Properties

Label 3800.2.a.j.1.1
Level $3800$
Weight $2$
Character 3800.1
Self dual yes
Analytic conductor $30.343$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) 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)
 
\(f(q)\) \(=\) \(q-2.73205 q^{3} +4.46410 q^{9} +O(q^{10})\) \(q-2.73205 q^{3} +4.46410 q^{9} +2.00000 q^{11} +0.732051 q^{13} -7.46410 q^{17} +1.00000 q^{19} +1.46410 q^{23} -4.00000 q^{27} -3.46410 q^{29} +4.00000 q^{31} -5.46410 q^{33} -7.66025 q^{37} -2.00000 q^{39} +0.535898 q^{41} +2.92820 q^{43} +2.92820 q^{47} -7.00000 q^{49} +20.3923 q^{51} +11.6603 q^{53} -2.73205 q^{57} +1.46410 q^{59} -6.53590 q^{61} +4.19615 q^{67} -4.00000 q^{69} +10.9282 q^{71} +10.3923 q^{73} -8.39230 q^{79} -2.46410 q^{81} +5.46410 q^{83} +9.46410 q^{87} -3.46410 q^{89} -10.9282 q^{93} +15.6603 q^{97} +8.92820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{9} + 4 q^{11} - 2 q^{13} - 8 q^{17} + 2 q^{19} - 4 q^{23} - 8 q^{27} + 8 q^{31} - 4 q^{33} + 2 q^{37} - 4 q^{39} + 8 q^{41} - 8 q^{43} - 8 q^{47} - 14 q^{49} + 20 q^{51} + 6 q^{53} - 2 q^{57} - 4 q^{59} - 20 q^{61} - 2 q^{67} - 8 q^{69} + 8 q^{71} + 4 q^{79} + 2 q^{81} + 4 q^{83} + 12 q^{87} - 8 q^{93} + 14 q^{97} + 4 q^{99}+O(q^{100}) \) 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
\(3\) −2.73205 −1.57735 −0.788675 0.614810i \(-0.789233\pi\)
−0.788675 + 0.614810i \(0.789233\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 4.46410 1.48803
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 0.732051 0.203034 0.101517 0.994834i \(-0.467630\pi\)
0.101517 + 0.994834i \(0.467630\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.46410 −1.81031 −0.905155 0.425081i \(-0.860246\pi\)
−0.905155 + 0.425081i \(0.860246\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.46410 0.305286 0.152643 0.988281i \(-0.451221\pi\)
0.152643 + 0.988281i \(0.451221\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −3.46410 −0.643268 −0.321634 0.946864i \(-0.604232\pi\)
−0.321634 + 0.946864i \(0.604232\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) −5.46410 −0.951178
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.66025 −1.25934 −0.629669 0.776864i \(-0.716809\pi\)
−0.629669 + 0.776864i \(0.716809\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 0.535898 0.0836933 0.0418466 0.999124i \(-0.486676\pi\)
0.0418466 + 0.999124i \(0.486676\pi\)
\(42\) 0 0
\(43\) 2.92820 0.446547 0.223273 0.974756i \(-0.428326\pi\)
0.223273 + 0.974756i \(0.428326\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.92820 0.427122 0.213561 0.976930i \(-0.431494\pi\)
0.213561 + 0.976930i \(0.431494\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 20.3923 2.85549
\(52\) 0 0
\(53\) 11.6603 1.60166 0.800830 0.598892i \(-0.204392\pi\)
0.800830 + 0.598892i \(0.204392\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.73205 −0.361869
\(58\) 0 0
\(59\) 1.46410 0.190610 0.0953049 0.995448i \(-0.469617\pi\)
0.0953049 + 0.995448i \(0.469617\pi\)
\(60\) 0 0
\(61\) −6.53590 −0.836836 −0.418418 0.908255i \(-0.637415\pi\)
−0.418418 + 0.908255i \(0.637415\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.19615 0.512642 0.256321 0.966592i \(-0.417490\pi\)
0.256321 + 0.966592i \(0.417490\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 10.9282 1.29694 0.648470 0.761241i \(-0.275409\pi\)
0.648470 + 0.761241i \(0.275409\pi\)
\(72\) 0 0
\(73\) 10.3923 1.21633 0.608164 0.793812i \(-0.291906\pi\)
0.608164 + 0.793812i \(0.291906\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.39230 −0.944208 −0.472104 0.881543i \(-0.656505\pi\)
−0.472104 + 0.881543i \(0.656505\pi\)
\(80\) 0 0
\(81\) −2.46410 −0.273789
\(82\) 0 0
\(83\) 5.46410 0.599763 0.299882 0.953976i \(-0.403053\pi\)
