Properties

Label 1520.2.d.k.609.3
Level $1520$
Weight $2$
Character 1520.609
Analytic conductor $12.137$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1520,2,Mod(609,1520)]
 
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([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1520.609");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.d (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: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 9 x^{10} - 8 x^{9} - 11 x^{8} + 60 x^{7} - 126 x^{6} + 180 x^{5} - 99 x^{4} + \cdots + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 760)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 609.3
Root \(0.588529 - 1.62900i\) of defining polynomial
Character \(\chi\) \(=\) 1520.609
Dual form 1520.2.d.k.609.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.43031i q^{3} +(-2.06801 + 0.850485i) q^{5} +3.60737i q^{7} -2.90640 q^{9} +2.37997 q^{11} -3.96613i q^{13} +(2.06694 + 5.02591i) q^{15} -4.28257i q^{17} +1.00000 q^{19} +8.76701 q^{21} +1.07377i q^{23} +(3.55335 - 3.51763i) q^{25} -0.227486i q^{27} -9.47021 q^{29} -6.38211 q^{31} -5.78405i q^{33} +(-3.06801 - 7.46008i) q^{35} -2.04540i q^{37} -9.63892 q^{39} -4.38211 q^{41} -7.86284i q^{43} +(6.01046 - 2.47185i) q^{45} +3.83485i q^{47} -6.01310 q^{49} -10.4080 q^{51} -11.7789i q^{53} +(-4.92180 + 2.02413i) q^{55} -2.43031i q^{57} +4.59593 q^{59} +6.62819 q^{61} -10.4844i q^{63} +(3.37314 + 8.20201i) q^{65} +7.02837i q^{67} +2.60959 q^{69} -4.99450 q^{71} -2.93216i q^{73} +(-8.54892 - 8.63573i) q^{75} +8.58541i q^{77} +0.860616 q^{79} -9.27205 q^{81} -11.9179i q^{83} +(3.64227 + 8.85642i) q^{85} +23.0155i q^{87} -13.6460 q^{89} +14.3073 q^{91} +15.5105i q^{93} +(-2.06801 + 0.850485i) q^{95} -18.5757i q^{97} -6.91712 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{5} - 16 q^{9} + 4 q^{11} + 12 q^{15} + 12 q^{19} + 36 q^{21} + 18 q^{25} + 4 q^{29} - 16 q^{31} - 6 q^{35} - 36 q^{39} + 8 q^{41} - 2 q^{45} - 4 q^{49} + 68 q^{51} + 18 q^{55} - 4 q^{59} + 20 q^{61}+ \cdots - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 2.43031i 1.40314i −0.712601 0.701569i \(-0.752483\pi\)
0.712601 0.701569i \(-0.247517\pi\)
\(4\) 0 0
\(5\) −2.06801 + 0.850485i −0.924843 + 0.380349i
\(6\) 0 0
\(7\) 3.60737i 1.36346i 0.731605 + 0.681728i \(0.238771\pi\)
−0.731605 + 0.681728i \(0.761229\pi\)
\(8\) 0 0
\(9\) −2.90640 −0.968799
\(10\) 0 0
\(11\) 2.37997 0.717587 0.358793 0.933417i \(-0.383188\pi\)
0.358793 + 0.933417i \(0.383188\pi\)
\(12\) 0 0
\(13\) 3.96613i 1.10001i −0.835162 0.550003i \(-0.814626\pi\)
0.835162 0.550003i \(-0.185374\pi\)
\(14\) 0 0
\(15\) 2.06694 + 5.02591i 0.533682 + 1.29768i
\(16\) 0 0
\(17\) 4.28257i 1.03868i −0.854569 0.519338i \(-0.826178\pi\)
0.854569 0.519338i \(-0.173822\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 8.76701 1.91312
\(22\) 0 0
\(23\) 1.07377i 0.223897i 0.993714 + 0.111948i \(0.0357091\pi\)
−0.993714 + 0.111948i \(0.964291\pi\)
\(24\) 0 0
\(25\) 3.55335 3.51763i 0.710670 0.703526i
\(26\) 0 0
\(27\) 0.227486i 0.0437796i
\(28\) 0 0
\(29\) −9.47021 −1.75857 −0.879287 0.476293i \(-0.841980\pi\)
−0.879287 + 0.476293i \(0.841980\pi\)
\(30\) 0 0
\(31\) −6.38211 −1.14626 −0.573130 0.819464i \(-0.694271\pi\)
−0.573130 + 0.819464i \(0.694271\pi\)
\(32\) 0 0
\(33\) 5.78405i 1.00687i
\(34\) 0 0
\(35\) −3.06801 7.46008i −0.518589 1.26098i
\(36\) 0 0
\(37\) 2.04540i 0.336262i −0.985765 0.168131i \(-0.946227\pi\)
0.985765 0.168131i \(-0.0537732\pi\)
\(38\) 0 0
\(39\) −9.63892 −1.54346
\(40\) 0 0
\(41\) −4.38211 −0.684370 −0.342185 0.939633i \(-0.611167\pi\)
−0.342185 + 0.939633i \(0.611167\pi\)
\(42\) 0 0
\(43\) 7.86284i 1.19907i −0.800348 0.599536i \(-0.795352\pi\)
0.800348 0.599536i \(-0.204648\pi\)
\(44\) 0 0
\(45\) 6.01046 2.47185i 0.895987 0.368481i
\(46\) 0 0
\(47\) 3.83485i 0.559371i 0.960092 + 0.279685i \(0.0902301\pi\)
−0.960092 + 0.279685i \(0.909770\pi\)
\(48\) 0 0
\(49\) −6.01310 −0.859014
\(50\) 0 0
\(51\) −10.4080 −1.45741
\(52\) 0 0
\(53\) 11.7789i 1.61796i −0.587836 0.808980i \(-0.700020\pi\)
0.587836 0.808980i \(-0.299980\pi\)
\(54\) 0 0
\(55\) −4.92180 + 2.02413i −0.663655 + 0.272933i
\(56\) 0 0
\(57\) 2.43031i 0.321902i
\(58\) 0 0
\(59\) 4.59593 0.598340 0.299170 0.954200i \(-0.403290\pi\)
0.299170 + 0.954200i \(0.403290\pi\)
\(60\) 0 0
\(61\) 6.62819 0.848653 0.424327 0.905509i \(-0.360511\pi\)
0.424327 + 0.905509i \(0.360511\pi\)
\(62\) 0 0
\(63\) 10.4844i 1.32092i
\(64\) 0 0
\(65\) 3.37314 + 8.20201i 0.418386 + 1.01733i
\(66\) 0 0
