Properties

Label 672.2.j.d.239.10
Level 672
Weight 2
Character 672.239
Analytic conductor 5.366
Analytic rank 0
Dimension 12
CM no
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 672.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.2593100598870016.2
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 239.10
Root \(-0.430469 + 1.34711i\)
Character \(\chi\) = 672.239
Dual form 672.2.j.d.239.12

$q$-expansion

\(f(q)\) \(=\) \(q+(1.46962 - 0.916638i) q^{3} +1.83328 q^{5} -1.00000i q^{7} +(1.31955 - 2.69421i) q^{9} +O(q^{10})\) \(q+(1.46962 - 0.916638i) q^{3} +1.83328 q^{5} -1.00000i q^{7} +(1.31955 - 2.69421i) q^{9} +5.57834i q^{13} +(2.69421 - 1.68045i) q^{15} -6.08565i q^{17} +6.93923 q^{19} +(-0.916638 - 1.46962i) q^{21} +0.697224 q^{23} -1.63910 q^{25} +(-0.530383 - 5.16901i) q^{27} -7.11030 q^{29} -1.83328i q^{35} +1.87847i q^{37} +(5.11331 + 8.19802i) q^{39} +4.69120i q^{41} -1.36090 q^{43} +(2.41910 - 4.93923i) q^{45} -3.44375 q^{47} -1.00000 q^{49} +(-5.57834 - 8.94358i) q^{51} +12.1713 q^{53} +(10.1980 - 6.36076i) q^{57} +8.94358i q^{59} -4.30013i q^{61} +(-2.69421 - 1.31955i) q^{63} +10.2266i q^{65} -4.51757 q^{67} +(1.02465 - 0.639102i) q^{69} -6.08565 q^{71} -6.00000 q^{73} +(-2.40885 + 1.50246i) q^{75} -11.7569i q^{79} +(-5.51757 - 7.11030i) q^{81} +0.438828i q^{83} -11.1567i q^{85} +(-10.4494 + 6.51757i) q^{87} +2.64190i q^{89} +5.57834 q^{91} +12.7215 q^{95} +11.0351 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 4q^{9} + O(q^{10}) \) \( 12q + 4q^{9} + 48q^{19} + 4q^{25} - 24q^{27} - 40q^{43} - 12q^{49} - 8q^{51} + 40q^{57} + 40q^{67} - 72q^{73} + 24q^{75} + 28q^{81} + 8q^{91} - 56q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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
\(3\) 1.46962 0.916638i 0.848484 0.529221i
\(4\) 0 0
\(5\) 1.83328 0.819866 0.409933 0.912116i \(-0.365552\pi\)
0.409933 + 0.912116i \(0.365552\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 1.31955 2.69421i 0.439850 0.898071i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 5.57834i 1.54715i 0.633703 + 0.773576i \(0.281534\pi\)
−0.633703 + 0.773576i \(0.718466\pi\)
\(14\) 0 0
\(15\) 2.69421 1.68045i 0.695643 0.433890i
\(16\) 0 0
\(17\) 6.08565i 1.47599i −0.674808 0.737994i \(-0.735773\pi\)
0.674808 0.737994i \(-0.264227\pi\)
\(18\) 0 0
\(19\) 6.93923 1.59197 0.795985 0.605317i \(-0.206953\pi\)
0.795985 + 0.605317i \(0.206953\pi\)
\(20\) 0 0
\(21\) −0.916638 1.46962i −0.200027 0.320697i
\(22\) 0 0
\(23\) 0.697224 0.145381 0.0726906 0.997355i \(-0.476841\pi\)
0.0726906 + 0.997355i \(0.476841\pi\)
\(24\) 0 0
\(25\) −1.63910 −0.327820
\(26\) 0 0
\(27\) −0.530383 5.16901i −0.102072 0.994777i
\(28\) 0 0
\(29\) −7.11030 −1.32035 −0.660175 0.751112i \(-0.729518\pi\)
−0.660175 + 0.751112i \(0.729518\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.83328i 0.309880i
\(36\) 0 0
\(37\) 1.87847i 0.308819i 0.988007 + 0.154409i \(0.0493474\pi\)
−0.988007 + 0.154409i \(0.950653\pi\)
\(38\) 0 0
\(39\) 5.11331 + 8.19802i 0.818785 + 1.31273i
\(40\) 0 0
\(41\) 4.69120i 0.732643i 0.930488 + 0.366321i \(0.119383\pi\)
−0.930488 + 0.366321i \(0.880617\pi\)
\(42\) 0 0
\(43\) −1.36090 −0.207535 −0.103768 0.994602i \(-0.533090\pi\)
−0.103768 + 0.994602i \(0.533090\pi\)
\(44\) 0 0
\(45\) 2.41910 4.93923i 0.360618 0.736298i
\(46\) 0 0
\(47\) −3.44375 −0.502323 −0.251161 0.967945i \(-0.580813\pi\)
−0.251161 + 0.967945i \(0.580813\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −5.57834 8.94358i −0.781123 1.25235i
\(52\) 0 0
\(53\) 12.1713 1.67186 0.835928 0.548839i \(-0.184930\pi\)
