Properties

Label 6160.2.a.t.1.1
Level $6160$
Weight $2$
Character 6160.1
Self dual yes
Analytic conductor $49.188$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(49.1878476451\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 6160.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.73205 q^{3} -1.00000 q^{5} -1.00000 q^{7} +4.46410 q^{9} +O(q^{10})\) \(q-2.73205 q^{3} -1.00000 q^{5} -1.00000 q^{7} +4.46410 q^{9} -1.00000 q^{11} -1.46410 q^{13} +2.73205 q^{15} -3.46410 q^{17} -6.73205 q^{19} +2.73205 q^{21} +8.19615 q^{23} +1.00000 q^{25} -4.00000 q^{27} -4.73205 q^{29} -2.00000 q^{31} +2.73205 q^{33} +1.00000 q^{35} +0.732051 q^{37} +4.00000 q^{39} -2.19615 q^{41} -2.00000 q^{43} -4.46410 q^{45} -6.92820 q^{47} +1.00000 q^{49} +9.46410 q^{51} -7.26795 q^{53} +1.00000 q^{55} +18.3923 q^{57} -6.92820 q^{59} -4.92820 q^{61} -4.46410 q^{63} +1.46410 q^{65} +4.00000 q^{67} -22.3923 q^{69} -9.46410 q^{71} -14.3923 q^{73} -2.73205 q^{75} +1.00000 q^{77} +12.1962 q^{79} -2.46410 q^{81} +16.3923 q^{83} +3.46410 q^{85} +12.9282 q^{87} +3.46410 q^{89} +1.46410 q^{91} +5.46410 q^{93} +6.73205 q^{95} +14.5885 q^{97} -4.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} + 4 q^{13} + 2 q^{15} - 10 q^{19} + 2 q^{21} + 6 q^{23} + 2 q^{25} - 8 q^{27} - 6 q^{29} - 4 q^{31} + 2 q^{33} + 2 q^{35} - 2 q^{37} + 8 q^{39} + 6 q^{41} - 4 q^{43} - 2 q^{45} + 2 q^{49} + 12 q^{51} - 18 q^{53} + 2 q^{55} + 16 q^{57} + 4 q^{61} - 2 q^{63} - 4 q^{65} + 8 q^{67} - 24 q^{69} - 12 q^{71} - 8 q^{73} - 2 q^{75} + 2 q^{77} + 14 q^{79} + 2 q^{81} + 12 q^{83} + 12 q^{87} - 4 q^{91} + 4 q^{93} + 10 q^{95} - 2 q^{97} - 2 q^{99} + O(q^{100}) \)

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
\(3\) −2.73205 −1.57735 −0.788675 0.614810i \(-0.789233\pi\)
−0.788675 + 0.614810i \(0.789233\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 4.46410 1.48803
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −1.46410 −0.406069 −0.203034 0.979172i \(-0.565080\pi\)
−0.203034 + 0.979172i \(0.565080\pi\)
\(14\) 0 0
\(15\) 2.73205 0.705412
\(16\) 0 0
\(17\) −3.46410 −0.840168 −0.420084 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 0 0
\(19\) −6.73205 −1.54444 −0.772219 0.635356i \(-0.780853\pi\)
−0.772219 + 0.635356i \(0.780853\pi\)
\(20\) 0 0
\(21\) 2.73205 0.596182
\(22\) 0 0
\(23\) 8.19615 1.70902 0.854508 0.519438i \(-0.173859\pi\)
0.854508 + 0.519438i \(0.173859\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −4.73205 −0.878720 −0.439360 0.898311i \(-0.644795\pi\)
−0.439360 + 0.898311i \(0.644795\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 2.73205 0.475589
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 0.732051 0.120348 0.0601742 0.998188i \(-0.480834\pi\)
0.0601742 + 0.998188i \(0.480834\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −2.19615 −0.342981 −0.171491 0.985186i \(-0.554858\pi\)
−0.171491 + 0.985186i \(0.554858\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) −4.46410 −0.665469
\(46\) 0 0
\(47\) −6.92820 −1.01058 −0.505291 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 9.46410 1.32524
\(52\) 0 0
\(53\) −7.26795 −0.998330 −0.499165 0.866507i \(-0.666360\pi\)
−0.499165 + 0.866507i \(0.666360\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 18.3923 2.43612
\(58\) 0 0
\(59\) −6.92820 −0.901975 −0.450988 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) 0 0
\(61\) −4.92820 −0.630992 −0.315496 0.948927i \(-0.602171\pi\)
−0.315496 + 0.948927i \(0.602171\pi\)
\(62\) 0 0
\(63\) −4.46410 −0.562424
\(64\) 0 0
\(65\) 1.46410 0.181599
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −22.3923 −2.69572
\(70\) 0 0
\(71\) −9.46410 −1.12318 −0.561591 0.827415i \(-0.689811\pi\)
−0.561591 + 0.827415i \(0.689811\pi\)
\(72\) 0 0
\(73\) −14.3923 −1.68449 −0.842246 0.539093i \(-0.818767\pi\)
−0.842246 + 0.539093i \(0.818767\pi\)
\(74\) 0 0
\(75\) −2.73205 −0.315470
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 12.1962 1.37217 0.686087 0.727519i \(-0.259327\pi\)
