Properties

Label 4680.2.l.f.2809.1
Level $4680$
Weight $2$
Character 4680.2809
Analytic conductor $37.370$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4680.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(37.3699881460\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1698758656.6
Defining polynomial: \( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 1560)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2809.1
Root \(2.16053i\) of defining polynomial
Character \(\chi\) \(=\) 4680.2809
Dual form 4680.2.l.f.2809.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-2.23483 - 0.0743018i) q^{5} -2.82843i q^{7} +O(q^{10})\) \(q+(-2.23483 - 0.0743018i) q^{5} -2.82843i q^{7} +6.46967 q^{11} -1.00000i q^{13} +3.49264i q^{17} +2.21016 q^{19} +7.14949i q^{23} +(4.98896 + 0.332104i) q^{25} +4.82843 q^{29} -9.65685 q^{31} +(-0.210157 + 6.32106i) q^{35} -6.93933i q^{37} +0.148604 q^{41} +3.03858i q^{43} +6.79073i q^{47} -1.00000 q^{49} -1.70279i q^{53} +(-14.4586 - 0.480708i) q^{55} -6.46967 q^{59} +2.66421 q^{61} +(-0.0743018 + 2.23483i) q^{65} +7.70279i q^{67} +11.0879 q^{71} +5.76776i q^{73} -18.2990i q^{77} +10.0239 q^{79} +2.27171i q^{83} +(0.259509 - 7.80546i) q^{85} -9.21104 q^{89} -2.82843 q^{91} +(-4.93933 - 0.164219i) q^{95} +8.11091i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{11} + 24 q^{19} - 4 q^{25} + 16 q^{29} - 32 q^{31} - 8 q^{35} + 8 q^{41} - 8 q^{49} - 36 q^{55} - 16 q^{59} + 24 q^{61} - 4 q^{65} + 24 q^{71} + 24 q^{79} - 40 q^{85} - 8 q^{89} + 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times\).

\(n\) \(937\) \(1081\) \(2081\) \(2341\) \(3511\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −2.23483 0.0743018i −0.999448 0.0332288i
\(6\) 0 0
\(7\) 2.82843i 1.06904i −0.845154 0.534522i \(-0.820491\pi\)
0.845154 0.534522i \(-0.179509\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.46967 1.95068 0.975339 0.220713i \(-0.0708383\pi\)
0.975339 + 0.220713i \(0.0708383\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.49264i 0.847089i 0.905875 + 0.423544i \(0.139214\pi\)
−0.905875 + 0.423544i \(0.860786\pi\)
\(18\) 0 0
\(19\) 2.21016 0.507045 0.253522 0.967330i \(-0.418411\pi\)
0.253522 + 0.967330i \(0.418411\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.14949i 1.49077i 0.666633 + 0.745386i \(0.267735\pi\)
−0.666633 + 0.745386i \(0.732265\pi\)
\(24\) 0 0
\(25\) 4.98896 + 0.332104i 0.997792 + 0.0664208i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.82843 0.896616 0.448308 0.893879i \(-0.352027\pi\)
0.448308 + 0.893879i \(0.352027\pi\)
\(30\) 0 0
\(31\) −9.65685 −1.73442 −0.867211 0.497941i \(-0.834090\pi\)
−0.867211 + 0.497941i \(0.834090\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.210157 + 6.32106i −0.0355231 + 1.06845i
\(36\) 0 0
\(37\) 6.93933i 1.14082i −0.821360 0.570410i \(-0.806784\pi\)
0.821360 0.570410i \(-0.193216\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.148604 0.0232080 0.0116040 0.999933i \(-0.496306\pi\)
0.0116040 + 0.999933i \(0.496306\pi\)
\(42\) 0 0
\(43\) 3.03858i 0.463380i 0.972790 + 0.231690i \(0.0744255\pi\)
−0.972790 + 0.231690i \(0.925575\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.79073i 0.990530i 0.868742 + 0.495265i \(0.164929\pi\)
−0.868742 + 0.495265i \(0.835071\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.70279i 0.233897i −0.993138 0.116948i \(-0.962689\pi\)
