Properties

Label 3800.2.a.bc.1.6
Level $3800$
Weight $2$
Character 3800.1
Self dual yes
Analytic conductor $30.343$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3800,2,Mod(1,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, 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("3800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.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: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 10x^{4} + 16x^{3} + 15x^{2} - 14x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(3.26143\) of defining polynomial
Character \(\chi\) \(=\) 3800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.26143 q^{3} +4.07225 q^{7} +7.63693 q^{9} +O(q^{10})\) \(q+3.26143 q^{3} +4.07225 q^{7} +7.63693 q^{9} -0.786366 q^{11} +1.07974 q^{13} -1.90793 q^{17} -1.00000 q^{19} +13.2813 q^{21} -1.41383 q^{23} +15.1230 q^{27} -7.26439 q^{29} +2.22003 q^{31} -2.56468 q^{33} +9.14283 q^{37} +3.52151 q^{39} -6.11703 q^{41} +8.40634 q^{43} +3.56468 q^{47} +9.58319 q^{49} -6.22257 q^{51} -8.57472 q^{53} -3.26143 q^{57} -13.4043 q^{59} +12.7768 q^{61} +31.0994 q^{63} -5.10008 q^{67} -4.61112 q^{69} +1.65535 q^{71} +10.3302 q^{73} -3.20228 q^{77} -16.2256 q^{79} +26.4119 q^{81} +9.35104 q^{83} -23.6923 q^{87} +3.10419 q^{89} +4.39698 q^{91} +7.24046 q^{93} +4.55874 q^{97} -6.00542 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{3} + 2 q^{7} + 6 q^{9} + 3 q^{11} - q^{13} - 14 q^{17} - 6 q^{19} + 15 q^{21} + 12 q^{23} + 8 q^{27} + 9 q^{29} + 5 q^{31} + 2 q^{33} - 8 q^{37} + 12 q^{39} + 3 q^{41} + 15 q^{43} + 4 q^{47} + 22 q^{49} + 33 q^{51} + 13 q^{53} - 2 q^{57} + 9 q^{61} + 21 q^{63} - 3 q^{67} - 11 q^{69} + 19 q^{71} + 3 q^{73} - 36 q^{77} - 16 q^{79} + 26 q^{81} + 31 q^{83} - 25 q^{87} + 14 q^{89} + 42 q^{91} + 39 q^{93} - 11 q^{97} - 6 q^{99}+O(q^{100}) \) 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
\(3\) 3.26143 1.88299 0.941494 0.337031i \(-0.109423\pi\)
0.941494 + 0.337031i \(0.109423\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.07225 1.53916 0.769582 0.638548i \(-0.220464\pi\)
0.769582 + 0.638548i \(0.220464\pi\)
\(8\) 0 0
\(9\) 7.63693 2.54564
\(10\) 0 0
\(11\) −0.786366 −0.237098 −0.118549 0.992948i \(-0.537824\pi\)
−0.118549 + 0.992948i \(0.537824\pi\)
\(12\) 0 0
\(13\) 1.07974 0.299467 0.149733 0.988726i \(-0.452158\pi\)
0.149733 + 0.988726i \(0.452158\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.90793 −0.462741 −0.231370 0.972866i \(-0.574321\pi\)
−0.231370 + 0.972866i \(0.574321\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 13.2813 2.89823
\(22\) 0 0
\(23\) −1.41383 −0.294805 −0.147402 0.989077i \(-0.547091\pi\)
−0.147402 + 0.989077i \(0.547091\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 15.1230 2.91042
\(28\) 0 0
\(29\) −7.26439 −1.34896 −0.674482 0.738291i \(-0.735633\pi\)
−0.674482 + 0.738291i \(0.735633\pi\)
\(30\) 0 0
\(31\) 2.22003 0.398728 0.199364 0.979925i \(-0.436112\pi\)
0.199364 + 0.979925i \(0.436112\pi\)
\(32\) 0 0
\(33\) −2.56468 −0.446453
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9.14283 1.50307 0.751536 0.659692i \(-0.229313\pi\)
0.751536 + 0.659692i \(0.229313\pi\)
\(38\) 0 0
\(39\) 3.52151 0.563893
\(40\) 0 0
\(41\) −6.11703 −0.955319 −0.477660 0.878545i \(-0.658515\pi\)
−0.477660 + 0.878545i \(0.658515\pi\)
\(42\) 0 0
\(43\) 8.40634 1.28195 0.640977 0.767560i \(-0.278529\pi\)
0.640977 + 0.767560i \(0.278529\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.56468 0.519962 0.259981 0.965614i \(-0.416284\pi\)
0.259981 + 0.965614i \(0.416284\pi\)
\(48\) 0 0
\(49\) 9.58319 1.36903
\(50\) 0 0
\(51\) −6.22257 −0.871335
\(52\) 0 0
\(53\) −8.57472 −1.17783 −0.588914 0.808195i \(-0.700445\pi\)
−0.588914 + 0.808195i \(0.700445\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.26143 −0.431987
\(58\) 0 0
\(59\) −13.4043 −1.74509 −0.872543 0.488537i \(-0.837531\pi\)
−0.872543 + 0.488537i \(0.837531\pi\)
\(60\) 0 0
\(61\) 12.7768 1.63590 0.817950 0.575289i \(-0.195110\pi\)
0.817950 + 0.575289i \(0.195110\pi\)
\(62\) 0 0
\(63\) 31.0994 3.91816