0.299882 + 0.953976i \(0.403053\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 9.46410 1.01466
\(88\) 0 0
\(89\) −3.46410 −0.367194 −0.183597 0.983002i \(-0.558774\pi\)
−0.183597 + 0.983002i \(0.558774\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −10.9282 −1.13320
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 15.6603 1.59006 0.795029 0.606571i \(-0.207456\pi\)
0.795029 + 0.606571i \(0.207456\pi\)
\(98\) 0 0
\(99\) 8.92820 0.897318
\(100\) 0 0
\(101\) 1.46410 0.145684 0.0728418 0.997344i \(-0.476793\pi\)
0.0728418 + 0.997344i \(0.476793\pi\)
\(102\) 0 0
\(103\) −5.66025 −0.557721 −0.278861 0.960332i \(-0.589957\pi\)
−0.278861 + 0.960332i \(0.589957\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −15.1244 −1.46213 −0.731063 0.682310i \(-0.760976\pi\)
−0.731063 + 0.682310i \(0.760976\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 20.9282 1.98642
\(112\) 0 0
\(113\) −16.7321 −1.57402 −0.787009 0.616941i \(-0.788372\pi\)
−0.787009 + 0.616941i \(0.788372\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 3.26795 0.302122
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) −1.46410 −0.132014
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −17.6603 −1.56709 −0.783547 0.621332i \(-0.786592\pi\)
−0.783547 + 0.621332i \(0.786592\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −0.535898 −0.0457849 −0.0228924 0.999738i \(-0.507288\pi\)
−0.0228924 + 0.999738i \(0.507288\pi\)
\(138\) 0 0
\(139\) 7.85641 0.666372 0.333186 0.942861i \(-0.391876\pi\)
0.333186 + 0.942861i \(0.391876\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 1.46410 0.122434
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 19.1244 1.57735
\(148\) 0 0
\(149\) 15.3205 1.25510 0.627552 0.778574i \(-0.284057\pi\)
0.627552 + 0.778574i \(0.284057\pi\)
\(150\) 0 0
\(151\) −5.46410 −0.444662 −0.222331 0.974971i \(-0.571367\pi\)
−0.222331 + 0.974971i \(0.571367\pi\)
\(152\) 0 0
\(153\) −33.3205 −2.69380
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 7.46410 0.595700 0.297850 0.954613i \(-0.403730\pi\)
0.297850 + 0.954613i \(0.403730\pi\)
\(158\) 0 0
\(159\) −31.8564 −2.52638
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.73205 0.520942 0.260471 0.965482i \(-0.416122\pi\)
0.260471 + 0.965482i \(0.416122\pi\)
\(168\) 0 0
\(169\) −12.4641 −0.958777
\(170\) 0 0
\(171\) 4.46410 0.341378
\(172\) 0 0
\(173\) 17.1244 1.30194 0.650970 0.759103i \(-0.274362\pi\)
0.650970 + 0.759103i \(0.274362\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) 0 0
\(179\) −9.46410 −0.707380 −0.353690 0.935363i \(-0.615073\pi\)
−0.353690 + 0.935363i \(0.615073\pi\)
\(180\) 0 0
\(181\) −15.8564 −1.17860 −0.589299 0.807915i \(-0.700596\pi\)
−0.589299 + 0.807915i \(0.700596\pi\)
\(182\) 0 0
\(183\) 17.8564 1.31998
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −14.9282 −1.09166
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.9282 −0.790737 −0.395369 0.918523i \(-0.629383\pi\)
−0.395369 + 0.918523i \(0.629383\pi\)
\(192\) 0 0
\(193\) −14.1962 −1.02186 −0.510931 0.859622i \(-0.670699\pi\)
−0.510931 + 0.859622i \(0.670699\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −11.8564 −0.844734 −0.422367 0.906425i \(-0.638801\pi\)
−0.422367 + 0.906425i \(0.638801\pi\)
\(198\) 0 0
\(199\) −18.9282 −1.34178 −0.670892 0.741555i \(-0.734089\pi\)
−0.670892 + 0.741555i \(0.734089\pi\)
\(200\) 0 0
\(201\) −11.4641 −0.808615
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 6.53590 0.454276
\(208\) 0 0
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) −16.3923 −1.12849 −0.564246 0.825606i \(-0.690833\pi\)
−0.564246 + 0.825606i \(0.690833\pi\)
\(212\) 0 0
\(213\) −29.8564 −2.04573
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −28.3923 −1.91857
\(220\) 0 0
