i \(0.537225\pi\)
\(84\) 0 0
\(85\) 5.58456 + 1.56145i 0.605730 + 0.169363i
\(86\) 0 0
\(87\) 19.8891 2.13233
\(88\) 0 0
\(89\) 8.67627i 0.919683i −0.888001 0.459841i \(-0.847906\pi\)
0.888001 0.459841i \(-0.152094\pi\)
\(90\) 0 0
\(91\) 0.281515 0.0295108
\(92\) 0 0
\(93\) 14.2089i 1.47340i
\(94\) 0 0
\(95\) 4.22285 + 8.78451i 0.433255 + 0.901271i
\(96\) 0 0
\(97\) 14.1549 1.43721 0.718605 0.695418i \(-0.244781\pi\)
0.718605 + 0.695418i \(0.244781\pi\)
\(98\) 0 0
\(99\) 24.2110i 2.43329i
\(100\) 0 0
\(101\) −3.29247 −0.327613 −0.163807 0.986492i \(-0.552377\pi\)
−0.163807 + 0.986492i \(0.552377\pi\)
\(102\) 0 0
\(103\) 14.7620i 1.45454i −0.686349 0.727272i \(-0.740788\pi\)
0.686349 0.727272i \(-0.259212\pi\)
\(104\) 0 0
\(105\) 3.59018 + 1.00382i 0.350366 + 0.0979627i
\(106\) 0 0
\(107\) 14.9715i 1.44734i 0.690144 + 0.723672i \(0.257547\pi\)
−0.690144 + 0.723672i \(0.742453\pi\)
\(108\) 0 0
\(109\) 3.74835i 0.359027i −0.983755 0.179514i \(-0.942548\pi\)
0.983755 0.179514i \(-0.0574524\pi\)
\(110\) 0 0
\(111\) 11.5295i 1.09433i
\(112\) 0 0
\(113\) −8.52285 −0.801763 −0.400881 0.916130i \(-0.631296\pi\)
−0.400881 + 0.916130i \(0.631296\pi\)
\(114\) 0 0
\(115\) 3.06726 10.9701i 0.286024 1.02297i
\(116\) 0 0
\(117\) −2.89083 −0.267257
\(118\) 0 0
\(119\) 1.45923i 0.133767i
\(120\) 0 0
\(121\) −6.55643 −0.596039
\(122\) 0 0
\(123\) −34.4895 −3.10982
\(124\) 0 0
\(125\) 8.15855 7.64448i 0.729723 0.683743i
\(126\) 0 0
\(127\) 0.395651i 0.0351083i 0.999846 + 0.0175542i \(0.00558795\pi\)
−0.999846 + 0.0175542i \(0.994412\pi\)
\(128\) 0 0
\(129\) 25.0667i 2.20700i
\(130\) 0 0
\(131\) 6.79842i 0.593980i −0.954881 0.296990i \(-0.904017\pi\)
0.954881 0.296990i \(-0.0959829\pi\)
\(132\) 0 0
\(133\) 1.84278 1.61866i 0.159789 0.140356i
\(134\) 0 0
\(135\) −17.7259 4.95619i −1.52561 0.426561i
\(136\) 0 0
\(137\) 0.834074i 0.0712598i 0.999365 + 0.0356299i \(0.0113437\pi\)
−0.999365 + 0.0356299i \(0.988656\pi\)
\(138\) 0 0
\(139\) 11.8211i 1.00265i 0.865258 + 0.501327i \(0.167155\pi\)
−0.865258 + 0.501327i \(0.832845\pi\)
\(140\) 0 0
\(141\) 30.4086i 2.56087i
\(142\) 0 0
\(143\) 2.09627i 0.175299i
\(144\) 0 0
\(145\) −14.4561 4.04194i −1.20052 0.335665i
\(146\) 0 0
\(147\) 19.8015i 1.63320i
\(148\) 0 0
\(149\) 6.63808 0.543813 0.271906 0.962324i \(-0.412346\pi\)
0.271906 + 0.962324i \(0.412346\pi\)
\(150\) 0 0
\(151\) 15.3667 1.25053 0.625264 0.780413i \(-0.284991\pi\)
0.625264 + 0.780413i \(0.284991\pi\)
\(152\) 0 0
\(153\) 14.9845i 1.21143i
\(154\) 0 0
\(155\) −2.88760 + 10.3276i −0.231938 + 0.829531i
\(156\) 0 0
\(157\) 8.42387i 0.672298i 0.941809 + 0.336149i \(0.109125\pi\)
−0.941809 + 0.336149i \(0.890875\pi\)
\(158\) 0 0
\(159\) 16.9893i 1.34734i
\(160\) 0 0
\(161\) −2.86646 −0.225909
\(162\) 0 0
\(163\) 13.1919 1.03327 0.516633 0.856207i \(-0.327185\pi\)
0.516633 + 0.856207i \(0.327185\pi\)
\(164\) 0 0
\(165\) 7.47481 26.7338i 0.581913 2.08123i
\(166\) 0 0
\(167\) 8.57495i 0.663550i −0.943359 0.331775i \(-0.892353\pi\)
0.943359 0.331775i \(-0.107647\pi\)
\(168\) 0 0
\(169\) −12.7497 −0.980746
\(170\) 0 0
\(171\) −18.9232 + 16.6217i −1.44709 + 1.27109i
\(172\) 0 0
\(173\) −19.3888 −1.47410 −0.737051 0.675837i \(-0.763782\pi\)
−0.737051 + 0.675837i \(0.763782\pi\)
\(174\) 0 0
\(175\) −2.40548 1.45923i −0.181837 0.110307i
\(176\) 0 0
\(177\) 20.2400i 1.52133i
\(178\) 0 0
\(179\) 24.1063 1.80179 0.900893 0.434040i \(-0.142912\pi\)
0.900893 + 0.434040i \(0.142912\pi\)
\(180\) 0 0
\(181\) 8.74983i 0.650370i 0.945650 + 0.325185i \(0.105427\pi\)
−0.945650 + 0.325185i \(0.894573\pi\)
\(182\) 0 0
\(183\) 21.5022i 1.58948i
\(184\) 0 0
\(185\) 2.34308 8.38008i 0.172267 0.616116i
\(186\) 0 0
\(187\) −10.8659 −0.794596
\(188\) 0 0
\(189\) 4.63172i 0.336908i
\(190\) 0 0
\(191\) 12.3043i 0.890311i −0.895453 0.445155i \(-0.853148\pi\)
0.895453 0.445155i \(-0.146852\pi\)
\(192\) 0 0
\(193\) 15.3598 1.10562 0.552812 0.833306i \(-0.313555\pi\)
0.552812 + 0.833306i \(0.313555\pi\)