0.835928 + 0.548839i \(0.184930\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 10.1980 6.36076i 1.35076 0.842504i
\(58\) 0 0
\(59\) 8.94358i 1.16435i 0.813062 + 0.582177i \(0.197799\pi\)
−0.813062 + 0.582177i \(0.802201\pi\)
\(60\) 0 0
\(61\) 4.30013i 0.550576i −0.961362 0.275288i \(-0.911227\pi\)
0.961362 0.275288i \(-0.0887732\pi\)
\(62\) 0 0
\(63\) −2.69421 1.31955i −0.339439 0.166248i
\(64\) 0 0
\(65\) 10.2266i 1.26846i
\(66\) 0 0
\(67\) −4.51757 −0.551909 −0.275955 0.961171i \(-0.588994\pi\)
−0.275955 + 0.961171i \(0.588994\pi\)
\(68\) 0 0
\(69\) 1.02465 0.639102i 0.123354 0.0769388i
\(70\) 0 0
\(71\) −6.08565 −0.722234 −0.361117 0.932521i \(-0.617604\pi\)
−0.361117 + 0.932521i \(0.617604\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) −2.40885 + 1.50246i −0.278150 + 0.173489i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 11.7569i 1.32276i −0.750051 0.661380i \(-0.769971\pi\)
0.750051 0.661380i \(-0.230029\pi\)
\(80\) 0 0
\(81\) −5.51757 7.11030i −0.613063 0.790034i
\(82\) 0 0
\(83\) 0.438828i 0.0481676i 0.999710 + 0.0240838i \(0.00766685\pi\)
−0.999710 + 0.0240838i \(0.992333\pi\)
\(84\) 0 0
\(85\) 11.1567i 1.21011i
\(86\) 0 0
\(87\) −10.4494 + 6.51757i −1.12030 + 0.698757i
\(88\) 0 0
\(89\) 2.64190i 0.280041i 0.990149 + 0.140020i \(0.0447168\pi\)
−0.990149 + 0.140020i \(0.955283\pi\)
\(90\) 0 0
\(91\) 5.57834 0.584769
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 12.7215 1.30520
\(96\) 0 0
\(97\) 11.0351 1.12045 0.560224 0.828341i \(-0.310715\pi\)
0.560224 + 0.828341i \(0.310715\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −10.3380 −1.02867 −0.514336 0.857589i \(-0.671962\pi\)
−0.514336 + 0.857589i \(0.671962\pi\)
\(102\) 0 0
\(103\) 11.7569i 1.15845i 0.815169 + 0.579223i \(0.196644\pi\)
−0.815169 + 0.579223i \(0.803356\pi\)
\(104\) 0 0
\(105\) −1.68045 2.69421i −0.163995 0.262928i
\(106\) 0 0
\(107\) 3.66655i 0.354459i 0.984170 + 0.177229i \(0.0567135\pi\)
−0.984170 + 0.177229i \(0.943287\pi\)
\(108\) 0 0
\(109\) 17.8785i 1.71245i 0.516606 + 0.856223i \(0.327195\pi\)
−0.516606 + 0.856223i \(0.672805\pi\)
\(110\) 0 0
\(111\) 1.72188 + 2.76063i 0.163433 + 0.262028i
\(112\) 0 0
\(113\) 5.38843i 0.506901i 0.967348 + 0.253450i \(0.0815654\pi\)
−0.967348 + 0.253450i \(0.918435\pi\)
\(114\) 0 0
\(115\) 1.27820 0.119193
\(116\) 0 0
\(117\) 15.0292 + 7.36090i 1.38945 + 0.680515i
\(118\) 0 0
\(119\) −6.08565 −0.557871
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 4.30013 + 6.89427i 0.387730 + 0.621636i
\(124\) 0 0
\(125\) −12.1713 −1.08863
\(126\) 0 0
\(127\) 8.39604i 0.745028i −0.928027 0.372514i \(-0.878496\pi\)
0.928027 0.372514i \(-0.121504\pi\)
\(128\) 0 0
\(129\) −2.00000 + 1.24745i −0.176090 + 0.109832i
\(130\) 0 0
\(131\) 6.89427i 0.602355i 0.953568 + 0.301178i \(0.0973797\pi\)
−0.953568 + 0.301178i \(0.902620\pi\)
\(132\) 0 0
\(133\) 6.93923i 0.601708i
\(134\) 0 0
\(135\) −0.972337 9.47622i −0.0836855 0.815583i
\(136\) 0 0
\(137\) 19.5044i 1.66637i 0.552992 + 0.833187i \(0.313486\pi\)
−0.552992 + 0.833187i \(0.686514\pi\)
\(138\) 0 0
\(139\) −12.2174 −1.03627 −0.518135 0.855299i \(-0.673373\pi\)
−0.518135 + 0.855299i \(0.673373\pi\)
\(140\) 0 0
\(141\) −5.06100 + 3.15667i −0.426213 + 0.265840i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −13.0351 −1.08251
\(146\) 0 0
\(147\) −1.46962 + 0.916638i −0.121212 + 0.0756030i
\(148\) 0 0
\(149\) −4.83820 −0.396361 −0.198180 0.980166i \(-0.563503\pi\)
−0.198180 + 0.980166i \(0.563503\pi\)
\(150\) 0 0
\(151\) 15.1178i 1.23027i 0.788421 + 0.615136i \(0.210899\pi\)
−0.788421 + 0.615136i \(0.789101\pi\)
\(152\) 0 0
\(153\) −16.3960 8.03033i −1.32554 0.649213i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 9.33528i 0.745036i −0.928025 0.372518i \(-0.878494\pi\)