0.686087 + 0.727519i \(0.259327\pi\)
\(80\) 0 0
\(81\) −2.46410 −0.273789
\(82\) 0 0
\(83\) 16.3923 1.79929 0.899645 0.436623i \(-0.143826\pi\)
0.899645 + 0.436623i \(0.143826\pi\)
\(84\) 0 0
\(85\) 3.46410 0.375735
\(86\) 0 0
\(87\) 12.9282 1.38605
\(88\) 0 0
\(89\) 3.46410 0.367194 0.183597 0.983002i \(-0.441226\pi\)
0.183597 + 0.983002i \(0.441226\pi\)
\(90\) 0 0
\(91\) 1.46410 0.153480
\(92\) 0 0
\(93\) 5.46410 0.566601
\(94\) 0 0
\(95\) 6.73205 0.690694
\(96\) 0 0
\(97\) 14.5885 1.48123 0.740617 0.671928i \(-0.234533\pi\)
0.740617 + 0.671928i \(0.234533\pi\)
\(98\) 0 0
\(99\) −4.46410 −0.448659
\(100\) 0 0
\(101\) −7.85641 −0.781742 −0.390871 0.920446i \(-0.627826\pi\)
−0.390871 + 0.920446i \(0.627826\pi\)
\(102\) 0 0
\(103\) −12.3923 −1.22105 −0.610525 0.791997i \(-0.709042\pi\)
−0.610525 + 0.791997i \(0.709042\pi\)
\(104\) 0 0
\(105\) −2.73205 −0.266621
\(106\) 0 0
\(107\) −7.85641 −0.759507 −0.379754 0.925088i \(-0.623991\pi\)
−0.379754 + 0.925088i \(0.623991\pi\)
\(108\) 0 0
\(109\) −15.6603 −1.49998 −0.749990 0.661449i \(-0.769942\pi\)
−0.749990 + 0.661449i \(0.769942\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) 7.85641 0.739069 0.369534 0.929217i \(-0.379517\pi\)
0.369534 + 0.929217i \(0.379517\pi\)
\(114\) 0 0
\(115\) −8.19615 −0.764295
\(116\) 0 0
\(117\) −6.53590 −0.604244
\(118\) 0 0
\(119\) 3.46410 0.317554
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 6.00000 0.541002
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −1.07180 −0.0951066 −0.0475533 0.998869i \(-0.515142\pi\)
−0.0475533 + 0.998869i \(0.515142\pi\)
\(128\) 0 0
\(129\) 5.46410 0.481087
\(130\) 0 0
\(131\) −5.66025 −0.494539 −0.247269 0.968947i \(-0.579533\pi\)
−0.247269 + 0.968947i \(0.579533\pi\)
\(132\) 0 0
\(133\) 6.73205 0.583743
\(134\) 0 0
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) −0.928203 −0.0793018 −0.0396509 0.999214i \(-0.512625\pi\)
−0.0396509 + 0.999214i \(0.512625\pi\)
\(138\) 0 0
\(139\) −13.6603 −1.15865 −0.579324 0.815097i \(-0.696683\pi\)
−0.579324 + 0.815097i \(0.696683\pi\)
\(140\) 0 0
\(141\) 18.9282 1.59404
\(142\) 0 0
\(143\) 1.46410 0.122434
\(144\) 0 0
\(145\) 4.73205 0.392975
\(146\) 0 0
\(147\) −2.73205 −0.225336
\(148\) 0 0
\(149\) −7.26795 −0.595414 −0.297707 0.954657i \(-0.596222\pi\)
−0.297707 + 0.954657i \(0.596222\pi\)
\(150\) 0 0
\(151\) −11.1244 −0.905287 −0.452644 0.891692i \(-0.649519\pi\)
−0.452644 + 0.891692i \(0.649519\pi\)
\(152\) 0 0
\(153\) −15.4641 −1.25020
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) 4.53590 0.362004 0.181002 0.983483i \(-0.442066\pi\)
0.181002 + 0.983483i \(0.442066\pi\)
\(158\) 0 0
\(159\) 19.8564 1.57472
\(160\) 0 0
\(161\) −8.19615 −0.645947
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) −2.73205 −0.212690
\(166\) 0 0
\(167\) −13.8564 −1.07224 −0.536120 0.844141i \(-0.680111\pi\)
−0.536120 + 0.844141i \(0.680111\pi\)
\(168\) 0 0
\(169\) −10.8564 −0.835108
\(170\) 0 0
\(171\) −30.0526 −2.29818
\(172\) 0 0
\(173\) 0.928203 0.0705700 0.0352850 0.999377i \(-0.488766\pi\)
0.0352850 + 0.999377i \(0.488766\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 18.9282 1.42273
\(178\) 0 0
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 13.4641 0.995295
\(184\) 0 0
\(185\) −0.732051 −0.0538214
\(186\) 0 0
\(187\) 3.46410 0.253320
\(188\) 0 0
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) 12.3923 0.892018 0.446009 0.895029i \(-0.352845\pi\)
0.446009 + 0.895029i \(0.352845\pi\)
\(194\) 0 0
\(195\) −4.00000 −0.286446
\(196\) 0 0
\(197\) −24.2487 −1.72765 −0.863825 0.503793i \(-0.831938\pi\)
−0.863825 + 0.503793i \(0.831938\pi\)
\(198\) 0 0
\(199\) −2.92820 −0.207575 −0.103787 0.994600i \(-0.533096\pi\)
−0.103787 + 0.994600i \(0.533096\pi\)
\(200\) 0 0
\(201\) −10.9282 −0.770816
\(202\) 0 0
\(203\) 4.73205 0.332125
\(204\) 0 0
\(205\) 2.19615 0.153386
\(206\) 0 0
\(207\) 36.5885 2.54307
\(208\) 0 0
\(209\) 6.73205 0.465666
\(210\) 0 0
\(211\) −26.9282 −1.85381 −0.926907 0.375291i \(-0.877543\pi\)
