Properties

Label 5520.2.be.a.1471.13
Level $5520$
Weight $2$
Character 5520.1471
Analytic conductor $44.077$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.be (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 4 x^{15} + 8 x^{14} + 28 x^{13} + 373 x^{12} - 920 x^{11} + 1088 x^{10} - 168 x^{9} + 16460 x^{8} - 45408 x^{7} + 62624 x^{6} - 18048 x^{5} + 2160 x^{4} - 1664 x^{3} + 6272 x^{2} - 896 x + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.13
Root \(-2.02116 - 2.02116i\) of defining polynomial
Character \(\chi\) \(=\) 5520.1471
Dual form 5520.2.be.a.1471.5

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} -1.00000i q^{5} +0.279423 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} -1.00000i q^{5} +0.279423 q^{7} -1.00000 q^{9} +0.628840 q^{11} -3.03226 q^{13} +1.00000 q^{15} -2.36579i q^{17} -1.48654 q^{19} +0.279423i q^{21} +(-1.08636 - 4.67117i) q^{23} -1.00000 q^{25} -1.00000i q^{27} +3.72600 q^{29} +8.99698i q^{31} +0.628840i q^{33} -0.279423i q^{35} +5.99375i q^{37} -3.03226i q^{39} -7.39498 q^{41} +8.69030 q^{43} +1.00000i q^{45} +2.41437i q^{47} -6.92192 q^{49} +2.36579 q^{51} -1.22938i q^{53} -0.628840i q^{55} -1.48654i q^{57} +7.12332i q^{59} +3.34390i q^{61} -0.279423 q^{63} +3.03226i q^{65} -8.37910 q^{67} +(4.67117 - 1.08636i) q^{69} -3.78611i q^{71} +15.1169 q^{73} -1.00000i q^{75} +0.175712 q^{77} +16.5375 q^{79} +1.00000 q^{81} +7.07387 q^{83} -2.36579 q^{85} +3.72600i q^{87} +12.8162i q^{89} -0.847281 q^{91} -8.99698 q^{93} +1.48654i q^{95} +12.7589i q^{97} -0.628840 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 8q^{7} - 16q^{9} + O(q^{10}) \) \( 16q - 8q^{7} - 16q^{9} + 8q^{11} + 8q^{13} + 16q^{15} + 12q^{23} - 16q^{25} - 4q^{29} + 4q^{41} + 20q^{49} - 4q^{51} + 8q^{63} - 16q^{67} + 40q^{73} + 24q^{77} + 32q^{79} + 16q^{81} + 4q^{85} - 48q^{91} - 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(1381\) \(1841\) \(4417\) \(4831\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(-1\)

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
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 0.279423 0.105612 0.0528059 0.998605i \(-0.483184\pi\)
0.0528059 + 0.998605i \(0.483184\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0.628840 0.189602 0.0948012 0.995496i \(-0.469778\pi\)
0.0948012 + 0.995496i \(0.469778\pi\)
\(12\) 0 0
\(13\) −3.03226 −0.840997 −0.420499 0.907293i \(-0.638145\pi\)
−0.420499 + 0.907293i \(0.638145\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 2.36579i 0.573787i −0.957962 0.286894i \(-0.907377\pi\)
0.957962 0.286894i \(-0.0926227\pi\)
\(18\) 0 0
\(19\) −1.48654 −0.341036 −0.170518 0.985355i \(-0.554544\pi\)
−0.170518 + 0.985355i \(0.554544\pi\)
\(20\) 0 0
\(21\) 0.279423i 0.0609750i
\(22\) 0 0
\(23\) −1.08636 4.67117i −0.226522 0.974006i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 3.72600 0.691900 0.345950 0.938253i \(-0.387557\pi\)
0.345950 + 0.938253i \(0.387557\pi\)
\(30\) 0 0
\(31\) 8.99698i 1.61591i 0.589247 + 0.807953i \(0.299425\pi\)
−0.589247 + 0.807953i \(0.700575\pi\)
\(32\) 0 0
\(33\) 0.628840i 0.109467i
\(34\) 0 0
\(35\) 0.279423i 0.0472310i
\(36\) 0 0
\(37\) 5.99375i 0.985366i 0.870209 + 0.492683i \(0.163984\pi\)
−0.870209 + 0.492683i \(0.836016\pi\)
\(38\) 0 0
\(39\) 3.03226i 0.485550i
\(40\) 0 0
\(41\) −7.39498 −1.15490 −0.577451 0.816425i \(-0.695953\pi\)
−0.577451 + 0.816425i \(0.695953\pi\)
\(42\) 0 0
\(43\) 8.69030 1.32526 0.662629 0.748948i \(-0.269441\pi\)