0.928025 0.372518i \(-0.121506\pi\)
\(158\) 0 0
\(159\) 17.8872 11.1567i 1.41854 0.884782i
\(160\) 0 0
\(161\) 0.697224i 0.0549489i
\(162\) 0 0
\(163\) −6.63910 −0.520015 −0.260007 0.965607i \(-0.583725\pi\)
−0.260007 + 0.965607i \(0.583725\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −19.5044 −1.50930 −0.754648 0.656130i \(-0.772192\pi\)
−0.754648 + 0.656130i \(0.772192\pi\)
\(168\) 0 0
\(169\) −18.1178 −1.39368
\(170\) 0 0
\(171\) 9.15667 18.6958i 0.700228 1.42970i
\(172\) 0 0
\(173\) −11.2157 −0.852712 −0.426356 0.904555i \(-0.640203\pi\)
−0.426356 + 0.904555i \(0.640203\pi\)
\(174\) 0 0
\(175\) 1.63910i 0.123904i
\(176\) 0 0
\(177\) 8.19802 + 13.1436i 0.616201 + 0.987936i
\(178\) 0 0
\(179\) 20.6761i 1.54540i 0.634771 + 0.772700i \(0.281094\pi\)
−0.634771 + 0.772700i \(0.718906\pi\)
\(180\) 0 0
\(181\) 4.30013i 0.319626i −0.987147 0.159813i \(-0.948911\pi\)
0.987147 0.159813i \(-0.0510892\pi\)
\(182\) 0 0
\(183\) −3.94166 6.31955i −0.291376 0.467155i
\(184\) 0 0
\(185\) 3.44375i 0.253190i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −5.16901 + 0.530383i −0.375990 + 0.0385797i
\(190\) 0 0
\(191\) 24.1956 1.75073 0.875366 0.483460i \(-0.160620\pi\)
0.875366 + 0.483460i \(0.160620\pi\)
\(192\) 0 0
\(193\) −6.39604 −0.460397 −0.230199 0.973144i \(-0.573938\pi\)
−0.230199 + 0.973144i \(0.573938\pi\)
\(194\) 0 0
\(195\) 9.37411 + 15.0292i 0.671294 + 1.07627i
\(196\) 0 0
\(197\) 3.44375 0.245357 0.122679 0.992446i \(-0.460852\pi\)
0.122679 + 0.992446i \(0.460852\pi\)
\(198\) 0 0
\(199\) 9.27820i 0.657714i −0.944380 0.328857i \(-0.893337\pi\)
0.944380 0.328857i \(-0.106663\pi\)
\(200\) 0 0
\(201\) −6.63910 + 4.14098i −0.468286 + 0.292082i
\(202\) 0 0
\(203\) 7.11030i 0.499045i
\(204\) 0 0
\(205\) 8.60027i 0.600669i
\(206\) 0 0
\(207\) 0.920022 1.87847i 0.0639460 0.130563i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −0.760632 −0.0523641 −0.0261820 0.999657i \(-0.508335\pi\)
−0.0261820 + 0.999657i \(0.508335\pi\)
\(212\) 0 0
\(213\) −8.94358 + 5.57834i −0.612804 + 0.382221i
\(214\) 0 0
\(215\) −2.49490 −0.170151
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −8.81770 + 5.49983i −0.595845 + 0.371644i
\(220\) 0 0
\(221\) 33.9478 2.28358
\(222\) 0 0
\(223\) 29.0351i 1.94434i −0.234283 0.972168i \(-0.575274\pi\)
0.234283 0.972168i \(-0.424726\pi\)
\(224\) 0 0
\(225\) −2.16288 + 4.41609i −0.144192 + 0.294406i
\(226\) 0 0
\(227\) 3.88258i 0.257696i 0.991664 + 0.128848i \(0.0411279\pi\)
−0.991664 + 0.128848i \(0.958872\pi\)
\(228\) 0 0
\(229\) 3.02193i 0.199695i 0.995003 + 0.0998474i \(0.0318355\pi\)
−0.995003 + 0.0998474i \(0.968165\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 12.1713i 0.797368i −0.917088 0.398684i \(-0.869467\pi\)
0.917088 0.398684i \(-0.130533\pi\)
\(234\) 0 0
\(235\) −6.31335 −0.411837
\(236\) 0 0
\(237\) −10.7769 17.2782i −0.700032 1.12234i
\(238\) 0 0
\(239\) −22.2509 −1.43929 −0.719647 0.694341i \(-0.755696\pi\)
−0.719647 + 0.694341i \(0.755696\pi\)
\(240\) 0 0
\(241\) 15.0351 0.968499 0.484249 0.874930i \(-0.339093\pi\)
0.484249 + 0.874930i \(0.339093\pi\)
\(242\) 0 0
\(243\) −14.6263 5.39181i −0.938277 0.345885i
\(244\) 0 0
\(245\) −1.83328 −0.117124
\(246\) 0 0
\(247\) 38.7094i 2.46302i
\(248\) 0 0
\(249\) 0.402246 + 0.644909i 0.0254913 + 0.0408694i
\(250\) 0 0
\(251\) 23.3870i 1.47617i −0.674706 0.738087i \(-0.735730\pi\)
0.674706 0.738087i \(-0.264270\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −10.2266 16.3960i −0.640416 1.02676i
\(256\) 0 0
\(257\) 20.3063i 1.26667i −0.773878 0.633335i \(-0.781686\pi\)
