Properties

Label 1680.2.t.i.1009.1
Level $1680$
Weight $2$
Character 1680.1009
Analytic conductor $13.415$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.t (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \(x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 840)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1009.1
Root \(-0.854638 - 0.854638i\) of defining polynomial
Character \(\chi\) \(=\) 1680.1009
Dual form 1680.2.t.i.1009.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +(-2.17009 - 0.539189i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +(-2.17009 - 0.539189i) q^{5} +1.00000i q^{7} -1.00000 q^{9} -5.41855 q^{11} +4.34017i q^{13} +(-0.539189 + 2.17009i) q^{15} -1.07838i q^{17} +4.34017 q^{19} +1.00000 q^{21} -6.34017i q^{23} +(4.41855 + 2.34017i) q^{25} +1.00000i q^{27} +8.83710 q^{29} +4.34017 q^{31} +5.41855i q^{33} +(0.539189 - 2.17009i) q^{35} +8.68035i q^{37} +4.34017 q^{39} +8.34017 q^{41} -6.15676i q^{43} +(2.17009 + 0.539189i) q^{45} -6.83710i q^{47} -1.00000 q^{49} -1.07838 q^{51} +6.18342i q^{53} +(11.7587 + 2.92162i) q^{55} -4.34017i q^{57} -6.83710 q^{59} -4.52359 q^{61} -1.00000i q^{63} +(2.34017 - 9.41855i) q^{65} -6.34017 q^{69} +14.0989 q^{71} -11.1773i q^{73} +(2.34017 - 4.41855i) q^{75} -5.41855i q^{77} +0.680346 q^{79} +1.00000 q^{81} +6.83710i q^{83} +(-0.581449 + 2.34017i) q^{85} -8.83710i q^{87} -6.49693 q^{89} -4.34017 q^{91} -4.34017i q^{93} +(-9.41855 - 2.34017i) q^{95} +10.4969i q^{97} +5.41855 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{5} - 6 q^{9} + O(q^{10}) \) \( 6 q - 2 q^{5} - 6 q^{9} - 4 q^{11} + 4 q^{19} + 6 q^{21} - 2 q^{25} - 4 q^{29} + 4 q^{31} + 4 q^{39} + 28 q^{41} + 2 q^{45} - 6 q^{49} + 20 q^{55} + 16 q^{59} + 4 q^{61} - 8 q^{65} - 16 q^{69} + 12 q^{71} - 8 q^{75} - 40 q^{79} + 6 q^{81} - 32 q^{85} - 4 q^{89} - 4 q^{91} - 28 q^{95} + 4 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) −2.17009 0.539189i −0.970492 0.241133i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −5.41855 −1.63375 −0.816877 0.576812i \(-0.804297\pi\)
−0.816877 + 0.576812i \(0.804297\pi\)
\(12\) 0 0
\(13\) 4.34017i 1.20375i 0.798591 + 0.601874i \(0.205579\pi\)
−0.798591 + 0.601874i \(0.794421\pi\)
\(14\) 0 0
\(15\) −0.539189 + 2.17009i −0.139218 + 0.560314i
\(16\) 0 0
\(17\) 1.07838i 0.261545i −0.991412 0.130773i \(-0.958254\pi\)
0.991412 0.130773i \(-0.0417457\pi\)
\(18\) 0 0
\(19\) 4.34017 0.995704 0.497852 0.867262i \(-0.334122\pi\)
0.497852 + 0.867262i \(0.334122\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 6.34017i 1.32202i −0.750378 0.661009i \(-0.770129\pi\)
0.750378 0.661009i \(-0.229871\pi\)
\(24\) 0 0
\(25\) 4.41855 + 2.34017i 0.883710 + 0.468035i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 8.83710 1.64101 0.820504 0.571640i \(-0.193693\pi\)
0.820504 + 0.571640i \(0.193693\pi\)
\(30\) 0 0
\(31\) 4.34017 0.779518 0.389759 0.920917i \(-0.372558\pi\)
0.389759 + 0.920917i \(0.372558\pi\)
\(32\) 0 0
\(33\) 5.41855i 0.943249i
\(34\) 0 0
\(35\) 0.539189 2.17009i 0.0911396 0.366812i
\(36\) 0 0
\(37\) 8.68035i 1.42704i 0.700635 + 0.713520i \(0.252900\pi\)
−0.700635 + 0.713520i \(0.747100\pi\)
\(38\) 0 0
\(39\) 4.34017 0.694984
\(40\) 0 0
\(41\) 8.34017 1.30252 0.651258 0.758856i \(-0.274242\pi\)
0.651258 + 0.758856i \(0.274242\pi\)
\(42\) 0 0
\(43\) 6.15676i 0.938896i −0.882960 0.469448i \(-0.844453\pi\)
0.882960 0.469448i \(-0.155547\pi\)
\(44\) 0 0
\(45\) 2.17009 + 0.539189i 0.323497 + 0.0803775i
\(46\) 0 0
\(47\) 6.83710i 0.997294i −0.866805 0.498647i \(-0.833830\pi\)
