Properties

Label 1680.2.t.j.1009.5
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 1009.5
Root \(1.45161 + 1.45161i\) of defining polynomial
Character \(\chi\) \(=\) 1680.1009
Dual form 1680.2.t.j.1009.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +(-0.311108 + 2.21432i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +(-0.311108 + 2.21432i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +5.05086 q^{11} +3.37778i q^{13} +(-2.21432 - 0.311108i) q^{15} -7.18421i q^{17} +8.23506 q^{19} +1.00000 q^{21} +6.23506i q^{23} +(-4.80642 - 1.37778i) q^{25} -1.00000i q^{27} +2.00000 q^{29} +4.62222 q^{31} +5.05086i q^{33} +(2.21432 + 0.311108i) q^{35} +4.85728i q^{37} -3.37778 q^{39} -3.37778 q^{41} -1.24443i q^{43} +(0.311108 - 2.21432i) q^{45} -1.00000 q^{49} +7.18421 q^{51} +4.62222i q^{53} +(-1.57136 + 11.1842i) q^{55} +8.23506i q^{57} -11.6128 q^{59} +0.488863 q^{61} +1.00000i q^{63} +(-7.47949 - 1.05086i) q^{65} +3.61285i q^{67} -6.23506 q^{69} +10.2953 q^{71} +16.2351i q^{73} +(1.37778 - 4.80642i) q^{75} -5.05086i q^{77} +1.24443 q^{79} +1.00000 q^{81} -11.6128i q^{83} +(15.9081 + 2.23506i) q^{85} +2.00000i q^{87} -6.99063 q^{89} +3.37778 q^{91} +4.62222i q^{93} +(-2.56199 + 18.2351i) q^{95} +8.23506i q^{97} -5.05086 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{5} - 6 q^{9} + O(q^{10}) \) \( 6 q - 2 q^{5} - 6 q^{9} + 4 q^{11} - 4 q^{19} + 6 q^{21} - 2 q^{25} + 12 q^{29} + 28 q^{31} - 20 q^{39} - 20 q^{41} + 2 q^{45} - 6 q^{49} + 16 q^{51} - 36 q^{55} - 16 q^{59} + 4 q^{61} + 8 q^{65} + 16 q^{69} + 36 q^{71} + 8 q^{75} + 8 q^{79} + 6 q^{81} + 16 q^{85} + 12 q^{89} + 20 q^{91} + 12 q^{95} - 4 q^{99} + O(q^{100}) \)

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
\(5\) −0.311108 + 2.21432i −0.139132 + 0.990274i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 5.05086 1.52289 0.761445 0.648229i \(-0.224490\pi\)
0.761445 + 0.648229i \(0.224490\pi\)
\(12\) 0 0
\(13\) 3.37778i 0.936829i 0.883509 + 0.468414i \(0.155175\pi\)
−0.883509 + 0.468414i \(0.844825\pi\)
\(14\) 0 0
\(15\) −2.21432 0.311108i −0.571735 0.0803277i
\(16\) 0 0
\(17\) 7.18421i 1.74243i −0.490905 0.871213i \(-0.663334\pi\)
0.490905 0.871213i \(-0.336666\pi\)
\(18\) 0 0
\(19\) 8.23506 1.88925 0.944627 0.328147i \(-0.106424\pi\)
0.944627 + 0.328147i \(0.106424\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 6.23506i 1.30010i 0.759891 + 0.650050i \(0.225252\pi\)
−0.759891 + 0.650050i \(0.774748\pi\)
\(24\) 0 0
\(25\) −4.80642 1.37778i −0.961285 0.275557i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 4.62222 0.830174 0.415087 0.909782i \(-0.363751\pi\)
0.415087 + 0.909782i \(0.363751\pi\)
\(32\) 0 0
\(33\) 5.05086i 0.879241i
\(34\) 0 0
\(35\) 2.21432 + 0.311108i 0.374288 + 0.0525868i
\(36\) 0 0
\(37\) 4.85728i 0.798532i 0.916835 + 0.399266i \(0.130735\pi\)
−0.916835 + 0.399266i \(0.869265\pi\)
\(38\) 0 0
\(39\) −3.37778 −0.540878
\(40\) 0 0
\(41\) −3.37778 −0.527521 −0.263761 0.964588i \(-0.584963\pi\)
−0.263761 + 0.964588i \(0.584963\pi\)
\(42\) 0 0
\(43\) 1.24443i 0.189774i −0.995488 0.0948870i \(-0.969751\pi\)
0.995488 0.0948870i \(-0.0302490\pi\)
\(44\) 0 0
\(45\) 0.311108 2.21432i 0.0463772 0.330091i
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 7.18421 1.00599
\(52\) 0 0
\(53\) 4.62222i 0.634910i 0.948273 + 0.317455i \(0.102828\pi\)
−0.948273 + 0.317455i \(0.897172\pi\)
\(54\) 0 0
\(55\) −1.57136 + 11.1842i −0.211882 + 1.50808i
\(56\) 0 0
\(57\) 8.23506i 1.09076i
\(58\) 0 0
\(59\) −11.6128 −1.51186 −0.755932 0.654650i \(-0.772816\pi\)