−0.926907 + 0.375291i \(0.877543\pi\)
\(212\) 0 0
\(213\) 25.8564 1.77165
\(214\) 0 0
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) 0 0
\(219\) 39.3205 2.65703
\(220\) 0 0
\(221\) 5.07180 0.341166
\(222\) 0 0
\(223\) 25.4641 1.70520 0.852601 0.522562i \(-0.175024\pi\)
0.852601 + 0.522562i \(0.175024\pi\)
\(224\) 0 0
\(225\) 4.46410 0.297607
\(226\) 0 0
\(227\) −6.92820 −0.459841 −0.229920 0.973209i \(-0.573847\pi\)
−0.229920 + 0.973209i \(0.573847\pi\)
\(228\) 0 0
\(229\) 24.3923 1.61189 0.805944 0.591991i \(-0.201658\pi\)
0.805944 + 0.591991i \(0.201658\pi\)
\(230\) 0 0
\(231\) −2.73205 −0.179756
\(232\) 0 0
\(233\) −7.85641 −0.514690 −0.257345 0.966320i \(-0.582848\pi\)
−0.257345 + 0.966320i \(0.582848\pi\)
\(234\) 0 0
\(235\) 6.92820 0.451946
\(236\) 0 0
\(237\) −33.3205 −2.16440
\(238\) 0 0
\(239\) −1.26795 −0.0820168 −0.0410084 0.999159i \(-0.513057\pi\)
−0.0410084 + 0.999159i \(0.513057\pi\)
\(240\) 0 0
\(241\) 3.26795 0.210507 0.105254 0.994445i \(-0.466435\pi\)
0.105254 + 0.994445i \(0.466435\pi\)
\(242\) 0 0
\(243\) 18.7321 1.20166
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 9.85641 0.627148
\(248\) 0 0
\(249\) −44.7846 −2.83811
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) −8.19615 −0.515288
\(254\) 0 0
\(255\) −9.46410 −0.592665
\(256\) 0 0
\(257\) −23.6603 −1.47589 −0.737943 0.674863i \(-0.764203\pi\)
−0.737943 + 0.674863i \(0.764203\pi\)
\(258\) 0 0
\(259\) −0.732051 −0.0454874
\(260\) 0 0
\(261\) −21.1244 −1.30756
\(262\) 0 0
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 7.26795 0.446467
\(266\) 0 0
\(267\) −9.46410 −0.579194
\(268\) 0 0
\(269\) 28.3923 1.73111 0.865555 0.500814i \(-0.166966\pi\)
0.865555 + 0.500814i \(0.166966\pi\)
\(270\) 0 0
\(271\) −0.392305 −0.0238308 −0.0119154 0.999929i \(-0.503793\pi\)
−0.0119154 + 0.999929i \(0.503793\pi\)
\(272\) 0 0
\(273\) −4.00000 −0.242091
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 7.07180 0.424903 0.212452 0.977172i \(-0.431855\pi\)
0.212452 + 0.977172i \(0.431855\pi\)
\(278\) 0 0
\(279\) −8.92820 −0.534518
\(280\) 0 0
\(281\) 22.3923 1.33581 0.667906 0.744245i \(-0.267191\pi\)
0.667906 + 0.744245i \(0.267191\pi\)
\(282\) 0 0
\(283\) 31.7128 1.88513 0.942566 0.334021i \(-0.108406\pi\)
0.942566 + 0.334021i \(0.108406\pi\)
\(284\) 0 0
\(285\) −18.3923 −1.08947
\(286\) 0 0
\(287\) 2.19615 0.129635
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) −39.8564 −2.33642
\(292\) 0 0
\(293\) 21.4641 1.25395 0.626973 0.779041i \(-0.284294\pi\)
0.626973 + 0.779041i \(0.284294\pi\)
\(294\) 0 0
\(295\) 6.92820 0.403376
\(296\) 0 0
\(297\) 4.00000 0.232104
\(298\) 0 0
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) 2.00000 0.115278
\(302\) 0 0
\(303\) 21.4641 1.23308
\(304\) 0 0
\(305\) 4.92820 0.282188
\(306\) 0 0
\(307\) −24.3923 −1.39214 −0.696071 0.717973i \(-0.745070\pi\)
−0.696071 + 0.717973i \(0.745070\pi\)
\(308\) 0 0
\(309\) 33.8564 1.92602
\(310\) 0 0
\(311\) −24.9282 −1.41355 −0.706774 0.707439i \(-0.749850\pi\)
−0.706774 + 0.707439i \(0.749850\pi\)
\(312\) 0 0
\(313\) 22.1962 1.25460 0.627300 0.778777i \(-0.284160\pi\)
0.627300 + 0.778777i \(0.284160\pi\)
\(314\) 0 0
\(315\) 4.46410 0.251524
\(316\) 0 0
\(317\) 30.5885 1.71802 0.859009 0.511960i \(-0.171080\pi\)
0.859009 + 0.511960i \(0.171080\pi\)
\(318\) 0 0
\(319\) 4.73205 0.264944
\(320\) 0 0
\(321\) 21.4641 1.19801
\(322\) 0 0
\(323\) 23.3205 1.29759
\(324\) 0 0
\(325\) −1.46410 −0.0812137
\(326\) 0 0
\(327\) 42.7846 2.36599
\(328\) 0 0
\(329\) 6.92820 0.381964
\(330\) 0 0
\(331\) 18.7846 1.03250 0.516248 0.856439i \(-0.327328\pi\)
0.516248 + 0.856439i \(0.327328\pi\)
\(332\) 0 0
\(333\) 3.26795 0.179083
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 22.7846 1.24116 0.620578 0.784144i \(-0.286898\pi\)
0.620578 + 0.784144i \(0.286898\pi\)
\(338\) 0 0
\(339\) −21.4641 −1.16577
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 22.3923 1.20556
\(346\) 0 0
\(347\) 23.0718 1.23856 0.619279 0.785171i \(-0.287425\pi\)