0.866805 0.498647i \(-0.166170\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −1.07838 −0.151003
\(52\) 0 0
\(53\) 6.18342i 0.849358i 0.905344 + 0.424679i \(0.139613\pi\)
−0.905344 + 0.424679i \(0.860387\pi\)
\(54\) 0 0
\(55\) 11.7587 + 2.92162i 1.58555 + 0.393951i
\(56\) 0 0
\(57\) 4.34017i 0.574870i
\(58\) 0 0
\(59\) −6.83710 −0.890115 −0.445057 0.895502i \(-0.646817\pi\)
−0.445057 + 0.895502i \(0.646817\pi\)
\(60\) 0 0
\(61\) −4.52359 −0.579186 −0.289593 0.957150i \(-0.593520\pi\)
−0.289593 + 0.957150i \(0.593520\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) 2.34017 9.41855i 0.290263 1.16823i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) −6.34017 −0.763267
\(70\) 0 0
\(71\) 14.0989 1.67323 0.836616 0.547790i \(-0.184531\pi\)
0.836616 + 0.547790i \(0.184531\pi\)
\(72\) 0 0
\(73\) 11.1773i 1.30820i −0.756408 0.654101i \(-0.773047\pi\)
0.756408 0.654101i \(-0.226953\pi\)
\(74\) 0 0
\(75\) 2.34017 4.41855i 0.270220 0.510210i
\(76\) 0 0
\(77\) 5.41855i 0.617501i
\(78\) 0 0
\(79\) 0.680346 0.0765449 0.0382724 0.999267i \(-0.487815\pi\)
0.0382724 + 0.999267i \(0.487815\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.83710i 0.750469i 0.926930 + 0.375235i \(0.122438\pi\)
−0.926930 + 0.375235i \(0.877562\pi\)
\(84\) 0 0
\(85\) −0.581449 + 2.34017i −0.0630670 + 0.253827i
\(86\) 0 0
\(87\) 8.83710i 0.947437i
\(88\) 0 0
\(89\) −6.49693 −0.688673 −0.344337 0.938846i \(-0.611896\pi\)
−0.344337 + 0.938846i \(0.611896\pi\)
\(90\) 0 0
\(91\) −4.34017 −0.454974
\(92\) 0 0
\(93\) 4.34017i 0.450055i
\(94\) 0 0
\(95\) −9.41855 2.34017i −0.966323 0.240097i
\(96\) 0 0
\(97\) 10.4969i 1.06580i 0.846178 + 0.532901i \(0.178898\pi\)
−0.846178 + 0.532901i \(0.821102\pi\)
\(98\) 0 0
\(99\) 5.41855 0.544585
\(100\) 0 0
\(101\) 18.8638 1.87701 0.938507 0.345259i \(-0.112209\pi\)
0.938507 + 0.345259i \(0.112209\pi\)
\(102\) 0 0
\(103\) 10.1568i 1.00077i −0.865802 0.500387i \(-0.833191\pi\)
0.865802 0.500387i \(-0.166809\pi\)
\(104\) 0 0
\(105\) −2.17009 0.539189i −0.211779 0.0526194i
\(106\) 0 0
\(107\) 14.6537i 1.41663i 0.705899 + 0.708313i \(0.250543\pi\)
−0.705899 + 0.708313i \(0.749457\pi\)
\(108\) 0 0
\(109\) 12.8371 1.22957 0.614786 0.788694i \(-0.289243\pi\)
0.614786 + 0.788694i \(0.289243\pi\)
\(110\) 0 0
\(111\) 8.68035 0.823902
\(112\) 0 0
\(113\) 1.50307i 0.141397i 0.997498 + 0.0706985i \(0.0225228\pi\)
−0.997498 + 0.0706985i \(0.977477\pi\)
\(114\) 0 0
\(115\) −3.41855 + 13.7587i −0.318781 + 1.28301i
\(116\) 0 0
\(117\) 4.34017i 0.401249i
\(118\) 0 0
\(119\) 1.07838 0.0988547
\(120\) 0 0
\(121\) 18.3607 1.66915
\(122\) 0 0
\(123\) 8.34017i 0.752008i
\(124\) 0 0
\(125\) −8.32684 7.46081i −0.744775 0.667315i
\(126\) 0 0
\(127\) 19.2039i 1.70407i 0.523482 + 0.852037i \(0.324633\pi\)
−0.523482 + 0.852037i \(0.675367\pi\)
\(128\) 0 0
\(129\) −6.15676 −0.542072
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 4.34017i 0.376341i
\(134\) 0 0
\(135\) 0.539189 2.17009i 0.0464060 0.186771i
\(136\) 0 0
\(137\) 8.65368i 0.739334i −0.929164 0.369667i \(-0.879472\pi\)
0.929164 0.369667i \(-0.120528\pi\)
\(138\) 0 0
\(139\) −6.18342 −0.524471 −0.262235 0.965004i \(-0.584460\pi\)
−0.262235 + 0.965004i \(0.584460\pi\)
\(140\) 0 0
\(141\) −6.83710 −0.575788
\(142\) 0 0
\(143\) 23.5174i 1.96663i
\(144\) 0 0
\(145\) −19.1773 4.76487i −1.59259 0.395701i
\(146\) 0 0
\(147\) 1.00000i 0.0824786i
\(148\) 0 0
\(149\) 13.2039 1.08171 0.540854 0.841116i \(-0.318101\pi\)
0.540854 + 0.841116i \(0.318101\pi\)
\(150\) 0 0
\(151\) 18.1568 1.47758 0.738788 0.673938i \(-0.235399\pi\)
0.738788 + 0.673938i \(0.235399\pi\)
\(152\) 0 0
\(153\) 1.07838i 0.0871817i
