Properties

Label 3744.2.m.h.1585.9
Level $3744$
Weight $2$
Character 3744.1585
Analytic conductor $29.896$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3744.m (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.8959905168\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 2 x^{14} - 16 x^{12} - 72 x^{10} + 26 x^{8} + 360 x^{6} + 725 x^{4} + 1000 x^{2} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 936)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1585.9
Root \(1.90184 + 0.0324487i\) of defining polynomial
Character \(\chi\) \(=\) 3744.1585
Dual form 3744.2.m.h.1585.11

$q$-expansion

\(f(q)\) \(=\) \(q+1.66251 q^{5} -3.57266i q^{7} +O(q^{10})\) \(q+1.66251 q^{5} -3.57266i q^{7} -1.02749 q^{11} +(-1.87826 - 3.07768i) q^{13} +5.05251 q^{17} +1.16083 q^{19} +8.17513 q^{23} -2.23607 q^{25} +4.29792i q^{29} +7.98872i q^{31} -5.93958i q^{35} +9.83470 q^{37} +1.62054i q^{41} +2.35114i q^{43} -7.73877i q^{47} -5.76393 q^{49} -11.2521i q^{53} -1.70820 q^{55} -9.73249 q^{59} -11.4127i q^{61} +(-3.12262 - 5.11667i) q^{65} -7.23901 q^{67} -10.5672i q^{71} -4.41606i q^{73} +3.67086i q^{77} +6.94427 q^{79} -2.29753 q^{83} +8.39984 q^{85} -9.02546i q^{89} +(-10.9955 + 6.71040i) q^{91} +1.92989 q^{95} -11.5614i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + O(q^{10}) \) \( 16 q - 128 q^{49} + 80 q^{55} - 32 q^{79} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(2017\) \(2081\) \(2341\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.66251 0.743496 0.371748 0.928334i \(-0.378759\pi\)
0.371748 + 0.928334i \(0.378759\pi\)
\(6\) 0 0
\(7\) 3.57266i 1.35034i −0.737662 0.675170i \(-0.764070\pi\)
0.737662 0.675170i \(-0.235930\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.02749 −0.309799 −0.154899 0.987930i \(-0.549505\pi\)
−0.154899 + 0.987930i \(0.549505\pi\)
\(12\) 0 0
\(13\) −1.87826 3.07768i −0.520936 0.853596i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.05251 1.22541 0.612707 0.790310i \(-0.290081\pi\)
0.612707 + 0.790310i \(0.290081\pi\)
\(18\) 0 0
\(19\) 1.16083 0.266312 0.133156 0.991095i \(-0.457489\pi\)
0.133156 + 0.991095i \(0.457489\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.17513 1.70463 0.852317 0.523026i \(-0.175197\pi\)
0.852317 + 0.523026i \(0.175197\pi\)
\(24\) 0 0
\(25\) −2.23607 −0.447214
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.29792i 0.798104i 0.916928 + 0.399052i \(0.130661\pi\)
−0.916928 + 0.399052i \(0.869339\pi\)
\(30\) 0 0
\(31\) 7.98872i 1.43482i 0.696653 + 0.717408i \(0.254672\pi\)
−0.696653 + 0.717408i \(0.745328\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.93958i 1.00397i
\(36\) 0 0
\(37\) 9.83470 1.61682 0.808408 0.588623i \(-0.200330\pi\)
0.808408 + 0.588623i \(0.200330\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.62054i 0.253087i 0.991961 + 0.126543i \(0.0403883\pi\)
−0.991961 + 0.126543i \(0.959612\pi\)
\(42\) 0 0
\(43\) 2.35114i 0.358546i 0.983799 + 0.179273i \(0.0573745\pi\)
−0.983799 + 0.179273i \(0.942626\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.73877i 1.12882i −0.825496 0.564408i \(-0.809105\pi\)
0.825496 0.564408i \(-0.190895\pi\)
\(48\) 0 0
\(49\) −5.76393 −0.823419
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.2521i 1.54560i −0.634652 0.772798i \(-0.718857\pi\)
0.634652 0.772798i \(-0.281143\pi\)