−0.755932 + 0.654650i \(0.772816\pi\)
\(60\) 0 0
\(61\) 0.488863 0.0625924 0.0312962 0.999510i \(-0.490036\pi\)
0.0312962 + 0.999510i \(0.490036\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) −7.47949 1.05086i −0.927717 0.130343i
\(66\) 0 0
\(67\) 3.61285i 0.441380i 0.975344 + 0.220690i \(0.0708308\pi\)
−0.975344 + 0.220690i \(0.929169\pi\)
\(68\) 0 0
\(69\) −6.23506 −0.750613
\(70\) 0 0
\(71\) 10.2953 1.22183 0.610913 0.791698i \(-0.290803\pi\)
0.610913 + 0.791698i \(0.290803\pi\)
\(72\) 0 0
\(73\) 16.2351i 1.90017i 0.311991 + 0.950085i \(0.399004\pi\)
−0.311991 + 0.950085i \(0.600996\pi\)
\(74\) 0 0
\(75\) 1.37778 4.80642i 0.159093 0.554998i
\(76\) 0 0
\(77\) 5.05086i 0.575598i
\(78\) 0 0
\(79\) 1.24443 0.140009 0.0700047 0.997547i \(-0.477699\pi\)
0.0700047 + 0.997547i \(0.477699\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 11.6128i 1.27468i −0.770585 0.637338i \(-0.780036\pi\)
0.770585 0.637338i \(-0.219964\pi\)
\(84\) 0 0
\(85\) 15.9081 + 2.23506i 1.72548 + 0.242427i
\(86\) 0 0
\(87\) 2.00000i 0.214423i
\(88\) 0 0
\(89\) −6.99063 −0.741006 −0.370503 0.928831i \(-0.620815\pi\)
−0.370503 + 0.928831i \(0.620815\pi\)
\(90\) 0 0
\(91\) 3.37778 0.354088
\(92\) 0 0
\(93\) 4.62222i 0.479301i
\(94\) 0 0
\(95\) −2.56199 + 18.2351i −0.262855 + 1.87088i
\(96\) 0 0
\(97\) 8.23506i 0.836144i 0.908414 + 0.418072i \(0.137294\pi\)
−0.908414 + 0.418072i \(0.862706\pi\)
\(98\) 0 0
\(99\) −5.05086 −0.507630
\(100\) 0 0
\(101\) −9.47949 −0.943245 −0.471622 0.881801i \(-0.656331\pi\)
−0.471622 + 0.881801i \(0.656331\pi\)
\(102\) 0 0
\(103\) 16.8573i 1.66100i −0.557021 0.830499i \(-0.688056\pi\)
0.557021 0.830499i \(-0.311944\pi\)
\(104\) 0 0
\(105\) −0.311108 + 2.21432i −0.0303610 + 0.216095i
\(106\) 0 0
\(107\) 15.4795i 1.49646i 0.663440 + 0.748230i \(0.269096\pi\)
−0.663440 + 0.748230i \(0.730904\pi\)
\(108\) 0 0
\(109\) 1.61285 0.154483 0.0772414 0.997012i \(-0.475389\pi\)
0.0772414 + 0.997012i \(0.475389\pi\)
\(110\) 0 0
\(111\) −4.85728 −0.461033
\(112\) 0 0
\(113\) 1.86665i 0.175599i 0.996138 + 0.0877997i \(0.0279835\pi\)
−0.996138 + 0.0877997i \(0.972016\pi\)
\(114\) 0 0
\(115\) −13.8064 1.93978i −1.28746 0.180885i
\(116\) 0 0
\(117\) 3.37778i 0.312276i
\(118\) 0 0
\(119\) −7.18421 −0.658575
\(120\) 0 0
\(121\) 14.5111 1.31919
\(122\) 0 0
\(123\) 3.37778i 0.304565i
\(124\) 0 0
\(125\) 4.54617 10.2143i 0.406622 0.913597i
\(126\) 0 0
\(127\) 12.8573i 1.14090i 0.821333 + 0.570450i \(0.193231\pi\)
−0.821333 + 0.570450i \(0.806769\pi\)
\(128\) 0 0
\(129\) 1.24443 0.109566
\(130\) 0 0
\(131\) −21.7146 −1.89721 −0.948605 0.316463i \(-0.897505\pi\)
−0.948605 + 0.316463i \(0.897505\pi\)
\(132\) 0 0
\(133\) 8.23506i 0.714071i
\(134\) 0 0
\(135\) 2.21432 + 0.311108i 0.190578 + 0.0267759i
\(136\) 0 0
\(137\) 9.47949i 0.809888i −0.914342 0.404944i \(-0.867291\pi\)
0.914342 0.404944i \(-0.132709\pi\)
\(138\) 0 0
\(139\) 10.1334 0.859500 0.429750 0.902948i \(-0.358602\pi\)
0.429750 + 0.902948i \(0.358602\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 17.0607i 1.42669i
\(144\) 0 0
\(145\) −0.622216 + 4.42864i −0.0516722 + 0.367778i
\(146\) 0 0
\(147\) 1.00000i 0.0824786i
\(148\) 0 0
\(149\) −7.24443 −0.593487 −0.296743 0.954957i \(-0.595901\pi\)
−0.296743 + 0.954957i \(0.595901\pi\)
\(150\) 0 0
\(151\) 8.85728 0.720795 0.360398 0.932799i \(-0.382641\pi\)
0.360398 + 0.932799i \(0.382641\pi\)
\(152\) 0 0
\(153\) 7.18421i 0.580809i
\(154\) 0 0
\(155\) −1.43801 + 10.2351i −0.115504 + 0.822100i
\(156\) 0 0
\(157\) 13.4795i 1.07578i −0.843015 0.537890i \(-0.819221\pi\)