\(154\) 0 0
\(155\) −9.41855 2.34017i −0.756516 0.187967i
\(156\) 0 0
\(157\) 15.1773i 1.21128i −0.795739 0.605639i \(-0.792917\pi\)
0.795739 0.605639i \(-0.207083\pi\)
\(158\) 0 0
\(159\) 6.18342 0.490377
\(160\) 0 0
\(161\) 6.34017 0.499676
\(162\) 0 0
\(163\) 2.83710i 0.222219i 0.993808 + 0.111109i \(0.0354404\pi\)
−0.993808 + 0.111109i \(0.964560\pi\)
\(164\) 0 0
\(165\) 2.92162 11.7587i 0.227448 0.915415i
\(166\) 0 0
\(167\) 13.3607i 1.03388i 0.856021 + 0.516941i \(0.172929\pi\)
−0.856021 + 0.516941i \(0.827071\pi\)
\(168\) 0 0
\(169\) −5.83710 −0.449008
\(170\) 0 0
\(171\) −4.34017 −0.331901
\(172\) 0 0
\(173\) 2.55479i 0.194237i 0.995273 + 0.0971184i \(0.0309626\pi\)
−0.995273 + 0.0971184i \(0.969037\pi\)
\(174\) 0 0
\(175\) −2.34017 + 4.41855i −0.176900 + 0.334011i
\(176\) 0 0
\(177\) 6.83710i 0.513908i
\(178\) 0 0
\(179\) −11.9421 −0.892598 −0.446299 0.894884i \(-0.647258\pi\)
−0.446299 + 0.894884i \(0.647258\pi\)
\(180\) 0 0
\(181\) 4.15676 0.308969 0.154485 0.987995i \(-0.450628\pi\)
0.154485 + 0.987995i \(0.450628\pi\)
\(182\) 0 0
\(183\) 4.52359i 0.334393i
\(184\) 0 0
\(185\) 4.68035 18.8371i 0.344106 1.38493i
\(186\) 0 0
\(187\) 5.84324i 0.427300i
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −6.09890 −0.441301 −0.220650 0.975353i \(-0.570818\pi\)
−0.220650 + 0.975353i \(0.570818\pi\)
\(192\) 0 0
\(193\) 12.6803i 0.912751i 0.889787 + 0.456376i \(0.150853\pi\)
−0.889787 + 0.456376i \(0.849147\pi\)
\(194\) 0 0
\(195\) −9.41855 2.34017i −0.674476 0.167583i
\(196\) 0 0
\(197\) 11.8576i 0.844820i 0.906405 + 0.422410i \(0.138816\pi\)
−0.906405 + 0.422410i \(0.861184\pi\)
\(198\) 0 0
\(199\) −5.50307 −0.390102 −0.195051 0.980793i \(-0.562487\pi\)
−0.195051 + 0.980793i \(0.562487\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 8.83710i 0.620243i
\(204\) 0 0
\(205\) −18.0989 4.49693i −1.26408 0.314079i
\(206\) 0 0
\(207\) 6.34017i 0.440672i
\(208\) 0 0
\(209\) −23.5174 −1.62674
\(210\) 0 0
\(211\) 19.1506 1.31838 0.659191 0.751975i \(-0.270899\pi\)
0.659191 + 0.751975i \(0.270899\pi\)
\(212\) 0 0
\(213\) 14.0989i 0.966040i
\(214\) 0 0
\(215\) −3.31965 + 13.3607i −0.226399 + 0.911192i
\(216\) 0 0
\(217\) 4.34017i 0.294630i
\(218\) 0 0
\(219\) −11.1773 −0.755290
\(220\) 0 0
\(221\) 4.68035 0.314834
\(222\) 0 0
\(223\) 12.3135i 0.824574i −0.911054 0.412287i \(-0.864730\pi\)
0.911054 0.412287i \(-0.135270\pi\)
\(224\) 0 0
\(225\) −4.41855 2.34017i −0.294570 0.156012i
\(226\) 0 0
\(227\) 15.2039i 1.00912i −0.863376 0.504560i \(-0.831655\pi\)
0.863376 0.504560i \(-0.168345\pi\)
\(228\) 0 0
\(229\) −5.20394 −0.343886 −0.171943 0.985107i \(-0.555004\pi\)
−0.171943 + 0.985107i \(0.555004\pi\)
\(230\) 0 0
\(231\) −5.41855 −0.356514
\(232\) 0 0
\(233\) 11.6598i 0.763861i 0.924191 + 0.381930i \(0.124741\pi\)
−0.924191 + 0.381930i \(0.875259\pi\)
\(234\) 0 0
\(235\) −3.68649 + 14.8371i −0.240480 + 0.967866i
\(236\) 0 0
\(237\) 0.680346i 0.0441932i
\(238\) 0 0
\(239\) 20.6225 1.33396 0.666979 0.745077i \(-0.267587\pi\)
0.666979 + 0.745077i \(0.267587\pi\)
\(240\) 0 0
\(241\) −20.3545 −1.31115 −0.655576 0.755129i \(-0.727574\pi\)
−0.655576 + 0.755129i \(0.727574\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 2.17009 + 0.539189i 0.138642 + 0.0344475i
\(246\) 0 0
\(247\) 18.8371i 1.19858i
\(248\) 0 0
\(249\) 6.83710 0.433284
\(250\) 0 0
\(251\) −10.5236 −0.664243 −0.332122 0.943237i \(-0.607764\pi\)
−0.332122 + 0.943237i \(0.607764\pi\)
\(252\) 0 0
\(253\) 34.3545i 2.15985i
\(254\) 0 0
\(255\) 2.34017 + 0.581449i 0.146547 + 0.0364118i
\(256\) 0 0
\(257\) 22.8059i 1.42259i −0.702892 0.711297i \(-0.748108\pi\)