0.773878 0.633335i \(-0.218314\pi\)
\(258\) 0 0
\(259\) 1.87847 0.116722
\(260\) 0 0
\(261\) −9.38241 + 19.1567i −0.580756 + 1.18577i
\(262\) 0 0
\(263\) 18.2570 1.12577 0.562886 0.826535i \(-0.309691\pi\)
0.562886 + 0.826535i \(0.309691\pi\)
\(264\) 0 0
\(265\) 22.3133 1.37070
\(266\) 0 0
\(267\) 2.42166 + 3.88258i 0.148203 + 0.237610i
\(268\) 0 0
\(269\) 0.216029 0.0131715 0.00658577 0.999978i \(-0.497904\pi\)
0.00658577 + 0.999978i \(0.497904\pi\)
\(270\) 0 0
\(271\) 10.5564i 0.641256i 0.947205 + 0.320628i \(0.103894\pi\)
−0.947205 + 0.320628i \(0.896106\pi\)
\(272\) 0 0
\(273\) 8.19802 5.11331i 0.496167 0.309472i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 6.72180i 0.403874i −0.979399 0.201937i \(-0.935276\pi\)
0.979399 0.201937i \(-0.0647236\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.83820i 0.288623i −0.989532 0.144311i \(-0.953903\pi\)
0.989532 0.144311i \(-0.0460967\pi\)
\(282\) 0 0
\(283\) −2.33897 −0.139037 −0.0695186 0.997581i \(-0.522146\pi\)
−0.0695186 + 0.997581i \(0.522146\pi\)
\(284\) 0 0
\(285\) 18.6958 11.6610i 1.10744 0.690740i
\(286\) 0 0
\(287\) 4.69120 0.276913
\(288\) 0 0
\(289\) −20.0351 −1.17854
\(290\) 0 0
\(291\) 16.2174 10.1152i 0.950683 0.592965i
\(292\) 0 0
\(293\) −11.2157 −0.655227 −0.327614 0.944812i \(-0.606244\pi\)
−0.327614 + 0.944812i \(0.606244\pi\)
\(294\) 0 0
\(295\) 16.3960i 0.954614i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.88935i 0.224927i
\(300\) 0 0
\(301\) 1.36090i 0.0784409i
\(302\) 0 0
\(303\) −15.1929 + 9.47622i −0.872812 + 0.544395i
\(304\) 0 0
\(305\) 7.88333i 0.451398i
\(306\) 0 0
\(307\) 1.73870 0.0992330 0.0496165 0.998768i \(-0.484200\pi\)
0.0496165 + 0.998768i \(0.484200\pi\)
\(308\) 0 0
\(309\) 10.7769 + 17.2782i 0.613074 + 0.982923i
\(310\) 0 0
\(311\) −10.7769 −0.611099 −0.305550 0.952176i \(-0.598840\pi\)
−0.305550 + 0.952176i \(0.598840\pi\)
\(312\) 0 0
\(313\) 16.4787 0.931433 0.465717 0.884934i \(-0.345797\pi\)
0.465717 + 0.884934i \(0.345797\pi\)
\(314\) 0 0
\(315\) −4.93923 2.41910i −0.278294 0.136301i
\(316\) 0 0
\(317\) 33.7250 1.89419 0.947093 0.320960i \(-0.104006\pi\)
0.947093 + 0.320960i \(0.104006\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 3.36090 + 5.38843i 0.187587 + 0.300753i
\(322\) 0 0
\(323\) 42.2298i 2.34973i
\(324\) 0 0
\(325\) 9.14346i 0.507188i
\(326\) 0 0
\(327\) 16.3881 + 26.2745i 0.906263 + 1.45298i
\(328\) 0 0
\(329\) 3.44375i 0.189860i
\(330\) 0 0
\(331\) 23.6742 1.30125 0.650627 0.759397i \(-0.274506\pi\)
0.650627 + 0.759397i \(0.274506\pi\)
\(332\) 0 0
\(333\) 5.06100 + 2.47874i 0.277341 + 0.135834i
\(334\) 0 0
\(335\) −8.28195 −0.452491
\(336\) 0 0
\(337\) 13.1178 0.714574 0.357287 0.933995i \(-0.383702\pi\)
0.357287 + 0.933995i \(0.383702\pi\)
\(338\) 0 0
\(339\) 4.93923 + 7.91893i 0.268262 + 0.430097i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 1.87847 1.17165i 0.101133 0.0630795i
\(346\) 0 0
\(347\) 28.8868i 1.55073i −0.631516 0.775363i \(-0.717567\pi\)
0.631516 0.775363i \(-0.282433\pi\)
\(348\) 0 0
\(349\) 8.05707i 0.431285i −0.976472 0.215643i \(-0.930815\pi\)
0.976472 0.215643i \(-0.0691846\pi\)
\(350\) 0 0
\(351\) 28.8345 2.95865i 1.53907 0.157921i
\(352\) 0 0
\(353\) 2.64190i 0.140614i −0.997525 0.0703070i \(-0.977602\pi\)
0.997525 0.0703070i \(-0.0223979\pi\)
\(354\) 0 0
\(355\) −11.1567 −0.592135
\(356\) 0 0
\(357\) −8.94358 + 5.57834i −0.473344 + 0.295237i
\(358\) 0 0
\(359\) 6.19028 0.326711 0.163355 0.986567i \(-0.447768\pi\)
0.163355 + 0.986567i \(0.447768\pi\)
\(360\) 0 0
\(361\) 29.1530 1.53437
\(362\) 0 0
\(363\) 16.1658 10.0830i 0.848484 0.529221i
\(364\) 0 0
\(365\) −10.9997 −0.575748