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −5.10008 −0.623073 −0.311537 0.950234i \(-0.600844\pi\)
−0.311537 + 0.950234i \(0.600844\pi\)
\(68\) 0 0
\(69\) −4.61112 −0.555114
\(70\) 0 0
\(71\) 1.65535 0.196454 0.0982268 0.995164i \(-0.468683\pi\)
0.0982268 + 0.995164i \(0.468683\pi\)
\(72\) 0 0
\(73\) 10.3302 1.20906 0.604532 0.796581i \(-0.293360\pi\)
0.604532 + 0.796581i \(0.293360\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.20228 −0.364933
\(78\) 0 0
\(79\) −16.2256 −1.82553 −0.912764 0.408488i \(-0.866056\pi\)
−0.912764 + 0.408488i \(0.866056\pi\)
\(80\) 0 0
\(81\) 26.4119 2.93465
\(82\) 0 0
\(83\) 9.35104 1.02641 0.513205 0.858266i \(-0.328458\pi\)
0.513205 + 0.858266i \(0.328458\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −23.6923 −2.54008
\(88\) 0 0
\(89\) 3.10419 0.329044 0.164522 0.986373i \(-0.447392\pi\)
0.164522 + 0.986373i \(0.447392\pi\)
\(90\) 0 0
\(91\) 4.39698 0.460929
\(92\) 0 0
\(93\) 7.24046 0.750801
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.55874 0.462870 0.231435 0.972850i \(-0.425658\pi\)
0.231435 + 0.972850i \(0.425658\pi\)
\(98\) 0 0
\(99\) −6.00542 −0.603567
\(100\) 0 0
\(101\) 0.743401 0.0739712 0.0369856 0.999316i \(-0.488224\pi\)
0.0369856 + 0.999316i \(0.488224\pi\)
\(102\) 0 0
\(103\) −10.3710 −1.02188 −0.510941 0.859616i \(-0.670703\pi\)
−0.510941 + 0.859616i \(0.670703\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −17.1400 −1.65698 −0.828491 0.560002i \(-0.810800\pi\)
−0.828491 + 0.560002i \(0.810800\pi\)
\(108\) 0 0
\(109\) −17.1878 −1.64629 −0.823144 0.567832i \(-0.807782\pi\)
−0.823144 + 0.567832i \(0.807782\pi\)
\(110\) 0 0
\(111\) 29.8187 2.83027
\(112\) 0 0
\(113\) −1.05696 −0.0994300 −0.0497150 0.998763i \(-0.515831\pi\)
−0.0497150 + 0.998763i \(0.515831\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 8.24592 0.762336
\(118\) 0 0
\(119\) −7.76956 −0.712234
\(120\) 0 0
\(121\) −10.3816 −0.943784
\(122\) 0 0
\(123\) −19.9503 −1.79885
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −5.22444 −0.463594 −0.231797 0.972764i \(-0.574461\pi\)
−0.231797 + 0.972764i \(0.574461\pi\)
\(128\) 0 0
\(129\) 27.4167 2.41390
\(130\) 0 0
\(131\) 3.00267 0.262344 0.131172 0.991360i \(-0.458126\pi\)
0.131172 + 0.991360i \(0.458126\pi\)
\(132\) 0 0
\(133\) −4.07225 −0.353109
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −23.0922 −1.97290 −0.986448 0.164072i \(-0.947537\pi\)
−0.986448 + 0.164072i \(0.947537\pi\)
\(138\) 0 0
\(139\) −11.9630 −1.01469 −0.507343 0.861744i \(-0.669372\pi\)
−0.507343 + 0.861744i \(0.669372\pi\)
\(140\) 0 0
\(141\) 11.6259 0.979082
\(142\) 0 0
\(143\) −0.849074 −0.0710031
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 31.2549 2.57786
\(148\) 0 0
\(149\) 23.5411 1.92856 0.964282 0.264879i \(-0.0853320\pi\)
0.964282 + 0.264879i \(0.0853320\pi\)
\(150\) 0 0
\(151\) 10.6136 0.863721 0.431860 0.901940i \(-0.357857\pi\)
0.431860 + 0.901940i \(0.357857\pi\)
\(152\) 0 0
\(153\) −14.5707 −1.17797
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 13.9997 1.11730 0.558648 0.829405i \(-0.311320\pi\)
0.558648 + 0.829405i \(0.311320\pi\)
\(158\) 0 0
\(159\) −27.9659 −2.21784
\(160\) 0 0
\(161\) −5.75748 −0.453753
\(162\) 0 0
\(163\) −10.6662 −0.835442 −0.417721 0.908575i \(-0.637171\pi\)
−0.417721 + 0.908575i \(0.637171\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3.97856 −0.307870 −0.153935 0.988081i \(-0.549195\pi\)
−0.153935 + 0.988081i \(0.549195\pi\)
\(168\) 0 0
\(169\) −11.8342 −0.910320
\(170\) 0 0
\(171\) −7.63693 −0.584010
\(172\) 0 0
\(173\) 22.1310 1.68259 0.841295 0.540577i \(-0.181794\pi\)
0.841295 + 0.540577i \(0.181794\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −43.7171 −3.28598
\(178\) 0 0
\(179\) 4.80086 0.358833 0.179417 0.983773i \(-0.442579\pi\)
0.179417 + 0.983773i \(0.442579\pi\)
\(180\) 0 0
\(181\) −16.4333 −1.22148 −0.610740 0.791831i \(-0.709128\pi\)
−0.610740 + 0.791831i \(0.709128\pi\)
\(182\) 0 0
\(183\) 41.6706 3.08038
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.50033 0.109715
\(188\) 0 0
\(189\) 61.5846 4.47962
\(190\) 0 0
\(191\) 5.45151 0.394457 0.197229 0.980358i \(-0.436806\pi\)