0.843015 0.537890i \(-0.180779\pi\)
\(158\) 0 0
\(159\) −4.62222 −0.366566
\(160\) 0 0
\(161\) 6.23506 0.491392
\(162\) 0 0
\(163\) 4.00000i 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) 0 0
\(165\) −11.1842 1.57136i −0.870689 0.122330i
\(166\) 0 0
\(167\) 2.10171i 0.162635i 0.996688 + 0.0813176i \(0.0259128\pi\)
−0.996688 + 0.0813176i \(0.974087\pi\)
\(168\) 0 0
\(169\) 1.59057 0.122352
\(170\) 0 0
\(171\) −8.23506 −0.629751
\(172\) 0 0
\(173\) 11.1842i 0.850320i 0.905118 + 0.425160i \(0.139782\pi\)
−0.905118 + 0.425160i \(0.860218\pi\)
\(174\) 0 0
\(175\) −1.37778 + 4.80642i −0.104151 + 0.363331i
\(176\) 0 0
\(177\) 11.6128i 0.872875i
\(178\) 0 0
\(179\) 20.6637 1.54448 0.772239 0.635332i \(-0.219137\pi\)
0.772239 + 0.635332i \(0.219137\pi\)
\(180\) 0 0
\(181\) 24.9590 1.85519 0.927594 0.373591i \(-0.121874\pi\)
0.927594 + 0.373591i \(0.121874\pi\)
\(182\) 0 0
\(183\) 0.488863i 0.0361378i
\(184\) 0 0
\(185\) −10.7556 1.51114i −0.790765 0.111101i
\(186\) 0 0
\(187\) 36.2864i 2.65352i
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −9.52098 −0.688914 −0.344457 0.938802i \(-0.611937\pi\)
−0.344457 + 0.938802i \(0.611937\pi\)
\(192\) 0 0
\(193\) 22.9590i 1.65262i −0.563212 0.826312i \(-0.690435\pi\)
0.563212 0.826312i \(-0.309565\pi\)
\(194\) 0 0
\(195\) 1.05086 7.47949i 0.0752533 0.535618i
\(196\) 0 0
\(197\) 19.8479i 1.41411i 0.707161 + 0.707053i \(0.249976\pi\)
−0.707161 + 0.707053i \(0.750024\pi\)
\(198\) 0 0
\(199\) −8.23506 −0.583768 −0.291884 0.956454i \(-0.594282\pi\)
−0.291884 + 0.956454i \(0.594282\pi\)
\(200\) 0 0
\(201\) −3.61285 −0.254831
\(202\) 0 0
\(203\) 2.00000i 0.140372i
\(204\) 0 0
\(205\) 1.05086 7.47949i 0.0733949 0.522391i
\(206\) 0 0
\(207\) 6.23506i 0.433367i
\(208\) 0 0
\(209\) 41.5941 2.87712
\(210\) 0 0
\(211\) 11.6128 0.799461 0.399731 0.916633i \(-0.369104\pi\)
0.399731 + 0.916633i \(0.369104\pi\)
\(212\) 0 0
\(213\) 10.2953i 0.705421i
\(214\) 0 0
\(215\) 2.75557 + 0.387152i 0.187928 + 0.0264036i
\(216\) 0 0
\(217\) 4.62222i 0.313776i
\(218\) 0 0
\(219\) −16.2351 −1.09706
\(220\) 0 0
\(221\) 24.2667 1.63236
\(222\) 0 0
\(223\) 2.48886i 0.166667i −0.996522 0.0833333i \(-0.973443\pi\)
0.996522 0.0833333i \(-0.0265566\pi\)
\(224\) 0 0
\(225\) 4.80642 + 1.37778i 0.320428 + 0.0918523i
\(226\) 0 0
\(227\) 0.857279i 0.0568996i −0.999595 0.0284498i \(-0.990943\pi\)
0.999595 0.0284498i \(-0.00905708\pi\)
\(228\) 0 0
\(229\) 14.4701 0.956213 0.478106 0.878302i \(-0.341323\pi\)
0.478106 + 0.878302i \(0.341323\pi\)
\(230\) 0 0
\(231\) 5.05086 0.332322
\(232\) 0 0
\(233\) 3.96836i 0.259976i −0.991516 0.129988i \(-0.958506\pi\)
0.991516 0.129988i \(-0.0414938\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.24443i 0.0808345i
\(238\) 0 0
\(239\) 1.90813 0.123427 0.0617135 0.998094i \(-0.480344\pi\)
0.0617135 + 0.998094i \(0.480344\pi\)
\(240\) 0 0
\(241\) 0.755569 0.0486705 0.0243352 0.999704i \(-0.492253\pi\)
0.0243352 + 0.999704i \(0.492253\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 0.311108 2.21432i 0.0198759 0.141468i
\(246\) 0 0
\(247\) 27.8163i 1.76991i
\(248\) 0 0
\(249\) 11.6128 0.735934
\(250\) 0 0
\(251\) 9.12399 0.575901 0.287950 0.957645i \(-0.407026\pi\)
0.287950 + 0.957645i \(0.407026\pi\)
\(252\) 0 0
\(253\) 31.4924i 1.97991i
\(254\) 0 0
\(255\) −2.23506 + 15.9081i −0.139965 + 0.996206i
\(256\) 0 0
\(257\) 1.28592i 0.0802134i 0.999195 + 0.0401067i \(0.0127698\pi\)
−0.999195 + 0.0401067i \(0.987230\pi\)
\(258\) 0 0
\(259\) 4.85728 0.301817
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) 7.00937i 0.432216i −0.976369 0.216108i \(-0.930664\pi\)