\(54\) 0 0
\(55\) −1.70820 −0.230334
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −9.73249 −1.26706 −0.633531 0.773717i \(-0.718395\pi\)
−0.633531 + 0.773717i \(0.718395\pi\)
\(60\) 0 0
\(61\) 11.4127i 1.46124i −0.682782 0.730622i \(-0.739230\pi\)
0.682782 0.730622i \(-0.260770\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.12262 5.11667i −0.387314 0.634645i
\(66\) 0 0
\(67\) −7.23901 −0.884386 −0.442193 0.896920i \(-0.645799\pi\)
−0.442193 + 0.896920i \(0.645799\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.5672i 1.25410i −0.778981 0.627048i \(-0.784263\pi\)
0.778981 0.627048i \(-0.215737\pi\)
\(72\) 0 0
\(73\) 4.41606i 0.516860i −0.966030 0.258430i \(-0.916795\pi\)
0.966030 0.258430i \(-0.0832052\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.67086i 0.418334i
\(78\) 0 0
\(79\) 6.94427 0.781292 0.390646 0.920541i \(-0.372252\pi\)
0.390646 + 0.920541i \(0.372252\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.29753 −0.252187 −0.126093 0.992018i \(-0.540244\pi\)
−0.126093 + 0.992018i \(0.540244\pi\)
\(84\) 0 0
\(85\) 8.39984 0.911090
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.02546i 0.956697i −0.878170 0.478349i \(-0.841236\pi\)
0.878170 0.478349i \(-0.158764\pi\)
\(90\) 0 0
\(91\) −10.9955 + 6.71040i −1.15264 + 0.703441i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.92989 0.198002
\(96\) 0 0
\(97\) 11.5614i 1.17388i −0.809630 0.586940i \(-0.800332\pi\)
0.809630 0.586940i \(-0.199668\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.96879i 0.792924i 0.918051 + 0.396462i \(0.129762\pi\)
−0.918051 + 0.396462i \(0.870238\pi\)
\(102\) 0 0
\(103\) 11.7082 1.15364 0.576822 0.816870i \(-0.304293\pi\)
0.576822 + 0.816870i \(0.304293\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.65626i 0.256791i 0.991723 + 0.128395i \(0.0409826\pi\)
−0.991723 + 0.128395i \(0.959017\pi\)
\(108\) 0 0
\(109\) −12.1564 −1.16437 −0.582184 0.813057i \(-0.697802\pi\)
−0.582184 + 0.813057i \(0.697802\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 5.05251 0.475300 0.237650 0.971351i \(-0.423623\pi\)
0.237650 + 0.971351i \(0.423623\pi\)
\(114\) 0 0
\(115\) 13.5912 1.26739
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 18.0509i 1.65473i
\(120\) 0 0
\(121\) −9.94427 −0.904025
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −12.0300 −1.07600
\(126\) 0 0
\(127\) 6.00000 0.532414 0.266207 0.963916i \(-0.414230\pi\)
0.266207 + 0.963916i \(0.414230\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 16.5646i 1.44726i −0.690189 0.723629i \(-0.742473\pi\)
0.690189 0.723629i \(-0.257527\pi\)
\(132\) 0 0
\(133\) 4.14725i 0.359612i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 13.6020i 1.16209i −0.813870 0.581047i \(-0.802643\pi\)
0.813870 0.581047i \(-0.197357\pi\)
\(138\) 0 0
\(139\) 19.3642i 1.64245i 0.570607 + 0.821223i \(0.306708\pi\)
−0.570607 + 0.821223i \(0.693292\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.92989 + 3.16228i 0.161385 + 0.264443i
\(144\) 0 0
\(145\) 7.14533i 0.593387i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.04250 0.576944 0.288472 0.957488i \(-0.406853\pi\)
0.288472 + 0.957488i \(0.406853\pi\)
\(150\) 0 0
\(151\) 3.57266i 0.290739i 0.989377 + 0.145370i \(0.0464371\pi\)
−0.989377 + 0.145370i \(0.953563\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 13.2813i 1.06678i
