Properties

Label 3420.2.f.c.1369.6
Level $3420$
Weight $2$
Character 3420.1369
Analytic conductor $27.309$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3420.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(27.3088374913\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.14077504.2
Defining polynomial: \( x^{6} + 9x^{4} + 14x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1369.6
Root \(1.23277i\) of defining polynomial
Character \(\chi\) \(=\) 3420.1369
Dual form 3420.2.f.c.1369.5

$q$-expansion

\(f(q)\) \(=\) \(q+(1.92411 + 1.13921i) q^{5} -4.74397i q^{7} +O(q^{10})\) \(q+(1.92411 + 1.13921i) q^{5} -4.74397i q^{7} -4.48028 q^{11} +0.843176i q^{13} +5.52315i q^{17} -1.00000 q^{19} +0.779187i q^{23} +(2.40439 + 4.38394i) q^{25} -10.6570 q^{29} -8.65699 q^{31} +(5.40439 - 9.12790i) q^{35} -1.62236i q^{37} -4.73588 q^{41} -9.67504i q^{43} +3.18559i q^{47} -15.5052 q^{49} +6.17922i q^{53} +(-8.62054 - 5.10399i) q^{55} +11.6964 q^{59} +6.48028 q^{61} +(-0.960558 + 1.62236i) q^{65} +14.8880i q^{67} -0.303566 q^{71} +10.0800i q^{73} +21.2543i q^{77} -4.00000 q^{79} -0.779187i q^{83} +(-6.29205 + 10.6271i) q^{85} -5.69643 q^{89} +4.00000 q^{91} +(-1.92411 - 1.13921i) q^{95} +6.17922i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{5} - 18 q^{11} - 6 q^{19} - 5 q^{25} - 4 q^{29} + 8 q^{31} + 13 q^{35} - 4 q^{41} - 12 q^{49} + q^{55} + 28 q^{59} + 30 q^{61} + 12 q^{65} - 44 q^{71} - 24 q^{79} - 15 q^{85} + 8 q^{89} + 24 q^{91} - q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times\).

\(n\) \(1711\) \(1901\) \(2737\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(5\) 1.92411 + 1.13921i 0.860487 + 0.509472i
\(6\) 0 0
\(7\) 4.74397i 1.79305i −0.442993 0.896525i \(-0.646083\pi\)
0.442993 0.896525i \(-0.353917\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.48028 −1.35085 −0.675427 0.737426i \(-0.736041\pi\)
−0.675427 + 0.737426i \(0.736041\pi\)
\(12\) 0 0
\(13\) 0.843176i 0.233855i 0.993140 + 0.116928i \(0.0373045\pi\)
−0.993140 + 0.116928i \(0.962695\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.52315i 1.33956i 0.742559 + 0.669781i \(0.233612\pi\)
−0.742559 + 0.669781i \(0.766388\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.779187i 0.162472i 0.996695 + 0.0812358i \(0.0258867\pi\)
−0.996695 + 0.0812358i \(0.974113\pi\)
\(24\) 0 0
\(25\) 2.40439 + 4.38394i 0.480877 + 0.876788i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −10.6570 −1.97895 −0.989477 0.144691i \(-0.953781\pi\)
−0.989477 + 0.144691i \(0.953781\pi\)
\(30\) 0 0
\(31\) −8.65699 −1.55484 −0.777421 0.628981i \(-0.783472\pi\)
−0.777421 + 0.628981i \(0.783472\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.40439 9.12790i 0.913508 1.54290i
\(36\) 0 0
\(37\) 1.62236i 0.266715i −0.991068 0.133357i \(-0.957424\pi\)
0.991068 0.133357i \(-0.0425758\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.73588 −0.739620 −0.369810 0.929107i \(-0.620577\pi\)
−0.369810 + 0.929107i \(0.620577\pi\)
\(42\) 0 0
\(43\) 9.67504i 1.47543i −0.675112 0.737715i \(-0.735905\pi\)
0.675112 0.737715i \(-0.264095\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.18559i 0.464666i 0.972636 + 0.232333i \(0.0746360\pi\)
−0.972636 + 0.232333i \(0.925364\pi\)
\(48\) 0 0
\(49\) −15.5052 −2.21503
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.17922i 0.848780i 0.905479 + 0.424390i \(0.139512\pi\)