0.976369 0.216108i \(-0.0693364\pi\)
\(264\) 0 0
\(265\) −10.2351 1.43801i −0.628735 0.0883361i
\(266\) 0 0
\(267\) 6.99063i 0.427820i
\(268\) 0 0
\(269\) −23.1941 −1.41417 −0.707083 0.707130i \(-0.749989\pi\)
−0.707083 + 0.707130i \(0.749989\pi\)
\(270\) 0 0
\(271\) −7.11108 −0.431967 −0.215984 0.976397i \(-0.569296\pi\)
−0.215984 + 0.976397i \(0.569296\pi\)
\(272\) 0 0
\(273\) 3.37778i 0.204433i
\(274\) 0 0
\(275\) −24.2766 6.95899i −1.46393 0.419643i
\(276\) 0 0
\(277\) 7.34614i 0.441387i −0.975343 0.220693i \(-0.929168\pi\)
0.975343 0.220693i \(-0.0708320\pi\)
\(278\) 0 0
\(279\) −4.62222 −0.276725
\(280\) 0 0
\(281\) −19.9813 −1.19198 −0.595991 0.802991i \(-0.703241\pi\)
−0.595991 + 0.802991i \(0.703241\pi\)
\(282\) 0 0
\(283\) 4.85728i 0.288735i −0.989524 0.144368i \(-0.953885\pi\)
0.989524 0.144368i \(-0.0461148\pi\)
\(284\) 0 0
\(285\) −18.2351 2.56199i −1.08015 0.151759i
\(286\) 0 0
\(287\) 3.37778i 0.199384i
\(288\) 0 0
\(289\) −34.6128 −2.03605
\(290\) 0 0
\(291\) −8.23506 −0.482748
\(292\) 0 0
\(293\) 11.1842i 0.653388i −0.945130 0.326694i \(-0.894065\pi\)
0.945130 0.326694i \(-0.105935\pi\)
\(294\) 0 0
\(295\) 3.61285 25.7146i 0.210348 1.49716i
\(296\) 0 0
\(297\) 5.05086i 0.293080i
\(298\) 0 0
\(299\) −21.0607 −1.21797
\(300\) 0 0
\(301\) −1.24443 −0.0717278
\(302\) 0 0
\(303\) 9.47949i 0.544583i
\(304\) 0 0
\(305\) −0.152089 + 1.08250i −0.00870859 + 0.0619837i
\(306\) 0 0
\(307\) 23.3461i 1.33243i −0.745758 0.666217i \(-0.767912\pi\)
0.745758 0.666217i \(-0.232088\pi\)
\(308\) 0 0
\(309\) 16.8573 0.958977
\(310\) 0 0
\(311\) 32.8573 1.86317 0.931583 0.363530i \(-0.118428\pi\)
0.931583 + 0.363530i \(0.118428\pi\)
\(312\) 0 0
\(313\) 28.8256i 1.62932i 0.579938 + 0.814661i \(0.303077\pi\)
−0.579938 + 0.814661i \(0.696923\pi\)
\(314\) 0 0
\(315\) −2.21432 0.311108i −0.124763 0.0175289i
\(316\) 0 0
\(317\) 25.3590i 1.42431i −0.702024 0.712153i \(-0.747720\pi\)
0.702024 0.712153i \(-0.252280\pi\)
\(318\) 0 0
\(319\) 10.1017 0.565587
\(320\) 0 0
\(321\) −15.4795 −0.863981
\(322\) 0 0
\(323\) 59.1624i 3.29188i
\(324\) 0 0
\(325\) 4.65386 16.2351i 0.258150 0.900559i
\(326\) 0 0
\(327\) 1.61285i 0.0891907i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.89829 −0.104339 −0.0521697 0.998638i \(-0.516614\pi\)
−0.0521697 + 0.998638i \(0.516614\pi\)
\(332\) 0 0
\(333\) 4.85728i 0.266177i
\(334\) 0 0
\(335\) −8.00000 1.12399i −0.437087 0.0614099i
\(336\) 0 0
\(337\) 23.2257i 1.26518i −0.774485 0.632592i \(-0.781991\pi\)
0.774485 0.632592i \(-0.218009\pi\)
\(338\) 0 0
\(339\) −1.86665 −0.101382
\(340\) 0 0
\(341\) 23.3461 1.26426
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 1.93978 13.8064i 0.104434 0.743313i
\(346\) 0 0
\(347\) 7.47949i 0.401520i 0.979640 + 0.200760i \(0.0643412\pi\)
−0.979640 + 0.200760i \(0.935659\pi\)
\(348\) 0 0
\(349\) −29.2257 −1.56442 −0.782208 0.623018i \(-0.785906\pi\)
−0.782208 + 0.623018i \(0.785906\pi\)
\(350\) 0 0
\(351\) 3.37778 0.180293
\(352\) 0 0
\(353\) 30.4099i 1.61856i −0.587426 0.809278i \(-0.699859\pi\)
0.587426 0.809278i \(-0.300141\pi\)
\(354\) 0 0
\(355\) −3.20294 + 22.7971i −0.169995 + 1.20994i
\(356\) 0 0
\(357\) 7.18421i 0.380229i
\(358\) 0 0
\(359\) −16.1936 −0.854664 −0.427332 0.904095i \(-0.640546\pi\)
−0.427332 + 0.904095i \(0.640546\pi\)
\(360\) 0 0
\(361\) 48.8163 2.56928
\(362\) 0 0
\(363\) 14.5111i 0.761637i
\(364\) 0 0
\(365\) −35.9496 5.05086i −1.88169 0.264374i
\(366\) 0 0
\(367\) 5.51114i 0.287679i −0.989601 0.143840i \(-0.954055\pi\)
0.989601 0.143840i \(-0.0459449\pi\)