\(156\) 0 0
\(157\) 0.555029i 0.0442961i −0.999755 0.0221481i \(-0.992949\pi\)
0.999755 0.0221481i \(-0.00705053\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 29.2070i 2.30183i
\(162\) 0 0
\(163\) −7.23901 −0.567003 −0.283501 0.958972i \(-0.591496\pi\)
−0.283501 + 0.958972i \(0.591496\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.32611i 0.566911i −0.958985 0.283456i \(-0.908519\pi\)
0.958985 0.283456i \(-0.0914809\pi\)
\(168\) 0 0
\(169\) −5.94427 + 11.5614i −0.457252 + 0.889337i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 11.2521i 0.855482i −0.903901 0.427741i \(-0.859310\pi\)
0.903901 0.427741i \(-0.140690\pi\)
\(174\) 0 0
\(175\) 7.98872i 0.603891i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 21.8772i 1.63518i 0.575804 + 0.817588i \(0.304689\pi\)
−0.575804 + 0.817588i \(0.695311\pi\)
\(180\) 0 0
\(181\) 12.8658i 0.956305i 0.878277 + 0.478152i \(0.158693\pi\)
−0.878277 + 0.478152i \(0.841307\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 16.3503 1.20210
\(186\) 0 0
\(187\) −5.19139 −0.379632
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.92989 0.139642 0.0698209 0.997560i \(-0.477757\pi\)
0.0698209 + 0.997560i \(0.477757\pi\)
\(192\) 0 0
\(193\) 4.41606i 0.317875i −0.987289 0.158937i \(-0.949193\pi\)
0.987289 0.158937i \(-0.0508068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −23.1825 −1.65168 −0.825841 0.563903i \(-0.809299\pi\)
−0.825841 + 0.563903i \(0.809299\pi\)
\(198\) 0 0
\(199\) −18.0000 −1.27599 −0.637993 0.770042i \(-0.720235\pi\)
−0.637993 + 0.770042i \(0.720235\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 15.3550 1.07771
\(204\) 0 0
\(205\) 2.69417i 0.188169i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1.19274 −0.0825033
\(210\) 0 0
\(211\) 10.5146i 0.723856i 0.932206 + 0.361928i \(0.117881\pi\)
−0.932206 + 0.361928i \(0.882119\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.90879i 0.266577i
\(216\) 0 0
\(217\) 28.5410 1.93749
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −9.48993 15.5500i −0.638362 1.04601i
\(222\) 0 0
\(223\) 7.98872i 0.534964i −0.963563 0.267482i \(-0.913808\pi\)
0.963563 0.267482i \(-0.0861916\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.35250 0.288886 0.144443 0.989513i \(-0.453861\pi\)
0.144443 + 0.989513i \(0.453861\pi\)
\(228\) 0 0
\(229\) 1.43486 0.0948185 0.0474092 0.998876i \(-0.484904\pi\)
0.0474092 + 0.998876i \(0.484904\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.3503 1.07114 0.535571 0.844490i \(-0.320096\pi\)
0.535571 + 0.844490i \(0.320096\pi\)
\(234\) 0 0
\(235\) 12.8658i 0.839270i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7.07107i 0.457389i 0.973498 + 0.228695i \(0.0734457\pi\)
−0.973498 + 0.228695i \(0.926554\pi\)
\(240\) 0 0
\(241\) 15.9774i 1.02920i 0.857431 + 0.514599i \(0.172059\pi\)
−0.857431 + 0.514599i \(0.827941\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −9.58258 −0.612209
\(246\) 0 0
\(247\) −2.18034 3.57266i −0.138732 0.227323i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 22.5042i 1.42045i −0.703973 0.710227i \(-0.748592\pi\)
0.703973 0.710227i \(-0.251408\pi\)
\(252\) 0 0
\(253\) −8.39984 −0.528093
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.5955 1.40947 0.704735 0.709471i \(-0.251066\pi\)