\(67\) 7.02837i 0.858652i 0.903150 + 0.429326i \(0.141249\pi\)
−0.903150 + 0.429326i \(0.858751\pi\)
\(68\) 0 0
\(69\) 2.60959 0.314158
\(70\) 0 0
\(71\) −4.99450 −0.592738 −0.296369 0.955074i \(-0.595776\pi\)
−0.296369 + 0.955074i \(0.595776\pi\)
\(72\) 0 0
\(73\) 2.93216i 0.343183i −0.985168 0.171592i \(-0.945109\pi\)
0.985168 0.171592i \(-0.0548910\pi\)
\(74\) 0 0
\(75\) −8.54892 8.63573i −0.987144 0.997169i
\(76\) 0 0
\(77\) 8.58541i 0.978398i
\(78\) 0 0
\(79\) 0.860616 0.0968268 0.0484134 0.998827i \(-0.484584\pi\)
0.0484134 + 0.998827i \(0.484584\pi\)
\(80\) 0 0
\(81\) −9.27205 −1.03023
\(82\) 0 0
\(83\) 11.9179i 1.30816i −0.756424 0.654081i \(-0.773055\pi\)
0.756424 0.654081i \(-0.226945\pi\)
\(84\) 0 0
\(85\) 3.64227 + 8.85642i 0.395059 + 0.960613i
\(86\) 0 0
\(87\) 23.0155i 2.46752i
\(88\) 0 0
\(89\) −13.6460 −1.44647 −0.723237 0.690600i \(-0.757346\pi\)
−0.723237 + 0.690600i \(0.757346\pi\)
\(90\) 0 0
\(91\) 14.3073 1.49981
\(92\) 0 0
\(93\) 15.5105i 1.60836i
\(94\) 0 0
\(95\) −2.06801 + 0.850485i −0.212174 + 0.0872579i
\(96\) 0 0
\(97\) 18.5757i 1.88608i −0.332681 0.943040i \(-0.607953\pi\)
0.332681 0.943040i \(-0.392047\pi\)
\(98\) 0 0
\(99\) −6.91712 −0.695197
\(100\) 0 0
\(101\) −9.82645 −0.977769 −0.488884 0.872349i \(-0.662596\pi\)
−0.488884 + 0.872349i \(0.662596\pi\)
\(102\) 0 0
\(103\) 8.85269i 0.872282i 0.899878 + 0.436141i \(0.143655\pi\)
−0.899878 + 0.436141i \(0.856345\pi\)
\(104\) 0 0
\(105\) −18.1303 + 7.45621i −1.76934 + 0.727652i
\(106\) 0 0
\(107\) 6.58269i 0.636372i 0.948028 + 0.318186i \(0.103074\pi\)
−0.948028 + 0.318186i \(0.896926\pi\)
\(108\) 0 0
\(109\) −3.40919 −0.326541 −0.163271 0.986581i \(-0.552204\pi\)
−0.163271 + 0.986581i \(0.552204\pi\)
\(110\) 0 0
\(111\) −4.97096 −0.471823
\(112\) 0 0
\(113\) 6.58065i 0.619055i −0.950890 0.309528i \(-0.899829\pi\)
0.950890 0.309528i \(-0.100171\pi\)
\(114\) 0 0
\(115\) −0.913226 2.22057i −0.0851588 0.207069i
\(116\) 0 0
\(117\) 11.5271i 1.06569i
\(118\) 0 0
\(119\) 15.4488 1.41619
\(120\) 0 0
\(121\) −5.33576 −0.485069
\(122\) 0 0
\(123\) 10.6499i 0.960267i
\(124\) 0 0
\(125\) −4.35668 + 10.2966i −0.389673 + 0.920953i
\(126\) 0 0
\(127\) 12.2826i 1.08991i −0.838466 0.544953i \(-0.816547\pi\)
0.838466 0.544953i \(-0.183453\pi\)
\(128\) 0 0
\(129\) −19.1091 −1.68246
\(130\) 0 0
\(131\) 10.0308 0.876395 0.438198 0.898879i \(-0.355617\pi\)
0.438198 + 0.898879i \(0.355617\pi\)
\(132\) 0 0
\(133\) 3.60737i 0.312798i
\(134\) 0 0
\(135\) 0.193473 + 0.470443i 0.0166515 + 0.0404893i
\(136\) 0 0
\(137\) 17.1273i 1.46329i 0.681687 + 0.731644i \(0.261247\pi\)
−0.681687 + 0.731644i \(0.738753\pi\)
\(138\) 0 0
\(139\) −5.50494 −0.466923 −0.233461 0.972366i \(-0.575005\pi\)
−0.233461 + 0.972366i \(0.575005\pi\)
\(140\) 0 0
\(141\) 9.31987 0.784875
\(142\) 0 0
\(143\) 9.43926i 0.789350i
\(144\) 0 0
\(145\) 19.5845 8.05427i 1.62641 0.668871i
\(146\) 0 0
\(147\) 14.6137i 1.20532i
\(148\) 0 0
\(149\) 21.4593 1.75801 0.879007 0.476808i \(-0.158206\pi\)
0.879007 + 0.476808i \(0.158206\pi\)
\(150\) 0 0
\(151\) 14.1805 1.15399 0.576996 0.816747i \(-0.304225\pi\)
0.576996 + 0.816747i \(0.304225\pi\)
\(152\) 0 0
\(153\) 12.4469i 1.00627i
\(154\) 0 0
\(155\) 13.1983 5.42789i 1.06011 0.435979i
\(156\) 0 0
\(157\) 2.69891i 0.215397i −0.994184 0.107698i \(-0.965652\pi\)
0.994184 0.107698i \(-0.0343481\pi\)
\(158\) 0 0
\(159\) −28.6264 −2.27022
\(160\) 0 0
\(161\) −3.87349 −0.305273
\(162\) 0 0
\(163\) 11.1955i 0.876898i 0.898756 + 0.438449i \(0.144472\pi\)
−0.898756 + 0.438449i \(0.855528\pi\)
\(164\) 0 0
\(165\) 4.91925 + 11.9615i 0.382963 + 0.931200i
\(166\) 0 0
\(167\) 9.71331i 0.751639i −0.926693 0.375819i \(-0.877361\pi\)
0.926693 0.375819i \(-0.122639\pi\)
\(168\) 0 0
\(169\) −2.73019 −0.210015
\(170\) 0 0
\(171\) −2.90640 −0.222258
\(172\) 0 0
\(173\) 14.2783i 1.08556i −0.839874 0.542781i \(-0.817371\pi\)
0.839874 0.542781i \(-0.182629\pi\)
\(174\) 0 0
\(175\) 12.6894 + 12.8182i 0.959227 + 0.968968i
\(176\) 0 0
\(177\) 11.1695i 0.839554i
\(178\) 0 0
\(179\) −17.1327 −1.28056 −0.640278 0.768143i \(-0.721181\pi\)
−0.640278 + 0.768143i \(0.721181\pi\)
\(180\) 0 0
\(181\) −23.6883 −1.76074 −0.880370 0.474288i \(-0.842705\pi\)
−0.880370 + 0.474288i \(0.842705\pi\)
\(182\) 0 0
\(183\) 16.1085i 1.19078i
\(184\) 0 0
\(185\) 1.73959 + 4.22992i 0.127897 + 0.310990i
\(186\) 0 0
\(187\) 10.1924i 0.745341i
\(188\) 0 0
\(189\) 0.820624 0.0596916
\(190\) 0 0
\(191\) −25.2506 −1.82707 −0.913535 0.406761i \(-0.866658\pi\)
−0.913535 + 0.406761i \(0.866658\pi\)
\(192\) 0 0