0.993138 0.116948i \(-0.0373112\pi\)
\(54\) 0 0
\(55\) −14.4586 0.480708i −1.94960 0.0648186i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.46967 −0.842279 −0.421139 0.906996i \(-0.638370\pi\)
−0.421139 + 0.906996i \(0.638370\pi\)
\(60\) 0 0
\(61\) 2.66421 0.341117 0.170558 0.985348i \(-0.445443\pi\)
0.170558 + 0.985348i \(0.445443\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.0743018 + 2.23483i −0.00921600 + 0.277197i
\(66\) 0 0
\(67\) 7.70279i 0.941046i 0.882388 + 0.470523i \(0.155935\pi\)
−0.882388 + 0.470523i \(0.844065\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.0879 1.31590 0.657948 0.753063i \(-0.271425\pi\)
0.657948 + 0.753063i \(0.271425\pi\)
\(72\) 0 0
\(73\) 5.76776i 0.675065i 0.941314 + 0.337533i \(0.109592\pi\)
−0.941314 + 0.337533i \(0.890408\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 18.2990i 2.08536i
\(78\) 0 0
\(79\) 10.0239 1.12777 0.563886 0.825853i \(-0.309306\pi\)
0.563886 + 0.825853i \(0.309306\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.27171i 0.249353i 0.992197 + 0.124676i \(0.0397892\pi\)
−0.992197 + 0.124676i \(0.960211\pi\)
\(84\) 0 0
\(85\) 0.259509 7.80546i 0.0281477 0.846621i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.21104 −0.976369 −0.488184 0.872741i \(-0.662341\pi\)
−0.488184 + 0.872741i \(0.662341\pi\)
\(90\) 0 0
\(91\) −2.82843 −0.296500
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.93933 0.164219i −0.506765 0.0168485i
\(96\) 0 0
\(97\) 8.11091i 0.823538i 0.911288 + 0.411769i \(0.135089\pi\)
−0.911288 + 0.411769i \(0.864911\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 16.1421 1.60620 0.803101 0.595843i \(-0.203182\pi\)
0.803101 + 0.595843i \(0.203182\pi\)
\(102\) 0 0
\(103\) 19.5815i 1.92942i −0.263318 0.964709i \(-0.584817\pi\)
0.263318 0.964709i \(-0.415183\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.35965i 0.131442i 0.997838 + 0.0657210i \(0.0209347\pi\)
−0.997838 + 0.0657210i \(0.979065\pi\)
\(108\) 0 0
\(109\) −3.49264 −0.334534 −0.167267 0.985912i \(-0.553494\pi\)
−0.167267 + 0.985912i \(0.553494\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.4320i 1.16950i 0.811213 + 0.584751i \(0.198808\pi\)
−0.811213 + 0.584751i \(0.801192\pi\)
\(114\) 0 0
\(115\) 0.531220 15.9779i 0.0495365 1.48995i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 9.87867 0.905576
\(120\) 0 0
\(121\) 30.8566 2.80514
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −11.1248 1.11289i −0.995034 0.0995396i
\(126\) 0 0
\(127\) 18.6200i 1.65226i −0.563478 0.826131i \(-0.690537\pi\)
0.563478 0.826131i \(-0.309463\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0.751258 0.0656378 0.0328189 0.999461i \(-0.489552\pi\)
0.0328189 + 0.999461i \(0.489552\pi\)
\(132\) 0 0
\(133\) 6.25127i 0.542054i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.08794i 0.0929487i 0.998919 + 0.0464743i \(0.0147986\pi\)
−0.998919 + 0.0464743i \(0.985201\pi\)
\(138\) 0 0
\(139\) −0.420314 −0.0356506 −0.0178253 0.999841i \(-0.505674\pi\)
−0.0178253 + 0.999841i \(0.505674\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.46967i 0.541021i
\(144\) 0 0
\(145\) −10.7907 0.358761i −0.896121 0.0297935i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −14.1265 −1.15729 −0.578645 0.815580i \(-0.696418\pi\)
−0.578645 + 0.815580i \(0.696418\pi\)
\(150\) 0 0
\(151\) 6.07717 0.494553 0.247276 0.968945i \(-0.420464\pi\)