0.619279 + 0.785171i \(0.287425\pi\)
\(348\) 0 0
\(349\) −5.60770 −0.300173 −0.150087 0.988673i \(-0.547955\pi\)
−0.150087 + 0.988673i \(0.547955\pi\)
\(350\) 0 0
\(351\) 5.85641 0.312592
\(352\) 0 0
\(353\) 14.1962 0.755585 0.377792 0.925890i \(-0.376683\pi\)
0.377792 + 0.925890i \(0.376683\pi\)
\(354\) 0 0
\(355\) 9.46410 0.502302
\(356\) 0 0
\(357\) −9.46410 −0.500893
\(358\) 0 0
\(359\) 34.0526 1.79723 0.898613 0.438743i \(-0.144576\pi\)
0.898613 + 0.438743i \(0.144576\pi\)
\(360\) 0 0
\(361\) 26.3205 1.38529
\(362\) 0 0
\(363\) −2.73205 −0.143395
\(364\) 0 0
\(365\) 14.3923 0.753328
\(366\) 0 0
\(367\) −12.3923 −0.646873 −0.323437 0.946250i \(-0.604838\pi\)
−0.323437 + 0.946250i \(0.604838\pi\)
\(368\) 0 0
\(369\) −9.80385 −0.510368
\(370\) 0 0
\(371\) 7.26795 0.377333
\(372\) 0 0
\(373\) 18.3923 0.952317 0.476159 0.879359i \(-0.342029\pi\)
0.476159 + 0.879359i \(0.342029\pi\)
\(374\) 0 0
\(375\) 2.73205 0.141082
\(376\) 0 0
\(377\) 6.92820 0.356821
\(378\) 0 0
\(379\) −6.14359 −0.315575 −0.157788 0.987473i \(-0.550436\pi\)
−0.157788 + 0.987473i \(0.550436\pi\)
\(380\) 0 0
\(381\) 2.92820 0.150016
\(382\) 0 0
\(383\) 33.4641 1.70994 0.854968 0.518681i \(-0.173577\pi\)
0.854968 + 0.518681i \(0.173577\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 0 0
\(387\) −8.92820 −0.453846
\(388\) 0 0
\(389\) −1.60770 −0.0815134 −0.0407567 0.999169i \(-0.512977\pi\)
−0.0407567 + 0.999169i \(0.512977\pi\)
\(390\) 0 0
\(391\) −28.3923 −1.43586
\(392\) 0 0
\(393\) 15.4641 0.780061
\(394\) 0 0
\(395\) −12.1962 −0.613655
\(396\) 0 0
\(397\) 30.3923 1.52535 0.762673 0.646784i \(-0.223887\pi\)
0.762673 + 0.646784i \(0.223887\pi\)
\(398\) 0 0
\(399\) −18.3923 −0.920767
\(400\) 0 0
\(401\) 2.53590 0.126637 0.0633184 0.997993i \(-0.479832\pi\)
0.0633184 + 0.997993i \(0.479832\pi\)
\(402\) 0 0
\(403\) 2.92820 0.145864
\(404\) 0 0
\(405\) 2.46410 0.122442
\(406\) 0 0
\(407\) −0.732051 −0.0362864
\(408\) 0 0
\(409\) 10.8756 0.537766 0.268883 0.963173i \(-0.413346\pi\)
0.268883 + 0.963173i \(0.413346\pi\)
\(410\) 0 0
\(411\) 2.53590 0.125087
\(412\) 0 0
\(413\) 6.92820 0.340915
\(414\) 0 0
\(415\) −16.3923 −0.804667
\(416\) 0 0
\(417\) 37.3205 1.82759
\(418\) 0 0
\(419\) −30.9282 −1.51094 −0.755471 0.655182i \(-0.772592\pi\)
−0.755471 + 0.655182i \(0.772592\pi\)
\(420\) 0 0
\(421\) −35.8564 −1.74753 −0.873767 0.486344i \(-0.838330\pi\)
−0.873767 + 0.486344i \(0.838330\pi\)
\(422\) 0 0
\(423\) −30.9282 −1.50378
\(424\) 0 0
\(425\) −3.46410 −0.168034
\(426\) 0 0
\(427\) 4.92820 0.238492
\(428\) 0 0
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) −3.12436 −0.150495 −0.0752475 0.997165i \(-0.523975\pi\)
−0.0752475 + 0.997165i \(0.523975\pi\)
\(432\) 0 0
\(433\) −18.1962 −0.874451 −0.437226 0.899352i \(-0.644039\pi\)
−0.437226 + 0.899352i \(0.644039\pi\)
\(434\) 0 0
\(435\) −12.9282 −0.619860
\(436\) 0 0
\(437\) −55.1769 −2.63947
\(438\) 0 0
\(439\) −26.2487 −1.25278 −0.626391 0.779509i \(-0.715469\pi\)
−0.626391 + 0.779509i \(0.715469\pi\)
\(440\) 0 0
\(441\) 4.46410 0.212576
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) −3.46410 −0.164214
\(446\) 0 0
\(447\) 19.8564 0.939176
\(448\) 0 0
\(449\) 40.3923 1.90623 0.953115 0.302607i \(-0.0978570\pi\)
0.953115 + 0.302607i \(0.0978570\pi\)
\(450\) 0 0
\(451\) 2.19615 0.103413
\(452\) 0 0
\(453\) 30.3923 1.42796
\(454\) 0 0
\(455\) −1.46410 −0.0686381
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 0 0
\(459\) 13.8564 0.646762
\(460\) 0 0
\(461\) 22.3923 1.04291 0.521457 0.853278i \(-0.325389\pi\)
0.521457 + 0.853278i \(0.325389\pi\)
\(462\) 0 0
\(463\) −27.5167 −1.27881 −0.639404 0.768871i \(-0.720819\pi\)
−0.639404 + 0.768871i \(0.720819\pi\)
\(464\) 0 0
\(465\) −5.46410 −0.253392
\(466\) 0 0
\(467\) 34.0526 1.57576 0.787882 0.615826i \(-0.211178\pi\)
0.787882 + 0.615826i \(0.211178\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) −12.3923 −0.571007
\(472\) 0 0
\(473\) 2.00000 0.0919601
\(474\) 0 0
\(475\) −6.73205 −0.308888
\(476\) 0 0
\(477\) −32.4449 −1.48555