\(221\) −5.46410 −0.367555
\(222\) 0 0
\(223\) −26.0526 −1.74461 −0.872304 0.488964i \(-0.837375\pi\)
−0.872304 + 0.488964i \(0.837375\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.26795 −0.0841567 −0.0420784 0.999114i \(-0.513398\pi\)
−0.0420784 + 0.999114i \(0.513398\pi\)
\(228\) 0 0
\(229\) 5.46410 0.361078 0.180539 0.983568i \(-0.442216\pi\)
0.180539 + 0.983568i \(0.442216\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 22.9282 1.48935
\(238\) 0 0
\(239\) −18.9282 −1.22436 −0.612182 0.790717i \(-0.709708\pi\)
−0.612182 + 0.790717i \(0.709708\pi\)
\(240\) 0 0
\(241\) −10.3923 −0.669427 −0.334714 0.942320i \(-0.608640\pi\)
−0.334714 + 0.942320i \(0.608640\pi\)
\(242\) 0 0
\(243\) 18.7321 1.20166
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0.732051 0.0465793
\(248\) 0 0
\(249\) −14.9282 −0.946036
\(250\) 0 0
\(251\) −1.85641 −0.117175 −0.0585877 0.998282i \(-0.518660\pi\)
−0.0585877 + 0.998282i \(0.518660\pi\)
\(252\) 0 0
\(253\) 2.92820 0.184095
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −24.7321 −1.54274 −0.771371 0.636385i \(-0.780429\pi\)
−0.771371 + 0.636385i \(0.780429\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −15.4641 −0.957204
\(262\) 0 0
\(263\) −19.3205 −1.19135 −0.595677 0.803224i \(-0.703116\pi\)
−0.595677 + 0.803224i \(0.703116\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 9.46410 0.579194
\(268\) 0 0
\(269\) −8.92820 −0.544362 −0.272181 0.962246i \(-0.587745\pi\)
−0.272181 + 0.962246i \(0.587745\pi\)
\(270\) 0 0
\(271\) 14.0000 0.850439 0.425220 0.905090i \(-0.360197\pi\)
0.425220 + 0.905090i \(0.360197\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 20.5359 1.23388 0.616941 0.787009i \(-0.288372\pi\)
0.616941 + 0.787009i \(0.288372\pi\)
\(278\) 0 0
\(279\) 17.8564 1.06904
\(280\) 0 0
\(281\) −22.3923 −1.33581 −0.667906 0.744245i \(-0.732809\pi\)
−0.667906 + 0.744245i \(0.732809\pi\)
\(282\) 0 0
\(283\) 9.46410 0.562582 0.281291 0.959622i \(-0.409237\pi\)
0.281291 + 0.959622i \(0.409237\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 38.7128 2.27722
\(290\) 0 0
\(291\) −42.7846 −2.50808
\(292\) 0 0
\(293\) −18.1962 −1.06303 −0.531515 0.847049i \(-0.678377\pi\)
−0.531515 + 0.847049i \(0.678377\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −8.00000 −0.464207
\(298\) 0 0
\(299\) 1.07180 0.0619836
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −4.00000 −0.229794
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −25.2679 −1.44212 −0.721059 0.692874i \(-0.756344\pi\)
−0.721059 + 0.692874i \(0.756344\pi\)
\(308\) 0 0
\(309\) 15.4641 0.879722
\(310\) 0 0
\(311\) 19.8564 1.12595 0.562977 0.826473i \(-0.309656\pi\)
0.562977 + 0.826473i \(0.309656\pi\)
\(312\) 0 0
\(313\) 15.8564 0.896257 0.448129 0.893969i \(-0.352091\pi\)
0.448129 + 0.893969i \(0.352091\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.58846 0.370045 0.185022 0.982734i \(-0.440764\pi\)
0.185022 + 0.982734i \(0.440764\pi\)
\(318\) 0 0
\(319\) −6.92820 −0.387905
\(320\) 0 0
\(321\) 41.3205 2.30629
\(322\) 0 0
\(323\) −7.46410 −0.415314
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 38.2487 2.11516
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 14.2487 0.783180 0.391590 0.920140i \(-0.371925\pi\)
0.391590 + 0.920140i \(0.371925\pi\)
\(332\) 0 0
\(333\) −34.1962 −1.87394
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −11.6603 −0.635175 −0.317587 0.948229i \(-0.602873\pi\)
−0.317587 + 0.948229i \(0.602873\pi\)
\(338\) 0 0
\(339\) 45.7128 2.48278
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8.39230 0.450523 0.225261 0.974298i \(-0.427676\pi\)
0.225261 + 0.974298i \(0.427676\pi\)
\(348\) 0 0
\(349\) −22.7846 −1.21963 −0.609816 0.792543i \(-0.708757\pi\)