−0.905479 + 0.424390i \(0.860488\pi\)
\(54\) 0 0
\(55\) −8.62054 5.10399i −1.16239 0.688222i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.6964 1.52275 0.761373 0.648314i \(-0.224526\pi\)
0.761373 + 0.648314i \(0.224526\pi\)
\(60\) 0 0
\(61\) 6.48028 0.829715 0.414857 0.909886i \(-0.363831\pi\)
0.414857 + 0.909886i \(0.363831\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.960558 + 1.62236i −0.119143 + 0.201229i
\(66\) 0 0
\(67\) 14.8880i 1.81885i 0.415864 + 0.909427i \(0.363479\pi\)
−0.415864 + 0.909427i \(0.636521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.303566 −0.0360266 −0.0180133 0.999838i \(-0.505734\pi\)
−0.0180133 + 0.999838i \(0.505734\pi\)
\(72\) 0 0
\(73\) 10.0800i 1.17978i 0.807485 + 0.589888i \(0.200828\pi\)
−0.807485 + 0.589888i \(0.799172\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 21.2543i 2.42215i
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.779187i 0.0855268i −0.999085 0.0427634i \(-0.986384\pi\)
0.999085 0.0427634i \(-0.0136162\pi\)
\(84\) 0 0
\(85\) −6.29205 + 10.6271i −0.682468 + 1.15268i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.69643 −0.603821 −0.301910 0.953336i \(-0.597624\pi\)
−0.301910 + 0.953336i \(0.597624\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.92411 1.13921i −0.197409 0.116881i
\(96\) 0 0
\(97\) 6.17922i 0.627404i 0.949521 + 0.313702i \(0.101569\pi\)
−0.949521 + 0.313702i \(0.898431\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.36794 −0.335122 −0.167561 0.985862i \(-0.553589\pi\)
−0.167561 + 0.985862i \(0.553589\pi\)
\(102\) 0 0
\(103\) 3.71368i 0.365919i 0.983120 + 0.182960i \(0.0585678\pi\)
−0.983120 + 0.182960i \(0.941432\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.2031i 0.986374i −0.869923 0.493187i \(-0.835832\pi\)
0.869923 0.493187i \(-0.164168\pi\)
\(108\) 0 0
\(109\) −2.96056 −0.283570 −0.141785 0.989897i \(-0.545284\pi\)
−0.141785 + 0.989897i \(0.545284\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 7.08638i 0.666631i −0.942815 0.333315i \(-0.891833\pi\)
0.942815 0.333315i \(-0.108167\pi\)
\(114\) 0 0
\(115\) −0.887659 + 1.49924i −0.0827747 + 0.139805i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 26.2016 2.40190
\(120\) 0 0
\(121\) 9.07290 0.824809
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −0.367938 + 11.1743i −0.0329094 + 0.999458i
\(126\) 0 0
\(127\) 9.79817i 0.869447i 0.900564 + 0.434723i \(0.143154\pi\)
−0.900564 + 0.434723i \(0.856846\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −12.5841 −1.09948 −0.549739 0.835337i \(-0.685273\pi\)
−0.549739 + 0.835337i \(0.685273\pi\)
\(132\) 0 0
\(133\) 4.74397i 0.411354i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.89586i 0.760024i −0.924981 0.380012i \(-0.875920\pi\)
0.924981 0.380012i \(-0.124080\pi\)
\(138\) 0 0
\(139\) −6.25560 −0.530593 −0.265296 0.964167i \(-0.585470\pi\)
−0.265296 + 0.964167i \(0.585470\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.77767i 0.315904i
\(144\) 0 0
\(145\) −20.5052 12.1406i −1.70286 1.00822i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.48028 0.203192 0.101596 0.994826i \(-0.467605\pi\)
0.101596 + 0.994826i \(0.467605\pi\)
\(150\) 0 0
\(151\) 3.69643 0.300812 0.150406 0.988624i \(-0.451942\pi\)
0.150406 + 0.988624i \(0.451942\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −16.6570 9.86216i −1.33792 0.792148i
