Properties

Label 1792.2.a.i.1.2
Level $1792$
Weight $2$
Character 1792.1
Self dual yes
Analytic conductor $14.309$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.3091920422\)
Analytic rank: \(1\)
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 448)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1792.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.732051 q^{3} +0.732051 q^{5} -1.00000 q^{7} -2.46410 q^{9} +O(q^{10})\) \(q+0.732051 q^{3} +0.732051 q^{5} -1.00000 q^{7} -2.46410 q^{9} -1.46410 q^{11} -3.26795 q^{13} +0.535898 q^{15} +2.00000 q^{17} +4.73205 q^{19} -0.732051 q^{21} -3.46410 q^{23} -4.46410 q^{25} -4.00000 q^{27} -5.46410 q^{29} +4.00000 q^{31} -1.07180 q^{33} -0.732051 q^{35} -5.46410 q^{37} -2.39230 q^{39} +2.00000 q^{41} -1.46410 q^{43} -1.80385 q^{45} -10.9282 q^{47} +1.00000 q^{49} +1.46410 q^{51} -12.0000 q^{53} -1.07180 q^{55} +3.46410 q^{57} +7.66025 q^{59} -13.1244 q^{61} +2.46410 q^{63} -2.39230 q^{65} -8.00000 q^{67} -2.53590 q^{69} +10.9282 q^{71} +0.928203 q^{73} -3.26795 q^{75} +1.46410 q^{77} +2.92820 q^{79} +4.46410 q^{81} -11.6603 q^{83} +1.46410 q^{85} -4.00000 q^{87} +15.8564 q^{89} +3.26795 q^{91} +2.92820 q^{93} +3.46410 q^{95} -4.92820 q^{97} +3.60770 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{5} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{5} - 2q^{7} + 2q^{9} + 4q^{11} - 10q^{13} + 8q^{15} + 4q^{17} + 6q^{19} + 2q^{21} - 2q^{25} - 8q^{27} - 4q^{29} + 8q^{31} - 16q^{33} + 2q^{35} - 4q^{37} + 16q^{39} + 4q^{41} + 4q^{43} - 14q^{45} - 8q^{47} + 2q^{49} - 4q^{51} - 24q^{53} - 16q^{55} - 2q^{59} - 2q^{61} - 2q^{63} + 16q^{65} - 16q^{67} - 12q^{69} + 8q^{71} - 12q^{73} - 10q^{75} - 4q^{77} - 8q^{79} + 2q^{81} - 6q^{83} - 4q^{85} - 8q^{87} + 4q^{89} + 10q^{91} - 8q^{93} + 4q^{97} + 28q^{99} + O(q^{100}) \)

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
\(3\) 0.732051 0.422650 0.211325 0.977416i \(-0.432222\pi\)
0.211325 + 0.977416i \(0.432222\pi\)
\(4\) 0 0
\(5\) 0.732051 0.327383 0.163692 0.986512i \(-0.447660\pi\)
0.163692 + 0.986512i \(0.447660\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −2.46410 −0.821367
\(10\) 0 0
\(11\) −1.46410 −0.441443 −0.220722 0.975337i \(-0.570841\pi\)
−0.220722 + 0.975337i \(0.570841\pi\)
\(12\) 0 0
\(13\) −3.26795 −0.906366 −0.453183 0.891417i \(-0.649712\pi\)
−0.453183 + 0.891417i \(0.649712\pi\)
\(14\) 0 0
\(15\) 0.535898 0.138368
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 4.73205 1.08561 0.542803 0.839860i \(-0.317363\pi\)
0.542803 + 0.839860i \(0.317363\pi\)
\(20\) 0 0
\(21\) −0.732051 −0.159747
\(22\) 0 0
\(23\) −3.46410 −0.722315 −0.361158 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) 0 0
\(25\) −4.46410 −0.892820
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −5.46410 −1.01466 −0.507329 0.861752i \(-0.669367\pi\)
−0.507329 + 0.861752i \(0.669367\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) −1.07180 −0.186576
\(34\) 0 0
\(35\) −0.732051 −0.123739
\(36\) 0 0
\(37\) −5.46410 −0.898293 −0.449146 0.893458i \(-0.648272\pi\)
−0.449146 + 0.893458i \(0.648272\pi\)
\(38\) 0 0
\(39\) −2.39230 −0.383075
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −1.46410 −0.223273 −0.111637 0.993749i \(-0.535609\pi\)
−0.111637 + 0.993749i \(0.535609\pi\)
\(44\) 0 0
\(45\) −1.80385 −0.268902
\(46\) 0 0
\(47\) −10.9282 −1.59404 −0.797021 0.603951i \(-0.793592\pi\)
−0.797021 + 0.603951i \(0.793592\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 1.46410 0.205015
\(52\) 0 0
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) −1.07180 −0.144521
\(56\) 0 0
\(57\) 3.46410 0.458831
\(58\) 0 0
\(59\) 7.66025 0.997280 0.498640 0.866809i \(-0.333833\pi\)
0.498640 + 0.866809i \(0.333833\pi\)
\(60\) 0 0
\(61\) −13.1244 −1.68040 −0.840201 0.542275i \(-0.817563\pi\)
−0.840201 + 0.542275i \(0.817563\pi\)
\(62\) 0 0
\(63\) 2.46410 0.310448
\(64\) 0 0
\(65\) −2.39230 −0.296729
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) −2.53590 −0.305286
\(70\) 0 0
\(71\) 10.9282 1.29694 0.648470 0.761241i \(-0.275409\pi\)
0.648470 + 0.761241i \(0.275409\pi\)
\(72\) 0 0
\(73\) 0.928203 0.108638 0.0543190 0.998524i \(-0.482701\pi\)
