Properties

Label 4400.2.a.cc.1.2
Level $4400$
Weight $2$
Character 4400.1
Self dual yes
Analytic conductor $35.134$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4400,2,Mod(1,4400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4400.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.792287\) of defining polynomial
Character \(\chi\) \(=\) 4400.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.792287 q^{3} +3.46410 q^{7} -2.37228 q^{9} +O(q^{10})\) \(q-0.792287 q^{3} +3.46410 q^{7} -2.37228 q^{9} +1.00000 q^{11} -1.58457 q^{17} -4.00000 q^{19} -2.74456 q^{21} +0.792287 q^{23} +4.25639 q^{27} +8.74456 q^{29} -3.37228 q^{31} -0.792287 q^{33} +1.08724 q^{37} +8.74456 q^{41} +3.46410 q^{43} -6.63325 q^{47} +5.00000 q^{49} +1.25544 q^{51} -10.0974 q^{53} +3.16915 q^{57} +7.37228 q^{59} -0.744563 q^{61} -8.21782 q^{63} +9.30506 q^{67} -0.627719 q^{69} +10.1168 q^{71} +6.92820 q^{73} +3.46410 q^{77} -1.25544 q^{79} +3.74456 q^{81} +6.63325 q^{83} -6.92820 q^{87} +1.37228 q^{89} +2.67181 q^{93} -5.84096 q^{97} -2.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{9} + 4 q^{11} - 16 q^{19} + 12 q^{21} + 12 q^{29} - 2 q^{31} + 12 q^{41} + 20 q^{49} + 28 q^{51} + 18 q^{59} + 20 q^{61} - 14 q^{69} + 6 q^{71} - 28 q^{79} - 8 q^{81} - 6 q^{89} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −0.792287 −0.457427 −0.228714 0.973494i \(-0.573452\pi\)
−0.228714 + 0.973494i \(0.573452\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.46410 1.30931 0.654654 0.755929i \(-0.272814\pi\)
0.654654 + 0.755929i \(0.272814\pi\)
\(8\) 0 0
\(9\) −2.37228 −0.790760
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.58457 −0.384316 −0.192158 0.981364i \(-0.561549\pi\)
−0.192158 + 0.981364i \(0.561549\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) −2.74456 −0.598913
\(22\) 0 0
\(23\) 0.792287 0.165203 0.0826016 0.996583i \(-0.473677\pi\)
0.0826016 + 0.996583i \(0.473677\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.25639 0.819142
\(28\) 0 0
\(29\) 8.74456 1.62382 0.811912 0.583779i \(-0.198427\pi\)
0.811912 + 0.583779i \(0.198427\pi\)
\(30\) 0 0
\(31\) −3.37228 −0.605680 −0.302840 0.953041i \(-0.597935\pi\)
−0.302840 + 0.953041i \(0.597935\pi\)
\(32\) 0 0
\(33\) −0.792287 −0.137919
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.08724 0.178741 0.0893706 0.995998i \(-0.471514\pi\)
0.0893706 + 0.995998i \(0.471514\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.74456 1.36567 0.682836 0.730572i \(-0.260747\pi\)
0.682836 + 0.730572i \(0.260747\pi\)
\(42\) 0 0
\(43\) 3.46410 0.528271 0.264135 0.964486i \(-0.414913\pi\)
0.264135 + 0.964486i \(0.414913\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.63325 −0.967559 −0.483779 0.875190i \(-0.660736\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 1.25544 0.175796
\(52\) 0 0
\(53\) −10.0974 −1.38698 −0.693489 0.720467i \(-0.743927\pi\)
−0.693489 + 0.720467i \(0.743927\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.16915 0.419764
\(58\) 0 0
\(59\) 7.37228 0.959789 0.479895 0.877326i \(-0.340675\pi\)
0.479895 + 0.877326i \(0.340675\pi\)
\(60\) 0 0
\(61\) −0.744563 −0.0953315 −0.0476657 0.998863i \(-0.515178\pi\)
−0.0476657 + 0.998863i \(0.515178\pi\)
\(62\) 0 0
\(63\) −8.21782 −1.03535
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 9.30506 1.13679 0.568397 0.822754i \(-0.307564\pi\)
0.568397 + 0.822754i \(0.307564\pi\)
\(68\) 0 0
\(69\) −0.627719 −0.0755684
\(70\) 0 0
\(71\) 10.1168 1.20065 0.600324 0.799757i \(-0.295038\pi\)
0.600324 + 0.799757i \(0.295038\pi\)
\(72\) 0 0
\(73\) 6.92820 0.810885 0.405442 0.914121i \(-0.367117\pi\)
0.405442 + 0.914121i \(0.367117\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.46410 0.394771
\(78\) 0 0
\(79\) −1.25544 −0.141248 −0.0706239 0.997503i \(-0.522499\pi\)
−0.0706239 + 0.997503i \(0.522499\pi\)
\(80\) 0 0
\(81\) 3.74456 0.416063
\(82\) 0 0
\(83\) 6.63325 0.728094 0.364047 0.931381i \(-0.381395\pi\)
0.364047 + 0.931381i \(0.381395\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −6.92820 −0.742781
\(88\) 0 0