\(366\) 0 0
\(367\) 10.4787i 0.546986i −0.961874 0.273493i \(-0.911821\pi\)
0.961874 0.273493i \(-0.0881790\pi\)
\(368\) 0 0
\(369\) 12.6391 + 6.19028i 0.657965 + 0.322253i
\(370\) 0 0
\(371\) 12.1713i 0.631902i
\(372\) 0 0
\(373\) 26.4787i 1.37102i −0.728065 0.685508i \(-0.759580\pi\)
0.728065 0.685508i \(-0.240420\pi\)
\(374\) 0 0
\(375\) −17.8872 + 11.1567i −0.923689 + 0.576128i
\(376\) 0 0
\(377\) 39.6637i 2.04278i
\(378\) 0 0
\(379\) 20.9524 1.07625 0.538127 0.842863i \(-0.319132\pi\)
0.538127 + 0.842863i \(0.319132\pi\)
\(380\) 0 0
\(381\) −7.69613 12.3390i −0.394285 0.632144i
\(382\) 0 0
\(383\) 8.72755 0.445957 0.222978 0.974823i \(-0.428422\pi\)
0.222978 + 0.974823i \(0.428422\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.79577 + 3.66655i −0.0912844 + 0.186381i
\(388\) 0 0
\(389\) −7.11030 −0.360507 −0.180253 0.983620i \(-0.557692\pi\)
−0.180253 + 0.983620i \(0.557692\pi\)
\(390\) 0 0
\(391\) 4.24306i 0.214581i
\(392\) 0 0
\(393\) 6.31955 + 10.1319i 0.318779 + 0.511089i
\(394\) 0 0
\(395\) 21.5537i 1.08448i
\(396\) 0 0
\(397\) 19.2137i 0.964310i 0.876086 + 0.482155i \(0.160146\pi\)
−0.876086 + 0.482155i \(0.839854\pi\)
\(398\) 0 0
\(399\) −6.36076 10.1980i −0.318436 0.510540i
\(400\) 0 0
\(401\) 17.5597i 0.876891i −0.898758 0.438445i \(-0.855529\pi\)
0.898758 0.438445i \(-0.144471\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −10.1152 13.0351i −0.502630 0.647721i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −7.20053 −0.356043 −0.178022 0.984027i \(-0.556970\pi\)
−0.178022 + 0.984027i \(0.556970\pi\)
\(410\) 0 0
\(411\) 17.8785 + 28.6640i 0.881880 + 1.41389i
\(412\) 0 0
\(413\) 8.94358 0.440085
\(414\) 0 0
\(415\) 0.804492i 0.0394910i
\(416\) 0 0
\(417\) −17.9550 + 11.1990i −0.879258 + 0.548416i
\(418\) 0 0
\(419\) 13.7818i 0.673284i 0.941633 + 0.336642i \(0.109291\pi\)
−0.941633 + 0.336642i \(0.890709\pi\)
\(420\) 0 0
\(421\) 13.0351i 0.635294i 0.948209 + 0.317647i \(0.102893\pi\)
−0.948209 + 0.317647i \(0.897107\pi\)
\(422\) 0 0
\(423\) −4.54421 + 9.27820i −0.220947 + 0.451122i
\(424\) 0 0
\(425\) 9.97500i 0.483859i
\(426\) 0 0
\(427\) −4.30013 −0.208098
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 30.5329 1.47072 0.735359 0.677678i \(-0.237014\pi\)
0.735359 + 0.677678i \(0.237014\pi\)
\(432\) 0 0
\(433\) 1.75694 0.0844331 0.0422166 0.999108i \(-0.486558\pi\)
0.0422166 + 0.999108i \(0.486558\pi\)
\(434\) 0 0
\(435\) −19.1567 + 11.9485i −0.918492 + 0.572887i
\(436\) 0 0
\(437\) 4.83820 0.231442
\(438\) 0 0
\(439\) 24.7921i 1.18326i −0.806209 0.591631i \(-0.798484\pi\)
0.806209 0.591631i \(-0.201516\pi\)
\(440\) 0 0
\(441\) −1.31955 + 2.69421i −0.0628358 + 0.128296i
\(442\) 0 0
\(443\) 17.4416i 0.828673i 0.910124 + 0.414337i \(0.135986\pi\)
−0.910124 + 0.414337i \(0.864014\pi\)
\(444\) 0 0
\(445\) 4.84333i 0.229596i
\(446\) 0 0
\(447\) −7.11030 + 4.43488i −0.336306 + 0.209762i
\(448\) 0 0
\(449\) 6.88750i 0.325041i −0.986705 0.162521i \(-0.948038\pi\)
0.986705 0.162521i \(-0.0519624\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 13.8576 + 22.2174i 0.651086 + 1.04387i
\(454\) 0 0
\(455\) 10.2266 0.479432
\(456\) 0 0
\(457\) −32.9524 −1.54145 −0.770725 0.637168i \(-0.780106\pi\)
−0.770725 + 0.637168i \(0.780106\pi\)
\(458\) 0 0
\(459\) −31.4568 + 3.22772i −1.46828 + 0.150657i
\(460\) 0 0
\(461\) −5.49983 −0.256152 −0.128076 0.991764i \(-0.540880\pi\)
−0.128076 + 0.991764i \(0.540880\pi\)
\(462\) 0 0
\(463\) 5.44359i 0.252985i −0.991968 0.126493i \(-0.959628\pi\)
0.991968 0.126493i \(-0.0403720\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 38.7792i 1.79449i −0.441534 0.897245i \(-0.645566\pi\)
