Properties

Label 420.2.k.a.169.2
Level $420$
Weight $2$
Character 420.169
Analytic conductor $3.354$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 169.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 420.169
Dual form 420.2.k.a.169.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +(-2.00000 - 1.00000i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +(-2.00000 - 1.00000i) q^{5} +1.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} +6.00000i q^{13} +(1.00000 - 2.00000i) q^{15} +2.00000i q^{17} -6.00000 q^{19} -1.00000 q^{21} -2.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -1.00000i q^{27} -6.00000 q^{29} -2.00000 q^{31} -4.00000i q^{33} +(1.00000 - 2.00000i) q^{35} -4.00000i q^{37} -6.00000 q^{39} +8.00000 q^{41} +4.00000i q^{43} +(2.00000 + 1.00000i) q^{45} +4.00000i q^{47} -1.00000 q^{49} -2.00000 q^{51} -6.00000i q^{53} +(8.00000 + 4.00000i) q^{55} -6.00000i q^{57} -4.00000 q^{59} +14.0000 q^{61} -1.00000i q^{63} +(6.00000 - 12.0000i) q^{65} +4.00000i q^{67} +2.00000 q^{69} +10.0000i q^{73} +(-4.00000 + 3.00000i) q^{75} -4.00000i q^{77} +1.00000 q^{81} +16.0000i q^{83} +(2.00000 - 4.00000i) q^{85} -6.00000i q^{87} -8.00000 q^{89} -6.00000 q^{91} -2.00000i q^{93} +(12.0000 + 6.00000i) q^{95} +10.0000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{5} - 2 q^{9} - 8 q^{11} + 2 q^{15} - 12 q^{19} - 2 q^{21} + 6 q^{25} - 12 q^{29} - 4 q^{31} + 2 q^{35} - 12 q^{39} + 16 q^{41} + 4 q^{45} - 2 q^{49} - 4 q^{51} + 16 q^{55} - 8 q^{59} + 28 q^{61} + 12 q^{65} + 4 q^{69} - 8 q^{75} + 2 q^{81} + 4 q^{85} - 16 q^{89} - 12 q^{91} + 24 q^{95} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/420\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(241\) \(281\) \(337\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) −2.00000 1.00000i −0.894427 0.447214i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 1.00000 2.00000i 0.258199 0.516398i
\(16\) 0 0
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 2.00000i 0.417029i −0.978019 0.208514i \(-0.933137\pi\)
0.978019 0.208514i \(-0.0668628\pi\)
\(24\) 0 0
\(25\) 3.00000 + 4.00000i 0.600000 + 0.800000i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 4.00000i 0.696311i
\(34\) 0 0
\(35\) 1.00000 2.00000i 0.169031 0.338062i
\(36\) 0 0
\(37\) 4.00000i 0.657596i −0.944400 0.328798i \(-0.893356\pi\)
0.944400 0.328798i \(-0.106644\pi\)
\(38\) 0 0
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 2.00000 + 1.00000i 0.298142 + 0.149071i
\(46\) 0 0
\(47\) 4.00000i 0.583460i 0.956501 + 0.291730i \(0.0942309\pi\)
−0.956501 + 0.291730i \(0.905769\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 6.00000i 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 8.00000 + 4.00000i 1.07872 + 0.539360i
\(56\) 0 0
\(57\) 6.00000i 0.794719i
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) 6.00000 12.0000i 0.744208 1.48842i
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) 0 0
\(75\) −4.00000 + 3.00000i −0.461880 + 0.346410i
\(76\) 0 0
\(77\) 4.00000i 0.455842i
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 16.0000i 1.75623i 0.478451 + 0.878114i \(0.341198\pi\)