0.0543190 + 0.998524i \(0.482701\pi\)
\(74\) 0 0
\(75\) −3.26795 −0.377350
\(76\) 0 0
\(77\) 1.46410 0.166850
\(78\) 0 0
\(79\) 2.92820 0.329449 0.164724 0.986340i \(-0.447327\pi\)
0.164724 + 0.986340i \(0.447327\pi\)
\(80\) 0 0
\(81\) 4.46410 0.496011
\(82\) 0 0
\(83\) −11.6603 −1.27988 −0.639940 0.768425i \(-0.721041\pi\)
−0.639940 + 0.768425i \(0.721041\pi\)
\(84\) 0 0
\(85\) 1.46410 0.158804
\(86\) 0 0
\(87\) −4.00000 −0.428845
\(88\) 0 0
\(89\) 15.8564 1.68078 0.840388 0.541985i \(-0.182327\pi\)
0.840388 + 0.541985i \(0.182327\pi\)
\(90\) 0 0
\(91\) 3.26795 0.342574
\(92\) 0 0
\(93\) 2.92820 0.303641
\(94\) 0 0
\(95\) 3.46410 0.355409
\(96\) 0 0
\(97\) −4.92820 −0.500383 −0.250192 0.968196i \(-0.580494\pi\)
−0.250192 + 0.968196i \(0.580494\pi\)
\(98\) 0 0
\(99\) 3.60770 0.362587
\(100\) 0 0
\(101\) −15.6603 −1.55825 −0.779127 0.626866i \(-0.784337\pi\)
−0.779127 + 0.626866i \(0.784337\pi\)
\(102\) 0 0
\(103\) 17.8564 1.75944 0.879722 0.475488i \(-0.157729\pi\)
0.879722 + 0.475488i \(0.157729\pi\)
\(104\) 0 0
\(105\) −0.535898 −0.0522983
\(106\) 0 0
\(107\) 18.9282 1.82986 0.914929 0.403614i \(-0.132246\pi\)
0.914929 + 0.403614i \(0.132246\pi\)
\(108\) 0 0
\(109\) 8.39230 0.803837 0.401919 0.915675i \(-0.368344\pi\)
0.401919 + 0.915675i \(0.368344\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) −13.4641 −1.26660 −0.633298 0.773908i \(-0.718299\pi\)
−0.633298 + 0.773908i \(0.718299\pi\)
\(114\) 0 0
\(115\) −2.53590 −0.236474
\(116\) 0 0
\(117\) 8.05256 0.744459
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) −8.85641 −0.805128
\(122\) 0 0
\(123\) 1.46410 0.132014
\(124\) 0 0
\(125\) −6.92820 −0.619677
\(126\) 0 0
\(127\) −4.53590 −0.402496 −0.201248 0.979540i \(-0.564500\pi\)
−0.201248 + 0.979540i \(0.564500\pi\)
\(128\) 0 0
\(129\) −1.07180 −0.0943664
\(130\) 0 0
\(131\) 7.26795 0.635004 0.317502 0.948258i \(-0.397156\pi\)
0.317502 + 0.948258i \(0.397156\pi\)
\(132\) 0 0
\(133\) −4.73205 −0.410321
\(134\) 0 0
\(135\) −2.92820 −0.252020
\(136\) 0 0
\(137\) 7.85641 0.671218 0.335609 0.942001i \(-0.391058\pi\)
0.335609 + 0.942001i \(0.391058\pi\)
\(138\) 0 0
\(139\) −12.7321 −1.07992 −0.539959 0.841691i \(-0.681560\pi\)
−0.539959 + 0.841691i \(0.681560\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 4.78461 0.400109
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) 0 0
\(147\) 0.732051 0.0603785
\(148\) 0 0
\(149\) −14.9282 −1.22297 −0.611483 0.791258i \(-0.709427\pi\)
−0.611483 + 0.791258i \(0.709427\pi\)
\(150\) 0 0
\(151\) −18.3923 −1.49674 −0.748372 0.663279i \(-0.769164\pi\)
−0.748372 + 0.663279i \(0.769164\pi\)
\(152\) 0 0
\(153\) −4.92820 −0.398422
\(154\) 0 0
\(155\) 2.92820 0.235199
\(156\) 0 0
\(157\) 1.80385 0.143963 0.0719814 0.997406i \(-0.477068\pi\)
0.0719814 + 0.997406i \(0.477068\pi\)
\(158\) 0 0
\(159\) −8.78461 −0.696665
\(160\) 0 0
\(161\) 3.46410 0.273009
\(162\) 0 0
\(163\) 6.53590 0.511931 0.255966 0.966686i \(-0.417607\pi\)
0.255966 + 0.966686i \(0.417607\pi\)
\(164\) 0 0
\(165\) −0.784610 −0.0610818
\(166\) 0 0
\(167\) 18.9282 1.46471 0.732354 0.680924i \(-0.238422\pi\)
0.732354 + 0.680924i \(0.238422\pi\)
\(168\) 0 0
\(169\) −2.32051 −0.178501
\(170\) 0 0
\(171\) −11.6603 −0.891682
\(172\) 0 0
\(173\) −14.1962 −1.07931 −0.539657 0.841885i \(-0.681446\pi\)
−0.539657 + 0.841885i \(0.681446\pi\)
\(174\) 0 0
\(175\) 4.46410 0.337454
\(176\) 0 0
\(177\) 5.60770 0.421500
\(178\) 0 0
\(179\) −10.9282 −0.816812 −0.408406 0.912800i \(-0.633915\pi\)
−0.408406 + 0.912800i \(0.633915\pi\)
\(180\) 0 0
\(181\) 7.26795 0.540222 0.270111 0.962829i \(-0.412940\pi\)
0.270111 + 0.962829i \(0.412940\pi\)
\(182\) 0 0
\(183\) −9.60770 −0.710221
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) −2.92820 −0.214131
\(188\) 0 0
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) 13.4641 0.969167 0.484584 0.874745i \(-0.338971\pi\)
0.484584 + 0.874745i \(0.338971\pi\)
\(194\) 0 0
\(195\) −1.75129 −0.125412
\(196\) 0 0
\(197\) 1.85641 0.132263 0.0661317 0.997811i \(-0.478934\pi\)
0.0661317 + 0.997811i \(0.478934\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) −5.85641 −0.413079
\(202\) 0 0