0.247276 + 0.968945i \(0.420464\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 21.5815 + 0.717522i 1.73346 + 0.0576327i
\(156\) 0 0
\(157\) 11.7266i 0.935888i 0.883758 + 0.467944i \(0.155005\pi\)
−0.883758 + 0.467944i \(0.844995\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 20.2218 1.59370
\(162\) 0 0
\(163\) 7.31371i 0.572854i 0.958102 + 0.286427i \(0.0924676\pi\)
−0.958102 + 0.286427i \(0.907532\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.0420i 1.62828i −0.580670 0.814139i \(-0.697209\pi\)
0.580670 0.814139i \(-0.302791\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 18.9706i 1.44231i 0.692776 + 0.721153i \(0.256387\pi\)
−0.692776 + 0.721153i \(0.743613\pi\)
\(174\) 0 0
\(175\) 0.939333 14.1109i 0.0710069 1.06668i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 20.4099 1.52551 0.762753 0.646690i \(-0.223847\pi\)
0.762753 + 0.646690i \(0.223847\pi\)
\(180\) 0 0
\(181\) 24.8934 1.85031 0.925156 0.379588i \(-0.123934\pi\)
0.925156 + 0.379588i \(0.123934\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.515605 + 15.5083i −0.0379080 + 1.14019i
\(186\) 0 0
\(187\) 22.5962i 1.65240i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 14.9853 1.08430 0.542148 0.840283i \(-0.317611\pi\)
0.542148 + 0.840283i \(0.317611\pi\)
\(192\) 0 0
\(193\) 18.7990i 1.35318i −0.736360 0.676590i \(-0.763457\pi\)
0.736360 0.676590i \(-0.236543\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.92857i 0.137405i 0.997637 + 0.0687023i \(0.0218859\pi\)
−0.997637 + 0.0687023i \(0.978114\pi\)
\(198\) 0 0
\(199\) 10.0239 0.710572 0.355286 0.934758i \(-0.384383\pi\)
0.355286 + 0.934758i \(0.384383\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 13.6569i 0.958523i
\(204\) 0 0
\(205\) −0.332104 0.0110415i −0.0231952 0.000771173i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 14.2990 0.989081
\(210\) 0 0
\(211\) 20.2751 1.39580 0.697899 0.716197i \(-0.254119\pi\)
0.697899 + 0.716197i \(0.254119\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.225772 6.79073i 0.0153975 0.463124i
\(216\) 0 0
\(217\) 27.3137i 1.85418i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.49264 0.234940
\(222\) 0 0
\(223\) 16.1881i 1.08403i 0.840368 + 0.542017i \(0.182339\pi\)
−0.840368 + 0.542017i \(0.817661\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 23.0879i 1.53240i −0.642602 0.766200i \(-0.722145\pi\)
0.642602 0.766200i \(-0.277855\pi\)
\(228\) 0 0
\(229\) 4.30886 0.284738 0.142369 0.989814i \(-0.454528\pi\)
0.142369 + 0.989814i \(0.454528\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.53857i 0.100795i 0.998729 + 0.0503977i \(0.0160489\pi\)
−0.998729 + 0.0503977i \(0.983951\pi\)
\(234\) 0 0
\(235\) 0.504563 15.1761i 0.0329141 0.989983i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −17.8514 −1.15471 −0.577355 0.816493i \(-0.695915\pi\)
−0.577355 + 0.816493i \(0.695915\pi\)
\(240\) 0 0
\(241\) 3.95406 0.254703 0.127352 0.991858i \(-0.459352\pi\)
0.127352 + 0.991858i \(0.459352\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.23483 + 0.0743018i 0.142778 + 0.00474697i
\(246\) 0 0
\(247\) 2.21016i 0.140629i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3.42284 −0.216048 −0.108024 0.994148i \(-0.534452\pi\)
−0.108024 + 0.994148i \(0.534452\pi\)
\(252\) 0 0
\(253\) 46.2548i 2.90802i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.14772i 0.196349i −0.995169 0.0981746i \(-0.968700\pi\)
0.995169 0.0981746i \(-0.0313004\pi\)