\(156\) 0 0
\(157\) 18.8479i 1.50422i 0.659035 + 0.752112i \(0.270965\pi\)
−0.659035 + 0.752112i \(0.729035\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.69643 0.291320
\(162\) 0 0
\(163\) 2.46554i 0.193116i −0.995327 0.0965580i \(-0.969217\pi\)
0.995327 0.0965580i \(-0.0307833\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.49134i 0.579697i 0.957072 + 0.289849i \(0.0936050\pi\)
−0.957072 + 0.289849i \(0.906395\pi\)
\(168\) 0 0
\(169\) 12.2891 0.945312
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 20.0653i 1.52554i −0.646673 0.762768i \(-0.723840\pi\)
0.646673 0.762768i \(-0.276160\pi\)
\(174\) 0 0
\(175\) 20.7973 11.4063i 1.57212 0.862238i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −3.69643 −0.276284 −0.138142 0.990412i \(-0.544113\pi\)
−0.138142 + 0.990412i \(0.544113\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.84822 3.12160i 0.135884 0.229505i
\(186\) 0 0
\(187\) 24.7453i 1.80955i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −0.887659 −0.0642288 −0.0321144 0.999484i \(-0.510224\pi\)
−0.0321144 + 0.999484i \(0.510224\pi\)
\(192\) 0 0
\(193\) 19.1581i 1.37903i −0.724271 0.689516i \(-0.757823\pi\)
0.724271 0.689516i \(-0.242177\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.37271i 0.240295i −0.992756 0.120148i \(-0.961663\pi\)
0.992756 0.120148i \(-0.0383368\pi\)
\(198\) 0 0
\(199\) −4.58409 −0.324958 −0.162479 0.986712i \(-0.551949\pi\)
−0.162479 + 0.986712i \(0.551949\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 50.5564i 3.54836i
\(204\) 0 0
\(205\) −9.11234 5.39517i −0.636433 0.376815i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4.48028 0.309907
\(210\) 0 0
\(211\) −14.8817 −1.02450 −0.512248 0.858837i \(-0.671187\pi\)
−0.512248 + 0.858837i \(0.671187\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 11.0219 18.6158i 0.751690 1.26959i
\(216\) 0 0
\(217\) 41.0685i 2.78791i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −4.65699 −0.313263
\(222\) 0 0
\(223\) 18.7839i 1.25786i −0.777461 0.628931i \(-0.783493\pi\)
0.777461 0.628931i \(-0.216507\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0.843176i 0.0559636i 0.999608 + 0.0279818i \(0.00890804\pi\)
−0.999608 + 0.0279818i \(0.991092\pi\)
\(228\) 0 0
\(229\) 4.86273 0.321338 0.160669 0.987008i \(-0.448635\pi\)
0.160669 + 0.987008i \(0.448635\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.90268i 0.321185i −0.987021 0.160593i \(-0.948659\pi\)
0.987021 0.160593i \(-0.0513405\pi\)
\(234\) 0 0
\(235\) −3.62907 + 6.12943i −0.236734 + 0.399840i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.20164 −0.401151 −0.200575 0.979678i \(-0.564281\pi\)
−0.200575 + 0.979678i \(0.564281\pi\)
\(240\) 0 0
\(241\) −3.26412 −0.210261 −0.105130 0.994458i \(-0.533526\pi\)
−0.105130 + 0.994458i \(0.533526\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −29.8337 17.6637i −1.90601 1.12849i
\(246\) 0 0
\(247\) 0.843176i 0.0536500i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −28.4263 −1.79425 −0.897127 0.441773i \(-0.854350\pi\)
−0.897127 + 0.441773i \(0.854350\pi\)
\(252\) 0 0
\(253\) 3.49097i 0.219476i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.55192i 0.595832i 0.954592 + 0.297916i \(0.0962916\pi\)
−0.954592 + 0.297916i \(0.903708\pi\)
\(258\) 0 0
\(259\) −7.69643 −0.478233
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.54706i 0.588697i −0.955698 0.294349i \(-0.904897\pi\)