\(193\) 16.6814i 1.20076i 0.799716 + 0.600378i \(0.204983\pi\)
−0.799716 + 0.600378i \(0.795017\pi\)
\(194\) 0 0
\(195\) 19.9334 8.19776i 1.42746 0.587054i
\(196\) 0 0
\(197\) 17.1471i 1.22168i −0.791755 0.610839i \(-0.790832\pi\)
0.791755 0.610839i \(-0.209168\pi\)
\(198\) 0 0
\(199\) 22.1273 1.56856 0.784280 0.620407i \(-0.213033\pi\)
0.784280 + 0.620407i \(0.213033\pi\)
\(200\) 0 0
\(201\) 17.0811 1.20481
\(202\) 0 0
\(203\) 34.1625i 2.39774i
\(204\) 0 0
\(205\) 9.06225 3.72692i 0.632935 0.260299i
\(206\) 0 0
\(207\) 3.12080i 0.216911i
\(208\) 0 0
\(209\) 2.37997 0.164626
\(210\) 0 0
\(211\) 24.6690 1.69828 0.849141 0.528166i \(-0.177120\pi\)
0.849141 + 0.528166i \(0.177120\pi\)
\(212\) 0 0
\(213\) 12.1382i 0.831694i
\(214\) 0 0
\(215\) 6.68723 + 16.2605i 0.456065 + 1.10895i
\(216\) 0 0
\(217\) 23.0226i 1.56288i
\(218\) 0 0
\(219\) −7.12605 −0.481534
\(220\) 0 0
\(221\) −16.9852 −1.14255
\(222\) 0 0
\(223\) 21.4787i 1.43832i −0.694843 0.719162i \(-0.744526\pi\)
0.694843 0.719162i \(-0.255474\pi\)
\(224\) 0 0
\(225\) −10.3274 + 10.2236i −0.688496 + 0.681575i
\(226\) 0 0
\(227\) 20.5568i 1.36440i −0.731164 0.682201i \(-0.761023\pi\)
0.731164 0.682201i \(-0.238977\pi\)
\(228\) 0 0
\(229\) 17.6465 1.16611 0.583057 0.812431i \(-0.301856\pi\)
0.583057 + 0.812431i \(0.301856\pi\)
\(230\) 0 0
\(231\) 20.8652 1.37283
\(232\) 0 0
\(233\) 15.5021i 1.01558i 0.861482 + 0.507788i \(0.169537\pi\)
−0.861482 + 0.507788i \(0.830463\pi\)
\(234\) 0 0
\(235\) −3.26149 7.93052i −0.212756 0.517330i
\(236\) 0 0
\(237\) 2.09156i 0.135862i
\(238\) 0 0
\(239\) −28.2186 −1.82531 −0.912656 0.408730i \(-0.865972\pi\)
−0.912656 + 0.408730i \(0.865972\pi\)
\(240\) 0 0
\(241\) 23.8653 1.53730 0.768649 0.639671i \(-0.220929\pi\)
0.768649 + 0.639671i \(0.220929\pi\)
\(242\) 0 0
\(243\) 21.8515i 1.40177i
\(244\) 0 0
\(245\) 12.4352 5.11405i 0.794453 0.326725i
\(246\) 0 0
\(247\) 3.96613i 0.252359i
\(248\) 0 0
\(249\) −28.9642 −1.83553
\(250\) 0 0
\(251\) −8.15793 −0.514924 −0.257462 0.966288i \(-0.582886\pi\)
−0.257462 + 0.966288i \(0.582886\pi\)
\(252\) 0 0
\(253\) 2.55554i 0.160665i
\(254\) 0 0
\(255\) 21.5238 8.85183i 1.34787 0.554323i
\(256\) 0 0
\(257\) 6.07899i 0.379197i −0.981862 0.189598i \(-0.939281\pi\)
0.981862 0.189598i \(-0.0607186\pi\)
\(258\) 0 0
\(259\) 7.37852 0.458479
\(260\) 0 0
\(261\) 27.5242 1.70370
\(262\) 0 0
\(263\) 8.47535i 0.522612i −0.965256 0.261306i \(-0.915847\pi\)
0.965256 0.261306i \(-0.0841532\pi\)
\(264\) 0 0
\(265\) 10.0178 + 24.3590i 0.615389 + 1.49636i
\(266\) 0 0
\(267\) 33.1640i 2.02960i
\(268\) 0 0
\(269\) 16.6667 1.01619 0.508093 0.861302i \(-0.330351\pi\)
0.508093 + 0.861302i \(0.330351\pi\)
\(270\) 0 0
\(271\) 11.8151 0.717718 0.358859 0.933392i \(-0.383166\pi\)
0.358859 + 0.933392i \(0.383166\pi\)
\(272\) 0 0
\(273\) 34.7711i 2.10444i
\(274\) 0 0
\(275\) 8.45685 8.37183i 0.509967 0.504841i
\(276\) 0 0
\(277\) 2.10858i 0.126692i 0.997992 + 0.0633462i \(0.0201772\pi\)
−0.997992 + 0.0633462i \(0.979823\pi\)
\(278\) 0 0
\(279\) 18.5489 1.11050
\(280\) 0 0
\(281\) −3.76859 −0.224815 −0.112408 0.993662i \(-0.535856\pi\)
−0.112408 + 0.993662i \(0.535856\pi\)
\(282\) 0 0
\(283\) 0.0720389i 0.00428227i −0.999998 0.00214114i \(-0.999318\pi\)
0.999998 0.00214114i \(-0.000681545\pi\)
\(284\) 0 0
\(285\) 2.06694 + 5.02591i 0.122435 + 0.297709i
\(286\) 0 0
\(287\) 15.8079i 0.933109i
\(288\) 0 0
\(289\) −1.34044 −0.0788494
\(290\) 0 0
\(291\) −45.1447 −2.64643
\(292\) 0 0
\(293\) 9.98245i 0.583181i −0.956543 0.291590i \(-0.905816\pi\)
0.956543 0.291590i \(-0.0941844\pi\)
\(294\) 0 0
\(295\) −9.50445 + 3.90877i −0.553370 + 0.227578i
\(296\) 0 0
\(297\) 0.541408i 0.0314157i
\(298\) 0 0
\(299\) 4.25872 0.246288
\(300\) 0 0
\(301\) 28.3642 1.63488
\(302\) 0 0
\(303\) 23.8813i 1.37195i
\(304\) 0 0
\(305\) −13.7072 + 5.63718i −0.784871 + 0.322784i
\(306\) 0 0
\(307\) 21.0679i 1.20241i 0.799096 + 0.601204i \(0.205312\pi\)
−0.799096 + 0.601204i \(0.794688\pi\)
\(308\) 0 0
\(309\) 21.5148 1.22393
\(310\) 0 0
\(311\) 29.9966 1.70095 0.850475 0.526015i \(-0.176314\pi\)
0.850475 + 0.526015i \(0.176314\pi\)
\(312\) 0 0
\(313\) 5.48121i 0.309817i 0.987929 + 0.154908i \(0.0495082\pi\)
−0.987929 + 0.154908i \(0.950492\pi\)
\(314\) 0 0
\(315\) 8.91686 + 21.6819i 0.502408 + 1.22164i
\(316\) 0 0
\(317\) 0.687909i 0.0386368i −0.999813 0.0193184i \(-0.993850\pi\)
0.999813 0.0193184i \(-0.00614962\pi\)
\(318\) 0 0
\(319\) −22.5388 −1.26193
\(320\) 0 0
\(321\) 15.9980 0.892919
\(322\) 0 0
\(323\) 4.28257i 0.238289i
\(324\) 0 0