\(258\) 0 0
\(259\) −19.6274 −1.21959
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 13.7144i 0.845669i −0.906207 0.422835i \(-0.861035\pi\)
0.906207 0.422835i \(-0.138965\pi\)
\(264\) 0 0
\(265\) −0.126521 + 3.80546i −0.00777210 + 0.233767i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 9.84316 0.600148 0.300074 0.953916i \(-0.402989\pi\)
0.300074 + 0.953916i \(0.402989\pi\)
\(270\) 0 0
\(271\) −2.86216 −0.173864 −0.0869320 0.996214i \(-0.527706\pi\)
−0.0869320 + 0.996214i \(0.527706\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 32.2769 + 2.14860i 1.94637 + 0.129566i
\(276\) 0 0
\(277\) 6.24389i 0.375159i −0.982249 0.187580i \(-0.939936\pi\)
0.982249 0.187580i \(-0.0600643\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 31.3869 1.87239 0.936193 0.351486i \(-0.114323\pi\)
0.936193 + 0.351486i \(0.114323\pi\)
\(282\) 0 0
\(283\) 2.56496i 0.152471i 0.997090 + 0.0762354i \(0.0242901\pi\)
−0.997090 + 0.0762354i \(0.975710\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.420314i 0.0248104i
\(288\) 0 0
\(289\) 4.80150 0.282441
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0.834895i 0.0487751i 0.999703 + 0.0243875i \(0.00776357\pi\)
−0.999703 + 0.0243875i \(0.992236\pi\)
\(294\) 0 0
\(295\) 14.4586 + 0.480708i 0.841814 + 0.0279879i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 7.14949 0.413466
\(300\) 0 0
\(301\) 8.59441 0.495374
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −5.95406 0.197955i −0.340929 0.0113349i
\(306\) 0 0
\(307\) 17.7556i 1.01336i 0.862133 + 0.506682i \(0.169128\pi\)
−0.862133 + 0.506682i \(0.830872\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.35965 0.0770985 0.0385493 0.999257i \(-0.487726\pi\)
0.0385493 + 0.999257i \(0.487726\pi\)
\(312\) 0 0
\(313\) 7.75611i 0.438401i −0.975680 0.219201i \(-0.929655\pi\)
0.975680 0.219201i \(-0.0703449\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.38514i 0.527122i −0.964643 0.263561i \(-0.915103\pi\)
0.964643 0.263561i \(-0.0848971\pi\)
\(318\) 0 0
\(319\) 31.2383 1.74901
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 7.71927i 0.429512i
\(324\) 0 0
\(325\) 0.332104 4.98896i 0.0184218 0.276738i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 19.2071 1.05892
\(330\) 0 0
\(331\) 16.0576 0.882605 0.441303 0.897358i \(-0.354517\pi\)
0.441303 + 0.897358i \(0.354517\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0.572331 17.2145i 0.0312698 0.940526i
\(336\) 0 0
\(337\) 11.0698i 0.603011i −0.953464 0.301506i \(-0.902511\pi\)
0.953464 0.301506i \(-0.0974892\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −62.4766 −3.38330
\(342\) 0 0
\(343\) 16.9706i 0.916324i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 29.6127i 1.58969i 0.606811 + 0.794846i \(0.292449\pi\)
−0.606811 + 0.794846i \(0.707551\pi\)
\(348\) 0 0
\(349\) −17.1183 −0.916319 −0.458160 0.888870i \(-0.651491\pi\)
−0.458160 + 0.888870i \(0.651491\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 18.3245i 0.975313i −0.873035 0.487657i \(-0.837852\pi\)
0.873035 0.487657i \(-0.162148\pi\)
\(354\) 0 0
\(355\) −24.7797 0.823854i −1.31517 0.0437256i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 27.1356 1.43216 0.716082 0.698016i \(-0.245933\pi\)
0.716082 + 0.698016i \(0.245933\pi\)
\(360\) 0 0
\(361\) −14.1152 −0.742906
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.428555 12.8900i 0.0224316 0.674692i