\(368\) 0 0
\(369\) 3.37778 0.175840
\(370\) 0 0
\(371\) 4.62222 0.239973
\(372\) 0 0
\(373\) 1.63158i 0.0844802i 0.999107 + 0.0422401i \(0.0134494\pi\)
−0.999107 + 0.0422401i \(0.986551\pi\)
\(374\) 0 0
\(375\) 10.2143 + 4.54617i 0.527465 + 0.234763i
\(376\) 0 0
\(377\) 6.75557i 0.347929i
\(378\) 0 0
\(379\) 20.8573 1.07137 0.535683 0.844419i \(-0.320054\pi\)
0.535683 + 0.844419i \(0.320054\pi\)
\(380\) 0 0
\(381\) −12.8573 −0.658698
\(382\) 0 0
\(383\) 20.0830i 1.02619i −0.858331 0.513096i \(-0.828498\pi\)
0.858331 0.513096i \(-0.171502\pi\)
\(384\) 0 0
\(385\) 11.1842 + 1.57136i 0.570000 + 0.0800839i
\(386\) 0 0
\(387\) 1.24443i 0.0632580i
\(388\) 0 0
\(389\) −33.2257 −1.68461 −0.842305 0.539001i \(-0.818802\pi\)
−0.842305 + 0.539001i \(0.818802\pi\)
\(390\) 0 0
\(391\) 44.7940 2.26533
\(392\) 0 0
\(393\) 21.7146i 1.09535i
\(394\) 0 0
\(395\) −0.387152 + 2.75557i −0.0194797 + 0.138648i
\(396\) 0 0
\(397\) 22.7239i 1.14048i −0.821478 0.570241i \(-0.806850\pi\)
0.821478 0.570241i \(-0.193150\pi\)
\(398\) 0 0
\(399\) 8.23506 0.412269
\(400\) 0 0
\(401\) 12.7556 0.636983 0.318491 0.947926i \(-0.396824\pi\)
0.318491 + 0.947926i \(0.396824\pi\)
\(402\) 0 0
\(403\) 15.6128i 0.777731i
\(404\) 0 0
\(405\) −0.311108 + 2.21432i −0.0154591 + 0.110030i
\(406\) 0 0
\(407\) 24.5334i 1.21608i
\(408\) 0 0
\(409\) −27.4479 −1.35721 −0.678604 0.734504i \(-0.737415\pi\)
−0.678604 + 0.734504i \(0.737415\pi\)
\(410\) 0 0
\(411\) 9.47949 0.467589
\(412\) 0 0
\(413\) 11.6128i 0.571431i
\(414\) 0 0
\(415\) 25.7146 + 3.61285i 1.26228 + 0.177348i
\(416\) 0 0
\(417\) 10.1334i 0.496232i
\(418\) 0 0
\(419\) 2.36842 0.115705 0.0578524 0.998325i \(-0.481575\pi\)
0.0578524 + 0.998325i \(0.481575\pi\)
\(420\) 0 0
\(421\) 39.3274 1.91670 0.958350 0.285596i \(-0.0921914\pi\)
0.958350 + 0.285596i \(0.0921914\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −9.89829 + 34.5303i −0.480138 + 1.67497i
\(426\) 0 0
\(427\) 0.488863i 0.0236577i
\(428\) 0 0
\(429\) −17.0607 −0.823698
\(430\) 0 0
\(431\) −8.19358 −0.394671 −0.197335 0.980336i \(-0.563229\pi\)
−0.197335 + 0.980336i \(0.563229\pi\)
\(432\) 0 0
\(433\) 14.6035i 0.701798i −0.936413 0.350899i \(-0.885876\pi\)
0.936413 0.350899i \(-0.114124\pi\)
\(434\) 0 0
\(435\) −4.42864 0.622216i −0.212337 0.0298330i
\(436\) 0 0
\(437\) 51.3461i 2.45622i
\(438\) 0 0
\(439\) 5.27607 0.251813 0.125907 0.992042i \(-0.459816\pi\)
0.125907 + 0.992042i \(0.459816\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 18.5018i 0.879046i −0.898231 0.439523i \(-0.855148\pi\)
0.898231 0.439523i \(-0.144852\pi\)
\(444\) 0 0
\(445\) 2.17484 15.4795i 0.103097 0.733798i
\(446\) 0 0
\(447\) 7.24443i 0.342650i
\(448\) 0 0
\(449\) −11.7146 −0.552844 −0.276422 0.961036i \(-0.589149\pi\)
−0.276422 + 0.961036i \(0.589149\pi\)
\(450\) 0 0
\(451\) −17.0607 −0.803357
\(452\) 0 0
\(453\) 8.85728i 0.416151i
\(454\) 0 0
\(455\) −1.05086 + 7.47949i −0.0492648 + 0.350644i
\(456\) 0 0
\(457\) 18.9590i 0.886864i 0.896308 + 0.443432i \(0.146239\pi\)
−0.896308 + 0.443432i \(0.853761\pi\)
\(458\) 0 0
\(459\) −7.18421 −0.335330
\(460\) 0 0
\(461\) −15.1111 −0.703793 −0.351897 0.936039i \(-0.614463\pi\)
−0.351897 + 0.936039i \(0.614463\pi\)
\(462\) 0 0
\(463\) 22.5718i 1.04900i 0.851410 + 0.524501i \(0.175748\pi\)
−0.851410 + 0.524501i \(0.824252\pi\)
\(464\) 0 0
\(465\) −10.2351 1.43801i −0.474640 0.0666860i
\(466\) 0 0
\(467\) 22.3684i 1.03509i −0.855657 0.517543i \(-0.826847\pi\)
0.855657 0.517543i \(-0.173153\pi\)
\(468\) 0 0
\(469\) 3.61285 0.166826
\(470\) 0 0
\(471\) 13.4795 0.621102