0.662629 + 0.748948i \(0.269441\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 2.41437i 0.352172i 0.984375 + 0.176086i \(0.0563437\pi\)
−0.984375 + 0.176086i \(0.943656\pi\)
\(48\) 0 0
\(49\) −6.92192 −0.988846
\(50\) 0 0
\(51\) 2.36579 0.331276
\(52\) 0 0
\(53\) 1.22938i 0.168869i −0.996429 0.0844343i \(-0.973092\pi\)
0.996429 0.0844343i \(-0.0269083\pi\)
\(54\) 0 0
\(55\) 0.628840i 0.0847927i
\(56\) 0 0
\(57\) 1.48654i 0.196897i
\(58\) 0 0
\(59\) 7.12332i 0.927377i 0.885998 + 0.463689i \(0.153474\pi\)
−0.885998 + 0.463689i \(0.846526\pi\)
\(60\) 0 0
\(61\) 3.34390i 0.428142i 0.976818 + 0.214071i \(0.0686724\pi\)
−0.976818 + 0.214071i \(0.931328\pi\)
\(62\) 0 0
\(63\) −0.279423 −0.0352039
\(64\) 0 0
\(65\) 3.03226i 0.376105i
\(66\) 0 0
\(67\) −8.37910 −1.02367 −0.511835 0.859084i \(-0.671034\pi\)
−0.511835 + 0.859084i \(0.671034\pi\)
\(68\) 0 0
\(69\) 4.67117 1.08636i 0.562343 0.130783i
\(70\) 0 0
\(71\) 3.78611i 0.449328i −0.974436 0.224664i \(-0.927872\pi\)
0.974436 0.224664i \(-0.0721285\pi\)
\(72\) 0 0
\(73\) 15.1169 1.76930 0.884651 0.466255i \(-0.154397\pi\)
0.884651 + 0.466255i \(0.154397\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) 0.175712 0.0200242
\(78\) 0 0
\(79\) 16.5375 1.86062 0.930309 0.366778i \(-0.119539\pi\)
0.930309 + 0.366778i \(0.119539\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.07387 0.776458 0.388229 0.921563i \(-0.373087\pi\)
0.388229 + 0.921563i \(0.373087\pi\)
\(84\) 0 0
\(85\) −2.36579 −0.256606
\(86\) 0 0
\(87\) 3.72600i 0.399469i
\(88\) 0 0
\(89\) 12.8162i 1.35852i 0.733899 + 0.679259i \(0.237699\pi\)
−0.733899 + 0.679259i \(0.762301\pi\)
\(90\) 0 0
\(91\) −0.847281 −0.0888192
\(92\) 0 0
\(93\) −8.99698 −0.932944
\(94\) 0 0
\(95\) 1.48654i 0.152516i
\(96\) 0 0
\(97\) 12.7589i 1.29547i 0.761866 + 0.647734i \(0.224283\pi\)
−0.761866 + 0.647734i \(0.775717\pi\)
\(98\) 0 0
\(99\) −0.628840 −0.0632008
\(100\) 0 0
\(101\) −11.6258 −1.15681 −0.578403 0.815751i \(-0.696324\pi\)
−0.578403 + 0.815751i \(0.696324\pi\)
\(102\) 0 0
\(103\) −10.8093 −1.06508 −0.532538 0.846406i \(-0.678762\pi\)
−0.532538 + 0.846406i \(0.678762\pi\)
\(104\) 0 0
\(105\) 0.279423 0.0272689
\(106\) 0 0
\(107\) 15.1779 1.46730 0.733651 0.679526i \(-0.237815\pi\)
0.733651 + 0.679526i \(0.237815\pi\)
\(108\) 0 0
\(109\) 8.74076i 0.837213i 0.908168 + 0.418607i \(0.137481\pi\)
−0.908168 + 0.418607i \(0.862519\pi\)
\(110\) 0 0
\(111\) −5.99375 −0.568901
\(112\) 0 0
\(113\) 19.5279i 1.83703i 0.395382 + 0.918517i \(0.370612\pi\)
−0.395382 + 0.918517i \(0.629388\pi\)
\(114\) 0 0
\(115\) −4.67117 + 1.08636i −0.435589 + 0.101304i
\(116\) 0 0
\(117\) 3.03226 0.280332
\(118\) 0 0
\(119\) 0.661054i 0.0605987i
\(120\) 0 0
\(121\) −10.6046 −0.964051
\(122\) 0 0
\(123\) 7.39498i 0.666783i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 3.45069i 0.306199i −0.988211 0.153100i \(-0.951074\pi\)
0.988211 0.153100i \(-0.0489256\pi\)
\(128\) 0 0
\(129\) 8.69030i 0.765138i
\(130\) 0 0
\(131\) 11.6225i 1.01546i −0.861516 0.507730i \(-0.830485\pi\)
0.861516 0.507730i \(-0.169515\pi\)
\(132\) 0 0
\(133\) −0.415374 −0.0360175
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 16.9576i 1.44879i 0.689385 + 0.724395i \(0.257881\pi\)
−0.689385 + 0.724395i \(0.742119\pi\)
\(138\) 0 0
\(139\) 5.39812i 0.457862i −0.973443 0.228931i \(-0.926477\pi\)
0.973443 0.228931i \(-0.0735231\pi\)
\(140\) 0 0
\(141\) −2.41437 −0.203327
\(142\) 0 0
\(143\) −1.90680 −0.159455
\(144\) 0 0
\(145\) 3.72600i 0.309427i
\(146\) 0 0