−0.478451 + 0.878114i \(0.658802\pi\)
\(84\) 0 0
\(85\) 2.00000 4.00000i 0.216930 0.433861i
\(86\) 0 0
\(87\) 6.00000i 0.643268i
\(88\) 0 0
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) 2.00000i 0.207390i
\(94\) 0 0
\(95\) 12.0000 + 6.00000i 1.23117 + 0.615587i
\(96\) 0 0
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 16.0000 1.59206 0.796030 0.605257i \(-0.206930\pi\)
0.796030 + 0.605257i \(0.206930\pi\)
\(102\) 0 0
\(103\) 16.0000i 1.57653i −0.615338 0.788263i \(-0.710980\pi\)
0.615338 0.788263i \(-0.289020\pi\)
\(104\) 0 0
\(105\) 2.00000 + 1.00000i 0.195180 + 0.0975900i
\(106\) 0 0
\(107\) 6.00000i 0.580042i −0.957020 0.290021i \(-0.906338\pi\)
0.957020 0.290021i \(-0.0936623\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) 18.0000i 1.69330i −0.532152 0.846649i \(-0.678617\pi\)
0.532152 0.846649i \(-0.321383\pi\)
\(114\) 0 0
\(115\) −2.00000 + 4.00000i −0.186501 + 0.373002i
\(116\) 0 0
\(117\) 6.00000i 0.554700i
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 8.00000i 0.721336i
\(124\) 0 0
\(125\) −2.00000 11.0000i −0.178885 0.983870i
\(126\) 0 0
\(127\) 16.0000i 1.41977i −0.704317 0.709885i \(-0.748747\pi\)
0.704317 0.709885i \(-0.251253\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 6.00000i 0.520266i
\(134\) 0 0
\(135\) −1.00000 + 2.00000i −0.0860663 + 0.172133i
\(136\) 0 0
\(137\) 18.0000i 1.53784i 0.639343 + 0.768922i \(0.279207\pi\)
−0.639343 + 0.768922i \(0.720793\pi\)
\(138\) 0 0
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 0 0
\(143\) 24.0000i 2.00698i
\(144\) 0 0
\(145\) 12.0000 + 6.00000i 0.996546 + 0.498273i
\(146\) 0 0
\(147\) 1.00000i 0.0824786i
\(148\) 0 0
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) 2.00000i 0.161690i
\(154\) 0 0
\(155\) 4.00000 + 2.00000i 0.321288 + 0.160644i
\(156\) 0 0
\(157\) 10.0000i 0.798087i 0.916932 + 0.399043i \(0.130658\pi\)
−0.916932 + 0.399043i \(0.869342\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) 0 0
\(163\) 8.00000i 0.626608i 0.949653 + 0.313304i \(0.101436\pi\)
−0.949653 + 0.313304i \(0.898564\pi\)
\(164\) 0 0
\(165\) −4.00000 + 8.00000i −0.311400 + 0.622799i
\(166\) 0 0
\(167\) 24.0000i 1.85718i 0.371113 + 0.928588i \(0.378976\pi\)
−0.371113 + 0.928588i \(0.621024\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) 0 0
\(173\) 2.00000i 0.152057i 0.997106 + 0.0760286i \(0.0242240\pi\)
−0.997106 + 0.0760286i \(0.975776\pi\)
\(174\) 0 0
\(175\) −4.00000 + 3.00000i −0.302372 + 0.226779i
\(176\) 0 0
\(177\) 4.00000i 0.300658i
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 14.0000i 1.03491i
\(184\) 0 0
\(185\) −4.00000 + 8.00000i −0.294086 + 0.588172i
\(186\) 0 0
\(187\) 8.00000i 0.585018i
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 8.00000i 0.575853i −0.957653 0.287926i \(-0.907034\pi\)
0.957653 0.287926i \(-0.0929658\pi\)
\(194\) 0 0
\(195\) 12.0000 + 6.00000i 0.859338 + 0.429669i
\(196\) 0 0
\(197\) 10.0000i 0.712470i 0.934396 + 0.356235i \(0.115940\pi\)
−0.934396 + 0.356235i \(0.884060\pi\)
\(198\) 0 0
\(199\) −18.0000 −1.27599 −0.637993 0.770042i \(-0.720235\pi\)
−0.637993 + 0.770042i \(0.720235\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) 6.00000i 0.421117i