\(366\) 0 0
\(367\) 15.1855i 0.792679i 0.918104 + 0.396340i \(0.129720\pi\)
−0.918104 + 0.396340i \(0.870280\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −4.81623 −0.250046
\(372\) 0 0
\(373\) 22.1776i 1.14831i 0.818745 + 0.574157i \(0.194670\pi\)
−0.818745 + 0.574157i \(0.805330\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.82843i 0.248677i
\(378\) 0 0
\(379\) −14.7604 −0.758191 −0.379096 0.925358i \(-0.623765\pi\)
−0.379096 + 0.925358i \(0.623765\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 9.33238i 0.476862i 0.971159 + 0.238431i \(0.0766331\pi\)
−0.971159 + 0.238431i \(0.923367\pi\)
\(384\) 0 0
\(385\) −1.35965 + 40.8952i −0.0692940 + 2.08421i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −3.09618 −0.156982 −0.0784912 0.996915i \(-0.525010\pi\)
−0.0784912 + 0.996915i \(0.525010\pi\)
\(390\) 0 0
\(391\) −24.9706 −1.26282
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −22.4016 0.744790i −1.12715 0.0374745i
\(396\) 0 0
\(397\) 7.26777i 0.364759i −0.983228 0.182379i \(-0.941620\pi\)
0.983228 0.182379i \(-0.0583799\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 8.40164 0.419558 0.209779 0.977749i \(-0.432726\pi\)
0.209779 + 0.977749i \(0.432726\pi\)
\(402\) 0 0
\(403\) 9.65685i 0.481042i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 44.8952i 2.22537i
\(408\) 0 0
\(409\) −11.5815 −0.572666 −0.286333 0.958130i \(-0.592436\pi\)
−0.286333 + 0.958130i \(0.592436\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 18.2990i 0.900434i
\(414\) 0 0
\(415\) 0.168792 5.07689i 0.00828568 0.249215i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −24.7071 −1.20702 −0.603510 0.797355i \(-0.706232\pi\)
−0.603510 + 0.797355i \(0.706232\pi\)
\(420\) 0 0
\(421\) −20.7769 −1.01260 −0.506302 0.862356i \(-0.668988\pi\)
−0.506302 + 0.862356i \(0.668988\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.15992 + 17.4246i −0.0562643 + 0.845218i
\(426\) 0 0
\(427\) 7.53552i 0.364669i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −16.6694 −0.802936 −0.401468 0.915873i \(-0.631500\pi\)
−0.401468 + 0.915873i \(0.631500\pi\)
\(432\) 0 0
\(433\) 12.8860i 0.619263i 0.950857 + 0.309631i \(0.100206\pi\)
−0.950857 + 0.309631i \(0.899794\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 15.8015i 0.755888i
\(438\) 0 0
\(439\) −37.1924 −1.77510 −0.887548 0.460716i \(-0.847593\pi\)
−0.887548 + 0.460716i \(0.847593\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 33.0693i 1.57117i 0.618755 + 0.785584i \(0.287637\pi\)
−0.618755 + 0.785584i \(0.712363\pi\)
\(444\) 0 0
\(445\) 20.5851 + 0.684397i 0.975829 + 0.0324435i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −10.7448 −0.507078 −0.253539 0.967325i \(-0.581595\pi\)
−0.253539 + 0.967325i \(0.581595\pi\)
\(450\) 0 0
\(451\) 0.961416 0.0452713
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 6.32106 + 0.210157i 0.296336 + 0.00985232i
\(456\) 0 0
\(457\) 21.9419i 1.02640i 0.858270 + 0.513198i \(0.171540\pi\)
−0.858270 + 0.513198i \(0.828460\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −22.0034 −1.02480 −0.512401 0.858747i \(-0.671244\pi\)
−0.512401 + 0.858747i \(0.671244\pi\)
\(462\) 0 0
\(463\) 32.0208i 1.48813i −0.668106 0.744066i \(-0.732895\pi\)
0.668106 0.744066i \(-0.267105\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 25.3596i 1.17350i −0.809767 0.586752i \(-0.800406\pi\)
0.809767 0.586752i \(-0.199594\pi\)