0.704735 + 0.709471i \(0.251066\pi\)
\(258\) 0 0
\(259\) 35.1361i 2.18325i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 16.3503 1.00820 0.504100 0.863645i \(-0.331824\pi\)
0.504100 + 0.863645i \(0.331824\pi\)
\(264\) 0 0
\(265\) 18.7067i 1.14914i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 7.58124i 0.462237i −0.972926 0.231118i \(-0.925762\pi\)
0.972926 0.231118i \(-0.0742384\pi\)
\(270\) 0 0
\(271\) 7.98872i 0.485280i −0.970116 0.242640i \(-0.921987\pi\)
0.970116 0.242640i \(-0.0780134\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.29753 0.138546
\(276\) 0 0
\(277\) 18.1231i 1.08891i 0.838790 + 0.544455i \(0.183263\pi\)
−0.838790 + 0.544455i \(0.816737\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6.19704i 0.369684i 0.982768 + 0.184842i \(0.0591774\pi\)
−0.982768 + 0.184842i \(0.940823\pi\)
\(282\) 0 0
\(283\) 2.90617i 0.172754i 0.996263 + 0.0863769i \(0.0275289\pi\)
−0.996263 + 0.0863769i \(0.972471\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.78966 0.341753
\(288\) 0 0
\(289\) 8.52786 0.501639
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0.877578 0.0512687 0.0256343 0.999671i \(-0.491839\pi\)
0.0256343 + 0.999671i \(0.491839\pi\)
\(294\) 0 0
\(295\) −16.1803 −0.942056
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −15.3550 25.1605i −0.888005 1.45507i
\(300\) 0 0
\(301\) 8.39984 0.484159
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 18.9737i 1.08643i
\(306\) 0 0
\(307\) 14.7521 0.841944 0.420972 0.907074i \(-0.361689\pi\)
0.420972 + 0.907074i \(0.361689\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 24.5254 1.39071 0.695354 0.718667i \(-0.255248\pi\)
0.695354 + 0.718667i \(0.255248\pi\)
\(312\) 0 0
\(313\) 6.47214 0.365827 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 31.1025 1.74689 0.873446 0.486921i \(-0.161880\pi\)
0.873446 + 0.486921i \(0.161880\pi\)
\(318\) 0 0
\(319\) 4.41606i 0.247252i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 5.86510 0.326343
\(324\) 0 0
\(325\) 4.19992 + 6.88191i 0.232970 + 0.381740i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −27.6480 −1.52428
\(330\) 0 0
\(331\) 10.9955 0.604369 0.302185 0.953249i \(-0.402284\pi\)
0.302185 + 0.953249i \(0.402284\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −12.0349 −0.657537
\(336\) 0 0
\(337\) 5.52786 0.301122 0.150561 0.988601i \(-0.451892\pi\)
0.150561 + 0.988601i \(0.451892\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8.20830i 0.444504i
\(342\) 0 0
\(343\) 4.41606i 0.238445i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 20.4750i 1.09916i 0.835442 + 0.549578i \(0.185212\pi\)
−0.835442 + 0.549578i \(0.814788\pi\)
\(348\) 0 0
\(349\) −13.5912 −0.727522 −0.363761 0.931492i \(-0.618507\pi\)
−0.363761 + 0.931492i \(0.618507\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 20.5942i 1.09612i 0.836439 + 0.548060i \(0.184633\pi\)
−0.836439 + 0.548060i \(0.815367\pi\)
\(354\) 0 0
\(355\) 17.5680i 0.932415i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 32.1142i 1.69493i −0.530855 0.847463i \(-0.678129\pi\)
0.530855 0.847463i \(-0.321871\pi\)
\(360\) 0 0
\(361\) −17.6525 −0.929078
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 7.34173i 0.384284i
\(366\) 0 0
\(367\) −31.1246 −1.62469 −0.812346 0.583176i \(-0.801810\pi\)