0.702892 0.711297i \(-0.251892\pi\)
\(258\) 0 0
\(259\) −8.68035 −0.539370
\(260\) 0 0
\(261\) −8.83710 −0.547003
\(262\) 0 0
\(263\) 28.0144i 1.72744i 0.503972 + 0.863720i \(0.331872\pi\)
−0.503972 + 0.863720i \(0.668128\pi\)
\(264\) 0 0
\(265\) 3.33403 13.4186i 0.204808 0.824295i
\(266\) 0 0
\(267\) 6.49693i 0.397606i
\(268\) 0 0
\(269\) −18.4969 −1.12778 −0.563889 0.825851i \(-0.690695\pi\)
−0.563889 + 0.825851i \(0.690695\pi\)
\(270\) 0 0
\(271\) 29.0205 1.76287 0.881435 0.472304i \(-0.156578\pi\)
0.881435 + 0.472304i \(0.156578\pi\)
\(272\) 0 0
\(273\) 4.34017i 0.262679i
\(274\) 0 0
\(275\) −23.9421 12.6803i −1.44377 0.764654i
\(276\) 0 0
\(277\) 8.68035i 0.521551i 0.965399 + 0.260776i \(0.0839783\pi\)
−0.965399 + 0.260776i \(0.916022\pi\)
\(278\) 0 0
\(279\) −4.34017 −0.259839
\(280\) 0 0
\(281\) 5.63317 0.336046 0.168023 0.985783i \(-0.446262\pi\)
0.168023 + 0.985783i \(0.446262\pi\)
\(282\) 0 0
\(283\) 2.47027i 0.146842i 0.997301 + 0.0734210i \(0.0233917\pi\)
−0.997301 + 0.0734210i \(0.976608\pi\)
\(284\) 0 0
\(285\) −2.34017 + 9.41855i −0.138620 + 0.557907i
\(286\) 0 0
\(287\) 8.34017i 0.492305i
\(288\) 0 0
\(289\) 15.8371 0.931594
\(290\) 0 0
\(291\) 10.4969 0.615341
\(292\) 0 0
\(293\) 7.60197i 0.444112i −0.975034 0.222056i \(-0.928723\pi\)
0.975034 0.222056i \(-0.0712767\pi\)
\(294\) 0 0
\(295\) 14.8371 + 3.68649i 0.863849 + 0.214636i
\(296\) 0 0
\(297\) 5.41855i 0.314416i
\(298\) 0 0
\(299\) 27.5174 1.59138
\(300\) 0 0
\(301\) 6.15676 0.354869
\(302\) 0 0
\(303\) 18.8638i 1.08369i
\(304\) 0 0
\(305\) 9.81658 + 2.43907i 0.562096 + 0.139661i
\(306\) 0 0
\(307\) 6.15676i 0.351385i 0.984445 + 0.175692i \(0.0562164\pi\)
−0.984445 + 0.175692i \(0.943784\pi\)
\(308\) 0 0
\(309\) −10.1568 −0.577798
\(310\) 0 0
\(311\) 1.52973 0.0867432 0.0433716 0.999059i \(-0.486190\pi\)
0.0433716 + 0.999059i \(0.486190\pi\)
\(312\) 0 0
\(313\) 11.9733i 0.676773i 0.941007 + 0.338387i \(0.109881\pi\)
−0.941007 + 0.338387i \(0.890119\pi\)
\(314\) 0 0
\(315\) −0.539189 + 2.17009i −0.0303799 + 0.122271i
\(316\) 0 0
\(317\) 23.5441i 1.32237i −0.750223 0.661184i \(-0.770054\pi\)
0.750223 0.661184i \(-0.229946\pi\)
\(318\) 0 0
\(319\) −47.8843 −2.68101
\(320\) 0 0
\(321\) 14.6537 0.817889
\(322\) 0 0
\(323\) 4.68035i 0.260421i
\(324\) 0 0
\(325\) −10.1568 + 19.1773i −0.563395 + 1.06376i
\(326\) 0 0
\(327\) 12.8371i 0.709893i
\(328\) 0 0
\(329\) 6.83710 0.376942
\(330\) 0 0
\(331\) 9.16290 0.503638 0.251819 0.967774i \(-0.418971\pi\)
0.251819 + 0.967774i \(0.418971\pi\)
\(332\) 0 0
\(333\) 8.68035i 0.475680i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 16.0000i 0.871576i −0.900049 0.435788i \(-0.856470\pi\)
0.900049 0.435788i \(-0.143530\pi\)
\(338\) 0 0
\(339\) 1.50307 0.0816356
\(340\) 0 0
\(341\) −23.5174 −1.27354
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 13.7587 + 3.41855i 0.740745 + 0.184049i
\(346\) 0 0
\(347\) 23.0205i 1.23581i −0.786254 0.617903i \(-0.787982\pi\)
0.786254 0.617903i \(-0.212018\pi\)
\(348\) 0 0
\(349\) 3.78992 0.202870 0.101435 0.994842i \(-0.467657\pi\)
0.101435 + 0.994842i \(0.467657\pi\)
\(350\) 0 0
\(351\) −4.34017 −0.231661
\(352\) 0 0
\(353\) 28.5958i 1.52200i 0.648751 + 0.761001i \(0.275292\pi\)
−0.648751 + 0.761001i \(0.724708\pi\)
\(354\) 0 0
\(355\) −30.5958 7.60197i −1.62386 0.403471i
\(356\) 0 0
\(357\) 1.07838i 0.0570738i
\(358\) 0 0
\(359\) −11.2618 −0.594375 −0.297187 0.954819i \(-0.596049\pi\)
−0.297187 + 0.954819i \(0.596049\pi\)
\(360\) 0 0
\(361\) −0.162899 −0.00857361
\(362\) 0 0
\(363\) 18.3607i 0.963686i
\(364\) 0 0