\(472\) 0 0
\(473\) 6.28544i 0.289005i
\(474\) 0 0
\(475\) −39.5812 11.3461i −1.81611 0.520597i
\(476\) 0 0
\(477\) 4.62222i 0.211637i
\(478\) 0 0
\(479\) −32.8573 −1.50129 −0.750644 0.660707i \(-0.770256\pi\)
−0.750644 + 0.660707i \(0.770256\pi\)
\(480\) 0 0
\(481\) −16.4068 −0.748088
\(482\) 0 0
\(483\) 6.23506i 0.283705i
\(484\) 0 0
\(485\) −18.2351 2.56199i −0.828012 0.116334i
\(486\) 0 0
\(487\) 3.40943i 0.154496i 0.997012 + 0.0772479i \(0.0246133\pi\)
−0.997012 + 0.0772479i \(0.975387\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 33.7877 1.52482 0.762409 0.647096i \(-0.224017\pi\)
0.762409 + 0.647096i \(0.224017\pi\)
\(492\) 0 0
\(493\) 14.3684i 0.647121i
\(494\) 0 0
\(495\) 1.57136 11.1842i 0.0706274 0.502693i
\(496\) 0 0
\(497\) 10.2953i 0.461807i
\(498\) 0 0
\(499\) 13.6316 0.610233 0.305117 0.952315i \(-0.401305\pi\)
0.305117 + 0.952315i \(0.401305\pi\)
\(500\) 0 0
\(501\) −2.10171 −0.0938975
\(502\) 0 0
\(503\) 7.34614i 0.327548i −0.986498 0.163774i \(-0.947633\pi\)
0.986498 0.163774i \(-0.0523668\pi\)
\(504\) 0 0
\(505\) 2.94914 20.9906i 0.131235 0.934071i
\(506\) 0 0
\(507\) 1.59057i 0.0706398i
\(508\) 0 0
\(509\) 15.9684 0.707785 0.353892 0.935286i \(-0.384858\pi\)
0.353892 + 0.935286i \(0.384858\pi\)
\(510\) 0 0
\(511\) 16.2351 0.718197
\(512\) 0 0
\(513\) 8.23506i 0.363587i
\(514\) 0 0
\(515\) 37.3274 + 5.24443i 1.64484 + 0.231097i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −11.1842 −0.490932
\(520\) 0 0
\(521\) −1.09234 −0.0478564 −0.0239282 0.999714i \(-0.507617\pi\)
−0.0239282 + 0.999714i \(0.507617\pi\)
\(522\) 0 0
\(523\) 17.5111i 0.765709i −0.923809 0.382854i \(-0.874941\pi\)
0.923809 0.382854i \(-0.125059\pi\)
\(524\) 0 0
\(525\) −4.80642 1.37778i −0.209770 0.0601314i
\(526\) 0 0
\(527\) 33.2070i 1.44652i
\(528\) 0 0
\(529\) −15.8760 −0.690262
\(530\) 0 0
\(531\) 11.6128 0.503955
\(532\) 0 0
\(533\) 11.4094i 0.494197i
\(534\) 0 0
\(535\) −34.2766 4.81579i −1.48190 0.208205i
\(536\) 0 0
\(537\) 20.6637i 0.891705i
\(538\) 0 0
\(539\) −5.05086 −0.217556
\(540\) 0 0
\(541\) −15.3274 −0.658977 −0.329488 0.944160i \(-0.606876\pi\)
−0.329488 + 0.944160i \(0.606876\pi\)
\(542\) 0 0
\(543\) 24.9590i 1.07109i
\(544\) 0 0
\(545\) −0.501770 + 3.57136i −0.0214934 + 0.152980i
\(546\) 0 0
\(547\) 5.32741i 0.227783i −0.993493 0.113892i \(-0.963668\pi\)
0.993493 0.113892i \(-0.0363317\pi\)
\(548\) 0 0
\(549\) −0.488863 −0.0208641
\(550\) 0 0
\(551\) 16.4701 0.701651
\(552\) 0 0
\(553\) 1.24443i 0.0529186i
\(554\) 0 0
\(555\) 1.51114 10.7556i 0.0641442 0.456548i
\(556\) 0 0
\(557\) 0.355509i 0.0150634i 0.999972 + 0.00753171i \(0.00239744\pi\)
−0.999972 + 0.00753171i \(0.997603\pi\)
\(558\) 0 0
\(559\) 4.20342 0.177786
\(560\) 0 0
\(561\) 36.2864 1.53201
\(562\) 0 0
\(563\) 16.4701i 0.694133i −0.937841 0.347067i \(-0.887178\pi\)
0.937841 0.347067i \(-0.112822\pi\)
\(564\) 0 0
\(565\) −4.13335 0.580728i −0.173891 0.0244314i
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) 20.9590 0.878647 0.439323 0.898329i \(-0.355218\pi\)
0.439323 + 0.898329i \(0.355218\pi\)
\(570\) 0 0
\(571\) −17.5111 −0.732818 −0.366409 0.930454i \(-0.619413\pi\)
−0.366409 + 0.930454i \(0.619413\pi\)
\(572\) 0 0
\(573\) 9.52098i 0.397745i
\(574\) 0 0
\(575\) 8.59057 29.9684i 0.358252 1.24977i
\(576\) 0 0
\(577\) 0.152089i 0.00633155i 0.999995 + 0.00316577i \(0.00100770\pi\)
−0.999995 + 0.00316577i \(0.998992\pi\)
\(578\) 0 0
\(579\) 22.9590 0.954143
\(580\) 0 0
\(581\) −11.6128 −0.481782
\(582\) 0 0
\(583\) 23.3461i 0.966898i
\(584\) 0 0