0.955698 0.294349i \(-0.0951027\pi\)
\(264\) 0 0
\(265\) −7.03944 + 11.8895i −0.432430 + 0.730365i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −14.3534 −0.875144 −0.437572 0.899183i \(-0.644161\pi\)
−0.437572 + 0.899183i \(0.644161\pi\)
\(270\) 0 0
\(271\) −12.1038 −0.735254 −0.367627 0.929973i \(-0.619830\pi\)
−0.367627 + 0.929973i \(0.619830\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −10.7723 19.6413i −0.649596 1.18441i
\(276\) 0 0
\(277\) 3.59055i 0.215735i 0.994165 + 0.107868i \(0.0344023\pi\)
−0.994165 + 0.107868i \(0.965598\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −26.7069 −1.59320 −0.796599 0.604509i \(-0.793369\pi\)
−0.796599 + 0.604509i \(0.793369\pi\)
\(282\) 0 0
\(283\) 20.4751i 1.21712i 0.793508 + 0.608559i \(0.208252\pi\)
−0.793508 + 0.608559i \(0.791748\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 22.4668i 1.32618i
\(288\) 0 0
\(289\) −13.5052 −0.794424
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 19.1581i 1.11923i 0.828753 + 0.559615i \(0.189051\pi\)
−0.828753 + 0.559615i \(0.810949\pi\)
\(294\) 0 0
\(295\) 22.5052 + 13.3247i 1.31030 + 0.775796i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −0.656992 −0.0379948
\(300\) 0 0
\(301\) −45.8981 −2.64552
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 12.4688 + 7.38242i 0.713959 + 0.422716i
\(306\) 0 0
\(307\) 32.0495i 1.82916i −0.404404 0.914580i \(-0.632521\pi\)
0.404404 0.914580i \(-0.367479\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 20.5301 1.16416 0.582079 0.813132i \(-0.302240\pi\)
0.582079 + 0.813132i \(0.302240\pi\)
\(312\) 0 0
\(313\) 14.7932i 0.836163i −0.908409 0.418082i \(-0.862703\pi\)
0.908409 0.418082i \(-0.137297\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 16.9485i 0.951925i −0.879466 0.475962i \(-0.842100\pi\)
0.879466 0.475962i \(-0.157900\pi\)
\(318\) 0 0
\(319\) 47.7463 2.67328
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 5.52315i 0.307316i
\(324\) 0 0
\(325\) −3.69643 + 2.02732i −0.205041 + 0.112456i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 15.1123 0.833170
\(330\) 0 0
\(331\) −22.1287 −1.21631 −0.608153 0.793820i \(-0.708089\pi\)
−0.608153 + 0.793820i \(0.708089\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −16.9606 + 28.6461i −0.926654 + 1.56510i
\(336\) 0 0
\(337\) 27.9948i 1.52498i −0.647002 0.762488i \(-0.723978\pi\)
0.647002 0.762488i \(-0.276022\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 38.7857 2.10037
\(342\) 0 0
\(343\) 40.3484i 2.17861i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 17.1024i 0.918105i −0.888409 0.459052i \(-0.848189\pi\)
0.888409 0.459052i \(-0.151811\pi\)
\(348\) 0 0
\(349\) 19.2411 1.02995 0.514976 0.857205i \(-0.327801\pi\)
0.514976 + 0.857205i \(0.327801\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7.42735i 0.395318i −0.980271 0.197659i \(-0.936666\pi\)
0.980271 0.197659i \(-0.0633339\pi\)
\(354\) 0 0
\(355\) −0.584094 0.345826i −0.0310005 0.0183545i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 28.1767 1.48711 0.743555 0.668675i \(-0.233138\pi\)
0.743555 + 0.668675i \(0.233138\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −11.4833 + 19.3950i −0.601062 + 1.01518i
\(366\) 0 0
\(367\) 26.1165i 1.36327i 0.731692 + 0.681636i \(0.238731\pi\)