\(147\) 6.92192i 0.570911i
\(148\) 0 0
\(149\) 2.66008i 0.217922i −0.994046 0.108961i \(-0.965248\pi\)
0.994046 0.108961i \(-0.0347523\pi\)
\(150\) 0 0
\(151\) 8.87377i 0.722137i −0.932539 0.361069i \(-0.882412\pi\)
0.932539 0.361069i \(-0.117588\pi\)
\(152\) 0 0
\(153\) 2.36579i 0.191262i
\(154\) 0 0
\(155\) 8.99698 0.722655
\(156\) 0 0
\(157\) 24.5886i 1.96239i −0.193031 0.981193i \(-0.561832\pi\)
0.193031 0.981193i \(-0.438168\pi\)
\(158\) 0 0
\(159\) 1.22938 0.0974963
\(160\) 0 0
\(161\) −0.303555 1.30523i −0.0239234 0.102867i
\(162\) 0 0
\(163\) 19.3751i 1.51757i 0.651339 + 0.758786i \(0.274207\pi\)
−0.651339 + 0.758786i \(0.725793\pi\)
\(164\) 0 0
\(165\) 0.628840 0.0489551
\(166\) 0 0
\(167\) 14.5746i 1.12782i 0.825838 + 0.563908i \(0.190703\pi\)
−0.825838 + 0.563908i \(0.809297\pi\)
\(168\) 0 0
\(169\) −3.80541 −0.292724
\(170\) 0 0
\(171\) 1.48654 0.113679
\(172\) 0 0
\(173\) 12.2951 0.934776 0.467388 0.884052i \(-0.345195\pi\)
0.467388 + 0.884052i \(0.345195\pi\)
\(174\) 0 0
\(175\) −0.279423 −0.0211224
\(176\) 0 0
\(177\) −7.12332 −0.535421
\(178\) 0 0
\(179\) 10.0284i 0.749557i 0.927114 + 0.374779i \(0.122281\pi\)
−0.927114 + 0.374779i \(0.877719\pi\)
\(180\) 0 0
\(181\) 12.4338i 0.924195i 0.886829 + 0.462097i \(0.152903\pi\)
−0.886829 + 0.462097i \(0.847097\pi\)
\(182\) 0 0
\(183\) −3.34390 −0.247188
\(184\) 0 0
\(185\) 5.99375 0.440669
\(186\) 0 0
\(187\) 1.48770i 0.108791i
\(188\) 0 0
\(189\) 0.279423i 0.0203250i
\(190\) 0 0
\(191\) −0.607809 −0.0439795 −0.0219898 0.999758i \(-0.507000\pi\)
−0.0219898 + 0.999758i \(0.507000\pi\)
\(192\) 0 0
\(193\) 6.47078 0.465777 0.232888 0.972503i \(-0.425182\pi\)
0.232888 + 0.972503i \(0.425182\pi\)
\(194\) 0 0
\(195\) −3.03226 −0.217145
\(196\) 0 0
\(197\) −14.5268 −1.03499 −0.517495 0.855686i \(-0.673135\pi\)
−0.517495 + 0.855686i \(0.673135\pi\)
\(198\) 0 0
\(199\) −23.1375 −1.64018 −0.820089 0.572237i \(-0.806076\pi\)
−0.820089 + 0.572237i \(0.806076\pi\)
\(200\) 0 0
\(201\) 8.37910i 0.591016i
\(202\) 0 0
\(203\) 1.04113 0.0730728
\(204\) 0 0
\(205\) 7.39498i 0.516488i
\(206\) 0 0
\(207\) 1.08636 + 4.67117i 0.0755075 + 0.324669i
\(208\) 0 0
\(209\) −0.934798 −0.0646613
\(210\) 0 0
\(211\) 14.4669i 0.995943i −0.867193 0.497971i \(-0.834078\pi\)
0.867193 0.497971i \(-0.165922\pi\)
\(212\) 0 0
\(213\) 3.78611 0.259420
\(214\) 0 0
\(215\) 8.69030i 0.592673i
\(216\) 0 0
\(217\) 2.51396i 0.170659i
\(218\) 0 0
\(219\) 15.1169i 1.02151i
\(220\) 0 0
\(221\) 7.17367i 0.482554i
\(222\) 0 0
\(223\) 22.1810i 1.48535i 0.669654 + 0.742673i \(0.266442\pi\)
−0.669654 + 0.742673i \(0.733558\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −0.753365 −0.0500026 −0.0250013 0.999687i \(-0.507959\pi\)
−0.0250013 + 0.999687i \(0.507959\pi\)
\(228\) 0 0
\(229\) 8.34911i 0.551725i 0.961197 + 0.275862i \(0.0889634\pi\)
−0.961197 + 0.275862i \(0.911037\pi\)
\(230\) 0 0
\(231\) 0.175712i 0.0115610i
\(232\) 0 0
\(233\) 23.4423 1.53575 0.767877 0.640597i \(-0.221313\pi\)
0.767877 + 0.640597i \(0.221313\pi\)
\(234\) 0 0
\(235\) 2.41437 0.157496
\(236\) 0 0
\(237\) 16.5375i 1.07423i
\(238\) 0 0
\(239\) 13.7015i 0.886279i −0.896453 0.443139i \(-0.853865\pi\)
0.896453 0.443139i \(-0.146135\pi\)
\(240\) 0 0
\(241\) 22.8899i 1.47447i 0.675636 + 0.737235i \(0.263869\pi\)
−0.675636 + 0.737235i \(0.736131\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 6.92192i 0.442225i
\(246\) 0 0
\(247\) 4.50758 0.286811
\(248\) 0 0
\(249\) 7.07387i 0.448288i
\(250\) 0 0