\(89\) 1.37228 0.145462 0.0727308 0.997352i \(-0.476829\pi\)
0.0727308 + 0.997352i \(0.476829\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 2.67181 0.277054
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −5.84096 −0.593060 −0.296530 0.955024i \(-0.595829\pi\)
−0.296530 + 0.955024i \(0.595829\pi\)
\(98\) 0 0
\(99\) −2.37228 −0.238423
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −10.3923 −1.02398 −0.511992 0.858990i \(-0.671092\pi\)
−0.511992 + 0.858990i \(0.671092\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.63325 −0.641260 −0.320630 0.947204i \(-0.603895\pi\)
−0.320630 + 0.947204i \(0.603895\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) −0.861407 −0.0817611
\(112\) 0 0
\(113\) 0.497333 0.0467852 0.0233926 0.999726i \(-0.492553\pi\)
0.0233926 + 0.999726i \(0.492553\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5.48913 −0.503187
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −6.92820 −0.624695
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −8.21782 −0.729214 −0.364607 0.931162i \(-0.618797\pi\)
−0.364607 + 0.931162i \(0.618797\pi\)
\(128\) 0 0
\(129\) −2.74456 −0.241645
\(130\) 0 0
\(131\) 2.74456 0.239794 0.119897 0.992786i \(-0.461744\pi\)
0.119897 + 0.992786i \(0.461744\pi\)
\(132\) 0 0
\(133\) −13.8564 −1.20150
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −14.3537 −1.22632 −0.613161 0.789958i \(-0.710102\pi\)
−0.613161 + 0.789958i \(0.710102\pi\)
\(138\) 0 0
\(139\) 16.2337 1.37692 0.688462 0.725273i \(-0.258286\pi\)
0.688462 + 0.725273i \(0.258286\pi\)
\(140\) 0 0
\(141\) 5.25544 0.442588
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −3.96143 −0.326734
\(148\) 0 0
\(149\) −11.4891 −0.941226 −0.470613 0.882340i \(-0.655967\pi\)
−0.470613 + 0.882340i \(0.655967\pi\)
\(150\) 0 0
\(151\) 12.2337 0.995563 0.497782 0.867302i \(-0.334148\pi\)
0.497782 + 0.867302i \(0.334148\pi\)
\(152\) 0 0
\(153\) 3.75906 0.303902
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 24.4511 1.95141 0.975705 0.219090i \(-0.0703087\pi\)
0.975705 + 0.219090i \(0.0703087\pi\)
\(158\) 0 0
\(159\) 8.00000 0.634441
\(160\) 0 0
\(161\) 2.74456 0.216302
\(162\) 0 0
\(163\) −3.46410 −0.271329 −0.135665 0.990755i \(-0.543317\pi\)
−0.135665 + 0.990755i \(0.543317\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −15.7359 −1.21768 −0.608842 0.793292i \(-0.708365\pi\)
−0.608842 + 0.793292i \(0.708365\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 9.48913 0.725652
\(172\) 0 0
\(173\) 8.51278 0.647214 0.323607 0.946192i \(-0.395104\pi\)
0.323607 + 0.946192i \(0.395104\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −5.84096 −0.439034
\(178\) 0 0
\(179\) −12.8614 −0.961307 −0.480653 0.876911i \(-0.659600\pi\)
−0.480653 + 0.876911i \(0.659600\pi\)
\(180\) 0 0
\(181\) 24.1168 1.79259 0.896295 0.443457i \(-0.146248\pi\)
0.896295 + 0.443457i \(0.146248\pi\)
\(182\) 0 0
\(183\) 0.589907 0.0436072
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1.58457 −0.115876
\(188\) 0 0
\(189\) 14.7446 1.07251
\(190\) 0 0
\(191\) 19.3723 1.40173 0.700865 0.713294i \(-0.252798\pi\)
0.700865 + 0.713294i \(0.252798\pi\)
\(192\) 0 0
\(193\) 16.4356 1.18306 0.591532 0.806282i \(-0.298523\pi\)
0.591532 + 0.806282i \(0.298523\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.51278 −0.606510 −0.303255 0.952909i \(-0.598073\pi\)
−0.303255 + 0.952909i \(0.598073\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) −7.37228 −0.520001
\(202\) 0 0
\(203\) 30.2921 2.12609
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.87953 −0.130636
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −1.48913 −0.102516 −0.0512578 0.998685i \(-0.516323\pi\)
−0.0512578 + 0.998685i \(0.516323\pi\)
\(212\) 0 0
\(213\) −8.01544 −0.549209
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −11.6819 −0.793021
\(218\) 0 0
\(219\) −5.48913 −0.370921
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −2.37686 −0.159166 −0.0795832 0.996828i \(-0.525359\pi\)
−0.0795832 + 0.996828i \(0.525359\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 16.7306 1.11045 0.555224 0.831701i \(-0.312632\pi\)