−0.609816 + 0.792543i \(0.708757\pi\)
\(350\) 0 0
\(351\) −2.92820 −0.156296
\(352\) 0 0
\(353\) −32.2487 −1.71643 −0.858213 0.513294i \(-0.828425\pi\)
−0.858213 + 0.513294i \(0.828425\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8.92820 0.471213 0.235606 0.971849i \(-0.424292\pi\)
0.235606 + 0.971849i \(0.424292\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 19.1244 1.00377
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −10.9282 −0.570448 −0.285224 0.958461i \(-0.592068\pi\)
−0.285224 + 0.958461i \(0.592068\pi\)
\(368\) 0 0
\(369\) 2.39230 0.124538
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −10.1962 −0.527937 −0.263968 0.964531i \(-0.585031\pi\)
−0.263968 + 0.964531i \(0.585031\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.53590 −0.130605
\(378\) 0 0
\(379\) −34.6410 −1.77939 −0.889695 0.456556i \(-0.849083\pi\)
−0.889695 + 0.456556i \(0.849083\pi\)
\(380\) 0 0
\(381\) 48.2487 2.47186
\(382\) 0 0
\(383\) −32.1962 −1.64515 −0.822573 0.568659i \(-0.807462\pi\)
−0.822573 + 0.568659i \(0.807462\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 13.0718 0.664477
\(388\) 0 0
\(389\) −15.8564 −0.803952 −0.401976 0.915650i \(-0.631676\pi\)
−0.401976 + 0.915650i \(0.631676\pi\)
\(390\) 0 0
\(391\) −10.9282 −0.552663
\(392\) 0 0
\(393\) −32.7846 −1.65376
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 19.4641 0.976875 0.488438 0.872599i \(-0.337567\pi\)
0.488438 + 0.872599i \(0.337567\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 14.0000 0.699127 0.349563 0.936913i \(-0.386330\pi\)
0.349563 + 0.936913i \(0.386330\pi\)
\(402\) 0 0
\(403\) 2.92820 0.145864
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −15.3205 −0.759409
\(408\) 0 0
\(409\) 15.0718 0.745252 0.372626 0.927982i \(-0.378457\pi\)
0.372626 + 0.927982i \(0.378457\pi\)
\(410\) 0 0
\(411\) 1.46410 0.0722188
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −21.4641 −1.05110
\(418\) 0 0
\(419\) 25.8564 1.26317 0.631584 0.775307i \(-0.282405\pi\)
0.631584 + 0.775307i \(0.282405\pi\)
\(420\) 0 0
\(421\) 16.2487 0.791914 0.395957 0.918269i \(-0.370413\pi\)
0.395957 + 0.918269i \(0.370413\pi\)
\(422\) 0 0
\(423\) 13.0718 0.635573
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) 28.3923 1.36761 0.683805 0.729665i \(-0.260324\pi\)
0.683805 + 0.729665i \(0.260324\pi\)
\(432\) 0 0
\(433\) 26.9808 1.29661 0.648306 0.761380i \(-0.275478\pi\)
0.648306 + 0.761380i \(0.275478\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.46410 0.0700375
\(438\) 0 0
\(439\) 26.2487 1.25278 0.626391 0.779509i \(-0.284531\pi\)
0.626391 + 0.779509i \(0.284531\pi\)
\(440\) 0 0
\(441\) −31.2487 −1.48803
\(442\) 0 0
\(443\) 33.4641 1.58993 0.794964 0.606657i \(-0.207490\pi\)
0.794964 + 0.606657i \(0.207490\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −41.8564 −1.97974
\(448\) 0 0
\(449\) −36.6410 −1.72920 −0.864598 0.502464i \(-0.832427\pi\)
−0.864598 + 0.502464i \(0.832427\pi\)
\(450\) 0 0
\(451\) 1.07180 0.0504689
\(452\) 0 0
\(453\) 14.9282 0.701388
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −28.9282 −1.35320 −0.676602 0.736349i \(-0.736548\pi\)
−0.676602 + 0.736349i \(0.736548\pi\)
\(458\) 0 0
\(459\) 29.8564 1.39358
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) −10.2487 −0.476298 −0.238149 0.971229i \(-0.576541\pi\)
−0.238149 + 0.971229i \(0.576541\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 35.3205 1.63444 0.817219 0.576327i \(-0.195515\pi\)
0.817219 + 0.576327i \(0.195515\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −20.3923 −0.939628
\(472\) 0 0
\(473\) 5.85641 0.269278
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 52.0526 2.38332
\(478\) 0 0
\(479\) 19.8564 0.907262 0.453631 0.891190i \(-0.350128\pi\)