\(203\) 5.46410 0.383505
\(204\) 0 0
\(205\) 1.46410 0.102257
\(206\) 0 0
\(207\) 8.53590 0.593286
\(208\) 0 0
\(209\) −6.92820 −0.479234
\(210\) 0 0
\(211\) 24.7846 1.70624 0.853121 0.521712i \(-0.174707\pi\)
0.853121 + 0.521712i \(0.174707\pi\)
\(212\) 0 0
\(213\) 8.00000 0.548151
\(214\) 0 0
\(215\) −1.07180 −0.0730959
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 0 0
\(219\) 0.679492 0.0459158
\(220\) 0 0
\(221\) −6.53590 −0.439652
\(222\) 0 0
\(223\) 6.92820 0.463947 0.231973 0.972722i \(-0.425482\pi\)
0.231973 + 0.972722i \(0.425482\pi\)
\(224\) 0 0
\(225\) 11.0000 0.733333
\(226\) 0 0
\(227\) −14.1962 −0.942232 −0.471116 0.882071i \(-0.656149\pi\)
−0.471116 + 0.882071i \(0.656149\pi\)
\(228\) 0 0
\(229\) 20.7321 1.37001 0.685006 0.728537i \(-0.259799\pi\)
0.685006 + 0.728537i \(0.259799\pi\)
\(230\) 0 0
\(231\) 1.07180 0.0705191
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) 0 0
\(237\) 2.14359 0.139241
\(238\) 0 0
\(239\) −19.4641 −1.25903 −0.629514 0.776989i \(-0.716746\pi\)
−0.629514 + 0.776989i \(0.716746\pi\)
\(240\) 0 0
\(241\) 22.7846 1.46769 0.733843 0.679319i \(-0.237725\pi\)
0.733843 + 0.679319i \(0.237725\pi\)
\(242\) 0 0
\(243\) 15.2679 0.979439
\(244\) 0 0
\(245\) 0.732051 0.0467690
\(246\) 0 0
\(247\) −15.4641 −0.983957
\(248\) 0 0
\(249\) −8.53590 −0.540941
\(250\) 0 0
\(251\) −26.1962 −1.65349 −0.826743 0.562579i \(-0.809809\pi\)
−0.826743 + 0.562579i \(0.809809\pi\)
\(252\) 0 0
\(253\) 5.07180 0.318861
\(254\) 0 0
\(255\) 1.07180 0.0671185
\(256\) 0 0
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) 5.46410 0.339523
\(260\) 0 0
\(261\) 13.4641 0.833407
\(262\) 0 0
\(263\) 2.92820 0.180561 0.0902804 0.995916i \(-0.471224\pi\)
0.0902804 + 0.995916i \(0.471224\pi\)
\(264\) 0 0
\(265\) −8.78461 −0.539634
\(266\) 0 0
\(267\) 11.6077 0.710379
\(268\) 0 0
\(269\) 18.5885 1.13336 0.566679 0.823939i \(-0.308228\pi\)
0.566679 + 0.823939i \(0.308228\pi\)
\(270\) 0 0
\(271\) 1.07180 0.0651070 0.0325535 0.999470i \(-0.489636\pi\)
0.0325535 + 0.999470i \(0.489636\pi\)
\(272\) 0 0
\(273\) 2.39230 0.144789
\(274\) 0 0
\(275\) 6.53590 0.394130
\(276\) 0 0
\(277\) 4.00000 0.240337 0.120168 0.992754i \(-0.461657\pi\)
0.120168 + 0.992754i \(0.461657\pi\)
\(278\) 0 0
\(279\) −9.85641 −0.590088
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) −14.5885 −0.867194 −0.433597 0.901107i \(-0.642756\pi\)
−0.433597 + 0.901107i \(0.642756\pi\)
\(284\) 0 0
\(285\) 2.53590 0.150214
\(286\) 0 0
\(287\) −2.00000 −0.118056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −3.60770 −0.211487
\(292\) 0 0
\(293\) −1.80385 −0.105382 −0.0526910 0.998611i \(-0.516780\pi\)
−0.0526910 + 0.998611i \(0.516780\pi\)
\(294\) 0 0
\(295\) 5.60770 0.326493
\(296\) 0 0
\(297\) 5.85641 0.339823
\(298\) 0 0
\(299\) 11.3205 0.654682
\(300\) 0 0
\(301\) 1.46410 0.0843894
\(302\) 0 0
\(303\) −11.4641 −0.658595
\(304\) 0 0
\(305\) −9.60770 −0.550135
\(306\) 0 0
\(307\) −22.5885 −1.28919 −0.644596 0.764524i \(-0.722974\pi\)
−0.644596 + 0.764524i \(0.722974\pi\)
\(308\) 0 0
\(309\) 13.0718 0.743629
\(310\) 0 0
\(311\) −21.8564 −1.23936 −0.619682 0.784853i \(-0.712738\pi\)
−0.619682 + 0.784853i \(0.712738\pi\)
\(312\) 0 0
\(313\) 7.85641 0.444070 0.222035 0.975039i \(-0.428730\pi\)
0.222035 + 0.975039i \(0.428730\pi\)
\(314\) 0 0
\(315\) 1.80385 0.101635
\(316\) 0 0
\(317\) 12.7846 0.718055 0.359028 0.933327i \(-0.383108\pi\)
0.359028 + 0.933327i \(0.383108\pi\)
\(318\) 0 0
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) 13.8564 0.773389
\(322\) 0 0
\(323\) 9.46410 0.526597
\(324\) 0 0
\(325\) 14.5885 0.809222
\(326\) 0 0
\(327\) 6.14359 0.339741
\(328\) 0 0
\(329\) 10.9282 0.602491
\(330\) 0 0
\(331\) 4.39230 0.241423 0.120711 0.992688i \(-0.461482\pi\)
0.120711 + 0.992688i \(0.461482\pi\)
\(332\) 0 0
\(333\) 13.4641 0.737828
\(334\) 0 0
\(335\) −5.85641 −0.319970
\(336\) 0 0
\(337\) −4.39230 −0.239264 −0.119632 0.992818i \(-0.538171\pi\)
−0.119632 + 0.992818i \(0.538171\pi\)
\(338\) 0 0
\(339\) −9.85641 −0.535327
\(340\) 0 0
\(341\) −5.85641 −0.317142
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −1.85641 −0.0999456
\(346\) 0 0