\(194\) 0 0
\(195\) 3.19206 + 0.892504i 0.228588 + 0.0639135i
\(196\) 0 0
\(197\) 8.30945i 0.592024i −0.955184 0.296012i \(-0.904343\pi\)
0.955184 0.296012i \(-0.0956568\pi\)
\(198\) 0 0
\(199\) 21.3137i 1.51089i −0.655212 0.755445i \(-0.727421\pi\)
0.655212 0.755445i \(-0.272579\pi\)
\(200\) 0 0
\(201\) −13.1384 −0.926709
\(202\) 0 0
\(203\) 3.77733i 0.265117i
\(204\) 0 0
\(205\) 25.0683 + 7.00911i 1.75084 + 0.489538i
\(206\) 0 0
\(207\) 29.4351 2.04588
\(208\) 0 0
\(209\) −12.0531 13.7220i −0.833733 0.949173i
\(210\) 0 0
\(211\) 10.7251 0.738345 0.369173 0.929361i \(-0.379641\pi\)
0.369173 + 0.929361i \(0.379641\pi\)
\(212\) 0 0
\(213\) 14.2089i 0.973580i
\(214\) 0 0
\(215\) 5.09416 18.2194i 0.347419 1.24255i
\(216\) 0 0
\(217\) 2.69856 0.183190
\(218\) 0 0
\(219\) 30.9376 2.09057
\(220\) 0 0
\(221\) 1.29741i 0.0872732i
\(222\) 0 0
\(223\) 6.74014i 0.451353i −0.974202 0.225676i \(-0.927541\pi\)
0.974202 0.225676i \(-0.0724592\pi\)
\(224\) 0 0
\(225\) 24.7014 + 14.9845i 1.64676 + 0.998967i
\(226\) 0 0
\(227\) 28.5558i 1.89531i −0.319293 0.947656i \(-0.603445\pi\)
0.319293 0.947656i \(-0.396555\pi\)
\(228\) 0 0
\(229\) −10.6030 −0.700664 −0.350332 0.936626i \(-0.613931\pi\)
−0.350332 + 0.936626i \(0.613931\pi\)
\(230\) 0 0
\(231\) −6.98546 −0.459609
\(232\) 0 0
\(233\) 2.71441i 0.177827i −0.996039 0.0889136i \(-0.971660\pi\)
0.996039 0.0889136i \(-0.0283395\pi\)
\(234\) 0 0
\(235\) 6.17977 22.1021i 0.403124 1.44178i
\(236\) 0 0
\(237\) 23.5984i 1.53288i
\(238\) 0 0
\(239\) 19.6023i 1.26797i −0.773347 0.633983i \(-0.781419\pi\)
0.773347 0.633983i \(-0.218581\pi\)
\(240\) 0 0
\(241\) 23.8871i 1.53870i −0.638825 0.769352i \(-0.720579\pi\)
0.638825 0.769352i \(-0.279421\pi\)
\(242\) 0 0
\(243\) 3.79688i 0.243570i
\(244\) 0 0
\(245\) 4.02415 14.3925i 0.257094 0.919502i
\(246\) 0 0
\(247\) 1.63843 1.43916i 0.104251 0.0915717i
\(248\) 0 0
\(249\) 6.29888i 0.399175i
\(250\) 0 0
\(251\) 4.57367i 0.288688i −0.989528 0.144344i \(-0.953893\pi\)
0.989528 0.144344i \(-0.0461071\pi\)
\(252\) 0 0
\(253\) 21.3447i 1.34193i
\(254\) 0 0
\(255\) 4.62627 16.5460i 0.289708 1.03615i
\(256\) 0 0
\(257\) 1.28776 0.0803282 0.0401641 0.999193i \(-0.487212\pi\)
0.0401641 + 0.999193i \(0.487212\pi\)
\(258\) 0 0
\(259\) −2.18969 −0.136060
\(260\) 0 0
\(261\) 38.7887i 2.40096i
\(262\) 0 0
\(263\) 0.921042 0.0567939 0.0283969 0.999597i \(-0.490960\pi\)
0.0283969 + 0.999597i \(0.490960\pi\)
\(264\) 0 0
\(265\) 3.45264 12.3485i 0.212094 0.758560i
\(266\) 0 0
\(267\) −25.7061 −1.57319
\(268\) 0 0
\(269\) 0.883370i 0.0538600i 0.999637 + 0.0269300i \(0.00857312\pi\)
−0.999637 + 0.0269300i \(0.991427\pi\)
\(270\) 0 0
\(271\) 1.61207i 0.0979263i 0.998801 + 0.0489631i \(0.0155917\pi\)
−0.998801 + 0.0489631i \(0.984408\pi\)
\(272\) 0 0
\(273\) 0.834074i 0.0504805i
\(274\) 0 0
\(275\) −10.8659 + 17.9121i −0.655240 + 1.08014i
\(276\) 0 0
\(277\) 24.1213i 1.44931i 0.689112 + 0.724655i \(0.258001\pi\)
−0.689112 + 0.724655i \(0.741999\pi\)
\(278\) 0 0
\(279\) −27.7110 −1.65901
\(280\) 0 0
\(281\) 19.1375i 1.14165i 0.821072 + 0.570825i \(0.193377\pi\)
−0.821072 + 0.570825i \(0.806623\pi\)
\(282\) 0 0
\(283\) 19.2965 1.14706 0.573529 0.819186i \(-0.305574\pi\)
0.573529 + 0.819186i \(0.305574\pi\)
\(284\) 0 0
\(285\) 26.0268 12.5115i 1.54169 0.741116i
\(286\) 0 0
\(287\) 6.55025i 0.386649i
\(288\) 0 0
\(289\) 10.2749 0.604407
\(290\) 0 0
\(291\) 41.9382i 2.45846i
\(292\) 0 0
\(293\) 28.3572 1.65665 0.828323 0.560251i \(-0.189295\pi\)
0.828323 + 0.560251i \(0.189295\pi\)
\(294\) 0 0
\(295\) −4.11325 + 14.7111i −0.239483 + 0.856516i
\(296\) 0 0
\(297\) 34.4895 2.00129
\(298\) 0 0
\(299\) −2.54859 −0.147389
\(300\) 0 0
\(301\) −4.76066 −0.274400
\(302\) 0 0
\(303\) 9.75496i 0.560408i
\(304\) 0 0
\(305\) −4.36976 + 15.6286i −0.250212 + 0.894888i
\(306\) 0 0
\(307\) 9.46104i 0.539970i −0.962865 0.269985i \(-0.912981\pi\)
0.962865 0.269985i \(-0.0870187\pi\)
\(308\) 0 0