0.453631 + 0.891190i \(0.350128\pi\)
\(480\) 0 0
\(481\) −5.60770 −0.255689
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −42.7321 −1.93637 −0.968187 0.250228i \(-0.919495\pi\)
−0.968187 + 0.250228i \(0.919495\pi\)
\(488\) 0 0
\(489\) −10.9282 −0.494190
\(490\) 0 0
\(491\) −1.85641 −0.0837785 −0.0418892 0.999122i \(-0.513338\pi\)
−0.0418892 + 0.999122i \(0.513338\pi\)
\(492\) 0 0
\(493\) 25.8564 1.16451
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 30.7846 1.37811 0.689054 0.724710i \(-0.258026\pi\)
0.689054 + 0.724710i \(0.258026\pi\)
\(500\) 0 0
\(501\) −18.3923 −0.821708
\(502\) 0 0
\(503\) −23.3205 −1.03981 −0.519905 0.854224i \(-0.674033\pi\)
−0.519905 + 0.854224i \(0.674033\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 34.0526 1.51233
\(508\) 0 0
\(509\) 2.39230 0.106037 0.0530185 0.998594i \(-0.483116\pi\)
0.0530185 + 0.998594i \(0.483116\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 5.85641 0.257564
\(518\) 0 0
\(519\) −46.7846 −2.05362
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 18.7321 0.819095 0.409548 0.912289i \(-0.365687\pi\)
0.409548 + 0.912289i \(0.365687\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −29.8564 −1.30057
\(528\) 0 0
\(529\) −20.8564 −0.906800
\(530\) 0 0
\(531\) 6.53590 0.283634
\(532\) 0 0
\(533\) 0.392305 0.0169926
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 25.8564 1.11579
\(538\) 0 0
\(539\) −14.0000 −0.603023
\(540\) 0 0
\(541\) −23.6077 −1.01497 −0.507487 0.861659i \(-0.669425\pi\)
−0.507487 + 0.861659i \(0.669425\pi\)
\(542\) 0 0
\(543\) 43.3205 1.85906
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −16.5885 −0.709271 −0.354636 0.935005i \(-0.615395\pi\)
−0.354636 + 0.935005i \(0.615395\pi\)
\(548\) 0 0
\(549\) −29.1769 −1.24524
\(550\) 0 0
\(551\) −3.46410 −0.147576
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −12.2487 −0.518995 −0.259497 0.965744i \(-0.583557\pi\)
−0.259497 + 0.965744i \(0.583557\pi\)
\(558\) 0 0
\(559\) 2.14359 0.0906643
\(560\) 0 0
\(561\) 40.7846 1.72193
\(562\) 0 0
\(563\) 8.58846 0.361960 0.180980 0.983487i \(-0.442073\pi\)
0.180980 + 0.983487i \(0.442073\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 14.7846 0.619803 0.309902 0.950769i \(-0.399704\pi\)
0.309902 + 0.950769i \(0.399704\pi\)
\(570\) 0 0
\(571\) −30.7846 −1.28830 −0.644148 0.764901i \(-0.722788\pi\)
−0.644148 + 0.764901i \(0.722788\pi\)
\(572\) 0 0
\(573\) 29.8564 1.24727
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −35.8564 −1.49272 −0.746361 0.665541i \(-0.768201\pi\)
−0.746361 + 0.665541i \(0.768201\pi\)
\(578\) 0 0
\(579\) 38.7846 1.61183
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 23.3205 0.965837
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −36.1051 −1.49022 −0.745109 0.666943i \(-0.767602\pi\)
−0.745109 + 0.666943i \(0.767602\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 32.3923 1.33244
\(592\) 0 0
\(593\) 35.8564 1.47245 0.736223 0.676739i \(-0.236607\pi\)
0.736223 + 0.676739i \(0.236607\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 51.7128 2.11646
\(598\) 0 0
\(599\) −43.0333 −1.75829 −0.879147 0.476551i \(-0.841887\pi\)
−0.879147 + 0.476551i \(0.841887\pi\)
\(600\) 0 0
\(601\) −36.5359 −1.49033 −0.745165 0.666880i \(-0.767629\pi\)
−0.745165 + 0.666880i \(0.767629\pi\)
\(602\) 0 0
\(603\) 18.7321 0.762828
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −38.4449 −1.56043 −0.780214 0.625512i \(-0.784890\pi\)
−0.780214 + 0.625512i \(0.784890\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.14359 0.0867205
\(612\) 0 0
\(613\) −44.2487 −1.78719 −0.893594 0.448875i \(-0.851825\pi\)
−0.893594 + 0.448875i \(0.851825\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −3.46410 −0.139459 −0.0697297 0.997566i \(-0.522214\pi\)