−0.812346 + 0.583176i \(0.801810\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −40.2000 −2.08708
\(372\) 0 0
\(373\) 26.0746i 1.35009i −0.737777 0.675045i \(-0.764124\pi\)
0.737777 0.675045i \(-0.235876\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 13.2276 8.07262i 0.681258 0.415761i
\(378\) 0 0
\(379\) 19.3954 0.996273 0.498137 0.867099i \(-0.334018\pi\)
0.498137 + 0.867099i \(0.334018\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.57494i 0.182671i 0.995820 + 0.0913354i \(0.0291135\pi\)
−0.995820 + 0.0913354i \(0.970886\pi\)
\(384\) 0 0
\(385\) 6.10284i 0.311030i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 35.3980i 1.79475i 0.441270 + 0.897374i \(0.354528\pi\)
−0.441270 + 0.897374i \(0.645472\pi\)
\(390\) 0 0
\(391\) 41.3050 2.08888
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 11.5449 0.580887
\(396\) 0 0
\(397\) −20.5562 −1.03169 −0.515843 0.856683i \(-0.672521\pi\)
−0.515843 + 0.856683i \(0.672521\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 30.4149i 1.51885i 0.650596 + 0.759424i \(0.274519\pi\)
−0.650596 + 0.759424i \(0.725481\pi\)
\(402\) 0 0
\(403\) 24.5868 15.0049i 1.22475 0.747447i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −10.1050 −0.500887
\(408\) 0 0
\(409\) 34.6842i 1.71502i 0.514466 + 0.857511i \(0.327990\pi\)
−0.514466 + 0.857511i \(0.672010\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 34.7709i 1.71097i
\(414\) 0 0
\(415\) −3.81966 −0.187500
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 7.96879i 0.389301i 0.980873 + 0.194650i \(0.0623572\pi\)
−0.980873 + 0.194650i \(0.937643\pi\)
\(420\) 0 0
\(421\) 8.39984 0.409383 0.204692 0.978827i \(-0.434381\pi\)
0.204692 + 0.978827i \(0.434381\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −11.2978 −0.548022
\(426\) 0 0
\(427\) −40.7737 −1.97318
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.82688i 0.0879975i 0.999032 + 0.0439987i \(0.0140098\pi\)
−0.999032 + 0.0439987i \(0.985990\pi\)
\(432\) 0 0
\(433\) −8.18034 −0.393122 −0.196561 0.980492i \(-0.562977\pi\)
−0.196561 + 0.980492i \(0.562977\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 9.48993 0.453965
\(438\) 0 0
\(439\) −1.81966 −0.0868476 −0.0434238 0.999057i \(-0.513827\pi\)
−0.0434238 + 0.999057i \(0.513827\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 11.2521i 0.534604i −0.963613 0.267302i \(-0.913868\pi\)
0.963613 0.267302i \(-0.0861321\pi\)
\(444\) 0 0
\(445\) 15.0049i 0.711301i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 4.86163i 0.229435i −0.993398 0.114717i \(-0.963404\pi\)
0.993398 0.114717i \(-0.0365962\pi\)
\(450\) 0 0
\(451\) 1.66509i 0.0784059i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −18.2802 + 11.1561i −0.856987 + 0.523005i
\(456\) 0 0
\(457\) 17.0199i 0.796159i −0.917351 0.398079i \(-0.869677\pi\)
0.917351 0.398079i \(-0.130323\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 21.9124 1.02056 0.510282 0.860007i \(-0.329541\pi\)
0.510282 + 0.860007i \(0.329541\pi\)
\(462\) 0 0
\(463\) 3.57266i 0.166036i −0.996548 0.0830179i \(-0.973544\pi\)
0.996548 0.0830179i \(-0.0264559\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.59584i 0.397768i 0.980023 + 0.198884i \(0.0637317\pi\)
−0.980023 + 0.198884i \(0.936268\pi\)
\(468\) 0 0
\(469\) 25.8626i 1.19422i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2.41577i 0.111077i