\(251\) 31.1381 1.96542 0.982711 0.185148i \(-0.0592764\pi\)
0.982711 + 0.185148i \(0.0592764\pi\)
\(252\) 0 0
\(253\) −0.683149 2.93742i −0.0429492 0.184674i
\(254\) 0 0
\(255\) 2.36579i 0.148151i
\(256\) 0 0
\(257\) −14.1422 −0.882166 −0.441083 0.897466i \(-0.645406\pi\)
−0.441083 + 0.897466i \(0.645406\pi\)
\(258\) 0 0
\(259\) 1.67479i 0.104066i
\(260\) 0 0
\(261\) −3.72600 −0.230633
\(262\) 0 0
\(263\) 26.4000 1.62789 0.813947 0.580939i \(-0.197314\pi\)
0.813947 + 0.580939i \(0.197314\pi\)
\(264\) 0 0
\(265\) −1.22938 −0.0755203
\(266\) 0 0
\(267\) −12.8162 −0.784340
\(268\) 0 0
\(269\) 9.45718 0.576615 0.288307 0.957538i \(-0.406908\pi\)
0.288307 + 0.957538i \(0.406908\pi\)
\(270\) 0 0
\(271\) 24.3332i 1.47814i −0.673629 0.739070i \(-0.735265\pi\)
0.673629 0.739070i \(-0.264735\pi\)
\(272\) 0 0
\(273\) 0.847281i 0.0512798i
\(274\) 0 0
\(275\) −0.628840 −0.0379205
\(276\) 0 0
\(277\) −9.92017 −0.596045 −0.298023 0.954559i \(-0.596327\pi\)
−0.298023 + 0.954559i \(0.596327\pi\)
\(278\) 0 0
\(279\) 8.99698i 0.538635i
\(280\) 0 0
\(281\) 21.7780i 1.29916i −0.760292 0.649582i \(-0.774944\pi\)
0.760292 0.649582i \(-0.225056\pi\)
\(282\) 0 0
\(283\) −22.9722 −1.36556 −0.682779 0.730625i \(-0.739229\pi\)
−0.682779 + 0.730625i \(0.739229\pi\)
\(284\) 0 0
\(285\) −1.48654 −0.0880552
\(286\) 0 0
\(287\) −2.06633 −0.121971
\(288\) 0 0
\(289\) 11.4031 0.670768
\(290\) 0 0
\(291\) −12.7589 −0.747939
\(292\) 0 0
\(293\) 15.7683i 0.921193i 0.887610 + 0.460596i \(0.152364\pi\)
−0.887610 + 0.460596i \(0.847636\pi\)
\(294\) 0 0
\(295\) 7.12332 0.414736
\(296\) 0 0
\(297\) 0.628840i 0.0364890i
\(298\) 0 0
\(299\) 3.29413 + 14.1642i 0.190505 + 0.819136i
\(300\) 0 0
\(301\) 2.42827 0.139963
\(302\) 0 0
\(303\) 11.6258i 0.667882i
\(304\) 0 0
\(305\) 3.34390 0.191471
\(306\) 0 0
\(307\) 14.9802i 0.854963i 0.904024 + 0.427481i \(0.140599\pi\)
−0.904024 + 0.427481i \(0.859401\pi\)
\(308\) 0 0
\(309\) 10.8093i 0.614922i
\(310\) 0 0
\(311\) 6.28014i 0.356114i 0.984020 + 0.178057i \(0.0569812\pi\)
−0.984020 + 0.178057i \(0.943019\pi\)
\(312\) 0 0
\(313\) 21.0909i 1.19213i 0.802937 + 0.596064i \(0.203270\pi\)
−0.802937 + 0.596064i \(0.796730\pi\)
\(314\) 0 0
\(315\) 0.279423i 0.0157437i
\(316\) 0 0
\(317\) 11.4813 0.644852 0.322426 0.946595i \(-0.395502\pi\)
0.322426 + 0.946595i \(0.395502\pi\)
\(318\) 0 0
\(319\) 2.34305 0.131186
\(320\) 0 0
\(321\) 15.1779i 0.847148i
\(322\) 0 0
\(323\) 3.51684i 0.195682i
\(324\) 0 0
\(325\) 3.03226 0.168199
\(326\) 0 0
\(327\) −8.74076 −0.483365
\(328\) 0 0
\(329\) 0.674629i 0.0371935i
\(330\) 0 0
\(331\) 5.43242i 0.298593i −0.988792 0.149297i \(-0.952299\pi\)
0.988792 0.149297i \(-0.0477009\pi\)
\(332\) 0 0
\(333\) 5.99375i 0.328455i
\(334\) 0 0
\(335\) 8.37910i 0.457799i
\(336\) 0 0
\(337\) 13.9809i 0.761588i 0.924660 + 0.380794i \(0.124349\pi\)
−0.924660 + 0.380794i \(0.875651\pi\)
\(338\) 0 0
\(339\) −19.5279 −1.06061
\(340\) 0 0
\(341\) 5.65766i 0.306380i
\(342\) 0 0
\(343\) −3.89010 −0.210046
\(344\) 0 0
\(345\) −1.08636 4.67117i −0.0584878 0.251487i
\(346\) 0 0
\(347\) 10.9072i 0.585530i 0.956184 + 0.292765i \(0.0945754\pi\)
−0.956184 + 0.292765i \(0.905425\pi\)
\(348\) 0 0
\(349\) −22.9947 −1.23088 −0.615440 0.788184i \(-0.711022\pi\)
−0.615440 + 0.788184i \(0.711022\pi\)
\(350\) 0 0
\(351\) 3.03226i 0.161850i
\(352\) 0 0
\(353\) −18.1073 −0.963752 −0.481876 0.876239i \(-0.660044\pi\)
−0.481876 + 0.876239i \(0.660044\pi\)
\(354\) 0 0
\(355\) −3.78611 −0.200946
\(356\) 0 0
\(357\) 0.661054 0.0349867