−0.0697297 + 0.997566i \(0.522214\pi\)
\(618\) 0 0
\(619\) −14.0000 −0.562708 −0.281354 0.959604i \(-0.590783\pi\)
−0.281354 + 0.959604i \(0.590783\pi\)
\(620\) 0 0
\(621\) −5.85641 −0.235009
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −5.46410 −0.218215
\(628\) 0 0
\(629\) 57.1769 2.27979
\(630\) 0 0
\(631\) 20.6410 0.821706 0.410853 0.911702i \(-0.365231\pi\)
0.410853 + 0.911702i \(0.365231\pi\)
\(632\) 0 0
\(633\) 44.7846 1.78003
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −5.12436 −0.203034
\(638\) 0 0
\(639\) 48.7846 1.92989
\(640\) 0 0
\(641\) 45.0333 1.77871 0.889355 0.457218i \(-0.151154\pi\)
0.889355 + 0.457218i \(0.151154\pi\)
\(642\) 0 0
\(643\) −1.75129 −0.0690641 −0.0345320 0.999404i \(-0.510994\pi\)
−0.0345320 + 0.999404i \(0.510994\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 47.3205 1.86036 0.930181 0.367102i \(-0.119650\pi\)
0.930181 + 0.367102i \(0.119650\pi\)
\(648\) 0 0
\(649\) 2.92820 0.114942
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 34.3923 1.34587 0.672937 0.739699i \(-0.265032\pi\)
0.672937 + 0.739699i \(0.265032\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 46.3923 1.80994
\(658\) 0 0
\(659\) 32.3923 1.26183 0.630913 0.775854i \(-0.282681\pi\)
0.630913 + 0.775854i \(0.282681\pi\)
\(660\) 0 0
\(661\) −16.5359 −0.643172 −0.321586 0.946880i \(-0.604216\pi\)
−0.321586 + 0.946880i \(0.604216\pi\)
\(662\) 0 0
\(663\) 14.9282 0.579763
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −5.07180 −0.196381
\(668\) 0 0
\(669\) 71.1769 2.75186
\(670\) 0 0
\(671\) −13.0718 −0.504631
\(672\) 0 0
\(673\) 33.8038 1.30304 0.651521 0.758630i \(-0.274131\pi\)
0.651521 + 0.758630i \(0.274131\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −29.9090 −1.14949 −0.574747 0.818331i \(-0.694900\pi\)
−0.574747 + 0.818331i \(0.694900\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 3.46410 0.132745
\(682\) 0 0
\(683\) 15.8038 0.604717 0.302359 0.953194i \(-0.402226\pi\)
0.302359 + 0.953194i \(0.402226\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −14.9282 −0.569546
\(688\) 0 0
\(689\) 8.53590 0.325192
\(690\) 0 0
\(691\) −38.7846 −1.47544 −0.737718 0.675109i \(-0.764097\pi\)
−0.737718 + 0.675109i \(0.764097\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) 0 0
\(699\) 38.2487 1.44670
\(700\) 0 0
\(701\) −22.2487 −0.840322 −0.420161 0.907450i \(-0.638026\pi\)
−0.420161 + 0.907450i \(0.638026\pi\)
\(702\) 0 0
\(703\) −7.66025 −0.288912
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 18.7846 0.705471 0.352735 0.935723i \(-0.385252\pi\)
0.352735 + 0.935723i \(0.385252\pi\)
\(710\) 0 0
\(711\) −37.4641 −1.40501
\(712\) 0 0
\(713\) 5.85641 0.219324
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 51.7128 1.93125
\(718\) 0 0
\(719\) −47.5692 −1.77403 −0.887016 0.461738i \(-0.847226\pi\)
−0.887016 + 0.461738i \(0.847226\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 28.3923 1.05592
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −21.0718 −0.781510 −0.390755 0.920495i \(-0.627786\pi\)
−0.390755 + 0.920495i \(0.627786\pi\)
\(728\) 0 0
\(729\) −43.7846 −1.62165
\(730\) 0 0
\(731\) −21.8564 −0.808388
\(732\) 0 0
\(733\) 16.1436 0.596277 0.298139 0.954523i \(-0.403634\pi\)
0.298139 + 0.954523i \(0.403634\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.39230 0.309135
\(738\) 0 0
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) −2.00000 −0.0734718
\(742\) 0 0
\(743\) −29.3731 −1.07759 −0.538797 0.842436i \(-0.681121\pi\)
−0.538797 + 0.842436i \(0.681121\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 24.3923 0.892468
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 25.4641 0.929198 0.464599 0.885521i \(-0.346198\pi\)