\(474\) 0 0
\(475\) −2.59569 −0.119099
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 29.2858i 1.33810i −0.743216 0.669052i \(-0.766700\pi\)
0.743216 0.669052i \(-0.233300\pi\)
\(480\) 0 0
\(481\) −18.4721 30.2681i −0.842257 1.38011i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 19.2209i 0.872776i
\(486\) 0 0
\(487\) 0.843392i 0.0382177i −0.999817 0.0191089i \(-0.993917\pi\)
0.999817 0.0191089i \(-0.00608291\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.02920i 0.0915767i 0.998951 + 0.0457883i \(0.0145800\pi\)
−0.998951 + 0.0457883i \(0.985420\pi\)
\(492\) 0 0
\(493\) 21.7153i 0.978008i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −37.7530 −1.69346
\(498\) 0 0
\(499\) 33.5347 1.50122 0.750609 0.660747i \(-0.229760\pi\)
0.750609 + 0.660747i \(0.229760\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 6.24525 0.278462 0.139231 0.990260i \(-0.455537\pi\)
0.139231 + 0.990260i \(0.455537\pi\)
\(504\) 0 0
\(505\) 13.2482i 0.589536i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −9.58258 −0.424740 −0.212370 0.977189i \(-0.568118\pi\)
−0.212370 + 0.977189i \(0.568118\pi\)
\(510\) 0 0
\(511\) −15.7771 −0.697937
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 19.4650 0.857729
\(516\) 0 0
\(517\) 7.95148i 0.349706i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.19274 −0.0522547 −0.0261274 0.999659i \(-0.508318\pi\)
−0.0261274 + 0.999659i \(0.508318\pi\)
\(522\) 0 0
\(523\) 24.0664i 1.05235i 0.850376 + 0.526176i \(0.176375\pi\)
−0.850376 + 0.526176i \(0.823625\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 40.3631i 1.75824i
\(528\) 0 0
\(529\) 43.8328 1.90577
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.98752 3.04381i 0.216034 0.131842i
\(534\) 0 0
\(535\) 4.41606i 0.190923i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.92236 0.255094
\(540\) 0 0
\(541\) 3.20845 0.137942 0.0689711 0.997619i \(-0.478028\pi\)
0.0689711 + 0.997619i \(0.478028\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −20.2100 −0.865703
\(546\) 0 0
\(547\) 42.1895i 1.80389i −0.431847 0.901947i \(-0.642138\pi\)
0.431847 0.901947i \(-0.357862\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 4.98915i 0.212545i
\(552\) 0 0
\(553\) 24.8096i 1.05501i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −24.4525 −1.03609 −0.518043 0.855355i \(-0.673339\pi\)
−0.518043 + 0.855355i \(0.673339\pi\)
\(558\) 0 0
\(559\) 7.23607 4.41606i 0.306053 0.186779i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0.627058i 0.0264274i 0.999913 + 0.0132137i \(0.00420617\pi\)
−0.999913 + 0.0132137i \(0.995794\pi\)
\(564\) 0 0
\(565\) 8.39984 0.353384
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −26.4553 −1.10906 −0.554532 0.832163i \(-0.687103\pi\)
−0.554532 + 0.832163i \(0.687103\pi\)
\(570\) 0 0
\(571\) 24.6215i 1.03038i 0.857077 + 0.515188i \(0.172278\pi\)
−0.857077 + 0.515188i \(0.827722\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −18.2802 −0.762335
\(576\) 0 0
\(577\) 44.5588i 1.85501i 0.373816 + 0.927503i \(0.378049\pi\)
−0.373816 + 0.927503i \(0.621951\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.20830i 0.340538i
\(582\) 0 0
\(583\) 11.5614i 0.478824i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −10.5174 −0.434100 −0.217050 0.976160i \(-0.569644\pi\)