\(585\) 7.47949 + 1.05086i 0.309239 + 0.0434475i
\(586\) 0 0
\(587\) 34.1847i 1.41095i −0.708733 0.705476i \(-0.750733\pi\)
0.708733 0.705476i \(-0.249267\pi\)
\(588\) 0 0
\(589\) 38.0642 1.56841
\(590\) 0 0
\(591\) −19.8479 −0.816434
\(592\) 0 0
\(593\) 19.3047i 0.792747i −0.918089 0.396374i \(-0.870269\pi\)
0.918089 0.396374i \(-0.129731\pi\)
\(594\) 0 0
\(595\) 2.23506 15.9081i 0.0916287 0.652170i
\(596\) 0 0
\(597\) 8.23506i 0.337039i
\(598\) 0 0
\(599\) −25.9081 −1.05858 −0.529289 0.848442i \(-0.677541\pi\)
−0.529289 + 0.848442i \(0.677541\pi\)
\(600\) 0 0
\(601\) −22.7368 −0.927455 −0.463727 0.885978i \(-0.653488\pi\)
−0.463727 + 0.885978i \(0.653488\pi\)
\(602\) 0 0
\(603\) 3.61285i 0.147127i
\(604\) 0 0
\(605\) −4.51453 + 32.1323i −0.183542 + 1.30636i
\(606\) 0 0
\(607\) 32.9403i 1.33700i 0.743711 + 0.668502i \(0.233064\pi\)
−0.743711 + 0.668502i \(0.766936\pi\)
\(608\) 0 0
\(609\) 2.00000 0.0810441
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 13.1240i 0.530073i 0.964238 + 0.265036i \(0.0853840\pi\)
−0.964238 + 0.265036i \(0.914616\pi\)
\(614\) 0 0
\(615\) 7.47949 + 1.05086i 0.301602 + 0.0423746i
\(616\) 0 0
\(617\) 18.2538i 0.734870i 0.930049 + 0.367435i \(0.119764\pi\)
−0.930049 + 0.367435i \(0.880236\pi\)
\(618\) 0 0
\(619\) 7.64449 0.307258 0.153629 0.988129i \(-0.450904\pi\)
0.153629 + 0.988129i \(0.450904\pi\)
\(620\) 0 0
\(621\) 6.23506 0.250204
\(622\) 0 0
\(623\) 6.99063i 0.280074i
\(624\) 0 0
\(625\) 21.2034 + 13.2444i 0.848137 + 0.529777i
\(626\) 0 0
\(627\) 41.5941i 1.66111i
\(628\) 0 0
\(629\) 34.8957 1.39138
\(630\) 0 0
\(631\) −28.6735 −1.14148 −0.570738 0.821132i \(-0.693343\pi\)
−0.570738 + 0.821132i \(0.693343\pi\)
\(632\) 0 0
\(633\) 11.6128i 0.461569i
\(634\) 0 0
\(635\) −28.4701 4.00000i −1.12980 0.158735i
\(636\) 0 0
\(637\) 3.37778i 0.133833i
\(638\) 0 0
\(639\) −10.2953 −0.407275
\(640\) 0 0
\(641\) 15.4479 0.610153 0.305077 0.952328i \(-0.401318\pi\)
0.305077 + 0.952328i \(0.401318\pi\)
\(642\) 0 0
\(643\) 25.8350i 1.01883i 0.860520 + 0.509417i \(0.170139\pi\)
−0.860520 + 0.509417i \(0.829861\pi\)
\(644\) 0 0
\(645\) −0.387152 + 2.75557i −0.0152441 + 0.108500i
\(646\) 0 0
\(647\) 23.6128i 0.928317i 0.885752 + 0.464158i \(0.153643\pi\)
−0.885752 + 0.464158i \(0.846357\pi\)
\(648\) 0 0
\(649\) −58.6548 −2.30240
\(650\) 0 0
\(651\) 4.62222 0.181159
\(652\) 0 0
\(653\) 36.7052i 1.43639i −0.695844 0.718193i \(-0.744970\pi\)
0.695844 0.718193i \(-0.255030\pi\)
\(654\) 0 0
\(655\) 6.75557 48.0830i 0.263962 1.87876i
\(656\) 0 0
\(657\) 16.2351i 0.633390i
\(658\) 0 0
\(659\) 37.9911 1.47992 0.739962 0.672649i \(-0.234844\pi\)
0.739962 + 0.672649i \(0.234844\pi\)
\(660\) 0 0
\(661\) −49.2257 −1.91466 −0.957329 0.289001i \(-0.906677\pi\)
−0.957329 + 0.289001i \(0.906677\pi\)
\(662\) 0 0
\(663\) 24.2667i 0.942441i
\(664\) 0 0
\(665\) 18.2351 + 2.56199i 0.707125 + 0.0993498i
\(666\) 0 0
\(667\) 12.4701i 0.482845i
\(668\) 0 0
\(669\) 2.48886 0.0962250
\(670\) 0 0
\(671\) 2.46917 0.0953214
\(672\) 0 0
\(673\) 11.2257i 0.432719i 0.976314 + 0.216359i \(0.0694183\pi\)
−0.976314 + 0.216359i \(0.930582\pi\)
\(674\) 0 0
\(675\) −1.37778 + 4.80642i −0.0530309 + 0.184999i
\(676\) 0 0
\(677\) 9.55262i 0.367137i 0.983007 + 0.183569i \(0.0587649\pi\)
−0.983007 + 0.183569i \(0.941235\pi\)
\(678\) 0 0
\(679\) 8.23506 0.316033
\(680\) 0 0
\(681\) 0.857279 0.0328510
\(682\) 0 0
\(683\) 17.9684i 0.687540i −0.939054 0.343770i \(-0.888296\pi\)
0.939054 0.343770i \(-0.111704\pi\)
\(684\) 0 0
\(685\) 20.9906 + 2.94914i 0.802011 + 0.112681i
\(686\) 0 0
\(687\) 14.4701i 0.552070i