0.555224 + 0.831701i \(0.312632\pi\)
\(228\) 0 0
\(229\) 14.6277 0.966627 0.483313 0.875447i \(-0.339433\pi\)
0.483313 + 0.875447i \(0.339433\pi\)
\(230\) 0 0
\(231\) −2.74456 −0.180579
\(232\) 0 0
\(233\) 3.75906 0.246264 0.123132 0.992390i \(-0.460706\pi\)
0.123132 + 0.992390i \(0.460706\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0.994667 0.0646105
\(238\) 0 0
\(239\) 14.7446 0.953746 0.476873 0.878972i \(-0.341770\pi\)
0.476873 + 0.878972i \(0.341770\pi\)
\(240\) 0 0
\(241\) 16.7446 1.07861 0.539306 0.842110i \(-0.318687\pi\)
0.539306 + 0.842110i \(0.318687\pi\)
\(242\) 0 0
\(243\) −15.7359 −1.00946
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −5.25544 −0.333050
\(250\) 0 0
\(251\) 22.1168 1.39600 0.698001 0.716096i \(-0.254073\pi\)
0.698001 + 0.716096i \(0.254073\pi\)
\(252\) 0 0
\(253\) 0.792287 0.0498107
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −10.6873 −0.666653 −0.333326 0.942811i \(-0.608171\pi\)
−0.333326 + 0.942811i \(0.608171\pi\)
\(258\) 0 0
\(259\) 3.76631 0.234027
\(260\) 0 0
\(261\) −20.7446 −1.28406
\(262\) 0 0
\(263\) 27.4179 1.69066 0.845329 0.534246i \(-0.179405\pi\)
0.845329 + 0.534246i \(0.179405\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −1.08724 −0.0665380
\(268\) 0 0
\(269\) −11.4891 −0.700504 −0.350252 0.936655i \(-0.613904\pi\)
−0.350252 + 0.936655i \(0.613904\pi\)
\(270\) 0 0
\(271\) −13.4891 −0.819406 −0.409703 0.912219i \(-0.634368\pi\)
−0.409703 + 0.912219i \(0.634368\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 11.6819 0.701899 0.350949 0.936394i \(-0.385859\pi\)
0.350949 + 0.936394i \(0.385859\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −0.510875 −0.0304762 −0.0152381 0.999884i \(-0.504851\pi\)
−0.0152381 + 0.999884i \(0.504851\pi\)
\(282\) 0 0
\(283\) 15.1460 0.900338 0.450169 0.892943i \(-0.351364\pi\)
0.450169 + 0.892943i \(0.351364\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 30.2921 1.78808
\(288\) 0 0
\(289\) −14.4891 −0.852301
\(290\) 0 0
\(291\) 4.62772 0.271282
\(292\) 0 0
\(293\) −3.16915 −0.185144 −0.0925718 0.995706i \(-0.529509\pi\)
−0.0925718 + 0.995706i \(0.529509\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.25639 0.246981
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 0 0
\(303\) −4.75372 −0.273094
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 31.5817 1.80246 0.901231 0.433340i \(-0.142665\pi\)
0.901231 + 0.433340i \(0.142665\pi\)
\(308\) 0 0
\(309\) 8.23369 0.468398
\(310\) 0 0
\(311\) −5.48913 −0.311260 −0.155630 0.987815i \(-0.549741\pi\)
−0.155630 + 0.987815i \(0.549741\pi\)
\(312\) 0 0
\(313\) 21.8719 1.23627 0.618135 0.786072i \(-0.287888\pi\)
0.618135 + 0.786072i \(0.287888\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −32.9639 −1.85144 −0.925718 0.378215i \(-0.876538\pi\)
−0.925718 + 0.378215i \(0.876538\pi\)
\(318\) 0 0
\(319\) 8.74456 0.489602
\(320\) 0 0
\(321\) 5.25544 0.293330
\(322\) 0 0
\(323\) 6.33830 0.352672
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −7.92287 −0.438136
\(328\) 0 0
\(329\) −22.9783 −1.26683
\(330\) 0 0
\(331\) 14.1168 0.775932 0.387966 0.921674i \(-0.373178\pi\)
0.387966 + 0.921674i \(0.373178\pi\)
\(332\) 0 0
\(333\) −2.57924 −0.141342
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 32.4665 1.76856 0.884282 0.466952i \(-0.154648\pi\)
0.884282 + 0.466952i \(0.154648\pi\)
\(338\) 0 0
\(339\) −0.394031 −0.0214008
\(340\) 0 0
\(341\) −3.37228 −0.182619
\(342\) 0 0
\(343\) −6.92820 −0.374088
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −22.6641 −1.21667 −0.608337 0.793679i \(-0.708163\pi\)
−0.608337 + 0.793679i \(0.708163\pi\)
\(348\) 0 0
\(349\) 15.4891 0.829114 0.414557 0.910023i \(-0.363937\pi\)
0.414557 + 0.910023i \(0.363937\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −25.0410 −1.33280 −0.666399 0.745595i \(-0.732165\pi\)
−0.666399 + 0.745595i \(0.732165\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 4.34896 0.230172
\(358\) 0 0
\(359\) −29.4891 −1.55638 −0.778188 0.628031i \(-0.783861\pi\)
−0.778188 + 0.628031i \(0.783861\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) −0.792287 −0.0415843