\(358\) 0 0
\(359\) 10.5753 0.558141 0.279071 0.960271i \(-0.409974\pi\)
0.279071 + 0.960271i \(0.409974\pi\)
\(360\) 0 0
\(361\) −16.7902 −0.883694
\(362\) 0 0
\(363\) 10.6046i 0.556595i
\(364\) 0 0
\(365\) 15.1169i 0.791256i
\(366\) 0 0
\(367\) −29.3289 −1.53096 −0.765479 0.643461i \(-0.777498\pi\)
−0.765479 + 0.643461i \(0.777498\pi\)
\(368\) 0 0
\(369\) 7.39498 0.384968
\(370\) 0 0
\(371\) 0.343517i 0.0178345i
\(372\) 0 0
\(373\) 28.1275i 1.45639i 0.685373 + 0.728193i \(0.259639\pi\)
−0.685373 + 0.728193i \(0.740361\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −11.2982 −0.581886
\(378\) 0 0
\(379\) 0.554605 0.0284881 0.0142441 0.999899i \(-0.495466\pi\)
0.0142441 + 0.999899i \(0.495466\pi\)
\(380\) 0 0
\(381\) 3.45069 0.176784
\(382\) 0 0
\(383\) −30.8384 −1.57577 −0.787883 0.615824i \(-0.788823\pi\)
−0.787883 + 0.615824i \(0.788823\pi\)
\(384\) 0 0
\(385\) 0.175712i 0.00895512i
\(386\) 0 0
\(387\) −8.69030 −0.441753
\(388\) 0 0
\(389\) 5.89847i 0.299065i −0.988757 0.149532i \(-0.952223\pi\)
0.988757 0.149532i \(-0.0477768\pi\)
\(390\) 0 0
\(391\) −11.0510 + 2.57010i −0.558872 + 0.129976i
\(392\) 0 0
\(393\) 11.6225 0.586277
\(394\) 0 0
\(395\) 16.5375i 0.832093i
\(396\) 0 0
\(397\) −4.49346 −0.225520 −0.112760 0.993622i \(-0.535969\pi\)
−0.112760 + 0.993622i \(0.535969\pi\)
\(398\) 0 0
\(399\) 0.415374i 0.0207947i
\(400\) 0 0
\(401\) 34.9973i 1.74768i 0.486211 + 0.873842i \(0.338379\pi\)
−0.486211 + 0.873842i \(0.661621\pi\)
\(402\) 0 0
\(403\) 27.2812i 1.35897i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 3.76911i 0.186828i
\(408\) 0 0
\(409\) 30.8491 1.52539 0.762696 0.646757i \(-0.223875\pi\)
0.762696 + 0.646757i \(0.223875\pi\)
\(410\) 0 0
\(411\) −16.9576 −0.836459
\(412\) 0 0
\(413\) 1.99042i 0.0979420i
\(414\) 0 0
\(415\) 7.07387i 0.347242i
\(416\) 0 0
\(417\) 5.39812 0.264347
\(418\) 0 0
\(419\) 21.6640 1.05835 0.529177 0.848511i \(-0.322501\pi\)
0.529177 + 0.848511i \(0.322501\pi\)
\(420\) 0 0
\(421\) 32.5964i 1.58865i −0.607494 0.794325i \(-0.707825\pi\)
0.607494 0.794325i \(-0.292175\pi\)
\(422\) 0 0
\(423\) 2.41437i 0.117391i
\(424\) 0 0
\(425\) 2.36579i 0.114757i
\(426\) 0 0
\(427\) 0.934360i 0.0452168i
\(428\) 0 0
\(429\) 1.90680i 0.0920614i
\(430\) 0 0
\(431\) 3.64945 0.175788 0.0878938 0.996130i \(-0.471986\pi\)
0.0878938 + 0.996130i \(0.471986\pi\)
\(432\) 0 0
\(433\) 23.4352i 1.12622i 0.826381 + 0.563111i \(0.190396\pi\)
−0.826381 + 0.563111i \(0.809604\pi\)
\(434\) 0 0
\(435\) 3.72600 0.178648
\(436\) 0 0
\(437\) 1.61493 + 6.94389i 0.0772524 + 0.332171i
\(438\) 0 0
\(439\) 7.38913i 0.352664i −0.984331 0.176332i \(-0.943577\pi\)
0.984331 0.176332i \(-0.0564232\pi\)
\(440\) 0 0
\(441\) 6.92192 0.329615
\(442\) 0 0
\(443\) 25.5242i 1.21269i −0.795202 0.606345i \(-0.792635\pi\)
0.795202 0.606345i \(-0.207365\pi\)
\(444\) 0 0
\(445\) 12.8162 0.607548
\(446\) 0 0
\(447\) 2.66008 0.125817
\(448\) 0 0
\(449\) −36.1882 −1.70783 −0.853914 0.520414i \(-0.825778\pi\)
−0.853914 + 0.520414i \(0.825778\pi\)
\(450\) 0 0
\(451\) −4.65026 −0.218972
\(452\) 0 0
\(453\) 8.87377 0.416926
\(454\) 0 0
\(455\) 0.847281i 0.0397212i
\(456\) 0 0
\(457\) 6.57098i 0.307378i −0.988119 0.153689i \(-0.950885\pi\)
0.988119 0.153689i \(-0.0491153\pi\)
\(458\) 0 0
\(459\) −2.36579 −0.110425
\(460\) 0 0
\(461\) −20.5992 −0.959401 −0.479701 0.877432i \(-0.659255\pi\)
−0.479701 + 0.877432i \(0.659255\pi\)
\(462\) 0 0
\(463\) 18.3552i 0.853040i 0.904478 + 0.426520i \(0.140261\pi\)