\(478\) 0 0
\(479\) 32.7846 1.49797 0.748984 0.662589i \(-0.230542\pi\)
0.748984 + 0.662589i \(0.230542\pi\)
\(480\) 0 0
\(481\) −1.07180 −0.0488697
\(482\) 0 0
\(483\) 22.3923 1.01889
\(484\) 0 0
\(485\) −14.5885 −0.662428
\(486\) 0 0
\(487\) −4.19615 −0.190146 −0.0950729 0.995470i \(-0.530308\pi\)
−0.0950729 + 0.995470i \(0.530308\pi\)
\(488\) 0 0
\(489\) −10.9282 −0.494190
\(490\) 0 0
\(491\) 27.7128 1.25066 0.625331 0.780360i \(-0.284964\pi\)
0.625331 + 0.780360i \(0.284964\pi\)
\(492\) 0 0
\(493\) 16.3923 0.738272
\(494\) 0 0
\(495\) 4.46410 0.200646
\(496\) 0 0
\(497\) 9.46410 0.424523
\(498\) 0 0
\(499\) −12.1436 −0.543622 −0.271811 0.962351i \(-0.587623\pi\)
−0.271811 + 0.962351i \(0.587623\pi\)
\(500\) 0 0
\(501\) 37.8564 1.69130
\(502\) 0 0
\(503\) −8.78461 −0.391686 −0.195843 0.980635i \(-0.562744\pi\)
−0.195843 + 0.980635i \(0.562744\pi\)
\(504\) 0 0
\(505\) 7.85641 0.349605
\(506\) 0 0
\(507\) 29.6603 1.31726
\(508\) 0 0
\(509\) 11.0718 0.490749 0.245374 0.969428i \(-0.421089\pi\)
0.245374 + 0.969428i \(0.421089\pi\)
\(510\) 0 0
\(511\) 14.3923 0.636678
\(512\) 0 0
\(513\) 26.9282 1.18891
\(514\) 0 0
\(515\) 12.3923 0.546070
\(516\) 0 0
\(517\) 6.92820 0.304702
\(518\) 0 0
\(519\) −2.53590 −0.111314
\(520\) 0 0
\(521\) 10.3923 0.455295 0.227648 0.973744i \(-0.426897\pi\)
0.227648 + 0.973744i \(0.426897\pi\)
\(522\) 0 0
\(523\) 29.1769 1.27582 0.637909 0.770112i \(-0.279800\pi\)
0.637909 + 0.770112i \(0.279800\pi\)
\(524\) 0 0
\(525\) 2.73205 0.119236
\(526\) 0 0
\(527\) 6.92820 0.301797
\(528\) 0 0
\(529\) 44.1769 1.92074
\(530\) 0 0
\(531\) −30.9282 −1.34217
\(532\) 0 0
\(533\) 3.21539 0.139274
\(534\) 0 0
\(535\) 7.85641 0.339662
\(536\) 0 0
\(537\) −16.3923 −0.707380
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −20.7321 −0.891340 −0.445670 0.895197i \(-0.647035\pi\)
−0.445670 + 0.895197i \(0.647035\pi\)
\(542\) 0 0
\(543\) −38.2487 −1.64141
\(544\) 0 0
\(545\) 15.6603 0.670812
\(546\) 0 0
\(547\) −28.7846 −1.23074 −0.615371 0.788238i \(-0.710994\pi\)
−0.615371 + 0.788238i \(0.710994\pi\)
\(548\) 0 0
\(549\) −22.0000 −0.938937
\(550\) 0 0
\(551\) 31.8564 1.35713
\(552\) 0 0
\(553\) −12.1962 −0.518633
\(554\) 0 0
\(555\) 2.00000 0.0848953
\(556\) 0 0
\(557\) −25.6077 −1.08503 −0.542516 0.840045i \(-0.682528\pi\)
−0.542516 + 0.840045i \(0.682528\pi\)
\(558\) 0 0
\(559\) 2.92820 0.123850
\(560\) 0 0
\(561\) −9.46410 −0.399575
\(562\) 0 0
\(563\) −5.07180 −0.213751 −0.106875 0.994272i \(-0.534085\pi\)
−0.106875 + 0.994272i \(0.534085\pi\)
\(564\) 0 0
\(565\) −7.85641 −0.330522
\(566\) 0 0
\(567\) 2.46410 0.103483
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −3.60770 −0.150977 −0.0754887 0.997147i \(-0.524052\pi\)
−0.0754887 + 0.997147i \(0.524052\pi\)
\(572\) 0 0
\(573\) −32.7846 −1.36960
\(574\) 0 0
\(575\) 8.19615 0.341803
\(576\) 0 0
\(577\) −34.5885 −1.43994 −0.719968 0.694007i \(-0.755844\pi\)
−0.719968 + 0.694007i \(0.755844\pi\)
\(578\) 0 0
\(579\) −33.8564 −1.40702
\(580\) 0 0
\(581\) −16.3923 −0.680067
\(582\) 0 0
\(583\) 7.26795 0.301008
\(584\) 0 0
\(585\) 6.53590 0.270226
\(586\) 0 0
\(587\) 11.4115 0.471005 0.235502 0.971874i \(-0.424326\pi\)
0.235502 + 0.971874i \(0.424326\pi\)
\(588\) 0 0
\(589\) 13.4641 0.554779
\(590\) 0 0
\(591\) 66.2487 2.72511
\(592\) 0 0
\(593\) 0.248711 0.0102133 0.00510667 0.999987i \(-0.498374\pi\)
0.00510667 + 0.999987i \(0.498374\pi\)
\(594\) 0 0
\(595\) −3.46410 −0.142014
\(596\) 0 0
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) −37.1769 −1.51901 −0.759504 0.650503i \(-0.774558\pi\)
−0.759504 + 0.650503i \(0.774558\pi\)
\(600\) 0 0
\(601\) −5.51666 −0.225029 −0.112515 0.993650i \(-0.535891\pi\)
−0.112515 + 0.993650i \(0.535891\pi\)
\(602\) 0 0
\(603\) 17.8564 0.727169
\(604\) 0 0
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) −20.9282 −0.849450 −0.424725 0.905323i \(-0.639629\pi\)
−0.424725 + 0.905323i \(0.639629\pi\)
\(608\) 0 0
\(609\) −12.9282 −0.523877
\(610\) 0 0
\(611\) 10.1436 0.410366