\(365\) −6.02666 + 24.2557i −0.315450 + 1.26960i
\(366\) 0 0
\(367\) 25.3607i 1.32382i −0.749584 0.661909i \(-0.769747\pi\)
0.749584 0.661909i \(-0.230253\pi\)
\(368\) 0 0
\(369\) −8.34017 −0.434172
\(370\) 0 0
\(371\) −6.18342 −0.321027
\(372\) 0 0
\(373\) 21.3074i 1.10325i −0.834091 0.551627i \(-0.814007\pi\)
0.834091 0.551627i \(-0.185993\pi\)
\(374\) 0 0
\(375\) −7.46081 + 8.32684i −0.385275 + 0.429996i
\(376\) 0 0
\(377\) 38.3545i 1.97536i
\(378\) 0 0
\(379\) −1.84324 −0.0946811 −0.0473406 0.998879i \(-0.515075\pi\)
−0.0473406 + 0.998879i \(0.515075\pi\)
\(380\) 0 0
\(381\) 19.2039 0.983847
\(382\) 0 0
\(383\) 4.99386i 0.255174i −0.991827 0.127587i \(-0.959277\pi\)
0.991827 0.127587i \(-0.0407232\pi\)
\(384\) 0 0
\(385\) −2.92162 + 11.7587i −0.148900 + 0.599280i
\(386\) 0 0
\(387\) 6.15676i 0.312965i
\(388\) 0 0
\(389\) −16.8371 −0.853675 −0.426837 0.904328i \(-0.640372\pi\)
−0.426837 + 0.904328i \(0.640372\pi\)
\(390\) 0 0
\(391\) −6.83710 −0.345767
\(392\) 0 0
\(393\) 4.00000i 0.201773i
\(394\) 0 0
\(395\) −1.47641 0.366835i −0.0742862 0.0184575i
\(396\) 0 0
\(397\) 36.8515i 1.84952i 0.380548 + 0.924761i \(0.375736\pi\)
−0.380548 + 0.924761i \(0.624264\pi\)
\(398\) 0 0
\(399\) 4.34017 0.217280
\(400\) 0 0
\(401\) −24.3545 −1.21621 −0.608104 0.793857i \(-0.708070\pi\)
−0.608104 + 0.793857i \(0.708070\pi\)
\(402\) 0 0
\(403\) 18.8371i 0.938343i
\(404\) 0 0
\(405\) −2.17009 0.539189i −0.107832 0.0267925i
\(406\) 0 0
\(407\) 47.0349i 2.33143i
\(408\) 0 0
\(409\) 28.0410 1.38654 0.693270 0.720678i \(-0.256169\pi\)
0.693270 + 0.720678i \(0.256169\pi\)
\(410\) 0 0
\(411\) −8.65368 −0.426855
\(412\) 0 0
\(413\) 6.83710i 0.336432i
\(414\) 0 0
\(415\) 3.68649 14.8371i 0.180963 0.728325i
\(416\) 0 0
\(417\) 6.18342i 0.302803i
\(418\) 0 0
\(419\) 0.482553 0.0235742 0.0117871 0.999931i \(-0.496248\pi\)
0.0117871 + 0.999931i \(0.496248\pi\)
\(420\) 0 0
\(421\) 21.1506 1.03082 0.515409 0.856944i \(-0.327640\pi\)
0.515409 + 0.856944i \(0.327640\pi\)
\(422\) 0 0
\(423\) 6.83710i 0.332431i
\(424\) 0 0
\(425\) 2.52359 4.76487i 0.122412 0.231130i
\(426\) 0 0
\(427\) 4.52359i 0.218912i
\(428\) 0 0
\(429\) −23.5174 −1.13543
\(430\) 0 0
\(431\) 28.7382 1.38427 0.692135 0.721768i \(-0.256670\pi\)
0.692135 + 0.721768i \(0.256670\pi\)
\(432\) 0 0
\(433\) 9.02052i 0.433498i 0.976227 + 0.216749i \(0.0695454\pi\)
−0.976227 + 0.216749i \(0.930455\pi\)
\(434\) 0 0
\(435\) −4.76487 + 19.1773i −0.228458 + 0.919480i
\(436\) 0 0
\(437\) 27.5174i 1.31634i
\(438\) 0 0
\(439\) −17.8166 −0.850339 −0.425170 0.905114i \(-0.639786\pi\)
−0.425170 + 0.905114i \(0.639786\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 11.7009i 0.555925i −0.960592 0.277962i \(-0.910341\pi\)
0.960592 0.277962i \(-0.0896591\pi\)
\(444\) 0 0
\(445\) 14.0989 + 3.50307i 0.668352 + 0.166062i
\(446\) 0 0
\(447\) 13.2039i 0.624525i
\(448\) 0 0
\(449\) −10.7337 −0.506553 −0.253277 0.967394i \(-0.581508\pi\)
−0.253277 + 0.967394i \(0.581508\pi\)
\(450\) 0 0
\(451\) −45.1917 −2.12799
\(452\) 0 0
\(453\) 18.1568i 0.853079i
\(454\) 0 0
\(455\) 9.41855 + 2.34017i 0.441548 + 0.109709i
\(456\) 0 0
\(457\) 20.9939i 0.982051i −0.871145 0.491026i \(-0.836622\pi\)
0.871145 0.491026i \(-0.163378\pi\)
\(458\) 0 0
\(459\) 1.07838 0.0503344
\(460\) 0 0
\(461\) −15.3751 −0.716088 −0.358044 0.933705i \(-0.616556\pi\)
−0.358044 + 0.933705i \(0.616556\pi\)
\(462\) 0 0
\(463\) 29.1917i 1.35665i 0.734762 + 0.678326i \(0.237294\pi\)
−0.734762 + 0.678326i \(0.762706\pi\)
\(464\) 0 0
\(465\) −2.34017 + 9.41855i −0.108523 + 0.436775i
\(466\) 0 0
\(467\) 15.2039i 0.703554i 0.936084 + 0.351777i \(0.114423\pi\)