−0.731692 + 0.681636i \(0.761269\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 29.3140 1.52191
\(372\) 0 0
\(373\) 0.438217i 0.0226900i −0.999936 0.0113450i \(-0.996389\pi\)
0.999936 0.0113450i \(-0.00361130\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.98572i 0.462788i
\(378\) 0 0
\(379\) 13.7753 0.707591 0.353795 0.935323i \(-0.384891\pi\)
0.353795 + 0.935323i \(0.384891\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 22.3153i 1.14026i 0.821555 + 0.570130i \(0.193107\pi\)
−0.821555 + 0.570130i \(0.806893\pi\)
\(384\) 0 0
\(385\) −24.2132 + 40.8956i −1.23402 + 2.08423i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 14.4263 0.731444 0.365722 0.930724i \(-0.380822\pi\)
0.365722 + 0.930724i \(0.380822\pi\)
\(390\) 0 0
\(391\) −4.30357 −0.217641
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −7.69643 4.55685i −0.387250 0.229280i
\(396\) 0 0
\(397\) 16.4512i 0.825662i 0.910808 + 0.412831i \(0.135460\pi\)
−0.910808 + 0.412831i \(0.864540\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2.96056 −0.147843 −0.0739216 0.997264i \(-0.523551\pi\)
−0.0739216 + 0.997264i \(0.523551\pi\)
\(402\) 0 0
\(403\) 7.29937i 0.363608i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.26864i 0.360293i
\(408\) 0 0
\(409\) −17.0893 −0.845012 −0.422506 0.906360i \(-0.638849\pi\)
−0.422506 + 0.906360i \(0.638849\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 55.4875i 2.73036i
\(414\) 0 0
\(415\) 0.887659 1.49924i 0.0435735 0.0735948i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −15.3929 −0.751991 −0.375995 0.926622i \(-0.622699\pi\)
−0.375995 + 0.926622i \(0.622699\pi\)
\(420\) 0 0
\(421\) −16.4323 −0.800862 −0.400431 0.916327i \(-0.631140\pi\)
−0.400431 + 0.916327i \(0.631140\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −24.2132 + 13.2798i −1.17451 + 0.644165i
\(426\) 0 0
\(427\) 30.7422i 1.48772i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −15.1852 −0.731447 −0.365724 0.930724i \(-0.619178\pi\)
−0.365724 + 0.930724i \(0.619178\pi\)
\(432\) 0 0
\(433\) 23.4380i 1.12636i 0.826335 + 0.563179i \(0.190422\pi\)
−0.826335 + 0.563179i \(0.809578\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.779187i 0.0372735i
\(438\) 0 0
\(439\) −0.511196 −0.0243980 −0.0121990 0.999926i \(-0.503883\pi\)
−0.0121990 + 0.999926i \(0.503883\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 19.7834i 0.939940i 0.882682 + 0.469970i \(0.155735\pi\)
−0.882682 + 0.469970i \(0.844265\pi\)
\(444\) 0 0
\(445\) −10.9606 6.48945i −0.519580 0.307630i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 31.1682 1.47092 0.735459 0.677569i \(-0.236967\pi\)
0.735459 + 0.677569i \(0.236967\pi\)
\(450\) 0 0
\(451\) 21.2180 0.999119
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 7.69643 + 4.55685i 0.360814 + 0.213629i
\(456\) 0 0
\(457\) 7.20950i 0.337246i 0.985681 + 0.168623i \(0.0539321\pi\)
−0.985681 + 0.168623i \(0.946068\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 5.41591 0.252244 0.126122 0.992015i \(-0.459747\pi\)
0.126122 + 0.992015i \(0.459747\pi\)
\(462\) 0 0
\(463\) 8.73715i 0.406050i 0.979174 + 0.203025i \(0.0650772\pi\)
−0.979174 + 0.203025i \(0.934923\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 17.3584i 0.803249i −0.915804 0.401624i \(-0.868446\pi\)
0.915804 0.401624i \(-0.131554\pi\)
\(468\) 0 0
\(469\) 70.6280 3.26130
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 43.3469i 1.99309i