\(309\) −43.7370 −2.48811
\(310\) 0 0
\(311\) 0.0624374i 0.00354050i −0.999998 0.00177025i \(-0.999437\pi\)
0.999998 0.00177025i \(-0.000563488\pi\)
\(312\) 0 0
\(313\) 9.06791i 0.512549i 0.966604 + 0.256274i \(0.0824951\pi\)
−0.966604 + 0.256274i \(0.917505\pi\)
\(314\) 0 0
\(315\) 1.95770 7.00176i 0.110304 0.394505i
\(316\) 0 0
\(317\) −34.3874 −1.93139 −0.965693 0.259686i \(-0.916381\pi\)
−0.965693 + 0.259686i \(0.916381\pi\)
\(318\) 0 0
\(319\) 28.1274 1.57483
\(320\) 0 0
\(321\) 44.3575 2.47579
\(322\) 0 0
\(323\) −7.45986 8.49277i −0.415078 0.472550i
\(324\) 0 0
\(325\) −2.13873 1.29741i −0.118635 0.0719673i
\(326\) 0 0
\(327\) −11.1056 −0.614144
\(328\) 0 0
\(329\) −5.77520 −0.318397
\(330\) 0 0
\(331\) 17.0633 0.937885 0.468942 0.883229i \(-0.344635\pi\)
0.468942 + 0.883229i \(0.344635\pi\)
\(332\) 0 0
\(333\) 22.4855 1.23220
\(334\) 0 0
\(335\) 9.54945 + 2.67004i 0.521742 + 0.145880i
\(336\) 0 0
\(337\) 10.7284 0.584413 0.292206 0.956355i \(-0.405611\pi\)
0.292206 + 0.956355i \(0.405611\pi\)
\(338\) 0 0
\(339\) 25.2516i 1.37148i
\(340\) 0 0
\(341\) 20.0945i 1.08818i
\(342\) 0 0
\(343\) −7.69957 −0.415738
\(344\) 0 0
\(345\) −32.5024 9.08771i −1.74987 0.489266i
\(346\) 0 0
\(347\) 9.38766 0.503956 0.251978 0.967733i \(-0.418919\pi\)
0.251978 + 0.967733i \(0.418919\pi\)
\(348\) 0 0
\(349\) 17.2894 0.925478 0.462739 0.886495i \(-0.346867\pi\)
0.462739 + 0.886495i \(0.346867\pi\)
\(350\) 0 0
\(351\) 4.11810i 0.219808i
\(352\) 0 0
\(353\) 30.1419i 1.60429i −0.597127 0.802147i \(-0.703691\pi\)
0.597127 0.802147i \(-0.296309\pi\)
\(354\) 0 0
\(355\) 2.88760 10.3276i 0.153258 0.548131i
\(356\) 0 0
\(357\) −4.32340 −0.228819
\(358\) 0 0
\(359\) 16.2449i 0.857374i −0.903453 0.428687i \(-0.858976\pi\)
0.903453 0.428687i \(-0.141024\pi\)
\(360\) 0 0
\(361\) 2.45017 18.8414i 0.128956 0.991650i
\(362\) 0 0
\(363\) 19.4254i 1.01957i
\(364\) 0 0
\(365\) −22.4866 6.28728i −1.17700 0.329091i
\(366\) 0 0
\(367\) −14.5651 −0.760291 −0.380146 0.924927i \(-0.624126\pi\)
−0.380146 + 0.924927i \(0.624126\pi\)
\(368\) 0 0
\(369\) 67.2633i 3.50159i
\(370\) 0 0
\(371\) −3.22661 −0.167517
\(372\) 0 0
\(373\) 11.5159 0.596268 0.298134 0.954524i \(-0.403636\pi\)
0.298134 + 0.954524i \(0.403636\pi\)
\(374\) 0 0
\(375\) −22.6491 24.1722i −1.16960 1.24825i
\(376\) 0 0
\(377\) 3.35846i 0.172969i
\(378\) 0 0
\(379\) −26.6549 −1.36917 −0.684584 0.728934i \(-0.740016\pi\)
−0.684584 + 0.728934i \(0.740016\pi\)
\(380\) 0 0
\(381\) 1.17224 0.0600555
\(382\) 0 0
\(383\) 11.3608i 0.580509i 0.956949 + 0.290254i \(0.0937400\pi\)
−0.956949 + 0.290254i \(0.906260\pi\)
\(384\) 0 0
\(385\) 5.07729 + 1.41961i 0.258762 + 0.0723503i
\(386\) 0 0
\(387\) 48.8863 2.48503
\(388\) 0 0
\(389\) −27.4651 −1.39253 −0.696267 0.717783i \(-0.745157\pi\)
−0.696267 + 0.717783i \(0.745157\pi\)
\(390\) 0 0
\(391\) 13.2106i 0.668087i
\(392\) 0 0
\(393\) −20.1424 −1.01605
\(394\) 0 0
\(395\) 4.79577 17.1522i 0.241301 0.863021i
\(396\) 0 0
\(397\) 10.8572i 0.544907i −0.962169 0.272454i \(-0.912165\pi\)
0.962169 0.272454i \(-0.0878351\pi\)
\(398\) 0 0
\(399\) −4.79577 5.45980i −0.240089 0.273332i
\(400\) 0 0
\(401\) 8.67627i 0.433272i 0.976252 + 0.216636i \(0.0695085\pi\)
−0.976252 + 0.216636i \(0.930491\pi\)
\(402\) 0 0
\(403\) 2.39931 0.119518
\(404\) 0 0
\(405\) −4.24679 + 15.1888i −0.211025 + 0.754735i
\(406\) 0 0
\(407\) 16.3052i 0.808219i
\(408\) 0 0
\(409\) 34.9225i 1.72681i 0.504514 + 0.863403i \(0.331672\pi\)
−0.504514 + 0.863403i \(0.668328\pi\)
\(410\) 0 0
\(411\) 2.47120 0.121895
\(412\) 0 0
\(413\) 3.84397 0.189149
\(414\) 0 0
\(415\) 1.28009 4.57826i 0.0628369 0.224738i
\(416\) 0 0
\(417\) 35.0237 1.71512
\(418\) 0 0
\(419\) 22.1341i 1.08132i 0.841240 + 0.540662i \(0.181826\pi\)
−0.841240 + 0.540662i \(0.818174\pi\)
\(420\) 0 0
\(421\) 33.9344i 1.65386i −0.562305 0.826930i \(-0.690085\pi\)
0.562305 0.826930i \(-0.309915\pi\)
\(422\) 0 0
\(423\) 59.3045 2.88348
\(424\) 0 0