0.197229 + 0.980358i \(0.436806\pi\)
\(192\) 0 0
\(193\) −16.3968 −1.18026 −0.590132 0.807306i \(-0.700924\pi\)
−0.590132 + 0.807306i \(0.700924\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 19.2546 1.37183 0.685916 0.727681i \(-0.259402\pi\)
0.685916 + 0.727681i \(0.259402\pi\)
\(198\) 0 0
\(199\) 7.95572 0.563966 0.281983 0.959419i \(-0.409008\pi\)
0.281983 + 0.959419i \(0.409008\pi\)
\(200\) 0 0
\(201\) −16.6335 −1.17324
\(202\) 0 0
\(203\) −29.5824 −2.07628
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −10.7973 −0.750468
\(208\) 0 0
\(209\) 0.786366 0.0543941
\(210\) 0 0
\(211\) 18.2270 1.25480 0.627400 0.778697i \(-0.284119\pi\)
0.627400 + 0.778697i \(0.284119\pi\)
\(212\) 0 0
\(213\) 5.39880 0.369920
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 9.04049 0.613709
\(218\) 0 0
\(219\) 33.6914 2.27665
\(220\) 0 0
\(221\) −2.06007 −0.138576
\(222\) 0 0
\(223\) −0.430994 −0.0288615 −0.0144307 0.999896i \(-0.504594\pi\)
−0.0144307 + 0.999896i \(0.504594\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −7.63334 −0.506643 −0.253321 0.967382i \(-0.581523\pi\)
−0.253321 + 0.967382i \(0.581523\pi\)
\(228\) 0 0
\(229\) 20.3914 1.34750 0.673750 0.738959i \(-0.264682\pi\)
0.673750 + 0.738959i \(0.264682\pi\)
\(230\) 0 0
\(231\) −10.4440 −0.687165
\(232\) 0 0
\(233\) −20.6308 −1.35157 −0.675785 0.737099i \(-0.736195\pi\)
−0.675785 + 0.737099i \(0.736195\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −52.9188 −3.43744
\(238\) 0 0
\(239\) −4.61247 −0.298356 −0.149178 0.988810i \(-0.547663\pi\)
−0.149178 + 0.988810i \(0.547663\pi\)
\(240\) 0 0
\(241\) 3.92363 0.252744 0.126372 0.991983i \(-0.459667\pi\)
0.126372 + 0.991983i \(0.459667\pi\)
\(242\) 0 0
\(243\) 40.7714 2.61549
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.07974 −0.0687024
\(248\) 0 0
\(249\) 30.4978 1.93272
\(250\) 0 0
\(251\) 23.9491 1.51165 0.755827 0.654771i \(-0.227235\pi\)
0.755827 + 0.654771i \(0.227235\pi\)
\(252\) 0 0
\(253\) 1.11179 0.0698978
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.05331 0.564730 0.282365 0.959307i \(-0.408881\pi\)
0.282365 + 0.959307i \(0.408881\pi\)
\(258\) 0 0
\(259\) 37.2319 2.31348
\(260\) 0 0
\(261\) −55.4776 −3.43398
\(262\) 0 0
\(263\) 15.4483 0.952583 0.476291 0.879287i \(-0.341981\pi\)
0.476291 + 0.879287i \(0.341981\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 10.1241 0.619586
\(268\) 0 0
\(269\) −4.44718 −0.271150 −0.135575 0.990767i \(-0.543288\pi\)
−0.135575 + 0.990767i \(0.543288\pi\)
\(270\) 0 0
\(271\) 23.8511 1.44885 0.724425 0.689354i \(-0.242105\pi\)
0.724425 + 0.689354i \(0.242105\pi\)
\(272\) 0 0
\(273\) 14.3404 0.867923
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 6.86751 0.412629 0.206314 0.978486i \(-0.433853\pi\)
0.206314 + 0.978486i \(0.433853\pi\)
\(278\) 0 0
\(279\) 16.9542 1.01502
\(280\) 0 0
\(281\) 26.7419 1.59529 0.797645 0.603128i \(-0.206079\pi\)
0.797645 + 0.603128i \(0.206079\pi\)
\(282\) 0 0
\(283\) −17.4698 −1.03847 −0.519235 0.854632i \(-0.673783\pi\)
−0.519235 + 0.854632i \(0.673783\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −24.9101 −1.47039
\(288\) 0 0
\(289\) −13.3598 −0.785871
\(290\) 0 0
\(291\) 14.8680 0.871579
\(292\) 0 0
\(293\) −14.8968 −0.870282 −0.435141 0.900362i \(-0.643302\pi\)
−0.435141 + 0.900362i \(0.643302\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −11.8922 −0.690057
\(298\) 0 0
\(299\) −1.52658 −0.0882843
\(300\) 0 0
\(301\) 34.2327 1.97314
\(302\) 0 0
\(303\) 2.42455 0.139287
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 11.2895 0.644326 0.322163 0.946684i \(-0.395590\pi\)
0.322163 + 0.946684i \(0.395590\pi\)
\(308\) 0 0
\(309\) −33.8242 −1.92419
\(310\) 0 0
\(311\) −5.65634 −0.320741 −0.160371 0.987057i \(-0.551269\pi\)
−0.160371 + 0.987057i \(0.551269\pi\)
\(312\) 0 0
\(313\) −29.9564 −1.69324 −0.846618 0.532201i \(-0.821365\pi\)
−0.846618 + 0.532201i \(0.821365\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.7992 0.662706 0.331353 0.943507i \(-0.392495\pi\)
0.331353 + 0.943507i \(0.392495\pi\)
\(318\) 0 0
\(319\) 5.71247 0.319837
\(320\) 0 0
\(321\) −55.9008 −3.12008