\(474\) 0 0
\(475\) −2.40439 4.38394i −0.110321 0.201149i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 7.44682 0.340254 0.170127 0.985422i \(-0.445582\pi\)
0.170127 + 0.985422i \(0.445582\pi\)
\(480\) 0 0
\(481\) 1.36794 0.0623726
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7.03944 + 11.8895i −0.319645 + 0.539874i
\(486\) 0 0
\(487\) 2.15530i 0.0976661i −0.998807 0.0488330i \(-0.984450\pi\)
0.998807 0.0488330i \(-0.0155502\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −29.3140 −1.32292 −0.661461 0.749980i \(-0.730063\pi\)
−0.661461 + 0.749980i \(0.730063\pi\)
\(492\) 0 0
\(493\) 58.8602i 2.65093i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.44011i 0.0645976i
\(498\) 0 0
\(499\) 23.5696 1.05512 0.527560 0.849518i \(-0.323107\pi\)
0.527560 + 0.849518i \(0.323107\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9.08297i 0.404990i 0.979283 + 0.202495i \(0.0649049\pi\)
−0.979283 + 0.202495i \(0.935095\pi\)
\(504\) 0 0
\(505\) −6.48028 3.83680i −0.288369 0.170735i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −18.5112 −0.820494 −0.410247 0.911974i \(-0.634558\pi\)
−0.410247 + 0.911974i \(0.634558\pi\)
\(510\) 0 0
\(511\) 47.8192 2.11540
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.23067 + 7.14552i −0.186425 + 0.314869i
\(516\) 0 0
\(517\) 14.2723i 0.627697i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 25.0893 1.09918 0.549591 0.835434i \(-0.314784\pi\)
0.549591 + 0.835434i \(0.314784\pi\)
\(522\) 0 0
\(523\) 28.5181i 1.24701i 0.781820 + 0.623504i \(0.214292\pi\)
−0.781820 + 0.623504i \(0.785708\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 47.8139i 2.08281i
\(528\) 0 0
\(529\) 22.3929 0.973603
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3.99318i 0.172964i
\(534\) 0 0
\(535\) 11.6235 19.6319i 0.502529 0.848762i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 69.4677 2.99218
\(540\) 0 0
\(541\) −15.6485 −0.672780 −0.336390 0.941723i \(-0.609206\pi\)
−0.336390 + 0.941723i \(0.609206\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5.69643 3.37271i −0.244008 0.144471i
\(546\) 0 0
\(547\) 10.8543i 0.464098i −0.972704 0.232049i \(-0.925457\pi\)
0.972704 0.232049i \(-0.0745429\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 10.6570 0.454003
\(552\) 0 0
\(553\) 18.9759i 0.806936i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.8763i 0.799814i −0.916556 0.399907i \(-0.869042\pi\)
0.916556 0.399907i \(-0.130958\pi\)
\(558\) 0 0
\(559\) 8.15777 0.345037
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 22.2846i 0.939183i −0.882884 0.469591i \(-0.844401\pi\)
0.882884 0.469591i \(-0.155599\pi\)
\(564\) 0 0
\(565\) 8.07290 13.6350i 0.339629 0.573627i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −7.11833 −0.298416 −0.149208 0.988806i \(-0.547672\pi\)
−0.149208 + 0.988806i \(0.547672\pi\)
\(570\) 0 0
\(571\) 1.21017 0.0506440 0.0253220 0.999679i \(-0.491939\pi\)
0.0253220 + 0.999679i \(0.491939\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3.41591 + 1.87347i −0.142453 + 0.0781289i
\(576\) 0 0
\(577\) 41.4143i 1.72410i −0.506823 0.862050i \(-0.669180\pi\)
0.506823 0.862050i \(-0.330820\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −3.69643 −0.153354
\(582\) 0 0
\(583\) 27.6846i 1.14658i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0.0591336i 0.00244071i 0.999999 + 0.00122035i \(0.000388450\pi\)