−0.936084 + 0.351777i \(0.885577\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −15.1773 −0.699332
\(472\) 0 0
\(473\) 33.3607i 1.53393i
\(474\) 0 0
\(475\) 19.1773 + 10.1568i 0.879914 + 0.466024i
\(476\) 0 0
\(477\) 6.18342i 0.283119i
\(478\) 0 0
\(479\) −18.7838 −0.858253 −0.429126 0.903244i \(-0.641178\pi\)
−0.429126 + 0.903244i \(0.641178\pi\)
\(480\) 0 0
\(481\) −37.6742 −1.71780
\(482\) 0 0
\(483\) 6.34017i 0.288488i
\(484\) 0 0
\(485\) 5.65983 22.7792i 0.257000 1.03435i
\(486\) 0 0
\(487\) 16.5113i 0.748199i 0.927389 + 0.374099i \(0.122048\pi\)
−0.927389 + 0.374099i \(0.877952\pi\)
\(488\) 0 0
\(489\) 2.83710 0.128298
\(490\) 0 0
\(491\) −39.4063 −1.77838 −0.889190 0.457538i \(-0.848731\pi\)
−0.889190 + 0.457538i \(0.848731\pi\)
\(492\) 0 0
\(493\) 9.52973i 0.429198i
\(494\) 0 0
\(495\) −11.7587 2.92162i −0.528515 0.131317i
\(496\) 0 0
\(497\) 14.0989i 0.632422i
\(498\) 0 0
\(499\) −31.5174 −1.41091 −0.705457 0.708752i \(-0.749258\pi\)
−0.705457 + 0.708752i \(0.749258\pi\)
\(500\) 0 0
\(501\) 13.3607 0.596912
\(502\) 0 0
\(503\) 20.3668i 0.908112i 0.890973 + 0.454056i \(0.150023\pi\)
−0.890973 + 0.454056i \(0.849977\pi\)
\(504\) 0 0
\(505\) −40.9360 10.1711i −1.82163 0.452609i
\(506\) 0 0
\(507\) 5.83710i 0.259235i
\(508\) 0 0
\(509\) −11.1773 −0.495424 −0.247712 0.968834i \(-0.579679\pi\)
−0.247712 + 0.968834i \(0.579679\pi\)
\(510\) 0 0
\(511\) 11.1773 0.494454
\(512\) 0 0
\(513\) 4.34017i 0.191623i
\(514\) 0 0
\(515\) −5.47641 + 22.0410i −0.241319 + 0.971244i
\(516\) 0 0
\(517\) 37.0472i 1.62933i
\(518\) 0 0
\(519\) 2.55479 0.112143
\(520\) 0 0
\(521\) 32.6537 1.43058 0.715292 0.698826i \(-0.246294\pi\)
0.715292 + 0.698826i \(0.246294\pi\)
\(522\) 0 0
\(523\) 15.6865i 0.685922i −0.939350 0.342961i \(-0.888570\pi\)
0.939350 0.342961i \(-0.111430\pi\)
\(524\) 0 0
\(525\) 4.41855 + 2.34017i 0.192841 + 0.102134i
\(526\) 0 0
\(527\) 4.68035i 0.203879i
\(528\) 0 0
\(529\) −17.1978 −0.747730
\(530\) 0 0
\(531\) 6.83710 0.296705
\(532\) 0 0
\(533\) 36.1978i 1.56790i
\(534\) 0 0
\(535\) 7.90110 31.7998i 0.341594 1.37482i
\(536\) 0 0
\(537\) 11.9421i 0.515341i
\(538\) 0 0
\(539\) 5.41855 0.233394
\(540\) 0 0
\(541\) 30.1978 1.29830 0.649152 0.760658i \(-0.275124\pi\)
0.649152 + 0.760658i \(0.275124\pi\)
\(542\) 0 0
\(543\) 4.15676i 0.178383i
\(544\) 0 0
\(545\) −27.8576 6.92162i −1.19329 0.296490i
\(546\) 0 0
\(547\) 9.36069i 0.400234i 0.979772 + 0.200117i \(0.0641323\pi\)
−0.979772 + 0.200117i \(0.935868\pi\)
\(548\) 0 0
\(549\) 4.52359 0.193062
\(550\) 0 0
\(551\) 38.3545 1.63396
\(552\) 0 0
\(553\) 0.680346i 0.0289313i
\(554\) 0 0
\(555\) −18.8371 4.68035i −0.799590 0.198670i
\(556\) 0 0
\(557\) 7.49079i 0.317395i 0.987327 + 0.158697i \(0.0507294\pi\)
−0.987327 + 0.158697i \(0.949271\pi\)
\(558\) 0 0
\(559\) 26.7214 1.13019
\(560\) 0 0
\(561\) 5.84324 0.246702
\(562\) 0 0
\(563\) 31.7152i 1.33664i −0.743875 0.668319i \(-0.767014\pi\)
0.743875 0.668319i \(-0.232986\pi\)
\(564\) 0 0
\(565\) 0.810439 3.26180i 0.0340954 0.137225i
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) −30.6803 −1.28619 −0.643094 0.765788i \(-0.722349\pi\)
−0.643094 + 0.765788i \(0.722349\pi\)
\(570\) 0 0
\(571\) 10.6393 0.445241 0.222621 0.974905i \(-0.428539\pi\)
0.222621 + 0.974905i \(0.428539\pi\)
\(572\) 0 0
\(573\) 6.09890i 0.254785i
\(574\) 0 0
\(575\) 14.8371 28.0144i 0.618750 1.16828i
\(576\) 0 0
\(577\) 30.0677i 1.25173i −0.779930 0.625867i \(-0.784745\pi\)
0.779930 0.625867i \(-0.215255\pi\)
\(578\) 0 0
\(579\) 12.6803 0.526977
\(580\) 0 0
\(581\) −6.83710 −0.283651