\(612\) 0 0
\(613\) −35.1769 −1.42078 −0.710391 0.703807i \(-0.751482\pi\)
−0.710391 + 0.703807i \(0.751482\pi\)
\(614\) 0 0
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) 17.3205 0.697297 0.348649 0.937253i \(-0.386641\pi\)
0.348649 + 0.937253i \(0.386641\pi\)
\(618\) 0 0
\(619\) −28.7846 −1.15695 −0.578476 0.815700i \(-0.696352\pi\)
−0.578476 + 0.815700i \(0.696352\pi\)
\(620\) 0 0
\(621\) −32.7846 −1.31560
\(622\) 0 0
\(623\) −3.46410 −0.138786
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −18.3923 −0.734518
\(628\) 0 0
\(629\) −2.53590 −0.101113
\(630\) 0 0
\(631\) 46.9282 1.86818 0.934091 0.357035i \(-0.116212\pi\)
0.934091 + 0.357035i \(0.116212\pi\)
\(632\) 0 0
\(633\) 73.5692 2.92411
\(634\) 0 0
\(635\) 1.07180 0.0425330
\(636\) 0 0
\(637\) −1.46410 −0.0580098
\(638\) 0 0
\(639\) −42.2487 −1.67133
\(640\) 0 0
\(641\) −35.5692 −1.40490 −0.702450 0.711733i \(-0.747910\pi\)
−0.702450 + 0.711733i \(0.747910\pi\)
\(642\) 0 0
\(643\) −33.2679 −1.31196 −0.655980 0.754778i \(-0.727744\pi\)
−0.655980 + 0.754778i \(0.727744\pi\)
\(644\) 0 0
\(645\) −5.46410 −0.215149
\(646\) 0 0
\(647\) 26.5359 1.04323 0.521617 0.853180i \(-0.325329\pi\)
0.521617 + 0.853180i \(0.325329\pi\)
\(648\) 0 0
\(649\) 6.92820 0.271956
\(650\) 0 0
\(651\) −5.46410 −0.214155
\(652\) 0 0
\(653\) 11.6603 0.456301 0.228151 0.973626i \(-0.426732\pi\)
0.228151 + 0.973626i \(0.426732\pi\)
\(654\) 0 0
\(655\) 5.66025 0.221164
\(656\) 0 0
\(657\) −64.2487 −2.50658
\(658\) 0 0
\(659\) −30.2487 −1.17832 −0.589161 0.808015i \(-0.700542\pi\)
−0.589161 + 0.808015i \(0.700542\pi\)
\(660\) 0 0
\(661\) −46.0000 −1.78919 −0.894596 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) 0 0
\(663\) −13.8564 −0.538138
\(664\) 0 0
\(665\) −6.73205 −0.261058
\(666\) 0 0
\(667\) −38.7846 −1.50175
\(668\) 0 0
\(669\) −69.5692 −2.68970
\(670\) 0 0
\(671\) 4.92820 0.190251
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) −4.14359 −0.159251 −0.0796256 0.996825i \(-0.525372\pi\)
−0.0796256 + 0.996825i \(0.525372\pi\)
\(678\) 0 0
\(679\) −14.5885 −0.559854
\(680\) 0 0
\(681\) 18.9282 0.725330
\(682\) 0 0
\(683\) −6.24871 −0.239100 −0.119550 0.992828i \(-0.538145\pi\)
−0.119550 + 0.992828i \(0.538145\pi\)
\(684\) 0 0
\(685\) 0.928203 0.0354648
\(686\) 0 0
\(687\) −66.6410 −2.54251
\(688\) 0 0
\(689\) 10.6410 0.405390
\(690\) 0 0
\(691\) 13.4641 0.512199 0.256099 0.966650i \(-0.417563\pi\)
0.256099 + 0.966650i \(0.417563\pi\)
\(692\) 0 0
\(693\) 4.46410 0.169577
\(694\) 0 0
\(695\) 13.6603 0.518163
\(696\) 0 0
\(697\) 7.60770 0.288162
\(698\) 0 0
\(699\) 21.4641 0.811847
\(700\) 0 0
\(701\) 41.9090 1.58288 0.791440 0.611247i \(-0.209332\pi\)
0.791440 + 0.611247i \(0.209332\pi\)
\(702\) 0 0
\(703\) −4.92820 −0.185871
\(704\) 0 0
\(705\) −18.9282 −0.712877
\(706\) 0 0
\(707\) 7.85641 0.295471
\(708\) 0 0
\(709\) 25.3205 0.950932 0.475466 0.879734i \(-0.342280\pi\)
0.475466 + 0.879734i \(0.342280\pi\)
\(710\) 0 0
\(711\) 54.4449 2.04184
\(712\) 0 0
\(713\) −16.3923 −0.613897
\(714\) 0 0
\(715\) −1.46410 −0.0547543
\(716\) 0 0
\(717\) 3.46410 0.129369
\(718\) 0 0
\(719\) 27.7128 1.03351 0.516757 0.856132i \(-0.327139\pi\)
0.516757 + 0.856132i \(0.327139\pi\)
\(720\) 0 0
\(721\) 12.3923 0.461514
\(722\) 0 0
\(723\) −8.92820 −0.332043
\(724\) 0 0
\(725\) −4.73205 −0.175744
\(726\) 0 0
\(727\) 4.00000 0.148352 0.0741759 0.997245i \(-0.476367\pi\)
0.0741759 + 0.997245i \(0.476367\pi\)
\(728\) 0 0
\(729\) −43.7846 −1.62165
\(730\) 0 0
\(731\) 6.92820 0.256249
\(732\) 0 0
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) 0 0
\(735\) 2.73205 0.100773
\(736\) 0 0
\(737\) −4.00000 −0.147342
\(738\) 0 0
\(739\) −11.7128 −0.430863 −0.215431 0.976519i \(-0.569116\pi\)
−0.215431 + 0.976519i \(0.569116\pi\)
\(740\) 0 0
\(741\) −26.9282 −0.989232
\(742\) 0 0
\(743\) 32.7846 1.20275 0.601375 0.798967i \(-0.294620\pi\)
0.601375 + 0.798967i \(0.294620\pi\)
\(744\) 0 0
\(745\) 7.26795 0.266277
\(746\) 0 0
\(747\) 73.1769 2.67740
\(748\) 0 0
\(749\) 7.85641 0.287067