−0.999999 + 0.00122035i \(0.999612\pi\)
\(588\) 0 0
\(589\) 8.65699 0.356705
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 42.8828i 1.76099i 0.474059 + 0.880493i \(0.342788\pi\)
−0.474059 + 0.880493i \(0.657212\pi\)
\(594\) 0 0
\(595\) 50.4148 + 29.8493i 2.06681 + 1.22370i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 15.3430 0.626898 0.313449 0.949605i \(-0.398515\pi\)
0.313449 + 0.949605i \(0.398515\pi\)
\(600\) 0 0
\(601\) −12.5781 −0.513072 −0.256536 0.966535i \(-0.582581\pi\)
−0.256536 + 0.966535i \(0.582581\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 17.4572 + 10.3360i 0.709738 + 0.420217i
\(606\) 0 0
\(607\) 2.65751i 0.107865i 0.998545 + 0.0539325i \(0.0171756\pi\)
−0.998545 + 0.0539325i \(0.982824\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.68602 −0.108665
\(612\) 0 0
\(613\) 34.1052i 1.37750i −0.725001 0.688748i \(-0.758161\pi\)
0.725001 0.688748i \(-0.241839\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 29.9226i 1.20464i −0.798255 0.602319i \(-0.794244\pi\)
0.798255 0.602319i \(-0.205756\pi\)
\(618\) 0 0
\(619\) −17.2102 −0.691735 −0.345868 0.938283i \(-0.612415\pi\)
−0.345868 + 0.938283i \(0.612415\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 27.0237i 1.08268i
\(624\) 0 0
\(625\) −13.4378 + 21.0814i −0.537514 + 0.843255i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 8.96056 0.357281
\(630\) 0 0
\(631\) −3.36195 −0.133837 −0.0669186 0.997758i \(-0.521317\pi\)
−0.0669186 + 0.997758i \(0.521317\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −11.1622 + 18.8527i −0.442958 + 0.748148i
\(636\) 0 0
\(637\) 13.0736i 0.517996i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 16.2745 0.642806 0.321403 0.946943i \(-0.395846\pi\)
0.321403 + 0.946943i \(0.395846\pi\)
\(642\) 0 0
\(643\) 35.7040i 1.40803i −0.710185 0.704015i \(-0.751389\pi\)
0.710185 0.704015i \(-0.248611\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 15.2881i 0.601036i −0.953776 0.300518i \(-0.902840\pi\)
0.953776 0.300518i \(-0.0971595\pi\)
\(648\) 0 0
\(649\) −52.4033 −2.05701
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 16.4512i 0.643785i 0.946776 + 0.321892i \(0.104319\pi\)
−0.946776 + 0.321892i \(0.895681\pi\)
\(654\) 0 0
\(655\) −24.2132 14.3360i −0.946087 0.560152i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 7.03944 0.274218 0.137109 0.990556i \(-0.456219\pi\)
0.137109 + 0.990556i \(0.456219\pi\)
\(660\) 0 0
\(661\) 6.35343 0.247120 0.123560 0.992337i \(-0.460569\pi\)
0.123560 + 0.992337i \(0.460569\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −5.40439 + 9.12790i −0.209573 + 0.353965i
\(666\) 0 0
\(667\) 8.30378i 0.321524i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −29.0335 −1.12082
\(672\) 0 0
\(673\) 7.08638i 0.273160i −0.990629 0.136580i \(-0.956389\pi\)
0.990629 0.136580i \(-0.0436111\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 17.6095i 0.676786i 0.941005 + 0.338393i \(0.109883\pi\)
−0.941005 + 0.338393i \(0.890117\pi\)
\(678\) 0 0
\(679\) 29.3140 1.12497
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 26.5855i 1.01726i −0.860984 0.508632i \(-0.830151\pi\)
0.860984 0.508632i \(-0.169849\pi\)
\(684\) 0 0
\(685\) 10.1343 17.1166i 0.387211 0.653992i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5.21017 −0.198492
\(690\) 0 0