\(425\) −6.72508 + 11.0860i −0.326214 + 0.537752i
\(426\) 0 0
\(427\) 4.08369 0.197623
\(428\) 0 0
\(429\) −6.21083 −0.299862
\(430\) 0 0
\(431\) −28.6904 −1.38197 −0.690985 0.722869i \(-0.742823\pi\)
−0.690985 + 0.722869i \(0.742823\pi\)
\(432\) 0 0
\(433\) 19.8818 0.955459 0.477729 0.878507i \(-0.341460\pi\)
0.477729 + 0.878507i \(0.341460\pi\)
\(434\) 0 0
\(435\) −11.9755 + 42.8307i −0.574181 + 2.05357i
\(436\) 0 0
\(437\) −16.6829 + 14.6539i −0.798054 + 0.700993i
\(438\) 0 0
\(439\) 4.23274 0.202018 0.101009 0.994886i \(-0.467793\pi\)
0.101009 + 0.994886i \(0.467793\pi\)
\(440\) 0 0
\(441\) 38.6180 1.83895
\(442\) 0 0
\(443\) 9.43556 0.448297 0.224149 0.974555i \(-0.428040\pi\)
0.224149 + 0.974555i \(0.428040\pi\)
\(444\) 0 0
\(445\) 18.6841 + 5.22410i 0.885713 + 0.247646i
\(446\) 0 0
\(447\) 19.6673i 0.930233i
\(448\) 0 0
\(449\) 19.5776i 0.923924i 0.886900 + 0.461962i \(0.152854\pi\)
−0.886900 + 0.461962i \(0.847146\pi\)
\(450\) 0 0
\(451\) −48.7756 −2.29675
\(452\) 0 0
\(453\) 45.5287i 2.13912i
\(454\) 0 0
\(455\) −0.169504 + 0.606236i −0.00794648 + 0.0284208i
\(456\) 0 0
\(457\) 19.9251i 0.932057i 0.884770 + 0.466029i \(0.154316\pi\)
−0.884770 + 0.466029i \(0.845684\pi\)
\(458\) 0 0
\(459\) 21.3461 0.996349
\(460\) 0 0
\(461\) 40.4234 1.88270 0.941352 0.337427i \(-0.109557\pi\)
0.941352 + 0.337427i \(0.109557\pi\)
\(462\) 0 0
\(463\) 19.3760 0.900481 0.450240 0.892907i \(-0.351338\pi\)
0.450240 + 0.892907i \(0.351338\pi\)
\(464\) 0 0
\(465\) 30.5986 + 8.55540i 1.41898 + 0.396747i
\(466\) 0 0
\(467\) −19.4450 −0.899808 −0.449904 0.893077i \(-0.648542\pi\)
−0.449904 + 0.893077i \(0.648542\pi\)
\(468\) 0 0
\(469\) 2.49524i 0.115219i
\(470\) 0 0
\(471\) 24.9583 1.15002
\(472\) 0 0
\(473\) 35.4496i 1.62998i
\(474\) 0 0
\(475\) −21.4599 + 3.80453i −0.984646 + 0.174564i
\(476\) 0 0
\(477\) 33.1335 1.51708
\(478\) 0 0
\(479\) 42.8444i 1.95761i 0.204795 + 0.978805i \(0.434347\pi\)
−0.204795 + 0.978805i \(0.565653\pi\)
\(480\) 0 0
\(481\) −1.94687 −0.0887695
\(482\) 0 0
\(483\) 8.49277i 0.386434i
\(484\) 0 0
\(485\) −8.52285 + 30.4822i −0.387003 + 1.38413i
\(486\) 0 0
\(487\) 40.7181i 1.84512i −0.385859 0.922558i \(-0.626095\pi\)
0.385859 0.922558i \(-0.373905\pi\)
\(488\) 0 0
\(489\) 39.0849i 1.76748i
\(490\) 0 0
\(491\) 9.85124i 0.444580i −0.974981 0.222290i \(-0.928647\pi\)
0.974981 0.222290i \(-0.0713532\pi\)
\(492\) 0 0
\(493\) 17.4085 0.784038
\(494\) 0 0
\(495\) −52.1377 14.5778i −2.34342 0.655221i
\(496\) 0 0
\(497\) −2.69856 −0.121047
\(498\) 0 0
\(499\) 32.4805i 1.45403i −0.686622 0.727014i \(-0.740907\pi\)
0.686622 0.727014i \(-0.259093\pi\)
\(500\) 0 0
\(501\) −25.4059 −1.13505
\(502\) 0 0
\(503\) 21.7655 0.970474 0.485237 0.874383i \(-0.338733\pi\)
0.485237 + 0.874383i \(0.338733\pi\)
\(504\) 0 0
\(505\) 1.98244 7.09026i 0.0882176 0.315512i
\(506\) 0 0
\(507\) 37.7749i 1.67764i
\(508\) 0 0
\(509\) 35.6015i 1.57801i 0.614387 + 0.789005i \(0.289403\pi\)
−0.614387 + 0.789005i \(0.710597\pi\)
\(510\) 0 0
\(511\) 5.87567i 0.259924i
\(512\) 0 0
\(513\) 23.6783 + 26.9569i 1.04542 + 1.19017i
\(514\) 0 0
\(515\) 31.7896 + 8.88842i 1.40082 + 0.391670i
\(516\) 0 0
\(517\) 43.0043i 1.89133i
\(518\) 0 0
\(519\) 57.4452i 2.52156i
\(520\) 0 0
\(521\) 9.26345i 0.405839i −0.979195 0.202920i \(-0.934957\pi\)
0.979195 0.202920i \(-0.0650430\pi\)
\(522\) 0 0
\(523\) 1.75263i 0.0766371i 0.999266 + 0.0383185i \(0.0122002\pi\)
−0.999266 + 0.0383185i \(0.987800\pi\)
\(524\) 0 0
\(525\) −4.32340 + 7.12696i −0.188689 + 0.311046i
\(526\) 0 0
\(527\) 12.4368i 0.541754i
\(528\) 0 0
\(529\) 2.95045 0.128281
\(530\) 0 0
\(531\) −39.4730 −1.71298
\(532\) 0 0
\(533\) 5.82388i 0.252260i
\(534\) 0 0
\(535\) −32.2407 9.01452i −1.39389 0.389732i
\(536\) 0 0
\(537\) 71.4222i 3.08209i
\(538\) 0 0
\(539\) 28.0036i 1.20620i
\(540\) 0 0
\(541\) 14.8621 0.638972 0.319486 0.947591i \(-0.396490\pi\)
0.319486 + 0.947591i \(0.396490\pi\)
\(542\) 0 0
\(543\) 25.9240 1.11251
\(544\) 0 0