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 25.7407 1.34365 0.671827 0.740708i \(-0.265510\pi\)
0.671827 + 0.740708i \(0.265510\pi\)
\(368\) 0 0
\(369\) −20.7446 −1.07992
\(370\) 0 0
\(371\) −34.9783 −1.81598
\(372\) 0 0
\(373\) 11.6819 0.604867 0.302434 0.953170i \(-0.402201\pi\)
0.302434 + 0.953170i \(0.402201\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0.627719 0.0322437 0.0161219 0.999870i \(-0.494868\pi\)
0.0161219 + 0.999870i \(0.494868\pi\)
\(380\) 0 0
\(381\) 6.51087 0.333562
\(382\) 0 0
\(383\) −10.8896 −0.556435 −0.278217 0.960518i \(-0.589744\pi\)
−0.278217 + 0.960518i \(0.589744\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.21782 −0.417735
\(388\) 0 0
\(389\) 18.8614 0.956311 0.478156 0.878275i \(-0.341305\pi\)
0.478156 + 0.878275i \(0.341305\pi\)
\(390\) 0 0
\(391\) −1.25544 −0.0634902
\(392\) 0 0
\(393\) −2.17448 −0.109688
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 16.4356 0.824881 0.412441 0.910984i \(-0.364676\pi\)
0.412441 + 0.910984i \(0.364676\pi\)
\(398\) 0 0
\(399\) 10.9783 0.549600
\(400\) 0 0
\(401\) −11.4891 −0.573740 −0.286870 0.957970i \(-0.592615\pi\)
−0.286870 + 0.957970i \(0.592615\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.08724 0.0538925
\(408\) 0 0
\(409\) 27.4891 1.35925 0.679625 0.733560i \(-0.262143\pi\)
0.679625 + 0.733560i \(0.262143\pi\)
\(410\) 0 0
\(411\) 11.3723 0.560953
\(412\) 0 0
\(413\) 25.5383 1.25666
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −12.8617 −0.629842
\(418\) 0 0
\(419\) 22.9783 1.12256 0.561280 0.827626i \(-0.310309\pi\)
0.561280 + 0.827626i \(0.310309\pi\)
\(420\) 0 0
\(421\) 8.51087 0.414795 0.207397 0.978257i \(-0.433501\pi\)
0.207397 + 0.978257i \(0.433501\pi\)
\(422\) 0 0
\(423\) 15.7359 0.765107
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −2.57924 −0.124818
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 25.7228 1.23902 0.619512 0.784987i \(-0.287330\pi\)
0.619512 + 0.784987i \(0.287330\pi\)
\(432\) 0 0
\(433\) −29.2048 −1.40349 −0.701747 0.712426i \(-0.747596\pi\)
−0.701747 + 0.712426i \(0.747596\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.16915 −0.151601
\(438\) 0 0
\(439\) −21.4891 −1.02562 −0.512810 0.858502i \(-0.671395\pi\)
−0.512810 + 0.858502i \(0.671395\pi\)
\(440\) 0 0
\(441\) −11.8614 −0.564829
\(442\) 0 0
\(443\) −31.6742 −1.50489 −0.752444 0.658656i \(-0.771125\pi\)
−0.752444 + 0.658656i \(0.771125\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 9.10268 0.430542
\(448\) 0 0
\(449\) 6.86141 0.323810 0.161905 0.986806i \(-0.448236\pi\)
0.161905 + 0.986806i \(0.448236\pi\)
\(450\) 0 0
\(451\) 8.74456 0.411765
\(452\) 0 0
\(453\) −9.69259 −0.455398
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 20.7846 0.972263 0.486132 0.873886i \(-0.338408\pi\)
0.486132 + 0.873886i \(0.338408\pi\)
\(458\) 0 0
\(459\) −6.74456 −0.314809
\(460\) 0 0
\(461\) 2.23369 0.104033 0.0520166 0.998646i \(-0.483435\pi\)
0.0520166 + 0.998646i \(0.483435\pi\)
\(462\) 0 0
\(463\) −30.0897 −1.39839 −0.699193 0.714933i \(-0.746457\pi\)
−0.699193 + 0.714933i \(0.746457\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −7.72049 −0.357262 −0.178631 0.983916i \(-0.557167\pi\)
−0.178631 + 0.983916i \(0.557167\pi\)
\(468\) 0 0
\(469\) 32.2337 1.48841
\(470\) 0 0
\(471\) −19.3723 −0.892628
\(472\) 0 0
\(473\) 3.46410 0.159280
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 23.9538 1.09677
\(478\) 0 0
\(479\) 5.48913 0.250805 0.125402 0.992106i \(-0.459978\pi\)
0.125402 + 0.992106i \(0.459978\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −2.17448 −0.0989423
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −7.13058 −0.323118 −0.161559 0.986863i \(-0.551652\pi\)
−0.161559 + 0.986863i \(0.551652\pi\)
\(488\) 0 0
\(489\) 2.74456 0.124113
\(490\) 0 0
\(491\) −6.51087 −0.293832 −0.146916 0.989149i \(-0.546935\pi\)
−0.146916 + 0.989149i \(0.546935\pi\)
\(492\) 0 0
\(493\) −13.8564 −0.624061
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 35.0458 1.57202