\(691\) 49.4907 1.88271 0.941357 0.337411i \(-0.109551\pi\)
0.941357 + 0.337411i \(0.109551\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −12.0364 7.12646i −0.456569 0.270322i
\(696\) 0 0
\(697\) 26.1570i 0.990766i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −29.3389 −1.10812 −0.554058 0.832478i \(-0.686921\pi\)
−0.554058 + 0.832478i \(0.686921\pi\)
\(702\) 0 0
\(703\) 1.62236i 0.0611886i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 15.9774i 0.600891i
\(708\) 0 0
\(709\) −24.7359 −0.928975 −0.464488 0.885580i \(-0.653762\pi\)
−0.464488 + 0.885580i \(0.653762\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 6.74541i 0.252618i
\(714\) 0 0
\(715\) 4.30357 7.26864i 0.160944 0.271832i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 26.7548 0.997786 0.498893 0.866663i \(-0.333740\pi\)
0.498893 + 0.866663i \(0.333740\pi\)
\(720\) 0 0
\(721\) 17.6175 0.656112
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −25.6235 46.7196i −0.951634 1.73512i
\(726\) 0 0
\(727\) 27.2108i 1.00919i 0.863355 + 0.504596i \(0.168359\pi\)
−0.863355 + 0.504596i \(0.831641\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 53.4367 1.97643
\(732\) 0 0
\(733\) 40.0026i 1.47753i 0.673963 + 0.738765i \(0.264591\pi\)
−0.673963 + 0.738765i \(0.735409\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 66.7022i 2.45701i
\(738\) 0 0
\(739\) −35.1123 −1.29163 −0.645814 0.763495i \(-0.723482\pi\)
−0.645814 + 0.763495i \(0.723482\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 25.6476i 0.940918i −0.882422 0.470459i \(-0.844088\pi\)
0.882422 0.470459i \(-0.155912\pi\)
\(744\) 0 0
\(745\) 4.77233 + 2.82557i 0.174844 + 0.103521i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −48.4033 −1.76862
\(750\) 0 0
\(751\) −13.2641 −0.484015 −0.242007 0.970274i \(-0.577806\pi\)
−0.242007 + 0.970274i \(0.577806\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 7.11234 + 4.21103i 0.258845 + 0.153255i
\(756\) 0 0
\(757\) 2.77092i 0.100711i −0.998731 0.0503554i \(-0.983965\pi\)
0.998731 0.0503554i \(-0.0160354\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −10.7838 −0.390914 −0.195457 0.980712i \(-0.562619\pi\)
−0.195457 + 0.980712i \(0.562619\pi\)
\(762\) 0 0
\(763\) 14.0448i 0.508455i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.86216i 0.356102i
\(768\) 0 0
\(769\) 40.6090 1.46440 0.732199 0.681090i \(-0.238494\pi\)
0.732199 + 0.681090i \(0.238494\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 17.4410i 0.627310i 0.949537 + 0.313655i \(0.101554\pi\)
−0.949537 + 0.313655i \(0.898446\pi\)
\(774\) 0 0
\(775\) −20.8148 37.9517i −0.747688 1.36327i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4.73588 0.169680
\(780\) 0 0
\(781\) 1.36006 0.0486668
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −21.4718 + 36.2654i −0.766360 + 1.29437i
\(786\) 0 0
\(787\) 30.8346i 1.09914i −0.835449 0.549568i \(-0.814792\pi\)
0.835449 0.549568i \(-0.185208\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −33.6175 −1.19530
\(792\) 0 0
\(793\) 5.46402i 0.194033i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 38.1340i 1.35077i −0.737463 0.675387i \(-0.763976\pi\)
0.737463 0.675387i \(-0.236024\pi\)
\(798\) 0 0
\(799\) −17.5945 −0.622449
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 45.1612i 1.59371i
\(804\) 0 0
\(805\) 7.11234 + 4.21103i 0.250677 + 0.148419i