\(325\) −13.9514 14.0930i −0.773883 0.781742i
\(326\) 0 0
\(327\) 8.28538i 0.458183i
\(328\) 0 0
\(329\) −13.8337 −0.762678
\(330\) 0 0
\(331\) −23.1679 −1.27342 −0.636712 0.771102i \(-0.719706\pi\)
−0.636712 + 0.771102i \(0.719706\pi\)
\(332\) 0 0
\(333\) 5.94475i 0.325771i
\(334\) 0 0
\(335\) −5.97752 14.5347i −0.326587 0.794118i
\(336\) 0 0
\(337\) 28.8512i 1.57162i −0.618465 0.785812i \(-0.712245\pi\)
0.618465 0.785812i \(-0.287755\pi\)
\(338\) 0 0
\(339\) −15.9930 −0.868620
\(340\) 0 0
\(341\) −15.1892 −0.822541
\(342\) 0 0
\(343\) 3.56013i 0.192229i
\(344\) 0 0
\(345\) −5.39667 + 2.21942i −0.290547 + 0.119490i
\(346\) 0 0
\(347\) 5.51365i 0.295988i 0.988988 + 0.147994i \(0.0472816\pi\)
−0.988988 + 0.147994i \(0.952718\pi\)
\(348\) 0 0
\(349\) 6.62869 0.354826 0.177413 0.984137i \(-0.443227\pi\)
0.177413 + 0.984137i \(0.443227\pi\)
\(350\) 0 0
\(351\) −0.902238 −0.0481579
\(352\) 0 0
\(353\) 10.0657i 0.535745i −0.963454 0.267872i \(-0.913679\pi\)
0.963454 0.267872i \(-0.0863206\pi\)
\(354\) 0 0
\(355\) 10.3287 4.24775i 0.548190 0.225447i
\(356\) 0 0
\(357\) 37.5454i 1.98711i
\(358\) 0 0
\(359\) −23.9747 −1.26534 −0.632668 0.774423i \(-0.718040\pi\)
−0.632668 + 0.774423i \(0.718040\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 12.9675i 0.680620i
\(364\) 0 0
\(365\) 2.49376 + 6.06374i 0.130529 + 0.317391i
\(366\) 0 0
\(367\) 2.10030i 0.109635i 0.998496 + 0.0548174i \(0.0174577\pi\)
−0.998496 + 0.0548174i \(0.982542\pi\)
\(368\) 0 0
\(369\) 12.7361 0.663017
\(370\) 0 0
\(371\) 42.4909 2.20602
\(372\) 0 0
\(373\) 19.2927i 0.998939i 0.866331 + 0.499470i \(0.166472\pi\)
−0.866331 + 0.499470i \(0.833528\pi\)
\(374\) 0 0
\(375\) 25.0238 + 10.5881i 1.29223 + 0.546766i
\(376\) 0 0
\(377\) 37.5601i 1.93444i
\(378\) 0 0
\(379\) 4.29783 0.220765 0.110382 0.993889i \(-0.464792\pi\)
0.110382 + 0.993889i \(0.464792\pi\)
\(380\) 0 0
\(381\) −29.8506 −1.52929
\(382\) 0 0
\(383\) 15.6099i 0.797628i 0.917032 + 0.398814i \(0.130578\pi\)
−0.917032 + 0.398814i \(0.869422\pi\)
\(384\) 0 0
\(385\) −7.30176 17.7547i −0.372132 0.904865i
\(386\) 0 0
\(387\) 22.8525i 1.16166i
\(388\) 0 0
\(389\) −15.4509 −0.783390 −0.391695 0.920095i \(-0.628111\pi\)
−0.391695 + 0.920095i \(0.628111\pi\)
\(390\) 0 0
\(391\) 4.59850 0.232556
\(392\) 0 0
\(393\) 24.3779i 1.22970i
\(394\) 0 0
\(395\) −1.77976 + 0.731941i −0.0895496 + 0.0368279i
\(396\) 0 0
\(397\) 0.852887i 0.0428051i −0.999771 0.0214026i \(-0.993187\pi\)
0.999771 0.0214026i \(-0.00681317\pi\)
\(398\) 0 0
\(399\) 8.76701 0.438900
\(400\) 0 0
\(401\) 38.8284 1.93900 0.969500 0.245091i \(-0.0788180\pi\)
0.969500 + 0.245091i \(0.0788180\pi\)
\(402\) 0 0
\(403\) 25.3123i 1.26089i
\(404\) 0 0
\(405\) 19.1747 7.88574i 0.952799 0.391846i
\(406\) 0 0
\(407\) 4.86799i 0.241297i
\(408\) 0 0
\(409\) −6.16871 −0.305023 −0.152512 0.988302i \(-0.548736\pi\)
−0.152512 + 0.988302i \(0.548736\pi\)
\(410\) 0 0
\(411\) 41.6247 2.05320
\(412\) 0 0
\(413\) 16.5792i 0.815810i
\(414\) 0 0
\(415\) 10.1360 + 24.6464i 0.497558 + 1.20985i
\(416\) 0 0
\(417\) 13.3787i 0.655157i
\(418\) 0 0
\(419\) −14.0712 −0.687422 −0.343711 0.939075i \(-0.611684\pi\)
−0.343711 + 0.939075i \(0.611684\pi\)
\(420\) 0 0
\(421\) 4.85298 0.236520 0.118260 0.992983i \(-0.462268\pi\)
0.118260 + 0.992983i \(0.462268\pi\)
\(422\) 0 0
\(423\) 11.1456i 0.541918i
\(424\) 0 0
\(425\) −15.0645 15.2175i −0.730736 0.738156i
\(426\) 0 0
\(427\) 23.9103i 1.15710i
\(428\) 0 0
\(429\) −22.9403 −1.10757
\(430\) 0 0
\(431\) 15.6171 0.752251 0.376126 0.926569i \(-0.377256\pi\)
0.376126 + 0.926569i \(0.377256\pi\)
\(432\) 0 0
\(433\) 25.3497i 1.21823i 0.793082 + 0.609114i \(0.208475\pi\)
−0.793082 + 0.609114i \(0.791525\pi\)
\(434\) 0 0
\(435\) −19.5744 47.5964i −0.938519 2.28207i
\(436\) 0 0
\(437\) 1.07377i 0.0513654i
\(438\) 0 0
\(439\) 23.4184 1.11770 0.558849 0.829269i \(-0.311243\pi\)
0.558849 + 0.829269i \(0.311243\pi\)
\(440\) 0 0
\(441\) 17.4764 0.832211
\(442\) 0 0
\(443\) 25.1133i 1.19317i 0.802551 + 0.596584i \(0.203476\pi\)
−0.802551 + 0.596584i \(0.796524\pi\)
\(444\) 0 0
\(445\) 28.2201 11.6057i 1.33776 0.550164i
\(446\) 0 0
\(447\) 52.1527i 2.46674i
\(448\) 0 0
\(449\) −6.37270 −0.300746 −0.150373 0.988629i \(-0.548047\pi\)
−0.150373 + 0.988629i \(0.548047\pi\)
\(450\) 0 0
\(451\) −10.4293 −0.491095
\(452\) 0 0
\(453\) 34.4630i 1.61921i
\(454\) 0 0
\(455\) −29.5876 + 12.1681i −1.38709 + 0.570451i
\(456\) 0 0
\(457\) 40.6893i 1.90336i 0.307085 + 0.951682i \(0.400646\pi\)
−0.307085 + 0.951682i \(0.599354\pi\)
\(458\) 0 0
\(459\) −0.974224 −0.0454729