\(204\) 0 0
\(205\) −16.0000 8.00000i −1.11749 0.558744i
\(206\) 0 0
\(207\) 2.00000i 0.139010i
\(208\) 0 0
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) −24.0000 −1.65223 −0.826114 0.563503i \(-0.809453\pi\)
−0.826114 + 0.563503i \(0.809453\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.00000 8.00000i 0.272798 0.545595i
\(216\) 0 0
\(217\) 2.00000i 0.135769i
\(218\) 0 0
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) 0 0
\(225\) −3.00000 4.00000i −0.200000 0.266667i
\(226\) 0 0
\(227\) 20.0000i 1.32745i 0.747978 + 0.663723i \(0.231025\pi\)
−0.747978 + 0.663723i \(0.768975\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 0 0
\(233\) 26.0000i 1.70332i 0.524097 + 0.851658i \(0.324403\pi\)
−0.524097 + 0.851658i \(0.675597\pi\)
\(234\) 0 0
\(235\) 4.00000 8.00000i 0.260931 0.521862i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 2.00000 + 1.00000i 0.127775 + 0.0638877i
\(246\) 0 0
\(247\) 36.0000i 2.29063i
\(248\) 0 0
\(249\) −16.0000 −1.01396
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 8.00000i 0.502956i
\(254\) 0 0
\(255\) 4.00000 + 2.00000i 0.250490 + 0.125245i
\(256\) 0 0
\(257\) 6.00000i 0.374270i 0.982334 + 0.187135i \(0.0599201\pi\)
−0.982334 + 0.187135i \(0.940080\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 10.0000i 0.616626i 0.951285 + 0.308313i \(0.0997645\pi\)
−0.951285 + 0.308313i \(0.900236\pi\)
\(264\) 0 0
\(265\) −6.00000 + 12.0000i −0.368577 + 0.737154i
\(266\) 0 0
\(267\) 8.00000i 0.489592i
\(268\) 0 0
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 0 0
\(271\) −14.0000 −0.850439 −0.425220 0.905090i \(-0.639803\pi\)
−0.425220 + 0.905090i \(0.639803\pi\)
\(272\) 0 0
\(273\) 6.00000i 0.363137i
\(274\) 0 0
\(275\) −12.0000 16.0000i −0.723627 0.964836i
\(276\) 0 0
\(277\) 28.0000i 1.68236i −0.540758 0.841178i \(-0.681862\pi\)
0.540758 0.841178i \(-0.318138\pi\)
\(278\) 0 0
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 28.0000i 1.66443i −0.554455 0.832214i \(-0.687073\pi\)
0.554455 0.832214i \(-0.312927\pi\)
\(284\) 0 0
\(285\) −6.00000 + 12.0000i −0.355409 + 0.710819i
\(286\) 0 0
\(287\) 8.00000i 0.472225i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 0 0
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 0 0
\(295\) 8.00000 + 4.00000i 0.465778 + 0.232889i
\(296\) 0 0
\(297\) 4.00000i 0.232104i
\(298\) 0 0
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) 16.0000i 0.919176i
\(304\) 0 0
\(305\) −28.0000 14.0000i −1.60328 0.801638i
\(306\) 0 0
\(307\) 12.0000i 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 14.0000i 0.791327i −0.918396 0.395663i \(-0.870515\pi\)
0.918396 0.395663i \(-0.129485\pi\)
\(314\) 0 0
\(315\) −1.00000 + 2.00000i −0.0563436 + 0.112687i
\(316\) 0 0
\(317\) 14.0000i 0.786318i 0.919470 + 0.393159i \(0.128618\pi\)
−0.919470 + 0.393159i \(0.871382\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) 12.0000i 0.667698i
\(324\) 0 0
\(325\) −24.0000 + 18.0000i −1.33128 + 0.998460i
\(326\) 0 0
\(327\) 14.0000i 0.774202i
\(328\) 0 0
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 0 0
\(333\) 4.00000i 0.219199i