\(750\) 0 0
\(751\) 11.6077 0.423571 0.211785 0.977316i \(-0.432072\pi\)
0.211785 + 0.977316i \(0.432072\pi\)
\(752\) 0 0
\(753\) 32.7846 1.19474
\(754\) 0 0
\(755\) 11.1244 0.404857
\(756\) 0 0
\(757\) −43.3731 −1.57642 −0.788210 0.615406i \(-0.788992\pi\)
−0.788210 + 0.615406i \(0.788992\pi\)
\(758\) 0 0
\(759\) 22.3923 0.812789
\(760\) 0 0
\(761\) −12.3397 −0.447315 −0.223658 0.974668i \(-0.571800\pi\)
−0.223658 + 0.974668i \(0.571800\pi\)
\(762\) 0 0
\(763\) 15.6603 0.566939
\(764\) 0 0
\(765\) 15.4641 0.559106
\(766\) 0 0
\(767\) 10.1436 0.366264
\(768\) 0 0
\(769\) −18.1962 −0.656170 −0.328085 0.944648i \(-0.606403\pi\)
−0.328085 + 0.944648i \(0.606403\pi\)
\(770\) 0 0
\(771\) 64.6410 2.32799
\(772\) 0 0
\(773\) −0.928203 −0.0333851 −0.0166926 0.999861i \(-0.505314\pi\)
−0.0166926 + 0.999861i \(0.505314\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) 0 0
\(777\) 2.00000 0.0717496
\(778\) 0 0
\(779\) 14.7846 0.529714
\(780\) 0 0
\(781\) 9.46410 0.338652
\(782\) 0 0
\(783\) 18.9282 0.676439
\(784\) 0 0
\(785\) −4.53590 −0.161893
\(786\) 0 0
\(787\) −18.1436 −0.646749 −0.323375 0.946271i \(-0.604817\pi\)
−0.323375 + 0.946271i \(0.604817\pi\)
\(788\) 0 0
\(789\) −65.5692 −2.33433
\(790\) 0 0
\(791\) −7.85641 −0.279342
\(792\) 0 0
\(793\) 7.21539 0.256226
\(794\) 0 0
\(795\) −19.8564 −0.704234
\(796\) 0 0
\(797\) −25.6077 −0.907071 −0.453536 0.891238i \(-0.649837\pi\)
−0.453536 + 0.891238i \(0.649837\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) 15.4641 0.546397
\(802\) 0 0
\(803\) 14.3923 0.507893
\(804\) 0 0
\(805\) 8.19615 0.288876
\(806\) 0 0
\(807\) −77.5692 −2.73057
\(808\) 0 0
\(809\) −31.1769 −1.09612 −0.548061 0.836438i \(-0.684634\pi\)
−0.548061 + 0.836438i \(0.684634\pi\)
\(810\) 0 0
\(811\) 43.1244 1.51430 0.757150 0.653241i \(-0.226591\pi\)
0.757150 + 0.653241i \(0.226591\pi\)
\(812\) 0 0
\(813\) 1.07180 0.0375896
\(814\) 0 0
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) 13.4641 0.471049
\(818\) 0 0
\(819\) 6.53590 0.228383
\(820\) 0 0
\(821\) 17.9090 0.625027 0.312514 0.949913i \(-0.398829\pi\)
0.312514 + 0.949913i \(0.398829\pi\)
\(822\) 0 0
\(823\) 0.875644 0.0305230 0.0152615 0.999884i \(-0.495142\pi\)
0.0152615 + 0.999884i \(0.495142\pi\)
\(824\) 0 0
\(825\) 2.73205 0.0951178
\(826\) 0 0
\(827\) 25.8564 0.899115 0.449558 0.893251i \(-0.351582\pi\)
0.449558 + 0.893251i \(0.351582\pi\)
\(828\) 0 0
\(829\) 2.24871 0.0781010 0.0390505 0.999237i \(-0.487567\pi\)
0.0390505 + 0.999237i \(0.487567\pi\)
\(830\) 0 0
\(831\) −19.3205 −0.670221
\(832\) 0 0
\(833\) −3.46410 −0.120024
\(834\) 0 0
\(835\) 13.8564 0.479521
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) 0 0
\(839\) 31.8564 1.09981 0.549903 0.835229i \(-0.314665\pi\)
0.549903 + 0.835229i \(0.314665\pi\)
\(840\) 0 0
\(841\) −6.60770 −0.227852
\(842\) 0 0
\(843\) −61.1769 −2.10704
\(844\) 0 0
\(845\) 10.8564 0.373472
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 0 0
\(849\) −86.6410 −2.97351
\(850\) 0 0
\(851\) 6.00000 0.205677
\(852\) 0 0
\(853\) −54.7846 −1.87579 −0.937895 0.346920i \(-0.887227\pi\)
−0.937895 + 0.346920i \(0.887227\pi\)
\(854\) 0 0
\(855\) 30.0526 1.02778
\(856\) 0 0
\(857\) −27.4641 −0.938156 −0.469078 0.883157i \(-0.655414\pi\)
−0.469078 + 0.883157i \(0.655414\pi\)
\(858\) 0 0
\(859\) 12.7846 0.436205 0.218103 0.975926i \(-0.430013\pi\)
0.218103 + 0.975926i \(0.430013\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) 7.51666 0.255870 0.127935 0.991783i \(-0.459165\pi\)
0.127935 + 0.991783i \(0.459165\pi\)
\(864\) 0 0
\(865\) −0.928203 −0.0315599
\(866\) 0 0
\(867\) 13.6603 0.463927
\(868\) 0 0
\(869\) −12.1962 −0.413726
\(870\) 0 0
\(871\) −5.85641 −0.198437
\(872\) 0 0
\(873\) 65.1244 2.20413
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −21.3205 −0.719942 −0.359971 0.932963i \(-0.617213\pi\)
−0.359971 + 0.932963i \(0.617213\pi\)
\(878\) 0 0
\(879\) −58.6410 −1.97791
\(880\) 0 0
\(881\) 11.0718 0.373018 0.186509 0.982453i \(-0.440283\pi\)