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) 12.4674 0.557001
\(502\) 0 0
\(503\) 13.5615 0.604675 0.302338 0.953201i \(-0.402233\pi\)
0.302338 + 0.953201i \(0.402233\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 10.2997 0.457427
\(508\) 0 0
\(509\) 22.6277 1.00296 0.501478 0.865170i \(-0.332790\pi\)
0.501478 + 0.865170i \(0.332790\pi\)
\(510\) 0 0
\(511\) 24.0000 1.06170
\(512\) 0 0
\(513\) −17.0256 −0.751697
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −6.63325 −0.291730
\(518\) 0 0
\(519\) −6.74456 −0.296053
\(520\) 0 0
\(521\) −21.6060 −0.946575 −0.473287 0.880908i \(-0.656933\pi\)
−0.473287 + 0.880908i \(0.656933\pi\)
\(522\) 0 0
\(523\) 29.0024 1.26819 0.634094 0.773256i \(-0.281373\pi\)
0.634094 + 0.773256i \(0.281373\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 5.34363 0.232772
\(528\) 0 0
\(529\) −22.3723 −0.972708
\(530\) 0 0
\(531\) −17.4891 −0.758963
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 10.1899 0.439728
\(538\) 0 0
\(539\) 5.00000 0.215365
\(540\) 0 0
\(541\) 34.2337 1.47182 0.735911 0.677079i \(-0.236754\pi\)
0.735911 + 0.677079i \(0.236754\pi\)
\(542\) 0 0
\(543\) −19.1075 −0.819980
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −29.0024 −1.24005 −0.620027 0.784580i \(-0.712878\pi\)
−0.620027 + 0.784580i \(0.712878\pi\)
\(548\) 0 0
\(549\) 1.76631 0.0753844
\(550\) 0 0
\(551\) −34.9783 −1.49012
\(552\) 0 0
\(553\) −4.34896 −0.184937
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0.994667 0.0421454 0.0210727 0.999778i \(-0.493292\pi\)
0.0210727 + 0.999778i \(0.493292\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 1.25544 0.0530046
\(562\) 0 0
\(563\) −18.9051 −0.796754 −0.398377 0.917222i \(-0.630426\pi\)
−0.398377 + 0.917222i \(0.630426\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 12.9715 0.544754
\(568\) 0 0
\(569\) −27.2554 −1.14261 −0.571304 0.820739i \(-0.693562\pi\)
−0.571304 + 0.820739i \(0.693562\pi\)
\(570\) 0 0
\(571\) −1.48913 −0.0623180 −0.0311590 0.999514i \(-0.509920\pi\)
−0.0311590 + 0.999514i \(0.509920\pi\)
\(572\) 0 0
\(573\) −15.3484 −0.641189
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −21.8719 −0.910537 −0.455269 0.890354i \(-0.650457\pi\)
−0.455269 + 0.890354i \(0.650457\pi\)
\(578\) 0 0
\(579\) −13.0217 −0.541165
\(580\) 0 0
\(581\) 22.9783 0.953298
\(582\) 0 0
\(583\) −10.0974 −0.418190
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −41.2743 −1.70357 −0.851786 0.523890i \(-0.824480\pi\)
−0.851786 + 0.523890i \(0.824480\pi\)
\(588\) 0 0
\(589\) 13.4891 0.555810
\(590\) 0 0
\(591\) 6.74456 0.277434
\(592\) 0 0
\(593\) 22.7739 0.935214 0.467607 0.883937i \(-0.345116\pi\)
0.467607 + 0.883937i \(0.345116\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −6.33830 −0.259409
\(598\) 0 0
\(599\) −10.9783 −0.448559 −0.224280 0.974525i \(-0.572003\pi\)
−0.224280 + 0.974525i \(0.572003\pi\)
\(600\) 0 0
\(601\) −38.4674 −1.56912 −0.784558 0.620055i \(-0.787110\pi\)
−0.784558 + 0.620055i \(0.787110\pi\)
\(602\) 0 0
\(603\) −22.0742 −0.898932
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 3.46410 0.140604 0.0703018 0.997526i \(-0.477604\pi\)
0.0703018 + 0.997526i \(0.477604\pi\)
\(608\) 0 0
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 4.34896 0.175653 0.0878265 0.996136i \(-0.472008\pi\)
0.0878265 + 0.996136i \(0.472008\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 17.0256 0.685423 0.342712 0.939441i \(-0.388655\pi\)
0.342712 + 0.939441i \(0.388655\pi\)
\(618\) 0 0
\(619\) −14.1168 −0.567404 −0.283702 0.958913i \(-0.591563\pi\)
−0.283702 + 0.958913i \(0.591563\pi\)
\(620\) 0 0
\(621\) 3.37228 0.135325
\(622\) 0 0
\(623\) 4.75372 0.190454
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 3.16915 0.126564
\(628\) 0 0
\(629\) −1.72281 −0.0686931
\(630\) 0 0
\(631\) −23.6060 −0.939739 −0.469869 0.882736i \(-0.655699\pi\)
−0.469869 + 0.882736i \(0.655699\pi\)
\(632\) 0 0
\(633\) 1.17981 0.0468934
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −24.0000 −0.949425
\(640\) 0 0
\(641\) −25.3723 −1.00214 −0.501072 0.865405i \(-0.667061\pi\)
−0.501072 + 0.865405i \(0.667061\pi\)