\(334\) 0 0
\(335\) 4.00000 8.00000i 0.218543 0.437087i
\(336\) 0 0
\(337\) 8.00000i 0.435788i 0.975972 + 0.217894i \(0.0699187\pi\)
−0.975972 + 0.217894i \(0.930081\pi\)
\(338\) 0 0
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) −4.00000 2.00000i −0.215353 0.107676i
\(346\) 0 0
\(347\) 18.0000i 0.966291i 0.875540 + 0.483145i \(0.160506\pi\)
−0.875540 + 0.483145i \(0.839494\pi\)
\(348\) 0 0
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 0 0
\(353\) 2.00000i 0.106449i 0.998583 + 0.0532246i \(0.0169499\pi\)
−0.998583 + 0.0532246i \(0.983050\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 2.00000i 0.105851i
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 5.00000i 0.262432i
\(364\) 0 0
\(365\) 10.0000 20.0000i 0.523424 1.04685i
\(366\) 0 0
\(367\) 16.0000i 0.835193i −0.908633 0.417597i \(-0.862873\pi\)
0.908633 0.417597i \(-0.137127\pi\)
\(368\) 0 0
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) 24.0000i 1.24267i 0.783544 + 0.621336i \(0.213410\pi\)
−0.783544 + 0.621336i \(0.786590\pi\)
\(374\) 0 0
\(375\) 11.0000 2.00000i 0.568038 0.103280i
\(376\) 0 0
\(377\) 36.0000i 1.85409i
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) 0 0
\(383\) 28.0000i 1.43073i −0.698749 0.715367i \(-0.746260\pi\)
0.698749 0.715367i \(-0.253740\pi\)
\(384\) 0 0
\(385\) −4.00000 + 8.00000i −0.203859 + 0.407718i
\(386\) 0 0
\(387\) 4.00000i 0.203331i
\(388\) 0 0
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) 12.0000i 0.605320i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 10.0000i 0.501886i −0.968002 0.250943i \(-0.919259\pi\)
0.968002 0.250943i \(-0.0807406\pi\)
\(398\) 0 0
\(399\) 6.00000 0.300376
\(400\) 0 0
\(401\) 22.0000 1.09863 0.549314 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) 0 0
\(403\) 12.0000i 0.597763i
\(404\) 0 0
\(405\) −2.00000 1.00000i −0.0993808 0.0496904i
\(406\) 0 0
\(407\) 16.0000i 0.793091i
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) 0 0
\(413\) 4.00000i 0.196827i
\(414\) 0 0
\(415\) 16.0000 32.0000i 0.785409 1.57082i
\(416\) 0 0
\(417\) 10.0000i 0.489702i
\(418\) 0 0
\(419\) 40.0000 1.95413 0.977064 0.212946i \(-0.0683059\pi\)
0.977064 + 0.212946i \(0.0683059\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) 4.00000i 0.194487i
\(424\) 0 0
\(425\) −8.00000 + 6.00000i −0.388057 + 0.291043i
\(426\) 0 0
\(427\) 14.0000i 0.677507i
\(428\) 0 0
\(429\) 24.0000 1.15873
\(430\) 0 0
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 0 0
\(433\) 10.0000i 0.480569i 0.970702 + 0.240285i \(0.0772408\pi\)
−0.970702 + 0.240285i \(0.922759\pi\)
\(434\) 0 0
\(435\) −6.00000 + 12.0000i −0.287678 + 0.575356i
\(436\) 0 0
\(437\) 12.0000i 0.574038i
\(438\) 0 0
\(439\) 30.0000 1.43182 0.715911 0.698192i \(-0.246012\pi\)
0.715911 + 0.698192i \(0.246012\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 30.0000i 1.42534i 0.701498 + 0.712672i \(0.252515\pi\)
−0.701498 + 0.712672i \(0.747485\pi\)
\(444\) 0 0
\(445\) 16.0000 + 8.00000i 0.758473 + 0.379236i
\(446\) 0 0
\(447\) 22.0000i 1.04056i
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −32.0000 −1.50682
\(452\) 0 0
\(453\) 16.0000i 0.751746i