\(545\) 8.07199 + 2.25694i 0.345766 + 0.0966766i
\(546\) 0 0
\(547\) 3.27398i 0.139985i −0.997548 0.0699926i \(-0.977702\pi\)
0.997548 0.0699926i \(-0.0222976\pi\)
\(548\) 0 0
\(549\) −41.9346 −1.78973
\(550\) 0 0
\(551\) 19.3105 + 21.9843i 0.822655 + 0.936561i
\(552\) 0 0
\(553\) −4.48181 −0.190586
\(554\) 0 0
\(555\) −24.8285 6.94209i −1.05391 0.294675i
\(556\) 0 0
\(557\) 37.9218i 1.60680i 0.595442 + 0.803399i \(0.296977\pi\)
−0.595442 + 0.803399i \(0.703023\pi\)
\(558\) 0 0
\(559\) −4.23274 −0.179026
\(560\) 0 0
\(561\) 32.1936i 1.35922i
\(562\) 0 0
\(563\) 31.6528i 1.33400i 0.745056 + 0.667002i \(0.232423\pi\)
−0.745056 + 0.667002i \(0.767577\pi\)
\(564\) 0 0
\(565\) 5.13173 18.3538i 0.215893 0.772148i
\(566\) 0 0
\(567\) 3.96877 0.166673
\(568\) 0 0
\(569\) 3.55695i 0.149115i −0.997217 0.0745575i \(-0.976246\pi\)
0.997217 0.0745575i \(-0.0237544\pi\)
\(570\) 0 0
\(571\) 4.42184i 0.185048i 0.995710 + 0.0925242i \(0.0294936\pi\)
−0.995710 + 0.0925242i \(0.970506\pi\)
\(572\) 0 0
\(573\) −36.4554 −1.52294
\(574\) 0 0
\(575\) 21.7771 + 13.2106i 0.908168 + 0.550919i
\(576\) 0 0
\(577\) 16.2037i 0.674569i 0.941403 + 0.337284i \(0.109508\pi\)
−0.941403 + 0.337284i \(0.890492\pi\)
\(578\) 0 0
\(579\) 45.5082i 1.89125i
\(580\) 0 0
\(581\) −1.19628 −0.0496302
\(582\) 0 0
\(583\) 24.0265i 0.995077i
\(584\) 0 0
\(585\) 1.74061 6.22532i 0.0719652 0.257385i
\(586\) 0 0
\(587\) −41.1470 −1.69832 −0.849159 0.528138i \(-0.822891\pi\)
−0.849159 + 0.528138i \(0.822891\pi\)
\(588\) 0 0
\(589\) 15.7058 13.7956i 0.647144 0.568437i
\(590\) 0 0
\(591\) −24.6193 −1.01270
\(592\) 0 0
\(593\) 36.2536i 1.48876i −0.667757 0.744379i \(-0.732746\pi\)
0.667757 0.744379i \(-0.267254\pi\)
\(594\) 0 0
\(595\) 3.14241 + 0.878620i 0.128826 + 0.0360199i
\(596\) 0 0
\(597\) −63.1485 −2.58449
\(598\) 0 0
\(599\) 10.5710 0.431918 0.215959 0.976402i \(-0.430712\pi\)
0.215959 + 0.976402i \(0.430712\pi\)
\(600\) 0 0
\(601\) 28.4010i 1.15850i 0.815150 + 0.579250i \(0.196655\pi\)
−0.815150 + 0.579250i \(0.803345\pi\)
\(602\) 0 0
\(603\) 25.6231i 1.04345i
\(604\) 0 0
\(605\) 3.94772 14.1191i 0.160498 0.574024i
\(606\) 0 0
\(607\) 3.70439i 0.150356i −0.997170 0.0751782i \(-0.976047\pi\)
0.997170 0.0751782i \(-0.0239526\pi\)
\(608\) 0 0
\(609\) 11.1915 0.453502
\(610\) 0 0
\(611\) −5.13478 −0.207731
\(612\) 0 0
\(613\) 39.6526i 1.60155i −0.598964 0.800776i \(-0.704421\pi\)
0.598964 0.800776i \(-0.295579\pi\)
\(614\) 0 0
\(615\) 20.7666 74.2724i 0.837391 2.99495i
\(616\) 0 0
\(617\) 13.6104i 0.547935i −0.961739 0.273968i \(-0.911664\pi\)
0.961739 0.273968i \(-0.0883361\pi\)
\(618\) 0 0
\(619\) 12.4083i 0.498732i 0.968409 + 0.249366i \(0.0802222\pi\)
−0.968409 + 0.249366i \(0.919778\pi\)
\(620\) 0 0
\(621\) 41.9316i 1.68266i
\(622\) 0 0
\(623\) 4.88210i 0.195597i
\(624\) 0 0
\(625\) 11.5498 + 22.1721i 0.461993 + 0.886883i
\(626\) 0 0
\(627\) −40.6557 + 35.7111i −1.62363 + 1.42616i
\(628\) 0 0
\(629\) 10.0915i 0.402376i
\(630\) 0 0
\(631\) 9.61505i 0.382769i −0.981515 0.191385i \(-0.938702\pi\)
0.981515 0.191385i \(-0.0612978\pi\)
\(632\) 0 0
\(633\) 31.7763i 1.26300i
\(634\) 0 0
\(635\) −0.852024 0.238227i −0.0338115 0.00945374i
\(636\) 0 0
\(637\) −3.34367 −0.132481
\(638\) 0 0
\(639\) 27.7110 1.09623
\(640\) 0 0
\(641\) 14.9752i 0.591483i −0.955268 0.295741i \(-0.904433\pi\)
0.955268 0.295741i \(-0.0955667\pi\)
\(642\) 0 0
\(643\) 32.6413 1.28725 0.643623 0.765342i \(-0.277430\pi\)
0.643623 + 0.765342i \(0.277430\pi\)
\(644\) 0 0
\(645\) −53.9805 15.0930i −2.12548 0.594286i
\(646\) 0 0
\(647\) −45.9678 −1.80718 −0.903591 0.428397i \(-0.859078\pi\)
−0.903591 + 0.428397i \(0.859078\pi\)
\(648\) 0 0
\(649\) 28.6236i 1.12358i
\(650\) 0 0
\(651\) 7.99531i 0.313361i
\(652\) 0 0
\(653\) 44.0635i 1.72434i −0.506619 0.862170i \(-0.669105\pi\)
0.506619 0.862170i \(-0.330895\pi\)
\(654\) 0 0
\(655\) 14.6402 + 4.09342i 0.572041 + 0.159943i
\(656\) 0 0
\(657\) 60.3361i 2.35394i
\(658\) 0 0