\(642\) 0 0
\(643\) −30.4944 −1.20258 −0.601292 0.799030i \(-0.705347\pi\)
−0.601292 + 0.799030i \(0.705347\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −21.9817 −0.864188 −0.432094 0.901829i \(-0.642225\pi\)
−0.432094 + 0.901829i \(0.642225\pi\)
\(648\) 0 0
\(649\) 7.37228 0.289387
\(650\) 0 0
\(651\) 9.25544 0.362749
\(652\) 0 0
\(653\) 30.7894 1.20488 0.602441 0.798163i \(-0.294195\pi\)
0.602441 + 0.798163i \(0.294195\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −16.4356 −0.641216
\(658\) 0 0
\(659\) −21.2554 −0.827994 −0.413997 0.910278i \(-0.635868\pi\)
−0.413997 + 0.910278i \(0.635868\pi\)
\(660\) 0 0
\(661\) −16.3505 −0.635962 −0.317981 0.948097i \(-0.603005\pi\)
−0.317981 + 0.948097i \(0.603005\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 6.92820 0.268261
\(668\) 0 0
\(669\) 1.88316 0.0728070
\(670\) 0 0
\(671\) −0.744563 −0.0287435
\(672\) 0 0
\(673\) −18.6101 −0.717368 −0.358684 0.933459i \(-0.616774\pi\)
−0.358684 + 0.933459i \(0.616774\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −50.0820 −1.92481 −0.962404 0.271623i \(-0.912440\pi\)
−0.962404 + 0.271623i \(0.912440\pi\)
\(678\) 0 0
\(679\) −20.2337 −0.776498
\(680\) 0 0
\(681\) −13.2554 −0.507949
\(682\) 0 0
\(683\) 17.9104 0.685323 0.342661 0.939459i \(-0.388672\pi\)
0.342661 + 0.939459i \(0.388672\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −11.5894 −0.442161
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −44.8614 −1.70661 −0.853304 0.521413i \(-0.825405\pi\)
−0.853304 + 0.521413i \(0.825405\pi\)
\(692\) 0 0
\(693\) −8.21782 −0.312169
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −13.8564 −0.524849
\(698\) 0 0
\(699\) −2.97825 −0.112648
\(700\) 0 0
\(701\) −12.5109 −0.472529 −0.236265 0.971689i \(-0.575923\pi\)
−0.236265 + 0.971689i \(0.575923\pi\)
\(702\) 0 0
\(703\) −4.34896 −0.164024
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 20.7846 0.781686
\(708\) 0 0
\(709\) 23.8832 0.896951 0.448475 0.893795i \(-0.351967\pi\)
0.448475 + 0.893795i \(0.351967\pi\)
\(710\) 0 0
\(711\) 2.97825 0.111693
\(712\) 0 0
\(713\) −2.67181 −0.100060
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −11.6819 −0.436269
\(718\) 0 0
\(719\) 30.3505 1.13188 0.565942 0.824445i \(-0.308513\pi\)
0.565942 + 0.824445i \(0.308513\pi\)
\(720\) 0 0
\(721\) −36.0000 −1.34071
\(722\) 0 0
\(723\) −13.2665 −0.493386
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 14.0588 0.521412 0.260706 0.965418i \(-0.416045\pi\)
0.260706 + 0.965418i \(0.416045\pi\)
\(728\) 0 0
\(729\) 1.23369 0.0456921
\(730\) 0 0
\(731\) −5.48913 −0.203023
\(732\) 0 0
\(733\) 9.50744 0.351165 0.175583 0.984465i \(-0.443819\pi\)
0.175583 + 0.984465i \(0.443819\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.30506 0.342756
\(738\) 0 0
\(739\) 10.7446 0.395245 0.197623 0.980278i \(-0.436678\pi\)
0.197623 + 0.980278i \(0.436678\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 11.3870 0.417747 0.208874 0.977943i \(-0.433020\pi\)
0.208874 + 0.977943i \(0.433020\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −15.7359 −0.575748
\(748\) 0 0
\(749\) −22.9783 −0.839607
\(750\) 0 0
\(751\) −27.3723 −0.998829 −0.499414 0.866363i \(-0.666451\pi\)
−0.499414 + 0.866363i \(0.666451\pi\)
\(752\) 0 0
\(753\) −17.5229 −0.638570
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −39.7995 −1.44654 −0.723269 0.690567i \(-0.757361\pi\)
−0.723269 + 0.690567i \(0.757361\pi\)
\(758\) 0 0
\(759\) −0.627719 −0.0227847
\(760\) 0 0
\(761\) 32.7446 1.18699 0.593495 0.804838i \(-0.297748\pi\)
0.593495 + 0.804838i \(0.297748\pi\)
\(762\) 0 0
\(763\) 34.6410 1.25409
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 29.2119 1.05341 0.526705 0.850048i \(-0.323427\pi\)
0.526705 + 0.850048i \(0.323427\pi\)
\(770\) 0 0
\(771\) 8.46738 0.304945
\(772\) 0 0
\(773\) 17.6155 0.633584 0.316792 0.948495i \(-0.397394\pi\)
0.316792 + 0.948495i \(0.397394\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −2.98400 −0.107050
\(778\) 0 0
\(779\) −34.9783 −1.25323
\(780\) 0 0
\(781\) 10.1168 0.362009
\(782\) 0 0