\(454\) 0 0
\(455\) 12.0000 + 6.00000i 0.562569 + 0.281284i
\(456\) 0 0
\(457\) 20.0000i 0.935561i 0.883845 + 0.467780i \(0.154946\pi\)
−0.883845 + 0.467780i \(0.845054\pi\)
\(458\) 0 0
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 24.0000i 1.11537i 0.830051 + 0.557687i \(0.188311\pi\)
−0.830051 + 0.557687i \(0.811689\pi\)
\(464\) 0 0
\(465\) −2.00000 + 4.00000i −0.0927478 + 0.185496i
\(466\) 0 0
\(467\) 12.0000i 0.555294i 0.960683 + 0.277647i \(0.0895545\pi\)
−0.960683 + 0.277647i \(0.910445\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 0 0
\(473\) 16.0000i 0.735681i
\(474\) 0 0
\(475\) −18.0000 24.0000i −0.825897 1.10120i
\(476\) 0 0
\(477\) 6.00000i 0.274721i
\(478\) 0 0
\(479\) 4.00000 0.182765 0.0913823 0.995816i \(-0.470871\pi\)
0.0913823 + 0.995816i \(0.470871\pi\)
\(480\) 0 0
\(481\) 24.0000 1.09431
\(482\) 0 0
\(483\) 2.00000i 0.0910032i
\(484\) 0 0
\(485\) 10.0000 20.0000i 0.454077 0.908153i
\(486\) 0 0
\(487\) 12.0000i 0.543772i −0.962329 0.271886i \(-0.912353\pi\)
0.962329 0.271886i \(-0.0876473\pi\)
\(488\) 0 0
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 12.0000i 0.540453i
\(494\) 0 0
\(495\) −8.00000 4.00000i −0.359573 0.179787i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 0 0
\(501\) −24.0000 −1.07224
\(502\) 0 0
\(503\) 36.0000i 1.60516i −0.596544 0.802580i \(-0.703460\pi\)
0.596544 0.802580i \(-0.296540\pi\)
\(504\) 0 0
\(505\) −32.0000 16.0000i −1.42398 0.711991i
\(506\) 0 0
\(507\) 23.0000i 1.02147i
\(508\) 0 0
\(509\) −12.0000 −0.531891 −0.265945 0.963988i \(-0.585684\pi\)
−0.265945 + 0.963988i \(0.585684\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) 0 0
\(513\) 6.00000i 0.264906i
\(514\) 0 0
\(515\) −16.0000 + 32.0000i −0.705044 + 1.41009i
\(516\) 0 0
\(517\) 16.0000i 0.703679i
\(518\) 0 0
\(519\) −2.00000 −0.0877903
\(520\) 0 0
\(521\) −44.0000 −1.92767 −0.963837 0.266491i \(-0.914136\pi\)
−0.963837 + 0.266491i \(0.914136\pi\)
\(522\) 0 0
\(523\) 20.0000i 0.874539i 0.899331 + 0.437269i \(0.144054\pi\)
−0.899331 + 0.437269i \(0.855946\pi\)
\(524\) 0 0
\(525\) −3.00000 4.00000i −0.130931 0.174574i
\(526\) 0 0
\(527\) 4.00000i 0.174243i
\(528\) 0 0
\(529\) 19.0000 0.826087
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 48.0000i 2.07911i
\(534\) 0 0
\(535\) −6.00000 + 12.0000i −0.259403 + 0.518805i
\(536\) 0 0
\(537\) 4.00000i 0.172613i
\(538\) 0 0
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 0 0
\(543\) 10.0000i 0.429141i
\(544\) 0 0
\(545\) −28.0000 14.0000i −1.19939 0.599694i
\(546\) 0 0
\(547\) 20.0000i 0.855138i 0.903983 + 0.427569i \(0.140630\pi\)
−0.903983 + 0.427569i \(0.859370\pi\)
\(548\) 0 0
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) 36.0000 1.53365
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −8.00000 4.00000i −0.339581 0.169791i
\(556\) 0 0
\(557\) 6.00000i 0.254228i 0.991888 + 0.127114i \(0.0405714\pi\)
−0.991888 + 0.127114i \(0.959429\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) 40.0000i 1.68580i −0.538071 0.842900i \(-0.680847\pi\)
0.538071 0.842900i \(-0.319153\pi\)
\(564\) 0 0
\(565\) −18.0000 + 36.0000i −0.757266 + 1.51453i