\(582\) 0 0
\(583\) 33.5052i 1.38764i
\(584\) 0 0
\(585\) −2.34017 + 9.41855i −0.0967542 + 0.389409i
\(586\) 0 0
\(587\) 10.0410i 0.414438i 0.978295 + 0.207219i \(0.0664413\pi\)
−0.978295 + 0.207219i \(0.933559\pi\)
\(588\) 0 0
\(589\) 18.8371 0.776169
\(590\) 0 0
\(591\) 11.8576 0.487757
\(592\) 0 0
\(593\) 24.2823i 0.997155i −0.866845 0.498578i \(-0.833856\pi\)
0.866845 0.498578i \(-0.166144\pi\)
\(594\) 0 0
\(595\) −2.34017 0.581449i −0.0959377 0.0238371i
\(596\) 0 0
\(597\) 5.50307i 0.225226i
\(598\) 0 0
\(599\) −9.90110 −0.404548 −0.202274 0.979329i \(-0.564833\pi\)
−0.202274 + 0.979329i \(0.564833\pi\)
\(600\) 0 0
\(601\) 17.6865 0.721447 0.360723 0.932673i \(-0.382530\pi\)
0.360723 + 0.932673i \(0.382530\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −39.8443 9.89988i −1.61990 0.402487i
\(606\) 0 0
\(607\) 28.3135i 1.14921i 0.818431 + 0.574605i \(0.194844\pi\)
−0.818431 + 0.574605i \(0.805156\pi\)
\(608\) 0 0
\(609\) 8.83710 0.358097
\(610\) 0 0
\(611\) 29.6742 1.20049
\(612\) 0 0
\(613\) 23.6865i 0.956688i −0.878172 0.478344i \(-0.841237\pi\)
0.878172 0.478344i \(-0.158763\pi\)
\(614\) 0 0
\(615\) −4.49693 + 18.0989i −0.181334 + 0.729818i
\(616\) 0 0
\(617\) 13.7009i 0.551576i 0.961218 + 0.275788i \(0.0889388\pi\)
−0.961218 + 0.275788i \(0.911061\pi\)
\(618\) 0 0
\(619\) −2.49693 −0.100360 −0.0501800 0.998740i \(-0.515980\pi\)
−0.0501800 + 0.998740i \(0.515980\pi\)
\(620\) 0 0
\(621\) 6.34017 0.254422
\(622\) 0 0
\(623\) 6.49693i 0.260294i
\(624\) 0 0
\(625\) 14.0472 + 20.6803i 0.561887 + 0.827214i
\(626\) 0 0
\(627\) 23.5174i 0.939196i
\(628\) 0 0
\(629\) 9.36069 0.373235
\(630\) 0 0
\(631\) −8.68035 −0.345559 −0.172780 0.984961i \(-0.555275\pi\)
−0.172780 + 0.984961i \(0.555275\pi\)
\(632\) 0 0
\(633\) 19.1506i 0.761169i
\(634\) 0 0
\(635\) 10.3545 41.6742i 0.410908 1.65379i
\(636\) 0 0
\(637\) 4.34017i 0.171964i
\(638\) 0 0
\(639\) −14.0989 −0.557744
\(640\) 0 0
\(641\) 3.30737 0.130633 0.0653166 0.997865i \(-0.479194\pi\)
0.0653166 + 0.997865i \(0.479194\pi\)
\(642\) 0 0
\(643\) 6.15676i 0.242799i 0.992604 + 0.121399i \(0.0387382\pi\)
−0.992604 + 0.121399i \(0.961262\pi\)
\(644\) 0 0
\(645\) 13.3607 + 3.31965i 0.526077 + 0.130711i
\(646\) 0 0
\(647\) 6.95282i 0.273344i −0.990616 0.136672i \(-0.956359\pi\)
0.990616 0.136672i \(-0.0436406\pi\)
\(648\) 0 0
\(649\) 37.0472 1.45423
\(650\) 0 0
\(651\) 4.34017 0.170105
\(652\) 0 0
\(653\) 38.7480i 1.51633i −0.652064 0.758164i \(-0.726097\pi\)
0.652064 0.758164i \(-0.273903\pi\)
\(654\) 0 0
\(655\) −8.68035 2.15676i −0.339169 0.0842714i
\(656\) 0 0
\(657\) 11.1773i 0.436067i
\(658\) 0 0
\(659\) 9.22076 0.359190 0.179595 0.983741i \(-0.442521\pi\)
0.179595 + 0.983741i \(0.442521\pi\)
\(660\) 0 0
\(661\) 25.8843 1.00678 0.503391 0.864059i \(-0.332086\pi\)
0.503391 + 0.864059i \(0.332086\pi\)
\(662\) 0 0
\(663\) 4.68035i 0.181770i
\(664\) 0 0
\(665\) 2.34017 9.41855i 0.0907480 0.365236i
\(666\) 0 0
\(667\) 56.0288i 2.16944i
\(668\) 0 0
\(669\) −12.3135 −0.476068
\(670\) 0 0
\(671\) 24.5113 0.946248
\(672\) 0 0
\(673\) 40.0821i 1.54505i −0.634984 0.772525i \(-0.718993\pi\)
0.634984 0.772525i \(-0.281007\pi\)
\(674\) 0 0
\(675\) −2.34017 + 4.41855i −0.0900733 + 0.170070i
\(676\) 0 0
\(677\) 33.5897i 1.29096i −0.763779 0.645478i \(-0.776658\pi\)
0.763779 0.645478i \(-0.223342\pi\)
\(678\) 0 0
\(679\) −10.4969 −0.402835
\(680\) 0 0
\(681\) −15.2039 −0.582616
\(682\) 0 0
\(683\) 18.7070i 0.715804i 0.933759 + 0.357902i \(0.116508\pi\)
−0.933759 + 0.357902i \(0.883492\pi\)
\(684\) 0 0
\(685\) −4.66597 + 18.7792i −0.178278 + 0.717518i
\(686\) 0 0