0.441534 0.897245i \(-0.354434\pi\)
\(468\) 0 0
\(469\) 4.51757i 0.208602i
\(470\) 0 0
\(471\) −8.55707 13.7193i −0.394289 0.632151i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −11.3741 −0.521880
\(476\) 0 0
\(477\) 16.0607 32.7921i 0.735367 1.50145i
\(478\) 0 0
\(479\) 14.2206 0.649756 0.324878 0.945756i \(-0.394677\pi\)
0.324878 + 0.945756i \(0.394677\pi\)
\(480\) 0 0
\(481\) −10.4787 −0.477789
\(482\) 0 0
\(483\) −0.639102 1.02465i −0.0290801 0.0466233i
\(484\) 0 0
\(485\) 20.2305 0.918618
\(486\) 0 0
\(487\) 33.6742i 1.52593i −0.646442 0.762963i \(-0.723744\pi\)
0.646442 0.762963i \(-0.276256\pi\)
\(488\) 0 0
\(489\) −9.75694 + 6.08565i −0.441224 + 0.275203i
\(490\) 0 0
\(491\) 32.5534i 1.46911i 0.678548 + 0.734556i \(0.262610\pi\)
−0.678548 + 0.734556i \(0.737390\pi\)
\(492\) 0 0
\(493\) 43.2708i 1.94882i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.08565i 0.272979i
\(498\) 0 0
\(499\) 18.9963 0.850392 0.425196 0.905101i \(-0.360205\pi\)
0.425196 + 0.905101i \(0.360205\pi\)
\(500\) 0 0
\(501\) −28.6640 + 17.8785i −1.28061 + 0.798751i
\(502\) 0 0
\(503\) −2.49490 −0.111242 −0.0556211 0.998452i \(-0.517714\pi\)
−0.0556211 + 0.998452i \(0.517714\pi\)
\(504\) 0 0
\(505\) −18.9524 −0.843373
\(506\) 0 0
\(507\) −26.6263 + 16.6075i −1.18252 + 0.737565i
\(508\) 0 0
\(509\) −33.0634 −1.46551 −0.732754 0.680493i \(-0.761766\pi\)
−0.732754 + 0.680493i \(0.761766\pi\)
\(510\) 0 0
\(511\) 6.00000i 0.265424i
\(512\) 0 0
\(513\) −3.68045 35.8690i −0.162496 1.58365i
\(514\) 0 0
\(515\) 21.5537i 0.949770i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −16.4828 + 10.2807i −0.723513 + 0.451273i
\(520\) 0 0
\(521\) 30.1343i 1.32021i −0.751175 0.660103i \(-0.770513\pi\)
0.751175 0.660103i \(-0.229487\pi\)
\(522\) 0 0
\(523\) 7.37411 0.322447 0.161224 0.986918i \(-0.448456\pi\)
0.161224 + 0.986918i \(0.448456\pi\)
\(524\) 0 0
\(525\) 1.50246 + 2.40885i 0.0655728 + 0.105131i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −22.5139 −0.978864
\(530\) 0 0
\(531\) 24.0959 + 11.8015i 1.04567 + 0.512142i
\(532\) 0 0
\(533\) −26.1691 −1.13351
\(534\) 0 0
\(535\) 6.72180i 0.290609i
\(536\) 0 0
\(537\) 18.9524 + 30.3859i 0.817858 + 1.31125i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0.792082i 0.0340543i −0.999855 0.0170271i \(-0.994580\pi\)
0.999855 0.0170271i \(-0.00542017\pi\)
\(542\) 0 0
\(543\) −3.94166 6.31955i −0.169153 0.271198i
\(544\) 0 0
\(545\) 32.7762i 1.40398i
\(546\) 0 0
\(547\) 9.36090 0.400243 0.200122 0.979771i \(-0.435866\pi\)
0.200122 + 0.979771i \(0.435866\pi\)
\(548\) 0 0
\(549\) −11.5855 5.67424i −0.494456 0.242171i
\(550\) 0 0
\(551\) −49.3401 −2.10196
\(552\) 0 0
\(553\) −11.7569 −0.499956
\(554\) 0 0
\(555\) 3.15667 + 5.06100i 0.133993 + 0.214827i
\(556\) 0 0
\(557\) 1.60371 0.0679513 0.0339756 0.999423i \(-0.489183\pi\)
0.0339756 + 0.999423i \(0.489183\pi\)
\(558\) 0 0
\(559\) 7.59155i 0.321088i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 8.94358i 0.376927i −0.982080 0.188463i \(-0.939649\pi\)
0.982080 0.188463i \(-0.0603507\pi\)
\(564\) 0 0
\(565\) 9.87847i 0.415590i
\(566\) 0 0
\(567\) −7.11030 + 5.51757i −0.298605 + 0.231716i
\(568\) 0 0
\(569\) 39.1134i 1.63972i 0.572564 + 0.819860i \(0.305949\pi\)
−0.572564 + 0.819860i \(0.694051\pi\)
\(570\) 0 0
\(571\) −14.6391 −0.612627 −0.306314 0.951931i \(-0.599096\pi\)
−0.306314 + 0.951931i \(0.599096\pi\)
\(572\) 0 0
\(573\) 35.5583 22.1786i 1.48547 0.926524i
\(574\) 0 0
\(575\) −1.14282 −0.0476589
\(576\) 0 0
\(577\) 28.2357 1.17547 0.587733 0.809055i \(-0.300020\pi\)
0.587733 + 0.809055i \(0.300020\pi\)
\(578\) 0 0