\(347\) −0.679492 −0.0364770 −0.0182385 0.999834i \(-0.505806\pi\)
−0.0182385 + 0.999834i \(0.505806\pi\)
\(348\) 0 0
\(349\) 24.0526 1.28750 0.643752 0.765234i \(-0.277377\pi\)
0.643752 + 0.765234i \(0.277377\pi\)
\(350\) 0 0
\(351\) 13.0718 0.697721
\(352\) 0 0
\(353\) −12.9282 −0.688099 −0.344049 0.938952i \(-0.611799\pi\)
−0.344049 + 0.938952i \(0.611799\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) 0 0
\(357\) −1.46410 −0.0774885
\(358\) 0 0
\(359\) 20.5359 1.08384 0.541922 0.840429i \(-0.317697\pi\)
0.541922 + 0.840429i \(0.317697\pi\)
\(360\) 0 0
\(361\) 3.39230 0.178542
\(362\) 0 0
\(363\) −6.48334 −0.340287
\(364\) 0 0
\(365\) 0.679492 0.0355662
\(366\) 0 0
\(367\) −20.7846 −1.08495 −0.542474 0.840073i \(-0.682512\pi\)
−0.542474 + 0.840073i \(0.682512\pi\)
\(368\) 0 0
\(369\) −4.92820 −0.256552
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) 17.0718 0.883944 0.441972 0.897029i \(-0.354279\pi\)
0.441972 + 0.897029i \(0.354279\pi\)
\(374\) 0 0
\(375\) −5.07180 −0.261906
\(376\) 0 0
\(377\) 17.8564 0.919652
\(378\) 0 0
\(379\) 34.2487 1.75924 0.879619 0.475679i \(-0.157798\pi\)
0.879619 + 0.475679i \(0.157798\pi\)
\(380\) 0 0
\(381\) −3.32051 −0.170115
\(382\) 0 0
\(383\) 8.78461 0.448873 0.224436 0.974489i \(-0.427946\pi\)
0.224436 + 0.974489i \(0.427946\pi\)
\(384\) 0 0
\(385\) 1.07180 0.0546238
\(386\) 0 0
\(387\) 3.60770 0.183389
\(388\) 0 0
\(389\) 11.3205 0.573973 0.286986 0.957935i \(-0.407347\pi\)
0.286986 + 0.957935i \(0.407347\pi\)
\(390\) 0 0
\(391\) −6.92820 −0.350374
\(392\) 0 0
\(393\) 5.32051 0.268384
\(394\) 0 0
\(395\) 2.14359 0.107856
\(396\) 0 0
\(397\) 29.1244 1.46171 0.730855 0.682533i \(-0.239122\pi\)
0.730855 + 0.682533i \(0.239122\pi\)
\(398\) 0 0
\(399\) −3.46410 −0.173422
\(400\) 0 0
\(401\) 6.53590 0.326387 0.163194 0.986594i \(-0.447820\pi\)
0.163194 + 0.986594i \(0.447820\pi\)
\(402\) 0 0
\(403\) −13.0718 −0.651153
\(404\) 0 0
\(405\) 3.26795 0.162386
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) 16.9282 0.837046 0.418523 0.908206i \(-0.362548\pi\)
0.418523 + 0.908206i \(0.362548\pi\)
\(410\) 0 0
\(411\) 5.75129 0.283690
\(412\) 0 0
\(413\) −7.66025 −0.376936
\(414\) 0 0
\(415\) −8.53590 −0.419011
\(416\) 0 0
\(417\) −9.32051 −0.456427
\(418\) 0 0
\(419\) −34.1962 −1.67059 −0.835296 0.549801i \(-0.814704\pi\)
−0.835296 + 0.549801i \(0.814704\pi\)
\(420\) 0 0
\(421\) −1.85641 −0.0904757 −0.0452379 0.998976i \(-0.514405\pi\)
−0.0452379 + 0.998976i \(0.514405\pi\)
\(422\) 0 0
\(423\) 26.9282 1.30929
\(424\) 0 0
\(425\) −8.92820 −0.433081
\(426\) 0 0
\(427\) 13.1244 0.635132
\(428\) 0 0
\(429\) 3.50258 0.169106
\(430\) 0 0
\(431\) −28.5359 −1.37453 −0.687263 0.726409i \(-0.741188\pi\)
−0.687263 + 0.726409i \(0.741188\pi\)
\(432\) 0 0
\(433\) −11.8564 −0.569783 −0.284891 0.958560i \(-0.591957\pi\)
−0.284891 + 0.958560i \(0.591957\pi\)
\(434\) 0 0
\(435\) −2.92820 −0.140397
\(436\) 0 0
\(437\) −16.3923 −0.784150
\(438\) 0 0
\(439\) −17.0718 −0.814792 −0.407396 0.913252i \(-0.633563\pi\)
−0.407396 + 0.913252i \(0.633563\pi\)
\(440\) 0 0
\(441\) −2.46410 −0.117338
\(442\) 0 0
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) 11.6077 0.550258
\(446\) 0 0
\(447\) −10.9282 −0.516886
\(448\) 0 0
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) −2.92820 −0.137884
\(452\) 0 0
\(453\) −13.4641 −0.632599
\(454\) 0 0
\(455\) 2.39230 0.112153
\(456\) 0 0
\(457\) 19.3205 0.903775 0.451888 0.892075i \(-0.350751\pi\)
0.451888 + 0.892075i \(0.350751\pi\)
\(458\) 0 0
\(459\) −8.00000 −0.373408
\(460\) 0 0
\(461\) −13.8038 −0.642909 −0.321455 0.946925i \(-0.604172\pi\)
−0.321455 + 0.946925i \(0.604172\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 0 0
\(465\) 2.14359 0.0994068
\(466\) 0 0
\(467\) −25.1244 −1.16262 −0.581308 0.813683i \(-0.697459\pi\)
−0.581308 + 0.813683i \(0.697459\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 1.32051 0.0608458
\(472\) 0 0
\(473\) 2.14359 0.0985625
\(474\) 0 0
\(475\) −21.1244 −0.969252
\(476\) 0 0
\(477\) 29.5692 1.35388
\(478\) 0 0
\(479\) 17.8564 0.815880 0.407940 0.913009i \(-0.366247\pi\)
0.407940 + 0.913009i \(0.366247\pi\)
\(480\) 0 0