\(468\) 0 0
\(469\) 21.7868 1.00602
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 19.6586i 0.903905i
\(474\) 0 0
\(475\) 11.0264 + 0.734003i 0.505925 + 0.0336783i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 9.46231 0.432344 0.216172 0.976355i \(-0.430643\pi\)
0.216172 + 0.976355i \(0.430643\pi\)
\(480\) 0 0
\(481\) −6.93933 −0.316406
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0.602655 18.1265i 0.0273651 0.823083i
\(486\) 0 0
\(487\) 22.3327i 1.01199i −0.862536 0.505996i \(-0.831125\pi\)
0.862536 0.505996i \(-0.168875\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −29.8762 −1.34829 −0.674146 0.738598i \(-0.735488\pi\)
−0.674146 + 0.738598i \(0.735488\pi\)
\(492\) 0 0
\(493\) 16.8639i 0.759513i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 31.3614i 1.40675i
\(498\) 0 0
\(499\) 24.7145 1.10637 0.553186 0.833058i \(-0.313412\pi\)
0.553186 + 0.833058i \(0.313412\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 17.2267i 0.768099i −0.923313 0.384049i \(-0.874529\pi\)
0.923313 0.384049i \(-0.125471\pi\)
\(504\) 0 0
\(505\) −36.0750 1.19939i −1.60532 0.0533721i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 26.1430 1.15877 0.579384 0.815055i \(-0.303293\pi\)
0.579384 + 0.815055i \(0.303293\pi\)
\(510\) 0 0
\(511\) 16.3137 0.721675
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.45494 + 43.7613i −0.0641122 + 1.92835i
\(516\) 0 0
\(517\) 43.9338i 1.93220i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 33.0165 1.44648 0.723240 0.690597i \(-0.242652\pi\)
0.723240 + 0.690597i \(0.242652\pi\)
\(522\) 0 0
\(523\) 32.2751i 1.41129i −0.708564 0.705646i \(-0.750657\pi\)
0.708564 0.705646i \(-0.249343\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 33.7279i 1.46921i
\(528\) 0 0
\(529\) −28.1152 −1.22240
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0.148604i 0.00643674i
\(534\) 0 0
\(535\) 0.101024 3.03858i 0.00436766 0.131369i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −6.46967 −0.278668
\(540\) 0 0
\(541\) −2.21193 −0.0950983 −0.0475491 0.998869i \(-0.515141\pi\)
−0.0475491 + 0.998869i \(0.515141\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.80546 + 0.259509i 0.334349 + 0.0111161i
\(546\) 0 0
\(547\) 31.4822i 1.34608i −0.739605 0.673041i \(-0.764988\pi\)
0.739605 0.673041i \(-0.235012\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 10.6716 0.454625
\(552\) 0 0
\(553\) 28.3517i 1.20564i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 35.1209i 1.48812i 0.668112 + 0.744061i \(0.267103\pi\)
−0.668112 + 0.744061i \(0.732897\pi\)
\(558\) 0 0
\(559\) 3.03858 0.128518
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 30.3467i 1.27896i −0.768808 0.639480i \(-0.779150\pi\)
0.768808 0.639480i \(-0.220850\pi\)
\(564\) 0 0
\(565\) 0.923718 27.7834i 0.0388611 1.16886i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 30.8327 1.29257 0.646287 0.763094i \(-0.276321\pi\)
0.646287 + 0.763094i \(0.276321\pi\)
\(570\) 0 0
\(571\) 21.1832 0.886490 0.443245 0.896400i \(-0.353827\pi\)
0.443245 + 0.896400i \(0.353827\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.37438 + 35.6685i −0.0990183 + 1.48748i
\(576\) 0 0
\(577\) 22.6594i 0.943322i 0.881780 + 0.471661i \(0.156345\pi\)
−0.881780 + 0.471661i \(0.843655\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 6.42537 0.266569
\(582\) 0 0