−0.217050 + 0.976160i \(0.569644\pi\)
\(588\) 0 0
\(589\) 9.27354i 0.382110i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 26.6637i 1.09495i 0.836823 + 0.547474i \(0.184411\pi\)
−0.836823 + 0.547474i \(0.815589\pi\)
\(594\) 0 0
\(595\) 30.0098i 1.23028i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −46.6653 −1.90669 −0.953347 0.301877i \(-0.902387\pi\)
−0.953347 + 0.301877i \(0.902387\pi\)
\(600\) 0 0
\(601\) 19.4164 0.792012 0.396006 0.918248i \(-0.370396\pi\)
0.396006 + 0.918248i \(0.370396\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −16.5324 −0.672139
\(606\) 0 0
\(607\) −15.7082 −0.637576 −0.318788 0.947826i \(-0.603276\pi\)
−0.318788 + 0.947826i \(0.603276\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −23.8175 + 14.5354i −0.963552 + 0.588040i
\(612\) 0 0
\(613\) −6.96497 −0.281313 −0.140656 0.990058i \(-0.544921\pi\)
−0.140656 + 0.990058i \(0.544921\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.36861i 0.135615i 0.997698 + 0.0678075i \(0.0216004\pi\)
−0.997698 + 0.0678075i \(0.978400\pi\)
\(618\) 0 0
\(619\) −29.7781 −1.19688 −0.598442 0.801166i \(-0.704213\pi\)
−0.598442 + 0.801166i \(0.704213\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −32.2450 −1.29187
\(624\) 0 0
\(625\) −8.81966 −0.352786
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 49.6899 1.98127
\(630\) 0 0
\(631\) 22.2794i 0.886928i 0.896292 + 0.443464i \(0.146251\pi\)
−0.896292 + 0.443464i \(0.853749\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 9.97505 0.395848
\(636\) 0 0
\(637\) 10.8262 + 17.7396i 0.428948 + 0.702867i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 43.9983 1.73783 0.868914 0.494963i \(-0.164818\pi\)
0.868914 + 0.494963i \(0.164818\pi\)
\(642\) 0 0
\(643\) −33.8734 −1.33584 −0.667918 0.744235i \(-0.732814\pi\)
−0.667918 + 0.744235i \(0.732814\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 36.5603 1.43733 0.718667 0.695354i \(-0.244753\pi\)
0.718667 + 0.695354i \(0.244753\pi\)
\(648\) 0 0
\(649\) 10.0000 0.392534
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 33.3688i 1.30582i 0.757435 + 0.652911i \(0.226452\pi\)
−0.757435 + 0.652911i \(0.773548\pi\)
\(654\) 0 0
\(655\) 27.5388i 1.07603i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 36.4126i 1.41843i −0.704991 0.709216i \(-0.749049\pi\)
0.704991 0.709216i \(-0.250951\pi\)
\(660\) 0 0
\(661\) −46.3038 −1.80101 −0.900504 0.434847i \(-0.856802\pi\)
−0.900504 + 0.434847i \(0.856802\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 6.89484i 0.267370i
\(666\) 0 0
\(667\) 35.1361i 1.36047i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 11.7264i 0.452692i
\(672\) 0 0
\(673\) −12.6525 −0.487717 −0.243859 0.969811i \(-0.578413\pi\)
−0.243859 + 0.969811i \(0.578413\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 17.8187i 0.684830i −0.939549 0.342415i \(-0.888755\pi\)
0.939549 0.342415i \(-0.111245\pi\)
\(678\) 0 0
\(679\) −41.3050 −1.58514
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −10.5174 −0.402438 −0.201219 0.979546i \(-0.564490\pi\)
−0.201219 + 0.979546i \(0.564490\pi\)
\(684\) 0 0
\(685\) 22.6134i 0.864012i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −34.6304 + 21.1344i −1.31931 + 0.805156i
\(690\) 0 0
\(691\) 4.91735 0.187065 0.0935324 0.995616i \(-0.470184\pi\)