\(322\) 0 0
\(323\) 1.90793 0.106160
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −56.0567 −3.09994
\(328\) 0 0
\(329\) 14.5163 0.800307
\(330\) 0 0
\(331\) −21.7197 −1.19382 −0.596912 0.802307i \(-0.703606\pi\)
−0.596912 + 0.802307i \(0.703606\pi\)
\(332\) 0 0
\(333\) 69.8231 3.82628
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −32.7221 −1.78249 −0.891245 0.453523i \(-0.850167\pi\)
−0.891245 + 0.453523i \(0.850167\pi\)
\(338\) 0 0
\(339\) −3.44718 −0.187225
\(340\) 0 0
\(341\) −1.74575 −0.0945378
\(342\) 0 0
\(343\) 10.5194 0.567994
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −28.5890 −1.53474 −0.767369 0.641206i \(-0.778435\pi\)
−0.767369 + 0.641206i \(0.778435\pi\)
\(348\) 0 0
\(349\) 22.4911 1.20392 0.601961 0.798525i \(-0.294386\pi\)
0.601961 + 0.798525i \(0.294386\pi\)
\(350\) 0 0
\(351\) 16.3290 0.871576
\(352\) 0 0
\(353\) −21.2381 −1.13039 −0.565196 0.824957i \(-0.691199\pi\)
−0.565196 + 0.824957i \(0.691199\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −25.3399 −1.34113
\(358\) 0 0
\(359\) −5.30705 −0.280095 −0.140048 0.990145i \(-0.544726\pi\)
−0.140048 + 0.990145i \(0.544726\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −33.8590 −1.77713
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1.12291 0.0586156 0.0293078 0.999570i \(-0.490670\pi\)
0.0293078 + 0.999570i \(0.490670\pi\)
\(368\) 0 0
\(369\) −46.7153 −2.43190
\(370\) 0 0
\(371\) −34.9184 −1.81287
\(372\) 0 0
\(373\) 3.99673 0.206943 0.103471 0.994632i \(-0.467005\pi\)
0.103471 + 0.994632i \(0.467005\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −7.84368 −0.403970
\(378\) 0 0
\(379\) 3.36846 0.173026 0.0865132 0.996251i \(-0.472428\pi\)
0.0865132 + 0.996251i \(0.472428\pi\)
\(380\) 0 0
\(381\) −17.0392 −0.872942
\(382\) 0 0
\(383\) 17.8476 0.911969 0.455985 0.889988i \(-0.349287\pi\)
0.455985 + 0.889988i \(0.349287\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 64.1986 3.26340
\(388\) 0 0
\(389\) −19.1690 −0.971907 −0.485953 0.873985i \(-0.661528\pi\)
−0.485953 + 0.873985i \(0.661528\pi\)
\(390\) 0 0
\(391\) 2.69750 0.136418
\(392\) 0 0
\(393\) 9.79298 0.493991
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 9.04085 0.453747 0.226874 0.973924i \(-0.427150\pi\)
0.226874 + 0.973924i \(0.427150\pi\)
\(398\) 0 0
\(399\) −13.2813 −0.664899
\(400\) 0 0
\(401\) −15.0634 −0.752230 −0.376115 0.926573i \(-0.622740\pi\)
−0.376115 + 0.926573i \(0.622740\pi\)
\(402\) 0 0
\(403\) 2.39706 0.119406
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −7.18961 −0.356376
\(408\) 0 0
\(409\) −7.22091 −0.357051 −0.178525 0.983935i \(-0.557133\pi\)
−0.178525 + 0.983935i \(0.557133\pi\)
\(410\) 0 0
\(411\) −75.3135 −3.71494
\(412\) 0 0
\(413\) −54.5855 −2.68597
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −39.0164 −1.91064
\(418\) 0 0
\(419\) −21.0968 −1.03065 −0.515323 0.856996i \(-0.672328\pi\)
−0.515323 + 0.856996i \(0.672328\pi\)
\(420\) 0 0
\(421\) 5.50321 0.268210 0.134105 0.990967i \(-0.457184\pi\)
0.134105 + 0.990967i \(0.457184\pi\)
\(422\) 0 0
\(423\) 27.2232 1.32364
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 52.0303 2.51792
\(428\) 0 0
\(429\) −2.76920 −0.133698
\(430\) 0 0
\(431\) −10.9748 −0.528640 −0.264320 0.964435i \(-0.585147\pi\)
−0.264320 + 0.964435i \(0.585147\pi\)
\(432\) 0 0
\(433\) 14.2313 0.683915 0.341957 0.939715i \(-0.388910\pi\)
0.341957 + 0.939715i \(0.388910\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.41383 0.0676329
\(438\) 0 0
\(439\) 12.0446 0.574859 0.287429 0.957802i \(-0.407199\pi\)
0.287429 + 0.957802i \(0.407199\pi\)
\(440\) 0 0
\(441\) 73.1861 3.48505
\(442\) 0 0
\(443\) 11.1718 0.530789 0.265394 0.964140i \(-0.414498\pi\)
0.265394 + 0.964140i \(0.414498\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 76.7777 3.63146
\(448\) 0 0
\(449\) −9.07650 −0.428347 −0.214173 0.976796i \(-0.568706\pi\)
−0.214173 + 0.976796i \(0.568706\pi\)
\(450\) 0 0
\(451\) 4.81023 0.226505
\(452\) 0 0
\(453\) 34.6154 1.62638
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −16.6795 −0.780235 −0.390118 0.920765i \(-0.627566\pi\)