\(481\) 17.8564 0.814182
\(482\) 0 0
\(483\) 2.53590 0.115387
\(484\) 0 0
\(485\) −3.60770 −0.163817
\(486\) 0 0
\(487\) 42.3923 1.92098 0.960489 0.278317i \(-0.0897765\pi\)
0.960489 + 0.278317i \(0.0897765\pi\)
\(488\) 0 0
\(489\) 4.78461 0.216368
\(490\) 0 0
\(491\) −8.78461 −0.396444 −0.198222 0.980157i \(-0.563517\pi\)
−0.198222 + 0.980157i \(0.563517\pi\)
\(492\) 0 0
\(493\) −10.9282 −0.492182
\(494\) 0 0
\(495\) 2.64102 0.118705
\(496\) 0 0
\(497\) −10.9282 −0.490197
\(498\) 0 0
\(499\) 26.9282 1.20547 0.602736 0.797941i \(-0.294077\pi\)
0.602736 + 0.797941i \(0.294077\pi\)
\(500\) 0 0
\(501\) 13.8564 0.619059
\(502\) 0 0
\(503\) −17.0718 −0.761194 −0.380597 0.924741i \(-0.624281\pi\)
−0.380597 + 0.924741i \(0.624281\pi\)
\(504\) 0 0
\(505\) −11.4641 −0.510146
\(506\) 0 0
\(507\) −1.69873 −0.0754432
\(508\) 0 0
\(509\) 21.1244 0.936321 0.468160 0.883644i \(-0.344917\pi\)
0.468160 + 0.883644i \(0.344917\pi\)
\(510\) 0 0
\(511\) −0.928203 −0.0410613
\(512\) 0 0
\(513\) −18.9282 −0.835701
\(514\) 0 0
\(515\) 13.0718 0.576012
\(516\) 0 0
\(517\) 16.0000 0.703679
\(518\) 0 0
\(519\) −10.3923 −0.456172
\(520\) 0 0
\(521\) −26.7846 −1.17346 −0.586728 0.809784i \(-0.699584\pi\)
−0.586728 + 0.809784i \(0.699584\pi\)
\(522\) 0 0
\(523\) 34.5885 1.51245 0.756224 0.654313i \(-0.227042\pi\)
0.756224 + 0.654313i \(0.227042\pi\)
\(524\) 0 0
\(525\) 3.26795 0.142625
\(526\) 0 0
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) 0 0
\(531\) −18.8756 −0.819133
\(532\) 0 0
\(533\) −6.53590 −0.283101
\(534\) 0 0
\(535\) 13.8564 0.599065
\(536\) 0 0
\(537\) −8.00000 −0.345225
\(538\) 0 0
\(539\) −1.46410 −0.0630633
\(540\) 0 0
\(541\) −4.00000 −0.171973 −0.0859867 0.996296i \(-0.527404\pi\)
−0.0859867 + 0.996296i \(0.527404\pi\)
\(542\) 0 0
\(543\) 5.32051 0.228325
\(544\) 0 0
\(545\) 6.14359 0.263163
\(546\) 0 0
\(547\) −2.24871 −0.0961480 −0.0480740 0.998844i \(-0.515308\pi\)
−0.0480740 + 0.998844i \(0.515308\pi\)
\(548\) 0 0
\(549\) 32.3397 1.38023
\(550\) 0 0
\(551\) −25.8564 −1.10152
\(552\) 0 0
\(553\) −2.92820 −0.124520
\(554\) 0 0
\(555\) −2.92820 −0.124295
\(556\) 0 0
\(557\) −31.7128 −1.34372 −0.671858 0.740680i \(-0.734503\pi\)
−0.671858 + 0.740680i \(0.734503\pi\)
\(558\) 0 0
\(559\) 4.78461 0.202367
\(560\) 0 0
\(561\) −2.14359 −0.0905026
\(562\) 0 0
\(563\) 4.05256 0.170795 0.0853975 0.996347i \(-0.472784\pi\)
0.0853975 + 0.996347i \(0.472784\pi\)
\(564\) 0 0
\(565\) −9.85641 −0.414662
\(566\) 0 0
\(567\) −4.46410 −0.187475
\(568\) 0 0
\(569\) −1.46410 −0.0613783 −0.0306892 0.999529i \(-0.509770\pi\)
−0.0306892 + 0.999529i \(0.509770\pi\)
\(570\) 0 0
\(571\) −37.1769 −1.55581 −0.777903 0.628385i \(-0.783716\pi\)
−0.777903 + 0.628385i \(0.783716\pi\)
\(572\) 0 0
\(573\) 11.7128 0.489310
\(574\) 0 0
\(575\) 15.4641 0.644898
\(576\) 0 0
\(577\) −44.9282 −1.87039 −0.935193 0.354139i \(-0.884774\pi\)
−0.935193 + 0.354139i \(0.884774\pi\)
\(578\) 0 0
\(579\) 9.85641 0.409618
\(580\) 0 0
\(581\) 11.6603 0.483749
\(582\) 0 0
\(583\) 17.5692 0.727643
\(584\) 0 0
\(585\) 5.89488 0.243723
\(586\) 0 0
\(587\) −36.7321 −1.51609 −0.758047 0.652200i \(-0.773846\pi\)
−0.758047 + 0.652200i \(0.773846\pi\)
\(588\) 0 0
\(589\) 18.9282 0.779923
\(590\) 0 0
\(591\) 1.35898 0.0559011
\(592\) 0 0
\(593\) 39.8564 1.63671 0.818353 0.574716i \(-0.194887\pi\)
0.818353 + 0.574716i \(0.194887\pi\)
\(594\) 0 0
\(595\) −1.46410 −0.0600223
\(596\) 0 0
\(597\) 2.92820 0.119843
\(598\) 0 0
\(599\) 27.7128 1.13231 0.566157 0.824297i \(-0.308429\pi\)
0.566157 + 0.824297i \(0.308429\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 19.7128 0.802768
\(604\) 0 0
\(605\) −6.48334 −0.263585
\(606\) 0 0
\(607\) 27.7128 1.12483 0.562414 0.826856i \(-0.309873\pi\)
0.562414 + 0.826856i \(0.309873\pi\)
\(608\) 0 0
\(609\) 4.00000 0.162088
\(610\) 0 0
\(611\) 35.7128 1.44479
\(612\) 0 0
\(613\) −4.67949 −0.189003 −0.0945014 0.995525i \(-0.530126\pi\)
−0.0945014 + 0.995525i \(0.530126\pi\)
\(614\) 0 0
\(615\) 1.07180 0.0432190
\(616\) 0 0
\(617\) 3.60770 0.145240 0.0726202 0.997360i \(-0.476864\pi\)
0.0726202 + 0.997360i \(0.476864\pi\)
\(618\) 0 0
\(619\) −13.8038 −0.554823 −0.277412 0.960751i \(-0.589477\pi\)