0.464599 + 0.885521i \(0.346198\pi\)
\(752\) 0 0
\(753\) 5.07180 0.184827
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.21539 0.0441741 0.0220871 0.999756i \(-0.492969\pi\)
0.0220871 + 0.999756i \(0.492969\pi\)
\(758\) 0 0
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) 24.3923 0.884220 0.442110 0.896961i \(-0.354230\pi\)
0.442110 + 0.896961i \(0.354230\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.07180 0.0387003
\(768\) 0 0
\(769\) 24.3923 0.879609 0.439805 0.898094i \(-0.355048\pi\)
0.439805 + 0.898094i \(0.355048\pi\)
\(770\) 0 0
\(771\) 67.5692 2.43345
\(772\) 0 0
\(773\) −10.1962 −0.366730 −0.183365 0.983045i \(-0.558699\pi\)
−0.183365 + 0.983045i \(0.558699\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0.535898 0.0192006
\(780\) 0 0
\(781\) 21.8564 0.782084
\(782\) 0 0
\(783\) 13.8564 0.495188
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0.196152 0.00699208 0.00349604 0.999994i \(-0.498887\pi\)
0.00349604 + 0.999994i \(0.498887\pi\)
\(788\) 0 0
\(789\) 52.7846 1.87918
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −4.78461 −0.169906
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −16.0526 −0.568611 −0.284305 0.958734i \(-0.591763\pi\)
−0.284305 + 0.958734i \(0.591763\pi\)
\(798\) 0 0
\(799\) −21.8564 −0.773224
\(800\) 0 0
\(801\) −15.4641 −0.546397
\(802\) 0 0
\(803\) 20.7846 0.733473
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 24.3923 0.858650
\(808\) 0 0
\(809\) 9.71281 0.341484 0.170742 0.985316i \(-0.445383\pi\)
0.170742 + 0.985316i \(0.445383\pi\)
\(810\) 0 0
\(811\) 13.8564 0.486564 0.243282 0.969956i \(-0.421776\pi\)
0.243282 + 0.969956i \(0.421776\pi\)
\(812\) 0 0
\(813\) −38.2487 −1.34144
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 2.92820 0.102445
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 22.7846 0.795188 0.397594 0.917561i \(-0.369845\pi\)
0.397594 + 0.917561i \(0.369845\pi\)
\(822\) 0 0
\(823\) 26.9282 0.938658 0.469329 0.883023i \(-0.344496\pi\)
0.469329 + 0.883023i \(0.344496\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −35.9090 −1.24868 −0.624339 0.781154i \(-0.714631\pi\)
−0.624339 + 0.781154i \(0.714631\pi\)
\(828\) 0 0
\(829\) −33.7128 −1.17089 −0.585447 0.810711i \(-0.699081\pi\)
−0.585447 + 0.810711i \(0.699081\pi\)
\(830\) 0 0
\(831\) −56.1051 −1.94626
\(832\) 0 0
\(833\) 52.2487 1.81031
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −16.0000 −0.553041
\(838\) 0 0
\(839\) 23.6077 0.815028 0.407514 0.913199i \(-0.366396\pi\)
0.407514 + 0.913199i \(0.366396\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) 0 0
\(843\) 61.1769 2.10704
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −25.8564 −0.887390
\(850\) 0 0
\(851\) −11.2154 −0.384459
\(852\) 0 0
\(853\) −14.7846 −0.506215 −0.253108 0.967438i \(-0.581453\pi\)
−0.253108 + 0.967438i \(0.581453\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 29.1244 0.994869 0.497435 0.867502i \(-0.334275\pi\)
0.497435 + 0.867502i \(0.334275\pi\)
\(858\) 0 0
\(859\) 30.1436 1.02849 0.514243 0.857644i \(-0.328073\pi\)
0.514243 + 0.857644i \(0.328073\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 25.2679 0.860131 0.430065 0.902798i \(-0.358490\pi\)
0.430065 + 0.902798i \(0.358490\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −105.765 −3.59198
\(868\) 0 0
\(869\) −16.7846 −0.569379
\(870\) 0 0
\(871\) 3.07180 0.104084
\(872\) 0 0
\(873\) 69.9090 2.36606
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 12.4449 0.420233 0.210117 0.977676i \(-0.432616\pi\)
0.210117 + 0.977676i \(0.432616\pi\)
\(878\) 0 0
\(879\) 49.7128 1.67677
\(880\) 0 0
\(881\) 12.3923 0.417507 0.208754 0.977968i \(-0.433059\pi\)
0.208754 + 0.977968i \(0.433059\pi\)
\(882\) 0 0