\(783\) 37.2203 1.33014
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −15.1460 −0.539898 −0.269949 0.962875i \(-0.587007\pi\)
−0.269949 + 0.962875i \(0.587007\pi\)
\(788\) 0 0
\(789\) −21.7228 −0.773353
\(790\) 0 0
\(791\) 1.72281 0.0612562
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −5.25106 −0.186002 −0.0930010 0.995666i \(-0.529646\pi\)
−0.0930010 + 0.995666i \(0.529646\pi\)
\(798\) 0 0
\(799\) 10.5109 0.371848
\(800\) 0 0
\(801\) −3.25544 −0.115025
\(802\) 0 0
\(803\) 6.92820 0.244491
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 9.10268 0.320430
\(808\) 0 0
\(809\) −32.7446 −1.15124 −0.575619 0.817718i \(-0.695239\pi\)
−0.575619 + 0.817718i \(0.695239\pi\)
\(810\) 0 0
\(811\) 0.233688 0.00820589 0.00410295 0.999992i \(-0.498694\pi\)
0.00410295 + 0.999992i \(0.498694\pi\)
\(812\) 0 0
\(813\) 10.6873 0.374819
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −13.8564 −0.484774
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) 56.0328 1.95318 0.976590 0.215111i \(-0.0690112\pi\)
0.976590 + 0.215111i \(0.0690112\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 28.4125 0.988000 0.494000 0.869462i \(-0.335534\pi\)
0.494000 + 0.869462i \(0.335534\pi\)
\(828\) 0 0
\(829\) −20.3505 −0.706803 −0.353402 0.935472i \(-0.614975\pi\)
−0.353402 + 0.935472i \(0.614975\pi\)
\(830\) 0 0
\(831\) −9.25544 −0.321068
\(832\) 0 0
\(833\) −7.92287 −0.274511
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −14.3537 −0.496138
\(838\) 0 0
\(839\) 10.1168 0.349272 0.174636 0.984633i \(-0.444125\pi\)
0.174636 + 0.984633i \(0.444125\pi\)
\(840\) 0 0
\(841\) 47.4674 1.63681
\(842\) 0 0
\(843\) 0.404759 0.0139407
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 3.46410 0.119028
\(848\) 0 0
\(849\) −12.0000 −0.411839
\(850\) 0 0
\(851\) 0.861407 0.0295286
\(852\) 0 0
\(853\) −35.0458 −1.19994 −0.599972 0.800021i \(-0.704822\pi\)
−0.599972 + 0.800021i \(0.704822\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 23.9538 0.818245 0.409122 0.912480i \(-0.365835\pi\)
0.409122 + 0.912480i \(0.365835\pi\)
\(858\) 0 0
\(859\) 6.11684 0.208704 0.104352 0.994540i \(-0.466723\pi\)
0.104352 + 0.994540i \(0.466723\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) 0 0
\(863\) −2.87419 −0.0978387 −0.0489194 0.998803i \(-0.515578\pi\)
−0.0489194 + 0.998803i \(0.515578\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 11.4795 0.389866
\(868\) 0 0
\(869\) −1.25544 −0.0425878
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 13.8564 0.468968
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 0 0
\(879\) 2.51087 0.0846897
\(880\) 0 0
\(881\) −6.86141 −0.231167 −0.115583 0.993298i \(-0.536874\pi\)
−0.115583 + 0.993298i \(0.536874\pi\)
\(882\) 0 0
\(883\) 24.2487 0.816034 0.408017 0.912974i \(-0.366220\pi\)
0.408017 + 0.912974i \(0.366220\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −27.4179 −0.920602 −0.460301 0.887763i \(-0.652258\pi\)
−0.460301 + 0.887763i \(0.652258\pi\)
\(888\) 0 0
\(889\) −28.4674 −0.954765
\(890\) 0 0
\(891\) 3.74456 0.125448
\(892\) 0 0
\(893\) 26.5330 0.887893
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −29.4891 −0.983517
\(900\) 0 0
\(901\) 16.0000 0.533037
\(902\) 0 0
\(903\) −9.50744 −0.316388
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −19.8997 −0.660760 −0.330380 0.943848i \(-0.607177\pi\)
−0.330380 + 0.943848i \(0.607177\pi\)
\(908\) 0 0
\(909\) −14.2337 −0.472102
\(910\) 0 0
\(911\) 53.4891 1.77217 0.886087 0.463519i \(-0.153413\pi\)
0.886087 + 0.463519i \(0.153413\pi\)
\(912\) 0 0
\(913\) 6.63325 0.219529
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 9.50744 0.313963
\(918\) 0 0
\(919\) 28.2337 0.931343 0.465672 0.884958i \(-0.345813\pi\)
0.465672 + 0.884958i \(0.345813\pi\)
\(920\) 0 0
\(921\) −25.0217 −0.824495
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 24.6535 0.809726
\(928\) 0 0
\(929\) −7.02175 −0.230376 −0.115188 0.993344i \(-0.536747\pi\)
−0.115188 + 0.993344i \(0.536747\pi\)
\(930\) 0 0
\(931\) −20.0000 −0.655474
\(932\) 0 0
\(933\) 4.34896 0.142379