−0.277412 + 0.960751i \(0.589477\pi\)
\(620\) 0 0
\(621\) 13.8564 0.556038
\(622\) 0 0
\(623\) −15.8564 −0.635274
\(624\) 0 0
\(625\) 17.2487 0.689948
\(626\) 0 0
\(627\) −5.07180 −0.202548
\(628\) 0 0
\(629\) −10.9282 −0.435736
\(630\) 0 0
\(631\) −33.5692 −1.33637 −0.668185 0.743995i \(-0.732928\pi\)
−0.668185 + 0.743995i \(0.732928\pi\)
\(632\) 0 0
\(633\) 18.1436 0.721143
\(634\) 0 0
\(635\) −3.32051 −0.131770
\(636\) 0 0
\(637\) −3.26795 −0.129481
\(638\) 0 0
\(639\) −26.9282 −1.06526
\(640\) 0 0
\(641\) 26.2487 1.03676 0.518381 0.855150i \(-0.326535\pi\)
0.518381 + 0.855150i \(0.326535\pi\)
\(642\) 0 0
\(643\) −31.6603 −1.24856 −0.624279 0.781201i \(-0.714607\pi\)
−0.624279 + 0.781201i \(0.714607\pi\)
\(644\) 0 0
\(645\) −0.784610 −0.0308940
\(646\) 0 0
\(647\) −25.8564 −1.01652 −0.508260 0.861204i \(-0.669711\pi\)
−0.508260 + 0.861204i \(0.669711\pi\)
\(648\) 0 0
\(649\) −11.2154 −0.440243
\(650\) 0 0
\(651\) −2.92820 −0.114765
\(652\) 0 0
\(653\) −45.4641 −1.77915 −0.889574 0.456791i \(-0.848999\pi\)
−0.889574 + 0.456791i \(0.848999\pi\)
\(654\) 0 0
\(655\) 5.32051 0.207889
\(656\) 0 0
\(657\) −2.28719 −0.0892317
\(658\) 0 0
\(659\) 31.3205 1.22007 0.610037 0.792373i \(-0.291155\pi\)
0.610037 + 0.792373i \(0.291155\pi\)
\(660\) 0 0
\(661\) 22.9808 0.893848 0.446924 0.894572i \(-0.352519\pi\)
0.446924 + 0.894572i \(0.352519\pi\)
\(662\) 0 0
\(663\) −4.78461 −0.185819
\(664\) 0 0
\(665\) −3.46410 −0.134332
\(666\) 0 0
\(667\) 18.9282 0.732903
\(668\) 0 0
\(669\) 5.07180 0.196087
\(670\) 0 0
\(671\) 19.2154 0.741802
\(672\) 0 0
\(673\) 31.8564 1.22797 0.613987 0.789316i \(-0.289565\pi\)
0.613987 + 0.789316i \(0.289565\pi\)
\(674\) 0 0
\(675\) 17.8564 0.687293
\(676\) 0 0
\(677\) −44.4449 −1.70815 −0.854077 0.520146i \(-0.825878\pi\)
−0.854077 + 0.520146i \(0.825878\pi\)
\(678\) 0 0
\(679\) 4.92820 0.189127
\(680\) 0 0
\(681\) −10.3923 −0.398234
\(682\) 0 0
\(683\) −38.6410 −1.47856 −0.739279 0.673400i \(-0.764833\pi\)
−0.739279 + 0.673400i \(0.764833\pi\)
\(684\) 0 0
\(685\) 5.75129 0.219745
\(686\) 0 0
\(687\) 15.1769 0.579035
\(688\) 0 0
\(689\) 39.2154 1.49399
\(690\) 0 0
\(691\) 13.1244 0.499274 0.249637 0.968339i \(-0.419689\pi\)
0.249637 + 0.968339i \(0.419689\pi\)
\(692\) 0 0
\(693\) −3.60770 −0.137045
\(694\) 0 0
\(695\) −9.32051 −0.353547
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 0 0
\(699\) −4.39230 −0.166132
\(700\) 0 0
\(701\) −40.3923 −1.52560 −0.762798 0.646637i \(-0.776175\pi\)
−0.762798 + 0.646637i \(0.776175\pi\)
\(702\) 0 0
\(703\) −25.8564 −0.975193
\(704\) 0 0
\(705\) −5.85641 −0.220565
\(706\) 0 0
\(707\) 15.6603 0.588964
\(708\) 0 0
\(709\) 28.1051 1.05551 0.527755 0.849397i \(-0.323034\pi\)
0.527755 + 0.849397i \(0.323034\pi\)
\(710\) 0 0
\(711\) −7.21539 −0.270598
\(712\) 0 0
\(713\) −13.8564 −0.518927
\(714\) 0 0
\(715\) 3.50258 0.130989
\(716\) 0 0
\(717\) −14.2487 −0.532128
\(718\) 0 0
\(719\) 34.9282 1.30260 0.651301 0.758819i \(-0.274223\pi\)
0.651301 + 0.758819i \(0.274223\pi\)
\(720\) 0 0
\(721\) −17.8564 −0.665007
\(722\) 0 0
\(723\) 16.6795 0.620317
\(724\) 0 0
\(725\) 24.3923 0.905907
\(726\) 0 0
\(727\) 2.92820 0.108601 0.0543005 0.998525i \(-0.482707\pi\)
0.0543005 + 0.998525i \(0.482707\pi\)
\(728\) 0 0
\(729\) −2.21539 −0.0820515
\(730\) 0 0
\(731\) −2.92820 −0.108304
\(732\) 0 0
\(733\) 3.66025 0.135195 0.0675973 0.997713i \(-0.478467\pi\)
0.0675973 + 0.997713i \(0.478467\pi\)
\(734\) 0 0
\(735\) 0.535898 0.0197669
\(736\) 0 0
\(737\) 11.7128 0.431447
\(738\) 0 0
\(739\) −31.3205 −1.15214 −0.576072 0.817399i \(-0.695415\pi\)
−0.576072 + 0.817399i \(0.695415\pi\)
\(740\) 0 0
\(741\) −11.3205 −0.415869
\(742\) 0 0
\(743\) 16.2487 0.596107 0.298054 0.954549i \(-0.403663\pi\)
0.298054 + 0.954549i \(0.403663\pi\)
\(744\) 0 0
\(745\) −10.9282 −0.400378
\(746\) 0 0
\(747\) 28.7321 1.05125
\(748\) 0 0
\(749\) −18.9282 −0.691621
\(750\) 0 0
\(751\) −24.2487 −0.884848 −0.442424 0.896806i \(-0.645881\pi\)
−0.442424 + 0.896806i \(0.645881\pi\)
\(752\) 0 0
\(753\) −19.1769 −0.698846
\(754\) 0 0
\(755\) −13.4641 −0.490009
\(756\) 0 0
\(757\) −38.2487 −1.39017 −0.695087 0.718926i \(-0.744634\pi\)