\(583\) 11.0165i 0.456257i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 13.6529i 0.563515i 0.959486 + 0.281758i \(0.0909174\pi\)
−0.959486 + 0.281758i \(0.909083\pi\)
\(588\) 0 0
\(589\) −21.3432 −0.879430
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 18.7448i 0.769756i −0.922967 0.384878i \(-0.874243\pi\)
0.922967 0.384878i \(-0.125757\pi\)
\(594\) 0 0
\(595\) −22.0772 0.734003i −0.905076 0.0300912i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −48.7885 −1.99345 −0.996723 0.0808922i \(-0.974223\pi\)
−0.996723 + 0.0808922i \(0.974223\pi\)
\(600\) 0 0
\(601\) 16.5870 0.676599 0.338300 0.941038i \(-0.390148\pi\)
0.338300 + 0.941038i \(0.390148\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −68.9593 2.29270i −2.80359 0.0932115i
\(606\) 0 0
\(607\) 29.0642i 1.17968i 0.807520 + 0.589840i \(0.200809\pi\)
−0.807520 + 0.589840i \(0.799191\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6.79073 0.274723
\(612\) 0 0
\(613\) 22.3926i 0.904430i −0.891909 0.452215i \(-0.850634\pi\)
0.891909 0.452215i \(-0.149366\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.63136i 0.0656760i 0.999461 + 0.0328380i \(0.0104545\pi\)
−0.999461 + 0.0328380i \(0.989545\pi\)
\(618\) 0 0
\(619\) 37.4485 1.50518 0.752591 0.658489i \(-0.228804\pi\)
0.752591 + 0.658489i \(0.228804\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 26.0528i 1.04378i
\(624\) 0 0
\(625\) 24.7794 + 3.31371i 0.991177 + 0.132548i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 24.2366 0.966375
\(630\) 0 0
\(631\) 10.7193 0.426728 0.213364 0.976973i \(-0.431558\pi\)
0.213364 + 0.976973i \(0.431558\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.38350 + 41.6127i −0.0549026 + 1.65135i
\(636\) 0 0
\(637\) 1.00000i 0.0396214i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 31.0183 1.22515 0.612574 0.790413i \(-0.290134\pi\)
0.612574 + 0.790413i \(0.290134\pi\)
\(642\) 0 0
\(643\) 12.4515i 0.491041i 0.969391 + 0.245520i \(0.0789589\pi\)
−0.969391 + 0.245520i \(0.921041\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 41.6538i 1.63758i −0.574093 0.818790i \(-0.694645\pi\)
0.574093 0.818790i \(-0.305355\pi\)
\(648\) 0 0
\(649\) −41.8566 −1.64301
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3.83095i 0.149917i 0.997187 + 0.0749584i \(0.0238824\pi\)
−0.997187 + 0.0749584i \(0.976118\pi\)
\(654\) 0 0
\(655\) −1.67894 0.0558199i −0.0656015 0.00218106i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −41.8926 −1.63191 −0.815953 0.578119i \(-0.803787\pi\)
−0.815953 + 0.578119i \(0.803787\pi\)
\(660\) 0 0
\(661\) 40.8523 1.58897 0.794485 0.607284i \(-0.207741\pi\)
0.794485 + 0.607284i \(0.207741\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −0.464480 + 13.9705i −0.0180118 + 0.541754i
\(666\) 0 0
\(667\) 34.5208i 1.33665i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 17.2365 0.665409
\(672\) 0 0
\(673\) 45.2401i 1.74388i −0.489615 0.871939i \(-0.662863\pi\)
0.489615 0.871939i \(-0.337137\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.58146i 0.137647i −0.997629 0.0688233i \(-0.978076\pi\)
0.997629 0.0688233i \(-0.0219245\pi\)
\(678\) 0 0
\(679\) 22.9411 0.880399
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 44.8402i 1.71576i −0.513848 0.857882i \(-0.671780\pi\)
0.513848 0.857882i \(-0.328220\pi\)
\(684\) 0 0
\(685\) 0.0808356 2.43136i 0.00308857 0.0928973i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1.70279 −0.0648712