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) −8.00000 −0.334790 −0.167395 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 8.00000 6.00000i 0.333623 0.250217i
\(576\) 0 0
\(577\) 38.0000i 1.58196i 0.611842 + 0.790980i \(0.290429\pi\)
−0.611842 + 0.790980i \(0.709571\pi\)
\(578\) 0 0
\(579\) 8.00000 0.332469
\(580\) 0 0
\(581\) −16.0000 −0.663792
\(582\) 0 0
\(583\) 24.0000i 0.993978i
\(584\) 0 0
\(585\) −6.00000 + 12.0000i −0.248069 + 0.496139i
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) −10.0000 −0.411345
\(592\) 0 0
\(593\) 18.0000i 0.739171i −0.929197 0.369586i \(-0.879500\pi\)
0.929197 0.369586i \(-0.120500\pi\)
\(594\) 0 0
\(595\) 4.00000 + 2.00000i 0.163984 + 0.0819920i
\(596\) 0 0
\(597\) 18.0000i 0.736691i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −34.0000 −1.38689 −0.693444 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 0 0
\(605\) −10.0000 5.00000i −0.406558 0.203279i
\(606\) 0 0
\(607\) 40.0000i 1.62355i 0.583970 + 0.811775i \(0.301498\pi\)
−0.583970 + 0.811775i \(0.698502\pi\)
\(608\) 0 0
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) 44.0000i 1.77714i −0.458738 0.888572i \(-0.651698\pi\)
0.458738 0.888572i \(-0.348302\pi\)
\(614\) 0 0
\(615\) 8.00000 16.0000i 0.322591 0.645182i
\(616\) 0 0
\(617\) 34.0000i 1.36879i −0.729112 0.684394i \(-0.760067\pi\)
0.729112 0.684394i \(-0.239933\pi\)
\(618\) 0 0
\(619\) −26.0000 −1.04503 −0.522514 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) 0 0
\(621\) −2.00000 −0.0802572
\(622\) 0 0
\(623\) 8.00000i 0.320513i
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) 0 0
\(627\) 24.0000i 0.958468i
\(628\) 0 0
\(629\) 8.00000 0.318981
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) 24.0000i 0.953914i
\(634\) 0 0
\(635\) −16.0000 + 32.0000i −0.634941 + 1.26988i
\(636\) 0 0
\(637\) 6.00000i 0.237729i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) 12.0000i 0.473234i 0.971603 + 0.236617i \(0.0760386\pi\)
−0.971603 + 0.236617i \(0.923961\pi\)
\(644\) 0 0
\(645\) 8.00000 + 4.00000i 0.315000 + 0.157500i
\(646\) 0 0
\(647\) 24.0000i 0.943537i −0.881722 0.471769i \(-0.843616\pi\)
0.881722 0.471769i \(-0.156384\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 2.00000 0.0783862
\(652\) 0 0
\(653\) 34.0000i 1.33052i 0.746611 + 0.665261i \(0.231680\pi\)
−0.746611 + 0.665261i \(0.768320\pi\)
\(654\) 0 0
\(655\) 24.0000 + 12.0000i 0.937758 + 0.468879i
\(656\) 0 0
\(657\) 10.0000i 0.390137i
\(658\) 0 0
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 0 0
\(663\) 12.0000i 0.466041i
\(664\) 0 0
\(665\) −6.00000 + 12.0000i −0.232670 + 0.465340i
\(666\) 0 0
\(667\) 12.0000i 0.464642i
\(668\) 0 0
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) −56.0000 −2.16186
\(672\) 0 0
\(673\) 12.0000i 0.462566i −0.972887 0.231283i \(-0.925708\pi\)
0.972887 0.231283i \(-0.0742923\pi\)
\(674\) 0 0
\(675\) 4.00000 3.00000i 0.153960 0.115470i
\(676\) 0 0
\(677\) 38.0000i 1.46046i 0.683202 + 0.730229i \(0.260587\pi\)
−0.683202 + 0.730229i \(0.739413\pi\)
\(678\) 0 0
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) 26.0000i 0.994862i −0.867503 0.497431i \(-0.834277\pi\)