\(460\) 0 0
\(461\) −11.5304 −0.537024 −0.268512 0.963276i \(-0.586532\pi\)
−0.268512 + 0.963276i \(0.586532\pi\)
\(462\) 0 0
\(463\) 32.0882i 1.49127i 0.666357 + 0.745633i \(0.267853\pi\)
−0.666357 + 0.745633i \(0.732147\pi\)
\(464\) 0 0
\(465\) −13.1914 32.0759i −0.611738 1.48748i
\(466\) 0 0
\(467\) 20.6254i 0.954429i 0.878787 + 0.477214i \(0.158353\pi\)
−0.878787 + 0.477214i \(0.841647\pi\)
\(468\) 0 0
\(469\) −25.3539 −1.17073
\(470\) 0 0
\(471\) −6.55919 −0.302231
\(472\) 0 0
\(473\) 18.7133i 0.860438i
\(474\) 0 0
\(475\) 3.55335 3.51763i 0.163039 0.161400i
\(476\) 0 0
\(477\) 34.2342i 1.56748i
\(478\) 0 0
\(479\) 10.8936 0.497740 0.248870 0.968537i \(-0.419941\pi\)
0.248870 + 0.968537i \(0.419941\pi\)
\(480\) 0 0
\(481\) −8.11234 −0.369891
\(482\) 0 0
\(483\) 9.41376i 0.428341i
\(484\) 0 0
\(485\) 15.7984 + 38.4148i 0.717367 + 1.74433i
\(486\) 0 0
\(487\) 5.05668i 0.229140i −0.993415 0.114570i \(-0.963451\pi\)
0.993415 0.114570i \(-0.0365491\pi\)
\(488\) 0 0
\(489\) 27.2085 1.23041
\(490\) 0 0
\(491\) 25.5833 1.15456 0.577279 0.816547i \(-0.304114\pi\)
0.577279 + 0.816547i \(0.304114\pi\)
\(492\) 0 0
\(493\) 40.5569i 1.82659i
\(494\) 0 0
\(495\) 14.3047 5.88291i 0.642948 0.264417i
\(496\) 0 0
\(497\) 18.0170i 0.808172i
\(498\) 0 0
\(499\) −34.0476 −1.52418 −0.762090 0.647471i \(-0.775827\pi\)
−0.762090 + 0.647471i \(0.775827\pi\)
\(500\) 0 0
\(501\) −23.6063 −1.05465
\(502\) 0 0
\(503\) 5.70145i 0.254215i 0.991889 + 0.127108i \(0.0405693\pi\)
−0.991889 + 0.127108i \(0.959431\pi\)
\(504\) 0 0
\(505\) 20.3212 8.35725i 0.904283 0.371893i
\(506\) 0 0
\(507\) 6.63521i 0.294680i
\(508\) 0 0
\(509\) 27.9604 1.23932 0.619661 0.784870i \(-0.287270\pi\)
0.619661 + 0.784870i \(0.287270\pi\)
\(510\) 0 0
\(511\) 10.5774 0.467916
\(512\) 0 0
\(513\) 0.227486i 0.0100437i
\(514\) 0 0
\(515\) −7.52909 18.3075i −0.331771 0.806724i
\(516\) 0 0
\(517\) 9.12682i 0.401397i
\(518\) 0 0
\(519\) −34.7008 −1.52319
\(520\) 0 0
\(521\) −0.419591 −0.0183826 −0.00919131 0.999958i \(-0.502926\pi\)
−0.00919131 + 0.999958i \(0.502926\pi\)
\(522\) 0 0
\(523\) 2.23302i 0.0976433i 0.998808 + 0.0488217i \(0.0155466\pi\)
−0.998808 + 0.0488217i \(0.984453\pi\)
\(524\) 0 0
\(525\) 31.1523 30.8391i 1.35960 1.34593i
\(526\) 0 0
\(527\) 27.3319i 1.19059i
\(528\) 0 0
\(529\) 21.8470 0.949870
\(530\) 0 0
\(531\) −13.3576 −0.579671
\(532\) 0 0
\(533\) 17.3800i 0.752812i
\(534\) 0 0
\(535\) −5.59848 13.6131i −0.242043 0.588545i
\(536\) 0 0
\(537\) 41.6377i 1.79680i
\(538\) 0 0
\(539\) −14.3110 −0.616417
\(540\) 0 0
\(541\) 39.9938 1.71947 0.859734 0.510741i \(-0.170629\pi\)
0.859734 + 0.510741i \(0.170629\pi\)
\(542\) 0 0
\(543\) 57.5699i 2.47056i
\(544\) 0 0
\(545\) 7.05025 2.89947i 0.301999 0.124199i
\(546\) 0 0
\(547\) 15.5651i 0.665516i 0.943012 + 0.332758i \(0.107979\pi\)
−0.943012 + 0.332758i \(0.892021\pi\)
\(548\) 0 0
\(549\) −19.2642 −0.822174
\(550\) 0 0
\(551\) −9.47021 −0.403444
\(552\) 0 0
\(553\) 3.10456i 0.132019i
\(554\) 0 0
\(555\) 10.2800 4.22773i 0.436362 0.179457i
\(556\) 0 0
\(557\) 2.68789i 0.113890i −0.998377 0.0569448i \(-0.981864\pi\)
0.998377 0.0569448i \(-0.0181359\pi\)
\(558\) 0 0
\(559\) −31.1851 −1.31899
\(560\) 0 0
\(561\) −24.7706 −1.04582
\(562\) 0 0
\(563\) 12.7089i 0.535618i −0.963472 0.267809i \(-0.913700\pi\)
0.963472 0.267809i \(-0.0862996\pi\)
\(564\) 0 0
\(565\) 5.59674 + 13.6089i 0.235457 + 0.572529i
\(566\) 0 0
\(567\) 33.4477i 1.40467i
\(568\) 0 0
\(569\) −28.9522 −1.21374 −0.606870 0.794801i \(-0.707575\pi\)
−0.606870 + 0.794801i \(0.707575\pi\)
\(570\) 0 0
\(571\) −8.89885 −0.372405 −0.186203 0.982511i \(-0.559618\pi\)
−0.186203 + 0.982511i \(0.559618\pi\)
\(572\) 0 0
\(573\) 61.3667i 2.56363i
\(574\) 0 0
\(575\) 3.77713 + 3.81548i 0.157517 + 0.159117i
\(576\) 0 0
\(577\) 25.3754i 1.05639i −0.849123 0.528196i \(-0.822869\pi\)
0.849123 0.528196i \(-0.177131\pi\)
\(578\) 0 0
\(579\) 40.5411 1.68483
\(580\) 0 0
\(581\) 42.9924 1.78362
\(582\) 0 0
\(583\) 28.0334i 1.16103i
\(584\) 0 0
\(585\) −9.80367 23.8383i −0.405332 0.985592i
\(586\) 0 0
\(587\) 38.7611i 1.59984i −0.600107 0.799920i \(-0.704875\pi\)
0.600107 0.799920i \(-0.295125\pi\)
\(588\) 0 0
\(589\) −6.38211 −0.262970
\(590\) 0 0
\(591\) −41.6727 −1.71418
\(592\) 0 0
\(593\) 35.5132i 1.45835i 0.684327 + 0.729176i \(0.260096\pi\)
−0.684327 + 0.729176i \(0.739904\pi\)
\(594\) 0 0
\(595\) −31.9483 + 13.1390i −1.30975 + 0.538646i
\(596\) 0 0
\(597\) 53.7761i 2.20091i
\(598\) 0 0
\(599\) −22.7248 −0.928512 −0.464256 0.885701i \(-0.653678\pi\)