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −53.6559 −1.75286 −0.876431 0.481527i \(-0.840082\pi\)
−0.876431 + 0.481527i \(0.840082\pi\)
\(938\) 0 0
\(939\) −17.3288 −0.565503
\(940\) 0 0
\(941\) −58.4674 −1.90598 −0.952991 0.302999i \(-0.902012\pi\)
−0.952991 + 0.302999i \(0.902012\pi\)
\(942\) 0 0
\(943\) 6.92820 0.225613
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −26.7354 −0.868783 −0.434392 0.900724i \(-0.643037\pi\)
−0.434392 + 0.900724i \(0.643037\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 26.1168 0.846897
\(952\) 0 0
\(953\) −31.2867 −1.01348 −0.506738 0.862100i \(-0.669149\pi\)
−0.506738 + 0.862100i \(0.669149\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −6.92820 −0.223957
\(958\) 0 0
\(959\) −49.7228 −1.60563
\(960\) 0 0
\(961\) −19.6277 −0.633152
\(962\) 0 0
\(963\) 15.7359 0.507083
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −26.4232 −0.849713 −0.424856 0.905261i \(-0.639675\pi\)
−0.424856 + 0.905261i \(0.639675\pi\)
\(968\) 0 0
\(969\) −5.02175 −0.161322
\(970\) 0 0
\(971\) 9.09509 0.291875 0.145938 0.989294i \(-0.453380\pi\)
0.145938 + 0.989294i \(0.453380\pi\)
\(972\) 0 0
\(973\) 56.2351 1.80282
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 50.5793 1.61818 0.809088 0.587687i \(-0.199962\pi\)
0.809088 + 0.587687i \(0.199962\pi\)
\(978\) 0 0
\(979\) 1.37228 0.0438583
\(980\) 0 0
\(981\) −23.7228 −0.757411
\(982\) 0 0
\(983\) −24.7460 −0.789276 −0.394638 0.918837i \(-0.629130\pi\)
−0.394638 + 0.918837i \(0.629130\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 18.2054 0.579483
\(988\) 0 0
\(989\) 2.74456 0.0872720
\(990\) 0 0
\(991\) −18.9783 −0.602864 −0.301432 0.953488i \(-0.597465\pi\)
−0.301432 + 0.953488i \(0.597465\pi\)
\(992\) 0 0
\(993\) −11.1846 −0.354932
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −2.17448 −0.0688665 −0.0344333 0.999407i \(-0.510963\pi\)
−0.0344333 + 0.999407i \(0.510963\pi\)
\(998\) 0 0
\(999\) 4.62772 0.146415
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 4400.2.a.cc.1.2 4
4.3 odd 2 275.2.a.h.1.4 4
5.2 odd 4 880.2.b.h.529.3 4
5.3 odd 4 880.2.b.h.529.2 4
5.4 even 2 inner 4400.2.a.cc.1.3 4
12.11 even 2 2475.2.a.bi.1.1 4
20.3 even 4 55.2.b.a.34.1 4
20.7 even 4 55.2.b.a.34.4 yes 4
20.19 odd 2 275.2.a.h.1.1 4
44.43 even 2 3025.2.a.ba.1.1 4
60.23 odd 4 495.2.c.a.199.4 4
60.47 odd 4 495.2.c.a.199.1 4
60.59 even 2 2475.2.a.bi.1.4 4
220.3 even 20 605.2.j.i.9.1 16
220.7 odd 20 605.2.j.j.269.4 16
220.27 even 20 605.2.j.i.124.1 16
220.43 odd 4 605.2.b.c.364.4 4
220.47 even 20 605.2.j.i.9.4 16
220.63 odd 20 605.2.j.j.9.4 16
220.83 odd 20 605.2.j.j.124.1 16
220.87 odd 4 605.2.b.c.364.1 4
220.103 even 20 605.2.j.i.269.4 16
220.107 odd 20 605.2.j.j.9.1 16
220.123 odd 20 605.2.j.j.444.4 16
220.127 odd 20 605.2.j.j.124.4 16
220.147 even 20 605.2.j.i.269.1 16
220.163 even 20 605.2.j.i.444.1 16
220.167 odd 20 605.2.j.j.444.1 16
220.183 odd 20 605.2.j.j.269.1 16
220.203 even 20 605.2.j.i.124.4 16
220.207 even 20 605.2.j.i.444.4 16
220.219 even 2 3025.2.a.ba.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.b.a.34.1 4 20.3 even 4
55.2.b.a.34.4 yes 4 20.7 even 4
275.2.a.h.1.1 4 20.19 odd 2
275.2.a.h.1.4 4 4.3 odd 2
495.2.c.a.199.1 4 60.47 odd 4
495.2.c.a.199.4 4 60.23 odd 4
605.2.b.c.364.1 4 220.87 odd 4
605.2.b.c.364.4 4 220.43 odd 4
605.2.j.i.9.1 16 220.3 even 20
605.2.j.i.9.4 16 220.47 even 20
605.2.j.i.124.1 16 220.27 even 20
605.2.j.i.124.4 16 220.203 even 20
605.2.j.i.269.1 16 220.147 even 20
605.2.j.i.269.4 16 220.103 even 20
605.2.j.i.444.1 16 220.163 even 20
605.2.j.i.444.4 16 220.207 even 20
605.2.j.j.9.1 16 220.107 odd 20
605.2.j.j.9.4 16 220.63 odd 20
605.2.j.j.124.1 16 220.83 odd 20
605.2.j.j.124.4 16 220.127 odd 20
605.2.j.j.269.1 16 220.183 odd 20
605.2.j.j.269.4 16 220.7 odd 20
605.2.j.j.444.1 16 220.167 odd 20
605.2.j.j.444.4 16 220.123 odd 20
880.2.b.h.529.2 4 5.3 odd 4
880.2.b.h.529.3 4 5.2 odd 4
2475.2.a.bi.1.1 4 12.11 even 2
2475.2.a.bi.1.4 4 60.59 even 2
3025.2.a.ba.1.1 4 44.43 even 2
3025.2.a.ba.1.4 4 220.219 even 2
4400.2.a.cc.1.2 4 1.1 even 1 trivial
4400.2.a.cc.1.3 4 5.4 even 2 inner