−0.390118 + 0.920765i \(0.627566\pi\)
\(458\) 0 0
\(459\) −28.8536 −1.34677
\(460\) 0 0
\(461\) −31.2657 −1.45619 −0.728095 0.685476i \(-0.759594\pi\)
−0.728095 + 0.685476i \(0.759594\pi\)
\(462\) 0 0
\(463\) −10.1802 −0.473116 −0.236558 0.971617i \(-0.576019\pi\)
−0.236558 + 0.971617i \(0.576019\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 33.0731 1.53044 0.765220 0.643769i \(-0.222630\pi\)
0.765220 + 0.643769i \(0.222630\pi\)
\(468\) 0 0
\(469\) −20.7688 −0.959012
\(470\) 0 0
\(471\) 45.6590 2.10385
\(472\) 0 0
\(473\) −6.61046 −0.303949
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −65.4845 −2.99833
\(478\) 0 0
\(479\) 7.80955 0.356827 0.178414 0.983956i \(-0.442903\pi\)
0.178414 + 0.983956i \(0.442903\pi\)
\(480\) 0 0
\(481\) 9.87191 0.450120
\(482\) 0 0
\(483\) −18.7776 −0.854412
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −4.37584 −0.198288 −0.0991442 0.995073i \(-0.531611\pi\)
−0.0991442 + 0.995073i \(0.531611\pi\)
\(488\) 0 0
\(489\) −34.7871 −1.57313
\(490\) 0 0
\(491\) 42.0821 1.89914 0.949568 0.313560i \(-0.101522\pi\)
0.949568 + 0.313560i \(0.101522\pi\)
\(492\) 0 0
\(493\) 13.8599 0.624220
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.74098 0.302374
\(498\) 0 0
\(499\) −8.59546 −0.384786 −0.192393 0.981318i \(-0.561625\pi\)
−0.192393 + 0.981318i \(0.561625\pi\)
\(500\) 0 0
\(501\) −12.9758 −0.579716
\(502\) 0 0
\(503\) 6.30722 0.281225 0.140612 0.990065i \(-0.455093\pi\)
0.140612 + 0.990065i \(0.455093\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −38.5963 −1.71412
\(508\) 0 0
\(509\) −31.0352 −1.37561 −0.687805 0.725896i \(-0.741426\pi\)
−0.687805 + 0.725896i \(0.741426\pi\)
\(510\) 0 0
\(511\) 42.0673 1.86095
\(512\) 0 0
\(513\) −15.1230 −0.667697
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −2.80314 −0.123282
\(518\) 0 0
\(519\) 72.1787 3.16830
\(520\) 0 0
\(521\) 32.8924 1.44104 0.720522 0.693432i \(-0.243902\pi\)
0.720522 + 0.693432i \(0.243902\pi\)
\(522\) 0 0
\(523\) −37.6201 −1.64501 −0.822506 0.568757i \(-0.807425\pi\)
−0.822506 + 0.568757i \(0.807425\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.23565 −0.184508
\(528\) 0 0
\(529\) −21.0011 −0.913090
\(530\) 0 0
\(531\) −102.367 −4.44236
\(532\) 0 0
\(533\) −6.60482 −0.286087
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 15.6577 0.675679
\(538\) 0 0
\(539\) −7.53590 −0.324594
\(540\) 0 0
\(541\) −33.6736 −1.44774 −0.723871 0.689936i \(-0.757639\pi\)
−0.723871 + 0.689936i \(0.757639\pi\)
\(542\) 0 0
\(543\) −53.5962 −2.30003
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 14.0295 0.599859 0.299930 0.953961i \(-0.403037\pi\)
0.299930 + 0.953961i \(0.403037\pi\)
\(548\) 0 0
\(549\) 97.5754 4.16442
\(550\) 0 0
\(551\) 7.26439 0.309474
\(552\) 0 0
\(553\) −66.0748 −2.80979
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −16.5434 −0.700968 −0.350484 0.936569i \(-0.613983\pi\)
−0.350484 + 0.936569i \(0.613983\pi\)
\(558\) 0 0
\(559\) 9.07669 0.383903
\(560\) 0 0
\(561\) 4.89322 0.206592
\(562\) 0 0
\(563\) 36.3727 1.53293 0.766463 0.642288i \(-0.222015\pi\)
0.766463 + 0.642288i \(0.222015\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 107.556 4.51691
\(568\) 0 0
\(569\) 21.8270 0.915035 0.457517 0.889201i \(-0.348739\pi\)
0.457517 + 0.889201i \(0.348739\pi\)
\(570\) 0 0
\(571\) 41.2781 1.72743 0.863717 0.503977i \(-0.168130\pi\)
0.863717 + 0.503977i \(0.168130\pi\)
\(572\) 0 0
\(573\) 17.7797 0.742758
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 23.6240 0.983479 0.491739 0.870742i \(-0.336361\pi\)
0.491739 + 0.870742i \(0.336361\pi\)
\(578\) 0 0
\(579\) −53.4769 −2.22242
\(580\) 0 0
\(581\) 38.0798 1.57981
\(582\) 0 0
\(583\) 6.74287 0.279261
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.48730 0.102662 0.0513309 0.998682i \(-0.483654\pi\)
0.0513309 + 0.998682i \(0.483654\pi\)
\(588\) 0 0
\(589\) −2.22003 −0.0914746
\(590\) 0 0
\(591\) 62.7975 2.58314
\(592\) 0 0
\(593\) −11.6543 −0.478587 −0.239293 0.970947i \(-0.576916\pi\)
−0.239293 + 0.970947i \(0.576916\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 25.9470 1.06194
\(598\) 0 0