\(690\) 0 0
\(691\) 18.2891 0.695750 0.347875 0.937541i \(-0.386903\pi\)
0.347875 + 0.937541i \(0.386903\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0.939333 + 0.0312301i 0.0356309 + 0.00118463i
\(696\) 0 0
\(697\) 0.519018i 0.0196592i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −36.0667 −1.36222 −0.681111 0.732180i \(-0.738503\pi\)
−0.681111 + 0.732180i \(0.738503\pi\)
\(702\) 0 0
\(703\) 15.3370i 0.578447i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 45.6569i 1.71710i
\(708\) 0 0
\(709\) 11.7769 0.442291 0.221146 0.975241i \(-0.429020\pi\)
0.221146 + 0.975241i \(0.429020\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 69.0416i 2.58563i
\(714\) 0 0
\(715\) −0.480708 + 14.4586i −0.0179775 + 0.540722i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −46.7261 −1.74259 −0.871295 0.490759i \(-0.836719\pi\)
−0.871295 + 0.490759i \(0.836719\pi\)
\(720\) 0 0
\(721\) −55.3847 −2.06264
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 24.0888 + 1.60354i 0.894636 + 0.0595540i
\(726\) 0 0
\(727\) 37.8038i 1.40207i −0.713129 0.701033i \(-0.752723\pi\)
0.713129 0.701033i \(-0.247277\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −10.6127 −0.392524
\(732\) 0 0
\(733\) 20.9411i 0.773477i −0.922189 0.386739i \(-0.873602\pi\)
0.922189 0.386739i \(-0.126398\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 49.8345i 1.83568i
\(738\) 0 0
\(739\) −31.5012 −1.15879 −0.579396 0.815046i \(-0.696711\pi\)
−0.579396 + 0.815046i \(0.696711\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.96483i 0.108769i −0.998520 0.0543845i \(-0.982680\pi\)
0.998520 0.0543845i \(-0.0173197\pi\)
\(744\) 0 0
\(745\) 31.5704 + 1.04963i 1.15665 + 0.0384553i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 3.84566 0.140517
\(750\) 0 0
\(751\) 10.0295 0.365980 0.182990 0.983115i \(-0.441422\pi\)
0.182990 + 0.983115i \(0.441422\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −13.5815 0.451545i −0.494280 0.0164334i
\(756\) 0 0
\(757\) 36.4731i 1.32564i 0.748780 + 0.662818i \(0.230640\pi\)
−0.748780 + 0.662818i \(0.769360\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 9.88085 0.358181 0.179090 0.983833i \(-0.442685\pi\)
0.179090 + 0.983833i \(0.442685\pi\)
\(762\) 0 0
\(763\) 9.87867i 0.357632i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.46967i 0.233606i
\(768\) 0 0
\(769\) −32.6274 −1.17657 −0.588287 0.808652i \(-0.700198\pi\)
−0.588287 + 0.808652i \(0.700198\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 44.2950i 1.59318i 0.604519 + 0.796591i \(0.293365\pi\)
−0.604519 + 0.796591i \(0.706635\pi\)
\(774\) 0 0
\(775\) −48.1776 3.20708i −1.73059 0.115202i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0.328437 0.0117675
\(780\) 0 0
\(781\) 71.7352 2.56689
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0.871311 26.2071i 0.0310984 0.935372i
\(786\) 0 0
\(787\) 24.0068i 0.855751i −0.903838 0.427875i \(-0.859262\pi\)
0.903838 0.427875i \(-0.140738\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 35.1629 1.25025
\(792\) 0 0
\(793\) 2.66421i 0.0946088i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 42.4271i 1.50285i 0.659821 + 0.751423i \(0.270632\pi\)
−0.659821 + 0.751423i \(0.729368\pi\)
\(798\) 0 0
\(799\) −23.7175 −0.839066
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 37.3155i 1.31683i
\(804\) 0 0
\(805\) −45.1924 1.50252i −1.59282 0.0529568i
\(806\) 0