−0.695087 + 0.718926i \(0.744634\pi\)
\(758\) 0 0
\(759\) 3.71281 0.134767
\(760\) 0 0
\(761\) −33.7128 −1.22209 −0.611044 0.791596i \(-0.709250\pi\)
−0.611044 + 0.791596i \(0.709250\pi\)
\(762\) 0 0
\(763\) −8.39230 −0.303822
\(764\) 0 0
\(765\) −3.60770 −0.130436
\(766\) 0 0
\(767\) −25.0333 −0.903901
\(768\) 0 0
\(769\) 30.7846 1.11012 0.555061 0.831810i \(-0.312695\pi\)
0.555061 + 0.831810i \(0.312695\pi\)
\(770\) 0 0
\(771\) −4.39230 −0.158185
\(772\) 0 0
\(773\) −45.1244 −1.62301 −0.811505 0.584345i \(-0.801351\pi\)
−0.811505 + 0.584345i \(0.801351\pi\)
\(774\) 0 0
\(775\) −17.8564 −0.641421
\(776\) 0 0
\(777\) 4.00000 0.143499
\(778\) 0 0
\(779\) 9.46410 0.339087
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 0 0
\(783\) 21.8564 0.781084
\(784\) 0 0
\(785\) 1.32051 0.0471310
\(786\) 0 0
\(787\) 10.5885 0.377438 0.188719 0.982031i \(-0.439567\pi\)
0.188719 + 0.982031i \(0.439567\pi\)
\(788\) 0 0
\(789\) 2.14359 0.0763140
\(790\) 0 0
\(791\) 13.4641 0.478728
\(792\) 0 0
\(793\) 42.8897 1.52306
\(794\) 0 0
\(795\) −6.43078 −0.228076
\(796\) 0 0
\(797\) −38.9808 −1.38077 −0.690385 0.723442i \(-0.742559\pi\)
−0.690385 + 0.723442i \(0.742559\pi\)
\(798\) 0 0
\(799\) −21.8564 −0.773224
\(800\) 0 0
\(801\) −39.0718 −1.38053
\(802\) 0 0
\(803\) −1.35898 −0.0479575
\(804\) 0 0
\(805\) 2.53590 0.0893787
\(806\) 0 0
\(807\) 13.6077 0.479014
\(808\) 0 0
\(809\) 11.3205 0.398008 0.199004 0.979999i \(-0.436229\pi\)
0.199004 + 0.979999i \(0.436229\pi\)
\(810\) 0 0
\(811\) −2.87564 −0.100978 −0.0504888 0.998725i \(-0.516078\pi\)
−0.0504888 + 0.998725i \(0.516078\pi\)
\(812\) 0 0
\(813\) 0.784610 0.0275175
\(814\) 0 0
\(815\) 4.78461 0.167598
\(816\) 0 0
\(817\) −6.92820 −0.242387
\(818\) 0 0
\(819\) −8.05256 −0.281379
\(820\) 0 0
\(821\) −6.92820 −0.241796 −0.120898 0.992665i \(-0.538577\pi\)
−0.120898 + 0.992665i \(0.538577\pi\)
\(822\) 0 0
\(823\) −10.9282 −0.380933 −0.190467 0.981694i \(-0.561000\pi\)
−0.190467 + 0.981694i \(0.561000\pi\)
\(824\) 0 0
\(825\) 4.78461 0.166579
\(826\) 0 0
\(827\) 13.8564 0.481834 0.240917 0.970546i \(-0.422552\pi\)
0.240917 + 0.970546i \(0.422552\pi\)
\(828\) 0 0
\(829\) −10.9808 −0.381378 −0.190689 0.981651i \(-0.561072\pi\)
−0.190689 + 0.981651i \(0.561072\pi\)
\(830\) 0 0
\(831\) 2.92820 0.101578
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 13.8564 0.479521
\(836\) 0 0
\(837\) −16.0000 −0.553041
\(838\) 0 0
\(839\) −15.7128 −0.542467 −0.271233 0.962514i \(-0.587431\pi\)
−0.271233 + 0.962514i \(0.587431\pi\)
\(840\) 0 0
\(841\) 0.856406 0.0295313
\(842\) 0 0
\(843\) −1.46410 −0.0504263
\(844\) 0 0
\(845\) −1.69873 −0.0584381
\(846\) 0 0
\(847\) 8.85641 0.304310
\(848\) 0 0
\(849\) −10.6795 −0.366519
\(850\) 0 0
\(851\) 18.9282 0.648850
\(852\) 0 0
\(853\) −46.9808 −1.60859 −0.804295 0.594230i \(-0.797457\pi\)
−0.804295 + 0.594230i \(0.797457\pi\)
\(854\) 0 0
\(855\) −8.53590 −0.291922
\(856\) 0 0
\(857\) −10.7846 −0.368395 −0.184198 0.982889i \(-0.558969\pi\)
−0.184198 + 0.982889i \(0.558969\pi\)
\(858\) 0 0
\(859\) 28.4449 0.970526 0.485263 0.874368i \(-0.338724\pi\)
0.485263 + 0.874368i \(0.338724\pi\)
\(860\) 0 0
\(861\) −1.46410 −0.0498964
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) −10.3923 −0.353349
\(866\) 0 0
\(867\) −9.51666 −0.323203
\(868\) 0 0
\(869\) −4.28719 −0.145433
\(870\) 0 0
\(871\) 26.1436 0.885842
\(872\) 0 0
\(873\) 12.1436 0.410998
\(874\) 0 0
\(875\) 6.92820 0.234216
\(876\) 0 0
\(877\) −0.392305 −0.0132472 −0.00662360 0.999978i \(-0.502108\pi\)
−0.00662360 + 0.999978i \(0.502108\pi\)
\(878\) 0 0
\(879\) −1.32051 −0.0445396
\(880\) 0 0
\(881\) −9.21539 −0.310474 −0.155237 0.987877i \(-0.549614\pi\)
−0.155237 + 0.987877i \(0.549614\pi\)
\(882\) 0 0
\(883\) −18.9282 −0.636985 −0.318492 0.947925i \(-0.603177\pi\)
−0.318492 + 0.947925i \(0.603177\pi\)
\(884\) 0 0
\(885\) 4.10512 0.137992
\(886\) 0 0
\(887\) −22.6410 −0.760211 −0.380105 0.924943i \(-0.624112\pi\)
−0.380105 + 0.924943i \(0.624112\pi\)
\(888\) 0 0
\(889\) 4.53590 0.152129
\(890\) 0 0
\(891\) −6.53590 −0.218961
\(892\) 0 0
\(893\) −51.7128 −1.73050
\(894\) 0 0
\(895\) −8.00000 −0.267411