\(883\) −38.6410 −1.30037 −0.650187 0.759774i \(-0.725309\pi\)
−0.650187 + 0.759774i \(0.725309\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 20.8756 0.700936 0.350468 0.936575i \(-0.386023\pi\)
0.350468 + 0.936575i \(0.386023\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −4.92820 −0.165101
\(892\) 0 0
\(893\) 2.92820 0.0979886
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −2.92820 −0.0977699
\(898\) 0 0
\(899\) −13.8564 −0.462137
\(900\) 0 0
\(901\) −87.0333 −2.89950
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 34.4449 1.14372 0.571861 0.820350i \(-0.306221\pi\)
0.571861 + 0.820350i \(0.306221\pi\)
\(908\) 0 0
\(909\) 6.53590 0.216782
\(910\) 0 0
\(911\) 49.4641 1.63882 0.819409 0.573209i \(-0.194302\pi\)
0.819409 + 0.573209i \(0.194302\pi\)
\(912\) 0 0
\(913\) 10.9282 0.361671
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 2.92820 0.0965925 0.0482963 0.998833i \(-0.484621\pi\)
0.0482963 + 0.998833i \(0.484621\pi\)
\(920\) 0 0
\(921\) 69.0333 2.27473
\(922\) 0 0
\(923\) 8.00000 0.263323
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −25.2679 −0.829908
\(928\) 0 0
\(929\) −50.0000 −1.64045 −0.820223 0.572043i \(-0.806151\pi\)
−0.820223 + 0.572043i \(0.806151\pi\)
\(930\) 0 0
\(931\) −7.00000 −0.229416
\(932\) 0 0
\(933\) −54.2487 −1.77602
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −29.6077 −0.967241 −0.483621 0.875278i \(-0.660679\pi\)
−0.483621 + 0.875278i \(0.660679\pi\)
\(938\) 0 0
\(939\) −43.3205 −1.41371
\(940\) 0 0
\(941\) 35.0718 1.14331 0.571654 0.820495i \(-0.306302\pi\)
0.571654 + 0.820495i \(0.306302\pi\)
\(942\) 0 0
\(943\) 0.784610 0.0255504
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 50.2487 1.63286 0.816432 0.577442i \(-0.195949\pi\)
0.816432 + 0.577442i \(0.195949\pi\)
\(948\) 0 0
\(949\) 7.60770 0.246956
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) 17.4115 0.564015 0.282008 0.959412i \(-0.409000\pi\)
0.282008 + 0.959412i \(0.409000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 18.9282 0.611862
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −67.5167 −2.17569
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 57.1769 1.83869 0.919343 0.393457i \(-0.128721\pi\)
0.919343 + 0.393457i \(0.128721\pi\)
\(968\) 0 0
\(969\) 20.3923 0.655095
\(970\) 0 0
\(971\) 6.92820 0.222337 0.111168 0.993802i \(-0.464541\pi\)
0.111168 + 0.993802i \(0.464541\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 8.44486 0.270175 0.135088 0.990834i \(-0.456868\pi\)
0.135088 + 0.990834i \(0.456868\pi\)
\(978\) 0 0
\(979\) −6.92820 −0.221426
\(980\) 0 0
\(981\) −62.4974 −1.99539
\(982\) 0 0
\(983\) 3.51666 0.112164 0.0560820 0.998426i \(-0.482139\pi\)
0.0560820 + 0.998426i \(0.482139\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 4.28719 0.136325
\(990\) 0 0
\(991\) −3.32051 −0.105479 −0.0527397 0.998608i \(-0.516795\pi\)
−0.0527397 + 0.998608i \(0.516795\pi\)
\(992\) 0 0
\(993\) −38.9282 −1.23535
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 38.3923 1.21590 0.607948 0.793977i \(-0.291993\pi\)
0.607948 + 0.793977i \(0.291993\pi\)
\(998\) 0 0
\(999\) 30.6410 0.969439
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.j.1.1 2
4.3 odd 2 7600.2.a.bd.1.2 2
5.2 odd 4 3800.2.d.h.3649.4 4
5.3 odd 4 3800.2.d.h.3649.1 4
5.4 even 2 760.2.a.g.1.2 2
15.14 odd 2 6840.2.a.y.1.1 2
20.19 odd 2 1520.2.a.k.1.1 2
40.19 odd 2 6080.2.a.bk.1.2 2
40.29 even 2 6080.2.a.ba.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.g.1.2 2 5.4 even 2
1520.2.a.k.1.1 2 20.19 odd 2
3800.2.a.j.1.1 2 1.1 even 1 trivial
3800.2.d.h.3649.1 4 5.3 odd 4
3800.2.d.h.3649.4 4 5.2 odd 4
6080.2.a.ba.1.1 2 40.29 even 2
6080.2.a.bk.1.2 2 40.19 odd 2
6840.2.a.y.1.1 2 15.14 odd 2
7600.2.a.bd.1.2 2 4.3 odd 2