\(599\) −0.149185 −0.00609555 −0.00304777 0.999995i \(-0.500970\pi\)
−0.00304777 + 0.999995i \(0.500970\pi\)
\(600\) 0 0
\(601\) 29.2954 1.19498 0.597492 0.801875i \(-0.296164\pi\)
0.597492 + 0.801875i \(0.296164\pi\)
\(602\) 0 0
\(603\) −38.9489 −1.58612
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −2.55466 −0.103691 −0.0518453 0.998655i \(-0.516510\pi\)
−0.0518453 + 0.998655i \(0.516510\pi\)
\(608\) 0 0
\(609\) −96.4809 −3.90960
\(610\) 0 0
\(611\) 3.84894 0.155711
\(612\) 0 0
\(613\) −42.6703 −1.72344 −0.861718 0.507387i \(-0.830611\pi\)
−0.861718 + 0.507387i \(0.830611\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −19.8058 −0.797353 −0.398676 0.917092i \(-0.630530\pi\)
−0.398676 + 0.917092i \(0.630530\pi\)
\(618\) 0 0
\(619\) −23.6180 −0.949287 −0.474643 0.880178i \(-0.657423\pi\)
−0.474643 + 0.880178i \(0.657423\pi\)
\(620\) 0 0
\(621\) −21.3814 −0.858007
\(622\) 0 0
\(623\) 12.6410 0.506453
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 2.56468 0.102423
\(628\) 0 0
\(629\) −17.4439 −0.695533
\(630\) 0 0
\(631\) 33.3007 1.32568 0.662841 0.748760i \(-0.269351\pi\)
0.662841 + 0.748760i \(0.269351\pi\)
\(632\) 0 0
\(633\) 59.4462 2.36277
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 10.3474 0.409979
\(638\) 0 0
\(639\) 12.6418 0.500100
\(640\) 0 0
\(641\) −37.5897 −1.48470 −0.742352 0.670010i \(-0.766290\pi\)
−0.742352 + 0.670010i \(0.766290\pi\)
\(642\) 0 0
\(643\) −2.61556 −0.103148 −0.0515739 0.998669i \(-0.516424\pi\)
−0.0515739 + 0.998669i \(0.516424\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 10.3725 0.407784 0.203892 0.978993i \(-0.434641\pi\)
0.203892 + 0.978993i \(0.434641\pi\)
\(648\) 0 0
\(649\) 10.5407 0.413757
\(650\) 0 0
\(651\) 29.4849 1.15561
\(652\) 0 0
\(653\) 0.949620 0.0371615 0.0185808 0.999827i \(-0.494085\pi\)
0.0185808 + 0.999827i \(0.494085\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 78.8913 3.07784
\(658\) 0 0
\(659\) 32.1009 1.25047 0.625237 0.780435i \(-0.285002\pi\)
0.625237 + 0.780435i \(0.285002\pi\)
\(660\) 0 0
\(661\) −32.8796 −1.27887 −0.639434 0.768846i \(-0.720831\pi\)
−0.639434 + 0.768846i \(0.720831\pi\)
\(662\) 0 0
\(663\) −6.71879 −0.260936
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 10.2707 0.397681
\(668\) 0 0
\(669\) −1.40566 −0.0543458
\(670\) 0 0
\(671\) −10.0472 −0.387869
\(672\) 0 0
\(673\) 25.9466 1.00017 0.500084 0.865977i \(-0.333302\pi\)
0.500084 + 0.865977i \(0.333302\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 47.0807 1.80946 0.904729 0.425987i \(-0.140073\pi\)
0.904729 + 0.425987i \(0.140073\pi\)
\(678\) 0 0
\(679\) 18.5643 0.712433
\(680\) 0 0
\(681\) −24.8956 −0.954002
\(682\) 0 0
\(683\) 29.9429 1.14573 0.572867 0.819648i \(-0.305831\pi\)
0.572867 + 0.819648i \(0.305831\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 66.5050 2.53733
\(688\) 0 0
\(689\) −9.25850 −0.352721
\(690\) 0 0
\(691\) −30.4205 −1.15725 −0.578625 0.815594i \(-0.696411\pi\)
−0.578625 + 0.815594i \(0.696411\pi\)
\(692\) 0 0
\(693\) −24.4556 −0.928990
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 11.6709 0.442065
\(698\) 0 0
\(699\) −67.2859 −2.54499
\(700\) 0 0
\(701\) 25.0769 0.947141 0.473570 0.880756i \(-0.342965\pi\)
0.473570 + 0.880756i \(0.342965\pi\)
\(702\) 0 0
\(703\) −9.14283 −0.344828
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3.02731 0.113854
\(708\) 0 0
\(709\) 13.5057 0.507219 0.253610 0.967307i \(-0.418382\pi\)
0.253610 + 0.967307i \(0.418382\pi\)
\(710\) 0 0
\(711\) −123.914 −4.64714
\(712\) 0 0
\(713\) −3.13875 −0.117547
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −15.0433 −0.561801
\(718\) 0 0
\(719\) −2.27969 −0.0850179 −0.0425090 0.999096i \(-0.513535\pi\)
−0.0425090 + 0.999096i \(0.513535\pi\)
\(720\) 0 0
\(721\) −42.2331 −1.57284
\(722\) 0 0
\(723\) 12.7967 0.475913
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0.925937 0.0343411 0.0171705 0.999853i \(-0.494534\pi\)
0.0171705 + 0.999853i \(0.494534\pi\)
\(728\) 0 0
\(729\) 53.7374 1.99028
\(730\) 0 0
\(731\) −16.0387 −0.593212
\(732\) 0 0
\(733\) 20.9455 0.773639 0.386820 0.922155i \(-0.373574\pi\)