0.186509 + 0.982453i \(0.440283\pi\)
\(882\) 0 0
\(883\) 35.6077 1.19829 0.599147 0.800639i \(-0.295506\pi\)
0.599147 + 0.800639i \(0.295506\pi\)
\(884\) 0 0
\(885\) −18.9282 −0.636265
\(886\) 0 0
\(887\) −13.8564 −0.465253 −0.232626 0.972566i \(-0.574732\pi\)
−0.232626 + 0.972566i \(0.574732\pi\)
\(888\) 0 0
\(889\) 1.07180 0.0359469
\(890\) 0 0
\(891\) 2.46410 0.0825505
\(892\) 0 0
\(893\) 46.6410 1.56078
\(894\) 0 0
\(895\) −6.00000 −0.200558
\(896\) 0 0
\(897\) 32.7846 1.09465
\(898\) 0 0
\(899\) 9.46410 0.315645
\(900\) 0 0
\(901\) 25.1769 0.838765
\(902\) 0 0
\(903\) −5.46410 −0.181834
\(904\) 0 0
\(905\) −14.0000 −0.465376
\(906\) 0 0
\(907\) 31.0333 1.03044 0.515222 0.857057i \(-0.327709\pi\)
0.515222 + 0.857057i \(0.327709\pi\)
\(908\) 0 0
\(909\) −35.0718 −1.16326
\(910\) 0 0
\(911\) −9.46410 −0.313560 −0.156780 0.987634i \(-0.550111\pi\)
−0.156780 + 0.987634i \(0.550111\pi\)
\(912\) 0 0
\(913\) −16.3923 −0.542506
\(914\) 0 0
\(915\) −13.4641 −0.445109
\(916\) 0 0
\(917\) 5.66025 0.186918
\(918\) 0 0
\(919\) 25.3731 0.836980 0.418490 0.908221i \(-0.362559\pi\)
0.418490 + 0.908221i \(0.362559\pi\)
\(920\) 0 0
\(921\) 66.6410 2.19590
\(922\) 0 0
\(923\) 13.8564 0.456089
\(924\) 0 0
\(925\) 0.732051 0.0240697
\(926\) 0 0
\(927\) −55.3205 −1.81696
\(928\) 0 0
\(929\) −25.6077 −0.840161 −0.420081 0.907487i \(-0.637998\pi\)
−0.420081 + 0.907487i \(0.637998\pi\)
\(930\) 0 0
\(931\) −6.73205 −0.220634
\(932\) 0 0
\(933\) 68.1051 2.22966
\(934\) 0 0
\(935\) −3.46410 −0.113288
\(936\) 0 0
\(937\) −1.21539 −0.0397051 −0.0198525 0.999803i \(-0.506320\pi\)
−0.0198525 + 0.999803i \(0.506320\pi\)
\(938\) 0 0
\(939\) −60.6410 −1.97894
\(940\) 0 0
\(941\) −14.7846 −0.481965 −0.240982 0.970530i \(-0.577470\pi\)
−0.240982 + 0.970530i \(0.577470\pi\)
\(942\) 0 0
\(943\) −18.0000 −0.586161
\(944\) 0 0
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) −47.3205 −1.53771 −0.768855 0.639423i \(-0.779173\pi\)
−0.768855 + 0.639423i \(0.779173\pi\)
\(948\) 0 0
\(949\) 21.0718 0.684019
\(950\) 0 0
\(951\) −83.5692 −2.70992
\(952\) 0 0
\(953\) 26.5359 0.859582 0.429791 0.902928i \(-0.358587\pi\)
0.429791 + 0.902928i \(0.358587\pi\)
\(954\) 0 0
\(955\) −12.0000 −0.388311
\(956\) 0 0
\(957\) −12.9282 −0.417909
\(958\) 0 0
\(959\) 0.928203 0.0299732
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −35.0718 −1.13017
\(964\) 0 0
\(965\) −12.3923 −0.398922
\(966\) 0 0
\(967\) −26.9282 −0.865953 −0.432976 0.901405i \(-0.642537\pi\)
−0.432976 + 0.901405i \(0.642537\pi\)
\(968\) 0 0
\(969\) −63.7128 −2.04675
\(970\) 0 0
\(971\) −42.9282 −1.37763 −0.688816 0.724936i \(-0.741869\pi\)
−0.688816 + 0.724936i \(0.741869\pi\)
\(972\) 0 0
\(973\) 13.6603 0.437928
\(974\) 0 0
\(975\) 4.00000 0.128103
\(976\) 0 0
\(977\) −33.7128 −1.07857 −0.539284 0.842124i \(-0.681305\pi\)
−0.539284 + 0.842124i \(0.681305\pi\)
\(978\) 0 0
\(979\) −3.46410 −0.110713
\(980\) 0 0
\(981\) −69.9090 −2.23202
\(982\) 0 0
\(983\) 37.1769 1.18576 0.592880 0.805291i \(-0.297991\pi\)
0.592880 + 0.805291i \(0.297991\pi\)
\(984\) 0 0
\(985\) 24.2487 0.772628
\(986\) 0 0
\(987\) −18.9282 −0.602491
\(988\) 0 0
\(989\) −16.3923 −0.521245
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 0 0
\(993\) −51.3205 −1.62861
\(994\) 0 0
\(995\) 2.92820 0.0928303
\(996\) 0 0
\(997\) 17.7128 0.560970 0.280485 0.959858i \(-0.409505\pi\)
0.280485 + 0.959858i \(0.409505\pi\)
\(998\) 0 0
\(999\) −2.92820 −0.0926443
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6160.2.a.t.1.1 2
4.3 odd 2 770.2.a.j.1.2 2
12.11 even 2 6930.2.a.bv.1.1 2
20.3 even 4 3850.2.c.x.1849.2 4
20.7 even 4 3850.2.c.x.1849.3 4
20.19 odd 2 3850.2.a.bd.1.1 2
28.27 even 2 5390.2.a.bs.1.1 2
44.43 even 2 8470.2.a.br.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.j.1.2 2 4.3 odd 2
3850.2.a.bd.1.1 2 20.19 odd 2
3850.2.c.x.1849.2 4 20.3 even 4
3850.2.c.x.1849.3 4 20.7 even 4
5390.2.a.bs.1.1 2 28.27 even 2
6160.2.a.t.1.1 2 1.1 even 1 trivial
6930.2.a.bv.1.1 2 12.11 even 2
8470.2.a.br.1.2 2 44.43 even 2