−0.904478 + 0.426520i \(0.859739\pi\)
\(464\) 0 0
\(465\) 8.99698i 0.417225i
\(466\) 0 0
\(467\) 28.2461 1.30707 0.653536 0.756895i \(-0.273285\pi\)
0.653536 + 0.756895i \(0.273285\pi\)
\(468\) 0 0
\(469\) −2.34131 −0.108112
\(470\) 0 0
\(471\) 24.5886 1.13298
\(472\) 0 0
\(473\) 5.46480 0.251272
\(474\) 0 0
\(475\) 1.48654 0.0682073
\(476\) 0 0
\(477\) 1.22938i 0.0562895i
\(478\) 0 0
\(479\) −22.7535 −1.03963 −0.519817 0.854277i \(-0.674000\pi\)
−0.519817 + 0.854277i \(0.674000\pi\)
\(480\) 0 0
\(481\) 18.1746i 0.828690i
\(482\) 0 0
\(483\) 1.30523 0.303555i 0.0593900 0.0138122i
\(484\) 0 0
\(485\) 12.7589 0.579351
\(486\) 0 0
\(487\) 30.0261i 1.36061i 0.732928 + 0.680306i \(0.238153\pi\)
−0.732928 + 0.680306i \(0.761847\pi\)
\(488\) 0 0
\(489\) −19.3751 −0.876171
\(490\) 0 0
\(491\) 19.2126i 0.867055i 0.901140 + 0.433527i \(0.142731\pi\)
−0.901140 + 0.433527i \(0.857269\pi\)
\(492\) 0 0
\(493\) 8.81491i 0.397003i
\(494\) 0 0
\(495\) 0.628840i 0.0282642i
\(496\) 0 0
\(497\) 1.05792i 0.0474544i
\(498\) 0 0
\(499\) 9.24353i 0.413797i −0.978362 0.206899i \(-0.933663\pi\)
0.978362 0.206899i \(-0.0663370\pi\)
\(500\) 0 0
\(501\) −14.5746 −0.651145
\(502\) 0 0
\(503\) 2.15661 0.0961583 0.0480791 0.998844i \(-0.484690\pi\)
0.0480791 + 0.998844i \(0.484690\pi\)
\(504\) 0 0
\(505\) 11.6258i 0.517339i
\(506\) 0 0
\(507\) 3.80541i 0.169004i
\(508\) 0 0
\(509\) −26.3925 −1.16983 −0.584913 0.811096i \(-0.698871\pi\)
−0.584913 + 0.811096i \(0.698871\pi\)
\(510\) 0 0
\(511\) 4.22401 0.186859
\(512\) 0 0
\(513\) 1.48654i 0.0656325i
\(514\) 0 0
\(515\) 10.8093i 0.476317i
\(516\) 0 0
\(517\) 1.51825i 0.0667726i
\(518\) 0 0
\(519\) 12.2951i 0.539693i
\(520\) 0 0
\(521\) 35.5317i 1.55667i 0.627848 + 0.778336i \(0.283936\pi\)
−0.627848 + 0.778336i \(0.716064\pi\)
\(522\) 0 0
\(523\) 6.64224 0.290445 0.145222 0.989399i \(-0.453610\pi\)
0.145222 + 0.989399i \(0.453610\pi\)
\(524\) 0 0
\(525\) 0.279423i 0.0121950i
\(526\) 0 0
\(527\) 21.2849 0.927187
\(528\) 0 0
\(529\) −20.6396 + 10.1492i −0.897375 + 0.441268i
\(530\) 0 0
\(531\) 7.12332i 0.309126i
\(532\) 0 0
\(533\) 22.4235 0.971270
\(534\) 0 0
\(535\) 15.1779i 0.656198i
\(536\) 0 0
\(537\) −10.0284 −0.432757
\(538\) 0 0
\(539\) −4.35278 −0.187488
\(540\) 0 0
\(541\) −15.5317 −0.667760 −0.333880 0.942616i \(-0.608358\pi\)
−0.333880 + 0.942616i \(0.608358\pi\)
\(542\) 0 0
\(543\) −12.4338 −0.533584
\(544\) 0 0
\(545\) 8.74076 0.374413
\(546\) 0 0
\(547\) 0.526997i 0.0225328i −0.999937 0.0112664i \(-0.996414\pi\)
0.999937 0.0112664i \(-0.00358628\pi\)
\(548\) 0 0
\(549\) 3.34390i 0.142714i
\(550\) 0 0
\(551\) −5.53885 −0.235963
\(552\) 0 0
\(553\) 4.62096 0.196503
\(554\) 0 0
\(555\) 5.99375i 0.254420i
\(556\) 0 0
\(557\) 5.19113i 0.219955i 0.993934 + 0.109978i \(0.0350779\pi\)
−0.993934 + 0.109978i \(0.964922\pi\)
\(558\) 0 0
\(559\) −26.3512 −1.11454
\(560\) 0 0
\(561\) 1.48770 0.0628108
\(562\) 0 0
\(563\) 45.5261 1.91869 0.959347 0.282228i \(-0.0910736\pi\)
0.959347 + 0.282228i \(0.0910736\pi\)
\(564\) 0 0
\(565\) 19.5279 0.821546
\(566\) 0 0
\(567\) 0.279423 0.0117346
\(568\) 0 0
\(569\) 4.59772i 0.192746i −0.995345 0.0963732i \(-0.969276\pi\)
0.995345 0.0963732i \(-0.0307242\pi\)
\(570\) 0 0
\(571\) −1.18402 −0.0495498 −0.0247749 0.999693i \(-0.507887\pi\)
−0.0247749 + 0.999693i \(0.507887\pi\)
\(572\) 0 0
\(573\) 0.607809i 0.0253916i
\(574\) 0 0
\(575\) 1.08636 + 4.67117i 0.0453045 + 0.194801i
\(576\) 0 0
\(577\) −42.3261 −1.76206 −0.881029 0.473063i \(-0.843148\pi\)