0.386820 + 0.922155i \(0.373574\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.01053 0.147730
\(738\) 0 0
\(739\) 36.9385 1.35880 0.679402 0.733766i \(-0.262239\pi\)
0.679402 + 0.733766i \(0.262239\pi\)
\(740\) 0 0
\(741\) −3.52151 −0.129366
\(742\) 0 0
\(743\) −48.3612 −1.77420 −0.887100 0.461577i \(-0.847284\pi\)
−0.887100 + 0.461577i \(0.847284\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 71.4132 2.61287
\(748\) 0 0
\(749\) −69.7981 −2.55037
\(750\) 0 0
\(751\) −19.9871 −0.729340 −0.364670 0.931137i \(-0.618818\pi\)
−0.364670 + 0.931137i \(0.618818\pi\)
\(752\) 0 0
\(753\) 78.1084 2.84643
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −36.3938 −1.32276 −0.661378 0.750053i \(-0.730028\pi\)
−0.661378 + 0.750053i \(0.730028\pi\)
\(758\) 0 0
\(759\) 3.62603 0.131617
\(760\) 0 0
\(761\) −24.2163 −0.877839 −0.438920 0.898526i \(-0.644639\pi\)
−0.438920 + 0.898526i \(0.644639\pi\)
\(762\) 0 0
\(763\) −69.9928 −2.53391
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −14.4732 −0.522596
\(768\) 0 0
\(769\) 30.7013 1.10712 0.553559 0.832810i \(-0.313269\pi\)
0.553559 + 0.832810i \(0.313269\pi\)
\(770\) 0 0
\(771\) 29.5267 1.06338
\(772\) 0 0
\(773\) 21.9986 0.791234 0.395617 0.918416i \(-0.370531\pi\)
0.395617 + 0.918416i \(0.370531\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 121.429 4.35625
\(778\) 0 0
\(779\) 6.11703 0.219165
\(780\) 0 0
\(781\) −1.30171 −0.0465788
\(782\) 0 0
\(783\) −109.859 −3.92606
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −20.9830 −0.747962 −0.373981 0.927436i \(-0.622007\pi\)
−0.373981 + 0.927436i \(0.622007\pi\)
\(788\) 0 0
\(789\) 50.3835 1.79370
\(790\) 0 0
\(791\) −4.30418 −0.153039
\(792\) 0 0
\(793\) 13.7957 0.489898
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −29.1150 −1.03131 −0.515653 0.856798i \(-0.672451\pi\)
−0.515653 + 0.856798i \(0.672451\pi\)
\(798\) 0 0
\(799\) −6.80115 −0.240607
\(800\) 0 0
\(801\) 23.7065 0.837628
\(802\) 0 0
\(803\) −8.12336 −0.286667
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −14.5042 −0.510571
\(808\) 0 0
\(809\) −47.5139 −1.67050 −0.835250 0.549871i \(-0.814677\pi\)
−0.835250 + 0.549871i \(0.814677\pi\)
\(810\) 0 0
\(811\) 11.7957 0.414204 0.207102 0.978319i \(-0.433597\pi\)
0.207102 + 0.978319i \(0.433597\pi\)
\(812\) 0 0
\(813\) 77.7886 2.72817
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −8.40634 −0.294101
\(818\) 0 0
\(819\) 33.5794 1.17336
\(820\) 0 0
\(821\) 28.0549 0.979123 0.489562 0.871969i \(-0.337157\pi\)
0.489562 + 0.871969i \(0.337157\pi\)
\(822\) 0 0
\(823\) 16.7826 0.585006 0.292503 0.956265i \(-0.405512\pi\)
0.292503 + 0.956265i \(0.405512\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12.1984 0.424180 0.212090 0.977250i \(-0.431973\pi\)
0.212090 + 0.977250i \(0.431973\pi\)
\(828\) 0 0
\(829\) 3.30978 0.114954 0.0574768 0.998347i \(-0.481694\pi\)
0.0574768 + 0.998347i \(0.481694\pi\)
\(830\) 0 0
\(831\) 22.3979 0.776975
\(832\) 0 0
\(833\) −18.2840 −0.633505
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 33.5735 1.16047
\(838\) 0 0
\(839\) −27.9925 −0.966408 −0.483204 0.875508i \(-0.660527\pi\)
−0.483204 + 0.875508i \(0.660527\pi\)
\(840\) 0 0
\(841\) 23.7714 0.819704
\(842\) 0 0
\(843\) 87.2169 3.00391
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −42.2766 −1.45264
\(848\) 0 0
\(849\) −56.9764 −1.95543
\(850\) 0 0
\(851\) −12.9265 −0.443113
\(852\) 0 0
\(853\) −11.5191 −0.394408 −0.197204 0.980362i \(-0.563186\pi\)
−0.197204 + 0.980362i \(0.563186\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 5.40616 0.184671 0.0923355 0.995728i \(-0.470567\pi\)
0.0923355 + 0.995728i \(0.470567\pi\)
\(858\) 0 0
\(859\) −17.4306 −0.594725 −0.297363 0.954765i \(-0.596107\pi\)
−0.297363 + 0.954765i \(0.596107\pi\)
\(860\) 0 0
\(861\) −81.2424 −2.76873
\(862\) 0 0
\(863\) −8.83048 −0.300593 −0.150297 0.988641i \(-0.548023\pi\)
−0.150297 + 0.988641i \(0.548023\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −43.5721 −1.47979
\(868\) 0 0
\(869\) 12.7593 0.432829
\(870\) 0 0
\(871\) −5.50677 −0.186590
\(872\) 0 0
\(873\) 34.8148 1.17830