−0.464256 + 0.885701i \(0.653678\pi\)
\(600\) 0 0
\(601\) −25.3951 −1.03589 −0.517944 0.855415i \(-0.673302\pi\)
−0.517944 + 0.855415i \(0.673302\pi\)
\(602\) 0 0
\(603\) 20.4272i 0.831861i
\(604\) 0 0
\(605\) 11.0344 4.53799i 0.448613 0.184495i
\(606\) 0 0
\(607\) 18.2509i 0.740779i −0.928876 0.370390i \(-0.879224\pi\)
0.928876 0.370390i \(-0.120776\pi\)
\(608\) 0 0
\(609\) −83.0254 −3.36436
\(610\) 0 0
\(611\) 15.2095 0.615312
\(612\) 0 0
\(613\) 3.43574i 0.138768i −0.997590 0.0693842i \(-0.977897\pi\)
0.997590 0.0693842i \(-0.0221034\pi\)
\(614\) 0 0
\(615\) −9.05756 22.0241i −0.365236 0.888096i
\(616\) 0 0
\(617\) 11.7976i 0.474952i −0.971393 0.237476i \(-0.923680\pi\)
0.971393 0.237476i \(-0.0763201\pi\)
\(618\) 0 0
\(619\) 27.6561 1.11159 0.555797 0.831318i \(-0.312413\pi\)
0.555797 + 0.831318i \(0.312413\pi\)
\(620\) 0 0
\(621\) 0.244267 0.00980211
\(622\) 0 0
\(623\) 49.2261i 1.97220i
\(624\) 0 0
\(625\) 0.252588 24.9987i 0.0101035 0.999949i
\(626\) 0 0
\(627\) 5.78405i 0.230993i
\(628\) 0 0
\(629\) −8.75959 −0.349268
\(630\) 0 0
\(631\) −7.23176 −0.287892 −0.143946 0.989586i \(-0.545979\pi\)
−0.143946 + 0.989586i \(0.545979\pi\)
\(632\) 0 0
\(633\) 59.9532i 2.38293i
\(634\) 0 0
\(635\) 10.4462 + 25.4006i 0.414544 + 1.00799i
\(636\) 0 0
\(637\) 23.8487i 0.944921i
\(638\) 0 0
\(639\) 14.5160 0.574244
\(640\) 0 0
\(641\) 20.1802 0.797070 0.398535 0.917153i \(-0.369519\pi\)
0.398535 + 0.917153i \(0.369519\pi\)
\(642\) 0 0
\(643\) 4.68579i 0.184790i −0.995722 0.0923948i \(-0.970548\pi\)
0.995722 0.0923948i \(-0.0294522\pi\)
\(644\) 0 0
\(645\) 39.5179 16.2520i 1.55602 0.639923i
\(646\) 0 0
\(647\) 2.17831i 0.0856383i −0.999083 0.0428191i \(-0.986366\pi\)
0.999083 0.0428191i \(-0.0136339\pi\)
\(648\) 0 0
\(649\) 10.9382 0.429361
\(650\) 0 0
\(651\) −55.9520 −2.19293
\(652\) 0 0
\(653\) 3.73150i 0.146025i 0.997331 + 0.0730124i \(0.0232613\pi\)
−0.997331 + 0.0730124i \(0.976739\pi\)
\(654\) 0 0
\(655\) −20.7438 + 8.53105i −0.810528 + 0.333336i
\(656\) 0 0
\(657\) 8.52202i 0.332476i
\(658\) 0 0
\(659\) −18.1674 −0.707700 −0.353850 0.935302i \(-0.615128\pi\)
−0.353850 + 0.935302i \(0.615128\pi\)
\(660\) 0 0
\(661\) 8.87307 0.345123 0.172561 0.984999i \(-0.444796\pi\)
0.172561 + 0.984999i \(0.444796\pi\)
\(662\) 0 0
\(663\) 41.2794i 1.60316i
\(664\) 0 0
\(665\) −3.06801 7.46008i −0.118972 0.289289i
\(666\) 0 0
\(667\) 10.1688i 0.393739i
\(668\) 0 0
\(669\) −52.2000 −2.01817
\(670\) 0 0
\(671\) 15.7749 0.608982
\(672\) 0 0
\(673\) 35.7318i 1.37736i 0.725065 + 0.688680i \(0.241809\pi\)
−0.725065 + 0.688680i \(0.758191\pi\)
\(674\) 0 0
\(675\) −0.800210 0.808336i −0.0308001 0.0311129i
\(676\) 0 0
\(677\) 21.0748i 0.809971i 0.914323 + 0.404985i \(0.132723\pi\)
−0.914323 + 0.404985i \(0.867277\pi\)
\(678\) 0 0
\(679\) 67.0095 2.57159
\(680\) 0 0
\(681\) −49.9594 −1.91445
\(682\) 0 0
\(683\) 9.90904i 0.379159i 0.981865 + 0.189579i \(0.0607124\pi\)
−0.981865 + 0.189579i \(0.939288\pi\)
\(684\) 0 0
\(685\) −14.5666 35.4196i −0.556559 1.35331i
\(686\) 0 0
\(687\) 42.8865i 1.63622i
\(688\) 0 0
\(689\) −46.7168 −1.77977
\(690\) 0 0
\(691\) 11.9960 0.456348 0.228174 0.973620i \(-0.426724\pi\)
0.228174 + 0.973620i \(0.426724\pi\)
\(692\) 0 0
\(693\) 24.9526i 0.947871i
\(694\) 0 0
\(695\) 11.3843 4.68187i 0.431830 0.177593i
\(696\) 0 0
\(697\) 18.7667i 0.710840i
\(698\) 0 0
\(699\) 37.6748 1.42499
\(700\) 0 0
\(701\) −42.9782 −1.62326 −0.811632 0.584169i \(-0.801420\pi\)
−0.811632 + 0.584169i \(0.801420\pi\)
\(702\) 0 0
\(703\) 2.04540i 0.0771439i
\(704\) 0 0
\(705\) −19.2736 + 7.92641i −0.725886 + 0.298526i
\(706\) 0 0
\(707\) 35.4476i 1.33314i
\(708\) 0 0
\(709\) 30.3524 1.13991 0.569954 0.821676i \(-0.306961\pi\)
0.569954 + 0.821676i \(0.306961\pi\)
\(710\) 0 0
\(711\) −2.50129 −0.0938057
\(712\) 0 0
\(713\) 6.85292i 0.256644i
\(714\) 0 0
\(715\) 8.02795 + 19.5205i 0.300228 + 0.730025i
\(716\) 0 0
\(717\) 68.5799i 2.56116i
\(718\) 0 0
\(719\) 38.0282 1.41821 0.709106 0.705102i \(-0.249099\pi\)
0.709106 + 0.705102i \(0.249099\pi\)
\(720\) 0 0
\(721\) −31.9349 −1.18932
\(722\) 0 0
\(723\) 58.0000i 2.15704i
\(724\) 0 0
\(725\) −33.6510 + 33.3127i −1.24977 + 1.23720i
\(726\) 0 0
\(727\) 17.1429i 0.635796i −0.948125 0.317898i \(-0.897023\pi\)
0.948125 0.317898i \(-0.102977\pi\)
\(728\) 0 0
\(729\) 25.2897 0.936654
\(730\) 0 0
\(731\) −33.6732 −1.24545
\(732\) 0 0
\(733\) 14.0860i 0.520277i −0.965571 0.260139i \(-0.916232\pi\)
0.965571 0.260139i \(-0.0837683\pi\)
\(734\) 0 0
\(735\) −12.4287 30.2213i −0.458440 1.11473i
\(736\) 0 0