\(579\) −9.39973 + 5.86285i −0.390640 + 0.243652i
\(580\) 0 0
\(581\) 0.438828 0.0182056
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 27.5527 + 13.4946i 1.13916 + 0.557931i
\(586\) 0 0
\(587\) 3.45052i 0.142418i 0.997461 + 0.0712091i \(0.0226858\pi\)
−0.997461 + 0.0712091i \(0.977314\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 5.06100 3.15667i 0.208182 0.129848i
\(592\) 0 0
\(593\) 4.03635i 0.165753i 0.996560 + 0.0828764i \(0.0264107\pi\)
−0.996560 + 0.0828764i \(0.973589\pi\)
\(594\) 0 0
\(595\) −11.1567 −0.457379
\(596\) 0 0
\(597\) −8.50475 13.6354i −0.348076 0.558060i
\(598\) 0 0
\(599\) 9.23541 0.377349 0.188674 0.982040i \(-0.439581\pi\)
0.188674 + 0.982040i \(0.439581\pi\)
\(600\) 0 0
\(601\) −38.7921 −1.58236 −0.791181 0.611582i \(-0.790533\pi\)
−0.791181 + 0.611582i \(0.790533\pi\)
\(602\) 0 0
\(603\) −5.96116 + 12.1713i −0.242757 + 0.495654i
\(604\) 0 0
\(605\) 20.1660 0.819866
\(606\) 0 0
\(607\) 6.31335i 0.256251i −0.991758 0.128125i \(-0.959104\pi\)
0.991758 0.128125i \(-0.0408960\pi\)
\(608\) 0 0
\(609\) 6.51757 + 10.4494i 0.264105 + 0.423432i
\(610\) 0 0
\(611\) 19.2104i 0.777170i
\(612\) 0 0
\(613\) 8.19182i 0.330864i 0.986221 + 0.165432i \(0.0529019\pi\)
−0.986221 + 0.165432i \(0.947098\pi\)
\(614\) 0 0
\(615\) 7.88333 + 12.6391i 0.317886 + 0.509658i
\(616\) 0 0
\(617\) 18.6602i 0.751231i −0.926776 0.375615i \(-0.877431\pi\)
0.926776 0.375615i \(-0.122569\pi\)
\(618\) 0 0
\(619\) −14.6962 −0.590689 −0.295345 0.955391i \(-0.595434\pi\)
−0.295345 + 0.955391i \(0.595434\pi\)
\(620\) 0 0
\(621\) −0.369795 3.60396i −0.0148394 0.144622i
\(622\) 0 0
\(623\) 2.64190 0.105845
\(624\) 0 0
\(625\) −14.1178 −0.564714
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 11.4317 0.455812
\(630\) 0 0
\(631\) 10.5564i 0.420244i 0.977675 + 0.210122i \(0.0673861\pi\)
−0.977675 + 0.210122i \(0.932614\pi\)
\(632\) 0 0
\(633\) −1.11784 + 0.697224i −0.0444301 + 0.0277122i
\(634\) 0 0
\(635\) 15.3923i 0.610823i
\(636\) 0 0
\(637\) 5.57834i 0.221022i
\(638\) 0 0
\(639\) −8.03033 + 16.3960i −0.317675 + 0.648617i
\(640\) 0 0
\(641\) 14.5616i 0.575148i −0.957758 0.287574i \(-0.907151\pi\)
0.957758 0.287574i \(-0.0928486\pi\)
\(642\) 0 0
\(643\) 14.3390 0.565474 0.282737 0.959198i \(-0.408758\pi\)
0.282737 + 0.959198i \(0.408758\pi\)
\(644\) 0 0
\(645\) −3.66655 + 2.28692i −0.144370 + 0.0900474i
\(646\) 0 0
\(647\) 34.3799 1.35161 0.675806 0.737080i \(-0.263796\pi\)
0.675806 + 0.737080i \(0.263796\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −15.8379 −0.619783 −0.309892 0.950772i \(-0.600293\pi\)
−0.309892 + 0.950772i \(0.600293\pi\)
\(654\) 0 0
\(655\) 12.6391i 0.493851i
\(656\) 0 0
\(657\) −7.91730 + 16.1653i −0.308883 + 0.630668i
\(658\) 0 0
\(659\) 34.4646i 1.34255i 0.741208 + 0.671275i \(0.234253\pi\)
−0.741208 + 0.671275i \(0.765747\pi\)
\(660\) 0 0
\(661\) 36.4919i 1.41937i −0.704518 0.709686i \(-0.748837\pi\)
0.704518 0.709686i \(-0.251163\pi\)
\(662\) 0 0
\(663\) 49.8903 31.1178i 1.93758 1.20852i
\(664\) 0 0
\(665\) 12.7215i 0.493320i
\(666\) 0 0
\(667\) −4.95747 −0.191954
\(668\) 0 0
\(669\) −26.6147 42.6706i −1.02898 1.64974i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 27.0351 1.04213 0.521064 0.853518i \(-0.325535\pi\)
0.521064 + 0.853518i \(0.325535\pi\)
\(674\) 0 0
\(675\) 0.869351 + 8.47254i 0.0334613 + 0.326108i
\(676\) 0 0
\(677\) −12.8329 −0.493209 −0.246605 0.969116i \(-0.579315\pi\)
−0.246605 + 0.969116i \(0.579315\pi\)
\(678\) 0 0
\(679\) 11.0351i 0.423490i
\(680\) 0 0
\(681\) 3.55892 + 5.70591i 0.136378 + 0.218651i
\(682\) 0 0
\(683\) 6.45545i 0.247011i −0.992344 0.123505i \(-0.960586\pi\)
0.992344 0.123505i \(-0.0394136\pi\)