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −0.840573 −0.0283841 −0.0141921 0.999899i \(-0.504518\pi\)
−0.0141921 + 0.999899i \(0.504518\pi\)
\(878\) 0 0
\(879\) −48.5850 −1.63873
\(880\) 0 0
\(881\) 6.78562 0.228613 0.114307 0.993446i \(-0.463535\pi\)
0.114307 + 0.993446i \(0.463535\pi\)
\(882\) 0 0
\(883\) −10.0799 −0.339217 −0.169608 0.985512i \(-0.554250\pi\)
−0.169608 + 0.985512i \(0.554250\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −9.41650 −0.316175 −0.158088 0.987425i \(-0.550533\pi\)
−0.158088 + 0.987425i \(0.550533\pi\)
\(888\) 0 0
\(889\) −21.2752 −0.713548
\(890\) 0 0
\(891\) −20.7694 −0.695801
\(892\) 0 0
\(893\) −3.56468 −0.119287
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −4.97883 −0.166238
\(898\) 0 0
\(899\) −16.1271 −0.537870
\(900\) 0 0
\(901\) 16.3600 0.545029
\(902\) 0 0
\(903\) 111.647 3.71540
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −23.2639 −0.772466 −0.386233 0.922401i \(-0.626224\pi\)
−0.386233 + 0.922401i \(0.626224\pi\)
\(908\) 0 0
\(909\) 5.67730 0.188304
\(910\) 0 0
\(911\) 15.4610 0.512245 0.256123 0.966644i \(-0.417555\pi\)
0.256123 + 0.966644i \(0.417555\pi\)
\(912\) 0 0
\(913\) −7.35335 −0.243360
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 12.2276 0.403791
\(918\) 0 0
\(919\) 20.1775 0.665594 0.332797 0.942998i \(-0.392008\pi\)
0.332797 + 0.942998i \(0.392008\pi\)
\(920\) 0 0
\(921\) 36.8199 1.21326
\(922\) 0 0
\(923\) 1.78735 0.0588314
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −79.2023 −2.60134
\(928\) 0 0
\(929\) 11.3148 0.371227 0.185613 0.982623i \(-0.440573\pi\)
0.185613 + 0.982623i \(0.440573\pi\)
\(930\) 0 0
\(931\) −9.58319 −0.314076
\(932\) 0 0
\(933\) −18.4477 −0.603952
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −16.8943 −0.551911 −0.275956 0.961170i \(-0.588994\pi\)
−0.275956 + 0.961170i \(0.588994\pi\)
\(938\) 0 0
\(939\) −97.7007 −3.18834
\(940\) 0 0
\(941\) −15.0272 −0.489871 −0.244936 0.969539i \(-0.578767\pi\)
−0.244936 + 0.969539i \(0.578767\pi\)
\(942\) 0 0
\(943\) 8.64847 0.281633
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −0.208895 −0.00678818 −0.00339409 0.999994i \(-0.501080\pi\)
−0.00339409 + 0.999994i \(0.501080\pi\)
\(948\) 0 0
\(949\) 11.1540 0.362075
\(950\) 0 0
\(951\) 38.4821 1.24787
\(952\) 0 0
\(953\) 18.0875 0.585910 0.292955 0.956126i \(-0.405361\pi\)
0.292955 + 0.956126i \(0.405361\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 18.6308 0.602249
\(958\) 0 0
\(959\) −94.0370 −3.03661
\(960\) 0 0
\(961\) −26.0715 −0.841016
\(962\) 0 0
\(963\) −130.897 −4.21808
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −40.5540 −1.30413 −0.652064 0.758164i \(-0.726097\pi\)
−0.652064 + 0.758164i \(0.726097\pi\)
\(968\) 0 0
\(969\) 6.22257 0.199898
\(970\) 0 0
\(971\) −51.7597 −1.66105 −0.830523 0.556984i \(-0.811959\pi\)
−0.830523 + 0.556984i \(0.811959\pi\)
\(972\) 0 0
\(973\) −48.7161 −1.56177
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −23.6104 −0.755364 −0.377682 0.925935i \(-0.623279\pi\)
−0.377682 + 0.925935i \(0.623279\pi\)
\(978\) 0 0
\(979\) −2.44103 −0.0780158
\(980\) 0 0
\(981\) −131.262 −4.19086
\(982\) 0 0
\(983\) 34.0532 1.08613 0.543065 0.839691i \(-0.317264\pi\)
0.543065 + 0.839691i \(0.317264\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 47.3437 1.50697
\(988\) 0 0
\(989\) −11.8852 −0.377926
\(990\) 0 0
\(991\) 42.7936 1.35938 0.679692 0.733497i \(-0.262113\pi\)
0.679692 + 0.733497i \(0.262113\pi\)
\(992\) 0 0
\(993\) −70.8373 −2.24796
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −30.1937 −0.956243 −0.478122 0.878294i \(-0.658682\pi\)
−0.478122 + 0.878294i \(0.658682\pi\)
\(998\) 0 0
\(999\) 138.267 4.37458
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 3800.2.a.bc.1.6 yes 6
4.3 odd 2 7600.2.a.ch.1.1 6
5.2 odd 4 3800.2.d.q.3649.1 12
5.3 odd 4 3800.2.d.q.3649.12 12
5.4 even 2 3800.2.a.ba.1.1 6
20.19 odd 2 7600.2.a.cl.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.2.a.ba.1.1 6 5.4 even 2
3800.2.a.bc.1.6 yes 6 1.1 even 1 trivial
3800.2.d.q.3649.1 12 5.2 odd 4
3800.2.d.q.3649.12 12 5.3 odd 4
7600.2.a.ch.1.1 6 4.3 odd 2
7600.2.a.cl.1.6 6 20.19 odd 2