\(688\) 0 0
\(689\) −15.6128 −0.594802
\(690\) 0 0
\(691\) −0.355509 −0.0135242 −0.00676211 0.999977i \(-0.502152\pi\)
−0.00676211 + 0.999977i \(0.502152\pi\)
\(692\) 0 0
\(693\) 5.05086i 0.191866i
\(694\) 0 0
\(695\) −3.15257 + 22.4385i −0.119584 + 0.851140i
\(696\) 0 0
\(697\) 24.2667i 0.919167i
\(698\) 0 0
\(699\) 3.96836 0.150097
\(700\) 0 0
\(701\) 13.2257 0.499528 0.249764 0.968307i \(-0.419647\pi\)
0.249764 + 0.968307i \(0.419647\pi\)
\(702\) 0 0
\(703\) 40.0000i 1.50863i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.47949i 0.356513i
\(708\) 0 0
\(709\) 10.9777 0.412277 0.206139 0.978523i \(-0.433910\pi\)
0.206139 + 0.978523i \(0.433910\pi\)
\(710\) 0 0
\(711\) −1.24443 −0.0466698
\(712\) 0 0
\(713\) 28.8198i 1.07931i
\(714\) 0 0
\(715\) −37.7778 5.30772i −1.41281 0.198497i
\(716\) 0 0
\(717\) 1.90813i 0.0712606i
\(718\) 0 0
\(719\) −21.9813 −0.819763 −0.409881 0.912139i \(-0.634430\pi\)
−0.409881 + 0.912139i \(0.634430\pi\)
\(720\) 0 0
\(721\) −16.8573 −0.627798
\(722\) 0 0
\(723\) 0.755569i 0.0280999i
\(724\) 0 0
\(725\) −9.61285 2.75557i −0.357012 0.102339i
\(726\) 0 0
\(727\) 13.0607i 0.484395i 0.970227 + 0.242197i \(0.0778681\pi\)
−0.970227 + 0.242197i \(0.922132\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −8.94025 −0.330667
\(732\) 0 0
\(733\) 13.5625i 0.500941i −0.968124 0.250471i \(-0.919415\pi\)
0.968124 0.250471i \(-0.0805853\pi\)
\(734\) 0 0
\(735\) 2.21432 + 0.311108i 0.0816764 + 0.0114754i
\(736\) 0 0
\(737\) 18.2480i 0.672173i
\(738\) 0 0
\(739\) 10.2854 0.378356 0.189178 0.981943i \(-0.439418\pi\)
0.189178 + 0.981943i \(0.439418\pi\)
\(740\) 0 0
\(741\) −27.8163 −1.02186
\(742\) 0 0
\(743\) 21.4608i 0.787319i 0.919256 + 0.393659i \(0.128791\pi\)
−0.919256 + 0.393659i \(0.871209\pi\)
\(744\) 0 0
\(745\) 2.25380 16.0415i 0.0825728 0.587715i
\(746\) 0 0
\(747\) 11.6128i 0.424892i
\(748\) 0 0
\(749\) 15.4795 0.565608
\(750\) 0 0
\(751\) −46.0642 −1.68091 −0.840454 0.541883i \(-0.817712\pi\)
−0.840454 + 0.541883i \(0.817712\pi\)
\(752\) 0 0
\(753\) 9.12399i 0.332497i
\(754\) 0 0
\(755\) −2.75557 + 19.6128i −0.100285 + 0.713785i
\(756\) 0 0
\(757\) 27.1052i 0.985157i 0.870268 + 0.492579i \(0.163946\pi\)
−0.870268 + 0.492579i \(0.836054\pi\)
\(758\) 0 0
\(759\) −31.4924 −1.14310
\(760\) 0 0
\(761\) 39.4608 1.43045 0.715226 0.698894i \(-0.246324\pi\)
0.715226 + 0.698894i \(0.246324\pi\)
\(762\) 0 0
\(763\) 1.61285i 0.0583890i
\(764\) 0 0
\(765\) −15.9081 2.23506i −0.575160 0.0808089i
\(766\) 0 0
\(767\) 39.2257i 1.41636i
\(768\) 0 0
\(769\) 7.51114 0.270859 0.135429 0.990787i \(-0.456759\pi\)
0.135429 + 0.990787i \(0.456759\pi\)
\(770\) 0 0
\(771\) −1.28592 −0.0463112
\(772\) 0 0
\(773\) 1.46965i 0.0528596i −0.999651 0.0264298i \(-0.991586\pi\)
0.999651 0.0264298i \(-0.00841385\pi\)
\(774\) 0 0
\(775\) −22.2163 6.36842i −0.798034 0.228760i
\(776\) 0 0
\(777\) 4.85728i 0.174254i
\(778\) 0 0
\(779\) −27.8163 −0.996621
\(780\) 0 0
\(781\) 52.0000 1.86071
\(782\) 0 0
\(783\) 2.00000i 0.0714742i
\(784\) 0 0
\(785\) 29.8479 + 4.19358i 1.06532 + 0.149675i
\(786\) 0 0
\(787\) 16.2034i 0.577590i 0.957391 + 0.288795i \(0.0932545\pi\)
−0.957391 + 0.288795i \(0.906745\pi\)
\(788\) 0 0
\(789\) 7.00937 0.249540
\(790\) 0 0
\(791\) 1.86665 0.0663703
\(792\) 0 0
\(793\) 1.65127i 0.0586384i
\(794\) 0 0
\(795\) 1.43801 10.2351i 0.0510009 0.363000i
\(796\) 0 0
\(797\) 0.161933i 0.00573597i −0.999996 0.00286799i \(-0.999087\pi\)
0.999996 0.00286799i \(-0.000912909\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 6.99063 0.247002
\(802\) 0 0