0.0935324 + 0.995616i \(0.470184\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 32.1931i 1.22115i
\(696\) 0 0
\(697\) 8.18782i 0.310136i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 26.8021i 1.01230i −0.862445 0.506151i \(-0.831068\pi\)
0.862445 0.506151i \(-0.168932\pi\)
\(702\) 0 0
\(703\) 11.4164 0.430578
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 28.4698 1.07072
\(708\) 0 0
\(709\) −8.39984 −0.315463 −0.157731 0.987482i \(-0.550418\pi\)
−0.157731 + 0.987482i \(0.550418\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 65.3089i 2.44584i
\(714\) 0 0
\(715\) 3.20845 + 5.25731i 0.119989 + 0.196612i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 12.0349 0.448826 0.224413 0.974494i \(-0.427953\pi\)
0.224413 + 0.974494i \(0.427953\pi\)
\(720\) 0 0
\(721\) 41.8295i 1.55781i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 9.61045i 0.356923i
\(726\) 0 0
\(727\) 22.3607 0.829312 0.414656 0.909978i \(-0.363902\pi\)
0.414656 + 0.909978i \(0.363902\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 11.8792i 0.439367i
\(732\) 0 0
\(733\) 46.8519 1.73051 0.865256 0.501330i \(-0.167156\pi\)
0.865256 + 0.501330i \(0.167156\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7.43798 0.273982
\(738\) 0 0
\(739\) 19.9434 0.733631 0.366816 0.930294i \(-0.380448\pi\)
0.366816 + 0.930294i \(0.380448\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 47.5918i 1.74597i 0.487744 + 0.872987i \(0.337820\pi\)
−0.487744 + 0.872987i \(0.662180\pi\)
\(744\) 0 0
\(745\) 11.7082 0.428955
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 9.48993 0.346755
\(750\) 0 0
\(751\) 54.5410 1.99023 0.995115 0.0987222i \(-0.0314755\pi\)
0.995115 + 0.0987222i \(0.0314755\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5.93958i 0.216164i
\(756\) 0 0
\(757\) 8.50651i 0.309174i −0.987979 0.154587i \(-0.950595\pi\)
0.987979 0.154587i \(-0.0494047\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 16.4304i 0.595601i 0.954628 + 0.297800i \(0.0962530\pi\)
−0.954628 + 0.297800i \(0.903747\pi\)
\(762\) 0 0
\(763\) 43.4306i 1.57229i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 18.2802 + 29.9535i 0.660058 + 1.08156i
\(768\) 0 0
\(769\) 24.1653i 0.871422i 0.900087 + 0.435711i \(0.143503\pi\)
−0.900087 + 0.435711i \(0.856497\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 30.3176 1.09045 0.545224 0.838290i \(-0.316445\pi\)
0.545224 + 0.838290i \(0.316445\pi\)
\(774\) 0 0
\(775\) 17.8633i 0.641669i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.88118i 0.0674001i
\(780\) 0 0
\(781\) 10.8576i 0.388517i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0.922740i 0.0329340i
\(786\) 0 0
\(787\) 40.4996 1.44366 0.721828 0.692072i \(-0.243302\pi\)
0.721828 + 0.692072i \(0.243302\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 18.0509i 0.641817i
\(792\) 0 0
\(793\) −35.1246 + 21.4360i −1.24731 + 0.761214i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 21.4896i 0.761201i 0.924740 + 0.380601i \(0.124283\pi\)
−0.924740 + 0.380601i \(0.875717\pi\)
\(798\) 0 0
\(799\) 39.1002i 1.38327i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.53744i 0.160123i
\(804\) 0 0
\(805\) 48.5569i 1.71141i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 27.6480 0.972053 0.486026 0.873944i \(-0.338446\pi\)
0.486026 + 0.873944i \(0.338446\pi\)
\(810\)