\(659\) 26.0144 1.01338 0.506689 0.862129i \(-0.330869\pi\)
0.506689 + 0.862129i \(0.330869\pi\)
\(660\) 0 0
\(661\) 40.4559i 1.57355i 0.617239 + 0.786775i \(0.288251\pi\)
−0.617239 + 0.786775i \(0.711749\pi\)
\(662\) 0 0
\(663\) −3.84397 −0.149287
\(664\) 0 0
\(665\) 2.37618 + 4.94300i 0.0921442 + 0.191681i
\(666\) 0 0
\(667\) 34.1967i 1.32410i
\(668\) 0 0
\(669\) −19.9697 −0.772074
\(670\) 0 0
\(671\) 30.4086i 1.17391i
\(672\) 0 0
\(673\) 19.1844 0.739506 0.369753 0.929130i \(-0.379442\pi\)
0.369753 + 0.929130i \(0.379442\pi\)
\(674\) 0 0
\(675\) 21.3461 35.1882i 0.821611 1.35439i
\(676\) 0 0
\(677\) 1.22823 0.0472045 0.0236023 0.999721i \(-0.492486\pi\)
0.0236023 + 0.999721i \(0.492486\pi\)
\(678\) 0 0
\(679\) 7.96489 0.305664
\(680\) 0 0
\(681\) −84.6051 −3.24208
\(682\) 0 0
\(683\) 19.8015i 0.757685i 0.925461 + 0.378842i \(0.123678\pi\)
−0.925461 + 0.378842i \(0.876322\pi\)
\(684\) 0 0
\(685\) −1.79616 0.502208i −0.0686277 0.0191884i
\(686\) 0 0
\(687\) 31.4145i 1.19854i
\(688\) 0 0
\(689\) −2.86881 −0.109293
\(690\) 0 0
\(691\) 14.1103i 0.536779i −0.963310 0.268390i \(-0.913509\pi\)
0.963310 0.268390i \(-0.0864915\pi\)
\(692\) 0 0
\(693\) 13.6234i 0.517510i
\(694\) 0 0
\(695\) −25.4565 7.11766i −0.965620 0.269988i
\(696\) 0 0
\(697\) −30.1879 −1.14345
\(698\) 0 0
\(699\) −8.04228 −0.304187
\(700\) 0 0
\(701\) −21.8986 −0.827097 −0.413549 0.910482i \(-0.635711\pi\)
−0.413549 + 0.910482i \(0.635711\pi\)
\(702\) 0 0
\(703\) −12.7441 + 11.1941i −0.480652 + 0.422194i
\(704\) 0 0
\(705\) −65.4843 18.3095i −2.46628 0.689574i
\(706\) 0 0
\(707\) −1.85266 −0.0696765
\(708\) 0 0
\(709\) 41.8947 1.57339 0.786694 0.617344i \(-0.211791\pi\)
0.786694 + 0.617344i \(0.211791\pi\)
\(710\) 0 0
\(711\) 46.0228 1.72599
\(712\) 0 0
\(713\) −24.4304 −0.914927
\(714\) 0 0
\(715\) 4.51426 + 1.26219i 0.168824 + 0.0472033i
\(716\) 0 0
\(717\) −58.0778 −2.16895
\(718\) 0 0
\(719\) 5.29226i 0.197368i −0.995119 0.0986840i \(-0.968537\pi\)
0.995119 0.0986840i \(-0.0314633\pi\)
\(720\) 0 0
\(721\) 8.30652i 0.309351i
\(722\) 0 0
\(723\) −70.7729 −2.63207
\(724\) 0 0
\(725\) 17.4085 28.6972i 0.646534 1.06579i
\(726\) 0 0
\(727\) 38.5396 1.42935 0.714677 0.699454i \(-0.246573\pi\)
0.714677 + 0.699454i \(0.246573\pi\)
\(728\) 0 0
\(729\) 32.4088 1.20033
\(730\) 0 0
\(731\) 21.9403i 0.811491i
\(732\) 0 0
\(733\) 19.5959i 0.723792i 0.932219 + 0.361896i \(0.117870\pi\)
−0.932219 + 0.361896i \(0.882130\pi\)
\(734\) 0 0
\(735\) −42.6421 11.9228i −1.57288 0.439779i
\(736\) 0 0
\(737\) −18.5805 −0.684420
\(738\) 0 0
\(739\) 28.0172i 1.03063i −0.857001 0.515314i \(-0.827675\pi\)
0.857001 0.515314i \(-0.172325\pi\)
\(740\) 0 0
\(741\) −4.26396 4.85436i −0.156640 0.178329i
\(742\) 0 0
\(743\) 6.94590i 0.254820i 0.991850 + 0.127410i \(0.0406664\pi\)
−0.991850 + 0.127410i \(0.959334\pi\)
\(744\) 0 0
\(745\) −3.99688 + 14.2949i −0.146434 + 0.523726i
\(746\) 0 0
\(747\) 12.2844 0.449463
\(748\) 0 0
\(749\) 8.42437i 0.307820i
\(750\) 0 0
\(751\) 23.5933 0.860930 0.430465 0.902607i \(-0.358350\pi\)
0.430465 + 0.902607i \(0.358350\pi\)
\(752\) 0 0
\(753\) −13.5509 −0.493822
\(754\) 0 0
\(755\) −9.25254 + 33.0919i −0.336734 + 1.20434i
\(756\) 0 0
\(757\) 35.5537i 1.29222i −0.763244 0.646110i \(-0.776395\pi\)
0.763244 0.646110i \(-0.223605\pi\)
\(758\) 0 0
\(759\) 63.2403 2.29548
\(760\) 0 0
\(761\) 22.7431 0.824437 0.412218 0.911085i \(-0.364754\pi\)
0.412218 + 0.911085i \(0.364754\pi\)
\(762\) 0 0
\(763\) 2.10918i 0.0763575i
\(764\) 0 0
\(765\) −32.2688 9.02239i −1.16668 0.326205i
\(766\) 0 0
\(767\) 3.41771 0.123406
\(768\) 0 0
\(769\) −28.6948 −1.03476 −0.517380 0.855756i \(-0.673093\pi\)
−0.517380 + 0.855756i \(0.673093\pi\)
\(770\) 0 0
\(771\) 3.81538i 0.137408i
\(772\) 0 0
\(773\) 39.0735 1.40538 0.702688 0.711498i \(-0.251983\pi\)
0.702688 + 0.711498i \(0.251983\pi\)
\(774\) 0 0
\(775\) −20.5015 12.4368i −0.736437 0.446742i
\(776\) 0 0
\(777\) 6.48761i 0.232742i
\(778\) 0 0