0.867503 0.497431i \(-0.165723\pi\)
\(684\) 0 0
\(685\) 18.0000 36.0000i 0.687745 1.37549i
\(686\) 0 0
\(687\) 2.00000i 0.0763048i
\(688\) 0 0
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) −2.00000 −0.0760836 −0.0380418 0.999276i \(-0.512112\pi\)
−0.0380418 + 0.999276i \(0.512112\pi\)
\(692\) 0 0
\(693\) 4.00000i 0.151947i
\(694\) 0 0
\(695\) 20.0000 + 10.0000i 0.758643 + 0.379322i
\(696\) 0 0
\(697\) 16.0000i 0.606043i
\(698\) 0 0
\(699\) −26.0000 −0.983410
\(700\) 0 0
\(701\) 38.0000 1.43524 0.717620 0.696435i \(-0.245231\pi\)
0.717620 + 0.696435i \(0.245231\pi\)
\(702\) 0 0
\(703\) 24.0000i 0.905177i
\(704\) 0 0
\(705\) 8.00000 + 4.00000i 0.301297 + 0.150649i
\(706\) 0 0
\(707\) 16.0000i 0.601742i
\(708\) 0 0
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4.00000i 0.149801i
\(714\) 0 0
\(715\) −24.0000 + 48.0000i −0.897549 + 1.79510i
\(716\) 0 0
\(717\) 16.0000i 0.597531i
\(718\) 0 0
\(719\) −44.0000 −1.64092 −0.820462 0.571702i \(-0.806283\pi\)
−0.820462 + 0.571702i \(0.806283\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) 10.0000i 0.371904i
\(724\) 0 0
\(725\) −18.0000 24.0000i −0.668503 0.891338i
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) 34.0000i 1.25582i −0.778287 0.627909i \(-0.783911\pi\)
0.778287 0.627909i \(-0.216089\pi\)
\(734\) 0 0
\(735\) −1.00000 + 2.00000i −0.0368856 + 0.0737711i
\(736\) 0 0
\(737\) 16.0000i 0.589368i
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 36.0000 1.32249
\(742\) 0 0
\(743\) 54.0000i 1.98107i 0.137268 + 0.990534i \(0.456168\pi\)
−0.137268 + 0.990534i \(0.543832\pi\)
\(744\) 0 0
\(745\) 44.0000 + 22.0000i 1.61204 + 0.806018i
\(746\) 0 0
\(747\) 16.0000i 0.585409i
\(748\) 0 0
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 0 0
\(753\) 12.0000i 0.437304i
\(754\) 0 0
\(755\) −32.0000 16.0000i −1.16460 0.582300i
\(756\) 0 0
\(757\) 8.00000i 0.290765i 0.989376 + 0.145382i \(0.0464413\pi\)
−0.989376 + 0.145382i \(0.953559\pi\)
\(758\) 0 0
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) −24.0000 −0.869999 −0.435000 0.900431i \(-0.643252\pi\)
−0.435000 + 0.900431i \(0.643252\pi\)
\(762\) 0 0
\(763\) 14.0000i 0.506834i
\(764\) 0 0
\(765\) −2.00000 + 4.00000i −0.0723102 + 0.144620i
\(766\) 0 0
\(767\) 24.0000i 0.866590i
\(768\) 0 0
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 0 0
\(773\) 6.00000i 0.215805i 0.994161 + 0.107903i \(0.0344134\pi\)
−0.994161 + 0.107903i \(0.965587\pi\)
\(774\) 0 0
\(775\) −6.00000 8.00000i −0.215526 0.287368i
\(776\) 0 0
\(777\) 4.00000i 0.143499i
\(778\) 0 0
\(779\) −48.0000 −1.71978
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000i 0.214423i
\(784\) 0 0
\(785\) 10.0000 20.0000i 0.356915 0.713831i
\(786\) 0 0
\(787\) 4.00000i 0.142585i −0.997455 0.0712923i \(-0.977288\pi\)
0.997455 0.0712923i \(-0.0227123\pi\)
\(788\) 0 0
\(789\) −10.0000 −0.356009
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) 84.0000i 2.98293i
\(794\) 0 0
\(795\) −12.0000 6.00000i −0.425596 0.212798i
\(796\) 0 0
\(797\) 34.0000i 1.20434i −0.798367 0.602171i \(-0.794303\pi\)
0.798367 0.602171i \(-0.205697\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 0