−0.881029 + 0.473063i \(0.843148\pi\)
\(578\) 0 0
\(579\) 6.47078i 0.268916i
\(580\) 0 0
\(581\) 1.97660 0.0820031
\(582\) 0 0
\(583\) 0.773084i 0.0320179i
\(584\) 0 0
\(585\) 3.03226i 0.125368i
\(586\) 0 0
\(587\) 25.7260i 1.06183i 0.847426 + 0.530914i \(0.178151\pi\)
−0.847426 + 0.530914i \(0.821849\pi\)
\(588\) 0 0
\(589\) 13.3744i 0.551083i
\(590\) 0 0
\(591\) 14.5268i 0.597552i
\(592\) 0 0
\(593\) 9.16615 0.376409 0.188204 0.982130i \(-0.439733\pi\)
0.188204 + 0.982130i \(0.439733\pi\)
\(594\) 0 0
\(595\) −0.661054 −0.0271006
\(596\) 0 0
\(597\) 23.1375i 0.946957i
\(598\) 0 0
\(599\) 3.47195i 0.141860i 0.997481 + 0.0709301i \(0.0225967\pi\)
−0.997481 + 0.0709301i \(0.977403\pi\)
\(600\) 0 0
\(601\) −20.5833 −0.839609 −0.419805 0.907615i \(-0.637901\pi\)
−0.419805 + 0.907615i \(0.637901\pi\)
\(602\) 0 0
\(603\) 8.37910 0.341223
\(604\) 0 0
\(605\) 10.6046i 0.431137i
\(606\) 0 0
\(607\) 18.8120i 0.763555i −0.924254 0.381778i \(-0.875312\pi\)
0.924254 0.381778i \(-0.124688\pi\)
\(608\) 0 0
\(609\) 1.04113i 0.0421886i
\(610\) 0 0
\(611\) 7.32099i 0.296176i
\(612\) 0 0
\(613\) 30.5205i 1.23271i −0.787469 0.616355i \(-0.788609\pi\)
0.787469 0.616355i \(-0.211391\pi\)
\(614\) 0 0
\(615\) −7.39498 −0.298195
\(616\) 0 0
\(617\) 6.93087i 0.279026i −0.990220 0.139513i \(-0.955446\pi\)
0.990220 0.139513i \(-0.0445538\pi\)
\(618\) 0 0
\(619\) −3.83752 −0.154243 −0.0771216 0.997022i \(-0.524573\pi\)
−0.0771216 + 0.997022i \(0.524573\pi\)
\(620\) 0 0
\(621\) −4.67117 + 1.08636i −0.187448 + 0.0435943i
\(622\) 0 0
\(623\) 3.58114i 0.143476i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0.934798i 0.0373322i
\(628\) 0 0
\(629\) 14.1799 0.565390
\(630\) 0 0
\(631\) −7.79137 −0.310169 −0.155085 0.987901i \(-0.549565\pi\)
−0.155085 + 0.987901i \(0.549565\pi\)
\(632\) 0 0
\(633\) 14.4669 0.575008
\(634\) 0 0
\(635\) −3.45069 −0.136937
\(636\) 0 0
\(637\) 20.9891 0.831617
\(638\) 0 0
\(639\) 3.78611i 0.149776i
\(640\) 0 0
\(641\) 30.7791i 1.21570i −0.794052 0.607850i \(-0.792032\pi\)
0.794052 0.607850i \(-0.207968\pi\)
\(642\) 0 0
\(643\) −30.8235 −1.21556 −0.607779 0.794106i \(-0.707939\pi\)
−0.607779 + 0.794106i \(0.707939\pi\)
\(644\) 0 0
\(645\) 8.69030 0.342180
\(646\) 0 0
\(647\) 40.3220i 1.58522i −0.609728 0.792611i \(-0.708721\pi\)
0.609728 0.792611i \(-0.291279\pi\)
\(648\) 0 0
\(649\) 4.47943i 0.175833i
\(650\) 0 0
\(651\) −2.51396 −0.0985299
\(652\) 0 0
\(653\) −6.20691 −0.242895 −0.121447 0.992598i \(-0.538754\pi\)
−0.121447 + 0.992598i \(0.538754\pi\)
\(654\) 0 0
\(655\) −11.6225 −0.454128
\(656\) 0 0
\(657\) −15.1169 −0.589767
\(658\) 0 0
\(659\) 32.7113 1.27425 0.637126 0.770759i \(-0.280123\pi\)
0.637126 + 0.770759i \(0.280123\pi\)
\(660\) 0 0
\(661\) 10.5475i 0.410250i −0.978736 0.205125i \(-0.934240\pi\)
0.978736 0.205125i \(-0.0657601\pi\)
\(662\) 0 0
\(663\) −7.17367 −0.278602
\(664\) 0 0
\(665\) 0.415374i 0.0161075i
\(666\) 0 0
\(667\) −4.04779 17.4048i −0.156731 0.673915i
\(668\) 0 0
\(669\) −22.1810 −0.857565
\(670\) 0 0
\(671\) 2.10277i 0.0811767i
\(672\) 0 0
\(673\) 37.5234 1.44642 0.723210 0.690629i \(-0.242666\pi\)
0.723210 + 0.690629i \(0.242666\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 3.31254i 0.127311i 0.997972 + 0.0636556i \(0.0202759\pi\)
−0.997972 + 0.0636556i \(0.979724\pi\)
\(678\) 0 0
\(679\) 3.56512i 0.136817i
\(680\) 0 0
\(681\) 0.753365i 0.0288690i
\(682\) 0 0
\(683\) 34.1195i 1.30555i 0.757554 + 0.652773i \(0.226394\pi\)
−0.757554 + 0.652773i \(0.773606\pi\)
\(684\) 0 0