\(687\) 5.20394i 0.198543i
\(688\) 0 0
\(689\) −26.8371 −1.02241
\(690\) 0 0
\(691\) 19.1773 0.729538 0.364769 0.931098i \(-0.381148\pi\)
0.364769 + 0.931098i \(0.381148\pi\)
\(692\) 0 0
\(693\) 5.41855i 0.205834i
\(694\) 0 0
\(695\) 13.4186 + 3.33403i 0.508995 + 0.126467i
\(696\) 0 0
\(697\) 8.99386i 0.340667i
\(698\) 0 0
\(699\) 11.6598 0.441015
\(700\) 0 0
\(701\) 21.4641 0.810689 0.405344 0.914164i \(-0.367152\pi\)
0.405344 + 0.914164i \(0.367152\pi\)
\(702\) 0 0
\(703\) 37.6742i 1.42091i
\(704\) 0 0
\(705\) 14.8371 + 3.68649i 0.558798 + 0.138841i
\(706\) 0 0
\(707\) 18.8638i 0.709445i
\(708\) 0 0
\(709\) −2.62702 −0.0986599 −0.0493299 0.998783i \(-0.515709\pi\)
−0.0493299 + 0.998783i \(0.515709\pi\)
\(710\) 0 0
\(711\) −0.680346 −0.0255150
\(712\) 0 0
\(713\) 27.5174i 1.03054i
\(714\) 0 0
\(715\) −12.6803 + 51.0349i −0.474218 + 1.90860i
\(716\) 0 0
\(717\) 20.6225i 0.770161i
\(718\) 0 0
\(719\) −4.36683 −0.162855 −0.0814277 0.996679i \(-0.525948\pi\)
−0.0814277 + 0.996679i \(0.525948\pi\)
\(720\) 0 0
\(721\) 10.1568 0.378257
\(722\) 0 0
\(723\) 20.3545i 0.756994i
\(724\) 0 0
\(725\) 39.0472 + 20.6803i 1.45018 + 0.768049i
\(726\) 0 0
\(727\) 28.1445i 1.04382i −0.853000 0.521910i \(-0.825220\pi\)
0.853000 0.521910i \(-0.174780\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −6.63931 −0.245564
\(732\) 0 0
\(733\) 26.7480i 0.987962i 0.869473 + 0.493981i \(0.164459\pi\)
−0.869473 + 0.493981i \(0.835541\pi\)
\(734\) 0 0
\(735\) 0.539189 2.17009i 0.0198883 0.0800448i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 19.0349 0.700210 0.350105 0.936710i \(-0.386146\pi\)
0.350105 + 0.936710i \(0.386146\pi\)
\(740\) 0 0
\(741\) 18.8371 0.691998
\(742\) 0 0
\(743\) 33.6598i 1.23486i 0.786626 + 0.617430i \(0.211826\pi\)
−0.786626 + 0.617430i \(0.788174\pi\)
\(744\) 0 0
\(745\) −28.6537 7.11942i −1.04979 0.260835i
\(746\) 0 0
\(747\) 6.83710i 0.250156i
\(748\) 0 0
\(749\) −14.6537 −0.535434
\(750\) 0 0
\(751\) 49.8720 1.81985 0.909927 0.414767i \(-0.136137\pi\)
0.909927 + 0.414767i \(0.136137\pi\)
\(752\) 0 0
\(753\) 10.5236i 0.383501i
\(754\) 0 0
\(755\) −39.4017 9.78992i −1.43398 0.356292i
\(756\) 0 0
\(757\) 7.63317i 0.277432i −0.990332 0.138716i \(-0.955702\pi\)
0.990332 0.138716i \(-0.0442975\pi\)
\(758\) 0 0
\(759\) 34.3545 1.24699
\(760\) 0 0
\(761\) 19.5974 0.710406 0.355203 0.934789i \(-0.384412\pi\)
0.355203 + 0.934789i \(0.384412\pi\)
\(762\) 0 0
\(763\) 12.8371i 0.464734i
\(764\) 0 0
\(765\) 0.581449 2.34017i 0.0210223 0.0846091i
\(766\) 0 0
\(767\) 29.6742i 1.07147i
\(768\) 0 0
\(769\) 32.4079 1.16866 0.584329 0.811517i \(-0.301358\pi\)
0.584329 + 0.811517i \(0.301358\pi\)
\(770\) 0 0
\(771\) −22.8059 −0.821335
\(772\) 0 0
\(773\) 12.0845i 0.434650i 0.976099 + 0.217325i \(0.0697331\pi\)
−0.976099 + 0.217325i \(0.930267\pi\)
\(774\) 0 0
\(775\) 19.1773 + 10.1568i 0.688868 + 0.364841i
\(776\) 0 0
\(777\) 8.68035i 0.311406i
\(778\) 0 0
\(779\) 36.1978 1.29692
\(780\) 0 0
\(781\) −76.3956 −2.73365
\(782\) 0 0
\(783\) 8.83710i 0.315812i
\(784\) 0 0
\(785\) −8.18342 + 32.9360i −0.292079 + 1.17554i
\(786\) 0 0
\(787\) 25.0472i 0.892836i −0.894825 0.446418i \(-0.852700\pi\)
0.894825 0.446418i \(-0.147300\pi\)
\(788\) 0 0
\(789\) 28.0144 0.997338
\(790\) 0 0
\(791\) −1.50307 −0.0534431
\(792\) 0 0
\(793\) 19.6332i 0.697194i
\(794\) 0 0
\(795\) −13.4186 3.33403i −0.475907 0.118246i
\(796\) 0 0
\(797\) 21.2762i 0.753641i −0.926286 0.376820i \(-0.877017\pi\)
0.926286 0.376820i \(-0.122983\pi\)
\(798\) 0 0
\(799\) −7.37298 −0.260837
\(800\) 0 0
\(801\) 6.49693 0.229558
\(802\) 0 0
\(803\) 60.5646i