\(896\) 0 0
\(897\) 8.28719 0.276701
\(898\) 0 0
\(899\) −21.8564 −0.728952
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) 0 0
\(903\) 1.07180 0.0356672
\(904\) 0 0
\(905\) 5.32051 0.176860
\(906\) 0 0
\(907\) 11.7128 0.388918 0.194459 0.980911i \(-0.437705\pi\)
0.194459 + 0.980911i \(0.437705\pi\)
\(908\) 0 0
\(909\) 38.5885 1.27990
\(910\) 0 0
\(911\) −11.4641 −0.379823 −0.189911 0.981801i \(-0.560820\pi\)
−0.189911 + 0.981801i \(0.560820\pi\)
\(912\) 0 0
\(913\) 17.0718 0.564994
\(914\) 0 0
\(915\) −7.03332 −0.232514
\(916\) 0 0
\(917\) −7.26795 −0.240009
\(918\) 0 0
\(919\) 48.7846 1.60926 0.804628 0.593779i \(-0.202365\pi\)
0.804628 + 0.593779i \(0.202365\pi\)
\(920\) 0 0
\(921\) −16.5359 −0.544876
\(922\) 0 0
\(923\) −35.7128 −1.17550
\(924\) 0 0
\(925\) 24.3923 0.802014
\(926\) 0 0
\(927\) −44.0000 −1.44515
\(928\) 0 0
\(929\) −17.7128 −0.581139 −0.290569 0.956854i \(-0.593845\pi\)
−0.290569 + 0.956854i \(0.593845\pi\)
\(930\) 0 0
\(931\) 4.73205 0.155087
\(932\) 0 0
\(933\) −16.0000 −0.523816
\(934\) 0 0
\(935\) −2.14359 −0.0701030
\(936\) 0 0
\(937\) 60.6410 1.98106 0.990528 0.137312i \(-0.0438463\pi\)
0.990528 + 0.137312i \(0.0438463\pi\)
\(938\) 0 0
\(939\) 5.75129 0.187686
\(940\) 0 0
\(941\) 6.98076 0.227566 0.113783 0.993506i \(-0.463703\pi\)
0.113783 + 0.993506i \(0.463703\pi\)
\(942\) 0 0
\(943\) −6.92820 −0.225613
\(944\) 0 0
\(945\) 2.92820 0.0952545
\(946\) 0 0
\(947\) 3.60770 0.117234 0.0586172 0.998281i \(-0.481331\pi\)
0.0586172 + 0.998281i \(0.481331\pi\)
\(948\) 0 0
\(949\) −3.03332 −0.0984658
\(950\) 0 0
\(951\) 9.35898 0.303486
\(952\) 0 0
\(953\) −7.85641 −0.254494 −0.127247 0.991871i \(-0.540614\pi\)
−0.127247 + 0.991871i \(0.540614\pi\)
\(954\) 0 0
\(955\) 11.7128 0.379018
\(956\) 0 0
\(957\) 5.85641 0.189311
\(958\) 0 0
\(959\) −7.85641 −0.253697
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −46.6410 −1.50299
\(964\) 0 0
\(965\) 9.85641 0.317289
\(966\) 0 0
\(967\) −47.1769 −1.51711 −0.758554 0.651611i \(-0.774094\pi\)
−0.758554 + 0.651611i \(0.774094\pi\)
\(968\) 0 0
\(969\) 6.92820 0.222566
\(970\) 0 0
\(971\) −10.4833 −0.336426 −0.168213 0.985751i \(-0.553800\pi\)
−0.168213 + 0.985751i \(0.553800\pi\)
\(972\) 0 0
\(973\) 12.7321 0.408171
\(974\) 0 0
\(975\) 10.6795 0.342017
\(976\) 0 0
\(977\) −55.8564 −1.78700 −0.893502 0.449058i \(-0.851759\pi\)
−0.893502 + 0.449058i \(0.851759\pi\)
\(978\) 0 0
\(979\) −23.2154 −0.741967
\(980\) 0 0
\(981\) −20.6795 −0.660245
\(982\) 0 0
\(983\) −7.71281 −0.246001 −0.123000 0.992407i \(-0.539252\pi\)
−0.123000 + 0.992407i \(0.539252\pi\)
\(984\) 0 0
\(985\) 1.35898 0.0433008
\(986\) 0 0
\(987\) 8.00000 0.254643
\(988\) 0 0
\(989\) 5.07180 0.161274
\(990\) 0 0
\(991\) −16.7846 −0.533181 −0.266590 0.963810i \(-0.585897\pi\)
−0.266590 + 0.963810i \(0.585897\pi\)
\(992\) 0 0
\(993\) 3.21539 0.102037
\(994\) 0 0
\(995\) 2.92820 0.0928303
\(996\) 0 0
\(997\) 8.73205 0.276547 0.138273 0.990394i \(-0.455845\pi\)
0.138273 + 0.990394i \(0.455845\pi\)
\(998\) 0 0
\(999\) 21.8564 0.691506
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 1792.2.a.i.1.2 2
4.3 odd 2 1792.2.a.q.1.1 2
8.3 odd 2 1792.2.a.k.1.2 2
8.5 even 2 1792.2.a.s.1.1 2
16.3 odd 4 448.2.b.c.225.2 4
16.5 even 4 448.2.b.d.225.2 yes 4
16.11 odd 4 448.2.b.c.225.3 yes 4
16.13 even 4 448.2.b.d.225.3 yes 4
48.5 odd 4 4032.2.c.n.2017.2 4
48.11 even 4 4032.2.c.k.2017.2 4
48.29 odd 4 4032.2.c.n.2017.3 4
48.35 even 4 4032.2.c.k.2017.3 4
112.13 odd 4 3136.2.b.h.1569.2 4
112.27 even 4 3136.2.b.g.1569.2 4
112.69 odd 4 3136.2.b.h.1569.3 4
112.83 even 4 3136.2.b.g.1569.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
448.2.b.c.225.2 4 16.3 odd 4
448.2.b.c.225.3 yes 4 16.11 odd 4
448.2.b.d.225.2 yes 4 16.5 even 4
448.2.b.d.225.3 yes 4 16.13 even 4
1792.2.a.i.1.2 2 1.1 even 1 trivial
1792.2.a.k.1.2 2 8.3 odd 2
1792.2.a.q.1.1 2 4.3 odd 2
1792.2.a.s.1.1 2 8.5 even 2
3136.2.b.g.1569.2 4 112.27 even 4
3136.2.b.g.1569.3 4 112.83 even 4
3136.2.b.h.1569.2 4 112.13 odd 4
3136.2.b.h.1569.3 4 112.69 odd 4
4032.2.c.k.2017.2 4 48.11 even 4
4032.2.c.k.2017.3 4 48.35 even 4
4032.2.c.n.2017.2 4 48.5 odd 4
4032.2.c.n.2017.3 4 48.29 odd 4