\(779\) −33.4862 38.1228i −1.19977 1.36589i
\(780\) 0 0
\(781\) 20.0945i 0.719037i
\(782\) 0 0
\(783\) −55.2562 −1.97469
\(784\) 0 0
\(785\) −18.1406 5.07213i −0.647466 0.181032i
\(786\) 0 0
\(787\) 31.5915i 1.12612i −0.826418 0.563058i \(-0.809625\pi\)
0.826418 0.563058i \(-0.190375\pi\)
\(788\) 0 0
\(789\) 2.72887i 0.0971502i
\(790\) 0 0
\(791\) −4.79577 −0.170518
\(792\) 0 0
\(793\) 3.63084 0.128935
\(794\) 0 0
\(795\) −36.5861 10.2295i −1.29758 0.362803i
\(796\) 0 0
\(797\) 35.3196 1.25109 0.625543 0.780190i \(-0.284878\pi\)
0.625543 + 0.780190i \(0.284878\pi\)
\(798\) 0 0
\(799\) 26.6160i 0.941606i
\(800\) 0 0
\(801\) 50.1334i 1.77138i
\(802\) 0 0
\(803\) 43.7524 1.54399
\(804\) 0 0
\(805\) 1.72594 6.17285i 0.0608313 0.217565i
\(806\) 0 0
\(807\) 2.61725 0.0921316
\(808\) 0 0
\(809\) −30.3527 −1.06714 −0.533572 0.845755i \(-0.679150\pi\)
−0.533572 + 0.845755i \(0.679150\pi\)
\(810\) 0 0
\(811\) 34.6197 1.21566 0.607832 0.794066i \(-0.292039\pi\)
0.607832 + 0.794066i \(0.292039\pi\)
\(812\) 0 0
\(813\) 4.77625 0.167510
\(814\) 0 0
\(815\) −7.94300 + 28.4083i −0.278231 + 0.995101i
\(816\) 0 0
\(817\) −27.7073 + 24.3375i −0.969355 + 0.851460i
\(818\) 0 0
\(819\) −1.62665 −0.0568399
\(820\) 0 0
\(821\) 18.7467 0.654264 0.327132 0.944979i \(-0.393918\pi\)
0.327132 + 0.944979i \(0.393918\pi\)
\(822\) 0 0
\(823\) −24.3454 −0.848627 −0.424314 0.905515i \(-0.639485\pi\)
−0.424314 + 0.905515i \(0.639485\pi\)
\(824\) 0 0
\(825\) 53.0700 + 32.1936i 1.84766 + 1.12084i
\(826\) 0 0
\(827\) 25.2470i 0.877925i 0.898505 + 0.438962i \(0.144654\pi\)
−0.898505 + 0.438962i \(0.855346\pi\)
\(828\) 0 0
\(829\) 22.9562i 0.797302i −0.917103 0.398651i \(-0.869478\pi\)
0.917103 0.398651i \(-0.130522\pi\)
\(830\) 0 0
\(831\) 71.4668 2.47915
\(832\) 0 0
\(833\) 17.3318i 0.600513i
\(834\) 0 0
\(835\) 18.4659 + 5.16310i 0.639041 + 0.178676i
\(836\) 0 0
\(837\) 39.4755i 1.36447i
\(838\) 0 0
\(839\) −48.8530 −1.68659 −0.843296 0.537450i \(-0.819388\pi\)
−0.843296 + 0.537450i \(0.819388\pi\)
\(840\) 0 0
\(841\) −16.0633 −0.553907
\(842\) 0 0
\(843\) 56.7008 1.95288
\(844\) 0 0
\(845\) 7.67678 27.4562i 0.264089 0.944521i
\(846\) 0 0
\(847\) −3.68928 −0.126765
\(848\) 0 0
\(849\) 57.1717i 1.96213i
\(850\) 0 0
\(851\) 19.8235 0.679541
\(852\) 0 0
\(853\) 18.3596i 0.628621i −0.949320 0.314311i \(-0.898227\pi\)
0.949320 0.314311i \(-0.101773\pi\)
\(854\) 0 0
\(855\) −24.4005 50.7588i −0.834481 1.73591i
\(856\) 0 0
\(857\) 37.9587 1.29664 0.648322 0.761367i \(-0.275471\pi\)
0.648322 + 0.761367i \(0.275471\pi\)
\(858\) 0 0
\(859\) 11.6267i 0.396700i 0.980131 + 0.198350i \(0.0635582\pi\)
−0.980131 + 0.198350i \(0.936442\pi\)
\(860\) 0 0
\(861\) −19.4071 −0.661393
\(862\) 0 0
\(863\) 28.1889i 0.959561i −0.877389 0.479781i \(-0.840716\pi\)
0.877389 0.479781i \(-0.159284\pi\)
\(864\) 0 0
\(865\) 11.6743 41.7533i 0.396937 1.41965i
\(866\) 0 0
\(867\) 30.4426i 1.03388i
\(868\) 0 0
\(869\) 33.3732i 1.13211i
\(870\) 0 0
\(871\) 2.21854i 0.0751722i
\(872\) 0 0
\(873\) −81.7900 −2.76817
\(874\) 0 0
\(875\) 4.59078 4.30152i 0.155197 0.145418i
\(876\) 0 0
\(877\) 54.0914 1.82654 0.913269 0.407357i \(-0.133549\pi\)
0.913269 + 0.407357i \(0.133549\pi\)
\(878\) 0 0
\(879\) 84.0169i 2.83382i
\(880\) 0 0
\(881\) −34.1111 −1.14923 −0.574616 0.818423i \(-0.694849\pi\)
−0.574616 + 0.818423i \(0.694849\pi\)
\(882\) 0 0
\(883\) 18.1513 0.610841 0.305421 0.952218i \(-0.401203\pi\)
0.305421 + 0.952218i \(0.401203\pi\)
\(884\) 0 0
\(885\) 43.5863 + 12.1868i 1.46514 + 0.409654i
\(886\) 0 0
\(887\) 11.3608i 0.381458i 0.981643 + 0.190729i \(0.0610851\pi\)
−0.981643 + 0.190729i \(0.938915\pi\)
\(888\) 0 0
\(889\) 0.222631i 0.00746680i
\(890\) 0 0
\(891\) 29.5529i 0.990060i
\(892\) 0 0
\(893\) −33.6120 + 29.5240i −1.12478 + 0.987984i
\(894\) 0 0
\(895\) −14.5147 + 51.9123i −0.485174 + 1.73524i
\(896\) 0 0
\(897\) 7.55099i 0.252120i
\(898\) 0 0
\(899\) 32.1936i 1.07372i
\(900\) 0 0
\(901\) 14.8704i 0.495404i
\(902\) 0 0