\(685\) 16.9576 0.647918
\(686\) 0 0
\(687\) −8.34911 −0.318538
\(688\) 0 0
\(689\) 3.72780i 0.142018i
\(690\) 0 0
\(691\) 41.4908i 1.57839i 0.614145 + 0.789193i \(0.289501\pi\)
−0.614145 + 0.789193i \(0.710499\pi\)
\(692\) 0 0
\(693\) −0.175712 −0.00667475
\(694\) 0 0
\(695\) −5.39812 −0.204762
\(696\) 0 0
\(697\) 17.4950i 0.662669i
\(698\) 0 0
\(699\) 23.4423i 0.886668i
\(700\) 0 0
\(701\) 44.9555i 1.69795i −0.528436 0.848973i \(-0.677221\pi\)
0.528436 0.848973i \(-0.322779\pi\)
\(702\) 0 0
\(703\) 8.90996i 0.336046i
\(704\) 0 0
\(705\) 2.41437i 0.0909304i
\(706\) 0 0
\(707\) −3.24850 −0.122172
\(708\) 0 0
\(709\) 21.8350i 0.820029i −0.912079 0.410015i \(-0.865524\pi\)
0.912079 0.410015i \(-0.134476\pi\)
\(710\) 0 0
\(711\) −16.5375 −0.620206
\(712\) 0 0
\(713\) 42.0264 9.77400i 1.57390 0.366039i
\(714\) 0 0
\(715\) 1.90680i 0.0713105i
\(716\) 0 0
\(717\) 13.7015 0.511693
\(718\) 0 0
\(719\) 24.5319i 0.914887i 0.889239 + 0.457443i \(0.151235\pi\)
−0.889239 + 0.457443i \(0.848765\pi\)
\(720\) 0 0
\(721\) −3.02037 −0.112485
\(722\) 0 0
\(723\) −22.8899 −0.851286
\(724\) 0 0
\(725\) −3.72600 −0.138380
\(726\) 0 0
\(727\) 27.5981 1.02356 0.511778 0.859118i \(-0.328987\pi\)
0.511778 + 0.859118i \(0.328987\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 20.5594i 0.760416i
\(732\) 0 0
\(733\) 16.4949i 0.609251i −0.952472 0.304626i \(-0.901469\pi\)
0.952472 0.304626i \(-0.0985314\pi\)
\(734\) 0 0
\(735\) −6.92192 −0.255319
\(736\) 0 0
\(737\) −5.26911 −0.194090
\(738\) 0 0
\(739\) 36.3941i 1.33878i −0.742912 0.669389i \(-0.766556\pi\)
0.742912 0.669389i \(-0.233444\pi\)
\(740\) 0 0
\(741\) 4.50758i 0.165590i
\(742\) 0 0
\(743\) 24.7192 0.906860 0.453430 0.891292i \(-0.350200\pi\)
0.453430 + 0.891292i \(0.350200\pi\)
\(744\) 0 0
\(745\) −2.66008 −0.0974576
\(746\) 0 0
\(747\) −7.07387 −0.258819
\(748\) 0 0
\(749\) 4.24105 0.154964
\(750\) 0 0
\(751\) −14.5578 −0.531223 −0.265611 0.964080i \(-0.585574\pi\)
−0.265611 + 0.964080i \(0.585574\pi\)
\(752\) 0 0
\(753\) 31.1381i 1.13474i
\(754\) 0 0
\(755\) −8.87377 −0.322950
\(756\) 0 0
\(757\) 19.6837i 0.715416i −0.933833 0.357708i \(-0.883558\pi\)
0.933833 0.357708i \(-0.116442\pi\)
\(758\) 0 0
\(759\) 2.93742 0.683149i 0.106621 0.0247967i
\(760\) 0 0
\(761\) −23.3192 −0.845322 −0.422661 0.906288i \(-0.638904\pi\)
−0.422661 + 0.906288i \(0.638904\pi\)
\(762\) 0 0
\(763\) 2.44237i 0.0884196i
\(764\) 0 0
\(765\) 2.36579 0.0855352
\(766\) 0 0
\(767\) 21.5997i 0.779921i
\(768\) 0 0
\(769\) 35.1863i 1.26885i 0.772985 + 0.634424i \(0.218763\pi\)
−0.772985 + 0.634424i \(0.781237\pi\)
\(770\) 0 0
\(771\) 14.1422i 0.509319i
\(772\) 0 0
\(773\) 40.4149i 1.45362i −0.686836 0.726812i \(-0.741001\pi\)
0.686836 0.726812i \(-0.258999\pi\)
\(774\) 0 0
\(775\) 8.99698i 0.323181i
\(776\) 0 0
\(777\) −1.67479 −0.0600827
\(778\) 0 0
\(779\) 10.9930 0.393864
\(780\) 0 0
\(781\) 2.38085i 0.0851937i
\(782\) 0 0
\(783\) 3.72600i 0.133156i
\(784\) 0 0
\(785\) −24.5886 −0.877605
\(786\) 0 0
\(787\) −6.91283 −0.246416 −0.123208 0.992381i \(-0.539318\pi\)
−0.123208 + 0.992381i \(0.539318\pi\)
\(788\) 0 0
\(789\) 26.4000i 0.939865i
\(790\) 0 0
\(791\) 5.45655i 0.194012i
\(792\) 0 0
\(793\) 10.1396i 0.360066i
\(794\) 0 0
\(795\) 1.22938i 0.0436017i
\(796\) 0 0
\(797\) 30.4929i 1.08011i −0.841629 0.540057i \(-0.818403\pi\)
0.841629 0.540057i \(-0.181597\pi\)
\(798\) 0 0
\(799\) 5.71188 0.202072
\(800\) 0 0
\(801\) 12.8162i 0.452839i
\(802\) 0 0
\(803\) 9.50612 0.335464
\(804\) 0 0