\(737\) 16.7273i 0.616157i
\(738\) 0 0
\(739\) −4.08166 −0.150146 −0.0750731 0.997178i \(-0.523919\pi\)
−0.0750731 + 0.997178i \(0.523919\pi\)
\(740\) 0 0
\(741\) −9.63892 −0.354095
\(742\) 0 0
\(743\) 22.5661i 0.827872i 0.910306 + 0.413936i \(0.135846\pi\)
−0.910306 + 0.413936i \(0.864154\pi\)
\(744\) 0 0
\(745\) −44.3781 + 18.2508i −1.62589 + 0.668658i
\(746\) 0 0
\(747\) 34.6382i 1.26735i
\(748\) 0 0
\(749\) −23.7462 −0.867666
\(750\) 0 0
\(751\) −39.5626 −1.44366 −0.721830 0.692070i \(-0.756699\pi\)
−0.721830 + 0.692070i \(0.756699\pi\)
\(752\) 0 0
\(753\) 19.8263i 0.722510i
\(754\) 0 0
\(755\) −29.3254 + 12.0603i −1.06726 + 0.438919i
\(756\) 0 0
\(757\) 33.4582i 1.21606i −0.793915 0.608029i \(-0.791960\pi\)
0.793915 0.608029i \(-0.208040\pi\)
\(758\) 0 0
\(759\) 6.21074 0.225436
\(760\) 0 0
\(761\) 4.30846 0.156181 0.0780907 0.996946i \(-0.475118\pi\)
0.0780907 + 0.996946i \(0.475118\pi\)
\(762\) 0 0
\(763\) 12.2982i 0.445225i
\(764\) 0 0
\(765\) −10.5859 25.7403i −0.382733 0.930641i
\(766\) 0 0
\(767\) 18.2281i 0.658178i
\(768\) 0 0
\(769\) 29.3601 1.05875 0.529376 0.848387i \(-0.322426\pi\)
0.529376 + 0.848387i \(0.322426\pi\)
\(770\) 0 0
\(771\) −14.7738 −0.532066
\(772\) 0 0
\(773\) 43.7346i 1.57303i −0.617574 0.786513i \(-0.711884\pi\)
0.617574 0.786513i \(-0.288116\pi\)
\(774\) 0 0
\(775\) −22.6779 + 22.4499i −0.814613 + 0.806424i
\(776\) 0 0
\(777\) 17.9321i 0.643310i
\(778\) 0 0
\(779\) −4.38211 −0.157005
\(780\) 0 0
\(781\) −11.8867 −0.425341
\(782\) 0 0
\(783\) 2.15434i 0.0769897i
\(784\) 0 0
\(785\) 2.29538 + 5.58138i 0.0819258 + 0.199208i
\(786\) 0 0
\(787\) 29.4309i 1.04910i −0.851380 0.524550i \(-0.824234\pi\)
0.851380 0.524550i \(-0.175766\pi\)
\(788\) 0 0
\(789\) −20.5977 −0.733298
\(790\) 0 0
\(791\) 23.7388 0.844055
\(792\) 0 0
\(793\) 26.2883i 0.933524i
\(794\) 0 0
\(795\) 59.1998 24.3463i 2.09960 0.863476i
\(796\) 0 0
\(797\) 21.2121i 0.751370i −0.926747 0.375685i \(-0.877407\pi\)
0.926747 0.375685i \(-0.122593\pi\)
\(798\) 0 0
\(799\) 16.4230 0.581005
\(800\) 0 0
\(801\) 39.6607 1.40134
\(802\) 0 0
\(803\) 6.97844i 0.246264i
\(804\) 0 0
\(805\) 8.01042 3.29434i 0.282330 0.116110i
\(806\) 0 0
\(807\) 40.5052i 1.42585i
\(808\) 0 0
\(809\) 23.3698 0.821638 0.410819 0.911717i \(-0.365243\pi\)
0.410819 + 0.911717i \(0.365243\pi\)
\(810\) 0 0
\(811\) 15.1194 0.530915 0.265458 0.964123i \(-0.414477\pi\)
0.265458 + 0.964123i \(0.414477\pi\)
\(812\) 0 0
\(813\) 28.7144i 1.00706i
\(814\) 0 0
\(815\) −9.52159 23.1524i −0.333527 0.810993i
\(816\) 0 0
\(817\) 7.86284i 0.275086i
\(818\) 0 0
\(819\) −41.5827 −1.45302
\(820\) 0 0
\(821\) 27.2818 0.952140 0.476070 0.879407i \(-0.342061\pi\)
0.476070 + 0.879407i \(0.342061\pi\)
\(822\) 0 0
\(823\) 18.4082i 0.641670i 0.947135 + 0.320835i \(0.103964\pi\)
−0.947135 + 0.320835i \(0.896036\pi\)
\(824\) 0 0
\(825\) −20.3461 20.5527i −0.708361 0.715555i
\(826\) 0 0
\(827\) 25.3875i 0.882811i −0.897308 0.441405i \(-0.854480\pi\)
0.897308 0.441405i \(-0.145520\pi\)
\(828\) 0 0
\(829\) 1.32188 0.0459108 0.0229554 0.999736i \(-0.492692\pi\)
0.0229554 + 0.999736i \(0.492692\pi\)
\(830\) 0 0
\(831\) 5.12450 0.177767
\(832\) 0 0
\(833\) 25.7515i 0.892238i
\(834\) 0 0
\(835\) 8.26103 + 20.0872i 0.285885 + 0.695148i
\(836\) 0 0
\(837\) 1.45184i 0.0501828i
\(838\) 0 0
\(839\) 22.9481 0.792257 0.396129 0.918195i \(-0.370353\pi\)
0.396129 + 0.918195i \(0.370353\pi\)
\(840\) 0 0
\(841\) 60.6849 2.09258
\(842\) 0 0
\(843\) 9.15884i 0.315447i
\(844\) 0 0
\(845\) 5.64607 2.32199i 0.194231 0.0798789i
\(846\) 0 0
\(847\) 19.2481i 0.661371i
\(848\) 0 0
\(849\) −0.175077 −0.00600862
\(850\) 0 0
\(851\) 2.19630 0.0752880
\(852\) 0 0
\(853\) 2.58912i 0.0886498i −0.999017 0.0443249i \(-0.985886\pi\)
0.999017 0.0443249i \(-0.0141137\pi\)
\(854\) 0 0
\(855\) 6.01046 2.47185i 0.205554 0.0845354i
\(856\) 0 0
\(857\) 2.62800i 0.0897709i −0.998992 0.0448855i \(-0.985708\pi\)
0.998992 0.0448855i \(-0.0142923\pi\)
\(858\) 0 0
\(859\) 4.38188 0.149508 0.0747539 0.997202i \(-0.476183\pi\)
0.0747539 + 0.997202i \(0.476183\pi\)
\(860\) 0 0
\(861\) −38.4180 −1.30928
\(862\) 0 0
\(863\) 25.8373i 0.879511i 0.898118 + 0.439755i \(0.144935\pi\)
−0.898118 + 0.439755i \(0.855065\pi\)
\(864\) 0 0
\(865\) 12.1435 + 29.5278i 0.412892 + 1.00397i
\(866\) 0 0
\(867\) 3.25768i 0.110637i
\(868\) 0 0
\(869\) 2.04824 0.0694816
\(870\) 0 0
\(871\) 27.8754 0.944523
\(872\) 0 0
\(873\) 53.9884i 1.82723i
\(874\) 0 0
\(875\) −37.1435 15.7161i −1.25568 0.531303i
\(876\) 0 0
\(877\) 6.72116i 0.226957i 0.993540 + 0.113479i \(0.0361994\pi\)