\(903\) 14.1049i 0.469382i
\(904\) 0 0
\(905\) −18.8425 5.26839i −0.626347 0.175127i
\(906\) 0 0
\(907\) 13.6572i 0.453481i −0.973955 0.226741i \(-0.927193\pi\)
0.973955 0.226741i \(-0.0728070\pi\)
\(908\) 0 0
\(909\) 19.0246 0.631007
\(910\) 0 0
\(911\) −1.70405 −0.0564576 −0.0282288 0.999601i \(-0.508987\pi\)
−0.0282288 + 0.999601i \(0.508987\pi\)
\(912\) 0 0
\(913\) 8.90796i 0.294811i
\(914\) 0 0
\(915\) 46.3044 + 12.9467i 1.53078 + 0.428006i
\(916\) 0 0
\(917\) 3.82544i 0.126327i
\(918\) 0 0
\(919\) 49.7791i 1.64206i 0.570884 + 0.821031i \(0.306601\pi\)
−0.570884 + 0.821031i \(0.693399\pi\)
\(920\) 0 0
\(921\) −28.0312 −0.923660
\(922\) 0 0
\(923\) −2.39931 −0.0789743
\(924\) 0 0
\(925\) 16.6355 + 10.0915i 0.546972 + 0.331807i
\(926\) 0 0
\(927\) 85.2981i 2.80156i
\(928\) 0 0
\(929\) −22.4229 −0.735671 −0.367836 0.929891i \(-0.619901\pi\)
−0.367836 + 0.929891i \(0.619901\pi\)
\(930\) 0 0
\(931\) −21.8875 + 19.2255i −0.717334 + 0.630090i
\(932\) 0 0
\(933\) −0.184990 −0.00605629
\(934\) 0 0
\(935\) 6.54253 23.3995i 0.213964 0.765246i
\(936\) 0 0
\(937\) 30.2982i 0.989800i −0.868950 0.494900i \(-0.835205\pi\)
0.868950 0.494900i \(-0.164795\pi\)
\(938\) 0 0
\(939\) 26.8665 0.876754
\(940\) 0 0
\(941\) 20.2859i 0.661301i −0.943753 0.330650i \(-0.892732\pi\)
0.943753 0.330650i \(-0.107268\pi\)
\(942\) 0 0
\(943\) 59.3003i 1.93108i
\(944\) 0 0
\(945\) −9.97431 2.78883i −0.324464 0.0907205i
\(946\) 0 0
\(947\) 6.88729 0.223807 0.111903 0.993719i \(-0.464305\pi\)
0.111903 + 0.993719i \(0.464305\pi\)
\(948\) 0 0
\(949\) 5.22410i 0.169582i
\(950\) 0 0
\(951\) 101.883i 3.30378i
\(952\) 0 0
\(953\) −13.2356 −0.428743 −0.214371 0.976752i \(-0.568770\pi\)
−0.214371 + 0.976752i \(0.568770\pi\)
\(954\) 0 0
\(955\) 26.4971 + 7.40861i 0.857426 + 0.239737i
\(956\) 0 0
\(957\) 83.3360i 2.69387i
\(958\) 0 0
\(959\) 0.469330i 0.0151555i
\(960\) 0 0
\(961\) −8.00057 −0.258083
\(962\) 0 0
\(963\) 86.5083i 2.78769i
\(964\) 0 0
\(965\) −9.24837 + 33.0770i −0.297715 + 1.06479i
\(966\) 0 0
\(967\) 2.65507 0.0853813 0.0426907 0.999088i \(-0.486407\pi\)
0.0426907 + 0.999088i \(0.486407\pi\)
\(968\) 0 0
\(969\) −25.1624 + 22.1021i −0.808333 + 0.710022i
\(970\) 0 0
\(971\) −11.1216 −0.356909 −0.178454 0.983948i \(-0.557110\pi\)
−0.178454 + 0.983948i \(0.557110\pi\)
\(972\) 0 0
\(973\) 6.65169i 0.213244i
\(974\) 0 0
\(975\) −3.84397 + 6.33664i −0.123106 + 0.202935i
\(976\) 0 0
\(977\) −48.3717 −1.54755 −0.773775 0.633461i \(-0.781634\pi\)
−0.773775 + 0.633461i \(0.781634\pi\)
\(978\) 0 0
\(979\) −36.3539 −1.16188
\(980\) 0 0
\(981\) 21.6588i 0.691512i
\(982\) 0 0
\(983\) 14.6603i 0.467590i 0.972286 + 0.233795i \(0.0751145\pi\)
−0.972286 + 0.233795i \(0.924885\pi\)
\(984\) 0 0
\(985\) 17.8942 + 5.00324i 0.570157 + 0.159416i
\(986\) 0 0
\(987\) 17.1108i 0.544643i
\(988\) 0 0
\(989\) 43.0989 1.37047
\(990\) 0 0
\(991\) −23.6331 −0.750729 −0.375364 0.926877i \(-0.622482\pi\)
−0.375364 + 0.926877i \(0.622482\pi\)
\(992\) 0 0
\(993\) 50.5553i 1.60432i
\(994\) 0 0
\(995\) 45.8986 + 12.8333i 1.45508 + 0.406843i
\(996\) 0 0
\(997\) 22.5810i 0.715148i 0.933885 + 0.357574i \(0.116396\pi\)
−0.933885 + 0.357574i \(0.883604\pi\)
\(998\) 0 0
\(999\) 32.0315i 1.01343i
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 1520.2.g.g.1519.2 yes 16
4.3 odd 2 1520.2.g.e.1519.15 yes 16
5.4 even 2 inner 1520.2.g.g.1519.15 yes 16
19.18 odd 2 1520.2.g.e.1519.16 yes 16
20.19 odd 2 1520.2.g.e.1519.2 yes 16
76.75 even 2 inner 1520.2.g.g.1519.1 yes 16
95.94 odd 2 1520.2.g.e.1519.1 16
380.379 even 2 inner 1520.2.g.g.1519.16 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1520.2.g.e.1519.1 16 95.94 odd 2
1520.2.g.e.1519.2 yes 16 20.19 odd 2
1520.2.g.e.1519.15 yes 16 4.3 odd 2
1520.2.g.e.1519.16 yes 16 19.18 odd 2
1520.2.g.g.1519.1 yes 16 76.75 even 2 inner
1520.2.g.g.1519.2 yes 16 1.1 even 1 trivial
1520.2.g.g.1519.15 yes 16 5.4 even 2 inner
1520.2.g.g.1519.16 yes 16 380.379 even 2 inner