\(684\) 0 0
\(685\) 35.7569i 1.36620i
\(686\) 0 0
\(687\) 2.77001 + 4.44108i 0.105683 + 0.169438i
\(688\) 0 0
\(689\) 67.8956i 2.58662i
\(690\) 0 0
\(691\) 1.90409 0.0724351 0.0362175 0.999344i \(-0.488469\pi\)
0.0362175 + 0.999344i \(0.488469\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −22.3979 −0.849602
\(696\) 0 0
\(697\) 28.5490 1.08137
\(698\) 0 0
\(699\) −11.1567 17.8872i −0.421984 0.676554i
\(700\) 0 0
\(701\) 8.50475 0.321220 0.160610 0.987018i \(-0.448654\pi\)
0.160610 + 0.987018i \(0.448654\pi\)
\(702\) 0 0
\(703\) 13.0351i 0.491630i
\(704\) 0 0
\(705\) −9.27820 + 5.78705i −0.349437 + 0.217953i
\(706\) 0 0
\(707\) 10.3380i 0.388801i
\(708\) 0 0
\(709\) 20.4349i 0.767448i −0.923448 0.383724i \(-0.874641\pi\)
0.923448 0.383724i \(-0.125359\pi\)
\(710\) 0 0
\(711\) −31.6757 15.5139i −1.18793 0.581816i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −32.7004 + 20.3960i −1.22122 + 0.761704i
\(718\) 0 0
\(719\) −30.2813 −1.12930 −0.564650 0.825330i \(-0.690989\pi\)
−0.564650 + 0.825330i \(0.690989\pi\)
\(720\) 0 0
\(721\) 11.7569 0.437851
\(722\) 0 0
\(723\) 22.0959 13.7818i 0.821756 0.512550i
\(724\) 0 0
\(725\) 11.6545 0.432838
\(726\) 0 0
\(727\) 34.4787i 1.27875i 0.768897 + 0.639373i \(0.220806\pi\)
−0.768897 + 0.639373i \(0.779194\pi\)
\(728\) 0 0
\(729\) −26.4374 + 5.48311i −0.979163 + 0.203078i
\(730\) 0 0
\(731\) 8.28195i 0.306319i
\(732\) 0 0
\(733\) 22.9707i 0.848442i −0.905559 0.424221i \(-0.860548\pi\)
0.905559 0.424221i \(-0.139452\pi\)
\(734\) 0 0
\(735\) −2.69421 + 1.68045i −0.0993776 + 0.0619843i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −6.80449 −0.250307 −0.125154 0.992137i \(-0.539942\pi\)
−0.125154 + 0.992137i \(0.539942\pi\)
\(740\) 0 0
\(741\) 35.4825 + 56.8880i 1.30348 + 2.08983i
\(742\) 0 0
\(743\) 32.8185 1.20399 0.601997 0.798498i \(-0.294372\pi\)
0.601997 + 0.798498i \(0.294372\pi\)
\(744\) 0 0
\(745\) −8.86975 −0.324963
\(746\) 0 0
\(747\) 1.18230 + 0.579055i 0.0432579 + 0.0211865i
\(748\) 0 0
\(749\) 3.66655 0.133973
\(750\) 0 0
\(751\) 14.3258i 0.522754i −0.965237 0.261377i \(-0.915823\pi\)
0.965237 0.261377i \(-0.0841766\pi\)
\(752\) 0 0
\(753\) −21.4374 34.3699i −0.781222 1.25251i
\(754\) 0 0
\(755\) 27.7152i 1.00866i
\(756\) 0 0
\(757\) 36.4349i 1.32425i −0.749394 0.662124i \(-0.769655\pi\)
0.749394 0.662124i \(-0.230345\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 14.3676i 0.520825i −0.965497 0.260413i \(-0.916141\pi\)
0.965497 0.260413i \(-0.0838586\pi\)
\(762\) 0 0
\(763\) 17.8785 0.647244
\(764\) 0 0
\(765\) −30.0585 14.7218i −1.08677 0.532268i
\(766\) 0 0
\(767\) −49.8903 −1.80143
\(768\) 0 0
\(769\) 12.4787 0.449995 0.224997 0.974359i \(-0.427763\pi\)
0.224997 + 0.974359i \(0.427763\pi\)
\(770\) 0 0
\(771\) −18.6135 29.8424i −0.670348 1.07475i
\(772\) 0 0
\(773\) 26.1759 0.941481 0.470740 0.882272i \(-0.343987\pi\)
0.470740 + 0.882272i \(0.343987\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 2.76063 1.72188i 0.0990371 0.0617720i
\(778\) 0 0
\(779\) 32.5534i 1.16635i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 3.77118 + 36.7532i 0.134771 + 1.31345i
\(784\) 0 0
\(785\) 17.1141i 0.610830i
\(786\) 0 0
\(787\) −20.2174 −0.720674 −0.360337 0.932822i \(-0.617338\pi\)
−0.360337 + 0.932822i \(0.617338\pi\)
\(788\) 0 0
\(789\) 26.8307 16.7350i 0.955199 0.595782i
\(790\) 0 0
\(791\) 5.38843 0.191590
\(792\) 0 0
\(793\) 23.9876 0.851824
\(794\) 0 0
\(795\) 32.7921 20.4533i 1.16302 0.725402i
\(796\) 0 0
\(797\) 24.1266 0.854607 0.427304 0.904108i \(-0.359464\pi\)
0.427304 + 0.904108i \(0.359464\pi\)
\(798\) 0 0
\(799\) 20.9575i 0.741422i
\(800\) 0 0