−0.993540 + 0.113479i \(0.963801\pi\)
\(878\) 0 0
\(879\) −24.2604 −0.818284
\(880\) 0 0
\(881\) −7.25064 −0.244280 −0.122140 0.992513i \(-0.538976\pi\)
−0.122140 + 0.992513i \(0.538976\pi\)
\(882\) 0 0
\(883\) 27.9309i 0.939949i −0.882680 0.469974i \(-0.844263\pi\)
0.882680 0.469974i \(-0.155737\pi\)
\(884\) 0 0
\(885\) 9.49952 + 23.0987i 0.319323 + 0.776455i
\(886\) 0 0
\(887\) 38.8032i 1.30288i 0.758699 + 0.651442i \(0.225836\pi\)
−0.758699 + 0.651442i \(0.774164\pi\)
\(888\) 0 0
\(889\) 44.3079 1.48604
\(890\) 0 0
\(891\) −22.0672 −0.739278
\(892\) 0 0
\(893\) 3.83485i 0.128328i
\(894\) 0 0
\(895\) 35.4306 14.5711i 1.18431 0.487057i
\(896\) 0 0
\(897\) 10.3500i 0.345576i
\(898\) 0 0
\(899\) 60.4399 2.01578
\(900\) 0 0
\(901\) −50.4441 −1.68054
\(902\) 0 0
\(903\) 68.9336i 2.29397i
\(904\) 0 0
\(905\) 48.9877 20.1466i 1.62841 0.669695i
\(906\) 0 0
\(907\) 31.8847i 1.05872i 0.848399 + 0.529358i \(0.177567\pi\)
−0.848399 + 0.529358i \(0.822433\pi\)
\(908\) 0 0
\(909\) 28.5596 0.947261
\(910\) 0 0
\(911\) 30.1437 0.998707 0.499353 0.866398i \(-0.333571\pi\)
0.499353 + 0.866398i \(0.333571\pi\)
\(912\) 0 0
\(913\) 28.3643i 0.938720i
\(914\) 0 0
\(915\) 13.7001 + 33.3127i 0.452911 + 1.10128i
\(916\) 0 0
\(917\) 36.1848i 1.19493i
\(918\) 0 0
\(919\) −55.2958 −1.82404 −0.912020 0.410147i \(-0.865478\pi\)
−0.912020 + 0.410147i \(0.865478\pi\)
\(920\) 0 0
\(921\) 51.2015 1.68715
\(922\) 0 0
\(923\) 19.8088i 0.652016i
\(924\) 0 0
\(925\) −7.19497 7.26804i −0.236569 0.238972i
\(926\) 0 0
\(927\) 25.7294i 0.845066i
\(928\) 0 0
\(929\) −21.1376 −0.693503 −0.346751 0.937957i \(-0.612715\pi\)
−0.346751 + 0.937957i \(0.612715\pi\)
\(930\) 0 0
\(931\) −6.01310 −0.197071
\(932\) 0 0
\(933\) 72.9009i 2.38667i
\(934\) 0 0
\(935\) 8.66847 + 21.0780i 0.283489 + 0.689323i
\(936\) 0 0
\(937\) 2.59000i 0.0846117i 0.999105 + 0.0423059i \(0.0134704\pi\)
−0.999105 + 0.0423059i \(0.986530\pi\)
\(938\) 0 0
\(939\) 13.3210 0.434716
\(940\) 0 0
\(941\) −35.1167 −1.14477 −0.572386 0.819984i \(-0.693982\pi\)
−0.572386 + 0.819984i \(0.693982\pi\)
\(942\) 0 0
\(943\) 4.70538i 0.153228i
\(944\) 0 0
\(945\) −1.69706 + 0.697929i −0.0552054 + 0.0227036i
\(946\) 0 0
\(947\) 27.8898i 0.906295i −0.891435 0.453148i \(-0.850301\pi\)
0.891435 0.453148i \(-0.149699\pi\)
\(948\) 0 0
\(949\) −11.6293 −0.377504
\(950\) 0 0
\(951\) −1.67183 −0.0542128
\(952\) 0 0
\(953\) 47.2402i 1.53026i −0.643875 0.765131i \(-0.722674\pi\)
0.643875 0.765131i \(-0.277326\pi\)
\(954\) 0 0
\(955\) 52.2186 21.4753i 1.68975 0.694923i
\(956\) 0 0
\(957\) 54.7762i 1.77066i
\(958\) 0 0
\(959\) −61.7846 −1.99513
\(960\) 0 0
\(961\) 9.73131 0.313913
\(962\) 0 0
\(963\) 19.1319i 0.616517i
\(964\) 0 0
\(965\) −14.1873 34.4974i −0.456706 1.11051i
\(966\) 0 0
\(967\) 29.1848i 0.938519i 0.883060 + 0.469259i \(0.155479\pi\)
−0.883060 + 0.469259i \(0.844521\pi\)
\(968\) 0 0
\(969\) −10.4080 −0.334352
\(970\) 0 0
\(971\) −43.4864 −1.39555 −0.697773 0.716319i \(-0.745826\pi\)
−0.697773 + 0.716319i \(0.745826\pi\)
\(972\) 0 0
\(973\) 19.8583i 0.636629i
\(974\) 0 0
\(975\) −34.2504 + 33.9061i −1.09689 + 1.08587i
\(976\) 0 0
\(977\) 38.2511i 1.22376i −0.790951 0.611880i \(-0.790413\pi\)
0.790951 0.611880i \(-0.209587\pi\)
\(978\) 0 0
\(979\) −32.4770 −1.03797
\(980\) 0 0
\(981\) 9.90846 0.316353
\(982\) 0 0
\(983\) 53.5820i 1.70900i 0.519451 + 0.854500i \(0.326136\pi\)
−0.519451 + 0.854500i \(0.673864\pi\)
\(984\) 0 0
\(985\) 14.5833 + 35.4604i 0.464664 + 1.12986i
\(986\) 0 0
\(987\) 33.6202i 1.07014i
\(988\) 0 0
\(989\) 8.44289 0.268468
\(990\) 0 0
\(991\) −2.34300 −0.0744277 −0.0372139 0.999307i \(-0.511848\pi\)
−0.0372139 + 0.999307i \(0.511848\pi\)
\(992\) 0 0
\(993\) 56.3051i 1.78679i
\(994\) 0 0
\(995\) −45.7595 + 18.8189i −1.45067 + 0.596600i
\(996\) 0 0
\(997\) 17.6187i 0.557991i −0.960292 0.278995i \(-0.909999\pi\)
0.960292 0.278995i \(-0.0900014\pi\)
\(998\) 0 0
\(999\) −0.465300 −0.0147214
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.d.k.609.3 12
4.3 odd 2 760.2.d.e.609.10 yes 12
5.2 odd 4 7600.2.a.cg.1.2 6
5.3 odd 4 7600.2.a.cn.1.5 6
5.4 even 2 inner 1520.2.d.k.609.10 12
20.3 even 4 3800.2.a.z.1.2 6
20.7 even 4 3800.2.a.be.1.5 6
20.19 odd 2 760.2.d.e.609.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.d.e.609.3 12 20.19 odd 2
760.2.d.e.609.10 yes 12 4.3 odd 2
1520.2.d.k.609.3 12 1.1 even 1 trivial
1520.2.d.k.609.10 12 5.4 even 2 inner
3800.2.a.z.1.2 6 20.3 even 4
3800.2.a.be.1.5 6 20.7 even 4
7600.2.a.cg.1.2 6 5.2 odd 4
7600.2.a.cn.1.5 6 5.3 odd 4